LMFDB - Embedded newform 1216.2.b.f.607.3

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:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} + 18x^{10} + 207x^{8} + 1014x^{6} + 1065x^{4} - 5508x^{2} + 8464 $ x^12 + 18x^10 + 207x^8 + 1014x^6 + 1065x^4 - 5508*x^2 + 8464
Coefficient ring:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$ 2^{14} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label			607.3
Root			$-1.13617 + 0.418247i$ of defining polynomial
Character	$\chi$	$=$	1216.607
Dual form			1216.2.b.f.607.10

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.27234i q^{3} -4.53419i q^{7} -2.16351 q^{9} +O(q^{10})$	$q-2.27234i q^{3} -4.53419i q^{7} -2.16351 q^{9} +6.13008 q^{13} +8.23189 q^{17} -4.35890i q^{19} -10.3032 q^{21} +2.86121i q^{23} +5.00000 q^{25} -1.90079i q^{27} -10.6747 q^{29} -8.71780 q^{37} -13.9296i q^{39} +6.00000i q^{47} -13.5589 q^{49} -18.7056i q^{51} -2.95926 q^{53} -9.90488 q^{57} +14.5325i q^{59} +9.80976i q^{63} +5.44315i q^{67} +6.50162 q^{69} +15.6273 q^{73} -11.3617i q^{75} -10.8098 q^{81} +24.2566i q^{87} -27.7950i q^{91} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q - 36 q^{9}+O(q^{10}) $ 12 * q - 36 * q^9	$ 12 q - 36 q^{9} + 60 q^{25} - 84 q^{49} + 108 q^{81}+O(q^{100}) $ 12 * q - 36 * q^9 + 60 * q^25 - 84 * q^49 + 108 * 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
$3$	− 2.27234i	− 1.31193i	−0.754790	−	0.655967i	$-0.772261\pi$
$3$	− 2.27234i	− 1.31193i	0.754790	−	0.655967i	$-0.227739\pi$
$4$	0	0
$5$	0	0	1.00000			$0$
$5$	0	0	−1.00000			$\pi$
$6$	0	0
$7$	− 4.53419i	− 1.71376i	−0.515513	−	0.856882i	$-0.672398\pi$
$7$	− 4.53419i	− 1.71376i	0.515513	−	0.856882i	$-0.327602\pi$
$8$	0	0
$9$	−2.16351	−0.721169
$10$	0	0
$11$	0	0		−	1.00000i	$-0.5\pi$
$11$	0	0			1.00000i	$0.5\pi$
$12$	0	0
$13$	6.13008	1.70018	0.850089	−	0.526639i	$-0.176548\pi$
$13$	6.13008	1.70018	0.850089	+	0.526639i	$0.176548\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	8.23189	1.99653	0.998264	−	0.0589035i	$-0.0187604\pi$
$17$	8.23189	1.99653	0.998264	+	0.0589035i	$0.0187604\pi$
$18$	0	0
$19$	− 4.35890i	− 1.00000i
$19$	− 4.35890i	− 1.00000i
$20$	0	0
$21$	−10.3032	−2.24834
$22$	0	0
$23$	2.86121i	0.596603i	0.954472	+	0.298301i	$0.0964201\pi$
$23$	2.86121i	0.596603i	−0.954472	+	0.298301i	$0.903580\pi$
$24$	0	0
$25$	5.00000	1.00000
$26$	0	0
$27$	− 1.90079i	− 0.365808i
$28$	0	0
$29$	−10.6747	−1.98225	−0.991126	−	0.132929i	$-0.957562\pi$
$29$	−10.6747	−1.98225	−0.991126	+	0.132929i	$0.957562\pi$
$30$	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
$35$	0	0
$36$	0	0
$37$	−8.71780	−1.43320	−0.716599	−	0.697486i	$-0.754302\pi$
$37$	−8.71780	−1.43320	−0.716599	+	0.697486i	$0.754302\pi$
$38$	0	0
$39$	− 13.9296i	− 2.23052i
$40$	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
$47$	6.00000i	0.875190i	0.899172	+	0.437595i	$0.144170\pi$
$47$	6.00000i	0.875190i	−0.899172	+	0.437595i	$0.855830\pi$
$48$	0	0
$49$	−13.5589	−1.93699
$50$	0	0
$51$	− 18.7056i	− 2.61931i
$52$	0	0
$53$	−2.95926	−0.406486	−0.203243	−	0.979128i	$-0.565148\pi$
$53$	−2.95926	−0.406486	−0.203243	+	0.979128i	$0.565148\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	−9.90488	−1.31193
$58$	0	0
$59$	14.5325i	1.89197i	0.324211	+	0.945985i	$0.394901\pi$
$59$	14.5325i	1.89197i	−0.324211	+	0.945985i	$0.605099\pi$
$60$	0	0
$61$	0	0	1.00000			$0$
