LMFDB - Embedded newform 144.4.l.a.35.19

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [144,4,Mod(35,144)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(144, base_ring=CyclotomicField(4)) chi = DirichletCharacter(H, H._module([2, 3, 2])) N = Newforms(chi, 4, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("144.35"); S:= CuspForms(chi, 4); N := Newforms(S);

Level:	$ N $	$=$	$ 144 = 2^{4} \cdot 3^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	144.l (of order $4$, degree $2$, minimal)

Newform invariants

comment:select newform

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

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

Self dual:	no
Analytic conductor:	$8.49627504083$
Analytic rank:	$0$
Dimension:	$48$
Relative dimension:	$24$ over $\Q(i)$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			35.19
Character	$\chi$	$=$	144.35
Dual form			144.4.l.a.107.19

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+(1.89081 + 2.10353i) q^{2} +(-0.849709 + 7.95475i) q^{4} +(-11.2962 - 11.2962i) q^{5} -19.1985 q^{7} +(-18.3397 + 13.2535i) q^{8} +(2.40303 - 45.1210i) q^{10} +(34.3321 - 34.3321i) q^{11} +(-59.4521 - 59.4521i) q^{13} +(-36.3006 - 40.3847i) q^{14} +(-62.5560 - 13.5184i) q^{16} +102.723i q^{17} +(-60.4132 + 60.4132i) q^{19} +(99.4571 - 80.2601i) q^{20} +(137.134 + 7.30341i) q^{22} -106.082i q^{23} +130.209i q^{25} +(12.6471 - 237.472i) q^{26} +(16.3131 - 152.719i) q^{28} +(30.6651 - 30.6651i) q^{29} +99.7709i q^{31} +(-89.8447 - 157.149i) q^{32} +(-216.082 + 194.230i) q^{34} +(216.870 + 216.870i) q^{35} +(-94.9220 + 94.9220i) q^{37} +(-241.311 - 12.8516i) q^{38} +(356.884 + 57.4552i) q^{40} +34.5075 q^{41} +(-198.078 - 198.078i) q^{43} +(243.931 + 302.276i) q^{44} +(223.147 - 200.581i) q^{46} -314.847 q^{47} +25.5819 q^{49} +(-273.900 + 246.201i) q^{50} +(523.443 - 422.409i) q^{52} +(187.087 + 187.087i) q^{53} -775.647 q^{55} +(352.095 - 254.447i) q^{56} +(122.487 + 6.52334i) q^{58} +(382.568 - 382.568i) q^{59} +(509.275 + 509.275i) q^{61} +(-209.871 + 188.647i) q^{62} +(160.690 - 486.130i) q^{64} +1343.17i q^{65} +(-107.017 + 107.017i) q^{67} +(-817.139 - 87.2851i) q^{68} +(-46.1345 + 866.254i) q^{70} +209.706i q^{71} +411.628i q^{73} +(-379.151 - 20.1926i) q^{74} +(-429.238 - 531.906i) q^{76} +(-659.125 + 659.125i) q^{77} -992.581i q^{79} +(553.939 + 859.354i) q^{80} +(65.2470 + 72.5877i) q^{82} +(-852.083 - 852.083i) q^{83} +(1160.39 - 1160.39i) q^{85} +(42.1367 - 791.189i) q^{86} +(-174.621 + 1084.66i) q^{88} -205.447 q^{89} +(1141.39 + 1141.39i) q^{91} +(843.856 + 90.1389i) q^{92} +(-595.315 - 662.292i) q^{94} +1364.88 q^{95} -775.052 q^{97} +(48.3705 + 53.8125i) q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 48 q - 120 q^{10} - 144 q^{16} - 48 q^{19} + 72 q^{22} + 72 q^{28} - 984 q^{34} - 1272 q^{40} + 864 q^{43} - 1416 q^{46} + 2352 q^{49} - 648 q^{52} - 576 q^{55} + 1128 q^{58} + 1824 q^{61} + 3024 q^{64}+ \cdots - 11304 q^{94}+O(q^{100}) $ 48 * q - 120 * q^10 - 144 * q^16 - 48 * q^19 + 72 * q^22 + 72 * q^28 - 984 * q^34 - 1272 * q^40 + 864 * q^43 - 1416 * q^46 + 2352 * q^49 - 648 * q^52 - 576 * q^55 + 1128 * q^58 + 1824 * q^61 + 3024 * q^64 + 816 * q^67 + 2664 * q^70 + 1920 * q^76 + 1200 * q^82 - 480 * q^85 - 4560 * q^88 - 3600 * q^91 - 11304 * q^94

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/144\mathbb{Z}\right)^\times$.

