LMFDB - Embedded newform 144.4.l.a.35.12

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			35.12
Character	$\chi$	$=$	144.35
Dual form			144.4.l.a.107.12

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+(-0.0720364 + 2.82751i) q^{2} +(-7.98962 - 0.407367i) q^{4} +(-2.40838 - 2.40838i) q^{5} -11.7205 q^{7} +(1.72738 - 22.5614i) q^{8} +(6.98322 - 6.63623i) q^{10} +(34.7409 - 34.7409i) q^{11} +(-3.17950 - 3.17950i) q^{13} +(0.844304 - 33.1399i) q^{14} +(63.6681 + 6.50942i) q^{16} -98.0797i q^{17} +(-15.9562 + 15.9562i) q^{19} +(18.2610 + 20.2232i) q^{20} +(95.7276 + 100.733i) q^{22} -69.6819i q^{23} -113.399i q^{25} +(9.21909 - 8.76101i) q^{26} +(93.6425 + 4.77456i) q^{28} +(15.9649 - 15.9649i) q^{29} +121.295i q^{31} +(-22.9919 + 179.553i) q^{32} +(277.321 + 7.06531i) q^{34} +(28.2275 + 28.2275i) q^{35} +(-37.0567 + 37.0567i) q^{37} +(-43.9670 - 46.2658i) q^{38} +(-58.4966 + 50.1763i) q^{40} -59.3202 q^{41} +(-241.737 - 241.737i) q^{43} +(-291.719 + 263.414i) q^{44} +(197.026 + 5.01964i) q^{46} +395.106 q^{47} -205.629 q^{49} +(320.638 + 8.16889i) q^{50} +(24.1077 + 26.6982i) q^{52} +(-458.194 - 458.194i) q^{53} -167.339 q^{55} +(-20.2458 + 264.431i) q^{56} +(43.9910 + 46.2911i) q^{58} +(257.629 - 257.629i) q^{59} +(-373.295 - 373.295i) q^{61} +(-342.962 - 8.73764i) q^{62} +(-506.032 - 77.9441i) q^{64} +15.3149i q^{65} +(-648.397 + 648.397i) q^{67} +(-39.9545 + 783.619i) q^{68} +(-81.8469 + 77.7801i) q^{70} +787.139i q^{71} -1074.96i q^{73} +(-102.109 - 107.447i) q^{74} +(133.984 - 120.984i) q^{76} +(-407.181 + 407.181i) q^{77} +382.146i q^{79} +(-137.660 - 169.014i) q^{80} +(4.27322 - 167.728i) q^{82} +(491.315 + 491.315i) q^{83} +(-236.213 + 236.213i) q^{85} +(700.927 - 666.099i) q^{86} +(-723.792 - 843.814i) q^{88} -624.448 q^{89} +(37.2653 + 37.2653i) q^{91} +(-28.3861 + 556.732i) q^{92} +(-28.4620 + 1117.17i) q^{94} +76.8575 q^{95} +1665.45 q^{97} +(14.8128 - 581.419i) q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	−0.0720364	+	2.82751i	−0.0254687	+	0.999676i
$2$	−0.0720364	+	2.82751i	−0.0254687	+	0.999676i
$3$		0			0
$3$		0			0
$4$	−7.98962	−	0.407367i	−0.998703	−	0.0509209i
$5$	−2.40838	−	2.40838i	−0.215412	−	0.215412i	0.591150	−	0.806562i	$-0.298674\pi$
$5$	−2.40838	−	2.40838i	−0.215412	−	0.215412i	−0.806562	+	0.591150i	$0.798674\pi$
$6$		0			0
$7$	−11.7205			−0.632848			−0.316424	−	0.948618i	$-0.602482\pi$
$7$	−11.7205			−0.632848			−0.316424	+	0.948618i	$0.602482\pi$
$8$	1.72738	−	22.5614i	0.0763401	−	0.997082i
$9$		0			0
$10$	6.98322	−	6.63623i	0.220829	−	0.209856i
$11$	34.7409	−	34.7409i	0.952252	−	0.952252i	−0.0466586	−	0.998911i	$-0.514857\pi$
$11$	34.7409	−	34.7409i	0.952252	−	0.952252i	0.998911	+	0.0466586i	$0.0148573\pi$
$12$		0			0
$13$	−3.17950	−	3.17950i	−0.0678333	−	0.0678333i	0.672376	−	0.740210i	$-0.265274\pi$
$13$	−3.17950	−	3.17950i	−0.0678333	−	0.0678333i	−0.740210	+	0.672376i	$0.765274\pi$
$14$	0.844304	−	33.1399i	0.0161178	−	0.632643i
$15$		0			0
$16$	63.6681	+	6.50942i	0.994814	+	0.101710i
$17$		−	98.0797i		−	1.39928i	−0.714494	−	0.699642i	$-0.753343\pi$
$17$		−	98.0797i		−	1.39928i	0.714494	−	0.699642i	$-0.246657\pi$
$18$		0			0
$19$	−15.9562	+	15.9562i	−0.192664	+	0.192664i	−0.796846	−	0.604182i	$-0.793500\pi$
$19$	−15.9562	+	15.9562i	−0.192664	+	0.192664i	0.604182	+	0.796846i	$0.293500\pi$
$20$	18.2610	+	20.2232i	0.204164	+	0.226102i
$21$		0			0
$22$	95.7276	+	100.733i	0.927691	+	0.976196i
$23$		−	69.6819i		−	0.631725i	−0.948805	−	0.315863i	$-0.897706\pi$
$23$		−	69.6819i		−	0.631725i	0.948805	−	0.315863i	$-0.102294\pi$
$24$		0			0
$25$		−	113.399i		−	0.907195i
$26$	9.21909	−	8.76101i	0.0695390	−	0.0660837i
$27$		0			0
$28$	93.6425	+	4.77456i	0.632027	+	0.0322252i
$29$	15.9649	−	15.9649i	0.102228	−	0.102228i	−0.654143	−	0.756371i	$-0.726971\pi$
$29$	15.9649	−	15.9649i	0.102228	−	0.102228i	0.756371	+	0.654143i	$0.226971\pi$
$30$		0			0
$31$			121.295i			0.702748i	0.936235	+	0.351374i	$0.114285\pi$
$31$			121.295i			0.702748i	−0.936235	+	0.351374i	$0.885715\pi$
$32$	−22.9919	+	179.553i	−0.127013	+	0.991901i
$33$		0			0
$34$	277.321	+	7.06531i	1.39883	+	0.0356380i
$35$	28.2275	+	28.2275i	0.136323	+	0.136323i
$36$		0			0
$37$	−37.0567	+	37.0567i	−0.164651	+	0.164651i	−0.784623	−	0.619973i	$-0.787144\pi$
$37$	−37.0567	+	37.0567i	−0.164651	+	0.164651i	0.619973	+	0.784623i	$0.287144\pi$
$38$	−43.9670	−	46.2658i	−0.187694	−	0.197508i
$39$		0			0
$40$	−58.4966	+	50.1763i	−0.231228	+	0.198339i
$41$	−59.3202			−0.225958			−0.112979	−	0.993597i	$-0.536039\pi$
$41$	−59.3202			−0.225958			−0.112979	+	0.993597i	$0.536039\pi$
$42$		0			0
$43$	−241.737	−	241.737i	−0.857314	−	0.857314i	0.133707	−	0.991021i	$-0.457312\pi$
$43$	−241.737	−	241.737i	−0.857314	−	0.857314i	−0.991021	+	0.133707i	$0.957312\pi$
$44$	−291.719	+	263.414i	−0.999506	+	0.902527i
$45$		0			0
$46$	197.026	+	5.01964i	0.631520	+	0.0160892i
$47$	395.106			1.22622			0.613108	−	0.789999i	$-0.289919\pi$
$47$	395.106			1.22622			0.613108	+	0.789999i	$0.289919\pi$
$48$		0			0
$49$	−205.629			−0.599503
$50$	320.638	+	8.16889i	0.906901	+	0.0231051i
$51$		0			0
$52$	24.1077	+	26.6982i	0.0642912	+	0.0711995i
$53$	−458.194	−	458.194i	−1.18751	−	1.18751i	−0.977755	−	0.209750i	$-0.932735\pi$
$53$	−458.194	−	458.194i	−1.18751	−	1.18751i	−0.209750	−	0.977755i	$-0.567265\pi$
$54$		0			0
$55$	−167.339			−0.410254
$56$	−20.2458	+	264.431i	−0.0483117	+	0.631002i
$57$		0			0
$58$	43.9910	+	46.2911i	0.0995913	+	0.104799i
$59$	257.629	−	257.629i	0.568483	−	0.568483i	−0.363220	−	0.931703i	$-0.618323\pi$
$59$	257.629	−	257.629i	0.568483	−	0.568483i	0.931703	+	0.363220i	$0.118323\pi$
$60$		0			0
$61$	−373.295	−	373.295i	−0.783533	−	0.783533i	0.196892	−	0.980425i	$-0.436915\pi$
$61$	−373.295	−	373.295i	−0.783533	−	0.783533i	−0.980425	+	0.196892i	$0.936915\pi$
$62$	−342.962	−	8.73764i	−0.702520	−	0.0178981i
$63$		0			0
$64$	−506.032	−	77.9441i	−0.988344	−	0.152235i
$65$			15.3149i			0.0292243i
$66$		0			0
$67$	−648.397	+	648.397i	−1.18230	+	1.18230i	−0.203157	+	0.979146i	$0.565120\pi$
$67$	−648.397	+	648.397i	−1.18230	+	1.18230i	−0.979146	+	0.203157i	$0.934880\pi$
$68$	−39.9545	+	783.619i	−0.0712528	+	1.39747i
$69$		0			0
$70$	−81.8469	+	77.7801i	−0.139751	+	0.132807i
$71$			787.139i			1.31572i	0.753140	+	0.657861i	$0.228538\pi$
$71$			787.139i			1.31572i	−0.753140	+	0.657861i	$0.771462\pi$
$72$		0			0
$73$		−	1074.96i		−	1.72349i	−0.507340	−	0.861746i	$-0.669371\pi$
$73$		−	1074.96i		−	1.72349i	0.507340	−	0.861746i	$-0.330629\pi$
$74$	−102.109	−	107.447i	−0.160404	−	0.168791i
$75$		0			0
$76$	133.984	−	120.984i	0.202224	−	0.182603i
$77$	−407.181	+	407.181i	−0.602631	+	0.602631i
$78$		0			0
$79$			382.146i			0.544238i	0.962264	+	0.272119i	$0.0877245\pi$
$79$			382.146i			0.544238i	−0.962264	+	0.272119i	$0.912276\pi$
$80$	−137.660	−	169.014i	−0.192386	−	0.236205i
$81$		0			0
$82$	4.27322	−	167.728i	0.00575485	−	0.225884i
$83$	491.315	+	491.315i	0.649744	+	0.649744i	0.952931	−	0.303187i	$-0.0980506\pi$
$83$	491.315	+	491.315i	0.649744	+	0.649744i	−0.303187	+	0.952931i	$0.598051\pi$
$84$		0			0
$85$	−236.213	+	236.213i	−0.301423	+	0.301423i
