LMFDB - Embedded newform 87.7.b.a.59.5

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [87,7,Mod(59,87)] 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, 0])) N = Newforms(chi, 7, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("87.59"); S:= CuspForms(chi, 7); N := Newforms(S);

Level:	$ N $	$=$	$ 87 = 3 \cdot 29 $
Weight:	$ k $	$=$	$ 7 $
Character orbit:	$[\chi]$	$=$	87.b (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

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

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

Self dual:	no
Analytic conductor:	$20.0147052749$
Analytic rank:	$0$
Dimension:	$56$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			59.5
Character	$\chi$	$=$	87.59
Dual form			87.7.b.a.59.52

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q-14.2052i q^{2} +(4.68709 - 26.5901i) q^{3} -137.789 q^{4} +131.841i q^{5} +(-377.718 - 66.5813i) q^{6} -416.698 q^{7} +1048.19i q^{8} +(-685.062 - 249.260i) q^{9} +1872.83 q^{10} -793.372i q^{11} +(-645.830 + 3663.82i) q^{12} +1434.39 q^{13} +5919.30i q^{14} +(3505.66 + 617.950i) q^{15} +6071.32 q^{16} -844.742i q^{17} +(-3540.80 + 9731.48i) q^{18} +8110.48 q^{19} -18166.2i q^{20} +(-1953.10 + 11080.0i) q^{21} -11270.0 q^{22} +13423.0i q^{23} +(27871.5 + 4912.97i) q^{24} -1757.04 q^{25} -20375.9i q^{26} +(-9838.79 + 17047.5i) q^{27} +57416.4 q^{28} -4528.92i q^{29} +(8778.14 - 49798.7i) q^{30} -33131.6 q^{31} -19160.4i q^{32} +(-21095.8 - 3718.61i) q^{33} -11999.8 q^{34} -54937.9i q^{35} +(94394.1 + 34345.3i) q^{36} -75174.2 q^{37} -115211. i q^{38} +(6723.14 - 38140.6i) q^{39} -138195. q^{40} +124511. i q^{41} +(157394. + 27744.3i) q^{42} +50576.9 q^{43} +109318. i q^{44} +(32862.7 - 90319.3i) q^{45} +190678. q^{46} +145412. i q^{47} +(28456.8 - 161437. i) q^{48} +55988.2 q^{49} +24959.2i q^{50} +(-22461.7 - 3959.38i) q^{51} -197644. q^{52} -158591. i q^{53} +(242165. + 139762. i) q^{54} +104599. q^{55} -436779. i q^{56} +(38014.6 - 215658. i) q^{57} -64334.5 q^{58} +208067. i q^{59} +(-483041. - 85146.8i) q^{60} +20791.7 q^{61} +470643. i q^{62} +(285464. + 103866. i) q^{63} +116387. q^{64} +189112. i q^{65} +(-52823.7 + 299671. i) q^{66} -451067. q^{67} +116396. i q^{68} +(356919. + 62915.0i) q^{69} -780406. q^{70} +216270. i q^{71} +(261272. - 718077. i) q^{72} -475219. q^{73} +1.06787e6i q^{74} +(-8235.39 + 46719.7i) q^{75} -1.11754e6 q^{76} +330597. i q^{77} +(-541797. - 95503.9i) q^{78} -816713. q^{79} +800449. i q^{80} +(407180. + 341517. i) q^{81} +1.76870e6 q^{82} -4923.46i q^{83} +(269116. - 1.52671e6i) q^{84} +111372. q^{85} -718458. i q^{86} +(-120424. - 21227.5i) q^{87} +831606. q^{88} -602331. i q^{89} +(-1.28301e6 - 466822. i) q^{90} -597709. q^{91} -1.84955e6i q^{92} +(-155291. + 880972. i) q^{93} +2.06562e6 q^{94} +1.06929e6i q^{95} +(-509475. - 89806.3i) q^{96} +708292. q^{97} -795326. i q^{98} +(-197756. + 543510. i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 56 q + 52 q^{3} - 1924 q^{4} - 160 q^{6} + 160 q^{7} - 1060 q^{9} - 3588 q^{10} - 2166 q^{12} - 1400 q^{13} - 6240 q^{15} + 56588 q^{16} - 5978 q^{18} + 25000 q^{19} + 7520 q^{21} + 20970 q^{22} + 1238 q^{24}+ \cdots + 4793544 q^{99}+O(q^{100}) $ 56 * q + 52 * q^3 - 1924 * q^4 - 160 * q^6 + 160 * q^7 - 1060 * q^9 - 3588 * q^10 - 2166 * q^12 - 1400 * q^13 - 6240 * q^15 + 56588 * q^16 - 5978 * q^18 + 25000 * q^19 + 7520 * q^21 + 20970 * q^22 + 1238 * q^24 - 213304 * q^25 + 25492 * q^27 + 7870 * q^28 + 105092 * q^30 - 78656 * q^31 + 18952 * q^33 - 96642 * q^34 - 40224 * q^36 + 52120 * q^37 - 112096 * q^39 + 343440 * q^40 - 341758 * q^42 - 286760 * q^43 + 68312 * q^45 + 255516 * q^46 + 386086 * q^48 + 1051872 * q^49 - 256656 * q^51 + 58606 * q^52 + 450124 * q^54 - 409632 * q^55 - 1141736 * q^57 + 817806 * q^60 + 309976 * q^61 + 567632 * q^63 - 2844730 * q^64 + 958426 * q^66 + 748696 * q^67 - 1323704 * q^69 - 2123748 * q^70 - 1026242 * q^72 - 333728 * q^73 - 331316 * q^75 + 2846728 * q^76 + 292446 * q^78 - 661184 * q^79 + 1614852 * q^81 + 376908 * q^82 - 1670584 * q^84 + 3342000 * q^85 - 927168 * q^88 + 835882 * q^90 + 2248016 * q^91 + 3960776 * q^93 - 1368294 * q^94 - 5811456 * q^96 - 3057296 * q^97 + 4793544 * q^99

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 $

$2$		−	14.2052i		−	1.77566i	−0.460176	−	0.887828i	$-0.652214\pi$
$2$		−	14.2052i		−	1.77566i	0.460176	−	0.887828i	$-0.347786\pi$
$3$	4.68709	−	26.5901i	0.173596	−	0.984817i
$3$	4.68709	−	26.5901i	0.173596	−	0.984817i
$4$	−137.789			−2.15295
$5$			131.841i			1.05473i	0.849640	+	0.527364i	$0.176820\pi$
$5$			131.841i			1.05473i	−0.849640	+	0.527364i	$0.823180\pi$
$6$	−377.718	−	66.5813i	−1.74870	−	0.308247i
$7$	−416.698			−1.21486			−0.607431	−	0.794372i	$-0.707800\pi$
$7$	−416.698			−1.21486			−0.607431	+	0.794372i	$0.707800\pi$
$8$			1048.19i			2.04725i
$9$	−685.062	−	249.260i	−0.939729	−	0.341920i
$10$	1872.83			1.87283
$11$		−	793.372i		−	0.596072i	−0.954555	−	0.298036i	$-0.903668\pi$
$11$		−	793.372i		−	0.596072i	0.954555	−	0.298036i	$-0.0963316\pi$
$12$	−645.830	+	3663.82i	−0.373744	+	2.12027i
$13$	1434.39			0.652888			0.326444	−	0.945217i	$-0.394150\pi$
$13$	1434.39			0.652888			0.326444	+	0.945217i	$0.394150\pi$
$14$			5919.30i			2.15718i
$15$	3505.66	+	617.950i	1.03871	+	0.183096i
$16$	6071.32			1.48226
$17$		−	844.742i		−	0.171940i	−0.996298	−	0.0859701i	$-0.972601\pi$
$17$		−	844.742i		−	0.171940i	0.996298	−	0.0859701i	$-0.0273989\pi$
$18$	−3540.80	+	9731.48i	−0.607133	+	1.66864i
$19$	8110.48			1.18246			0.591229	−	0.806504i	$-0.298643\pi$
$19$	8110.48			1.18246			0.591229	+	0.806504i	$0.298643\pi$
$20$		−	18166.2i		−	2.27078i
$21$	−1953.10	+	11080.0i	−0.210895	+	1.19642i
$22$	−11270.0			−1.05842
$23$			13423.0i			1.10323i	0.834098	+	0.551616i	$0.185989\pi$
$23$			13423.0i			1.10323i	−0.834098	+	0.551616i	$0.814011\pi$
$24$	27871.5	+	4912.97i	2.01617	+	0.355394i
$25$	−1757.04			−0.112450
$26$		−	20375.9i		−	1.15930i
$27$	−9838.79	+	17047.5i	−0.499862	+	0.866105i
$28$	57416.4			2.61554
$29$		−	4528.92i		−	0.185695i
$29$		−	4528.92i		−	0.185695i
$30$	8778.14	−	49798.7i	0.325116	−	1.84440i
$31$	−33131.6			−1.11213			−0.556067	−	0.831137i	$-0.687690\pi$
$31$	−33131.6			−1.11213			−0.556067	+	0.831137i	$0.687690\pi$
$32$		−	19160.4i		−	0.584728i
$33$	−21095.8	−	3718.61i	−0.587022	−	0.103476i
$34$	−11999.8			−0.305307
$35$		−	54937.9i		−	1.28135i
$36$	94394.1	+	34345.3i	2.02319	+	0.736139i
$37$	−75174.2			−1.48410			−0.742051	−	0.670344i	$-0.766147\pi$
$37$	−75174.2			−1.48410			−0.742051	+	0.670344i	$0.766147\pi$
$38$		−	115211.i		−	2.09964i
$39$	6723.14	−	38140.6i	0.113339	−	0.642975i
$40$	−138195.			−2.15929
$41$			124511.i			1.80657i	0.429043	+	0.903284i	$0.358851\pi$
$41$			124511.i			1.80657i	−0.429043	+	0.903284i	$0.641149\pi$
$42$	157394.	+	27744.3i	2.12443	+	0.374477i
$43$	50576.9			0.636132			0.318066	−	0.948069i	$-0.396967\pi$
$43$	50576.9			0.636132			0.318066	+	0.948069i	$0.396967\pi$
$44$			109318.i			1.28332i
$45$	32862.7	−	90319.3i	0.360633	−	0.991158i
$46$	190678.			1.95896
$47$			145412.i			1.40058i	0.713858	+	0.700290i	$0.246946\pi$
$47$			145412.i			1.40058i	−0.713858	+	0.700290i	$0.753054\pi$
$48$	28456.8	−	161437.i	0.257314	−	1.45975i
$49$	55988.2			0.475892
$50$			24959.2i			0.199673i
$51$	−22461.7	−	3959.38i	−0.169330	−	0.0298481i
$52$	−197644.			−1.40564
$53$		−	158591.i		−	1.06525i	−0.846352	−	0.532624i	$-0.821206\pi$
$53$		−	158591.i		−	1.06525i	0.846352	−	0.532624i	$-0.178794\pi$
$54$	242165.	+	139762.i	1.53790	+	0.887583i
$55$	104599.			0.628694
$56$		−	436779.i		−	2.48713i
$57$	38014.6	−	215658.i	0.205270	−	1.16451i
$58$	−64334.5			−0.329731
$59$			208067.i			1.01309i	0.862214	+	0.506544i	$0.169077\pi$
$59$			208067.i			1.01309i	−0.862214	+	0.506544i	$0.830923\pi$
