LMFDB - Embedded newform 420.2.bv.c.317.3

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			317.3
Character	$\chi$	$=$	420.317
Dual form			420.2.bv.c.53.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-1.56806 - 0.735650i) q^{3} +(-1.07475 + 1.96085i) q^{5} +(-2.28090 - 1.34070i) q^{7} +(1.91764 + 2.30709i) q^{9} +O(q^{10})$	\(q+(-1.56806 - 0.735650i) q^{3} +(-1.07475 + 1.96085i) q^{5} +(-2.28090 - 1.34070i) q^{7} +(1.91764 + 2.30709i) q^{9} +(4.85929 - 2.80551i) q^{11} +(0.354801 - 0.354801i) q^{13} +(3.12777 - 2.28409i) q^{15} +(3.57956 + 0.959140i) q^{17} +(2.40962 + 1.39119i) q^{19} +(2.59031 + 3.78025i) q^{21} +(3.95939 - 1.06092i) q^{23} +(-2.68984 - 4.21483i) q^{25} +(-1.30977 - 5.02837i) q^{27} +8.44181 q^{29} +(-0.730482 - 1.26523i) q^{31} +(-9.68354 + 0.824484i) q^{33} +(5.08030 - 3.03159i) q^{35} +(-7.29673 + 1.95515i) q^{37} +(-0.817359 + 0.295341i) q^{39} -1.89242i q^{41} +(7.19023 - 7.19023i) q^{43} +(-6.58482 + 1.28066i) q^{45} +(2.88649 + 10.7725i) q^{47} +(3.40505 + 6.11602i) q^{49} +(-4.90738 - 4.13729i) q^{51} +(0.636707 - 2.37622i) q^{53} +(0.278670 + 12.5435i) q^{55} +(-2.75500 - 3.95411i) q^{57} +(-5.64924 - 9.78477i) q^{59} +(1.67588 - 2.90270i) q^{61} +(-1.28084 - 7.83323i) q^{63} +(0.314389 + 1.07703i) q^{65} +(-0.921448 + 3.43889i) q^{67} +(-6.98904 - 1.24914i) q^{69} +10.5611i q^{71} +(10.4253 + 2.79345i) q^{73} +(1.11720 + 8.58789i) q^{75} +(-14.8449 - 0.115747i) q^{77} +(-1.42483 - 0.822628i) q^{79} +(-1.64532 + 8.84833i) q^{81} +(-0.481281 - 0.481281i) q^{83} +(-5.72785 + 5.98813i) q^{85} +(-13.2373 - 6.21021i) q^{87} +(-7.37227 + 12.7691i) q^{89} +(-1.28495 + 0.333585i) q^{91} +(0.214674 + 2.52134i) q^{93} +(-5.31764 + 3.22971i) q^{95} +(-5.38134 - 5.38134i) q^{97} +(15.7909 + 5.83085i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	−1.56806	−	0.735650i	−0.905321	−	0.424727i
$3$	−1.56806	−	0.735650i	−0.905321	−	0.424727i
$4$		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
$9$	1.91764	+	2.30709i	0.639213	+	0.769030i
$10$		0			0
$11$	4.85929	−	2.80551i	1.46513	−	0.845893i	0.465889	−	0.884843i	$-0.345735\pi$
$11$	4.85929	−	2.80551i	1.46513	−	0.845893i	0.999241	+	0.0389498i	$0.0124012\pi$
$12$		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
$15$	3.12777	−	2.28409i	0.807586	−	0.589750i
$16$		0			0
$17$	3.57956	+	0.959140i	0.868171	+	0.232626i	0.665297	−	0.746579i	$-0.268305\pi$
$17$	3.57956	+	0.959140i	0.868171	+	0.232626i	0.202874	+	0.979205i	$0.434972\pi$
$18$		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
$21$	2.59031	+	3.78025i	0.565253	+	0.824918i
$22$		0			0
$23$	3.95939	−	1.06092i	0.825591	−	0.221216i	0.178802	−	0.983885i	$-0.442778\pi$
$23$	3.95939	−	1.06092i	0.825591	−	0.221216i	0.646789	+	0.762669i	$0.276111\pi$
$24$		0			0
$25$	−2.68984	−	4.21483i	−0.537967	−	0.842966i
$26$		0			0
$27$	−1.30977	−	5.02837i	−0.252065	−	0.967710i
$28$		0			0
$29$	8.44181			1.56760			0.783802	−	0.621010i	$-0.213278\pi$
$29$	8.44181			1.56760			0.783802	+	0.621010i	$0.213278\pi$
$30$		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
$33$	−9.68354	+	0.824484i	−1.68569	+	0.143524i
$34$		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
$39$	−0.817359	+	0.295341i	−0.130882	+	0.0472924i
$40$		0			0
$41$		−	1.89242i		−	0.295547i	−0.989021	−	0.147773i	$-0.952789\pi$
$41$		−	1.89242i		−	0.295547i	0.989021	−	0.147773i	$-0.0472106\pi$
$42$		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
$45$	−6.58482	+	1.28066i	−0.981608	+	0.190909i
$46$		0			0
$47$	2.88649	+	10.7725i	0.421038	+	1.57133i	0.772426	+	0.635104i	$0.219043\pi$
$47$	2.88649	+	10.7725i	0.421038	+	1.57133i	−0.351389	+	0.936230i	$0.614290\pi$
$48$		0			0
$49$	3.40505	+	6.11602i	0.486435	+	0.873717i
$50$		0			0
$51$	−4.90738	−	4.13729i	−0.687171	−	0.579337i
$52$		0			0
$53$	0.636707	−	2.37622i	0.0874584	−	0.326399i	−0.908310	−	0.418298i	$-0.862627\pi$
$53$	0.636707	−	2.37622i	0.0874584	−	0.326399i	0.995768	+	0.0918986i	$0.0292936\pi$
$54$		0			0
$55$	0.278670	+	12.5435i	0.0375759	+	1.69137i
$56$		0			0
$57$	−2.75500	−	3.95411i	−0.364908	−	0.523734i
$58$		0			0
$59$	−5.64924	−	9.78477i	−0.735468	−	1.27387i	−0.954518	−	0.298155i	$-0.903629\pi$
$59$	−5.64924	−	9.78477i	−0.735468	−	1.27387i	0.219049	−	0.975714i	$-0.429704\pi$
$60$		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
$63$	−1.28084	−	7.83323i	−0.161370	−	0.986894i
