LMFDB - Embedded newform 420.2.bv.c.317.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [420,2,Mod(53,420)]

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

lf = mfeigenbasis(mf)

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")

//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: f = N[0] # Warning: the index may be different

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			317.4
Character	$\chi$	$=$	420.317
Dual form			420.2.bv.c.53.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-1.20150 - 1.24755i) q^{3} +(2.05928 - 0.871404i) q^{5} +(2.63382 + 0.250938i) q^{7} +(-0.112777 + 2.99788i) q^{9} +O(q^{10})$	\(q+(-1.20150 - 1.24755i) q^{3} +(2.05928 - 0.871404i) q^{5} +(2.63382 + 0.250938i) q^{7} +(-0.112777 + 2.99788i) q^{9} +(1.95574 - 1.12915i) q^{11} +(-1.60840 + 1.60840i) q^{13} +(-3.56136 - 1.52207i) q^{15} +(4.12942 + 1.10647i) q^{17} +(-6.82987 - 3.94323i) q^{19} +(-2.85149 - 3.58734i) q^{21} +(8.63467 - 2.31365i) q^{23} +(3.48131 - 3.58894i) q^{25} +(3.87552 - 3.46127i) q^{27} -3.34184 q^{29} +(-2.63173 - 4.55830i) q^{31} +(-3.75850 - 1.08321i) q^{33} +(5.64246 - 1.77837i) q^{35} +(1.41453 - 0.379022i) q^{37} +(3.93905 + 0.0740656i) q^{39} +5.07849i q^{41} +(-4.16728 + 4.16728i) q^{43} +(2.38012 + 6.27176i) q^{45} +(-3.26390 - 12.1810i) q^{47} +(6.87406 + 1.32186i) q^{49} +(-3.58112 - 6.48110i) q^{51} +(-1.11448 + 4.15930i) q^{53} +(3.04348 - 4.02947i) q^{55} +(3.28673 + 13.2584i) q^{57} +(-0.733066 - 1.26971i) q^{59} +(-4.02291 + 6.96788i) q^{61} +(-1.04932 + 7.86759i) q^{63} +(-1.91058 + 4.71371i) q^{65} +(-2.91714 + 10.8869i) q^{67} +(-13.2610 - 7.99235i) q^{69} +9.99909i q^{71} +(-1.31856 - 0.353306i) q^{73} +(-8.66020 - 0.0309927i) q^{75} +(5.43442 - 2.48320i) q^{77} +(-5.66226 - 3.26911i) q^{79} +(-8.97456 - 0.676187i) q^{81} +(7.33432 + 7.33432i) q^{83} +(9.46783 - 1.31985i) q^{85} +(4.01523 + 4.16912i) q^{87} +(4.63460 - 8.02736i) q^{89} +(-4.63984 + 3.83262i) q^{91} +(-2.52468 + 8.76004i) q^{93} +(-17.5008 - 2.16865i) q^{95} +(7.06487 + 7.06487i) q^{97} +(3.16448 + 5.99041i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 48 q - 2 q^{3} - 8 q^{7}+O(q^{10}) $ 48 * q - 2 * q^3 - 8 * q^7	$ 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}+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

Display 100 coefficients 10 coefficients

Character values

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

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

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
$3$	−1.20150	−	1.24755i	−0.693689	−	0.720275i
$3$	−1.20150	−	1.24755i	−0.693689	−	0.720275i
$4$		0			0
$5$	2.05928	−	0.871404i	0.920940	−	0.389704i
$5$	2.05928	−	0.871404i	0.920940	−	0.389704i
$6$		0			0
$7$	2.63382	+	0.250938i	0.995492	+	0.0948458i
$7$	2.63382	+	0.250938i	0.995492	+	0.0948458i
$8$		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	−0.894052	−	0.447963i	$-0.852150\pi$
$13$	−1.60840	+	1.60840i	−0.446089	+	0.446089i	0.447963	+	0.894052i	$0.352150\pi$
$14$		0			0
$15$	−3.56136	−	1.52207i	−0.919540	−	0.392997i
$16$		0			0
$17$	4.12942	+	1.10647i	1.00153	+	0.268359i	0.722087	−	0.691803i	$-0.243183\pi$
$17$	4.12942	+	1.10647i	1.00153	+	0.268359i	0.279444	+	0.960162i	$0.409850\pi$
$18$		0			0
$19$	−6.82987	−	3.94323i	−1.56688	−	0.904639i	−0.996530	−	0.0832371i	$-0.973474\pi$
$19$	−6.82987	−	3.94323i	−1.56688	−	0.904639i	−0.570350	−	0.821402i	$-0.693193\pi$
$20$		0			0
$21$	−2.85149	−	3.58734i	−0.622246	−	0.782822i
$22$		0			0
$23$	8.63467	−	2.31365i	1.80045	−	0.482430i	0.806405	−	0.591364i	$-0.201410\pi$
$23$	8.63467	−	2.31365i	1.80045	−	0.482430i	0.994049	+	0.108933i	$0.0347435\pi$
$24$		0			0
$25$	3.48131	−	3.58894i	0.696262	−	0.717788i
$26$		0			0
$27$	3.87552	−	3.46127i	0.745843	−	0.666121i
$28$		0			0
$29$	−3.34184			−0.620563			−0.310282	−	0.950645i	$-0.600423\pi$
$29$	−3.34184			−0.620563			−0.310282	+	0.950645i	$0.600423\pi$
$30$		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
$33$	−3.75850	−	1.08321i	−0.654270	−	0.188563i
$34$		0			0
$35$	5.64246	−	1.77837i	0.953750	−	0.300600i
$36$		0			0
$37$	1.41453	−	0.379022i	0.232547	−	0.0623108i	−0.140663	−	0.990057i	$-0.544924\pi$
$37$	1.41453	−	0.379022i	0.232547	−	0.0623108i	0.373211	+	0.927747i	$0.378257\pi$
$38$		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	−0.949443	−	0.313939i	$-0.898351\pi$
$43$	−4.16728	+	4.16728i	−0.635504	+	0.635504i	0.313939	+	0.949443i	$0.398351\pi$
$44$		0			0
$45$	2.38012	+	6.27176i	0.354808	+	0.934939i
$46$		0			0
$47$	−3.26390	−	12.1810i	−0.476089	−	1.77679i	−0.617214	−	0.786795i	$-0.711739\pi$
$47$	−3.26390	−	12.1810i	−0.476089	−	1.77679i	0.141126	−	0.989992i	$-0.454928\pi$
$48$		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
$53$	−1.11448	+	4.15930i	−0.153086	+	0.571324i	0.846176	+	0.532904i	$0.178899\pi$
$53$	−1.11448	+	4.15930i	−0.153086	+	0.571324i	−0.999262	+	0.0384204i	$0.987767\pi$
$54$		0			0
$55$	3.04348	−	4.02947i	0.410383	−	0.543334i
$56$		0			0
$57$	3.28673	+	13.2584i	0.435338	+	1.75612i
$58$		0			0
$59$	−0.733066	−	1.26971i	−0.0954370	−	0.165302i	0.814354	−	0.580369i	$-0.197092\pi$
$59$	−0.733066	−	1.26971i	−0.0954370	−	0.165302i	−0.909791	+	0.415067i	$0.863758\pi$
$60$		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
$63$	−1.04932	+	7.86759i	−0.132202	+	0.991223i
$64$		0			0
$65$	−1.91058	+	4.71371i	−0.236979	+	0.584664i
$66$		0			0
$67$	−2.91714	+	10.8869i	−0.356386	+	1.33005i	0.522346	+	0.852734i	$0.325057\pi$
$67$	−2.91714	+	10.8869i	−0.356386	+	1.33005i	−0.878732	+	0.477316i	$0.841610\pi$
