LMFDB - Embedded newform 468.2.bw.a.245.8

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [468,2,Mod(149,468)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("468.149"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(468, base_ring=CyclotomicField(12)) chi = DirichletCharacter(H, H._module([0, 10, 5])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 468 = 2^{2} \cdot 3^{2} \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	468.bw (of order $12$, degree $4$, 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:	$3.73699881460$
Analytic rank:	$0$
Dimension:	$56$
Relative dimension:	$14$ over $\Q(\zeta_{12})$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{12}]$

Embedding invariants

Embedding label			245.8
Character	$\chi$	$=$	468.245
Dual form			468.2.bw.a.149.8

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+(0.0136673 - 1.73200i) q^{3} +(2.03032 + 0.544024i) q^{5} +(-4.33109 - 1.16051i) q^{7} +(-2.99963 - 0.0473436i) q^{9} +(-3.26962 - 3.26962i) q^{11} +(-3.37661 - 1.26431i) q^{13} +(0.969996 - 3.50908i) q^{15} +(-1.12575 + 1.94986i) q^{17} +(6.39637 - 1.71390i) q^{19} +(-2.06920 + 7.48558i) q^{21} +(3.34811 - 5.79909i) q^{23} +(-0.503873 - 0.290911i) q^{25} +(-0.122996 + 5.19470i) q^{27} +0.297717i q^{29} +(0.00538704 - 0.0201047i) q^{31} +(-5.70765 + 5.61828i) q^{33} +(-8.16218 - 4.71244i) q^{35} +(5.91919 + 1.58604i) q^{37} +(-2.23594 + 5.83100i) q^{39} +(-2.03104 - 7.57993i) q^{41} +(0.683630 - 0.394694i) q^{43} +(-6.06446 - 1.72799i) q^{45} +(-10.7709 + 2.88605i) q^{47} +(11.3494 + 6.55259i) q^{49} +(3.36176 + 1.97645i) q^{51} -4.32687i q^{53} +(-4.85963 - 8.41713i) q^{55} +(-2.88105 - 11.1019i) q^{57} +(-2.56815 - 2.56815i) q^{59} +(7.34891 + 12.7287i) q^{61} +(12.9367 + 3.68616i) q^{63} +(-6.16780 - 4.40393i) q^{65} +(4.99170 - 1.33752i) q^{67} +(-9.99825 - 5.87817i) q^{69} +(2.28190 + 8.51617i) q^{71} +(2.45364 - 2.45364i) q^{73} +(-0.510744 + 0.868731i) q^{75} +(10.3666 + 17.9554i) q^{77} +(4.23744 - 7.33946i) q^{79} +(8.99552 + 0.284026i) q^{81} +(-1.57541 - 5.87950i) q^{83} +(-3.34641 + 3.34641i) q^{85} +(0.515645 + 0.00406899i) q^{87} +(4.04724 - 15.1045i) q^{89} +(13.1572 + 9.39447i) q^{91} +(-0.0347477 - 0.00960512i) q^{93} +13.9191 q^{95} +(0.573740 - 2.14123i) q^{97} +(9.65283 + 9.96242i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 56 q - 2 q^{7} + 4 q^{19} - 24 q^{21} + 8 q^{31} + 12 q^{33} - 66 q^{35} + 2 q^{37} + 24 q^{41} + 12 q^{43} - 30 q^{47} + 30 q^{57} + 84 q^{63} + 6 q^{65} + 28 q^{67} - 36 q^{69} + 24 q^{71} - 14 q^{73}+ \cdots - 42 q^{99}+O(q^{100}) $ 56 * q - 2 * q^7 + 4 * q^19 - 24 * q^21 + 8 * q^31 + 12 * q^33 - 66 * q^35 + 2 * q^37 + 24 * q^41 + 12 * q^43 - 30 * q^47 + 30 * q^57 + 84 * q^63 + 6 * q^65 + 28 * q^67 - 36 * q^69 + 24 * q^71 - 14 * q^73 - 24 * q^77 + 12 * q^79 - 12 * q^81 - 60 * q^83 + 36 * q^85 + 4 * q^91 - 54 * q^93 + 26 * q^97 - 42 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/468\mathbb{Z}\right)^\times$.

$n$	$145$	$209$	$235$
$\chi(n)$	$e\left(\frac{7}{12}\right)$	$e\left(\frac{1}{6}\right)$	$1$

Coefficient data

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

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

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

$2$		0			0
$2$		0			0
$3$	0.0136673	−	1.73200i	0.00789084	−	0.999969i
$3$	0.0136673	−	1.73200i	0.00789084	−	0.999969i
$4$		0			0
$5$	2.03032	+	0.544024i	0.907989	+	0.243295i	0.682444	−	0.730938i	$-0.260917\pi$
$5$	2.03032	+	0.544024i	0.907989	+	0.243295i	0.225545	+	0.974233i	$0.427584\pi$
$6$		0			0
$7$	−4.33109	−	1.16051i	−1.63700	−	0.438633i	−0.681068	−	0.732220i	$-0.738484\pi$
$7$	−4.33109	−	1.16051i	−1.63700	−	0.438633i	−0.955932	+	0.293588i	$0.905151\pi$
$8$		0			0
$9$	−2.99963	−	0.0473436i	−0.999875	−	0.0157812i
$10$		0			0
$11$	−3.26962	−	3.26962i	−0.985826	−	0.985826i	0.0140749	−	0.999901i	$-0.495520\pi$
$11$	−3.26962	−	3.26962i	−0.985826	−	0.985826i	−0.999901	+	0.0140749i	$0.995520\pi$
$12$		0			0
$13$	−3.37661	−	1.26431i	−0.936504	−	0.350658i
$13$	−3.37661	−	1.26431i	−0.936504	−	0.350658i
$14$		0			0
$15$	0.969996	−	3.50908i	0.250452	−	0.906040i
$16$		0			0
$17$	−1.12575	+	1.94986i	−0.273035	+	0.472910i	−0.969637	−	0.244547i	$-0.921361\pi$
$17$	−1.12575	+	1.94986i	−0.273035	+	0.472910i	0.696603	+	0.717457i	$0.254694\pi$
$18$		0			0
$19$	6.39637	−	1.71390i	1.46743	−	0.393196i	0.565380	−	0.824831i	$-0.308730\pi$
$19$	6.39637	−	1.71390i	1.46743	−	0.393196i	0.902049	+	0.431634i	$0.142063\pi$
$20$		0			0
$21$	−2.06920	+	7.48558i	−0.451536	+	1.63349i
$22$		0			0
$23$	3.34811	−	5.79909i	0.698129	−	1.20919i	−0.270986	−	0.962583i	$-0.587350\pi$
$23$	3.34811	−	5.79909i	0.698129	−	1.20919i	0.969115	−	0.246611i	$-0.0793170\pi$
$24$		0			0
$25$	−0.503873	−	0.290911i	−0.100775	−	0.0581823i
$26$		0			0
$27$	−0.122996	+	5.19470i	−0.0236705	+	0.999720i
$28$		0			0
$29$			0.297717i			0.0552846i	0.999618	+	0.0276423i	$0.00879994\pi$
$29$			0.297717i			0.0552846i	−0.999618	+	0.0276423i	$0.991200\pi$
$30$		0			0
$31$	0.00538704	−	0.0201047i	0.000967541	−	0.00361091i	−0.965440	−	0.260624i	$-0.916072\pi$
$31$	0.00538704	−	0.0201047i	0.000967541	−	0.00361091i	0.966408	+	0.257013i	$0.0827384\pi$
$32$		0			0
$33$	−5.70765	+	5.61828i	−0.993574	+	0.978016i
$34$		0			0
$35$	−8.16218	−	4.71244i	−1.37966	−	0.796547i
$36$		0			0
$37$	5.91919	+	1.58604i	0.973109	+	0.260744i	0.710140	−	0.704061i	$-0.248632\pi$
$37$	5.91919	+	1.58604i	0.973109	+	0.260744i	0.262969	+	0.964804i	$0.415298\pi$
$38$		0			0
$39$	−2.23594	+	5.83100i	−0.358037	+	0.933708i
$40$		0			0
$41$	−2.03104	−	7.57993i	−0.317195	−	1.18379i	−0.921929	−	0.387359i	$-0.873387\pi$
$41$	−2.03104	−	7.57993i	−0.317195	−	1.18379i	0.604734	−	0.796427i	$-0.293279\pi$
$42$		0			0
$43$	0.683630	−	0.394694i	0.104253	−	0.0601903i	−0.446967	−	0.894550i	$-0.647496\pi$
$43$	0.683630	−	0.394694i	0.104253	−	0.0601903i	0.551220	+	0.834360i	$0.314163\pi$
$44$		0			0
$45$	−6.06446	−	1.72799i	−0.904036	−	0.257594i
$46$		0			0
$47$	−10.7709	+	2.88605i	−1.57109	+	0.420973i	−0.936155	−	0.351589i	$-0.885642\pi$
$47$	−10.7709	+	2.88605i	−1.57109	+	0.420973i	−0.634939	+	0.772562i	$0.718975\pi$
$48$		0			0
$49$	11.3494	+	6.55259i	1.62134	+	0.936084i
$50$		0			0
$51$	3.36176	+	1.97645i	0.470741	+	0.276758i
$52$		0			0
$53$		−	4.32687i		−	0.594342i	−0.954824	−	0.297171i	$-0.903957\pi$
$53$		−	4.32687i		−	0.594342i	0.954824	−	0.297171i	$-0.0960431\pi$
$54$		0			0
$55$	−4.85963	−	8.41713i	−0.655272	−	1.13497i
$56$		0			0
$57$	−2.88105	−	11.1019i	−0.381605	−	1.47049i
$58$		0			0
$59$	−2.56815	−	2.56815i	−0.334345	−	0.334345i	0.519889	−	0.854234i	$-0.325973\pi$
$59$	−2.56815	−	2.56815i	−0.334345	−	0.334345i	−0.854234	+	0.519889i	$0.825973\pi$
$60$		0			0
$61$	7.34891	+	12.7287i	0.940931	+	1.62974i	0.763700	+	0.645571i	$0.223381\pi$
$61$	7.34891	+	12.7287i	0.940931	+	1.62974i	0.177231	+	0.984169i	$0.443286\pi$
$62$		0			0
