LMFDB - Embedded newform 1584.2.o.e.703.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1584,2,Mod(703,1584)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1584, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 0, 0, 1]))

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("1584.703");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1584 = 2^{4} \cdot 3^{2} \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1584.o (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:	no
Analytic conductor:	$12.6483036802$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\sqrt{-3}, \sqrt{-11})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 2x^{2} - 3x + 9 $ x^4 - x^3 - 2x^2 - 3x + 9
Coefficient ring:	$\Z[a_1, \ldots, a_{23}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 176)
Sato-Tate group:	$\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label			703.1
Root			$-1.18614 + 1.26217i$ of defining polynomial
Character	$\chi$	$=$	1584.703
Dual form			1584.2.o.e.703.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.37228 q^{5} +O(q^{10})$	$q-1.37228 q^{5} -3.31662i q^{11} +6.13592i q^{23} -3.11684 q^{25} -9.30506i q^{31} -12.1168 q^{37} +6.63325i q^{47} -7.00000 q^{49} -6.00000 q^{53} +4.55134i q^{55} -14.6487i q^{59} -16.2333i q^{67} -10.8896i q^{71} -18.8614 q^{89} -0.116844 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 6 q^{5}+O(q^{10}) $ 4 * q + 6 * q^5	$ 4 q + 6 q^{5} + 22 q^{25} - 14 q^{37} - 28 q^{49} - 24 q^{53} - 18 q^{89} + 34 q^{97}+O(q^{100}) $ 4 * q + 6 * q^5 + 22 * q^25 - 14 * q^37 - 28 * q^49 - 24 * q^53 - 18 * q^89 + 34 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1584\mathbb{Z}\right)^\times$.

$n$	$145$	$353$	$991$	$1189$
$\chi(n)$	$-1$	$1$	$-1$	$1$

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
$2$	0	0
$3$	0	0
$3$	0	0
$4$	0	0
$5$	−1.37228	−0.613703	−0.306851	−	0.951757i	$-0.599275\pi$
$5$	−1.37228	−0.613703	−0.306851	+	0.951757i	$0.599275\pi$
$6$	0	0
$7$	0	0		−	1.00000i	$-0.5\pi$
$7$	0	0			1.00000i	$0.5\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	− 3.31662i	− 1.00000i
$11$	− 3.31662i	− 1.00000i
$12$	0	0
$13$	0	0	1.00000			$0$
$13$	0	0	−1.00000			$\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	0	0	1.00000			$0$
$17$	0	0	−1.00000			$\pi$
$18$	0	0
$19$	0	0		−	1.00000i	$-0.5\pi$
$19$	0	0			1.00000i	$0.5\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	6.13592i	1.27943i	0.768613	+	0.639713i	$0.220947\pi$
$23$	6.13592i	1.27943i	−0.768613	+	0.639713i	$0.779053\pi$
$24$	0	0
$25$	−3.11684	−0.623369
$26$	0	0
$27$	0	0
$28$	0	0
$29$	0	0	1.00000			$0$
$29$	0	0	−1.00000			$\pi$
$30$	0	0
$31$	− 9.30506i	− 1.67124i	−0.549309	−	0.835619i	$-0.685109\pi$
$31$	− 9.30506i	− 1.67124i	0.549309	−	0.835619i	$-0.314891\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−12.1168	−1.99200	−0.995998	−	0.0893706i	$-0.971514\pi$
$37$	−12.1168	−1.99200	−0.995998	+	0.0893706i	$0.971514\pi$
$38$	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		−	1.00000i	$-0.5\pi$
$43$	0	0			1.00000i	$0.5\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	6.63325i	0.967559i	0.875190	+	0.483779i	$0.160736\pi$
$47$	6.63325i	0.967559i	−0.875190	+	0.483779i	$0.839264\pi$
$48$	0	0
$49$	−7.00000	−1.00000
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−6.00000	−0.824163	−0.412082	−	0.911147i	$-0.635198\pi$
$53$	−6.00000	−0.824163	−0.412082	+	0.911147i	$0.635198\pi$
$54$	0	0
$55$	4.55134i	0.613703i
$56$	0	0
$57$	0	0
$58$	0	0
$59$	− 14.6487i	− 1.90710i	−0.301239	−	0.953549i	$-0.597400\pi$
$59$	− 14.6487i	− 1.90710i	0.301239	−	0.953549i	$-0.402600\pi$
