[N,k,chi] = [2,38,Mod(1,2)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2, base_ring=CyclotomicField(1))
chi = DirichletCharacter(H, H._module([]))
N = Newforms(chi, 38, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("2.1");
S:= CuspForms(chi, 38);
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 \(\beta = 17280\sqrt{3026574721}\).
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
This newform does not admit any (nontrivial ) inner twists .
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{2} - 423071208T_{3} - 858983057411401584 \)
T3^2 - 423071208*T3 - 858983057411401584
acting on \(S_{38}^{\mathrm{new}}(\Gamma_0(2))\).
$p$
$F_p(T)$
$2$
\( (T + 262144)^{2} \)
(T + 262144)^2
$3$
\( T^{2} - 423071208 T - 85\!\cdots\!84 \)
T^2 - 423071208*T - 858983057411401584
$5$
\( T^{2} + 13507530555540 T + 34\!\cdots\!00 \)
T^2 + 13507530555540*T + 34894076611143696643462500
$7$
\( T^{2} + \cdots - 21\!\cdots\!16 \)
T^2 - 3106174162962256*T - 21340880402412489355338134561216
$11$
\( T^{2} + \cdots - 57\!\cdots\!96 \)
T^2 - 4002878661541032504*T - 572924086717461351777145337359835514096
$13$
\( T^{2} + \cdots + 42\!\cdots\!96 \)
T^2 + 417717305951174459972*T + 42163030619935434852932587360136461502596
$17$
\( T^{2} + \cdots - 48\!\cdots\!76 \)
T^2 - 924888511508274921636*T - 4824675936062845560548781350742083029779244476
$19$
\( T^{2} + \cdots + 55\!\cdots\!00 \)
T^2 + 1502800666863954423438200*T + 558294968489684915592295528874099875820759376400
$23$
\( T^{2} + \cdots - 16\!\cdots\!84 \)
T^2 - 4977294427767526272845808*T - 163984902977942675247252227414282441053143590453184
$29$
\( T^{2} + \cdots - 34\!\cdots\!00 \)
T^2 - 1174284943679786618741874780*T - 343598038505549149964173133414420334391909308214458300
$31$
\( T^{2} + \cdots + 69\!\cdots\!84 \)
T^2 + 5529622970549622212915224256*T + 6955867069316120449733932206840101176382726336141353984
$37$
\( T^{2} + \cdots + 26\!\cdots\!84 \)
T^2 + 115976327866820065024925346644*T + 2603901057295244152117261262487307179402209395250838394084
$41$
\( T^{2} + \cdots - 44\!\cdots\!16 \)
T^2 - 479903809827585250071775594644*T - 44478022907018776425640423161197883670607777570679122434716
$43$
\( T^{2} + \cdots + 90\!\cdots\!76 \)
T^2 + 3191808379897050809392290082952*T + 907923743659928456046944637170510538529839124425791163240976
$47$
\( T^{2} + \cdots - 20\!\cdots\!96 \)
T^2 + 11214072446606967601132239964704*T - 20040548881137146723125846690496717942262163174100001591549696
$53$
\( T^{2} + \cdots + 62\!\cdots\!96 \)
T^2 - 53695242990369729463598618231628*T + 625127054876617069169927625172904461274462169575468888343000996
$59$
\( T^{2} + \cdots - 60\!\cdots\!00 \)
T^2 - 58132414930279452353014089908760*T - 600457187623873291956952780721674050817566084824492652255231289200
$61$
\( T^{2} + \cdots - 32\!\cdots\!36 \)
T^2 - 949379410209793628730917745257884*T - 321747581855552036234910088163411634054725444972396501575112946236
$67$
\( T^{2} + \cdots + 26\!\cdots\!04 \)
T^2 - 3280488273924335151706569871726696*T + 2657372696300700565375448280710824634898394765248589408640345090704
$71$
\( T^{2} + \cdots + 21\!\cdots\!44 \)
T^2 + 36632636242259358919270327956284976*T + 218702176467438382158235578902960964749891920285334066609711070548544
$73$
\( T^{2} + \cdots - 13\!\cdots\!04 \)
T^2 + 10722559667485328549057125275606572*T - 1307214393958286249199620225261461280073263440608169435180600950155804
$79$
\( T^{2} + \cdots - 98\!\cdots\!00 \)
T^2 - 41775981111170500771329248250783520*T - 9838563969714934103476039830674794451580474996600352228580432664416000
$83$
\( T^{2} + \cdots - 18\!\cdots\!64 \)
T^2 - 173069025965519133350295483333594888*T - 18877250690778647954070722224402407367371399695935696845740056557306864
$89$
\( T^{2} + \cdots - 23\!\cdots\!00 \)
T^2 + 770863557920750582452392321302299020*T - 233312153285683476680637060796379451481707899838744929250060155124999900
$97$
\( T^{2} + \cdots - 19\!\cdots\!16 \)
T^2 + 6502221928957848822710145562502442044*T - 19061728272172494224054989606634929511913335126809230650347416452885435516
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