LMFDB - Embedded newform 3380.1.cs.a.687.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3380,1,Mod(7,3380)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3380, base_ring=CyclotomicField(156))

chi = DirichletCharacter(H, H._module([78, 39, 107]))

N = Newforms(chi, 1, names="a")

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

chi := DirichletCharacter("3380.7");

S:= CuspForms(chi, 1);

N := Newforms(S);

Level:	$N$	$=$	$3380 = 2^{2} \cdot 5 \cdot 13^{2}$
Weight:	$k$	$=$	$1$
Character orbit:	$[\chi]$	$=$	3380.cs (of order $156$ , degree $48$ , 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:	$1.68683974270$
Analytic rank:	$0$
Dimension:	$48$
Coefficient field:	$\Q(\zeta_{156})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$x^{48} + x^{46} - x^{42} - x^{40} + x^{36} + x^{34} - x^{30} - x^{28} + x^{24} - x^{20} - x^{18} + \cdots + 1$ x^48 + x^46 - x^42 - x^40 + x^36 + x^34 - x^30 - x^28 + x^24 - x^20 - x^18 + x^14 + x^12 - x^8 - x^6 + x^2 + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$1$
Twist minimal:	yes
Projective image:	$D_{156}$
Projective field:	Galois closure of $\mathbb{Q}[x]/(x^{156} - \cdots)$

Embedding invariants

Embedding label			687.1
Root			$-0.600742 + 0.799443i$ of defining polynomial
Character	$\chi$	$=$	3380.687
Dual form			3380.1.cs.a.123.1

$q$ -expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(0.960518 - 0.278217i) q^{2} +(0.845190 - 0.534466i) q^{4} +(-0.200026 - 0.979791i) q^{5} +(0.663123 - 0.748511i) q^{8} +(-0.721202 - 0.692724i) q^{9} +(-0.464723 - 0.885456i) q^{10} +(-0.948536 - 0.316668i) q^{13} +(0.428693 - 0.903450i) q^{16} +(0.196337 - 0.393130i) q^{17} +(-0.885456 - 0.464723i) q^{18} +(-0.692724 - 0.721202i) q^{20} +(-0.919979 + 0.391967i) q^{25} +(-0.999189 - 0.0402659i) q^{26} +(-1.53576 + 0.444838i) q^{29} +(0.160411 - 0.987050i) q^{32} +(0.0792096 - 0.432233i) q^{34} +(-0.979791 - 0.200026i) q^{36} +(0.158849 - 0.0258155i) q^{37} +(-0.866025 - 0.500000i) q^{40} +(1.62515 + 0.654068i) q^{41} +(-0.534466 + 0.845190i) q^{45} +(0.799443 - 0.600742i) q^{49} +(-0.774605 + 0.632445i) q^{50} +(-0.970942 + 0.239316i) q^{52} +(1.78600 - 0.108033i) q^{53} +(-1.35136 + 0.854550i) q^{58} +(0.156807 + 0.768090i) q^{61} +(-0.120537 - 0.992709i) q^{64} +(-0.120537 + 0.992709i) q^{65} +(-0.0441724 - 0.437205i) q^{68} +(-0.996757 + 0.0804666i) q^{72} +(-0.239316 - 0.970942i) q^{73} +(0.145395 - 0.0689908i) q^{74} +(-0.970942 - 0.239316i) q^{80} +(0.0402659 + 0.999189i) q^{81} +(1.74296 + 0.176098i) q^{82} +(-0.424457 - 0.113733i) q^{85} +(0.0933069 - 0.348226i) q^{89} +(-0.278217 + 0.960518i) q^{90} +(-0.150475 - 1.86397i) q^{97} +(0.600742 - 0.799443i) q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$48 q - 2 q^{4} + 2 q^{5} - 2 q^{13} + 2 q^{16} + 4 q^{17} + 4 q^{18} - 2 q^{20} + 2 q^{25} + 2 q^{34} - 2 q^{41} + 2 q^{49} - 4 q^{52} - 2 q^{53} - 2 q^{58} + 4 q^{64} + 4 q^{65} + 2 q^{68} + 2 q^{72} + 20 q^{74}+ \cdots - 2 q^{90}+O(q^{100})$ 48 * q - 2 * q^4 + 2 * q^5 - 2 * q^13 + 2 * q^16 + 4 * q^17 + 4 * q^18 - 2 * q^20 + 2 * q^25 + 2 * q^34 - 2 * q^41 + 2 * q^49 - 4 * q^52 - 2 * q^53 - 2 * q^58 + 4 * q^64 + 4 * q^65 + 2 * q^68 + 2 * q^72 + 20 * q^74 - 4 * q^80 - 2 * q^81 - 2 * q^82 - 2 * q^85 - 2 * q^89 - 2 * q^90

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/3380\mathbb{Z}\right)^\times$ .

$n$	$677$	$1691$	$1861$
$\chi(n)$	$e\left(\frac{1}{4}\right)$	$-1$	$e\left(\frac{103}{156}\right)$

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

)

