LMFDB - Embedded newform 420.2.bv.c.233.10

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [420,2,Mod(53,420)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(420, base_ring=CyclotomicField(12)) chi = DirichletCharacter(H, H._module([0, 6, 9, 8])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("420.53"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 420 = 2^{2} \cdot 3 \cdot 5 \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	420.bv (of order $12$, degree $4$, minimal)

Newform invariants

comment:select newform

sage:traces = [48,0,-2] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$3.35371688489$
Analytic rank:	$0$
Dimension:	$48$
Relative dimension:	$12$ over $\Q(\zeta_{12})$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{12}]$

Embedding invariants

Embedding label			233.10
Character	$\chi$	$=$	420.233
Dual form			420.2.bv.c.137.10

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+(1.24755 - 1.20150i) q^{3} +(0.274984 + 2.21910i) q^{5} +(0.250938 - 2.63382i) q^{7} +(0.112777 - 2.99788i) q^{9} +(1.95574 - 1.12915i) q^{11} +(-1.60840 - 1.60840i) q^{13} +(3.00931 + 2.43804i) q^{15} +(1.10647 - 4.12942i) q^{17} +(6.82987 + 3.94323i) q^{19} +(-2.85149 - 3.58734i) q^{21} +(2.31365 + 8.63467i) q^{23} +(-4.84877 + 1.22043i) q^{25} +(-3.46127 - 3.87552i) q^{27} +3.34184 q^{29} +(-2.63173 - 4.55830i) q^{31} +(1.08321 - 3.75850i) q^{33} +(5.91371 - 0.167404i) q^{35} +(-0.379022 - 1.41453i) q^{37} +(-3.93905 - 0.0740656i) q^{39} +5.07849i q^{41} +(-4.16728 - 4.16728i) q^{43} +(6.68359 - 0.574105i) q^{45} +(-12.1810 + 3.26390i) q^{47} +(-6.87406 - 1.32186i) q^{49} +(-3.58112 - 6.48110i) q^{51} +(-4.15930 - 1.11448i) q^{53} +(3.04348 + 4.02947i) q^{55} +(13.2584 - 3.28673i) q^{57} +(0.733066 + 1.26971i) q^{59} +(-4.02291 + 6.96788i) q^{61} +(-7.86759 - 1.04932i) q^{63} +(3.12690 - 4.01147i) q^{65} +(10.8869 + 2.91714i) q^{67} +(13.2610 + 7.99235i) q^{69} +9.99909i q^{71} +(0.353306 - 1.31856i) q^{73} +(-4.58274 + 7.34837i) q^{75} +(-2.48320 - 5.43442i) q^{77} +(5.66226 + 3.26911i) q^{79} +(-8.97456 - 0.676187i) q^{81} +(-7.33432 + 7.33432i) q^{83} +(9.46783 + 1.31985i) q^{85} +(4.16912 - 4.01523i) q^{87} +(-4.63460 + 8.02736i) q^{89} +(-4.63984 + 3.83262i) q^{91} +(-8.76004 - 2.52468i) q^{93} +(-6.87229 + 16.2405i) q^{95} +(7.06487 - 7.06487i) q^{97} +(-3.16448 - 5.99041i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 48 q - 2 q^{3} - 8 q^{7} + 32 q^{13} - 16 q^{21} + 16 q^{25} - 32 q^{27} - 64 q^{31} + 34 q^{33} - 40 q^{37} - 24 q^{43} + 30 q^{45} + 36 q^{51} + 92 q^{57} - 18 q^{63} + 28 q^{67} - 8 q^{73} - 38 q^{75}+ \cdots + 56 q^{97}+O(q^{100}) $ 48 * q - 2 * q^3 - 8 * q^7 + 32 * q^13 - 16 * q^21 + 16 * q^25 - 32 * q^27 - 64 * q^31 + 34 * q^33 - 40 * q^37 - 24 * q^43 + 30 * q^45 + 36 * q^51 + 92 * q^57 - 18 * q^63 + 28 * q^67 - 8 * q^73 - 38 * q^75 + 20 * q^81 - 56 * q^85 - 24 * q^87 - 24 * q^91 + 6 * q^93 + 56 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/420\mathbb{Z}\right)^\times$.

$n$	$211$	$241$	$281$	$337$
$\chi(n)$	$1$	$e\left(\frac{1}{3}\right)$	$-1$	$e\left(\frac{3}{4}\right)$

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			0
$2$		0			0
$3$	1.24755	−	1.20150i	0.720275	−	0.693689i
$3$	1.24755	−	1.20150i	0.720275	−	0.693689i
$4$		0			0
$5$	0.274984	+	2.21910i	0.122977	+	0.992410i
$5$	0.274984	+	2.21910i	0.122977	+	0.992410i
$6$		0			0
$7$	0.250938	−	2.63382i	0.0948458	−	0.995492i
$7$	0.250938	−	2.63382i	0.0948458	−	0.995492i
$8$		0			0
$9$	0.112777	−	2.99788i	0.0375925	−	0.999293i
$10$		0			0
$11$	1.95574	−	1.12915i	0.589677	−	0.340450i	−0.175293	−	0.984516i	$-0.556087\pi$
$11$	1.95574	−	1.12915i	0.589677	−	0.340450i	0.764970	+	0.644066i	$0.222754\pi$
$12$		0			0
$13$	−1.60840	−	1.60840i	−0.446089	−	0.446089i	0.447963	−	0.894052i	$-0.352150\pi$
$13$	−1.60840	−	1.60840i	−0.446089	−	0.446089i	−0.894052	+	0.447963i	$0.852150\pi$
$14$		0			0
$15$	3.00931	+	2.43804i	0.777000	+	0.629500i
$16$		0			0
$17$	1.10647	−	4.12942i	0.268359	−	1.00153i	−0.691803	−	0.722087i	$-0.743183\pi$
$17$	1.10647	−	4.12942i	0.268359	−	1.00153i	0.960162	−	0.279444i	$-0.0901501\pi$
$18$		0			0
$19$	6.82987	+	3.94323i	1.56688	+	0.904639i	0.996530	+	0.0832371i	$0.0265259\pi$
$19$	6.82987	+	3.94323i	1.56688	+	0.904639i	0.570350	+	0.821402i	$0.306807\pi$
$20$		0			0
$21$	−2.85149	−	3.58734i	−0.622246	−	0.782822i
$22$		0			0
$23$	2.31365	+	8.63467i	0.482430	+	1.80045i	0.591364	+	0.806405i	$0.298590\pi$
$23$	2.31365	+	8.63467i	0.482430	+	1.80045i	−0.108933	+	0.994049i	$0.534744\pi$
$24$		0			0
$25$	−4.84877	+	1.22043i	−0.969753	+	0.244086i
$26$		0			0
$27$	−3.46127	−	3.87552i	−0.666121	−	0.745843i
$28$		0			0
$29$	3.34184			0.620563			0.310282	−	0.950645i	$-0.399577\pi$
$29$	3.34184			0.620563			0.310282	+	0.950645i	$0.399577\pi$
$30$		0			0
$31$	−2.63173	−	4.55830i	−0.472673	−	0.818694i	0.526838	−	0.849966i	$-0.323378\pi$
$31$	−2.63173	−	4.55830i	−0.472673	−	0.818694i	−0.999511	+	0.0312718i	$0.990044\pi$
$32$		0			0
$33$	1.08321	−	3.75850i	0.188563	−	0.654270i
$34$		0			0
$35$	5.91371	−	0.167404i	0.999600	−	0.0282964i
$36$		0			0
$37$	−0.379022	−	1.41453i	−0.0623108	−	0.232547i	0.927747	−	0.373211i	$-0.121743\pi$
$37$	−0.379022	−	1.41453i	−0.0623108	−	0.232547i	−0.990057	+	0.140663i	$0.955076\pi$
$38$		0			0
$39$	−3.93905	−	0.0740656i	−0.630753	−	0.0118600i
$40$		0			0
$41$			5.07849i			0.793127i	0.918007	+	0.396564i	$0.129797\pi$
$41$			5.07849i			0.793127i	−0.918007	+	0.396564i	$0.870203\pi$
$42$		0			0
$43$	−4.16728	−	4.16728i	−0.635504	−	0.635504i	0.313939	−	0.949443i	$-0.398351\pi$
$43$	−4.16728	−	4.16728i	−0.635504	−	0.635504i	−0.949443	+	0.313939i	$0.898351\pi$
$44$		0			0
$45$	6.68359	−	0.574105i	0.996331	−	0.0855826i
$46$		0			0
$47$	−12.1810	+	3.26390i	−1.77679	+	0.476089i	−0.989992	−	0.141126i	$-0.954928\pi$
$47$	−12.1810	+	3.26390i	−1.77679	+	0.476089i	−0.786795	+	0.617214i	$0.788261\pi$
$48$		0			0
$49$	−6.87406	−	1.32186i	−0.982009	−	0.188836i
$50$		0			0
$51$	−3.58112	−	6.48110i	−0.501458	−	0.907535i
$52$		0			0
$53$	−4.15930	−	1.11448i	−0.571324	−	0.153086i	−0.0384204	−	0.999262i	$-0.512233\pi$
$53$	−4.15930	−	1.11448i	−0.571324	−	0.153086i	−0.532904	+	0.846176i	$0.678899\pi$
$54$		0			0
$55$	3.04348	+	4.02947i	0.410383	+	0.543334i
$56$		0			0
$57$	13.2584	−	3.28673i	1.75612	−	0.435338i
$58$		0			0
$59$	0.733066	+	1.26971i	0.0954370	+	0.165302i	0.909791	−	0.415067i	$-0.136242\pi$
$59$	0.733066	+	1.26971i	0.0954370	+	0.165302i	−0.814354	+	0.580369i	$0.802908\pi$
$60$		0			0
$61$	−4.02291	+	6.96788i	−0.515081	+	0.892146i	0.484766	+	0.874644i	$0.338905\pi$
$61$	−4.02291	+	6.96788i	−0.515081	+	0.892146i	−0.999847	+	0.0175021i	$0.994429\pi$
$62$		0			0
$63$	−7.86759	−	1.04932i	−0.991223	−	0.132202i
$64$		0			0
$65$	3.12690	−	4.01147i	0.387844	−	0.497561i
$66$		0			0
$67$	10.8869	+	2.91714i	1.33005	+	0.356386i	0.852734	−	0.522346i	$-0.174943\pi$