$68$		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
$73$	−1.31856	−	0.353306i	−0.154325	−	0.0413514i	0.180829	−	0.983515i	$-0.442122\pi$
$73$	−1.31856	−	0.353306i	−0.154325	−	0.0413514i	−0.335155	+	0.942163i	$0.608789\pi$
$74$		0			0
$75$	−8.66020	−	0.0309927i	−0.999994	−	0.00357873i
$76$		0			0
$77$	5.43442	−	2.48320i	0.619309	−	0.282987i
$78$		0			0
$79$	−5.66226	−	3.26911i	−0.637054	−	0.367804i	0.146425	−	0.989222i	$-0.453223\pi$
$79$	−5.66226	−	3.26911i	−0.637054	−	0.367804i	−0.783479	+	0.621418i	$0.786557\pi$
$80$		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	$-0.0572324\pi$
$83$	7.33432	+	7.33432i	0.805046	+	0.805046i	−0.178834	+	0.983879i	$0.557232\pi$
$84$		0			0
$85$	9.46783	−	1.31985i	1.02693	−	0.143158i
$86$		0			0
$87$	4.01523	+	4.16912i	0.430478	+	0.446976i
$88$		0			0
$89$	4.63460	−	8.02736i	0.491267	−	0.850899i	−0.508683	−	0.860954i	$-0.669867\pi$
$89$	4.63460	−	8.02736i	0.491267	−	0.850899i	0.999949	+	0.0100553i	$0.00320075\pi$
$90$		0			0
$91$	−4.63984	+	3.83262i	−0.486387	+	0.401768i
$92$		0			0
$93$	−2.52468	+	8.76004i	−0.261797	+	0.908374i
$94$		0			0
$95$	−17.5008	−	2.16865i	−1.79554	−	0.222499i
$96$		0			0
$97$	7.06487	+	7.06487i	0.717328	+	0.717328i	0.968057	−	0.250729i	$-0.0806703\pi$
$97$	7.06487	+	7.06487i	0.717328	+	0.717328i	−0.250729	+	0.968057i	$0.580670\pi$
$98$		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
$103$	1.12816	+	4.21034i	0.111161	+	0.414857i	0.998971	−	0.0453523i	$-0.0144410\pi$
$103$	1.12816	+	4.21034i	0.111161	+	0.414857i	−0.887810	+	0.460209i	$0.847774\pi$
$104$		0			0
$105$	−8.99805	−	4.90255i	−0.878120	−	0.478440i
$106$		0			0
$107$	2.26966	+	8.47050i	0.219417	+	0.818874i	0.984565	+	0.175020i	$0.0559990\pi$
$107$	2.26966	+	8.47050i	0.219417	+	0.818874i	−0.765148	+	0.643854i	$0.777334\pi$
$108$		0			0
$109$	2.06149	−	1.19020i	0.197455	−	0.114000i	−0.398013	−	0.917380i	$-0.630300\pi$
$109$	2.06149	−	1.19020i	0.197455	−	0.114000i	0.595468	+	0.803379i	$0.296967\pi$
$110$		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	$-0.333113\pi$
$113$	−3.88087	−	3.88087i	−0.365082	−	0.365082i	−0.865680	+	0.500598i	$0.833113\pi$
$114$		0			0
$115$	15.7651	−	12.2888i	1.47011	−	1.14593i
$116$		0			0
$117$	−4.64039	−	5.00317i	−0.429004	−	0.462543i
$118$		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
$123$	6.33569	−	6.10183i	0.571270	−	0.550183i
$124$		0			0
$125$	4.04159	−	10.4243i	0.361491	−	0.932376i
$126$		0			0
$127$	2.33279	+	2.33279i	0.207001	+	0.207001i	0.802992	−	0.595990i	$-0.203240\pi$
$127$	2.33279	+	2.33279i	0.207001	+	0.207001i	−0.595990	+	0.802992i	$0.703240\pi$
$128$		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
$133$	−16.9992	−	12.0996i	−1.47402	−	1.04917i
$134$		0			0
$135$	4.96463	−	10.5049i	0.427287	−	0.904116i
$136$		0			0
$137$	5.79751	+	1.55344i	0.495315	+	0.132719i	0.497825	−	0.867278i	$-0.334132\pi$
$137$	5.79751	+	1.55344i	0.495315	+	0.132719i	−0.00250999	+	0.999997i	$0.500799\pi$
$138$		0			0
$139$		−	1.40112i		−	0.118841i	−0.998233	−	0.0594207i	$-0.981075\pi$
$139$		−	1.40112i		−	0.118841i	0.998233	−	0.0594207i	$-0.0189253\pi$
$140$		0			0
$141$	−11.2749	+	18.7074i	−0.949518	+	1.57545i
$142$		0			0
$143$	−1.32949	+	4.96172i	−0.111177	+	0.414920i
$144$		0			0
$145$	−6.88179	+	2.91209i	−0.571502	+	0.241836i
$146$		0			0
$147$	−6.61012	−	10.1640i	−0.545194	−	0.838310i
$148$		0			0
$149$	−0.789937	+	1.36821i	−0.0647142	+	0.112088i	−0.896567	−	0.442908i	$-0.853947\pi$
$149$	−0.789937	+	1.36821i	−0.0647142	+	0.112088i	0.831853	+	0.554996i	$0.187280\pi$
$150$		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
$153$	−3.78278	+	12.2547i	−0.305820	+	0.990734i
$154$		0			0
$155$	−9.39161	−	7.09353i	−0.754352	−	0.569766i
$156$		0			0
$157$	2.50127	−	9.33487i	0.199623	−	0.745003i	−0.791398	−	0.611301i	$-0.790647\pi$
$157$	2.50127	−	9.33487i	0.199623	−	0.745003i	0.991021	−	0.133703i	$-0.0426867\pi$
$158$		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
$163$	−1.39612	−	5.21038i	−0.109352	−	0.408108i	0.889450	−	0.457032i	$-0.151088\pi$
$163$	−1.39612	−	5.21038i	−0.109352	−	0.408108i	−0.998803	+	0.0489238i	$0.984421\pi$
$164$		0			0
$165$	−8.68374	+	1.04452i	−0.676028	+	0.0813161i
$166$		0			0
$167$	−12.9779	+	12.9779i	−1.00426	+	1.00426i	−0.00426571	+	0.999991i	$0.501358\pi$
$167$	−12.9779	+	12.9779i	−1.00426	+	1.00426i	−0.999991	+	0.00426571i	$0.998642\pi$
$168$		0			0
$169$			7.82612i			0.602010i
$170$		0			0
$171$	12.5916	−	20.0304i	0.962902	−	1.53176i
$172$		0			0
$173$	−6.16566	+	1.65208i	−0.468766	+	0.125606i	−0.485467	−	0.874255i	$-0.661350\pi$
$173$	−6.16566	+	1.65208i	−0.468766	+	0.125606i	0.0167007	+	0.999861i	$0.494684\pi$
$174$		0			0
$175$	10.0698	−	8.57904i	0.761202	−	0.648515i
$176$		0			0
$177$	−0.703246	+	2.44010i	−0.0528592	+	0.183409i
$178$		0			0
$179$	−5.16769	−	8.95070i	−0.386251	−	0.669006i	0.605691	−	0.795700i	$-0.292897\pi$
$179$	−5.16769	−	8.95070i	−0.386251	−	0.669006i	−0.991942	+	0.126694i	$0.959564\pi$
$180$		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
$183$	13.5263	−	3.35314i	0.999896	−	0.247872i
$184$		0			0
$185$	2.58264	−	2.01314i	0.189879	−	0.148009i
$186$		0			0
$187$	9.32543	−	2.49874i	0.681943	−	0.182726i
$188$		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
$193$	−15.6110	−	4.18295i	−1.12370	−	0.301095i	−0.351321	−	0.936255i	$-0.614268\pi$
$193$	−15.6110	−	4.18295i	−1.12370	−	0.301095i	−0.772381	+	0.635160i	$0.780934\pi$
$194$		0			0
$195$	8.17617	−	3.27999i	0.585508	−	0.234885i
$196$		0			0
$197$	4.87326	−	4.87326i	0.347205	−	0.347205i	−0.511862	−	0.859068i	$-0.671044\pi$
