LMFDB - Embedded newform 2816.2.g.c.1407.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2816,2,Mod(1407,2816)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2816, 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("2816.1407");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2816 = 2^{8} \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2816.g (of order $2$, degree $1$, not 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:	$22.4858732092$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.0.303595776.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + 5x^{6} + 16x^{4} + 45x^{2} + 81 $ x^8 + 5x^6 + 16x^4 + 45*x^2 + 81
Coefficient ring:	$\Z[a_1, \ldots, a_{23}]$
Coefficient ring index:	$ 2^{6} $
Twist minimal:	no (minimal twist has level 176)
Sato-Tate group:	$\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label			1407.4
Root			$-1.26217 - 1.18614i$ of defining polynomial
Character	$\chi$	$=$	2816.1407
Dual form			2816.2.g.c.1407.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.792287 q^{3} +1.37228i q^{5} -2.37228 q^{9} +O(q^{10})$	$q-0.792287 q^{3} +1.37228i q^{5} -2.37228 q^{9} +3.31662 q^{11} -1.08724i q^{15} -6.13592i q^{23} +3.11684 q^{25} +4.25639 q^{27} +9.30506i q^{31} -2.62772 q^{33} -12.1168i q^{37} -3.25544i q^{45} +6.63325i q^{47} -7.00000 q^{49} +6.00000i q^{53} +4.55134i q^{55} +14.6487 q^{59} +16.2333 q^{67} +4.86141i q^{69} +10.8896i q^{71} -2.46943 q^{75} +3.74456 q^{81} -18.8614 q^{89} -7.37228i q^{93} -0.116844 q^{97} -7.86797 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 4 q^{9}+O(q^{10}) $ 8 * q + 4 * q^9	$ 8 q + 4 q^{9} - 44 q^{25} - 44 q^{33} - 56 q^{49} - 16 q^{81} - 36 q^{89} + 68 q^{97}+O(q^{100}) $ 8 * q + 4 * q^9 - 44 * q^25 - 44 * q^33 - 56 * q^49 - 16 * q^81 - 36 * q^89 + 68 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2816\mathbb{Z}\right)^\times$.

$n$	$1025$	$1541$	$2047$
$\chi(n)$	$-1$	$-1$	$-1$

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
$3$	−0.792287	−0.457427	−0.228714	−	0.973494i	$-0.573452\pi$
$3$	−0.792287	−0.457427	−0.228714	+	0.973494i	$0.573452\pi$
$4$	0	0
$5$	1.37228i	0.613703i	0.951757	+	0.306851i	$0.0992755\pi$
$5$	1.37228i	0.613703i	−0.951757	+	0.306851i	$0.900725\pi$
$6$	0	0
$7$	0	0		−	1.00000i	$-0.5\pi$
$7$	0	0			1.00000i	$0.5\pi$
$8$	0	0
$9$	−2.37228	−0.790760
$10$	0	0
$11$	3.31662	1.00000
$11$	3.31662	1.00000
$12$	0	0
$13$	0	0		−	1.00000i	$-0.5\pi$
$13$	0	0			1.00000i	$0.5\pi$
$14$	0	0
$15$	− 1.08724i	− 0.280724i
$16$	0	0
$17$	0	0	1.00000			$0$
$17$	0	0	−1.00000			$\pi$
$18$	0	0
$19$	0	0	1.00000			$0$
$19$	0	0	−1.00000			$\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	− 6.13592i	− 1.27943i	−0.768613	−	0.639713i	$-0.779053\pi$
$23$	− 6.13592i	− 1.27943i	0.768613	−	0.639713i	$-0.220947\pi$
$24$	0	0
$25$	3.11684	0.623369
$26$	0	0
$27$	4.25639	0.819142
$28$	0	0
$29$	0	0		−	1.00000i	$-0.5\pi$
$29$	0	0			1.00000i	$0.5\pi$
$30$	0	0
$31$	9.30506i	1.67124i	0.549309	+	0.835619i	$0.314891\pi$
$31$	9.30506i	1.67124i	−0.549309	+	0.835619i	$0.685109\pi$
$32$	0	0
$33$	−2.62772	−0.457427
$34$	0	0
