LMFDB - Embedded newform 87.7.b.a.59.14

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			59.14
Character	$\chi$	$=$	87.59
Dual form			87.7.b.a.59.43

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q-9.12250i q^{2} +(26.3711 + 5.79335i) q^{3} -19.2200 q^{4} +194.806i q^{5} +(52.8499 - 240.571i) q^{6} -587.861 q^{7} -408.506i q^{8} +(661.874 + 305.555i) q^{9} +1777.12 q^{10} +1305.70i q^{11} +(-506.853 - 111.348i) q^{12} -2280.71 q^{13} +5362.76i q^{14} +(-1128.58 + 5137.26i) q^{15} -4956.67 q^{16} -743.497i q^{17} +(2787.42 - 6037.95i) q^{18} -7675.22 q^{19} -3744.17i q^{20} +(-15502.6 - 3405.69i) q^{21} +11911.2 q^{22} +18427.6i q^{23} +(2366.62 - 10772.8i) q^{24} -22324.4 q^{25} +20805.7i q^{26} +(15684.2 + 11892.3i) q^{27} +11298.7 q^{28} +4528.92i q^{29} +(46864.6 + 10295.5i) q^{30} +32164.9 q^{31} +19072.9i q^{32} +(-7564.37 + 34432.7i) q^{33} -6782.55 q^{34} -114519. i q^{35} +(-12721.2 - 5872.75i) q^{36} -42287.8 q^{37} +70017.2i q^{38} +(-60144.8 - 13212.9i) q^{39} +79579.4 q^{40} -24710.1i q^{41} +(-31068.4 + 141422. i) q^{42} +117724. q^{43} -25095.5i q^{44} +(-59523.9 + 128937. i) q^{45} +168105. q^{46} +70765.2i q^{47} +(-130713. - 28715.8i) q^{48} +227931. q^{49} +203654. i q^{50} +(4307.34 - 19606.9i) q^{51} +43835.1 q^{52} +65859.1i q^{53} +(108487. - 143079. i) q^{54} -254358. q^{55} +240144. i q^{56} +(-202404. - 44465.3i) q^{57} +41315.1 q^{58} -360284. i q^{59} +(21691.3 - 98738.0i) q^{60} +311577. q^{61} -293424. i q^{62} +(-389090. - 179624. i) q^{63} -143235. q^{64} -444296. i q^{65} +(314112. + 69005.9i) q^{66} +13894.5 q^{67} +14290.0i q^{68} +(-106757. + 485956. i) q^{69} -1.04470e6 q^{70} +5083.31i q^{71} +(124821. - 270379. i) q^{72} -69853.2 q^{73} +385771. i q^{74} +(-588720. - 129333. i) q^{75} +147517. q^{76} -767568. i q^{77} +(-120535. + 548671. i) q^{78} -557621. q^{79} -965590. i q^{80} +(344714. + 404478. i) q^{81} -225418. q^{82} +271037. i q^{83} +(297959. + 65457.2i) q^{84} +144838. q^{85} -1.07394e6i q^{86} +(-26237.7 + 119433. i) q^{87} +533385. q^{88} +823262. i q^{89} +(1.17623e6 + 543007. i) q^{90} +1.34074e6 q^{91} -354177. i q^{92} +(848226. + 186343. i) q^{93} +645555. q^{94} -1.49518e6i q^{95} +(-110496. + 502973. i) q^{96} -1.70159e6 q^{97} -2.07930e6i q^{98} +(-398962. + 864207. 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 $

$2$		−	9.12250i		−	1.14031i	−0.821536	−	0.570156i	$-0.806883\pi$
$2$		−	9.12250i		−	1.14031i	0.821536	−	0.570156i	$-0.193117\pi$
$3$	26.3711	+	5.79335i	0.976709	+	0.214569i
$3$	26.3711	+	5.79335i	0.976709	+	0.214569i
$4$	−19.2200			−0.300312
$5$			194.806i			1.55845i	0.626745	+	0.779224i	$0.284387\pi$
$5$			194.806i			1.55845i	−0.626745	+	0.779224i	$0.715613\pi$
$6$	52.8499	−	240.571i	0.244675	−	1.11375i
$7$	−587.861			−1.71388			−0.856940	−	0.515416i	$-0.827637\pi$
$7$	−587.861			−1.71388			−0.856940	+	0.515416i	$0.827637\pi$
$8$		−	408.506i		−	0.797863i
$9$	661.874	+	305.555i	0.907921	+	0.419142i
$10$	1777.12			1.77712
$11$			1305.70i			0.980990i	0.871444	+	0.490495i	$0.163184\pi$
$11$			1305.70i			0.980990i	−0.871444	+	0.490495i	$0.836816\pi$
$12$	−506.853	−	111.348i	−0.293317	−	0.0644376i
$13$	−2280.71			−1.03810			−0.519050	−	0.854744i	$-0.673714\pi$
$13$	−2280.71			−1.03810			−0.519050	+	0.854744i	$0.673714\pi$
$14$			5362.76i			1.95436i
$15$	−1128.58	+	5137.26i	−0.334394	+	1.52215i
$16$	−4956.67			−1.21012
$17$		−	743.497i		−	0.151333i	−0.997133	−	0.0756663i	$-0.975892\pi$
$17$		−	743.497i		−	0.151333i	0.997133	−	0.0756663i	$-0.0241084\pi$
$18$	2787.42	−	6037.95i	0.477953	−	1.03531i
$19$	−7675.22			−1.11900			−0.559500	−	0.828831i	$-0.689007\pi$
$19$	−7675.22			−1.11900			−0.559500	+	0.828831i	$0.689007\pi$
$20$		−	3744.17i		−	0.468021i
$21$	−15502.6	−	3405.69i	−1.67396	−	0.367745i
$22$	11911.2			1.11863
$23$			18427.6i			1.51455i	0.653094	+	0.757277i	$0.273471\pi$
$23$			18427.6i			1.51455i	−0.653094	+	0.757277i	$0.726529\pi$
$24$	2366.62	−	10772.8i	0.171196	−	0.779280i
$25$	−22324.4			−1.42876
$26$			20805.7i			1.18376i
$27$	15684.2	+	11892.3i	0.796839	+	0.604191i
$28$	11298.7			0.514699
$29$			4528.92i			0.185695i
$29$			4528.92i			0.185695i
$30$	46864.6	+	10295.5i	1.73573	+	0.381314i
$31$	32164.9			1.07969			0.539843	−	0.841766i	$-0.318484\pi$
$31$	32164.9			1.07969			0.539843	+	0.841766i	$0.318484\pi$
$32$			19072.9i			0.582057i
$33$	−7564.37	+	34432.7i	−0.210490	+	0.958141i
$34$	−6782.55			−0.172566
$35$		−	114519.i		−	2.67099i
$36$	−12721.2	−	5872.75i	−0.272660	−	0.125873i
$37$	−42287.8			−0.834853			−0.417427	−	0.908711i	$-0.637068\pi$
$37$	−42287.8			−0.834853			−0.417427	+	0.908711i	$0.637068\pi$
$38$			70017.2i			1.27601i
$39$	−60144.8	−	13212.9i	−1.01392	−	0.222744i
$40$	79579.4			1.24343
$41$		−	24710.1i		−	0.358528i	−0.983801	−	0.179264i	$-0.942628\pi$
$41$		−	24710.1i		−	0.358528i	0.983801	−	0.179264i	$-0.0573717\pi$
$42$	−31068.4	+	141422.i	−0.419344	+	1.90884i
$43$	117724.			1.48067			0.740337	−	0.672236i	$-0.234666\pi$
$43$	117724.			1.48067			0.740337	+	0.672236i	$0.234666\pi$
$44$		−	25095.5i		−	0.294603i
$45$	−59523.9	+	128937.i	−0.653212	+	1.41495i
$46$	168105.			1.72706
$47$			70765.2i			0.681595i	0.940137	+	0.340797i	$0.110697\pi$
$47$			70765.2i			0.681595i	−0.940137	+	0.340797i	$0.889303\pi$
$48$	−130713.	−	28715.8i	−1.18194	−	0.259655i
$49$	227931.			1.93738
$50$			203654.i			1.62924i
$51$	4307.34	−	19606.9i	0.0324712	−	0.147808i
$52$	43835.1			0.311754
$53$			65859.1i			0.442373i	0.975232	+	0.221186i	$0.0709929\pi$
$53$			65859.1i			0.442373i	−0.975232	+	0.221186i	$0.929007\pi$
$54$	108487.	−	143079.i	0.688967	−	0.908646i
$55$	−254358.			−1.52882
$56$			240144.i			1.36744i
$57$	−202404.	−	44465.3i	−1.09294	−	0.240102i
$58$	41315.1			0.211751
$59$		−	360284.i		−	1.75424i	−0.480273	−	0.877119i	$-0.659462\pi$
$59$		−	360284.i		−	1.75424i	0.480273	−	0.877119i	$-0.340538\pi$
$60$	21691.3	−	98738.0i	0.100423	−	0.457120i
$61$	311577.			1.37270			0.686350	−	0.727272i	$-0.259212\pi$
$61$	311577.			1.37270			0.686350	+	0.727272i	$0.259212\pi$
$62$		−	293424.i		−	1.23118i
$63$	−389090.	−	179624.i	−1.55607	−	0.718359i
$64$	−143235.			−0.546398
$65$		−	444296.i		−	1.61783i
$66$	314112.	+	69005.9i	1.09258	+	0.240024i
$67$	13894.5			0.0461976			0.0230988	−	0.999733i	$-0.492647\pi$
$67$	13894.5			0.0461976			0.0230988	+	0.999733i	$0.492647\pi$
$68$			14290.0i			0.0454470i
$69$	−106757.	+	485956.i	−0.324976	+	1.47928i
$70$	−1.04470e6			−3.04577
$71$			5083.31i			0.0142027i	0.999975	+	0.00710136i	$0.00226045\pi$
$71$			5083.31i			0.0142027i	−0.999975	+	0.00710136i	$0.997740\pi$
$72$	124821.	−	270379.i	0.334418	−	0.724396i
$73$	−69853.2			−0.179563			−0.0897816	−	0.995961i	$-0.528617\pi$
$73$	−69853.2			−0.179563			−0.0897816	+	0.995961i	$0.528617\pi$
$74$			385771.i			0.951994i
$75$	−588720.	−	129333.i	−1.39548	−	0.306568i
$76$	147517.			0.336049
$77$		−	767568.i		−	1.68130i
$78$	−120535.	+	548671.i	−0.253998	+	1.15619i
$79$	−557621.			−1.13099			−0.565494	−	0.824753i	$-0.691314\pi$
$79$	−557621.			−1.13099			−0.565494	+	0.824753i	$0.691314\pi$
$80$		−	965590.i		−	1.88592i
$81$	344714.	+	404478.i	0.648639	+	0.761096i
$82$	−225418.			−0.408834
$83$			271037.i			0.474018i	0.971507	+	0.237009i	$0.0761670\pi$
$83$			271037.i			0.474018i	−0.971507	+	0.237009i	$0.923833\pi$
$84$	297959.	+	65457.2i	0.502711	+	0.110438i
$85$	144838.			0.235844