$61$	0	0	−1.00000			$\pi$
$62$	0	0
$63$	9.80976i	1.23591i
$64$	0	0
$65$	0	0
$66$	0	0
$67$	5.44315i	0.664987i	0.943106	+	0.332493i	$0.107890\pi$
$67$	5.44315i	0.664987i	−0.943106	+	0.332493i	$0.892110\pi$
$68$	0	0
$69$	6.50162	0.782703
$70$	0	0
$71$	0	0		−	1.00000i	$-0.5\pi$
$71$	0	0			1.00000i	$0.5\pi$
$72$	0	0
$73$	15.6273	1.82904	0.914518	−	0.404545i	$-0.132570\pi$
$73$	15.6273	1.82904	0.914518	+	0.404545i	$0.132570\pi$
$74$	0	0
$75$	− 11.3617i	− 1.31193i
$76$	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
$81$	−10.8098	−1.20108
$82$	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
$87$	24.2566i	2.60058i
$88$	0	0
$89$	0	0	1.00000			$0$
$89$	0	0	−1.00000			$\pi$
$90$	0	0
$91$	− 27.7950i	− 2.91370i
$92$	0	0
$93$	0	0
$94$	0	0
$95$	0	0
$96$	0	0
$97$	0	0	1.00000			$0$
$97$	0	0	−1.00000			$\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	0	0	1.00000			$0$
$101$	0	0	−1.00000			$\pi$
$102$	0	0
$103$	0	0		−	1.00000i	$-0.5\pi$
$103$	0	0			1.00000i	$0.5\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	− 0.898482i	− 0.0868595i	−0.999056	−	0.0434298i	$-0.986172\pi$
$107$	− 0.898482i	− 0.0868595i	0.999056	−	0.0434298i	$-0.0138285\pi$
$108$	0	0
$109$	7.50393	0.718746	0.359373	−	0.933194i	$-0.382991\pi$
$109$	7.50393	0.718746	0.359373	+	0.933194i	$0.382991\pi$
$110$	0	0
$111$	19.8098i	1.88026i
$112$	0	0
$113$	0	0	1.00000			$0$
$113$	0	0	−1.00000			$\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	−13.2625	−1.22612
$118$	0	0
$119$	− 37.3250i	− 3.42158i
$120$	0	0
$121$	−11.0000	−1.00000
$122$	0	0
$123$	0	0
$124$	0	0
$125$	0	0
$126$	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
$133$	−19.7641	−1.71376
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−13.9543	−1.19220	−0.596098	−	0.802911i	$-0.703283\pi$
$137$	−13.9543	−1.19220	−0.596098	+	0.802911i	$0.703283\pi$
$138$	0	0
$139$	0	0		−	1.00000i	$-0.5\pi$
$139$	0	0			1.00000i	$0.5\pi$
$140$	0	0
$141$	13.6340	1.14819
$142$	0	0
$143$	0	0
$144$	0	0
$145$	0	0
$146$	0	0
$147$	30.8104i	2.54120i
$148$	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
$153$	−17.8098	−1.43983
$154$	0	0
$155$	0	0
$156$	0	0
$157$	0	0	1.00000			$0$
$157$	0	0	−1.00000			$\pi$
$158$	0	0
$159$	6.72443i	0.533282i
$160$	0	0
$161$	12.9733	1.02244
$162$	0	0
$163$	0	0		−	1.00000i	$-0.5\pi$
$163$	0	0			1.00000i	$0.5\pi$
$164$	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
$169$	24.5779	1.89061
$170$	0	0
$171$	9.43051i	0.721169i
$172$	0	0
$173$	26.1534	1.98841	0.994203	−	0.107521i	$-0.0342912\pi$
$173$	26.1534	1.98841	0.994203	+	0.107521i	$0.0342912\pi$
$174$	0	0
$175$	− 22.6710i	− 1.71376i
$176$	0	0
$177$	33.0227	2.48214
$178$	0	0
$179$	26.1534i	1.95480i	0.211407	+	0.977398i	$0.432196\pi$
$179$	26.1534i	1.95480i	−0.211407	+	0.977398i	$0.567804\pi$
$180$	0	0
$181$	−8.71780	−0.647989	−0.323994	−	0.946059i	$-0.605026\pi$
$181$	−8.71780	−0.647989	−0.323994	+	0.946059i	$0.605026\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	0	0
$186$	0	0
$187$	0	0
$188$	0	0
$189$	−8.61856	−0.626908
$190$	0	0
$191$	− 7.88017i	− 0.570189i	−0.958499	−	0.285094i	$-0.907975\pi$
$191$	− 7.88017i	− 0.570189i	0.958499	−	0.285094i	$-0.0920249\pi$
$192$	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
$199$	17.6520i	1.25132i	0.780097	+	0.625659i	$0.215170\pi$
$199$	17.6520i	1.25132i	−0.780097	+	0.625659i	$0.784830\pi$
