LMFDB - Embedded newform 420.2.bv.c.53.9

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

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

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

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

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

Newform invariants

comment:select newform

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

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

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

Embedding invariants

Embedding label			53.9
Character	$\chi$	$=$	420.53
Dual form			420.2.bv.c.317.9

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+(0.774760 - 1.54911i) q^{3} +(2.17211 + 0.530994i) q^{5} +(1.78762 - 1.95049i) q^{7} +(-1.79949 - 2.40038i) q^{9} +(-2.07693 - 1.19911i) q^{11} +(3.37848 + 3.37848i) q^{13} +(2.50543 - 2.95344i) q^{15} +(-4.74353 + 1.27103i) q^{17} +(0.151250 - 0.0873244i) q^{19} +(-1.63655 - 4.28039i) q^{21} +(1.21053 + 0.324361i) q^{23} +(4.43609 + 2.30675i) q^{25} +(-5.11263 + 0.927900i) q^{27} +4.65176 q^{29} +(1.26323 - 2.18799i) q^{31} +(-3.46668 + 2.28836i) q^{33} +(4.91860 - 3.28746i) q^{35} +(-10.9974 - 2.94674i) q^{37} +(7.85116 - 2.61614i) q^{39} +1.78650i q^{41} +(-2.46081 - 2.46081i) q^{43} +(-2.63411 - 6.16940i) q^{45} +(2.07195 - 7.73263i) q^{47} +(-0.608831 - 6.97347i) q^{49} +(-1.70614 + 8.33300i) q^{51} +(2.92005 + 10.8978i) q^{53} +(-3.87458 - 3.70744i) q^{55} +(-0.0180926 - 0.301959i) q^{57} +(-2.06606 + 3.57852i) q^{59} +(3.47828 + 6.02455i) q^{61} +(-7.89873 - 0.781068i) q^{63} +(5.54447 + 9.13238i) q^{65} +(3.12231 + 11.6526i) q^{67} +(1.44034 - 1.62395i) q^{69} +15.9938i q^{71} +(-5.23288 + 1.40215i) q^{73} +(7.01032 - 5.08482i) q^{75} +(-6.05161 + 1.90747i) q^{77} +(5.52266 - 3.18851i) q^{79} +(-2.52364 + 8.63894i) q^{81} +(3.05025 - 3.05025i) q^{83} +(-10.9784 + 0.242015i) q^{85} +(3.60400 - 7.20610i) q^{87} +(2.58127 + 4.47089i) q^{89} +(12.6291 - 0.550258i) q^{91} +(-2.41073 - 3.65205i) q^{93} +(0.374900 - 0.109365i) q^{95} +(-2.70738 + 2.70738i) q^{97} +(0.859088 + 7.14321i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 48 q - 2 q^{3} - 8 q^{7} + 32 q^{13} - 16 q^{21} + 16 q^{25} - 32 q^{27} - 64 q^{31} + 34 q^{33} - 40 q^{37} - 24 q^{43} + 30 q^{45} + 36 q^{51} + 92 q^{57} - 18 q^{63} + 28 q^{67} - 8 q^{73} - 38 q^{75}+ \cdots + 56 q^{97}+O(q^{100}) $ 48 * q - 2 * q^3 - 8 * q^7 + 32 * q^13 - 16 * q^21 + 16 * q^25 - 32 * q^27 - 64 * q^31 + 34 * q^33 - 40 * q^37 - 24 * q^43 + 30 * q^45 + 36 * q^51 + 92 * q^57 - 18 * q^63 + 28 * q^67 - 8 * q^73 - 38 * q^75 + 20 * q^81 - 56 * q^85 - 24 * q^87 - 24 * q^91 + 6 * q^93 + 56 * q^97

