LMFDB - Embedded newform 420.2.bv.c.317.5

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			317.5
Character	$\chi$	$=$	420.317
Dual form			420.2.bv.c.53.5

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+(-0.795708 + 1.53846i) q^{3} +(0.746108 + 2.10792i) q^{5} +(-2.40557 + 1.10146i) q^{7} +(-1.73370 - 2.44833i) q^{9} +(-2.63993 + 1.52416i) q^{11} +(0.512041 - 0.512041i) q^{13} +(-3.83663 - 0.529433i) q^{15} +(3.88873 + 1.04198i) q^{17} +(-5.78739 - 3.34135i) q^{19} +(0.219587 - 4.57731i) q^{21} +(2.29376 - 0.614611i) q^{23} +(-3.88664 + 3.14547i) q^{25} +(5.14616 - 0.719065i) q^{27} -8.51556 q^{29} +(0.921008 + 1.59523i) q^{31} +(-0.244246 - 5.27420i) q^{33} +(-4.11661 - 4.24895i) q^{35} +(2.78938 - 0.747413i) q^{37} +(0.380318 + 1.19519i) q^{39} +8.91355i q^{41} +(-7.54399 + 7.54399i) q^{43} +(3.86735 - 5.48121i) q^{45} +(1.95084 + 7.28063i) q^{47} +(4.57357 - 5.29929i) q^{49} +(-4.69734 + 5.15352i) q^{51} +(-2.81412 + 10.5024i) q^{53} +(-5.18248 - 4.42756i) q^{55} +(9.74561 - 6.24491i) q^{57} +(-1.06765 - 1.84923i) q^{59} +(7.12318 - 12.3377i) q^{61} +(6.86727 + 3.98003i) q^{63} +(1.46138 + 0.697303i) q^{65} +(1.03593 - 3.86614i) q^{67} +(-0.879611 + 4.01790i) q^{69} +3.41895i q^{71} +(11.0196 + 2.95269i) q^{73} +(-1.74654 - 8.48231i) q^{75} +(4.67173 - 6.57426i) q^{77} +(12.1569 + 7.01879i) q^{79} +(-2.98859 + 8.48931i) q^{81} +(0.995323 + 0.995323i) q^{83} +(0.705000 + 8.97455i) q^{85} +(6.77590 - 13.1008i) q^{87} +(0.706248 - 1.22326i) q^{89} +(-0.667760 + 1.79575i) q^{91} +(-3.18705 + 0.147591i) q^{93} +(2.72528 - 14.6924i) q^{95} +(-0.556234 - 0.556234i) q^{97} +(8.30847 + 3.82096i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $\left(\mathbb{Z}/420\mathbb{Z}\right)^\times$.

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

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
$3$	−0.795708	+	1.53846i	−0.459402	+	0.888228i
$3$	−0.795708	+	1.53846i	−0.459402	+	0.888228i
$4$		0			0
$5$	0.746108	+	2.10792i	0.333670	+	0.942690i
$5$	0.746108	+	2.10792i	0.333670	+	0.942690i
$6$		0			0
$7$	−2.40557	+	1.10146i	−0.909221	+	0.416313i
$7$	−2.40557	+	1.10146i	−0.909221	+	0.416313i
$8$		0			0
$9$	−1.73370	−	2.44833i	−0.577899	−	0.816108i
$10$		0			0
$11$	−2.63993	+	1.52416i	−0.795968	+	0.459552i	−0.842059	−	0.539385i	$-0.818657\pi$
$11$	−2.63993	+	1.52416i	−0.795968	+	0.459552i	0.0460916	+	0.998937i	$0.485323\pi$
$12$		0			0
$13$	0.512041	−	0.512041i	0.142015	−	0.142015i	−0.632525	−	0.774540i	$-0.717982\pi$
$13$	0.512041	−	0.512041i	0.142015	−	0.142015i	0.774540	+	0.632525i	$0.217982\pi$
$14$		0			0
$15$	−3.83663	−	0.529433i	−0.990613	−	0.136699i
$16$		0			0
$17$	3.88873	+	1.04198i	0.943155	+	0.252718i	0.697455	−	0.716628i	$-0.254316\pi$
$17$	3.88873	+	1.04198i	0.943155	+	0.252718i	0.245700	+	0.969346i	$0.420982\pi$
$18$		0			0
$19$	−5.78739	−	3.34135i	−1.32772	−	0.766559i	−0.342773	−	0.939418i	$-0.611366\pi$
$19$	−5.78739	−	3.34135i	−1.32772	−	0.766559i	−0.984947	+	0.172859i	$0.944699\pi$
$20$		0			0
$21$	0.219587	−	4.57731i	0.0479178	−	0.998851i
$22$		0			0
$23$	2.29376	−	0.614611i	0.478282	−	0.128155i	−0.0116206	−	0.999932i	$-0.503699\pi$
$23$	2.29376	−	0.614611i	0.478282	−	0.128155i	0.489902	+	0.871777i	$0.337032\pi$
$24$		0			0
$25$	−3.88664	+	3.14547i	−0.777329	+	0.629094i
$26$		0			0
$27$	5.14616	−	0.719065i	0.990379	−	0.138384i
$28$		0			0
$29$	−8.51556			−1.58130			−0.790650	−	0.612269i	$-0.790257\pi$
$29$	−8.51556			−1.58130			−0.790650	+	0.612269i	$0.790257\pi$
$30$		0			0
$31$	0.921008	+	1.59523i	0.165418	+	0.286512i	0.936804	−	0.349856i	$-0.113769\pi$
$31$	0.921008	+	1.59523i	0.165418	+	0.286512i	−0.771386	+	0.636368i	$0.780436\pi$
$32$		0			0
$33$	−0.244246	−	5.27420i	−0.0425178	−	0.918120i
$34$		0			0
$35$	−4.11661	−	4.24895i	−0.695834	−	0.718203i
$36$		0			0
$37$	2.78938	−	0.747413i	0.458572	−	0.122874i	−0.0221357	−	0.999755i	$-0.507047\pi$
$37$	2.78938	−	0.747413i	0.458572	−	0.122874i	0.480707	+	0.876881i	$0.340380\pi$
$38$		0			0
$39$	0.380318	+	1.19519i	0.0608996	+	0.191383i
$40$		0			0
$41$			8.91355i			1.39206i	0.718012	+	0.696031i	$0.245052\pi$
$41$			8.91355i			1.39206i	−0.718012	+	0.696031i	$0.754948\pi$
$42$		0			0
$43$	−7.54399	+	7.54399i	−1.15045	+	1.15045i	−0.163984	+	0.986463i	$0.552435\pi$
$43$	−7.54399	+	7.54399i	−1.15045	+	1.15045i	−0.986463	+	0.163984i	$0.947565\pi$
$44$		0			0
$45$	3.86735	−	5.48121i	0.576510	−	0.817090i
$46$		0			0
$47$	1.95084	+	7.28063i	0.284559	+	1.06199i	0.949161	+	0.314791i	$0.101934\pi$
$47$	1.95084	+	7.28063i	0.284559	+	1.06199i	−0.664602	+	0.747198i	$0.731399\pi$
$48$		0			0
$49$	4.57357	−	5.29929i	0.653367	−	0.757041i
$50$		0			0
$51$	−4.69734	+	5.15352i	−0.657759	+	0.721638i
$52$		0			0
$53$	−2.81412	+	10.5024i	−0.386549	+	1.44262i	0.449161	+	0.893451i	$0.351723\pi$
$53$	−2.81412	+	10.5024i	−0.386549	+	1.44262i	−0.835710	+	0.549171i	$0.814944\pi$
$54$		0			0
$55$	−5.18248	−	4.42756i	−0.698806	−	0.597012i
$56$		0			0
$57$	9.74561	−	6.24491i	1.29084	−	0.827159i
$58$		0			0
$59$	−1.06765	−	1.84923i	−0.138996	−	0.240749i	0.788121	−	0.615521i	$-0.211054\pi$
$59$	−1.06765	−	1.84923i	−0.138996	−	0.240749i	−0.927117	+	0.374772i	$0.877721\pi$
$60$		0			0
$61$	7.12318	−	12.3377i	0.912029	−	1.57968i	0.100836	−	0.994903i	$-0.467848\pi$
$61$	7.12318	−	12.3377i	0.912029	−	1.57968i	0.811193	−	0.584778i	$-0.198818\pi$
$62$		0			0
$63$	6.86727	+	3.98003i	0.865194	+	0.501437i
$64$		0			0
$65$	1.46138	+	0.697303i	0.181262	+	0.0864898i
$66$		0			0
$67$	1.03593	−	3.86614i	0.126559	−	0.472325i	−0.873331	−	0.487126i	$-0.838045\pi$
$67$	1.03593	−	3.86614i	0.126559	−	0.472325i	0.999890	+	0.0148018i	$0.00471174\pi$
$68$		0			0
$69$	−0.879611	+	4.01790i	−0.105893	+	0.483698i
$70$		0			0
$71$			3.41895i			0.405755i	0.979204	+	0.202878i	$0.0650293\pi$
$71$			3.41895i			0.405755i	−0.979204	+	0.202878i	$0.934971\pi$
$72$		0			0
$73$	11.0196	+	2.95269i	1.28975	+	0.345586i	0.837564	−	0.546340i	$-0.183979\pi$
$73$	11.0196	+	2.95269i	1.28975	+	0.345586i	0.452182	+	0.891926i	$0.350646\pi$
$74$		0			0
$75$	−1.74654	−	8.48231i	−0.201673	−	0.979453i
$76$		0			0
$77$	4.67173	−	6.57426i	0.532393	−	0.749206i
$78$		0			0
$79$	12.1569	+	7.01879i	1.36776	+	0.789675i	0.990641	−	0.136490i	$-0.0435823\pi$
$79$	12.1569	+	7.01879i	1.36776	+	0.789675i	0.377117	+	0.926166i	$0.376916\pi$
$80$		0			0
$81$	−2.98859	+	8.48931i	−0.332066	+	0.943256i
$82$		0			0
$83$	0.995323	+	0.995323i	0.109251	+	0.109251i	0.759619	−	0.650368i	$-0.225385\pi$
$83$	0.995323	+	0.995323i	0.109251	+	0.109251i	−0.650368	+	0.759619i	$0.725385\pi$
$84$		0			0
$85$	0.705000	+	8.97455i	0.0764680	+	0.973427i
$86$		0			0
$87$	6.77590	−	13.1008i	0.726453	−	1.40456i
$88$		0			0
