LMFDB - Embedded newform 3380.1.cs.a.1783.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [3380,1,Mod(7,3380)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage: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")

magma://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:traces = [] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(0)] == traces)

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			1783.1
Root			$-0.960518 - 0.278217i$ of defining polynomial
Character	$\chi$	$=$	3380.1783
Dual form			3380.1.cs.a.527.1

$q$ -expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+(0.534466 - 0.845190i) q^{2} +(-0.428693 - 0.903450i) q^{4} +(-0.919979 - 0.391967i) q^{5} +(-0.992709 - 0.120537i) q^{8} +(-0.999189 - 0.0402659i) q^{9} +(-0.822984 + 0.568065i) q^{10} +(-0.799443 + 0.600742i) q^{13} +(-0.632445 + 0.774605i) q^{16} +(0.0943346 + 0.664749i) q^{17} +(-0.568065 + 0.822984i) q^{18} +(0.0402659 + 0.999189i) q^{20} +(0.692724 + 0.721202i) q^{25} +(0.0804666 + 0.996757i) q^{26} +(-0.297395 + 0.470293i) q^{29} +(0.316668 + 0.948536i) q^{32} +(0.612258 + 0.275555i) q^{34} +(0.391967 + 0.919979i) q^{36} +(-0.304312 - 0.101594i) q^{37} +(0.866025 + 0.500000i) q^{40} +(0.439341 + 0.00884883i) q^{41} +(0.903450 + 0.428693i) q^{45} +(0.278217 + 0.960518i) q^{49} +(0.979791 - 0.200026i) q^{50} +(0.885456 + 0.464723i) q^{52} +(-1.55875 + 1.22120i) q^{53} +(0.238540 + 0.502711i) q^{58} +(-1.32698 - 0.565375i) q^{61} +(0.970942 + 0.239316i) q^{64} +(0.970942 - 0.239316i) q^{65} +(0.560127 - 0.370200i) q^{68} +(0.987050 + 0.160411i) q^{72} +(0.464723 - 0.885456i) q^{73} +(-0.248511 + 0.202903i) q^{74} +(0.885456 - 0.464723i) q^{80} +(0.996757 + 0.0804666i) q^{81} +(0.242292 - 0.366598i) q^{82} +(0.173773 - 0.648531i) q^{85} +(-1.76166 - 0.472034i) q^{89} +(0.845190 - 0.534466i) q^{90} +(-0.212745 + 1.30907i) q^{97} +(0.960518 + 0.278217i) q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$48 q - 2 q^{4} + 2 q^{5} - 2 q^{13} + 2 q^{16} + 4 q^{17} + 4 q^{18} - 2 q^{20} + 2 q^{25} + 2 q^{34} - 2 q^{41} + 2 q^{49} - 4 q^{52} - 2 q^{53} - 2 q^{58} + 4 q^{64} + 4 q^{65} + 2 q^{68} + 2 q^{72} + 20 q^{74}+ \cdots - 2 q^{90}+O(q^{100})$ 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 $\left(\mathbb{Z}/3380\mathbb{Z}\right)^\times$ .

$n$	$677$	$1691$	$1861$
$\chi(n)$	$e\left(\frac{3}{4}\right)$	$-1$	$e\left(\frac{145}{156}\right)$

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

)