$60$	0	0
$61$	0	0	1.00000			$0$
$61$	0	0	−1.00000			$\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	0	0
$66$	0	0
$67$	− 16.2333i	− 1.98321i	−0.129307	−	0.991605i	$-0.541275\pi$
$67$	− 16.2333i	− 1.98321i	0.129307	−	0.991605i	$-0.458725\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	− 10.8896i	− 1.29236i	−0.763184	−	0.646181i	$-0.776365\pi$
$71$	− 10.8896i	− 1.29236i	0.763184	−	0.646181i	$-0.223635\pi$
$72$	0	0
$73$	0	0	1.00000			$0$
$73$	0	0	−1.00000			$\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	0	0
$78$	0	0
$79$	0	0		−	1.00000i	$-0.5\pi$
$79$	0	0			1.00000i	$0.5\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	0	0		−	1.00000i	$-0.5\pi$
$83$	0	0			1.00000i	$0.5\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−18.8614	−1.99931	−0.999653	−	0.0263586i	$-0.991609\pi$
$89$	−18.8614	−1.99931	−0.999653	+	0.0263586i	$0.991609\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	0	0
$94$	0	0
$95$	0	0
$96$	0	0
$97$	−0.116844	−0.0118637	−0.00593185	−	0.999982i	$-0.501888\pi$
$97$	−0.116844	−0.0118637	−0.00593185	+	0.999982i	$0.501888\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	0	0	1.00000			$0$
$101$	0	0	−1.00000			$\pi$
$102$	0	0
$103$	19.8997i	1.96078i	0.197066	+	0.980390i	$0.436859\pi$
$103$	19.8997i	1.96078i	−0.197066	+	0.980390i	$0.563141\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	0	0		−	1.00000i	$-0.5\pi$
$107$	0	0			1.00000i	$0.5\pi$
$108$	0	0
$109$	0	0	1.00000			$0$
$109$	0	0	−1.00000			$\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−13.3723	−1.25796	−0.628979	−	0.777422i	$-0.716527\pi$
$113$	−13.3723	−1.25796	−0.628979	+	0.777422i	$0.716527\pi$
$114$	0	0
$115$	− 8.42020i	− 0.785188i
$116$	0	0
$117$	0	0
$118$	0	0
$119$	0	0
$120$	0	0
$121$	−11.0000	−1.00000
$122$	0	0
$123$	0	0
$124$	0	0
$125$	11.1386	0.996266
$126$	0	0
$127$	0	0		−	1.00000i	$-0.5\pi$
$127$	0	0			1.00000i	$0.5\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	0	0		−	1.00000i	$-0.5\pi$
$131$	0	0			1.00000i	$0.5\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	0	0
$136$	0	0
$137$	21.6060	1.84592	0.922961	−	0.384893i	$-0.125762\pi$
$137$	21.6060	1.84592	0.922961	+	0.384893i	$0.125762\pi$
$138$	0	0
$139$	0	0		−	1.00000i	$-0.5\pi$
$139$	0	0			1.00000i	$0.5\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	0	0
$144$	0	0
$145$	0	0
$146$	0	0
$147$	0	0
$148$	0	0
$149$	0	0	1.00000			$0$
$149$	0	0	−1.00000			$\pi$
$150$	0	0
$151$	0	0		−	1.00000i	$-0.5\pi$
$151$	0	0			1.00000i	$0.5\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	12.7692i	1.02564i
$156$	0	0
$157$	−20.1168	−1.60550	−0.802749	−	0.596316i	$-0.796630\pi$
$157$	−20.1168	−1.60550	−0.802749	+	0.596316i	$0.796630\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	0	0
$162$	0	0
$163$	− 19.8997i	− 1.55867i	−0.626608	−	0.779334i	$-0.715557\pi$
$163$	− 19.8997i	− 1.55867i	0.626608	−	0.779334i	$-0.284443\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	0	0		−	1.00000i	$-0.5\pi$
$167$	0	0			1.00000i	$0.5\pi$
$168$	0	0
$169$	13.0000	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	0
$178$	0	0
$179$	9.89497i	0.739585i	0.929114	+	0.369792i	$0.120571\pi$
$179$	9.89497i	0.739585i	−0.929114	+	0.369792i	$0.879429\pi$
$180$	0	0
$181$	3.88316	0.288633	0.144316	−	0.989532i	$-0.453902\pi$
$181$	3.88316	0.288633	0.144316	+	0.989532i	$0.453902\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	16.6277	1.22249
$186$	0	0
$187$	0	0
$188$	0	0
$189$	0	0
$190$	0	0
$191$	− 1.38219i	− 0.100012i	−0.998749	−	0.0500060i	$-0.984076\pi$
$191$	− 1.38219i	− 0.100012i	0.998749	−	0.0500060i	$-0.0159241\pi$
