LMFDB - Embedded newform 87.7.b.a.59.17

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			59.17
Character	$\chi$	$=$	87.59
Dual form			87.7.b.a.59.40

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q-7.77900i q^{2} +(-23.4760 + 13.3371i) q^{3} +3.48715 q^{4} +148.308i q^{5} +(103.749 + 182.620i) q^{6} +172.143 q^{7} -524.983i q^{8} +(373.244 - 626.203i) q^{9} +1153.69 q^{10} +775.451i q^{11} +(-81.8643 + 46.5084i) q^{12} -2698.75 q^{13} -1339.10i q^{14} +(-1978.00 - 3481.68i) q^{15} -3860.66 q^{16} +1712.63i q^{17} +(-4871.23 - 2903.47i) q^{18} -3288.45 q^{19} +517.173i q^{20} +(-4041.22 + 2295.88i) q^{21} +6032.24 q^{22} +4756.34i q^{23} +(7001.74 + 12324.5i) q^{24} -6370.36 q^{25} +20993.6i q^{26} +(-410.554 + 19678.7i) q^{27} +600.288 q^{28} -4528.92i q^{29} +(-27084.0 + 15386.9i) q^{30} -36696.0 q^{31} -3566.79i q^{32} +(-10342.3 - 18204.5i) q^{33} +13322.5 q^{34} +25530.2i q^{35} +(1301.56 - 2183.66i) q^{36} -35742.7 q^{37} +25580.8i q^{38} +(63355.8 - 35993.5i) q^{39} +77859.3 q^{40} +113883. i q^{41} +(17859.7 + 31436.7i) q^{42} -121960. q^{43} +2704.11i q^{44} +(92871.1 + 55355.2i) q^{45} +36999.6 q^{46} +82119.8i q^{47} +(90632.9 - 51490.0i) q^{48} -88015.9 q^{49} +49555.1i q^{50} +(-22841.5 - 40205.6i) q^{51} -9410.94 q^{52} -88459.6i q^{53} +(153081. + 3193.70i) q^{54} -115006. q^{55} -90371.9i q^{56} +(77199.6 - 43858.3i) q^{57} -35230.5 q^{58} -18795.0i q^{59} +(-6897.59 - 12141.2i) q^{60} -176481. q^{61} +285458. i q^{62} +(64251.3 - 107796. i) q^{63} -274828. q^{64} -400247. i q^{65} +(-141613. + 80452.5i) q^{66} -93073.0 q^{67} +5972.19i q^{68} +(-63435.8 - 111660. i) q^{69} +198599. q^{70} +107624. i q^{71} +(-328746. - 195947. i) q^{72} -291342. q^{73} +278042. i q^{74} +(149551. - 84962.1i) q^{75} -11467.3 q^{76} +133488. i q^{77} +(-279993. - 492845. i) q^{78} +943880. q^{79} -572568. i q^{80} +(-252819. - 467453. i) q^{81} +885897. q^{82} -798424. i q^{83} +(-14092.3 + 8006.09i) q^{84} -253997. q^{85} +948727. i q^{86} +(60402.7 + 106321. i) q^{87} +407098. q^{88} +386603. i q^{89} +(430608. - 722444. i) q^{90} -464570. q^{91} +16586.1i q^{92} +(861476. - 489418. i) q^{93} +638810. q^{94} -487704. i q^{95} +(47570.6 + 83734.0i) q^{96} +1.69517e6 q^{97} +684676. i q^{98} +(485590. + 289433. 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 $

$2$		−	7.77900i		−	0.972375i	−0.873854	−	0.486188i	$-0.838387\pi$
$2$		−	7.77900i		−	0.972375i	0.873854	−	0.486188i	$-0.161613\pi$
$3$	−23.4760	+	13.3371i	−0.869481	+	0.493966i
$3$	−23.4760	+	13.3371i	−0.869481	+	0.493966i
$4$	3.48715			0.0544867
$5$			148.308i			1.18647i	0.805031	+	0.593233i	$0.202149\pi$
$5$			148.308i			1.18647i	−0.805031	+	0.593233i	$0.797851\pi$
$6$	103.749	+	182.620i	0.480320	+	0.845462i
$7$	172.143			0.501874			0.250937	−	0.968003i	$-0.419261\pi$
$7$	172.143			0.501874			0.250937	+	0.968003i	$0.419261\pi$
$8$		−	524.983i		−	1.02536i
$9$	373.244	−	626.203i	0.511995	−	0.858989i
$10$	1153.69			1.15369
$11$			775.451i			0.582608i	0.956631	+	0.291304i	$0.0940891\pi$
$11$			775.451i			0.582608i	−0.956631	+	0.291304i	$0.905911\pi$
$12$	−81.8643	+	46.5084i	−0.0473752	+	0.0269146i
$13$	−2698.75			−1.22838			−0.614190	−	0.789159i	$-0.710517\pi$
$13$	−2698.75			−1.22838			−0.614190	+	0.789159i	$0.710517\pi$
$14$		−	1339.10i		−	0.488010i
$15$	−1978.00	−	3481.68i	−0.586075	−	1.03161i
$16$	−3860.66			−0.942544
$17$			1712.63i			0.348591i	0.984693	+	0.174295i	$0.0557648\pi$
$17$			1712.63i			0.348591i	−0.984693	+	0.174295i	$0.944235\pi$
$18$	−4871.23	−	2903.47i	−0.835259	−	0.497851i
$19$	−3288.45			−0.479436			−0.239718	−	0.970843i	$-0.577055\pi$
$19$	−3288.45			−0.479436			−0.239718	+	0.970843i	$0.577055\pi$
$20$			517.173i			0.0646467i
$21$	−4041.22	+	2295.88i	−0.436370	+	0.247909i
$22$	6032.24			0.566513
$23$			4756.34i			0.390922i	0.980712	+	0.195461i	$0.0626202\pi$
$23$			4756.34i			0.390922i	−0.980712	+	0.195461i	$0.937380\pi$
$24$	7001.74	+	12324.5i	0.506492	+	0.891528i
$25$	−6370.36			−0.407703
$26$			20993.6i			1.19445i
$27$	−410.554	+	19678.7i	−0.0208583	+	0.999782i
$28$	600.288			0.0273455
$29$		−	4528.92i		−	0.185695i
$29$		−	4528.92i		−	0.185695i
$30$	−27084.0	+	15386.9i	−1.00311	+	0.569884i
$31$	−36696.0			−1.23178			−0.615891	−	0.787831i	$-0.711204\pi$
$31$	−36696.0			−1.23178			−0.615891	+	0.787831i	$0.711204\pi$
$32$		−	3566.79i		−	0.108850i
$33$	−10342.3	−	18204.5i	−0.287789	−	0.506567i
$34$	13322.5			0.338961
$35$			25530.2i			0.595457i
$36$	1301.56	−	2183.66i	0.0278969	−	0.0468035i
$37$	−35742.7			−0.705638			−0.352819	−	0.935692i	$-0.614777\pi$
$37$	−35742.7			−0.705638			−0.352819	+	0.935692i	$0.614777\pi$
$38$			25580.8i			0.466191i
$39$	63355.8	−	35993.5i	1.06805	−	0.606778i
$40$	77859.3			1.21655
$41$			113883.i			1.65237i	0.563397	+	0.826186i	$0.309494\pi$
$41$			113883.i			1.65237i	−0.563397	+	0.826186i	$0.690506\pi$
$42$	17859.7	+	31436.7i	0.241060	+	0.424315i
$43$	−121960.			−1.53395			−0.766976	−	0.641676i	$-0.778240\pi$
$43$	−121960.			−1.53395			−0.766976	+	0.641676i	$0.778240\pi$
$44$			2704.11i			0.0317444i
$45$	92871.1	+	55355.2i	1.01916	+	0.607465i
$46$	36999.6			0.380122
$47$			82119.8i			0.790959i	0.918475	+	0.395480i	$0.129422\pi$
$47$			82119.8i			0.790959i	−0.918475	+	0.395480i	$0.870578\pi$
$48$	90632.9	−	51490.0i	0.819525	−	0.465585i
$49$	−88015.9			−0.748123
$50$			49555.1i			0.396440i
$51$	−22841.5	−	40205.6i	−0.172192	−	0.303093i
$52$	−9410.94			−0.0669303
$53$		−	88459.6i		−	0.594179i	−0.954850	−	0.297090i	$-0.903984\pi$
$53$		−	88459.6i		−	0.594179i	0.954850	−	0.297090i	$-0.0960160\pi$
$54$	153081.	+	3193.70i	0.972164	+	0.0202821i
$55$	−115006.			−0.691245
$56$		−	90371.9i		−	0.514600i
$57$	77199.6	−	43858.3i	0.416860	−	0.236825i
$58$	−35230.5			−0.180566
$59$		−	18795.0i		−	0.0915136i	−0.998953	−	0.0457568i	$-0.985430\pi$
$59$		−	18795.0i		−	0.0915136i	0.998953	−	0.0457568i	$-0.0145699\pi$
$60$	−6897.59	−	12141.2i	−0.0319333	−	0.0562091i
$61$	−176481.			−0.777513			−0.388756	−	0.921341i	$-0.627095\pi$
$61$	−176481.			−0.777513			−0.388756	+	0.921341i	$0.627095\pi$
$62$			285458.i			1.19775i
$63$	64251.3	−	107796.i	0.256957	−	0.431104i
$64$	−274828.			−1.04839
$65$		−	400247.i		−	1.45743i
$66$	−141613.	+	80452.5i	−0.492573	+	0.279839i
$67$	−93073.0			−0.309456			−0.154728	−	0.987957i	$-0.549450\pi$
$67$	−93073.0			−0.309456			−0.154728	+	0.987957i	$0.549450\pi$
$68$			5972.19i			0.0189936i
$69$	−63435.8	−	111660.i	−0.193102	−	0.339899i
$70$	198599.			0.579007
$71$			107624.i			0.300700i	0.988633	+	0.150350i	$0.0480400\pi$
$71$			107624.i			0.300700i	−0.988633	+	0.150350i	$0.951960\pi$
$72$	−328746.	−	195947.i	−0.880770	−	0.524977i
$73$	−291342.			−0.748919			−0.374459	−	0.927243i	$-0.622172\pi$
$73$	−291342.			−0.748919			−0.374459	+	0.927243i	$0.622172\pi$
$74$			278042.i			0.686145i
$75$	149551.	−	84962.1i	0.354490	−	0.201392i
$76$	−11467.3			−0.0261229
$77$			133488.i			0.292396i
$78$	−279993.	−	492845.i	−0.590016	−	1.03855i
$79$	943880.			1.91441			0.957206	−	0.289408i	$-0.0934585\pi$
$79$	943880.			1.91441			0.957206	+	0.289408i	$0.0934585\pi$
$80$		−	572568.i		−	1.11830i
$81$	−252819.	−	467453.i	−0.475723	−	0.879595i
$82$	885897.			1.60673
$83$		−	798424.i		−	1.39637i	−0.715920	−	0.698183i	$-0.753992\pi$
$83$		−	798424.i		−	1.39637i	0.715920	−	0.698183i	$-0.246008\pi$
$84$	−14092.3	+	8006.09i	−0.0237764	+	0.0135077i
$85$	−253997.			−0.413592
