LMFDB - Embedded newform 1331.1.b.a.1330.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1331,1,Mod(1330,1331)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1331, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1]))

N = Newforms(chi, 1, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("1331.1330");

S:= CuspForms(chi, 1);

N := Newforms(S);

Level:	$ N $	$=$	$ 1331 = 11^{3} $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	1331.b (of order $2$, degree $1$, minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$0.664255531815$
Analytic rank:	$0$
Dimension:	$5$
Coefficient field:	$\Q(\zeta_{22})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - x^{4} - 4x^{3} + 3x^{2} + 3x - 1 $ x^5 - x^4 - 4x^3 + 3x^2 + 3*x - 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Projective image:	$D_{11}$
Projective field:	Galois closure of 11.1.4177248169415651.1

Embedding invariants

Embedding label			1330.5
Root			$0.284630$ of defining polynomial
Character	$\chi$	$=$	1331.1330

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.68251 q^{3} +1.00000 q^{4} -1.30972 q^{5} +1.83083 q^{9} +1.68251 q^{12} -2.20362 q^{15} +1.00000 q^{16} -1.30972 q^{20} -1.91899 q^{23} +0.715370 q^{25} +1.39788 q^{27} -1.91899 q^{31} +1.83083 q^{36} +0.830830 q^{37} -2.39788 q^{45} +0.830830 q^{47} +1.68251 q^{48} +1.00000 q^{49} -0.284630 q^{53} -0.284630 q^{59} -2.20362 q^{60} +1.00000 q^{64} -1.30972 q^{67} -3.22871 q^{69} +0.830830 q^{71} +1.20362 q^{75} -1.30972 q^{80} +0.521109 q^{81} -0.284630 q^{89} -1.91899 q^{92} -3.22871 q^{93} -1.91899 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q - q^{3} + 5 q^{4} - q^{5} + 4 q^{9} - q^{12} - 2 q^{15} + 5 q^{16} - q^{20} - q^{23} + 4 q^{25} - 2 q^{27} - q^{31} + 4 q^{36} - q^{37} - 3 q^{45} - q^{47} - q^{48} + 5 q^{49} - q^{53} - q^{59}+ \cdots - q^{97}+O(q^{100}) $ 5 * q - q^3 + 5 * q^4 - q^5 + 4 * q^9 - q^12 - 2 * q^15 + 5 * q^16 - q^20 - q^23 + 4 * q^25 - 2 * q^27 - q^31 + 4 * q^36 - q^37 - 3 * q^45 - q^47 - q^48 + 5 * q^49 - q^53 - q^59 - 2 * q^60 + 5 * q^64 - q^67 - 2 * q^69 - q^71 - 3 * q^75 - q^80 + 3 * q^81 - q^89 - q^92 - 2 * q^93 - q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1331\mathbb{Z}\right)^\times$.

$n$	$2$
$\chi(n)$	$-1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0	1.00000			$0$
$2$	0	0	−1.00000			$\pi$
$3$	1.68251	1.68251	0.841254	−	0.540641i	$-0.181818\pi$
$3$	1.68251	1.68251	0.841254	+	0.540641i	$0.181818\pi$
$4$	1.00000	1.00000
$5$	−1.30972	−1.30972	−0.654861	−	0.755750i	$-0.727273\pi$
$5$	−1.30972	−1.30972	−0.654861	+	0.755750i	$0.727273\pi$
$6$	0	0
$7$	0	0	1.00000			$0$
$7$	0	0	−1.00000			$\pi$
$8$	0	0
$9$	1.83083	1.83083
$10$	0	0
$11$	0	0
$11$	0	0
$12$	1.68251	1.68251
$13$	0	0	1.00000			$0$
$13$	0	0	−1.00000			$\pi$
$14$	0	0
$15$	−2.20362	−2.20362
$16$	1.00000	1.00000
$17$	0	0	1.00000			$0$
$17$	0	0	−1.00000			$\pi$
$18$	0	0
$19$	0	0	1.00000			$0$
$19$	0	0	−1.00000			$\pi$
$20$	−1.30972	−1.30972
$21$	0	0
$22$	0	0
$23$	−1.91899	−1.91899	−0.959493	−	0.281733i	$-0.909091\pi$
$23$	−1.91899	−1.91899	−0.959493	+	0.281733i	$0.909091\pi$
$24$	0	0
$25$	0.715370	0.715370
$26$	0	0
$27$	1.39788	1.39788
$28$	0	0
$29$	0	0	1.00000			$0$
$29$	0	0	−1.00000			$\pi$
$30$	0	0
$31$	−1.91899	−1.91899	−0.959493	−	0.281733i	$-0.909091\pi$
$31$	−1.91899	−1.91899	−0.959493	+	0.281733i	$0.909091\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	0	0
$36$	1.83083	1.83083
$37$	0.830830	0.830830	0.415415	−	0.909632i	$-0.363636\pi$
$37$	0.830830	0.830830	0.415415	+	0.909632i	$0.363636\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	0	0	1.00000			$0$
$41$	0	0	−1.00000			$\pi$
$42$	0	0
$43$	0	0	1.00000			$0$
$43$	0	0	−1.00000			$\pi$
$44$	0	0
$45$	−2.39788	−2.39788
$46$	0	0
