LMFDB - Embedded newform 144.4.l.a.35.7

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			35.7
Character	$\chi$	$=$	144.35
Dual form			144.4.l.a.107.7

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+(-1.80581 + 2.17694i) q^{2} +(-1.47812 - 7.86226i) q^{4} +(-6.31601 - 6.31601i) q^{5} +16.2772 q^{7} +(19.7849 + 10.9799i) q^{8} +(25.1551 - 2.34407i) q^{10} +(-48.1287 + 48.1287i) q^{11} +(-8.61642 - 8.61642i) q^{13} +(-29.3935 + 35.4345i) q^{14} +(-59.6303 + 23.2428i) q^{16} +53.2206i q^{17} +(-55.5604 + 55.5604i) q^{19} +(-40.3223 + 58.9940i) q^{20} +(-17.8621 - 191.684i) q^{22} -66.9842i q^{23} -45.2160i q^{25} +(34.3170 - 3.19782i) q^{26} +(-24.0597 - 127.976i) q^{28} +(-126.481 + 126.481i) q^{29} +121.117i q^{31} +(57.0828 - 171.783i) q^{32} +(-115.858 - 96.1061i) q^{34} +(-102.807 - 102.807i) q^{35} +(-250.289 + 250.289i) q^{37} +(-20.6202 - 221.283i) q^{38} +(-55.6119 - 194.311i) q^{40} -402.012 q^{41} +(187.233 + 187.233i) q^{43} +(449.541 + 307.261i) q^{44} +(145.821 + 120.961i) q^{46} +96.1703 q^{47} -78.0518 q^{49} +(98.4325 + 81.6514i) q^{50} +(-55.0084 + 80.4807i) q^{52} +(90.3337 + 90.3337i) q^{53} +607.963 q^{55} +(322.043 + 178.723i) q^{56} +(-46.9409 - 503.740i) q^{58} +(488.025 - 488.025i) q^{59} +(378.467 + 378.467i) q^{61} +(-263.664 - 218.714i) q^{62} +(270.882 + 434.473i) q^{64} +108.843i q^{65} +(-223.231 + 223.231i) q^{67} +(418.434 - 78.6665i) q^{68} +(409.455 - 38.1550i) q^{70} -231.902i q^{71} -265.600i q^{73} +(-92.8902 - 996.839i) q^{74} +(518.955 + 354.705i) q^{76} +(-783.402 + 783.402i) q^{77} +604.662i q^{79} +(523.427 + 229.824i) q^{80} +(725.956 - 875.155i) q^{82} +(-351.298 - 351.298i) q^{83} +(336.142 - 336.142i) q^{85} +(-745.702 + 69.4881i) q^{86} +(-1480.67 + 423.769i) q^{88} +1365.36 q^{89} +(-140.251 - 140.251i) q^{91} +(-526.648 + 99.0109i) q^{92} +(-173.665 + 209.357i) q^{94} +701.839 q^{95} -1854.47 q^{97} +(140.947 - 169.914i) q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	−1.80581	+	2.17694i	−0.638449	+	0.769664i
$2$	−1.80581	+	2.17694i	−0.638449	+	0.769664i
$3$		0			0
$3$		0			0
$4$	−1.47812	−	7.86226i	−0.184765	−	0.982783i
$5$	−6.31601	−	6.31601i	−0.564921	−	0.564921i	0.365780	−	0.930701i	$-0.380802\pi$
$5$	−6.31601	−	6.31601i	−0.564921	−	0.564921i	−0.930701	+	0.365780i	$0.880802\pi$
$6$		0			0
$7$	16.2772			0.878888			0.439444	−	0.898270i	$-0.355176\pi$
$7$	16.2772			0.878888			0.439444	+	0.898270i	$0.355176\pi$
$8$	19.7849	+	10.9799i	0.874376	+	0.485250i
$9$		0			0
$10$	25.1551	−	2.34407i	0.795473	−	0.0741260i
$11$	−48.1287	+	48.1287i	−1.31921	+	1.31921i	−0.404816	+	0.914398i	$0.632665\pi$
$11$	−48.1287	+	48.1287i	−1.31921	+	1.31921i	−0.914398	+	0.404816i	$0.867335\pi$
$12$		0			0
$13$	−8.61642	−	8.61642i	−0.183828	−	0.183828i	0.609194	−	0.793022i	$-0.291493\pi$
$13$	−8.61642	−	8.61642i	−0.183828	−	0.183828i	−0.793022	+	0.609194i	$0.791493\pi$
$14$	−29.3935	+	35.4345i	−0.561125	+	0.676448i
$15$		0			0
$16$	−59.6303	+	23.2428i	−0.931724	+	0.363168i
$17$			53.2206i			0.759287i	0.925133	+	0.379644i	$0.123953\pi$
$17$			53.2206i			0.759287i	−0.925133	+	0.379644i	$0.876047\pi$
$18$		0			0
$19$	−55.5604	+	55.5604i	−0.670864	+	0.670864i	−0.957915	−	0.287051i	$-0.907325\pi$
$19$	−55.5604	+	55.5604i	−0.670864	+	0.670864i	0.287051	+	0.957915i	$0.407325\pi$
$20$	−40.3223	+	58.9940i	−0.450817	+	0.659572i
$21$		0			0
$22$	−17.8621	−	191.684i	−0.173100	−	1.85760i
$23$		−	66.9842i		−	0.607269i	−0.952789	−	0.303634i	$-0.901800\pi$
$23$		−	66.9842i		−	0.607269i	0.952789	−	0.303634i	$-0.0982001\pi$
$24$		0			0
$25$		−	45.2160i		−	0.361728i
$26$	34.3170	−	3.19782i	0.258851	−	0.0241210i
$27$		0			0
$28$	−24.0597	−	127.976i	−0.162388	−	0.863756i
$29$	−126.481	+	126.481i	−0.809892	+	0.809892i	−0.984617	−	0.174726i	$-0.944096\pi$
$29$	−126.481	+	126.481i	−0.809892	+	0.809892i	0.174726	+	0.984617i	$0.444096\pi$
$30$		0			0
$31$			121.117i			0.701717i	0.936429	+	0.350858i	$0.114110\pi$
$31$			121.117i			0.701717i	−0.936429	+	0.350858i	$0.885890\pi$
$32$	57.0828	−	171.783i	0.315341	−	0.948979i
$33$		0			0
$34$	−115.858	−	96.1061i	−0.584396	−	0.484766i
$35$	−102.807	−	102.807i	−0.496502	−	0.496502i
$36$		0			0
$37$	−250.289	+	250.289i	−1.11209	+	1.11209i	−0.119222	+	0.992868i	$0.538040\pi$
$37$	−250.289	+	250.289i	−1.11209	+	1.11209i	−0.992868	+	0.119222i	$0.961960\pi$
$38$	−20.6202	−	221.283i	−0.0880273	−	0.944653i
$39$		0			0
$40$	−55.6119	−	194.311i	−0.219825	−	0.768081i
$41$	−402.012			−1.53131			−0.765655	−	0.643252i	$-0.777585\pi$
$41$	−402.012			−1.53131			−0.765655	+	0.643252i	$0.777585\pi$
$42$		0			0
$43$	187.233	+	187.233i	0.664018	+	0.664018i	0.956325	−	0.292307i	$-0.0944228\pi$
$43$	187.233	+	187.233i	0.664018	+	0.664018i	−0.292307	+	0.956325i	$0.594423\pi$
$44$	449.541	+	307.261i	1.54025	+	1.05276i
$45$		0			0
$46$	145.821	+	120.961i	0.467393	+	0.387710i
$47$	96.1703			0.298465			0.149233	−	0.988802i	$-0.452320\pi$
$47$	96.1703			0.298465			0.149233	+	0.988802i	$0.452320\pi$
$48$		0			0
$49$	−78.0518			−0.227556
$50$	98.4325	+	81.6514i	0.278409	+	0.230945i
$51$		0			0
$52$	−55.0084	+	80.4807i	−0.146698	+	0.214628i
$53$	90.3337	+	90.3337i	0.234119	+	0.234119i	0.814409	−	0.580291i	$-0.197061\pi$
$53$	90.3337	+	90.3337i	0.234119	+	0.234119i	−0.580291	+	0.814409i	$0.697061\pi$
$54$		0			0
$55$	607.963			1.49050
$56$	322.043	+	178.723i	0.768478	+	0.426480i
$57$		0			0
$58$	−46.9409	−	503.740i	−0.106270	−	1.14042i
$59$	488.025	−	488.025i	1.07687	−	1.07687i	0.0800840	−	0.996788i	$-0.474481\pi$
$59$	488.025	−	488.025i	1.07687	−	1.07687i	0.996788	−	0.0800840i	$-0.0255188\pi$
$60$		0			0
$61$	378.467	+	378.467i	0.794389	+	0.794389i	0.982204	−	0.187815i	$-0.0601405\pi$
$61$	378.467	+	378.467i	0.794389	+	0.794389i	−0.187815	+	0.982204i	$0.560141\pi$
$62$	−263.664	−	218.714i	−0.540086	−	0.448010i
$63$		0			0
$64$	270.882	+	434.473i	0.529066	+	0.848581i
$65$			108.843i			0.207697i
$66$		0			0
$67$	−223.231	+	223.231i	−0.407045	+	0.407045i	−0.880707	−	0.473662i	$-0.842932\pi$
$67$	−223.231	+	223.231i	−0.407045	+	0.407045i	0.473662	+	0.880707i	$0.342932\pi$
$68$	418.434	−	78.6665i	0.746214	−	0.140290i
$69$		0			0
$70$	409.455	−	38.1550i	0.699131	−	0.0651484i
$71$		−	231.902i		−	0.387630i	−0.981038	−	0.193815i	$-0.937914\pi$
$71$		−	231.902i		−	0.387630i	0.981038	−	0.193815i	$-0.0620862\pi$
$72$		0			0
$73$		−	265.600i		−	0.425838i	−0.977070	−	0.212919i	$-0.931703\pi$
$73$		−	265.600i		−	0.425838i	0.977070	−	0.212919i	$-0.0682970\pi$
$74$	−92.8902	−	996.839i	−0.145923	−	1.56595i
$75$		0			0
$76$	518.955	+	354.705i	0.783266	+	0.535361i
$77$	−783.402	+	783.402i	−1.15944	+	1.15944i
$78$		0			0
$79$			604.662i			0.861137i	0.902558	+	0.430569i	$0.141687\pi$
$79$			604.662i			0.861137i	−0.902558	+	0.430569i	$0.858313\pi$
$80$	523.427	+	229.824i	0.731512	+	0.321189i
$81$		0			0
$82$	725.956	−	875.155i	0.977663	−	1.17859i
$83$	−351.298	−	351.298i	−0.464577	−	0.464577i	0.435575	−	0.900152i	$-0.356545\pi$
$83$	−351.298	−	351.298i	−0.464577	−	0.464577i	−0.900152	+	0.435575i	$0.856545\pi$
$84$		0			0
$85$	336.142	−	336.142i	0.428937	−	0.428937i