$64$		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	$0.215589\pi$
$71$			10.5611i			1.25337i	−0.779273	+	0.626685i	$0.784411\pi$
$72$		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
$75$	1.11720	+	8.58789i	0.129003	+	0.991644i
$76$		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
$81$	−1.64532	+	8.84833i	−0.182813	+	0.983148i
$82$		0			0
$83$	−0.481281	−	0.481281i	−0.0528275	−	0.0528275i	0.680200	−	0.733027i	$-0.261893\pi$
$83$	−0.481281	−	0.481281i	−0.0528275	−	0.0528275i	−0.733027	+	0.680200i	$0.761893\pi$
$84$		0			0
$85$	−5.72785	+	5.98813i	−0.621272	+	0.649504i
$86$		0			0
$87$	−13.2373	−	6.21021i	−1.41919	−	0.665805i
$88$		0			0
$89$	−7.37227	+	12.7691i	−0.781459	+	1.35353i	0.149633	+	0.988742i	$0.452191\pi$
$89$	−7.37227	+	12.7691i	−0.781459	+	1.35353i	−0.931092	+	0.364785i	$0.881142\pi$
$90$		0			0
$91$	−1.28495	+	0.333585i	−0.134699	+	0.0349692i
$92$		0			0
$93$	0.214674	+	2.52134i	0.0222607	+	0.261451i
$94$		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	−0.467790	−	0.883840i	$-0.654950\pi$
$101$	0.640605	−	0.369854i	0.0637426	−	0.0368018i	0.531533	+	0.847038i	$0.321616\pi$
$102$		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
$105$	−10.1964	+	1.01640i	−0.995068	+	0.0991902i
$106$		0			0
$107$	−3.68897	−	13.7674i	−0.356626	−	1.33095i	−0.878426	−	0.477879i	$-0.841406\pi$
$107$	−3.68897	−	13.7674i	−0.356626	−	1.33095i	0.521800	−	0.853068i	$-0.325261\pi$
$108$		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	0.810133	−	0.586246i	$-0.199395\pi$
$113$	2.37995	+	2.37995i	0.223887	+	0.223887i	−0.586246	+	0.810133i	$0.699395\pi$
$114$		0			0
$115$	−2.17505	+	8.90398i	−0.202825	+	0.830301i
$116$		0			0
$117$	1.49894	+	0.138177i	0.138577	+	0.0127745i
$118$		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
$123$	−1.39216	+	2.96743i	−0.125527	+	0.267565i
$124$		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
$129$	−16.5642	+	5.98524i	−1.45840	+	0.526971i
$130$		0			0
$131$	6.97835	+	4.02895i	0.609701	+	0.352011i	0.772848	−	0.634591i	$-0.218831\pi$
$131$	6.97835	+	4.02895i	0.609701	+	0.352011i	−0.163147	+	0.986602i	$0.552165\pi$
$132$		0			0
$133$	−3.63093	−	6.40375i	−0.314841	−	0.555275i
$134$		0			0
$135$	11.2675	+	2.83597i	0.969755	+	0.244081i
$136$		0			0
$137$	11.6723	+	3.12759i	0.997235	+	0.267208i	0.720287	−	0.693677i	$-0.244010\pi$
$137$	11.6723	+	3.12759i	0.997235	+	0.267208i	0.276948	+	0.960885i	$0.410677\pi$
$138$		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
$147$	−0.840078	−	12.0952i	−0.0692885	−	0.997597i
$148$		0			0
$149$	4.18906	−	7.25567i	0.343181	−	0.594407i	−0.641840	−	0.766838i	$-0.721829\pi$
$149$	4.18906	−	7.25567i	0.343181	−	0.594407i	0.985022	+	0.172431i	$0.0551621\pi$
$150$		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
$153$	4.65148	+	10.0976i	0.376050	+	0.816346i
$154$		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
$159$	−2.74646	+	3.25767i	−0.217809	+	0.258350i
$160$		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
$165$	8.79067	−	19.8740i	0.684353	−	1.54719i
$166$		0			0
$167$	−4.96146	+	4.96146i	−0.383929	+	0.383929i	−0.872516	−	0.488586i	$-0.837513\pi$
$167$	−4.96146	+	4.96146i	−0.383929	+	0.383929i	0.488586	+	0.872516i	$0.337513\pi$
$168$		0			0
$169$			12.7482i			0.980633i
$170$		0			0
$171$	1.41117	+	8.22700i	0.107915	+	0.629134i
$172$		0			0
$173$	−6.20080	+	1.66150i	−0.471438	+	0.126322i	−0.486713	−	0.873562i	$-0.661804\pi$
$173$	−6.20080	+	1.66150i	−0.471438	+	0.126322i	0.0152749	+	0.999883i	$0.495138\pi$
$174$		0			0
$175$	0.484438	+	13.2199i	0.0366200	+	0.999329i
$176$		0			0
$177$	1.66020	+	19.4990i	0.124788	+	1.46563i
$178$		0			0
$179$	10.8624	+	18.8143i	0.811896	+	1.40625i	0.911535	+	0.411222i	$0.134898\pi$
$179$	10.8624	+	18.8143i	0.811896	+	1.40625i	−0.0996392	+	0.995024i	$0.531769\pi$
$180$		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
$183$	−4.76325	+	3.31876i	−0.352110	+	0.245330i
$184$		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
$189$	−3.75408	+	13.2252i	−0.273069	+	0.961994i
$190$		0			0
$191$	−7.73525	−	4.46595i	−0.559703	−	0.323145i	0.193323	−	0.981135i	$-0.438073\pi$
$191$	−7.73525	−	4.46595i	−0.559703	−	0.323145i	−0.753026	+	0.657990i	$0.771407\pi$
$192$		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
$195$	0.299336	−	1.92013i	0.0214359	−	0.137503i
$196$		0			0
$197$	6.77027	−	6.77027i	0.482362	−	0.482362i	−0.423524	−	0.905885i	$-0.639207\pi$
