LMFDB - Embedded newform 420.2.bv.c.317.1

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			317.1
Character	$\chi$	$=$	420.317
Dual form			420.2.bv.c.53.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-1.72581 - 0.146940i) q^{3} +(1.07475 - 1.96085i) q^{5} +(-2.28090 - 1.34070i) q^{7} +(2.95682 + 0.507180i) q^{9} +O(q^{10})$	\(q+(-1.72581 - 0.146940i) q^{3} +(1.07475 - 1.96085i) q^{5} +(-2.28090 - 1.34070i) q^{7} +(2.95682 + 0.507180i) q^{9} +(-4.85929 + 2.80551i) q^{11} +(0.354801 - 0.354801i) q^{13} +(-2.14293 + 3.22612i) q^{15} +(-3.57956 - 0.959140i) q^{17} +(2.40962 + 1.39119i) q^{19} +(3.73940 + 2.64895i) q^{21} +(-3.95939 + 1.06092i) q^{23} +(-2.68984 - 4.21483i) q^{25} +(-5.02837 - 1.30977i) q^{27} -8.44181 q^{29} +(-0.730482 - 1.26523i) q^{31} +(8.79843 - 4.12775i) q^{33} +(-5.08030 + 3.03159i) q^{35} +(-7.29673 + 1.95515i) q^{37} +(-0.664452 + 0.560183i) q^{39} +1.89242i q^{41} +(7.19023 - 7.19023i) q^{43} +(4.17233 - 5.25277i) q^{45} +(-2.88649 - 10.7725i) q^{47} +(3.40505 + 6.11602i) q^{49} +(6.03669 + 2.18127i) q^{51} +(-0.636707 + 2.37622i) q^{53} +(0.278670 + 12.5435i) q^{55} +(-3.95411 - 2.75500i) q^{57} +(5.64924 + 9.78477i) q^{59} +(1.67588 - 2.90270i) q^{61} +(-6.06424 - 5.12103i) q^{63} +(-0.314389 - 1.07703i) q^{65} +(-0.921448 + 3.43889i) q^{67} +(6.98904 - 1.24914i) q^{69} -10.5611i q^{71} +(10.4253 + 2.79345i) q^{73} +(4.02281 + 7.66922i) q^{75} +(14.8449 + 0.115747i) q^{77} +(-1.42483 - 0.822628i) q^{79} +(8.48554 + 2.99928i) q^{81} +(0.481281 + 0.481281i) q^{83} +(-5.72785 + 5.98813i) q^{85} +(14.5689 + 1.24044i) q^{87} +(7.37227 - 12.7691i) q^{89} +(-1.28495 + 0.333585i) q^{91} +(1.07476 + 2.29088i) q^{93} +(5.31764 - 3.22971i) q^{95} +(-5.38134 - 5.38134i) q^{97} +(-15.7909 + 5.83085i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	−1.72581	−	0.146940i	−0.996395	−	0.0848359i
$3$	−1.72581	−	0.146940i	−0.996395	−	0.0848359i
$4$		0			0
$5$	1.07475	−	1.96085i	0.480642	−	0.876917i
$5$	1.07475	−	1.96085i	0.480642	−	0.876917i
$6$		0			0
$7$	−2.28090	−	1.34070i	−0.862101	−	0.506737i
$7$	−2.28090	−	1.34070i	−0.862101	−	0.506737i
$8$		0			0
$9$	2.95682	+	0.507180i	0.985606	+	0.169060i
$10$		0			0
$11$	−4.85929	+	2.80551i	−1.46513	+	0.845893i	−0.999241	−	0.0389498i	$-0.987599\pi$
$11$	−4.85929	+	2.80551i	−1.46513	+	0.845893i	−0.465889	+	0.884843i	$0.654265\pi$
$12$		0			0
$13$	0.354801	−	0.354801i	0.0984040	−	0.0984040i	−0.656191	−	0.754595i	$-0.727833\pi$
$13$	0.354801	−	0.354801i	0.0984040	−	0.0984040i	0.754595	+	0.656191i	$0.227833\pi$
$14$		0			0
$15$	−2.14293	+	3.22612i	−0.553303	+	0.832980i
$16$		0			0
$17$	−3.57956	−	0.959140i	−0.868171	−	0.232626i	−0.202874	−	0.979205i	$-0.565028\pi$
$17$	−3.57956	−	0.959140i	−0.868171	−	0.232626i	−0.665297	+	0.746579i	$0.731695\pi$
$18$		0			0
$19$	2.40962	+	1.39119i	0.552804	+	0.319161i	0.750252	−	0.661152i	$-0.229932\pi$
$19$	2.40962	+	1.39119i	0.552804	+	0.319161i	−0.197448	+	0.980313i	$0.563265\pi$
$20$		0			0
$21$	3.73940	+	2.64895i	0.816003	+	0.578047i
$22$		0			0
$23$	−3.95939	+	1.06092i	−0.825591	+	0.221216i	−0.646789	−	0.762669i	$-0.723889\pi$
$23$	−3.95939	+	1.06092i	−0.825591	+	0.221216i	−0.178802	+	0.983885i	$0.557222\pi$
$24$		0			0
$25$	−2.68984	−	4.21483i	−0.537967	−	0.842966i
$26$		0			0
$27$	−5.02837	−	1.30977i	−0.967710	−	0.252065i
$28$		0			0
$29$	−8.44181			−1.56760			−0.783802	−	0.621010i	$-0.786722\pi$
$29$	−8.44181			−1.56760			−0.783802	+	0.621010i	$0.786722\pi$
$30$		0			0
$31$	−0.730482	−	1.26523i	−0.131198	−	0.227242i	0.792940	−	0.609299i	$-0.208549\pi$
$31$	−0.730482	−	1.26523i	−0.131198	−	0.227242i	−0.924139	+	0.382057i	$0.875216\pi$
$32$		0			0
$33$	8.79843	−	4.12775i	1.53161	−	0.718548i
$34$		0			0
$35$	−5.08030	+	3.03159i	−0.858728	+	0.512432i
$36$		0			0
$37$	−7.29673	+	1.95515i	−1.19958	+	0.321425i	−0.802664	−	0.596432i	$-0.796585\pi$
$37$	−7.29673	+	1.95515i	−1.19958	+	0.321425i	−0.396911	+	0.917857i	$0.629918\pi$
$38$		0			0
$39$	−0.664452	+	0.560183i	−0.106397	+	0.0897011i
$40$		0			0
$41$			1.89242i			0.295547i	0.989021	+	0.147773i	$0.0472106\pi$
$41$			1.89242i			0.295547i	−0.989021	+	0.147773i	$0.952789\pi$
$42$		0			0
$43$	7.19023	−	7.19023i	1.09650	−	1.09650i	0.101682	−	0.994817i	$-0.467578\pi$
$43$	7.19023	−	7.19023i	1.09650	−	1.09650i	0.994817	−	0.101682i	$-0.0324225\pi$
$44$		0			0
$45$	4.17233	−	5.25277i	0.621975	−	0.783037i
$46$		0			0
$47$	−2.88649	−	10.7725i	−0.421038	−	1.57133i	−0.772426	−	0.635104i	$-0.780957\pi$
$47$	−2.88649	−	10.7725i	−0.421038	−	1.57133i	0.351389	−	0.936230i	$-0.385710\pi$
$48$		0			0
$49$	3.40505	+	6.11602i	0.486435	+	0.873717i
$50$		0			0
$51$	6.03669	+	2.18127i	0.845306	+	0.305439i
$52$		0			0
$53$	−0.636707	+	2.37622i	−0.0874584	+	0.326399i	−0.995768	−	0.0918986i	$-0.970706\pi$
$53$	−0.636707	+	2.37622i	−0.0874584	+	0.326399i	0.908310	+	0.418298i	$0.137373\pi$
$54$		0			0
$55$	0.278670	+	12.5435i	0.0375759	+	1.69137i
$56$		0			0
$57$	−3.95411	−	2.75500i	−0.523734	−	0.364908i
$58$		0			0
$59$	5.64924	+	9.78477i	0.735468	+	1.27387i	0.954518	+	0.298155i	$0.0963712\pi$
$59$	5.64924	+	9.78477i	0.735468	+	1.27387i	−0.219049	+	0.975714i	$0.570296\pi$
$60$		0			0
$61$	1.67588	−	2.90270i	0.214574	−	0.371653i	−0.738567	−	0.674180i	$-0.764497\pi$
$61$	1.67588	−	2.90270i	0.214574	−	0.371653i	0.953141	+	0.302527i	$0.0978304\pi$
$62$		0			0
$63$	−6.06424	−	5.12103i	−0.764022	−	0.645190i
$64$		0			0
$65$	−0.314389	−	1.07703i	−0.0389951	−	0.133589i
$66$		0			0
$67$	−0.921448	+	3.43889i	−0.112573	+	0.420127i	−0.999094	−	0.0425609i	$-0.986448\pi$
$67$	−0.921448	+	3.43889i	−0.112573	+	0.420127i	0.886521	+	0.462688i	$0.153115\pi$
$68$		0			0
$69$	6.98904	−	1.24914i	0.841382	−	0.150379i
$70$		0			0
$71$		−	10.5611i		−	1.25337i	−0.779273	−	0.626685i	$-0.784411\pi$
$71$		−	10.5611i		−	1.25337i	0.779273	−	0.626685i	$-0.215589\pi$
$72$		0			0
$73$	10.4253	+	2.79345i	1.22019	+	0.326948i	0.810751	−	0.585391i	$-0.199059\pi$
$73$	10.4253	+	2.79345i	1.22019	+	0.326948i	0.409436	+	0.912339i	$0.365726\pi$
$74$		0			0
$75$	4.02281	+	7.66922i	0.464514	+	0.885566i
$76$		0			0
$77$	14.8449	+	0.115747i	1.69174	+	0.0131907i
$78$		0			0
$79$	−1.42483	−	0.822628i	−0.160306	−	0.0925529i	0.417701	−	0.908585i	$-0.362836\pi$
$79$	−1.42483	−	0.822628i	−0.160306	−	0.0925529i	−0.578007	+	0.816032i	$0.696169\pi$
$80$		0			0
$81$	8.48554	+	2.99928i	0.942837	+	0.333253i
$82$		0			0
$83$	0.481281	+	0.481281i	0.0528275	+	0.0528275i	0.733027	−	0.680200i	$-0.238107\pi$
$83$	0.481281	+	0.481281i	0.0528275	+	0.0528275i	−0.680200	+	0.733027i	$0.738107\pi$
$84$		0			0
$85$	−5.72785	+	5.98813i	−0.621272	+	0.649504i
$86$		0			0
$87$	14.5689	+	1.24044i	1.56195	+	0.132989i
$88$		0			0