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

$2$	0.960518	−	0.278217i	0.960518	−	0.278217i
$2$	0.960518	−	0.278217i	0.960518	−	0.278217i
$3$		0			0		0.373361	−	0.927686i	$-0.378205\pi$
$3$		0			0		−0.373361	+	0.927686i	$0.621795\pi$
$4$	0.845190	−	0.534466i	0.845190	−	0.534466i
$5$	−0.200026	−	0.979791i	−0.200026	−	0.979791i
$5$	−0.200026	−	0.979791i	−0.200026	−	0.979791i
$6$		0			0
$7$		0			0		0.948536	−	0.316668i	$-0.102564\pi$
$7$		0			0		−0.948536	+	0.316668i	$0.897436\pi$
$8$	0.663123	−	0.748511i	0.663123	−	0.748511i
$9$	−0.721202	−	0.692724i	−0.721202	−	0.692724i
$10$	−0.464723	−	0.885456i	−0.464723	−	0.885456i
$11$		0			0		−0.999797	−	0.0201371i	$-0.993590\pi$
$11$		0			0		0.999797	+	0.0201371i	$0.00641026\pi$
$12$		0			0
$13$	−0.948536	−	0.316668i	−0.948536	−	0.316668i
$13$	−0.948536	−	0.316668i	−0.948536	−	0.316668i
$14$		0			0
$15$		0			0
$16$	0.428693	−	0.903450i	0.428693	−	0.903450i
$17$	0.196337	−	0.393130i	0.196337	−	0.393130i	−0.774605	−	0.632445i	$-0.782051\pi$
$17$	0.196337	−	0.393130i	0.196337	−	0.393130i	0.970942	+	0.239316i	$0.0769231\pi$
$18$	−0.885456	−	0.464723i	−0.885456	−	0.464723i
$19$		0			0		−0.258819	−	0.965926i	$-0.583333\pi$
$19$		0			0		0.258819	+	0.965926i	$0.416667\pi$
$20$	−0.692724	−	0.721202i	−0.692724	−	0.721202i
$21$		0			0
$22$		0			0
$23$		0			0		0.258819	−	0.965926i	$-0.416667\pi$
$23$		0			0		−0.258819	+	0.965926i	$0.583333\pi$
$24$		0			0
$25$	−0.919979	+	0.391967i	−0.919979	+	0.391967i
$26$	−0.999189	−	0.0402659i	−0.999189	−	0.0402659i
$27$		0			0
$28$		0			0
$29$	−1.53576	+	0.444838i	−1.53576	+	0.444838i	−0.935016	−	0.354605i	$-0.884615\pi$
$29$	−1.53576	+	0.444838i	−1.53576	+	0.444838i	−0.600742	+	0.799443i	$0.705128\pi$
$30$		0			0
$31$		0			0		0.911900	−	0.410413i	$-0.134615\pi$
$31$		0			0		−0.911900	+	0.410413i	$0.865385\pi$
$32$	0.160411	−	0.987050i	0.160411	−	0.987050i
$33$		0			0
$34$	0.0792096	−	0.432233i	0.0792096	−	0.432233i
$35$		0			0
$36$	−0.979791	−	0.200026i	−0.979791	−	0.200026i
$37$	0.158849	−	0.0258155i	0.158849	−	0.0258155i	−0.0804666	−	0.996757i	$-0.525641\pi$
$37$	0.158849	−	0.0258155i	0.158849	−	0.0258155i	0.239316	+	0.970942i	$0.423077\pi$
$38$		0			0
$39$		0			0
$40$	−0.866025	−	0.500000i	−0.866025	−	0.500000i
$41$	1.62515	+	0.654068i	1.62515	+	0.654068i	0.992709	−	0.120537i	$-0.0384615\pi$
$41$	1.62515	+	0.654068i	1.62515	+	0.654068i	0.632445	+	0.774605i	$0.282051\pi$
$42$		0			0
$43$		0			0		−0.584522	−	0.811378i	$-0.698718\pi$
$43$		0			0		0.584522	+	0.811378i	$0.301282\pi$
$44$		0			0
$45$	−0.534466	+	0.845190i	−0.534466	+	0.845190i
$46$		0			0
$47$		0			0		0.885456	−	0.464723i	$-0.153846\pi$
$47$		0			0		−0.885456	+	0.464723i	$0.846154\pi$
$48$		0			0
$49$	0.799443	−	0.600742i	0.799443	−	0.600742i
$50$	−0.774605	+	0.632445i	−0.774605	+	0.632445i
$51$		0			0
$52$	−0.970942	+	0.239316i	−0.970942	+	0.239316i
$53$	1.78600	−	0.108033i	1.78600	−	0.108033i	0.866025	−	0.500000i	$-0.166667\pi$
$53$	1.78600	−	0.108033i	1.78600	−	0.108033i	0.919979	+	0.391967i	$0.128205\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$		0			0
$58$	−1.35136	+	0.854550i	−1.35136	+	0.854550i
$59$		0			0		0.335705	−	0.941967i	$-0.391026\pi$
$59$		0			0		−0.335705	+	0.941967i	$0.608974\pi$
$60$		0			0
$61$	0.156807	+	0.768090i	0.156807	+	0.768090i	0.979791	+	0.200026i	$0.0641026\pi$
$61$	0.156807	+	0.768090i	0.156807	+	0.768090i	−0.822984	+	0.568065i	$0.807692\pi$
$62$		0			0
$63$		0			0
$64$	−0.120537	−	0.992709i	−0.120537	−	0.992709i
$65$	−0.120537	+	0.992709i	−0.120537	+	0.992709i
$66$		0			0
$67$		0			0		0.534466	−	0.845190i	$-0.320513\pi$
$67$		0			0		−0.534466	+	0.845190i	$0.679487\pi$
$68$	−0.0441724	−	0.437205i	−0.0441724	−	0.437205i
$69$		0			0
$70$		0			0
$71$		0			0		−0.140502	−	0.990080i	$-0.544872\pi$
$71$		0			0		0.140502	+	0.990080i	$0.455128\pi$
$72$	−0.996757	+	0.0804666i	−0.996757	+	0.0804666i
$73$	−0.239316	−	0.970942i	−0.239316	−	0.970942i	−0.960518	−	0.278217i	$-0.910256\pi$
$73$	−0.239316	−	0.970942i	−0.239316	−	0.970942i	0.721202	−	0.692724i	$-0.243590\pi$
$74$	0.145395	−	0.0689908i	0.145395	−	0.0689908i
$75$		0			0
$76$		0			0
$77$		0			0
$78$		0			0
$79$		0			0		−0.464723	−	0.885456i	$-0.653846\pi$
$79$		0			0		0.464723	+	0.885456i	$0.346154\pi$
$80$	−0.970942	−	0.239316i	−0.970942	−	0.239316i
$81$	0.0402659	+	0.999189i	0.0402659	+	0.999189i
$82$	1.74296	+	0.176098i	1.74296	+	0.176098i
$83$		0			0		−0.120537	−	0.992709i	$-0.538462\pi$
$83$		0			0		0.120537	+	0.992709i	$0.461538\pi$
$84$		0			0
$85$	−0.424457	−	0.113733i	−0.424457	−	0.113733i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	0.0933069	−	0.348226i	0.0933069	−	0.348226i	−0.903450	−	0.428693i	$-0.858974\pi$
$89$	0.0933069	−	0.348226i	0.0933069	−	0.348226i	0.996757	+	0.0804666i	$0.0256410\pi$
$90$	−0.278217	+	0.960518i	−0.278217	+	0.960518i
$91$		0			0
$92$		0			0
$93$		0			0
$94$		0			0
$95$		0			0
$96$		0			0
$97$	−0.150475	−	1.86397i	−0.150475	−	1.86397i	−0.428693	−	0.903450i	$-0.641026\pi$
$97$	−0.150475	−	1.86397i	−0.150475	−	1.86397i	0.278217	−	0.960518i	$-0.410256\pi$
$98$	0.600742	−	0.799443i	0.600742	−	0.799443i
$99$		0			0
$100$	−0.568065	+	0.822984i	−0.568065	+	0.822984i
$101$	0.248511	+	0.202903i	0.248511	+	0.202903i	0.748511	−	0.663123i	$-0.230769\pi$
$101$	0.248511	+	0.202903i	0.248511	+	0.202903i	−0.500000	+	0.866025i	$0.666667\pi$
$102$		0			0
$103$		0			0		0.787183	−	0.616719i	$-0.211538\pi$
$103$		0			0		−0.787183	+	0.616719i	$0.788462\pi$
$104$	−0.866025	+	0.500000i	−0.866025	+	0.500000i
$105$		0			0
$106$	1.68543	−	0.600666i	1.68543	−	0.600666i
$107$		0			0		−0.647915	−	0.761712i	$-0.724359\pi$
$107$		0			0		0.647915	+	0.761712i	$0.275641\pi$
$108$		0			0
$109$	−0.542586	−	0.244198i	−0.542586	−	0.244198i	0.120537	−	0.992709i	$-0.461538\pi$
$109$	−0.542586	−	0.244198i	−0.542586	−	0.244198i	−0.663123	+	0.748511i	$0.730769\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	0.622857	−	0.250678i	0.622857	−	0.250678i	−0.0402659	−	0.999189i	$-0.512821\pi$
$113$	0.622857	−	0.250678i	0.622857	−	0.250678i	0.663123	+	0.748511i	$0.269231\pi$
$114$		0			0
$115$		0			0
$116$	−1.06026	+	1.19678i	−1.06026	+	1.19678i
$117$	0.464723	+	0.885456i	0.464723	+	0.885456i
$118$		0			0
$119$		0			0
$120$		0			0
$121$	0.999189	+	0.0402659i	0.999189	+	0.0402659i
$122$	0.364312	+	0.694138i	0.364312	+	0.694138i
$123$		0			0
$124$		0			0
$125$	0.568065	+	0.822984i	0.568065	+	0.822984i
$126$		0			0
$127$		0			0		0.975564	−	0.219715i	$-0.0705128\pi$
$127$		0			0		−0.975564	+	0.219715i	$0.929487\pi$
$128$	−0.391967	−	0.919979i	−0.391967	−	0.919979i
$129$		0			0
$130$	0.160411	+	0.987050i	0.160411	+	0.987050i
$131$		0			0		−0.970942	−	0.239316i	$-0.923077\pi$
$131$		0			0		0.970942	+	0.239316i	$0.0769231\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$		0			0
$136$	−0.164066	−	0.407653i	−0.164066	−	0.407653i
$137$	0.846282	+	0.137534i	0.846282	+	0.137534i	0.568065	−	0.822984i	$-0.307692\pi$
$137$	0.846282	+	0.137534i	0.846282	+	0.137534i	0.278217	+	0.960518i	$0.410256\pi$
$138$		0			0