$192$	0	0
$193$	0	0	1.00000			$0$
$193$	0	0	−1.00000			$\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	0	0	1.00000			$0$
$197$	0	0	−1.00000			$\pi$
$198$	0	0
$199$	19.8997i	1.41066i	0.708881	+	0.705328i	$0.249200\pi$
$199$	19.8997i	1.41066i	−0.708881	+	0.705328i	$0.750800\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	0	0
$204$	0	0
$205$	0	0
$206$	0	0
$207$	0	0
$208$	0	0
$209$	0	0
$210$	0	0
$211$	0	0		−	1.00000i	$-0.5\pi$
$211$	0	0			1.00000i	$0.5\pi$
$212$	0	0
$213$	0	0
$214$	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
$223$	14.0588i	0.941446i	0.882281	+	0.470723i	$0.156007\pi$
$223$	14.0588i	0.941446i	−0.882281	+	0.470723i	$0.843993\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	0	0		−	1.00000i	$-0.5\pi$
$227$	0	0			1.00000i	$0.5\pi$
$228$	0	0
$229$	28.3505	1.87346	0.936728	−	0.350058i	$-0.113838\pi$
$229$	28.3505	1.87346	0.936728	+	0.350058i	$0.113838\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	0	0	1.00000			$0$
$233$	0	0	−1.00000			$\pi$
$234$	0	0
$235$	− 9.10268i	− 0.593794i
$236$	0	0
$237$	0	0
$238$	0	0
$239$	0	0		−	1.00000i	$-0.5\pi$
$239$	0	0			1.00000i	$0.5\pi$
$240$	0	0
$241$	0	0	1.00000			$0$
$241$	0	0	−1.00000			$\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	9.60597	0.613703
$246$	0	0
$247$	0	0
$248$	0	0
$249$	0	0
$250$	0	0
$251$	− 31.6742i	− 1.99926i	−0.0271858	−	0.999630i	$-0.508655\pi$
$251$	− 31.6742i	− 1.99926i	0.0271858	−	0.999630i	$-0.491345\pi$
$252$	0	0
$253$	20.3505	1.27943
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−18.0000	−1.12281	−0.561405	−	0.827541i	$-0.689739\pi$
$257$	−18.0000	−1.12281	−0.561405	+	0.827541i	$0.689739\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	0	0
$262$	0	0
$263$	0	0		−	1.00000i	$-0.5\pi$
$263$	0	0			1.00000i	$0.5\pi$
$264$	0	0
$265$	8.23369	0.505791
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−30.0000	−1.82913	−0.914566	−	0.404436i	$-0.867468\pi$
$269$	−30.0000	−1.82913	−0.914566	+	0.404436i	$0.867468\pi$
$270$	0	0
$271$	0	0		−	1.00000i	$-0.5\pi$
$271$	0	0			1.00000i	$0.5\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	10.3374i	0.623369i
$276$	0	0
$277$	0	0	1.00000			$0$
$277$	0	0	−1.00000			$\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	0	0	1.00000			$0$
$281$	0	0	−1.00000			$\pi$
$282$	0	0
$283$	0	0		−	1.00000i	$-0.5\pi$
$283$	0	0			1.00000i	$0.5\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	0	0
$288$	0	0
$289$	17.0000	1.00000
$290$	0	0
$291$	0	0
$292$	0	0
$293$	0	0	1.00000			$0$
$293$	0	0	−1.00000			$\pi$
$294$	0	0
$295$	20.1021i	1.17039i
$296$	0	0
$297$	0	0
$298$	0	0
$299$	0	0
$300$	0	0
$301$	0	0
$302$	0	0
$303$	0	0
$304$	0	0
$305$	0	0
$306$	0	0
$307$	0	0		−	1.00000i	$-0.5\pi$
$307$	0	0			1.00000i	$0.5\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	33.1662i	1.88069i	0.340229	+	0.940343i	$0.389495\pi$
$311$	33.1662i	1.88069i	−0.340229	+	0.940343i	$0.610505\pi$
$312$	0	0
$313$	16.3505	0.924187	0.462093	−	0.886831i	$-0.347098\pi$
$313$	16.3505	0.924187	0.462093	+	0.886831i	$0.347098\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	33.6060	1.88750	0.943750	−	0.330661i	$-0.107272\pi$
$317$	33.6060	1.88750	0.943750	+	0.330661i	$0.107272\pi$
$318$	0	0
$319$	0	0
$320$	0	0
$321$	0	0
$322$	0	0
$323$	0	0
$324$	0	0
$325$	0	0
$326$	0	0
$327$	0	0
$328$	0	0
$329$	0	0
$330$	0	0
$331$	25.3360i	1.39259i	0.717756	+	0.696295i	$0.245169\pi$
$331$	25.3360i	1.39259i	−0.717756	+	0.696295i	$0.754831\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	22.2766i	1.21710i
$336$	0	0
$337$	0	0	1.00000			$0$