$89$	7.37227	−	12.7691i	0.781459	−	1.35353i	−0.149633	−	0.988742i	$-0.547809\pi$
$89$	7.37227	−	12.7691i	0.781459	−	1.35353i	0.931092	−	0.364785i	$-0.118858\pi$
$90$		0			0
$91$	−1.28495	+	0.333585i	−0.134699	+	0.0349692i
$92$		0			0
$93$	1.07476	+	2.29088i	0.111447	+	0.237553i
$94$		0			0
$95$	5.31764	−	3.22971i	0.545578	−	0.331361i
$96$		0			0
$97$	−5.38134	−	5.38134i	−0.546392	−	0.546392i	0.379003	−	0.925395i	$-0.376267\pi$
$97$	−5.38134	−	5.38134i	−0.546392	−	0.546392i	−0.925395	+	0.379003i	$0.876267\pi$
$98$		0			0
$99$	−15.7909	+	5.83085i	−1.58705	+	0.586022i
$100$		0			0
$101$	−0.640605	+	0.369854i	−0.0637426	+	0.0368018i	−0.531533	−	0.847038i	$-0.678384\pi$
$101$	−0.640605	+	0.369854i	−0.0637426	+	0.0368018i	0.467790	+	0.883840i	$0.345050\pi$
$102$		0			0
$103$	−1.16067	−	4.33168i	−0.114364	−	0.426813i	0.884874	−	0.465830i	$-0.154244\pi$
$103$	−1.16067	−	4.33168i	−0.114364	−	0.426813i	−0.999239	+	0.0390165i	$0.987578\pi$
$104$		0			0
$105$	9.21308	−	4.48544i	0.899105	−	0.437734i
$106$		0			0
$107$	3.68897	+	13.7674i	0.356626	+	1.33095i	0.878426	+	0.477879i	$0.158594\pi$
$107$	3.68897	+	13.7674i	0.356626	+	1.33095i	−0.521800	+	0.853068i	$0.674739\pi$
$108$		0			0
$109$	0.798292	−	0.460894i	0.0764625	−	0.0441456i	−0.461281	−	0.887254i	$-0.652610\pi$
$109$	0.798292	−	0.460894i	0.0764625	−	0.0441456i	0.537744	+	0.843108i	$0.319277\pi$
$110$		0			0
$111$	12.8800	−	2.30203i	1.22252	−	0.218499i
$112$		0			0
$113$	−2.37995	−	2.37995i	−0.223887	−	0.223887i	0.586246	−	0.810133i	$-0.300605\pi$
$113$	−2.37995	−	2.37995i	−0.223887	−	0.223887i	−0.810133	+	0.586246i	$0.800605\pi$
$114$		0			0
$115$	−2.17505	+	8.90398i	−0.202825	+	0.830301i
$116$		0			0
$117$	1.22903	−	0.869133i	0.113624	−	0.0803514i
$118$		0			0
$119$	6.87871	+	6.98682i	0.630570	+	0.640481i
$120$		0			0
$121$	10.2418	−	17.7393i	0.931071	−	1.61266i
$122$		0			0
$123$	0.278072	−	3.26595i	0.0250729	−	0.294481i
$124$		0			0
$125$	−11.1555	+	0.744482i	−0.997781	+	0.0665885i
$126$		0			0
$127$	−10.0614	−	10.0614i	−0.892806	−	0.892806i	0.101980	−	0.994786i	$-0.467482\pi$
$127$	−10.0614	−	10.0614i	−0.892806	−	0.892806i	−0.994786	+	0.101980i	$0.967482\pi$
$128$		0			0
$129$	−13.4655	+	11.3524i	−1.18557	+	0.999524i
$130$		0			0
$131$	−6.97835	−	4.02895i	−0.609701	−	0.352011i	0.163147	−	0.986602i	$-0.447835\pi$
$131$	−6.97835	−	4.02895i	−0.609701	−	0.352011i	−0.772848	+	0.634591i	$0.781169\pi$
$132$		0			0
$133$	−3.63093	−	6.40375i	−0.314841	−	0.555275i
$134$		0			0
$135$	−7.97248	+	8.45219i	−0.686162	+	0.727449i
$136$		0			0
$137$	−11.6723	−	3.12759i	−0.997235	−	0.267208i	−0.276948	−	0.960885i	$-0.589323\pi$
$137$	−11.6723	−	3.12759i	−0.997235	−	0.267208i	−0.720287	+	0.693677i	$0.755990\pi$
$138$		0			0
$139$		−	16.3972i		−	1.39079i	−0.718628	−	0.695395i	$-0.755229\pi$
$139$		−	16.3972i		−	1.39079i	0.718628	−	0.695395i	$-0.244771\pi$
$140$		0			0
$141$	3.39861	+	19.0154i	0.286214	+	1.60139i
$142$		0			0
$143$	−0.728682	+	2.71948i	−0.0609354	+	0.227414i
$144$		0			0
$145$	−9.07281	+	16.5531i	−0.753456	+	1.37466i
$146$		0			0
$147$	−4.97776	−	11.0554i	−0.410559	−	0.911834i
$148$		0			0
$149$	−4.18906	+	7.25567i	−0.343181	+	0.594407i	−0.985022	−	0.172431i	$-0.944838\pi$
$149$	−4.18906	+	7.25567i	−0.343181	+	0.594407i	0.641840	+	0.766838i	$0.278171\pi$
$150$		0			0
$151$	−0.350603	−	0.607261i	−0.0285316	−	0.0494182i	0.851407	−	0.524506i	$-0.175750\pi$
$151$	−0.350603	−	0.607261i	−0.0285316	−	0.0494182i	−0.879939	+	0.475087i	$0.842416\pi$
$152$		0			0
$153$	−10.0976	−	4.65148i	−0.816346	−	0.376050i
$154$		0			0
$155$	−3.26601	+	0.0725584i	−0.262332	+	0.00582803i
$156$		0			0
$157$	−0.209367	+	0.781369i	−0.0167093	+	0.0623600i	−0.973777	−	0.227504i	$-0.926943\pi$
$157$	−0.209367	+	0.781369i	−0.0167093	+	0.0623600i	0.957068	+	0.289864i	$0.0936101\pi$
$158$		0			0
$159$	1.44799	−	4.00734i	0.114833	−	0.317803i
$160$		0			0
$161$	10.4534	+	2.88851i	0.823841	+	0.227647i
$162$		0			0
$163$	0.911747	+	3.40268i	0.0714135	+	0.266519i	0.992396	−	0.123085i	$-0.0392787\pi$
$163$	0.911747	+	3.40268i	0.0714135	+	0.266519i	−0.920983	+	0.389604i	$0.872612\pi$
$164$		0			0
$165$	1.36222	−	21.6887i	0.106048	−	1.68846i
$166$		0			0
$167$	4.96146	−	4.96146i	0.383929	−	0.383929i	−0.488586	−	0.872516i	$-0.662487\pi$
$167$	4.96146	−	4.96146i	0.383929	−	0.383929i	0.872516	+	0.488586i	$0.162487\pi$
$168$		0			0
$169$			12.7482i			0.980633i
$170$		0			0
$171$	6.41921	+	5.33561i	0.490889	+	0.408024i
$172$		0			0
$173$	6.20080	−	1.66150i	0.471438	−	0.126322i	−0.0152749	−	0.999883i	$-0.504862\pi$
$173$	6.20080	−	1.66150i	0.471438	−	0.126322i	0.486713	+	0.873562i	$0.338196\pi$
$174$		0			0
$175$	0.484438	+	13.2199i	0.0366200	+	0.999329i
$176$		0			0
$177$	−8.31172	−	17.7167i	−0.624747	−	1.33167i
$178$		0			0
$179$	−10.8624	−	18.8143i	−0.811896	−	1.40625i	−0.911535	−	0.411222i	$-0.865102\pi$
$179$	−10.8624	−	18.8143i	−0.811896	−	1.40625i	0.0996392	−	0.995024i	$-0.468231\pi$
$180$		0			0
$181$	6.70675			0.498509			0.249254	−	0.968438i	$-0.419814\pi$
$181$	6.70675			0.498509			0.249254	+	0.968438i	$0.419814\pi$
$182$		0			0
$183$	−3.31876	+	4.76325i	−0.245330	+	0.352110i
$184$		0			0
$185$	−4.00839	+	16.4091i	−0.294702	+	1.20642i
$186$		0			0
$187$	20.0850	−	5.38176i	1.46876	−	0.393553i
$188$		0			0
$189$	9.71322	+	9.72900i	0.706533	+	0.707680i
$190$		0			0
$191$	7.73525	+	4.46595i	0.559703	+	0.323145i	0.753026	−	0.657990i	$-0.228593\pi$
$191$	7.73525	+	4.46595i	0.559703	+	0.323145i	−0.193323	+	0.981135i	$0.561927\pi$
$192$		0			0
$193$	20.2556	+	5.42747i	1.45803	+	0.390678i	0.898809	−	0.438340i	$-0.144434\pi$
$193$	20.2556	+	5.42747i	1.45803	+	0.390678i	0.559221	+	0.829018i	$0.311100\pi$
$194$		0			0
$195$	0.384315	+	1.90494i	0.0275214	+	0.136416i
$196$		0			0
$197$	−6.77027	+	6.77027i	−0.482362	+	0.482362i	−0.905885	−	0.423524i	$-0.860793\pi$
$197$	−6.77027	+	6.77027i	−0.482362	+	0.482362i	0.423524	+	0.905885i	$0.360793\pi$
$198$		0			0
$199$	6.41199	−	3.70197i	0.454534	−	0.262425i	−0.255209	−	0.966886i	$-0.582144\pi$
$199$	6.41199	−	3.70197i	0.454534	−	0.262425i	0.709743	+	0.704461i	$0.248811\pi$
$200$		0			0
$201$	2.09555	−	5.79946i	0.147809	−	0.409062i
$202$		0			0
$203$	19.2550	+	11.3179i	1.35143	+	0.794363i
$204$		0			0
$205$	3.71075	+	2.03387i	0.259170	+	0.142052i
$206$		0			0
$207$	−12.2453	+	1.12881i	−0.851106	+	0.0784577i
$208$		0			0
$209$	−15.6120			−1.07991
$210$		0			0
$211$	−15.0795			−1.03812			−0.519058	−	0.854739i	$-0.673717\pi$
$211$	−15.0795			−1.03812			−0.519058	+	0.854739i	$0.673717\pi$
$212$		0			0
$213$	−1.55184	+	18.2264i	−0.106331	+	1.24885i
$214$		0			0
$215$	−6.37125	−	21.8266i	−0.434516	−	1.48856i
$216$		0			0
$217$	−0.0301376	+	3.86523i	−0.00204587	+	0.262389i
$218$		0			0