$86$		−	1.07394e6i		−	1.68843i
$87$	−26237.7	+	119433.i	−0.0398444	+	0.181370i
$88$	533385.			0.782695
$89$			823262.i			1.16780i	0.811826	+	0.583899i	$0.198474\pi$
$89$			823262.i			1.16780i	−0.811826	+	0.583899i	$0.801526\pi$
$90$	1.17623e6	+	543007.i	1.61348	+	0.744865i
$91$	1.34074e6			1.77918
$92$		−	354177.i		−	0.454839i
$93$	848226.	+	186343.i	1.05454	+	0.231667i
$94$	645555.			0.777231
$95$		−	1.49518e6i		−	1.74390i
$96$	−110496.	+	502973.i	−0.124891	+	0.568501i
$97$	−1.70159e6			−1.86440			−0.932202	−	0.361938i	$-0.882115\pi$
$97$	−1.70159e6			−1.86440			−0.932202	+	0.361938i	$0.882115\pi$
$98$		−	2.07930e6i		−	2.20922i
$99$	−398962.	+	864207.i	−0.411174	+	0.890661i
$100$	429075.			0.429075
$101$			956245.i			0.928122i	0.885803	+	0.464061i	$0.153608\pi$
$101$			956245.i			0.928122i	−0.885803	+	0.464061i	$0.846392\pi$
$102$	−178863.	−	39293.7i	−0.168547	−	0.0370273i
$103$	−440067.			−0.402724			−0.201362	−	0.979517i	$-0.564537\pi$
$103$	−440067.			−0.402724			−0.201362	+	0.979517i	$0.564537\pi$
$104$			931682.i			0.828262i
$105$	663448.	−	3.01999e6i	0.573112	−	2.60878i
$106$	600800.			0.504443
$107$		−	282753.i		−	0.230811i	−0.993318	−	0.115405i	$-0.963183\pi$
$107$		−	282753.i		−	0.230811i	0.993318	−	0.115405i	$-0.0368167\pi$
$108$	−301450.	−	228570.i	−0.239300	−	0.181446i
$109$	1.38314e6			1.06804			0.534020	−	0.845472i	$-0.320681\pi$
$109$	1.38314e6			1.06804			0.534020	+	0.845472i	$0.320681\pi$
$110$			2.32038e6i			1.74333i
$111$	−1.11518e6	−	244988.i	−0.815409	−	0.179133i
$112$	2.91383e6			2.07401
$113$			1.58436e6i			1.09804i	0.835808	+	0.549022i	$0.185000\pi$
$113$			1.58436e6i			1.09804i	−0.835808	+	0.549022i	$0.815000\pi$
$114$	−405634.	+	1.84643e6i	−0.273792	+	1.24629i
$115$	−3.58980e6			−2.36035
$116$		−	87045.8i		−	0.0557666i
$117$	−1.50954e6	−	696881.i	−0.942513	−	0.435112i
$118$	−3.28669e6			−2.00038
$119$			437072.i			0.259366i
$120$	2.09860e6	+	461032.i	1.21447	+	0.266801i
$121$	66715.8			0.0376593
$122$		−	2.84236e6i		−	1.56531i
$123$	143155.	−	651634.i	0.0769289	−	0.350178i
$124$	−618209.			−0.324243
$125$		−	1.30509e6i		−	0.668204i
$126$	−1.63862e6	+	3.54947e6i	−0.819154	+	1.77440i
$127$	394547.			0.192614			0.0963070	−	0.995352i	$-0.469297\pi$
$127$	394547.			0.192614			0.0963070	+	0.995352i	$0.469297\pi$
$128$			2.52732e6i			1.20512i
$129$	3.10451e6	+	682016.i	1.44619	+	0.317706i
$130$	−4.05309e6			−1.84483
$131$		−	1.65849e6i		−	0.737732i	−0.929483	−	0.368866i	$-0.879746\pi$
$131$		−	1.65849e6i		−	0.737732i	0.929483	−	0.368866i	$-0.120254\pi$
$132$	145387.	−	661796.i	0.0632126	−	0.287741i
$133$	4.51196e6			1.91783
$134$		−	126753.i		−	0.0526797i
$135$	−2.31669e6	+	3.05538e6i	−0.941601	+	1.24183i
$136$	−303723.			−0.120743
$137$		−	2.67518e6i		−	1.04038i	−0.854051	−	0.520189i	$-0.825861\pi$
$137$		−	2.67518e6i		−	1.04038i	0.854051	−	0.520189i	$-0.174139\pi$
$138$	4.43313e6	+	973895.i	1.68684	+	0.370574i
$139$	1.98293e6			0.738351			0.369176	−	0.929360i	$-0.379640\pi$
$139$	1.98293e6			0.738351			0.369176	+	0.929360i	$0.379640\pi$
$140$			2.20105e6i			0.802132i
$141$	−409968.	+	1.86616e6i	−0.146249	+	0.665719i
$142$	46372.5			0.0161955
$143$		−	2.97791e6i		−	1.01837i
$144$	−3.28069e6	−	1.51453e6i	−1.09870	−	0.507214i
$145$	−882262.			−0.289397
$146$			637235.i			0.204758i
$147$	6.01081e6	+	1.32049e6i	1.89226	+	0.415702i
$148$	812771.			0.250717
$149$		−	217205.i		−	0.0656614i	−0.999461	−	0.0328307i	$-0.989548\pi$
$149$		−	217205.i		−	0.0656614i	0.999461	−	0.0328307i	$-0.0104522\pi$
$150$	−1.17984e6	+	5.37060e6i	−0.349583	+	1.59129i
$151$	258466.			0.0750710			0.0375355	−	0.999295i	$-0.488049\pi$
$151$	258466.			0.0750710			0.0375355	+	0.999295i	$0.488049\pi$
$152$			3.13537e6i			0.892808i
$153$	227179.	−	492101.i	0.0634299	−	0.137398i
$154$	−7.00214e6			−1.91721
$155$			6.26592e6i			1.68264i
$156$	1.15598e6	+	253952.i	0.304493	+	0.0668927i
$157$	−4.16166e6			−1.07540			−0.537698	−	0.843138i	$-0.680706\pi$
$157$	−4.16166e6			−1.07540			−0.537698	+	0.843138i	$0.680706\pi$
$158$			5.08689e6i			1.28968i
$159$	−381545.	+	1.73678e6i	−0.0949193	+	0.432069i
$160$	−3.71551e6			−0.907107
$161$		−	1.08328e7i		−	2.59576i
$162$	3.68985e6	−	3.14465e6i	0.867887	−	0.739652i
$163$	571883.			0.132052			0.0660259	−	0.997818i	$-0.478968\pi$
$163$	571883.			0.132052			0.0660259	+	0.997818i	$0.478968\pi$
$164$			474928.i			0.107670i
$165$	−6.70770e6	−	1.47358e6i	−1.49321	−	0.328037i
$166$	2.47254e6			0.540528
$167$			8.88593e6i			1.90789i	0.299982	+	0.953945i	$0.403019\pi$
$167$			8.88593e6i			1.90789i	−0.299982	+	0.953945i	$0.596981\pi$
$168$	−1.39124e6	+	6.33288e6i	−0.293410	+	1.33559i
$169$	374815.			0.0776527
$170$		−	1.32128e6i		−	0.268936i
$171$	−5.08003e6	−	2.34520e6i	−1.01596	−	0.469020i
$172$	−2.26265e6			−0.444664
$173$		−	117606.i		−	0.0227139i	−0.999936	−	0.0113569i	$-0.996385\pi$
$173$		−	117606.i		−	0.0227139i	0.999936	−	0.0113569i	$-0.00361511\pi$
$174$	1.08953e6	+	239353.i	0.206819	+	0.0454351i
$175$	1.31236e7			2.44873
$176$		−	6.47191e6i		−	1.18712i
$177$	2.08725e6	−	9.50109e6i	0.376405	−	1.71338i
$178$	7.51021e6			1.33166
$179$		−	2.17791e6i		−	0.379735i	−0.981810	−	0.189867i	$-0.939194\pi$
$179$		−	2.17791e6i		−	0.379735i	0.981810	−	0.189867i	$-0.0608058\pi$
$180$	1.14405e6	−	2.47817e6i	0.196167	−	0.424926i
$181$	6.05935e6			1.02186			0.510929	−	0.859623i	$-0.329302\pi$
$181$	6.05935e6			1.02186			0.510929	+	0.859623i	$0.329302\pi$
$182$		−	1.22309e7i		−	2.02882i
$183$	8.21663e6	+	1.80507e6i	1.34073	+	0.294538i
$184$	7.52777e6			1.20841
$185$		−	8.23793e6i		−	1.30108i
$186$	1.69991e6	−	7.73794e6i	0.264172	−	1.20250i
$187$	970781.			0.148456
$188$		−	1.36011e6i		−	0.204691i
$189$	−9.22012e6	−	6.99102e6i	−1.36569	−	1.03551i
$190$	−1.36398e7			−1.98859
$191$		−	6.78023e6i		−	0.973070i	−0.873661	−	0.486535i	$-0.838261\pi$
$191$		−	6.78023e6i		−	0.973070i	0.873661	−	0.486535i	$-0.161739\pi$
$192$	−3.77727e6	−	829810.i	−0.533671	−	0.117240i
$193$	−2.47779e6			−0.344662			−0.172331	−	0.985039i	$-0.555130\pi$
$193$	−2.47779e6			−0.344662			−0.172331	+	0.985039i	$0.555130\pi$
$194$			1.55228e7i			2.12600i
$195$	2.57396e6	−	1.17166e7i	0.347135	−	1.58015i
$196$	−4.38083e6			−0.581820
$197$			6.80350e6i			0.889884i	0.895559	+	0.444942i	$0.146776\pi$
$197$			6.80350e6i			0.889884i	−0.895559	+	0.444942i	$0.853224\pi$
$198$	7.88373e6	+	3.63953e6i	1.01563	+	0.468867i
$199$	−48087.3			−0.00610199			−0.00305099	−	0.999995i	$-0.500971\pi$
$199$	−48087.3			−0.00610199			−0.00305099	+	0.999995i	$0.500971\pi$
$200$			9.11965e6i			1.13996i
$201$	366415.	+	80495.9i	0.0451216	+	0.00991256i
$202$	8.72334e6			1.05835
$203$		−	2.66238e6i		−	0.318259i
$204$	−82786.9	+	376843.i	−0.00975150	+	0.0443885i
$205$	4.81368e6			0.558748
$206$			4.01451e6i			0.459231i
$207$	−5.63063e6	+	1.21967e7i	−0.634813	+	1.37509i
$208$	1.13047e7			1.25623
$209$		−	1.00215e7i		−	1.09773i
$210$	−2.75499e7	−	6.05231e6i	−2.97483	−	0.653526i
$211$	−9.79326e6			−1.04251			−0.521254	−	0.853401i	$-0.674536\pi$
$211$	−9.79326e6			−1.04251			−0.521254	+	0.853401i	$0.674536\pi$
$212$		−	1.26581e6i		−	0.132850i
$213$	−29449.4	+	134053.i	−0.00304746	+	0.0138719i
$214$	−2.57941e6			−0.263196
$215$			2.29333e7i			2.30755i
$216$	4.85807e6	−	6.40708e6i	0.482062	−	0.635768i
$217$	−1.89085e7			−1.85045
$218$		−	1.26177e7i		−	1.21790i
$219$	−1.84211e6	−	404684.i	−0.175381	−	0.0385287i
