[was@form vatsal]$ [was@form vatsal]$ Magma V2.8-10 Thu May 2 2002 16:00:23 [Seed = 4112012115] Type ? for help. Type -D to quit. Loading startup file "/home/was/magma/local/emacs.m" Loading "/home/was/magma/local/init.m" > M := ModularSymbols(65); > L, phi := Lattice(M); > L; Standard Lattice of rank 13 and degree 13 > phi; Mapping from: Lat: L to ModSym: M given by a rule [no inverse] > L, phi, psi := Lattice(M); >> L, phi, psi := Lattice(M); ^ Runtime error in :=: Expected to assign 3 value(s) but only computed 2 value(s) > psi := hom L | x :-> L!Eltseq(x)>; > psi(M.1); (1 0 0 0 0 0 0 0 0 0 0 0 0) > Parent($1); Standard Lattice of rank 13 and degree 13 > TwistedWindingElementModular; >> ElementModular; ^ User error: bad syntax > TwistedWindingElementModular; >> TwistedWindingElement; ^ User error: Identifier 'TwistedWindingElementModular' has not been declared or assigned > > TwistedWindingElement; Intrinsic 'TwistedWindingElement' Signatures: ( M, i, eps) -> ModSymElt The element sum_{a in (Z/mZ)^*} . > eps := DirichletGroup(11).1; > e := TwistedWindingElement(M,1,eps); > e; 2*{-1/20, 0} + 4*{-1/30, 0} + -3*{-1/35, 0} + -1*{-1/40, 0} + -2*{-1/45, 0} + {-1/52, 0} + -2*{-1/55, 0} + {-1/60, 0} + {-2/5, -5/13} > psi(e); ( 0 0 2 0 4 -3 -1 -2 0 1 -2 1 1) > gens := [psi(TwistedWindingElement(M,1,DirichletGroup(p).1)) : p in PrimeSeq(17,200)]; > gens; [ ( 0 -2 2 0 0 0 0 2 0 -2 0 0 -2), ( 0 -2 6 4 -2 -3 1 -6 2 -5 10 -5 -1), ( 0 2 1 -2 5 -3 0 -1 -2 2 -4 2 0), ( 0 0 10 -9 -1 1 -11 10 -4 6 0 -2 -3), ( 0 4 2 -4 4 -3 3 -2 -4 1 -2 1 -3), ( 0 0 12 -7 -5 -5 -7 12 2 -2 0 0 -9), ( 0 -2 2 0 0 0 0 2 0 -2 0 0 -2), ( 0 -2 5 2 1 -3 0 -5 2 -2 4 -2 0), ( 0 0 0 -1 1 -1 1 0 0 0 0 0 -1), ( 0 -8 8 0 0 0 0 8 0 -8 0 0 -8), ( 0 -6 10 4 6 -7 -3 -10 6 -1 2 -1 3), ( 0 0 8 -5 -3 -7 -1 8 0 -4 0 4 -9), ( 0 0 0 -9 9 -9 9 0 0 0 0 0 -9), ( 0 10 -2 -4 10 -7 1 2 -10 3 -6 3 -1), ( 0 0 4 -7 3 -5 1 4 -6 -2 0 8 -9), (0 0 0 0 0 0 0 0 0 0 0 0 0), ( 0 0 0 -1 1 -1 1 0 0 0 0 0 -1), ( 0 -8 8 0 0 0 0 8 0 -8 0 0 -8), ( 0 0 6 -10 4 -8 2 6 -10 -2 0 12 -12), ( 0 0 6 -8 2 -4 -2 6 -8 2 0 6 -6), ( 0 12 5 -16 9 -5 12 -5 -12 4 -8 4 -12), ( 0 2 1 -2 5 -3 0 -1 -2 2 -4 2 0), ( 0 -2 2 0 0 0 0 2 0 -2 0 0 -2), ( 0 -2 2 0 0 0 0 2 0 -2 0 0 -2), ( 0 12 5 -16 9 -5 12 -5 -12 4 -8 4 -12), ( 0 0 0 -4 4 -4 4 0 0 0 0 0 -4), ( 0 0 8 -18 10 -4 -4 8 -14 0 0 14 -18), (0 0 0 0 0 0 0 0 0 0 0 0 0), ( 0 -2 2 0 0 0 0 2 0 -2 0 0 -2), ( 0 10 -2 -4 10 -7 1 2 -10 3 -6 3 -1), ( 0 -2 2 0 0 0 0 2 0 -2 0 0 -2), ( 0 0 0 -25 25 -25 25 0 0 0 0 0 -25), ( 0 0 0 -1 1 -1 1 0 0 0 0 0 -1), ( 0 -8 8 0 0 0 0 8 0 -8 0 0 -8), ( 0 0 0 -4 4 -4 4 0 0 0 0 0 -4), ( 0 0 6 -5 -1 -11 5 6 -12 2 0 10 -3), ( 0 0 0 -4 4 -4 4 0 0 0 0 0 -4), ( 0 0 16 -10 -6 -20 4 16 -6 -8 0 14 -18), ( 0 0 12 -10 -2 -4 -8 12 -6 4 0 2 -6), ( 0 0 0 -4 4 -4 4 0 0 0 0 0 -4) ] > W := Lattice(gens); >> W := Lattice(gens); ^ Runtime error in 'Lattice': Bad argument types Argument types given: SeqEnum[LatElt] > W := sub; > W; Lattice of rank 10 and degree 13 Basis: ( 0 0 0 1 -1 1 -1 0 0 0 0 0 1) ( 0 0 0 1 1 -2 0 0 0 -1 2 -1 0) ( 0 0 0 0 0 0 0 0 0 2 0 -2 2) ( 0 2 0 1 1 -1 -1 0 -2 0 0 0 1) ( 0 2 0 -1 -1 -1 -1 0 2 0 0 0 -1) ( 0 0 2 -1 -1 1 1 -2 0 0 0 0 -1) ( 0 0 0 1 -1 -1 1 0 -2 0 0 2 1) ( 0 2 -2 0 0 0 0 -2 0 0 0 2 0) ( 0 2 0 1 -1 0 0 0 2 -1 -2 -1 0) ( 0 2 -1 0 -1 0 1 1 -2 -1 2 -1 -1) > L/W; Abelian Group isomorphic to Z/2 + Z/2 + Z/2 + Z/4 + Z/4 + Z/12 + Z/12 + Z + Z + Z Defined on 10 generators Relations: 2*$.1 = 0 2*$.2 = 0 2*$.3 = 0 4*$.4 = 0 4*$.5 = 0 12*$.6 = 0 12*$.7 = 0 > gens := gens cat [psi(TwistedWindingElement(M,1,DirichletGroup(4*p).1*DirichletGroup(4*p).2)) : p in PrimeSeq(17,200)]; *( x: $.1, y: $.2 ) In file "/home/was/magma/packages/modsym/code/dirichlet.m", line 704, column 48: >> return initGrpDrchElt(Parent(x),x`Element+y`Element); ^ Runtime error in '+': Arguments are not compatible Argument types given: GrpAbElt, GrpAbElt > chi := []; > for p in PrimeSeq(17,200) do G := DirichletGroup(4*p); e := G.1*G.2; gens := gens cat psi(TwistedWindingElement(M,1,e)); end for; >> for p in PrimeSeq(17,200) do G := DirichletGroup(4*p); e = G.1*G.2; gens : ^ Runtime error in elt< ... >: LHS and RHS of relation constructor are not compatible > for p in PrimeSeq(17,200) do G := DirichletGroup(4*p); e := G.1*G.2; gens := gens cat psi(TwistedWindingElement(M,1,e)); end for; >> e := G.1*G.2; gens := gens cat psi(TwistedWindingElement(M,1,e)); end for; ^ Runtime error in 'cat': Bad argument types Argument types given: SeqEnum[LatElt], LatElt > for p in PrimeSeq(17,200) do