[was@modular jochnowitz]$ [was@modular jochnowitz]$ Magma V2.8-10     Wed Apr 24 2002 18:14:32 on modular  [Seed = 384631911]
Type ? for help.  Type <Ctrl>-D to quit.

Loading startup file "/home/was/magma/local/emacs.m"

Loading "/home/was/magma/local/init.m"
> BernoulliNumber(4);
-1/30
> Basis(ModularForms(1,4));
[
1 + 240*q + 2160*q^2 + 6720*q^3 + 17520*q^4 + 30240*q^5 + 60480*q^6 + 82560*q^7 + O(q^8)
]
> factor(82560);
[ <2, 7>, <3, 1>, <5, 1>, <43, 1> ]
1
> S := CuspForms(1,12);
> S;
Space of modular forms on Gamma_0(1) of weight 12 and dimension 1 over Integer Ring.
> Basis(S);
[
q - 24*q^2 + 252*q^3 - 1472*q^4 + 4830*q^5 - 6048*q^6 - 16744*q^7 + O(q^8)
]
> factor(16744);
[ <2, 3>, <7, 1>, <13, 1>, <23, 1> ]
1
> g := PowerSeries(Basis(ES(ModularForms(1,12)))[1],20);
[
691 + 65520*q + 134250480*q^2 + 11606736960*q^3 + 274945048560*q^4 + 3199218815520*q^5 + 23782204031040*q^6 + 129554448266880*q^7 + O(q^8)
]
> factor(691);
[ <691, 1> ]
1
> factor(129554448266880);
[ <2, 7>, <3, 2>, <5, 1>, <7, 1>, <13, 1>, <23, 1>, <10746341, 1> ]
1
> M := ModularForms(1,12);
> E := ES(M);
> S := CS(M);
> S := CS(M);
> CongruenceGroup(E,S,30);
Abelian Group isomorphic to Z/691
Defined on 1 generator
Relations:
691*$.1 = 0
> Basis(BaseExtend(E,GF(7)));
[
5 + O(q^8)
]
> Basis(BaseExtend(S,GF(7)));
[
q + 4*q^2 + 5*q^4 + O(q^8)
]
> (1/2)*Basis(BaseExtend(ModularForms(1,4),GF(7)))[1];
[
1 + 2*q + 4*q^2 + 6*q^4 + 2*q^7 + O(q^8)
]
> factor(65520);
[ <2, 4>, <3, 2>, <5, 1>, <7, 1>, <13, 1> ]
1
> g := PowerSeries(Basis(ES(ModularForms(1,12)))[1],20);
> g;
691 + 65520*q + 134250480*q^2 + 11606736960*q^3 + 274945048560*q^4 + 3199218815520*q^5 + 23782204031040*q^6 + 129554448266880*q^7 + 563087459516400*q^8 + 2056098632318640*q^9 + 6555199353000480*q^10 + 18693620658498240*q^11 + 48705965462306880*q^12 + 117422349017369760*q^13 + 265457064498837120*q^14 + 566735214731736960*q^15 + 1153203117089652720*q^16 + 2245494646076179680*q^17 + 4212946097620893360*q^18 + 7632441763011374400*q^19 + O(q^20)
> Parent(g);
Power series ring in q over Integer Ring
> g := PowerSeries(Basis(BaseExtend(ES(ModularForms(1,12)),RationalField()))[1],20);
> g;
691 + 65520*q + 134250480*q^2 + 11606736960*q^3 + 274945048560*q^4 + 3199218815520*q^5 + 23782204031040*q^6 + 129554448266880*q^7 + 563087459516400*q^8 + 2056098632318640*q^9 + 6555199353000480*q^10 + 18693620658498240*q^11 + 48705965462306880*q^12 + 117422349017369760*q^13 + 265457064498837120*q^14 + 566735214731736960*q^15 + 1153203117089652720*q^16 + 2245494646076179680*q^17 + 4212946097620893360*q^18 + 7632441763011374400*q^19 + O(q^20)
> Parent(g);
Power series ring in q over Rational Field
> 
