LMFDB - Embedded newform 420.2.bv.c.53.11

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			53.11
Character	$\chi$	$=$	420.53
Dual form			420.2.bv.c.317.11

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+(1.44552 + 0.954190i) q^{3} +(-2.17211 - 0.530994i) q^{5} +(1.78762 - 1.95049i) q^{7} +(1.17904 + 2.75860i) q^{9} +(2.07693 + 1.19911i) q^{11} +(3.37848 + 3.37848i) q^{13} +(-2.63315 - 2.84016i) q^{15} +(4.74353 - 1.27103i) q^{17} +(0.151250 - 0.0873244i) q^{19} +(4.44518 - 1.11374i) q^{21} +(-1.21053 - 0.324361i) q^{23} +(4.43609 + 2.30675i) q^{25} +(-0.927900 + 5.11263i) q^{27} -4.65176 q^{29} +(1.26323 - 2.18799i) q^{31} +(1.85805 + 3.71512i) q^{33} +(-4.91860 + 3.28746i) q^{35} +(-10.9974 - 2.94674i) q^{37} +(1.65994 + 8.10737i) q^{39} -1.78650i q^{41} +(-2.46081 - 2.46081i) q^{43} +(-1.09621 - 6.61803i) q^{45} +(-2.07195 + 7.73263i) q^{47} +(-0.608831 - 6.97347i) q^{49} +(8.06966 + 2.68894i) q^{51} +(-2.92005 - 10.8978i) q^{53} +(-3.87458 - 3.70744i) q^{55} +(0.301959 + 0.0180926i) q^{57} +(2.06606 - 3.57852i) q^{59} +(3.47828 + 6.02455i) q^{61} +(7.48830 + 2.63161i) q^{63} +(-5.54447 - 9.13238i) q^{65} +(3.12231 + 11.6526i) q^{67} +(-1.44034 - 1.62395i) q^{69} -15.9938i q^{71} +(-5.23288 + 1.40215i) q^{73} +(4.21137 + 7.56732i) q^{75} +(6.05161 - 1.90747i) q^{77} +(5.52266 - 3.18851i) q^{79} +(-6.21972 + 6.50501i) q^{81} +(-3.05025 + 3.05025i) q^{83} +(-10.9784 + 0.242015i) q^{85} +(-6.72421 - 4.43867i) q^{87} +(-2.58127 - 4.47089i) q^{89} +(12.6291 - 0.550258i) q^{91} +(3.91378 - 1.95741i) q^{93} +(-0.374900 + 0.109365i) q^{95} +(-2.70738 + 2.70738i) q^{97} +(-0.859088 + 7.14321i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	1.44552	+	0.954190i	0.834570	+	0.550902i
$3$	1.44552	+	0.954190i	0.834570	+	0.550902i
$4$		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
$9$	1.17904	+	2.75860i	0.393014	+	0.919532i
$10$		0			0
$11$	2.07693	+	1.19911i	0.626217	+	0.361546i	0.779285	−	0.626669i	$-0.215582\pi$
$11$	2.07693	+	1.19911i	0.626217	+	0.361546i	−0.153069	+	0.988216i	$0.548916\pi$
$12$		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
$15$	−2.63315	−	2.84016i	−0.679876	−	0.733327i
$16$		0			0
$17$	4.74353	−	1.27103i	1.15048	−	0.308269i	0.367320	−	0.930095i	$-0.380275\pi$
$17$	4.74353	−	1.27103i	1.15048	−	0.308269i	0.783156	+	0.621826i	$0.213609\pi$
$18$		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
$21$	4.44518	−	1.11374i	0.970017	−	0.243038i
$22$		0			0
$23$	−1.21053	−	0.324361i	−0.252413	−	0.0676339i	0.130394	−	0.991462i	$-0.458376\pi$
$23$	−1.21053	−	0.324361i	−0.252413	−	0.0676339i	−0.382807	+	0.923828i	$0.625043\pi$
$24$		0			0
$25$	4.43609	+	2.30675i	0.887218	+	0.461350i
$26$		0			0
$27$	−0.927900	+	5.11263i	−0.178574	+	0.983926i
$28$		0			0
$29$	−4.65176			−0.863811			−0.431905	−	0.901919i	$-0.642159\pi$
$29$	−4.65176			−0.863811			−0.431905	+	0.901919i	$0.642159\pi$
$30$		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
$33$	1.85805	+	3.71512i	0.323445	+	0.646720i
$34$		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
$39$	1.65994	+	8.10737i	0.265803	+	1.29822i
$40$		0			0
$41$		−	1.78650i		−	0.279005i	−0.990222	−	0.139503i	$-0.955450\pi$
$41$		−	1.78650i		−	0.279005i	0.990222	−	0.139503i	$-0.0445504\pi$
$42$		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
$45$	−1.09621	−	6.61803i	−0.163413	−	0.986558i
$46$		0			0
$47$	−2.07195	+	7.73263i	−0.302225	+	1.12792i	0.633082	+	0.774084i	$0.281789\pi$
$47$	−2.07195	+	7.73263i	−0.302225	+	1.12792i	−0.935308	+	0.353835i	$0.884877\pi$
$48$		0			0
$49$	−0.608831	−	6.97347i	−0.0869758	−	0.996210i
$50$		0			0
$51$	8.06966	+	2.68894i	1.12998	+	0.376527i
$52$		0			0
$53$	−2.92005	−	10.8978i	−0.401100	−	1.49692i	−0.811137	−	0.584856i	$-0.801151\pi$
$53$	−2.92005	−	10.8978i	−0.401100	−	1.49692i	0.410037	−	0.912069i	$-0.365516\pi$
$54$		0			0
$55$	−3.87458	−	3.70744i	−0.522448	−	0.499911i
$56$		0			0
$57$	0.301959	+	0.0180926i	0.0399954	+	0.00239642i
$58$		0			0
$59$	2.06606	−	3.57852i	0.268978	−	0.465884i	−0.699620	−	0.714515i	$-0.746647\pi$
$59$	2.06606	−	3.57852i	0.268978	−	0.465884i	0.968598	+	0.248631i	$0.0799807\pi$
$60$		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
$63$	7.48830	+	2.63161i	0.943437	+	0.331552i
$64$		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	$-0.602042\pi$
$71$		−	15.9938i		−	1.89811i	0.315111	−	0.949055i	$-0.397958\pi$