$197$	6.77027	−	6.77027i	0.482362	−	0.482362i	0.905885	+	0.423524i	$0.139207\pi$
$198$		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
$201$	3.97471	−	4.71453i	0.280354	−	0.332537i
$202$		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
$207$	10.0403	+	7.10022i	0.697850	+	0.493499i
$208$		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
$213$	7.76925	−	16.5604i	0.532340	−	1.13470i
$214$		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
$219$	−14.2925	−	12.0497i	−0.965798	−	0.814240i
$220$		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
$225$	4.56585	−	14.2882i	0.304390	−	0.952548i
$226$		0			0
$227$	11.4363	+	3.06435i	0.759055	+	0.203388i	0.617531	−	0.786547i	$-0.288133\pi$
$227$	11.4363	+	3.06435i	0.759055	+	0.203388i	0.141524	+	0.989935i	$0.454800\pi$
$228$		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
$231$	23.1926	+	11.1022i	1.52596	+	0.730468i
$232$		0			0
$233$	7.01982	−	1.88095i	0.459884	−	0.123225i	−0.0214368	−	0.999770i	$-0.506824\pi$
$233$	7.01982	−	1.88095i	0.459884	−	0.123225i	0.481320	+	0.876545i	$0.340157\pi$
$234$		0			0
$235$	−24.2255	−	5.91778i	−1.58030	−	0.386033i
$236$		0			0
$237$	1.62906	+	2.33811i	0.105819	+	0.151877i
$238$		0			0
$239$	−22.9991			−1.48769			−0.743845	−	0.668352i	$-0.767000\pi$
$239$	−22.9991			−1.48769			−0.743845	+	0.668352i	$0.767000\pi$
$240$		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
$243$	9.08923	−	12.6644i	0.583074	−	0.812419i
$244$		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
$249$	0.400625	+	1.10873i	0.0253886	+	0.0702631i
$250$		0			0
$251$		−	27.8607i		−	1.75855i	−0.476313	−	0.879276i	$-0.658027\pi$
$251$		−	27.8607i		−	1.75855i	0.476313	−	0.879276i	$-0.341973\pi$
$252$		0			0
$253$	16.2634	−	16.2634i	1.02247	−	1.02247i
$254$		0			0
$255$	13.3868	−	5.17608i	0.838313	−	0.324139i
$256$		0			0
$257$	−2.05198	−	7.65810i	−0.127999	−	0.477699i	0.871930	−	0.489631i	$-0.162869\pi$
$257$	−2.05198	−	7.65810i	−0.127999	−	0.477699i	−0.999929	+	0.0119320i	$0.996202\pi$
$258$		0			0
$259$	19.2644	+	5.32321i	1.19703	+	0.330768i
$260$		0			0
$261$	16.1883	+	19.4760i	1.00203	+	1.20553i
$262$		0			0
$263$	−5.28947	+	19.7406i	−0.326163	+	1.21726i	0.586975	+	0.809605i	$0.300319\pi$
$263$	−5.28947	+	19.7406i	−0.326163	+	1.21726i	−0.913138	+	0.407651i	$0.866348\pi$
$264$		0			0
$265$	3.97511	+	3.80232i	0.244189	+	0.233575i
$266$		0			0
$267$	20.9538	−	14.5994i	1.28235	−	0.893470i
$268$		0			0
$269$	9.59686	+	16.6223i	0.585131	+	1.01348i	0.994859	+	0.101269i	$0.0322902\pi$
$269$	9.59686	+	16.6223i	0.585131	+	1.01348i	−0.409728	+	0.912208i	$0.634376\pi$
$270$		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
$273$	2.26028	+	0.422189i	0.136798	+	0.0255521i
$274$		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	$0.0255796\pi$
$281$			2.69129i			0.160549i	−0.996773	+	0.0802743i	$0.974420\pi$
$282$		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
$285$	10.7143	−	1.15246i	0.634662	−	0.0682658i
$286$		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
$291$	4.47949	+	12.3971i	0.262593	+	0.726728i
$292$		0			0
$293$	4.68014	+	4.68014i	0.273417	+	0.273417i	0.830474	−	0.557057i	$-0.188070\pi$
$293$	4.68014	+	4.68014i	0.273417	+	0.273417i	−0.557057	+	0.830474i	$0.688070\pi$
$294$		0			0
$295$	25.2579	−	0.561137i	1.47057	−	0.0326706i
$296$		0			0
$297$	−20.4717	−	20.7597i	−1.18789	−	1.20460i
$298$		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
$303$	−1.27659	+	0.108693i	−0.0733383	+	0.00624423i
$304$		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	−0.776101	−	0.630608i	$-0.782805\pi$
$311$	−10.8991	+	6.29258i	−0.618029	+	0.356819i	0.158072	+	0.987428i	$0.449472\pi$
$312$		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
$315$	16.7363	+	5.90722i	0.942985	+	0.332834i
$316$		0			0
$317$	−2.92155	−	10.9034i	−0.164091	−	0.612394i	−0.998154	−	0.0607264i	$-0.980658\pi$
$317$	−2.92155	−	10.9034i	−0.164091	−	0.612394i	0.834064	−	0.551668i	$-0.186008\pi$
$318$		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
$327$	−1.59083	+	0.135448i	−0.0879730	+	0.00749026i
$328$		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
$333$	−18.5032	−	13.0849i	−1.01397	−	0.717050i
$334$		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
$339$	−1.98110	−	5.48272i	−0.107599	−	0.297781i
$340$		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