$47$	0.830830	0.830830	0.415415	−	0.909632i	$-0.363636\pi$
$47$	0.830830	0.830830	0.415415	+	0.909632i	$0.363636\pi$
$48$	1.68251	1.68251
$49$	1.00000	1.00000
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−0.284630	−0.284630	−0.142315	−	0.989821i	$-0.545455\pi$
$53$	−0.284630	−0.284630	−0.142315	+	0.989821i	$0.545455\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−0.284630	−0.284630	−0.142315	−	0.989821i	$-0.545455\pi$
$59$	−0.284630	−0.284630	−0.142315	+	0.989821i	$0.545455\pi$
$60$	−2.20362	−2.20362
$61$	0	0	1.00000			$0$
$61$	0	0	−1.00000			$\pi$
$62$	0	0
$63$	0	0
$64$	1.00000	1.00000
$65$	0	0
$66$	0	0
$67$	−1.30972	−1.30972	−0.654861	−	0.755750i	$-0.727273\pi$
$67$	−1.30972	−1.30972	−0.654861	+	0.755750i	$0.727273\pi$
$68$	0	0
$69$	−3.22871	−3.22871
$70$	0	0
$71$	0.830830	0.830830	0.415415	−	0.909632i	$-0.363636\pi$
$71$	0.830830	0.830830	0.415415	+	0.909632i	$0.363636\pi$
$72$	0	0
$73$	0	0	1.00000			$0$
$73$	0	0	−1.00000			$\pi$
$74$	0	0
$75$	1.20362	1.20362
$76$	0	0
$77$	0	0
$78$	0	0
$79$	0	0	1.00000			$0$
$79$	0	0	−1.00000			$\pi$
$80$	−1.30972	−1.30972
$81$	0.521109	0.521109
$82$	0	0
$83$	0	0	1.00000			$0$
$83$	0	0	−1.00000			$\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−0.284630	−0.284630	−0.142315	−	0.989821i	$-0.545455\pi$
$89$	−0.284630	−0.284630	−0.142315	+	0.989821i	$0.545455\pi$
$90$	0	0
$91$	0	0
$92$	−1.91899	−1.91899
$93$	−3.22871	−3.22871
$94$	0	0
$95$	0	0
$96$	0	0
$97$	−1.91899	−1.91899	−0.959493	−	0.281733i	$-0.909091\pi$
$97$	−1.91899	−1.91899	−0.959493	+	0.281733i	$0.909091\pi$
$98$	0	0
$99$	0	0
$100$	0.715370	0.715370
$101$	0	0	1.00000			$0$
$101$	0	0	−1.00000			$\pi$
$102$	0	0
$103$	−1.30972	−1.30972	−0.654861	−	0.755750i	$-0.727273\pi$
$103$	−1.30972	−1.30972	−0.654861	+	0.755750i	$0.727273\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	0	0	1.00000			$0$
$107$	0	0	−1.00000			$\pi$
$108$	1.39788	1.39788
$109$	0	0	1.00000			$0$
$109$	0	0	−1.00000			$\pi$
$110$	0	0
$111$	1.39788	1.39788
$112$	0	0
$113$	1.68251	1.68251	0.841254	−	0.540641i	$-0.181818\pi$
$113$	1.68251	1.68251	0.841254	+	0.540641i	$0.181818\pi$
$114$	0	0
$115$	2.51334	2.51334
$116$	0	0
$117$	0	0
$118$	0	0
$119$	0	0
$120$	0	0
$121$	0	0
$122$	0	0
$123$	0	0
$124$	−1.91899	−1.91899
$125$	0.372786	0.372786
$126$	0	0
$127$	0	0	1.00000			$0$
$127$	0	0	−1.00000			$\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	0	0	1.00000			$0$
$131$	0	0	−1.00000			$\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	−1.83083	−1.83083
$136$	0	0
$137$	−1.91899	−1.91899	−0.959493	−	0.281733i	$-0.909091\pi$
$137$	−1.91899	−1.91899	−0.959493	+	0.281733i	$0.909091\pi$
$138$	0	0
$139$	0	0	1.00000			$0$
$139$	0	0	−1.00000			$\pi$
$140$	0	0
$141$	1.39788	1.39788
$142$	0	0
$143$	0	0
$144$	1.83083	1.83083
$145$	0	0
$146$	0	0
$147$	1.68251	1.68251
$148$	0.830830	0.830830
$149$	0	0	1.00000			$0$
$149$	0	0	−1.00000			$\pi$
$150$	0	0
$151$	0	0	1.00000			$0$
$151$	0	0	−1.00000			$\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	2.51334	2.51334
$156$	0	0
$157$	−0.284630	−0.284630	−0.142315	−	0.989821i	$-0.545455\pi$
$157$	−0.284630	−0.284630	−0.142315	+	0.989821i	$0.545455\pi$
$158$	0	0
$159$	−0.478891	−0.478891
$160$	0	0
$161$	0	0
$162$	0	0
$163$	1.68251	1.68251	0.841254	−	0.540641i	$-0.181818\pi$
$163$	1.68251	1.68251	0.841254	+	0.540641i	$0.181818\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	0	0	1.00000			$0$
$167$	0	0	−1.00000			$\pi$
$168$	0	0
$169$	1.00000	1.00000
$170$	0	0
$171$	0	0
$172$	0	0
$173$	0	0	1.00000			$0$
$173$	0	0	−1.00000			$\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−0.478891	−0.478891
$178$	0	0
$179$	−1.91899	−1.91899	−0.959493	−	0.281733i	$-0.909091\pi$
$179$	−1.91899	−1.91899	−0.959493	+	0.281733i	$0.909091\pi$