$60$	−483041.	−	85146.8i	−2.23630	−	0.394198i
$61$	20791.7			0.0916009			0.0458004	−	0.998951i	$-0.485416\pi$
$61$	20791.7			0.0916009			0.0458004	+	0.998951i	$0.485416\pi$
$62$			470643.i			1.97477i
$63$	285464.	+	103866.i	1.14164	+	0.415386i
$64$	116387.			0.443981
$65$			189112.i			0.688619i
$66$	−52823.7	+	299671.i	−0.183737	+	1.04235i
$67$	−451067.			−1.49974			−0.749870	−	0.661585i	$-0.769884\pi$
$67$	−451067.			−1.49974			−0.749870	+	0.661585i	$0.769884\pi$
$68$			116396.i			0.370179i
$69$	356919.	+	62915.0i	1.08648	+	0.191517i
$70$	−780406.			−2.27524
$71$			216270.i			0.604258i	0.953267	+	0.302129i	$0.0976973\pi$
$71$			216270.i			0.604258i	−0.953267	+	0.302129i	$0.902303\pi$
$72$	261272.	−	718077.i	0.699996	−	1.92386i
$73$	−475219.			−1.22159			−0.610795	−	0.791789i	$-0.709150\pi$
$73$	−475219.			−1.22159			−0.610795	+	0.791789i	$0.709150\pi$
$74$			1.06787e6i			2.63525i
$75$	−8235.39	+	46719.7i	−0.0195209	+	0.110743i
$76$	−1.11754e6			−2.54578
$77$			330597.i			0.724146i
$78$	−541797.	−	95503.9i	−1.14170	−	0.201251i
$79$	−816713.			−1.65649			−0.828244	−	0.560367i	$-0.810660\pi$
$79$	−816713.			−1.65649			−0.828244	+	0.560367i	$0.810660\pi$
$80$			800449.i			1.56338i
$81$	407180.	+	341517.i	0.766181	+	0.642625i
$82$	1.76870e6			3.20784
$83$		−	4923.46i		−	0.00861065i	−0.999991	−	0.00430533i	$-0.998630\pi$
$83$		−	4923.46i		−	0.00861065i	0.999991	−	0.00430533i	$-0.00137043\pi$
$84$	269116.	−	1.52671e6i	0.454048	−	2.57583i
$85$	111372.			0.181350
$86$		−	718458.i		−	1.12955i
$87$	−120424.	−	21227.5i	−0.182876	−	0.0322360i
$88$	831606.			1.22031
$89$		−	602331.i		−	0.854407i	−0.904155	−	0.427204i	$-0.859499\pi$
$89$		−	602331.i		−	0.854407i	0.904155	−	0.427204i	$-0.140501\pi$
$90$	−1.28301e6	−	466822.i	−1.75996	−	0.640360i
$91$	−597709.			−0.793169
$92$		−	1.84955e6i		−	2.37521i
$93$	−155291.	+	880972.i	−0.193062	+	1.09525i
$94$	2.06562e6			2.48695
$95$			1.06929e6i			1.24717i
$96$	−509475.	−	89806.3i	−0.575850	−	0.101506i
$97$	708292.			0.776064			0.388032	−	0.921646i	$-0.373155\pi$
$97$	708292.			0.776064			0.388032	+	0.921646i	$0.373155\pi$
$98$		−	795326.i		−	0.845020i
$99$	−197756.	+	543510.i	−0.203809	+	0.560146i
$100$	242101.			0.242101
$101$		−	120774.i		−	0.117222i	−0.998281	−	0.0586112i	$-0.981333\pi$
$101$		−	120774.i		−	0.117222i	0.998281	−	0.0586112i	$-0.0186672\pi$
$102$	−56244.0	+	319075.i	−0.0530000	+	0.300671i
$103$	1.79234e6			1.64025			0.820124	−	0.572186i	$-0.193905\pi$
$103$	1.79234e6			1.64025			0.820124	+	0.572186i	$0.193905\pi$
$104$			1.50352e6i			1.33662i
$105$	−1.46080e6	−	257499.i	−1.26189	−	0.222437i
$106$	−2.25283e6			−1.89152
$107$			318528.i			0.260014i	0.991513	+	0.130007i	$0.0415000\pi$
$107$			318528.i			0.260014i	−0.991513	+	0.130007i	$0.958500\pi$
$108$	1.35568e6	−	2.34896e6i	1.07618	−	1.86468i
$109$	−1.48543e6			−1.14702			−0.573511	−	0.819198i	$-0.694419\pi$
$109$	−1.48543e6			−1.14702			−0.573511	+	0.819198i	$0.694419\pi$
$110$		−	1.48585e6i		−	1.11634i
$111$	−352348.	+	1.99889e6i	−0.257634	+	1.46157i
$112$	−2.52991e6			−1.80074
$113$		−	2.69955e6i		−	1.87092i	−0.353432	−	0.935460i	$-0.614985\pi$
$113$		−	2.69955e6i		−	1.87092i	0.353432	−	0.935460i	$-0.385015\pi$
$114$	−3.06348e6	−	540006.i	−2.06776	−	0.364489i
$115$	−1.76971e6			−1.16361
$116$			624036.i			0.399793i
$117$	−982650.	−	357537.i	−0.613538	−	0.223236i
$118$	2.95564e6			1.79890
$119$			352002.i			0.208884i
$120$	−647730.	+	3.67460e6i	−0.374844	+	2.12651i
$121$	1.14212e6			0.644698
$122$		−	295351.i		−	0.162652i
$123$	3.31074e6	+	583592.i	1.77914	+	0.313613i
$124$	4.56517e6			2.39438
$125$			1.82837e6i			0.936123i
$126$	1.47544e6	−	4.05509e6i	0.737583	−	2.02716i
$127$	942313.			0.460028			0.230014	−	0.973187i	$-0.426123\pi$
$127$	942313.			0.460028			0.230014	+	0.973187i	$0.426123\pi$
$128$		−	2.87957e6i		−	1.37309i
$129$	237059.	−	1.34484e6i	0.110430	−	0.626473i
$130$	2.68638e6			1.22275
$131$			2.86732e6i			1.27545i	0.770266	+	0.637723i	$0.220123\pi$
$131$			2.86732e6i			1.27545i	−0.770266	+	0.637723i	$0.779877\pi$
$132$	2.90677e6	+	512383.i	1.26383	+	0.222778i
$133$	−3.37962e6			−1.43652
$134$			6.40751e6i			2.66302i
$135$	−2.24756e6	−	1.29716e6i	−0.913505	−	0.527218i
$136$	885451.			0.352004
$137$		−	2.37552e6i		−	0.923840i	−0.886922	−	0.461920i	$-0.847161\pi$
$137$		−	2.37552e6i		−	0.923840i	0.886922	−	0.461920i	$-0.152839\pi$
$138$	893723.	−	5.07013e6i	0.340068	−	1.92922i
$139$	−1.34164e6			−0.499564			−0.249782	−	0.968302i	$-0.580359\pi$
$139$	−1.34164e6			−0.499564			−0.249782	+	0.968302i	$0.580359\pi$
$140$			7.56983e6i			2.75869i
$141$	3.86652e6	+	681561.i	1.37932	+	0.243135i
$142$	3.07218e6			1.07295
$143$		−	1.13801e6i		−	0.389169i
$144$	−4.15923e6	−	1.51334e6i	−1.39292	−	0.506814i
$145$	597098.			0.195858
$146$			6.75061e6i			2.16912i
$147$	262422.	−	1.48873e6i	0.0826129	−	0.468666i
$148$	1.03582e7			3.19520
$149$		−	2.57614e6i		−	0.778772i	−0.921075	−	0.389386i	$-0.872687\pi$
$149$		−	2.57614e6i		−	0.778772i	0.921075	−	0.389386i	$-0.127313\pi$
$150$	663665.	+	116986.i	0.196642	+	0.0346625i
$151$	−3.05370e6			−0.886941			−0.443471	−	0.896289i	$-0.646253\pi$
$151$	−3.05370e6			−0.886941			−0.443471	+	0.896289i	$0.646253\pi$
$152$			8.50134e6i			2.42079i
$153$	−210560.	+	578701.i	−0.0587899	+	0.161577i
$154$	4.69621e6			1.28583
$155$		−	4.36810e6i		−	1.17300i
$156$	−926375.	+	5.25536e6i	−0.244013	+	1.38430i
$157$	−3.56846e6			−0.922107			−0.461054	−	0.887372i	$-0.652528\pi$
$157$	−3.56846e6			−0.922107			−0.461054	+	0.887372i	$0.652528\pi$
$158$			1.16016e7i			2.94135i
$159$	−4.21695e6	−	743331.i	−1.04908	−	0.184923i
$160$	2.52612e6			0.616729
$161$		−	5.59335e6i		−	1.34028i
$162$	4.85134e6	−	5.78409e6i	1.14108	−	1.36047i
$163$	1.40370e6			0.324124			0.162062	−	0.986781i	$-0.448185\pi$
$163$	1.40370e6			0.324124			0.162062	+	0.986781i	$0.448185\pi$
$164$		−	1.71562e7i		−	3.88946i
$165$	490265.	−	2.78129e6i	0.109139	−	0.619149i
$166$	−69939.0			−0.0152896
$167$			7.06551e6i			1.51703i	0.651656	+	0.758514i	$0.274074\pi$
$167$			7.06551e6i			1.51703i	−0.651656	+	0.758514i	$0.725926\pi$
$168$	−1.16140e7	−	2.04722e6i	−2.44936	−	0.431755i
$169$	−2.76932e6			−0.573737
$170$		−	1.58206e6i		−	0.322015i
$171$	−5.55619e6	−	2.02162e6i	−1.11119	−	0.404307i
$172$	−6.96895e6			−1.36956
$173$			2.59546e6i			0.501275i	0.968081	+	0.250638i	$0.0806403\pi$
$173$			2.59546e6i			0.501275i	−0.968081	+	0.250638i	$0.919360\pi$
$174$	−301542.	+	1.71066e6i	−0.0572400	+	0.324725i
$175$	732154.			0.136612
$176$		−	4.81682e6i		−	0.883532i
$177$	5.53252e6	+	975229.i	0.997707	+	0.175868i
$178$	−8.55626e6			−1.51713
$179$		−	3.41957e6i		−	0.596229i	−0.954530	−	0.298114i	$-0.903642\pi$
$179$		−	3.41957e6i		−	0.596229i	0.954530	−	0.298114i	$-0.0963577\pi$
$180$	−4.52812e6	+	1.24450e7i	−0.776426	+	2.13392i
$181$	−7.44865e6			−1.25615			−0.628075	−	0.778153i	$-0.716157\pi$
$181$	−7.44865e6			−1.25615			−0.628075	+	0.778153i	$0.716157\pi$
$182$			8.49061e6i			1.40840i
$183$	97452.4	−	552851.i	0.0159015	−	0.0902101i
$184$	−1.40699e7			−2.25859
$185$		−	9.91104e6i		−	1.56532i
$186$	1.25144e7	+	2.20595e6i	1.94479	+	0.342812i
$187$	−670195.			−0.102489
$188$		−	2.00362e7i		−	3.01538i
$189$	4.09980e6	−	7.10368e6i	0.607264	−	1.05220i
$190$	1.51896e7			2.21455
$191$			5.36234e6i			0.769580i	0.923004	+	0.384790i	$0.125726\pi$
$191$			5.36234e6i			0.769580i	−0.923004	+	0.384790i	$0.874274\pi$
$192$	545516.	−	3.09474e6i	0.0770733	−	0.437240i
$193$	−1.30801e7			−1.81944			−0.909720	−	0.415222i	$-0.863704\pi$
$193$	−1.30801e7			−1.81944			−0.909720	+	0.415222i	$0.863704\pi$
$194$		−	1.00615e7i		−	1.37802i
$195$	5.02850e6	+	886385.i	0.678164	+	0.119541i
$196$	−7.71456e6			−1.02457
$197$		−	9.49346e6i		−	1.24173i	−0.783919	−	0.620863i	$-0.786782\pi$