Character values

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

$n$	$211$	$241$	$281$	$337$
$\chi(n)$	$1$	$e\left(\frac{2}{3}\right)$	$-1$	$e\left(\frac{3}{4}\right)$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$		0			0
$2$		0			0
$3$	0.774760	−	1.54911i	0.447308	−	0.894380i
$3$	0.774760	−	1.54911i	0.447308	−	0.894380i
$4$		0			0
$5$	2.17211	+	0.530994i	0.971395	+	0.237468i
$5$	2.17211	+	0.530994i	0.971395	+	0.237468i
$6$		0			0
$7$	1.78762	−	1.95049i	0.675657	−	0.737216i
$7$	1.78762	−	1.95049i	0.675657	−	0.737216i
$8$		0			0
$9$	−1.79949	−	2.40038i	−0.599831	−	0.800126i
$10$		0			0
$11$	−2.07693	−	1.19911i	−0.626217	−	0.361546i	0.153069	−	0.988216i	$-0.451084\pi$
$11$	−2.07693	−	1.19911i	−0.626217	−	0.361546i	−0.779285	+	0.626669i	$0.784418\pi$
$12$		0			0
$13$	3.37848	+	3.37848i	0.937023	+	0.937023i	0.998131	−	0.0611083i	$-0.0194635\pi$
$13$	3.37848	+	3.37848i	0.937023	+	0.937023i	−0.0611083	+	0.998131i	$0.519464\pi$
$14$		0			0
$15$	2.50543	−	2.95344i	0.646899	−	0.762575i
$16$		0			0
$17$	−4.74353	+	1.27103i	−1.15048	+	0.308269i	−0.783156	−	0.621826i	$-0.786391\pi$
$17$	−4.74353	+	1.27103i	−1.15048	+	0.308269i	−0.367320	+	0.930095i	$0.619725\pi$
$18$		0			0
$19$	0.151250	−	0.0873244i	0.0346992	−	0.0200336i	−0.482550	−	0.875868i	$-0.660289\pi$
$19$	0.151250	−	0.0873244i	0.0346992	−	0.0200336i	0.517249	+	0.855835i	$0.326956\pi$
$20$		0			0
$21$	−1.63655	−	4.28039i	−0.357125	−	0.934057i
$22$		0			0
$23$	1.21053	+	0.324361i	0.252413	+	0.0676339i	0.382807	−	0.923828i	$-0.374957\pi$
$23$	1.21053	+	0.324361i	0.252413	+	0.0676339i	−0.130394	+	0.991462i	$0.541624\pi$
$24$		0			0
$25$	4.43609	+	2.30675i	0.887218	+	0.461350i
$26$		0			0
$27$	−5.11263	+	0.927900i	−0.983926	+	0.178574i
$28$		0			0
$29$	4.65176			0.863811			0.431905	−	0.901919i	$-0.357841\pi$
$29$	4.65176			0.863811			0.431905	+	0.901919i	$0.357841\pi$
$30$		0			0
$31$	1.26323	−	2.18799i	0.226884	−	0.392974i	−0.729999	−	0.683448i	$-0.760480\pi$
$31$	1.26323	−	2.18799i	0.226884	−	0.392974i	0.956883	+	0.290474i	$0.0938130\pi$
$32$		0			0
$33$	−3.46668	+	2.28836i	−0.603471	+	0.398353i
$34$		0			0
$35$	4.91860	−	3.28746i	0.831395	−	0.555682i
$36$		0			0
$37$	−10.9974	−	2.94674i	−1.80796	−	0.484441i	−0.812783	−	0.582566i	$-0.802049\pi$
$37$	−10.9974	−	2.94674i	−1.80796	−	0.484441i	−0.995174	+	0.0981254i	$0.968715\pi$
$38$		0			0
$39$	7.85116	−	2.61614i	1.25719	−	0.418917i
$40$		0			0
$41$			1.78650i			0.279005i	0.990222	+	0.139503i	$0.0445504\pi$
$41$			1.78650i			0.279005i	−0.990222	+	0.139503i	$0.955450\pi$
$42$		0			0
$43$	−2.46081	−	2.46081i	−0.375271	−	0.375271i	0.494122	−	0.869393i	$-0.335490\pi$
$43$	−2.46081	−	2.46081i	−0.375271	−	0.375271i	−0.869393	+	0.494122i	$0.835490\pi$
$44$		0			0
$45$	−2.63411	−	6.16940i	−0.392669	−	0.919680i
$46$		0			0
$47$	2.07195	−	7.73263i	0.302225	−	1.12792i	−0.633082	−	0.774084i	$-0.718211\pi$
$47$	2.07195	−	7.73263i	0.302225	−	1.12792i	0.935308	−	0.353835i	$-0.115123\pi$
$48$		0			0
$49$	−0.608831	−	6.97347i	−0.0869758	−	0.996210i
$50$		0			0
$51$	−1.70614	+	8.33300i	−0.238907	+	1.16685i
$52$		0			0
$53$	2.92005	+	10.8978i	0.401100	+	1.49692i	0.811137	+	0.584856i	$0.198849\pi$
$53$	2.92005	+	10.8978i	0.401100	+	1.49692i	−0.410037	+	0.912069i	$0.634484\pi$
$54$		0			0
$55$	−3.87458	−	3.70744i	−0.522448	−	0.499911i
$56$		0			0
$57$	−0.0180926	−	0.301959i	−0.00239642	−	0.0399954i
$58$		0			0
$59$	−2.06606	+	3.57852i	−0.268978	+	0.465884i	−0.968598	−	0.248631i	$-0.920019\pi$
$59$	−2.06606	+	3.57852i	−0.268978	+	0.465884i	0.699620	+	0.714515i	$0.253353\pi$
$60$		0			0
$61$	3.47828	+	6.02455i	0.445348	+	0.771365i	0.998076	−	0.0619964i	$-0.0197467\pi$
$61$	3.47828	+	6.02455i	0.445348	+	0.771365i	−0.552729	+	0.833361i	$0.686413\pi$
$62$		0			0
$63$	−7.89873	−	0.781068i	−0.995146	−	0.0984053i
$64$		0			0
$65$	5.54447	+	9.13238i	0.687707	+	1.13273i
$66$		0			0
$67$	3.12231	+	11.6526i	0.381451	+	1.42359i	0.843686	+	0.536837i	$0.180381\pi$
$67$	3.12231	+	11.6526i	0.381451	+	1.42359i	−0.462235	+	0.886757i	$0.652953\pi$
$68$		0			0
$69$	1.44034	−	1.62395i	0.173397	−	0.195500i
$70$		0			0
$71$			15.9938i			1.89811i	0.315111	+	0.949055i	$0.397958\pi$
$71$			15.9938i			1.89811i	−0.315111	+	0.949055i	$0.602042\pi$
$72$		0			0
$73$	−5.23288	+	1.40215i	−0.612462	+	0.164109i	−0.551699	−	0.834043i	$-0.686020\pi$
$73$	−5.23288	+	1.40215i	−0.612462	+	0.164109i	−0.0607633	+	0.998152i	$0.519353\pi$
$74$		0			0
$75$	7.01032	−	5.08482i	0.809482	−	0.587145i
$76$		0			0
$77$	−6.05161	+	1.90747i	−0.689645	+	0.217376i
$78$		0			0
$79$	5.52266	−	3.18851i	0.621348	−	0.358735i	−0.156046	−	0.987750i	$-0.549875\pi$
$79$	5.52266	−	3.18851i	0.621348	−	0.358735i	0.777394	+	0.629014i	$0.216541\pi$
$80$		0			0
$81$	−2.52364	+	8.63894i	−0.280405	+	0.959882i
$82$		0			0
$83$	3.05025	−	3.05025i	0.334808	−	0.334808i	−0.519601	−	0.854409i	$-0.673919\pi$
$83$	3.05025	−	3.05025i	0.334808	−	0.334808i	0.854409	+	0.519601i	$0.173919\pi$
$84$		0			0
$85$	−10.9784	+	0.242015i	−1.19077	+	0.0262502i
$86$		0			0
$87$	3.60400	−	7.20610i	0.386389	−	0.772575i
$88$		0			0
$89$	2.58127	+	4.47089i	0.273614	+	0.473914i	0.969785	−	0.243963i	$-0.0784475\pi$
$89$	2.58127	+	4.47089i	0.273614	+	0.473914i	−0.696170	+	0.717877i	$0.745114\pi$
$90$		0			0
$91$	12.6291	−	0.550258i	1.32389	−	0.0576827i
$92$		0			0
$93$	−2.41073	−	3.65205i	−0.249981	−	0.378700i
$94$		0			0
$95$	0.374900	−	0.109365i	0.0384640	−	0.0112206i
$96$		0			0
$97$	−2.70738	+	2.70738i	−0.274893	+	0.274893i	−0.831066	−	0.556173i	$-0.812269\pi$
$97$	−2.70738	+	2.70738i	−0.274893	+	0.274893i	0.556173	+	0.831066i	$0.312269\pi$
$98$		0			0
$99$	0.859088	+	7.14321i	0.0863416	+	0.717919i
$100$		0			0
$101$	−1.03830	−	0.599462i	−0.103315	−	0.0596487i	0.447452	−	0.894308i	$-0.352331\pi$
$101$	−1.03830	−	0.599462i	−0.103315	−	0.0596487i	−0.550767	+	0.834659i	$0.685665\pi$
$102$		0			0
$103$	−1.39827	+	5.21841i	−0.137775	+	0.514185i	0.862196	+	0.506575i	$0.169089\pi$
$103$	−1.39827	+	5.21841i	−0.137775	+	0.514185i	−0.999971	+	0.00760935i	$0.997578\pi$
$104$		0			0
$105$	−1.28191	−	10.1665i	−0.125101	−	0.992144i
$106$		0			0
$107$	2.56642	−	9.57801i	0.248105	−	0.925941i	−0.723692	−	0.690123i	$-0.757556\pi$
$107$	2.56642	−	9.57801i	0.248105	−	0.925941i	0.971797	−	0.235818i	$-0.0757769\pi$
$108$		0			0
$109$	−13.1440	−	7.58867i	−1.25896	−	0.726863i	−0.286090	−	0.958203i	$-0.592356\pi$
$109$	−13.1440	−	7.58867i	−1.25896	−	0.726863i	−0.972873	+	0.231340i	$0.925689\pi$
$110$		0			0
$111$	−13.0852	+	14.7531i	−1.24199	+	1.40031i
$112$		0			0
$113$	−11.3407	+	11.3407i	−1.06685	+	1.06685i	−0.0692464	+	0.997600i	$0.522059\pi$
$113$	−11.3407	+	11.3407i	−1.06685	+	1.06685i	−0.997600	+	0.0692464i	$0.977941\pi$
$114$		0			0
$115$	2.45717	+	1.34733i	0.229132	+	0.125639i
$116$		0			0
$117$	2.03008	−	14.1892i	0.187681	−	1.31179i
$118$		0			0
$119$	−6.00051	+	11.5243i	−0.550066	+	1.05643i
$120$		0			0
$121$	−2.62425	−	4.54534i	−0.238569	−	0.413213i
$122$		0			0
$123$	2.76750	+	1.38411i	0.249537	+	0.124801i
$124$		0			0
$125$	8.41079	+	7.36605i	0.752284	+	0.658839i
$126$		0			0
$127$	−14.7150	+	14.7150i	−1.30574	+	1.30574i	−0.381287	+	0.924457i	$0.624519\pi$
$127$	−14.7150	+	14.7150i	−1.30574	+	1.30574i	−0.924457	+	0.381287i	$0.875481\pi$
$128$		0			0
$129$	−5.71861	+	1.90554i	−0.503496	+	0.167773i
$130$		0			0
$131$	11.6665	−	6.73566i	1.01931	−	0.588497i	0.105404	−	0.994429i	$-0.466386\pi$
$131$	11.6665	−	6.73566i	1.01931	−	0.588497i	0.913903	+	0.405932i	$0.133053\pi$
$132$		0			0
$133$	0.100053	−	0.451115i	0.00867565	−	0.0391166i
$134$		0			0
$135$	−11.5979	−	0.699280i	−0.998187	−	0.0601845i
$136$		0			0
$137$	12.8468	−	3.44229i	1.09758	−	0.294095i	0.335799	−	0.941934i	$-0.390994\pi$
$137$	12.8468	−	3.44229i	1.09758	−	0.294095i	0.761777	+	0.647839i	$0.224327\pi$
$138$		0			0
$139$		−	4.58221i		−	0.388658i	−0.980936	−	0.194329i	$-0.937747\pi$
$139$		−	4.58221i		−	0.388658i	0.980936	−	0.194329i	$-0.0622530\pi$
$140$		0			0
$141$	−10.3734	−	9.20061i	−0.873601	−	0.774832i
$142$		0			0
$143$	−2.96567	−	11.0680i	−0.248002	−	0.925556i
$144$		0			0
$145$	10.1041	+	2.47006i	0.839102	+	0.205127i
$146$		0			0
$147$	−11.2744	−	4.45962i	−0.929896	−	0.367823i
$148$		0			0
$149$	3.58754	+	6.21379i	0.293902	+	0.509054i	0.974729	−	0.223391i	$-0.0717126\pi$