$72$		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
$75$	4.21137	+	7.56732i	0.486287	+	0.873799i
$76$		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
$81$	−6.21972	+	6.50501i	−0.691080	+	0.722778i
$82$		0			0
$83$	−3.05025	+	3.05025i	−0.334808	+	0.334808i	−0.854409	−	0.519601i	$-0.826081\pi$
$83$	−3.05025	+	3.05025i	−0.334808	+	0.334808i	0.519601	+	0.854409i	$0.326081\pi$
$84$		0			0
$85$	−10.9784	+	0.242015i	−1.19077	+	0.0262502i
$86$		0			0
$87$	−6.72421	−	4.43867i	−0.720910	−	0.475875i
$88$		0			0
$89$	−2.58127	−	4.47089i	−0.273614	−	0.473914i	0.696170	−	0.717877i	$-0.254886\pi$
$89$	−2.58127	−	4.47089i	−0.273614	−	0.473914i	−0.969785	+	0.243963i	$0.921553\pi$
$90$		0			0
$91$	12.6291	−	0.550258i	1.32389	−	0.0576827i
$92$		0			0
$93$	3.91378	−	1.95741i	0.405840	−	0.202974i
$94$		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	0.550767	−	0.834659i	$-0.314335\pi$
$101$	1.03830	+	0.599462i	0.103315	+	0.0596487i	−0.447452	+	0.894308i	$0.647669\pi$
$102$		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
$105$	−10.2468	+	0.0587995i	−0.999984	+	0.00573824i
$106$		0			0
$107$	−2.56642	+	9.57801i	−0.248105	+	0.925941i	0.723692	+	0.690123i	$0.242444\pi$
$107$	−2.56642	+	9.57801i	−0.248105	+	0.925941i	−0.971797	+	0.235818i	$0.924223\pi$
$108$		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	$-0.477941\pi$
$113$	11.3407	−	11.3407i	1.06685	−	1.06685i	0.997600	−	0.0692464i	$-0.0220595\pi$
$114$		0			0
$115$	2.45717	+	1.34733i	0.229132	+	0.125639i
$116$		0			0
$117$	−5.33650	+	13.3033i	−0.493360	+	1.22989i
$118$		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
$123$	1.70467	−	2.58242i	0.153705	−	0.232849i
$124$		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
$129$	−1.20907	−	5.90523i	−0.106452	−	0.519927i
$130$		0			0
$131$	−11.6665	+	6.73566i	−1.01931	+	0.588497i	−0.913903	−	0.405932i	$-0.866947\pi$
$131$	−11.6665	+	6.73566i	−1.01931	+	0.588497i	−0.105404	+	0.994429i	$0.533614\pi$
$132$		0			0
$133$	0.100053	−	0.451115i	0.00867565	−	0.0391166i
$134$		0			0
$135$	4.73027	−	10.6125i	0.407117	−	0.913376i
$136$		0			0
$137$	−12.8468	+	3.44229i	−1.09758	+	0.294095i	−0.761777	−	0.647839i	$-0.775673\pi$
$137$	−12.8468	+	3.44229i	−1.09758	+	0.294095i	−0.335799	+	0.941934i	$0.609006\pi$
$138$		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
$147$	5.77394	−	10.6612i	0.476227	−	0.879322i
$148$		0			0
$149$	−3.58754	−	6.21379i	−0.293902	−	0.509054i	0.680827	−	0.732445i	$-0.261621\pi$
$149$	−3.58754	−	6.21379i	−0.293902	−	0.509054i	−0.974729	+	0.223391i	$0.928287\pi$
$150$		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
$153$	9.09907	+	11.5869i	0.735616	+	0.936745i
$154$		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
$159$	6.17757	−	18.5392i	0.489913	−	1.47026i
$160$		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
$165$	−2.06317	−	9.05625i	−0.160618	−	0.705028i
$166$		0			0
$167$	16.1499	+	16.1499i	1.24972	+	1.24972i	0.955844	+	0.293876i	$0.0949452\pi$
$167$	16.1499	+	16.1499i	1.24972	+	1.24972i	0.293876	+	0.955844i	$0.405055\pi$
$168$		0			0
$169$			9.82831i			0.756024i
$170$		0			0
$171$	0.419223	+	0.314279i	0.0320588	+	0.0240335i
$172$		0			0
$173$	24.8433	+	6.65674i	1.88880	+	0.506102i	0.998736	+	0.0502667i	$0.0160071\pi$
$173$	24.8433	+	6.65674i	1.88880	+	0.506102i	0.890064	+	0.455836i	$0.150660\pi$
$174$		0			0
$175$	12.4293	−	4.52896i	0.939570	−	0.342357i
$176$		0			0
$177$	6.40112	−	3.20140i	0.481137	−	0.240632i
$178$		0			0
$179$	−9.95927	+	17.2500i	−0.744391	+	1.28932i	0.206088	+	0.978533i	$0.433927\pi$
$179$	−9.95927	+	17.2500i	−0.744391	+	1.28932i	−0.950479	+	0.310789i	$0.899407\pi$
$180$		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
$183$	−0.720658	+	12.0275i	−0.0532726	+	0.889101i
$184$		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
$189$	8.31341	+	10.9493i	0.604712	+	0.796445i
$190$		0			0
$191$	1.56368	−	0.902790i	0.113144	−	0.0653236i	−0.442360	−	0.896837i	$-0.645859\pi$
$191$	1.56368	−	0.902790i	0.113144	−	0.0653236i	0.555504	+	0.831514i	$0.312525\pi$
$192$		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
$195$	0.699398	−	18.4915i	0.0500849	−	1.32420i
$196$		0			0
$197$	−15.1248	−	15.1248i	−1.07760	−	1.07760i	−0.996724	−	0.0808739i	$-0.974229\pi$
$197$	−15.1248	−	15.1248i	−1.07760	−	1.07760i	−0.0808739	−	0.996724i	$-0.525771\pi$
