LMFDB - Embedded newform 87.7.b.a.59.20

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			59.20
Character	$\chi$	$=$	87.59
Dual form			87.7.b.a.59.37

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q-5.40547i q^{2} +(14.0726 + 23.0426i) q^{3} +34.7809 q^{4} +110.739i q^{5} +(124.556 - 76.0688i) q^{6} +411.141 q^{7} -533.957i q^{8} +(-332.926 + 648.538i) q^{9} +598.599 q^{10} -606.117i q^{11} +(489.456 + 801.443i) q^{12} +1441.19 q^{13} -2222.41i q^{14} +(-2551.73 + 1558.39i) q^{15} -660.314 q^{16} +4583.06i q^{17} +(3505.65 + 1799.62i) q^{18} -4731.81 q^{19} +3851.62i q^{20} +(5785.81 + 9473.78i) q^{21} -3276.35 q^{22} +10056.6i q^{23} +(12303.8 - 7514.14i) q^{24} +3361.77 q^{25} -7790.33i q^{26} +(-19629.1 + 1455.09i) q^{27} +14299.9 q^{28} +4528.92i q^{29} +(8423.82 + 13793.3i) q^{30} -10674.4 q^{31} -30604.0i q^{32} +(13966.5 - 8529.61i) q^{33} +24773.6 q^{34} +45529.6i q^{35} +(-11579.5 + 22556.7i) q^{36} +63422.5 q^{37} +25577.6i q^{38} +(20281.3 + 33208.9i) q^{39} +59130.1 q^{40} +13030.7i q^{41} +(51210.2 - 31275.0i) q^{42} +86610.9 q^{43} -21081.3i q^{44} +(-71818.7 - 36868.1i) q^{45} +54360.6 q^{46} -38515.5i q^{47} +(-9292.30 - 15215.4i) q^{48} +51388.2 q^{49} -18171.9i q^{50} +(-105606. + 64495.4i) q^{51} +50126.0 q^{52} -107324. i q^{53} +(7865.44 + 106105. i) q^{54} +67121.1 q^{55} -219532. i q^{56} +(-66588.6 - 109033. i) q^{57} +24481.0 q^{58} +128345. i q^{59} +(-88751.4 + 54202.1i) q^{60} -171773. q^{61} +57700.0i q^{62} +(-136880. + 266641. i) q^{63} -207689. q^{64} +159597. i q^{65} +(-46106.6 - 75495.7i) q^{66} +43758.7 q^{67} +159403. i q^{68} +(-231730. + 141522. i) q^{69} +246109. q^{70} -22981.7i q^{71} +(346291. + 177768. i) q^{72} -549858. q^{73} -342828. i q^{74} +(47308.7 + 77464.0i) q^{75} -164576. q^{76} -249200. i q^{77} +(179510. - 109630. i) q^{78} -781292. q^{79} -73122.8i q^{80} +(-309761. - 431830. i) q^{81} +70437.1 q^{82} -155460. i q^{83} +(201236. + 329506. i) q^{84} -507526. q^{85} -468173. i q^{86} +(-104358. + 63733.5i) q^{87} -323640. q^{88} -782740. i q^{89} +(-199289. + 388214. i) q^{90} +592534. q^{91} +349777. i q^{92} +(-150216. - 245966. i) q^{93} -208194. q^{94} -523998. i q^{95} +(705196. - 430676. i) q^{96} +1.39477e6 q^{97} -277777. i q^{98} +(393090. + 201792. 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 $

$2$		−	5.40547i		−	0.675684i	−0.941203	−	0.337842i	$-0.890303\pi$
$2$		−	5.40547i		−	0.675684i	0.941203	−	0.337842i	$-0.109697\pi$
$3$	14.0726	+	23.0426i	0.521206	+	0.853431i
$3$	14.0726	+	23.0426i	0.521206	+	0.853431i
$4$	34.7809			0.543451
$5$			110.739i			0.885916i	0.896542	+	0.442958i	$0.146071\pi$
$5$			110.739i			0.885916i	−0.896542	+	0.442958i	$0.853929\pi$
$6$	124.556	−	76.0688i	0.576650	−	0.352170i
$7$	411.141			1.19866			0.599331	−	0.800501i	$-0.295433\pi$
$7$	411.141			1.19866			0.599331	+	0.800501i	$0.295433\pi$
$8$		−	533.957i		−	1.04289i
$9$	−332.926	+	648.538i	−0.456689	+	0.889626i
$10$	598.599			0.598599
$11$		−	606.117i		−	0.455385i	−0.973733	−	0.227692i	$-0.926882\pi$
$11$		−	606.117i		−	0.455385i	0.973733	−	0.227692i	$-0.0731180\pi$
$12$	489.456	+	801.443i	0.283250	+	0.463798i
$13$	1441.19			0.655983			0.327991	−	0.944681i	$-0.393628\pi$
$13$	1441.19			0.655983			0.327991	+	0.944681i	$0.393628\pi$
$14$		−	2222.41i		−	0.809917i
$15$	−2551.73	+	1558.39i	−0.756068	+	0.461745i
$16$	−660.314			−0.161209
$17$			4583.06i			0.932844i	0.884562	+	0.466422i	$0.154457\pi$
$17$			4583.06i			0.932844i	−0.884562	+	0.466422i	$0.845543\pi$
$18$	3505.65	+	1799.62i	0.601106	+	0.308577i
$19$	−4731.81			−0.689868			−0.344934	−	0.938627i	$-0.612099\pi$
$19$	−4731.81			−0.689868			−0.344934	+	0.938627i	$0.612099\pi$
$20$			3851.62i			0.481452i
$21$	5785.81	+	9473.78i	0.624750	+	1.02298i
$22$	−3276.35			−0.307696
$23$			10056.6i			0.826547i	0.910607	+	0.413273i	$0.135615\pi$
$23$			10056.6i			0.826547i	−0.910607	+	0.413273i	$0.864385\pi$
$24$	12303.8	−	7514.14i	0.890030	−	0.543558i
$25$	3361.77			0.215153
$26$		−	7790.33i		−	0.443237i
$27$	−19629.1	+	1455.09i	−0.997264	+	0.0739262i
$28$	14299.9			0.651415
$29$			4528.92i			0.185695i
$29$			4528.92i			0.185695i
$30$	8423.82	+	13793.3i	0.311993	+	0.510863i
$31$	−10674.4			−0.358309			−0.179154	−	0.983821i	$-0.557336\pi$
$31$	−10674.4			−0.358309			−0.179154	+	0.983821i	$0.557336\pi$
$32$		−	30604.0i		−	0.933959i
$33$	13966.5	−	8529.61i	0.388639	−	0.237349i
$34$	24773.6			0.630307
$35$			45529.6i			1.06191i
$36$	−11579.5	+	22556.7i	−0.248188	+	0.483469i
$37$	63422.5			1.25210			0.626049	−	0.779784i	$-0.284671\pi$
$37$	63422.5			1.25210			0.626049	+	0.779784i	$0.284671\pi$
$38$			25577.6i			0.466133i
$39$	20281.3	+	33208.9i	0.341902	+	0.559836i
$40$	59130.1			0.923908
$41$			13030.7i			0.189067i	0.995522	+	0.0945336i	$0.0301360\pi$
$41$			13030.7i			0.189067i	−0.995522	+	0.0945336i	$0.969864\pi$
$42$	51210.2	−	31275.0i	0.691208	−	0.422134i
$43$	86610.9			1.08935			0.544675	−	0.838647i	$-0.316653\pi$
$43$	86610.9			1.08935			0.544675	+	0.838647i	$0.316653\pi$
$44$		−	21081.3i		−	0.247479i
$45$	−71818.7	−	36868.1i	−0.788134	−	0.404588i
$46$	54360.6			0.558484
$47$		−	38515.5i		−	0.370972i	−0.982647	−	0.185486i	$-0.940614\pi$
$47$		−	38515.5i		−	0.370972i	0.982647	−	0.185486i	$-0.0593860\pi$
$48$	−9292.30	−	15215.4i	−0.0840233	−	0.137581i
$49$	51388.2			0.436792
$50$		−	18171.9i		−	0.145376i
$51$	−105606.	+	64495.4i	−0.796118	+	0.486204i
$52$	50126.0			0.356495
$53$		−	107324.i		−	0.720889i	−0.932781	−	0.360445i	$-0.882625\pi$
$53$		−	107324.i		−	0.720889i	0.932781	−	0.360445i	$-0.117375\pi$
$54$	7865.44	+	106105.i	0.0499507	+	0.673835i
$55$	67121.1			0.403432
$56$		−	219532.i		−	1.25007i
$57$	−66588.6	−	109033.i	−0.359564	−	0.588755i
$58$	24481.0			0.125471
$59$			128345.i			0.624919i	0.949931	+	0.312459i	$0.101153\pi$
$59$			128345.i			0.624919i	−0.949931	+	0.312459i	$0.898847\pi$
$60$	−88751.4	+	54202.1i	−0.410886	+	0.250936i
$61$	−171773.			−0.756774			−0.378387	−	0.925648i	$-0.623521\pi$
$61$	−171773.			−0.756774			−0.378387	+	0.925648i	$0.623521\pi$
$62$			57700.0i			0.242103i
$63$	−136880.	+	266641.i	−0.547416	+	1.06636i
$64$	−207689.			−0.792270
$65$			159597.i			0.581146i
$66$	−46106.6	−	75495.7i	−0.160373	−	0.262597i
$67$	43758.7			0.145492			0.0727462	−	0.997350i	$-0.476824\pi$
$67$	43758.7			0.145492			0.0727462	+	0.997350i	$0.476824\pi$
$68$			159403.i			0.506955i
$69$	−231730.	+	141522.i	−0.705401	+	0.430801i
$70$	246109.			0.717518
$71$		−	22981.7i		−	0.0642105i	−0.999484	−	0.0321053i	$-0.989779\pi$
$71$		−	22981.7i		−	0.0642105i	0.999484	−	0.0321053i	$-0.0102212\pi$
$72$	346291.	+	177768.i	0.927778	+	0.476274i
$73$	−549858.			−1.41346			−0.706728	−	0.707486i	$-0.749829\pi$
$73$	−549858.			−1.41346			−0.706728	+	0.707486i	$0.749829\pi$
$74$		−	342828.i		−	0.846022i
$75$	47308.7	+	77464.0i	0.112139	+	0.183618i
$76$	−164576.			−0.374910
$77$		−	249200.i		−	0.545852i
$78$	179510.	−	109630.i	0.378272	−	0.231018i
$79$	−781292.			−1.58465			−0.792323	−	0.610102i	$-0.791129\pi$
$79$	−781292.			−1.58465			−0.792323	+	0.610102i	$0.791129\pi$
$80$		−	73122.8i		−	0.142818i
$81$	−309761.	−	431830.i	−0.582871	−	0.812565i
$82$	70437.1			0.127750
$83$		−	155460.i		−	0.271885i	−0.990717	−	0.135942i	$-0.956594\pi$
$83$		−	155460.i		−	0.271885i	0.990717	−	0.135942i	$-0.0434062\pi$
$84$	201236.	+	329506.i	0.339521	+	0.555938i
$85$	−507526.			−0.826421
$86$		−	468173.i		−	0.736056i
