LMFDB - Embedded newform 832.2.p.a.337.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [832,2,Mod(337,832)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(832, base_ring=CyclotomicField(4))

chi = DirichletCharacter(H, H._module([0, 3, 2]))

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

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

chi := DirichletCharacter("832.337");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 832 = 2^{6} \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	832.p (of order $4$, degree $2$, 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:	$6.64355344817$
Analytic rank:	$0$
Dimension:	$8$
Relative dimension:	$4$ over $\Q(i)$
Coefficient field:	8.0.959512576.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + 7x^{4} + 81 $ x^8 + 7*x^4 + 81
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 208)
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			337.1
Root			$0.819051 + 1.52616i$ of defining polynomial
Character	$\chi$	$=$	832.337
Dual form			832.2.p.a.753.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q+(-2.15831 - 2.15831i) q^{3} +(-0.707107 - 0.707107i) q^{5} +0.223888 q^{7} +6.31662i q^{9} +(-1.41421 - 1.41421i) q^{11} +(-3.34521 + 1.34521i) q^{13} +3.05231i q^{15} +3.00000 q^{17} +(-4.46653 + 4.46653i) q^{19} +(-0.483219 - 0.483219i) q^{21} +6.31662i q^{23} -4.00000i q^{25} +(7.15831 - 7.15831i) q^{27} +(0.316625 + 0.316625i) q^{29} -4.24264i q^{31} +6.10463i q^{33} +(-0.158312 - 0.158312i) q^{35} +(6.58785 + 6.58785i) q^{37} +(10.1234 + 4.31662i) q^{39} +(5.47494 - 5.47494i) q^{43} +(4.46653 - 4.46653i) q^{45} +10.5712i q^{47} -6.94987 q^{49} +(-6.47494 - 6.47494i) q^{51} +(3.63325 - 3.63325i) q^{53} +2.00000i q^{55} +19.2803 q^{57} +(7.74273 + 7.74273i) q^{59} +(4.00000 + 4.00000i) q^{61} +1.41421i q^{63} +(3.31662 + 1.41421i) q^{65} +(4.69042 - 4.69042i) q^{67} +(13.6332 - 13.6332i) q^{69} -0.671663 q^{71} -3.79487 q^{73} +(-8.63325 + 8.63325i) q^{75} +(-0.316625 - 0.316625i) q^{77} +10.9499 q^{79} -11.9499 q^{81} +(-11.9854 + 11.9854i) q^{83} +(-2.12132 - 2.12132i) q^{85} -1.36675i q^{87} -14.0712 q^{89} +(-0.748950 + 0.301175i) q^{91} +(-9.15694 + 9.15694i) q^{93} +6.31662 q^{95} +8.48528i q^{97} +(8.93306 - 8.93306i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 4 q^{3} - 8 q^{13} + 24 q^{17} + 44 q^{27} - 24 q^{29} + 12 q^{35} + 4 q^{43} + 24 q^{49} - 12 q^{51} - 24 q^{53} + 32 q^{61} + 56 q^{69} - 16 q^{75} + 24 q^{77} + 8 q^{79} - 16 q^{81} - 44 q^{91}+ \cdots + 24 q^{95}+O(q^{100}) $ 8 * q - 4 * q^3 - 8 * q^13 + 24 * q^17 + 44 * q^27 - 24 * q^29 + 12 * q^35 + 4 * q^43 + 24 * q^49 - 12 * q^51 - 24 * q^53 + 32 * q^61 + 56 * q^69 - 16 * q^75 + 24 * q^77 + 8 * q^79 - 16 * q^81 - 44 * q^91 + 24 * q^95

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/832\mathbb{Z}\right)^\times$.