$n$	$37$	$65$	$127$
$\chi(n)$	$e\left(\frac{3}{4}\right)$	$-1$	$-1$

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 $

$2$	1.89081	+	2.10353i	0.668501	+	0.743712i
$2$	1.89081	+	2.10353i	0.668501	+	0.743712i
$3$		0			0
$3$		0			0
$4$	−0.849709	+	7.95475i	−0.106214	+	0.994343i
$5$	−11.2962	−	11.2962i	−1.01037	−	1.01037i	−0.999946	−	0.0104195i	$-0.996683\pi$
$5$	−11.2962	−	11.2962i	−1.01037	−	1.01037i	−0.0104195	−	0.999946i	$-0.503317\pi$
$6$		0			0
$7$	−19.1985			−1.03662			−0.518310	−	0.855193i	$-0.673439\pi$
$7$	−19.1985			−1.03662			−0.518310	+	0.855193i	$0.673439\pi$
$8$	−18.3397	+	13.2535i	−0.810508	+	0.585727i
$9$		0			0
$10$	2.40303	−	45.1210i	0.0759904	−	1.42685i
$11$	34.3321	−	34.3321i	0.941047	−	0.941047i	−0.0573091	−	0.998356i	$-0.518252\pi$
$11$	34.3321	−	34.3321i	0.941047	−	0.941047i	0.998356	+	0.0573091i	$0.0182521\pi$
$12$		0			0
$13$	−59.4521	−	59.4521i	−1.26839	−	1.26839i	−0.946920	−	0.321468i	$-0.895824\pi$
$13$	−59.4521	−	59.4521i	−1.26839	−	1.26839i	−0.321468	−	0.946920i	$-0.604176\pi$
$14$	−36.3006	−	40.3847i	−0.692982	−	0.770947i
$15$		0			0
$16$	−62.5560	−	13.5184i	−0.977437	−	0.211226i
$17$			102.723i			1.46554i	0.680479	+	0.732768i	$0.261772\pi$
$17$			102.723i			1.46554i	−0.680479	+	0.732768i	$0.738228\pi$
$18$		0			0
$19$	−60.4132	+	60.4132i	−0.729460	+	0.729460i	−0.970512	−	0.241052i	$-0.922508\pi$
$19$	−60.4132	+	60.4132i	−0.729460	+	0.729460i	0.241052	+	0.970512i	$0.422508\pi$
$20$	99.4571	−	80.2601i	1.11196	−	0.897335i
$21$		0			0
$22$	137.134	+	7.30341i	1.32896	+	0.0707770i
$23$		−	106.082i		−	0.961723i	−0.876796	−	0.480862i	$-0.840324\pi$
$23$		−	106.082i		−	0.961723i	0.876796	−	0.480862i	$-0.159676\pi$
$24$		0			0
$25$			130.209i			1.04168i
$26$	12.6471	−	237.472i	0.0953965	−	1.79123i
$27$		0			0
$28$	16.3131	−	152.719i	0.110103	−	1.03076i
$29$	30.6651	−	30.6651i	0.196358	−	0.196358i	−0.602079	−	0.798437i	$-0.705661\pi$
$29$	30.6651	−	30.6651i	0.196358	−	0.196358i	0.798437	+	0.602079i	$0.205661\pi$
$30$		0			0
$31$			99.7709i			0.578044i	0.957322	+	0.289022i	$0.0933301\pi$
$31$			99.7709i			0.578044i	−0.957322	+	0.289022i	$0.906670\pi$
$32$	−89.8447	−	157.149i	−0.496327	−	0.868136i
$33$		0			0
$34$	−216.082	+	194.230i	−1.08994	+	0.979711i
$35$	216.870	+	216.870i	1.04737	+	1.04737i
$36$		0			0
$37$	−94.9220	+	94.9220i	−0.421759	+	0.421759i	−0.885809	−	0.464050i	$-0.846396\pi$
$37$	−94.9220	+	94.9220i	−0.421759	+	0.421759i	0.464050	+	0.885809i	$0.346396\pi$
$38$	−241.311	−	12.8516i	−1.03015	−	0.0548633i
$39$		0			0
$40$	356.884	+	57.4552i	1.41071	+	0.227112i
$41$	34.5075			0.131443			0.0657216	−	0.997838i	$-0.479065\pi$
$41$	34.5075			0.131443			0.0657216	+	0.997838i	$0.479065\pi$
$42$		0			0
$43$	−198.078	−	198.078i	−0.702478	−	0.702478i	0.262464	−	0.964942i	$-0.415465\pi$
$43$	−198.078	−	198.078i	−0.702478	−	0.702478i	−0.964942	+	0.262464i	$0.915465\pi$
$44$	243.931	+	302.276i	0.835772	+	1.03568i
$45$		0			0
$46$	223.147	−	200.581i	0.715245	−	0.642913i
$47$	−314.847			−0.977131			−0.488566	−	0.872527i	$-0.662480\pi$
$47$	−314.847			−0.977131			−0.488566	+	0.872527i	$0.662480\pi$
$48$		0			0
$49$	25.5819			0.0745829
$50$	−273.900	+	246.201i	−0.774706	+	0.696361i
$51$		0			0
$52$	523.443	−	422.409i	1.39593	−	1.12649i
$53$	187.087	+	187.087i	0.484876	+	0.484876i	0.906685	−	0.421809i	$-0.138605\pi$
$53$	187.087	+	187.087i	0.484876	+	0.484876i	−0.421809	+	0.906685i	$0.638605\pi$
$54$		0			0
$55$	−775.647			−1.90160
$56$	352.095	−	254.447i	0.840190	−	0.607177i
$57$		0			0
$58$	122.487	+	6.52334i	0.277299	+	0.0147682i
$59$	382.568	−	382.568i	0.844172	−	0.844172i	−0.145226	−	0.989398i	$-0.546391\pi$
$59$	382.568	−	382.568i	0.844172	−	0.844172i	0.989398	+	0.145226i	$0.0463909\pi$
$60$		0			0
$61$	509.275	+	509.275i	1.06895	+	1.06895i	0.997440	+	0.0715101i	$0.0227818\pi$
$61$	509.275	+	509.275i	1.06895	+	1.06895i	0.0715101	+	0.997440i	$0.477218\pi$
$62$	−209.871	+	188.647i	−0.429898	+	0.386423i
$63$		0			0
$64$	160.690	−	486.130i	0.313848	−	0.949473i
$65$			1343.17i			2.56307i
$66$		0			0
$67$	−107.017	+	107.017i	−0.195137	+	0.195137i	−0.797912	−	0.602774i	$-0.794062\pi$
$67$	−107.017	+	107.017i	−0.195137	+	0.195137i	0.602774	+	0.797912i	$0.294062\pi$
$68$	−817.139	−	87.2851i	−1.45725	−	0.155660i
$69$		0			0
$70$	−46.1345	+	866.254i	−0.0787732	+	1.47910i
$71$			209.706i			0.350529i	0.984521	+	0.175265i	$0.0560781\pi$
$71$			209.706i			0.350529i	−0.984521	+	0.175265i	$0.943922\pi$
$72$		0			0
$73$			411.628i			0.659965i	0.943987	+	0.329982i	$0.107043\pi$
$73$			411.628i			0.659965i	−0.943987	+	0.329982i	$0.892957\pi$
$74$	−379.151	−	20.1926i	−0.595613	−	0.0317208i
$75$		0			0
$76$	−429.238	−	531.906i	−0.647855	−	0.802813i
$77$	−659.125	+	659.125i	−0.975509	+	0.975509i
$78$		0			0
$79$		−	992.581i		−	1.41360i	−0.707415	−	0.706798i	$-0.750139\pi$
$79$		−	992.581i		−	1.41360i	0.707415	−	0.706798i	$-0.249861\pi$
$80$	553.939	+	859.354i	0.774154	+	1.20098i
$81$		0			0
$82$	65.2470	+	72.5877i	0.0878698	+	0.0977558i
$83$	−852.083	−	852.083i	−1.12685	−	1.12685i	−0.990687	−	0.136160i	$-0.956524\pi$
$83$	−852.083	−	852.083i	−1.12685	−	1.12685i	−0.136160	−	0.990687i	$-0.543476\pi$
$84$		0			0
$85$	1160.39	−	1160.39i	1.48073	−	1.48073i
$86$	42.1367	−	791.189i	0.0528339	−	0.992048i
$87$		0			0
$88$	−174.621	+	1084.66i	−0.211530	+	1.31392i
$89$	−205.447			−0.244689			−0.122345	−	0.992488i	$-0.539041\pi$
$89$	−205.447			−0.244689			−0.122345	+	0.992488i	$0.539041\pi$
$90$		0			0
$91$	1141.39	+	1141.39i	1.31484	+	1.31484i
$92$	843.856	+	90.1389i	0.956283	+	0.102148i
$93$		0			0
$94$	−595.315	−	662.292i	−0.653213	−	0.726704i
$95$	1364.88			1.47404
$96$		0			0
$97$	−775.052			−0.811285			−0.405642	−	0.914032i	$-0.632952\pi$
$97$	−775.052			−0.811285			−0.405642	+	0.914032i	$0.632952\pi$
$98$	48.3705	+	53.8125i	0.0498587	+	0.0554682i
$99$		0			0
$100$	−1035.78	−	110.640i	−1.03578	−	0.110640i
$101$	−424.287	−	424.287i	−0.418001	−	0.418001i	0.466513	−	0.884514i	$-0.345510\pi$
$101$	−424.287	−	424.287i	−0.418001	−	0.418001i	−0.884514	+	0.466513i	$0.845510\pi$
$102$		0			0
$103$	−108.709			−0.103994			−0.0519972	−	0.998647i	$-0.516559\pi$
$103$	−108.709			−0.103994			−0.0519972	+	0.998647i	$0.516559\pi$
$104$	1878.28	+	302.387i	1.77097	+	0.285110i
$105$		0			0
$106$	−39.7988	+	747.291i	−0.0364679	+	0.684748i
$107$	1030.41	−	1030.41i	0.930964	−	0.930964i	−0.0668021	−	0.997766i	$-0.521280\pi$
$107$	1030.41	−	1030.41i	0.930964	−	0.930964i	0.997766	+	0.0668021i	$0.0212796\pi$
$108$		0			0
$109$	632.898	+	632.898i	0.556153	+	0.556153i	0.928210	−	0.372057i	$-0.121347\pi$
$109$	632.898	+	632.898i	0.556153	+	0.556153i	−0.372057	+	0.928210i	$0.621347\pi$
$110$	−1466.60	−	1631.60i	−1.27122	−	1.41424i
$111$		0			0
$112$	1200.98	+	259.534i	1.01323	+	0.218961i
$113$		−	1025.75i		−	0.853929i	−0.904268	−	0.426965i	$-0.859583\pi$
$113$		−	1025.75i		−	0.853929i	0.904268	−	0.426965i	$-0.140417\pi$
$114$		0			0
$115$	−1198.33	+	1198.33i	−0.971692	+	0.971692i
$116$	217.877	+	269.990i	0.174391	+	0.216103i
$117$		0			0
$118$	1528.11	+	81.3832i	1.19215	+	0.0634909i
$119$		−	1972.13i		−	1.51920i
$120$		0			0
$121$		−	1026.39i		−	0.771140i
$122$	−108.337	+	2034.22i	−0.0803966	+	1.50958i
$123$		0			0
$124$	−793.652	−	84.7762i	−0.574775	−	0.0613962i
$125$	58.8473	−	58.8473i	0.0421077	−	0.0421077i
$126$		0			0
$127$		−	2285.63i		−	1.59699i	−0.602004	−	0.798493i	$-0.705631\pi$
$127$		−	2285.63i		−	1.59699i	0.602004	−	0.798493i	$-0.294369\pi$
$128$	1326.43	−	581.161i	0.915942	−	0.401311i
$129$		0			0
$130$	−2825.40	+	2539.67i	−1.90619	+	1.71341i
$131$	−277.305	−	277.305i	−0.184948	−	0.184948i	0.608560	−	0.793508i	$-0.291748\pi$
$131$	−277.305	−	277.305i	−0.184948	−	0.184948i	−0.793508	+	0.608560i	$0.791748\pi$
$132$		0			0
$133$	1159.84	−	1159.84i	0.756174	−	0.756174i
$134$	−427.462	−	22.7655i	−0.275575	−	0.0146764i
$135$		0			0
$136$	−1361.44	−	1883.92i	−0.858403	−	1.18783i
$137$	733.827			0.457628			0.228814	−	0.973470i	$-0.426515\pi$
$137$	733.827			0.457628			0.228814	+	0.973470i	$0.426515\pi$
$138$		0			0
$139$	−687.229	−	687.229i	−0.419352	−	0.419352i	0.465628	−	0.884981i	$-0.345828\pi$
$139$	−687.229	−	687.229i	−0.419352	−	0.419352i	−0.884981	+	0.465628i	$0.845828\pi$
$140$	−1909.43	+	1540.87i	−1.15269	+	0.930197i
$141$		0			0
$142$	−441.124	+	396.514i	−0.260692	+	0.234329i
$143$	−4082.23			−2.38723
$144$		0			0
$145$	−692.800			−0.396786
$146$	−865.873	+	778.309i	−0.490823	+	0.441187i
$147$		0			0
$148$	−674.424	−	835.736i	−0.374576	−	0.464170i
$149$	−1380.56	−	1380.56i	−0.759061	−	0.759061i	0.217091	−	0.976151i	$-0.430343\pi$
$149$	−1380.56	−	1380.56i	−0.759061	−	0.759061i	−0.976151	+	0.217091i	$0.930343\pi$
$150$		0			0
