LMFDB - Embedded newform 1216.2.b.e.607.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1216,2,Mod(607,1216)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1216, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("1216.607");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1216 = 2^{6} \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1216.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:	no
Analytic conductor:	$9.70980888579$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.0.2702336256.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + 9x^{6} + 56x^{4} + 225x^{2} + 625 $ x^8 + 9x^6 + 56x^4 + 225*x^2 + 625
Coefficient ring:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$ 2^{8} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label			607.5
Root			$-1.52274 + 1.63746i$ of defining polynomial
Character	$\chi$	$=$	1216.607
Dual form			1216.2.b.e.607.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.31342i q^{5} -2.27492i q^{7} +3.00000 q^{9} +O(q^{10})$	$q+1.31342i q^{5} -2.27492i q^{7} +3.00000 q^{9} -2.15068 q^{11} +0.274917 q^{17} +4.35890 q^{19} -4.00000i q^{23} +3.27492 q^{25} +2.98793 q^{35} +7.40437 q^{43} +3.94027i q^{45} -10.2749i q^{47} +1.82475 q^{49} -2.82475i q^{55} +15.1698i q^{61} -6.82475i q^{63} +16.8248 q^{73} +4.89261i q^{77} +9.00000 q^{81} +8.71780 q^{83} +0.361083i q^{85} +5.72508i q^{95} -6.45203 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 24 q^{9}+O(q^{10}) $ 8 * q + 24 * q^9	$ 8 q + 24 q^{9} - 28 q^{17} - 4 q^{25} - 76 q^{49} + 44 q^{73} + 72 q^{81}+O(q^{100}) $ 8 * q + 24 * q^9 - 28 * q^17 - 4 * q^25 - 76 * q^49 + 44 * q^73 + 72 * q^81

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1216\mathbb{Z}\right)^\times$.

$n$	$191$	$705$	$837$
$\chi(n)$	$-1$	$-1$	$-1$

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	1.00000			$0$
$3$	0	0	−1.00000			$\pi$
$4$	0	0
$5$	1.31342i	0.587381i	0.955901	+	0.293691i	$0.0948835\pi$
$5$	1.31342i	0.587381i	−0.955901	+	0.293691i	$0.905116\pi$
$6$	0	0
$7$	− 2.27492i	− 0.859838i	−0.902867	−	0.429919i	$-0.858542\pi$
$7$	− 2.27492i	− 0.859838i	0.902867	−	0.429919i	$-0.141458\pi$
$8$	0	0
$9$	3.00000	1.00000
$10$	0	0
$11$	−2.15068	−0.648454	−0.324227	−	0.945979i	$-0.605104\pi$
$11$	−2.15068	−0.648454	−0.324227	+	0.945979i	$0.605104\pi$
$12$	0	0
$13$	0	0		−	1.00000i	$-0.5\pi$
$13$	0	0			1.00000i	$0.5\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	0.274917	0.0666772	0.0333386	−	0.999444i	$-0.489386\pi$
$17$	0.274917	0.0666772	0.0333386	+	0.999444i	$0.489386\pi$
$18$	0	0
$19$	4.35890	1.00000
$19$	4.35890	1.00000
$20$	0	0
$21$	0	0
$22$	0	0
$23$	− 4.00000i	− 0.834058i	−0.908893	−	0.417029i	$-0.863071\pi$
$23$	− 4.00000i	− 0.834058i	0.908893	−	0.417029i	$-0.136929\pi$
$24$	0	0
$25$	3.27492	0.654983
$26$	0	0
$27$	0	0
$28$	0	0
$29$	0	0		−	1.00000i	$-0.5\pi$
$29$	0	0			1.00000i	$0.5\pi$
$30$	0	0
$31$	0	0		−	1.00000i	$-0.5\pi$
$31$	0	0			1.00000i	$0.5\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	2.98793	0.505053
$36$	0	0
$37$	0	0		−	1.00000i	$-0.5\pi$
$37$	0	0			1.00000i	$0.5\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	0	0	1.00000			$0$
$41$	0	0	−1.00000			$\pi$
$42$	0	0
$43$	7.40437	1.12916	0.564578	−	0.825380i	$-0.309039\pi$
$43$	7.40437	1.12916	0.564578	+	0.825380i	$0.309039\pi$
$44$	0	0
$45$	3.94027i	0.587381i
$46$	0	0
$47$	− 10.2749i	− 1.49875i	−0.662145	−	0.749375i	$-0.730354\pi$
$47$	− 10.2749i	− 1.49875i	0.662145	−	0.749375i	$-0.269646\pi$
$48$	0	0
$49$	1.82475	0.260679
$50$	0	0
$51$	0	0
$52$	0	0
$53$	0	0		−	1.00000i	$-0.5\pi$
$53$	0	0			1.00000i	$0.5\pi$
$54$	0	0