$197$	4.87326	−	4.87326i	0.347205	−	0.347205i	0.859068	+	0.511862i	$0.171044\pi$
$198$		0			0
$199$	−10.9994	+	6.35052i	−0.779729	+	0.450176i	−0.836334	−	0.548220i	$-0.815306\pi$
$199$	−10.9994	+	6.35052i	−0.779729	+	0.450176i	0.0566055	+	0.998397i	$0.481972\pi$
$200$		0			0
$201$	17.0870	−	9.44139i	1.20522	−	0.665944i
$202$		0			0
$203$	−8.80181	−	0.838595i	−0.617766	−	0.0588578i
$204$		0			0
$205$	4.42542	+	10.4581i	0.309085	+	0.730423i
$206$		0			0
$207$	5.96226	+	26.1466i	0.414406	+	1.81732i
$208$		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
$213$	12.4744	−	12.0139i	0.854731	−	0.823182i
$214$		0			0
$215$	−4.95023	+	12.2130i	−0.337603	+	0.832919i
$216$		0			0
$217$	−5.78767	−	12.6662i	−0.392893	−	0.859835i
$218$		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	−0.935559	−	0.353171i	$-0.885103\pi$
$223$	−8.69690	+	8.69690i	−0.582387	+	0.582387i	0.353171	+	0.935559i	$0.385103\pi$
$224$		0			0
$225$	10.3666	+	10.8413i	0.691106	+	0.722753i
$226$		0			0
$227$	12.6513	+	3.38991i	0.839697	+	0.224996i	0.652940	−	0.757410i	$-0.273535\pi$
$227$	12.6513	+	3.38991i	0.839697	+	0.224996i	0.186757	+	0.982406i	$0.440202\pi$
$228$		0			0
$229$	−4.89414	−	2.82564i	−0.323414	−	0.186723i	0.329499	−	0.944156i	$-0.393120\pi$
$229$	−4.89414	−	2.82564i	−0.323414	−	0.186723i	−0.652913	+	0.757433i	$0.726453\pi$
$230$		0			0
$231$	−9.62740	−	3.79615i	−0.633437	−	0.249768i
$232$		0			0
$233$	−0.397381	+	0.106478i	−0.0260333	+	0.00697559i	−0.271812	−	0.962350i	$-0.587623\pi$
$233$	−0.397381	+	0.106478i	−0.0260333	+	0.00697559i	0.245779	+	0.969326i	$0.420956\pi$
$234$		0			0
$235$	−17.3359	−	22.2400i	−1.13087	−	1.45078i
$236$		0			0
$237$	2.72484	+	10.9918i	0.176998	+	0.713996i
$238$		0			0
$239$	−0.602538			−0.0389749			−0.0194875	−	0.999810i	$-0.506203\pi$
$239$	−0.602538			−0.0389749			−0.0194875	+	0.999810i	$0.506203\pi$
$240$		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
$243$	9.93939	+	12.0087i	0.637612	+	0.770357i
$244$		0			0
$245$	15.3075	−	3.26801i	0.977961	−	0.208785i
$246$		0			0
$247$	17.3274	−	4.64287i	1.10252	−	0.295418i
$248$		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
$255$	−13.0222	−	10.2258i	−0.815483	−	0.640366i
$256$		0			0
$257$	−6.31782	−	23.5784i	−0.394095	−	1.47078i	−0.823317	−	0.567582i	$-0.807879\pi$
$257$	−6.31782	−	23.5784i	−0.394095	−	1.47078i	0.429222	−	0.903199i	$-0.358788\pi$
$258$		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
$263$	4.10523	−	15.3209i	0.253139	−	0.944729i	−0.715977	−	0.698124i	$-0.754018\pi$
$263$	4.10523	−	15.3209i	0.253139	−	0.944729i	0.969116	−	0.246605i	$-0.0793150\pi$
$264$		0			0
$265$	1.32940	+	9.53636i	0.0816644	+	0.585814i
$266$		0			0
$267$	−15.5830	+	3.86300i	−0.953667	+	0.236412i
$268$		0			0
$269$	12.1007	+	20.9591i	0.737796	+	1.27790i	0.953486	+	0.301438i	$0.0974665\pi$
$269$	12.1007	+	20.9591i	0.737796	+	1.27790i	−0.215690	+	0.976462i	$0.569200\pi$
$270$		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
$273$	10.3562	+	1.18354i	0.626785	+	0.0716308i
$274$		0			0
$275$	2.75609	−	10.9499i	0.166199	−	0.660306i
$276$		0			0
$277$	−1.45148	+	5.41699i	−0.0872108	+	0.325475i	−0.995724	−	0.0923818i	$-0.970552\pi$
$277$	−1.45148	+	5.41699i	−0.0872108	+	0.325475i	0.908513	+	0.417857i	$0.137219\pi$
$278$		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
$283$	−4.31710	−	1.15676i	−0.256625	−	0.0687624i	0.128213	−	0.991747i	$-0.459076\pi$
$283$	−4.31710	−	1.15676i	−0.256625	−	0.0687624i	−0.384838	+	0.922984i	$0.625743\pi$
$284$		0			0
$285$	18.3218	+	24.4388i	1.08529	+	1.44763i
$286$		0			0
$287$	−1.27439	+	13.3759i	−0.0752248	+	0.789552i
$288$		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	$-0.172848\pi$
$293$	5.81011	+	5.81011i	0.339430	+	0.339430i	−0.516723	+	0.856153i	$0.672848\pi$
$294$		0			0
$295$	−2.61602	−	1.97589i	−0.152311	−	0.115041i
$296$		0			0
$297$	3.67122	−	11.1454i	0.213026	−	0.646719i
$298$		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
$303$	9.84006	+	2.83595i	0.565297	+	0.162921i
$304$		0			0
$305$	−2.21247	+	17.8544i	−0.126686	+	1.02234i
$306$		0			0
$307$	−14.0973	−	14.0973i	−0.804576	−	0.804576i	0.179231	−	0.983807i	$-0.442639\pi$
$307$	−14.0973	−	14.0973i	−0.804576	−	0.804576i	−0.983807	+	0.179231i	$0.942639\pi$
$308$		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
$313$	−3.26411	−	12.1818i	−0.184498	−	0.688558i	−0.994737	−	0.102458i	$-0.967329\pi$
$313$	−3.26411	−	12.1818i	−0.184498	−	0.688558i	0.810239	−	0.586100i	$-0.199337\pi$
$314$		0			0
$315$	4.69500	+	17.1160i	0.264533	+	0.964377i
$316$		0			0
$317$	1.22273	+	4.56328i	0.0686752	+	0.256299i	0.991725	−	0.128382i	$-0.0409782\pi$
$317$	1.22273	+	4.56328i	0.0686752	+	0.256299i	−0.923050	+	0.384681i	$0.874312\pi$
$318$		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
$325$	0.173114	+	11.3718i	0.00960261	+	0.630792i
$326$		0			0
$327$	−3.96172	−	1.14178i	−0.219084	−	0.0631408i
$328$		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
$333$	0.976735	+	4.28333i	0.0535248	+	0.234725i
$334$		0			0
$335$	3.47969	+	24.9613i	0.190116	+	1.36378i
$336$		0			0
$337$	−23.9082	−	23.9082i	−1.30237	−	1.30237i	−0.926793	−	0.375572i	$-0.877446\pi$
$337$	−23.9082	−	23.9082i	−1.30237	−	1.30237i	−0.375572	−	0.926793i	$-0.622554\pi$
$338$		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
$343$	17.7734	+	5.20650i	0.959671	+	0.281125i
$344$		0			0
$345$	−34.2727	−	4.90283i	−1.84518	−	0.263960i
$346$		0			0
$347$	−6.21434	−	1.66513i	−0.333603	−	0.0893887i	0.0881292	−	0.996109i	$-0.471911\pi$
$347$	−6.21434	−	1.66513i	−0.333603	−	0.0893887i	−0.421732	+	0.906720i	$0.638578\pi$