$67$	10.8869	+	2.91714i	1.33005	+	0.356386i	0.477316	+	0.878732i	$0.341610\pi$
$68$		0			0
$69$	13.2610	+	7.99235i	1.59644	+	0.962166i
$70$		0			0
$71$			9.99909i			1.18667i	0.804954	+	0.593337i	$0.202190\pi$
$71$			9.99909i			1.18667i	−0.804954	+	0.593337i	$0.797810\pi$
$72$		0			0
$73$	0.353306	−	1.31856i	0.0413514	−	0.154325i	−0.942163	−	0.335155i	$-0.891211\pi$
$73$	0.353306	−	1.31856i	0.0413514	−	0.154325i	0.983515	+	0.180829i	$0.0578781\pi$
$74$		0			0
$75$	−4.58274	+	7.34837i	−0.529169	+	0.848516i
$76$		0			0
$77$	−2.48320	−	5.43442i	−0.282987	−	0.619309i
$78$		0			0
$79$	5.66226	+	3.26911i	0.637054	+	0.367804i	0.783479	−	0.621418i	$-0.213443\pi$
$79$	5.66226	+	3.26911i	0.637054	+	0.367804i	−0.146425	+	0.989222i	$0.546777\pi$
$80$		0			0
$81$	−8.97456	−	0.676187i	−0.997174	−	0.0751318i
$82$		0			0
$83$	−7.33432	+	7.33432i	−0.805046	+	0.805046i	−0.983879	−	0.178834i	$-0.942768\pi$
$83$	−7.33432	+	7.33432i	−0.805046	+	0.805046i	0.178834	+	0.983879i	$0.442768\pi$
$84$		0			0
$85$	9.46783	+	1.31985i	1.02693	+	0.143158i
$86$		0			0
$87$	4.16912	−	4.01523i	0.446976	−	0.430478i
$88$		0			0
$89$	−4.63460	+	8.02736i	−0.491267	+	0.850899i	−0.999949	−	0.0100553i	$-0.996799\pi$
$89$	−4.63460	+	8.02736i	−0.491267	+	0.850899i	0.508683	+	0.860954i	$0.330133\pi$
$90$		0			0
$91$	−4.63984	+	3.83262i	−0.486387	+	0.401768i
$92$		0			0
$93$	−8.76004	−	2.52468i	−0.908374	−	0.261797i
$94$		0			0
$95$	−6.87229	+	16.2405i	−0.705082	+	1.66624i
$96$		0			0
$97$	7.06487	−	7.06487i	0.717328	−	0.717328i	−0.250729	−	0.968057i	$-0.580670\pi$
$97$	7.06487	−	7.06487i	0.717328	−	0.717328i	0.968057	+	0.250729i	$0.0806703\pi$
$98$		0			0
$99$	−3.16448	−	5.99041i	−0.318042	−	0.602059i
$100$		0			0
$101$	−5.12029	+	2.95620i	−0.509488	+	0.294153i	−0.732623	−	0.680635i	$-0.761704\pi$
$101$	−5.12029	+	2.95620i	−0.509488	+	0.294153i	0.223135	+	0.974787i	$0.428371\pi$
$102$		0			0
$103$	−4.21034	+	1.12816i	−0.414857	+	0.111161i	−0.460209	−	0.887810i	$-0.652226\pi$
$103$	−4.21034	+	1.12816i	−0.414857	+	0.111161i	0.0453523	+	0.998971i	$0.485559\pi$
$104$		0			0
$105$	7.17653	−	7.31419i	0.700358	−	0.713792i
$106$		0			0
$107$	8.47050	−	2.26966i	0.818874	−	0.219417i	0.175020	−	0.984565i	$-0.444001\pi$
$107$	8.47050	−	2.26966i	0.818874	−	0.219417i	0.643854	+	0.765148i	$0.277334\pi$
$108$		0			0
$109$	−2.06149	+	1.19020i	−0.197455	+	0.114000i	−0.595468	−	0.803379i	$-0.703033\pi$
$109$	−2.06149	+	1.19020i	−0.197455	+	0.114000i	0.398013	+	0.917380i	$0.369700\pi$
$110$		0			0
$111$	−2.17241	−	1.30930i	−0.206196	−	0.124274i
$112$		0			0
$113$	3.88087	−	3.88087i	0.365082	−	0.365082i	−0.500598	−	0.865680i	$-0.666887\pi$
$113$	3.88087	−	3.88087i	0.365082	−	0.365082i	0.865680	+	0.500598i	$0.166887\pi$
$114$		0			0
$115$	−18.5249	+	7.50862i	−1.72746	+	0.700182i
$116$		0			0
$117$	−5.00317	+	4.64039i	−0.462543	+	0.429004i
$118$		0			0
$119$	−10.5985	−	3.95049i	−0.971563	−	0.362141i
$120$		0			0
$121$	−2.95006	+	5.10965i	−0.268187	+	0.464514i
$122$		0			0
$123$	6.10183	+	6.33569i	0.550183	+	0.571270i
$124$		0			0
$125$	−4.04159	−	10.4243i	−0.361491	−	0.932376i
$126$		0			0
$127$	2.33279	−	2.33279i	0.207001	−	0.207001i	−0.595990	−	0.802992i	$-0.703240\pi$
$127$	2.33279	−	2.33279i	0.207001	−	0.207001i	0.802992	+	0.595990i	$0.203240\pi$
$128$		0			0
$129$	−10.2059	−	0.191900i	−0.898579	−	0.0168959i
$130$		0			0
$131$	−14.7871	−	8.53732i	−1.29195	−	0.745909i	−0.312952	−	0.949769i	$-0.601318\pi$
$131$	−14.7871	−	8.53732i	−1.29195	−	0.745909i	−0.979000	+	0.203860i	$0.934651\pi$
$132$		0			0
$133$	12.0996	−	16.9992i	1.04917	−	1.47402i
$134$		0			0
$135$	7.64835	−	8.74659i	0.658265	−	0.752786i
$136$		0			0
$137$	1.55344	−	5.79751i	0.132719	−	0.495315i	−0.867278	−	0.497825i	$-0.834132\pi$
$137$	1.55344	−	5.79751i	0.132719	−	0.495315i	0.999997	+	0.00250999i	$0.000798955\pi$
$138$		0			0
$139$			1.40112i			0.118841i	0.998233	+	0.0594207i	$0.0189253\pi$
$139$			1.40112i			0.118841i	−0.998233	+	0.0594207i	$0.981075\pi$
$140$		0			0
$141$	−11.2749	+	18.7074i	−0.949518	+	1.57545i
$142$		0			0
$143$	−4.96172	−	1.32949i	−0.414920	−	0.111177i
$144$		0			0
$145$	0.918952	+	7.41585i	0.0763148	+	0.615853i
$146$		0			0
$147$	−10.1640	+	6.61012i	−0.838310	+	0.545194i
$148$		0			0
$149$	0.789937	−	1.36821i	0.0647142	−	0.112088i	−0.831853	−	0.554996i	$-0.812720\pi$
$149$	0.789937	−	1.36821i	0.0647142	−	0.112088i	0.896567	+	0.442908i	$0.146053\pi$
$150$		0			0
$151$	7.36338	+	12.7538i	0.599224	+	1.03789i	0.992936	+	0.118652i	$0.0378573\pi$
$151$	7.36338	+	12.7538i	0.599224	+	1.03789i	−0.393712	+	0.919234i	$0.628809\pi$
$152$		0			0
$153$	−12.2547	−	3.78278i	−0.990734	−	0.305820i
$154$		0			0
$155$	9.39161	−	7.09353i	0.754352	−	0.569766i
$156$		0			0
$157$	−9.33487	−	2.50127i	−0.745003	−	0.199623i	−0.133703	−	0.991021i	$-0.542687\pi$
$157$	−9.33487	−	2.50127i	−0.745003	−	0.199623i	−0.611301	+	0.791398i	$0.709353\pi$
$158$		0			0
$159$	−6.52801	+	3.60704i	−0.517705	+	0.286057i
$160$		0			0
$161$	23.3228	−	3.92699i	1.83809	−	0.309490i
$162$		0			0
$163$	5.21038	−	1.39612i	0.408108	−	0.109352i	−0.0489238	−	0.998803i	$-0.515579\pi$
$163$	5.21038	−	1.39612i	0.408108	−	0.109352i	0.457032	+	0.889450i	$0.348912\pi$
$164$		0			0
$165$	8.63833	+	1.37023i	0.672493	+	0.106672i
$166$		0			0
$167$	12.9779	+	12.9779i	1.00426	+	1.00426i	0.999991	+	0.00426571i	$0.00135782\pi$
$167$	12.9779	+	12.9779i	1.00426	+	1.00426i	0.00426571	+	0.999991i	$0.498642\pi$
$168$		0			0
$169$		−	7.82612i		−	0.602010i
$170$		0			0
$171$	12.5916	−	20.0304i	0.962902	−	1.53176i
$172$		0			0
$173$	−1.65208	−	6.16566i	−0.125606	−	0.468766i	0.874255	−	0.485467i	$-0.161350\pi$
$173$	−1.65208	−	6.16566i	−0.125606	−	0.468766i	−0.999861	+	0.0167007i	$0.994684\pi$
$174$		0			0
$175$	1.99766	+	13.0771i	0.151009	+	0.988532i
$176$		0			0
$177$	2.44010	+	0.703246i	0.183409	+	0.0528592i
$178$		0			0
$179$	5.16769	+	8.95070i	0.386251	+	0.669006i	0.991942	−	0.126694i	$-0.0404365\pi$
$179$	5.16769	+	8.95070i	0.386251	+	0.669006i	−0.605691	+	0.795700i	$0.707103\pi$
$180$		0			0
$181$	2.42214			0.180037			0.0900183	−	0.995940i	$-0.471307\pi$
$181$	2.42214			0.180037			0.0900183	+	0.995940i	$0.471307\pi$
$182$		0			0
$183$	3.35314	+	13.5263i	0.247872	+	0.999896i
$184$		0			0
$185$	3.03475	−	1.23006i	0.223119	−	0.0904357i
$186$		0			0
$187$	−2.49874	−	9.32543i	−0.182726	−	0.681943i
$188$		0			0
$189$	−11.0760	+	8.14385i	−0.805660	+	0.592378i
$190$		0			0
$191$	15.4811	+	8.93801i	1.12017	+	0.646732i	0.941445	−	0.337167i	$-0.109469\pi$
$191$	15.4811	+	8.93801i	1.12017	+	0.646732i	0.178727	+	0.983899i	$0.442802\pi$
$192$		0			0
$193$	4.18295	−	15.6110i	0.301095	−	1.12370i	−0.635160	−	0.772381i	$-0.719066\pi$
$193$	4.18295	−	15.6110i	0.301095	−	1.12370i	0.936255	−	0.351321i	$-0.114268\pi$
$194$		0			0
$195$	−0.918818	−	8.76150i	−0.0657980	−	0.627424i
$196$		0			0
$197$	−4.87326	−	4.87326i	−0.347205	−	0.347205i	0.511862	−	0.859068i	$-0.328956\pi$