$198$		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
$201$	−6.60546	+	19.8233i	−0.465913	+	1.39823i
$202$		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
$207$	−0.532487	−	3.72180i	−0.0370104	−	0.258683i
$208$		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
$213$	15.2611	−	23.1193i	1.04567	−	1.58411i
$214$		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
$219$	−8.90213	−	2.96633i	−0.601550	−	0.200446i
$220$		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
$225$	−1.13306	+	14.9571i	−0.0755373	+	0.997143i
$226$		0			0
$227$	6.32132	−	1.69379i	0.419561	−	0.112421i	−0.0428611	−	0.999081i	$-0.513647\pi$
$227$	6.32132	−	1.69379i	0.419561	−	0.112421i	0.462422	+	0.886660i	$0.346981\pi$
$228$		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
$231$	10.5678	+	3.01711i	0.695310	+	0.198512i
$232$		0			0
$233$	−1.94292	−	0.520604i	−0.127285	−	0.0341059i	0.194614	−	0.980880i	$-0.437655\pi$
$233$	−1.94292	−	0.520604i	−0.127285	−	0.0341059i	−0.321899	+	0.946774i	$0.604321\pi$
$234$		0			0
$235$	8.60648	−	15.6959i	0.561425	−	1.02389i
$236$		0			0
$237$	11.0256	+	0.660622i	0.716187	+	0.0429120i
$238$		0			0
$239$	10.1453			0.656245			0.328123	−	0.944635i	$-0.393584\pi$
$239$	10.1453			0.656245			0.328123	+	0.944635i	$0.393584\pi$
$240$		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
$243$	−15.1977	+	3.46831i	−0.974935	+	0.222492i
$244$		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
$249$	−7.31971	+	1.49867i	−0.463868	+	0.0949744i
$250$		0			0
$251$			4.37814i			0.276346i	0.990408	+	0.138173i	$0.0441230\pi$
$251$			4.37814i			0.276346i	−0.990408	+	0.138173i	$0.955877\pi$
$252$		0			0
$253$	−2.12524	−	2.12524i	−0.133613	−	0.133613i
$254$		0			0
$255$	−16.1003	−	10.1256i	−1.00824	−	0.634090i
$256$		0			0
$257$	0.639388	−	2.38623i	0.0398839	−	0.148849i	−0.943112	−	0.332474i	$-0.892117\pi$
$257$	0.639388	−	2.38623i	0.0398839	−	0.148849i	0.982996	+	0.183625i	$0.0587833\pi$
$258$		0			0
$259$	−25.4067	+	16.1826i	−1.57870	+	1.00554i
$260$		0			0
$261$	−5.48463	−	12.8323i	−0.339490	−	0.794302i
$262$		0			0
$263$	−0.0785232	−	0.293053i	−0.00484195	−	0.0180704i	0.963463	−	0.267843i	$-0.0863108\pi$
$263$	−0.0785232	−	0.293053i	−0.00484195	−	0.0180704i	−0.968305	+	0.249773i	$0.919644\pi$
$264$		0			0
$265$	0.556004	+	25.2217i	0.0341551	+	1.54935i
$266$		0			0
$267$	0.534809	−	8.92578i	0.0327298	−	0.546249i
$268$		0			0
$269$	−8.59823	+	14.8926i	−0.524244	+	0.908017i	0.475358	+	0.879792i	$0.342319\pi$
$269$	−8.59823	+	14.8926i	−0.524244	+	0.908017i	−0.999602	+	0.0282241i	$0.991015\pi$
$270$		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
$273$	18.7807	+	11.2552i	1.13666	+	0.681196i
$274$		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	$-0.739515\pi$
$281$		−	24.4744i		−	1.46002i	0.683436	−	0.730011i	$-0.260485\pi$
$282$		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
$285$	−0.646280	−	0.199637i	−0.0382823	−	0.0118255i
$286$		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
$291$	−6.49693	+	1.33021i	−0.380857	+	0.0779784i
$292$		0			0
$293$	−19.9960	+	19.9960i	−1.16818	+	1.16818i	−0.185541	+	0.982637i	$0.559404\pi$
$293$	−19.9960	+	19.9960i	−1.16818	+	1.16818i	−0.982637	+	0.185541i	$0.940596\pi$
$294$		0			0
$295$	−6.38787	+	6.67586i	−0.371916	+	0.388684i
$296$		0			0
$297$	−8.05780	+	9.50590i	−0.467561	+	0.551588i
$298$		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
$303$	0.928878	+	1.85727i	0.0533626	+	0.106697i
$304$		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	0.999848	−	0.0174217i	$-0.00554578\pi$
$311$	26.1827	+	15.1166i	1.48468	+	0.857183i	0.484836	+	0.874605i	$0.338879\pi$
$312$		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
$315$	−14.8680	−	9.69238i	−0.837717	−	0.546104i
$316$		0			0
$317$	−2.75178	+	10.2698i	−0.154555	+	0.576808i	0.844588	+	0.535417i	$0.179846\pi$
$317$	−2.75178	+	10.2698i	−0.154555	+	0.576808i	−0.999143	+	0.0413908i	$0.986821\pi$
$318$		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
$327$	−11.7588	−	23.5114i	−0.650263	−	1.30018i
$328$		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
$333$	−4.83751	−	33.8117i	−0.265094	−	1.85287i
$334$		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
$339$	27.2144	−	5.57201i	1.47809	−	0.302630i
$340$		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
$345$	2.26627	+	4.29220i	0.122012	+	0.231084i
$346$		0			0
