LMFDB - Embedded newform 1116.2.g.c.991.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1116,2,Mod(991,1116)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1116, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 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("1116.991");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1116 = 2^{2} \cdot 3^{2} \cdot 31 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1116.g (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:	$8.91130486557$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\sqrt{-2}, \sqrt{31})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 32x^{2} + 225 $ x^4 + 32*x^2 + 225
Coefficient ring:	$\Z[a_1, \ldots, a_{31}]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label			991.1
Root			$4.64411i$ of defining polynomial
Character	$\chi$	$=$	1116.991
Dual form			1116.2.g.c.991.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.41421i q^{2} -2.00000 q^{4} +2.82843i q^{8} +O(q^{10})$	$q-1.41421i q^{2} -2.00000 q^{4} +2.82843i q^{8} +4.00000 q^{16} -7.87401i q^{17} -5.00000 q^{25} -7.87401i q^{29} -5.56776 q^{31} -5.65685i q^{32} -11.1355 q^{34} -11.1355 q^{43} +1.41421i q^{47} +7.00000 q^{49} +7.07107i q^{50} -7.87401i q^{53} -11.1355 q^{58} -7.07107i q^{59} +7.87401i q^{62} -8.00000 q^{64} +15.7480i q^{68} +9.89949i q^{71} -11.1355 q^{79} +15.7480i q^{86} -7.87401i q^{89} +2.00000 q^{94} -4.00000 q^{97} -9.89949i q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 8 q^{4}+O(q^{10}) $ 4 * q - 8 * q^4	$ 4 q - 8 q^{4} + 16 q^{16} - 20 q^{25} + 28 q^{49} - 32 q^{64} + 8 q^{94} - 16 q^{97}+O(q^{100}) $ 4 * q - 8 * q^4 + 16 * q^16 - 20 * q^25 + 28 * q^49 - 32 * q^64 + 8 * q^94 - 16 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1116\mathbb{Z}\right)^\times$.

$n$	$497$	$559$	$685$
$\chi(n)$	$1$	$-1$	$-1$

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 $

$2$	− 1.41421i	− 1.00000i
$2$	− 1.41421i	− 1.00000i
$3$	0	0
$3$	0	0
$4$	−2.00000	−1.00000
$5$	0	0		−	1.00000i	$-0.5\pi$
$5$	0	0			1.00000i	$0.5\pi$
$6$	0	0
$7$	0	0	1.00000			$0$
$7$	0	0	−1.00000			$\pi$
$8$	2.82843i	1.00000i
$9$	0	0
$10$	0	0
$11$	0	0		−	1.00000i	$-0.5\pi$
$11$	0	0			1.00000i	$0.5\pi$
$12$	0	0
$13$	0	0	1.00000			$0$
$13$	0	0	−1.00000			$\pi$
$14$	0	0
$15$	0	0
$16$	4.00000	1.00000
$17$	− 7.87401i	− 1.90973i	−0.297044	−	0.954864i	$-0.596001\pi$
$17$	− 7.87401i	− 1.90973i	0.297044	−	0.954864i	$-0.403999\pi$
$18$	0	0
$19$	0	0	1.00000			$0$
$19$	0	0	−1.00000			$\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	0	0		−	1.00000i	$-0.5\pi$
$23$	0	0			1.00000i	$0.5\pi$
$24$	0	0
$25$	−5.00000	−1.00000
$26$	0	0
$27$	0	0
$28$	0	0
$29$	− 7.87401i	− 1.46217i	−0.682288	−	0.731083i	$-0.739015\pi$
$29$	− 7.87401i	− 1.46217i	0.682288	−	0.731083i	$-0.260985\pi$
$30$	0	0
$31$	−5.56776	−1.00000
$31$	−5.56776	−1.00000
$32$	− 5.65685i	− 1.00000i
$33$	0	0
$34$	−11.1355	−1.90973
$35$	0	0
$36$	0	0
$37$	0	0	1.00000			$0$
$37$	0	0	−1.00000			$\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	0	0		−	1.00000i	$-0.5\pi$
$41$	0	0			1.00000i	$0.5\pi$
$42$	0	0
$43$	−11.1355	−1.69815	−0.849076	−	0.528271i	$-0.822841\pi$
$43$	−11.1355	−1.69815	−0.849076	+	0.528271i	$0.822841\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	1.41421i	0.206284i	0.994667	+	0.103142i	$0.0328896\pi$
$47$	1.41421i	0.206284i	−0.994667	+	0.103142i	$0.967110\pi$
$48$	0	0
$49$	7.00000	1.00000
$50$	7.07107i	1.00000i
$51$	0	0
$52$	0	0
$53$	− 7.87401i	− 1.08158i	−0.841158	−	0.540789i	$-0.818126\pi$
$53$	− 7.87401i	− 1.08158i	0.841158	−	0.540789i	$-0.181874\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	0	0
