LMFDB - Embedded newform 420.2.bv.c.317.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [420,2,Mod(53,420)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(420, base_ring=CyclotomicField(12))

chi = DirichletCharacter(H, H._module([0, 6, 9, 8]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("420.53");

S:= CuspForms(chi, 2);

N := Newforms(S);

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

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

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

Embedding invariants

Embedding label			317.6
Character	$\chi$	$=$	420.317
Dual form			420.2.bv.c.53.6

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-0.302208 + 1.70548i) q^{3} +(-1.92665 - 1.13491i) q^{5} +(-0.225753 - 2.63610i) q^{7} +(-2.81734 - 1.03082i) q^{9} +O(q^{10})$	\(q+(-0.302208 + 1.70548i) q^{3} +(-1.92665 - 1.13491i) q^{5} +(-0.225753 - 2.63610i) q^{7} +(-2.81734 - 1.03082i) q^{9} +(1.86244 - 1.07528i) q^{11} +(3.89727 - 3.89727i) q^{13} +(2.51782 - 2.94289i) q^{15} +(-4.10211 - 1.09916i) q^{17} +(-0.631006 - 0.364311i) q^{19} +(4.56405 + 0.411633i) q^{21} +(5.55616 - 1.48877i) q^{23} +(2.42395 + 4.37315i) q^{25} +(2.60947 - 4.49340i) q^{27} -3.95947 q^{29} +(-2.33251 - 4.04003i) q^{31} +(1.27103 + 3.50133i) q^{33} +(-2.55680 + 5.33505i) q^{35} +(2.23265 - 0.598238i) q^{37} +(5.46894 + 7.82450i) q^{39} +4.95231i q^{41} +(3.57830 - 3.57830i) q^{43} +(4.25814 + 5.18346i) q^{45} +(-2.50295 - 9.34112i) q^{47} +(-6.89807 + 1.19022i) q^{49} +(3.11428 - 6.66390i) q^{51} +(1.66848 - 6.22687i) q^{53} +(-4.80863 - 0.0420191i) q^{55} +(0.812022 - 0.966072i) q^{57} +(-4.01205 - 6.94907i) q^{59} +(-6.20108 + 10.7406i) q^{61} +(-2.08132 + 7.65951i) q^{63} +(-11.9317 + 3.08561i) q^{65} +(0.162395 - 0.606066i) q^{67} +(0.859954 + 9.92585i) q^{69} +4.00247i q^{71} +(-10.4387 - 2.79703i) q^{73} +(-8.19087 + 2.81240i) q^{75} +(-3.25501 - 4.66685i) q^{77} +(6.61039 + 3.81651i) q^{79} +(6.87482 + 5.80834i) q^{81} +(-6.56196 - 6.56196i) q^{83} +(6.65587 + 6.77322i) q^{85} +(1.19658 - 6.75282i) q^{87} +(0.656714 - 1.13746i) q^{89} +(-11.1534 - 9.39378i) q^{91} +(7.59510 - 2.75713i) q^{93} +(0.802265 + 1.41804i) q^{95} +(10.0207 + 10.0207i) q^{97} +(-6.35557 + 1.10959i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 48 q - 2 q^{3} - 8 q^{7}+O(q^{10}) $ 48 * q - 2 * q^3 - 8 * q^7	$ 48 q - 2 q^{3} - 8 q^{7} + 32 q^{13} - 16 q^{21} + 16 q^{25} - 32 q^{27} - 64 q^{31} + 34 q^{33} - 40 q^{37} - 24 q^{43} + 30 q^{45} + 36 q^{51} + 92 q^{57} - 18 q^{63} + 28 q^{67} - 8 q^{73} - 38 q^{75} + 20 q^{81} - 56 q^{85} - 24 q^{87} - 24 q^{91} + 6 q^{93} + 56 q^{97}+O(q^{100}) $ 48 * q - 2 * q^3 - 8 * q^7 + 32 * q^13 - 16 * q^21 + 16 * q^25 - 32 * q^27 - 64 * q^31 + 34 * q^33 - 40 * q^37 - 24 * q^43 + 30 * q^45 + 36 * q^51 + 92 * q^57 - 18 * q^63 + 28 * q^67 - 8 * q^73 - 38 * q^75 + 20 * q^81 - 56 * q^85 - 24 * q^87 - 24 * q^91 + 6 * q^93 + 56 * q^97