$n$	$261$	$703$	$769$
$\chi(n)$	$e\left(\frac{3}{4}\right)$	$1$	$-1$

Coefficient data

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

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

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

$2$		0			0
$2$		0			0
$3$	−2.15831	−	2.15831i	−1.24610	−	1.24610i	−0.957427	−	0.288675i	$-0.906785\pi$
$3$	−2.15831	−	2.15831i	−1.24610	−	1.24610i	−0.288675	−	0.957427i	$-0.593215\pi$
$4$		0			0
$5$	−0.707107	−	0.707107i	−0.316228	−	0.316228i	0.531089	−	0.847316i	$-0.321783\pi$
$5$	−0.707107	−	0.707107i	−0.316228	−	0.316228i	−0.847316	+	0.531089i	$0.821783\pi$
$6$		0			0
$7$	0.223888			0.0846215			0.0423108	−	0.999104i	$-0.486528\pi$
$7$	0.223888			0.0846215			0.0423108	+	0.999104i	$0.486528\pi$
$8$		0			0
$9$			6.31662i			2.10554i
$10$		0			0
$11$	−1.41421	−	1.41421i	−0.426401	−	0.426401i	0.460999	−	0.887401i	$-0.347491\pi$
$11$	−1.41421	−	1.41421i	−0.426401	−	0.426401i	−0.887401	+	0.460999i	$0.847491\pi$
$12$		0			0
$13$	−3.34521	+	1.34521i	−0.927794	+	0.373094i
$13$	−3.34521	+	1.34521i	−0.927794	+	0.373094i
$14$		0			0
$15$			3.05231i			0.788104i
$16$		0			0
$17$	3.00000			0.727607			0.363803	−	0.931476i	$-0.381478\pi$
$17$	3.00000			0.727607			0.363803	+	0.931476i	$0.381478\pi$
$18$		0			0
$19$	−4.46653	+	4.46653i	−1.02469	+	1.02469i	−0.0250045	+	0.999687i	$0.507960\pi$
$19$	−4.46653	+	4.46653i	−1.02469	+	1.02469i	−0.999687	+	0.0250045i	$0.992040\pi$
$20$		0			0
$21$	−0.483219	−	0.483219i	−0.105447	−	0.105447i
$22$		0			0
$23$			6.31662i			1.31711i	0.752534	+	0.658554i	$0.228831\pi$
$23$			6.31662i			1.31711i	−0.752534	+	0.658554i	$0.771169\pi$
$24$		0			0
$25$		−	4.00000i		−	0.800000i
$26$		0			0
$27$	7.15831	−	7.15831i	1.37762	−	1.37762i
$28$		0			0
$29$	0.316625	+	0.316625i	0.0587957	+	0.0587957i	0.735893	−	0.677098i	$-0.236762\pi$
$29$	0.316625	+	0.316625i	0.0587957	+	0.0587957i	−0.677098	+	0.735893i	$0.736762\pi$
$30$		0			0
$31$		−	4.24264i		−	0.762001i	−0.924575	−	0.381000i	$-0.875580\pi$
$31$		−	4.24264i		−	0.762001i	0.924575	−	0.381000i	$-0.124420\pi$
$32$		0			0
$33$			6.10463i			1.06268i
$34$		0			0
$35$	−0.158312	−	0.158312i	−0.0267597	−	0.0267597i
$36$		0			0
$37$	6.58785	+	6.58785i	1.08304	+	1.08304i	0.996225	+	0.0868108i	$0.0276676\pi$
$37$	6.58785	+	6.58785i	1.08304	+	1.08304i	0.0868108	+	0.996225i	$0.472332\pi$
$38$		0			0
$39$	10.1234	+	4.31662i	1.62104	+	0.691213i
$40$		0			0
$41$		0			0			−	1.00000i	$-0.5\pi$
$41$		0			0				1.00000i	$0.5\pi$
$42$		0			0
$43$	5.47494	−	5.47494i	0.834920	−	0.834920i	−0.153265	−	0.988185i	$-0.548979\pi$
$43$	5.47494	−	5.47494i	0.834920	−	0.834920i	0.988185	+	0.153265i	$0.0489788\pi$
$44$		0			0
$45$	4.46653	−	4.46653i	0.665831	−	0.665831i
$46$		0			0
$47$			10.5712i			1.54196i	0.636858	+	0.770981i	$0.280234\pi$
$47$			10.5712i			1.54196i	−0.636858	+	0.770981i	$0.719766\pi$
$48$		0			0
$49$	−6.94987			−0.992839
$50$		0			0
$51$	−6.47494	−	6.47494i	−0.906673	−	0.906673i
$52$		0			0
$53$	3.63325	−	3.63325i	0.499065	−	0.499065i	−0.412082	−	0.911147i	$-0.635198\pi$
$53$	3.63325	−	3.63325i	0.499065	−	0.499065i	0.911147	+	0.412082i	$0.135198\pi$
$54$		0			0
$55$			2.00000i			0.269680i
$56$		0			0
$57$	19.2803			2.55374
$58$		0			0
$59$	7.74273	+	7.74273i	1.00802	+	1.00802i	0.999968	+	0.00805004i	$0.00256244\pi$
$59$	7.74273	+	7.74273i	1.00802	+	1.00802i	0.00805004	+	0.999968i	$0.497438\pi$
$60$		0			0
$61$	4.00000	+	4.00000i	0.512148	+	0.512148i	0.915184	−	0.403036i	$-0.132045\pi$
$61$	4.00000	+	4.00000i	0.512148	+	0.512148i	−0.403036	+	0.915184i	$0.632045\pi$
$62$		0			0
$63$			1.41421i			0.178174i
$64$		0			0
$65$	3.31662	+	1.41421i	0.411377	+	0.175412i
$66$		0			0
$67$	4.69042	−	4.69042i	0.573025	−	0.573025i	−0.359947	−	0.932973i	$-0.617205\pi$
$67$	4.69042	−	4.69042i	0.573025	−	0.573025i	0.932973	+	0.359947i	$0.117205\pi$
$68$		0			0
$69$	13.6332	−	13.6332i	1.64125	−	1.64125i
$70$		0			0
$71$	−0.671663			−0.0797117			−0.0398558	−	0.999205i	$-0.512690\pi$
$71$	−0.671663			−0.0797117			−0.0398558	+	0.999205i	$0.512690\pi$
$72$		0			0
$73$	−3.79487			−0.444155			−0.222078	−	0.975029i	$-0.571284\pi$
$73$	−3.79487			−0.444155			−0.222078	+	0.975029i	$0.571284\pi$
$74$		0			0
$75$	−8.63325	+	8.63325i	−0.996882	+	0.996882i
$76$		0			0
$77$	−0.316625	−	0.316625i	−0.0360827	−	0.0360827i
$78$		0			0
$79$	10.9499			1.23196			0.615979	−	0.787763i	$-0.288761\pi$
$79$	10.9499			1.23196			0.615979	+	0.787763i	$0.288761\pi$
$80$		0			0
$81$	−11.9499			−1.32776
$82$		0			0
$83$	−11.9854	+	11.9854i	−1.31557	+	1.31557i	−0.398318	+	0.917247i	$0.630406\pi$
$83$	−11.9854	+	11.9854i	−1.31557	+	1.31557i	−0.917247	+	0.398318i	$0.869594\pi$
$84$		0			0
$85$	−2.12132	−	2.12132i	−0.230089	−	0.230089i
$86$		0			0
$87$		−	1.36675i		−	0.146531i
$88$		0			0
$89$	−14.0712			−1.49155			−0.745775	−	0.666198i	$-0.767920\pi$
$89$	−14.0712			−1.49155			−0.745775	+	0.666198i	$0.767920\pi$
$90$		0			0
$91$	−0.748950	+	0.301175i	−0.0785113	+	0.0315717i
$92$		0			0
$93$	−9.15694	+	9.15694i	−0.949531	+	0.949531i
$94$		0			0
$95$	6.31662			0.648072
$96$		0			0
$97$			8.48528i			0.861550i	0.902459	+	0.430775i	$0.141760\pi$
$97$			8.48528i			0.861550i	−0.902459	+	0.430775i	$0.858240\pi$
$98$		0			0
$99$	8.93306	−	8.93306i	0.897806	−	0.897806i
$100$		0			0
$101$	−6.31662	+	6.31662i	−0.628528	+	0.628528i	−0.947698	−	0.319170i	$-0.896596\pi$
$101$	−6.31662	+	6.31662i	−0.628528	+	0.628528i	0.319170	+	0.947698i	$0.396596\pi$
$102$		0			0
$103$			0.949874i			0.0935939i	0.998904	+	0.0467970i	$0.0149014\pi$
$103$			0.949874i			0.0935939i	−0.998904	+	0.0467970i	$0.985099\pi$
$104$		0			0
$105$			0.683375i			0.0666906i
$106$		0			0
$107$	−9.63325	+	9.63325i	−0.931281	+	0.931281i	−0.997786	−	0.0665047i	$-0.978815\pi$
$107$	−9.63325	+	9.63325i	−0.931281	+	0.931281i	0.0665047	+	0.997786i	$0.478815\pi$
$108$		0			0
$109$	2.34521	−	2.34521i	0.224630	−	0.224630i	−0.585815	−	0.810445i	$-0.699225\pi$
$109$	2.34521	−	2.34521i	0.224630	−	0.224630i	0.810445	+	0.585815i	$0.199225\pi$
$110$		0			0
$111$		−	28.4373i		−	2.69915i
$112$		0			0
$113$	−6.00000			−0.564433			−0.282216	−	0.959351i	$-0.591070\pi$
$113$	−6.00000			−0.564433			−0.282216	+	0.959351i	$0.591070\pi$
$114$		0			0
$115$	4.46653	−	4.46653i	0.416506	−	0.416506i
$116$		0			0
$117$	−8.49717	−	21.1304i	−0.785564	−	1.95351i
$118$		0			0
$119$	0.671663			0.0615712
$120$		0			0
$121$		−	7.00000i		−	0.636364i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	−6.36396	+	6.36396i	−0.569210	+	0.569210i
$126$		0			0
$127$	−2.94987			−0.261759			−0.130880	−	0.991398i	$-0.541780\pi$
$127$	−2.94987			−0.261759			−0.130880	+	0.991398i	$0.541780\pi$
$128$		0			0
$129$	−23.6332			−2.08079
$130$		0			0
$131$	−2.84169	−	2.84169i	−0.248279	−	0.248279i	0.571985	−	0.820264i	$-0.306174\pi$
$131$	−2.84169	−	2.84169i	−0.248279	−	0.248279i	−0.820264	+	0.571985i	$0.806174\pi$
$132$		0			0
$133$	−1.00000	+	1.00000i	−0.0867110	+	0.0867110i
$134$		0			0
$135$	−10.1234			−0.871282
$136$		0			0
$137$	−14.0712			−1.20219			−0.601094	−	0.799178i	$-0.705268\pi$
$137$	−14.0712			−1.20219			−0.601094	+	0.799178i	$0.705268\pi$
$138$		0			0
$139$	2.47494	−	2.47494i	0.209921	−	0.209921i	−0.594313	−	0.804234i	$-0.702576\pi$
$139$	2.47494	−	2.47494i	0.209921	−	0.209921i	0.804234	+	0.594313i	$0.202576\pi$
$140$		0			0
$141$	22.8159	−	22.8159i	1.92144	−	1.92144i
$142$		0			0
$143$	6.63325	+	2.82843i	0.554700	+	0.236525i