$58$	−11.1355	−1.46217
$59$	− 7.07107i	− 0.920575i	−0.887770	−	0.460287i	$-0.847746\pi$
$59$	− 7.07107i	− 0.920575i	0.887770	−	0.460287i	$-0.152254\pi$
$60$	0	0
$61$	0	0	1.00000			$0$
$61$	0	0	−1.00000			$\pi$
$62$	7.87401i	1.00000i
$63$	0	0
$64$	−8.00000	−1.00000
$65$	0	0
$66$	0	0
$67$	0	0	1.00000			$0$
$67$	0	0	−1.00000			$\pi$
$68$	15.7480i	1.90973i
$69$	0	0
$70$	0	0
$71$	9.89949i	1.17485i	0.809277	+	0.587427i	$0.199859\pi$
$71$	9.89949i	1.17485i	−0.809277	+	0.587427i	$0.800141\pi$
$72$	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
$79$	−11.1355	−1.25284	−0.626422	−	0.779484i	$-0.715481\pi$
$79$	−11.1355	−1.25284	−0.626422	+	0.779484i	$0.715481\pi$
$80$	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
$86$	15.7480i	1.69815i
$87$	0	0
$88$	0	0
$89$	− 7.87401i	− 0.834643i	−0.908759	−	0.417322i	$-0.862969\pi$
$89$	− 7.87401i	− 0.834643i	0.908759	−	0.417322i	$-0.137031\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	0	0
$94$	2.00000	0.206284
$95$	0	0
$96$	0	0
$97$	−4.00000	−0.406138	−0.203069	−	0.979164i	$-0.565092\pi$
$97$	−4.00000	−0.406138	−0.203069	+	0.979164i	$0.565092\pi$
$98$	− 9.89949i	− 1.00000i
$99$	0	0
$100$	10.0000	1.00000
$101$	0	0		−	1.00000i	$-0.5\pi$
$101$	0	0			1.00000i	$0.5\pi$
$102$	0	0
$103$	0	0	1.00000			$0$
$103$	0	0	−1.00000			$\pi$
$104$	0	0
$105$	0	0
$106$	−11.1355	−1.08158
$107$	− 15.5563i	− 1.50389i	−0.659226	−	0.751945i	$-0.729116\pi$
$107$	− 15.5563i	− 1.50389i	0.659226	−	0.751945i	$-0.270884\pi$
$108$	0	0
$109$	8.00000	0.766261	0.383131	−	0.923694i	$-0.374846\pi$
$109$	8.00000	0.766261	0.383131	+	0.923694i	$0.374846\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	0	0		−	1.00000i	$-0.5\pi$
$113$	0	0			1.00000i	$0.5\pi$
$114$	0	0
$115$	0	0
$116$	15.7480i	1.46217i
$117$	0	0
$118$	−10.0000	−0.920575
$119$	0	0
$120$	0	0
$121$	−11.0000	−1.00000
$122$	0	0
$123$	0	0
$124$	11.1355	1.00000
$125$	0	0
$126$	0	0
$127$	22.2711	1.97624	0.988118	−	0.153695i	$-0.0491172\pi$
$127$	22.2711	1.97624	0.988118	+	0.153695i	$0.0491172\pi$
$128$	11.3137i	1.00000i
$129$	0	0
$130$	0	0
$131$	18.3848i	1.60629i	0.595787	+	0.803143i	$0.296840\pi$
$131$	18.3848i	1.60629i	−0.595787	+	0.803143i	$0.703160\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	0	0
$136$	22.2711	1.90973
$137$	− 7.87401i	− 0.672722i	−0.941733	−	0.336361i	$-0.890804\pi$
$137$	− 7.87401i	− 0.672722i	0.941733	−	0.336361i	$-0.109196\pi$
$138$	0	0
$139$	−11.1355	−0.944503	−0.472252	−	0.881464i	$-0.656559\pi$
$139$	−11.1355	−0.944503	−0.472252	+	0.881464i	$0.656559\pi$
$140$	0	0
$141$	0	0
$142$	14.0000	1.17485
$143$	0	0
$144$	0	0
$145$	0	0
$146$	0	0
$147$	0	0
$148$	0	0
$149$	0	0		−	1.00000i	$-0.5\pi$
$149$	0	0			1.00000i	$0.5\pi$
$150$	0	0
$151$	22.2711	1.81239	0.906196	−	0.422857i	$-0.138973\pi$
$151$	22.2711	1.81239	0.906196	+	0.422857i	$0.138973\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	0	0
$156$	0	0
$157$	−16.0000	−1.27694	−0.638470	−	0.769647i	$-0.720432\pi$
$157$	−16.0000	−1.27694	−0.638470	+	0.769647i	$0.720432\pi$
$158$	15.7480i	1.25284i
$159$	0	0
$160$	0	0
$161$	0	0
$162$	0	0
$163$	0	0	1.00000			$0$
$163$	0	0	−1.00000			$\pi$
$164$	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
$172$	22.2711	1.69815
$173$	0	0		−	1.00000i	$-0.5\pi$
$173$	0	0			1.00000i	$0.5\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	0	0
$178$	−11.1355	−0.834643
$179$	0	0		−	1.00000i	$-0.5\pi$
$179$	0	0			1.00000i	$0.5\pi$
$180$	0	0
$181$	0	0	1.00000			$0$
$181$	0	0	−1.00000			$\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	0	0