$89$	0.706248	−	1.22326i	0.0748622	−	0.129665i	−0.826164	−	0.563429i	$-0.809482\pi$
$89$	0.706248	−	1.22326i	0.0748622	−	0.129665i	0.901026	+	0.433764i	$0.142815\pi$
$90$		0			0
$91$	−0.667760	+	1.79575i	−0.0700003	+	0.188245i
$92$		0			0
$93$	−3.18705	+	0.147591i	−0.330482	+	0.0153045i
$94$		0			0
$95$	2.72528	−	14.6924i	0.279608	−	1.50741i
$96$		0			0
$97$	−0.556234	−	0.556234i	−0.0564770	−	0.0564770i	0.678304	−	0.734781i	$-0.262715\pi$
$97$	−0.556234	−	0.556234i	−0.0564770	−	0.0564770i	−0.734781	+	0.678304i	$0.762715\pi$
$98$		0			0
$99$	8.30847	+	3.82096i	0.835033	+	0.384021i
$100$		0			0
$101$	−2.90475	+	1.67706i	−0.289034	+	0.166874i	−0.637506	−	0.770445i	$-0.720034\pi$
$101$	−2.90475	+	1.67706i	−0.289034	+	0.166874i	0.348472	+	0.937319i	$0.386701\pi$
$102$		0			0
$103$	0.448028	+	1.67206i	0.0441456	+	0.164753i	0.984480	−	0.175499i	$-0.0561540\pi$
$103$	0.448028	+	1.67206i	0.0441456	+	0.164753i	−0.940334	+	0.340253i	$0.889487\pi$
$104$		0			0
$105$	9.81244	−	2.95230i	0.957596	−	0.288115i
$106$		0			0
$107$	2.06095	+	7.69158i	0.199240	+	0.743573i	0.991128	+	0.132908i	$0.0424313\pi$
$107$	2.06095	+	7.69158i	0.199240	+	0.743573i	−0.791889	+	0.610666i	$0.790902\pi$
$108$		0			0
$109$	2.12083	−	1.22446i	0.203138	−	0.117282i	−0.394980	−	0.918690i	$-0.629249\pi$
$109$	2.12083	−	1.22446i	0.203138	−	0.117282i	0.598119	+	0.801408i	$0.295915\pi$
$110$		0			0
$111$	−1.06967	+	4.88607i	−0.101529	+	0.463765i
$112$		0			0
$113$	5.01803	+	5.01803i	0.472056	+	0.472056i	0.902580	−	0.430523i	$-0.141671\pi$
$113$	5.01803	+	5.01803i	0.472056	+	0.472056i	−0.430523	+	0.902580i	$0.641671\pi$
$114$		0			0
$115$	3.00694	+	4.37649i	0.280399	+	0.408110i
$116$		0			0
$117$	−2.14137	−	0.365919i	−0.197969	−	0.0338292i
$118$		0			0
$119$	−10.5023	+	1.77671i	−0.962746	+	0.162871i
$120$		0			0
$121$	−0.853861	+	1.47893i	−0.0776237	+	0.134448i
$122$		0			0
$123$	−13.7131	−	7.09258i	−1.23647	−	0.639517i
$124$		0			0
$125$	−9.53026	−	5.84587i	−0.852412	−	0.522870i
$126$		0			0
$127$	−5.74490	−	5.74490i	−0.509777	−	0.509777i	0.404681	−	0.914458i	$-0.367383\pi$
$127$	−5.74490	−	5.74490i	−0.509777	−	0.509777i	−0.914458	+	0.404681i	$0.867383\pi$
$128$		0			0
$129$	−5.60328	−	17.6089i	−0.493342	−	1.55038i
$130$		0			0
$131$	1.12384	+	0.648847i	0.0981900	+	0.0566900i	0.548291	−	0.836288i	$-0.315279\pi$
$131$	1.12384	+	0.648847i	0.0981900	+	0.0566900i	−0.450101	+	0.892978i	$0.648612\pi$
$132$		0			0
$133$	17.6024	+	1.66329i	1.52632	+	0.144226i
$134$		0			0
$135$	5.35532	+	10.3112i	0.460913	+	0.887445i
$136$		0			0
$137$	0.507990	+	0.136116i	0.0434006	+	0.0116291i	0.280454	−	0.959867i	$-0.409515\pi$
$137$	0.507990	+	0.136116i	0.0434006	+	0.0116291i	−0.237053	+	0.971497i	$0.576182\pi$
$138$		0			0
$139$			17.6744i			1.49912i	0.661936	+	0.749561i	$0.269735\pi$
$139$			17.6744i			1.49912i	−0.661936	+	0.749561i	$0.730265\pi$
$140$		0			0
$141$	−12.7532	−	2.79198i	−1.07402	−	0.235127i
$142$		0			0
$143$	−0.571317	+	2.13218i	−0.0477759	+	0.178302i
$144$		0			0
$145$	−6.35353	−	17.9501i	−0.527632	−	1.49068i
$146$		0			0
$147$	4.51349	+	11.2529i	0.372267	+	0.928126i
$148$		0			0
$149$	10.3981	−	18.0100i	0.851842	−	1.47543i	−0.0277017	−	0.999616i	$-0.508819\pi$
$149$	10.3981	−	18.0100i	0.851842	−	1.47543i	0.879544	−	0.475818i	$-0.157848\pi$
$150$		0			0
$151$	−5.44655	−	9.43371i	−0.443234	−	0.767704i	0.554693	−	0.832055i	$-0.312836\pi$
$151$	−5.44655	−	9.43371i	−0.443234	−	0.767704i	−0.997927	+	0.0643508i	$0.979502\pi$
$152$		0			0
$153$	−4.19077	−	11.3273i	−0.338803	−	0.915762i
$154$		0			0
$155$	−2.67545	+	3.13163i	−0.214897	+	0.251538i
$156$		0			0
$157$	−2.58508	+	9.64763i	−0.206312	+	0.769965i	0.782734	+	0.622356i	$0.213824\pi$
$157$	−2.58508	+	9.64763i	−0.206312	+	0.769965i	−0.989046	+	0.147609i	$0.952842\pi$
$158$		0			0
$159$	−13.9183	−	12.6863i	−1.10380	−	1.00609i
$160$		0			0
$161$	−4.84084	+	4.00498i	−0.381511	+	0.315636i
$162$		0			0
$163$	−0.883109	−	3.29581i	−0.0691705	−	0.258148i	0.922678	−	0.385572i	$-0.125996\pi$
$163$	−0.883109	−	3.29581i	−0.0691705	−	0.258148i	−0.991848	+	0.127424i	$0.959329\pi$
$164$		0			0
$165$	10.9354	−	4.44998i	0.851316	−	0.346430i
$166$		0			0
$167$	10.6613	−	10.6613i	0.824994	−	0.824994i	−0.161825	−	0.986819i	$-0.551738\pi$
$167$	10.6613	−	10.6613i	0.824994	−	0.824994i	0.986819	+	0.161825i	$0.0517381\pi$
$168$		0			0
$169$			12.4756i			0.959664i
$170$		0			0
$171$	1.85287	+	19.9623i	0.141692	+	1.52656i
$172$		0			0
$173$	−16.7718	+	4.49399i	−1.27514	+	0.341672i	−0.831996	−	0.554781i	$-0.812802\pi$
$173$	−16.7718	+	4.49399i	−1.27514	+	0.341672i	−0.443140	+	0.896453i	$0.646135\pi$
$174$		0			0
$175$	5.88500	−	11.8476i	0.444864	−	0.895598i
$176$		0			0
$177$	3.69450	−	0.171091i	0.277695	−	0.0128600i
$178$		0			0
$179$	7.14090	+	12.3684i	0.533736	+	0.924458i	0.999223	+	0.0394031i	$0.0125456\pi$
$179$	7.14090	+	12.3684i	0.533736	+	0.924458i	−0.465488	+	0.885054i	$0.654121\pi$
$180$		0			0
$181$	−5.90010			−0.438551			−0.219275	−	0.975663i	$-0.570369\pi$
$181$	−5.90010			−0.438551			−0.219275	+	0.975663i	$0.570369\pi$
$182$		0			0
$183$	13.3130	+	20.7759i	0.984129	+	1.53580i
$184$		0			0
$185$	3.65667	+	5.32214i	0.268844	+	0.391292i
$186$		0			0
$187$	−11.8541	+	3.17630i	−0.866858	+	0.232274i
$188$		0			0
$189$	−11.5874	+	7.39805i	−0.842863	+	0.538129i
$190$		0			0
$191$	−9.55692	−	5.51769i	−0.691514	−	0.399246i	0.112665	−	0.993633i	$-0.464061\pi$
$191$	−9.55692	−	5.51769i	−0.691514	−	0.399246i	−0.804179	+	0.594387i	$0.797395\pi$
$192$		0			0
$193$	−13.6035	−	3.64505i	−0.979203	−	0.262377i	−0.266494	−	0.963837i	$-0.585865\pi$
$193$	−13.6035	−	3.64505i	−0.979203	−	0.262377i	−0.712709	+	0.701460i	$0.752532\pi$
$194$		0			0
$195$	−2.23560	+	1.69342i	−0.160095	+	0.121268i
$196$		0			0
$197$	−1.63139	+	1.63139i	−0.116232	+	0.116232i	−0.762830	−	0.646599i	$-0.776191\pi$
$197$	−1.63139	+	1.63139i	−0.116232	+	0.116232i	0.646599	+	0.762830i	$0.276191\pi$
$198$		0			0
$199$	9.65354	−	5.57347i	0.684321	−	0.395093i	−0.117160	−	0.993113i	$-0.537379\pi$
$199$	9.65354	−	5.57347i	0.684321	−	0.395093i	0.801481	+	0.598020i	$0.204046\pi$
$200$		0			0
$201$	5.12360	+	4.67006i	0.361391	+	0.329400i
$202$		0			0
$203$	20.4848	−	9.37955i	1.43775	−	0.658315i
$204$		0			0
$205$	−18.7890	+	6.65047i	−1.31228	+	0.464489i
$206$		0			0
$207$	−5.48145	−	4.55032i	−0.380987	−	0.316269i
$208$		0			0
$209$	20.3711			1.40910
$210$		0			0
$211$	2.72171			0.187370			0.0936852	−	0.995602i	$-0.470135\pi$
$211$	2.72171			0.187370			0.0936852	+	0.995602i	$0.470135\pi$
$212$		0			0
$213$	−5.25991	−	2.72049i	−0.360403	−	0.186405i
$214$		0			0
$215$	−21.5307	−	10.2735i	−1.46838	−	0.700646i
$216$		0			0
$217$	−3.97264	−	2.82300i	−0.269680	−	0.191638i
$218$		0			0