$144$		0			0
$145$		−	0.447775i		−	0.0371857i
$146$		0			0
$147$	15.0000	+	15.0000i	1.23718	+	1.23718i
$148$		0			0
$149$	−7.07107	−	7.07107i	−0.579284	−	0.579284i	0.355422	−	0.934706i	$-0.384337\pi$
$149$	−7.07107	−	7.07107i	−0.579284	−	0.579284i	−0.934706	+	0.355422i	$0.884337\pi$
$150$		0			0
$151$	4.46653			0.363481			0.181740	−	0.983347i	$-0.441827\pi$
$151$	4.46653			0.363481			0.181740	+	0.983347i	$0.441827\pi$
$152$		0			0
$153$			18.9499i			1.53201i
$154$		0			0
$155$	−3.00000	+	3.00000i	−0.240966	+	0.240966i
$156$		0			0
$157$	7.94987	+	7.94987i	0.634469	+	0.634469i	0.949186	−	0.314717i	$-0.101909\pi$
$157$	7.94987	+	7.94987i	0.634469	+	0.634469i	−0.314717	+	0.949186i	$0.601909\pi$
$158$		0			0
$159$	−15.6834			−1.24377
$160$		0			0
$161$			1.41421i			0.111456i
$162$		0			0
$163$	−8.70917	+	8.70917i	−0.682155	+	0.682155i	−0.960485	−	0.278331i	$-0.910219\pi$
$163$	−8.70917	+	8.70917i	−0.682155	+	0.682155i	0.278331	+	0.960485i	$0.410219\pi$
$164$		0			0
$165$	4.31662	−	4.31662i	0.336049	−	0.336049i
$166$		0			0
$167$	−14.0712			−1.08887			−0.544433	−	0.838804i	$-0.683255\pi$
$167$	−14.0712			−1.08887			−0.544433	+	0.838804i	$0.683255\pi$
$168$		0			0
$169$	9.38083	−	9.00000i	0.721602	−	0.692308i
$170$		0			0
$171$	−28.2134	−	28.2134i	−2.15753	−	2.15753i
$172$		0			0
$173$	15.3166	+	15.3166i	1.16450	+	1.16450i	0.983479	+	0.181022i	$0.0579407\pi$
$173$	15.3166	+	15.3166i	1.16450	+	1.16450i	0.181022	+	0.983479i	$0.442059\pi$
$174$		0			0
$175$		−	0.895550i		−	0.0676972i
$176$		0			0
$177$		−	33.4225i		−	2.51219i
$178$		0			0
$179$	−6.79156	−	6.79156i	−0.507625	−	0.507625i	0.406172	−	0.913797i	$-0.366864\pi$
$179$	−6.79156	−	6.79156i	−0.507625	−	0.507625i	−0.913797	+	0.406172i	$0.866864\pi$
$180$		0			0
$181$	−17.9499	+	17.9499i	−1.33420	+	1.33420i	−0.432634	+	0.901570i	$0.642416\pi$
$181$	−17.9499	+	17.9499i	−1.33420	+	1.33420i	−0.901570	+	0.432634i	$0.857584\pi$
$182$		0			0
$183$		−	17.2665i		−	1.27638i
$184$		0			0
$185$		−	9.31662i		−	0.684972i
$186$		0			0
$187$	−4.24264	−	4.24264i	−0.310253	−	0.310253i
$188$		0			0
$189$	1.60266	−	1.60266i	0.116576	−	0.116576i
$190$		0			0
$191$	18.0000			1.30243			0.651217	−	0.758891i	$-0.274259\pi$
$191$	18.0000			1.30243			0.651217	+	0.758891i	$0.274259\pi$
$192$		0			0
$193$			12.7279i			0.916176i	0.888907	+	0.458088i	$0.151466\pi$
$193$			12.7279i			0.916176i	−0.888907	+	0.458088i	$0.848534\pi$
$194$		0			0
$195$	−4.10600	−	10.2106i	−0.294037	−	0.731198i
$196$		0			0
$197$	2.86387	+	2.86387i	0.204042	+	0.204042i	0.801729	−	0.597687i	$-0.203913\pi$
$197$	2.86387	+	2.86387i	0.204042	+	0.204042i	−0.597687	+	0.801729i	$0.703913\pi$
$198$		0			0
$199$			18.0000i			1.27599i	0.770042	+	0.637993i	$0.220235\pi$
$199$			18.0000i			1.27599i	−0.770042	+	0.637993i	$0.779765\pi$
$200$		0			0
$201$	−20.2468			−1.42810
$202$		0			0
$203$	0.0708883	+	0.0708883i	0.00497539	+	0.00497539i
$204$		0			0
$205$		0			0
$206$		0			0
$207$	−39.8997			−2.77322
$208$		0			0
$209$	12.6332			0.873860
$210$		0			0
$211$	−6.52506	−	6.52506i	−0.449204	−	0.449204i	0.445886	−	0.895090i	$-0.352889\pi$
$211$	−6.52506	−	6.52506i	−0.449204	−	0.449204i	−0.895090	+	0.445886i	$0.852889\pi$
$212$		0			0
$213$	1.44966	+	1.44966i	0.0993289	+	0.0993289i
$214$		0			0
$215$	−7.74273			−0.528050
$216$		0			0
$217$		−	0.949874i		−	0.0644817i
$218$		0			0
$219$	8.19051	+	8.19051i	0.553463	+	0.553463i
$220$		0			0
$221$	−10.0356	+	4.03562i	−0.675069	+	0.271465i
$222$		0			0
$223$			23.2282i			1.55547i	0.628589	+	0.777737i	$0.283633\pi$
$223$			23.2282i			1.55547i	−0.628589	+	0.777737i	$0.716367\pi$
$224$		0			0
$225$	25.2665			1.68443
$226$		0			0
$227$	1.41421	−	1.41421i	0.0938647	−	0.0938647i	−0.658615	−	0.752480i	$-0.728857\pi$
$227$	1.41421	−	1.41421i	0.0938647	−	0.0938647i	0.752480	+	0.658615i	$0.228857\pi$
$228$		0			0
$229$	1.67355	+	1.67355i	0.110591	+	0.110591i	0.760237	−	0.649646i	$-0.225083\pi$
$229$	1.67355	+	1.67355i	0.110591	+	0.110591i	−0.649646	+	0.760237i	$0.725083\pi$
$230$		0			0
$231$			1.36675i			0.0899256i
$232$		0			0
$233$			22.5831i			1.47947i	0.672899	+	0.739735i	$0.265049\pi$
$233$			22.5831i			1.47947i	−0.672899	+	0.739735i	$0.734951\pi$
$234$		0			0
$235$	7.47494	−	7.47494i	0.487611	−	0.487611i
$236$		0			0
$237$	−23.6332	−	23.6332i	−1.53514	−	1.53514i
$238$		0			0
$239$		−	11.9854i		−	0.775269i	−0.921813	−	0.387635i	$-0.873292\pi$
$239$		−	11.9854i		−	0.775269i	0.921813	−	0.387635i	$-0.126708\pi$
$240$		0			0
$241$		−	12.7279i		−	0.819878i	−0.912113	−	0.409939i	$-0.865550\pi$
$241$		−	12.7279i		−	0.819878i	0.912113	−	0.409939i	$-0.134450\pi$
$242$		0			0
$243$	4.31662	+	4.31662i	0.276912	+	0.276912i
$244$		0			0
$245$	4.91430	+	4.91430i	0.313963	+	0.313963i
$246$		0			0
$247$	8.93306	−	20.9499i	0.568397	−	1.33301i
$248$		0			0
$249$	51.7364			3.27866
$250$		0			0
$251$	−3.63325	+	3.63325i	−0.229329	+	0.229329i	−0.812412	−	0.583084i	$-0.801846\pi$
$251$	−3.63325	+	3.63325i	−0.229329	+	0.229329i	0.583084	+	0.812412i	$0.301846\pi$
$252$		0			0
$253$	8.93306	−	8.93306i	0.561616	−	0.561616i
$254$		0			0
$255$			9.15694i			0.573430i
$256$		0			0
$257$	15.0000			0.935674			0.467837	−	0.883815i	$-0.345033\pi$
$257$	15.0000			0.935674			0.467837	+	0.883815i	$0.345033\pi$
$258$		0			0
$259$	1.47494	+	1.47494i	0.0916481	+	0.0916481i
$260$		0			0
$261$	−2.00000	+	2.00000i	−0.123797	+	0.123797i
$262$		0			0
$263$		−	0.633250i		−	0.0390478i	−0.999809	−	0.0195239i	$-0.993785\pi$
$263$		−	0.633250i		−	0.0390478i	0.999809	−	0.0195239i	$-0.00621505\pi$
$264$		0			0
$265$	−5.13819			−0.315637
$266$		0			0
$267$	30.3701	+	30.3701i	1.85862	+	1.85862i
$268$		0			0
$269$	−12.6332	−	12.6332i	−0.770263	−	0.770263i	0.207889	−	0.978152i	$-0.433341\pi$
$269$	−12.6332	−	12.6332i	−0.770263	−	0.770263i	−0.978152	+	0.207889i	$0.933341\pi$
$270$		0			0
$271$		−	23.2282i		−	1.41101i	−0.708704	−	0.705506i	$-0.750720\pi$
$271$		−	23.2282i		−	1.41101i	0.708704	−	0.705506i	$-0.249280\pi$
$272$		0			0
$273$	2.26650	+	0.966438i	0.137175	+	0.0584915i
$274$		0			0
$275$	−5.65685	+	5.65685i	−0.341121	+	0.341121i
$276$		0			0
$277$	−14.9499	+	14.9499i	−0.898251	+	0.898251i	−0.995281	−	0.0970305i	$-0.969066\pi$
$277$	−14.9499	+	14.9499i	−0.898251	+	0.898251i	0.0970305	+	0.995281i	$0.469066\pi$
$278$		0			0
$279$	26.7992			1.60442
$280$		0			0
$281$	12.7279			0.759284			0.379642	−	0.925133i	$-0.376047\pi$
$281$	12.7279			0.759284			0.379642	+	0.925133i	$0.376047\pi$
$282$		0			0
$283$	11.0000	−	11.0000i	0.653882	−	0.653882i	−0.300043	−	0.953926i	$-0.597001\pi$
$283$	11.0000	−	11.0000i	0.653882	−	0.653882i	0.953926	+	0.300043i	$0.0970012\pi$
$284$		0			0
$285$	−13.6332	−	13.6332i	−0.807564	−	0.807564i
$286$		0			0
$287$		0			0
$288$		0			0
$289$	−8.00000			−0.470588
$290$		0			0
$291$	18.3139	−	18.3139i	1.07358	−	1.07358i
$292$		0			0
$293$	12.6925	+	12.6925i	0.741502	+	0.741502i	0.972867	−	0.231365i	$-0.0743190\pi$
$293$	12.6925	+	12.6925i	0.741502	+	0.741502i	−0.231365	+	0.972867i	$0.574319\pi$
$294$		0			0
$295$		−	10.9499i		−	0.637526i
$296$		0			0
$297$	−20.2468			−1.17484
$298$		0			0
$299$	−8.49717	−	21.1304i	−0.491404	−	1.22200i
$300$		0			0
$301$	1.22577	−	1.22577i	0.0706522	−	0.0706522i
$302$		0			0
$303$	27.2665			1.56642
$304$		0			0
$305$		−	5.65685i		−	0.323911i
$306$		0			0
$307$	12.5040	−	12.5040i	0.713643	−	0.713643i	−0.253652	−	0.967295i	$-0.581632\pi$