$186$	0	0
$187$	0	0
$188$	− 2.82843i	− 0.206284i
$189$	0	0
$190$	0	0
$191$	− 24.0416i	− 1.73959i	−0.493412	−	0.869796i	$-0.664251\pi$
$191$	− 24.0416i	− 1.73959i	0.493412	−	0.869796i	$-0.335749\pi$
$192$	0	0
$193$	20.0000	1.43963	0.719816	−	0.694165i	$-0.244226\pi$
$193$	20.0000	1.43963	0.719816	+	0.694165i	$0.244226\pi$
$194$	5.65685i	0.406138i
$195$	0	0
$196$	−14.0000	−1.00000
$197$	− 7.87401i	− 0.561000i	−0.959854	−	0.280500i	$-0.909500\pi$
$197$	− 7.87401i	− 0.561000i	0.959854	−	0.280500i	$-0.0905002\pi$
$198$	0	0
$199$	22.2711	1.57875	0.789377	−	0.613909i	$-0.210404\pi$
$199$	22.2711	1.57875	0.789377	+	0.613909i	$0.210404\pi$
$200$	− 14.1421i	− 1.00000i
$201$	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	1.00000			$0$
$211$	0	0	−1.00000			$\pi$
$212$	15.7480i	1.08158i
$213$	0	0
$214$	−22.0000	−1.50389
$215$	0	0
$216$	0	0
$217$	0	0
$218$	− 11.3137i	− 0.766261i
$219$	0	0
$220$	0	0
$221$	0	0
$222$	0	0
$223$	−11.1355	−0.745690	−0.372845	−	0.927894i	$-0.621618\pi$
$223$	−11.1355	−0.745690	−0.372845	+	0.927894i	$0.621618\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	26.8701i	1.78343i	0.452599	+	0.891714i	$0.350497\pi$
$227$	26.8701i	1.78343i	−0.452599	+	0.891714i	$0.649503\pi$
$228$	0	0
$229$	0	0	1.00000			$0$
$229$	0	0	−1.00000			$\pi$
$230$	0	0
$231$	0	0
$232$	22.2711	1.46217
$233$	0	0		−	1.00000i	$-0.5\pi$
$233$	0	0			1.00000i	$0.5\pi$
$234$	0	0
$235$	0	0
$236$	14.1421i	0.920575i
$237$	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			$\pi$
$242$	15.5563i	1.00000i
$243$	0	0
$244$	0	0
$245$	0	0
$246$	0	0
$247$	0	0
$248$	− 15.7480i	− 1.00000i
$249$	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
$254$	− 31.4960i	− 1.97624i
$255$	0	0
$256$	16.0000	1.00000
$257$	0	0		−	1.00000i	$-0.5\pi$
$257$	0	0			1.00000i	$0.5\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	0	0
$262$	26.0000	1.60629
$263$	0	0		−	1.00000i	$-0.5\pi$
$263$	0	0			1.00000i	$0.5\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	0	0
$268$	0	0
$269$	− 7.87401i	− 0.480087i	−0.970762	−	0.240043i	$-0.922838\pi$
$269$	− 7.87401i	− 0.480087i	0.970762	−	0.240043i	$-0.0771617\pi$
$270$	0	0
$271$	22.2711	1.35287	0.676435	−	0.736502i	$-0.263524\pi$
$271$	22.2711	1.35287	0.676435	+	0.736502i	$0.263524\pi$
$272$	− 31.4960i	− 1.90973i
$273$	0	0
$274$	−11.1355	−0.672722
$275$	0	0
$276$	0	0
$277$	0	0	1.00000			$0$
$277$	0	0	−1.00000			$\pi$
$278$	15.7480i	0.944503i
$279$	0	0
$280$	0	0
$281$	0	0		−	1.00000i	$-0.5\pi$
$281$	0	0			1.00000i	$0.5\pi$
$282$	0	0
$283$	0	0	1.00000			$0$
$283$	0	0	−1.00000			$\pi$
$284$	− 19.7990i	− 1.17485i
$285$	0	0
$286$	0	0
$287$	0	0
$288$	0	0
$289$	−45.0000	−2.64706
$290$	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
$301$	0	0
$302$	− 31.4960i	− 1.81239i
$303$	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
$309$	0	0
$310$	0	0
$311$	− 32.5269i	− 1.84443i	−0.386675	−	0.922216i	$-0.626377\pi$
$311$	− 32.5269i	− 1.84443i	0.386675	−	0.922216i	$-0.373623\pi$
$312$	0	0
$313$	0	0	1.00000			$0$
$313$	0	0	−1.00000			$\pi$
$314$	22.6274i	1.27694i
$315$	0	0
$316$	22.2711	1.25284
$317$	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
$323$	0	0
$324$	0	0
$325$	0	0
$326$	0	0
$327$	0	0
$328$	0	0
$329$	0	0
$330$	0	0
$331$	−11.1355	−0.612064	−0.306032	−	0.952021i	$-0.599001\pi$
$331$	−11.1355	−0.612064	−0.306032	+	0.952021i	$0.599001\pi$
$332$	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			$\pi$
$338$	− 18.3848i	− 1.00000i
$339$	0	0
$340$	0	0
$341$	0	0
$342$	0	0
$343$	0	0
$344$	− 31.4960i	− 1.69815i
$345$	0	0
$346$	0	0
$347$	0	0		−	1.00000i	$-0.5\pi$