$220$	4.88875e6			0.459124
$221$			1.69570e6i			0.157098i
$222$	−2.23491e6	+	1.01732e7i	−0.204268	+	0.929821i
$223$	−1.70498e7			−1.53746			−0.768729	−	0.639574i	$-0.779111\pi$
$223$	−1.70498e7			−1.53746			−0.768729	+	0.639574i	$0.779111\pi$
$224$		−	1.12122e7i		−	0.997576i
$225$	−1.47759e7	−	6.82133e6i	−1.29720	−	0.598855i
$226$	1.44534e7			1.25211
$227$		−	1.75406e7i		−	1.49957i	−0.661682	−	0.749785i	$-0.730157\pi$
$227$		−	1.75406e7i		−	1.49957i	0.661682	−	0.749785i	$-0.269843\pi$
$228$	3.89020e6	+	854621.i	0.328222	+	0.0721056i
$229$	2.32066e7			1.93244			0.966220	−	0.257719i	$-0.0829709\pi$
$229$	2.32066e7			1.93244			0.966220	+	0.257719i	$0.0829709\pi$
$230$			3.27480e7i			2.69154i
$231$	4.44679e6	−	2.02416e7i	0.360754	−	1.64214i
$232$	1.85009e6			0.148159
$233$			1.58997e6i			0.125696i	0.998023	+	0.0628480i	$0.0200183\pi$
$233$			1.58997e6i			0.125696i	−0.998023	+	0.0628480i	$0.979982\pi$
$234$	−6.35729e6	+	1.37708e7i	−0.496163	+	1.07476i
$235$	−1.37855e7			−1.06223
$236$			6.92464e6i			0.526819i
$237$	−1.47051e7	−	3.23049e6i	−1.10465	−	0.242674i
$238$	3.98719e6			0.295758
$239$		−	1.25468e7i		−	0.919054i	−0.888164	−	0.459527i	$-0.848019\pi$
$239$		−	1.25468e7i		−	0.919054i	0.888164	−	0.459527i	$-0.151981\pi$
$240$	5.59400e6	−	2.54637e7i	0.404659	−	1.84199i
$241$	1.79779e7			1.28436			0.642181	−	0.766553i	$-0.278030\pi$
$241$	1.79779e7			1.28436			0.642181	+	0.766553i	$0.278030\pi$
$242$		−	608615.i		−	0.0429434i
$243$	6.74721e6	+	1.26636e7i	0.470225	+	0.882547i
$244$	−5.98850e6			−0.412238
$245$			4.44024e7i			3.01931i
$246$	−5.94453e6	−	1.30593e6i	−0.399312	−	0.0877230i
$247$	1.75049e7			1.16163
$248$		−	1.31396e7i		−	0.861441i
$249$	−1.57021e6	+	7.14756e6i	−0.101709	+	0.462977i
$250$	−1.19056e7			−0.761961
$251$			5.20275e6i			0.329012i	0.986376	+	0.164506i	$0.0526030\pi$
$251$			5.20275e6i			0.329012i	−0.986376	+	0.164506i	$0.947397\pi$
$252$	7.47830e6	+	3.45236e6i	0.467306	+	0.215732i
$253$	−2.40608e7			−1.48576
$254$		−	3.59926e6i		−	0.219640i
$255$	3.81953e6	+	839096.i	0.230351	+	0.0506047i
$256$	1.38885e7			0.827817
$257$			1.82042e7i			1.07244i	0.844080	+	0.536218i	$0.180148\pi$
$257$			1.82042e7i			1.07244i	−0.844080	+	0.536218i	$0.819852\pi$
$258$	6.22169e6	−	2.83209e7i	0.362284	−	1.64910i
$259$	2.48594e7			1.43084
$260$			8.53935e6i			0.485853i
$261$	−1.38383e6	+	2.99758e6i	−0.0778328	+	0.168597i
$262$	−1.51296e7			−0.841245
$263$			1.58130e7i			0.869256i	0.900610	+	0.434628i	$0.143120\pi$
$263$			1.58130e7i			0.869256i	−0.900610	+	0.434628i	$0.856880\pi$
$264$	1.40660e7	+	3.09009e6i	0.764465	+	0.167942i
$265$	−1.28298e7			−0.689415
$266$		−	4.11603e7i		−	2.18693i
$267$	−4.76945e6	+	2.17104e7i	−0.250573	+	1.14060i
$268$	−267052.			−0.0138737
$269$			1.84429e7i			0.947487i	0.880663	+	0.473743i	$0.157098\pi$
$269$			1.84429e7i			0.947487i	−0.880663	+	0.473743i	$0.842902\pi$
$270$	2.78727e7	+	2.11340e7i	1.41608	+	1.07372i
$271$	−1.33168e7			−0.669103			−0.334552	−	0.942377i	$-0.608585\pi$
$271$	−1.33168e7			−0.669103			−0.334552	+	0.942377i	$0.608585\pi$
$272$			3.68527e6i			0.183131i
$273$	3.53568e7	+	7.76737e6i	1.73774	+	0.381756i
$274$	−2.44043e7			−1.18636
$275$		−	2.91489e7i		−	1.40160i
$276$	2.05188e6	−	9.34006e6i	0.0975941	−	0.444245i
$277$	2.28018e6			0.107283			0.0536415	−	0.998560i	$-0.482917\pi$
$277$	2.28018e6			0.107283			0.0536415	+	0.998560i	$0.482917\pi$
$278$		−	1.80893e7i		−	0.841951i
$279$	2.12891e7	+	9.82814e6i	0.980269	+	0.452542i
$280$	−4.67816e7			−2.13109
$281$			3.69455e7i			1.66511i	0.553943	+	0.832554i	$0.313122\pi$
$281$			3.69455e7i			1.66511i	−0.553943	+	0.832554i	$0.686878\pi$
$282$	1.70240e7	+	3.73993e6i	0.759128	+	0.166769i
$283$	1.74974e7			0.771995			0.385997	−	0.922500i	$-0.373857\pi$
$283$	1.74974e7			0.771995			0.385997	+	0.922500i	$0.373857\pi$
$284$		−	97701.1i		−	0.00426525i
$285$	8.66210e6	−	3.94296e7i	0.374187	−	1.70329i
$286$	−2.71660e7			−1.16126
$287$			1.45261e7i			0.614474i
$288$	−5.82780e6	+	1.26238e7i	−0.243965	+	0.528462i
$289$	2.35848e7			0.977098
$290$			8.04843e6i			0.330003i
$291$	−4.48729e7	−	9.85792e6i	−1.82098	−	0.400043i
$292$	1.34258e6			0.0539250
$293$			1.85147e7i			0.736059i	0.929814	+	0.368030i	$0.119968\pi$
$293$			1.85147e7i			0.736059i	−0.929814	+	0.368030i	$0.880032\pi$
$294$	1.20461e7	−	5.48336e7i	0.474030	−	2.15777i
$295$	7.01855e7			2.73389
$296$			1.72748e7i			0.666098i
$297$	−1.55277e7	+	2.04788e7i	−0.592705	+	0.781691i
$298$	−1.98145e6			−0.0748745
$299$		−	4.20279e7i		−	1.57226i
$300$	1.13152e7	+	2.48578e6i	0.419081	+	0.0920660i
$301$	−6.92053e7			−2.53770
$302$		−	2.35785e6i		−	0.0856044i
$303$	−5.53986e6	+	2.52173e7i	−0.199146	+	0.906505i
$304$	3.80435e7			1.35413
$305$			6.06970e7i			2.13928i
$306$	−4.48919e6	−	2.07244e6i	−0.156677	−	0.0723298i
$307$	−3.72431e7			−1.28715			−0.643577	−	0.765381i	$-0.722550\pi$
$307$	−3.72431e7			−1.28715			−0.643577	+	0.765381i	$0.722550\pi$
$308$			1.47526e7i			0.504914i
$309$	−1.16051e7	−	2.54947e6i	−0.393344	−	0.0864119i
$310$	5.71609e7			1.91873
$311$			8.61024e6i			0.286242i	0.989705	+	0.143121i	$0.0457139\pi$
$311$			8.61024e6i			0.286242i	−0.989705	+	0.143121i	$0.954286\pi$
$312$	−5.39756e6	+	2.45695e7i	−0.177719	+	0.808971i
$313$	3.91504e7			1.27674			0.638371	−	0.769729i	$-0.279609\pi$
$313$	3.91504e7			1.27674			0.638371	+	0.769729i	$0.279609\pi$
$314$			3.79648e7i			1.22629i
$315$	3.49918e7	−	7.57971e7i	1.11953	−	2.42505i
$316$	1.07175e7			0.339649
$317$		−	1.97849e7i		−	0.621091i	−0.950559	−	0.310545i	$-0.899488\pi$
$317$		−	1.97849e7i		−	0.621091i	0.950559	−	0.310545i	$-0.100512\pi$
$318$	1.58438e7	+	3.48065e6i	0.492694	+	0.108238i
$319$	−5.91340e6			−0.182165
$320$		−	2.79030e7i		−	0.851533i
$321$	1.63809e6	−	7.45652e6i	0.0495248	−	0.225435i
$322$	−9.88226e7			−2.95998
$323$			5.70650e6i			0.169341i
$324$	−6.62539e6	−	7.77405e6i	−0.194794	−	0.228566i
$325$	5.09154e7			1.48320
$326$		−	5.21700e6i		−	0.150580i
$327$	3.64750e7	+	8.01303e6i	1.04316	+	0.229168i
$328$	−1.00942e7			−0.286056
$329$		−	4.16001e7i		−	1.16817i
$330$	−1.34428e7	+	6.11910e7i	−0.374065	+	1.70273i
$331$	−7.15608e7			−1.97329			−0.986645	−	0.162884i	$-0.947920\pi$
$331$	−7.15608e7			−1.97329			−0.986645	+	0.162884i	$0.947920\pi$
$332$		−	5.20933e6i		−	0.142353i
$333$	−2.79892e7	−	1.29212e7i	−0.757981	−	0.349922i
$334$	8.10618e7			2.17559
$335$			2.70674e6i			0.0719966i
$336$	7.68411e7	+	1.68809e7i	2.02570	+	0.445017i
$337$	−1.69062e7			−0.441729			−0.220864	−	0.975305i	$-0.570888\pi$
$337$	−1.69062e7			−0.441729			−0.220864	+	0.975305i	$0.570888\pi$
$338$		−	3.41925e6i		−	0.0885483i
$339$	−9.17879e6	+	4.17815e7i	−0.235606	+	1.07247i
$340$	−2.78378e6			−0.0708268
$341$			4.19976e7i			1.05916i
$342$	−2.13941e7	+	4.63425e7i	−0.534829	+	1.15851i
$343$	−6.48306e7			−1.60656
$344$		−	4.80909e7i		−	1.18137i
$345$	−9.46672e7	−	2.07970e7i	−2.30538	−	0.506458i
$346$	−1.07286e6			−0.0259009
$347$		−	8.34376e6i		−	0.199698i	−0.995003	−	0.0998489i	$-0.968164\pi$
$347$		−	8.34376e6i		−	0.199698i	0.995003	−	0.0998489i	$-0.0318359\pi$
$348$	504287.	−	2.29550e6i	0.0119658	−	0.0544677i
$349$	2.28039e7			0.536455			0.268227	−	0.963356i	$-0.413562\pi$
$349$	2.28039e7			0.536455			0.268227	+	0.963356i	$0.413562\pi$
$350$		−	1.19720e8i		−	2.79231i
$351$	−3.57710e7	−	2.71228e7i	−0.827199	−	0.627211i
$352$	−2.49034e7			−0.570992
$353$			1.71293e7i			0.389418i	0.980861	+	0.194709i	$0.0623763\pi$