$86$			948727.i			1.49158i
$87$	60402.7	+	106321.i	0.0917272	+	0.161459i
$88$	407098.			0.597381
$89$			386603.i			0.548397i	0.961673	+	0.274198i	$0.0884125\pi$
$89$			386603.i			0.548397i	−0.961673	+	0.274198i	$0.911588\pi$
$90$	430608.	−	722444.i	0.590683	−	0.991007i
$91$	−464570.			−0.616491
$92$			16586.1i			0.0213000i
$93$	861476.	−	489418.i	1.07101	−	0.608459i
$94$	638810.			0.769109
$95$		−	487704.i		−	0.568834i
$96$	47570.6	+	83734.0i	0.0537681	+	0.0946429i
$97$	1.69517e6			1.85737			0.928684	−	0.370873i	$-0.120941\pi$
$97$	1.69517e6			1.85737			0.928684	+	0.370873i	$0.120941\pi$
$98$			684676.i			0.727456i
$99$	485590.	+	289433.i	0.500454	+	0.298292i
$100$	−22214.4			−0.0222144
$101$			14549.7i			0.0141218i	0.999975	+	0.00706089i	$0.00224757\pi$
$101$			14549.7i			0.0141218i	−0.999975	+	0.00706089i	$0.997752\pi$
$102$	−312760.	+	177684.i	−0.294720	+	0.167435i
$103$	−954200.			−0.873228			−0.436614	−	0.899649i	$-0.643822\pi$
$103$	−954200.			−0.873228			−0.436614	+	0.899649i	$0.643822\pi$
$104$			1.41680e6i			1.25953i
$105$	−340499.	−	599347.i	−0.294135	−	0.517738i
$106$	−688127.			−0.577765
$107$		−	1.57391e6i		−	1.28478i	−0.766379	−	0.642388i	$-0.777944\pi$
$107$		−	1.57391e6i		−	1.28478i	0.766379	−	0.642388i	$-0.222056\pi$
$108$	−1431.66	+	68622.6i	−0.00113650	+	0.0544749i
$109$	187660.			0.144908			0.0724539	−	0.997372i	$-0.476917\pi$
$109$	187660.			0.144908			0.0724539	+	0.997372i	$0.476917\pi$
$110$			894631.i			0.672149i
$111$	839095.	−	476704.i	0.613539	−	0.348562i
$112$	−664585.			−0.473038
$113$			1.24151e6i			0.860431i	0.902726	+	0.430215i	$0.141562\pi$
$113$			1.24151e6i			0.860431i	−0.902726	+	0.430215i	$0.858438\pi$
$114$	−341174.	−	600536.i	−0.230283	−	0.405344i
$115$	−705405.			−0.463815
$116$		−	15793.0i		−	0.0101179i
$117$	−1.00729e6	+	1.68996e6i	−0.628924	+	1.05516i
$118$	−146206.			−0.0889855
$119$			294816.i			0.174949i
$120$	−1.82782e6	+	1.03842e6i	−1.05777	+	0.600935i
$121$	1.17024e6			0.660568
$122$			1.37284e6i			0.756034i
$123$	−1.51887e6	−	2.67352e6i	−0.816216	−	1.43671i
$124$	−127965.			−0.0671158
$125$			1.37254e6i			0.702740i
$126$	−838547.	−	499811.i	−0.419195	−	0.249858i
$127$	263141.			0.128463			0.0642315	−	0.997935i	$-0.479540\pi$
$127$	263141.			0.128463			0.0642315	+	0.997935i	$0.479540\pi$
$128$			1.90962e6i			0.910576i
$129$	2.86313e6	−	1.62659e6i	1.33374	−	0.757721i
$130$	−3.11352e6			−1.41717
$131$			1.92152e6i			0.854734i	0.904078	+	0.427367i	$0.140559\pi$
$131$			1.92152e6i			0.854734i	−0.904078	+	0.427367i	$0.859441\pi$
$132$	−36065.0	−	63481.8i	−0.0156807	−	0.0276012i
$133$	−566083.			−0.240616
$134$			724015.i			0.300908i
$135$	−2.91852e6	−	60888.6i	−1.18621	−	0.0247477i
$136$	899100.			0.357430
$137$			257927.i			0.100308i	0.998742	+	0.0501539i	$0.0159712\pi$
$137$			257927.i			0.100308i	−0.998742	+	0.0501539i	$0.984029\pi$
$138$	−868602.	+	493467.i	−0.330509	+	0.187768i
$139$	−978990.			−0.364531			−0.182265	−	0.983249i	$-0.558343\pi$
$139$	−978990.			−0.364531			−0.182265	+	0.983249i	$0.558343\pi$
$140$			89027.7i			0.0324445i
$141$	−1.09524e6	−	1.92784e6i	−0.390707	−	0.687724i
$142$	837205.			0.292393
$143$		−	2.09275e6i		−	0.715663i
$144$	−1.44097e6	+	2.41756e6i	−0.482578	+	0.809635i
$145$	671677.			0.220321
$146$			2.26635e6i			0.728230i
$147$	2.06626e6	−	1.17388e6i	0.650478	−	0.369547i
$148$	−124640.			−0.0384479
$149$			3.83651e6i			1.15979i	0.814693	+	0.579893i	$0.196905\pi$
$149$			3.83651e6i			1.15979i	−0.814693	+	0.579893i	$0.803095\pi$
$150$	−660920.	−	1.16335e6i	−0.195828	−	0.344698i
$151$	4.96928e6			1.44332			0.721661	−	0.692247i	$-0.243379\pi$
$151$	4.96928e6			1.44332			0.721661	+	0.692247i	$0.243379\pi$
$152$			1.72638e6i			0.491592i
$153$	1.07245e6	+	639228.i	0.299436	+	0.178477i
$154$	1.03841e6			0.284318
$155$		−	5.44233e6i		−	1.46147i
$156$	220931.	−	125515.i	0.0581947	−	0.0330613i
$157$	−4.81541e6			−1.24433			−0.622163	−	0.782888i	$-0.713746\pi$
$157$	−4.81541e6			−1.24433			−0.622163	+	0.782888i	$0.713746\pi$
$158$		−	7.34244e6i		−	1.86153i
$159$	1.17979e6	+	2.07668e6i	0.293504	+	0.516628i
$160$	528985.			0.129147
$161$			818770.i			0.196193i
$162$	−3.63632e6	+	1.96668e6i	−0.855296	+	0.462581i
$163$	2.65119e6			0.612177			0.306089	−	0.952003i	$-0.400980\pi$
$163$	2.65119e6			0.612177			0.306089	+	0.952003i	$0.400980\pi$
$164$			397128.i			0.0900323i
$165$	2.69988e6	−	1.53384e6i	0.601024	−	0.341452i
$166$	−6.21094e6			−1.35779
$167$		−	2.12850e6i		−	0.457007i	−0.973543	−	0.228504i	$-0.926617\pi$
$167$		−	2.12850e6i		−	0.457007i	0.973543	−	0.228504i	$-0.0733833\pi$
$168$	1.20530e6	+	2.12157e6i	0.254195	+	0.447435i
$169$	2.45644e6			0.508915
$170$			1.97584e6i			0.402166i
$171$	−1.22739e6	+	2.05924e6i	−0.245468	+	0.411830i
$172$	−425293.			−0.0835800
$173$			4.89172e6i			0.944763i	0.881394	+	0.472382i	$0.156606\pi$
$173$			4.89172e6i			0.944763i	−0.881394	+	0.472382i	$0.843394\pi$
$174$	827071.	−	469872.i	0.156998	−	0.0891933i
$175$	−1.09661e6			−0.204616
$176$		−	2.99376e6i		−	0.549134i
$177$	250670.	+	441230.i	0.0452046	+	0.0795693i
$178$	3.00738e6			0.533247
$179$			1.87481e6i			0.326888i	0.986553	+	0.163444i	$0.0522603\pi$
$179$			1.87481e6i			0.326888i	−0.986553	+	0.163444i	$0.947740\pi$
$180$	323855.	+	193032.i	0.0555308	+	0.0330988i
$181$	8.27467e6			1.39545			0.697726	−	0.716365i	$-0.254195\pi$
$181$	8.27467e6			1.39545			0.697726	+	0.716365i	$0.254195\pi$
$182$			3.61389e6i			0.599461i
$183$	4.14306e6	−	2.35374e6i	0.676033	−	0.384065i
$184$	2.49700e6			0.400834
$185$		−	5.30094e6i		−	0.837216i
$186$	−3.80718e6	−	6.70142e6i	−0.591650	−	1.04142i
$187$	−1.32806e6			−0.203092
$188$			286364.i			0.0430968i
$189$	−70674.0	+	3.38755e6i	−0.0104682	+	0.501765i
$190$	−3.79385e6			−0.553120
$191$			4.07627e6i			0.585009i	0.956264	+	0.292505i	$0.0944887\pi$
$191$			4.07627e6i			0.585009i	−0.956264	+	0.292505i	$0.905511\pi$
$192$	6.45187e6	−	3.66541e6i	0.911553	−	0.517868i
$193$	−4.65906e6			−0.648076			−0.324038	−	0.946044i	$-0.605041\pi$
$193$	−4.65906e6			−0.648076			−0.324038	+	0.946044i	$0.605041\pi$
$194$		−	1.31867e7i		−	1.80606i
$195$	5.33813e6	+	9.39619e6i	0.719922	+	1.26721i
$196$	−306925.			−0.0407627
$197$		−	3.82855e6i		−	0.500767i	−0.968147	−	0.250383i	$-0.919443\pi$
$197$		−	3.82855e6i		−	0.500767i	0.968147	−	0.250383i	$-0.0805567\pi$
$198$	2.25150e6	−	3.77740e6i	0.290052	−	0.486629i
$199$	8.07380e6			1.02452			0.512258	−	0.858832i	$-0.328809\pi$
$199$	8.07380e6			1.02452			0.512258	+	0.858832i	$0.328809\pi$
$200$			3.34433e6i			0.418041i
$201$	2.18498e6	−	1.24132e6i	0.269066	−	0.152861i
$202$	113182.			0.0137317
$203$		−	779621.i		−	0.0931956i
$204$	−79651.6	−	140203.i	−0.00938219	−	0.0165146i
$205$	−1.68898e7			−1.96048
$206$			7.42272e6i			0.849105i
$207$	2.97843e6	+	1.77528e6i	0.335797	+	0.200150i
$208$	1.04190e7			1.15780
$209$		−	2.55003e6i		−	0.279323i
$210$	−4.66232e6	+	2.64874e6i	−0.503436	+	0.286010i
$211$	−4.47301e6			−0.476159			−0.238080	−	0.971246i	$-0.576518\pi$
$211$	−4.47301e6			−0.476159			−0.238080	+	0.971246i	$0.576518\pi$
$212$		−	308472.i		−	0.0323749i
$213$	−1.43539e6	−	2.52657e6i	−0.148535	−	0.261453i
$214$	−1.22434e7			−1.24928
$215$		−	1.80877e7i		−	1.81998i
$216$	1.03310e7	+	215534.i	1.02513	+	0.0213872i
$217$	−6.31695e6			−0.618199
$218$		−	1.45981e6i		−	0.140905i
$219$	6.83954e6	−	3.88565e6i	0.651170	−	0.369940i
$220$	−401043.			−0.0376637