$200$	0	0
$201$	12.3687	0.872418
$202$	0	0
$203$	48.4014i	3.39711i
$204$	0	0
$205$	0	0
$206$	0	0
$207$	− 6.19024i	− 0.430252i
$208$	0	0
$209$	0	0
$210$	0	0
$211$	− 17.7033i	− 1.21875i	−0.792884	−	0.609373i	$-0.791421\pi$
$211$	− 17.7033i	− 1.21875i	0.792884	−	0.609373i	$-0.208579\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	0	0
$216$	0	0
$217$	0	0
$218$	0	0
$219$	− 35.5104i	− 2.39957i
$220$	0	0
$221$	50.4622	3.39445
$222$	0	0
$223$	0	0		−	1.00000i	$-0.5\pi$
$223$	0	0			1.00000i	$0.5\pi$
$224$	0	0
$225$	−10.8175	−0.721169
$226$	0	0
$227$	− 28.1665i	− 1.86948i	−0.355337	−	0.934738i	$-0.615634\pi$
$227$	− 28.1665i	− 1.86948i	0.355337	−	0.934738i	$-0.384366\pi$
$228$	0	0
$229$	0	0	1.00000			$0$
$229$	0	0	−1.00000			$\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	18.0000	1.17922	0.589610	−	0.807688i	$-0.299282\pi$
$233$	18.0000	1.17922	0.589610	+	0.807688i	$0.299282\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	0	0
$238$	0	0
$239$	− 30.0664i	− 1.94483i	−0.233254	−	0.972416i	$-0.574937\pi$
$239$	− 30.0664i	− 1.94483i	0.233254	−	0.972416i	$-0.425063\pi$
$240$	0	0
$241$	0	0	1.00000			$0$
$241$	0	0	−1.00000			$\pi$
$242$	0	0
$243$	18.8610i	1.20993i
$244$	0	0
$245$	0	0
$246$	0	0
$247$	− 26.7204i	− 1.70018i
$248$	0	0
$249$	0	0
$250$	0	0
$251$	0	0		−	1.00000i	$-0.5\pi$
$251$	0	0			1.00000i	$0.5\pi$
$252$	0	0
$253$	0	0
$254$	0	0
$255$	0	0
$256$	0	0
$257$	0	0	1.00000			$0$
$257$	0	0	−1.00000			$\pi$
$258$	0	0
$259$	39.5282i	2.45616i
$260$	0	0
$261$	23.0949	1.42954
$262$	0	0
$263$	30.0000i	1.84988i	0.380114	+	0.924940i	$0.375885\pi$
$263$	30.0000i	1.84988i	−0.380114	+	0.924940i	$0.624115\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	0	0
$268$	0	0
$269$	26.1534	1.59460	0.797300	−	0.603583i	$-0.206261\pi$
$269$	26.1534	1.59460	0.797300	+	0.603583i	$0.206261\pi$
$270$	0	0
$271$	− 0.484765i	− 0.0294474i	−0.999892	−	0.0147237i	$-0.995313\pi$
$271$	− 0.484765i	− 0.0294474i	0.999892	−	0.0147237i	$-0.00468686\pi$
$272$	0	0
$273$	−63.1595	−3.82258
$274$	0	0
$275$	0	0
$276$	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
$289$	50.7641	2.98612
$290$	0	0
$291$	0	0
$292$	0	0
$293$	16.5933	0.969389	0.484695	−	0.874683i	$-0.338931\pi$
$293$	16.5933	0.969389	0.484695	+	0.874683i	$0.338931\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	0	0
$298$	0	0
$299$	17.5394i	1.01433i
$300$	0	0
$301$	0	0
$302$	0	0
$303$	0	0
$304$	0	0
$305$	0	0
$306$	0	0
$307$	− 8.71780i	− 0.497551i	−0.968561	−	0.248776i	$-0.919972\pi$
$307$	− 8.71780i	− 0.497551i	0.968561	−	0.248776i	$-0.0800281\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	24.3440i	1.38042i	0.723610	+	0.690209i	$0.242482\pi$
$311$	24.3440i	1.38042i	−0.723610	+	0.690209i	$0.757518\pi$
$312$	0	0
$313$	−20.6463	−1.16700	−0.583498	−	0.812115i	$-0.698316\pi$
$313$	−20.6463	−1.16700	−0.583498	+	0.812115i	$0.698316\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−26.1057	−1.46624	−0.733122	−	0.680097i	$-0.761937\pi$
$317$	−26.1057	−1.46624	−0.733122	+	0.680097i	$0.761937\pi$
$318$	0	0
$319$	0	0
$320$	0	0
$321$	−2.04165	−0.113954
$322$	0	0
$323$	− 35.8820i	− 1.99653i
$324$	0	0
$325$	30.6504	1.70018
$326$	0	0
$327$	− 17.0514i	− 0.942947i
$328$	0	0
$329$	27.2052	1.49987
$330$	0	0
$331$	32.7112i	1.79797i	0.437981	+	0.898984i	$0.355694\pi$
$331$	32.7112i	1.79797i	−0.437981	+	0.898984i	$0.644306\pi$
