LMFDB - Embedded newform 87.7.b.a.59.16

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			59.16
Character	$\chi$	$=$	87.59
Dual form			87.7.b.a.59.41

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q-8.21957i q^{2} +(1.43004 - 26.9621i) q^{3} -3.56134 q^{4} +27.4201i q^{5} +(-221.617 - 11.7543i) q^{6} +526.273 q^{7} -496.780i q^{8} +(-724.910 - 77.1139i) q^{9} +225.382 q^{10} -993.883i q^{11} +(-5.09287 + 96.0213i) q^{12} -10.0183 q^{13} -4325.74i q^{14} +(739.304 + 39.2119i) q^{15} -4311.24 q^{16} -9477.19i q^{17} +(-633.843 + 5958.45i) q^{18} -4564.09 q^{19} -97.6524i q^{20} +(752.593 - 14189.4i) q^{21} -8169.29 q^{22} +19187.9i q^{23} +(-13394.2 - 710.416i) q^{24} +14873.1 q^{25} +82.3464i q^{26} +(-3115.80 + 19434.8i) q^{27} -1874.24 q^{28} +4528.92i q^{29} +(322.305 - 6076.76i) q^{30} +14172.3 q^{31} +3642.66i q^{32} +(-26797.2 - 1421.29i) q^{33} -77898.4 q^{34} +14430.5i q^{35} +(2581.65 + 274.629i) q^{36} -47186.1 q^{37} +37514.9i q^{38} +(-14.3266 + 270.115i) q^{39} +13621.8 q^{40} +8542.42i q^{41} +(-116631. - 6185.99i) q^{42} -72470.5 q^{43} +3539.56i q^{44} +(2114.47 - 19877.1i) q^{45} +157716. q^{46} -64474.3i q^{47} +(-6165.26 + 116240. i) q^{48} +159314. q^{49} -122251. i q^{50} +(-255525. - 13552.8i) q^{51} +35.6787 q^{52} +195048. i q^{53} +(159746. + 25610.6i) q^{54} +27252.4 q^{55} -261442. i q^{56} +(-6526.84 + 123058. i) q^{57} +37225.8 q^{58} -122318. i q^{59} +(-2632.91 - 139.647i) q^{60} +18733.9 q^{61} -116490. i q^{62} +(-381501. - 40583.0i) q^{63} -245978. q^{64} -274.704i q^{65} +(-11682.4 + 220261. i) q^{66} -45066.5 q^{67} +33751.5i q^{68} +(517345. + 27439.5i) q^{69} +118612. q^{70} -352226. i q^{71} +(-38308.6 + 360121. i) q^{72} +733499. q^{73} +387849. i q^{74} +(21269.2 - 401011. i) q^{75} +16254.3 q^{76} -523054. i q^{77} +(2220.23 + 117.759i) q^{78} +791148. q^{79} -118215. i q^{80} +(519548. + 111801. i) q^{81} +70215.0 q^{82} -533839. i q^{83} +(-2680.24 + 50533.4i) q^{84} +259866. q^{85} +595676. i q^{86} +(122109. + 6476.55i) q^{87} -493741. q^{88} +69045.2i q^{89} +(-163381. - 17380.1i) q^{90} -5272.38 q^{91} -68334.5i q^{92} +(20266.9 - 382114. i) q^{93} -529951. q^{94} -125148. i q^{95} +(98213.7 + 5209.15i) q^{96} -1.39339e6 q^{97} -1.30950e6i q^{98} +(-76642.2 + 720476. 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 $

$2$		−	8.21957i		−	1.02745i	−0.857956	−	0.513723i	$-0.828266\pi$
$2$		−	8.21957i		−	1.02745i	0.857956	−	0.513723i	$-0.171734\pi$
$3$	1.43004	−	26.9621i	0.0529645	−	0.998596i
$3$	1.43004	−	26.9621i	0.0529645	−	0.998596i
$4$	−3.56134			−0.0556460
$5$			27.4201i			0.219361i	0.993967	+	0.109680i	$0.0349828\pi$
$5$			27.4201i			0.219361i	−0.993967	+	0.109680i	$0.965017\pi$
$6$	−221.617	−	11.7543i	−1.02600	−	0.0544182i
$7$	526.273			1.53432			0.767162	−	0.641453i	$-0.221668\pi$
$7$	526.273			1.53432			0.767162	+	0.641453i	$0.221668\pi$
$8$		−	496.780i		−	0.970273i
$9$	−724.910	−	77.1139i	−0.994390	−	0.105780i
$10$	225.382			0.225382
$11$		−	993.883i		−	0.746719i	−0.927687	−	0.373359i	$-0.878206\pi$
$11$		−	993.883i		−	0.746719i	0.927687	−	0.373359i	$-0.121794\pi$
$12$	−5.09287	+	96.0213i	−0.00294726	+	0.0555679i
$13$	−10.0183			−0.00456001			−0.00228000	−	0.999997i	$-0.500726\pi$
$13$	−10.0183			−0.00456001			−0.00228000	+	0.999997i	$0.500726\pi$
$14$		−	4325.74i		−	1.57644i
$15$	739.304	+	39.2119i	0.219053	+	0.0116183i
$16$	−4311.24			−1.05255
$17$		−	9477.19i		−	1.92900i	−0.264080	−	0.964501i	$-0.585068\pi$
$17$		−	9477.19i		−	1.92900i	0.264080	−	0.964501i	$-0.414932\pi$
$18$	−633.843	+	5958.45i	−0.108684	+	1.02168i
$19$	−4564.09			−0.665417			−0.332708	−	0.943030i	$-0.607962\pi$
$19$	−4564.09			−0.665417			−0.332708	+	0.943030i	$0.607962\pi$
$20$		−	97.6524i		−	0.0122066i
$21$	752.593	−	14189.4i	0.0812647	−	1.53217i
$22$	−8169.29			−0.767214
$23$			19187.9i			1.57704i	0.615008	+	0.788521i	$0.289153\pi$
$23$			19187.9i			1.57704i	−0.615008	+	0.788521i	$0.710847\pi$
$24$	−13394.2	−	710.416i	−0.968911	−	0.0513901i
$25$	14873.1			0.951881
$26$			82.3464i			0.00468516i
$27$	−3115.80	+	19434.8i	−0.158299	+	0.987391i
$28$	−1874.24			−0.0853790
$29$			4528.92i			0.185695i
$29$			4528.92i			0.185695i
$30$	322.305	−	6076.76i	0.0119372	−	0.225065i
$31$	14172.3			0.475723			0.237861	−	0.971299i	$-0.423554\pi$
$31$	14172.3			0.475723			0.237861	+	0.971299i	$0.423554\pi$
$32$			3642.66i			0.111165i
$33$	−26797.2	−	1421.29i	−0.745671	−	0.0395496i
$34$	−77898.4			−1.98195
$35$			14430.5i			0.336571i
$36$	2581.65	+	274.629i	0.0553338	+	0.00588625i
$37$	−47186.1			−0.931556			−0.465778	−	0.884902i	$-0.654225\pi$
$37$	−47186.1			−0.931556			−0.465778	+	0.884902i	$0.654225\pi$
$38$			37514.9i			0.683680i
$39$	−14.3266	+	270.115i	−0.000241519	+	0.00455361i
$40$	13621.8			0.212840
$41$			8542.42i			0.123945i	0.998078	+	0.0619726i	$0.0197391\pi$
$41$			8542.42i			0.123945i	−0.998078	+	0.0619726i	$0.980261\pi$
$42$	−116631.	−	6185.99i	−1.57422	−	0.0834952i
$43$	−72470.5			−0.911498			−0.455749	−	0.890108i	$-0.650629\pi$