$139$		0			0		−0.0804666	−	0.996757i	$-0.525641\pi$
$139$		0			0		0.0804666	+	0.996757i	$0.474359\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		0			0
$144$	−0.935016	+	0.354605i	−0.935016	+	0.354605i
$145$	0.743039	+	1.41574i	0.743039	+	1.41574i
$146$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$147$		0			0
$148$	0.120460	−	0.106718i	0.120460	−	0.106718i
$149$	1.34925	+	1.46257i	1.34925	+	1.46257i	0.748511	+	0.663123i	$0.230769\pi$
$149$	1.34925	+	1.46257i	1.34925	+	1.46257i	0.600742	+	0.799443i	$0.294872\pi$
$150$		0			0
$151$		0			0		0.954721	−	0.297503i	$-0.0961538\pi$
$151$		0			0		−0.954721	+	0.297503i	$0.903846\pi$
$152$		0			0
$153$	−0.413929	+	0.147519i	−0.413929	+	0.147519i
$154$		0			0
$155$		0			0
$156$		0			0
$157$	−0.197751	+	0.0362392i	−0.197751	+	0.0362392i	−0.278217	−	0.960518i	$-0.589744\pi$
$157$	−0.197751	+	0.0362392i	−0.197751	+	0.0362392i	0.0804666	+	0.996757i	$0.474359\pi$
$158$		0			0
$159$		0			0
$160$	−0.999189	+	0.0402659i	−0.999189	+	0.0402659i
$161$		0			0
$162$	0.316668	+	0.948536i	0.316668	+	0.948536i
$163$		0			0		0.774605	−	0.632445i	$-0.217949\pi$
$163$		0			0		−0.774605	+	0.632445i	$0.782051\pi$
$164$	1.72314	−	0.315777i	1.72314	−	0.315777i
$165$		0			0
$166$		0			0
$167$		0			0		−0.278217	−	0.960518i	$-0.589744\pi$
$167$		0			0		0.278217	+	0.960518i	$0.410256\pi$
$168$		0			0
$169$	0.799443	+	0.600742i	0.799443	+	0.600742i
$170$	−0.439341	+	0.00884883i	−0.439341	+	0.00884883i
$171$		0			0
$172$		0			0
$173$	−1.53590	−	0.345912i	−1.53590	−	0.345912i	−0.632445	−	0.774605i	$-0.717949\pi$
$173$	−1.53590	−	0.345912i	−1.53590	−	0.345912i	−0.903450	+	0.428693i	$0.858974\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$		0			0
$178$	−0.00725961	−	0.360437i	−0.00725961	−	0.360437i
$179$		0			0		−0.948536	−	0.316668i	$-0.897436\pi$
$179$		0			0		0.948536	+	0.316668i	$0.102564\pi$
$180$			1.00000i			1.00000i
$181$	−0.329236	−	0.227255i	−0.329236	−	0.227255i	0.391967	−	0.919979i	$-0.371795\pi$
$181$	−0.329236	−	0.227255i	−0.329236	−	0.227255i	−0.721202	+	0.692724i	$0.756410\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−0.0570677	−	0.150475i	−0.0570677	−	0.150475i
$186$		0			0
$187$		0			0
$188$		0			0
$189$		0			0
$190$		0			0
$191$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$191$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$192$		0			0
$193$	−0.490585	+	1.46948i	−0.490585	+	1.46948i	0.354605	+	0.935016i	$0.384615\pi$
$193$	−0.490585	+	1.46948i	−0.490585	+	1.46948i	−0.845190	+	0.534466i	$0.820513\pi$
$194$	−0.663123	−	1.74851i	−0.663123	−	1.74851i
$195$		0			0
$196$	0.354605	−	0.935016i	0.354605	−	0.935016i
$197$	1.53791	+	1.25567i	1.53791	+	1.25567i	0.845190	+	0.534466i	$0.179487\pi$
$197$	1.53791	+	1.25567i	1.53791	+	1.25567i	0.692724	+	0.721202i	$0.256410\pi$
$198$		0			0
$199$		0			0		0.996757	−	0.0804666i	$-0.0256410\pi$
$199$		0			0		−0.996757	+	0.0804666i	$0.974359\pi$
$200$	−0.316668	+	0.948536i	−0.316668	+	0.948536i
$201$		0			0
$202$	0.295150	+	0.125752i	0.295150	+	0.125752i
$203$		0			0
$204$		0			0
$205$	0.315777	−	1.72314i	0.315777	−	1.72314i
$206$		0			0
$207$		0			0
$208$	−0.692724	+	0.721202i	−0.692724	+	0.721202i
$209$		0			0
$210$		0			0
$211$		0			0		−0.845190	−	0.534466i	$-0.820513\pi$
$211$		0			0		0.845190	+	0.534466i	$0.179487\pi$
$212$	1.45177	−	1.04587i	1.45177	−	1.04587i
$213$		0			0
$214$		0			0
$215$		0			0
$216$		0			0
$217$		0			0
$218$	−0.589104	−	0.0835998i	−0.589104	−	0.0835998i
$219$		0			0
$220$		0			0
$221$	−0.310724	+	0.310724i	−0.310724	+	0.310724i
$222$		0			0
$223$		0			0		−0.996757	−	0.0804666i	$-0.974359\pi$
$223$		0			0		0.996757	+	0.0804666i	$0.0256410\pi$
$224$		0			0
$225$	0.935016	+	0.354605i	0.935016	+	0.354605i
$226$	0.528522	−	0.414071i	0.528522	−	0.414071i
$227$		0			0		−0.534466	−	0.845190i	$-0.679487\pi$
$227$		0			0		0.534466	+	0.845190i	$0.320513\pi$
$228$		0			0
$229$	0.819328	−	1.82047i	0.819328	−	1.82047i	0.354605	−	0.935016i	$-0.384615\pi$
$229$	0.819328	−	1.82047i	0.819328	−	1.82047i	0.464723	−	0.885456i	$-0.346154\pi$
$230$		0			0
$231$		0			0
$232$	−0.685430	+	1.44451i	−0.685430	+	1.44451i
$233$	1.12477	−	1.43566i	1.12477	−	1.43566i	0.239316	−	0.970942i	$-0.423077\pi$
$233$	1.12477	−	1.43566i	1.12477	−	1.43566i	0.885456	−	0.464723i	$-0.153846\pi$
$234$	0.692724	+	0.721202i	0.692724	+	0.721202i
$235$		0			0
$236$		0			0
$237$		0			0
$238$		0			0
$239$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$239$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$240$		0			0
$241$	−1.20212	+	1.41326i	−1.20212	+	1.41326i	−0.316668	+	0.948536i	$0.602564\pi$
$241$	−1.20212	+	1.41326i	−1.20212	+	1.41326i	−0.885456	+	0.464723i	$0.846154\pi$
$242$	0.970942	−	0.239316i	0.970942	−	0.239316i
$243$		0			0
$244$	0.543050	+	0.565375i	0.543050	+	0.565375i
$245$	−0.748511	−	0.663123i	−0.748511	−	0.663123i
$246$		0			0
$247$		0			0
$248$		0			0
$249$		0			0
$250$	0.774605	+	0.632445i	0.774605	+	0.632445i
$251$		0			0		0.774605	−	0.632445i	$-0.217949\pi$
$251$		0			0		−0.774605	+	0.632445i	$0.782051\pi$
$252$		0			0
$253$		0			0
$254$		0			0
$255$		0			0
$256$	−0.632445	−	0.774605i	−0.632445	−	0.774605i
$257$	−1.22637	−	1.13135i	−1.22637	−	1.13135i	−0.987050	−	0.160411i	$-0.948718\pi$
$257$	−1.22637	−	1.13135i	−1.22637	−	1.13135i	−0.239316	−	0.970942i	$-0.576923\pi$
$258$		0			0
$259$		0			0
$260$	0.428693	+	0.903450i	0.428693	+	0.903450i
$261$	1.41574	+	0.743039i	1.41574	+	0.743039i
$262$		0			0
$263$		0			0		−0.999797	−	0.0201371i	$-0.993590\pi$
$263$		0			0		0.999797	+	0.0201371i	$0.00641026\pi$
$264$		0			0
$265$	−0.463097	−	1.72830i	−0.463097	−	1.72830i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−0.988802	+	1.31586i	−0.988802	+	1.31586i	−0.0402659	+	0.999189i	$0.512821\pi$
$269$	−0.988802	+	1.31586i	−0.988802	+	1.31586i	−0.948536	+	0.316668i	$0.897436\pi$
$270$		0			0
$271$		0			0		0.975564	−	0.219715i	$-0.0705128\pi$
$271$		0			0		−0.975564	+	0.219715i	$0.929487\pi$
$272$	−0.271005	−	0.345912i	−0.271005	−	0.345912i
$273$		0			0
$274$	0.851134	−	0.103346i	0.851134	−	0.103346i
$275$		0			0
$276$		0			0
$277$	−0.431843	−	0.285414i	−0.431843	−	0.285414i	0.316668	−	0.948536i	$-0.397436\pi$
$277$	−0.431843	−	0.285414i	−0.431843	−	0.285414i	−0.748511	+	0.663123i	$0.769231\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	0.551098	+	0.703425i	0.551098	+	0.703425i	0.979791	−	0.200026i	$-0.0641026\pi$
$281$	0.551098	+	0.703425i	0.551098	+	0.703425i	−0.428693	+	0.903450i	$0.641026\pi$
$282$		0			0
$283$		0			0		−0.811378	−	0.584522i	$-0.801282\pi$
$283$		0			0		0.811378	+	0.584522i	$0.198718\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$	−0.799443	+	0.600742i	−0.799443	+	0.600742i
$289$	0.484739	+	0.645071i	0.484739	+	0.645071i
$290$	1.10759	+	1.15312i	1.10759	+	1.15312i
$291$		0			0
$292$	−0.721202	−	0.692724i	−0.721202	−	0.692724i
$293$	−1.93421	+	0.394871i	−1.93421	+	0.394871i	−0.935016	+	0.354605i	$0.884615\pi$
$293$	−1.93421	+	0.394871i	−1.93421	+	0.394871i	−0.999189	+	0.0402659i	$0.987179\pi$