Display 100 coefficients 10 coefficients

Character values

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

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

Coefficient data

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

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

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

$2$		0			0
$2$		0			0
$3$	−0.302208	+	1.70548i	−0.174480	+	0.984661i
$3$	−0.302208	+	1.70548i	−0.174480	+	0.984661i
$4$		0			0
$5$	−1.92665	−	1.13491i	−0.861623	−	0.507548i
$5$	−1.92665	−	1.13491i	−0.861623	−	0.507548i
$6$		0			0
$7$	−0.225753	−	2.63610i	−0.0853265	−	0.996353i
$7$	−0.225753	−	2.63610i	−0.0853265	−	0.996353i
$8$		0			0
$9$	−2.81734	−	1.03082i	−0.939114	−	0.343607i
$10$		0			0
$11$	1.86244	−	1.07528i	0.561548	−	0.324210i	−0.192218	−	0.981352i	$-0.561568\pi$
$11$	1.86244	−	1.07528i	0.561548	−	0.324210i	0.753767	+	0.657142i	$0.228235\pi$
$12$		0			0
$13$	3.89727	−	3.89727i	1.08091	−	1.08091i	0.0844823	−	0.996425i	$-0.473076\pi$
$13$	3.89727	−	3.89727i	1.08091	−	1.08091i	0.996425	−	0.0844823i	$-0.0269237\pi$
$14$		0			0
$15$	2.51782	−	2.94289i	0.650099	−	0.759850i
$16$		0			0
$17$	−4.10211	−	1.09916i	−0.994907	−	0.266584i	−0.275596	−	0.961273i	$-0.588875\pi$
$17$	−4.10211	−	1.09916i	−0.994907	−	0.266584i	−0.719310	+	0.694689i	$0.755542\pi$
$18$		0			0
$19$	−0.631006	−	0.364311i	−0.144763	−	0.0835788i	0.425869	−	0.904785i	$-0.359968\pi$
$19$	−0.631006	−	0.364311i	−0.144763	−	0.0835788i	−0.570632	+	0.821206i	$0.693302\pi$
$20$		0			0
$21$	4.56405	+	0.411633i	0.995957	+	0.0898257i
$22$		0			0
$23$	5.55616	−	1.48877i	1.15854	−	0.310430i	0.372157	−	0.928170i	$-0.378618\pi$
$23$	5.55616	−	1.48877i	1.15854	−	0.310430i	0.786382	+	0.617740i	$0.211952\pi$
$24$		0			0
$25$	2.42395	+	4.37315i	0.484790	+	0.874631i
$26$		0			0
$27$	2.60947	−	4.49340i	0.502192	−	0.864756i
$28$		0			0
$29$	−3.95947			−0.735256			−0.367628	−	0.929973i	$-0.619830\pi$
$29$	−3.95947			−0.735256			−0.367628	+	0.929973i	$0.619830\pi$
$30$		0			0
$31$	−2.33251	−	4.04003i	−0.418931	−	0.725610i	0.576901	−	0.816814i	$-0.304262\pi$
$31$	−2.33251	−	4.04003i	−0.418931	−	0.725610i	−0.995832	+	0.0912037i	$0.970929\pi$
$32$		0			0
$33$	1.27103	+	3.50133i	0.221258	+	0.609503i
$34$		0			0
$35$	−2.55680	+	5.33505i	−0.432178	+	0.901788i
$36$		0			0
$37$	2.23265	−	0.598238i	0.367046	−	0.0983497i	−0.0705816	−	0.997506i	$-0.522486\pi$
$37$	2.23265	−	0.598238i	0.367046	−	0.0983497i	0.437628	+	0.899156i	$0.355819\pi$
$38$		0			0
$39$	5.46894	+	7.82450i	0.875731	+	1.25292i
$40$		0			0
$41$			4.95231i			0.773420i	0.922201	+	0.386710i	$0.126389\pi$
$41$			4.95231i			0.773420i	−0.922201	+	0.386710i	$0.873611\pi$
$42$		0			0
$43$	3.57830	−	3.57830i	0.545686	−	0.545686i	−0.379504	−	0.925190i	$-0.623905\pi$
$43$	3.57830	−	3.57830i	0.545686	−	0.545686i	0.925190	+	0.379504i	$0.123905\pi$
$44$		0			0
$45$	4.25814	+	5.18346i	0.634765	+	0.772705i
$46$		0			0
$47$	−2.50295	−	9.34112i	−0.365092	−	1.36254i	−0.867296	−	0.497793i	$-0.834144\pi$
$47$	−2.50295	−	9.34112i	−0.365092	−	1.36254i	0.502204	−	0.864749i	$-0.332523\pi$
$48$		0			0
$49$	−6.89807	+	1.19022i	−0.985439	+	0.170031i
$50$		0			0
$51$	3.11428	−	6.66390i	0.436086	−	0.933132i
$52$		0			0
$53$	1.66848	−	6.22687i	0.229184	−	0.855326i	−0.751501	−	0.659732i	$-0.770670\pi$
$53$	1.66848	−	6.22687i	0.229184	−	0.855326i	0.980685	−	0.195594i	$-0.0626635\pi$
$54$		0			0
$55$	−4.80863	−	0.0420191i	−0.648395	−	0.00566586i
$56$		0			0
$57$	0.812022	−	0.966072i	0.107555	−	0.127959i
$58$		0			0
$59$	−4.01205	−	6.94907i	−0.522324	−	0.904692i	−0.999663	−	0.0259723i	$-0.991732\pi$
$59$	−4.01205	−	6.94907i	−0.522324	−	0.904692i	0.477339	−	0.878719i	$-0.341601\pi$
$60$		0			0
$61$	−6.20108	+	10.7406i	−0.793967	+	1.37519i	0.129526	+	0.991576i	$0.458654\pi$
$61$	−6.20108	+	10.7406i	−0.793967	+	1.37519i	−0.923493	+	0.383615i	$0.874679\pi$
$62$		0			0
$63$	−2.08132	+	7.65951i	−0.262222	+	0.965008i
$64$		0			0
$65$	−11.9317	+	3.08561i	−1.47995	+	0.382722i
$66$		0			0
$67$	0.162395	−	0.606066i	0.0198397	−	0.0740427i	−0.955296	−	0.295650i	$-0.904464\pi$
$67$	0.162395	−	0.606066i	0.0198397	−	0.0740427i	0.975136	+	0.221608i	$0.0711304\pi$
$68$		0			0
$69$	0.859954	+	9.92585i	0.103526	+	1.19493i
$70$		0			0
$71$			4.00247i			0.475006i	0.971387	+	0.237503i	$0.0763290\pi$
$71$			4.00247i			0.475006i	−0.971387	+	0.237503i	$0.923671\pi$
$72$		0			0
$73$	−10.4387	−	2.79703i	−1.22175	−	0.327368i	−0.410391	−	0.911910i	$-0.634608\pi$
$73$	−10.4387	−	2.79703i	−1.22175	−	0.327368i	−0.811364	+	0.584542i	$0.801274\pi$
$74$		0			0
$75$	−8.19087	+	2.81240i	−0.945801	+	0.324748i
$76$		0			0
$77$	−3.25501	−	4.66685i	−0.370943	−	0.531837i
$78$		0			0
$79$	6.61039	+	3.81651i	0.743727	+	0.429391i	0.823423	−	0.567428i	$-0.192062\pi$
$79$	6.61039	+	3.81651i	0.743727	+	0.429391i	−0.0796961	+	0.996819i	$0.525395\pi$
$80$		0			0
$81$	6.87482	+	5.80834i	0.763869	+	0.645371i
$82$		0			0
$83$	−6.56196	−	6.56196i	−0.720268	−	0.720268i	0.248392	−	0.968660i	$-0.420098\pi$
$83$	−6.56196	−	6.56196i	−0.720268	−	0.720268i	−0.968660	+	0.248392i	$0.920098\pi$
$84$		0			0
$85$	6.65587	+	6.77322i	0.721930	+	0.734658i
$86$		0			0
$87$	1.19658	−	6.75282i	0.128287	−	0.723978i
$88$		0			0
$89$	0.656714	−	1.13746i	0.0696116	−	0.120571i	−0.829119	−	0.559072i	$-0.811157\pi$
$89$	0.656714	−	1.13746i	0.0696116	−	0.120571i	0.898730	+	0.438502i	$0.144491\pi$
$90$		0			0
$91$	−11.1534	−	9.39378i	−1.16920	−	0.984735i
$92$		0			0
$93$	7.59510	−	2.75713i	0.787575	−	0.285901i
$94$		0			0
$95$	0.802265	+	1.41804i	0.0823107	+	0.145487i
$96$		0			0
$97$	10.0207	+	10.0207i	1.01745	+	1.01745i	0.999845	+	0.0176018i	$0.00560313\pi$
$97$	10.0207	+	10.0207i	1.01745	+	1.01745i	0.0176018	+	0.999845i	$0.494397\pi$
$98$		0			0
$99$	−6.35557	+	1.10959i	−0.638758	+	0.111518i
$100$		0			0
$101$	6.74304	−	3.89310i	0.670958	−	0.387378i	−0.125482	−	0.992096i	$-0.540048\pi$
$101$	6.74304	−	3.89310i	0.670958	−	0.387378i	0.796439	+	0.604718i	$0.206714\pi$
$102$		0			0
$103$	5.15875	+	19.2527i	0.508307	+	1.89703i	0.436731	+	0.899592i	$0.356136\pi$
$103$	5.15875	+	19.2527i	0.508307	+	1.89703i	0.0715756	+	0.997435i	$0.477197\pi$
$104$		0			0
$105$	−8.32615	−	5.97287i	−0.812549	−	0.582892i
$106$		0			0
$107$	−1.79006	−	6.68059i	−0.173052	−	0.645837i	−0.996875	−	0.0789913i	$-0.974830\pi$
$107$	−1.79006	−	6.68059i	−0.173052	−	0.645837i	0.823824	−	0.566846i	$-0.191837\pi$
$108$		0			0
$109$	5.29779	−	3.05868i	0.507436	−	0.292968i	−0.224343	−	0.974510i	$-0.572024\pi$
$109$	5.29779	−	3.05868i	0.507436	−	0.292968i	0.731779	+	0.681542i	$0.238690\pi$
$110$		0			0
$111$	0.345559	+	3.98854i	0.0327990	+	0.378576i
$112$		0			0
$113$	13.8572	+	13.8572i	1.30357	+	1.30357i	0.925967	+	0.377605i	$0.123252\pi$
$113$	13.8572	+	13.8572i	1.30357	+	1.30357i	0.377605	+	0.925967i	$0.376748\pi$
$114$		0			0
$115$	−12.3944	−	3.43742i	−1.15578	−	0.320541i
$116$		0			0
$117$	−14.9973	+	6.96255i	−1.38650	+	0.643688i
$118$		0			0
$119$	−1.97143	+	11.0617i	−0.180720	+	1.01403i
$120$		0			0
$121$	−3.18753	+	5.52097i	−0.289776	+	0.501906i
$122$		0			0
$123$	−8.44607	−	1.49663i	−0.761557	−	0.134946i
$124$		0			0
$125$	0.293050	−	11.1765i	0.0262112	−	0.999656i
$126$		0			0
$127$	4.34338	+	4.34338i	0.385412	+	0.385412i	0.873047	−	0.487635i	$-0.162140\pi$
$127$	4.34338	+	4.34338i	0.385412	+	0.385412i	−0.487635	+	0.873047i	$0.662140\pi$
$128$		0			0
$129$	5.02134	+	7.18412i	0.442104	+	0.632527i
$130$		0			0
$131$	18.6405	+	10.7621i	1.62863	+	0.940289i	0.984503	+	0.175370i	$0.0561121\pi$
$131$	18.6405	+	10.7621i	1.62863	+	0.940289i	0.644126	+	0.764919i	$0.277221\pi$
$132$		0			0
$133$	−0.817911	+	1.74564i	−0.0709219	+	0.151366i
$134$		0			0
$135$	−10.1271	+	5.69569i	−0.871606	+	0.490207i
$136$		0			0
$137$	−15.3138	−	4.10332i	−1.30835	−	0.350570i	−0.463747	−	0.885968i	$-0.653495\pi$
$137$	−15.3138	−	4.10332i	−1.30835	−	0.350570i	−0.844600	+	0.535397i	$0.820162\pi$
$138$		0			0
$139$		−	11.4342i		−	0.969835i	−0.874560	−	0.484917i	$-0.838850\pi$
$139$		−	11.4342i		−	0.969835i	0.874560	−	0.484917i	$-0.161150\pi$
$140$		0			0
$141$	16.6875	−	1.44577i	1.40534	−	0.121756i
$142$		0			0
$143$	3.06778	−	11.4491i	0.256541	−	0.957423i
$144$		0			0
$145$	7.62852	+	4.49366i	0.633514	+	0.373178i
$146$		0			0
$147$	0.0547598	−	12.1242i	0.00451651	−	0.999990i
