LMFDB - Embedded newform 87.5.d.c.86.12

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			86.12
Character	$\chi$	$=$	87.86
Dual form			87.5.d.c.86.11

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q-3.29168 q^{2} +(-1.45168 + 8.88215i) q^{3} -5.16484 q^{4} -6.84103i q^{5} +(4.77846 - 29.2372i) q^{6} -58.5203 q^{7} +69.6679 q^{8} +(-76.7853 - 25.7881i) q^{9} +22.5185i q^{10} -62.4927 q^{11} +(7.49769 - 45.8749i) q^{12} +135.505 q^{13} +192.630 q^{14} +(60.7630 + 9.93097i) q^{15} -146.687 q^{16} +362.853 q^{17} +(252.753 + 84.8861i) q^{18} +169.000i q^{19} +35.3328i q^{20} +(84.9526 - 519.786i) q^{21} +205.706 q^{22} -889.439i q^{23} +(-101.135 + 618.801i) q^{24} +578.200 q^{25} -446.040 q^{26} +(340.521 - 644.582i) q^{27} +302.248 q^{28} +(375.007 - 752.762i) q^{29} +(-200.013 - 32.6896i) q^{30} +156.038i q^{31} -631.839 q^{32} +(90.7193 - 555.069i) q^{33} -1194.40 q^{34} +400.339i q^{35} +(396.583 + 133.191i) q^{36} +1496.22i q^{37} -556.293i q^{38} +(-196.710 + 1203.58i) q^{39} -476.600i q^{40} -90.8657 q^{41} +(-279.637 + 1710.97i) q^{42} -2749.96i q^{43} +322.764 q^{44} +(-176.417 + 525.290i) q^{45} +2927.75i q^{46} -2050.73 q^{47} +(212.943 - 1302.90i) q^{48} +1023.62 q^{49} -1903.25 q^{50} +(-526.746 + 3222.92i) q^{51} -699.863 q^{52} +2798.90i q^{53} +(-1120.89 + 2121.76i) q^{54} +427.514i q^{55} -4076.98 q^{56} +(-1501.08 - 245.333i) q^{57} +(-1234.40 + 2477.85i) q^{58} -5400.05i q^{59} +(-313.831 - 51.2919i) q^{60} -1215.04i q^{61} -513.626i q^{62} +(4493.49 + 1509.12i) q^{63} +4426.81 q^{64} -926.996i q^{65} +(-298.619 + 1827.11i) q^{66} -2421.80 q^{67} -1874.08 q^{68} +(7900.13 + 1291.18i) q^{69} -1317.79i q^{70} -1596.45i q^{71} +(-5349.47 - 1796.60i) q^{72} -7003.97i q^{73} -4925.09i q^{74} +(-839.361 + 5135.66i) q^{75} -872.857i q^{76} +3657.09 q^{77} +(647.507 - 3961.80i) q^{78} -4229.21i q^{79} +1003.49i q^{80} +(5230.95 + 3960.29i) q^{81} +299.101 q^{82} -10621.7i q^{83} +(-438.766 + 2684.61i) q^{84} -2482.29i q^{85} +9051.99i q^{86} +(6141.76 + 4423.64i) q^{87} -4353.73 q^{88} +5030.78 q^{89} +(580.708 - 1729.09i) q^{90} -7929.81 q^{91} +4593.81i q^{92} +(-1385.95 - 226.517i) q^{93} +6750.34 q^{94} +1156.13 q^{95} +(917.228 - 5612.09i) q^{96} +13293.3i q^{97} -3369.43 q^{98} +(4798.51 + 1611.56i) 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 $

$2$	−3.29168			−0.822920			−0.411460	−	0.911428i	$-0.634981\pi$
$2$	−3.29168			−0.822920			−0.411460	+	0.911428i	$0.634981\pi$
$3$	−1.45168	+	8.88215i	−0.161298	+	0.986906i
$3$	−1.45168	+	8.88215i	−0.161298	+	0.986906i
$4$	−5.16484			−0.322802
$5$		−	6.84103i		−	0.273641i	−0.990596	−	0.136821i	$-0.956312\pi$
$5$		−	6.84103i		−	0.273641i	0.990596	−	0.136821i	$-0.0436884\pi$
$6$	4.77846	−	29.2372i	0.132735	−	0.812145i
$7$	−58.5203			−1.19429			−0.597145	−	0.802133i	$-0.703699\pi$
$7$	−58.5203			−1.19429			−0.597145	+	0.802133i	$0.703699\pi$
$8$	69.6679			1.08856
$9$	−76.7853	−	25.7881i	−0.947966	−	0.318371i
$10$			22.5185i			0.225185i
$11$	−62.4927			−0.516468			−0.258234	−	0.966082i	$-0.583141\pi$
$11$	−62.4927			−0.516468			−0.258234	+	0.966082i	$0.583141\pi$
$12$	7.49769	−	45.8749i	0.0520673	−	0.318575i
$13$	135.505			0.801807			0.400903	−	0.916120i	$-0.368696\pi$
$13$	135.505			0.801807			0.400903	+	0.916120i	$0.368696\pi$
$14$	192.630			0.982806
$15$	60.7630	+	9.93097i	0.270058	+	0.0441377i
$16$	−146.687			−0.572996
$17$	362.853			1.25555			0.627773	−	0.778396i	$-0.283966\pi$
$17$	362.853			1.25555			0.627773	+	0.778396i	$0.283966\pi$
$18$	252.753	+	84.8861i	0.780100	+	0.261994i
$19$			169.000i			0.468144i	0.972219	+	0.234072i	$0.0752051\pi$
$19$			169.000i			0.468144i	−0.972219	+	0.234072i	$0.924795\pi$
$20$			35.3328i			0.0883320i
$21$	84.9526	−	519.786i	0.192636	−	1.17865i
$22$	205.706			0.425012
$23$		−	889.439i		−	1.68136i	−0.541533	−	0.840680i	$-0.682156\pi$
$23$		−	889.439i		−	1.68136i	0.541533	−	0.840680i	$-0.317844\pi$
$24$	−101.135	+	618.801i	−0.175582	+	1.07431i
$25$	578.200			0.925121
$26$	−446.040			−0.659823
$27$	340.521	−	644.582i	0.467107	−	0.884201i
$28$	302.248			0.385520
$29$	375.007	−	752.762i	0.445906	−	0.895080i
$29$	375.007	−	752.762i	0.445906	−	0.895080i
$30$	−200.013	−	32.6896i	−0.222236	−	0.0363218i
$31$			156.038i			0.162370i	0.996699	+	0.0811851i	$0.0258705\pi$
$31$			156.038i			0.162370i	−0.996699	+	0.0811851i	$0.974130\pi$
$32$	−631.839			−0.617031
$33$	90.7193	−	555.069i	0.0833051	−	0.509706i
$34$	−1194.40			−1.03321
$35$			400.339i			0.326807i
$36$	396.583	+	133.191i	0.306006	+	0.102771i
$37$			1496.22i			1.09293i	0.837481	+	0.546466i	$0.184027\pi$
$37$			1496.22i			1.09293i	−0.837481	+	0.546466i	$0.815973\pi$
$38$		−	556.293i		−	0.385245i
$39$	−196.710	+	1203.58i	−0.129330	+	0.791308i
$40$		−	476.600i		−	0.297875i
$41$	−90.8657			−0.0540545			−0.0270273	−	0.999635i	$-0.508604\pi$
$41$	−90.8657			−0.0540545			−0.0270273	+	0.999635i	$0.508604\pi$
$42$	−279.637	+	1710.97i	−0.158524	+	0.969937i
$43$		−	2749.96i		−	1.48727i	−0.668586	−	0.743635i	$-0.733100\pi$
$43$		−	2749.96i		−	1.48727i	0.668586	−	0.743635i	$-0.266900\pi$
$44$	322.764			0.166717
$45$	−176.417	+	525.290i	−0.0871194	+	0.259402i
$46$			2927.75i			1.38362i
$47$	−2050.73			−0.928352			−0.464176	−	0.885743i	$-0.653649\pi$
$47$	−2050.73			−0.928352			−0.464176	+	0.885743i	$0.653649\pi$
$48$	212.943	−	1302.90i	0.0924230	−	0.565493i
$49$	1023.62			0.426331
$50$	−1903.25			−0.761300
$51$	−526.746	+	3222.92i	−0.202517	+	1.23911i
$52$	−699.863			−0.258825
$53$			2798.90i			0.996404i	0.867061	+	0.498202i	$0.166006\pi$
$53$			2798.90i			0.996404i	−0.867061	+	0.498202i	$0.833994\pi$
$54$	−1120.89	+	2121.76i	−0.384392	+	0.727627i
$55$			427.514i			0.141327i
$56$	−4076.98			−1.30006
$57$	−1501.08	−	245.333i	−0.462014	−	0.0755105i
$58$	−1234.40	+	2477.85i	−0.366945	+	0.736579i
$59$		−	5400.05i		−	1.55129i	−0.631167	−	0.775647i	$-0.717424\pi$
$59$		−	5400.05i		−	1.55129i	0.631167	−	0.775647i	$-0.282576\pi$
$60$	−313.831	−	51.2919i	−0.0871753	−	0.0142477i
$61$		−	1215.04i		−	0.326536i	−0.986582	−	0.163268i	$-0.947796\pi$
$61$		−	1215.04i		−	0.326536i	0.986582	−	0.163268i	$-0.0522035\pi$
$62$		−	513.626i		−	0.133618i
$63$	4493.49	+	1509.12i	1.13215	+	0.380228i
$64$	4426.81			1.08076
$65$		−	926.996i		−	0.219407i
$66$	−298.619	+	1827.11i	−0.0685535	+	0.419447i
$67$	−2421.80			−0.539496			−0.269748	−	0.962931i	$-0.586940\pi$
$67$	−2421.80			−0.539496			−0.269748	+	0.962931i	$0.586940\pi$
$68$	−1874.08			−0.405293
$69$	7900.13	+	1291.18i	1.65934	+	0.271199i
$70$		−	1317.79i		−	0.268936i
$71$		−	1596.45i		−	0.316694i	−0.987384	−	0.158347i	$-0.949384\pi$
$71$		−	1596.45i		−	0.316694i	0.987384	−	0.158347i	$-0.0506164\pi$
$72$	−5349.47	−	1796.60i	−1.03192	−	0.346566i
$73$		−	7003.97i		−	1.31431i	−0.753754	−	0.657156i	$-0.771759\pi$
$73$		−	7003.97i		−	1.31431i	0.753754	−	0.657156i	$-0.228241\pi$
$74$		−	4925.09i		−	0.899396i
$75$	−839.361	+	5135.66i	−0.149220	+	0.913007i
$76$		−	872.857i		−	0.151118i
$77$	3657.09			0.616813
$78$	647.507	−	3961.80i	0.106428	−	0.651183i
$79$		−	4229.21i		−	0.677649i	−0.940850	−	0.338825i	$-0.889971\pi$
$79$		−	4229.21i		−	0.677649i	0.940850	−	0.338825i	$-0.110029\pi$