$197$	−4.87326	−	4.87326i	−0.347205	−	0.347205i	−0.859068	+	0.511862i	$0.828956\pi$
$198$		0			0
$199$	10.9994	−	6.35052i	0.779729	−	0.450176i	−0.0566055	−	0.998397i	$-0.518028\pi$
$199$	10.9994	−	6.35052i	0.779729	−	0.450176i	0.836334	+	0.548220i	$0.184694\pi$
$200$		0			0
$201$	17.0870	−	9.44139i	1.20522	−	0.665944i
$202$		0			0
$203$	0.838595	−	8.80181i	0.0588578	−	0.617766i
$204$		0			0
$205$	−11.2697	+	1.39650i	−0.787107	+	0.0975361i
$206$		0			0
$207$	26.1466	−	5.96226i	1.81732	−	0.414406i
$208$		0			0
$209$	17.8099			1.23194
$210$		0			0
$211$	−2.65439			−0.182736			−0.0913680	−	0.995817i	$-0.529124\pi$
$211$	−2.65439			−0.182736			−0.0913680	+	0.995817i	$0.529124\pi$
$212$		0			0
$213$	12.0139	+	12.4744i	0.823182	+	0.854731i
$214$		0			0
$215$	8.10165	−	10.3935i	0.552528	−	0.708832i
$216$		0			0
$217$	−12.6662	+	5.78767i	−0.859835	+	0.392893i
$218$		0			0
$219$	−1.14348	−	2.06947i	−0.0772694	−	0.139842i
$220$		0			0
$221$	−8.42139	+	4.86209i	−0.566484	+	0.327060i
$222$		0			0
$223$	−8.69690	−	8.69690i	−0.582387	−	0.582387i	0.353171	−	0.935559i	$-0.385103\pi$
$223$	−8.69690	−	8.69690i	−0.582387	−	0.582387i	−0.935559	+	0.353171i	$0.885103\pi$
$224$		0			0
$225$	3.11188	+	14.6737i	0.207458	+	0.978244i
$226$		0			0
$227$	3.38991	−	12.6513i	0.224996	−	0.839697i	−0.757410	−	0.652940i	$-0.773535\pi$
$227$	3.38991	−	12.6513i	0.224996	−	0.839697i	0.982406	−	0.186757i	$-0.0597978\pi$
$228$		0			0
$229$	4.89414	+	2.82564i	0.323414	+	0.186723i	0.652913	−	0.757433i	$-0.273547\pi$
$229$	4.89414	+	2.82564i	0.323414	+	0.186723i	−0.329499	+	0.944156i	$0.606880\pi$
$230$		0			0
$231$	−9.62740	−	3.79615i	−0.633437	−	0.249768i
$232$		0			0
$233$	−0.106478	−	0.397381i	−0.00697559	−	0.0260333i	0.962350	−	0.271812i	$-0.0876229\pi$
$233$	−0.106478	−	0.397381i	−0.00697559	−	0.0260333i	−0.969326	+	0.245779i	$0.920956\pi$
$234$		0			0
$235$	−10.5925	−	26.1334i	−0.690978	−	1.70475i
$236$		0			0
$237$	10.9918	−	2.72484i	0.713996	−	0.176998i
$238$		0			0
$239$	0.602538			0.0389749			0.0194875	−	0.999810i	$-0.493797\pi$
$239$	0.602538			0.0389749			0.0194875	+	0.999810i	$0.493797\pi$
$240$		0			0
$241$	2.80979	+	4.86670i	0.180994	+	0.313491i	0.942219	−	0.334996i	$-0.108735\pi$
$241$	2.80979	+	4.86670i	0.180994	+	0.313491i	−0.761225	+	0.648488i	$0.775402\pi$
$242$		0			0
$243$	−12.0087	+	9.93939i	−0.770357	+	0.637612i
$244$		0			0
$245$	1.04307	−	15.6177i	0.0666390	−	0.997777i
$246$		0			0
$247$	−4.64287	−	17.3274i	−0.295418	−	1.10252i
$248$		0			0
$249$	−0.337740	+	17.9622i	−0.0214034	+	1.13831i
$250$		0			0
$251$		−	14.7240i		−	0.929370i	−0.885476	−	0.464685i	$-0.846168\pi$
$251$		−	14.7240i		−	0.929370i	0.885476	−	0.464685i	$-0.153832\pi$
$252$		0			0
$253$	14.2747	+	14.2747i	0.897444	+	0.897444i
$254$		0			0
$255$	13.3974	−	9.72906i	0.838979	−	0.609257i
$256$		0			0
$257$	−23.5784	+	6.31782i	−1.47078	+	0.394095i	−0.903199	−	0.429222i	$-0.858788\pi$
$257$	−23.5784	+	6.31782i	−1.47078	+	0.394095i	−0.567582	+	0.823317i	$0.692121\pi$
$258$		0			0
$259$	−3.82073	+	0.643317i	−0.237409	+	0.0399738i
$260$		0			0
$261$	0.376884	−	10.0184i	0.0233285	−	0.620125i
$262$		0			0
$263$	15.3209	+	4.10523i	0.944729	+	0.253139i	0.698124	−	0.715977i	$-0.254018\pi$
$263$	15.3209	+	4.10523i	0.944729	+	0.253139i	0.246605	+	0.969116i	$0.420685\pi$
$264$		0			0
$265$	1.32940	−	9.53636i	0.0816644	−	0.585814i
$266$		0			0
$267$	3.86300	+	15.5830i	0.236412	+	0.953667i
$268$		0			0
$269$	−12.1007	−	20.9591i	−0.737796	−	1.27790i	−0.953486	−	0.301438i	$-0.902533\pi$
$269$	−12.1007	−	20.9591i	−0.737796	−	1.27790i	0.215690	−	0.976462i	$-0.430800\pi$
$270$		0			0
$271$	−2.17200	+	3.76201i	−0.131939	+	0.228526i	−0.924424	−	0.381366i	$-0.875454\pi$
$271$	−2.17200	+	3.76201i	−0.131939	+	0.228526i	0.792485	+	0.609892i	$0.208787\pi$
$272$		0			0
$273$	−1.18354	+	10.3562i	−0.0716308	+	0.626785i
$274$		0			0
$275$	−8.10488	+	7.86181i	−0.488742	+	0.474085i
$276$		0			0
$277$	5.41699	+	1.45148i	0.325475	+	0.0872108i	0.417857	−	0.908513i	$-0.362781\pi$
$277$	5.41699	+	1.45148i	0.325475	+	0.0872108i	−0.0923818	+	0.995724i	$0.529448\pi$
$278$		0			0
$279$	−13.9620	+	7.37555i	−0.835884	+	0.441562i
$280$		0			0
$281$		−	22.7222i		−	1.35550i	−0.735294	−	0.677748i	$-0.762956\pi$
$281$		−	22.7222i		−	1.35550i	0.735294	−	0.677748i	$-0.237044\pi$
$282$		0			0
$283$	1.15676	−	4.31710i	0.0687624	−	0.256625i	−0.922984	−	0.384838i	$-0.874257\pi$
$283$	1.15676	−	4.31710i	0.0687624	−	0.256625i	0.991747	+	0.128213i	$0.0409241\pi$
$284$		0			0
$285$	10.9394	+	28.5179i	0.647996	+	1.68926i
$286$		0			0
$287$	13.3759	+	1.27439i	0.789552	+	0.0752248i
$288$		0			0
$289$	−1.10537	−	0.638183i	−0.0650215	−	0.0375402i
$290$		0			0
$291$	0.325332	−	17.3023i	0.0190713	−	1.01428i
$292$		0			0
$293$	−5.81011	+	5.81011i	−0.339430	+	0.339430i	−0.856153	−	0.516723i	$-0.827152\pi$
$293$	−5.81011	+	5.81011i	−0.339430	+	0.339430i	0.516723	+	0.856153i	$0.327152\pi$
$294$		0			0
$295$	−2.61602	+	1.97589i	−0.152311	+	0.115041i
$296$		0			0
$297$	−11.1454	−	3.67122i	−0.646719	−	0.213026i
$298$		0			0
$299$	10.1667	−	17.6092i	0.587956	−	1.01837i
$300$		0			0
$301$	−12.0216	+	9.93015i	−0.692914	+	0.572364i
$302$		0			0
$303$	−2.83595	+	9.84006i	−0.162921	+	0.565297i
$304$		0			0
$305$	−16.5686	−	7.01116i	−0.948717	−	0.401458i
$306$		0			0
$307$	−14.0973	+	14.0973i	−0.804576	+	0.804576i	−0.983807	−	0.179231i	$-0.942639\pi$
$307$	−14.0973	+	14.0973i	−0.804576	+	0.804576i	0.179231	+	0.983807i	$0.442639\pi$
$308$		0			0
$309$	−3.89714	+	6.46617i	−0.221700	+	0.367848i
$310$		0			0
$311$	−26.4066	+	15.2459i	−1.49738	+	0.864513i	−0.999995	−	0.00301684i	$-0.999040\pi$
$311$	−26.4066	+	15.2459i	−1.49738	+	0.864513i	−0.497385	+	0.867530i	$0.665706\pi$
$312$		0			0
$313$	12.1818	−	3.26411i	0.688558	−	0.184498i	0.102458	−	0.994737i	$-0.467329\pi$
$313$	12.1818	−	3.26411i	0.688558	−	0.184498i	0.586100	+	0.810239i	$0.300663\pi$
$314$		0			0
$315$	0.165078	−	17.7475i	0.00930107	−	0.999957i
$316$		0			0
$317$	4.56328	−	1.22273i	0.256299	−	0.0686752i	−0.128382	−	0.991725i	$-0.540978\pi$
$317$	4.56328	−	1.22273i	0.256299	−	0.0686752i	0.384681	+	0.923050i	$0.374312\pi$
$318$		0			0
$319$	6.53576	−	3.77342i	0.365932	−	0.211271i
$320$		0			0
$321$	7.84039	−	13.0089i	0.437608	−	0.726084i
$322$		0			0
$323$	23.8403	−	23.8403i	1.32651	−	1.32651i
$324$		0			0
$325$	9.76168	+	5.83580i	0.541480	+	0.323712i
$326$		0			0
$327$	−1.14178	+	3.96172i	−0.0631408	+	0.219084i
$328$		0			0
$329$	5.53985	+	32.9017i	0.305422	+	1.81393i
$330$		0			0
$331$	5.49408	−	9.51603i	0.301982	−	0.523048i	−0.674603	−	0.738181i	$-0.735685\pi$
$331$	5.49408	−	9.51603i	0.301982	−	0.523048i	0.976585	+	0.215133i	$0.0690184\pi$
$332$		0			0
$333$	−4.28333	+	0.976735i	−0.234725	+	0.0535248i
$334$		0			0
$335$	−3.47969	+	24.9613i	−0.190116	+	1.36378i
$336$		0			0
$337$	−23.9082	+	23.9082i	−1.30237	+	1.30237i	−0.375572	+	0.926793i	$0.622554\pi$
