LMFDB - Embedded newform 87.5.d.c.86.16

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [87,5,Mod(86,87)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(87, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 1])) N = Newforms(chi, 5, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("87.86"); S:= CuspForms(chi, 5); N := Newforms(S);

Level:	$ N $	$=$	$ 87 = 3 \cdot 29 $
Weight:	$ k $	$=$	$ 5 $
Character orbit:	$[\chi]$	$=$	87.d (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

sage:traces = [32,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)

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

Self dual:	no
Analytic conductor:	$8.99318678829$
Analytic rank:	$0$
Dimension:	$32$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			86.16
Character	$\chi$	$=$	87.86
Dual form			87.5.d.c.86.15

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q-0.630141 q^{2} +(-7.48299 + 5.00048i) q^{3} -15.6029 q^{4} +33.7696i q^{5} +(4.71534 - 3.15101i) q^{6} -48.1347 q^{7} +19.9143 q^{8} +(30.9904 - 74.8371i) q^{9} -21.2796i q^{10} +157.719 q^{11} +(116.757 - 78.0221i) q^{12} -194.493 q^{13} +30.3316 q^{14} +(-168.864 - 252.698i) q^{15} +237.098 q^{16} -111.468 q^{17} +(-19.5283 + 47.1580i) q^{18} -542.924i q^{19} -526.905i q^{20} +(360.191 - 240.697i) q^{21} -99.3854 q^{22} +419.833i q^{23} +(-149.019 + 99.5811i) q^{24} -515.387 q^{25} +122.558 q^{26} +(142.321 + 714.973i) q^{27} +751.042 q^{28} +(357.228 - 761.360i) q^{29} +(106.408 + 159.235i) q^{30} +340.678i q^{31} -468.034 q^{32} +(-1180.21 + 788.672i) q^{33} +70.2405 q^{34} -1625.49i q^{35} +(-483.540 + 1167.68i) q^{36} -1476.51i q^{37} +342.118i q^{38} +(1455.39 - 972.561i) q^{39} +672.498i q^{40} +205.263 q^{41} +(-226.971 + 151.673i) q^{42} +2137.37i q^{43} -2460.88 q^{44} +(2527.22 + 1046.53i) q^{45} -264.554i q^{46} -2204.79 q^{47} +(-1774.20 + 1185.60i) q^{48} -84.0530 q^{49} +324.767 q^{50} +(834.113 - 557.393i) q^{51} +3034.66 q^{52} -1418.53i q^{53} +(-89.6824 - 450.534i) q^{54} +5326.12i q^{55} -958.568 q^{56} +(2714.88 + 4062.69i) q^{57} +(-225.104 + 479.764i) q^{58} +44.2369i q^{59} +(2634.78 + 3942.82i) q^{60} -1309.06i q^{61} -214.675i q^{62} +(-1491.71 + 3602.26i) q^{63} -3498.64 q^{64} -6567.97i q^{65} +(743.700 - 496.975i) q^{66} -3054.68 q^{67} +1739.22 q^{68} +(-2099.37 - 3141.61i) q^{69} +1024.29i q^{70} +350.225i q^{71} +(617.151 - 1490.33i) q^{72} -10245.9i q^{73} +930.411i q^{74} +(3856.64 - 2577.19i) q^{75} +8471.19i q^{76} -7591.76 q^{77} +(-917.103 + 612.850i) q^{78} -4691.27i q^{79} +8006.71i q^{80} +(-4640.20 - 4638.46i) q^{81} -129.345 q^{82} -3227.89i q^{83} +(-5620.04 + 3755.57i) q^{84} -3764.23i q^{85} -1346.84i q^{86} +(1134.04 + 7483.56i) q^{87} +3140.87 q^{88} +12721.8 q^{89} +(-1592.51 - 659.463i) q^{90} +9361.87 q^{91} -6550.63i q^{92} +(-1703.55 - 2549.29i) q^{93} +1389.33 q^{94} +18334.3 q^{95} +(3502.29 - 2340.40i) q^{96} -14053.2i q^{97} +52.9653 q^{98} +(4887.78 - 11803.3i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 32 q + 188 q^{4} - 36 q^{6} - 84 q^{7} - 452 q^{9} - 224 q^{13} - 52 q^{16} - 4216 q^{22} - 832 q^{24} - 7684 q^{25} - 396 q^{28} + 3384 q^{30} - 3308 q^{33} + 9124 q^{34} - 3680 q^{36} + 19764 q^{42} + 44 q^{45}+ \cdots + 13884 q^{96}+O(q^{100}) $ 32 * q + 188 * q^4 - 36 * q^6 - 84 * q^7 - 452 * q^9 - 224 * q^13 - 52 * q^16 - 4216 * q^22 - 832 * q^24 - 7684 * q^25 - 396 * q^28 + 3384 * q^30 - 3308 * q^33 + 9124 * q^34 - 3680 * q^36 + 19764 * q^42 + 44 * q^45 + 4796 * q^49 - 7052 * q^51 + 34048 * q^52 + 3008 * q^54 + 7976 * q^57 + 5936 * q^58 - 22848 * q^63 - 14668 * q^64 - 1872 * q^67 + 15192 * q^78 - 41088 * q^81 - 27028 * q^82 - 35572 * q^87 - 60032 * q^88 - 34272 * q^91 + 38000 * q^93 + 59636 * q^94 + 13884 * q^96

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/87\mathbb{Z}\right)^\times$.

$n$	$31$	$59$
$\chi(n)$	$-1$	$-1$

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 $

$2$	−0.630141			−0.157535			−0.0787676	−	0.996893i	$-0.525099\pi$
$2$	−0.630141			−0.157535			−0.0787676	+	0.996893i	$0.525099\pi$
$3$	−7.48299	+	5.00048i	−0.831444	+	0.555609i
$3$	−7.48299	+	5.00048i	−0.831444	+	0.555609i
$4$	−15.6029			−0.975183
$5$			33.7696i			1.35078i	0.737458	+	0.675392i	$0.236026\pi$
$5$			33.7696i			1.35078i	−0.737458	+	0.675392i	$0.763974\pi$
$6$	4.71534	−	3.15101i	0.130982	−	0.0875280i
$7$	−48.1347			−0.982340			−0.491170	−	0.871064i	$-0.663431\pi$
$7$	−48.1347			−0.982340			−0.491170	+	0.871064i	$0.663431\pi$
$8$	19.9143			0.311161
$9$	30.9904	−	74.8371i	0.382597	−	0.923915i
$10$		−	21.2796i		−	0.212796i
$11$	157.719			1.30346			0.651732	−	0.758449i	$-0.274043\pi$
$11$	157.719			1.30346			0.651732	+	0.758449i	$0.274043\pi$
$12$	116.757	−	78.0221i	0.810809	−	0.541820i
$13$	−194.493			−1.15085			−0.575424	−	0.817855i	$-0.695163\pi$
$13$	−194.493			−1.15085			−0.575424	+	0.817855i	$0.695163\pi$
$14$	30.3316			0.154753
$15$	−168.864	−	252.698i	−0.750508	−	1.12310i
$16$	237.098			0.926164
$17$	−111.468			−0.385702			−0.192851	−	0.981228i	$-0.561773\pi$
$17$	−111.468			−0.385702			−0.192851	+	0.981228i	$0.561773\pi$
$18$	−19.5283	+	47.1580i	−0.0602725	+	0.145549i
$19$		−	542.924i		−	1.50394i	−0.659195	−	0.751972i	$-0.729103\pi$
$19$		−	542.924i		−	1.50394i	0.659195	−	0.751972i	$-0.270897\pi$
$20$		−	526.905i		−	1.31726i
$21$	360.191	−	240.697i	0.816761	−	0.545797i
$22$	−99.3854			−0.205342
$23$			419.833i			0.793636i	0.917897	+	0.396818i	$0.129886\pi$
$23$			419.833i			0.793636i	−0.917897	+	0.396818i	$0.870114\pi$
$24$	−149.019	+	99.5811i	−0.258713	+	0.172884i
$25$	−515.387			−0.824620
$26$	122.558			0.181299
$27$	142.321	+	714.973i	0.195228	+	0.980758i
$28$	751.042			0.957961
$29$	357.228	−	761.360i	0.424765	−	0.905303i
$29$	357.228	−	761.360i	0.424765	−	0.905303i
$30$	106.408	+	159.235i	0.118232	+	0.176928i
$31$			340.678i			0.354504i	0.984166	+	0.177252i	$0.0567207\pi$
$31$			340.678i			0.354504i	−0.984166	+	0.177252i	$0.943279\pi$
$32$	−468.034			−0.457064
$33$	−1180.21	+	788.672i	−1.08376	+	0.724217i
$34$	70.2405			0.0607616
$35$		−	1625.49i		−	1.32693i
$36$	−483.540	+	1167.68i	−0.373102	+	0.900986i
$37$		−	1476.51i		−	1.07853i	−0.842135	−	0.539267i	$-0.818702\pi$
$37$		−	1476.51i		−	1.07853i	0.842135	−	0.539267i	$-0.181298\pi$
$38$			342.118i			0.236924i
$39$	1455.39	−	972.561i	0.956865	−	0.639422i
$40$			672.498i			0.420311i
$41$	205.263			0.122108			0.0610538	−	0.998134i	$-0.480554\pi$
$41$	205.263			0.122108			0.0610538	+	0.998134i	$0.480554\pi$
$42$	−226.971	+	151.673i	−0.128669	+	0.0859823i
$43$			2137.37i			1.15596i	0.816052	+	0.577979i	$0.196158\pi$
$43$			2137.37i			1.15596i	−0.816052	+	0.577979i	$0.803842\pi$
$44$	−2460.88			−1.27112
$45$	2527.22	+	1046.53i	1.24801	+	0.516806i
$46$		−	264.554i		−	0.125026i
$47$	−2204.79			−0.998095			−0.499047	−	0.866575i	$-0.666317\pi$
$47$	−2204.79			−0.998095			−0.499047	+	0.866575i	$0.666317\pi$
$48$	−1774.20	+	1185.60i	−0.770053	+	0.514585i
$49$	−84.0530			−0.0350075
$50$	324.767			0.129907
$51$	834.113	−	557.393i	0.320689	−	0.214299i
$52$	3034.66			1.12229
$53$		−	1418.53i		−	0.504994i	−0.967598	−	0.252497i	$-0.918748\pi$
$53$		−	1418.53i		−	0.504994i	0.967598	−	0.252497i	$-0.0812517\pi$
$54$	−89.6824	−	450.534i	−0.0307553	−	0.154504i
$55$			5326.12i			1.76070i
$56$	−958.568			−0.305666
$57$	2714.88	+	4062.69i	0.835605	+	1.25044i
$58$	−225.104	+	479.764i	−0.0669155	+	0.142617i
$59$			44.2369i			0.0127081i	0.999980	+	0.00635405i	$0.00202257\pi$