$43$	−72470.5			−0.911498			−0.455749	+	0.890108i	$0.650629\pi$
$44$			3539.56i			0.0415519i
$45$	2114.47	−	19877.1i	0.0232041	−	0.218130i
$46$	157716.			1.62033
$47$		−	64474.3i		−	0.621002i	−0.950573	−	0.310501i	$-0.899503\pi$
$47$		−	64474.3i		−	0.621002i	0.950573	−	0.310501i	$-0.100497\pi$
$48$	−6165.26	+	116240.i	−0.0557478	+	1.05107i
$49$	159314.			1.35415
$50$		−	122251.i		−	0.978006i
$51$	−255525.	−	13552.8i	−1.92629	−	0.102169i
$52$	35.6787			0.000253746
$53$			195048.i			1.31013i	0.755573	+	0.655064i	$0.227358\pi$
$53$			195048.i			1.31013i	−0.755573	+	0.655064i	$0.772642\pi$
$54$	159746.	+	25610.6i	1.01449	+	0.162644i
$55$	27252.4			0.163801
$56$		−	261442.i		−	1.48871i
$57$	−6526.84	+	123058.i	−0.0352435	+	0.664483i
$58$	37225.8			0.190792
$59$		−	122318.i		−	0.595574i	−0.954632	−	0.297787i	$-0.903752\pi$
$59$		−	122318.i		−	0.595574i	0.954632	−	0.297787i	$-0.0962485\pi$
$60$	−2632.91	−	139.647i	−0.0121894	−	0.000646514i
$61$	18733.9			0.0825349			0.0412675	−	0.999148i	$-0.486860\pi$
$61$	18733.9			0.0825349			0.0412675	+	0.999148i	$0.486860\pi$
$62$		−	116490.i		−	0.488779i
$63$	−381501.	−	40583.0i	−1.52572	−	0.162301i
$64$	−245978.			−0.938333
$65$		−	274.704i		−	0.00100029i
$66$	−11682.4	+	220261.i	−0.0406351	+	0.766137i
$67$	−45066.5			−0.149841			−0.0749203	−	0.997190i	$-0.523870\pi$
$67$	−45066.5			−0.149841			−0.0749203	+	0.997190i	$0.523870\pi$
$68$			33751.5i			0.107341i
$69$	517345.	+	27439.5i	1.57483	+	0.0835272i
$70$	118612.			0.345808
$71$		−	352226.i		−	0.984117i	−0.870562	−	0.492058i	$-0.836245\pi$
$71$		−	352226.i		−	0.984117i	0.870562	−	0.492058i	$-0.163755\pi$
$72$	−38308.6	+	360121.i	−0.102636	+	0.964829i
$73$	733499.			1.88552			0.942759	−	0.333474i	$-0.108221\pi$
$73$	733499.			1.88552			0.942759	+	0.333474i	$0.108221\pi$
$74$			387849.i			0.957124i
$75$	21269.2	−	401011.i	0.0504159	−	0.950545i
$76$	16254.3			0.0370278
$77$		−	523054.i		−	1.14571i
$78$	2220.23	+	117.759i	0.00467858	+	0.000248147i
$79$	791148.			1.60463			0.802317	−	0.596898i	$-0.203600\pi$
$79$	791148.			1.60463			0.802317	+	0.596898i	$0.203600\pi$
$80$		−	118215.i		−	0.230888i
$81$	519548.	+	111801.i	0.977621	+	0.210374i
$82$	70215.0			0.127347
$83$		−	533839.i		−	0.933632i	−0.884355	−	0.466816i	$-0.845401\pi$
$83$		−	533839.i		−	0.933632i	0.884355	−	0.466816i	$-0.154599\pi$
$84$	−2680.24	+	50533.4i	−0.00452206	+	0.0852591i
$85$	259866.			0.423148
$86$			595676.i			0.936516i
$87$	122109.	+	6476.55i	0.185435	+	0.00983527i
$88$	−493741.			−0.724521
$89$			69045.2i			0.0979407i	0.998800	+	0.0489703i	$0.0155940\pi$
$89$			69045.2i			0.0979407i	−0.998800	+	0.0489703i	$0.984406\pi$
$90$	−163381.	−	17380.1i	−0.224117	−	0.0238410i
$91$	−5272.38			−0.00699653
$92$		−	68334.5i		−	0.0877560i
$93$	20266.9	−	382114.i	0.0251964	−	0.475055i
$94$	−529951.			−0.638046
$95$		−	125148.i		−	0.145966i
$96$	98213.7	+	5209.15i	0.111009	+	0.00588780i
$97$	−1.39339e6			−1.52672			−0.763359	−	0.645974i	$-0.776451\pi$
$97$	−1.39339e6			−1.52672			−0.763359	+	0.645974i	$0.776451\pi$
$98$		−	1.30950e6i		−	1.39132i
$99$	−76642.2	+	720476.i	−0.0789882	+	0.742529i
$100$	−52968.3			−0.0529683
$101$			645302.i			0.626324i	0.949700	+	0.313162i	$0.101388\pi$
$101$			645302.i			0.626324i	−0.949700	+	0.313162i	$0.898612\pi$
$102$	−111398.	+	2.10030e6i	−0.104973	+	1.97916i
$103$	927503.			0.848797			0.424398	−	0.905476i	$-0.360486\pi$
$103$	927503.			0.848797			0.424398	+	0.905476i	$0.360486\pi$
$104$			4976.91i			0.00442445i
$105$	389076.	+	20636.2i	0.336098	+	0.0178263i
$106$	1.60321e6			1.34609
$107$			2.21917e6i			1.81150i	0.423813	+	0.905750i	$0.360692\pi$
$107$			2.21917e6i			1.81150i	−0.423813	+	0.905750i	$0.639308\pi$
$108$	11096.4	−	69214.0i	0.00880872	−	0.0549443i
$109$	1.34130e6			1.03573			0.517865	−	0.855463i	$-0.326727\pi$
$109$	1.34130e6			1.03573			0.517865	+	0.855463i	$0.326727\pi$
$110$		−	224003.i		−	0.168297i
$111$	−67478.1	+	1.27224e6i	−0.0493394	+	0.930248i
$112$	−2.26889e6			−1.61495
$113$		−	2.16316e6i		−	1.49918i	−0.661904	−	0.749589i	$-0.730251\pi$
$113$		−	2.16316e6i		−	1.49918i	0.661904	−	0.749589i	$-0.269749\pi$
$114$	1.01148e6	+	53647.9i	0.682720	+	0.0362108i
$115$	−526134.			−0.345941
$116$		−	16129.0i		−	0.0103332i
$117$	7262.39	+	772.553i	0.00453442	+	0.000482359i
$118$	−1.00540e6			−0.611920
$119$		−	4.98759e6i		−	2.95971i
$120$	19479.7	−	367271.i	0.0112730	−	0.212541i
$121$	783758.			0.442411
$122$		−	153984.i		−	0.0848002i
$123$	230322.	+	12216.0i	0.123771	+	0.00656469i
$124$	−50472.2			−0.0264720
$125$			836263.i			0.428166i
$126$	−333575.	+	3.13577e6i	−0.166756	+	1.56759i
$127$	−3.35966e6			−1.64015			−0.820076	−	0.572255i	$-0.806069\pi$
$127$	−3.35966e6			−1.64015			−0.820076	+	0.572255i	$0.806069\pi$
$128$			2.25497e6i			1.07525i
$129$	−103636.	+	1.95396e6i	−0.0482771	+	0.910219i
$130$	−2257.95			−0.00102774
$131$			160070.i			0.0712028i	0.999366	+	0.0356014i	$0.0113347\pi$
$131$			160070.i			0.0712028i	−0.999366	+	0.0356014i	$0.988665\pi$
$132$	95433.9	+	5061.72i	0.0414936	+	0.00220078i
$133$	−2.40196e6			−1.02096