$294$		0			0
$295$		0			0
$296$	0.0860133	−	0.136019i	0.0860133	−	0.136019i
$297$		0			0
$298$	1.70289	+	1.02943i	1.70289	+	1.02943i
$299$		0			0
$300$		0			0
$301$		0			0
$302$		0			0
$303$		0			0
$304$		0			0
$305$	0.721202	−	0.307276i	0.721202	−	0.307276i
$306$	−0.356544	+	0.256857i	−0.356544	+	0.256857i
$307$		0			0		0.354605	−	0.935016i	$-0.384615\pi$
$307$		0			0		−0.354605	+	0.935016i	$0.615385\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$		0			0		−0.935016	−	0.354605i	$-0.884615\pi$
$311$		0			0		0.935016	+	0.354605i	$0.115385\pi$
$312$		0			0
$313$	−1.56077	−	0.702447i	−1.56077	−	0.702447i	−0.568065	−	0.822984i	$-0.692308\pi$
$313$	−1.56077	−	0.702447i	−1.56077	−	0.702447i	−0.992709	+	0.120537i	$0.961538\pi$
$314$	−0.179861	+	0.0898262i	−0.179861	+	0.0898262i
$315$		0			0
$316$		0			0
$317$	1.25799	+	1.41998i	1.25799	+	1.41998i	0.866025	+	0.500000i	$0.166667\pi$
$317$	1.25799	+	1.41998i	1.25799	+	1.41998i	0.391967	+	0.919979i	$0.371795\pi$
$318$		0			0
$319$		0			0
$320$	−0.948536	+	0.316668i	−0.948536	+	0.316668i
$321$		0			0
$322$		0			0
$323$		0			0
$324$	0.568065	+	0.822984i	0.568065	+	0.822984i
$325$	0.996757	−	0.0804666i	0.996757	−	0.0804666i
$326$		0			0
$327$		0			0
$328$	1.56725	−	0.782718i	1.56725	−	0.782718i
$329$		0			0
$330$		0			0
$331$		0			0		−0.894635	−	0.446798i	$-0.852564\pi$
$331$		0			0		0.894635	+	0.446798i	$0.147436\pi$
$332$		0			0
$333$	−0.132445	−	0.0914204i	−0.132445	−	0.0914204i
$334$		0			0
$335$		0			0
$336$		0			0
$337$	0.826639	−	0.826639i	0.826639	−	0.826639i	−0.160411	−	0.987050i	$-0.551282\pi$
$337$	0.826639	−	0.826639i	0.826639	−	0.826639i	0.987050	+	0.160411i	$0.0512821\pi$
$338$	0.935016	+	0.354605i	0.935016	+	0.354605i
$339$		0			0
$340$	−0.419533	+	0.130732i	−0.419533	+	0.130732i
$341$		0			0
$342$		0			0
$343$		0			0
$344$		0			0
$345$		0			0
$346$	−1.57149	+	0.0950579i	−1.57149	+	0.0950579i
$347$		0			0		0.999797	−	0.0201371i	$-0.00641026\pi$
$347$		0			0		−0.999797	+	0.0201371i	$0.993590\pi$
$348$		0			0
$349$	−0.0396144	+	1.96684i	−0.0396144	+	1.96684i	0.160411	+	0.987050i	$0.448718\pi$
$349$	−0.0396144	+	1.96684i	−0.0396144	+	1.96684i	−0.200026	+	0.979791i	$0.564103\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−0.391967	+	1.91998i	−0.391967	+	1.91998i			1.00000i	$0.5\pi$
$353$	−0.391967	+	1.91998i	−0.391967	+	1.91998i	−0.391967	+	0.919979i	$0.628205\pi$
$354$		0			0
$355$		0			0
$356$	−0.107253	−	0.344186i	−0.107253	−	0.344186i
$357$		0			0
$358$		0			0
$359$		0			0		−0.998176	−	0.0603785i	$-0.980769\pi$
$359$		0			0		0.998176	+	0.0603785i	$0.0192308\pi$
$360$	0.278217	+	0.960518i	0.278217	+	0.960518i
$361$	−0.866025	+	0.500000i	−0.866025	+	0.500000i
$362$	−0.379463	−	0.126683i	−0.379463	−	0.126683i
$363$		0			0
$364$		0			0
$365$	−0.903450	+	0.428693i	−0.903450	+	0.428693i
$366$		0			0
$367$		0			0		0.482459	−	0.875918i	$-0.339744\pi$
$367$		0			0		−0.482459	+	0.875918i	$0.660256\pi$
$368$		0			0
$369$	−0.718976	−	1.59750i	−0.718976	−	1.59750i
$370$	−0.0966793	−	0.128657i	−0.0966793	−	0.128657i
$371$		0			0
$372$		0			0
$373$	−0.465534	−	0.845190i	−0.465534	−	0.845190i	0.534466	−	0.845190i	$-0.320513\pi$
$373$	−0.465534	−	0.845190i	−0.465534	−	0.845190i	−1.00000			$\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	1.59759	+	0.0643806i	1.59759	+	0.0643806i
$378$		0			0
$379$		0			0		0.373361	−	0.927686i	$-0.378205\pi$
$379$		0			0		−0.373361	+	0.927686i	$0.621795\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$		0			0		−0.200026	−	0.979791i	$-0.564103\pi$
$383$		0			0		0.200026	+	0.979791i	$0.435897\pi$
$384$		0			0
$385$		0			0
$386$	−0.0623804	+	1.54795i	−0.0623804	+	1.54795i
$387$		0			0
$388$	−1.12341	−	1.49498i	−1.12341	−	1.49498i
$389$	−1.49217	+	1.32194i	−1.49217	+	1.32194i	−0.692724	+	0.721202i	$0.743590\pi$
$389$	−1.49217	+	1.32194i	−1.49217	+	1.32194i	−0.799443	+	0.600742i	$0.794872\pi$
$390$		0			0
$391$		0			0
$392$	0.0804666	−	0.996757i	0.0804666	−	0.996757i
$393$		0			0
$394$	1.82654	+	0.778217i	1.82654	+	0.778217i
$395$		0			0
$396$		0			0
$397$	−0.205186	−	0.432420i	−0.205186	−	0.432420i	0.774605	−	0.632445i	$-0.217949\pi$
$397$	−0.205186	−	0.432420i	−0.205186	−	0.432420i	−0.979791	+	0.200026i	$0.935897\pi$
$398$		0			0
$399$		0			0
$400$	−0.0402659	+	0.999189i	−0.0402659	+	0.999189i
$401$	0.544766	+	1.52858i	0.544766	+	1.52858i	0.822984	+	0.568065i	$0.192308\pi$
$401$	0.544766	+	1.52858i	0.544766	+	1.52858i	−0.278217	+	0.960518i	$0.589744\pi$
$402$		0			0
$403$		0			0
$404$	0.318483	+	0.0386709i	0.318483	+	0.0386709i
$405$	0.970942	−	0.239316i	0.970942	−	0.239316i
$406$		0			0
$407$		0			0
$408$		0			0
$409$	1.91269	−	0.271430i	1.91269	−	0.271430i	0.919979	−	0.391967i	$-0.128205\pi$
$409$	1.91269	−	0.271430i	1.91269	−	0.271430i	0.992709	+	0.120537i	$0.0384615\pi$
$410$	−0.176098	−	1.74296i	−0.176098	−	1.74296i
$411$		0			0
$412$		0			0
$413$		0			0
$414$		0			0
$415$		0			0
$416$	−0.464723	+	0.885456i	−0.464723	+	0.885456i
$417$		0			0
$418$		0			0
$419$		0			0		0.999189	−	0.0402659i	$-0.0128205\pi$
$419$		0			0		−0.999189	+	0.0402659i	$0.987179\pi$
$420$		0			0
$421$	0.460518	−	0.587808i	0.460518	−	0.587808i	−0.500000	−	0.866025i	$-0.666667\pi$
$421$	0.460518	−	0.587808i	0.460518	−	0.587808i	0.960518	+	0.278217i	$0.0897436\pi$
$422$		0			0
$423$		0			0
$424$	1.10348	−	1.40848i	1.10348	−	1.40848i
$425$	−0.0265322	+	0.438629i	−0.0265322	+	0.438629i
$426$		0			0
$427$		0			0
$428$		0			0
$429$		0			0
$430$		0			0
$431$		0			0		0.678061	−	0.735006i	$-0.262821\pi$
$431$		0			0		−0.678061	+	0.735006i	$0.737179\pi$
$432$		0			0
$433$	0.435016	−	1.22063i	0.435016	−	1.22063i	−0.500000	−	0.866025i	$-0.666667\pi$
$433$	0.435016	−	1.22063i	0.435016	−	1.22063i	0.935016	−	0.354605i	$-0.115385\pi$
$434$		0			0
$435$		0			0
$436$	−0.589104	+	0.0835998i	−0.589104	+	0.0835998i
$437$		0			0
$438$		0			0
$439$		0			0		0.632445	−	0.774605i	$-0.282051\pi$
$439$		0			0		−0.632445	+	0.774605i	$0.717949\pi$
$440$		0			0
$441$	−0.992709	−	0.120537i	−0.992709	−	0.120537i
$442$	−0.212007	+	0.384905i	−0.212007	+	0.384905i
$443$		0			0		−0.787183	−	0.616719i	$-0.788462\pi$
$443$		0			0		0.787183	+	0.616719i	$0.211538\pi$
$444$		0			0
$445$	−0.359852	−	0.0217671i	−0.359852	−	0.0217671i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	1.00939	+	0.227334i	1.00939	+	0.227334i	0.692724	−	0.721202i	$-0.256410\pi$
$449$	1.00939	+	0.227334i	1.00939	+	0.227334i	0.316668	+	0.948536i	$0.397436\pi$
$450$	0.996757	+	0.0804666i	0.996757	+	0.0804666i
$451$		0			0
$452$	0.392453	−	0.544766i	0.392453	−	0.544766i
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$	0.380472	+	0.506316i	0.380472	+	0.506316i	0.948536	−	0.316668i	$-0.102564\pi$
$457$	0.380472	+	0.506316i	0.380472	+	0.506316i	−0.568065	+	0.822984i	$0.692308\pi$
$458$	0.280492	−	1.97655i	0.280492	−	1.97655i
$459$		0			0
$460$		0			0
$461$	1.88355	−	0.0379369i	1.88355	−	0.0379369i	0.935016	−	0.354605i	$-0.115385\pi$
$461$	1.88355	−	0.0379369i	1.88355	−	0.0379369i	0.948536	+	0.316668i	$0.102564\pi$
$462$		0			0
