[N,k,chi] = [2,48,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, 48, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("2.1");
S:= CuspForms(chi, 48);
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 \(\beta = 8640\sqrt{23589383914321}\).
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
\( T_{3}^{2} - 122289844824T_{3} - 40284750299492923142256 \)
T3^2 - 122289844824*T3 - 40284750299492923142256
acting on \(S_{48}^{\mathrm{new}}(\Gamma_0(2))\).
$p$
$F_p(T)$
$2$
\( (T + 8388608)^{2} \)
(T + 8388608)^2
$3$
\( T^{2} - 122289844824 T - 40\!\cdots\!56 \)
T^2 - 122289844824*T - 40284750299492923142256
$5$
\( T^{2} + \cdots - 29\!\cdots\!00 \)
T^2 - 18178497839194140*T - 291472519478814040743990593437500
$7$
\( T^{2} + \cdots + 56\!\cdots\!56 \)
T^2 - 161018545456026827632*T + 5678914641051595864736540497602405921856
$11$
\( T^{2} + \cdots - 10\!\cdots\!36 \)
T^2 - 1148414200601353937498184*T - 10916338079468376301291205622853735256523389815536
$13$
\( T^{2} + \cdots - 33\!\cdots\!76 \)
T^2 + 91147460388086216079419636*T - 33258305616195728177429480115633727942896269876966876
$17$
\( T^{2} + \cdots - 58\!\cdots\!64 \)
T^2 + 102841736272610788276309928988*T - 5817875270181664287116049814481949528749514213820513883964
$19$
\( T^{2} + \cdots - 18\!\cdots\!00 \)
T^2 - 1021269902513170084335814340920*T - 184083551344210409802767314842191748319502181654718920028400
$23$
\( T^{2} + \cdots - 81\!\cdots\!56 \)
T^2 - 8761047668524985915668413593424*T - 816521283887426672325611809324510053828626176073542417759429056
$29$
\( T^{2} + \cdots + 43\!\cdots\!00 \)
T^2 - 44866738270461738647546902365332460*T + 43550206442626160140530707194237059799292257513069993772019272652900
$31$
\( T^{2} + \cdots + 62\!\cdots\!04 \)
T^2 - 207627610718284877414441666427987904*T + 6248413694072820396848834771203840649749175765467310099049450363618304
$37$
\( T^{2} + \cdots + 49\!\cdots\!76 \)
T^2 - 15454503825836181610788445034143257052*T + 49839428305731469959830762025456398682346167403453517291523793370425572676
$41$
\( T^{2} + \cdots + 82\!\cdots\!64 \)
T^2 + 183644075219801436287443355635335693516*T + 8279586657111518425973535270422749691521723262921428173190323073486683750564
$43$
\( T^{2} + \cdots - 43\!\cdots\!96 \)
T^2 - 251792238934935641547265550793932228104*T - 43905566863939881624293422914548172834823012102611608752509801975911170701296
$47$
\( T^{2} + \cdots - 22\!\cdots\!64 \)
T^2 - 2877259596300283378159171629753736631712*T - 2210904571016427776001025062795811922895831131415435297727194443060336218347264
$53$
\( T^{2} + \cdots + 30\!\cdots\!24 \)
T^2 - 36237003736012302609225684974144068811964*T + 301572939368384355896187096267648908823482756204187068333856119786017399921224324
$59$
\( T^{2} + \cdots - 22\!\cdots\!00 \)
T^2 - 216631413518842627379871131067900548684520*T - 226898024619263567488541974306219654225028961478719848471673561621593375420541932400
$61$
\( T^{2} + \cdots + 13\!\cdots\!04 \)
T^2 - 2343345794410501150323838534047225181209004*T + 1363715601562481933866000321819854516169228398417121378122722483907522343677311628004
$67$
\( T^{2} + \cdots - 16\!\cdots\!84 \)
T^2 - 9679359518522646693994124509775830904939992*T - 16265097916410211119563100564976515431801326827901607685318855724925895743443606799984
$71$
\( T^{2} + \cdots + 30\!\cdots\!84 \)
T^2 + 34699689890171309465034074226599979171319056*T + 300515895631199275915581291863768191368949087028983608055409313840564564247126349342784
$73$
\( T^{2} + \cdots + 23\!\cdots\!24 \)
T^2 + 44919947676390510667705279890416107043412236*T + 233259621999333310665778418811668888818517502534599469927534644349250568210734247269924
$79$
\( T^{2} + \cdots - 11\!\cdots\!00 \)
T^2 - 459051391676893300046865817924586171479631200*T - 115593368576838518707696208418412018458353771551721570630118914851127179087354277302880000
$83$
\( T^{2} + \cdots - 24\!\cdots\!36 \)
T^2 + 1357110492338922730441331533770737425416063816*T - 2411164331125473560223551862123179991039640729865190846404604170488664724671609363749879536
$89$
\( T^{2} + \cdots + 37\!\cdots\!00 \)
T^2 - 4559737044037194815580171398227946101567633620*T + 3755478690617220842784881326888128716121478459555144501649461982886147670099604487062576100
$97$
\( T^{2} + \cdots - 35\!\cdots\!84 \)
T^2 - 53827580940974934190192198959367945947750881092*T - 357808429925493074380456101830528107721888035350555933000498779123309220649928309079955221884
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