LMFDB - Embedded newform 960.2.m.a.479.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [960,2,Mod(479,960)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(960, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 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("960.479");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 960 = 2^{6} \cdot 3 \cdot 5 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	960.m (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:	$7.66563859404$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.0.12960000.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 3x^{6} + 8x^{4} - 3x^{2} + 1 $ x^8 - 3x^6 + 8x^4 - 3*x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{12} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label			479.1
Root			$-0.535233 + 0.309017i$ of defining polynomial
Character	$\chi$	$=$	960.479
Dual form			960.2.m.a.479.6

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.73205i q^{3} -2.23607 q^{5} -3.00000 q^{9} +O(q^{10})$	$q-1.73205i q^{3} -2.23607 q^{5} -3.00000 q^{9} -4.47214i q^{11} +3.46410 q^{13} +3.87298i q^{15} -7.74597 q^{17} +7.74597i q^{23} +5.00000 q^{25} +5.19615i q^{27} -4.47214 q^{29} +2.00000i q^{31} -7.74597 q^{33} -10.3923 q^{37} -6.00000i q^{39} -3.46410i q^{43} +6.70820 q^{45} +7.74597i q^{47} -7.00000 q^{49} +13.4164i q^{51} +10.0000i q^{55} -4.47214i q^{59} -7.74597 q^{65} +10.3923i q^{67} +13.4164 q^{69} -8.66025i q^{75} -14.0000i q^{79} +9.00000 q^{81} +17.3205 q^{85} +7.74597i q^{87} +3.46410 q^{93} +13.4164i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 24 q^{9}+O(q^{10}) $ 8 * q - 24 * q^9	$ 8 q - 24 q^{9} + 40 q^{25} - 56 q^{49} + 72 q^{81}+O(q^{100}) $ 8 * q - 24 * q^9 + 40 * q^25 - 56 * q^49 + 72 * q^81

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/960\mathbb{Z}\right)^\times$.

$n$	$511$	$577$	$641$	$901$
$\chi(n)$	$-1$	$-1$	$-1$	$-1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	− 1.73205i	− 1.00000i
$3$	− 1.73205i	− 1.00000i
$4$	0	0
$5$	−2.23607	−1.00000
$5$	−2.23607	−1.00000
$6$	0	0
$7$	0	0		−	1.00000i	$-0.5\pi$
$7$	0	0			1.00000i	$0.5\pi$
$8$	0	0
$9$	−3.00000	−1.00000
$10$	0	0
$11$	− 4.47214i	− 1.34840i	−0.738549	−	0.674200i	$-0.764489\pi$
$11$	− 4.47214i	− 1.34840i	0.738549	−	0.674200i	$-0.235511\pi$
$12$	0	0
$13$	3.46410	0.960769	0.480384	−	0.877058i	$-0.340497\pi$
$13$	3.46410	0.960769	0.480384	+	0.877058i	$0.340497\pi$
$14$	0	0
$15$	3.87298i	1.00000i
$16$	0	0
$17$	−7.74597	−1.87867	−0.939336	−	0.342997i	$-0.888558\pi$
$17$	−7.74597	−1.87867	−0.939336	+	0.342997i	$0.888558\pi$
$18$	0	0
$19$	0	0		−	1.00000i	$-0.5\pi$
$19$	0	0			1.00000i	$0.5\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	7.74597i	1.61515i	0.589768	+	0.807573i	$0.299219\pi$
$23$	7.74597i	1.61515i	−0.589768	+	0.807573i	$0.700781\pi$
$24$	0	0
$25$	5.00000	1.00000
$26$	0	0
$27$	5.19615i	1.00000i
$28$	0	0
$29$	−4.47214	−0.830455	−0.415227	−	0.909718i	$-0.636298\pi$
$29$	−4.47214	−0.830455	−0.415227	+	0.909718i	$0.636298\pi$
$30$	0	0
$31$	2.00000i	0.359211i	0.983739	+	0.179605i	$0.0574821\pi$
$31$	2.00000i	0.359211i	−0.983739	+	0.179605i	$0.942518\pi$
$32$	0	0
$33$	−7.74597	−1.34840
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−10.3923	−1.70848	−0.854242	−	0.519875i	$-0.825978\pi$
$37$	−10.3923	−1.70848	−0.854242	+	0.519875i	$0.825978\pi$
$38$	0	0
$39$	− 6.00000i	− 0.960769i
$40$	0	0
$41$	0	0	1.00000			$0$
$41$	0	0	−1.00000			$\pi$
$42$	0	0
$43$	− 3.46410i	− 0.528271i	−0.964486	−	0.264135i	$-0.914913\pi$
$43$	− 3.46410i	− 0.528271i	0.964486	−	0.264135i	$-0.0850865\pi$
$44$	0	0
$45$	6.70820	1.00000
$46$	0	0
$47$	7.74597i	1.12987i	0.825137	+	0.564933i	$0.191098\pi$