$332$	0	0
$333$	18.8610	1.03358
$334$	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
$343$	29.7394i	1.60577i
$344$	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
$351$	− 11.6520i	− 0.621938i
$352$	0	0
$353$	−13.2509	−0.705272	−0.352636	−	0.935761i	$-0.614715\pi$
$353$	−13.2509	−0.705272	−0.352636	+	0.935761i	$0.614715\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	−84.8149	−4.48888
$358$	0	0
$359$	25.0474i	1.32195i	0.750407	+	0.660976i	$0.229857\pi$
$359$	25.0474i	1.32195i	−0.750407	+	0.660976i	$0.770143\pi$
$360$	0	0
$361$	−19.0000	−1.00000
$362$	0	0
$363$	24.9957i	1.31193i
$364$	0	0
$365$	0	0
$366$	0	0
$367$	− 10.0000i	− 0.521996i	−0.965339	−	0.260998i	$-0.915948\pi$
$367$	− 10.0000i	− 0.521996i	0.965339	−	0.260998i	$-0.0840516\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	13.4179i	0.696621i
$372$	0	0
$373$	−17.0164	−0.881075	−0.440537	−	0.897734i	$-0.645212\pi$
$373$	−17.0164	−0.881075	−0.440537	+	0.897734i	$0.645212\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−65.4371	−3.37018
$378$	0	0
$379$	4.06930i	0.209026i	0.994524	+	0.104513i	$0.0333284\pi$
$379$	4.06930i	0.209026i	−0.994524	+	0.104513i	$0.966672\pi$
$380$	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
$385$	0	0
$386$	0	0
$387$	0	0
$388$	0	0
$389$	0	0	1.00000			$0$
$389$	0	0	−1.00000			$\pi$
$390$	0	0
$391$	23.5532i	1.19113i
$392$	0	0
$393$	0	0
$394$	0	0
$395$	0	0
$396$	0	0
$397$	0	0	1.00000			$0$
$397$	0	0	−1.00000			$\pi$
$398$	0	0
$399$	44.9106i	2.24834i
$400$	0	0
$401$	0	0	1.00000			$0$
$401$	0	0	−1.00000			$\pi$
$402$	0	0
$403$	0	0
$404$	0	0
$405$	0	0
$406$	0	0
$407$	0	0
$408$	0	0
$409$	0	0	1.00000			$0$
$409$	0	0	−1.00000			$\pi$
$410$	0	0
$411$	31.7089i	1.56408i
$412$	0	0
$413$	65.8931	3.24239
$414$	0	0
$415$	0	0
$416$	0	0
$417$	0	0
$418$	0	0
$419$	0	0		−	1.00000i	$-0.5\pi$
$419$	0	0			1.00000i	$0.5\pi$
$420$	0	0
$421$	−21.1379	−1.03020	−0.515100	−	0.857130i	$-0.672245\pi$
$421$	−21.1379	−1.03020	−0.515100	+	0.857130i	$0.672245\pi$
$422$	0	0
$423$	− 12.9810i	− 0.631160i
$424$	0	0
$425$	41.1595	1.99653
$426$	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
$433$	0	0	1.00000			$0$
$433$	0	0	−1.00000			$\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	12.4717	0.596603
$438$	0	0
$439$	0	0		−	1.00000i	$-0.5\pi$
$439$	0	0			1.00000i	$0.5\pi$
$440$	0	0
$441$	29.3348	1.39689
$442$	0	0
$443$	0	0		−	1.00000i	$-0.5\pi$
$443$	0	0			1.00000i	$0.5\pi$
$444$	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
$457$	−28.7451	−1.34464	−0.672320	−	0.740261i	$-0.734702\pi$
$457$	−28.7451	−1.34464	−0.672320	+	0.740261i	$0.734702\pi$
$458$	0	0
$459$	− 15.6471i	− 0.730345i
$460$	0	0
$461$	0	0	1.00000			$0$
$461$	0	0	−1.00000			$\pi$
$462$	0	0
$463$	22.0000i	1.02243i	0.859454	+	0.511213i	$0.170804\pi$
$463$	22.0000i	1.02243i	−0.859454	+	0.511213i	$0.829196\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	0	0		−	1.00000i	$-0.5\pi$
$467$	0	0			1.00000i	$0.5\pi$
$468$	0	0
$469$	24.6803	1.13963
$470$	0	0
$471$	0	0
$472$	0	0
$473$	0	0
$474$	0	0
$475$	− 21.7945i	− 1.00000i
$476$	0	0
$477$	6.40238	0.293145
$478$	0	0
$479$	− 42.0000i	− 1.91903i	−0.281659	−	0.959514i	$-0.590885\pi$
$479$	− 42.0000i	− 1.91903i	0.281659	−	0.959514i	$-0.409115\pi$
$480$	0	0
$481$	−53.4408	−2.43669
$482$	0	0
$483$	− 29.4796i	− 1.34137i