$345$	9.96083	−	12.3619i	0.536273	−	0.665543i
$346$		0			0
$347$	−17.6803	−	4.73743i	−0.949130	−	0.254319i	−0.249137	−	0.968468i	$-0.580147\pi$
$347$	−17.6803	−	4.73743i	−0.949130	−	0.254319i	−0.699993	+	0.714150i	$0.746814\pi$
$348$		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	−0.965367	−	0.260895i	$-0.915982\pi$
$353$	0.0209074	−	0.0780276i	0.00111279	−	0.00415299i	0.966480	+	0.256742i	$0.0826490\pi$
$354$		0			0
$355$	−20.7086	−	11.3505i	−1.09910	−	0.602421i
$356$		0			0
$357$	5.64640	+	16.0161i	0.298839	+	0.847662i
$358$		0			0
$359$	−2.75276	+	4.76793i	−0.145285	+	0.251642i	−0.929479	−	0.368874i	$-0.879743\pi$
$359$	−2.75276	+	4.76793i	−0.145285	+	0.251642i	0.784194	+	0.620516i	$0.213077\pi$
$360$		0			0
$361$	−5.62917	−	9.75001i	−0.296272	−	0.513158i
$362$		0			0
$363$	−29.1097	+	20.2819i	−1.52786	+	1.06453i
$364$		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
$369$	4.36598	−	3.62898i	0.227284	−	0.188917i
$370$		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
$375$	−18.0402	−	7.03916i	−0.931594	−	0.363501i
$376$		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
$381$	8.37525	+	23.1786i	0.429077	+	1.18748i
$382$		0			0
$383$	0.878346	−	0.235352i	0.0448814	−	0.0120259i	−0.236308	−	0.971678i	$-0.575938\pi$
$383$	0.878346	−	0.235352i	0.0448814	−	0.0120259i	0.281190	+	0.959652i	$0.409271\pi$
$384$		0			0
$385$	16.1815	−	28.9842i	0.824685	−	1.47717i
$386$		0			0
$387$	30.3767	+	2.80023i	1.54414	+	0.142344i
$388$		0			0
$389$	−14.2750	−	24.7251i	−0.723772	−	1.25361i	−0.959477	−	0.281785i	$-0.909073\pi$
$389$	−14.2750	−	24.7251i	−0.723772	−	1.25361i	0.235705	−	0.971825i	$-0.424260\pi$
$390$		0			0
$391$	15.1905			0.768214
$392$		0			0
$393$	−7.97859	−	11.4513i	−0.402466	−	0.577640i
$394$		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
$399$	0.982610	+	12.7126i	0.0491920	+	0.636424i
$400$		0			0
$401$	0.256628	+	0.148164i	0.0128154	+	0.00739896i	0.506394	−	0.862302i	$-0.330978\pi$
$401$	0.256628	+	0.148164i	0.0128154	+	0.00739896i	−0.493579	+	0.869701i	$0.664311\pi$
$402$		0			0
$403$	−0.708081	−	0.189730i	−0.0352720	−	0.00945111i
$404$		0			0
$405$	−15.5819	−	12.7359i	−0.774272	−	0.632854i
$406$		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
$411$	−16.0021	−	13.4910i	−0.789327	−	0.665462i
$412$		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
$417$	−12.0626	+	25.7118i	−0.590707	+	1.25911i
$418$		0			0
$419$	−24.1241			−1.17854			−0.589269	−	0.807937i	$-0.700584\pi$
$419$	−24.1241			−1.17854			−0.589269	+	0.807937i	$0.700584\pi$
$420$		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
$423$	−19.3179	+	27.3172i	−0.939270	+	1.32821i
$424$		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
$429$	−3.14320	+	3.72825i	−0.151755	+	0.180002i
$430$		0			0
$431$	−11.8878	+	6.86340i	−0.572613	+	0.330598i	−0.758192	−	0.652031i	$-0.773917\pi$
$431$	−11.8878	+	6.86340i	−0.572613	+	0.330598i	0.185579	+	0.982629i	$0.440584\pi$
$432$		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
$435$	26.4040	−	19.2819i	1.26598	−	0.924495i
$436$		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
$441$	−7.58055	+	19.5841i	−0.360978	+	0.932574i
$442$		0			0
$443$	−21.3126	+	5.71070i	−1.01259	+	0.271323i	−0.726712	−	0.686942i	$-0.758953\pi$
$443$	−21.3126	+	5.71070i	−1.01259	+	0.271323i	−0.285880	+	0.958265i	$0.592286\pi$
$444$		0			0
$445$	−17.1150	−	28.1795i	−0.811329	−	1.33584i
$446$		0			0
$447$	−11.9063	+	8.29566i	−0.563151	+	0.392371i
$448$		0			0
$449$	0.0268595			0.00126758			0.000633788	−	1.00000i	$-0.499798\pi$
$449$	0.0268595			0.00126758			0.000633788		1.00000i	$0.499798\pi$
$450$		0			0
$451$	−5.30921	−	9.19582i	−0.250001	−	0.433014i
$452$		0			0
$453$	0.103035	+	1.21014i	0.00484101	+	0.0568575i
$454$		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
$459$	0.134513	−	19.2556i	0.00627853	−	0.898775i
$460$		0			0
$461$			27.4449i			1.27824i	0.769108	+	0.639119i	$0.220701\pi$
$461$			27.4449i			1.27824i	−0.769108	+	0.639119i	$0.779299\pi$
$462$		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
$465$	−5.17468	−	2.28886i	−0.239970	−	0.106143i
$466$		0			0
$467$	−1.68325	−	6.28196i	−0.0778914	−	0.290695i	0.915982	−	0.401219i	$-0.131414\pi$
$467$	−1.68325	−	6.28196i	−0.0778914	−	0.290695i	−0.993873	+	0.110525i	$0.964747\pi$
$468$		0			0
$469$	6.71225	−	6.60839i	0.309943	−	0.305147i
$470$		0			0
$471$	0.903114	−	1.07121i	0.0416133	−	0.0493589i