$307$	12.5040	−	12.5040i	0.713643	−	0.713643i	0.967295	+	0.253652i	$0.0816320\pi$
$308$		0			0
$309$	2.05013	−	2.05013i	0.116628	−	0.116628i
$310$		0			0
$311$			18.3166i			1.03864i	0.854580	+	0.519320i	$0.173815\pi$
$311$			18.3166i			1.03864i	−0.854580	+	0.519320i	$0.826185\pi$
$312$		0			0
$313$		−	9.94987i		−	0.562400i	−0.959649	−	0.281200i	$-0.909268\pi$
$313$		−	9.94987i		−	0.562400i	0.959649	−	0.281200i	$-0.0907325\pi$
$314$		0			0
$315$	1.00000	−	1.00000i	0.0563436	−	0.0563436i
$316$		0			0
$317$	−15.4855	+	15.4855i	−0.869750	+	0.869750i	−0.992445	−	0.122694i	$-0.960847\pi$
$317$	−15.4855	+	15.4855i	−0.869750	+	0.869750i	0.122694	+	0.992445i	$0.460847\pi$
$318$		0			0
$319$		−	0.895550i		−	0.0501412i
$320$		0			0
$321$	41.5831			2.32094
$322$		0			0
$323$	−13.3996	+	13.3996i	−0.745573	+	0.745573i
$324$		0			0
$325$	5.38083	+	13.3808i	0.298475	+	0.742235i
$326$		0			0
$327$	−10.1234			−0.559824
$328$		0			0
$329$			2.36675i			0.130483i
$330$		0			0
$331$	5.36208	+	5.36208i	0.294726	+	0.294726i	0.838944	−	0.544218i	$-0.183173\pi$
$331$	5.36208	+	5.36208i	0.294726	+	0.294726i	−0.544218	+	0.838944i	$0.683173\pi$
$332$		0			0
$333$	−41.6130	+	41.6130i	−2.28038	+	2.28038i
$334$		0			0
$335$	−6.63325			−0.362413
$336$		0			0
$337$	5.94987			0.324110			0.162055	−	0.986782i	$-0.448188\pi$
$337$	5.94987			0.324110			0.162055	+	0.986782i	$0.448188\pi$
$338$		0			0
$339$	12.9499	+	12.9499i	0.703341	+	0.703341i
$340$		0			0
$341$	−6.00000	+	6.00000i	−0.324918	+	0.324918i
$342$		0			0
$343$	−3.12320			−0.168637
$344$		0			0
$345$	−19.2803			−1.03802
$346$		0			0
$347$	5.84169	−	5.84169i	0.313598	−	0.313598i	−0.532704	−	0.846302i	$-0.678824\pi$
$347$	5.84169	−	5.84169i	0.313598	−	0.313598i	0.846302	+	0.532704i	$0.178824\pi$
$348$		0			0
$349$	3.01687	−	3.01687i	0.161489	−	0.161489i	−0.621737	−	0.783226i	$-0.713573\pi$
$349$	3.01687	−	3.01687i	0.161489	−	0.161489i	0.783226	+	0.621737i	$0.213573\pi$
$350$		0			0
$351$	−14.3166	+	33.5755i	−0.764165	+	1.79213i
$352$		0			0
$353$		−	11.2428i		−	0.598395i	−0.954191	−	0.299197i	$-0.903281\pi$
$353$		−	11.2428i		−	0.598395i	0.954191	−	0.299197i	$-0.0967189\pi$
$354$		0			0
$355$	0.474937	+	0.474937i	0.0252070	+	0.0252070i
$356$		0			0
$357$	−1.44966	−	1.44966i	−0.0767240	−	0.0767240i
$358$		0			0
$359$	−12.7279			−0.671754			−0.335877	−	0.941906i	$-0.609033\pi$
$359$	−12.7279			−0.671754			−0.335877	+	0.941906i	$0.609033\pi$
$360$		0			0
$361$		−	20.8997i		−	1.09999i
$362$		0			0
$363$	−15.1082	+	15.1082i	−0.792974	+	0.792974i
$364$		0			0
$365$	2.68338	+	2.68338i	0.140454	+	0.140454i
$366$		0			0
$367$	−27.8997			−1.45636			−0.728178	−	0.685389i	$-0.759632\pi$
$367$	−27.8997			−1.45636			−0.728178	+	0.685389i	$0.759632\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	0.813439	−	0.813439i	0.0422317	−	0.0422317i
$372$		0			0
$373$	−2.94987	+	2.94987i	−0.152739	+	0.152739i	−0.779340	−	0.626601i	$-0.784445\pi$
$373$	−2.94987	+	2.94987i	−0.152739	+	0.152739i	0.626601	+	0.779340i	$0.284445\pi$
$374$		0			0
$375$	27.4708			1.41859
$376$		0			0
$377$	−1.48510	−	0.633250i	−0.0764866	−	0.0326140i
$378$		0			0
$379$	−17.1945	−	17.1945i	−0.883220	−	0.883220i	0.110641	−	0.993860i	$-0.464710\pi$
$379$	−17.1945	−	17.1945i	−0.883220	−	0.883220i	−0.993860	+	0.110641i	$0.964710\pi$
$380$		0			0
$381$	6.36675	+	6.36675i	0.326179	+	0.326179i
$382$		0			0
$383$			2.08588i			0.106583i	0.998579	+	0.0532916i	$0.0169713\pi$
$383$			2.08588i			0.106583i	−0.998579	+	0.0532916i	$0.983029\pi$
$384$		0			0
$385$			0.447775i			0.0228207i
$386$		0			0
$387$	34.5831	+	34.5831i	1.75796	+	1.75796i
$388$		0			0
$389$	−3.31662	+	3.31662i	−0.168160	+	0.168160i	−0.786170	−	0.618010i	$-0.787939\pi$
$389$	−3.31662	+	3.31662i	−0.168160	+	0.168160i	0.618010	+	0.786170i	$0.287939\pi$
$390$		0			0
$391$			18.9499i			0.958336i
$392$		0			0
$393$			12.2665i			0.618763i
$394$		0			0
$395$	−7.74273	−	7.74273i	−0.389579	−	0.389579i
$396$		0			0
$397$	27.6947	−	27.6947i	1.38996	−	1.38996i	0.564579	−	0.825379i	$-0.309038\pi$
$397$	27.6947	−	27.6947i	1.38996	−	1.38996i	0.825379	−	0.564579i	$-0.190962\pi$
$398$		0			0
$399$	4.31662			0.216102
$400$		0			0
$401$			26.7283i			1.33475i	0.744723	+	0.667373i	$0.232581\pi$
$401$			26.7283i			1.33475i	−0.744723	+	0.667373i	$0.767419\pi$
$402$		0			0
$403$	5.70723	+	14.1925i	0.284298	+	0.706980i
$404$		0			0
$405$	8.44984	+	8.44984i	0.419876	+	0.419876i
$406$		0			0
$407$		−	18.6332i		−	0.923616i
$408$		0			0
$409$	−30.5940			−1.51278			−0.756389	−	0.654122i	$-0.773038\pi$
$409$	−30.5940			−1.51278			−0.756389	+	0.654122i	$0.773038\pi$
$410$		0			0
$411$	30.3701	+	30.3701i	1.49805	+	1.49805i
$412$		0			0
$413$	1.73350	+	1.73350i	0.0853000	+	0.0853000i
$414$		0			0
$415$	16.9499			0.832037
$416$		0			0
$417$	−10.6834			−0.523167
$418$		0			0
$419$	−5.84169	−	5.84169i	−0.285385	−	0.285385i	0.549867	−	0.835252i	$-0.314678\pi$
$419$	−5.84169	−	5.84169i	−0.285385	−	0.285385i	−0.835252	+	0.549867i	$0.814678\pi$
$420$		0			0
$421$	15.7448	+	15.7448i	0.767354	+	0.767354i	0.977640	−	0.210286i	$-0.0674394\pi$
$421$	15.7448	+	15.7448i	0.767354	+	0.767354i	−0.210286	+	0.977640i	$0.567439\pi$
$422$		0			0
$423$	−66.7740			−3.24666
$424$		0			0
$425$		−	12.0000i		−	0.582086i
$426$		0			0
$427$	0.895550	+	0.895550i	0.0433387	+	0.0433387i
$428$		0			0
$429$	−8.21200	−	20.4213i	−0.396479	−	0.985947i
$430$		0			0
$431$			6.32852i			0.304834i	0.988316	+	0.152417i	$0.0487057\pi$
$431$			6.32852i			0.304834i	−0.988316	+	0.152417i	$0.951294\pi$
$432$		0			0
$433$	10.0501			0.482978			0.241489	−	0.970404i	$-0.422364\pi$
$433$	10.0501			0.482978			0.241489	+	0.970404i	$0.422364\pi$
$434$		0			0
$435$	−0.966438	+	0.966438i	−0.0463372	+	0.0463372i
$436$		0			0
$437$	−28.2134	−	28.2134i	−1.34963	−	1.34963i
$438$		0			0
$439$			0.949874i			0.0453350i	0.999743	+	0.0226675i	$0.00721591\pi$
$439$			0.949874i			0.0453350i	−0.999743	+	0.0226675i	$0.992784\pi$
$440$		0			0
$441$		−	43.8997i		−	2.09046i
$442$		0			0
$443$	−22.1082	+	22.1082i	−1.05039	+	1.05039i	−0.0517306	+	0.998661i	$0.516474\pi$
$443$	−22.1082	+	22.1082i	−1.05039	+	1.05039i	−0.998661	+	0.0517306i	$0.983526\pi$
$444$		0			0
$445$	9.94987	+	9.94987i	0.471669	+	0.471669i
$446$		0			0
$447$			30.5231i			1.44370i
$448$		0			0
$449$		−	4.31353i		−	0.203568i	−0.994807	−	0.101784i	$-0.967545\pi$
$449$		−	4.31353i		−	0.203568i	0.994807	−	0.101784i	$-0.0324551\pi$
$450$		0			0
$451$		0			0
$452$		0			0
$453$	−9.64016	−	9.64016i	−0.452934	−	0.452934i
$454$		0			0
$455$	0.742551	+	0.316625i	0.0348113	+	0.0148436i
$456$		0			0
$457$	14.5190			0.679171			0.339586	−	0.940575i	$-0.389713\pi$
$457$	14.5190			0.679171			0.339586	+	0.940575i	$0.389713\pi$
$458$		0			0
$459$	21.4749	−	21.4749i	1.00236	−	1.00236i
$460$		0			0
$461$	4.27808	−	4.27808i	0.199250	−	0.199250i	−0.600428	−	0.799679i	$-0.705003\pi$
$461$	4.27808	−	4.27808i	0.199250	−	0.199250i	0.799679	+	0.600428i	$0.205003\pi$
$462$		0			0
$463$		−	12.7279i		−	0.591517i	−0.955263	−	0.295758i	$-0.904428\pi$
$463$		−	12.7279i		−	0.591517i	0.955263	−	0.295758i	$-0.0955723\pi$
$464$		0			0
$465$	12.9499			0.600536
$466$		0			0
$467$	3.63325	+	3.63325i	0.168127	+	0.168127i	0.786155	−	0.618029i	$-0.212068\pi$
$467$	3.63325	+	3.63325i	0.168127	+	0.168127i	−0.618029	+	0.786155i	$0.712068\pi$
$468$		0			0
$469$	1.05013	−	1.05013i	0.0484903	−	0.0484903i
$470$		0			0
$471$		−	34.3166i		−	1.58123i