$47$	7.74597i	1.12987i	−0.825137	+	0.564933i	$0.808902\pi$
$48$	0	0
$49$	−7.00000	−1.00000
$50$	0	0
$51$	13.4164i	1.87867i
$52$	0	0
$53$	0	0	1.00000			$0$
$53$	0	0	−1.00000			$\pi$
$54$	0	0
$55$	10.0000i	1.34840i
$56$	0	0
$57$	0	0
$58$	0	0
$59$	− 4.47214i	− 0.582223i	−0.956689	−	0.291111i	$-0.905975\pi$
$59$	− 4.47214i	− 0.582223i	0.956689	−	0.291111i	$-0.0940250\pi$
$60$	0	0
$61$	0	0	1.00000			$0$
$61$	0	0	−1.00000			$\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−7.74597	−0.960769
$66$	0	0
$67$	10.3923i	1.26962i	0.772667	+	0.634811i	$0.218922\pi$
$67$	10.3923i	1.26962i	−0.772667	+	0.634811i	$0.781078\pi$
$68$	0	0
$69$	13.4164	1.61515
$70$	0	0
$71$	0	0		−	1.00000i	$-0.5\pi$
$71$	0	0			1.00000i	$0.5\pi$
$72$	0	0
$73$	0	0	1.00000			$0$
$73$	0	0	−1.00000			$\pi$
$74$	0	0
$75$	− 8.66025i	− 1.00000i
$76$	0	0
$77$	0	0
$78$	0	0
$79$	− 14.0000i	− 1.57512i	−0.616236	−	0.787562i	$-0.711343\pi$
$79$	− 14.0000i	− 1.57512i	0.616236	−	0.787562i	$-0.288657\pi$
$80$	0	0
$81$	9.00000	1.00000
$82$	0	0
$83$	0	0		−	1.00000i	$-0.5\pi$
$83$	0	0			1.00000i	$0.5\pi$
$84$	0	0
$85$	17.3205	1.87867
$86$	0	0
$87$	7.74597i	0.830455i
$88$	0	0
$89$	0	0	1.00000			$0$
$89$	0	0	−1.00000			$\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	3.46410	0.359211
$94$	0	0
$95$	0	0
$96$	0	0
$97$	0	0	1.00000			$0$
$97$	0	0	−1.00000			$\pi$
$98$	0	0
$99$	13.4164i	1.34840i
$100$	0	0
$101$	−4.47214	−0.444994	−0.222497	−	0.974933i	$-0.571421\pi$
$101$	−4.47214	−0.444994	−0.222497	+	0.974933i	$0.571421\pi$
$102$	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		−	1.00000i	$-0.5\pi$
$107$	0	0			1.00000i	$0.5\pi$
$108$	0	0
$109$	0	0	1.00000			$0$
$109$	0	0	−1.00000			$\pi$
$110$	0	0
$111$	18.0000i	1.70848i
$112$	0	0
$113$	−7.74597	−0.728679	−0.364340	−	0.931266i	$-0.618705\pi$
$113$	−7.74597	−0.728679	−0.364340	+	0.931266i	$0.618705\pi$
$114$	0	0
$115$	− 17.3205i	− 1.61515i
$116$	0	0
$117$	−10.3923	−0.960769
$118$	0	0
$119$	0	0
$120$	0	0
$121$	−9.00000	−0.818182
$122$	0	0
$123$	0	0
$124$	0	0
$125$	−11.1803	−1.00000
$126$	0	0
$127$	0	0		−	1.00000i	$-0.5\pi$
$127$	0	0			1.00000i	$0.5\pi$
$128$	0	0
$129$	−6.00000	−0.528271
$130$	0	0
$131$	− 22.3607i	− 1.95366i	−0.214013	−	0.976831i	$-0.568653\pi$
$131$	− 22.3607i	− 1.95366i	0.214013	−	0.976831i	$-0.431347\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	− 11.6190i	− 1.00000i
$136$	0	0
$137$	23.2379	1.98535	0.992674	−	0.120824i	$-0.0385538\pi$
$137$	23.2379	1.98535	0.992674	+	0.120824i	$0.0385538\pi$
$138$	0	0
$139$	0	0		−	1.00000i	$-0.5\pi$
$139$	0	0			1.00000i	$0.5\pi$
$140$	0	0
$141$	13.4164	1.12987
$142$	0	0
$143$	− 15.4919i	− 1.29550i
$144$	0	0
$145$	10.0000	0.830455
$146$	0	0
$147$	12.1244i	1.00000i
$148$	0	0
$149$	−22.3607	−1.83186	−0.915929	−	0.401340i	$-0.868545\pi$
$149$	−22.3607	−1.83186	−0.915929	+	0.401340i	$0.868545\pi$
$150$	0	0
$151$	− 22.0000i	− 1.79033i	−0.445730	−	0.895167i	$-0.647056\pi$
$151$	− 22.0000i	− 1.79033i	0.445730	−	0.895167i	$-0.352944\pi$
$152$	0	0
$153$	23.2379	1.87867
$154$	0	0
$155$	− 4.47214i	− 0.359211i
$156$	0	0
$157$	−24.2487	−1.93526	−0.967629	−	0.252377i	$-0.918788\pi$
$157$	−24.2487	−1.93526	−0.967629	+	0.252377i	$0.918788\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	0	0
$162$	0	0
$163$	− 3.46410i	− 0.271329i	−0.990755	−	0.135665i	$-0.956683\pi$
$163$	− 3.46410i	− 0.271329i	0.990755	−	0.135665i	$-0.0433170\pi$
$164$	0	0
$165$	17.3205	1.34840
$166$	0	0
$167$	− 23.2379i	− 1.79820i	−0.437741	−	0.899101i	$-0.644221\pi$
$167$	− 23.2379i	− 1.79820i	0.437741	−	0.899101i	$-0.355779\pi$
$168$	0	0
$169$	−1.00000	−0.0769231
$170$	0	0
$171$	0	0
$172$	0	0
$173$	0	0	1.00000			$0$
$173$	0	0	−1.00000			$\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−7.74597	−0.582223