$148$		0			0
$149$	−4.30668	+	7.45938i	−0.352817	+	0.611096i	−0.986742	−	0.162298i	$-0.948109\pi$
$149$	−4.30668	+	7.45938i	−0.352817	+	0.611096i	0.633925	+	0.773394i	$0.281443\pi$
$150$		0			0
$151$	4.98113	+	8.62757i	0.405358	+	0.702101i	0.994363	−	0.106028i	$-0.0338134\pi$
$151$	4.98113	+	8.62757i	0.405358	+	0.702101i	−0.589005	+	0.808130i	$0.700480\pi$
$152$		0			0
$153$	10.4240	+	7.32523i	0.842730	+	0.592210i
$154$		0			0
$155$	−0.0911482	+	10.4309i	−0.00732120	+	0.837831i
$156$		0			0
$157$	−0.341811	+	1.27566i	−0.0272795	+	0.101808i	−0.978223	−	0.207555i	$-0.933449\pi$
$157$	−0.341811	+	1.27566i	−0.0272795	+	0.101808i	0.950944	+	0.309364i	$0.100116\pi$
$158$		0			0
$159$	10.1156	+	4.72738i	0.802218	+	0.374905i
$160$		0			0
$161$	−5.17886	−	14.3105i	−0.408152	−	1.12783i
$162$		0			0
$163$	−5.77269	−	21.5440i	−0.452152	−	1.68745i	−0.696328	−	0.717724i	$-0.745184\pi$
$163$	−5.77269	−	21.5440i	−0.452152	−	1.68745i	0.244176	−	0.969731i	$-0.421483\pi$
$164$		0			0
$165$	1.52487	−	8.18833i	0.118711	−	0.637461i
$166$		0			0
$167$	−5.00089	+	5.00089i	−0.386980	+	0.386980i	−0.873609	−	0.486629i	$-0.838226\pi$
$167$	−5.00089	+	5.00089i	−0.386980	+	0.386980i	0.486629	+	0.873609i	$0.338226\pi$
$168$		0			0
$169$		−	17.3774i		−	1.33672i
$170$		0			0
$171$	1.40222	+	1.67684i	0.107230	+	0.128231i
$172$		0			0
$173$	2.69465	−	0.722028i	0.204870	−	0.0548948i	−0.154925	−	0.987926i	$-0.549513\pi$
$173$	2.69465	−	0.722028i	0.204870	−	0.0548948i	0.359795	+	0.933032i	$0.382847\pi$
$174$		0			0
$175$	10.9809	−	7.37703i	0.830076	−	0.557651i
$176$		0			0
$177$	13.0640	−	4.74241i	0.981949	−	0.356462i
$178$		0			0
$179$	7.09783	+	12.2938i	0.530517	+	0.918882i	0.999366	+	0.0356039i	$0.0113355\pi$
$179$	7.09783	+	12.2938i	0.530517	+	0.918882i	−0.468849	+	0.883278i	$0.655331\pi$
$180$		0			0
$181$	−10.0280			−0.745372			−0.372686	−	0.927958i	$-0.621563\pi$
$181$	−10.0280			−0.745372			−0.372686	+	0.927958i	$0.621563\pi$
$182$		0			0
$183$	−16.4439	−	13.8217i	−1.21557	−	1.02173i
$184$		0			0
$185$	−4.98049	−	1.38127i	−0.366173	−	0.101553i
$186$		0			0
$187$	−8.82185	+	2.36381i	−0.645117	+	0.172859i
$188$		0			0
$189$	−12.4342	−	5.86443i	−0.904453	−	0.426574i
$190$		0			0
$191$	5.89882	+	3.40569i	0.426824	+	0.246427i	0.697993	−	0.716105i	$-0.254077\pi$
$191$	5.89882	+	3.40569i	0.426824	+	0.246427i	−0.271169	+	0.962532i	$0.587410\pi$
$192$		0			0
$193$	8.23125	+	2.20556i	0.592498	+	0.158759i	0.542595	−	0.839995i	$-0.317442\pi$
$193$	8.23125	+	2.20556i	0.592498	+	0.158759i	0.0499036	+	0.998754i	$0.484109\pi$
$194$		0			0
$195$	−1.65659	−	21.2818i	−0.118631	−	1.52402i
$196$		0			0
$197$	15.8809	−	15.8809i	1.13147	−	1.13147i	0.141537	−	0.989933i	$-0.454796\pi$
$197$	15.8809	−	15.8809i	1.13147	−	1.13147i	0.989933	−	0.141537i	$-0.0452044\pi$
$198$		0			0
$199$	7.68076	−	4.43449i	0.544475	−	0.314353i	−0.202416	−	0.979300i	$-0.564879\pi$
$199$	7.68076	−	4.43449i	0.544475	−	0.314353i	0.746891	+	0.664947i	$0.231546\pi$
$200$		0			0
$201$	0.984557	+	0.460119i	0.0694453	+	0.0324543i
$202$		0			0
$203$	0.893863	+	10.4376i	0.0627369	+	0.732575i
$204$		0			0
$205$	5.62043	−	9.54135i	0.392548	−	0.666397i
$206$		0			0
$207$	−17.1882	−	1.53303i	−1.19467	−	0.106553i
$208$		0			0
$209$	−1.56695			−0.108388
$210$		0			0
$211$	14.4567			0.995243			0.497621	−	0.867394i	$-0.334207\pi$
$211$	14.4567			0.995243			0.497621	+	0.867394i	$0.334207\pi$
$212$		0			0
$213$	−6.82615	−	1.20958i	−0.467720	−	0.0828789i
$214$		0			0
$215$	−10.9552	+	2.83307i	−0.747138	+	0.193214i
$216$		0			0
$217$	−10.1234	+	7.06079i	−0.687218	+	0.479317i
$218$		0			0
$219$	7.92494	−	16.9577i	0.535518	−	1.14589i
$220$		0			0
$221$	−20.2707	+	11.7033i	−1.36355	+	0.787249i
$222$		0			0
$223$	17.3260	−	17.3260i	1.16023	−	1.16023i	0.175810	−	0.984424i	$-0.443746\pi$
$223$	17.3260	−	17.3260i	1.16023	−	1.16023i	0.984424	−	0.175810i	$-0.0562545\pi$
$224$		0			0
$225$	−2.32115	−	14.8193i	−0.154744	−	0.987955i
$226$		0			0
$227$	21.6202	+	5.79310i	1.43498	+	0.384502i	0.890773	−	0.454449i	$-0.150164\pi$
$227$	21.6202	+	5.79310i	1.43498	+	0.384502i	0.544207	+	0.838951i	$0.316830\pi$
$228$		0			0
$229$	−18.9575	−	10.9451i	−1.25274	−	0.723272i	−0.281091	−	0.959681i	$-0.590696\pi$
$229$	−18.9575	−	10.9451i	−1.25274	−	0.723272i	−0.971654	+	0.236409i	$0.924030\pi$
$230$		0			0
$231$	8.94291	−	4.14100i	0.588401	−	0.272458i
$232$		0			0
$233$	−22.6220	+	6.06156i	−1.48202	+	0.397106i	−0.907034	−	0.421058i	$-0.861659\pi$
$233$	−22.6220	+	6.06156i	−1.48202	+	0.397106i	−0.574986	+	0.818163i	$0.694992\pi$
$234$		0			0
$235$	−5.77905	+	20.8377i	−0.376984	+	1.35930i
$236$		0			0
$237$	−8.50670	+	10.1205i	−0.552569	+	0.657398i
$238$		0			0
$239$	19.7365			1.27665			0.638323	−	0.769769i	$-0.279628\pi$
$239$	19.7365			1.27665			0.638323	+	0.769769i	$0.279628\pi$
$240$		0			0
$241$	−6.49458	−	11.2489i	−0.418353	−	0.724608i	0.577421	−	0.816446i	$-0.304059\pi$
$241$	−6.49458	−	11.2489i	−0.418353	−	0.724608i	−0.995774	+	0.0918384i	$0.970726\pi$
$242$		0			0
$243$	−11.9837	+	9.96956i	−0.768752	+	0.639547i
$244$		0			0
$245$	14.6409	+	5.53558i	0.935376	+	0.353655i
$246$		0			0
$247$	−3.87902	+	1.03938i	−0.246816	+	0.0661341i
$248$		0			0
$249$	13.1744	−	9.20823i	0.834892	−	0.583548i
$250$		0			0
$251$		−	1.84736i		−	0.116604i	−0.998299	−	0.0583022i	$-0.981431\pi$
$251$		−	1.84736i		−	0.116604i	0.998299	−	0.0583022i	$-0.0185687\pi$
$252$		0			0
$253$	8.74719	−	8.74719i	0.549931	−	0.549931i
$254$		0			0
$255$	−13.5631	+	9.30455i	−0.849352	+	0.582674i
$256$		0			0
$257$	6.42263	+	23.9696i	0.400633	+	1.49518i	0.811970	+	0.583699i	$0.198395\pi$
$257$	6.42263	+	23.9696i	0.400633	+	1.49518i	−0.411337	+	0.911483i	$0.634938\pi$
$258$		0			0
$259$	−2.08104	−	5.75045i	−0.129310	−	0.357316i
$260$		0			0
$261$	11.1552	+	4.08151i	0.690489	+	0.252639i
$262$		0			0
$263$	3.10469	−	11.5869i	0.191444	−	0.714477i	−0.801715	−	0.597706i	$-0.796079\pi$
$263$	3.10469	−	11.5869i	0.191444	−	0.714477i	0.993159	−	0.116771i	$-0.0372544\pi$
$264$		0			0
$265$	−10.2815	+	10.1034i	−0.631589	+	0.620647i
$266$		0			0
$267$	1.74146	+	1.46376i	0.106575	+	0.0895809i
$268$		0			0
$269$	−1.64773	−	2.85395i	−0.100464	−	0.174008i	0.811412	−	0.584474i	$-0.198699\pi$
$269$	−1.64773	−	2.85395i	−0.100464	−	0.174008i	−0.911876	+	0.410466i	$0.865366\pi$
$270$		0			0
$271$	−4.06164	+	7.03497i	−0.246727	+	0.427344i	−0.962616	−	0.270870i	$-0.912689\pi$
$271$	−4.06164	+	7.03497i	−0.246727	+	0.427344i	0.715889	+	0.698215i	$0.246022\pi$
$272$		0			0
$273$	19.3916	−	16.1831i	1.17363	−	0.979444i
$274$		0			0
$275$	9.21685	+	5.53833i	0.555797	+	0.333974i
$276$		0			0
$277$	7.34355	−	27.4065i	0.441231	−	1.64670i	−0.284467	−	0.958686i	$-0.591817\pi$
$277$	7.34355	−	27.4065i	0.441231	−	1.64670i	0.725699	−	0.688012i	$-0.241517\pi$
$278$		0			0
$279$	2.40694	+	13.7865i	0.144100	+	0.825378i
$280$		0			0
$281$		−	14.5343i		−	0.867042i	−0.901144	−	0.433521i	$-0.857271\pi$
$281$		−	14.5343i		−	0.867042i	0.901144	−	0.433521i	$-0.142729\pi$
$282$		0			0
$283$	2.81028	+	0.753013i	0.167054	+	0.0447620i	0.341376	−	0.939927i	$-0.389107\pi$
$283$	2.81028	+	0.753013i	0.167054	+	0.0447620i	−0.174323	+	0.984689i	$0.555774\pi$
$284$		0			0
$285$	−2.66089	+	0.939707i	−0.157617	+	0.0556635i
$286$		0			0
$287$	13.0548	−	1.11800i	0.770600	−	0.0659933i
$288$		0			0
$289$	0.896693	+	0.517706i	0.0527466	+	0.0304533i
$290$		0			0
$291$	−20.1184	+	14.0618i	−1.17936	+	0.824316i
$292$		0			0
$293$	11.9716	+	11.9716i	0.699387	+	0.699387i	0.964278	−	0.264891i	$-0.0853361\pi$
$293$	11.9716	+	11.9716i	0.699387	+	0.699387i	−0.264891	+	0.964278i	$0.585336\pi$
$294$		0			0
$295$	−0.156780	+	17.9417i	−0.00912808	+	1.04461i
$296$		0			0
$297$	0.0283094	−	11.1746i	0.00164268	−	0.648418i
$298$		0			0
$299$	15.8517	−	27.4560i	0.916728	−	1.58782i
$300$		0			0
$301$	−10.2406	−	8.62496i	−0.590257	−	0.497134i
$302$		0			0
$303$	4.60181	+	12.6767i	0.264367	+	0.728255i
$304$		0			0
$305$	24.1369	−	13.6556i	1.38208	−	0.781920i
$306$		0			0
$307$	5.89474	+	5.89474i	0.336431	+	0.336431i	0.855022	−	0.518591i	$-0.173543\pi$
$307$	5.89474	+	5.89474i	0.336431	+	0.336431i	−0.518591	+	0.855022i	$0.673543\pi$
$308$		0			0
$309$	−34.3942	+	2.97984i	−1.95662	+	0.169517i
$310$		0			0
$311$	0.794450	−	0.458676i	0.0450491	−	0.0260091i	−0.477306	−	0.878737i	$-0.658387\pi$