$197$		−	9.49346e6i		−	1.24173i	0.783919	−	0.620863i	$-0.213218\pi$
$198$	7.72069e6	+	2.80917e6i	0.994627	+	0.361895i
$199$	1.20213e7			1.52543			0.762717	−	0.646732i	$-0.223865\pi$
$199$	1.20213e7			1.52543			0.762717	+	0.646732i	$0.223865\pi$
$200$		−	1.84171e6i		−	0.230214i
$201$	−2.11419e6	+	1.19939e7i	−0.260349	+	1.47697i
$202$	−1.71563e6			−0.208147
$203$			1.88719e6i			0.225594i
$204$	3.09498e6	+	545559.i	0.364559	+	0.0642616i
$205$	−1.64156e7			−1.90544
$206$		−	2.54607e7i		−	2.91252i
$207$	3.34583e6	−	9.19562e6i	0.377218	−	1.03674i
$208$	8.70867e6			0.967747
$209$		−	6.43463e6i		−	0.704831i
$210$	−3.65783e6	+	2.07510e7i	−0.394972	+	2.24069i
$211$	−9.01533e6			−0.959697			−0.479848	−	0.877351i	$-0.659308\pi$
$211$	−9.01533e6			−0.959697			−0.479848	+	0.877351i	$0.659308\pi$
$212$			2.18521e7i			2.29343i
$213$	5.75064e6	+	1.01368e6i	0.595083	+	0.104897i
$214$	4.52477e6			0.461695
$215$			6.66811e6i			0.670946i
$216$	−1.78691e7	−	1.03129e7i	−1.77313	−	1.02334i
$217$	1.38059e7			1.35109
$218$			2.11009e7i			2.03672i
$219$	−2.22740e6	+	1.26361e7i	−0.212063	+	1.20304i
$220$	−1.44126e7			−1.35355
$221$		−	1.21169e6i		−	0.112258i
$222$	2.83947e7	+	5.00519e6i	2.59524	+	0.457469i
$223$	723489.			0.0652405			0.0326203	−	0.999468i	$-0.489615\pi$
$223$	723489.			0.0652405			0.0326203	+	0.999468i	$0.489615\pi$
$224$			7.98408e6i			0.710364i
$225$	1.20368e6	+	437959.i	0.105673	+	0.0384491i
$226$	−3.83477e7			−3.32211
$227$			7.58378e6i			0.648348i	0.945998	+	0.324174i	$0.105086\pi$
$227$			7.58378e6i			0.648348i	−0.945998	+	0.324174i	$0.894914\pi$
$228$	−5.23799e6	+	2.97153e7i	−0.441937	+	2.50713i
$229$	−5.06241e6			−0.421552			−0.210776	−	0.977534i	$-0.567599\pi$
$229$	−5.06241e6			−0.421552			−0.210776	+	0.977534i	$0.567599\pi$
$230$			2.51391e7i			2.06617i
$231$	8.79058e6	+	1.54954e6i	0.713151	+	0.125709i
$232$	4.74718e6			0.380165
$233$			1.61121e7i			1.27375i	0.770967	+	0.636875i	$0.219773\pi$
$233$			1.61121e7i			1.27375i	−0.770967	+	0.636875i	$0.780227\pi$
$234$	−5.07891e6	+	1.39588e7i	−0.396390	+	1.08943i
$235$	−1.91713e7			−1.47723
$236$		−	2.86694e7i		−	2.18113i
$237$	−3.82801e6	+	2.17165e7i	−0.287560	+	1.63134i
$238$	5.00028e6			0.370906
$239$			9.76363e6i			0.715184i	0.933878	+	0.357592i	$0.116402\pi$
$239$			9.76363e6i			0.715184i	−0.933878	+	0.357592i	$0.883598\pi$
$240$	2.12840e7	+	3.75178e6i	1.53964	+	0.271396i
$241$	7.79681e6			0.557013			0.278507	−	0.960434i	$-0.410161\pi$
$241$	7.79681e6			0.557013			0.278507	+	0.960434i	$0.410161\pi$
$242$		−	1.62241e7i		−	1.14476i
$243$	1.09895e7	−	9.22621e6i	0.765874	−	0.642991i
$244$	−2.86486e6			−0.197212
$245$			7.38154e6i			0.501936i
$246$	8.29007e6	−	4.70299e7i	0.556869	−	3.15914i
$247$	1.16336e7			0.772013
$248$		−	3.47283e7i		−	2.27682i
$249$	−130915.	−	23076.7i	−0.00847992	−	0.00149477i
$250$	2.59724e7			1.66223
$251$		−	1.03771e7i		−	0.656228i	−0.944638	−	0.328114i	$-0.893587\pi$
$251$		−	1.03771e7i		−	0.656228i	0.944638	−	0.328114i	$-0.106413\pi$
$252$	−3.93338e7	−	1.43116e7i	−2.45790	−	0.894308i
$253$	1.06495e7			0.657607
$254$		−	1.33858e7i		−	0.816851i
$255$	522009.	−	2.96138e6i	0.0314816	−	0.178597i
$256$	−3.34562e7			−1.99415
$257$		−	8.95978e6i		−	0.527835i	−0.964545	−	0.263918i	$-0.914985\pi$
$257$		−	8.95978e6i		−	0.527835i	0.964545	−	0.263918i	$-0.0850147\pi$
$258$	−1.91038e7	−	3.36748e6i	−1.11240	−	0.196085i
$259$	3.13249e7			1.80298
$260$		−	2.60576e7i		−	1.48256i
$261$	−1.12888e6	+	3.10260e6i	−0.0634930	+	0.174503i
$262$	4.07310e7			2.26475
$263$			1.41502e7i			0.777851i	0.921269	+	0.388925i	$0.127154\pi$
$263$			1.41502e7i			0.777851i	−0.921269	+	0.388925i	$0.872846\pi$
$264$	3.89781e6	−	2.21125e7i	0.211841	−	1.20178i
$265$	2.09088e7			1.12355
$266$			4.80084e7i			2.55077i
$267$	−1.60160e7	−	2.82318e6i	−0.841435	−	0.148322i
$268$	6.21520e7			3.22887
$269$			3.12154e6i			0.160366i	0.996780	+	0.0801830i	$0.0255505\pi$
$269$			3.12154e6i			0.160366i	−0.996780	+	0.0801830i	$0.974450\pi$
$270$	−1.84264e7	+	3.19272e7i	−0.936159	+	1.62207i
$271$	−2.10289e7			−1.05659			−0.528297	−	0.849059i	$-0.677169\pi$
$271$	−2.10289e7			−1.05659			−0.528297	+	0.849059i	$0.677169\pi$
$272$		−	5.12870e6i		−	0.254859i
$273$	−2.80152e6	+	1.58931e7i	−0.137691	+	0.781127i
$274$	−3.37448e7			−1.64042
$275$			1.39398e6i			0.0670286i
$276$	−4.91796e7	−	8.66899e6i	−2.33915	−	0.412327i
$277$	−1.00837e7			−0.474440			−0.237220	−	0.971456i	$-0.576236\pi$
$277$	−1.00837e7			−0.474440			−0.237220	+	0.971456i	$0.576236\pi$
$278$			1.90583e7i			0.887054i
$279$	2.26972e7	+	8.25839e6i	1.04511	+	0.380262i
$280$	5.75854e7			2.62324
$281$		−	2.48557e7i		−	1.12023i	−0.828415	−	0.560114i	$-0.810757\pi$
$281$		−	2.48557e7i		−	1.12023i	0.828415	−	0.560114i	$-0.189243\pi$
$282$	9.68175e6	−	5.49249e7i	0.431724	−	2.44919i
$283$	521274.			0.0229989			0.0114994	−	0.999934i	$-0.496340\pi$
$283$	521274.			0.0229989			0.0114994	+	0.999934i	$0.496340\pi$
$284$		−	2.97997e7i		−	1.30094i
$285$	2.84326e7	+	5.01188e6i	1.22824	+	0.216504i
$286$	−1.61657e7			−0.691029
$287$		−	5.18833e7i		−	2.19473i
$288$	−4.77591e6	+	1.31260e7i	−0.199930	+	0.549486i
$289$	2.34240e7			0.970437
$290$		−	8.48192e6i		−	0.347776i
$291$	3.31983e6	−	1.88335e7i	0.134722	−	0.764281i
$292$	6.54800e7			2.63003
$293$			4.05435e7i			1.61183i	0.592034	+	0.805913i	$0.298325\pi$
$293$			4.05435e7i			1.61183i	−0.592034	+	0.805913i	$0.701675\pi$
$294$	−2.11478e7	−	3.72777e6i	−0.832190	−	0.146692i
$295$	−2.74318e7			−1.06853
$296$		−	7.87970e7i		−	3.03833i
$297$	1.35250e7	+	7.80582e6i	0.516261	+	0.297954i
$298$	−3.65947e7			−1.38283
$299$			1.92539e7i			0.720288i
$300$	1.13475e6	−	6.43747e6i	0.0420277	−	0.238425i
$301$	−2.10753e7			−0.772813
$302$			4.33785e7i			1.57490i
$303$	−3.21140e6	−	566081.i	−0.115443	−	0.0203493i
$304$	4.92413e7			1.75271
$305$			2.74119e6i			0.0966140i
$306$	8.22059e6	+	2.99106e6i	0.286905	+	0.104391i
$307$	−5.79208e6			−0.200179			−0.100090	−	0.994978i	$-0.531913\pi$
$307$	−5.79208e6			−0.200179			−0.100090	+	0.994978i	$0.531913\pi$
$308$		−	4.55526e7i		−	1.55905i
$309$	8.40087e6	−	4.76585e7i	0.284740	−	1.61534i
$310$	−6.20500e7			−2.08284
$311$			2.90695e7i			0.966399i	0.875510	+	0.483200i	$0.160525\pi$
$311$			2.90695e7i			0.966399i	−0.875510	+	0.483200i	$0.839475\pi$
$312$	3.99787e7	+	7.04714e6i	1.31633	+	0.232033i
$313$	1.22823e6			0.0400540			0.0200270	−	0.999799i	$-0.493625\pi$
$313$	1.22823e6			0.0400540			0.0200270	+	0.999799i	$0.493625\pi$
$314$			5.06908e7i			1.63734i
$315$	−1.36938e7	+	3.76359e7i	−0.438120	+	1.20412i
$316$	1.12534e8			3.56634
$317$			5.77468e7i			1.81280i	0.422419	+	0.906401i	$0.361181\pi$
$317$			5.77468e7i			1.81280i	−0.422419	+	0.906401i	$0.638819\pi$
$318$	−1.05592e7	+	5.99028e7i	−0.328359	+	1.86280i
$319$	−3.59312e6			−0.110688
$320$			1.53446e7i			0.468279i
$321$	8.46969e6	+	1.49297e6i	0.256066	+	0.0451374i
$322$	−7.94549e7			−2.37987
$323$		−	6.85126e6i		−	0.203312i
$324$	−5.61049e7	−	4.70573e7i	−1.64955	−	1.38354i
$325$	−2.52029e6			−0.0734175
$326$		−	1.99399e7i		−	0.575533i
$327$	−6.96233e6	+	3.94976e7i	−0.199118	+	1.12961i
$328$	−1.30511e8			−3.69850
$329$		−	6.05931e7i		−	1.70151i
$330$	−3.95089e7	−	6.96433e6i	−1.09939	−	0.193793i
$331$	−5.04285e7			−1.39057			−0.695283	−	0.718736i	$-0.744721\pi$
$331$	−5.04285e7			−1.39057			−0.695283	+	0.718736i	$0.744721\pi$
$332$			678399.i			0.0185383i
$333$	5.14990e7	+	1.87379e7i	1.39465	+	0.507445i
$334$	1.00367e8			2.69372
$335$		−	5.94690e7i		−	1.58182i
$336$	−1.18579e7	+	6.72704e7i	−0.312601	+	1.77340i
$337$	9.40636e6			0.245772			0.122886	−	0.992421i	$-0.460785\pi$
$337$	9.40636e6			0.245772			0.122886	+	0.992421i	$0.460785\pi$
$338$			3.93389e7i			1.01876i
$339$	−7.17811e7	−	1.26530e7i	−1.84251	−	0.324784i
$340$	−1.53458e7			−0.390438