$149$	3.58754	+	6.21379i	0.293902	+	0.509054i	−0.680827	+	0.732445i	$0.738379\pi$
$150$		0			0
$151$	2.15084	−	3.72536i	0.175033	−	0.303166i	−0.765140	−	0.643864i	$-0.777330\pi$
$151$	2.15084	−	3.72536i	0.175033	−	0.303166i	0.940173	+	0.340698i	$0.110663\pi$
$152$		0			0
$153$	11.5869	+	9.09907i	0.936745	+	0.735616i
$154$		0			0
$155$	3.90569	−	4.08177i	0.313712	−	0.327855i
$156$		0			0
$157$	−2.30081	−	8.58676i	−0.183625	−	0.685298i	−0.994921	−	0.100662i	$-0.967904\pi$
$157$	−2.30081	−	8.58676i	−0.183625	−	0.685298i	0.811296	−	0.584636i	$-0.198763\pi$
$158$		0			0
$159$	19.1442	+	3.91968i	1.51823	+	0.310851i
$160$		0			0
$161$	2.79663	−	1.78130i	0.220406	−	0.140386i
$162$		0			0
$163$	−1.53272	+	5.72018i	−0.120052	+	0.448039i	−0.999615	−	0.0277426i	$-0.991168\pi$
$163$	−1.53272	+	5.72018i	−0.120052	+	0.448039i	0.879563	+	0.475782i	$0.157835\pi$
$164$		0			0
$165$	−8.74510	+	3.12978i	−0.680805	+	0.243653i
$166$		0			0
$167$	−16.1499	−	16.1499i	−1.24972	−	1.24972i	−0.955844	−	0.293876i	$-0.905055\pi$
$167$	−16.1499	−	16.1499i	−1.24972	−	1.24972i	−0.293876	−	0.955844i	$-0.594945\pi$
$168$		0			0
$169$			9.82831i			0.756024i
$170$		0			0
$171$	−0.481785	−	0.205918i	−0.0368431	−	0.0157470i
$172$		0			0
$173$	−24.8433	−	6.65674i	−1.88880	−	0.506102i	−0.998736	−	0.0502667i	$-0.983993\pi$
$173$	−24.8433	−	6.65674i	−1.88880	−	0.506102i	−0.890064	−	0.455836i	$-0.849340\pi$
$174$		0			0
$175$	12.4293	−	4.52896i	0.939570	−	0.342357i
$176$		0			0
$177$	3.94283	+	5.97305i	0.296361	+	0.448962i
$178$		0			0
$179$	9.95927	−	17.2500i	0.744391	−	1.28932i	−0.206088	−	0.978533i	$-0.566073\pi$
$179$	9.95927	−	17.2500i	0.744391	−	1.28932i	0.950479	−	0.310789i	$-0.100593\pi$
$180$		0			0
$181$	16.4609			1.22353			0.611765	−	0.791040i	$-0.290460\pi$
$181$	16.4609			1.22353			0.611765	+	0.791040i	$0.290460\pi$
$182$		0			0
$183$	12.0275	−	0.720658i	0.889101	−	0.0532726i
$184$		0			0
$185$	−22.3228	−	12.2402i	−1.64120	−	0.899915i
$186$		0			0
$187$	11.3761	+	3.04821i	0.831900	+	0.222907i
$188$		0			0
$189$	−7.32958	+	11.6309i	−0.533149	+	0.846022i
$190$		0			0
$191$	−1.56368	+	0.902790i	−0.113144	+	0.0653236i	−0.555504	−	0.831514i	$-0.687475\pi$
$191$	−1.56368	+	0.902790i	−0.113144	+	0.0653236i	0.442360	+	0.896837i	$0.354141\pi$
$192$		0			0
$193$	−5.16003	+	1.38263i	−0.371427	+	0.0995236i	−0.439704	−	0.898143i	$-0.644917\pi$
$193$	−5.16003	+	1.38263i	−0.371427	+	0.0995236i	0.0682768	+	0.997666i	$0.478250\pi$
$194$		0			0
$195$	18.4427	−	1.51360i	1.32071	−	0.108391i
$196$		0			0
$197$	15.1248	+	15.1248i	1.07760	+	1.07760i	0.996724	+	0.0808739i	$0.0257711\pi$
$197$	15.1248	+	15.1248i	1.07760	+	1.07760i	0.0808739	+	0.996724i	$0.474229\pi$
$198$		0			0
$199$	2.10884	+	1.21754i	0.149491	+	0.0863090i	0.572880	−	0.819639i	$-0.305826\pi$
$199$	2.10884	+	1.21754i	0.149491	+	0.0863090i	−0.423388	+	0.905948i	$0.639159\pi$
$200$		0			0
$201$	20.4702	+	4.19117i	1.44386	+	0.295623i
$202$		0			0
$203$	8.31558	−	9.07322i	0.583640	−	0.636815i
$204$		0			0
$205$	−0.948624	+	3.88048i	−0.0662547	+	0.271024i
$206$		0			0
$207$	−1.39976	−	3.48942i	−0.0972897	−	0.242532i
$208$		0			0
$209$	−0.418847			−0.0289723
$210$		0			0
$211$	13.8080			0.950581			0.475291	−	0.879829i	$-0.342343\pi$
$211$	13.8080			0.950581			0.475291	+	0.879829i	$0.342343\pi$
$212$		0			0
$213$	24.7761	+	12.3913i	1.69763	+	0.849039i
$214$		0			0
$215$	−4.03847	−	6.65183i	−0.275421	−	0.453651i
$216$		0			0
$217$	−2.00946	−	6.37521i	−0.136411	−	0.432778i
$218$		0			0
$219$	−1.88214	+	9.19264i	−0.127184	+	0.621181i
$220$		0			0
$221$	−20.3201	−	11.7318i	−1.36688	−	0.789167i
$222$		0			0
$223$	−17.6103	−	17.6103i	−1.17928	−	1.17928i	−0.979929	−	0.199347i	$-0.936118\pi$
$223$	−17.6103	−	17.6103i	−1.17928	−	1.17928i	−0.199347	−	0.979929i	$-0.563882\pi$
$224$		0			0
$225$	−2.44564	−	14.7993i	−0.163043	−	0.986619i
$226$		0			0
$227$	−6.32132	+	1.69379i	−0.419561	+	0.112421i	−0.462422	−	0.886660i	$-0.653019\pi$
$227$	−6.32132	+	1.69379i	−0.419561	+	0.112421i	0.0428611	+	0.999081i	$0.486353\pi$
$228$		0			0
$229$	20.8421	−	12.0332i	1.37729	−	0.795177i	0.385454	−	0.922727i	$-0.374045\pi$
$229$	20.8421	−	12.0332i	1.37729	−	0.795177i	0.991832	+	0.127550i	$0.0407115\pi$
$230$		0			0
$231$	−1.73367	+	10.8525i	−0.114067	+	0.714039i
$232$		0			0
$233$	1.94292	+	0.520604i	0.127285	+	0.0341059i	0.321899	−	0.946774i	$-0.395679\pi$
$233$	1.94292	+	0.520604i	0.127285	+	0.0341059i	−0.194614	+	0.980880i	$0.562345\pi$
$234$		0			0
$235$	8.60648	−	15.6959i	0.561425	−	1.02389i
$236$		0			0
$237$	−0.660622	−	11.0256i	−0.0429120	−	0.716187i
$238$		0			0
$239$	−10.1453			−0.656245			−0.328123	−	0.944635i	$-0.606416\pi$
$239$	−10.1453			−0.656245			−0.328123	+	0.944635i	$0.606416\pi$
$240$		0			0
$241$	11.1599	−	19.3295i	0.718871	−	1.24512i	−0.242577	−	0.970132i	$-0.577993\pi$
$241$	11.1599	−	19.3295i	0.718871	−	1.24512i	0.961448	−	0.274988i	$-0.0886740\pi$
$242$		0			0
$243$	11.4275	+	10.6025i	0.733072	+	0.680151i
$244$		0			0
$245$	2.38043	−	15.4704i	0.152080	−	0.988368i
$246$		0			0
$247$	0.806020	+	0.215973i	0.0512858	+	0.0137420i
$248$		0			0
$249$	−2.36197	−	7.08839i	−0.149684	−	0.449208i
$250$		0			0
$251$		−	4.37814i		−	0.276346i	−0.990408	−	0.138173i	$-0.955877\pi$
$251$		−	4.37814i		−	0.276346i	0.990408	−	0.138173i	$-0.0441230\pi$
$252$		0			0
$253$	−2.12524	−	2.12524i	−0.133613	−	0.133613i
$254$		0			0
$255$	−8.13069	+	17.1942i	−0.509163	+	1.07674i
$256$		0			0
$257$	−0.639388	+	2.38623i	−0.0398839	+	0.148849i	−0.982996	−	0.183625i	$-0.941217\pi$
$257$	−0.639388	+	2.38623i	−0.0398839	+	0.148849i	0.943112	+	0.332474i	$0.107883\pi$
$258$		0			0
$259$	−25.4067	+	16.1826i	−1.57870	+	1.00554i
$260$		0			0
$261$	−8.37082	−	11.1660i	−0.518141	−	0.691158i
$262$		0			0
$263$	0.0785232	+	0.293053i	0.00484195	+	0.0180704i	0.968305	−	0.249773i	$-0.0803558\pi$
$263$	0.0785232	+	0.293053i	0.00484195	+	0.0180704i	−0.963463	+	0.267843i	$0.913689\pi$
$264$		0			0
$265$	0.556004	+	25.2217i	0.0341551	+	1.54935i
$266$		0			0
$267$	8.92578	−	0.534809i	0.546249	−	0.0327298i
$268$		0			0
$269$	8.59823	−	14.8926i	0.524244	−	0.908017i	−0.475358	−	0.879792i	$-0.657681\pi$
$269$	8.59823	−	14.8926i	0.524244	−	0.908017i	0.999602	−	0.0282241i	$-0.00898522\pi$
$270$		0			0
$271$	−4.92042	−	8.52242i	−0.298894	−	0.517700i	0.676989	−	0.735993i	$-0.263284\pi$
$271$	−4.92042	−	8.52242i	−0.298894	−	0.517700i	−0.975883	+	0.218293i	$0.929951\pi$
$272$		0			0
$273$	8.93215	−	19.9903i	0.540598	−	1.20987i
$274$		0			0
$275$	−6.44737	−	10.1103i	−0.388791	−	0.609676i
$276$		0			0
$277$	3.48534	+	13.0075i	0.209414	+	0.781543i	0.988059	+	0.154078i	$0.0492407\pi$
$277$	3.48534	+	13.0075i	0.209414	+	0.781543i	−0.778645	+	0.627465i	$0.784093\pi$
$278$		0			0
$279$	−7.52518	+	0.905026i	−0.450521	+	0.0541825i
$280$		0			0
$281$			24.4744i			1.46002i	0.683436	+	0.730011i	$0.260485\pi$
$281$			24.4744i			1.46002i	−0.683436	+	0.730011i	$0.739515\pi$
$282$		0			0
$283$	−6.55682	+	1.75689i	−0.389763	+	0.104437i	−0.448378	−	0.893844i	$-0.647998\pi$
$283$	−6.55682	+	1.75689i	−0.389763	+	0.104437i	0.0586154	+	0.998281i	$0.481331\pi$
$284$		0			0
$285$	0.121039	−	0.665494i	0.00716976	−	0.0394205i
$286$		0			0
$287$	3.48456	+	3.19359i	0.205687	+	0.188512i
$288$		0			0
$289$	6.16315	−	3.55830i	0.362538	−	0.209312i
$290$		0			0
$291$	2.09647	+	6.29161i	0.122897	+	0.368821i
$292$		0			0
$293$	19.9960	−	19.9960i	1.16818	−	1.16818i	0.185541	−	0.982637i	$-0.440596\pi$
$293$	19.9960	−	19.9960i	1.16818	−	1.16818i	0.982637	−	0.185541i	$-0.0594037\pi$
$294$		0			0
$295$	−6.38787	+	6.67586i	−0.371916	+	0.388684i
$296$		0			0
$297$	11.7312	+	4.20345i	0.680714	+	0.243909i
$298$		0			0
$299$	2.99391	+	5.18561i	0.173142	+	0.299892i
$300$		0			0
$301$	−9.19879	+	0.400796i	−0.530210	+	0.0231015i
$302$		0			0
$303$	−1.73307	+	1.14400i	−0.0995620	+	0.0657211i
$304$		0			0
$305$	4.35618	+	14.9329i	0.249434	+	0.855056i
$306$		0			0
$307$	−5.61961	+	5.61961i	−0.320728	+	0.320728i	−0.849046	−	0.528318i	$-0.822823\pi$
$307$	−5.61961	+	5.61961i	−0.320728	+	0.320728i	0.528318	+	0.849046i	$0.322823\pi$
$308$		0			0
$309$	7.00057	+	6.20908i	0.398249	+	0.353222i
$310$		0			0
$311$	−26.1827	−	15.1166i	−1.48468	−	0.857183i	−0.484836	−	0.874605i	$-0.661121\pi$
$311$	−26.1827	−	15.1166i	−1.48468	−	0.857183i	−0.999848	+	0.0174217i	$0.994454\pi$
$312$		0			0