$151$	579.048			0.312068			0.156034	−	0.987752i	$-0.450129\pi$
$151$	579.048			0.312068			0.156034	+	0.987752i	$0.450129\pi$
$152$	307.275	−	1908.65i	0.163969	−	1.01850i
$153$		0			0
$154$	−2632.77	−	140.214i	−1.37763	−	0.0733689i
$155$	1127.03	−	1127.03i	0.584036	−	0.584036i
$156$		0			0
$157$	339.614	+	339.614i	0.172638	+	0.172638i	0.788137	−	0.615499i	$-0.211046\pi$
$157$	339.614	+	339.614i	0.172638	+	0.172638i	−0.615499	+	0.788137i	$0.711046\pi$
$158$	2087.93	−	1876.78i	1.05131	−	0.944990i
$159$		0			0
$160$	−760.289	+	2790.10i	−0.375663	+	1.37861i
$161$			2036.61i			0.996942i
$162$		0			0
$163$	1700.16	−	1700.16i	0.816974	−	0.816974i	−0.168694	−	0.985668i	$-0.553955\pi$
$163$	1700.16	−	1700.16i	0.816974	−	0.816974i	0.985668	+	0.168694i	$0.0539550\pi$
$164$	−29.3214	+	274.499i	−0.0139611	+	0.130700i
$165$		0			0
$166$	181.262	−	3403.51i	0.0847511	−	1.59135i
$167$			1641.78i			0.760747i	0.924833	+	0.380374i	$0.124205\pi$
$167$			1641.78i			0.760747i	−0.924833	+	0.380374i	$0.875795\pi$
$168$		0			0
$169$			4872.11i			2.21762i
$170$	4634.98	+	246.847i	2.09110	+	0.111367i
$171$		0			0
$172$	1743.96	−	1407.35i	0.773117	−	0.623891i
$173$	106.718	−	106.718i	0.0468995	−	0.0468995i	−0.683268	−	0.730168i	$-0.739442\pi$
$173$	106.718	−	106.718i	0.0468995	−	0.0468995i	0.730168	+	0.683268i	$0.239442\pi$
$174$		0			0
$175$		−	2499.82i		−	1.07982i
$176$	−2611.80	+	1683.56i	−1.11859	+	0.721041i
$177$		0			0
$178$	−388.460	−	432.165i	−0.163575	−	0.181978i
$179$	2620.63	+	2620.63i	1.09427	+	1.09427i	0.995067	+	0.0992073i	$0.0316307\pi$
$179$	2620.63	+	2620.63i	1.09427	+	1.09427i	0.0992073	+	0.995067i	$0.468369\pi$
$180$		0			0
$181$	2928.95	−	2928.95i	1.20280	−	1.20280i	0.229489	−	0.973311i	$-0.426294\pi$
$181$	2928.95	−	2928.95i	1.20280	−	1.20280i	0.973311	−	0.229489i	$-0.0737057\pi$
$182$	−242.806	+	4559.10i	−0.0988900	+	1.85683i
$183$		0			0
$184$	1405.96	+	1945.51i	0.563307	+	0.779485i
$185$	2144.52			0.852261
$186$		0			0
$187$	3526.71	+	3526.71i	1.37914	+	1.37914i
$188$	267.528	−	2504.53i	0.103785	−	0.971604i
$189$		0			0
$190$	2580.73	+	2871.08i	0.985399	+	1.09626i
$191$	−1753.27			−0.664199			−0.332100	−	0.943244i	$-0.607757\pi$
$191$	−1753.27			−0.664199			−0.332100	+	0.943244i	$0.607757\pi$
$192$		0			0
$193$	−967.380			−0.360796			−0.180398	−	0.983594i	$-0.557739\pi$
$193$	−967.380			−0.360796			−0.180398	+	0.983594i	$0.557739\pi$
$194$	−1465.47	−	1630.35i	−0.542344	−	0.603362i
$195$		0			0
$196$	−21.7372	+	203.498i	−0.00792172	+	0.0741610i
$197$	−1798.77	−	1798.77i	−0.650542	−	0.650542i	0.302581	−	0.953124i	$-0.402152\pi$
$197$	−1798.77	−	1798.77i	−0.650542	−	0.650542i	−0.953124	+	0.302581i	$0.902152\pi$
$198$		0			0
$199$	−4873.66			−1.73610			−0.868052	−	0.496474i	$-0.834628\pi$
$199$	−4873.66			−1.73610			−0.868052	+	0.496474i	$0.834628\pi$
$200$	−1725.73	−	2388.00i	−0.610137	−	0.844287i
$201$		0			0
$202$	90.2578	−	1694.75i	0.0314382	−	0.590306i
$203$	−588.724	+	588.724i	−0.203548	+	0.203548i
$204$		0			0
$205$	−389.805	−	389.805i	−0.132806	−	0.132806i
$206$	−205.548	−	228.673i	−0.0695203	−	0.0773418i
$207$		0			0
$208$	2915.39	+	4522.79i	0.971854	+	1.50769i
$209$			4148.23i			1.37291i
$210$		0			0
$211$	−1612.73	+	1612.73i	−0.526184	+	0.526184i	−0.919432	−	0.393248i	$-0.871351\pi$
$211$	−1612.73	+	1612.73i	−0.526184	+	0.526184i	0.393248	+	0.919432i	$0.371351\pi$
$212$	−1647.20	+	1329.26i	−0.533634	+	0.430633i
$213$		0			0
$214$	4115.79	+	219.197i	1.31472	+	0.0700186i
$215$			4475.06i			1.41952i
$216$		0			0
$217$		−	1915.45i		−	0.599213i
$218$	−134.635	+	2528.01i	−0.0418287	+	0.785406i
$219$		0			0
$220$	659.074	−	6170.07i	0.201976	−	1.89085i
$221$	6107.13	−	6107.13i	1.85887	−	1.85887i
$222$		0			0
$223$			3969.43i			1.19199i	0.802990	+	0.595993i	$0.203241\pi$
$223$			3969.43i			1.19199i	−0.802990	+	0.595993i	$0.796759\pi$
$224$	1724.88	+	3017.03i	0.514502	+	0.899928i
$225$		0			0
$226$	2157.69	−	1939.48i	0.635077	−	0.570852i
$227$	498.814	+	498.814i	0.145848	+	0.145848i	0.776260	−	0.630412i	$-0.217114\pi$
$227$	498.814	+	498.814i	0.145848	+	0.145848i	−0.630412	+	0.776260i	$0.717114\pi$
$228$		0			0
$229$	−3714.57	+	3714.57i	−1.07190	+	1.07190i	−0.0746968	+	0.997206i	$0.523799\pi$
$229$	−3714.57	+	3714.57i	−1.07190	+	1.07190i	−0.997206	+	0.0746968i	$0.976201\pi$
$230$	−4786.52	−	254.918i	−1.37223	−	0.0730817i
$231$		0			0
$232$	−155.970	+	968.809i	−0.0441376	+	0.274161i
$233$	−4041.77			−1.13642			−0.568209	−	0.822884i	$-0.692364\pi$
$233$	−4041.77			−1.13642			−0.568209	+	0.822884i	$0.692364\pi$
$234$		0			0
$235$	3556.58	+	3556.58i	0.987259	+	0.987259i
$236$	2718.16	+	3368.31i	0.749735	+	0.929060i
$237$		0			0
$238$	4148.45	−	3728.92i	1.12985	−	1.01559i
$239$	−98.4276			−0.0266391			−0.0133196	−	0.999911i	$-0.504240\pi$
$239$	−98.4276			−0.0266391			−0.0133196	+	0.999911i	$0.504240\pi$
$240$		0			0
$241$	−6987.33			−1.86761			−0.933804	−	0.357785i	$-0.883532\pi$
$241$	−6987.33			−1.86761			−0.933804	+	0.357785i	$0.883532\pi$
$242$	2159.04	−	1940.70i	0.573506	−	0.515508i
$243$		0			0
$244$	−4483.89	+	3618.42i	−1.17644	+	0.949366i
$245$	−288.979	−	288.979i	−0.0753560	−	0.0753560i
$246$		0			0
$247$	7183.39			1.85048
$248$	−1322.31	−	1829.77i	−0.338576	−	0.468510i
$249$		0			0
$250$	235.056	+	12.5185i	0.0594650	+	0.00316695i
$251$	1054.33	−	1054.33i	0.265134	−	0.265134i	−0.562002	−	0.827136i	$-0.689969\pi$
$251$	1054.33	−	1054.33i	0.265134	−	0.265134i	0.827136	+	0.562002i	$0.189969\pi$
$252$		0			0
$253$	−3642.02	−	3642.02i	−0.905027	−	0.905027i
$254$	4807.91	−	4321.69i	1.18770	−	1.06759i
$255$		0			0
$256$	3730.50	+	1691.32i	0.910767	+	0.412920i
$257$		−	159.320i		−	0.0386696i	−0.999813	−	0.0193348i	$-0.993845\pi$
$257$		−	159.320i		−	0.0386696i	0.999813	−	0.0193348i	$-0.00615485\pi$
$258$		0			0
$259$	1822.36	−	1822.36i	0.437204	−	0.437204i
$260$	−10684.6	−	1141.30i	−2.54857	−	0.272233i
$261$		0			0
$262$	58.9906	−	1107.65i	0.0139101	−	0.261186i
$263$			5388.56i			1.26339i	0.775215	+	0.631697i	$0.217641\pi$
$263$			5388.56i			1.26339i	−0.775215	+	0.631697i	$0.782359\pi$
$264$		0			0
$265$		−	4226.76i		−	0.979804i
$266$	4632.81	+	246.731i	1.06788	+	0.0568725i
$267$		0			0
$268$	−760.359	−	942.225i	−0.173307	−	0.214760i
$269$	−1342.95	+	1342.95i	−0.304390	+	0.304390i	−0.842729	−	0.538338i	$-0.819052\pi$
$269$	−1342.95	+	1342.95i	−0.304390	+	0.304390i	0.538338	+	0.842729i	$0.319052\pi$
$270$		0			0
$271$		−	985.086i		−	0.220811i	−0.993887	−	0.110405i	$-0.964785\pi$
$271$		−	985.086i		−	0.220811i	0.993887	−	0.110405i	$-0.0352149\pi$
$272$	1388.66	−	6425.97i	0.309559	−	1.43247i
$273$		0			0
$274$	1387.52	+	1543.63i	0.305925	+	0.340344i
$275$	4470.37	+	4470.37i	0.980266	+	0.980266i
$276$		0			0
$277$	−2005.49	+	2005.49i	−0.435012	+	0.435012i	−0.890329	−	0.455317i	$-0.849526\pi$
$277$	−2005.49	+	2005.49i	−0.435012	+	0.435012i	0.455317	+	0.890329i	$0.349526\pi$
$278$	146.193	−	2745.03i	0.0315398	−	0.592215i
$279$		0			0
$280$	−6851.63	−	1103.05i	−1.46237	−	0.235429i
$281$	5647.53			1.19895			0.599473	−	0.800395i	$-0.295377\pi$
$281$	5647.53			1.19895			0.599473	+	0.800395i	$0.295377\pi$
$282$		0			0
$283$	−2391.89	−	2391.89i	−0.502415	−	0.502415i	0.409773	−	0.912188i	$-0.365608\pi$
$283$	−2391.89	−	2391.89i	−0.502415	−	0.502415i	−0.912188	+	0.409773i	$0.865608\pi$
$284$	−1668.16	−	178.189i	−0.348546	−	0.0372310i
$285$		0			0
$286$	−7718.71	−	8587.11i	−1.59586	−	1.77541i
$287$	−662.492			−0.136257
$288$		0			0
$289$	−5639.11			−1.14779
$290$	−1309.95	−	1457.33i	−0.265251	−	0.295094i
$291$		0			0
$292$	−3274.40	−	349.764i	−0.656231	−	0.0700972i
$293$	−720.156	−	720.156i	−0.143590	−	0.143590i	0.631657	−	0.775248i	$-0.282375\pi$
$293$	−720.156	−	720.156i	−0.143590	−	0.143590i	−0.775248	+	0.631657i	$0.782375\pi$
$294$		0			0
$295$	−8643.16			−1.70584
$296$	482.795	−	2998.89i	0.0948036	−	0.588875i
$297$		0			0
$298$	293.685	−	5514.43i	0.0570896	−	1.07195i
$299$	−6306.80	+	6306.80i	−1.21984	+	1.21984i
$300$		0			0
$301$	3802.79	+	3802.79i	0.728203	+	0.728203i
$302$	1094.87	+	1218.05i	0.208618	+	0.232088i
$303$		0			0
$304$	4595.90	−	2962.52i	0.867083	−	0.558921i
$305$		−	11505.8i		−	2.16006i
$306$		0			0
$307$	−5876.19	+	5876.19i	−1.09242	+	1.09242i	−0.0971468	+	0.995270i	$0.530972\pi$
$307$	−5876.19	+	5876.19i	−1.09242	+	1.09242i	−0.995270	+	0.0971468i	$0.969028\pi$
$308$	−4683.10	−	5803.23i	−0.866379	−	1.07360i
$309$		0			0
$310$	4501.76	+	239.752i	0.824783	+	0.0439258i
$311$		−	3603.64i		−	0.657054i	−0.944495	−	0.328527i	$-0.893448\pi$
$311$		−	3603.64i		−	0.657054i	0.944495	−	0.328527i	$-0.106552\pi$
$312$		0			0
$313$			8101.77i			1.46306i	0.681807	+	0.731532i	$0.261194\pi$