$180$	−2.39788	−2.39788
$181$	1.68251	1.68251	0.841254	−	0.540641i	$-0.181818\pi$
$181$	1.68251	1.68251	0.841254	+	0.540641i	$0.181818\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−1.08816	−1.08816
$186$	0	0
$187$	0	0
$188$	0.830830	0.830830
$189$	0	0
$190$	0	0
$191$	1.68251	1.68251	0.841254	−	0.540641i	$-0.181818\pi$
$191$	1.68251	1.68251	0.841254	+	0.540641i	$0.181818\pi$
$192$	1.68251	1.68251
$193$	0	0	1.00000			$0$
$193$	0	0	−1.00000			$\pi$
$194$	0	0
$195$	0	0
$196$	1.00000	1.00000
$197$	0	0	1.00000			$0$
$197$	0	0	−1.00000			$\pi$
$198$	0	0
$199$	−0.284630	−0.284630	−0.142315	−	0.989821i	$-0.545455\pi$
$199$	−0.284630	−0.284630	−0.142315	+	0.989821i	$0.545455\pi$
$200$	0	0
$201$	−2.20362	−2.20362
$202$	0	0
$203$	0	0
$204$	0	0
$205$	0	0
$206$	0	0
$207$	−3.51334	−3.51334
$208$	0	0
$209$	0	0
$210$	0	0
$211$	0	0	1.00000			$0$
$211$	0	0	−1.00000			$\pi$
$212$	−0.284630	−0.284630
$213$	1.39788	1.39788
$214$	0	0
$215$	0	0
$216$	0	0
$217$	0	0
$218$	0	0
$219$	0	0
$220$	0	0
$221$	0	0
$222$	0	0
$223$	0.830830	0.830830	0.415415	−	0.909632i	$-0.363636\pi$
$223$	0.830830	0.830830	0.415415	+	0.909632i	$0.363636\pi$
$224$	0	0
$225$	1.30972	1.30972
$226$	0	0
$227$	0	0	1.00000			$0$
$227$	0	0	−1.00000			$\pi$
$228$	0	0
$229$	−1.30972	−1.30972	−0.654861	−	0.755750i	$-0.727273\pi$
$229$	−1.30972	−1.30972	−0.654861	+	0.755750i	$0.727273\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	0	0	1.00000			$0$
$233$	0	0	−1.00000			$\pi$
$234$	0	0
$235$	−1.08816	−1.08816
$236$	−0.284630	−0.284630
$237$	0	0
$238$	0	0
$239$	0	0	1.00000			$0$
$239$	0	0	−1.00000			$\pi$
$240$	−2.20362	−2.20362
$241$	0	0	1.00000			$0$
$241$	0	0	−1.00000			$\pi$
$242$	0	0
$243$	−0.521109	−0.521109
$244$	0	0
$245$	−1.30972	−1.30972
$246$	0	0
$247$	0	0
$248$	0	0
$249$	0	0
$250$	0	0
$251$	1.68251	1.68251	0.841254	−	0.540641i	$-0.181818\pi$
$251$	1.68251	1.68251	0.841254	+	0.540641i	$0.181818\pi$
$252$	0	0
$253$	0	0
$254$	0	0
$255$	0	0
$256$	1.00000	1.00000
$257$	0.830830	0.830830	0.415415	−	0.909632i	$-0.363636\pi$
$257$	0.830830	0.830830	0.415415	+	0.909632i	$0.363636\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	0	0
$262$	0	0
$263$	0	0	1.00000			$0$
$263$	0	0	−1.00000			$\pi$
$264$	0	0
$265$	0.372786	0.372786
$266$	0	0
$267$	−0.478891	−0.478891
$268$	−1.30972	−1.30972
$269$	−1.91899	−1.91899	−0.959493	−	0.281733i	$-0.909091\pi$
$269$	−1.91899	−1.91899	−0.959493	+	0.281733i	$0.909091\pi$
$270$	0	0
$271$	0	0	1.00000			$0$
$271$	0	0	−1.00000			$\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	0	0
$276$	−3.22871	−3.22871
$277$	0	0	1.00000			$0$
$277$	0	0	−1.00000			$\pi$
$278$	0	0
$279$	−3.51334	−3.51334
$280$	0	0
$281$	0	0	1.00000			$0$
$281$	0	0	−1.00000			$\pi$
$282$	0	0
$283$	0	0	1.00000			$0$
$283$	0	0	−1.00000			$\pi$
$284$	0.830830	0.830830
$285$	0	0
$286$	0	0
$287$	0	0
$288$	0	0
$289$	1.00000	1.00000
$290$	0	0
$291$	−3.22871	−3.22871
$292$	0	0
$293$	0	0	1.00000			$0$
$293$	0	0	−1.00000			$\pi$
$294$	0	0
$295$	0.372786	0.372786
$296$	0	0
$297$	0	0
$298$	0	0
$299$	0	0
$300$	1.20362	1.20362
$301$	0	0
$302$	0	0
$303$	0	0
$304$	0	0
$305$	0	0
$306$	0	0
$307$	0	0	1.00000			$0$
$307$	0	0	−1.00000			$\pi$
$308$	0	0
$309$	−2.20362	−2.20362
$310$	0	0
$311$	1.68251	1.68251	0.841254	−	0.540641i	$-0.181818\pi$
$311$	1.68251	1.68251	0.841254	+	0.540641i	$0.181818\pi$
$312$	0	0
$313$	−0.284630	−0.284630	−0.142315	−	0.989821i	$-0.545455\pi$
$313$	−0.284630	−0.284630	−0.142315	+	0.989821i	$0.545455\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−0.284630	−0.284630	−0.142315	−	0.989821i	$-0.545455\pi$
$317$	−0.284630	−0.284630	−0.142315	+	0.989821i	$0.545455\pi$
$318$	0	0
$319$	0	0
$320$	−1.30972	−1.30972
$321$	0	0
$322$	0	0
$323$	0	0
$324$	0.521109	0.521109
$325$	0	0
$326$	0	0