$63$	12.9367	+	3.68616i	1.62987	+	0.464412i
$64$		0			0
$65$	−6.16780	−	4.40393i	−0.765021	−	0.546240i
$66$		0			0
$67$	4.99170	−	1.33752i	0.609833	−	0.163404i	0.0593315	−	0.998238i	$-0.481103\pi$
$67$	4.99170	−	1.33752i	0.609833	−	0.163404i	0.550502	+	0.834834i	$0.314436\pi$
$68$		0			0
$69$	−9.99825	−	5.87817i	−1.20365	−	0.707648i
$70$		0			0
$71$	2.28190	+	8.51617i	0.270812	+	1.01068i	0.958596	+	0.284769i	$0.0919168\pi$
$71$	2.28190	+	8.51617i	0.270812	+	1.01068i	−0.687784	+	0.725915i	$0.741417\pi$
$72$		0			0
$73$	2.45364	−	2.45364i	0.287176	−	0.287176i	−0.548786	−	0.835963i	$-0.684910\pi$
$73$	2.45364	−	2.45364i	0.287176	−	0.287176i	0.835963	+	0.548786i	$0.184910\pi$
$74$		0			0
$75$	−0.510744	+	0.868731i	−0.0589756	+	0.100312i
$76$		0			0
$77$	10.3666	+	17.9554i	1.18138	+	2.04621i
$78$		0			0
$79$	4.23744	−	7.33946i	0.476749	−	0.825753i	−0.522896	−	0.852396i	$-0.675148\pi$
$79$	4.23744	−	7.33946i	0.476749	−	0.825753i	0.999645	+	0.0266431i	$0.00848178\pi$
$80$		0			0
$81$	8.99552	+	0.284026i	0.999502	+	0.0315584i
$82$		0			0
$83$	−1.57541	−	5.87950i	−0.172923	−	0.645359i	−0.996896	−	0.0787294i	$-0.974914\pi$
$83$	−1.57541	−	5.87950i	−0.172923	−	0.645359i	0.823973	−	0.566630i	$-0.191753\pi$
$84$		0			0
$85$	−3.34641	+	3.34641i	−0.362969	+	0.362969i
$86$		0			0
$87$	0.515645	+	0.00406899i	0.0552829	+	0.000436242i
$88$		0			0
$89$	4.04724	−	15.1045i	0.429007	−	1.60108i	−0.326008	−	0.945367i	$-0.605704\pi$
$89$	4.04724	−	15.1045i	0.429007	−	1.60108i	0.755014	−	0.655708i	$-0.227630\pi$
$90$		0			0
$91$	13.1572	+	9.39447i	1.37925	+	0.984808i
$92$		0			0
$93$	−0.0347477	−	0.00960512i	−0.00360317	−	0.000996004i
$94$		0			0
$95$	13.9191			1.42807
$96$		0			0
$97$	0.573740	−	2.14123i	0.0582544	−	0.217409i	−0.930662	−	0.365879i	$-0.880768\pi$
$97$	0.573740	−	2.14123i	0.0582544	−	0.217409i	0.988917	+	0.148471i	$0.0474350\pi$
$98$		0			0
$99$	9.65283	+	9.96242i	0.970146	+	1.00126i
$100$		0			0
$101$	4.37617			0.435445			0.217723	−	0.976011i	$-0.430137\pi$
$101$	4.37617			0.435445			0.217723	+	0.976011i	$0.430137\pi$
$102$		0			0
$103$	3.36780	−	1.94440i	0.331839	−	0.191587i	−0.324818	−	0.945776i	$-0.605303\pi$
$103$	3.36780	−	1.94440i	0.331839	−	0.191587i	0.656657	+	0.754189i	$0.271970\pi$
$104$		0			0
$105$	−8.27348	+	14.0725i	−0.807409	+	1.37333i
$106$		0			0
$107$	−2.82575	+	1.63145i	−0.273176	+	0.157718i	−0.630330	−	0.776327i	$-0.717080\pi$
$107$	−2.82575	+	1.63145i	−0.273176	+	0.157718i	0.357154	+	0.934045i	$0.383747\pi$
$108$		0			0
$109$	−0.398131	−	0.398131i	−0.0381340	−	0.0381340i	0.687783	−	0.725917i	$-0.258584\pi$
$109$	−0.398131	−	0.398131i	−0.0381340	−	0.0381340i	−0.725917	+	0.687783i	$0.758584\pi$
$110$		0			0
$111$	2.82792	−	10.2303i	0.268414	−	0.971021i
$112$		0			0
$113$			12.9836i			1.22139i	0.791865	+	0.610696i	$0.209110\pi$
$113$			12.9836i			1.22139i	−0.791865	+	0.610696i	$0.790890\pi$
$114$		0			0
$115$	9.95259	−	9.95259i	0.928083	−	0.928083i
$116$		0			0
$117$	10.0687	+	3.95233i	0.930853	+	0.365393i
$118$		0			0
$119$	7.13857	−	7.13857i	0.654391	−	0.654391i
$120$		0			0
$121$			10.3808i			0.943706i
$122$		0			0
$123$	−13.1562	+	3.41415i	−1.18625	+	0.307844i
$124$		0			0
$125$	−8.29626	−	8.29626i	−0.742040	−	0.742040i
$126$		0			0
$127$	−14.3435	+	8.28122i	−1.27278	+	0.734839i	−0.975510	−	0.219955i	$-0.929409\pi$
$127$	−14.3435	+	8.28122i	−1.27278	+	0.734839i	−0.297268	+	0.954794i	$0.596076\pi$
$128$		0			0
$129$	−0.674266	−	1.18944i	−0.0593658	−	0.104724i
$130$		0			0
$131$	−1.15647	+	0.667686i	−0.101041	+	0.0583360i	−0.549669	−	0.835383i	$-0.685246\pi$
$131$	−1.15647	+	0.667686i	−0.101041	+	0.0583360i	0.448628	+	0.893719i	$0.351913\pi$
$132$		0			0
$133$	−29.6923			−2.57465
$134$		0			0
$135$	−3.07576	+	10.4800i	−0.264719	+	0.901975i
$136$		0			0
$137$	4.87994	−	18.2122i	0.416921	−	1.55597i	−0.364036	−	0.931385i	$-0.618601\pi$
$137$	4.87994	−	18.2122i	0.416921	−	1.55597i	0.780957	−	0.624585i	$-0.214732\pi$
$138$		0			0
$139$	−2.66207			−0.225794			−0.112897	−	0.993607i	$-0.536013\pi$
$139$	−2.66207			−0.225794			−0.112897	+	0.993607i	$0.536013\pi$
$140$		0			0
$141$	4.85142	+	18.6946i	0.408563	+	1.57437i
$142$		0			0
$143$	6.90640	+	15.1740i	0.577542	+	1.26892i
$144$		0			0
$145$	−0.161965	+	0.604462i	−0.0134505	+	0.0501978i
$146$		0			0
$147$	11.5042	−	19.5676i	0.948848	−	1.61391i
$148$		0			0
$149$	5.66297	−	5.66297i	0.463929	−	0.463929i	−0.436012	−	0.899941i	$-0.643610\pi$
$149$	5.66297	−	5.66297i	0.463929	−	0.463929i	0.899941	+	0.436012i	$0.143610\pi$
$150$		0			0
$151$	1.17884	+	4.39949i	0.0959326	+	0.358025i	0.997159	−	0.0753229i	$-0.0239987\pi$
$151$	1.17884	+	4.39949i	0.0959326	+	0.358025i	−0.901227	+	0.433348i	$0.857332\pi$
$152$		0			0
$153$	3.46914	−	5.79555i	0.280464	−	0.468542i
$154$		0			0
$155$	0.0218749	−	0.0378884i	0.00175703	−	0.00304327i
$156$		0			0
$157$	−3.77808	−	6.54383i	−0.301524	−	0.522254i	0.674958	−	0.737857i	$-0.264162\pi$
$157$	−3.77808	−	6.54383i	−0.301524	−	0.522254i	−0.976481	+	0.215602i	$0.930829\pi$
$158$		0			0
$159$	−7.49413	−	0.0591368i	−0.594323	−	0.00468985i
$160$		0			0
$161$	−21.2309	+	21.2309i	−1.67323	+	1.67323i
$162$		0			0
$163$	−6.42642	−	23.9837i	−0.503356	−	1.87855i	−0.477009	−	0.878898i	$-0.658279\pi$
$163$	−6.42642	−	23.9837i	−0.503356	−	1.87855i	−0.0263474	−	0.999653i	$-0.508388\pi$
$164$		0			0
$165$	−14.6449	+	8.30182i	−1.14010	+	0.646296i
$166$		0			0
$167$	−9.16934	+	2.45692i	−0.709545	+	0.190122i	−0.595502	−	0.803354i	$-0.703047\pi$
$167$	−9.16934	+	2.45692i	−0.709545	+	0.190122i	−0.114043	+	0.993476i	$0.536380\pi$
$168$		0			0
$169$	9.80302	+	8.53820i	0.754078	+	0.656785i
$170$		0			0
$171$	−19.2679	+	4.83824i	−1.47345	+	0.369990i
$172$		0			0
$173$	8.11178	+	14.0500i	0.616727	+	1.06820i	0.990079	+	0.140513i	$0.0448752\pi$
$173$	8.11178	+	14.0500i	0.616727	+	1.06820i	−0.373352	+	0.927690i	$0.621791\pi$
$174$		0			0
$175$	1.84472	+	1.84472i	0.139447	+	0.139447i
$176$		0			0
$177$	−4.48313	+	4.41293i	−0.336972	+	0.331696i
$178$		0			0
$179$	0.516398	+	0.894428i	0.0385974	+	0.0668527i	0.884679	−	0.466201i	$-0.154378\pi$
$179$	0.516398	+	0.894428i	0.0385974	+	0.0668527i	−0.846081	+	0.533054i	$0.821044\pi$
$180$		0			0
$181$			2.32376i			0.172724i	0.996264	+	0.0863620i	$0.0275242\pi$
$181$			2.32376i			0.172724i	−0.996264	+	0.0863620i	$0.972476\pi$
$182$		0			0
$183$	22.1465	−	12.5543i	1.63711	−	0.928042i
$184$		0			0
$185$	11.1550	+	6.44036i	0.820134	+	0.473504i
$186$		0			0
$187$	10.0561	−	2.69451i	0.735372	−	0.197042i
$188$		0			0
$189$	6.56122	−	22.3560i	0.477259	−	1.62616i
$190$		0			0
$191$	20.5074	−	11.8399i	1.48386	−	0.856707i	0.484028	−	0.875052i	$-0.339173\pi$
$191$	20.5074	−	11.8399i	1.48386	−	0.856707i	0.999832	+	0.0183456i	$0.00583991\pi$
$192$		0			0
$193$	−0.266914	−	0.996138i	−0.0192129	−	0.0717036i	0.955654	−	0.294492i	$-0.0951506\pi$
$193$	−0.266914	−	0.996138i	−0.0192129	−	0.0717036i	−0.974867	+	0.222789i	$0.928484\pi$