$35$	0	0
$36$	0	0
$37$	− 12.1168i	− 1.99200i	−0.0893706	−	0.995998i	$-0.528486\pi$
$37$	− 12.1168i	− 1.99200i	0.0893706	−	0.995998i	$-0.471514\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	0	0	1.00000			$0$
$41$	0	0	−1.00000			$\pi$
$42$	0	0
$43$	0	0	1.00000			$0$
$43$	0	0	−1.00000			$\pi$
$44$	0	0
$45$	− 3.25544i	− 0.485292i
$46$	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
$53$	6.00000i	0.824163i	0.911147	+	0.412082i	$0.135198\pi$
$53$	6.00000i	0.824163i	−0.911147	+	0.412082i	$0.864802\pi$
$54$	0	0
$55$	4.55134i	0.613703i
$56$	0	0
$57$	0	0
$58$	0	0
$59$	14.6487	1.90710	0.953549	−	0.301239i	$-0.0974001\pi$
$59$	14.6487	1.90710	0.953549	+	0.301239i	$0.0974001\pi$
$60$	0	0
$61$	0	0		−	1.00000i	$-0.5\pi$
$61$	0	0			1.00000i	$0.5\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	0	0
$66$	0	0
$67$	16.2333	1.98321	0.991605	−	0.129307i	$-0.0412752\pi$
$67$	16.2333	1.98321	0.991605	+	0.129307i	$0.0412752\pi$
$68$	0	0
$69$	4.86141i	0.585245i
$70$	0	0
$71$	10.8896i	1.29236i	0.763184	+	0.646181i	$0.223635\pi$
$71$	10.8896i	1.29236i	−0.763184	+	0.646181i	$0.776365\pi$
$72$	0	0
$73$	0	0	1.00000			$0$
$73$	0	0	−1.00000			$\pi$
$74$	0	0
$75$	−2.46943	−0.285146
$76$	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
$81$	3.74456	0.416063
$82$	0	0
$83$	0	0	1.00000			$0$
$83$	0	0	−1.00000			$\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−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
$93$	− 7.37228i	− 0.764470i
$94$	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
$99$	−7.86797	−0.790760
$100$	0	0
$101$	0	0		−	1.00000i	$-0.5\pi$
$101$	0	0			1.00000i	$0.5\pi$
$102$	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	1.00000			$0$
$107$	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
$111$	9.60002i	0.911193i
$112$	0	0
$113$	13.3723	1.25796	0.628979	−	0.777422i	$-0.283473\pi$
$113$	13.3723	1.25796	0.628979	+	0.777422i	$0.283473\pi$
$114$	0	0
$115$	8.42020	0.785188
$116$	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
$125$	11.1386i	0.996266i
$126$	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	1.00000			$0$
$131$	0	0	−1.00000			$\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	5.84096i	0.502710i
$136$	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	1.00000			$0$
$139$	0	0	−1.00000			$\pi$
$140$	0	0
$141$	− 5.25544i	− 0.442588i
$142$	0	0
$143$	0	0
$144$	0	0
$145$	0	0
$146$	0	0
$147$	5.54601	0.457427
$148$	0	0
$149$	0	0		−	1.00000i	$-0.5\pi$
$149$	0	0			1.00000i	$0.5\pi$
$150$	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
$155$	−12.7692	−1.02564
$156$	0	0
$157$	20.1168i	1.60550i	0.596316	+	0.802749i	$0.296630\pi$
$157$	20.1168i	1.60550i	−0.596316	+	0.802749i	$0.703370\pi$
$158$	0	0
$159$	− 4.75372i	− 0.376995i
$160$	0	0
$161$	0	0
$162$	0	0
$163$	19.8997	1.55867	0.779334	−	0.626608i	$-0.215557\pi$
$163$	19.8997	1.55867	0.779334	+	0.626608i	$0.215557\pi$
$164$	0	0
$165$	− 3.60597i	− 0.280724i
$166$	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		−	1.00000i	$-0.5\pi$
$173$	0	0			1.00000i	$0.5\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−11.6060	−0.872358