$337$	−23.9082	+	23.9082i	−1.30237	+	1.30237i	−0.926793	+	0.375572i	$0.877446\pi$
$338$		0			0
$339$	0.178712	−	9.50448i	0.00970629	−	0.516213i
$340$		0			0
$341$	−10.2940	−	5.94322i	−0.557450	−	0.321844i
$342$		0			0
$343$	−5.20650	+	17.7734i	−0.281125	+	0.959671i
$344$		0			0
$345$	−14.0892	+	31.6252i	−0.758538	+	1.70264i
$346$		0			0
$347$	−1.66513	+	6.21434i	−0.0893887	+	0.333603i	−0.996109	−	0.0881292i	$-0.971911\pi$
$347$	−1.66513	+	6.21434i	−0.0893887	+	0.333603i	0.906720	+	0.421732i	$0.138578\pi$
$348$		0			0
$349$			15.3799i			0.823266i	0.911350	+	0.411633i	$0.135041\pi$
$349$			15.3799i			0.823266i	−0.911350	+	0.411633i	$0.864959\pi$
$350$		0			0
$351$	−0.666276	+	11.8005i	−0.0355632	+	0.629862i
$352$		0			0
$353$	−10.3958	−	2.78555i	−0.553313	−	0.148260i	−0.0286804	−	0.999589i	$-0.509131\pi$
$353$	−10.3958	−	2.78555i	−0.553313	−	0.148260i	−0.524632	+	0.851329i	$0.675797\pi$
$354$		0			0
$355$	−22.1889	+	2.74959i	−1.17767	+	0.145933i
$356$		0			0
$357$	−17.9687	+	7.80570i	−0.951005	+	0.413121i
$358$		0			0
$359$	16.2835	−	28.2038i	0.859410	−	1.48854i	−0.0130825	−	0.999914i	$-0.504164\pi$
$359$	16.2835	−	28.2038i	0.859410	−	1.48854i	0.872493	−	0.488627i	$-0.162502\pi$
$360$		0			0
$361$	21.5981	+	37.4090i	1.13674	+	1.96890i
$362$		0			0
$363$	2.45891	+	9.91906i	0.129059	+	0.520616i
$364$		0			0
$365$	3.02316	+	0.421438i	0.158239	+	0.0220591i
$366$		0			0
$367$	12.4872	+	3.34594i	0.651828	+	0.174657i	0.569555	−	0.821953i	$-0.307115\pi$
$367$	12.4872	+	3.34594i	0.651828	+	0.174657i	0.0822728	+	0.996610i	$0.473782\pi$
$368$		0			0
$369$	15.2247	+	0.572739i	0.792566	+	0.0298156i
$370$		0			0
$371$	−3.97908	+	10.6752i	−0.206583	+	0.554229i
$372$		0			0
$373$	−1.39327	+	0.373327i	−0.0721410	+	0.0193301i	−0.294709	−	0.955587i	$-0.595223\pi$
$373$	−1.39327	+	0.373327i	−0.0721410	+	0.0193301i	0.222568	+	0.974917i	$0.428556\pi$
$374$		0			0
$375$	−17.5669	−	8.14886i	−0.907151	−	0.420805i
$376$		0			0
$377$	−5.37500	−	5.37500i	−0.276826	−	0.276826i
$378$		0			0
$379$		−	36.9758i		−	1.89932i	−0.313284	−	0.949659i	$-0.601429\pi$
$379$		−	36.9758i		−	1.89932i	0.313284	−	0.949659i	$-0.398571\pi$
$380$		0			0
$381$	0.107423	−	5.71313i	0.00550346	−	0.292692i
$382$		0			0
$383$	−6.06426	−	22.6321i	−0.309869	−	1.15645i	−0.928672	−	0.370901i	$-0.879049\pi$
$383$	−6.06426	−	22.6321i	−0.309869	−	1.15645i	0.618803	−	0.785546i	$-0.287618\pi$
$384$		0			0
$385$	11.3767	−	7.00484i	0.579808	−	0.357000i
$386$		0			0
$387$	−12.9630	+	12.0230i	−0.658945	+	0.611165i
$388$		0			0
$389$	−10.0292	−	17.3711i	−0.508502	−	0.880752i	−0.999952	−	0.00984566i	$-0.996866\pi$
$389$	−10.0292	−	17.3711i	−0.508502	−	0.880752i	0.491449	−	0.870906i	$-0.336467\pi$
$390$		0			0
$391$	38.2162			1.93267
$392$		0			0
$393$	−28.7053	+	7.11596i	−1.44799	+	0.358953i
$394$		0			0
$395$	−5.69743	+	13.4641i	−0.286669	+	0.677450i
$396$		0			0
$397$	−9.30700	−	34.7342i	−0.467105	−	1.74326i	−0.649815	−	0.760092i	$-0.725154\pi$
$397$	−9.30700	−	34.7342i	−0.467105	−	1.74326i	0.182710	−	0.983167i	$-0.441513\pi$
$398$		0			0
$399$	−5.32962	−	35.7451i	−0.266815	−	1.78950i
$400$		0			0
$401$	−11.0543	−	6.38223i	−0.552027	−	0.318713i	0.197912	−	0.980220i	$-0.436584\pi$
$401$	−11.0543	−	6.38223i	−0.552027	−	0.318713i	−0.749939	+	0.661507i	$0.769917\pi$
$402$		0			0
$403$	−3.09868	+	11.5644i	−0.154356	+	0.576065i
$404$		0			0
$405$	−0.967340	−	20.1013i	−0.0480675	−	0.998844i
$406$		0			0
$407$	−2.33848	−	2.33848i	−0.115914	−	0.115914i
$408$		0			0
$409$	10.3607	−	5.98176i	0.512304	−	0.295779i	−0.221476	−	0.975166i	$-0.571087\pi$
$409$	10.3607	−	5.98176i	0.512304	−	0.295779i	0.733780	+	0.679387i	$0.237754\pi$
$410$		0			0
$411$	−5.02773	−	9.09916i	−0.248000	−	0.448829i
$412$		0			0
$413$	3.52814	−	1.61215i	0.173608	−	0.0793286i
$414$		0			0
$415$	−18.2924	−	14.2587i	−0.897937	−	0.699933i
$416$		0			0
$417$	1.68345	+	1.74797i	0.0824389	+	0.0855985i
$418$		0			0
$419$	5.09737			0.249023			0.124511	−	0.992218i	$-0.460264\pi$
$419$	5.09737			0.249023			0.124511	+	0.992218i	$0.460264\pi$
$420$		0			0
$421$	18.8316			0.917796			0.458898	−	0.888489i	$-0.348244\pi$
$421$	18.8316			0.917796			0.458898	+	0.888489i	$0.348244\pi$
$422$		0			0
$423$	8.41103	+	36.8854i	0.408958	+	1.79343i
$424$		0			0
$425$	−0.325363	+	21.3730i	−0.0157824	+	1.03674i
$426$		0			0
$427$	17.3427	+	12.3441i	0.839271	+	0.597375i
$428$		0			0
$429$	−7.78739	+	4.30291i	−0.375979	+	0.207747i
$430$		0			0
$431$	2.55511	−	1.47519i	0.123075	−	0.0710575i	−0.437198	−	0.899365i	$-0.644029\pi$
$431$	2.55511	−	1.47519i	0.123075	−	0.0710575i	0.560274	+	0.828308i	$0.310696\pi$
$432$		0			0
$433$	−5.74628	−	5.74628i	−0.276149	−	0.276149i	0.555421	−	0.831569i	$-0.312557\pi$
$433$	−5.74628	−	5.74628i	−0.276149	−	0.276149i	−0.831569	+	0.555421i	$0.812557\pi$
$434$		0			0
$435$	10.0566	+	8.14755i	0.482178	+	0.390645i
$436$		0			0
$437$	−18.2465	+	68.0970i	−0.872850	+	3.25752i
$438$		0			0
$439$	35.1852	+	20.3142i	1.67930	+	0.969544i	0.962109	+	0.272666i	$0.0879053\pi$
$439$	35.1852	+	20.3142i	1.67930	+	0.969544i	0.717190	+	0.696878i	$0.245428\pi$
$440$		0			0
$441$	−4.73800	+	20.4585i	−0.225619	+	0.974216i
$442$		0			0
$443$	4.71442	+	17.5944i	0.223989	+	0.835937i	0.982807	+	0.184635i	$0.0591103\pi$
$443$	4.71442	+	17.5944i	0.223989	+	0.835937i	−0.758818	+	0.651302i	$0.774223\pi$
$444$		0			0
$445$	−19.0879	−	8.07722i	−0.904854	−	0.382897i
$446$		0			0
$447$	−0.658423	−	2.65603i	−0.0311423	−	0.125626i
$448$		0			0
$449$	−29.2170			−1.37883			−0.689417	−	0.724364i	$-0.742133\pi$
$449$	−29.2170			−1.37883			−0.689417	+	0.724364i	$0.742133\pi$
$450$		0			0
$451$	5.73436	+	9.93220i	0.270020	+	0.467689i
$452$		0			0
$453$	24.5099	+	7.06385i	1.15158	+	0.331889i
$454$		0			0
$455$	−9.78084	−	9.24234i	−0.458533	−	0.433287i
$456$		0			0
$457$	6.29669	+	23.4995i	0.294547	+	1.09926i	0.941577	+	0.336798i	$0.109344\pi$
$457$	6.29669	+	23.4995i	0.294547	+	1.09926i	−0.647030	+	0.762464i	$0.723989\pi$
$458$		0			0
$459$	−19.8334	+	10.0049i	−0.925745	+	0.466987i
$460$		0			0
$461$		−	15.0369i		−	0.700340i	−0.936686	−	0.350170i	$-0.886124\pi$
$461$		−	15.0369i		−	0.700340i	0.936686	−	0.350170i	$-0.113876\pi$
$462$		0			0
$463$	9.45318	+	9.45318i	0.439327	+	0.439327i	0.891785	−	0.452459i	$-0.149453\pi$
$463$	9.45318	+	9.45318i	0.439327	+	0.439327i	−0.452459	+	0.891785i	$0.649453\pi$
$464$		0			0
$465$	3.19363	−	20.1336i	0.148101	−	0.933674i
$466$		0			0
$467$	24.1420	−	6.46882i	1.11716	−	0.299341i	0.347425	−	0.937708i	$-0.387056\pi$
$467$	24.1420	−	6.46882i	1.11716	−	0.299341i	0.769733	+	0.638366i	$0.220390\pi$
$468$		0			0
$469$	10.4152	−	27.9422i	0.480929	−	1.29025i
$470$		0			0
$471$	−14.6510	+	8.09541i	−0.675084	+	0.373017i
$472$		0			0
$473$	−12.8556	−	3.44464i	−0.591100	−	0.158385i
$474$		0			0
$475$	−37.9289	−	10.7844i	−1.74030	−	0.494822i
$476$		0			0
$477$	−3.81016	+	12.3434i	−0.174455	+	0.565166i
$478$		0			0