$472$		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
$477$	6.70313	−	3.08780i	0.306915	−	0.141381i
$478$		0			0
$479$	14.1306	+	24.4749i	0.645643	+	1.11829i	0.984153	+	0.177324i	$0.0567440\pi$
$479$	14.1306	+	24.4749i	0.645643	+	1.11829i	−0.338509	+	0.940963i	$0.609923\pi$
$480$		0			0
$481$	−1.89520	+	3.28258i	−0.0864135	+	0.149673i
$482$		0			0
$483$	14.2666	+	12.2194i	0.649153	+	0.556001i
$484$		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	$0.146656\pi$
$491$			19.7036i			0.889213i	−0.895726	+	0.444606i	$0.853344\pi$
$492$		0			0
$493$	30.2180	+	8.09688i	1.36095	+	0.364665i
$494$		0			0
$495$	−28.4047	+	24.6969i	−1.27669	+	1.11004i
$496$		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
$501$	11.4298	−	4.12998i	0.510645	−	0.184514i
$502$		0			0
$503$	−1.01611	−	1.01611i	−0.0453059	−	0.0453059i	0.684091	−	0.729397i	$-0.260199\pi$
$503$	−1.01611	−	1.01611i	−0.0453059	−	0.0453059i	−0.729397	+	0.684091i	$0.760199\pi$
$504$		0			0
$505$	0.0367374	+	1.65363i	0.00163479	+	0.0735855i
$506$		0			0
$507$	9.37823	−	19.9900i	0.416502	−	0.887788i
$508$		0			0
$509$	0.920899	−	1.59504i	0.0408181	−	0.0706991i	−0.844895	−	0.534933i	$-0.820337\pi$
$509$	0.920899	−	1.59504i	0.0408181	−	0.0706991i	0.885713	+	0.464234i	$0.153670\pi$
$510$		0			0
$511$	−20.0339	−	20.3488i	−0.886247	−	0.900176i
$512$		0			0
$513$	3.83939	−	13.9386i	0.169513	−	0.615403i
$514$		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	0.463851	−	0.885913i	$-0.346467\pi$
$521$	33.3936	−	19.2798i	1.46300	−	0.844664i	0.999149	+	0.0412494i	$0.0131338\pi$
$522$		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
$525$	8.96557	−	21.0860i	0.391290	−	0.920268i
$526$		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
$537$	−3.19225	−	37.4929i	−0.137756	−	1.61794i
$538$		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
$543$	−10.5166	−	4.93382i	−0.451311	−	0.211730i
$544$		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
$549$	9.91052	−	1.69994i	0.422971	−	0.0725518i
$550$		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
$555$	−18.3567	+	22.7817i	−0.779199	+	0.967028i
$556$		0			0
$557$	28.4418	+	7.62097i	1.20512	+	0.322911i	0.804845	−	0.593485i	$-0.202248\pi$
$557$	28.4418	+	7.62097i	1.20512	+	0.322911i	0.400274	+	0.916395i	$0.368915\pi$
$558$		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	−0.934374	−	0.356293i	$-0.884041\pi$
$563$	1.25671	−	4.69011i	0.0529640	−	0.197664i	0.987338	+	0.158628i	$0.0507072\pi$
$564$		0			0
$565$	−7.22457	+	2.10887i	−0.303940	+	0.0887210i
$566$		0			0
$567$	15.6158	−	17.9763i	0.655801	−	0.754934i
$568$		0			0
$569$	−7.18242	+	12.4403i	−0.301103	+	0.521525i	−0.976386	−	0.216033i	$-0.930688\pi$
$569$	−7.18242	+	12.4403i	−0.301103	+	0.521525i	0.675283	+	0.737558i	$0.264021\pi$
$570$		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
$573$	8.84398	+	12.6933i	0.369463	+	0.530271i
$574$		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
$579$	−27.7693	−	23.4116i	−1.15405	−	0.972955i
$580$		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
$585$	−1.88192	+	2.79068i	−0.0778079	+	0.115380i
$586$		0			0
$587$	32.7234	−	32.7234i	1.35064	−	1.35064i	0.465697	−	0.884944i	$-0.345804\pi$
$587$	32.7234	−	32.7234i	1.35064	−	1.35064i	0.884944	−	0.465697i	$-0.154196\pi$
$588$		0			0
$589$		−	4.06496i		−	0.167494i
$590$		0			0
$591$	−15.5967	+	5.63566i	−0.641564	+	0.231820i
$592$		0			0
$593$	4.74225	−	1.27068i	0.194741	−	0.0521807i	−0.160130	−	0.987096i	$-0.551191\pi$
$593$	4.74225	−	1.27068i	0.194741	−	0.0521807i	0.354871	+	0.934915i	$0.384525\pi$
$594$		0			0
$595$	21.0930	−	5.97903i	0.864727	−	0.245116i
$596$		0			0
$597$	−12.7778	+	1.08793i	−0.522959	+	0.0445262i
$598$		0			0
$599$	−11.3589	−	19.6742i	−0.464113	−	0.803868i	0.535048	−	0.844822i	$-0.320294\pi$
$599$	−11.3589	−	19.6742i	−0.464113	−	0.803868i	−0.999161	+	0.0409541i	$0.986960\pi$
$600$		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
$603$	−9.70083	+	4.46869i	−0.395048	+	0.181979i
$604$		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
$609$	21.8669	+	31.9121i	0.886093	+	1.29314i
$610$		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
$615$	−4.32247	−	5.91905i	−0.174299	−	0.238679i
$616$		0			0
$617$	−26.0352	+	26.0352i	−1.04814	+	1.04814i	−0.0493563	+	0.998781i	$0.515717\pi$
$617$	−26.0352	+	26.0352i	−1.04814	+	1.04814i	−0.998781	+	0.0493563i	$0.984283\pi$