$80$			1003.49i			0.156795i
$81$	5230.95	+	3960.29i	0.797280	+	0.603610i
$82$	299.101			0.0444826
$83$		−	10621.7i		−	1.54183i	−0.636937	−	0.770916i	$-0.719799\pi$
$83$		−	10621.7i		−	1.54183i	0.636937	−	0.770916i	$-0.280201\pi$
$84$	−438.766	+	2684.61i	−0.0621835	+	0.380472i
$85$		−	2482.29i		−	0.343569i
$86$			9051.99i			1.22390i
$87$	6141.76	+	4423.64i	0.811436	+	0.584442i
$88$	−4353.73			−0.562207
$89$	5030.78			0.635119			0.317560	−	0.948238i	$-0.397137\pi$
$89$	5030.78			0.635119			0.317560	+	0.948238i	$0.397137\pi$
$90$	580.708	−	1729.09i	0.0716923	−	0.213468i
$91$	−7929.81			−0.957591
$92$			4593.81i			0.542747i
$93$	−1385.95	−	226.517i	−0.160244	−	0.0261899i
$94$	6750.34			0.763959
$95$	1156.13			0.128103
$96$	917.228	−	5612.09i	0.0995256	−	0.608951i
$97$			13293.3i			1.41283i	0.707797	+	0.706416i	$0.249689\pi$
$97$			13293.3i			1.41283i	−0.707797	+	0.706416i	$0.750311\pi$
$98$	−3369.43			−0.350836
$99$	4798.51	+	1611.56i	0.489594	+	0.164429i
$100$	−2986.31			−0.298631
$101$	19006.3			1.86318			0.931590	−	0.363510i	$-0.118422\pi$
$101$	19006.3			1.86318			0.931590	+	0.363510i	$0.118422\pi$
$102$	1733.88	−	10608.8i	0.166655	−	1.01969i
$103$	4398.72			0.414621			0.207311	−	0.978275i	$-0.433529\pi$
$103$	4398.72			0.414621			0.207311	+	0.978275i	$0.433529\pi$
$104$	9440.37			0.872816
$105$	−3555.87	−	581.163i	−0.322528	−	0.0527132i
$106$		−	9213.08i		−	0.819961i
$107$		−	5006.16i		−	0.437257i	−0.975808	−	0.218629i	$-0.929842\pi$
$107$		−	5006.16i		−	0.437257i	0.975808	−	0.218629i	$-0.0701583\pi$
$108$	−1758.74	+	3329.16i	−0.150783	+	0.285422i
$109$	−4613.46			−0.388306			−0.194153	−	0.980971i	$-0.562196\pi$
$109$	−4613.46			−0.388306			−0.194153	+	0.980971i	$0.562196\pi$
$110$		−	1407.24i		−	0.116301i
$111$	−13289.7	−	2172.04i	−1.07862	−	0.176287i
$112$	8584.16			0.684324
$113$	−5445.52			−0.426464			−0.213232	−	0.977002i	$-0.568399\pi$
$113$	−5445.52			−0.426464			−0.213232	+	0.977002i	$0.568399\pi$
$114$	4941.08	+	807.559i	0.380200	+	0.0621391i
$115$	−6084.68			−0.460089
$116$	−1936.85	+	3887.89i	−0.143940	+	0.288934i
$117$	−10404.8	−	3494.42i	−0.760086	−	0.255272i
$118$			17775.3i			1.27659i
$119$	−21234.3			−1.49949
$120$	4233.23	+	691.870i	0.293975	+	0.0480465i
$121$	−10735.7			−0.733261
$122$			3999.52i			0.268713i
$123$	131.908	−	807.083i	0.00871887	−	0.0533467i
$124$		−	805.910i		−	0.0524135i
$125$		−	8231.13i		−	0.526792i
$126$	−14791.1	−	4967.56i	−0.931667	−	0.312897i
$127$			1068.66i			0.0662571i	0.999451	+	0.0331286i	$0.0105471\pi$
$127$			1068.66i			0.0662571i	−0.999451	+	0.0331286i	$0.989453\pi$
$128$	−4462.20			−0.272351
$129$	24425.6	+	3992.06i	1.46779	+	0.239893i
$130$			3051.37i			0.180555i
$131$	3556.94			0.207269			0.103634	−	0.994615i	$-0.466953\pi$
$131$	3556.94			0.207269			0.103634	+	0.994615i	$0.466953\pi$
$132$	−468.550	+	2866.84i	−0.0268911	+	0.164534i
$133$		−	9889.91i		−	0.559100i
$134$	7971.78			0.443962
$135$	−4409.61	−	2329.51i	−0.241954	−	0.127820i
$136$	25279.2			1.36674
$137$	−6623.73			−0.352908			−0.176454	−	0.984309i	$-0.556463\pi$
$137$	−6623.73			−0.352908			−0.176454	+	0.984309i	$0.556463\pi$
$138$	−26004.7	−	4250.15i	−1.36551	−	0.223175i
$139$	10755.9			0.556693			0.278347	−	0.960481i	$-0.410214\pi$
$139$	10755.9			0.556693			0.278347	+	0.960481i	$0.410214\pi$
$140$		−	2067.68i		−	0.105494i
$141$	2977.00	−	18214.9i	0.149741	−	0.916196i
$142$			5255.02i			0.260614i
$143$	−8468.09			−0.414108
$144$	11263.4	+	3782.78i	0.543181	+	0.182426i
$145$	−5149.67	−	2565.43i	−0.244931	−	0.122018i
$146$			23054.8i			1.08157i
$147$	−1485.97	+	9091.95i	−0.0687662	+	0.420748i
$148$		−	7727.75i		−	0.352801i
$149$			31213.5i			1.40595i	0.711215	+	0.702974i	$0.248145\pi$
$149$			31213.5i			1.40595i	−0.711215	+	0.702974i	$0.751855\pi$
$150$	2762.91	−	16905.0i	0.122796	−	0.751332i
$151$	30770.5			1.34952			0.674761	−	0.738036i	$-0.264247\pi$
$151$	30770.5			1.34952			0.674761	+	0.738036i	$0.264247\pi$
$152$			11773.9i			0.509603i
$153$	−27861.8	−	9357.28i	−1.19022	−	0.399730i
$154$	−12038.0			−0.507588
$155$	1067.46			0.0444312
$156$	1015.98	−	6216.29i	0.0417479	−	0.255436i
$157$		−	12044.0i		−	0.488619i	−0.969697	−	0.244309i	$-0.921439\pi$
$157$		−	12044.0i		−	0.488619i	0.969697	−	0.244309i	$-0.0785612\pi$
$158$			13921.2i			0.557651i
$159$	−24860.2	−	4063.10i	−0.983357	−	0.160718i
$160$			4322.43i			0.168845i
$161$			52050.2i			2.00803i
$162$	−17218.6	−	13036.0i	−0.656097	−	0.496723i
$163$			25980.7i			0.977857i	0.872324	+	0.488928i	$0.162612\pi$
$163$			25980.7i			0.977857i	−0.872324	+	0.488928i	$0.837388\pi$
$164$	469.307			0.0174489
$165$	−3797.24	−	620.613i	−0.139476	−	0.0227957i
$166$			34963.2i			1.26880i
$167$			13532.2i			0.485215i	0.970125	+	0.242608i	$0.0780027\pi$
$167$			13532.2i			0.485215i	−0.970125	+	0.242608i	$0.921997\pi$
$168$	5918.47	−	36212.4i	0.209696	−	1.28304i
$169$	−10199.3			−0.357106
$170$			8170.90i			0.282730i
$171$	4358.18	−	12976.7i	0.149043	−	0.443784i
$172$			14203.1i			0.480094i
$173$		−	24080.1i		−	0.804575i	−0.915513	−	0.402288i	$-0.868215\pi$
$173$		−	24080.1i		−	0.804575i	0.915513	−	0.402288i	$-0.131785\pi$
$174$	−20216.7	−	14561.2i	−0.667747	−	0.480949i
$175$	−33836.4			−1.10486
$176$	9166.86			0.295934
$177$	47964.1	+	7839.14i	1.53098	+	0.250220i
$178$	−16559.7			−0.522653
$179$		−	34092.2i		−	1.06402i	−0.846739	−	0.532008i	$-0.821437\pi$
$179$		−	34092.2i		−	1.06402i	0.846739	−	0.532008i	$-0.178563\pi$
$180$	911.164	−	2713.04i	0.0281224	−	0.0837357i
$181$	53640.8			1.63734			0.818668	−	0.574267i	$-0.194713\pi$
$181$	53640.8			1.63734			0.818668	+	0.574267i	$0.194713\pi$
$182$	26102.4			0.788021
$183$	10792.2	+	1763.85i	0.322260	+	0.0526695i
$184$		−	61965.3i		−	1.83026i
$185$	10235.7			0.299071
$186$	4562.11	+	745.621i	0.131868	+	0.0215522i
$187$	−22675.6			−0.648450
$188$	10591.7			0.299674
$189$	−19927.4	+	37721.1i	−0.557862	+	1.05599i
$190$	−3805.62			−0.105419
$191$	−25164.7			−0.689802			−0.344901	−	0.938639i	$-0.612087\pi$
$191$	−25164.7			−0.689802			−0.344901	+	0.938639i	$0.612087\pi$
$192$	−6426.30	+	39319.6i	−0.174325	+	1.06661i
$193$			16800.3i			0.451028i	0.974240	+	0.225514i	$0.0724061\pi$
$193$			16800.3i			0.451028i	−0.974240	+	0.225514i	$0.927594\pi$
$194$		−	43757.4i		−	1.16265i
$195$	8233.72	+	1345.70i	0.216534	+	0.0353899i
$196$	−5286.83			−0.137621
$197$		−	38965.9i		−	1.00404i	−0.864855	−	0.502021i	$-0.832590\pi$
$197$		−	38965.9i		−	1.00404i	0.864855	−	0.502021i	$-0.167410\pi$
$198$	−15795.2	−	5304.76i	−0.402897	−	0.135312i
$199$	6910.50			0.174503			0.0872516	−	0.996186i	$-0.472192\pi$
$199$	6910.50			0.174503			0.0872516	+	0.996186i	$0.472192\pi$
$200$	40282.0			1.00705
$201$	3515.67	−	21510.8i	0.0870194	−	0.532432i
$202$	−62562.7			−1.53325
$203$	−21945.5	+	44051.8i	−0.532542	+	1.06899i
$204$	2720.56	−	16645.8i	0.0653729	−	0.399986i
$205$			621.615i			0.0147915i
$206$	−14479.2			−0.341200
$207$	−22936.9	+	68295.8i	−0.535296	+	1.59387i
$208$	−19876.9			−0.459432
$209$		−	10561.2i		−	0.241781i
$210$	11704.8	+	1913.00i	0.265415	+	0.0433788i
$211$		−	39202.4i		−	0.880538i	−0.897866	−	0.440269i	$-0.854883\pi$
$211$		−	39202.4i		−	0.880538i	0.897866	−	0.440269i	$-0.145117\pi$