$472$		0			0
$473$	−15.4855			−0.712022
$474$		0			0
$475$	17.8661	+	17.8661i	0.819753	+	0.819753i
$476$		0			0
$477$	22.9499	+	22.9499i	1.05080	+	1.05080i
$478$		0			0
$479$		−	16.2280i		−	0.741477i	−0.928737	−	0.370738i	$-0.879105\pi$
$479$		−	16.2280i		−	0.741477i	0.928737	−	0.370738i	$-0.120895\pi$
$480$		0			0
$481$	−30.8997	−	13.1757i	−1.40891	−	0.600760i
$482$		0			0
$483$	3.05231	−	3.05231i	0.138885	−	0.138885i
$484$		0			0
$485$	6.00000	−	6.00000i	0.272446	−	0.272446i
$486$		0			0
$487$	29.2507			1.32548			0.662738	−	0.748851i	$-0.269394\pi$
$487$	29.2507			1.32548			0.662738	+	0.748851i	$0.269394\pi$
$488$		0			0
$489$	37.5942			1.70007
$490$		0			0
$491$	−21.1583	+	21.1583i	−0.954861	+	0.954861i	−0.999024	−	0.0441631i	$-0.985938\pi$
$491$	−21.1583	+	21.1583i	−0.954861	+	0.954861i	0.0441631	+	0.999024i	$0.485938\pi$
$492$		0			0
$493$	0.949874	+	0.949874i	0.0427802	+	0.0427802i
$494$		0			0
$495$	−12.6332			−0.567822
$496$		0			0
$497$	−0.150377			−0.00674533
$498$		0			0
$499$	17.4183	−	17.4183i	0.779752	−	0.779752i	−0.200037	−	0.979788i	$-0.564106\pi$
$499$	17.4183	−	17.4183i	0.779752	−	0.779752i	0.979788	+	0.200037i	$0.0641061\pi$
$500$		0			0
$501$	30.3701	+	30.3701i	1.35684	+	1.35684i
$502$		0			0
$503$			10.4169i			0.464466i	0.972660	+	0.232233i	$0.0746031\pi$
$503$			10.4169i			0.464466i	−0.972660	+	0.232233i	$0.925397\pi$
$504$		0			0
$505$	8.93306			0.397516
$506$		0			0
$507$	−39.6716	−	0.821953i	−1.76188	−	0.0365042i
$508$		0			0
$509$	−5.65685	+	5.65685i	−0.250736	+	0.250736i	−0.821272	−	0.570537i	$-0.806735\pi$
$509$	−5.65685	+	5.65685i	−0.250736	+	0.250736i	0.570537	+	0.821272i	$0.306735\pi$
$510$		0			0
$511$	−0.849623			−0.0375851
$512$		0			0
$513$			63.9456i			2.82327i
$514$		0			0
$515$	0.671663	−	0.671663i	0.0295970	−	0.0295970i
$516$		0			0
$517$	14.9499	−	14.9499i	0.657495	−	0.657495i
$518$		0			0
$519$		−	66.1161i		−	2.90218i
$520$		0			0
$521$		−	7.41688i		−	0.324939i	−0.986714	−	0.162470i	$-0.948054\pi$
$521$		−	7.41688i		−	0.324939i	0.986714	−	0.162470i	$-0.0519459\pi$
$522$		0			0
$523$	−0.0501256	+	0.0501256i	−0.00219184	+	0.00219184i	−0.708202	−	0.706010i	$-0.750493\pi$
$523$	−0.0501256	+	0.0501256i	−0.00219184	+	0.00219184i	0.706010	+	0.708202i	$0.250493\pi$
$524$		0			0
$525$	−1.93288	+	1.93288i	−0.0843577	+	0.0843577i
$526$		0			0
$527$		−	12.7279i		−	0.554437i
$528$		0			0
$529$	−16.8997			−0.734772
$530$		0			0
$531$	−48.9079	+	48.9079i	−2.12242	+	2.12242i
$532$		0			0
$533$		0			0
$534$		0			0
$535$	13.6235			0.588994
$536$		0			0
$537$			29.3166i			1.26511i
$538$		0			0
$539$	9.82861	+	9.82861i	0.423348	+	0.423348i
$540$		0			0
$541$	−14.6254	+	14.6254i	−0.628793	+	0.628793i	−0.947764	−	0.318971i	$-0.896663\pi$
$541$	−14.6254	+	14.6254i	−0.628793	+	0.628793i	0.318971	+	0.947764i	$0.396663\pi$
$542$		0			0
$543$	77.4829			3.32511
$544$		0			0
$545$	−3.31662			−0.142069
$546$		0			0
$547$	−5.42481	−	5.42481i	−0.231948	−	0.231948i	0.581557	−	0.813505i	$-0.302444\pi$
$547$	−5.42481	−	5.42481i	−0.231948	−	0.231948i	−0.813505	+	0.581557i	$0.802444\pi$
$548$		0			0
$549$	−25.2665	+	25.2665i	−1.07835	+	1.07835i
$550$		0			0
$551$	−2.82843			−0.120495
$552$		0			0
$553$	2.45154			0.104250
$554$		0			0
$555$	−20.1082	+	20.1082i	−0.853545	+	0.853545i
$556$		0			0
$557$	8.52073	−	8.52073i	0.361035	−	0.361035i	−0.503159	−	0.864194i	$-0.667829\pi$
$557$	8.52073	−	8.52073i	0.361035	−	0.361035i	0.864194	+	0.503159i	$0.167829\pi$
$558$		0			0
$559$	−10.9499	+	25.6797i	−0.463130	+	1.08614i
$560$		0			0
$561$			18.3139i			0.773213i
$562$		0			0
$563$	24.1583	+	24.1583i	1.01815	+	1.01815i	0.999832	+	0.0183193i	$0.00583154\pi$
$563$	24.1583	+	24.1583i	1.01815	+	1.01815i	0.0183193	+	0.999832i	$0.494168\pi$
$564$		0			0
$565$	4.24264	+	4.24264i	0.178489	+	0.178489i
$566$		0			0
$567$	−2.67543			−0.112357
$568$		0			0
$569$			21.6332i			0.906913i	0.891278	+	0.453457i	$0.149809\pi$
$569$			21.6332i			0.906913i	−0.891278	+	0.453457i	$0.850191\pi$
$570$		0			0
$571$	21.4248	−	21.4248i	0.896600	−	0.896600i	−0.0985333	−	0.995134i	$-0.531415\pi$
$571$	21.4248	−	21.4248i	0.896600	−	0.896600i	0.995134	+	0.0985333i	$0.0314151\pi$
$572$		0			0
$573$	−38.8496	−	38.8496i	−1.62297	−	1.62297i
$574$		0			0
$575$	25.2665			1.05369
$576$		0			0
$577$		−	29.6985i		−	1.23636i	−0.786035	−	0.618182i	$-0.787869\pi$
$577$		−	29.6985i		−	1.23636i	0.786035	−	0.618182i	$-0.212131\pi$
$578$		0			0
$579$	27.4708	−	27.4708i	1.14165	−	1.14165i
$580$		0			0
$581$	−2.68338	+	2.68338i	−0.111325	+	0.111325i
$582$		0			0
$583$	−10.2764			−0.425604
$584$		0			0
$585$	−8.93306	+	20.9499i	−0.369336	+	0.866171i
$586$		0			0
$587$	10.6420	+	10.6420i	0.439244	+	0.439244i	0.891758	−	0.452513i	$-0.149473\pi$
$587$	10.6420	+	10.6420i	0.439244	+	0.439244i	−0.452513	+	0.891758i	$0.649473\pi$
$588$		0			0
$589$	18.9499	+	18.9499i	0.780816	+	0.780816i
$590$		0			0
$591$		−	12.3623i		−	0.508515i
$592$		0			0
$593$		−	4.31353i		−	0.177135i	−0.996070	−	0.0885677i	$-0.971771\pi$
$593$		−	4.31353i		−	0.177135i	0.996070	−	0.0885677i	$-0.0282290\pi$
$594$		0			0
$595$	−0.474937	−	0.474937i	−0.0194705	−	0.0194705i
$596$		0			0
$597$	38.8496	−	38.8496i	1.59001	−	1.59001i
$598$		0			0
$599$		−	11.6834i		−	0.477370i	−0.971097	−	0.238685i	$-0.923284\pi$
$599$		−	11.6834i		−	0.477370i	0.971097	−	0.238685i	$-0.0767163\pi$
$600$		0			0
$601$		−	28.8997i		−	1.17885i	−0.807825	−	0.589423i	$-0.799355\pi$
$601$		−	28.8997i		−	1.17885i	0.807825	−	0.589423i	$-0.200645\pi$
$602$		0			0
$603$	29.6276	+	29.6276i	1.20653	+	1.20653i
$604$		0			0
$605$	−4.94975	+	4.94975i	−0.201236	+	0.201236i
$606$		0			0
$607$	−32.9499			−1.33739			−0.668697	−	0.743535i	$-0.733148\pi$
$607$	−32.9499			−1.33739			−0.668697	+	0.743535i	$0.733148\pi$
$608$		0			0
$609$		−	0.305998i		−	0.0123997i
$610$		0			0
$611$	−14.2204	−	35.3627i	−0.575296	−	1.43062i
$612$		0			0
$613$	−27.2469	−	27.2469i	−1.10049	−	1.10049i	−0.994351	−	0.106143i	$-0.966150\pi$
$613$	−27.2469	−	27.2469i	−1.10049	−	1.10049i	−0.106143	−	0.994351i	$-0.533850\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−1.34333			−0.0540802			−0.0270401	−	0.999634i	$-0.508608\pi$
$617$	−1.34333			−0.0540802			−0.0270401	+	0.999634i	$0.508608\pi$
$618$		0			0
$619$	−5.13819	−	5.13819i	−0.206521	−	0.206521i	0.596266	−	0.802787i	$-0.296650\pi$
$619$	−5.13819	−	5.13819i	−0.206521	−	0.206521i	−0.802787	+	0.596266i	$0.796650\pi$
$620$		0			0
$621$	45.2164	+	45.2164i	1.81447	+	1.81447i
$622$		0			0
$623$	−3.15038			−0.126217
$624$		0			0
$625$	−11.0000			−0.440000
$626$		0			0
$627$	−27.2665	−	27.2665i	−1.08892	−	1.08892i
$628$		0			0
$629$	19.7635	+	19.7635i	0.788024	+	0.788024i
$630$		0			0
$631$	5.80985			0.231287			0.115643	−	0.993291i	$-0.463107\pi$
$631$	5.80985			0.231287			0.115643	+	0.993291i	$0.463107\pi$
$632$		0			0
$633$			28.1662i			1.11951i
$634$		0			0
$635$	2.08588	+	2.08588i	0.0827755	+	0.0827755i
$636$		0			0
$637$	23.2488	−	9.34903i	0.921150	−	0.370422i
$638$		0			0
$639$		−	4.24264i		−	0.167836i
$640$		0			0
$641$	12.0000			0.473972			0.236986	−	0.971513i	$-0.423841\pi$
$641$	12.0000			0.473972			0.236986	+	0.971513i	$0.423841\pi$
$642$		0			0
$643$	8.93306	−	8.93306i	0.352285	−	0.352285i	−0.508674	−	0.860959i	$-0.669864\pi$
$643$	8.93306	−	8.93306i	0.352285	−	0.352285i	0.860959	+	0.508674i	$0.169864\pi$
$644$		0			0
$645$	16.7112	+	16.7112i	0.658004	+	0.658004i