$55$	− 2.82475i	− 0.380889i
$56$	0	0
$57$	0	0
$58$	0	0
$59$	0	0	1.00000			$0$
$59$	0	0	−1.00000			$\pi$
$60$	0	0
$61$	15.1698i	1.94230i	0.238474	+	0.971149i	$0.423353\pi$
$61$	15.1698i	1.94230i	−0.238474	+	0.971149i	$0.576647\pi$
$62$	0	0
$63$	− 6.82475i	− 0.859838i
$64$	0	0
$65$	0	0
$66$	0	0
$67$	0	0	1.00000			$0$
$67$	0	0	−1.00000			$\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	0	0		−	1.00000i	$-0.5\pi$
$71$	0	0			1.00000i	$0.5\pi$
$72$	0	0
$73$	16.8248	1.96919	0.984594	−	0.174855i	$-0.0559458\pi$
$73$	16.8248	1.96919	0.984594	+	0.174855i	$0.0559458\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	4.89261i	0.557565i
$78$	0	0
$79$	0	0		−	1.00000i	$-0.5\pi$
$79$	0	0			1.00000i	$0.5\pi$
$80$	0	0
$81$	9.00000	1.00000
$82$	0	0
$83$	8.71780	0.956903	0.478451	−	0.878114i	$-0.341198\pi$
$83$	8.71780	0.956903	0.478451	+	0.878114i	$0.341198\pi$
$84$	0	0
$85$	0.361083i	0.0391649i
$86$	0	0
$87$	0	0
$88$	0	0
$89$	0	0	1.00000			$0$
$89$	0	0	−1.00000			$\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	0	0
$94$	0	0
$95$	5.72508i	0.587381i
$96$	0	0
$97$	0	0	1.00000			$0$
$97$	0	0	−1.00000			$\pi$
$98$	0	0
$99$	−6.45203	−0.648454
$100$	0	0
$101$	− 17.4356i	− 1.73491i	−0.497519	−	0.867453i	$-0.665755\pi$
$101$	− 17.4356i	− 1.73491i	0.497519	−	0.867453i	$-0.334245\pi$
$102$	0	0
$103$	0	0		−	1.00000i	$-0.5\pi$
$103$	0	0			1.00000i	$0.5\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	0	0	1.00000			$0$
$107$	0	0	−1.00000			$\pi$
$108$	0	0
$109$	0	0		−	1.00000i	$-0.5\pi$
$109$	0	0			1.00000i	$0.5\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	0	0	1.00000			$0$
$113$	0	0	−1.00000			$\pi$
$114$	0	0
$115$	5.25370	0.489910
$116$	0	0
$117$	0	0
$118$	0	0
$119$	− 0.625414i	− 0.0573316i
$120$	0	0
$121$	−6.37459	−0.579508
$122$	0	0
$123$	0	0
$124$	0	0
$125$	10.8685i	0.972106i
$126$	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
$131$	−16.9594	−1.48175	−0.740876	−	0.671642i	$-0.765589\pi$
$131$	−16.9594	−1.48175	−0.740876	+	0.671642i	$0.765589\pi$
$132$	0	0
$133$	− 9.91613i	− 0.859838i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−7.72508	−0.659998	−0.329999	−	0.943981i	$-0.607048\pi$
$137$	−7.72508	−0.659998	−0.329999	+	0.943981i	$0.607048\pi$
$138$	0	0
$139$	−3.10302	−0.263195	−0.131597	−	0.991303i	$-0.542011\pi$
$139$	−3.10302	−0.263195	−0.131597	+	0.991303i	$0.542011\pi$
$140$	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
$149$	− 20.4235i	− 1.67316i	−0.547844	−	0.836580i	$-0.684551\pi$
$149$	− 20.4235i	− 1.67316i	0.547844	−	0.836580i	$-0.315449\pi$
$150$	0	0
$151$	0	0		−	1.00000i	$-0.5\pi$
$151$	0	0			1.00000i	$0.5\pi$
$152$	0	0
$153$	0.824752	0.0666772
$154$	0	0
$155$	0	0
$156$	0	0
$157$	17.4356i	1.39151i	0.718278	+	0.695756i	$0.244931\pi$
$157$	17.4356i	1.39151i	−0.718278	+	0.695756i	$0.755069\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−9.09967	−0.717154
$162$	0	0
$163$	−8.71780	−0.682831	−0.341415	−	0.939913i	$-0.610906\pi$
$163$	−8.71780	−0.682831	−0.341415	+	0.939913i	$0.610906\pi$
$164$	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
$171$	13.0767	1.00000
$172$	0	0
$173$	0	0		−	1.00000i	$-0.5\pi$
$173$	0	0			1.00000i	$0.5\pi$
$174$	0	0
$175$	− 7.45017i	− 0.563180i
$176$	0	0
$177$	0	0
$178$	0	0
$179$	0	0	1.00000			$0$
$179$	0	0	−1.00000			$\pi$
$180$	0	0
$181$	0	0		−	1.00000i	$-0.5\pi$
$181$	0	0			1.00000i	$0.5\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	0	0
$186$	0	0
$187$	−0.591258	−0.0432371
$188$	0	0
$189$	0	0
$190$	0	0
$191$	27.3746i	1.98076i	0.138390	+	0.990378i	$0.455807\pi$
$191$	27.3746i	1.98076i	−0.138390	+	0.990378i	$0.544193\pi$