$347$	0	0			1.00000i	$0.5\pi$
$348$	0	0
$349$	32.0000	1.71292	0.856460	−	0.516213i	$-0.172659\pi$
$349$	32.0000	1.71292	0.856460	+	0.516213i	$0.172659\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	− 7.87401i	− 0.419091i	−0.977799	−	0.209546i	$-0.932802\pi$
$353$	− 7.87401i	− 0.419091i	0.977799	−	0.209546i	$-0.0671985\pi$
$354$	0	0
$355$	0	0
$356$	15.7480i	0.834643i
$357$	0	0
$358$	0	0
$359$	35.3553i	1.86598i	0.359898	+	0.932992i	$0.382812\pi$
$359$	35.3553i	1.86598i	−0.359898	+	0.932992i	$0.617188\pi$
$360$	0	0
$361$	19.0000	1.00000
$362$	0	0
$363$	0	0
$364$	0	0
$365$	0	0
$366$	0	0
$367$	22.2711	1.16254	0.581270	−	0.813711i	$-0.302556\pi$
$367$	22.2711	1.16254	0.581270	+	0.813711i	$0.302556\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	0	0
$372$	0	0
$373$	2.00000	0.103556	0.0517780	−	0.998659i	$-0.483511\pi$
$373$	2.00000	0.103556	0.0517780	+	0.998659i	$0.483511\pi$
$374$	0	0
$375$	0	0
$376$	−4.00000	−0.206284
$377$	0	0
$378$	0	0
$379$	0	0	1.00000			$0$
$379$	0	0	−1.00000			$\pi$
$380$	0	0
$381$	0	0
$382$	−34.0000	−1.73959
$383$	0	0		−	1.00000i	$-0.5\pi$
$383$	0	0			1.00000i	$0.5\pi$
$384$	0	0
$385$	0	0
$386$	− 28.2843i	− 1.43963i
$387$	0	0
$388$	8.00000	0.406138
$389$	39.3700i	1.99614i	0.0620970	+	0.998070i	$0.480221\pi$
$389$	39.3700i	1.99614i	−0.0620970	+	0.998070i	$0.519779\pi$
$390$	0	0
$391$	0	0
$392$	19.7990i	1.00000i
$393$	0	0
$394$	−11.1355	−0.561000
$395$	0	0
$396$	0	0
$397$	−10.0000	−0.501886	−0.250943	−	0.968002i	$-0.580741\pi$
$397$	−10.0000	−0.501886	−0.250943	+	0.968002i	$0.580741\pi$
$398$	− 31.4960i	− 1.57875i
$399$	0	0
$400$	−20.0000	−1.00000
$401$	39.3700i	1.96605i	0.183483	+	0.983023i	$0.441263\pi$
$401$	39.3700i	1.96605i	−0.183483	+	0.983023i	$0.558737\pi$
$402$	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
$411$	0	0
$412$	0	0
$413$	0	0
$414$	0	0
$415$	0	0
$416$	0	0
$417$	0	0
$418$	0	0
$419$	1.41421i	0.0690889i	0.999403	+	0.0345444i	$0.0109980\pi$
$419$	1.41421i	0.0690889i	−0.999403	+	0.0345444i	$0.989002\pi$
$420$	0	0
$421$	14.0000	0.682318	0.341159	−	0.940006i	$-0.389181\pi$
$421$	14.0000	0.682318	0.341159	+	0.940006i	$0.389181\pi$
$422$	0	0
$423$	0	0
$424$	22.2711	1.08158
$425$	39.3700i	1.90973i
$426$	0	0
$427$	0	0
$428$	31.1127i	1.50389i
$429$	0	0
$430$	0	0
$431$	− 7.07107i	− 0.340601i	−0.985392	−	0.170301i	$-0.945526\pi$
$431$	− 7.07107i	− 0.340601i	0.985392	−	0.170301i	$-0.0544739\pi$
$432$	0	0
$433$	0	0	1.00000			$0$
$433$	0	0	−1.00000			$\pi$
$434$	0	0
$435$	0	0
$436$	−16.0000	−0.766261
$437$	0	0
$438$	0	0
$439$	0	0	1.00000			$0$
$439$	0	0	−1.00000			$\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	9.89949i	0.470339i	0.971954	+	0.235170i	$0.0755646\pi$
$443$	9.89949i	0.470339i	−0.971954	+	0.235170i	$0.924435\pi$
$444$	0	0
$445$	0	0
$446$	15.7480i	0.745690i
$447$	0	0
$448$	0	0
$449$	− 7.87401i	− 0.371597i	−0.982588	−	0.185799i	$-0.940513\pi$
$449$	− 7.87401i	− 0.371597i	0.982588	−	0.185799i	$-0.0594872\pi$
$450$	0	0
$451$	0	0
$452$	0	0
$453$	0	0
$454$	38.0000	1.78343
$455$	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
$461$	39.3700i	1.83365i	0.399294	+	0.916823i	$0.369255\pi$
$461$	39.3700i	1.83365i	−0.399294	+	0.916823i	$0.630745\pi$
$462$	0	0
$463$	−11.1355	−0.517512	−0.258756	−	0.965943i	$-0.583313\pi$
$463$	−11.1355	−0.517512	−0.258756	+	0.965943i	$0.583313\pi$
$464$	− 31.4960i	− 1.46217i
$465$	0	0
$466$	0	0
$467$	− 41.0122i	− 1.89782i	−0.315550	−	0.948909i	$-0.602189\pi$
$467$	− 41.0122i	− 1.89782i	0.315550	−	0.948909i	$-0.397811\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	0	0
$472$	20.0000	0.920575