$178$	0	0
$179$	9.89497	0.739585	0.369792	−	0.929114i	$-0.379429\pi$
$179$	9.89497	0.739585	0.369792	+	0.929114i	$0.379429\pi$
$180$	0	0
$181$	3.88316i	0.288633i	0.989532	+	0.144316i	$0.0460983\pi$
$181$	3.88316i	0.288633i	−0.989532	+	0.144316i	$0.953902\pi$
$182$	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		−	1.00000i	$-0.5\pi$
$197$	0	0			1.00000i	$0.5\pi$
$198$	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
$201$	−12.8614	−0.907174
$202$	0	0
$203$	0	0
$204$	0	0
$205$	0	0
$206$	0	0
$207$	14.5561i	1.01172i
$208$	0	0
$209$	0	0
$210$	0	0
$211$	0	0	1.00000			$0$
$211$	0	0	−1.00000			$\pi$
$212$	0	0
$213$	− 8.62772i	− 0.591162i
$214$	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	$-0.843993\pi$
$223$	− 14.0588i	− 0.941446i	0.882281	−	0.470723i	$-0.156007\pi$
$224$	0	0
$225$	−7.39403	−0.492935
$226$	0	0
$227$	0	0	1.00000			$0$
$227$	0	0	−1.00000			$\pi$
$228$	0	0
$229$	28.3505i	1.87346i	0.350058	+	0.936728i	$0.386162\pi$
$229$	28.3505i	1.87346i	−0.350058	+	0.936728i	$0.613838\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	0	0	1.00000			$0$
$233$	0	0	−1.00000			$\pi$
$234$	0	0
$235$	−9.10268	−0.593794
$236$	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
$243$	−15.7359	−1.00946
$244$	0	0
$245$	− 9.60597i	− 0.613703i
$246$	0	0
$247$	0	0
$248$	0	0
$249$	0	0
$250$	0	0
$251$	31.6742	1.99926	0.999630	−	0.0271858i	$-0.00865456\pi$
$251$	31.6742	1.99926	0.999630	+	0.0271858i	$0.00865456\pi$
$252$	0	0
$253$	− 20.3505i	− 1.27943i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	18.0000	1.12281	0.561405	−	0.827541i	$-0.310261\pi$
$257$	18.0000	1.12281	0.561405	+	0.827541i	$0.310261\pi$
$258$	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
$267$	14.9436	0.914536
$268$	0	0
$269$	− 30.0000i	− 1.82913i	−0.404436	−	0.914566i	$-0.632532\pi$
$269$	− 30.0000i	− 1.82913i	0.404436	−	0.914566i	$-0.367468\pi$
$270$	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
$275$	10.3374	0.623369
$276$	0	0
$277$	0	0		−	1.00000i	$-0.5\pi$
$277$	0	0			1.00000i	$0.5\pi$
$278$	0	0
$279$	− 22.0742i	− 1.32155i
$280$	0	0
$281$	0	0	1.00000			$0$
$281$	0	0	−1.00000			$\pi$
$282$	0	0
$283$	0	0	1.00000			$0$
$283$	0	0	−1.00000			$\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	0	0
$288$	0	0
$289$	17.0000	1.00000
$290$	0	0
$291$	0.0925740	0.00542678
$292$	0	0
$293$	0	0		−	1.00000i	$-0.5\pi$
$293$	0	0			1.00000i	$0.5\pi$
$294$	0	0
$295$	20.1021i	1.17039i
$296$	0	0
$297$	14.1168	0.819142
$298$	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	1.00000			$0$
$307$	0	0	−1.00000			$\pi$
$308$	0	0
$309$	− 15.7663i	− 0.896914i
$310$	0	0
$311$	− 33.1662i	− 1.88069i	−0.340229	−	0.940343i	$-0.610505\pi$
$311$	− 33.1662i	− 1.88069i	0.340229	−	0.940343i	$-0.389495\pi$
$312$	0	0
$313$	−16.3505	−0.924187	−0.462093	−	0.886831i	$-0.652902\pi$
$313$	−16.3505	−0.924187	−0.462093	+	0.886831i	$0.652902\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	33.6060i	1.88750i	0.330661	+	0.943750i	$0.392728\pi$
$317$	33.6060i	1.88750i	−0.330661	+	0.943750i	$0.607272\pi$
$318$	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