$311$	0.794450	−	0.458676i	0.0450491	−	0.0260091i	0.522355	+	0.852728i	$0.325053\pi$
$312$		0			0
$313$	0.320387	+	1.19570i	0.0181094	+	0.0675850i	0.974389	−	0.224868i	$-0.0721951\pi$
$313$	0.320387	+	1.19570i	0.0181094	+	0.0675850i	−0.956280	+	0.292453i	$0.905528\pi$
$314$		0			0
$315$	12.7029	−	12.3951i	0.715725	−	0.698383i
$316$		0			0
$317$	4.20581	+	15.6963i	0.236222	+	0.881591i	0.977594	+	0.210498i	$0.0675084\pi$
$317$	4.20581	+	15.6963i	0.236222	+	0.881591i	−0.741373	+	0.671093i	$0.765825\pi$
$318$		0			0
$319$	−7.37430	+	4.25756i	−0.412882	+	0.238377i
$320$		0			0
$321$	11.9346	−	1.03399i	0.666125	−	0.0577116i
$322$		0			0
$323$	2.18802	+	2.18802i	0.121745	+	0.121745i
$324$		0			0
$325$	26.4901	+	7.59657i	1.46941	+	0.421382i
$326$		0			0
$327$	3.61549	+	9.95964i	0.199937	+	0.550769i
$328$		0			0
$329$	−24.0591	+	8.70680i	−1.32642	+	0.480022i
$330$		0			0
$331$	−5.43665	+	9.41656i	−0.298825	+	0.517581i	−0.975868	−	0.218364i	$-0.929928\pi$
$331$	−5.43665	+	9.41656i	−0.298825	+	0.517581i	0.677042	+	0.735944i	$0.263261\pi$
$332$		0			0
$333$	−6.90682	−	0.616025i	−0.378491	−	0.0337579i
$334$		0			0
$335$	−1.00071	+	0.983371i	−0.0546746	+	0.0537273i
$336$		0			0
$337$	−2.89002	−	2.89002i	−0.157430	−	0.157430i	0.623997	−	0.781427i	$-0.285508\pi$
$337$	−2.89002	−	2.89002i	−0.157430	−	0.157430i	−0.781427	+	0.623997i	$0.785508\pi$
$338$		0			0
$339$	−27.8209	+	19.4454i	−1.51102	+	1.05613i
$340$		0			0
$341$	−8.68835	−	5.01622i	−0.470500	−	0.271643i
$342$		0			0
$343$	4.69479	+	17.9153i	0.253495	+	0.967337i
$344$		0			0
$345$	9.60814	−	20.0996i	0.517285	−	1.08213i
$346$		0			0
$347$	1.08593	+	0.290975i	0.0582960	+	0.0156204i	0.287849	−	0.957676i	$-0.407060\pi$
$347$	1.08593	+	0.290975i	0.0582960	+	0.0156204i	−0.229553	+	0.973296i	$0.573726\pi$
$348$		0			0
$349$			33.7351i			1.80580i	0.429852	+	0.902899i	$0.358566\pi$
$349$			33.7351i			1.80580i	−0.429852	+	0.902899i	$0.641434\pi$
$350$		0			0
$351$	−7.34220	−	27.6818i	−0.391898	−	1.47754i
$352$		0			0
$353$	−1.77829	+	6.63668i	−0.0946489	+	0.353235i	−0.996966	−	0.0778384i	$-0.975198\pi$
$353$	−1.77829	+	6.63668i	−0.0946489	+	0.353235i	0.902317	+	0.431073i	$0.141865\pi$
$354$		0			0
$355$	4.54245	−	7.71136i	0.241088	−	0.409276i
$356$		0			0
$357$	−18.2698	−	6.70517i	−0.966939	−	0.354875i
$358$		0			0
$359$	−1.02137	+	1.76907i	−0.0539060	+	0.0933680i	−0.891719	−	0.452589i	$-0.850500\pi$
$359$	−1.02137	+	1.76907i	−0.0539060	+	0.0933680i	0.837813	+	0.545957i	$0.183834\pi$
$360$		0			0
$361$	−9.23455	−	15.9947i	−0.486029	−	0.841827i
$362$		0			0
$363$	−8.45262	−	7.10476i	−0.443647	−	0.372903i
$364$		0			0
$365$	16.9373	+	17.2359i	0.886537	+	0.902167i
$366$		0			0
$367$	−1.94879	+	7.27299i	−0.101726	+	0.379647i	−0.997953	−	0.0639485i	$-0.979631\pi$
$367$	−1.94879	+	7.27299i	−0.101726	+	0.379647i	0.896227	+	0.443596i	$0.146297\pi$
$368$		0			0
$369$	5.10494	−	13.9523i	0.265752	−	0.726329i
$370$		0			0
$371$	−16.7913	−	2.99256i	−0.871762	−	0.155366i
$372$		0			0
$373$	−0.374011	−	1.39583i	−0.0193655	−	0.0722732i	0.955567	−	0.294774i	$-0.0952445\pi$
$373$	−0.374011	−	1.39583i	−0.0193655	−	0.0722732i	−0.974932	+	0.222501i	$0.928578\pi$
$374$		0			0
$375$	18.9728	+	3.87742i	0.979749	+	0.200229i
$376$		0			0
$377$	−15.4311	+	15.4311i	−0.794744	+	0.794744i
$378$		0			0
$379$			1.83298i			0.0941538i	0.998891	+	0.0470769i	$0.0149906\pi$
$379$			1.83298i			0.0941538i	−0.998891	+	0.0470769i	$0.985009\pi$
$380$		0			0
$381$	−8.72015	+	6.09495i	−0.446747	+	0.312254i
$382$		0			0
$383$	−16.8876	+	4.52502i	−0.862916	+	0.231218i	−0.663022	−	0.748600i	$-0.730726\pi$
$383$	−16.8876	+	4.52502i	−0.862916	+	0.231218i	−0.199894	+	0.979817i	$0.564060\pi$
$384$		0			0
$385$	0.974795	+	12.6855i	0.0496801	+	0.646514i
$386$		0			0
$387$	−13.7699	+	6.39271i	−0.699962	+	0.324960i
$388$		0			0
$389$	11.9965	+	20.7786i	0.608249	+	1.05352i	0.991529	+	0.129886i	$0.0414610\pi$
$389$	11.9965	+	20.7786i	0.608249	+	1.05352i	−0.383280	+	0.923632i	$0.625206\pi$
$390$		0			0
$391$	−24.4283			−1.23539
$392$		0			0
$393$	−23.9879	+	28.5387i	−1.21003	+	1.43959i
$394$		0			0
$395$	−8.40449	−	14.8553i	−0.422876	−	0.747450i
$396$		0			0
$397$	22.9338	−	6.14509i	1.15101	−	0.308413i	0.367643	−	0.929967i	$-0.380165\pi$
$397$	22.9338	−	6.14509i	1.15101	−	0.308413i	0.783372	+	0.621554i	$0.213498\pi$
$398$		0			0
$399$	−2.72998	−	1.92248i	−0.136670	−	0.0962443i
$400$		0			0
$401$	−2.79781	−	1.61532i	−0.139716	−	0.0806650i	0.428513	−	0.903536i	$-0.359038\pi$
$401$	−2.79781	−	1.61532i	−0.139716	−	0.0806650i	−0.568229	+	0.822871i	$0.692371\pi$
$402$		0			0
$403$	−24.8355	−	6.65465i	−1.23714	−	0.331492i
$404$		0			0
$405$	−6.65340	−	18.9930i	−0.330610	−	0.943767i
$406$		0			0
$407$	3.51492	−	3.51492i	0.174228	−	0.174228i
$408$		0			0
$409$	4.14956	−	2.39575i	0.205183	−	0.118462i	−0.393888	−	0.919158i	$-0.628870\pi$
$409$	4.14956	−	2.39575i	0.205183	−	0.118462i	0.599071	+	0.800696i	$0.295537\pi$
$410$		0			0
$411$	11.6261	−	24.8774i	0.573473	−	1.22711i
$412$		0			0
$413$	−17.4127	+	12.1449i	−0.856824	+	0.597613i
$414$		0			0
$415$	5.19534	+	20.0898i	0.255029	+	0.986171i
$416$		0			0
$417$	19.5008	+	3.45550i	0.954958	+	0.169217i
$418$		0			0
$419$	20.3805			0.995651			0.497826	−	0.867277i	$-0.334132\pi$
$419$	20.3805			0.995651			0.497826	+	0.867277i	$0.334132\pi$
$420$		0			0
$421$	−33.9866			−1.65641			−0.828203	−	0.560428i	$-0.810637\pi$
$421$	−33.9866			−1.65641			−0.828203	+	0.560428i	$0.810637\pi$
$422$		0			0
$423$	−2.57736	+	28.8972i	−0.125316	+	1.40503i
$424$		0			0
$425$	−5.13651	−	20.6034i	−0.249158	−	0.999413i
$426$		0			0
$427$	29.7132	+	13.9220i	1.43792	+	0.673731i
$428$		0			0
$429$	18.5991	+	8.69205i	0.897975	+	0.419656i
$430$		0			0
$431$	1.88165	−	1.08637i	0.0906357	−	0.0523285i	−0.453997	−	0.891003i	$-0.650002\pi$
$431$	1.88165	−	1.08637i	0.0906357	−	0.0523285i	0.544633	+	0.838675i	$0.316669\pi$
$432$		0			0
$433$	1.81661	−	1.81661i	0.0873009	−	0.0873009i	−0.662108	−	0.749409i	$-0.730338\pi$
$433$	1.81661	−	1.81661i	0.0873009	−	0.0873009i	0.749409	+	0.662108i	$0.230338\pi$
$434$		0			0
$435$	−9.96925	+	11.6523i	−0.477989	+	0.558684i
$436$		0			0
$437$	−4.04834	−	1.08475i	−0.193659	−	0.0518907i
$438$		0			0
$439$	8.95736	+	5.17153i	0.427511	+	0.246824i	0.698286	−	0.715819i	$-0.253946\pi$
$439$	8.95736	+	5.17153i	0.427511	+	0.246824i	−0.270775	+	0.962643i	$0.587280\pi$
$440$		0			0
$441$	20.6611	+	3.75743i	0.983863	+	0.178925i
$442$		0			0
$443$	−19.6578	+	5.26730i	−0.933971	+	0.250257i	−0.693547	−	0.720411i	$-0.743953\pi$
$443$	−19.6578	+	5.26730i	−0.933971	+	0.250257i	−0.240424	+	0.970668i	$0.577286\pi$
$444$		0			0
$445$	−2.55618	+	1.44618i	−0.121174	+	0.0685554i
$446$		0			0
$447$	−11.4203	−	9.59925i	−0.540163	−	0.454029i
$448$		0			0
$449$	−14.6156			−0.689754			−0.344877	−	0.938648i	$-0.612079\pi$
$449$	−14.6156			−0.689754			−0.344877	+	0.938648i	$0.612079\pi$
$450$		0			0
$451$	5.32513	+	9.22340i	0.250751	+	0.434313i
$452$		0			0
$453$	−16.2195	+	5.88791i	−0.762058	+	0.276638i
$454$		0			0
$455$	10.8276	+	30.7566i	0.507605	+	1.44189i
$456$		0			0
$457$	6.40237	−	1.71551i	0.299490	−	0.0802481i	−0.105945	−	0.994372i	$-0.533787\pi$
$457$	6.40237	−	1.71551i	0.299490	−	0.0802481i	0.405435	+	0.914124i	$0.367120\pi$
$458$		0			0
$459$	−15.6433	+	15.5642i	−0.730165	+	0.726475i
$460$		0			0
$461$		−	29.3424i		−	1.36661i	−0.730132	−	0.683306i	$-0.760541\pi$
$461$		−	29.3424i		−	1.36661i	0.730132	−	0.683306i	$-0.239459\pi$
$462$		0			0
$463$	−3.11379	+	3.11379i	−0.144710	+	0.144710i	−0.775750	−	0.631040i	$-0.782628\pi$
$463$	−3.11379	+	3.11379i	−0.144710	+	0.144710i	0.631040	+	0.775750i	$0.282628\pi$
$464$		0			0
$465$	−17.7622	−	3.30775i	−0.823702	−	0.153393i
$466$		0			0
$467$	6.99718	+	26.1138i	0.323791	+	1.20840i	0.915522	+	0.402268i	$0.131778\pi$
$467$	6.99718	+	26.1138i	0.323791	+	1.20840i	−0.591731	+	0.806135i	$0.701555\pi$
$468$		0			0
$469$	−1.63431	−	0.291268i	−0.0754655	−	0.0134495i
$470$		0			0
$471$	−2.07231	−	0.968466i	−0.0954871	−	0.0446246i
$472$		0			0
$473$	2.81670	−	10.5121i	0.129512	−	0.483346i
$474$		0			0
$475$	0.0636642	−	3.64256i	0.00292112	−	0.167132i
$476$		0			0
$477$	−11.1195	+	15.8233i	−0.509125	+	0.724499i
$478$		0			0
$479$	−18.8746	−	32.6918i	−0.862404	−	1.49373i	−0.869602	−	0.493753i	$-0.835625\pi$
