[was@descent doc]$ [was@descent doc]$ Magma V2.7-1 Fri Jun 30 2000 06:12:15 on descent [Seed = 1294487087] Type ? for help. Type -D to quit. Loading startup file "/home/was/modsym/init-magma.m" C IndexGamma0 R factormod padiccharpoly CS MS Tn fcp qexp DC ND Z fn x ES NS charpoly idxG0 F Q ellap modcharpoly > M:=MS(571,2); > M; Full Modular symbols space of level 571, weight 2, and dimension 95 > C:=CS(M); > C; Modular symbols space of level 571, weight 2, and dimension 94 > time D:=DC(C,7); Time: 12.839 > D; [ Modular symbols space of level 571, weight 2, and dimension 4, Modular symbols space of level 571, weight 2, and dimension 2, Modular symbols space of level 571, weight 2, and dimension 2, Modular symbols space of level 571, weight 2, and dimension 4, Modular symbols space of level 571, weight 2, and dimension 6, Modular symbols space of level 571, weight 2, and dimension 8, Modular symbols space of level 571, weight 2, and dimension 12, Modular symbols space of level 571, weight 2, and dimension 20, Modular symbols space of level 571, weight 2, and dimension 36 ] > D:=SortDecomposition(D); > D; [ Modular symbols space of level 571, weight 2, and dimension 2, Modular symbols space of level 571, weight 2, and dimension 4, Modular symbols space of level 571, weight 2, and dimension 2, Modular symbols space of level 571, weight 2, and dimension 4, Modular symbols space of level 571, weight 2, and dimension 6, Modular symbols space of level 571, weight 2, and dimension 8, Modular symbols space of level 571, weight 2, and dimension 12, Modular symbols space of level 571, weight 2, and dimension 20, Modular symbols space of level 571, weight 2, and dimension 36 ] > A:=D[4]; > A; Modular symbols space of level 571, weight 2, and dimension 4 > time G:=ModularKernel(A);G; Time: 1.520 Abelian Group isomorphic to Z/6 + Z/6 + Z/762 + Z/762 Defined on 4 generators Relations: 6*G.1 = 0 6*G.2 = 0 762*G.3 = 0 762*G.4 = 0 > factor(762); [ <2, 1>, <3, 1>, <127, 1> ] 1 > f:=qEigenform(A,20); > f; q + q^2 - 2*q^3 - q^4 + (a - 3)*q^5 - 2*q^6 + 2*q^7 - 3*q^8 + q^9 + (a - 3)*q^10 + (-a + 4)*q^11 + 2*q^12 + a*q^13 + 2*q^14 + (-2*a + 6)*q^15 - q^16 + 4*q^17 + q^18 + (-2*a + 6)*q^19 + O(q^20) > Parent(f); Power series ring in q over Univariate Quotient Polynomial Algebra in a over Rational Field with modulus a^2 - 7*a + 3 > Discriminant(x^2-7*x+3); 37 > ClassNumber(37); 1 > factormod(x^2-37,3); [ <$.1 + 1, 1>, <$.1 + 2, 1> ] 1 > factormod(x^2-37,127); [ <$.1 + 52, 1>, <$.1 + 75, 1> ] 1 > factormod(x^2-7*x+3,127); [ <$.1 + 34, 1>, <$.1 + 86, 1> ] 1 > Tn(A,5); [ 3 7 -15 11] [ 3 7 -9 3] [ 9/2 15/2 -15 8] [ 9/2 9/2 -12 7] > IntegralBasis(A); [ {-1/440, 0} + -1*{-1/348, 0} + {-1/554, 0} + -1*{-1/465, 0} + -1*{-1/305, 0} + 2*{-1/241, 0} + -2*{-1/567, 0} + {-1/382, 0} + {-1/490, 0} + -2*{-1/270, 0} + 2*{-1/562, 0} + {-1/122, 0} + -2*{-1/507, 0} + -1*{-1/255, 0} + -2*{-1/424, 0} + 2*{-1/177, 0} + -2*{-1/494, 0} + 3*{-1/279, 0} + -1*{-1/537, 0} + {-1/399, 0} + -1*{-1/159, 0} + -1*{-1/482, 0} + -1*{-1/193, 0} + {-1/448, 0} + -1*{-1/455, 0} + -1*{-1/145, 0} + -1*{-1/234, 0} + {-1/295, 0} + {-1/300, 0} + {-1/218, 0} + -3*{-1/74, 0} + -1*{-1/515, 0} + 2*{-1/354, 0} + -1*{-1/226, 0} + {-1/328, 0} + -1*{-1/558, 0} + 2*{-1/503, 0} + -2*{-1/376, 0} + 2*{-1/157, 0} + -1*{-1/173, 0} + 2*{-1/339, 0} + {-1/469, 0} + {-1/137, 0} + -1*{-1/422, 0} + -1*{-1/314, 0} + {-1/30, 0} + -1*{-1/107, 0} + {-1/38, 0} + -1*{-1/527, 0} + -2*{-1/333, 0} + 3*{-1/519, 0} + -1*{-1/57, 0} + -1*{-1/163, 0} + -1*{-1/114, 0} + 2*{-1/428, 0} + {-1/190, 0} + -2*{-1/285, 0}, {-1/297, 0} + -2*{-1/532, 0} + {-1/440, 0} + -1*{-1/348, 0} + -1*{-1/554, 0} + {-1/465, 0} + 2*{-1/305, 0} + {-1/241, 0} + {-1/382, 0} + -1*{-1/270, 0} + 2*{-1/562, 0} + -1*{-1/358, 0} + -1*{-1/122, 0} + -1*{-1/507, 0} + -3*{-1/255, 0} + {-1/424, 0} + 2*{-1/177, 0} + {-1/248, 0} + -2*{-1/134, 0} + -1*{-1/320, 0} + {-1/387, 0} + {-1/279, 0} + -2*{-1/309, 0} + -1*{-1/537, 0} + 2*{-1/399, 0} + -1*{-1/430, 0} + {-1/159, 0} + -1*{-1/482, 0} + 2*{-1/193, 0} + {-1/529, 0} + -1*{-1/196, 0} + -1*{-1/448, 0} + 2*{-1/145, 0} + -1*{-1/396, 0} + -1*{-1/295, 0} + {-1/300, 0} + -2*{-1/218, 0} + {-1/74, 0} + {-1/515, 0} + 2*{-1/354, 0} + {-1/211, 0} + -2*{-1/203, 0} + {-1/558, 0} + -1*{-1/239, 0} + -1*{-1/376, 0} + {-1/157, 0} + 2*{-1/261, 0} + -2*{-1/173, 0} + -1*{-1/339, 0} + {-1/469, 0} + -2*{-1/549, 0} + {-1/137, 0} + -2*{-1/422, 0} + 2*{-1/545, 0} + -1*{-1/435, 0} + {-1/314, 0} + -1*{-1/30, 0} + 3*{-1/222, 0} + -1*{-1/403, 0} + -3*{-1/107, 