[N,k,chi] = [19,8,Mod(1,19)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(19, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0]))
N = Newforms(chi, 8, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("19.1");
S:= CuspForms(chi, 8);
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
\(19\)
\(-1\)
This newform does not admit any (nontrivial ) inner twists .
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{4} + 9T_{2}^{3} - 234T_{2}^{2} - 396T_{2} + 3240 \)
T2^4 + 9*T2^3 - 234*T2^2 - 396*T2 + 3240
acting on \(S_{8}^{\mathrm{new}}(\Gamma_0(19))\).
$p$
$F_p(T)$
$2$
\( T^{4} + 9 T^{3} - 234 T^{2} + \cdots + 3240 \)
T^4 + 9*T^3 - 234*T^2 - 396*T + 3240
$3$
\( T^{4} + 14 T^{3} - 1827 T^{2} + \cdots + 83700 \)
T^4 + 14*T^3 - 1827*T^2 + 18900*T + 83700
$5$
\( T^{4} + 222 T^{3} + \cdots + 983232000 \)
T^4 + 222*T^3 - 119655*T^2 + 3224700*T + 983232000
$7$
\( T^{4} + 1246 T^{3} + \cdots + 30962663047 \)
T^4 + 1246*T^3 - 837048*T^2 - 958460846*T + 30962663047
$11$
\( T^{4} + \cdots - 235640376975204 \)
T^4 + 8718*T^3 + 5399757*T^2 - 112537710072*T - 235640376975204
$13$
\( T^{4} + \cdots - 256119966125456 \)
T^4 + 4480*T^3 - 115242465*T^2 + 373254914200*T - 256119966125456
$17$
\( T^{4} + 4440 T^{3} + \cdots - 30\!\cdots\!31 \)
T^4 + 4440*T^3 - 861485130*T^2 - 10236888310680*T - 30693507380216631
$19$
\( (T - 6859)^{4} \)
(T - 6859)^4
$23$
\( T^{4} + 30528 T^{3} + \cdots + 79\!\cdots\!00 \)
T^4 + 30528*T^3 - 6852928689*T^2 - 30515183390592*T + 7978094538928171200
$29$
\( T^{4} + 254244 T^{3} + \cdots - 58\!\cdots\!40 \)
T^4 + 254244*T^3 - 18473070861*T^2 - 8998207267267044*T - 582178603345234904340
$31$
\( T^{4} + 303460 T^{3} + \cdots + 52\!\cdots\!32 \)
T^4 + 303460*T^3 - 33188622816*T^2 - 11416233406324352*T + 52178733228863736832
$37$
\( T^{4} + 270460 T^{3} + \cdots + 92\!\cdots\!00 \)
T^4 + 270460*T^3 - 189544884816*T^2 - 43507917730125680*T + 921025727539819428400
$41$
\( T^{4} + 828564 T^{3} + \cdots - 25\!\cdots\!00 \)
T^4 + 828564*T^3 - 163207258092*T^2 - 220463034427427040*T - 25225611910025769907200
$43$
\( T^{4} - 37454 T^{3} + \cdots - 47\!\cdots\!04 \)
T^4 - 37454*T^3 - 341276991087*T^2 + 89147058869387884*T - 4763669949167924278304
$47$
\( T^{4} - 335670 T^{3} + \cdots + 57\!\cdots\!44 \)
T^4 - 335670*T^3 - 1945493085735*T^2 + 593010721459834560*T + 570094233095073100791744
$53$
\( T^{4} - 76728 T^{3} + \cdots + 18\!\cdots\!00 \)
T^4 - 76728*T^3 - 1247166391833*T^2 - 186336043185492360*T + 185918958682688611159200
$59$
\( T^{4} + 3191334 T^{3} + \cdots - 29\!\cdots\!00 \)
T^4 + 3191334*T^3 - 382531350915*T^2 - 5442661402656268500*T - 2918255829042824621437500
$61$
\( T^{4} - 346550 T^{3} + \cdots + 48\!\cdots\!72 \)
T^4 - 346550*T^3 - 2024831419419*T^2 - 233933942729834456*T + 486202470131499435226972
$67$
\( T^{4} + 270322 T^{3} + \cdots - 35\!\cdots\!32 \)
T^4 + 270322*T^3 - 15216114092439*T^2 - 15063131738013392552*T - 3552144795187925156467232
$71$
\( T^{4} + 2066124 T^{3} + \cdots - 80\!\cdots\!68 \)
T^4 + 2066124*T^3 - 18256724434368*T^2 - 24907275834032823984*T - 8079982254199016783524368
$73$
\( T^{4} + 416044 T^{3} + \cdots + 61\!\cdots\!53 \)
T^4 + 416044*T^3 - 16768962526026*T^2 - 6161580140890980596*T + 61687490519304246536020753
$79$
\( T^{4} - 16025864 T^{3} + \cdots + 96\!\cdots\!20 \)
T^4 - 16025864*T^3 + 89120773501980*T^2 - 189305091389847250112*T + 96763272229532098807894720
$83$
\( T^{4} - 8524128 T^{3} + \cdots - 58\!\cdots\!92 \)
T^4 - 8524128*T^3 + 10594248858396*T^2 + 41765922989454706848*T - 58354473089388296179536192
$89$
\( T^{4} - 2899092 T^{3} + \cdots + 24\!\cdots\!80 \)
T^4 - 2899092*T^3 - 37145859132384*T^2 + 38927393078768258688*T + 245343398528404802668538880
$97$
\( T^{4} + 4766908 T^{3} + \cdots + 38\!\cdots\!56 \)
T^4 + 4766908*T^3 - 212881382777388*T^2 - 397963609143821356352*T + 3864169762282210632420998656
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