$219$	−13.3110	+	14.6037i	−0.899472	+	0.986825i
$220$		0			0
$221$	2.52473	−	1.45765i	0.169831	−	0.0980522i
$222$		0			0
$223$	13.0957	−	13.0957i	0.876950	−	0.876950i	−0.116268	−	0.993218i	$-0.537093\pi$
$223$	13.0957	−	13.0957i	0.876950	−	0.876950i	0.993218	+	0.116268i	$0.0370931\pi$
$224$		0			0
$225$	14.4394	+	4.06247i	0.962627	+	0.270832i
$226$		0			0
$227$	13.2362	+	3.54662i	0.878515	+	0.235397i	0.669766	−	0.742572i	$-0.266394\pi$
$227$	13.2362	+	3.54662i	0.878515	+	0.235397i	0.208748	+	0.977969i	$0.433061\pi$
$228$		0			0
$229$	−10.3326	−	5.96554i	−0.682799	−	0.394214i	0.118110	−	0.993001i	$-0.462316\pi$
$229$	−10.3326	−	5.96554i	−0.682799	−	0.394214i	−0.800909	+	0.598787i	$0.795650\pi$
$230$		0			0
$231$	6.39687	+	12.4184i	0.420883	+	0.817074i
$232$		0			0
$233$	−9.76217	+	2.61577i	−0.639541	+	0.171365i	−0.563996	−	0.825778i	$-0.690736\pi$
$233$	−9.76217	+	2.61577i	−0.639541	+	0.171365i	−0.0755455	+	0.997142i	$0.524070\pi$
$234$		0			0
$235$	−13.8914	+	9.54435i	−0.906177	+	0.622605i
$236$		0			0
$237$	−20.4714	+	13.1180i	−1.32976	+	0.852103i
$238$		0			0
$239$	−21.4204			−1.38557			−0.692784	−	0.721145i	$-0.743616\pi$
$239$	−21.4204			−1.38557			−0.692784	+	0.721145i	$0.743616\pi$
$240$		0			0
$241$	−5.02811	−	8.70894i	−0.323889	−	0.560992i	0.657398	−	0.753544i	$-0.271657\pi$
$241$	−5.02811	−	8.70894i	−0.323889	−	0.560992i	−0.981287	+	0.192552i	$0.938324\pi$
$242$		0			0
$243$	−10.6824	−	11.3528i	−0.685275	−	0.728284i
$244$		0			0
$245$	14.5828	+	5.68688i	0.931664	+	0.363321i
$246$		0			0
$247$	−4.67429	+	1.25247i	−0.297418	+	0.0796930i
$248$		0			0
$249$	−2.32325	+	0.739275i	−0.147230	+	0.0468496i
$250$		0			0
$251$		−	5.13237i		−	0.323952i	−0.986795	−	0.161976i	$-0.948213\pi$
$251$		−	5.13237i		−	0.323952i	0.986795	−	0.161976i	$-0.0517867\pi$
$252$		0			0
$253$	−5.11859	+	5.11859i	−0.321803	+	0.321803i
$254$		0			0
$255$	−14.3679	−	6.05651i	−0.899755	−	0.379274i
$256$		0			0
$257$	2.90270	+	10.8330i	0.181065	+	0.675745i	0.995439	+	0.0954037i	$0.0304142\pi$
$257$	2.90270	+	10.8330i	0.181065	+	0.675745i	−0.814373	+	0.580341i	$0.802919\pi$
$258$		0			0
$259$	−5.88682	+	4.87035i	−0.365789	+	0.302629i
$260$		0			0
$261$	14.7634	+	20.8489i	0.913831	+	1.29051i
$262$		0			0
$263$	3.68895	−	13.7673i	0.227470	−	0.848931i	−0.753930	−	0.656955i	$-0.771844\pi$
$263$	3.68895	−	13.7673i	0.227470	−	0.848931i	0.981400	−	0.191975i	$-0.0614893\pi$
$264$		0			0
$265$	−24.2379	+	1.90402i	−1.48893	+	0.116963i
$266$		0			0
$267$	1.31996	+	2.05989i	0.0807803	+	0.126063i
$268$		0			0
$269$	7.17746	+	12.4317i	0.437618	+	0.757976i	0.997505	−	0.0705927i	$-0.0224891\pi$
$269$	7.17746	+	12.4317i	0.437618	+	0.757976i	−0.559888	+	0.828568i	$0.689156\pi$
$270$		0			0
$271$	0.840888	−	1.45646i	0.0510803	−	0.0884737i	−0.839355	−	0.543584i	$-0.817067\pi$
$271$	0.840888	−	1.45646i	0.0510803	−	0.0884737i	0.890435	+	0.455110i	$0.150400\pi$
$272$		0			0
$273$	−2.23133	−	2.45621i	−0.135046	−	0.148657i
$274$		0			0
$275$	5.46624	−	14.2277i	0.329627	−	0.857962i
$276$		0			0
$277$	−2.03258	+	7.58569i	−0.122126	+	0.455780i	−0.999721	−	0.0236225i	$-0.992480\pi$
$277$	−2.03258	+	7.58569i	−0.122126	+	0.455780i	0.877595	+	0.479403i	$0.159147\pi$
$278$		0			0
$279$	2.30890	−	5.02058i	0.138230	−	0.300574i
$280$		0			0
$281$		−	8.01304i		−	0.478018i	−0.971017	−	0.239009i	$-0.923177\pi$
$281$		−	8.01304i		−	0.478018i	0.971017	−	0.239009i	$-0.0768225\pi$
$282$		0			0
$283$	1.35822	+	0.363933i	0.0807376	+	0.0216336i	0.298962	−	0.954265i	$-0.403360\pi$
$283$	1.35822	+	0.363933i	0.0807376	+	0.0216336i	−0.218224	+	0.975899i	$0.570026\pi$
$284$		0			0
$285$	20.4350	+	15.8836i	1.21047	+	0.940861i
$286$		0			0
$287$	−9.81791	−	21.4422i	−0.579533	−	1.26569i
$288$		0			0
$289$	−0.685955	−	0.396036i	−0.0403503	−	0.0232963i
$290$		0			0
$291$	1.29834	−	0.413142i	0.0761102	−	0.0242188i
$292$		0			0
$293$	−3.47074	−	3.47074i	−0.202763	−	0.202763i	0.598420	−	0.801183i	$-0.295795\pi$
$293$	−3.47074	−	3.47074i	−0.202763	−	0.202763i	−0.801183	+	0.598420i	$0.795795\pi$
$294$		0			0
$295$	3.10144	−	3.63025i	0.180573	−	0.211361i
$296$		0			0
$297$	−12.4895	+	9.74186i	−0.724715	+	0.565280i
$298$		0			0
$299$	0.859793	−	1.48920i	0.0497231	−	0.0861229i
$300$		0			0
$301$	9.83822	−	26.4570i	0.567066	−	1.52496i
$302$		0			0
$303$	−0.268748	−	5.80329i	−0.0154392	−	0.333390i
$304$		0			0
$305$	31.3215	+	5.80981i	1.79347	+	0.332669i
$306$		0			0
$307$	19.7303	+	19.7303i	1.12607	+	1.12607i	0.990811	+	0.135254i	$0.0431852\pi$
$307$	19.7303	+	19.7303i	1.12607	+	1.12607i	0.135254	+	0.990811i	$0.456815\pi$
$308$		0			0
$309$	−2.92890	−	0.641204i	−0.166619	−	0.0364768i
$310$		0			0
$311$	23.7702	−	13.7238i	1.34789	−	0.778203i	0.359937	−	0.932977i	$-0.382798\pi$
$311$	23.7702	−	13.7238i	1.34789	−	0.778203i	0.987950	+	0.154774i	$0.0494649\pi$
$312$		0			0
$313$	6.10435	+	22.7817i	0.345038	+	1.28770i	0.892568	+	0.450913i	$0.148902\pi$
$313$	6.10435	+	22.7817i	0.345038	+	1.28770i	−0.547530	+	0.836786i	$0.684432\pi$
$314$		0			0
$315$	−3.26585	+	17.4452i	−0.184010	+	0.982924i
$316$		0			0
$317$	−8.31005	−	31.0135i	−0.466739	−	1.74189i	−0.651059	−	0.759027i	$-0.725675\pi$
$317$	−8.31005	−	31.0135i	−0.466739	−	1.74189i	0.184320	−	0.982866i	$-0.440992\pi$
$318$		0			0
$319$	22.4804	−	12.9791i	1.25866	−	0.726690i
$320$		0			0
$321$	−13.4731	−	2.94957i	−0.751994	−	0.164629i
$322$		0			0
$323$	−19.0240	−	19.0240i	−1.05852	−	1.05852i
$324$		0			0
$325$	−0.379511	+	3.60073i	−0.0210515	+	0.199733i
$326$		0			0
$327$	0.196219	+	4.23711i	0.0108509	+	0.234313i
$328$		0			0
$329$	−12.7122	−	15.3653i	−0.700847	−	0.847117i
$330$		0			0
$331$	13.9114	−	24.0952i	0.764637	−	1.32439i	−0.175801	−	0.984426i	$-0.556252\pi$
$331$	13.9114	−	24.0952i	0.764637	−	1.32439i	0.940438	−	0.339965i	$-0.110415\pi$
$332$		0			0
$333$	−6.66585	−	5.53353i	−0.365287	−	0.303236i
$334$		0			0
$335$	8.92243	−	0.700906i	0.487485	−	0.0382946i
$336$		0			0
$337$	−5.16934	−	5.16934i	−0.281592	−	0.281592i	0.552152	−	0.833744i	$-0.313807\pi$
$337$	−5.16934	−	5.16934i	−0.281592	−	0.281592i	−0.833744	+	0.552152i	$0.813807\pi$
$338$		0			0
$339$	−11.7129	+	3.72713i	−0.636158	+	0.202430i
$340$		0			0
$341$	−4.86279	−	2.80753i	−0.263335	−	0.152036i
$342$		0			0
$343$	−5.16511	+	17.7854i	−0.278890	+	0.960323i
$344$		0			0
$345$	−9.12569	+	1.14364i	−0.491311	+	0.0615715i
$346$		0			0
$347$	−24.8906	−	6.66941i	−1.33620	−	0.358033i	−0.481175	−	0.876625i	$-0.659790\pi$
$347$	−24.8906	−	6.66941i	−1.33620	−	0.358033i	−0.855022	+	0.518592i	$0.826457\pi$
$348$		0			0
$349$		−	18.4866i		−	0.989563i	−0.869017	−	0.494782i	$-0.835248\pi$
$349$		−	18.4866i		−	0.989563i	0.869017	−	0.494782i	$-0.164752\pi$
$350$		0			0
$351$	2.26685	−	3.00323i	0.120996	−	0.160301i