$86$	−745.702	+	69.4881i	−0.935013	+	0.0871290i
$87$		0			0
$88$	−1480.67	+	423.769i	−1.79364	+	0.513341i
$89$	1365.36			1.62615			0.813077	−	0.582156i	$-0.197791\pi$
$89$	1365.36			1.62615			0.813077	+	0.582156i	$0.197791\pi$
$90$		0			0
$91$	−140.251	−	140.251i	−0.161564	−	0.161564i
$92$	−526.648	+	99.0109i	−0.596813	+	0.112202i
$93$		0			0
$94$	−173.665	+	209.357i	−0.190555	+	0.229718i
$95$	701.839			0.757971
$96$		0			0
$97$	−1854.47			−1.94117			−0.970584	−	0.240762i	$-0.922603\pi$
$97$	−1854.47			−1.94117			−0.970584	+	0.240762i	$0.922603\pi$
$98$	140.947	−	169.914i	0.145283	−	0.175142i
$99$		0			0
$100$	−355.500	+	66.8348i	−0.355500	+	0.0668348i
$101$	−755.210	−	755.210i	−0.744022	−	0.744022i	0.229328	−	0.973349i	$-0.426347\pi$
$101$	−755.210	−	755.210i	−0.744022	−	0.744022i	−0.973349	+	0.229328i	$0.926347\pi$
$102$		0			0
$103$	1428.89			1.36692			0.683461	−	0.729987i	$-0.260474\pi$
$103$	1428.89			1.36692			0.683461	+	0.729987i	$0.260474\pi$
$104$	−75.8668	−	265.082i	−0.0715323	−	0.249937i
$105$		0			0
$106$	−359.776	+	33.5257i	−0.329665	+	0.0307198i
$107$	1013.69	−	1013.69i	0.915859	−	0.915859i	−0.0808660	−	0.996725i	$-0.525769\pi$
$107$	1013.69	−	1013.69i	0.915859	−	0.915859i	0.996725	+	0.0808660i	$0.0257686\pi$
$108$		0			0
$109$	−1069.24	−	1069.24i	−0.939579	−	0.939579i	0.0586964	−	0.998276i	$-0.481306\pi$
$109$	−1069.24	−	1069.24i	−0.939579	−	0.939579i	−0.998276	+	0.0586964i	$0.981306\pi$
$110$	−1097.86	+	1323.50i	−0.951611	+	1.14719i
$111$		0			0
$112$	−970.616	+	378.328i	−0.818880	+	0.319184i
$113$			1280.42i			1.06595i	0.846132	+	0.532974i	$0.178926\pi$
$113$			1280.42i			1.06595i	−0.846132	+	0.532974i	$0.821074\pi$
$114$		0			0
$115$	−423.073	+	423.073i	−0.343059	+	0.343059i
$116$	1181.38	+	807.470i	0.945587	+	0.646308i
$117$		0			0
$118$	181.121	+	1943.68i	0.141302	+	1.51636i
$119$			866.283i			0.667328i
$120$		0			0
$121$		−	3301.75i		−	2.48065i
$122$	−1507.34	+	140.461i	−1.11859	+	0.104236i
$123$		0			0
$124$	952.252	−	179.025i	0.689635	−	0.129653i
$125$	−1075.09	+	1075.09i	−0.769269	+	0.769269i
$126$		0			0
$127$		−	2778.90i		−	1.94163i	−0.239828	−	0.970815i	$-0.577091\pi$
$127$		−	2778.90i		−	1.94163i	0.239828	−	0.970815i	$-0.422909\pi$
$128$	−1434.98	−	194.883i	−0.990904	−	0.134573i
$129$		0			0
$130$	−236.944	−	196.549i	−0.159857	−	0.132604i
$131$	−938.133	−	938.133i	−0.625687	−	0.625687i	0.321293	−	0.946980i	$-0.395883\pi$
$131$	−938.133	−	938.133i	−0.625687	−	0.625687i	−0.946980	+	0.321293i	$0.895883\pi$
$132$		0			0
$133$	−904.369	+	904.369i	−0.589614	+	0.589614i
$134$	−82.8481	−	889.073i	−0.0534103	−	0.573165i
$135$		0			0
$136$	−584.359	+	1052.96i	−0.368444	+	0.663902i
$137$	−96.6385			−0.0602656			−0.0301328	−	0.999546i	$-0.509593\pi$
$137$	−96.6385			−0.0602656			−0.0301328	+	0.999546i	$0.509593\pi$
$138$		0			0
$139$	−211.002	−	211.002i	−0.128755	−	0.128755i	0.639793	−	0.768548i	$-0.279020\pi$
$139$	−211.002	−	211.002i	−0.128755	−	0.128755i	−0.768548	+	0.639793i	$0.779020\pi$
$140$	−656.335	+	960.258i	−0.396217	+	0.579690i
$141$		0			0
$142$	504.837	+	418.771i	0.298345	+	0.247482i
$143$	829.395			0.485017
$144$		0			0
$145$	1597.71			0.915050
$146$	578.196	+	479.623i	0.327752	+	0.271876i
$147$		0			0
$148$	2337.80	+	1597.88i	1.29842	+	0.887467i
$149$	1113.17	+	1113.17i	0.612042	+	0.612042i	0.943478	−	0.331435i	$-0.107533\pi$
$149$	1113.17	+	1113.17i	0.612042	+	0.612042i	−0.331435	+	0.943478i	$0.607533\pi$
$150$		0			0
$151$	285.222			0.153716			0.0768578	−	0.997042i	$-0.475511\pi$
$151$	285.222			0.153716			0.0768578	+	0.997042i	$0.475511\pi$
$152$	−1709.30	+	489.204i	−0.912124	+	0.261051i
$153$		0			0
$154$	−290.745	−	3120.09i	−0.152136	−	1.63262i
$155$	764.975	−	764.975i	0.396414	−	0.396414i
$156$		0			0
$157$	2536.76	+	2536.76i	1.28953	+	1.28953i	0.935073	+	0.354454i	$0.115333\pi$
$157$	2536.76	+	2536.76i	1.28953	+	1.28953i	0.354454	+	0.935073i	$0.384667\pi$
$158$	−1316.31	−	1091.90i	−0.662786	−	0.549792i
$159$		0			0
$160$	−1445.52	+	724.451i	−0.714241	+	0.357955i
$161$		−	1090.32i		−	0.533721i
$162$		0			0
$163$	−1268.22	+	1268.22i	−0.609414	+	0.609414i	−0.942793	−	0.333379i	$-0.891811\pi$
$163$	−1268.22	+	1268.22i	−0.609414	+	0.609414i	0.333379	+	0.942793i	$0.391811\pi$
$164$	594.222	+	3160.72i	0.282933	+	1.50494i
$165$		0			0
$166$	1399.13	−	130.378i	0.654178	−	0.0609594i
$167$			2766.46i			1.28189i	0.767588	+	0.640943i	$0.221457\pi$
$167$			2766.46i			1.28189i	−0.767588	+	0.640943i	$0.778543\pi$
$168$		0			0
$169$		−	2048.51i		−	0.932415i
$170$	124.753	+	1338.77i	0.0562829	+	0.603992i
$171$		0			0
$172$	1195.32	−	1748.83i	0.529898	−	0.775273i
$173$	346.606	−	346.606i	0.152324	−	0.152324i	−0.626831	−	0.779155i	$-0.715649\pi$
$173$	346.606	−	346.606i	0.152324	−	0.152324i	0.779155	+	0.626831i	$0.215649\pi$
$174$		0			0
$175$		−	735.992i		−	0.317919i
$176$	1751.29	−	3988.58i	0.750046	−	1.70824i
$177$		0			0
$178$	−2465.57	+	2972.30i	−1.03822	+	1.25159i
$179$	106.140	+	106.140i	0.0443198	+	0.0443198i	0.728919	−	0.684600i	$-0.240023\pi$
$179$	106.140	+	106.140i	0.0443198	+	0.0443198i	−0.684600	+	0.728919i	$0.740023\pi$
$180$		0			0
$181$	−1553.97	+	1553.97i	−0.638154	+	0.638154i	−0.950100	−	0.311946i	$-0.899019\pi$
$181$	−1553.97	+	1553.97i	−0.638154	+	0.638154i	0.311946	+	0.950100i	$0.399019\pi$
$182$	558.586	−	52.0517i	0.227501	−	0.0211996i
$183$		0			0
$184$	735.483	−	1325.27i	0.294677	−	0.530981i
$185$	3161.66			1.25649
$186$		0			0
$187$	−2561.44	−	2561.44i	−1.00166	−	1.00166i
$188$	−142.151	−	756.116i	−0.0551460	−	0.293327i
$189$		0			0
$190$	−1267.39	+	1527.86i	−0.483926	+	0.583383i
$191$	1311.80			0.496955			0.248478	−	0.968638i	$-0.420070\pi$
$191$	1311.80			0.496955			0.248478	+	0.968638i	$0.420070\pi$
$192$		0			0
$193$	−1692.72			−0.631319			−0.315659	−	0.948873i	$-0.602226\pi$
$193$	−1692.72			−0.631319			−0.315659	+	0.948873i	$0.602226\pi$
$194$	3348.82	−	4037.08i	1.23934	−	1.49405i
$195$		0			0
$196$	115.370	+	613.664i	0.0420445	+	0.223638i
$197$	−2419.50	−	2419.50i	−0.875037	−	0.875037i	0.117979	−	0.993016i	$-0.462358\pi$
$197$	−2419.50	−	2419.50i	−0.875037	−	0.875037i	−0.993016	+	0.117979i	$0.962358\pi$
$198$		0			0
$199$	4687.76			1.66988			0.834941	−	0.550339i	$-0.185502\pi$
$199$	4687.76			1.66988			0.834941	+	0.550339i	$0.185502\pi$
$200$	496.470	−	894.593i	0.175529	−	0.316286i
$201$		0			0
$202$	3007.81	−	280.282i	1.04767	−	0.0976267i
$203$	−2058.75	+	2058.75i	−0.711804	+	0.711804i
$204$		0			0
$205$	2539.11	+	2539.11i	0.865069	+	0.865069i
$206$	−2580.30	+	3110.61i	−0.872710	+	1.05207i
$207$		0			0
$208$	714.069	+	313.530i	0.238037	+	0.104516i
$209$		−	5348.10i		−	1.77003i
$210$		0			0
$211$	−1994.60	+	1994.60i	−0.650777	+	0.650777i	−0.953180	−	0.302403i	$-0.902211\pi$
$211$	−1994.60	+	1994.60i	−0.650777	+	0.650777i	0.302403	+	0.953180i	$0.402211\pi$
$212$	576.703	−	843.751i	0.186831	−	0.273345i
$213$		0			0
$214$	376.211	+	4037.26i	0.120174	+	1.28963i
$215$		−	2365.13i		−	0.750236i
$216$		0			0
$217$			1971.45i			0.616730i
$218$	4258.49	−	396.827i	1.32303	−	0.123287i
$219$		0			0
$220$	−898.644	−	4779.96i	−0.275393	−	1.46484i