$59$			44.2369i			0.0127081i	−0.999980	+	0.00635405i	$0.997977\pi$
$60$	2634.78	+	3942.82i	0.731883	+	1.09523i
$61$		−	1309.06i		−	0.351804i	−0.984408	−	0.175902i	$-0.943716\pi$
$61$		−	1309.06i		−	0.351804i	0.984408	−	0.175902i	$-0.0562842\pi$
$62$		−	214.675i		−	0.0558468i
$63$	−1491.71	+	3602.26i	−0.375840	+	0.907599i
$64$	−3498.64			−0.854160
$65$		−	6567.97i		−	1.55455i
$66$	743.700	−	496.975i	0.170730	−	0.114090i
$67$	−3054.68			−0.680481			−0.340240	−	0.940338i	$-0.610508\pi$
$67$	−3054.68			−0.680481			−0.340240	+	0.940338i	$0.610508\pi$
$68$	1739.22			0.376130
$69$	−2099.37	−	3141.61i	−0.440951	−	0.659864i
$70$			1024.29i			0.209038i
$71$			350.225i			0.0694752i	0.999396	+	0.0347376i	$0.0110596\pi$
$71$			350.225i			0.0694752i	−0.999396	+	0.0347376i	$0.988940\pi$
$72$	617.151	−	1490.33i	0.119049	−	0.287486i
$73$		−	10245.9i		−	1.92267i	−0.275375	−	0.961337i	$-0.588802\pi$
$73$		−	10245.9i		−	1.92267i	0.275375	−	0.961337i	$-0.411198\pi$
$74$			930.411i			0.169907i
$75$	3856.64	−	2577.19i	0.685625	−	0.458166i
$76$			8471.19i			1.46662i
$77$	−7591.76			−1.28045
$78$	−917.103	+	612.850i	−0.150740	+	0.100731i
$79$		−	4691.27i		−	0.751685i	−0.926683	−	0.375843i	$-0.877353\pi$
$79$		−	4691.27i		−	0.751685i	0.926683	−	0.375843i	$-0.122647\pi$
$80$			8006.71i			1.25105i
$81$	−4640.20	−	4638.46i	−0.707239	−	0.706974i
$82$	−129.345			−0.0192363
$83$		−	3227.89i		−	0.468558i	−0.972169	−	0.234279i	$-0.924727\pi$
$83$		−	3227.89i		−	0.468558i	0.972169	−	0.234279i	$-0.0752729\pi$
$84$	−5620.04	+	3755.57i	−0.796491	+	0.532252i
$85$		−	3764.23i		−	0.521000i
$86$		−	1346.84i		−	0.182104i
$87$	1134.04	+	7483.56i	0.149826	+	0.988712i
$88$	3140.87			0.405587
$89$	12721.8			1.60608			0.803042	−	0.595923i	$-0.203214\pi$
$89$	12721.8			1.60608			0.803042	+	0.595923i	$0.203214\pi$
$90$	−1592.51	−	659.463i	−0.196606	−	0.0814152i
$91$	9361.87			1.13052
$92$		−	6550.63i		−	0.773940i
$93$	−1703.55	−	2549.29i	−0.196965	−	0.294750i
$94$	1389.33			0.157235
$95$	18334.3			2.03150
$96$	3502.29	−	2340.40i	0.380023	−	0.253949i
$97$		−	14053.2i		−	1.49359i	−0.665054	−	0.746796i	$-0.731591\pi$
$97$		−	14053.2i		−	1.49359i	0.665054	−	0.746796i	$-0.268409\pi$
$98$	52.9653			0.00551492
$99$	4887.78	−	11803.3i	0.498702	−	1.20429i
$100$	8041.55			0.804155
$101$	−19596.4			−1.92102			−0.960512	−	0.278238i	$-0.910250\pi$
$101$	−19596.4			−1.92102			−0.960512	+	0.278238i	$0.910250\pi$
$102$	−525.609	+	351.236i	−0.0505199	+	0.0337597i
$103$	−20485.3			−1.93094			−0.965470	−	0.260513i	$-0.916108\pi$
$103$	−20485.3			−1.93094			−0.965470	+	0.260513i	$0.916108\pi$
$104$	−3873.20			−0.358099
$105$	8128.23	+	12163.5i	0.737255	+	1.10327i
$106$			893.872i			0.0795543i
$107$		−	10010.8i		−	0.874385i	−0.899368	−	0.437193i	$-0.855973\pi$
$107$		−	10010.8i		−	0.874385i	0.899368	−	0.437193i	$-0.144027\pi$
$108$	−2220.63	−	11155.7i	−0.190383	−	0.956418i
$109$	7617.87			0.641181			0.320591	−	0.947218i	$-0.396119\pi$
$109$	7617.87			0.641181			0.320591	+	0.947218i	$0.396119\pi$
$110$		−	3356.21i		−	0.277372i
$111$	7383.27	+	11048.7i	0.599243	+	0.896739i
$112$	−11412.6			−0.909808
$113$	1379.19			0.108011			0.0540056	−	0.998541i	$-0.482801\pi$
$113$	1379.19			0.108011			0.0540056	+	0.998541i	$0.482801\pi$
$114$	−1710.76	−	2560.07i	−0.131637	−	0.196989i
$115$	−14177.6			−1.07203
$116$	−5573.80	+	11879.4i	−0.414224	+	0.882836i
$117$	−6027.42	+	14555.3i	−0.440311	+	1.06329i
$118$		−	27.8755i		−	0.00200197i
$119$	5365.47			0.378890
$120$	−3362.82	−	5032.30i	−0.233529	−	0.349465i
$121$	10234.4			0.699020
$122$			824.895i			0.0554216i
$123$	−1535.98	+	1026.41i	−0.101526	+	0.0678441i
$124$		−	5315.57i		−	0.345706i
$125$			3701.57i			0.236901i
$126$	939.988	−	2269.93i	0.0592081	−	0.142979i
$127$			13207.4i			0.818862i	0.912341	+	0.409431i	$0.134273\pi$
$127$			13207.4i			0.818862i	−0.912341	+	0.409431i	$0.865727\pi$
$128$	9693.18			0.591625
$129$	−10687.9	−	15993.9i	−0.642261	−	0.961114i
$130$			4138.75i			0.244896i
$131$	−10148.6			−0.591377			−0.295688	−	0.955284i	$-0.595549\pi$
$131$	−10148.6			−0.591377			−0.295688	+	0.955284i	$0.595549\pi$
$132$	18414.8	−	12305.6i	1.05686	−	0.706244i
$133$			26133.5i			1.47738i
$134$	1924.88			0.107200
$135$	−24144.4	+	4806.13i	−1.32479	+	0.263711i
$136$	−2219.80			−0.120015
$137$	29238.4			1.55780			0.778902	−	0.627146i	$-0.215777\pi$
$137$	29238.4			1.55780			0.778902	+	0.627146i	$0.215777\pi$
$138$	1322.90	+	1979.66i	0.0694654	+	0.103952i
$139$	−21851.0			−1.13095			−0.565473	−	0.824767i	$-0.691306\pi$
$139$	−21851.0			−1.13095			−0.565473	+	0.824767i	$0.691306\pi$
$140$			25362.4i			1.29400i
$141$	16498.4	−	11025.0i	0.829859	−	0.554551i
$142$		−	220.691i		−	0.0109448i
$143$	−30675.3			−1.50009
$144$	7347.75	−	17743.7i	0.354347	−	0.855697i
$145$	25710.8	+	12063.4i	1.22287	+	0.573767i
$146$			6456.38i			0.302889i
$147$	628.968	−	420.306i	0.0291068	−	0.0194505i
$148$			23037.9i			1.05177i
$149$			37875.8i			1.70604i	0.521876	+	0.853021i	$0.325232\pi$
$149$			37875.8i			1.70604i	−0.521876	+	0.853021i	$0.674768\pi$
$150$	−2430.23	+	1623.99i	−0.108010	+	0.0721774i
$151$	33435.0			1.46638			0.733191	−	0.680022i	$-0.238030\pi$
$151$	33435.0			1.46638			0.733191	+	0.680022i	$0.238030\pi$
$152$		−	10811.9i		−	0.467968i
$153$	−3454.43	+	8341.93i	−0.147568	+	0.356356i
$154$	4783.88			0.201715
$155$	−11504.6			−0.478858
$156$	−22708.4	+	15174.8i	−0.933119	+	0.623553i
$157$		−	35624.2i		−	1.44526i	−0.691235	−	0.722630i	$-0.742933\pi$
$157$		−	35624.2i		−	1.44526i	0.691235	−	0.722630i	$-0.257067\pi$
$158$			2956.16i			0.118417i
$159$	7093.32	+	10614.8i	0.280579	+	0.419874i
$160$		−	15805.3i		−	0.617396i
$161$		−	20208.5i		−	0.779621i
$162$	2923.98	+	2922.88i	0.111415	+	0.111373i
$163$			21894.7i			0.824070i	0.911168	+	0.412035i	$0.135182\pi$
$163$			21894.7i			0.824070i	−0.911168	+	0.412035i	$0.864818\pi$
$164$	−3202.70			−0.119077
$165$	−26633.2	−	39855.3i	−0.978261	−	1.46392i
$166$			2034.03i			0.0738144i
$167$			27520.9i			0.986803i	0.869802	+	0.493401i	$0.164247\pi$
$167$			27520.9i			0.986803i	−0.869802	+	0.493401i	$0.835753\pi$
$168$	7172.96	−	4793.30i	0.254144	−	0.169831i
$169$	9266.67			0.324452
$170$			2371.99i			0.0820759i
$171$	−40630.8	−	16825.4i	−1.38952	−	0.575404i
$172$		−	33349.2i		−	1.12727i
$173$			44616.5i			1.49075i	0.666648	+	0.745373i	$0.267729\pi$
$173$			44616.5i			1.49075i	−0.666648	+	0.745373i	$0.732271\pi$
$174$	−714.602	−	4715.70i	−0.0236029	−	0.155757i
$175$	24808.0			0.810057
$176$	37394.9			1.20722
$177$	−221.206	−	331.024i	−0.00706073	−	0.0105661i
$178$	−8016.52			−0.253015
$179$		−	4659.03i		−	0.145408i	−0.997354	−	0.0727042i	$-0.976837\pi$
$179$		−	4659.03i		−	0.145408i	0.997354	−	0.0727042i	$-0.0231629\pi$
$180$	−39432.0	−	16329.0i	−1.21704	−	0.503980i
$181$	−25894.6			−0.790410			−0.395205	−	0.918593i	$-0.629326\pi$
$181$	−25894.6			−0.790410			−0.395205	+	0.918593i	$0.629326\pi$
$182$	−5899.30			−0.178098
$183$	6545.95	+	9795.71i	0.195466	+	0.292505i
$184$			8360.69i			0.246949i
$185$	49861.2			1.45687
$186$	1073.48	+	1606.41i	0.0310290	+	0.0464335i
$187$	−17580.6			−0.502749
$188$	34401.2			0.973325
$189$	−6850.58	−	34415.0i	−0.191780	−	0.963438i
$190$	−11553.2			−0.320034
$191$	−30806.0			−0.844440			−0.422220	−	0.906493i	$-0.638749\pi$
$191$	−30806.0			−0.844440			−0.422220	+	0.906493i	$0.638749\pi$
$192$	26180.3	−	17494.9i	0.710186	−	0.474579i
$193$			12256.4i			0.329040i	0.986374	+	0.164520i	$0.0526075\pi$
$193$			12256.4i			0.329040i	−0.986374	+	0.164520i	$0.947392\pi$