$352$		0			0
$353$	−6.80767	+	25.4066i	−0.362336	+	1.35226i	0.508661	+	0.860967i	$0.330141\pi$
$353$	−6.80767	+	25.4066i	−0.362336	+	1.35226i	−0.870997	+	0.491289i	$0.836526\pi$
$354$		0			0
$355$	−7.20688	+	2.55091i	−0.382501	+	0.135388i
$356$		0			0
$357$	5.62339	−	17.5711i	0.297621	−	0.929962i
$358$		0			0
$359$	4.69610	−	8.13389i	0.247851	−	0.429290i	−0.715078	−	0.699044i	$-0.753609\pi$
$359$	4.69610	−	8.13389i	0.247851	−	0.429290i	0.962929	+	0.269754i	$0.0869424\pi$
$360$		0			0
$361$	12.8293	+	22.2210i	0.675226	+	1.16953i
$362$		0			0
$363$	−1.59585	−	2.49043i	−0.0837602	−	0.130713i
$364$		0			0
$365$	1.99778	+	25.4314i	0.104568	+	1.33114i
$366$		0			0
$367$	3.54474	−	13.2291i	0.185034	−	0.690555i	−0.809590	−	0.586996i	$-0.800310\pi$
$367$	3.54474	−	13.2291i	0.185034	−	0.690555i	0.994623	−	0.103559i	$-0.0330231\pi$
$368$		0			0
$369$	21.8233	−	15.4534i	1.13607	−	0.804471i
$370$		0			0
$371$	−4.79845	−	28.3641i	−0.249123	−	1.47259i
$372$		0			0
$373$	7.60558	+	28.3844i	0.393802	+	1.46969i	0.823811	+	0.566865i	$0.191844\pi$
$373$	7.60558	+	28.3844i	0.393802	+	1.46969i	−0.430008	+	0.902825i	$0.641489\pi$
$374$		0			0
$375$	16.5769	−	10.0103i	0.856028	−	0.516929i
$376$		0			0
$377$	−4.36032	+	4.36032i	−0.224568	+	0.224568i
$378$		0			0
$379$		−	0.342505i		−	0.0175933i	−0.999961	−	0.00879664i	$-0.997200\pi$
$379$		−	0.342505i		−	0.0175933i	0.999961	−	0.00879664i	$-0.00280009\pi$
$380$		0			0
$381$	13.4095	−	4.26701i	0.686991	−	0.218606i
$382$		0			0
$383$	25.0334	−	6.70768i	1.27915	−	0.342746i	0.445618	−	0.895223i	$-0.352984\pi$
$383$	25.0334	−	6.70768i	1.27915	−	0.342746i	0.833528	+	0.552477i	$0.186317\pi$
$384$		0			0
$385$	17.3436	+	4.94253i	0.883913	+	0.251895i
$386$		0			0
$387$	31.5491	+	5.39115i	1.60373	+	0.274047i
$388$		0			0
$389$	14.6458	+	25.3673i	0.742572	+	1.28617i	0.951321	+	0.308203i	$0.0997275\pi$
$389$	14.6458	+	25.3673i	0.742572	+	1.28617i	−0.208749	+	0.977969i	$0.566939\pi$
$390$		0			0
$391$	9.56022			0.483481
$392$		0			0
$393$	−1.89247	+	1.21268i	−0.0954624	+	0.0611716i
$394$		0			0
$395$	−5.72467	+	30.8625i	−0.288040	+	1.55286i
$396$		0			0
$397$	27.5081	−	7.37078i	1.38059	−	0.369929i	0.509257	−	0.860615i	$-0.329920\pi$
$397$	27.5081	−	7.37078i	1.38059	−	0.369929i	0.871337	+	0.490686i	$0.163254\pi$
$398$		0			0
$399$	−16.5653	+	25.7570i	−0.829300	+	1.28946i
$400$		0			0
$401$	22.3395	+	12.8977i	1.11558	+	0.644081i	0.940269	−	0.340432i	$-0.110573\pi$
$401$	22.3395	+	12.8977i	1.11558	+	0.644081i	0.175311	+	0.984513i	$0.443907\pi$
$402$		0			0
$403$	1.28842	+	0.345231i	0.0641807	+	0.0171972i
$404$		0			0
$405$	−20.1246	+	0.0342349i	−0.999999	+	0.00170114i
$406$		0			0
$407$	−6.22459	+	6.22459i	−0.308541	+	0.308541i
$408$		0			0
$409$	25.1282	−	14.5078i	1.24251	−	0.717364i	0.272905	−	0.962041i	$-0.412015\pi$
$409$	25.1282	−	14.5078i	1.24251	−	0.717364i	0.969605	+	0.244677i	$0.0786820\pi$
$410$		0			0
$411$	−0.613620	+	0.673213i	−0.0302677	+	0.0332071i
$412$		0			0
$413$	4.60517	+	3.27248i	0.226605	+	0.161028i
$414$		0			0
$415$	−1.35544	+	2.84068i	−0.0665360	+	0.139443i
$416$		0			0
$417$	−27.1913	−	14.0636i	−1.33156	−	0.688700i
$418$		0			0
$419$	20.0060			0.977359			0.488679	−	0.872463i	$-0.337479\pi$
$419$	20.0060			0.977359			0.488679	+	0.872463i	$0.337479\pi$
$420$		0			0
$421$	−0.405902			−0.0197825			−0.00989124	−	0.999951i	$-0.503149\pi$
$421$	−0.405902			−0.0197825			−0.00989124	+	0.999951i	$0.503149\pi$
$422$		0			0
$423$	14.4432	−	17.3987i	0.702252	−	0.845953i
$424$		0			0
$425$	−18.3916	+	8.18207i	−0.892125	+	0.396889i
$426$		0			0
$427$	−3.54584	+	37.5251i	−0.171595	+	1.81597i
$428$		0			0
$429$	−2.82567	−	2.57554i	−0.136425	−	0.124348i
$430$		0			0
$431$	−34.8654	+	20.1295i	−1.67940	+	0.969605i	−0.717364	+	0.696698i	$0.754652\pi$
$431$	−34.8654	+	20.1295i	−1.67940	+	0.969605i	−0.962041	+	0.272907i	$0.912015\pi$
$432$		0			0
$433$	−9.70713	+	9.70713i	−0.466495	+	0.466495i	−0.900777	−	0.434282i	$-0.857002\pi$
$433$	−9.70713	+	9.70713i	−0.466495	+	0.466495i	0.434282	+	0.900777i	$0.357002\pi$
$434$		0			0
$435$	32.6710	+	4.50842i	1.56646	+	0.216162i
$436$		0			0
$437$	−15.3285	−	4.10727i	−0.733263	−	0.196477i
$438$		0			0
$439$	10.6016	+	6.12083i	0.505987	+	0.292131i	0.731182	−	0.682182i	$-0.238969\pi$
$439$	10.6016	+	6.12083i	0.505987	+	0.292131i	−0.225196	+	0.974314i	$0.572302\pi$
$440$		0			0
$441$	−20.9036	−	2.01023i	−0.995408	−	0.0957255i
$442$		0			0
$443$	10.5633	−	2.83043i	0.501879	−	0.134478i	0.00100695	−	0.999999i	$-0.499679\pi$
$443$	10.5633	−	2.83043i	0.501879	−	0.134478i	0.500872	+	0.865521i	$0.333013\pi$
$444$		0			0
$445$	3.10547	+	0.576031i	0.147213	+	0.0273065i
$446$		0			0
$447$	19.4337	+	30.3276i	0.919184	+	1.43445i
$448$		0			0
$449$	−22.4970			−1.06170			−0.530849	−	0.847467i	$-0.678127\pi$
$449$	−22.4970			−1.06170			−0.530849	+	0.847467i	$0.678127\pi$
$450$		0			0
$451$	−13.5857	−	23.5311i	−0.639725	−	1.10804i
$452$		0			0
$453$	18.8472	−	0.872807i	0.885520	−	0.0410080i
$454$		0			0
$455$	−4.28351	−	0.0677631i	−0.200814	−	0.00317678i
$456$		0			0
$457$	−17.9896	+	4.82030i	−0.841517	+	0.225484i	−0.653732	−	0.756726i	$-0.726798\pi$
$457$	−17.9896	+	4.82030i	−0.841517	+	0.225484i	−0.187785	+	0.982210i	$0.560131\pi$
$458$		0			0
$459$	20.7613	+	2.56596i	0.969053	+	0.119769i
$460$		0			0
$461$		−	5.92233i		−	0.275830i	−0.990444	−	0.137915i	$-0.955960\pi$
$461$		−	5.92233i		−	0.275830i	0.990444	−	0.137915i	$-0.0440402\pi$
$462$		0			0
$463$	26.3657	−	26.3657i	1.22532	−	1.22532i	0.259605	−	0.965715i	$-0.416408\pi$
$463$	26.3657	−	26.3657i	1.22532	−	1.22532i	0.965715	−	0.259605i	$-0.0835922\pi$
$464$		0			0
$465$	−2.68899	−	6.60792i	−0.124699	−	0.306435i
$466$		0			0
$467$	10.4566	+	39.0245i	0.483873	+	1.80584i	0.585083	+	0.810973i	$0.301062\pi$
$467$	10.4566	+	39.0245i	0.483873	+	1.80584i	−0.101210	+	0.994865i	$0.532271\pi$
$468$		0			0
$469$	1.76640	+	10.4413i	0.0815646	+	0.482136i
$470$		0			0
$471$	−12.7855	−	11.6537i	−0.589125	−	0.536975i
$472$		0			0
$473$	8.41731	−	31.4138i	0.387028	−	1.44441i
$474$		0			0
$475$	33.0037	−	5.21743i	1.51431	−	0.239392i
$476$		0			0
$477$	30.5922	−	11.3182i	1.40072	−	0.518224i
$478$		0			0
$479$	0.363532	+	0.629656i	0.0166102	+	0.0287697i	0.874211	−	0.485546i	$-0.161379\pi$
$479$	0.363532	+	0.629656i	0.0166102	+	0.0287697i	−0.857601	+	0.514316i	$0.828046\pi$
$480$		0			0
$481$	1.04557	−	1.81098i	0.0476740	−	0.0825738i
$482$		0			0
$483$	−2.30959	−	10.6342i	−0.105090	−	0.483873i
$484$		0			0
$485$	0.757485	−	1.58751i	0.0343956	−	0.0720850i
$486$		0			0
$487$	4.72938	−	17.6503i	0.214309	−	0.799812i	−0.772100	−	0.635501i	$-0.780793\pi$
$487$	4.72938	−	17.6503i	0.214309	−	0.799812i	0.986409	−	0.164310i	$-0.0525398\pi$
