[N,k,chi] = [112,10,Mod(65,112)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(112, base_ring=CyclotomicField(6))
chi = DirichletCharacter(H, H._module([0, 0, 2]))
N = Newforms(chi, 10, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("112.65");
S:= CuspForms(chi, 10);
N := Newforms(S);
Newform invariants
sage: f = N[0] # Warning: the index may be different
gp: f = lf[1] \\ Warning: the index may be different
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/112\mathbb{Z}\right)^\times\).
\(n\)
\(15\)
\(17\)
\(85\)
\(\chi(n)\)
\(1\)
\(\beta_{1}\)
\(1\)
For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.
For more information on an embedded modular form you can click on its label.
Refresh table
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{6} + 71T_{3}^{5} + 36598T_{3}^{4} + 2541603T_{3}^{3} + 1165610574T_{3}^{2} + 75455153775T_{3} + 5717239655625 \)
T3^6 + 71*T3^5 + 36598*T3^4 + 2541603*T3^3 + 1165610574*T3^2 + 75455153775*T3 + 5717239655625
acting on \(S_{10}^{\mathrm{new}}(112, [\chi])\).
$p$
$F_p(T)$
$2$
\( T^{6} \)
T^6
$3$
\( T^{6} + 71 T^{5} + \cdots + 5717239655625 \)
T^6 + 71*T^5 + 36598*T^4 + 2541603*T^3 + 1165610574*T^2 + 75455153775*T + 5717239655625
$5$
\( T^{6} + 1085 T^{5} + \cdots + 18\!\cdots\!25 \)
T^6 + 1085*T^5 + 3950086*T^4 - 3870845895*T^3 + 7220964872646*T^2 - 1195507526641155*T + 185886748283681025
$7$
\( T^{6} - 6796 T^{5} + \cdots + 65\!\cdots\!43 \)
T^6 - 6796*T^5 - 31140235*T^4 + 432947916632*T^3 - 1256620805077645*T^2 - 11066698811399411404*T + 65712362363534280139543
$11$
\( T^{6} + 2555 T^{5} + \cdots + 31\!\cdots\!41 \)
T^6 + 2555*T^5 + 4097877742*T^4 - 121888868945193*T^3 + 16596783693336655494*T^2 - 227960740179750869966493*T + 3104466016834613449364938641
$13$
\( (T^{3} - 18070 T^{2} + \cdots + 368974841338200)^{2} \)
(T^3 - 18070*T^2 - 10506518276*T + 368974841338200)^2
$17$
\( T^{6} - 20759 T^{5} + \cdots + 11\!\cdots\!25 \)
T^6 - 20759*T^5 + 108332373958*T^4 + 9098508778892973*T^3 + 11571531635440060323894*T^2 + 370025474577985532568003705*T + 11760039609007551056826954687225
$19$
\( T^{6} + 1220649 T^{5} + \cdots + 37\!\cdots\!25 \)
T^6 + 1220649*T^5 + 1170655553358*T^4 + 428263259498975557*T^3 + 125453083436679866481174*T^2 - 6143133859620337759329932175*T + 370087819691800041403239196425625
$23$
\( T^{6} - 1960903 T^{5} + \cdots + 97\!\cdots\!61 \)
T^6 - 1960903*T^5 + 9093716677654*T^4 - 9499973607852847203*T^3 + 46952570941769185371838782*T^2 - 51939705023425631186770228706655*T + 97930046414068460487812373647637953961
$29$
\( (T^{3} + 2606146 T^{2} + \cdots - 11\!\cdots\!32)^{2} \)
(T^3 + 2606146*T^2 - 4849005669540*T - 1136027665699119432)^2
$31$
\( T^{6} - 9377989 T^{5} + \cdots + 38\!\cdots\!09 \)
T^6 - 9377989*T^5 + 101258784791214*T^4 - 269592060839948120329*T^3 + 2026705682674900330965214966*T^2 - 2625366205207012085830956119849229*T + 38894319247678181161351219869360782711409
$37$
\( T^{6} + 25814913 T^{5} + \cdots + 22\!\cdots\!69 \)