$178$	0	0
$179$	− 22.3607i	− 1.67132i	−0.549250	−	0.835658i	$-0.685087\pi$
$179$	− 22.3607i	− 1.67132i	0.549250	−	0.835658i	$-0.314913\pi$
$180$	0	0
$181$	0	0	1.00000			$0$
$181$	0	0	−1.00000			$\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	23.2379	1.70848
$186$	0	0
$187$	34.6410i	2.53320i
$188$	0	0
$189$	0	0
$190$	0	0
$191$	0	0		−	1.00000i	$-0.5\pi$
$191$	0	0			1.00000i	$0.5\pi$
$192$	0	0
$193$	0	0	1.00000			$0$
$193$	0	0	−1.00000			$\pi$
$194$	0	0
$195$	13.4164i	0.960769i
$196$	0	0
$197$	0	0	1.00000			$0$
$197$	0	0	−1.00000			$\pi$
$198$	0	0
$199$	26.0000i	1.84309i	0.388270	+	0.921546i	$0.373073\pi$
$199$	26.0000i	1.84309i	−0.388270	+	0.921546i	$0.626927\pi$
$200$	0	0
$201$	18.0000	1.26962
$202$	0	0
$203$	0	0
$204$	0	0
$205$	0	0
$206$	0	0
$207$	− 23.2379i	− 1.61515i
$208$	0	0
$209$	0	0
$210$	0	0
$211$	0	0		−	1.00000i	$-0.5\pi$
$211$	0	0			1.00000i	$0.5\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	7.74597i	0.528271i
$216$	0	0
$217$	0	0
$218$	0	0
$219$	0	0
$220$	0	0
$221$	−26.8328	−1.80497
$222$	0	0
$223$	0	0		−	1.00000i	$-0.5\pi$
$223$	0	0			1.00000i	$0.5\pi$
$224$	0	0
$225$	−15.0000	−1.00000
$226$	0	0
$227$	0	0		−	1.00000i	$-0.5\pi$
$227$	0	0			1.00000i	$0.5\pi$
$228$	0	0
$229$	0	0	1.00000			$0$
$229$	0	0	−1.00000			$\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	23.2379	1.52237	0.761183	−	0.648537i	$-0.224619\pi$
$233$	23.2379	1.52237	0.761183	+	0.648537i	$0.224619\pi$
$234$	0	0
$235$	− 17.3205i	− 1.12987i
$236$	0	0
$237$	−24.2487	−1.57512
$238$	0	0
$239$	0	0		−	1.00000i	$-0.5\pi$
$239$	0	0			1.00000i	$0.5\pi$
$240$	0	0
$241$	−22.0000	−1.41714	−0.708572	−	0.705638i	$-0.750660\pi$
$241$	−22.0000	−1.41714	−0.708572	+	0.705638i	$0.750660\pi$
$242$	0	0
$243$	− 15.5885i	− 1.00000i
$244$	0	0
$245$	15.6525	1.00000
$246$	0	0
$247$	0	0
$248$	0	0
$249$	0	0
$250$	0	0
$251$	31.3050i	1.97595i	0.154610	+	0.987976i	$0.450588\pi$
$251$	31.3050i	1.97595i	−0.154610	+	0.987976i	$0.549412\pi$
$252$	0	0
$253$	34.6410	2.17786
$254$	0	0
$255$	− 30.0000i	− 1.87867i
$256$	0	0
$257$	−7.74597	−0.483180	−0.241590	−	0.970378i	$-0.577669\pi$
$257$	−7.74597	−0.483180	−0.241590	+	0.970378i	$0.577669\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	13.4164	0.830455
$262$	0	0
$263$	− 23.2379i	− 1.43291i	−0.697633	−	0.716455i	$-0.745763\pi$
$263$	− 23.2379i	− 1.43291i	0.697633	−	0.716455i	$-0.254237\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	0	0
$268$	0	0
$269$	31.3050	1.90870	0.954348	−	0.298696i	$-0.0965517\pi$
$269$	31.3050	1.90870	0.954348	+	0.298696i	$0.0965517\pi$
$270$	0	0
$271$	2.00000i	0.121491i	0.998153	+	0.0607457i	$0.0193479\pi$
$271$	2.00000i	0.121491i	−0.998153	+	0.0607457i	$0.980652\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	− 22.3607i	− 1.34840i
$276$	0	0
$277$	−10.3923	−0.624413	−0.312207	−	0.950014i	$-0.601068\pi$
$277$	−10.3923	−0.624413	−0.312207	+	0.950014i	$0.601068\pi$
$278$	0	0
$279$	− 6.00000i	− 0.359211i
$280$	0	0
$281$	0	0	1.00000			$0$
$281$	0	0	−1.00000			$\pi$
$282$	0	0
$283$	− 31.1769i	− 1.85328i	−0.375956	−	0.926638i	$-0.622686\pi$
$283$	− 31.1769i	− 1.85328i	0.375956	−	0.926638i	$-0.377314\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	0	0
$288$	0	0
$289$	43.0000	2.52941
$290$	0	0
$291$	0	0
$292$	0	0
$293$	0	0	1.00000			$0$
$293$	0	0	−1.00000			$\pi$
$294$	0	0
$295$	10.0000i	0.582223i
$296$	0	0
$297$	23.2379	1.34840
$298$	0	0
$299$	26.8328i	1.55178i
$300$	0	0
$301$	0	0
$302$	0	0
$303$	7.74597i	0.444994i
$304$	0	0
$305$	0	0
$306$	0	0
$307$	24.2487i	1.38395i	0.721923	+	0.691974i	$0.243259\pi$
$307$	24.2487i	1.38395i	−0.721923	+	0.691974i	$0.756741\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	0	0		−	1.00000i	$-0.5\pi$