$473$	0	0
$474$	0	0
$475$	0	0
$476$	0	0
$477$	0	0
$478$	0	0
$479$	− 15.5563i	− 0.710788i	−0.934717	−	0.355394i	$-0.884347\pi$
$479$	− 15.5563i	− 0.710788i	0.934717	−	0.355394i	$-0.115653\pi$
$480$	0	0
$481$	0	0
$482$	0	0
$483$	0	0
$484$	22.0000	1.00000
$485$	0	0
$486$	0	0
$487$	22.2711	1.00920	0.504599	−	0.863354i	$-0.331640\pi$
$487$	22.2711	1.00920	0.504599	+	0.863354i	$0.331640\pi$
$488$	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
$493$	−62.0000	−2.79234
$494$	0	0
$495$	0	0
$496$	−22.2711	−1.00000
$497$	0	0
$498$	0	0
$499$	−44.5421	−1.99398	−0.996989	−	0.0775372i	$-0.975294\pi$
$499$	−44.5421	−1.99398	−0.996989	+	0.0775372i	$0.975294\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	18.3848i	0.819737i	0.912145	+	0.409868i	$0.134425\pi$
$503$	18.3848i	0.819737i	−0.912145	+	0.409868i	$0.865575\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	0	0
$508$	−44.5421	−1.97624
$509$	39.3700i	1.74505i	0.488573	+	0.872523i	$0.337518\pi$
$509$	39.3700i	1.74505i	−0.488573	+	0.872523i	$0.662482\pi$
$510$	0	0
$511$	0	0
$512$	− 22.6274i	− 1.00000i
$513$	0	0
$514$	0	0
$515$	0	0
$516$	0	0
$517$	0	0
$518$	0	0
$519$	0	0
$520$	0	0
$521$	0	0		−	1.00000i	$-0.5\pi$
$521$	0	0			1.00000i	$0.5\pi$
$522$	0	0
$523$	−44.5421	−1.94769	−0.973845	−	0.227212i	$-0.927039\pi$
$523$	−44.5421	−1.94769	−0.973845	+	0.227212i	$0.927039\pi$
$524$	− 36.7696i	− 1.60629i
$525$	0	0
$526$	0	0
$527$	43.8406i	1.90973i
$528$	0	0
$529$	−23.0000	−1.00000
$530$	0	0
$531$	0	0
$532$	0	0
$533$	0	0
$534$	0	0
$535$	0	0
$536$	0	0
$537$	0	0
$538$	−11.1355	−0.480087
$539$	0	0
$540$	0	0
$541$	26.0000	1.11783	0.558914	−	0.829226i	$-0.311218\pi$
$541$	26.0000	1.11783	0.558914	+	0.829226i	$0.311218\pi$
$542$	− 31.4960i	− 1.35287i
$543$	0	0
$544$	−44.5421	−1.90973
$545$	0	0
$546$	0	0
$547$	0	0	1.00000			$0$
$547$	0	0	−1.00000			$\pi$
$548$	15.7480i	0.672722i
$549$	0	0
$550$	0	0
$551$	0	0
$552$	0	0
$553$	0	0
$554$	0	0
$555$	0	0
$556$	22.2711	0.944503
$557$	− 7.87401i	− 0.333632i	−0.985988	−	0.166816i	$-0.946651\pi$
$557$	− 7.87401i	− 0.333632i	0.985988	−	0.166816i	$-0.0533487\pi$
$558$	0	0
$559$	0	0
$560$	0	0
$561$	0	0
$562$	0	0
$563$	− 24.0416i	− 1.01323i	−0.862171	−	0.506617i	$-0.830896\pi$
$563$	− 24.0416i	− 1.01323i	0.862171	−	0.506617i	$-0.169104\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	0	0
$568$	−28.0000	−1.17485
$569$	39.3700i	1.65048i	0.564784	+	0.825239i	$0.308960\pi$
$569$	39.3700i	1.65048i	−0.564784	+	0.825239i	$0.691040\pi$
$570$	0	0
$571$	−44.5421	−1.86403	−0.932015	−	0.362420i	$-0.881950\pi$
$571$	−44.5421	−1.86403	−0.932015	+	0.362420i	$0.881950\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	0	0
$576$	0	0
$577$	44.0000	1.83174	0.915872	−	0.401470i	$-0.131501\pi$
$577$	44.0000	1.83174	0.915872	+	0.401470i	$0.131501\pi$
$578$	63.6396i	2.64706i
$579$	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
$593$	0	0		−	1.00000i	$-0.5\pi$
$593$	0	0			1.00000i	$0.5\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	0	0
$598$	0	0
$599$	26.8701i	1.09788i	0.835861	+	0.548940i	$0.184969\pi$
$599$	26.8701i	1.09788i	−0.835861	+	0.548940i	$0.815031\pi$
$600$	0	0
$601$	0	0	1.00000			$0$
$601$	0	0	−1.00000			$\pi$
$602$	0	0
$603$	0	0
$604$	−44.5421	−1.81239
$605$	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
$617$	0	0		−	1.00000i	$-0.5\pi$
$617$	0	0			1.00000i	$0.5\pi$
$618$	0	0
$619$	−11.1355	−0.447575	−0.223787	−	0.974638i	$-0.571842\pi$
$619$	−11.1355	−0.447575	−0.223787	+	0.974638i	$0.571842\pi$
$620$	0	0
$621$	0	0
$622$	−46.0000	−1.84443
$623$	0	0
$624$	0	0
$625$	25.0000	1.00000