$331$	25.3360	1.39259	0.696295	−	0.717756i	$-0.254831\pi$
$331$	25.3360	1.39259	0.696295	+	0.717756i	$0.254831\pi$
$332$	0	0
$333$	28.7446i	1.57519i
$334$	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
$339$	−10.5947	−0.575424
$340$	0	0
$341$	30.8614i	1.67124i
$342$	0	0
$343$	0	0
$344$	0	0
$345$	−6.67122	−0.359166
$346$	0	0
$347$	0	0	1.00000			$0$
$347$	0	0	−1.00000			$\pi$
$348$	0	0
$349$	0	0		−	1.00000i	$-0.5\pi$
$349$	0	0			1.00000i	$0.5\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−27.0951	−1.44213	−0.721063	−	0.692869i	$-0.756346\pi$
$353$	−27.0951	−1.44213	−0.721063	+	0.692869i	$0.756346\pi$
$354$	0	0
$355$	−14.9436	−0.793126
$356$	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
$363$	−8.71516	−0.457427
$364$	0	0
$365$	0	0
$366$	0	0
$367$	37.0179i	1.93232i	0.257948	+	0.966159i	$0.416954\pi$
$367$	37.0179i	1.93232i	−0.257948	+	0.966159i	$0.583046\pi$
$368$	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
$375$	− 8.82496i	− 0.455719i
$376$	0	0
$377$	0	0
$378$	0	0
$379$	−6.72582	−0.345482	−0.172741	−	0.984967i	$-0.555262\pi$
$379$	−6.72582	−0.345482	−0.172741	+	0.984967i	$0.555262\pi$
$380$	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
$389$	− 39.0951i	− 1.98220i	−0.133120	−	0.991100i	$-0.542499\pi$
$389$	− 39.0951i	− 1.98220i	0.133120	−	0.991100i	$-0.457501\pi$
$390$	0	0
$391$	0	0
$392$	0	0
$393$	0	0
$394$	0	0
$395$	0	0
$396$	0	0
$397$	2.00000i	0.100377i	0.998740	+	0.0501886i	$0.0159822\pi$
$397$	2.00000i	0.100377i	−0.998740	+	0.0501886i	$0.984018\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−30.0000	−1.49813	−0.749064	−	0.662497i	$-0.769497\pi$
$401$	−30.0000	−1.49813	−0.749064	+	0.662497i	$0.769497\pi$
$402$	0	0
$403$	0	0
$404$	0	0
$405$	5.13859i	0.255339i
$406$	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
$411$	−17.1181	−0.844375
$412$	0	0
$413$	0	0
$414$	0	0
$415$	0	0
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−33.1662	−1.62028	−0.810139	−	0.586238i	$-0.800608\pi$
$419$	−33.1662	−1.62028	−0.810139	+	0.586238i	$0.800608\pi$
$420$	0	0
$421$	− 10.0000i	− 0.487370i	−0.969854	−	0.243685i	$-0.921644\pi$
$421$	− 10.0000i	− 0.487370i	0.969854	−	0.243685i	$-0.0783563\pi$
$422$	0	0
$423$	− 15.7359i	− 0.765107i
$424$	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
$441$	16.6060	0.790760
$442$	0	0
$443$	−36.4280	−1.73075	−0.865373	−	0.501129i	$-0.832918\pi$
$443$	−36.4280	−1.73075	−0.865373	+	0.501129i	$0.832918\pi$
$444$	0	0
$445$	− 25.8832i	− 1.22698i
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−5.13859	−0.242505	−0.121253	−	0.992622i	$-0.538691\pi$
$449$	−5.13859	−0.242505	−0.121253	+	0.992622i	$0.538691\pi$
$450$	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		−	1.00000i	$-0.5\pi$
$461$	0	0			1.00000i	$0.5\pi$
$462$	0	0
$463$	− 41.7716i	− 1.94129i	−0.240515	−	0.970645i	$-0.577316\pi$
$463$	− 41.7716i	− 1.94129i	0.240515	−	0.970645i	$-0.422684\pi$
$464$	0	0
$465$	10.1168	0.469157
$466$	0	0
$467$	−24.1561	−1.11781	−0.558906	−	0.829231i	$-0.688779\pi$