1/7 + 9360/691*q + 19178640/691*q^2 + 1658105280/691*q^3 + 39277864080/691*q^4 + 457031259360/691*q^5 + 3397457718720/691*q^6 + 18507778323840/691*q^7 + 80441065645200/691*q^8 + 293728376045520/691*q^9 + 936457050428640/691*q^10 + 2670517236928320/691*q^11 + 6957995066043840/691*q^12 + 16774621288195680/691*q^13 + 37922437785548160/691*q^14 + 80962173533105280/691*q^15 + 164743302441378960/691*q^16 + 320784949439454240/691*q^17 + 601849442517270480/691*q^18 + 1090348823287339200/691*q^19 + O(q^20)
> g/7;
691/7 + 9360*q + 19178640*q^2 + 1658105280*q^3 + 39277864080*q^4 + 457031259360*q^5 + 3397457718720*q^6 + 18507778323840*q^7 + 80441065645200*q^8 + 293728376045520*q^9 + 936457050428640*q^10 + 2670517236928320*q^11 + 6957995066043840*q^12 + 16774621288195680*q^13 + 37922437785548160*q^14 + 80962173533105280*q^15 + 164743302441378960*q^16 + 320784949439454240*q^17 + 601849442517270480*q^18 + 1090348823287339200*q^19 + O(q^20)
> (1/2)*Basis(BaseExtend(ModularForms(1,4),GF(7)))[1];

*(
x: 1/2,
f: 1 + 2*q + 4*q^2 + 6*q^4 + 2*q^7 + O(q^8)
)
In file "/home/was/magma/packages/modform/code/arithmetic.m", line 176, column 14:
>>    require x in BaseRing(M) : "Argument 1 must be in the base ring of the p
                ^
Runtime error in 'in': Bad argument types
> (GF(7)!(1/2))*Basis(BaseExtend(ModularForms(1,4),GF(7)))[1];
4 + q + 2*q^2 + 3*q^4 + q^7 + O(q^8)
> PowerSeries(Basis(BaseExtend(S,GF(7)))[1],30);
[
q + 4*q^2 + 5*q^4 + O(q^8)
]
> PowerSeries(Basis(BaseExtend(S,GF(7)))[1],30);
q + 4*q^2 + 5*q^4 + 4*q^8 + 2*q^9 + q^11 + 3*q^16 + q^18 + 4*q^22 + 4*q^23 + 4*q^25 + 2*q^29 + O(q^30)
> (GF(7)!(1/2))*PowerSeries(Basis(BaseExtend(ModularForms(1,4),GF(7)))[1],30);
4 + q + 2*q^2 + 3*q^4 + q^7 + 4*q^8 + q^9 + 2*q^11 + 2*q^14 + 5*q^16 + 2*q^18 + 4*q^22 + 2*q^23 + q^25 + 3*q^28 + 2*q^29 + O(q^30)
> h := $1;
> q := Parent(h).1;
> thetah := &+[i*q^i*Coefficient(h,i) : i in [1..29]] + O(q^30);
> thetah;
q + 4*q^2 + 5*q^4 + 4*q^8 + 2*q^9 + q^11 + 3*q^16 + q^18 + 4*q^22 + 4*q^23 + 4*q^25 + 2*q^29 + O(q^30)
> f11 := ModularForm(EC("11A"));
> f11;
q - 2*q^2 - q^3 + 2*q^4 + q^5 + 2*q^6 - 2*q^7 + O(q^8)
> Parent(h)!PowerSeries(f11,20);
q + 5*q^2 + 6*q^3 + 2*q^4 + q^5 + 2*q^6 + 5*q^7 + 5*q^9 + 5*q^10 + q^11 + 5*q^12 + 4*q^13 + 4*q^14 + 6*q^15 + 3*q^16 + 5*q^17 + 4*q^18 + O(q^20)
> f7_4 := ModularForms(7,4).1;
> Parent(h)!PowerSeries(f7_4,20);
1 + 2*q^7 + 4*q^14 + O(q^20)
> f7 := ModularForms(7,6).1;
> Parent(h)!PowerSeries(f7,20);
1 + O(q^20)
> f7 := ModularForms(7,6).1;
> Parent(h)!PowerSeries(f7,20);
1 + O(q^20)
> q := Parent(h).1; f7 := q - 3*q^2 + 5*q^4 - 7*q^7 - 3*q^8 + 9*q^9 - 6*q^11 + 21*q^14 - 11*q^16 - 27*q^18 ;
> f7;
q + 4*q^2 + 5*q^4 + 4*q^8 + 2*q^9 + q^11 + 3*q^16 + q^18
> thetah
> ;
q + 4*q^2 + 5*q^4 + 4*q^8 + 2*q^9 + q^11 + 3*q^16 + q^18 + 4*q^22 + 4*q^23 + 4*q^25 + 2*q^29 + O(q^30)
> M7 := ModularForms(Gamma1(7),3);
Space of modular forms on Gamma_1(7) of weight 3 and dimension 7 over Integer Ring.
> ES($1);
Space of modular forms on Gamma_1(7) of weight 3 and dimension 6 over Integer Ring.
> M7 := ModularForms(Gamma1(7),3);
> CongruenceGroup(ES(M7),CS(M7),50);
Abelian Group isomorphic to Z/8