$194$		0			0
$195$	−7.71188	+	10.6224i	−0.552259	+	0.760687i
$196$		0			0
$197$	−15.5162	−	4.15756i	−1.10548	−	0.296214i	−0.340488	−	0.940249i	$-0.610592\pi$
$197$	−15.5162	−	4.15756i	−1.10548	−	0.296214i	−0.764996	+	0.644035i	$0.777259\pi$
$198$		0			0
$199$	17.6176	+	10.1715i	1.24888	+	0.721040i	0.970886	−	0.239543i	$-0.0769977\pi$
$199$	17.6176	+	10.1715i	1.24888	+	0.721040i	0.277992	+	0.960583i	$0.410331\pi$
$200$		0			0
$201$	−2.24836	−	8.66389i	−0.158587	−	0.611104i
$202$		0			0
$203$	0.345504	−	1.28944i	0.0242496	−	0.0905009i
$204$		0			0
$205$		−	16.4947i		−	1.15204i
$206$		0			0
$207$	−10.3176	+	17.2366i	−0.717124	+	1.19803i
$208$		0			0
$209$	−26.5175	−	15.3099i	−1.83425	−	1.05901i
$210$		0			0
$211$	5.29402	−	9.16950i	0.364455	−	0.631255i	−0.624233	−	0.781238i	$-0.714589\pi$
$211$	5.29402	−	9.16950i	0.364455	−	0.631255i	0.988689	+	0.149983i	$0.0479219\pi$
$212$		0			0
$213$	14.7812	−	3.83585i	1.01279	−	0.262828i
$214$		0			0
$215$	1.60271	−	0.429446i	0.109304	−	0.0292880i
$216$		0			0
$217$	−0.0466636	+	0.0808237i	−0.00316773	+	0.00548667i
$218$		0			0
$219$	−4.21616	−	4.28322i	−0.284901	−	0.289434i
$220$		0			0
$221$	6.26646	−	5.16061i	0.421527	−	0.347140i
$222$		0			0
$223$	12.5450	+	12.5450i	0.840075	+	0.840075i	0.988868	−	0.148794i	$-0.0475390\pi$
$223$	12.5450	+	12.5450i	0.840075	+	0.840075i	−0.148794	+	0.988868i	$0.547539\pi$
$224$		0			0
$225$	1.49766	+	0.896480i	0.0998439	+	0.0597654i
$226$		0			0
$227$	−25.3042	−	6.78025i	−1.67950	−	0.450021i	−0.711853	−	0.702328i	$-0.752144\pi$
$227$	−25.3042	−	6.78025i	−1.67950	−	0.450021i	−0.967647	+	0.252307i	$0.918811\pi$
$228$		0			0
$229$	8.41036	+	2.25355i	0.555772	+	0.148919i	0.525763	−	0.850631i	$-0.323780\pi$
$229$	8.41036	+	2.25355i	0.555772	+	0.148919i	0.0300088	+	0.999550i	$0.490446\pi$
$230$		0			0
$231$	31.2405	−	17.7095i	2.05547	−	1.16520i
$232$		0			0
$233$	8.30872			0.544322			0.272161	−	0.962252i	$-0.412262\pi$
$233$	8.30872			0.544322			0.272161	+	0.962252i	$0.412262\pi$
$234$		0			0
$235$	−23.4384			−1.52896
$236$		0			0
$237$	−12.6540	−	7.43954i	−0.821966	−	0.483250i
$238$		0			0
$239$	−23.5124	−	6.30014i	−1.52089	−	0.407522i	−0.600857	−	0.799357i	$-0.705174\pi$
$239$	−23.5124	−	6.30014i	−1.52089	−	0.407522i	−0.920036	+	0.391835i	$0.871840\pi$
$240$		0			0
$241$	−4.09572	−	1.09744i	−0.263829	−	0.0706926i	0.124480	−	0.992222i	$-0.460274\pi$
$241$	−4.09572	−	1.09744i	−0.263829	−	0.0706926i	−0.388308	+	0.921529i	$0.626940\pi$
$242$		0			0
$243$	0.614877	−	15.5763i	0.0394444	−	0.999222i
$244$		0			0
$245$	19.4782	+	19.4782i	1.24442	+	1.24442i
$246$		0			0
$247$	−23.7650	−	2.29984i	−1.51213	−	0.146335i
$248$		0			0
$249$	−10.2048	+	2.64824i	−0.646703	+	0.167826i
$250$		0			0
$251$	−1.74440	+	3.02138i	−0.110105	+	0.190708i	−0.915813	−	0.401606i	$-0.868452\pi$
$251$	−1.74440	+	3.02138i	−0.110105	+	0.190708i	0.805707	+	0.592314i	$0.201786\pi$
$252$		0			0
$253$	−29.9078	+	8.01378i	−1.88029	+	0.503822i
$254$		0			0
$255$	5.75023	+	5.84170i	0.360093	+	0.365822i
$256$		0			0
$257$	11.0876	−	19.2043i	0.691626	−	1.19793i	−0.279678	−	0.960094i	$-0.590228\pi$
$257$	11.0876	−	19.2043i	0.691626	−	1.19793i	0.971305	−	0.237838i	$-0.0764388\pi$
$258$		0			0
$259$	−23.7959	−	13.7386i	−1.47861	−	0.853675i
$260$		0			0
$261$	0.0140950	−	0.893039i	0.000872457	−	0.0552777i
$262$		0			0
$263$			8.17887i			0.504331i	0.967684	+	0.252166i	$0.0811427\pi$
$263$			8.17887i			0.504331i	−0.967684	+	0.252166i	$0.918857\pi$
$264$		0			0
$265$	2.35392	−	8.78496i	0.144600	−	0.539656i
$266$		0			0
$267$	−26.1057	−	7.21625i	−1.59764	−	0.441627i
$268$		0			0
$269$	3.02873	+	1.74864i	0.184665	+	0.106616i	0.589483	−	0.807781i	$-0.299332\pi$
$269$	3.02873	+	1.74864i	0.184665	+	0.106616i	−0.404818	+	0.914397i	$0.632665\pi$
$270$		0			0
$271$	4.08370	+	1.09422i	0.248067	+	0.0664693i	0.380710	−	0.924695i	$-0.375680\pi$
$271$	4.08370	+	1.09422i	0.248067	+	0.0664693i	−0.132643	+	0.991164i	$0.542346\pi$
$272$		0			0
$273$	16.4510	−	22.6598i	0.995661	−	1.37143i
$274$		0			0
$275$	0.696303	+	2.59864i	0.0419887	+	0.156704i
$276$		0			0
$277$	−18.6202	+	10.7504i	−1.11878	+	0.645929i	−0.941090	−	0.338157i	$-0.890196\pi$
$277$	−18.6202	+	10.7504i	−1.11878	+	0.645929i	−0.177692	+	0.984086i	$0.556863\pi$
$278$		0			0
$279$	−0.0171109	+	0.0600516i	−0.00102441	+	0.00359519i
$280$		0			0
$281$	6.44983	−	1.72823i	0.384765	−	0.103097i	−0.0612514	−	0.998122i	$-0.519509\pi$
$281$	6.44983	−	1.72823i	0.384765	−	0.103097i	0.446016	+	0.895025i	$0.352842\pi$
$282$		0			0
$283$	−18.4334	−	10.6425i	−1.09575	−	0.632633i	−0.160650	−	0.987011i	$-0.551359\pi$
$283$	−18.4334	−	10.6425i	−1.09575	−	0.632633i	−0.935102	+	0.354379i	$0.884692\pi$
$284$		0			0
$285$	0.190237	−	24.1079i	0.0112687	−	1.42803i
$286$		0			0
$287$			35.1865i			2.07699i
$288$		0			0
$289$	5.96537	+	10.3323i	0.350904	+	0.607784i
$290$		0			0
$291$	−3.70076	−	1.02298i	−0.216942	−	0.0599682i
$292$		0			0
$293$	1.95642	+	1.95642i	0.114295	+	0.114295i	0.761941	−	0.647646i	$-0.224246\pi$
$293$	1.95642	+	1.95642i	0.114295	+	0.114295i	−0.647646	+	0.761941i	$0.724246\pi$
$294$		0			0
$295$	−3.81704	−	6.61131i	−0.222237	−	0.384925i
$296$		0			0
$297$	17.3868	−	16.5825i	1.00888	−	0.962215i
$298$		0			0
$299$	−18.6371	+	15.3482i	−1.07781	+	0.887611i
$300$		0			0
$301$	−3.41892	+	0.916096i	−0.197063	+	0.0528029i
$302$		0			0
$303$	0.0598106	−	7.57951i	0.00343603	−	0.435432i
$304$		0			0
$305$	7.99596	+	29.8413i	0.457847	+	1.70871i
$306$		0			0
$307$	−18.0252	+	18.0252i	−1.02875	+	1.02875i	−0.0291785	+	0.999574i	$0.509289\pi$
$307$	−18.0252	+	18.0252i	−1.02875	+	1.02875i	−0.999574	+	0.0291785i	$0.990711\pi$
$308$		0			0
$309$	−3.32166	−	5.85959i	−0.188963	−	0.333340i
$310$		0			0
$311$	3.31606	+	5.74359i	0.188037	+	0.325689i	0.944596	−	0.328236i	$-0.106454\pi$
$311$	3.31606	+	5.74359i	0.188037	+	0.325689i	−0.756559	+	0.653926i	$0.773121\pi$
$312$		0			0
$313$	4.21383	−	7.29857i	0.238180	−	0.412539i	−0.722012	−	0.691880i	$-0.756783\pi$
$313$	4.21383	−	7.29857i	0.238180	−	0.412539i	0.960192	+	0.279341i	$0.0901159\pi$
$314$		0			0
$315$	24.2604	+	14.5220i	1.36692	+	0.818221i
$316$		0			0
$317$	−4.76158	−	17.7705i	−0.267437	−	0.998089i	−0.960742	−	0.277444i	$-0.910513\pi$
$317$	−4.76158	−	17.7705i	−0.267437	−	0.998089i	0.693305	−	0.720645i	$-0.256154\pi$
$318$		0			0
$319$	0.973419	−	0.973419i	0.0545010	−	0.0545010i
$320$		0			0
$321$	2.78705	+	4.91650i	0.155558	+	0.274412i
$322$		0			0
$323$	−3.85886	+	14.4014i	−0.214712	+	0.801318i
$324$		0			0
$325$	1.33358	+	1.61935i	0.0739737	+	0.0898253i
$326$		0			0
$327$	−0.695002	+	0.684120i	−0.0384337	+	0.0378319i
$328$		0			0
$329$	49.9990			2.75653
$330$		0			0
$331$	−1.64708	+	6.14697i	−0.0905315	+	0.337868i	−0.996304	−	0.0858958i	$-0.972625\pi$
$331$	−1.64708	+	6.14697i	−0.0905315	+	0.337868i	0.905773	+	0.423764i	$0.139291\pi$
$332$		0			0
$333$	−17.6803	−	5.03777i	−0.968873	−	0.276068i
$334$		0			0