$313$			8101.77i			1.46306i	−0.681807	+	0.731532i	$0.738806\pi$
$314$	−72.2455	+	1356.53i	−0.0129842	+	0.243801i
$315$		0			0
$316$	7895.73	+	843.406i	1.40560	+	0.150143i
$317$	−4409.48	+	4409.48i	−0.781265	+	0.781265i	−0.980044	−	0.198779i	$-0.936302\pi$
$317$	−4409.48	+	4409.48i	−0.781265	+	0.781265i	0.198779	+	0.980044i	$0.436302\pi$
$318$		0			0
$319$		−	2105.60i		−	0.369563i
$320$	−7306.63	+	3676.25i	−1.27642	+	0.642214i
$321$		0			0
$322$	−4284.09	+	3850.84i	−0.741438	+	0.666457i
$323$	−6205.86	−	6205.86i	−1.06905	−	1.06905i
$324$		0			0
$325$	7741.23	−	7741.23i	1.32125	−	1.32125i
$326$	6791.02	+	361.672i	1.15374	+	0.0614453i
$327$		0			0
$328$	−632.858	+	457.345i	−0.106536	+	0.0769898i
$329$	6044.59			1.01291
$330$		0			0
$331$	133.348	+	133.348i	0.0221434	+	0.0221434i	0.718092	−	0.695948i	$-0.245016\pi$
$331$	133.348	+	133.348i	0.0221434	+	0.0221434i	−0.695948	+	0.718092i	$0.745016\pi$
$332$	7502.13	−	6054.08i	1.24016	−	1.00079i
$333$		0			0
$334$	−3453.54	+	3104.29i	−0.565776	+	0.508560i
$335$	2417.77			0.394320
$336$		0			0
$337$	−365.268			−0.0590427			−0.0295214	−	0.999564i	$-0.509398\pi$
$337$	−365.268			−0.0590427			−0.0295214	+	0.999564i	$0.509398\pi$
$338$	−10248.6	+	9212.21i	−1.64927	+	1.48248i
$339$		0			0
$340$	8244.60	+	10216.6i	1.31508	+	1.62962i
$341$	3425.34	+	3425.34i	0.543967	+	0.543967i
$342$		0			0
$343$	6093.95			0.959307
$344$	6257.90	+	1007.47i	0.980824	+	0.157904i
$345$		0			0
$346$	426.267	+	22.7019i	0.0662320	+	0.00352735i
$347$	6172.89	−	6172.89i	0.954980	−	0.954980i	−0.0440491	−	0.999029i	$-0.514026\pi$
$347$	6172.89	−	6172.89i	0.954980	−	0.954980i	0.999029	+	0.0440491i	$0.0140258\pi$
$348$		0			0
$349$	158.317	+	158.317i	0.0242822	+	0.0242822i	0.719144	−	0.694861i	$-0.244534\pi$
$349$	158.317	+	158.317i	0.0242822	+	0.0242822i	−0.694861	+	0.719144i	$0.744534\pi$
$350$	5258.47	−	4726.68i	0.803077	−	0.721862i
$351$		0			0
$352$	−8479.83	−	2310.71i	−1.28402	−	0.349890i
$353$		−	8796.93i		−	1.32638i	−0.748449	−	0.663192i	$-0.769201\pi$
$353$		−	8796.93i		−	1.32638i	0.748449	−	0.663192i	$-0.230799\pi$
$354$		0			0
$355$	2368.89	−	2368.89i	0.354162	−	0.354162i
$356$	174.570	−	1634.28i	0.0259893	−	0.243305i
$357$		0			0
$358$	−557.482	+	10467.7i	−0.0823013	+	1.54535i
$359$		−	1994.39i		−	0.293203i	−0.989196	−	0.146601i	$-0.953167\pi$
$359$		−	1994.39i		−	0.293203i	0.989196	−	0.146601i	$-0.0468335\pi$
$360$		0			0
$361$		−	440.519i		−	0.0642249i
$362$	11699.2	+	623.070i	1.69861	+	0.0904636i
$363$		0			0
$364$	−10049.3	+	8109.62i	−1.44705	+	1.16775i
$365$	4649.84	−	4649.84i	0.666805	−	0.666805i
$366$		0			0
$367$		−	11626.5i		−	1.65367i	−0.562445	−	0.826835i	$-0.690139\pi$
$367$		−	11626.5i		−	1.65367i	0.562445	−	0.826835i	$-0.309861\pi$
$368$	−1434.06	+	6636.07i	−0.203141	+	0.940024i
$369$		0			0
$370$	4054.87	+	4511.07i	0.569737	+	0.633836i
$371$	−3591.80	−	3591.80i	−0.502633	−	0.502633i
$372$		0			0
$373$	−6474.18	+	6474.18i	−0.898713	+	0.898713i	−0.995322	−	0.0966090i	$-0.969200\pi$
$373$	−6474.18	+	6474.18i	−0.898713	+	0.898713i	0.0966090	+	0.995322i	$0.469200\pi$
$374$	−750.232	+	14086.9i	−0.103726	+	1.94764i
$375$		0			0
$376$	5774.21	−	4172.82i	0.791973	−	0.572332i
$377$	−3646.21			−0.498115
$378$		0			0
$379$	2278.55	+	2278.55i	0.308816	+	0.308816i	0.844450	−	0.535634i	$-0.179927\pi$
$379$	2278.55	+	2278.55i	0.308816	+	0.308816i	−0.535634	+	0.844450i	$0.679927\pi$
$380$	−1159.75	+	10857.3i	−0.156563	+	1.46570i
$381$		0			0
$382$	−3315.09	−	3688.06i	−0.444018	−	0.493973i
$383$	6258.43			0.834963			0.417481	−	0.908685i	$-0.362913\pi$
$383$	6258.43			0.834963			0.417481	+	0.908685i	$0.362913\pi$
$384$		0			0
$385$	14891.2			1.97124
$386$	−1829.13	−	2034.92i	−0.241192	−	0.268328i
$387$		0			0
$388$	658.569	−	6165.34i	0.0861695	−	0.806695i
$389$	−4476.87	−	4476.87i	−0.583512	−	0.583512i	0.352354	−	0.935867i	$-0.385381\pi$
$389$	−4476.87	−	4476.87i	−0.583512	−	0.583512i	−0.935867	+	0.352354i	$0.885381\pi$
$390$		0			0
$391$	10897.1			1.40944
$392$	−469.165	+	339.050i	−0.0604501	+	0.0436852i
$393$		0			0
$394$	382.649	−	7184.88i	0.0489278	−	0.918703i
$395$	−11212.4	+	11212.4i	−1.42825	+	1.42825i
$396$		0			0
$397$	−4550.42	−	4550.42i	−0.575263	−	0.575263i	0.358332	−	0.933594i	$-0.383346\pi$
$397$	−4550.42	−	4550.42i	−0.575263	−	0.575263i	−0.933594	+	0.358332i	$0.883346\pi$
$398$	−9215.14	−	10251.9i	−1.16059	−	1.29116i
$399$		0			0
$400$	1760.23	−	8145.38i	0.220029	−	1.01817i
$401$			10156.3i			1.26479i	0.774645	+	0.632396i	$0.217929\pi$
$401$			10156.3i			1.26479i	−0.774645	+	0.632396i	$0.782071\pi$
$402$		0			0
$403$	5931.59	−	5931.59i	0.733185	−	0.733185i
$404$	3735.61	−	3014.57i	0.460034	−	0.371239i
$405$		0			0
$406$	−2351.56	−	125.238i	−0.287453	−	0.0153090i
$407$			6517.74i			0.793790i
$408$		0			0
$409$		−	8099.91i		−	0.979253i	−0.871932	−	0.489627i	$-0.837133\pi$
$409$		−	8099.91i		−	0.979253i	0.871932	−	0.489627i	$-0.162867\pi$
$410$	82.9225	−	1557.01i	0.00998842	−	0.187550i
$411$		0			0
$412$	92.3711	−	864.753i	0.0110456	−	0.103406i
$413$	−7344.74	+	7344.74i	−0.875087	+	0.875087i
$414$		0			0
$415$			19250.7i			2.27705i
$416$	−4001.41	+	14684.3i	−0.471599	+	1.73067i
$417$		0			0
$418$	−8725.94	+	7843.49i	−1.02105	+	0.917794i
$419$	3174.43	+	3174.43i	0.370122	+	0.370122i	0.867522	−	0.497400i	$-0.165712\pi$
$419$	3174.43	+	3174.43i	0.370122	+	0.370122i	−0.497400	+	0.867522i	$0.665712\pi$
$420$		0			0
$421$	−1382.13	+	1382.13i	−0.160002	+	0.160002i	−0.782568	−	0.622565i	$-0.786090\pi$
$421$	−1382.13	+	1382.13i	−0.160002	+	0.160002i	0.622565	+	0.782568i	$0.286090\pi$
$422$	−6441.79	−	343.073i	−0.743083	−	0.0395747i
$423$		0			0
$424$	−5910.69	−	951.569i	−0.677001	−	0.108991i
$425$	−13375.6			−1.52661
$426$		0			0
$427$	−9777.30	−	9777.30i	−1.10810	−	1.10810i
$428$	7321.08	+	9072.17i	0.826817	+	1.02458i
$429$		0			0
$430$	−9413.43	+	8461.46i	−1.05571	+	0.948949i
$431$	−5143.97			−0.574887			−0.287444	−	0.957798i	$-0.592805\pi$
$431$	−5143.97			−0.574887			−0.287444	+	0.957798i	$0.592805\pi$
$432$		0			0
$433$	14855.1			1.64871			0.824354	−	0.566075i	$-0.191539\pi$
$433$	14855.1			1.64871			0.824354	+	0.566075i	$0.191539\pi$
$434$	4029.21	−	3621.74i	0.445642	−	0.400574i
$435$		0			0
$436$	−5572.32	+	4496.77i	−0.612078	+	0.493936i
$437$	6408.76	+	6408.76i	0.701539	+	0.701539i
$438$		0			0
$439$	6820.25			0.741487			0.370743	−	0.928735i	$-0.379103\pi$
$439$	6820.25			0.741487			0.370743	+	0.928735i	$0.379103\pi$
$440$	14225.1	−	10280.0i	1.54127	−	1.11382i
$441$		0			0
$442$	24393.9	+	1299.16i	2.62512	+	0.139807i
$443$	3410.66	−	3410.66i	0.365791	−	0.365791i	−0.500149	−	0.865940i	$-0.666721\pi$
$443$	3410.66	−	3410.66i	0.365791	−	0.365791i	0.865940	+	0.500149i	$0.166721\pi$
$444$		0			0
$445$	2320.78	+	2320.78i	0.247226	+	0.247226i
$446$	−8349.84	+	7505.43i	−0.886494	+	0.796844i
$447$		0			0
$448$	−3085.01	+	9332.97i	−0.325341	+	0.984244i
$449$			11673.0i			1.22692i	0.789728	+	0.613458i	$0.210222\pi$
$449$			11673.0i			1.22692i	−0.789728	+	0.613458i	$0.789778\pi$
$450$		0			0
$451$	1184.72	−	1184.72i	0.123694	−	0.123694i
$452$	8159.54	+	871.585i	0.849099	+	0.0906989i
$453$		0			0
$454$	−106.112	+	1992.43i	−0.0109693	+	0.205968i
$455$		−	25786.8i		−	2.65693i
$456$		0			0
$457$		−	18011.9i		−	1.84368i	−0.387572	−	0.921839i	$-0.626686\pi$
$457$		−	18011.9i		−	1.84368i	0.387572	−	0.921839i	$-0.373314\pi$
$458$	−14837.3	−	790.195i	−1.51375	−	0.0806187i
$459$		0			0
$460$	−8514.16	−	10550.6i	−0.862988	−	1.06940i
$461$	−817.679	+	817.679i	−0.0826098	+	0.0826098i	−0.747204	−	0.664594i	$-0.768604\pi$
$461$	−817.679	+	817.679i	−0.0826098	+	0.0826098i	0.664594	+	0.747204i	$0.268604\pi$
$462$		0			0
$463$			15535.3i			1.55936i	0.626177	+	0.779681i	$0.284619\pi$
$463$			15535.3i			1.55936i	−0.626177	+	0.779681i	$0.715381\pi$
$464$	−2332.83	+	1503.74i	−0.233403	+	0.150451i
$465$		0			0
$466$	−7642.21	−	8502.01i	−0.759696	−	0.845167i
$467$	−4070.59	−	4070.59i	−0.403350	−	0.403350i	0.476062	−	0.879412i	$-0.342064\pi$
$467$	−4070.59	−	4070.59i	−0.403350	−	0.403350i	−0.879412	+	0.476062i	$0.842064\pi$
$468$		0			0
$469$	2054.56	−	2054.56i	0.202283	−	0.202283i
$470$	−756.586	+	14206.2i	−0.0742526	+	1.39422i
$471$		0			0
$472$	−1945.83	+	12086.6i	−0.189754	+	1.17866i
$473$	−13600.8			−1.32213
$474$		0			0
$475$	−7866.38	−	7866.38i	−0.759861	−	0.759861i
$476$	15687.8	+	1675.74i	1.51061	+	0.161360i
$477$		0			0
$478$	−186.107	−	207.046i	−0.0178083	−	0.0198118i
$479$	13672.3			1.30418			0.652092	−	0.758140i	$-0.273892\pi$
$479$	13672.3			1.30418			0.652092	+	0.758140i	$0.273892\pi$
$480$		0			0
$481$	11286.6			1.06991
$482$	−13211.7	−	14698.1i	−1.24850	−	1.38896i
$483$		0			0
$484$	8164.65	+	872.131i	0.766778	+	0.0819056i