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

$2$	0.534466	−	0.845190i	0.534466	−	0.845190i
$2$	0.534466	−	0.845190i	0.534466	−	0.845190i
$3$		0			0		0.0201371	−	0.999797i	$-0.493590\pi$
$3$		0			0		−0.0201371	+	0.999797i	$0.506410\pi$
$4$	−0.428693	−	0.903450i	−0.428693	−	0.903450i
$5$	−0.919979	−	0.391967i	−0.919979	−	0.391967i
$5$	−0.919979	−	0.391967i	−0.919979	−	0.391967i
$6$		0			0
$7$		0			0		−0.799443	−	0.600742i	$-0.794872\pi$
$7$		0			0		0.799443	+	0.600742i	$0.205128\pi$
$8$	−0.992709	−	0.120537i	−0.992709	−	0.120537i
$9$	−0.999189	−	0.0402659i	−0.999189	−	0.0402659i
$10$	−0.822984	+	0.568065i	−0.822984	+	0.568065i
$11$		0			0		0.678061	−	0.735006i	$-0.262821\pi$
$11$		0			0		−0.678061	+	0.735006i	$0.737179\pi$
$12$		0			0
$13$	−0.799443	+	0.600742i	−0.799443	+	0.600742i
$13$	−0.799443	+	0.600742i	−0.799443	+	0.600742i
$14$		0			0
$15$		0			0
$16$	−0.632445	+	0.774605i	−0.632445	+	0.774605i
$17$	0.0943346	+	0.664749i	0.0943346	+	0.664749i	0.979791	+	0.200026i	$0.0641026\pi$
$17$	0.0943346	+	0.664749i	0.0943346	+	0.664749i	−0.885456	+	0.464723i	$0.846154\pi$
$18$	−0.568065	+	0.822984i	−0.568065	+	0.822984i
$19$		0			0		0.965926	−	0.258819i	$-0.0833333\pi$
$19$		0			0		−0.965926	+	0.258819i	$0.916667\pi$
$20$	0.0402659	+	0.999189i	0.0402659	+	0.999189i
$21$		0			0
$22$		0			0
$23$		0			0		−0.965926	−	0.258819i	$-0.916667\pi$
$23$		0			0		0.965926	+	0.258819i	$0.0833333\pi$
$24$		0			0
$25$	0.692724	+	0.721202i	0.692724	+	0.721202i
$26$	0.0804666	+	0.996757i	0.0804666	+	0.996757i
$27$		0			0
$28$		0			0
$29$	−0.297395	+	0.470293i	−0.297395	+	0.470293i	−0.960518	−	0.278217i	$-0.910256\pi$
$29$	−0.297395	+	0.470293i	−0.297395	+	0.470293i	0.663123	+	0.748511i	$0.269231\pi$
$30$		0			0
$31$		0			0		0.0603785	−	0.998176i	$-0.480769\pi$
$31$		0			0		−0.0603785	+	0.998176i	$0.519231\pi$
$32$	0.316668	+	0.948536i	0.316668	+	0.948536i
$33$		0			0
$34$	0.612258	+	0.275555i	0.612258	+	0.275555i
$35$		0			0
$36$	0.391967	+	0.919979i	0.391967	+	0.919979i
$37$	−0.304312	−	0.101594i	−0.304312	−	0.101594i	0.160411	−	0.987050i	$-0.448718\pi$
$37$	−0.304312	−	0.101594i	−0.304312	−	0.101594i	−0.464723	+	0.885456i	$0.653846\pi$
$38$		0			0
$39$		0			0
$40$	0.866025	+	0.500000i	0.866025	+	0.500000i
$41$	0.439341	+	0.00884883i	0.439341	+	0.00884883i	0.239316	−	0.970942i	$-0.423077\pi$
$41$	0.439341	+	0.00884883i	0.439341	+	0.00884883i	0.200026	+	0.979791i	$0.435897\pi$
$42$		0			0
$43$		0			0		0.446798	−	0.894635i	$-0.352564\pi$
$43$		0			0		−0.446798	+	0.894635i	$0.647436\pi$
$44$		0			0
$45$	0.903450	+	0.428693i	0.903450	+	0.428693i
$46$		0			0
$47$		0			0		−0.568065	−	0.822984i	$-0.692308\pi$
$47$		0			0		0.568065	+	0.822984i	$0.307692\pi$
$48$		0			0
$49$	0.278217	+	0.960518i	0.278217	+	0.960518i
$50$	0.979791	−	0.200026i	0.979791	−	0.200026i
$51$		0			0
$52$	0.885456	+	0.464723i	0.885456	+	0.464723i
$53$	−1.55875	+	1.22120i	−1.55875	+	1.22120i	−0.692724	+	0.721202i	$0.743590\pi$
$53$	−1.55875	+	1.22120i	−1.55875	+	1.22120i	−0.866025	+	0.500000i	$0.833333\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$		0			0
$58$	0.238540	+	0.502711i	0.238540	+	0.502711i
$59$		0			0		−0.994935	−	0.100522i	$-0.967949\pi$
$59$		0			0		0.994935	+	0.100522i	$0.0320513\pi$
$60$		0			0
$61$	−1.32698	−	0.565375i	−1.32698	−	0.565375i	−0.391967	−	0.919979i	$-0.628205\pi$
$61$	−1.32698	−	0.565375i	−1.32698	−	0.565375i	−0.935016	+	0.354605i	$0.884615\pi$
$62$		0			0
$63$		0			0
$64$	0.970942	+	0.239316i	0.970942	+	0.239316i
$65$	0.970942	−	0.239316i	0.970942	−	0.239316i
$66$		0			0
$67$		0			0		−0.903450	−	0.428693i	$-0.858974\pi$
$67$		0			0		0.903450	+	0.428693i	$0.141026\pi$
$68$	0.560127	−	0.370200i	0.560127	−	0.370200i
$69$		0			0
$70$		0			0
$71$		0			0		−0.482459	−	0.875918i	$-0.660256\pi$
$71$		0			0		0.482459	+	0.875918i	$0.339744\pi$
$72$	0.987050	+	0.160411i	0.987050	+	0.160411i
$73$	0.464723	−	0.885456i	0.464723	−	0.885456i	−0.534466	−	0.845190i	$-0.679487\pi$
$73$	0.464723	−	0.885456i	0.464723	−	0.885456i	0.999189	−	0.0402659i	$-0.0128205\pi$
$74$	−0.248511	+	0.202903i	−0.248511	+	0.202903i
$75$		0			0
$76$		0			0
$77$		0			0
$78$		0			0
$79$		0			0		0.822984	−	0.568065i	$-0.192308\pi$
$79$		0			0		−0.822984	+	0.568065i	$0.807692\pi$
$80$	0.885456	−	0.464723i	0.885456	−	0.464723i
$81$	0.996757	+	0.0804666i	0.996757	+	0.0804666i
$82$	0.242292	−	0.366598i	0.242292	−	0.366598i
$83$		0			0		−0.970942	−	0.239316i	$-0.923077\pi$
$83$		0			0		0.970942	+	0.239316i	$0.0769231\pi$
$84$		0			0
$85$	0.173773	−	0.648531i	0.173773	−	0.648531i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−1.76166	−	0.472034i	−1.76166	−	0.472034i	−0.774605	−	0.632445i	$-0.782051\pi$
$89$	−1.76166	−	0.472034i	−1.76166	−	0.472034i	−0.987050	+	0.160411i	$0.948718\pi$
$90$	0.845190	−	0.534466i	0.845190	−	0.534466i
$91$		0			0
$92$		0			0
$93$		0			0
$94$		0			0
$95$		0			0
$96$		0			0
$97$	−0.212745	+	1.30907i	−0.212745	+	1.30907i	0.632445	+	0.774605i	$0.282051\pi$
$97$	−0.212745	+	1.30907i	−0.212745	+	1.30907i	−0.845190	+	0.534466i	$0.820513\pi$
$98$	0.960518	+	0.278217i	0.960518	+	0.278217i
$99$		0			0
$100$	0.354605	−	0.935016i	0.354605	−	0.935016i
$101$	−0.620537	−	0.126683i	−0.620537	−	0.126683i	−0.120537	−	0.992709i	$-0.538462\pi$
$101$	−0.620537	−	0.126683i	−0.620537	−	0.126683i	−0.500000	+	0.866025i	$0.666667\pi$
$102$		0			0
$103$		0			0		0.517338	−	0.855781i	$-0.326923\pi$
$103$		0			0		−0.517338	+	0.855781i	$0.673077\pi$
$104$	0.866025	−	0.500000i	0.866025	−	0.500000i
$105$		0			0
$106$	0.199050	+	1.97013i	0.199050	+	1.97013i
$107$		0			0		0.584522	−	0.811378i	$-0.301282\pi$
$107$		0			0		−0.584522	+	0.811378i	$0.698718\pi$
$108$		0			0
$109$	0.0217671	+	0.359852i	0.0217671	+	0.359852i	0.992709	+	0.120537i	$0.0384615\pi$
$109$	0.0217671	+	0.359852i	0.0217671	+	0.359852i	−0.970942	+	0.239316i	$0.923077\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−1.98947	+	0.0400701i	−1.98947	+	0.0400701i	−0.996757	−	0.0804666i	$-0.974359\pi$
$113$	−1.98947	+	0.0400701i	−1.98947	+	0.0400701i	−0.992709	+	0.120537i	$0.961538\pi$
$114$		0			0
$115$		0			0
$116$	0.552378	+	0.0670708i	0.552378	+	0.0670708i