$192$	0	0
$193$	0	0	1.00000			$0$
$193$	0	0	−1.00000			$\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	− 17.4356i	− 1.24223i	−0.783718	−	0.621117i	$-0.786679\pi$
$197$	− 17.4356i	− 1.24223i	0.783718	−	0.621117i	$-0.213321\pi$
$198$	0	0
$199$	1.17525i	0.0833111i	0.999132	+	0.0416556i	$0.0132632\pi$
$199$	1.17525i	0.0833111i	−0.999132	+	0.0416556i	$0.986737\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	0	0
$204$	0	0
$205$	0	0
$206$	0	0
$207$	− 12.0000i	− 0.834058i
$208$	0	0
$209$	−9.37459	−0.648454
$210$	0	0
$211$	0	0	1.00000			$0$
$211$	0	0	−1.00000			$\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	9.72508i	0.663245i
$216$	0	0
$217$	0	0
$218$	0	0
$219$	0	0
$220$	0	0
$221$	0	0
$222$	0	0
$223$	0	0		−	1.00000i	$-0.5\pi$
$223$	0	0			1.00000i	$0.5\pi$
$224$	0	0
$225$	9.82475	0.654983
$226$	0	0
$227$	0	0	1.00000			$0$
$227$	0	0	−1.00000			$\pi$
$228$	0	0
$229$	7.28929i	0.481690i	0.970564	+	0.240845i	$0.0774245\pi$
$229$	7.28929i	0.481690i	−0.970564	+	0.240845i	$0.922576\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−25.9244	−1.69837	−0.849183	−	0.528099i	$-0.822905\pi$
$233$	−25.9244	−1.69837	−0.849183	+	0.528099i	$0.822905\pi$
$234$	0	0
$235$	13.4953	0.880338
$236$	0	0
$237$	0	0
$238$	0	0
$239$	23.9244i	1.54754i	0.633465	+	0.773771i	$0.281632\pi$
$239$	23.9244i	1.54754i	−0.633465	+	0.773771i	$0.718368\pi$
$240$	0	0
$241$	0	0	1.00000			$0$
$241$	0	0	−1.00000			$\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	2.39667i	0.153118i
$246$	0	0
$247$	0	0
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−30.8158	−1.94508	−0.972539	−	0.232740i	$-0.925231\pi$
$251$	−30.8158	−1.94508	−0.972539	+	0.232740i	$0.925231\pi$
$252$	0	0
$253$	8.60271i	0.540848i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	0	0	1.00000			$0$
$257$	0	0	−1.00000			$\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	0	0
$262$	0	0
$263$	31.9244i	1.96854i	0.176659	+	0.984272i	$0.443471\pi$
$263$	31.9244i	1.96854i	−0.176659	+	0.984272i	$0.556529\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	0	0
$268$	0	0
$269$	0	0		−	1.00000i	$-0.5\pi$
$269$	0	0			1.00000i	$0.5\pi$
$270$	0	0
$271$	20.0000i	1.21491i	0.794353	+	0.607457i	$0.207810\pi$
$271$	20.0000i	1.21491i	−0.794353	+	0.607457i	$0.792190\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−7.04329	−0.424726
$276$	0	0
$277$	− 26.3994i	− 1.58619i	−0.609101	−	0.793093i	$-0.708470\pi$
$277$	− 26.3994i	− 1.58619i	0.609101	−	0.793093i	$-0.291530\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	0	0	1.00000			$0$
$281$	0	0	−1.00000			$\pi$
$282$	0	0
$283$	−20.3084	−1.20721	−0.603606	−	0.797283i	$-0.706270\pi$
$283$	−20.3084	−1.20721	−0.603606	+	0.797283i	$0.706270\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	0	0
$288$	0	0
$289$	−16.9244	−0.995554
$290$	0	0
$291$	0	0
$292$	0	0
$293$	0	0		−	1.00000i	$-0.5\pi$
$293$	0	0			1.00000i	$0.5\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	0	0
$298$	0	0
$299$	0	0
$300$	0	0
$301$	− 16.8443i	− 0.970891i
$302$	0	0
$303$	0	0
$304$	0	0
$305$	−19.9244	−1.14087
$306$	0	0
$307$	0	0	1.00000			$0$
$307$	0	0	−1.00000			$\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	13.7251i	0.778278i	0.921179	+	0.389139i	$0.127227\pi$
$311$	13.7251i	0.778278i	−0.921179	+	0.389139i	$0.872773\pi$
$312$	0	0
$313$	−6.00000	−0.339140	−0.169570	−	0.985518i	$-0.554238\pi$
$313$	−6.00000	−0.339140	−0.169570	+	0.985518i	$0.554238\pi$
$314$	0	0
$315$	8.96379	0.505053
$316$	0	0
$317$	0	0		−	1.00000i	$-0.5\pi$
$317$	0	0			1.00000i	$0.5\pi$
$318$	0	0
$319$	0	0
$320$	0	0
$321$	0	0
$322$	0	0
$323$	1.19834	0.0666772
