LMFDB - Embedded newform 87.5.d.c.86.10

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			86.10
Character	$\chi$	$=$	87.86
Dual form			87.5.d.c.86.9

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q-3.96406 q^{2} +(6.34485 + 6.38301i) q^{3} -0.286250 q^{4} -7.48339i q^{5} +(-25.1514 - 25.3026i) q^{6} +18.5615 q^{7} +64.5596 q^{8} +(-0.485670 + 80.9985i) q^{9} +29.6646i q^{10} +141.778 q^{11} +(-1.81621 - 1.82714i) q^{12} -227.588 q^{13} -73.5790 q^{14} +(47.7665 - 47.4810i) q^{15} -251.338 q^{16} +50.8285 q^{17} +(1.92522 - 321.083i) q^{18} +628.112i q^{19} +2.14212i q^{20} +(117.770 + 118.479i) q^{21} -562.017 q^{22} +898.114i q^{23} +(409.621 + 412.085i) q^{24} +568.999 q^{25} +902.171 q^{26} +(-520.096 + 510.824i) q^{27} -5.31324 q^{28} +(-605.272 - 583.889i) q^{29} +(-189.349 + 188.217i) q^{30} +1348.35i q^{31} -36.6355 q^{32} +(899.563 + 904.973i) q^{33} -201.487 q^{34} -138.903i q^{35} +(0.139023 - 23.1858i) q^{36} -550.479i q^{37} -2489.87i q^{38} +(-1444.01 - 1452.70i) q^{39} -483.125i q^{40} +1775.07 q^{41} +(-466.848 - 469.656i) q^{42} +916.159i q^{43} -40.5840 q^{44} +(606.143 + 3.63446i) q^{45} -3560.17i q^{46} +1685.01 q^{47} +(-1594.70 - 1604.29i) q^{48} -2056.47 q^{49} -2255.54 q^{50} +(322.500 + 324.439i) q^{51} +65.1469 q^{52} -2492.27i q^{53} +(2061.69 - 2024.94i) q^{54} -1060.98i q^{55} +1198.33 q^{56} +(-4009.24 + 3985.28i) q^{57} +(2399.33 + 2314.57i) q^{58} -3146.25i q^{59} +(-13.6732 + 13.5914i) q^{60} -4513.21i q^{61} -5344.95i q^{62} +(-9.01479 + 1503.46i) q^{63} +4166.63 q^{64} +1703.13i q^{65} +(-3565.92 - 3587.36i) q^{66} +5132.45 q^{67} -14.5497 q^{68} +(-5732.67 + 5698.40i) q^{69} +550.620i q^{70} +4512.12i q^{71} +(-31.3547 + 5229.24i) q^{72} +7190.98i q^{73} +2182.13i q^{74} +(3610.21 + 3631.93i) q^{75} -179.797i q^{76} +2631.62 q^{77} +(5724.14 + 5758.57i) q^{78} -1900.62i q^{79} +1880.86i q^{80} +(-6560.53 - 78.6772i) q^{81} -7036.47 q^{82} +1667.81i q^{83} +(-33.7117 - 33.9144i) q^{84} -380.370i q^{85} -3631.71i q^{86} +(-113.389 - 7568.15i) q^{87} +9153.16 q^{88} +5572.12 q^{89} +(-2402.79 - 14.4072i) q^{90} -4224.38 q^{91} -257.085i q^{92} +(-8606.56 + 8555.11i) q^{93} -6679.46 q^{94} +4700.40 q^{95} +(-232.447 - 233.845i) q^{96} -14488.2i q^{97} +8151.96 q^{98} +(-68.8575 + 11483.8i) 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 $

$2$	−3.96406			−0.991014			−0.495507	−	0.868604i	$-0.665018\pi$
$2$	−3.96406			−0.991014			−0.495507	+	0.868604i	$0.665018\pi$
$3$	6.34485	+	6.38301i	0.704984	+	0.709223i
$3$	6.34485	+	6.38301i	0.704984	+	0.709223i
$4$	−0.286250			−0.0178906
$5$		−	7.48339i		−	0.299335i	−0.988736	−	0.149668i	$-0.952180\pi$
$5$		−	7.48339i		−	0.299335i	0.988736	−	0.149668i	$-0.0478204\pi$
$6$	−25.1514	−	25.3026i	−0.698649	−	0.702851i
$7$	18.5615			0.378807			0.189403	−	0.981899i	$-0.439345\pi$
$7$	18.5615			0.378807			0.189403	+	0.981899i	$0.439345\pi$
$8$	64.5596			1.00874
$9$	−0.485670	+	80.9985i	−0.00599593	+	0.999982i
$10$			29.6646i			0.296646i
$11$	141.778			1.17172			0.585861	−	0.810412i	$-0.300757\pi$
$11$	141.778			1.17172			0.585861	+	0.810412i	$0.300757\pi$
$12$	−1.81621	−	1.82714i	−0.0126126	−	0.0126884i
$13$	−227.588			−1.34667			−0.673336	−	0.739336i	$-0.735139\pi$
$13$	−227.588			−1.34667			−0.673336	+	0.739336i	$0.735139\pi$
$14$	−73.5790			−0.375403
$15$	47.7665	−	47.4810i	0.212296	−	0.211027i
$16$	−251.338			−0.981789
$17$	50.8285			0.175877			0.0879387	−	0.996126i	$-0.471972\pi$
$17$	50.8285			0.175877			0.0879387	+	0.996126i	$0.471972\pi$
$18$	1.92522	−	321.083i	0.00594205	−	0.990997i
$19$			628.112i			1.73992i	0.493121	+	0.869961i	$0.335856\pi$
$19$			628.112i			1.73992i	−0.493121	+	0.869961i	$0.664144\pi$
$20$			2.14212i			0.00535529i
$21$	117.770	+	118.479i	0.267053	+	0.268659i
$22$	−562.017			−1.16119
$23$			898.114i			1.69776i	0.528588	+	0.848879i	$0.322722\pi$
$23$			898.114i			1.69776i	−0.528588	+	0.848879i	$0.677278\pi$
$24$	409.621	+	412.085i	0.711148	+	0.715425i
$25$	568.999			0.910398
$26$	902.171			1.33457
$27$	−520.096	+	510.824i	−0.713438	+	0.700719i
$28$	−5.31324			−0.00677709
$29$	−605.272	−	583.889i	−0.719705	−	0.694280i
$29$	−605.272	−	583.889i	−0.719705	−	0.694280i
$30$	−189.349	+	188.217i	−0.210388	+	0.209130i
$31$			1348.35i			1.40307i	0.712633	+	0.701537i	$0.247502\pi$
$31$			1348.35i			1.40307i	−0.712633	+	0.701537i	$0.752498\pi$
$32$	−36.6355			−0.0357769
$33$	899.563	+	904.973i	0.826045	+	0.831013i
$34$	−201.487			−0.174297
$35$		−	138.903i		−	0.113390i
$36$	0.139023	−	23.1858i	0.000107271	−	0.0178903i
$37$		−	550.479i		−	0.402103i	−0.979581	−	0.201052i	$-0.935564\pi$
$37$		−	550.479i		−	0.402103i	0.979581	−	0.201052i	$-0.0644359\pi$
$38$		−	2489.87i		−	1.72429i
$39$	−1444.01	−	1452.70i	−0.949382	−	0.955092i
$40$		−	483.125i		−	0.301953i
$41$	1775.07			1.05596			0.527980	−	0.849257i	$-0.322950\pi$
$41$	1775.07			1.05596			0.527980	+	0.849257i	$0.322950\pi$
$42$	−466.848	−	469.656i	−0.264653	−	0.266245i
$43$			916.159i			0.495489i	0.968825	+	0.247744i	$0.0796893\pi$
$43$			916.159i			0.495489i	−0.968825	+	0.247744i	$0.920311\pi$
$44$	−40.5840			−0.0209628
$45$	606.143	+	3.63446i	0.299330	+	0.00179479i
$46$		−	3560.17i		−	1.68250i
$47$	1685.01			0.762792			0.381396	−	0.924412i	$-0.375443\pi$
$47$	1685.01			0.762792			0.381396	+	0.924412i	$0.375443\pi$
$48$	−1594.70	−	1604.29i	−0.692145	−	0.696308i
$49$	−2056.47			−0.856505
$50$	−2255.54			−0.902218
$51$	322.500	+	324.439i	0.123991	+	0.124736i
$52$	65.1469			0.0240928
$53$		−	2492.27i		−	0.887244i	−0.896214	−	0.443622i	$-0.853693\pi$
$53$		−	2492.27i		−	0.887244i	0.896214	−	0.443622i	$-0.146307\pi$
$54$	2061.69	−	2024.94i	0.707027	−	0.694422i
$55$		−	1060.98i		−	0.350738i
$56$	1198.33			0.382119
$57$	−4009.24	+	3985.28i	−1.23399	+	1.22662i
$58$	2399.33	+	2314.57i	0.713238	+	0.688041i
$59$		−	3146.25i		−	0.903835i	−0.892060	−	0.451917i	$-0.850740\pi$
$59$		−	3146.25i		−	0.903835i	0.892060	−	0.451917i	$-0.149260\pi$
$60$	−13.6732	+	13.5914i	−0.00379810	+	0.00377539i
$61$		−	4513.21i		−	1.21290i	−0.795121	−	0.606451i	$-0.792592\pi$
$61$		−	4513.21i		−	1.21290i	0.795121	−	0.606451i	$-0.207408\pi$
$62$		−	5344.95i		−	1.39047i
$63$	−9.01479	+	1503.46i	−0.00227130	+	0.378800i
$64$	4166.63			1.01724
$65$			1703.13i			0.403107i
$66$	−3565.92	−	3587.36i	−0.818622	−	0.823545i
$67$	5132.45			1.14334			0.571670	−	0.820484i	$-0.306296\pi$
$67$	5132.45			1.14334			0.571670	+	0.820484i	$0.306296\pi$
$68$	−14.5497			−0.00314655
$69$	−5732.67	+	5698.40i	−1.20409	+	1.19689i
$70$			550.620i			0.112371i
$71$			4512.12i			0.895084i	0.894263	+	0.447542i	$0.147700\pi$
$71$			4512.12i			0.895084i	−0.894263	+	0.447542i	$0.852300\pi$
$72$	−31.3547	+	5229.24i	−0.00604836	+	1.00873i
$73$			7190.98i			1.34941i	0.738090	+	0.674703i	$0.235728\pi$
$73$			7190.98i			1.34941i	−0.738090	+	0.674703i	$0.764272\pi$
$74$			2182.13i			0.398490i
$75$	3610.21	+	3631.93i	0.641816	+	0.645676i
$76$		−	179.797i		−	0.0311283i
$77$	2631.62			0.443856
$78$	5724.14	+	5758.57i	0.940852	+	0.946510i
$79$		−	1900.62i		−	0.304538i	−0.988339	−	0.152269i	$-0.951342\pi$
$79$		−	1900.62i		−	0.304538i	0.988339	−	0.152269i	$-0.0486580\pi$
$80$			1880.86i			0.293884i