$348$		0			0
$349$		−	15.3799i		−	0.823266i	−0.911350	−	0.411633i	$-0.864959\pi$
$349$		−	15.3799i		−	0.823266i	0.911350	−	0.411633i	$-0.135041\pi$
$350$		0			0
$351$	−0.666276	+	11.8005i	−0.0355632	+	0.629862i
$352$		0			0
$353$	−2.78555	+	10.3958i	−0.148260	+	0.553313i	0.851329	+	0.524632i	$0.175797\pi$
$353$	−2.78555	+	10.3958i	−0.148260	+	0.553313i	−0.999589	+	0.0286804i	$0.990869\pi$
$354$		0			0
$355$	8.71325	+	20.5910i	0.462451	+	1.09286i
$356$		0			0
$357$	−7.80570	−	17.9687i	−0.413121	−	0.951005i
$358$		0			0
$359$	−16.2835	+	28.2038i	−0.859410	+	1.48854i	0.0130825	+	0.999914i	$0.495836\pi$
$359$	−16.2835	+	28.2038i	−0.859410	+	1.48854i	−0.872493	+	0.488627i	$0.837498\pi$
$360$		0			0
$361$	21.5981	+	37.4090i	1.13674	+	1.96890i
$362$		0			0
$363$	9.91906	−	2.45891i	0.520616	−	0.129059i
$364$		0			0
$365$	−3.02316	+	0.421438i	−0.158239	+	0.0220591i
$366$		0			0
$367$	−3.34594	+	12.4872i	−0.174657	+	0.651828i	0.821953	+	0.569555i	$0.192885\pi$
$367$	−3.34594	+	12.4872i	−0.174657	+	0.651828i	−0.996610	+	0.0822728i	$0.973782\pi$
$368$		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
$373$	0.373327	+	1.39327i	0.0193301	+	0.0721410i	0.974917	−	0.222568i	$-0.0714439\pi$
$373$	0.373327	+	1.39327i	0.0193301	+	0.0721410i	−0.955587	+	0.294709i	$0.904777\pi$
$374$		0			0
$375$	−17.8608	+	7.48271i	−0.922329	+	0.386406i
$376$		0			0
$377$	5.37500	−	5.37500i	0.276826	−	0.276826i
$378$		0			0
$379$			36.9758i			1.89932i	0.313284	+	0.949659i	$0.398571\pi$
$379$			36.9758i			1.89932i	−0.313284	+	0.949659i	$0.601429\pi$
$380$		0			0
$381$	0.107423	−	5.71313i	0.00550346	−	0.292692i
$382$		0			0
$383$	−22.6321	+	6.06426i	−1.15645	+	0.309869i	−0.785546	−	0.618803i	$-0.787618\pi$
$383$	−22.6321	+	6.06426i	−1.15645	+	0.309869i	−0.370901	+	0.928672i	$0.620951\pi$
$384$		0			0
$385$	9.02714	−	9.84920i	0.460066	−	0.501962i
$386$		0			0
$387$	−12.0230	−	12.9630i	−0.611165	−	0.658945i
$388$		0			0
$389$	10.0292	+	17.3711i	0.508502	+	0.880752i	0.999952	+	0.00984566i	$0.00313402\pi$
$389$	10.0292	+	17.3711i	0.508502	+	0.880752i	−0.491449	+	0.870906i	$0.663533\pi$
$390$		0			0
$391$	38.2162			1.93267
$392$		0			0
$393$	7.11596	+	28.7053i	0.358953	+	1.44799i
$394$		0			0
$395$	−14.5089	−	1.79791i	−0.730023	−	0.0904625i
$396$		0			0
$397$	34.7342	−	9.30700i	1.74326	−	0.467105i	0.760092	−	0.649815i	$-0.225154\pi$
$397$	34.7342	−	9.30700i	1.74326	−	0.467105i	0.983167	+	0.182710i	$0.0584871\pi$
$398$		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
$403$	11.5644	+	3.09868i	0.576065	+	0.154356i
$404$		0			0
$405$	−19.0704	+	6.42801i	−0.947616	+	0.319411i
$406$		0			0
$407$	2.33848	−	2.33848i	0.115914	−	0.115914i
$408$		0			0
$409$	−10.3607	+	5.98176i	−0.512304	+	0.295779i	−0.733780	−	0.679387i	$-0.762246\pi$
$409$	−10.3607	+	5.98176i	−0.512304	+	0.295779i	0.221476	+	0.975166i	$0.428913\pi$
$410$		0			0
$411$	−5.02773	−	9.09916i	−0.248000	−	0.448829i
$412$		0			0
$413$	−1.61215	−	3.52814i	−0.0793286	−	0.173608i
$414$		0			0
$415$	21.4946	+	8.71229i	1.05513	+	0.427670i
$416$		0			0
$417$	−1.74797	+	1.68345i	−0.0855985	+	0.0824389i
$418$		0			0
$419$	−5.09737			−0.249023			−0.124511	−	0.992218i	$-0.539736\pi$
$419$	−5.09737			−0.249023			−0.124511	+	0.992218i	$0.539736\pi$
$420$		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
$423$	36.8854	−	8.41103i	1.79343	−	0.408958i
$424$		0			0
$425$	18.3468	−	10.9683i	0.889953	−	0.532038i
$426$		0			0
$427$	−12.3441	+	17.3427i	−0.597375	+	0.839271i
$428$		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	−0.831569	−	0.555421i	$-0.812557\pi$
$433$	−5.74628	+	5.74628i	−0.276149	+	0.276149i	0.555421	+	0.831569i	$0.312557\pi$
$434$		0			0
$435$	11.9015	+	5.08651i	0.570633	+	0.243880i
$436$		0			0
$437$	−68.0970	−	18.2465i	−3.25752	−	0.872850i
$438$		0			0
$439$	−35.1852	−	20.3142i	−1.67930	−	0.969544i	−0.962109	−	0.272666i	$-0.912095\pi$
$439$	−35.1852	−	20.3142i	−1.67930	−	0.969544i	−0.717190	−	0.696878i	$-0.754572\pi$
$440$		0			0
$441$	−4.73800	+	20.4585i	−0.225619	+	0.974216i
$442$		0			0
$443$	17.5944	−	4.71442i	0.835937	−	0.223989i	0.184635	−	0.982807i	$-0.440890\pi$
$443$	17.5944	−	4.71442i	0.835937	−	0.223989i	0.651302	+	0.758818i	$0.274223\pi$
$444$		0			0
$445$	2.54888	−	20.5692i	0.120829	−	0.975075i
$446$		0			0
$447$	2.65603	−	0.658423i	0.125626	−	0.0311423i
$448$		0			0
$449$	29.2170			1.37883			0.689417	−	0.724364i	$-0.257867\pi$
$449$	29.2170			1.37883			0.689417	+	0.724364i	$0.257867\pi$
$450$		0			0
$451$	5.73436	+	9.93220i	0.270020	+	0.467689i
$452$		0			0
$453$	7.06385	−	24.5099i	0.331889	−	1.15158i
$454$		0			0
$455$	−6.21499	+	11.9356i	−0.291363	+	0.559552i
$456$		0			0
$457$	−23.4995	+	6.29669i	−1.09926	+	0.294547i	−0.762464	−	0.647030i	$-0.776011\pi$
$457$	−23.4995	+	6.29669i	−1.09926	+	0.294547i	−0.336798	+	0.941577i	$0.609344\pi$
$458$		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	−0.452459	−	0.891785i	$-0.649453\pi$
$463$	9.45318	−	9.45318i	0.439327	−	0.439327i	0.891785	+	0.452459i	$0.149453\pi$
$464$		0			0
$465$	2.43450	+	20.2394i	0.112897	+	0.938581i
$466$		0			0
$467$	6.46882	+	24.1420i	0.299341	+	1.11716i	0.937708	+	0.347425i	$0.112944\pi$
$467$	6.46882	+	24.1420i	0.299341	+	1.11716i	−0.638366	+	0.769733i	$0.720390\pi$
$468$		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
$473$	−3.44464	+	12.8556i	−0.158385	+	0.591100i
$474$		0			0
$475$	−37.9289	+	10.7844i	−1.74030	+	0.494822i
$476$		0			0
$477$	−12.3434	−	3.81016i	−0.565166	−	0.174455i
$478$		0			0
$479$	−6.08042	−	10.5316i	−0.277821	−	0.481201i	0.693022	−	0.720917i	$-0.256279\pi$