$463$		0			0		0.239316	−	0.970942i	$-0.423077\pi$
$463$		0			0		−0.239316	+	0.970942i	$0.576923\pi$
$464$	−0.256479	+	1.57818i	−0.256479	+	1.57818i
$465$		0			0
$466$	0.680937	−	1.69191i	0.680937	−	1.69191i
$467$		0			0		0.517338	−	0.855781i	$-0.326923\pi$
$467$		0			0		−0.517338	+	0.855781i	$0.673077\pi$
$468$	0.866025	+	0.500000i	0.866025	+	0.500000i
$469$		0			0
$470$		0			0
$471$		0			0
$472$		0			0
$473$		0			0
$474$		0			0
$475$		0			0
$476$		0			0
$477$	−1.36291	−	1.15930i	−1.36291	−	1.15930i
$478$		0			0
$479$		0			0		0.994935	−	0.100522i	$-0.0320513\pi$
$479$		0			0		−0.994935	+	0.100522i	$0.967949\pi$
$480$		0			0
$481$	−0.158849	−	0.0258155i	−0.158849	−	0.0258155i
$482$	−0.761468	+	1.69191i	−0.761468	+	1.69191i
$483$		0			0
$484$	0.866025	−	0.500000i	0.866025	−	0.500000i
$485$	−1.79620	+	0.520276i	−1.79620	+	0.520276i
$486$		0			0
$487$		0			0		−0.999189	−	0.0402659i	$-0.987179\pi$
$487$		0			0		0.999189	+	0.0402659i	$0.0128205\pi$
$488$	0.678906	+	0.391967i	0.678906	+	0.391967i
$489$		0			0
$490$	−0.903450	−	0.428693i	−0.903450	−	0.428693i
$491$		0			0		−0.903450	−	0.428693i	$-0.858974\pi$
$491$		0			0		0.903450	+	0.428693i	$0.141026\pi$
$492$		0			0
$493$	−0.126647	+	0.691090i	−0.126647	+	0.691090i
$494$		0			0
$495$		0			0
$496$		0			0
$497$		0			0
$498$		0			0
$499$		0			0		0.998176	−	0.0603785i	$-0.0192308\pi$
$499$		0			0		−0.998176	+	0.0603785i	$0.980769\pi$
$500$	0.919979	+	0.391967i	0.919979	+	0.391967i
$501$		0			0
$502$		0			0
$503$		0			0		0.219715	−	0.975564i	$-0.429487\pi$
$503$		0			0		−0.219715	+	0.975564i	$0.570513\pi$
$504$		0			0
$505$	0.149094	−	0.284074i	0.149094	−	0.284074i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	0.246137	+	0.135573i	0.246137	+	0.135573i	0.600742	−	0.799443i	$-0.294872\pi$
$509$	0.246137	+	0.135573i	0.246137	+	0.135573i	−0.354605	+	0.935016i	$0.615385\pi$
$510$		0			0
$511$		0			0
$512$	−0.822984	−	0.568065i	−0.822984	−	0.568065i
$513$		0			0
$514$	−1.49271	−	0.745489i	−1.49271	−	0.745489i
$515$		0			0
$516$		0			0
$517$		0			0
$518$		0			0
$519$		0			0
$520$	0.663123	+	0.748511i	0.663123	+	0.748511i
$521$	1.09127	+	1.58098i	1.09127	+	1.58098i	0.774605	+	0.632445i	$0.217949\pi$
$521$	1.09127	+	1.58098i	1.09127	+	1.58098i	0.316668	+	0.948536i	$0.397436\pi$
$522$	1.56657	+	0.319818i	1.56657	+	0.319818i
$523$		0			0		−0.335705	−	0.941967i	$-0.608974\pi$
$523$		0			0		0.335705	+	0.941967i	$0.391026\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$		0			0
$528$		0			0
$529$	−0.866025	−	0.500000i	−0.866025	−	0.500000i
$530$	−0.925657	−	1.53122i	−0.925657	−	1.53122i
$531$		0			0
$532$		0			0
$533$	−1.33440	−	1.13504i	−1.33440	−	1.13504i
$534$		0			0
$535$		0			0
$536$		0			0
$537$		0			0
$538$	−0.583668	+	1.53901i	−0.583668	+	1.53901i
$539$		0			0
$540$		0			0
$541$	−1.08469	−	0.198777i	−1.08469	−	0.198777i	−0.391967	−	0.919979i	$-0.628205\pi$
$541$	−1.08469	−	0.198777i	−1.08469	−	0.198777i	−0.692724	+	0.721202i	$0.743590\pi$
$542$		0			0
$543$		0			0
$544$	−0.356544	−	0.256857i	−0.356544	−	0.256857i
$545$	−0.130732	+	0.580467i	−0.130732	+	0.580467i
$546$		0			0
$547$		0			0		−0.855781	−	0.517338i	$-0.826923\pi$
$547$		0			0		0.855781	+	0.517338i	$0.173077\pi$
$548$	0.788777	−	0.336066i	0.788777	−	0.336066i
$549$	0.418986	−	0.662573i	0.418986	−	0.662573i
$550$		0			0
$551$		0			0
$552$		0			0
$553$		0			0
$554$	−0.494200	−	0.153999i	−0.494200	−	0.153999i
$555$		0			0
$556$		0			0
$557$	−1.38468	+	1.04052i	−1.38468	+	1.04052i	−0.391967	+	0.919979i	$0.628205\pi$
$557$	−1.38468	+	1.04052i	−1.38468	+	1.04052i	−0.992709	+	0.120537i	$0.961538\pi$
$558$		0			0
$559$		0			0
$560$		0			0
$561$		0			0
$562$	0.725045	+	0.522327i	0.725045	+	0.522327i
$563$		0			0		−0.373361	−	0.927686i	$-0.621795\pi$
$563$		0			0		0.373361	+	0.927686i	$0.378205\pi$
$564$		0			0
$565$	−0.370200	−	0.560127i	−0.370200	−	0.560127i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	0.926432	+	0.0747894i	0.926432	+	0.0747894i	0.534466	−	0.845190i	$-0.320513\pi$
$569$	0.926432	+	0.0747894i	0.926432	+	0.0747894i	0.391967	+	0.919979i	$0.371795\pi$
$570$		0			0
$571$		0			0		0.992709	−	0.120537i	$-0.0384615\pi$
$571$		0			0		−0.992709	+	0.120537i	$0.961538\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$	−0.600742	+	0.799443i	−0.600742	+	0.799443i
$577$		−	0.0805319i		−	0.0805319i	−0.999189	−	0.0402659i	$-0.987179\pi$
$577$		−	0.0805319i		−	0.0805319i	0.999189	−	0.0402659i	$-0.0128205\pi$
$578$	0.645071	+	0.484739i	0.645071	+	0.484739i
$579$		0			0
$580$	1.38468	+	0.799443i	1.38468	+	0.799443i
$581$		0			0
$582$		0			0
$583$		0			0
$584$	−0.885456	−	0.464723i	−0.885456	−	0.464723i
$585$	0.774605	−	0.632445i	0.774605	−	0.632445i
$586$	−1.74798	+	0.917410i	−1.74798	+	0.917410i
$587$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$587$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$588$		0			0
$589$		0			0
$590$		0			0
$591$		0			0
$592$	0.0447744	−	0.154579i	0.0447744	−	0.154579i
$593$	−1.98381	+	0.240878i	−1.98381	+	0.240878i	−0.987050	+	0.160411i	$0.948718\pi$
$593$	−1.98381	+	0.240878i	−1.98381	+	0.240878i	−0.996757	+	0.0804666i	$0.974359\pi$
$594$		0			0
$595$		0			0
$596$	1.92207	+	0.515016i	1.92207	+	0.515016i
$597$		0			0
$598$		0			0
$599$		0			0		0.464723	−	0.885456i	$-0.346154\pi$
$599$		0			0		−0.464723	+	0.885456i	$0.653846\pi$
$600$		0			0
$601$	−1.27458	−	1.32698i	−1.27458	−	1.32698i	−0.919979	−	0.391967i	$-0.871795\pi$
$601$	−1.27458	−	1.32698i	−1.27458	−	1.32698i	−0.354605	−	0.935016i	$-0.615385\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−0.160411	−	0.987050i	−0.160411	−	0.987050i
$606$		0			0
$607$		0			0		−0.140502	−	0.990080i	$-0.544872\pi$
$607$		0			0		0.140502	+	0.990080i	$0.455128\pi$
$608$		0			0
$609$		0			0
$610$	0.607239	−	0.495795i	0.607239	−	0.495795i
$611$		0			0
$612$	−0.271005	+	0.345912i	−0.271005	+	0.345912i
$613$	−0.788777	+	1.66231i	−0.788777	+	1.66231i	−0.0402659	+	0.999189i	$0.512821\pi$
$613$	−0.788777	+	1.66231i	−0.788777	+	1.66231i	−0.748511	+	0.663123i	$0.769231\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	1.08548	−	0.515067i	1.08548	−	0.515067i	0.200026	−	0.979791i	$-0.435897\pi$
$617$	1.08548	−	0.515067i	1.08548	−	0.515067i	0.885456	+	0.464723i	$0.153846\pi$
$618$		0			0
$619$		0			0		0.787183	−	0.616719i	$-0.211538\pi$
$619$		0			0		−0.787183	+	0.616719i	$0.788462\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$		0			0
$625$	0.692724	−	0.721202i	0.692724	−	0.721202i
$626$	−1.69458	−	0.240479i	−1.69458	−	0.240479i
$627$		0			0
$628$	−0.147768	+	0.136320i	−0.147768	+	0.136320i
$629$	0.0210391	−	0.0675168i	0.0210391	−	0.0675168i
$630$		0			0
$631$		0			0		−0.834256	−	0.551377i	$-0.814103\pi$
$631$		0			0		0.834256	+	0.551377i	$0.185897\pi$
$632$		0			0
$633$		0			0
$634$	1.60339	+	1.01392i	1.60339	+	1.01392i
$635$		0			0
$636$		0			0
$637$	−0.948536	+	0.316668i	−0.948536	+	0.316668i
$638$		0			0
$639$		0			0
$640$	−0.822984	+	0.568065i	−0.822984	+	0.568065i
$641$	−1.66367	−	0.481887i	−1.66367	−	0.481887i	−0.692724	−	0.721202i	$-0.743590\pi$