$337$	0	0	−1.00000			$\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	−30.8614	−1.67124
$342$	0	0
$343$	0	0
$344$	0	0
$345$	0	0
$346$	0	0
$347$	0	0		−	1.00000i	$-0.5\pi$
$347$	0	0			1.00000i	$0.5\pi$
$348$	0	0
$349$	0	0	1.00000			$0$
$349$	0	0	−1.00000			$\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	27.0951	1.44213	0.721063	−	0.692869i	$-0.243654\pi$
$353$	27.0951	1.44213	0.721063	+	0.692869i	$0.243654\pi$
$354$	0	0
$355$	14.9436i	0.793126i
$356$	0	0
$357$	0	0
$358$	0	0
$359$	0	0		−	1.00000i	$-0.5\pi$
$359$	0	0			1.00000i	$0.5\pi$
$360$	0	0
$361$	−19.0000	−1.00000
$362$	0	0
$363$	0	0
$364$	0	0
$365$	0	0
$366$	0	0
$367$	− 37.0179i	− 1.93232i	−0.257948	−	0.966159i	$-0.583046\pi$
$367$	− 37.0179i	− 1.93232i	0.257948	−	0.966159i	$-0.416954\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	0	0
$372$	0	0
$373$	0	0	1.00000			$0$
$373$	0	0	−1.00000			$\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	0	0
$378$	0	0
$379$	− 6.72582i	− 0.345482i	−0.984967	−	0.172741i	$-0.944738\pi$
$379$	− 6.72582i	− 0.345482i	0.984967	−	0.172741i	$-0.0552624\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	− 35.4333i	− 1.81056i	−0.424818	−	0.905279i	$-0.639662\pi$
$383$	− 35.4333i	− 1.81056i	0.424818	−	0.905279i	$-0.360338\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	0	0
$388$	0	0
$389$	39.0951	1.98220	0.991100	−	0.133120i	$-0.0424994\pi$
$389$	39.0951	1.98220	0.991100	+	0.133120i	$0.0424994\pi$
$390$	0	0
$391$	0	0
$392$	0	0
$393$	0	0
$394$	0	0
$395$	0	0
$396$	0	0
$397$	−2.00000	−0.100377	−0.0501886	−	0.998740i	$-0.515982\pi$
$397$	−2.00000	−0.100377	−0.0501886	+	0.998740i	$0.515982\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	30.0000	1.49813	0.749064	−	0.662497i	$-0.230503\pi$
$401$	30.0000	1.49813	0.749064	+	0.662497i	$0.230503\pi$
$402$	0	0
$403$	0	0
$404$	0	0
$405$	0	0
$406$	0	0
$407$	40.1870i	1.99200i
$408$	0	0
$409$	0	0	1.00000			$0$
$409$	0	0	−1.00000			$\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	0	0
$414$	0	0
$415$	0	0
$416$	0	0
$417$	0	0
$418$	0	0
$419$	− 33.1662i	− 1.62028i	−0.586238	−	0.810139i	$-0.699392\pi$
$419$	− 33.1662i	− 1.62028i	0.586238	−	0.810139i	$-0.300608\pi$
$420$	0	0
$421$	−10.0000	−0.487370	−0.243685	−	0.969854i	$-0.578356\pi$
$421$	−10.0000	−0.487370	−0.243685	+	0.969854i	$0.578356\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	0	0
$426$	0	0
$427$	0	0
$428$	0	0
$429$	0	0
$430$	0	0
$431$	0	0		−	1.00000i	$-0.5\pi$
$431$	0	0			1.00000i	$0.5\pi$
$432$	0	0
$433$	40.3505	1.93912	0.969561	−	0.244848i	$-0.0787382\pi$
$433$	40.3505	1.93912	0.969561	+	0.244848i	$0.0787382\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	0	0
$438$	0	0
$439$	0	0		−	1.00000i	$-0.5\pi$
$439$	0	0			1.00000i	$0.5\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	36.4280i	1.73075i	0.501129	+	0.865373i	$0.332918\pi$
$443$	36.4280i	1.73075i	−0.501129	+	0.865373i	$0.667082\pi$
$444$	0	0
$445$	25.8832	1.22698
$446$	0	0
$447$	0	0
$448$	0	0
$449$	5.13859	0.242505	0.121253	−	0.992622i	$-0.461309\pi$
$449$	5.13859	0.242505	0.121253	+	0.992622i	$0.461309\pi$
$450$	0	0
$451$	0	0
$452$	0	0
$453$	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
$460$	0	0
$461$	0	0	1.00000			$0$
$461$	0	0	−1.00000			$\pi$
$462$	0	0
$463$	41.7716i	1.94129i	0.240515	+	0.970645i	$0.422684\pi$
$463$	41.7716i	1.94129i	−0.240515	+	0.970645i	$0.577316\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	− 24.1561i	− 1.11781i	−0.829231	−	0.558906i	$-0.811221\pi$
$467$	− 24.1561i	− 1.11781i	0.829231	−	0.558906i	$-0.188779\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	0	0
$472$	0	0
$473$	0	0
$474$	0	0