$479$	−18.8746	−	32.6918i	−0.862404	−	1.49373i	0.00719862	−	0.999974i	$-0.497709\pi$
$480$		0			0
$481$	6.36975	−	11.0327i	0.290436	−	0.503050i
$482$		0			0
$483$	25.9714	−	4.50771i	1.18174	−	0.205108i
$484$		0			0
$485$	−7.93374	−	30.6790i	−0.360253	−	1.39306i
$486$		0			0
$487$	−7.63106	+	28.4795i	−0.345796	+	1.29053i	0.545883	+	0.837861i	$0.316194\pi$
$487$	−7.63106	+	28.4795i	−0.345796	+	1.29053i	−0.891679	+	0.452668i	$0.850472\pi$
$488$		0			0
$489$	38.4874	−	3.33447i	1.74046	−	0.150790i
$490$		0			0
$491$			31.9025i			1.43974i	0.694110	+	0.719869i	$0.255798\pi$
$491$			31.9025i			1.43974i	−0.694110	+	0.719869i	$0.744202\pi$
$492$		0			0
$493$	16.2422	+	4.35208i	0.731511	+	0.196008i
$494$		0			0
$495$	13.5042	+	5.07521i	0.606970	+	0.228114i
$496$		0			0
$497$	10.5509	−	0.903569i	0.473274	−	0.0405306i
$498$		0			0
$499$	−13.9880	−	8.07598i	−0.626189	−	0.361531i	0.153086	−	0.988213i	$-0.451079\pi$
$499$	−13.9880	−	8.07598i	−0.626189	−	0.361531i	−0.779275	+	0.626682i	$0.784412\pi$
$500$		0			0
$501$	−7.01762	−	10.0402i	−0.313524	−	0.448565i
$502$		0			0
$503$	−25.0418	−	25.0418i	−1.11656	−	1.11656i	−0.992243	−	0.124316i	$-0.960326\pi$
$503$	−25.0418	−	25.0418i	−1.11656	−	1.11656i	−0.124316	−	0.992243i	$-0.539674\pi$
$504$		0			0
$505$	−17.4098	−	0.152132i	−0.774726	−	0.00676977i
$506$		0			0
$507$	29.6368	+	5.25158i	1.31622	+	0.233231i
$508$		0			0
$509$	21.6658	−	37.5262i	0.960319	−	1.66332i	0.238621	−	0.971113i	$-0.423305\pi$
$509$	21.6658	−	37.5262i	0.960319	−	1.66332i	0.721698	−	0.692208i	$-0.243362\pi$
$510$		0			0
$511$	−5.01671	+	28.1488i	−0.221926	+	1.24523i
$512$		0			0
$513$	−3.28359	+	1.88471i	−0.144974	+	0.0832118i
$514$		0			0
$515$	11.9111	−	42.9480i	0.524864	−	1.89251i
$516$		0			0
$517$	−14.7059	−	14.7059i	−0.646767	−	0.646767i
$518$		0			0
$519$	0.417063	+	4.81387i	0.0183071	+	0.211306i
$520$		0			0
$521$	21.1824	−	12.2297i	0.928018	−	0.535791i	0.0418336	−	0.999125i	$-0.486680\pi$
$521$	21.1824	−	12.2297i	0.928018	−	0.535791i	0.886184	+	0.463333i	$0.153347\pi$
$522$		0			0
$523$	−7.55434	−	28.1932i	−0.330328	−	1.23280i	−0.908846	−	0.417132i	$-0.863035\pi$
$523$	−7.55434	−	28.1932i	−0.330328	−	1.23280i	0.578518	−	0.815670i	$-0.303631\pi$
$524$		0			0
$525$	9.26289	+	20.9571i	0.404266	+	0.914642i
$526$		0			0
$527$	5.12759	+	19.1364i	0.223361	+	0.833595i
$528$		0			0
$529$	8.73588	−	5.04366i	0.379821	−	0.219290i
$530$		0			0
$531$	4.14006	+	23.7136i	0.179663	+	1.02908i
$532$		0			0
$533$	19.3005	+	19.3005i	0.835996	+	0.835996i
$534$		0			0
$535$	−4.13307	+	14.9027i	−0.178688	+	0.644301i
$536$		0			0
$537$	−23.1119	+	8.38994i	−0.997352	+	0.362053i
$538$		0			0
$539$	−11.5675	+	9.63409i	−0.498246	+	0.414970i
$540$		0			0
$541$	4.55156	−	7.88354i	0.195687	−	0.338940i	−0.751439	−	0.659803i	$-0.770640\pi$
$541$	4.55156	−	7.88354i	0.195687	−	0.338940i	0.947125	+	0.320863i	$0.103973\pi$
$542$		0			0
$543$	3.03053	−	17.1025i	0.130052	−	0.733938i
$544$		0			0
$545$	−13.6783	−	0.119525i	−0.585914	−	0.00511988i
$546$		0			0
$547$	−17.5854	−	17.5854i	−0.751900	−	0.751900i	0.222934	−	0.974834i	$-0.428437\pi$
$547$	−17.5854	−	17.5854i	−0.751900	−	0.751900i	−0.974834	+	0.222934i	$0.928437\pi$
$548$		0			0
$549$	28.5422	−	23.8677i	1.21815	−	1.01865i
$550$		0			0
$551$	2.49845	+	1.44248i	0.106438	+	0.0614518i
$552$		0			0
$553$	8.56839	−	18.2872i	0.364365	−	0.777653i
$554$		0			0
$555$	3.86088	−	8.07670i	0.163885	−	0.342837i
$556$		0			0
$557$	−10.3211	−	2.76552i	−0.437318	−	0.117179i	0.0334413	−	0.999441i	$-0.489353\pi$
$557$	−10.3211	−	2.76552i	−0.437318	−	0.117179i	−0.470759	+	0.882262i	$0.656020\pi$
$558$		0			0
$559$		−	27.8912i		−	1.17967i
$560$		0			0
$561$	−1.36540	−	15.7599i	−0.0576473	−	0.665382i
$562$		0			0
$563$	0.963101	−	3.59434i	0.0405898	−	0.151483i	−0.942657	−	0.333764i	$-0.891681\pi$
$563$	0.963101	−	3.59434i	0.0405898	−	0.151483i	0.983247	+	0.182280i	$0.0583479\pi$
$564$		0			0
$565$	−10.9712	−	42.4245i	−0.461562	−	1.78481i
$566$		0			0
$567$	13.7594	−	19.4340i	0.577840	−	0.816150i
$568$		0			0
$569$	1.12664	−	1.95139i	0.0472311	−	0.0818066i	−0.841443	−	0.540345i	$-0.818294\pi$
$569$	1.12664	−	1.95139i	0.0472311	−	0.0818066i	0.888674	+	0.458539i	$0.151627\pi$
$570$		0			0
$571$	2.03062	+	3.51713i	0.0849787	+	0.147187i	0.905382	−	0.424598i	$-0.139584\pi$
$571$	2.03062	+	3.51713i	0.0849787	+	0.147187i	−0.820403	+	0.571785i	$0.806251\pi$
$572$		0			0
$573$	−7.59101	+	9.03111i	−0.317119	+	0.377280i
$574$		0			0
$575$	19.9785	+	20.6892i	0.833159	+	0.862801i
$576$		0			0
$577$	−4.34025	+	16.1980i	−0.180687	+	0.674332i	0.814826	+	0.579706i	$0.196832\pi$
$577$	−4.34025	+	16.1980i	−0.180687	+	0.674332i	−0.995513	+	0.0946268i	$0.969834\pi$
$578$		0			0
$579$	−6.24909	+	13.3717i	−0.259703	+	0.555710i
$580$		0			0
$581$	−15.8166	+	18.7794i	−0.656183	+	0.779099i
$582$		0			0
$583$	−3.58818	−	13.3913i	−0.148607	−	0.554610i
$584$		0			0
$585$	36.7964	+	3.60625i	1.52134	+	0.149100i
$586$		0			0
$587$	11.2728	−	11.2728i	0.465279	−	0.465279i	−0.435102	−	0.900381i	$-0.643288\pi$
$587$	11.2728	−	11.2728i	0.465279	−	0.465279i	0.900381	+	0.435102i	$0.143288\pi$
$588$		0			0
$589$			3.39904i			0.140055i
$590$		0			0
$591$	22.2853	+	31.8840i	0.916696	+	1.31153i
$592$		0			0
$593$	15.1364	−	4.05578i	0.621577	−	0.166551i	0.0657322	−	0.997837i	$-0.479062\pi$
$593$	15.1364	−	4.05578i	0.621577	−	0.166551i	0.555844	+	0.831286i	$0.312395\pi$
$594$		0			0
$595$	16.3523	−	19.0746i	0.670379	−	0.781983i
$596$		0			0
$597$	5.24176	+	14.4395i	0.214531	+	0.590971i
$598$		0			0
$599$	16.5463	+	28.6589i	0.676062	+	1.17097i	0.976157	+	0.217064i	$0.0696480\pi$
$599$	16.5463	+	28.6589i	0.676062	+	1.17097i	−0.300096	+	0.953909i	$0.597019\pi$
$600$		0			0
$601$	25.2056			1.02816			0.514079	−	0.857743i	$-0.328134\pi$
$601$	25.2056			1.02816			0.514079	+	0.857743i	$0.328134\pi$
$602$		0			0
$603$	−1.08227	+	1.54009i	−0.0440733	+	0.0627175i
$604$		0			0
$605$	12.4071	−	7.01940i	0.504419	−	0.285379i
$606$		0			0
$607$	33.2974	−	8.92202i	1.35150	−	0.362134i	0.490812	−	0.871265i	$-0.336700\pi$
$607$	33.2974	−	8.92202i	1.35150	−	0.362134i	0.860689	+	0.509132i	$0.170033\pi$
$608$		0			0
$609$	−18.0712	−	1.62985i	−0.732284	−	0.0660449i
$610$		0			0
$611$	−46.1595	−	26.6502i	−1.86741	−	1.07815i
$612$		0			0
$613$	19.2254	+	5.15144i	0.776509	+	0.208065i	0.625245	−	0.780429i	$-0.284999\pi$
$613$	19.2254	+	5.15144i	0.776509	+	0.208065i	0.151264	+	0.988493i	$0.451666\pi$
$614$		0			0
$615$	14.5741	+	12.4690i	0.587683	+	0.502799i
$616$		0			0
$617$	−12.3161	+	12.3161i	−0.495829	+	0.495829i	−0.910137	−	0.414308i	$-0.864024\pi$
$617$	−12.3161	+	12.3161i	−0.495829	+	0.495829i	0.414308	+	0.910137i	$0.364024\pi$
$618$		0			0
$619$	23.0037	−	13.2812i	0.924597	−	0.533816i	0.0394980	−	0.999220i	$-0.487424\pi$
$619$	23.0037	−	13.2812i	0.924597	−	0.533816i	0.885099	+	0.465404i	$0.154091\pi$
$620$		0			0
$621$	7.80898	−	28.8510i	0.313364	−	1.15775i
$622$		0			0
$623$	−3.14672	−	1.47438i	−0.126071	−	0.0590698i
$624$		0			0
$625$	−13.2489	+	21.2006i	−0.529958	+	0.848024i
$626$		0			0
$627$	0.473545	−	2.67241i	0.0189116	−	0.106726i
$628$		0			0
$629$	−9.81614			−0.391395
$630$		0			0
$631$	8.52290			0.339291			0.169646	−	0.985505i	$-0.445738\pi$
$631$	8.52290			0.339291			0.169646	+	0.985505i	$0.445738\pi$
$632$		0			0
$633$	−4.36894	+	24.6557i	−0.173650	+	0.979977i
$634$		0			0
$635$	−3.43881	−	13.2975i	−0.136465	−	0.527696i
$636$		0			0
$637$	−22.2450	+	31.5222i	−0.881381	+	1.24896i
$638$		0			0
$639$	4.12583	−	11.2763i	0.163215	−	0.446085i
$640$		0			0
$641$	−28.8842	+	16.6763i	−1.14086	+	0.658674i	−0.946643	−	0.322286i	$-0.895549\pi$
$641$	−28.8842	+	16.6763i	−1.14086	+	0.658674i	−0.194214	+	0.980959i	$0.562216\pi$
$642$		0			0
$643$	−6.59026	+	6.59026i	−0.259895	+	0.259895i	−0.825011	−	0.565117i	$-0.808831\pi$
$643$	−6.59026	+	6.59026i	−0.259895	+	0.259895i	0.565117	+	0.825011i	$0.308831\pi$
$644$		0			0
$645$	−1.52101	−	19.5401i	−0.0598897	−	0.769389i
$646$		0			0
$647$	−24.4118	−	6.54113i	−0.959729	−	0.257159i	−0.255243	−	0.966877i	$-0.582156\pi$
$647$	−24.4118	−	6.54113i	−0.959729	−	0.257159i	−0.704485	+	0.709718i	$0.748822\pi$
$648$		0			0
$649$	−14.9444	−	8.62817i	−0.586620	−	0.338685i
$650$		0			0
$651$	−8.98269	−	19.3990i	−0.352059	−	0.760308i
$652$		0			0
$653$	12.7291	−	3.41075i	0.498129	−	0.133473i	−0.00100275	−	0.999999i	$-0.500319\pi$