$327$	0	0
$328$	0	0
$329$	0	0
$330$	0	0
$331$	−1.30972	−1.30972	−0.654861	−	0.755750i	$-0.727273\pi$
$331$	−1.30972	−1.30972	−0.654861	+	0.755750i	$0.727273\pi$
$332$	0	0
$333$	1.52111	1.52111
$334$	0	0
$335$	1.71537	1.71537
$336$	0	0
$337$	0	0	1.00000			$0$
$337$	0	0	−1.00000			$\pi$
$338$	0	0
$339$	2.83083	2.83083
$340$	0	0
$341$	0	0
$342$	0	0
$343$	0	0
$344$	0	0
$345$	4.22871	4.22871
$346$	0	0
$347$	0	0	1.00000			$0$
$347$	0	0	−1.00000			$\pi$
$348$	0	0
$349$	0	0	1.00000			$0$
$349$	0	0	−1.00000			$\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	2.00000	2.00000	1.00000			$0$
$353$	2.00000	2.00000	1.00000			$0$
$354$	0	0
$355$	−1.08816	−1.08816
$356$	−0.284630	−0.284630
$357$	0	0
$358$	0	0
$359$	0	0	1.00000			$0$
$359$	0	0	−1.00000			$\pi$
$360$	0	0
$361$	1.00000	1.00000
$362$	0	0
$363$	0	0
$364$	0	0
$365$	0	0
$366$	0	0
$367$	0.830830	0.830830	0.415415	−	0.909632i	$-0.363636\pi$
$367$	0.830830	0.830830	0.415415	+	0.909632i	$0.363636\pi$
$368$	−1.91899	−1.91899
$369$	0	0
$370$	0	0
$371$	0	0
$372$	−3.22871	−3.22871
$373$	0	0	1.00000			$0$
$373$	0	0	−1.00000			$\pi$
$374$	0	0
$375$	0.627214	0.627214
$376$	0	0
$377$	0	0
$378$	0	0
$379$	−0.284630	−0.284630	−0.142315	−	0.989821i	$-0.545455\pi$
$379$	−0.284630	−0.284630	−0.142315	+	0.989821i	$0.545455\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	0.830830	0.830830	0.415415	−	0.909632i	$-0.363636\pi$
$383$	0.830830	0.830830	0.415415	+	0.909632i	$0.363636\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	0	0
$388$	−1.91899	−1.91899
$389$	2.00000	2.00000	1.00000			$0$
$389$	2.00000	2.00000	1.00000			$0$
$390$	0	0
$391$	0	0
$392$	0	0
$393$	0	0
$394$	0	0
$395$	0	0
$396$	0	0
$397$	−1.91899	−1.91899	−0.959493	−	0.281733i	$-0.909091\pi$
$397$	−1.91899	−1.91899	−0.959493	+	0.281733i	$0.909091\pi$
$398$	0	0
$399$	0	0
$400$	0.715370	0.715370
$401$	1.68251	1.68251	0.841254	−	0.540641i	$-0.181818\pi$
$401$	1.68251	1.68251	0.841254	+	0.540641i	$0.181818\pi$
$402$	0	0
$403$	0	0
$404$	0	0
$405$	−0.682507	−0.682507
$406$	0	0
$407$	0	0
$408$	0	0
$409$	0	0	1.00000			$0$
$409$	0	0	−1.00000			$\pi$
$410$	0	0
$411$	−3.22871	−3.22871
$412$	−1.30972	−1.30972
$413$	0	0
$414$	0	0
$415$	0	0
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−1.91899	−1.91899	−0.959493	−	0.281733i	$-0.909091\pi$
$419$	−1.91899	−1.91899	−0.959493	+	0.281733i	$0.909091\pi$
$420$	0	0
$421$	1.68251	1.68251	0.841254	−	0.540641i	$-0.181818\pi$
$421$	1.68251	1.68251	0.841254	+	0.540641i	$0.181818\pi$
$422$	0	0
$423$	1.52111	1.52111
$424$	0	0
$425$	0	0
$426$	0	0
$427$	0	0
$428$	0	0
$429$	0	0
$430$	0	0
$431$	0	0	1.00000			$0$
$431$	0	0	−1.00000			$\pi$
$432$	1.39788	1.39788
$433$	−1.91899	−1.91899	−0.959493	−	0.281733i	$-0.909091\pi$
$433$	−1.91899	−1.91899	−0.959493	+	0.281733i	$0.909091\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	0	0
$438$	0	0
$439$	0	0	1.00000			$0$
$439$	0	0	−1.00000			$\pi$
$440$	0	0
$441$	1.83083	1.83083
$442$	0	0
$443$	2.00000	2.00000	1.00000			$0$
$443$	2.00000	2.00000	1.00000			$0$
$444$	1.39788	1.39788
$445$	0.372786	0.372786
$446$	0	0
$447$	0	0
$448$	0	0
$449$	1.68251	1.68251	0.841254	−	0.540641i	$-0.181818\pi$
$449$	1.68251	1.68251	0.841254	+	0.540641i	$0.181818\pi$
$450$	0	0
$451$	0	0
$452$	1.68251	1.68251
$453$	0	0
$454$	0	0
$455$	0	0
$456$	0	0
$457$	0	0	1.00000			$0$
$457$	0	0	−1.00000			$\pi$
$458$	0	0
$459$	0	0
$460$	2.51334	2.51334
$461$	0	0	1.00000			$0$
$461$	0	0	−1.00000			$\pi$
$462$	0	0
$463$	1.68251	1.68251	0.841254	−	0.540641i	$-0.181818\pi$
$463$	1.68251	1.68251	0.841254	+	0.540641i	$0.181818\pi$
$464$	0	0
$465$	4.22871	4.22871
$466$	0	0
$467$	−0.284630	−0.284630	−0.142315	−	0.989821i	$-0.545455\pi$
$467$	−0.284630	−0.284630	−0.142315	+	0.989821i	$0.545455\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	−0.478891	−0.478891