$86$	700.927	−	666.099i	0.878871	−	0.835201i
$87$		0			0
$88$	−723.792	−	843.814i	−0.876778	−	1.02217i
$89$	−624.448			−0.743723			−0.371862	−	0.928288i	$-0.621280\pi$
$89$	−624.448			−0.743723			−0.371862	+	0.928288i	$0.621280\pi$
$90$		0			0
$91$	37.2653	+	37.2653i	0.0429282	+	0.0429282i
$92$	−28.3861	+	556.732i	−0.0321680	+	0.630906i
$93$		0			0
$94$	−28.4620	+	1117.17i	−0.0312301	+	1.22582i
$95$	76.8575			0.0830043
$96$		0			0
$97$	1665.45			1.74331			0.871654	−	0.490122i	$-0.163048\pi$
$97$	1665.45			1.74331			0.871654	+	0.490122i	$0.163048\pi$
$98$	14.8128	−	581.419i	0.0152686	−	0.599308i
$99$		0			0
$100$	−46.1952	+	906.018i	−0.0461952	+	0.906018i
$101$	−625.952	−	625.952i	−0.616678	−	0.616678i	0.327999	−	0.944678i	$-0.393626\pi$
$101$	−625.952	−	625.952i	−0.616678	−	0.616678i	−0.944678	+	0.327999i	$0.893626\pi$
$102$		0			0
$103$	1641.52			1.57033			0.785163	−	0.619289i	$-0.212579\pi$
$103$	1641.52			1.57033			0.785163	+	0.619289i	$0.212579\pi$
$104$	−77.2260	+	66.2416i	−0.0728138	+	0.0624570i
$105$		0			0
$106$	1328.55	−	1262.54i	1.21736	−	1.15688i
$107$	956.649	−	956.649i	0.864325	−	0.864325i	−0.127512	−	0.991837i	$-0.540699\pi$
$107$	956.649	−	956.649i	0.864325	−	0.864325i	0.991837	+	0.127512i	$0.0406991\pi$
$108$		0			0
$109$	1221.44	+	1221.44i	1.07333	+	1.07333i	0.997089	+	0.0762403i	$0.0242916\pi$
$109$	1221.44	+	1221.44i	1.07333	+	1.07333i	0.0762403	+	0.997089i	$0.475708\pi$
$110$	12.0545	−	473.152i	0.0104486	−	0.410121i
$111$		0			0
$112$	−746.223	−	76.2938i	−0.629567	−	0.0643668i
$113$			360.983i			0.300517i	0.988647	+	0.150258i	$0.0480105\pi$
$113$			360.983i			0.300517i	−0.988647	+	0.150258i	$0.951989\pi$
$114$		0			0
$115$	−167.821	+	167.821i	−0.136081	+	0.136081i
$116$	−134.057	+	121.050i	−0.107301	+	0.0968899i
$117$		0			0
$118$	709.891	+	747.008i	0.553820	+	0.582777i
$119$			1149.54i			0.885534i
$120$		0			0
$121$		−	1082.86i		−	0.813569i
$122$	1082.39	−	1028.60i	0.803235	−	0.763323i
$123$		0			0
$124$	49.4115	−	969.099i	0.0357846	−	0.701836i
$125$	−574.157	+	574.157i	−0.410833	+	0.410833i
$126$		0			0
$127$			1713.36i			1.19713i	0.801073	+	0.598566i	$0.204263\pi$
$127$			1713.36i			1.19713i	−0.801073	+	0.598566i	$0.795737\pi$
$128$	256.841	−	1425.20i	0.177357	−	0.984147i
$129$		0			0
$130$	−43.3030	−	1.10323i	−0.0292148	−	0.000744305i
$131$	−191.155	−	191.155i	−0.127491	−	0.127491i	0.640482	−	0.767973i	$-0.278735\pi$
$131$	−191.155	−	191.155i	−0.127491	−	0.127491i	−0.767973	+	0.640482i	$0.778735\pi$
$132$		0			0
$133$	187.015	−	187.015i	0.121927	−	0.121927i
$134$	−1786.64	−	1880.06i	−1.15181	−	1.21203i
$135$		0			0
$136$	−2212.81	−	169.421i	−1.39520	−	0.106821i
$137$	2130.50			1.32862			0.664311	−	0.747456i	$-0.268725\pi$
$137$	2130.50			1.32862			0.664311	+	0.747456i	$0.268725\pi$
$138$		0			0
$139$	−423.046	−	423.046i	−0.258146	−	0.258146i	0.566154	−	0.824300i	$-0.308431\pi$
$139$	−423.046	−	423.046i	−0.258146	−	0.258146i	−0.824300	+	0.566154i	$0.808431\pi$
$140$	−214.028	−	237.026i	−0.129205	−	0.143088i
$141$		0			0
$142$	−2225.64	−	56.7027i	−1.31529	−	0.0335097i
$143$	−220.917			−0.129189
$144$		0			0
$145$	−76.8994			−0.0440424
$146$	3039.47	+	77.4365i	1.72293	+	0.0438951i
$147$		0			0
$148$	311.164	−	280.973i	0.172821	−	0.156053i
$149$	−2020.44	−	2020.44i	−1.11088	−	1.11088i	−0.993032	−	0.117844i	$-0.962402\pi$
$149$	−2020.44	−	2020.44i	−1.11088	−	1.11088i	−0.117844	−	0.993032i	$-0.537598\pi$
$150$		0			0
$151$	2605.57			1.40423			0.702115	−	0.712064i	$-0.252239\pi$
$151$	2605.57			1.40423			0.702115	+	0.712064i	$0.252239\pi$
$152$	332.432	+	387.557i	0.177394	+	0.206810i
$153$		0			0
$154$	−1121.98	−	1180.64i	−0.587088	−	0.617784i
$155$	292.124	−	292.124i	0.151381	−	0.151381i
$156$		0			0
$157$	815.970	+	815.970i	0.414787	+	0.414787i	0.883402	−	0.468615i	$-0.155247\pi$
$157$	815.970	+	815.970i	0.414787	+	0.414787i	−0.468615	+	0.883402i	$0.655247\pi$
$158$	−1080.52	−	27.5285i	−0.544062	−	0.0138611i
$159$		0			0
$160$	487.806	−	377.060i	0.241028	−	0.186307i
$161$			816.708i			0.399786i
$162$		0			0
$163$	268.825	−	268.825i	0.129178	−	0.129178i	−0.639562	−	0.768740i	$-0.720884\pi$
$163$	268.825	−	268.825i	0.129178	−	0.129178i	0.768740	+	0.639562i	$0.220884\pi$
$164$	473.946	+	24.1651i	0.225664	+	0.0115060i
$165$		0			0
$166$	−1424.59	+	1353.80i	−0.666082	+	0.632986i
$167$			1537.53i			0.712440i	0.934402	+	0.356220i	$0.115935\pi$
$167$			1537.53i			0.712440i	−0.934402	+	0.356220i	$0.884065\pi$
$168$		0			0
$169$		−	2176.78i		−	0.990797i
$170$	−650.880	−	684.912i	−0.293648	−	0.309002i
$171$		0			0
$172$	1832.91	+	2029.86i	0.812547	+	0.899857i
$173$	−598.582	+	598.582i	−0.263060	+	0.263060i	−0.826296	−	0.563236i	$-0.809556\pi$
$173$	−598.582	+	598.582i	−0.263060	+	0.263060i	0.563236	+	0.826296i	$0.309556\pi$
$174$		0			0
$175$			1329.10i			0.574117i
$176$	2438.03	−	1985.74i	1.04417	−	0.850461i
$177$		0			0
$178$	44.9830	−	1765.63i	0.0189417	−	0.743482i
$179$	−285.351	−	285.351i	−0.119152	−	0.119152i	0.645017	−	0.764168i	$-0.276850\pi$
$179$	−285.351	−	285.351i	−0.119152	−	0.119152i	−0.764168	+	0.645017i	$0.776850\pi$
$180$		0			0
$181$	−2919.93	+	2919.93i	−1.19910	+	1.19910i	−0.224660	+	0.974437i	$0.572127\pi$
$181$	−2919.93	+	2919.93i	−1.19910	+	1.19910i	−0.974437	+	0.224660i	$0.927873\pi$
$182$	−108.053	+	102.684i	−0.0440076	+	0.0418210i
$183$		0			0
$184$	−1572.12	−	120.367i	−0.629882	−	0.0482260i
$185$	178.493			0.0709356
$186$		0			0
$187$	−3407.38	−	3407.38i	−1.33247	−	1.33247i
$188$	−3156.75	−	160.953i	−1.22462	−	0.0624400i
$189$		0			0
$190$	−5.53654	+	217.315i	−0.00211401	+	0.0829774i
$191$	−1085.25			−0.411129			−0.205564	−	0.978644i	$-0.565903\pi$
$191$	−1085.25			−0.411129			−0.205564	+	0.978644i	$0.565903\pi$
$192$		0			0
$193$	1150.76			0.429191			0.214595	−	0.976703i	$-0.431157\pi$
$193$	1150.76			0.429191			0.214595	+	0.976703i	$0.431157\pi$
$194$	−119.973	+	4709.08i	−0.0443998	+	1.74274i
$195$		0			0
$196$	1642.90	+	83.7667i	0.598725	+	0.0305272i
$197$	2302.47	+	2302.47i	0.832713	+	0.832713i	0.987887	−	0.155174i	$-0.0495939\pi$
$197$	2302.47	+	2302.47i	0.832713	+	0.832713i	−0.155174	+	0.987887i	$0.549594\pi$
$198$		0			0
$199$	1300.97			0.463432			0.231716	−	0.972783i	$-0.425566\pi$
$199$	1300.97			0.463432			0.231716	+	0.972783i	$0.425566\pi$
$200$	−2558.45	−	195.884i	−0.904548	−	0.0692554i
$201$		0			0
$202$	1814.98	−	1724.79i	0.632184	−	0.600772i
$203$	−187.117	+	187.117i	−0.0646949	+	0.0646949i
$204$		0			0
$205$	142.866	+	142.866i	0.0486740	+	0.0486740i
$206$	−118.249	+	4641.41i	−0.0399942	+	1.56982i
$207$		0			0
$208$	−181.736	−	223.129i	−0.0605822	−	0.0743809i
$209$			1108.67i			0.366929i
$210$		0			0
$211$	−1702.90	+	1702.90i	−0.555604	+	0.555604i	−0.928053	−	0.372448i	$-0.878518\pi$
$211$	−1702.90	+	1702.90i	−0.555604	+	0.555604i	0.372448	+	0.928053i	$0.378518\pi$
$212$	3474.14	+	3847.45i	1.12550	+	1.24643i
$213$		0			0
$214$	2636.02	+	2773.85i	0.842031	+	0.886058i
$215$			1164.39i			0.369352i
$216$		0			0
$217$		−	1421.64i		−	0.444733i
$218$	−3541.63	+	3365.65i	−1.10032	+	1.04565i
$219$		0			0
$220$	1336.97	+	68.1684i	0.409722	+	0.0208905i