$347$	31.3558	−	8.40176i	1.68327	−	0.451030i	0.714629	−	0.699504i	$-0.246596\pi$
$347$	31.3558	−	8.40176i	1.68327	−	0.451030i	0.968639	+	0.248473i	$0.0799289\pi$
$348$		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	$-0.998629\pi$
$353$	−2.55758	−	9.54503i	−0.136126	−	0.508031i	0.863864	−	0.503725i	$-0.168037\pi$
$354$		0			0
$355$	−8.49259	+	34.7401i	−0.450740	+	1.84381i
$356$		0			0
$357$	19.6702	−	10.9330i	1.04106	−	0.578635i
$358$		0			0
$359$	5.54047	+	9.59637i	0.292415	+	0.506477i	0.974380	−	0.224907i	$-0.0722079\pi$
$359$	5.54047	+	9.59637i	0.292415	+	0.506477i	−0.681966	+	0.731384i	$0.738875\pi$
$360$		0			0
$361$	−9.48475	+	16.4281i	−0.499197	+	0.864635i
$362$		0			0
$363$	0.543715	−	9.07441i	0.0285376	−	0.476283i
$364$		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
$369$	4.92825	−	2.10636i	0.256554	−	0.109653i
$370$		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
$375$	−5.12933	−	18.6732i	−0.264878	−	0.964282i
$376$		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
$381$	−35.3117	+	7.22988i	−1.80907	+	0.370398i
$382$		0			0
$383$	12.3453	+	3.30792i	0.630817	+	0.169027i	0.560040	−	0.828465i	$-0.310786\pi$
$383$	12.3453	+	3.30792i	0.630817	+	0.169027i	0.0707764	+	0.997492i	$0.477452\pi$
$384$		0			0
$385$	−14.1576	+	0.929847i	−0.721538	+	0.0473894i
$386$		0			0
$387$	3.88699	−	9.68980i	0.197587	−	0.492560i
$388$		0			0
$389$	−17.5206	+	30.3465i	−0.888327	+	1.53863i	−0.0464754	+	0.998919i	$0.514799\pi$
$389$	−17.5206	+	30.3465i	−0.888327	+	1.53863i	−0.841852	+	0.539709i	$0.818534\pi$
$390$		0			0
$391$	−6.15447			−0.311245
$392$		0			0
$393$	−23.2912	−	1.39555i	−1.17489	−	0.0703962i
$394$		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
$399$	0.575077	−	0.556625i	0.0287899	−	0.0278661i
$400$		0			0
$401$	−13.0417	+	7.52965i	−0.651273	+	0.376013i	−0.788944	−	0.614465i	$-0.789372\pi$
$401$	−13.0417	+	7.52965i	−0.651273	+	0.376013i	0.137671	+	0.990478i	$0.456038\pi$
$402$		0			0
$403$	11.6599	−	3.12426i	0.580820	−	0.155630i
$404$		0			0
$405$	16.9640	−	10.8269i	0.842948	−	0.537994i
$406$		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
$411$	−21.8549	−	7.28240i	−1.07802	−	0.359214i
$412$		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
$417$	4.37230	−	6.62367i	0.214113	−	0.324362i
$418$		0			0
$419$	20.7118			1.01184			0.505919	−	0.862581i	$-0.331153\pi$
$419$	20.7118			1.01184			0.505919	+	0.862581i	$0.331153\pi$
$420$		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
$423$	−23.7741	+	3.40142i	−1.15594	+	0.165383i
$424$		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
$429$	−6.27408	+	18.8289i	−0.302916	+	0.909066i
$430$		0			0
$431$	0.725507	+	0.418872i	0.0349464	+	0.0201763i	0.517371	−	0.855761i	$-0.326911\pi$
$431$	0.725507	+	0.418872i	0.0349464	+	0.0201763i	−0.482425	+	0.875937i	$0.660244\pi$
$432$		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
$435$	12.2488	+	13.2118i	0.587284	+	0.633456i
$436$		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
$441$	18.5192	−	9.90154i	0.881865	−	0.471502i
$442$		0			0
$443$	−32.0596	−	8.59035i	−1.52320	−	0.408140i	−0.602405	−	0.798191i	$-0.705791\pi$
$443$	−32.0596	−	8.59035i	−1.52320	−	0.408140i	−0.920793	+	0.390051i	$0.872457\pi$
$444$		0			0
$445$	3.23278	+	11.0819i	0.153248	+	0.525332i
$446$		0			0
$447$	0.743295	−	12.4053i	0.0351566	−	0.586752i
$448$		0			0
$449$	32.3879			1.52848			0.764239	−	0.644933i	$-0.223115\pi$
$449$	32.3879			1.52848			0.764239	+	0.644933i	$0.223115\pi$
$450$		0			0
$451$	2.14222	−	3.71044i	0.100873	−	0.174718i
$452$		0			0
$453$	6.66378	−	3.33277i	0.313092	−	0.156587i
$454$		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
$459$	2.09676	+	25.4313i	0.0978685	+	1.18703i
$460$		0			0
$461$			24.9156i			1.16044i	0.814461	+	0.580218i	$0.197033\pi$
$461$			24.9156i			1.16044i	−0.814461	+	0.580218i	$0.802967\pi$
$462$		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
$465$	−9.54052	+	2.17350i	−0.442431	+	0.100794i
$466$		0			0
$467$	3.14017	−	11.7193i	0.145310	−	0.542303i	−0.854432	−	0.519563i	$-0.826095\pi$
$467$	3.14017	−	11.7193i	0.145310	−	0.542303i	0.999741	−	0.0227391i	$-0.00723871\pi$
$468$		0			0
$469$	28.3098	+	14.7404i	1.30723	+	0.680649i
$470$		0			0
$471$	4.86753	−	14.6077i	0.224284	−	0.673088i
$472$		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
$477$	26.6197	−	20.9042i	1.21883	−	0.957137i
$478$		0			0