0} + {-1/38, 0} + {-1/367, 0} + -1*{-1/527, 0} + -1*{-1/333, 0} + {-1/519, 0} + {-1/57, 0} + -1*{-1/214, 0} + {-1/163, 0} + -2*{-1/95, 0} + {-1/114, 0} + 3*{-1/428, 0} + -1*{-1/190, 0}, {-1/297, 0} + {-1/532, 0} + -2*{-1/348, 0} + 2*{-1/465, 0} + {-1/241, 0} + 2*{-1/567, 0} + -1*{-1/490, 0} + -1*{-1/358, 0} + {-1/507, 0} + {-1/177, 0} + {-1/248, 0} + -2*{-1/264, 0} + 2*{-1/391, 0} + -1*{-1/320, 0} + -1*{-1/387, 0} + -2*{-1/537, 0} + 3*{-1/399, 0} + -1*{-1/430, 0} + 2*{-1/482, 0} + {-1/193, 0} + -1*{-1/529, 0} + -1*{-1/196, 0} + 2*{-1/448, 0} + -1*{-1/455, 0} + 2*{-1/145, 0} + -3*{-1/396, 0} + 2*{-1/234, 0} + -2*{-1/295, 0} + {-1/300, 0} + -2*{-1/218, 0} + 2*{-1/74, 0} + -2*{-1/515, 0} + 2*{-1/354, 0} + -1*{-1/268, 0} + {-1/226, 0} + -1*{-1/328, 0} + -1*{-1/211, 0} + {-1/203, 0} + -2*{-1/239, 0} + {-1/376, 0} + {-1/157, 0} + 2*{-1/261, 0} + -2*{-1/487, 0} + -2*{-1/173, 0} + -2*{-1/148, 0} + -1*{-1/422, 0} + {-1/435, 0} + 2*{-1/314, 0} + -4*{-1/30, 0} + {-1/222, 0} + -1*{-1/403, 0} + {-1/38, 0} + -1*{-1/367, 0} + 3*{-1/519, 0} + -2*{-1/57, 0} + -1*{-1/444, 0} + 2*{-1/214, 0} + {-1/163, 0} + {-1/428, 0} + -2*{-1/190, 0} + {-1/285, 0}, {-1/297, 0} + -1*{-1/532, 0} + 5*{-1/554, 0} + -2*{-1/465, 0} + -2*{-1/241, 0} + -3*{-1/567, 0} + -1*{-1/382, 0} + {-1/490, 0} + -4*{-1/270, 0} + 5*{-1/562, 0} + -1*{-1/358, 0} + 3*{-1/150, 0} + {-1/122, 0} + -1*{-1/507, 0} + -2*{-1/255, 0} + -1*{-1/177, 0} + {-1/248, 0} + -3*{-1/134, 0} + 4*{-1/264, 0} + -3*{-1/391, 0} + -1*{-1/320, 0} + {-1/387, 0} + 4*{-1/279, 0} + {-1/405, 0} + -2*{-1/309, 0} + 2*{-1/537, 0} + -2*{-1/399, 0} + -4*{-1/430, 0} + {-1/159, 0} + 3*{-1/183, 0} + {-1/482, 0} + -3*{-1/193, 0} + 4*{-1/416, 0} + -3*{-1/529, 0} + {-1/196, 0} + 3*{-1/372, 0} + -3*{-1/455, 0} + -1*{-1/145, 0} + 2*{-1/396, 0} + 2*{-1/234, 0} + -2*{-1/295, 0} + 2*{-1/300, 0} + -1*{-1/218, 0} + {-1/74, 0} + -8*{-1/560, 0} + {-1/515, 0} + 4*{-1/354, 0} + -2*{-1/226, 0} + 4*{-1/328, 0} + -1*{-1/211, 0} + {-1/203, 0} + 4*{-1/558, 0} + -1*{-1/239, 0} + -1*{-1/376, 0} + -6*{-1/157, 0} + 2*{-1/261, 0} + 3*{-1/173, 0} + -1*{-1/339, 0} + -1*{-1/469, 0} + -2*{-1/148, 0} + -2*{-1/549, 0} + 5*{-1/137, 0} + 3*{-1/452, 0} + -3*{-1/422, 0} + -4*{-1/545, 0} + -2*{-1/314, 0} + {-1/30, 0} + 3*{-1/222, 0} + 2*{-1/403, 0} + -2*{-1/107, 0} + -2*{-1/38, 0} + {-1/367, 0} + 6*{-1/527, 0} + -3*{-1/333, 0} + -4*{-1/519, 0} + {-1/57, 0} + 3*{-1/444, 0} + -4*{-1/214, 0} + {-1/163, 0} + -2*{-1/95, 0} + 3*{-1/114, 0} + -2*{-1/428, 0} + 2*{-1/190, 0} + -1*{-1/285, 0} ] > Basis(A); [ {-1/297, 0} + 12*{-1/554, 0} + -6*{-1/465, 0} + -3*{-1/305, 0} + -3*{-1/241, 0} + -8*{-1/567, 0} + -2*{-1/382, 0} + 3*{-1/490, 0} + -9*{-1/270, 0} + 10*{-1/562, 0} + -1*{-1/358, 0} + 6*{-1/150, 0} + 4*{-1/122, 0} + -3*{-1/507, 0} + -2*{-1/255, 0} + -3*{-1/424, 0} + -2*{-1/177, 0} + {-1/248, 0} + -4*{-1/134, 0} + 8*{-1/264, 0} + -6*{-1/391, 0} + -1*{-1/320, 0} + {-1/387, 0} + -2*{-1/494, 0} + 10*{-1/279, 0} + 2*{-1/405, 0} + -2*{-1/309, 0} + 4*{-1/537, 0} + -5*{-1/399, 0} + -7*{-1/430, 0} + 6*{-1/183, 0} + 2*{-1/482, 0} + -9*{-1/193, 0} + 8*{-1/416, 0} + -7*{-1/529, 0} + 3*{-1/196, 0} + 6*{-1/372, 0} + 2*{-1/448, 0} + -7*{-1/455, 0} + -5*{-1/145, 0} + 5*{-1/396, 0} + 3*{-1/234, 0} + -2*{-1/295, 0} + 4*{-1/300, 0} + {-1/218, 0} + -2*{-1/74, 0} + -16*{-1/560, 0} + 8*{-1/354, 0} + -5*{-1/226, 0} + 9*{-1/328, 0} + -3*{-1/211, 0} + 4*{-1/203, 0} + 6*{-1/558, 0} + -1*{-1/239, 0} + 2*{-1/503, 0} + -3*{-1/376, 0} + -11*{-1/157, 0} + 2*{-1/261, 0} + 7*{-1/173, 0} + {-1/339, 0} + -2*{-1/469, 0} + -4*{-1/148, 0} + -2*{-1/549, 0} + 10*{-1/137, 0} + 6*{-1/452, 0} + -5*{-1/422, 0} + -10*{-1/545, 0} + {-1/435, 0} + -6*{-1/314, 0} + 4*{-1/30, 0} + 3*{-1/222, 0} + 5*{-1/403, 0} + -2*{-1/107, 0} + -4*{-1/38, 0} + {-1/367, 0} + 12*{-1/527, 0} + -7*{-1/333, 0} + -6*{-1/519, 0} + 6*{-1/444, 0} + -7*{-1/214, 0} + -2*{-1/95, 0} + 4*{-1/114, 0} + -5*{-1/428, 0} + 6*{-1/190, 0} + -4*{-1/285, 0}, {-1/532, 0} + 7*{-1/554, 0} + -4*{-1/465, 0} + -3*{-1/305, 0} + -1*{-1/241, 0} + -5*{-1/567, 0} + -1*{-1/382, 0} + 2*{-1/490, 0} + -5*{-1/270, 0} + 5*{-1/562, 0} + 3*{-1/150, 0} + 3*{-1/122, 0} + -2*{-1/507, 0} + -3*{-1/424, 0} + -1*{-1/177, 0} + -1*{-1/134, 0} + 4*{-1/264, 0} + -3*{-1/391, 0} + -2*{-1/494, 0} + 6*{-1/279, 0} + {-1/405, 0} + 2*{-1/537, 0} + -3*{-1/399, 0} + -3*{-1/430, 0} + -1*{-1/159, 0} + 3*{-1/183, 0} + {-1/482, 0} + -6*{-1/193, 0} + 4*{-1/416, 0} + -4*{-1/529, 0} + 2*{-1/196, 0} + 3*{-1/372, 0} + 2*{-1/448, 0} + -4*{-1/455, 0} + -4*{-1/145, 0} + 3*{-1/396, 0} + {-1/234, 0} + 2*{-1/300, 