$221$		−	4.62195e6i		−	0.428202i
$222$	−3.70828e6	−	6.52732e6i	−0.338933	−	0.596590i
$223$	2.02868e7			1.82936			0.914680	−	0.404178i	$-0.132442\pi$
$223$	2.02868e7			1.82936			0.914680	+	0.404178i	$0.132442\pi$
$224$		−	613997.i		−	0.0546289i
$225$	−2.37770e6	+	3.98914e6i	−0.208742	+	0.350212i
$226$	9.65773e6			0.836662
$227$		−	1.38977e7i		−	1.18813i	−0.804416	−	0.594066i	$-0.797522\pi$
$227$		−	1.38977e7i		−	1.18813i	0.804416	−	0.594066i	$-0.202478\pi$
$228$	269206.	−	152941.i	0.0227133	−	0.0129038i
$229$	−1.84534e7			−1.53663			−0.768315	−	0.640072i	$-0.778904\pi$
$229$	−1.84534e7			−1.53663			−0.768315	+	0.640072i	$0.778904\pi$
$230$			5.48735e6i			0.451003i
$231$	−1.78035e6	−	3.13377e6i	−0.144434	−	0.254233i
$232$	−2.37761e6			−0.190404
$233$		−	1.66303e7i		−	1.31472i	−0.753577	−	0.657360i	$-0.771673\pi$
$233$		−	1.66303e7i		−	1.31472i	0.753577	−	0.657360i	$-0.228327\pi$
$234$	1.31462e7	+	7.83573e6i	1.02601	+	0.611550i
$235$	−1.21790e7			−0.938447
$236$		−	65540.9i		−	0.00498627i
$237$	−2.21585e7	+	1.25886e7i	−1.66454	+	0.945655i
$238$	2.29338e6			0.170116
$239$		−	1.19983e7i		−	0.878869i	−0.898275	−	0.439435i	$-0.855179\pi$
$239$		−	1.19983e7i		−	0.878869i	0.898275	−	0.439435i	$-0.144821\pi$
$240$	7.63640e6	+	1.34416e7i	0.552401	+	0.972339i
$241$	−1.61481e7			−1.15364			−0.576820	−	0.816872i	$-0.695706\pi$
$241$	−1.61481e7			−1.15364			−0.576820	+	0.816872i	$0.695706\pi$
$242$		−	9.10327e6i		−	0.642320i
$243$	1.21696e7	+	7.60206e6i	0.848122	+	0.529800i
$244$	−615414.			−0.0423641
$245$		−	1.30535e7i		−	0.887623i
$246$	−2.07973e7	+	1.18153e7i	−1.39702	+	0.793668i
$247$	8.87470e6			0.588929
$248$			1.92648e7i			1.26302i
$249$	1.06486e7	+	1.87438e7i	0.689757	+	1.21411i
$250$	1.06770e7			0.683327
$251$		−	7.80364e6i		−	0.493487i	−0.969081	−	0.246744i	$-0.920639\pi$
$251$		−	7.80364e6i		−	0.493487i	0.969081	−	0.246744i	$-0.0793605\pi$
$252$	224054.	−	375902.i	0.0140007	−	0.0234894i
$253$	−3.68831e6			−0.227754
$254$		−	2.04698e6i		−	0.124914i
$255$	5.96283e6	−	3.38758e6i	0.359610	−	0.204300i
$256$	−2.73412e6			−0.162966
$257$			2.85578e7i			1.68239i	0.540735	+	0.841193i	$0.318146\pi$
$257$			2.85578e7i			1.68239i	−0.540735	+	0.841193i	$0.681854\pi$
$258$	−1.26532e7	−	2.22723e7i	−0.736789	−	1.29690i
$259$	−6.15285e6			−0.354141
$260$		−	1.39572e6i		−	0.0794106i
$261$	−2.83602e6	−	1.69039e6i	−0.159510	−	0.0950750i
$262$	1.49475e7			0.831122
$263$			9.85496e6i			0.541736i	0.962617	+	0.270868i	$0.0873106\pi$
$263$			9.85496e6i			0.541736i	−0.962617	+	0.270868i	$0.912689\pi$
$264$	−9.55704e6	+	5.42951e6i	−0.519411	+	0.295086i
$265$	1.31193e7			0.704974
$266$			4.40356e6i			0.233969i
$267$	−5.15615e6	−	9.07588e6i	−0.270889	−	0.476821i
$268$	−324560.			−0.0168613
$269$			3.47508e7i			1.78529i	0.450763	+	0.892643i	$0.351152\pi$
$269$			3.47508e7i			1.78529i	−0.450763	+	0.892643i	$0.648848\pi$
$270$	−473653.	+	2.27032e7i	−0.0240641	+	1.15344i
$271$	4.11527e6			0.206772			0.103386	−	0.994641i	$-0.467032\pi$
$271$	4.11527e6			0.206772			0.103386	+	0.994641i	$0.467032\pi$
$272$		−	6.61188e6i		−	0.328563i
$273$	1.09062e7	−	6.19601e6i	0.536028	−	0.304526i
$274$	2.00641e6			0.0975367
$275$		−	4.93991e6i		−	0.237531i
$276$	−221210.	−	389375.i	−0.0105215	−	0.0185200i
$277$	−3.99466e7			−1.87949			−0.939746	−	0.341874i	$-0.888938\pi$
$277$	−3.99466e7			−1.87949			−0.939746	+	0.341874i	$0.888938\pi$
$278$			7.61556e6i			0.354460i
$279$	−1.36966e7	+	2.29792e7i	−0.630666	+	1.05809i
$280$	1.34029e7			0.610555
$281$			4.19054e6i			0.188865i	0.995531	+	0.0944324i	$0.0301036\pi$
$281$			4.19054e6i			0.188865i	−0.995531	+	0.0944324i	$0.969896\pi$
$282$	−1.49967e7	+	8.51986e6i	−0.668726	+	0.379914i
$283$	3.19996e7			1.41184			0.705921	−	0.708291i	$-0.250534\pi$
$283$	3.19996e7			1.41184			0.705921	+	0.708291i	$0.250534\pi$
$284$			375300.i			0.0163841i
$285$	6.50456e6	+	1.14493e7i	0.280985	+	0.494591i
$286$	−1.62795e7			−0.695893
$287$			1.96042e7i			0.829282i
$288$	−2.23353e6	−	1.33128e6i	−0.0935008	−	0.0557305i
$289$	2.12045e7			0.878484
$290$		−	5.22498e6i		−	0.214235i
$291$	−3.97958e7	+	2.26086e7i	−1.61495	+	0.917477i
$292$	−1.01595e6			−0.0408061
$293$			2.12784e6i			0.0845934i	0.999105	+	0.0422967i	$0.0134675\pi$
$293$			2.12784e6i			0.0845934i	−0.999105	+	0.0422967i	$0.986533\pi$
$294$	−9.13158e6	−	1.60734e7i	−0.359339	−	0.632509i
$295$	2.78745e6			0.108578
$296$			1.87643e7i			0.723531i
$297$	−1.52599e7	−	318365.i	−0.582481	−	0.0121522i
$298$	2.98442e7			1.12775
$299$		−	1.28362e7i		−	0.480200i
$300$	521505.	−	296276.i	0.0193150	−	0.0109732i
$301$	−2.09945e7			−0.769851
$302$		−	3.86561e7i		−	1.40345i
$303$	−194050.	−	341568.i	−0.00697568	−	0.0122786i
$304$	1.26956e7			0.451889
$305$		−	2.61735e7i		−	0.922493i
$306$	4.97256e6	−	8.34260e6i	0.173546	−	0.291164i
$307$	1.30973e7			0.452654			0.226327	−	0.974051i	$-0.427328\pi$
$307$	1.30973e7			0.452654			0.226327	+	0.974051i	$0.427328\pi$
$308$			465494.i			0.0159317i
$309$	2.24008e7	−	1.27263e7i	0.759255	−	0.431345i
$310$	−4.23359e7			−1.42110
$311$			4.12146e7i			1.37015i	0.728471	+	0.685077i	$0.240231\pi$
$311$			4.12146e7i			1.37015i	−0.728471	+	0.685077i	$0.759769\pi$
$312$	−1.88959e7	−	3.32607e7i	−0.622164	−	1.09513i
$313$	3.09792e7			1.01027			0.505134	−	0.863041i	$-0.331443\pi$
$313$	3.09792e7			1.01027			0.505134	+	0.863041i	$0.331443\pi$
$314$			3.74591e7i			1.20995i
$315$	1.59871e7	+	9.52900e6i	0.511490	+	0.304871i
$316$	3.29145e6			0.104310
$317$		−	4.03629e7i		−	1.26708i	−0.773709	−	0.633542i	$-0.781601\pi$
$317$		−	4.03629e7i		−	1.26708i	0.773709	−	0.633542i	$-0.218399\pi$
$318$	1.61545e7	−	9.17762e6i	0.502356	−	0.285396i
$319$	3.51196e6			0.108188
$320$		−	4.07594e7i		−	1.24388i
$321$	2.09913e7	+	3.69490e7i	0.634636	+	1.11709i
$322$	6.36921e6			0.190773
$323$		−	5.63189e6i		−	0.167127i
$324$	−881616.	−	1.63008e6i	−0.0259206	−	0.0479263i
$325$	1.71920e7			0.500814
$326$		−	2.06236e7i		−	0.595266i
$327$	−4.40550e6	+	2.50283e6i	−0.125995	+	0.0715795i
$328$	5.97867e7			1.69427
$329$			1.41363e7i			0.396962i
$330$	−1.19318e7	−	2.10023e7i	−0.332019	−	0.584421i
$331$	−8.40105e6			−0.231659			−0.115830	−	0.993269i	$-0.536953\pi$
$331$	−8.40105e6			−0.231659			−0.115830	+	0.993269i	$0.536953\pi$
$332$		−	2.78422e6i		−	0.0760834i
$333$	−1.33408e7	+	2.23822e7i	−0.361283	+	0.606135i
$334$	−1.65576e7			−0.444383
$335$		−	1.38035e7i		−	0.367160i
$336$	1.56018e7	−	8.86363e6i	0.411298	−	0.233665i
$337$	−4.21547e7			−1.10143			−0.550714	−	0.834694i	$-0.685645\pi$
$337$	−4.21547e7			−1.10143			−0.550714	+	0.834694i	$0.685645\pi$
$338$		−	1.91086e7i		−	0.494856i
$339$	−1.65582e7	−	2.91457e7i	−0.425024	−	0.748128i
$340$	−885725.			−0.0225352
$341$		−	2.84560e7i		−	0.717646i
$342$	1.60188e7	+	9.54790e6i	0.400453	+	0.238687i
$343$	−3.54037e7			−0.877337
$344$			6.40268e7i			1.57285i
$345$	1.65601e7	−	9.40805e6i	0.403279	−	0.229109i
$346$	3.80527e7			0.918664
$347$			5.55094e7i			1.32855i	0.747488	+	0.664275i	$0.231260\pi$
$347$			5.55094e7i			1.32855i	−0.747488	+	0.664275i	$0.768740\pi$
$348$	210633.	+	370757.i	0.00499792	+	0.00879735i
$349$	−4.56421e7			−1.07371			−0.536857	−	0.843673i	$-0.680389\pi$
$349$	−4.56421e7			−1.07371			−0.536857	+	0.843673i	$0.680389\pi$
$350$			8.53054e6i			0.198963i
$351$	1.10798e6	−	5.31079e7i	0.0256219	−	1.22811i
$352$	2.76587e6			0.0634168
$353$			6.13280e7i			1.39423i	0.716959	+	0.697115i	$0.245533\pi$