$221$	458.571	−	458.571i	0.139578	−	0.139578i
$222$		0			0
$223$			3668.79i			1.10170i	0.834603	+	0.550852i	$0.185697\pi$
$223$			3668.79i			1.10170i	−0.834603	+	0.550852i	$0.814303\pi$
$224$	929.149	−	2796.16i	0.277149	−	0.834046i
$225$		0			0
$226$	−2787.40	−	2312.20i	−0.820422	−	0.680554i
$227$	−2455.95	−	2455.95i	−0.718092	−	0.718092i	0.250123	−	0.968214i	$-0.419529\pi$
$227$	−2455.95	−	2455.95i	−0.718092	−	0.718092i	−0.968214	+	0.250123i	$0.919529\pi$
$228$		0			0
$229$	1201.57	−	1201.57i	0.346733	−	0.346733i	−0.512158	−	0.858891i	$-0.671154\pi$
$229$	1201.57	−	1201.57i	0.346733	−	0.346733i	0.858891	+	0.512158i	$0.171154\pi$
$230$	−157.016	−	1684.99i	−0.0450144	−	0.483066i
$231$		0			0
$232$	−3891.15	+	1113.65i	−1.10115	+	0.315150i
$233$	1560.55			0.438776			0.219388	−	0.975638i	$-0.429594\pi$
$233$	1560.55			0.438776			0.219388	+	0.975638i	$0.429594\pi$
$234$		0			0
$235$	−607.412	−	607.412i	−0.168609	−	0.168609i
$236$	−4558.34	−	3115.62i	−1.25730	−	0.859363i
$237$		0			0
$238$	−1885.85	−	1564.34i	−0.513618	−	0.426055i
$239$	−3543.70			−0.959091			−0.479546	−	0.877517i	$-0.659198\pi$
$239$	−3543.70			−0.959091			−0.479546	+	0.877517i	$0.659198\pi$
$240$		0			0
$241$	1481.98			0.396110			0.198055	−	0.980191i	$-0.436538\pi$
$241$	1481.98			0.396110			0.198055	+	0.980191i	$0.436538\pi$
$242$	7187.70	+	5962.32i	1.90927	+	1.58377i
$243$		0			0
$244$	2416.19	−	3535.03i	0.633937	−	0.927488i
$245$	492.976	+	492.976i	0.128551	+	0.128551i
$246$		0			0
$247$	957.462			0.246647
$248$	−1329.86	+	2396.28i	−0.340508	+	0.613564i
$249$		0			0
$250$	−398.998	−	4281.79i	−0.100939	−	1.08322i
$251$	−2357.87	+	2357.87i	−0.592938	+	0.592938i	−0.938424	−	0.345486i	$-0.887714\pi$
$251$	−2357.87	+	2357.87i	−0.592938	+	0.592938i	0.345486	+	0.938424i	$0.387714\pi$
$252$		0			0
$253$	3223.87	+	3223.87i	0.801118	+	0.801118i
$254$	6049.48	+	5018.15i	1.49440	+	1.23963i
$255$		0			0
$256$	3015.55	−	2771.95i	0.736218	−	0.676745i
$257$			2117.93i			0.514059i	0.966404	+	0.257029i	$0.0827437\pi$
$257$			2117.93i			0.514059i	−0.966404	+	0.257029i	$0.917256\pi$
$258$		0			0
$259$	−4074.02	+	4074.02i	−0.977402	+	0.977402i
$260$	855.750	−	160.883i	0.204121	−	0.0383751i
$261$		0			0
$262$	3736.34	−	348.170i	0.881038	−	0.0820994i
$263$			7741.72i			1.81511i	0.419929	+	0.907557i	$0.362055\pi$
$263$			7741.72i			1.81511i	−0.419929	+	0.907557i	$0.637945\pi$
$264$		0			0
$265$		−	1141.10i		−	0.264517i
$266$	−335.640	−	3601.87i	−0.0773661	−	0.830244i
$267$		0			0
$268$	2085.06	+	1425.14i	0.475245	+	0.324829i
$269$	−1111.14	+	1111.14i	−0.251849	+	0.251849i	−0.821728	−	0.569880i	$-0.806990\pi$
$269$	−1111.14	+	1111.14i	−0.251849	+	0.251849i	0.569880	+	0.821728i	$0.306990\pi$
$270$		0			0
$271$			1728.54i			0.387458i	0.981055	+	0.193729i	$0.0620583\pi$
$271$			1728.54i			0.387458i	−0.981055	+	0.193729i	$0.937942\pi$
$272$	−1236.99	−	3173.56i	−0.275749	−	0.707446i
$273$		0			0
$274$	174.510	−	210.376i	0.0384765	−	0.0463842i
$275$	2176.19	+	2176.19i	0.477197	+	0.477197i
$276$		0			0
$277$	−1977.36	+	1977.36i	−0.428910	+	0.428910i	−0.888257	−	0.459347i	$-0.848083\pi$
$277$	−1977.36	+	1977.36i	−0.428910	+	0.428910i	0.459347	+	0.888257i	$0.348083\pi$
$278$	840.366	−	78.3093i	0.181301	−	0.0168945i
$279$		0			0
$280$	−905.208	−	3162.84i	−0.193202	−	0.675057i
$281$	−1583.09			−0.336083			−0.168041	−	0.985780i	$-0.553744\pi$
$281$	−1583.09			−0.336083			−0.168041	+	0.985780i	$0.553744\pi$
$282$		0			0
$283$	−987.718	−	987.718i	−0.207469	−	0.207469i	0.595722	−	0.803191i	$-0.296866\pi$
$283$	−987.718	−	987.718i	−0.207469	−	0.207469i	−0.803191	+	0.595722i	$0.796866\pi$
$284$	−1823.28	+	342.780i	−0.380956	+	0.0716206i
$285$		0			0
$286$	−1497.73	+	1805.54i	−0.309659	+	0.373300i
$287$	−6543.64			−1.34585
$288$		0			0
$289$	2080.57			0.423483
$290$	−2885.15	+	3478.11i	−0.584213	+	0.704281i
$291$		0			0
$292$	−2088.22	+	392.590i	−0.418506	+	0.0786800i
$293$	140.498	+	140.498i	0.0280136	+	0.0280136i	0.720975	−	0.692961i	$-0.243694\pi$
$293$	140.498	+	140.498i	0.0280136	+	0.0280136i	−0.692961	+	0.720975i	$0.743694\pi$
$294$		0			0
$295$	−6164.74			−1.21670
$296$	−7700.10	+	2203.78i	−1.51203	+	0.432743i
$297$		0			0
$298$	−4433.47	+	413.132i	−0.861825	+	0.0803090i
$299$	−577.164	+	577.164i	−0.111633	+	0.111633i
$300$		0			0
$301$	3047.64	+	3047.64i	0.583597	+	0.583597i
$302$	−515.057	+	620.912i	−0.0981397	+	0.118309i
$303$		0			0
$304$	2021.71	−	4604.46i	0.381423	−	0.868696i
$305$		−	4780.80i		−	0.897535i
$306$		0			0
$307$	5197.73	−	5197.73i	0.966288	−	0.966288i	−0.0331621	−	0.999450i	$-0.510558\pi$
$307$	5197.73	−	5197.73i	0.966288	−	0.966288i	0.999450	+	0.0331621i	$0.0105578\pi$
$308$	7317.28	+	5001.35i	1.35370	+	0.925254i
$309$		0			0
$310$	283.906	+	3046.70i	0.0520154	+	0.558196i
$311$			8112.26i			1.47911i	0.673094	+	0.739557i	$0.264965\pi$
$311$			8112.26i			1.47911i	−0.673094	+	0.739557i	$0.735035\pi$
$312$		0			0
$313$			3307.27i			0.597246i	0.954371	+	0.298623i	$0.0965273\pi$
$313$			3307.27i			0.597246i	−0.954371	+	0.298623i	$0.903473\pi$
$314$	−10103.3	+	941.473i	−1.81580	+	0.169205i
$315$		0			0
$316$	4754.01	−	893.765i	0.846311	−	0.159108i
$317$	2991.50	−	2991.50i	0.530030	−	0.530030i	−0.390551	−	0.920581i	$-0.627716\pi$
$317$	2991.50	−	2991.50i	0.530030	−	0.530030i	0.920581	+	0.390551i	$0.127716\pi$
$318$		0			0
$319$		−	12174.7i		−	2.13684i
$320$	1033.25	−	4455.03i	0.180501	−	0.778262i
$321$		0			0
$322$	2373.56	+	1968.90i	0.410786	+	0.340754i
$323$	−2956.95	−	2956.95i	−0.509379	−	0.509379i
$324$		0			0
$325$	−389.600	+	389.600i	−0.0664958	+	0.0664958i
$326$	−470.676	−	5050.99i	−0.0799641	−	0.858124i
$327$		0			0
$328$	−7953.75	−	4414.07i	−1.33894	−	0.743067i
$329$	1565.39			0.262318
$330$		0			0
$331$	3171.15	+	3171.15i	0.526592	+	0.526592i	0.919555	−	0.392962i	$-0.128550\pi$
$331$	3171.15	+	3171.15i	0.526592	+	0.526592i	−0.392962	+	0.919555i	$0.628550\pi$
$332$	−2242.73	+	3281.25i	−0.370741	+	0.542416i
$333$		0			0
$334$	−6022.41	−	4995.69i	−0.986622	−	0.818419i
$335$	2819.86			0.459897
$336$		0			0
$337$	−2213.85			−0.357852			−0.178926	−	0.983863i	$-0.557262\pi$
$337$	−2213.85			−0.357852			−0.178926	+	0.983863i	$0.557262\pi$
$338$	4459.49	+	3699.22i	0.717646	+	0.595299i
$339$		0			0
$340$	−3139.69	−	2145.97i	−0.500805	−	0.342299i
$341$	−5829.20	−	5829.20i	−0.925715	−	0.925715i
$342$		0			0
$343$	−6853.56			−1.07888
$344$	1648.57	+	5760.19i	0.258387	+	0.902816i
$345$		0			0
$346$	128.636	+	1380.44i	0.0199871	+	0.214489i
$347$	344.315	−	344.315i	0.0532674	−	0.0532674i	−0.679971	−	0.733239i	$-0.738008\pi$
$347$	344.315	−	344.315i	0.0532674	−	0.0532674i	0.733239	+	0.679971i	$0.238008\pi$
$348$		0			0
$349$	−8129.29	−	8129.29i	−1.24685	−	1.24685i	−0.957103	−	0.289747i	$-0.906429\pi$
$349$	−8129.29	−	8129.29i	−1.24685	−	1.24685i	−0.289747	−	0.957103i	$-0.593571\pi$
$350$	1602.21	+	1329.06i	0.244690	+	0.202975i
$351$		0			0
$352$	5520.40	+	11015.0i	0.835904	+	1.66791i
$353$			3235.69i			0.487871i	0.969791	+	0.243935i	$0.0784385\pi$
$353$			3235.69i			0.487871i	−0.969791	+	0.243935i	$0.921561\pi$
$354$		0			0