$467$	−24.1561	−1.11781	−0.558906	+	0.829231i	$0.688779\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	− 15.9383i	− 0.734399i
$472$	0	0
$473$	0	0
$474$	0	0
$475$	0	0
$476$	0	0
$477$	− 14.2337i	− 0.651716i
$478$	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
$485$	− 0.160343i	− 0.00728079i
$486$	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
$489$	−15.7663	−0.712977
$490$	0	0
$491$	0	0	1.00000			$0$
$491$	0	0	−1.00000			$\pi$
$492$	0	0
$493$	0	0
$494$	0	0
$495$	− 10.7971i	− 0.485292i
$496$	0	0
$497$	0	0
$498$	0	0
$499$	19.8997	0.890835	0.445418	−	0.895323i	$-0.353055\pi$
$499$	19.8997	0.890835	0.445418	+	0.895323i	$0.353055\pi$
$500$	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
$507$	10.2997	0.457427
$508$	0	0
$509$	− 25.3723i	− 1.12461i	−0.826931	−	0.562303i	$-0.809915\pi$
$509$	− 25.3723i	− 1.12461i	0.826931	−	0.562303i	$-0.190085\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	0	0
$514$	0	0
$515$	−27.3081	−1.20334
$516$	0	0
$517$	22.0000i	0.967559i
$518$	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	1.00000			$0$
$523$	0	0	−1.00000			$\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	0	0
$528$	0	0
$529$	−14.6495	−0.636933
$530$	0	0
$531$	−34.7508	−1.50806
$532$	0	0
$533$	0	0
$534$	0	0
$535$	0	0
$536$	0	0
$537$	−7.83966	−0.338306
$538$	0	0
$539$	−23.2164	−1.00000
$540$	0	0
$541$	0	0		−	1.00000i	$-0.5\pi$
$541$	0	0			1.00000i	$0.5\pi$
$542$	0	0
$543$	− 3.07657i	− 0.132028i
$544$	0	0
$545$	0	0
$546$	0	0
$547$	0	0	1.00000			$0$
$547$	0	0	−1.00000			$\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	0	0
$552$	0	0
$553$	0	0
$554$	0	0
$555$	−13.1739	−0.559202
$556$	0	0
$557$	0	0		−	1.00000i	$-0.5\pi$
$557$	0	0			1.00000i	$0.5\pi$
$558$	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
$565$	18.3505i	0.772013i
$566$	0	0
$567$	0	0
$568$	0	0
$569$	0	0	1.00000			$0$
$569$	0	0	−1.00000			$\pi$
$570$	0	0
$571$	0	0	1.00000			$0$
$571$	0	0	−1.00000			$\pi$
$572$	0	0
$573$	1.09509i	0.0457482i
$574$	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
$587$	6.63325	0.273784	0.136892	−	0.990586i	$-0.456289\pi$
$587$	6.63325	0.273784	0.136892	+	0.990586i	$0.456289\pi$
$588$	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
$597$	− 15.7663i	− 0.645272i
$598$	0	0
$599$	− 33.1662i	− 1.35514i	−0.735460	−	0.677568i	$-0.763034\pi$
$599$	− 33.1662i	− 1.35514i	0.735460	−	0.677568i	$-0.236966\pi$
$600$	0	0
$601$	0	0	1.00000			$0$
$601$	0	0	−1.00000			$\pi$
$602$	0	0
$603$	−38.5099	−1.56824
$604$	0	0
$605$	15.0951i	0.613703i
$606$	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		−	1.00000i	$-0.5\pi$
$613$	0	0			1.00000i	$0.5\pi$
$614$	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
$619$	−25.7407	−1.03461	−0.517303	−	0.855802i	$-0.673064\pi$
$619$	−25.7407	−1.03461	−0.517303	+	0.855802i	$0.673064\pi$
$620$	0	0
$621$	− 26.1168i	− 1.04803i
$622$	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
$639$	− 25.8333i	− 1.02195i
$640$	0	0
$641$	2.39403	0.0945585	0.0472793	−	0.998882i	$-0.484945\pi$
$641$	2.39403	0.0945585	0.0472793	+	0.998882i	$0.484945\pi$
$642$	0	0
$643$	20.5822	0.811684	0.405842	−	0.913943i	$-0.366978\pi$