$353$			6.13280e7i			1.39423i	−0.716959	+	0.697115i	$0.754467\pi$
$354$	3.43233e6	−	1.94996e6i	0.0773712	−	0.0439558i
$355$	−1.59615e7			−0.356770
$356$			1.34814e6i			0.0298803i
$357$	−3.93199e6	−	6.92111e6i	−0.0864188	−	0.152115i
$358$	1.45842e7			0.317857
$359$		−	8.89647e7i		−	1.92280i	−0.275154	−	0.961400i	$-0.588729\pi$
$359$		−	8.89647e7i		−	1.92280i	0.275154	−	0.961400i	$-0.411271\pi$
$360$	2.90605e7	−	4.87557e7i	0.622868	−	1.04500i
$361$	−3.62320e7			−0.770142
$362$		−	6.43686e7i		−	1.35690i
$363$	−2.74725e7	+	1.56075e7i	−0.574351	+	0.326298i
$364$	−1.62003e6			−0.0335906
$365$		−	4.32085e7i		−	0.888567i
$366$	−1.83097e7	−	3.22288e7i	−0.373455	−	0.657357i
$367$	−6.40374e7			−1.29549			−0.647747	−	0.761855i	$-0.724289\pi$
$367$	−6.40374e7			−1.29549			−0.647747	+	0.761855i	$0.724289\pi$
$368$		−	1.83626e7i		−	0.368461i
$369$	7.13139e7	+	4.25062e7i	1.41937	+	0.846006i
$370$	−4.12360e7			−0.814088
$371$		−	1.52277e7i		−	0.298203i
$372$	3.00409e6	−	1.70667e6i	0.0583559	−	0.0331529i
$373$	−4.27481e7			−0.823741			−0.411871	−	0.911242i	$-0.635124\pi$
$373$	−4.27481e7			−0.823741			−0.411871	+	0.911242i	$0.635124\pi$
$374$			1.03310e7i			0.197481i
$375$	−1.83057e7	−	3.22217e7i	−0.347130	−	0.611019i
$376$	4.31114e7			0.811015
$377$			1.22224e7i			0.228104i
$378$	2.63517e7	+	549773.i	0.487903	+	0.0101791i
$379$	2.25392e7			0.414020			0.207010	−	0.978339i	$-0.433627\pi$
$379$	2.25392e7			0.414020			0.207010	+	0.978339i	$0.433627\pi$
$380$		−	1.70070e6i		−	0.0309939i
$381$	−6.17750e6	+	3.50954e6i	−0.111696	+	0.0634564i
$382$	3.17093e7			0.568849
$383$		−	4.61805e7i		−	0.821982i	−0.911639	−	0.410991i	$-0.865183\pi$
$383$		−	4.61805e7i		−	0.821982i	0.911639	−	0.410991i	$-0.134817\pi$
$384$	−2.54687e7	−	4.48301e7i	−0.449794	−	0.791729i
$385$	−1.97974e7			−0.346918
$386$			3.62428e7i			0.630173i
$387$	−4.55208e7	+	7.63717e7i	−0.785375	+	1.31765i
$388$	5.91131e6			0.101202
$389$			6.14289e6i			0.104358i	0.998638	+	0.0521788i	$0.0166166\pi$
$389$			6.14289e6i			0.104358i	−0.998638	+	0.0521788i	$0.983383\pi$
$390$	7.30930e7	−	4.15253e7i	1.23220	−	0.700034i
$391$	−8.14584e6			−0.136272
$392$			4.62068e7i			0.767092i
$393$	−2.56275e7	−	4.51096e7i	−0.422210	−	0.743175i
$394$	−2.97823e7			−0.486933
$395$			1.39985e8i			2.27139i
$396$	1.69332e6	+	1.00929e6i	0.0272681	+	0.0162530i
$397$	9.98785e7			1.59625			0.798125	−	0.602492i	$-0.205826\pi$
$397$	9.98785e7			1.59625			0.798125	+	0.602492i	$0.205826\pi$
$398$		−	6.28061e7i		−	0.996214i
$399$	1.32893e7	−	7.54989e6i	0.209211	−	0.118856i
$400$	2.45938e7			0.384278
$401$			1.18952e8i			1.84475i	0.386291	+	0.922377i	$0.373756\pi$
$401$			1.18952e8i			1.84475i	−0.386291	+	0.922377i	$0.626244\pi$
$402$	−9.65625e6	−	1.69970e7i	−0.148638	−	0.261634i
$403$	9.90334e7			1.51310
$404$			50736.9i			0.000769449i
$405$	6.93272e7	−	3.74951e7i	1.04361	−	0.564429i
$406$	−6.06467e6			−0.0906211
$407$		−	2.77167e7i		−	0.411111i
$408$	−2.11073e7	+	1.19914e7i	−0.310779	+	0.176558i
$409$	7.73616e7			1.13072			0.565360	−	0.824844i	$-0.308737\pi$
$409$	7.73616e7			1.13072			0.565360	+	0.824844i	$0.308737\pi$
$410$			1.31386e8i			1.90633i
$411$	−3.43999e6	−	6.05508e6i	−0.0495486	−	0.0872157i
$412$	−3.32744e6			−0.0475793
$413$		−	3.23542e6i		−	0.0459283i
$414$	1.38099e7	−	2.31692e7i	0.194621	−	0.326521i
$415$	1.18413e8			1.65674
$416$			9.62587e6i			0.133709i
$417$	2.29828e7	−	1.30569e7i	0.316952	−	0.180066i
$418$	−1.98367e7			−0.271607
$419$			2.08271e7i			0.283130i	0.989929	+	0.141565i	$0.0452135\pi$
$419$			2.08271e7i			0.283130i	−0.989929	+	0.141565i	$0.954787\pi$
$420$	−1.18737e6	−	2.09001e6i	−0.0160265	−	0.0282099i
$421$	4.70216e6			0.0630160			0.0315080	−	0.999503i	$-0.489969\pi$
$421$	4.70216e6			0.0630160			0.0315080	+	0.999503i	$0.489969\pi$
$422$			3.47955e7i			0.463006i
$423$	5.14236e7	+	3.06507e7i	0.679425	+	0.404967i
$424$	−4.64398e7			−0.609245
$425$		−	1.09101e7i		−	0.142122i
$426$	−1.96542e7	+	1.11659e7i	−0.254230	+	0.144432i
$427$	−3.03799e7			−0.390213
$428$		−	5.48845e6i		−	0.0700032i
$429$	2.79112e7	+	4.91293e7i	0.353514	+	0.622256i
$430$	−1.40704e8			−1.76971
$431$			1.55276e7i			0.193942i	0.995287	+	0.0969711i	$0.0309155\pi$
$431$			1.55276e7i			0.193942i	−0.995287	+	0.0969711i	$0.969085\pi$
$432$	1.58501e6	−	7.59729e7i	0.0196599	−	0.942339i
$433$	6.51193e7			0.802132			0.401066	−	0.916049i	$-0.368640\pi$
$433$	6.51193e7			0.802132			0.401066	+	0.916049i	$0.368640\pi$
$434$			4.91396e7i			0.601122i
$435$	−1.57683e7	+	8.95822e6i	−0.191565	+	0.108831i
$436$	654398.			0.00789555
$437$		−	1.56410e7i		−	0.187422i
$438$	−3.02265e7	−	5.32048e7i	−0.359721	−	0.633182i
$439$	5.30078e7			0.626536			0.313268	−	0.949665i	$-0.398576\pi$
$439$	5.30078e7			0.626536			0.313268	+	0.949665i	$0.398576\pi$
$440$			6.03761e7i			0.708773i
$441$	−3.28514e7	+	5.51158e7i	−0.383035	+	0.642629i
$442$	−3.59542e7			−0.416373
$443$		−	8.07988e7i		−	0.929381i	−0.885473	−	0.464691i	$-0.846166\pi$
$443$		−	8.07988e7i		−	0.929381i	0.885473	−	0.464691i	$-0.153834\pi$
$444$	2.92605e6	−	1.66234e6i	0.0334297	−	0.0189920i
$445$	−5.73364e7			−0.650654
$446$		−	1.57811e8i		−	1.77882i
$447$	−5.11679e7	−	9.00659e7i	−0.572895	−	1.00841i
$448$	−4.73097e7			−0.526158
$449$			5.74725e7i			0.634923i	0.948271	+	0.317461i	$0.102830\pi$
$449$			5.74725e7i			0.634923i	−0.948271	+	0.317461i	$0.897170\pi$
$450$	3.10315e7	+	1.84961e7i	0.340538	+	0.202975i
$451$	−8.83108e7			−0.962685
$452$			4.32934e6i			0.0468821i
$453$	−1.16659e8	+	6.62758e7i	−1.25494	+	0.712952i
$454$	−1.08110e8			−1.15531
$455$		−	6.88996e7i		−	0.731447i
$456$	−2.30249e7	−	4.05284e7i	−0.242830	−	0.427430i
$457$	−9.65094e7			−1.01116			−0.505581	−	0.862779i	$-0.668722\pi$
$457$	−9.65094e7			−1.01116			−0.505581	+	0.862779i	$0.668722\pi$
$458$			1.43549e8i			1.49418i
$459$	−3.37023e7	−	703127.i	−0.348515	−	0.00727102i
$460$	−2.45985e6			−0.0252718
$461$			5.75187e7i			0.587092i	0.955945	+	0.293546i	$0.0948354\pi$
$461$			5.75187e7i			0.587092i	−0.955945	+	0.293546i	$0.905165\pi$
$462$	−2.43776e7	+	1.38493e7i	−0.247209	+	0.140444i
$463$	5.33333e7			0.537348			0.268674	−	0.963231i	$-0.413415\pi$
$463$	5.33333e7			0.537348			0.268674	+	0.963231i	$0.413415\pi$
$464$			1.74846e7i			0.175026i
$465$	7.25848e7	+	1.27764e8i	0.721916	+	1.27072i
$466$	−1.29367e8			−1.27840
$467$		−	1.94390e8i		−	1.90864i	−0.298785	−	0.954320i	$-0.596581\pi$
$467$		−	1.94390e8i		−	1.90864i	0.298785	−	0.954320i	$-0.403419\pi$
$468$	−3.51258e6	+	5.89316e6i	−0.0342680	+	0.0574924i
$469$	−1.60218e7			−0.155308
$470$			9.47408e7i			0.912522i
$471$	1.13047e8	−	6.42236e7i	1.08192	−	0.614655i
$472$	−9.86703e6			−0.0938341
$473$		−	9.45740e7i		−	0.893693i
$474$	9.79268e7	+	1.72371e8i	0.919531	+	1.61856i
$475$	2.09486e7			0.195467
$476$			1.02807e6i			0.00953238i
$477$	−5.53936e7	−	3.30170e7i	−0.510393	−	0.304217i
$478$	−9.33344e7			−0.854591
$479$		−	1.27719e7i		−	0.116212i	−0.998310	−	0.0581058i	$-0.981494\pi$
$479$		−	1.27719e7i		−	0.116212i	0.998310	−	0.0581058i	$-0.0185061\pi$
$480$	−1.24184e7	+	7.05512e6i	−0.112291	+	0.0637941i
$481$	9.64606e7			0.866791
$482$			1.25616e8i			1.12177i
$483$	−1.09200e7	−	1.92214e7i	−0.0969129	−	0.170586i
$484$	4.08079e6			0.0359922
$485$			2.51408e8i			2.20370i
$486$	5.91364e7	−	9.46676e7i	0.515165	−	0.824693i
$487$	−5.90986e7			−0.511671			−0.255835	−	0.966720i	$-0.582351\pi$
$487$	−5.90986e7			−0.511671			−0.255835	+	0.966720i	$0.582351\pi$