$311$	0	0			1.00000i	$0.5\pi$
$312$	0	0
$313$	0	0	1.00000			$0$
$313$	0	0	−1.00000			$\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	0	0	1.00000			$0$
$317$	0	0	−1.00000			$\pi$
$318$	0	0
$319$	20.0000i	1.11979i
$320$	0	0
$321$	0	0
$322$	0	0
$323$	0	0
$324$	0	0
$325$	17.3205	0.960769
$326$	0	0
$327$	0	0
$328$	0	0
$329$	0	0
$330$	0	0
$331$	0	0		−	1.00000i	$-0.5\pi$
$331$	0	0			1.00000i	$0.5\pi$
$332$	0	0
$333$	31.1769	1.70848
$334$	0	0
$335$	− 23.2379i	− 1.26962i
$336$	0	0
$337$	0	0	1.00000			$0$
$337$	0	0	−1.00000			$\pi$
$338$	0	0
$339$	13.4164i	0.728679i
$340$	0	0
$341$	8.94427	0.484359
$342$	0	0
$343$	0	0
$344$	0	0
$345$	−30.0000	−1.61515
$346$	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
$351$	18.0000i	0.960769i
$352$	0	0
$353$	−7.74597	−0.412276	−0.206138	−	0.978523i	$-0.566090\pi$
$353$	−7.74597	−0.412276	−0.206138	+	0.978523i	$0.566090\pi$
$354$	0	0
$355$	0	0
$356$	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
$363$	15.5885i	0.818182i
$364$	0	0
$365$	0	0
$366$	0	0
$367$	0	0		−	1.00000i	$-0.5\pi$
$367$	0	0			1.00000i	$0.5\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	0	0
$372$	0	0
$373$	−38.1051	−1.97301	−0.986504	−	0.163737i	$-0.947645\pi$
$373$	−38.1051	−1.97301	−0.986504	+	0.163737i	$0.947645\pi$
$374$	0	0
$375$	19.3649i	1.00000i
$376$	0	0
$377$	−15.4919	−0.797875
$378$	0	0
$379$	0	0		−	1.00000i	$-0.5\pi$
$379$	0	0			1.00000i	$0.5\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	38.7298i	1.97900i	0.144526	+	0.989501i	$0.453834\pi$
$383$	38.7298i	1.97900i	−0.144526	+	0.989501i	$0.546166\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	10.3923i	0.528271i
$388$	0	0
$389$	−4.47214	−0.226746	−0.113373	−	0.993552i	$-0.536166\pi$
$389$	−4.47214	−0.226746	−0.113373	+	0.993552i	$0.536166\pi$
$390$	0	0
$391$	− 60.0000i	− 3.03433i
$392$	0	0
$393$	−38.7298	−1.95366
$394$	0	0
$395$	31.3050i	1.57512i
$396$	0	0
$397$	−24.2487	−1.21701	−0.608504	−	0.793551i	$-0.708230\pi$
$397$	−24.2487	−1.21701	−0.608504	+	0.793551i	$0.708230\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	0	0	1.00000			$0$
$401$	0	0	−1.00000			$\pi$
$402$	0	0
$403$	6.92820i	0.345118i
$404$	0	0
$405$	−20.1246	−1.00000
$406$	0	0
$407$	46.4758i	2.30372i
$408$	0	0
$409$	34.0000	1.68119	0.840596	−	0.541663i	$-0.182205\pi$
$409$	34.0000	1.68119	0.840596	+	0.541663i	$0.182205\pi$
$410$	0	0
$411$	− 40.2492i	− 1.98535i
$412$	0	0
$413$	0	0
$414$	0	0
$415$	0	0
$416$	0	0
$417$	0	0
$418$	0	0
$419$	− 22.3607i	− 1.09239i	−0.837658	−	0.546195i	$-0.816076\pi$
$419$	− 22.3607i	− 1.09239i	0.837658	−	0.546195i	$-0.183924\pi$
$420$	0	0
$421$	0	0	1.00000			$0$
$421$	0	0	−1.00000			$\pi$
$422$	0	0
$423$	− 23.2379i	− 1.12987i
$424$	0	0
$425$	−38.7298	−1.87867
$426$	0	0
$427$	0	0
$428$	0	0
$429$	−26.8328	−1.29550
$430$	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
$435$	− 17.3205i	− 0.830455i
$436$	0	0
$437$	0	0
$438$	0	0
$439$	26.0000i	1.24091i	0.784241	+	0.620456i	$0.213053\pi$
$439$	26.0000i	1.24091i	−0.784241	+	0.620456i	$0.786947\pi$
$440$	0	0
$441$	21.0000	1.00000
$442$	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
$447$	38.7298i	1.83186i
$448$	0	0
$449$	0	0	1.00000			$0$
$449$	0	0	−1.00000			$\pi$
$450$	0	0
$451$	0	0
$452$	0	0
$453$	−38.1051	−1.79033
$454$	0	0
$455$	0	0
$456$	0	0
$457$	0	0	1.00000			$0$
$457$	0	0	−1.00000			$\pi$
$458$	0	0
$459$	− 40.2492i	− 1.87867i
$460$	0	0
$461$	31.3050	1.45802	0.729008	−	0.684505i	$-0.239981\pi$
$461$	31.3050	1.45802	0.729008	+	0.684505i	$0.239981\pi$
$462$	0	0
$463$	0	0		−	1.00000i	$-0.5\pi$
$463$	0	0			1.00000i	$0.5\pi$
$464$	0	0
$465$	−7.74597	−0.359211
$466$	0	0
$467$	0	0		−	1.00000i	$-0.5\pi$
$467$	0	0			1.00000i	$0.5\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	42.0000i	1.93526i