0} + 2*{-1/218, 0} + -3*{-1/74, 0} + -8*{-1/560, 0} + -1*{-1/515, 0} + 4*{-1/354, 0} + -3*{-1/226, 0} + 5*{-1/328, 0} + -2*{-1/211, 0} + 3*{-1/203, 0} + 2*{-1/558, 0} + 2*{-1/503, 0} + -2*{-1/376, 0} + -5*{-1/157, 0} + 4*{-1/173, 0} + 2*{-1/339, 0} + -1*{-1/469, 0} + -2*{-1/148, 0} + 5*{-1/137, 0} + 3*{-1/452, 0} + -2*{-1/422, 0} + -6*{-1/545, 0} + {-1/435, 0} + -4*{-1/314, 0} + 3*{-1/30, 0} + 3*{-1/403, 0} + -2*{-1/38, 0} + 6*{-1/527, 0} + -4*{-1/333, 0} + -2*{-1/519, 0} + -1*{-1/57, 0} + 3*{-1/444, 0} + -3*{-1/214, 0} + -1*{-1/163, 0} + {-1/114, 0} + -3*{-1/428, 0} + 4*{-1/190, 0} + -3*{-1/285, 0}, {-1/440, 0} + 21/2*{-1/554, 0} + -7*{-1/465, 0} + -4*{-1/305, 0} + -1/2*{-1/241, 0} + -19/2*{-1/567, 0} + -1/2*{-1/382, 0} + 4*{-1/490, 0} + -9*{-1/270, 0} + 19/2*{-1/562, 0} + 9/2*{-1/150, 0} + 9/2*{-1/122, 0} + -5*{-1/507, 0} + -2*{-1/255, 0} + -5*{-1/424, 0} + -5/2*{-1/134, 0} + 7*{-1/264, 0} + -11/2*{-1/391, 0} + {-1/387, 0} + -4*{-1/494, 0} + 11*{-1/279, 0} + 3/2*{-1/405, 0} + -1*{-1/309, 0} + 3*{-1/537, 0} + -9/2*{-1/399, 0} + -9/2*{-1/430, 0} + -3/2*{-1/159, 0} + 9/2*{-1/183, 0} + -1/2*{-1/482, 0} + -9*{-1/193, 0} + 6*{-1/416, 0} + -5*{-1/529, 0} + 3*{-1/196, 0} + 9/2*{-1/372, 0} + 2*{-1/448, 0} + -6*{-1/455, 0} + -13/2*{-1/145, 0} + 11/2*{-1/396, 0} + {-1/295, 0} + 7/2*{-1/300, 0} + 7/2*{-1/218, 0} + -13/2*{-1/74, 0} + -12*{-1/560, 0} + -1/2*{-1/515, 0} + 7*{-1/354, 0} + 1/2*{-1/268, 0} + -11/2*{-1/226, 0} + 17/2*{-1/328, 0} + -2*{-1/211, 0} + 3*{-1/203, 0} + 3*{-1/558, 0} + 1/2*{-1/239, 0} + 4*{-1/503, 0} + -5*{-1/376, 0} + -13/2*{-1/157, 0} + {-1/487, 0} + 11/2*{-1/173, 0} + 7/2*{-1/339, 0} + -1/2*{-1/469, 0} + -2*{-1/148, 0} + -1*{-1/549, 0} + 17/2*{-1/137, 0} + 9/2*{-1/452, 0} + -4*{-1/422, 0} + -8*{-1/545, 0} + 1/2*{-1/435, 0} + -7*{-1/314, 0} + 13/2*{-1/30, 0} + {-1/222, 0} + 9/2*{-1/403, 0} + -2*{-1/107, 0} + -5/2*{-1/38, 0} + {-1/367, 0} + 8*{-1/527, 0} + -15/2*{-1/333, 0} + -5/2*{-1/519, 0} + -1/2*{-1/57, 0} + 5*{-1/444, 0} + -6*{-1/214, 0} + -2*{-1/163, 0} + -1*{-1/95, 0} + 3/2*{-1/114, 0} + -5/2*{-1/428, 0} + 7*{-1/190, 0} + -6*{-1/285, 0}, {-1/348, 0} + 19/2*{-1/554, 0} + -6*{-1/465, 0} + -3*{-1/305, 0} + -5/2*{-1/241, 0} + -15/2*{-1/567, 0} + -3/2*{-1/382, 0} + 3*{-1/490, 0} + -7*{-1/270, 0} + 15/2*{-1/562, 0} + 9/2*{-1/150, 0} + 7/2*{-1/122, 0} + -3*{-1/507, 0} + -1*{-1/255, 0} + -3*{-1/424, 0} + -2*{-1/177, 0} + -5/2*{-1/134, 0} + 7*{-1/264, 0} + -11/2*{-1/391, 0} + {-1/387, 0} + -2*{-1/494, 0} + 8*{-1/279, 0} + 3/2*{-1/405, 0} + -1*{-1/309, 0} + 4*{-1/537, 0} + -11/2*{-1/399, 0} + -9/2*{-1/430, 0} + -1/2*{-1/159, 0} + 9/2*{-1/183, 0} + 1/2*{-1/482, 0} + -8*{-1/193, 0} + 6*{-1/416, 0} + -5*{-1/529, 0} + 3*{-1/196, 0} + 9/2*{-1/372, 0} + {-1/448, 0} + -5*{-1/455, 0} + -11/2*{-1/145, 0} + 11/2*{-1/396, 0} + {-1/234, 0} + 5/2*{-1/300, 0} + 5/2*{-1/218, 0} + -7/2*{-1/74, 0} + -12*{-1/560, 0} + 1/2*{-1/515, 0} + 5*{-1/354, 0} + 1/2*{-1/268, 0} + -9/2*{-1/226, 0} + 15/2*{-1/328, 0} + -2*{-1/211, 0} + 3*{-1/203, 0} + 4*{-1/558, 0} + 1/2*{-1/239, 0} + 2*{-1/503, 0} + -3*{-1/376, 0} + -17/2*{-1/157, 0} + {-1/487, 0} + 13/2*{-1/173, 0} + 3/2*{-1/339, 0} + -3/2*{-1/469, 0} + -2*{-1/148, 0} + -1*{-1/549, 0} + 15/2*{-1/137, 0} + 9/2*{-1/452, 0} + -3*{-1/422, 0} + -8*{-1/545, 0} + 1/2*{-1/435, 0} + -6*{-1/314, 0} + 11/2*{-1/30, 0} + {-1/222, 0} + 9/2*{-1/403, 0} + -1*{-1/107, 0} + -7/2*{-1/38, 0} + {-1/367, 0} + 9*{-1/527, 0} + -11/2*{-1/333, 0} + -11/2*{-1/519, 0} + 1/2*{-1/57, 0} + 5*{-1/444, 0} + -6*{-1/214, 0} + -1*{-1/163, 0} + -1*{-1/95, 0} + 5/2*{-1/114, 0} + -9/2*{-1/428, 0} + 6*{-1/190, 0} + -4*{-1/285, 0} ] > Basis(A)[1]*Tn(A,5); Runtime error in 'eq': Arguments have incompatible dimensions. > Basis(A)[1]*Tn(M,5); 3*{-1/297, 0} + 7*{-1/532, 0} + -15*{-1/440, 0} + 11*{-1/348, 0} + 32*{-1/554, 0} + -7*{-1/465, 0} + -3*{-1/305, 0} + -36*{-1/241, 0} + {-1/567, 0} + -22*{-1/382, 0} + -4*{-1/490, 0} + -4*{-1/270, 0} + 5*{-1/562, 0} + -3*{-1/358, 0} + 21*{-1/150, 0} + 4*{-1/122, 0} + 19*{-1/507, 0} + 13*{-1/255, 0} + 12*{-1/424, 0} + -35*{-1/177, 0} + 3*{-1/248, 0} + -9*{-1/134, 0} + 24*{-1/264, 0} + -17*{-1/391, 0} + -3*{-1/320, 0} + -1*{-1/387, 0} + 18*{-1/494, 0} + -5*{-1/279, 0} + 7*{-1/405, 0} + -2*{-1/309, 0} + 25*{-1/537, 0} + -29*{-1/399, 0} + -24*{-1/430, 0} + 10*{-1/159, 0} + 21*{-1/183, 0} + 26*{-1/482, 0} + -22*{-1/193, 0} + 28*{-1/416, 0} + -29*{-1/529, 0} + 11*{-1/196, 0} + 21*{-1/372, 0} + {-1/448, 0} + -14*{-1/455, 0} + -6*{-1/145, 0} + 14*{-1/396, 0} + 27*{-1/234, 0} + -21*{-1/295, 0} + {-1/300, 0} + -8*{-1/218, 