$134$			370427.i			0.153953i
$135$	−532905.	−	85435.7i	−0.216595	−	0.0347247i
$136$	−4.70807e6			−1.87166
$137$		−	1.11773e6i		−	0.434684i	−0.976096	−	0.217342i	$-0.930261\pi$
$137$		−	1.11773e6i		−	0.434684i	0.976096	−	0.217342i	$-0.0697386\pi$
$138$	225541.	−	4.25235e6i	0.0858198	−	1.61805i
$139$	348897.			0.129913			0.0649566	−	0.997888i	$-0.479309\pi$
$139$	348897.			0.129913			0.0649566	+	0.997888i	$0.479309\pi$
$140$		−	51391.9i		−	0.0187288i
$141$	−1.73836e6	−	92200.9i	−0.620130	−	0.0328911i
$142$	−2.89515e6			−1.01113
$143$			9957.05i			0.00340504i
$144$	3.12526e6	+	332457.i	1.04664	+	0.111339i
$145$	−124184.			−0.0407343
$146$		−	6.02905e6i		−	1.93727i
$147$	227826.	−	4.29545e6i	0.0717219	−	1.35225i
$148$	168046.			0.0518373
$149$		−	4.09116e6i		−	1.23677i	−0.785877	−	0.618383i	$-0.787788\pi$
$149$		−	4.09116e6i		−	1.23677i	0.785877	−	0.618383i	$-0.212212\pi$
$150$	−3.29614e6	−	174824.i	−0.976634	−	0.0517996i
$151$	5.39153e6			1.56596			0.782981	−	0.622046i	$-0.213698\pi$
$151$	5.39153e6			1.56596			0.782981	+	0.622046i	$0.213698\pi$
$152$			2.26735e6i			0.645636i
$153$	−730823.	+	6.87011e6i	−0.204051	+	1.91818i
$154$	−4.29928e6			−1.17715
$155$			388605.i			0.104355i
$156$	51.0221	−	961.973i	1.34395e−5	−	0.000253390i
$157$	5.24153e6			1.35444			0.677219	−	0.735782i	$-0.263185\pi$
$157$	5.24153e6			1.35444			0.677219	+	0.735782i	$0.263185\pi$
$158$		−	6.50289e6i		−	1.64868i
$159$	5.25890e6	+	278927.i	1.30829	+	0.0693903i
$160$	−99882.1			−0.0243853
$161$			1.00981e7i			2.41969i
$162$	918958.	−	4.27046e6i	0.216148	−	1.00445i
$163$	3.41116e6			0.787660			0.393830	−	0.919183i	$-0.371150\pi$
$163$	3.41116e6			0.787660			0.393830	+	0.919183i	$0.371150\pi$
$164$		−	30422.5i		−	0.00689705i
$165$	38972.1	−	734782.i	0.00867564	−	0.163571i
$166$	−4.38792e6			−0.959257
$167$		−	587384.i		−	0.126117i	−0.998010	−	0.0630584i	$-0.979915\pi$
$167$		−	587384.i		−	0.126117i	0.998010	−	0.0630584i	$-0.0200854\pi$
$168$	−7.04902e6	−	373873.i	−1.48662	−	0.0788490i
$169$	−4.82671e6			−0.999979
$170$		−	2.13598e6i		−	0.434762i
$171$	3.30856e6	+	351955.i	0.661683	+	0.0703880i
$172$	258092.			0.0507212
$173$			6.26741e6i			1.21046i	0.796051	+	0.605229i	$0.206918\pi$
$173$			6.26741e6i			1.21046i	−0.796051	+	0.605229i	$0.793082\pi$
$174$	53234.5	−	1.00369e6i	0.0101052	−	0.190524i
$175$	7.82733e6			1.46049
$176$			4.28487e6i			0.785959i
$177$	−3.29796e6	−	174920.i	−0.594738	−	0.0315443i
$178$	567522.			0.100629
$179$			7.15708e6i			1.24789i	0.781468	+	0.623946i	$0.214471\pi$
$179$			7.15708e6i			1.24789i	−0.781468	+	0.623946i	$0.785529\pi$
$180$	−7530.36	+	70789.2i	−0.00129121	+	0.0121381i
$181$	7.02443e6			1.18461			0.592305	−	0.805714i	$-0.298218\pi$
$181$	7.02443e6			1.18461			0.592305	+	0.805714i	$0.298218\pi$
$182$			43336.7i			0.00718856i
$183$	26790.2	−	505104.i	0.00437142	−	0.0824191i
$184$	9.53214e6			1.53016
$185$		−	1.29385e6i		−	0.204347i
$186$	−3.14081e6	−	166585.i	−0.488093	−	0.0258880i
$187$	−9.41921e6			−1.44042
$188$			229615.i			0.0345562i
$189$	−1.63976e6	+	1.02280e7i	−0.242882	+	1.51498i
$190$	−1.02866e6			−0.149973
$191$		−	1.05522e6i		−	0.151441i	−0.997129	−	0.0757204i	$-0.975874\pi$
$191$		−	1.05522e6i		−	0.151441i	0.997129	−	0.0757204i	$-0.0241257\pi$
$192$	−351760.	+	6.63210e6i	−0.0496984	+	0.937016i
$193$	5.19727e6			0.722942			0.361471	−	0.932383i	$-0.382275\pi$
$193$	5.19727e6			0.722942			0.361471	+	0.932383i	$0.382275\pi$
$194$			1.14531e7i			1.56862i
$195$	−7406.59	−	392.838i	−0.000998883	−	5.29797e-5i
$196$	−567373.			−0.0753530
$197$		−	4.27750e6i		−	0.559488i	−0.960075	−	0.279744i	$-0.909750\pi$
$197$		−	4.27750e6i		−	0.559488i	0.960075	−	0.279744i	$-0.0902496\pi$
$198$	5.92200e6	+	629966.i	0.762909	+	0.0811561i
$199$	9.29138e6			1.17902			0.589510	−	0.807761i	$-0.299321\pi$
$199$	9.29138e6			1.17902			0.589510	+	0.807761i	$0.299321\pi$
$200$		−	7.38867e6i		−	0.923584i
$201$	−64447.0	+	1.21509e6i	−0.00793623	+	0.149630i
$202$	5.30410e6			0.643514
$203$			2.38345e6i			0.284917i
$204$	910011.	+	48266.1i	0.107191	+	0.00568527i
$205$	−234234.			−0.0271887
$206$		−	7.62368e6i		−	0.872093i
$207$	1.47965e6	−	1.39095e7i	0.166820	−	1.56819i
$208$	43191.5			0.00479963
$209$			4.53617e6i			0.496879i
$210$	169621.	−	3.19804e6i	0.0183156	−	0.345323i
$211$	−1.86792e6			−0.198844			−0.0994219	−	0.995045i	$-0.531699\pi$
$211$	−1.86792e6			−0.198844			−0.0994219	+	0.995045i	$0.531699\pi$
$212$		−	694633.i		−	0.0729034i
$213$	−9.49676e6	−	503698.i	−0.982735	−	0.0521233i
$214$	1.82406e7			1.86122
$215$		−	1.98715e6i		−	0.199947i
$216$	9.65483e6	+	1.54787e6i	0.958039	+	0.153594i
$217$	7.45848e6			0.729913
$218$		−	1.10249e7i		−	1.06416i
$219$	1.04893e6	−	1.97767e7i	0.0998656	−	1.88287i
$220$	−97055.1			−0.00911486
$221$			94945.6i			0.00879626i
$222$	1.04572e7	+	554641.i	0.955780	+	0.0506936i
$223$	9.29137e6			0.837848			0.418924	−	0.908021i	$-0.362407\pi$
$223$	9.29137e6			0.837848			0.418924	+	0.908021i	$0.362407\pi$
$224$			1.91703e6i			0.170563i
$225$	−1.07817e7	−	1.14693e6i	−0.946540	−	0.100690i