$219$	−17.5816	−	6.35284i	−1.18805	−	0.429285i
$220$		0			0
$221$	−1.61033	+	0.929727i	−0.108323	+	0.0625402i
$222$		0			0
$223$	−11.8910	+	11.8910i	−0.796283	+	0.796283i	−0.982507	−	0.186224i	$-0.940375\pi$
$223$	−11.8910	+	11.8910i	−0.796283	+	0.796283i	0.186224	+	0.982507i	$0.440375\pi$
$224$		0			0
$225$	−5.81568	−	13.8267i	−0.387712	−	0.921781i
$226$		0			0
$227$	−11.4363	−	3.06435i	−0.759055	−	0.203388i	−0.141524	−	0.989935i	$-0.545200\pi$
$227$	−11.4363	−	3.06435i	−0.759055	−	0.203388i	−0.617531	+	0.786547i	$0.711867\pi$
$228$		0			0
$229$	−14.0898	−	8.13475i	−0.931080	−	0.537559i	−0.0439269	−	0.999035i	$-0.513987\pi$
$229$	−14.0898	−	8.13475i	−0.931080	−	0.537559i	−0.887153	+	0.461476i	$0.847320\pi$
$230$		0			0
$231$	−25.6024	−	2.38107i	−1.68452	−	0.156663i
$232$		0			0
$233$	−7.01982	+	1.88095i	−0.459884	+	0.123225i	−0.481320	−	0.876545i	$-0.659843\pi$
$233$	−7.01982	+	1.88095i	−0.459884	+	0.123225i	0.0214368	+	0.999770i	$0.493176\pi$
$234$		0			0
$235$	−24.2255	−	5.91778i	−1.58030	−	0.386033i
$236$		0			0
$237$	2.33811	+	1.62906i	0.151877	+	0.105819i
$238$		0			0
$239$	22.9991			1.48769			0.743845	−	0.668352i	$-0.233000\pi$
$239$	22.9991			1.48769			0.743845	+	0.668352i	$0.233000\pi$
$240$		0			0
$241$	−12.7879	−	22.1493i	−0.823740	−	1.42676i	−0.902878	−	0.429897i	$-0.858550\pi$
$241$	−12.7879	−	22.1493i	−0.823740	−	1.42676i	0.0791378	−	0.996864i	$-0.474783\pi$
$242$		0			0
$243$	−14.2037	−	6.42304i	−0.911167	−	0.412038i
$244$		0			0
$245$	15.6521	−	0.103598i	0.999978	−	0.00661864i
$246$		0			0
$247$	1.34853	−	0.361337i	0.0858049	−	0.0229913i
$248$		0			0
$249$	−0.759879	−	0.901318i	−0.0481554	−	0.0571187i
$250$		0			0
$251$			27.8607i			1.75855i	0.476313	+	0.879276i	$0.341973\pi$
$251$			27.8607i			1.75855i	−0.476313	+	0.879276i	$0.658027\pi$
$252$		0			0
$253$	16.2634	−	16.2634i	1.02247	−	1.02247i
$254$		0			0
$255$	10.7651	−	9.49271i	0.674134	−	0.594457i
$256$		0			0
$257$	2.05198	+	7.65810i	0.127999	+	0.477699i	0.999929	−	0.0119320i	$-0.00379816\pi$
$257$	2.05198	+	7.65810i	0.127999	+	0.477699i	−0.871930	+	0.489631i	$0.837131\pi$
$258$		0			0
$259$	19.2644	+	5.32321i	1.19703	+	0.330768i
$260$		0			0
$261$	−24.9609	−	4.28152i	−1.54504	−	0.265019i
$262$		0			0
$263$	5.28947	−	19.7406i	0.326163	−	1.21726i	−0.586975	−	0.809605i	$-0.699681\pi$
$263$	5.28947	−	19.7406i	0.326163	−	1.21726i	0.913138	−	0.407651i	$-0.133652\pi$
$264$		0			0
$265$	3.97511	+	3.80232i	0.244189	+	0.233575i
$266$		0			0
$267$	−14.5994	+	20.9538i	−0.893470	+	1.28235i
$268$		0			0
$269$	−9.59686	−	16.6223i	−0.585131	−	1.01348i	−0.994859	−	0.101269i	$-0.967710\pi$
$269$	−9.59686	−	16.6223i	−0.585131	−	1.01348i	0.409728	−	0.912208i	$-0.365624\pi$
$270$		0			0
$271$	2.31420	−	4.00831i	0.140578	−	0.243488i	−0.787137	−	0.616779i	$-0.788437\pi$
$271$	2.31420	−	4.00831i	0.140578	−	0.243488i	0.927714	+	0.373291i	$0.121771\pi$
$272$		0			0
$273$	2.26659	−	0.386893i	0.137180	−	0.0234158i
$274$		0			0
$275$	24.8954	+	12.9347i	1.50125	+	0.779991i
$276$		0			0
$277$	−7.48322	+	27.9277i	−0.449623	+	1.67802i	0.253810	+	0.967254i	$0.418316\pi$
$277$	−7.48322	+	27.9277i	−0.449623	+	1.67802i	−0.703433	+	0.710762i	$0.748350\pi$
$278$		0			0
$279$	−1.51820	−	4.11154i	−0.0908923	−	0.246152i
$280$		0			0
$281$		−	2.69129i		−	0.160549i	−0.996773	−	0.0802743i	$-0.974420\pi$
$281$		−	2.69129i		−	0.160549i	0.996773	−	0.0802743i	$-0.0255796\pi$
$282$		0			0
$283$	−13.8845	−	3.72033i	−0.825346	−	0.221151i	−0.178664	−	0.983910i	$-0.557177\pi$
$283$	−13.8845	−	3.72033i	−0.825346	−	0.221151i	−0.646682	+	0.762759i	$0.723844\pi$
$284$		0			0
$285$	−9.65179	+	4.79247i	−0.571723	+	0.283882i
$286$		0			0
$287$	2.53717	−	4.31643i	0.149764	−	0.254791i
$288$		0			0
$289$	−2.82913	−	1.63340i	−0.166420	−	0.0960825i
$290$		0			0
$291$	8.49641	+	10.0779i	0.498069	+	0.590776i
$292$		0			0
$293$	−4.68014	−	4.68014i	−0.273417	−	0.273417i	0.557057	−	0.830474i	$-0.311930\pi$
$293$	−4.68014	−	4.68014i	−0.273417	−	0.273417i	−0.830474	+	0.557057i	$0.811930\pi$
$294$		0			0
$295$	25.2579	−	0.561137i	1.47057	−	0.0326706i
$296$		0			0
$297$	28.1089	−	7.74260i	1.63104	−	0.449271i
$298$		0			0
$299$	−1.02838	+	1.78121i	−0.0594729	+	0.103010i
$300$		0			0
$301$	−26.0402	+	6.76028i	−1.50093	+	0.389656i
$302$		0			0
$303$	1.15991	−	0.544165i	0.0666349	−	0.0312615i
$304$		0			0
$305$	−3.89061	−	6.40581i	−0.222776	−	0.366796i
$306$		0			0
$307$	−0.468035	−	0.468035i	−0.0267122	−	0.0267122i	0.693625	−	0.720337i	$-0.256013\pi$
$307$	−0.468035	−	0.468035i	−0.0267122	−	0.0267122i	−0.720337	+	0.693625i	$0.756013\pi$
$308$		0			0
$309$	1.36660	+	7.64620i	0.0777430	+	0.434977i
$310$		0			0
$311$	10.8991	−	6.29258i	0.618029	−	0.356819i	−0.158072	−	0.987428i	$-0.550528\pi$
$311$	10.8991	−	6.29258i	0.618029	−	0.356819i	0.776101	+	0.630608i	$0.217195\pi$
$312$		0			0
$313$	4.14514	+	15.4699i	0.234297	+	0.874408i	0.978465	+	0.206415i	$0.0661797\pi$
$313$	4.14514	+	15.4699i	0.234297	+	0.874408i	−0.744168	+	0.667993i	$0.767154\pi$
$314$		0			0
$315$	−16.5591	+	6.38722i	−0.932999	+	0.359879i
$316$		0			0
$317$	2.92155	+	10.9034i	0.164091	+	0.612394i	0.998154	+	0.0607264i	$0.0193417\pi$
$317$	2.92155	+	10.9034i	0.164091	+	0.612394i	−0.834064	+	0.551668i	$0.813992\pi$
$318$		0			0
$319$	41.0212	−	23.6836i	2.29675	−	1.32603i
$320$		0			0
$321$	−4.34346	−	24.3020i	−0.242428	−	1.35640i
$322$		0			0
$323$	−7.29101	−	7.29101i	−0.405683	−	0.405683i
$324$		0			0
$325$	−2.44978	−	0.541068i	−0.135889	−	0.0300131i
$326$		0			0
$327$	−1.44542	+	0.678113i	−0.0799320	+	0.0374997i
$328$		0			0
$329$	−7.85892	+	28.4410i	−0.433276	+	1.56800i
$330$		0			0
$331$	−15.2057	+	26.3371i	−0.835782	+	1.44762i	0.0576098	+	0.998339i	$0.481652\pi$
$331$	−15.2057	+	26.3371i	−0.835782	+	1.44762i	−0.893392	+	0.449278i	$0.851681\pi$
$332$		0			0
$333$	−22.5667	+	2.08027i	−1.23665	+	0.113998i
$334$		0			0
$335$	5.75281	+	5.50275i	0.314310	+	0.300648i
$336$		0			0
$337$	6.99764	+	6.99764i	0.381186	+	0.381186i	0.871529	−	0.490343i	$-0.163129\pi$
$337$	6.99764	+	6.99764i	0.381186	+	0.381186i	−0.490343	+	0.871529i	$0.663129\pi$
$338$		0			0
$339$	3.75763	+	4.45705i	0.204086	+	0.242074i
$340$		0			0
$341$	7.09924	+	4.09875i	0.384445	+	0.221960i
$342$		0			0
$343$	0.433165	−	18.5152i	0.0233887	−	0.999726i
$344$		0			0
$345$	5.06207	−	15.0469i	0.272533	−	0.810100i
$346$		0			0
$347$	17.6803	+	4.73743i	0.949130	+	0.254319i	0.699993	−	0.714150i	$-0.253186\pi$
$347$	17.6803	+	4.73743i	0.949130	+	0.254319i	0.249137	+	0.968468i	$0.419853\pi$
$348$		0			0
$349$		−	21.3176i		−	1.14110i	−0.821262	−	0.570552i	$-0.806729\pi$
$349$		−	21.3176i		−	1.14110i	0.821262	−	0.570552i	$-0.193271\pi$
$350$		0			0
$351$	−2.24878	+	1.31936i	−0.120031	+	0.0704224i