$117$	0.822984	−	0.568065i	0.822984	−	0.568065i
$118$		0			0
$119$		0			0
$120$		0			0
$121$	−0.0804666	−	0.996757i	−0.0804666	−	0.996757i
$122$	−1.18708	+	0.819379i	−1.18708	+	0.819379i
$123$		0			0
$124$		0			0
$125$	−0.354605	−	0.935016i	−0.354605	−	0.935016i
$126$		0			0
$127$		0			0		0.941967	−	0.335705i	$-0.108974\pi$
$127$		0			0		−0.941967	+	0.335705i	$0.891026\pi$
$128$	0.721202	−	0.692724i	0.721202	−	0.692724i
$129$		0			0
$130$	0.316668	−	0.948536i	0.316668	−	0.948536i
$131$		0			0		0.885456	−	0.464723i	$-0.153846\pi$
$131$		0			0		−0.885456	+	0.464723i	$0.846154\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$		0			0
$136$	−0.0135202	−	0.671273i	−0.0135202	−	0.671273i
$137$	−1.19979	+	0.400550i	−1.19979	+	0.400550i	−0.845190	−	0.534466i	$-0.820513\pi$
$137$	−1.19979	+	0.400550i	−1.19979	+	0.400550i	−0.354605	+	0.935016i	$0.615385\pi$
$138$		0			0
$139$		0			0		0.160411	−	0.987050i	$-0.448718\pi$
$139$		0			0		−0.160411	+	0.987050i	$0.551282\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		0			0
$144$	0.663123	−	0.748511i	0.663123	−	0.748511i
$145$	0.457937	−	0.316091i	0.457937	−	0.316091i
$146$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$147$		0			0
$148$	0.0386709	+	0.318483i	0.0386709	+	0.318483i
$149$	0.839981	+	0.714491i	0.839981	+	0.714491i	0.960518	−	0.278217i	$-0.0897436\pi$
$149$	0.839981	+	0.714491i	0.839981	+	0.714491i	−0.120537	+	0.992709i	$0.538462\pi$
$150$		0			0
$151$		0			0		0.983620	−	0.180255i	$-0.0576923\pi$
$151$		0			0		−0.983620	+	0.180255i	$0.942308\pi$
$152$		0			0
$153$	−0.0674914	−	0.668008i	−0.0674914	−	0.668008i
$154$		0			0
$155$		0			0
$156$		0			0
$157$	0.684779	+	1.52152i	0.684779	+	1.52152i	0.845190	+	0.534466i	$0.179487\pi$
$157$	0.684779	+	1.52152i	0.684779	+	1.52152i	−0.160411	+	0.987050i	$0.551282\pi$
$158$		0			0
$159$		0			0
$160$	0.0804666	−	0.996757i	0.0804666	−	0.996757i
$161$		0			0
$162$	0.600742	−	0.799443i	0.600742	−	0.799443i
$163$		0			0		0.979791	−	0.200026i	$-0.0641026\pi$
$163$		0			0		−0.979791	+	0.200026i	$0.935897\pi$
$164$	−0.180348	−	0.400717i	−0.180348	−	0.400717i
$165$		0			0
$166$		0			0
$167$		0			0		−0.845190	−	0.534466i	$-0.820513\pi$
$167$		0			0		0.845190	+	0.534466i	$0.179487\pi$
$168$		0			0
$169$	0.278217	−	0.960518i	0.278217	−	0.960518i
$170$	−0.455256	−	0.493489i	−0.455256	−	0.493489i
$171$		0			0
$172$		0			0
$173$	−0.974631	−	0.347345i	−0.974631	−	0.347345i	−0.200026	−	0.979791i	$-0.564103\pi$
$173$	−0.974631	−	0.347345i	−0.974631	−	0.347345i	−0.774605	+	0.632445i	$0.782051\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$		0			0
$178$	−1.34050	+	1.23665i	−1.34050	+	1.23665i
$179$		0			0		0.799443	−	0.600742i	$-0.205128\pi$
$179$		0			0		−0.799443	+	0.600742i	$0.794872\pi$
$180$		−	1.00000i		−	1.00000i
$181$	−1.72039	−	0.652458i	−1.72039	−	0.652458i	−0.721202	−	0.692724i	$-0.756410\pi$
$181$	−1.72039	−	0.652458i	−1.72039	−	0.652458i	−0.999189	+	0.0402659i	$0.987179\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	0.240139	+	0.212745i	0.240139	+	0.212745i
$186$		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
$193$	1.17720	+	1.56657i	1.17720	+	1.56657i	0.748511	+	0.663123i	$0.230769\pi$
$193$	1.17720	+	1.56657i	1.17720	+	1.56657i	0.428693	+	0.903450i	$0.358974\pi$
$194$	0.992709	+	0.879463i	0.992709	+	0.879463i
$195$		0			0
$196$	0.748511	−	0.663123i	0.748511	−	0.663123i
$197$	−0.468959	−	0.0957386i	−0.468959	−	0.0957386i	−0.0402659	−	0.999189i	$-0.512821\pi$
$197$	−0.468959	−	0.0957386i	−0.468959	−	0.0957386i	−0.428693	+	0.903450i	$0.641026\pi$
$198$		0			0
$199$		0			0		−0.987050	−	0.160411i	$-0.948718\pi$
$199$		0			0		0.987050	+	0.160411i	$0.0512821\pi$
$200$	−0.600742	−	0.799443i	−0.600742	−	0.799443i
$201$		0			0
$202$	−0.438727	+	0.456763i	−0.438727	+	0.456763i
$203$		0			0
$204$		0			0
$205$	−0.400717	−	0.180348i	−0.400717	−	0.180348i
$206$		0			0
$207$		0			0
$208$	0.0402659	−	0.999189i	0.0402659	−	0.999189i
$209$		0			0
$210$		0			0
$211$		0			0		0.428693	−	0.903450i	$-0.358974\pi$
$211$		0			0		−0.428693	+	0.903450i	$0.641026\pi$
$212$	1.77152	+	0.884733i	1.77152	+	0.884733i
$213$		0			0
$214$		0			0
$215$		0			0
$216$		0			0
$217$		0			0
$218$	0.315777	+	0.173931i	0.315777	+	0.173931i
$219$		0			0
$220$		0			0
$221$	−0.474758	−	0.474758i	−0.474758	−	0.474758i
$222$		0			0
$223$		0			0		0.987050	−	0.160411i	$-0.0512821\pi$
$223$		0			0		−0.987050	+	0.160411i	$0.948718\pi$
$224$		0			0
$225$	−0.663123	−	0.748511i	−0.663123	−	0.748511i
$226$	−1.02943	+	1.70289i	−1.02943	+	1.70289i
$227$		0			0		0.903450	−	0.428693i	$-0.141026\pi$
$227$		0			0		−0.903450	+	0.428693i	$0.858974\pi$
$228$		0			0
$229$	1.57149	−	0.0950579i	1.57149	−	0.0950579i	0.748511	−	0.663123i	$-0.230769\pi$
$229$	1.57149	−	0.0950579i	1.57149	−	0.0950579i	0.822984	+	0.568065i	$0.192308\pi$
$230$		0			0
$231$		0			0
$232$	0.351915	−	0.431017i	0.351915	−	0.431017i
$233$	0.103342	−	0.0624722i	0.103342	−	0.0624722i	−0.464723	−	0.885456i	$-0.653846\pi$
$233$	0.103342	−	0.0624722i	0.103342	−	0.0624722i	0.568065	+	0.822984i	$0.307692\pi$
$234$	−0.0402659	−	0.999189i	−0.0402659	−	0.999189i
$235$		0			0
$236$		0			0
$237$		0			0
$238$		0			0
$239$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$239$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$240$		0			0
$241$	−1.16881	−	1.62243i	−1.16881	−	1.62243i	−0.600742	−	0.799443i	$-0.705128\pi$
$241$	−1.16881	−	1.62243i	−1.16881	−	1.62243i	−0.568065	−	0.822984i	$-0.692308\pi$
$242$	−0.885456	−	0.464723i	−0.885456	−	0.464723i
$243$		0			0
$244$	0.0580798	+	1.44124i	0.0580798	+	1.44124i
$245$	0.120537	−	0.992709i	0.120537	−	0.992709i
$246$		0			0
$247$		0			0
$248$		0			0
$249$		0			0
$250$	−0.979791	−	0.200026i	−0.979791	−	0.200026i
$251$		0			0		0.979791	−	0.200026i	$-0.0641026\pi$
$251$		0			0		−0.979791	+	0.200026i	$0.935897\pi$
$252$		0			0
$253$		0			0
$254$		0			0
$255$		0			0
$256$	−0.200026	−	0.979791i	−0.200026	−	0.979791i
$257$	−0.483813	−	0.568788i	−0.483813	−	0.568788i	0.464723	−	0.885456i	$-0.346154\pi$
$257$	−0.483813	−	0.568788i	−0.483813	−	0.568788i	−0.948536	+	0.316668i	$0.897436\pi$
$258$		0			0
$259$		0			0