$194$			8855.50i			0.235293i
$195$	32843.0	+	49148.1i	0.863721	+	1.29252i
$196$	1311.47			0.0341387
$197$		−	58435.3i		−	1.50571i	−0.658184	−	0.752857i	$-0.728675\pi$
$197$		−	58435.3i		−	1.50571i	0.658184	−	0.752857i	$-0.271325\pi$
$198$	−3079.99	+	7437.72i	−0.0785631	+	0.189718i
$199$	−2598.08			−0.0656064			−0.0328032	−	0.999462i	$-0.510443\pi$
$199$	−2598.08			−0.0656064			−0.0328032	+	0.999462i	$0.510443\pi$
$200$	−10263.6			−0.256590
$201$	22858.1	−	15274.9i	0.565782	−	0.378081i
$202$	12348.5			0.302629
$203$	−17195.0	+	36647.8i	−0.417264	+	0.889316i
$204$	−13014.6	+	8696.96i	−0.312731	+	0.208981i
$205$			6931.65i			0.164941i
$206$	12908.7			0.304191
$207$	31419.1	+	13010.8i	0.733253	+	0.303643i
$208$	−46114.0			−1.06587
$209$		−	85629.5i		−	1.96034i
$210$	−5121.93	−	7664.74i	−0.116144	−	0.173804i
$211$			23410.2i			0.525824i	0.964820	+	0.262912i	$0.0846829\pi$
$211$			23410.2i			0.525824i	−0.964820	+	0.262912i	$0.915317\pi$
$212$			22133.2i			0.492461i
$213$	−1751.29	−	2620.73i	−0.0386011	−	0.0577647i
$214$			6308.24i			0.137746i
$215$	−72178.0			−1.56145
$216$	2834.23	+	14238.2i	0.0607473	+	0.305174i
$217$		−	16398.4i		−	0.348243i
$218$	−4800.34			−0.101009
$219$	51234.6	+	76670.2i	1.06826	+	1.59859i
$220$		−	83103.0i		−	1.71700i
$221$	21679.8			0.443884
$222$	−4652.50	−	6962.26i	−0.0944019	−	0.141268i
$223$	−18380.7			−0.369618			−0.184809	−	0.982775i	$-0.559167\pi$
$223$	−18380.7			−0.369618			−0.184809	+	0.982775i	$0.559167\pi$
$224$	22528.7			0.448993
$225$	−15972.0	+	38570.1i	−0.315497	+	0.761879i
$226$	−869.087			−0.0170156
$227$		−	51625.7i		−	1.00188i	−0.865483	−	0.500938i	$-0.832988\pi$
$227$		−	51625.7i		−	1.00188i	0.865483	−	0.500938i	$-0.167012\pi$
$228$	−42360.1	−	63389.9i	−0.814867	−	1.21941i
$229$		−	83982.0i		−	1.60146i	−0.599027	−	0.800729i	$-0.704446\pi$
$229$		−	83982.0i		−	1.60146i	0.599027	−	0.800729i	$-0.295554\pi$
$230$	8933.90			0.168883
$231$	56809.1	−	37962.5i	1.06462	−	0.711427i
$232$	7113.94	−	15162.0i	0.132170	−	0.281695i
$233$			22097.2i			0.407029i	0.979072	+	0.203515i	$0.0652365\pi$
$233$			22097.2i			0.407029i	−0.979072	+	0.203515i	$0.934764\pi$
$234$	3798.12	−	9171.91i	0.0693645	−	0.167505i
$235$		−	74455.0i		−	1.34821i
$236$		−	690.225i		−	0.0123927i
$237$	23458.6	+	35104.7i	0.417643	+	0.624984i
$238$	−3381.00			−0.0596886
$239$		−	96321.0i		−	1.68626i	−0.537708	−	0.843131i	$-0.680710\pi$
$239$		−	96321.0i		−	1.68626i	0.537708	−	0.843131i	$-0.319290\pi$
$240$	−40037.4	−	59914.1i	−0.695094	−	1.04018i
$241$	−43690.0			−0.752226			−0.376113	−	0.926574i	$-0.622740\pi$
$241$	−43690.0			−0.752226			−0.376113	+	0.926574i	$0.622740\pi$
$242$	−6449.09			−0.110120
$243$	57917.1	+	11506.3i	0.980831	+	0.194861i
$244$			20425.2i			0.343073i
$245$		−	2838.44i		−	0.0472876i
$246$	967.885	−	646.786i	0.0159939	−	0.0106878i
$247$			105595.i			1.73081i
$248$			6784.36i			0.110308i
$249$	16141.0	+	24154.3i	0.260335	+	0.389579i
$250$		−	2332.51i		−	0.0373202i
$251$	73276.9			1.16311			0.581553	−	0.813508i	$-0.302445\pi$
$251$	73276.9			1.16311			0.581553	+	0.813508i	$0.302445\pi$
$252$	23275.0	−	56205.8i	0.366513	−	0.885075i
$253$			66215.8i			1.03448i
$254$		−	8322.55i		−	0.129000i
$255$	18822.9	+	28167.7i	0.289472	+	0.433182i
$256$	49870.2			0.760958
$257$			28955.2i			0.438390i	0.975681	+	0.219195i	$0.0703430\pi$
$257$			28955.2i			0.438390i	−0.975681	+	0.219195i	$0.929657\pi$
$258$	6734.86	+	10078.4i	0.101179	+	0.151409i
$259$			71071.4i			1.05949i
$260$			102479.i			1.51597i
$261$	−45907.4	−	50328.7i	−0.673910	−	0.738814i
$262$	6395.06			0.0931627
$263$	−10055.4			−0.145374			−0.0726868	−	0.997355i	$-0.523157\pi$
$263$	−10055.4			−0.145374			−0.0726868	+	0.997355i	$0.523157\pi$
$264$	−23503.1	+	15705.9i	−0.337223	+	0.225348i
$265$	47903.1			0.682138
$266$		−	16467.8i		−	0.232740i
$267$	−95197.0	+	63615.1i	−1.33537	+	0.892355i
$268$	47661.9			0.663593
$269$	−73382.9			−1.01412			−0.507061	−	0.861910i	$-0.669268\pi$
$269$	−73382.9			−1.01412			−0.507061	+	0.861910i	$0.669268\pi$
$270$	15214.3	−	3028.54i	0.208702	−	0.0415438i
$271$			39697.2i			0.540532i	0.962786	+	0.270266i	$0.0871116\pi$
$271$			39697.2i			0.540532i	−0.962786	+	0.270266i	$0.912888\pi$
$272$	−26428.8			−0.357223
$273$	−70054.8	+	46813.9i	−0.939968	+	0.628130i
$274$	−18424.3			−0.245409
$275$	−81286.5			−1.07486
$276$	32756.3	+	49018.3i	0.430008	+	0.643488i
$277$	−7560.73			−0.0985380			−0.0492690	−	0.998786i	$-0.515689\pi$
$277$	−7560.73			−0.0985380			−0.0492690	+	0.998786i	$0.515689\pi$
$278$	13769.2			0.178164
$279$	25495.4	+	10557.7i	0.327531	+	0.135632i
$280$		−	32370.5i		−	0.412889i
$281$		−	19737.3i		−	0.249963i	−0.992159	−	0.124982i	$-0.960113\pi$
$281$		−	19737.3i		−	0.249963i	0.992159	−	0.124982i	$-0.0398872\pi$
$282$	−10396.3	+	6947.32i	−0.130732	+	0.0873613i
$283$	76861.1			0.959696			0.479848	−	0.877352i	$-0.340692\pi$
$283$	76861.1			0.959696			0.479848	+	0.877352i	$0.340692\pi$
$284$		−	5464.53i		−	0.0677510i
$285$	−137196.	+	91680.5i	−1.68908	+	1.12872i
$286$	19329.8			0.236317
$287$	−9880.27			−0.119951
$288$	−14504.5	+	35026.3i	−0.174871	+	0.422289i
$289$	−71095.9			−0.851234
$290$	−16201.5	−	7601.67i	−0.192645	−	0.0903885i
$291$	70272.8	+	105160.i	0.829853	+	1.24184i
$292$			159866.i			1.87496i
$293$	−102689.			−1.19616			−0.598079	−	0.801437i	$-0.704069\pi$
$293$	−102689.			−1.19616			−0.598079	+	0.801437i	$0.704069\pi$
$294$	−396.339	+	264.852i	−0.00458534	+	0.00306414i
$295$	−1493.86			−0.0171659
$296$		−	29403.7i		−	0.335597i
$297$	22446.8	+	112765.i	0.254473	+	1.27838i
$298$		−	23867.1i		−	0.268762i
$299$		−	81654.8i		−	0.913355i
$300$	−60174.9	+	40211.6i	−0.668610	+	0.446796i
$301$		−	102881.i		−	1.13554i
$302$	−21068.8			−0.231007
$303$	146639.	−	97991.3i	1.59722	−	1.06734i
$304$		−	128726.i		−	1.39290i
$305$	44206.6			0.475212
$306$	2176.78	−	5256.59i	0.0232472	−	0.0561386i
$307$			108734.i			1.15369i	0.816853	+	0.576846i	$0.195717\pi$
$307$			108734.i			1.15369i	−0.816853	+	0.576846i	$0.804283\pi$
$308$	118454.			1.24867
$309$	153292.	−	102437.i	1.60547	−	1.07285i
$310$	7249.50			0.0754370
$311$	−50640.2			−0.523570			−0.261785	−	0.965126i	$-0.584311\pi$
$311$	−50640.2			−0.523570			−0.261785	+	0.965126i	$0.584311\pi$
$312$	28983.1	−	19367.9i	0.297739	−	0.198963i
$313$	−67839.5			−0.692459			−0.346230	−	0.938150i	$-0.612538\pi$
$313$	−67839.5			−0.692459			−0.346230	+	0.938150i	$0.612538\pi$
$314$			22448.3i			0.227679i
$315$	−121647.	−	50374.5i	−1.22597	−	0.507680i
$316$			73197.5i			0.733031i
$317$	−102874.			−1.02373			−0.511866	−	0.859065i	$-0.671046\pi$
$317$	−102874.			−1.02373			−0.511866	+	0.859065i	$0.671046\pi$
$318$	−4469.79	−	6688.84i	−0.0442011	−	0.0661449i
$319$	56341.7	−	120081.i	0.553667	−	1.18003i
$320$		−	118148.i		−	1.15379i
$321$	50059.0	+	74911.0i	0.485816	+	0.727002i
$322$			12734.2i			0.122818i
$323$			60518.5i			0.580074i
$324$	72400.6	+	72373.5i	0.689687	+	0.689429i
$325$	100239.			0.949012
$326$		−	13796.8i		−	0.129820i
$327$	−57004.5	+	38093.0i	−0.533106	+	0.356246i
$328$	4087.67			0.0379951
$329$	106127.			0.980469
$330$	16782.7	+	25114.5i	0.154111	+	0.230620i
$331$			68418.8i			0.624482i	0.950003	+	0.312241i	$0.101080\pi$
$331$			68418.8i			0.624482i	−0.950003	+	0.312241i	$0.898920\pi$
$332$			50364.6i			0.456929i
$333$	−110498.	−	45757.6i	−0.996473	−	0.412644i
$334$		−	17342.1i		−	0.155456i
$335$		−	103155.i		−	0.919183i
$336$	85400.6	−	57068.7i	0.756454	−	0.505498i
$337$		−	25491.6i		−	0.224459i	−0.993682	−	0.112230i	$-0.964201\pi$