$352$		0			0
$353$	−0.0209074	+	0.0780276i	−0.00111279	+	0.00415299i	−0.966480	−	0.256742i	$-0.917351\pi$
$353$	−0.0209074	+	0.0780276i	−0.00111279	+	0.00415299i	0.965367	+	0.260895i	$0.0840176\pi$
$354$		0			0
$355$	−20.7086	−	11.3505i	−1.09910	−	0.602421i
$356$		0			0
$357$	−10.8447	−	13.0687i	−0.573961	−	0.691667i
$358$		0			0
$359$	2.75276	−	4.76793i	0.145285	−	0.251642i	−0.784194	−	0.620516i	$-0.786923\pi$
$359$	2.75276	−	4.76793i	0.145285	−	0.251642i	0.929479	+	0.368874i	$0.120257\pi$
$360$		0			0
$361$	−5.62917	−	9.75001i	−0.296272	−	0.513158i
$362$		0			0
$363$	−20.2819	+	29.1097i	−1.06453	+	1.52786i
$364$		0			0
$365$	16.6821	−	17.4401i	0.873179	−	0.912858i
$366$		0			0
$367$	−6.29334	+	23.4871i	−0.328510	+	1.22601i	0.582227	+	0.813026i	$0.302182\pi$
$367$	−6.29334	+	23.4871i	−0.328510	+	1.22601i	−0.910736	+	0.412988i	$0.864485\pi$
$368$		0			0
$369$	−0.959798	+	5.59554i	−0.0499651	+	0.291292i
$370$		0			0
$371$	4.63807	−	4.56630i	0.240796	−	0.237070i
$372$		0			0
$373$	5.07391	+	18.9361i	0.262717	+	0.980474i	0.963633	+	0.267230i	$0.0861082\pi$
$373$	5.07391	+	18.9361i	0.262717	+	0.980474i	−0.700916	+	0.713244i	$0.747225\pi$
$374$		0			0
$375$	19.3617	+	0.354361i	0.999833	+	0.0182991i
$376$		0			0
$377$	−2.99516	+	2.99516i	−0.154259	+	0.154259i
$378$		0			0
$379$			15.7804i			0.810583i	0.914188	+	0.405291i	$0.132830\pi$
$379$			15.7804i			0.810583i	−0.914188	+	0.405291i	$0.867170\pi$
$380$		0			0
$381$	15.8856	+	18.8425i	0.813845	+	0.965329i
$382$		0			0
$383$	−0.878346	+	0.235352i	−0.0448814	+	0.0120259i	−0.281190	−	0.959652i	$-0.590729\pi$
$383$	−0.878346	+	0.235352i	−0.0448814	+	0.0120259i	0.236308	+	0.971678i	$0.424062\pi$
$384$		0			0
$385$	16.1815	−	28.9842i	0.824685	−	1.47717i
$386$		0			0
$387$	24.9069	−	17.6134i	1.26609	−	0.895342i
$388$		0			0
$389$	14.2750	+	24.7251i	0.723772	+	1.25361i	0.959477	+	0.281785i	$0.0909266\pi$
$389$	14.2750	+	24.7251i	0.723772	+	1.25361i	−0.235705	+	0.971825i	$0.575740\pi$
$390$		0			0
$391$	15.1905			0.768214
$392$		0			0
$393$	11.4513	+	7.97859i	0.577640	+	0.402466i
$394$		0			0
$395$	−3.14438	+	1.90976i	−0.158211	+	0.0960906i
$396$		0			0
$397$	27.7352	−	7.43161i	1.39199	−	0.372982i	0.516527	−	0.856271i	$-0.327225\pi$
$397$	27.7352	−	7.43161i	1.39199	−	0.372982i	0.875461	+	0.483289i	$0.160558\pi$
$398$		0			0
$399$	5.32531	+	11.5852i	0.266599	+	0.579983i
$400$		0			0
$401$	−0.256628	−	0.148164i	−0.0128154	−	0.00739896i	0.493579	−	0.869701i	$-0.335689\pi$
$401$	−0.256628	−	0.148164i	−0.0128154	−	0.00739896i	−0.506394	+	0.862302i	$0.669022\pi$
$402$		0			0
$403$	−0.708081	−	0.189730i	−0.0352720	−	0.00945111i
$404$		0			0
$405$	15.0009	−	13.4154i	0.745402	−	0.666615i
$406$		0			0
$407$	29.9717	−	29.9717i	1.48564	−	1.48564i
$408$		0			0
$409$	−20.8175	+	12.0190i	−1.02936	+	0.594301i	−0.916801	−	0.399344i	$-0.869238\pi$
$409$	−20.8175	+	12.0190i	−1.02936	+	0.594301i	−0.112558	+	0.993645i	$0.535905\pi$
$410$		0			0
$411$	19.6846	+	7.11275i	0.970971	+	0.350846i
$412$		0			0
$413$	0.233072	−	29.8921i	0.0114687	−	1.47089i
$414$		0			0
$415$	1.46097	−	0.426463i	0.0717164	−	0.0209342i
$416$		0			0
$417$	−2.40940	+	28.2983i	−0.117989	+	1.38578i
$418$		0			0
$419$	24.1241			1.17854			0.589269	−	0.807937i	$-0.299416\pi$
$419$	24.1241			1.17854			0.589269	+	0.807937i	$0.299416\pi$
$420$		0			0
$421$	−25.5203			−1.24378			−0.621891	−	0.783104i	$-0.713635\pi$
$421$	−25.5203			−1.24378			−0.621891	+	0.783104i	$0.713635\pi$
$422$		0			0
$423$	−3.07121	−	33.3164i	−0.149327	−	1.61990i
$424$		0			0
$425$	5.58582	+	17.6672i	0.270952	+	0.856983i
$426$		0			0
$427$	−7.71417	+	4.37394i	−0.373315	+	0.211670i
$428$		0			0
$429$	1.65716	−	4.58622i	0.0800086	−	0.221425i
$430$		0			0
$431$	11.8878	−	6.86340i	0.572613	−	0.330598i	−0.185579	−	0.982629i	$-0.559416\pi$
$431$	11.8878	−	6.86340i	0.572613	−	0.330598i	0.758192	+	0.652031i	$0.226083\pi$
$432$		0			0
$433$	−1.57386	+	1.57386i	−0.0756350	+	0.0756350i	−0.743912	−	0.668277i	$-0.767032\pi$
$433$	−1.57386	+	1.57386i	−0.0756350	+	0.0756350i	0.668277	+	0.743912i	$0.267032\pi$
$434$		0			0
$435$	18.0902	−	27.2343i	0.867360	−	1.30578i
$436$		0			0
$437$	−11.0166	−	2.95188i	−0.526993	−	0.141207i
$438$		0			0
$439$	−3.53021	−	2.03816i	−0.168487	−	0.0972763i	0.413385	−	0.910556i	$-0.364346\pi$
$439$	−3.53021	−	2.03816i	−0.168487	−	0.0972763i	−0.581872	+	0.813280i	$0.697680\pi$
$440$		0			0
$441$	6.96617	+	19.8109i	0.331723	+	0.943377i
$442$		0			0
$443$	21.3126	−	5.71070i	1.01259	−	0.271323i	0.285880	−	0.958265i	$-0.407714\pi$
$443$	21.3126	−	5.71070i	1.01259	−	0.271323i	0.726712	+	0.686942i	$0.241047\pi$
$444$		0			0
$445$	−17.1150	−	28.1795i	−0.811329	−	1.33584i
$446$		0			0
$447$	8.29566	−	11.9063i	0.392371	−	0.563151i
$448$		0			0
$449$	−0.0268595			−0.00126758			−0.000633788	−	1.00000i	$-0.500202\pi$
$449$	−0.0268595			−0.00126758			−0.000633788		1.00000i	$0.500202\pi$
$450$		0			0
$451$	−5.30921	−	9.19582i	−0.250001	−	0.433014i
$452$		0			0
$453$	0.515841	+	1.09953i	0.0242363	+	0.0516606i
$454$		0			0
$455$	−0.726885	+	2.87811i	−0.0340769	+	0.134928i
$456$		0			0
$457$	1.69590	−	0.454415i	0.0793309	−	0.0212566i	−0.218935	−	0.975739i	$-0.570258\pi$
$457$	1.69590	−	0.454415i	0.0793309	−	0.0212566i	0.298266	+	0.954483i	$0.403592\pi$
$458$		0			0
$459$	16.7431	+	9.51131i	0.781501	+	0.443950i
$460$		0			0
$461$		−	27.4449i		−	1.27824i	−0.769108	−	0.639119i	$-0.779299\pi$
$461$		−	27.4449i		−	1.27824i	0.769108	−	0.639119i	$-0.220701\pi$
$462$		0			0
$463$	−15.0804	+	15.0804i	−0.700846	+	0.700846i	−0.964592	−	0.263746i	$-0.915042\pi$
$463$	−15.0804	+	15.0804i	−0.700846	+	0.700846i	0.263746	+	0.964592i	$0.415042\pi$
$464$		0			0
$465$	5.64716	+	0.354685i	0.261881	+	0.0164481i
$466$		0			0
$467$	1.68325	+	6.28196i	0.0778914	+	0.290695i	0.993873	−	0.110525i	$-0.0352532\pi$
$467$	1.68325	+	6.28196i	0.0778914	+	0.290695i	−0.915982	+	0.401219i	$0.868586\pi$
$468$		0			0
$469$	6.71225	−	6.60839i	0.309943	−	0.305147i
$470$		0			0
$471$	0.476142	−	1.31773i	0.0219394	−	0.0607177i
$472$		0			0
$473$	−14.7671	+	55.1116i	−0.678993	+	2.53404i
$474$		0			0
$475$	−0.617836	−	13.8982i	−0.0283483	−	0.637693i
$476$		0			0
$477$	−3.08780	+	6.70313i	−0.141381	+	0.306915i
$478$		0			0
$479$	−14.1306	−	24.4749i	−0.645643	−	1.11829i	−0.984153	−	0.177324i	$-0.943256\pi$
$479$	−14.1306	−	24.4749i	−0.645643	−	1.11829i	0.338509	−	0.940963i	$-0.390077\pi$
$480$		0			0
$481$	−1.89520	+	3.28258i	−0.0864135	+	0.149673i
$482$		0			0
$483$	−17.6161	−	6.52103i	−0.801558	−	0.296717i
$484$		0			0
$485$	−16.3356	+	4.76840i	−0.741759	+	0.216522i
$486$		0			0
$487$	−5.59776	+	20.8911i	−0.253659	+	0.946667i	0.715173	+	0.698947i	$0.246348\pi$