$475$	0	0
$476$	0	0
$477$	0	0
$478$	0	0
$479$	0	0		−	1.00000i	$-0.5\pi$
$479$	0	0			1.00000i	$0.5\pi$
$480$	0	0
$481$	0	0
$482$	0	0
$483$	0	0
$484$	0	0
$485$	0.160343	0.00728079
$486$	0	0
$487$	32.2642i	1.46203i	0.682362	+	0.731014i	$0.260953\pi$
$487$	32.2642i	1.46203i	−0.682362	+	0.731014i	$0.739047\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	0	0		−	1.00000i	$-0.5\pi$
$491$	0	0			1.00000i	$0.5\pi$
$492$	0	0
$493$	0	0
$494$	0	0
$495$	0	0
$496$	0	0
$497$	0	0
$498$	0	0
$499$	− 19.8997i	− 0.890835i	−0.895323	−	0.445418i	$-0.853055\pi$
$499$	− 19.8997i	− 0.890835i	0.895323	−	0.445418i	$-0.146945\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	0	0		−	1.00000i	$-0.5\pi$
$503$	0	0			1.00000i	$0.5\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−25.3723	−1.12461	−0.562303	−	0.826931i	$-0.690085\pi$
$509$	−25.3723	−1.12461	−0.562303	+	0.826931i	$0.690085\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	0	0
$514$	0	0
$515$	− 27.3081i	− 1.20334i
$516$	0	0
$517$	22.0000	0.967559
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−29.8397	−1.30730	−0.653650	−	0.756797i	$-0.726763\pi$
$521$	−29.8397	−1.30730	−0.653650	+	0.756797i	$0.726763\pi$
$522$	0	0
$523$	0	0		−	1.00000i	$-0.5\pi$
$523$	0	0			1.00000i	$0.5\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	0	0
$528$	0	0
$529$	−14.6495	−0.636933
$530$	0	0
$531$	0	0
$532$	0	0
$533$	0	0
$534$	0	0
$535$	0	0
$536$	0	0
$537$	0	0
$538$	0	0
$539$	23.2164i	1.00000i
$540$	0	0
$541$	0	0	1.00000			$0$
$541$	0	0	−1.00000			$\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	0	0
$546$	0	0
$547$	0	0		−	1.00000i	$-0.5\pi$
$547$	0	0			1.00000i	$0.5\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	0	0
$552$	0	0
$553$	0	0
$554$	0	0
$555$	0	0
$556$	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		−	1.00000i	$-0.5\pi$
$563$	0	0			1.00000i	$0.5\pi$
$564$	0	0
$565$	18.3505	0.772013
$566$	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		−	1.00000i	$-0.5\pi$
$571$	0	0			1.00000i	$0.5\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	− 19.1247i	− 0.797555i
$576$	0	0
$577$	−32.1168	−1.33704	−0.668521	−	0.743693i	$-0.733072\pi$
$577$	−32.1168	−1.33704	−0.668521	+	0.743693i	$0.733072\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	0	0
$582$	0	0
$583$	19.8997i	0.824163i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	− 6.63325i	− 0.273784i	−0.990586	−	0.136892i	$-0.956289\pi$
$587$	− 6.63325i	− 0.273784i	0.990586	−	0.136892i	$-0.0437113\pi$
$588$	0	0
$589$	0	0
$590$	0	0
$591$	0	0
$592$	0	0
$593$	0	0	1.00000			$0$
$593$	0	0	−1.00000			$\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	0	0
$598$	0	0
$599$	33.1662i	1.35514i	0.735460	+	0.677568i	$0.236966\pi$
$599$	33.1662i	1.35514i	−0.735460	+	0.677568i	$0.763034\pi$
$600$	0	0
$601$	0	0	1.00000			$0$
$601$	0	0	−1.00000			$\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	15.0951	0.613703
$606$	0	0
$607$	0	0		−	1.00000i	$-0.5\pi$
$607$	0	0			1.00000i	$0.5\pi$
$608$	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
$617$	−42.0000	−1.69086	−0.845428	−	0.534089i	$-0.820655\pi$
$617$	−42.0000	−1.69086	−0.845428	+	0.534089i	$0.820655\pi$
$618$	0	0
$619$	− 25.7407i	− 1.03461i	−0.855802	−	0.517303i	$-0.826936\pi$
$619$	− 25.7407i	− 1.03461i	0.855802	−	0.517303i	$-0.173064\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	0	0
$624$	0	0
$625$	0.298936	0.0119574
$626$	0	0
$627$	0	0
$628$	0	0
$629$	0	0
$630$	0	0
$631$	− 18.8125i	− 0.748914i	−0.927244	−	0.374457i	$-0.877829\pi$
$631$	− 18.8125i	− 0.748914i	0.927244	−	0.374457i	$-0.122171\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	0	0