$341$			2.62857e7i			0.662913i
$342$	−2.87176e7	+	7.89270e7i	−0.717910	+	1.97309i
$343$	2.56939e7			0.636720
$344$			5.30143e7i			1.30232i
$345$	−8.29477e6	+	4.70566e7i	−0.201998	+	1.14594i
$346$	3.68692e7			0.890093
$347$		−	1.08775e7i		−	0.260341i	−0.991492	−	0.130170i	$-0.958448\pi$
$347$		−	1.08775e7i		−	0.260341i	0.991492	−	0.130170i	$-0.0415524\pi$
$348$	1.65932e7	+	2.92491e6i	0.393723	+	0.0694025i
$349$	5.37912e7			1.26542			0.632710	−	0.774389i	$-0.281943\pi$
$349$	5.37912e7			1.26542			0.632710	+	0.774389i	$0.281943\pi$
$350$		−	1.04004e7i		−	0.242576i
$351$	−1.41127e7	+	2.44529e7i	−0.326354	+	0.565470i
$352$	−1.52013e7			−0.348540
$353$		−	6.11813e7i		−	1.39090i	−0.718576	−	0.695448i	$-0.755206\pi$
$353$		−	6.11813e7i		−	1.39090i	0.718576	−	0.695448i	$-0.244794\pi$
$354$	1.38534e7	−	7.85908e7i	0.312281	−	1.77158i
$355$	−2.85133e7			−0.637327
$356$			8.29946e7i			1.83950i
$357$	9.35976e6	+	1.64987e6i	0.205712	+	0.0362614i
$358$	−4.85759e7			−1.05870
$359$		−	5.67180e7i		−	1.22585i	−0.790141	−	0.612926i	$-0.789993\pi$
$359$		−	5.67180e7i		−	1.22585i	0.790141	−	0.612926i	$-0.210007\pi$
$360$	9.46719e7	+	3.44464e7i	2.02915	+	0.738306i
$361$	1.87341e7			0.398208
$362$			1.05810e8i			2.23049i
$363$	5.35323e6	−	3.03691e7i	0.111917	−	0.634909i
$364$	8.23578e7			1.70766
$365$		−	6.26534e7i		−	1.28844i
$366$	−7.85339e6	−	1.38433e6i	−0.160182	−	0.0282357i
$367$	7.66618e7			1.55089			0.775445	−	0.631415i	$-0.217526\pi$
$367$	7.66618e7			1.55089			0.775445	+	0.631415i	$0.217526\pi$
$368$			8.14956e7i			1.63527i
$369$	3.10355e7	−	8.52975e7i	0.617703	−	1.69768i
$370$	−1.40789e8			−2.77948
$371$			6.60846e7i			1.29413i
$372$	2.13974e7	−	1.21388e8i	0.415654	−	2.35802i
$373$	4.41383e7			0.850528			0.425264	−	0.905069i	$-0.360181\pi$
$373$	4.41383e7			0.850528			0.425264	+	0.905069i	$0.360181\pi$
$374$			9.52028e6i			0.181985i
$375$	4.86163e7	+	8.56971e6i	0.921910	+	0.162507i
$376$	−1.52420e8			−2.86734
$377$		−	6.49627e6i		−	0.121238i
$378$	−1.00909e8	−	5.82387e7i	−1.86834	−	1.07829i
$379$	−447294.			−0.00821628			−0.00410814	−	0.999992i	$-0.501308\pi$
$379$	−447294.			−0.00821628			−0.00410814	+	0.999992i	$0.501308\pi$
$380$		−	1.47337e8i		−	2.68510i
$381$	4.41671e6	−	2.50562e7i	0.0798589	−	0.453043i
$382$	7.61733e7			1.36651
$383$			1.76079e7i			0.313409i	0.987646	+	0.156704i	$0.0500870\pi$
$383$			1.76079e7i			0.313409i	−0.987646	+	0.156704i	$0.949913\pi$
$384$	−7.65679e7	−	1.34968e7i	−1.35224	−	0.238362i
$385$	−4.35862e7			−0.763777
$386$			1.85805e8i			3.23070i
$387$	−3.46483e7	−	1.26068e7i	−0.597791	−	0.217506i
$388$	−9.75949e7			−1.67083
$389$		−	3.61476e7i		−	0.614087i	−0.951695	−	0.307044i	$-0.900660\pi$
$389$		−	3.61476e7i		−	0.614087i	0.951695	−	0.307044i	$-0.0993398\pi$
$390$	1.25913e7	−	7.14311e7i	0.212265	−	1.20419i
$391$	1.13390e7			0.189690
$392$			5.86864e7i			0.974269i
$393$	7.62422e7	+	1.34394e7i	1.25608	+	0.221412i
$394$	−1.34857e8			−2.20488
$395$		−	1.07676e8i		−	1.74714i
$396$	2.72486e7	−	7.48897e7i	0.438792	−	1.20597i
$397$	8.97996e7			1.43517			0.717584	−	0.696472i	$-0.245248\pi$
$397$	8.97996e7			1.43517			0.717584	+	0.696472i	$0.245248\pi$
$398$		−	1.70766e8i		−	2.70865i
$399$	−1.58406e7	+	8.98643e7i	−0.249375	+	1.41471i
$400$	−1.06675e7			−0.166680
$401$		−	4.60800e7i		−	0.714627i	−0.933984	−	0.357314i	$-0.883693\pi$
$401$		−	4.60800e7i		−	0.714627i	0.933984	−	0.357314i	$-0.116307\pi$
$402$	1.70376e8	+	3.00326e7i	2.62259	+	0.462290i
$403$	−4.75238e7			−0.726100
$404$			1.66414e7i			0.252375i
$405$	−4.50260e7	+	5.36830e7i	−0.677794	+	0.808112i
$406$	2.68080e7			0.400578
$407$			5.96411e7i			0.884632i
$408$	4.15019e6	−	2.35442e7i	0.0611065	−	0.346660i
$409$	8.28049e7			1.21028			0.605140	−	0.796119i	$-0.293117\pi$
$409$	8.28049e7			1.21028			0.605140	+	0.796119i	$0.293117\pi$
$410$			2.33187e8i			3.38340i
$411$	−6.31652e7	−	1.11343e7i	−0.909814	−	0.160375i
$412$	−2.46965e8			−3.53138
$413$		−	8.67011e7i		−	1.23076i
$414$	−1.30626e8	−	4.75283e7i	−1.84089	−	0.669809i
$415$	649114.			0.00908189
$416$		−	2.74835e7i		−	0.381762i
$417$	−6.28838e6	+	3.56742e7i	−0.0867223	+	0.491979i
$418$	−9.14055e7			−1.25154
$419$			1.33556e7i			0.181560i	0.995871	+	0.0907800i	$0.0289360\pi$
$419$			1.33556e7i			0.181560i	−0.995871	+	0.0907800i	$0.971064\pi$
$420$	2.01282e8	+	3.54805e7i	2.71680	+	0.478897i
$421$	−1.41520e8			−1.89658			−0.948291	−	0.317402i	$-0.897190\pi$
$421$	−1.41520e8			−1.89658			−0.948291	+	0.317402i	$0.897190\pi$
$422$			1.28065e8i			1.70409i
$423$	3.62455e7	−	9.96166e7i	0.478887	−	1.31617i
$424$	1.66234e8			2.18083
$425$			1.48424e6i			0.0193347i
$426$	1.43996e7	−	8.16893e7i	0.186260	−	1.05666i
$427$	−8.66384e6			−0.111282
$428$		−	4.38897e7i		−	0.559798i
$429$	−3.02597e7	−	5.33395e6i	−0.383260	−	0.0675581i
$430$	9.47221e7			1.19137
$431$		−	8.49447e7i		−	1.06097i	−0.847693	−	0.530487i	$-0.822009\pi$
$431$		−	8.49447e7i		−	1.06097i	0.847693	−	0.530487i	$-0.177991\pi$
$432$	−5.97344e7	+	1.03501e8i	−0.740924	+	1.28379i
$433$	−1.40924e8			−1.73589			−0.867945	−	0.496661i	$-0.834559\pi$
$433$	−1.40924e8			−1.73589			−0.867945	+	0.496661i	$0.834559\pi$
$434$		−	1.96116e8i		−	2.39907i
$435$	2.79865e6	−	1.58769e7i	0.0340002	−	0.192884i
$436$	2.04676e8			2.46949
$437$			1.08867e8i			1.30453i
$438$	1.79499e8	+	3.16407e7i	2.13619	+	0.376551i
$439$	−2.80817e7			−0.331917			−0.165959	−	0.986133i	$-0.553072\pi$
$439$	−2.80817e7			−0.331917			−0.165959	+	0.986133i	$0.553072\pi$
$440$			1.09640e8i			1.28709i
$441$	−3.83554e7	−	1.39556e7i	−0.447209	−	0.162717i
$442$	−1.72124e7			−0.199331
$443$		−	1.15813e8i		−	1.33213i	−0.745894	−	0.666064i	$-0.767978\pi$
$443$		−	1.15813e8i		−	1.33213i	0.745894	−	0.666064i	$-0.232022\pi$
$444$	4.85497e7	−	2.75425e8i	0.554674	−	3.14669i
$445$	7.94119e7			0.901167
$446$		−	1.02773e7i		−	0.115845i
$447$	−6.84996e7	−	1.20746e7i	−0.766947	−	0.135192i
$448$	−4.84982e7			−0.539376
$449$			1.69002e7i			0.186703i	0.995633	+	0.0933516i	$0.0297581\pi$
$449$			1.69002e7i			0.186703i	−0.995633	+	0.0933516i	$0.970242\pi$
$450$	6.22132e6	−	1.70986e7i	0.0682724	−	0.187639i
$451$	9.87832e7			1.07685
$452$			3.71968e8i			4.02801i
$453$	−1.43129e7	+	8.11979e7i	−0.153969	+	0.873475i
$454$	1.07729e8			1.15124
$455$		−	7.88026e7i		−	0.836578i
$456$	2.26051e8	+	3.98465e7i	2.38403	+	0.420239i
$457$	3.24252e7			0.339730			0.169865	−	0.985467i	$-0.445667\pi$
$457$	3.24252e7			0.339730			0.169865	+	0.985467i	$0.445667\pi$
$458$			7.19128e7i			0.748531i
$459$	1.44008e7	+	8.31124e6i	0.148918	+	0.0859464i
$460$	2.43846e8			2.50520
$461$		−	1.75464e7i		−	0.179096i	−0.995983	−	0.0895479i	$-0.971458\pi$
$461$		−	1.75464e7i		−	0.179096i	0.995983	−	0.0895479i	$-0.0285422\pi$
$462$	2.20115e7	−	1.24872e8i	0.223216	−	1.26631i
$463$	−6.29362e7			−0.634100			−0.317050	−	0.948409i	$-0.602692\pi$
$463$	−6.29362e7			−0.634100			−0.317050	+	0.948409i	$0.602692\pi$
$464$		−	2.74966e7i		−	0.275248i
$465$	−1.16148e8	−	2.04737e7i	−1.15519	−	0.203628i
$466$	2.28876e8			2.26174
$467$			9.75511e7i			0.957815i	0.877865	+	0.478907i	$0.158967\pi$
$467$			9.75511e7i			0.957815i	−0.877865	+	0.478907i	$0.841033\pi$
$468$	1.35398e8	+	4.92647e7i	1.32092	+	0.480616i
$469$	1.87959e8			1.82198
$470$			2.72333e8i			2.62305i
$471$	−1.67257e7	+	9.48855e7i	−0.160074	+	0.908107i
$472$	−2.18094e8			−2.07404
$473$		−	4.01263e7i		−	0.379181i
$474$	3.08488e8	+	5.43778e7i	2.89669	+	0.510607i
$475$	−1.42504e7			−0.132968
$476$		−	4.85021e7i		−	0.449717i
$477$	−3.95304e7	+	1.08645e8i	−0.364230	+	1.00105i
$478$	1.38695e8			1.26992
$479$			1.06227e8i			0.966562i	0.875465	+	0.483281i	$0.160555\pi$
$479$			1.06227e8i			0.966562i	−0.875465	+	0.483281i	$0.839445\pi$
$480$	1.18402e7	−	6.71697e7i	0.107062	−	0.607365i
$481$	−1.07830e8			−0.968952