$212$		−	14455.9i		−	0.321641i
$213$	14179.9	+	2317.54i	0.312547	+	0.0510820i
$214$			16478.7i			0.359828i
$215$	−18812.6			−0.406978
$216$	23723.4	−	44906.7i	0.508474	−	0.962506i
$217$		−	9131.37i		−	0.193917i
$218$	15186.0			0.319545
$219$	62210.3	+	10167.5i	1.29710	+	0.211995i
$220$		−	2208.04i		−	0.0456207i
$221$	49168.5			1.00671
$222$	43745.4	+	7149.65i	0.887619	+	0.145070i
$223$	67649.0			1.36035			0.680176	−	0.733049i	$-0.261903\pi$
$223$	67649.0			1.36035			0.680176	+	0.733049i	$0.261903\pi$
$224$	36975.4			0.736914
$225$	−44397.3	−	14910.7i	−0.876983	−	0.294532i
$226$	17924.9			0.350946
$227$		−	10600.2i		−	0.205714i	−0.994696	−	0.102857i	$-0.967202\pi$
$227$		−	10600.2i		−	0.205714i	0.994696	−	0.102857i	$-0.0327984\pi$
$228$	7752.84	+	1267.11i	0.149139	+	0.0243750i
$229$		−	84616.3i		−	1.61355i	−0.590857	−	0.806776i	$-0.701210\pi$
$229$		−	84616.3i		−	1.61355i	0.590857	−	0.806776i	$-0.298790\pi$
$230$	20028.8			0.378617
$231$	−5308.92	+	32482.8i	−0.0994905	+	0.608737i
$232$	26125.9	−	52443.3i	0.485396	−	0.974349i
$233$		−	81121.3i		−	1.49425i	−0.664684	−	0.747124i	$-0.731434\pi$
$233$		−	81121.3i		−	1.49425i	0.664684	−	0.747124i	$-0.268566\pi$
$234$	34249.3	+	11502.5i	0.625490	+	0.210069i
$235$			14029.1i			0.254035i
$236$			27890.4i			0.500761i
$237$	37564.5	+	6139.46i	0.668776	+	0.109303i
$238$	69896.4			1.23396
$239$			70001.3i			1.22549i	0.790280	+	0.612745i	$0.209935\pi$
$239$			70001.3i			1.22549i	−0.790280	+	0.612745i	$0.790065\pi$
$240$	−8913.15	−	1456.75i	−0.154742	−	0.0252907i
$241$	−60412.6			−1.04014			−0.520072	−	0.854123i	$-0.674095\pi$
$241$	−60412.6			−1.04014			−0.520072	+	0.854123i	$0.674095\pi$
$242$	35338.4			0.603415
$243$	−42769.5	+	40713.0i	−0.724306	+	0.689479i
$244$			6275.48i			0.105407i
$245$		−	7002.62i		−	0.116662i
$246$	−434.198	+	2656.66i	−0.00717494	+	0.0439001i
$247$			22900.4i			0.375361i
$248$			10870.8i			0.176750i
$249$	94343.4	+	15419.3i	1.52164	+	0.248694i
$250$			27094.2i			0.433508i
$251$	−81052.7			−1.28653			−0.643265	−	0.765643i	$-0.722421\pi$
$251$	−81052.7			−1.28653			−0.643265	+	0.765643i	$0.722421\pi$
$252$	−23208.2	−	7794.38i	−0.365460	−	0.122738i
$253$			55583.4i			0.868369i
$254$		−	3517.69i		−	0.0545243i
$255$	22048.1	+	3603.48i	0.339070	+	0.0554169i
$256$	−56140.7			−0.856640
$257$		−	83939.8i		−	1.27087i	−0.772154	−	0.635435i	$-0.780821\pi$
$257$		−	83939.8i		−	1.27087i	0.772154	−	0.635435i	$-0.219179\pi$
$258$	−80401.2	−	13140.6i	−1.20788	−	0.197413i
$259$		−	87559.4i		−	1.30528i
$260$			4787.78i			0.0708252i
$261$	−48207.3	+	48130.3i	−0.707672	+	0.706542i
$262$	−11708.3			−0.170566
$263$	−125244.			−1.81070			−0.905350	−	0.424667i	$-0.860391\pi$
$263$	−125244.			−1.81070			−0.905350	+	0.424667i	$0.860391\pi$
$264$	6320.22	−	38670.5i	0.0906827	−	0.554845i
$265$	19147.3			0.272657
$266$			32554.4i			0.460094i
$267$	−7303.08	+	44684.2i	−0.102443	+	0.626803i
$268$	12508.2			0.174151
$269$	14562.1			0.201242			0.100621	−	0.994925i	$-0.467917\pi$
$269$	14562.1			0.201242			0.100621	+	0.994925i	$0.467917\pi$
$270$	14515.0	+	7668.02i	0.199109	+	0.105185i
$271$			107677.i			1.46617i	0.680139	+	0.733083i	$0.261919\pi$
$271$			107677.i			1.46617i	−0.680139	+	0.733083i	$0.738081\pi$
$272$	−53225.8			−0.719424
$273$	11511.5	−	70433.8i	0.154457	−	0.945052i
$274$	21803.2			0.290415
$275$	−36133.3			−0.477795
$276$	−40802.9	−	6668.74i	−0.535640	−	0.0875438i
$277$	−129146.			−1.68315			−0.841575	−	0.540141i	$-0.818371\pi$
$277$	−129146.			−1.68315			−0.841575	+	0.540141i	$0.818371\pi$
$278$	−35404.9			−0.458114
$279$	4023.91	−	11981.4i	0.0516940	−	0.153921i
$280$			27890.7i			0.355749i
$281$		−	117136.i		−	1.48346i	−0.670697	−	0.741731i	$-0.734005\pi$
$281$		−	117136.i		−	1.48346i	0.670697	−	0.741731i	$-0.265995\pi$
$282$	−9799.33	+	59957.6i	−0.123225	+	0.753956i
$283$	28457.6			0.355324			0.177662	−	0.984092i	$-0.443147\pi$
$283$	28457.6			0.355324			0.177662	+	0.984092i	$0.443147\pi$
$284$			8245.42i			0.102230i
$285$	−1678.33	+	10268.9i	−0.0206628	+	0.126426i
$286$	27874.3			0.340778
$287$	5317.48			0.0645569
$288$	48515.9	+	16293.9i	0.584924	+	0.196445i
$289$	48141.3			0.576398
$290$	16951.1	+	8444.59i	0.201558	+	0.100411i
$291$	−118073.	−	19297.6i	−1.39433	−	0.227886i
$292$			36174.4i			0.424263i
$293$	−17402.0			−0.202705			−0.101352	−	0.994851i	$-0.532317\pi$
$293$	−17402.0			−0.202705			−0.101352	+	0.994851i	$0.532317\pi$
$294$	4891.33	−	29927.8i	0.0565891	−	0.346242i
$295$	−36941.9			−0.424498
$296$			104239.i			1.18972i
$297$	−21280.1	+	40281.7i	−0.241246	+	0.456662i
$298$		−	102745.i		−	1.15698i
$299$		−	120524.i		−	1.34813i
$300$	4335.16	−	26524.9i	0.0481685	−	0.294721i
$301$			160928.i			1.77623i
$302$	−101287.			−1.11055
$303$	−27591.1	+	168817.i	−0.300527	+	1.83878i
$304$		−	24790.1i		−	0.268245i
$305$	−8312.12			−0.0893536
$306$	91712.0	+	30801.2i	0.979453	+	0.328946i
$307$			106206.i			1.12687i	0.826160	+	0.563435i	$0.190520\pi$
$307$			106206.i			1.12687i	−0.826160	+	0.563435i	$0.809480\pi$
$308$	−18888.3			−0.199109
$309$	−6385.52	+	39070.1i	−0.0668774	+	0.409192i
$310$	−3513.73			−0.0365633
$311$	45599.8			0.471457			0.235729	−	0.971819i	$-0.424252\pi$
$311$	45599.8			0.471457			0.235729	+	0.971819i	$0.424252\pi$
$312$	−13704.4	+	83850.8i	−0.140783	+	0.861387i
$313$	−20128.0			−0.205452			−0.102726	−	0.994710i	$-0.532757\pi$
$313$	−20128.0			−0.205452			−0.102726	+	0.994710i	$0.532757\pi$
$314$			39644.9i			0.402094i
$315$	10324.0	−	30740.1i	0.104046	−	0.309802i
$316$			21843.2i			0.218747i
$317$	−95526.6			−0.950617			−0.475309	−	0.879819i	$-0.657664\pi$
$317$	−95526.6			−0.950617			−0.475309	+	0.879819i	$0.657664\pi$
$318$	81832.0	+	13374.4i	0.809224	+	0.132258i
$319$	−23435.2	+	47042.1i	−0.230296	+	0.462280i
$320$		−	30283.9i		−	0.295741i
$321$	44465.4	+	7267.33i	0.431532	+	0.0705285i
$322$		−	171333.i		−	1.65245i
$323$			61322.1i			0.587776i
$324$	−27017.0	−	20454.2i	−0.257364	−	0.194847i
$325$	78349.2			0.741768
$326$		−	85520.1i		−	0.804698i
$327$	6697.26	−	40977.5i	0.0626328	−	0.383221i
$328$	−6330.42			−0.0588417
$329$	120009.			1.10872
$330$	12499.3	+	2042.86i	0.114778	+	0.0187590i
$331$		−	175080.i		−	1.59802i	−0.601319	−	0.799009i	$-0.705358\pi$
$331$		−	175080.i		−	1.59802i	0.601319	−	0.799009i	$-0.294642\pi$
$332$			54859.2i			0.497707i
$333$	38584.7	−	114888.i	0.347958	−	1.03606i
$334$		−	44543.6i		−	0.399293i
$335$			16567.6i			0.147628i
$336$	−12461.5	+	76245.9i	−0.110380	+	0.675364i
$337$			46431.7i			0.408842i	0.978883	+	0.204421i	$0.0655311\pi$
$337$			46431.7i			0.408842i	−0.978883	+	0.204421i	$0.934469\pi$
$338$	33572.8			0.293870
$339$	7905.15	−	48368.0i	0.0687877	−	0.420880i
$340$			12820.6i			0.110905i
$341$		−	9751.21i		−	0.0838590i
$342$	−14345.7	+	42715.1i	−0.122651	+	0.365199i
$343$	80604.6			0.685128
$344$		−	191584.i		−	1.61898i
$345$	8833.00	−	54045.0i	0.0742113	−	0.454065i
$346$			79264.1i			0.662101i
$347$			118276.i			0.982285i	0.871079	+	0.491142i	$0.163421\pi$
$347$			118276.i			0.982285i	−0.871079	+	0.491142i	$0.836579\pi$
$348$	−31721.2	−	22847.4i	−0.261933	−	0.188659i
$349$	125769.			1.03258			0.516291	−	0.856413i	$-0.327312\pi$