$472$	0	0
$473$	−15.4919	−0.712320
$474$	0	0
$475$	0	0
$476$	0	0
$477$	0	0
$478$	0	0
$479$	0	0		−	1.00000i	$-0.5\pi$
$479$	0	0			1.00000i	$0.5\pi$
$480$	0	0
$481$	−36.0000	−1.64146
$482$	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
$489$	−6.00000	−0.271329
$490$	0	0
$491$	− 4.47214i	− 0.201825i	−0.994895	−	0.100912i	$-0.967824\pi$
$491$	− 4.47214i	− 0.201825i	0.994895	−	0.100912i	$-0.0321762\pi$
$492$	0	0
$493$	34.6410	1.56015
$494$	0	0
$495$	− 30.0000i	− 1.34840i
$496$	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
$501$	−40.2492	−1.79820
$502$	0	0
$503$	38.7298i	1.72688i	0.504453	+	0.863439i	$0.331694\pi$
$503$	38.7298i	1.72688i	−0.504453	+	0.863439i	$0.668306\pi$
$504$	0	0
$505$	10.0000	0.444994
$506$	0	0
$507$	1.73205i	0.0769231i
$508$	0	0
$509$	−22.3607	−0.991120	−0.495560	−	0.868574i	$-0.665037\pi$
$509$	−22.3607	−0.991120	−0.495560	+	0.868574i	$0.665037\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	0	0
$514$	0	0
$515$	0	0
$516$	0	0
$517$	34.6410	1.52351
$518$	0	0
$519$	0	0
$520$	0	0
$521$	0	0	1.00000			$0$
$521$	0	0	−1.00000			$\pi$
$522$	0	0
$523$	38.1051i	1.66622i	0.553107	+	0.833110i	$0.313442\pi$
$523$	38.1051i	1.66622i	−0.553107	+	0.833110i	$0.686558\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	− 15.4919i	− 0.674839i
$528$	0	0
$529$	−37.0000	−1.60870
$530$	0	0
$531$	13.4164i	0.582223i
$532$	0	0
$533$	0	0
$534$	0	0
$535$	0	0
$536$	0	0
$537$	−38.7298	−1.67132
$538$	0	0
$539$	31.3050i	1.34840i
$540$	0	0
$541$	0	0	1.00000			$0$
$541$	0	0	−1.00000			$\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	0	0
$546$	0	0
$547$	− 45.0333i	− 1.92549i	−0.270418	−	0.962743i	$-0.587162\pi$
$547$	− 45.0333i	− 1.92549i	0.270418	−	0.962743i	$-0.412838\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	0	0
$552$	0	0
$553$	0	0
$554$	0	0
$555$	− 40.2492i	− 1.70848i
$556$	0	0
$557$	0	0	1.00000			$0$
$557$	0	0	−1.00000			$\pi$
$558$	0	0
$559$	− 12.0000i	− 0.507546i
$560$	0	0
$561$	60.0000	2.53320
$562$	0	0
$563$	0	0		−	1.00000i	$-0.5\pi$
$563$	0	0			1.00000i	$0.5\pi$
$564$	0	0
$565$	17.3205	0.728679
$566$	0	0
$567$	0	0
$568$	0	0
$569$	0	0	1.00000			$0$
$569$	0	0	−1.00000			$\pi$
$570$	0	0
$571$	0	0		−	1.00000i	$-0.5\pi$
$571$	0	0			1.00000i	$0.5\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	38.7298i	1.61515i
$576$	0	0
$577$	0	0	1.00000			$0$
$577$	0	0	−1.00000			$\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	0	0
$582$	0	0
$583$	0	0
$584$	0	0
$585$	23.2379	0.960769
$586$	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
$593$	−7.74597	−0.318089	−0.159044	−	0.987271i	$-0.550841\pi$
$593$	−7.74597	−0.318089	−0.159044	+	0.987271i	$0.550841\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	45.0333	1.84309
$598$	0	0
$599$	0	0		−	1.00000i	$-0.5\pi$
$599$	0	0			1.00000i	$0.5\pi$
$600$	0	0
$601$	−22.0000	−0.897399	−0.448699	−	0.893683i	$-0.648113\pi$
$601$	−22.0000	−0.897399	−0.448699	+	0.893683i	$0.648113\pi$
$602$	0	0
$603$	− 31.1769i	− 1.26962i
$604$	0	0
$605$	20.1246	0.818182
$606$	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
$611$	26.8328i	1.08554i
$612$	0	0
$613$	−38.1051	−1.53905	−0.769526	−	0.638616i	$-0.779507\pi$
$613$	−38.1051	−1.53905	−0.769526	+	0.638616i	$0.779507\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−38.7298	−1.55920	−0.779602	−	0.626275i	$-0.784579\pi$
$617$	−38.7298	−1.55920	−0.779602	+	0.626275i	$0.784579\pi$
$618$	0	0
$619$	0	0		−	1.00000i	$-0.5\pi$
$619$	0	0			1.00000i	$0.5\pi$
$620$	0	0
$621$	−40.2492	−1.61515
$622$	0	0
$623$	0	0
$624$	0	0
$625$	25.0000	1.00000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	80.4984	3.20968
$630$	0	0