$479$	−6.08042	−	10.5316i	−0.277821	−	0.481201i	−0.970843	+	0.239716i	$0.922946\pi$
$480$		0			0
$481$	−1.66551	+	2.88474i	−0.0759405	+	0.131533i
$482$		0			0
$483$	−32.9216	−	24.3781i	−1.49798	−	1.10924i
$484$		0			0
$485$	20.7049	+	8.39222i	0.940162	+	0.381071i
$486$		0			0
$487$	8.94495	−	33.3830i	0.405334	−	1.51273i	−0.398104	−	0.917340i	$-0.630332\pi$
$487$	8.94495	−	33.3830i	0.405334	−	1.51273i	0.803438	−	0.595388i	$-0.203002\pi$
$488$		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
$493$	−13.7998	−	3.69765i	−0.621513	−	0.166534i
$494$		0			0
$495$	11.7366	+	9.57842i	0.527523	+	0.430518i
$496$		0			0
$497$	−2.50915	+	26.3358i	−0.112551	+	1.18132i
$498$		0			0
$499$	−26.3227	−	15.1974i	−1.17837	−	0.680329i	−0.222729	−	0.974880i	$-0.571497\pi$
$499$	−26.3227	−	15.1974i	−1.17837	−	0.680329i	−0.955636	+	0.294551i	$0.904830\pi$
$500$		0			0
$501$	31.7835	+	0.597622i	1.41998	+	0.0266998i
$502$		0			0
$503$	23.4649	+	23.4649i	1.04625	+	1.04625i	0.998877	+	0.0473718i	$0.0150845\pi$
$503$	23.4649	+	23.4649i	1.04625	+	1.04625i	0.0473718	+	0.998877i	$0.484915\pi$
$504$		0			0
$505$	−7.96809	+	10.5495i	−0.354575	+	0.469447i
$506$		0			0
$507$	9.76351	−	9.40312i	0.433613	−	0.417607i
$508$		0			0
$509$	0.122791	−	0.212680i	0.00544260	−	0.00942686i	−0.863291	−	0.504706i	$-0.831601\pi$
$509$	0.122791	−	0.212680i	0.00544260	−	0.00942686i	0.868734	+	0.495279i	$0.164934\pi$
$510$		0			0
$511$	−3.38419	−	1.26142i	−0.149708	−	0.0558021i
$512$		0			0
$513$	−40.1178	+	8.35797i	−1.77125	+	0.369013i
$514$		0			0
$515$	5.99211	+	7.68721i	0.264044	+	0.338739i
$516$		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
$523$	5.81090	+	21.6866i	0.254093	+	0.948288i	0.968593	+	0.248652i	$0.0799876\pi$
$523$	5.81090	+	21.6866i	0.254093	+	0.948288i	−0.714500	+	0.699636i	$0.753346\pi$
$524$		0			0
$525$	−22.8017	−	2.25481i	−0.995146	−	0.0984078i
$526$		0			0
$527$	−5.82389	−	21.7350i	−0.253693	−	0.946794i
$528$		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
$535$	12.0551	+	15.4654i	0.521188	+	0.668626i
$536$		0			0
$537$	−4.95747	+	17.2013i	−0.213931	+	0.742289i
$538$		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
$543$	−2.91022	−	3.02175i	−0.124889	−	0.129676i
$544$		0			0
$545$	3.20804	−	4.24735i	0.137417	−	0.181936i
$546$		0			0
$547$	−11.0461	−	11.0461i	−0.472296	−	0.472296i	0.430361	−	0.902657i	$-0.358386\pi$
$547$	−11.0461	−	11.0461i	−0.472296	−	0.472296i	−0.902657	+	0.430361i	$0.858386\pi$
$548$		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
$553$	−14.0931	−	10.0311i	−0.599298	−	0.426567i
$554$		0			0
$555$	−5.61455	−	0.803180i	−0.238324	−	0.0340931i
$556$		0			0
$557$	36.2006	+	9.69992i	1.53387	+	0.410999i	0.924279	−	0.381719i	$-0.124668\pi$
$557$	36.2006	+	9.69992i	1.53387	+	0.410999i	0.609589	+	0.792717i	$0.291334\pi$
$558$		0			0
$559$		−	13.4053i		−	0.566982i
$560$		0			0
$561$	−14.3219	−	8.63172i	−0.604669	−	0.364432i
$562$		0			0
$563$	0.742806	−	2.77219i	0.0313056	−	0.116834i	−0.948505	−	0.316762i	$-0.897404\pi$
$563$	0.742806	−	2.77219i	0.0313056	−	0.116834i	0.979811	+	0.199928i	$0.0640709\pi$
$564$		0			0
$565$	−11.3736	−	4.61001i	−0.478492	−	0.193945i
$566$		0			0
$567$	−23.4677	−	4.03302i	−0.985552	−	0.169371i
$568$		0			0
$569$	17.2252	−	29.8350i	0.722119	−	1.25075i	−0.238029	−	0.971258i	$-0.576501\pi$
$569$	17.2252	−	29.8350i	0.722119	−	1.25075i	0.960149	−	0.279490i	$-0.0901652\pi$
$570$		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
$573$	−7.44994	−	30.0525i	−0.311226	−	1.25546i
$574$		0			0
$575$	21.7564	−	39.0439i	0.907305	−	1.62824i
$576$		0			0
$577$	2.86608	−	10.6964i	0.119317	−	0.445296i	−0.880257	−	0.474497i	$-0.842630\pi$
$577$	2.86608	−	10.6964i	0.119317	−	0.445296i	0.999574	+	0.0292017i	$0.00929650\pi$
$578$		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
$583$	2.51683	+	9.39293i	0.104236	+	0.389015i
$584$		0			0
$585$	−13.9157	−	6.25930i	−0.575342	−	0.258790i
$586$		0			0
$587$	3.07192	−	3.07192i	0.126792	−	0.126792i	−0.640863	−	0.767655i	$-0.721423\pi$
$587$	3.07192	−	3.07192i	0.126792	−	0.126792i	0.767655	+	0.640863i	$0.221423\pi$
$588$		0			0
$589$			41.5101i			1.71039i
$590$		0			0
$591$	−11.9349	−	0.224410i	−0.490936	−	0.00923101i
$592$		0			0
$593$	33.6650	−	9.02050i	1.38245	−	0.370428i	0.510442	−	0.859912i	$-0.329482\pi$
$593$	33.6650	−	9.02050i	1.38245	−	0.370428i	0.872012	+	0.489484i	$0.162815\pi$
$594$		0			0
$595$	25.2678	−	1.10040i	1.03588	−	0.0451121i
$596$		0			0
$597$	21.1385	+	6.09219i	0.865140	+	0.249337i
$598$		0			0
$599$	−13.7809	−	23.8692i	−0.563073	−	0.975270i	−0.997226	−	0.0744315i	$-0.976286\pi$
$599$	−13.7809	−	23.8692i	−0.563073	−	0.975270i	0.434153	−	0.900839i	$-0.357048\pi$
$600$		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
$603$	−32.3087	−	9.97304i	−1.31571	−	0.406134i
$604$		0			0
$605$	−1.62244	+	13.0929i	−0.0659615	+	0.532303i
$606$		0			0
$607$	32.5881	−	8.73196i	1.32271	−	0.354419i	0.472718	−	0.881214i	$-0.343273\pi$
$607$	32.5881	−	8.73196i	1.32271	−	0.354419i	0.849993	+	0.526794i	$0.176606\pi$
$608$		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
$613$	−12.1300	−	3.25023i	−0.489927	−	0.131276i	0.00539398	−	0.999985i	$-0.498283\pi$
$613$	−12.1300	−	3.25023i	−0.489927	−	0.131276i	−0.495321	+	0.868710i	$0.664950\pi$
$614$		0			0
$615$	7.72983	−	18.0863i	0.311697	−	0.729312i
$616$		0			0
$617$	−4.35042	+	4.35042i	−0.175141	+	0.175141i	−0.789234	−	0.614093i	$-0.789522\pi$
$617$	−4.35042	+	4.35042i	−0.175141	+	0.175141i	0.614093	+	0.789234i	$0.289522\pi$
$618$		0			0
$619$	−7.59365	+	4.38420i	−0.305215	+	0.176216i	−0.644783	−	0.764366i	$-0.723052\pi$