$349$	125769.			1.03258			0.516291	+	0.856413i	$0.327312\pi$
$350$	111379.			0.909214
$351$	46142.4	−	87344.4i	0.374530	−	0.708958i
$352$	39485.3			0.318677
$353$		−	22537.2i		−	0.180863i	−0.995903	−	0.0904317i	$-0.971175\pi$
$353$		−	22537.2i		−	0.180863i	0.995903	−	0.0904317i	$-0.0288247\pi$
$354$	−157883.	−	25804.0i	−1.25988	−	0.205911i
$355$	−10921.4			−0.0866605
$356$	−25983.2			−0.205018
$357$	30825.3	−	188606.i	0.241864	−	1.47985i
$358$			112221.i			0.875601i
$359$	223306.			1.73265			0.866325	−	0.499480i	$-0.166476\pi$
$359$	223306.			1.73265			0.866325	+	0.499480i	$0.166476\pi$
$360$	−12290.6	+	36595.8i	−0.0948348	+	0.282375i
$361$	101760.			0.780842
$362$	−176568.			−1.34740
$363$	15584.7	−	95355.8i	0.118273	−	0.723659i
$364$	40956.2			0.309113
$365$	−47914.3			−0.359650
$366$	−35524.4	−	5806.03i	−0.265194	−	0.0433428i
$367$		−	3611.56i		−	0.0268141i	−0.999910	−	0.0134070i	$-0.995732\pi$
$367$		−	3611.56i		−	0.0268141i	0.999910	−	0.0134070i	$-0.00426772\pi$
$368$			130469.i			0.963413i
$369$	6977.15	+	2343.25i	0.0512419	+	0.0172094i
$370$	−33692.7			−0.246112
$371$		−	163792.i		−	1.19000i
$372$	7158.21	+	1169.92i	0.0517272	+	0.00845417i
$373$	−147889.			−1.06296			−0.531480	−	0.847071i	$-0.678364\pi$
$373$	−147889.			−1.06296			−0.531480	+	0.847071i	$0.678364\pi$
$374$	74641.0			0.533623
$375$	73110.1	+	11949.0i	0.519894	+	0.0849703i
$376$	−142870.			−1.01057
$377$	50815.5	−	102003.i	0.357531	−	0.717681i
$378$	65594.6	−	124166.i	0.459076	−	0.868998i
$379$			218265.i			1.51952i	0.650204	+	0.759760i	$0.274683\pi$
$379$			218265.i			1.51952i	−0.650204	+	0.759760i	$0.725317\pi$
$380$	−5971.24			−0.0413520
$381$	−9492.01	−	1551.35i	−0.0653895	−	0.0106871i
$382$	82834.0			0.567652
$383$			44333.7i			0.302229i	0.988516	+	0.151115i	$0.0482863\pi$
$383$			44333.7i			0.302229i	−0.988516	+	0.151115i	$0.951714\pi$
$384$	6477.69	−	39634.0i	0.0439296	−	0.268785i
$385$		−	25018.2i		−	0.168785i
$386$		−	55301.3i		−	0.371160i
$387$	−70916.2	+	211156.i	−0.473504	+	1.40988i
$388$		−	68657.9i		−	0.456065i
$389$	277875.			1.83633			0.918164	−	0.396201i	$-0.129672\pi$
$389$	277875.			1.83633			0.918164	+	0.396201i	$0.129672\pi$
$390$	−27102.8	−	4429.62i	−0.178190	−	0.0291231i
$391$		−	322736.i		−	2.11103i
$392$	71313.5			0.464087
$393$	−5163.54	+	31593.3i	−0.0334320	+	0.204555i
$394$			128263.i			0.826247i
$395$	−28932.1			−0.185433
$396$	−24783.5	−	8323.47i	−0.158042	−	0.0530779i
$397$	−234013.			−1.48477			−0.742385	−	0.669973i	$-0.766306\pi$
$397$	−234013.			−1.48477			−0.742385	+	0.669973i	$0.766306\pi$
$398$	−22747.2			−0.143602
$399$	87843.7	+	14357.0i	0.551779	+	0.0901815i
$400$	−84814.5			−0.530091
$401$			55664.1i			0.346168i	0.984907	+	0.173084i	$0.0553731\pi$
$401$			55664.1i			0.346168i	−0.984907	+	0.173084i	$0.944627\pi$
$402$	−11572.5	+	70806.6i	−0.0716101	+	0.438149i
$403$			21144.0i			0.130190i
$404$	−98164.5			−0.601439
$405$	27092.4	−	35785.1i	0.165173	−	0.218168i
$406$	72237.6	−	145005.i	0.438239	−	0.879690i
$407$		−	93503.0i		−	0.564465i
$408$	−36697.3	+	224534.i	−0.220452	+	1.34884i
$409$		−	286073.i		−	1.71013i	−0.518519	−	0.855066i	$-0.673516\pi$
$409$		−	286073.i		−	1.71013i	0.518519	−	0.855066i	$-0.326484\pi$
$410$		−	2046.16i		−	0.0121723i
$411$	9615.53	−	58833.0i	0.0569232	−	0.348287i
$412$	−22718.7			−0.133841
$413$			316013.i			1.85270i
$414$	75501.0	−	224808.i	0.440506	−	1.31163i
$415$	−72663.2			−0.421908
$416$	−85617.6			−0.494739
$417$	−15614.1	+	95535.3i	−0.0897933	+	0.549404i
$418$			34764.3i			0.198967i
$419$			11486.2i			0.0654258i	0.999465	+	0.0327129i	$0.0104147\pi$
$419$			11486.2i			0.0654258i	−0.999465	+	0.0327129i	$0.989585\pi$
$420$	18365.5	+	3001.61i	0.104113	+	0.0170159i
$421$		−	6592.84i		−	0.0371971i	−0.999827	−	0.0185985i	$-0.994080\pi$
$421$		−	6592.84i		−	0.0371971i	0.999827	−	0.0185985i	$-0.00592044\pi$
$422$			129042.i			0.724613i
$423$	157466.	+	52884.3i	0.880046	+	0.295560i
$424$			194993.i			1.08465i
$425$	209802.			1.16153
$426$	−46675.9	−	7628.60i	−0.257201	−	0.0420364i
$427$			71104.5i			0.389979i
$428$			25856.0i			0.141148i
$429$	12292.9	−	75214.9i	0.0667946	−	0.408685i
$430$	61924.9			0.334910
$431$			106206.i			0.571735i	0.958269	+	0.285867i	$0.0922817\pi$
$431$			106206.i			0.571735i	−0.958269	+	0.285867i	$0.907718\pi$
$432$	−49950.0	+	94551.9i	−0.267651	+	0.506644i
$433$		−	47436.0i		−	0.253007i	−0.991966	−	0.126504i	$-0.959624\pi$
$433$		−	47436.0i		−	0.253007i	0.991966	−	0.126504i	$-0.0403755\pi$
$434$			30057.6i			0.159578i
$435$	30262.2	−	42015.9i	0.159927	−	0.222042i
$436$	23827.8			0.125346
$437$	150315.			0.787118
$438$	−204777.	−	33468.2i	−1.06741	−	0.174455i
$439$	7137.50			0.0370354			0.0185177	−	0.999829i	$-0.494105\pi$
$439$	7137.50			0.0370354			0.0185177	+	0.999829i	$0.494105\pi$
$440$			29784.0i			0.153843i
$441$	−78599.0	−	26397.2i	−0.404147	−	0.135731i
$442$	−161847.			−0.828439
$443$	−197515.			−1.00645			−0.503225	−	0.864155i	$-0.667853\pi$
$443$	−197515.			−1.00645			−0.503225	+	0.864155i	$0.667853\pi$
$444$	68639.1	+	11218.2i	0.348181	+	0.0569060i
$445$		−	34415.7i		−	0.173795i
$446$	−222679.			−1.11946
$447$	−277243.	−	45311.9i	−1.38754	−	0.226776i
$448$	−259058.			−1.29075
$449$	248523.			1.23275			0.616373	−	0.787454i	$-0.288601\pi$
$449$	248523.			1.23275			0.616373	+	0.787454i	$0.288601\pi$
$450$	146142.	+	49081.2i	0.721687	+	0.242376i
$451$	5678.44			0.0279175
$452$	28125.2			0.137664
$453$	−44668.8	+	273308.i	−0.217675	+	1.33185i
$454$			34892.6i			0.169286i
$455$			54248.0i			0.262036i
$456$	−104577.	−	17091.9i	−0.502930	−	0.0821977i
$457$	270855.			1.29689			0.648446	−	0.761261i	$-0.275419\pi$
$457$	270855.			1.29689			0.648446	+	0.761261i	$0.275419\pi$
$458$			278530.i			1.32783i
$459$	123559.	−	233889.i	0.586475	−	1.11016i
$460$	31426.4			0.148518
$461$	−263209.			−1.23851			−0.619254	−	0.785191i	$-0.712565\pi$
$461$	−263209.			−1.23851			−0.619254	+	0.785191i	$0.712565\pi$
$462$	17475.3	−	106923.i	0.0818728	−	0.500942i
$463$	214906.			1.00251			0.501253	−	0.865301i	$-0.332873\pi$
$463$	214906.			1.00251			0.501253	+	0.865301i	$0.332873\pi$
$464$	−55008.7	+	110420.i	−0.255503	+	0.512877i
$465$	−1549.61	+	9481.33i	−0.00716664	+	0.0438494i
$466$			267025.i			1.22965i
$467$	−344789.			−1.58095			−0.790477	−	0.612492i	$-0.790167\pi$
$467$	−344789.			−1.58095			−0.790477	+	0.612492i	$0.790167\pi$
$468$	53739.2	+	18048.1i	0.245357	+	0.0824025i
$469$	141724.			0.644315
$470$		−	46179.3i		−	0.209051i
$471$	106976.	+	17484.0i	0.482221	+	0.0788130i
$472$		−	376210.i		−	1.68868i
$473$			171852.i			0.768127i
$474$	−123650.	−	20209.1i	−0.550349	−	0.0899479i
$475$			97715.8i			0.433089i
$476$	109671.			0.484038
$477$	72178.2	−	214914.i	0.317226	−	0.944557i
$478$		−	230422.i		−	1.00848i
$479$	−212339.			−0.925464			−0.462732	−	0.886498i	$-0.653131\pi$
$479$	−212339.			−0.925464			−0.462732	+	0.886498i	$0.653131\pi$
$480$	−38392.5	−	6274.78i	−0.166634	−	0.0272343i
$481$			202746.i			0.876320i
$482$	198859.			0.855955
$483$	−462318.	−	75560.2i	−1.98174	−	0.323891i
$484$	55448.0			0.236698
$485$	90940.0			0.386609
$486$	140784.	−	134014.i	0.596046	−	0.567386i