$226$	−1.77802e7			−1.54032
$227$			1.45920e7i			1.24749i	0.781628	+	0.623744i	$0.214389\pi$
$227$			1.45920e7i			1.24749i	−0.781628	+	0.623744i	$0.785611\pi$
$228$	23244.3	−	438250.i	0.00196116	−	0.0369758i
$229$	−145812.			−0.0121419			−0.00607094	−	0.999982i	$-0.501932\pi$
$229$	−145812.			−0.0121419			−0.00607094	+	0.999982i	$0.501932\pi$
$230$			4.32459e6i			0.355436i
$231$	−1.41026e7	−	747989.i	−1.14410	−	0.0606819i
$232$	2.24988e6			0.180175
$233$		−	6.52761e6i		−	0.516044i	−0.966139	−	0.258022i	$-0.916929\pi$
$233$		−	6.52761e6i		−	0.516044i	0.966139	−	0.258022i	$-0.0830707\pi$
$234$	6350.05	−	59693.7i	0.000495598	−	0.00465887i
$235$	1.76789e6			0.136224
$236$			435618.i			0.0331413i
$237$	1.13137e6	−	2.13310e7i	0.0849887	−	1.60238i
$238$	−4.09958e7			−3.04095
$239$			6.85727e6i			0.502294i	0.967949	+	0.251147i	$0.0808077\pi$
$239$			6.85727e6i			0.502294i	−0.967949	+	0.251147i	$0.919192\pi$
$240$	−3.18732e6	−	169052.i	−0.230564	−	0.0122289i
$241$	413659.			0.0295523			0.0147762	−	0.999891i	$-0.495296\pi$
$241$	413659.			0.0295523			0.0147762	+	0.999891i	$0.495296\pi$
$242$		−	6.44215e6i		−	0.454554i
$243$	3.75737e6	−	1.38482e7i	0.261858	−	0.965106i
$244$	−66717.7			−0.00459274
$245$			4.36842e6i			0.297048i
$246$	100410.	−	1.89314e6i	0.00674487	−	0.127168i
$247$	45724.6			0.00303430
$248$		−	7.04049e6i		−	0.461581i
$249$	−1.43934e7	−	763412.i	−0.932321	−	0.0494494i
$250$	6.87372e6			0.439918
$251$		−	5.84846e6i		−	0.369845i	−0.982753	−	0.184923i	$-0.940797\pi$
$251$		−	5.84846e6i		−	0.369845i	0.982753	−	0.184923i	$-0.0592034\pi$
$252$	1.35865e6	+	144530.i	0.0848999	+	0.00903142i
$253$	1.90705e7			1.17761
$254$			2.76150e7i			1.68517i
$255$	371619.	−	7.00652e6i	0.0224118	−	0.422554i
$256$	2.79224e6			0.166431
$257$			1.29196e7i			0.761113i	0.924758	+	0.380556i	$0.124268\pi$
$257$			1.29196e7i			0.761113i	−0.924758	+	0.380556i	$0.875732\pi$
$258$	1.60607e7	+	851842.i	0.935201	+	0.0496021i
$259$	−2.48328e7			−1.42931
$260$			978.315i			5.56620e-5i
$261$	349243.	−	3.28306e6i	0.0196429	−	0.184653i
$262$	1.31571e6			0.0731570
$263$			7.82292e6i			0.430033i	0.976610	+	0.215016i	$0.0689805\pi$
$263$			7.82292e6i			0.430033i	−0.976610	+	0.215016i	$0.931020\pi$
$264$	−706070.	+	1.33123e7i	−0.0383739	+	0.723504i
$265$	−5.34824e6			−0.287391
$266$			1.97431e7i			1.04899i
$267$	1.86160e6	+	98737.5i	0.0978032	+	0.00518738i
$268$	160497.			0.00833802
$269$		−	121998.i		−	0.00626754i	−0.999995	−	0.00313377i	$-0.999002\pi$
$269$		−	121998.i		−	0.00626754i	0.999995	−	0.00313377i	$-0.000997511\pi$
$270$	−702245.	+	4.38025e6i	−0.0356777	+	0.222540i
$271$	−1.40479e7			−0.705834			−0.352917	−	0.935655i	$-0.614810\pi$
$271$	−1.40479e7			−0.705834			−0.352917	+	0.935655i	$0.614810\pi$
$272$			4.08584e7i			2.03037i
$273$	−7539.72	+	142154.i	−0.000370568	+	0.00698671i
$274$	−9.18722e6			−0.446614
$275$		−	1.47822e7i		−	0.710787i
$276$	−1.84244e6	−	97721.3i	−0.0876328	−	0.00464795i
$277$	2.84627e7			1.33917			0.669586	−	0.742734i	$-0.266471\pi$
$277$	2.84627e7			1.33917			0.669586	+	0.742734i	$0.266471\pi$
$278$		−	2.86779e6i		−	0.133479i
$279$	−1.02736e7	−	1.09288e6i	−0.473054	−	0.0503221i
$280$	7.16877e6			0.326566
$281$		−	2.60655e7i		−	1.17475i	−0.809314	−	0.587376i	$-0.800161\pi$
$281$		−	2.60655e7i		−	1.17475i	0.809314	−	0.587376i	$-0.199839\pi$
$282$	−757852.	+	1.42886e7i	−0.0337938	+	0.637150i
$283$	6.00133e6			0.264782			0.132391	−	0.991198i	$-0.457735\pi$
$283$	6.00133e6			0.264782			0.132391	+	0.991198i	$0.457735\pi$
$284$			1.25440e6i			0.0547621i
$285$	−3.37425e6	−	178967.i	−0.145762	−	0.00773104i
$286$	81842.7			0.00349850
$287$			4.49565e6i			0.190172i
$288$	280899.	−	2.64060e6i	0.0117591	−	0.110541i
$289$	−6.56795e7			−2.72105
$290$			1.02074e6i			0.0418523i
$291$	−1.99261e6	+	3.75688e7i	−0.0808619	+	1.52458i
$292$	−2.61224e6			−0.104922
$293$			2.32761e7i			0.925354i	0.886527	+	0.462677i	$0.153111\pi$
$293$			2.32761e7i			0.925354i	−0.886527	+	0.462677i	$0.846889\pi$
$294$	−3.53068e7	−	1.87264e6i	−1.38936	−	0.0736904i
$295$	3.35398e6			0.130646
$296$			2.34411e7i			0.903864i
$297$	1.93159e7	+	3.09674e6i	0.737304	+	0.118205i
$298$	−3.36275e7			−1.27071
$299$		−	192230.i		−	0.00719132i
$300$	−75746.9	+	1.42814e6i	−0.00280544	+	0.0528940i
$301$	−3.81393e7			−1.39853
$302$		−	4.43161e7i		−	1.60894i
$303$	1.73987e7	+	922809.i	0.625444	+	0.0331729i
$304$	1.96769e7			0.700384
$305$			513685.i			0.0181049i
$306$	5.64693e7	+	6.00705e6i	1.97083	+	0.209651i
$307$	−4.64492e7			−1.60532			−0.802662	−	0.596434i	$-0.796584\pi$
$307$	−4.64492e7			−1.60532			−0.802662	+	0.596434i	$0.796584\pi$
$308$			1.86277e6i			0.0637541i
$309$	1.32637e6	−	2.50074e7i	0.0449561	−	0.847605i
$310$	3.19417e6			0.107219
$311$		−	9.60311e6i		−	0.319250i	−0.987178	−	0.159625i	$-0.948972\pi$
$311$		−	9.60311e6i		−	0.319250i	0.987178	−	0.159625i	$-0.0510285\pi$
$312$	134188.	+	7117.18i	0.00441824	+	0.000234339i
$313$	−4.34567e7			−1.41718			−0.708589	−	0.705622i	$-0.750668\pi$
$313$	−4.34567e7			−1.41718			−0.708589	+	0.705622i	$0.750668\pi$