$472$	0	0
$473$	0	0
$474$	0	0
$475$	0	0
$476$	0	0
$477$	−0.521109	−0.521109
$478$	0	0
$479$	0	0	1.00000			$0$
$479$	0	0	−1.00000			$\pi$
$480$	0	0
$481$	0	0
$482$	0	0
$483$	0	0
$484$	0	0
$485$	2.51334	2.51334
$486$	0	0
$487$	−0.284630	−0.284630	−0.142315	−	0.989821i	$-0.545455\pi$
$487$	−0.284630	−0.284630	−0.142315	+	0.989821i	$0.545455\pi$
$488$	0	0
$489$	2.83083	2.83083
$490$	0	0
$491$	0	0	1.00000			$0$
$491$	0	0	−1.00000			$\pi$
$492$	0	0
$493$	0	0
$494$	0	0
$495$	0	0
$496$	−1.91899	−1.91899
$497$	0	0
$498$	0	0
$499$	−1.30972	−1.30972	−0.654861	−	0.755750i	$-0.727273\pi$
$499$	−1.30972	−1.30972	−0.654861	+	0.755750i	$0.727273\pi$
$500$	0.372786	0.372786
$501$	0	0
$502$	0	0
$503$	0	0	1.00000			$0$
$503$	0	0	−1.00000			$\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	1.68251	1.68251
$508$	0	0
$509$	1.68251	1.68251	0.841254	−	0.540641i	$-0.181818\pi$
$509$	1.68251	1.68251	0.841254	+	0.540641i	$0.181818\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	0	0
$514$	0	0
$515$	1.71537	1.71537
$516$	0	0
$517$	0	0
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−1.91899	−1.91899	−0.959493	−	0.281733i	$-0.909091\pi$
$521$	−1.91899	−1.91899	−0.959493	+	0.281733i	$0.909091\pi$
$522$	0	0
$523$	0	0	1.00000			$0$
$523$	0	0	−1.00000			$\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	0	0
$528$	0	0
$529$	2.68251	2.68251
$530$	0	0
$531$	−0.521109	−0.521109
$532$	0	0
$533$	0	0
$534$	0	0
$535$	0	0
$536$	0	0
$537$	−3.22871	−3.22871
$538$	0	0
$539$	0	0
$540$	−1.83083	−1.83083
$541$	0	0	1.00000			$0$
$541$	0	0	−1.00000			$\pi$
$542$	0	0
$543$	2.83083	2.83083
$544$	0	0
$545$	0	0
$546$	0	0
$547$	0	0	1.00000			$0$
$547$	0	0	−1.00000			$\pi$
$548$	−1.91899	−1.91899
$549$	0	0
$550$	0	0
$551$	0	0
$552$	0	0
$553$	0	0
$554$	0	0
$555$	−1.83083	−1.83083
$556$	0	0
$557$	0	0	1.00000			$0$
$557$	0	0	−1.00000			$\pi$
$558$	0	0
$559$	0	0
$560$	0	0
$561$	0	0
$562$	0	0
$563$	0	0	1.00000			$0$
$563$	0	0	−1.00000			$\pi$
$564$	1.39788	1.39788
$565$	−2.20362	−2.20362
$566$	0	0
$567$	0	0
$568$	0	0
$569$	0	0	1.00000			$0$
$569$	0	0	−1.00000			$\pi$
$570$	0	0
$571$	0	0	1.00000			$0$
$571$	0	0	−1.00000			$\pi$
$572$	0	0
$573$	2.83083	2.83083
$574$	0	0
$575$	−1.37279	−1.37279
$576$	1.83083	1.83083
$577$	1.68251	1.68251	0.841254	−	0.540641i	$-0.181818\pi$
$577$	1.68251	1.68251	0.841254	+	0.540641i	$0.181818\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	0	0
$582$	0	0
$583$	0	0
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−1.91899	−1.91899	−0.959493	−	0.281733i	$-0.909091\pi$
$587$	−1.91899	−1.91899	−0.959493	+	0.281733i	$0.909091\pi$
$588$	1.68251	1.68251
$589$	0	0
$590$	0	0
$591$	0	0
$592$	0.830830	0.830830
$593$	0	0	1.00000			$0$
$593$	0	0	−1.00000			$\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−0.478891	−0.478891
$598$	0	0
$599$	−1.30972	−1.30972	−0.654861	−	0.755750i	$-0.727273\pi$
$599$	−1.30972	−1.30972	−0.654861	+	0.755750i	$0.727273\pi$
$600$	0	0
$601$	0	0	1.00000			$0$
$601$	0	0	−1.00000			$\pi$
$602$	0	0
$603$	−2.39788	−2.39788
$604$	0	0
$605$	0	0
$606$	0	0
$607$	0	0	1.00000			$0$
$607$	0	0	−1.00000			$\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	0	0
$612$	0	0
$613$	0	0	1.00000			$0$
$613$	0	0	−1.00000			$\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−1.30972	−1.30972	−0.654861	−	0.755750i	$-0.727273\pi$
$617$	−1.30972	−1.30972	−0.654861	+	0.755750i	$0.727273\pi$
$618$	0	0
$619$	−1.30972	−1.30972	−0.654861	−	0.755750i	$-0.727273\pi$
$619$	−1.30972	−1.30972	−0.654861	+	0.755750i	$0.727273\pi$
$620$	2.51334	2.51334
$621$	−2.68251	−2.68251
$622$	0	0
$623$	0	0
$624$	0	0
$625$	−1.20362	−1.20362
$626$	0	0
$627$	0	0
$628$	−0.284630	−0.284630
$629$	0	0
$630$	0	0