$355$	−1464.70	+	1464.70i	−0.218980	+	0.218980i
$356$	−2018.17	−	10734.8i	−0.300457	−	1.59816i
$357$		0			0
$358$	−422.727	+	39.3917i	−0.0624073	+	0.00581541i
$359$		−	4710.91i		−	0.692569i	−0.938129	−	0.346285i	$-0.887443\pi$
$359$		−	4710.91i		−	0.692569i	0.938129	−	0.346285i	$-0.112557\pi$
$360$		0			0
$361$			685.094i			0.0998825i
$362$	−576.728	−	6189.08i	−0.0837353	−	0.898594i
$363$		0			0
$364$	−895.385	+	1310.00i	−0.128931	+	0.188634i
$365$	−1677.53	+	1677.53i	−0.240565	+	0.240565i
$366$		0			0
$367$			1585.57i			0.225521i	0.993622	+	0.112761i	$0.0359693\pi$
$367$			1585.57i			0.225521i	−0.993622	+	0.112761i	$0.964031\pi$
$368$	1556.90	+	3994.29i	0.220541	+	0.565807i
$369$		0			0
$370$	−5709.35	+	6882.74i	−0.802202	+	0.967072i
$371$	1470.38	+	1470.38i	0.205764	+	0.205764i
$372$		0			0
$373$	2078.83	−	2078.83i	0.288573	−	0.288573i	−0.547943	−	0.836516i	$-0.684589\pi$
$373$	2078.83	−	2078.83i	0.288573	−	0.288573i	0.836516	+	0.547943i	$0.184589\pi$
$374$	10201.6	−	950.630i	1.41045	−	0.131433i
$375$		0			0
$376$	1902.72	+	1055.94i	0.260971	+	0.144830i
$377$	2179.62			0.297762
$378$		0			0
$379$	−3704.15	−	3704.15i	−0.502031	−	0.502031i	0.410038	−	0.912068i	$-0.365515\pi$
$379$	−3704.15	−	3704.15i	−0.502031	−	0.502031i	−0.912068	+	0.410038i	$0.865515\pi$
$380$	−1037.40	−	5518.05i	−0.140047	−	0.744920i
$381$		0			0
$382$	−2368.86	+	2855.70i	−0.317281	+	0.382488i
$383$	7560.19			1.00864			0.504318	−	0.863518i	$-0.331744\pi$
$383$	7560.19			1.00864			0.504318	+	0.863518i	$0.331744\pi$
$384$		0			0
$385$	9895.95			1.30999
$386$	3056.72	−	3684.94i	0.403065	−	0.485903i
$387$		0			0
$388$	2741.14	+	14580.4i	0.358660	+	1.90775i
$389$	4868.51	+	4868.51i	0.634559	+	0.634559i	0.949208	−	0.314649i	$-0.101887\pi$
$389$	4868.51	+	4868.51i	0.634559	+	0.634559i	−0.314649	+	0.949208i	$0.601887\pi$
$390$		0			0
$391$	3564.94			0.461091
$392$	−1544.24	−	857.005i	−0.198970	−	0.110422i
$393$		0			0
$394$	9636.25	−	897.953i	1.23215	−	0.114818i
$395$	3819.05	−	3819.05i	0.486474	−	0.486474i
$396$		0			0
$397$	−2442.83	−	2442.83i	−0.308822	−	0.308822i	0.535631	−	0.844452i	$-0.320074\pi$
$397$	−2442.83	−	2442.83i	−0.308822	−	0.308822i	−0.844452	+	0.535631i	$0.820074\pi$
$398$	−8465.19	+	10205.0i	−1.06613	+	1.28525i
$399$		0			0
$400$	1050.95	+	2696.25i	0.131368	+	0.337031i
$401$			10348.7i			1.28876i	0.764707	+	0.644378i	$0.222883\pi$
$401$			10348.7i			1.28876i	−0.764707	+	0.644378i	$0.777117\pi$
$402$		0			0
$403$	1043.59	−	1043.59i	0.128995	−	0.128995i
$404$	−4821.37	+	7053.95i	−0.593742	+	0.868681i
$405$		0			0
$406$	−764.068	−	8199.49i	−0.0933992	−	1.00230i
$407$		−	24092.2i		−	2.93417i
$408$		0			0
$409$			3378.72i			0.408477i	0.978921	+	0.204238i	$0.0654717\pi$
$409$			3378.72i			0.408477i	−0.978921	+	0.204238i	$0.934528\pi$
$410$	−10112.6	+	942.343i	−1.21811	+	0.113510i
$411$		0			0
$412$	−2112.08	−	11234.3i	−0.252560	−	1.34339i
$413$	7943.70	−	7943.70i	0.946450	−	0.946450i
$414$		0			0
$415$			4437.60i			0.524899i
$416$	−1972.01	+	988.309i	−0.232417	+	0.116480i
$417$		0			0
$418$	11642.5	+	9657.63i	1.36233	+	1.13007i
$419$	8945.75	+	8945.75i	1.04303	+	1.04303i	0.999032	+	0.0439961i	$0.0140089\pi$
$419$	8945.75	+	8945.75i	1.04303	+	1.04303i	0.0439961	+	0.999032i	$0.485991\pi$
$420$		0			0
$421$	10686.4	−	10686.4i	1.23711	−	1.23711i	0.275933	−	0.961177i	$-0.411013\pi$
$421$	10686.4	−	10686.4i	1.23711	−	1.23711i	0.961177	−	0.275933i	$-0.0889867\pi$
$422$	−740.259	−	7943.99i	−0.0853916	−	0.916368i
$423$		0			0
$424$	795.380	+	2779.10i	0.0911016	+	0.318313i
$425$	2406.42			0.274656
$426$		0			0
$427$	6160.40	+	6160.40i	0.698179	+	0.698179i
$428$	−9468.23	−	6471.53i	−1.06931	−	0.730871i
$429$		0			0
$430$	5148.75	+	4270.97i	0.577429	+	0.478987i
$431$	4267.29			0.476910			0.238455	−	0.971154i	$-0.423359\pi$
$431$	4267.29			0.476910			0.238455	+	0.971154i	$0.423359\pi$
$432$		0			0
$433$	−15686.4			−1.74097			−0.870483	−	0.492198i	$-0.836194\pi$
$433$	−15686.4			−1.74097			−0.870483	+	0.492198i	$0.836194\pi$
$434$	−4291.72	−	3560.05i	−0.474675	−	0.393751i
$435$		0			0
$436$	−6826.15	+	9987.07i	−0.749801	+	1.09700i
$437$	3721.67	+	3721.67i	0.407395	+	0.407395i
$438$		0			0
$439$	5963.75			0.648370			0.324185	−	0.945994i	$-0.394910\pi$
$439$	5963.75			0.648370			0.324185	+	0.945994i	$0.394910\pi$
$440$	12028.5	+	6675.40i	1.30326	+	0.723267i
$441$		0			0
$442$	170.190	+	1826.37i	0.0183147	+	0.196542i
$443$	−1512.40	+	1512.40i	−0.162204	+	0.162204i	−0.783542	−	0.621338i	$-0.786589\pi$
$443$	−1512.40	+	1512.40i	−0.162204	+	0.162204i	0.621338	+	0.783542i	$0.286589\pi$
$444$		0			0
$445$	−8623.62	−	8623.62i	−0.918649	−	0.918649i
$446$	−7986.72	−	6625.12i	−0.847942	−	0.703382i
$447$		0			0
$448$	4409.20	+	7072.02i	0.464989	+	0.745807i
$449$		−	5201.31i		−	0.546693i	−0.961916	−	0.273346i	$-0.911869\pi$
$449$		−	5201.31i		−	0.546693i	0.961916	−	0.273346i	$-0.0881305\pi$
$450$		0			0
$451$	19348.3	−	19348.3i	2.02013	−	2.02013i
$452$	10067.0	−	1892.62i	1.04760	−	0.196950i
$453$		0			0
$454$	9781.40	−	911.478i	1.01115	−	0.0942242i
$455$			1771.66i			0.182542i
$456$		0			0
$457$		−	6308.57i		−	0.645738i	−0.946444	−	0.322869i	$-0.895353\pi$
$457$		−	6308.57i		−	0.645738i	0.946444	−	0.322869i	$-0.104647\pi$
$458$	445.940	+	4785.54i	0.0454965	+	0.488239i
$459$		0			0
$460$	3951.67	+	2700.96i	0.400538	+	0.273767i
$461$	−1200.54	+	1200.54i	−0.121290	+	0.121290i	−0.765146	−	0.643856i	$-0.777333\pi$
$461$	−1200.54	+	1200.54i	−0.121290	+	0.121290i	0.643856	+	0.765146i	$0.277333\pi$
$462$		0			0
$463$			15194.4i			1.52515i	0.646901	+	0.762574i	$0.276065\pi$
$463$			15194.4i			1.52515i	−0.646901	+	0.762574i	$0.723935\pi$
$464$	4602.32	−	10481.8i	0.460468	−	1.04872i
$465$		0			0
$466$	−2818.05	+	3397.21i	−0.280136	+	0.337710i
$467$	4068.55	+	4068.55i	0.403148	+	0.403148i	0.879341	−	0.476193i	$-0.157984\pi$
$467$	4068.55	+	4068.55i	0.403148	+	0.403148i	−0.476193	+	0.879341i	$0.657984\pi$
$468$		0			0
$469$	−3633.58	+	3633.58i	−0.357747	+	0.357747i
$470$	2419.17	−	225.430i	0.237421	−	0.0221240i
$471$		0			0
$472$	15014.0	−	4297.02i	1.46414	−	0.419039i
$473$	−18022.6			−1.75196
$474$		0			0
$475$	2512.22	+	2512.22i	0.242671	+	0.242671i
$476$	6810.95	−	1280.47i	0.655839	−	0.123299i
$477$		0			0
$478$	6399.24	−	7714.41i	0.612331	−	0.738178i
$479$	−3340.15			−0.318612			−0.159306	−	0.987229i	$-0.550926\pi$
$479$	−3340.15			−0.318612			−0.159306	+	0.987229i	$0.550926\pi$
$480$		0			0
$481$	4313.20			0.408867
$482$	−2676.16	+	3226.17i	−0.252896	+	0.304872i
$483$		0			0
$484$	−25959.2	+	4880.39i	−2.43794	+	0.458338i
$485$	11712.9	+	11712.9i	1.09661	+	1.09661i
$486$		0			0
$487$	−10788.7			−1.00386			−0.501932	−	0.864907i	$-0.667377\pi$
$487$	−10788.7			−1.00386			−0.501932	+	0.864907i	$0.667377\pi$
$488$	3332.37	+	11643.5i	0.309118	+	1.08007i
$489$		0			0
$490$	−1963.40	+	182.959i	−0.181015	+	0.0168678i
$491$	4711.73	−	4711.73i	0.433071	−	0.433071i	−0.456601	−	0.889672i	$-0.650933\pi$
$491$	4711.73	−	4711.73i	0.433071	−	0.433071i	0.889672	+	0.456601i	$0.150933\pi$
$492$		0			0
$493$	−6731.37	−	6731.37i	−0.614940	−	0.614940i