$353$			1.71293e7i			0.389418i	−0.980861	+	0.194709i	$0.937624\pi$
$354$	−8.66737e7	−	1.90409e7i	−1.95379	−	0.429219i
$355$	−990260.			−0.0221342
$356$		−	1.58231e7i		−	0.350704i
$357$	−2.53212e6	+	1.15261e7i	−0.0556518	+	0.253325i
$358$	−1.98680e7			−0.433016
$359$			5.65246e6i			0.122167i	0.998133	+	0.0610835i	$0.0194556\pi$
$359$			5.65246e6i			0.122167i	−0.998133	+	0.0610835i	$0.980544\pi$
$360$	5.26715e7	+	2.43159e7i	1.12893	+	0.521173i
$361$	1.18631e7			0.252160
$362$		−	5.52764e7i		−	1.16524i
$363$	1.75937e6	+	386508.i	0.0367822	+	0.00808051i
$364$	−2.57689e7			−0.534309
$365$		−	1.36078e7i		−	0.279840i
$366$	1.64668e7	−	7.49562e7i	0.335866	−	1.52885i
$367$	1.86788e7			0.377878			0.188939	−	0.981989i	$-0.439495\pi$
$367$	1.86788e7			0.377878			0.188939	+	0.981989i	$0.439495\pi$
$368$		−	9.13394e7i		−	1.83280i
$369$	7.55030e6	−	1.63550e7i	0.150274	−	0.325515i
$370$	−7.51505e7			−1.48363
$371$		−	3.87160e7i		−	0.758174i
$372$	−1.63029e7	−	3.58150e6i	−0.316691	−	0.0695723i
$373$	2.91963e7			0.562602			0.281301	−	0.959620i	$-0.409234\pi$
$373$	2.91963e7			0.562602			0.281301	+	0.959620i	$0.409234\pi$
$374$		−	8.85595e6i		−	0.169286i
$375$	7.56083e6	−	3.44166e7i	0.143376	−	0.652641i
$376$	2.89080e7			0.543819
$377$		−	1.03291e7i		−	0.192770i
$378$	−6.37755e7	+	8.41105e7i	−1.18081	+	1.55731i
$379$	7.99094e6			0.146785			0.0733923	−	0.997303i	$-0.476617\pi$
$379$	7.99094e6			0.146785			0.0733923	+	0.997303i	$0.476617\pi$
$380$			2.87373e7i			0.523715i
$381$	1.04047e7	+	2.28575e6i	0.188128	+	0.0413290i
$382$	−6.18526e7			−1.10960
$383$			6.05876e7i			1.07842i	0.842172	+	0.539210i	$0.181277\pi$
$383$			6.05876e7i			1.07842i	−0.842172	+	0.539210i	$0.818723\pi$
$384$	−1.46417e7	+	6.66484e7i	−0.258581	+	1.17705i
$385$	1.49527e8			2.62022
$386$			2.26037e7i			0.393022i
$387$	7.79184e7	+	3.59711e7i	1.34433	+	0.620613i
$388$	3.27045e7			0.559903
$389$			9.58451e7i			1.62825i	0.580690	+	0.814125i	$0.302783\pi$
$389$			9.58451e7i			1.62825i	−0.580690	+	0.814125i	$0.697217\pi$
$390$	−1.06884e8	−	2.34810e7i	−1.80186	−	0.395842i
$391$	1.37008e7			0.229201
$392$		−	9.31112e7i		−	1.54577i
$393$	9.60821e6	−	4.37362e7i	0.158294	−	0.720549i
$394$	6.20649e7			1.01475
$395$		−	1.08628e8i		−	1.76259i
$396$	7.66804e6	−	1.66100e7i	0.123481	−	0.267476i
$397$	3.13807e7			0.501523			0.250762	−	0.968049i	$-0.419319\pi$
$397$	3.13807e7			0.501523			0.250762	+	0.968049i	$0.419319\pi$
$398$			438676.i			0.00695817i
$399$	1.18986e8	+	2.61394e7i	1.87316	+	0.411506i
$400$	1.10655e8			1.72898
$401$			1.63069e7i			0.252894i	0.991973	+	0.126447i	$0.0403574\pi$
$401$			1.63069e7i			0.252894i	−0.991973	+	0.126447i	$0.959643\pi$
$402$	734324.	−	3.34262e6i	0.0113034	−	0.0514527i
$403$	−7.33588e7			−1.12082
$404$		−	1.83790e7i		−	0.278726i
$405$	−7.87947e7	+	6.71523e7i	−1.18613	+	1.01087i
$406$	−2.42875e7			−0.362915
$407$		−	5.52151e7i		−	0.818983i
$408$	−8.00951e6	−	1.75957e6i	−0.117930	−	0.0259076i
$409$	8.03034e7			1.17372			0.586860	−	0.809689i	$-0.300364\pi$
$409$	8.03034e7			1.17372			0.586860	+	0.809689i	$0.300364\pi$
$410$		−	4.39128e7i		−	0.637147i
$411$	1.54983e7	−	7.05475e7i	0.223233	−	1.01615i
$412$	8.45808e6			0.120943
$413$			2.11797e8i			3.00655i
$414$	1.11265e8	+	5.13654e7i	1.56804	+	0.723886i
$415$	−5.27997e7			−0.738732
$416$		−	4.34996e7i		−	0.604234i
$417$	5.22921e7	+	1.14878e7i	0.721154	+	0.158427i
$418$	−9.14212e7			−1.25175
$419$		−	9.24892e7i		−	1.25733i	−0.777676	−	0.628665i	$-0.783602\pi$
$419$		−	9.24892e7i		−	1.25733i	0.777676	−	0.628665i	$-0.216398\pi$
$420$	−1.27515e7	+	5.80442e7i	−0.172112	+	0.783449i
$421$	−8.38644e6			−0.112391			−0.0561955	−	0.998420i	$-0.517897\pi$
$421$	−8.38644e6			−0.112391			−0.0561955	+	0.998420i	$0.517897\pi$
$422$			8.93390e7i			1.18879i
$423$	−2.16226e7	+	4.68376e7i	−0.285685	+	0.618834i
$424$	2.69038e7			0.352953
$425$			1.65981e7i			0.216218i
$426$	1.22290e6	+	268652.i	0.0158183	+	0.00347506i
$427$	−1.83164e8			−2.35264
$428$			5.43451e6i			0.0693153i
$429$	1.72521e7	−	7.85309e7i	0.218509	−	0.994647i
$430$	2.09209e8			2.63133
$431$		−	1.33165e8i		−	1.66326i	−0.555333	−	0.831628i	$-0.687409\pi$
$431$		−	1.33165e8i		−	1.66326i	0.555333	−	0.831628i	$-0.312591\pi$
$432$	−7.77414e7	−	5.89462e7i	−0.964275	−	0.731147i
$433$	1.34574e8			1.65766			0.828831	−	0.559499i	$-0.189006\pi$
$433$	1.34574e8			1.65766			0.828831	+	0.559499i	$0.189006\pi$
$434$			1.72493e8i			2.11009i
$435$	−2.32663e7	−	5.11126e6i	−0.282656	−	0.0620955i
$436$	−2.65839e7			−0.320745
$437$		−	1.41436e8i		−	1.69478i
$438$	−3.69173e6	+	1.68046e7i	−0.0439347	+	0.199989i
$439$	−1.16634e8			−1.37858			−0.689289	−	0.724486i	$-0.742077\pi$
$439$	−1.16634e8			−1.37858			−0.689289	+	0.724486i	$0.742077\pi$
$440$			1.03907e8i			1.21979i
$441$	1.50862e8	+	6.96455e7i	1.75899	+	0.812039i
$442$	1.54690e7			0.179141
$443$			9.23728e6i			0.106251i	0.998588	+	0.0531255i	$0.0169183\pi$
$443$			9.23728e6i			0.106251i	−0.998588	+	0.0531255i	$0.983082\pi$
$444$	2.14337e7	+	4.70867e6i	0.244877	+	0.0537959i
$445$	−1.60376e8			−1.81995
$446$			1.55536e8i			1.75318i
$447$	1.25834e6	−	5.72793e6i	0.0140889	−	0.0641321i
$448$	8.42021e7			0.936460
$449$		−	1.37104e8i		−	1.51465i	−0.653038	−	0.757325i	$-0.726506\pi$
$449$		−	1.37104e8i		−	1.51465i	0.653038	−	0.757325i	$-0.273494\pi$
$450$	−6.22276e7	+	1.34794e8i	−0.682881	+	1.47922i
$451$	3.22639e7			0.351713
$452$		−	3.04514e7i		−	0.329756i
$453$	6.81604e6	+	1.49738e6i	0.0733225	+	0.0161079i
$454$	−1.60014e8			−1.70998
$455$			2.61184e8i			2.77276i
$456$	−1.81643e7	+	8.26833e7i	−0.191569	+	0.872014i
$457$	2.70765e7			0.283689			0.141845	−	0.989889i	$-0.454697\pi$
$457$	2.70765e7			0.283689			0.141845	+	0.989889i	$0.454697\pi$
$458$		−	2.11703e8i		−	2.20358i
$459$	8.84188e6	−	1.16611e7i	0.0914338	−	0.120588i
$460$	6.89959e7			0.708843
$461$			3.75813e7i			0.383591i	0.981435	+	0.191796i	$0.0614310\pi$
$461$			3.75813e7i			0.383591i	−0.981435	+	0.191796i	$0.938569\pi$
$462$	−1.84654e8	−	4.05659e7i	−1.87255	−	0.411372i
$463$	1.36458e8			1.37485			0.687425	−	0.726255i	$-0.258741\pi$
$463$	1.36458e8			1.37485			0.687425	+	0.726255i	$0.258741\pi$
$464$		−	2.24484e7i		−	0.224715i
$465$	−3.63007e7	+	1.65240e8i	−0.361041	+	1.64344i
$466$	1.45045e7			0.143333
$467$			1.22105e8i			1.19890i	0.800411	+	0.599452i	$0.204615\pi$
$467$			1.22105e8i			1.19890i	−0.800411	+	0.599452i	$0.795385\pi$
$468$	2.90133e7	+	1.33940e7i	0.283048	+	0.130669i
$469$	−8.16805e6			−0.0791771
$470$			1.25758e8i			1.21127i
$471$	−1.09748e8	−	2.41100e7i	−1.05035	−	0.230746i
$472$	−1.47178e8			−1.39964
$473$			1.53712e8i			1.45253i
$474$	−2.94702e7	+	1.34147e8i	−0.276725	+	1.25964i
$475$	1.71345e8			1.59878
$476$		−	8.40052e6i		−	0.0778907i
$477$	−2.01236e7	+	4.35904e7i	−0.185417	+	0.401639i
$478$	−1.14459e8			−1.04801
$479$			8.36984e7i			0.761571i	0.924663	+	0.380786i	$0.124346\pi$
$479$			8.36984e7i			0.761571i	−0.924663	+	0.380786i	$0.875654\pi$
$480$	−9.79822e7	−	2.15253e7i	−0.885979	−	0.194637i
$481$	9.64461e7			0.866662
$482$		−	1.64003e8i		−	1.46457i
$483$	6.27585e7	−	2.85675e8i	0.556969	−	2.53530i
$484$	−1.28228e6			−0.0113095
$485$		−	3.31480e8i		−	2.90558i
$486$	1.15523e8	−	6.15514e7i	1.00638	−	0.536203i
$487$	−2.22785e8			−1.92886			−0.964428	−	0.264347i	$-0.914844\pi$
$487$	−2.22785e8			−1.92886			−0.964428	+	0.264347i	$0.914844\pi$