$488$		0			0
$489$	5.77316	+	1.26388i	0.261071	+	0.0571545i
$490$		0			0
$491$		−	37.6269i		−	1.69808i	−0.528330	−	0.849039i	$-0.677182\pi$
$491$		−	37.6269i		−	1.69808i	0.528330	−	0.849039i	$-0.322818\pi$
$492$		0			0
$493$	−33.1147	−	8.87305i	−1.49141	−	0.399622i
$494$		0			0
$495$	−1.85526	+	20.3644i	−0.0833875	+	0.915314i
$496$		0			0
$497$	−3.76584	−	8.22455i	−0.168921	−	0.368921i
$498$		0			0
$499$	3.10886	+	1.79490i	0.139172	+	0.0803509i	0.567969	−	0.823050i	$-0.307729\pi$
$499$	3.10886	+	1.79490i	0.139172	+	0.0803509i	−0.428797	+	0.903401i	$0.641063\pi$
$500$		0			0
$501$	7.91864	+	24.8852i	0.353779	+	1.11179i
$502$		0			0
$503$	−11.0730	−	11.0730i	−0.493722	−	0.493722i	0.415754	−	0.909477i	$-0.363518\pi$
$503$	−11.0730	−	11.0730i	−0.493722	−	0.493722i	−0.909477	+	0.415754i	$0.863518\pi$
$504$		0			0
$505$	−5.70237	−	4.87172i	−0.253752	−	0.216789i
$506$		0			0
$507$	−19.1932	−	9.92696i	−0.852400	−	0.440872i
$508$		0			0
$509$	17.1754	−	29.7487i	0.761287	−	1.31859i	−0.180900	−	0.983502i	$-0.557901\pi$
$509$	17.1754	−	29.7487i	0.761287	−	1.31859i	0.942187	−	0.335087i	$-0.108766\pi$
$510$		0			0
$511$	−29.7607	+	5.03472i	−1.31654	+	0.222723i
$512$		0			0
$513$	−32.1855	−	13.0336i	−1.42102	−	0.575449i
$514$		0			0
$515$	−3.19030	+	2.19195i	−0.140581	+	0.0965888i
$516$		0			0
$517$	−16.2469	−	16.2469i	−0.714539	−	0.714539i
$518$		0			0
$519$	6.43165	−	29.3786i	0.282318	−	1.28958i
$520$		0			0
$521$	23.1158	−	13.3459i	1.01272	−	0.584696i	0.100736	−	0.994913i	$-0.467880\pi$
$521$	23.1158	−	13.3459i	1.01272	−	0.584696i	0.911988	+	0.410217i	$0.134547\pi$
$522$		0			0
$523$	6.89247	+	25.7231i	0.301387	+	1.12479i	0.936011	+	0.351970i	$0.114488\pi$
$523$	6.89247	+	25.7231i	0.301387	+	1.12479i	−0.634625	+	0.772821i	$0.718845\pi$
$524$		0			0
$525$	13.5443	+	18.4811i	0.591124	+	0.806581i
$526$		0			0
$527$	1.91935	+	7.16310i	0.0836080	+	0.312029i
$528$		0			0
$529$	−15.0350	+	8.68046i	−0.653696	+	0.377411i
$530$		0			0
$531$	−2.67653	+	5.81996i	−0.116151	+	0.252565i
$532$		0			0
$533$	4.56410	+	4.56410i	0.197693	+	0.197693i
$534$		0			0
$535$	−14.6755	+	10.0831i	−0.634479	+	0.435929i
$536$		0			0
$537$	−24.7103	+	1.14432i	−1.06633	+	0.0493812i
$538$		0			0
$539$	−3.99692	+	20.9606i	−0.172160	+	0.902836i
$540$		0			0
$541$	4.10742	−	7.11425i	0.176592	−	0.305866i	−0.764119	−	0.645075i	$-0.776826\pi$
$541$	4.10742	−	7.11425i	0.176592	−	0.305866i	0.940711	+	0.339209i	$0.110159\pi$
$542$		0			0
$543$	4.69475	−	9.07704i	0.201471	−	0.389533i
$544$		0			0
$545$	4.16343	+	3.55695i	0.178342	+	0.152363i
$546$		0			0
$547$	−22.4125	−	22.4125i	−0.958288	−	0.958288i	0.0408761	−	0.999164i	$-0.486985\pi$
$547$	−22.4125	−	22.4125i	−0.958288	−	0.958288i	−0.999164	+	0.0408761i	$0.986985\pi$
$548$		0			0
$549$	−42.5561	+	3.94998i	−1.81625	+	0.168581i
$550$		0			0
$551$	49.2829	+	28.4535i	2.09952	+	1.21216i
$552$		0			0
$553$	−36.9752	−	3.49388i	−1.57235	−	0.148575i
$554$		0			0
$555$	−11.0975	+	1.39075i	−0.471064	+	0.0590341i
$556$		0			0
$557$	−35.9630	−	9.63625i	−1.52380	−	0.408301i	−0.602809	−	0.797886i	$-0.705952\pi$
$557$	−35.9630	−	9.63625i	−1.52380	−	0.408301i	−0.920991	+	0.389585i	$0.872618\pi$
$558$		0			0
$559$			7.72566i			0.326761i
$560$		0			0
$561$	4.54581	−	20.7644i	0.191924	−	0.876675i
$562$		0			0
$563$	5.68038	−	21.1995i	0.239399	−	0.893450i	−0.736717	−	0.676201i	$-0.763625\pi$
$563$	5.68038	−	21.1995i	0.239399	−	0.893450i	0.976116	−	0.217249i	$-0.0697084\pi$
$564$		0			0
$565$	−6.83360	+	14.3216i	−0.287492	+	0.602514i
$566$		0			0
$567$	−2.16135	−	23.7135i	−0.0907683	−	0.995872i
$568$		0			0
$569$	−13.0396	+	22.5852i	−0.546648	+	0.946821i	0.451854	+	0.892092i	$0.350763\pi$
$569$	−13.0396	+	22.5852i	−0.546648	+	0.946821i	−0.998501	+	0.0547292i	$0.982570\pi$
$570$		0			0
$571$	0.341798	+	0.592011i	0.0143038	+	0.0247749i	0.873089	−	0.487561i	$-0.162113\pi$
$571$	0.341798	+	0.592011i	0.0143038	+	0.0247749i	−0.858785	+	0.512336i	$0.828780\pi$
$572$		0			0
$573$	16.0932	−	10.3124i	0.672305	−	0.430808i
$574$		0			0
$575$	−6.98178	+	9.60373i	−0.291161	+	0.400503i
$576$		0			0
$577$	−7.48627	+	27.9391i	−0.311657	+	1.16312i	0.615404	+	0.788212i	$0.288993\pi$
$577$	−7.48627	+	27.9391i	−0.311657	+	1.16312i	−0.927061	+	0.374910i	$0.877674\pi$
$578$		0			0
$579$	16.4322	−	18.0280i	0.682898	−	0.749219i
$580$		0			0
$581$	−3.49063	−	1.29801i	−0.144816	−	0.0538507i
$582$		0			0
$583$	−8.57836	−	32.0149i	−0.355279	−	1.32592i
$584$		0			0
$585$	−0.826364	−	4.78684i	−0.0341660	−	0.197912i
$586$		0			0
$587$	−32.5547	+	32.5547i	−1.34367	+	1.34367i	−0.451304	+	0.892370i	$0.649041\pi$
$587$	−32.5547	+	32.5547i	−1.34367	+	1.34367i	−0.892370	+	0.451304i	$0.850959\pi$
$588$		0			0
$589$		−	12.3097i		−	0.507211i
$590$		0			0
$591$	−1.21171	−	3.80793i	−0.0498431	−	0.156637i
$592$		0			0
$593$	−35.3500	+	9.47201i	−1.45165	+	0.388969i	−0.896598	−	0.442846i	$-0.853969\pi$
$593$	−35.3500	+	9.47201i	−1.45165	+	0.388969i	−0.555053	+	0.831815i	$0.687302\pi$
$594$		0			0
$595$	−11.5810	−	20.8124i	−0.474776	−	0.853226i
$596$		0			0
$597$	0.893145	+	19.2864i	0.0365540	+	0.789340i
$598$		0			0
$599$	16.8528	+	29.1898i	0.688585	+	1.19266i	0.972296	+	0.233754i	$0.0751010\pi$
$599$	16.8528	+	29.1898i	0.688585	+	1.19266i	−0.283711	+	0.958910i	$0.591566\pi$
$600$		0			0
$601$	7.76968			0.316932			0.158466	−	0.987364i	$-0.449345\pi$
$601$	7.76968			0.316932			0.158466	+	0.987364i	$0.449345\pi$
$602$		0			0
$603$	−11.2616	+	4.16643i	−0.458606	+	0.169670i
$604$		0			0
$605$	−3.75454	−	0.696427i	−0.152644	−	0.0283138i
$606$		0			0
$607$	0.00880159	−	0.00235838i	0.000357246	−	9.57237e-5i	−0.258640	−	0.965974i	$-0.583274\pi$
$607$	0.00880159	−	0.00235838i	0.000357246	−	9.57237e-5i	0.258998	+	0.965878i	$0.416608\pi$
$608$		0			0
$609$	−1.86990	+	38.9784i	−0.0757724	+	1.57948i
$610$		0			0
$611$	4.72689	+	2.72907i	0.191229	+	0.110406i
$612$		0			0
$613$	36.9113	+	9.89035i	1.49083	+	0.399467i	0.910018	−	0.414568i	$-0.136067\pi$
$613$	36.9113	+	9.89035i	1.49083	+	0.399467i	0.580815	+	0.814036i	$0.302734\pi$
$614$		0			0
$615$	4.71913	−	34.1979i	0.190294	−	1.37899i
$616$		0			0
$617$	−13.3739	+	13.3739i	−0.538413	+	0.538413i	−0.923063	−	0.384650i	$-0.874322\pi$
$617$	−13.3739	+	13.3739i	−0.538413	+	0.538413i	0.384650	+	0.923063i	$0.374322\pi$
$618$		0			0
$619$	−35.2461	+	20.3493i	−1.41666	+	0.817908i	−0.996004	−	0.0893126i	$-0.971533\pi$
$619$	−35.2461	+	20.3493i	−1.41666	+	0.817908i	−0.420655	+	0.907221i	$0.638200\pi$
$620$		0			0
$621$	11.3621	−	4.81225i	0.455945	−	0.193109i
$622$		0			0
$623$	−0.351563	+	3.72054i	−0.0140851	+	0.149060i
$624$		0			0
$625$	5.21201	−	24.4507i	0.208480	−	0.978027i
$626$		0			0
$627$	−16.2094	+	31.3400i	−0.647342	+	1.25160i