$479$	6.08042	+	10.5316i	0.277821	+	0.481201i	0.970843	−	0.239716i	$-0.0770544\pi$
$479$	6.08042	+	10.5316i	0.277821	+	0.481201i	−0.693022	+	0.720917i	$0.743721\pi$
$480$		0			0
$481$	−1.66551	+	2.88474i	−0.0759405	+	0.131533i
$482$		0			0
$483$	24.3781	−	32.9216i	1.10924	−	1.49798i
$484$		0			0
$485$	17.6203	+	13.7349i	0.800098	+	0.623669i
$486$		0			0
$487$	−33.3830	−	8.94495i	−1.51273	−	0.405334i	−0.595388	−	0.803438i	$-0.703002\pi$
$487$	−33.3830	−	8.94495i	−1.51273	−	0.405334i	−0.917340	+	0.398104i	$0.869668\pi$
$488$		0			0
$489$	4.82278	−	8.00202i	0.218094	−	0.361864i
$490$		0			0
$491$		−	28.8464i		−	1.30182i	−0.759154	−	0.650911i	$-0.774387\pi$
$491$		−	28.8464i		−	1.30182i	0.759154	−	0.650911i	$-0.225613\pi$
$492$		0			0
$493$	3.69765	−	13.7998i	0.166534	−	0.621513i
$494$		0			0
$495$	12.4231	−	8.66955i	0.558377	−	0.389667i
$496$		0			0
$497$	26.3358	+	2.50915i	1.18132	+	0.112551i
$498$		0			0
$499$	26.3227	+	15.1974i	1.17837	+	0.680329i	0.955636	−	0.294551i	$-0.0951701\pi$
$499$	26.3227	+	15.1974i	1.17837	+	0.680329i	0.222729	+	0.974880i	$0.428503\pi$
$500$		0			0
$501$	31.7835	+	0.597622i	1.41998	+	0.0266998i
$502$		0			0
$503$	−23.4649	+	23.4649i	−1.04625	+	1.04625i	−0.0473718	+	0.998877i	$0.515085\pi$
$503$	−23.4649	+	23.4649i	−1.04625	+	1.04625i	−0.998877	+	0.0473718i	$0.984915\pi$
$504$		0			0
$505$	−7.96809	−	10.5495i	−0.354575	−	0.469447i
$506$		0			0
$507$	−9.40312	−	9.76351i	−0.417607	−	0.433613i
$508$		0			0
$509$	−0.122791	+	0.212680i	−0.00544260	+	0.00942686i	−0.868734	−	0.495279i	$-0.835066\pi$
$509$	−0.122791	+	0.212680i	−0.00544260	+	0.00942686i	0.863291	+	0.504706i	$0.168399\pi$
$510$		0			0
$511$	−3.38419	−	1.26142i	−0.149708	−	0.0558021i
$512$		0			0
$513$	−8.35797	−	40.1178i	−0.369013	−	1.77125i
$514$		0			0
$515$	−3.66126	−	9.03292i	−0.161335	−	0.398038i
$516$		0			0
$517$	−20.1375	+	20.1375i	−0.885647	+	0.885647i
$518$		0			0
$519$	−9.46912	−	5.70700i	−0.415648	−	0.250510i
$520$		0			0
$521$	18.1819	−	10.4973i	0.796561	−	0.459895i	−0.0457060	−	0.998955i	$-0.514554\pi$
$521$	18.1819	−	10.4973i	0.796561	−	0.459895i	0.842267	+	0.539060i	$0.181220\pi$
$522$		0			0
$523$	−21.6866	+	5.81090i	−0.948288	+	0.254093i	−0.699636	−	0.714500i	$-0.746654\pi$
$523$	−21.6866	+	5.81090i	−0.948288	+	0.254093i	−0.248652	+	0.968593i	$0.579988\pi$
$524$		0			0
$525$	18.2043	+	13.9141i	0.794502	+	0.607262i
$526$		0			0
$527$	−21.7350	+	5.82389i	−0.946794	+	0.253693i
$528$		0			0
$529$	−49.2860	+	28.4553i	−2.14287	+	1.23719i
$530$		0			0
$531$	3.88910	−	2.05445i	0.168773	−	0.0891555i
$532$		0			0
$533$	8.16822	−	8.16822i	0.353805	−	0.353805i
$534$		0			0
$535$	7.36585	+	18.1727i	0.318454	+	0.785675i
$536$		0			0
$537$	17.2013	+	4.95747i	0.742289	+	0.213931i
$538$		0			0
$539$	−14.9364	+	5.17662i	−0.643358	+	0.222973i
$540$		0			0
$541$	8.28329	−	14.3471i	0.356126	−	0.616829i	−0.631184	−	0.775633i	$-0.717431\pi$
$541$	8.28329	−	14.3471i	0.356126	−	0.616829i	0.987310	+	0.158804i	$0.0507639\pi$
$542$		0			0
$543$	3.02175	−	2.91022i	0.129676	−	0.124889i
$544$		0			0
$545$	−3.20804	−	4.24735i	−0.137417	−	0.181936i
$546$		0			0
$547$	−11.0461	+	11.0461i	−0.472296	+	0.472296i	−0.902657	−	0.430361i	$-0.858386\pi$
$547$	−11.0461	+	11.0461i	−0.472296	+	0.472296i	0.430361	+	0.902657i	$0.358386\pi$
$548$		0			0
$549$	20.4352	+	12.8460i	0.872152	+	0.548255i
$550$		0			0
$551$	22.8243	+	13.1776i	0.972348	+	0.561386i
$552$		0			0
$553$	10.0311	−	14.0931i	0.426567	−	0.599298i
$554$		0			0
$555$	2.30809	−	5.18083i	0.0979730	−	0.219914i
$556$		0			0
$557$	9.69992	−	36.2006i	0.410999	−	1.53387i	−0.381719	−	0.924279i	$-0.624668\pi$
$557$	9.69992	−	36.2006i	0.410999	−	1.53387i	0.792717	−	0.609589i	$-0.208666\pi$
$558$		0			0
$559$			13.4053i			0.566982i
$560$		0			0
$561$	−14.3219	−	8.63172i	−0.604669	−	0.364432i
$562$		0			0
$563$	2.77219	+	0.742806i	0.116834	+	0.0313056i	0.316762	−	0.948505i	$-0.397404\pi$
$563$	2.77219	+	0.742806i	0.116834	+	0.0313056i	−0.199928	+	0.979811i	$0.564071\pi$
$564$		0			0
$565$	9.67921	+	7.54485i	0.407207	+	0.317414i
$566$		0			0
$567$	−4.03302	+	23.4677i	−0.169371	+	0.985552i
$568$		0			0
$569$	−17.2252	+	29.8350i	−0.722119	+	1.25075i	0.238029	+	0.971258i	$0.423499\pi$
$569$	−17.2252	+	29.8350i	−0.722119	+	1.25075i	−0.960149	+	0.279490i	$0.909835\pi$
$570$		0			0
$571$	−6.99523	−	12.1161i	−0.292741	−	0.507043i	0.681716	−	0.731617i	$-0.261234\pi$
$571$	−6.99523	−	12.1161i	−0.292741	−	0.507043i	−0.974457	+	0.224574i	$0.927901\pi$
$572$		0			0
$573$	30.0525	−	7.44994i	1.25546	−	0.311226i
$574$		0			0
$575$	−21.7564	−	39.0439i	−0.907305	−	1.62824i
$576$		0			0
$577$	−10.6964	−	2.86608i	−0.445296	−	0.119317i	0.0292017	−	0.999574i	$-0.490703\pi$
$577$	−10.6964	−	2.86608i	−0.445296	−	0.119317i	−0.474497	+	0.880257i	$0.657370\pi$
$578$		0			0
$579$	−13.5382	−	24.5013i	−0.562628	−	1.01824i
$580$		0			0
$581$	17.4768	+	21.1578i	0.725061	+	0.877772i
$582$		0			0
$583$	−9.39293	+	2.51683i	−0.389015	+	0.104236i
$584$		0			0
$585$	−11.6733	−	9.82647i	−0.482630	−	0.406275i
$586$		0			0
$587$	−3.07192	−	3.07192i	−0.126792	−	0.126792i	0.640863	−	0.767655i	$-0.278577\pi$
$587$	−3.07192	−	3.07192i	−0.126792	−	0.126792i	−0.767655	+	0.640863i	$0.778577\pi$
$588$		0			0
$589$		−	41.5101i		−	1.71039i
$590$		0			0
$591$	−11.9349	−	0.224410i	−0.490936	−	0.00923101i
$592$		0			0
$593$	9.02050	+	33.6650i	0.370428	+	1.38245i	0.859912	+	0.510442i	$0.170518\pi$
$593$	9.02050	+	33.6650i	0.370428	+	1.38245i	−0.489484	+	0.872012i	$0.662815\pi$
$594$		0			0
$595$	5.85209	−	24.6054i	0.239912	−	1.00872i
$596$		0			0
$597$	6.09219	−	21.1385i	0.249337	−	0.865140i
$598$		0			0
$599$	13.7809	+	23.8692i	0.563073	+	0.975270i	0.997226	+	0.0744315i	$0.0237142\pi$
$599$	13.7809	+	23.8692i	0.563073	+	0.975270i	−0.434153	+	0.900839i	$0.642952\pi$
$600$		0			0
$601$	−22.5471			−0.919717			−0.459858	−	0.887992i	$-0.652100\pi$
$601$	−22.5471			−0.919717			−0.459858	+	0.887992i	$0.652100\pi$
$602$		0			0
$603$	9.97304	−	32.3087i	0.406134	−	1.31571i
$604$		0			0
$605$	−12.1500	−	5.14139i	−0.493968	−	0.209027i
$606$		0			0
$607$	−8.73196	−	32.5881i	−0.354419	−	1.32271i	−0.881214	−	0.472718i	$-0.843273\pi$
$607$	−8.73196	−	32.5881i	−0.354419	−	1.32271i	0.526794	−	0.849993i	$-0.323394\pi$
$608$		0			0
$609$	−9.52921	−	11.9883i	−0.386143	−	0.485790i
$610$		0			0
$611$	24.8416	+	14.3423i	1.00498	+	0.580227i
$612$		0			0
$613$	3.25023	−	12.1300i	0.131276	−	0.489927i	−0.868710	−	0.495321i	$-0.835050\pi$
$613$	3.25023	−	12.1300i	0.131276	−	0.489927i	0.999985	+	0.00539398i	$0.00171697\pi$
$614$		0			0
$615$	−12.3816	+	15.2827i	−0.499274	+	0.616260i
$616$		0			0
$617$	4.35042	+	4.35042i	0.175141	+	0.175141i	0.789234	−	0.614093i	$-0.210478\pi$
$617$	4.35042	+	4.35042i	0.175141	+	0.175141i	−0.614093	+	0.789234i	$0.710478\pi$
$618$		0			0
$619$	7.59365	−	4.38420i	0.305215	−	0.176216i	−0.339568	−	0.940581i	$-0.610281\pi$
$619$	7.59365	−	4.38420i	0.305215	−	0.176216i	0.644783	+	0.764366i	$0.276948\pi$