$335$	10.8624			0.593477
$336$		0			0
$337$	−17.8689	+	10.3166i	−0.973381	+	0.561982i	−0.900265	−	0.435342i	$-0.856628\pi$
$337$	−17.8689	+	10.3166i	−0.973381	+	0.561982i	−0.0731159	+	0.997323i	$0.523294\pi$
$338$		0			0
$339$	22.4875	+	0.177451i	1.22135	+	0.00963781i
$340$		0			0
$341$	−0.0833482	+	0.0481211i	−0.00451356	+	0.00260590i
$342$		0			0
$343$	−19.3569	−	19.3569i	−1.04518	−	1.04518i
$344$		0			0
$345$	−17.1018	−	17.3739i	−0.920731	−	0.935378i
$346$		0			0
$347$			13.8268i			0.742264i	0.928580	+	0.371132i	$0.121030\pi$
$347$			13.8268i			0.742264i	−0.928580	+	0.371132i	$0.878970\pi$
$348$		0			0
$349$	−4.74820	+	4.74820i	−0.254166	+	0.254166i	−0.822676	−	0.568510i	$-0.807520\pi$
$349$	−4.74820	+	4.74820i	−0.254166	+	0.254166i	0.568510	+	0.822676i	$0.307520\pi$
$350$		0			0
$351$	6.98304	−	17.3850i	0.372727	−	0.927941i
$352$		0			0
$353$	6.47776	−	6.47776i	0.344776	−	0.344776i	−0.513383	−	0.858160i	$-0.671608\pi$
$353$	6.47776	−	6.47776i	0.344776	−	0.344776i	0.858160	+	0.513383i	$0.171608\pi$
$354$		0			0
$355$			18.5320i			0.983576i
$356$		0			0
$357$	−12.2664	−	12.4615i	−0.649207	−	0.659535i
$358$		0			0
$359$	5.81523	+	5.81523i	0.306916	+	0.306916i	0.843712	−	0.536796i	$-0.180366\pi$
$359$	5.81523	+	5.81523i	0.306916	+	0.306916i	−0.536796	+	0.843712i	$0.680366\pi$
$360$		0			0
$361$	21.5217	−	12.4255i	1.13272	−	0.653975i
$362$		0			0
$363$	17.9795	+	0.141877i	0.943677	+	0.00744663i
$364$		0			0
$365$	6.31651	−	3.64684i	0.330621	−	0.190884i
$366$		0			0
$367$	34.5708			1.80458			0.902291	−	0.431127i	$-0.141884\pi$
$367$	34.5708			1.80458			0.902291	+	0.431127i	$0.141884\pi$
$368$		0			0
$369$	5.73349	+	22.8331i	0.298474	+	1.18864i
$370$		0			0
$371$	−5.02139	+	18.7401i	−0.260698	+	0.972937i
$372$		0			0
$373$	−1.15565			−0.0598372			−0.0299186	−	0.999552i	$-0.509525\pi$
$373$	−1.15565			−0.0598372			−0.0299186	+	0.999552i	$0.509525\pi$
$374$		0			0
$375$	−14.4825	+	14.2557i	−0.747873	+	0.736162i
$376$		0			0
$377$	0.376408	−	1.00527i	0.0193860	−	0.0517743i
$378$		0			0
$379$	2.85386	−	10.6507i	0.146593	−	0.547092i	−0.853087	−	0.521770i	$-0.825272\pi$
$379$	2.85386	−	10.6507i	0.146593	−	0.547092i	0.999679	−	0.0253224i	$-0.00806123\pi$
$380$		0			0
$381$	14.1470	+	24.9561i	0.724773	+	1.27854i
$382$		0			0
$383$	−19.6227	+	19.6227i	−1.00267	+	1.00267i	−0.00267678	+	0.999996i	$0.500852\pi$
$383$	−19.6227	+	19.6227i	−1.00267	+	1.00267i	−0.999996	+	0.00267678i	$0.999148\pi$
$384$		0			0
$385$	11.2793	+	42.0950i	0.574848	+	2.14536i
$386$		0			0
$387$	−2.06932	+	1.15157i	−0.105190	+	0.0585376i
$388$		0			0
$389$	10.4710	−	18.1363i	0.530901	−	0.919547i	−0.468449	−	0.883491i	$-0.655187\pi$
$389$	10.4710	−	18.1363i	0.530901	−	0.919547i	0.999350	−	0.0360565i	$-0.0114796\pi$
$390$		0			0
$391$	7.53827	+	13.0567i	0.381227	+	0.660304i
$392$		0			0
$393$	1.14062	+	2.01212i	0.0575369	+	0.101498i
$394$		0			0
$395$	12.5962	−	12.5962i	0.633784	−	0.633784i
$396$		0			0
$397$	0.349912	+	1.30589i	0.0175616	+	0.0655407i	0.974151	−	0.225900i	$-0.0725322\pi$
$397$	0.349912	+	1.30589i	0.0175616	+	0.0655407i	−0.956589	+	0.291440i	$0.905866\pi$
$398$		0			0
$399$	−0.405815	+	51.4270i	−0.0203161	+	2.57457i
$400$		0			0
$401$	−32.5628	+	8.72519i	−1.62611	+	0.435715i	−0.952789	−	0.303634i	$-0.901800\pi$
$401$	−32.5628	+	8.72519i	−1.62611	+	0.435715i	−0.673322	+	0.739349i	$0.735133\pi$
$402$		0			0
$403$	−0.0436086	+	0.0610749i	−0.00217230	+	0.00304236i
$404$		0			0
$405$	18.1093	+	5.47044i	0.899858	+	0.271828i
$406$		0			0
$407$	−14.1677	−	24.5392i	−0.702268	−	1.21636i
$408$		0			0
$409$	5.48526	+	5.48526i	0.271228	+	0.271228i	0.829595	−	0.558366i	$-0.188572\pi$
$409$	5.48526	+	5.48526i	0.271228	+	0.271228i	−0.558366	+	0.829595i	$0.688572\pi$
$410$		0			0
$411$	−31.4767	−	8.70094i	−1.55263	−	0.429186i
$412$		0			0
$413$	8.14253	+	14.1033i	0.400668	+	0.693977i
$414$		0			0
$415$		−	12.7944i		−	0.628050i
$416$		0			0
$417$	−0.0363835	+	4.61070i	−0.00178171	+	0.225787i
$418$		0			0
$419$	−32.6192	−	18.8327i	−1.59355	−	0.920037i	−0.992691	−	0.120683i	$-0.961492\pi$
$419$	−32.6192	−	18.8327i	−1.59355	−	0.920037i	−0.600860	−	0.799354i	$-0.705175\pi$
$420$		0			0
$421$	8.56415	−	2.29476i	0.417391	−	0.111840i	−0.0440103	−	0.999031i	$-0.514013\pi$
$421$	8.56415	−	2.29476i	0.417391	−	0.111840i	0.461402	+	0.887191i	$0.347347\pi$
$422$		0			0
$423$	32.4452	−	8.14713i	1.57754	−	0.396127i
$424$		0			0
$425$	1.13447	−	0.654987i	0.0550299	−	0.0317715i
$426$		0			0
$427$	−17.0570	−	63.6576i	−0.825447	−	3.08061i
$428$		0			0
$429$	26.3758	−	11.7545i	1.27344	−	0.567511i
$430$		0			0
$431$	4.98825	+	1.33660i	0.240275	+	0.0643816i	0.376947	−	0.926235i	$-0.376974\pi$
$431$	4.98825	+	1.33660i	0.240275	+	0.0643816i	−0.136672	+	0.990616i	$0.543641\pi$
$432$		0			0
$433$	11.0611	+	6.38615i	0.531564	+	0.306899i	0.741653	−	0.670784i	$-0.234042\pi$
$433$	11.0611	+	6.38615i	0.531564	+	0.306899i	−0.210089	+	0.977682i	$0.567375\pi$
$434$		0			0
$435$	1.04471	+	0.288784i	0.0500901	+	0.0138461i
$436$		0			0
$437$	11.4767	−	42.8315i	0.549003	−	2.04891i
$438$		0			0
$439$		−	16.6598i		−	0.795130i	−0.917574	−	0.397565i	$-0.869855\pi$
$439$		−	16.6598i		−	0.795130i	0.917574	−	0.397565i	$-0.130145\pi$
$440$		0			0
$441$	−33.7338	−	20.1926i	−1.60637	−	0.961554i
$442$		0			0
$443$	14.6094	+	8.43471i	0.694111	+	0.400745i	0.805150	−	0.593071i	$-0.202085\pi$
$443$	14.6094	+	8.43471i	0.694111	+	0.400745i	−0.111039	+	0.993816i	$0.535418\pi$
$444$		0			0
$445$	16.4344	−	28.4653i	0.779067	−	1.34938i
$446$		0			0
$447$	−9.73085	−	9.88564i	−0.460253	−	0.467575i
$448$		0			0
$449$	5.25531	−	1.40816i	0.248014	−	0.0664550i	−0.132671	−	0.991160i	$-0.542355\pi$
$449$	5.25531	−	1.40816i	0.248014	−	0.0664550i	0.380684	+	0.924705i	$0.375689\pi$
$450$		0			0
$451$	−18.1428	+	31.4242i	−0.854309	+	1.47971i
$452$		0			0
$453$	7.63601	−	1.98162i	0.358771	−	0.0931045i
$454$		0			0
$455$	21.6025	+	26.2316i	1.01274	+	1.22976i
$456$		0			0
$457$	3.63248	+	3.63248i	0.169920	+	0.169920i	0.786944	−	0.617024i	$-0.211662\pi$
$457$	3.63248	+	3.63248i	0.169920	+	0.169920i	−0.617024	+	0.786944i	$0.711662\pi$
$458$		0			0
$459$	−9.99045	−	6.08776i	−0.466314	−	0.284152i
$460$		0			0
$461$	17.8341	+	4.77863i	0.830616	+	0.222563i	0.648982	−	0.760803i	$-0.275195\pi$
$461$	17.8341	+	4.77863i	0.830616	+	0.222563i	0.181634	+	0.983366i	$0.441862\pi$
$462$		0			0
$463$	19.5812	+	5.24676i	0.910014	+	0.243838i	0.683312	−	0.730127i	$-0.260539\pi$
$463$	19.5812	+	5.24676i	0.910014	+	0.243838i	0.226702	+	0.973964i	$0.427206\pi$
$464$		0			0
$465$	−0.0653236	−	0.0384051i	−0.00302931	−	0.00178099i
$466$		0			0
$467$	20.4136			0.944630			0.472315	−	0.881430i	$-0.343418\pi$
$467$	20.4136			0.944630			0.472315	+	0.881430i	$0.343418\pi$
$468$		0			0
$469$	−23.1717			−1.06997
$470$		0			0
$471$	−11.3855	+	6.45419i	−0.524617	+	0.297393i
$472$		0			0
$473$	−3.52571	−	0.944710i	−0.162112	−	0.0434378i
$474$		0			0