$653$	12.7291	−	3.41075i	0.498129	−	0.133473i	0.499131	+	0.866526i	$0.333653\pi$
$654$		0			0
$655$	−23.6997	−	41.8901i	−0.926022	−	1.63678i
$656$		0			0
$657$	26.5261	+	18.6406i	1.03488	+	0.727239i
$658$		0			0
$659$	−34.5112			−1.34437			−0.672183	−	0.740385i	$-0.734643\pi$
$659$	−34.5112			−1.34437			−0.672183	+	0.740385i	$0.734643\pi$
$660$		0			0
$661$	5.75162	+	9.96211i	0.223712	+	0.387481i	0.955932	−	0.293587i	$-0.0948491\pi$
$661$	5.75162	+	9.96211i	0.223712	+	0.387481i	−0.732220	+	0.681068i	$0.761516\pi$
$662$		0			0
$663$	−13.8338	−	38.1082i	−0.537260	−	1.48000i
$664$		0			0
$665$	3.55698	−	2.43498i	0.137934	−	0.0944244i
$666$		0			0
$667$	−21.9995	+	5.89474i	−0.851823	+	0.228245i
$668$		0			0
$669$	24.3131	+	34.7852i	0.940000	+	1.34487i
$670$		0			0
$671$			26.6717i			1.02965i
$672$		0			0
$673$	2.07846	−	2.07846i	0.0801187	−	0.0801187i	−0.665912	−	0.746030i	$-0.731957\pi$
$673$	2.07846	−	2.07846i	0.0801187	−	0.0801187i	0.746030	+	0.665912i	$0.231957\pi$
$674$		0			0
$675$	25.9756	+	0.519827i	0.999800	+	0.0200081i
$676$		0			0
$677$	4.59302	+	17.1414i	0.176524	+	0.658797i	0.996287	+	0.0860941i	$0.0274386\pi$
$677$	4.59302	+	17.1414i	0.176524	+	0.658797i	−0.819763	+	0.572703i	$0.805895\pi$
$678$		0			0
$679$	24.1534	−	28.6778i	0.926921	−	1.10055i
$680$		0			0
$681$	−16.4138	+	35.1221i	−0.628979	+	1.34588i
$682$		0			0
$683$	2.97965	−	11.1202i	0.114013	−	0.425503i	−0.885198	−	0.465215i	$-0.845977\pi$
$683$	2.97965	−	11.1202i	0.114013	−	0.425503i	0.999211	+	0.0397112i	$0.0126438\pi$
$684$		0			0
$685$	24.8474	+	25.2855i	0.949371	+	0.966109i
$686$		0			0
$687$	24.3958	−	29.0239i	0.930757	−	1.10733i
$688$		0			0
$689$	−17.7652	−	30.7703i	−0.676801	−	1.17225i
$690$		0			0
$691$	−2.37495	+	4.11354i	−0.0903475	+	0.156486i	−0.907657	−	0.419712i	$-0.862131\pi$
$691$	−2.37495	+	4.11354i	−0.0903475	+	0.156486i	0.817310	+	0.576198i	$0.195464\pi$
$692$		0			0
$693$	4.35979	+	16.5034i	0.165615	+	0.626913i
$694$		0			0
$695$	−12.9768	+	22.0297i	−0.492238	+	0.835632i
$696$		0			0
$697$	5.44336	−	20.3149i	0.206182	−	0.769481i
$698$		0			0
$699$	−3.50132	−	40.4134i	−0.132432	−	1.52857i
$700$		0			0
$701$		−	29.9516i		−	1.13126i	−0.824660	−	0.565628i	$-0.808634\pi$
$701$		−	29.9516i		−	1.13126i	0.824660	−	0.565628i	$-0.191366\pi$
$702$		0			0
$703$	−1.62676	−	0.435890i	−0.0613545	−	0.0164399i
$704$		0			0
$705$	−33.7918	−	16.1534i	−1.27267	−	0.608372i
$706$		0			0
$707$	−11.7849	−	16.8965i	−0.443215	−	0.635457i
$708$		0			0
$709$	4.99216	+	2.88222i	0.187484	+	0.108244i	0.590804	−	0.806815i	$-0.298811\pi$
$709$	4.99216	+	2.88222i	0.187484	+	0.108244i	−0.403320	+	0.915059i	$0.632144\pi$
$710$		0			0
$711$	−14.6896	−	17.5665i	−0.550902	−	0.658796i
$712$		0			0
$713$	−18.9745	−	18.9745i	−0.710599	−	0.710599i
$714$		0			0
$715$	−18.9043	+	18.5767i	−0.706979	+	0.694731i
$716$		0			0
$717$	−5.96451	+	33.6602i	−0.222749	+	1.25706i
$718$		0			0
$719$	18.4248	−	31.9127i	0.687129	−	1.19014i	−0.285633	−	0.958339i	$-0.592204\pi$
$719$	18.4248	−	31.9127i	0.687129	−	1.19014i	0.972763	−	0.231804i	$-0.0744627\pi$
$720$		0			0
$721$	49.5875	−	17.9454i	1.84674	−	0.668320i
$722$		0			0
$723$	21.1476	−	7.67687i	0.786487	−	0.285506i
$724$		0			0
$725$	−9.59756	−	17.3154i	−0.356445	−	0.643078i
$726$		0			0
$727$	−5.34727	−	5.34727i	−0.198319	−	0.198319i	0.600960	−	0.799279i	$-0.294785\pi$
$727$	−5.34727	−	5.34727i	−0.198319	−	0.198319i	−0.799279	+	0.600960i	$0.794785\pi$
$728$		0			0
$729$	−13.3814	−	23.4508i	−0.495606	−	0.868548i
$730$		0			0
$731$	−18.6117	+	10.7455i	−0.688378	+	0.397435i
$732$		0			0
$733$	−4.79803	−	17.9065i	−0.177219	−	0.661391i	−0.996163	−	0.0875174i	$-0.972107\pi$
$733$	−4.79803	−	17.9065i	−0.177219	−	0.661391i	0.818944	−	0.573874i	$-0.194560\pi$
$734$		0			0
$735$	−13.8654	+	23.2970i	−0.511435	+	0.859322i
$736$		0			0
$737$	−0.349241	−	1.30338i	−0.0128644	−	0.0480108i
$738$		0			0
$739$	6.24688	−	3.60664i	0.229795	−	0.132672i	−0.380682	−	0.924706i	$-0.624311\pi$
$739$	6.24688	−	3.60664i	0.229795	−	0.132672i	0.610478	+	0.792034i	$0.290978\pi$
$740$		0			0
$741$	−0.600375	−	6.92971i	−0.0220553	−	0.254569i
$742$		0			0
$743$	9.61154	+	9.61154i	0.352613	+	0.352613i	0.861081	−	0.508468i	$-0.169788\pi$
$743$	9.61154	+	9.61154i	0.352613	+	0.352613i	−0.508468	+	0.861081i	$0.669788\pi$
$744$		0			0
$745$	16.7632	−	9.48391i	0.614156	−	0.347464i
$746$		0			0
$747$	11.7231	+	25.2515i	0.428925	+	0.923903i
$748$		0			0
$749$	−17.2066	+	6.22694i	−0.628716	+	0.227528i
$750$		0			0
$751$	8.51460	−	14.7477i	0.310702	−	0.538152i	−0.667812	−	0.744330i	$-0.732769\pi$
$751$	8.51460	−	14.7477i	0.310702	−	0.538152i	0.978515	+	0.206178i	$0.0661025\pi$
$752$		0			0
$753$	3.15064	+	0.558287i	0.114816	+	0.0203451i
$754$		0			0
$755$	0.194649	−	22.2754i	0.00708400	−	0.810686i
$756$		0			0
$757$	−18.5494	−	18.5494i	−0.674191	−	0.674191i	0.284488	−	0.958679i	$-0.408176\pi$
$757$	−18.5494	−	18.5494i	−0.674191	−	0.674191i	−0.958679	+	0.284488i	$0.908176\pi$
$758$		0			0
$759$	12.2747	+	17.5616i	0.445544	+	0.637447i
$760$		0			0
$761$	33.0626	+	19.0887i	1.19852	+	0.691965i	0.960225	−	0.279228i	$-0.0900784\pi$
$761$	33.0626	+	19.0887i	1.19852	+	0.691965i	0.238294	+	0.971193i	$0.423412\pi$
$762$		0			0
$763$	−9.25898	−	13.2750i	−0.335198	−	0.480587i
$764$		0			0
$765$	−11.7699	−	25.9435i	−0.425541	−	0.937988i
$766$		0			0
$767$	−42.7184	−	11.4464i	−1.54247	−	0.413304i
$768$		0			0
$769$			16.7248i			0.603110i	0.953449	+	0.301555i	$0.0975057\pi$
$769$			16.7248i			0.603110i	−0.953449	+	0.301555i	$0.902494\pi$
$770$		0			0
$771$	−42.8207	+	3.70989i	−1.54215	+	0.133608i
$772$		0			0
$773$	1.57137	−	5.86442i	0.0565182	−	0.210929i	−0.931892	−	0.362736i	$-0.881843\pi$
$773$	1.57137	−	5.86442i	0.0565182	−	0.210929i	0.988410	+	0.151807i	$0.0485093\pi$
$774$		0			0
$775$	12.0138	−	19.9932i	0.431548	−	0.718179i
$776$		0			0
$777$	10.4362	−	1.81135i	0.374397	−	0.0649819i
$778$		0			0
$779$	1.80418	−	3.12493i	0.0646415	−	0.111962i
$780$		0			0
$781$	4.30379	+	7.45438i	0.154002	+	0.266739i
$782$		0			0
$783$	−10.3321	+	17.7915i	−0.369240	+	0.635817i
$784$		0			0
$785$	2.10631	−	2.06982i	0.0751774	−	0.0738749i
$786$		0			0
$787$	−3.20956	+	11.9782i	−0.114409	+	0.426978i	−0.999242	−	0.0389290i	$-0.987605\pi$
$787$	−3.20956	+	11.9782i	−0.114409	+	0.426978i	0.884833	+	0.465907i	$0.154272\pi$
$788$		0			0
$789$	18.8229	+	8.79664i	0.670115	+	0.313169i
$790$		0			0
$791$	33.4006	−	39.6572i	1.18759	−	1.41005i
$792$		0			0
$793$	17.6917	+	66.0262i	0.628249	+	2.34466i
$794$		0			0
$795$	−14.1240	−	20.5883i	−0.500927	−	0.730191i
$796$		0			0
$797$	−17.2195	+	17.2195i	−0.609945	+	0.609945i	−0.942932	−	0.332986i	$-0.891944\pi$
$797$	−17.2195	+	17.2195i	−0.609945	+	0.609945i	0.332986	+	0.942932i	$0.391944\pi$
$798$		0			0
$799$			41.0694i			1.45293i
$800$		0			0
$801$	−3.02271	+	2.52766i	−0.106802	+	0.0893106i
$802$		0			0
$803$	−22.4491	+	6.01521i	−0.792210	+	0.212272i
$804$		0			0
$805$	−6.26332	+	33.4489i	−0.220753	+	1.17892i
$806$		0			0
$807$	5.36531	−	1.94768i	0.188868	−	0.0685617i
$808$		0			0
$809$	−0.599532	−	1.03842i	−0.0210784	−	0.0365089i	0.855294	−	0.518143i	$-0.173377\pi$
$809$	−0.599532	−	1.03842i	−0.0210784	−	0.0365089i	−0.876372	+	0.481634i	$0.840043\pi$
$810$		0			0
$811$	−24.0455			−0.844353			−0.422176	−	0.906514i	$-0.638734\pi$
$811$	−24.0455			−0.844353			−0.422176	+	0.906514i	$0.638734\pi$
$812$		0			0
$813$	−10.7706	−	9.05309i	−0.377740	−	0.317506i
$814$		0			0
$815$	−13.3286	+	48.0592i	−0.466880	+	1.68344i
$816$		0			0
$817$	−3.56155	+	0.954314i	−0.124603	+	0.0333872i
$818$		0			0
$819$	21.7397	+	37.9626i	0.759646	+	1.32652i
$820$		0			0
$821$	33.8794	+	19.5603i	1.18240	+	0.682659i	0.956568	−	0.291508i	$-0.0941569\pi$
$821$	33.8794	+	19.5603i	1.18240	+	0.682659i	0.225831	+	0.974166i	$0.427490\pi$
$822$		0			0
$823$	1.79793	+	0.481753i	0.0626718	+	0.0167929i	0.290019	−	0.957021i	$-0.406338\pi$
$823$	1.79793	+	0.481753i	0.0626718	+	0.0167929i	−0.227347	+	0.973814i	$0.573005\pi$
$824$		0			0
$825$	−12.2309	+	14.0454i	−0.425826	+	0.489000i
$826$		0			0
$827$	23.1571	−	23.1571i	0.805251	−	0.805251i	−0.178660	−	0.983911i	$-0.557176\pi$
$827$	23.1571	−	23.1571i	0.805251	−	0.805251i	0.983911	+	0.178660i	$0.0571763\pi$
$828$		0			0
$829$	−27.5833	+	15.9252i	−0.958008	+	0.553106i	−0.895559	−	0.444942i	$-0.853224\pi$