$626$	0	0
$627$	0	0
$628$	32.0000	1.27694
$629$	0	0
$630$	0	0
$631$	22.2711	0.886597	0.443298	−	0.896374i	$-0.353808\pi$
$631$	22.2711	0.886597	0.443298	+	0.896374i	$0.353808\pi$
$632$	− 31.4960i	− 1.25284i
$633$	0	0
$634$	0	0
$635$	0	0
$636$	0	0
$637$	0	0
$638$	0	0
$639$	0	0
$640$	0	0
$641$	39.3700i	1.55502i	0.628869	+	0.777512i	$0.283518\pi$
$641$	39.3700i	1.55502i	−0.628869	+	0.777512i	$0.716482\pi$
$642$	0	0
$643$	−44.5421	−1.75657	−0.878285	−	0.478138i	$-0.841312\pi$
$643$	−44.5421	−1.75657	−0.878285	+	0.478138i	$0.841312\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	0	0		−	1.00000i	$-0.5\pi$
$647$	0	0			1.00000i	$0.5\pi$
$648$	0	0
$649$	0	0
$650$	0	0
$651$	0	0
$652$	0	0
$653$	0	0		−	1.00000i	$-0.5\pi$
$653$	0	0			1.00000i	$0.5\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	0	0
$658$	0	0
$659$	− 49.4975i	− 1.92815i	−0.265634	−	0.964074i	$-0.585581\pi$
$659$	− 49.4975i	− 1.92815i	0.265634	−	0.964074i	$-0.414419\pi$
$660$	0	0
$661$	−34.0000	−1.32245	−0.661223	−	0.750189i	$-0.729962\pi$
$661$	−34.0000	−1.32245	−0.661223	+	0.750189i	$0.729962\pi$
$662$	15.7480i	0.612064i
$663$	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
$676$	−26.0000	−1.00000
$677$	− 7.87401i	− 0.302623i	−0.988486	−	0.151311i	$-0.951650\pi$
$677$	− 7.87401i	− 0.302623i	0.988486	−	0.151311i	$-0.0483496\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	0	0
$682$	0	0
$683$	− 32.5269i	− 1.24461i	−0.782776	−	0.622304i	$-0.786197\pi$
$683$	− 32.5269i	− 1.24461i	0.782776	−	0.622304i	$-0.213803\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	0	0
$688$	−44.5421	−1.69815
$689$	0	0
$690$	0	0
$691$	0	0	1.00000			$0$
$691$	0	0	−1.00000			$\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	0	0
$696$	0	0
$697$	0	0
$698$	− 45.2548i	− 1.71292i
$699$	0	0
$700$	0	0
$701$	0	0		−	1.00000i	$-0.5\pi$
$701$	0	0			1.00000i	$0.5\pi$
$702$	0	0
$703$	0	0
$704$	0	0
$705$	0	0
$706$	−11.1355	−0.419091
$707$	0	0
$708$	0	0
$709$	0	0	1.00000			$0$
$709$	0	0	−1.00000			$\pi$
$710$	0	0
$711$	0	0
$712$	22.2711	0.834643
$713$	0	0
$714$	0	0
$715$	0	0
$716$	0	0
$717$	0	0
$718$	50.0000	1.86598
$719$	0	0		−	1.00000i	$-0.5\pi$
$719$	0	0			1.00000i	$0.5\pi$
$720$	0	0
$721$	0	0
$722$	− 26.8701i	− 1.00000i
$723$	0	0
$724$	0	0
$725$	39.3700i	1.46217i
$726$	0	0
$727$	0	0	1.00000			$0$
$727$	0	0	−1.00000			$\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	87.6812i	3.24301i
$732$	0	0
$733$	38.0000	1.40356	0.701781	−	0.712393i	$-0.252388\pi$
$733$	38.0000	1.40356	0.701781	+	0.712393i	$0.252388\pi$
$734$	− 31.4960i	− 1.16254i
$735$	0	0
$736$	0	0
$737$	0	0
$738$	0	0
$739$	−44.5421	−1.63851	−0.819254	−	0.573431i	$-0.805612\pi$
$739$	−44.5421	−1.63851	−0.819254	+	0.573431i	$0.805612\pi$
$740$	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
$746$	− 2.82843i	− 0.103556i
$747$	0	0
$748$	0	0
$749$	0	0
$750$	0	0
$751$	0	0	1.00000			$0$
$751$	0	0	−1.00000			$\pi$
$752$	5.65685i	0.206284i
$753$	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
$761$	− 55.1181i	− 1.99803i	−0.0443970	−	0.999014i	$-0.514137\pi$
$761$	− 55.1181i	− 1.99803i	0.0443970	−	0.999014i	$-0.485863\pi$
$762$	0	0
$763$	0	0
$764$	48.0833i	1.73959i
$765$	0	0
$766$	0	0
$767$	0	0
$768$	0	0
$769$	−52.0000	−1.87517	−0.937584	−	0.347759i	$-0.886943\pi$
$769$	−52.0000	−1.87517	−0.937584	+	0.347759i	$0.886943\pi$
$770$	0	0
$771$	0	0
$772$	−40.0000	−1.43963
$773$	− 55.1181i	− 1.98246i	−0.132153	−	0.991229i	$-0.542189\pi$
$773$	− 55.1181i	− 1.98246i	0.132153	−	0.991229i	$-0.457811\pi$
$774$	0	0
$775$	27.8388	1.00000
$776$	− 11.3137i	− 0.406138i
$777$	0	0