$488$		−	1.27281e8i		−	1.09523i
$489$	1.50812e7	+	3.31312e6i	0.128976	+	0.0283342i
$490$	4.05061e8			3.44296
$491$		−	6.94921e7i		−	0.587071i	−0.955948	−	0.293536i	$-0.905168\pi$
$491$		−	6.94921e7i		−	0.587071i	0.955948	−	0.293536i	$-0.0948320\pi$
$492$	−2.75143e6	+	1.25244e7i	−0.0231027	+	0.105163i
$493$	3.36724e6			0.0281017
$494$		−	1.59689e8i		−	1.32463i
$495$	−1.68353e8	−	7.77202e7i	−1.38805	−	0.640794i
$496$	−1.59431e8			−1.30655
$497$		−	2.98828e6i		−	0.0243418i
$498$	6.52036e7	+	1.43243e7i	0.527939	+	0.115980i
$499$	4.82519e7			0.388341			0.194170	−	0.980968i	$-0.437799\pi$
$499$	4.82519e7			0.388341			0.194170	+	0.980968i	$0.437799\pi$
$500$			2.50837e7i			0.200670i
$501$	−5.14793e7	+	2.34332e8i	−0.409373	+	1.86345i
$502$	4.74621e7			0.375177
$503$		−	1.46399e8i		−	1.15036i	−0.818027	−	0.575180i	$-0.804932\pi$
$503$		−	1.46399e8i		−	1.15036i	0.818027	−	0.575180i	$-0.195068\pi$
$504$	−7.33773e7	+	1.58945e8i	−0.573152	+	1.24153i
$505$	−1.86282e8			−1.44643
$506$			2.19495e8i			1.69423i
$507$	9.88429e6	+	2.17143e6i	0.0758440	+	0.0166618i
$508$	−7.58319e6			−0.0578443
$509$			2.22925e8i			1.69047i	0.534398	+	0.845233i	$0.320538\pi$
$509$			2.22925e8i			1.69047i	−0.534398	+	0.845233i	$0.679462\pi$
$510$	7.65465e6	−	3.48437e7i	0.0577052	−	0.262672i
$511$	4.10639e7			0.307750
$512$			3.50512e7i			0.261151i
$513$	−1.20380e8	−	9.12760e7i	−0.891663	−	0.676090i
$514$	1.66067e8			1.22291
$515$		−	8.57278e7i		−	0.627625i
$516$	−5.96687e7	−	1.31083e7i	−0.434307	−	0.0954110i
$517$	−9.23979e7			−0.668637
$518$		−	2.26779e8i		−	1.63160i
$519$	681334.	−	3.10141e6i	0.00487369	−	0.0221849i
$520$	−1.81497e8			−1.29080
$521$			8.71251e7i			0.616070i	0.951375	+	0.308035i	$0.0996714\pi$
$521$			8.71251e7i			0.616070i	−0.951375	+	0.308035i	$0.900329\pi$
$522$	2.73454e7	+	1.26240e7i	0.192253	+	0.0887537i
$523$	−6.65644e7			−0.465304			−0.232652	−	0.972560i	$-0.574740\pi$
$523$	−6.65644e7			−0.465304			−0.232652	+	0.972560i	$0.574740\pi$
$524$			3.18761e7i			0.221550i
$525$	3.46085e8	+	7.60299e7i	2.39169	+	0.525420i
$526$	1.44254e8			0.991223
$527$		−	2.39145e7i		−	0.163392i
$528$	3.74941e7	−	1.70672e8i	0.254719	−	1.15947i
$529$	−1.91540e8			−1.29387
$530$			1.17039e8i			0.786149i
$531$	1.10086e8	−	2.38462e8i	0.735275	−	1.59271i
$532$	−8.67197e7			−0.575948
$533$			5.63566e7i			0.372188i
$534$	1.98053e8	+	4.35093e7i	1.30064	+	0.285732i
$535$	5.50820e7			0.359707
$536$		−	5.67599e6i		−	0.0368593i
$537$	1.26174e7	−	5.74339e7i	0.0814792	−	0.370891i
$538$	1.68246e8			1.08043
$539$			2.97609e8i			1.90055i
$540$	4.45268e7	−	5.87242e7i	0.282774	−	0.372937i
$541$	−9.33589e7			−0.589609			−0.294804	−	0.955558i	$-0.595254\pi$
$541$	−9.33589e7			−0.589609			−0.294804	+	0.955558i	$0.595254\pi$
$542$			1.21483e8i			0.762986i
$543$	1.59792e8	+	3.51040e7i	0.998058	+	0.219259i
$544$	1.41806e7			0.0880842
$545$			2.69444e8i			1.66448i
$546$	7.08578e7	−	3.22542e8i	0.435321	−	1.98157i
$547$	−2.14928e8			−1.31320			−0.656599	−	0.754240i	$-0.728006\pi$
$547$	−2.14928e8			−1.31320			−0.656599	+	0.754240i	$0.728006\pi$
$548$			5.14169e7i			0.312438i
$549$	2.06225e8	+	9.52037e7i	1.24630	+	0.575356i
$550$	−2.65911e8			−1.59826
$551$		−	3.47605e7i		−	0.207793i
$552$	1.98516e8	+	4.36110e7i	1.18026	+	0.259286i
$553$	3.27803e8			1.93838
$554$		−	2.08010e7i		−	0.122336i
$555$	4.77252e7	−	2.17244e8i	0.279170	−	1.27077i
$556$	−3.81119e7			−0.221736
$557$		−	2.58422e8i		−	1.49542i	−0.664023	−	0.747712i	$-0.731152\pi$
$557$		−	2.58422e8i		−	1.49542i	0.664023	−	0.747712i	$-0.268848\pi$
$558$	8.96572e7	−	1.94210e8i	0.516039	−	1.11781i
$559$	−2.68494e8			−1.53709
$560$			5.67632e8i			3.23224i
$561$	2.56006e7	+	5.62408e6i	0.144998	+	0.0318539i
$562$	3.37035e8			1.89874
$563$			2.69601e8i			1.51076i	0.655287	+	0.755380i	$0.272548\pi$
$563$			2.69601e8i			1.51076i	−0.655287	+	0.755380i	$0.727452\pi$
$564$	7.87957e6	−	3.58675e7i	0.0439203	−	0.199924i
$565$	−3.08644e8			−1.71125
$566$		−	1.59620e8i		−	0.880315i
$567$	−2.02644e8	−	2.37776e8i	−1.11169	−	1.30443i
$568$	2.07656e6			0.0113318
$569$		−	2.64432e8i		−	1.43541i	−0.696346	−	0.717706i	$-0.745192\pi$
$569$		−	2.64432e8i		−	1.43541i	0.696346	−	0.717706i	$-0.254808\pi$
$570$	−3.59696e8	−	7.90200e7i	−1.94228	−	0.426690i
$571$	−6.34371e7			−0.340749			−0.170375	−	0.985379i	$-0.554498\pi$
$571$	−6.34371e7			−0.340749			−0.170375	+	0.985379i	$0.554498\pi$
$572$			5.72354e7i			0.305828i
$573$	3.92803e7	−	1.78802e8i	0.208790	−	0.950406i
$574$	1.32514e8			0.700693
$575$		−	4.11385e8i		−	2.16394i
$576$	−9.48034e7	−	4.37661e7i	−0.496086	−	0.229018i
$577$	7.18925e7			0.374245			0.187122	−	0.982337i	$-0.440084\pi$
$577$	7.18925e7			0.374245			0.187122	+	0.982337i	$0.440084\pi$
$578$		−	2.15152e8i		−	1.11420i
$579$	−6.53422e7	−	1.43547e7i	−0.336634	−	0.0739536i
$580$	1.69570e7			0.0869093
$581$		−	1.59332e8i		−	0.812409i
$582$	−8.99289e7	+	4.09353e8i	−0.456174	+	2.07649i
$583$	−8.59921e7			−0.433963
$584$			2.85354e7i			0.143267i
$585$	1.35757e8	−	2.94068e8i	0.678099	−	1.46886i
$586$	1.68900e8			0.839338
$587$		−	3.14372e8i		−	1.55428i	−0.629328	−	0.777140i	$-0.716670\pi$
$587$		−	3.14372e8i		−	1.55428i	0.629328	−	0.777140i	$-0.283330\pi$
$588$	−1.15528e8	−	2.53797e7i	−0.568269	−	0.124840i
$589$	−2.46873e8			−1.20817
$590$		−	6.40267e8i		−	3.11749i
$591$	−3.94151e7	+	1.79416e8i	−0.190941	+	0.869158i
$592$	2.09607e8			1.01028
$593$			2.34855e8i			1.12625i	0.826371	+	0.563126i	$0.190401\pi$
$593$			2.34855e8i			1.12625i	−0.826371	+	0.563126i	$0.809599\pi$
$594$	1.86818e8	+	1.41652e8i	0.891372	+	0.675869i
$595$	−8.51444e7			−0.404208
$596$			4.17467e6i			0.0197189i
$597$	−1.26812e6	−	278587.i	−0.00595986	−	0.00130930i
$598$	−3.83399e8			−1.79287
$599$			1.58036e8i			0.735317i	0.929961	+	0.367658i	$0.119840\pi$
$599$			1.58036e8i			0.735317i	−0.929961	+	0.367658i	$0.880160\pi$
$600$	−5.28334e7	+	2.40496e8i	−0.244599	+	1.11341i
$601$	1.63119e8			0.751419			0.375710	−	0.926737i	$-0.377399\pi$
$601$	1.63119e8			0.751419			0.375710	+	0.926737i	$0.377399\pi$
$602$			6.31325e8i			2.89377i
$603$	9.19643e6	+	4.24554e6i	0.0419438	+	0.0193634i
$604$	−4.96771e6			−0.0225447
$605$			1.29966e7i			0.0586901i
$606$	2.30044e8	+	5.05374e7i	1.03370	+	0.227088i
$607$	9.07470e7			0.405757			0.202879	−	0.979204i	$-0.434970\pi$
$607$	9.07470e7			0.405757			0.202879	+	0.979204i	$0.434970\pi$
$608$		−	1.46388e8i		−	0.651322i
$609$	1.54241e7	−	7.02099e7i	0.0682885	−	0.310847i
$610$	5.53709e8			2.43945
$611$		−	1.61395e8i		−	0.707564i
$612$	−4.36637e6	+	9.45817e6i	−0.0190488	+	0.0412622i
$613$	−1.95742e8			−0.849774			−0.424887	−	0.905246i	$-0.639686\pi$
$613$	−1.95742e8			−0.849774			−0.424887	+	0.905246i	$0.639686\pi$
$614$			3.39750e8i			1.46776i
$615$	1.26942e8	+	2.78874e7i	0.545734	+	0.119890i
$616$	−3.13556e8			−1.34145
$617$		−	769703.i		−	0.00327693i	−0.999999	−	0.00163847i	$-0.999478\pi$
$617$		−	769703.i		−	0.00327693i	0.999999	−	0.00163847i	$-0.000521541\pi$
$618$	−2.32575e7	+	1.05867e8i	−0.0985366	+	0.448535i
$619$	2.12497e8			0.895944			0.447972	−	0.894048i	$-0.352146\pi$
$619$	2.12497e8			0.895944			0.447972	+	0.894048i	$0.352146\pi$
$620$		−	1.20431e8i		−	0.505316i
$621$	−2.19146e8	+	2.89021e8i	−0.915080	+	1.20686i
$622$	7.85469e7			0.326406
$623$		−	4.83963e8i		−	2.00147i
$624$	2.98118e8	+	6.54922e7i	1.22697	+	0.269548i
$625$	−9.45802e7			−0.387401
$626$		−	3.57149e8i		−	1.45588i