$488$			9.26492e7i			0.797228i
$489$	−6.22392e7	+	3.53591e7i	−0.532277	+	0.302395i
$490$	−1.01543e8			−0.863102
$491$			8.13817e7i			0.687515i	0.939058	+	0.343757i	$0.111700\pi$
$491$			8.13817e7i			0.687515i	−0.939058	+	0.343757i	$0.888300\pi$
$492$	−5.29653e6	−	9.32296e6i	−0.0444729	−	0.0782814i
$493$	7.75636e6			0.0647317
$494$		−	6.90363e7i		−	0.572659i
$495$	−4.29253e7	+	7.20170e7i	−0.353914	+	0.593772i
$496$	1.41671e8			1.16101
$497$			1.85266e7i			0.150913i
$498$	1.45808e8	−	8.28358e7i	1.18057	−	0.670703i
$499$	−7.26690e7			−0.584854			−0.292427	−	0.956288i	$-0.594463\pi$
$499$	−7.26690e7			−0.584854			−0.292427	+	0.956288i	$0.594463\pi$
$500$			4.78625e6i			0.0382900i
$501$	2.83879e7	+	4.99685e7i	0.225746	+	0.397359i
$502$	−6.07045e7			−0.479855
$503$		−	1.49264e8i		−	1.17288i	−0.809994	−	0.586438i	$-0.800530\pi$
$503$		−	1.49264e8i		−	1.17288i	0.809994	−	0.586438i	$-0.199470\pi$
$504$	−5.65912e7	−	3.37308e7i	−0.442035	−	0.263472i
$505$	−2.15784e6			−0.0167550
$506$			2.86914e7i			0.221462i
$507$	−5.76673e7	+	3.27617e7i	−0.442492	+	0.251387i
$508$	917613.			0.00699953
$509$			1.58558e7i			0.120236i	0.998191	+	0.0601181i	$0.0191477\pi$
$509$			1.58558e7i			0.120236i	−0.998191	+	0.0601181i	$0.980852\pi$
$510$	−2.63520e7	−	4.63849e7i	−0.198657	−	0.349676i
$511$	−5.01524e7			−0.375863
$512$			1.43484e8i			1.06904i
$513$	1.35009e6	−	6.47124e7i	0.0100002	−	0.479331i
$514$	2.22151e8			1.63591
$515$		−	1.41516e8i		−	1.03606i
$516$	9.98417e6	−	5.67217e6i	0.0726713	−	0.0412857i
$517$	−6.36799e7			−0.460819
$518$			4.78630e7i			0.344358i
$519$	−6.52412e7	−	1.14838e8i	−0.466681	−	0.821454i
$520$	−2.10123e8			−1.49439
$521$		−	1.69131e8i		−	1.19594i	−0.801518	−	0.597970i	$-0.795974\pi$
$521$		−	1.69131e8i		−	1.19594i	0.801518	−	0.597970i	$-0.204026\pi$
$522$	−1.31496e7	+	2.20614e7i	−0.0924486	+	0.155104i
$523$	5.42865e7			0.379478			0.189739	−	0.981835i	$-0.439236\pi$
$523$	5.42865e7			0.379478			0.189739	+	0.981835i	$0.439236\pi$
$524$			6.70063e6i			0.0465716i
$525$	2.57440e7	−	1.46256e7i	0.177909	−	0.101073i
$526$	7.66617e7			0.526770
$527$		−	6.28466e7i		−	0.429388i
$528$	3.99280e7	+	7.02814e7i	0.271254	+	0.477462i
$529$	1.25413e8			0.847180
$530$		−	1.02055e8i		−	0.685499i
$531$	−1.17695e7	−	7.01511e6i	−0.0786091	−	0.0468545i
$532$	−1.97401e6			−0.0131104
$533$		−	3.07342e8i		−	2.02974i
$534$	−7.06013e7	+	4.01097e7i	−0.463648	+	0.263406i
$535$	2.33423e8			1.52434
$536$			4.88617e7i			0.317303i
$537$	−2.50045e7	−	4.40131e7i	−0.161471	−	0.284223i
$538$	2.70327e8			1.73597
$539$		−	6.82520e7i		−	0.435862i
$540$	−1.01773e7	−	212328.i	−0.0646326	−	0.00134842i
$541$	2.02371e8			1.27808			0.639038	−	0.769175i	$-0.279332\pi$
$541$	2.02371e8			1.27808			0.639038	+	0.769175i	$0.279332\pi$
$542$		−	3.20127e7i		−	0.201060i
$543$	−1.94256e8	+	1.10360e8i	−1.21332	+	0.689306i
$544$	6.10859e6			0.0379441
$545$			2.78315e7i			0.171928i
$546$	−4.81988e7	−	8.48397e7i	−0.296113	−	0.521220i
$547$	7.58476e7			0.463425			0.231713	−	0.972784i	$-0.425567\pi$
$547$	7.58476e7			0.463425			0.231713	+	0.972784i	$0.425567\pi$
$548$			899429.i			0.00546544i
$549$	−6.58704e7	+	1.10513e8i	−0.398082	+	0.667875i
$550$	−3.84275e7			−0.230969
$551$			1.48931e7i			0.0890289i
$552$	−5.86195e7	+	3.33027e7i	−0.348518	+	0.197998i
$553$	1.62482e8			0.960793
$554$			3.10745e8i			1.82757i
$555$	7.06991e7	+	1.24445e8i	0.413557	+	0.727944i
$556$	−3.41389e6			−0.0198621
$557$		−	3.34792e8i		−	1.93736i	−0.248318	−	0.968679i	$-0.579878\pi$
$557$		−	3.34792e8i		−	1.93736i	0.248318	−	0.968679i	$-0.420122\pi$
$558$	1.78755e8	+	1.06546e8i	1.02886	+	0.613244i
$559$	3.29139e8			1.88428
$560$		−	9.85635e7i		−	0.561244i
$561$	3.11775e7	−	1.77124e7i	0.176585	−	0.100321i
$562$	3.25982e7			0.183647
$563$		−	2.84555e8i		−	1.59456i	−0.603610	−	0.797280i	$-0.706271\pi$
$563$		−	2.84555e8i		−	1.59456i	0.603610	−	0.797280i	$-0.293729\pi$
$564$	−3.81926e6	−	6.72268e6i	−0.0212883	−	0.0374718i
$565$	−1.84127e8			−1.02087
$566$		−	2.48925e8i		−	1.37284i
$567$	−4.35209e7	−	8.04686e7i	−0.238753	−	0.441446i
$568$	5.65006e7			0.308324
$569$			2.15770e8i			1.17126i	0.810578	+	0.585630i	$0.199153\pi$
$569$			2.15770e8i			1.17126i	−0.810578	+	0.585630i	$0.800847\pi$
$570$	8.90644e7	−	5.05989e7i	0.480928	−	0.273223i
$571$	1.48881e8			0.799708			0.399854	−	0.916579i	$-0.369061\pi$
$571$	1.48881e8			0.799708			0.399854	+	0.916579i	$0.369061\pi$
$572$		−	7.29773e6i		−	0.0389942i
$573$	−5.43656e7	−	9.56945e7i	−0.288975	−	0.508655i
$574$	1.52501e8			0.806374
$575$		−	3.02996e7i		−	0.159380i
$576$	−1.02578e8	+	1.72098e8i	−0.536769	+	0.900553i
$577$	−2.39721e8			−1.24790			−0.623949	−	0.781465i	$-0.714473\pi$
$577$	−2.39721e8			−1.24790			−0.623949	+	0.781465i	$0.714473\pi$
$578$		−	1.64950e8i		−	0.854216i
$579$	1.09376e8	−	6.21383e7i	0.563490	−	0.320128i
$580$	2.34224e6			0.0120046
$581$		−	1.37443e8i		−	0.700799i
$582$	1.75872e8	+	3.09571e8i	0.892132	+	1.57033i
$583$	6.85961e7			0.346173
$584$			1.52949e8i			0.767909i
$585$	−2.50636e8	−	1.49390e8i	−1.25192	−	0.746197i
$586$	1.65525e7			0.0822565
$587$		−	2.91617e8i		−	1.44178i	−0.693050	−	0.720889i	$-0.743734\pi$
$587$		−	2.91617e8i		−	1.44178i	0.693050	−	0.720889i	$-0.256266\pi$
$588$	7.20536e6	−	4.09348e6i	0.0354424	−	0.0201354i
$589$	1.20673e8			0.590560
$590$		−	2.16836e7i		−	0.105578i
$591$	5.10617e7	+	8.98790e7i	0.247362	+	0.435407i
$592$	1.37990e8			0.665096
$593$			1.25993e8i			0.604204i	0.953276	+	0.302102i	$0.0976883\pi$
$593$			1.25993e8i			0.604204i	−0.953276	+	0.302102i	$0.902312\pi$
$594$	−2.47656e6	+	1.18707e8i	−0.0118165	+	0.566390i
$595$	−4.37237e7			−0.207571
$596$			1.33785e7i			0.0631929i
$597$	−1.89540e8	+	1.07681e8i	−0.890798	+	0.506076i
$598$	−9.98526e7			−0.466934
$599$			3.55722e8i			1.65512i	0.561377	+	0.827560i	$0.310272\pi$
$599$			3.55722e8i			1.65512i	−0.561377	+	0.827560i	$0.689728\pi$
$600$	−4.46036e7	−	7.85115e7i	−0.206498	−	0.363479i
$601$	−1.21331e8			−0.558920			−0.279460	−	0.960157i	$-0.590155\pi$
$601$	−1.21331e8			−0.558920			−0.279460	+	0.960157i	$0.590155\pi$
$602$			1.63316e8i			0.748584i
$603$	−3.47390e7	+	5.82826e7i	−0.158440	+	0.265819i
$604$	1.73286e7			0.0786418
$605$			1.73556e8i			0.783742i
$606$	−2.65706e6	+	1.50952e6i	−0.0119394	+	0.00678298i
$607$	−3.95525e8			−1.76851			−0.884255	−	0.467004i	$-0.845333\pi$
$607$	−3.95525e8			−1.76851			−0.884255	+	0.467004i	$0.845333\pi$
$608$			1.17292e7i			0.0521865i
$609$	1.03979e7	+	1.83024e7i	0.0460355	+	0.0810318i
$610$	−2.03604e8			−0.897009
$611$		−	2.21621e8i		−	0.971598i
$612$	3.73980e6	+	2.22908e6i	0.0163153	+	0.00972461i
$613$	−1.63324e8			−0.709035			−0.354517	−	0.935049i	$-0.615355\pi$
$613$	−1.63324e8			−0.709035			−0.354517	+	0.935049i	$0.615355\pi$
$614$		−	1.01884e8i		−	0.440149i
$615$	3.96505e8	−	2.25261e8i	1.70460	−	0.968413i
$616$	7.00790e7			0.299810
$617$			1.19047e8i			0.506829i	0.967358	+	0.253414i	$0.0815536\pi$
$617$			1.19047e8i			0.506829i	−0.967358	+	0.253414i	$0.918446\pi$
$618$	−9.89975e7	−	1.74256e8i	−0.419429	−	0.738281i
$619$	−1.49505e8			−0.630354			−0.315177	−	0.949033i	$-0.602064\pi$
$619$	−1.49505e8			−0.630354			−0.315177	+	0.949033i	$0.602064\pi$
$620$		−	1.89782e7i		−	0.0796306i
$621$	−9.35987e7	−	1.95274e6i	−0.390836	−	0.00815397i
$622$	3.20608e8			1.33230
$623$			6.65508e7i			0.275226i
$624$	−2.44595e8	+	1.38959e8i	−1.00669	+	0.571915i
$625$	−3.03096e8			−1.24148