$221$	−311.844	+	311.844i	−0.0949180	+	0.0949180i
$222$		0			0
$223$		−	5316.70i		−	1.59656i	−0.602288	−	0.798279i	$-0.705744\pi$
$223$		−	5316.70i		−	1.59656i	0.602288	−	0.798279i	$-0.294256\pi$
$224$	269.477	−	2104.46i	0.0803802	−	0.627723i
$225$		0			0
$226$	−1020.68	−	26.0039i	−0.300419	−	0.00765378i
$227$	−3132.85	−	3132.85i	−0.916013	−	0.916013i	0.0807239	−	0.996737i	$-0.474277\pi$
$227$	−3132.85	−	3132.85i	−0.916013	−	0.916013i	−0.996737	+	0.0807239i	$0.974277\pi$
$228$		0			0
$229$	−4530.55	+	4530.55i	−1.30737	+	1.30737i	−0.384059	+	0.923308i	$0.625474\pi$
$229$	−4530.55	+	4530.55i	−1.30737	+	1.30737i	−0.923308	+	0.384059i	$0.874526\pi$
$230$	−462.425	−	486.604i	−0.132571	−	0.139503i
$231$		0			0
$232$	−332.614	−	387.769i	−0.0941257	−	0.109734i
$233$	5381.47			1.51310			0.756549	−	0.653937i	$-0.226884\pi$
$233$	5381.47			1.51310			0.756549	+	0.653937i	$0.226884\pi$
$234$		0			0
$235$	−951.566	−	951.566i	−0.264142	−	0.264142i
$236$	−2163.31	+	1953.41i	−0.596693	+	0.538798i
$237$		0			0
$238$	−3250.35	−	82.8091i	−0.885247	−	0.0225534i
$239$	−4955.44			−1.34117			−0.670587	−	0.741831i	$-0.733958\pi$
$239$	−4955.44			−1.34117			−0.670587	+	0.741831i	$0.733958\pi$
$240$		0			0
$241$	−589.625			−0.157598			−0.0787989	−	0.996891i	$-0.525109\pi$
$241$	−589.625			−0.157598			−0.0787989	+	0.996891i	$0.525109\pi$
$242$	3061.80	+	78.0054i	0.813305	+	0.0207206i
$243$		0			0
$244$	2830.42	+	3134.55i	0.742619	+	0.822415i
$245$	495.234	+	495.234i	0.129140	+	0.129140i
$246$		0			0
$247$	101.466			0.0261381
$248$	2736.58	+	209.522i	0.700697	+	0.0536478i
$249$		0			0
$250$	−1582.07	−	1664.79i	−0.400237	−	0.421163i
$251$	2418.09	−	2418.09i	0.608082	−	0.608082i	−0.334362	−	0.942445i	$-0.608521\pi$
$251$	2418.09	−	2418.09i	0.608082	−	0.608082i	0.942445	+	0.334362i	$0.108521\pi$
$252$		0			0
$253$	−2420.81	−	2420.81i	−0.601562	−	0.601562i
$254$	−4844.53	−	123.424i	−1.19674	−	0.0304894i
$255$		0			0
$256$	4011.25	+	828.885i	0.979310	+	0.202365i
$257$		−	976.607i		−	0.237039i	−0.992952	−	0.118520i	$-0.962185\pi$
$257$		−	976.607i		−	0.237039i	0.992952	−	0.118520i	$-0.0378148\pi$
$258$		0			0
$259$	434.323	−	434.323i	0.104199	−	0.104199i
$260$	6.23878	−	122.360i	0.00148813	−	0.0291864i
$261$		0			0
$262$	554.264	−	526.724i	0.130697	−	0.124203i
$263$		−	1820.40i		−	0.426809i	−0.976964	−	0.213404i	$-0.931545\pi$
$263$		−	1820.40i		−	0.426809i	0.976964	−	0.213404i	$-0.0684552\pi$
$264$		0			0
$265$			2207.01i			0.511606i
$266$	515.316	+	542.260i	0.118782	+	0.124993i
$267$		0			0
$268$	5444.58	−	4916.31i	1.24097	−	1.12057i
$269$	4795.34	−	4795.34i	1.08690	−	1.08690i	0.0910580	−	0.995846i	$-0.470975\pi$
$269$	4795.34	−	4795.34i	1.08690	−	1.08690i	0.995846	−	0.0910580i	$-0.0290249\pi$
$270$		0			0
$271$			1294.24i			0.290108i	0.989424	+	0.145054i	$0.0463356\pi$
$271$			1294.24i			0.290108i	−0.989424	+	0.145054i	$0.953664\pi$
$272$	638.442	−	6244.55i	0.142321	−	1.39203i
$273$		0			0
$274$	−153.474	+	6024.02i	−0.0338383	+	1.32819i
$275$	−3939.60	−	3939.60i	−0.863879	−	0.863879i
$276$		0			0
$277$	701.252	−	701.252i	0.152109	−	0.152109i	−0.626950	−	0.779059i	$-0.715697\pi$
$277$	701.252	−	701.252i	0.152109	−	0.152109i	0.779059	+	0.626950i	$0.215697\pi$
$278$	1226.64	−	1165.69i	0.264637	−	0.251487i
$279$		0			0
$280$	685.611	−	588.092i	0.146332	−	0.125519i
$281$	5755.44			1.22185			0.610927	−	0.791687i	$-0.290797\pi$
$281$	5755.44			1.22185			0.610927	+	0.791687i	$0.290797\pi$
$282$		0			0
$283$	−3908.33	−	3908.33i	−0.820939	−	0.820939i	0.165303	−	0.986243i	$-0.447140\pi$
$283$	−3908.33	−	3908.33i	−0.820939	−	0.820939i	−0.986243	+	0.165303i	$0.947140\pi$
$284$	320.655	−	6288.94i	0.0669977	−	1.31401i
$285$		0			0
$286$	15.9141	−	624.645i	0.00329028	−	0.129147i
$287$	695.264			0.142997
$288$		0			0
$289$	−4706.62			−0.957993
$290$	5.53956	−	217.434i	0.00112170	−	0.0440281i
$291$		0			0
$292$	−437.905	+	8588.55i	−0.0877618	+	1.72126i
$293$	−3513.72	−	3513.72i	−0.700592	−	0.700592i	0.263946	−	0.964538i	$-0.414976\pi$
$293$	−3513.72	−	3513.72i	−0.700592	−	0.700592i	−0.964538	+	0.263946i	$0.914976\pi$
$294$		0			0
$295$	−1240.94			−0.244916
$296$	772.039	+	900.060i	0.151601	+	0.176740i
$297$		0			0
$298$	5858.34	−	5567.25i	1.13881	−	1.08222i
$299$	−221.553	+	221.553i	−0.0428520	+	0.0428520i
$300$		0			0
$301$	2833.28	+	2833.28i	0.542550	+	0.542550i
$302$	−187.696	+	7367.29i	−0.0357639	+	1.40377i
$303$		0			0
$304$	−1119.77	+	912.037i	−0.211260	+	0.172069i
$305$			1798.07i			0.337565i
$306$		0			0
$307$	2537.12	−	2537.12i	0.471666	−	0.471666i	−0.430788	−	0.902453i	$-0.641764\pi$
$307$	2537.12	−	2537.12i	0.471666	−	0.471666i	0.902453	+	0.430788i	$0.141764\pi$
$308$	3419.10	−	3087.35i	0.632536	−	0.571163i
$309$		0			0
$310$	804.941	+	847.028i	0.147476	+	0.155187i
$311$		−	7063.50i		−	1.28789i	−0.765071	−	0.643946i	$-0.777296\pi$
$311$		−	7063.50i		−	1.28789i	0.765071	−	0.643946i	$-0.222704\pi$
$312$		0			0
$313$		−	579.160i		−	0.104588i	−0.998632	−	0.0522940i	$-0.983347\pi$
$313$		−	579.160i		−	0.104588i	0.998632	−	0.0522940i	$-0.0166533\pi$
$314$	−2365.94	+	2248.38i	−0.425216	+	0.404088i
$315$		0			0
$316$	155.674	−	3053.21i	0.0277131	−	0.543532i
$317$	−340.991	+	340.991i	−0.0604164	+	0.0604164i	−0.736669	−	0.676253i	$-0.763603\pi$
$317$	−340.991	+	340.991i	−0.0604164	+	0.0604164i	0.676253	+	0.736669i	$0.263603\pi$
$318$		0			0
$319$		−	1109.27i		−	0.194694i
$320$	1031.00	+	1406.44i	0.180108	+	0.245695i
$321$		0			0
$322$	−2309.25	−	58.8327i	−0.399657	−	0.0101820i
$323$	1564.98	+	1564.98i	0.269591	+	0.269591i
$324$		0			0
$325$	−360.553	+	360.553i	−0.0615381	+	0.0615381i
$326$	740.739	+	779.469i	0.125846	+	0.132426i
$327$		0			0
$328$	−102.468	+	1338.35i	−0.0172496	+	0.225298i
$329$	−4630.85			−0.776009
$330$		0			0
$331$	2509.44	+	2509.44i	0.416711	+	0.416711i	0.884068	−	0.467358i	$-0.154794\pi$
$331$	2509.44	+	2509.44i	0.416711	+	0.416711i	−0.467358	+	0.884068i	$0.654794\pi$
$332$	−3725.27	−	4125.56i	−0.615816	−	0.681987i
$333$		0			0
$334$	−4347.37	−	110.758i	−0.712209	−	0.0181449i
$335$	3123.18			0.509365
$336$		0			0
$337$	1805.84			0.291899			0.145950	−	0.989292i	$-0.453376\pi$
$337$	1805.84			0.291899			0.145950	+	0.989292i	$0.453376\pi$
$338$	6154.87	+	156.808i	0.990476	+	0.0252343i
$339$		0			0
$340$	1983.48	−	1791.03i	0.316381	−	0.285683i
$341$	4213.89	+	4213.89i	0.669193	+	0.669193i
$342$		0			0
$343$	6430.22			1.01224
$344$	−5871.49	+	5036.35i	−0.920260	+	0.789365i
$345$		0			0
$346$	−1649.38	−	1735.62i	−0.256275	−	0.269674i
$347$	−3463.32	+	3463.32i	−0.535795	+	0.535795i	−0.922291	−	0.386496i	$-0.873685\pi$
$347$	−3463.32	+	3463.32i	−0.535795	+	0.535795i	0.386496	+	0.922291i	$0.373685\pi$
$348$		0			0
$349$	5797.35	+	5797.35i	0.889183	+	0.889183i	0.994445	−	0.105261i	$-0.0335679\pi$
$349$	5797.35	+	5797.35i	0.889183	+	0.889183i	−0.105261	+	0.994445i	$0.533568\pi$
$350$	−3758.04	−	95.7436i	−0.573931	−	0.0146220i
$351$		0			0
$352$	5439.08	+	7036.60i	0.823591	+	1.06549i
$353$		−	2793.32i		−	0.421172i	−0.977575	−	0.210586i	$-0.932463\pi$
$353$		−	2793.32i		−	0.421172i	0.977575	−	0.210586i	$-0.0675372\pi$
$354$		0			0
