[N,k,chi] = [78,14,Mod(1,78)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(78, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0]))
N = Newforms(chi, 14, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("78.1");
S:= CuspForms(chi, 14);
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
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
\( p \)
Sign
\(2\)
\(1\)
\(3\)
\(1\)
\(13\)
\(-1\)
This newform does not admit any (nontrivial ) inner twists .
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{4} - 17302T_{5}^{3} - 3263891324T_{5}^{2} + 2093196490200T_{5} + 1813895761795020000 \)
T5^4 - 17302*T5^3 - 3263891324*T5^2 + 2093196490200*T5 + 1813895761795020000
acting on \(S_{14}^{\mathrm{new}}(\Gamma_0(78))\).
$p$
$F_p(T)$
$2$
\( (T + 64)^{4} \)
(T + 64)^4
$3$
\( (T + 729)^{4} \)
(T + 729)^4
$5$
\( T^{4} - 17302 T^{3} + \cdots + 18\!\cdots\!00 \)
T^4 - 17302*T^3 - 3263891324*T^2 + 2093196490200*T + 1813895761795020000
$7$
\( T^{4} - 110778 T^{3} + \cdots + 37\!\cdots\!84 \)
T^4 - 110778*T^3 - 110726682056*T^2 - 2319990416306976*T + 371204693074763949184
$11$
\( T^{4} - 5029228 T^{3} + \cdots + 14\!\cdots\!96 \)
T^4 - 5029228*T^3 - 117462958186640*T^2 + 467563370214924303936*T + 1456325935581989992292061696
$13$
\( (T - 4826809)^{4} \)
(T - 4826809)^4
$17$
\( T^{4} - 178554136 T^{3} + \cdots - 47\!\cdots\!04 \)
T^4 - 178554136*T^3 - 15794285401292168*T^2 + 3178025446931642817164064*T - 47675651566291364604466142005104
$19$
\( T^{4} - 330495078 T^{3} + \cdots - 28\!\cdots\!56 \)
T^4 - 330495078*T^3 + 20964645785412304*T^2 + 440238502995758187432384*T - 28862428743403849870848726189056
$23$
\( T^{4} - 483904200 T^{3} + \cdots - 15\!\cdots\!00 \)
T^4 - 483904200*T^3 - 1094852837413262592*T^2 + 295041072710616565237954560*T - 15619861340135615559508061921280000
$29$
\( T^{4} - 3450624948 T^{3} + \cdots + 49\!\cdots\!92 \)
T^4 - 3450624948*T^3 - 12644344450284341328*T^2 + 23906421825981637971334878288*T + 49298862252892322325589671287692114992
$31$
\( T^{4} + 1760560078 T^{3} + \cdots + 29\!\cdots\!48 \)
T^4 + 1760560078*T^3 - 112452549044754859464*T^2 - 132627874070618413516597182368*T + 2931311350942324030518160247195955384448
$37$
\( T^{4} + 21294929844 T^{3} + \cdots - 27\!\cdots\!52 \)
T^4 + 21294929844*T^3 - 86296208945250458240*T^2 - 1742183061105749119560576736848*T - 2756507190220683334135590655067303777552
$41$
\( T^{4} + 70606057250 T^{3} + \cdots - 11\!\cdots\!40 \)
T^4 + 70606057250*T^3 + 1002089158474967145916*T^2 - 5629038007639546997384664547848*T - 117801728977125356481833871721268989847040
$43$
\( T^{4} + 75558364632 T^{3} + \cdots - 33\!\cdots\!84 \)
T^4 + 75558364632*T^3 - 991326433823362520192*T^2 - 180755464719636201350833653301632*T - 3350936238863985021391685361252133077526784
$47$
\( T^{4} + 112183555592 T^{3} + \cdots + 25\!\cdots\!16 \)
T^4 + 112183555592*T^3 - 7432908945898970755376*T^2 - 503195680465595589849460322965632*T + 2517004353298264664665559054203036733078016
$53$
\( T^{4} + 104575883144 T^{3} + \cdots + 21\!\cdots\!40 \)
T^4 + 104575883144*T^3 - 104347041514301594662376*T^2 - 8114208378581090044296686262792672*T + 2108863548252170164141883990990166696254234640
$59$
\( T^{4} + 953392600312 T^{3} + \cdots - 19\!\cdots\!60 \)
T^4 + 953392600312*T^3 + 89217091431037324567888*T^2 - 100983589900826941551950554880611968*T - 19237868319762019682161852749049784974010204160
$61$
\( T^{4} + 173990512884 T^{3} + \cdots - 15\!\cdots\!92 \)
T^4 + 173990512884*T^3 - 95974885113134792825888*T^2 - 25637034982533416464491535025891280*T - 1575720578488363714831818487137570193767880592
$67$
\( T^{4} + 163049862070 T^{3} + \cdots + 72\!\cdots\!72 \)
T^4 + 163049862070*T^3 - 1112264052547416612192048*T^2 + 62667310461843054360644596657975360*T + 72029841896447043612361016285879733237435907072
$71$
\( T^{4} - 1141535539628 T^{3} + \cdots + 28\!\cdots\!16 \)
T^4 - 1141535539628*T^3 - 3750988443832097097706400*T^2 + 3029785136114200792597797024889393536*T + 2869921910750281413160307917592654948457074082816
$73$
\( T^{4} - 2505023734484 T^{3} + \cdots - 12\!\cdots\!64 \)
T^4 - 2505023734484*T^3 - 542953179521466471047424*T^2 + 3746776555382290591417350521812813264*T - 1233489165296109037169892869199144144576709742864
$79$
\( T^{4} - 2114849719736 T^{3} + \cdots + 24\!\cdots\!72 \)
T^4 - 2114849719736*T^3 - 4828576772918658294207552*T^2 + 1162187186934588651435072382600728064*T + 2425656251653431121869261048688561080958779523072
$83$
\( T^{4} - 3245912192784 T^{3} + \cdots - 32\!\cdots\!28 \)
T^4 - 3245912192784*T^3 - 16838789728584846316778352*T^2 + 62987148099983740436764670517404057856*T - 32609230557559661471633719561601985130301051440128
$89$
\( T^{4} + 4151539197470 T^{3} + \cdots - 35\!\cdots\!28 \)
T^4 + 4151539197470*T^3 - 13861799954566121504623628*T^2 - 66895082445470351123040308535101362680*T - 35322607823716796295501184325748733756181068309728
$97$
\( T^{4} - 7366767632100 T^{3} + \cdots - 17\!\cdots\!40 \)
T^4 - 7366767632100*T^3 - 31310652986702307766277696*T^2 - 27046348809594632330159956897149142896*T - 1773708955496851509695391863525057154744433287440
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