$260$	−0.632445	−	0.774605i	−0.632445	−	0.774605i
$261$	0.316091	−	0.457937i	0.316091	−	0.457937i
$262$		0			0
$263$		0			0		0.678061	−	0.735006i	$-0.262821\pi$
$263$		0			0		−0.678061	+	0.735006i	$0.737179\pi$
$264$		0			0
$265$	1.91269	−	0.512503i	1.91269	−	0.512503i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−1.79620	−	0.520276i	−1.79620	−	0.520276i	−0.799443	−	0.600742i	$-0.794872\pi$
$269$	−1.79620	−	0.520276i	−1.79620	−	0.520276i	−0.996757	+	0.0804666i	$0.974359\pi$
$270$		0			0
$271$		0			0		0.941967	−	0.335705i	$-0.108974\pi$
$271$		0			0		−0.941967	+	0.335705i	$0.891026\pi$
$272$	−0.574579	−	0.347345i	−0.574579	−	0.347345i
$273$		0			0
$274$	−0.302708	+	1.22814i	−0.302708	+	1.22814i
$275$		0			0
$276$		0			0
$277$	0.721279	+	1.79215i	0.721279	+	1.79215i	0.600742	+	0.799443i	$0.294872\pi$
$277$	0.721279	+	1.79215i	0.721279	+	1.79215i	0.120537	+	0.992709i	$0.461538\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	0.240479	+	0.145374i	0.240479	+	0.145374i	0.632445	−	0.774605i	$-0.282051\pi$
$281$	0.240479	+	0.145374i	0.240479	+	0.145374i	−0.391967	+	0.919979i	$0.628205\pi$
$282$		0			0
$283$		0			0		0.894635	−	0.446798i	$-0.147436\pi$
$283$		0			0		−0.894635	+	0.446798i	$0.852564\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$	−0.278217	−	0.960518i	−0.278217	−	0.960518i
$289$	0.527526	−	0.152800i	0.527526	−	0.152800i
$290$	−0.0224054	−	0.555984i	−0.0224054	−	0.555984i
$291$		0			0
$292$	−0.999189	−	0.0402659i	−0.999189	−	0.0402659i
$293$	0.743589	−	1.74527i	0.743589	−	1.74527i	0.0804666	−	0.996757i	$-0.474359\pi$
$293$	0.743589	−	1.74527i	0.743589	−	1.74527i	0.663123	−	0.748511i	$-0.269231\pi$
$294$		0			0
$295$		0			0
$296$	0.289847	+	0.137534i	0.289847	+	0.137534i
$297$		0			0
$298$	1.05282	−	0.328073i	1.05282	−	0.328073i
$299$		0			0
$300$		0			0
$301$		0			0
$302$		0			0
$303$		0			0
$304$		0			0
$305$	0.999189	+	1.04027i	0.999189	+	1.04027i
$306$	−0.600666	−	0.299985i	−0.600666	−	0.299985i
$307$		0			0		0.748511	−	0.663123i	$-0.230769\pi$
$307$		0			0		−0.748511	+	0.663123i	$0.769231\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$		0			0		−0.663123	−	0.748511i	$-0.730769\pi$
$311$		0			0		0.663123	+	0.748511i	$0.269231\pi$
$312$		0			0
$313$	0.115289	+	1.90596i	0.115289	+	1.90596i	0.354605	+	0.935016i	$0.384615\pi$
$313$	0.115289	+	1.90596i	0.115289	+	1.90596i	−0.239316	+	0.970942i	$0.576923\pi$
$314$	1.65196	+	0.234430i	1.65196	+	0.234430i
$315$		0			0
$316$		0			0
$317$	−1.58723	+	0.192724i	−1.58723	+	0.192724i	−0.866025	−	0.500000i	$-0.833333\pi$
$317$	−1.58723	+	0.192724i	−1.58723	+	0.192724i	−0.721202	+	0.692724i	$0.756410\pi$
$318$		0			0
$319$		0			0
$320$	−0.799443	−	0.600742i	−0.799443	−	0.600742i
$321$		0			0
$322$		0			0
$323$		0			0
$324$	−0.354605	−	0.935016i	−0.354605	−	0.935016i
$325$	−0.987050	−	0.160411i	−0.987050	−	0.160411i
$326$		0			0
$327$		0			0
$328$	−0.435071	−	0.0617411i	−0.435071	−	0.0617411i
$329$		0			0
$330$		0			0
$331$		0			0		0.990080	−	0.140502i	$-0.0448718\pi$
$331$		0			0		−0.990080	+	0.140502i	$0.955128\pi$
$332$		0			0
$333$	0.299974	+	0.113765i	0.299974	+	0.113765i
$334$		0			0
$335$		0			0
$336$		0			0
$337$	0.631868	+	0.631868i	0.631868	+	0.631868i	0.948536	−	0.316668i	$-0.102564\pi$
$337$	0.631868	+	0.631868i	0.631868	+	0.631868i	−0.316668	+	0.948536i	$0.602564\pi$
$338$	−0.663123	−	0.748511i	−0.663123	−	0.748511i
$339$		0			0
$340$	−0.660411	+	0.121025i	−0.660411	+	0.121025i
$341$		0			0
$342$		0			0
$343$		0			0
$344$		0			0
$345$		0			0
$346$	−0.814480	+	0.638104i	−0.814480	+	0.638104i
$347$		0			0		−0.678061	−	0.735006i	$-0.737179\pi$
$347$		0			0		0.678061	+	0.735006i	$0.262821\pi$
$348$		0			0
$349$	−0.603311	−	0.556570i	−0.603311	−	0.556570i	0.316668	−	0.948536i	$-0.397436\pi$
$349$	−0.603311	−	0.556570i	−0.603311	−	0.556570i	−0.919979	+	0.391967i	$0.871795\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	0.721202	−	0.307276i	0.721202	−	0.307276i		−	1.00000i	$-0.5\pi$
$353$	0.721202	−	0.307276i	0.721202	−	0.307276i	0.721202	+	0.692724i	$0.243590\pi$
$354$		0			0
$355$		0			0
$356$	0.328749	+	1.79393i	0.328749	+	1.79393i
$357$		0			0
$358$		0			0
$359$		0			0		−0.787183	−	0.616719i	$-0.788462\pi$
$359$		0			0		0.787183	+	0.616719i	$0.211538\pi$
$360$	−0.845190	−	0.534466i	−0.845190	−	0.534466i
$361$	0.866025	−	0.500000i	0.866025	−	0.500000i
$362$	−1.47094	+	1.10534i	−1.47094	+	1.10534i
$363$		0			0
$364$		0			0
$365$	−0.774605	+	0.632445i	−0.774605	+	0.632445i
$366$		0			0
$367$		0			0		0.975564	−	0.219715i	$-0.0705128\pi$
$367$		0			0		−0.975564	+	0.219715i	$0.929487\pi$
$368$		0			0
$369$	−0.438629	−	0.0265322i	−0.438629	−	0.0265322i
$370$	0.308156	−	0.0892584i	0.308156	−	0.0892584i
$371$		0			0
$372$		0			0
$373$	−1.90345	−	0.428693i	−1.90345	−	0.428693i	−0.903450	−	0.428693i	$-0.858974\pi$
$373$	−1.90345	−	0.428693i	−1.90345	−	0.428693i	−1.00000			$\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−0.0447744	−	0.554631i	−0.0447744	−	0.554631i
$378$		0			0
$379$		0			0		0.0201371	−	0.999797i	$-0.493590\pi$
$379$		0			0		−0.0201371	+	0.999797i	$0.506410\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$		0			0		−0.919979	−	0.391967i	$-0.871795\pi$
$383$		0			0		0.919979	+	0.391967i	$0.128205\pi$
$384$		0			0
$385$		0			0
$386$	1.95323	−	0.157681i	1.95323	−	0.157681i
$387$		0			0
$388$	1.27388	−	0.368985i	1.27388	−	0.368985i
$389$	−0.237952	−	1.95971i	−0.237952	−	1.95971i	−0.278217	−	0.960518i	$-0.589744\pi$
$389$	−0.237952	−	1.95971i	−0.237952	−	1.95971i	0.0402659	−	0.999189i	$-0.487179\pi$
$390$		0			0
$391$		0			0
$392$	−0.160411	−	0.987050i	−0.160411	−	0.987050i
$393$		0			0
$394$	−0.331560	+	0.345190i	−0.331560	+	0.345190i
$395$		0			0
$396$		0			0
$397$	−0.587824	−	0.719954i	−0.587824	−	0.719954i	0.391967	−	0.919979i	$-0.371795\pi$
$397$	−0.587824	−	0.719954i	−0.587824	−	0.719954i	−0.979791	+	0.200026i	$0.935897\pi$
$398$		0			0
$399$		0			0
$400$	−0.996757	+	0.0804666i	−0.996757	+	0.0804666i
$401$	1.78021	−	0.179861i	1.78021	−	0.179861i	0.845190	−	0.534466i	$-0.179487\pi$
$401$	1.78021	−	0.179861i	1.78021	−	0.179861i	0.935016	+	0.354605i	$0.115385\pi$