$484$	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
$491$	0	0		−	1.00000i	$-0.5\pi$
$491$	0	0			1.00000i	$0.5\pi$
$492$	0	0
$493$	−87.8734	−3.95762
$494$	0	0
$495$	0	0
$496$	0	0
$497$	0	0
$498$	0	0
$499$	0	0		−	1.00000i	$-0.5\pi$
$499$	0	0			1.00000i	$0.5\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	2.15775i	0.0962094i	0.998842	+	0.0481047i	$0.0153181\pi$
$503$	2.15775i	0.0962094i	−0.998842	+	0.0481047i	$0.984682\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	− 55.8491i	− 2.48035i
$508$	0	0
$509$	26.1534	1.15923	0.579614	−	0.814891i	$-0.303203\pi$
$509$	26.1534	1.15923	0.579614	+	0.814891i	$0.303203\pi$
$510$	0	0
$511$	− 70.8572i	− 3.13454i
$512$	0	0
$513$	−8.28536	−0.365808
$514$	0	0
$515$	0	0
$516$	0	0
$517$	0	0
$518$	0	0
$519$	− 59.4293i	− 2.60866i
$520$	0	0
$521$	0	0	1.00000			$0$
$521$	0	0	−1.00000			$\pi$
$522$	0	0
$523$	− 44.9713i	− 1.96646i	−0.182373	−	0.983230i	$-0.558378\pi$
$523$	− 44.9713i	− 1.96646i	0.182373	−	0.983230i	$-0.441622\pi$
$524$	0	0
$525$	−51.5160	−2.24834
$526$	0	0
$527$	0	0
$528$	0	0
$529$	14.8135	0.644065
$530$	0	0
$531$	− 31.4411i	− 1.36443i
$532$	0	0
$533$	0	0
$534$	0	0
$535$	0	0
$536$	0	0
$537$	59.4293	2.56456
$538$	0	0
$539$	0	0
$540$	0	0
$541$	0	0	1.00000			$0$
$541$	0	0	−1.00000			$\pi$
$542$	0	0
$543$	19.8098i	0.850118i
$544$	0	0
$545$	0	0
$546$	0	0
$547$	− 43.5890i	− 1.86373i	−0.362804	−	0.931865i	$-0.618181\pi$
$547$	− 43.5890i	− 1.86373i	0.362804	−	0.931865i	$-0.381819\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	46.5302i	1.98225i
$552$	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
$563$	26.1534i	1.10223i	0.834428	+	0.551117i	$0.185798\pi$
$563$	26.1534i	1.10223i	−0.834428	+	0.551117i	$0.814202\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	49.0135i	2.05837i
$568$	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
$573$	−17.9064	−0.748050
$574$	0	0
$575$	14.3060i	0.596603i
$576$	0	0
$577$	1.53995	0.0641089	0.0320545	−	0.999486i	$-0.489795\pi$
$577$	1.53995	0.0641089	0.0320545	+	0.999486i	$0.489795\pi$
$578$	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
$587$	0	0		−	1.00000i	$-0.5\pi$
$587$	0	0			1.00000i	$0.5\pi$
$588$	0	0
$589$	0	0
$590$	0	0
$591$	0	0
$592$	0	0
$593$	42.0000	1.72473	0.862367	−	0.506284i	$-0.168981\pi$
$593$	42.0000	1.72473	0.862367	+	0.506284i	$0.168981\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	40.1113	1.64165
$598$	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
$603$	− 11.7763i	− 0.479568i
$604$	0	0
$605$	0	0
$606$	0	0
$607$	0	0		−	1.00000i	$-0.5\pi$
$607$	0	0			1.00000i	$0.5\pi$
$608$	0	0
$609$	109.984	4.45678
$610$	0	0
$611$	36.7805i	1.48798i
$612$	0	0
$613$	0	0	1.00000			$0$
$613$	0	0	−1.00000			$\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−6.00000	−0.241551	−0.120775	−	0.992680i	$-0.538538\pi$
$617$	−6.00000	−0.241551	−0.120775	+	0.992680i	$0.538538\pi$
$618$	0	0
$619$	0	0		−	1.00000i	$-0.5\pi$
$619$	0	0			1.00000i	$0.5\pi$
$620$	0	0
$621$	5.43856	0.218242
$622$	0	0
$623$	0	0
$624$	0	0
$625$	25.0000	1.00000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−71.7640	−2.86142
$630$	0	0
$631$	− 34.0000i	− 1.35352i	−0.736204	−	0.676759i	$-0.763384\pi$
$631$	− 34.0000i	− 1.35352i	0.736204	−	0.676759i	$-0.236616\pi$
$632$	0	0
$633$	−40.2279	−1.59891
$634$	0	0
$635$	0	0
$636$	0	0
$637$	−83.1172	−3.29322