$619$	−7.59365	+	4.38420i	−0.305215	+	0.176216i	0.339568	+	0.940581i	$0.389719\pi$
$620$		0			0
$621$	25.4556	−	38.8535i	1.02150	−	1.55914i
$622$		0			0
$623$	14.2211	−	19.9797i	0.569756	−	0.800468i
$624$		0			0
$625$	−0.760979	−	24.9884i	−0.0304392	−	0.999537i
$626$		0			0
$627$	21.3987	+	22.2188i	0.854582	+	0.887335i
$628$		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
$633$	3.18926	+	3.31150i	0.126762	+	0.131620i
$634$		0			0
$635$	6.83667	+	2.77107i	0.271305	+	0.109967i
$636$		0			0
$637$	−13.1823	+	8.93014i	−0.522301	+	0.353825i
$638$		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	0.267335	−	0.963604i	$-0.413857\pi$
$643$	31.2135	−	31.2135i	1.23094	−	1.23094i	0.963604	−	0.267335i	$-0.0861430\pi$
$644$		0			0
$645$	21.1841	−	8.49829i	0.834122	−	0.334620i
$646$		0			0
$647$	2.86702	+	0.768215i	0.112714	+	0.0302016i	0.314735	−	0.949180i	$-0.398084\pi$
$647$	2.86702	+	0.768215i	0.112714	+	0.0302016i	−0.202021	+	0.979381i	$0.564751\pi$
$648$		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
$653$	8.08279	−	2.16578i	0.316304	−	0.0847534i	−0.0971743	−	0.995267i	$-0.530980\pi$
$653$	8.08279	−	2.16578i	0.316304	−	0.0847534i	0.413478	+	0.910514i	$0.364314\pi$
$654$		0			0
$655$	−37.8903	−	4.69525i	−1.48049	−	0.183459i
$656$		0			0
$657$	1.20787	−	3.91303i	0.0471236	−	0.152662i
$658$		0			0
$659$	35.1050			1.36750			0.683749	−	0.729717i	$-0.260348\pi$
$659$	35.1050			1.36750			0.683749	+	0.729717i	$0.260348\pi$
$660$		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
$663$	16.1840	+	4.66431i	0.628536	+	0.181147i
$664$		0			0
$665$	−45.5498	−	10.1035i	−1.76635	−	0.391796i
$666$		0			0
$667$	−28.8557	+	7.73185i	−1.11730	+	0.299379i
$668$		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	−0.264284	−	0.964445i	$-0.585135\pi$
$673$	18.1638	−	18.1638i	0.700162	−	0.700162i	0.964445	+	0.264284i	$0.0851355\pi$
$674$		0			0
$675$	1.06959	−	25.9587i	0.0411685	−	0.999152i
$676$		0			0
$677$	4.81492	+	17.9695i	0.185053	+	0.690626i	0.994619	+	0.103598i	$0.0330355\pi$
$677$	4.81492	+	17.9695i	0.185053	+	0.690626i	−0.809567	+	0.587028i	$0.800298\pi$
$678$		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
$683$	6.39675	−	23.8730i	0.244765	−	0.913475i	−0.728736	−	0.684794i	$-0.759892\pi$
$683$	6.39675	−	23.8730i	0.244765	−	0.913475i	0.973501	−	0.228681i	$-0.0734413\pi$
$684$		0			0
$685$	13.2924	−	1.85300i	0.507876	−	0.0707996i
$686$		0			0
$687$	2.35520	+	9.50072i	0.0898566	+	0.362475i
$688$		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
$693$	6.83146	+	16.5718i	0.259506	+	0.629510i
$694$		0			0
$695$	−1.22094	−	2.88530i	−0.0463130	−	0.109446i
$696$		0			0
$697$	−5.61922	+	20.9712i	−0.212843	+	0.794341i
$698$		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
$703$	−11.1556	−	2.98914i	−0.420742	−	0.112738i
$704$		0			0
$705$	−6.91649	+	48.3490i	−0.260490	+	1.82093i
$706$		0			0
$707$	−14.2278	+	6.50123i	−0.535090	+	0.244504i
$708$		0			0
$709$	31.3009	+	18.0716i	1.17553	+	0.678693i	0.954977	−	0.296681i	$-0.0958799\pi$
$709$	31.3009	+	18.0716i	1.17553	+	0.678693i	0.220555	+	0.975375i	$0.429213\pi$
$710$		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
$717$	0.723951	+	0.751698i	0.0270365	+	0.0280727i
$718$		0			0
$719$	−11.1558	+	19.3224i	−0.416041	+	0.720604i	−0.995537	−	0.0943709i	$-0.969916\pi$
$719$	−11.1558	+	19.3224i	−0.416041	+	0.720604i	0.579496	+	0.814975i	$0.303249\pi$
$720$		0			0
$721$	1.91483	+	11.3724i	0.0713121	+	0.423530i
$722$		0			0
$723$	2.69549	−	9.35271i	0.100246	−	0.347831i
$724$		0			0
$725$	−11.6340	+	11.9936i	−0.432075	+	0.445433i
$726$		0			0
$727$	32.5217	+	32.5217i	1.20616	+	1.20616i	0.972260	+	0.233902i	$0.0751494\pi$
$727$	32.5217	+	32.5217i	1.20616	+	1.20616i	0.233902	+	0.972260i	$0.424851\pi$
$728$		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
$733$	4.93222	+	18.4073i	0.182176	+	0.679889i	0.995217	+	0.0976839i	$0.0311434\pi$
$733$	4.93222	+	18.4073i	0.182176	+	0.679889i	−0.813042	+	0.582205i	$0.802190\pi$
$734$		0			0
$735$	−22.4691	−	15.1704i	−0.828784	−	0.559569i
$736$		0			0
$737$	6.58776	+	24.5859i	0.242663	+	0.905632i
$738$		0			0
$739$	−2.96973	+	1.71458i	−0.109243	+	0.0630717i	−0.553626	−	0.832765i	$-0.686756\pi$
$739$	−2.96973	+	1.71458i	−0.109243	+	0.0630717i	0.444383	+	0.895837i	$0.353423\pi$
$740$		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	$-0.105712\pi$
$743$	16.8816	+	16.8816i	0.619326	+	0.619326i	−0.326033	+	0.945358i	$0.605712\pi$
$744$		0			0
$745$	−0.434440	+	3.50589i	−0.0159167	+	0.128446i
$746$		0			0
$747$	−22.8145	+	21.1602i	−0.834740	+	0.774213i
$748$		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
$753$	−18.3690	+	17.6909i	−0.669402	+	0.644693i
$754$		0			0
$755$	26.2770	+	19.8471i	0.956317	+	0.722311i
$756$		0			0
$757$	−28.5534	−	28.5534i	−1.03779	−	1.03779i	−0.999257	−	0.0385344i	$-0.987731\pi$
$757$	−28.5534	−	28.5534i	−1.03779	−	1.03779i	−0.0385344	−	0.999257i	$-0.512269\pi$
$758$		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
$763$	5.72826	−	2.61747i	0.207377	−	0.0947588i
$764$		0			0
$765$	2.88898	+	28.5323i	0.104451	+	1.03159i
$766$		0			0
$767$	3.22125	+	0.863132i	0.116313	+	0.0311659i
$768$		0			0
$769$			16.1547i			0.582552i	0.956639	+	0.291276i	$0.0940799\pi$
$769$			16.1547i			0.582552i	−0.956639	+	0.291276i	$0.905920\pi$
$770$		0			0
$771$	−21.8244	+	36.2114i	−0.785988	+	1.30412i
$772$		0			0
$773$	−2.83854	+	10.5936i	−0.102095	+	0.381024i	−0.997999	−	0.0632229i	$-0.979862\pi$
$773$	−2.83854	+	10.5936i	−0.102095	+	0.381024i	0.895904	+	0.444247i	$0.146529\pi$
$774$		0			0