$314$		−	4.30831e7i		−	1.39161i
$315$	1.11279e6	−	1.04608e7i	0.0356026	−	0.334683i
$316$	−2.81755e6			−0.0892915
$317$			96332.8i			0.00302410i	0.999999	+	0.00151205i	$0.000481301\pi$
$317$			96332.8i			0.00302410i	−0.999999	+	0.00151205i	$0.999519\pi$
$318$	2.29266e6	−	4.32259e7i	0.0712948	−	1.34420i
$319$	4.50122e6			0.138662
$320$		−	6.74476e6i		−	0.205834i
$321$	5.98334e7	+	3.17350e6i	1.80896	+	0.0959452i
$322$	8.30017e7			2.48610
$323$			4.32547e7i			1.28359i
$324$	−1.85029e6	−	398162.i	−0.0544007	−	0.0117065i
$325$	−149004.			−0.00434058
$326$		−	2.80382e7i		−	0.809278i
$327$	1.91811e6	−	3.61642e7i	0.0548569	−	1.03428i
$328$	4.24370e6			0.120261
$329$		−	3.39311e7i		−	0.952818i
$330$	−6.03959e6	−	320334.i	−0.168060	−	0.00891375i
$331$	2.91261e7			0.803154			0.401577	−	0.915825i	$-0.368462\pi$
$331$	2.91261e7			0.803154			0.401577	+	0.915825i	$0.368462\pi$
$332$			1.90118e6i			0.0519528i
$333$	3.42057e7	+	3.63870e6i	0.926329	+	0.0985403i
$334$	−4.82805e6			−0.129578
$335$		−	1.23573e6i		−	0.0328692i
$336$	−3.24461e6	+	6.11741e7i	−0.0855352	+	1.61269i
$337$	1.33666e7			0.349245			0.174623	−	0.984635i	$-0.444129\pi$
$337$	1.33666e7			0.349245			0.174623	+	0.984635i	$0.444129\pi$
$338$			3.96735e7i			1.02742i
$339$	−5.83233e7	−	3.09341e6i	−1.49707	−	0.0794032i
$340$	−925470.			−0.0235465
$341$		−	1.40856e7i		−	0.355231i
$342$	2.89292e6	−	2.71949e7i	0.0723199	−	0.679844i
$343$	2.19274e7			0.543382
$344$			3.60019e7i			0.884402i
$345$	−752393.	+	1.41857e7i	−0.0183226	+	0.345456i
$346$	5.15154e7			1.24368
$347$		−	3.03380e7i		−	0.726103i	−0.931769	−	0.363052i	$-0.881735\pi$
$347$		−	3.03380e7i		−	0.726103i	0.931769	−	0.363052i	$-0.118265\pi$
$348$	−434873.	−	23065.2i	−0.0103187	−	0.000547293i
$349$	5.99937e6			0.141133			0.0705666	−	0.997507i	$-0.477519\pi$
$349$	5.99937e6			0.141133			0.0705666	+	0.997507i	$0.477519\pi$
$350$		−	6.43373e7i		−	1.50058i
$351$	31215.2	−	194704.i	0.000721845	−	0.00450251i
$352$	3.62037e6			0.0830090
$353$		−	1.47501e6i		−	0.0335329i	−0.999859	−	0.0167665i	$-0.994663\pi$
$353$		−	1.47501e6i		−	0.0335329i	0.999859	−	0.0167665i	$-0.00533718\pi$
$354$	−1.43777e6	+	2.71078e7i	−0.0324101	+	0.611061i
$355$	9.65808e6			0.215877
$356$		−	245893.i		−	0.00545001i
$357$	−1.34476e8	−	7.13246e6i	−2.95556	−	0.156760i
$358$	5.88281e7			1.28214
$359$			2.78598e7i			0.602136i	0.953603	+	0.301068i	$0.0973432\pi$
$359$			2.78598e7i			0.602136i	−0.953603	+	0.301068i	$0.902657\pi$
$360$	−9.87455e6	−	1.05043e6i	−0.211646	−	0.0225143i
$361$	−2.62149e7			−0.557221
$362$		−	5.77378e7i		−	1.21712i
$363$	1.12081e6	−	2.11318e7i	0.0234321	−	0.441790i
$364$	18776.7			0.000389329
$365$			2.01126e7i			0.413609i
$366$	−4.15174e6	−	220204.i	−0.0846812	−	0.00449140i
$367$	4.54760e7			0.919992			0.459996	−	0.887921i	$-0.347851\pi$
$367$	4.54760e7			0.919992			0.459996	+	0.887921i	$0.347851\pi$
$368$		−	8.27235e7i		−	1.65991i
$369$	658739.	−	6.19249e6i	0.0131110	−	0.123250i
$370$	−1.06349e7			−0.209956
$371$			1.02649e8i			2.01016i
$372$	−72177.4	+	1.36084e6i	−0.00140208	+	0.0264349i
$373$	4.87094e7			0.938612			0.469306	−	0.883036i	$-0.344504\pi$
$373$	4.87094e7			0.938612			0.469306	+	0.883036i	$0.344504\pi$
$374$			7.74219e7i			1.47996i
$375$	2.25474e7	+	1.19589e6i	0.427565	+	0.0226776i
$376$	−3.20295e7			−0.602541
$377$		−	45372.3i		−	0.000846772i
$378$	8.40700e7	+	1.34782e7i	1.55656	+	0.249549i
$379$	−3.95431e7			−0.726362			−0.363181	−	0.931719i	$-0.618309\pi$
$379$	−3.95431e7			−0.726362			−0.363181	+	0.931719i	$0.618309\pi$
$380$			445695.i			0.00812244i
$381$	−4.80445e6	+	9.05835e7i	−0.0868699	+	1.63785i
$382$	−8.67346e6			−0.155597
$383$		−	7.92576e7i		−	1.41073i	−0.708843	−	0.705366i	$-0.750783\pi$
$383$		−	7.92576e7i		−	1.41073i	0.708843	−	0.705366i	$-0.249217\pi$
$384$	6.07987e7	+	3.22470e6i	1.07374	+	0.0569502i
$385$	1.43422e7			0.251324
$386$		−	4.27194e7i		−	0.742785i
$387$	5.25346e7	+	5.58848e6i	0.906384	+	0.0964186i
$388$	4.96235e6			0.0849557
$389$			9.14835e7i			1.55415i	0.629406	+	0.777077i	$0.283298\pi$
$389$			9.14835e7i			1.55415i	−0.629406	+	0.777077i	$0.716702\pi$
$390$	−3228.96	+	60879.0i	−5.44338e−5	+	0.00102630i
$391$	1.81847e8			3.04212
$392$		−	7.91442e7i		−	1.31390i
$393$	4.31583e6	+	228907.i	0.0711028	+	0.00377122i
$394$	−3.51592e7			−0.574844
$395$			2.16934e7i			0.351994i
$396$	272949.	−	2.56586e6i	0.00439537	−	0.0413188i
$397$	−8.41857e7			−1.34545			−0.672724	−	0.739894i	$-0.734876\pi$
$397$	−8.41857e7			−1.34545			−0.672724	+	0.739894i	$0.734876\pi$
$398$		−	7.63712e7i		−	1.21138i
$399$	−3.43490e6	+	6.47619e7i	−0.0540749	+	1.01953i
$400$	−6.41217e7			−1.00190
$401$			4.17015e7i			0.646724i	0.946275	+	0.323362i	$0.104813\pi$
$401$			4.17015e7i			0.646724i	−0.946275	+	0.323362i	$0.895187\pi$
$402$	9.98750e6	+	529726.i	0.153737	+	0.00815405i
$403$	−141982.			−0.00216930
$404$		−	2.29814e6i		−	0.0348524i
$405$	−3.06560e6	+	1.42461e7i	−0.0461478	+	0.214452i
$406$	1.95909e7			0.292737
$407$			4.68974e7i			0.695610i