$355$	1895.73	−	1895.73i	0.283423	−	0.283423i
$356$	4989.10	+	254.380i	0.742758	+	0.0378711i
$357$		0			0
$358$	827.388	−	786.277i	0.122148	−	0.116078i
$359$			6896.95i			1.01395i	0.861962	+	0.506974i	$0.169236\pi$
$359$			6896.95i			1.01395i	−0.861962	+	0.506974i	$0.830764\pi$
$360$		0			0
$361$			6349.80i			0.925761i
$362$	−8045.78	−	8466.47i	−1.16817	−	1.22925i
$363$		0			0
$364$	−282.555	−	312.917i	−0.0406866	−	0.0450585i
$365$	−2588.92	+	2588.92i	−0.371261	+	0.371261i
$366$		0			0
$367$			1724.22i			0.245242i	0.992454	+	0.122621i	$0.0391299\pi$
$367$			1724.22i			0.245242i	−0.992454	+	0.122621i	$0.960870\pi$
$368$	453.589	−	4436.51i	0.0642526	−	0.628449i
$369$		0			0
$370$	−12.8580	+	504.691i	−0.00180664	+	0.0709126i
$371$	5370.27	+	5370.27i	0.751511	+	0.751511i
$372$		0			0
$373$	5937.62	−	5937.62i	0.824231	−	0.824231i	−0.162481	−	0.986712i	$-0.551949\pi$
$373$	5937.62	−	5937.62i	0.824231	−	0.824231i	0.986712	+	0.162481i	$0.0519495\pi$
$374$	9879.84	−	9388.93i	1.36597	−	1.29810i
$375$		0			0
$376$	682.498	−	8914.14i	0.0936094	−	1.22264i
$377$	−101.521			−0.0138689
$378$		0			0
$379$	8009.60	+	8009.60i	1.08556	+	1.08556i	0.995980	+	0.0895751i	$0.0285509\pi$
$379$	8009.60	+	8009.60i	1.08556	+	1.08556i	0.0895751	+	0.995980i	$0.471449\pi$
$380$	−614.062	−	31.3092i	−0.0828966	−	0.00422666i
$381$		0			0
$382$	78.1772	−	3068.54i	0.0104709	−	0.410995i
$383$	−12372.3			−1.65064			−0.825319	−	0.564667i	$-0.809005\pi$
$383$	−12372.3			−1.65064			−0.825319	+	0.564667i	$0.809005\pi$
$384$		0			0
$385$	1961.30			0.259628
$386$	−82.8970	+	3253.80i	−0.0109309	+	0.429052i
$387$		0			0
$388$	−13306.3	−	678.450i	−1.74105	−	0.0887709i
$389$	3950.83	+	3950.83i	0.514948	+	0.514948i	0.916039	−	0.401090i	$-0.131369\pi$
$389$	3950.83	+	3950.83i	0.514948	+	0.514948i	−0.401090	+	0.916039i	$0.631369\pi$
$390$		0			0
$391$	−6834.38			−0.883962
$392$	−355.200	+	4639.29i	−0.0457661	+	0.597753i
$393$		0			0
$394$	−6676.13	+	6344.40i	−0.853651	+	0.811235i
$395$	920.355	−	920.355i	0.117236	−	0.117236i
$396$		0			0
$397$	−9644.39	−	9644.39i	−1.21924	−	1.21924i	−0.967899	−	0.251341i	$-0.919129\pi$
$397$	−9644.39	−	9644.39i	−1.21924	−	1.21924i	−0.251341	−	0.967899i	$-0.580871\pi$
$398$	−93.7169	+	3678.49i	−0.0118030	+	0.463282i
$399$		0			0
$400$	738.165	−	7219.92i	0.0922706	−	0.902490i
$401$			3647.28i			0.454206i	0.973871	+	0.227103i	$0.0729254\pi$
$401$			3647.28i			0.454206i	−0.973871	+	0.227103i	$0.927075\pi$
$402$		0			0
$403$	385.656	−	385.656i	0.0476697	−	0.0476697i
$404$	4746.12	+	5256.11i	0.584477	+	0.647280i
$405$		0			0
$406$	−515.597	−	542.555i	−0.0630262	−	0.0663216i
$407$			2574.76i			0.313578i
$408$		0			0
$409$		−	860.719i		−	0.104058i	−0.998646	−	0.0520291i	$-0.983431\pi$
$409$		−	860.719i		−	0.104058i	0.998646	−	0.0520291i	$-0.0165689\pi$
$410$	−414.246	+	393.663i	−0.0498979	+	0.0474186i
$411$		0			0
$412$	−13115.1	−	668.701i	−1.56829	−	0.0799625i
$413$	−3019.55	+	3019.55i	−0.359763	+	0.359763i
$414$		0			0
$415$		−	2366.55i		−	0.279926i
$416$	643.991	−	497.786i	0.0758997	−	0.0586682i
$417$		0			0
$418$	−3134.77	−	79.8645i	−0.366810	−	0.00934521i
$419$	−1158.11	−	1158.11i	−0.135029	−	0.135029i	0.636362	−	0.771391i	$-0.280439\pi$
$419$	−1158.11	−	1158.11i	−0.135029	−	0.135029i	−0.771391	+	0.636362i	$0.780439\pi$
$420$		0			0
$421$	2410.72	−	2410.72i	0.279076	−	0.279076i	−0.553664	−	0.832740i	$-0.686771\pi$
$421$	2410.72	−	2410.72i	0.279076	−	0.279076i	0.832740	+	0.553664i	$0.186771\pi$
$422$	−4692.30	−	4937.64i	−0.541274	−	0.569575i
$423$		0			0
$424$	−11129.0	+	9546.01i	−1.27469	+	1.09339i
$425$	−11122.2			−1.26942
$426$		0			0
$427$	4375.21	+	4375.21i	0.495858	+	0.495858i
$428$	−8032.97	+	7253.56i	−0.907216	+	0.819192i
$429$		0			0
$430$	−3292.32	−	83.8784i	−0.369232	−	0.00940692i
$431$	10544.5			1.17844			0.589221	−	0.807972i	$-0.299435\pi$
$431$	10544.5			1.17844			0.589221	+	0.807972i	$0.299435\pi$
$432$		0			0
$433$	1205.31			0.133773			0.0668864	−	0.997761i	$-0.478693\pi$
$433$	1205.31			0.133773			0.0668864	+	0.997761i	$0.478693\pi$
$434$	4019.69	+	102.410i	0.444589	+	0.0113268i
$435$		0			0
$436$	−9261.29	−	10256.4i	−1.01728	−	1.12659i
$437$	1111.86	+	1111.86i	0.121711	+	0.121711i
$438$		0			0
$439$	−16643.3			−1.80943			−0.904716	−	0.426015i	$-0.859917\pi$
$439$	−16643.3			−1.80943			−0.904716	+	0.426015i	$0.859917\pi$
$440$	−289.057	+	3775.39i	−0.0313188	+	0.409057i
$441$		0			0
$442$	−859.277	−	904.206i	−0.0924698	−	0.0973047i
$443$	−7621.94	+	7621.94i	−0.817448	+	0.817448i	−0.985738	−	0.168290i	$-0.946176\pi$
$443$	−7621.94	+	7621.94i	−0.817448	+	0.817448i	0.168290	+	0.985738i	$0.446176\pi$
$444$		0			0
$445$	1503.91	+	1503.91i	0.160207	+	0.160207i
$446$	15033.0	+	382.996i	1.59604	+	0.0406623i
$447$		0			0
$448$	5930.96	+	913.546i	0.625472	+	0.0963415i
$449$			1505.28i			0.158215i	0.996866	+	0.0791077i	$0.0252071\pi$
$449$			1505.28i			0.158215i	−0.996866	+	0.0791077i	$0.974793\pi$
$450$		0			0
$451$	−2060.84	+	2060.84i	−0.215169	+	0.215169i
$452$	147.053	−	2884.11i	0.0153026	−	0.300127i
$453$		0			0
$454$	9083.86	−	8632.50i	0.939045	−	0.892386i
$455$		−	179.498i		−	0.0184945i
$456$		0			0
$457$			6918.12i			0.708131i	0.935221	+	0.354065i	$0.115201\pi$
$457$			6918.12i			0.708131i	−0.935221	+	0.354065i	$0.884799\pi$
$458$	−12483.8	−	13136.5i	−1.27365	−	1.34024i
$459$		0			0
$460$	1409.19	−	1272.46i	0.142834	−	0.128975i
$461$	−3348.19	+	3348.19i	−0.338266	+	0.338266i	−0.855715	−	0.517448i	$-0.826882\pi$
$461$	−3348.19	+	3348.19i	−0.338266	+	0.338266i	0.517448	+	0.855715i	$0.326882\pi$
$462$		0			0
$463$		−	2122.03i		−	0.213000i	−0.994313	−	0.106500i	$-0.966036\pi$
$463$		−	2122.03i		−	0.213000i	0.994313	−	0.106500i	$-0.0339644\pi$
$464$	1120.38	−	912.535i	0.112096	−	0.0913003i
$465$		0			0
$466$	−387.662	+	15216.2i	−0.0385367	+	1.51261i
$467$	12230.3	+	12230.3i	1.21189	+	1.21189i	0.970405	+	0.241482i	$0.0776334\pi$
$467$	12230.3	+	12230.3i	1.21189	+	1.21189i	0.241482	+	0.970405i	$0.422367\pi$
$468$		0			0
$469$	7599.55	−	7599.55i	0.748219	−	0.748219i
$470$	2759.11	−	2622.02i	0.270784	−	0.257329i
$471$		0			0
$472$	−5367.45	−	6257.50i	−0.523426	−	0.610222i
$473$	−16796.3			−1.63276
$474$		0			0
$475$	1809.43	+	1809.43i	0.174784	+	0.174784i
$476$	468.287	−	9184.43i	0.0450922	−	0.884385i
$477$		0			0
$478$	356.972	−	14011.5i	0.0341580	−	1.34074i
$479$	20589.9			1.96404			0.982020	−	0.188779i	$-0.0604529\pi$
$479$	20589.9			1.96404			0.982020	+	0.188779i	$0.0604529\pi$
$480$		0			0
$481$	235.643			0.0223376
$482$	42.4745	−	1667.17i	0.00401382	−	0.157547i
$483$		0			0
$484$	−441.122	+	8651.64i	−0.0414277	+	0.812513i
$485$	−4011.04	−	4011.04i	−0.375530	−	0.375530i
$486$		0			0
$487$	8672.88			0.806994			0.403497	−	0.914981i	$-0.367795\pi$
$487$	8672.88			0.806994			0.403497	+	0.914981i	$0.367795\pi$
$488$	−9066.87	+	7777.23i	−0.841062	+	0.721432i
$489$		0			0
$490$	−1435.96	+	1364.61i	−0.132387	+	0.125809i
$491$	−1815.32	+	1815.32i	−0.166852	+	0.166852i	−0.785594	−	0.618742i	$-0.787643\pi$
$491$	−1815.32	+	1815.32i	−0.166852	+	0.166852i	0.618742	+	0.785594i	$0.287643\pi$
$492$		0			0
$493$	−1565.84	−	1565.84i	−0.143046	−	0.143046i