$337$		−	25491.6i		−	0.224459i	0.993682	−	0.112230i	$-0.0357992\pi$
$338$	−5839.31			−0.0511126
$339$	−10320.5	+	6896.63i	−0.0898051	+	0.0600120i
$340$			58732.9i			0.508070i
$341$			53731.5i			0.462083i
$342$	25603.2	+	10602.4i	0.218898	+	0.0906465i
$343$	119617.			1.01673
$344$			42564.1i			0.359689i
$345$	106091.	−	70894.9i	0.891334	−	0.595631i
$346$		−	28114.7i		−	0.234845i
$347$		−	65699.4i		−	0.545635i	−0.962066	−	0.272818i	$-0.912044\pi$
$347$		−	65699.4i		−	0.545635i	0.962066	−	0.272818i	$-0.0879556\pi$
$348$	−17694.3	−	116765.i	−0.146108	−	0.964175i
$349$	−187726.			−1.54125			−0.770624	−	0.637290i	$-0.780055\pi$
$349$	−187726.			−1.54125			−0.770624	+	0.637290i	$0.780055\pi$
$350$	−15632.5			−0.127613
$351$	−27680.5	−	139057.i	−0.224678	−	1.12870i
$352$	−73818.0			−0.595767
$353$		−	240553.i		−	1.93046i	−0.261402	−	0.965230i	$-0.584185\pi$
$353$		−	240553.i		−	1.93046i	0.261402	−	0.965230i	$-0.415815\pi$
$354$	139.391	+	208.592i	0.00111231	+	0.00166453i
$355$	−11827.0			−0.0938461
$356$	−198497.			−1.56622
$357$	−40149.8	+	26829.9i	−0.315026	+	0.210515i
$358$			2935.85i			0.0229070i
$359$	138776.			1.07678			0.538388	−	0.842697i	$-0.319034\pi$
$359$	138776.			1.07678			0.538388	+	0.842697i	$0.319034\pi$
$360$	50327.9	+	20841.0i	0.388332	+	0.160810i
$361$	−164445.			−1.26185
$362$	16317.3			0.124517
$363$	−76583.6	+	51176.7i	−0.581196	+	0.388382i
$364$	−146073.			−1.10247
$365$	346001.			2.59712
$366$	−4124.87	−	6172.68i	−0.0307927	−	0.0460799i
$367$		−	13869.3i		−	0.102972i	−0.998674	−	0.0514862i	$-0.983604\pi$
$367$		−	13869.3i		−	0.102972i	0.998674	−	0.0514862i	$-0.0163958\pi$
$368$			99541.7i			0.735037i
$369$	6361.17	−	15361.3i	0.0467180	−	0.112817i
$370$	−31419.6			−0.229508
$371$			68280.3i			0.496076i
$372$	26580.4	+	39776.4i	0.192077	+	0.287435i
$373$	174276.			1.25262			0.626310	−	0.779574i	$-0.284564\pi$
$373$	174276.			1.25262			0.626310	+	0.779574i	$0.284564\pi$
$374$	11078.3			0.0792006
$375$	−18509.7	−	27698.9i	−0.131624	−	0.196970i
$376$	−43906.9			−0.310568
$377$	−69478.4	+	148080.i	−0.488841	+	1.04187i
$378$	4316.83	+	21686.3i	0.0302122	+	0.151775i
$379$		−	97262.7i		−	0.677124i	−0.940944	−	0.338562i	$-0.890060\pi$
$379$		−	97262.7i		−	0.677124i	0.940944	−	0.338562i	$-0.109940\pi$
$380$	−286069.			−1.98109
$381$	−66043.5	−	98831.1i	−0.454967	−	0.680838i
$382$	19412.1			0.133029
$383$			198169.i			1.35095i	0.737385	+	0.675473i	$0.236060\pi$
$383$			198169.i			1.35095i	−0.737385	+	0.675473i	$0.763940\pi$
$384$	−72534.0	+	48470.6i	−0.491903	+	0.328712i
$385$		−	256371.i		−	1.72961i
$386$		−	7723.27i		−	0.0518354i
$387$	159954.	+	66237.7i	1.06801	+	0.442266i
$388$			219271.i			1.45652i
$389$	−46348.1			−0.306290			−0.153145	−	0.988204i	$-0.548940\pi$
$389$	−46348.1			−0.306290			−0.153145	+	0.988204i	$0.548940\pi$
$390$	−20695.7	−	30970.2i	−0.136067	−	0.203617i
$391$		−	46797.9i		−	0.306107i
$392$	−1673.86			−0.0108930
$393$	75942.0	−	50748.0i	0.491697	−	0.328574i
$394$			36822.5i			0.237203i
$395$	158422.			1.01537
$396$	−76263.6	+	184165.i	−0.486325	+	1.17440i
$397$	−25656.5			−0.162786			−0.0813928	−	0.996682i	$-0.525937\pi$
$397$	−25656.5			−0.162786			−0.0813928	+	0.996682i	$0.525937\pi$
$398$	1637.16			0.0103353
$399$	−130680.	−	195556.i	−0.820848	−	1.22836i
$400$	−122197.			−0.763733
$401$		−	215361.i		−	1.33930i	−0.742677	−	0.669650i	$-0.766444\pi$
$401$		−	215361.i		−	1.33930i	0.742677	−	0.669650i	$-0.233556\pi$
$402$	−14403.9	+	9625.32i	−0.0891305	+	0.0595612i
$403$		−	66259.6i		−	0.407980i
$404$	305761.			1.87335
$405$	156639.	−	156698.i	0.954970	−	0.955328i
$406$	10835.3	−	23093.3i	0.0657338	−	0.140099i
$407$		−	232874.i		−	1.40583i
$408$	16610.8	−	11100.1i	0.0997860	−	0.0666816i
$409$		−	281069.i		−	1.68022i	−0.542416	−	0.840110i	$-0.682490\pi$
$409$		−	281069.i		−	1.68022i	0.542416	−	0.840110i	$-0.317510\pi$
$410$		−	4367.92i		−	0.0259841i
$411$	−218791.	+	146206.i	−1.29523	+	0.865530i
$412$	319631.			1.88302
$413$		−	2129.33i		−	0.0124837i
$414$	−19798.5	−	8198.63i	−0.115513	−	0.0478344i
$415$	109005.			0.632921
$416$	91029.5			0.526012
$417$	163511.	−	109266.i	0.940318	−	0.628364i
$418$			53958.7i			0.308822i
$419$			260243.i			1.48235i	0.671312	+	0.741175i	$0.265731\pi$
$419$			260243.i			1.48235i	−0.671312	+	0.741175i	$0.734269\pi$
$420$	−126824.	−	189787.i	−0.718958	−	1.07589i
$421$		−	92253.6i		−	0.520498i	−0.965542	−	0.260249i	$-0.916195\pi$
$421$		−	92253.6i		−	0.520498i	0.965542	−	0.260249i	$-0.0838047\pi$
$422$		−	14751.7i		−	0.0828359i
$423$	−68327.3	+	165000.i	−0.381868	+	0.922155i
$424$		−	28249.0i		−	0.157134i
$425$	57449.1			0.318057
$426$	1103.56	+	1651.43i	0.00608103	+	0.00909998i
$427$			63011.3i			0.345591i
$428$			156198.i			0.852685i
$429$	229543.	−	153392.i	1.24724	−	0.833464i
$430$	45482.3			0.245983
$431$			216475.i			1.16534i	0.812709	+	0.582669i	$0.197992\pi$
$431$			216475.i			1.16534i	−0.812709	+	0.582669i	$0.802008\pi$
$432$	33744.1	+	169519.i	0.180813	+	0.908342i
$433$			16748.2i			0.0893289i	0.999002	+	0.0446644i	$0.0142219\pi$
$433$			16748.2i			0.0893289i	−0.999002	+	0.0446644i	$0.985778\pi$
$434$			10333.3i			0.0548606i
$435$	−252717.	+	38295.9i	−1.33554	+	0.202383i
$436$	−118861.			−0.625269
$437$	227937.			1.19358
$438$	−32285.0	−	48313.0i	−0.168288	−	0.251835i
$439$	51198.0			0.265658			0.132829	−	0.991139i	$-0.457594\pi$
$439$	51198.0			0.265658			0.132829	+	0.991139i	$0.457594\pi$
$440$			106066.i			0.547861i
$441$	−2604.83	+	6290.29i	−0.0133938	+	0.0323440i
$442$	−13661.3			−0.0699274
$443$	−307994.			−1.56940			−0.784702	−	0.619873i	$-0.787184\pi$
$443$	−307994.			−1.56940			−0.784702	+	0.619873i	$0.787184\pi$
$444$	−115201.	−	172392.i	−0.584371	−	0.874485i
$445$			429610.i			2.16947i
$446$	11582.4			0.0582278
$447$	−189397.	−	283425.i	−0.947893	−	1.41848i
$448$	168406.			0.839076
$449$	−281767.			−1.39765			−0.698824	−	0.715294i	$-0.746293\pi$
$449$	−281767.			−1.39765			−0.698824	+	0.715294i	$0.746293\pi$
$450$	10064.6	−	24304.6i	0.0497019	−	0.120023i
$451$	32373.9			0.159163
$452$	−21519.5			−0.105331
$453$	−250194.	+	167191.i	−1.21921	+	0.814736i
$454$			32531.5i			0.157831i
$455$			316147.i			1.52710i
$456$	54064.9	+	80905.7i	0.260008	+	0.389089i
$457$	−95866.1			−0.459021			−0.229510	−	0.973306i	$-0.573713\pi$
$457$	−95866.1			−0.459021			−0.229510	+	0.973306i	$0.573713\pi$
$458$			52920.5i			0.252286i
$459$	−15864.2	−	79696.4i	−0.0752998	−	0.378280i
$460$	221212.			1.04543
$461$	−75535.7			−0.355427			−0.177713	−	0.984082i	$-0.556870\pi$
$461$	−75535.7			−0.355427			−0.177713	+	0.984082i	$0.556870\pi$
$462$	−35797.8	+	23921.7i	−0.167715	+	0.112075i
$463$	−219935.			−1.02596			−0.512982	−	0.858399i	$-0.671459\pi$
$463$	−219935.			−1.02596			−0.512982	+	0.858399i	$0.671459\pi$
$464$	84698.0	−	180517.i	0.393402	−	0.838459i
$465$	86088.6	−	57528.4i	0.398143	−	0.266058i
$466$		−	13924.4i		−	0.0641215i
$467$	−265329.			−1.21661			−0.608305	−	0.793703i	$-0.708150\pi$
$467$	−265329.			−1.21661			−0.608305	+	0.793703i	$0.708150\pi$
$468$	94045.3	−	227106.i	0.429384	−	1.03690i
$469$	147036.			0.668464
$470$			46917.1i			0.212391i
$471$	178138.	+	266576.i	0.802999	+	1.20165i
$472$			880.947i			0.00395426i
$473$			337104.i			1.50675i
$474$	−14782.2	−	22120.9i	−0.0657935	−	0.0984570i
$475$			279816.i			1.24018i
$476$	−83717.0			−0.369487
$477$	−106159.	−	43960.7i	−0.466571	−	0.193209i
$478$			60695.8i			0.265646i
$479$	83970.4			0.365978			0.182989	−	0.983115i	$-0.441423\pi$
