[N,k,chi] = [9,32,Mod(1,9)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(9, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0]))
N = Newforms(chi, 32, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("9.1");
S:= CuspForms(chi, 32);
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 \(\beta = 12\sqrt{18295489}\).
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
\(3\)
\(-1\)
This newform does not admit any (nontrivial ) inner twists .
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{2} + 39960T_{2} - 2235350016 \)
T2^2 + 39960*T2 - 2235350016
acting on \(S_{32}^{\mathrm{new}}(\Gamma_0(9))\).
$p$
$F_p(T)$
$2$
\( T^{2} + 39960 T - 2235350016 \)
T^2 + 39960*T - 2235350016
$3$
\( T^{2} \)
T^2
$5$
\( T^{2} - 19391218020 T - 52\!\cdots\!00 \)
T^2 - 19391218020*T - 5207533830370075837500
$7$
\( T^{2} - 30257527577200 T + 21\!\cdots\!64 \)
T^2 - 30257527577200*T + 215248958321503343099673664
$11$
\( T^{2} + \cdots + 12\!\cdots\!44 \)
T^2 - 7782353745118776*T + 12925461622179405522665416694544
$13$
\( T^{2} + \cdots - 55\!\cdots\!04 \)
T^2 - 74708953050260620*T - 55840446248564766859168952706602204
$17$
\( T^{2} + \cdots + 68\!\cdots\!24 \)
T^2 + 17224607828987089380*T + 68816558085487278109255355211944035524
$19$
\( T^{2} + \cdots - 27\!\cdots\!00 \)
T^2 + 12370563328022164040*T - 27349625681839823756754216365982057200
$23$
\( T^{2} + \cdots - 73\!\cdots\!24 \)
T^2 + 1897344841989911219280*T - 7320789774164772118783844443128838117824
$29$
\( T^{2} + \cdots + 39\!\cdots\!00 \)
T^2 + 128576144217217055807340*T + 3988461504574857245463376159085403051965523300
$31$
\( T^{2} + \cdots - 11\!\cdots\!76 \)
T^2 - 125733527517961838793664*T - 11382805413133685253581281370331459411205323776
$37$
\( T^{2} + \cdots - 25\!\cdots\!56 \)
T^2 + 833815207054016911025060*T - 2561910230639750316088168823481574918712779239356
$41$
\( T^{2} + \cdots - 70\!\cdots\!36 \)
T^2 + 8724924335662925840671284*T - 70412100180933090684963737379633805837990111807836
$43$
\( T^{2} + \cdots - 22\!\cdots\!64 \)
T^2 + 18397105293779438708372600*T - 224744625618288919560448240483378192111482942804464
$47$
\( T^{2} + \cdots + 15\!\cdots\!04 \)
T^2 + 95450963964856190793148320*T + 1582516295602393343949151969077682217870833535387904
$53$
\( T^{2} + \cdots + 56\!\cdots\!16 \)
T^2 + 194822508473721098983660860*T + 5669456386269946229894588471187975779922419145648516
$59$
\( T^{2} + \cdots - 51\!\cdots\!00 \)
T^2 - 198723263547765513990746520*T - 5106130251385620156703043876512306275679756183066258800
$61$
\( T^{2} + \cdots + 36\!\cdots\!44 \)
T^2 + 12056218201113004361157656276*T + 36002055256799213986728060478774063685786261227804547044
$67$
\( T^{2} + \cdots + 26\!\cdots\!24 \)
T^2 + 9688140802872256994032247720*T + 2609823606313430294561304878158722768094085812212647824
$71$
\( T^{2} + \cdots - 16\!\cdots\!16 \)
T^2 + 55784576625034657512878287344*T - 1619113558799962439757154641063694527169347926257166356416
$73$
\( T^{2} + \cdots + 90\!\cdots\!76 \)
T^2 - 62234095932086098750552955380*T + 901750243634718265920960629620255610952850535031976944676
$79$
\( T^{2} + \cdots - 53\!\cdots\!00 \)
T^2 + 119164191093299704964212710560*T - 53777624289831823819259548596309958903379081142820880787200
$83$
\( T^{2} + \cdots - 86\!\cdots\!44 \)
T^2 - 264863373681569145625357596360*T - 86079799395595721746499756931995789333131931253865756053744
$89$
\( T^{2} + \cdots - 62\!\cdots\!00 \)
T^2 - 2154414614246670291602721895980*T - 629919768069727638550675160656965945559168595702629212150300
$97$
\( T^{2} + \cdots + 19\!\cdots\!04 \)
T^2 + 9060767994874032615529957205180*T + 19801392873969376170123170208547004916023170619961754039900804
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