$627$	5.80582e7	−	2.64279e8i	0.235538	−	1.07216i
$628$	7.99871e7			0.322954
$629$			3.14409e7i			0.126340i
$630$	−6.91459e8	−	3.19212e8i	−2.76531	−	1.27661i
$631$	8.58903e7			0.341866			0.170933	−	0.985283i	$-0.445322\pi$
$631$	8.58903e7			0.341866			0.170933	+	0.985283i	$0.445322\pi$
$632$			2.27791e8i			0.902373i
$633$	−2.58259e8	−	5.67358e7i	−1.01823	−	0.223690i
$634$	−1.80487e8			−0.708237
$635$			7.68602e7i			0.300179i
$636$	7.33329e6	−	3.33809e7i	0.0285054	−	0.129756i
$637$	−5.19844e8			−2.01120
$638$			5.39450e7i			0.207725i
$639$	−1.55323e6	+	3.36451e6i	−0.00595296	+	0.0128949i
$640$	−4.92338e8			−1.87812
$641$			2.69595e8i			1.02362i	0.859100	+	0.511808i	$0.171024\pi$
$641$			2.69595e8i			1.02362i	−0.859100	+	0.511808i	$0.828976\pi$
$642$	−6.80221e7	−	1.49435e7i	−0.257066	−	0.0564737i
$643$	2.00268e8			0.753318			0.376659	−	0.926352i	$-0.377073\pi$
$643$	2.00268e8			0.753318			0.376659	+	0.926352i	$0.377073\pi$
$644$			2.08207e8i			0.779539i
$645$	−1.32861e8	+	6.04778e8i	−0.495129	+	2.25381i
$646$	5.20575e7			0.193102
$647$		−	1.74016e8i		−	0.642503i	−0.946994	−	0.321251i	$-0.895897\pi$
$647$		−	1.74016e8i		−	0.642503i	0.946994	−	0.321251i	$-0.104103\pi$
$648$	1.65231e8	−	1.40817e8i	0.607250	−	0.517525i
$649$	4.70421e8			1.72089
$650$		−	4.64476e8i		−	1.69131i
$651$	−4.98639e8	−	1.09544e8i	−1.80735	−	0.397049i
$652$	−1.09916e7			−0.0396567
$653$			1.72162e7i			0.0618297i	0.999522	+	0.0309149i	$0.00984208\pi$
$653$			1.72162e7i			0.0618297i	−0.999522	+	0.0309149i	$0.990158\pi$
$654$	7.30989e7	−	3.32743e8i	0.261323	−	1.18953i
$655$	3.23084e8			1.14972
$656$			1.22480e8i			0.433864i
$657$	−4.62340e7	−	2.13440e7i	−0.163029	−	0.0752626i
$658$	−3.79497e8			−1.33208
$659$			4.49244e8i			1.56973i	0.619665	+	0.784867i	$0.287269\pi$
$659$			4.49244e8i			1.56973i	−0.619665	+	0.784867i	$0.712731\pi$
$660$	1.28922e8	+	2.83223e7i	0.448430	+	0.0985136i
$661$	5.18005e7			0.179362			0.0896808	−	0.995971i	$-0.471415\pi$
$661$	5.18005e7			0.179362			0.0896808	+	0.995971i	$0.471415\pi$
$662$			6.52813e8i			2.25017i
$663$	−9.82378e6	+	4.47175e7i	−0.0337084	+	0.153439i
$664$	1.10720e8			0.378201
$665$			8.78957e8i			2.98884i
$666$	−1.17874e8	+	2.55332e8i	−0.399021	+	0.864335i
$667$	−8.34571e7			−0.281246
$668$		−	1.70787e8i		−	0.572962i
$669$	−4.49621e8	−	9.87752e7i	−1.50165	−	0.329890i
$670$	2.46922e7			0.0820986
$671$			4.06825e8i			1.34660i
$672$	6.49562e7	−	2.95678e8i	0.214049	−	0.974342i
$673$	−8.49447e7			−0.278670			−0.139335	−	0.990245i	$-0.544497\pi$
$673$	−8.49447e7			−0.278670			−0.139335	+	0.990245i	$0.544497\pi$
$674$			1.54227e8i			0.503709i
$675$	−3.50140e8	−	2.65489e8i	−1.13849	−	0.863246i
$676$	−7.20393e6			−0.0233200
$677$			5.07006e8i			1.63398i	0.576651	+	0.816990i	$0.304359\pi$
$677$			5.07006e8i			1.63398i	−0.576651	+	0.816990i	$0.695641\pi$
$678$	3.81152e8	+	8.37335e7i	1.22295	+	0.268664i
$679$	1.00030e9			3.19537
$680$		−	5.91670e7i		−	0.188171i
$681$	1.01619e8	−	4.62565e8i	0.321761	−	1.46464i
$682$	3.83123e8			1.20777
$683$		−	3.23030e8i		−	1.01387i	−0.861985	−	0.506934i	$-0.830779\pi$
$683$		−	3.23030e8i		−	1.01387i	0.861985	−	0.506934i	$-0.169221\pi$
$684$	9.76380e7	+	4.50747e7i	0.305106	+	0.140852i
$685$	5.21141e8			1.62138
$686$			5.91417e8i			1.83198i
$687$	6.11986e8	+	1.34444e8i	1.88743	+	0.414641i
$688$	−5.83519e8			−1.79180
$689$		−	1.50205e8i		−	0.459227i
$690$	−1.89721e8	+	8.63601e8i	−0.577520	+	2.62885i
$691$	2.02104e8			0.612550			0.306275	−	0.951943i	$-0.400917\pi$
$691$	2.02104e8			0.612550			0.306275	+	0.951943i	$0.400917\pi$
$692$			2.26039e6i			0.00682126i
$693$	2.34534e8	−	5.08033e8i	0.704703	−	1.52649i
$694$	−7.61159e7			−0.227718
$695$			3.86287e8i			1.15068i
$696$	4.87890e7	+	1.07182e7i	0.144709	+	0.0317904i
$697$	−1.83719e7			−0.0542570
$698$		−	2.08029e8i		−	0.611726i
$699$	−9.21126e6	+	4.19293e7i	−0.0269704	+	0.122768i
$700$	−2.52236e8			−0.735382
$701$		−	1.07309e8i		−	0.311518i	−0.987795	−	0.155759i	$-0.950218\pi$
$701$		−	1.07309e8i		−	0.311518i	0.987795	−	0.155759i	$-0.0497824\pi$
$702$	−2.47428e8	+	3.26321e8i	−0.715217	+	0.943265i
$703$	3.24568e8			0.934201
$704$		−	1.87021e8i		−	0.536010i
$705$	−3.63539e8	−	7.98642e7i	−1.03749	−	0.227921i
$706$	1.56262e8			0.444058
$707$		−	5.62139e8i		−	1.59069i
$708$	−4.01169e7	+	1.82611e8i	−0.113039	+	0.514549i
$709$	−1.73929e8			−0.488015			−0.244007	−	0.969773i	$-0.578462\pi$
$709$	−1.73929e8			−0.488015			−0.244007	+	0.969773i	$0.578462\pi$
$710$			9.03365e6i			0.0252399i
$711$	−3.69075e8	−	1.70384e8i	−1.02685	−	0.474045i
$712$	3.36307e8			0.931743
$713$			5.92721e8i			1.63524i
$714$	1.05147e8	+	2.30992e7i	0.288869	+	0.0634604i
$715$	5.80115e8			1.58707
$716$			4.18593e7i			0.114039i
$717$	7.26883e7	−	3.30875e8i	0.197200	−	0.897648i
$718$	5.15645e7			0.139309
$719$		−	1.25042e8i		−	0.336410i	−0.985752	−	0.168205i	$-0.946203\pi$
$719$		−	1.25042e8i		−	0.336410i	0.985752	−	0.168205i	$-0.0537971\pi$
$720$	2.95040e8	−	6.39099e8i	0.790468	−	1.71226i
$721$	2.58698e8			0.690220
$722$		−	1.08221e8i		−	0.287541i
$723$	4.74097e8	+	1.04152e8i	1.25445	+	0.275584i
$724$	−1.16461e8			−0.306876
$725$		−	1.01106e8i		−	0.265315i
$726$	3.52592e6	−	1.60499e7i	0.00921431	−	0.0419432i
$727$	5.68882e8			1.48053			0.740267	−	0.672312i	$-0.234699\pi$
$727$	5.68882e8			1.48053			0.740267	+	0.672312i	$0.234699\pi$
$728$		−	5.47699e8i		−	1.41954i
$729$	1.04567e8	+	3.73042e8i	0.269906	+	0.962887i
$730$	−1.24137e8			−0.319105
$731$		−	8.75273e7i		−	0.224074i
$732$	−1.57923e8	−	3.46935e7i	−0.402637	−	0.0884534i
$733$	−6.01854e7			−0.152820			−0.0764098	−	0.997076i	$-0.524346\pi$
$733$	−6.01854e7			−0.152820			−0.0764098	+	0.997076i	$0.524346\pi$
$734$		−	1.70398e8i		−	0.430899i
$735$	−2.57239e8	+	1.17094e9i	−0.647850	+	2.94899i
$736$	−3.51466e8			−0.881557
$737$			1.81420e7i			0.0453194i
$738$	−1.49198e8	−	6.88776e7i	−0.371189	−	0.171360i
$739$	1.90784e8			0.472726			0.236363	−	0.971665i	$-0.424045\pi$
$739$	1.90784e8			0.472726			0.236363	+	0.971665i	$0.424045\pi$
$740$			1.58333e8i			0.390729i
$741$	4.61625e8	+	1.01412e8i	1.13458	+	0.249250i
$742$	−3.53187e8			−0.864555
$743$		−	6.26696e8i		−	1.52788i	−0.645285	−	0.763942i	$-0.723261\pi$
$743$		−	6.26696e8i		−	1.52788i	0.645285	−	0.763942i	$-0.276739\pi$
$744$	7.61221e7	−	3.46505e8i	0.184838	−	0.841377i
$745$	4.23128e7			0.102330
$746$		−	2.66343e8i		−	0.641541i
$747$	−8.28167e7	+	1.79392e8i	−0.198681	+	0.430370i
$748$	−1.86584e7			−0.0445830
$749$			1.66219e8i			0.395582i
$750$	−3.13965e8	−	6.89736e7i	−0.744214	−	0.163493i
$751$	−1.77754e8			−0.419662			−0.209831	−	0.977738i	$-0.567291\pi$
$751$	−1.77754e8			−0.419662			−0.209831	+	0.977738i	$0.567291\pi$
$752$		−	3.50760e8i		−	0.824814i
$753$	−3.01414e7	+	1.37203e8i	−0.0705957	+	0.321349i
$754$	−9.42276e7			−0.219818
$755$			5.03507e7i			0.116994i
$756$	1.77210e8	+	1.34367e8i	0.410132	+	0.310977i
$757$	−3.39594e8			−0.782839			−0.391420	−	0.920212i	$-0.628016\pi$
$757$	−3.39594e8			−0.782839			−0.391420	+	0.920212i	$0.628016\pi$
$758$		−	7.28974e7i		−	0.167380i
$759$	−6.34512e8	−	1.39393e8i	−1.45116	−	0.318798i
$760$	−6.10789e8			−1.39140
$761$		−	6.71681e8i		−	1.52408i	−0.647527	−	0.762042i	$-0.724197\pi$
$761$		−	6.71681e8i		−	1.52408i	0.647527	−	0.762042i	$-0.275803\pi$
$762$	2.08518e7	−	9.49165e7i	0.0471279	−	0.214525i
$763$	−8.13095e8			−1.83049
$764$			1.30316e8i			0.292225i