$479$	83970.4			0.365978			0.182989	+	0.983115i	$0.441423\pi$
$480$	79034.3	+	118271.i	0.343031	+	0.513330i
$481$			287172.i			1.24123i
$482$	27530.9			0.118502
$483$	101052.	+	151220.i	0.433164	+	0.648211i
$484$	−159686.			−0.681672
$485$	474571.			2.01752
$486$	−36495.9	−	7250.62i	−0.154515	−	0.0306975i
$487$	−106418.			−0.448703			−0.224352	−	0.974508i	$-0.572026\pi$
$487$	−106418.			−0.448703			−0.224352	+	0.974508i	$0.572026\pi$
$488$		−	26069.1i		−	0.109468i
$489$	−109484.	−	163838.i	−0.457861	−	0.685168i
$490$			1788.62i			0.00744947i
$491$	−160380.			−0.665252			−0.332626	−	0.943059i	$-0.607935\pi$
$491$	−160380.			−0.665252			−0.332626	+	0.943059i	$0.607935\pi$
$492$	23965.8	−	16015.1i	0.0990061	−	0.0661604i
$493$	−39819.4	+	84867.2i	−0.163833	+	0.349177i
$494$		−	66539.8i		−	0.272664i
$495$	398592.	+	165058.i	1.62674	+	0.673639i
$496$			80774.0i			0.328328i
$497$		−	16857.9i		−	0.0682483i
$498$	−10171.1	−	15220.6i	−0.0410119	−	0.0613725i
$499$	−182487.			−0.732875			−0.366438	−	0.930443i	$-0.619423\pi$
$499$	−182487.			−0.732875			−0.366438	+	0.930443i	$0.619423\pi$
$500$		−	57755.4i		−	0.231021i
$501$	−137618.	−	205939.i	−0.548277	−	0.820471i
$502$	−46174.8			−0.183230
$503$	280094.			1.10705			0.553525	−	0.832832i	$-0.313282\pi$
$503$	280094.			1.10705			0.553525	+	0.832832i	$0.313282\pi$
$504$	−29706.4	+	71736.5i	−0.116947	+	0.282409i
$505$		−	661762.i		−	2.59489i
$506$		−	41725.3i		−	0.162967i
$507$	−69342.4	+	46337.8i	−0.269763	+	0.180268i
$508$		−	206075.i		−	0.798540i
$509$			372754.i			1.43875i	0.694620	+	0.719376i	$0.255572\pi$
$509$			372754.i			1.43875i	−0.694620	+	0.719376i	$0.744428\pi$
$510$	−11861.1	−	17749.6i	−0.0456021	−	0.0682415i
$511$			493184.i			1.88872i
$512$	−186516.			−0.711502
$513$	388175.	−	77269.5i	1.47500	−	0.293612i
$514$		−	18245.9i		−	0.0690618i
$515$		−	691782.i		−	2.60829i
$516$	166762.	+	249551.i	0.626321	+	0.937261i
$517$	−347738.			−1.30098
$518$		−	44785.0i		−	0.166906i
$519$	−223104.	−	333865.i	−0.828272	−	1.23947i
$520$		−	130796.i		−	0.483715i
$521$			59484.7i			0.219144i	0.993979	+	0.109572i	$0.0349480\pi$
$521$			59484.7i			0.219144i	−0.993979	+	0.109572i	$0.965052\pi$
$522$	28928.1	+	31714.2i	0.106165	+	0.116389i
$523$	122539.			0.447994			0.223997	−	0.974590i	$-0.428089\pi$
$523$	122539.			0.447994			0.223997	+	0.974590i	$0.428089\pi$
$524$	158348.			0.576700
$525$	−185638.	+	124052.i	−0.673517	+	0.450075i
$526$	6336.29			0.0229015
$527$		−	37974.6i		−	0.136733i
$528$	−279826.	+	186993.i	−1.00374	+	0.670743i
$529$	103581.			0.370142
$530$	−30185.7			−0.107461
$531$	3310.56	+	1370.92i	0.0117412	+	0.00486208i
$532$		−	407758.i		−	1.44072i
$533$	−39922.3			−0.140527
$534$	59987.6	−	40086.5i	0.210368	−	0.140577i
$535$	338062.			1.18111
$536$	−60831.8			−0.211739
$537$	23297.4	+	34863.5i	0.0807902	+	0.120899i
$538$	46241.6			0.159760
$539$	−13256.8			−0.0456310
$540$	376722.	−	74989.7i	1.29192	−	0.257166i
$541$			233288.i			0.797074i	0.917152	+	0.398537i	$0.130482\pi$
$541$			233288.i			0.797074i	−0.917152	+	0.398537i	$0.869518\pi$
$542$		−	25014.8i		−	0.0851528i
$543$	193769.	−	129486.i	0.657181	−	0.439159i
$544$	52170.7			0.176291
$545$			257253.i			0.866098i
$546$	44144.4	−	29499.4i	0.148078	−	0.0989526i
$547$	−164676.			−0.550372			−0.275186	−	0.961391i	$-0.588739\pi$
$547$	−164676.			−0.550372			−0.275186	+	0.961391i	$0.588739\pi$
$548$	−456205.			−1.51914
$549$	−97966.6	−	40568.3i	−0.325037	−	0.134599i
$550$	51222.0			0.169329
$551$	−413360.	−	193947.i	−1.36153	−	0.638823i
$552$	−41807.5	−	62563.0i	−0.137207	−	0.205324i
$553$			225813.i			0.738411i
$554$	4764.32			0.0155232
$555$	−373111.	+	249330.i	−1.21130	+	0.809448i
$556$	340940.			1.10288
$557$		−	148476.i		−	0.478570i	−0.970949	−	0.239285i	$-0.923087\pi$
$557$		−	148476.i		−	0.478570i	0.970949	−	0.239285i	$-0.0769130\pi$
$558$	−16065.7	−	6652.86i	−0.0515977	−	0.0213668i
$559$		−	415703.i		−	1.33033i
$560$		−	385400.i		−	1.22896i
$561$	131556.	−	87911.6i	0.418007	−	0.279332i
$562$			12437.3i			0.0393780i
$563$	9184.31			0.0289754			0.0144877	−	0.999895i	$-0.495388\pi$
$563$	9184.31			0.0289754			0.0144877	+	0.999895i	$0.495388\pi$
$564$	−257424.	+	172023.i	−0.809265	+	0.540788i
$565$			46574.9i			0.145900i
$566$	−48433.3			−0.151186
$567$	223354.	+	223271.i	0.694749	+	0.694490i
$568$			6974.48i			0.0216180i
$569$	448012.			1.38377			0.691887	−	0.722006i	$-0.256780\pi$
$569$	448012.			1.38377			0.691887	+	0.722006i	$0.256780\pi$
$570$	86452.6	−	57771.6i	0.266090	−	0.177814i
$571$	107032.			0.328279			0.164139	−	0.986437i	$-0.447515\pi$
$571$	107032.			0.328279			0.164139	+	0.986437i	$0.447515\pi$
$572$	478625.			1.46286
$573$	230521.	−	154045.i	0.702104	−	0.469178i
$574$	6225.96			0.0188966
$575$		−	216377.i		−	0.654448i
$576$	−108424.	+	261828.i	−0.326799	+	0.789172i
$577$			5581.80i			0.0167657i	0.999965	+	0.00838287i	$0.00266838\pi$
$577$			5581.80i			0.0167657i	−0.999965	+	0.00838287i	$0.997332\pi$
$578$	44800.5			0.134099
$579$	−61288.0	−	91714.6i	−0.182818	−	0.273578i
$580$	−401164.	−	188225.i	−1.19252	−	0.559527i
$581$			155374.i			0.460283i
$582$	−44281.8	−	66265.6i	−0.130731	−	0.195633i
$583$		−	223729.i		−	0.658241i
$584$		−	204040.i		−	0.598261i
$585$	−491528.	−	203544.i	−1.43627	−	0.594766i
$586$	64708.5			0.188437
$587$			333889.i			0.969006i	0.874790	+	0.484503i	$0.160999\pi$
$587$			333889.i			0.969006i	−0.874790	+	0.484503i	$0.839001\pi$
$588$	−9813.74	+	6558.00i	−0.0283844	+	0.0189678i
$589$	184962.			0.533153
$590$	941.344			0.00270424
$591$	292205.	+	437271.i	0.836589	+	1.25192i
$592$		−	350078.i		−	0.998898i
$593$		−	240831.i		−	0.684863i	−0.939543	−	0.342432i	$-0.888749\pi$
$593$		−	240831.i		−	0.684863i	0.939543	−	0.342432i	$-0.111251\pi$
$594$	−14144.6	−	71057.8i	−0.0400884	−	0.201390i
$595$			181190.i			0.511799i
$596$		−	590974.i		−	1.66370i
$597$	19441.4	−	12991.6i	0.0545480	−	0.0364515i
$598$			51454.1i			0.143886i
$599$	138765.			0.386746			0.193373	−	0.981125i	$-0.438057\pi$
$599$	138765.			0.386746			0.193373	+	0.981125i	$0.438057\pi$
$600$	76802.3	−	51322.9i	0.213340	−	0.142563i
$601$			117210.i			0.324499i	0.986750	+	0.162250i	$0.0518750\pi$
$601$			117210.i			0.324499i	−0.986750	+	0.162250i	$0.948125\pi$
$602$			64829.8i			0.178888i
$603$	−94665.6	+	228603.i	−0.260350	+	0.628707i
$604$	−521684.			−1.42999
$605$			345610.i			0.944226i
$606$	−92403.6	+	61748.3i	−0.251619	+	0.168143i
$607$		−	202092.i		−	0.548494i	−0.961659	−	0.274247i	$-0.911571\pi$
$607$		−	202092.i		−	0.548494i	0.961659	−	0.274247i	$-0.0884286\pi$
$608$			254107.i			0.687399i
$609$	−54586.4	−	360219.i	−0.147180	−	0.971252i
$610$	−27856.4			−0.0748626
$611$	428817.			1.14866
$612$	53899.2	−	130159.i	0.143906	−	0.347512i
$613$	483368.			1.28634			0.643171	−	0.765722i	$-0.277618\pi$
$613$	483368.			1.28634			0.643171	+	0.765722i	$0.277618\pi$
$614$		−	68518.0i		−	0.181747i
$615$	−34661.6	−	51869.5i	−0.0916428	−	0.137139i
$616$	−151185.			−0.398425
$617$	152864.			0.401547			0.200773	−	0.979638i	$-0.435654\pi$
$617$	152864.			0.401547			0.200773	+	0.979638i	$0.435654\pi$
$618$	−96595.4	+	64549.5i	−0.252918	+	0.169011i
$619$		−	406789.i		−	1.06166i	−0.847477	−	0.530832i	$-0.821879\pi$
$619$		−	406789.i		−	1.06166i	0.847477	−	0.530832i	$-0.178121\pi$
$620$	179505.			0.466974
$621$	−300169.	+	59751.2i	−0.778365	+	0.154940i
$622$	31910.5			0.0824807
$623$	−612359.			−1.57772
$624$	345071.	−	230592.i	0.886214	−	0.592209i
$625$	−447118.			−1.14462