$636$	0	0
$637$	0	0
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−2.39403	−0.0945585	−0.0472793	−	0.998882i	$-0.515055\pi$
$641$	−2.39403	−0.0945585	−0.0472793	+	0.998882i	$0.515055\pi$
$642$	0	0
$643$	− 20.5822i	− 0.811684i	−0.913943	−	0.405842i	$-0.866978\pi$
$643$	− 20.5822i	− 0.811684i	0.913943	−	0.405842i	$-0.133022\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	− 44.9407i	− 1.76680i	−0.468617	−	0.883402i	$-0.655247\pi$
$647$	− 44.9407i	− 1.76680i	0.468617	−	0.883402i	$-0.344753\pi$
$648$	0	0
$649$	−48.5842	−1.90710
$650$	0	0
$651$	0	0
$652$	0	0
$653$	22.6277	0.885491	0.442746	−	0.896647i	$-0.354005\pi$
$653$	22.6277	0.885491	0.442746	+	0.896647i	$0.354005\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	0	0
$658$	0	0
$659$	0	0		−	1.00000i	$-0.5\pi$
$659$	0	0			1.00000i	$0.5\pi$
$660$	0	0
$661$	−36.5842	−1.42296	−0.711481	−	0.702706i	$-0.751975\pi$
$661$	−36.5842	−1.42296	−0.711481	+	0.702706i	$0.751975\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	0	0
$666$	0	0
$667$	0	0
$668$	0	0
$669$	0	0
$670$	0	0
$671$	0	0
$672$	0	0
$673$	0	0	1.00000			$0$
$673$	0	0	−1.00000			$\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	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
$683$	46.4327i	1.77670i	0.459167	+	0.888350i	$0.348148\pi$
$683$	46.4327i	1.77670i	−0.459167	+	0.888350i	$0.651852\pi$
$684$	0	0
$685$	−29.6495	−1.13285
$686$	0	0
$687$	0	0
$688$	0	0
$689$	0	0
$690$	0	0
$691$	− 39.5971i	− 1.50635i	−0.657823	−	0.753173i	$-0.728522\pi$
$691$	− 39.5971i	− 1.50635i	0.657823	−	0.753173i	$-0.271478\pi$
$692$	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
$705$	0	0
$706$	0	0
$707$	0	0
$708$	0	0
$709$	−52.5842	−1.97484	−0.987421	−	0.158114i	$-0.949459\pi$
$709$	−52.5842	−1.97484	−0.987421	+	0.158114i	$0.949459\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	57.0951	2.13823
$714$	0	0
$715$	0	0
$716$	0	0
$717$	0	0
$718$	0	0
$719$	− 52.4589i	− 1.95639i	−0.207700	−	0.978193i	$-0.566598\pi$
$719$	− 52.4589i	− 1.95639i	0.207700	−	0.978193i	$-0.433402\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	0	0
$724$	0	0
$725$	0	0
$726$	0	0
$727$	− 50.8743i	− 1.88682i	−0.331625	−	0.943411i	$-0.607597\pi$
$727$	− 50.8743i	− 1.88682i	0.331625	−	0.943411i	$-0.392403\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	0	0
$732$	0	0
$733$	0	0	1.00000			$0$
$733$	0	0	−1.00000			$\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−53.8397	−1.98321
$738$	0	0
$739$	0	0		−	1.00000i	$-0.5\pi$
$739$	0	0			1.00000i	$0.5\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	0	0		−	1.00000i	$-0.5\pi$
$743$	0	0			1.00000i	$0.5\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	0	0
$748$	0	0
$749$	0	0
$750$	0	0
$751$	− 4.95610i	− 0.180851i	−0.995903	−	0.0904254i	$-0.971177\pi$
$751$	− 4.95610i	− 0.180851i	0.995903	−	0.0904254i	$-0.0288227\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	0	0
$756$	0	0
$757$	38.0000	1.38113	0.690567	−	0.723269i	$-0.257361\pi$
$757$	38.0000	1.38113	0.690567	+	0.723269i	$0.257361\pi$
$758$	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
$764$	0	0
$765$	0	0
$766$	0	0
$767$	0	0
$768$	0	0
$769$	0	0	1.00000			$0$
$769$	0	0	−1.00000			$\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	−54.0000	−1.94225	−0.971123	−	0.238581i	$-0.923318\pi$
$773$	−54.0000	−1.94225	−0.971123	+	0.238581i	$0.923318\pi$
$774$	0	0
$775$	29.0024i	1.04180i
$776$	0	0
$777$	0	0
$778$	0	0
$779$	0	0
$780$	0	0
$781$	−36.1168	−1.29236
$782$	0	0
$783$	0	0
$784$	0	0
$785$	27.6060	0.985299
$786$	0	0