$87$	−104358.	+	63733.5i	−0.158478	+	0.0967855i
$88$	−323640.			−0.474914
$89$		−	782740.i		−	1.11032i	−0.831744	−	0.555159i	$-0.812657\pi$
$89$		−	782740.i		−	1.11032i	0.831744	−	0.555159i	$-0.187343\pi$
$90$	−199289.	+	388214.i	−0.273373	+	0.532530i
$91$	592534.			0.786302
$92$			349777.i			0.449188i
$93$	−150216.	−	245966.i	−0.186753	−	0.305792i
$94$	−208194.			−0.250660
$95$		−	523998.i		−	0.611165i
$96$	705196.	−	430676.i	0.797069	−	0.486785i
$97$	1.39477e6			1.52822			0.764110	−	0.645086i	$-0.223178\pi$
$97$	1.39477e6			1.52822			0.764110	+	0.645086i	$0.223178\pi$
$98$		−	277777.i		−	0.295133i
$99$	393090.	+	201792.i	0.405122	+	0.207969i
$100$	116925.			0.116925
$101$		−	1.97695e6i		−	1.91881i	−0.282032	−	0.959405i	$-0.591008\pi$
$101$		−	1.97695e6i		−	1.91881i	0.282032	−	0.959405i	$-0.408992\pi$
$102$	348628.	+	570849.i	0.328520	+	0.537924i
$103$	1.16265e6			1.06399			0.531993	−	0.846749i	$-0.321443\pi$
$103$	1.16265e6			1.06399			0.531993	+	0.846749i	$0.321443\pi$
$104$		−	769536.i		−	0.684115i
$105$	−1.04912e6	+	640718.i	−0.906270	+	0.553476i
$106$	−580136.			−0.487093
$107$			5836.50i			0.00476432i	0.999997	+	0.00238216i	$0.000758266\pi$
$107$			5836.50i			0.00476432i	−0.999997	+	0.00238216i	$0.999242\pi$
$108$	−682719.	+	50609.3i	−0.541964	+	0.0401753i
$109$	−844806.			−0.652345			−0.326173	−	0.945310i	$-0.605759\pi$
$109$	−844806.			−0.652345			−0.326173	+	0.945310i	$0.605759\pi$
$110$		−	362821.i		−	0.272593i
$111$	892517.	+	1.46142e6i	0.652600	+	1.06858i
$112$	−271482.			−0.193236
$113$			115920.i			0.0803383i	0.999193	+	0.0401692i	$0.0127897\pi$
$113$			115920.i			0.0803383i	−0.999193	+	0.0401692i	$0.987210\pi$
$114$	−589376.	+	359943.i	−0.397812	+	0.242951i
$115$	−1.11366e6			−0.732251
$116$			157520.i			0.100916i
$117$	−479811.	+	934669.i	−0.299580	+	0.583580i
$118$	693766.			0.422248
$119$			1.88429e6i			1.11816i
$120$	832112.	+	1.36251e6i	0.481547	+	0.788492i
$121$	1.40418e6			0.792625
$122$			928515.i			0.511340i
$123$	−300262.	+	183375.i	−0.161356	+	0.0985429i
$124$	−371264.			−0.194723
$125$			2.10258e6i			1.07652i
$126$	1.44132e6	+	739899.i	0.720524	+	0.369880i
$127$	−2.38417e6			−1.16393			−0.581964	−	0.813215i	$-0.697716\pi$
$127$	−2.38417e6			−1.16393			−0.581964	+	0.813215i	$0.697716\pi$
$128$		−	835997.i		−	0.398634i
$129$	1.21884e6	+	1.99574e6i	0.567775	+	0.929685i
$130$	862697.			0.392671
$131$		−	4.37080e6i		−	1.94423i	−0.234513	−	0.972113i	$-0.575349\pi$
$131$		−	4.37080e6i		−	1.94423i	0.234513	−	0.972113i	$-0.424651\pi$
$132$	485768.	−	296668.i	0.211207	−	0.128988i
$133$	−1.94544e6			−0.826920
$134$		−	236536.i		−	0.0983068i
$135$	−161136.	−	2.17372e6i	−0.0654924	−	0.883492i
$136$	2.44716e6			0.972849
$137$		−	902098.i		−	0.350826i	−0.984495	−	0.175413i	$-0.943874\pi$
$137$		−	902098.i		−	0.350826i	0.984495	−	0.175413i	$-0.0561261\pi$
$138$	764993.	+	1.25261e6i	0.291085	+	0.476628i
$139$	4.40643e6			1.64075			0.820375	−	0.571826i	$-0.193765\pi$
$139$	4.40643e6			1.64075			0.820375	+	0.571826i	$0.193765\pi$
$140$			1.58356e6i			0.577099i
$141$	887498.	−	542011.i	0.316599	−	0.193353i
$142$	−124227.			−0.0433860
$143$		−	873532.i		−	0.298724i
$144$	219836.	−	428238.i	0.0736225	−	0.143416i
$145$	−501531.			−0.164510
$146$			2.97224e6i			0.955049i
$147$	723163.	+	1.18412e6i	0.227659	+	0.372772i
$148$	2.20589e6			0.680454
$149$		−	1.95066e6i		−	0.589690i	−0.955545	−	0.294845i	$-0.904732\pi$
$149$		−	1.95066e6i		−	0.589690i	0.955545	−	0.294845i	$-0.0952680\pi$
$150$	418729.	−	255726.i	0.124068	−	0.0757706i
$151$	−6.07887e6			−1.76560			−0.882799	−	0.469751i	$-0.844344\pi$
$151$	−6.07887e6			−1.76560			−0.882799	+	0.469751i	$0.844344\pi$
$152$			2.52658e6i			0.719454i
$153$	−2.97229e6	−	1.52582e6i	−0.829882	−	0.426019i
$154$	−1.34704e6			−0.368824
$155$		−	1.18207e6i		−	0.317431i
$156$	705401.	+	1.15504e6i	0.185807	+	0.304244i
$157$	5.18617e6			1.34013			0.670067	−	0.742301i	$-0.266265\pi$
$157$	5.18617e6			1.34013			0.670067	+	0.742301i	$0.266265\pi$
$158$			4.22325e6i			1.07072i
$159$	2.47302e6	−	1.51032e6i	0.615229	−	0.375732i
$160$	3.38907e6			0.827409
$161$			4.13468e6i			0.990751i
$162$	−2.33425e6	+	1.67441e6i	−0.549037	+	0.393836i
$163$	−3.25647e6			−0.751943			−0.375971	−	0.926631i	$-0.622691\pi$
$163$	−3.25647e6			−0.751943			−0.375971	+	0.926631i	$0.622691\pi$
$164$			453219.i			0.102749i
$165$	944565.	+	1.54665e6i	0.210271	+	0.344302i
$166$	−840336.			−0.183708
$167$		−	1.51890e6i		−	0.326122i	−0.986616	−	0.163061i	$-0.947863\pi$
$167$		−	1.51890e6i		−	0.326122i	0.986616	−	0.163061i	$-0.0521368\pi$
$168$	5.05859e6	−	3.08938e6i	1.06685	−	0.651543i
$169$	−2.74977e6			−0.569687
$170$			2.74342e6i			0.558399i
$171$	1.57534e6	−	3.06876e6i	0.315055	−	0.613725i
$172$	3.01240e6			0.592008
$173$		−	1.00818e7i		−	1.94715i	−0.228369	−	0.973575i	$-0.573339\pi$
$173$		−	1.00818e7i		−	1.94715i	0.228369	−	0.973575i	$-0.426661\pi$
$174$	344510.	+	564106.i	0.0653964	+	0.107081i
$175$	1.38216e6			0.257896
$176$			400227.i			0.0734123i
$177$	−2.95741e6	+	1.80615e6i	−0.533325	+	0.325711i
$178$	−4.23108e6			−0.750224
$179$		−	4.08978e6i		−	0.713084i	−0.934279	−	0.356542i	$-0.883956\pi$
$179$		−	4.08978e6i		−	0.713084i	0.934279	−	0.356542i	$-0.116044\pi$
$180$	−2.49792e6	−	1.28230e6i	−0.428313	−	0.219874i
$181$	−4.70671e6			−0.793747			−0.396873	−	0.917873i	$-0.629905\pi$
$181$	−4.70671e6			−0.793747			−0.396873	+	0.917873i	$0.629905\pi$
$182$		−	3.20293e6i		−	0.531292i
$183$	−2.41729e6	−	3.95811e6i	−0.394435	−	0.645854i
$184$	5.36979e6			0.861994
$185$			7.02337e6i			1.10925i
$186$	−1.32956e6	+	811987.i	−0.206619	+	0.126186i
$187$	2.77787e6			0.424803
$188$		−	1.33960e6i		−	0.201605i
$189$	−8.07035e6	+	598247.i	−1.19538	+	0.0886125i
$190$	−2.83246e6			−0.412955
$191$		−	3.73212e6i		−	0.535619i	−0.963472	−	0.267809i	$-0.913700\pi$
$191$		−	3.73212e6i		−	0.535619i	0.963472	−	0.267809i	$-0.0862998\pi$
$192$	−2.92271e6	−	4.78570e6i	−0.412936	−	0.676148i
$193$	3.61636e6			0.503037			0.251518	−	0.967853i	$-0.419070\pi$
$193$	3.61636e6			0.503037			0.251518	+	0.967853i	$0.419070\pi$
$194$		−	7.53936e6i		−	1.03259i
$195$	−3.67754e6	+	2.24594e6i	−0.495968	+	0.302896i
$196$	1.78733e6			0.237375
$197$			3.77076e6i			0.493208i	0.969116	+	0.246604i	$0.0793147\pi$
$197$			3.77076e6i			0.493208i	−0.969116	+	0.246604i	$0.920685\pi$
$198$	1.09078e6	−	2.12483e6i	0.140521	−	0.273735i
$199$	532296.			0.0675451			0.0337725	−	0.999430i	$-0.489248\pi$
$199$	532296.			0.0675451			0.0337725	+	0.999430i	$0.489248\pi$
$200$		−	1.79504e6i		−	0.224380i
$201$	615797.	+	1.00832e6i	0.0758315	+	0.124168i
$202$	−1.06864e7			−1.29651
$203$			1.86203e6i			0.222586i
$204$	−3.67306e6	+	2.24321e6i	−0.432651	+	0.264228i
$205$	−1.44301e6			−0.167498
$206$		−	6.28465e6i		−	0.718918i
$207$	−6.52208e6	−	3.34810e6i	−0.735318	−	0.377475i
$208$	−951640.			−0.105751
$209$			2.86803e6i			0.314155i
$210$	3.46338e6	+	5.67100e6i	0.373975	+	0.612352i
$211$	−1.46909e7			−1.56388			−0.781938	−	0.623356i	$-0.785769\pi$
$211$	−1.46909e7			−1.56388			−0.781938	+	0.623356i	$0.785769\pi$
$212$		−	3.73282e6i		−	0.391768i
$213$	529558.	−	323411.i	0.0547992	−	0.0334669i
$214$	31549.0			0.00321917
$215$			9.59125e6i			0.965072i
$216$	776955.	+	1.04811e7i	0.0770965	+	1.04003i
$217$	−4.38868e6			−0.429491
$218$			4.56657e6i			0.440779i
$219$	−7.73791e6	−	1.26702e7i	−0.736701	−	1.20629i
$220$	2.33453e6			0.219246