$475$	−3.72155	−	0.997188i	−0.170757	−	0.0457541i
$476$		0			0
$477$	−0.204850	+	12.9790i	−0.00937942	+	0.594268i
$478$		0			0
$479$	23.5105	+	23.5105i	1.07422	+	1.07422i	0.997015	+	0.0772046i	$0.0245995\pi$
$479$	23.5105	+	23.5105i	1.07422	+	1.07422i	0.0772046	+	0.997015i	$0.475401\pi$
$480$		0			0
$481$	−17.9815	−	12.8392i	−0.819888	−	0.585415i
$482$		0			0
$483$	36.4817	+	37.0620i	1.65997	+	1.68638i
$484$		0			0
$485$	2.32976	−	4.03525i	0.105789	−	0.183231i
$486$		0			0
$487$	21.2583	−	5.69613i	0.963303	−	0.258116i	0.257305	−	0.966330i	$-0.417165\pi$
$487$	21.2583	−	5.69613i	0.963303	−	0.258116i	0.705998	+	0.708214i	$0.250499\pi$
$488$		0			0
$489$	−41.6276	+	10.8028i	−1.88246	+	0.488517i
$490$		0			0
$491$	−3.92229	+	6.79361i	−0.177011	+	0.306591i	−0.940855	−	0.338809i	$-0.889976\pi$
$491$	−3.92229	+	6.79361i	−0.177011	+	0.306591i	0.763845	+	0.645400i	$0.223309\pi$
$492$		0			0
$493$	−0.580505	−	0.335155i	−0.0261446	−	0.0150946i
$494$		0			0
$495$	14.1786	+	25.4783i	0.637280	+	1.14516i
$496$		0			0
$497$		−	39.5325i		−	1.77328i
$498$		0			0
$499$	2.47222	−	9.22644i	0.110672	−	0.413032i	−0.888255	−	0.459350i	$-0.848082\pi$
$499$	2.47222	−	9.22644i	0.110672	−	0.413032i	0.998927	+	0.0463184i	$0.0147489\pi$
$500$		0			0
$501$	4.13005	+	15.9149i	0.184517	+	0.711023i
$502$		0			0
$503$	−31.3713	−	18.1122i	−1.39878	−	0.807584i	−0.404512	−	0.914533i	$-0.632559\pi$
$503$	−31.3713	−	18.1122i	−1.39878	−	0.807584i	−0.994265	+	0.106948i	$0.965892\pi$
$504$		0			0
$505$	8.88505	+	2.38074i	0.395379	+	0.105942i
$506$		0			0
$507$	14.9221	−	16.8621i	0.662714	−	0.748872i
$508$		0			0
$509$	6.35239	+	23.7074i	0.281565	+	1.05081i	0.951313	+	0.308226i	$0.0997352\pi$
$509$	6.35239	+	23.7074i	0.281565	+	1.05081i	−0.669749	+	0.742588i	$0.733598\pi$
$510$		0			0
$511$	−13.4744	+	7.77945i	−0.596073	+	0.344143i
$512$		0			0
$513$	8.11648	+	33.4380i	0.358351	+	1.47632i
$514$		0			0
$515$	7.89552	−	2.11560i	0.347918	−	0.0932244i
$516$		0			0
$517$	44.6529	+	25.7804i	1.96383	+	1.13382i
$518$		0			0
$519$	24.4454	−	13.8576i	1.07304	−	0.608279i
$520$		0			0
$521$		−	2.76734i		−	0.121239i	−0.998161	−	0.0606196i	$-0.980692\pi$
$521$		−	2.76734i		−	0.121239i	0.998161	−	0.0606196i	$-0.0193077\pi$
$522$		0			0
$523$	15.1711	+	26.2772i	0.663388	+	1.14902i	0.979720	+	0.200373i	$0.0642154\pi$
$523$	15.1711	+	26.2772i	0.663388	+	1.14902i	−0.316332	+	0.948649i	$0.602451\pi$
$524$		0			0
$525$	3.22025	−	3.16983i	0.140543	−	0.138343i
$526$		0			0
$527$	0.0331369	+	0.0331369i	0.00144346	+	0.00144346i
$528$		0			0
$529$	−10.9196	−	18.9134i	−0.474767	−	0.822321i
$530$		0			0
$531$	7.58190	+	7.82507i	0.329027	+	0.339579i
$532$		0			0
$533$	−2.72540	+	28.1624i	−0.118050	+	1.21985i
$534$		0			0
$535$	−6.62475	+	1.77510i	−0.286413	+	0.0767441i
$536$		0			0
$537$	1.55620	−	0.882176i	0.0671552	−	0.0380687i
$538$		0			0
$539$	−15.6838	−	58.5326i	−0.675548	−	2.52118i
$540$		0			0
$541$	20.9305	−	20.9305i	0.899874	−	0.899874i	−0.0955505	−	0.995425i	$-0.530461\pi$
$541$	20.9305	−	20.9305i	0.899874	−	0.899874i	0.995425	+	0.0955505i	$0.0304611\pi$
$542$		0			0
$543$	4.02475	+	0.0317597i	0.172719	+	0.00136294i
$544$		0			0
$545$	−0.591742	−	1.02493i	−0.0253474	−	0.0439030i
$546$		0			0
$547$	−3.75811	+	6.50923i	−0.160685	+	0.278315i	−0.935115	−	0.354346i	$-0.884704\pi$
$547$	−3.75811	+	6.50923i	−0.160685	+	0.278315i	0.774430	+	0.632660i	$0.218037\pi$
$548$		0			0
$549$	−21.4414	−	38.5292i	−0.915095	−	1.64439i
$550$		0			0
$551$	0.510258	+	1.90431i	0.0217377	+	0.0811262i
$552$		0			0
$553$	−26.8703	+	26.8703i	−1.14264	+	1.14264i
$554$		0			0
$555$	11.3071	−	19.2324i	0.479961	−	0.816372i
$556$		0			0
$557$	1.14680	−	4.27992i	0.0485916	−	0.181346i	−0.937365	−	0.348349i	$-0.886742\pi$
$557$	1.14680	−	4.27992i	0.0485916	−	0.181346i	0.985956	+	0.167003i	$0.0534090\pi$
$558$		0			0
$559$	−2.80737	+	0.468405i	−0.118739	+	0.0198114i
$560$		0			0
$561$	−4.52945	−	17.4539i	−0.191233	−	0.736903i
$562$		0			0
$563$	18.4321			0.776820			0.388410	−	0.921487i	$-0.373024\pi$
$563$	18.4321			0.776820			0.388410	+	0.921487i	$0.373024\pi$
$564$		0			0
$565$	−7.06337	+	26.3609i	−0.297158	+	1.10901i
$566$		0			0
$567$	−38.6308	−	11.6696i	−1.62234	−	0.490075i
$568$		0			0
$569$	−22.1697			−0.929403			−0.464702	−	0.885467i	$-0.653838\pi$
$569$	−22.1697			−0.929403			−0.464702	+	0.885467i	$0.653838\pi$
$570$		0			0
$571$	−23.2580	+	13.4280i	−0.973316	+	0.561944i	−0.900245	−	0.435383i	$-0.856613\pi$
$571$	−23.2580	+	13.4280i	−0.973316	+	0.561944i	−0.0730703	+	0.997327i	$0.523280\pi$
$572$		0			0
$573$	−20.2264	−	35.6805i	−0.844971	−	1.49057i
$574$		0			0
$575$	−3.37404	+	1.94800i	−0.140707	+	0.0812374i
$576$		0			0
$577$	3.18909	+	3.18909i	0.132763	+	0.132763i	0.770366	−	0.637602i	$-0.220074\pi$
$577$	3.18909	+	3.18909i	0.132763	+	0.132763i	−0.637602	+	0.770366i	$0.720074\pi$
$578$		0			0
$579$	−1.72896	+	0.448680i	−0.0718529	+	0.0186465i
$580$		0			0
$581$			27.2930i			1.13230i
$582$		0			0
$583$	−14.1472	+	14.1472i	−0.585918	+	0.585918i
$584$		0			0
$585$	18.2926	+	13.5021i	0.756306	+	0.558245i
$586$		0			0
$587$	10.3678	−	10.3678i	0.427926	−	0.427926i	−0.459995	−	0.887921i	$-0.652149\pi$
$587$	10.3678	−	10.3678i	0.427926	−	0.427926i	0.887921	+	0.459995i	$0.152149\pi$
$588$		0			0
$589$		−	0.137830i		−	0.00567919i
$590$		0			0
$591$	−7.41294	+	26.8172i	−0.304928	+	1.10311i
$592$		0			0
$593$	−15.2003	−	15.2003i	−0.624200	−	0.624200i	0.322403	−	0.946603i	$-0.395509\pi$
$593$	−15.2003	−	15.2003i	−0.624200	−	0.624200i	−0.946603	+	0.322403i	$0.895509\pi$
$594$		0			0
$595$	18.3772	−	10.6101i	0.753390	−	0.434970i
$596$		0			0
$597$	17.8578	−	30.3746i	0.730872	−	1.24315i
$598$		0			0
$599$	−27.3111	+	15.7681i	−1.11590	+	0.644266i	−0.940351	−	0.340205i	$-0.889504\pi$
$599$	−27.3111	+	15.7681i	−1.11590	+	0.644266i	−0.175550	+	0.984471i	$0.556170\pi$
$600$		0			0
$601$	20.1704			0.822768			0.411384	−	0.911462i	$-0.365045\pi$
$601$	20.1704			0.822768			0.411384	+	0.911462i	$0.365045\pi$
$602$		0			0
$603$	−15.0366	+	3.77574i	−0.612336	+	0.153760i
$604$		0			0
$605$	−5.64738	+	21.0763i	−0.229599	+	0.856874i
$606$		0			0
$607$	33.6095			1.36417			0.682084	−	0.731274i	$-0.261074\pi$
$607$	33.6095			1.36417			0.682084	+	0.731274i	$0.261074\pi$
$608$		0			0
$609$	−2.22858	−	0.616036i	−0.0903068	−	0.0249630i
$610$		0			0
$611$	40.0179	+	3.87271i	1.61895	+	0.156673i
$612$		0			0
$613$	5.07932	−	18.9563i	0.205152	−	0.765637i	−0.784252	−	0.620443i	$-0.786953\pi$
$613$	5.07932	−	18.9563i	0.205152	−	0.765637i	0.989403	−	0.145194i	$-0.0463805\pi$
$614$		0			0
$615$	−28.5687	−	0.225438i	−1.15200	−	0.00909053i
$616$		0			0
$617$	4.44777	−	4.44777i	0.179061	−	0.179061i	−0.611886	−	0.790946i	$-0.709589\pi$
$617$	4.44777	−	4.44777i	0.179061	−	0.179061i	0.790946	+	0.611886i	$0.209589\pi$
$618$		0			0
$619$	5.28084	+	19.7084i	0.212255	+	0.792146i	0.987115	+	0.160013i	$0.0511537\pi$