$620$		0			0
$621$	25.4556	−	38.8535i	1.02150	−	1.55914i
$622$		0			0
$623$	19.9797	+	14.2211i	0.800468	+	0.569756i
$624$		0			0
$625$	22.0211	−	11.8352i	0.880844	−	0.473407i
$626$		0			0
$627$	22.2188	−	21.3987i	0.887335	−	0.854582i
$628$		0			0
$629$	−6.26056			−0.249625
$630$		0			0
$631$	−28.4353			−1.13199			−0.565995	−	0.824409i	$-0.691508\pi$
$631$	−28.4353			−1.13199			−0.565995	+	0.824409i	$0.691508\pi$
$632$		0			0
$633$	−3.31150	+	3.18926i	−0.131620	+	0.126762i
$634$		0			0
$635$	5.81816	+	4.53520i	0.230886	+	0.179974i
$636$		0			0
$637$	8.93014	+	13.1823i	0.353825	+	0.522301i
$638$		0			0
$639$	29.9761	+	1.12767i	1.18583	+	0.0446100i
$640$		0			0
$641$	16.8799	−	9.74560i	0.666715	−	0.384928i	−0.128116	−	0.991759i	$-0.540893\pi$
$641$	16.8799	−	9.74560i	0.666715	−	0.384928i	0.794831	+	0.606831i	$0.207560\pi$
$642$		0			0
$643$	31.2135	+	31.2135i	1.23094	+	1.23094i	0.963604	+	0.267335i	$0.0861430\pi$
$643$	31.2135	+	31.2135i	1.23094	+	1.23094i	0.267335	+	0.963604i	$0.413857\pi$
$644$		0			0
$645$	−2.38062	−	22.7006i	−0.0937366	−	0.893837i
$646$		0			0
$647$	0.768215	−	2.86702i	0.0302016	−	0.112714i	−0.949180	−	0.314735i	$-0.898084\pi$
$647$	0.768215	−	2.86702i	0.0302016	−	0.112714i	0.979381	+	0.202021i	$0.0647510\pi$
$648$		0			0
$649$	2.86737	+	1.65548i	0.112554	+	0.0649832i
$650$		0			0
$651$	−8.84779	+	22.4389i	−0.346772	+	0.879448i
$652$		0			0
$653$	2.16578	+	8.08279i	0.0847534	+	0.316304i	0.995267	−	0.0971743i	$-0.0309804\pi$
$653$	2.16578	+	8.08279i	0.0847534	+	0.316304i	−0.910514	+	0.413478i	$0.864314\pi$
$654$		0			0
$655$	14.8789	−	35.1616i	0.581367	−	1.37388i
$656$		0			0
$657$	−3.91303	−	1.20787i	−0.152662	−	0.0471236i
$658$		0			0
$659$	−35.1050			−1.36750			−0.683749	−	0.729717i	$-0.739652\pi$
$659$	−35.1050			−1.36750			−0.683749	+	0.729717i	$0.739652\pi$
$660$		0			0
$661$	5.30983	+	9.19689i	0.206528	+	0.357718i	0.950619	−	0.310361i	$-0.100450\pi$
$661$	5.30983	+	9.19689i	0.206528	+	0.357718i	−0.744090	+	0.668079i	$0.767117\pi$
$662$		0			0
$663$	−4.66431	+	16.1840i	−0.181147	+	0.628536i
$664$		0			0
$665$	41.0500	+	22.1758i	1.59185	+	0.859939i
$666$		0			0
$667$	7.73185	+	28.8557i	0.299379	+	1.11730i
$668$		0			0
$669$	−21.2992	−	0.400486i	−0.823474	−	0.0154837i
$670$		0			0
$671$			18.1698i			0.701438i
$672$		0			0
$673$	18.1638	+	18.1638i	0.700162	+	0.700162i	0.964445	−	0.264284i	$-0.0851355\pi$
$673$	18.1638	+	18.1638i	0.700162	+	0.700162i	−0.264284	+	0.964445i	$0.585135\pi$
$674$		0			0
$675$	21.5127	+	14.5672i	0.828024	+	0.560693i
$676$		0			0
$677$	17.9695	−	4.81492i	0.690626	−	0.185053i	0.103598	−	0.994619i	$-0.466965\pi$
$677$	17.9695	−	4.81492i	0.690626	−	0.185053i	0.587028	+	0.809567i	$0.300298\pi$
$678$		0			0
$679$	−16.8348	−	20.3805i	−0.646059	−	0.782130i
$680$		0			0
$681$	−10.9715	−	19.8562i	−0.420429	−	0.760890i
$682$		0			0
$683$	23.8730	+	6.39675i	0.913475	+	0.244765i	0.684794	−	0.728736i	$-0.259892\pi$
$683$	23.8730	+	6.39675i	0.913475	+	0.244765i	0.228681	+	0.973501i	$0.426559\pi$
$684$		0			0
$685$	13.2924	+	1.85300i	0.507876	+	0.0707996i
$686$		0			0
$687$	9.50072	−	2.35520i	0.362475	−	0.0898566i
$688$		0			0
$689$	4.89728	+	8.48234i	0.186571	+	0.323151i
$690$		0			0
$691$	22.6542	−	39.2382i	0.861806	−	1.49269i	−0.00837711	−	0.999965i	$-0.502667\pi$
$691$	22.6542	−	39.2382i	0.861806	−	1.49269i	0.870184	−	0.492728i	$-0.164000\pi$
$692$		0			0
$693$	−16.5718	+	6.83146i	−0.629510	+	0.259506i
$694$		0			0
$695$	−3.10922	+	0.385286i	−0.117939	+	0.0146147i
$696$		0			0
$697$	20.9712	+	5.61922i	0.794341	+	0.212843i
$698$		0			0
$699$	−0.610291	−	0.367820i	−0.0230833	−	0.0139122i
$700$		0			0
$701$			7.92195i			0.299208i	0.988746	+	0.149604i	$0.0477999\pi$
$701$			7.92195i			0.299208i	−0.988746	+	0.149604i	$0.952200\pi$
$702$		0			0
$703$	2.98914	−	11.1556i	0.112738	−	0.420742i
$704$		0			0
$705$	−44.6140	−	19.8758i	−1.68026	−	0.748567i
$706$		0			0
$707$	6.50123	+	14.2278i	0.244504	+	0.535090i
$708$		0			0
$709$	−31.3009	−	18.0716i	−1.17553	−	0.678693i	−0.220555	−	0.975375i	$-0.570787\pi$
$709$	−31.3009	−	18.0716i	−1.17553	−	0.678693i	−0.954977	+	0.296681i	$0.904120\pi$
$710$		0			0
$711$	10.4390	−	16.6061i	0.391492	−	0.622777i
$712$		0			0
$713$	33.2705	−	33.2705i	1.24599	−	1.24599i
$714$		0			0
$715$	1.58587	−	11.3761i	0.0593081	−	0.425442i
$716$		0			0
$717$	0.751698	−	0.723951i	0.0280727	−	0.0270365i
$718$		0			0
$719$	11.1558	−	19.3224i	0.416041	−	0.720604i	−0.579496	−	0.814975i	$-0.696751\pi$
$719$	11.1558	−	19.3224i	0.416041	−	0.720604i	0.995537	+	0.0943709i	$0.0300839\pi$
$720$		0			0
$721$	1.91483	+	11.3724i	0.0713121	+	0.423530i
$722$		0			0
$723$	9.35271	+	2.69549i	0.347831	+	0.100246i
$724$		0			0
$725$	−16.2038	+	4.07848i	−0.601793	+	0.151471i
$726$		0			0
$727$	32.5217	−	32.5217i	1.20616	−	1.20616i	0.233902	−	0.972260i	$-0.424851\pi$
$727$	32.5217	−	32.5217i	1.20616	−	1.20616i	0.972260	−	0.233902i	$-0.0751494\pi$
$728$		0			0
$729$	−3.03925	+	26.8284i	−0.112565	+	0.993644i
$730$		0			0
$731$	−21.8194	+	12.5974i	−0.807020	+	0.465933i
$732$		0			0
$733$	−18.4073	+	4.93222i	−0.679889	+	0.182176i	−0.582205	−	0.813042i	$-0.697810\pi$
$733$	−18.4073	+	4.93222i	−0.679889	+	0.182176i	−0.0976839	+	0.995217i	$0.531143\pi$
$734$		0			0
$735$	−17.4634	−	20.7371i	−0.644148	−	0.764901i
$736$		0			0
$737$	24.5859	−	6.58776i	0.905632	−	0.242663i
$738$		0			0
$739$	2.96973	−	1.71458i	0.109243	−	0.0630717i	−0.444383	−	0.895837i	$-0.646577\pi$
$739$	2.96973	−	1.71458i	0.109243	−	0.0630717i	0.553626	+	0.832765i	$0.313244\pi$
$740$		0			0
$741$	−26.6112	−	16.0384i	−0.977586	−	0.589187i
$742$		0			0
$743$	−16.8816	+	16.8816i	−0.619326	+	0.619326i	−0.945358	−	0.326033i	$-0.894288\pi$
$743$	−16.8816	+	16.8816i	−0.619326	+	0.619326i	0.326033	+	0.945358i	$0.394288\pi$
$744$		0			0
$745$	3.25341	+	1.37671i	0.119196	+	0.0504387i
$746$		0			0
$747$	21.1602	+	22.8145i	0.774213	+	0.834740i
$748$		0			0
$749$	−3.85232	−	22.8793i	−0.140761	−	0.835993i
$750$		0			0
$751$	−17.7576	+	30.7571i	−0.647985	+	1.12234i	0.335618	+	0.941998i	$0.391055\pi$
$751$	−17.7576	+	30.7571i	−0.647985	+	1.12234i	−0.983603	+	0.180345i	$0.942279\pi$
$752$		0			0
$753$	−17.6909	−	18.3690i	−0.644693	−	0.669402i
$754$		0			0
$755$	−26.2770	+	19.8471i	−0.956317	+	0.722311i
$756$		0			0
$757$	−28.5534	+	28.5534i	−1.03779	+	1.03779i	−0.0385344	+	0.999257i	$0.512269\pi$
$757$	−28.5534	+	28.5534i	−1.03779	+	1.03779i	−0.999257	+	0.0385344i	$0.987731\pi$
$758$		0			0
$759$	34.9596	+	0.657341i	1.26895	+	0.0238600i
$760$		0			0
$761$	14.9877	+	8.65316i	0.543304	+	0.313677i	0.746417	−	0.665479i	$-0.231773\pi$
$761$	14.9877	+	8.65316i	0.543304	+	0.313677i	−0.203113	+	0.979155i	$0.565106\pi$
$762$		0			0
$763$	2.61747	+	5.72826i	0.0947588	+	0.207377i
$764$		0			0
$765$	5.02450	−	28.2346i	0.181661	−	1.02082i
$766$		0			0
$767$	0.863132	−	3.22125i	0.0311659	−	0.116313i
$768$		0			0