$631$	− 38.0000i	− 1.51276i	−0.654135	−	0.756378i	$-0.726967\pi$
$631$	− 38.0000i	− 1.51276i	0.654135	−	0.756378i	$-0.273033\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	0	0
$636$	0	0
$637$	−24.2487	−0.960769
$638$	0	0
$639$	0	0
$640$	0	0
$641$	0	0	1.00000			$0$
$641$	0	0	−1.00000			$\pi$
$642$	0	0
$643$	− 3.46410i	− 0.136611i	−0.997664	−	0.0683054i	$-0.978241\pi$
$643$	− 3.46410i	− 0.136611i	0.997664	−	0.0683054i	$-0.0217592\pi$
$644$	0	0
$645$	13.4164	0.528271
$646$	0	0
$647$	− 23.2379i	− 0.913576i	−0.889576	−	0.456788i	$-0.849000\pi$
$647$	− 23.2379i	− 0.913576i	0.889576	−	0.456788i	$-0.151000\pi$
$648$	0	0
$649$	−20.0000	−0.785069
$650$	0	0
$651$	0	0
$652$	0	0
$653$	0	0	1.00000			$0$
$653$	0	0	−1.00000			$\pi$
$654$	0	0
$655$	50.0000i	1.95366i
$656$	0	0
$657$	0	0
$658$	0	0
$659$	49.1935i	1.91631i	0.286256	+	0.958153i	$0.407589\pi$
$659$	49.1935i	1.91631i	−0.286256	+	0.958153i	$0.592411\pi$
$660$	0	0
$661$	0	0	1.00000			$0$
$661$	0	0	−1.00000			$\pi$
$662$	0	0
$663$	46.4758i	1.80497i
$664$	0	0
$665$	0	0
$666$	0	0
$667$	− 34.6410i	− 1.34131i
$668$	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
$675$	25.9808i	1.00000i
$676$	0	0
$677$	0	0	1.00000			$0$
$677$	0	0	−1.00000			$\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	0	0
$682$	0	0
$683$	0	0		−	1.00000i	$-0.5\pi$
$683$	0	0			1.00000i	$0.5\pi$
$684$	0	0
$685$	−51.9615	−1.98535
$686$	0	0
$687$	0	0
$688$	0	0
$689$	0	0
$690$	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
$699$	− 40.2492i	− 1.52237i
$700$	0	0
$701$	49.1935	1.85801	0.929006	−	0.370064i	$-0.120664\pi$
$701$	49.1935	1.85801	0.929006	+	0.370064i	$0.120664\pi$
$702$	0	0
$703$	0	0
$704$	0	0
$705$	−30.0000	−1.12987
$706$	0	0
$707$	0	0
$708$	0	0
$709$	0	0	1.00000			$0$
$709$	0	0	−1.00000			$\pi$
$710$	0	0
$711$	42.0000i	1.57512i
$712$	0	0
$713$	−15.4919	−0.580177
$714$	0	0
$715$	34.6410i	1.29550i
$716$	0	0
$717$	0	0
$718$	0	0
$719$	0	0		−	1.00000i	$-0.5\pi$
$719$	0	0			1.00000i	$0.5\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	38.1051i	1.41714i
$724$	0	0
$725$	−22.3607	−0.830455
$726$	0	0
$727$	0	0		−	1.00000i	$-0.5\pi$
$727$	0	0			1.00000i	$0.5\pi$
$728$	0	0
$729$	−27.0000	−1.00000
$730$	0	0
$731$	26.8328i	0.992448i
$732$	0	0
$733$	31.1769	1.15155	0.575773	−	0.817610i	$-0.304701\pi$
$733$	31.1769	1.15155	0.575773	+	0.817610i	$0.304701\pi$
$734$	0	0
$735$	− 27.1109i	− 1.00000i
$736$	0	0
$737$	46.4758	1.71196
$738$	0	0
$739$	0	0		−	1.00000i	$-0.5\pi$
$739$	0	0			1.00000i	$0.5\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	− 54.2218i	− 1.98920i	−0.103765	−	0.994602i	$-0.533089\pi$
$743$	− 54.2218i	− 1.98920i	0.103765	−	0.994602i	$-0.466911\pi$
$744$	0	0
$745$	50.0000	1.83186
$746$	0	0
$747$	0	0
$748$	0	0
$749$	0	0
$750$	0	0
$751$	2.00000i	0.0729810i	0.999334	+	0.0364905i	$0.0116179\pi$
$751$	2.00000i	0.0729810i	−0.999334	+	0.0364905i	$0.988382\pi$
$752$	0	0
$753$	54.2218	1.97595
$754$	0	0
$755$	49.1935i	1.79033i
$756$	0	0
$757$	45.0333	1.63676	0.818382	−	0.574675i	$-0.194871\pi$
$757$	45.0333	1.63676	0.818382	+	0.574675i	$0.194871\pi$
$758$	0	0
$759$	− 60.0000i	− 2.17786i
$760$	0	0
$761$	0	0	1.00000			$0$
$761$	0	0	−1.00000			$\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	−51.9615	−1.87867
$766$	0	0
$767$	− 15.4919i	− 0.559381i
$768$	0	0
$769$	34.0000	1.22607	0.613036	−	0.790055i	$-0.289948\pi$
$769$	34.0000	1.22607	0.613036	+	0.790055i	$0.289948\pi$
$770$	0	0
$771$	13.4164i	0.483180i
$772$	0	0
$773$	0	0	1.00000			$0$
$773$	0	0	−1.00000			$\pi$
$774$	0	0
$775$	10.0000i	0.359211i
$776$	0	0
$777$	0	0
$778$	0	0
$779$	0	0
$780$	0	0
$781$	0	0
$782$	0	0
$783$	− 23.2379i	− 0.830455i
$784$	0	0
$785$	54.2218	1.93526
$786$	0	0
$787$	24.2487i	0.864373i	0.901784	+	0.432187i	$0.142258\pi$