$81$	−6560.53	−	78.6772i	−0.999928	−	0.0119916i
$82$	−7036.47			−1.04647
$83$			1667.81i			0.242097i	0.992647	+	0.121049i	$0.0386257\pi$
$83$			1667.81i			0.242097i	−0.992647	+	0.121049i	$0.961374\pi$
$84$	−33.7117	−	33.9144i	−0.00477774	−	0.00480647i
$85$		−	380.370i		−	0.0526463i
$86$		−	3631.71i		−	0.491036i
$87$	−113.389	−	7568.15i	−0.0149807	−	0.999888i
$88$	9153.16			1.18197
$89$	5572.12			0.703462			0.351731	−	0.936101i	$-0.385593\pi$
$89$	5572.12			0.703462			0.351731	+	0.936101i	$0.385593\pi$
$90$	−2402.79	−	14.4072i	−0.296640	−	0.00177867i
$91$	−4224.38			−0.510129
$92$		−	257.085i		−	0.0303739i
$93$	−8606.56	+	8555.11i	−0.995093	+	0.989144i
$94$	−6679.46			−0.755938
$95$	4700.40			0.520820
$96$	−232.447	−	233.845i	−0.0252221	−	0.0253738i
$97$		−	14488.2i		−	1.53983i	−0.638148	−	0.769914i	$-0.720299\pi$
$97$		−	14488.2i		−	1.53983i	0.638148	−	0.769914i	$-0.279701\pi$
$98$	8151.96			0.848809
$99$	−68.8575	+	11483.8i	−0.00702556	+	1.17170i
$100$	−162.876			−0.0162876
$101$	2130.39			0.208842			0.104421	−	0.994533i	$-0.466701\pi$
$101$	2130.39			0.208842			0.104421	+	0.994533i	$0.466701\pi$
$102$	−1278.41	−	1286.10i	−0.122877	−	0.123615i
$103$	−4238.51			−0.399520			−0.199760	−	0.979845i	$-0.564016\pi$
$103$	−4238.51			−0.399520			−0.199760	+	0.979845i	$0.564016\pi$
$104$	−14693.0			−1.35845
$105$	886.620	−	881.320i	0.0804191	−	0.0799383i
$106$			9879.50i			0.879272i
$107$		−	2002.81i		−	0.174933i	−0.996167	−	0.0874665i	$-0.972123\pi$
$107$		−	2002.81i		−	0.174933i	0.996167	−	0.0874665i	$-0.0278771\pi$
$108$	148.877	−	146.223i	0.0127638	−	0.0125363i
$109$	−13769.4			−1.15894			−0.579472	−	0.814992i	$-0.696741\pi$
$109$	−13769.4			−1.15894			−0.579472	+	0.814992i	$0.696741\pi$
$110$			4205.79i			0.347586i
$111$	3513.72	−	3492.71i	0.285181	−	0.283476i
$112$	−4665.22			−0.371909
$113$	13310.3			1.04239			0.521194	−	0.853438i	$-0.325487\pi$
$113$	13310.3			1.04239			0.521194	+	0.853438i	$0.325487\pi$
$114$	15892.9	−	15797.9i	1.22290	−	1.21559i
$115$	6720.93			0.508199
$116$	173.259	+	167.138i	0.0128760	+	0.0124211i
$117$	110.533	−	18434.3i	0.00807456	−	1.34665i
$118$			12471.9i			0.895713i
$119$	943.456			0.0666235
$120$	3083.79	−	3065.35i	0.214152	−	0.212872i
$121$	5460.09			0.372932
$122$			17890.6i			1.20200i
$123$	11262.5	+	11330.3i	0.744434	+	0.748911i
$124$		−	385.966i		−	0.0251018i
$125$		−	8935.15i		−	0.571850i
$126$	35.7351	−	5959.79i	0.00225089	−	0.375396i
$127$			11188.9i			0.693716i	0.937918	+	0.346858i	$0.112751\pi$
$127$			11188.9i			0.693716i	−0.937918	+	0.346858i	$0.887249\pi$
$128$	−15930.6			−0.972327
$129$	−5847.85	+	5812.89i	−0.351412	+	0.349311i
$130$		−	6751.29i		−	0.399485i
$131$	−6843.15			−0.398762			−0.199381	−	0.979922i	$-0.563893\pi$
$131$	−6843.15			−0.398762			−0.199381	+	0.979922i	$0.563893\pi$
$132$	−257.500	−	259.048i	−0.0147784	−	0.0148673i
$133$			11658.7i			0.659094i
$134$	−20345.3			−1.13307
$135$	3822.69	+	3892.08i	0.209750	+	0.213557i
$136$	3281.47			0.177415
$137$	−5158.03			−0.274816			−0.137408	−	0.990515i	$-0.543877\pi$
$137$	−5158.03			−0.274816			−0.137408	+	0.990515i	$0.543877\pi$
$138$	22724.6	−	22588.8i	1.19327	−	1.18614i
$139$	36925.2			1.91114			0.955571	−	0.294760i	$-0.0952396\pi$
$139$	36925.2			1.91114			0.955571	+	0.294760i	$0.0952396\pi$
$140$			39.7610i			0.00202862i
$141$	10691.1	+	10755.4i	0.537756	+	0.540990i
$142$		−	17886.3i		−	0.887041i
$143$	−32267.0			−1.57793
$144$	122.067	−	20358.0i	0.00588674	−	0.981772i
$145$	−4369.47	+	4529.48i	−0.207823	+	0.215433i
$146$		−	28505.5i		−	1.33728i
$147$	−13048.0	−	13126.5i	−0.603822	−	0.607454i
$148$			157.575i			0.00719387i
$149$			17763.8i			0.800137i	0.916485	+	0.400068i	$0.131014\pi$
$149$			17763.8i			0.800137i	−0.916485	+	0.400068i	$0.868986\pi$
$150$	−14311.1	−	14397.2i	−0.636049	−	0.639874i
$151$	−19720.7			−0.864905			−0.432453	−	0.901657i	$-0.642352\pi$
$151$	−19720.7			−0.864905			−0.432453	+	0.901657i	$0.642352\pi$
$152$			40550.7i			1.75514i
$153$	−24.6859	+	4117.04i	−0.00105455	+	0.175874i
$154$	−10431.9			−0.439868
$155$	10090.3			0.419990
$156$	413.348	+	415.834i	0.0169850	+	0.0170872i
$157$		−	21898.4i		−	0.888410i	−0.895925	−	0.444205i	$-0.853486\pi$
$157$		−	21898.4i		−	0.888410i	0.895925	−	0.444205i	$-0.146514\pi$
$158$			7534.17i			0.301802i
$159$	15908.2	−	15813.1i	0.629255	−	0.625493i
$160$			274.158i			0.0107093i
$161$			16670.4i			0.643122i
$162$	26006.3	+	311.881i	0.990943	+	0.0118839i
$163$		−	2843.91i		−	0.107039i	−0.998567	−	0.0535194i	$-0.982956\pi$
$163$		−	2843.91i		−	0.107039i	0.998567	−	0.0535194i	$-0.0170439\pi$
$164$	−508.113			−0.0188918
$165$	6772.26	−	6731.77i	0.248751	−	0.247264i
$166$		−	6611.28i		−	0.239922i
$167$		−	53431.2i		−	1.91585i	−0.287018	−	0.957925i	$-0.592664\pi$
$167$		−	53431.2i		−	1.91585i	0.287018	−	0.957925i	$-0.407336\pi$
$168$	7603.20	+	7648.93i	0.269388	+	0.271008i
$169$	23235.2			0.813528
$170$			1507.81i			0.0521732i
$171$	−50876.1	−	305.055i	−1.73989	−	0.0104324i
$172$		−	262.250i		−	0.00886459i
$173$		−	9975.60i		−	0.333309i	−0.986015	−	0.166654i	$-0.946704\pi$
$173$		−	9975.60i		−	0.333309i	0.986015	−	0.166654i	$-0.0532964\pi$
$174$	449.482	+	30000.6i	0.0148461	+	0.990903i
$175$	10561.5			0.344865
$176$	−35634.3			−1.15038
$177$	20082.5	−	19962.5i	0.641021	−	0.637189i
$178$	−22088.2			−0.697141
$179$		−	4165.62i		−	0.130009i	−0.997885	−	0.0650046i	$-0.979294\pi$
$179$		−	4165.62i		−	0.130009i	0.997885	−	0.0650046i	$-0.0207062\pi$
$180$	−173.508	−	1.04036i	−0.00535520	−	3.21100e-5i
$181$	−47906.0			−1.46229			−0.731143	−	0.682224i	$-0.761013\pi$
$181$	−47906.0			−1.46229			−0.731143	+	0.682224i	$0.761013\pi$
$182$	16745.7			0.505545
$183$	28807.9	−	28635.7i	0.860219	−	0.855077i
$184$			57981.9i			1.71260i
$185$	−4119.45			−0.120364
$186$	34116.9	−	33912.9i	0.986151	−	0.980256i
$187$	7206.39			0.206079
$188$	−482.333			−0.0136468
$189$	−9653.79	+	9481.68i	−0.270255	+	0.265437i
$190$	−18632.7			−0.516140
$191$	5121.88			0.140398			0.0701992	−	0.997533i	$-0.477637\pi$
$191$	5121.88			0.140398			0.0701992	+	0.997533i	$0.477637\pi$
$192$	26436.7	+	26595.7i	0.717141	+	0.721454i
$193$		−	23714.6i		−	0.636651i	−0.947981	−	0.318326i	$-0.896879\pi$
$193$		−	23714.6i		−	0.636651i	0.947981	−	0.318326i	$-0.103121\pi$
$194$			57432.2i			1.52599i
$195$	−10871.1	+	10806.1i	−0.285893	+	0.284184i
$196$	588.664			0.0153234
$197$			1734.75i			0.0446998i	0.999750	+	0.0223499i	$0.00711478\pi$
$197$			1734.75i			0.0446998i	−0.999750	+	0.0223499i	$0.992885\pi$
$198$	272.955	−	45522.6i	0.00696243	−	1.16117i
$199$	20797.7			0.525180			0.262590	−	0.964907i	$-0.415423\pi$
$199$	20797.7			0.525180			0.262590	+	0.964907i	$0.415423\pi$
$200$	36734.4			0.918359
$201$	32564.7	+	32760.5i	0.806036	+	0.810883i
$202$	−8445.00			−0.206965
$203$	−11234.8	−	10837.9i	−0.272629	−	0.262998i
$204$	−92.3154	−	92.8706i	−0.00221827	−	0.00223161i
$205$		−	13283.5i		−	0.316086i
$206$	16801.7			0.395930
$207$	−72745.9	−	436.187i	−1.69773	−	0.0101796i
$208$	57201.5			1.32215
$209$			89052.6i			2.03870i
$210$	−3514.61	+	3493.60i	−0.0796965	+	0.0792200i
$211$		−	30443.4i		−	0.683799i	−0.939737	−	0.341899i	$-0.888930\pi$
$211$		−	30443.4i		−	0.683799i	0.939737	−	0.341899i	$-0.111070\pi$