$631$	1.68251	1.68251	0.841254	−	0.540641i	$-0.181818\pi$
$631$	1.68251	1.68251	0.841254	+	0.540641i	$0.181818\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	0	0
$636$	−0.478891	−0.478891
$637$	0	0
$638$	0	0
$639$	1.52111	1.52111
$640$	0	0
$641$	−1.30972	−1.30972	−0.654861	−	0.755750i	$-0.727273\pi$
$641$	−1.30972	−1.30972	−0.654861	+	0.755750i	$0.727273\pi$
$642$	0	0
$643$	−0.284630	−0.284630	−0.142315	−	0.989821i	$-0.545455\pi$
$643$	−0.284630	−0.284630	−0.142315	+	0.989821i	$0.545455\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−1.30972	−1.30972	−0.654861	−	0.755750i	$-0.727273\pi$
$647$	−1.30972	−1.30972	−0.654861	+	0.755750i	$0.727273\pi$
$648$	0	0
$649$	0	0
$650$	0	0
$651$	0	0
$652$	1.68251	1.68251
$653$	−0.284630	−0.284630	−0.142315	−	0.989821i	$-0.545455\pi$
$653$	−0.284630	−0.284630	−0.142315	+	0.989821i	$0.545455\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	0	0
$658$	0	0
$659$	0	0	1.00000			$0$
$659$	0	0	−1.00000			$\pi$
$660$	0	0
$661$	0.830830	0.830830	0.415415	−	0.909632i	$-0.363636\pi$
$661$	0.830830	0.830830	0.415415	+	0.909632i	$0.363636\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	0	0
$666$	0	0
$667$	0	0
$668$	0	0
$669$	1.39788	1.39788
$670$	0	0
$671$	0	0
$672$	0	0
$673$	0	0	1.00000			$0$
$673$	0	0	−1.00000			$\pi$
$674$	0	0
$675$	1.00000	1.00000
$676$	1.00000	1.00000
$677$	0	0	1.00000			$0$
$677$	0	0	−1.00000			$\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−1.30972	−1.30972	−0.654861	−	0.755750i	$-0.727273\pi$
$683$	−1.30972	−1.30972	−0.654861	+	0.755750i	$0.727273\pi$
$684$	0	0
$685$	2.51334	2.51334
$686$	0	0
$687$	−2.20362	−2.20362
$688$	0	0
$689$	0	0
$690$	0	0
$691$	−0.284630	−0.284630	−0.142315	−	0.989821i	$-0.545455\pi$
$691$	−0.284630	−0.284630	−0.142315	+	0.989821i	$0.545455\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	0	0
$696$	0	0
$697$	0	0
$698$	0	0
$699$	0	0
$700$	0	0
$701$	0	0	1.00000			$0$
$701$	0	0	−1.00000			$\pi$
$702$	0	0
$703$	0	0
$704$	0	0
$705$	−1.83083	−1.83083
$706$	0	0
$707$	0	0
$708$	−0.478891	−0.478891
$709$	−1.91899	−1.91899	−0.959493	−	0.281733i	$-0.909091\pi$
$709$	−1.91899	−1.91899	−0.959493	+	0.281733i	$0.909091\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	3.68251	3.68251
$714$	0	0
$715$	0	0
$716$	−1.91899	−1.91899
$717$	0	0
$718$	0	0
$719$	−1.30972	−1.30972	−0.654861	−	0.755750i	$-0.727273\pi$
$719$	−1.30972	−1.30972	−0.654861	+	0.755750i	$0.727273\pi$
$720$	−2.39788	−2.39788
$721$	0	0
$722$	0	0
$723$	0	0
$724$	1.68251	1.68251
$725$	0	0
$726$	0	0
$727$	−1.30972	−1.30972	−0.654861	−	0.755750i	$-0.727273\pi$
$727$	−1.30972	−1.30972	−0.654861	+	0.755750i	$0.727273\pi$
$728$	0	0
$729$	−1.39788	−1.39788
$730$	0	0
$731$	0	0
$732$	0	0
$733$	0	0	1.00000			$0$
$733$	0	0	−1.00000			$\pi$
$734$	0	0
$735$	−2.20362	−2.20362
$736$	0	0
$737$	0	0
$738$	0	0
$739$	0	0	1.00000			$0$
$739$	0	0	−1.00000			$\pi$
$740$	−1.08816	−1.08816
$741$	0	0
$742$	0	0
$743$	0	0	1.00000			$0$
$743$	0	0	−1.00000			$\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	0	0
$748$	0	0
$749$	0	0
$750$	0	0
$751$	−1.30972	−1.30972	−0.654861	−	0.755750i	$-0.727273\pi$
$751$	−1.30972	−1.30972	−0.654861	+	0.755750i	$0.727273\pi$
$752$	0.830830	0.830830
$753$	2.83083	2.83083
$754$	0	0
$755$	0	0
$756$	0	0
$757$	0.830830	0.830830	0.415415	−	0.909632i	$-0.363636\pi$
$757$	0.830830	0.830830	0.415415	+	0.909632i	$0.363636\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	0	0	1.00000			$0$
$761$	0	0	−1.00000			$\pi$
$762$	0	0
$763$	0	0
$764$	1.68251	1.68251
$765$	0	0
$766$	0	0
$767$	0	0
$768$	1.68251	1.68251
$769$	0	0	1.00000			$0$
$769$	0	0	−1.00000			$\pi$
$770$	0	0
$771$	1.39788	1.39788
$772$	0	0
$773$	−1.30972	−1.30972	−0.654861	−	0.755750i	$-0.727273\pi$
$773$	−1.30972	−1.30972	−0.654861	+	0.755750i	$0.727273\pi$
