[N,k,chi] = [38,8,Mod(1,38)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(38, 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("38.1");
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
\(2\)
\(-1\)
\(19\)
\(-1\)
This newform does not admit any (nontrivial ) inner twists .
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{4} - 12T_{3}^{3} - 9043T_{3}^{2} + 164994T_{3} + 9954648 \)
T3^4 - 12*T3^3 - 9043*T3^2 + 164994*T3 + 9954648
acting on \(S_{8}^{\mathrm{new}}(\Gamma_0(38))\).
$p$
$F_p(T)$
$2$
\( (T - 8)^{4} \)
(T - 8)^4
$3$
\( T^{4} - 12 T^{3} - 9043 T^{2} + \cdots + 9954648 \)
T^4 - 12*T^3 - 9043*T^2 + 164994*T + 9954648
$5$
\( T^{4} + 279 T^{3} + \cdots + 6648480000 \)
T^4 + 279*T^3 - 173610*T^2 - 31638600*T + 6648480000
$7$
\( T^{4} - 2485 T^{3} + \cdots - 74716579192 \)
T^4 - 2485*T^3 + 1751247*T^2 - 214155299*T - 74716579192
$11$
\( T^{4} - 5269 T^{3} + \cdots + 88406758195632 \)
T^4 - 5269*T^3 - 14443560*T^2 + 44780357412*T + 88406758195632
$13$
\( T^{4} - 5406 T^{3} + \cdots - 31\!\cdots\!80 \)
T^4 - 5406*T^3 - 152473893*T^2 + 1420048205146*T - 3110537408743080
$17$
\( T^{4} - 22885 T^{3} + \cdots + 18\!\cdots\!58 \)
T^4 - 22885*T^3 - 901600659*T^2 + 8379333931725*T + 186464917523314458
$19$
\( (T - 6859)^{4} \)
(T - 6859)^4
$23$
\( T^{4} - 3364 T^{3} + \cdots + 28\!\cdots\!76 \)
T^4 - 3364*T^3 - 12072948381*T^2 - 14333869035720*T + 28662175856445077376
$29$
\( T^{4} + 122136 T^{3} + \cdots + 27\!\cdots\!20 \)
T^4 + 122136*T^3 - 13898936313*T^2 + 218031219002556*T + 2775856276186444620
$31$
\( T^{4} - 225480 T^{3} + \cdots + 32\!\cdots\!28 \)
T^4 - 225480*T^3 - 40955486928*T^2 + 3473955963466624*T + 323110415225499414528
$37$
\( T^{4} - 154096 T^{3} + \cdots + 11\!\cdots\!28 \)
T^4 - 154096*T^3 - 335762746632*T^2 + 66645207745528768*T + 11396813950122541990928
$41$
\( T^{4} + 1054628 T^{3} + \cdots - 43\!\cdots\!00 \)
T^4 + 1054628*T^3 + 315974762100*T^2 + 12178490485622400*T - 4325964277988597760000
$43$
\( T^{4} + 840795 T^{3} + \cdots + 32\!\cdots\!56 \)
T^4 + 840795*T^3 - 207747337704*T^2 - 154742908167258992*T + 32640642367940057033856
$47$
\( T^{4} + 1021877 T^{3} + \cdots - 57\!\cdots\!80 \)
T^4 + 1021877*T^3 - 54019190952*T^2 - 266978614446350208*T - 57891124954117677588480
$53$
\( T^{4} + 326842 T^{3} + \cdots + 18\!\cdots\!48 \)
T^4 + 326842*T^3 - 347890289085*T^2 - 35704786356945846*T + 18709950481247520979848
$59$
\( T^{4} - 421384 T^{3} + \cdots - 33\!\cdots\!00 \)
T^4 - 421384*T^3 - 5538000319011*T^2 + 8271239778108549774*T - 3302238646893360441551400
$61$
\( T^{4} - 116825 T^{3} + \cdots + 23\!\cdots\!60 \)
T^4 - 116825*T^3 - 3251116328970*T^2 - 59553768706005308*T + 2330991955052985877433560
$67$
\( T^{4} - 5794566 T^{3} + \cdots + 26\!\cdots\!80 \)
T^4 - 5794566*T^3 + 11419584052881*T^2 - 9210308051941505432*T + 2616528632045115659204880
$71$
\( T^{4} - 10590626 T^{3} + \cdots + 14\!\cdots\!04 \)
T^4 - 10590626*T^3 + 36394168513356*T^2 - 43544330388552637368*T + 14454059305454778160528704
$73$
\( T^{4} - 3971389 T^{3} + \cdots + 13\!\cdots\!58 \)
T^4 - 3971389*T^3 - 32853962452815*T^2 + 110325479713313166673*T + 13022210324470434835321658
$79$
\( T^{4} - 5597800 T^{3} + \cdots + 85\!\cdots\!00 \)
T^4 - 5597800*T^3 - 29059967927988*T^2 + 30842618200472950720*T + 85146096600208554445721600
$83$
\( T^{4} - 4665800 T^{3} + \cdots + 88\!\cdots\!88 \)
T^4 - 4665800*T^3 - 18352798021476*T^2 + 33307217683949825280*T + 88825376924693638386631488
$89$
\( T^{4} + 2794214 T^{3} + \cdots + 25\!\cdots\!80 \)
T^4 + 2794214*T^3 - 78161657660496*T^2 - 25710653689707752352*T + 253495159649154562031182080
$97$
\( T^{4} + 14445130 T^{3} + \cdots - 28\!\cdots\!80 \)
T^4 + 14445130*T^3 - 35804303115060*T^2 - 445273526593930340264*T - 280256871884960159987094080
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