$626$		−	2.40987e8i		−	0.982360i
$627$	3.40100e7	+	5.98645e7i	0.137976	+	0.242866i
$628$	−1.67921e7			−0.0677993
$629$		−	6.12139e7i		−	0.245979i
$630$	7.41261e7	−	1.24364e8i	0.296449	−	0.497361i
$631$	2.61517e8			1.04091			0.520453	−	0.853890i	$-0.325763\pi$
$631$	2.61517e8			1.04091			0.520453	+	0.853890i	$0.325763\pi$
$632$		−	4.95520e8i		−	1.96295i
$633$	1.05008e8	−	5.96569e7i	0.414012	−	0.235207i
$634$	−3.13983e8			−1.23208
$635$			3.90261e7i			0.152417i
$636$	4.11412e6	+	7.24168e6i	0.0159921	+	0.0281493i
$637$	2.37533e8			0.918978
$638$		−	2.73195e7i		−	0.105199i
$639$	6.73943e7	+	4.01699e7i	0.258298	+	0.153957i
$640$	−2.83212e8			−1.08037
$641$		−	4.43231e7i		−	0.168289i	−0.996454	−	0.0841445i	$-0.973184\pi$
$641$		−	4.43231e7i		−	0.168289i	0.996454	−	0.0841445i	$-0.0268157\pi$
$642$	2.87426e8	−	1.63292e8i	1.08623	−	0.617104i
$643$	−4.59510e8			−1.72847			−0.864236	−	0.503086i	$-0.832198\pi$
$643$	−4.59510e8			−1.72847			−0.864236	+	0.503086i	$0.832198\pi$
$644$			2.85517e6i			0.0106899i
$645$	2.41237e8	+	4.24626e8i	0.899010	+	1.58244i
$646$	−4.38105e7			−0.162510
$647$			1.10104e8i			0.406528i	0.979124	+	0.203264i	$0.0651549\pi$
$647$			1.10104e8i			0.406528i	−0.979124	+	0.203264i	$0.934845\pi$
$648$	−2.45405e8	+	1.32725e8i	−0.901899	+	0.487786i
$649$	1.45746e7			0.0533165
$650$		−	1.33737e8i		−	0.486979i
$651$	1.48297e8	−	8.42498e7i	0.537513	−	0.305370i
$652$	9.24508e6			0.0333555
$653$			2.50924e7i			0.0901161i	0.998984	+	0.0450581i	$0.0143473\pi$
$653$			2.50924e7i			0.0901161i	−0.998984	+	0.0450581i	$0.985653\pi$
$654$	1.94696e7	+	3.42704e7i	0.0696022	+	0.122514i
$655$	−2.84977e8			−1.01411
$656$		−	4.39664e8i		−	1.55743i
$657$	−1.08742e8	+	1.82439e8i	−0.383442	+	0.643312i
$658$	1.09966e8			0.385996
$659$			2.81480e8i			0.983539i	0.870725	+	0.491770i	$0.163650\pi$
$659$			2.81480e8i			0.983539i	−0.870725	+	0.491770i	$0.836350\pi$
$660$	9.41487e6	−	5.34874e6i	0.0327478	−	0.0186046i
$661$	−2.43675e8			−0.843738			−0.421869	−	0.906657i	$-0.638626\pi$
$661$	−2.43675e8			−0.843738			−0.421869	+	0.906657i	$0.638626\pi$
$662$			6.53518e7i			0.225260i
$663$	6.16434e7	+	1.08505e8i	0.211517	+	0.372313i
$664$	−4.19159e8			−1.43177
$665$		−	8.39548e7i		−	0.285483i
$666$	1.74111e8	+	1.03778e8i	0.589391	+	0.351303i
$667$	2.15411e7			0.0725923
$668$		−	7.42238e6i		−	0.0249008i
$669$	−4.76253e8	+	2.70567e8i	−1.59059	+	0.903642i
$670$	−1.07377e8			−0.357017
$671$		−	1.36852e8i		−	0.452985i
$672$	8.18894e6	+	1.44142e7i	0.0269848	+	0.0474988i
$673$	1.82038e8			0.597197			0.298599	−	0.954379i	$-0.403481\pi$
$673$	1.82038e8			0.597197			0.298599	+	0.954379i	$0.403481\pi$
$674$			3.27922e8i			1.07100i
$675$	2.61538e6	−	1.25361e8i	0.00850401	−	0.407615i
$676$	8.56596e6			0.0277291
$677$		−	3.09228e8i		−	0.996581i	−0.867010	−	0.498291i	$-0.833961\pi$
$677$		−	3.09228e8i		−	0.996581i	0.867010	−	0.498291i	$-0.166039\pi$
$678$	−2.26725e8	+	1.28806e8i	−0.727461	+	0.413283i
$679$	2.91811e8			0.932164
$680$			1.33344e8i			0.424079i
$681$	1.85355e8	+	3.26262e8i	0.586897	+	1.03306i
$682$	−2.21359e8			−0.697821
$683$			3.68914e8i			1.15788i	0.815371	+	0.578939i	$0.196533\pi$
$683$			3.68914e8i			1.15788i	−0.815371	+	0.578939i	$0.803467\pi$
$684$	−4.28011e6	+	7.18086e6i	−0.0133748	+	0.0224392i
$685$	−3.82527e7			−0.119012
$686$			2.75406e8i			0.853101i
$687$	4.33211e8	−	2.46114e8i	1.33607	−	0.759043i
$688$	4.70846e8			1.44582
$689$			2.38730e8i			0.729877i
$690$	−7.31852e7	−	1.28821e8i	−0.222780	−	0.392138i
$691$	−7.67725e7			−0.232687			−0.116343	−	0.993209i	$-0.537117\pi$
$691$	−7.67725e7			−0.232687			−0.116343	+	0.993209i	$0.537117\pi$
$692$			1.70581e7i			0.0514770i
$693$	8.35907e7	+	4.98237e7i	0.251165	+	0.149705i
$694$	4.31807e8			1.29185
$695$		−	1.45192e8i		−	0.432503i
$696$	5.58167e7	−	3.17103e7i	0.165553	−	0.0940531i
$697$	−1.95039e8			−0.576002
$698$			3.55050e8i			1.04405i
$699$	2.21800e8	+	3.90413e8i	0.649427	+	1.14312i
$700$	−3.82405e6			−0.0111488
$701$			6.10355e8i			1.77186i	0.463822	+	0.885928i	$0.346478\pi$
$701$			6.10355e8i			1.77186i	−0.463822	+	0.885928i	$0.653522\pi$
$702$	−4.13127e8	−	8.61900e6i	−1.19419	−	0.0249141i
$703$	1.17538e8			0.338308
$704$		−	2.13116e8i		−	0.610799i
$705$	2.85915e8	−	1.62433e8i	0.815962	−	0.463561i
$706$	4.77070e8			1.35571
$707$			2.50462e6i			0.00708735i
$708$	874124.	+	1.53864e6i	0.00246305	+	0.00433547i
$709$	1.80648e8			0.506869			0.253434	−	0.967353i	$-0.418440\pi$
$709$	1.80648e8			0.506869			0.253434	+	0.967353i	$0.418440\pi$
$710$			1.24164e8i			0.346914i
$711$	3.52298e8	−	5.91060e8i	0.980169	−	1.64446i
$712$	2.02960e8			0.562302
$713$		−	1.74539e8i		−	0.481530i
$714$	−5.38393e7	+	3.05870e7i	−0.147912	+	0.0840314i
$715$	3.10372e8			0.849111
$716$			6.53775e6i			0.0178110i
$717$	1.60022e8	+	2.81671e8i	0.434132	+	0.764160i
$718$	−6.92056e8			−1.86968
$719$		−	2.27771e8i		−	0.612790i	−0.951905	−	0.306395i	$-0.900877\pi$
$719$		−	2.27771e8i		−	0.612790i	0.951905	−	0.306395i	$-0.0991228\pi$
$720$	−3.58544e8	−	2.13708e8i	−0.960605	−	0.572562i
$721$	−1.64259e8			−0.438250
$722$			2.81849e8i			0.748866i
$723$	3.79092e8	−	2.15369e8i	1.00307	−	0.569859i
$724$	2.88550e7			0.0760336
$725$			2.88509e7i			0.0757086i
$726$	1.21411e8	+	2.13708e8i	0.317284	+	0.558485i
$727$	5.76922e8			1.50146			0.750730	−	0.660609i	$-0.229702\pi$
$727$	5.76922e8			1.50146			0.750730	+	0.660609i	$0.229702\pi$
$728$			2.43891e8i			0.632124i
$729$	−3.87083e8	−	1.61584e7i	−0.999130	−	0.0417076i
$730$	−3.36119e8			−0.864020
$731$		−	2.08872e8i		−	0.534722i
$732$	1.44475e7	−	8.20784e6i	0.0368348	−	0.0209264i
$733$	−4.81312e8			−1.22212			−0.611061	−	0.791584i	$-0.709257\pi$
$733$	−4.81312e8			−1.22212			−0.611061	+	0.791584i	$0.709257\pi$
$734$			4.98147e8i			1.25971i
$735$	1.74096e8	+	3.06444e8i	0.438456	+	0.771771i
$736$	1.69649e7			0.0425517
$737$		−	7.21736e7i		−	0.180292i
$738$	3.30656e8	−	5.54751e8i	0.822635	−	1.38016i
$739$	−5.14739e8			−1.27542			−0.637711	−	0.770276i	$-0.720119\pi$
$739$	−5.14739e8			−1.27542			−0.637711	+	0.770276i	$0.720119\pi$
$740$		−	1.84852e7i		−	0.0456172i
$741$	−2.08342e8	+	1.18363e8i	−0.512062	+	0.290911i
$742$	−1.18456e8			−0.289965
$743$		−	2.40835e8i		−	0.587156i	−0.955935	−	0.293578i	$-0.905154\pi$
$743$		−	2.40835e8i		−	0.587156i	0.955935	−	0.293578i	$-0.0948460\pi$
$744$	−2.56936e8	−	4.52260e8i	−0.623887	−	1.09817i
$745$	−5.68986e8			−1.37605
$746$			3.32538e8i			0.800985i
$747$	−4.99975e8	−	2.98007e8i	−1.19946	−	0.714932i
$748$	−4.63114e6			−0.0110658
$749$		−	2.70937e8i		−	0.644796i
$750$	−2.50653e8	+	1.42400e8i	−0.594140	+	0.337541i
$751$	−1.53712e8			−0.362901			−0.181451	−	0.983400i	$-0.558079\pi$
$751$	−1.53712e8			−0.362901			−0.181451	+	0.983400i	$0.558079\pi$
$752$		−	3.17037e8i		−	0.745514i
$753$	1.04078e8	+	1.83198e8i	0.243766	+	0.429078i
$754$	9.50783e7			0.221803
$755$			7.36986e8i			1.71245i
$756$	−246451.	+	1.18129e7i	−0.000570380	+	0.0273395i
$757$	2.38030e8			0.548712			0.274356	−	0.961628i	$-0.411535\pi$
$757$	2.38030e8			0.548712			0.274356	+	0.961628i	$0.411535\pi$
$758$		−	1.75333e8i		−	0.402583i
$759$	8.65868e7	−	4.91913e7i	0.198028	−	0.112503i
$760$	−2.56036e8			−0.583258
$761$			4.89141e8i			1.10989i	0.831887	+	0.554945i	$0.187261\pi$
$761$			4.89141e8i			1.10989i	−0.831887	+	0.554945i	$0.812739\pi$
$762$	2.73007e7	+	4.80548e7i	0.0617034	+	0.108611i
$763$	3.23043e7			0.0727254
$764$			1.42146e7i			0.0318752i
