[N,k,chi] = [15,10,Mod(1,15)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(15, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0]))
N = Newforms(chi, 10, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("15.1");
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
sage: f.q_expansion() # note that sage often uses an isomorphic number field
Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \frac{1}{2}(-1 + 3\sqrt{241})\).
We also show the integral \(q\)-expansion of the trace form .
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
\(3\)
\(1\)
\(5\)
\(-1\)
This newform does not admit any (nontrivial ) inner twists .
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{2} - 31T_{2} - 302 \)
T2^2 - 31*T2 - 302
acting on \(S_{10}^{\mathrm{new}}(\Gamma_0(15))\).
$p$
$F_p(T)$
$2$
\( T^{2} - 31T - 302 \)
T^2 - 31*T - 302
$3$
\( (T + 81)^{2} \)
(T + 81)^2
$5$
\( (T - 625)^{2} \)
(T - 625)^2
$7$
\( T^{2} - 14112 T + 22579200 \)
T^2 - 14112*T + 22579200
$11$
\( T^{2} + 21512 T - 2924934128 \)
T^2 + 21512*T - 2924934128
$13$
\( T^{2} - 24284 T - 15338329532 \)
T^2 - 24284*T - 15338329532
$17$
\( T^{2} + 156956 T - 24505657916 \)
T^2 + 156956*T - 24505657916
$19$
\( T^{2} + 95896 T - 15492203120 \)
T^2 + 95896*T - 15492203120
$23$
\( T^{2} + 735264 T - 239117414400 \)
T^2 + 735264*T - 239117414400
$29$
\( T^{2} + 2678212 T - 13616383922300 \)
T^2 + 2678212*T - 13616383922300
$31$
\( T^{2} - 10782432 T + 16415447040000 \)
T^2 - 10782432*T + 16415447040000
$37$
\( T^{2} - 21968332 T + 99286893737380 \)
T^2 - 21968332*T + 99286893737380
$41$
\( T^{2} - 26060372 T + 38475315093220 \)
T^2 - 26060372*T + 38475315093220
$43$
\( T^{2} + \cdots - 254550637865456 \)
T^2 + 7191160*T - 254550637865456
$47$
\( T^{2} + 31580240 T - 18\!\cdots\!24 \)
T^2 + 31580240*T - 1856218340076224
$53$
\( T^{2} - 3131116 T - 14677210114460 \)
T^2 - 3131116*T - 14677210114460
$59$
\( T^{2} + 35494664 T - 99\!\cdots\!20 \)
T^2 + 35494664*T - 9937984740980720
$61$
\( T^{2} - 341497340 T + 28\!\cdots\!96 \)
T^2 - 341497340*T + 28665513361108996
$67$
\( T^{2} + 288195816 T + 20\!\cdots\!64 \)
T^2 + 288195816*T + 20192770248606864
$71$
\( T^{2} + \cdots - 147594805309376 \)
T^2 - 210286064*T - 147594805309376
$73$
\( T^{2} + 232663084 T + 12\!\cdots\!60 \)
T^2 + 232663084*T + 12793033974476260
$79$
\( T^{2} + 24755040 T - 19\!\cdots\!00 \)
T^2 + 24755040*T - 196612497485568000
$83$
\( T^{2} + 372082152 T + 60\!\cdots\!92 \)
T^2 + 372082152*T + 6028318254901392
$89$
\( T^{2} + 427639116 T + 13\!\cdots\!20 \)
T^2 + 427639116*T + 1315495255181220
$97$
\( T^{2} - 1771658884 T + 68\!\cdots\!64 \)
T^2 - 1771658884*T + 686586583127112964
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