$402$		0			0
$403$		0			0
$404$	0.151567	+	0.614932i	0.151567	+	0.614932i
$405$	−0.885456	−	0.464723i	−0.885456	−	0.464723i
$406$		0			0
$407$		0			0
$408$		0			0
$409$	−0.453409	+	0.249739i	−0.453409	+	0.249739i	−0.692724	−	0.721202i	$-0.743590\pi$
$409$	−0.453409	+	0.249739i	−0.453409	+	0.249739i	0.239316	+	0.970942i	$0.423077\pi$
$410$	−0.366598	+	0.242292i	−0.366598	+	0.242292i
$411$		0			0
$412$		0			0
$413$		0			0
$414$		0			0
$415$		0			0
$416$	−0.822984	−	0.568065i	−0.822984	−	0.568065i
$417$		0			0
$418$		0			0
$419$		0			0		0.0804666	−	0.996757i	$-0.474359\pi$
$419$		0			0		−0.0804666	+	0.996757i	$0.525641\pi$
$420$		0			0
$421$	0.0344658	−	0.0208353i	0.0344658	−	0.0208353i	−0.500000	−	0.866025i	$-0.666667\pi$
$421$	0.0344658	−	0.0208353i	0.0344658	−	0.0208353i	0.534466	+	0.845190i	$0.320513\pi$
$422$		0			0
$423$		0			0
$424$	1.69458	−	1.02441i	1.69458	−	1.02441i
$425$	−0.414071	+	0.528522i	−0.414071	+	0.528522i
$426$		0			0
$427$		0			0
$428$		0			0
$429$		0			0
$430$		0			0
$431$		0			0		0.761712	−	0.647915i	$-0.224359\pi$
$431$		0			0		−0.761712	+	0.647915i	$0.775641\pi$
$432$		0			0
$433$	−1.16312	−	0.117515i	−1.16312	−	0.117515i	−0.500000	−	0.866025i	$-0.666667\pi$
$433$	−1.16312	−	0.117515i	−1.16312	−	0.117515i	−0.663123	+	0.748511i	$0.730769\pi$
$434$		0			0
$435$		0			0
$436$	0.315777	−	0.173931i	0.315777	−	0.173931i
$437$		0			0
$438$		0			0
$439$		0			0		0.200026	−	0.979791i	$-0.435897\pi$
$439$		0			0		−0.200026	+	0.979791i	$0.564103\pi$
$440$		0			0
$441$	−0.239316	−	0.970942i	−0.239316	−	0.970942i
$442$	−0.655003	+	0.147519i	−0.655003	+	0.147519i
$443$		0			0		−0.517338	−	0.855781i	$-0.673077\pi$
$443$		0			0		0.517338	+	0.855781i	$0.326923\pi$
$444$		0			0
$445$	1.43566	+	1.12477i	1.43566	+	1.12477i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	0.560476	+	0.199746i	0.560476	+	0.199746i	0.600742	−	0.799443i	$-0.294872\pi$
$449$	0.560476	+	0.199746i	0.560476	+	0.199746i	−0.0402659	+	0.999189i	$0.512821\pi$
$450$	−0.987050	+	0.160411i	−0.987050	+	0.160411i
$451$		0			0
$452$	0.889071	+	1.78021i	0.889071	+	1.78021i
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$	1.15405	−	0.334274i	1.15405	−	0.334274i	0.354605	−	0.935016i	$-0.384615\pi$
$457$	1.15405	−	0.334274i	1.15405	−	0.334274i	0.799443	+	0.600742i	$0.205128\pi$
$458$	0.759568	−	1.37902i	0.759568	−	1.37902i
$459$		0			0
$460$		0			0
$461$	0.136320	+	0.147768i	0.136320	+	0.147768i	0.799443	−	0.600742i	$-0.205128\pi$
$461$	0.136320	+	0.147768i	0.136320	+	0.147768i	−0.663123	+	0.748511i	$0.730769\pi$
$462$		0			0
$463$		0			0		−0.464723	−	0.885456i	$-0.653846\pi$
$463$		0			0		0.464723	+	0.885456i	$0.346154\pi$
$464$	−0.176205	−	0.527799i	−0.176205	−	0.527799i
$465$		0			0
$466$	0.00243169	−	0.120733i	0.00243169	−	0.120733i
$467$		0			0		−0.297503	−	0.954721i	$-0.596154\pi$
$467$		0			0		0.297503	+	0.954721i	$0.403846\pi$
$468$	−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
$477$	1.60666	−	1.15745i	1.60666	−	1.15745i
$478$		0			0
$479$		0			0		−0.551377	−	0.834256i	$-0.685897\pi$
$479$		0			0		0.551377	+	0.834256i	$0.314103\pi$
$480$		0			0
$481$	0.304312	−	0.101594i	0.304312	−	0.101594i
$482$	−1.99595	+	0.120733i	−1.99595	+	0.120733i
$483$		0			0
$484$	−0.866025	+	0.500000i	−0.866025	+	0.500000i
$485$	0.708833	−	1.12093i	0.708833	−	1.12093i
$486$		0			0
$487$		0			0		−0.0804666	−	0.996757i	$-0.525641\pi$
$487$		0			0		0.0804666	+	0.996757i	$0.474359\pi$
$488$	1.24916	+	0.721202i	1.24916	+	0.721202i
$489$		0			0
$490$	−0.774605	−	0.632445i	−0.774605	−	0.632445i
$491$		0			0		−0.774605	−	0.632445i	$-0.782051\pi$
$491$		0			0		0.774605	+	0.632445i	$0.217949\pi$
$492$		0			0
$493$	−0.340682	−	0.153328i	−0.340682	−	0.153328i
$494$		0			0
$495$		0			0
$496$		0			0
$497$		0			0
$498$		0			0
$499$		0			0		0.787183	−	0.616719i	$-0.211538\pi$
$499$		0			0		−0.787183	+	0.616719i	$0.788462\pi$
$500$	−0.692724	+	0.721202i	−0.692724	+	0.721202i
$501$		0			0
$502$		0			0
$503$		0			0		0.335705	−	0.941967i	$-0.391026\pi$
$503$		0			0		−0.335705	+	0.941967i	$0.608974\pi$
$504$		0			0
$505$	0.521225	+	0.359776i	0.521225	+	0.359776i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	0.212007	+	0.941340i	0.212007	+	0.941340i	0.960518	+	0.278217i	$0.0897436\pi$
$509$	0.212007	+	0.941340i	0.212007	+	0.941340i	−0.748511	+	0.663123i	$0.769231\pi$
$510$		0			0
$511$		0			0
$512$	−0.935016	−	0.354605i	−0.935016	−	0.354605i
$513$		0			0
$514$	−0.739316	+	0.104916i	−0.739316	+	0.104916i
$515$		0			0
$516$		0			0
$517$		0			0
$518$		0			0
$519$		0			0
$520$	−0.992709	+	0.120537i	−0.992709	+	0.120537i
$521$	−0.379048	−	0.999468i	−0.379048	−	0.999468i	−0.979791	−	0.200026i	$-0.935897\pi$
$521$	−0.379048	−	0.999468i	−0.379048	−	0.999468i	0.600742	−	0.799443i	$-0.294872\pi$
$522$	−0.218104	−	0.511909i	−0.218104	−	0.511909i
$523$		0			0		0.994935	−	0.100522i	$-0.0320513\pi$
$523$		0			0		−0.994935	+	0.100522i	$0.967949\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$		0			0
$528$		0			0
$529$	0.866025	+	0.500000i	0.866025	+	0.500000i
$530$	0.589104	−	1.89050i	0.589104	−	1.89050i
$531$		0			0
$532$		0			0
$533$	−0.356544	+	0.256857i	−0.356544	+	0.256857i
$534$		0			0
$535$		0			0
$536$		0			0
$537$		0			0
$538$	−1.39974	+	1.24006i	−1.39974	+	1.24006i
$539$		0			0
$540$		0			0
$541$	0.761468	−	1.69191i	0.761468	−	1.69191i	0.0402659	−	0.999189i	$-0.487179\pi$
$541$	0.761468	−	1.69191i	0.761468	−	1.69191i	0.721202	−	0.692724i	$-0.243590\pi$
$542$		0			0
$543$		0			0
$544$	−0.600666	+	0.299985i	−0.600666	+	0.299985i
$545$	0.121025	−	0.339589i	0.121025	−	0.339589i
$546$		0			0
$547$		0			0		0.954721	−	0.297503i	$-0.0961538\pi$
$547$		0			0		−0.954721	+	0.297503i	$0.903846\pi$
$548$	0.876221	+	0.912242i	0.876221	+	0.912242i
$549$	1.30314	+	0.618348i	1.30314	+	0.618348i
$550$		0			0
$551$		0			0
$552$		0			0
$553$		0			0
$554$	1.90021	+	0.348226i	1.90021	+	0.348226i
$555$		0			0
$556$		0			0
$557$	0.481887	+	1.66367i	0.481887	+	1.66367i	0.721202	+	0.692724i	$0.243590\pi$
$557$	0.481887	+	1.66367i	0.481887	+	1.66367i	−0.239316	+	0.970942i	$0.576923\pi$
$558$		0			0
$559$		0			0
$560$		0			0
$561$		0			0