$765$	9.58643e7	+	4.42558e7i	0.214128	+	0.0988522i
$766$	5.52711e8			1.22973
$767$			8.21701e8i			1.82108i
$768$	3.66255e8	+	8.04608e7i	0.808536	+	0.177624i
$769$	−1.79932e8			−0.395667			−0.197834	−	0.980236i	$-0.563391\pi$
$769$	−1.79932e8			−0.395667			−0.197834	+	0.980236i	$0.563391\pi$
$770$		−	1.36406e9i		−	2.98787i
$771$	−1.05463e8	+	4.80065e8i	−0.230111	+	1.04746i
$772$	4.76231e7			0.103506
$773$			6.38103e6i			0.0138150i	0.999976	+	0.00690752i	$0.00219875\pi$
$773$			6.38103e6i			0.0138150i	−0.999976	+	0.00690752i	$0.997801\pi$
$774$	3.28146e8	−	7.10811e8i	0.707693	−	1.53296i
$775$	−7.18063e8			−1.54261
$776$			6.95110e8i			1.48754i
$777$	6.55570e8	+	1.44019e8i	1.39751	+	0.307013i
$778$	8.74346e8			1.85671
$779$			1.89656e8i			0.401193i
$780$	−4.94715e7	+	2.25192e8i	−0.104249	+	0.474537i
$781$	−6.63727e6			−0.0139327
$782$		−	1.24986e8i		−	0.261361i
$783$	−5.38593e7	+	7.10325e7i	−0.112196	+	0.147969i
$784$	−1.12978e9			−2.34448
$785$		−	8.10717e8i		−	1.67595i
$786$	−3.98984e8	−	8.76509e7i	−0.821651	−	0.180505i
$787$	−1.92659e8			−0.395243			−0.197621	−	0.980278i	$-0.563322\pi$
$787$	−1.92659e8			−0.395243			−0.197621	+	0.980278i	$0.563322\pi$
$788$		−	1.30763e8i		−	0.267243i
$789$	−9.16104e7	+	4.17007e8i	−0.186515	+	0.849010i
$790$	−9.90958e8			−2.00990
$791$		−	9.31386e8i		−	1.88192i
$792$	3.53034e8	+	1.62978e8i	0.710625	+	0.328061i
$793$	−7.10615e8			−1.42500
$794$		−	2.86270e8i		−	0.571893i
$795$	−3.38335e8	−	7.43273e7i	−0.673358	−	0.147927i
$796$	924237.			0.00183250
$797$			2.79199e8i			0.551491i	0.961231	+	0.275745i	$0.0889246\pi$
$797$			2.79199e8i			0.551491i	−0.961231	+	0.275745i	$0.911075\pi$
$798$	2.38456e8	−	1.08545e9i	0.469246	−	2.13599i
$799$	5.26137e7			0.103147
$800$		−	4.25790e8i		−	0.831622i
$801$	−2.51552e8	+	5.44896e8i	−0.489474	+	1.06027i
$802$	1.48760e8			0.288378
$803$		−	9.12071e7i		−	0.176150i
$804$	−7.04248e6	−	1.54713e6i	−0.0135506	−	0.00297686i
$805$	2.11030e9			4.04536
$806$			6.69215e8i			1.27809i
$807$	−1.06846e8	+	4.86361e8i	−0.203301	+	0.925419i
$808$	3.90631e8			0.740514
$809$			2.73189e8i			0.515963i	0.966150	+	0.257981i	$0.0830573\pi$
$809$			2.73189e8i			0.515963i	−0.966150	+	0.257981i	$0.916943\pi$
$810$	6.12597e8	+	7.18804e8i	1.15271	+	1.35256i
$811$	5.28457e8			0.990710			0.495355	−	0.868691i	$-0.335038\pi$
$811$	5.28457e8			0.990710			0.495355	+	0.868691i	$0.335038\pi$
$812$			5.11708e7i			0.0955772i
$813$	−3.51180e8	−	7.71491e7i	−0.653519	−	0.143569i
$814$	−5.03700e8			−0.933896
$815$			1.11406e8i			0.205796i
$816$	−2.13501e7	+	9.71847e7i	−0.0392942	+	0.178866i
$817$	−9.03557e8			−1.65687
$818$		−	7.32568e8i		−	1.33841i
$819$	8.87400e8	+	4.09669e8i	1.61535	+	0.745729i
$820$	−9.25189e7			−0.167799
$821$		−	9.56772e7i		−	0.172894i	−0.996256	−	0.0864468i	$-0.972449\pi$
$821$		−	9.56772e7i		−	0.172894i	0.996256	−	0.0864468i	$-0.0275513\pi$
$822$	−6.43570e8	−	1.41383e8i	−1.15872	−	0.254555i
$823$	−9.42714e6			−0.0169114			−0.00845572	−	0.999964i	$-0.502692\pi$
$823$	−9.42714e6			−0.0169114			−0.00845572	+	0.999964i	$0.502692\pi$
$824$			1.79770e8i			0.321318i
$825$	1.68870e8	−	7.68690e8i	0.300740	−	1.36896i
$826$	1.93211e9			3.42841
$827$			1.05436e8i			0.186412i	0.995647	+	0.0932059i	$0.0297115\pi$
$827$			1.05436e8i			0.186412i	−0.995647	+	0.0932059i	$0.970289\pi$
$828$	1.08221e8	−	2.34421e8i	0.190642	−	0.412957i
$829$	−1.62632e8			−0.285459			−0.142729	−	0.989762i	$-0.545588\pi$
$829$	−1.62632e8			−0.285459			−0.142729	+	0.989762i	$0.545588\pi$
$830$			4.81665e8i			0.842385i
$831$	6.01311e7	+	1.32099e7i	0.104784	+	0.0230196i
$832$	3.26677e8			0.567216
$833$		−	1.69466e8i		−	0.293189i
$834$	1.04798e8	−	4.77035e8i	0.180656	−	0.822341i
$835$	−1.73103e9			−2.97335
$836$			1.92613e8i			0.329661i
$837$	5.04481e8	+	3.82515e8i	0.860336	+	0.652337i
$838$	−8.43733e8			−1.43375
$839$			3.09555e8i			0.524146i	0.965048	+	0.262073i	$0.0844060\pi$
$839$			3.09555e8i			0.524146i	−0.965048	+	0.262073i	$0.915594\pi$
$840$	−1.23368e9	−	2.71022e8i	−2.08145	−	0.457264i
$841$	−2.05111e7			−0.0344828
$842$			7.65053e7i			0.128161i
$843$	−2.14038e8	+	9.74295e8i	−0.357280	+	1.62633i
$844$	1.88226e8			0.313078
$845$			7.30162e7i			0.121018i
$846$	4.27276e8	+	1.97252e8i	0.705664	+	0.325770i
$847$	−3.92196e7			−0.0645436
$848$		−	3.26442e8i		−	0.535326i
$849$	4.61427e8	+	1.01369e8i	0.754014	+	0.165646i
$850$	1.51416e8			0.246556
$851$		−	7.79262e8i		−	1.26443i
$852$	566017.	−	2.57649e6i	0.000915189	−	0.00416591i
$853$	4.61296e8			0.743247			0.371623	−	0.928384i	$-0.378801\pi$
$853$	4.61296e8			0.743247			0.371623	+	0.928384i	$0.378801\pi$
$854$			1.67091e9i			2.68275i
$855$	4.56859e8	−	9.89620e8i	0.730944	−	1.58333i
$856$	−1.15506e8			−0.184155
$857$		−	1.93865e8i		−	0.308005i	−0.988070	−	0.154002i	$-0.950784\pi$
$857$		−	1.93865e8i		−	0.308005i	0.988070	−	0.154002i	$-0.0492163\pi$
$858$	−7.16398e8	−	1.57382e8i	−1.13421	−	0.249169i
$859$	−8.49186e7			−0.133975			−0.0669874	−	0.997754i	$-0.521339\pi$
$859$	−8.49186e7			−0.133975			−0.0669874	+	0.997754i	$0.521339\pi$
$860$		−	4.40778e8i		−	0.692986i
$861$	−8.41549e7	+	3.83070e8i	−0.131847	+	0.600163i
$862$	−1.21480e9			−1.89663
$863$		−	1.05217e9i		−	1.63702i	−0.574492	−	0.818510i	$-0.694800\pi$
$863$		−	1.05217e9i		−	1.63702i	0.574492	−	0.818510i	$-0.305200\pi$
$864$	−2.26820e8	+	2.99142e8i	−0.351674	+	0.463806i
$865$	2.29104e7			0.0353984
$866$		−	1.22765e9i		−	1.89025i
$867$	6.21958e8	+	1.36635e8i	0.954341	+	0.209655i
$868$	3.63421e8			0.555713
$869$		−	7.28084e8i		−	1.10949i
$870$	−4.66274e7	+	2.12246e8i	−0.0708082	+	0.322316i
$871$	−3.16893e7			−0.0479578
$872$		−	5.65021e8i		−	0.852149i
$873$	−1.12624e9	−	5.19929e8i	−1.69273	−	0.781451i
$874$	−1.29025e9			−1.93258
$875$			7.67209e8i			1.14522i
$876$	3.54053e7	+	7.77802e6i	0.0526690	+	0.0115706i
$877$	−4.17613e8			−0.619120			−0.309560	−	0.950880i	$-0.600182\pi$
$877$	−4.17613e8			−0.619120			−0.309560	+	0.950880i	$0.600182\pi$
$878$			1.06399e9i			1.57201i
$879$	−1.07262e8	+	4.88253e8i	−0.157935	+	0.718916i
$880$	1.26077e9			1.85007
$881$		−	1.09238e9i		−	1.59751i	−0.601654	−	0.798757i	$-0.705491\pi$
$881$		−	1.09238e9i		−	1.59751i	0.601654	−	0.798757i	$-0.294509\pi$
$882$	6.35341e8	−	1.37624e9i	0.925979	−	2.00580i
$883$	−4.37097e8			−0.634886			−0.317443	−	0.948277i	$-0.602824\pi$
$883$	−4.37097e8			−0.634886			−0.317443	+	0.948277i	$0.602824\pi$
$884$		−	3.25913e7i		−	0.0471785i
$885$	1.85087e9	+	4.06609e8i	2.67022	+	0.586607i
$886$	8.42670e7			0.121159
$887$		−	9.33714e8i		−	1.33796i	−0.743280	−	0.668980i	$-0.766731\pi$
$887$		−	9.33714e8i		−	1.33796i	0.743280	−	0.668980i	$-0.233269\pi$
$888$	−1.00079e8	+	4.55557e8i	−0.142924	+	0.650584i
$889$	−2.31939e8			−0.330117
$890$			1.46303e9i			2.07532i
$891$	−5.28125e8	+	4.50092e8i	−0.746627	+	0.636309i
$892$	3.27696e8			0.461717
$893$		−	5.43138e8i		−	0.762704i
$894$	−5.22531e7	−	1.14792e7i	−0.0731306	−	0.0160657i
$895$	4.24270e8			0.591797
$896$		−	1.48571e9i		−	2.06543i
$897$	2.43482e8	−	1.10832e9i	0.337357	−	1.53564i
$898$	−1.25073e9			−1.72717
$899$			1.45672e8i			0.200493i
$900$	2.83993e8	+	1.31106e8i	0.389566	+	0.179843i
$901$	4.89660e7			0.0669454
$902$		−	2.94328e8i		−	0.401062i
$903$	−1.82502e9	−	4.00931e8i	−2.47859	−	0.544510i
$904$	6.47222e8			0.876089
$905$			1.18040e9i			1.59251i
$906$	1.36599e7	−	6.21793e7i	0.0183680	−	0.0836106i
$907$	5.65229e7			0.0757535			0.0378767	−	0.999282i	$-0.487941\pi$