$626$	42748.5			0.109087
$627$	428189.	+	640765.i	1.08918	+	1.62991i
$628$			555842.i			1.40939i
$629$			164584.i			0.415992i
$630$	76654.8	+	31743.0i	0.193134	+	0.0799774i
$631$	−599217.			−1.50496			−0.752480	−	0.658615i	$-0.771143\pi$
$631$	−599217.			−1.50496			−0.752480	+	0.658615i	$0.771143\pi$
$632$		−	93423.3i		−	0.233895i
$633$	−117062.	−	175178.i	−0.292153	−	0.437193i
$634$	64825.1			0.161274
$635$	−446010.			−1.10611
$636$	−110677.	−	165622.i	−0.273616	−	0.409454i
$637$	16347.8			0.0402883
$638$	−35503.2	+	75668.1i	−0.0872220	+	0.185897i
$639$	26209.8	+	10853.6i	0.0641892	+	0.0265810i
$640$			327335.i			0.799158i
$641$	−18851.1			−0.0458798			−0.0229399	−	0.999737i	$-0.507303\pi$
$641$	−18851.1			−0.0458798			−0.0229399	+	0.999737i	$0.507303\pi$
$642$	−31544.2	−	47204.5i	−0.0765332	−	0.114528i
$643$	298232.			0.721326			0.360663	−	0.932696i	$-0.382550\pi$
$643$	298232.			0.721326			0.360663	+	0.932696i	$0.382550\pi$
$644$			315312.i			0.760273i
$645$	540108.	−	360925.i	1.29826	−	0.867556i
$646$		−	38135.2i		−	0.0913821i
$647$		−	235747.i		−	0.563167i	−0.959537	−	0.281583i	$-0.909140\pi$
$647$		−	235747.i		−	0.563167i	0.959537	−	0.281583i	$-0.0908596\pi$
$648$	−92406.2	−	92371.7i	−0.220065	−	0.219983i
$649$			6977.01i			0.0165646i
$650$	−63165.0			−0.149503
$651$	82000.0	+	122709.i	0.193487	+	0.289545i
$652$		−	341622.i		−	0.803619i
$653$	12502.0			0.0293192			0.0146596	−	0.999893i	$-0.495334\pi$
$653$	12502.0			0.0293192			0.0146596	+	0.999893i	$0.495334\pi$
$654$	35920.9	−	24004.0i	0.0839830	−	0.0561213i
$655$		−	342715.i		−	0.798823i
$656$	48667.4			0.113092
$657$	−766776.	−	317525.i	−1.77639	−	0.735609i
$658$	−66874.9			−0.154458
$659$	421411.			0.970365			0.485183	−	0.874413i	$-0.338753\pi$
$659$	421411.			0.970365			0.485183	+	0.874413i	$0.338753\pi$
$660$	415555.	+	621859.i	0.953983	+	1.42759i
$661$	328543.			0.751952			0.375976	−	0.926629i	$-0.377308\pi$
$661$	328543.			0.751952			0.375976	+	0.926629i	$0.377308\pi$
$662$		−	43113.5i		−	0.0983779i
$663$	−162229.	+	108409.i	−0.369065	+	0.246626i
$664$		−	64281.3i		−	0.145797i
$665$	−882517.			−1.99563
$666$	69629.3	+	28833.8i	0.156980	+	0.0650059i
$667$	319644.	+	149976.i	0.718481	+	0.337109i
$668$		−	429407.i		−	0.962313i
$669$	137543.	−	91912.4i	0.307316	−	0.205363i
$670$			65002.4i			0.144804i
$671$		−	206464.i		−	0.458564i
$672$	−168582.	+	112654.i	−0.373312	+	0.249464i
$673$	318120.			0.702362			0.351181	−	0.936308i	$-0.385780\pi$
$673$	318120.			0.702362			0.351181	+	0.936308i	$0.385780\pi$
$674$			16063.3i			0.0353602i
$675$	−73350.6	−	368488.i	−0.160989	−	0.808753i
$676$	−144587.			−0.316400
$677$	−805095.			−1.75659			−0.878294	−	0.478121i	$-0.841318\pi$
$677$	−805095.			−1.75659			−0.878294	+	0.478121i	$0.841318\pi$
$678$	6503.37	−	4345.85i	0.0141475	−	0.00945400i
$679$			676446.i			1.46721i
$680$		−	74961.9i		−	0.162115i
$681$	258153.	+	386315.i	0.556652	+	0.833004i
$682$		−	33858.4i		−	0.0727944i
$683$			172224.i			0.369191i	0.982815	+	0.184596i	$0.0590976\pi$
$683$			172224.i			0.369191i	−0.982815	+	0.184596i	$0.940902\pi$
$684$	633960.	+	262525.i	1.35503	+	0.561124i
$685$			987370.i			2.10426i
$686$	−75375.7			−0.160171
$687$	419951.	+	628437.i	0.889785	+	1.33152i
$688$			506765.i			1.07061i
$689$			275894.i			0.581171i
$690$	−66852.3	+	44673.8i	−0.140417	+	0.0938328i
$691$	842962.			1.76544			0.882718	−	0.469904i	$-0.155711\pi$
$691$	842962.			1.76544			0.882718	+	0.469904i	$0.155711\pi$
$692$		−	696148.i		−	1.45375i
$693$	−235271.	+	568146.i	−0.489895	+	1.18302i
$694$			41399.9i			0.0859568i
$695$		−	737900.i		−	1.52767i
$696$	22583.5	+	149030.i	0.0466201	+	0.307649i
$697$	−22880.2			−0.0470972
$698$	118294.			0.242801
$699$	−110497.	−	165353.i	−0.226149	−	0.338422i
$700$	−387077.			−0.789954
$701$			181247.i			0.368838i	0.982848	+	0.184419i	$0.0590403\pi$
$701$			181247.i			0.368838i	−0.982848	+	0.184419i	$0.940960\pi$
$702$	17442.6	+	87625.8i	0.0353947	+	0.177811i
$703$	−801633.			−1.62205
$704$	−551803.			−1.11337
$705$	372311.	+	557146.i	0.749079	+	1.12096i
$706$			151582.i			0.304116i
$707$	943265.			1.88710
$708$	3451.46	+	5164.95i	0.00688551	+	0.0103038i
$709$	−503485.			−1.00160			−0.500800	−	0.865563i	$-0.666961\pi$
$709$	−503485.			−1.00160			−0.500800	+	0.865563i	$0.666961\pi$
$710$	7452.65			0.0147841
$711$	−351081.	−	145384.i	−0.694494	−	0.287593i
$712$	253345.			0.499750
$713$	−143028.			−0.281347
$714$	25300.0	−	16906.6i	0.0496277	−	0.0331635i
$715$		−	1.03589e6i		−	2.02630i
$716$			72694.5i			0.141800i
$717$	481651.	+	720769.i	0.936903	+	1.40203i
$718$	−87448.4			−0.169630
$719$		−	653680.i		−	1.26447i	−0.774778	−	0.632234i	$-0.782138\pi$
$719$		−	653680.i		−	1.26447i	0.774778	−	0.632234i	$-0.217862\pi$
$720$	599199.	+	248131.i	1.15586	+	0.478647i
$721$	986056.			1.89684
$722$	103624.			0.198785
$723$	326932.	−	218471.i	0.625433	−	0.417943i
$724$	404032.			0.770794
$725$	−184111.	+	392395.i	−0.350270	+	0.746531i
$726$	48258.5	−	32248.5i	0.0915589	−	0.0611839i
$727$			763045.i			1.44371i	0.692042	+	0.721857i	$0.256711\pi$
$727$			763045.i			1.44371i	−0.692042	+	0.721857i	$0.743289\pi$
$728$	186435.			0.351775
$729$	−490930.	+	203511.i	−0.923772	+	0.382943i
$730$	−218030.			−0.409138
$731$		−	238248.i		−	0.445855i
$732$	−102136.	−	152842.i	−0.190615	−	0.285246i
$733$			119072.i			0.221617i	0.993842	+	0.110809i	$0.0353440\pi$
$733$			119072.i			0.221617i	−0.993842	+	0.110809i	$0.964656\pi$
$734$			8739.59i			0.0162218i
$735$	14193.6	+	21240.0i	0.0262734	+	0.0393170i
$736$		−	196496.i		−	0.362743i
$737$	−481782.			−0.886983
$738$	−4008.44	+	9679.78i	−0.00735974	+	0.0177727i
$739$		−	387109.i		−	0.708833i	−0.935088	−	0.354417i	$-0.884679\pi$
$739$		−	387109.i		−	0.708833i	0.935088	−	0.354417i	$-0.115321\pi$
$740$	−777981.			−1.42071
$741$	−528026.	−	790167.i	−0.961654	−	1.43907i
$742$		−	43026.2i		−	0.0781494i
$743$	627893.			1.13739			0.568693	−	0.822550i	$-0.307449\pi$
$743$	627893.			1.13739			0.568693	+	0.822550i	$0.307449\pi$
$744$	−33925.1	−	50767.3i	−0.0612879	−	0.0917146i
$745$	−1.27905e6			−2.30450
$746$	−109818.			−0.197332
$747$	−241566.	−	100034.i	−0.432908	−	0.179269i
$748$	274309.			0.490272
$749$			481868.i			0.858944i
$750$	11663.7	+	17454.2i	0.0207355	+	0.0310297i
$751$		−	707998.i		−	1.25531i	−0.778490	−	0.627657i	$-0.784014\pi$
$751$		−	707998.i		−	1.25531i	0.778490	−	0.627657i	$-0.215986\pi$
$752$	−522751.			−0.924399
$753$	−548330.	+	366420.i	−0.967057	+	0.646232i
$754$	43781.2	−	93311.0i	0.0770096	−	0.164131i
$755$			1.12909e6i			1.98077i
$756$	106889.	+	536974.i	0.187021	+	0.939528i
$757$		−	192253.i		−	0.335491i	−0.985830	−	0.167745i	$-0.946351\pi$
$757$		−	192253.i		−	0.335491i	0.985830	−	0.167745i	$-0.0536487\pi$
$758$			61289.2i			0.106671i
$759$	−331111.	−	495492.i	−0.574765	−	0.860109i
$760$	365115.			0.632125
$761$			428996.i			0.740770i	0.928878	+	0.370385i	$0.120774\pi$
$761$			428996.i			0.740770i	−0.928878	+	0.370385i	$0.879226\pi$
$762$	41616.7	+	62277.5i	0.0716734	+	0.107256i
$763$	−366684.			−0.629858
$764$	480664.			0.823483
$765$	−281704.	−	116655.i	−0.481360	−	0.199333i
$766$		−	124874.i		−	0.212822i
$767$		−	8603.78i		−	0.0146251i
$768$	−373178.	+	249375.i	−0.632694	+	0.422795i
$769$			203801.i			0.344630i	0.985042	+	0.172315i	$0.0551248\pi$
$769$			203801.i			0.344630i	−0.985042	+	0.172315i	$0.944875\pi$
$770$			161550.i			0.272474i
$771$	−144790.	−	216672.i	−0.243573	−	0.364496i
$772$		−	191236.i		−	0.320874i