$221$			6.60508e6i			0.611929i
$222$	7.89967e6	−	4.82447e6i	0.722021	−	0.440952i
$223$	1.72965e7			1.55971			0.779855	−	0.625960i	$-0.215293\pi$
$223$	1.72965e7			1.55971			0.779855	+	0.625960i	$0.215293\pi$
$224$		−	1.25825e7i		−	1.11950i
$225$	−1.11922e6	+	2.18023e6i	−0.0982581	+	0.191406i
$226$	626602.			0.0542833
$227$			6.67519e6i			0.570671i	0.958428	+	0.285336i	$0.0921051\pi$
$227$			6.67519e6i			0.570671i	−0.958428	+	0.285336i	$0.907895\pi$
$228$	−2.31601e6	−	3.79228e6i	−0.195405	−	0.319960i
$229$	4.19604e6			0.349408			0.174704	−	0.984621i	$-0.444103\pi$
$229$	4.19604e6			0.349408			0.174704	+	0.984621i	$0.444103\pi$
$230$			6.01987e6i			0.494770i
$231$	5.74222e6	−	3.50688e6i	0.465847	−	0.284502i
$232$	2.41825e6			0.193659
$233$		−	4.92660e6i		−	0.389475i	−0.980855	−	0.194737i	$-0.937615\pi$
$233$		−	4.92660e6i		−	0.389475i	0.980855	−	0.194737i	$-0.0623855\pi$
$234$	5.05233e6	+	2.59361e6i	0.394315	+	0.202421i
$235$	4.26518e6			0.328650
$236$			4.46396e6i			0.339613i
$237$	−1.09948e7	−	1.80030e7i	−0.825927	−	1.35239i
$238$	1.01855e7			0.755526
$239$			2.32737e7i			1.70479i	0.522897	+	0.852396i	$0.324851\pi$
$239$			2.32737e7i			1.70479i	−0.522897	+	0.852396i	$0.675149\pi$
$240$	1.68494e6	−	1.02902e6i	0.121885	−	0.0744375i
$241$	1.74137e6			0.124406			0.0622028	−	0.998064i	$-0.480187\pi$
$241$	1.74137e6			0.124406			0.0622028	+	0.998064i	$0.480187\pi$
$242$		−	7.59027e6i		−	0.535564i
$243$	5.59137e6	−	1.32147e7i	0.389673	−	0.920953i
$244$	−5.97443e6			−0.411270
$245$			5.69070e6i			0.386961i
$246$	991230.	+	1.62306e6i	0.0665839	+	0.109025i
$247$	−6.81945e6			−0.452542
$248$			5.69966e6i			0.373675i
$249$	3.58221e6	−	2.18772e6i	0.232035	−	0.141708i
$250$	1.13655e7			0.727390
$251$			2.93560e7i			1.85642i	0.372061	+	0.928208i	$0.378651\pi$
$251$			2.93560e7i			1.85642i	−0.372061	+	0.928208i	$0.621349\pi$
$252$	−4.76080e6	+	9.27400e6i	−0.297494	+	0.579516i
$253$	6.09547e6			0.376397
$254$			1.28876e7i			0.786447i
$255$	−7.14219e6	−	1.16947e7i	−0.430735	−	0.705293i
$256$	−1.78110e7			−1.06162
$257$			4.11410e6i			0.242368i	0.992630	+	0.121184i	$0.0386691\pi$
$257$			4.11410e6i			0.242368i	−0.992630	+	0.121184i	$0.961331\pi$
$258$	1.07879e7	−	6.58839e6i	0.628173	−	0.383637i
$259$	2.60756e7			1.50084
$260$			5.55093e6i			0.315824i
$261$	−2.93718e6	−	1.50780e6i	−0.165199	−	0.0848050i
$262$	−2.36262e7			−1.31368
$263$		−	1.80841e7i		−	0.994097i	−0.867723	−	0.497049i	$-0.834417\pi$
$263$		−	1.80841e7i		−	0.994097i	0.867723	−	0.497049i	$-0.165583\pi$
$264$	−4.55445e6	−	7.45753e6i	−0.247528	−	0.405306i
$265$	1.18850e7			0.638647
$266$			1.05160e7i			0.558736i
$267$	1.80364e7	−	1.10152e7i	0.947580	−	0.578704i
$268$	1.52197e6			0.0790680
$269$			1.03874e6i			0.0533640i	0.999644	+	0.0266820i	$0.00849415\pi$
$269$			1.03874e6i			0.0533640i	−0.999644	+	0.0266820i	$0.991506\pi$
$270$	−1.17500e7	+	871015.i	−0.596961	+	0.0442521i
$271$	−4.05967e6			−0.203978			−0.101989	−	0.994786i	$-0.532521\pi$
$271$	−4.05967e6			−0.203978			−0.101989	+	0.994786i	$0.532521\pi$
$272$		−	3.02626e6i		−	0.150383i
$273$	8.33848e6	+	1.36536e7i	0.409825	+	0.671055i
$274$	−4.87627e6			−0.237048
$275$		−	2.03762e6i		−	0.0979774i
$276$	−8.05979e6	+	4.92226e6i	−0.383351	+	0.234119i
$277$	−2.26727e7			−1.06675			−0.533376	−	0.845878i	$-0.679077\pi$
$277$	−2.26727e7			−1.06675			−0.533376	+	0.845878i	$0.679077\pi$
$278$		−	2.38188e7i		−	1.10863i
$279$	3.55378e6	−	6.92274e6i	0.163636	−	0.318761i
$280$	2.43108e7			1.10745
$281$			2.33305e7i			1.05149i	0.850642	+	0.525745i	$0.176213\pi$
$281$			2.33305e7i			1.05149i	−0.850642	+	0.525745i	$0.823787\pi$
$282$	−2.92983e6	−	4.79735e6i	−0.130646	−	0.213921i
$283$	2.46029e7			1.08549			0.542746	−	0.839897i	$-0.317384\pi$
$283$	2.46029e7			1.08549			0.542746	+	0.839897i	$0.317384\pi$
$284$		−	799322.i		−	0.0348953i
$285$	1.20743e7	−	7.37399e6i	0.521587	−	0.318543i
$286$	−4.72185e6			−0.201843
$287$			5.35746e6i			0.226628i
$288$	1.98478e7	+	1.01889e7i	0.830874	+	0.426528i
$289$	3.13312e6			0.129803
$290$			2.71101e6i			0.111157i
$291$	1.96279e7	+	3.21391e7i	0.796517	+	1.30423i
$292$	−1.91245e7			−0.768144
$293$			1.24717e7i			0.495820i	0.968783	+	0.247910i	$0.0797437\pi$
$293$			1.24717e7i			0.495820i	−0.968783	+	0.247910i	$0.920256\pi$
$294$	6.40072e6	−	3.90904e6i	0.251876	−	0.153825i
$295$	−1.42129e7			−0.553625
$296$		−	3.38649e7i		−	1.30579i
$297$	881954.	+	1.18976e7i	0.0336648	+	0.454138i
$298$	−1.05443e7			−0.398444
$299$			1.44935e7i			0.542201i
$300$	1.64544e6	+	2.69427e6i	0.0609421	+	0.0997877i
$301$	3.56093e7			1.30576
$302$			3.28591e7i			1.19299i
$303$	4.55542e7	−	2.78208e7i	1.63757	−	1.00010i
$304$	3.12448e6			0.111213
$305$		−	1.90221e7i		−	0.670438i
$306$	−8.24778e6	+	1.60666e7i	−0.287854	+	0.560738i
$307$	−3.38989e7			−1.17158			−0.585788	−	0.810465i	$-0.699215\pi$
$307$	−3.38989e7			−1.17158			−0.585788	+	0.810465i	$0.699215\pi$
$308$		−	8.66738e6i		−	0.296644i
$309$	1.63614e7	+	2.67904e7i	0.554556	+	0.908039i
$310$	−6.38967e6			−0.214483
$311$		−	2.40534e7i		−	0.799641i	−0.916594	−	0.399820i	$-0.869073\pi$
$311$		−	2.40534e7i		−	0.799641i	0.916594	−	0.399820i	$-0.130927\pi$
$312$	1.77321e7	−	1.08293e7i	0.583845	−	0.356565i
$313$	−1.24234e7			−0.405144			−0.202572	−	0.979267i	$-0.564930\pi$
$313$	−1.24234e7			−0.405144			−0.202572	+	0.979267i	$0.564930\pi$
$314$		−	2.80337e7i		−	0.905507i
$315$	−2.95276e7	−	1.51580e7i	−0.944707	−	0.484964i
$316$	−2.71740e7			−0.861178
$317$			4.60354e7i			1.44515i	0.691290	+	0.722577i	$0.257043\pi$
$317$			4.60354e7i			1.44515i	−0.691290	+	0.722577i	$0.742957\pi$
$318$	−8.16400e6	−	1.33679e7i	−0.253876	−	0.415700i
$319$	2.74506e6			0.0845628
$320$		−	2.29994e7i		−	0.701885i
$321$	−134488.	+	82134.4i	−0.00406602	+	0.00248319i
$322$	2.23499e7			0.669434
$323$		−	2.16862e7i		−	0.643539i
$324$	−1.07738e7	−	1.50194e7i	−0.316762	−	0.441590i
$325$	4.84496e6			0.141137
$326$			1.76028e7i			0.508075i
$327$	−1.18886e7	−	1.94666e7i	−0.340006	−	0.556732i
$328$	6.95783e6			0.197175
$329$		−	1.58353e7i		−	0.444671i
$330$	8.36035e6	−	5.10582e6i	0.232639	−	0.142077i
$331$	5.93267e7			1.63594			0.817968	−	0.575263i	$-0.195100\pi$
$331$	5.93267e7			1.63594			0.817968	+	0.575263i	$0.195100\pi$
$332$		−	5.40704e6i		−	0.147756i
$333$	−2.11150e7	+	4.11319e7i	−0.571819	+	1.11390i
$334$	−8.21038e6			−0.220356
$335$			4.84582e6i			0.128894i
$336$	−3.82045e6	−	6.25567e6i	−0.100716	−	0.164913i
$337$	3.08452e7			0.805931			0.402965	−	0.915215i	$-0.367979\pi$
$337$	3.08452e7			0.805931			0.402965	+	0.915215i	$0.367979\pi$
$338$			1.48638e7i			0.384928i
$339$	−2.67110e6	+	1.63129e6i	−0.0685632	+	0.0418728i
$340$	−1.76522e7			−0.449120
$341$			6.46992e6i			0.163168i
$342$	−1.65881e7	−	8.51547e6i	−0.414684	−	0.212878i
$343$	−2.72426e7			−0.675096
$344$		−	4.62465e7i		−	1.13607i
$345$	−1.56721e7	−	2.56617e7i	−0.381654	−	0.624926i
$346$	−5.44968e7			−1.31566
$347$			6.59809e7i			1.57917i	0.613639	+	0.789586i	$0.289705\pi$
$347$			6.59809e7i			1.57917i	−0.613639	+	0.789586i	$0.710295\pi$
$348$	−3.62968e6	+	2.21671e6i	−0.0861252	+	0.0525982i
$349$	5.60177e7			1.31780			0.658900	−	0.752231i	$-0.271022\pi$
$349$	5.60177e7			1.31780			0.658900	+	0.752231i	$0.271022\pi$
$350$		−	7.47124e6i		−	0.174256i
$351$	−2.82894e7	+	2.09707e6i	−0.654188	+	0.0484943i
$352$	−1.85496e7			−0.425310
$353$		−	1.90588e6i		−	0.0433283i	−0.999765	−	0.0216642i	$-0.993104\pi$