$769$		−	16.1547i		−	0.582552i	−0.956639	−	0.291276i	$-0.905920\pi$
$769$		−	16.1547i		−	0.582552i	0.956639	−	0.291276i	$-0.0940799\pi$
$770$		0			0
$771$	−21.8244	+	36.2114i	−0.785988	+	1.30412i
$772$		0			0
$773$	−10.5936	−	2.83854i	−0.381024	−	0.102095i	0.0632229	−	0.997999i	$-0.479862\pi$
$773$	−10.5936	−	2.83854i	−0.381024	−	0.102095i	−0.444247	+	0.895904i	$0.646529\pi$
$774$		0			0
$775$	18.3238	+	18.8903i	0.658209	+	0.678558i
$776$		0			0
$777$	−3.99362	+	5.39320i	−0.143270	+	0.193480i
$778$		0			0
$779$	−20.0257	+	34.6854i	−0.717493	+	1.24274i
$780$		0			0
$781$	11.2904	+	19.5556i	0.404003	+	0.699754i
$782$		0			0
$783$	−11.5670	−	12.9513i	−0.413370	−	0.462843i
$784$		0			0
$785$	2.98362	−	21.4028i	0.106490	−	0.763897i
$786$		0			0
$787$	−8.74047	−	2.34200i	−0.311564	−	0.0834834i	0.0996487	−	0.995023i	$-0.468228\pi$
$787$	−8.74047	−	2.34200i	−0.311564	−	0.0834834i	−0.411213	+	0.911539i	$0.634895\pi$
$788$		0			0
$789$	24.0461	−	13.2867i	0.856065	−	0.473018i
$790$		0			0
$791$	−9.24768	−	11.1954i	−0.328810	−	0.398063i
$792$		0			0
$793$	17.6775	−	4.73668i	0.627748	−	0.168205i
$794$		0			0
$795$	−9.79947	−	13.4944i	−0.347551	−	0.478597i
$796$		0			0
$797$	−16.3203	−	16.3203i	−0.578093	−	0.578093i	0.356284	−	0.934378i	$-0.384043\pi$
$797$	−16.3203	−	16.3203i	−0.578093	−	0.578093i	−0.934378	+	0.356284i	$0.884043\pi$
$798$		0			0
$799$			53.9120i			1.90727i
$800$		0			0
$801$	23.5424	+	14.7993i	0.831829	+	0.522907i
$802$		0			0
$803$	−0.797869	−	2.97769i	−0.0281562	−	0.105080i
$804$		0			0
$805$	15.1278	+	50.6757i	0.533183	+	1.78608i
$806$		0			0
$807$	−40.2788	−	11.6085i	−1.41788	−	0.408639i
$808$		0			0
$809$	20.0240	+	34.6826i	0.704006	+	1.21937i	0.967049	+	0.254591i	$0.0819407\pi$
$809$	20.0240	+	34.6826i	0.704006	+	1.21937i	−0.263043	+	0.964784i	$0.584726\pi$
$810$		0			0
$811$	27.7465			0.974312			0.487156	−	0.873315i	$-0.338034\pi$
$811$	27.7465			0.974312			0.487156	+	0.873315i	$0.338034\pi$
$812$		0			0
$813$	1.81039	+	7.30297i	0.0634930	+	0.256126i
$814$		0			0
$815$	4.53089	+	11.1784i	0.158710	+	0.391563i
$816$		0			0
$817$	−12.0294	−	44.8945i	−0.420857	−	1.57066i
$818$		0			0
$819$	10.9665	+	14.3419i	0.383200	+	0.501147i
$820$		0			0
$821$	−28.5058	−	16.4578i	−0.994858	−	0.574381i	−0.0881350	−	0.996109i	$-0.528091\pi$
$821$	−28.5058	−	16.4578i	−0.994858	−	0.574381i	−0.906723	+	0.421727i	$0.861424\pi$
$822$		0			0
$823$	6.01032	−	22.4308i	0.209506	−	0.781889i	−0.778522	−	0.627617i	$-0.784030\pi$
$823$	6.01032	−	22.4308i	0.209506	−	0.781889i	0.988028	−	0.154272i	$-0.0493031\pi$
$824$		0			0
$825$	−0.665265	+	19.5461i	−0.0231616	+	0.680507i
$826$		0			0
$827$	−21.1976	−	21.1976i	−0.737112	−	0.737112i	0.234906	−	0.972018i	$-0.424522\pi$
$827$	−21.1976	−	21.1976i	−0.737112	−	0.737112i	−0.972018	+	0.234906i	$0.924522\pi$
$828$		0			0
$829$	−21.2241	+	12.2537i	−0.737143	+	0.425590i	−0.821030	−	0.570886i	$-0.806600\pi$
$829$	−21.2241	+	12.2537i	−0.737143	+	0.425590i	0.0838866	+	0.996475i	$0.473267\pi$
$830$		0			0
$831$	8.50193	−	4.69773i	0.294929	−	0.162963i
$832$		0			0
$833$	−13.0645	+	26.9233i	−0.452657	+	0.932836i
$834$		0			0
$835$	−25.2304	+	32.3678i	−0.873134	+	1.12013i
$836$		0			0
$837$	−8.55662	+	25.9768i	−0.295760	+	0.897890i
$838$		0			0
$839$	−16.2459			−0.560870			−0.280435	−	0.959873i	$-0.590479\pi$
$839$	−16.2459			−0.560870			−0.280435	+	0.959873i	$0.590479\pi$
$840$		0			0
$841$	−17.8321			−0.614901
$842$		0			0
$843$	−27.3009	−	28.3472i	−0.940292	−	0.976330i
$844$		0			0
$845$	17.3669	−	2.15206i	0.597440	−	0.0740331i
$846$		0			0
$847$	12.7176	+	9.05214i	0.436983	+	0.311035i
$848$		0			0
$849$	−3.74389	−	6.77566i	−0.128490	−	0.232540i
$850$		0			0
$851$	11.3371	−	6.54546i	0.388630	−	0.224376i
$852$		0			0
$853$	−36.9068	−	36.9068i	−1.26367	−	1.26367i	−0.949303	−	0.314364i	$-0.898209\pi$
$853$	−36.9068	−	36.9068i	−1.26367	−	1.26367i	−0.314364	−	0.949303i	$-0.601791\pi$
$854$		0			0
$855$	47.9119	+	22.4339i	1.63855	+	0.767222i
$856$		0			0
$857$	−0.970024	+	3.62018i	−0.0331354	+	0.123663i	−0.980512	−	0.196457i	$-0.937056\pi$
$857$	−0.970024	+	3.62018i	−0.0331354	+	0.123663i	0.947377	+	0.320120i	$0.103723\pi$
$858$		0			0
$859$	3.27915	+	1.89322i	0.111883	+	0.0645958i	0.554897	−	0.831919i	$-0.312758\pi$
$859$	3.27915	+	1.89322i	0.111883	+	0.0645958i	−0.443014	+	0.896515i	$0.646091\pi$
$860$		0			0
$861$	18.2183	−	14.4813i	0.620877	−	0.493520i
$862$		0			0
$863$	−4.11255	−	15.3482i	−0.139993	−	0.522460i	−0.999927	−	0.0120559i	$-0.996162\pi$
$863$	−4.11255	−	15.3482i	−0.139993	−	0.522460i	0.859934	−	0.510404i	$-0.170504\pi$
$864$		0			0
$865$	13.2279	−	5.36159i	0.449762	−	0.182299i
$866$		0			0
$867$	−2.14578	+	0.531934i	−0.0728746	+	0.0180654i
$868$		0			0
$869$	14.7652			0.500875
$870$		0			0
$871$	−12.8186	−	22.2024i	−0.434341	−	0.752300i
$872$		0			0
$873$	−20.3829	−	21.9764i	−0.689855	−	0.743788i
$874$		0			0
$875$	−28.4699	+	8.02898i	−0.962458	+	0.271429i
$876$		0			0
$877$	−2.47546	−	9.23854i	−0.0835903	−	0.311963i	0.911453	−	0.411404i	$-0.134961\pi$
$877$	−2.47546	−	9.23854i	−0.0835903	−	0.311963i	−0.995044	+	0.0994405i	$0.968295\pi$
$878$		0			0
$879$	−0.267552	+	14.2293i	−0.00902429	+	0.479942i
$880$		0			0
$881$			47.9942i			1.61697i	0.588519	+	0.808483i	$0.299711\pi$
$881$			47.9942i			1.61697i	−0.588519	+	0.808483i	$0.700289\pi$
$882$		0			0
$883$	7.51781	+	7.51781i	0.252994	+	0.252994i	0.822197	−	0.569203i	$-0.192748\pi$
$883$	7.51781	+	7.51781i	0.252994	+	0.252994i	−0.569203	+	0.822197i	$0.692748\pi$
$884$		0			0
$885$	−0.889582	+	5.60819i	−0.0299030	+	0.188517i
$886$		0			0
$887$	−18.2956	+	4.90230i	−0.614307	+	0.164603i	−0.552538	−	0.833487i	$-0.686341\pi$
$887$	−18.2956	+	4.90230i	−0.614307	+	0.164603i	−0.0617686	+	0.998090i	$0.519674\pi$
$888$		0			0
$889$	−5.55877	−	6.72954i	−0.186435	−	0.225701i
$890$		0			0
$891$	−18.3154	+	8.81115i	−0.613589	+	0.295185i
$892$		0			0
$893$	−96.0652	−	25.7406i	−3.21470	−	0.861376i
$894$		0			0
$895$	−18.4414	+	13.9289i	−0.616429	+	0.465591i
$896$		0			0
$897$	−8.47407	−	34.1838i	−0.282941	−	1.14136i
$898$		0			0
$899$	−8.79482	−	15.2331i	−0.293324	−	0.508052i
$900$		0			0
$901$	−9.20432	+	15.9424i	−0.306640	+	0.531117i
$902$		0			0
$903$	−3.06648	+	26.8324i	−0.102046	+	0.892926i
$904$		0			0
$905$	0.666051	+	5.37497i	0.0221403	+	0.178670i
$906$		0			0
$907$	37.1996	+	9.96761i	1.23519	+	0.330969i	0.816599	−	0.577205i	$-0.195857\pi$
$907$	37.1996	+	9.96761i	1.23519	+	0.330969i	0.418594	+	0.908174i	$0.362523\pi$
$908$		0			0
$909$	8.28488	+	15.6834i	0.274792	+	0.520186i
$910$		0			0
$911$		−	11.7562i		−	0.389501i	−0.980853	−	0.194751i	$-0.937610\pi$
$911$		−	11.7562i		−	0.389501i	0.980853	−	0.194751i	$-0.0623897\pi$
$912$		0			0
$913$	−6.06249	+	22.6255i	−0.200639	+	0.748795i
$914$		0			0
$915$	−29.0942	+	11.1605i	−0.961824	+	0.368954i
$916$		0			0
$917$	−26.1964	+	36.8042i	−0.865083	+	1.21538i
$918$		0			0