$313$	2.77697	−	10.3638i	0.156964	−	0.585796i	−0.841966	−	0.539531i	$-0.818602\pi$
$313$	2.77697	−	10.3638i	0.156964	−	0.585796i	0.998929	−	0.0462649i	$-0.0147318\pi$
$314$		0			0
$315$	−16.7421	−	5.89074i	−0.943313	−	0.331906i
$316$		0			0
$317$	2.75178	−	10.2698i	0.154555	−	0.576808i	−0.844588	−	0.535417i	$-0.820154\pi$
$317$	2.75178	−	10.2698i	0.154555	−	0.576808i	0.999143	−	0.0413908i	$-0.0131789\pi$
$318$		0			0
$319$	−9.66136	−	5.57799i	−0.540933	−	0.312308i
$320$		0			0
$321$	−12.8490	−	11.3963i	−0.717164	−	0.636081i
$322$		0			0
$323$	−0.606469	+	0.606469i	−0.0337448	+	0.0337448i
$324$		0			0
$325$	7.19394	+	22.7806i	0.399048	+	1.26364i
$326$		0			0
$327$	−21.9391	+	14.4821i	−1.21324	+	0.800860i
$328$		0			0
$329$	−11.3786	−	17.8643i	−0.627320	−	0.984892i
$330$		0			0
$331$	−0.207490	−	0.359384i	−0.0114047	−	0.0197535i	0.860267	−	0.509844i	$-0.170297\pi$
$331$	−0.207490	−	0.359384i	−0.0114047	−	0.0197535i	−0.871671	+	0.490091i	$0.836964\pi$
$332$		0			0
$333$	12.7164	+	31.7005i	0.696856	+	1.73718i
$334$		0			0
$335$	0.594516	+	26.9686i	0.0324819	+	1.47345i
$336$		0			0
$337$	12.7807	−	12.7807i	0.696210	−	0.696210i	−0.267381	−	0.963591i	$-0.586158\pi$
$337$	12.7807	−	12.7807i	0.696210	−	0.696210i	0.963591	+	0.267381i	$0.0861582\pi$
$338$		0			0
$339$	8.78171	+	26.3544i	0.476957	+	1.43137i
$340$		0			0
$341$	−5.24729	+	3.02952i	−0.284156	+	0.164058i
$342$		0			0
$343$	−14.6901	−	11.2784i	−0.793188	−	0.608976i
$344$		0			0
$345$	3.99088	−	2.76257i	0.214862	−	0.148732i
$346$		0			0
$347$	−31.3558	+	8.40176i	−1.68327	+	0.451030i	−0.968639	−	0.248473i	$-0.920071\pi$
$347$	−31.3558	+	8.40176i	−1.68327	+	0.451030i	−0.714629	+	0.699504i	$0.753404\pi$
$348$		0			0
$349$		−	13.1679i		−	0.704860i	−0.935838	−	0.352430i	$-0.885355\pi$
$349$		−	13.1679i		−	0.704860i	0.935838	−	0.352430i	$-0.114645\pi$
$350$		0			0
$351$	−20.4078	−	14.1380i	−1.08929	−	0.754633i
$352$		0			0
$353$	2.55758	+	9.54503i	0.136126	+	0.508031i	0.999991	+	0.00430620i	$0.00137071\pi$
$353$	2.55758	+	9.54503i	0.136126	+	0.508031i	−0.863864	+	0.503725i	$0.831963\pi$
$354$		0			0
$355$	−8.49259	+	34.7401i	−0.450740	+	1.84381i
$356$		0			0
$357$	13.2035	+	18.2240i	0.698804	+	0.964519i
$358$		0			0
$359$	−5.54047	−	9.59637i	−0.292415	−	0.506477i	0.681966	−	0.731384i	$-0.261125\pi$
$359$	−5.54047	−	9.59637i	−0.292415	−	0.506477i	−0.974380	+	0.224907i	$0.927792\pi$
$360$		0			0
$361$	−9.48475	+	16.4281i	−0.499197	+	0.864635i
$362$		0			0
$363$	−9.07441	+	0.543715i	−0.476283	+	0.0285376i
$364$		0			0
$365$	−12.1109	+	0.266981i	−0.633913	+	0.0139744i
$366$		0			0
$367$	−2.03172	−	7.58247i	−0.106055	−	0.395802i	0.892408	−	0.451230i	$-0.149014\pi$
$367$	−2.03172	−	7.58247i	−0.106055	−	0.395802i	−0.998463	+	0.0554277i	$0.982348\pi$
$368$		0			0
$369$	4.28829	−	3.21481i	0.223239	−	0.167356i
$370$		0			0
$371$	26.4760	+	13.7855i	1.37456	+	0.715710i
$372$		0			0
$373$	−2.82512	+	10.5435i	−0.146279	+	0.545921i	0.853416	+	0.521230i	$0.174527\pi$
$373$	−2.82512	+	10.5435i	−0.146279	+	0.545921i	−0.999695	+	0.0246907i	$0.992140\pi$
$374$		0			0
$375$	17.9272	−	7.32233i	0.925755	−	0.378124i
$376$		0			0
$377$	15.7159	+	15.7159i	0.809410	+	0.809410i
$378$		0			0
$379$		−	16.7144i		−	0.858560i	−0.903172	−	0.429280i	$-0.858767\pi$
$379$		−	16.7144i		−	0.858560i	0.903172	−	0.429280i	$-0.141233\pi$
$380$		0			0
$381$	11.3946	+	34.1957i	0.583762	+	1.75190i
$382$		0			0
$383$	−12.3453	−	3.30792i	−0.630817	−	0.169027i	−0.0707764	−	0.997492i	$-0.522548\pi$
$383$	−12.3453	−	3.30792i	−0.630817	−	0.169027i	−0.560040	+	0.828465i	$0.689214\pi$
$384$		0			0
$385$	−14.1576	+	0.929847i	−0.721538	+	0.0473894i
$386$		0			0
$387$	−1.47867	+	10.3351i	−0.0751648	+	0.525363i
$388$		0			0
$389$	17.5206	−	30.3465i	0.888327	−	1.53863i	0.0464754	−	0.998919i	$-0.485201\pi$
$389$	17.5206	−	30.3465i	0.888327	−	1.53863i	0.841852	−	0.539709i	$-0.181466\pi$
$390$		0			0
$391$	−6.15447			−0.311245
$392$		0			0
$393$	−1.39555	−	23.2912i	−0.0703962	−	1.17489i
$394$		0			0
$395$	13.6889	−	3.99328i	0.688763	−	0.200924i
$396$		0			0
$397$	1.97792	+	0.529982i	0.0992690	+	0.0265990i	0.308112	−	0.951350i	$-0.400303\pi$
$397$	1.97792	+	0.529982i	0.0992690	+	0.0265990i	−0.208843	+	0.977949i	$0.566970\pi$
$398$		0			0
$399$	−0.621311	−	0.504498i	−0.0311044	−	0.0252565i
$400$		0			0
$401$	13.0417	−	7.52965i	0.651273	−	0.376013i	−0.137671	−	0.990478i	$-0.543962\pi$
$401$	13.0417	−	7.52965i	0.651273	−	0.376013i	0.788944	+	0.614465i	$0.210628\pi$
$402$		0			0
$403$	11.6599	−	3.12426i	0.580820	−	0.155630i
$404$		0			0
$405$	−10.0688	+	17.4247i	−0.500325	+	0.865838i
$406$		0			0
$407$	19.3073	+	19.3073i	0.957025	+	0.957025i
$408$		0			0
$409$	−8.10578	−	4.67987i	−0.400805	−	0.231405i	0.286026	−	0.958222i	$-0.407666\pi$
$409$	−8.10578	−	4.67987i	−0.400805	−	0.231405i	−0.686831	+	0.726817i	$0.740999\pi$
$410$		0			0
$411$	4.62070	−	22.5681i	0.227922	−	1.11320i
$412$		0			0
$413$	3.28654	+	10.4269i	0.161720	+	0.513073i
$414$		0			0
$415$	8.24513	−	5.00580i	0.404738	−	0.245725i
$416$		0			0
$417$	−7.09836	−	3.55011i	−0.347608	−	0.173850i
$418$		0			0
$419$	−20.7118			−1.01184			−0.505919	−	0.862581i	$-0.668847\pi$
$419$	−20.7118			−1.01184			−0.505919	+	0.862581i	$0.668847\pi$
$420$		0			0
$421$	18.3198			0.892851			0.446426	−	0.894821i	$-0.352697\pi$
$421$	18.3198			0.892851			0.446426	+	0.894821i	$0.352697\pi$
$422$		0			0
$423$	−22.2897	+	8.94135i	−1.08376	+	0.434743i
$424$		0			0
$425$	−23.9747	−	5.30376i	−1.16294	−	0.257270i
$426$		0			0
$427$	17.9687	+	3.98526i	0.869565	+	0.192860i
$428$		0			0
$429$	−19.4433	−	3.98092i	−0.938732	−	0.192200i
$430$		0			0
$431$	−0.725507	−	0.418872i	−0.0349464	−	0.0201763i	0.482425	−	0.875937i	$-0.339756\pi$
$431$	−0.725507	−	0.418872i	−0.0349464	−	0.0201763i	−0.517371	+	0.855761i	$0.673089\pi$
$432$		0			0
$433$	11.5356	+	11.5356i	0.554366	+	0.554366i	0.927698	−	0.373332i	$-0.121785\pi$
$433$	11.5356	+	11.5356i	0.554366	+	0.554366i	−0.373332	+	0.927698i	$0.621785\pi$
$434$		0			0
$435$	11.6547	−	13.7387i	0.558798	−	0.658721i
$436$		0			0
$437$	0.211418	−	0.0566492i	0.0101135	−	0.00270990i
$438$		0			0
$439$	−27.5755	+	15.9207i	−1.31611	+	0.759855i	−0.983100	−	0.183069i	$-0.941397\pi$
$439$	−27.5755	+	15.9207i	−1.31611	+	0.759855i	−0.333008	+	0.942924i	$0.608063\pi$
$440$		0			0
$441$	−15.6434	+	14.0101i	−0.744923	+	0.667150i
$442$		0			0
$443$	32.0596	+	8.59035i	1.52320	+	0.408140i	0.920793	−	0.390051i	$-0.127543\pi$
$443$	32.0596	+	8.59035i	1.52320	+	0.408140i	0.602405	+	0.798191i	$0.294209\pi$
$444$		0			0
$445$	3.23278	+	11.0819i	0.153248	+	0.525332i
$446$		0			0
$447$	12.4053	−	0.743295i	0.586752	−	0.0351566i
$448$		0			0
$449$	−32.3879			−1.52848			−0.764239	−	0.644933i	$-0.776885\pi$
$449$	−32.3879			−1.52848			−0.764239	+	0.644933i	$0.776885\pi$
$450$		0			0
$451$	2.14222	−	3.71044i	0.100873	−	0.174718i
$452$		0			0
$453$	−4.10462	−	6.21815i	−0.192852	−	0.292154i
$454$		0			0
$455$	27.7240	+	5.51078i	1.29972	+	0.258350i
$456$		0			0
$457$	18.4000	+	4.93026i	0.860716	+	0.230628i	0.662069	−	0.749443i	$-0.269679\pi$
$457$	18.4000	+	4.93026i	0.860716	+	0.230628i	0.198647	+	0.980071i	$0.436345\pi$
$458$		0			0
$459$	23.0725	−	10.8998i	1.07693	−	0.508759i
$460$		0			0
$461$		−	24.9156i		−	1.16044i	−0.814461	−	0.580218i	$-0.802967\pi$
$461$		−	24.9156i		−	1.16044i	0.814461	−	0.580218i	$-0.197033\pi$
$462$		0			0
$463$	−12.6420	−	12.6420i	−0.587524	−	0.587524i	0.349436	−	0.936960i	$-0.386373\pi$
$463$	−12.6420	−	12.6420i	−0.587524	−	0.587524i	−0.936960	+	0.349436i	$0.886373\pi$
$464$		0			0
$465$	−3.29714	−	9.21273i	−0.152901	−	0.427230i
$466$		0			0
$467$	−3.14017	+	11.7193i	−0.145310	+	0.542303i	0.854432	+	0.519563i	$0.173905\pi$
$467$	−3.14017	+	11.7193i	−0.145310	+	0.542303i	−0.999741	+	0.0227391i	$0.992761\pi$
$468$		0			0
$469$	28.3098	+	14.7404i	1.30723	+	0.680649i
$470$		0			0
$471$	−15.0844	−	3.08846i	−0.695053	−	0.142309i
$472$		0			0
$473$	2.16013	+	8.06172i	0.0993229	+	0.370678i
$474$		0			0
$475$	0.872395	−	0.0384821i	0.0400282	−	0.00176568i
$476$		0			0
$477$	20.9042	−	26.6197i	0.957137	−	1.21883i
$478$		0			0
$479$	−3.14571	+	5.44853i	−0.143731	+	0.248950i	−0.928899	−	0.370333i	$-0.879243\pi$
$479$	−3.14571	+	5.44853i	−0.143731	+	0.248950i	0.785168	+	0.619283i	$0.212577\pi$
$480$		0			0