$829$	−27.5833	+	15.9252i	−0.958008	+	0.553106i	−0.0624485	+	0.998048i	$0.519891\pi$
$830$		0			0
$831$	44.5221	+	20.8068i	1.54445	+	0.721779i
$832$		0			0
$833$	29.6048	+	2.69967i	1.02575	+	0.0935379i
$834$		0			0
$835$	15.3105	−	3.95938i	0.529842	−	0.137020i
$836$		0			0
$837$	−24.2401	−	0.0614089i	−0.837860	−	0.00212260i
$838$		0			0
$839$	25.9027			0.894262			0.447131	−	0.894469i	$-0.352446\pi$
$839$	25.9027			0.894262			0.447131	+	0.894469i	$0.352446\pi$
$840$		0			0
$841$	−13.3226			−0.459399
$842$		0			0
$843$	24.7879	+	4.39237i	0.853742	+	0.151281i
$844$		0			0
$845$	−19.7218	+	33.4801i	−0.678450	+	1.15175i
$846$		0			0
$847$	15.2734	+	7.15629i	0.524801	+	0.245893i
$848$		0			0
$849$	−2.13354	+	4.56532i	−0.0732229	+	0.156681i
$850$		0			0
$851$	11.5143	−	6.64781i	0.394706	−	0.227884i
$852$		0			0
$853$	−25.9730	+	25.9730i	−0.889299	+	0.889299i	−0.994456	−	0.105157i	$-0.966465\pi$
$853$	−25.9730	+	25.9730i	−0.889299	+	0.889299i	0.105157	+	0.994456i	$0.466465\pi$
$854$		0			0
$855$	−0.798514	−	4.82208i	−0.0273086	−	0.164912i
$856$		0			0
$857$	17.0839	+	4.57762i	0.583575	+	0.156368i	0.538515	−	0.842616i	$-0.318985\pi$
$857$	17.0839	+	4.57762i	0.583575	+	0.156368i	0.0450600	+	0.998984i	$0.485652\pi$
$858$		0			0
$859$	14.7883	+	8.53804i	0.504571	+	0.291314i	0.730599	−	0.682806i	$-0.239241\pi$
$859$	14.7883	+	8.53804i	0.504571	+	0.291314i	−0.226028	+	0.974121i	$0.572574\pi$
$860$		0			0
$861$	−2.03853	+	22.6026i	−0.0694730	+	0.770294i
$862$		0			0
$863$	4.54605	−	1.21811i	0.154749	−	0.0414650i	−0.180613	−	0.983554i	$-0.557808\pi$
$863$	4.54605	−	1.21811i	0.154749	−	0.0414650i	0.335362	+	0.942089i	$0.391141\pi$
$864$		0			0
$865$	−6.01107	−	1.66709i	−0.204383	−	0.0566828i
$866$		0			0
$867$	−1.15393	+	1.37284i	−0.0391894	+	0.0466241i
$868$		0			0
$869$	16.4153			0.556851
$870$		0			0
$871$	−1.72910	−	2.99489i	−0.0585884	−	0.101478i
$872$		0			0
$873$	−17.9022	−	38.5612i	−0.605897	−	1.30510i
$874$		0			0
$875$	−29.5286	+	1.75062i	−0.998247	+	0.0591817i
$876$		0			0
$877$	21.3601	−	5.72343i	0.721281	−	0.193267i	0.120538	−	0.992709i	$-0.461538\pi$
$877$	21.3601	−	5.72343i	0.721281	−	0.193267i	0.600743	+	0.799442i	$0.294872\pi$
$878$		0			0
$879$	−24.0352	+	16.7994i	−0.810688	+	0.566630i
$880$		0			0
$881$			49.8630i			1.67993i	0.542642	+	0.839964i	$0.317424\pi$
$881$			49.8630i			1.67993i	−0.542642	+	0.839964i	$0.682576\pi$
$882$		0			0
$883$	−17.7492	+	17.7492i	−0.597308	+	0.597308i	−0.939595	−	0.342287i	$-0.888798\pi$
$883$	−17.7492	+	17.7492i	−0.597308	+	0.597308i	0.342287	+	0.939595i	$0.388798\pi$
$884$		0			0
$885$	−30.5519	−	5.68952i	−1.02699	−	0.191251i
$886$		0			0
$887$	−6.64206	−	24.7885i	−0.223019	−	0.832316i	−0.983189	−	0.182592i	$-0.941551\pi$
$887$	−6.64206	−	24.7885i	−0.223019	−	0.832316i	0.760170	−	0.649724i	$-0.225116\pi$
$888$		0			0
$889$	10.4691	−	12.4301i	0.351121	−	0.416893i
$890$		0			0
$891$	19.0496	+	3.42534i	0.638185	+	0.114753i
$892$		0			0
$893$	−1.82370	+	6.80615i	−0.0610279	+	0.227759i
$894$		0			0
$895$	0.277364	−	31.7413i	0.00927125	−	1.06099i
$896$		0			0
$897$	42.0351	+	35.3322i	1.40351	+	1.17971i
$898$		0			0
$899$	9.23552	+	15.9964i	0.308022	+	0.533509i
$900$		0			0
$901$	−13.6886	+	23.7093i	−0.456033	+	0.789872i
$902$		0			0
$903$	17.8045	−	14.8586i	0.592497	−	0.494463i
$904$		0			0
$905$	19.3203	+	11.3808i	0.642230	+	0.378312i
$906$		0			0
$907$	0.432502	−	1.61412i	0.0143610	−	0.0535960i	−0.958374	−	0.285517i	$-0.907835\pi$
$907$	0.432502	−	1.61412i	0.0143610	−	0.0535960i	0.972735	+	0.231921i	$0.0745012\pi$
$908$		0			0
$909$	−23.0105	+	4.01732i	−0.763211	+	0.133246i
$910$		0			0
$911$		−	40.8752i		−	1.35426i	−0.735865	−	0.677129i	$-0.763224\pi$
$911$		−	40.8752i		−	1.35426i	0.735865	−	0.677129i	$-0.236776\pi$
$912$		0			0
$913$	−19.2772	−	5.16532i	−0.637983	−	0.170947i
$914$		0			0
$915$	15.9951	+	45.2919i	0.528782	+	1.49731i
$916$		0			0
$917$	24.1619	−	51.5679i	0.797895	−	1.70292i
$918$		0			0
$919$	−17.0980	−	9.87155i	−0.564012	−	0.325632i	0.190742	−	0.981640i	$-0.438911\pi$
$919$	−17.0980	−	9.87155i	−0.564012	−	0.325632i	−0.754754	+	0.656008i	$0.772244\pi$
$920$		0			0
$921$	−11.8348	+	8.27194i	−0.389970	+	0.272570i
$922$		0			0
$923$	15.5987	+	15.5987i	0.513437	+	0.513437i
$924$		0			0
$925$	8.02802	+	8.31364i	0.263960	+	0.273351i
$926$		0			0
$927$	5.31213	−	59.5592i	0.174473	−	1.95618i
$928$		0			0
$929$	−25.3654	+	43.9342i	−0.832212	+	1.44143i	0.0640690	+	0.997945i	$0.479592\pi$
$929$	−25.3654	+	43.9342i	−0.832212	+	1.44143i	−0.896281	+	0.443487i	$0.853741\pi$
$930$		0			0
$931$	4.78633	+	1.76201i	0.156866	+	0.0577477i
$932$		0			0
$933$	0.542175	+	1.49354i	0.0177500	+	0.0488962i
$934$		0			0
$935$	19.6793	+	5.45780i	0.643582	+	0.178489i
$936$		0			0
$937$	−26.5787	−	26.5787i	−0.868287	−	0.868287i	0.123995	−	0.992283i	$-0.460429\pi$
$937$	−26.5787	−	26.5787i	−0.868287	−	0.868287i	−0.992283	+	0.123995i	$0.960429\pi$
$938$		0			0
$939$	−2.13607	+	0.185065i	−0.0697080	+	0.00603935i
$940$		0			0
$941$	23.6212	−	13.6377i	0.770030	−	0.444577i	−0.0628556	−	0.998023i	$-0.520021\pi$
$941$	23.6212	−	13.6377i	0.770030	−	0.444577i	0.832885	+	0.553446i	$0.186687\pi$
$942$		0			0
$943$	7.37283	+	27.5158i	0.240093	+	0.896037i
$944$		0			0
$945$	17.3007	+	25.4104i	0.562790	+	0.826600i
$946$		0			0
$947$	0.412833	+	1.54071i	0.0134153	+	0.0500664i	0.972309	−	0.233699i	$-0.0750830\pi$
$947$	0.412833	+	1.54071i	0.0134153	+	0.0500664i	−0.958894	+	0.283765i	$0.908416\pi$
$948$		0			0
$949$	−51.5831	+	29.7815i	−1.67446	+	0.966748i
$950$		0			0
$951$	−28.0408	+	2.42939i	−0.909284	+	0.0787784i
$952$		0			0
$953$	−17.6192	−	17.6192i	−0.570741	−	0.570741i	0.361594	−	0.932336i	$-0.382233\pi$
$953$	−17.6192	−	17.6192i	−0.570741	−	0.570741i	−0.932336	+	0.361594i	$0.882233\pi$
$954$		0			0
$955$	−7.49980	−	13.2562i	−0.242688	−	0.428961i
$956$		0			0
$957$	−5.03262	−	13.8634i	−0.162681	−	0.448140i
$958$		0			0
$959$	−7.35964	+	41.2951i	−0.237655	+	1.33349i
$960$		0			0
$961$	4.61878	−	7.99997i	0.148993	−	0.258063i
$962$		0			0
$963$	−1.84328	+	20.6667i	−0.0593989	+	0.665976i
$964$		0			0
$965$	−13.3556	−	13.5911i	−0.429932	−	0.437512i
$966$		0			0
$967$	−8.13992	−	8.13992i	−0.261762	−	0.261762i	0.564007	−	0.825770i	$-0.309259\pi$
$967$	−8.13992	−	8.13992i	−0.261762	−	0.261762i	−0.825770	+	0.564007i	$0.809259\pi$
$968$		0			0
$969$	−4.39286	+	3.07039i	−0.141119	+	0.0986351i
$970$		0			0
$971$	3.45490	+	1.99469i	0.110873	+	0.0640125i	0.554411	−	0.832243i	$-0.312944\pi$
$971$	3.45490	+	1.99469i	0.110873	+	0.0640125i	−0.443538	+	0.896256i	$0.646277\pi$
$972$		0			0
$973$	−30.1417	+	2.58130i	−0.966298	+	0.0827527i
$974$		0			0
$975$	−20.9613	+	42.8827i	−0.671300	+	1.37335i
$976$		0			0
$977$	−45.5422	−	12.2030i	−1.45702	−	0.390408i	−0.558563	−	0.829462i	$-0.688647\pi$
$977$	−45.5422	−	12.2030i	−1.45702	−	0.390408i	−0.898461	+	0.439054i	$0.855314\pi$
$978$		0			0
$979$		−	2.82461i		−	0.0902751i
$980$		0			0
$981$	−18.0786	+	3.15627i	−0.577206	+	0.100772i
$982$		0			0
$983$	9.41853	−	35.1504i	0.300404	−	1.12112i	−0.636425	−	0.771338i	$-0.719588\pi$
$983$	9.41853	−	35.1504i	0.300404	−	1.12112i	0.936830	−	0.349786i	$-0.113746\pi$
$984$		0			0
$985$	−48.6205	+	12.5735i	−1.54918	+	0.400625i
$986$		0			0
$987$	−7.57845	−	43.6636i	−0.241225	−	1.38983i
$988$		0			0
$989$	14.5543	−	25.2089i	0.462801	−	0.801596i
$990$		0			0
$991$	−7.14153	−	12.3695i	−0.226858	−	0.392930i	0.730017	−	0.683429i	$-0.239512\pi$
$991$	−7.14153	−	12.3695i	−0.226858	−	0.392930i	−0.956875	+	0.290499i	$0.906179\pi$
$992$		0			0
$993$	−14.4168	−	12.1179i	−0.457503	−	0.384549i
$994$		0			0
$995$	−19.8309	−	0.173288i	−0.628681	−	0.00549359i
$996$		0			0
$997$	−9.15638	+	34.1721i	−0.289985	+	1.08224i	0.655133	+	0.755513i	$0.272612\pi$
$997$	−9.15638	+	34.1721i	−0.289985	+	1.08224i	−0.945119	+	0.326727i	$0.894054\pi$
$998$		0			0
$999$	3.13791	−	11.5933i	0.0992792	−	0.366796i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	420.2.bv.c.317.6	yes	48
3.2	odd	2	inner	420.2.bv.c.317.8	yes	48
5.3	odd	4	inner	420.2.bv.c.233.1	yes	48
7.4	even	3	inner	420.2.bv.c.137.11	yes	48
15.8	even	4	inner	420.2.bv.c.233.11	yes	48
21.11	odd	6	inner	420.2.bv.c.137.1	yes	48
35.18	odd	12	inner	420.2.bv.c.53.8	yes	48
105.53	even	12	inner	420.2.bv.c.53.6	✓	48