$212$			713.411i			0.0158733i
$213$	−28800.9	+	28628.7i	−0.634814	+	0.631019i
$214$			7939.25i			0.173361i
$215$	6855.97			0.148317
$216$	−33577.2	+	32978.6i	−0.719676	+	0.706846i
$217$			25027.5i			0.531494i
$218$	54582.8			1.14853
$219$	−45900.1	+	45625.7i	−0.957030	+	0.951309i
$220$			303.706i			0.00627491i
$221$	−11568.0			−0.236849
$222$	−13928.6	+	13845.3i	−0.282619	+	0.280929i
$223$	46878.8			0.942685			0.471343	−	0.881950i	$-0.343770\pi$
$223$	46878.8			0.942685			0.471343	+	0.881950i	$0.343770\pi$
$224$	−680.012			−0.0135525
$225$	−276.346	+	46088.1i	−0.00545868	+	0.910382i
$226$	−52762.6			−1.03302
$227$		−	97468.1i		−	1.89152i	−0.324866	−	0.945760i	$-0.605319\pi$
$227$		−	97468.1i		−	1.89152i	0.324866	−	0.945760i	$-0.394681\pi$
$228$	1147.64	−	1140.78i	0.0220769	−	0.0219449i
$229$			49269.9i			0.939530i	0.882792	+	0.469765i	$0.155661\pi$
$229$			49269.9i			0.939530i	−0.882792	+	0.469765i	$0.844339\pi$
$230$	−26642.2			−0.503632
$231$	16697.3	+	16797.7i	0.312911	+	0.314793i
$232$	−39076.1	−	37695.7i	−0.725998	−	0.700351i
$233$		−	62695.1i		−	1.15484i	−0.816448	−	0.577420i	$-0.804060\pi$
$233$		−	62695.1i		−	1.15484i	0.816448	−	0.577420i	$-0.195940\pi$
$234$	−438.158	+	73074.5i	−0.00800200	+	1.33455i
$235$		−	12609.6i		−	0.228331i
$236$			900.613i			0.0161702i
$237$	12131.7	−	12059.2i	0.215986	−	0.214694i
$238$	−3739.91			−0.0660249
$239$			77087.8i			1.34955i	0.738022	+	0.674777i	$0.235760\pi$
$239$			77087.8i			1.34955i	−0.738022	+	0.674777i	$0.764240\pi$
$240$	−12005.5	+	11933.8i	−0.208430	+	0.207184i
$241$	68992.2			1.18786			0.593931	−	0.804516i	$-0.297575\pi$
$241$	68992.2			1.18786			0.593931	+	0.804516i	$0.297575\pi$
$242$	−21644.1			−0.369581
$243$	−41123.4	−	42375.1i	−0.696428	−	0.717626i
$244$			1291.91i			0.0216996i
$245$			15389.4i			0.256382i
$246$	−44645.4	−	44913.9i	−0.737745	−	0.742182i
$247$		−	142950.i		−	2.34311i
$248$			87049.2i			1.41534i
$249$	−10645.6	+	10582.0i	−0.171701	+	0.170675i
$250$			35419.5i			0.566711i
$251$	22173.6			0.351956			0.175978	−	0.984394i	$-0.443691\pi$
$251$	22173.6			0.351956			0.175978	+	0.984394i	$0.443691\pi$
$252$	2.58048	−	430.364i	4.06349e−5	−	0.00677696i
$253$			127333.i			1.98930i
$254$		−	44353.6i		−	0.687483i
$255$	2427.90	−	2413.39i	0.0373380	−	0.0371148i
$256$	−3516.30			−0.0536545
$257$		−	91364.9i		−	1.38329i	−0.722238	−	0.691645i	$-0.756886\pi$
$257$		−	91364.9i		−	1.38329i	0.722238	−	0.691645i	$-0.243114\pi$
$258$	23181.2	−	23042.6i	0.348255	−	0.346173i
$259$		−	10217.7i		−	0.152320i
$260$		−	487.520i		−	0.00721183i
$261$	47588.2	−	48742.6i	0.698583	−	0.715529i
$262$	27126.6			0.395179
$263$	−14052.8			−0.203166			−0.101583	−	0.994827i	$-0.532391\pi$
$263$	−14052.8			−0.203166			−0.101583	+	0.994827i	$0.532391\pi$
$264$	58075.4	+	58424.7i	0.833268	+	0.838279i
$265$	−18650.6			−0.265584
$266$		−	46215.8i		−	0.653172i
$267$	35354.3	+	35566.9i	0.495929	+	0.498912i
$268$	−1469.16			−0.0204550
$269$	132853.			1.83598			0.917990	−	0.396604i	$-0.129811\pi$
$269$	132853.			1.83598			0.917990	+	0.396604i	$0.129811\pi$
$270$	−15153.4	−	15428.4i	−0.207865	−	0.211638i
$271$		−	3176.64i		−	0.0432544i	−0.999766	−	0.0216272i	$-0.993115\pi$
$271$		−	3176.64i		−	0.0432544i	0.999766	−	0.0216272i	$-0.00688468\pi$
$272$	−12775.1			−0.172674
$273$	−26803.1	−	26964.3i	−0.359633	−	0.361796i
$274$	20446.7			0.272347
$275$	80671.7			1.06673
$276$	1640.98	−	1631.17i	0.0215419	−	0.0214131i
$277$	−66770.4			−0.870211			−0.435105	−	0.900380i	$-0.643289\pi$
$277$	−66770.4			−0.870211			−0.435105	+	0.900380i	$0.643289\pi$
$278$	−146374.			−1.89397
$279$	−109215.	−	654.855i	−1.40305	−	0.00841273i
$280$		−	8967.54i		−	0.114382i
$281$		−	84043.8i		−	1.06437i	−0.846628	−	0.532185i	$-0.821371\pi$
$281$		−	84043.8i		−	1.06437i	0.846628	−	0.532185i	$-0.178629\pi$
$282$	−42380.2	−	42635.1i	−0.532924	−	0.536129i
$283$	25654.4			0.320323			0.160162	−	0.987091i	$-0.448798\pi$
$283$	25654.4			0.320323			0.160162	+	0.987091i	$0.448798\pi$
$284$		−	1291.59i		−	0.0160136i
$285$	29823.4	+	30002.7i	0.367170	+	0.369378i
$286$	127908.			1.56375
$287$	32948.0			0.400005
$288$	17.7928	−	2967.43i	0.000214516	−	0.0357763i
$289$	−80937.5			−0.969067
$290$	17320.8	−	17955.1i	0.205955	−	0.213497i
$291$	92478.6	−	91925.7i	1.09208	−	1.08555i
$292$		−	2058.42i		−	0.0241417i
$293$	−53324.1			−0.621138			−0.310569	−	0.950551i	$-0.600520\pi$
$293$	−53324.1			−0.621138			−0.310569	+	0.950551i	$0.600520\pi$
$294$	51723.0	+	52034.1i	0.598397	+	0.601995i
$295$	−23544.6			−0.270550
$296$		−	35538.7i		−	0.405619i
$297$	−73738.4	+	72423.7i	−0.835951	+	0.821047i
$298$		−	70416.9i		−	0.792947i
$299$		−	204400.i		−	2.28632i
$300$	−1033.42	−	1039.64i	−0.0114825	−	0.0115515i
$301$			17005.3i			0.187695i
$302$	78174.0			0.857133
$303$	13517.0	+	13598.3i	0.147230	+	0.148115i
$304$		−	157868.i		−	1.70824i
$305$	−33774.1			−0.363065
$306$	97.8564	−	16320.2i	0.00104507	−	0.174294i
$307$			58822.0i			0.624113i	0.950064	+	0.312056i	$0.101018\pi$
$307$			58822.0i			0.624113i	−0.950064	+	0.312056i	$0.898982\pi$
$308$	−753.302			−0.00794086
$309$	−26892.7	−	27054.4i	−0.281655	−	0.283349i
$310$	−39998.3			−0.416216
$311$	−28239.0			−0.291964			−0.145982	−	0.989287i	$-0.546634\pi$
$311$	−28239.0			−0.291964			−0.145982	+	0.989287i	$0.546634\pi$
$312$	−93224.8	−	93785.5i	−0.957684	−	0.963444i
$313$	71547.7			0.730310			0.365155	−	0.930947i	$-0.381016\pi$
$313$	71547.7			0.730310			0.365155	+	0.930947i	$0.381016\pi$
$314$			86806.6i			0.880427i
$315$	11251.0	+	67.4611i	0.113388	+	0.000679880i
$316$			544.053i			0.00544837i
$317$	108594.			1.08066			0.540328	−	0.841454i	$-0.318300\pi$
$317$	108594.			1.08066			0.540328	+	0.841454i	$0.318300\pi$
$318$	−63061.0	+	62684.0i	−0.623600	+	0.619872i
$319$	−85814.4	−	82782.9i	−0.843294	−	0.813503i
$320$		−	31180.5i		−	0.304497i
$321$	12784.0	−	12707.5i	0.124067	−	0.123325i
$322$		−	66082.3i		−	0.637343i
$323$			31926.0i			0.306013i
$324$	1877.95	+	22.5213i	0.0178893	+	0.000214538i
$325$	−129497.			−1.22601
$326$			11273.4i			0.106077i
$327$	−87365.0	−	87890.4i	−0.817037	−	0.821951i
$328$	114598.			1.06519
$329$	31276.3			0.288951
$330$	−26845.6	+	26685.1i	−0.246516	+	0.245043i
$331$			51045.9i			0.465913i	0.972487	+	0.232956i	$0.0748399\pi$
$331$			51045.9i			0.465913i	−0.972487	+	0.232956i	$0.925160\pi$
$332$		−	477.409i		−	0.00433127i
$333$	44588.0	+	267.351i	0.402096	+	0.00241098i
$334$			211804.i			1.89864i
$335$		−	38408.1i		−	0.342242i
$336$	−29600.1	−	29778.2i	−0.262190	−	0.263766i
$337$			209472.i			1.84444i	0.386662	+	0.922221i	$0.373628\pi$
$337$			209472.i			1.84444i	−0.386662	+	0.922221i	$0.626372\pi$
$338$	−92105.6			−0.806218
$339$	84451.6	+	84959.5i	0.734867	+	0.739286i
$340$			108.881i			0.000941874i
$341$			191167.i			1.64401i
$342$	201676.	+	1209.26i	1.72426	+	0.0103387i
$343$	−82737.5			−0.703257
$344$			59146.9i			0.499821i
$345$	42643.3	+	42899.8i	0.358272	+	0.360427i
$346$			39543.8i			0.330314i
$347$			13540.2i			0.112452i	0.998418	+	0.0562259i	$0.0179067\pi$
$347$			13540.2i			0.112452i	−0.998418	+	0.0562259i	$0.982093\pi$
$348$	32.4576	+	2166.38i	0.000268015	+	0.0178886i
$349$	85154.7			0.699129			0.349565	−	0.936912i	$-0.386329\pi$