$494$	−7.30922	+	286.895i	−0.000665703	+	0.0261296i
$495$		0			0
$496$	−789.559	+	7722.61i	−0.0714763	+	0.699103i
$497$		−	9225.68i		−	0.832652i
$498$		0			0
$499$	−2716.12	+	2716.12i	−0.243668	+	0.243668i	−0.818366	−	0.574698i	$-0.805120\pi$
$499$	−2716.12	+	2716.12i	−0.243668	+	0.243668i	0.574698	+	0.818366i	$0.305120\pi$
$500$	4821.19	−	4353.40i	0.431220	−	0.389380i
$501$		0			0
$502$	6662.99	+	7011.37i	0.592398	+	0.623372i
$503$		−	17354.3i		−	1.53835i	−0.639040	−	0.769173i	$-0.720668\pi$
$503$		−	17354.3i		−	1.53835i	0.639040	−	0.769173i	$-0.279332\pi$
$504$		0			0
$505$			3015.06i			0.265680i
$506$	7019.26	−	6670.48i	0.616688	−	0.586046i
$507$		0			0
$508$	697.966	−	13689.1i	0.0609591	−	1.19558i
$509$	8211.14	−	8211.14i	0.715034	−	0.715034i	−0.252550	−	0.967584i	$-0.581269\pi$
$509$	8211.14	−	8211.14i	0.715034	−	0.715034i	0.967584	+	0.252550i	$0.0812692\pi$
$510$		0			0
$511$			12599.1i			1.09071i
$512$	−2632.64	+	11282.2i	−0.227241	+	0.973839i
$513$		0			0
$514$	2761.37	+	70.3513i	0.236962	+	0.00603708i
$515$	−3953.41	−	3953.41i	−0.338268	−	0.338268i
$516$		0			0
$517$	13726.3	−	13726.3i	1.16767	−	1.16767i
$518$	1196.77	+	1259.34i	0.101511	+	0.106819i
$519$		0			0
$520$	345.525	+	26.4546i	0.0291390	+	0.00223098i
$521$	−9350.18			−0.786255			−0.393127	−	0.919484i	$-0.628607\pi$
$521$	−9350.18			−0.786255			−0.393127	+	0.919484i	$0.628607\pi$
$522$		0			0
$523$	11818.7	+	11818.7i	0.988140	+	0.988140i	0.999930	−	0.0117907i	$-0.00375319\pi$
$523$	11818.7	+	11818.7i	0.988140	+	0.988140i	−0.0117907	+	0.999930i	$0.503753\pi$
$524$	1449.39	+	1605.13i	0.120834	+	0.133818i
$525$		0			0
$526$	5147.20	+	131.135i	0.426671	+	0.0108703i
$527$	11896.5			0.983343
$528$		0			0
$529$	7311.43			0.600923
$530$	−6240.35	−	158.985i	−0.511440	−	0.0130300i
$531$		0			0
$532$	−1570.37	+	1418.00i	−0.127977	+	0.115560i
$533$	188.608	+	188.608i	0.0153275	+	0.0153275i
$534$		0			0
$535$	−4607.96			−0.372373
$536$	13508.7	+	15748.8i	1.08860	+	1.26911i
$537$		0			0
$538$	13213.4	+	13904.3i	1.05887	+	1.11423i
$539$	−7143.75	+	7143.75i	−0.570878	+	0.570878i
$540$		0			0
$541$	−10764.1	−	10764.1i	−0.855422	−	0.855422i	0.135373	−	0.990795i	$-0.456777\pi$
$541$	−10764.1	−	10764.1i	−0.855422	−	0.855422i	−0.990795	+	0.135373i	$0.956777\pi$
$542$	−3659.47	−	93.2322i	−0.290014	−	0.00738868i
$543$		0			0
$544$	17610.5	+	2255.04i	1.38795	+	0.177728i
$545$		−	5883.40i		−	0.462417i
$546$		0			0
$547$	2093.77	−	2093.77i	0.163662	−	0.163662i	−0.620525	−	0.784187i	$-0.713080\pi$
$547$	2093.77	−	2093.77i	0.163662	−	0.163662i	0.784187	+	0.620525i	$0.213080\pi$
$548$	−17021.9	−	867.898i	−1.32690	−	0.0676547i
$549$		0			0
$550$	11423.0	−	10855.5i	0.885600	−	0.841596i
$551$			509.481i			0.0393913i
$552$		0			0
$553$		−	4478.95i		−	0.344420i
$554$	1932.28	+	2033.31i	0.148186	+	0.155934i
$555$		0			0
$556$	3207.64	+	3552.31i	0.244666	+	0.270956i
$557$	5245.63	−	5245.63i	0.399038	−	0.399038i	−0.478855	−	0.877894i	$-0.658948\pi$
$557$	5245.63	−	5245.63i	0.399038	−	0.399038i	0.877894	+	0.478855i	$0.158948\pi$
$558$		0			0
$559$			1537.20i			0.116309i
$560$	1613.45	+	1980.94i	0.121751	+	0.149482i
$561$		0			0
$562$	−414.601	+	16273.6i	−0.0311190	+	1.22146i
$563$	7593.93	+	7593.93i	0.568465	+	0.568465i	0.931698	−	0.363233i	$-0.118327\pi$
$563$	7593.93	+	7593.93i	0.568465	+	0.568465i	−0.363233	+	0.931698i	$0.618327\pi$
$564$		0			0
$565$	869.384	−	869.384i	0.0647350	−	0.0647350i
$566$	11332.4	−	10769.3i	0.841581	−	0.799765i
$567$		0			0
$568$	17758.9	+	1359.69i	1.31188	+	0.100442i
$569$	3416.36			0.251707			0.125853	−	0.992049i	$-0.459833\pi$
$569$	3416.36			0.251707			0.125853	+	0.992049i	$0.459833\pi$
$570$		0			0
$571$	1823.75	+	1823.75i	0.133663	+	0.133663i	0.770773	−	0.637110i	$-0.219870\pi$
$571$	1823.75	+	1823.75i	0.133663	+	0.133663i	−0.637110	+	0.770773i	$0.719870\pi$
$572$	1765.04	+	89.9944i	0.129021	+	0.00657842i
$573$		0			0
$574$	−50.0843	+	1965.86i	−0.00364195	+	0.142951i
$575$	−7901.88			−0.573098
$576$		0			0
$577$	−23143.8			−1.66982			−0.834912	−	0.550383i	$-0.814482\pi$
$577$	−23143.8			−1.66982			−0.834912	+	0.550383i	$0.814482\pi$
$578$	339.048	−	13308.0i	0.0243989	−	0.957682i
$579$		0			0
$580$	614.397	+	31.3263i	0.0439852	+	0.00224268i
$581$	−5758.46	−	5758.46i	−0.411190	−	0.411190i
$582$		0			0
$583$	−31836.1			−2.26161
$584$	−24252.7	−	1856.87i	−1.71846	−	0.131572i
$585$		0			0
$586$	10188.2	−	9681.95i	0.718208	−	0.682522i
$587$	16398.7	−	16398.7i	1.15306	−	1.15306i	0.167122	−	0.985936i	$-0.446553\pi$
$587$	16398.7	−	16398.7i	1.15306	−	1.15306i	0.985936	−	0.167122i	$-0.0534473\pi$
$588$		0			0
$589$	−1935.41	−	1935.41i	−0.135394	−	0.135394i
$590$	89.3929	−	3508.77i	0.00623771	−	0.244837i
$591$		0			0
$592$	−2600.54	+	2118.11i	−0.180543	+	0.147050i
$593$		−	13256.2i		−	0.917989i	−0.888439	−	0.458994i	$-0.848210\pi$
$593$		−	13256.2i		−	0.917989i	0.888439	−	0.458994i	$-0.151790\pi$
$594$		0			0
$595$	2768.54	−	2768.54i	0.190755	−	0.190755i
$596$	15319.5	+	16965.6i	1.05287	+	1.16600i
$597$		0			0
$598$	−610.484	−	642.404i	−0.0417467	−	0.0439295i
$599$		−	9973.40i		−	0.680304i	−0.940370	−	0.340152i	$-0.889521\pi$
$599$		−	9973.40i		−	0.680304i	0.940370	−	0.340152i	$-0.110479\pi$
$600$		0			0
$601$			22549.7i			1.53048i	0.643744	+	0.765241i	$0.277380\pi$
$601$			22549.7i			1.53048i	−0.643744	+	0.765241i	$0.722620\pi$
$602$	−8215.23	+	7807.03i	−0.556192	+	0.528556i
$603$		0			0
$604$	−20817.6	−	1061.43i	−1.40241	−	0.0715047i
$605$	−2607.94	+	2607.94i	−0.175253	+	0.175253i
$606$		0			0
$607$		−	21635.2i		−	1.44670i	−0.690482	−	0.723350i	$-0.742602\pi$
$607$		−	21635.2i		−	1.44670i	0.690482	−	0.723350i	$-0.257398\pi$
$608$	−2498.13	−	3231.86i	−0.166633	−	0.215574i
$609$		0			0
$610$	−5084.07	−	129.527i	−0.337456	−	0.00859736i
$611$	−1256.24	−	1256.24i	−0.0831783	−	0.0831783i
$612$		0			0
$613$	2977.25	−	2977.25i	0.196166	−	0.196166i	−0.602188	−	0.798354i	$-0.705704\pi$
$613$	2977.25	−	2977.25i	0.196166	−	0.196166i	0.798354	+	0.602188i	$0.205704\pi$
$614$	6990.98	+	7356.51i	0.459500	+	0.483525i
$615$		0			0
$616$	8483.22	+	9889.93i	0.554868	+	0.646878i
$617$	−6649.18			−0.433851			−0.216925	−	0.976188i	$-0.569603\pi$
$617$	−6649.18			−0.433851			−0.216925	+	0.976188i	$0.569603\pi$
$618$		0			0
$619$	−2920.44	−	2920.44i	−0.189632	−	0.189632i	0.605905	−	0.795537i	$-0.292811\pi$
$619$	−2920.44	−	2920.44i	−0.189632	−	0.189632i	−0.795537	+	0.605905i	$0.792811\pi$
$620$	−2452.96	+	2214.96i	−0.158893	+	0.143476i
$621$		0			0
$622$	19972.1	+	508.829i	1.28747	+	0.0328009i
$623$	7318.86			0.470664
$624$		0			0
$625$	−11409.3			−0.730198
$626$	1637.58	+	41.7206i	0.104554	+	0.00266372i
$627$		0			0
$628$	−6186.89	−	6851.69i	−0.393127	−	0.435370i
$629$	3634.50	+	3634.50i	0.230393	+	0.230393i
$630$		0			0
$631$	−3289.01			−0.207501			−0.103751	−	0.994603i	$-0.533084\pi$
$631$	−3289.01			−0.207501			−0.103751	+	0.994603i	$0.533084\pi$
$632$	8621.75	+	660.112i	0.542650	+	0.0415472i
$633$		0			0
$634$	−939.593	−	988.720i	−0.0588580	−	0.0619355i
$635$	4126.42	−	4126.42i	0.257877	−	0.257877i
$636$		0			0
$637$	653.798	+	653.798i	0.0406663	+	0.0406663i