$408$	−6.73275e6	+	1.26940e8i	−0.0991315	+	1.86903i
$409$	−3.70766e7			−0.541913			−0.270957	−	0.962592i	$-0.587340\pi$
$409$	−3.70766e7			−0.541913			−0.270957	+	0.962592i	$0.587340\pi$
$410$			1.92530e6i			0.0279350i
$411$	−3.01362e7	−	1.59839e6i	−0.434074	−	0.0230228i
$412$	−3.30316e6			−0.0472321
$413$		−	6.43729e7i		−	0.913803i
$414$	−1.14330e8	−	1.21621e7i	−1.61123	−	0.171399i
$415$	1.46379e7			0.204802
$416$		−	36493.3i		−	0.000506913i
$417$	498938.	−	9.40700e6i	0.00688079	−	0.129731i
$418$	3.72854e7			0.510517
$419$			5.85726e7i			0.796256i	0.917330	+	0.398128i	$0.130340\pi$
$419$			5.85726e7i			0.796256i	−0.917330	+	0.398128i	$0.869660\pi$
$420$	−1.38563e6	−	73492.5i	−0.0187025	−	0.000991963i
$421$	−7.79458e7			−1.04459			−0.522295	−	0.852765i	$-0.674924\pi$
$421$	−7.79458e7			−1.04459			−0.522295	+	0.852765i	$0.674924\pi$
$422$			1.53535e7i			0.204301i
$423$	−4.97186e6	+	4.67380e7i	−0.0656898	+	0.617518i
$424$	9.68959e7			1.27118
$425$		−	1.40955e8i		−	1.83618i
$426$	−4.14018e6	+	7.80593e7i	−0.0535539	+	1.00971i
$427$	9.85913e6			0.126635
$428$		−	7.90321e6i		−	0.100803i
$429$	268463.	+	14239.0i	0.00340026	+	0.000180346i
$430$	−1.63335e7			−0.205435
$431$		−	8.22894e7i		−	1.02781i	−0.857847	−	0.513904i	$-0.828199\pi$
$431$		−	8.22894e7i		−	1.02781i	0.857847	−	0.513904i	$-0.171801\pi$
$432$	1.34330e7	−	8.37882e7i	0.166618	−	1.03928i
$433$	−6.82966e7			−0.841269			−0.420635	−	0.907230i	$-0.638193\pi$
$433$	−6.82966e7			−0.841269			−0.420635	+	0.907230i	$0.638193\pi$
$434$		−	6.13055e7i		−	0.749946i
$435$	−177588.	+	3.34825e6i	−0.00215747	+	0.0406771i
$436$	−4.77683e6			−0.0576342
$437$		−	8.75752e7i		−	1.04939i
$438$	−1.62556e8	−	8.62179e6i	−1.93455	−	0.102607i
$439$	8.73631e7			1.03261			0.516303	−	0.856406i	$-0.327308\pi$
$439$	8.73631e7			1.03261			0.516303	+	0.856406i	$0.327308\pi$
$440$		−	1.35384e7i		−	0.158932i
$441$	−1.15489e8	−	1.22854e7i	−1.34655	−	0.143243i
$442$	780412.			0.00903768
$443$		−	1.08767e7i		−	0.125108i	−0.998042	−	0.0625541i	$-0.980075\pi$
$443$		−	1.08767e7i		−	0.125108i	0.998042	−	0.0625541i	$-0.0199246\pi$
$444$	240313.	−	4.53087e6i	0.00274554	−	0.0517646i
$445$	−1.89323e6			−0.0214844
$446$		−	7.63711e7i		−	0.860844i
$447$	−1.10306e8	−	5.85053e6i	−1.23503	−	0.0655047i
$448$	−1.29452e8			−1.43971
$449$			1.41021e8i			1.55792i	0.627072	+	0.778961i	$0.284253\pi$
$449$			1.41021e8i			1.55792i	−0.627072	+	0.778961i	$0.715747\pi$
$450$	−9.42723e6	+	8.86208e7i	−0.103454	+	0.972519i
$451$	8.49017e6			0.0925522
$452$			7.70375e6i			0.0834232i
$453$	7.71011e6	−	1.45367e8i	0.0829404	−	1.56376i
$454$	1.19940e8			1.28173
$455$		−	144569.i		−	0.00153476i
$456$	6.11325e7	+	3.24240e6i	0.644730	+	0.0341958i
$457$	−1.35918e7			−0.142406			−0.0712032	−	0.997462i	$-0.522684\pi$
$457$	−1.35918e7			−0.142406			−0.0712032	+	0.997462i	$0.522684\pi$
$458$			1.19851e6i			0.0124751i
$459$	1.84187e8	+	2.95291e7i	1.90468	+	0.305360i
$460$	1.87374e6			0.0192502
$461$			1.40829e8i			1.43744i	0.695298	+	0.718722i	$0.255272\pi$
$461$			1.40829e8i			1.43744i	−0.695298	+	0.718722i	$0.744728\pi$
$462$	−6.14815e6	+	1.15918e8i	−0.0623474	+	1.17550i
$463$	−5.76881e7			−0.581223			−0.290612	−	0.956841i	$-0.593859\pi$
$463$	−5.76881e7			−0.581223			−0.290612	+	0.956841i	$0.593859\pi$
$464$		−	1.95253e7i		−	0.195454i
$465$	1.04776e7	+	555721.i	0.104208	+	0.00552711i
$466$	−5.36542e7			−0.530208
$467$			6.46004e7i			0.634285i	0.948378	+	0.317143i	$0.102723\pi$
$467$			6.46004e7i			0.634285i	−0.948378	+	0.317143i	$0.897277\pi$
$468$	−25863.9	−	2751.32i	−0.000252322	−	2.68413e-5i
$469$	−2.37173e7			−0.229904
$470$		−	1.45313e7i		−	0.139962i
$471$	7.49561e6	−	1.41323e8i	0.0717372	−	1.35254i
$472$	−6.07653e7			−0.577869
$473$			7.20272e7i			0.680633i
$474$	−1.75332e8	−	9.29941e6i	−1.64636	−	0.0873214i
$475$	−6.78824e7			−0.633397
$476$			1.77625e7i			0.164696i
$477$	1.50409e7	−	1.41392e8i	0.138586	−	1.30278i
$478$	5.63638e7			0.516080
$479$			1.88584e8i			1.71593i	0.513711	+	0.857963i	$0.328270\pi$
$479$			1.88584e8i			1.71593i	−0.513711	+	0.857963i	$0.671730\pi$
$480$	−142836.	+	2.69303e6i	−0.00129155	+	0.0243510i
$481$	472726.			0.00424790
$482$		−	3.40010e6i		−	0.0303634i
$483$	2.72265e8	+	1.44406e7i	2.41630	+	0.128158i
$484$	−2.79123e6			−0.0246184
$485$		−	3.82070e7i		−	0.334902i
$486$	−1.13826e8	−	3.08840e7i	−0.991595	−	0.269045i
$487$	−1.60078e8			−1.38594			−0.692970	−	0.720967i	$-0.743698\pi$
$487$	−1.60078e8			−1.38594			−0.692970	+	0.720967i	$0.743698\pi$
$488$		−	9.30660e6i		−	0.0800814i
$489$	4.87810e6	−	9.19719e7i	0.0417180	−	0.786554i
$490$	3.59066e7			0.305201
$491$		−	2.05811e8i		−	1.73870i	−0.494196	−	0.869350i	$-0.664538\pi$
$491$		−	2.05811e8i		−	1.73870i	0.494196	−	0.869350i	$-0.335462\pi$
$492$	−820254.	−	43505.4i	−0.00688737	−	0.000365299i
$493$	4.29214e7			0.358207
$494$		−	375837.i		−	0.00311758i
$495$	−1.97555e7	−	2.10154e6i	−0.162882	−	0.0173269i
$496$	−6.11000e7			−0.500722
$497$		−	1.85367e8i		−	1.50995i
$498$	−6.27492e6	+	1.18308e8i	−0.0508066	+	0.957910i
$499$	−1.81955e8			−1.46441			−0.732206	−	0.681083i	$-0.761509\pi$