$775$	−25.5213	−	6.42370i	−0.916753	−	0.230746i
$776$		0			0
$777$	−5.39320	−	3.99362i	−0.193480	−	0.143270i
$778$		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
$783$	−12.9513	+	11.5670i	−0.462843	+	0.413370i
$784$		0			0
$785$	−2.98362	−	21.4028i	−0.106490	−	0.763897i
$786$		0			0
$787$	2.34200	−	8.74047i	0.0834834	−	0.311564i	−0.911539	−	0.411213i	$-0.865105\pi$
$787$	2.34200	−	8.74047i	0.0834834	−	0.311564i	0.995023	+	0.0996487i	$0.0317719\pi$
$788$		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
$793$	−4.73668	−	17.6775i	−0.168205	−	0.627748i
$794$		0			0
$795$	10.2998	−	13.1165i	0.365297	−	0.465193i
$796$		0			0
$797$	16.3203	−	16.3203i	0.578093	−	0.578093i	−0.356284	−	0.934378i	$-0.615957\pi$
$797$	16.3203	−	16.3203i	0.578093	−	0.578093i	0.934378	+	0.356284i	$0.115957\pi$
$798$		0			0
$799$		−	53.9120i		−	1.90727i
$800$		0			0
$801$	23.5424	+	14.7993i	0.831829	+	0.522907i
$802$		0			0
$803$	−2.97769	+	0.797869i	−0.105080	+	0.0281562i
$804$		0			0
$805$	44.6063	−	28.4104i	1.57217	−	1.00133i
$806$		0			0
$807$	11.6085	−	40.2788i	0.408639	−	1.41788i
$808$		0			0
$809$	−20.0240	−	34.6826i	−0.704006	−	1.21937i	−0.967049	−	0.254591i	$-0.918059\pi$
$809$	−20.0240	−	34.6826i	−0.704006	−	1.21937i	0.263043	−	0.964784i	$-0.415274\pi$
$810$		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
$813$	7.30297	−	1.81039i	0.256126	−	0.0634930i
$814$		0			0
$815$	−7.41535	−	9.51307i	−0.259748	−	0.333228i
$816$		0			0
$817$	44.8945	−	12.0294i	1.57066	−	0.420857i
$818$		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
$823$	−22.4308	−	6.01032i	−0.781889	−	0.209506i	−0.154272	−	0.988028i	$-0.549303\pi$
$823$	−22.4308	−	6.01032i	−0.781889	−	0.209506i	−0.627617	+	0.778522i	$0.715970\pi$
$824$		0			0
$825$	−16.9721	+	9.71802i	−0.590892	+	0.338338i
$826$		0			0
$827$	21.1976	−	21.1976i	0.737112	−	0.737112i	−0.234906	−	0.972018i	$-0.575478\pi$
$827$	21.1976	−	21.1976i	0.737112	−	0.737112i	0.972018	+	0.234906i	$0.0754782\pi$
$828$		0			0
$829$	21.2241	−	12.2537i	0.737143	−	0.425590i	−0.0838866	−	0.996475i	$-0.526733\pi$
$829$	21.2241	−	12.2537i	0.737143	−	0.425590i	0.821030	+	0.570886i	$0.193400\pi$
$830$		0			0
$831$	8.50193	−	4.69773i	0.294929	−	0.162963i
$832$		0			0
$833$	26.9233	+	13.0645i	0.932836	+	0.452657i
$834$		0			0
$835$	−15.4161	+	38.0341i	−0.533498	+	1.31622i
$836$		0			0
$837$	−25.9768	−	8.55662i	−0.897890	−	0.295760i
$838$		0			0
$839$	16.2459			0.560870			0.280435	−	0.959873i	$-0.409521\pi$
$839$	16.2459			0.560870			0.280435	+	0.959873i	$0.409521\pi$
$840$		0			0
$841$	−17.8321			−0.614901
$842$		0			0
$843$	−28.3472	+	27.3009i	−0.976330	+	0.940292i
$844$		0			0
$845$	6.81972	+	16.1162i	0.234605	+	0.554415i
$846$		0			0
$847$	−9.05214	+	12.7176i	−0.311035	+	0.436983i
$848$		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	−0.314364	+	0.949303i	$0.601791\pi$
$853$	−36.9068	+	36.9068i	−1.26367	+	1.26367i	−0.949303	+	0.314364i	$0.898209\pi$
$854$		0			0
$855$	8.47505	−	52.2207i	0.289841	−	1.78591i
$856$		0			0
$857$	−3.62018	−	0.970024i	−0.123663	−	0.0331354i	0.196457	−	0.980512i	$-0.437056\pi$
$857$	−3.62018	−	0.970024i	−0.123663	−	0.0331354i	−0.320120	+	0.947377i	$0.603723\pi$
$858$		0			0
$859$	−3.27915	−	1.89322i	−0.111883	−	0.0645958i	0.443014	−	0.896515i	$-0.353909\pi$
$859$	−3.27915	−	1.89322i	−0.111883	−	0.0645958i	−0.554897	+	0.831919i	$0.687242\pi$
$860$		0			0
$861$	18.2183	−	14.4813i	0.620877	−	0.493520i
$862$		0			0
$863$	−15.3482	+	4.11255i	−0.522460	+	0.139993i	−0.510404	−	0.859934i	$-0.670504\pi$
$863$	−15.3482	+	4.11255i	−0.522460	+	0.139993i	−0.0120559	+	0.999927i	$0.503838\pi$
$864$		0			0
$865$	−11.2572	+	8.77489i	−0.382757	+	0.298355i
$866$		0			0
$867$	−0.531934	−	2.14578i	−0.0180654	−	0.0728746i
$868$		0			0
$869$	−14.7652			−0.500875
$870$		0			0
$871$	−12.8186	−	22.2024i	−0.434341	−	0.752300i
$872$		0			0
$873$	−21.9764	+	20.3829i	−0.743788	+	0.689855i
$874$		0			0
$875$	13.2607	−	26.4415i	0.448293	−	0.893887i
$876$		0			0
$877$	9.23854	−	2.47546i	0.311963	−	0.0835903i	−0.0994405	−	0.995044i	$-0.531705\pi$
$877$	9.23854	−	2.47546i	0.311963	−	0.0835903i	0.411404	+	0.911453i	$0.365039\pi$
$878$		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	−0.569203	−	0.822197i	$-0.692748\pi$
$883$	7.51781	−	7.51781i	0.252994	−	0.252994i	0.822197	+	0.569203i	$0.192748\pi$
$884$		0			0
$885$	0.678127	+	5.63766i	0.0227950	+	0.189508i
$886$		0			0
$887$	−4.90230	−	18.2956i	−0.164603	−	0.614307i	−0.998090	−	0.0617686i	$-0.980326\pi$
$887$	−4.90230	−	18.2956i	−0.164603	−	0.614307i	0.833487	−	0.552538i	$-0.186341\pi$
$888$		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
$893$	−25.7406	+	96.0652i	−0.861376	+	3.21470i
$894$		0			0
$895$	−18.4414	−	13.9289i	−0.616429	−	0.465591i
$896$		0			0
$897$	34.1838	−	8.47407i	1.14136	−	0.282941i
$898$		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
$903$	26.8324	+	3.06648i	0.892926	+	0.102046i
$904$		0			0
$905$	4.98788	−	2.11067i	0.165803	−	0.0701609i
$906$		0			0
$907$	−9.96761	+	37.1996i	−0.330969	+	1.23519i	0.577205	+	0.816599i	$0.304143\pi$
$907$	−9.96761	+	37.1996i	−0.330969	+	1.23519i	−0.908174	+	0.418594i	$0.862523\pi$
$908$		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
$913$	22.6255	+	6.06249i	0.748795	+	0.200639i
$914$		0			0
$915$	24.9326	−	18.6920i	0.824248	−	0.617938i
$916$		0			0
$917$	−36.8042	−	26.1964i	−1.21538	−	0.865083i
$918$		0			0
$919$	−7.47875	−	4.31786i	−0.246701	−	0.142433i	0.371552	−	0.928412i	$-0.378826\pi$
$919$	−7.47875	−	4.31786i	−0.246701	−	0.142433i	−0.618253	+	0.785979i	$0.712159\pi$