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
420.2.bv.c.53.6	✓	48	105.53	even	12	inner
420.2.bv.c.53.8	yes	48	35.18	odd	12	inner
420.2.bv.c.137.1	yes	48	21.11	odd	6	inner
420.2.bv.c.137.11	yes	48	7.4	even	3	inner
420.2.bv.c.233.1	yes	48	5.3	odd	4	inner
420.2.bv.c.233.11	yes	48	15.8	even	4	inner
420.2.bv.c.317.6	yes	48	1.1	even	1	trivial
420.2.bv.c.317.8	yes	48	3.2	odd	2	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q+(-0.302208 + 1.70548i) q^{3} +(-1.92665 - 1.13491i) q^{5} +(-0.225753 - 2.63610i) q^{7} +(-2.81734 - 1.03082i) q^{9} +O(q^{10})\)	\(q+(-0.302208 + 1.70548i) q^{3} +(-1.92665 - 1.13491i) q^{5} +(-0.225753 - 2.63610i) q^{7} +(-2.81734 - 1.03082i) q^{9} +(1.86244 - 1.07528i) q^{11} +(3.89727 - 3.89727i) q^{13} +(2.51782 - 2.94289i) q^{15} +(-4.10211 - 1.09916i) q^{17} +(-0.631006 - 0.364311i) q^{19} +(4.56405 + 0.411633i) q^{21} +(5.55616 - 1.48877i) q^{23} +(2.42395 + 4.37315i) q^{25} +(2.60947 - 4.49340i) q^{27} -3.95947 q^{29} +(-2.33251 - 4.04003i) q^{31} +(1.27103 + 3.50133i) q^{33} +(-2.55680 + 5.33505i) q^{35} +(2.23265 - 0.598238i) q^{37} +(5.46894 + 7.82450i) q^{39} +4.95231i q^{41} +(3.57830 - 3.57830i) q^{43} +(4.25814 + 5.18346i) q^{45} +(-2.50295 - 9.34112i) q^{47} +(-6.89807 + 1.19022i) q^{49} +(3.11428 - 6.66390i) q^{51} +(1.66848 - 6.22687i) q^{53} +(-4.80863 - 0.0420191i) q^{55} +(0.812022 - 0.966072i) q^{57} +(-4.01205 - 6.94907i) q^{59} +(-6.20108 + 10.7406i) q^{61} +(-2.08132 + 7.65951i) q^{63} +(-11.9317 + 3.08561i) q^{65} +(0.162395 - 0.606066i) q^{67} +(0.859954 + 9.92585i) q^{69} +4.00247i q^{71} +(-10.4387 - 2.79703i) q^{73} +(-8.19087 + 2.81240i) q^{75} +(-3.25501 - 4.66685i) q^{77} +(6.61039 + 3.81651i) q^{79} +(6.87482 + 5.80834i) q^{81} +(-6.56196 - 6.56196i) q^{83} +(6.65587 + 6.77322i) q^{85} +(1.19658 - 6.75282i) q^{87} +(0.656714 - 1.13746i) q^{89} +(-11.1534 - 9.39378i) q^{91} +(7.59510 - 2.75713i) q^{93} +(0.802265 + 1.41804i) q^{95} +(10.0207 + 10.0207i) q^{97} +(-6.35557 + 1.10959i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 48 q - 2 q^{3} - 8 q^{7}+O(q^{10}) \) 48 * q - 2 * q^3 - 8 * q^7	\( 48 q - 2 q^{3} - 8 q^{7} + 32 q^{13} - 16 q^{21} + 16 q^{25} - 32 q^{27} - 64 q^{31} + 34 q^{33} - 40 q^{37} - 24 q^{43} + 30 q^{45} + 36 q^{51} + 92 q^{57} - 18 q^{63} + 28 q^{67} - 8 q^{73} - 38 q^{75} + 20 q^{81} - 56 q^{85} - 24 q^{87} - 24 q^{91} + 6 q^{93} + 56 q^{97}+O(q^{100}) \) 48 * q - 2 * q^3 - 8 * q^7 + 32 * q^13 - 16 * q^21 + 16 * q^25 - 32 * q^27 - 64 * q^31 + 34 * q^33 - 40 * q^37 - 24 * q^43 + 30 * q^45 + 36 * q^51 + 92 * q^57 - 18 * q^63 + 28 * q^67 - 8 * q^73 - 38 * q^75 + 20 * q^81 - 56 * q^85 - 24 * q^87 - 24 * q^91 + 6 * q^93 + 56 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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