$638$	3136.48	+	79.9080i	0.194631	+	0.00495860i
$639$		0			0
$640$	−4050.99	+	2813.85i	−0.250202	+	0.173792i
$641$		−	9161.75i		−	0.564536i	−0.959336	−	0.282268i	$-0.908913\pi$
$641$		−	9161.75i		−	0.564536i	0.959336	−	0.282268i	$-0.0910867\pi$
$642$		0			0
$643$	6165.52	−	6165.52i	0.378141	−	0.378141i	−0.492290	−	0.870431i	$-0.663840\pi$
$643$	6165.52	−	6165.52i	0.378141	−	0.378141i	0.870431	+	0.492290i	$0.163840\pi$
$644$	332.700	−	6525.19i	0.0203575	−	0.399268i
$645$		0			0
$646$	−4537.74	+	4312.27i	−0.276370	+	0.262638i
$647$		−	6481.62i		−	0.393846i	−0.980419	−	0.196923i	$-0.936905\pi$
$647$		−	6481.62i		−	0.393846i	0.980419	−	0.196923i	$-0.0630950\pi$
$648$		0			0
$649$		−	17900.5i		−	1.08268i
$650$	−993.494	−	1045.44i	−0.0599508	−	0.0630854i
$651$		0			0
$652$	−2257.32	+	2038.30i	−0.135588	+	0.122432i
$653$	−772.442	+	772.442i	−0.0462910	+	0.0462910i	−0.729873	−	0.683582i	$-0.760421\pi$
$653$	−772.442	+	772.442i	−0.0462910	+	0.0462910i	0.683582	+	0.729873i	$0.260421\pi$
$654$		0			0
$655$			920.751i			0.0549263i
$656$	−3776.81	−	386.140i	−0.224786	−	0.0229821i
$657$		0			0
$658$	333.590	−	13093.8i	0.0197639	−	0.775757i
$659$	−1416.83	−	1416.83i	−0.0837510	−	0.0837510i	0.663990	−	0.747741i	$-0.268862\pi$
$659$	−1416.83	−	1416.83i	−0.0837510	−	0.0837510i	−0.747741	+	0.663990i	$0.768862\pi$
$660$		0			0
$661$	−7384.66	+	7384.66i	−0.434539	+	0.434539i	−0.890169	−	0.455630i	$-0.849414\pi$
$661$	−7384.66	+	7384.66i	−0.434539	+	0.434539i	0.455630	+	0.890169i	$0.349414\pi$
$662$	−7276.23	+	6914.69i	−0.427188	+	0.405962i
$663$		0			0
$664$	11933.4	−	10236.1i	0.697450	−	0.598247i
$665$	−900.809			−0.0525291
$666$		0			0
$667$	−1112.47	−	1112.47i	−0.0645801	−	0.0645801i
$668$	626.338	−	12284.3i	0.0362781	−	0.711515i
$669$		0			0
$670$	−224.983	+	8830.82i	−0.0129729	+	0.509200i
$671$	−25937.2			−1.49224
$672$		0			0
$673$	8695.15			0.498029			0.249014	−	0.968500i	$-0.419893\pi$
$673$	8695.15			0.498029			0.249014	+	0.968500i	$0.419893\pi$
$674$	−130.086	+	5106.02i	−0.00743431	+	0.291805i
$675$		0			0
$676$	−886.750	+	17391.7i	−0.0504523	+	0.989512i
$677$	19700.8	+	19700.8i	1.11841	+	1.11841i	0.991975	+	0.126436i	$0.0403540\pi$
$677$	19700.8	+	19700.8i	1.11841	+	1.11841i	0.126436	+	0.991975i	$0.459646\pi$
$678$		0			0
$679$	−19519.9			−1.10325
$680$	4921.27	+	5737.33i	0.277533	+	0.323554i
$681$		0			0
$682$	−12218.4	+	11611.3i	−0.686020	+	0.651933i
$683$	−6263.20	+	6263.20i	−0.350885	+	0.350885i	−0.860439	−	0.509554i	$-0.829811\pi$
$683$	−6263.20	+	6263.20i	−0.350885	+	0.350885i	0.509554	+	0.860439i	$0.329811\pi$
$684$		0			0
$685$	−5131.07	−	5131.07i	−0.286202	−	0.286202i
$686$	−463.210	+	18181.5i	−0.0257805	+	1.01191i
$687$		0			0
$688$	−13817.4	−	16964.5i	−0.765671	−	0.940065i
$689$			2913.65i			0.161105i
$690$		0			0
$691$	−21784.7	+	21784.7i	−1.19932	+	1.19932i	−0.224945	+	0.974371i	$0.572220\pi$
$691$	−21784.7	+	21784.7i	−1.19932	+	1.19932i	−0.974371	+	0.224945i	$0.927780\pi$
$692$	5026.29	−	4538.60i	0.276114	−	0.249323i
$693$		0			0
$694$	−9543.09	−	10042.1i	−0.521975	−	0.549267i
$695$			2037.71i			0.111216i
$696$		0			0
$697$			5818.11i			0.316179i
$698$	−16809.7	+	15974.4i	−0.911541	+	0.866248i
$699$		0			0
$700$	541.432	−	10619.0i	0.0292346	−	0.573372i
$701$	6306.61	−	6306.61i	0.339797	−	0.339797i	−0.516494	−	0.856291i	$-0.672763\pi$
$701$	6306.61	−	6306.61i	0.339797	−	0.339797i	0.856291	+	0.516494i	$0.172763\pi$
$702$		0			0
$703$		−	1182.57i		−	0.0634445i
$704$	−20287.9	+	14872.2i	−1.08612	+	0.796187i
$705$		0			0
$706$	7898.15	+	201.221i	0.421035	+	0.0107267i
$707$	7336.48	+	7336.48i	0.390264	+	0.390264i
$708$		0			0
$709$	6829.13	−	6829.13i	0.361740	−	0.361740i	−0.502713	−	0.864453i	$-0.667665\pi$
$709$	6829.13	−	6829.13i	0.361740	−	0.361740i	0.864453	+	0.502713i	$0.167665\pi$
$710$	5223.64	+	5496.76i	0.276112	+	0.290549i
$711$		0			0
$712$	−1078.66	+	14088.4i	−0.0567759	+	0.741553i
$713$	8452.05			0.443944
$714$		0			0
$715$	532.053	+	532.053i	0.0278289	+	0.0278289i
$716$	2163.60	+	2396.09i	0.112930	+	0.125064i
$717$		0			0
$718$	−19501.2	−	496.832i	−1.01362	−	0.0258239i
$719$	3688.19			0.191302			0.0956511	−	0.995415i	$-0.469507\pi$
$719$	3688.19			0.191302			0.0956511	+	0.995415i	$0.469507\pi$
$720$		0			0
$721$	−19239.4			−0.993779
$722$	−17954.1	−	457.417i	−0.925461	−	0.0235780i
$723$		0			0
$724$	24518.6	−	22139.6i	1.25860	−	1.13648i
$725$	−1810.41	−	1810.41i	−0.0927408	−	0.0927408i
$726$		0			0
$727$	28693.9			1.46382			0.731911	−	0.681400i	$-0.238629\pi$
$727$	28693.9			1.46382			0.731911	+	0.681400i	$0.238629\pi$
$728$	905.129	−	776.386i	0.0460801	−	0.0395258i
$729$		0			0
$730$	−7133.71	−	7506.70i	−0.361685	−	0.380597i
$731$	−23709.5	+	23709.5i	−1.19963	+	1.19963i
$732$		0			0
$733$	−4890.55	−	4890.55i	−0.246434	−	0.246434i	0.573071	−	0.819506i	$-0.305752\pi$
$733$	−4890.55	−	4890.55i	−0.246434	−	0.246434i	−0.819506	+	0.573071i	$0.805752\pi$
$734$	−4875.26	−	124.207i	−0.245162	−	0.00624599i
$735$		0			0
$736$	12511.6	+	1602.12i	0.626609	+	0.0802376i
$737$			45051.8i			2.25170i
$738$		0			0
$739$	23911.6	−	23911.6i	1.19026	−	1.19026i	0.213269	−	0.976993i	$-0.431589\pi$
$739$	23911.6	−	23911.6i	1.19026	−	1.19026i	0.976993	−	0.213269i	$-0.0684111\pi$
$740$	−1426.09	−	72.7123i	−0.0708436	−	0.00361211i
$741$		0			0
$742$	−15571.3	+	14797.6i	−0.770407	+	0.732127i
$743$			14426.5i			0.712323i	0.934424	+	0.356162i	$0.115915\pi$
$743$			14426.5i			0.712323i	−0.934424	+	0.356162i	$0.884085\pi$
$744$		0			0
$745$			9731.96i			0.478593i
$746$	16360.9	+	17216.4i	0.802972	+	0.844956i
$747$		0			0
$748$	25835.6	+	28611.7i	1.26289	+	1.39859i
$749$	−11212.4	+	11212.4i	−0.546987	+	0.546987i
$750$		0			0
$751$			17255.7i			0.838439i	0.907885	+	0.419220i	$0.137696\pi$
$751$			17255.7i			0.838439i	−0.907885	+	0.419220i	$0.862304\pi$
$752$	25155.6	+	2571.91i	1.21986	+	0.124718i
$753$		0			0
$754$	7.31320	−	287.051i	0.000353224	−	0.0138644i
$755$	−6275.22	−	6275.22i	−0.302488	−	0.302488i
$756$		0			0
$757$	−12874.2	+	12874.2i	−0.618124	+	0.618124i	−0.945050	−	0.326926i	$-0.893987\pi$
$757$	−12874.2	+	12874.2i	−0.618124	+	0.618124i	0.326926	+	0.945050i	$0.393987\pi$
$758$	−23224.2	+	22070.2i	−1.11285	+	1.05756i
$759$		0			0
$760$	132.762	−	1734.01i	0.00633656	−	0.0827621i
$761$	4694.94			0.223642			0.111821	−	0.993728i	$-0.464332\pi$
$761$	4694.94			0.223642			0.111821	+	0.993728i	$0.464332\pi$
$762$		0			0
$763$	−14315.9	−	14315.9i	−0.679255	−	0.679255i
$764$	8670.70	+	442.094i	0.410595	+	0.0209351i
$765$		0			0
$766$	891.255	−	34982.8i	0.0420396	−	1.65010i
$767$	−1638.26			−0.0771242
$768$		0			0
$769$	27268.7			1.27872			0.639359	−	0.768908i	$-0.279200\pi$
$769$	27268.7			1.27872			0.639359	+	0.768908i	$0.279200\pi$
$770$	−141.285	+	5545.59i	−0.00661241	+	0.259544i
$771$		0			0
$772$	−9194.17	−	468.784i	−0.428634	−	0.0218548i
$773$	−10010.2	−	10010.2i	−0.465772	−	0.465772i	0.434769	−	0.900542i	$-0.356830\pi$
$773$	−10010.2	−	10010.2i	−0.465772	−	0.465772i	−0.900542	+	0.434769i	$0.856830\pi$
$774$		0			0
$775$	13754.8			0.637529
$776$	2876.86	−	37574.9i	0.133084	−	1.73822i
$777$		0			0