$485$	8755.16	+	8755.16i	0.819694	+	0.819694i
$486$		0			0
$487$	−16464.2			−1.53196			−0.765981	−	0.642863i	$-0.777747\pi$
$487$	−16464.2			−1.53196			−0.765981	+	0.642863i	$0.777747\pi$
$488$	−16089.6	−	2590.29i	−1.49251	−	0.240280i
$489$		0			0
$490$	61.4741	−	1154.28i	0.00566758	−	0.106419i
$491$	11659.2	−	11659.2i	1.07163	−	1.07163i	0.0744061	−	0.997228i	$-0.476294\pi$
$491$	11659.2	−	11659.2i	1.07163	−	1.07163i	0.997228	−	0.0744061i	$-0.0237061\pi$
$492$		0			0
$493$	3150.03	+	3150.03i	0.287769	+	0.287769i
$494$	13582.4	+	15110.5i	1.23705	+	1.37622i
$495$		0			0
$496$	1348.75	−	6241.27i	0.122098	−	0.565002i
$497$		−	4026.04i		−	0.363366i
$498$		0			0
$499$	5942.66	−	5942.66i	0.533126	−	0.533126i	−0.388375	−	0.921501i	$-0.626963\pi$
$499$	5942.66	−	5942.66i	0.533126	−	0.533126i	0.921501	+	0.388375i	$0.126963\pi$
$500$	418.112	+	518.118i	0.0373971	+	0.0463419i
$501$		0			0
$502$	4211.35	+	224.286i	0.374426	+	0.0199410i
$503$		−	9432.81i		−	0.836160i	−0.908410	−	0.418080i	$-0.862703\pi$
$503$		−	9432.81i		−	0.836160i	0.908410	−	0.418080i	$-0.137297\pi$
$504$		0			0
$505$			9585.68i			0.844667i
$506$	774.761	−	14547.5i	0.0680678	−	1.27809i
$507$		0			0
$508$	18181.6	+	1942.12i	1.58795	+	0.169622i
$509$	−360.977	+	360.977i	−0.0314343	+	0.0314343i	−0.722649	−	0.691215i	$-0.757076\pi$
$509$	−360.977	+	360.977i	−0.0314343	+	0.0314343i	0.691215	+	0.722649i	$0.257076\pi$
$510$		0			0
$511$		−	7902.64i		−	0.684133i
$512$	3495.91	+	11045.2i	0.301756	+	0.953385i
$513$		0			0
$514$	335.134	−	301.243i	0.0287590	−	0.0258507i
$515$	1228.00	+	1228.00i	0.105072	+	0.105072i
$516$		0			0
$517$	−10809.4	+	10809.4i	−0.919527	+	0.919527i
$518$	7279.12	+	387.667i	0.617425	+	0.0328825i
$519$		0			0
$520$	−17801.7	−	24633.3i	−1.50126	−	2.07739i
$521$	7038.01			0.591825			0.295913	−	0.955215i	$-0.404376\pi$
$521$	7038.01			0.591825			0.295913	+	0.955215i	$0.404376\pi$
$522$		0			0
$523$	−7004.77	−	7004.77i	−0.585655	−	0.585655i	0.350797	−	0.936452i	$-0.385911\pi$
$523$	−7004.77	−	7004.77i	−0.585655	−	0.585655i	−0.936452	+	0.350797i	$0.885911\pi$
$524$	2441.52	−	1970.26i	0.203546	−	0.164258i
$525$		0			0
$526$	−11335.0	+	10188.7i	−0.939601	+	0.844580i
$527$	−10248.8			−0.847144
$528$		0			0
$529$	913.601			0.0750884
$530$	8891.14	−	7991.99i	0.728692	−	0.655000i
$531$		0			0
$532$	8240.73	+	10211.8i	0.671580	+	0.832212i
$533$	−2051.54	−	2051.54i	−0.166721	−	0.166721i
$534$		0			0
$535$	−23279.4			−1.88123
$536$	544.312	−	3381.00i	0.0438632	−	0.272457i
$537$		0			0
$538$	−5364.19	−	285.683i	−0.429864	−	0.0228935i
$539$	878.282	−	878.282i	0.0701860	−	0.0701860i
$540$		0			0
$541$	54.6730	+	54.6730i	0.00434487	+	0.00434487i	0.709276	−	0.704931i	$-0.249022\pi$
$541$	54.6730	+	54.6730i	0.00434487	+	0.00434487i	−0.704931	+	0.709276i	$0.749022\pi$
$542$	2072.16	−	1862.61i	0.164220	−	0.147612i
$543$		0			0
$544$	16142.9	−	9229.16i	1.27228	−	0.727384i
$545$		−	14298.7i		−	1.12383i
$546$		0			0
$547$	8483.80	−	8483.80i	0.663146	−	0.663146i	−0.292974	−	0.956120i	$-0.594645\pi$
$547$	8483.80	−	8483.80i	0.663146	−	0.663146i	0.956120	+	0.292974i	$0.0946449\pi$
$548$	−623.540	+	5837.41i	−0.0486064	+	0.455040i
$549$		0			0
$550$	−950.973	+	17856.2i	−0.0737266	+	1.38434i
$551$			3705.16i			0.286470i
$552$		0			0
$553$			19056.1i			1.46536i
$554$	−8010.62	−	426.625i	−0.614330	−	0.0327176i
$555$		0			0
$556$	6050.68	−	4882.79i	0.461521	−	0.372439i
$557$	1467.75	−	1467.75i	0.111653	−	0.111653i	−0.649073	−	0.760726i	$-0.724843\pi$
$557$	1467.75	−	1467.75i	0.111653	−	0.111653i	0.760726	+	0.649073i	$0.224843\pi$
$558$		0			0
$559$			23552.3i			1.78203i
$560$	−10634.8	−	16498.3i	−0.802504	−	1.24496i
$561$		0			0
$562$	10678.4	+	11879.8i	0.801496	+	0.891670i
$563$	−11322.6	−	11322.6i	−0.847589	−	0.847589i	0.142243	−	0.989832i	$-0.454569\pi$
$563$	−11322.6	−	11322.6i	−0.847589	−	0.847589i	−0.989832	+	0.142243i	$0.954569\pi$
$564$		0			0
$565$	−11587.1	+	11587.1i	−0.862780	+	0.862780i
$566$	508.824	−	9554.04i	0.0377870	−	0.709516i
$567$		0			0
$568$	−2779.34	−	3845.95i	−0.205314	−	0.284107i
$569$	7736.59			0.570008			0.285004	−	0.958526i	$-0.408005\pi$
$569$	7736.59			0.570008			0.285004	+	0.958526i	$0.408005\pi$
$570$		0			0
$571$	−9925.92	−	9925.92i	−0.727472	−	0.727472i	0.242643	−	0.970116i	$-0.421986\pi$
$571$	−9925.92	−	9925.92i	−0.727472	−	0.727472i	−0.970116	+	0.242643i	$0.921986\pi$
$572$	3468.71	−	32473.1i	0.253556	−	2.37372i
$573$		0			0
$574$	−1252.64	−	1393.57i	−0.0910877	−	0.101336i
$575$	13812.9			1.00180
$576$		0			0
$577$	4772.65			0.344347			0.172173	−	0.985067i	$-0.444921\pi$
$577$	4772.65			0.344347			0.172173	+	0.985067i	$0.444921\pi$
$578$	−10662.5	−	11862.1i	−0.767300	−	0.853627i
$579$		0			0
$580$	588.679	−	5511.05i	0.0421441	−	0.394541i
$581$	16358.7	+	16358.7i	1.16811	+	1.16811i
$582$		0			0
$583$	12846.2			0.912583
$584$	−5455.51	−	7549.14i	−0.386559	−	0.534907i
$585$		0			0
$586$	153.198	−	2876.55i	0.0107995	−	0.202780i
$587$	−4567.88	+	4567.88i	−0.321187	+	0.321187i	−0.849222	−	0.528036i	$-0.822929\pi$
$587$	−4567.88	+	4567.88i	−0.321187	+	0.321187i	0.528036	+	0.849222i	$0.322929\pi$
$588$		0			0
$589$	−6027.48	−	6027.48i	−0.421661	−	0.421661i
$590$	−16342.5	−	18181.2i	−1.14036	−	1.26866i
$591$		0			0
$592$	7221.13	−	4654.74i	0.501329	−	0.323156i
$593$			23559.5i			1.63149i	0.578411	+	0.815746i	$0.303673\pi$
$593$			23559.5i			1.63149i	−0.578411	+	0.815746i	$0.696327\pi$
$594$		0			0
$595$	−22277.7	+	22277.7i	−1.53495	+	1.53495i
$596$	12155.1	−	9808.94i	0.835389	−	0.674144i
$597$		0			0
$598$	−25191.5	−	1341.64i	−1.72267	−	0.0917451i
$599$			2962.86i			0.202102i	0.994881	+	0.101051i	$0.0322205\pi$
$599$			2962.86i			0.202102i	−0.994881	+	0.101051i	$0.967779\pi$
$600$		0			0
$601$		−	2867.08i		−	0.194594i	−0.995255	−	0.0972969i	$-0.968980\pi$
$601$		−	2867.08i		−	0.194594i	0.995255	−	0.0972969i	$-0.0310196\pi$
$602$	−808.961	+	15189.6i	−0.0547688	+	1.02838i
$603$		0			0
$604$	−492.022	+	4606.18i	−0.0331459	+	0.310303i
$605$	−11594.3	+	11594.3i	−0.779133	+	0.779133i
$606$		0			0
$607$		−	14432.9i		−	0.965094i	−0.875870	−	0.482547i	$-0.839712\pi$
$607$		−	14432.9i		−	0.965094i	0.875870	−	0.482547i	$-0.160288\pi$
$608$	14921.7	+	4066.09i	0.995321	+	0.271220i
$609$		0			0
$610$	24202.8	−	21755.2i	1.60646	−	1.44400i
$611$	18718.3	+	18718.3i	1.23938	+	1.23938i
$612$		0			0
$613$	−3242.15	+	3242.15i	−0.213620	+	0.213620i	−0.805803	−	0.592183i	$-0.798266\pi$
$613$	−3242.15	+	3242.15i	−0.213620	+	0.213620i	0.592183	+	0.805803i	$0.298266\pi$
$614$	−23471.5	−	1250.03i	−1.54272	−	0.0821616i
$615$		0			0
$616$	3352.46	−	20823.9i	0.219277	−	1.36204i
$617$	10722.8			0.699651			0.349826	−	0.936815i	$-0.386241\pi$
$617$	10722.8			0.699651			0.349826	+	0.936815i	$0.386241\pi$
$618$		0			0
$619$	−12578.8	−	12578.8i	−0.816779	−	0.816779i	0.168861	−	0.985640i	$-0.445991\pi$
$619$	−12578.8	−	12578.8i	−0.816779	−	0.816779i	−0.985640	+	0.168861i	$0.945991\pi$
$620$	8007.62	+	9922.92i	0.518700	+	0.642765i
$621$		0			0
$622$	7580.39	−	6813.79i	0.488659	−	0.439241i
$623$	3944.27			0.253650
$624$		0			0
$625$	14946.7			0.956587
$626$	−17042.4	+	15318.9i	−1.08810	+	0.978060i
$627$		0			0
$628$	−2990.12	+	2412.97i	−0.189998	+	0.153325i
$629$	−9750.71	−	9750.71i	−0.618102	−	0.618102i
$630$		0			0
$631$	16416.5			1.03570			0.517852	−	0.855470i	$-0.326732\pi$
$631$	16416.5			1.03570			0.517852	+	0.855470i	$0.326732\pi$
$632$	13155.2	+	18203.7i	0.827982	+	1.14573i
$633$		0			0
$634$	−17613.0	−	938.021i	−1.10331	−	0.0587596i
$635$	−25819.0	+	25819.0i	−1.61354	+	1.61354i
$636$		0			0
$637$	−1520.90	−	1520.90i	−0.0946001	−	0.0946001i
$638$	4429.19	−	3981.27i	0.274849	−	0.247053i
$639$		0			0
$640$	−21548.5	−	8418.68i	−1.33091	−	0.519965i
$641$		−	19386.0i		−	1.19454i	−0.802040	−	0.597271i	$-0.796252\pi$
$641$		−	19386.0i		−	1.19454i	0.802040	−	0.597271i	$-0.203748\pi$
$642$		0			0
$643$	−16044.8	+	16044.8i	−0.984050	+	0.984050i	−0.999875	−	0.0158251i	$-0.994963\pi$
$643$	−16044.8	+	16044.8i	−0.984050	+	0.984050i	0.0158251	+	0.999875i	$0.494963\pi$
$644$	−16200.8	−	1730.53i	−0.991303	−	0.105889i
$645$		0			0
$646$	1320.16	−	24788.3i	0.0804041	−	1.50973i
$647$		−	6318.29i		−	0.383922i	−0.981403	−	0.191961i	$-0.938515\pi$
$647$		−	6318.29i		−	0.383922i	0.981403	−	0.191961i	$-0.0614847\pi$
$648$		0			0
$649$		−	26268.8i		−	1.58881i
$650$	30921.1	+	1646.78i	1.86588	+	0.0993723i
$651$		0			0
$652$	12079.7	+	14969.0i	0.725579	+	0.899127i
$653$	−6586.42	+	6586.42i	−0.394711	+	0.394711i	−0.876363	−	0.481652i	$-0.840037\pi$
$653$	−6586.42	+	6586.42i	−0.394711	+	0.394711i	0.481652	+	0.876363i	$0.340037\pi$
$654$		0			0
$655$			6265.00i			0.373731i
$656$	−2158.65	−	466.488i	−0.128477	−	0.0277642i