$641$	−1.66367	−	0.481887i	−1.66367	−	0.481887i	−0.970942	+	0.239316i	$0.923077\pi$
$642$		0			0
$643$		0			0		−0.919979	−	0.391967i	$-0.871795\pi$
$643$		0			0		0.919979	+	0.391967i	$0.128205\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		0			0		−0.875918	−	0.482459i	$-0.839744\pi$
$647$		0			0		0.875918	+	0.482459i	$0.160256\pi$
$648$	0.774605	+	0.632445i	0.774605	+	0.632445i
$649$		0			0
$650$	0.935016	−	0.354605i	0.935016	−	0.354605i
$651$		0			0
$652$		0			0
$653$	0.116642	+	0.0312542i	0.116642	+	0.0312542i	0.316668	−	0.948536i	$-0.397436\pi$
$653$	0.116642	+	0.0312542i	0.116642	+	0.0312542i	−0.200026	+	0.979791i	$0.564103\pi$
$654$		0			0
$655$		0			0
$656$	1.28761	−	1.18785i	1.28761	−	1.18785i
$657$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$658$		0			0
$659$		0			0		−0.316668	−	0.948536i	$-0.602564\pi$
$659$		0			0		0.316668	+	0.948536i	$0.397436\pi$
$660$		0			0
$661$	−0.810531	−	1.22637i	−0.810531	−	1.22637i	−0.970942	−	0.239316i	$-0.923077\pi$
$661$	−0.810531	−	1.22637i	−0.810531	−	1.22637i	0.160411	−	0.987050i	$-0.448718\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$	−0.152651	−	0.0509624i	−0.152651	−	0.0509624i
$667$		0			0
$668$		0			0
$669$		0			0
$670$		0			0
$671$		0			0
$672$		0			0
$673$	1.02301	+	0.411726i	1.02301	+	0.411726i	0.822984	−	0.568065i	$-0.192308\pi$
$673$	1.02301	+	0.411726i	1.02301	+	0.411726i	0.200026	+	0.979791i	$0.435897\pi$
$674$	0.564016	−	1.02399i	0.564016	−	1.02399i
$675$		0			0
$676$	0.996757	+	0.0804666i	0.996757	+	0.0804666i
$677$	−1.35018	−	1.35018i	−1.35018	−	1.35018i	−0.885456	−	0.464723i	$-0.846154\pi$
$677$	−1.35018	−	1.35018i	−1.35018	−	1.35018i	−0.464723	−	0.885456i	$-0.653846\pi$
$678$		0			0
$679$		0			0
$680$	−0.366598	+	0.242292i	−0.366598	+	0.242292i
$681$		0			0
$682$		0			0
$683$		0			0		−0.316668	−	0.948536i	$-0.602564\pi$
$683$		0			0		0.316668	+	0.948536i	$0.397436\pi$
$684$		0			0
$685$	−0.0345234	−	0.856690i	−0.0345234	−	0.856690i
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−1.72830	−	0.463097i	−1.72830	−	0.463097i
$690$		0			0
$691$		0			0		0.834256	−	0.551377i	$-0.185897\pi$
$691$		0			0		−0.834256	+	0.551377i	$0.814103\pi$
$692$	−1.48300	+	0.528522i	−1.48300	+	0.528522i
$693$		0			0
$694$		0			0
$695$		0			0
$696$		0			0
$697$	0.576211	−	0.510479i	0.576211	−	0.510479i
$698$	0.509159	+	1.90021i	0.509159	+	1.90021i
$699$		0			0
$700$		0			0
$701$	−1.24006	+	0.470293i	−1.24006	+	0.470293i	−0.885456	−	0.464723i	$-0.846154\pi$
$701$	−1.24006	+	0.470293i	−1.24006	+	0.470293i	−0.354605	+	0.935016i	$0.615385\pi$
$702$		0			0
$703$		0			0
$704$		0			0
$705$		0			0
$706$	0.157681	+	1.95323i	0.157681	+	1.95323i
$707$		0			0
$708$		0			0
$709$	0.506324	+	1.25806i	0.506324	+	1.25806i	0.935016	+	0.354605i	$0.115385\pi$
$709$	0.506324	+	1.25806i	0.506324	+	1.25806i	−0.428693	+	0.903450i	$0.641026\pi$
$710$		0			0
$711$		0			0
$712$	−0.198777	−	0.300758i	−0.198777	−	0.300758i
$713$		0			0
$714$		0			0
$715$		0			0
$716$		0			0
$717$		0			0
$718$		0			0
$719$		0			0		−0.987050	−	0.160411i	$-0.948718\pi$
$719$		0			0		0.987050	+	0.160411i	$0.0512821\pi$
$720$	0.534466	+	0.845190i	0.534466	+	0.845190i
$721$		0			0
$722$	−0.692724	+	0.721202i	−0.692724	+	0.721202i
$723$		0			0
$724$	−0.399727	−	0.0161084i	−0.399727	−	0.0161084i
$725$	1.23850	−	1.01121i	1.23850	−	1.01121i
$726$		0			0
$727$		0			0		0.0603785	−	0.998176i	$-0.480769\pi$
$727$		0			0		−0.0603785	+	0.998176i	$0.519231\pi$
$728$		0			0
$729$	0.663123	−	0.748511i	0.663123	−	0.748511i
$730$	−0.748511	+	0.663123i	−0.748511	+	0.663123i
$731$		0			0
$732$		0			0
$733$	0.152466	−	1.25567i	0.152466	−	1.25567i	−0.692724	−	0.721202i	$-0.743590\pi$
$733$	0.152466	−	1.25567i	0.152466	−	1.25567i	0.845190	−	0.534466i	$-0.179487\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		0			0
$738$	−1.13504	−	1.33440i	−1.13504	−	1.33440i
$739$		0			0		0.941967	−	0.335705i	$-0.108974\pi$
$739$		0			0		−0.941967	+	0.335705i	$0.891026\pi$
$740$	−0.128657	−	0.0966793i	−0.128657	−	0.0966793i
$741$		0			0
$742$		0			0
$743$		0			0		−0.845190	−	0.534466i	$-0.820513\pi$
$743$		0			0		0.845190	+	0.534466i	$0.179487\pi$
$744$		0			0
$745$	1.16312	−	1.61454i	1.16312	−	1.61454i
$746$	−0.682301	−	0.682301i	−0.682301	−	0.682301i
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$		0			0		0.721202	−	0.692724i	$-0.243590\pi$
$751$		0			0		−0.721202	+	0.692724i	$0.756410\pi$
$752$		0			0
$753$		0			0
$754$	1.55242	−	0.382638i	1.55242	−	0.382638i
$755$		0			0
$756$		0			0
$757$	−0.836346	+	0.906584i	−0.836346	+	0.906584i	−0.996757	−	0.0804666i	$-0.974359\pi$
$757$	−0.836346	+	0.906584i	−0.836346	+	0.906584i	0.160411	+	0.987050i	$0.448718\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	0.0582603	+	0.105773i	0.0582603	+	0.105773i	0.903450	−	0.428693i	$-0.141026\pi$
$761$	0.0582603	+	0.105773i	0.0582603	+	0.105773i	−0.845190	+	0.534466i	$0.820513\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$	0.227334	+	0.376056i	0.227334	+	0.376056i
$766$		0			0
$767$		0			0
$768$		0			0
$769$	0.906584	+	0.836346i	0.906584	+	0.836346i	0.987050	−	0.160411i	$-0.0512821\pi$
$769$	0.906584	+	0.836346i	0.906584	+	0.836346i	−0.0804666	+	0.996757i	$0.525641\pi$
$770$		0			0
$771$		0			0
$772$	0.370750	+	1.50419i	0.370750	+	1.50419i
$773$	0.926432	−	0.0747894i	0.926432	−	0.0747894i	0.391967	−	0.919979i	$-0.371795\pi$
$773$	0.926432	−	0.0747894i	0.926432	−	0.0747894i	0.534466	+	0.845190i	$0.320513\pi$
$774$		0			0
$775$		0			0
$776$	−1.49498	−	1.12341i	−1.49498	−	1.12341i
$777$		0			0
$778$	−1.06547	+	1.68490i	−1.06547	+	1.68490i
$779$		0			0
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$	−0.200026	−	0.979791i	−0.200026	−	0.979791i
$785$	0.0750621	+	0.186506i	0.0750621	+	0.186506i
$786$		0			0
$787$		0			0		0.845190	−	0.534466i	$-0.179487\pi$
$787$		0			0		−0.845190	+	0.534466i	$0.820513\pi$
$788$	1.97094	+	0.239316i	1.97094	+	0.239316i
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$	0.0944927	−	0.778217i	0.0944927	−	0.778217i
$794$	−0.317391	−	0.358261i	−0.317391	−	0.358261i
$795$		0			0
$796$		0			0
$797$	0.701573	+	0.760492i	0.701573	+	0.760492i	0.979791	−	0.200026i	$-0.0641026\pi$
$797$	0.701573	+	0.760492i	0.701573	+	0.760492i	−0.278217	+	0.960518i	$0.589744\pi$
$798$		0			0
$799$		0			0
$800$	0.239316	+	0.970942i	0.239316	+	0.970942i
$801$	−0.308518	+	0.186505i	−0.308518	+	0.186505i
$802$	0.948536	+	1.31667i	0.948536	+	1.31667i
$803$		0			0
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$	0.316668	−	0.0514636i	0.316668	−	0.0514636i
$809$	−1.23933	−	0.253011i	−1.23933	−	0.253011i	−0.464723	−	0.885456i	$-0.653846\pi$
$809$	−1.23933	−	0.253011i	−1.23933	−	0.253011i	−0.774605	+	0.632445i	$0.782051\pi$
$810$	0.866025	−	0.500000i	0.866025	−	0.500000i
$811$		0			0		0.180255	−	0.983620i	$-0.442308\pi$
$811$		0			0		−0.180255	+	0.983620i	$0.557692\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$		0			0
$816$		0			0
$817$		0			0
$818$	1.76166	−	0.792857i	1.76166	−	0.792857i
$819$		0			0