$638$	0	0
$639$	0	0
$640$	0	0
$641$	0	0	1.00000			$0$
$641$	0	0	−1.00000			$\pi$
$642$	0	0
$643$	0	0		−	1.00000i	$-0.5\pi$
$643$	0	0			1.00000i	$0.5\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	− 41.5112i	− 1.63197i	−0.578071	−	0.815987i	$-0.696194\pi$
$647$	− 41.5112i	− 1.63197i	0.578071	−	0.815987i	$-0.303806\pi$
$648$	0	0
$649$	0	0
$650$	0	0
$651$	0	0
$652$	0	0
$653$	0	0	1.00000			$0$
$653$	0	0	−1.00000			$\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	−33.8098	−1.31904
$658$	0	0
$659$	49.5160i	1.92887i	0.264321	+	0.964435i	$0.414852\pi$
$659$	49.5160i	1.92887i	−0.264321	+	0.964435i	$0.585148\pi$
$660$	0	0
$661$	29.2765	1.13873	0.569363	−	0.822086i	$-0.307190\pi$
$661$	29.2765	1.13873	0.569363	+	0.822086i	$0.307190\pi$
$662$	0	0
$663$	− 114.667i	− 4.45329i
$664$	0	0
$665$	0	0
$666$	0	0
$667$	− 30.5427i	− 1.18262i
$668$	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
$675$	− 9.50396i	− 0.365808i
$676$	0	0
$677$	47.4552	1.82385	0.911926	−	0.410354i	$-0.134595\pi$
$677$	47.4552	1.82385	0.911926	+	0.410354i	$0.134595\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	−64.0037	−2.45263
$682$	0	0
$683$	26.1534i	1.00073i	0.865814	+	0.500366i	$0.166801\pi$
$683$	26.1534i	1.00073i	−0.865814	+	0.500366i	$0.833199\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	0	0
$688$	0	0
$689$	−18.1405	−0.691098
$690$	0	0
$691$	0	0		−	1.00000i	$-0.5\pi$
$691$	0	0			1.00000i	$0.5\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	0	0
$696$	0	0
$697$	0	0
$698$	0	0
$699$	− 40.9020i	− 1.54706i
$700$	0	0
$701$	0	0	1.00000			$0$
$701$	0	0	−1.00000			$\pi$
$702$	0	0
$703$	38.0000i	1.43320i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	0	0
$708$	0	0
$709$	0	0	1.00000			$0$
$709$	0	0	−1.00000			$\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	0	0
$714$	0	0
$715$	0	0
$716$	0	0
$717$	−68.3209	−2.55149
$718$	0	0
$719$	− 52.2526i	− 1.94869i	−0.225055	−	0.974346i	$-0.572256\pi$
$719$	− 52.2526i	− 1.94869i	0.225055	−	0.974346i	$-0.427744\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	0	0
$724$	0	0
$725$	−53.3737	−1.98225
$726$	0	0
$727$	53.9256i	1.99999i	0.00345931	+	0.999994i	$0.498899\pi$
$727$	53.9256i	1.99999i	−0.00345931	+	0.999994i	$0.501101\pi$
$728$	0	0
$729$	10.4293	0.386269
$730$	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
$737$	0	0
$738$	0	0
$739$	0	0		−	1.00000i	$-0.5\pi$
$739$	0	0			1.00000i	$0.5\pi$
$740$	0	0
$741$	−60.7177	−2.23052
$742$	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
$749$	−4.07389	−0.148857
$750$	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	0	1.00000			$0$
$757$	0	0	−1.00000			$\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	8.93535	0.323906	0.161953	−	0.986798i	$-0.448221\pi$
$761$	8.93535	0.323906	0.161953	+	0.986798i	$0.448221\pi$
$762$	0	0
$763$	− 34.0243i	− 1.23176i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	89.0853i	3.21668i
$768$	0	0
$769$	−38.7830	−1.39855	−0.699276	−	0.714852i	$-0.746494\pi$
$769$	−38.7830	−1.39855	−0.699276	+	0.714852i	$0.746494\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	1.16230	0.0418050	0.0209025	−	0.999782i	$-0.493346\pi$
$773$	1.16230	0.0418050	0.0209025	+	0.999782i	$0.493346\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	89.8213	3.22232
$778$	0	0
$779$	0	0
$780$	0	0
$781$	0	0
$782$	0	0