$562$	0.251397	−	0.125553i	0.251397	−	0.125553i
$563$		0			0		−0.0201371	−	0.999797i	$-0.506410\pi$
$563$		0			0		0.0201371	+	0.999797i	$0.493590\pi$
$564$		0			0
$565$	1.84597	+	0.742941i	1.84597	+	0.742941i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−1.62465	+	0.264032i	−1.62465	+	0.264032i	−0.903450	−	0.428693i	$-0.858974\pi$
$569$	−1.62465	+	0.264032i	−1.62465	+	0.264032i	−0.721202	+	0.692724i	$0.756410\pi$
$570$		0			0
$571$		0			0		0.239316	−	0.970942i	$-0.423077\pi$
$571$		0			0		−0.239316	+	0.970942i	$0.576923\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$	−0.960518	−	0.278217i	−0.960518	−	0.278217i
$577$			1.99351i			1.99351i	0.0804666	+	0.996757i	$0.474359\pi$
$577$			1.99351i			1.99351i	−0.0804666	+	0.996757i	$0.525641\pi$
$578$	0.152800	−	0.527526i	0.152800	−	0.527526i
$579$		0			0
$580$	−0.481887	−	0.278217i	−0.481887	−	0.278217i
$581$		0			0
$582$		0			0
$583$		0			0
$584$	−0.568065	+	0.822984i	−0.568065	+	0.822984i
$585$	−0.979791	+	0.200026i	−0.979791	+	0.200026i
$586$	−1.07766	−	1.56126i	−1.07766	−	1.56126i
$587$		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
$592$	0.271156	−	0.171469i	0.271156	−	0.171469i
$593$	0.0385138	−	0.156257i	0.0385138	−	0.156257i	−0.948536	−	0.316668i	$-0.897436\pi$
$593$	0.0385138	−	0.156257i	0.0385138	−	0.156257i	0.987050	+	0.160411i	$0.0512821\pi$
$594$		0			0
$595$		0			0
$596$	0.285414	−	1.06518i	0.285414	−	1.06518i
$597$		0			0
$598$		0			0
$599$		0			0		−0.822984	−	0.568065i	$-0.807692\pi$
$599$		0			0		0.822984	+	0.568065i	$0.192308\pi$
$600$		0			0
$601$	−0.0557864	−	1.38433i	−0.0557864	−	1.38433i	−0.748511	−	0.663123i	$-0.769231\pi$
$601$	−0.0557864	−	1.38433i	−0.0557864	−	1.38433i	0.692724	−	0.721202i	$-0.256410\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−0.316668	+	0.948536i	−0.316668	+	0.948536i
$606$		0			0
$607$		0			0		−0.482459	−	0.875918i	$-0.660256\pi$
$607$		0			0		0.482459	+	0.875918i	$0.339744\pi$
$608$		0			0
$609$		0			0
$610$	1.41325	−	0.288518i	1.41325	−	0.288518i
$611$		0			0
$612$	−0.574579	+	0.347345i	−0.574579	+	0.347345i
$613$	−0.876221	+	1.07318i	−0.876221	+	1.07318i	0.120537	+	0.992709i	$0.461538\pi$
$613$	−0.876221	+	1.07318i	−0.876221	+	1.07318i	−0.996757	+	0.0804666i	$0.974359\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	1.48804	−	1.21495i	1.48804	−	1.21495i	0.568065	−	0.822984i	$-0.307692\pi$
$617$	1.48804	−	1.21495i	1.48804	−	1.21495i	0.919979	−	0.391967i	$-0.128205\pi$
$618$		0			0
$619$		0			0		0.517338	−	0.855781i	$-0.326923\pi$
$619$		0			0		−0.517338	+	0.855781i	$0.673077\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$		0			0
$625$	−0.0402659	+	0.999189i	−0.0402659	+	0.999189i
$626$	1.67252	+	0.921228i	1.67252	+	0.921228i
$627$		0			0
$628$	1.08105	−	1.27093i	1.08105	−	1.27093i
$629$	0.0388275	−	0.211875i	0.0388275	−	0.211875i
$630$		0			0
$631$		0			0		−0.373361	−	0.927686i	$-0.621795\pi$
$631$		0			0		0.373361	+	0.927686i	$0.378205\pi$
$632$		0			0
$633$		0			0
$634$	−0.685430	+	1.44451i	−0.685430	+	1.44451i
$635$		0			0
$636$		0			0
$637$	−0.799443	−	0.600742i	−0.799443	−	0.600742i
$638$		0			0
$639$		0			0
$640$	−0.935016	+	0.354605i	−0.935016	+	0.354605i
$641$	0.925722	+	1.46391i	0.925722	+	1.46391i	0.885456	+	0.464723i	$0.153846\pi$
$641$	0.925722	+	1.46391i	0.925722	+	1.46391i	0.0402659	+	0.999189i	$0.487179\pi$
$642$		0			0
$643$		0			0		0.692724	−	0.721202i	$-0.256410\pi$
$643$		0			0		−0.692724	+	0.721202i	$0.743590\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		0			0		−0.219715	−	0.975564i	$-0.570513\pi$
$647$		0			0		0.219715	+	0.975564i	$0.429487\pi$
$648$	−0.979791	−	0.200026i	−0.979791	−	0.200026i
$649$		0			0
$650$	−0.663123	+	0.748511i	−0.663123	+	0.748511i
$651$		0			0
$652$		0			0
$653$	−0.319237	+	1.19141i	−0.319237	+	1.19141i	0.600742	+	0.799443i	$0.294872\pi$
$653$	−0.319237	+	1.19141i	−0.319237	+	1.19141i	−0.919979	+	0.391967i	$0.871795\pi$
$654$		0			0
$655$		0			0
$656$	−0.284714	+	0.334720i	−0.284714	+	0.334720i
$657$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$658$		0			0
$659$		0			0		0.600742	−	0.799443i	$-0.294872\pi$
$659$		0			0		−0.600742	+	0.799443i	$0.705128\pi$
$660$		0			0
$661$	1.20212	+	0.483813i	1.20212	+	0.483813i	0.885456	−	0.464723i	$-0.153846\pi$
$661$	1.20212	+	0.483813i	1.20212	+	0.483813i	0.316668	+	0.948536i	$0.397436\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$	0.256479	−	0.192732i	0.256479	−	0.192732i
$667$		0			0
$668$		0			0
$669$		0			0
$670$		0			0
$671$		0			0
$672$		0			0
$673$	1.85500	+	0.0373617i	1.85500	+	0.0373617i	0.935016	−	0.354605i	$-0.115385\pi$
$673$	1.85500	+	0.0373617i	1.85500	+	0.0373617i	0.919979	+	0.391967i	$0.128205\pi$
$674$	0.871761	−	0.196337i	0.871761	−	0.196337i
$675$		0			0
$676$	−0.987050	+	0.160411i	−0.987050	+	0.160411i
$677$	−1.39105	+	1.39105i	−1.39105	+	1.39105i	−0.568065	+	0.822984i	$0.692308\pi$
$677$	−1.39105	+	1.39105i	−1.39105	+	1.39105i	−0.822984	+	0.568065i	$0.807692\pi$
$678$		0			0
$679$		0			0
$680$	−0.250678	+	0.622857i	−0.250678	+	0.622857i
$681$		0			0
$682$		0			0
$683$		0			0		0.600742	−	0.799443i	$-0.294872\pi$
$683$		0			0		−0.600742	+	0.799443i	$0.705128\pi$
$684$		0			0
$685$	1.26079	+	0.101781i	1.26079	+	0.101781i
$686$		0			0
$687$		0			0
$688$		0			0
$689$	0.512503	−	1.91269i	0.512503	−	1.91269i
$690$		0			0
$691$		0			0		0.373361	−	0.927686i	$-0.378205\pi$
$691$		0			0		−0.373361	+	0.927686i	$0.621795\pi$
$692$	0.104008	+	1.02943i	0.104008	+	1.02943i
$693$		0			0
$694$		0			0
$695$		0			0
$696$		0			0
$697$	0.0355629	+	0.292886i	0.0355629	+	0.292886i
$698$	−0.792857	+	0.212445i	−0.792857	+	0.212445i
$699$		0			0
$700$		0			0
$701$	−1.31658	+	1.48611i	−1.31658	+	1.48611i	−0.568065	+	0.822984i	$0.692308\pi$
$701$	−1.31658	+	1.48611i	−1.31658	+	1.48611i	−0.748511	+	0.663123i	$0.769231\pi$
$702$		0			0
$703$		0			0
$704$		0			0
$705$		0			0
$706$	0.125752	−	0.773781i	0.125752	−	0.773781i
$707$		0			0
$708$		0			0
$709$	−0.0306773	−	1.52312i	−0.0306773	−	1.52312i	−0.663123	−	0.748511i	$-0.730769\pi$
$709$	−0.0306773	−	1.52312i	−0.0306773	−	1.52312i	0.632445	−	0.774605i	$-0.282051\pi$
$710$		0			0
$711$		0			0