G := DirichletGroup(4*p); e := G.1*G.2; Append(~gens,psi(TwistedWindingElement(M,1,e))); end for; > gens; [ ( 0 -2 2 0 0 0 0 2 0 -2 0 0 -2), ( 0 -2 6 4 -2 -3 1 -6 2 -5 10 -5 -1), ( 0 2 1 -2 5 -3 0 -1 -2 2 -4 2 0), ( 0 0 10 -9 -1 1 -11 10 -4 6 0 -2 -3), ( 0 4 2 -4 4 -3 3 -2 -4 1 -2 1 -3), ( 0 0 12 -7 -5 -5 -7 12 2 -2 0 0 -9), ( 0 -2 2 0 0 0 0 2 0 -2 0 0 -2), ( 0 -2 5 2 1 -3 0 -5 2 -2 4 -2 0), ( 0 0 0 -1 1 -1 1 0 0 0 0 0 -1), ( 0 -8 8 0 0 0 0 8 0 -8 0 0 -8), ( 0 -6 10 4 6 -7 -3 -10 6 -1 2 -1 3), ( 0 0 8 -5 -3 -7 -1 8 0 -4 0 4 -9), ( 0 0 0 -9 9 -9 9 0 0 0 0 0 -9), ( 0 10 -2 -4 10 -7 1 2 -10 3 -6 3 -1), ( 0 0 4 -7 3 -5 1 4 -6 -2 0 8 -9), (0 0 0 0 0 0 0 0 0 0 0 0 0), ( 0 0 0 -1 1 -1 1 0 0 0 0 0 -1), ( 0 -8 8 0 0 0 0 8 0 -8 0 0 -8), ( 0 0 6 -10 4 -8 2 6 -10 -2 0 12 -12), ( 0 0 6 -8 2 -4 -2 6 -8 2 0 6 -6), ( 0 12 5 -16 9 -5 12 -5 -12 4 -8 4 -12), ( 0 2 1 -2 5 -3 0 -1 -2 2 -4 2 0), ( 0 -2 2 0 0 0 0 2 0 -2 0 0 -2), ( 0 -2 2 0 0 0 0 2 0 -2 0 0 -2), ( 0 12 5 -16 9 -5 12 -5 -12 4 -8 4 -12), ( 0 0 0 -4 4 -4 4 0 0 0 0 0 -4), ( 0 0 8 -18 10 -4 -4 8 -14 0 0 14 -18), (0 0 0 0 0 0 0 0 0 0 0 0 0), ( 0 -2 2 0 0 0 0 2 0 -2 0 0 -2), ( 0 10 -2 -4 10 -7 1 2 -10 3 -6 3 -1), ( 0 -2 2 0 0 0 0 2 0 -2 0 0 -2), ( 0 0 0 -25 25 -25 25 0 0 0 0 0 -25), ( 0 0 0 -1 1 -1 1 0 0 0 0 0 -1), ( 0 -8 8 0 0 0 0 8 0 -8 0 0 -8), ( 0 0 0 -4 4 -4 4 0 0 0 0 0 -4), ( 0 0 6 -5 -1 -11 5 6 -12 2 0 10 -3), ( 0 0 0 -4 4 -4 4 0 0 0 0 0 -4), ( 0 0 16 -10 -6 -20 4 16 -6 -8 0 14 -18), ( 0 0 12 -10 -2 -4 -8 12 -6 4 0 2 -6), ( 0 0 0 -4 4 -4 4 0 0 0 0 0 -4), ( 0 0 8 -4 12 -8 0 -8 0 4 -8 4 0), ( 0 -2 2 0 0 0 0 2 0 -2 0 0 -2), ( 0 -2 2 0 0 0 0 2 0 -2 0 0 -2), ( 0 0 0 -4 4 -4 4 0 0 0 0 0 -4), ( 0 -2 2 0 0 0 0 2 0 -2 0 0 -2), ( 0 0 0 -4 4 -4 4 0 0 0 0 0 -4), ( 0 8 4 -4 8 -8 4 -4 -8 0 0 0 -4), ( 0 -2 2 0 0 0 0 2 0 -2 0 0 -2), ( 0 0 6 -16 10 -4 -2 6 -12 -2 0 14 -18), ( 0 8 4 -8 20 -12 0 -4 -8 8 -16 8 0), ( 0 -18 18 0 0 0 0 18 0 -18 0 0 -18), ( 0 0 0 -4 4 -4 4 0 0 0 0 0 -4), ( 0 0 14 -12 -2 -16 2 14 -20 6 0 14 -6), ( 0 -2 2 0 0 0 0 2 0 -2 0 0 -2), ( 0 0 0 -16 16 -16 16 0 0 0 0 0 -16), ( 0 0 32 -20 -12 -28 -4 32 0 -16 0 16 -36), ( 0 0 6 -16 10 -4 -2 6 -12 -2 0 14 -18), ( 0 8 16 -8 32 -24 0 -16 -8 8 -16 8 0), ( 0 0 0 -4 4 -4 4 0 0 0 0 0 -4), (0 0 0 0 0 0 0 0 0 0 0 0 0), ( 0 -18 18 0 