$907$	5.65229e7			0.0757535			0.0378767	+	0.999282i	$0.487941\pi$
$908$			3.37130e8i			0.450339i
$909$	−2.92185e8	+	6.32914e8i	−0.389015	+	0.842661i
$910$	2.38265e9			3.16181
$911$			6.53925e8i			0.864914i	0.901655	+	0.432457i	$0.142353\pi$
$911$			6.53925e8i			0.864914i	−0.901655	+	0.432457i	$0.857647\pi$
$912$	1.00325e9	+	2.20400e8i	1.32259	+	0.290554i
$913$	−3.53892e8			−0.465006
$914$		−	2.47005e8i		−	0.323495i
$915$	−3.51639e8	+	1.60065e9i	−0.459023	+	2.08946i
$916$	−4.46031e8			−0.580335
$917$			9.74960e8i			1.26438i
$918$	−1.06379e8	−	8.06601e7i	−0.137508	−	0.104263i
$919$	5.41770e8			0.698021			0.349011	−	0.937119i	$-0.386518\pi$
$919$	5.41770e8			0.698021			0.349011	+	0.937119i	$0.386518\pi$
$920$			1.46646e9i			1.88324i
$921$	−9.82142e8	−	2.15762e8i	−1.25717	−	0.276183i
$922$	3.42835e8			0.437414
$923$		−	1.15935e7i		−	0.0147439i
$924$	−8.54673e7	+	3.89044e8i	−0.108339	+	0.493154i
$925$	9.44051e8			1.19281
$926$		−	1.24484e9i		−	1.56776i
$927$	−2.91269e8	−	1.34465e8i	−0.365641	−	0.168799i
$928$	−8.63795e7			−0.108085
$929$			4.49857e7i			0.0561083i	0.999606	+	0.0280542i	$0.00893109\pi$
$929$			4.49857e7i			0.0561083i	−0.999606	+	0.0280542i	$0.991069\pi$
$930$	1.50740e9	+	3.31153e8i	1.87404	+	0.411699i
$931$	−1.74942e9			−2.16793
$932$		−	3.05592e7i		−	0.0377480i
$933$	−4.98821e7	+	2.27062e8i	−0.0614186	+	0.279575i
$934$	1.11391e9			1.36712
$935$			1.89114e8i			0.231360i
$936$	−2.84680e8	+	6.16656e8i	−0.347160	+	0.751996i
$937$	1.23725e9			1.50396			0.751982	−	0.659184i	$-0.229098\pi$
$937$	1.23725e9			1.50396			0.751982	+	0.659184i	$0.229098\pi$
$938$			7.45130e7i			0.0902867i
$939$	1.03244e9	+	2.26812e8i	1.24701	+	0.273949i
$940$	2.64957e8			0.319001
$941$			6.82579e7i			0.0819189i	0.999161	+	0.0409594i	$0.0130414\pi$
$941$			6.82579e7i			0.0819189i	−0.999161	+	0.0409594i	$0.986959\pi$
$942$	−2.19943e8	+	1.00117e9i	−0.263123	+	1.19772i
$943$	4.55348e8			0.543010
$944$			1.78581e9i			2.12285i
$945$	1.36189e9	−	1.79614e9i	1.61379	−	2.12835i
$946$	1.40224e9			1.65633
$947$		−	7.47615e8i		−	0.880295i	−0.897925	−	0.440147i	$-0.854926\pi$
$947$		−	7.47615e8i		−	0.880295i	0.897925	−	0.440147i	$-0.145074\pi$
$948$	2.82632e8	+	6.20900e7i	0.331738	+	0.0728781i
$949$	1.59315e8			0.186405
$950$		−	1.56309e9i		−	1.82311i
$951$	1.14621e8	−	5.21749e8i	0.133267	−	0.606625i
$952$	1.78547e8			0.206938
$953$			8.63508e8i			0.997672i	0.866696	+	0.498836i	$0.166239\pi$
$953$			8.63508e8i			0.997672i	−0.866696	+	0.498836i	$0.833761\pi$
$954$	3.97654e8	+	1.83577e8i	0.457994	+	0.211433i
$955$	1.32083e9			1.51648
$956$			2.41150e8i			0.276003i
$957$	−1.55943e8	−	3.42584e7i	−0.177922	−	0.0390869i
$958$	7.63538e8			0.868429
$959$			1.57263e9i			1.78308i
$960$	1.61652e8	−	7.35834e8i	0.182712	−	0.831699i
$961$	1.47079e8			0.165722
$962$		−	8.79830e8i		−	0.988265i
$963$	8.63966e7	−	1.87147e8i	0.0967426	−	0.209558i
$964$	−3.45534e8			−0.385709
$965$		−	4.82689e8i		−	0.537138i
$966$	−2.60607e9	−	5.72514e8i	−2.89104	−	0.635119i
$967$	−3.23453e8			−0.357710			−0.178855	−	0.983875i	$-0.557239\pi$
$967$	−3.23453e8			−0.357710			−0.178855	+	0.983875i	$0.557239\pi$
$968$		−	2.72538e7i		−	0.0300470i
$969$	−3.30598e7	+	1.50487e8i	−0.0363353	+	0.165397i
$970$	−3.02393e9			−3.31327
$971$		−	1.06429e9i		−	1.16252i	−0.813716	−	0.581262i	$-0.802559\pi$
$971$		−	1.06429e9i		−	1.16252i	0.813716	−	0.581262i	$-0.197441\pi$
$972$	−1.29681e8	−	2.43394e8i	−0.141214	−	0.265039i
$973$	−1.16569e9			−1.26545
$974$			2.03236e9i			2.19950i
$975$	1.34270e9	+	2.94971e8i	1.44865	+	0.318248i
$976$	−1.54438e9			−1.66114
$977$			1.38873e8i			0.148914i	0.997224	+	0.0744568i	$0.0237223\pi$
$977$			1.38873e8i			0.148914i	−0.997224	+	0.0744568i	$0.976278\pi$
$978$	3.02239e7	−	1.37578e8i	0.0323098	−	0.147073i
$979$	−1.07493e9			−1.14560
$980$		−	8.53413e8i		−	0.906736i
$981$	9.15466e8	+	4.22625e8i	0.969695	+	0.447660i
$982$	−6.33942e8			−0.669445
$983$			7.97139e8i			0.839216i	0.907706	+	0.419608i	$0.137832\pi$
$983$			7.97139e8i			0.839216i	−0.907706	+	0.419608i	$0.862168\pi$
$984$	−2.66196e8	−	5.84794e7i	−0.279394	−	0.0613787i
$985$	−1.32536e9			−1.38684
$986$		−	3.07176e7i		−	0.0320448i
$987$	2.41004e8	−	1.09704e9i	0.250653	−	1.14096i
$988$	−3.36444e8			−0.348853
$989$			2.16937e9i			2.24256i
$990$	−7.09003e8	+	1.53580e9i	−0.730705	+	1.58281i
$991$	1.48866e9			1.52959			0.764793	−	0.644276i	$-0.222841\pi$
$991$	1.48866e9			1.52959			0.764793	+	0.644276i	$0.222841\pi$
$992$			6.13477e8i			0.628439i
$993$	−1.88714e9	−	4.14577e8i	−1.92733	−	0.423406i
$994$	−2.72606e7			−0.0277572
$995$		−	9.36770e6i		−	0.00950963i
$996$	3.01795e7	−	1.37376e8i	0.0305445	−	0.139038i
$997$	−2.77324e8			−0.279835			−0.139917	−	0.990163i	$-0.544684\pi$
$997$	−2.77324e8			−0.279835			−0.139917	+	0.990163i	$0.544684\pi$
$998$		−	4.40178e8i		−	0.442830i
$999$	−6.63250e8	−	5.02899e8i	−0.665244	−	0.504411i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	87.7.b.a.59.14	✓	56
3.2	odd	2	inner	87.7.b.a.59.43	yes	56

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
87.7.b.a.59.14	✓	56	1.1	even	1	trivial
87.7.b.a.59.43	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-9.12250i q^{2} +(26.3711 + 5.79335i) q^{3} -19.2200 q^{4} +194.806i q^{5} +(52.8499 - 240.571i) q^{6} -587.861 q^{7} -408.506i q^{8} +(661.874 + 305.555i) q^{9} +1777.12 q^{10} +1305.70i q^{11} +(-506.853 - 111.348i) q^{12} -2280.71 q^{13} +5362.76i q^{14} +(-1128.58 + 5137.26i) q^{15} -4956.67 q^{16} -743.497i q^{17} +(2787.42 - 6037.95i) q^{18} -7675.22 q^{19} -3744.17i q^{20} +(-15502.6 - 3405.69i) q^{21} +11911.2 q^{22} +18427.6i q^{23} +(2366.62 - 10772.8i) q^{24} -22324.4 q^{25} +20805.7i q^{26} +(15684.2 + 11892.3i) q^{27} +11298.7 q^{28} +4528.92i q^{29} +(46864.6 + 10295.5i) q^{30} +32164.9 q^{31} +19072.9i q^{32} +(-7564.37 + 34432.7i) q^{33} -6782.55 q^{34} -114519. i q^{35} +(-12721.2 - 5872.75i) q^{36} -42287.8 q^{37} +70017.2i q^{38} +(-60144.8 - 13212.9i) q^{39} +79579.4 q^{40} -24710.1i q^{41} +(-31068.4 + 141422. i) q^{42} +117724. q^{43} -25095.5i q^{44} +(-59523.9 + 128937. i) q^{45} +168105. q^{46} +70765.2i q^{47} +(-130713. - 28715.8i) q^{48} +227931. q^{49} +203654. i q^{50} +(4307.34 - 19606.9i) q^{51} +43835.1 q^{52} +65859.1i q^{53} +(108487. - 143079. i) q^{54} -254358. q^{55} +240144. i q^{56} +(-202404. - 44465.3i) q^{57} +41315.1 q^{58} -360284. i q^{59} +(21691.3 - 98738.0i) q^{60} +311577. q^{61} -293424. i q^{62} +(-389090. - 179624. i) q^{63} -143235. q^{64} -444296. i q^{65} +(314112. + 69005.9i) q^{66} +13894.5 q^{67} +14290.0i q^{68} +(-106757. + 485956. i) q^{69} -1.04470e6 q^{70} +5083.31i q^{71} +(124821. - 270379. i) q^{72} -69853.2 q^{73} +385771. i q^{74} +(-588720. - 129333. i) q^{75} +147517. q^{76} -767568. i q^{77} +(-120535. + 548671. i) q^{78} -557621. q^{79} -965590. i q^{80} +(344714. + 404478. i) q^{81} -225418. q^{82} +271037. i q^{83} +(297959. + 65457.2i) q^{84} +144838. q^{85} -1.07394e6i q^{86} +(-26237.7 + 119433. i) q^{87} +533385. q^{88} +823262. i q^{89} +(1.17623e6 + 543007. i) q^{90} +1.34074e6 q^{91} -354177. i q^{92} +(848226. + 186343. i) q^{93} +645555. q^{94} -1.49518e6i q^{95} +(-110496. + 502973. i) q^{96} -1.70159e6 q^{97} -2.07930e6i q^{98} +(-398962. + 864207. 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

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