$324$	0	0
$325$	0	0
$326$	0	0
$327$	0	0
$328$	0	0
$329$	−23.3746	−1.28868
$330$	0	0
$331$	0	0	1.00000			$0$
$331$	0	0	−1.00000			$\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	0	0
$336$	0	0
$337$	0	0	1.00000			$0$
$337$	0	0	−1.00000			$\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	0	0
$342$	0	0
$343$	− 20.0756i	− 1.08398i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−29.8635	−1.60316	−0.801578	−	0.597890i	$-0.796006\pi$
$347$	−29.8635	−1.60316	−0.801578	+	0.597890i	$0.796006\pi$
$348$	0	0
$349$	− 23.7725i	− 1.27251i	−0.771477	−	0.636257i	$-0.780482\pi$
$349$	− 23.7725i	− 1.27251i	0.771477	−	0.636257i	$-0.219518\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	14.0000	0.745145	0.372572	−	0.928003i	$-0.378476\pi$
$353$	14.0000	0.745145	0.372572	+	0.928003i	$0.378476\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	0	0
$358$	0	0
$359$	3.37459i	0.178104i	0.996027	+	0.0890519i	$0.0283837\pi$
$359$	3.37459i	0.178104i	−0.996027	+	0.0890519i	$0.971616\pi$
$360$	0	0
$361$	19.0000	1.00000
$362$	0	0
$363$	0	0
$364$	0	0
$365$	22.0980i	1.15666i
$366$	0	0
$367$	− 28.0000i	− 1.46159i	−0.682598	−	0.730794i	$-0.739150\pi$
$367$	− 28.0000i	− 1.46159i	0.682598	−	0.730794i	$-0.260850\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	0	0
$372$	0	0
$373$	0	0		−	1.00000i	$-0.5\pi$
$373$	0	0			1.00000i	$0.5\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	0	0
$378$	0	0
$379$	0	0	1.00000			$0$
$379$	0	0	−1.00000			$\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	0	0		−	1.00000i	$-0.5\pi$
$383$	0	0			1.00000i	$0.5\pi$
$384$	0	0
$385$	−6.42608	−0.327503
$386$	0	0
$387$	22.2131	1.12916
$388$	0	0
$389$	− 36.9068i	− 1.87125i	−0.352998	−	0.935624i	$-0.614838\pi$
$389$	− 36.9068i	− 1.87125i	0.352998	−	0.935624i	$-0.385162\pi$
$390$	0	0
$391$	− 1.09967i	− 0.0556126i
$392$	0	0
$393$	0	0
$394$	0	0
$395$	0	0
$396$	0	0
$397$	25.6772i	1.28870i	0.764730	+	0.644351i	$0.222873\pi$
$397$	25.6772i	1.28870i	−0.764730	+	0.644351i	$0.777127\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	0	0	1.00000			$0$
$401$	0	0	−1.00000			$\pi$
$402$	0	0
$403$	0	0
$404$	0	0
$405$	11.8208i	0.587381i
$406$	0	0
$407$	0	0
$408$	0	0
$409$	0	0	1.00000			$0$
$409$	0	0	−1.00000			$\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	0	0
$414$	0	0
$415$	11.4502i	0.562067i
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−8.71780	−0.425892	−0.212946	−	0.977064i	$-0.568306\pi$
$419$	−8.71780	−0.425892	−0.212946	+	0.977064i	$0.568306\pi$
$420$	0	0
$421$	0	0		−	1.00000i	$-0.5\pi$
$421$	0	0			1.00000i	$0.5\pi$
$422$	0	0
$423$	− 30.8248i	− 1.49875i
$424$	0	0
$425$	0.900331	0.0436725
$426$	0	0
$427$	34.5101	1.67006
$428$	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
$433$	0	0	1.00000			$0$
$433$	0	0	−1.00000			$\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	− 17.4356i	− 0.834058i
$438$	0	0
$439$	0	0		−	1.00000i	$-0.5\pi$
$439$	0	0			1.00000i	$0.5\pi$
$440$	0	0
$441$	5.47425	0.260679
$442$	0	0
$443$	−40.3709	−1.91808	−0.959039	−	0.283273i	$-0.908580\pi$
$443$	−40.3709	−1.91808	−0.959039	+	0.283273i	$0.908580\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	0	0
$448$	0	0
$449$	0	0	1.00000			$0$
$449$	0	0	−1.00000			$\pi$
$450$	0	0
$451$	0	0
$452$	0	0
$453$	0	0
$454$	0	0
$455$	0	0
$456$	0	0
$457$	42.4743	1.98686	0.993431	−	0.114433i	$-0.0365053\pi$
$457$	42.4743	1.98686	0.993431	+	0.114433i	$0.0365053\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	21.1457i	0.984853i	0.870354	+	0.492427i	$0.163890\pi$
$461$	21.1457i	0.984853i	−0.870354	+	0.492427i	$0.836110\pi$
$462$	0	0