$646$		0			0
$647$		−	19.5831i		−	0.769892i	−0.922939	−	0.384946i	$-0.874220\pi$
$647$		−	19.5831i		−	0.769892i	0.922939	−	0.384946i	$-0.125780\pi$
$648$		0			0
$649$		−	21.8997i		−	0.859640i
$650$		0			0
$651$	−2.05013	+	2.05013i	−0.0803508	+	0.0803508i
$652$		0			0
$653$	−13.5831	−	13.5831i	−0.531549	−	0.531549i	0.389484	−	0.921033i	$-0.372653\pi$
$653$	−13.5831	−	13.5831i	−0.531549	−	0.531549i	−0.921033	+	0.389484i	$0.872653\pi$
$654$		0			0
$655$			4.01875i			0.157026i
$656$		0			0
$657$		−	23.9707i		−	0.935188i
$658$		0			0
$659$	8.68338	+	8.68338i	0.338256	+	0.338256i	0.855711	−	0.517454i	$-0.173120\pi$
$659$	8.68338	+	8.68338i	0.338256	+	0.338256i	−0.517454	+	0.855711i	$0.673120\pi$
$660$		0			0
$661$	−17.4183	−	17.4183i	−0.677495	−	0.677495i	0.281938	−	0.959433i	$-0.409023\pi$
$661$	−17.4183	−	17.4183i	−0.677495	−	0.677495i	−0.959433	+	0.281938i	$0.909023\pi$
$662$		0			0
$663$	30.3701	+	12.9499i	1.17948	+	0.502931i
$664$		0			0
$665$	1.41421			0.0548408
$666$		0			0
$667$	−2.00000	+	2.00000i	−0.0774403	+	0.0774403i
$668$		0			0
$669$	50.1337	−	50.1337i	1.93828	−	1.93828i
$670$		0			0
$671$		−	11.3137i		−	0.436761i
$672$		0			0
$673$	18.8997			0.728532			0.364266	−	0.931295i	$-0.381320\pi$
$673$	18.8997			0.728532			0.364266	+	0.931295i	$0.381320\pi$
$674$		0			0
$675$	−28.6332	−	28.6332i	−1.10209	−	1.10209i
$676$		0			0
$677$	8.68338	−	8.68338i	0.333729	−	0.333729i	−0.520272	−	0.854001i	$-0.674169\pi$
$677$	8.68338	−	8.68338i	0.333729	−	0.333729i	0.854001	+	0.520272i	$0.174169\pi$
$678$		0			0
$679$			1.89975i			0.0729057i
$680$		0			0
$681$	−6.10463			−0.233930
$682$		0			0
$683$	−10.5712	−	10.5712i	−0.404494	−	0.404494i	0.475319	−	0.879813i	$-0.342333\pi$
$683$	−10.5712	−	10.5712i	−0.404494	−	0.404494i	−0.879813	+	0.475319i	$0.842333\pi$
$684$		0			0
$685$	9.94987	+	9.94987i	0.380165	+	0.380165i
$686$		0			0
$687$		−	7.22407i		−	0.275615i
$688$		0			0
$689$	−7.26650	+	17.0415i	−0.276832	+	0.649228i
$690$		0			0
$691$	3.34709	−	3.34709i	0.127329	−	0.127329i	−0.640570	−	0.767900i	$-0.721302\pi$
$691$	3.34709	−	3.34709i	0.127329	−	0.127329i	0.767900	+	0.640570i	$0.221302\pi$
$692$		0			0
$693$	2.00000	−	2.00000i	0.0759737	−	0.0759737i
$694$		0			0
$695$	−3.50009			−0.132766
$696$		0			0
$697$		0			0
$698$		0			0
$699$	48.7414	−	48.7414i	1.84357	−	1.84357i
$700$		0			0
$701$	11.2164	+	11.2164i	0.423637	+	0.423637i	0.886454	−	0.462817i	$-0.153161\pi$
$701$	11.2164	+	11.2164i	0.423637	+	0.423637i	−0.462817	+	0.886454i	$0.653161\pi$
$702$		0			0
$703$	−58.8496			−2.21956
$704$		0			0
$705$	−32.2665			−1.21523
$706$		0			0
$707$	−1.41421	+	1.41421i	−0.0531870	+	0.0531870i
$708$		0			0
$709$	−3.34709	−	3.34709i	−0.125703	−	0.125703i	0.641457	−	0.767159i	$-0.278330\pi$
$709$	−3.34709	−	3.34709i	−0.125703	−	0.125703i	−0.767159	+	0.641457i	$0.778330\pi$
$710$		0			0
$711$			69.1662i			2.59394i
$712$		0			0
$713$	26.7992			1.00364
$714$		0			0
$715$	−2.69042	−	6.69042i	−0.100616	−	0.250207i
$716$		0			0
$717$	−25.8682	+	25.8682i	−0.966065	+	0.966065i
$718$		0			0
$719$	6.00000			0.223762			0.111881	−	0.993722i	$-0.464312\pi$
$719$	6.00000			0.223762			0.111881	+	0.993722i	$0.464312\pi$
$720$		0			0
$721$			0.212665i			0.00792006i
$722$		0			0
$723$	−27.4708	+	27.4708i	−1.02165	+	1.02165i
$724$		0			0
$725$	1.26650	−	1.26650i	0.0470366	−	0.0470366i
$726$		0			0
$727$			37.8997i			1.40562i	0.711376	+	0.702812i	$0.248072\pi$
$727$			37.8997i			1.40562i	−0.711376	+	0.702812i	$0.751928\pi$
$728$		0			0
$729$			17.2164i			0.637643i
$730$		0			0
$731$	16.4248	−	16.4248i	0.607494	−	0.607494i
$732$		0			0
$733$	−6.81174	+	6.81174i	−0.251597	+	0.251597i	−0.821625	−	0.570028i	$-0.806932\pi$
$733$	−6.81174	+	6.81174i	−0.251597	+	0.251597i	0.570028	+	0.821625i	$0.306932\pi$
$734$		0			0
$735$		−	21.2132i		−	0.782461i
$736$		0			0
$737$	−13.2665			−0.488678
$738$		0			0
$739$	18.7617	−	18.7617i	0.690159	−	0.690159i	−0.272108	−	0.962267i	$-0.587721\pi$
$739$	18.7617	−	18.7617i	0.690159	−	0.690159i	0.962267	+	0.272108i	$0.0877207\pi$
$740$		0			0
$741$	−64.4967	+	25.9360i	−2.36935	+	0.952784i
$742$		0			0
$743$	12.0563			0.442301			0.221151	−	0.975240i	$-0.429019\pi$
$743$	12.0563			0.442301			0.221151	+	0.975240i	$0.429019\pi$
$744$		0			0
$745$			10.0000i			0.366372i
$746$		0			0
$747$	−75.7071	−	75.7071i	−2.76998	−	2.76998i
$748$		0			0
$749$	−2.15676	+	2.15676i	−0.0788065	+	0.0788065i
$750$		0			0
$751$	−15.8997			−0.580190			−0.290095	−	0.956998i	$-0.593687\pi$
$751$	−15.8997			−0.580190			−0.290095	+	0.956998i	$0.593687\pi$
$752$		0			0
$753$	15.6834			0.571534
$754$		0			0
$755$	−3.15831	−	3.15831i	−0.114943	−	0.114943i
$756$		0			0
$757$	20.8997	−	20.8997i	0.759614	−	0.759614i	−0.216638	−	0.976252i	$-0.569509\pi$
$757$	20.8997	−	20.8997i	0.759614	−	0.759614i	0.976252	+	0.216638i	$0.0695091\pi$
$758$		0			0
$759$	−38.5607			−1.39966
$760$		0			0
$761$	−12.7279			−0.461387			−0.230693	−	0.973026i	$-0.574099\pi$
$761$	−12.7279			−0.461387			−0.230693	+	0.973026i	$0.574099\pi$
$762$		0			0
$763$	0.525063	−	0.525063i	0.0190086	−	0.0190086i
$764$		0			0
$765$	13.3996	−	13.3996i	0.484463	−	0.484463i
$766$		0			0
$767$	−36.3166	−	15.4855i	−1.31132	−	0.559148i
$768$		0			0
$769$		−	25.4558i		−	0.917961i	−0.888446	−	0.458981i	$-0.848215\pi$
$769$		−	25.4558i		−	0.917961i	0.888446	−	0.458981i	$-0.151785\pi$
$770$		0			0
$771$	−32.3747	−	32.3747i	−1.16595	−	1.16595i
$772$		0			0
$773$	−23.2636	−	23.2636i	−0.836735	−	0.836735i	0.151693	−	0.988428i	$-0.451528\pi$
$773$	−23.2636	−	23.2636i	−0.836735	−	0.836735i	−0.988428	+	0.151693i	$0.951528\pi$
$774$		0			0
$775$	−16.9706			−0.609601
$776$		0			0
$777$		−	6.36675i		−	0.228406i
$778$		0			0
$779$		0			0
$780$		0			0
$781$	0.949874	+	0.949874i	0.0339892	+	0.0339892i
$782$		0			0
$783$	4.53300			0.161996
$784$		0			0
$785$		−	11.2428i		−	0.401273i
$786$		0			0
$787$	−25.0081	+	25.0081i	−0.891441	+	0.891441i	−0.994659	−	0.103217i	$-0.967086\pi$
$787$	−25.0081	+	25.0081i	−0.891441	+	0.891441i	0.103217	+	0.994659i	$0.467086\pi$
$788$		0			0
$789$	−1.36675	+	1.36675i	−0.0486576	+	0.0486576i
$790$		0			0
$791$	−1.34333			−0.0477631
$792$		0			0
$793$	−18.7617	−	8.00000i	−0.666246	−	0.284088i
$794$		0			0
$795$	11.0898	+	11.0898i	0.393315	+	0.393315i
$796$		0			0
$797$	−34.5831	−	34.5831i	−1.22500	−	1.22500i	−0.965833	−	0.259164i	$-0.916553\pi$
$797$	−34.5831	−	34.5831i	−1.22500	−	1.22500i	−0.259164	−	0.965833i	$-0.583447\pi$
$798$		0			0
$799$			31.7135i			1.12194i
$800$		0			0
$801$		−	88.8828i		−	3.14052i
$802$		0			0
$803$	5.36675	+	5.36675i	0.189389	+	0.189389i
$804$		0			0
$805$	1.00000	−	1.00000i	0.0352454	−	0.0352454i
$806$		0			0
$807$			54.5330i			1.91965i
$808$		0			0
$809$			22.5831i			0.793980i	0.917823	+	0.396990i	$0.129945\pi$
$809$			22.5831i			0.793980i	−0.917823	+	0.396990i	$0.870055\pi$
$810$		0			0
$811$	−27.6947	−	27.6947i	−0.972493	−	0.972493i	0.0271385	−	0.999632i	$-0.491360\pi$
$811$	−27.6947	−	27.6947i	−0.972493	−	0.972493i	−0.999632	+	0.0271385i	$0.991360\pi$
$812$		0			0
$813$	−50.1337	+	50.1337i	−1.75827	+	1.75827i
$814$		0			0
$815$	12.3166			0.431433
$816$		0			0
$817$			48.9079i			1.71107i
$818$		0			0
$819$	−1.90241	−	4.73084i	−0.0664756	−	0.165309i
$820$		0			0
$821$	−17.6777	−	17.6777i	−0.616955	−	0.616955i	0.327794	−	0.944749i	$-0.393695\pi$
$821$	−17.6777	−	17.6777i	−0.616955	−	0.616955i	−0.944749	+	0.327794i	$0.893695\pi$