$479$	3.14571	−	5.44853i	0.143731	−	0.248950i	−0.785168	−	0.619283i	$-0.787423\pi$
$479$	3.14571	−	5.44853i	0.143731	−	0.248950i	0.928899	+	0.370333i	$0.120757\pi$
$480$		0			0
$481$	−27.1989	−	47.1100i	−1.24017	−	2.14803i
$482$		0			0
$483$	−5.74228	−	0.0936239i	−0.261283	−	0.00426004i
$484$		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	$0.138559\pi$
$491$			18.6876i			0.843358i	−0.906745	+	0.421679i	$0.861441\pi$
$492$		0			0
$493$	−22.0658	+	5.91251i	−0.993793	+	0.266286i
$494$		0			0
$495$	5.65903	−	15.0596i	0.254355	−	0.676880i
$496$		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
$501$	7.93491	+	38.7551i	0.354505	+	1.73145i
$502$		0			0
$503$	8.52544	−	8.52544i	0.380131	−	0.380131i	−0.491018	−	0.871149i	$-0.663375\pi$
$503$	8.52544	−	8.52544i	0.380131	−	0.380131i	0.871149	+	0.491018i	$0.163375\pi$
$504$		0			0
$505$	−1.93698	−	1.85342i	−0.0861946	−	0.0824763i
$506$		0			0
$507$	−9.37807	+	14.2070i	−0.416495	+	0.630955i
$508$		0			0
$509$	−16.6925	−	28.9123i	−0.739883	−	1.28151i	−0.952548	−	0.304389i	$-0.901548\pi$
$509$	−16.6925	−	28.9123i	−0.739883	−	1.28151i	0.212665	−	0.977125i	$-0.431786\pi$
$510$		0			0
$511$	−6.61952	+	12.7132i	−0.292831	+	0.562398i
$512$		0			0
$513$	0.306112	+	0.854315i	0.0135152	+	0.0377189i
$514$		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	0.152760	−	0.988263i	$-0.451184\pi$
$521$	−14.3052	−	8.25910i	−0.626722	−	0.361838i	−0.779481	+	0.626425i	$0.784517\pi$
$522$		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
$525$	22.2883	+	5.31326i	0.972742	+	0.231890i
$526$		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
$537$	−30.8560	+	15.4321i	−1.33154	+	0.665944i
$538$		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
$543$	23.7945	+	15.7068i	1.02112	+	0.674045i
$544$		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
$549$	−12.5183	+	16.6984i	−0.534267	+	0.712669i
$550$		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
$555$	20.5885	+	38.9935i	0.873933	+	1.65518i
$556$		0			0
$557$	19.2930	−	5.16955i	0.817472	−	0.219041i	0.174231	−	0.984705i	$-0.444256\pi$
$557$	19.2930	−	5.16955i	0.817472	−	0.219041i	0.643241	+	0.765664i	$0.277589\pi$
$558$		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	$0.206055\pi$
$563$	9.69046	+	36.1653i	0.408404	+	1.52419i	−0.389285	+	0.921117i	$0.627278\pi$
$564$		0			0
$565$	−30.6551	+	18.6114i	−1.28967	+	0.782988i
$566$		0			0
$567$	1.56946	+	23.7600i	0.0659113	+	0.997825i
$568$		0			0
$569$	−8.83961	−	15.3107i	−0.370576	−	0.641856i	0.619078	−	0.785329i	$-0.287506\pi$
$569$	−8.83961	−	15.3107i	−0.370576	−	0.641856i	−0.989654	+	0.143473i	$0.954173\pi$
$570$		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
$573$	3.12176	+	0.187047i	0.130413	+	0.00781401i
$574$		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
$579$	−8.77820	−	2.92504i	−0.364810	−	0.121560i
$580$		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
$585$	18.6554	−	26.0624i	0.771306	−	1.07755i
$586$		0			0
$587$	16.5786	+	16.5786i	0.684270	+	0.684270i	0.960959	−	0.276689i	$-0.0892372\pi$
$587$	16.5786	+	16.5786i	0.684270	+	0.684270i	−0.276689	+	0.960959i	$0.589237\pi$
$588$		0			0
$589$		−	0.441244i		−	0.0181812i
$590$		0			0
$591$	−7.43124	−	36.2951i	−0.305680	−	1.49298i
$592$		0			0
$593$	−13.4136	−	3.59418i	−0.550832	−	0.147595i	−0.0273412	−	0.999626i	$-0.508704\pi$
$593$	−13.4136	−	3.59418i	−0.550832	−	0.147595i	−0.523491	+	0.852031i	$0.675371\pi$
$594$		0			0
$595$	−19.1531	+	21.8458i	−0.785200	+	0.895592i
$596$		0			0
$597$	1.88660	+	3.77220i	0.0772133	+	0.154386i
$598$		0			0
$599$	1.15232	−	1.99588i	0.0470827	−	0.0815496i	−0.841524	−	0.540220i	$-0.818341\pi$
$599$	1.15232	−	1.99588i	0.0470827	−	0.0815496i	0.888606	+	0.458671i	$0.151674\pi$
$600$		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
$603$	−28.4635	+	22.3521i	−1.15913	+	0.910249i
$604$		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
$609$	−20.6779	+	5.18085i	−0.837911	+	0.209939i
$610$		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
$615$	−5.07397	+	4.70413i	−0.204602	+	0.189689i
$616$		0			0
$617$	21.0456	+	21.0456i	0.847265	+	0.847265i	0.989791	−	0.142526i	$-0.0455224\pi$
$617$	21.0456	+	21.0456i	0.847265	+	0.847265i	−0.142526	+	0.989791i	$0.545522\pi$
$618$		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
$621$	2.78159	−	5.88803i	0.111621	−	0.236278i