$778$	−11455.6	+	10886.4i	−0.527896	+	0.501666i
$779$	946.527	−	946.527i	0.0435338	−	0.0435338i
$780$		0			0
$781$	27345.9	+	27345.9i	1.25290	+	1.25290i
$782$	492.324	−	19324.3i	0.0225134	−	0.883676i
$783$		0			0
$784$	−13092.0	−	1338.53i	−0.596394	−	0.0609753i
$785$		−	3930.34i		−	0.178700i
$786$		0			0
$787$	1703.37	−	1703.37i	0.0771521	−	0.0771521i	−0.667478	−	0.744630i	$-0.732626\pi$
$787$	1703.37	−	1703.37i	0.0771521	−	0.0771521i	0.744630	+	0.667478i	$0.232626\pi$
$788$	−17457.9	−	19333.8i	−0.789230	−	0.874035i
$789$		0			0
$790$	2536.01	+	2668.61i	0.114212	+	0.120183i
$791$		−	4230.90i		−	0.190182i
$792$		0			0
$793$			2373.78i			0.106299i
$794$	27964.4	−	26574.9i	1.24990	−	1.18779i
$795$		0			0
$796$	−10394.2	−	529.971i	−0.462831	−	0.0235984i
$797$	13767.6	−	13767.6i	0.611886	−	0.611886i	−0.331551	−	0.943437i	$-0.607572\pi$
$797$	13767.6	−	13767.6i	0.611886	−	0.611886i	0.943437	+	0.331551i	$0.107572\pi$
$798$		0			0
$799$		−	38751.9i		−	1.71582i
$800$	20361.2	+	2607.26i	0.899848	+	0.115226i
$801$		0			0
$802$	−10312.7	−	262.737i	−0.454059	−	0.0115680i
$803$	−37345.2	−	37345.2i	−1.64120	−	1.64120i
$804$		0			0
$805$	1966.95	−	1966.95i	0.0861189	−	0.0861189i
$806$	1062.67	+	1118.23i	0.0464402	+	0.0488684i
$807$		0			0
$808$	−15203.6	+	13041.1i	−0.661956	+	0.567802i
$809$	34122.6			1.48292			0.741462	−	0.670995i	$-0.234133\pi$
$809$	34122.6			1.48292			0.741462	+	0.670995i	$0.234133\pi$
$810$		0			0
$811$	19921.8	+	19921.8i	0.862576	+	0.862576i	0.991637	−	0.129061i	$-0.0411962\pi$
$811$	19921.8	+	19921.8i	0.862576	+	0.862576i	−0.129061	+	0.991637i	$0.541196\pi$
$812$	1571.22	−	1418.77i	0.0679053	−	0.0613166i
$813$		0			0
$814$	−7280.17	−	185.477i	−0.313476	−	0.00798643i
$815$	−1294.87			−0.0556529
$816$		0			0
$817$	7714.42			0.330347
$818$	2433.69	+	62.0031i	0.104025	+	0.00265023i
$819$		0			0
$820$	−1083.24	−	1199.64i	−0.0461324	−	0.0510894i
$821$	3546.97	+	3546.97i	0.150780	+	0.150780i	0.778466	−	0.627687i	$-0.215998\pi$
$821$	3546.97	+	3546.97i	0.150780	+	0.150780i	−0.627687	+	0.778466i	$0.715998\pi$
$822$		0			0
$823$	33931.8			1.43717			0.718583	−	0.695442i	$-0.244791\pi$
$823$	33931.8			1.43717			0.718583	+	0.695442i	$0.244791\pi$
$824$	2835.52	−	37034.9i	0.119879	−	1.56574i
$825$		0			0
$826$	−8320.29	−	8755.32i	−0.350484	−	0.368809i
$827$	5818.78	−	5818.78i	0.244666	−	0.244666i	−0.574111	−	0.818777i	$-0.694652\pi$
$827$	5818.78	−	5818.78i	0.244666	−	0.244666i	0.818777	+	0.574111i	$0.194652\pi$
$828$		0			0
$829$	7895.08	+	7895.08i	0.330769	+	0.330769i	0.852878	−	0.522110i	$-0.174855\pi$
$829$	7895.08	+	7895.08i	0.330769	+	0.330769i	−0.522110	+	0.852878i	$0.674855\pi$
$830$	6691.44	+	170.478i	0.279835	+	0.00712936i
$831$		0			0
$832$	1361.10	+	1856.75i	0.0567161	+	0.0773693i
$833$			20168.1i			0.838874i
$834$		0			0
$835$	3702.95	−	3702.95i	0.153468	−	0.153468i
$836$	451.635	−	8857.84i	0.0186844	−	0.366453i
$837$		0			0
$838$	3357.99	−	3191.14i	0.138425	−	0.131547i
$839$			23123.0i			0.951485i	0.879585	+	0.475743i	$0.157821\pi$
$839$			23123.0i			0.951485i	−0.879585	+	0.475743i	$0.842179\pi$
$840$		0			0
$841$			23879.2i			0.979099i
$842$	6642.66	+	6989.98i	0.271878	+	0.286093i
$843$		0			0
$844$	14299.2	−	12911.8i	0.583176	−	0.526592i
$845$	−5242.52	+	5242.52i	−0.213430	+	0.213430i
$846$		0			0
$847$			12691.7i			0.514866i
$848$	−26189.8	−	32154.9i	−1.06057	−	1.30213i
$849$		0			0
$850$	801.202	−	31448.1i	0.0323306	−	1.26901i
$851$	2582.18	+	2582.18i	0.104014	+	0.104014i
$852$		0			0
$853$	−10555.7	+	10555.7i	−0.423707	+	0.423707i	−0.886478	−	0.462771i	$-0.846855\pi$
$853$	−10555.7	+	10555.7i	−0.423707	+	0.423707i	0.462771	+	0.886478i	$0.346855\pi$
$854$	−12686.1	+	12055.8i	−0.508326	+	0.483068i
$855$		0			0
$856$	−19930.8	−	23235.8i	−0.795820	−	0.927786i
$857$	14835.9			0.591349			0.295674	−	0.955289i	$-0.404456\pi$
$857$	14835.9			0.591349			0.295674	+	0.955289i	$0.404456\pi$
$858$		0			0
$859$	−21200.1	−	21200.1i	−0.842069	−	0.842069i	0.147059	−	0.989128i	$-0.453019\pi$
$859$	−21200.1	−	21200.1i	−0.842069	−	0.842069i	−0.989128	+	0.147059i	$0.953019\pi$
$860$	474.334	−	9303.03i	0.0188077	−	0.368873i
$861$		0			0
$862$	−759.585	+	29814.6i	−0.0300134	+	1.17806i
$863$	−38729.6			−1.52766			−0.763830	−	0.645418i	$-0.776683\pi$
$863$	−38729.6			−1.52766			−0.763830	+	0.645418i	$0.776683\pi$
$864$		0			0
$865$	2883.23			0.113333
$866$	−86.8264	+	3408.03i	−0.00340702	+	0.133729i
$867$		0			0
$868$	−579.129	+	11358.3i	−0.0226462	+	0.444156i
$869$	13276.1	+	13276.1i	0.518252	+	0.518252i
$870$		0			0
$871$	4123.15			0.160399
$872$	29667.3	−	25447.5i	1.15214	−	0.988260i
$873$		0			0
$874$	−3223.89	+	3063.70i	−0.124771	+	0.118571i
$875$	6729.42	−	6729.42i	0.259995	−	0.259995i
$876$		0			0
$877$	3724.13	+	3724.13i	0.143392	+	0.143392i	0.775159	−	0.631766i	$-0.217670\pi$
$877$	3724.13	+	3724.13i	0.143392	+	0.143392i	−0.631766	+	0.775159i	$0.717670\pi$
$878$	1198.92	−	47059.0i	0.0460839	−	1.80884i
$879$		0			0
$880$	−10654.1	−	1089.28i	−0.408126	−	0.0417268i
$881$		−	10798.5i		−	0.412951i	−0.978452	−	0.206475i	$-0.933801\pi$
$881$		−	10798.5i		−	0.412951i	0.978452	−	0.206475i	$-0.0661993\pi$
$882$		0			0
$883$	3539.05	−	3539.05i	0.134879	−	0.134879i	−0.636444	−	0.771323i	$-0.719595\pi$
$883$	3539.05	−	3539.05i	0.134879	−	0.134879i	0.771323	+	0.636444i	$0.219595\pi$
$884$	2618.55	−	2364.48i	0.0996282	−	0.0899616i
$885$		0			0
$886$	−21002.1	−	22100.2i	−0.796363	−	0.838002i
$887$		−	22802.5i		−	0.863172i	−0.902072	−	0.431586i	$-0.857954\pi$
$887$		−	22802.5i		−	0.863172i	0.902072	−	0.431586i	$-0.142046\pi$
$888$		0			0
$889$		−	20081.4i		−	0.757604i
$890$	−4360.66	+	4143.98i	−0.164235	+	0.156075i
$891$		0			0
$892$	−2165.85	+	42478.4i	−0.0812982	+	1.59449i
$893$	−6304.40	+	6304.40i	−0.236247	+	0.236247i
$894$		0			0
$895$			1374.47i			0.0513334i
$896$	−3010.30	+	16704.0i	−0.112240	+	0.622816i
$897$		0			0
$898$	−4256.20	−	108.435i	−0.158164	−	0.00402954i
$899$	1936.46	+	1936.46i	0.0718406	+	0.0718406i
$900$		0			0
$901$	−44939.5	+	44939.5i	−1.66166	+	1.66166i
$902$	−5678.58	−	5975.49i	−0.209619	−	0.220579i
$903$		0			0
$904$	8144.27	+	623.554i	0.299640	+	0.0229415i
$905$	14064.6			0.516600
$906$		0			0
$907$	−6649.86	−	6649.86i	−0.243445	−	0.243445i	0.574829	−	0.818274i	$-0.305069\pi$
$907$	−6649.86	−	6649.86i	−0.243445	−	0.243445i	−0.818274	+	0.574829i	$0.805069\pi$
$908$	23754.1	+	26306.5i	0.868180	+	0.961468i
$909$		0			0
$910$	507.533	+	12.9304i	0.0184885	+	0.000471032i
$911$	2309.48			0.0839917			0.0419958	−	0.999118i	$-0.486628\pi$
$911$	2309.48			0.0839917			0.0419958	+	0.999118i	$0.486628\pi$
$912$		0			0
$913$	34137.4			1.23744
$914$	−19561.0	−	498.356i	−0.707901	−	0.0180352i
$915$		0			0
$916$	38043.0	−	34351.8i	1.37224	−	1.23910i
$917$	2240.44	+	2240.44i	0.0806825	+	0.0806825i
$918$		0			0
$919$	48567.3			1.74329			0.871646	−	0.490135i	$-0.163053\pi$
$919$	48567.3			1.74329			0.871646	+	0.490135i	$0.163053\pi$
$920$	3496.38	+	4076.16i	0.125296	+	0.146073i
$921$		0			0
$922$	−9225.85	−	9708.23i	−0.329541	−	0.346772i
$923$	2502.70	−	2502.70i	0.0892498	−	0.0892498i
$924$		0			0