$481$	−27.1989	−	47.1100i	−1.24017	−	2.14803i
$482$		0			0
$483$	−0.592709	−	5.71238i	−0.0269692	−	0.259922i
$484$		0			0
$485$	−7.31833	+	4.44312i	−0.332308	+	0.201752i
$486$		0			0
$487$	5.64485	+	21.0669i	0.255793	+	0.954632i	0.967648	+	0.252305i	$0.0811887\pi$
$487$	5.64485	+	21.0669i	0.255793	+	0.954632i	−0.711855	+	0.702326i	$0.752145\pi$
$488$		0			0
$489$	7.67371	+	6.80612i	0.347017	+	0.307783i
$490$		0			0
$491$		−	18.6876i		−	0.843358i	−0.906745	−	0.421679i	$-0.861441\pi$
$491$		−	18.6876i		−	0.843358i	0.906745	−	0.421679i	$-0.138559\pi$
$492$		0			0
$493$	−22.0658	+	5.91251i	−0.993793	+	0.266286i
$494$		0			0
$495$	−1.92697	+	15.9720i	−0.0866108	+	0.717887i
$496$		0			0
$497$	31.1957	+	28.5908i	1.39932	+	1.28247i
$498$		0			0
$499$	−9.45427	+	5.45843i	−0.423231	+	0.244353i	−0.696459	−	0.717597i	$-0.745242\pi$
$499$	−9.45427	+	5.45843i	−0.423231	+	0.244353i	0.273228	+	0.961949i	$0.411909\pi$
$500$		0			0
$501$	−37.5304	+	12.5057i	−1.67673	+	0.558715i
$502$		0			0
$503$	−8.52544	+	8.52544i	−0.380131	+	0.380131i	−0.871149	−	0.491018i	$-0.836625\pi$
$503$	−8.52544	+	8.52544i	−0.380131	+	0.380131i	0.491018	+	0.871149i	$0.336625\pi$
$504$		0			0
$505$	−1.93698	−	1.85342i	−0.0861946	−	0.0824763i
$506$		0			0
$507$	15.2251	+	7.61458i	0.676172	+	0.338175i
$508$		0			0
$509$	16.6925	+	28.9123i	0.739883	+	1.28151i	0.952548	+	0.304389i	$0.0984524\pi$
$509$	16.6925	+	28.9123i	0.739883	+	1.28151i	−0.212665	+	0.977125i	$0.568214\pi$
$510$		0			0
$511$	−6.61952	+	12.7132i	−0.292831	+	0.562398i
$512$		0			0
$513$	−0.692258	+	0.586802i	−0.0305640	+	0.0259080i
$514$		0			0
$515$	−5.80813	+	10.5925i	−0.255937	+	0.466760i
$516$		0			0
$517$	−13.5756	+	13.5756i	−0.597054	+	0.597054i
$518$		0			0
$519$	−29.5596	+	33.3276i	−1.29752	+	1.46292i
$520$		0			0
$521$	14.3052	+	8.25910i	0.626722	+	0.361838i	0.779481	−	0.626425i	$-0.215483\pi$
$521$	14.3052	+	8.25910i	0.626722	+	0.361838i	−0.152760	+	0.988263i	$0.548816\pi$
$522$		0			0
$523$	0.449698	−	1.67830i	0.0196639	−	0.0733868i	−0.955397	−	0.295326i	$-0.904572\pi$
$523$	0.449698	−	1.67830i	0.0196639	−	0.0733868i	0.975061	+	0.221939i	$0.0712385\pi$
$524$		0			0
$525$	2.61389	−	22.7633i	0.114079	−	0.993472i
$526$		0			0
$527$	−3.21120	+	11.9844i	−0.139882	+	0.522048i
$528$		0			0
$529$	−18.5584	−	10.7147i	−0.806887	−	0.465857i
$530$		0			0
$531$	12.3077	−	1.48020i	0.534107	−	0.0642352i
$532$		0			0
$533$	−6.03568	+	6.03568i	−0.261434	+	0.261434i
$534$		0			0
$535$	10.6604	−	19.4417i	0.460889	−	0.840538i
$536$		0			0
$537$	−19.0061	−	28.7926i	−0.820173	−	1.24249i
$538$		0			0
$539$	−7.09749	+	15.2134i	−0.305710	+	0.655289i
$540$		0			0
$541$	−1.62294	−	2.81101i	−0.0697756	−	0.120855i	0.829027	−	0.559209i	$-0.188895\pi$
$541$	−1.62294	−	2.81101i	−0.0697756	−	0.120855i	−0.898802	+	0.438354i	$0.855562\pi$
$542$		0			0
$543$	12.7532	−	25.4998i	0.547294	−	1.09430i
$544$		0			0
$545$	−24.5205	−	23.4628i	−1.05034	−	1.00503i
$546$		0			0
$547$	17.2003	−	17.2003i	0.735434	−	0.735434i	−0.236257	−	0.971691i	$-0.575921\pi$
$547$	17.2003	−	17.2003i	0.735434	−	0.735434i	0.971691	+	0.236257i	$0.0759207\pi$
$548$		0			0
$549$	8.20207	−	19.1903i	0.350056	−	0.819023i
$550$		0			0
$551$	0.703580	−	0.406212i	0.0299735	−	0.0173052i
$552$		0			0
$553$	3.65326	−	16.4717i	0.155352	−	0.700450i
$554$		0			0
$555$	−36.2562	+	25.0973i	−1.53899	+	1.06532i
$556$		0			0
$557$	−19.2930	+	5.16955i	−0.817472	+	0.219041i	−0.643241	−	0.765664i	$-0.722411\pi$
$557$	−19.2930	+	5.16955i	−0.817472	+	0.219041i	−0.174231	+	0.984705i	$0.555744\pi$
$558$		0			0
$559$		−	16.6276i		−	0.703274i
$560$		0			0
$561$	13.5357	−	15.2612i	0.571479	−	0.644327i
$562$		0			0
$563$	−9.69046	−	36.1653i	−0.408404	−	1.52419i	−0.797690	−	0.603068i	$-0.793945\pi$
$563$	−9.69046	−	36.1653i	−0.408404	−	1.52419i	0.389285	−	0.921117i	$-0.372722\pi$
$564$		0			0
$565$	−30.6551	+	18.6114i	−1.28967	+	0.782988i
$566$		0			0
$567$	12.3389	+	20.3655i	0.518183	+	0.855270i
$568$		0			0
$569$	8.83961	+	15.3107i	0.370576	+	0.641856i	0.989654	−	0.143473i	$-0.0458270\pi$
$569$	8.83961	+	15.3107i	0.370576	+	0.641856i	−0.619078	+	0.785329i	$0.712494\pi$
$570$		0			0
$571$	9.16309	−	15.8709i	0.383463	−	0.664178i	−0.608092	−	0.793867i	$-0.708065\pi$
$571$	9.16309	−	15.8709i	0.383463	−	0.664178i	0.991555	+	0.129689i	$0.0413980\pi$
$572$		0			0
$573$	0.187047	+	3.12176i	0.00781401	+	0.130413i
$574$		0			0
$575$	4.62181	+	4.23129i	0.192743	+	0.176457i
$576$		0			0
$577$	−2.68675	−	10.0271i	−0.111851	−	0.417433i	0.887181	−	0.461421i	$-0.152660\pi$
$577$	−2.68675	−	10.0271i	−0.111851	−	0.417433i	−0.999032	+	0.0439883i	$0.985994\pi$
$578$		0			0
$579$	−1.85594	+	9.06466i	−0.0771303	+	0.376715i
$580$		0			0
$581$	−0.496798	−	11.4022i	−0.0206107	−	0.473042i
$582$		0			0
$583$	7.00294	−	26.1353i	0.290032	−	1.08242i
$584$		0			0
$585$	11.9439	−	29.7425i	0.493821	−	1.22970i
$586$		0			0
$587$	−16.5786	−	16.5786i	−0.684270	−	0.684270i	0.276689	−	0.960959i	$-0.410763\pi$
$587$	−16.5786	−	16.5786i	−0.684270	−	0.684270i	−0.960959	+	0.276689i	$0.910763\pi$
$588$		0			0
$589$		−	0.441244i		−	0.0181812i
$590$		0			0
$591$	35.1481	−	11.7119i	1.44580	−	0.481764i
$592$		0			0
$593$	13.4136	+	3.59418i	0.550832	+	0.147595i	0.523491	−	0.852031i	$-0.324629\pi$
$593$	13.4136	+	3.59418i	0.550832	+	0.147595i	0.0273412	+	0.999626i	$0.491296\pi$
$594$		0			0
$595$	−19.1531	+	21.8458i	−0.785200	+	0.895592i
$596$		0			0
$597$	3.51994	−	2.32352i	0.144062	−	0.0950955i
$598$		0			0
$599$	−1.15232	+	1.99588i	−0.0470827	+	0.0815496i	−0.888606	−	0.458671i	$-0.848326\pi$
$599$	−1.15232	+	1.99588i	−0.0470827	+	0.0815496i	0.841524	+	0.540220i	$0.181659\pi$
$600$		0			0
$601$	−7.55165			−0.308038			−0.154019	−	0.988068i	$-0.549222\pi$
$601$	−7.55165			−0.308038			−0.154019	+	0.988068i	$0.549222\pi$
$602$		0			0
$603$	22.3521	−	28.4635i	0.910249	−	1.15913i
$604$		0			0
$605$	−3.28661	−	11.2664i	−0.133620	−	0.458045i
$606$		0			0
$607$	−31.7660	−	8.51167i	−1.28934	−	0.345478i	−0.451932	−	0.892053i	$-0.649265\pi$
$607$	−31.7660	−	8.51167i	−1.28934	−	0.345478i	−0.837411	+	0.546574i	$0.815932\pi$
$608$		0			0
$609$	−7.61285	−	19.9113i	−0.308488	−	0.806848i
$610$		0			0
$611$	33.1246	−	19.1245i	1.34008	−	0.773695i
$612$		0			0
$613$	19.8867	−	5.32864i	0.803218	−	0.215222i	0.166222	−	0.986088i	$-0.446843\pi$
$613$	19.8867	−	5.32864i	0.803218	−	0.215222i	0.636996	+	0.770867i	$0.280177\pi$
$614$		0			0
$615$	5.27634	+	4.47596i	0.212763	+	0.180488i
$616$		0			0
$617$	−21.0456	−	21.0456i	−0.847265	−	0.847265i	0.142526	−	0.989791i	$-0.454478\pi$
$617$	−21.0456	−	21.0456i	−0.847265	−	0.847265i	−0.989791	+	0.142526i	$0.954478\pi$
$618$		0			0
$619$	24.1068	+	13.9181i	0.968933	+	0.559414i	0.898911	−	0.438132i	$-0.144360\pi$
$619$	24.1068	+	13.9181i	0.968933	+	0.559414i	0.0700222	+	0.997545i	$0.477693\pi$
$620$		0			0
$621$	−6.48998	−	0.535086i	−0.260434	−	0.0214723i
$622$		0			0
$623$	13.3348	+	2.95751i	0.534246	+	0.118490i
$624$		0			0
$625$	14.3578	+	20.4659i	0.574312	+	0.818637i
$626$		0			0
$627$	−0.324506	+	0.648841i	−0.0129595	+	0.0259122i
$628$		0			0
$629$	55.9118			2.22935
$630$		0			0
$631$	43.7136			1.74021			0.870105	−	0.492867i	$-0.164051\pi$
$631$	43.7136			1.74021			0.870105	+	0.492867i	$0.164051\pi$
$632$		0			0
$633$	10.6979	−	21.3901i	0.425202	−	0.850181i
$634$		0			0
$635$	−39.7761	+	24.1489i	−1.57847	+	0.958321i
$636$		0			0
$637$	21.5028	−	25.6167i	0.851974	−	1.01497i
$638$		0			0
$639$	38.3911	−	28.7807i	1.51873	−	1.13855i
$640$		0			0
$641$	−3.97235	−	2.29344i	−0.156898	−	0.0905854i	0.419495	−	0.907758i	$-0.362207\pi$
$641$	−3.97235	−	2.29344i	−0.156898	−	0.0905854i	−0.576394	+	0.817172i	$0.695540\pi$
$642$		0			0
$643$	18.3311	+	18.3311i	0.722906	+	0.722906i	0.969196	−	0.246290i	$-0.0792115\pi$
$643$	18.3311	+	18.3311i	0.722906	+	0.722906i	−0.246290	+	0.969196i	$0.579212\pi$
$644$		0			0
$645$	−13.4333	+	1.10247i	−0.528934	+	0.0434099i
$646$		0			0
$647$	42.1467	−	11.2932i	1.65696	−	0.443980i	0.695408	−	0.718615i	$-0.255224\pi$
$647$	42.1467	−	11.2932i	1.65696	−	0.443980i	0.961549	+	0.274635i	$0.0885569\pi$
$648$		0			0
$649$	8.58210	−	4.95488i	0.336877	−	0.194496i
$650$		0			0
$651$	−11.4328	−	1.82637i	−0.448086	−	0.0715813i
$652$		0			0
$653$	14.0659	+	3.76895i	0.550442	+	0.147490i	0.523312	−	0.852141i	$-0.324696\pi$