$712$	1.69191	+	0.680937i	1.69191	+	0.680937i
$713$		0			0
$714$		0			0
$715$		0			0
$716$		0			0
$717$		0			0
$718$		0			0
$719$		0			0		0.948536	−	0.316668i	$-0.102564\pi$
$719$		0			0		−0.948536	+	0.316668i	$0.897436\pi$
$720$	−0.903450	+	0.428693i	−0.903450	+	0.428693i
$721$		0			0
$722$	0.0402659	−	0.999189i	0.0402659	−	0.999189i
$723$		0			0
$724$	0.148055	+	1.83399i	0.148055	+	1.83399i
$725$	−0.545190	+	0.111301i	−0.545190	+	0.111301i
$726$		0			0
$727$		0			0		0.616719	−	0.787183i	$-0.288462\pi$
$727$		0			0		−0.616719	+	0.787183i	$0.711538\pi$
$728$		0			0
$729$	−0.992709	−	0.120537i	−0.992709	−	0.120537i
$730$	0.120537	+	0.992709i	0.120537	+	0.992709i
$731$		0			0
$732$		0			0
$733$	−0.388427	+	0.0957386i	−0.388427	+	0.0957386i	−0.428693	−	0.903450i	$-0.641026\pi$
$733$	−0.388427	+	0.0957386i	−0.388427	+	0.0957386i	0.0402659	+	0.999189i	$0.487179\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		0			0
$738$	−0.256857	+	0.356544i	−0.256857	+	0.356544i
$739$		0			0		−0.100522	−	0.994935i	$-0.532051\pi$
$739$		0			0		0.100522	+	0.994935i	$0.467949\pi$
$740$	0.0892584	−	0.308156i	0.0892584	−	0.308156i
$741$		0			0
$742$		0			0
$743$		0			0		0.428693	−	0.903450i	$-0.358974\pi$
$743$		0			0		−0.428693	+	0.903450i	$0.641026\pi$
$744$		0			0
$745$	−0.492709	−	0.986562i	−0.492709	−	0.986562i
$746$	−1.37966	+	1.37966i	−1.37966	+	1.37966i
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$		0			0		0.999189	−	0.0402659i	$-0.0128205\pi$
$751$		0			0		−0.999189	+	0.0402659i	$0.987179\pi$
$752$		0			0
$753$		0			0
$754$	−0.492699	−	0.258588i	−0.492699	−	0.258588i
$755$		0			0
$756$		0			0
$757$	1.30372	−	1.10895i	1.30372	−	1.10895i	0.316668	−	0.948536i	$-0.397436\pi$
$757$	1.30372	−	1.10895i	1.30372	−	1.10895i	0.987050	−	0.160411i	$-0.0512821\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	1.20330	+	0.271005i	1.20330	+	0.271005i	0.774605	−	0.632445i	$-0.217949\pi$
$761$	1.20330	+	0.271005i	1.20330	+	0.271005i	0.428693	+	0.903450i	$0.358974\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$	−0.199746	+	0.641008i	−0.199746	+	0.641008i
$766$		0			0
$767$		0			0
$768$		0			0
$769$	1.10895	+	1.30372i	1.10895	+	1.30372i	0.948536	+	0.316668i	$0.102564\pi$
$769$	1.10895	+	1.30372i	1.10895	+	1.30372i	0.160411	+	0.987050i	$0.448718\pi$
$770$		0			0
$771$		0			0
$772$	0.910663	−	1.73512i	0.910663	−	1.73512i
$773$	−1.62465	−	0.264032i	−1.62465	−	0.264032i	−0.721202	−	0.692724i	$-0.756410\pi$
$773$	−1.62465	−	0.264032i	−1.62465	−	0.264032i	−0.903450	+	0.428693i	$0.858974\pi$
$774$		0			0
$775$		0			0
$776$	0.368985	−	1.27388i	0.368985	−	1.27388i
$777$		0			0
$778$	−1.78350	−	0.846282i	−1.78350	−	0.846282i
$779$		0			0
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$	−0.919979	−	0.391967i	−0.919979	−	0.391967i
$785$	−0.0335989	−	1.66817i	−0.0335989	−	1.66817i
$786$		0			0
$787$		0			0		−0.428693	−	0.903450i	$-0.641026\pi$
$787$		0			0		0.428693	+	0.903450i	$0.358974\pi$
$788$	0.114544	+	0.464723i	0.114544	+	0.464723i
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$	1.40049	−	0.345190i	1.40049	−	0.345190i
$794$	−0.922670	+	0.112032i	−0.922670	+	0.112032i
$795$		0			0
$796$		0			0
$797$	0.453223	+	0.385514i	0.453223	+	0.385514i	0.845190	−	0.534466i	$-0.179487\pi$
$797$	0.453223	+	0.385514i	0.453223	+	0.385514i	−0.391967	+	0.919979i	$0.628205\pi$
$798$		0			0
$799$		0			0
$800$	−0.464723	+	0.885456i	−0.464723	+	0.885456i
$801$	1.74122	+	0.542586i	1.74122	+	0.542586i
$802$	0.799443	−	1.60074i	0.799443	−	1.60074i
$803$		0			0
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$	0.600742	+	0.200557i	0.600742	+	0.200557i
$809$	0.156807	+	0.368039i	0.156807	+	0.368039i	0.979791	−	0.200026i	$-0.0641026\pi$
$809$	0.156807	+	0.368039i	0.156807	+	0.368039i	−0.822984	+	0.568065i	$0.807692\pi$
$810$	−0.866025	+	0.500000i	−0.866025	+	0.500000i
$811$		0			0		−0.911900	−	0.410413i	$-0.865385\pi$
$811$		0			0		0.911900	+	0.410413i	$0.134615\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$		0			0
$816$		0			0
$817$		0			0
$818$	−0.0312542	+	0.516694i	−0.0312542	+	0.516694i
$819$		0			0
$820$	0.00884883	+	0.439341i	0.00884883	+	0.439341i
$821$	0.280492	−	1.97655i	0.280492	−	1.97655i	0.0804666	−	0.996757i	$-0.474359\pi$
$821$	0.280492	−	1.97655i	0.280492	−	1.97655i	0.200026	−	0.979791i	$-0.435897\pi$
$822$		0			0
$823$		0			0		−0.258819	−	0.965926i	$-0.583333\pi$
$823$		0			0		0.258819	+	0.965926i	$0.416667\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		0			0		0.568065	−	0.822984i	$-0.307692\pi$
$827$		0			0		−0.568065	+	0.822984i	$0.692308\pi$
$828$		0			0
$829$	0.759873	−	0.930676i	0.759873	−	0.930676i	−0.239316	−	0.970942i	$-0.576923\pi$
$829$	0.759873	−	0.930676i	0.759873	−	0.930676i	0.999189	+	0.0402659i	$0.0128205\pi$
$830$		0			0
$831$		0			0
$832$	−0.919979	+	0.391967i	−0.919979	+	0.391967i
$833$	−0.612258	+	0.275555i	−0.612258	+	0.275555i
$834$		0			0
$835$		0			0
$836$		0			0
$837$		0			0
$838$		0			0
$839$		0			0		−0.834256	−	0.551377i	$-0.814103\pi$
$839$		0			0		0.834256	+	0.551377i	$0.185897\pi$
$840$		0			0
$841$	0.295961	+	0.623724i	0.295961	+	0.623724i
$842$	0.000811002	−	0.0402659i	0.000811002	−	0.0402659i
$843$		0			0
$844$		0			0
$845$	−0.632445	+	0.774605i	−0.632445	+	0.774605i
$846$		0			0
$847$		0			0
$848$	0.0398746	−	1.97976i	0.0398746	−	1.97976i
$849$		0			0
$850$	0.225395	+	0.632445i	0.225395	+	0.632445i
$851$		0			0
$852$		0			0
$853$	−0.851134	−	0.103346i	−0.851134	−	0.103346i	−0.316668	−	0.948536i	$-0.602564\pi$
$853$	−0.851134	−	0.103346i	−0.851134	−	0.103346i	−0.534466	+	0.845190i	$0.679487\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−0.0367260	+	0.0165290i	−0.0367260	+	0.0165290i	−0.428693	−	0.903450i	$-0.641026\pi$
$857$	−0.0367260	+	0.0165290i	−0.0367260	+	0.0165290i	0.391967	+	0.919979i	$0.371795\pi$
$858$		0			0
$859$		0			0		0.935016	−	0.354605i	$-0.115385\pi$
$859$		0			0		−0.935016	+	0.354605i	$0.884615\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		0			0		0.568065	−	0.822984i	$-0.307692\pi$
$863$		0			0		−0.568065	+	0.822984i	$0.692308\pi$
$864$		0			0
$865$	0.760492	+	0.701573i	0.760492	+	0.701573i