$778$	55.6776	1.99614
$779$	0	0
$780$	0	0
$781$	0	0
$782$	0	0
$783$	0	0
$784$	28.0000	1.00000
$785$	0	0
$786$	0	0
$787$	55.6776	1.98469	0.992347	−	0.123482i	$-0.0394061\pi$
$787$	55.6776	1.98469	0.992347	+	0.123482i	$0.0394061\pi$
$788$	15.7480i	0.561000i
$789$	0	0
$790$	0	0
$791$	0	0
$792$	0	0
$793$	0	0
$794$	14.1421i	0.501886i
$795$	0	0
$796$	−44.5421	−1.57875
$797$	− 55.1181i	− 1.95238i	−0.216913	−	0.976191i	$-0.569599\pi$
$797$	− 55.1181i	− 1.95238i	0.216913	−	0.976191i	$-0.430401\pi$
$798$	0	0
$799$	11.1355	0.393947
$800$	28.2843i	1.00000i
$801$	0	0
$802$	55.6776	1.96605
$803$	0	0
$804$	0	0
$805$	0	0
$806$	0	0
$807$	0	0
$808$	0	0
$809$	− 7.87401i	− 0.276835i	−0.990374	−	0.138418i	$-0.955798\pi$
$809$	− 7.87401i	− 0.276835i	0.990374	−	0.138418i	$-0.0442016\pi$
$810$	0	0
$811$	0	0	1.00000			$0$
$811$	0	0	−1.00000			$\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	0	0
$816$	0	0
$817$	0	0
$818$	0	0
$819$	0	0
$820$	0	0
$821$	39.3700i	1.37402i	0.726646	+	0.687012i	$0.241078\pi$
$821$	39.3700i	1.37402i	−0.726646	+	0.687012i	$0.758922\pi$
$822$	0	0
$823$	55.6776	1.94080	0.970400	−	0.241502i	$-0.0776400\pi$
$823$	55.6776	1.94080	0.970400	+	0.241502i	$0.0776400\pi$
$824$	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
$829$	0	0	1.00000			$0$
$829$	0	0	−1.00000			$\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	− 55.1181i	− 1.90973i
$834$	0	0
$835$	0	0
$836$	0	0
$837$	0	0
$838$	2.00000	0.0690889
$839$	− 41.0122i	− 1.41590i	−0.706263	−	0.707949i	$-0.749621\pi$
$839$	− 41.0122i	− 1.41590i	0.706263	−	0.707949i	$-0.250379\pi$
$840$	0	0
$841$	−33.0000	−1.13793
$842$	− 19.7990i	− 0.682318i
$843$	0	0
$844$	0	0
$845$	0	0
$846$	0	0
$847$	0	0
$848$	− 31.4960i	− 1.08158i
$849$	0	0
$850$	55.6776	1.90973
$851$	0	0
$852$	0	0
$853$	8.00000	0.273915	0.136957	−	0.990577i	$-0.456268\pi$
$853$	8.00000	0.273915	0.136957	+	0.990577i	$0.456268\pi$
$854$	0	0
$855$	0	0
$856$	44.0000	1.50389
$857$	0	0		−	1.00000i	$-0.5\pi$
$857$	0	0			1.00000i	$0.5\pi$
$858$	0	0
$859$	−44.5421	−1.51976	−0.759879	−	0.650065i	$-0.774742\pi$
$859$	−44.5421	−1.51976	−0.759879	+	0.650065i	$0.774742\pi$
$860$	0	0
$861$	0	0
$862$	−10.0000	−0.340601
$863$	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
$872$	22.6274i	0.766261i
$873$	0	0
$874$	0	0
$875$	0	0
$876$	0	0
$877$	56.0000	1.89099	0.945493	−	0.325643i	$-0.105581\pi$
$877$	56.0000	1.89099	0.945493	+	0.325643i	$0.105581\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	− 55.1181i	− 1.85697i	−0.371364	−	0.928487i	$-0.621110\pi$
$881$	− 55.1181i	− 1.85697i	0.371364	−	0.928487i	$-0.378890\pi$
$882$	0	0
$883$	55.6776	1.87370	0.936851	−	0.349729i	$-0.113726\pi$
$883$	55.6776	1.87370	0.936851	+	0.349729i	$0.113726\pi$
$884$	0	0
$885$	0	0
$886$	14.0000	0.470339
$887$	− 57.9828i	− 1.94687i	−0.228963	−	0.973435i	$-0.573533\pi$
$887$	− 57.9828i	− 1.94687i	0.228963	−	0.973435i	$-0.426467\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	0	0
$892$	22.2711	0.745690
$893$	0	0
$894$	0	0
$895$	0	0
$896$	0	0
$897$	0	0
$898$	−11.1355	−0.371597
$899$	43.8406i	1.46217i
$900$	0	0
$901$	−62.0000	−2.06552
$902$	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			$\pi$
$908$	− 53.7401i	− 1.78343i
$909$	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
$919$	0	0	1.00000			$0$
$919$	0	0	−1.00000			$\pi$
$920$	0	0
$921$	0	0
$922$	55.6776	1.83365
$923$	0	0
$924$	0	0
$925$	0	0
$926$	15.7480i	0.517512i
$927$	0	0
$928$	−44.5421	−1.46217
$929$	39.3700i	1.29169i	0.763469	+	0.645845i	$0.223495\pi$
$929$	39.3700i	1.29169i	−0.763469	+	0.645845i	$0.776505\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	0	0