$653$	14.0659	+	3.76895i	0.550442	+	0.147490i	0.0271304	+	0.999632i	$0.491363\pi$
$654$		0			0
$655$	28.9175	−	8.43572i	1.12990	−	0.329611i
$656$		0			0
$657$	12.7822	+	10.0377i	0.498682	+	0.391610i
$658$		0			0
$659$	−19.4487			−0.757615			−0.378808	−	0.925475i	$-0.623666\pi$
$659$	−19.4487			−0.757615			−0.378808	+	0.925475i	$0.623666\pi$
$660$		0			0
$661$	−15.9402	+	27.6092i	−0.620001	+	1.07387i	0.369483	+	0.929237i	$0.379535\pi$
$661$	−15.9402	+	27.6092i	−0.620001	+	1.07387i	−0.989485	+	0.144637i	$0.953799\pi$
$662$		0			0
$663$	−33.9171	+	22.3887i	−1.31723	+	0.869507i
$664$		0			0
$665$	0.456864	−	0.926742i	0.0177164	−	0.0359375i
$666$		0			0
$667$	5.63111	+	1.50885i	0.218037	+	0.0584229i
$668$		0			0
$669$	−40.9242	+	13.6366i	−1.58222	+	0.527222i
$670$		0			0
$671$		−	16.6834i		−	0.644055i
$672$		0			0
$673$	13.2491	+	13.2491i	0.510715	+	0.510715i	0.914745	−	0.404031i	$-0.132391\pi$
$673$	13.2491	+	13.2491i	0.510715	+	0.510715i	−0.404031	+	0.914745i	$0.632391\pi$
$674$		0			0
$675$	−24.8205	−	7.67732i	−0.955343	−	0.295500i
$676$		0			0
$677$	−2.45368	+	9.15724i	−0.0943024	+	0.351941i	−0.996913	−	0.0785172i	$-0.974981\pi$
$677$	−2.45368	+	9.15724i	−0.0943024	+	0.351941i	0.902610	+	0.430459i	$0.141648\pi$
$678$		0			0
$679$	0.440955	+	10.1205i	0.0169223	+	0.388389i
$680$		0			0
$681$	−2.27363	+	11.1047i	−0.0871257	+	0.425533i
$682$		0			0
$683$	0.869046	+	3.24333i	0.0332531	+	0.124102i	0.980557	−	0.196233i	$-0.0628710\pi$
$683$	0.869046	+	3.24333i	0.0332531	+	0.124102i	−0.947304	+	0.320336i	$0.896204\pi$
$684$		0			0
$685$	29.7325	−	0.655444i	1.13602	−	0.0250432i
$686$		0			0
$687$	−2.49314	−	41.6096i	−0.0951192	−	1.58751i
$688$		0			0
$689$	−26.9526	+	46.6833i	−1.02681	+	1.77849i
$690$		0			0
$691$	0.681637	+	1.18063i	0.0259307	+	0.0449133i	0.878700	−	0.477375i	$-0.158412\pi$
$691$	0.681637	+	1.18063i	0.0259307	+	0.0449133i	−0.852769	+	0.522289i	$0.825078\pi$
$692$		0			0
$693$	15.4685	+	11.0937i	0.587599	+	0.421415i
$694$		0			0
$695$	2.43313	−	9.95305i	0.0922938	−	0.377541i
$696$		0			0
$697$	−2.27069	−	8.47434i	−0.0860086	−	0.320989i
$698$		0			0
$699$	2.31177	−	2.60646i	0.0874391	−	0.0985852i
$700$		0			0
$701$			16.8670i			0.637057i	0.947913	+	0.318528i	$0.103189\pi$
$701$			16.8670i			0.637057i	−0.947913	+	0.318528i	$0.896811\pi$
$702$		0			0
$703$	−1.92068	+	0.514644i	−0.0724397	+	0.0194102i
$704$		0			0
$705$	−17.6467	−	25.4929i	−0.664615	−	0.960120i
$706$		0			0
$707$	−3.02533	+	0.953582i	−0.113779	+	0.0358631i
$708$		0			0
$709$	−6.24998	+	3.60843i	−0.234723	+	0.135517i	−0.612749	−	0.790278i	$-0.709936\pi$
$709$	−6.24998	+	3.60843i	−0.234723	+	0.135517i	0.378026	+	0.925795i	$0.376603\pi$
$710$		0			0
$711$	−17.5916	−	7.51878i	−0.659738	−	0.281976i
$712$		0			0
$713$	2.23888	−	2.23888i	0.0838468	−	0.0838468i
$714$		0			0
$715$	−0.564691	−	25.6157i	−0.0211183	−	0.957974i
$716$		0			0
$717$	−7.86018	+	15.7162i	−0.293544	+	0.586933i
$718$		0			0
$719$	12.0886	+	20.9381i	0.450829	+	0.780858i	0.998438	−	0.0558760i	$-0.0177952\pi$
$719$	12.0886	+	20.9381i	0.450829	+	0.780858i	−0.547609	+	0.836734i	$0.684462\pi$
$720$		0			0
$721$	7.67888	+	12.0558i	0.285977	+	0.448983i
$722$		0			0
$723$	−21.2973	−	32.2636i	−0.792055	−	1.19990i
$724$		0			0
$725$	20.6356	+	10.7305i	0.766388	+	0.398519i
$726$		0			0
$727$	20.9329	−	20.9329i	0.776359	−	0.776359i	−0.202850	−	0.979210i	$-0.565021\pi$
$727$	20.9329	−	20.9329i	0.776359	−	0.776359i	0.979210	+	0.202850i	$0.0650205\pi$
$728$		0			0
$729$	25.2780	−	9.48802i	0.936222	−	0.351408i
$730$		0			0
$731$	14.8007	+	8.54519i	0.547424	+	0.316055i
$732$		0			0
$733$	4.64570	−	17.3380i	0.171593	−	0.640393i	−0.825514	−	0.564381i	$-0.809115\pi$
$733$	4.64570	−	17.3380i	0.171593	−	0.640393i	0.997107	−	0.0760117i	$-0.0242186\pi$
$734$		0			0
$735$	−22.1211	−	15.6734i	−0.815950	−	0.578122i
$736$		0			0
$737$	7.48801	−	27.9456i	0.275824	−	1.02939i
$738$		0			0
$739$	−19.9418	−	11.5134i	−0.733569	−	0.423526i	0.0861572	−	0.996282i	$-0.472541\pi$
$739$	−19.9418	−	11.5134i	−0.733569	−	0.423526i	−0.819727	+	0.572755i	$0.805875\pi$
$740$		0			0
$741$	0.959038	−	1.08129i	0.0352311	−	0.0397221i
$742$		0			0
$743$	14.5081	−	14.5081i	0.532251	−	0.532251i	−0.388990	−	0.921242i	$-0.627176\pi$
$743$	14.5081	−	14.5081i	0.532251	−	0.532251i	0.921242	+	0.388990i	$0.127176\pi$
$744$		0			0
$745$	4.49302	+	15.4020i	0.164612	+	0.564285i
$746$		0			0
$747$	−12.8107	−	1.83285i	−0.468718	−	0.0670605i
$748$		0			0
$749$	−14.0940	−	22.1276i	−0.514985	−	0.808526i
$750$		0			0
$751$	15.6139	+	27.0440i	0.569758	+	0.986850i	0.996590	+	0.0825187i	$0.0262964\pi$
$751$	15.6139	+	27.0440i	0.569758	+	0.986850i	−0.426831	+	0.904331i	$0.640370\pi$
$752$		0			0
$753$	−6.78223	−	3.39201i	−0.247158	−	0.123612i
$754$		0			0
$755$	6.65000	−	6.94980i	0.242018	−	0.252929i
$756$		0			0
$757$	−8.35485	+	8.35485i	−0.303662	+	0.303662i	−0.842445	−	0.538783i	$-0.818884\pi$
$757$	−8.35485	+	8.35485i	−0.303662	+	0.303662i	0.538783	+	0.842445i	$0.318884\pi$
$758$		0			0
$759$	−4.93878	+	1.64568i	−0.179266	+	0.0597345i
$760$		0			0
$761$	−16.1307	+	9.31307i	−0.584738	+	0.337598i	−0.763014	−	0.646382i	$-0.776281\pi$
$761$	−16.1307	+	9.31307i	−0.584738	+	0.337598i	0.178276	+	0.983980i	$0.442948\pi$
$762$		0			0
$763$	−38.2980	+	12.0715i	−1.38648	+	0.437018i
$764$		0			0
$765$	20.3364	+	25.9167i	0.735265	+	0.937021i
$766$		0			0
$767$	−19.0701	+	5.10982i	−0.688582	+	0.184505i
$768$		0			0
$769$		−	6.68431i		−	0.241042i	−0.992711	−	0.120521i	$-0.961543\pi$
$769$		−	6.68431i		−	0.241042i	0.992711	−	0.120521i	$-0.0384566\pi$
$770$		0			0
$771$	3.20116	+	2.83924i	0.115287	+	0.102253i
$772$		0			0
$773$	−0.503766	−	1.88008i	−0.0181192	−	0.0676217i	0.956274	−	0.292471i	$-0.0944776\pi$
$773$	−0.503766	−	1.88008i	−0.0181192	−	0.0676217i	−0.974393	+	0.224850i	$0.927811\pi$
$774$		0			0
$775$	10.6510	−	6.79213i	0.382594	−	0.243981i
$776$		0			0
$777$	5.38461	+	51.8955i	0.193172	+	1.86174i
$778$		0			0
$779$	0.156005	+	0.270209i	0.00558947	+	0.00968125i
$780$		0			0
$781$	19.1783	−	33.2178i	0.686254	−	1.18863i
$782$		0			0
$783$	−23.7828	+	4.31637i	−0.849926	+	0.154255i
$784$		0			0
$785$	−0.438096	−	19.8731i	−0.0156363	−	0.709300i
$786$		0			0
$787$	9.08117	+	33.8914i	0.323709	+	1.20810i	0.915604	+	0.402082i	$0.131713\pi$
$787$	9.08117	+	33.8914i	0.323709	+	1.20810i	−0.591895	+	0.806015i	$0.701620\pi$
$788$		0			0
$789$	0.514808	+	0.105404i	0.0183277	+	0.00375249i
$790$		0			0
$791$	1.84708	+	42.3929i	0.0656746	+	1.50732i
$792$		0			0
$793$	−8.60255	+	32.1052i	−0.305486	+	1.14009i
$794$		0			0
$795$	39.5019	+	18.6794i	1.40099	+	0.662491i
$796$		0			0
$797$	−9.84195	−	9.84195i	−0.348620	−	0.348620i	0.510975	−	0.859595i	$-0.329284\pi$
$797$	−9.84195	−	9.84195i	−0.348620	−	0.348620i	−0.859595	+	0.510975i	$0.829284\pi$
$798$		0			0
$799$			39.3135i			1.39081i
$800$		0			0
$801$	6.08686	−	14.2414i	0.215069	−	0.503194i
$802$		0			0
$803$	12.5496	+	3.36266i	0.442867	+	0.118666i
$804$		0			0
$805$	7.02044	−	2.38417i	0.247438	−	0.0840310i
$806$		0			0
$807$	−16.4087	−	24.8578i	−0.577614	−	0.875036i
$808$		0			0
$809$	−3.50324	+	6.06779i	−0.123167	+	0.213332i	−0.921015	−	0.389527i	$-0.872639\pi$
$809$	−3.50324	+	6.06779i	−0.123167	+	0.213332i	0.797848	+	0.602859i	$0.205972\pi$
$810$		0			0
$811$	−49.1753			−1.72678			−0.863390	−	0.504538i	$-0.831663\pi$
$811$	−49.1753			−1.72678			−0.863390	+	0.504538i	$0.831663\pi$
$812$		0			0
$813$	−17.0143	+	1.01945i	−0.596718	+	0.0357538i
$814$		0			0
$815$	−6.36661	+	11.6110i	−0.223013	+	0.406715i
$816$		0			0
$817$	−0.587087	−	0.157310i	−0.0205396	−	0.00550357i
$818$		0			0
$819$	−24.0469	−	29.3246i	−0.840267	−	1.02468i
$820$		0			0
$821$	5.37009	−	3.10042i	0.187417	−	0.108206i	−0.403356	−	0.915043i	$-0.632156\pi$
$821$	5.37009	−	3.10042i	0.187417	−	0.108206i	0.590773	+	0.806838i	$0.298823\pi$
$822$		0			0
$823$	−11.2275	+	3.00839i	−0.391365	+	0.104866i	−0.449135	−	0.893464i	$-0.648268\pi$
$823$	−11.2275	+	3.00839i	−0.391365	+	0.104866i	0.0577701	+	0.998330i	$0.481601\pi$
$824$		0			0
$825$	−20.6572	+	2.15462i	−0.719191	+	0.0750144i
$826$		0			0
$827$	30.1289	+	30.1289i	1.04769	+	1.04769i	0.998805	+	0.0488806i	$0.0155654\pi$
$827$	30.1289	+	30.1289i	1.04769	+	1.04769i	0.0488806	+	0.998805i	$0.484435\pi$
$828$		0			0