$783$	20.2905i	0.725123i
$784$	0	0
$785$	0	0
$786$	0	0
$787$	− 46.3452i	− 1.65203i	−0.563650	−	0.826014i	$-0.690603\pi$
$787$	− 46.3452i	− 1.65203i	0.563650	−	0.826014i	$-0.309397\pi$
$788$	0	0
$789$	68.1701	2.42692
$790$	0	0
$791$	0	0
$792$	0	0
$793$	0	0
$794$	0	0
$795$	0	0
$796$	0	0
$797$	20.1872	0.715067	0.357534	−	0.933900i	$-0.383618\pi$
$797$	20.1872	0.715067	0.357534	+	0.933900i	$0.383618\pi$
$798$	0	0
$799$	49.3914i	1.74734i
$800$	0	0
$801$	0	0
$802$	0	0
$803$	0	0
$804$	0	0
$805$	0	0
$806$	0	0
$807$	− 59.4293i	− 2.09201i
$808$	0	0
$809$	−46.1784	−1.62355	−0.811773	−	0.583972i	$-0.801498\pi$
$809$	−46.1784	−1.62355	−0.811773	+	0.583972i	$0.801498\pi$
$810$	0	0
$811$	− 42.2236i	− 1.48267i	−0.671134	−	0.741336i	$-0.734193\pi$
$811$	− 42.2236i	− 1.48267i	0.671134	−	0.741336i	$-0.265807\pi$
$812$	0	0
$813$	−1.10155	−0.0386330
$814$	0	0
$815$	0	0
$816$	0	0
$817$	0	0
$818$	0	0
$819$	60.1346i	2.10127i
$820$	0	0
$821$	0	0	1.00000			$0$
$821$	0	0	−1.00000			$\pi$
$822$	0	0
$823$	39.8382i	1.38867i	0.719651	+	0.694336i	$0.244302\pi$
$823$	39.8382i	1.38867i	−0.719651	+	0.694336i	$0.755698\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	37.6790i	1.31023i	0.755531	+	0.655113i	$0.227379\pi$
$827$	37.6790i	1.31023i	−0.755531	+	0.655113i	$0.772621\pi$
$828$	0	0
$829$	56.5446	1.96387	0.981937	−	0.189209i	$-0.0605923\pi$
$829$	56.5446	1.96387	0.981937	+	0.189209i	$0.0605923\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	−111.615	−3.86725
$834$	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
$841$	84.9503	2.92932
$842$	0	0
$843$	0	0
$844$	0	0
$845$	0	0
$846$	0	0
$847$	49.8761i	1.71376i
$848$	0	0
$849$	0	0
$850$	0	0
$851$	− 24.9434i	− 0.855050i
$852$	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
$859$	0	0		−	1.00000i	$-0.5\pi$
$859$	0	0			1.00000i	$0.5\pi$
$860$	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
$867$	− 115.353i	− 3.91759i
$868$	0	0
$869$	0	0
$870$	0	0
$871$	33.3670i	1.13060i
$872$	0	0
$873$	0	0
$874$	0	0
$875$	0	0
$876$	0	0
$877$	57.9184	1.95577	0.977883	−	0.209153i	$-0.0670706\pi$
$877$	57.9184	1.95577	0.977883	+	0.209153i	$0.0670706\pi$
$878$	0	0
$879$	− 37.7055i	− 1.27177i
$880$	0	0
$881$	−54.0000	−1.81931	−0.909653	−	0.415369i	$-0.863653\pi$
$881$	−54.0000	−1.81931	−0.909653	+	0.415369i	$0.863653\pi$
$882$	0	0
$883$	0	0		−	1.00000i	$-0.5\pi$
$883$	0	0			1.00000i	$0.5\pi$
$884$	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
$893$	26.1534	0.875190
$894$	0	0
$895$	0	0
$896$	0	0
$897$	39.8555	1.33073
$898$	0	0
$899$	0	0
$900$	0	0
$901$	−24.3603	−0.811560
$902$	0	0
$903$	0	0
$904$	0	0
$905$	0	0
$906$	0	0
$907$	55.8576i	1.85472i	0.374167	+	0.927361i	$0.377929\pi$
$907$	55.8576i	1.85472i	−0.374167	+	0.927361i	$0.622071\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	0	0		−	1.00000i	$-0.5\pi$
$911$	0	0			1.00000i	$0.5\pi$
$912$	0	0
$913$	0	0
$914$	0	0
$915$	0	0
$916$	0	0
$917$	0	0
$918$	0	0
$919$	− 48.9066i	− 1.61328i	−0.591043	−	0.806640i	$-0.701284\pi$
$919$	− 48.9066i	− 1.61328i	0.591043	−	0.806640i	$-0.298716\pi$
$920$	0	0
$921$	−19.8098	−0.652754
$922$	0	0
$923$	0	0
$924$	0	0
$925$	−43.5890	−1.43320
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−25.3991	−0.833319	−0.416659	−	0.909063i	$-0.636799\pi$
$929$	−25.3991	−0.833319	−0.416659	+	0.909063i	$0.636799\pi$
$930$	0	0
$931$	59.1019i	1.93699i