$622$		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
$627$	0.605451	+	0.399660i	0.0241794	+	0.0159609i
$628$		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
$633$	19.9597	+	13.1754i	0.793326	+	0.523677i
$634$		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
$639$	44.1203	−	18.8573i	1.74537	−	0.745984i
$640$		0			0
$641$	3.97235	+	2.29344i	0.156898	+	0.0905854i	0.576394	−	0.817172i	$-0.304460\pi$
$641$	3.97235	+	2.29344i	0.156898	+	0.0905854i	−0.419495	+	0.907758i	$0.637793\pi$
$642$		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
$645$	−0.509426	+	13.4688i	−0.0200586	+	0.530333i
$646$		0			0
$647$	−42.1467	+	11.2932i	−1.65696	+	0.443980i	−0.961549	−	0.274635i	$-0.911443\pi$
$647$	−42.1467	+	11.2932i	−1.65696	+	0.443980i	−0.695408	+	0.718615i	$0.744776\pi$
$648$		0			0
$649$	8.58210	−	4.95488i	0.336877	−	0.194496i
$650$		0			0
$651$	3.17845	−	11.1329i	0.124573	−	0.436332i
$652$		0			0
$653$	−14.0659	−	3.76895i	−0.550442	−	0.147490i	−0.0271304	−	0.999632i	$-0.508637\pi$
$653$	−14.0659	−	3.76895i	−0.550442	−	0.147490i	−0.523312	+	0.852141i	$0.675304\pi$
$654$		0			0
$655$	28.9175	−	8.43572i	1.12990	−	0.329611i
$656$		0			0
$657$	−10.0377	−	12.7822i	−0.391610	−	0.498682i
$658$		0			0
$659$	19.4487			0.757615			0.378808	−	0.925475i	$-0.376334\pi$
$659$	19.4487			0.757615			0.378808	+	0.925475i	$0.376334\pi$
$660$		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
$663$	18.1787	+	36.3478i	0.706001	+	1.41163i
$664$		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
$669$	−8.65245	−	42.2597i	−0.334523	−	1.63385i
$670$		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
$675$	−15.9098	+	20.5397i	−0.612369	+	0.790572i
$676$		0			0
$677$	2.45368	−	9.15724i	0.0943024	−	0.351941i	−0.902610	−	0.430459i	$-0.858352\pi$
$677$	2.45368	−	9.15724i	0.0943024	−	0.351941i	0.996913	+	0.0785172i	$0.0250185\pi$
$678$		0			0
$679$	0.440955	+	10.1205i	0.0169223	+	0.388389i
$680$		0			0
$681$	10.7538	+	3.58333i	0.412086	+	0.137314i
$682$		0			0
$683$	−0.869046	−	3.24333i	−0.0332531	−	0.124102i	0.947304	−	0.320336i	$-0.103796\pi$
$683$	−0.869046	−	3.24333i	−0.0332531	−	0.124102i	−0.980557	+	0.196233i	$0.937129\pi$
$684$		0			0
$685$	29.7325	−	0.655444i	1.13602	−	0.0250432i
$686$		0			0
$687$	41.6096	+	2.49314i	1.58751	+	0.0951192i
$688$		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
$693$	12.3970	+	14.4450i	0.470924	+	0.548719i
$694$		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	$-0.896811\pi$
$701$		−	16.8670i		−	0.637057i	0.947913	−	0.318528i	$-0.103189\pi$
$702$		0			0
$703$	−1.92068	+	0.514644i	−0.0724397	+	0.0194102i
$704$		0			0
$705$	27.4177	−	14.4765i	1.03261	−	0.545216i
$706$		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
$711$	15.3073	+	11.4754i	0.574067	+	0.430362i
$712$		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
$717$	14.6652	+	9.68055i	0.547683	+	0.361527i
$718$		0			0
$719$	−12.0886	−	20.9381i	−0.450829	−	0.780858i	0.547609	−	0.836734i	$-0.315538\pi$
$719$	−12.0886	−	20.9381i	−0.450829	−	0.780858i	−0.998438	+	0.0558760i	$0.982205\pi$
$720$		0			0
$721$	7.67888	+	12.0558i	0.285977	+	0.448983i
$722$		0			0
$723$	34.5758	−	17.2924i	1.28589	−	0.643113i
$724$		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
$735$	−18.2027	+	20.0914i	−0.671415	+	0.741081i
$736$		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	−0.921242	−	0.388990i	$-0.872824\pi$
$743$	−14.5081	+	14.5081i	−0.532251	+	0.532251i	0.388990	+	0.921242i	$0.372824\pi$
$744$		0			0
$745$	4.49302	+	15.4020i	0.164612	+	0.564285i
$746$		0			0
$747$	−12.0108	−	4.81804i	−0.439452	−	0.176283i
$748$		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
$753$	−4.17758	+	6.32868i	−0.152239	+	0.230630i
$754$		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
$759$	−1.04419	−	5.09995i	−0.0379016	−	0.185116i
$760$		0			0
$761$	16.1307	−	9.31307i	0.584738	−	0.337598i	−0.178276	−	0.983980i	$-0.557052\pi$
$761$	16.1307	−	9.31307i	0.584738	−	0.337598i	0.763014	+	0.646382i	$0.223719\pi$
$762$		0			0
$763$	−38.2980	+	12.0715i	−1.38648	+	0.437018i
$764$		0			0
$765$	−13.6116	−	29.9995i	−0.492127	−	1.08464i
$766$		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	0.974393	−	0.224850i	$-0.0721891\pi$
$773$	0.503766	+	1.88008i	0.0181192	+	0.0676217i	−0.956274	+	0.292471i	$0.905522\pi$
$774$		0			0
$775$	10.6510	−	6.79213i	0.382594	−	0.243981i
$776$		0			0