$494$	−1728.99	+	2084.34i	−0.157472	+	0.189836i
$495$		0			0
$496$	−2815.09	−	7222.23i	−0.254841	−	0.653806i
$497$		−	3774.73i		−	0.340683i
$498$		0			0
$499$	−5656.27	+	5656.27i	−0.507433	+	0.507433i	−0.913738	−	0.406304i	$-0.866817\pi$
$499$	−5656.27	+	5656.27i	−0.507433	+	0.507433i	0.406304	+	0.913738i	$0.366817\pi$
$500$	10041.7	+	6863.50i	0.898158	+	0.613890i
$501$		0			0
$502$	−875.080	−	9390.80i	−0.0778022	−	0.834924i
$503$		−	4974.64i		−	0.440971i	−0.975390	−	0.220485i	$-0.929236\pi$
$503$		−	4974.64i		−	0.440971i	0.975390	−	0.220485i	$-0.0707641\pi$
$504$		0			0
$505$			9539.83i			0.840627i
$506$	−12839.8	+	1196.48i	−1.12806	+	0.105118i
$507$		0			0
$508$	−21848.4	+	4107.55i	−1.90820	+	0.358746i
$509$	−5549.92	+	5549.92i	−0.483292	+	0.483292i	−0.906181	−	0.422889i	$-0.861016\pi$
$509$	−5549.92	+	5549.92i	−0.483292	+	0.483292i	0.422889	+	0.906181i	$0.361016\pi$
$510$		0			0
$511$		−	4323.24i		−	0.374264i
$512$	588.859	+	11570.3i	0.0508284	+	0.998707i
$513$		0			0
$514$	−4610.61	−	3824.58i	−0.395653	−	0.328201i
$515$	−9024.90	−	9024.90i	−0.772203	−	0.772203i
$516$		0			0
$517$	−4628.55	+	4628.55i	−0.393740	+	0.393740i
$518$	−1512.00	−	16225.8i	−0.128250	−	1.37629i
$519$		0			0
$520$	−1195.09	+	2153.44i	−0.100785	+	0.181605i
$521$	−21737.7			−1.82792			−0.913961	−	0.405802i	$-0.866992\pi$
$521$	−21737.7			−1.82792			−0.913961	+	0.405802i	$0.866992\pi$
$522$		0			0
$523$	2235.12	+	2235.12i	0.186874	+	0.186874i	0.794343	−	0.607469i	$-0.207815\pi$
$523$	2235.12	+	2235.12i	0.186874	+	0.186874i	−0.607469	+	0.794343i	$0.707815\pi$
$524$	−5989.17	+	8762.52i	−0.499309	+	0.730520i
$525$		0			0
$526$	−16853.2	−	13980.1i	−1.39703	−	1.15886i
$527$	−6445.90			−0.532804
$528$		0			0
$529$	7680.11			0.631225
$530$	2484.10	+	2060.60i	0.203589	+	0.168881i
$531$		0			0
$532$	8447.15	+	5773.62i	0.688403	+	0.470522i
$533$	3463.90	+	3463.90i	0.281498	+	0.281498i
$534$		0			0
$535$	−12804.9			−1.03478
$536$	−6867.66	+	1965.53i	−0.553429	+	0.158392i
$537$		0			0
$538$	−412.378	−	4425.38i	−0.0330463	−	0.354631i
$539$	3756.53	−	3756.53i	0.300196	−	0.300196i
$540$		0			0
$541$	−3371.85	−	3371.85i	−0.267961	−	0.267961i	0.560317	−	0.828278i	$-0.310679\pi$
$541$	−3371.85	−	3371.85i	−0.267961	−	0.267961i	−0.828278	+	0.560317i	$0.810679\pi$
$542$	−3762.92	−	3121.40i	−0.298213	−	0.247372i
$543$		0			0
$544$	9142.41	+	3037.98i	0.720547	+	0.239434i
$545$			13506.6i			1.06158i
$546$		0			0
$547$	3242.62	−	3242.62i	0.253463	−	0.253463i	−0.568926	−	0.822389i	$-0.692641\pi$
$547$	3242.62	−	3242.62i	0.253463	−	0.253463i	0.822389	+	0.568926i	$0.192641\pi$
$548$	142.843	+	759.797i	0.0111350	+	0.0592280i
$549$		0			0
$550$	−8667.21	+	807.653i	−0.671948	+	0.0626153i
$551$		−	14054.6i		−	1.08665i
$552$		0			0
$553$			9842.23i			0.756843i
$554$	−733.861	−	7875.32i	−0.0562793	−	0.603954i
$555$		0			0
$556$	−1347.06	+	1970.84i	−0.102749	+	0.150327i
$557$	4967.01	−	4967.01i	0.377844	−	0.377844i	−0.492480	−	0.870324i	$-0.663909\pi$
$557$	4967.01	−	4967.01i	0.377844	−	0.377844i	0.870324	+	0.492480i	$0.163909\pi$
$558$		0			0
$559$		−	3226.56i		−	0.244130i
$560$	8519.94	+	3740.90i	0.642917	+	0.282289i
$561$		0			0
$562$	2858.76	−	3446.29i	0.214572	−	0.258671i
$563$	10845.8	+	10845.8i	0.811891	+	0.811891i	0.984917	−	0.173026i	$-0.0553545\pi$
$563$	10845.8	+	10845.8i	0.811891	+	0.811891i	−0.173026	+	0.984917i	$0.555355\pi$
$564$		0			0
$565$	8087.17	−	8087.17i	0.602177	−	0.602177i
$566$	3933.83	−	366.573i	0.292140	−	0.0272230i
$567$		0			0
$568$	2546.28	−	4588.16i	0.188097	−	0.338934i
$569$	−18374.6			−1.35378			−0.676892	−	0.736082i	$-0.736674\pi$
$569$	−18374.6			−1.35378			−0.676892	+	0.736082i	$0.736674\pi$
$570$		0			0
$571$	4779.72	+	4779.72i	0.350306	+	0.350306i	0.860224	−	0.509917i	$-0.170324\pi$
$571$	4779.72	+	4779.72i	0.350306	+	0.350306i	−0.509917	+	0.860224i	$0.670324\pi$
$572$	−1225.95	−	6520.92i	−0.0896143	−	0.476666i
$573$		0			0
$574$	11816.5	−	14245.1i	0.859256	−	1.03585i
$575$	−3028.76			−0.219666
$576$		0			0
$577$	−14300.3			−1.03176			−0.515882	−	0.856659i	$-0.672536\pi$
$577$	−14300.3			−1.03176			−0.515882	+	0.856659i	$0.672536\pi$
$578$	−3757.11	+	4529.28i	−0.270372	+	0.325940i
$579$		0			0
$580$	−2361.60	−	12561.6i	−0.169069	−	0.899295i
$581$	−5718.15	−	5718.15i	−0.408311	−	0.408311i
$582$		0			0
$583$	−8695.29			−0.617705
$584$	2916.28	−	5254.87i	0.206638	−	0.372342i
$585$		0			0
$586$	−559.569	+	52.1433i	−0.0394464	+	0.00367580i
$587$	−7852.73	+	7852.73i	−0.552158	+	0.552158i	−0.927063	−	0.374905i	$-0.877675\pi$
$587$	−7852.73	+	7852.73i	−0.552158	+	0.552158i	0.374905	+	0.927063i	$0.377675\pi$
$588$		0			0
$589$	−6729.29	−	6729.29i	−0.470756	−	0.470756i
$590$	11132.3	−	13420.3i	0.776798	−	0.936447i
$591$		0			0
$592$	9107.42	−	20742.2i	0.632285	−	1.44004i
$593$			27409.5i			1.89810i	0.315122	+	0.949051i	$0.397954\pi$
$593$			27409.5i			1.89810i	−0.315122	+	0.949051i	$0.602046\pi$
$594$		0			0
$595$	5471.45	−	5471.45i	0.376988	−	0.376988i
$596$	7106.62	−	10397.4i	0.488421	−	0.714589i
$597$		0			0
$598$	−214.204	−	2298.70i	−0.0146479	−	0.157192i
$599$		−	8983.16i		−	0.612758i	−0.951910	−	0.306379i	$-0.900883\pi$
$599$		−	8983.16i		−	0.612758i	0.951910	−	0.306379i	$-0.0991174\pi$
$600$		0			0
$601$			1686.23i			0.114447i	0.998361	+	0.0572236i	$0.0182248\pi$
$601$			1686.23i			0.114447i	−0.998361	+	0.0572236i	$0.981775\pi$
$602$	−12138.0	+	1131.07i	−0.821771	+	0.0765766i
$603$		0			0
$604$	−421.593	−	2242.49i	−0.0284013	−	0.151069i
$605$	−20853.9	+	20853.9i	−1.40137	+	1.40137i
$606$		0			0
$607$			19571.7i			1.30872i	0.756185	+	0.654358i	$0.227061\pi$
$607$			19571.7i			1.30872i	−0.756185	+	0.654358i	$0.772939\pi$
$608$	6372.81	+	12715.9i	0.425085	+	0.848186i
$609$		0			0
$610$	10407.5	+	8633.21i	0.690800	+	0.573030i
$611$	−828.643	−	828.643i	−0.0548663	−	0.0548663i
$612$		0			0
$613$	−16131.0	+	16131.0i	−1.06285	+	1.06285i	−0.0649592	+	0.997888i	$0.520692\pi$
$613$	−16131.0	+	16131.0i	−1.06285	+	1.06285i	−0.997888	+	0.0649592i	$0.979308\pi$
$614$	1929.04	+	20701.3i	0.126791	+	1.36064i
$615$		0			0
$616$	−24101.2	+	6897.79i	−1.57641	+	0.451169i
$617$	14419.1			0.940829			0.470415	−	0.882445i	$-0.344104\pi$
$617$	14419.1			0.940829			0.470415	+	0.882445i	$0.344104\pi$
$618$		0			0
$619$	14348.2	+	14348.2i	0.931665	+	0.931665i	0.997810	−	0.0661446i	$-0.0210699\pi$
$619$	14348.2	+	14348.2i	0.931665	+	0.931665i	−0.0661446	+	0.997810i	$0.521070\pi$
$620$	−7145.16	−	4883.71i	−0.462833	−	0.316346i
$621$		0			0
$622$	−17659.9	−	14649.2i	−1.13842	−	0.944339i
$623$	22224.3			1.42921
$624$		0			0
$625$	7928.50			0.507424
$626$	−7199.73	−	5972.29i	−0.459679	−	0.381311i
$627$		0			0
$628$	16195.1	−	23694.4i	1.02907	−	1.50559i
$629$	−13320.5	−	13320.5i	−0.844395	−	0.844395i
$630$		0			0
$631$	28650.9			1.80757			0.903785	−	0.427987i	$-0.140777\pi$
$631$	28650.9			1.80757			0.903785	+	0.427987i	$0.140777\pi$
$632$	−6639.16	+	11963.2i	−0.417866	+	0.752957i
$633$		0			0
$634$	1110.24	+	11914.4i	0.0695478	+	0.746342i
$635$	−17551.5	+	17551.5i	−1.09687	+	1.09687i
$636$		0			0
$637$	672.527	+	672.527i	0.0418312	+	0.0418312i