$934$	−58.0000	−1.89782
$935$	0	0
$936$	0	0
$937$	20.0000	0.653372	0.326686	−	0.945133i	$-0.394068\pi$
$937$	20.0000	0.653372	0.326686	+	0.945133i	$0.394068\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	− 55.1181i	− 1.79680i	−0.439181	−	0.898399i	$-0.644731\pi$
$941$	− 55.1181i	− 1.79680i	0.439181	−	0.898399i	$-0.355269\pi$
$942$	0	0
$943$	0	0
$944$	− 28.2843i	− 0.920575i
$945$	0	0
$946$	0	0
$947$	0	0		−	1.00000i	$-0.5\pi$
$947$	0	0			1.00000i	$0.5\pi$
$948$	0	0
$949$	0	0
$950$	0	0
$951$	0	0
$952$	0	0
$953$	− 7.87401i	− 0.255064i	−0.991834	−	0.127532i	$-0.959294\pi$
$953$	− 7.87401i	− 0.255064i	0.991834	−	0.127532i	$-0.0407056\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	0	0
$958$	−22.0000	−0.710788
$959$	0	0
$960$	0	0
$961$	31.0000	1.00000
$962$	0	0
$963$	0	0
$964$	0	0
$965$	0	0
$966$	0	0
$967$	55.6776	1.79047	0.895236	−	0.445592i	$-0.147007\pi$
$967$	55.6776	1.79047	0.895236	+	0.445592i	$0.147007\pi$
$968$	− 31.1127i	− 1.00000i
$969$	0	0
$970$	0	0
$971$	60.8112i	1.95152i	0.218835	+	0.975762i	$0.429774\pi$
$971$	60.8112i	1.95152i	−0.218835	+	0.975762i	$0.570226\pi$
$972$	0	0
$973$	0	0
$974$	− 31.4960i	− 1.00920i
$975$	0	0
$976$	0	0
$977$	0	0		−	1.00000i	$-0.5\pi$
$977$	0	0			1.00000i	$0.5\pi$
$978$	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
$985$	0	0
$986$	87.6812i	2.79234i
$987$	0	0
$988$	0	0
$989$	0	0
$990$	0	0
$991$	22.2711	0.707463	0.353732	−	0.935347i	$-0.384913\pi$
$991$	22.2711	0.707463	0.353732	+	0.935347i	$0.384913\pi$
$992$	31.4960i	1.00000i
$993$	0	0
$994$	0	0
$995$	0	0
$996$	0	0
$997$	50.0000	1.58352	0.791758	−	0.610835i	$-0.209166\pi$
$997$	50.0000	1.58352	0.791758	+	0.610835i	$0.209166\pi$
$998$	62.9921i	1.99398i
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1116.2.g.c.991.1	✓	4
3.2	odd	2	inner	1116.2.g.c.991.4	yes	4
4.3	odd	2	inner	1116.2.g.c.991.3	yes	4
12.11	even	2	inner	1116.2.g.c.991.2	yes	4
31.30	odd	2	inner	1116.2.g.c.991.2	yes	4
93.92	even	2	inner	1116.2.g.c.991.3	yes	4
124.123	even	2	inner	1116.2.g.c.991.4	yes	4
372.371	odd	2	CM	1116.2.g.c.991.1	✓	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1116.2.g.c.991.1	✓	4	1.1	even	1	trivial
1116.2.g.c.991.1	✓	4	372.371	odd	2	CM
1116.2.g.c.991.2	yes	4	12.11	even	2	inner
1116.2.g.c.991.2	yes	4	31.30	odd	2	inner
1116.2.g.c.991.3	yes	4	4.3	odd	2	inner
1116.2.g.c.991.3	yes	4	93.92	even	2	inner
1116.2.g.c.991.4	yes	4	3.2	odd	2	inner
1116.2.g.c.991.4	yes	4	124.123	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:	\( N \)	\(=\)	\( 1116 = 2^{2} \cdot 3^{2} \cdot 31 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1116.g (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-1.41421i q^{2} -2.00000 q^{4} +2.82843i q^{8} +O(q^{10})\)	\(q-1.41421i q^{2} -2.00000 q^{4} +2.82843i q^{8} +4.00000 q^{16} -7.87401i q^{17} -5.00000 q^{25} -7.87401i q^{29} -5.56776 q^{31} -5.65685i q^{32} -11.1355 q^{34} -11.1355 q^{43} +1.41421i q^{47} +7.00000 q^{49} +7.07107i q^{50} -7.87401i q^{53} -11.1355 q^{58} -7.07107i q^{59} +7.87401i q^{62} -8.00000 q^{64} +15.7480i q^{68} +9.89949i q^{71} -11.1355 q^{79} +15.7480i q^{86} -7.87401i q^{89} +2.00000 q^{94} -4.00000 q^{97} -9.89949i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 8 q^{4}+O(q^{10}) \) 4 * q - 8 * q^4	\( 4 q - 8 q^{4} + 16 q^{16} - 20 q^{25} + 28 q^{49} - 32 q^{64} + 8 q^{94} - 16 q^{97}+O(q^{100}) \) 4 * q - 8 * q^4 + 16 * q^16 - 20 * q^25 + 28 * q^49 - 32 * q^64 + 8 * q^94 - 16 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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