$349$	85154.7			0.699129			0.349565	+	0.936912i	$0.386329\pi$
$350$	−41866.4			−0.341766
$351$	118367.	−	116257.i	0.960767	−	0.943639i
$352$	−5194.13			−0.0419206
$353$			102903.i			0.825806i	0.910775	+	0.412903i	$0.135485\pi$
$353$			102903.i			0.825806i	−0.910775	+	0.412903i	$0.864515\pi$
$354$	−79608.4	+	79132.5i	−0.635261	+	0.631463i
$355$	33765.9			0.267930
$356$	−1595.02			−0.0125854
$357$	5986.09	+	6022.09i	0.0469685	+	0.0472510i
$358$			16512.8i			0.128841i
$359$	−194404.			−1.50840			−0.754201	−	0.656644i	$-0.771975\pi$
$359$	−194404.			−1.50840			−0.754201	+	0.656644i	$0.771975\pi$
$360$	39132.4	+	234.639i	0.301947	+	0.00181049i
$361$	−264203.			−2.02733
$362$	189902.			1.44915
$363$	34643.5	+	34851.8i	0.262911	+	0.264492i
$364$	1209.23			0.00912652
$365$	53812.9			0.403925
$366$	−114196.	+	113513.i	−0.852489	+	0.847393i
$367$		−	25078.0i		−	0.186192i	−0.995657	−	0.0930959i	$-0.970324\pi$
$367$		−	25078.0i		−	0.186192i	0.995657	−	0.0930959i	$-0.0296763\pi$
$368$		−	225730.i		−	1.66684i
$369$	−862.098	+	143778.i	−0.00633146	+	1.05594i
$370$	16329.7			0.119282
$371$		−	46260.4i		−	0.336094i
$372$	2463.62	−	2448.90i	0.0178028	−	0.0176964i
$373$	58824.9			0.422808			0.211404	−	0.977399i	$-0.432196\pi$
$373$	58824.9			0.422808			0.211404	+	0.977399i	$0.432196\pi$
$374$	−28566.5			−0.204227
$375$	57033.2	−	56692.2i	0.405569	−	0.403145i
$376$	108783.			0.769462
$377$	137752.	+	132886.i	0.969207	+	0.934968i
$378$	38268.2	−	37585.9i	0.267827	−	0.263052i
$379$			267968.i			1.86554i	0.360469	+	0.932771i	$0.382617\pi$
$379$			267968.i			1.86554i	−0.360469	+	0.932771i	$0.617383\pi$
$380$	−1345.49			−0.00931779
$381$	−71419.2	+	70992.2i	−0.492000	+	0.489059i
$382$	−20303.4			−0.139137
$383$			2302.04i			0.0156934i	0.999969	+	0.00784668i	$0.00249770\pi$
$383$			2302.04i			0.0156934i	−0.999969	+	0.00784668i	$0.997502\pi$
$384$	−101077.	−	101685.i	−0.685475	−	0.689597i
$385$		−	19693.5i		−	0.132862i
$386$			94006.1i			0.630930i
$387$	−74207.5	−	444.951i	−0.495480	−	0.00297092i
$388$			4147.25i			0.0275484i
$389$	218702.			1.44528			0.722642	−	0.691223i	$-0.242928\pi$
$389$	218702.			1.44528			0.722642	+	0.691223i	$0.242928\pi$
$390$	43093.6	−	42836.0i	0.283324	−	0.281630i
$391$			45649.8i			0.298597i
$392$	−132765.			−0.863995
$393$	−43418.8	−	43679.9i	−0.281121	−	0.282811i
$394$		−	6876.66i		−	0.0442981i
$395$	−14223.1			−0.0911590
$396$	19.7104	−	3287.25i	0.000125692	−	0.0209624i
$397$	80294.1			0.509452			0.254726	−	0.967013i	$-0.418015\pi$
$397$	80294.1			0.509452			0.254726	+	0.967013i	$0.418015\pi$
$398$	−82443.1			−0.520461
$399$	−74417.7	+	73972.9i	−0.467445	+	0.464651i
$400$	−143011.			−0.893819
$401$			218911.i			1.36138i	0.732571	+	0.680690i	$0.238320\pi$
$401$			218911.i			1.36138i	−0.732571	+	0.680690i	$0.761680\pi$
$402$	−129088.	−	129864.i	−0.798793	−	0.803597i
$403$		−	306869.i		−	1.88948i
$404$	−609.824			−0.00373630
$405$	−588.772	+	49095.0i	−0.00358952	+	0.299314i
$406$	44535.3	+	42962.0i	0.270180	+	0.260635i
$407$		−	78046.0i		−	0.471153i
$408$	20820.5	+	20945.7i	0.125075	+	0.125827i
$409$		−	72150.8i		−	0.431315i	−0.976469	−	0.215658i	$-0.930811\pi$
$409$		−	72150.8i		−	0.431315i	0.976469	−	0.215658i	$-0.0691895\pi$
$410$			52656.6i			0.313246i
$411$	−32726.9	−	32923.8i	−0.193741	−	0.194906i
$412$	1213.27			0.00714766
$413$		−	58399.2i		−	0.342379i
$414$	288369.	+	1729.07i	1.68247	+	0.0100882i
$415$	12480.8			0.0724683
$416$	8337.80			0.0481798
$417$	234285.	+	235694.i	1.34732	+	1.35543i
$418$		−	353010.i		−	2.02038i
$419$			235144.i			1.33939i	0.742638	+	0.669694i	$0.233575\pi$
$419$			235144.i			1.33939i	−0.742638	+	0.669694i	$0.766425\pi$
$420$	−253.795	+	252.278i	−0.00143875	+	0.00143015i
$421$			56537.7i			0.318988i	0.987199	+	0.159494i	$0.0509862\pi$
$421$			56537.7i			0.318988i	−0.987199	+	0.159494i	$0.949014\pi$
$422$			120679.i			0.677654i
$423$	−818.358	+	136483.i	−0.00457365	+	0.762778i
$424$		−	160900.i		−	0.895003i
$425$	28921.4			0.160118
$426$	114168.	−	113486.i	0.629110	−	0.625349i
$427$		−	83772.1i		−	0.459456i
$428$			573.303i			0.00312966i
$429$	−204729.	−	205961.i	−1.11241	−	1.11910i
$430$	−27177.4			−0.146985
$431$		−	237846.i		−	1.28039i	−0.768214	−	0.640193i	$-0.778855\pi$
$431$		−	237846.i		−	1.28039i	0.768214	−	0.640193i	$-0.221145\pi$
$432$	130720.	−	128389.i	0.700446	−	0.687958i
$433$		−	213192.i		−	1.13709i	−0.822652	−	0.568546i	$-0.807506\pi$
$433$		−	213192.i		−	1.13709i	0.822652	−	0.568546i	$-0.192494\pi$
$434$		−	99210.5i		−	0.526718i
$435$	−56635.4	+	848.536i	−0.299302	+	0.00448427i
$436$	3941.49			0.0207342
$437$	−564116.			−2.95396
$438$	181951.	−	180863.i	0.948430	−	0.942760i
$439$	−170822.			−0.886367			−0.443184	−	0.896431i	$-0.646151\pi$
$439$	−170822.			−0.886367			−0.443184	+	0.896431i	$0.646151\pi$
$440$		−	68496.6i		−	0.353805i
$441$	998.766	−	166571.i	0.00513555	−	0.856490i
$442$	45856.0			0.234721
$443$	110340.			0.562244			0.281122	−	0.959672i	$-0.409293\pi$
$443$	110340.			0.562244			0.281122	+	0.959672i	$0.409293\pi$
$444$	−1005.80	+	999.788i	−0.00510206	+	0.00507156i
$445$		−	41698.4i		−	0.210571i
$446$	−185830.			−0.934215
$447$	−113387.	+	112709.i	−0.567476	+	0.564083i
$448$	77339.2			0.385339
$449$	−204649.			−1.01512			−0.507559	−	0.861617i	$-0.669452\pi$
$449$	−204649.			−1.01512			−0.507559	+	0.861617i	$0.669452\pi$
$450$	1095.45	−	182696.i	0.00540963	−	0.902202i
$451$	251666.			1.23729
$452$	−3810.06			−0.0186490
$453$	−125125.	−	125877.i	−0.609744	−	0.613411i
$454$			386369.i			1.87452i
$455$			31612.7i			0.152700i
$456$	−258835.	+	257288.i	−1.24478	+	1.23734i
$457$	95123.8			0.455467			0.227733	−	0.973724i	$-0.426869\pi$
$457$	95123.8			0.455467			0.227733	+	0.973724i	$0.426869\pi$
$458$		−	195309.i		−	0.931088i
$459$	−26435.7	+	25964.4i	−0.125478	+	0.123241i
$460$	−1923.86			−0.00909199
$461$	39889.1			0.187695			0.0938475	−	0.995587i	$-0.470083\pi$
$461$	39889.1			0.187695			0.0938475	+	0.995587i	$0.470083\pi$
$462$	−66188.9	−	66587.0i	−0.310100	−	0.311965i
$463$	71426.4			0.333194			0.166597	−	0.986025i	$-0.446722\pi$
$463$	71426.4			0.333194			0.166597	+	0.986025i	$0.446722\pi$
$464$	152128.	+	146754.i	0.706599	+	0.681637i
$465$	64021.2	+	64406.2i	0.296086	+	0.297867i
$466$			248527.i			1.14446i
$467$	119947.			0.549993			0.274996	−	0.961445i	$-0.411323\pi$
$467$	119947.			0.549993			0.274996	+	0.961445i	$0.411323\pi$
$468$	−31.6399	+	5276.81i	−0.000144459	+	0.0240924i
$469$	95266.2			0.433105
$470$			49985.0i			0.226279i
$471$	139778.	−	138942.i	0.630081	−	0.626315i
$472$		−	203121.i		−	0.911738i
$473$			129891.i			0.580575i
$474$	−48090.7	+	47803.2i	−0.214045	+	0.212765i
$475$			357395.i			1.58402i
$476$	−270.064			−0.00119194
$477$	201870.	+	1210.42i	0.887228	+	0.00531985i
$478$		−	305581.i		−	1.33743i
$479$	219847.			0.958186			0.479093	−	0.877764i	$-0.340966\pi$
$479$	219847.			0.958186			0.479093	+	0.877764i	$0.340966\pi$
$480$	−1749.95	+	1739.49i	−0.00759528	+	0.00754988i
$481$			125282.i			0.541502i
$482$	−273489.			−1.17719
$483$	−106407.	+	105771.i	−0.456117	+	0.453391i
$484$	−1562.95			−0.00667197
$485$	−108421.			−0.460925
$486$	163015.	+	167977.i	0.690170	+	0.711178i