$638$	26503.6	+	21985.2i	1.64465	+	1.36426i
$639$		0			0
$640$	7832.48	+	10294.2i	0.483759	+	0.635806i
$641$		−	11361.1i		−	0.700060i	−0.936739	−	0.350030i	$-0.886171\pi$
$641$		−	11361.1i		−	0.700060i	0.936739	−	0.350030i	$-0.113829\pi$
$642$		0			0
$643$	1801.19	−	1801.19i	0.110470	−	0.110470i	−0.649711	−	0.760181i	$-0.725110\pi$
$643$	1801.19	−	1801.19i	0.110470	−	0.110470i	0.760181	+	0.649711i	$0.225110\pi$
$644$	−8572.36	+	1611.62i	−0.524532	+	0.0986131i
$645$		0			0
$646$	11776.8	−	1097.42i	0.717263	−	0.0668380i
$647$			3042.62i			0.184881i	0.995718	+	0.0924403i	$0.0294667\pi$
$647$			3042.62i			0.184881i	−0.995718	+	0.0924403i	$0.970533\pi$
$648$		0			0
$649$			46976.1i			2.84125i
$650$	−144.593	−	1551.68i	−0.00872523	−	0.0936336i
$651$		0			0
$652$	11845.6	+	8096.48i	0.711520	+	0.486323i
$653$	15755.2	−	15755.2i	0.944176	−	0.944176i	−0.0543465	−	0.998522i	$-0.517308\pi$
$653$	15755.2	−	15755.2i	0.944176	−	0.944176i	0.998522	+	0.0543465i	$0.0173076\pi$
$654$		0			0
$655$			11850.5i			0.706928i
$656$	23972.1	−	9343.86i	1.42676	−	0.556123i
$657$		0			0
$658$	−2826.78	+	3407.75i	−0.167476	+	0.201896i
$659$	3650.15	+	3650.15i	0.215766	+	0.215766i	0.806711	−	0.590946i	$-0.201245\pi$
$659$	3650.15	+	3650.15i	0.215766	+	0.215766i	−0.590946	+	0.806711i	$0.701245\pi$
$660$		0			0
$661$	17498.6	−	17498.6i	1.02968	−	1.02968i	0.0301299	−	0.999546i	$-0.490408\pi$
$661$	17498.6	−	17498.6i	1.02968	−	1.02968i	0.999546	−	0.0301299i	$-0.00959208\pi$
$662$	−12629.9	+	1176.91i	−0.741502	+	0.0690967i
$663$		0			0
$664$	−3093.15	−	10807.6i	−0.180779	−	0.631651i
$665$	11424.0			0.666171
$666$		0			0
$667$	8472.21	+	8472.21i	0.491822	+	0.491822i
$668$	21750.6	−	4089.16i	1.25982	−	0.236848i
$669$		0			0
$670$	−5092.12	+	6138.66i	−0.293621	+	0.353966i
$671$	−36430.3			−2.09594
$672$		0			0
$673$	5976.34			0.342305			0.171152	−	0.985245i	$-0.445251\pi$
$673$	5976.34			0.342305			0.171152	+	0.985245i	$0.445251\pi$
$674$	3997.78	−	4819.41i	0.228470	−	0.275425i
$675$		0			0
$676$	−16106.0	+	3027.95i	−0.916361	+	0.172278i
$677$	−5706.93	−	5706.93i	−0.323981	−	0.323981i	0.526311	−	0.850292i	$-0.323575\pi$
$677$	−5706.93	−	5706.93i	−0.323981	−	0.323981i	−0.850292	+	0.526311i	$0.823575\pi$
$678$		0			0
$679$	−30185.7			−1.70607
$680$	10341.3	−	2959.70i	0.583194	−	0.166911i
$681$		0			0
$682$	23216.2	−	2163.40i	1.30351	−	0.121467i
$683$	18585.6	−	18585.6i	1.04123	−	1.04123i	0.0421122	−	0.999113i	$-0.486591\pi$
$683$	18585.6	−	18585.6i	1.04123	−	1.04123i	0.999113	−	0.0421122i	$-0.0134087\pi$
$684$		0			0
$685$	610.370	+	610.370i	0.0340453	+	0.0340453i
$686$	12376.2	−	14919.8i	0.688813	−	0.830378i
$687$		0			0
$688$	−15516.6	−	6812.95i	−0.859832	−	0.377531i
$689$		−	1556.71i		−	0.0860751i
$690$		0			0
$691$	−7076.12	+	7076.12i	−0.389563	+	0.389563i	−0.874532	−	0.484968i	$-0.838831\pi$
$691$	−7076.12	+	7076.12i	−0.389563	+	0.389563i	0.484968	+	0.874532i	$0.338831\pi$
$692$	−3237.44	−	2212.78i	−0.177845	−	0.121557i
$693$		0			0
$694$	127.786	+	1371.32i	0.00698947	+	0.0750065i
$695$			2665.38i			0.145473i
$696$		0			0
$697$		−	21395.3i		−	1.16270i
$698$	32376.9	−	3017.03i	1.75571	−	0.163605i
$699$		0			0
$700$	−5786.56	+	1087.89i	−0.312445	+	0.0587403i
$701$	11574.5	−	11574.5i	0.623626	−	0.623626i	−0.322831	−	0.946457i	$-0.604634\pi$
$701$	11574.5	−	11574.5i	0.623626	−	0.623626i	0.946457	+	0.322831i	$0.104634\pi$
$702$		0			0
$703$		−	27812.3i		−	1.49212i
$704$	−33947.8	−	7873.47i	−1.81741	−	0.421509i
$705$		0			0
$706$	−7043.90	−	5843.03i	−0.375497	−	0.311481i
$707$	−12292.7	−	12292.7i	−0.653912	−	0.653912i
$708$		0			0
$709$	11104.7	−	11104.7i	0.588216	−	0.588216i	−0.348932	−	0.937148i	$-0.613456\pi$
$709$	11104.7	−	11104.7i	0.588216	−	0.588216i	0.937148	+	0.348932i	$0.113456\pi$
$710$	−543.595	−	5833.52i	−0.0287335	−	0.308349i
$711$		0			0
$712$	27013.4	+	14991.6i	1.42187	+	0.789091i
$713$	8112.92			0.426131
$714$		0			0
$715$	−5238.46	−	5238.46i	−0.273996	−	0.273996i
$716$	677.609	−	991.384i	0.0353680	−	0.0517455i
$717$		0			0
$718$	10255.4	+	8507.00i	0.533046	+	0.442170i
$719$	9398.56			0.487493			0.243746	−	0.969839i	$-0.421624\pi$
$719$	9398.56			0.487493			0.243746	+	0.969839i	$0.421624\pi$
$720$		0			0
$721$	23258.4			1.20137
$722$	−1491.41	−	1237.15i	−0.0768760	−	0.0637699i
$723$		0			0
$724$	14514.7	+	9920.78i	0.745076	+	0.509258i
$725$	5718.95	+	5718.95i	0.292961	+	0.292961i
$726$		0			0
$727$	−16800.4			−0.857071			−0.428536	−	0.903525i	$-0.640970\pi$
$727$	−16800.4			−0.857071			−0.428536	+	0.903525i	$0.640970\pi$
$728$	−1234.90	−	4314.81i	−0.0628688	−	0.219667i
$729$		0			0
$730$	−622.586	−	6681.19i	−0.0315657	−	0.338742i
$731$	−9964.65	+	9964.65i	−0.504180	+	0.504180i
$732$		0			0
$733$	5536.14	+	5536.14i	0.278966	+	0.278966i	0.832696	−	0.553730i	$-0.186796\pi$
$733$	5536.14	+	5536.14i	0.278966	+	0.278966i	−0.553730	+	0.832696i	$0.686796\pi$
$734$	−3451.70	−	2863.24i	−0.173576	−	0.143984i
$735$		0			0
$736$	−11506.8	−	3823.65i	−0.576285	−	0.191497i
$737$		−	21487.7i		−	1.07396i
$738$		0			0
$739$	−775.030	+	775.030i	−0.0385791	+	0.0385791i	−0.726133	−	0.687554i	$-0.758684\pi$
$739$	−775.030	+	775.030i	−0.0385791	+	0.0385791i	0.687554	+	0.726133i	$0.258684\pi$
$740$	−4673.32	−	24857.8i	−0.232155	−	1.23485i
$741$		0			0
$742$	−5856.16	+	545.705i	−0.289739	+	0.0269993i
$743$			21471.2i			1.06016i	0.847947	+	0.530081i	$0.177838\pi$
$743$			21471.2i			1.06016i	−0.847947	+	0.530081i	$0.822162\pi$
$744$		0			0
$745$		−	14061.6i		−	0.691511i
$746$	771.519	+	8279.45i	0.0378650	+	0.406343i
$747$		0			0
$748$	−16352.6	+	23924.8i	−0.799344	+	1.16949i
$749$	16500.0	−	16500.0i	0.804937	−	0.804937i
$750$		0			0
$751$			2293.60i			0.111444i	0.998446	+	0.0557221i	$0.0177461\pi$
$751$			2293.60i			0.111444i	−0.998446	+	0.0557221i	$0.982254\pi$
$752$	−5734.66	+	2235.26i	−0.278087	+	0.108393i
$753$		0			0
$754$	−3935.97	+	4744.90i	−0.190106	+	0.229176i
$755$	−1801.47	−	1801.47i	−0.0868372	−	0.0868372i
$756$		0			0
$757$	−14151.4	+	14151.4i	−0.679448	+	0.679448i	−0.959875	−	0.280427i	$-0.909524\pi$
$757$	−14151.4	+	14151.4i	−0.679448	+	0.679448i	0.280427	+	0.959875i	$0.409524\pi$
$758$	14752.7	−	1374.73i	0.706916	−	0.0658738i
$759$		0			0
$760$	13885.8	+	7706.16i	0.662751	+	0.367805i
$761$	18852.9			0.898054			0.449027	−	0.893518i	$-0.351771\pi$
$761$	18852.9			0.898054			0.449027	+	0.893518i	$0.351771\pi$
$762$		0			0
$763$	−17404.2	−	17404.2i	−0.825785	−	0.825785i
$764$	−1939.00	−	10313.7i	−0.0918200	−	0.488399i
$765$		0			0
$766$	−13652.2	+	16458.1i	−0.643963	+	0.776311i
$767$	−8410.06			−0.395919
$768$		0			0
$769$	−12172.9			−0.570828			−0.285414	−	0.958404i	$-0.592131\pi$
$769$	−12172.9			−0.570828			−0.285414	+	0.958404i	$0.592131\pi$
$770$	−17870.2	+	21542.9i	−0.836359	+	1.00825i
$771$		0			0
$772$	2502.04	+	13308.6i	0.116646	+	0.620449i
$773$	4971.12	+	4971.12i	0.231305	+	0.231305i	0.813237	−	0.581932i	$-0.197703\pi$
$773$	4971.12	+	4971.12i	0.231305	+	0.231305i	−0.581932	+	0.813237i	$0.697703\pi$
$774$		0			0
$775$	5476.42			0.253831
$776$	−36690.5	−	20362.0i	−1.69731	−	0.941951i
$777$		0			0