$487$	−168100.			−0.708776			−0.354388	−	0.935099i	$-0.615311\pi$
$487$	−168100.			−0.708776			−0.354388	+	0.935099i	$0.615311\pi$
$488$		−	84649.3i		−	0.355454i
$489$	−230764.	−	37715.6i	−0.965052	−	0.157726i
$490$			23050.4i			0.0960032i
$491$	16636.2			0.0690068			0.0345034	−	0.999405i	$-0.489015\pi$
$491$	16636.2			0.0690068			0.0345034	+	0.999405i	$0.489015\pi$
$492$	−681.282	+	4168.45i	−0.00281447	+	0.0172205i
$493$	136072.	−	273142.i	0.559856	−	1.12381i
$494$		−	75380.7i		−	0.308892i
$495$	11024.8	−	32826.8i	0.0449944	−	0.133973i
$496$		−	22888.7i		−	0.0930375i
$497$			93424.9i			0.378225i
$498$	−310548.	−	50755.3i	−1.25219	−	0.204655i
$499$	−284293.			−1.14174			−0.570868	−	0.821042i	$-0.693393\pi$
$499$	−284293.			−1.14174			−0.570868	+	0.821042i	$0.693393\pi$
$500$			42512.4i			0.170050i
$501$	−120195.	−	19644.4i	−0.478862	−	0.0782641i
$502$	266800.			1.05871
$503$	−133654.			−0.528257			−0.264128	−	0.964488i	$-0.585084\pi$
$503$	−133654.			−0.528257			−0.264128	+	0.964488i	$0.585084\pi$
$504$	313052.	+	105138.i	1.23241	+	0.413901i
$505$		−	130023.i		−	0.509843i
$506$		−	182963.i		−	0.714598i
$507$	14806.1	−	90591.7i	0.0576003	−	0.352430i
$508$		−	5519.46i		−	0.0213879i
$509$			61972.9i			0.239203i	0.992822	+	0.119601i	$0.0381617\pi$
$509$			61972.9i			0.239203i	−0.992822	+	0.119601i	$0.961838\pi$
$510$	−72575.2	−	11861.5i	−0.279028	−	0.0456037i
$511$			409874.i			1.56967i
$512$	256193.			0.977298
$513$	108934.	+	57548.0i	0.413933	+	0.218673i
$514$			276303.i			1.04583i
$515$		−	30091.7i		−	0.113457i
$516$	−126154.	−	20618.3i	−0.473807	−	0.0774380i
$517$	128155.			0.479464
$518$			288218.i			1.07414i
$519$	213883.	+	34956.6i	0.794040	+	0.129776i
$520$		−	64581.8i		−	0.238838i
$521$		−	420119.i		−	1.54773i	−0.633348	−	0.773867i	$-0.718320\pi$
$521$		−	420119.i		−	1.54773i	0.633348	−	0.773867i	$-0.281680\pi$
$522$	158683.	−	158430.i	0.582357	−	0.581427i
$523$	−17899.9			−0.0654405			−0.0327203	−	0.999465i	$-0.510417\pi$
$523$	−17899.9			−0.0654405			−0.0327203	+	0.999465i	$0.510417\pi$
$524$	−18371.0			−0.0669069
$525$	49119.6	−	300540.i	0.178212	−	1.09040i
$526$	412264.			1.49006
$527$			56618.8i			0.203863i
$528$	−13307.3	+	81421.5i	−0.0477335	+	0.292059i
$529$	−511261.			−1.82697
$530$	−63026.9			−0.224375
$531$	−139257.	+	414645.i	−0.493887	+	1.47057i
$532$			51079.8i			0.180479i
$533$	−12312.8			−0.0433413
$534$	24039.4	−	147086.i	0.0843026	−	0.515809i
$535$	−34247.3			−0.119652
$536$	−168722.			−0.587274
$537$	302812.	+	49490.9i	1.05008	+	0.171623i
$538$	−47933.8			−0.165606
$539$	−63968.8			−0.220186
$540$	22774.9	+	12031.6i	0.0781032	+	0.0412605i
$541$			74674.6i			0.255140i	0.991830	+	0.127570i	$0.0407177\pi$
$541$			74674.6i			0.255140i	−0.991830	+	0.127570i	$0.959282\pi$
$542$		−	354437.i		−	1.20654i
$543$	−77869.2	+	476446.i	−0.264099	+	1.61590i
$544$	−229265.			−0.774711
$545$			31560.8i			0.106256i
$546$	−37892.3	+	231845.i	−0.127106	+	0.777702i
$547$	312881.			1.04569			0.522847	−	0.852426i	$-0.324870\pi$
$547$	312881.			1.04569			0.522847	+	0.852426i	$0.324870\pi$
$548$	34210.5			0.113919
$549$	−31333.5	+	93297.2i	−0.103960	+	0.309545i
$550$	118939.			0.393187
$551$	127217.	+	63376.1i	0.419026	+	0.208748i
$552$	550386.	+	89953.8i	1.80630	+	0.295217i
$553$			247494.i			0.809311i
$554$	425109.			1.38510
$555$	−14859.0	+	90915.1i	−0.0482395	+	0.295155i
$556$	−55552.3			−0.179702
$557$			179993.i			0.580158i	0.957003	+	0.290079i	$0.0936816\pi$
$557$			179993.i			0.580158i	−0.957003	+	0.290079i	$0.906318\pi$
$558$	−13245.4	+	39438.9i	−0.0425400	+	0.126665i
$559$		−	372634.i		−	1.19250i
$560$		−	58724.5i		−	0.187259i
$561$	32917.8	−	201409.i	0.104593	−	0.639959i
$562$			385573.i			1.22077i
$563$	237023.			0.747779			0.373889	−	0.927473i	$-0.378024\pi$
$563$	237023.			0.747779			0.373889	+	0.927473i	$0.378024\pi$
$564$	−15375.7	+	94076.9i	−0.0483367	+	0.295750i
$565$			37253.0i			0.116698i
$566$	−93673.2			−0.292404
$567$	−306117.	−	231757.i	−0.952184	−	0.720886i
$568$		−	111222.i		−	0.344741i
$569$	5731.38			0.0177025			0.00885124	−	0.999961i	$-0.497183\pi$
$569$	5731.38			0.0177025			0.00885124	+	0.999961i	$0.497183\pi$
$570$	5524.54	−	33802.1i	0.0170038	−	0.104038i
$571$	−545446.			−1.67294			−0.836469	−	0.548015i	$-0.815384\pi$
$571$	−545446.			−1.67294			−0.836469	+	0.548015i	$0.815384\pi$
$572$	43736.3			0.133675
$573$	36531.0	−	223516.i	0.111263	−	0.680769i
$574$	−17503.5			−0.0531251
$575$		−	514274.i		−	1.55546i
$576$	−339913.	−	114159.i	−1.02453	−	0.344084i
$577$		−	494416.i		−	1.48505i	−0.669819	−	0.742525i	$-0.733628\pi$
$577$		−	494416.i		−	1.48505i	0.669819	−	0.742525i	$-0.266372\pi$
$578$	−158466.			−0.474329
$579$	−149223.	−	24388.7i	−0.445122	−	0.0727497i
$580$	26597.2	+	13250.0i	0.0790642	+	0.0393878i
$581$			621583.i			1.84140i
$582$	388660.	+	63521.7i	1.14742	+	0.187532i
$583$		−	174911.i		−	0.514611i
$584$		−	487952.i		−	1.43071i
$585$	−23905.4	+	71179.6i	−0.0698530	+	0.207991i
$586$	57281.8			0.166810
$587$		−	428251.i		−	1.24286i	−0.783470	−	0.621430i	$-0.786552\pi$
$587$		−	428251.i		−	1.24286i	0.783470	−	0.621430i	$-0.213448\pi$
$588$	7674.78	−	46958.5i	0.0221979	−	0.135819i
$589$	−26370.3			−0.0760125
$590$	121601.			0.349328
$591$	346101.	+	56565.9i	0.990895	+	0.161950i
$592$		−	219477.i		−	0.626246i
$593$			356829.i			1.01473i	0.861731	+	0.507366i	$0.169381\pi$
$593$			356829.i			1.01473i	−0.861731	+	0.507366i	$0.830619\pi$
$594$	70047.2	−	132594.i	0.198526	−	0.375796i
$595$			145264.i			0.410322i
$596$		−	161212.i		−	0.453844i
$597$	−10031.8	+	61380.1i	−0.0281470	+	0.172218i
$598$			396726.i			1.10940i
$599$	517685.			1.44282			0.721410	−	0.692508i	$-0.243494\pi$
$599$	517685.			1.44282			0.721410	+	0.692508i	$0.243494\pi$
$600$	−58476.5	+	357791.i	−0.162435	+	0.993863i
$601$			619888.i			1.71619i	0.513494	+	0.858093i	$0.328351\pi$
$601$			619888.i			1.71619i	−0.513494	+	0.858093i	$0.671649\pi$
$602$		−	529725.i		−	1.46170i
$603$	185958.	+	62453.5i	0.511424	+	0.171760i
$604$	−158924.			−0.435629
$605$			73443.0i			0.200650i
$606$	90821.0	−	555691.i	0.247310	−	1.51317i
$607$		−	187769.i		−	0.509619i	−0.966991	−	0.254810i	$-0.917987\pi$
$607$		−	187769.i		−	0.509619i	0.966991	−	0.254810i	$-0.0820128\pi$
$608$		−	106781.i		−	0.288859i
$609$	−359417.	−	258872.i	−0.969090	−	0.697993i
$610$	27360.9			0.0735309
$611$	−277885.			−0.744359
$612$	143901.	+	48328.8i	0.384204	+	0.129034i
$613$	−189907.			−0.505383			−0.252692	−	0.967547i	$-0.581316\pi$
$613$	−189907.			−0.505383			−0.252692	+	0.967547i	$0.581316\pi$
$614$		−	349597.i		−	0.927324i
$615$	−5521.28	−	902.385i	−0.0145979	−	0.00238584i
$616$	254781.			0.671439
$617$	41392.7			0.108731			0.0543656	−	0.998521i	$-0.482686\pi$
$617$	41392.7			0.108731			0.0543656	+	0.998521i	$0.482686\pi$
$618$	21019.1	−	128606.i	0.0550348	−	0.336732i
$619$			195178.i			0.509390i	0.967021	+	0.254695i	$0.0819750\pi$
$619$			195178.i			0.509390i	−0.967021	+	0.254695i	$0.918025\pi$
$620$	−5513.25			−0.0143425
$621$	−573317.	−	302873.i	−1.48666	−	0.785375i
$622$	−150100.			−0.387972
$623$	−294403.			−0.758517
$624$	28854.9	−	176550.i	0.0741054	−	0.453417i