$487$	−5.59776	+	20.8911i	−0.253659	+	0.946667i	−0.968832	+	0.247720i	$0.920319\pi$
$488$		0			0
$489$	−1.07351	−	6.00635i	−0.0485457	−	0.271616i
$490$		0			0
$491$		−	19.7036i		−	0.889213i	−0.895726	−	0.444606i	$-0.853344\pi$
$491$		−	19.7036i		−	0.889213i	0.895726	−	0.444606i	$-0.146656\pi$
$492$		0			0
$493$	30.2180	+	8.09688i	1.36095	+	0.364665i
$494$		0			0
$495$	−5.53785	+	37.2303i	−0.248908	+	1.67338i
$496$		0			0
$497$	−14.1592	+	24.0888i	−0.635129	+	1.08053i
$498$		0			0
$499$	−10.1127	−	5.83855i	−0.452705	−	0.261370i	0.256267	−	0.966606i	$-0.417507\pi$
$499$	−10.1127	−	5.83855i	−0.452705	−	0.261370i	−0.708972	+	0.705237i	$0.750841\pi$
$500$		0			0
$501$	−9.29156	+	7.83349i	−0.415116	+	0.349974i
$502$		0			0
$503$	1.01611	+	1.01611i	0.0453059	+	0.0453059i	0.729397	−	0.684091i	$-0.239801\pi$
$503$	1.01611	+	1.01611i	0.0453059	+	0.0453059i	−0.684091	+	0.729397i	$0.739801\pi$
$504$		0			0
$505$	0.0367374	+	1.65363i	0.00163479	+	0.0735855i
$506$		0			0
$507$	1.87323	−	22.0010i	0.0831929	−	0.977098i
$508$		0			0
$509$	−0.920899	+	1.59504i	−0.0408181	+	0.0706991i	−0.885713	−	0.464234i	$-0.846330\pi$
$509$	−0.920899	+	1.59504i	−0.0408181	+	0.0706991i	0.844895	+	0.534933i	$0.179663\pi$
$510$		0			0
$511$	−20.0339	−	20.3488i	−0.886247	−	0.900176i
$512$		0			0
$513$	−10.2943	−	10.1515i	−0.454504	−	0.448198i
$514$		0			0
$515$	−9.74119	−	2.37957i	−0.429248	−	0.104856i
$516$		0			0
$517$	44.2487	+	44.2487i	1.94606	+	1.94606i
$518$		0			0
$519$	−10.9455	+	1.95628i	−0.480455	+	0.0858712i
$520$		0			0
$521$	−33.3936	+	19.2798i	−1.46300	+	0.844664i	−0.999149	−	0.0412494i	$-0.986866\pi$
$521$	−33.3936	+	19.2798i	−1.46300	+	0.844664i	−0.463851	+	0.885913i	$0.653533\pi$
$522$		0			0
$523$	9.20544	+	34.3552i	0.402526	+	1.50225i	0.808574	+	0.588395i	$0.200240\pi$
$523$	9.20544	+	34.3552i	0.402526	+	1.50225i	−0.406048	+	0.913852i	$0.633093\pi$
$524$		0			0
$525$	1.10648	−	22.8861i	0.0482909	−	0.998833i
$526$		0			0
$527$	1.40127	+	5.22960i	0.0610402	+	0.227805i
$528$		0			0
$529$	−5.36732	+	3.09883i	−0.233362	+	0.134732i
$530$		0			0
$531$	11.7411	+	31.7970i	0.509522	+	1.37987i
$532$		0			0
$533$	0.671433	+	0.671433i	0.0290830	+	0.0290830i
$534$		0			0
$535$	30.9605	+	7.56300i	1.33854	+	0.326977i
$536$		0			0
$537$	15.9819	+	34.0659i	0.689669	+	1.47005i
$538$		0			0
$539$	−33.7046	−	20.1666i	−1.45176	−	0.868637i
$540$		0			0
$541$	11.4999	−	19.9184i	0.494420	−	0.856361i	−0.505559	−	0.862792i	$-0.668714\pi$
$541$	11.4999	−	19.9184i	0.494420	−	0.856361i	0.999979	+	0.00643114i	$0.00204711\pi$
$542$		0			0
$543$	−11.5746	−	0.985490i	−0.496712	−	0.0422914i
$544$		0			0
$545$	−0.0457804	−	2.06067i	−0.00196102	−	0.0882695i
$546$		0			0
$547$	−11.2947	−	11.2947i	−0.482926	−	0.482926i	0.423139	−	0.906065i	$-0.360928\pi$
$547$	−11.2947	−	11.2947i	−0.482926	−	0.482926i	−0.906065	+	0.423139i	$0.860928\pi$
$548$		0			0
$549$	6.42745	−	7.73279i	0.274317	−	0.330028i
$550$		0			0
$551$	−20.3415	−	11.7442i	−0.866578	−	0.500319i
$552$		0			0
$553$	2.14701	+	3.78661i	0.0913002	+	0.161023i
$554$		0			0
$555$	9.32885	−	27.7299i	0.395988	−	1.17707i
$556$		0			0
$557$	−28.4418	−	7.62097i	−1.20512	−	0.322911i	−0.400274	−	0.916395i	$-0.631085\pi$
$557$	−28.4418	−	7.62097i	−1.20512	−	0.322911i	−0.804845	+	0.593485i	$0.797752\pi$
$558$		0			0
$559$		−	5.10220i		−	0.215800i
$560$		0			0
$561$	−35.4536	+	6.33658i	−1.49685	+	0.267531i
$562$		0			0
$563$	−1.25671	+	4.69011i	−0.0529640	+	0.197664i	−0.987338	−	0.158628i	$-0.949293\pi$
$563$	−1.25671	+	4.69011i	−0.0529640	+	0.197664i	0.934374	+	0.356293i	$0.115959\pi$
$564$		0			0
$565$	−7.22457	+	2.10887i	−0.303940	+	0.0887210i
$566$		0			0
$567$	−15.3336	−	18.2176i	−0.643949	−	0.765068i
$568$		0			0
$569$	7.18242	−	12.4403i	0.301103	−	0.521525i	−0.675283	−	0.737558i	$-0.735979\pi$
$569$	7.18242	−	12.4403i	0.301103	−	0.521525i	0.976386	+	0.216033i	$0.0693120\pi$
$570$		0			0
$571$	13.1239	+	22.7312i	0.549218	+	0.951273i	0.998328	+	0.0577969i	$0.0184076\pi$
$571$	13.1239	+	22.7312i	0.549218	+	0.951273i	−0.449111	+	0.893476i	$0.648259\pi$
$572$		0			0
$573$	−12.6933	−	8.84398i	−0.530271	−	0.369463i
$574$		0			0
$575$	15.1217	+	13.8345i	0.630619	+	0.576938i
$576$		0			0
$577$	2.94280	−	10.9827i	0.122510	−	0.457215i	−0.877228	−	0.480073i	$-0.840610\pi$
$577$	2.94280	−	10.9827i	0.122510	−	0.457215i	0.999739	+	0.0228586i	$0.00727674\pi$
$578$		0			0
$579$	−34.1597	−	12.3431i	−1.41963	−	0.512963i
$580$		0			0
$581$	−0.452503	−	1.74301i	−0.0187730	−	0.0723123i
$582$		0			0
$583$	−3.57257	−	13.3330i	−0.147961	−	0.552198i
$584$		0			0
$585$	−0.383341	−	3.34404i	−0.0158492	−	0.138259i
$586$		0			0
$587$	−32.7234	+	32.7234i	−1.35064	+	1.35064i	−0.465697	+	0.884944i	$0.654196\pi$
$587$	−32.7234	+	32.7234i	−1.35064	+	1.35064i	−0.884944	+	0.465697i	$0.845804\pi$
$588$		0			0
$589$		−	4.06496i		−	0.167494i
$590$		0			0
$591$	12.6790	−	10.6893i	0.521544	−	0.439701i
$592$		0			0
$593$	−4.74225	+	1.27068i	−0.194741	+	0.0521807i	−0.354871	−	0.934915i	$-0.615475\pi$
$593$	−4.74225	+	1.27068i	−0.194741	+	0.0521807i	0.160130	+	0.987096i	$0.448809\pi$
$594$		0			0
$595$	21.0930	−	5.97903i	0.864727	−	0.245116i
$596$		0			0
$597$	−11.6098	+	5.44670i	−0.475159	+	0.222919i
$598$		0			0
$599$	11.3589	+	19.6742i	0.464113	+	0.803868i	0.999161	−	0.0409541i	$-0.0130398\pi$
$599$	11.3589	+	19.6742i	0.464113	+	0.803868i	−0.535048	+	0.844822i	$0.679706\pi$
$600$		0			0
$601$	−38.7931			−1.58240			−0.791202	−	0.611555i	$-0.790544\pi$
$601$	−38.7931			−1.58240			−0.791202	+	0.611555i	$0.790544\pi$
$602$		0			0
$603$	−4.46869	+	9.70083i	−0.181979	+	0.395048i
$604$		0			0
$605$	−23.7767	−	39.1478i	−0.966660	−	1.59158i
$606$		0			0
$607$	18.9786	−	5.08531i	0.770319	−	0.206406i	0.147806	−	0.989016i	$-0.452779\pi$
$607$	18.9786	−	5.08531i	0.770319	−	0.206406i	0.622512	+	0.782610i	$0.286112\pi$
$608$		0			0
$609$	−31.5673	−	22.3619i	−1.27917	−	0.906150i
$610$		0			0
$611$	−4.84623	−	2.79797i	−0.196057	−	0.113194i
$612$		0			0
$613$	−22.5674	−	6.04692i	−0.911490	−	0.244233i	−0.227546	−	0.973767i	$-0.573070\pi$
$613$	−22.5674	−	6.04692i	−0.911490	−	0.244233i	−0.683944	+	0.729535i	$0.739737\pi$
$614$		0			0
$615$	−6.10518	−	4.05533i	−0.246184	−	0.163527i
$616$		0			0
$617$	26.0352	−	26.0352i	1.04814	−	1.04814i	0.0493563	−	0.998781i	$-0.484283\pi$
$617$	26.0352	−	26.0352i	1.04814	−	1.04814i	0.998781	−	0.0493563i	$-0.0157170\pi$
$618$		0			0
$619$	8.79293	−	5.07660i	0.353418	−	0.204046i	−0.312772	−	0.949828i	$-0.601257\pi$
$619$	8.79293	−	5.07660i	0.353418	−	0.204046i	0.666190	+	0.745782i	$0.267924\pi$
$620$		0			0
$621$	21.2989	−	0.148786i	0.854694	−	0.00597060i
$622$		0			0
$623$	−33.9350	+	19.2412i	−1.35958	+	0.770882i
$624$		0			0
$625$	−10.5296	+	22.6744i	−0.421182	+	0.906976i