$482$		−	1.10756e8i		−	0.989064i
$483$	−1.48728e8	−	2.62165e7i	−1.31993	−	0.232667i
$484$	−1.57372e8			−1.38800
$485$			9.33820e7i			0.818536i
$486$	−1.31061e8	−	1.56108e8i	−1.14173	−	1.35993i
$487$	4.42572e6			0.0383175			0.0191587	−	0.999816i	$-0.493901\pi$
$487$	4.42572e6			0.0383175			0.0191587	+	0.999816i	$0.493901\pi$
$488$			2.17936e7i			0.187530i
$489$	6.57927e6	−	3.73245e7i	0.0562667	−	0.319203i
$490$	1.04857e8			0.891266
$491$			7.65408e6i			0.0646619i	0.999477	+	0.0323309i	$0.0102931\pi$
$491$			7.65408e6i			0.0646619i	−0.999477	+	0.0323309i	$0.989707\pi$
$492$	−4.56184e8	−	8.04126e7i	−3.83040	−	0.675194i
$493$	−3.82577e6			−0.0319285
$494$		−	1.65259e8i		−	1.37083i
$495$	−7.16568e7	−	2.60723e7i	−0.590802	−	0.214963i
$496$	−2.01153e8			−1.64847
$497$		−	9.01195e7i		−	0.734090i
$498$	−327810.	+	1.85968e6i	−0.00265421	+	0.0150574i
$499$	−1.89730e8			−1.52698			−0.763492	−	0.645818i	$-0.776517\pi$
$499$	−1.89730e8			−1.52698			−0.763492	+	0.645818i	$0.776517\pi$
$500$		−	2.51929e8i		−	2.01543i
$501$	1.87872e8	+	3.31167e7i	1.49400	+	0.263350i
$502$	−1.47409e8			−1.16524
$503$			6.14799e7i			0.483091i	0.970389	+	0.241546i	$0.0776543\pi$
$503$			6.14799e7i			0.483091i	−0.970389	+	0.241546i	$0.922346\pi$
$504$	−1.08872e8	+	2.99221e8i	−0.850400	+	2.33723i
$505$	1.59230e7			0.123638
$506$		−	1.51278e8i		−	1.16768i
$507$	−1.29801e7	+	7.36364e7i	−0.0995985	+	0.565026i
$508$	−1.29840e8			−0.990418
$509$		−	5.20433e7i		−	0.394649i	−0.980338	−	0.197325i	$-0.936775\pi$
$509$		−	5.20433e7i		−	0.394649i	0.980338	−	0.197325i	$-0.0632253\pi$
$510$	−4.20671e7	−	7.41526e6i	−0.317126	−	0.0559005i
$511$	1.98023e8			1.48406
$512$			2.90961e8i			2.16783i
$513$	−7.97973e7	+	1.38264e8i	−0.591066	+	1.02413i
$514$	−1.27276e8			−0.937253
$515$			2.36304e8i			1.73001i
$516$	−3.26641e7	+	1.85305e8i	−0.237750	+	1.34877i
$517$	1.15366e8			0.834847
$518$		−	4.44978e8i		−	3.20147i
$519$	6.90135e7	+	1.21652e7i	0.493665	+	0.0870194i
$520$	−1.98226e8			−1.40977
$521$			1.39429e8i			0.985915i	0.870053	+	0.492958i	$0.164084\pi$
$521$			1.39429e8i			0.985915i	−0.870053	+	0.492958i	$0.835916\pi$
$522$	4.40731e7	+	1.60360e7i	0.309858	+	0.112742i
$523$	7.11063e7			0.497053			0.248527	−	0.968625i	$-0.420054\pi$
$523$	7.11063e7			0.497053			0.248527	+	0.968625i	$0.420054\pi$
$524$		−	3.95085e8i		−	2.74598i
$525$	3.43167e6	−	1.94680e7i	0.0237153	−	0.134538i
$526$	2.01007e8			1.38119
$527$			2.79877e7i			0.191221i
$528$	−1.28079e8	−	2.25769e7i	−0.870117	−	0.153378i
$529$	−3.21420e7			−0.217123
$530$		−	2.97015e8i		−	1.99503i
$531$	5.18628e7	−	1.42539e8i	0.346396	−	0.952029i
$532$	4.65675e8			3.09277
$533$			1.78597e8i			1.17949i
$534$	−4.01040e7	+	2.27511e8i	−0.263368	+	1.49410i
$535$	−4.19951e7			−0.274244
$536$		−	4.72804e8i		−	3.07034i
$537$	−9.09267e7	−	1.60279e7i	−0.587176	−	0.103503i
$538$	4.43422e7			0.284755
$539$		−	4.44195e7i		−	0.283666i
$540$	3.09690e8	+	1.78734e8i	1.96673	+	1.13508i
$541$	−1.64328e8			−1.03782			−0.518908	−	0.854830i	$-0.673661\pi$
$541$	−1.64328e8			−1.03782			−0.518908	+	0.854830i	$0.673661\pi$
$542$			2.98721e8i			1.87615i
$543$	−3.49125e7	+	1.98060e8i	−0.218063	+	1.23708i
$544$	−1.61856e7			−0.100538
$545$		−	1.95840e8i		−	1.20980i
$546$	2.25766e8	+	3.97963e7i	1.38701	+	0.244492i
$547$	1.07276e8			0.655449			0.327725	−	0.944773i	$-0.393718\pi$
$547$	1.07276e8			0.655449			0.327725	+	0.944773i	$0.393718\pi$
$548$			3.27321e8i			1.98899i
$549$	−1.42436e7	−	5.18253e6i	−0.0860800	−	0.0313202i
$550$	1.98019e7			0.119020
$551$		−	3.67318e7i		−	0.219577i
$552$	−6.59469e7	+	3.74120e8i	−0.392082	+	2.22430i
$553$	3.40323e8			2.01241
$554$			1.43242e8i			0.842442i
$555$	−2.63535e8	−	4.64539e7i	−1.54156	−	0.271734i
$556$	1.84863e8			1.07554
$557$			1.08061e8i			0.625319i	0.949865	+	0.312659i	$0.101220\pi$
$557$			1.08061e8i			0.625319i	−0.949865	+	0.312659i	$0.898780\pi$
$558$	1.17312e8	−	3.22420e8i	0.675214	−	1.85575i
$559$	7.25473e7			0.415323
$560$		−	3.33545e8i		−	1.89929i
$561$	−3.14126e6	+	1.78205e7i	−0.0177916	+	0.100933i
$562$	−3.53081e8			−1.98914
$563$		−	3.11154e8i		−	1.74361i	−0.489849	−	0.871807i	$-0.662948\pi$
$563$		−	3.11154e8i		−	1.74361i	0.489849	−	0.871807i	$-0.337052\pi$
$564$	−5.32765e8	−	9.39117e7i	−2.96960	−	0.523458i
$565$	3.55911e8			1.97331
$566$		−	7.40483e6i		−	0.0408381i
$567$	−1.69671e8	−	1.42310e8i	−0.930805	−	0.780701i
$568$	−2.26693e8			−1.23707
$569$			1.59533e8i			0.865989i	0.901397	+	0.432995i	$0.142543\pi$
$569$			1.59533e8i			0.865989i	−0.901397	+	0.432995i	$0.857457\pi$
$570$	7.11949e7	−	4.03892e8i	0.384436	−	2.18092i
$571$	−1.20037e8			−0.644771			−0.322386	−	0.946608i	$-0.604485\pi$
$571$	−1.20037e8			−0.644771			−0.322386	+	0.946608i	$0.604485\pi$
$572$			1.56805e8i			0.837862i
$573$	1.42585e8	+	2.51338e7i	0.757896	+	0.133596i
$574$	−7.37015e8			−3.89709
$575$		−	2.35848e7i		−	0.124059i
$576$	−7.97323e7	−	2.90106e7i	−0.417222	−	0.151806i
$577$	1.65071e8			0.859295			0.429648	−	0.902997i	$-0.358638\pi$
$577$	1.65071e8			0.859295			0.429648	+	0.902997i	$0.358638\pi$
$578$		−	3.32743e8i		−	1.72316i
$579$	−6.13074e7	+	3.47800e8i	−0.315847	+	1.79182i
$580$	−8.22735e7			−0.421673
$581$			2.05160e6i			0.0104608i
$582$	−2.67535e8	−	4.71590e7i	−1.35710	−	0.239219i
$583$	−1.25822e8			−0.634965
$584$		−	4.98121e8i		−	2.50090i
$585$	4.71381e7	−	1.29554e8i	0.235453	−	0.647115i
$586$	5.75930e8			2.86205
$587$		−	3.29747e8i		−	1.63029i	−0.579253	−	0.815147i	$-0.696656\pi$
$587$		−	3.29747e8i		−	1.63029i	0.579253	−	0.815147i	$-0.303344\pi$
$588$	−3.61588e7	+	2.05131e8i	−0.177862	+	1.00902i
$589$	−2.68713e8			−1.31505
$590$			3.89675e8i			1.89735i
$591$	−2.52432e8	−	4.44967e7i	−1.22287	−	0.215559i
$592$	−4.56407e8			−2.19982
$593$			1.06205e8i			0.509309i	0.967032	+	0.254654i	$0.0819617\pi$
$593$			1.06205e8i			0.509309i	−0.967032	+	0.254654i	$0.918038\pi$
$594$	1.10884e8	−	1.92127e8i	0.529064	−	0.916702i
$595$	−4.64083e7			−0.220315
$596$			3.54963e8i			1.67666i
$597$	5.63451e7	−	3.19648e8i	0.264809	−	1.50227i
$598$	2.73507e8			1.27898
$599$			3.84048e7i			0.178692i	0.996001	+	0.0893461i	$0.0284777\pi$
$599$			3.84048e7i			0.178692i	−0.996001	+	0.0893461i	$0.971522\pi$
$600$	−4.89712e7	−	8.63227e6i	−0.226719	−	0.0399642i
$601$	1.23593e8			0.569340			0.284670	−	0.958626i	$-0.408116\pi$
$601$	1.23593e8			0.569340			0.284670	+	0.958626i	$0.408116\pi$
$602$			2.99380e8i			1.37225i
$603$	3.09009e8	+	1.12433e8i	1.40935	+	0.512792i
$604$	4.20766e8			1.90954
$605$			1.50578e8i			0.679981i
$606$	−8.04132e6	+	4.56187e7i	−0.0361334	+	0.204986i
$607$	−3.71983e8			−1.66325			−0.831624	−	0.555339i	$-0.812589\pi$
$607$	−3.71983e8			−1.66325			−0.831624	+	0.555339i	$0.812589\pi$
$608$		−	1.55400e8i		−	0.691416i
$609$	5.01806e7	+	8.84545e6i	0.222169	+	0.0391623i
$610$	3.89393e7			0.171553
$611$			2.08579e8i			0.914422i
$612$	2.90129e7	−	7.97386e7i	0.126572	−	0.347868i
$613$	5.72915e7			0.248719			0.124360	−	0.992237i	$-0.460312\pi$
$613$	5.72915e7			0.248719			0.124360	+	0.992237i	$0.460312\pi$
$614$			8.22779e7i			0.355450i
$615$	−7.69413e7	+	4.36491e8i	−0.330776	+	1.87651i
$616$	−3.46529e8			−1.48251
$617$		−	3.90063e8i		−	1.66065i	−0.557277	−	0.830327i	$-0.688154\pi$
$617$		−	3.90063e8i		−	1.66065i	0.557277	−	0.830327i	$-0.311846\pi$
$618$	−6.77001e8	−	1.19336e8i	−2.86829	−	0.505601i
$619$	−3.46859e8			−1.46245			−0.731225	−	0.682136i	$-0.761051\pi$
$619$	−3.46859e8			−1.46245			−0.731225	+	0.682136i	$0.761051\pi$
$620$			6.01877e8i			2.52541i
$621$	−2.28830e8	−	1.32066e8i	−0.955515	−	0.551464i
$622$	4.12940e8			1.71599
$623$			2.50990e8i			1.03799i
$624$	4.08183e7	−	2.31564e8i	0.167997	−	0.953054i
$625$	−2.68507e8			−1.09981
$626$		−	1.74473e7i		−	0.0711221i
$627$	−1.71097e8	−	3.01597e7i	−0.694129	−	0.122356i
$628$	4.91694e8			1.98525