$777$	−52.1672	−	0.850550i	−1.87149	−	0.0305133i
$778$		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
$783$	4.31637	−	23.7828i	0.154255	−	0.849926i
$784$		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
$789$	0.166121	−	0.498539i	0.00591407	−	0.0177485i
$790$		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
$795$	−23.2625	+	36.9889i	−0.825037	+	1.31186i
$796$		0			0
$797$	9.84195	+	9.84195i	0.348620	+	0.348620i	0.859595	−	0.510975i	$-0.170716\pi$
$797$	9.84195	+	9.84195i	0.348620	+	0.348620i	−0.510975	+	0.859595i	$0.670716\pi$
$798$		0			0
$799$			39.3135i			1.39081i
$800$		0			0
$801$	9.28997	−	12.3921i	0.328245	−	0.437852i
$802$		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
$807$	−26.6393	+	13.3231i	−0.937746	+	0.468997i
$808$		0			0
$809$	3.50324	−	6.06779i	0.123167	−	0.213332i	−0.797848	−	0.602859i	$-0.794028\pi$
$809$	3.50324	−	6.06779i	0.123167	−	0.213332i	0.921015	+	0.389527i	$0.127361\pi$
$810$		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
$813$	1.01945	−	17.0143i	0.0357538	−	0.596718i
$814$		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
$819$	16.4082	+	34.1900i	0.573350	+	1.19469i
$820$		0			0
$821$	−5.37009	+	3.10042i	−0.187417	+	0.108206i	−0.590773	−	0.806838i	$-0.701177\pi$
$821$	−5.37009	+	3.10042i	−0.187417	+	0.108206i	0.403356	+	0.915043i	$0.367844\pi$
$822$		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
$825$	−0.327383	+	20.7667i	−0.0113980	+	0.723003i
$826$		0			0
$827$	−30.1289	−	30.1289i	−1.04769	−	1.04769i	−0.998805	−	0.0488806i	$-0.984435\pi$
$827$	−30.1289	−	30.1289i	−1.04769	−	1.04769i	−0.0488806	−	0.998805i	$-0.515565\pi$
$828$		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
$831$	−7.37347	+	22.1282i	−0.255783	+	0.767619i
$832$		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
$837$	10.0142	+	8.48868i	0.346142	+	0.293412i
$838$		0			0
$839$	−6.48689			−0.223952			−0.111976	−	0.993711i	$-0.535718\pi$
$839$	−6.48689			−0.223952			−0.111976	+	0.993711i	$0.535718\pi$
$840$		0			0
$841$	−7.36110			−0.253831
$842$		0			0
$843$	23.3533	−	35.3782i	0.804329	−	1.21849i
$844$		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
$849$	−11.1544	−	3.71683i	−0.382818	−	0.127561i
$850$		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
$855$	−0.743717	−	0.905253i	−0.0254346	−	0.0309590i
$856$		0			0
$857$	20.5175	−	5.49764i	0.700864	−	0.187796i	0.109246	−	0.994015i	$-0.465156\pi$
$857$	20.5175	−	5.49764i	0.700864	−	0.187796i	0.591618	+	0.806219i	$0.298490\pi$
$858$		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
$861$	−1.98970	−	7.94133i	−0.0678089	−	0.270640i
$862$		0			0
$863$	34.2780	+	9.18477i	1.16684	+	0.312653i	0.789694	−	0.613501i	$-0.210239\pi$
$863$	34.2780	+	9.18477i	1.16684	+	0.312653i	0.377144	+	0.926155i	$0.376906\pi$
$864$		0			0
$865$	−50.4276	−	27.6508i	−1.71459	−	0.940155i
$866$		0			0
$867$	12.3042	+	0.737237i	0.417874	+	0.0250379i
$868$		0			0
$869$	15.2935			0.518798
$870$		0			0
$871$	−28.8195	+	49.9168i	−0.976512	+	1.69137i
$872$		0			0
$873$	−10.6607	−	4.27646i	−0.360810	−	0.144736i
$874$		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
$879$	−47.9845	+	9.82457i	−1.61848	+	0.331375i
$880$		0			0
$881$		−	24.3717i		−	0.821105i	−0.911837	−	0.410552i	$-0.865336\pi$
$881$		−	24.3717i		−	0.821105i	0.911837	−	0.410552i	$-0.134664\pi$
$882$		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
$885$	−15.6038	+	3.55483i	−0.524517	+	0.119494i
$886$		0			0
$887$	−10.8301	+	40.4184i	−0.363638	+	1.35712i	0.505619	+	0.862757i	$0.331264\pi$
$887$	−10.8301	+	40.4184i	−0.363638	+	1.35712i	−0.869258	+	0.494360i	$0.835403\pi$
$888$		0			0
$889$	2.39665	+	55.0062i	0.0803810	+	1.84485i
$890$		0			0
$891$	−20.7181	+	6.05226i	−0.694083	+	0.202758i
$892$		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
$897$	0.620304	−	10.3527i	0.0207113	−	0.345665i
$898$		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
$903$	−13.6795	−	8.19804i	−0.455224	−	0.272814i
$904$		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	$-0.983298\pi$
$911$		−	3.16590i		−	0.104891i	0.998624	−	0.0524454i	$-0.0167016\pi$
$912$		0			0
$913$	−9.99274	+	2.67755i	−0.330711	+	0.0886138i
$914$		0			0
$915$	7.95190	−	25.7424i	0.262882	−	0.851018i
$916$		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
$921$	−13.4854	+	2.76107i	−0.444360	+	0.0909803i
$922$		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