$657$		0			0
$658$	11429.1	+	12715.0i	0.677134	+	0.753316i
$659$	−19630.8	−	19630.8i	−1.16041	−	1.16041i	−0.984386	−	0.176021i	$-0.943677\pi$
$659$	−19630.8	−	19630.8i	−1.16041	−	1.16041i	−0.176021	−	0.984386i	$-0.556323\pi$
$660$		0			0
$661$	−1685.31	+	1685.31i	−0.0991697	+	0.0991697i	−0.754951	−	0.655781i	$-0.772339\pi$
$661$	−1685.31	+	1685.31i	−0.0991697	+	0.0991697i	0.655781	+	0.754951i	$0.272339\pi$
$662$	−28.3668	+	532.636i	−0.00166542	+	0.0312711i
$663$		0			0
$664$	26920.0	+	4333.89i	1.57334	+	0.253294i
$665$	−26203.7			−1.52802
$666$		0			0
$667$	−3253.02	−	3253.02i	−0.188842	−	0.188842i
$668$	−13059.9	−	1395.04i	−0.756444	−	0.0808017i
$669$		0			0
$670$	4571.54	+	5085.87i	0.263603	+	0.293260i
$671$	34969.0			2.01187
$672$		0			0
$673$	−27572.2			−1.57924			−0.789621	−	0.613595i	$-0.789723\pi$
$673$	−27572.2			−1.57924			−0.789621	+	0.613595i	$0.789723\pi$
$674$	−690.650	−	768.353i	−0.0394701	−	0.0439108i
$675$		0			0
$676$	−38756.4	−	4139.87i	−2.20507	−	0.235541i
$677$	18179.4	+	18179.4i	1.03204	+	1.03204i	0.999470	+	0.0325680i	$0.0103686\pi$
$677$	18179.4	+	18179.4i	1.03204	+	1.03204i	0.0325680	+	0.999470i	$0.489631\pi$
$678$		0			0
$679$	14879.8			0.840994
$680$	−5901.99	+	36660.4i	−0.332840	+	2.06744i
$681$		0			0
$682$	−728.668	+	13682.0i	−0.0409122	+	0.768197i
$683$	−6405.32	+	6405.32i	−0.358848	+	0.358848i	−0.863388	−	0.504540i	$-0.831662\pi$
$683$	−6405.32	+	6405.32i	−0.358848	+	0.358848i	0.504540	+	0.863388i	$0.331662\pi$
$684$		0			0
$685$	−8289.48	−	8289.48i	−0.462372	−	0.462372i
$686$	11522.5	+	12818.8i	0.641297	+	0.713447i
$687$		0			0
$688$	9713.24	+	15068.6i	0.538247	+	0.835009i
$689$		−	22245.5i		−	1.23002i
$690$		0			0
$691$	798.974	−	798.974i	0.0439861	−	0.0439861i	−0.684772	−	0.728758i	$-0.740098\pi$
$691$	798.974	−	798.974i	0.0439861	−	0.0439861i	0.728758	+	0.684772i	$0.240098\pi$
$692$	758.234	+	939.592i	0.0416528	+	0.0516155i
$693$		0			0
$694$	24656.6	+	1313.15i	1.34863	+	0.0718249i
$695$			15526.2i			0.847398i
$696$		0			0
$697$			3544.73i			0.192635i
$698$	−33.6785	+	632.371i	−0.00182629	+	0.0342917i
$699$		0			0
$700$	19885.5	+	2124.12i	1.07371	+	0.114692i
$701$	2571.67	−	2571.67i	0.138560	−	0.138560i	−0.634425	−	0.772985i	$-0.718763\pi$
$701$	2571.67	−	2571.67i	0.138560	−	0.138560i	0.772985	+	0.634425i	$0.218763\pi$
$702$		0			0
$703$		−	11469.1i		−	0.615313i
$704$	−11173.0	−	22206.7i	−0.598153	−	1.18885i
$705$		0			0
$706$	18504.6	−	16633.3i	0.986447	−	0.886688i
$707$	8145.66	+	8145.66i	0.433309	+	0.433309i
$708$		0			0
$709$	−1247.12	+	1247.12i	−0.0660601	+	0.0660601i	−0.739365	−	0.673305i	$-0.764874\pi$
$709$	−1247.12	+	1247.12i	−0.0660601	+	0.0660601i	0.673305	+	0.739365i	$0.264874\pi$
$710$	9462.15	+	503.930i	0.500152	+	0.0266368i
$711$		0			0
$712$	3767.84	−	2722.89i	0.198323	−	0.143321i
$713$	10583.9			0.555919
$714$		0			0
$715$	46113.8	+	46113.8i	2.41197	+	2.41197i
$716$	−23073.2	+	18619.7i	−1.20431	+	0.971857i
$717$		0			0
$718$	4195.27	−	3771.00i	0.218058	−	0.196006i
$719$	9209.29			0.477675			0.238838	−	0.971059i	$-0.423234\pi$
$719$	9209.29			0.477675			0.238838	+	0.971059i	$0.423234\pi$
$720$		0			0
$721$	2087.05			0.107803
$722$	926.646	−	832.935i	0.0477648	−	0.0429344i
$723$		0			0
$724$	20810.3	+	25787.8i	1.06824	+	1.32375i
$725$	3992.89	+	3992.89i	0.204541	+	0.204541i
$726$		0			0
$727$	1546.22			0.0788807			0.0394403	−	0.999222i	$-0.487442\pi$
$727$	1546.22			0.0788807			0.0394403	+	0.999222i	$0.487442\pi$
$728$	−36060.2	−	5805.37i	−1.83582	−	0.295551i
$729$		0			0
$730$	18573.1	+	989.153i	0.941671	+	0.0501510i
$731$	20347.2	−	20347.2i	1.02951	−	1.02951i
$732$		0			0
$733$	11592.2	+	11592.2i	0.584132	+	0.584132i	0.936036	−	0.351904i	$-0.114466\pi$
$733$	11592.2	+	11592.2i	0.584132	+	0.584132i	−0.351904	+	0.936036i	$0.614466\pi$
$734$	24456.7	−	21983.4i	1.22985	−	1.10548i
$735$		0			0
$736$	−16670.7	+	9530.91i	−0.834906	+	0.477329i
$737$			7348.23i			0.367267i
$738$		0			0
$739$	17265.6	−	17265.6i	0.859438	−	0.859438i	−0.131834	−	0.991272i	$-0.542086\pi$
$739$	17265.6	−	17265.6i	0.859438	−	0.859438i	0.991272	+	0.131834i	$0.0420865\pi$
$740$	−1822.22	+	17059.1i	−0.0905217	+	0.847440i
$741$		0			0
$742$	764.077	−	14346.9i	0.0378034	−	0.709824i
$743$			6931.89i			0.342269i	0.985248	+	0.171135i	$0.0547433\pi$
$743$			6931.89i			0.342269i	−0.985248	+	0.171135i	$0.945257\pi$
$744$		0			0
$745$			31190.3i			1.53386i
$746$	−25860.1	−	1377.24i	−1.26917	−	0.0675930i
$747$		0			0
$748$	−31050.8	+	25057.4i	−1.51782	+	1.22485i
$749$	−19782.2	+	19782.2i	−0.965057	+	0.965057i
$750$		0			0
$751$			29358.2i			1.42649i	0.700915	+	0.713245i	$0.252775\pi$
$751$			29358.2i			1.42649i	−0.700915	+	0.713245i	$0.747225\pi$
$752$	19695.6	+	4256.24i	0.955085	+	0.206395i
$753$		0			0
$754$	−6894.28	−	7669.93i	−0.332990	−	0.370454i
$755$	−6541.05	−	6541.05i	−0.315302	−	0.315302i
$756$		0			0
$757$	11711.4	−	11711.4i	0.562296	−	0.562296i	−0.367663	−	0.929959i	$-0.619842\pi$
$757$	11711.4	−	11711.4i	0.562296	−	0.562296i	0.929959	+	0.367663i	$0.119842\pi$
$758$	−484.712	+	9101.30i	−0.0232263	+	0.436114i
$759$		0			0
$760$	−25031.6	+	18089.5i	−1.19472	+	0.863386i
$761$	6035.74			0.287510			0.143755	−	0.989613i	$-0.454082\pi$
$761$	6035.74			0.287510			0.143755	+	0.989613i	$0.454082\pi$
$762$		0			0
$763$	−12150.7	−	12150.7i	−0.576520	−	0.576520i
$764$	1489.77	−	13946.8i	0.0705470	−	0.660442i
$765$		0			0
$766$	11833.5	+	13164.8i	0.558173	+	0.620972i
$767$	−45489.0			−2.14148
$768$		0			0
$769$	−28555.0			−1.33904			−0.669519	−	0.742795i	$-0.733500\pi$
$769$	−28555.0			−1.33904			−0.669519	+	0.742795i	$0.733500\pi$
$770$	28156.4	+	31324.2i	1.31778	+	1.46603i
$771$		0			0
$772$	821.992	−	7695.27i	0.0383214	−	0.358755i
$773$	−19821.5	−	19821.5i	−0.922289	−	0.922289i	0.0749018	−	0.997191i	$-0.476136\pi$
$773$	−19821.5	−	19821.5i	−0.922289	−	0.922289i	−0.997191	+	0.0749018i	$0.976136\pi$
$774$		0			0
$775$	−12991.1			−0.602135
$776$	14214.2	−	10272.1i	0.657553	−	0.475191i
$777$		0			0
$778$	952.357	−	17882.1i	0.0438864	−	0.824043i
$779$	−2084.71	+	2084.71i	−0.0958826	+	0.0958826i
$780$		0			0
$781$	7199.66	+	7199.66i	0.329864	+	0.329864i
$782$	20604.3	+	22922.4i	0.942211	+	1.04822i
$783$		0			0
$784$	−1600.30	−	345.828i	−0.0729001	−	0.0157538i
$785$		−	7672.71i		−	0.348855i
$786$		0			0
$787$	23418.5	−	23418.5i	1.06071	−	1.06071i	0.0626785	−	0.998034i	$-0.480036\pi$
$787$	23418.5	−	23418.5i	1.06071	−	1.06071i	0.998034	−	0.0626785i	$-0.0199643\pi$
$788$	15837.2	−	12780.3i	0.715959	−	0.577766i
$789$		0			0
$790$	−44786.2	−	2385.20i	−2.01699	−	0.107420i
$791$			19692.8i			0.885201i
$792$		0			0
$793$		−	60554.9i		−	2.71169i
$794$	968.004	−	18175.9i	0.0432660	−	0.812393i
$795$		0			0
$796$	4141.19	−	38768.7i	0.184398	−	1.72628i
$797$	28091.3	−	28091.3i	1.24849	−	1.24849i	0.292100	−	0.956388i	$-0.405646\pi$
$797$	28091.3	−	28091.3i	1.24849	−	1.24849i	0.956388	−	0.292100i	$-0.0943541\pi$
$798$		0			0
$799$		−	32342.2i		−	1.43202i
$800$	20462.3	−	11698.6i	0.904316	−	0.517011i
$801$		0			0
$802$	−21364.1	+	19203.6i	−0.940641	+	0.845515i
$803$	14132.1	+	14132.1i	0.621058	+	0.621058i
$804$		0			0
$805$	23006.1	−	23006.1i	1.00728	−	1.00728i
$806$	23692.8	+	1261.82i	1.03541	+	0.0551434i
$807$		0			0
$808$	13404.6	+	2158.02i	0.583628	+	0.0939589i
$809$	11636.8			0.505721			0.252861	−	0.967503i	$-0.418629\pi$
$809$	11636.8			0.505721			0.252861	+	0.967503i	$0.418629\pi$
$810$		0			0
$811$	26276.4	+	26276.4i	1.13772	+	1.13772i	0.988858	+	0.148859i	$0.0475601\pi$
$811$	26276.4	+	26276.4i	1.13772	+	1.13772i	0.148859	+	0.988858i	$0.452440\pi$
$812$	−4182.90	−	5183.39i	−0.180777	−	0.224017i
$813$		0			0
$814$	−13710.3	+	12323.8i	−0.590351	+	0.530649i
$815$	−38410.8			−1.65088
$816$		0			0
$817$	23933.0			1.02486
$818$	17038.4	−	15315.3i	0.728282	−	0.654632i
$819$		0			0
$820$	3432.02	−	2769.58i	0.146160	−	0.117949i
$821$	−25422.7	−	25422.7i	−1.08071	−	1.08071i	−0.996444	−	0.0842629i	$-0.973146\pi$
$821$	−25422.7	−	25422.7i	−1.08071	−	1.08071i	−0.0842629	−	0.996444i	$-0.526854\pi$
$822$		0			0
$823$	−2843.64			−0.120441			−0.0602206	−	0.998185i	$-0.519180\pi$
$823$	−2843.64			−0.120441			−0.0602206	+	0.998185i	$0.519180\pi$
$824$	1993.69	−	1440.77i	0.0842883	−	0.0609123i
$825$		0			0
$826$	−29337.4	−	1562.43i	−1.23581	−	0.0658160i
$827$	−7433.12	+	7433.12i	−0.312545	+	0.312545i	−0.845895	−	0.533350i	$-0.820933\pi$
$827$	−7433.12	+	7433.12i	−0.312545	+	0.312545i	0.533350	+	0.845895i	$0.320933\pi$
$828$		0			0
$829$	3375.43	+	3375.43i	0.141415	+	0.141415i	0.774270	−	0.632855i	$-0.218117\pi$
$829$	3375.43	+	3375.43i	0.141415	+	0.141415i	−0.632855	+	0.774270i	$0.718117\pi$