$628$		0			0
$629$	11.6259			0.463557
$630$		0			0
$631$	12.9314			0.514789			0.257395	−	0.966306i	$-0.417136\pi$
$631$	12.9314			0.514789			0.257395	+	0.966306i	$0.417136\pi$
$632$		0			0
$633$	−2.16569	+	4.18724i	−0.0860784	+	0.166428i
$634$		0			0
$635$	7.82346	−	16.3961i	0.310465	−	0.650659i
$636$		0			0
$637$	−0.371596	−	5.05531i	−0.0147232	−	0.200299i
$638$		0			0
$639$	8.37071	−	5.92743i	0.331140	−	0.234485i
$640$		0			0
$641$	−1.71838	+	0.992107i	−0.0678719	+	0.0391859i	−0.533552	−	0.845767i	$-0.679143\pi$
$641$	−1.71838	+	0.992107i	−0.0678719	+	0.0391859i	0.465680	+	0.884953i	$0.345810\pi$
$642$		0			0
$643$	−3.02169	+	3.02169i	−0.119164	+	0.119164i	−0.764174	−	0.645010i	$-0.776853\pi$
$643$	−3.02169	+	3.02169i	−0.119164	+	0.119164i	0.645010	+	0.764174i	$0.276853\pi$
$644$		0			0
$645$	32.9375	−	24.9494i	1.29691	−	0.982383i
$646$		0			0
$647$	20.3963	+	5.46518i	0.801862	+	0.214858i	0.636401	−	0.771358i	$-0.280422\pi$
$647$	20.3963	+	5.46518i	0.801862	+	0.214858i	0.165461	+	0.986216i	$0.447089\pi$
$648$		0			0
$649$	5.63704	+	3.25455i	0.221273	+	0.127752i
$650$		0			0
$651$	7.50412	−	3.86545i	0.294110	−	0.151499i
$652$		0			0
$653$	1.47134	−	0.394244i	0.0575779	−	0.0154279i	−0.229915	−	0.973211i	$-0.573845\pi$
$653$	1.47134	−	0.394244i	0.0575779	−	0.0154279i	0.287493	+	0.957783i	$0.407178\pi$
$654$		0			0
$655$	−0.529213	+	2.85306i	−0.0206781	+	0.111478i
$656$		0			0
$657$	−11.8755	−	32.0986i	−0.463307	−	1.25229i
$658$		0			0
$659$	−28.8543			−1.12400			−0.562002	−	0.827136i	$-0.689969\pi$
$659$	−28.8543			−1.12400			−0.562002	+	0.827136i	$0.689969\pi$
$660$		0			0
$661$	−5.34839	−	9.26369i	−0.208028	−	0.360316i	0.743065	−	0.669219i	$-0.233371\pi$
$661$	−5.34839	−	9.26369i	−0.208028	−	0.360316i	−0.951093	+	0.308903i	$0.900038\pi$
$662$		0			0
$663$	0.233588	+	5.04405i	0.00907179	+	0.195894i
$664$		0			0
$665$	9.62720	+	38.3454i	0.373327	+	1.48697i
$666$		0			0
$667$	−19.5326	+	5.23376i	−0.756307	+	0.202652i
$668$		0			0
$669$	9.72678	+	30.5674i	0.376059	+	1.18180i
$670$		0			0
$671$			43.4275i			1.67650i
$672$		0			0
$673$	25.9024	−	25.9024i	0.998464	−	0.998464i	−0.00153521	−	0.999999i	$-0.500489\pi$
$673$	25.9024	−	25.9024i	0.998464	−	0.998464i	0.999999	+	0.00153521i	$0.000488674\pi$
$674$		0			0
$675$	−17.7395	+	18.9818i	−0.682793	+	0.730612i
$676$		0			0
$677$	−3.01599	−	11.2558i	−0.115914	−	0.432597i	0.883440	−	0.468545i	$-0.155222\pi$
$677$	−3.01599	−	11.2558i	−0.115914	−	0.432597i	−0.999354	+	0.0359478i	$0.988555\pi$
$678$		0			0
$679$	1.95073	+	0.725393i	0.0748622	+	0.0278380i
$680$		0			0
$681$	−15.9884	+	17.5412i	−0.612678	+	0.672179i
$682$		0			0
$683$	−4.94076	+	18.4392i	−0.189053	+	0.705556i	0.804673	+	0.593718i	$0.202340\pi$
$683$	−4.94076	+	18.4392i	−0.189053	+	0.705556i	−0.993726	+	0.111838i	$0.964326\pi$
$684$		0			0
$685$	0.0920952	+	1.17236i	0.00351878	+	0.0447936i
$686$		0			0
$687$	17.3995	−	11.1495i	0.663831	−	0.425378i
$688$		0			0
$689$	3.93674	+	6.81863i	0.149978	+	0.259769i
$690$		0			0
$691$	−21.0814	+	36.5141i	−0.801976	+	1.38906i	0.116338	+	0.993210i	$0.462885\pi$
$691$	−21.0814	+	36.5141i	−0.801976	+	1.38906i	−0.918314	+	0.395854i	$0.870449\pi$
$692$		0			0
$693$	−24.1953	−	0.0401546i	−0.919103	−	0.00152535i
$694$		0			0
$695$	−37.2562	+	13.1870i	−1.41321	+	0.500211i
$696$		0			0
$697$	−9.28775	+	34.6624i	−0.351799	+	1.31293i
$698$		0			0
$699$	3.74360	−	17.1001i	0.141596	−	0.646784i
$700$		0			0
$701$			0.0274102i			0.00103527i	1.00000		0.000517635i	$0.000164768\pi$
$701$			0.0274102i			0.00103527i	−1.00000		0.000517635i	$0.999835\pi$
$702$		0			0
$703$	−18.6406	−	4.99474i	−0.703045	−	0.188380i
$704$		0			0
$705$	−3.63003	−	28.9659i	−0.136715	−	1.09092i
$706$		0			0
$707$	5.14039	−	7.23376i	0.193324	−	0.272054i
$708$		0			0
$709$	−24.5893	−	14.1966i	−0.923471	−	0.533166i	−0.0387299	−	0.999250i	$-0.512331\pi$
$709$	−24.5893	−	14.1966i	−0.923471	−	0.533166i	−0.884741	+	0.466084i	$0.845665\pi$
$710$		0			0
$711$	−3.89210	−	41.9325i	−0.145965	−	1.57259i
$712$		0			0
$713$	3.09302	+	3.09302i	0.115834	+	0.115834i
$714$		0			0
$715$	−4.92074	+	0.386550i	−0.184025	+	0.0144562i
$716$		0			0
$717$	17.0444	−	32.9543i	0.636533	−	1.23070i
$718$		0			0
$719$	−6.78490	+	11.7518i	−0.253034	+	0.438268i	−0.964360	−	0.264595i	$-0.914762\pi$
$719$	−6.78490	+	11.7518i	−0.253034	+	0.438268i	0.711326	+	0.702863i	$0.248095\pi$
$720$		0			0
$721$	−2.91948	−	3.52879i	−0.108727	−	0.131419i
$722$		0			0
$723$	17.3992	−	0.805751i	0.647084	−	0.0299662i
$724$		0			0
$725$	33.0970	−	26.7855i	1.22919	−	0.994787i
$726$		0			0
$727$	−11.5270	−	11.5270i	−0.427515	−	0.427515i	0.460266	−	0.887781i	$-0.347754\pi$
$727$	−11.5270	−	11.5270i	−0.427515	−	0.427515i	−0.887781	+	0.460266i	$0.847754\pi$
$728$		0			0
$729$	25.9659	−	7.40084i	0.961700	−	0.274105i
$730$		0			0
$731$	−37.1972	+	21.4758i	−1.37579	+	0.794312i
$732$		0			0
$733$	3.45107	+	12.8796i	0.127468	+	0.475718i	0.999916	−	0.0129901i	$-0.00413499\pi$
$733$	3.45107	+	12.8796i	0.127468	+	0.475718i	−0.872447	+	0.488708i	$0.837468\pi$
$734$		0			0
$735$	−20.3527	+	17.9100i	−0.750721	+	0.660620i
$736$		0			0
$737$	3.15785	+	11.7853i	0.116321	+	0.434116i
$738$		0			0
$739$	25.3089	−	14.6121i	0.931003	−	0.537515i	0.0438740	−	0.999037i	$-0.486030\pi$
$739$	25.3089	−	14.6121i	0.931003	−	0.537515i	0.887129	+	0.461522i	$0.152697\pi$
$740$		0			0
$741$	1.79250	−	8.18780i	0.0658491	−	0.300786i
$742$		0			0
$743$	12.8645	+	12.8645i	0.471952	+	0.471952i	0.902546	−	0.430594i	$-0.141696\pi$
$743$	12.8645	+	12.8645i	0.471952	+	0.471952i	−0.430594	+	0.902546i	$0.641696\pi$
$744$		0			0
$745$	45.7216	+	8.48088i	1.67511	+	0.310715i
$746$		0			0
$747$	0.711286	−	4.16246i	0.0260246	−	0.152297i
$748$		0			0
$749$	−13.4297	−	16.2326i	−0.490712	−	0.593127i
$750$		0			0
$751$	−10.6527	+	18.4510i	−0.388721	+	0.673285i	−0.992278	−	0.124035i	$-0.960416\pi$
$751$	−10.6527	+	18.4510i	−0.388721	+	0.673285i	0.603556	+	0.797320i	$0.293750\pi$
$752$		0			0
$753$	7.89592	+	4.08387i	0.287743	+	0.148824i
$754$		0			0
$755$	15.8218	−	18.5195i	0.575813	−	0.673992i
$756$		0			0
$757$	18.4939	+	18.4939i	0.672174	+	0.672174i	0.958217	−	0.286043i	$-0.0923401\pi$
$757$	18.4939	+	18.4939i	0.672174	+	0.672174i	−0.286043	+	0.958217i	$0.592340\pi$
$758$		0			0
$759$	−3.80182	−	11.9476i	−0.137997	−	0.433671i
$760$		0			0
$761$	29.7628	+	17.1835i	1.07890	+	0.622903i	0.930599	−	0.366039i	$-0.119286\pi$
$761$	29.7628	+	17.1835i	1.07890	+	0.622903i	0.148300	+	0.988942i	$0.452620\pi$
$762$		0			0
$763$	−3.75311	+	5.28153i	−0.135872	+	0.191204i
$764$		0			0
$765$	20.7504	−	17.2852i	0.750231	−	0.624949i
$766$		0			0
$767$	−1.49356	−	0.400199i	−0.0539294	−	0.0144503i
$768$		0			0