$829$	11.1605	+	6.44350i	0.387619	+	0.223792i	0.681128	−	0.732164i	$-0.261490\pi$
$829$	11.1605	+	6.44350i	0.387619	+	0.223792i	−0.293509	+	0.955956i	$0.594823\pi$
$830$		0			0
$831$	22.8503	+	4.67848i	0.792669	+	0.162295i
$832$		0			0
$833$	11.7515	+	32.3050i	0.407164	+	1.11930i
$834$		0			0
$835$	−26.5039	−	43.6549i	−0.917204	−	1.51074i
$836$		0			0
$837$	−4.42822	+	12.3585i	−0.153062	+	0.427173i
$838$		0			0
$839$	6.48689			0.223952			0.111976	−	0.993711i	$-0.464282\pi$
$839$	6.48689			0.223952			0.111976	+	0.993711i	$0.464282\pi$
$840$		0			0
$841$	−7.36110			−0.253831
$842$		0			0
$843$	37.9136	+	18.9618i	1.30581	+	0.653079i
$844$		0			0
$845$	−5.21877	+	21.3481i	−0.179531	+	0.734398i
$846$		0			0
$847$	−13.5568	−	3.00676i	−0.465818	−	0.103313i
$848$		0			0
$849$	−2.35834	+	11.5184i	−0.0809379	+	0.395311i
$850$		0			0
$851$	−12.3569	−	7.13424i	−0.423588	−	0.244559i
$852$		0			0
$853$	−14.9986	−	14.9986i	−0.513542	−	0.513542i	0.402068	−	0.915610i	$-0.368292\pi$
$853$	−14.9986	−	14.9986i	−0.513542	−	0.513542i	−0.915610	+	0.402068i	$0.868292\pi$
$854$		0			0
$855$	−0.937148	−	0.703101i	−0.0320498	−	0.0240456i
$856$		0			0
$857$	−20.5175	+	5.49764i	−0.700864	+	0.187796i	−0.591618	−	0.806219i	$-0.701510\pi$
$857$	−20.5175	+	5.49764i	−0.700864	+	0.187796i	−0.109246	+	0.994015i	$0.534844\pi$
$858$		0			0
$859$	29.8226	−	17.2181i	1.01753	−	0.587474i	0.104146	−	0.994562i	$-0.466789\pi$
$859$	29.8226	−	17.2181i	1.01753	−	0.587474i	0.913389	+	0.407088i	$0.133456\pi$
$860$		0			0
$861$	7.64693	−	2.92371i	0.260607	−	0.0996397i
$862$		0			0
$863$	−34.2780	−	9.18477i	−1.16684	−	0.312653i	−0.377144	−	0.926155i	$-0.623094\pi$
$863$	−34.2780	−	9.18477i	−1.16684	−	0.312653i	−0.789694	+	0.613501i	$0.789761\pi$
$864$		0			0
$865$	−50.4276	−	27.6508i	−1.71459	−	0.940155i
$866$		0			0
$867$	−0.737237	−	12.3042i	−0.0250379	−	0.417874i
$868$		0			0
$869$	−15.2935			−0.518798
$870$		0			0
$871$	−28.8195	+	49.9168i	−0.976512	+	1.69137i
$872$		0			0
$873$	11.3707	+	1.62683i	0.384839	+	0.0550597i
$874$		0			0
$875$	29.4027	−	3.23748i	0.993993	−	0.109447i
$876$		0			0
$877$	−45.5843	−	12.2143i	−1.53927	−	0.412446i	−0.613241	−	0.789896i	$-0.710134\pi$
$877$	−45.5843	−	12.2143i	−1.53927	−	0.412446i	−0.926030	+	0.377450i	$0.876801\pi$
$878$		0			0
$879$	−15.4839	−	46.4681i	−0.522260	−	1.56733i
$880$		0			0
$881$			24.3717i			0.821105i	0.911837	+	0.410552i	$0.134664\pi$
$881$			24.3717i			0.821105i	−0.911837	+	0.410552i	$0.865336\pi$
$882$		0			0
$883$	7.59214	+	7.59214i	0.255496	+	0.255496i	0.823219	−	0.567723i	$-0.192176\pi$
$883$	7.59214	+	7.59214i	0.255496	+	0.255496i	−0.567723	+	0.823219i	$0.692176\pi$
$884$		0			0
$885$	5.39259	+	15.0677i	0.181270	+	0.506496i
$886$		0			0
$887$	10.8301	−	40.4184i	0.363638	−	1.35712i	−0.505619	−	0.862757i	$-0.668736\pi$
$887$	10.8301	−	40.4184i	0.363638	−	1.35712i	0.869258	−	0.494360i	$-0.164597\pi$
$888$		0			0
$889$	2.39665	+	55.0062i	0.0803810	+	1.84485i
$890$		0			0
$891$	15.6005	−	14.9163i	0.522636	−	0.499715i
$892$		0			0
$893$	−0.361864	−	1.35049i	−0.0121093	−	0.0451925i
$894$		0			0
$895$	30.7922	−	32.1804i	1.02927	−	1.07567i
$896$		0			0
$897$	10.3527	−	0.620304i	0.345665	−	0.0207113i
$898$		0			0
$899$	5.87627	−	10.1780i	0.195984	−	0.339455i
$900$		0			0
$901$	−27.7027	−	47.9825i	−0.922911	−	1.59853i
$902$		0			0
$903$	−6.50598	+	14.5605i	−0.216505	+	0.484542i
$904$		0			0
$905$	35.7548	+	8.74064i	1.18853	+	0.290549i
$906$		0			0
$907$	8.06620	+	30.1035i	0.267834	+	0.999570i	0.960493	+	0.278304i	$0.0897724\pi$
$907$	8.06620	+	30.1035i	0.267834	+	0.999570i	−0.692659	+	0.721265i	$0.743561\pi$
$908$		0			0
$909$	0.429476	+	3.57104i	0.0142448	+	0.118444i
$910$		0			0
$911$			3.16590i			0.104891i	0.998624	+	0.0524454i	$0.0167016\pi$
$911$			3.16590i			0.104891i	−0.998624	+	0.0524454i	$0.983298\pi$
$912$		0			0
$913$	−9.99274	+	2.67755i	−0.330711	+	0.0886138i
$914$		0			0
$915$	26.5077	+	4.82120i	0.876319	+	0.159384i
$916$		0			0
$917$	7.71743	−	34.7962i	0.254852	−	1.14907i
$918$		0			0
$919$	20.7512	−	11.9807i	0.684520	−	0.395208i	−0.117036	−	0.993128i	$-0.537339\pi$
$919$	20.7512	−	11.9807i	0.684520	−	0.395208i	0.801556	+	0.597920i	$0.204006\pi$
$920$		0			0
$921$	4.35155	+	13.0593i	0.143389	+	0.430317i
$922$		0			0
$923$	−54.0346	+	54.0346i	−1.77857	+	1.77857i
$924$		0			0
$925$	−41.9880	−	38.4402i	−1.38056	−	1.26391i
$926$		0			0
$927$	15.0423	−	6.03412i	0.494055	−	0.198186i
$928$		0			0
$929$	2.59475	+	4.49423i	0.0851309	+	0.147451i	0.905447	−	0.424459i	$-0.139536\pi$
$929$	2.59475	+	4.49423i	0.0851309	+	0.147451i	−0.820316	+	0.571910i	$0.806202\pi$
$930$		0			0
$931$	−0.701040	−	1.00157i	−0.0229757	−	0.0328252i
$932$		0			0
$933$	−43.7026	+	28.8482i	−1.43076	+	0.944448i
$934$		0			0
$935$	23.0914	+	12.6617i	0.755171	+	0.414080i
$936$		0			0
$937$	−22.7185	+	22.7185i	−0.742180	+	0.742180i	−0.972997	−	0.230817i	$-0.925860\pi$
$937$	−22.7185	+	22.7185i	−0.742180	+	0.742180i	0.230817	+	0.972997i	$0.425860\pi$
$938$		0			0
$939$	−13.9032	−	12.3313i	−0.453713	−	0.402416i
$940$		0			0
$941$	14.4705	+	8.35457i	0.471726	+	0.272351i	0.716962	−	0.697112i	$-0.245532\pi$
$941$	14.4705	+	8.35457i	0.471726	+	0.272351i	−0.245236	+	0.969463i	$0.578865\pi$
$942$		0			0
$943$	−0.579473	+	2.16262i	−0.0188702	+	0.0704246i
$944$		0			0
$945$	−22.0966	+	21.3715i	−0.718801	+	0.695216i
$946$		0			0
$947$	−7.67673	+	28.6499i	−0.249460	+	0.930998i	0.721629	+	0.692280i	$0.243394\pi$
$947$	−7.67673	+	28.6499i	−0.249460	+	0.930998i	−0.971089	+	0.238718i	$0.923273\pi$
$948$		0			0
$949$	−22.4163	−	12.9421i	−0.727665	−	0.420117i
$950$		0			0
$951$	−13.7771	−	12.2194i	−0.446752	−	0.396242i
$952$		0			0
$953$	2.59553	−	2.59553i	0.0840774	−	0.0840774i	−0.663817	−	0.747895i	$-0.731065\pi$
$953$	2.59553	−	2.59553i	0.0840774	−	0.0840774i	0.747895	+	0.663817i	$0.231065\pi$
$954$		0			0
$955$	−3.87585	+	1.13065i	−0.125420	+	0.0365870i
$956$		0			0
$957$	−16.1262	+	10.6449i	−0.521285	+	0.344102i
$958$		0			0
$959$	16.2510	−	31.2111i	0.524774	−	1.00786i
$960$		0			0
$961$	12.3085	+	21.3189i	0.397048	+	0.687707i
$962$		0			0
$963$	−27.6091	+	11.0752i	−0.889691	+	0.356893i
$964$		0			0
$965$	−11.9423	+	0.263264i	−0.384436	+	0.00847478i
$966$		0			0
$967$	−25.3567	+	25.3567i	−0.815418	+	0.815418i	−0.985440	−	0.170022i	$-0.945616\pi$
$967$	−25.3567	+	25.3567i	−0.815418	+	0.815418i	0.170022	+	0.985440i	$0.445616\pi$
$968$		0			0
$969$	0.469620	+	1.40936i	0.0150864	+	0.0452750i
$970$		0			0
$971$	−11.9980	+	6.92703i	−0.385033	+	0.222299i	−0.680006	−	0.733207i	$-0.738023\pi$
$971$	−11.9980	+	6.92703i	−0.385033	+	0.222299i	0.294973	+	0.955506i	$0.404689\pi$
$972$		0			0
$973$	−8.93756	−	8.19125i	−0.286525	−	0.262600i
$974$		0			0
$975$	40.8632	+	6.50527i	1.30867	+	0.208335i
$976$		0			0
$977$	−12.0931	+	3.24035i	−0.386894	+	0.103668i	−0.447022	−	0.894523i	$-0.647516\pi$
$977$	−12.0931	+	3.24035i	−0.386894	+	0.103668i	0.0601286	+	0.998191i	$0.480849\pi$
$978$		0			0
$979$		−	12.3810i		−	0.395697i
$980$		0			0
$981$	5.43680	+	45.2063i	0.173584	+	1.44332i
$982$		0			0
$983$	0.827204	+	3.08717i	0.0263837	+	0.0984653i	0.977862	−	0.209250i	$-0.0671022\pi$
$983$	0.827204	+	3.08717i	0.0263837	+	0.0984653i	−0.951478	+	0.307715i	$0.900436\pi$
$984$		0			0
$985$	24.8215	+	40.8839i	0.790879	+	1.30267i
$986$		0			0
$987$	−36.4895	+	3.78610i	−1.16147	+	0.120513i
$988$		0			0
$989$	−2.18070	−	3.77708i	−0.0693422	−	0.120104i
$990$		0			0
$991$	−18.1183	+	31.3818i	−0.575547	+	0.996877i	0.420435	+	0.907323i	$0.361878\pi$
$991$	−18.1183	+	31.3818i	−0.575547	+	0.996877i	−0.995982	+	0.0895543i	$0.971456\pi$
$992$		0			0
$993$	−0.717480	+	0.0429895i	−0.0227685	+	0.00136423i
$994$		0			0
$995$	3.93411	+	3.76440i	0.124720	+	0.119340i
$996$		0			0
$997$	−3.42696	−	12.7896i	−0.108533	−	0.405051i	0.890189	−	0.455591i	$-0.150572\pi$
$997$	−3.42696	−	12.7896i	−0.108533	−	0.405051i	−0.998722	+	0.0505407i	$0.983906\pi$
$998$		0			0
$999$	58.9598	+	4.86112i	1.86541	+	0.153799i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	420.2.bv.c.53.9	✓	48
3.2	odd	2	inner	420.2.bv.c.53.11	yes	48
5.2	odd	4	inner	420.2.bv.c.137.2	yes	48
7.2	even	3	inner	420.2.bv.c.233.9	yes	48
15.2	even	4	inner	420.2.bv.c.137.9	yes	48
21.2	odd	6	inner	420.2.bv.c.233.2	yes	48
35.2	odd	12	inner	420.2.bv.c.317.11	yes	48
105.2	even	12	inner	420.2.bv.c.317.9	yes	48