$487$	275259.			1.16060			0.580302	−	0.814401i	$-0.302935\pi$
$487$	275259.			1.16060			0.580302	+	0.814401i	$0.302935\pi$
$488$		−	291371.i		−	1.22351i
$489$	18152.7	−	18044.2i	0.0759144	−	0.0754606i
$490$		−	61004.3i		−	0.254079i
$491$	−366481.			−1.52016			−0.760079	−	0.649831i	$-0.774840\pi$
$491$	−366481.			−1.52016			−0.760079	+	0.649831i	$0.774840\pi$
$492$	−3223.90	−	3243.29i	−0.0133184	−	0.0133985i
$493$	−30765.1	−	29678.2i	−0.126580	−	0.122108i
$494$			566664.i			2.32205i
$495$	85938.0	+	515.287i	0.350731	+	0.00210300i
$496$		−	338893.i		−	1.37752i
$497$			83751.8i			0.339064i
$498$	42199.9	−	41947.6i	0.170158	−	0.169141i
$499$	280703.			1.12732			0.563658	−	0.826008i	$-0.309394\pi$
$499$	280703.			1.12732			0.563658	+	0.826008i	$0.309394\pi$
$500$			2557.69i			0.0102307i
$501$	341052.	−	339013.i	1.35877	−	1.35064i
$502$	−87897.4			−0.348794
$503$	−167460.			−0.661874			−0.330937	−	0.943653i	$-0.607365\pi$
$503$	−167460.			−0.661874			−0.330937	+	0.943653i	$0.607365\pi$
$504$	−581.991	+	97062.7i	−0.00229116	+	0.382112i
$505$		−	15942.6i		−	0.0625137i
$506$		−	504756.i		−	1.97142i
$507$	147424.	+	148310.i	0.573524	+	0.576973i
$508$		−	3202.83i		−	0.0124110i
$509$		−	88725.9i		−	0.342464i	−0.985231	−	0.171232i	$-0.945225\pi$
$509$		−	88725.9i		−	0.342464i	0.985231	−	0.171232i	$-0.0547747\pi$
$510$	−9624.35	+	9566.81i	−0.0370025	+	0.0367813i
$511$			133476.i			0.511164i
$512$	268829.			1.02550
$513$	−320854.	−	326678.i	−1.21920	−	1.24133i
$514$			362176.i			1.37086i
$515$			31718.4i			0.119590i
$516$	1673.95	−	1663.94i	0.00628698	−	0.00624939i
$517$	238897.			0.893780
$518$			40503.7i			0.150951i
$519$	63674.3	−	63293.7i	0.236390	−	0.234977i
$520$			109953.i			0.406632i
$521$			352376.i			1.29817i	0.760717	+	0.649084i	$0.224848\pi$
$521$			352376.i			1.29817i	−0.760717	+	0.649084i	$0.775152\pi$
$522$	−188642.	+	193218.i	−0.692305	+	0.709100i
$523$	−198158.			−0.724451			−0.362226	−	0.932090i	$-0.617983\pi$
$523$	−198158.			−0.724451			−0.362226	+	0.932090i	$0.617983\pi$
$524$	1958.85			0.00713409
$525$	67011.1	+	67414.2i	0.243124	+	0.244587i
$526$	55706.0			0.201340
$527$			68534.9i			0.246769i
$528$	−226094.	−	227454.i	−0.811002	−	0.815879i
$529$	−526767.			−1.88238
$530$	73932.1			0.263197
$531$	254842.	+	1528.04i	0.903819	+	0.00541933i
$532$		−	3337.31i		−	0.0117916i
$533$	−403984.			−1.42203
$534$	−140147.	−	140989.i	−0.491473	−	0.494429i
$535$	−14987.8			−0.0523637
$536$	331349.			1.15334
$537$	26589.2	−	26430.3i	0.0922056	−	0.0916544i
$538$	−526638.			−1.81948
$539$	−291563.			−1.00359
$540$	−1094.24	−	1114.11i	−0.00375255	−	0.00382067i
$541$		−	290944.i		−	0.994067i	−0.867731	−	0.497033i	$-0.834423\pi$
$541$		−	290944.i		−	0.994067i	0.867731	−	0.497033i	$-0.165577\pi$
$542$			12592.4i			0.0428657i
$543$	−303956.	−	305784.i	−1.03089	−	1.03709i
$544$	−1862.13			−0.00629234
$545$			103042.i			0.346913i
$546$	106249.	+	106888.i	0.356401	+	0.358545i
$547$	87466.4			0.292326			0.146163	−	0.989261i	$-0.453308\pi$
$547$	87466.4			0.292326			0.146163	+	0.989261i	$0.453308\pi$
$548$	1476.48			0.00491663
$549$	365564.	+	2191.93i	1.21288	+	0.00727248i
$550$	−319787.			−1.05715
$551$	366748.	−	380178.i	1.20799	−	1.25223i
$552$	−370099.	+	367887.i	−1.21462	+	1.20736i
$553$		−	35278.5i		−	0.115361i
$554$	264682.			0.862391
$555$	−26137.3	−	26294.5i	−0.0848545	−	0.0853648i
$556$	−10569.8			−0.0341915
$557$		−	578992.i		−	1.86622i	−0.359595	−	0.933109i	$-0.617085\pi$
$557$		−	578992.i		−	1.86622i	0.359595	−	0.933109i	$-0.382915\pi$
$558$	432933.	+	2595.88i	1.39044	+	0.00833714i
$559$		−	208506.i		−	0.667261i
$560$			34911.7i			0.111325i
$561$	45723.5	+	45998.4i	0.145283	+	0.146156i
$562$			333154.i			1.05481i
$563$	548574.			1.73069			0.865343	−	0.501180i	$-0.167100\pi$
$563$	548574.			1.73069			0.865343	+	0.501180i	$0.167100\pi$
$564$	−3060.33	−	3078.74i	−0.00962078	−	0.00967864i
$565$		−	99605.8i		−	0.312024i
$566$	−101695.			−0.317445
$567$	−121774.	−	1460.37i	−0.378780	−	0.00454252i
$568$			291301.i			0.902910i
$569$	267083.			0.824939			0.412470	−	0.910971i	$-0.364666\pi$
$569$	267083.			0.824939			0.412470	+	0.910971i	$0.364666\pi$
$570$	−118222.	−	118932.i	−0.363870	−	0.366059i
$571$	−222961.			−0.683844			−0.341922	−	0.939728i	$-0.611078\pi$
$571$	−222961.			−0.683844			−0.341922	+	0.939728i	$0.611078\pi$
$572$	9236.42			0.0282301
$573$	32497.5	+	32693.0i	0.0989786	+	0.0995739i
$574$	−130608.			−0.396410
$575$			511026.i			1.54564i
$576$	−2023.61	+	337491.i	−0.00609933	+	1.01723i
$577$			77553.5i			0.232943i	0.993194	+	0.116472i	$0.0371584\pi$
$577$			77553.5i			0.232943i	−0.993194	+	0.116472i	$0.962842\pi$
$578$	320841.			0.960359
$579$	151371.	−	150466.i	0.451528	−	0.448829i
$580$	1250.76	−	1296.56i	0.00371807	−	0.00385423i
$581$			30957.1i			0.0917081i
$582$	−366590.	+	364399.i	−1.08227	+	1.07580i
$583$		−	353350.i		−	1.03960i
$584$			464247.i			1.36120i
$585$	−137951.	−	827.158i	−0.403100	−	0.00241700i
$586$	211380.			0.615557
$587$		−	221620.i		−	0.643179i	−0.946879	−	0.321590i	$-0.895783\pi$
$587$		−	221620.i		−	0.643179i	0.946879	−	0.321590i	$-0.104217\pi$
$588$	3734.99	+	3757.45i	0.0108027	+	0.0108677i
$589$	−846917.			−2.44124
$590$	93332.1			0.268119
$591$	−11073.0	+	11006.8i	−0.0317021	+	0.0315126i
$592$			138356.i			0.394781i
$593$		−	210667.i		−	0.599084i	−0.954083	−	0.299542i	$-0.903166\pi$
$593$		−	210667.i		−	0.599084i	0.954083	−	0.299542i	$-0.0968339\pi$
$594$	292303.	−	287092.i	0.828439	−	0.813669i
$595$		−	7060.25i		−	0.0199428i
$596$		−	5084.89i		−	0.0143149i
$597$	131958.	+	132752.i	0.370243	+	0.372470i
$598$			810252.i			2.26578i
$599$	123316.			0.343690			0.171845	−	0.985124i	$-0.445027\pi$
$599$	123316.			0.343690			0.171845	+	0.985124i	$0.445027\pi$
$600$	233074.	+	234476.i	0.647428	+	0.651322i
$601$			283900.i			0.785989i	0.919541	+	0.392994i	$0.128561\pi$
$601$			283900.i			0.785989i	−0.919541	+	0.392994i	$0.871439\pi$
$602$		−	67410.0i		−	0.186008i
$603$	−2492.68	+	415721.i	−0.00685538	+	1.14332i
$604$	5645.05			0.0154737
$605$		−	40860.0i		−	0.111632i
$606$	−53582.3	−	53904.5i	−0.145907	−	0.146784i
$607$		−	323187.i		−	0.877156i	−0.898693	−	0.438578i	$-0.855482\pi$
$607$		−	323187.i		−	0.877156i	0.898693	−	0.438578i	$-0.144518\pi$
$608$		−	23011.2i		−	0.0622490i
$609$	−2104.68	−	140477.i	−0.00567481	−	0.378764i
$610$	133882.			0.359802
$611$	−383487.			−1.02723
$612$	7.06634	−	1178.50i	1.88665e−5	−	0.00314650i
$613$	714415.			1.90121			0.950604	−	0.310407i	$-0.100465\pi$
$613$	714415.			1.90121			0.950604	+	0.310407i	$0.100465\pi$
$614$		−	233174.i		−	0.618505i
$615$	84788.8	−	84281.9i	0.224176	−	0.222835i
$616$	169897.			0.447737
$617$	−80086.4			−0.210372			−0.105186	−	0.994453i	$-0.533544\pi$
$617$	−80086.4			−0.210372			−0.105186	+	0.994453i	$0.533544\pi$
$618$	106604.	+	107245.i	0.279124	+	0.280803i
$619$			419531.i			1.09492i	0.836831	+	0.547461i	$0.184406\pi$
$619$			419531.i			1.09492i	−0.836831	+	0.547461i	$0.815594\pi$
$620$	−2888.33			−0.00751387
$621$	−458778.	−	467105.i	−1.18965	−	1.21124i
$622$	111941.			0.289340
$623$	103427.			0.266476
$624$	362935.	+	365118.i	0.932094	+	0.937699i
$625$	288759.			0.739223