$619$	5.28084	+	19.7084i	0.212255	+	0.792146i	−0.774860	+	0.632133i	$0.782180\pi$
$620$		0			0
$621$	29.7127	+	18.1057i	1.19233	+	0.726555i
$622$		0			0
$623$	−35.0580	+	60.7222i	−1.40457	+	2.43278i
$624$		0			0
$625$	−10.8762	−	18.8381i	−0.435047	−	0.753524i
$626$		0			0
$627$	−26.8791	+	45.7190i	−1.07345	+	1.82584i
$628$		0			0
$629$	−9.75608	+	9.75608i	−0.389001	+	0.389001i
$630$		0			0
$631$	2.93820	+	10.9655i	0.116968	+	0.436531i	0.999427	−	0.0338595i	$-0.0107799\pi$
$631$	2.93820	+	10.9655i	0.116968	+	0.436531i	−0.882459	+	0.470390i	$0.844113\pi$
$632$		0			0
$633$	−15.8092	−	9.29454i	−0.628359	−	0.369425i
$634$		0			0
$635$	−33.6271	+	9.01035i	−1.33445	+	0.357565i
$636$		0			0
$637$	−30.0380	−	36.4748i	−1.19015	−	1.44518i
$638$		0			0
$639$	−6.44167	−	25.6534i	−0.254828	−	1.01483i
$640$		0			0
$641$	−13.9974	−	24.2442i	−0.552865	−	0.957589i	−0.998066	−	0.0621592i	$-0.980201\pi$
$641$	−13.9974	−	24.2442i	−0.552865	−	0.957589i	0.445202	−	0.895430i	$-0.353132\pi$
$642$		0			0
$643$	−2.84425	−	2.84425i	−0.112166	−	0.112166i	0.648796	−	0.760962i	$-0.275273\pi$
$643$	−2.84425	−	2.84425i	−0.112166	−	0.112166i	−0.760962	+	0.648796i	$0.775273\pi$
$644$		0			0
$645$	−0.721894	−	2.78177i	−0.0284246	−	0.109532i
$646$		0			0
$647$	−19.7162	−	34.1494i	−0.775122	−	1.34255i	−0.934726	−	0.355370i	$-0.884355\pi$
$647$	−19.7162	−	34.1494i	−0.775122	−	1.34255i	0.159604	−	0.987181i	$-0.448978\pi$
$648$		0			0
$649$			16.7937i			0.659211i
$650$		0			0
$651$	0.139349	+	0.0819258i	0.00546150	+	0.00321093i
$652$		0			0
$653$	−7.77544	−	4.48915i	−0.304277	−	0.175674i	0.340086	−	0.940394i	$-0.389544\pi$
$653$	−7.77544	−	4.48915i	−0.304277	−	0.175674i	−0.644362	+	0.764720i	$0.722877\pi$
$654$		0			0
$655$	−2.71124	+	0.726474i	−0.105937	+	0.0283857i
$656$		0			0
$657$	−7.47616	+	7.24383i	−0.291673	+	0.282609i
$658$		0			0
$659$	−1.38440	+	0.799281i	−0.0539284	+	0.0311356i	−0.526722	−	0.850038i	$-0.676579\pi$
$659$	−1.38440	+	0.799281i	−0.0539284	+	0.0311356i	0.472793	+	0.881173i	$0.343246\pi$
$660$		0			0
$661$	12.4085	+	46.3090i	0.482633	+	1.80121i	0.590491	+	0.807044i	$0.298934\pi$
$661$	12.4085	+	46.3090i	0.482633	+	1.80121i	−0.107858	+	0.994166i	$0.534399\pi$
$662$		0			0
$663$	−8.85251	−	10.9240i	−0.343803	−	0.424254i
$664$		0			0
$665$	−60.2850	−	16.1533i	−2.33775	−	0.626399i
$666$		0			0
$667$	1.72649	+	0.996788i	0.0668498	+	0.0385958i
$668$		0			0
$669$	21.8993	−	21.5564i	0.846678	−	0.833420i
$670$		0			0
$671$	17.5898	−	65.6460i	0.679046	−	2.53424i
$672$		0			0
$673$		−	51.0740i		−	1.96876i	−0.176061	−	0.984379i	$-0.556335\pi$
$673$		−	51.0740i		−	1.96876i	0.176061	−	0.984379i	$-0.443665\pi$
$674$		0			0
$675$	1.57317	−	2.58169i	0.0605513	−	0.0993692i
$676$		0			0
$677$	2.72541	+	1.57351i	0.104746	+	0.0604751i	0.551458	−	0.834203i	$-0.314072\pi$
$677$	2.72541	+	1.57351i	0.104746	+	0.0604751i	−0.446712	+	0.894678i	$0.647405\pi$
$678$		0			0
$679$	−4.96984	+	8.60802i	−0.190725	+	0.330345i
$680$		0			0
$681$	−12.0892	+	43.7342i	−0.463259	+	1.67590i
$682$		0			0
$683$	26.4271	−	7.08112i	1.01121	−	0.270952i	0.285073	−	0.958506i	$-0.407982\pi$
$683$	26.4271	−	7.08112i	1.01121	−	0.270952i	0.726133	+	0.687554i	$0.241316\pi$
$684$		0			0
$685$	19.8157	−	34.3218i	0.757119	−	1.31137i
$686$		0			0
$687$	4.01809	−	14.5359i	0.153300	−	0.554580i
$688$		0			0
$689$	−5.47053	+	14.6102i	−0.208411	+	0.556603i
$690$		0			0
$691$	0.378153	+	0.378153i	0.0143856	+	0.0143856i	0.714263	−	0.699877i	$-0.246762\pi$
$691$	0.378153	+	0.378153i	0.0143856	+	0.0143856i	−0.699877	+	0.714263i	$0.746762\pi$
$692$		0			0
$693$	−30.2458	−	54.3504i	−1.14894	−	2.06460i
$694$		0			0
$695$	−5.40487	−	1.44823i	−0.205019	−	0.0549345i
$696$		0			0
$697$	17.0662	+	4.57288i	0.646430	+	0.173210i
$698$		0			0
$699$	0.113558	−	14.3907i	0.00429516	−	0.544305i
$700$		0			0
$701$	3.78591			0.142992			0.0714960	−	0.997441i	$-0.477223\pi$
$701$	3.78591			0.142992			0.0714960	+	0.997441i	$0.477223\pi$
$702$		0			0
$703$	40.5797			1.53049
$704$		0			0
$705$	−0.320341	+	40.5953i	−0.0120647	+	1.52891i
$706$		0			0
$707$	−18.9536	−	5.07860i	−0.712824	−	0.191001i
$708$		0			0
$709$	−17.1053	−	4.58335i	−0.642403	−	0.172131i	−0.0771116	−	0.997022i	$-0.524570\pi$
$709$	−17.1053	−	4.58335i	−0.642403	−	0.172131i	−0.565292	+	0.824891i	$0.691236\pi$
$710$		0			0
$711$	−13.0582	+	21.8150i	−0.489721	+	0.818127i
$712$		0			0
$713$	−0.0985527	−	0.0985527i	−0.00369083	−	0.00369083i
$714$		0			0
$715$	5.76719	+	34.5655i	0.215681	+	1.29268i
$716$		0			0
$717$	−11.2332	+	40.6373i	−0.419510	+	1.51763i
$718$		0			0
$719$	17.5430	−	30.3853i	0.654242	−	1.13318i	−0.327841	−	0.944733i	$-0.606321\pi$
$719$	17.5430	−	30.3853i	0.654242	−	1.13318i	0.982083	−	0.188448i	$-0.0603457\pi$
$720$		0			0
$721$	−16.8428	+	4.51300i	−0.627257	+	0.168073i
$722$		0			0
$723$	−1.95675	+	7.07878i	−0.0727723	+	0.263263i
$724$		0			0
$725$	0.0866092	−	0.150011i	0.00321658	−	0.00557129i
$726$		0			0
$727$	27.7648	+	16.0300i	1.02974	+	0.594521i	0.916910	−	0.399093i	$-0.130675\pi$
$727$	27.7648	+	16.0300i	1.02974	+	0.594521i	0.112830	+	0.993614i	$0.464008\pi$
$728$		0			0
$729$	−26.9697	−	1.27785i	−0.998879	−	0.0473278i
$730$		0			0
$731$			1.77731i			0.0657362i
$732$		0			0
$733$	5.51036	−	20.5649i	0.203530	−	0.759583i	−0.786363	−	0.617765i	$-0.788038\pi$
$733$	5.51036	−	20.5649i	0.203530	−	0.759583i	0.989893	−	0.141818i	$-0.0452949\pi$
$734$		0			0
$735$	34.0024	−	33.4700i	1.25420	−	1.23456i
$736$		0			0
$737$	−20.6941	−	11.9478i	−0.762278	−	0.440101i
$738$		0			0
$739$	−22.7510	−	6.09611i	−0.836908	−	0.224249i	−0.185183	−	0.982704i	$-0.559288\pi$
$739$	−22.7510	−	6.09611i	−0.836908	−	0.224249i	−0.651725	+	0.758455i	$0.725954\pi$
$740$		0			0
$741$	−4.30812	+	41.1294i	−0.158263	+	1.51093i
$742$		0			0
$743$	−8.80587	−	32.8639i	−0.323056	−	1.20566i	−0.916252	−	0.400603i	$-0.868801\pi$
$743$	−8.80587	−	32.8639i	−0.323056	−	1.20566i	0.593196	−	0.805058i	$-0.297866\pi$
$744$		0			0
$745$	14.5785	−	8.41687i	0.534113	−	0.308370i
$746$		0			0
$747$	4.44728	+	17.7109i	0.162717	+	0.648008i
$748$		0			0
$749$	14.1319	−	3.78664i	0.516370	−	0.138361i
$750$		0			0
$751$	−7.55905	−	4.36422i	−0.275834	−	0.159253i	0.355702	−	0.934599i	$-0.384242\pi$
$751$	−7.55905	−	4.36422i	−0.275834	−	0.159253i	−0.631536	+	0.775347i	$0.717575\pi$
$752$		0			0
$753$	5.20919	+	3.06259i	0.189833	+	0.111607i
$754$		0			0
$755$			9.57370i			0.348423i
$756$		0			0
$757$	−3.10306	−	5.37465i	−0.112783	−	0.195345i	0.804109	−	0.594482i	$-0.202643\pi$
$757$	−3.10306	−	5.37465i	−0.112783	−	0.195345i	−0.916891	+	0.399137i	$0.869310\pi$
$758$		0			0
$759$	13.4711	+	51.9098i	0.488969	+	1.88421i
$760$		0			0
$761$	4.06137	+	4.06137i	0.147225	+	0.147225i	0.776877	−	0.629652i	$-0.216803\pi$
$761$	4.06137	+	4.06137i	0.147225	+	0.147225i	−0.629652	+	0.776877i	$0.716803\pi$
$762$		0			0
$763$	1.26231	+	2.18638i	0.0456985	+	0.0791522i
$764$		0			0
$765$	10.1964	−	9.87954i	0.368652	−	0.357196i
$766$		0			0