0} + 32*{-1/74, 0} + -56*{-1/560, 0} + 6*{-1/515, 0} + 2*{-1/354, 0} + -2*{-1/268, 0} + -3*{-1/226, 0} + 17*{-1/328, 0} + -15*{-1/211, 0} + 21*{-1/203, 0} + 31*{-1/558, 0} + -5*{-1/239, 0} + -18*{-1/503, 0} + 19*{-1/376, 0} + -64*{-1/157, 0} + 6*{-1/261, 0} + -4*{-1/487, 0} + 38*{-1/173, 0} + -19*{-1/339, 0} + -22*{-1/469, 0} + -18*{-1/148, 0} + -2*{-1/549, 0} + 20*{-1/137, 0} + 21*{-1/452, 0} + -2*{-1/422, 0} + -40*{-1/545, 0} + 8*{-1/435, 0} + -7*{-1/314, 0} + -4*{-1/30, 0} + 5*{-1/222, 0} + 18*{-1/403, 0} + 13*{-1/107, 0} + -27*{-1/38, 0} + -1*{-1/367, 0} + 57*{-1/527, 0} + 3*{-1/333, 0} + -55*{-1/519, 0} + 6*{-1/57, 0} + 19*{-1/444, 0} + -18*{-1/214, 0} + 12*{-1/163, 0} + -2*{-1/95, 0} + 24*{-1/114, 0} + -48*{-1/428, 0} + 7*{-1/190, 0} + 13*{-1/285, 0} > v:=Basis(A); > 3*v[1]+7*v[2]-15*v[3]+11*v[4]; 3*{-1/297, 0} + 7*{-1/532, 0} + -15*{-1/440, 0} + 11*{-1/348, 0} + 32*{-1/554, 0} + -7*{-1/465, 0} + -3*{-1/305, 0} + -36*{-1/241, 0} + {-1/567, 0} + -22*{-1/382, 0} + -4*{-1/490, 0} + -4*{-1/270, 0} + 5*{-1/562, 0} + -3*{-1/358, 0} + 21*{-1/150, 0} + 4*{-1/122, 0} + 19*{-1/507, 0} + 13*{-1/255, 0} + 12*{-1/424, 0} + -35*{-1/177, 0} + 3*{-1/248, 0} + -9*{-1/134, 0} + 24*{-1/264, 0} + -17*{-1/391, 0} + -3*{-1/320, 0} + -1*{-1/387, 0} + 18*{-1/494, 0} + -5*{-1/279, 0} + 7*{-1/405, 0} + -2*{-1/309, 0} + 25*{-1/537, 0} + -29*{-1/399, 0} + -24*{-1/430, 0} + 10*{-1/159, 0} + 21*{-1/183, 0} + 26*{-1/482, 0} + -22*{-1/193, 0} + 28*{-1/416, 0} + -29*{-1/529, 0} + 11*{-1/196, 0} + 21*{-1/372, 0} + {-1/448, 0} + -14*{-1/455, 0} + -6*{-1/145, 0} + 14*{-1/396, 0} + 27*{-1/234, 0} + -21*{-1/295, 0} + {-1/300, 0} + -8*{-1/218, 0} + 32*{-1/74, 0} + -56*{-1/560, 0} + 6*{-1/515, 0} + 2*{-1/354, 0} + -2*{-1/268, 0} + -3*{-1/226, 0} + 17*{-1/328, 0} + -15*{-1/211, 0} + 21*{-1/203, 0} + 31*{-1/558, 0} + -5*{-1/239, 0} + -18*{-1/503, 0} + 19*{-1/376, 0} + -64*{-1/157, 0} + 6*{-1/261, 0} + -4*{-1/487, 0} + 38*{-1/173, 0} + -19*{-1/339, 0} + -22*{-1/469, 0} + -18*{-1/148, 0} + -2*{-1/549, 0} + 20*{-1/137, 0} + 21*{-1/452, 0} + -2*{-1/422, 0} + -40*{-1/545, 0} + 8*{-1/435, 0} + -7*{-1/314, 0} + -4*{-1/30, 0} + 5*{-1/222, 0} + 18*{-1/403, 0} + 13*{-1/107, 0} + -27*{-1/38, 0} + -1*{-1/367, 0} + 57*{-1/527, 0} + 3*{-1/333, 0} + -55*{-1/519, 0} + 6*{-1/57, 0} + 19*{-1/444, 0} + -18*{-1/214, 0} + 12*{-1/163, 0} + -2*{-1/95, 0} + 24*{-1/114, 0} + -48*{-1/428, 0} + 7*{-1/190, 0} + 13*{-1/285, 0} > Representation(Basis(A)[2]*Tn(M,5)); 3*{-1/297, 0} + 7*{-1/532, 0} + -9*{-1/440, 0} + 3*{-1/348, 0} + 19*{-1/554, 0} + -1*{-1/465, 0} + -3*{-1/305, 0} + -19*{-1/241, 0} + 4*{-1/567, 0} + -13*{-1/382, 0} + -4*{-1/490, 0} + -2*{-1/270, 0} + 2*{-1/562, 0} + -3*{-1/358, 0} + 12*{-1/150, 0} + 3*{-1/122, 0} + 13*{-1/507, 0} + 9*{-1/255, 0} + 6*{-1/424, 0} + -19*{-1/177, 0} + 3*{-1/248, 0} + -4*{-1/134, 0} + 10*{-1/264, 0} + -6*{-1/391, 0} + -3*{-1/320, 0} + -3*{-1/387, 0} + 10*{-1/494, 0} + -3*{-1/279, 0} + 4*{-1/405, 0} + 11*{-1/537, 0} + -12*{-1/399, 0} + -15*{-1/430, 0} + 5*{-1/159, 0} + 12*{-1/183, 0} + 19*{-1/482, 0} + -12*{-1/193, 0} + 16*{-1/416, 0} + -19*{-1/529, 0} + 5*{-1/196, 0} + 12*{-1/372, 0} + 5*{-1/448, 0} + -10*{-1/455, 0} + -1*{-1/145, 0} + 3*{-1/396, 0} + 19*{-1/234, 0} + -15*{-1/295, 0} + 2*{-1/300, 0} + -7*{-1/218, 0} + 21*{-1/74, 0} + -32*{-1/560, 0} + -1*{-1/515, 0} + 4*{-1/354, 0} + -3*{-1/268, 0} + 8*{-1/328, 0} + -11*{-1/211, 0} + 15*{-1/203, 0} + 17*{-1/558, 0} + -6*{-1/239, 0} + -10*{-1/503, 0} + 13*{-1/376, 0} + -35*{-1/157, 0} + 6*{-1/261, 0} + -6*{-1/487, 0} + 19*{-1/173, 0} + -10*{-1/339, 0} + -13*{-1/469, 0} + -14*{-1/148, 0} + 11*{-1/137, 0} + 12*{-1/452, 0} + -2*{-1/422, 0} + -24*{-1/545, 0} + 7*{-1/435, 0} + -1*{-1/314, 0} + -9*{-1/30, 0} + 3*{-1/222, 0} + 9*{-1/403, 0} + 9*{-1/107, 0} + -14*{-1/38, 0} + -3*{-1/367, 0} + 33*{-1/527, 0} + 2*{-1/333, 0} + -26*{-1/519, 0} + -1*{-1/57, 0} + 9*{-1/444, 0} + -6*{-1/214, 0} + 8*{-1/163, 0} + 13*{-1/114, 0} + -27*{-1/428, 0} + {-1/190, 0} + 9*{-1/285, 0} > Representation(Basis(A)[2]*Tn(M,5)); (3 7 -9 3 19 -1 -3 -19 4 -13 -4 -2 2 -3 12 3 13 9 6 -19 3 -4 10 -6 -3 -3 10 -3 4 0 11 -12 -15 5 12 19 -12 16 -19 5 12 5 -10 -1 3 19 -15 2 -7 21 -32 -1 4 -3 0 8 -11 15 17 -6 -10 13 -35 6 -6 19 -10 -13 -14 0 11 12 -2 -24 7 -1 -9 3 9 9 -14 -3 33 2 -26 -1 9 -6 8 0 13 -27 1 9 0) > Representation(Basis(A)[1]*Tn(M,5)); (3 7 -15 11 32 -7 -3 -36 1 -22 -4 -4 5 -3 21 4 19 13 12 -35 3 -9 24 -17 -3 -1 18 -5 7 -2 25 -29 -24 10 21 26 -22 28 -29 11 21 1 -14 -6 14 27 -21 1 -8 32 -56 6 2 -2 -3 17 -15 21 31 -5 -18 19 -64 6 -4 38 -19 -22 -18 -2 20 21 -2 -40 8 -7 -4 5 18 13 -27 -1 57 3 -55 6 19 -18 12 -2 24 -48 7 13 0) > Representation(Basis(A)[2]*Tn(M,5)); (3 7 -9 3 19 -1 -3 -19 4 -13 -4 -2 2 -3 12 3 13 9 6 -19 3 -4 10 -6 -3 -3 10 -3 4 0 11 -12 -15 5 12 19 -12 16 -19 5 12 5 -10 -1 3 19 -15 2 -7 21 -32 -1 4 -3 0 8 -11 15 17 -6 -10 13 -35 6 -6 19 -10 -13 -14 0 11 12 -2 -24 7 -1 -9 3 9 9 -14 -3 33 2 -26 -1 9 -6 8 0 13 -27 1 9 0) > Representation(Basis(A)[3]*Tn(M,5)); (9/2 15/2 -15 8 25 0 0 -67/2 9 -21 -15/2 1 0 -9/2 18 1 45/2 13 15 -65/2 9/2 -8 17 -11 -9/2 -5/2 20 -11 6 -2 20 -43/2 -45/2 11 18 28 -29/2 24 -53/2 15/2 18 2 -23/2 1 13/2 29 -24 1/2 -13 38 -48 4 1 -7/2 3/2 21/2 -29/2 39/2 29 -8 -20 45/2 -115/2 9 -7 31 -21 -21 -19 -2 15 18 -3/2 -34 17/2 0 -13 13/2 27/2 13 -47/2 -5/2 51 7 -97/2 4 29/2 -12 29/2 -2 23 -87/2 0 35/2 0) > Representation(Basis(A)[4]*Tn(M,5)); (9/2 9/2 -12 7 26 -3 0 -59/2 3 -18 -9/2 -4 6 -9/2 18 2 33/2 8 12 -55/2 9/2 -10 19 -13 -9/2 -1/2 16 -4 6 -4 19 -41/2 -45/2 10 18 23 -31/2 24 -49/2 15/2 18 1 -25/2 -1 17/2 25 -21 5/2 -11 31 -48 5 5 -5/2 -3/2 27/2 -25/2 33/2 28 -7 -16 33/2 -107/2 9 -5 29 -18 -18 -17 -4 18 18 -9/2 -32 13/2 -3 -8 17/2 27/2 8 -43/2 -1/2 48 2 -89/2 5 31/2 -15 25/2 -4 22 -75/2 3 25/2 0) > for i in [1..4] do Representation(IntegralBasis(A)[i]*Tn(M,5)); end for; (0 3 -3 1 -1 3 0 -4 6 -3 -3 5 -6 0 0 -1 6 5 3 -5 0 2 -2 2 0 -2 4 -7 0 2 1 -1 0 1 0 5 1 0 -2 0 0 1 1 2 -2 4 -3 -2 -2 7 0 -1 -4 -1 3 -3 -2 3 1 -1 -4 6 -4 0 -2 2 -3 -3 -2 2 -3 0 3 -2 2 3 -5 -2 0 5 -2 -2 3 5 -4 -1 -1 3 2 2 1 -6 -3 5 0) (-3 -4 0 6 -7 -2 3 -2 -1 1 1 5 -5 3 -3 -3 -1 0 3 -2 -3 1 2 -3 3 3 2 -6 -1 0 4 -6 6 1 -3 -7 3 -4 7 1 -3 -8 7 -2 6 -7 6 -5 4 -3 8 7 -10 3 0 -2 5 -6 -2 6 -2 -1 2 -6 6 2 -2 1 8 0 -5 -3 5 6 -4 -2 9 -3 0 0 -1 3 -6 4 -7 7 0 -3 -2 0 -1 0 2 0 0) (-3 5 0 0 -1 -2 -6 4 -1 1 1 2 -5 3 -3 3 -1 6 -6 1 -3 7 -4 3 3 -3 -4 0 -1 6 -2 0 6 -5 -3 -1 -3 -4 1 1 -3 4 1 -5 0 -4 6 -2 7 -9 8 -5 -4 0 0 -2 -1 3 -8 3 4 -1 8 -6 0 -1 7 1 2 6 -5 -3 5 0 2 -2 3 -9 0 6 2 -3 -6 1 8 -5 -3 6 -5 6 -7 0 2 -3 0) (0 0 -6 8 13 -6 0 -17 -3 -9 0 -2 3 0 9 1 6 4 6 -16 0 -5 14 -11 0 2 8 -2 3 -2 14 -17 -9 5 9 7 -10 12 -10 6 9 -4 -4 -5 11 8 -6 -1 -1 11 -24 7 -2 1 -3 9 -4 6 14 1 -8 6 -29 0 2 19 -9 -9 -4 -2 9 9 0 -16 1 -6 5 2 9 4 -13 2 24 1 -29 7 10 -12 4 -2 11 -21 6 4 0) > for i in [1..4] do Representation(IntegralBasis(A)[i]); end for; (0 0 1 -1 1 -1 -1 2 -2 1 1 -2 2 0 0 1 -2 -1 -2 2 0 0 0 0 0 0 -2 3 0 0 -1 1 0 -1 0 -1 -1 0 0 0 0 1 -1 -1 0 -1 1 1 1 -3 0 -1 2 0 -1 1 0 0 -1 0 2 -2 2 0 0 -1 2 1 0 0 1 0 -1 0 0 -1 1 0 0 -1 1 0 -1 -2 3 -1 0 0 -1 0 -1 2 1 -2 0) (1 -2 1 -1 -1 1 2 1 0 1 0 -1 2 -1 0 -1 -1 -3 1 2 1 -2 0 0 -1 1 0 1 0 -2 -1 2 -1 1 0 -1 2 0 1 -1 0 -1 0 2 -1 0 -1 1 -2 1 0 1 2 0 0 0 1 -2 1 -1 0 -1 1 2 0 -2 -1 1 0 -2 1 0 -2 2 -1 1 -1 3 -1 -3 1 1 -1 -1 1 1 0 -1 1 -2 1 3 -1 0 0) (1 1 0 -2 0 2 0 1 2 0 -1 0 0 -1 0 0 1 0 0 1 1 0 -2 2 -1 -1 0 0 0 0 -2 3 -1 0 0 2 1 0 -1 -1 0 2 -1 2 -3 2 -2 1 -2 2 0 -2 2 -1 1 -1 -1 1 0 -2 0 1 1 2 -2 -2 0 0 -2 0 0 0 -1 0 1 2 -4 1 -1 0 1 -1 0 0 3 -2 -1 2 1 0 0 1 -2 1 0) (1 -1 0 0 5 -2 0 -2 -3 -1 1 -4 5 -1 3 1 -1 -2 0 -1 1 -3 4 -3 -1 1 0 4 1 -2 2 -2 -4 1 3 1 -3 4 -3 1 3 0 -3 -1 2 2 -2 2 -1 1 -8 1 4 0 -2 4 -1 1 4 -1 0 -1 -6 2 0 3 -1 -1 -2 -2 5 3 -3 -4 0 -2 1 3 2 -2 -2 1 6 -3 -4 1 3 -4 1 -2 3 -2 2 -1 0) > // Restriction of A to x. function Restrict(A,x) F := BaseRing(Parent(A)); if Type(x) eq SeqEnum then B := x; else if Dimension(x) eq 0 then return MatrixAlgebra(F,0)!0; end if; Z := Basis(x); V := VectorSpace(F, Degree(x)); B := [V!Z[i] : i in [1..Dimension(x)]]; end if; S := RSpaceWithBasis(B); v := [Coordinates(S, S.i*A) : i in [1..#B]]; return MatrixAlgebra(F,#B) ! (&cat v); end function; > function> function> function|if> function|if> function|if> function|if|if> function|if|if> function|if> function|if> function|if> function|if> function> function> function> function> > > > > Restrict; function(A, x) ... end function > B:=[Representation(IntegralBasis(A)[i]) : i in [1..4]]; > Restrict(Tn(M,5),B); [-2 -1 1 0] [-1 1 -3 -1] [ 2 -2 0 -1] [-4 -2 -1 3] > T5:=Restrict(Tn(M,5),B); > factor(charpoly(T5)); [ ] 1 > factormod(x^2 - x - 9,127); [ <$.1 + 37, 1>, <$.1 + 89, 1> ] 1 > Discriminant(x^2 - x - 9); 37 > Restrict(Tn(M,11),B); [ 3 1 -1 0] [ 1 0 3 1] [-2 2 1 1] [ 4 2 1 -2] > O:=MaximalOrder(NumberField(x^2 - x - 9)); > O:=MaximalOrder(NumberField(x^2 - x - 9)); > O:=MaximalOrder(NumberField(x^2 - x - 9)); > O; Maximal Equation Order with defining polynomial x^2 - x - 9 over Z > Factorization(Ideal(O,127)); > Z:=Factorization(ideal); [ , ] > IsPrincipal(ideal); true > Z:=Factorization(ideal); > IsPrincipal(Z[1][1]); true > z0,z1:=IsPrincipal(Z[1][1]); > z1; 3*$.1 - 16 > Norm(z1); 127 > > S:=3*T5-16; > S; [-22 -3 3 0] [ -3 -13 -9 -3] [ 6 -6 -16 -3] [-12 -6 -3 -7] > fcp(S); [ ] 1 > G; Abelian Group isomorphic to Z/6 + Z/6 + Z/762 + Z/762 Defined on 4 generators Relations: 6*G.1 = 0 6*G.2 = 0 762*G.3 = 0 762*G.4 = 0 > D; [ Modular symbols space of level 571, weight 2, and dimension 2, Modular symbols space of level 571, weight 2, and dimension 4, Modular symbols space of level 571, weight 2, and dimension 2, Modular symbols space of level 571, weight 2, and dimension 4, Modular symbols space of level 571, weight 2, and dimension 6, Modular symbols space of level 571, weight 2, and dimension 8, Modular symbols space of level 571, weight 2, and dimension 12, Modular symbols space of level 571, weight 2, and dimension 20, Modular symbols space of level 571, weight 2, and dimension 36 ] > [#IntersectionGroup(D[i],A) : i in [1..#D] | D[i] ne A]; [ 1, 16, 9, 1, 9, 1, 1, 16129 ] > factor(16129); [ <127, 2> ] 1 > B:=D[#D]; > B; Modular symbols space of level 571, weight 2, and dimension 36 > IntersectionGroup(A,B); Abelian Group isomorphic to Z/127 + Z/127 Defined on 2 generators Relations: 127*$.1 = 0 127*$.2 = 0 > IntegralBasis(A); [ {-1/440, 0} + -1*{-1/348, 0} + {-1/554, 0} + -1*{-1/465, 0} + -1*{-1/305, 0} + 2*{-1/241, 0} + -2*{-1/567, 0} + {-1/382, 0} + {-1/490, 0} + -2*{-1/270, 0} + 2*{-1/562, 0} + {-1/122, 0} + -2*{-1/507, 0} + -1*{-1/255, 0} + -2*{-1/424, 0} + 2*{-1/177, 0} + -2*{-1/494, 0} + 3*{-1/279, 0} + -1*{-1/537, 0} + {-1/399, 0} + -1*{-1/159, 0} + -1*{-1/482, 0} + -1*{-1/193, 0} + {-1/448, 0} + -1*{-1/455, 0} + -1*{-1/145, 0} + -1*{-1/234, 0} + {-1/295, 0} + {-1/300, 0} + {-1/218, 0} + -3*{-1/74, 0} + -1*{-1/515, 0} + 2*{-1/354, 0} + -1*{-1/226, 0} + {-1/328, 0} + -1*{-1/558, 0} + 2*{-1/503, 0} + -2*{-1/376, 0} + 2*{-1/157, 0} + -1*{-1/173, 0} + 2*{-1/339, 0} + {-1/469, 0} + {-1/137, 0} + -1*{-1/422, 0} + -1*{-1/314, 0} + {-1/30, 0} + -1*{-1/107, 0} + {-1/38, 0} + -1*{-1/527, 0} + -2*{-1/333, 0} + 3*{-1/519, 0} + -1*{-1/57, 0} + -1*{-1/163, 0} + -1*{-1/114, 0} + 2*{-1/428, 0} + {-1/190, 0} + -2*{-1/285, 0}, {-1/297, 0} + -2*{-1/532, 0} + {-1/440, 0} + -1*{-1/348, 0} + -1*{-1/554, 0} + {-1/465, 0} + 2*{-1/305, 0} + {-1/241, 0} + {-1/382, 0} + -1*{-1/270, 0} + 2*{-1/562, 0} + -1*{-1/358, 0} + -1*{-1/122, 0} + -1*{-1/507, 0} + -3*{-1/255, 0} + {-1/424, 0} + 2*{-1/177, 0} + {-1/248, 0} + -2*{-1/134, 0} + -1*{-1/320, 0} + {-1/387, 0} + {-1/279, 0} + -2*{-1/309, 0} + -1*{-1/537, 0} + 2*{-1/399, 0} + -1*{-1/430, 0} + {-1/159, 0} + -1*{-1/482, 0} + 2*{-1/193, 0} + {-1/529, 0} + -1*{-1/196, 0} + -1*{-1/448, 0} + 2*{-1/145, 0} + -1*{-1/396, 0} + -1*{-1/295, 0} + {-1/300, 0} + -2*{-1/218, 0} + {-1/74, 0} + {-1/515, 0} + 2*{-1/354, 0} + {-1/211, 0} + -2*{-1/203, 0} + {-1/558, 0} + -1*{-1/239, 0} + -1*{-1/376, 0} + {-1/157, 0} + 2*{-1/261, 0} + -2*{-1/173, 0} + -1*{-1/339, 0} + {-1/469, 0} + -2*{-1/549, 0} + {-1/137, 0} + -2*{-1/422, 0} + 2*{-1/545, 0} + -1*{-1/435, 0} + {-1/314, 0} + -1*{-1/30, 0} + 3*{-1/222, 0} + -1*{-1/403, 0} + -3*{-1/107, 0} + {-1/38, 0} + {-1/367, 0} + -1*{-1/527, 0} + -1*{-1/333, 0} + {-1/519, 0} + {-1/57, 0} + -1*{-1/214, 0} + {-1/163, 0} + -2*{-1/95, 0} + {-1/114, 0} + 3*{-1/428, 0} + -1*{-1/190, 0}, {-1/297, 0} + {-1/532, 0} + -2*{-1/348, 0} + 2*{-1/465, 0} + {-1/241, 0} + 2*{-1/567, 0} + -1*{-1/490, 0} + -1*{-1/358, 0} + {-1/507, 0} + {-1/177, 0} + {-1/248, 0} + -2*{-1/264, 0} + 2*{-1/391, 0} + -1*{-1/320, 0} + -1*{-1/387, 0} + -2*{-1/537, 0} + 3*{-1/399, 0} + -1*{-1/430, 0} + 2*{-1/482, 0} + {-1/193, 0} + -1*{-1/529, 0} + -1*{-1/196, 0} + 2*{-1/448, 0} + -1*{-1/455, 0} + 2*{-1/145, 0} + -3*{-1/396, 0} + 2*{-1/234, 0} + -2*{-1/295, 0} + {-1/300, 0} + -2*{-1/218, 0} + 2*{-1/74, 0} + -2*{-1/515, 0} + 2*{-1/354, 0} + -1*{-1/268, 0} + {-1/226, 0} + -1*{-1/328, 0} + -1*{-1/211, 0} + {-1/203, 0} + -2*{-1/239, 0} + {-1/376, 0} + {-1/157, 0} + 2*{-1/261, 0} + -2*{-1/487, 0} + -2*{-1/173, 0} + -2*{-1/148, 0} + -1*{-1/422, 0} + {-1/435, 0} + 2*{-1/314, 0} + -4*{-1/30, 0} + {-1/222, 0} + -1*{-1/403, 0} + {-1/38, 0} + -1*{-1/367, 0} + 3*{-1/519, 0} + -2*{-1/57, 0} + -1*{-1/444, 0} + 2*{-1/214, 0} + {-1/163, 0} + {-1/428, 0} + -2*{-1/190, 0} + {-1/285, 0}, {-1/297, 0} + -1*{-1/532, 0} + 5*{-1/554, 0} + -2*{-1/465, 0} + -2*{-1/241, 0} + -3*{-1/567, 0} + -1*{-1/382, 0} + {-1/490, 0} + -4*{-1/270, 0} + 5*{-1/562, 0} + -1*{-1/358, 0} + 3*{-1/150, 0} + {-1/122, 0} + -1*{-1/507, 0} + -2*{-1/255, 0} + -1*{-1/177, 