$626$	−283619.			−0.723748
$627$	−568424.	+	565026.i	−1.44590	+	1.43725i
$628$			6268.42i			0.0158942i
$629$		−	27980.1i		−	0.0707208i
$630$	−44599.4	−	267.420i	−0.112369	−	0.000673771i
$631$	−459198.			−1.15330			−0.576649	−	0.816992i	$-0.695640\pi$
$631$	−459198.			−1.15330			−0.576649	+	0.816992i	$0.695640\pi$
$632$		−	122703.i		−	0.307201i
$633$	194321.	−	193159.i	0.484966	−	0.482067i
$634$	−430473.			−1.07095
$635$	83731.2			0.207654
$636$	−4553.71	+	4526.49i	−0.0112577	+	0.0111904i
$637$	468027.			1.15343
$638$	340173.	+	328156.i	0.835716	+	0.806193i
$639$	−365475.	−	2191.40i	−0.895067	−	0.00536686i
$640$			119215.i			0.291052i
$641$	273881.			0.666571			0.333286	−	0.942826i	$-0.391843\pi$
$641$	273881.			0.666571			0.333286	+	0.942826i	$0.391843\pi$
$642$	−50676.3	+	50373.4i	−0.122952	+	0.122217i
$643$	350395.			0.847492			0.423746	−	0.905781i	$-0.360715\pi$
$643$	350395.			0.847492			0.423746	+	0.905781i	$0.360715\pi$
$644$		−	4771.89i		−	0.0115059i
$645$	43500.1	+	43761.7i	0.104561	+	0.105190i
$646$		−	126556.i		−	0.303263i
$647$			328068.i			0.783711i	0.920027	+	0.391855i	$0.128167\pi$
$647$			328068.i			0.783711i	−0.920027	+	0.391855i	$0.871833\pi$
$648$	−423545.	−	5079.37i	−1.00867	−	0.0120965i
$649$		−	446070.i		−	1.05904i
$650$	513334.			1.21499
$651$	−159751.	+	158796.i	−0.376948	+	0.374695i
$652$			814.069i			0.00191499i
$653$	−703894.			−1.65075			−0.825374	−	0.564587i	$-0.809036\pi$
$653$	−703894.			−1.65075			−0.825374	+	0.564587i	$0.809036\pi$
$654$	346320.	+	348402.i	0.809695	+	0.814565i
$655$			51209.9i			0.119364i
$656$	−446142.			−1.03673
$657$	−582459.	−	3492.45i	−1.34938	−	0.00809094i
$658$	−123981.			−0.286354
$659$	251723.			0.579631			0.289815	−	0.957083i	$-0.406406\pi$
$659$	251723.			0.579631			0.289815	+	0.957083i	$0.406406\pi$
$660$	−1938.56	+	1926.97i	−0.00445032	+	0.00442371i
$661$	−269075.			−0.615845			−0.307922	−	0.951412i	$-0.599634\pi$
$661$	−269075.			−0.615845			−0.307922	+	0.951412i	$0.599634\pi$
$662$		−	202349.i		−	0.461726i
$663$	−73397.0	−	73838.4i	−0.166975	−	0.167979i
$664$			107673.i			0.244214i
$665$	87246.7			0.197290
$666$	−176749.	−	1059.80i	−0.398483	−	0.00238932i
$667$	524399.	−	543603.i	1.17872	−	1.22188i
$668$			15294.7i			0.0342757i
$669$	297439.	+	299228.i	0.664578	+	0.668575i
$670$			152252.i			0.339167i
$671$		−	639875.i		−	1.42118i
$672$	−4314.58	−	4340.52i	−0.00955432	−	0.00961178i
$673$	486318.			1.07372			0.536859	−	0.843672i	$-0.319611\pi$
$673$	486318.			1.07372			0.536859	+	0.843672i	$0.319611\pi$
$674$		−	830357.i		−	1.82787i
$675$	−295934.	+	290658.i	−0.649513	+	0.637933i
$676$	−6651.06			−0.0145545
$677$	−184479.			−0.402503			−0.201251	−	0.979540i	$-0.564501\pi$
$677$	−184479.			−0.402503			−0.201251	+	0.979540i	$0.564501\pi$
$678$	−334771.	−	336784.i	−0.728264	−	0.732643i
$679$		−	268924.i		−	0.583297i
$680$		−	24556.5i		−	0.0531067i
$681$	622140.	−	618421.i	1.34151	−	1.33349i
$682$		−	757798.i		−	1.62924i
$683$		−	707110.i		−	1.51581i	−0.652363	−	0.757907i	$-0.726222\pi$
$683$		−	707110.i		−	1.51581i	0.652363	−	0.757907i	$-0.273778\pi$
$684$	14563.3	+	87.3220i	0.0311277	+	0.000186643i
$685$			38599.5i			0.0822623i
$686$	327976.			0.696938
$687$	−314490.	+	312610.i	−0.666337	+	0.662353i
$688$		−	230266.i		−	0.486465i
$689$			567210.i			1.19483i
$690$	−169041.	−	170057.i	−0.355053	−	0.357188i
$691$	−740845.			−1.55157			−0.775785	−	0.630998i	$-0.782646\pi$
$691$	−740845.			−1.55157			−0.775785	+	0.630998i	$0.782646\pi$
$692$			2855.51i			0.00596309i
$693$	−1278.10	+	213158.i	−0.00266133	+	0.443848i
$694$		−	53674.2i		−	0.111441i
$695$		−	276325.i		−	0.572073i
$696$	−7320.37	−	488597.i	−0.0151117	−	1.00863i
$697$	90224.1			0.185719
$698$	−337558.			−0.692847
$699$	400183.	−	397791.i	0.819039	−	0.814143i
$700$	−3023.23			−0.00616985
$701$			538341.i			1.09552i	0.836635	+	0.547761i	$0.184520\pi$
$701$			538341.i			1.09552i	−0.836635	+	0.547761i	$0.815480\pi$
$702$	−469216.	+	460850.i	−0.952134	+	0.935160i
$703$	345762.			0.699628
$704$	590738.			1.19193
$705$	80486.9	−	80005.8i	0.161937	−	0.160969i
$706$		−	407913.i		−	0.818385i
$707$	39543.4			0.0791106
$708$	−5748.62	+	5714.26i	−0.0114683	+	0.0113997i
$709$	218109.			0.433892			0.216946	−	0.976184i	$-0.430390\pi$
$709$	218109.			0.433892			0.216946	+	0.976184i	$0.430390\pi$
$710$	−133850.			−0.265523
$711$	153948.	+	923.076i	0.304533	+	0.00182599i
$712$	359734.			0.709613
$713$	−1.21098e6			−2.38208
$714$	−23729.2	−	23871.9i	−0.0465465	−	0.0468264i
$715$			241466.i			0.472329i
$716$			1192.41i			0.00232594i
$717$	−492053.	+	489111.i	−0.957135	+	0.951413i
$718$	770630.			1.49485
$719$		−	193372.i		−	0.374055i	−0.982355	−	0.187027i	$-0.940115\pi$
$719$		−	193372.i		−	0.374055i	0.982355	−	0.187027i	$-0.0598853\pi$
$720$	−152347.	−	913.478i	−0.293879	−	0.00176211i
$721$	−78673.2			−0.151341
$722$	1.04732e6			2.00911
$723$	437746.	+	440378.i	0.837424	+	0.842460i
$724$	13713.1			0.0261612
$725$	−344399.	−	332232.i	−0.655218	−	0.632071i
$726$	−137329.	−	138155.i	−0.260548	−	0.262115i
$727$			262302.i			0.496286i	0.968723	+	0.248143i	$0.0798204\pi$
$727$			262302.i			0.496286i	−0.968723	+	0.248143i	$0.920180\pi$
$728$	−272724.			−0.514590
$729$	9558.99	−	531355.i	0.0179869	−	0.999838i
$730$	−213317.			−0.400295
$731$			46567.0i			0.0871452i
$732$	−8246.25	+	8196.95i	−0.0153898	+	0.0152978i
$733$		−	870273.i		−	1.61975i	−0.586604	−	0.809874i	$-0.699536\pi$
$733$		−	870273.i		−	1.61975i	0.586604	−	0.809874i	$-0.300464\pi$
$734$			99410.6i			0.184519i
$735$	−98230.4	+	97643.2i	−0.181832	+	0.180745i
$736$		−	32902.9i		−	0.0607405i
$737$	727670.			1.33968
$738$	3417.40	−	569944.i	0.00627457	−	1.04645i
$739$		−	489851.i		−	0.896964i	−0.893792	−	0.448482i	$-0.851965\pi$
$739$		−	489851.i		−	0.896964i	0.893792	−	0.448482i	$-0.148035\pi$
$740$	1179.19			0.00215338
$741$	912455.	−	907000.i	1.66179	−	1.65185i
$742$			183379.i			0.333074i
$743$	−11606.1			−0.0210236			−0.0105118	−	0.999945i	$-0.503346\pi$
$743$	−11606.1			−0.0210236			−0.0105118	+	0.999945i	$0.503346\pi$
$744$	−555636.	+	552315.i	−1.00379	+	0.997793i
$745$	132934.			0.239509
$746$	−233185.			−0.419009
$747$	−135090.	−	810.005i	−0.242093	−	0.00145160i
$748$	−2062.83			−0.00368688
$749$		−	37175.2i		−	0.0662659i
$750$	−226083.	+	224731.i	−0.401925	+	0.399522i
$751$			350136.i			0.620807i	0.950605	+	0.310403i	$0.100464\pi$
$751$			350136.i			0.620807i	−0.950605	+	0.310403i	$0.899536\pi$
$752$	−423506.			−0.748901
$753$	140688.	+	141534.i	0.248123	+	0.249616i
$754$	−546059.	−	526768.i	−0.960498	−	0.926567i
$755$			147578.i			0.258897i
$756$	2763.39	−	2714.13i	0.00483503	−	0.00474883i
$757$			250011.i			0.436282i	0.975917	+	0.218141i	$0.0699993\pi$
$757$			250011.i			0.436282i	−0.975917	+	0.218141i	$0.930001\pi$
$758$		−	1.06224e6i		−	1.84878i
$759$	−812768.	+	807910.i	−1.41086	+	1.40242i
$760$	303456.			0.525374
$761$			477243.i			0.824081i	0.911165	+	0.412041i	$0.135184\pi$
$761$			477243.i			0.824081i	−0.911165	+	0.412041i	$0.864816\pi$
$762$	283110.	−	281417.i	0.487579	−	0.484664i
$763$	−255582.			−0.439016
$764$	−1466.14			−0.00251181
$765$	30809.4	+	184.734i	0.0526454	+	0.000315664i
$766$		−	9125.43i		−	0.0155523i