$625$	305066.			0.780969
$626$	66254.8			0.169071
$627$	93806.6	+	15331.5i	0.238615	+	0.0389988i
$628$			62205.1i			0.157727i
$629$			542909.i			1.37223i
$630$	−33983.2	+	101187.i	−0.0856215	+	0.254942i
$631$	544068.			1.36645			0.683227	−	0.730206i	$-0.260576\pi$
$631$	544068.			1.36645			0.683227	+	0.730206i	$0.260576\pi$
$632$		−	294640.i		−	0.737663i
$633$	348202.	+	56909.4i	0.869008	+	0.142029i
$634$	314443.			0.782282
$635$	7310.74			0.0181307
$636$	128399.	+	20985.3i	0.317430	+	0.0518800i
$637$	138706.			0.341835
$638$	77141.2	−	154848.i	0.189516	−	0.380420i
$639$	−41169.5	+	122584.i	−0.100826	+	0.300215i
$640$			30526.1i			0.0745265i
$641$	−474728.			−1.15539			−0.577695	−	0.816253i	$-0.696048\pi$
$641$	−474728.			−1.15539			−0.577695	+	0.816253i	$0.696048\pi$
$642$	−146366.	−	23921.7i	−0.355116	−	0.0580394i
$643$	552280.			1.33579			0.667893	−	0.744257i	$-0.267196\pi$
$643$	552280.			1.33579			0.667893	+	0.744257i	$0.267196\pi$
$644$		−	268831.i		−	0.648198i
$645$	27309.8	−	167096.i	0.0656446	−	0.401649i
$646$		−	201853.i		−	0.483693i
$647$			234263.i			0.559624i	0.960055	+	0.279812i	$0.0902721\pi$
$647$			234263.i			0.559624i	−0.960055	+	0.279812i	$0.909728\pi$
$648$	364429.	+	275905.i	0.867887	+	0.657066i
$649$			337464.i			0.801194i
$650$	−257901.			−0.610416
$651$	81106.2	+	13255.8i	0.191378	+	0.0312784i
$652$		−	134186.i		−	0.315654i
$653$	−504311.			−1.18269			−0.591347	−	0.806417i	$-0.701403\pi$
$653$	−504311.			−1.18269			−0.591347	+	0.806417i	$0.701403\pi$
$654$	−22045.2	+	134885.i	−0.0515418	+	0.315360i
$655$		−	24333.1i		−	0.0567173i
$656$	13328.8			0.0309731
$657$	−180619.	+	537802.i	−0.418439	+	1.24592i
$658$	−395032.			−0.912390
$659$	623688.			1.43614			0.718070	−	0.695971i	$-0.245026\pi$
$659$	623688.			1.43614			0.718070	+	0.695971i	$0.245026\pi$
$660$	19612.1	+	3205.37i	0.0450233	+	0.00735851i
$661$	−293810.			−0.672455			−0.336228	−	0.941781i	$-0.609151\pi$
$661$	−293810.			−0.672455			−0.336228	+	0.941781i	$0.609151\pi$
$662$			576309.i			1.31504i
$663$	−71376.9	+	436722.i	−0.162379	+	0.993524i
$664$		−	739990.i		−	1.67838i
$665$	−67657.2			−0.152993
$666$	−127009.	+	378174.i	−0.286342	+	0.852597i
$667$	−669536.	−	333546.i	−1.50495	−	0.749728i
$668$		−	69891.4i		−	0.156629i
$669$	−98204.6	+	600868.i	−0.219422	+	1.34254i
$670$		−	54535.2i		−	0.121486i
$671$			75931.1i			0.168645i
$672$	−53676.4	+	328421.i	−0.118862	+	0.727265i
$673$	−17987.3			−0.0397133			−0.0198566	−	0.999803i	$-0.506321\pi$
$673$	−17987.3			−0.0397133			−0.0198566	+	0.999803i	$0.506321\pi$
$674$		−	152838.i		−	0.336444i
$675$	196889.	−	372698.i	0.432130	−	0.817992i
$676$	52677.7			0.115275
$677$	−62501.1			−0.136367			−0.0681837	−	0.997673i	$-0.521720\pi$
$677$	−62501.1			−0.136367			−0.0681837	+	0.997673i	$0.521720\pi$
$678$	−26021.2	+	159212.i	−0.0566068	+	0.346351i
$679$		−	777929.i		−	1.68733i
$680$		−	172936.i		−	0.373996i
$681$	94152.9	+	15388.1i	0.203020	+	0.0331812i
$682$			32097.9i			0.0690093i
$683$		−	648548.i		−	1.39027i	−0.718877	−	0.695137i	$-0.755344\pi$
$683$		−	648548.i		−	1.39027i	0.718877	−	0.695137i	$-0.244656\pi$
$684$	−22509.3	+	67022.5i	−0.0481116	+	0.143255i
$685$			45313.1i			0.0965701i
$686$	−265325.			−0.563806
$687$	751575.	+	122836.i	1.59242	+	0.260262i
$688$			403384.i			0.852200i
$689$			379266.i			0.798923i
$690$	−29075.4	+	177899.i	−0.0610700	+	0.373659i
$691$	31695.8			0.0663814			0.0331907	−	0.999449i	$-0.489433\pi$
$691$	31695.8			0.0663814			0.0331907	+	0.999449i	$0.489433\pi$
$692$			124370.i			0.259719i
$693$	−280810.	−	94309.2i	−0.584718	−	0.196376i
$694$		−	389327.i		−	0.808342i
$695$		−	73581.2i		−	0.152334i
$696$	427883.	+	308186.i	0.883297	+	0.636200i
$697$	−32970.9			−0.0678680
$698$	−413993.			−0.849732
$699$	720531.	+	117762.i	1.47468	+	0.241019i
$700$	174760.			0.356652
$701$			459917.i			0.935931i	0.883747	+	0.467965i	$0.155013\pi$
$701$			459917.i			0.935931i	−0.883747	+	0.467965i	$0.844987\pi$
$702$	−151886.	+	287510.i	−0.308208	+	0.583416i
$703$	−252862.			−0.511649
$704$	−276643.			−0.558180
$705$	−124609.	−	20365.7i	−0.250709	−	0.0409753i
$706$			74185.3i			0.148836i
$707$	−1.11225e6			−2.22518
$708$	−247727.	−	40487.9i	−0.494204	−	0.0807716i
$709$	126864.			0.252374			0.126187	−	0.992006i	$-0.459726\pi$
$709$	126864.			0.252374			0.126187	+	0.992006i	$0.459726\pi$
$710$	35949.7			0.0713146
$711$	−109063.	+	324741.i	−0.215744	+	0.642389i
$712$	350484.			0.691366
$713$	138786.			0.273003
$714$	−101467.	+	620830.i	−0.199035	+	1.21780i
$715$			57930.4i			0.113317i
$716$			176080.i			0.343467i
$717$	−621762.	−	101619.i	−1.20944	−	0.197669i
$718$	−735051.			−1.42583
$719$			786756.i			1.52189i	0.648819	+	0.760943i	$0.275263\pi$
$719$			786756.i			1.52189i	−0.648819	+	0.760943i	$0.724737\pi$
$720$	25878.1	−	77053.2i	0.0499191	−	0.148637i
$721$	−257414.			−0.495178
$722$	−334962.			−0.642570
$723$	87699.6	−	536594.i	0.167773	−	1.02652i
$724$	−277046.			−0.528536
$725$	216829.	−	435247.i	0.412517	−	0.828057i
$726$	−51300.0	+	313881.i	−0.0973294	+	0.595514i
$727$			488110.i			0.923526i	0.887003	+	0.461763i	$0.152783\pi$
$727$			488110.i			0.923526i	−0.887003	+	0.461763i	$0.847217\pi$
$728$	−552453.			−1.04240
$729$	−299532.	−	438988.i	−0.563622	−	0.826033i
$730$	157719.			0.295963
$731$		−	997831.i		−	1.86734i
$732$	−55739.8	−	9109.99i	−0.104026	−	0.0170018i
$733$			44492.3i			0.0828088i	0.999142	+	0.0414044i	$0.0131832\pi$
$733$			44492.3i			0.0828088i	−0.999142	+	0.0414044i	$0.986817\pi$
$734$			11888.1i			0.0220658i
$735$	62198.3	+	10165.5i	0.115134	+	0.0188173i
$736$			561983.i			1.03745i
$737$	151345.			0.278633
$738$	−22966.5	−	7713.23i	−0.0421680	−	0.0141620i
$739$		−	76270.1i		−	0.139658i	−0.997559	−	0.0698289i	$-0.977755\pi$
$739$		−	76270.1i		−	0.139658i	0.997559	−	0.0698289i	$-0.0222453\pi$
$740$	−52865.8			−0.0965408
$741$	−203405.	−	33244.0i	−0.370446	−	0.0605448i
$742$			539152.i			0.979272i
$743$	918379.			1.66358			0.831791	−	0.555089i	$-0.187316\pi$
$743$	918379.			1.66358			0.831791	+	0.555089i	$0.187316\pi$
$744$	−96556.3	−	15780.9i	−0.174435	−	0.0285093i
$745$	213532.			0.384725
$746$	486802.			0.874732
$747$	−273913.	+	815588.i	−0.490875	+	1.46160i
$748$	117116.			0.209321
$749$			292962.i			0.522212i
$750$	−240655.	−	39332.1i	−0.427831	−	0.0699238i
$751$			482653.i			0.855766i	0.903834	+	0.427883i	$0.140740\pi$
$751$			482653.i			0.855766i	−0.903834	+	0.427883i	$0.859260\pi$
$752$	300815.			0.531942
$753$	117663.	−	719923.i	0.207514	−	1.26968i
$754$	−167268.	+	335762.i	−0.294219	+	0.590594i
$755$		−	210502.i		−	0.369285i
$756$	102922.	−	194823.i	0.180079	−	0.340877i
$757$		−	1.05634e6i		−	1.84337i	−0.387935	−	0.921687i	$-0.626811\pi$
$757$		−	1.05634e6i		−	1.84337i	0.387935	−	0.921687i	$-0.373189\pi$
$758$		−	718459.i		−	1.25044i
$759$	−493700.	−	80689.3i	−0.856998	−	0.140066i
$760$	80545.3			0.139448
$761$		−	254574.i		−	0.439587i	−0.975546	−	0.219793i	$-0.929462\pi$
$761$		−	254574.i		−	0.439587i	0.975546	−	0.219793i	$-0.0705383\pi$
$762$	31244.7	+	5106.56i	0.0538104	+	0.00879464i
$763$	269981.			0.463750
$764$	129971.			0.222670
$765$	−64013.4	+	190603.i	−0.109383	+	0.325692i