$920$		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
$927$	−12.7493	+	2.90725i	−0.418743	+	0.0954865i
$928$		0			0
$929$	−24.2232	+	41.9559i	−0.794739	+	1.37653i	0.128266	+	0.991740i	$0.459059\pi$
$929$	−24.2232	+	41.9559i	−0.794739	+	1.37653i	−0.923005	+	0.384788i	$0.874275\pi$
$930$		0			0
$931$	−41.7366	−	36.1341i	−1.36786	−	1.18425i
$932$		0			0
$933$	50.7476	+	14.6257i	1.66140	+	0.478823i
$934$		0			0
$935$	17.0263	−	13.2718i	0.556820	−	0.434036i
$936$		0			0
$937$	41.6088	+	41.6088i	1.35930	+	1.35930i	0.874779	+	0.484521i	$0.161006\pi$
$937$	41.6088	+	41.6088i	1.35930	+	1.35930i	0.484521	+	0.874779i	$0.338994\pi$
$938$		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
$943$	11.7499	+	43.8511i	0.382628	+	1.42799i
$944$		0			0
$945$	15.7120	−	26.4222i	0.511113	−	0.859514i
$946$		0			0
$947$	−4.48228	−	16.7281i	−0.145655	−	0.543591i	−0.999725	−	0.0234337i	$-0.992540\pi$
$947$	−4.48228	−	16.7281i	−0.145655	−	0.543591i	0.854071	−	0.520157i	$-0.174127\pi$
$948$		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	$-0.340424\pi$
$953$	−12.2360	−	12.2360i	−0.396362	−	0.396362i	−0.876948	+	0.480586i	$0.840424\pi$
$954$		0			0
$955$	39.6686	+	4.91562i	1.28365	+	0.159066i
$956$		0			0
$957$	12.5603	+	3.61992i	0.406016	+	0.117016i
$958$		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
$963$	−25.6495	+	5.84890i	−0.826544	+	0.188478i
$964$		0			0
$965$	−35.7925	+	4.98959i	−1.15220	+	0.160621i
$966$		0			0
$967$	−22.2486	−	22.2486i	−0.715466	−	0.715466i	0.252207	−	0.967673i	$-0.418844\pi$
$967$	−22.2486	−	22.2486i	−0.715466	−	0.715466i	−0.967673	+	0.252207i	$0.918844\pi$
$968$		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
$973$	0.351595	−	3.69030i	0.0112716	−	0.118306i
$974$		0			0
$975$	13.9789	−	13.8792i	0.447682	−	0.444489i
$976$		0			0
$977$	4.25230	+	1.13940i	0.136043	+	0.0364526i	0.326198	−	0.945301i	$-0.394232\pi$
$977$	4.25230	+	1.13940i	0.136043	+	0.0364526i	−0.190155	+	0.981754i	$0.560899\pi$
$978$		0			0
$979$		−	20.9326i		−	0.669008i
$980$		0			0
$981$	3.33559	+	6.31431i	0.106497	+	0.201601i
$982$		0			0
$983$	2.59711	−	9.69254i	0.0828349	−	0.309144i	−0.912060	−	0.410056i	$-0.865509\pi$
$983$	2.59711	−	9.69254i	0.0828349	−	0.309144i	0.994895	+	0.100912i	$0.0321759\pi$
$984$		0			0
$985$	5.78885	−	14.2820i	0.184448	−	0.455063i
$986$		0			0
$987$	−34.3905	+	46.4428i	−1.09466	+	1.47829i
$988$		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
$993$	−18.4729	+	4.57939i	−0.586220	+	0.145322i
$994$		0			0
$995$	−17.1171	+	22.6625i	−0.542648	+	0.718449i
$996$		0			0
$997$	−13.3811	+	49.9388i	−0.423783	+	1.58158i	0.342784	+	0.939414i	$0.388630\pi$
$997$	−13.3811	+	49.9388i	−0.423783	+	1.58158i	−0.766567	+	0.642164i	$0.778037\pi$
$998$		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.317.4	yes	48
3.2	odd	2	inner	420.2.bv.c.317.2	yes	48
5.3	odd	4	inner	420.2.bv.c.233.10	yes	48
7.4	even	3	inner	420.2.bv.c.137.5	yes	48
15.8	even	4	inner	420.2.bv.c.233.5	yes	48
21.11	odd	6	inner	420.2.bv.c.137.10	yes	48
35.18	odd	12	inner	420.2.bv.c.53.2	✓	48
105.53	even	12	inner	420.2.bv.c.53.4	yes	48

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
420.2.bv.c.53.2	✓	48	35.18	odd	12	inner
420.2.bv.c.53.4	yes	48	105.53	even	12	inner
420.2.bv.c.137.5	yes	48	7.4	even	3	inner
420.2.bv.c.137.10	yes	48	21.11	odd	6	inner
420.2.bv.c.233.5	yes	48	15.8	even	4	inner
420.2.bv.c.233.10	yes	48	5.3	odd	4	inner
420.2.bv.c.317.2	yes	48	3.2	odd	2	inner
420.2.bv.c.317.4	yes	48	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q+(-1.20150 - 1.24755i) q^{3} +(2.05928 - 0.871404i) q^{5} +(2.63382 + 0.250938i) q^{7} +(-0.112777 + 2.99788i) q^{9} +O(q^{10})\)	\(q+(-1.20150 - 1.24755i) q^{3} +(2.05928 - 0.871404i) q^{5} +(2.63382 + 0.250938i) q^{7} +(-0.112777 + 2.99788i) q^{9} +(1.95574 - 1.12915i) q^{11} +(-1.60840 + 1.60840i) q^{13} +(-3.56136 - 1.52207i) q^{15} +(4.12942 + 1.10647i) q^{17} +(-6.82987 - 3.94323i) q^{19} +(-2.85149 - 3.58734i) q^{21} +(8.63467 - 2.31365i) q^{23} +(3.48131 - 3.58894i) q^{25} +(3.87552 - 3.46127i) q^{27} -3.34184 q^{29} +(-2.63173 - 4.55830i) q^{31} +(-3.75850 - 1.08321i) q^{33} +(5.64246 - 1.77837i) q^{35} +(1.41453 - 0.379022i) q^{37} +(3.93905 + 0.0740656i) q^{39} +5.07849i q^{41} +(-4.16728 + 4.16728i) q^{43} +(2.38012 + 6.27176i) q^{45} +(-3.26390 - 12.1810i) q^{47} +(6.87406 + 1.32186i) q^{49} +(-3.58112 - 6.48110i) q^{51} +(-1.11448 + 4.15930i) q^{53} +(3.04348 - 4.02947i) q^{55} +(3.28673 + 13.2584i) q^{57} +(-0.733066 - 1.26971i) q^{59} +(-4.02291 + 6.96788i) q^{61} +(-1.04932 + 7.86759i) q^{63} +(-1.91058 + 4.71371i) q^{65} +(-2.91714 + 10.8869i) q^{67} +(-13.2610 - 7.99235i) q^{69} +9.99909i q^{71} +(-1.31856 - 0.353306i) q^{73} +(-8.66020 - 0.0309927i) q^{75} +(5.43442 - 2.48320i) q^{77} +(-5.66226 - 3.26911i) q^{79} +(-8.97456 - 0.676187i) q^{81} +(7.33432 + 7.33432i) q^{83} +(9.46783 - 1.31985i) q^{85} +(4.01523 + 4.16912i) q^{87} +(4.63460 - 8.02736i) q^{89} +(-4.63984 + 3.83262i) q^{91} +(-2.52468 + 8.76004i) q^{93} +(-17.5008 - 2.16865i) q^{95} +(7.06487 + 7.06487i) q^{97} +(3.16448 + 5.99041i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 48 q - 2 q^{3} - 8 q^{7}+O(q^{10}) \) 48 * q - 2 * q^3 - 8 * q^7	\( 48 q - 2 q^{3} - 8 q^{7} + 32 q^{13} - 16 q^{21} + 16 q^{25} - 32 q^{27} - 64 q^{31} + 34 q^{33} - 40 q^{37} - 24 q^{43} + 30 q^{45} + 36 q^{51} + 92 q^{57} - 18 q^{63} + 28 q^{67} - 8 q^{73} - 38 q^{75} + 20 q^{81} - 56 q^{85} - 24 q^{87} - 24 q^{91} + 6 q^{93} + 56 q^{97}+O(q^{100}) \) 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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