$769$		−	8.44100i		−	0.304390i	−0.988350	−	0.152195i	$-0.951366\pi$
$769$		−	8.44100i		−	0.304390i	0.988350	−	0.152195i	$-0.0486342\pi$
$770$		0			0
$771$	−18.9758	−	4.15425i	−0.683398	−	0.149612i
$772$		0			0
$773$	−6.28472	+	23.4549i	−0.226046	+	0.843614i	0.755937	+	0.654644i	$0.227182\pi$
$773$	−6.28472	+	23.4549i	−0.226046	+	0.843614i	−0.981983	+	0.188970i	$0.939485\pi$
$774$		0			0
$775$	−8.59739	−	3.30310i	−0.308827	−	0.118651i
$776$		0			0
$777$	−2.80863	−	12.9320i	−0.100759	−	0.463933i
$778$		0			0
$779$	29.7833	−	51.5862i	1.06710	−	1.84827i
$780$		0			0
$781$	−5.21104	−	9.02578i	−0.186466	−	0.322968i
$782$		0			0
$783$	−43.8224	+	6.12324i	−1.56609	+	0.218827i
$784$		0			0
$785$	−22.2652	+	1.74905i	−0.794678	+	0.0624263i
$786$		0			0
$787$	0.522746	−	1.95091i	0.0186339	−	0.0695426i	−0.955983	−	0.293423i	$-0.905206\pi$
$787$	0.522746	−	1.95091i	0.0186339	−	0.0695426i	0.974617	+	0.223880i	$0.0718724\pi$
$788$		0			0
$789$	18.2451	+	16.6301i	0.649544	+	0.592046i
$790$		0			0
$791$	−17.5984	−	6.54408i	−0.625727	−	0.232681i
$792$		0			0
$793$	−2.67005	−	9.96477i	−0.0948163	−	0.353859i
$794$		0			0
$795$	16.3571	−	38.8041i	0.580126	−	1.37624i
$796$		0			0
$797$	−10.5999	+	10.5999i	−0.375468	+	0.375468i	−0.869464	−	0.493996i	$-0.835536\pi$
$797$	−10.5999	+	10.5999i	−0.375468	+	0.375468i	0.493996	+	0.869464i	$0.335536\pi$
$798$		0			0
$799$			30.3451i			1.07353i
$800$		0			0
$801$	−4.21935	+	0.391633i	−0.149084	+	0.0138377i
$802$		0			0
$803$	−33.5913	+	9.00076i	−1.18541	+	0.317630i
$804$		0			0
$805$	−12.0540	−	7.21595i	−0.424846	−	0.254329i
$806$		0			0
$807$	−24.8368	+	1.15018i	−0.874298	+	0.0404884i
$808$		0			0
$809$	−2.50317	−	4.33563i	−0.0880069	−	0.152432i	0.818662	−	0.574276i	$-0.194716\pi$
$809$	−2.50317	−	4.33563i	−0.0880069	−	0.152432i	−0.906669	+	0.421844i	$0.861383\pi$
$810$		0			0
$811$	6.45712			0.226740			0.113370	−	0.993553i	$-0.463835\pi$
$811$	6.45712			0.226740			0.113370	+	0.993553i	$0.463835\pi$
$812$		0			0
$813$	1.57160	+	2.45259i	0.0551184	+	0.0860160i
$814$		0			0
$815$	6.28840	−	4.32055i	0.220273	−	0.151342i
$816$		0			0
$817$	68.8672	−	18.4529i	2.40936	−	0.645585i
$818$		0			0
$819$	5.55426	−	1.47838i	0.194082	−	0.0516589i
$820$		0			0
$821$	−7.96631	−	4.59935i	−0.278026	−	0.160518i	0.354503	−	0.935055i	$-0.384650\pi$
$821$	−7.96631	−	4.59935i	−0.278026	−	0.160518i	−0.632530	+	0.774536i	$0.717983\pi$
$822$		0			0
$823$	12.6038	+	3.37717i	0.439340	+	0.117721i	0.471707	−	0.881755i	$-0.343638\pi$
$823$	12.6038	+	3.37717i	0.439340	+	0.117721i	−0.0323671	+	0.999476i	$0.510305\pi$
$824$		0			0
$825$	17.5391	+	19.7307i	0.610635	+	0.686934i
$826$		0			0
$827$	8.73603	−	8.73603i	0.303782	−	0.303782i	−0.538710	−	0.842491i	$-0.681088\pi$
$827$	8.73603	−	8.73603i	0.303782	−	0.303782i	0.842491	+	0.538710i	$0.181088\pi$
$828$		0			0
$829$	−23.6216	+	13.6379i	−0.820411	+	0.473665i	−0.850558	−	0.525881i	$-0.823736\pi$
$829$	−23.6216	+	13.6379i	−0.820411	+	0.473665i	0.0301471	+	0.999545i	$0.490402\pi$
$830$		0			0
$831$	−10.0529	−	9.16304i	−0.348732	−	0.317862i
$832$		0			0
$833$	23.3071	−	15.8419i	0.807544	−	0.548889i
$834$		0			0
$835$	30.4276	+	14.5186i	1.05299	+	0.502438i
$836$		0			0
$837$	5.88673	+	7.54706i	0.203475	+	0.260864i
$838$		0			0
$839$	−22.7675			−0.786022			−0.393011	−	0.919534i	$-0.628567\pi$
$839$	−22.7675			−0.786022			−0.393011	+	0.919534i	$0.628567\pi$
$840$		0			0
$841$	43.5147			1.50051
$842$		0			0
$843$	12.3277	+	6.37604i	0.424589	+	0.219602i
$844$		0			0
$845$	−26.2976	+	9.30817i	−0.904665	+	0.320211i
$846$		0			0
$847$	0.425043	−	4.49817i	0.0146046	−	0.154559i
$848$		0			0
$849$	−1.64064	+	1.79997i	−0.0563066	+	0.0617749i
$850$		0			0
$851$	5.93881	−	3.42877i	0.203580	−	0.117537i
$852$		0			0
$853$	7.75252	−	7.75252i	0.265441	−	0.265441i	−0.561819	−	0.827260i	$-0.689898\pi$
$853$	7.75252	−	7.75252i	0.265441	−	0.265441i	0.827260	+	0.561819i	$0.189898\pi$
$854$		0			0
$855$	−40.6965	+	18.7997i	−1.39179	+	0.642938i
$856$		0			0
$857$	54.0758	+	14.4896i	1.84719	+	0.494954i	0.999376	−	0.0353310i	$-0.0112485\pi$
$857$	54.0758	+	14.4896i	1.84719	+	0.494954i	0.847819	+	0.530285i	$0.177915\pi$
$858$		0			0
$859$	14.4633	+	8.35040i	0.493482	+	0.284912i	0.726018	−	0.687676i	$-0.241369\pi$
$859$	14.4633	+	8.35040i	0.493482	+	0.284912i	−0.232536	+	0.972588i	$0.574702\pi$
$860$		0			0
$861$	40.8001	+	1.95730i	1.39046	+	0.0667045i
$862$		0			0
$863$	35.0073	−	9.38019i	1.19166	−	0.319305i	0.392121	−	0.919914i	$-0.371742\pi$
$863$	35.0073	−	9.38019i	1.19166	−	0.319305i	0.799543	+	0.600609i	$0.205075\pi$
$864$		0			0
$865$	−21.9865	−	32.0006i	−0.747565	−	1.08805i
$866$		0			0
$867$	1.15511	−	0.740183i	0.0392294	−	0.0251379i
$868$		0			0
$869$	−42.7911			−1.45159
$870$		0			0
$871$	−1.44919	−	2.51006i	−0.0491038	−	0.0850502i
$872$		0			0
$873$	−0.397501	+	2.32618i	−0.0134534	+	0.0787294i
$874$		0			0
$875$	29.3647	+	3.56547i	0.992709	+	0.120535i
$876$		0			0
$877$	36.2427	−	9.71119i	1.22383	−	0.327924i	0.411654	−	0.911340i	$-0.364951\pi$
$877$	36.2427	−	9.71119i	1.22383	−	0.327924i	0.812173	+	0.583416i	$0.198284\pi$
$878$		0			0
$879$	8.10128	−	2.57788i	0.273249	−	0.0869499i
$880$		0			0
$881$		−	38.3229i		−	1.29113i	−0.763704	−	0.645566i	$-0.776621\pi$
$881$		−	38.3229i		−	1.29113i	0.763704	−	0.645566i	$-0.223379\pi$
$882$		0			0
$883$	14.7904	−	14.7904i	0.497735	−	0.497735i	−0.412997	−	0.910732i	$-0.635518\pi$
$883$	14.7904	−	14.7904i	0.497735	−	0.497735i	0.910732	+	0.412997i	$0.135518\pi$
$884$		0			0
$885$	3.11714	+	7.66005i	0.104781	+	0.257490i
$886$		0			0
$887$	−0.221528	−	0.826754i	−0.00743818	−	0.0277597i	0.962107	−	0.272672i	$-0.0879074\pi$
$887$	−0.221528	−	0.826754i	−0.00743818	−	0.0277597i	−0.969545	+	0.244912i	$0.921241\pi$
$888$		0			0
$889$	20.1475	+	7.49200i	0.675727	+	0.251274i
$890$		0			0
$891$	−5.04942	−	26.9662i	−0.169162	−	0.903403i
$892$		0			0
$893$	13.0369	−	48.6543i	0.436263	−	1.62815i
$894$		0			0
$895$	−20.7437	+	24.2806i	−0.693385	+	0.811611i
$896$		0			0
$897$	1.60693	+	2.50773i	0.0536539	+	0.0837305i
$898$		0			0
$899$	−7.84290	−	13.5843i	−0.261575	−	0.453062i
$900$		0			0
$901$	−21.8867	+	37.9089i	−0.729152	+	1.26293i
$902$		0			0
$903$	32.8746	+	36.1877i	1.09400	+	1.20425i
$904$		0			0
$905$	−4.40211	−	12.4369i	−0.146331	−	0.413417i
$906$		0			0
$907$	12.2353	−	45.6627i	0.406266	−	1.51621i	−0.395443	−	0.918491i	$-0.629409\pi$
$907$	12.2353	−	45.6627i	0.406266	−	1.51621i	0.801709	−	0.597715i	$-0.203925\pi$
$908$		0			0
$909$	9.14195	+	4.20427i	0.303219	+	0.139447i
$910$		0			0
$911$			32.5777i			1.07935i	0.841875	+	0.539673i	$0.181452\pi$
$911$			32.5777i			1.07935i	−0.841875	+	0.539673i	$0.818548\pi$
$912$		0			0
$913$	−4.14461	−	1.11055i	−0.137167	−	0.0367537i