$787$	24.2487i	0.864373i	−0.901784	+	0.432187i	$0.857742\pi$
$788$	0	0
$789$	−40.2492	−1.43291
$790$	0	0
$791$	0	0
$792$	0	0
$793$	0	0
$794$	0	0
$795$	0	0
$796$	0	0
$797$	0	0	1.00000			$0$
$797$	0	0	−1.00000			$\pi$
$798$	0	0
$799$	− 60.0000i	− 2.12265i
$800$	0	0
$801$	0	0
$802$	0	0
$803$	0	0
$804$	0	0
$805$	0	0
$806$	0	0
$807$	− 54.2218i	− 1.90870i
$808$	0	0
$809$	0	0	1.00000			$0$
$809$	0	0	−1.00000			$\pi$
$810$	0	0
$811$	0	0		−	1.00000i	$-0.5\pi$
$811$	0	0			1.00000i	$0.5\pi$
$812$	0	0
$813$	3.46410	0.121491
$814$	0	0
$815$	7.74597i	0.271329i
$816$	0	0
$817$	0	0
$818$	0	0
$819$	0	0
$820$	0	0
$821$	49.1935	1.71686	0.858432	−	0.512927i	$-0.171439\pi$
$821$	49.1935	1.71686	0.858432	+	0.512927i	$0.171439\pi$
$822$	0	0
$823$	0	0		−	1.00000i	$-0.5\pi$
$823$	0	0			1.00000i	$0.5\pi$
$824$	0	0
$825$	−38.7298	−1.34840
$826$	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
$831$	18.0000i	0.624413i
$832$	0	0
$833$	54.2218	1.87867
$834$	0	0
$835$	51.9615i	1.79820i
$836$	0	0
$837$	−10.3923	−0.359211
$838$	0	0
$839$	0	0		−	1.00000i	$-0.5\pi$
$839$	0	0			1.00000i	$0.5\pi$
$840$	0	0
$841$	−9.00000	−0.310345
$842$	0	0
$843$	0	0
$844$	0	0
$845$	2.23607	0.0769231
$846$	0	0
$847$	0	0
$848$	0	0
$849$	−54.0000	−1.85328
$850$	0	0
$851$	− 80.4984i	− 2.75945i
$852$	0	0
$853$	−38.1051	−1.30469	−0.652347	−	0.757920i	$-0.726216\pi$
$853$	−38.1051	−1.30469	−0.652347	+	0.757920i	$0.726216\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	23.2379	0.793792	0.396896	−	0.917864i	$-0.370087\pi$
$857$	23.2379	0.793792	0.396896	+	0.917864i	$0.370087\pi$
$858$	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
$863$	− 54.2218i	− 1.84573i	−0.385123	−	0.922865i	$-0.625841\pi$
$863$	− 54.2218i	− 1.84573i	0.385123	−	0.922865i	$-0.374159\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	− 74.4782i	− 2.52941i
$868$	0	0
$869$	−62.6099	−2.12390
$870$	0	0
$871$	36.0000i	1.21981i
$872$	0	0
$873$	0	0
$874$	0	0
$875$	0	0
$876$	0	0
$877$	58.8897	1.98856	0.994282	−	0.106783i	$-0.0340549\pi$
$877$	58.8897	1.98856	0.994282	+	0.106783i	$0.0340549\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	0	0	1.00000			$0$
$881$	0	0	−1.00000			$\pi$
$882$	0	0
$883$	− 31.1769i	− 1.04919i	−0.851353	−	0.524593i	$-0.824217\pi$
$883$	− 31.1769i	− 1.04919i	0.851353	−	0.524593i	$-0.175783\pi$
$884$	0	0
$885$	17.3205	0.582223
$886$	0	0
$887$	38.7298i	1.30042i	0.759754	+	0.650210i	$0.225319\pi$
$887$	38.7298i	1.30042i	−0.759754	+	0.650210i	$0.774681\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	− 40.2492i	− 1.34840i
$892$	0	0
$893$	0	0
$894$	0	0
$895$	50.0000i	1.67132i
$896$	0	0
$897$	46.4758	1.55178
$898$	0	0
$899$	− 8.94427i	− 0.298308i
$900$	0	0
$901$	0	0
$902$	0	0
$903$	0	0
$904$	0	0
$905$	0	0
$906$	0	0
$907$	− 58.8897i	− 1.95540i	−0.210003	−	0.977701i	$-0.567348\pi$
$907$	− 58.8897i	− 1.95540i	0.210003	−	0.977701i	$-0.432652\pi$
$908$	0	0
$909$	13.4164	0.444994
$910$	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
$919$	26.0000i	0.857661i	0.903385	+	0.428830i	$0.141074\pi$
$919$	26.0000i	0.857661i	−0.903385	+	0.428830i	$0.858926\pi$
$920$	0	0
$921$	42.0000	1.38395
$922$	0	0
$923$	0	0
$924$	0	0
$925$	−51.9615	−1.70848
$926$	0	0
$927$	0	0
$928$	0	0
$929$	0	0	1.00000			$0$
$929$	0	0	−1.00000			$\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	0	0
$934$	0	0
$935$	− 77.4597i	− 2.53320i
$936$	0	0
$937$	0	0	1.00000			$0$
$937$	0	0	−1.00000			$\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−58.1378	−1.89524	−0.947619	−	0.319404i	$-0.896517\pi$
$941$	−58.1378	−1.89524	−0.947619	+	0.319404i	$0.896517\pi$
$942$	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
$949$	0	0