$820$	−0.654068	−	1.62515i	−0.654068	−	1.62515i
$821$	−0.366744	−	0.734339i	−0.366744	−	0.734339i	0.632445	−	0.774605i	$-0.282051\pi$
$821$	−0.366744	−	0.734339i	−0.366744	−	0.734339i	−0.999189	+	0.0402659i	$0.987179\pi$
$822$		0			0
$823$		0			0		0.965926	−	0.258819i	$-0.0833333\pi$
$823$		0			0		−0.965926	+	0.258819i	$0.916667\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		0			0		−0.885456	−	0.464723i	$-0.846154\pi$
$827$		0			0		0.885456	+	0.464723i	$0.153846\pi$
$828$		0			0
$829$	−0.271506	+	0.572188i	−0.271506	+	0.572188i	−0.992709	−	0.120537i	$-0.961538\pi$
$829$	−0.271506	+	0.572188i	−0.271506	+	0.572188i	0.721202	+	0.692724i	$0.243590\pi$
$830$		0			0
$831$		0			0
$832$	−0.200026	+	0.979791i	−0.200026	+	0.979791i
$833$	−0.0792096	−	0.432233i	−0.0792096	−	0.432233i
$834$		0			0
$835$		0			0
$836$		0			0
$837$		0			0
$838$		0			0
$839$		0			0		0.100522	−	0.994935i	$-0.467949\pi$
$839$		0			0		−0.100522	+	0.994935i	$0.532051\pi$
$840$		0			0
$841$	1.31548	−	0.831861i	1.31548	−	0.831861i
$842$	0.278798	−	0.692724i	0.278798	−	0.692724i
$843$		0			0
$844$		0			0
$845$	0.428693	−	0.903450i	0.428693	−	0.903450i
$846$		0			0
$847$		0			0
$848$	0.668044	−	1.65988i	0.668044	−	1.65988i
$849$		0			0
$850$	0.0965496	+	0.428693i	0.0965496	+	0.428693i
$851$		0			0
$852$		0			0
$853$	−1.12093	+	1.26527i	−1.12093	+	1.26527i	−0.160411	+	0.987050i	$0.551282\pi$
$853$	−1.12093	+	1.26527i	−1.12093	+	1.26527i	−0.960518	+	0.278217i	$0.910256\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−0.134601	−	0.734492i	−0.134601	−	0.734492i	−0.979791	−	0.200026i	$-0.935897\pi$
$857$	−0.134601	−	0.734492i	−0.134601	−	0.734492i	0.845190	−	0.534466i	$-0.179487\pi$
$858$		0			0
$859$		0			0		0.822984	−	0.568065i	$-0.192308\pi$
$859$		0			0		−0.822984	+	0.568065i	$0.807692\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		0			0		−0.885456	−	0.464723i	$-0.846154\pi$
$863$		0			0		0.885456	+	0.464723i	$0.153846\pi$
$864$		0			0
$865$	−0.0317031	+	1.57405i	−0.0317031	+	1.57405i
$866$	0.0782403	−	1.29347i	0.0782403	−	1.29347i
$867$		0			0
$868$		0			0
$869$		0			0
$870$		0			0
$871$		0			0
$872$	−0.542586	+	0.244198i	−0.542586	+	0.244198i
$873$	−1.18269	+	1.44854i	−1.18269	+	1.44854i
$874$		0			0
$875$		0			0
$876$		0			0
$877$	0.308156	−	1.89616i	0.308156	−	1.89616i	−0.120537	−	0.992709i	$-0.538462\pi$
$877$	0.308156	−	1.89616i	0.308156	−	1.89616i	0.428693	−	0.903450i	$-0.358974\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	0.545190	+	0.111301i	0.545190	+	0.111301i	0.464723	−	0.885456i	$-0.346154\pi$
$881$	0.545190	+	0.111301i	0.545190	+	0.111301i	0.0804666	+	0.996757i	$0.474359\pi$
$882$	−0.987050	+	0.160411i	−0.987050	+	0.160411i
$883$		0			0		−0.517338	−	0.855781i	$-0.673077\pi$
$883$		0			0		0.517338	+	0.855781i	$0.326923\pi$
$884$	−0.0965496	+	0.428693i	−0.0965496	+	0.428693i
$885$		0			0
$886$		0			0
$887$		0			0		−0.219715	−	0.975564i	$-0.570513\pi$
$887$		0			0		0.219715	+	0.975564i	$0.429487\pi$
$888$		0			0
$889$		0			0
$890$	−0.351701	+	0.0792096i	−0.351701	+	0.0792096i
$891$		0			0
$892$		0			0
$893$		0			0
$894$		0			0
$895$		0			0
$896$		0			0
$897$		0			0
$898$	1.03279	−	0.0624722i	1.03279	−	0.0624722i
$899$		0			0
$900$	0.979791	−	0.200026i	0.979791	−	0.200026i
$901$	0.308187	−	0.723343i	0.308187	−	0.723343i
$902$		0			0
$903$		0			0
$904$	0.225395	−	0.632445i	0.225395	−	0.632445i
$905$	−0.156807	+	0.368039i	−0.156807	+	0.368039i
$906$		0			0
$907$		0			0		0.761712	−	0.647915i	$-0.224359\pi$
$907$		0			0		−0.761712	+	0.647915i	$0.775641\pi$
$908$		0			0
$909$	−0.0386709	−	0.318483i	−0.0386709	−	0.318483i
$910$		0			0
$911$		0			0		0.120537	−	0.992709i	$-0.461538\pi$
$911$		0			0		−0.120537	+	0.992709i	$0.538462\pi$
$912$		0			0
$913$		0			0
$914$	0.506316	+	0.380472i	0.506316	+	0.380472i
$915$		0			0
$916$	−0.280492	−	1.97655i	−0.280492	−	1.97655i
$917$		0			0
$918$		0			0
$919$		0			0		0.903450	−	0.428693i	$-0.141026\pi$
$919$		0			0		−0.903450	+	0.428693i	$0.858974\pi$
$920$		0			0
$921$		0			0
$922$	1.79863	−	0.560476i	1.79863	−	0.560476i
$923$		0			0
$924$		0			0
$925$	−0.136019	+	0.0860133i	−0.136019	+	0.0860133i
$926$		0			0
$927$		0			0
$928$	0.192724	+	1.58723i	0.192724	+	1.58723i
$929$	0.941340	+	1.70903i	0.941340	+	1.70903i	0.663123	+	0.748511i	$0.269231\pi$
$929$	0.941340	+	1.70903i	0.941340	+	1.70903i	0.278217	+	0.960518i	$0.410256\pi$
$930$		0			0
$931$		0			0
$932$	0.183332	−	1.81456i	0.183332	−	1.81456i
$933$		0			0
$934$		0			0
$935$		0			0
$936$	0.970942	+	0.239316i	0.970942	+	0.239316i
$937$	−0.574732	−	1.84438i	−0.574732	−	1.84438i	−0.534466	−	0.845190i	$-0.679487\pi$
$937$	−0.574732	−	1.84438i	−0.574732	−	1.84438i	−0.0402659	−	0.999189i	$-0.512821\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−1.05555	−	0.638104i	−1.05555	−	0.638104i	−0.120537	−	0.992709i	$-0.538462\pi$
$941$	−1.05555	−	0.638104i	−1.05555	−	0.638104i	−0.935016	+	0.354605i	$0.884615\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$		0			0
$946$		0			0
$947$		0			0		−0.845190	−	0.534466i	$-0.820513\pi$
$947$		0			0		0.845190	+	0.534466i	$0.179487\pi$
$948$		0			0
$949$	−0.0804666	+	0.996757i	−0.0804666	+	0.996757i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	0.680089	−	1.02900i	0.680089	−	1.02900i	−0.316668	−	0.948536i	$-0.602564\pi$
$953$	0.680089	−	1.02900i	0.680089	−	1.02900i	0.996757	−	0.0804666i	$-0.0256410\pi$
$954$	−1.63163	−	0.734339i	−1.63163	−	0.734339i
$955$		0			0
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	0.663123	−	0.748511i	0.663123	−	0.748511i
$962$	−0.159760	+	0.0193983i	−0.159760	+	0.0193983i
$963$		0			0
$964$	−0.260684	+	1.83697i	−0.260684	+	1.83697i
$965$	1.53791	+	0.186737i	1.53791	+	0.186737i
$966$		0			0
$967$		0			0		−0.464723	−	0.885456i	$-0.653846\pi$
$967$		0			0		0.464723	+	0.885456i	$0.346154\pi$
$968$	0.692724	−	0.721202i	0.692724	−	0.721202i
$969$		0			0
$970$	−1.58053	+	0.999468i	−1.58053	+	0.999468i
$971$		0			0		−0.987050	−	0.160411i	$-0.948718\pi$
$971$		0			0		0.987050	+	0.160411i	$0.0512821\pi$
$972$		0			0
$973$		0			0
$974$		0			0
$975$		0			0
$976$	0.761154	+	0.187607i	0.761154	+	0.187607i
$977$	−0.202903	−	1.24851i	−0.202903	−	1.24851i	−0.866025	−	0.500000i	$-0.833333\pi$
$977$	−0.202903	−	1.24851i	−0.202903	−	1.24851i	0.663123	−	0.748511i	$-0.269231\pi$
$978$		0			0
$979$		0			0
$980$	−0.987050	−	0.160411i	−0.987050	−	0.160411i
$981$	0.222152	+	0.551979i	0.222152	+	0.551979i
$982$		0			0
$983$		0			0		−0.935016	−	0.354605i	$-0.884615\pi$
$983$		0			0		0.935016	+	0.354605i	$0.115385\pi$
$984$		0			0
$985$	0.922670	−	1.75800i	0.922670	−	1.75800i
$986$	0.0706267	+	0.699040i	0.0706267	+	0.699040i
$987$		0			0
$988$		0			0
$989$		0			0
$990$		0			0
$991$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$991$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$		0			0
$996$		0			0
$997$	−1.68543	−	0.841739i	−1.68543	−	0.841739i	−0.992709	−	0.120537i	$-0.961538\pi$
$997$	−1.68543	−	0.841739i	−1.68543	−	0.841739i	−0.692724	−	0.721202i	$-0.743590\pi$
$998$		0			0
$999$		0			0