$914$		0			0
$915$	−33.8610	+	43.5639i	−1.11941	+	1.44018i
$916$		0			0
$917$	−3.41815	−	0.322989i	−0.112877	−	0.0106660i
$918$		0			0
$919$	17.0821	+	9.86237i	0.563487	+	0.325329i	0.754544	−	0.656250i	$-0.227858\pi$
$919$	17.0821	+	9.86237i	0.563487	+	0.325329i	−0.191057	+	0.981579i	$0.561191\pi$
$920$		0			0
$921$	−46.0537	+	14.6546i	−1.51752	+	0.482886i
$922$		0			0
$923$	1.75064	+	1.75064i	0.0576232	+	0.0576232i
$924$		0			0
$925$	−8.49037	+	11.6789i	−0.279162	+	0.383998i
$926$		0			0
$927$	3.31701	−	3.99577i	0.108945	−	0.131238i
$928$		0			0
$929$	22.4160	−	38.8256i	0.735444	−	1.27383i	−0.219085	−	0.975706i	$-0.570307\pi$
$929$	22.4160	−	38.8256i	0.735444	−	1.27383i	0.954528	−	0.298120i	$-0.0963595\pi$
$930$		0			0
$931$	−44.1759	+	15.3871i	−1.44781	+	0.504293i
$932$		0			0
$933$	2.19922	+	47.4896i	0.0719993	+	1.55474i
$934$		0			0
$935$	−15.5398	−	22.6176i	−0.508206	−	0.739675i
$936$		0			0
$937$	−20.1180	−	20.1180i	−0.657228	−	0.657228i	0.297495	−	0.954723i	$-0.403849\pi$
$937$	−20.1180	−	20.1180i	−0.657228	−	0.657228i	−0.954723	+	0.297495i	$0.903849\pi$
$938$		0			0
$939$	−39.9060	−	8.73634i	−1.30228	−	0.285100i
$940$		0			0
$941$	−30.7949	+	17.7794i	−1.00389	+	0.579593i	−0.909395	−	0.415933i	$-0.863455\pi$
$941$	−30.7949	+	17.7794i	−1.00389	+	0.579593i	−0.0944896	+	0.995526i	$0.530122\pi$
$942$		0			0
$943$	5.47836	+	20.4455i	0.178400	+	0.665798i
$944$		0			0
$945$	−24.2400	−	18.9056i	−0.788527	−	0.615001i
$946$		0			0
$947$	2.02075	+	7.54155i	0.0656657	+	0.245068i	0.990955	−	0.134194i	$-0.0428446\pi$
$947$	2.02075	+	7.54155i	0.0656657	+	0.245068i	−0.925289	+	0.379262i	$0.876178\pi$
$948$		0			0
$949$	7.15438	−	4.13058i	0.232241	−	0.134084i
$950$		0			0
$951$	54.3253	+	11.8931i	1.76162	+	0.385659i
$952$		0			0
$953$	9.66000	+	9.66000i	0.312918	+	0.312918i	0.846039	−	0.533121i	$-0.178981\pi$
$953$	9.66000	+	9.66000i	0.312918	+	0.312918i	−0.533121	+	0.846039i	$0.678981\pi$
$954$		0			0
$955$	4.50034	−	24.2620i	0.145628	−	0.785100i
$956$		0			0
$957$	2.07989	+	44.9128i	0.0672333	+	1.45182i
$958$		0			0
$959$	−1.37193	+	0.232095i	−0.0443021	+	0.00749474i
$960$		0			0
$961$	13.8035	−	23.9083i	0.445274	−	0.771237i
$962$		0			0
$963$	15.2584	−	18.3807i	0.491696	−	0.592311i
$964$		0			0
$965$	−2.46623	−	31.3947i	−0.0793906	−	1.01063i
$966$		0			0
$967$	−11.2478	−	11.2478i	−0.361704	−	0.361704i	0.502736	−	0.864440i	$-0.332327\pi$
$967$	−11.2478	−	11.2478i	−0.361704	−	0.361704i	−0.864440	+	0.502736i	$0.832327\pi$
$968$		0			0
$969$	44.4051	−	14.1300i	1.42650	−	0.453922i
$970$		0			0
$971$	−23.4447	−	13.5358i	−0.752375	−	0.434384i	0.0741766	−	0.997245i	$-0.476367\pi$
$971$	−23.4447	−	13.5358i	−0.752375	−	0.434384i	−0.826551	+	0.562861i	$0.809700\pi$
$972$		0			0
$973$	−19.4676	−	42.5170i	−0.624103	−	1.36303i
$974$		0			0
$975$	−5.23759	−	3.44899i	−0.167737	−	0.110456i
$976$		0			0
$977$	30.8489	+	8.26594i	0.986944	+	0.264451i	0.715966	−	0.698135i	$-0.245986\pi$
$977$	30.8489	+	8.26594i	0.986944	+	0.264451i	0.270978	+	0.962586i	$0.412653\pi$
$978$		0			0
$979$			4.30575i			0.137612i
$980$		0			0
$981$	−6.67475	−	3.06963i	−0.213108	−	0.0980058i
$982$		0			0
$983$	−7.40257	+	27.6268i	−0.236105	+	0.881157i	0.741542	+	0.670906i	$0.234095\pi$
$983$	−7.40257	+	27.6268i	−0.236105	+	0.881157i	−0.977648	+	0.210251i	$0.932572\pi$
$984$		0			0
$985$	−4.65603	−	2.22164i	−0.148353	−	0.0707874i
$986$		0			0
$987$	33.7541	−	7.33087i	1.07440	−	0.233344i
$988$		0			0
$989$	−12.6675	+	21.9407i	−0.402802	+	0.697674i
$990$		0			0
$991$	9.37664	+	16.2408i	0.297859	+	0.515907i	0.975646	−	0.219352i	$-0.0703942\pi$
$991$	9.37664	+	16.2408i	0.297859	+	0.515907i	−0.677787	+	0.735258i	$0.737061\pi$
$992$		0			0
$993$	26.0000	+	40.5747i	0.825085	+	1.28760i
$994$		0			0
$995$	18.9510	+	16.1905i	0.600787	+	0.513272i
$996$		0			0
$997$	−7.26414	+	27.1101i	−0.230058	+	0.858587i	0.750257	+	0.661146i	$0.229930\pi$
$997$	−7.26414	+	27.1101i	−0.230058	+	0.858587i	−0.980315	+	0.197441i	$0.936737\pi$
$998$		0			0
$999$	13.8172	−	5.85205i	0.437156	−	0.185151i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	420.2.bv.c.317.5	yes	48
3.2	odd	2	inner	420.2.bv.c.317.7	yes	48
5.3	odd	4	inner	420.2.bv.c.233.3	yes	48
7.4	even	3	inner	420.2.bv.c.137.12	yes	48
15.8	even	4	inner	420.2.bv.c.233.12	yes	48
21.11	odd	6	inner	420.2.bv.c.137.3	yes	48
35.18	odd	12	inner	420.2.bv.c.53.7	yes	48
105.53	even	12	inner	420.2.bv.c.53.5	✓	48

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
420.2.bv.c.53.5	✓	48	105.53	even	12	inner
420.2.bv.c.53.7	yes	48	35.18	odd	12	inner
420.2.bv.c.137.3	yes	48	21.11	odd	6	inner
420.2.bv.c.137.12	yes	48	7.4	even	3	inner
420.2.bv.c.233.3	yes	48	5.3	odd	4	inner
420.2.bv.c.233.12	yes	48	15.8	even	4	inner
420.2.bv.c.317.5	yes	48	1.1	even	1	trivial
420.2.bv.c.317.7	yes	48	3.2	odd	2	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)

\(f(q)\)	\(=\)	\(q+(-0.795708 + 1.53846i) q^{3} +(0.746108 + 2.10792i) q^{5} +(-2.40557 + 1.10146i) q^{7} +(-1.73370 - 2.44833i) q^{9} +(-2.63993 + 1.52416i) q^{11} +(0.512041 - 0.512041i) q^{13} +(-3.83663 - 0.529433i) q^{15} +(3.88873 + 1.04198i) q^{17} +(-5.78739 - 3.34135i) q^{19} +(0.219587 - 4.57731i) q^{21} +(2.29376 - 0.614611i) q^{23} +(-3.88664 + 3.14547i) q^{25} +(5.14616 - 0.719065i) q^{27} -8.51556 q^{29} +(0.921008 + 1.59523i) q^{31} +(-0.244246 - 5.27420i) q^{33} +(-4.11661 - 4.24895i) q^{35} +(2.78938 - 0.747413i) q^{37} +(0.380318 + 1.19519i) q^{39} +8.91355i q^{41} +(-7.54399 + 7.54399i) q^{43} +(3.86735 - 5.48121i) q^{45} +(1.95084 + 7.28063i) q^{47} +(4.57357 - 5.29929i) q^{49} +(-4.69734 + 5.15352i) q^{51} +(-2.81412 + 10.5024i) q^{53} +(-5.18248 - 4.42756i) q^{55} +(9.74561 - 6.24491i) q^{57} +(-1.06765 - 1.84923i) q^{59} +(7.12318 - 12.3377i) q^{61} +(6.86727 + 3.98003i) q^{63} +(1.46138 + 0.697303i) q^{65} +(1.03593 - 3.86614i) q^{67} +(-0.879611 + 4.01790i) q^{69} +3.41895i q^{71} +(11.0196 + 2.95269i) q^{73} +(-1.74654 - 8.48231i) q^{75} +(4.67173 - 6.57426i) q^{77} +(12.1569 + 7.01879i) q^{79} +(-2.98859 + 8.48931i) q^{81} +(0.995323 + 0.995323i) q^{83} +(0.705000 + 8.97455i) q^{85} +(6.77590 - 13.1008i) q^{87} +(0.706248 - 1.22326i) q^{89} +(-0.667760 + 1.79575i) q^{91} +(-3.18705 + 0.147591i) q^{93} +(2.72528 - 14.6924i) q^{95} +(-0.556234 - 0.556234i) q^{97} +(8.30847 + 3.82096i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 48 q - 2 q^{3} - 8 q^{7} + 32 q^{13} - 16 q^{21} + 16 q^{25} - 32 q^{27} - 64 q^{31} + 34 q^{33} - 40 q^{37} - 24 q^{43} + 30 q^{45} + 36 q^{51} + 92 q^{57} - 18 q^{63} + 28 q^{67} - 8 q^{73} - 38 q^{75}+ \cdots + 56 q^{97}+O(q^{100}) \) 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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