$919$	7.47875	+	4.31786i	0.246701	+	0.142433i	0.618253	−	0.785979i	$-0.287841\pi$
$919$	7.47875	+	4.31786i	0.246701	+	0.142433i	−0.371552	+	0.928412i	$0.621174\pi$
$920$		0			0
$921$	−0.649172	+	34.5251i	−0.0213909	+	1.13764i
$922$		0			0
$923$	16.0825	−	16.0825i	0.529362	−	0.529362i
$924$		0			0
$925$	3.56413	+	6.39615i	0.117188	+	0.210304i
$926$		0			0
$927$	2.90725	+	12.7493i	0.0954865	+	0.418743i
$928$		0			0
$929$	24.2232	−	41.9559i	0.794739	−	1.37653i	−0.128266	−	0.991740i	$-0.540941\pi$
$929$	24.2232	−	41.9559i	0.794739	−	1.37653i	0.923005	−	0.384788i	$-0.125725\pi$
$930$		0			0
$931$	−41.7366	−	36.1341i	−1.36786	−	1.18425i
$932$		0			0
$933$	−14.6257	+	50.7476i	−0.478823	+	1.66140i
$934$		0			0
$935$	20.0069	−	8.10929i	0.654296	−	0.265202i
$936$		0			0
$937$	41.6088	−	41.6088i	1.35930	−	1.35930i	0.484521	−	0.874779i	$-0.338994\pi$
$937$	41.6088	−	41.6088i	1.35930	−	1.35930i	0.874779	−	0.484521i	$-0.161006\pi$
$938$		0			0
$939$	11.2756	−	18.7087i	0.367966	−	0.610534i
$940$		0			0
$941$	−38.9456	+	22.4852i	−1.26959	+	0.732997i	−0.974910	−	0.222599i	$-0.928546\pi$
$941$	−38.9456	+	22.4852i	−1.26959	+	0.732997i	−0.294679	+	0.955596i	$0.595213\pi$
$942$		0			0
$943$	−43.8511	+	11.7499i	−1.42799	+	0.382628i
$944$		0			0
$945$	−21.1177	−	22.3393i	−0.686959	−	0.726696i
$946$		0			0
$947$	−16.7281	+	4.48228i	−0.543591	+	0.145655i	−0.520157	−	0.854071i	$-0.674127\pi$
$947$	−16.7281	+	4.48228i	−0.543591	+	0.145655i	−0.0234337	+	0.999725i	$0.507460\pi$
$948$		0			0
$949$	−2.68902	+	1.55251i	−0.0872892	+	0.0503964i
$950$		0			0
$951$	4.22382	−	7.00821i	0.136967	−	0.227257i
$952$		0			0
$953$	12.2360	−	12.2360i	0.396362	−	0.396362i	−0.480586	−	0.876948i	$-0.659576\pi$
$953$	12.2360	−	12.2360i	0.396362	−	0.396362i	0.876948	+	0.480586i	$0.159576\pi$
$954$		0			0
$955$	−15.5772	+	36.8118i	−0.504068	+	1.19120i
$956$		0			0
$957$	3.61992	−	12.5603i	0.117016	−	0.406016i
$958$		0			0
$959$	−14.8798	−	5.54630i	−0.480494	−	0.179099i
$960$		0			0
$961$	1.64796	−	2.85434i	0.0531599	−	0.0920756i
$962$		0			0
$963$	−5.84890	−	25.6495i	−0.188478	−	0.826544i
$964$		0			0
$965$	35.7925	+	4.98959i	1.15220	+	0.160621i
$966$		0			0
$967$	−22.2486	+	22.2486i	−0.715466	+	0.715466i	−0.967673	−	0.252207i	$-0.918844\pi$
$967$	−22.2486	+	22.2486i	−0.715466	+	0.715466i	0.252207	+	0.967673i	$0.418844\pi$
$968$		0			0
$969$	1.09783	−	58.3863i	0.0352674	−	1.87564i
$970$		0			0
$971$	−40.2070	−	23.2135i	−1.29030	−	0.744957i	−0.311595	−	0.950215i	$-0.600863\pi$
$971$	−40.2070	−	23.2135i	−1.29030	−	0.744957i	−0.978708	+	0.205258i	$0.934197\pi$
$972$		0			0
$973$	3.69030	+	0.351595i	0.118306	+	0.0112716i
$974$		0			0
$975$	19.1899	−	4.44822i	0.614570	−	0.142457i
$976$		0			0
$977$	1.13940	−	4.25230i	0.0364526	−	0.136043i	−0.945301	−	0.326198i	$-0.894232\pi$
$977$	1.13940	−	4.25230i	0.0364526	−	0.136043i	0.981754	+	0.190155i	$0.0608991\pi$
$978$		0			0
$979$			20.9326i			0.669008i
$980$		0			0
$981$	3.33559	+	6.31431i	0.106497	+	0.201601i
$982$		0			0
$983$	9.69254	+	2.59711i	0.309144	+	0.0828349i	0.410056	−	0.912060i	$-0.365509\pi$
$983$	9.69254	+	2.59711i	0.309144	+	0.0828349i	−0.100912	+	0.994895i	$0.532176\pi$
$984$		0			0
$985$	9.47415	−	12.1543i	0.301872	−	0.387268i
$986$		0			0
$987$	46.4428	+	34.3905i	1.47829	+	1.09466i
$988$		0			0
$989$	26.3414	−	45.6247i	0.837609	−	1.45078i
$990$		0			0
$991$	1.27892	+	2.21515i	0.0406261	+	0.0703666i	0.885624	−	0.464404i	$-0.153731\pi$
$991$	1.27892	+	2.21515i	0.0406261	+	0.0703666i	−0.844997	+	0.534770i	$0.820398\pi$
$992$		0			0
$993$	−4.57939	−	18.4729i	−0.145322	−	0.586220i
$994$		0			0
$995$	17.1171	+	22.6625i	0.542648	+	0.718449i
$996$		0			0
$997$	49.9388	+	13.3811i	1.58158	+	0.423783i	0.939414	−	0.342784i	$-0.111370\pi$
$997$	49.9388	+	13.3811i	1.58158	+	0.423783i	0.642164	+	0.766567i	$0.278037\pi$
$998$		0			0
$999$	−4.17013	+	6.36497i	−0.131937	+	0.201379i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	420.2.bv.c.233.10	yes	48
3.2	odd	2	inner	420.2.bv.c.233.5	yes	48
5.2	odd	4	inner	420.2.bv.c.317.4	yes	48
7.4	even	3	inner	420.2.bv.c.53.2	✓	48
15.2	even	4	inner	420.2.bv.c.317.2	yes	48
21.11	odd	6	inner	420.2.bv.c.53.4	yes	48
35.32	odd	12	inner	420.2.bv.c.137.5	yes	48
105.32	even	12	inner	420.2.bv.c.137.10	yes	48

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
420.2.bv.c.53.2	✓	48	7.4	even	3	inner
420.2.bv.c.53.4	yes	48	21.11	odd	6	inner
420.2.bv.c.137.5	yes	48	35.32	odd	12	inner
420.2.bv.c.137.10	yes	48	105.32	even	12	inner
420.2.bv.c.233.5	yes	48	3.2	odd	2	inner
420.2.bv.c.233.10	yes	48	1.1	even	1	trivial
420.2.bv.c.317.2	yes	48	15.2	even	4	inner
420.2.bv.c.317.4	yes	48	5.2	odd	4	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 420 = 2^{2} \cdot 3 \cdot 5 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	420.bv (of order \(12\), degree \(4\), minimal)

\(f(q)\)	\(=\)	\(q+(1.24755 - 1.20150i) q^{3} +(0.274984 + 2.21910i) q^{5} +(0.250938 - 2.63382i) q^{7} +(0.112777 - 2.99788i) q^{9} +(1.95574 - 1.12915i) q^{11} +(-1.60840 - 1.60840i) q^{13} +(3.00931 + 2.43804i) q^{15} +(1.10647 - 4.12942i) q^{17} +(6.82987 + 3.94323i) q^{19} +(-2.85149 - 3.58734i) q^{21} +(2.31365 + 8.63467i) q^{23} +(-4.84877 + 1.22043i) q^{25} +(-3.46127 - 3.87552i) q^{27} +3.34184 q^{29} +(-2.63173 - 4.55830i) q^{31} +(1.08321 - 3.75850i) q^{33} +(5.91371 - 0.167404i) q^{35} +(-0.379022 - 1.41453i) q^{37} +(-3.93905 - 0.0740656i) q^{39} +5.07849i q^{41} +(-4.16728 - 4.16728i) q^{43} +(6.68359 - 0.574105i) q^{45} +(-12.1810 + 3.26390i) q^{47} +(-6.87406 - 1.32186i) q^{49} +(-3.58112 - 6.48110i) q^{51} +(-4.15930 - 1.11448i) q^{53} +(3.04348 + 4.02947i) q^{55} +(13.2584 - 3.28673i) q^{57} +(0.733066 + 1.26971i) q^{59} +(-4.02291 + 6.96788i) q^{61} +(-7.86759 - 1.04932i) q^{63} +(3.12690 - 4.01147i) q^{65} +(10.8869 + 2.91714i) q^{67} +(13.2610 + 7.99235i) q^{69} +9.99909i q^{71} +(0.353306 - 1.31856i) q^{73} +(-4.58274 + 7.34837i) q^{75} +(-2.48320 - 5.43442i) q^{77} +(5.66226 + 3.26911i) q^{79} +(-8.97456 - 0.676187i) q^{81} +(-7.33432 + 7.33432i) q^{83} +(9.46783 + 1.31985i) q^{85} +(4.16912 - 4.01523i) q^{87} +(-4.63460 + 8.02736i) q^{89} +(-4.63984 + 3.83262i) q^{91} +(-8.76004 - 2.52468i) q^{93} +(-6.87229 + 16.2405i) q^{95} +(7.06487 - 7.06487i) q^{97} +(-3.16448 - 5.99041i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 48 q - 2 q^{3} - 8 q^{7} + 32 q^{13} - 16 q^{21} + 16 q^{25} - 32 q^{27} - 64 q^{31} + 34 q^{33} - 40 q^{37} - 24 q^{43} + 30 q^{45} + 36 q^{51} + 92 q^{57} - 18 q^{63} + 28 q^{67} - 8 q^{73} - 38 q^{75}+ \cdots + 56 q^{97}+O(q^{100}) \) 48 * q - 2 * q^3 - 8 * q^7 + 32 * q^13 - 16 * q^21 + 16 * q^25 - 32 * q^27 - 64 * q^31 + 34 * q^33 - 40 * q^37 - 24 * q^43 + 30 * q^45 + 36 * q^51 + 92 * q^57 - 18 * q^63 + 28 * q^67 - 8 * q^73 - 38 * q^75 + 20 * q^81 - 56 * q^85 - 24 * q^87 - 24 * q^91 + 6 * q^93 + 56 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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