$767$			716048.i			1.21717i
$768$	−22310.4	−	22444.6i	−0.0378256	−	0.0380531i
$769$			79299.9i			0.134097i	0.997750	+	0.0670487i	$0.0213583\pi$
$769$			79299.9i			0.134097i	−0.997750	+	0.0670487i	$0.978642\pi$
$770$			78066.0i			0.131668i
$771$	583183.	−	579697.i	0.981062	−	0.975197i
$772$			6788.30i			0.0113901i
$773$	677150.			1.13325			0.566625	−	0.823976i	$-0.308249\pi$
$773$	677150.			1.13325			0.566625	+	0.823976i	$0.308249\pi$
$774$	294163.	+	1763.81i	0.491028	+	0.00294422i
$775$			767212.i			1.27736i
$776$		−	935355.i		−	1.55329i
$777$	65220.0	−	64830.1i	0.108029	−	0.107383i
$778$	−866946.			−1.43230
$779$			1.11494e6i			1.83729i
$780$	3111.84	−	3093.24i	0.00511480	−	0.00508422i
$781$			639720.i			1.04879i
$782$		−	180958.i		−	0.295914i
$783$	613064.	−	5508.74i	0.999960	−	0.00898522i
$784$	516869.			0.840908
$785$	−163874.			−0.265933
$786$	172115.	+	173150.i	0.278595	+	0.280270i
$787$	9089.46			0.0146754			0.00733768	−	0.999973i	$-0.497664\pi$
$787$	9089.46			0.0146754			0.00733768	+	0.999973i	$0.497664\pi$
$788$		−	496.573i		−	0.000799706i
$789$	−89162.8	−	89699.1i	−0.143229	−	0.144090i
$790$	56381.1			0.0903399
$791$	247059.			0.394864
$792$	−4445.42	+	741392.i	−0.00708699	+	1.18195i
$793$			1.02715e6i			1.63338i
$794$	−318291.			−0.504874
$795$	−118335.	−	119047.i	−0.187232	−	0.188358i
$796$	−5953.32			−0.00939579
$797$	−257197.			−0.404901			−0.202451	−	0.979292i	$-0.564891\pi$
$797$	−257197.			−0.404901			−0.202451	+	0.979292i	$0.564891\pi$
$798$	294996.	−	293233.i	0.463245	−	0.460476i
$799$	85646.4			0.134158
$800$	−20845.6			−0.0325712
$801$	−2706.22	+	451334.i	−0.00421791	+	0.703450i
$802$		−	867777.i		−	1.34915i
$803$			1.01953e6i			1.58113i
$804$	−9321.62	−	9377.68i	−0.0144205	−	0.0145072i
$805$	124751.			0.192509
$806$			1.21645e6i			1.87250i
$807$	842935.	+	848004.i	1.29434	+	1.30212i
$808$	137537.			0.210668
$809$	−364643.			−0.557148			−0.278574	−	0.960415i	$-0.589862\pi$
$809$	−364643.			−0.557148			−0.278574	+	0.960415i	$0.589862\pi$
$810$	2333.92	−	194615.i	0.00355727	−	0.296624i
$811$	−545176.			−0.828886			−0.414443	−	0.910075i	$-0.636024\pi$
$811$	−545176.			−0.828886			−0.414443	+	0.910075i	$0.636024\pi$
$812$	3215.95	+	3102.34i	0.00487750	+	0.00470520i
$813$	20276.6	−	20155.3i	0.0306770	−	0.0304936i
$814$			309379.i			0.466919i
$815$	−21282.1			−0.0320405
$816$	−81056.4	−	81543.9i	−0.121733	−	0.122465i
$817$	−575450.			−0.862111
$818$			286010.i			0.427439i
$819$	2051.66	−	342169.i	0.00305870	−	0.510120i
$820$			3802.40i			0.00565497i
$821$		−	143771.i		−	0.213296i	−0.994297	−	0.106648i	$-0.965988\pi$
$821$		−	143771.i		−	0.213296i	0.994297	−	0.106648i	$-0.0340119\pi$
$822$	129731.	+	130512.i	0.192000	+	0.193155i
$823$			1.14960e6i			1.69725i	0.528993	+	0.848626i	$0.322570\pi$
$823$			1.14960e6i			1.69725i	−0.528993	+	0.848626i	$0.677430\pi$
$824$	−273637.			−0.403014
$825$	511850.	+	514928.i	0.752030	+	0.756552i
$826$			231498.i			0.339302i
$827$	625204.			0.914135			0.457068	−	0.889432i	$-0.348900\pi$
$827$	625204.			0.914135			0.457068	+	0.889432i	$0.348900\pi$
$828$	20823.5	+	124.858i	0.0303734	+	0.000182120i
$829$		−	194809.i		−	0.283465i	−0.989905	−	0.141733i	$-0.954733\pi$
$829$		−	194809.i		−	0.283465i	0.989905	−	0.141733i	$-0.0452673\pi$
$830$	−49474.8			−0.0718171
$831$	−423648.	−	426196.i	−0.613484	−	0.617174i
$832$	−948275.			−1.36990
$833$	−104527.			−0.150640
$834$	−928719.	−	934304.i	−1.33522	−	1.34325i
$835$	−399846.			−0.573482
$836$		−	25491.3i		−	0.0364736i
$837$	−688771.	−	701274.i	−0.983160	−	1.00101i
$838$		−	932125.i		−	1.32735i
$839$	−905638.			−1.28656			−0.643281	−	0.765630i	$-0.722427\pi$
$839$	−905638.			−1.28656			−0.643281	+	0.765630i	$0.722427\pi$
$840$	57239.9	−	56897.7i	0.0811223	−	0.0806373i
$841$	25427.3	+	706824.i	0.0359508	+	0.999354i
$842$		−	224119.i		−	0.316121i
$843$	536452.	−	533245.i	0.754877	−	0.750364i
$844$			8714.41i			0.0122336i
$845$		−	173878.i		−	0.243518i
$846$	3244.02	−	541027.i	0.00453255	−	0.755924i
$847$	101348.			0.141269
$848$			626402.i			0.871087i
$849$	162773.	+	163752.i	0.225823	+	0.227181i
$850$	−114646.			−0.158680
$851$	494393.			0.682674
$852$	8244.25	−	8194.96i	0.0113572	−	0.0112893i
$853$		−	958778.i		−	1.31771i	−0.752270	−	0.658855i	$-0.771041\pi$
$853$		−	958778.i		−	1.31771i	0.752270	−	0.658855i	$-0.228959\pi$
$854$			332078.i			0.455327i
$855$	−2282.85	+	380726.i	−0.00312280	+	0.520811i
$856$		−	129301.i		−	0.176463i
$857$		−	417813.i		−	0.568879i	−0.958694	−	0.284440i	$-0.908192\pi$
$857$		−	417813.i		−	0.568879i	0.958694	−	0.284440i	$-0.0918075\pi$
$858$	811559.	+	816440.i	1.10242	+	1.10905i
$859$			31309.1i			0.0424311i	0.999775	+	0.0212155i	$0.00675362\pi$
$859$			31309.1i			0.0424311i	−0.999775	+	0.0212155i	$0.993246\pi$
$860$	−1962.52			−0.00265349
$861$	209050.	+	210307.i	0.281997	+	0.283693i
$862$			942834.i			1.26888i
$863$			630699.i			0.846838i	0.905934	+	0.423419i	$0.139170\pi$
$863$			630699.i			0.846838i	−0.905934	+	0.423419i	$0.860830\pi$
$864$	19054.0	−	18714.3i	0.0255246	−	0.0250695i
$865$	−74651.2			−0.0997711
$866$			845106.i			1.12687i
$867$	−513536.	−	516625.i	−0.683177	−	0.687285i
$868$		−	7164.12i		−	0.00950875i
$869$		−	269467.i		−	0.356834i
$870$	224506.	−	3363.64i	0.296612	−	0.00444397i
$871$	−1.16808e6			−1.53970
$872$	−888949.			−1.16908
$873$	1.17353e6	+	7036.51i	1.53980	+	0.00923270i
$874$	2.23619e6			2.92742
$875$		−	165850.i		−	0.216621i
$876$	13138.9	−	13060.3i	0.0171218	−	0.0170195i
$877$	726795.			0.944959			0.472479	−	0.881342i	$-0.343359\pi$
$877$	726795.			0.944959			0.472479	+	0.881342i	$0.343359\pi$
$878$	677147.			0.878403
$879$	−338333.	−	340368.i	−0.437892	−	0.440526i
$880$			266665.i			0.344351i
$881$	−948050.			−1.22146			−0.610730	−	0.791839i	$-0.709124\pi$
$881$	−948050.			−1.22146			−0.610730	+	0.791839i	$0.709124\pi$
$882$	−3959.17	+	660297.i	−0.00508940	+	0.848794i
$883$	−1.41990e6			−1.82111			−0.910556	−	0.413386i	$-0.864346\pi$
$883$	−1.41990e6			−1.82111			−0.910556	+	0.413386i	$0.864346\pi$
$884$	3311.32			0.00423738
$885$	−149387.	−	150285.i	−0.190733	−	0.191880i
$886$	−437394.			−0.557192
$887$	471163.			0.598859			0.299429	−	0.954118i	$-0.403204\pi$
$887$	471163.			0.598859			0.299429	+	0.954118i	$0.403204\pi$
$888$	226844.	−	225488.i	0.287675	−	0.285955i
$889$			207684.i			0.262785i
$890$			165295.i			0.208679i
$891$	−930141.	−	11154.7i	−1.17164	−	0.0140509i
$892$	−13419.0			−0.0168652
$893$			1.05837e6i			1.32720i
$894$	449472.	−	446785.i	0.562377	−	0.559015i
$895$	−31173.0			−0.0389164
$896$	−295697.			−0.368324
$897$	1.30469e6	−	1.29689e6i	1.62151	−	1.61182i
$898$	811239.			1.00600
$899$	787290.	−	816121.i	0.974126	−	1.00980i
$900$	79.1039	−	13192.7i	9.76592e−5	−	0.0162873i
$901$		−	126678.i		−	0.156046i
$902$	−997619.			−1.22617
$903$	−108545.	+	107896.i	−0.133117	+	0.132322i
$904$	859305.			1.05150
$905$			358499.i			0.437714i
$906$	496003.	+	498986.i	0.604265	+	0.607899i
$907$			1.02103e6i			1.24115i	0.784149	+	0.620573i	$0.213100\pi$
$907$			1.02103e6i			1.24115i	−0.784149	+	0.620573i	$0.786900\pi$
$908$			27900.2i			0.0338404i
$909$	−1034.67	+	172559.i	−0.00125220	+	0.208838i
$910$		−	125314.i		−	0.151328i