$618$		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
$621$	−10.5206	−	18.5197i	−0.422176	−	0.743172i
$622$		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
$627$	−24.4806	−	11.4850i	−0.977662	−	0.458666i
$628$		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
$633$	23.6456	+	11.0932i	0.939828	+	0.440916i
$634$		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
$639$	−24.3653	+	20.2523i	−0.963878	+	0.801170i
$640$		0			0
$641$	27.2771	−	15.7485i	1.07738	−	0.622026i	0.147193	−	0.989108i	$-0.452976\pi$
$641$	27.2771	−	15.7485i	1.07738	−	0.622026i	0.930189	+	0.367081i	$0.119643\pi$
$642$		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
$645$	6.06621	−	38.9125i	0.238857	−	1.53218i
$646$		0			0
$647$	36.3105	+	9.72936i	1.42751	+	0.382501i	0.888142	−	0.459569i	$-0.151996\pi$
$647$	36.3105	+	9.72936i	1.42751	+	0.382501i	0.539369	+	0.842069i	$0.318663\pi$
$648$		0			0
$649$	−54.9026	−	31.6980i	−2.15511	−	1.24426i
$650$		0			0
$651$	2.89071	−	6.03875i	0.113296	−	0.236677i
$652$		0			0
$653$	2.27720	−	0.610173i	0.0891135	−	0.0238779i	−0.213987	−	0.976837i	$-0.568645\pi$
$653$	2.27720	−	0.610173i	0.0891135	−	0.0238779i	0.303100	+	0.952959i	$0.401978\pi$
$654$		0			0
$655$	−15.4001	+	9.35336i	−0.601732	+	0.365466i
$656$		0			0
$657$	13.5472	+	29.4089i	0.528527	+	1.14735i
$658$		0			0
$659$	−0.552796			−0.0215339			−0.0107669	−	0.999942i	$-0.503427\pi$
$659$	−0.552796			−0.0215339			−0.0107669	+	0.999942i	$0.503427\pi$
$660$		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
$663$	−3.20906	+	0.273228i	−0.124629	+	0.0106113i
$664$		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
$669$	27.3935	−	9.89826i	1.05910	−	0.382689i
$670$		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
$675$	−17.6706	+	19.0459i	−0.680144	+	0.733079i
$676$		0			0
$677$	1.99434	+	7.44300i	0.0766489	+	0.286058i	0.993602	−	0.112937i	$-0.0360259\pi$
$677$	1.99434	+	7.44300i	0.0766489	+	0.286058i	−0.916953	+	0.398995i	$0.869359\pi$
$678$		0			0
$679$	5.05955	+	19.4891i	0.194168	+	0.747922i
$680$		0			0
$681$	−15.6786	−	13.2182i	−0.600804	−	0.506523i
$682$		0			0
$683$	−6.06492	+	22.6346i	−0.232068	+	0.866089i	0.747381	+	0.664396i	$0.231311\pi$
$683$	−6.06492	+	22.6346i	−0.232068	+	0.866089i	−0.979449	+	0.201693i	$0.935356\pi$
$684$		0			0
$685$	−18.6775	+	19.5263i	−0.713632	+	0.746061i
$686$		0			0
$687$	16.1094	+	23.1209i	0.614610	+	0.882119i
$688$		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
$693$	−28.2002	−	34.4705i	−1.07124	−	1.30943i
$694$		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	$0.220288\pi$
$701$			33.7903i			1.27624i	−0.769937	+	0.638120i	$0.779712\pi$
$702$		0			0
$703$	−20.3023	−	5.43999i	−0.765716	−	0.205173i
$704$		0			0
$705$	33.6337	+	27.1009i	1.26672	+	1.02068i
$706$		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
$711$	−0.834441	−	4.86472i	−0.0312940	−	0.182441i
$712$		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
$717$	36.0641	+	16.9193i	1.34684	+	0.631863i
$718$		0			0
$719$	22.3999	−	38.7978i	0.835376	−	1.44691i	−0.0583487	−	0.998296i	$-0.518584\pi$
$719$	22.3999	−	38.7978i	0.835376	−	1.44691i	0.893724	−	0.448617i	$-0.148083\pi$
$720$		0			0
$721$	−3.16011	+	11.4363i	−0.117689	+	0.425909i
$722$		0			0
$723$	3.75811	+	44.1389i	0.139765	+	1.64154i
$724$		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
$735$	24.6197	+	11.3520i	0.908113	+	0.418726i
$736$		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	0.333829	−	0.942634i	$-0.391659\pi$
$743$	−16.5948	−	16.5948i	−0.608805	−	0.608805i	−0.942634	+	0.333829i	$0.891659\pi$
$744$		0			0
$745$	9.72507	+	16.0121i	0.356299	+	0.586639i
$746$		0			0
$747$	0.187435	−	2.03328i	0.00685788	−	0.0743939i
$748$		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
$753$	−20.4957	+	43.6873i	−0.746905	+	1.59205i
$754$		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
$759$	−37.4662	+	13.5379i	−1.35994	+	0.491394i
$760$		0			0
$761$	−8.45249	−	4.88005i	−0.306403	−	0.176902i	0.338913	−	0.940818i	$-0.389941\pi$
$761$	−8.45249	−	4.88005i	−0.306403	−	0.176902i	−0.645316	+	0.763916i	$0.723274\pi$
$762$		0			0
$763$	−2.43875	−	0.0190152i	−0.0882886	−	0.000688396i
$764$		0			0
$765$	−24.7991	−	1.73157i	−0.896613	−	0.0626052i
$766$		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	$0.203444\pi$
$773$	−5.30264	+	19.7897i	−0.190723	+	0.711787i	−0.993333	+	0.115283i	$0.963223\pi$