$932$	0	0
$933$	55.3176	1.81102
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−16.5968	−0.542195	−0.271097	−	0.962552i	$-0.587387\pi$
$937$	−16.5968	−0.542195	−0.271097	+	0.962552i	$0.587387\pi$
$938$	0	0
$939$	46.9152i	1.53102i
$940$	0	0
$941$	−61.0892	−1.99145	−0.995726	−	0.0923562i	$-0.970560\pi$
$941$	−61.0892	−1.99145	−0.995726	+	0.0923562i	$0.970560\pi$
$942$	0	0
$943$	0	0
$944$	0	0
$945$	0	0
$946$	0	0
$947$	0	0		−	1.00000i	$-0.5\pi$
$947$	0	0			1.00000i	$0.5\pi$
$948$	0	0
$949$	95.7965	3.10969
$950$	0	0
$951$	59.3210i	1.92361i
$952$	0	0
$953$	0	0	1.00000			$0$
$953$	0	0	−1.00000			$\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	0	0
$958$	0	0
$959$	63.2715i	2.04314i
$960$	0	0
$961$	−31.0000	−1.00000
$962$	0	0
$963$	1.94387i	0.0626404i
$964$	0	0
$965$	0	0
$966$	0	0
$967$	− 50.0000i	− 1.60789i	−0.594703	−	0.803946i	$-0.702730\pi$
$967$	− 50.0000i	− 1.60789i	0.594703	−	0.803946i	$-0.297270\pi$
$968$	0	0
$969$	−81.5359	−2.61931
$970$	0	0
$971$	26.1534i	0.839302i	0.907685	+	0.419651i	$0.137848\pi$
$971$	26.1534i	0.839302i	−0.907685	+	0.419651i	$0.862152\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	− 69.6480i	− 2.23052i
$976$	0	0
$977$	0	0	1.00000			$0$
$977$	0	0	−1.00000			$\pi$
$978$	0	0
$979$	0	0
$980$	0	0
$981$	−16.2348	−0.518338
$982$	0	0
$983$	0	0		−	1.00000i	$-0.5\pi$
$983$	0	0			1.00000i	$0.5\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	− 61.8192i	− 1.96773i
$988$	0	0
$989$	0	0
$990$	0	0
$991$	0	0		−	1.00000i	$-0.5\pi$
$991$	0	0			1.00000i	$0.5\pi$
$992$	0	0
$993$	74.3307	2.35882
$994$	0	0
$995$	0	0
$996$	0	0
$997$	0	0	1.00000			$0$
$997$	0	0	−1.00000			$\pi$
$998$	0	0
$999$	16.5707i	0.524275i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1216.2.b.f.607.3	✓	12
4.3	odd	2	inner	1216.2.b.f.607.10	yes	12
8.3	odd	2	inner	1216.2.b.f.607.4	yes	12
8.5	even	2	inner	1216.2.b.f.607.9	yes	12
19.18	odd	2	inner	1216.2.b.f.607.9	yes	12
76.75	even	2	inner	1216.2.b.f.607.4	yes	12
152.37	odd	2	CM	1216.2.b.f.607.3	✓	12
152.75	even	2	inner	1216.2.b.f.607.10	yes	12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1216.2.b.f.607.3	✓	12	1.1	even	1	trivial
1216.2.b.f.607.3	✓	12	152.37	odd	2	CM
1216.2.b.f.607.4	yes	12	8.3	odd	2	inner
1216.2.b.f.607.4	yes	12	76.75	even	2	inner
1216.2.b.f.607.9	yes	12	8.5	even	2	inner
1216.2.b.f.607.9	yes	12	19.18	odd	2	inner
1216.2.b.f.607.10	yes	12	4.3	odd	2	inner
1216.2.b.f.607.10	yes	12	152.75	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)

\(f(q)\)	\(=\)	\(q-2.27234i q^{3} -4.53419i q^{7} -2.16351 q^{9} +O(q^{10})\)	\(q-2.27234i q^{3} -4.53419i q^{7} -2.16351 q^{9} +6.13008 q^{13} +8.23189 q^{17} -4.35890i q^{19} -10.3032 q^{21} +2.86121i q^{23} +5.00000 q^{25} -1.90079i q^{27} -10.6747 q^{29} -8.71780 q^{37} -13.9296i q^{39} +6.00000i q^{47} -13.5589 q^{49} -18.7056i q^{51} -2.95926 q^{53} -9.90488 q^{57} +14.5325i q^{59} +9.80976i q^{63} +5.44315i q^{67} +6.50162 q^{69} +15.6273 q^{73} -11.3617i q^{75} -10.8098 q^{81} +24.2566i q^{87} -27.7950i q^{91} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 36 q^{9}+O(q^{10}) \) 12 * q - 36 * q^9	\( 12 q - 36 q^{9} + 60 q^{25} - 84 q^{49} + 108 q^{81}+O(q^{100}) \) 12 * q - 36 * q^9 + 60 * q^25 - 84 * q^49 + 108 * q^81

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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