$911$	827126.			0.996632			0.498316	−	0.866995i	$-0.333952\pi$
$911$	827126.			0.996632			0.498316	+	0.866995i	$0.333952\pi$
$912$	1.00768e6	−	1.00165e6i	1.21152	−	1.20428i
$913$			236459.i			0.283671i
$914$	−377076.			−0.451374
$915$	−214292.	−	215580.i	−0.255955	−	0.257494i
$916$		−	14103.5i		−	0.0168088i
$917$	−127019.			−0.151054
$918$	104793.	−	102924.i	0.124350	−	0.122133i
$919$	−420286.			−0.497638			−0.248819	−	0.968550i	$-0.580043\pi$
$919$	−420286.			−0.497638			−0.248819	+	0.968550i	$0.580043\pi$
$920$	433901.			0.512643
$921$	−375462.	+	373217.i	−0.442635	+	0.439989i
$922$	−158123.			−0.186009
$923$		−	1.02690e6i		−	1.20538i
$924$	−4779.59	−	4808.33i	−0.00559818	−	0.00563184i
$925$		−	313222.i		−	0.366074i
$926$	−283138.			−0.330200
$927$	2058.52	−	343313.i	0.00239549	−	0.399513i
$928$	22174.5	+	21391.1i	0.0257488	+	0.0248392i
$929$		−	1.11924e6i		−	1.29685i	−0.761278	−	0.648425i	$-0.775428\pi$
$929$		−	1.11924e6i		−	1.29685i	0.761278	−	0.648425i	$-0.224572\pi$
$930$	−253784.	−	255310.i	−0.293425	−	0.295190i
$931$		−	1.29169e6i		−	1.49025i
$932$			17946.4i			0.0206608i
$933$	−179173.	−	180250.i	−0.205830	−	0.207068i
$934$	−475478.			−0.545051
$935$		−	53928.2i		−	0.0616868i
$936$	7135.94	−	1.19011e6i	0.00814516	−	1.35842i
$937$	−542618.			−0.618038			−0.309019	−	0.951056i	$-0.600001\pi$
$937$	−542618.			−0.618038			−0.309019	+	0.951056i	$0.600001\pi$
$938$	−377641.			−0.429213
$939$	453960.	+	456690.i	0.514857	+	0.517953i
$940$			3609.48i			0.00408497i
$941$		−	556077.i		−	0.627994i	−0.949424	−	0.313997i	$-0.898332\pi$
$941$		−	556077.i		−	0.627994i	0.949424	−	0.313997i	$-0.101668\pi$
$942$	−554088.	+	550775.i	−0.624420	+	0.620687i
$943$			1.59421e6i			1.79276i
$944$			790772.i			0.887375i
$945$	70955.0	+	72243.0i	0.0794547	+	0.0808969i
$946$		−	514897.i		−	0.575358i
$947$	70866.5			0.0790207			0.0395104	−	0.999219i	$-0.487420\pi$
$947$	70866.5			0.0790207			0.0395104	+	0.999219i	$0.487420\pi$
$948$	−3472.69	+	3451.93i	−0.00386411	+	0.00384101i
$949$		−	1.63658e6i		−	1.81721i
$950$		−	1.41673e6i		−	1.56979i
$951$	689014.	+	693157.i	0.761845	+	0.766427i
$952$	60909.2			0.0672061
$953$			23783.2i			0.0261869i	0.999914	+	0.0130934i	$0.00416789\pi$
$953$			23783.2i			0.0261869i	−0.999914	+	0.0130934i	$0.995832\pi$
$954$	−800225.	−	4798.18i	−0.879256	−	0.00527205i
$955$		−	38329.0i		−	0.0420262i
$956$		−	22066.4i		−	0.0241443i
$957$	−16076.1	−	1.07300e6i	−0.0175533	−	1.17159i
$958$	−871487.			−0.949576
$959$	−95741.0			−0.104102
$960$	199026.	−	197836.i	0.215957	−	0.214666i
$961$	−894537.			−0.968616
$962$		−	496626.i		−	0.536636i
$963$	162225.	+	972.705i	0.174930	+	0.00104889i
$964$	−19749.0			−0.0212516
$965$	−177466.			−0.190572
$966$	421804.	−	419283.i	0.452019	−	0.449317i
$967$		−	1.37267e6i		−	1.46796i	−0.679170	−	0.733981i	$-0.737660\pi$
$967$		−	1.37267e6i		−	1.46796i	0.679170	−	0.733981i	$-0.262340\pi$
$968$	352502.			0.376193
$969$	−203784.	+	202566.i	−0.217031	+	0.215734i
$970$	429787.			0.456783
$971$	−371550.			−0.394075			−0.197038	−	0.980396i	$-0.563132\pi$
$971$	−371550.			−0.394075			−0.197038	+	0.980396i	$0.563132\pi$
$972$	11771.6	+	12129.9i	0.0124595	+	0.0128388i
$973$	685388.			0.723954
$974$	−1.09114e6			−1.15018
$975$	−821641.	−	826582.i	−0.864316	−	0.869514i
$976$			1.13434e6i			1.19081i
$977$			1.14608e6i			1.20068i	0.799746	+	0.600338i	$0.204967\pi$
$977$			1.14608e6i			1.20068i	−0.799746	+	0.600338i	$0.795033\pi$
$978$	−71958.4	+	71528.3i	−0.0752322	+	0.0747825i
$979$	790006.			0.824262
$980$		−	4405.20i		−	0.00458684i
$981$	6687.40	−	1.11530e6i	0.00694895	−	1.15892i
$982$	1.45275e6			1.50650
$983$	1.30592e6			1.35148			0.675739	−	0.737141i	$-0.263825\pi$
$983$	1.30592e6			1.35148			0.675739	+	0.737141i	$0.263825\pi$
$984$	727106.	+	731478.i	0.750944	+	0.755460i
$985$	12981.8			0.0133802
$986$	121955.	+	117646.i	0.125442	+	0.121011i
$987$	198444.	+	199637.i	0.203706	+	0.204931i
$988$			40919.5i			0.0419196i
$989$	−822815.			−0.841220
$990$	−340663.	−	2042.63i	−0.347580	−	0.00208410i
$991$	890092.			0.906333			0.453166	−	0.891426i	$-0.350294\pi$
$991$	890092.			0.906333			0.453166	+	0.891426i	$0.350294\pi$
$992$		−	49397.7i		−	0.0501976i
$993$	−325826.	+	323879.i	−0.330436	+	0.328461i
$994$		−	331997.i		−	0.336017i
$995$		−	155637.i		−	0.157205i
$996$	3047.31	−	3029.09i	0.00307184	−	0.00305347i
$997$		−	1.17021e6i		−	1.17726i	−0.808401	−	0.588632i	$-0.799667\pi$
$997$		−	1.17021e6i		−	1.17726i	0.808401	−	0.588632i	$-0.200333\pi$
$998$	−1.11272e6			−1.11719
$999$	281198.	+	286302.i	0.281761	+	0.286876i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	87.5.d.c.86.10	yes	32
3.2	odd	2	inner	87.5.d.c.86.24	yes	32
29.28	even	2	inner	87.5.d.c.86.23	yes	32
87.86	odd	2	inner	87.5.d.c.86.9	✓	32

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
87.5.d.c.86.9	✓	32	87.86	odd	2	inner
87.5.d.c.86.10	yes	32	1.1	even	1	trivial
87.5.d.c.86.23	yes	32	29.28	even	2	inner
87.5.d.c.86.24	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-3.96406 q^{2} +(6.34485 + 6.38301i) q^{3} -0.286250 q^{4} -7.48339i q^{5} +(-25.1514 - 25.3026i) q^{6} +18.5615 q^{7} +64.5596 q^{8} +(-0.485670 + 80.9985i) q^{9} +29.6646i q^{10} +141.778 q^{11} +(-1.81621 - 1.82714i) q^{12} -227.588 q^{13} -73.5790 q^{14} +(47.7665 - 47.4810i) q^{15} -251.338 q^{16} +50.8285 q^{17} +(1.92522 - 321.083i) q^{18} +628.112i q^{19} +2.14212i q^{20} +(117.770 + 118.479i) q^{21} -562.017 q^{22} +898.114i q^{23} +(409.621 + 412.085i) q^{24} +568.999 q^{25} +902.171 q^{26} +(-520.096 + 510.824i) q^{27} -5.31324 q^{28} +(-605.272 - 583.889i) q^{29} +(-189.349 + 188.217i) q^{30} +1348.35i q^{31} -36.6355 q^{32} +(899.563 + 904.973i) q^{33} -201.487 q^{34} -138.903i q^{35} +(0.139023 - 23.1858i) q^{36} -550.479i q^{37} -2489.87i q^{38} +(-1444.01 - 1452.70i) q^{39} -483.125i q^{40} +1775.07 q^{41} +(-466.848 - 469.656i) q^{42} +916.159i q^{43} -40.5840 q^{44} +(606.143 + 3.63446i) q^{45} -3560.17i q^{46} +1685.01 q^{47} +(-1594.70 - 1604.29i) q^{48} -2056.47 q^{49} -2255.54 q^{50} +(322.500 + 324.439i) q^{51} +65.1469 q^{52} -2492.27i q^{53} +(2061.69 - 2024.94i) q^{54} -1060.98i q^{55} +1198.33 q^{56} +(-4009.24 + 3985.28i) q^{57} +(2399.33 + 2314.57i) q^{58} -3146.25i q^{59} +(-13.6732 + 13.5914i) q^{60} -4513.21i q^{61} -5344.95i q^{62} +(-9.01479 + 1503.46i) q^{63} +4166.63 q^{64} +1703.13i q^{65} +(-3565.92 - 3587.36i) q^{66} +5132.45 q^{67} -14.5497 q^{68} +(-5732.67 + 5698.40i) q^{69} +550.620i q^{70} +4512.12i q^{71} +(-31.3547 + 5229.24i) q^{72} +7190.98i q^{73} +2182.13i q^{74} +(3610.21 + 3631.93i) q^{75} -179.797i q^{76} +2631.62 q^{77} +(5724.14 + 5758.57i) q^{78} -1900.62i q^{79} +1880.86i q^{80} +(-6560.53 - 78.6772i) q^{81} -7036.47 q^{82} +1667.81i q^{83} +(-33.7117 - 33.9144i) q^{84} -380.370i q^{85} -3631.71i q^{86} +(-113.389 - 7568.15i) q^{87} +9153.16 q^{88} +5572.12 q^{89} +(-2402.79 - 14.4072i) q^{90} -4224.38 q^{91} -257.085i q^{92} +(-8606.56 + 8555.11i) q^{93} -6679.46 q^{94} +4700.40 q^{95} +(-232.447 - 233.845i) q^{96} -14488.2i q^{97} +8151.96 q^{98} +(-68.8575 + 11483.8i) 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

Related objects

Downloads

Learn more