$629$			6.35028e7i			0.255177i
$630$	5.34627e8	+	1.94524e8i	2.13810	+	0.777950i
$631$	−2.54482e8			−1.01291			−0.506453	−	0.862268i	$-0.669044\pi$
$631$	−2.54482e8			−1.01291			−0.506453	+	0.862268i	$0.669044\pi$
$632$		−	8.56072e8i		−	3.39124i
$633$	−4.22557e7	+	2.39718e8i	−0.166600	+	0.945126i
$634$	8.20308e8			3.21891
$635$			1.24235e8i			0.485204i
$636$	5.81049e8	+	1.02423e8i	2.25861	+	0.398130i
$637$	8.03092e7			0.310704
$638$			5.10412e7i			0.196544i
$639$	5.39076e7	−	1.48159e8i	0.206608	−	0.567838i
$640$	3.79645e8			1.44823
$641$			3.73699e8i			1.41889i	0.704762	+	0.709444i	$0.251054\pi$
$641$			3.73699e8i			1.41889i	−0.704762	+	0.709444i	$0.748946\pi$
$642$	2.12080e7	−	1.20314e8i	0.0801484	−	0.454685i
$643$	3.06960e7			0.115464			0.0577322	−	0.998332i	$-0.481613\pi$
$643$	3.06960e7			0.115464			0.0577322	+	0.998332i	$0.481613\pi$
$644$			7.70702e8i			2.88555i
$645$	1.77305e8	+	3.12540e7i	0.660759	+	0.116473i
$646$	−9.73239e7			−0.361012
$647$			1.13945e8i			0.420710i	0.977625	+	0.210355i	$0.0674620\pi$
$647$			1.13945e8i			0.420710i	−0.977625	+	0.210355i	$0.932538\pi$
$648$	−3.57976e8	+	4.26803e8i	−1.31561	+	1.56856i
$649$	1.65075e8			0.603874
$650$			3.58013e7i			0.130364i
$651$	6.47094e7	−	3.67099e8i	0.234544	−	1.33058i
$652$	−1.93415e8			−0.697825
$653$		−	2.80341e8i		−	1.00681i	−0.864050	−	0.503405i	$-0.832080\pi$
$653$		−	2.80341e8i		−	1.00681i	0.864050	−	0.503405i	$-0.167920\pi$
$654$	5.61073e8	+	9.89016e7i	2.00579	+	0.353566i
$655$	−3.78030e8			−1.34525
$656$			7.55943e8i			2.67780i
$657$	3.25555e8	+	1.18453e8i	1.14796	+	0.417687i
$658$	−8.60739e8			−3.02130
$659$		−	3.41665e8i		−	1.19384i	−0.802302	−	0.596918i	$-0.796392\pi$
$659$		−	3.41665e8i		−	1.19384i	0.802302	−	0.596918i	$-0.203608\pi$
$660$	−6.75531e7	+	3.83232e8i	−0.234971	+	1.33300i
$661$	−1.62348e8			−0.562138			−0.281069	−	0.959688i	$-0.590689\pi$
$661$	−1.62348e8			−0.562138			−0.281069	+	0.959688i	$0.590689\pi$
$662$			7.16349e8i			2.46917i
$663$	−3.22190e7	−	5.67932e6i	−0.110553	−	0.0194875i
$664$	5.16073e6			0.0176282
$665$		−	4.45573e8i		−	1.51514i
$666$	2.66177e8	−	7.31556e8i	0.901047	−	2.47642i
$667$	6.07919e7			0.204865
$668$		−	9.73549e8i		−	3.26609i
$669$	3.39106e6	−	1.92376e7i	0.0113255	−	0.0642500i
$670$	−8.44772e8			−2.80876
$671$		−	1.64955e7i		−	0.0546007i
$672$	2.12297e8	+	3.74221e7i	0.699579	+	0.123316i
$673$	3.69184e8			1.21115			0.605575	−	0.795788i	$-0.292943\pi$
$673$	3.69184e8			1.21115			0.605575	+	0.795788i	$0.292943\pi$
$674$		−	1.33620e8i		−	0.436406i
$675$	1.72871e7	−	2.99532e7i	0.0562097	−	0.0973938i
$676$	3.81582e8			1.23523
$677$		−	1.99823e8i		−	0.643989i	−0.946741	−	0.321995i	$-0.895647\pi$
$677$		−	1.99823e8i		−	0.643989i	0.946741	−	0.321995i	$-0.104353\pi$
$678$	−1.79739e8	+	1.01967e9i	−0.576705	+	3.27167i
$679$	−2.95144e8			−0.942811
$680$			1.16739e8i			0.371269i
$681$	2.01653e8	+	3.55458e7i	0.638504	+	0.112551i
$682$	3.73395e8			1.17711
$683$		−	2.53072e8i		−	0.794295i	−0.917755	−	0.397147i	$-0.870000\pi$
$683$		−	2.53072e8i		−	0.794295i	0.917755	−	0.397147i	$-0.130000\pi$
$684$	7.65582e8	+	2.78557e8i	2.39234	+	0.870454i
$685$	3.13191e8			0.974400
$686$		−	3.64989e8i		−	1.13059i
$687$	−2.37280e7	+	1.34610e8i	−0.0731796	+	0.415151i
$688$	3.07069e8			0.942910
$689$		−	2.27482e8i		−	0.695488i
$690$	6.68450e8	+	1.17829e8i	2.03480	+	0.358679i
$691$	−9.31069e7			−0.282194			−0.141097	−	0.989996i	$-0.545063\pi$
$691$	−9.31069e7			−0.282194			−0.141097	+	0.989996i	$0.545063\pi$
$692$		−	3.57626e8i		−	1.07922i
$693$	8.24045e7	−	2.26479e8i	0.247600	−	0.680501i
$694$	−1.54518e8			−0.462275
$695$		−	1.76883e8i		−	0.526904i
$696$	2.22505e7	−	1.26228e8i	0.0659950	−	0.374393i
$697$	1.05179e8			0.310622
$698$		−	7.64117e8i		−	2.24695i
$699$	4.28422e8	+	7.55188e7i	1.25441	+	0.221118i
$700$	−1.00883e8			−0.294119
$701$		−	2.30462e8i		−	0.669029i	−0.942391	−	0.334515i	$-0.891428\pi$
$701$		−	2.30462e8i		−	0.669029i	0.942391	−	0.334515i	$-0.108572\pi$
$702$	3.47360e8	+	2.00474e8i	1.00408	+	0.579492i
$703$	−6.09699e8			−1.75489
$704$		−	9.23382e7i		−	0.264645i
$705$	−8.98577e7	+	5.09766e8i	−0.256441	+	1.45480i
$706$	−8.69096e8			−2.46975
$707$			5.03265e7i			0.142409i
$708$	−7.62320e8	−	1.34376e8i	−2.14802	−	0.378636i
$709$	2.98965e8			0.838844			0.419422	−	0.907791i	$-0.362233\pi$
$709$	2.98965e8			0.838844			0.419422	+	0.907791i	$0.362233\pi$
$710$			4.05038e8i			1.13167i
$711$	5.59500e8	+	2.03574e8i	1.55665	+	0.566387i
$712$	6.31358e8			1.74918
$713$		−	4.44727e8i		−	1.22694i
$714$	2.34368e7	−	1.32958e8i	0.0643877	−	0.365274i
$715$	1.50036e8			0.410467
$716$			4.71180e8i			1.28365i
$717$	2.59615e8	+	4.57630e7i	0.704325	+	0.124153i
$718$	−8.05693e8			−2.17669
$719$			1.64291e8i			0.442004i	0.975273	+	0.221002i	$0.0709327\pi$
$719$			1.64291e8i			0.442004i	−0.975273	+	0.221002i	$0.929067\pi$
$720$	1.99520e8	−	5.48357e8i	0.534550	−	1.46915i
$721$	−7.46866e8			−1.99268
$722$		−	2.66122e8i		−	0.707081i
$723$	3.65443e7	−	2.07318e8i	0.0966953	−	0.548556i
$724$	1.02634e9			2.70443
$725$			7.95749e6i			0.0208815i
$726$	−4.31400e8	−	7.60439e7i	−1.12738	−	0.198726i
$727$	3.45785e8			0.899917			0.449959	−	0.893049i	$-0.351439\pi$
$727$	3.45785e8			0.899917			0.449959	+	0.893049i	$0.351439\pi$
$728$		−	6.26514e8i		−	1.62382i
$729$	−1.93817e8	−	3.35454e8i	−0.500276	−	0.865866i
$730$	−8.90007e8			−2.28783
$731$		−	4.27244e7i		−	0.109377i
$732$	−1.34279e7	+	7.61768e7i	−0.0342353	+	0.194218i
$733$	4.47397e7			0.113601			0.0568004	−	0.998386i	$-0.481910\pi$
$733$	4.47397e7			0.113601			0.0568004	+	0.998386i	$0.481910\pi$
$734$		−	1.08900e9i		−	2.75385i
$735$	1.96275e8	+	3.45979e7i	0.494315	+	0.0871341i
$736$	2.57190e8			0.645091
$737$			3.57864e8i			0.893954i
$738$	−1.21167e9	−	4.40867e8i	−3.01450	−	1.09683i
$739$	2.83190e8			0.701689			0.350845	−	0.936434i	$-0.385895\pi$
$739$	2.83190e8			0.701689			0.350845	+	0.936434i	$0.385895\pi$
$740$			1.36563e9i			3.37007i
$741$	5.45279e7	−	3.09339e8i	0.134018	−	0.760291i
$742$	9.38748e8			2.29793
$743$		−	5.52126e7i		−	0.134608i	−0.997733	−	0.0673042i	$-0.978560\pi$
$743$		−	5.52126e7i		−	0.134608i	0.997733	−	0.0673042i	$-0.0214398\pi$
$744$	−9.23427e8	−	1.62775e8i	−2.24225	−	0.395246i
$745$	3.39640e8			0.821392
$746$		−	6.26995e8i		−	1.51025i
$747$	−1.22722e6	+	3.37288e6i	−0.00294416	+	0.00809168i
$748$	9.23455e7			0.220654
$749$		−	1.32730e8i		−	0.315881i
$750$	1.21735e8	−	6.90607e8i	0.288557	−	1.63699i
$751$	−7.58420e7			−0.179057			−0.0895283	−	0.995984i	$-0.528536\pi$
$751$	−7.58420e7			−0.179057			−0.0895283	+	0.995984i	$0.528536\pi$
$752$			8.82846e8i			2.07602i
$753$	−2.75928e8	−	4.86384e7i	−0.646265	−	0.113919i
$754$	−9.22811e7			−0.215277
$755$		−	4.02602e8i		−	0.935482i
$756$	−5.64908e8	+	9.78809e8i	−1.30741	+	2.26534i
$757$	−7.46787e8			−1.72151			−0.860754	−	0.509021i	$-0.830007\pi$
$757$	−7.46787e8			−1.72151			−0.860754	+	0.509021i	$0.830007\pi$
$758$			6.35392e6i			0.0145893i
$759$	4.99150e7	−	2.83170e8i	0.114158	−	0.647622i
$760$	−1.12082e9			−2.55327
$761$			9.60100e7i			0.217853i	0.994050	+	0.108926i	$0.0347412\pi$
$761$			9.60100e7i			0.217853i	−0.994050	+	0.108926i	$0.965259\pi$
$762$	−3.55929e8	−	6.27404e7i	−0.804449	−	0.141802i
$763$	6.18974e8			1.39347
$764$		−	7.38871e8i		−	1.65687i
$765$	−7.62965e7	−	2.77605e7i	−0.170420	−	0.0620073i
$766$	2.50125e8			0.556506
$767$			2.98450e8i			0.661433i
$768$	−1.56812e8	+	8.89603e8i	−0.346176	+	1.96387i
$769$	−2.43157e8			−0.534696			−0.267348	−	0.963600i	$-0.586147\pi$
$769$	−2.43157e8			−0.534696			−0.267348	+	0.963600i	$0.586147\pi$
$770$			6.19152e8i			1.35621i
$771$	−2.38241e8	−	4.19953e7i	−0.519821	−	0.0916300i
$772$	1.80229e9			3.91717
$773$			2.34450e8i			0.507588i	0.967258	+	0.253794i	$0.0816785\pi$
$773$			2.34450e8i			0.507588i	−0.967258	+	0.253794i	$0.918321\pi$