$866$	−0.720972	+	0.920252i	−0.720972	+	0.920252i
$867$		0			0
$868$		0			0
$869$		0			0
$870$		0			0
$871$		0			0
$872$	0.0217671	−	0.359852i	0.0217671	−	0.359852i
$873$	0.265283	−	1.29944i	0.265283	−	1.29944i
$874$		0			0
$875$		0			0
$876$		0			0
$877$	0.338496	+	1.01392i	0.338496	+	1.01392i	0.970942	+	0.239316i	$0.0769231\pi$
$877$	0.338496	+	1.01392i	0.338496	+	1.01392i	−0.632445	+	0.774605i	$0.717949\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	0.662573	+	1.55512i	0.662573	+	1.55512i	0.822984	+	0.568065i	$0.192308\pi$
$881$	0.662573	+	1.55512i	0.662573	+	1.55512i	−0.160411	+	0.987050i	$0.551282\pi$
$882$	−0.948536	−	0.316668i	−0.948536	−	0.316668i
$883$		0			0		0.297503	−	0.954721i	$-0.403846\pi$
$883$		0			0		−0.297503	+	0.954721i	$0.596154\pi$
$884$	−0.225395	+	0.632445i	−0.225395	+	0.632445i
$885$		0			0
$886$		0			0
$887$		0			0		−0.335705	−	0.941967i	$-0.608974\pi$
$887$		0			0		0.335705	+	0.941967i	$0.391026\pi$
$888$		0			0
$889$		0			0
$890$	1.71796	−	0.612258i	1.71796	−	0.612258i
$891$		0			0
$892$		0			0
$893$		0			0
$894$		0			0
$895$		0			0
$896$		0			0
$897$		0			0
$898$	0.468379	−	0.366951i	0.468379	−	0.366951i
$899$		0			0
$900$	−0.391967	+	0.919979i	−0.391967	+	0.919979i
$901$	−0.958837	−	0.920975i	−0.958837	−	0.920975i
$902$		0			0
$903$		0			0
$904$	1.97979	+	0.200026i	1.97979	+	0.200026i
$905$	1.32698	+	1.27458i	1.32698	+	1.27458i
$906$		0			0
$907$		0			0		−0.811378	−	0.584522i	$-0.801282\pi$
$907$		0			0		0.811378	+	0.584522i	$0.198718\pi$
$908$		0			0
$909$	0.614932	+	0.151567i	0.614932	+	0.151567i
$910$		0			0
$911$		0			0		0.970942	−	0.239316i	$-0.0769231\pi$
$911$		0			0		−0.970942	+	0.239316i	$0.923077\pi$
$912$		0			0
$913$		0			0
$914$	0.334274	−	1.15405i	0.334274	−	1.15405i
$915$		0			0
$916$	−0.759568	−	1.37902i	−0.759568	−	1.37902i
$917$		0			0
$918$		0			0
$919$		0			0		0.774605	−	0.632445i	$-0.217949\pi$
$919$		0			0		−0.774605	+	0.632445i	$0.782051\pi$
$920$		0			0
$921$		0			0
$922$	0.197751	−	0.0362392i	0.197751	−	0.0362392i
$923$		0			0
$924$		0			0
$925$	−0.137534	−	0.289847i	−0.137534	−	0.289847i
$926$		0			0
$927$		0			0
$928$	−0.540266	−	0.133164i	−0.540266	−	0.133164i
$929$	−1.83790	−	0.413929i	−1.83790	−	0.413929i	−0.845190	−	0.534466i	$-0.820513\pi$
$929$	−1.83790	−	0.413929i	−1.83790	−	0.413929i	−0.992709	+	0.120537i	$0.961538\pi$
$930$		0			0
$931$		0			0
$932$	−0.100742	−	0.0665826i	−0.100742	−	0.0665826i
$933$		0			0
$934$		0			0
$935$		0			0
$936$	−0.885456	+	0.464723i	−0.885456	+	0.464723i
$937$	−0.0933069	−	0.509159i	−0.0933069	−	0.509159i	−0.996757	−	0.0804666i	$-0.974359\pi$
$937$	−0.0933069	−	0.509159i	−0.0933069	−	0.509159i	0.903450	−	0.428693i	$-0.141026\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	1.63406	−	0.509195i	1.63406	−	0.509195i	0.663123	−	0.748511i	$-0.269231\pi$
$941$	1.63406	−	0.509195i	1.63406	−	0.509195i	0.970942	+	0.239316i	$0.0769231\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$		0			0
$946$		0			0
$947$		0			0		0.428693	−	0.903450i	$-0.358974\pi$
$947$		0			0		−0.428693	+	0.903450i	$0.641026\pi$
$948$		0			0
$949$	0.160411	+	0.987050i	0.160411	+	0.987050i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	−1.58779	+	0.639031i	−1.58779	+	0.639031i	−0.987050	−	0.160411i	$-0.948718\pi$
$953$	−1.58779	+	0.639031i	−1.58779	+	0.639031i	−0.600742	+	0.799443i	$0.705128\pi$
$954$	−0.119559	−	1.97655i	−0.119559	−	1.97655i
$955$		0			0
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	−0.992709	−	0.120537i	−0.992709	−	0.120537i
$962$	0.0767779	−	0.311500i	0.0767779	−	0.311500i
$963$		0			0
$964$	−0.964723	+	1.75148i	−0.964723	+	1.75148i
$965$	−0.468959	−	1.90264i	−0.468959	−	1.90264i
$966$		0			0
$967$		0			0		0.822984	−	0.568065i	$-0.192308\pi$
$967$		0			0		−0.822984	+	0.568065i	$0.807692\pi$
$968$	−0.0402659	+	0.999189i	−0.0402659	+	0.999189i
$969$		0			0
$970$	−0.568552	−	1.19820i	−0.568552	−	1.19820i
$971$		0			0		0.948536	−	0.316668i	$-0.102564\pi$
$971$		0			0		−0.948536	+	0.316668i	$0.897436\pi$
$972$		0			0
$973$		0			0
$974$		0			0
$975$		0			0
$976$	1.27719	−	0.670319i	1.27719	−	0.670319i
$977$	−0.126683	+	0.379463i	−0.126683	+	0.379463i	−0.992709	−	0.120537i	$-0.961538\pi$
$977$	−0.126683	+	0.379463i	−0.126683	+	0.379463i	0.866025	+	0.500000i	$0.166667\pi$
$978$		0			0
$979$		0			0
$980$	−0.948536	+	0.316668i	−0.948536	+	0.316668i
$981$	−0.00725961	−	0.360437i	−0.00725961	−	0.360437i
$982$		0			0
$983$		0			0		−0.663123	−	0.748511i	$-0.730769\pi$
$983$		0			0		0.663123	+	0.748511i	$0.269231\pi$
$984$		0			0
$985$	0.393906	+	0.271894i	0.393906	+	0.271894i
$986$	−0.311674	+	0.205992i	−0.311674	+	0.205992i
$987$		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
$997$	−0.199050	+	0.0282472i	−0.199050	+	0.0282472i	−0.239316	−	0.970942i	$-0.576923\pi$
$997$	−0.199050	+	0.0282472i	−0.199050	+	0.0282472i	0.0402659	+	0.999189i	$0.487179\pi$
$998$		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.1783.1	yes	48
4.3	odd	2	CM	3380.1.cs.a.1783.1	yes	48
5.2	odd	4		3380.1.cz.a.1107.1	yes	48
20.7	even	4		3380.1.cz.a.1107.1	yes	48
169.20	odd	156		3380.1.cz.a.1203.1	yes	48
676.527	even	156		3380.1.cz.a.1203.1	yes	48
845.527	even	156	inner	3380.1.cs.a.527.1	✓	48
3380.527	odd	156	inner	3380.1.cs.a.527.1	✓	48

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3380.1.cs.a.527.1	✓	48	845.527	even	156	inner
3380.1.cs.a.527.1	✓	48	3380.527	odd	156	inner
3380.1.cs.a.1783.1	yes	48	1.1	even	1	trivial
3380.1.cs.a.1783.1	yes	48	4.3	odd	2	CM
3380.1.cz.a.1107.1	yes	48	5.2	odd	4
3380.1.cz.a.1107.1	yes	48	20.7	even	4
3380.1.cz.a.1203.1	yes	48	169.20	odd	156
3380.1.cz.a.1203.1	yes	48	676.527	even	156

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 $\F_{q}$
Belyi maps

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more

Newspace parameters

Newform invariants

Embedding invariants

$q$ -expansion

Character values

Coefficient data

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.1783.1

Level	$3380$
Weight	$1$
Character	3380.1783
Analytic conductor	$1.687$
Analytic rank	$0$
Dimension	$48$
Projective image	$D_{156}$
CM discriminant	-4
Inner twists	$4$