$463$	9.17525i	0.426410i	0.977007	+	0.213205i	$0.0683902\pi$
$463$	9.17525i	0.426410i	−0.977007	+	0.213205i	$0.931610\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	39.4185	1.82407	0.912036	−	0.410110i	$-0.134510\pi$
$467$	39.4185	1.82407	0.912036	+	0.410110i	$0.134510\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	0	0
$472$	0	0
$473$	−15.9244	−0.732206
$474$	0	0
$475$	14.2750	0.654983
$476$	0	0
$477$	0	0
$478$	0	0
$479$	4.00000i	0.182765i	0.995816	+	0.0913823i	$0.0291285\pi$
$479$	4.00000i	0.182765i	−0.995816	+	0.0913823i	$0.970871\pi$
$480$	0	0
$481$	0	0
$482$	0	0
$483$	0	0
$484$	0	0
$485$	0	0
$486$	0	0
$487$	0	0		−	1.00000i	$-0.5\pi$
$487$	0	0			1.00000i	$0.5\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	43.5890	1.96714	0.983572	−	0.180517i	$-0.0577772\pi$
$491$	43.5890	1.96714	0.983572	+	0.180517i	$0.0577772\pi$
$492$	0	0
$493$	0	0
$494$	0	0
$495$	− 8.47425i	− 0.380889i
$496$	0	0
$497$	0	0
$498$	0	0
$499$	−44.6722	−1.99980	−0.999902	−	0.0139987i	$-0.995544\pi$
$499$	−44.6722	−1.99980	−0.999902	+	0.0139987i	$0.995544\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	44.0000i	1.96186i	0.194354	+	0.980932i	$0.437739\pi$
$503$	44.0000i	1.96186i	−0.194354	+	0.980932i	$0.562261\pi$
$504$	0	0
$505$	22.9003	1.01905
$506$	0	0
$507$	0	0
$508$	0	0
$509$	0	0		−	1.00000i	$-0.5\pi$
$509$	0	0			1.00000i	$0.5\pi$
$510$	0	0
$511$	− 38.2749i	− 1.69318i
$512$	0	0
$513$	0	0
$514$	0	0
$515$	0	0
$516$	0	0
$517$	22.0980i	0.971870i
$518$	0	0
$519$	0	0
$520$	0	0
$521$	0	0	1.00000			$0$
$521$	0	0	−1.00000			$\pi$
$522$	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
$529$	7.00000	0.304348
$530$	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
$539$	−3.92445	−0.169038
$540$	0	0
$541$	− 2.03559i	− 0.0875168i	−0.999042	−	0.0437584i	$-0.986067\pi$
$541$	− 2.03559i	− 0.0875168i	0.999042	−	0.0437584i	$-0.0139332\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	0	0
$546$	0	0
$547$	0	0	1.00000			$0$
$547$	0	0	−1.00000			$\pi$
$548$	0	0
$549$	45.5095i	1.94230i
$550$	0	0
$551$	0	0
$552$	0	0
$553$	0	0
$554$	0	0
$555$	0	0
$556$	0	0
$557$	42.8826i	1.81700i	0.417889	+	0.908498i	$0.362770\pi$
$557$	42.8826i	1.81700i	−0.417889	+	0.908498i	$0.637230\pi$
$558$	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			$\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	− 20.4743i	− 0.859838i
$568$	0	0
$569$	0	0	1.00000			$0$
$569$	0	0	−1.00000			$\pi$
$570$	0	0
$571$	−26.1534	−1.09449	−0.547243	−	0.836974i	$-0.684323\pi$
$571$	−26.1534	−1.09449	−0.547243	+	0.836974i	$0.684323\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	− 13.0997i	− 0.546294i
$576$	0	0
$577$	43.0241	1.79112	0.895558	−	0.444945i	$-0.146777\pi$
$577$	43.0241	1.79112	0.895558	+	0.444945i	$0.146777\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	− 19.8323i	− 0.822781i
$582$	0	0
$583$	0	0
$584$	0	0
$585$	0	0
$586$	0	0
$587$	17.9118	0.739298	0.369649	−	0.929172i	$-0.379478\pi$
$587$	17.9118	0.739298	0.369649	+	0.929172i	$0.379478\pi$
$588$	0	0
$589$	0	0
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−34.0000	−1.39621	−0.698106	−	0.715994i	$-0.745974\pi$
$593$	−34.0000	−1.39621	−0.698106	+	0.715994i	$0.745974\pi$
$594$	0	0
$595$	0.821434	0.0336755
$596$	0	0
$597$	0	0
$598$	0	0
$599$	0	0		−	1.00000i	$-0.5\pi$
$599$	0	0			1.00000i	$0.5\pi$
$600$	0	0
$601$	0	0	1.00000			$0$
$601$	0	0	−1.00000			$\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	− 8.37253i	− 0.340392i
$606$	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
$613$	18.5188i	0.747969i	0.927435	+	0.373985i	$0.122009\pi$