$822$		0			0
$823$			18.9499i			0.660551i	0.943885	+	0.330276i	$0.107142\pi$
$823$			18.9499i			0.660551i	−0.943885	+	0.330276i	$0.892858\pi$
$824$		0			0
$825$	24.4185			0.850144
$826$		0			0
$827$	3.50009	+	3.50009i	0.121710	+	0.121710i	0.765338	−	0.643628i	$-0.222572\pi$
$827$	3.50009	+	3.50009i	0.121710	+	0.121710i	−0.643628	+	0.765338i	$0.722572\pi$
$828$		0			0
$829$	−8.00000	−	8.00000i	−0.277851	−	0.277851i	0.554399	−	0.832251i	$-0.312948\pi$
$829$	−8.00000	−	8.00000i	−0.277851	−	0.277851i	−0.832251	+	0.554399i	$0.812948\pi$
$830$		0			0
$831$	64.5330			2.23862
$832$		0			0
$833$	−20.8496			−0.722397
$834$		0			0
$835$	9.94987	+	9.94987i	0.344330	+	0.344330i
$836$		0			0
$837$	−30.3701	−	30.3701i	−1.04975	−	1.04975i
$838$		0			0
$839$	15.4146			0.532170			0.266085	−	0.963950i	$-0.414270\pi$
$839$	15.4146			0.532170			0.266085	+	0.963950i	$0.414270\pi$
$840$		0			0
$841$		−	28.7995i		−	0.993086i
$842$		0			0
$843$	−27.4708	−	27.4708i	−0.946146	−	0.946146i
$844$		0			0
$845$	−12.9972	−	0.269289i	−0.447118	−	0.00926381i
$846$		0			0
$847$		−	1.56721i		−	0.0538501i
$848$		0			0
$849$	−47.4829			−1.62961
$850$		0			0
$851$	−41.6130	+	41.6130i	−1.42647	+	1.42647i
$852$		0			0
$853$	19.9874	+	19.9874i	0.684357	+	0.684357i	0.960979	−	0.276622i	$-0.0892150\pi$
$853$	19.9874	+	19.9874i	0.684357	+	0.684357i	−0.276622	+	0.960979i	$0.589215\pi$
$854$		0			0
$855$			39.8997i			1.36454i
$856$		0			0
$857$			36.6332i			1.25137i	0.780077	+	0.625684i	$0.215180\pi$
$857$			36.6332i			1.25137i	−0.780077	+	0.625684i	$0.784820\pi$
$858$		0			0
$859$	−25.9499	+	25.9499i	−0.885398	+	0.885398i	−0.994077	−	0.108679i	$-0.965338\pi$
$859$	−25.9499	+	25.9499i	−0.885398	+	0.885398i	0.108679	+	0.994077i	$0.465338\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		−	20.4707i		−	0.696829i	−0.937341	−	0.348415i	$-0.886720\pi$
$863$		−	20.4707i		−	0.696829i	0.937341	−	0.348415i	$-0.113280\pi$
$864$		0			0
$865$		−	21.6610i		−	0.736495i
$866$		0			0
$867$	17.2665	+	17.2665i	0.586401	+	0.586401i
$868$		0			0
$869$	−15.4855	−	15.4855i	−0.525308	−	0.525308i
$870$		0			0
$871$	−9.38083	+	22.0000i	−0.317857	+	0.745442i
$872$		0			0
$873$	−53.5983			−1.81403
$874$		0			0
$875$	−1.42481	+	1.42481i	−0.0481674	+	0.0481674i
$876$		0			0
$877$	1.67355	−	1.67355i	0.0565116	−	0.0565116i	−0.678286	−	0.734798i	$-0.737277\pi$
$877$	1.67355	−	1.67355i	0.0565116	−	0.0565116i	0.734798	+	0.678286i	$0.237277\pi$
$878$		0			0
$879$		−	54.7887i		−	1.84798i
$880$		0			0
$881$	21.0000			0.707508			0.353754	−	0.935339i	$-0.384905\pi$
$881$	21.0000			0.707508			0.353754	+	0.935339i	$0.384905\pi$
$882$		0			0
$883$	−1.47494	−	1.47494i	−0.0496356	−	0.0496356i	0.681853	−	0.731489i	$-0.261174\pi$
$883$	−1.47494	−	1.47494i	−0.0496356	−	0.0496356i	−0.731489	+	0.681853i	$0.761174\pi$
$884$		0			0
$885$	−23.6332	+	23.6332i	−0.794423	+	0.794423i
$886$		0			0
$887$			39.1662i			1.31507i	0.753422	+	0.657537i	$0.228402\pi$
$887$			39.1662i			1.31507i	−0.753422	+	0.657537i	$0.771598\pi$
$888$		0			0
$889$	−0.660440			−0.0221505
$890$		0			0
$891$	16.8997	+	16.8997i	0.566160	+	0.566160i
$892$		0			0
$893$	−47.2164	−	47.2164i	−1.58004	−	1.58004i
$894$		0			0
$895$			9.60472i			0.321050i
$896$		0			0
$897$	−27.2665	+	63.9456i	−0.910402	+	2.13508i
$898$		0			0
$899$	1.34333	−	1.34333i	0.0448024	−	0.0448024i
$900$		0			0
$901$	10.8997	−	10.8997i	0.363123	−	0.363123i
$902$		0			0
$903$	−5.29119			−0.176080
$904$		0			0
$905$	25.3850			0.843824
$906$		0			0
$907$	25.3747	−	25.3747i	0.842553	−	0.842553i	−0.146638	−	0.989190i	$-0.546845\pi$
$907$	25.3747	−	25.3747i	0.842553	−	0.842553i	0.989190	+	0.146638i	$0.0468451\pi$
$908$		0			0
$909$	−39.8997	−	39.8997i	−1.32339	−	1.32339i
$910$		0			0
$911$	30.0000			0.993944			0.496972	−	0.867766i	$-0.334445\pi$
$911$	30.0000			0.993944			0.496972	+	0.867766i	$0.334445\pi$
$912$		0			0
$913$	33.8997			1.12192
$914$		0			0
$915$	−12.2093	+	12.2093i	−0.403626	+	0.403626i
$916$		0			0
$917$	−0.636218	−	0.636218i	−0.0210098	−	0.0210098i
$918$		0			0
$919$		−	36.0000i		−	1.18753i	−0.804638	−	0.593765i	$-0.797641\pi$
$919$		−	36.0000i		−	1.18753i	0.804638	−	0.593765i	$-0.202359\pi$
$920$		0			0
$921$	−53.9752			−1.77854
$922$		0			0
$923$	2.24685	−	0.903526i	0.0739560	−	0.0297399i
$924$		0			0
$925$	26.3514	−	26.3514i	0.866429	−	0.866429i
$926$		0			0
$927$	−6.00000			−0.197066
$928$		0			0
$929$		−	25.3141i		−	0.830528i	−0.909701	−	0.415264i	$-0.863689\pi$
$929$		−	25.3141i		−	0.830528i	0.909701	−	0.415264i	$-0.136311\pi$
$930$		0			0
$931$	31.0418	−	31.0418i	1.01735	−	1.01735i
$932$		0			0
$933$	39.5330	−	39.5330i	1.29425	−	1.29425i
$934$		0			0
$935$			6.00000i			0.196221i
$936$		0			0
$937$			16.1003i			0.525972i	0.964800	+	0.262986i	$0.0847074\pi$
$937$			16.1003i			0.525972i	−0.964800	+	0.262986i	$0.915293\pi$
$938$		0			0
$939$	−21.4749	+	21.4749i	−0.700808	+	0.700808i
$940$		0			0
$941$	16.1217	−	16.1217i	0.525552	−	0.525552i	−0.393691	−	0.919243i	$-0.628802\pi$
$941$	16.1217	−	16.1217i	0.525552	−	0.525552i	0.919243	+	0.393691i	$0.128802\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$	−2.26650			−0.0737292
$946$		0			0
$947$	40.9413	−	40.9413i	1.33041	−	1.33041i	0.425415	−	0.904998i	$-0.360128\pi$
$947$	40.9413	−	40.9413i	1.33041	−	1.33041i	0.904998	−	0.425415i	$-0.139872\pi$
$948$		0			0
$949$	12.6946	−	5.10488i	0.412085	−	0.165712i
$950$		0			0
$951$	66.8449			2.16760
$952$		0			0
$953$			49.4327i			1.60128i	0.599143	+	0.800642i	$0.295508\pi$
$953$			49.4327i			1.60128i	−0.599143	+	0.800642i	$0.704492\pi$
$954$		0			0
$955$	−12.7279	−	12.7279i	−0.411866	−	0.411866i
$956$		0			0
$957$	−1.93288	+	1.93288i	−0.0624810	+	0.0624810i
$958$		0			0
$959$	−3.15038			−0.101731
$960$		0			0
$961$	13.0000			0.419355
$962$		0			0
$963$	−60.8496	−	60.8496i	−1.96085	−	1.96085i
$964$		0			0
$965$	9.00000	−	9.00000i	0.289720	−	0.289720i
$966$		0			0
$967$	7.36584			0.236870			0.118435	−	0.992962i	$-0.462212\pi$
$967$	7.36584			0.236870			0.118435	+	0.992962i	$0.462212\pi$
$968$		0			0
$969$	57.8410			1.85812
$970$		0			0
$971$	15.7916	−	15.7916i	0.506775	−	0.506775i	−0.406760	−	0.913535i	$-0.633341\pi$
$971$	15.7916	−	15.7916i	0.506775	−	0.506775i	0.913535	+	0.406760i	$0.133341\pi$
$972$		0			0
$973$	0.554108	−	0.554108i	0.0177639	−	0.0177639i
$974$		0			0
$975$	17.2665	−	40.4935i	0.552971	−	1.29683i
$976$		0			0
$977$			9.97038i			0.318981i	0.987199	+	0.159490i	$0.0509851\pi$
$977$			9.97038i			0.318981i	−0.987199	+	0.159490i	$0.949015\pi$
$978$		0			0
$979$	19.8997	+	19.8997i	0.635999	+	0.635999i
$980$		0			0
$981$	14.8138	+	14.8138i	0.472968	+	0.472968i
$982$		0			0
$983$	−51.5834			−1.64525			−0.822627	−	0.568582i	$-0.807492\pi$
$983$	−51.5834			−1.64525			−0.822627	+	0.568582i	$0.807492\pi$
$984$		0			0
$985$		−	4.05013i		−	0.129048i
$986$		0			0
$987$	5.10819	−	5.10819i	0.162595	−	0.162595i
$988$		0			0
$989$	34.5831	+	34.5831i	1.09968	+	1.09968i
$990$		0			0
$991$	−20.9499			−0.665495			−0.332747	−	0.943016i	$-0.607976\pi$
$991$	−20.9499			−0.665495			−0.332747	+	0.943016i	$0.607976\pi$
$992$		0			0
$993$		−	23.1461i		−	0.734519i
$994$		0			0
$995$	12.7279	−	12.7279i	0.403502	−	0.403502i
$996$		0			0
$997$	40.9499	−	40.9499i	1.29690	−	1.29690i	0.366463	−	0.930433i	$-0.380569\pi$
$997$	40.9499	−	40.9499i	1.29690	−	1.29690i	0.930433	−	0.366463i	$-0.119431\pi$
$998$		0			0
$999$	94.3158			2.98402