$353$		−	1.90588e6i		−	0.0433283i	0.999765	−	0.0216642i	$-0.00689646\pi$
$354$	9.76307e6	+	1.59862e7i	0.220078	+	0.360359i
$355$	2.54498e6			0.0568851
$356$		−	2.72244e7i		−	0.603404i
$357$	−4.34189e7	+	2.65167e7i	−0.954277	+	0.582794i
$358$	−2.21072e7			−0.481819
$359$			3.33216e7i			0.720183i	0.932917	+	0.360092i	$0.117255\pi$
$359$			3.33216e7i			0.720183i	−0.932917	+	0.360092i	$0.882745\pi$
$360$	−1.96860e7	+	3.83481e7i	−0.421939	+	0.821933i
$361$	−2.46559e7			−0.524082
$362$			2.54420e7i			0.536322i
$363$	1.97605e7	+	3.23561e7i	0.413121	+	0.676451i
$364$	2.06089e7			0.427317
$365$		−	6.08910e7i		−	1.25220i
$366$	−2.13954e7	+	1.30666e7i	−0.436393	+	0.266513i
$367$	−8.55252e7			−1.73020			−0.865099	−	0.501602i	$-0.832744\pi$
$367$	−8.55252e7			−1.73020			−0.865099	+	0.501602i	$0.832744\pi$
$368$		−	6.64051e6i		−	0.133247i
$369$	−8.45090e6	−	4.33826e6i	−0.168199	−	0.0863449i
$370$	3.79646e7			0.749504
$371$		−	4.41253e7i		−	0.864103i
$372$	−5.22464e6	−	8.55491e6i	−0.101491	−	0.166183i
$373$	−5.89028e7			−1.13504			−0.567518	−	0.823361i	$-0.692096\pi$
$373$	−5.89028e7			−1.13504			−0.567518	+	0.823361i	$0.692096\pi$
$374$		−	1.50157e7i		−	0.287032i
$375$	−4.84491e7	+	2.95887e7i	−0.918738	+	0.561090i
$376$	−2.05656e7			−0.386882
$377$			6.52706e6i			0.121813i
$378$	3.23381e6	+	4.36240e7i	0.0598741	+	0.807701i
$379$	−8.48316e7			−1.55826			−0.779131	−	0.626862i	$-0.784339\pi$
$379$	−8.48316e7			−1.55826			−0.779131	+	0.626862i	$0.784339\pi$
$380$		−	1.82251e7i		−	0.332139i
$381$	−3.35514e7	−	5.49376e7i	−0.606646	−	0.993332i
$382$	−2.01739e7			−0.361909
$383$		−	7.09006e7i		−	1.26198i	−0.775790	−	0.630991i	$-0.782648\pi$
$383$		−	7.09006e7i		−	1.26198i	0.775790	−	0.630991i	$-0.217352\pi$
$384$	1.92636e7	−	1.17646e7i	0.340207	−	0.207771i
$385$	2.75962e7			0.483579
$386$		−	1.95481e7i		−	0.339894i
$387$	−2.88350e7	+	5.61704e7i	−0.497494	+	0.969114i
$388$	4.85112e7			0.830513
$389$			7.37316e7i			1.25258i	0.779591	+	0.626289i	$0.215427\pi$
$389$			7.37316e7i			1.25258i	−0.779591	+	0.626289i	$0.784573\pi$
$390$	1.21404e7	+	1.98788e7i	0.204662	+	0.335117i
$391$	−4.60900e7			−0.771039
$392$		−	2.74391e7i		−	0.455524i
$393$	1.00715e8	−	6.15083e7i	1.65926	−	1.01334i
$394$	2.03827e7			0.333253
$395$		−	8.65199e7i		−	1.40386i
$396$	1.36720e7	+	7.01851e6i	0.220164	+	0.113021i
$397$	8.23252e7			1.31571			0.657857	−	0.753143i	$-0.271463\pi$
$397$	8.23252e7			1.31571			0.657857	+	0.753143i	$0.271463\pi$
$398$		−	2.87731e6i		−	0.0456391i
$399$	−2.73773e7	−	4.48281e7i	−0.430995	−	0.705719i
$400$	−2.21982e6			−0.0346847
$401$		−	8.66361e7i		−	1.34359i	−0.740739	−	0.671793i	$-0.765524\pi$
$401$		−	8.66361e7i		−	1.34359i	0.740739	−	0.671793i	$-0.234476\pi$
$402$	5.45042e6	−	3.32867e6i	0.0838981	−	0.0512381i
$403$	−1.53838e7			−0.235044
$404$		−	6.87601e7i		−	1.04278i
$405$	4.78207e7	−	3.43028e7i	0.719864	−	0.516374i
$406$	1.00651e7			0.150398
$407$		−	3.84414e7i		−	0.570186i
$408$	3.44378e7	+	5.63890e7i	0.507055	+	0.830259i
$409$	1.14283e8			1.67037			0.835186	−	0.549968i	$-0.185360\pi$
$409$	1.14283e8			1.67037			0.835186	+	0.549968i	$0.185360\pi$
$410$			7.80016e6i			0.113175i
$411$	2.07867e7	−	1.26948e7i	0.299406	−	0.182853i
$412$	4.04379e7			0.578225
$413$			5.27680e7i			0.749067i
$414$	−1.80981e7	+	3.52549e7i	−0.255054	+	0.496843i
$415$	1.72156e7			0.240867
$416$		−	4.41062e7i		−	0.612661i
$417$	6.20098e7	+	1.01536e8i	0.855169	+	1.40027i
$418$	1.55030e7			0.212270
$419$			1.01970e8i			1.38621i	0.720836	+	0.693106i	$0.243758\pi$
$419$			1.01970e8i			1.38621i	−0.720836	+	0.693106i	$0.756242\pi$
$420$	−3.64894e7	+	2.22847e7i	−0.492514	+	0.300787i
$421$	1.21575e8			1.62929			0.814646	−	0.579958i	$-0.196931\pi$
$421$	1.21575e8			1.62929			0.814646	+	0.579958i	$0.196931\pi$
$422$			7.94115e7i			1.05669i
$423$	2.49787e7	+	1.28228e7i	0.330027	+	0.169419i
$424$	−5.73063e7			−0.751805
$425$			1.54072e7i			0.200704i
$426$	−1.74819e6	−	2.86251e6i	−0.0226130	−	0.0370270i
$427$	−7.06231e7			−0.907117
$428$			202999.i			0.00258918i
$429$	2.01285e7	−	1.22928e7i	0.254941	−	0.155697i
$430$	5.18452e7			0.652084
$431$			1.60824e7i			0.200871i	0.994944	+	0.100436i	$0.0320237\pi$
$431$			1.60824e7i			0.200871i	−0.994944	+	0.100436i	$0.967976\pi$
$432$	1.29614e7	−	960815.i	0.160768	−	0.0119176i
$433$	−1.58213e7			−0.194885			−0.0974425	−	0.995241i	$-0.531066\pi$
$433$	−1.58213e7			−0.194885			−0.0974425	+	0.995241i	$0.531066\pi$
$434$			2.37229e7i			0.290200i
$435$	−7.05782e6	−	1.15566e7i	−0.0857438	−	0.140398i
$436$	−2.93831e7			−0.354518
$437$		−	4.75859e7i		−	0.570209i
$438$	−6.84883e7	+	4.18270e7i	−0.815068	+	0.497777i
$439$	2.69744e7			0.318829			0.159415	−	0.987212i	$-0.449039\pi$
$439$	2.69744e7			0.318829			0.159415	+	0.987212i	$0.449039\pi$
$440$		−	3.58398e7i		−	0.420734i
$441$	−1.71085e7	+	3.33272e7i	−0.199478	+	0.388582i
$442$	3.57036e7			0.413471
$443$			5.90195e6i			0.0678866i	0.999424	+	0.0339433i	$0.0108066\pi$
$443$			5.90195e6i			0.0678866i	−0.999424	+	0.0339433i	$0.989193\pi$
$444$	3.10425e7	+	5.08295e7i	0.354657	+	0.580720i
$445$	8.66802e7			0.983649
$446$		−	9.34958e7i		−	1.05387i
$447$	4.49485e7	−	2.74508e7i	0.503260	−	0.307350i
$448$	−8.53895e7			−0.949665
$449$		−	7.31592e7i		−	0.808221i	−0.914710	−	0.404110i	$-0.867581\pi$
$449$		−	7.31592e7i		−	0.808221i	0.914710	−	0.404110i	$-0.132419\pi$
$450$	1.17852e7	+	6.04991e6i	0.129330	+	0.0663914i
$451$	7.89813e6			0.0860983
$452$			4.03180e6i			0.0436600i
$453$	−8.55452e7	−	1.40073e8i	−0.920240	−	1.50682i
$454$	3.60825e7			0.385593
$455$			6.56170e7i			0.696597i
$456$	−5.82191e7	+	3.55555e7i	−0.614004	+	0.374983i
$457$	−2.06356e7			−0.216206			−0.108103	−	0.994140i	$-0.534478\pi$
$457$	−2.06356e7			−0.216206			−0.108103	+	0.994140i	$0.534478\pi$
$458$		−	2.26816e7i		−	0.236090i
$459$	−6.66876e6	−	8.99615e7i	−0.0689616	−	0.930291i
$460$	−3.87342e7			−0.397943
$461$			1.93944e8i			1.97958i	0.142529	+	0.989791i	$0.454476\pi$
$461$			1.93944e8i			1.97958i	−0.142529	+	0.989791i	$0.545524\pi$
$462$	−1.89563e7	−	3.10394e7i	−0.192233	−	0.314766i
$463$	−3.32063e7			−0.334563			−0.167281	−	0.985909i	$-0.553499\pi$
$463$	−3.32063e7			−0.334563			−0.167281	+	0.985909i	$0.553499\pi$
$464$		−	2.99051e6i		−	0.0299358i
$465$	2.72381e7	−	1.66348e7i	0.270906	−	0.165447i
$466$	−2.66306e7			−0.263162
$467$			1.51133e8i			1.48391i	0.670449	+	0.741955i	$0.266101\pi$
$467$			1.51133e8i			1.48391i	−0.670449	+	0.741955i	$0.733899\pi$
$468$	−1.66883e7	+	3.25086e7i	−0.162807	+	0.317147i
$469$	1.79910e7			0.174396
$470$		−	2.30553e7i		−	0.222064i
$471$	7.29827e7	+	1.19503e8i	0.698486	+	1.14371i
$472$	6.85308e7			0.651718
$473$		−	5.24963e7i		−	0.496073i
$474$	−9.73149e7	+	5.94320e7i	−0.913786	+	0.558066i
$475$	−1.59072e7			−0.148427
$476$			6.55371e7i			0.607668i
$477$	6.96036e7	+	3.57309e7i	0.641322	+	0.329222i
$478$	1.25805e8			1.15190
$479$		−	7.76341e7i		−	0.706392i	−0.935549	−	0.353196i	$-0.885095\pi$
$479$		−	7.76341e7i		−	0.706392i	0.935549	−	0.353196i	$-0.114905\pi$
$480$	4.76928e7	+	7.80930e7i	0.431250	+	0.706136i
$481$	9.14041e7			0.821354
$482$		−	9.41292e6i		−	0.0840588i
$483$	−9.52740e7	+	5.81856e7i	−0.845538	+	0.516385i
$484$	4.88387e7			0.430753
$485$			1.54456e8i			1.35387i
$486$	−7.14315e7	−	3.02240e7i	−0.622273	−	0.263295i
$487$	−1.60691e8			−1.39125			−0.695623	−	0.718407i	$-0.744872\pi$
$487$	−1.60691e8			−1.39125			−0.695623	+	0.718407i	$0.744872\pi$
$488$			9.17196e7i			0.789228i