$774$	−1.79083e8	+	4.92188e8i	−0.386217	+	1.06147i
$775$	5.82135e7			0.125060
$776$			7.42426e8i			1.58880i
$777$	1.46823e8	−	8.32932e8i	0.312990	−	1.77561i
$778$	−5.13485e8			−1.09041
$779$			1.00984e9i			2.13619i
$780$	−6.92872e8	−	1.22134e8i	−1.46006	−	0.257367i
$781$	1.71583e8			0.360181
$782$		−	1.61073e8i		−	0.336824i
$783$	7.72070e7	+	4.45591e7i	0.160832	+	0.0928221i
$784$	3.39922e8			0.705394
$785$		−	4.70469e8i		−	0.972572i
$786$	1.90910e8	−	1.08304e9i	0.393152	−	2.23037i
$787$	3.56198e8			0.730748			0.365374	−	0.930861i	$-0.380941\pi$
$787$	3.56198e8			0.730748			0.365374	+	0.930861i	$0.380941\pi$
$788$			1.30809e9i			2.67338i
$789$	3.76255e8	+	6.63234e7i	0.766040	+	0.135032i
$790$	−1.52957e9			−3.10233
$791$			1.12490e9i			2.27291i
$792$	−5.69702e8	−	2.07286e8i	−1.14676	−	0.417248i
$793$	2.98234e7			0.0598051
$794$		−	1.27562e9i		−	2.54836i
$795$	9.80014e7	−	5.55966e8i	0.195043	−	1.10649i
$796$	−1.65641e9			−3.28419
$797$			1.51015e8i			0.298295i	0.988815	+	0.149147i	$0.0476529\pi$
$797$			1.51015e8i			0.298295i	−0.988815	+	0.149147i	$0.952347\pi$
$798$	1.27655e9	+	2.25020e8i	2.51205	+	0.442804i
$799$	1.22836e8			0.240816
$800$			3.36655e7i			0.0657529i
$801$	−1.50137e8	+	4.12634e8i	−0.292139	+	0.802911i
$802$	−6.54578e8			−1.26893
$803$			3.77026e8i			0.728156i
$804$	2.91312e8	−	1.65263e9i	0.560519	−	3.17985i
$805$	7.37433e8			1.41363
$806$			6.75088e8i			1.28930i
$807$	8.30019e7	+	1.46309e7i	0.157931	+	0.0278389i
$808$	1.26595e8			0.239984
$809$			7.57544e8i			1.43074i	0.698743	+	0.715372i	$0.253743\pi$
$809$			7.57544e8i			1.43074i	−0.698743	+	0.715372i	$0.746257\pi$
$810$	7.62580e8	+	6.39605e8i	1.43493	+	1.20353i
$811$	−4.00774e8			−0.751341			−0.375671	−	0.926753i	$-0.622588\pi$
$811$	−4.00774e8			−0.751341			−0.375671	+	0.926753i	$0.622588\pi$
$812$		−	2.60035e8i		−	0.485694i
$813$	−9.85643e7	+	5.59159e8i	−0.183421	+	1.04055i
$814$	8.47217e8			1.57080
$815$			1.85065e8i			0.341863i
$816$	−1.36372e8	−	2.40387e7i	−0.250990	−	0.0442425i
$817$	4.10203e8			0.752199
$818$		−	1.17626e9i		−	2.14904i
$819$	4.09468e8	+	1.48985e8i	0.745364	+	0.271201i
$820$	2.26189e9			4.10232
$821$			1.04305e8i			0.188485i	0.995549	+	0.0942427i	$0.0300430\pi$
$821$			1.04305e8i			0.188485i	−0.995549	+	0.0942427i	$0.969957\pi$
$822$	−1.58165e8	+	8.97277e8i	−0.284771	+	1.61552i
$823$	1.06473e9			1.91003			0.955014	−	0.296562i	$-0.0958402\pi$
$823$	1.06473e9			1.91003			0.955014	+	0.296562i	$0.0958402\pi$
$824$			1.87872e9i			3.35800i
$825$	3.70661e7	+	6.53373e6i	0.0660109	+	0.0116359i
$826$	−1.23161e9			−2.18541
$827$		−	4.57990e8i		−	0.809728i	−0.914377	−	0.404864i	$-0.867319\pi$
$827$		−	4.57990e8i		−	0.809728i	0.914377	−	0.404864i	$-0.132681\pi$
$828$	−4.61018e8	+	1.26706e9i	−0.812133	+	2.23205i
$829$	−4.06079e8			−0.712766			−0.356383	−	0.934340i	$-0.615990\pi$
$829$	−4.06079e8			−0.712766			−0.356383	+	0.934340i	$0.615990\pi$
$830$		−	9.22082e6i		−	0.0161263i
$831$	−4.72633e7	+	2.68127e8i	−0.0823609	+	0.467237i
$832$	1.66945e8			0.289870
$833$		−	4.72956e7i		−	0.0818249i
$834$	5.06761e8	+	8.93280e7i	0.873586	+	0.153989i
$835$	−9.31523e8			−1.60005
$836$			8.86622e8i			1.51747i
$837$	3.25975e8	−	5.64813e8i	0.555914	−	0.963226i
$838$	1.89719e8			0.322388
$839$			2.28417e8i			0.386760i	0.981124	+	0.193380i	$0.0619451\pi$
$839$			2.28417e8i			0.386760i	−0.981124	+	0.193380i	$0.938055\pi$
$840$	2.69908e8	−	1.53120e9i	0.455384	−	2.58341i
$841$	−2.05111e7			−0.0344828
$842$			2.01033e9i			3.36768i
$843$	−6.60914e8	−	1.16501e8i	−1.10322	−	0.194467i
$844$	1.24221e9			2.06618
$845$		−	3.65110e8i		−	0.605137i
$846$	−1.41508e9	−	5.14876e8i	−2.33706	−	0.850338i
$847$	−4.75920e8			−0.783219
$848$		−	9.62857e8i		−	1.57897i
$849$	2.44326e6	−	1.38607e7i	0.00399251	−	0.0226497i
$850$	2.10840e7			0.0343318
$851$		−	1.00907e9i		−	1.63731i
$852$	−7.92376e8	−	1.39674e8i	−1.28119	−	0.225838i
$853$	4.71971e8			0.760446			0.380223	−	0.924895i	$-0.375847\pi$
$853$	4.71971e8			0.760446			0.380223	+	0.924895i	$0.375847\pi$
$854$			1.23072e8i			0.197599i
$855$	2.66532e8	−	7.32533e8i	0.426434	−	1.17200i
$856$	−3.33879e8			−0.532313
$857$		−	6.64002e8i		−	1.05494i	−0.849574	−	0.527469i	$-0.823141\pi$
$857$		−	6.64002e8i		−	1.05494i	0.849574	−	0.527469i	$-0.176859\pi$
$858$	−7.57701e7	+	4.29847e8i	−0.119960	+	0.680537i
$859$	−6.40919e8			−1.01117			−0.505584	−	0.862777i	$-0.668723\pi$
$859$	−6.40919e8			−1.01117			−0.505584	+	0.862777i	$0.668723\pi$
$860$		−	9.18792e8i		−	1.44452i
$861$	−1.37958e9	−	2.43182e8i	−2.16141	−	0.380997i
$862$	−1.20666e9			−1.88392
$863$			3.00381e8i			0.467348i	0.972315	+	0.233674i	$0.0750749\pi$
$863$			3.00381e8i			0.467348i	−0.972315	+	0.233674i	$0.924925\pi$
$864$	3.26637e8	+	1.88515e8i	0.506436	+	0.292283i
$865$	−3.42188e8			−0.528709
$866$			2.00186e9i			3.08234i
$867$	1.09790e8	−	6.22845e8i	0.168464	−	0.955702i
$868$	−1.90230e9			−2.90884
$869$			6.47958e8i			0.987387i
$870$	−2.25535e8	−	3.97555e7i	−0.342496	−	0.0603726i
$871$	−6.47008e8			−0.979163
$872$		−	1.55701e9i		−	2.34824i
$873$	−4.85225e8	−	1.76549e8i	−0.729290	−	0.265352i
$874$	1.54649e9			2.31639
$875$		−	7.61876e8i		−	1.13726i
$876$	3.06911e8	−	1.74112e9i	0.456562	−	2.59010i
$877$	4.33548e8			0.642744			0.321372	−	0.946953i	$-0.395856\pi$
$877$	4.33548e8			0.642744			0.321372	+	0.946953i	$0.395856\pi$
$878$			3.98908e8i			0.589371i
$879$	1.07805e9	+	1.90031e8i	1.58735	+	0.279807i
$880$	6.35054e8			0.931886
$881$			7.04743e8i			1.03063i	0.857001	+	0.515315i	$0.172325\pi$
$881$			7.04743e8i			1.03063i	−0.857001	+	0.515315i	$0.827675\pi$
$882$	−1.98243e8	+	5.44848e8i	−0.288930	+	0.794090i
$883$	2.67242e8			0.388171			0.194086	−	0.980985i	$-0.437826\pi$
$883$	2.67242e8			0.388171			0.194086	+	0.980985i	$0.437826\pi$
$884$			1.66958e8i			0.241686i
$885$	−1.28575e8	+	7.29412e8i	−0.185493	+	1.05231i
$886$	−1.64515e9			−2.36540
$887$			1.38012e9i			1.97764i	0.149117	+	0.988820i	$0.452357\pi$
$887$			1.38012e9i			1.97764i	−0.149117	+	0.988820i	$0.547643\pi$
$888$	−2.09522e9	−	3.69329e8i	−2.99219	−	0.527441i
$889$	−3.92660e8			−0.558871
$890$		−	1.12807e9i		−	1.60016i
$891$	2.70950e8	−	3.23045e8i	0.383051	−	0.456699i
$892$	−9.96889e7			−0.140460
$893$			1.17936e9i			1.65613i
$894$	−1.71522e8	+	9.73054e8i	−0.240054	+	1.36183i
$895$	4.50840e8			0.628859
$896$			1.19991e9i			1.66811i
$897$	5.11963e8	+	9.02449e7i	0.709351	+	0.125039i
$898$	2.40071e8			0.331521
$899$			1.50051e8i			0.206518i
$900$	−1.65854e8	−	6.03460e7i	−0.227509	−	0.0827791i
$901$	−1.33969e8			−0.183159
$902$		−	1.40324e9i		−	1.91211i
$903$	−9.87818e7	+	5.60393e8i	−0.134157	+	0.761079i
$904$	2.82964e9			3.83024
$905$		−	9.82037e8i		−	1.32490i
$906$	1.15344e9	+	2.03319e8i	1.55099	+	0.273397i
$907$	5.76817e7			0.0773066			0.0386533	−	0.999253i	$-0.487693\pi$
$907$	5.76817e7			0.0773066			0.0386533	+	0.999253i	$0.487693\pi$
$908$		−	1.04496e9i		−	1.39586i
$909$	−3.01042e7	+	8.27380e7i	−0.0400808	+	0.110157i
$910$	−1.11941e9			−1.48547
$911$			1.21669e8i			0.160925i	0.996758	+	0.0804625i	$0.0256397\pi$
$911$			1.21669e8i			0.160925i	−0.996758	+	0.0804625i	$0.974360\pi$
$912$	2.30799e8	−	1.30933e9i	0.304263	−	1.72610i
$913$	−3.90614e6			−0.00513257
$914$		−	4.60608e8i		−	0.603244i
$915$	7.28884e7	+	1.28482e7i	0.0951471	+	0.0167718i
$916$	6.97544e8			0.907581
$917$		−	1.19481e9i		−	1.54949i
$918$	1.18063e8	−	2.04567e8i	0.152611	−	0.264427i
$919$	8.63064e8			1.11198			0.555989	−	0.831189i	$-0.312340\pi$
$919$	8.63064e8			1.11198			0.555989	+	0.831189i	$0.312340\pi$
$920$		−	1.85499e9i		−	2.38220i
$921$	−2.71480e7	+	1.54012e8i	−0.0347503	+	0.197140i
$922$	−2.49251e8			−0.318013
$923$			3.10217e8i			0.394513i
$924$	−1.21125e9	−	2.13509e8i	−1.53538	−	0.270645i
$925$	1.32084e8			0.166888