$774$	0	0
$775$	−1.37279	−1.37279
$776$	0	0
$777$	0	0
$778$	0	0
$779$	0	0
$780$	0	0
$781$	0	0
$782$	0	0
$783$	0	0
$784$	1.00000	1.00000
$785$	0.372786	0.372786
$786$	0	0
$787$	0	0	1.00000			$0$
$787$	0	0	−1.00000			$\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	0	0
$792$	0	0
$793$	0	0
$794$	0	0
$795$	0.627214	0.627214
$796$	−0.284630	−0.284630
$797$	0.830830	0.830830	0.415415	−	0.909632i	$-0.363636\pi$
$797$	0.830830	0.830830	0.415415	+	0.909632i	$0.363636\pi$
$798$	0	0
$799$	0	0
$800$	0	0
$801$	−0.521109	−0.521109
$802$	0	0
$803$	0	0
$804$	−2.20362	−2.20362
$805$	0	0
$806$	0	0
$807$	−3.22871	−3.22871
$808$	0	0
$809$	0	0	1.00000			$0$
$809$	0	0	−1.00000			$\pi$
$810$	0	0
$811$	0	0	1.00000			$0$
$811$	0	0	−1.00000			$\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−2.20362	−2.20362
$816$	0	0
$817$	0	0
$818$	0	0
$819$	0	0
$820$	0	0
$821$	0	0	1.00000			$0$
$821$	0	0	−1.00000			$\pi$
$822$	0	0
$823$	−1.30972	−1.30972	−0.654861	−	0.755750i	$-0.727273\pi$
$823$	−1.30972	−1.30972	−0.654861	+	0.755750i	$0.727273\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	0	0	1.00000			$0$
$827$	0	0	−1.00000			$\pi$
$828$	−3.51334	−3.51334
$829$	1.68251	1.68251	0.841254	−	0.540641i	$-0.181818\pi$
$829$	1.68251	1.68251	0.841254	+	0.540641i	$0.181818\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	0	0
$834$	0	0
$835$	0	0
$836$	0	0
$837$	−2.68251	−2.68251
$838$	0	0
$839$	2.00000	2.00000	1.00000			$0$
$839$	2.00000	2.00000	1.00000			$0$
$840$	0	0
$841$	1.00000	1.00000
$842$	0	0
$843$	0	0
$844$	0	0
$845$	−1.30972	−1.30972
$846$	0	0
$847$	0	0
$848$	−0.284630	−0.284630
$849$	0	0
$850$	0	0
$851$	−1.59435	−1.59435
$852$	1.39788	1.39788
$853$	0	0	1.00000			$0$
$853$	0	0	−1.00000			$\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	0	0	1.00000			$0$
$857$	0	0	−1.00000			$\pi$
$858$	0	0
$859$	0.830830	0.830830	0.415415	−	0.909632i	$-0.363636\pi$
$859$	0.830830	0.830830	0.415415	+	0.909632i	$0.363636\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	1.68251	1.68251	0.841254	−	0.540641i	$-0.181818\pi$
$863$	1.68251	1.68251	0.841254	+	0.540641i	$0.181818\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	1.68251	1.68251
$868$	0	0
$869$	0	0
$870$	0	0
$871$	0	0
$872$	0	0
$873$	−3.51334	−3.51334
$874$	0	0
$875$	0	0
$876$	0	0
$877$	0	0	1.00000			$0$
$877$	0	0	−1.00000			$\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−0.284630	−0.284630	−0.142315	−	0.989821i	$-0.545455\pi$
$881$	−0.284630	−0.284630	−0.142315	+	0.989821i	$0.545455\pi$
$882$	0	0
$883$	−1.91899	−1.91899	−0.959493	−	0.281733i	$-0.909091\pi$
$883$	−1.91899	−1.91899	−0.959493	+	0.281733i	$0.909091\pi$
$884$	0	0
$885$	0.627214	0.627214
$886$	0	0
$887$	0	0	1.00000			$0$
$887$	0	0	−1.00000			$\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	0	0
$892$	0.830830	0.830830
$893$	0	0
$894$	0	0
$895$	2.51334	2.51334
$896$	0	0
$897$	0	0
$898$	0	0
$899$	0	0
$900$	1.30972	1.30972
$901$	0	0
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−2.20362	−2.20362
$906$	0	0
$907$	−1.91899	−1.91899	−0.959493	−	0.281733i	$-0.909091\pi$
$907$	−1.91899	−1.91899	−0.959493	+	0.281733i	$0.909091\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−1.30972	−1.30972	−0.654861	−	0.755750i	$-0.727273\pi$
$911$	−1.30972	−1.30972	−0.654861	+	0.755750i	$0.727273\pi$
$912$	0	0
$913$	0	0
$914$	0	0
$915$	0	0
$916$	−1.30972	−1.30972
$917$	0	0
$918$	0	0
$919$	0	0	1.00000			$0$
$919$	0	0	−1.00000			$\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	0	0
$924$	0	0
$925$	0.594351	0.594351
$926$	0	0
$927$	−2.39788	−2.39788
$928$	0	0
$929$	0.830830	0.830830	0.415415	−	0.909632i	$-0.363636\pi$
$929$	0.830830	0.830830	0.415415	+	0.909632i	$0.363636\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	2.83083	2.83083