$489$	−4.58269e7	−	7.50377e7i	−0.391917	−	0.641731i
$490$	3.07609e7			0.261463
$491$		−	1.04606e8i		−	0.883715i	−0.897085	−	0.441858i	$-0.854320\pi$
$491$		−	1.04606e8i		−	0.883715i	0.897085	−	0.441858i	$-0.145680\pi$
$492$	−1.04434e7	+	6.37795e6i	−0.0876890	+	0.0535533i
$493$	−2.07563e7			−0.173225
$494$			3.68624e7i			0.305775i
$495$	−2.23464e7	+	4.35305e7i	−0.184243	+	0.358904i
$496$	7.04844e6			0.0577627
$497$		−	9.44871e6i		−	0.0769668i
$498$	−1.18257e7	−	1.93636e7i	−0.0957498	−	0.156782i
$499$	−7.83811e7			−0.630826			−0.315413	−	0.948954i	$-0.602143\pi$
$499$	−7.83811e7			−0.630826			−0.315413	+	0.948954i	$0.602143\pi$
$500$			7.31298e7i			0.585038i
$501$	3.49995e7	−	2.13748e7i	0.278323	−	0.169977i
$502$	1.58683e8			1.25435
$503$		−	1.40702e8i		−	1.10560i	−0.833315	−	0.552799i	$-0.813560\pi$
$503$		−	1.40702e8i		−	1.10560i	0.833315	−	0.552799i	$-0.186440\pi$
$504$	1.42375e8	+	7.30879e7i	1.11209	+	0.570892i
$505$	2.18927e8			1.69990
$506$		−	3.29489e7i		−	0.254325i
$507$	−3.86963e7	−	6.33619e7i	−0.296924	−	0.486188i
$508$	−8.29235e7			−0.632538
$509$			4.88289e7i			0.370274i	0.982713	+	0.185137i	$0.0592729\pi$
$509$			4.88289e7i			0.370274i	−0.982713	+	0.185137i	$0.940727\pi$
$510$	−6.32155e7	+	3.86069e7i	−0.476555	+	0.291041i
$511$	−2.26069e8			−1.69426
$512$			4.27733e7i			0.318686i
$513$	9.28813e7	−	6.88520e6i	0.687981	−	0.0509993i
$514$	2.22386e7			0.163764
$515$			1.28751e8i			0.942602i
$516$	4.23922e7	+	6.94137e7i	0.308558	+	0.505238i
$517$	−2.33449e7			−0.168935
$518$		−	1.40951e8i		−	1.01409i
$519$	2.32311e8	−	1.41877e8i	1.66176	−	1.01487i
$520$	8.52180e7			0.606068
$521$		−	2.89559e6i		−	0.0204750i	−0.999948	−	0.0102375i	$-0.996741\pi$
$521$		−	2.89559e6i		−	0.0204750i	0.999948	−	0.0102375i	$-0.00325875\pi$
$522$	−8.15035e6	+	1.58768e7i	−0.0573014	+	0.111623i
$523$	−1.55624e8			−1.08785			−0.543927	−	0.839133i	$-0.683063\pi$
$523$	−1.55624e8			−1.08785			−0.543927	+	0.839133i	$0.683063\pi$
$524$		−	1.52020e8i		−	1.05659i
$525$	1.94506e7	+	3.18487e7i	0.134417	+	0.220097i
$526$	−9.77529e7			−0.671695
$527$		−	4.89213e7i		−	0.334246i
$528$	−9.22229e6	+	5.63222e6i	−0.0626523	+	0.0382629i
$529$	4.69008e7			0.316820
$530$		−	6.42439e7i		−	0.431524i
$531$	−8.32367e7	−	4.27295e7i	−0.555944	−	0.285393i
$532$	−6.76642e7			−0.449391
$533$			1.87798e7i			0.124025i
$534$	−5.95421e7	−	9.74952e7i	−0.391021	−	0.640265i
$535$	−646331.			−0.00422079
$536$		−	2.33653e7i		−	0.151732i
$537$	9.42393e7	−	5.75537e7i	0.608568	−	0.371664i
$538$	5.61486e6			0.0360572
$539$		−	3.11472e7i		−	0.198908i
$540$	−5.60444e6	−	7.56039e7i	−0.0355919	−	0.480135i
$541$	−2.67737e8			−1.69089			−0.845446	−	0.534060i	$-0.820665\pi$
$541$	−2.67737e8			−1.69089			−0.845446	+	0.534060i	$0.820665\pi$
$542$			2.19444e7i			0.137824i
$543$	−6.62355e7	−	1.08455e8i	−0.413706	−	0.677408i
$544$	1.40260e8			0.871237
$545$		−	9.35534e7i		−	0.577923i
$546$	7.38039e7	−	4.50734e7i	0.453421	−	0.276912i
$547$	−1.59781e8			−0.976256			−0.488128	−	0.872772i	$-0.662320\pi$
$547$	−1.59781e8			−0.976256			−0.488128	+	0.872772i	$0.662320\pi$
$548$		−	3.13758e7i		−	0.190657i
$549$	5.71878e7	−	1.11401e8i	0.345610	−	0.673246i
$550$	−1.10143e7			−0.0662018
$551$		−	2.14300e7i		−	0.128105i
$552$	7.55667e7	+	1.23734e8i	0.449276	+	0.735652i
$553$	−3.21222e8			−1.89946
$554$			1.22557e8i			0.720787i
$555$	−1.61837e8	+	9.88368e7i	−0.946671	+	0.578149i
$556$	1.53260e8			0.891668
$557$			1.89403e8i			1.09603i	0.836469	+	0.548014i	$0.184616\pi$
$557$			1.89403e8i			1.09603i	−0.836469	+	0.548014i	$0.815384\pi$
$558$	−3.74206e7	−	1.92098e7i	−0.215382	−	0.110566i
$559$	1.24823e8			0.714595
$560$		−	3.00638e7i		−	0.171191i
$561$	3.90917e7	+	6.40095e7i	0.221410	+	0.362540i
$562$	1.26112e8			0.710474
$563$		−	2.86962e8i		−	1.60805i	−0.594595	−	0.804026i	$-0.702688\pi$
$563$		−	2.86962e8i		−	1.60805i	0.594595	−	0.804026i	$-0.297312\pi$
$564$	3.08680e7	−	1.88516e7i	0.172056	−	0.105078i
$565$	−1.28369e7			−0.0711730
$566$		−	1.32990e8i		−	0.733450i
$567$	−1.27356e8	−	1.77543e8i	−0.698665	−	0.973991i
$568$	−1.22712e7			−0.0669642
$569$			2.51884e7i			0.136730i	0.997660	+	0.0683650i	$0.0217782\pi$
$569$			2.51884e7i			0.136730i	−0.997660	+	0.0683650i	$0.978222\pi$
$570$	−3.98599e7	−	6.52672e7i	−0.215234	−	0.352428i
$571$	−1.90245e8			−1.02189			−0.510946	−	0.859613i	$-0.670705\pi$
$571$	−1.90245e8			−1.02189			−0.510946	+	0.859613i	$0.670705\pi$
$572$		−	3.03822e7i		−	0.162342i
$573$	8.59980e7	−	5.25205e7i	0.457114	−	0.279168i
$574$	2.89596e7			0.153129
$575$			3.38079e7i			0.177834i
$576$	6.91451e7	−	1.34694e8i	0.361821	−	0.704824i
$577$	−9.15925e7			−0.476796			−0.238398	−	0.971168i	$-0.576622\pi$
$577$	−9.15925e7			−0.476796			−0.238398	+	0.971168i	$0.576622\pi$
$578$		−	1.69360e7i		−	0.0877056i
$579$	5.08914e7	+	8.33305e7i	0.262186	+	0.429307i
$580$	−1.74437e7			−0.0894034
$581$		−	6.39161e7i		−	0.325898i
$582$	1.73727e8	−	1.06098e8i	0.881247	−	0.538194i
$583$	−6.50508e7			−0.328282
$584$			2.93601e8i			1.47407i
$585$	−1.03505e8	−	5.31340e7i	−0.517002	−	0.265403i
$586$	6.74156e7			0.335017
$587$			1.03240e8i			0.510427i	0.966885	+	0.255213i	$0.0821457\pi$
$587$			1.03240e8i			0.510427i	−0.966885	+	0.255213i	$0.917854\pi$
$588$	2.51523e7	+	4.11847e7i	0.123721	+	0.202583i
$589$	5.05091e7			0.247186
$590$			7.68273e7i			0.374076i
$591$	−8.68882e7	+	5.30642e7i	−0.420919	+	0.257063i
$592$	−4.18787e7			−0.201850
$593$			2.99476e8i			1.43615i	0.695968	+	0.718073i	$0.254975\pi$
$593$			2.99476e8i			1.43615i	−0.695968	+	0.718073i	$0.745025\pi$
$594$	6.43119e7	−	4.76738e6i	0.306854	−	0.0227468i
$595$	−2.08665e8			−0.990600
$596$		−	6.78458e7i		−	0.320468i
$597$	7.49076e6	+	1.22655e7i	0.0352049	+	0.0576450i
$598$	7.83442e7			0.366356
$599$			8.51341e6i			0.0396117i	0.999804	+	0.0198058i	$0.00630480\pi$
$599$			8.51341e6i			0.0396117i	−0.999804	+	0.0198058i	$0.993695\pi$
$600$	4.13625e7	−	2.52608e7i	0.191493	−	0.116948i
$601$	−8.75497e7			−0.403303			−0.201651	−	0.979457i	$-0.564631\pi$
$601$	−8.75497e7			−0.403303			−0.201651	+	0.979457i	$0.564631\pi$
$602$		−	1.92485e8i		−	0.882283i
$603$	−1.45684e7	+	2.83792e7i	−0.0664447	+	0.129434i
$604$	−2.11428e8			−0.959516
$605$			1.55499e8i			0.702199i
$606$	−1.50384e8	−	2.46242e8i	−0.675748	−	1.10648i
$607$	2.52937e8			1.13096			0.565479	−	0.824763i	$-0.308691\pi$
$607$	2.52937e8			1.13096			0.565479	+	0.824763i	$0.308691\pi$
$608$			1.44812e8i			0.644309i
$609$	−4.29060e7	+	2.62035e7i	−0.189962	+	0.116013i
$610$	−1.02823e8			−0.453004
$611$		−	5.55083e7i		−	0.243352i
$612$	−1.03379e8	−	5.30694e7i	−0.451001	−	0.231521i
$613$	−1.80203e8			−0.782311			−0.391155	−	0.920325i	$-0.627925\pi$
$613$	−1.80203e8			−0.782311			−0.391155	+	0.920325i	$0.627925\pi$
$614$			1.83239e8i			0.791615i
$615$	−2.03069e7	−	3.32508e7i	−0.0873007	−	0.142948i
$616$	−1.33062e8			−0.569261
$617$			1.57922e8i			0.672337i	0.941802	+	0.336168i	$0.109131\pi$
$617$			1.57922e8i			0.672337i	−0.941802	+	0.336168i	$0.890869\pi$
$618$	1.44815e8	−	8.84411e7i	0.613547	−	0.374704i
$619$	7.33915e7			0.309438			0.154719	−	0.987959i	$-0.450553\pi$
$619$	7.33915e7			0.309438			0.154719	+	0.987959i	$0.450553\pi$
$620$		−	4.11136e7i		−	0.172509i
$621$	−1.46332e7	−	1.97402e8i	−0.0611035	−	0.824285i
$622$	−1.30020e8			−0.540304
$623$		−	3.21817e8i		−	1.33090i
$624$	−1.33920e7	−	2.19283e7i	−0.0551178	−	0.0902508i
$625$	−1.80312e8			−0.738556