$765$	−9.48029e7	+	1.59054e8i	−0.211757	+	0.355270i
$766$	−3.59238e8			−0.799275
$767$			5.07229e7i			0.112413i
$768$	6.41861e7	−	3.64652e7i	0.141696	−	0.0804997i
$769$	−4.74097e7			−0.104253			−0.0521265	−	0.998640i	$-0.516600\pi$
$769$	−4.74097e7			−0.104253			−0.0521265	+	0.998640i	$0.516600\pi$
$770$			1.54004e8i			0.337334i
$771$	−3.80878e8	−	6.70423e8i	−0.831042	−	1.46280i
$772$	−1.62468e7			−0.0353116
$773$			8.97869e8i			1.94390i	0.235184	+	0.971951i	$0.424431\pi$
$773$			8.97869e8i			1.94390i	−0.235184	+	0.971951i	$0.575569\pi$
$774$	5.94095e8	+	3.54107e8i	1.28125	+	0.763680i
$775$	2.33767e8			0.502202
$776$		−	8.89934e8i		−	1.90446i
$777$	1.44444e8	−	8.20611e7i	0.307919	−	0.174934i
$778$	4.77855e7			0.101475
$779$		−	3.74499e8i		−	0.792206i
$780$	1.86149e7	+	3.27659e7i	0.0392262	+	0.0690460i
$781$	−8.34569e7			−0.175190
$782$			6.33665e7i			0.132507i
$783$	8.91234e7	+	1.85937e6i	0.185655	+	0.00387329i
$784$	3.39800e8			0.705139
$785$		−	7.14165e8i		−	1.47635i
$786$	−3.50907e8	+	1.99356e8i	−0.722645	+	0.410546i
$787$	3.40052e8			0.697624			0.348812	−	0.937193i	$-0.386585\pi$
$787$	3.40052e8			0.697624			0.348812	+	0.937193i	$0.386585\pi$
$788$		−	1.33507e7i		−	0.0272851i
$789$	−1.31436e8	−	2.31355e8i	−0.267599	−	0.471029i
$790$	1.08895e9			2.20864
$791$			2.13717e8i			0.431828i
$792$	1.51947e8	−	2.54926e8i	0.305856	−	0.513143i
$793$	4.76277e8			0.955080
$794$		−	7.76955e8i		−	1.55215i
$795$	−3.07988e8	+	1.74973e8i	−0.612961	+	0.348233i
$796$	2.81546e7			0.0558225
$797$			6.65198e8i			1.31394i	0.753916	+	0.656971i	$0.228163\pi$
$797$			6.65198e8i			1.31394i	−0.753916	+	0.656971i	$0.771837\pi$
$798$	−5.87306e7	−	1.03378e8i	−0.115573	−	0.203432i
$799$	−1.40641e8			−0.275721
$800$			2.27218e7i			0.0443784i
$801$	2.42092e8	+	1.44297e8i	0.471067	+	0.280776i
$802$	9.25327e8			1.79379
$803$		−	2.25922e8i		−	0.436326i
$804$	7.61936e6	−	4.32868e6i	0.0146605	−	0.00832889i
$805$	−1.21430e8			−0.232777
$806$		−	7.70381e8i		−	1.47130i
$807$	−4.63475e8	−	8.15810e8i	−0.881872	−	1.55227i
$808$	7.63833e6			0.0144799
$809$		−	6.19262e8i		−	1.16958i	−0.811186	−	0.584789i	$-0.801177\pi$
$809$		−	6.19262e8i		−	1.16958i	0.811186	−	0.584789i	$-0.198823\pi$
$810$	−2.91674e8	−	5.39296e8i	−0.548837	−	1.01478i
$811$	9.11717e8			1.70922			0.854609	−	0.519271i	$-0.173797\pi$
$811$	9.11717e8			1.70922			0.854609	+	0.519271i	$0.173797\pi$
$812$		−	2.71866e6i		−	0.00507792i
$813$	−9.66101e7	+	5.48858e7i	−0.179784	+	0.102138i
$814$	−2.15608e8			−0.399754
$815$			3.93193e8i			0.726328i
$816$	8.81832e7	+	1.55220e8i	0.162299	+	0.285679i
$817$	4.01059e8			0.735431
$818$		−	6.01796e8i		−	1.09948i
$819$	−1.73398e8	+	2.90915e8i	−0.315640	+	0.529559i
$820$	−5.88973e7			−0.106820
$821$		−	1.98609e7i		−	0.0358896i	−0.999839	−	0.0179448i	$-0.994288\pi$
$821$		−	1.98609e7i		−	0.0358896i	0.999839	−	0.0179448i	$-0.00571231\pi$
$822$	−4.71025e7	+	2.67597e7i	−0.0848063	+	0.0481799i
$823$	−1.07498e9			−1.92841			−0.964204	−	0.265162i	$-0.914575\pi$
$823$	−1.07498e9			−1.92841			−0.964204	+	0.265162i	$0.914575\pi$
$824$			5.00938e8i			0.895370i
$825$	6.58840e7	+	1.15969e8i	0.117332	+	0.206529i
$826$	−2.51683e7			−0.0446595
$827$			3.48689e8i			0.616483i	0.951308	+	0.308242i	$0.0997406\pi$
$827$			3.48689e8i			0.616483i	−0.951308	+	0.308242i	$0.900259\pi$
$828$	1.03862e7	+	6.19066e6i	0.0182965	+	0.0109055i
$829$	−8.52363e8			−1.49610			−0.748051	−	0.663642i	$-0.769010\pi$
$829$	−8.52363e8			−1.49610			−0.748051	+	0.663642i	$0.769010\pi$
$830$		−	9.21134e8i		−	1.61097i
$831$	9.37786e8	−	5.32771e8i	1.63418	−	0.928405i
$832$	7.41693e8			1.28782
$833$		−	1.50738e8i		−	0.260789i
$834$	−1.01569e8	−	1.78783e8i	−0.175091	−	0.308197i
$835$	3.15674e8			0.542224
$836$		−	8.89234e6i		−	0.0152194i
$837$	1.50657e7	−	7.22131e8i	0.0256929	−	1.23151i
$838$	1.62014e8			0.275309
$839$			4.00851e8i			0.678730i	0.940655	+	0.339365i	$0.110212\pi$
$839$			4.00851e8i			0.678730i	−0.940655	+	0.339365i	$0.889788\pi$
$840$	−3.14647e8	+	1.78756e8i	−0.530866	+	0.301594i
$841$	−2.05111e7			−0.0344828
$842$		−	3.65781e7i		−	0.0612752i
$843$	−5.58896e7	−	9.83771e7i	−0.0932929	−	0.164214i
$844$	−1.55981e7			−0.0259444
$845$			3.64310e8i			0.603811i
$846$	2.38432e8	−	4.00024e8i	0.393780	−	0.660656i
$847$	2.01448e8			0.331522
$848$			3.41513e8i			0.560040i
$849$	−7.51223e8	+	4.26782e8i	−1.22757	+	0.697402i
$850$	−8.48694e7			−0.138196
$851$		−	1.70005e8i		−	0.275849i
$852$	−5.00541e6	−	8.81054e6i	−0.00809321	−	0.0142457i
$853$	6.79694e8			1.09513			0.547566	−	0.836763i	$-0.315555\pi$
$853$	6.79694e8			1.09513			0.547566	+	0.836763i	$0.315555\pi$
$854$			2.36325e8i			0.379434i
$855$	−3.05402e8	−	1.82033e8i	−0.488622	−	0.291240i
$856$	−8.26273e8			−1.31735
$857$			5.82064e8i			0.924759i	0.886682	+	0.462379i	$0.153004\pi$
$857$			5.82064e8i			0.924759i	−0.886682	+	0.462379i	$0.846996\pi$
$858$	3.82177e8	−	2.17121e8i	0.605066	−	0.343748i
$859$	−7.84287e8			−1.23736			−0.618679	−	0.785644i	$-0.712332\pi$
$859$	−7.84287e8			−1.23736			−0.618679	+	0.785644i	$0.712332\pi$
$860$		−	6.30744e7i		−	0.0991649i
$861$	−2.61462e8	−	4.60227e8i	−0.409638	−	0.721045i
$862$	1.20789e8			0.188585
$863$		−	2.47974e8i		−	0.385810i	−0.981217	−	0.192905i	$-0.938209\pi$
$863$		−	2.47974e8i		−	0.385810i	0.981217	−	0.192905i	$-0.0617909\pi$
$864$	7.01899e7	+	1.46436e6i	0.108826	+	0.00227043i
$865$	−7.25482e8			−1.12093
$866$		−	5.06563e8i		−	0.779974i
$867$	−4.97796e8	+	2.82806e8i	−0.763825	+	0.433942i
$868$	−2.20282e7			−0.0336837
$869$			7.31933e8i			1.11535i
$870$	6.96860e7	+	1.22661e8i	0.105825	+	0.186273i
$871$	2.51181e8			0.380130
$872$		−	9.85181e7i		−	0.148582i
$873$	6.32712e8	−	1.06152e9i	0.950962	−	1.59546i
$874$	−1.21671e8			−0.182244
$875$			2.36273e8i			0.352687i
$876$	2.38505e7	−	1.35499e7i	0.0354801	−	0.0201568i
$877$	5.77617e8			0.856329			0.428165	−	0.903701i	$-0.359160\pi$
$877$	5.77617e8			0.856329			0.428165	+	0.903701i	$0.359160\pi$
$878$		−	4.12347e8i		−	0.609228i
$879$	−2.83792e7	−	4.99532e7i	−0.0417863	−	0.0735523i
$880$	4.43999e8			0.651529
$881$			6.53462e8i			0.955636i	0.878459	+	0.477818i	$0.158572\pi$
$881$			6.53462e8i			0.955636i	−0.878459	+	0.477818i	$0.841428\pi$
$882$	4.28746e8	+	2.55551e8i	0.624876	+	0.372453i
$883$	−1.13453e8			−0.164791			−0.0823957	−	0.996600i	$-0.526257\pi$
$883$	−1.13453e8			−0.164791			−0.0823957	+	0.996600i	$0.526257\pi$
$884$		−	1.61174e7i		−	0.0233313i
$885$	−6.54382e7	+	3.71765e7i	−0.0944064	+	0.0536338i
$886$	−6.28534e8			−0.903707
$887$			3.07529e8i			0.440671i	0.975424	+	0.220336i	$0.0707152\pi$
$887$			3.07529e8i			0.440671i	−0.975424	+	0.220336i	$0.929285\pi$
$888$	−2.50261e8	−	4.40510e8i	−0.357400	−	0.629096i
$889$	4.52979e7			0.0644722
$890$			4.46020e8i			0.632680i
$891$	3.62487e8	−	1.96049e8i	0.512459	−	0.277160i
$892$	7.07432e7			0.0996759
$893$		−	2.70047e8i		−	0.379214i
$894$	−7.00622e8	+	3.98035e8i	−0.980554	+	0.557069i
$895$	−2.78050e8			−0.387841
$896$			3.28727e8i			0.456994i
$897$	1.71197e8	+	3.01342e8i	0.237203	+	0.417525i
$898$	4.47078e8			0.617383
$899$			1.66194e8i			0.228736i
$900$	−8.29140e6	+	1.39107e7i	−0.0113737	+	0.0190819i
$901$	1.51498e8			0.207125
$902$			6.86970e8i			0.936091i
$903$	4.92867e8	−	2.80006e8i	0.669371	−	0.380280i
$904$	6.51773e8			0.882249
$905$			1.22720e9i			1.65566i
$906$	5.15559e8	+	9.07489e8i	0.693257	+	1.22027i
$907$	−2.29374e7			−0.0307413			−0.0153707	−	0.999882i	$-0.504893\pi$