$773$	515215.			0.862242			0.431121	−	0.902294i	$-0.358118\pi$
$773$	515215.			0.862242			0.431121	+	0.902294i	$0.358118\pi$
$774$	−100794.	−	41739.1i	−0.168249	−	0.0696725i
$775$		−	175581.i		−	0.292331i
$776$		−	279860.i		−	0.464747i
$777$	−355391.	−	531827.i	−0.588660	−	0.880903i
$778$	29205.8			0.0482515
$779$		−	111442.i		−	0.183643i
$780$	−512447.	−	766853.i	−0.842286	−	1.26044i
$781$			55237.2i			0.0905585i
$782$			29489.3i			0.0482226i
$783$	595193.	+	147050.i	0.970810	+	0.239852i
$784$	−19928.8			−0.0324227
$785$	1.20302e6			1.95223
$786$	−47854.2	+	31978.4i	−0.0774595	+	0.0517621i
$787$	305990.			0.494035			0.247017	−	0.969011i	$-0.420549\pi$
$787$	305990.			0.494035			0.247017	+	0.969011i	$0.420549\pi$
$788$			911761.i			1.46835i
$789$	75244.1	−	50281.6i	0.120870	−	0.0807709i
$790$	−99828.5			−0.159956
$791$	−66387.0			−0.106104
$792$	97336.6	−	235054.i	0.155176	−	0.374728i
$793$			254604.i			0.404873i
$794$	16167.2			0.0256445
$795$	−358459.	+	239539.i	−0.567159	+	0.379002i
$796$	40537.6			0.0639782
$797$	−17838.2			−0.0280825			−0.0140412	−	0.999901i	$-0.504470\pi$
$797$	−17838.2			−0.0280825			−0.0140412	+	0.999901i	$0.504470\pi$
$798$	82346.7	+	123228.i	0.129313	+	0.193510i
$799$	245763.			0.384967
$800$	241219.			0.376904
$801$	394253.	−	952062.i	0.614483	−	1.48389i
$802$			135708.i			0.210987i
$803$		−	1.61598e6i		−	2.50614i
$804$	−356654.	+	238333.i	−0.551740	+	0.368698i
$805$	682435.			1.05310
$806$			41752.9i			0.0642712i
$807$	549124.	−	366950.i	0.843186	−	0.563456i
$808$	−390248.			−0.597748
$809$	981369.			1.49946			0.749731	−	0.661743i	$-0.230183\pi$
$809$	981369.			1.49946			0.749731	+	0.661743i	$0.230183\pi$
$810$	−98704.7	+	98741.6i	−0.150442	+	0.150498i
$811$	−212489.			−0.323068			−0.161534	−	0.986867i	$-0.551644\pi$
$811$	−212489.			−0.323068			−0.161534	+	0.986867i	$0.551644\pi$
$812$	268293.	−	571813.i	0.406909	−	0.867246i
$813$	−198505.	−	297054.i	−0.300324	−	0.449422i
$814$			146744.i			0.221468i
$815$	−739376.			−1.11314
$816$	197766.	−	132157.i	0.297011	−	0.198476i
$817$	1.16043e6			1.73850
$818$			177113.i			0.264694i
$819$	290128.	−	700616.i	0.432535	−	1.04451i
$820$		−	108154.i		−	0.160848i
$821$			436417.i			0.647463i	0.946149	+	0.323732i	$0.104938\pi$
$821$			436417.i			0.647463i	−0.946149	+	0.323732i	$0.895062\pi$
$822$	137869.	−	92130.5i	0.204044	−	0.136351i
$823$			414627.i			0.612151i	0.952007	+	0.306075i	$0.0990159\pi$
$823$			414627.i			0.612151i	−0.952007	+	0.306075i	$0.900984\pi$
$824$	−407951.			−0.600833
$825$	608266.	−	406472.i	0.893688	−	0.597204i
$826$			1341.78i			0.00196662i
$827$	−520899.			−0.761628			−0.380814	−	0.924652i	$-0.624356\pi$
$827$	−520899.			−0.761628			−0.380814	+	0.924652i	$0.624356\pi$
$828$	−490230.	−	203006.i	−0.715055	−	0.296107i
$829$			41779.1i			0.0607925i	0.999538	+	0.0303962i	$0.00967691\pi$
$829$			41779.1i			0.0607925i	−0.999538	+	0.0303962i	$0.990323\pi$
$830$	−68688.4			−0.0997073
$831$	56576.8	−	37807.3i	0.0819288	−	0.0547486i
$832$	680462.			0.983009
$833$	9369.21			0.0135025
$834$	−103035.	+	68852.7i	−0.148133	+	0.0989895i
$835$	−929372.			−1.33296
$836$			1.33607e6i			1.91169i
$837$	−243575.	+	48485.7i	−0.347682	+	0.0692090i
$838$		−	163990.i		−	0.233522i
$839$	628323.			0.892605			0.446303	−	0.894882i	$-0.352741\pi$
$839$	628323.			0.892605			0.446303	+	0.894882i	$0.352741\pi$
$840$	161868.	+	242228.i	0.229405	+	0.343294i
$841$	−452058.	−	543958.i	−0.639149	−	0.769083i
$842$			58132.8i			0.0819968i
$843$	98696.3	+	147694.i	0.138882	+	0.207830i
$844$		−	365268.i		−	0.512775i
$845$			312932.i			0.438265i
$846$	43055.8	−	103973.i	0.0601577	−	0.145272i
$847$	−492627.			−0.686676
$848$		−	336330.i		−	0.467707i
$849$	−575151.	+	384342.i	−0.797933	+	0.533216i
$850$	−36201.0			−0.0501053
$851$	619889.			0.855963
$852$	27325.3	+	40891.0i	0.0376431	+	0.0563312i
$853$		−	690533.i		−	0.949045i	−0.880243	−	0.474522i	$-0.842621\pi$
$853$		−	690533.i		−	0.949045i	0.880243	−	0.474522i	$-0.157379\pi$
$854$		−	39706.0i		−	0.0544428i
$855$	568187.	−	1.37209e6i	0.777247	−	1.87694i
$856$		−	199359.i		−	0.272074i
$857$			470602.i			0.640755i	0.947290	+	0.320377i	$0.103810\pi$
$857$			470602.i			0.640755i	−0.947290	+	0.320377i	$0.896190\pi$
$858$	−144645.	+	96658.3i	−0.196484	+	0.131300i
$859$		−	1.29875e6i		−	1.76011i	−0.474870	−	0.880056i	$-0.657505\pi$
$859$		−	1.29875e6i		−	1.76011i	0.474870	−	0.880056i	$-0.342495\pi$
$860$	1.12619e6			1.52270
$861$	73934.0	−	49406.1i	0.0997327	−	0.0666460i
$862$		−	136409.i		−	0.183582i
$863$			1.22107e6i			1.63953i	0.572702	+	0.819764i	$0.305895\pi$
$863$			1.22107e6i			1.63953i	−0.572702	+	0.819764i	$0.694105\pi$
$864$	−66611.1	−	334631.i	−0.0892318	−	0.448270i
$865$	−1.50668e6			−2.01368
$866$		−	10553.7i		−	0.0140724i
$867$	532010.	−	355514.i	0.707753	−	0.472953i
$868$			255863.i			0.339601i
$869$		−	739903.i		−	0.979795i
$870$	159247.	−	24131.8i	0.210394	−	0.0318825i
$871$	594115.			0.783130
$872$	151705.			0.199511
$873$	−1.05170e6	−	435514.i	−1.37995	−	0.571443i
$874$	−143633.			−0.188032
$875$		−	178174.i		−	0.232717i
$876$	−799409.	−	1.19628e6i	−1.04174	−	1.55892i
$877$	−695909.			−0.904802			−0.452401	−	0.891815i	$-0.649432\pi$
$877$	−695909.			−0.904802			−0.452401	+	0.891815i	$0.649432\pi$
$878$	−32261.9			−0.0418506
$879$	768421.	−	513494.i	0.994538	−	0.664596i
$880$			1.26281e6i			1.63070i
$881$	−205174.			−0.264344			−0.132172	−	0.991227i	$-0.542195\pi$
$881$	−205174.			−0.264344			−0.132172	+	0.991227i	$0.542195\pi$
$882$	1641.41	−	3963.77i	0.00210999	−	0.00509532i
$883$	−942359.			−1.20863			−0.604317	−	0.796744i	$-0.706554\pi$
$883$	−942359.			−1.20863			−0.604317	+	0.796744i	$0.706554\pi$
$884$	−338267.			−0.432868
$885$	11178.6	−	7470.04i	0.0142725	−	0.00953753i
$886$	194080.			0.247237
$887$	−205689.			−0.261435			−0.130718	−	0.991420i	$-0.541728\pi$
$887$	−205689.			−0.261435			−0.130718	+	0.991420i	$0.541728\pi$
$888$	147033.	+	220028.i	0.186461	+	0.279030i
$889$		−	635735.i		−	0.804402i
$890$		−	270715.i		−	0.341769i
$891$	−731848.	−	731574.i	−0.921861	−	0.921516i
$892$	286793.			0.360445
$893$			1.19703e6i			1.50108i
$894$	119347.	+	178598.i	0.149327	+	0.223460i
$895$	157334.			0.196415
$896$	−466578.			−0.581177
$897$	408313.	+	611022.i	0.507468	+	0.759403i
$898$	177553.			0.220179
$899$	259379.	+	121700.i	0.320933	+	0.150581i
$900$	249211.	−	601807.i	0.307667	−	0.742971i
$901$			158120.i			0.194777i
$902$	−20400.1			−0.0250738
$903$	514457.	+	769861.i	0.630919	+	0.944141i
$904$	27465.7			0.0336088
$905$		−	874452.i		−	1.06767i
$906$	157657.	−	105354.i	0.192069	−	0.128350i
$907$			861662.i			1.04742i	0.851895	+	0.523712i	$0.175453\pi$
$907$			861662.i			1.04742i	−0.851895	+	0.523712i	$0.824547\pi$
$908$			805512.i			0.977013i
$909$	−607299.	+	1.46654e6i	−0.734978	+	1.77486i
$910$		−	199217.i		−	0.240571i
$911$	265544.			0.319964			0.159982	−	0.987120i	$-0.448856\pi$
$911$	265544.			0.319964			0.159982	+	0.987120i	$0.448856\pi$
$912$	643692.	+	963256.i	0.773907	+	1.15812i
$913$		−	509101.i		−	0.610749i
$914$	60409.1			0.0723120
$915$	−330798.	+	221054.i	−0.395112	+	0.264032i
$916$			1.31037e6i			1.56171i
$917$	488500.			0.580933
$918$	9996.70	+	50220.0i	0.0118624	+	0.0595925i
$919$	290403.			0.343850			0.171925	−	0.985110i	$-0.445001\pi$
$919$	290403.			0.343850			0.171925	+	0.985110i	$0.445001\pi$
$920$	−282337.			−0.333574
$921$	−543724.	−	813659.i	−0.641002	−	0.959231i
$922$	47598.1			0.0559923
$923$		−	68116.4i		−	0.0799555i