$626$			6.71546e7i			0.273749i
$627$	−6.60869e7	+	4.03605e7i	−0.268110	+	0.163740i
$628$	1.80380e8			0.728297
$629$			2.90669e8i			1.16801i
$630$	−8.19361e7	+	1.59611e8i	−0.327683	+	0.638323i
$631$	3.73260e8			1.48567			0.742836	−	0.669473i	$-0.233480\pi$
$631$	3.73260e8			1.48567			0.742836	+	0.669473i	$0.233480\pi$
$632$			4.17177e8i			1.65260i
$633$	−2.06739e8	−	3.38518e8i	−0.815101	−	1.33466i
$634$	2.48843e8			0.976467
$635$		−	2.64022e8i		−	1.03114i
$636$	8.60140e7	−	5.25303e7i	0.334347	−	0.204192i
$637$	7.40603e7			0.286528
$638$		−	1.48383e7i		−	0.0571377i
$639$	1.49045e7	+	7.65119e6i	0.0571234	+	0.0293242i
$640$	9.25779e7			0.353157
$641$			4.17603e8i			1.58559i	0.609491	+	0.792793i	$0.291374\pi$
$641$			4.17603e8i			1.58559i	−0.609491	+	0.792793i	$0.708626\pi$
$642$	443975.	+	726972.i	0.00167785	+	0.00274734i
$643$	−2.22049e7			−0.0835248			−0.0417624	−	0.999128i	$-0.513297\pi$
$643$	−2.22049e7			−0.0835248			−0.0417624	+	0.999128i	$0.513297\pi$
$644$			1.43808e8i			0.538425i
$645$	−2.21008e8	+	1.34973e8i	−0.823622	+	0.503001i
$646$	−1.17224e8			−0.434829
$647$			2.62438e8i			0.968978i	0.874797	+	0.484489i	$0.160994\pi$
$647$			2.62438e8i			0.968978i	−0.874797	+	0.484489i	$0.839006\pi$
$648$	−2.30579e8	+	1.65399e8i	−0.847412	+	0.607867i
$649$	7.77922e7			0.284578
$650$		−	2.61893e7i		−	0.0953639i
$651$	−6.17599e7	−	1.01127e8i	−0.223853	−	0.366541i
$652$	−1.13263e8			−0.408644
$653$			2.48516e8i			0.892513i	0.894905	+	0.446257i	$0.147243\pi$
$653$			2.48516e8i			0.892513i	−0.894905	+	0.446257i	$0.852757\pi$
$654$	−1.05226e8	+	6.42634e7i	−0.376175	+	0.229737i
$655$	4.84020e8			1.72242
$656$		−	8.60435e6i		−	0.0304794i
$657$	1.83062e8	−	3.56604e8i	0.645509	−	1.25745i
$658$	−8.55973e7			−0.300457
$659$			2.06875e8i			0.722856i	0.932400	+	0.361428i	$0.117711\pi$
$659$			2.06875e8i			0.722856i	−0.932400	+	0.361428i	$0.882289\pi$
$660$	3.28528e7	+	5.37937e7i	0.114272	+	0.187111i
$661$	4.85124e8			1.67976			0.839882	−	0.542769i	$-0.182624\pi$
$661$	4.85124e8			1.67976			0.839882	+	0.542769i	$0.182624\pi$
$662$		−	3.20689e8i		−	1.10538i
$663$	−1.52198e8	+	9.29504e7i	−0.522239	+	0.318941i
$664$	−8.30091e7			−0.283545
$665$		−	2.15437e8i		−	0.732581i
$666$	2.22337e8	+	1.14137e8i	0.752644	+	0.386369i
$667$	−4.55456e7			−0.153486
$668$		−	5.28288e7i		−	0.177232i
$669$	2.43406e8	+	3.98557e8i	0.812931	+	1.33111i
$670$	2.61939e7			0.0870916
$671$			1.04115e8i			0.344623i
$672$	2.89935e8	−	1.77069e8i	0.955417	−	0.583491i
$673$	1.58232e8			0.519096			0.259548	−	0.965730i	$-0.416426\pi$
$673$	1.58232e8			0.519096			0.259548	+	0.965730i	$0.416426\pi$
$674$		−	1.66733e8i		−	0.544554i
$675$	−6.59886e7	+	4.89167e6i	−0.214564	+	0.0159055i
$676$	−9.56394e7			−0.309597
$677$			3.87813e7i			0.124985i	0.998045	+	0.0624923i	$0.0199049\pi$
$677$			3.87813e7i			0.124985i	−0.998045	+	0.0624923i	$0.980095\pi$
$678$	8.81789e6	+	1.44386e7i	0.0282928	+	0.0463271i
$679$	5.73446e8			1.83182
$680$			2.70997e8i			0.861862i
$681$	−1.53814e8	+	9.39370e7i	−0.487029	+	0.297437i
$682$	3.49730e7			0.110250
$683$		−	3.73234e8i		−	1.17144i	−0.810514	−	0.585719i	$-0.800812\pi$
$683$		−	3.73234e8i		−	1.17144i	0.810514	−	0.585719i	$-0.199188\pi$
$684$	5.47918e7	−	1.06734e8i	0.171217	−	0.333530i
$685$	9.98979e7			0.310803
$686$			1.47259e8i			0.456152i
$687$	5.90490e7	+	9.66879e7i	0.182114	+	0.298196i
$688$	−5.71904e7			−0.175613
$689$		−	1.54674e8i		−	0.472891i
$690$	−1.38714e8	+	8.47150e7i	−0.422252	+	0.257877i
$691$	1.63986e8			0.497019			0.248510	−	0.968629i	$-0.420059\pi$
$691$	1.63986e8			0.497019			0.248510	+	0.968629i	$0.420059\pi$
$692$		−	3.50653e8i		−	1.05818i
$693$	1.61615e8	+	8.29651e7i	0.485605	+	0.249285i
$694$	3.56658e8			1.06702
$695$			4.87966e8i			1.45357i
$696$	3.40310e7	+	5.57229e7i	0.100936	+	0.165275i
$697$	−5.97205e7			−0.176370
$698$		−	3.02802e8i		−	0.890416i
$699$	1.13522e8	−	6.93298e7i	0.332390	−	0.202997i
$700$	4.80728e7			0.140154
$701$		−	2.58341e8i		−	0.749963i	−0.927032	−	0.374981i	$-0.877649\pi$
$701$		−	2.58341e8i		−	0.749963i	0.927032	−	0.374981i	$-0.122351\pi$
$702$	1.13356e7	+	1.52918e8i	0.0327668	+	0.442024i
$703$	−3.00103e8			−0.863782
$704$			1.25884e8i			0.360788i
$705$	6.00220e7	+	9.82811e7i	0.171295	+	0.280480i
$706$	−1.03022e7			−0.0292763
$707$		−	8.12807e8i		−	2.30001i
$708$	−1.02861e8	+	6.28193e7i	−0.289836	+	0.177008i
$709$	3.35366e7			0.0940980			0.0470490	−	0.998893i	$-0.485018\pi$
$709$	3.35366e7			0.0940980			0.0470490	+	0.998893i	$0.485018\pi$
$710$		−	1.37568e7i		−	0.0384364i
$711$	2.60113e8	−	5.06698e8i	0.723690	−	1.40974i
$712$	−4.17950e8			−1.15793
$713$		−	1.07348e8i		−	0.296159i
$714$	1.43335e8	+	2.34700e8i	0.393785	+	0.644789i
$715$	9.67345e7			0.264645
$716$		−	1.42246e8i		−	0.387526i
$717$	−5.36287e8	+	3.27520e8i	−1.45492	+	0.888547i
$718$	1.80119e8			0.486616
$719$		−	1.53668e8i		−	0.413424i	−0.978402	−	0.206712i	$-0.933724\pi$
$719$		−	1.53668e8i		−	0.413424i	0.978402	−	0.206712i	$-0.0662764\pi$
$720$	4.74229e7	+	2.43445e7i	0.127055	+	0.0652234i
$721$	4.78012e8			1.27536
$722$			1.33277e8i			0.354113i
$723$	2.45055e7	+	4.01257e7i	0.0648409	+	0.106172i
$724$	−1.63704e8			−0.431363
$725$			1.52252e7i			0.0399529i
$726$	1.74900e8	−	1.06815e8i	0.457067	−	0.279139i
$727$	1.60024e8			0.416468			0.208234	−	0.978079i	$-0.433228\pi$
$727$	1.60024e8			0.416468			0.208234	+	0.978079i	$0.433228\pi$
$728$		−	3.16388e8i		−	0.820023i
$729$	3.83186e8	−	5.71243e7i	0.989070	−	0.147448i
$730$	−3.29145e8			−0.846093
$731$			3.96943e8i			1.01619i
$732$	−8.40755e7	−	1.37667e8i	−0.214356	−	0.350990i
$733$	−1.11592e8			−0.283348			−0.141674	−	0.989913i	$-0.545248\pi$
$733$	−1.11592e8			−0.283348			−0.141674	+	0.989913i	$0.545248\pi$
$734$			4.62304e8i			1.16907i
$735$	−1.31129e8	+	8.00827e7i	−0.330245	+	0.201686i
$736$	3.07772e8			0.771961
$737$		−	2.65229e7i		−	0.0662550i
$738$	−2.34503e7	+	4.56811e7i	−0.0583418	+	0.113649i
$739$	−6.24800e8			−1.54813			−0.774066	−	0.633105i	$-0.781780\pi$
$739$	−6.24800e8			−1.54813			−0.774066	+	0.633105i	$0.781780\pi$
$740$			2.44279e8i			0.602825i
$741$	−9.59672e7	−	1.57138e8i	−0.235867	−	0.386213i
$742$	−2.38518e8			−0.583860
$743$		−	3.52156e8i		−	0.858557i	−0.903172	−	0.429279i	$-0.858768\pi$
$743$		−	3.52156e8i		−	0.858557i	0.903172	−	0.429279i	$-0.141232\pi$
$744$	−1.31335e8	+	8.02088e7i	−0.318906	+	0.194762i
$745$	2.16016e8			0.522416
$746$			3.18397e8i			0.766925i
$747$	1.00822e8	+	5.17568e7i	0.241876	+	0.124167i
$748$	9.66168e7			0.230860
$749$			2.39962e6i			0.00571081i
$750$	1.59941e8	+	2.61890e8i	0.379120	+	0.620777i
$751$	1.80788e8			0.426826			0.213413	−	0.976962i	$-0.431542\pi$
$751$	1.80788e8			0.426826			0.213413	+	0.976962i	$0.431542\pi$
$752$			2.54323e7i			0.0598042i
$753$	−6.76439e8	+	4.13114e8i	−1.58432	+	0.967575i
$754$	3.52818e7			0.0823070
$755$		−	6.73171e8i		−	1.56417i
$756$	−2.80694e8	+	2.08076e7i	−0.649632	+	0.0481566i
$757$	−1.67808e8			−0.386834			−0.193417	−	0.981117i	$-0.561957\pi$
$757$	−1.67808e8			−0.386834			−0.193417	+	0.981117i	$0.561957\pi$
$758$			4.58555e8i			1.05289i
$759$	8.57789e7	+	1.40456e8i	0.196180	+	0.321229i
$760$	−2.79792e8			−0.637375
$761$		−	1.22351e8i		−	0.277621i	−0.990319	−	0.138810i	$-0.955672\pi$
$761$		−	1.22351e8i		−	0.277621i	0.990319	−	0.138810i	$-0.0443279\pi$
$762$	−2.96963e8	+	1.81361e8i	−0.671179	+	0.409901i
$763$	−3.47335e8			−0.781942
$764$		−	1.29807e8i		−	0.291083i