$925$	4202.20	+	4202.20i	0.149370	+	0.149370i
$926$	6000.05	+	152.863i	0.212931	+	0.00542483i
$927$		0			0
$928$	2499.49	+	3233.62i	0.0884158	+	0.114384i
$929$		−	55572.0i		−	1.96260i	−0.192477	−	0.981302i	$-0.561652\pi$
$929$		−	55572.0i		−	1.96260i	0.192477	−	0.981302i	$-0.438348\pi$
$930$		0			0
$931$	3281.07	−	3281.07i	0.115502	−	0.115502i
$932$	−42995.9	−	2192.24i	−1.51114	−	0.0770484i
$933$		0			0
$934$	−35462.4	+	33700.3i	−1.24236	+	1.18063i
$935$			16412.5i			0.574061i
$936$		0			0
$937$		−	46921.8i		−	1.63593i	−0.575268	−	0.817965i	$-0.695102\pi$
$937$		−	46921.8i		−	1.63593i	0.575268	−	0.817965i	$-0.304898\pi$
$938$	20940.4	+	22035.3i	0.728920	+	0.767032i
$939$		0			0
$940$	7215.02	+	7990.29i	0.250349	+	0.277250i
$941$	15833.3	−	15833.3i	0.548514	−	0.548514i	−0.377497	−	0.926011i	$-0.623215\pi$
$941$	15833.3	−	15833.3i	0.548514	−	0.548514i	0.926011	+	0.377497i	$0.123215\pi$
$942$		0			0
$943$			4133.55i			0.142743i
$944$	18079.8	−	14725.8i	0.623355	−	0.507714i
$945$		0			0
$946$	1209.95	−	47491.7i	0.0415843	−	1.63223i
$947$	−4616.34	−	4616.34i	−0.158406	−	0.158406i	0.623454	−	0.781860i	$-0.285729\pi$
$947$	−4616.34	−	4616.34i	−0.158406	−	0.158406i	−0.781860	+	0.623454i	$0.785729\pi$
$948$		0			0
$949$	−3417.84	+	3417.84i	−0.116910	+	0.116910i
$950$	−5246.52	+	4985.83i	−0.179178	+	0.170275i
$951$		0			0
$952$	25935.3	+	1985.70i	0.882950	+	0.0676018i
$953$	−7702.58			−0.261816			−0.130908	−	0.991394i	$-0.541789\pi$
$953$	−7702.58			−0.261816			−0.130908	+	0.991394i	$0.541789\pi$
$954$		0			0
$955$	2613.69	+	2613.69i	0.0885622	+	0.0885622i
$956$	39592.1	+	2018.68i	1.33943	+	0.0682938i
$957$		0			0
$958$	−1483.22	+	58218.0i	−0.0500216	+	1.96340i
$959$	−24970.6			−0.840817
$960$		0			0
$961$	15078.6			0.506145
$962$	−16.9749	+	666.283i	−0.000568910	+	0.0223304i
$963$		0			0
$964$	4710.88	+	240.194i	0.157393	+	0.00802503i
$965$	−2771.48	−	2771.48i	−0.0924530	−	0.0924530i
$966$		0			0
$967$	−3885.39			−0.129210			−0.0646048	−	0.997911i	$-0.520579\pi$
$967$	−3885.39			−0.129210			−0.0646048	+	0.997911i	$0.520579\pi$
$968$	−24430.8	−	1870.51i	−0.811195	−	0.0621079i
$969$		0			0
$970$	11630.2	−	11052.3i	0.384973	−	0.365844i
$971$	27634.4	−	27634.4i	0.913315	−	0.913315i	−0.0832167	−	0.996531i	$-0.526519\pi$
$971$	27634.4	−	27634.4i	0.913315	−	0.913315i	0.996531	+	0.0832167i	$0.0265194\pi$
$972$		0			0
$973$	4958.31	+	4958.31i	0.163367	+	0.163367i
$974$	−624.764	+	24522.7i	−0.0205531	+	0.806732i
$975$		0			0
$976$	−21337.1	−	26196.9i	−0.699777	−	0.859163i
$977$			59115.6i			1.93580i	0.251336	+	0.967900i	$0.419130\pi$
$977$			59115.6i			1.93580i	−0.251336	+	0.967900i	$0.580870\pi$
$978$		0			0
$979$	−21693.9	+	21693.9i	−0.708212	+	0.708212i
$980$	−3754.99	−	4158.48i	−0.122397	−	0.135549i
$981$		0			0
$982$	−5002.07	−	5263.61i	−0.162548	−	0.171047i
$983$			13689.3i			0.444172i	0.975027	+	0.222086i	$0.0712866\pi$
$983$			13689.3i			0.444172i	−0.975027	+	0.222086i	$0.928713\pi$
$984$		0			0
$985$		−	11090.5i		−	0.358753i
$986$	4540.21	−	4314.62i	0.146643	−	0.139356i
$987$		0			0
$988$	−810.671	−	41.3338i	−0.0261041	−	0.00133097i
$989$	−16844.7	+	16844.7i	−0.541587	+	0.541587i
$990$		0			0
$991$			37624.8i			1.20605i	0.797724	+	0.603023i	$0.206037\pi$
$991$			37624.8i			1.20605i	−0.797724	+	0.603023i	$0.793963\pi$
$992$	−21778.9	−	2788.79i	−0.697056	−	0.0892584i
$993$		0			0
$994$	26085.7	+	664.585i	0.832382	+	0.0212066i
$995$	−3133.22	−	3133.22i	−0.0998290	−	0.0998290i
$996$		0			0
$997$	11609.6	−	11609.6i	0.368786	−	0.368786i	−0.498248	−	0.867034i	$-0.666023\pi$
$997$	11609.6	−	11609.6i	0.368786	−	0.368786i	0.867034	+	0.498248i	$0.166023\pi$
$998$	−7484.19	−	7875.51i	−0.237383	−	0.249794i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	144.4.l.a.35.12	✓	48
3.2	odd	2	inner	144.4.l.a.35.13	yes	48
4.3	odd	2		576.4.l.a.431.12		48
8.3	odd	2		1152.4.l.a.863.13		48
8.5	even	2		1152.4.l.b.863.13		48
12.11	even	2		576.4.l.a.431.13		48
16.3	odd	4		1152.4.l.b.287.12		48
16.5	even	4		576.4.l.a.143.13		48
16.11	odd	4	inner	144.4.l.a.107.13	yes	48
16.13	even	4		1152.4.l.a.287.12		48
24.5	odd	2		1152.4.l.b.863.12		48
24.11	even	2		1152.4.l.a.863.12		48
48.5	odd	4		576.4.l.a.143.12		48
48.11	even	4	inner	144.4.l.a.107.12	yes	48
48.29	odd	4		1152.4.l.a.287.13		48
48.35	even	4		1152.4.l.b.287.13		48

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
144.4.l.a.35.12	✓	48	1.1	even	1	trivial
144.4.l.a.35.13	yes	48	3.2	odd	2	inner
144.4.l.a.107.12	yes	48	48.11	even	4	inner
144.4.l.a.107.13	yes	48	16.11	odd	4	inner
576.4.l.a.143.12		48	48.5	odd	4
576.4.l.a.143.13		48	16.5	even	4
576.4.l.a.431.12		48	4.3	odd	2
576.4.l.a.431.13		48	12.11	even	2
1152.4.l.a.287.12		48	16.13	even	4
1152.4.l.a.287.13		48	48.29	odd	4
1152.4.l.a.863.12		48	24.11	even	2
1152.4.l.a.863.13		48	8.3	odd	2
1152.4.l.b.287.12		48	16.3	odd	4
1152.4.l.b.287.13		48	48.35	even	4
1152.4.l.b.863.12		48	24.5	odd	2
1152.4.l.b.863.13		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)

\(f(q)\)	\(=\)	\(q+(-0.0720364 + 2.82751i) q^{2} +(-7.98962 - 0.407367i) q^{4} +(-2.40838 - 2.40838i) q^{5} -11.7205 q^{7} +(1.72738 - 22.5614i) q^{8} +(6.98322 - 6.63623i) q^{10} +(34.7409 - 34.7409i) q^{11} +(-3.17950 - 3.17950i) q^{13} +(0.844304 - 33.1399i) q^{14} +(63.6681 + 6.50942i) q^{16} -98.0797i q^{17} +(-15.9562 + 15.9562i) q^{19} +(18.2610 + 20.2232i) q^{20} +(95.7276 + 100.733i) q^{22} -69.6819i q^{23} -113.399i q^{25} +(9.21909 - 8.76101i) q^{26} +(93.6425 + 4.77456i) q^{28} +(15.9649 - 15.9649i) q^{29} +121.295i q^{31} +(-22.9919 + 179.553i) q^{32} +(277.321 + 7.06531i) q^{34} +(28.2275 + 28.2275i) q^{35} +(-37.0567 + 37.0567i) q^{37} +(-43.9670 - 46.2658i) q^{38} +(-58.4966 + 50.1763i) q^{40} -59.3202 q^{41} +(-241.737 - 241.737i) q^{43} +(-291.719 + 263.414i) q^{44} +(197.026 + 5.01964i) q^{46} +395.106 q^{47} -205.629 q^{49} +(320.638 + 8.16889i) q^{50} +(24.1077 + 26.6982i) q^{52} +(-458.194 - 458.194i) q^{53} -167.339 q^{55} +(-20.2458 + 264.431i) q^{56} +(43.9910 + 46.2911i) q^{58} +(257.629 - 257.629i) q^{59} +(-373.295 - 373.295i) q^{61} +(-342.962 - 8.73764i) q^{62} +(-506.032 - 77.9441i) q^{64} +15.3149i q^{65} +(-648.397 + 648.397i) q^{67} +(-39.9545 + 783.619i) q^{68} +(-81.8469 + 77.7801i) q^{70} +787.139i q^{71} -1074.96i q^{73} +(-102.109 - 107.447i) q^{74} +(133.984 - 120.984i) q^{76} +(-407.181 + 407.181i) q^{77} +382.146i q^{79} +(-137.660 - 169.014i) q^{80} +(4.27322 - 167.728i) q^{82} +(491.315 + 491.315i) q^{83} +(-236.213 + 236.213i) q^{85} +(700.927 - 666.099i) q^{86} +(-723.792 - 843.814i) q^{88} -624.448 q^{89} +(37.2653 + 37.2653i) q^{91} +(-28.3861 + 556.732i) q^{92} +(-28.4620 + 1117.17i) q^{94} +76.8575 q^{95} +1665.45 q^{97} +(14.8128 - 581.419i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 48 q - 120 q^{10} - 144 q^{16} - 48 q^{19} + 72 q^{22} + 72 q^{28} - 984 q^{34} - 1272 q^{40} + 864 q^{43} - 1416 q^{46} + 2352 q^{49} - 648 q^{52} - 576 q^{55} + 1128 q^{58} + 1824 q^{61} + 3024 q^{64}+ \cdots - 11304 q^{94}+O(q^{100}) \) 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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