$767$	5.42470	+	11.9186i	0.195874	+	0.430355i
$768$		0			0
$769$	43.4667	−	11.6469i	1.56745	−	0.419996i	0.632436	−	0.774613i	$-0.282055\pi$
$769$	43.4667	−	11.6469i	1.56745	−	0.419996i	0.935012	+	0.354616i	$0.115389\pi$
$770$		0			0
$771$	−33.1103	−	19.4662i	−1.19244	−	0.701058i
$772$		0			0
$773$	−6.09459	−	22.7453i	−0.219207	−	0.818093i	−0.984643	−	0.174580i	$-0.944143\pi$
$773$	−6.09459	−	22.7453i	−0.219207	−	0.818093i	0.765436	−	0.643513i	$-0.222524\pi$
$774$		0			0
$775$	−0.00856307	+	0.00856307i	−0.000307595	+	0.000307595i
$776$		0			0
$777$	−24.1204	+	41.0267i	−0.865316	+	1.47183i
$778$		0			0
$779$	−25.9825	−	45.0031i	−0.930921	−	1.61240i
$780$		0			0
$781$	20.3837	−	35.3055i	0.729385	−	1.26333i
$782$		0			0
$783$	−1.54655	−	0.0366179i	−0.0552691	−	0.00130862i
$784$		0			0
$785$	−4.11073	−	15.3415i	−0.146718	−	0.547560i
$786$		0			0
$787$	23.1222	−	23.1222i	0.824216	−	0.824216i	−0.162493	−	0.986710i	$-0.551954\pi$
$787$	23.1222	−	23.1222i	0.824216	−	0.824216i	0.986710	+	0.162493i	$0.0519536\pi$
$788$		0			0
$789$	14.1658	+	0.111783i	0.504315	+	0.00397959i
$790$		0			0
$791$	15.0676	−	56.2331i	0.535743	−	1.99942i
$792$		0			0
$793$	−8.72135	−	52.2711i	−0.309704	−	1.85620i
$794$		0			0
$795$	−15.1833	−	4.19705i	−0.538498	−	0.148854i
$796$		0			0
$797$	25.2623			0.894838			0.447419	−	0.894325i	$-0.352343\pi$
$797$	25.2623			0.894838			0.447419	+	0.894325i	$0.352343\pi$
$798$		0			0
$799$	6.49794	−	24.2506i	0.229881	−	0.857926i
$800$		0			0
$801$	−12.8553	+	45.1163i	−0.454220	+	1.59411i
$802$		0			0
$803$	−16.0449			−0.566212
$804$		0			0
$805$	−54.6557	+	31.5555i	−1.92636	+	1.11218i
$806$		0			0
$807$	3.07003	−	5.22186i	0.108070	−	0.183818i
$808$		0			0
$809$	9.97416	−	5.75858i	0.350673	−	0.202461i	−0.314309	−	0.949321i	$-0.601773\pi$
$809$	9.97416	−	5.75858i	0.350673	−	0.202461i	0.664982	+	0.746860i	$0.268439\pi$
$810$		0			0
$811$	7.70784	+	7.70784i	0.270659	+	0.270659i	0.829365	−	0.558706i	$-0.188702\pi$
$811$	7.70784	+	7.70784i	0.270659	+	0.270659i	−0.558706	+	0.829365i	$0.688702\pi$
$812$		0			0
$813$	1.95100	−	7.05799i	0.0684247	−	0.247535i
$814$		0			0
$815$		−	52.1909i		−	1.82817i
$816$		0			0
$817$	3.69629	−	3.69629i	0.129317	−	0.129317i
$818$		0			0
$819$	−39.0218	−	28.8028i	−1.36353	−	1.00645i
$820$		0			0
$821$	−0.527911	+	0.527911i	−0.0184242	+	0.0184242i	−0.716259	−	0.697835i	$-0.754147\pi$
$821$	−0.527911	+	0.527911i	−0.0184242	+	0.0184242i	0.697835	+	0.716259i	$0.254147\pi$
$822$		0			0
$823$			47.0444i			1.63987i	0.572459	+	0.819933i	$0.305989\pi$
$823$			47.0444i			1.63987i	−0.572459	+	0.819933i	$0.694011\pi$
$824$		0			0
$825$	4.51035	−	1.17048i	0.157030	−	0.0407508i
$826$		0			0
$827$	19.2596	+	19.2596i	0.669721	+	0.669721i	0.957651	−	0.287930i	$-0.0929671\pi$
$827$	19.2596	+	19.2596i	0.669721	+	0.669721i	−0.287930	+	0.957651i	$0.592967\pi$
$828$		0			0
$829$	−47.6707	+	27.5227i	−1.65567	+	0.955902i	−0.680992	+	0.732291i	$0.738451\pi$
$829$	−47.6707	+	27.5227i	−1.65567	+	0.955902i	−0.974679	+	0.223610i	$0.928216\pi$
$830$		0			0
$831$	18.3652	+	32.3971i	0.637080	+	1.12384i
$832$		0			0
$833$	−25.5532	+	14.7532i	−0.885366	+	0.511167i
$834$		0			0
$835$	−19.9534			−0.690514
$836$		0			0
$837$	0.103775	+	0.0304568i	0.00358700	+	0.00105274i
$838$		0			0
$839$	3.88992	−	14.5174i	0.134295	−	0.501195i	−0.865705	−	0.500555i	$-0.833129\pi$
$839$	3.88992	−	14.5174i	0.134295	−	0.501195i	1.00000		0.000640633i	$-0.000203920\pi$
$840$		0			0
$841$	28.9114			0.996944
$842$		0			0
$843$	−2.90513	−	11.1947i	−0.100058	−	0.385566i
$844$		0			0
$845$	15.2583	+	22.6684i	0.524902	+	0.779816i
$846$		0			0
$847$	12.0470	−	44.9601i	0.413940	−	1.54485i
$848$		0			0
$849$	−18.6848	+	31.7811i	−0.641259	+	1.09073i
$850$		0			0
$851$	29.0157	−	29.0157i	0.994645	−	0.994645i
$852$		0			0
$853$	4.44839	+	16.6016i	0.152310	+	0.568429i	0.999321	+	0.0368528i	$0.0117333\pi$
$853$	4.44839	+	16.6016i	0.152310	+	0.568429i	−0.847011	+	0.531576i	$0.821600\pi$
$854$		0			0
$855$	−41.7521	−	0.658980i	−1.42789	−	0.0225367i
$856$		0			0
$857$	−8.80280	+	15.2469i	−0.300698	+	0.520824i	−0.976294	−	0.216448i	$-0.930553\pi$
$857$	−8.80280	+	15.2469i	−0.300698	+	0.520824i	0.675596	+	0.737272i	$0.263886\pi$
$858$		0			0
$859$	−17.4097	−	30.1545i	−0.594012	−	1.02886i	−0.993685	−	0.112202i	$-0.964210\pi$
$859$	−17.4097	−	30.1545i	−0.594012	−	1.02886i	0.399673	−	0.916658i	$-0.369124\pi$
$860$		0			0
$861$	60.9428	+	0.480905i	2.07693	+	0.0163892i
$862$		0			0
$863$	3.98704	−	3.98704i	0.135720	−	0.135720i	−0.635983	−	0.771703i	$-0.719405\pi$
$863$	3.98704	−	3.98704i	0.135720	−	0.135720i	0.771703	+	0.635983i	$0.219405\pi$
$864$		0			0
$865$	8.82600	+	32.9391i	0.300093	+	1.11996i
$866$		0			0
$867$	17.9771	−	10.1908i	0.610534	−	0.346097i
$868$		0			0
$869$	−37.8520	+	10.1424i	−1.28404	+	0.344058i
$870$		0			0
$871$	−18.5461	−	1.79479i	−0.628410	−	0.0608140i
$872$		0			0
$873$	−1.82238	+	6.39571i	−0.0616782	+	0.216462i
$874$		0			0
$875$	26.3040	+	45.5598i	0.889237	+	1.54020i
$876$		0			0
$877$	−9.98693	−	9.98693i	−0.337235	−	0.337235i	0.518091	−	0.855326i	$-0.326643\pi$
$877$	−9.98693	−	9.98693i	−0.337235	−	0.337235i	−0.855326	+	0.518091i	$0.826643\pi$
$878$		0			0
$879$	3.41525	−	3.36177i	0.115194	−	0.113390i
$880$		0			0
$881$	23.5198	+	40.7374i	0.792401	+	1.37248i	0.924477	+	0.381239i	$0.124502\pi$
$881$	23.5198	+	40.7374i	0.792401	+	1.37248i	−0.132076	+	0.991240i	$0.542164\pi$
$882$		0			0
$883$			3.99445i			0.134424i	0.997739	+	0.0672120i	$0.0214104\pi$
$883$			3.99445i			0.134424i	−0.997739	+	0.0672120i	$0.978590\pi$
$884$		0			0
$885$	−11.5029	+	6.52074i	−0.386667	+	0.219192i
$886$		0			0
$887$	24.0314	+	13.8746i	0.806897	+	0.465862i	0.845877	−	0.533378i	$-0.179078\pi$
$887$	24.0314	+	13.8746i	0.806897	+	0.465862i	−0.0389804	+	0.999240i	$0.512411\pi$
$888$		0			0
$889$	71.7335	−	19.2209i	2.40586	−	0.644649i
$890$		0			0
$891$	−28.4832	−	30.3405i	−0.954224	−	1.01645i
$892$		0			0
$893$	−63.9481	+	36.9205i	−2.13994	+	1.23550i
$894$		0			0
$895$	0.561866	+	2.09691i	0.0187811	+	0.0700920i
$896$		0			0
$897$	26.3284	+	32.4892i	0.879078	+	1.08478i
$898$		0			0
$899$	0.00598551	+	0.00160381i	0.000199628	+	5.34902e-5i
$900$		0			0
$901$	8.43679	+	4.87098i	0.281070	+	0.162276i
$902$		0			0
$903$	1.53995	+	5.93407i	0.0512462	+	0.197474i
$904$		0			0
$905$	−1.26418	+	4.71799i	−0.0420228	+	0.156831i
$906$		0			0
$907$			2.59570i			0.0861887i	0.999071	+	0.0430944i	$0.0137216\pi$
$907$			2.59570i			0.0861887i	−0.999071	+	0.0430944i	$0.986278\pi$
$908$		0			0
$909$	−13.1269	−	0.207183i	−0.435391	−	0.00687184i
$910$		0			0
$911$	−4.04154	−	2.33338i	−0.133902	−	0.0773084i	0.431553	−	0.902088i	$-0.357966\pi$
$911$	−4.04154	−	2.33338i	−0.133902	−	0.0773084i	−0.565455	+	0.824779i	$0.691299\pi$
$912$		0			0
$913$	−14.0727	+	24.3747i	−0.465739	+	0.806684i
$914$		0			0
$915$	51.7944	−	13.4411i	1.71227	−	0.444350i
$916$		0			0
$917$	5.78362	−	1.54972i	0.190992	−	0.0511762i