$626$		0			0
$627$	26.9433	+	2.29403i	1.07601	+	0.0916147i
$628$		0			0
$629$	27.9943			1.11621
$630$		0			0
$631$	10.7854			0.429360			0.214680	−	0.976684i	$-0.431129\pi$
$631$	10.7854			0.429360			0.214680	+	0.976684i	$0.431129\pi$
$632$		0			0
$633$	26.0243	+	2.21578i	1.03437	+	0.0880694i
$634$		0			0
$635$	−30.5424	+	8.91541i	−1.21204	+	0.353797i
$636$		0			0
$637$	3.37808	+	0.961855i	0.133844	+	0.0381101i
$638$		0			0
$639$	5.35637	−	31.2272i	0.211895	−	1.23533i
$640$		0			0
$641$	−27.2771	+	15.7485i	−1.07738	+	0.622026i	−0.930189	−	0.367081i	$-0.880357\pi$
$641$	−27.2771	+	15.7485i	−1.07738	+	0.622026i	−0.147193	+	0.989108i	$0.547024\pi$
$642$		0			0
$643$	2.24828	−	2.24828i	0.0886636	−	0.0886636i	−0.661384	−	0.750048i	$-0.730031\pi$
$643$	2.24828	−	2.24828i	0.0886636	−	0.0886636i	0.750048	+	0.661384i	$0.230031\pi$
$644$		0			0
$645$	7.78835	+	38.6047i	0.306666	+	1.52006i
$646$		0			0
$647$	−36.3105	−	9.72936i	−1.42751	−	0.382501i	−0.539369	−	0.842069i	$-0.681337\pi$
$647$	−36.3105	−	9.72936i	−1.42751	−	0.382501i	−0.888142	+	0.459569i	$0.848004\pi$
$648$		0			0
$649$	−54.9026	−	31.6980i	−2.15511	−	1.24426i
$650$		0			0
$651$	0.619968	−	6.66621i	0.0242985	−	0.261269i
$652$		0			0
$653$	−2.27720	+	0.610173i	−0.0891135	+	0.0238779i	−0.303100	−	0.952959i	$-0.598022\pi$
$653$	−2.27720	+	0.610173i	−0.0891135	+	0.0238779i	0.213987	+	0.976837i	$0.431355\pi$
$654$		0			0
$655$	−15.4001	+	9.35336i	−0.601732	+	0.365466i
$656$		0			0
$657$	29.4089	+	13.5472i	1.14735	+	0.528527i
$658$		0			0
$659$	0.552796			0.0215339			0.0107669	−	0.999942i	$-0.496573\pi$
$659$	0.552796			0.0215339			0.0107669	+	0.999942i	$0.496573\pi$
$660$		0			0
$661$	−14.6334	−	25.3458i	−0.569173	−	0.985837i	−0.996648	−	0.0818098i	$-0.973930\pi$
$661$	−14.6334	−	25.3458i	−0.569173	−	0.985837i	0.427475	−	0.904027i	$-0.359403\pi$
$662$		0			0
$663$	2.91574	−	1.36791i	0.113238	−	0.0531251i
$664$		0			0
$665$	−16.4591	+	0.237285i	−0.638256	+	0.00920151i
$666$		0			0
$667$	33.4245	−	8.95606i	1.29420	−	0.346780i
$668$		0			0
$669$	22.2689	−	18.7744i	0.860966	−	0.725859i
$670$		0			0
$671$			18.8068i			0.726027i
$672$		0			0
$673$	13.7634	−	13.7634i	0.530542	−	0.530542i	−0.390192	−	0.920734i	$-0.627591\pi$
$673$	13.7634	−	13.7634i	0.530542	−	0.530542i	0.920734	+	0.390192i	$0.127591\pi$
$674$		0			0
$675$	8.00504	+	24.7168i	0.308114	+	0.951349i
$676$		0			0
$677$	−1.99434	−	7.44300i	−0.0766489	−	0.286058i	0.916953	−	0.398995i	$-0.130641\pi$
$677$	−1.99434	−	7.44300i	−0.0766489	−	0.286058i	−0.993602	+	0.112937i	$0.963974\pi$
$678$		0			0
$679$	5.05955	+	19.4891i	0.194168	+	0.747922i
$680$		0			0
$681$	19.2866	+	6.96893i	0.739064	+	0.267050i
$682$		0			0
$683$	6.06492	−	22.6346i	0.232068	−	0.866089i	−0.747381	−	0.664396i	$-0.768689\pi$
$683$	6.06492	−	22.6346i	0.232068	−	0.866089i	0.979449	−	0.201693i	$-0.0646443\pi$
$684$		0			0
$685$	−18.6775	+	19.5263i	−0.713632	+	0.746061i
$686$		0			0
$687$	23.1209	+	16.1094i	0.882119	+	0.614610i
$688$		0			0
$689$	0.617181	+	1.06899i	0.0235127	+	0.0407252i
$690$		0			0
$691$	−0.318591	+	0.551815i	−0.0121198	+	0.0209920i	−0.872022	−	0.489467i	$-0.837191\pi$
$691$	−0.318591	+	0.551815i	−0.0121198	+	0.0209920i	0.859902	+	0.510459i	$0.170525\pi$
$692$		0			0
$693$	43.8350	+	7.87129i	1.66515	+	0.299006i
$694$		0			0
$695$	−32.1523	−	17.6228i	−1.21961	−	0.668471i
$696$		0			0
$697$	1.81510	−	6.77404i	0.0687517	−	0.256585i
$698$		0			0
$699$	12.3912	−	2.21467i	0.468680	−	0.0837666i
$700$		0			0
$701$		−	33.7903i		−	1.27624i	−0.769937	−	0.638120i	$-0.779712\pi$
$701$		−	33.7903i		−	1.27624i	0.769937	−	0.638120i	$-0.220288\pi$
$702$		0			0
$703$	−20.3023	−	5.43999i	−0.765716	−	0.205173i
$704$		0			0
$705$	40.9390	+	13.7726i	1.54185	+	0.518708i
$706$		0			0
$707$	1.95702	+	0.0152591i	0.0736014	+	0.000573878i
$708$		0			0
$709$	9.01238	+	5.20330i	0.338467	+	0.195414i	0.659594	−	0.751622i	$-0.270728\pi$
$709$	9.01238	+	5.20330i	0.338467	+	0.195414i	−0.321127	+	0.947036i	$0.604062\pi$
$710$		0			0
$711$	−3.79575	−	3.15501i	−0.142352	−	0.118322i
$712$		0			0
$713$	4.23457	+	4.23457i	0.158586	+	0.158586i
$714$		0			0
$715$	4.54933	+	4.35158i	0.170135	+	0.162740i
$716$		0			0
$717$	−39.6921	−	3.37949i	−1.48233	−	0.126210i
$718$		0			0
$719$	−22.3999	+	38.7978i	−0.835376	+	1.44691i	0.0583487	+	0.998296i	$0.481416\pi$
$719$	−22.3999	+	38.7978i	−0.835376	+	1.44691i	−0.893724	+	0.448617i	$0.851917\pi$
$720$		0			0
$721$	−3.16011	+	11.4363i	−0.117689	+	0.425909i
$722$		0			0
$723$	18.8148	+	40.1044i	0.699730	+	1.49150i
$724$		0			0
$725$	22.7071	+	35.5808i	0.843320	+	1.32144i
$726$		0			0
$727$	−29.2716	−	29.2716i	−1.08562	−	1.08562i	−0.995973	−	0.0896504i	$-0.971425\pi$
$727$	−29.2716	−	29.2716i	−1.08562	−	1.08562i	−0.0896504	−	0.995973i	$-0.528575\pi$
$728$		0			0
$729$	23.5690	+	13.1720i	0.872926	+	0.487852i
$730$		0			0
$731$	−32.6343	+	18.8414i	−1.20702	+	0.696875i
$732$		0			0
$733$	5.04628	+	18.8330i	0.186388	+	0.695611i	0.994329	+	0.106347i	$0.0339155\pi$
$733$	5.04628	+	18.8330i	0.186388	+	0.695611i	−0.807941	+	0.589264i	$0.799418\pi$
$734$		0			0
$735$	−27.0278	−	2.12113i	−0.996935	−	0.0782392i
$736$		0			0
$737$	−5.17026	−	19.2957i	−0.190449	−	0.710766i
$738$		0			0
$739$	33.6491	−	19.4273i	1.23780	−	0.714645i	0.269156	−	0.963096i	$-0.413255\pi$
$739$	33.6491	−	19.4273i	1.23780	−	0.714645i	0.968644	+	0.248452i	$0.0799218\pi$
$740$		0			0
$741$	−2.38040	+	0.425445i	−0.0874460	+	0.0156291i
$742$		0			0
$743$	16.5948	+	16.5948i	0.608805	+	0.608805i	0.942634	−	0.333829i	$-0.108341\pi$
$743$	16.5948	+	16.5948i	0.608805	+	0.608805i	−0.333829	+	0.942634i	$0.608341\pi$
$744$		0			0
$745$	9.72507	+	16.0121i	0.356299	+	0.586639i
$746$		0			0
$747$	1.17896	+	1.66716i	0.0431361	+	0.0609981i
$748$		0			0
$749$	10.0438	−	36.3480i	0.366993	−	1.32813i
$750$		0			0
$751$	7.45444	−	12.9115i	0.272017	−	0.471146i	−0.697362	−	0.716720i	$-0.745643\pi$
$751$	7.45444	−	12.9115i	0.272017	−	0.471146i	0.969378	+	0.245573i	$0.0789761\pi$
$752$		0			0
$753$	4.09385	−	48.0822i	0.149188	−	1.75221i
$754$		0			0
$755$	−1.56756	+	0.0348252i	−0.0570492	+	0.00126742i
$756$		0			0
$757$	−24.0812	−	24.0812i	−0.875244	−	0.875244i	0.117794	−	0.993038i	$-0.462418\pi$
$757$	−24.0812	−	24.0812i	−0.875244	−	0.875244i	−0.993038	+	0.117794i	$0.962418\pi$
$758$		0			0
$759$	−30.4573	+	25.6778i	−1.10553	+	0.932044i
$760$		0			0
$761$	8.45249	+	4.88005i	0.306403	+	0.176902i	0.645316	−	0.763916i	$-0.276726\pi$
$761$	8.45249	+	4.88005i	0.306403	+	0.176902i	−0.338913	+	0.940818i	$0.610059\pi$
$762$		0			0
$763$	−2.43875	−	0.0190152i	−0.0882886	−	0.000688396i
$764$		0			0
$765$	−19.9733	+	14.8008i	−0.722135	+	0.535123i
$766$		0			0
$767$	5.47600	+	1.46729i	0.197727	+	0.0529808i