$613$	18.5188i	0.747969i	−0.927435	+	0.373985i	$0.877991\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	35.0241	1.41002	0.705008	−	0.709199i	$-0.250943\pi$
$617$	35.0241	1.41002	0.705008	+	0.709199i	$0.250943\pi$
$618$	0	0
$619$	43.5890	1.75199	0.875995	−	0.482321i	$-0.160206\pi$
$619$	43.5890	1.75199	0.875995	+	0.482321i	$0.160206\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	0	0
$624$	0	0
$625$	2.09967	0.0839868
$626$	0	0
$627$	0	0
$628$	0	0
$629$	0	0
$630$	0	0
$631$	− 49.0241i	− 1.95162i	−0.218624	−	0.975809i	$-0.570157\pi$
$631$	− 49.0241i	− 1.95162i	0.218624	−	0.975809i	$-0.429843\pi$
$632$	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
$641$	0	0	1.00000			$0$
$641$	0	0	−1.00000			$\pi$
$642$	0	0
$643$	48.9736	1.93133	0.965665	−	0.259791i	$-0.0836535\pi$
$643$	48.9736	1.93133	0.965665	+	0.259791i	$0.0836535\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	− 33.0241i	− 1.29831i	−0.760656	−	0.649155i	$-0.775122\pi$
$647$	− 33.0241i	− 1.29831i	0.760656	−	0.649155i	$-0.224878\pi$
$648$	0	0
$649$	0	0
$650$	0	0
$651$	0	0
$652$	0	0
$653$	− 50.7632i	− 1.98652i	−0.115920	−	0.993259i	$-0.536982\pi$
$653$	− 50.7632i	− 1.98652i	0.115920	−	0.993259i	$-0.463018\pi$
$654$	0	0
$655$	− 22.2749i	− 0.870353i
$656$	0	0
$657$	50.4743	1.96919
$658$	0	0
$659$	0	0	1.00000			$0$
$659$	0	0	−1.00000			$\pi$
$660$	0	0
$661$	0	0		−	1.00000i	$-0.5\pi$
$661$	0	0			1.00000i	$0.5\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	13.0241	0.505053
$666$	0	0
$667$	0	0
$668$	0	0
$669$	0	0
$670$	0	0
$671$	− 32.6254i	− 1.25949i
$672$	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		−	1.00000i	$-0.5\pi$
$677$	0	0			1.00000i	$0.5\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	0	0
$682$	0	0
$683$	0	0	1.00000			$0$
$683$	0	0	−1.00000			$\pi$
$684$	0	0
$685$	− 10.1463i	− 0.387671i
$686$	0	0
$687$	0	0
$688$	0	0
$689$	0	0
$690$	0	0
$691$	49.9259	1.89927	0.949636	−	0.313355i	$-0.101453\pi$
$691$	49.9259	1.89927	0.949636	+	0.313355i	$0.101453\pi$
$692$	0	0
$693$	14.6778i	0.557565i
$694$	0	0
$695$	− 4.07558i	− 0.154596i
$696$	0	0
$697$	0	0
$698$	0	0
$699$	0	0
$700$	0	0
$701$	17.4356i	0.658533i	0.944237	+	0.329267i	$0.106802\pi$
$701$	17.4356i	0.658533i	−0.944237	+	0.329267i	$0.893198\pi$
$702$	0	0
$703$	0	0
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−39.6645	−1.49174
$708$	0	0
$709$	52.3068i	1.96442i	0.187779	+	0.982211i	$0.439871\pi$
$709$	52.3068i	1.96442i	−0.187779	+	0.982211i	$0.560129\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	0	0
$714$	0	0
$715$	0	0
$716$	0	0
$717$	0	0
$718$	0	0
$719$	43.3746i	1.61760i	0.588084	+	0.808800i	$0.299882\pi$
$719$	43.3746i	1.61760i	−0.588084	+	0.808800i	$0.700118\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	0	0
$724$	0	0
$725$	0	0
$726$	0	0
$727$	− 52.4743i	− 1.94616i	−0.230463	−	0.973081i	$-0.574024\pi$
$727$	− 52.4743i	− 1.94616i	0.230463	−	0.973081i	$-0.425976\pi$
$728$	0	0
$729$	27.0000	1.00000
$730$	0	0
$731$	2.03559	0.0752890
$732$	0	0
$733$	− 52.3068i	− 1.93200i	−0.258551	−	0.965998i	$-0.583245\pi$
$733$	− 52.3068i	− 1.93200i	0.258551	−	0.965998i	$-0.416755\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	0	0
$738$	0	0
$739$	−54.2273	−1.99478	−0.997392	−	0.0721811i	$-0.977004\pi$
$739$	−54.2273	−1.99478	−0.997392	+	0.0721811i	$0.977004\pi$
$740$	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
$745$	26.8248	0.982783
$746$	0	0
$747$	26.1534	0.956903
$748$	0	0
$749$	0	0
$750$	0	0
$751$	0	0		−	1.00000i	$-0.5\pi$
$751$	0	0			1.00000i	$0.5\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	0	0
$756$	0	0
$757$	0.591258i	0.0214896i	0.999942	+	0.0107448i	$0.00342025\pi$