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

)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3380.1.cs.a.687.1	yes	48
4.3	odd	2	CM	3380.1.cs.a.687.1	yes	48
5.3	odd	4		3380.1.cz.a.1363.1	yes	48
20.3	even	4		3380.1.cz.a.1363.1	yes	48
169.123	odd	156		3380.1.cz.a.2827.1	yes	48
676.123	even	156		3380.1.cz.a.2827.1	yes	48
845.123	even	156	inner	3380.1.cs.a.123.1	✓	48
3380.123	odd	156	inner	3380.1.cs.a.123.1	✓	48

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3380.1.cs.a.123.1	✓	48	845.123	even	156	inner
3380.1.cs.a.123.1	✓	48	3380.123	odd	156	inner
3380.1.cs.a.687.1	yes	48	1.1	even	1	trivial
3380.1.cs.a.687.1	yes	48	4.3	odd	2	CM
3380.1.cz.a.1363.1	yes	48	5.3	odd	4
3380.1.cz.a.1363.1	yes	48	20.3	even	4
3380.1.cz.a.2827.1	yes	48	169.123	odd	156
3380.1.cz.a.2827.1	yes	48	676.123	even	156

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Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

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Twists

This project is supported by grants from the US National Science Foundation, the UK Engineering and Physical Sciences Research Council, and the Simons Foundation.

Label	3380.1.cs.a.687.1

Level	$3380$
Weight	$1$
Character	3380.687
Analytic conductor	$1.687$
Analytic rank	$0$
Dimension	$48$
Projective image	$D_{156}$
CM discriminant	-4
Inner twists	$4$