Defined on 1 generator
Relations:
8*$.1 = 0
> MS7 := ModularSymbols(DirichletGroup(7).1,3);
> IntersectionGroup(ES(MS7),CS(MS7));
Abelian Group isomorphic to Z/4 + Z/8
Defined on 2 generators
Relations:
4*$.1 = 0
8*$.2 = 0
> 120 mod 7;
1
> [4+8*i : i in [1..40] |i mod 7 eq 1];
[ 12, 68, 124, 180, 236, 292 ]
>  [4+8*i : i in [1..100] |i mod 7 eq 1];
[ 12, 68, 124, 180, 236, 292, 348, 404, 460, 516, 572, 628, 684, 740, 796 ]
> M := ModularForms(1,410);
> M := ModularForms(1,180);
> M;
Space of modular forms on Gamma_0(1) of weight 180 and dimension 16 over Integer Ring.
> ES(M);
Space of modular forms on Gamma_0(1) of weight 180 and dimension 1 over Integer Ring.
> Basis($1);
[
6173136454016248924640522272263470960199559328290655337530202055853397791747341312347030141906500993752700612233695954532816018207721731818225290076670213481102834647254685911917265818955932383093313 + 2601256998740400*q + 1993207375607924035375992553407878950327000512533893326217640348295600*q^2 + 66052286583264428846250157156925311091779135346918414132743050370492532000392693727492209179502267200*q^3 + 1527290707570072215904546573728212677026084235110799922821086346005937971451137685354051567840623768478215181599092024353200*q^4 + 339479878325863288137717960598361453301619922665810384313186342533780547975245196641443109168831057664839079279772704467177393653503092490400*q^5 + 50612417326424257434533455153290993414291447751910699399734179201753809772237189595345726002256386698745657536285098885842966361349889443862803806928980800*q^6 + 48722820854893927810268131502758916792020377970470261863314422724268263981224597463541717715862424856472382486286048270829285168223061595613437351748259738771559017600*q^7 + O(q^8)
]
> Valuation(Coefficient($1[1],7),7);
1
> time f7 := HeckePolynomial(M,7);
Time: 0.770
> NewtonSlopes(f7,2);
[* 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 *]
> f7;
$.1^16 - 18730491019721178760387354371114745047443028824998645259304527116270420929911508038943577371138551443844871489863899492248649148814573058834478937215249*$.1^15 + 22156047622666174938668972809739535234724835930901943891364417559205538307657246652706982139872029349624048868858300972058834093448188397865759454914043335599868273090798942896158483917514470602238859*$.1^14 - 13294198847027406805048371774714904171329845279531652541577473824399991966380948457670244673729700237439051759732997645749869071226766623893006720659718344213482950754963936809127922102191682847219002*$.1^13 - 70617845766393400524489079111319155870408302786442872727656385517707957071259230652607747819760815250448066449882540075667260422557005452878320551581531733286265435025239818584435209917954013159410834*$.1^12 - 40893103167991235207954728309048594632732237579013078187655624394634428604084235919837538790149928974515600076390764999877227962983842939338553620339427571269595177159514686911345934892565811264348663*$.1^11 - 80293168101358070617042932834818327908416214945288919026313214147510640781897454572885594870038641327427938406245560117616540811587520817412761904033565225736137683719029989250903572536885485319240053*$.1^10 - 47047733082087262341620246073279817462468687044889639162136677853155631842893531367734486785694452265566855798935004485760664023540982867300015227553936964081427217592257498561368270750182740999829933*$.1^9 - 122168385891752401839276814225654619818110832763539703345865692330013552632623433804950464844464632964918437309558738255672707611449237650224788479190414192475319205211923286987260409646550152736426550*$.1^8 - 15608342218150188923833977600041255145146703701213085339778492627420012630544104316972544257247549276837954369628132803431254424212327119657503206199801337578992562056796738934024032520268714583768628*$.1^7 + 122336335992047624381822596222607454320115627348669581222622753408985931797753711358418502193562385966388519401758156853497302721821766826921342442668114583290280319869645126167442548154592259363061687*$.1^6 - 110325700583179168813369028242207939974858276484020954013653372366310459031823398575051309174321002824626958044718614965367558740997874324302327734463617727407751107324849730080777130866425863186255547*$.1^5 - 48147303255110441229924849680529769892868085891097583168822667972845109545763328322484134407110176897620058355141498272934794680217606302232582500267990609476912901189780779550958198602162137162579289*$.1^4 - 131085721809173471028500172648261027000358161028568933407667164588702089194834987767400827660577972426510078491957011328850359441024824294206777473203602047110337881648575933449060155355088026707744406*$.1^3 + 117352431155082092804934123242495794358028242075938044936599788360449408394638242356047763572590046766671782960731896431816269982698617729176975808231158684609858103855505393797800909659270568317727335*$.1^2 + 101759951036941361627170299271255229151901751741092478663271054617881468888033478888082697914240645133385546175403917231292050145857669685378057000435336271337165456209503378123715558547840371162289209*$.1 - 88083568035815050164663006209611183008878569177263661901389498328540467059988975263853983169854547722907794795827811849310470978974197214093428947675535181200074988832654594446973066383906475428280280
> time f2 := HeckePolynomial(M,2);
Time: 0.080
> NewtonSlopes(f7,2);
[* 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 3 *]
> MM := ModularSymbols(1,180,+1);
> time fmm := CharacteristicPolynomial(HeckeOperator(MM,7));
> NewtonSlopes(fmm,7);
[* 0, 3, 9, 9, 16, 21, 25, 25, 29, 35, 42, 42, 48, 51, 57, 57 *]
> MM := ModularSymbols(1,348,+1);
time fmm := CharacteristicPolynomial(HeckeOperator(MM,7));
NewtonSlopes(fmm,7);
> 

Time: 499.500
> [* 0, 1, 3, 4, 5, 7, 8, 11, 12, 14, 15, 17, 19, 20, 22, 23, 25, 26, 26, 26, 27, 29, 30, 32, 33, 35, 37, 38, 40, 41 *]
> > > > > > 
> 
> 
>