$768$		0			0
$769$		−	21.7405i		−	0.783983i	−0.919969	−	0.391992i	$-0.871786\pi$
$769$		−	21.7405i		−	0.783983i	0.919969	−	0.391992i	$-0.128214\pi$
$770$		0			0
$771$	−2.41604	−	13.5179i	−0.0870116	−	0.486836i
$772$		0			0
$773$	5.30264	−	19.7897i	0.190723	−	0.711787i	−0.802610	−	0.596504i	$-0.796556\pi$
$773$	5.30264	−	19.7897i	0.190723	−	0.711787i	0.993333	−	0.115283i	$-0.0367774\pi$
$774$		0			0
$775$	−3.36786	+	6.48212i	−0.120977	+	0.232845i
$776$		0			0
$777$	−32.4645	−	12.0175i	−1.16466	−	0.431127i
$778$		0			0
$779$	−2.63272	+	4.56001i	−0.0943270	+	0.163379i
$780$		0			0
$781$	29.6292	+	51.3193i	1.06022	+	1.83635i
$782$		0			0
$783$	42.4485	+	11.0568i	1.51699	+	0.395139i
$784$		0			0
$785$	1.30713	+	1.25031i	0.0466534	+	0.0446255i
$786$		0			0
$787$	1.51536	−	5.65540i	0.0540167	−	0.201593i	−0.933644	−	0.358203i	$-0.883390\pi$
$787$	1.51536	−	5.65540i	0.0540167	−	0.201593i	0.987661	+	0.156609i	$0.0500564\pi$
$788$		0			0
$789$	−12.0293	+	33.2912i	−0.428254	+	1.18520i
$790$		0			0
$791$	2.23764	+	8.61924i	0.0795613	+	0.306465i
$792$		0			0
$793$	−0.435279	−	1.62448i	−0.0154572	−	0.0576871i
$794$		0			0
$795$	−6.30155	−	7.14617i	−0.223493	−	0.253449i
$796$		0			0
$797$	23.9839	−	23.9839i	0.849554	−	0.849554i	−0.140523	−	0.990077i	$-0.544878\pi$
$797$	23.9839	−	23.9839i	0.849554	−	0.849554i	0.990077	+	0.140523i	$0.0448785\pi$
$798$		0			0
$799$			41.3294i			1.46213i
$800$		0			0
$801$	28.2747	−	34.0170i	0.999038	−	1.20193i
$802$		0			0
$803$	−58.4965	+	15.6741i	−2.06430	+	0.553127i
$804$		0			0
$805$	16.8987	−	17.3930i	0.595599	−	0.613024i
$806$		0			0
$807$	14.1199	+	30.0970i	0.497042	+	1.05946i
$808$		0			0
$809$	−3.46923	−	6.00888i	−0.121972	−	0.211261i	0.798573	−	0.601897i	$-0.205588\pi$
$809$	−3.46923	−	6.00888i	−0.121972	−	0.211261i	−0.920545	+	0.390636i	$0.872255\pi$
$810$		0			0
$811$	21.4448			0.753029			0.376515	−	0.926411i	$-0.377122\pi$
$811$	21.4448			0.753029			0.376515	+	0.926411i	$0.377122\pi$
$812$		0			0
$813$	−4.58284	+	6.57752i	−0.160727	+	0.230684i
$814$		0			0
$815$	7.65204	+	1.86923i	0.268039	+	0.0654763i
$816$		0			0
$817$	27.3287	−	7.32269i	0.956109	−	0.256189i
$818$		0			0
$819$	−3.96854	+	0.334650i	−0.138672	+	0.0116936i
$820$		0			0
$821$	40.9600	+	23.6483i	1.42951	+	0.825330i	0.997082	−	0.0763370i	$-0.0243225\pi$
$821$	40.9600	+	23.6483i	1.42951	+	0.825330i	0.432431	+	0.901667i	$0.357656\pi$
$822$		0			0
$823$	−37.2064	−	9.96943i	−1.29693	−	0.347512i	−0.456644	−	0.889650i	$-0.650949\pi$
$823$	−37.2064	−	9.96943i	−1.29693	−	0.347512i	−0.840290	+	0.542137i	$0.817615\pi$
$824$		0			0
$825$	−41.0641	−	25.9809i	−1.42967	−	0.904540i
$826$		0			0
$827$	17.2898	−	17.2898i	0.601225	−	0.601225i	−0.339413	−	0.940638i	$-0.610228\pi$
$827$	17.2898	−	17.2898i	0.601225	−	0.601225i	0.940638	+	0.339413i	$0.110228\pi$
$828$		0			0
$829$	−5.14316	+	2.96940i	−0.178629	+	0.103132i	−0.586648	−	0.809842i	$-0.699553\pi$
$829$	−5.14316	+	2.96940i	−0.178629	+	0.103132i	0.408019	+	0.912973i	$0.366220\pi$
$830$		0			0
$831$	17.0183	−	47.0983i	0.590358	−	1.63382i
$832$		0			0
$833$	−6.32244	−	25.1586i	−0.219060	−	0.871693i
$834$		0			0
$835$	−4.39635	−	15.0610i	−0.152142	−	0.521207i
$836$		0			0
$837$	2.01597	+	7.31881i	0.0696821	+	0.252975i
$838$		0			0
$839$	−46.8893			−1.61880			−0.809400	−	0.587258i	$-0.800207\pi$
$839$	−46.8893			−1.61880			−0.809400	+	0.587258i	$0.800207\pi$
$840$		0			0
$841$	42.2642			1.45738
$842$		0			0
$843$	−0.395458	+	4.64464i	−0.0136203	+	0.159970i
$844$		0			0
$845$	24.9973	+	13.7011i	0.859934	+	0.471333i
$846$		0			0
$847$	−47.1436	+	26.7304i	−1.61987	+	0.918469i
$848$		0			0
$849$	23.4152	+	8.46076i	0.803609	+	0.290372i
$850$		0			0
$851$	26.8164	−	15.4824i	0.919254	−	0.530731i
$852$		0			0
$853$	17.1149	−	17.1149i	0.586004	−	0.586004i	−0.350543	−	0.936547i	$-0.614003\pi$
$853$	17.1149	−	17.1149i	0.586004	−	0.586004i	0.936547	+	0.350543i	$0.114003\pi$
$854$		0			0
$855$	17.3613	−	6.85265i	0.593745	−	0.234356i
$856$		0			0
$857$	36.6943	+	9.83221i	1.25345	+	0.335862i	0.823670	−	0.567070i	$-0.191923\pi$
$857$	36.6943	+	9.83221i	1.25345	+	0.335862i	0.429784	+	0.902932i	$0.358590\pi$
$858$		0			0
$859$	−40.6274	−	23.4562i	−1.38619	−	0.800317i	−0.393306	−	0.919408i	$-0.628669\pi$
$859$	−40.6274	−	23.4562i	−1.38619	−	0.800317i	−0.992883	+	0.119091i	$0.962002\pi$
$860$		0			0
$861$	−5.01292	+	7.07651i	−0.170840	+	0.241167i
$862$		0			0
$863$	−24.1896	+	6.48160i	−0.823425	+	0.220636i	−0.645843	−	0.763470i	$-0.723494\pi$
$863$	−24.1896	+	6.48160i	−0.823425	+	0.220636i	−0.177582	+	0.984106i	$0.556827\pi$
$864$		0			0
$865$	3.40635	−	13.9445i	0.115819	−	0.474128i
$866$		0			0
$867$	4.64253	+	3.23465i	0.157669	+	0.109854i
$868$		0			0
$869$	9.23157			0.313160
$870$		0			0
$871$	0.893190	+	1.54705i	0.0302646	+	0.0524198i
$872$		0			0
$873$	−13.1823	−	18.6409i	−0.446154	−	0.630900i
$874$		0			0
$875$	26.4428	+	13.2581i	0.893930	+	0.448206i
$876$		0			0
$877$	−25.9927	+	6.96471i	−0.877710	+	0.235182i	−0.669419	−	0.742885i	$-0.733457\pi$
$877$	−25.9927	+	6.96471i	−0.877710	+	0.235182i	−0.208291	+	0.978067i	$0.566790\pi$
$878$		0			0
$879$	7.38932	+	8.76472i	0.249236	+	0.295627i
$880$		0			0
$881$		−	23.4482i		−	0.789991i	−0.918683	−	0.394996i	$-0.870746\pi$
$881$		−	23.4482i		−	0.789991i	0.918683	−	0.394996i	$-0.129254\pi$
$882$		0			0
$883$	13.3238	−	13.3238i	0.448382	−	0.448382i	−0.446434	−	0.894816i	$-0.647306\pi$
$883$	13.3238	−	13.3238i	0.448382	−	0.448382i	0.894816	+	0.446434i	$0.147306\pi$
$884$		0			0
$885$	−43.6728	−	2.74299i	−1.46804	−	0.0922045i
$886$		0			0
$887$	10.9695	+	40.9386i	0.368319	+	1.37458i	0.862866	+	0.505433i	$0.168667\pi$
$887$	10.9695	+	40.9386i	0.368319	+	1.37458i	−0.494547	+	0.869151i	$0.664666\pi$
$888$		0			0
$889$	9.45978	+	36.4385i	0.317271	+	1.22211i
$890$		0			0
$891$	−49.6482	+	9.23191i	−1.66328	+	0.309281i
$892$		0			0
$893$	8.03132	−	29.9733i	0.268758	−	1.00302i
$894$		0			0
$895$	−48.5663	+	1.07896i	−1.62339	+	0.0360657i
$896$		0			0
$897$	2.03652	−	2.92291i	0.0679974	−	0.0975933i
$898$		0			0
$899$	6.16659	+	10.6808i	0.205667	+	0.356226i
$900$		0			0
$901$	4.55826	−	7.89513i	0.151858	−	0.263025i
$902$		0			0
$903$	45.9336	−	7.84059i	1.52858	−	0.260918i
$904$		0			0
$905$	7.20806	−	13.1509i	0.239604	−	0.437151i
$906$		0			0
$907$	2.40049	−	8.95873i	0.0797068	−	0.297470i	−0.914552	−	0.404467i	$-0.867457\pi$
$907$	2.40049	−	8.95873i	0.0797068	−	0.297470i	0.994259	+	0.106997i	$0.0341236\pi$
$908$		0			0
$909$	−2.08174	+	0.768687i	−0.0690468	+	0.0254958i
$910$		0			0
$911$			25.2501i			0.836573i	0.908315	+	0.418286i	$0.137369\pi$
$911$			25.2501i			0.836573i	−0.908315	+	0.418286i	$0.862631\pi$
$912$		0			0