0} + {-1/248, 0} + -3*{-1/134, 0} + 4*{-1/264, 0} + -3*{-1/391, 0} + -1*{-1/320, 0} + {-1/387, 0} + 4*{-1/279, 0} + {-1/405, 0} + -2*{-1/309, 0} + 2*{-1/537, 0} + -2*{-1/399, 0} + -4*{-1/430, 0} + {-1/159, 0} + 3*{-1/183, 0} + {-1/482, 0} + -3*{-1/193, 0} + 4*{-1/416, 0} + -3*{-1/529, 0} + {-1/196, 0} + 3*{-1/372, 0} + -3*{-1/455, 0} + -1*{-1/145, 0} + 2*{-1/396, 0} + 2*{-1/234, 0} + -2*{-1/295, 0} + 2*{-1/300, 0} + -1*{-1/218, 0} + {-1/74, 0} + -8*{-1/560, 0} + {-1/515, 0} + 4*{-1/354, 0} + -2*{-1/226, 0} + 4*{-1/328, 0} + -1*{-1/211, 0} + {-1/203, 0} + 4*{-1/558, 0} + -1*{-1/239, 0} + -1*{-1/376, 0} + -6*{-1/157, 0} + 2*{-1/261, 0} + 3*{-1/173, 0} + -1*{-1/339, 0} + -1*{-1/469, 0} + -2*{-1/148, 0} + -2*{-1/549, 0} + 5*{-1/137, 0} + 3*{-1/452, 0} + -3*{-1/422, 0} + -4*{-1/545, 0} + -2*{-1/314, 0} + {-1/30, 0} + 3*{-1/222, 0} + 2*{-1/403, 0} + -2*{-1/107, 0} + -2*{-1/38, 0} + {-1/367, 0} + 6*{-1/527, 0} + -3*{-1/333, 0} + -4*{-1/519, 0} + {-1/57, 0} + 3*{-1/444, 0} + -4*{-1/214, 0} + {-1/163, 0} + -2*{-1/95, 0} + 3*{-1/114, 0} + -2*{-1/428, 0} + 2*{-1/190, 0} + -1*{-1/285, 0} ] > L1L2:=IntegralBasis(A+B); > L1:=IntegralBasis(A); > L2:=IntegralBasis(B); > bL1:=[Representation(L1[i]) : i in [1..#L1]]; > bL1; [ (0 0 1 -1 1 -1 -1 2 -2 1 1 -2 2 0 0 1 -2 -1 -2 2 0 0 0 0 0 0 -2 3 0 0 -1 1 0 -1 0 -1 -1 0 0 0 0 1 -1 -1 0 -1 1 1 1 -3 0 -1 2 0 -1 1 0 0 -1 0 2 -2 2 0 0 -1 2 1 0 0 1 0 -1 0 0 -1 1 0 0 -1 1 0 -1 -2 3 -1 0 0 -1 0 -1 2 1 -2 0), (1 -2 1 -1 -1 1 2 1 0 1 0 -1 2 -1 0 -1 -1 -3 1 2 1 -2 0 0 -1 1 0 1 0 -2 -1 2 -1 1 0 -1 2 0 1 -1 0 -1 0 2 -1 0 -1 1 -2 1 0 1 2 0 0 0 1 -2 1 -1 0 -1 1 2 0 -2 -1 1 0 -2 1 0 -2 2 -1 1 -1 3 -1 -3 1 1 -1 -1 1 1 0 -1 1 -2 1 3 -1 0 0), (1 1 0 -2 0 2 0 1 2 0 -1 0 0 -1 0 0 1 0 0 1 1 0 -2 2 -1 -1 0 0 0 0 -2 3 -1 0 0 2 1 0 -1 -1 0 2 -1 2 -3 2 -2 1 -2 2 0 -2 2 -1 1 -1 -1 1 0 -2 0 1 1 2 -2 -2 0 0 -2 0 0 0 -1 0 1 2 -4 1 -1 0 1 -1 0 0 3 -2 -1 2 1 0 0 1 -2 1 0), (1 -1 0 0 5 -2 0 -2 -3 -1 1 -4 5 -1 3 1 -1 -2 0 -1 1 -3 4 -3 -1 1 0 4 1 -2 2 -2 -4 1 3 1 -3 4 -3 1 3 0 -3 -1 2 2 -2 2 -1 1 -8 1 4 0 -2 4 -1 1 4 -1 0 -1 -6 2 0 3 -1 -1 -2 -2 5 3 -3 -4 0 -2 1 3 2 -2 -2 1 6 -3 -4 1 3 -4 1 -2 3 -2 2 -1 0) ] > bL2:=[Representation(L2[i]) : i in [1..#L2]]; > bL1L2:=[Representation(L1L2[i]) : i in [1..#L1L2]]; > v:=bL1L2[1]; > v; (0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 -1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 -1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 -1 0 1 0 0 0) > basis:=bL1 cat bL2; > Coordinates(VectorSpaceWithBasis(basis),v); [ 3/127, 10/127, 4/127, -6/127, -26/127, -26/127, -46/127, 19/127, -21/127, 4/127, 113/127, -14/127, 2/127, -54/127, -3/127, -10/127, -19/127, 32/127, -20/127, 14/127, -80/127, -11/127, -12/127, -8/127, -5/127, 44/127, -68/127, 2/127, 59/127, -24/127, 12/127, -8/127, 76/127, 25/127, 6/127, 18/127, 15/127, 40/127, -2/127, 3/127 ] > Coordinates(VectorSpaceWithBasis(basis),bL1L2[2]); [ -5/127, -16/127, 16/127, 0, 97/127, -20/127, 51/127, 21/127, 53/127, -39/127, 52/127, 16/127, 48/127, -60/127, 15/127, 3/127, 21/127, -29/127, 1/127, 18/127, -35/127, -11/127, 21/127, 0, -29/127, 39/127, -24/127, 26/127, 54/127, 0, 55/127, 0, 44/127, 21/127, -42/127, -65/127, -53/127, 26/127, 9/127, 6/127 ] > a:=(1/127)*(VectorSpace(Q,4)![3,10,4,-6]); > b:=(1/127)*(VectorSpace(Q,4)![-5,-16/127,16/127,0]); > a; ( 3/127 10/127 4/127 -6/127) > b; ( -5/127 -16/16129 16/16129 0) > T5; [-2 -1 1 0] [-1 1 -3 -1] [ 2 -2 0 -1] [-4 -2 -1 3] > a*T5; ( 16/127 11/127 -21/127 -32/127) > S; [-22 -3 3 0] [ -3 -13 -9 -3] [ 6 -6 -16 -3] [-12 -6 -3 -7] > a*S; ( 0 -1 -1 0) > b*S; (14114/16129 2017/16129 -2017/16129 0) > factor(16129); [ <127, 2> ] 1 > Sprime:=-29-S; > MinimalPolynomial(Sprime); x^2 + 29*x + 127 > MinimalPolynomial(S); x^2 + 29*x + 127 > a*Sprime; ( -87/127 -163/127 11/127 174/127) > b*Sprime; ( 4301/16129 -1553/16129 1553/16129 0) > Sprime; [ -7 3 -3 0] [ 3 -16 9 3] [ -6 6 -13 3] [ 12 6 3 -22] > a; ( 3/127 10/127 4/127 -6/127) > b; ( -5/127 -16/16129 16/16129 0) > b:=(1/127)*(VectorSpace(Q,4)![-5,-16,16,0]); > b*S; ( 2 1 -1 0) > S; [-22 -3 3 0] [ -3 -13 -9 -3] [ 6 -6 -16 -3] [-12 -6 -3 -7] > Roots(PolynomialRing(RealField())!x^2 + 29*x + 127); [ <-23.624143795447329533499526367803100593, 1>, <-5.3758562045526704665004736321968994069, 1> ] > Roots(PolynomialRing(RealField())!x^2 + 29*x + 127); [ <-23.624143795447329533499526367803100593, 1>, <-5.3758562045526704665004736321968994069, 1> ] > MinimalPolynomial(-S); x^2 - 29*x + 127 > Roots(PolynomialRing(RealField())!x^2 - 29*x + 127); [ <5.3758562045526704665004736321968994069, 1>, <23.624143795447329533499526367803100593, 1> ] > a*S; ( 0 -1 -1 0) > b*S; ( 2 1 -1 0) > /// J_0(43) > M:=MS(43); > C:=CS(M); > D:=ND(C); > D; [ Modular symbols space of level 43, weight 2, and dimension 2, Modular symbols space of level 43, weight 2, and dimension 4 ] > A:=D[2]; > ModularKernel(A); Abelian Group isomorphic to Z/2 + Z/2 Defined on 2 generators Relations: 2*$.1 = 0 2*$.2 = 0 > // However, however says "However, there are no endos of degree 4 that are totally positive." > // However, with thought, however says, "However, I'm totally wrong! 