$778$	−19390.0	+	1806.86i	−0.893530	+	0.0832635i
$779$	22335.9	−	22335.9i	1.02730	−	1.02730i
$780$		0			0
$781$	11161.2	+	11161.2i	0.511367	+	0.511367i
$782$	−6437.59	+	7760.65i	−0.294383	+	0.354885i
$783$		0			0
$784$	4654.25	−	1814.14i	0.212020	−	0.0826412i
$785$		−	32044.5i		−	1.45696i
$786$		0			0
$787$	−3776.32	+	3776.32i	−0.171043	+	0.171043i	−0.787438	−	0.616394i	$-0.788593\pi$
$787$	−3776.32	+	3776.32i	−0.171043	+	0.171043i	0.616394	+	0.787438i	$0.288593\pi$
$788$	−15446.4	+	22599.1i	−0.698295	+	1.02165i
$789$		0			0
$790$	1417.37	+	15210.3i	0.0638326	+	0.685011i
$791$			20841.8i			0.936849i
$792$		0			0
$793$		−	6522.06i		−	0.292062i
$794$	9729.19	−	906.612i	0.434856	−	0.0405220i
$795$		0			0
$796$	−6929.08	−	36856.4i	−0.308536	−	1.64113i
$797$	1305.71	−	1305.71i	0.0580311	−	0.0580311i	−0.677496	−	0.735527i	$-0.736935\pi$
$797$	1305.71	−	1305.71i	0.0580311	−	0.0580311i	0.735527	+	0.677496i	$0.236935\pi$
$798$		0			0
$799$			5118.23i			0.226621i
$800$	−7767.37	−	2581.06i	−0.343272	−	0.114068i
$801$		0			0
$802$	−22528.5	−	18687.8i	−0.991908	−	0.822805i
$803$	12783.0	+	12783.0i	0.561771	+	0.561771i
$804$		0			0
$805$	−6886.46	+	6886.46i	−0.301510	+	0.301510i
$806$	387.310	+	4156.36i	0.0169261	+	0.181640i
$807$		0			0
$808$	−6649.56	−	23233.9i	−0.289518	−	1.01159i
$809$	−16741.4			−0.727560			−0.363780	−	0.931485i	$-0.618514\pi$
$809$	−16741.4			−0.727560			−0.363780	+	0.931485i	$0.618514\pi$
$810$		0			0
$811$	−17759.3	−	17759.3i	−0.768946	−	0.768946i	0.208975	−	0.977921i	$-0.432987\pi$
$811$	−17759.3	−	17759.3i	−0.768946	−	0.768946i	−0.977921	+	0.208975i	$0.932987\pi$
$812$	19229.5	+	13143.4i	0.831065	+	0.568032i
$813$		0			0
$814$	52447.3	+	43505.9i	2.25832	+	1.87332i
$815$	16020.2			0.688542
$816$		0			0
$817$	−20805.5			−0.890932
$818$	−7355.26	−	6101.32i	−0.314390	−	0.260792i
$819$		0			0
$820$	16210.0	−	23716.3i	0.690340	−	1.01001i
$821$	30920.3	+	30920.3i	1.31440	+	1.31440i	0.918135	+	0.396269i	$0.129695\pi$
$821$	30920.3	+	30920.3i	1.31440	+	1.31440i	0.396269	+	0.918135i	$0.370305\pi$
$822$		0			0
$823$	19433.1			0.823080			0.411540	−	0.911392i	$-0.364991\pi$
$823$	19433.1			0.823080			0.411540	+	0.911392i	$0.364991\pi$
$824$	28270.4	+	15689.2i	1.19520	+	0.663298i
$825$		0			0
$826$	2948.16	+	31637.7i	0.124188	+	1.33271i
$827$	−22294.5	+	22294.5i	−0.937431	+	0.937431i	−0.998155	−	0.0607235i	$-0.980659\pi$
$827$	−22294.5	+	22294.5i	−0.937431	+	0.937431i	0.0607235	+	0.998155i	$0.480659\pi$
$828$		0			0
$829$	16674.7	+	16674.7i	0.698595	+	0.698595i	0.964107	−	0.265513i	$-0.0855412\pi$
$829$	16674.7	+	16674.7i	0.698595	+	0.698595i	−0.265513	+	0.964107i	$0.585541\pi$
$830$	−9660.38	−	8013.45i	−0.403996	−	0.335121i
$831$		0			0
$832$	1409.58	−	6077.63i	0.0587359	−	0.253250i
$833$		−	4153.96i		−	0.172781i
$834$		0			0
$835$	17473.0	−	17473.0i	0.724164	−	0.724164i
$836$	−42048.1	+	7905.14i	−1.73955	+	0.327040i
$837$		0			0
$838$	−35628.7	+	3320.05i	−1.46870	+	0.136861i
$839$			30412.9i			1.25145i	0.780042	+	0.625727i	$0.215198\pi$
$839$			30412.9i			1.25145i	−0.780042	+	0.625727i	$0.784802\pi$
$840$		0			0
$841$		−	7605.69i		−	0.311849i
$842$	3966.06	+	42561.2i	0.162327	+	1.74199i
$843$		0			0
$844$	18630.3	+	12733.8i	0.759813	+	0.519331i
$845$	−12938.4	+	12938.4i	−0.526741	+	0.526741i
$846$		0			0
$847$		−	53743.3i		−	2.18022i
$848$	−7486.23	−	3287.02i	−0.303158	−	0.133109i
$849$		0			0
$850$	−4345.54	+	5238.63i	−0.175354	+	0.211393i
$851$	16765.4	+	16765.4i	0.675337	+	0.675337i
$852$		0			0
$853$	15673.4	−	15673.4i	0.629128	−	0.629128i	−0.318721	−	0.947849i	$-0.603253\pi$
$853$	15673.4	−	15673.4i	0.629128	−	0.629128i	0.947849	+	0.318721i	$0.103253\pi$
$854$	−24535.3	+	2286.32i	−0.983115	+	0.0916114i
$855$		0			0
$856$	31185.9	−	8925.44i	1.24523	−	0.356385i
$857$	7458.42			0.297287			0.148643	−	0.988891i	$-0.452509\pi$
$857$	7458.42			0.297287			0.148643	+	0.988891i	$0.452509\pi$
$858$		0			0
$859$	−22482.2	−	22482.2i	−0.892996	−	0.892996i	0.101808	−	0.994804i	$-0.467537\pi$
$859$	−22482.2	−	22482.2i	−0.892996	−	0.892996i	−0.994804	+	0.101808i	$0.967537\pi$
$860$	−18595.3	+	3495.95i	−0.737319	+	0.138617i
$861$		0			0
$862$	−7705.90	+	9289.63i	−0.304483	+	0.367060i
$863$	−27444.8			−1.08254			−0.541270	−	0.840849i	$-0.682056\pi$
$863$	−27444.8			−1.08254			−0.541270	+	0.840849i	$0.682056\pi$
$864$		0			0
$865$	−4378.34			−0.172102
$866$	28326.5	−	34148.2i	1.11152	−	1.33996i
$867$		0			0
$868$	15500.0	−	2914.04i	0.606112	−	0.113950i
$869$	−29101.6	−	29101.6i	−1.13602	−	1.13602i
$870$		0			0
$871$	3846.90			0.149653
$872$	−9414.53	−	32894.8i	−0.365615	−	1.27748i
$873$		0			0
$874$	−14822.5	+	1381.23i	−0.573658	+	0.0534562i
$875$	−17499.4	+	17499.4i	−0.676101	+	0.676101i
$876$		0			0
$877$	−167.934	−	167.934i	−0.00646604	−	0.00646604i	0.703866	−	0.710332i	$-0.251455\pi$
$877$	−167.934	−	167.934i	−0.00646604	−	0.00646604i	−0.710332	+	0.703866i	$0.751455\pi$
$878$	−10769.4	+	12982.7i	−0.413951	+	0.499027i
$879$		0			0
$880$	−36253.0	+	14130.7i	−1.38874	+	0.541304i
$881$			20968.8i			0.801882i	0.916104	+	0.400941i	$0.131317\pi$
$881$			20968.8i			0.801882i	−0.916104	+	0.400941i	$0.868683\pi$
$882$		0			0
$883$	−22968.4	+	22968.4i	−0.875366	+	0.875366i	−0.993051	−	0.117685i	$-0.962453\pi$
$883$	−22968.4	+	22968.4i	−0.875366	+	0.875366i	0.117685	+	0.993051i	$0.462453\pi$
$884$	−4283.23	−	2927.58i	−0.162964	−	0.111386i
$885$		0			0
$886$	−561.301	−	6023.52i	−0.0212836	−	0.228402i
$887$		−	41314.4i		−	1.56393i	−0.623325	−	0.781963i	$-0.714218\pi$
$887$		−	41314.4i		−	1.56393i	0.623325	−	0.781963i	$-0.285782\pi$
$888$		0			0
$889$		−	45232.7i		−	1.70648i
$890$	34345.7	−	3200.50i	1.29356	−	0.120540i
$891$		0			0
$892$	28845.0	−	5422.91i	1.08274	−	0.203557i
$893$	−5343.25	+	5343.25i	−0.200230	+	0.200230i
$894$		0			0
$895$		−	1340.76i		−	0.0500743i
$896$	−23357.5	−	3172.15i	−0.870893	−	0.118275i
$897$		0			0
$898$	11322.9	+	9392.56i	0.420770	+	0.349036i
$899$	−15318.9	−	15318.9i	−0.568314	−	0.568314i
$900$		0			0
$901$	−4807.61	+	4807.61i	−0.177763	+	0.177763i
$902$	7180.77	+	77059.4i	0.265070	+	2.84456i
$903$		0			0
$904$	−14059.0	+	25333.0i	−0.517251	+	0.932039i
$905$	19629.8			0.721014
$906$		0			0
$907$	−28396.2	−	28396.2i	−1.03956	−	1.03956i	−0.999185	−	0.0403761i	$-0.987144\pi$
$907$	−28396.2	−	28396.2i	−1.03956	−	1.03956i	−0.0403761	−	0.999185i	$-0.512856\pi$
$908$	−15679.1	+	22939.5i	−0.573050	+	0.838406i
$909$		0			0
$910$	−3856.79	−	3199.27i	−0.140496	−	0.116544i
$911$	42779.4			1.55581			0.777906	−	0.628380i	$-0.216282\pi$
$911$	42779.4			1.55581			0.777906	+	0.628380i	$0.216282\pi$
$912$		0			0
$913$	33815.0			1.22575
$914$	13733.4	+	11392.1i	0.497001	+	0.412271i
$915$		0			0
$916$	−11223.1	−	7670.98i	−0.404827	−	0.276699i
$917$	−15270.2	−	15270.2i	−0.549909	−	0.549909i
$918$		0			0
$919$	39163.0			1.40573			0.702866	−	0.711323i	$-0.251904\pi$
$919$	39163.0			1.40573			0.702866	+	0.711323i	$0.251904\pi$
$920$	−13015.8	+	3725.12i	−0.466432	+	0.133493i
$921$		0			0
$922$	−445.557	−	4781.44i	−0.0159150	−	0.170790i
$923$	−1998.17	+	1998.17i	−0.0712573	+	0.0712573i
$924$		0			0