$787$	0	0		−	1.00000i	$-0.5\pi$
$787$	0	0			1.00000i	$0.5\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	0	0
$792$	0	0
$793$	0	0
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−47.3288	−1.67647	−0.838236	−	0.545308i	$-0.816413\pi$
$797$	−47.3288	−1.67647	−0.838236	+	0.545308i	$0.816413\pi$
$798$	0	0
$799$	0	0
$800$	0	0
$801$	0	0
$802$	0	0
$803$	0	0
$804$	0	0
$805$	0	0
$806$	0	0
$807$	0	0
$808$	0	0
$809$	0	0	1.00000			$0$
$809$	0	0	−1.00000			$\pi$
$810$	0	0
$811$	0	0		−	1.00000i	$-0.5\pi$
$811$	0	0			1.00000i	$0.5\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	27.3081i	0.956559i
$816$	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
$823$	− 27.5104i	− 0.958953i	−0.877555	−	0.479477i	$-0.840826\pi$
$823$	− 27.5104i	− 0.958953i	0.877555	−	0.479477i	$-0.159174\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	0	0		−	1.00000i	$-0.5\pi$
$827$	0	0			1.00000i	$0.5\pi$
$828$	0	0
$829$	−28.5842	−0.992771	−0.496385	−	0.868102i	$-0.665340\pi$
$829$	−28.5842	−0.992771	−0.496385	+	0.868102i	$0.665340\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	0	0
$834$	0	0
$835$	0	0
$836$	0	0
$837$	0	0
$838$	0	0
$839$	57.2126i	1.97520i	0.156999	+	0.987599i	$0.449818\pi$
$839$	57.2126i	1.97520i	−0.156999	+	0.987599i	$0.550182\pi$
$840$	0	0
$841$	29.0000	1.00000
$842$	0	0
$843$	0	0
$844$	0	0
$845$	−17.8397	−0.613703
$846$	0	0
$847$	0	0
$848$	0	0
$849$	0	0
$850$	0	0
$851$	− 74.3479i	− 2.54861i
$852$	0	0
$853$	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
$859$	1.97210i	0.0672872i	0.999434	+	0.0336436i	$0.0107111\pi$
$859$	1.97210i	0.0672872i	−0.999434	+	0.0336436i	$0.989289\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	− 46.4327i	− 1.58059i	−0.612727	−	0.790295i	$-0.709928\pi$
$863$	− 46.4327i	− 1.58059i	0.612727	−	0.790295i	$-0.290072\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	0	0
$868$	0	0
$869$	0	0
$870$	0	0
$871$	0	0
$872$	0	0
$873$	0	0
$874$	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
$881$	−42.8614	−1.44404	−0.722019	−	0.691873i	$-0.756786\pi$
$881$	−42.8614	−1.44404	−0.722019	+	0.691873i	$0.756786\pi$
$882$	0	0
$883$	− 19.8997i	− 0.669680i	−0.942275	−	0.334840i	$-0.891318\pi$
$883$	− 19.8997i	− 0.669680i	0.942275	−	0.334840i	$-0.108682\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	0	0		−	1.00000i	$-0.5\pi$
$887$	0	0			1.00000i	$0.5\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	0	0
$892$	0	0
$893$	0	0
$894$	0	0
$895$	− 13.5787i	− 0.453885i
$896$	0	0
$897$	0	0
$898$	0	0
$899$	0	0
$900$	0	0
$901$	0	0
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−5.32878	−0.177135
$906$	0	0
$907$	59.6992i	1.98228i	0.132818	+	0.991140i	$0.457597\pi$
$907$	59.6992i	1.98228i	−0.132818	+	0.991140i	$0.542403\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	6.63325i	0.219769i	0.993944	+	0.109885i	$0.0350482\pi$
$911$	6.63325i	0.219769i	−0.993944	+	0.109885i	$0.964952\pi$
$912$	0	0
$913$	0	0
$914$	0	0
$915$	0	0
$916$	0	0
$917$	0	0
$918$	0	0
$919$	0	0		−	1.00000i	$-0.5\pi$
$919$	0	0			1.00000i	$0.5\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	0	0
$924$	0	0
$925$	37.7663	1.24175
$926$	0	0
$927$	0	0
$928$	0	0
$929$	30.0000	0.984268	0.492134	−	0.870519i	$-0.336217\pi$
$929$	30.0000	0.984268	0.492134	+	0.870519i	$0.336217\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	0	0
$934$	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	0
$940$	0	0
$941$	0	0	1.00000			$0$
$941$	0	0	−1.00000			$\pi$
$942$	0	0
$943$	0	0
$944$	0	0
$945$	0	0
$946$	0	0
$947$	60.9716i	1.98131i	0.136385	+	0.990656i	$0.456452\pi$
$947$	60.9716i	1.98131i	−0.136385	+	0.990656i	$0.543548\pi$