$830$	−40494.4	+	36399.2i	−1.69347	+	1.52221i
$831$		0			0
$832$	−38454.8	+	19348.1i	−1.60238	+	0.806220i
$833$			2627.86i			0.109304i
$834$		0			0
$835$	18545.9	−	18545.9i	0.768632	−	0.768632i
$836$	−32998.1	−	3524.79i	−1.36515	−	0.145822i
$837$		0			0
$838$	−675.291	+	12679.8i	−0.0278372	+	0.522691i
$839$			3873.13i			0.159375i	0.996820	+	0.0796873i	$0.0253922\pi$
$839$			3873.13i			0.159375i	−0.996820	+	0.0796873i	$0.974608\pi$
$840$		0			0
$841$			22508.3i			0.922887i
$842$	−5520.71	−	294.019i	−0.225957	−	0.0120339i
$843$		0			0
$844$	−11458.5	−	14199.2i	−0.467320	−	0.579095i
$845$	55036.4	−	55036.4i	2.24060	−	2.24060i
$846$		0			0
$847$			19705.1i			0.799380i
$848$	−9174.31	−	14232.6i	−0.371518	−	0.576354i
$849$		0			0
$850$	−25290.6	−	28136.0i	−1.02054	−	1.13536i
$851$	10069.5	+	10069.5i	0.405615	+	0.405615i
$852$		0			0
$853$	−9011.15	+	9011.15i	−0.361707	+	0.361707i	−0.864441	−	0.502734i	$-0.832327\pi$
$853$	−9011.15	+	9011.15i	−0.361707	+	0.361707i	0.502734	+	0.864441i	$0.332327\pi$
$854$	2079.91	−	39053.9i	0.0833408	−	1.56487i
$855$		0			0
$856$	−5240.88	+	32553.8i	−0.209264	+	1.29985i
$857$	19088.6			0.760857			0.380429	−	0.924810i	$-0.375776\pi$
$857$	19088.6			0.760857			0.380429	+	0.924810i	$0.375776\pi$
$858$		0			0
$859$	−30431.6	−	30431.6i	−1.20875	−	1.20875i	−0.971433	−	0.237312i	$-0.923734\pi$
$859$	−30431.6	−	30431.6i	−1.20875	−	1.20875i	−0.237312	−	0.971433i	$-0.576266\pi$
$860$	−35597.9	−	3802.50i	−1.41149	−	0.150772i
$861$		0			0
$862$	−9726.25	−	10820.5i	−0.384313	−	0.427550i
$863$	−25203.6			−0.994136			−0.497068	−	0.867712i	$-0.665590\pi$
$863$	−25203.6			−0.994136			−0.497068	+	0.867712i	$0.665590\pi$
$864$		0			0
$865$	−2411.02			−0.0947712
$866$	28088.1	+	31248.2i	1.10216	+	1.22616i
$867$		0			0
$868$	15236.9	+	1627.58i	0.595823	+	0.0636446i
$869$	−34077.4	−	34077.4i	−1.33026	−	1.33026i
$870$		0			0
$871$	12724.8			0.495019
$872$	−19995.3	−	3219.06i	−0.776520	−	0.125013i
$873$		0			0
$874$	−1363.32	+	25598.8i	−0.0527633	+	0.990722i
$875$	−1129.78	+	1129.78i	−0.0436497	+	0.0436497i
$876$		0			0
$877$	−29609.6	−	29609.6i	−1.14008	−	1.14008i	−0.988436	−	0.151640i	$-0.951544\pi$
$877$	−29609.6	−	29609.6i	−1.14008	−	1.14008i	−0.151640	−	0.988436i	$-0.548456\pi$
$878$	12895.8	+	14346.6i	0.495684	+	0.551452i
$879$		0			0
$880$	48521.3	+	10485.5i	1.85870	+	0.401667i
$881$		−	32388.0i		−	1.23857i	−0.785166	−	0.619285i	$-0.787422\pi$
$881$		−	32388.0i		−	1.23857i	0.785166	−	0.619285i	$-0.212578\pi$
$882$		0			0
$883$	−15752.4	+	15752.4i	−0.600353	+	0.600353i	−0.940406	−	0.340053i	$-0.889555\pi$
$883$	−15752.4	+	15752.4i	−0.600353	+	0.600353i	0.340053	+	0.940406i	$0.389555\pi$
$884$	43391.4	+	53769.9i	1.65092	+	2.04579i
$885$		0			0
$886$	13623.3	+	725.544i	0.516574	+	0.0275114i
$887$			30496.9i			1.15444i	0.816589	+	0.577219i	$0.195862\pi$
$887$			30496.9i			1.15444i	−0.816589	+	0.577219i	$0.804138\pi$
$888$		0			0
$889$			43880.7i			1.65547i
$890$	−493.695	+	9269.97i	−0.0185940	+	0.349135i
$891$		0			0
$892$	−31575.8	−	3372.86i	−1.18524	−	0.126605i
$893$	19020.9	−	19020.9i	0.712779	−	0.712779i
$894$		0			0
$895$		−	59206.5i		−	2.21123i
$896$	−25465.4	+	11157.4i	−0.949484	+	0.416007i
$897$		0			0
$898$	−24554.6	+	22071.5i	−0.912471	+	0.820194i
$899$	3059.48	+	3059.48i	0.113503	+	0.113503i
$900$		0			0
$901$	−19218.3	+	19218.3i	−0.710603	+	0.710603i
$902$	4732.16	+	252.023i	0.174682	+	0.00930314i
$903$		0			0
$904$	13594.7	+	18811.9i	0.500169	+	0.692117i
$905$	−66172.1			−2.43054
$906$		0			0
$907$	18365.1	+	18365.1i	0.672331	+	0.672331i	0.958253	−	0.285922i	$-0.0922997\pi$
$907$	18365.1	+	18365.1i	0.672331	+	0.672331i	−0.285922	+	0.958253i	$0.592300\pi$
$908$	−4391.79	+	3544.09i	−0.160514	+	0.129532i
$909$		0			0
$910$	54243.4	−	48757.8i	1.97599	−	1.77616i
$911$	20989.1			0.763338			0.381669	−	0.924299i	$-0.375349\pi$
$911$	20989.1			0.763338			0.381669	+	0.924299i	$0.375349\pi$
$912$		0			0
$913$	−58507.6			−2.12083
$914$	37888.6	−	34057.0i	1.37117	−	1.23250i
$915$		0			0
$916$	−26392.2	−	32704.8i	−0.951989	−	1.17969i
$917$	5323.83	+	5323.83i	0.191721	+	0.191721i
$918$		0			0
$919$	14926.2			0.535768			0.267884	−	0.963451i	$-0.413676\pi$
$919$	14926.2			0.535768			0.267884	+	0.963451i	$0.413676\pi$
$920$	6094.96	−	37859.0i	0.218418	−	1.35671i
$921$		0			0
$922$	−3266.09	−	173.944i	−0.116663	−	0.00621315i
$923$	12467.5	−	12467.5i	0.444607	−	0.444607i
$924$		0			0
$925$	−12359.7	−	12359.7i	−0.439336	−	0.439336i
$926$	−32678.9	+	29374.1i	−1.15972	+	1.04243i
$927$		0			0
$928$	−7574.10	−	2063.91i	−0.267922	−	0.0730076i
$929$			31117.8i			1.09897i	0.835504	+	0.549484i	$0.185176\pi$
$929$			31117.8i			1.09897i	−0.835504	+	0.549484i	$0.814824\pi$
$930$		0			0
$931$	−1545.49	+	1545.49i	−0.0544053	+	0.0544053i
$932$	3434.33	−	32151.3i	0.120703	−	1.12999i
$933$		0			0
$934$	865.931	−	16259.3i	0.0303363	−	0.569616i
$935$		−	79677.1i		−	2.78687i
$936$		0			0
$937$		−	48051.8i		−	1.67533i	−0.546184	−	0.837665i	$-0.683920\pi$
$937$		−	48051.8i		−	1.67533i	0.546184	−	0.837665i	$-0.316080\pi$
$938$	8206.62	+	437.064i	0.285667	+	0.0152139i
$939$		0			0
$940$	−31313.8	+	25269.7i	−1.08654	+	0.876814i
$941$	1453.30	−	1453.30i	0.0503468	−	0.0503468i	−0.681485	−	0.731832i	$-0.738666\pi$
$941$	1453.30	−	1453.30i	0.0503468	−	0.0503468i	0.731832	+	0.681485i	$0.238666\pi$
$942$		0			0
$943$		−	3660.63i		−	0.126412i
$944$	−29103.7	+	18760.2i	−1.00344	+	0.646815i
$945$		0			0
$946$	−25716.5	−	28609.8i	−0.883844	−	0.983283i
$947$	−33015.9	−	33015.9i	−1.13292	−	1.13292i	−0.989690	−	0.143229i	$-0.954252\pi$
$947$	−33015.9	−	33015.9i	−1.13292	−	1.13292i	−0.143229	−	0.989690i	$-0.545748\pi$
$948$		0			0
$949$	24472.2	−	24472.2i	0.837091	−	0.837091i
$950$	1673.40	−	31421.0i	0.0571498	−	1.07309i
$951$		0			0
$952$	26137.7	+	36168.4i	0.889839	+	1.23133i
$953$	19443.8			0.660908			0.330454	−	0.943822i	$-0.392798\pi$
$953$	19443.8			0.660908			0.330454	+	0.943822i	$0.392798\pi$
$954$		0			0
$955$	19805.3	+	19805.3i	0.671084	+	0.671084i
$956$	83.6348	−	782.967i	0.00282944	−	0.0264884i
$957$		0			0
$958$	25851.7	+	28760.2i	0.871847	+	0.969936i
$959$	−14088.4			−0.474387
$960$		0			0
$961$	19836.8			0.665865
$962$	21340.8	+	23741.8i	0.715234	+	0.795703i
$963$		0			0
$964$	5937.20	−	55582.5i	0.198365	−	1.85704i
$965$	10927.7	+	10927.7i	0.364535	+	0.364535i
$966$		0			0
$967$	−28069.0			−0.933441			−0.466720	−	0.884405i	$-0.654565\pi$
$967$	−28069.0			−0.933441			−0.466720	+	0.884405i	$0.654565\pi$
$968$	13603.2	+	18823.7i	0.451678	+	0.625016i
$969$		0			0
$970$	−1862.47	+	34971.1i	−0.0616498	+	1.15758i
$971$	−82.8628	+	82.8628i	−0.00273861	+	0.00273861i	−0.708475	−	0.705736i	$-0.750616\pi$
$971$	−82.8628	+	82.8628i	−0.00273861	+	0.00273861i	0.705736	+	0.708475i	$0.250616\pi$
$972$		0			0
$973$	13193.8	+	13193.8i	0.434709	+	0.434709i
$974$	−31130.7	−	34633.1i	−1.02412	−	1.13934i
$975$		0			0
$976$	−24973.6	−	38742.8i	−0.819042	−	1.27062i
$977$			55424.4i			1.81493i	0.420131	+	0.907464i	$0.361984\pi$
$977$			55424.4i			1.81493i	−0.420131	+	0.907464i	$0.638016\pi$
$978$		0			0
$979$	−7053.43	+	7053.43i	−0.230264	+	0.230264i
$980$	2544.31	−	2053.21i	0.0829335	−	0.0669259i
$981$		0			0
$982$	46570.8	+	2480.24i	1.51337	+	0.0805985i
$983$			49024.3i			1.59067i	0.606167	+	0.795337i	$0.292706\pi$
$983$			49024.3i			1.59067i	−0.606167	+	0.795337i	$0.707294\pi$
$984$		0			0
$985$			40638.5i			1.31457i
$986$	−670.100	+	12582.3i	−0.0216433	+	0.406391i
$987$		0			0
$988$	−6103.79	+	57142.0i	−0.196546	+	1.84001i
$989$	−21012.5	+	21012.5i	−0.675589	+	0.675589i
$990$		0			0
$991$			21281.2i			0.682160i	0.940034	+	0.341080i	$0.110793\pi$
$991$			21281.2i			0.682160i	−0.940034	+	0.341080i	$0.889207\pi$
$992$	15678.9	−	8963.88i	0.501821	−	0.286899i
$993$		0			0
$994$	8468.92	−	7612.47i	0.270239	−	0.242910i
$995$	55054.0	+	55054.0i	1.75410	+	1.75410i
$996$		0			0
$997$	16897.3	−	16897.3i	0.536752	−	0.536752i	−0.385822	−	0.922573i	$-0.626082\pi$
$997$	16897.3	−	16897.3i	0.536752	−	0.536752i	0.922573	+	0.385822i	$0.126082\pi$
$998$	23737.0	+	1264.17i	0.752888	+	0.0400969i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	144.4.l.a.35.19	yes	48
3.2	odd	2	inner	144.4.l.a.35.6	✓	48
4.3	odd	2		576.4.l.a.431.4		48
8.3	odd	2		1152.4.l.a.863.21		48
8.5	even	2		1152.4.l.b.863.21		48
12.11	even	2		576.4.l.a.431.21		48
16.3	odd	4		1152.4.l.b.287.4		48
16.5	even	4		576.4.l.a.143.21		48
16.11	odd	4	inner	144.4.l.a.107.6	yes	48
16.13	even	4		1152.4.l.a.287.4		48
24.5	odd	2		1152.4.l.b.863.4		48
24.11	even	2		1152.4.l.a.863.4		48
48.5	odd	4		576.4.l.a.143.4		48
48.11	even	4	inner	144.4.l.a.107.19	yes	48
48.29	odd	4		1152.4.l.a.287.21		48
48.35	even	4		1152.4.l.b.287.21		48