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−38.7298	−1.25458	−0.627291	−	0.778785i	$-0.715836\pi$
$953$	−38.7298	−1.25458	−0.627291	+	0.778785i	$0.715836\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	34.6410	1.11979
$958$	0	0
$959$	0	0
$960$	0	0
$961$	27.0000	0.870968
$962$	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
$971$	31.3050i	1.00462i	0.864687	+	0.502312i	$0.167517\pi$
$971$	31.3050i	1.00462i	−0.864687	+	0.502312i	$0.832483\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	− 30.0000i	− 0.960769i
$976$	0	0
$977$	54.2218	1.73471	0.867354	−	0.497692i	$-0.165819\pi$
$977$	54.2218	1.73471	0.867354	+	0.497692i	$0.165819\pi$
$978$	0	0
$979$	0	0
$980$	0	0
$981$	0	0
$982$	0	0
$983$	7.74597i	0.247058i	0.992341	+	0.123529i	$0.0394212\pi$
$983$	7.74597i	0.247058i	−0.992341	+	0.123529i	$0.960579\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	0	0
$988$	0	0
$989$	26.8328	0.853234
$990$	0	0
$991$	− 62.0000i	− 1.96949i	−0.173990	−	0.984747i	$-0.555666\pi$
$991$	− 62.0000i	− 1.96949i	0.173990	−	0.984747i	$-0.444334\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	− 58.1378i	− 1.84309i
$996$	0	0
$997$	45.0333	1.42622	0.713110	−	0.701052i	$-0.247286\pi$
$997$	45.0333	1.42622	0.713110	+	0.701052i	$0.247286\pi$
$998$	0	0
$999$	− 54.0000i	− 1.70848i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	960.2.m.a.479.1	✓	8
3.2	odd	2	inner	960.2.m.a.479.4	yes	8
4.3	odd	2	inner	960.2.m.a.479.6	yes	8
5.4	even	2	inner	960.2.m.a.479.5	yes	8
8.3	odd	2	inner	960.2.m.a.479.3	yes	8
8.5	even	2	inner	960.2.m.a.479.8	yes	8
12.11	even	2	inner	960.2.m.a.479.7	yes	8
15.14	odd	2	inner	960.2.m.a.479.8	yes	8
20.19	odd	2	inner	960.2.m.a.479.2	yes	8
24.5	odd	2	inner	960.2.m.a.479.5	yes	8
24.11	even	2	inner	960.2.m.a.479.2	yes	8
40.19	odd	2	inner	960.2.m.a.479.7	yes	8
40.29	even	2	inner	960.2.m.a.479.4	yes	8
60.59	even	2	inner	960.2.m.a.479.3	yes	8
120.29	odd	2	CM	960.2.m.a.479.1	✓	8
120.59	even	2	inner	960.2.m.a.479.6	yes	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
960.2.m.a.479.1	✓	8	1.1	even	1	trivial
960.2.m.a.479.1	✓	8	120.29	odd	2	CM
960.2.m.a.479.2	yes	8	20.19	odd	2	inner
960.2.m.a.479.2	yes	8	24.11	even	2	inner
960.2.m.a.479.3	yes	8	8.3	odd	2	inner
960.2.m.a.479.3	yes	8	60.59	even	2	inner
960.2.m.a.479.4	yes	8	3.2	odd	2	inner
960.2.m.a.479.4	yes	8	40.29	even	2	inner
960.2.m.a.479.5	yes	8	5.4	even	2	inner
960.2.m.a.479.5	yes	8	24.5	odd	2	inner
960.2.m.a.479.6	yes	8	4.3	odd	2	inner
960.2.m.a.479.6	yes	8	120.59	even	2	inner
960.2.m.a.479.7	yes	8	12.11	even	2	inner
960.2.m.a.479.7	yes	8	40.19	odd	2	inner
960.2.m.a.479.8	yes	8	8.5	even	2	inner
960.2.m.a.479.8	yes	8	15.14	odd	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:	\( N \)	\(=\)	\( 960 = 2^{6} \cdot 3 \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	960.m (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-1.73205i q^{3} -2.23607 q^{5} -3.00000 q^{9} +O(q^{10})\)	\(q-1.73205i q^{3} -2.23607 q^{5} -3.00000 q^{9} -4.47214i q^{11} +3.46410 q^{13} +3.87298i q^{15} -7.74597 q^{17} +7.74597i q^{23} +5.00000 q^{25} +5.19615i q^{27} -4.47214 q^{29} +2.00000i q^{31} -7.74597 q^{33} -10.3923 q^{37} -6.00000i q^{39} -3.46410i q^{43} +6.70820 q^{45} +7.74597i q^{47} -7.00000 q^{49} +13.4164i q^{51} +10.0000i q^{55} -4.47214i q^{59} -7.74597 q^{65} +10.3923i q^{67} +13.4164 q^{69} -8.66025i q^{75} -14.0000i q^{79} +9.00000 q^{81} +17.3205 q^{85} +7.74597i q^{87} +3.46410 q^{93} +13.4164i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 24 q^{9}+O(q^{10}) \) 8 * q - 24 * q^9	\( 8 q - 24 q^{9} + 40 q^{25} - 56 q^{49} + 72 q^{81}+O(q^{100}) \) 8 * q - 24 * q^9 + 40 * q^25 - 56 * q^49 + 72 * q^81

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more