$757$	0.591258i	0.0214896i	−0.999942	+	0.0107448i	$0.996580\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−23.7251	−0.860034	−0.430017	−	0.902821i	$-0.641492\pi$
$761$	−23.7251	−0.860034	−0.430017	+	0.902821i	$0.641492\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	1.08325i	0.0391649i
$766$	0	0
$767$	0	0
$768$	0	0
$769$	6.62541	0.238919	0.119459	−	0.992839i	$-0.461884\pi$
$769$	6.62541	0.238919	0.119459	+	0.992839i	$0.461884\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	0	0		−	1.00000i	$-0.5\pi$
$773$	0	0			1.00000i	$0.5\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	0	0
$778$	0	0
$779$	0	0
$780$	0	0
$781$	0	0
$782$	0	0
$783$	0	0
$784$	0	0
$785$	−22.9003	−0.817348
$786$	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	0
$796$	0	0
$797$	0	0		−	1.00000i	$-0.5\pi$
$797$	0	0			1.00000i	$0.5\pi$
$798$	0	0
$799$	− 2.82475i	− 0.0999325i
$800$	0	0
$801$	0	0
$802$	0	0
$803$	−36.1846	−1.27693
$804$	0	0
$805$	− 11.9517i	− 0.421243i
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−51.5739	−1.81324	−0.906621	−	0.421945i	$-0.861347\pi$
$809$	−51.5739	−1.81324	−0.906621	+	0.421945i	$0.861347\pi$
$810$	0	0
$811$	0	0	1.00000			$0$
$811$	0	0	−1.00000			$\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	− 11.4502i	− 0.401082i
$816$	0	0
$817$	32.2749	1.12916
$818$	0	0
$819$	0	0
$820$	0	0
$821$	35.0021i	1.22158i	0.791792	+	0.610791i	$0.209148\pi$
$821$	35.0021i	1.22158i	−0.791792	+	0.610791i	$0.790852\pi$
$822$	0	0
$823$	53.5739i	1.86747i	0.357966	+	0.933735i	$0.383471\pi$
$823$	53.5739i	1.86747i	−0.357966	+	0.933735i	$0.616529\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	0	0	1.00000			$0$
$827$	0	0	−1.00000			$\pi$
$828$	0	0
$829$	0	0		−	1.00000i	$-0.5\pi$
$829$	0	0			1.00000i	$0.5\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	0.501656	0.0173813
$834$	0	0
$835$	0	0
$836$	0	0
$837$	0	0
$838$	0	0
$839$	0	0		−	1.00000i	$-0.5\pi$
$839$	0	0			1.00000i	$0.5\pi$
$840$	0	0
$841$	−29.0000	−1.00000
$842$	0	0
$843$	0	0
$844$	0	0
$845$	− 17.0745i	− 0.587381i
$846$	0	0
$847$	14.5017i	0.498283i
$848$	0	0
$849$	0	0
$850$	0	0
$851$	0	0
$852$	0	0
$853$	52.3068i	1.79095i	0.445112	+	0.895475i	$0.353164\pi$
$853$	52.3068i	1.79095i	−0.445112	+	0.895475i	$0.646836\pi$
$854$	0	0
$855$	17.1752i	0.587381i
$856$	0	0
$857$	0	0	1.00000			$0$
$857$	0	0	−1.00000			$\pi$
$858$	0	0
$859$	−41.3232	−1.40993	−0.704965	−	0.709242i	$-0.749037\pi$
$859$	−41.3232	−1.40993	−0.704965	+	0.709242i	$0.749037\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	0	0		−	1.00000i	$-0.5\pi$
$863$	0	0			1.00000i	$0.5\pi$
$864$	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
$875$	24.7249	0.835854
$876$	0	0
$877$	0	0		−	1.00000i	$-0.5\pi$
$877$	0	0			1.00000i	$0.5\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	59.0241	1.98857	0.994286	−	0.106749i	$-0.0340440\pi$
$881$	59.0241	1.98857	0.994286	+	0.106749i	$0.0340440\pi$
$882$	0	0
$883$	28.9111	0.972938	0.486469	−	0.873698i	$-0.338285\pi$
$883$	28.9111	0.972938	0.486469	+	0.873698i	$0.338285\pi$
$884$	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
$891$	−19.3561	−0.648454
$892$	0	0
$893$	− 44.7873i	− 1.49875i
$894$	0	0
$895$	0	0
$896$	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
$905$	0	0
$906$	0	0
$907$	0	0	1.00000			$0$
$907$	0	0	−1.00000			$\pi$
$908$	0	0
$909$	− 52.3068i	− 1.73491i
$910$	0	0
$911$	0	0		−	1.00000i	$-0.5\pi$
$911$	0	0			1.00000i	$0.5\pi$
$912$	0	0
$913$	−18.7492	−0.620507
$914$	0	0
$915$	0	0
$916$	0	0
$917$	38.5813i	1.27407i
$918$	0	0
$919$	60.0000i	1.97922i	0.143787	+	0.989609i	$0.454072\pi$