2-sqrt(2)". > L1:=IntegralBasis(A); bL1:=[Representation(L1[i]) : i in [1..#L1]]; > bL1; [ (0 1 0 0 0 0 0), (0 0 0 1 0 0 0), (0 0 0 0 0 1 0), ( 1 0 -1 0 1 0 0) ] > Basis(A); > Basis(A); [ {-1/25, 0} + -1*{-1/17, 0} + {-1/14, 0}, {-1/39, 0}, {-1/32, 0}, {-1/21, 0} ] > qEigenform(A,17); q + a*q^2 - a*q^3 + (-a + 2)*q^5 - 2*q^6 + (a - 2)*q^7 - 2*a*q^8 - q^9 + (2*a - 2)*q^10 + (2*a - 1)*q^11 + (2*a + 1)*q^13 + (-2*a + 2)*q^14 + (-2*a + 2)*q^15 - 4*q^16 + O(q^17) > Parent($1); Power series ring in q over Univariate Quotient Polynomial Algebra in a over Rational Field with modulus a^2 - 2 > Restrict(Tn(M,2),bL1); [ 1 0 0 -1] [ 0 1 1 -1] [-1 1 -1 0] [-1 0 0 -1] > T2:=$1; > S:=2-T2; > MinimalPolynomial(S); x^2 - 4*x + 2 > Roots(PolynomialRing(RealField())!x^2 - 4*x + 2); [ <0.58578643762690495119831127579030192143, 1>, <3.4142135623730950488016887242096980785, 1> ] > B:=D[1]; > IntersectionGroup(A,B); Abelian Group isomorphic to Z/2 + Z/2 Defined on 2 generators Relations: 2*$.1 = 0 2*$.2 = 0 > L1L2:=IntegralBasis(A+B); > bL1L2:=[Representation(L1L2[i]) : i in [1..#L1L2]]; > L2:=IntegralBasis(B); > bL1:=[Representation(L1[i]) : i in [1..#L1]]; > bL2:=[Representation(L2[i]) : i in [1..#L2]]; > basis:=bL1 cat bL2; > Coordinates(VectorSpaceWithBasis(basis),bL1L2[1]); [ 1/2, 0, 0, 1/2, 1/2, 0 ] > Coordinates(VectorSpaceWithBasis(basis),bL1L2[2]); [ 1, 0, 0, 0, 0, 0 ] > Coordinates(VectorSpaceWithBasis(basis),bL1L2[3]); [ 0, 1/2, 1/2, -1/2, 0, 1/2 ] > a:=VectorSpace(Q,4)![ 1/2, 0, 0, 1/2]; > b:=VectorSpace(Q,4)![ 0, 1/2, 1/2, -1/2];a:=VectorSpace(Q,4)![ 1/2, 0, 0, 1/2]; > a; (1/2 0 0 1/2) > b; ( 0 1/2 1/2 -1/2) > a*S; (1 0 0 2) > b*S; ( 0 0 1 -1) > LRatio(A,1); 2/7 > TamagawaNumber(A,43); 7 > TorsionBound(A,11); 7 [ 196, 196, 9604, 25921 ] > Periods(A,97); [ (0.5250281159132219433729491620 + 0.8066018577029307230283142376*i -0.2259499583067642118739519173 - 1.766644676299599532273333141*i), (0.8241062742261960348649172089 - 0.1534409622571770568748354978*i 0.5981563162241222986475767259 - 1.920085638612119493276485634*i), (-0.2990781583129740914919680469 - 0.9600428199601077799031497354*i -0.8241062745308865105215286433 - 0.1534409623125199610031524930*i), (-0.8241062742261960348649172090 - 0.1534409622571770568748354978*i -0.5981563162241222986475767260 - 1.920085638612119493276485634*i) ] > IntegralBasis(A); [ {-1/39, 0}, {-1/32, 0}, {-1/21, 0}, {-1/25, 0} + -1*{-1/17, 0} + {-1/14, 0} ] > 20^7*8; 10240000000 > 10^7*8; 80000000 > 20^6; 64000000 > C:=HyperellipticCurve(x^6+2*x+1); > C; Hyperelliptic Curve defined by y^2 = x^6 + 2*x + 1 over Rational Field > J:=Jacobian(C); > EulerFactor(J,7); 49*T^4 + 7*T^3 + 9*T^2 + T + 1 > EulerFactor(J,97); 9409*T^4 + 582*T^3 + 26*T^2 + 6*T + 1 > time C:=HyperellipticCurve(x^6+12*x+191); time J:=Jacobian(C); time [EulerFactor(J,p) : p in [11,17,37,97]]; Time: 0.009 Time: 0.000 [ 121*T^4 + 77*T^3 + 28*T^2 + 7*T + 1, 289*T^4 - 17*T^3 + 16*T^2 - T + 1, 1369*T^4 - 74*T^3 + 46*T^2 - 2*T + 1, 9409*T^4 - 1649*T^3 + 187*T^2 - 17*T + 1 ] Time: 0.039 > 0.1*20^6; 6399999.999999999999999999999 > $1/(3600); 1777.777777777777777777777777 > 1777/24.0; 74.04166666666666666666666666 > save "showing.session"; Saving Magma state to "showing.session" > quit; Total time: 34.369 seconds [was@descent doc]$ pwd /home/was/modsym/doc [was@descent doc]$ ls NumThy handbook limits overview paragraph showing.session tutorial [was@descent doc]$ ls *.session showing.session [was@descent doc]$ /usr/bin/ls bash: /usr/bin/ls: No such file or directory [was@descent doc]$ /bin/ls NumThy handbook limits overview paragraph showing.session tutorial [was@descent doc]$ cp *.session /home/was/people/howe [was@descent doc]$ mv *.session mv: missing file argument Try `mv --help' for more information. [was@descent doc]$ /bin/ls NumThy handbook limits overview paragraph showing.session tutorial [was@descent doc]$ rm showing.session [was@descent doc]$ exit exit Process magma finished