$765$	1.68969e8	−	3.29150e8i	0.377417	−	0.735206i
$766$	−3.83251e8			−0.852701
$767$			1.84970e8i			0.409936i
$768$	−2.50647e8	−	4.10413e8i	−0.553323	−	0.906020i
$769$	6.44941e8			1.41821			0.709106	−	0.705102i	$-0.249099\pi$
$769$	6.44941e8			1.41821			0.709106	+	0.705102i	$0.249099\pi$
$770$		−	1.49171e8i		−	0.326747i
$771$	−9.47996e7	+	5.78959e7i	−0.206844	+	0.126324i
$772$	1.25780e8			0.273376
$773$			7.89133e8i			1.70849i	0.519873	+	0.854243i	$0.325979\pi$
$773$			7.89133e8i			1.70849i	−0.519873	+	0.854243i	$0.674021\pi$
$774$	3.03628e8	+	1.55867e8i	0.654815	+	0.336148i
$775$	−3.58848e7			−0.0770913
$776$		−	7.44745e8i		−	1.59376i
$777$	3.66950e8	+	6.00851e8i	0.782248	+	1.28087i
$778$	3.98554e8			0.846347
$779$		−	6.16588e7i		−	0.130431i
$780$	−1.27908e8	+	7.81158e7i	−0.269534	+	0.164609i
$781$	−1.39296e7			−0.0292405
$782$			2.49138e8i			0.520979i
$783$	−6.58999e6	−	8.88989e7i	−0.0137277	−	0.185187i
$784$	−3.39323e7			−0.0704150
$785$			5.74314e8i			1.18725i
$786$	−3.32481e8	−	5.44410e8i	−0.684699	−	1.12114i
$787$	1.42367e8			0.292068			0.146034	−	0.989280i	$-0.453349\pi$
$787$	1.42367e8			0.292068			0.146034	+	0.989280i	$0.453349\pi$
$788$			1.31150e8i			0.268035i
$789$	4.16705e8	−	2.54489e8i	0.848393	−	0.518129i
$790$	−4.67681e8			−0.948568
$791$			4.76595e7i			0.0962986i
$792$	1.07748e8	−	2.09893e8i	0.216888	−	0.422496i
$793$	−2.47559e8			−0.496431
$794$		−	4.45006e8i		−	0.889006i
$795$	1.67252e8	+	2.73861e8i	0.332867	+	0.545041i
$796$	1.85137e7			0.0367075
$797$		−	8.62596e8i		−	1.70385i	−0.523661	−	0.851927i	$-0.675434\pi$
$797$		−	8.62596e8i		−	1.70385i	0.523661	−	0.851927i	$-0.324566\pi$
$798$	−2.42317e8	+	1.47987e8i	−0.476843	+	0.291217i
$799$	1.76519e8			0.346059
$800$		−	1.02883e8i		−	0.200944i
$801$	5.07636e8	+	2.60595e8i	0.987769	+	0.507070i
$802$	−4.68309e8			−0.907840
$803$			3.33278e8i			0.643666i
$804$	2.14180e7	+	3.50701e7i	0.0412107	+	0.0674791i
$805$	−4.57873e8			−0.877722
$806$			8.31570e7i			0.158816i
$807$	−2.39352e7	+	1.46177e7i	−0.0455425	+	0.0278136i
$808$	−1.05561e9			−2.00110
$809$		−	9.90552e8i		−	1.87082i	−0.353567	−	0.935409i	$-0.615031\pi$
$809$		−	9.90552e8i		−	1.87082i	0.353567	−	0.935409i	$-0.384969\pi$
$810$	−1.85423e8	−	2.58493e8i	−0.348906	−	0.486401i
$811$	−5.18957e8			−0.972902			−0.486451	−	0.873708i	$-0.661709\pi$
$811$	−5.18957e8			−0.972902			−0.486451	+	0.873708i	$0.661709\pi$
$812$			6.47630e7i			0.120965i
$813$	−5.71299e7	−	9.35454e7i	−0.106314	−	0.174081i
$814$	−2.07794e8			−0.385265
$815$		−	3.60620e8i		−	0.666158i
$816$	6.97329e7	−	4.25872e7i	0.128342	−	0.0783806i
$817$	−4.09826e8			−0.751508
$818$		−	6.17755e8i		−	1.12864i
$819$	−1.97270e8	+	3.84281e8i	−0.359095	+	0.699515i
$820$	−5.01893e7			−0.0910268
$821$			5.88367e8i			1.06321i	0.846993	+	0.531605i	$0.178411\pi$
$821$			5.88367e8i			1.06321i	−0.846993	+	0.531605i	$0.821589\pi$
$822$	−6.86215e7	−	1.12362e8i	−0.123551	−	0.202304i
$823$	−3.38718e8			−0.607630			−0.303815	−	0.952731i	$-0.598260\pi$
$823$	−3.38718e8			−0.607630			−0.303815	+	0.952731i	$0.598260\pi$
$824$		−	6.20803e8i		−	1.10962i
$825$	4.69522e7	−	2.86746e7i	0.0836170	−	0.0510664i
$826$	2.85236e8			0.506132
$827$		−	4.72136e8i		−	0.834739i	−0.908737	−	0.417370i	$-0.862952\pi$
$827$		−	4.72136e8i		−	0.834739i	0.908737	−	0.417370i	$-0.137048\pi$
$828$	−2.26844e8	−	1.16450e8i	−0.399610	−	0.205139i
$829$	7.96835e8			1.39864			0.699318	−	0.714811i	$-0.253487\pi$
$829$	7.96835e8			1.39864			0.699318	+	0.714811i	$0.253487\pi$
$830$		−	9.30583e7i		−	0.162750i
$831$	−3.19063e8	−	5.22438e8i	−0.555998	−	0.910400i
$832$	−2.99320e8			−0.519716
$833$			2.35515e8i			0.407459i
$834$	5.48849e8	−	3.35192e8i	0.946138	−	0.577824i
$835$	1.68202e8			0.288917
$836$			9.97526e7i			0.170728i
$837$	2.09529e8	−	1.55322e7i	0.357328	−	0.0264884i
$838$	5.51195e8			0.936641
$839$			9.69710e7i			0.164194i	0.996624	+	0.0820968i	$0.0261617\pi$
$839$			9.69710e7i			0.164194i	−0.996624	+	0.0820968i	$0.973838\pi$
$840$	3.42116e8	+	5.60186e8i	0.577212	+	0.945136i
$841$	−2.05111e7			−0.0344828
$842$		−	6.57172e8i		−	1.10089i
$843$	−5.37596e8	+	3.28320e8i	−0.897374	+	0.548042i
$844$	−5.10964e8			−0.849891
$845$		−	3.04508e8i		−	0.504694i
$846$	6.93133e7	−	1.35022e8i	0.114474	−	0.222994i
$847$	5.77318e8			0.950090
$848$			7.08674e7i			0.116214i
$849$	3.46226e8	+	5.66916e8i	0.565765	+	0.926393i
$850$	8.32831e7			0.135613
$851$			6.37814e8i			1.03492i
$852$	1.84185e7	−	1.12485e7i	0.0297807	−	0.0181876i
$853$	−6.22970e8			−1.00374			−0.501868	−	0.864944i	$-0.667354\pi$
$853$	−6.22970e8			−1.00374			−0.501868	+	0.864944i	$0.667354\pi$
$854$			3.81751e8i			0.612924i
$855$	3.39832e8	+	1.74453e8i	0.543709	+	0.279112i
$856$	3.11644e6			0.00496864
$857$		−	2.79046e8i		−	0.443337i	−0.975122	−	0.221669i	$-0.928850\pi$
$857$		−	2.79046e8i		−	0.443337i	0.975122	−	0.221669i	$-0.0711503\pi$
$858$	−6.64485e7	−	1.08804e8i	−0.105202	−	0.172259i
$859$	−6.71460e7			−0.105935			−0.0529676	−	0.998596i	$-0.516868\pi$
$859$	−6.71460e7			−0.105935			−0.0529676	+	0.998596i	$0.516868\pi$
$860$			3.33592e8i			0.524470i
$861$	−1.23450e8	+	7.53931e7i	−0.193411	+	0.118120i
$862$	8.69328e7			0.135726
$863$		−	5.16619e8i		−	0.803782i	−0.915688	−	0.401891i	$-0.868353\pi$
$863$		−	5.16619e8i		−	0.803782i	0.915688	−	0.401891i	$-0.131647\pi$
$864$	4.45315e7	+	6.00729e8i	0.0690440	+	0.931403i
$865$	1.11645e9			1.72501
$866$			8.55216e7i			0.131681i
$867$	4.40911e7	+	7.21954e7i	0.0676540	+	0.110778i
$868$	−1.52642e8			−0.233408
$869$			4.73554e8i			0.721623i
$870$	−6.24688e7	+	3.81508e7i	−0.0948649	+	0.0579357i
$871$	6.30648e7			0.0954405
$872$			4.51090e8i			0.680321i
$873$	−4.64354e8	+	9.04558e8i	−0.697921	+	1.35955i
$874$	−2.57224e8			−0.385281
$875$			8.64459e8i			1.29039i
$876$	−2.69131e8	−	4.40680e8i	−0.400361	−	0.655558i
$877$	−1.43572e8			−0.212848			−0.106424	−	0.994321i	$-0.533940\pi$
$877$	−1.43572e8			−0.212848			−0.106424	+	0.994321i	$0.533940\pi$
$878$		−	1.45809e8i		−	0.215428i
$879$	−2.87381e8	+	1.75509e8i	−0.423148	+	0.258424i
$880$	−4.43209e7			−0.0650371
$881$		−	1.24774e9i		−	1.82472i	−0.409393	−	0.912358i	$-0.634260\pi$
$881$		−	1.24774e9i		−	1.82472i	0.409393	−	0.912358i	$-0.365740\pi$
$882$	1.80149e8	+	9.24793e7i	0.262559	+	0.134784i
$883$	−9.65905e7			−0.140298			−0.0701492	−	0.997537i	$-0.522348\pi$
$883$	−9.65905e7			−0.140298			−0.0701492	+	0.997537i	$0.522348\pi$
$884$			2.29731e8i			0.332554i
$885$	−2.00012e8	−	3.27502e8i	−0.288553	−	0.472481i
$886$	3.19028e7			0.0458699
$887$			2.90029e8i			0.415596i	0.978172	+	0.207798i	$0.0666296\pi$
$887$			2.90029e8i			0.415596i	−0.978172	+	0.207798i	$0.933370\pi$
$888$	7.80336e8	−	4.76566e8i	1.11440	−	0.680587i
$889$	−9.80231e8			−1.39516
$890$		−	4.68547e8i		−	0.664635i
$891$	−2.61740e8	+	1.87752e8i	−0.370030	+	0.265430i
$892$	6.01588e8			0.847627
$893$			1.82248e8i			0.255922i
$894$	−1.48385e8	−	2.42968e8i	−0.207671	−	0.340044i
$895$	4.52900e8			0.631732
$896$		−	3.43713e8i		−	0.477828i
$897$	−3.33969e8	+	2.03961e8i	−0.462731	+	0.282598i
$898$	−3.95460e8			−0.546102
$899$		−	4.83434e7i		−	0.0665363i
$900$	−3.89275e7	+	7.58305e7i	−0.0533985	+	0.104020i
$901$	4.91872e8			0.672477
$902$		−	4.26931e7i		−	0.0581752i
$903$	5.01114e8	+	8.20533e8i	0.680571	+	1.11438i
$904$	6.18963e7			0.0837837
$905$		−	5.21219e8i		−	0.703193i
$906$	−7.57161e8	+	4.62412e8i	−1.01813	+	0.621791i
$907$	−7.19914e8			−0.964848			−0.482424	−	0.875938i	$-0.660244\pi$