$643$	20.5822	0.811684	0.405842	+	0.913943i	$0.366978\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	44.9407i	1.76680i	0.468617	+	0.883402i	$0.344753\pi$
$647$	44.9407i	1.76680i	−0.468617	+	0.883402i	$0.655247\pi$
$648$	0	0
$649$	48.5842	1.90710
$650$	0	0
$651$	0	0
$652$	0	0
$653$	22.6277i	0.885491i	0.896647	+	0.442746i	$0.145995\pi$
$653$	22.6277i	0.885491i	−0.896647	+	0.442746i	$0.854005\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	0	0
$658$	0	0
$659$	0	0	1.00000			$0$
$659$	0	0	−1.00000			$\pi$
$660$	0	0
$661$	− 36.5842i	− 1.42296i	−0.702706	−	0.711481i	$-0.748025\pi$
$661$	− 36.5842i	− 1.42296i	0.702706	−	0.711481i	$-0.251975\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	0	0
$666$	0	0
$667$	0	0
$668$	0	0
$669$	11.1386i	0.430643i
$670$	0	0
$671$	0	0
$672$	0	0
$673$	0	0	1.00000			$0$
$673$	0	0	−1.00000			$\pi$
$674$	0	0
$675$	13.2665	0.510628
$676$	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
$683$	−46.4327	−1.77670	−0.888350	−	0.459167i	$-0.848148\pi$
$683$	−46.4327	−1.77670	−0.888350	+	0.459167i	$0.848148\pi$
$684$	0	0
$685$	29.6495i	1.13285i
$686$	0	0
$687$	− 22.4618i	− 0.856970i
$688$	0	0
$689$	0	0
$690$	0	0
$691$	39.5971	1.50635	0.753173	−	0.657823i	$-0.228522\pi$
$691$	39.5971	1.50635	0.753173	+	0.657823i	$0.228522\pi$
$692$	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		−	1.00000i	$-0.5\pi$
$701$	0	0			1.00000i	$0.5\pi$
$702$	0	0
$703$	0	0
$704$	0	0
$705$	7.21194	0.271617
$706$	0	0
$707$	0	0
$708$	0	0
$709$	− 52.5842i	− 1.97484i	−0.158114	−	0.987421i	$-0.550541\pi$
$709$	− 52.5842i	− 1.97484i	0.158114	−	0.987421i	$-0.449459\pi$
$710$	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
$729$	1.23369	0.0456921
$730$	0	0
$731$	0	0
$732$	0	0
$733$	0	0		−	1.00000i	$-0.5\pi$
$733$	0	0			1.00000i	$0.5\pi$
$734$	0	0
$735$	7.61068i	0.280724i
$736$	0	0
$737$	53.8397	1.98321
$738$	0	0
$739$	0	0	1.00000			$0$
$739$	0	0	−1.00000			$\pi$
$740$	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	$0.0288227\pi$
$751$	4.95610i	0.180851i	−0.995903	+	0.0904254i	$0.971177\pi$
$752$	0	0
$753$	−25.0951	−0.914516
$754$	0	0
$755$	0	0
$756$	0	0
$757$	38.0000i	1.38113i	0.723269	+	0.690567i	$0.242639\pi$
$757$	38.0000i	1.38113i	−0.723269	+	0.690567i	$0.757361\pi$
$758$	0	0
$759$	16.1235i	0.585245i
$760$	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
$771$	−14.2612	−0.513603
$772$	0	0
$773$	54.0000i	1.94225i	0.238581	+	0.971123i	$0.423318\pi$
$773$	54.0000i	1.94225i	−0.238581	+	0.971123i	$0.576682\pi$
$774$	0	0
$775$	29.0024i	1.04180i
$776$	0	0
$777$	0	0
$778$	0	0
$779$	0	0
$780$	0	0
$781$	36.1168i	1.29236i
$782$	0	0
$783$	0	0
$784$	0	0
$785$	−27.6060	−0.985299
$786$	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
$795$	6.52344	0.231363
$796$	0	0
$797$	− 47.3288i	− 1.67647i	−0.545308	−	0.838236i	$-0.683587\pi$
$797$	− 47.3288i	− 1.67647i	0.545308	−	0.838236i	$-0.316413\pi$
$798$	0	0
$799$	0	0
$800$	0	0
$801$	44.7446	1.58097
$802$	0	0
$803$	0	0
$804$	0	0
$805$	0	0
$806$	0	0
$807$	23.7686i	0.836695i
$808$	0	0
$809$	0	0	1.00000			$0$
$809$	0	0	−1.00000			$\pi$
$810$	0	0