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	832.2.p.a.337.1		8
4.3	odd	2		208.2.p.a.77.4	yes	8
13.12	even	2	inner	832.2.p.a.337.2		8
16.5	even	4	inner	832.2.p.a.753.2		8
16.11	odd	4		208.2.p.a.181.4	yes	8
52.51	odd	2		208.2.p.a.77.2	✓	8
208.155	odd	4		208.2.p.a.181.2	yes	8
208.181	even	4	inner	832.2.p.a.753.1		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
208.2.p.a.77.2	✓	8	52.51	odd	2
208.2.p.a.77.4	yes	8	4.3	odd	2
208.2.p.a.181.2	yes	8	208.155	odd	4
208.2.p.a.181.4	yes	8	16.11	odd	4
832.2.p.a.337.1		8	1.1	even	1	trivial
832.2.p.a.337.2		8	13.12	even	2	inner
832.2.p.a.753.1		8	208.181	even	4	inner
832.2.p.a.753.2		8	16.5	even	4	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:	\( N \)	\(=\)	\( 832 = 2^{6} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	832.p (of order \(4\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(-2.15831 - 2.15831i) q^{3} +(-0.707107 - 0.707107i) q^{5} +0.223888 q^{7} +6.31662i q^{9} +(-1.41421 - 1.41421i) q^{11} +(-3.34521 + 1.34521i) q^{13} +3.05231i q^{15} +3.00000 q^{17} +(-4.46653 + 4.46653i) q^{19} +(-0.483219 - 0.483219i) q^{21} +6.31662i q^{23} -4.00000i q^{25} +(7.15831 - 7.15831i) q^{27} +(0.316625 + 0.316625i) q^{29} -4.24264i q^{31} +6.10463i q^{33} +(-0.158312 - 0.158312i) q^{35} +(6.58785 + 6.58785i) q^{37} +(10.1234 + 4.31662i) q^{39} +(5.47494 - 5.47494i) q^{43} +(4.46653 - 4.46653i) q^{45} +10.5712i q^{47} -6.94987 q^{49} +(-6.47494 - 6.47494i) q^{51} +(3.63325 - 3.63325i) q^{53} +2.00000i q^{55} +19.2803 q^{57} +(7.74273 + 7.74273i) q^{59} +(4.00000 + 4.00000i) q^{61} +1.41421i q^{63} +(3.31662 + 1.41421i) q^{65} +(4.69042 - 4.69042i) q^{67} +(13.6332 - 13.6332i) q^{69} -0.671663 q^{71} -3.79487 q^{73} +(-8.63325 + 8.63325i) q^{75} +(-0.316625 - 0.316625i) q^{77} +10.9499 q^{79} -11.9499 q^{81} +(-11.9854 + 11.9854i) q^{83} +(-2.12132 - 2.12132i) q^{85} -1.36675i q^{87} -14.0712 q^{89} +(-0.748950 + 0.301175i) q^{91} +(-9.15694 + 9.15694i) q^{93} +6.31662 q^{95} +8.48528i q^{97} +(8.93306 - 8.93306i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 4 q^{3} - 8 q^{13} + 24 q^{17} + 44 q^{27} - 24 q^{29} + 12 q^{35} + 4 q^{43} + 24 q^{49} - 12 q^{51} - 24 q^{53} + 32 q^{61} + 56 q^{69} - 16 q^{75} + 24 q^{77} + 8 q^{79} - 16 q^{81} - 44 q^{91}+ \cdots + 24 q^{95}+O(q^{100}) \) 8 * q - 4 * q^3 - 8 * q^13 + 24 * q^17 + 44 * q^27 - 24 * q^29 + 12 * q^35 + 4 * q^43 + 24 * q^49 - 12 * q^51 - 24 * q^53 + 32 * q^61 + 56 * q^69 - 16 * q^75 + 24 * q^77 + 8 * q^79 - 16 * q^81 - 44 * q^91 + 24 * q^95

Introduction

L-functions

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