$774$		0			0
$775$	−3.36786	+	6.48212i	−0.120977	+	0.232845i
$776$		0			0
$777$	−26.2918	−	22.5190i	−0.943213	−	0.807864i
$778$		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
$783$	−11.0568	−	42.4485i	−0.395139	−	1.51699i
$784$		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
$789$	22.8164	−	27.0633i	0.812285	−	0.963478i
$790$		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
$795$	−3.43604	−	8.88656i	−0.121864	−	0.315174i
$796$		0			0
$797$	−23.9839	+	23.9839i	−0.849554	+	0.849554i	−0.990077	−	0.140523i	$-0.955122\pi$
$797$	−23.9839	+	23.9839i	−0.849554	+	0.849554i	0.140523	+	0.990077i	$0.455122\pi$
$798$		0			0
$799$			41.3294i			1.46213i
$800$		0			0
$801$	−43.5969	+	7.47814i	−1.54042	+	0.264227i
$802$		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
$807$	−2.82033	−	33.1247i	−0.0992802	−	1.16604i
$808$		0			0
$809$	3.46923	+	6.00888i	0.121972	+	0.211261i	0.920545	−	0.390636i	$-0.127745\pi$
$809$	3.46923	+	6.00888i	0.121972	+	0.211261i	−0.798573	+	0.601897i	$0.794412\pi$
$810$		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
$813$	−6.57752	+	4.58284i	−0.230684	+	0.160727i
$814$		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
$819$	−3.23368	−	2.32479i	−0.112994	−	0.0812349i
$820$		0			0
$821$	−40.9600	−	23.6483i	−1.42951	−	0.825330i	−0.432431	−	0.901667i	$-0.642344\pi$
$821$	−40.9600	−	23.6483i	−1.42951	−	0.825330i	−0.997082	+	0.0763370i	$0.975678\pi$
$822$		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
$825$	29.5222	+	38.5967i	1.02783	+	1.34377i
$826$		0			0
$827$	−17.2898	+	17.2898i	−0.601225	+	0.601225i	−0.940638	−	0.339413i	$-0.889772\pi$
$827$	−17.2898	+	17.2898i	−0.601225	+	0.601225i	0.339413	+	0.940638i	$0.389772\pi$
$828$		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
$831$	32.2792	−	38.2874i	1.11975	−	1.32818i
$832$		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
$837$	−5.40529	+	5.33029i	−0.186834	+	0.184242i
$838$		0			0
$839$	46.8893			1.61880			0.809400	−	0.587258i	$-0.199793\pi$
$839$	46.8893			1.61880			0.809400	+	0.587258i	$0.199793\pi$
$840$		0			0
$841$	42.2642			1.45738
$842$		0			0
$843$	1.97984	−	4.22010i	0.0681894	−	0.145348i
$844$		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
$849$	19.0349	+	16.0478i	0.653274	+	0.550760i
$850$		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
$855$	−17.6485	−	6.07486i	−0.603567	−	0.207756i
$856$		0			0
$857$	−36.6943	−	9.83221i	−1.25345	−	0.335862i	−0.429784	−	0.902932i	$-0.641410\pi$
$857$	−36.6943	−	9.83221i	−1.25345	−	0.335862i	−0.823670	+	0.567070i	$0.808077\pi$
$858$		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
$861$	7.15382	−	4.90197i	0.243802	−	0.167059i
$862$		0			0
$863$	24.1896	−	6.48160i	0.823425	−	0.220636i	0.177582	−	0.984106i	$-0.443173\pi$
$863$	24.1896	−	6.48160i	0.823425	−	0.220636i	0.645843	+	0.763470i	$0.276506\pi$
$864$		0			0
$865$	3.40635	−	13.9445i	0.115819	−	0.474128i
$866$		0			0
$867$	3.23465	+	4.64253i	0.109854	+	0.157669i
$868$		0			0
$869$	−9.23157			−0.313160
$870$		0			0
$871$	0.893190	+	1.54705i	0.0302646	+	0.0524198i
$872$		0			0
$873$	2.09576	−	22.7347i	0.0709307	−	0.769453i
$874$		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
$879$	−3.89581	−	10.7817i	−0.131402	−	0.363658i
$880$		0			0
$881$			23.4482i			0.789991i	0.918683	+	0.394996i	$0.129254\pi$
$881$			23.4482i			0.789991i	−0.918683	+	0.394996i	$0.870746\pi$
$882$		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
$885$	−40.0188	−	17.7011i	−1.34522	−	0.595016i
$886$		0			0
$887$	−10.9695	−	40.9386i	−0.368319	−	1.37458i	−0.862866	−	0.505433i	$-0.831333\pi$
$887$	−10.9695	−	40.9386i	−0.368319	−	1.37458i	0.494547	−	0.869151i	$-0.335334\pi$
$888$		0			0
$889$	9.45978	+	36.4385i	0.317271	+	1.22211i
$890$		0			0
$891$	16.8290	+	47.6125i	0.563793	+	1.59508i
$892$		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
$897$	−2.92291	+	2.03652i	−0.0975933	+	0.0679974i
$898$		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
$903$	45.8058	+	8.55589i	1.52432	+	0.284722i
$904$		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	$-0.862631\pi$
$911$		−	25.2501i		−	0.836573i	0.908315	−	0.418286i	$-0.137369\pi$
$912$		0			0
$913$	−3.68893	−	0.988444i	−0.122086	−	0.0327127i
$914$		0			0
$915$	−1.38829	−	12.9068i	−0.0458955	−	0.426687i
$916$		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