$766$		−	145932.i		−	0.248711i
$767$		−	731736.i		−	1.24384i
$768$	81498.3	−	498651.i	0.138174	−	0.845423i
$769$		−	432584.i		−	0.731506i	−0.930712	−	0.365753i	$-0.880811\pi$
$769$		−	432584.i		−	0.731506i	0.930712	−	0.365753i	$-0.119189\pi$
$770$			82352.0i			0.138897i
$771$	745566.	+	121854.i	1.25423	+	0.204989i
$772$		−	86771.0i		−	0.145593i
$773$	−262908.			−0.439992			−0.219996	−	0.975501i	$-0.570604\pi$
$773$	−262908.			−0.439992			−0.219996	+	0.975501i	$0.570604\pi$
$774$	233433.	−	695060.i	0.389656	−	1.16022i
$775$			90221.1i			0.150212i
$776$			926118.i			1.53795i
$777$	777716.	+	127108.i	1.28819	+	0.210538i
$778$	−914676.			−1.51115
$779$		−	15356.3i		−	0.0253053i
$780$	−42525.8	−	6950.32i	−0.0698978	−	0.0114239i
$781$			99766.6i			0.163562i
$782$			1.06234e6i			1.73721i
$783$	−357519.	−	498054.i	−0.583144	−	0.812369i
$784$	−150152.			−0.244286
$785$	−82393.1			−0.133706
$786$	16996.7	−	103995.i	0.0275119	−	0.168332i
$787$	−31127.4			−0.0502567			−0.0251283	−	0.999684i	$-0.507999\pi$
$787$	−31127.4			−0.0502567			−0.0251283	+	0.999684i	$0.507999\pi$
$788$			201252.i			0.324107i
$789$	181814.	−	1.11244e6i	0.292062	−	1.78699i
$790$	95235.4			0.152596
$791$	318673.			0.509323
$792$	334302.	+	112274.i	0.532953	+	0.178991i
$793$		−	164644.i		−	0.261819i
$794$	770297.			1.22185
$795$	−27795.8	+	170070.i	−0.0439789	+	0.269087i
$796$	−35691.6			−0.0563300
$797$	−494844.			−0.779025			−0.389512	−	0.921021i	$-0.627356\pi$
$797$	−494844.			−0.779025			−0.389512	+	0.921021i	$0.627356\pi$
$798$	−289153.	−	47258.6i	−0.454070	−	0.0742121i
$799$	−744113.			−1.16559
$800$	−365330.			−0.570828
$801$	−386290.	−	129734.i	−0.602072	−	0.202204i
$802$		−	183228.i		−	0.284868i
$803$			437697.i			0.678801i
$804$	−18157.9	+	111100.i	−0.0280901	+	0.171870i
$805$	356077.			0.549480
$806$		−	69599.1i		−	0.107136i
$807$	−21139.5	+	129343.i	−0.0324599	+	0.198607i
$808$	1.32413e6			2.02819
$809$	837412.			1.27951			0.639753	−	0.768581i	$-0.279037\pi$
$809$	837412.			1.27951			0.639753	+	0.768581i	$0.279037\pi$
$810$	−89179.6	+	117793.i	−0.135924	+	0.179535i
$811$	546447.			0.830819			0.415410	−	0.909634i	$-0.363638\pi$
$811$	546447.			0.830819			0.415410	+	0.909634i	$0.363638\pi$
$812$	113345.	−	227521.i	0.171906	−	0.345071i
$813$	−956401.	−	156312.i	−1.44697	−	0.236489i
$814$			307782.i			0.464509i
$815$	177734.			0.267582
$816$	77266.8	−	472760.i	0.116041	−	0.710003i
$817$	464743.			0.696255
$818$			941660.i			1.40730i
$819$	608892.	+	204494.i	0.907764	+	0.304869i
$820$		−	3210.54i		−	0.00477475i
$821$			899223.i			1.33408i	0.745023	+	0.667039i	$0.232439\pi$
$821$			899223.i			1.33408i	−0.745023	+	0.667039i	$0.767561\pi$
$822$	−31651.2	+	193659.i	−0.0468433	+	0.286612i
$823$		−	257868.i		−	0.380712i	−0.981715	−	0.190356i	$-0.939036\pi$
$823$		−	257868.i		−	0.380712i	0.981715	−	0.190356i	$-0.0609643\pi$
$824$	306449.			0.451340
$825$	52453.9	−	320941.i	0.0770673	−	0.471539i
$826$		−	1.04021e6i		−	1.52462i
$827$	979284.			1.43185			0.715925	−	0.698177i	$-0.246005\pi$
$827$	979284.			1.43185			0.715925	+	0.698177i	$0.246005\pi$
$828$	118465.	−	352737.i	0.172795	−	0.514506i
$829$			498418.i			0.725245i	0.931936	+	0.362623i	$0.118119\pi$
$829$			498418.i			0.725245i	−0.931936	+	0.362623i	$0.881881\pi$
$830$	239184.			0.347197
$831$	187479.	−	1.14710e6i	0.271488	−	1.66111i
$832$	599856.			0.866563
$833$	371424.			0.535278
$834$	51396.5	−	314472.i	0.0738927	−	0.452115i
$835$	92573.9			0.132775
$836$			54547.1i			0.0780476i
$837$	100579.	+	53134.1i	0.143568	+	0.0758443i
$838$		−	37809.0i		−	0.0538402i
$839$	−501245.			−0.712076			−0.356038	−	0.934472i	$-0.615873\pi$
$839$	−501245.			−0.712076			−0.356038	+	0.934472i	$0.615873\pi$
$840$	−247730.	−	40488.4i	−0.351091	−	0.0573815i
$841$	−426020.	−	564582.i	−0.602335	−	0.798243i
$842$			21701.5i			0.0306102i
$843$	1.04042e6	+	170043.i	1.46404	+	0.239279i
$844$			202474.i			0.284240i
$845$			69773.7i			0.0977188i
$846$	−518327.	−	174078.i	−0.724208	−	0.243223i
$847$	628254.			0.875726
$848$		−	410562.i		−	0.570936i
$849$	−41311.2	+	252764.i	−0.0573130	+	0.350672i
$850$	−690600.			−0.955848
$851$	1.33080e6			1.83761
$852$	−73237.1	−	11969.7i	−0.100891	−	0.0164894i
$853$			1.13716e6i			1.56288i	0.623983	+	0.781438i	$0.285513\pi$
$853$			1.13716e6i			1.56288i	−0.623983	+	0.781438i	$0.714487\pi$
$854$		−	234053.i		−	0.320922i
$855$	−88773.9	−	29814.4i	−0.121438	−	0.0407844i
$856$		−	348768.i		−	0.475981i
$857$		−	988346.i		−	1.34570i	−0.739780	−	0.672848i	$-0.765071\pi$
$857$		−	988346.i		−	1.34570i	0.739780	−	0.672848i	$-0.234929\pi$
$858$	−40464.5	+	247583.i	−0.0549666	+	0.336315i
$859$			1.02564e6i			1.38998i	0.719021	+	0.694989i	$0.244591\pi$
$859$			1.02564e6i			1.38998i	−0.719021	+	0.694989i	$0.755409\pi$
$860$	97163.8			0.131373
$861$	−7719.28	+	47230.7i	−0.0104129	+	0.0637115i
$862$		−	349596.i		−	0.470492i
$863$		−	551506.i		−	0.740507i	−0.928931	−	0.370253i	$-0.879271\pi$
$863$		−	551506.i		−	0.740507i	0.928931	−	0.370253i	$-0.120729\pi$
$864$	−215155.	+	407272.i	−0.288219	+	0.545579i
$865$	−164733.			−0.220165
$866$			156144.i			0.208205i
$867$	−69885.7	+	427598.i	−0.0929716	+	0.568850i
$868$			47162.0i			0.0625969i
$869$			264295.i			0.349984i
$870$	−99613.6	+	138303.i	−0.131607	+	0.182723i
$871$	−328167.			−0.432572
$872$	−321410.			−0.422694
$873$	342809.	−	1.02073e6i	0.449805	−	1.33932i
$874$	−494789.			−0.647735
$875$			481688.i			0.629143i
$876$	−321306.	−	52513.6i	−0.418708	−	0.0684326i
$877$	190425.			0.247586			0.123793	−	0.992308i	$-0.460494\pi$
$877$	190425.			0.247586			0.123793	+	0.992308i	$0.460494\pi$
$878$	−23494.4			−0.0304772
$879$	25262.1	−	154567.i	0.0326958	−	0.200050i
$880$		−	62710.8i		−	0.0809798i
$881$	564525.			0.727329			0.363665	−	0.931530i	$-0.381525\pi$
$881$	564525.			0.727329			0.363665	+	0.931530i	$0.381525\pi$
$882$	258723.	+	86891.1i	0.332581	+	0.111696i
$883$	370331.			0.474972			0.237486	−	0.971391i	$-0.423677\pi$
$883$	370331.			0.474972			0.237486	+	0.971391i	$0.423677\pi$
$884$	−253947.			−0.324967
$885$	53627.8	−	328124.i	0.0684705	−	0.418939i
$886$	650156.			0.828228
$887$	−1.27730e6			−1.62348			−0.811740	−	0.584019i	$-0.801480\pi$
$887$	−1.27730e6			−1.62348			−0.811740	+	0.584019i	$0.801480\pi$
$888$	−925865.	−	151321.i	−1.17414	−	0.191900i
$889$		−	62538.3i		−	0.0791303i
$890$			113286.i			0.143019i
$891$	−326896.	−	247489.i	−0.411770	−	0.311746i
$892$	−349396.			−0.439125
$893$		−	346573.i		−	0.434602i
$894$	912595.	+	149152.i	1.14183	+	0.186619i
$895$	−233225.			−0.291159
$896$	261129.			0.325267
$897$	1.07051e6	+	174962.i	1.33047	+	0.217450i
$898$	−818058.			−1.01445
$899$	117459.	+	58515.3i	0.145334	+	0.0724019i
$900$	229305.	+	77011.2i	0.283092	+	0.0950755i
$901$			1.01559e6i			1.25103i
$902$	−18691.6			−0.0229738
$903$	−1.42939e6	−	233616.i	−1.75297	−	0.286502i
$904$	−379378.			−0.464232
$905$		−	366958.i		−	0.448043i
$906$	147035.	−	899642.i	0.179129	−	1.09601i
$907$		−	1.05109e6i		−	1.27769i	−0.769335	−	0.638845i	$-0.779412\pi$
$907$		−	1.05109e6i		−	1.27769i	0.769335	−	0.638845i	$-0.220588\pi$
$908$			54748.5i			0.0664050i
$909$	−1.45940e6	−	490136.i	−1.76623	−	0.593183i