$927$	−16.0441	+	2.29546i	−0.526957	+	0.0753929i
$928$		0			0
$929$	−2.59475	−	4.49423i	−0.0851309	−	0.147451i	0.820316	−	0.571910i	$-0.193798\pi$
$929$	−2.59475	−	4.49423i	−0.0851309	−	0.147451i	−0.905447	+	0.424459i	$0.860464\pi$
$930$		0			0
$931$	−0.701040	−	1.00157i	−0.0229757	−	0.0328252i
$932$		0			0
$933$	23.4235	+	46.8346i	0.766849	+	1.53329i
$934$		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	0.245236	−	0.969463i	$-0.421135\pi$
$941$	−14.4705	−	8.35457i	−0.471726	−	0.272351i	−0.716962	+	0.697112i	$0.754468\pi$
$942$		0			0
$943$	−0.579473	+	2.16262i	−0.0188702	+	0.0704246i
$944$		0			0
$945$	−12.2436	−	28.1974i	−0.398284	−	0.917262i
$946$		0			0
$947$	7.67673	−	28.6499i	0.249460	−	0.930998i	−0.721629	−	0.692280i	$-0.756606\pi$
$947$	7.67673	−	28.6499i	0.249460	−	0.930998i	0.971089	−	0.238718i	$-0.0767271\pi$
$948$		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	−0.747895	−	0.663817i	$-0.768935\pi$
$953$	−2.59553	+	2.59553i	−0.0840774	+	0.0840774i	0.663817	+	0.747895i	$0.268935\pi$
$954$		0			0
$955$	−3.87585	+	1.13065i	−0.125420	+	0.0365870i
$956$		0			0
$957$	−8.64321	−	17.2819i	−0.279395	−	0.558643i
$958$		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
$963$	−29.4478	+	4.21316i	−0.948942	+	0.135767i
$964$		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
$969$	1.45535	−	0.297975i	0.0467525	−	0.00957233i
$970$		0			0
$971$	11.9980	−	6.92703i	0.385033	−	0.222299i	−0.294973	−	0.955506i	$-0.595311\pi$
$971$	11.9980	−	6.92703i	0.385033	−	0.222299i	0.680006	+	0.733207i	$0.261977\pi$
$972$		0			0
$973$	−8.93756	−	8.19125i	−0.286525	−	0.262600i
$974$		0			0
$975$	−11.3380	+	39.7941i	−0.363108	+	1.27443i
$976$		0			0
$977$	12.0931	−	3.24035i	0.386894	−	0.103668i	−0.0601286	−	0.998191i	$-0.519151\pi$
$977$	12.0931	−	3.24035i	0.386894	−	0.103668i	0.447022	+	0.894523i	$0.352484\pi$
$978$		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	0.951478	−	0.307715i	$-0.0995644\pi$
$983$	−0.827204	−	3.08717i	−0.0263837	−	0.0984653i	−0.977862	+	0.209250i	$0.932898\pi$
$984$		0			0
$985$	24.8215	+	40.8839i	0.790879	+	1.30267i
$986$		0			0
$987$	−0.598050	+	36.6805i	−0.0190362	+	1.16755i
$988$		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
$993$	0.0429895	−	0.717480i	0.00136423	−	0.0227685i
$994$		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
$999$	25.2700	−	53.4913i	0.799509	−	1.69239i

Twists

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

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

\(f(q)\)	\(=\)	\(q+(1.44552 + 0.954190i) q^{3} +(-2.17211 - 0.530994i) q^{5} +(1.78762 - 1.95049i) q^{7} +(1.17904 + 2.75860i) q^{9} +(2.07693 + 1.19911i) q^{11} +(3.37848 + 3.37848i) q^{13} +(-2.63315 - 2.84016i) q^{15} +(4.74353 - 1.27103i) q^{17} +(0.151250 - 0.0873244i) q^{19} +(4.44518 - 1.11374i) q^{21} +(-1.21053 - 0.324361i) q^{23} +(4.43609 + 2.30675i) q^{25} +(-0.927900 + 5.11263i) q^{27} -4.65176 q^{29} +(1.26323 - 2.18799i) q^{31} +(1.85805 + 3.71512i) q^{33} +(-4.91860 + 3.28746i) q^{35} +(-10.9974 - 2.94674i) q^{37} +(1.65994 + 8.10737i) q^{39} -1.78650i q^{41} +(-2.46081 - 2.46081i) q^{43} +(-1.09621 - 6.61803i) q^{45} +(-2.07195 + 7.73263i) q^{47} +(-0.608831 - 6.97347i) q^{49} +(8.06966 + 2.68894i) q^{51} +(-2.92005 - 10.8978i) q^{53} +(-3.87458 - 3.70744i) q^{55} +(0.301959 + 0.0180926i) q^{57} +(2.06606 - 3.57852i) q^{59} +(3.47828 + 6.02455i) q^{61} +(7.48830 + 2.63161i) q^{63} +(-5.54447 - 9.13238i) q^{65} +(3.12231 + 11.6526i) q^{67} +(-1.44034 - 1.62395i) q^{69} -15.9938i q^{71} +(-5.23288 + 1.40215i) q^{73} +(4.21137 + 7.56732i) q^{75} +(6.05161 - 1.90747i) q^{77} +(5.52266 - 3.18851i) q^{79} +(-6.21972 + 6.50501i) q^{81} +(-3.05025 + 3.05025i) q^{83} +(-10.9784 + 0.242015i) q^{85} +(-6.72421 - 4.43867i) q^{87} +(-2.58127 - 4.47089i) q^{89} +(12.6291 - 0.550258i) q^{91} +(3.91378 - 1.95741i) q^{93} +(-0.374900 + 0.109365i) q^{95} +(-2.70738 + 2.70738i) q^{97} +(-0.859088 + 7.14321i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 48 q - 2 q^{3} - 8 q^{7} + 32 q^{13} - 16 q^{21} + 16 q^{25} - 32 q^{27} - 64 q^{31} + 34 q^{33} - 40 q^{37} - 24 q^{43} + 30 q^{45} + 36 q^{51} + 92 q^{57} - 18 q^{63} + 28 q^{67} - 8 q^{73} - 38 q^{75}+ \cdots + 56 q^{97}+O(q^{100}) \) 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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