T^6 + 25814913*T^5 + 467514029705910*T^4 + 4176175130774770517093*T^3 + 27190283355289462291047687750*T^2 + 95300891338309528343803570903899633*T + 229584794821087263975310535723940413923369
$41$
\( (T^{3} - 209250 T^{2} + \cdots + 12\!\cdots\!24)^{2} \)
(T^3 - 209250*T^2 - 231960363901092*T + 1231904599729712698824)^2
$43$
\( (T^{3} + 2944308 T^{2} + \cdots - 85\!\cdots\!92)^{2} \)
(T^3 + 2944308*T^2 - 57381514505424*T - 85536171172243734592)^2
$47$
\( T^{6} + 48391269 T^{5} + \cdots + 21\!\cdots\!25 \)
T^6 + 48391269*T^5 + 3206409899672718*T^4 + 51590172679651495988937*T^3 + 3008388948671697384743938611414*T^2 + 40395895164805441044947654879032809645*T + 2182471561008350750763931473853502325903450225
$53$
\( T^{6} - 102186411 T^{5} + \cdots + 10\!\cdots\!21 \)
T^6 - 102186411*T^5 + 7664409827848422*T^4 - 263539284420663228202911*T^3 + 6678209682251704415699644420422*T^2 - 28191901541784806914994704192980438411*T + 103013189215749005018345980061427993704379921
$59$
\( T^{6} + 144220135 T^{5} + \cdots + 88\!\cdots\!25 \)
T^6 + 144220135*T^5 + 14210689521501526*T^4 + 761786634026879736566115*T^3 + 29822954550465258123371187699726*T^2 + 620808930268775044838303155654766403375*T + 8877870833577157451063799414440505309752015625
$61$
\( T^{6} - 280936871 T^{5} + \cdots + 31\!\cdots\!25 \)
T^6 - 280936871*T^5 + 71629017810652758*T^4 - 5578058109929154317284403*T^3 + 548839768805293713462836512626694*T^2 + 12871769462411493806124703719462785501865*T + 3112048940600992892598872288885128249528816124025
$67$
\( T^{6} + 170710399 T^{5} + \cdots + 29\!\cdots\!25 \)
T^6 + 170710399*T^5 + 76733529465554342*T^4 + 2679203207459047361648731*T^3 + 3187090310570127170395156082797886*T^2 + 257078880492622037284074273289097870424295*T + 29179255837545996541239807135758655238517655000025
$71$
\( (T^{3} + 469758688 T^{2} + \cdots + 12\!\cdots\!40)^{2} \)
(T^3 + 469758688*T^2 + 55775894095701696*T + 1286588505667388060067840)^2
$73$
\( T^{6} - 613838539 T^{5} + \cdots + 24\!\cdots\!41 \)
T^6 - 613838539*T^5 + 299769435047295782*T^4 - 57172987091662279140322879*T^3 + 8968804714604670026557017514882502*T^2 + 380906474323460493441198395566957642348181*T + 24453210810326806837216169108327427139940719797841
$79$
\( T^{6} + 197445809 T^{5} + \cdots + 77\!\cdots\!25 \)
T^6 + 197445809*T^5 + 285731562527913078*T^4 + 7107972215477320236036197*T^3 + 66395352570775563672490650534934174*T^2 + 6887573927926776078970190372939952816206745*T + 779165631141627003189520596303112240295324108367225
$83$
\( (T^{3} - 1074181436 T^{2} + \cdots + 87\!\cdots\!00)^{2} \)
(T^3 - 1074181436*T^2 + 126447385266753648*T + 87991061919101183276568000)^2
$89$
\( T^{6} - 805730427 T^{5} + \cdots + 11\!\cdots\!25 \)
T^6 - 805730427*T^5 + 1107424475695806822*T^4 + 155204567222106529474670961*T^3 + 296181274679336351365499057283403974*T^2 - 49029766738097999302499499661201593974106075*T + 11448958241287429346824994272938611302614016265000625
$97$
\( (T^{3} + 2262918094 T^{2} + \cdots + 14\!\cdots\!40)^{2} \)
(T^3 + 2262918094*T^2 + 1399099087623178268*T + 143508336204051088511912840)^2
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