$907$	−7.19914e8			−0.964848			−0.482424	+	0.875938i	$0.660244\pi$
$908$			2.32169e8i			0.310132i
$909$	1.28213e9	+	6.58179e8i	1.70702	+	0.876299i
$910$	3.54691e8			0.470680
$911$		−	1.28992e9i		−	1.70611i	−0.521824	−	0.853053i	$-0.674748\pi$
$911$		−	1.28992e9i		−	1.70611i	0.521824	−	0.853053i	$-0.325252\pi$
$912$	4.39694e7	+	7.19962e7i	0.0579650	+	0.0949128i
$913$	−9.42271e7			−0.123812
$914$			1.11545e8i			0.146087i
$915$	4.38319e8	−	2.67689e8i	0.572172	−	0.349436i
$916$	1.45942e8			0.189886
$917$		−	1.79702e9i		−	2.33047i
$918$	−4.86285e8	+	3.60478e7i	−0.628583	+	0.0465962i
$919$	9.33470e7			0.120269			0.0601345	−	0.998190i	$-0.480847\pi$
$919$	9.33470e7			0.120269			0.0601345	+	0.998190i	$0.480847\pi$
$920$			5.94648e8i			0.763654i
$921$	−4.77044e8	−	7.81120e8i	−0.610632	−	0.999859i
$922$	1.04836e9			1.33757
$923$		−	3.31210e7i		−	0.0421210i
$924$	1.99719e8	−	1.21972e8i	0.253165	−	0.154613i
$925$	2.13212e8			0.269393
$926$			1.79496e8i			0.226059i
$927$	−3.87075e8	+	7.54020e8i	−0.485910	+	0.946550i
$928$	1.38603e8			0.173432
$929$			1.12424e9i			1.40221i	0.713060	+	0.701103i	$0.247309\pi$
$929$			1.12424e9i			1.40221i	−0.713060	+	0.701103i	$0.752691\pi$
$930$	−8.99190e7	−	1.47235e8i	−0.111790	−	0.183047i
$931$	−2.43159e8			−0.301329
$932$		−	1.71351e8i		−	0.211661i
$933$	5.54253e8	−	3.38493e8i	0.682438	−	0.416777i
$934$	8.16943e8			1.00265
$935$			3.07620e8i			0.376339i
$936$	4.99073e8	+	2.56199e8i	0.608607	+	0.312428i
$937$	−4.35346e8			−0.529195			−0.264598	−	0.964359i	$-0.585239\pi$
$937$	−4.35346e8			−0.529195			−0.264598	+	0.964359i	$0.585239\pi$
$938$		−	9.72499e7i		−	0.117837i
$939$	−1.74830e8	−	2.86269e8i	−0.211163	−	0.345762i
$940$	1.48347e8			0.178605
$941$			4.80148e8i			0.576244i	0.957594	+	0.288122i	$0.0930309\pi$
$941$			4.80148e8i			0.576244i	−0.957594	+	0.288122i	$0.906969\pi$
$942$	6.45971e8	−	3.94506e8i	0.772788	−	0.471955i
$943$	−1.31044e8			−0.156273
$944$		−	8.47481e7i		−	0.100743i
$945$	−6.62496e7	−	8.93706e8i	−0.0785033	−	1.05901i
$946$	−2.83767e8			−0.335188
$947$		−	2.10412e7i		−	0.0247754i	−0.999923	−	0.0123877i	$-0.996057\pi$
$947$		−	2.10412e7i		−	0.0247754i	0.999923	−	0.0123877i	$-0.00394322\pi$
$948$	−3.82408e8	−	6.26162e8i	−0.448851	−	0.734956i
$949$	−7.92452e8			−0.927202
$950$			8.59861e7i			0.100290i
$951$	−1.06078e9	+	6.47836e8i	−1.23334	+	0.753223i
$952$	1.00613e9			1.16612
$953$			8.25428e8i			0.953675i	0.878991	+	0.476838i	$0.158217\pi$
$953$			8.25428e8i			0.953675i	−0.878991	+	0.476838i	$0.841783\pi$
$954$	1.93142e8	−	3.76240e8i	0.222450	−	0.433331i
$955$	4.13293e8			0.474513
$956$			8.09479e8i			0.926471i
$957$	3.86300e7	+	6.32533e7i	0.0440746	+	0.0721685i
$958$	−4.19649e8			−0.477298
$959$		−	3.70890e8i		−	0.420522i
$960$	5.29966e8	−	3.23660e8i	0.599010	−	0.365826i
$961$	−7.73561e8			−0.871615
$962$		−	4.94082e8i		−	0.554976i
$963$	−3.78519e6	−	1.94312e6i	−0.00423847	−	0.00217581i
$964$	6.05664e7			0.0676083
$965$			4.00474e8i			0.445648i
$966$	3.14520e8	+	5.15001e8i	0.348913	+	0.571316i
$967$	7.12481e8			0.787941			0.393971	−	0.919123i	$-0.371101\pi$
$967$	7.12481e8			0.787941			0.393971	+	0.919123i	$0.371101\pi$
$968$		−	7.49774e8i		−	0.826617i
$969$	4.99706e8	−	3.05180e8i	0.549216	−	0.335417i
$970$	8.34905e8			0.914791
$971$			1.44153e9i			1.57459i	0.616578	+	0.787293i	$0.288518\pi$
$971$			1.44153e9i			1.57459i	−0.616578	+	0.787293i	$0.711482\pi$
$972$	1.94473e8	−	4.59618e8i	0.211768	−	0.500493i
$973$	1.81167e9			1.96671
$974$			8.68609e8i			0.940043i
$975$	6.81810e7	+	1.11641e8i	0.0735613	+	0.120451i
$976$	1.13424e8			0.121999
$977$			1.58483e8i			0.169941i	0.996383	+	0.0849707i	$0.0270797\pi$
$977$			1.58483e8i			0.169941i	−0.996383	+	0.0849707i	$0.972920\pi$
$978$	−4.05614e8	+	2.47716e8i	−0.433607	+	0.264812i
$979$	−4.74432e8			−0.505622
$980$			1.97928e8i			0.210295i
$981$	2.81258e8	−	5.47889e8i	0.297919	−	0.580344i
$982$	−5.65445e8			−0.597112
$983$		−	1.03811e9i		−	1.09290i	−0.837491	−	0.546450i	$-0.815979\pi$
$983$		−	1.03811e9i		−	1.09290i	0.837491	−	0.546450i	$-0.184021\pi$
$984$	9.79145e7	+	1.60327e8i	0.102769	+	0.168276i
$985$	−4.17572e8			−0.436941
$986$			1.12198e8i			0.117045i
$987$	3.64887e8	−	2.22843e8i	0.379496	−	0.231765i
$988$	−2.37187e8			−0.245934
$989$			8.71011e8i			0.900398i
$990$	2.35303e8	+	1.20793e8i	0.242506	+	0.124490i
$991$	1.75227e9			1.80044			0.900222	−	0.435432i	$-0.143404\pi$
$991$	1.75227e9			1.80044			0.900222	+	0.435432i	$0.143404\pi$
$992$			3.26678e8i			0.334646i
$993$	8.34879e8	+	1.36704e9i	0.852660	+	1.39616i
$994$	−5.10747e7			−0.0520052
$995$			5.89461e7i			0.0598392i
$996$	1.24593e8	−	7.60910e7i	0.126100	−	0.0770114i
$997$	8.39850e8			0.847454			0.423727	−	0.905790i	$-0.360722\pi$
$997$	8.39850e8			0.847454			0.423727	+	0.905790i	$0.360722\pi$
$998$			4.23687e8i			0.426239i
$999$	−1.24493e9	+	9.22854e7i	−1.24867	+	0.0925628i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	87.7.b.a.59.20	✓	56
3.2	odd	2	inner	87.7.b.a.59.37	yes	56

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
87.7.b.a.59.20	✓	56	1.1	even	1	trivial
87.7.b.a.59.37	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-5.40547i q^{2} +(14.0726 + 23.0426i) q^{3} +34.7809 q^{4} +110.739i q^{5} +(124.556 - 76.0688i) q^{6} +411.141 q^{7} -533.957i q^{8} +(-332.926 + 648.538i) q^{9} +598.599 q^{10} -606.117i q^{11} +(489.456 + 801.443i) q^{12} +1441.19 q^{13} -2222.41i q^{14} +(-2551.73 + 1558.39i) q^{15} -660.314 q^{16} +4583.06i q^{17} +(3505.65 + 1799.62i) q^{18} -4731.81 q^{19} +3851.62i q^{20} +(5785.81 + 9473.78i) q^{21} -3276.35 q^{22} +10056.6i q^{23} +(12303.8 - 7514.14i) q^{24} +3361.77 q^{25} -7790.33i q^{26} +(-19629.1 + 1455.09i) q^{27} +14299.9 q^{28} +4528.92i q^{29} +(8423.82 + 13793.3i) q^{30} -10674.4 q^{31} -30604.0i q^{32} +(13966.5 - 8529.61i) q^{33} +24773.6 q^{34} +45529.6i q^{35} +(-11579.5 + 22556.7i) q^{36} +63422.5 q^{37} +25577.6i q^{38} +(20281.3 + 33208.9i) q^{39} +59130.1 q^{40} +13030.7i q^{41} +(51210.2 - 31275.0i) q^{42} +86610.9 q^{43} -21081.3i q^{44} +(-71818.7 - 36868.1i) q^{45} +54360.6 q^{46} -38515.5i q^{47} +(-9292.30 - 15215.4i) q^{48} +51388.2 q^{49} -18171.9i q^{50} +(-105606. + 64495.4i) q^{51} +50126.0 q^{52} -107324. i q^{53} +(7865.44 + 106105. i) q^{54} +67121.1 q^{55} -219532. i q^{56} +(-66588.6 - 109033. i) q^{57} +24481.0 q^{58} +128345. i q^{59} +(-88751.4 + 54202.1i) q^{60} -171773. q^{61} +57700.0i q^{62} +(-136880. + 266641. i) q^{63} -207689. q^{64} +159597. i q^{65} +(-46106.6 - 75495.7i) q^{66} +43758.7 q^{67} +159403. i q^{68} +(-231730. + 141522. i) q^{69} +246109. q^{70} -22981.7i q^{71} +(346291. + 177768. i) q^{72} -549858. q^{73} -342828. i q^{74} +(47308.7 + 77464.0i) q^{75} -164576. q^{76} -249200. i q^{77} +(179510. - 109630. i) q^{78} -781292. q^{79} -73122.8i q^{80} +(-309761. - 431830. i) q^{81} +70437.1 q^{82} -155460. i q^{83} +(201236. + 329506. i) q^{84} -507526. q^{85} -468173. i q^{86} +(-104358. + 63733.5i) q^{87} -323640. q^{88} -782740. i q^{89} +(-199289. + 388214. i) q^{90} +592534. q^{91} +349777. i q^{92} +(-150216. - 245966. i) q^{93} -208194. q^{94} -523998. i q^{95} +(705196. - 430676. i) q^{96} +1.39477e6 q^{97} -277777. i q^{98} +(393090. + 201792. 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

Modular forms

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Database

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