$948$	0	0
$949$	0	0
$950$	0	0
$951$	0	0
$952$	0	0
$953$	0	0	1.00000			$0$
$953$	0	0	−1.00000			$\pi$
$954$	0	0
$955$	1.89676i	0.0613777i
$956$	0	0
$957$	0	0
$958$	0	0
$959$	0	0
$960$	0	0
$961$	−55.5842	−1.79304
$962$	0	0
$963$	0	0
$964$	0	0
$965$	0	0
$966$	0	0
$967$	0	0		−	1.00000i	$-0.5\pi$
$967$	0	0			1.00000i	$0.5\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	17.4131i	0.558812i	0.960173	+	0.279406i	$0.0901376\pi$
$971$	17.4131i	0.558812i	−0.960173	+	0.279406i	$0.909862\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−35.3288	−1.13027	−0.565134	−	0.824999i	$-0.691176\pi$
$977$	−35.3288	−1.13027	−0.565134	+	0.824999i	$0.691176\pi$
$978$	0	0
$979$	62.5562i	1.99931i
$980$	0	0
$981$	0	0
$982$	0	0
$983$	− 25.9259i	− 0.826906i	−0.910525	−	0.413453i	$-0.864323\pi$
$983$	− 25.9259i	− 0.826906i	0.910525	−	0.413453i	$-0.135677\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	0	0
$988$	0	0
$989$	0	0
$990$	0	0
$991$	− 59.6992i	− 1.89641i	−0.317660	−	0.948205i	$-0.602897\pi$
$991$	− 59.6992i	− 1.89641i	0.317660	−	0.948205i	$-0.397103\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	− 27.3081i	− 0.865723i
$996$	0	0
$997$	0	0	1.00000			$0$
$997$	0	0	−1.00000			$\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1584.2.o.e.703.1		4
3.2	odd	2		176.2.e.b.175.3	yes	4
4.3	odd	2	inner	1584.2.o.e.703.2		4
11.10	odd	2	CM	1584.2.o.e.703.1		4
12.11	even	2		176.2.e.b.175.2	✓	4
24.5	odd	2		704.2.e.c.703.2		4
24.11	even	2		704.2.e.c.703.3		4
33.32	even	2		176.2.e.b.175.3	yes	4
44.43	even	2	inner	1584.2.o.e.703.2		4
48.5	odd	4		2816.2.g.c.1407.6		8
48.11	even	4		2816.2.g.c.1407.4		8
48.29	odd	4		2816.2.g.c.1407.3		8
48.35	even	4		2816.2.g.c.1407.5		8
132.131	odd	2		176.2.e.b.175.2	✓	4
264.131	odd	2		704.2.e.c.703.3		4
264.197	even	2		704.2.e.c.703.2		4
528.131	odd	4		2816.2.g.c.1407.5		8
528.197	even	4		2816.2.g.c.1407.6		8
528.395	odd	4		2816.2.g.c.1407.4		8
528.461	even	4		2816.2.g.c.1407.3		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
176.2.e.b.175.2	✓	4	12.11	even	2
176.2.e.b.175.2	✓	4	132.131	odd	2
176.2.e.b.175.3	yes	4	3.2	odd	2
176.2.e.b.175.3	yes	4	33.32	even	2
704.2.e.c.703.2		4	24.5	odd	2
704.2.e.c.703.2		4	264.197	even	2
704.2.e.c.703.3		4	24.11	even	2
704.2.e.c.703.3		4	264.131	odd	2
1584.2.o.e.703.1		4	1.1	even	1	trivial
1584.2.o.e.703.1		4	11.10	odd	2	CM
1584.2.o.e.703.2		4	4.3	odd	2	inner
1584.2.o.e.703.2		4	44.43	even	2	inner
2816.2.g.c.1407.3		8	48.29	odd	4
2816.2.g.c.1407.3		8	528.461	even	4
2816.2.g.c.1407.4		8	48.11	even	4
2816.2.g.c.1407.4		8	528.395	odd	4
2816.2.g.c.1407.5		8	48.35	even	4
2816.2.g.c.1407.5		8	528.131	odd	4
2816.2.g.c.1407.6		8	48.5	odd	4
2816.2.g.c.1407.6		8	528.197	even	4

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:	\( N \)	\(=\)	\( 1584 = 2^{4} \cdot 3^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1584.o (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-1.37228 q^{5} +O(q^{10})\)	\(q-1.37228 q^{5} -3.31662i q^{11} +6.13592i q^{23} -3.11684 q^{25} -9.30506i q^{31} -12.1168 q^{37} +6.63325i q^{47} -7.00000 q^{49} -6.00000 q^{53} +4.55134i q^{55} -14.6487i q^{59} -16.2333i q^{67} -10.8896i q^{71} -18.8614 q^{89} -0.116844 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 6 q^{5}+O(q^{10}) \) 4 * q + 6 * q^5	\( 4 q + 6 q^{5} + 22 q^{25} - 14 q^{37} - 28 q^{49} - 24 q^{53} - 18 q^{89} + 34 q^{97}+O(q^{100}) \) 4 * q + 6 * q^5 + 22 * q^25 - 14 * q^37 - 28 * q^49 - 24 * q^53 - 18 * q^89 + 34 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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