$925$	11317.1	+	11317.1i	0.402274	+	0.402274i
$926$	−33077.3	−	27438.2i	−1.17385	−	0.973730i
$927$		0			0
$928$	14507.4	+	28947.1i	0.513178	+	1.02396i
$929$		−	22120.8i		−	0.781225i	−0.920555	−	0.390613i	$-0.872263\pi$
$929$		−	22120.8i		−	0.781225i	0.920555	−	0.390613i	$-0.127737\pi$
$930$		0			0
$931$	4336.59	−	4336.59i	0.152659	−	0.152659i
$932$	−2306.68	−	12269.4i	−0.0810705	−	0.431221i
$933$		0			0
$934$	−16204.0	+	1509.97i	−0.567677	+	0.0528989i
$935$			32356.1i			1.13172i
$936$		0			0
$937$		−	13514.2i		−	0.471172i	−0.971853	−	0.235586i	$-0.924299\pi$
$937$		−	13514.2i		−	0.471172i	0.971853	−	0.235586i	$-0.0757010\pi$
$938$	−1348.54	−	14471.6i	−0.0469417	−	0.503748i
$939$		0			0
$940$	−3877.80	+	5673.46i	−0.134553	+	0.196860i
$941$	−3813.59	+	3813.59i	−0.132114	+	0.132114i	−0.770072	−	0.637957i	$-0.779780\pi$
$941$	−3813.59	+	3813.59i	−0.132114	+	0.132114i	0.637957	+	0.770072i	$0.279780\pi$
$942$		0			0
$943$			26928.5i			0.929916i
$944$	−17758.0	+	40444.1i	−0.612261	+	1.39443i
$945$		0			0
$946$	32545.3	−	39234.0i	1.11854	−	1.34842i
$947$	−15811.7	−	15811.7i	−0.542567	−	0.542567i	0.381714	−	0.924281i	$-0.375334\pi$
$947$	−15811.7	−	15811.7i	−0.542567	−	0.542567i	−0.924281	+	0.381714i	$0.875334\pi$
$948$		0			0
$949$	−2288.52	+	2288.52i	−0.0782809	+	0.0782809i
$950$	−10005.5	+	932.363i	−0.341708	+	0.0318420i
$951$		0			0
$952$	−9511.74	+	17139.3i	−0.323821	+	0.583496i
$953$	48996.7			1.66544			0.832718	−	0.553697i	$-0.186784\pi$
$953$	48996.7			1.66544			0.832718	+	0.553697i	$0.186784\pi$
$954$		0			0
$955$	−8285.33	−	8285.33i	−0.280740	−	0.280740i
$956$	5238.02	+	27861.5i	0.177207	+	0.942578i
$957$		0			0
$958$	6031.66	−	7271.29i	0.203418	−	0.245224i
$959$	−1573.01			−0.0529667
$960$		0			0
$961$	15121.7			0.507594
$962$	−7788.80	+	9389.56i	−0.261040	+	0.314690i
$963$		0			0
$964$	−2190.54	−	11651.7i	−0.0731874	−	0.389290i
$965$	10691.2	+	10691.2i	0.356645	+	0.356645i
$966$		0			0
$967$	−17758.0			−0.590548			−0.295274	−	0.955413i	$-0.595411\pi$
$967$	−17758.0			−0.590548			−0.295274	+	0.955413i	$0.595411\pi$
$968$	36253.0	−	65324.6i	1.20374	−	2.16902i
$969$		0			0
$970$	−46649.4	+	4347.02i	−1.54415	+	0.143891i
$971$	278.185	−	278.185i	0.00919401	−	0.00919401i	−0.702495	−	0.711689i	$-0.747931\pi$
$971$	278.185	−	278.185i	0.00919401	−	0.00919401i	0.711689	+	0.702495i	$0.247931\pi$
$972$		0			0
$973$	−3434.52	−	3434.52i	−0.113161	−	0.113161i
$974$	19482.3	−	23486.3i	0.640917	−	0.772639i
$975$		0			0
$976$	−31364.7	−	13771.5i	−1.02865	−	0.451654i
$977$			2036.82i			0.0666978i	0.999444	+	0.0333489i	$0.0106173\pi$
$977$			2036.82i			0.0666978i	−0.999444	+	0.0333489i	$0.989383\pi$
$978$		0			0
$979$	−65713.0	+	65713.0i	−2.14525	+	2.14525i
$980$	3147.23	−	4604.59i	0.102586	−	0.150090i
$981$		0			0
$982$	1748.67	+	18765.6i	0.0568252	+	0.609812i
$983$			4969.20i			0.161234i	0.996745	+	0.0806169i	$0.0256890\pi$
$983$			4969.20i			0.161234i	−0.996745	+	0.0806169i	$0.974311\pi$
$984$		0			0
$985$			30563.2i			0.988654i
$986$	26809.3	−	2498.22i	0.865906	−	0.0806893i
$987$		0			0
$988$	−1415.25	−	7527.82i	−0.0455718	−	0.242401i
$989$	12541.7	−	12541.7i	0.403237	−	0.403237i
$990$		0			0
$991$		−	43195.4i		−	1.38461i	−0.721606	−	0.692304i	$-0.756596\pi$
$991$		−	43195.4i		−	1.38461i	0.721606	−	0.692304i	$-0.243404\pi$
$992$	20805.9	+	6913.68i	0.665914	+	0.221280i
$993$		0			0
$994$	8217.35	+	6816.43i	0.262212	+	0.217509i
$995$	−29607.9	−	29607.9i	−0.943352	−	0.943352i
$996$		0			0
$997$	−22458.2	+	22458.2i	−0.713400	+	0.713400i	−0.967245	−	0.253845i	$-0.918305\pi$
$997$	−22458.2	+	22458.2i	−0.713400	+	0.713400i	0.253845	+	0.967245i	$0.418305\pi$
$998$	−2099.22	−	22527.5i	−0.0665827	−	0.714523i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	144.4.l.a.35.7	✓	48
3.2	odd	2	inner	144.4.l.a.35.18	yes	48
4.3	odd	2		576.4.l.a.431.7		48
8.3	odd	2		1152.4.l.a.863.18		48
8.5	even	2		1152.4.l.b.863.18		48
12.11	even	2		576.4.l.a.431.18		48
16.3	odd	4		1152.4.l.b.287.7		48
16.5	even	4		576.4.l.a.143.18		48
16.11	odd	4	inner	144.4.l.a.107.18	yes	48
16.13	even	4		1152.4.l.a.287.7		48
24.5	odd	2		1152.4.l.b.863.7		48
24.11	even	2		1152.4.l.a.863.7		48
48.5	odd	4		576.4.l.a.143.7		48
48.11	even	4	inner	144.4.l.a.107.7	yes	48
48.29	odd	4		1152.4.l.a.287.18		48
48.35	even	4		1152.4.l.b.287.18		48

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
144.4.l.a.35.7	✓	48	1.1	even	1	trivial
144.4.l.a.35.18	yes	48	3.2	odd	2	inner
144.4.l.a.107.7	yes	48	48.11	even	4	inner
144.4.l.a.107.18	yes	48	16.11	odd	4	inner
576.4.l.a.143.7		48	48.5	odd	4
576.4.l.a.143.18		48	16.5	even	4
576.4.l.a.431.7		48	4.3	odd	2
576.4.l.a.431.18		48	12.11	even	2
1152.4.l.a.287.7		48	16.13	even	4
1152.4.l.a.287.18		48	48.29	odd	4
1152.4.l.a.863.7		48	24.11	even	2
1152.4.l.a.863.18		48	8.3	odd	2
1152.4.l.b.287.7		48	16.3	odd	4
1152.4.l.b.287.18		48	48.35	even	4
1152.4.l.b.863.7		48	24.5	odd	2
1152.4.l.b.863.18		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)

\(f(q)\)	\(=\)	\(q+(-1.80581 + 2.17694i) q^{2} +(-1.47812 - 7.86226i) q^{4} +(-6.31601 - 6.31601i) q^{5} +16.2772 q^{7} +(19.7849 + 10.9799i) q^{8} +(25.1551 - 2.34407i) q^{10} +(-48.1287 + 48.1287i) q^{11} +(-8.61642 - 8.61642i) q^{13} +(-29.3935 + 35.4345i) q^{14} +(-59.6303 + 23.2428i) q^{16} +53.2206i q^{17} +(-55.5604 + 55.5604i) q^{19} +(-40.3223 + 58.9940i) q^{20} +(-17.8621 - 191.684i) q^{22} -66.9842i q^{23} -45.2160i q^{25} +(34.3170 - 3.19782i) q^{26} +(-24.0597 - 127.976i) q^{28} +(-126.481 + 126.481i) q^{29} +121.117i q^{31} +(57.0828 - 171.783i) q^{32} +(-115.858 - 96.1061i) q^{34} +(-102.807 - 102.807i) q^{35} +(-250.289 + 250.289i) q^{37} +(-20.6202 - 221.283i) q^{38} +(-55.6119 - 194.311i) q^{40} -402.012 q^{41} +(187.233 + 187.233i) q^{43} +(449.541 + 307.261i) q^{44} +(145.821 + 120.961i) q^{46} +96.1703 q^{47} -78.0518 q^{49} +(98.4325 + 81.6514i) q^{50} +(-55.0084 + 80.4807i) q^{52} +(90.3337 + 90.3337i) q^{53} +607.963 q^{55} +(322.043 + 178.723i) q^{56} +(-46.9409 - 503.740i) q^{58} +(488.025 - 488.025i) q^{59} +(378.467 + 378.467i) q^{61} +(-263.664 - 218.714i) q^{62} +(270.882 + 434.473i) q^{64} +108.843i q^{65} +(-223.231 + 223.231i) q^{67} +(418.434 - 78.6665i) q^{68} +(409.455 - 38.1550i) q^{70} -231.902i q^{71} -265.600i q^{73} +(-92.8902 - 996.839i) q^{74} +(518.955 + 354.705i) q^{76} +(-783.402 + 783.402i) q^{77} +604.662i q^{79} +(523.427 + 229.824i) q^{80} +(725.956 - 875.155i) q^{82} +(-351.298 - 351.298i) q^{83} +(336.142 - 336.142i) q^{85} +(-745.702 + 69.4881i) q^{86} +(-1480.67 + 423.769i) q^{88} +1365.36 q^{89} +(-140.251 - 140.251i) q^{91} +(-526.648 + 99.0109i) q^{92} +(-173.665 + 209.357i) q^{94} +701.839 q^{95} -1854.47 q^{97} +(140.947 - 169.914i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 48 q - 120 q^{10} - 144 q^{16} - 48 q^{19} + 72 q^{22} + 72 q^{28} - 984 q^{34} - 1272 q^{40} + 864 q^{43} - 1416 q^{46} + 2352 q^{49} - 648 q^{52} - 576 q^{55} + 1128 q^{58} + 1824 q^{61} + 3024 q^{64}+ \cdots - 11304 q^{94}+O(q^{100}) \) 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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