$924$	−886388.	+	592326.i	−1.03820	+	0.693772i
$925$			760976.i			0.889380i
$926$	138590.			0.161625
$927$	−634848.	+	1.53306e6i	−0.738772	+	1.78403i
$928$	−167195.	+	356342.i	−0.194145	+	0.413782i
$929$			1.38809e6i			1.60837i	0.594377	+	0.804186i	$0.297399\pi$
$929$			1.38809e6i			1.60837i	−0.594377	+	0.804186i	$0.702601\pi$
$930$	−54247.9	+	36251.0i	−0.0627216	+	0.0419135i
$931$			45634.4i			0.0526493i
$932$		−	344781.i		−	0.396928i
$933$	378940.	−	253226.i	0.435319	−	0.290900i
$934$	167195.			0.191659
$935$		−	593691.i		−	0.679105i
$936$	−120032.	+	289859.i	−0.137008	+	0.330853i
$937$	−24450.5			−0.0278490			−0.0139245	−	0.999903i	$-0.504432\pi$
$937$	−24450.5			−0.0278490			−0.0139245	+	0.999903i	$0.504432\pi$
$938$	−92653.4			−0.105307
$939$	507643.	−	339230.i	0.575741	−	0.384737i
$940$			1.16172e6i			1.31475i
$941$		−	771463.i		−	0.871236i	−0.900132	−	0.435618i	$-0.856530\pi$
$941$		−	771463.i		−	0.871236i	0.900132	−	0.435618i	$-0.143470\pi$
$942$	−112252.	−	167980.i	−0.126501	−	0.189303i
$943$			86176.3i			0.0969091i
$944$			10488.5i			0.0117698i
$945$	1.16218e6	−	231342.i	1.30140	−	0.259054i
$946$		−	212423.i		−	0.237366i
$947$	−1.35877e6			−1.51512			−0.757560	−	0.652765i	$-0.773609\pi$
$947$	−1.35877e6			−1.51512			−0.757560	+	0.652765i	$0.773609\pi$
$948$	−366023.	−	547736.i	−0.407278	−	0.609474i
$949$			1.99277e6i			2.21271i
$950$		−	176324.i		−	0.195372i
$951$	769805.	−	514419.i	0.851176	−	0.568795i
$952$	106850.			0.117896
$953$		−	755735.i		−	0.832116i	−0.909338	−	0.416058i	$-0.863411\pi$
$953$		−	755735.i		−	0.832116i	0.909338	−	0.416058i	$-0.136589\pi$
$954$	66894.8	+	27701.4i	0.0735014	+	0.0304372i
$955$		−	1.04031e6i		−	1.14066i
$956$			1.50289e6i			1.64441i
$957$	178859.	+	1.18030e6i	0.195293	+	1.28875i
$958$	−52913.2			−0.0576545
$959$	−1.40738e6			−1.53029
$960$	590796.	+	884099.i	0.641054	+	0.959308i
$961$	807460.			0.874327
$962$		−	180959.i		−	0.195537i
$963$	−749182.	−	310239.i	−0.807858	−	0.334537i
$964$	681692.			0.733557
$965$	−413894.			−0.444462
$966$	−63677.3	−	95290.2i	−0.0682387	−	0.102116i
$967$		−	142980.i		−	0.152905i	−0.997073	−	0.0764527i	$-0.975641\pi$
$967$		−	142980.i		−	0.152905i	0.997073	−	0.0764527i	$-0.0243594\pi$
$968$	203810.			0.217508
$969$	−302622.	−	452860.i	−0.322294	−	0.482299i
$970$	−299047.			−0.317831
$971$	38982.8			0.0413461			0.0206731	−	0.999786i	$-0.493419\pi$
$971$	38982.8			0.0413461			0.0206731	+	0.999786i	$0.493419\pi$
$972$	−903676.	−	179533.i	−0.956489	−	0.190025i
$973$	1.05179e6			1.11097
$974$	67058.7			0.0706866
$975$	−750091.	+	501246.i	−0.789050	+	0.527280i
$976$		−	310376.i		−	0.325828i
$977$		−	518384.i		−	0.543078i	−0.962427	−	0.271539i	$-0.912467\pi$
$977$		−	518384.i		−	0.543078i	0.962427	−	0.271539i	$-0.0875327\pi$
$978$	68990.4	+	103241.i	0.0721292	+	0.107938i
$979$	2.00647e6			2.09347
$980$			44287.9i			0.0461141i
$981$	236081.	−	570100.i	0.245314	−	0.592397i
$982$	101062.			0.104801
$983$	1.61151e6			1.66773			0.833865	−	0.551968i	$-0.186123\pi$
$983$	1.61151e6			1.66773			0.833865	+	0.551968i	$0.186123\pi$
$984$	−30588.0	+	20440.3i	−0.0315908	+	0.0211104i
$985$	1.97334e6			2.03390
$986$	25091.8	−	53478.3i	0.0258094	−	0.0550077i
$987$	−794147.	+	530686.i	−0.815204	+	0.544757i
$988$		−	1.64759e6i		−	1.68786i
$989$	−897338.			−0.917410
$990$	−251169.	−	104010.i	−0.256269	−	0.106122i
$991$	−1.21858e6			−1.24081			−0.620405	−	0.784282i	$-0.713032\pi$
$991$	−1.21858e6			−1.24081			−0.620405	+	0.784282i	$0.713032\pi$
$992$		−	159449.i		−	0.162031i
$993$	−342127.	−	511978.i	−0.346968	−	0.519221i
$994$			10622.9i			0.0107515i
$995$		−	87736.1i		−	0.0886201i
$996$	−251847.	−	376878.i	−0.253874	−	0.379911i
$997$			454057.i			0.456794i	0.973568	+	0.228397i	$0.0733484\pi$
$997$			454057.i			0.456794i	−0.973568	+	0.228397i	$0.926652\pi$
$998$	114992.			0.115454
$999$	1.05567e6	−	210139.i	1.05778	−	0.210560i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	87.5.d.c.86.16	yes	32
3.2	odd	2	inner	87.5.d.c.86.18	yes	32
29.28	even	2	inner	87.5.d.c.86.17	yes	32
87.86	odd	2	inner	87.5.d.c.86.15	✓	32

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
87.5.d.c.86.15	✓	32	87.86	odd	2	inner
87.5.d.c.86.16	yes	32	1.1	even	1	trivial
87.5.d.c.86.17	yes	32	29.28	even	2	inner
87.5.d.c.86.18	yes	32	3.2	odd	2	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 87 = 3 \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 5 \)
Character orbit:	\([\chi]\)	\(=\)	87.d (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-0.630141 q^{2} +(-7.48299 + 5.00048i) q^{3} -15.6029 q^{4} +33.7696i q^{5} +(4.71534 - 3.15101i) q^{6} -48.1347 q^{7} +19.9143 q^{8} +(30.9904 - 74.8371i) q^{9} -21.2796i q^{10} +157.719 q^{11} +(116.757 - 78.0221i) q^{12} -194.493 q^{13} +30.3316 q^{14} +(-168.864 - 252.698i) q^{15} +237.098 q^{16} -111.468 q^{17} +(-19.5283 + 47.1580i) q^{18} -542.924i q^{19} -526.905i q^{20} +(360.191 - 240.697i) q^{21} -99.3854 q^{22} +419.833i q^{23} +(-149.019 + 99.5811i) q^{24} -515.387 q^{25} +122.558 q^{26} +(142.321 + 714.973i) q^{27} +751.042 q^{28} +(357.228 - 761.360i) q^{29} +(106.408 + 159.235i) q^{30} +340.678i q^{31} -468.034 q^{32} +(-1180.21 + 788.672i) q^{33} +70.2405 q^{34} -1625.49i q^{35} +(-483.540 + 1167.68i) q^{36} -1476.51i q^{37} +342.118i q^{38} +(1455.39 - 972.561i) q^{39} +672.498i q^{40} +205.263 q^{41} +(-226.971 + 151.673i) q^{42} +2137.37i q^{43} -2460.88 q^{44} +(2527.22 + 1046.53i) q^{45} -264.554i q^{46} -2204.79 q^{47} +(-1774.20 + 1185.60i) q^{48} -84.0530 q^{49} +324.767 q^{50} +(834.113 - 557.393i) q^{51} +3034.66 q^{52} -1418.53i q^{53} +(-89.6824 - 450.534i) q^{54} +5326.12i q^{55} -958.568 q^{56} +(2714.88 + 4062.69i) q^{57} +(-225.104 + 479.764i) q^{58} +44.2369i q^{59} +(2634.78 + 3942.82i) q^{60} -1309.06i q^{61} -214.675i q^{62} +(-1491.71 + 3602.26i) q^{63} -3498.64 q^{64} -6567.97i q^{65} +(743.700 - 496.975i) q^{66} -3054.68 q^{67} +1739.22 q^{68} +(-2099.37 - 3141.61i) q^{69} +1024.29i q^{70} +350.225i q^{71} +(617.151 - 1490.33i) q^{72} -10245.9i q^{73} +930.411i q^{74} +(3856.64 - 2577.19i) q^{75} +8471.19i q^{76} -7591.76 q^{77} +(-917.103 + 612.850i) q^{78} -4691.27i q^{79} +8006.71i q^{80} +(-4640.20 - 4638.46i) q^{81} -129.345 q^{82} -3227.89i q^{83} +(-5620.04 + 3755.57i) q^{84} -3764.23i q^{85} -1346.84i q^{86} +(1134.04 + 7483.56i) q^{87} +3140.87 q^{88} +12721.8 q^{89} +(-1592.51 - 659.463i) q^{90} +9361.87 q^{91} -6550.63i q^{92} +(-1703.55 - 2549.29i) q^{93} +1389.33 q^{94} +18334.3 q^{95} +(3502.29 - 2340.40i) q^{96} -14053.2i q^{97} +52.9653 q^{98} +(4887.78 - 11803.3i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 32 q + 188 q^{4} - 36 q^{6} - 84 q^{7} - 452 q^{9} - 224 q^{13} - 52 q^{16} - 4216 q^{22} - 832 q^{24} - 7684 q^{25} - 396 q^{28} + 3384 q^{30} - 3308 q^{33} + 9124 q^{34} - 3680 q^{36} + 19764 q^{42} + 44 q^{45}+ \cdots + 13884 q^{96}+O(q^{100}) \) 32 * q + 188 * q^4 - 36 * q^6 - 84 * q^7 - 452 * q^9 - 224 * q^13 - 52 * q^16 - 4216 * q^22 - 832 * q^24 - 7684 * q^25 - 396 * q^28 + 3384 * q^30 - 3308 * q^33 + 9124 * q^34 - 3680 * q^36 + 19764 * q^42 + 44 * q^45 + 4796 * q^49 - 7052 * q^51 + 34048 * q^52 + 3008 * q^54 + 7976 * q^57 + 5936 * q^58 - 22848 * q^63 - 14668 * q^64 - 1872 * q^67 + 15192 * q^78 - 41088 * q^81 - 27028 * q^82 - 35572 * q^87 - 60032 * q^88 - 34272 * q^91 + 38000 * q^93 + 59636 * q^94 + 13884 * q^96

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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