$499$	−1.81955e8			−1.46441			−0.732206	+	0.681083i	$0.761509\pi$
$500$		−	2.97822e6i		−	0.0238257i
$501$	−1.58371e7	−	839984.i	−0.125940	−	0.00667972i
$502$	−4.80718e7			−0.379996
$503$		−	2.76416e7i		−	0.217200i	−0.994086	−	0.108600i	$-0.965363\pi$
$503$		−	2.76416e7i		−	0.217200i	0.994086	−	0.108600i	$-0.0346368\pi$
$504$	−2.01608e7	+	1.89522e8i	−0.157477	+	1.48036i
$505$	−1.76943e7			−0.137391
$506$		−	1.56751e8i		−	1.20993i
$507$	−6.90240e6	+	1.30138e8i	−0.0529634	+	0.998576i
$508$	1.19649e7			0.0912679
$509$		−	1.36494e6i		−	0.0103505i	−0.999987	−	0.00517525i	$-0.998353\pi$
$509$		−	1.36494e6i		−	0.0103505i	0.999987	−	0.00517525i	$-0.00164734\pi$
$510$	−5.75906e7	−	3.05455e6i	−0.434151	−	0.0230269i
$511$	3.86021e8			2.89300
$512$			1.21367e8i			0.904254i
$513$	1.42208e7	−	8.87023e7i	0.105335	−	0.657026i
$514$	1.06193e8			0.782003
$515$			2.54323e7i			0.186193i
$516$	369083.	−	6.95871e6i	0.00268642	−	0.0506500i
$517$	−6.40799e7			−0.463714
$518$			2.04115e8i			1.46854i
$519$	1.68983e8	+	8.96266e6i	1.20876	+	0.0641113i
$520$	−136467.			−0.000970552
$521$			1.63080e8i			1.15315i	0.817044	+	0.576576i	$0.195611\pi$
$521$			1.63080e8i			1.15315i	−0.817044	+	0.576576i	$0.804389\pi$
$522$	−2.69854e7	−	2.87063e6i	−0.189722	−	0.0201820i
$523$	3.30308e7			0.230895			0.115447	−	0.993314i	$-0.463170\pi$
$523$	3.30308e7			0.230895			0.115447	+	0.993314i	$0.463170\pi$
$524$		−	570065.i		−	0.00396215i
$525$	1.11934e7	−	2.11041e8i	0.0773543	−	1.45844i
$526$	6.43010e7			0.441836
$527$		−	1.34313e8i		−	0.917670i
$528$	1.15529e8	+	6.12754e6i	0.784855	+	0.0416279i
$529$	−2.20138e8			−1.48706
$530$			4.39602e7i			0.295279i
$531$	−9.43244e6	+	8.86698e7i	−0.0630000	+	0.592232i
$532$	8.55420e6			0.0568126
$533$		−	85580.8i		−	0.000565190i
$534$	811580.	−	1.53016e7i	0.00532976	−	0.100488i
$535$	−6.08498e7			−0.397372
$536$			2.23881e7i			0.145386i
$537$	1.92970e8	+	1.02349e7i	1.24614	+	0.0660940i
$538$	−1.00277e6			−0.00643956
$539$		−	1.58340e8i		−	1.01117i
$540$	1.89786e6	+	304266.i	0.0120526	+	0.00193229i
$541$	−1.35354e8			−0.854827			−0.427413	−	0.904056i	$-0.640575\pi$
$541$	−1.35354e8			−0.854827			−0.427413	+	0.904056i	$0.640575\pi$
$542$			1.15468e8i			0.725207i
$543$	1.00452e7	−	1.89393e8i	0.0627423	−	1.18295i
$544$	3.45221e7			0.214438
$545$			3.67786e7i			0.227199i
$546$	1.16845e6	+	61973.3i	0.00717847	+	0.000380738i
$547$	2.16134e8			1.32057			0.660284	−	0.751016i	$-0.270436\pi$
$547$	2.16134e8			1.32057			0.660284	+	0.751016i	$0.270436\pi$
$548$			3.98060e6i			0.0241884i
$549$	−1.35804e7	−	1.44464e6i	−0.0820719	−	0.00873057i
$550$	−1.21503e8			−0.730296
$551$		−	2.06704e7i		−	0.123565i
$552$	1.36314e7	−	2.57007e8i	0.0810442	−	1.52801i
$553$	4.16360e8			2.46203
$554$		−	2.33951e8i		−	1.37593i
$555$	−3.48849e7	−	1.85026e6i	−0.204060	−	0.0108231i
$556$	−1.24254e6			−0.00722914
$557$			9.72802e7i			0.562936i	0.959571	+	0.281468i	$0.0908213\pi$
$557$			9.72802e7i			0.562936i	−0.959571	+	0.281468i	$0.909179\pi$
$558$	−8.98298e6	+	8.44446e7i	−0.0517033	+	0.486037i
$559$	726034.			0.00415644
$560$		−	6.22133e7i		−	0.354257i
$561$	−1.34699e7	+	2.53962e8i	−0.0762913	+	1.43840i
$562$	−2.14247e8			−1.20700
$563$			2.58047e8i			1.44602i	0.690838	+	0.723010i	$0.257242\pi$
$563$			2.58047e8i			1.44602i	−0.690838	+	0.723010i	$0.742758\pi$
$564$	6.19090e6	+	328359.i	0.0345077	+	0.00183026i
$565$	5.93141e7			0.328861
$566$		−	4.93283e7i		−	0.272049i
$567$	2.73424e8	+	5.88380e7i	1.49999	+	0.322782i
$568$	−1.74979e8			−0.954862
$569$			3.26240e8i			1.77093i	0.464708	+	0.885464i	$0.346159\pi$
$569$			3.26240e8i			1.77093i	−0.464708	+	0.885464i	$0.653841\pi$
$570$	−1.47103e6	+	2.77349e7i	−0.00794323	+	0.149762i
$571$	−1.74709e8			−0.938440			−0.469220	−	0.883081i	$-0.655465\pi$
$571$	−1.74709e8			−0.938440			−0.469220	+	0.883081i	$0.655465\pi$
$572$		−	35460.5i		−	0.000189477i
$573$	−2.84510e7	−	1.50901e6i	−0.151228	−	0.00802099i
$574$	3.69523e7			0.195392
$575$			2.85384e8i			1.50116i
$576$	1.78312e8	+	1.89684e7i	0.933069	+	0.0992572i
$577$	−2.05275e8			−1.06858			−0.534292	−	0.845300i	$-0.679422\pi$
$577$	−2.05275e8			−1.06858			−0.534292	+	0.845300i	$0.679422\pi$
$578$			5.39857e8i			2.79573i
$579$	7.43232e6	−	1.40129e8i	0.0382903	−	0.721928i
$580$	442260.			0.00226670
$581$		−	2.80945e8i		−	1.43249i
$582$	3.08800e8	+	1.63784e7i	1.56642	+	0.0830813i
$583$	1.93855e8			0.978297
$584$		−	3.64387e8i		−	1.82947i
$585$	−21183.5	+	199136.i	−0.000105811	+	0.000994675i
$586$	1.91320e8			0.950751
$587$		−	1.29163e8i		−	0.638592i	−0.947655	−	0.319296i	$-0.896554\pi$
$587$		−	1.29163e8i		−	0.638592i	0.947655	−	0.319296i	$-0.103446\pi$
$588$	−811368.	+	1.52976e7i	−0.00399104	+	0.0752473i
$589$	−6.46835e7			−0.316554
$590$		−	2.75683e7i		−	0.134231i
$591$	−1.15330e8	−	6.11700e6i	−0.558703	−	0.0296330i
$592$	2.03431e8			0.980509
$593$		−	4.34746e6i		−	0.0208483i	−0.999946	−	0.0104242i	$-0.996682\pi$
$593$		−	4.34746e6i		−	0.0208483i	0.999946	−	0.0104242i	$-0.00331817\pi$
$594$	2.54539e7	−	1.58769e8i	0.121449	−	0.757540i