$913$	−3.68893	−	0.988444i	−0.122086	−	0.0327127i
$914$		0			0
$915$	5.77317	+	11.6269i	0.190855	+	0.384373i
$916$		0			0
$917$	10.5153	+	18.5455i	0.347247	+	0.612427i
$918$		0			0
$919$	16.7217	+	9.65429i	0.551599	+	0.318466i	0.749767	−	0.661702i	$-0.230166\pi$
$919$	16.7217	+	9.65429i	0.551599	+	0.318466i	−0.198168	+	0.980168i	$0.563499\pi$
$920$		0			0
$921$	0.738966	+	0.876512i	0.0243497	+	0.0288820i
$922$		0			0
$923$	−3.74708	−	3.74708i	−0.123337	−	0.123337i
$924$		0			0
$925$	27.8676	+	25.4954i	0.916283	+	0.838284i
$926$		0			0
$927$	−1.23495	−	13.3967i	−0.0405610	−	0.440004i
$928$		0			0
$929$	19.3847	−	33.5754i	0.635993	−	1.10157i	−0.350311	−	0.936633i	$-0.613924\pi$
$929$	19.3847	−	33.5754i	0.635993	−	1.10157i	0.986304	−	0.164938i	$-0.0527425\pi$
$930$		0			0
$931$	−0.303705	+	19.4743i	−0.00995354	+	0.638245i
$932$		0			0
$933$	−19.7343	+	9.25826i	−0.646072	+	0.303102i
$934$		0			0
$935$	11.0335	−	45.1676i	0.360834	−	1.47714i
$936$		0			0
$937$	5.52036	+	5.52036i	0.180342	+	0.180342i	0.791505	−	0.611163i	$-0.209298\pi$
$937$	5.52036	+	5.52036i	0.180342	+	0.180342i	−0.611163	+	0.791505i	$0.709298\pi$
$938$		0			0
$939$	−4.88056	−	27.3071i	−0.159271	−	0.891132i
$940$		0			0
$941$	11.0638	−	6.38770i	0.360670	−	0.208233i	−0.308704	−	0.951158i	$-0.599895\pi$
$941$	11.0638	−	6.38770i	0.360670	−	0.208233i	0.669375	+	0.742925i	$0.266562\pi$
$942$		0			0
$943$	−2.00770	−	7.49284i	−0.0653797	−	0.244001i
$944$		0			0
$945$	29.5163	−	8.58992i	0.960166	−	0.279430i
$946$		0			0
$947$	−6.79014	−	25.3411i	−0.220650	−	0.823477i	−0.984101	−	0.177611i	$-0.943163\pi$
$947$	−6.79014	−	25.3411i	−0.220650	−	0.823477i	0.763451	−	0.645866i	$-0.223503\pi$
$948$		0			0
$949$	4.69002	−	2.70778i	0.152244	−	0.0878983i
$950$		0			0
$951$	−3.43989	−	19.2464i	−0.111546	−	0.624107i
$952$		0			0
$953$	40.2920	+	40.2920i	1.30518	+	1.30518i	0.924848	+	0.380336i	$0.124192\pi$
$953$	40.2920	+	40.2920i	1.30518	+	1.30518i	0.380336	+	0.924848i	$0.375808\pi$
$954$		0			0
$955$	17.0705	−	10.3679i	0.552388	−	0.335496i
$956$		0			0
$957$	−74.2747	+	34.8456i	−2.40096	+	1.12640i
$958$		0			0
$959$	22.4303	+	22.7828i	0.724312	+	0.735696i
$960$		0			0
$961$	14.4328	−	24.9983i	0.465574	−	0.806398i
$962$		0			0
$963$	3.92505	+	42.5787i	0.126483	+	1.37208i
$964$		0			0
$965$	32.4121	−	33.8850i	1.04338	−	1.09080i
$966$		0			0
$967$	−36.8309	−	36.8309i	−1.18440	−	1.18440i	−0.978592	−	0.205810i	$-0.934017\pi$
$967$	−36.8309	−	36.8309i	−1.18440	−	1.18440i	−0.205810	−	0.978592i	$-0.565983\pi$
$968$		0			0
$969$	11.5115	+	13.6542i	0.369804	+	0.438637i
$970$		0			0
$971$	−2.79105	−	1.61141i	−0.0895690	−	0.0517127i	0.454547	−	0.890723i	$-0.349801\pi$
$971$	−2.79105	−	1.61141i	−0.0895690	−	0.0517127i	−0.544115	+	0.839010i	$0.683135\pi$
$972$		0			0
$973$	−21.9837	+	37.4004i	−0.704765	+	1.19900i
$974$		0			0
$975$	4.14834	+	1.29375i	0.132853	+	0.0414332i
$976$		0			0
$977$	3.08185	+	0.825778i	0.0985970	+	0.0264190i	0.307780	−	0.951457i	$-0.400414\pi$
$977$	3.08185	+	0.825778i	0.0985970	+	0.0264190i	−0.209183	+	0.977876i	$0.567081\pi$
$978$		0			0
$979$			82.7319i			2.64412i
$980$		0			0
$981$	2.59416	−	0.957901i	0.0828251	−	0.0305834i
$982$		0			0
$983$	−2.63692	+	9.84111i	−0.0841046	+	0.313883i	−0.995143	−	0.0984385i	$-0.968615\pi$
$983$	−2.63692	+	9.84111i	−0.0841046	+	0.313883i	0.911038	+	0.412321i	$0.135282\pi$
$984$		0			0
$985$	5.99913	+	20.5518i	0.191148	+	0.654834i
$986$		0			0
$987$	17.7421	−	47.9289i	0.564737	−	1.52559i
$988$		0			0
$989$	−20.8407	+	36.0972i	−0.662696	+	1.14782i
$990$		0			0
$991$	20.4323	+	35.3898i	0.649054	+	1.12419i	0.983349	+	0.181726i	$0.0581684\pi$
$991$	20.4323	+	35.3898i	0.649054	+	1.12419i	−0.334295	+	0.942468i	$0.608498\pi$
$992$		0			0
$993$	30.1121	−	43.2184i	0.955579	−	1.37149i
$994$		0			0
$995$	−0.367715	−	16.5516i	−0.0116573	−	0.524721i
$996$		0			0
$997$	5.34061	−	19.9314i	0.169139	−	0.631234i	−0.828337	−	0.560230i	$-0.810713\pi$
$997$	5.34061	−	19.9314i	0.169139	−	0.631234i	0.997476	−	0.0710045i	$-0.0226205\pi$
$998$		0			0
$999$	39.2515	−	0.274197i	1.24186	−	0.00867522i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	420.2.bv.c.317.1	yes	48
3.2	odd	2	inner	420.2.bv.c.317.3	yes	48
5.3	odd	4	inner	420.2.bv.c.233.6	yes	48
7.4	even	3	inner	420.2.bv.c.137.8	yes	48
15.8	even	4	inner	420.2.bv.c.233.8	yes	48
21.11	odd	6	inner	420.2.bv.c.137.6	yes	48
35.18	odd	12	inner	420.2.bv.c.53.3	yes	48
105.53	even	12	inner	420.2.bv.c.53.1	✓	48

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
420.2.bv.c.53.1	✓	48	105.53	even	12	inner
420.2.bv.c.53.3	yes	48	35.18	odd	12	inner
420.2.bv.c.137.6	yes	48	21.11	odd	6	inner
420.2.bv.c.137.8	yes	48	7.4	even	3	inner
420.2.bv.c.233.6	yes	48	5.3	odd	4	inner
420.2.bv.c.233.8	yes	48	15.8	even	4	inner
420.2.bv.c.317.1	yes	48	1.1	even	1	trivial
420.2.bv.c.317.3	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 $\F_{q}$

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)

\(f(q)\)	\(=\)	\(q+(-1.72581 - 0.146940i) q^{3} +(1.07475 - 1.96085i) q^{5} +(-2.28090 - 1.34070i) q^{7} +(2.95682 + 0.507180i) q^{9} +O(q^{10})\)	\(q+(-1.72581 - 0.146940i) q^{3} +(1.07475 - 1.96085i) q^{5} +(-2.28090 - 1.34070i) q^{7} +(2.95682 + 0.507180i) q^{9} +(-4.85929 + 2.80551i) q^{11} +(0.354801 - 0.354801i) q^{13} +(-2.14293 + 3.22612i) q^{15} +(-3.57956 - 0.959140i) q^{17} +(2.40962 + 1.39119i) q^{19} +(3.73940 + 2.64895i) q^{21} +(-3.95939 + 1.06092i) q^{23} +(-2.68984 - 4.21483i) q^{25} +(-5.02837 - 1.30977i) q^{27} -8.44181 q^{29} +(-0.730482 - 1.26523i) q^{31} +(8.79843 - 4.12775i) q^{33} +(-5.08030 + 3.03159i) q^{35} +(-7.29673 + 1.95515i) q^{37} +(-0.664452 + 0.560183i) q^{39} +1.89242i q^{41} +(7.19023 - 7.19023i) q^{43} +(4.17233 - 5.25277i) q^{45} +(-2.88649 - 10.7725i) q^{47} +(3.40505 + 6.11602i) q^{49} +(6.03669 + 2.18127i) q^{51} +(-0.636707 + 2.37622i) q^{53} +(0.278670 + 12.5435i) q^{55} +(-3.95411 - 2.75500i) q^{57} +(5.64924 + 9.78477i) q^{59} +(1.67588 - 2.90270i) q^{61} +(-6.06424 - 5.12103i) q^{63} +(-0.314389 - 1.07703i) q^{65} +(-0.921448 + 3.43889i) q^{67} +(6.98904 - 1.24914i) q^{69} -10.5611i q^{71} +(10.4253 + 2.79345i) q^{73} +(4.02281 + 7.66922i) q^{75} +(14.8449 + 0.115747i) q^{77} +(-1.42483 - 0.822628i) q^{79} +(8.48554 + 2.99928i) q^{81} +(0.481281 + 0.481281i) q^{83} +(-5.72785 + 5.98813i) q^{85} +(14.5689 + 1.24044i) q^{87} +(7.37227 - 12.7691i) q^{89} +(-1.28495 + 0.333585i) q^{91} +(1.07476 + 2.29088i) q^{93} +(5.31764 - 3.22971i) q^{95} +(-5.38134 - 5.38134i) q^{97} +(-15.7909 + 5.83085i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 48 q - 2 q^{3} - 8 q^{7}+O(q^{10}) \) 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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