$934$	0	0
$935$	0	0
$936$	0	0
$937$	0	0	1.00000			$0$
$937$	0	0	−1.00000			$\pi$
$938$	0	0
$939$	−0.478891	−0.478891
$940$	−1.08816	−1.08816
$941$	0	0	1.00000			$0$
$941$	0	0	−1.00000			$\pi$
$942$	0	0
$943$	0	0
$944$	−0.284630	−0.284630
$945$	0	0
$946$	0	0
$947$	0.830830	0.830830	0.415415	−	0.909632i	$-0.363636\pi$
$947$	0.830830	0.830830	0.415415	+	0.909632i	$0.363636\pi$
$948$	0	0
$949$	0	0
$950$	0	0
$951$	−0.478891	−0.478891
$952$	0	0
$953$	0	0	1.00000			$0$
$953$	0	0	−1.00000			$\pi$
$954$	0	0
$955$	−2.20362	−2.20362
$956$	0	0
$957$	0	0
$958$	0	0
$959$	0	0
$960$	−2.20362	−2.20362
$961$	2.68251	2.68251
$962$	0	0
$963$	0	0
$964$	0	0
$965$	0	0
$966$	0	0
$967$	0	0	1.00000			$0$
$967$	0	0	−1.00000			$\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	0.830830	0.830830	0.415415	−	0.909632i	$-0.363636\pi$
$971$	0.830830	0.830830	0.415415	+	0.909632i	$0.363636\pi$
$972$	−0.521109	−0.521109
$973$	0	0
$974$	0	0
$975$	0	0
$976$	0	0
$977$	1.68251	1.68251	0.841254	−	0.540641i	$-0.181818\pi$
$977$	1.68251	1.68251	0.841254	+	0.540641i	$0.181818\pi$
$978$	0	0
$979$	0	0
$980$	−1.30972	−1.30972
$981$	0	0
$982$	0	0
$983$	2.00000	2.00000	1.00000			$0$
$983$	2.00000	2.00000	1.00000			$0$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	0	0
$988$	0	0
$989$	0	0
$990$	0	0
$991$	0.830830	0.830830	0.415415	−	0.909632i	$-0.363636\pi$
$991$	0.830830	0.830830	0.415415	+	0.909632i	$0.363636\pi$
$992$	0	0
$993$	−2.20362	−2.20362
$994$	0	0
$995$	0.372786	0.372786
$996$	0	0
$997$	0	0	1.00000			$0$
$997$	0	0	−1.00000			$\pi$
$998$	0	0
$999$	1.16140	1.16140

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1331.1.b.a.1330.5	✓	5
11.2	odd	10		1331.1.d.a.161.5		20
11.3	even	5		1331.1.d.a.596.1		20
11.4	even	5		1331.1.d.a.699.1		20
11.5	even	5		1331.1.d.a.1207.5		20
11.6	odd	10		1331.1.d.a.1207.5		20
11.7	odd	10		1331.1.d.a.699.1		20
11.8	odd	10		1331.1.d.a.596.1		20
11.9	even	5		1331.1.d.a.161.5		20
11.10	odd	2	CM	1331.1.b.a.1330.5	✓	5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1331.1.b.a.1330.5	✓	5	1.1	even	1	trivial
1331.1.b.a.1330.5	✓	5	11.10	odd	2	CM
1331.1.d.a.161.5		20	11.2	odd	10
1331.1.d.a.161.5		20	11.9	even	5
1331.1.d.a.596.1		20	11.3	even	5
1331.1.d.a.596.1		20	11.8	odd	10
1331.1.d.a.699.1		20	11.4	even	5
1331.1.d.a.699.1		20	11.7	odd	10
1331.1.d.a.1207.5		20	11.5	even	5
1331.1.d.a.1207.5		20	11.6	odd	10

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 1331 = 11^{3} \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	1331.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+1.68251 q^{3} +1.00000 q^{4} -1.30972 q^{5} +1.83083 q^{9} +1.68251 q^{12} -2.20362 q^{15} +1.00000 q^{16} -1.30972 q^{20} -1.91899 q^{23} +0.715370 q^{25} +1.39788 q^{27} -1.91899 q^{31} +1.83083 q^{36} +0.830830 q^{37} -2.39788 q^{45} +0.830830 q^{47} +1.68251 q^{48} +1.00000 q^{49} -0.284630 q^{53} -0.284630 q^{59} -2.20362 q^{60} +1.00000 q^{64} -1.30972 q^{67} -3.22871 q^{69} +0.830830 q^{71} +1.20362 q^{75} -1.30972 q^{80} +0.521109 q^{81} -0.284630 q^{89} -1.91899 q^{92} -3.22871 q^{93} -1.91899 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q - q^{3} + 5 q^{4} - q^{5} + 4 q^{9} - q^{12} - 2 q^{15} + 5 q^{16} - q^{20} - q^{23} + 4 q^{25} - 2 q^{27} - q^{31} + 4 q^{36} - q^{37} - 3 q^{45} - q^{47} - q^{48} + 5 q^{49} - q^{53} - q^{59}+ \cdots - q^{97}+O(q^{100}) \) 5 * q - q^3 + 5 * q^4 - q^5 + 4 * q^9 - q^12 - 2 * q^15 + 5 * q^16 - q^20 - q^23 + 4 * q^25 - 2 * q^27 - q^31 + 4 * q^36 - q^37 - 3 * q^45 - q^47 - q^48 + 5 * q^49 - q^53 - q^59 - 2 * q^60 + 5 * q^64 - q^67 - 2 * q^69 - q^71 - 3 * q^75 - q^80 + 3 * q^81 - q^89 - q^92 - 2 * q^93 - q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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