$918$		0			0
$919$	−8.81604	+	15.2698i	−0.290814	+	0.503705i	−0.974002	−	0.226537i	$-0.927259\pi$
$919$	−8.81604	+	15.2698i	−0.290814	+	0.503705i	0.683188	+	0.730242i	$0.260593\pi$
$920$		0			0
$921$	30.9732	+	31.4660i	1.02060	+	1.03684i
$922$		0			0
$923$	3.06202	−	31.6409i	0.100788	−	1.04147i
$924$		0			0
$925$	−2.52112	−	2.52112i	−0.0828940	−	0.0828940i
$926$		0			0
$927$	−10.1942	+	5.67303i	−0.334821	+	0.186327i
$928$		0			0
$929$	46.1467	+	12.3650i	1.51402	+	0.405681i	0.917769	−	0.397114i	$-0.129988\pi$
$929$	46.1467	+	12.3650i	1.51402	+	0.405681i	0.596254	+	0.802796i	$0.296655\pi$
$930$		0			0
$931$	83.8256	+	22.4610i	2.74727	+	0.736129i
$932$		0			0
$933$	9.99320	−	5.66491i	0.327163	−	0.185461i
$934$		0			0
$935$	21.8829			0.715648
$936$		0			0
$937$	5.58504			0.182455			0.0912277	−	0.995830i	$-0.470921\pi$
$937$	5.58504			0.182455			0.0912277	+	0.995830i	$0.470921\pi$
$938$		0			0
$939$	−12.5835	−	7.39809i	−0.410647	−	0.241428i
$940$		0			0
$941$	−27.8948	−	7.47440i	−0.909346	−	0.243659i	−0.226320	−	0.974053i	$-0.572670\pi$
$941$	−27.8948	−	7.47440i	−0.909346	−	0.243659i	−0.683026	+	0.730394i	$0.739336\pi$
$942$		0			0
$943$	−50.7569	−	13.6003i	−1.65287	−	0.442885i
$944$		0			0
$945$	25.4836	−	41.8204i	0.828981	−	1.36042i
$946$		0			0
$947$	−13.5058	−	13.5058i	−0.438880	−	0.438880i	0.452755	−	0.891635i	$-0.350441\pi$
$947$	−13.5058	−	13.5058i	−0.438880	−	0.438880i	−0.891635	+	0.452755i	$0.850441\pi$
$948$		0			0
$949$	−11.3871	+	5.18281i	−0.369642	+	0.168241i
$950$		0			0
$951$	−30.8435	+	8.00417i	−1.00017	+	0.259553i
$952$		0			0
$953$	16.1381	−	27.9520i	0.522764	−	0.905455i	−0.476885	−	0.878966i	$-0.658234\pi$
$953$	16.1381	−	27.9520i	0.522764	−	0.905455i	0.999649	−	0.0264887i	$-0.00843260\pi$
$954$		0			0
$955$	48.0778	−	12.8824i	1.55576	−	0.416865i
$956$		0			0
$957$	−1.67266	−	1.69926i	−0.0540693	−	0.0549294i
$958$		0			0
$959$	−42.2709	+	73.2154i	−1.36500	+	2.36425i
$960$		0			0
$961$	26.8464	+	15.4998i	0.866013	+	0.499993i
$962$		0			0
$963$	8.55345	−	4.75996i	0.275631	−	0.153388i
$964$		0			0
$965$		−	2.16769i		−	0.0697804i
$966$		0			0
$967$	−8.60269	+	32.1057i	−0.276644	+	1.03245i	0.678088	+	0.734981i	$0.262809\pi$
$967$	−8.60269	+	32.1057i	−0.276644	+	1.03245i	−0.954732	+	0.297468i	$0.903858\pi$
$968$		0			0
$969$	24.8905	+	6.88035i	0.799599	+	0.221029i
$970$		0			0
$971$	42.4814	+	24.5267i	1.36329	+	0.787098i	0.990061	−	0.140640i	$-0.0449159\pi$
$971$	42.4814	+	24.5267i	1.36329	+	0.787098i	0.373233	+	0.927738i	$0.378249\pi$
$972$		0			0
$973$	11.5297	+	3.08937i	0.369625	+	0.0990407i
$974$		0			0
$975$	2.82293	−	2.28763i	0.0904062	−	0.0732626i
$976$		0			0
$977$	10.7433	+	40.0945i	0.343708	+	1.28274i	0.894114	+	0.447839i	$0.147806\pi$
$977$	10.7433	+	40.0945i	0.343708	+	1.28274i	−0.550407	+	0.834897i	$0.685527\pi$
$978$		0			0
$979$	−62.6189	+	36.1530i	−2.00131	+	1.15546i
$980$		0			0
$981$	1.17539	+	1.21309i	0.0375275	+	0.0387311i
$982$		0			0
$983$	−11.2829	+	3.02323i	−0.359867	+	0.0964261i	−0.434222	−	0.900806i	$-0.642977\pi$
$983$	−11.2829	+	3.02323i	−0.359867	+	0.0964261i	0.0743552	+	0.997232i	$0.476310\pi$
$984$		0			0
$985$	−29.2411	−	16.8824i	−0.931700	−	0.537917i
$986$		0			0
$987$	0.683353	−	86.5981i	0.0217514	−	2.75645i
$988$		0			0
$989$		−	5.28591i		−	0.168082i
$990$		0			0
$991$	2.87270	+	4.97567i	0.0912545	+	0.158057i	0.908039	−	0.418885i	$-0.137579\pi$
$991$	2.87270	+	4.97567i	0.0912545	+	0.158057i	−0.816785	+	0.576942i	$0.804246\pi$
$992$		0			0
$993$	10.6240	+	2.93674i	0.337143	+	0.0931948i
$994$		0			0
$995$	30.2359	+	30.2359i	0.958542	+	0.958542i
$996$		0			0
$997$	12.4545	+	21.5719i	0.394440	+	0.683189i	0.993029	−	0.117866i	$-0.0376054\pi$
$997$	12.4545	+	21.5719i	0.394440	+	0.683189i	−0.598590	+	0.801056i	$0.704272\pi$
$998$		0			0
$999$	−8.96704	+	30.5533i	−0.283705	+	0.966664i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	468.2.bw.a.245.8	yes	56
3.2	odd	2		1404.2.bz.a.89.3		56
9.4	even	3		1404.2.cc.a.557.3		56
9.5	odd	6		468.2.bz.a.401.2	yes	56
13.6	odd	12		468.2.bz.a.461.2	yes	56
39.32	even	12		1404.2.cc.a.305.3		56
117.32	even	12	inner	468.2.bw.a.149.8	✓	56
117.58	odd	12		1404.2.bz.a.773.3		56

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
468.2.bw.a.149.8	✓	56	117.32	even	12	inner
468.2.bw.a.245.8	yes	56	1.1	even	1	trivial
468.2.bz.a.401.2	yes	56	9.5	odd	6
468.2.bz.a.461.2	yes	56	13.6	odd	12
1404.2.bz.a.89.3		56	3.2	odd	2
1404.2.bz.a.773.3		56	117.58	odd	12
1404.2.cc.a.305.3		56	39.32	even	12
1404.2.cc.a.557.3		56	9.4	even	3

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

Level:	\( N \)	\(=\)	\( 468 = 2^{2} \cdot 3^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	468.bw (of order \(12\), degree \(4\), minimal)

\(f(q)\)	\(=\)	\(q+(0.0136673 - 1.73200i) q^{3} +(2.03032 + 0.544024i) q^{5} +(-4.33109 - 1.16051i) q^{7} +(-2.99963 - 0.0473436i) q^{9} +(-3.26962 - 3.26962i) q^{11} +(-3.37661 - 1.26431i) q^{13} +(0.969996 - 3.50908i) q^{15} +(-1.12575 + 1.94986i) q^{17} +(6.39637 - 1.71390i) q^{19} +(-2.06920 + 7.48558i) q^{21} +(3.34811 - 5.79909i) q^{23} +(-0.503873 - 0.290911i) q^{25} +(-0.122996 + 5.19470i) q^{27} +0.297717i q^{29} +(0.00538704 - 0.0201047i) q^{31} +(-5.70765 + 5.61828i) q^{33} +(-8.16218 - 4.71244i) q^{35} +(5.91919 + 1.58604i) q^{37} +(-2.23594 + 5.83100i) q^{39} +(-2.03104 - 7.57993i) q^{41} +(0.683630 - 0.394694i) q^{43} +(-6.06446 - 1.72799i) q^{45} +(-10.7709 + 2.88605i) q^{47} +(11.3494 + 6.55259i) q^{49} +(3.36176 + 1.97645i) q^{51} -4.32687i q^{53} +(-4.85963 - 8.41713i) q^{55} +(-2.88105 - 11.1019i) q^{57} +(-2.56815 - 2.56815i) q^{59} +(7.34891 + 12.7287i) q^{61} +(12.9367 + 3.68616i) q^{63} +(-6.16780 - 4.40393i) q^{65} +(4.99170 - 1.33752i) q^{67} +(-9.99825 - 5.87817i) q^{69} +(2.28190 + 8.51617i) q^{71} +(2.45364 - 2.45364i) q^{73} +(-0.510744 + 0.868731i) q^{75} +(10.3666 + 17.9554i) q^{77} +(4.23744 - 7.33946i) q^{79} +(8.99552 + 0.284026i) q^{81} +(-1.57541 - 5.87950i) q^{83} +(-3.34641 + 3.34641i) q^{85} +(0.515645 + 0.00406899i) q^{87} +(4.04724 - 15.1045i) q^{89} +(13.1572 + 9.39447i) q^{91} +(-0.0347477 - 0.00960512i) q^{93} +13.9191 q^{95} +(0.573740 - 2.14123i) q^{97} +(9.65283 + 9.96242i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 56 q - 2 q^{7} + 4 q^{19} - 24 q^{21} + 8 q^{31} + 12 q^{33} - 66 q^{35} + 2 q^{37} + 24 q^{41} + 12 q^{43} - 30 q^{47} + 30 q^{57} + 84 q^{63} + 6 q^{65} + 28 q^{67} - 36 q^{69} + 24 q^{71} - 14 q^{73}+ \cdots - 42 q^{99}+O(q^{100}) \) 56 * q - 2 * q^7 + 4 * q^19 - 24 * q^21 + 8 * q^31 + 12 * q^33 - 66 * q^35 + 2 * q^37 + 24 * q^41 + 12 * q^43 - 30 * q^47 + 30 * q^57 + 84 * q^63 + 6 * q^65 + 28 * q^67 - 36 * q^69 + 24 * q^71 - 14 * q^73 - 24 * q^77 + 12 * q^79 - 12 * q^81 - 60 * q^83 + 36 * q^85 + 4 * q^91 - 54 * q^93 + 26 * q^97 - 42 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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