$910$		−	178567.i		−	0.215635i
$911$	446552.			0.538066			0.269033	−	0.963131i	$-0.413296\pi$
$911$	446552.			0.538066			0.269033	+	0.963131i	$0.413296\pi$
$912$	220189.	+	35987.2i	0.264732	+	0.0432672i
$913$			663777.i			0.796307i
$914$	−891567.			−1.06724
$915$	12066.5	−	73829.5i	0.0144125	−	0.0881836i
$916$			437030.i			0.520859i
$917$	−208153.			−0.247539
$918$	−406717.	+	769887.i	−0.482622	+	0.913569i
$919$	441423.			0.522666			0.261333	−	0.965249i	$-0.415838\pi$
$919$	441423.			0.522666			0.261333	+	0.965249i	$0.415838\pi$
$920$	−423907.			−0.500835
$921$	−943341.	−	154178.i	−1.11211	−	0.181761i
$922$	866400.			1.01919
$923$		−	216328.i		−	0.253927i
$924$	27419.7	−	167768.i	0.0321158	−	0.196502i
$925$			865117.i			1.01109i
$926$	−707403.			−0.824983
$927$	−337757.	−	113434.i	−0.393047	−	0.132003i
$928$	−236944.	+	475625.i	−0.275138	+	0.552292i
$929$			1.59665e6i			1.85003i	0.379930	+	0.925015i	$0.375948\pi$
$929$			1.59665e6i			1.85003i	−0.379930	+	0.925015i	$0.624052\pi$
$930$	5100.81	−	31209.5i	0.00589757	−	0.0360845i
$931$			172992.i			0.199584i
$932$			418978.i			0.482347i
$933$	−66196.3	+	405025.i	−0.0760450	+	0.465284i
$934$	1.13493e6			1.30100
$935$			155125.i			0.177443i
$936$	−724881.	−	243449.i	−0.827400	−	0.277879i
$937$	−1.55026e6			−1.76573			−0.882865	−	0.469627i	$-0.844389\pi$
$937$	−1.55026e6			−1.76573			−0.882865	+	0.469627i	$0.844389\pi$
$938$	−466511.			−0.530220
$939$	29219.3	−	178780.i	0.0331390	−	0.202762i
$940$		−	72458.0i		−	0.0820031i
$941$			938876.i			1.06030i	0.847904	+	0.530150i	$0.177865\pi$
$941$			938876.i			1.06030i	−0.847904	+	0.530150i	$0.822135\pi$
$942$	−352132.	−	57551.6i	−0.396829	−	0.0648568i
$943$			80819.5i			0.0908851i
$944$			792118.i			0.888886i
$945$	258051.	+	136324.i	0.288963	+	0.152654i
$946$		−	565683.i		−	0.632107i
$947$	447015.			0.498451			0.249225	−	0.968446i	$-0.419824\pi$
$947$	447015.			0.498451			0.249225	+	0.968446i	$0.419824\pi$
$948$	−194014.	−	31709.3i	−0.215883	−	0.0352834i
$949$		−	949076.i		−	1.05382i
$950$		−	321649.i		−	0.356398i
$951$	138674.	−	848482.i	0.153332	−	0.938170i
$952$	−1.47935e6			−1.63228
$953$			627403.i			0.690813i	0.938453	+	0.345407i	$0.112259\pi$
$953$			627403.i			0.690813i	−0.938453	+	0.345407i	$0.887741\pi$
$954$	−237587.	+	707429.i	−0.261052	+	0.777295i
$955$			172152.i			0.188758i
$956$		−	361545.i		−	0.395591i
$957$	−383815.	−	276445.i	−0.419081	−	0.301846i
$958$	698953.			0.761583
$959$	387622.			0.421475
$960$	268986.	+	43962.5i	0.291869	+	0.0477024i
$961$	899173.			0.973636
$962$		−	667376.i		−	0.721142i
$963$	−129099.	+	384399.i	−0.139210	+	0.414505i
$964$	312021.			0.335761
$965$	114932.			0.123420
$966$	1.52180e6	+	248720.i	1.63081	+	0.266536i
$967$			1.27111e6i			1.35935i	0.733514	+	0.679675i	$0.237879\pi$
$967$			1.27111e6i			1.35935i	−0.733514	+	0.679675i	$0.762121\pi$
$968$	−747931.			−0.798199
$969$	−544672.	−	89020.0i	−0.580080	−	0.0948069i
$970$	−299346.			−0.318148
$971$	835783.			0.886452			0.443226	−	0.896410i	$-0.353834\pi$
$971$	835783.			0.886452			0.443226	+	0.896410i	$0.353834\pi$
$972$	220898.	−	210276.i	0.233808	−	0.222565i
$973$	−629436.			−0.664854
$974$	553330.			0.583266
$975$	−113738.	+	695910.i	−0.119645	+	0.732055i
$976$			178231.i			0.187104i
$977$		−	193547.i		−	0.202767i	−0.994847	−	0.101384i	$-0.967673\pi$
$977$		−	193547.i		−	0.202767i	0.994847	−	0.101384i	$-0.0323270\pi$
$978$	759602.	+	124148.i	0.794161	+	0.129796i
$979$	−314387.			−0.328019
$980$			36167.4i			0.0376587i
$981$	354246.	+	118972.i	0.368101	+	0.123625i
$982$	−54761.2			−0.0567871
$983$	−473793.			−0.490322			−0.245161	−	0.969482i	$-0.578841\pi$
$983$	−473793.			−0.490322			−0.245161	+	0.969482i	$0.578841\pi$
$984$	9189.74	−	56227.8i	0.00949102	−	0.0580712i
$985$	−266567.			−0.274747
$986$	−447907.	+	899096.i	−0.460717	+	0.924810i
$987$	−174215.	+	1.06594e6i	−0.178834	+	1.09420i
$988$		−	118277.i		−	0.121167i
$989$	−2.44592e6			−2.50063
$990$	−36290.0	+	108055.i	−0.0370268	+	0.110249i
$991$	872411.			0.888329			0.444164	−	0.895945i	$-0.353501\pi$
$991$	872411.			0.888329			0.444164	+	0.895945i	$0.353501\pi$
$992$		−	98590.8i		−	0.100187i
$993$	1.55509e6	+	254161.i	1.57709	+	0.257757i
$994$		−	307525.i		−	0.311249i
$995$		−	47274.9i		−	0.0477512i
$996$	−487268.	−	79638.0i	−0.491190	−	0.0802789i
$997$		−	590291.i		−	0.593848i	−0.954901	−	0.296924i	$-0.904039\pi$
$997$		−	590291.i		−	0.593848i	0.954901	−	0.296924i	$-0.0959609\pi$
$998$	935802.			0.939557
$999$	964440.	+	509496.i	0.966371	+	0.510516i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	87.5.d.c.86.12	yes	32
3.2	odd	2	inner	87.5.d.c.86.22	yes	32
29.28	even	2	inner	87.5.d.c.86.21	yes	32
87.86	odd	2	inner	87.5.d.c.86.11	✓	32

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
87.5.d.c.86.11	✓	32	87.86	odd	2	inner
87.5.d.c.86.12	yes	32	1.1	even	1	trivial
87.5.d.c.86.21	yes	32	29.28	even	2	inner
87.5.d.c.86.22	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-3.29168 q^{2} +(-1.45168 + 8.88215i) q^{3} -5.16484 q^{4} -6.84103i q^{5} +(4.77846 - 29.2372i) q^{6} -58.5203 q^{7} +69.6679 q^{8} +(-76.7853 - 25.7881i) q^{9} +22.5185i q^{10} -62.4927 q^{11} +(7.49769 - 45.8749i) q^{12} +135.505 q^{13} +192.630 q^{14} +(60.7630 + 9.93097i) q^{15} -146.687 q^{16} +362.853 q^{17} +(252.753 + 84.8861i) q^{18} +169.000i q^{19} +35.3328i q^{20} +(84.9526 - 519.786i) q^{21} +205.706 q^{22} -889.439i q^{23} +(-101.135 + 618.801i) q^{24} +578.200 q^{25} -446.040 q^{26} +(340.521 - 644.582i) q^{27} +302.248 q^{28} +(375.007 - 752.762i) q^{29} +(-200.013 - 32.6896i) q^{30} +156.038i q^{31} -631.839 q^{32} +(90.7193 - 555.069i) q^{33} -1194.40 q^{34} +400.339i q^{35} +(396.583 + 133.191i) q^{36} +1496.22i q^{37} -556.293i q^{38} +(-196.710 + 1203.58i) q^{39} -476.600i q^{40} -90.8657 q^{41} +(-279.637 + 1710.97i) q^{42} -2749.96i q^{43} +322.764 q^{44} +(-176.417 + 525.290i) q^{45} +2927.75i q^{46} -2050.73 q^{47} +(212.943 - 1302.90i) q^{48} +1023.62 q^{49} -1903.25 q^{50} +(-526.746 + 3222.92i) q^{51} -699.863 q^{52} +2798.90i q^{53} +(-1120.89 + 2121.76i) q^{54} +427.514i q^{55} -4076.98 q^{56} +(-1501.08 - 245.333i) q^{57} +(-1234.40 + 2477.85i) q^{58} -5400.05i q^{59} +(-313.831 - 51.2919i) q^{60} -1215.04i q^{61} -513.626i q^{62} +(4493.49 + 1509.12i) q^{63} +4426.81 q^{64} -926.996i q^{65} +(-298.619 + 1827.11i) q^{66} -2421.80 q^{67} -1874.08 q^{68} +(7900.13 + 1291.18i) q^{69} -1317.79i q^{70} -1596.45i q^{71} +(-5349.47 - 1796.60i) q^{72} -7003.97i q^{73} -4925.09i q^{74} +(-839.361 + 5135.66i) q^{75} -872.857i q^{76} +3657.09 q^{77} +(647.507 - 3961.80i) q^{78} -4229.21i q^{79} +1003.49i q^{80} +(5230.95 + 3960.29i) q^{81} +299.101 q^{82} -10621.7i q^{83} +(-438.766 + 2684.61i) q^{84} -2482.29i q^{85} +9051.99i q^{86} +(6141.76 + 4423.64i) q^{87} -4353.73 q^{88} +5030.78 q^{89} +(580.708 - 1729.09i) q^{90} -7929.81 q^{91} +4593.81i q^{92} +(-1385.95 - 226.517i) q^{93} +6750.34 q^{94} +1156.13 q^{95} +(917.228 - 5612.09i) q^{96} +13293.3i q^{97} -3369.43 q^{98} +(4798.51 + 1611.56i) 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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