0 0 0 18 0 -18 0 0 -18), ( 0 -18 18 0 0 0 0 18 0 -18 0 0 -18), ( 0 8 4 4 8 -12 0 -4 -8 -4 8 -4 0), ( 0 8 16 -20 44 -24 0 -16 -8 20 -40 20 0), ( 0 -2 2 0 0 0 0 2 0 -2 0 0 -2), ( 0 0 8 -16 8 -8 0 8 -8 -8 0 16 -24), ( 0 0 0 -4 4 -4 4 0 0 0 0 0 -4), ( 0 0 24 -20 -4 -44 20 24 -48 8 0 40 -12), ( 0 0 4 -4 8 -4 0 -4 0 4 -8 4 0), ( 0 -2 2 0 0 0 0 2 0 -2 0 0 -2), ( 0 24 20 -28 24 -20 24 -20 -24 4 -8 4 -24), ( 0 0 46 -28 -18 -32 -14 46 12 -26 0 14 -54), ( 0 0 14 -12 -2 -16 2 14 -20 6 0 14 -6), ( 0 24 4 -8 36 -28 0 -4 -24 8 -16 8 0), ( 0 0 40 -60 20 4 -44 40 -40 24 0 16 -36), ( 0 0 0 -4 4 -4 4 0 0 0 0 0 -4), ( 0 0 40 -60 20 4 -44 40 -40 24 0 16 -36), ( 0 0 0 -36 36 -36 36 0 0 0 0 0 -36), ( 0 0 0 -16 16 -16 16 0 0 0 0 0 -16), ( 0 0 16 -12 -4 -12 -4 16 -8 0 0 8 -12) ] > W := sub; > W; Lattice of rank 10 and degree 13 Basis: ( 0 0 1 0 1 -1 0 -1 0 0 0 0 0) ( 0 0 0 1 -1 1 -1 0 0 0 0 0 1) ( 0 0 1 -1 0 1 0 -1 0 1 -2 1 0) ( 0 0 1 0 -1 0 1 -1 0 -1 2 -1 -1) ( 0 2 1 0 -1 1 0 -1 -2 0 0 0 0) ( 0 0 0 1 1 0 2 0 0 -1 -2 -1 0) ( 0 0 0 0 0 0 0 0 0 2 0 -2 2) ( 0 2 0 -1 -1 -1 -1 0 2 0 0 0 -1) ( 0 0 0 1 -1 -1 1 0 -2 0 0 2 1) ( 0 0 1 0 1 -1 0 3 -2 0 0 -2 0) > #gens; 80 > EisensteinSubspace(M); Modular symbols space of level 65, weight 2, and dimension 3 > L/W; Abelian Group isomorphic to Z/2 + Z/2 + Z/2 + Z/4 + Z/4 + Z/4 + Z/12 + Z + Z + Z Defined on 10 generators Relations: 2*$.1 = 0 2*$.2 = 0 2*$.3 = 0 4*$.4 = 0 4*$.5 = 0 4*$.6 = 0 12*$.7 = 0 > IntersectionGroup(EisensteinSubspace(M), CuspidalSubspace(M)); Abelian Group isomorphic to Z/2 + Z/84 Defined on 2 generators Relations: 2*$.1 = 0 84*$.2 = 0 > T := sub; > T; Lattice of rank 10 and degree 13 Basis: ( 0 1 0 0 0 0 0 0 0 0 0 -1 0) ( 0 0 1 0 0 0 0 0 0 0 0 -1 0) ( 0 0 0 1 0 0 0 0 0 -1 0 0 0) ( 0 0 0 0 1 0 0 0 0 0 0 -1 0) ( 0 0 0 0 0 1 0 0 0 0 0 -1 0) ( 0 0 0 0 0 0 1 0 0 0 0 -1 0) ( 0 0 0 0 0 0 0 1 0 0 0 -1 0) ( 0 0 0 0 0 0 0 0 1 0 0 -1 0) ( 0 0 0 0 0 0 0 0 0 0 1 -1 0) ( 0 0 0 1 0 0 0 0 0 0 0 -1 1) > T eq W; false > T/W; Abelian Group isomorphic to Z/2 + Z/2 + Z/2 + Z/4 + Z/4 + Z/4 + Z/12 Defined on 7 generators Relations: 2*$.1 = 0 2*$.2 = 0 2*$.3 = 0 4*$.4 = 0 4*$.5 = 0 4*$.6 = 0 12*$.7 = 0 > W subset T; true > quit; Total time: 48.480 seconds [was@form vatsal]$ exit exit Process magma finished