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
420.2.bv.c.53.9	✓	48	1.1	even	1	trivial
420.2.bv.c.53.11	yes	48	3.2	odd	2	inner
420.2.bv.c.137.2	yes	48	5.2	odd	4	inner
420.2.bv.c.137.9	yes	48	15.2	even	4	inner
420.2.bv.c.233.2	yes	48	21.2	odd	6	inner
420.2.bv.c.233.9	yes	48	7.2	even	3	inner
420.2.bv.c.317.9	yes	48	105.2	even	12	inner
420.2.bv.c.317.11	yes	48	35.2	odd	12	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q+(0.774760 - 1.54911i) q^{3} +(2.17211 + 0.530994i) q^{5} +(1.78762 - 1.95049i) q^{7} +(-1.79949 - 2.40038i) q^{9} +(-2.07693 - 1.19911i) q^{11} +(3.37848 + 3.37848i) q^{13} +(2.50543 - 2.95344i) q^{15} +(-4.74353 + 1.27103i) q^{17} +(0.151250 - 0.0873244i) q^{19} +(-1.63655 - 4.28039i) q^{21} +(1.21053 + 0.324361i) q^{23} +(4.43609 + 2.30675i) q^{25} +(-5.11263 + 0.927900i) q^{27} +4.65176 q^{29} +(1.26323 - 2.18799i) q^{31} +(-3.46668 + 2.28836i) q^{33} +(4.91860 - 3.28746i) q^{35} +(-10.9974 - 2.94674i) q^{37} +(7.85116 - 2.61614i) q^{39} +1.78650i q^{41} +(-2.46081 - 2.46081i) q^{43} +(-2.63411 - 6.16940i) q^{45} +(2.07195 - 7.73263i) q^{47} +(-0.608831 - 6.97347i) q^{49} +(-1.70614 + 8.33300i) q^{51} +(2.92005 + 10.8978i) q^{53} +(-3.87458 - 3.70744i) q^{55} +(-0.0180926 - 0.301959i) q^{57} +(-2.06606 + 3.57852i) q^{59} +(3.47828 + 6.02455i) q^{61} +(-7.89873 - 0.781068i) q^{63} +(5.54447 + 9.13238i) q^{65} +(3.12231 + 11.6526i) q^{67} +(1.44034 - 1.62395i) q^{69} +15.9938i q^{71} +(-5.23288 + 1.40215i) q^{73} +(7.01032 - 5.08482i) q^{75} +(-6.05161 + 1.90747i) q^{77} +(5.52266 - 3.18851i) q^{79} +(-2.52364 + 8.63894i) q^{81} +(3.05025 - 3.05025i) q^{83} +(-10.9784 + 0.242015i) q^{85} +(3.60400 - 7.20610i) q^{87} +(2.58127 + 4.47089i) q^{89} +(12.6291 - 0.550258i) q^{91} +(-2.41073 - 3.65205i) q^{93} +(0.374900 - 0.109365i) q^{95} +(-2.70738 + 2.70738i) q^{97} +(0.859088 + 7.14321i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 48 q - 2 q^{3} - 8 q^{7} + 32 q^{13} - 16 q^{21} + 16 q^{25} - 32 q^{27} - 64 q^{31} + 34 q^{33} - 40 q^{37} - 24 q^{43} + 30 q^{45} + 36 q^{51} + 92 q^{57} - 18 q^{63} + 28 q^{67} - 8 q^{73} - 38 q^{75}+ \cdots + 56 q^{97}+O(q^{100}) \) 48 * q - 2 * q^3 - 8 * q^7 + 32 * q^13 - 16 * q^21 + 16 * q^25 - 32 * q^27 - 64 * q^31 + 34 * q^33 - 40 * q^37 - 24 * q^43 + 30 * q^45 + 36 * q^51 + 92 * q^57 - 18 * q^63 + 28 * q^67 - 8 * q^73 - 38 * q^75 + 20 * q^81 - 56 * q^85 - 24 * q^87 - 24 * q^91 + 6 * q^93 + 56 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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