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
144.4.l.a.35.6	✓	48	3.2	odd	2	inner
144.4.l.a.35.19	yes	48	1.1	even	1	trivial
144.4.l.a.107.6	yes	48	16.11	odd	4	inner
144.4.l.a.107.19	yes	48	48.11	even	4	inner
576.4.l.a.143.4		48	48.5	odd	4
576.4.l.a.143.21		48	16.5	even	4
576.4.l.a.431.4		48	4.3	odd	2
576.4.l.a.431.21		48	12.11	even	2
1152.4.l.a.287.4		48	16.13	even	4
1152.4.l.a.287.21		48	48.29	odd	4
1152.4.l.a.863.4		48	24.11	even	2
1152.4.l.a.863.21		48	8.3	odd	2
1152.4.l.b.287.4		48	16.3	odd	4
1152.4.l.b.287.21		48	48.35	even	4
1152.4.l.b.863.4		48	24.5	odd	2
1152.4.l.b.863.21		48	8.5	even	2

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:	\( N \)	\(=\)	\( 144 = 2^{4} \cdot 3^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	144.l (of order \(4\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(1.89081 + 2.10353i) q^{2} +(-0.849709 + 7.95475i) q^{4} +(-11.2962 - 11.2962i) q^{5} -19.1985 q^{7} +(-18.3397 + 13.2535i) q^{8} +(2.40303 - 45.1210i) q^{10} +(34.3321 - 34.3321i) q^{11} +(-59.4521 - 59.4521i) q^{13} +(-36.3006 - 40.3847i) q^{14} +(-62.5560 - 13.5184i) q^{16} +102.723i q^{17} +(-60.4132 + 60.4132i) q^{19} +(99.4571 - 80.2601i) q^{20} +(137.134 + 7.30341i) q^{22} -106.082i q^{23} +130.209i q^{25} +(12.6471 - 237.472i) q^{26} +(16.3131 - 152.719i) q^{28} +(30.6651 - 30.6651i) q^{29} +99.7709i q^{31} +(-89.8447 - 157.149i) q^{32} +(-216.082 + 194.230i) q^{34} +(216.870 + 216.870i) q^{35} +(-94.9220 + 94.9220i) q^{37} +(-241.311 - 12.8516i) q^{38} +(356.884 + 57.4552i) q^{40} +34.5075 q^{41} +(-198.078 - 198.078i) q^{43} +(243.931 + 302.276i) q^{44} +(223.147 - 200.581i) q^{46} -314.847 q^{47} +25.5819 q^{49} +(-273.900 + 246.201i) q^{50} +(523.443 - 422.409i) q^{52} +(187.087 + 187.087i) q^{53} -775.647 q^{55} +(352.095 - 254.447i) q^{56} +(122.487 + 6.52334i) q^{58} +(382.568 - 382.568i) q^{59} +(509.275 + 509.275i) q^{61} +(-209.871 + 188.647i) q^{62} +(160.690 - 486.130i) q^{64} +1343.17i q^{65} +(-107.017 + 107.017i) q^{67} +(-817.139 - 87.2851i) q^{68} +(-46.1345 + 866.254i) q^{70} +209.706i q^{71} +411.628i q^{73} +(-379.151 - 20.1926i) q^{74} +(-429.238 - 531.906i) q^{76} +(-659.125 + 659.125i) q^{77} -992.581i q^{79} +(553.939 + 859.354i) q^{80} +(65.2470 + 72.5877i) q^{82} +(-852.083 - 852.083i) q^{83} +(1160.39 - 1160.39i) q^{85} +(42.1367 - 791.189i) q^{86} +(-174.621 + 1084.66i) q^{88} -205.447 q^{89} +(1141.39 + 1141.39i) q^{91} +(843.856 + 90.1389i) q^{92} +(-595.315 - 662.292i) q^{94} +1364.88 q^{95} -775.052 q^{97} +(48.3705 + 53.8125i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 48 q - 120 q^{10} - 144 q^{16} - 48 q^{19} + 72 q^{22} + 72 q^{28} - 984 q^{34} - 1272 q^{40} + 864 q^{43} - 1416 q^{46} + 2352 q^{49} - 648 q^{52} - 576 q^{55} + 1128 q^{58} + 1824 q^{61} + 3024 q^{64}+ \cdots - 11304 q^{94}+O(q^{100}) \) 48 * q - 120 * q^10 - 144 * q^16 - 48 * q^19 + 72 * q^22 + 72 * q^28 - 984 * q^34 - 1272 * q^40 + 864 * q^43 - 1416 * q^46 + 2352 * q^49 - 648 * q^52 - 576 * q^55 + 1128 * q^58 + 1824 * q^61 + 3024 * q^64 + 816 * q^67 + 2664 * q^70 + 1920 * q^76 + 1200 * q^82 - 480 * q^85 - 4560 * q^88 - 3600 * q^91 - 11304 * q^94

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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