$811$	0	0	1.00000			$0$
$811$	0	0	−1.00000			$\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	27.3081i	0.956559i
$816$	0	0
$817$	0	0
$818$	0	0
$819$	0	0
$820$	0	0
$821$	0	0		−	1.00000i	$-0.5\pi$
$821$	0	0			1.00000i	$0.5\pi$
$822$	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
$825$	−8.19019	−0.285146
$826$	0	0
$827$	0	0	1.00000			$0$
$827$	0	0	−1.00000			$\pi$
$828$	0	0
$829$	28.5842i	0.992771i	0.868102	+	0.496385i	$0.165340\pi$
$829$	28.5842i	0.992771i	−0.868102	+	0.496385i	$0.834660\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	0	0
$834$	0	0
$835$	0	0
$836$	0	0
$837$	39.6060i	1.36898i
$838$	0	0
$839$	− 57.2126i	− 1.97520i	−0.156999	−	0.987599i	$-0.550182\pi$
$839$	− 57.2126i	− 1.97520i	0.156999	−	0.987599i	$-0.449818\pi$
$840$	0	0
$841$	−29.0000	−1.00000
$842$	0	0
$843$	0	0
$844$	0	0
$845$	− 17.8397i	− 0.613703i
$846$	0	0
$847$	0	0
$848$	0	0
$849$	0	0
$850$	0	0
$851$	−74.3479	−2.54861
$852$	0	0
$853$	0	0		−	1.00000i	$-0.5\pi$
$853$	0	0			1.00000i	$0.5\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	0	0	1.00000			$0$
$857$	0	0	−1.00000			$\pi$
$858$	0	0
$859$	1.97210	0.0672872	0.0336436	−	0.999434i	$-0.489289\pi$
$859$	1.97210	0.0672872	0.0336436	+	0.999434i	$0.489289\pi$
$860$	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
$867$	−13.4689	−0.457427
$868$	0	0
$869$	0	0
$870$	0	0
$871$	0	0
$872$	0	0
$873$	0.277187	0.00938135
$874$	0	0
$875$	0	0
$876$	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
$881$	42.8614	1.44404	0.722019	−	0.691873i	$-0.243214\pi$
$881$	42.8614	1.44404	0.722019	+	0.691873i	$0.243214\pi$
$882$	0	0
$883$	19.8997	0.669680	0.334840	−	0.942275i	$-0.391318\pi$
$883$	19.8997	0.669680	0.334840	+	0.942275i	$0.391318\pi$
$884$	0	0
$885$	− 15.9267i	− 0.535369i
$886$	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
$891$	12.4193	0.416063
$892$	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
$907$	59.6992	1.98228	0.991140	−	0.132818i	$-0.0424025\pi$
$907$	59.6992	1.98228	0.991140	+	0.132818i	$0.0424025\pi$
$908$	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
$925$	− 37.7663i	− 1.24175i
$926$	0	0
$927$	− 47.2078i	− 1.55051i
$928$	0	0
$929$	−30.0000	−0.984268	−0.492134	−	0.870519i	$-0.663783\pi$
$929$	−30.0000	−0.984268	−0.492134	+	0.870519i	$0.663783\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	26.2772i	0.860276i
$934$	0	0
$935$	0	0
$936$	0	0
$937$	0	0	1.00000			$0$
$937$	0	0	−1.00000			$\pi$
$938$	0	0
$939$	12.9543	0.422748
$940$	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
$947$	60.9716	1.98131	0.990656	−	0.136385i	$-0.0435483\pi$
$947$	60.9716	1.98131	0.990656	+	0.136385i	$0.0435483\pi$
$948$	0	0
$949$	0	0
$950$	0	0
$951$	− 26.6256i	− 0.863393i
$952$	0	0
$953$	0	0	1.00000			$0$
$953$	0	0	−1.00000			$\pi$
$954$	0	0
$955$	1.89676	0.0613777
$956$	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
$971$	−17.4131	−0.558812	−0.279406	−	0.960173i	$-0.590138\pi$
$971$	−17.4131	−0.558812	−0.279406	+	0.960173i	$0.590138\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	0	0
$976$	0	0
$977$	35.3288	1.13027	0.565134	−	0.824999i	$-0.308824\pi$