$921$	0.389599	+	1.07822i	0.0128377	+	0.0355285i
$922$		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
$927$	7.76783	−	10.9844i	0.255129	−	0.360774i
$928$		0			0
$929$	−19.3847	+	33.5754i	−0.635993	+	1.10157i	0.350311	+	0.936633i	$0.386076\pi$
$929$	−19.3847	+	33.5754i	−0.635993	+	1.10157i	−0.986304	+	0.164938i	$0.947258\pi$
$930$		0			0
$931$	−0.303705	+	19.4743i	−0.00995354	+	0.638245i
$932$		0			0
$933$	21.7195	−	1.84926i	0.711066	−	0.0605421i
$934$		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	−0.669375	−	0.742925i	$-0.733438\pi$
$941$	−11.0638	+	6.38770i	−0.360670	+	0.208233i	0.308704	+	0.951158i	$0.400105\pi$
$942$		0			0
$943$	−2.00770	−	7.49284i	−0.0653797	−	0.244001i
$944$		0			0
$945$	−21.8980	−	21.5750i	−0.712341	−	0.701833i
$946$		0			0
$947$	6.79014	+	25.3411i	0.220650	+	0.823477i	0.984101	+	0.177611i	$0.0568368\pi$
$947$	6.79014	+	25.3411i	0.220650	+	0.823477i	−0.763451	+	0.645866i	$0.776497\pi$
$948$		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	$-0.875808\pi$
$953$	−40.2920	−	40.2920i	−1.30518	−	1.30518i	−0.380336	−	0.924848i	$-0.624192\pi$
$954$		0			0
$955$	17.0705	−	10.3679i	0.552388	−	0.335496i
$956$		0			0
$957$	−81.7466	+	6.96013i	−2.64249	+	0.224989i
$958$		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
$963$	24.6886	−	34.9117i	0.795577	−	1.12501i
$964$		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
$969$	−6.06913	−	16.7964i	−0.194969	−	0.539578i
$970$		0			0
$971$	2.79105	+	1.61141i	0.0895690	+	0.0517127i	0.544115	−	0.839010i	$-0.316865\pi$
$971$	2.79105	+	1.61141i	0.0895690	+	0.0517127i	−0.454547	+	0.890723i	$0.650199\pi$
$972$		0			0
$973$	−21.9837	+	37.4004i	−0.704765	+	1.19900i
$974$		0			0
$975$	3.44337	+	2.65061i	0.110276	+	0.0848874i
$976$		0			0
$977$	−3.08185	−	0.825778i	−0.0985970	−	0.0264190i	0.209183	−	0.977876i	$-0.432919\pi$
$977$	−3.08185	−	0.825778i	−0.0985970	−	0.0264190i	−0.307780	+	0.951457i	$0.599586\pi$
$978$		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	−0.911038	−	0.412321i	$-0.864718\pi$
$983$	2.63692	−	9.84111i	0.0841046	−	0.313883i	0.995143	+	0.0984385i	$0.0313848\pi$
$984$		0			0
$985$	5.99913	+	20.5518i	0.191148	+	0.654834i
$986$		0			0
$987$	−33.2459	+	38.8159i	−1.05823	+	1.23552i
$988$		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
$993$	43.2184	−	30.1121i	1.37149	−	0.955579i
$994$		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
$999$	19.3883	+	34.1299i	0.613418	+	1.07982i

Twists

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

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

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q+(-1.56806 - 0.735650i) q^{3} +(-1.07475 + 1.96085i) q^{5} +(-2.28090 - 1.34070i) q^{7} +(1.91764 + 2.30709i) q^{9} +O(q^{10})\)	\(q+(-1.56806 - 0.735650i) q^{3} +(-1.07475 + 1.96085i) q^{5} +(-2.28090 - 1.34070i) q^{7} +(1.91764 + 2.30709i) q^{9} +(4.85929 - 2.80551i) q^{11} +(0.354801 - 0.354801i) q^{13} +(3.12777 - 2.28409i) q^{15} +(3.57956 + 0.959140i) q^{17} +(2.40962 + 1.39119i) q^{19} +(2.59031 + 3.78025i) q^{21} +(3.95939 - 1.06092i) q^{23} +(-2.68984 - 4.21483i) q^{25} +(-1.30977 - 5.02837i) q^{27} +8.44181 q^{29} +(-0.730482 - 1.26523i) q^{31} +(-9.68354 + 0.824484i) q^{33} +(5.08030 - 3.03159i) q^{35} +(-7.29673 + 1.95515i) q^{37} +(-0.817359 + 0.295341i) q^{39} -1.89242i q^{41} +(7.19023 - 7.19023i) q^{43} +(-6.58482 + 1.28066i) q^{45} +(2.88649 + 10.7725i) q^{47} +(3.40505 + 6.11602i) q^{49} +(-4.90738 - 4.13729i) q^{51} +(0.636707 - 2.37622i) q^{53} +(0.278670 + 12.5435i) q^{55} +(-2.75500 - 3.95411i) q^{57} +(-5.64924 - 9.78477i) q^{59} +(1.67588 - 2.90270i) q^{61} +(-1.28084 - 7.83323i) q^{63} +(0.314389 + 1.07703i) q^{65} +(-0.921448 + 3.43889i) q^{67} +(-6.98904 - 1.24914i) q^{69} +10.5611i q^{71} +(10.4253 + 2.79345i) q^{73} +(1.11720 + 8.58789i) q^{75} +(-14.8449 - 0.115747i) q^{77} +(-1.42483 - 0.822628i) q^{79} +(-1.64532 + 8.84833i) q^{81} +(-0.481281 - 0.481281i) q^{83} +(-5.72785 + 5.98813i) q^{85} +(-13.2373 - 6.21021i) q^{87} +(-7.37227 + 12.7691i) q^{89} +(-1.28495 + 0.333585i) q^{91} +(0.214674 + 2.52134i) q^{93} +(-5.31764 + 3.22971i) q^{95} +(-5.38134 - 5.38134i) q^{97} +(15.7909 + 5.83085i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 48 q - 2 q^{3} - 8 q^{7}+O(q^{10}) \) 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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