$926$			8.94025e8i			1.12594i
$927$	−1.22787e9	−	4.46759e8i	−1.54139	−	0.560834i
$928$	−8.67758e7			−0.108581
$929$		−	6.97473e8i		−	0.869922i	−0.900449	−	0.434961i	$-0.856762\pi$
$929$		−	6.97473e8i		−	0.869922i	0.900449	−	0.434961i	$-0.143238\pi$
$930$	−2.90834e8	+	1.64991e9i	−0.361573	+	2.05122i
$931$	4.54091e8			0.562722
$932$		−	2.22007e9i		−	2.74233i
$933$	7.72960e8	+	1.36251e8i	0.951726	+	0.167763i
$934$	1.38574e9			1.70075
$935$		−	8.83591e7i		−	0.108098i
$936$	3.74768e8	−	1.03001e9i	0.457019	−	1.25606i
$937$	9.72064e8			1.18161			0.590807	−	0.806813i	$-0.298809\pi$
$937$	9.72064e8			1.18161			0.590807	+	0.806813i	$0.298809\pi$
$938$		−	2.67000e9i		−	3.23521i
$939$	5.75682e6	−	3.26587e7i	0.00695321	−	0.0394459i
$940$	2.64160e9			3.18041
$941$		−	1.37520e9i		−	1.65043i	−0.564815	−	0.825217i	$-0.691053\pi$
$941$		−	1.37520e9i		−	1.65043i	0.564815	−	0.825217i	$-0.308947\pi$
$942$	1.34787e9	+	2.37592e8i	1.61249	+	0.284236i
$943$	−1.67131e9			−1.99307
$944$			1.26324e9i			1.50166i
$945$	9.36556e8	+	5.40522e8i	1.10978	+	0.640498i
$946$	−5.70004e8			−0.673294
$947$			1.81899e8i			0.214181i	0.994249	+	0.107091i	$0.0341535\pi$
$947$			1.81899e8i			0.214181i	−0.994249	+	0.107091i	$0.965847\pi$
$948$	5.27458e8	−	2.99229e9i	0.619103	−	3.51220i
$949$	−6.81652e8			−0.797562
$950$			2.02431e8i			0.236105i
$951$	1.53549e9	+	2.70665e8i	1.78528	+	0.314695i
$952$	−3.68966e8			−0.427637
$953$			8.29322e8i			0.958174i	0.877767	+	0.479087i	$0.159032\pi$
$953$			8.29322e8i			0.958174i	−0.877767	+	0.479087i	$0.840968\pi$
$954$	1.54333e9	+	5.61539e8i	1.77751	+	0.646748i
$955$	−7.06975e8			−0.811697
$956$		−	1.34532e9i		−	1.53976i
$957$	−1.68413e7	+	9.55413e7i	−0.0192150	+	0.109007i
$958$	1.50899e9			1.71628
$959$			9.89874e8i			1.12234i
$960$	4.08013e8	+	7.19214e7i	0.461169	+	0.0812913i
$961$	2.10200e8			0.236844
$962$			1.53174e9i			1.72053i
$963$	7.93964e7	−	2.18212e8i	0.0889041	−	0.244343i
$964$	−1.07431e9			−1.19922
$965$		−	1.72449e9i		−	1.91901i
$966$	−3.72412e8	+	2.11271e9i	−0.413136	+	2.34374i
$967$	3.64802e8			0.403439			0.201720	−	0.979443i	$-0.435347\pi$
$967$	3.64802e8			0.403439			0.201720	+	0.979443i	$0.435347\pi$
$968$			1.19716e9i			1.31986i
$969$	−1.82176e8	−	3.21125e7i	−0.200225	−	0.0352942i
$970$	1.32651e9			1.45344
$971$			6.92406e8i			0.756315i	0.925741	+	0.378158i	$0.123442\pi$
$971$			6.92406e8i			0.756315i	−0.925741	+	0.378158i	$0.876558\pi$
$972$	−1.51423e9	+	1.27127e9i	−1.64889	+	1.38433i
$973$	5.59058e8			0.606902
$974$		−	6.28685e7i		−	0.0680387i
$975$	−1.18128e7	+	6.70145e7i	−0.0127450	+	0.0723028i
$976$	1.26233e8			0.135776
$977$		−	6.71361e8i		−	0.719900i	−0.932972	−	0.359950i	$-0.882794\pi$
$977$		−	6.71361e8i		−	0.719900i	0.932972	−	0.359950i	$-0.117206\pi$
$978$	−5.30204e8	−	9.34602e7i	−0.566795	−	0.0999103i
$979$	−4.77873e8			−0.509289
$980$		−	1.01709e9i		−	1.08065i
$981$	1.01761e9	+	3.70257e8i	1.07789	+	0.392190i
$982$	1.08728e8			0.114817
$983$			5.98255e8i			0.629834i	0.949119	+	0.314917i	$0.101977\pi$
$983$			5.98255e8i			0.629834i	−0.949119	+	0.314917i	$0.898023\pi$
$984$	−6.11716e8	+	3.47029e9i	−0.642044	+	3.64234i
$985$	1.25163e9			1.30968
$986$			5.43460e7i			0.0566940i
$987$	−1.61117e9	−	2.84005e8i	−1.67568	−	0.295376i
$988$	−1.60299e9			−1.66211
$989$			6.78896e8i			0.701801i
$990$	−3.70364e8	+	1.01790e9i	−0.381701	+	1.04906i
$991$	1.23376e9			1.26768			0.633841	−	0.773464i	$-0.281478\pi$
$991$	1.23376e9			1.26768			0.633841	+	0.773464i	$0.281478\pi$
$992$			6.34814e8i			0.650296i
$993$	−2.36363e8	+	1.34090e9i	−0.241397	+	1.36945i
$994$	−1.28017e9			−1.30349
$995$			1.58490e9i			1.60892i
$996$	1.80387e7	+	3.17972e6i	0.0182569	+	0.00321818i
$997$	−2.31670e8			−0.233767			−0.116884	−	0.993146i	$-0.537290\pi$
$997$	−2.31670e8			−0.233767			−0.116884	+	0.993146i	$0.537290\pi$
$998$			2.69516e9i			2.71140i
$999$	7.39623e8	−	1.28154e9i	0.741846	−	1.28539i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	87.7.b.a.59.5	✓	56
3.2	odd	2	inner	87.7.b.a.59.52	yes	56

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
87.7.b.a.59.5	✓	56	1.1	even	1	trivial
87.7.b.a.59.52	yes	56	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:	\( k \)	\(=\)	\( 7 \)
Character orbit:	\([\chi]\)	\(=\)	87.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-14.2052i q^{2} +(4.68709 - 26.5901i) q^{3} -137.789 q^{4} +131.841i q^{5} +(-377.718 - 66.5813i) q^{6} -416.698 q^{7} +1048.19i q^{8} +(-685.062 - 249.260i) q^{9} +1872.83 q^{10} -793.372i q^{11} +(-645.830 + 3663.82i) q^{12} +1434.39 q^{13} +5919.30i q^{14} +(3505.66 + 617.950i) q^{15} +6071.32 q^{16} -844.742i q^{17} +(-3540.80 + 9731.48i) q^{18} +8110.48 q^{19} -18166.2i q^{20} +(-1953.10 + 11080.0i) q^{21} -11270.0 q^{22} +13423.0i q^{23} +(27871.5 + 4912.97i) q^{24} -1757.04 q^{25} -20375.9i q^{26} +(-9838.79 + 17047.5i) q^{27} +57416.4 q^{28} -4528.92i q^{29} +(8778.14 - 49798.7i) q^{30} -33131.6 q^{31} -19160.4i q^{32} +(-21095.8 - 3718.61i) q^{33} -11999.8 q^{34} -54937.9i q^{35} +(94394.1 + 34345.3i) q^{36} -75174.2 q^{37} -115211. i q^{38} +(6723.14 - 38140.6i) q^{39} -138195. q^{40} +124511. i q^{41} +(157394. + 27744.3i) q^{42} +50576.9 q^{43} +109318. i q^{44} +(32862.7 - 90319.3i) q^{45} +190678. q^{46} +145412. i q^{47} +(28456.8 - 161437. i) q^{48} +55988.2 q^{49} +24959.2i q^{50} +(-22461.7 - 3959.38i) q^{51} -197644. q^{52} -158591. i q^{53} +(242165. + 139762. i) q^{54} +104599. q^{55} -436779. i q^{56} +(38014.6 - 215658. i) q^{57} -64334.5 q^{58} +208067. i q^{59} +(-483041. - 85146.8i) q^{60} +20791.7 q^{61} +470643. i q^{62} +(285464. + 103866. i) q^{63} +116387. q^{64} +189112. i q^{65} +(-52823.7 + 299671. i) q^{66} -451067. q^{67} +116396. i q^{68} +(356919. + 62915.0i) q^{69} -780406. q^{70} +216270. i q^{71} +(261272. - 718077. i) q^{72} -475219. q^{73} +1.06787e6i q^{74} +(-8235.39 + 46719.7i) q^{75} -1.11754e6 q^{76} +330597. i q^{77} +(-541797. - 95503.9i) q^{78} -816713. q^{79} +800449. i q^{80} +(407180. + 341517. i) q^{81} +1.76870e6 q^{82} -4923.46i q^{83} +(269116. - 1.52671e6i) q^{84} +111372. q^{85} -718458. i q^{86} +(-120424. - 21227.5i) q^{87} +831606. q^{88} -602331. i q^{89} +(-1.28301e6 - 466822. i) q^{90} -597709. q^{91} -1.84955e6i q^{92} +(-155291. + 880972. i) q^{93} +2.06562e6 q^{94} +1.06929e6i q^{95} +(-509475. - 89806.3i) q^{96} +708292. q^{97} -795326. i q^{98} +(-197756. + 543510. i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 56 q + 52 q^{3} - 1924 q^{4} - 160 q^{6} + 160 q^{7} - 1060 q^{9} - 3588 q^{10} - 2166 q^{12} - 1400 q^{13} - 6240 q^{15} + 56588 q^{16} - 5978 q^{18} + 25000 q^{19} + 7520 q^{21} + 20970 q^{22} + 1238 q^{24}+ \cdots + 4793544 q^{99}+O(q^{100}) \) 56 * q + 52 * q^3 - 1924 * q^4 - 160 * q^6 + 160 * q^7 - 1060 * q^9 - 3588 * q^10 - 2166 * q^12 - 1400 * q^13 - 6240 * q^15 + 56588 * q^16 - 5978 * q^18 + 25000 * q^19 + 7520 * q^21 + 20970 * q^22 + 1238 * q^24 - 213304 * q^25 + 25492 * q^27 + 7870 * q^28 + 105092 * q^30 - 78656 * q^31 + 18952 * q^33 - 96642 * q^34 - 40224 * q^36 + 52120 * q^37 - 112096 * q^39 + 343440 * q^40 - 341758 * q^42 - 286760 * q^43 + 68312 * q^45 + 255516 * q^46 + 386086 * q^48 + 1051872 * q^49 - 256656 * q^51 + 58606 * q^52 + 450124 * q^54 - 409632 * q^55 - 1141736 * q^57 + 817806 * q^60 + 309976 * q^61 + 567632 * q^63 - 2844730 * q^64 + 958426 * q^66 + 748696 * q^67 - 1323704 * q^69 - 2123748 * q^70 - 1026242 * q^72 - 333728 * q^73 - 331316 * q^75 + 2846728 * q^76 + 292446 * q^78 - 661184 * q^79 + 1614852 * q^81 + 376908 * q^82 - 1670584 * q^84 + 3342000 * q^85 - 927168 * q^88 + 835882 * q^90 + 2248016 * q^91 + 3960776 * q^93 - 1368294 * q^94 - 5811456 * q^96 - 3057296 * q^97 + 4793544 * q^99

Introduction

L-functions

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