$977$	35.3288	1.13027	0.565134	+	0.824999i	$0.308824\pi$
$978$	0	0
$979$	−62.5562	−1.99931
$980$	0	0
$981$	0	0
$982$	0	0
$983$	25.9259i	0.826906i	0.910525	+	0.413453i	$0.135677\pi$
$983$	25.9259i	0.826906i	−0.910525	+	0.413453i	$0.864323\pi$
$984$	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	$0.397103\pi$
$991$	59.6992i	1.89641i	−0.317660	+	0.948205i	$0.602897\pi$
$992$	0	0
$993$	−20.0733	−0.637008
$994$	0	0
$995$	−27.3081	−0.865723
$996$	0	0
$997$	0	0		−	1.00000i	$-0.5\pi$
$997$	0	0			1.00000i	$0.5\pi$
$998$	0	0
$999$	− 51.5740i	− 1.63173i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2816.2.g.c.1407.4		8
4.3	odd	2	inner	2816.2.g.c.1407.6		8
8.3	odd	2	inner	2816.2.g.c.1407.3		8
8.5	even	2	inner	2816.2.g.c.1407.5		8
11.10	odd	2	CM	2816.2.g.c.1407.4		8
16.3	odd	4		176.2.e.b.175.3	yes	4
16.5	even	4		704.2.e.c.703.3		4
16.11	odd	4		704.2.e.c.703.2		4
16.13	even	4		176.2.e.b.175.2	✓	4
44.43	even	2	inner	2816.2.g.c.1407.6		8
48.29	odd	4		1584.2.o.e.703.2		4
48.35	even	4		1584.2.o.e.703.1		4
88.21	odd	2	inner	2816.2.g.c.1407.5		8
88.43	even	2	inner	2816.2.g.c.1407.3		8
176.21	odd	4		704.2.e.c.703.3		4
176.43	even	4		704.2.e.c.703.2		4
176.109	odd	4		176.2.e.b.175.2	✓	4
176.131	even	4		176.2.e.b.175.3	yes	4
528.131	odd	4		1584.2.o.e.703.1		4
528.461	even	4		1584.2.o.e.703.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
176.2.e.b.175.2	✓	4	16.13	even	4
176.2.e.b.175.2	✓	4	176.109	odd	4
176.2.e.b.175.3	yes	4	16.3	odd	4
176.2.e.b.175.3	yes	4	176.131	even	4
704.2.e.c.703.2		4	16.11	odd	4
704.2.e.c.703.2		4	176.43	even	4
704.2.e.c.703.3		4	16.5	even	4
704.2.e.c.703.3		4	176.21	odd	4
1584.2.o.e.703.1		4	48.35	even	4
1584.2.o.e.703.1		4	528.131	odd	4
1584.2.o.e.703.2		4	48.29	odd	4
1584.2.o.e.703.2		4	528.461	even	4
2816.2.g.c.1407.3		8	8.3	odd	2	inner
2816.2.g.c.1407.3		8	88.43	even	2	inner
2816.2.g.c.1407.4		8	1.1	even	1	trivial
2816.2.g.c.1407.4		8	11.10	odd	2	CM
2816.2.g.c.1407.5		8	8.5	even	2	inner
2816.2.g.c.1407.5		8	88.21	odd	2	inner
2816.2.g.c.1407.6		8	4.3	odd	2	inner
2816.2.g.c.1407.6		8	44.43	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:	\( N \)	\(=\)	\( 2816 = 2^{8} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2816.g (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-0.792287 q^{3} +1.37228i q^{5} -2.37228 q^{9} +O(q^{10})\)	\(q-0.792287 q^{3} +1.37228i q^{5} -2.37228 q^{9} +3.31662 q^{11} -1.08724i q^{15} -6.13592i q^{23} +3.11684 q^{25} +4.25639 q^{27} +9.30506i q^{31} -2.62772 q^{33} -12.1168i q^{37} -3.25544i q^{45} +6.63325i q^{47} -7.00000 q^{49} +6.00000i q^{53} +4.55134i q^{55} +14.6487 q^{59} +16.2333 q^{67} +4.86141i q^{69} +10.8896i q^{71} -2.46943 q^{75} +3.74456 q^{81} -18.8614 q^{89} -7.37228i q^{93} -0.116844 q^{97} -7.86797 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 4 q^{9}+O(q^{10}) \) 8 * q + 4 * q^9	\( 8 q + 4 q^{9} - 44 q^{25} - 44 q^{33} - 56 q^{49} - 16 q^{81} - 36 q^{89} + 68 q^{97}+O(q^{100}) \) 8 * q + 4 * q^9 - 44 * q^25 - 44 * q^33 - 56 * q^49 - 16 * q^81 - 36 * q^89 + 68 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more