$919$	60.0000i	1.97922i	−0.143787	+	0.989609i	$0.545928\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	0	0
$924$	0	0
$925$	0	0
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−50.0000	−1.64045	−0.820223	−	0.572043i	$-0.806151\pi$
$929$	−50.0000	−1.64045	−0.820223	+	0.572043i	$0.806151\pi$
$930$	0	0
$931$	7.95391	0.260679
$932$	0	0
$933$	0	0
$934$	0	0
$935$	− 0.776573i	− 0.0253966i
$936$	0	0
$937$	10.4743	0.342179	0.171089	−	0.985255i	$-0.445271\pi$
$937$	10.4743	0.342179	0.171089	+	0.985255i	$0.445271\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	0	0		−	1.00000i	$-0.5\pi$
$941$	0	0			1.00000i	$0.5\pi$
$942$	0	0
$943$	0	0
$944$	0	0
$945$	0	0
$946$	0	0
$947$	61.0246	1.98303	0.991516	−	0.129983i	$-0.0414921\pi$
$947$	61.0246	1.98303	0.991516	+	0.129983i	$0.0414921\pi$
$948$	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
$955$	−35.9544	−1.16346
$956$	0	0
$957$	0	0
$958$	0	0
$959$	17.5739i	0.567492i
$960$	0	0
$961$	−31.0000	−1.00000
$962$	0	0
$963$	0	0
$964$	0	0
$965$	0	0
$966$	0	0
$967$	12.0000i	0.385894i	0.981209	+	0.192947i	$0.0618045\pi$
$967$	12.0000i	0.385894i	−0.981209	+	0.192947i	$0.938195\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	0	0	1.00000			$0$
$971$	0	0	−1.00000			$\pi$
$972$	0	0
$973$	7.05911i	0.226305i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	0	0	1.00000			$0$
$977$	0	0	−1.00000			$\pi$
$978$	0	0
$979$	0	0
$980$	0	0
$981$	0	0
$982$	0	0
$983$	0	0		−	1.00000i	$-0.5\pi$
$983$	0	0			1.00000i	$0.5\pi$
$984$	0	0
$985$	22.9003	0.729665
$986$	0	0
$987$	0	0
$988$	0	0
$989$	− 29.6175i	− 0.941782i
$990$	0	0
$991$	0	0		−	1.00000i	$-0.5\pi$
$991$	0	0			1.00000i	$0.5\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	−1.54360	−0.0489354
$996$	0	0
$997$	56.7390i	1.79694i	0.439031	+	0.898472i	$0.355322\pi$
$997$	56.7390i	1.79694i	−0.439031	+	0.898472i	$0.644678\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1216.2.b.e.607.5	yes	8
4.3	odd	2	inner	1216.2.b.e.607.6	yes	8
8.3	odd	2	inner	1216.2.b.e.607.4	yes	8
8.5	even	2	inner	1216.2.b.e.607.3	✓	8
19.18	odd	2	CM	1216.2.b.e.607.5	yes	8
76.75	even	2	inner	1216.2.b.e.607.6	yes	8
152.37	odd	2	inner	1216.2.b.e.607.3	✓	8
152.75	even	2	inner	1216.2.b.e.607.4	yes	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1216.2.b.e.607.3	✓	8	8.5	even	2	inner
1216.2.b.e.607.3	✓	8	152.37	odd	2	inner
1216.2.b.e.607.4	yes	8	8.3	odd	2	inner
1216.2.b.e.607.4	yes	8	152.75	even	2	inner
1216.2.b.e.607.5	yes	8	1.1	even	1	trivial
1216.2.b.e.607.5	yes	8	19.18	odd	2	CM
1216.2.b.e.607.6	yes	8	4.3	odd	2	inner
1216.2.b.e.607.6	yes	8	76.75	even	2	inner

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:	\( N \)	\(=\)	\( 1216 = 2^{6} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1216.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+1.31342i q^{5} -2.27492i q^{7} +3.00000 q^{9} +O(q^{10})\)	\(q+1.31342i q^{5} -2.27492i q^{7} +3.00000 q^{9} -2.15068 q^{11} +0.274917 q^{17} +4.35890 q^{19} -4.00000i q^{23} +3.27492 q^{25} +2.98793 q^{35} +7.40437 q^{43} +3.94027i q^{45} -10.2749i q^{47} +1.82475 q^{49} -2.82475i q^{55} +15.1698i q^{61} -6.82475i q^{63} +16.8248 q^{73} +4.89261i q^{77} +9.00000 q^{81} +8.71780 q^{83} +0.361083i q^{85} +5.72508i q^{95} -6.45203 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 24 q^{9}+O(q^{10}) \) 8 * q + 24 * q^9	\( 8 q + 24 q^{9} - 28 q^{17} - 4 q^{25} - 76 q^{49} + 44 q^{73} + 72 q^{81}+O(q^{100}) \) 8 * q + 24 * q^9 - 28 * q^17 - 4 * q^25 - 76 * q^49 + 44 * q^73 + 72 * q^81

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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