$907$	−2.29374e7			−0.0307413			−0.0153707	+	0.999882i	$0.504893\pi$
$908$		−	4.84633e7i		−	0.0647374i
$909$	9.11105e6	+	5.43058e6i	0.0121304	+	0.00723028i
$910$	−5.35970e8			−0.711240
$911$			1.46929e8i			0.194336i	0.995268	+	0.0971679i	$0.0309784\pi$
$911$			1.46929e8i			0.194336i	−0.995268	+	0.0971679i	$0.969022\pi$
$912$	−2.98041e8	+	1.69322e8i	−0.392909	+	0.223218i
$913$	6.19139e8			0.813534
$914$			7.50747e8i			0.983229i
$915$	3.49079e8	+	6.14450e8i	0.455680	+	0.802090i
$916$	−6.43497e7			−0.0837259
$917$			3.30776e8i			0.428969i
$918$	−5.46962e6	+	2.62170e8i	−0.00707016	+	0.338887i
$919$	−4.76218e8			−0.613563			−0.306782	−	0.951780i	$-0.599252\pi$
$919$	−4.76218e8			−0.613563			−0.306782	+	0.951780i	$0.599252\pi$
$920$			3.70325e8i			0.475576i
$921$	−3.07472e8	+	1.74680e8i	−0.393574	+	0.223596i
$922$	4.47438e8			0.570873
$923$		−	2.90449e8i		−	0.369373i
$924$	−6.20833e6	−	1.09279e7i	−0.00786971	−	0.0138523i
$925$	2.27694e8			0.287691
$926$		−	4.14880e8i		−	0.522504i
$927$	−3.56150e8	+	5.97523e8i	−0.447088	+	0.750093i
$928$	−1.61537e7			−0.0202129
$929$		−	7.42276e8i		−	0.925802i	−0.886410	−	0.462901i	$-0.846808\pi$
$929$		−	7.42276e8i		−	0.925802i	0.886410	−	0.462901i	$-0.153192\pi$
$930$	9.93876e8	−	5.64637e8i	1.23562	−	0.701973i
$931$	2.89436e8			0.358677
$932$		−	5.79924e7i		−	0.0716347i
$933$	−5.49682e8	−	9.67552e8i	−0.676810	−	1.19132i
$934$	−1.51216e9			−1.85591
$935$		−	1.96962e8i		−	0.240962i
$936$	8.87202e8	+	5.28811e8i	1.08192	+	0.644871i
$937$	−1.13544e8			−0.138021			−0.0690105	−	0.997616i	$-0.521984\pi$
$937$	−1.13544e8			−0.138021			−0.0690105	+	0.997616i	$0.521984\pi$
$938$			1.24634e8i			0.151018i
$939$	−7.27267e8	+	4.13172e8i	−0.878409	+	0.499039i
$940$	−4.24702e7			−0.0511329
$941$			1.11849e9i			1.34235i	0.741300	+	0.671174i	$0.234209\pi$
$941$			1.11849e9i			1.34235i	−0.741300	+	0.671174i	$0.765791\pi$
$942$	−4.99595e8	−	8.79389e8i	−0.597675	−	1.05203i
$943$	−5.41667e8			−0.645948
$944$			7.25610e7i			0.0862556i
$945$	−5.02402e8	−	1.04815e7i	−0.595327	−	0.0124202i
$946$	−7.35691e8			−0.869005
$947$			6.36992e8i			0.750040i	0.927017	+	0.375020i	$0.122364\pi$
$947$			6.36992e8i			0.750040i	−0.927017	+	0.375020i	$0.877636\pi$
$948$	−7.72700e7	+	4.38984e7i	−0.0906956	+	0.0515256i
$949$	7.86259e8			0.919956
$950$		−	1.62959e8i		−	0.190068i
$951$	5.38324e8	+	9.47560e8i	0.625896	+	1.10170i
$952$	1.54773e8			0.179385
$953$		−	1.42387e9i		−	1.64510i	−0.568695	−	0.822548i	$-0.692552\pi$
$953$		−	1.42387e9i		−	1.64510i	0.568695	−	0.822548i	$-0.307448\pi$
$954$	−2.56839e8	+	4.30907e8i	−0.295813	+	0.496294i
$955$	−6.04545e8			−0.694094
$956$		−	4.18397e7i		−	0.0478867i
$957$	−8.24467e7	+	4.68393e7i	−0.0940671	+	0.0534410i
$958$	−9.93528e7			−0.113001
$959$			4.44002e7i			0.0503418i
$960$	5.43611e8	+	9.56866e8i	0.614433	+	1.08153i
$961$	4.59095e8			0.517288
$962$		−	7.50367e8i		−	0.842846i
$963$	−9.85584e8	−	5.87451e8i	−1.10361	−	0.657799i
$964$	−5.63108e7			−0.0628580
$965$		−	6.90977e8i		−	0.768921i
$966$	−1.49524e8	+	8.49467e7i	−0.165874	+	0.0942357i
$967$	−5.36219e8			−0.593011			−0.296506	−	0.955031i	$-0.595821\pi$
$967$	−5.36219e8			−0.593011			−0.296506	+	0.955031i	$0.595821\pi$
$968$		−	6.14354e8i		−	0.677318i
$969$	7.51130e7	+	1.32214e8i	0.0825551	+	0.145314i
$970$	1.95570e9			2.14283
$971$		−	4.06107e7i		−	0.0443591i	−0.999754	−	0.0221795i	$-0.992939\pi$
$971$		−	4.06107e7i		−	0.0443591i	0.999754	−	0.0221795i	$-0.00706054\pi$
$972$	4.24373e7	+	2.65095e7i	0.0462114	+	0.0288671i
$973$	−1.68526e8			−0.182948
$974$			4.59728e8i			0.497536i
$975$	−4.03599e8	+	2.29291e8i	−0.435448	+	0.247385i
$976$	6.81332e8			0.732840
$977$			1.13933e9i			1.22170i	0.791745	+	0.610851i	$0.209173\pi$
$977$			1.13933e9i			1.22170i	−0.791745	+	0.610851i	$0.790827\pi$
$978$	2.75058e8	+	4.84159e8i	0.294041	+	0.517573i
$979$	−2.99792e8			−0.319500
$980$		−	4.55195e7i		−	0.0483636i
$981$	7.00429e7	−	1.17513e8i	0.0741920	−	0.124474i
$982$	6.33068e8			0.668522
$983$		−	1.80605e9i		−	1.90138i	−0.310144	−	0.950690i	$-0.600377\pi$
$983$		−	1.80605e9i		−	1.90138i	0.310144	−	0.950690i	$-0.399623\pi$
$984$	−1.40355e9	+	7.97380e8i	−1.47314	+	0.836912i
$985$	5.67806e8			0.594143
$986$		−	6.03367e7i		−	0.0629435i
$987$	−1.88537e8	−	3.31864e8i	−0.196086	−	0.345151i
$988$	3.09474e7			0.0320888
$989$		−	5.80083e8i		−	0.599655i
$990$	5.60220e8	+	3.33916e8i	0.577369	+	0.344137i
$991$	−1.44945e9			−1.48930			−0.744650	−	0.667455i	$-0.767384\pi$
$991$	−1.44945e9			−1.48930			−0.744650	+	0.667455i	$0.767384\pi$
$992$			1.30887e8i			0.134079i
$993$	1.97223e8	−	1.12046e8i	0.201423	−	0.114432i
$994$	1.44119e8			0.146744
$995$			1.19741e9i			1.21555i
$996$	3.71334e7	+	6.53624e7i	0.0375826	+	0.0661531i
$997$	−9.23758e8			−0.932122			−0.466061	−	0.884753i	$-0.654327\pi$
$997$	−9.23758e8			−0.932122			−0.466061	+	0.884753i	$0.654327\pi$
$998$			5.65292e8i			0.568697i
$999$	1.46743e7	−	7.03371e8i	0.0147184	−	0.705485i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	87.7.b.a.59.17	✓	56
3.2	odd	2	inner	87.7.b.a.59.40	yes	56

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
87.7.b.a.59.17	✓	56	1.1	even	1	trivial
87.7.b.a.59.40	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-7.77900i q^{2} +(-23.4760 + 13.3371i) q^{3} +3.48715 q^{4} +148.308i q^{5} +(103.749 + 182.620i) q^{6} +172.143 q^{7} -524.983i q^{8} +(373.244 - 626.203i) q^{9} +1153.69 q^{10} +775.451i q^{11} +(-81.8643 + 46.5084i) q^{12} -2698.75 q^{13} -1339.10i q^{14} +(-1978.00 - 3481.68i) q^{15} -3860.66 q^{16} +1712.63i q^{17} +(-4871.23 - 2903.47i) q^{18} -3288.45 q^{19} +517.173i q^{20} +(-4041.22 + 2295.88i) q^{21} +6032.24 q^{22} +4756.34i q^{23} +(7001.74 + 12324.5i) q^{24} -6370.36 q^{25} +20993.6i q^{26} +(-410.554 + 19678.7i) q^{27} +600.288 q^{28} -4528.92i q^{29} +(-27084.0 + 15386.9i) q^{30} -36696.0 q^{31} -3566.79i q^{32} +(-10342.3 - 18204.5i) q^{33} +13322.5 q^{34} +25530.2i q^{35} +(1301.56 - 2183.66i) q^{36} -35742.7 q^{37} +25580.8i q^{38} +(63355.8 - 35993.5i) q^{39} +77859.3 q^{40} +113883. i q^{41} +(17859.7 + 31436.7i) q^{42} -121960. q^{43} +2704.11i q^{44} +(92871.1 + 55355.2i) q^{45} +36999.6 q^{46} +82119.8i q^{47} +(90632.9 - 51490.0i) q^{48} -88015.9 q^{49} +49555.1i q^{50} +(-22841.5 - 40205.6i) q^{51} -9410.94 q^{52} -88459.6i q^{53} +(153081. + 3193.70i) q^{54} -115006. q^{55} -90371.9i q^{56} +(77199.6 - 43858.3i) q^{57} -35230.5 q^{58} -18795.0i q^{59} +(-6897.59 - 12141.2i) q^{60} -176481. q^{61} +285458. i q^{62} +(64251.3 - 107796. i) q^{63} -274828. q^{64} -400247. i q^{65} +(-141613. + 80452.5i) q^{66} -93073.0 q^{67} +5972.19i q^{68} +(-63435.8 - 111660. i) q^{69} +198599. q^{70} +107624. i q^{71} +(-328746. - 195947. i) q^{72} -291342. q^{73} +278042. i q^{74} +(149551. - 84962.1i) q^{75} -11467.3 q^{76} +133488. i q^{77} +(-279993. - 492845. i) q^{78} +943880. q^{79} -572568. i q^{80} +(-252819. - 467453. i) q^{81} +885897. q^{82} -798424. i q^{83} +(-14092.3 + 8006.09i) q^{84} -253997. q^{85} +948727. i q^{86} +(60402.7 + 106321. i) q^{87} +407098. q^{88} +386603. i q^{89} +(430608. - 722444. i) q^{90} -464570. q^{91} +16586.1i q^{92} +(861476. - 489418. i) q^{93} +638810. q^{94} -487704. i q^{95} +(47570.6 + 83734.0i) q^{96} +1.69517e6 q^{97} +684676. i q^{98} +(485590. + 289433. 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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