$595$	1.36760e8			0.649246
$596$			1.45700e7i			0.0688210i
$597$	1.32871e7	−	2.50515e8i	0.0624462	−	1.17737i
$598$	−1.58005e6			−0.00738869
$599$		−	2.07726e8i		−	0.966517i	−0.875478	−	0.483259i	$-0.839453\pi$
$599$		−	2.07726e8i		−	0.966517i	0.875478	−	0.483259i	$-0.160547\pi$
$600$	−1.99214e8	−	1.05661e7i	−0.922288	−	0.0489172i
$601$	9.97958e7			0.459715			0.229858	−	0.973224i	$-0.426174\pi$
$601$	9.97958e7			0.459715			0.229858	+	0.973224i	$0.426174\pi$
$602$			3.13489e8i			1.43692i
$603$	3.26691e7	+	3.47525e6i	0.149000	+	0.0158502i
$604$	−1.92011e7			−0.0871395
$605$			2.14907e7i			0.0970477i
$606$	7.58509e6	−	1.43010e8i	0.0340834	−	0.642611i
$607$	1.03441e8			0.462514			0.231257	−	0.972893i	$-0.425716\pi$
$607$	1.03441e8			0.462514			0.231257	+	0.972893i	$0.425716\pi$
$608$		−	1.66254e7i		−	0.0739711i
$609$	6.42629e7	+	3.40844e6i	0.284517	+	0.0150905i
$610$	4.22227e6			0.0186019
$611$			645925.i			0.00283177i
$612$	2.60271e6	−	2.44668e7i	0.0113546	−	0.106739i
$613$	−5.88853e7			−0.255638			−0.127819	−	0.991798i	$-0.540798\pi$
$613$	−5.88853e7			−0.255638			−0.127819	+	0.991798i	$0.540798\pi$
$614$			3.81792e8i			1.64939i
$615$	−334965.	+	6.31545e6i	−0.00144004	+	0.0271506i
$616$	−2.59843e8			−1.11165
$617$		−	5.11701e7i		−	0.217852i	−0.994050	−	0.108926i	$-0.965259\pi$
$617$		−	5.11701e7i		−	0.217852i	0.994050	−	0.108926i	$-0.0347411\pi$
$618$	−2.05550e8	−	1.09022e7i	−0.870869	−	0.0461900i
$619$	2.21985e8			0.935947			0.467974	−	0.883742i	$-0.344984\pi$
$619$	2.21985e8			0.935947			0.467974	+	0.883742i	$0.344984\pi$
$620$		−	1.38395e6i		−	0.00580693i
$621$	−3.72913e8	−	5.97856e7i	−1.55716	−	0.249644i
$622$	−7.89335e7			−0.328012
$623$			3.63366e7i			0.150273i
$624$	61765.6	−	1.16453e6i	0.000254210	−	0.00479289i
$625$	2.09462e8			0.857958
$626$			3.57196e8i			1.45607i
$627$	1.22305e8	+	6.48692e6i	0.496182	+	0.0263170i
$628$	−1.86669e7			−0.0753690
$629$			4.47191e8i			1.79697i
$630$	−8.59832e7	−	9.14666e6i	−0.343868	−	0.0365797i
$631$	−2.92143e8			−1.16281			−0.581404	−	0.813615i	$-0.697496\pi$
$631$	−2.92143e8			−1.16281			−0.581404	+	0.813615i	$0.697496\pi$
$632$		−	3.93026e8i		−	1.55693i
$633$	−2.67121e6	+	5.03632e7i	−0.0105317	+	0.198565i
$634$	791814.			0.00310710
$635$		−	9.21223e7i		−	0.359785i
$636$	−1.87288e7	−	993354.i	−0.0728010	−	0.00386129i
$637$	−1.59607e6			−0.00617493
$638$		−	3.69981e7i		−	0.142468i
$639$	−2.71615e7	+	2.55332e8i	−0.104100	+	0.978595i
$640$	−6.18315e7			−0.235868
$641$		−	1.18663e8i		−	0.450547i	−0.974296	−	0.225274i	$-0.927672\pi$
$641$		−	1.18663e8i		−	0.450547i	0.974296	−	0.225274i	$-0.0723276\pi$
$642$	2.60848e7	−	4.91805e8i	0.0985786	−	1.85861i
$643$	3.82833e8			1.44004			0.720022	−	0.693951i	$-0.244132\pi$
$643$	3.82833e8			1.44004			0.720022	+	0.693951i	$0.244132\pi$
$644$		−	3.59626e7i		−	0.134646i
$645$	−5.35777e7	−	2.84171e6i	−0.199667	−	0.0105901i
$646$	3.55535e8			1.31882
$647$		−	3.75784e8i		−	1.38747i	−0.720228	−	0.693737i	$-0.755963\pi$
$647$		−	3.75784e8i		−	1.38747i	0.720228	−	0.693737i	$-0.244037\pi$
$648$	5.55406e7	−	2.58101e8i	0.204120	−	0.948559i
$649$	−1.21570e8			−0.444726
$650$			1.22475e6i			0.00445971i
$651$	1.06659e7	−	2.01096e8i	0.0386595	−	0.728888i
$652$	−1.21483e7			−0.0438301
$653$			3.78212e8i			1.35830i	0.733999	+	0.679151i	$0.237652\pi$
$653$			3.78212e8i			1.35830i	−0.733999	+	0.679151i	$0.762348\pi$
$654$	−2.97255e8	−	1.57661e7i	−1.06266	−	0.0563625i
$655$	−4.38915e6			−0.0156191
$656$		−	3.68285e7i		−	0.130458i
$657$	−5.31721e8	−	5.65629e7i	−1.87494	−	0.199451i
$658$	−2.78899e8			−0.978969
$659$		−	1.02913e7i		−	0.0359595i	−0.999838	−	0.0179798i	$-0.994277\pi$
$659$		−	1.02913e7i		−	0.0359595i	0.999838	−	0.0179798i	$-0.00572344\pi$
$660$	−138793.	+	2.61681e6i	−0.000482764	+	0.00910207i
$661$	6.79279e7			0.235204			0.117602	−	0.993061i	$-0.462479\pi$
$661$	6.79279e7			0.235204			0.117602	+	0.993061i	$0.462479\pi$
$662$		−	2.39404e8i		−	0.825198i
$663$	2.55993e6	+	135776.i	0.00878391	+	0.000465890i
$664$	−2.65200e8			−0.905878
$665$		−	6.58620e7i		−	0.223960i
$666$	2.99086e7	−	2.81156e8i	0.101245	−	0.951754i
$667$	−8.69004e7			−0.292849
$668$			2.09188e6i			0.00701789i
$669$	1.32871e7	−	2.50515e8i	0.0443762	−	0.836672i
$670$	−1.01572e7			−0.0337713
$671$		−	1.86193e7i		−	0.0616304i
$672$	5.16872e7	+	2.74144e6i	0.170324	+	0.00903380i
$673$	−2.86900e8			−0.941207			−0.470604	−	0.882345i	$-0.655964\pi$
$673$	−2.86900e8			−0.941207			−0.470604	+	0.882345i	$0.655964\pi$
$674$		−	1.09868e8i		−	0.358831i
$675$	−4.63418e7	+	2.89057e8i	−0.150682	+	0.939879i
$676$	1.71896e7			0.0556448
$677$		−	1.31298e8i		−	0.423149i	−0.977362	−	0.211574i	$-0.932141\pi$
$677$		−	1.31298e8i		−	0.423149i	0.977362	−	0.211574i	$-0.0678590\pi$
$678$	−2.54265e7	+	4.79392e8i	−0.0815825	+	1.53816i
$679$	−7.33306e8			−2.34248
$680$		−	1.29096e8i		−	0.410569i
$681$	3.93430e8	+	2.08671e7i	1.24574	+	0.0660727i
$682$	−1.15777e8			−0.364981
$683$			9.64308e7i			0.302659i	0.988483	+	0.151330i	$0.0483555\pi$
$683$			9.64308e7i			0.302659i	−0.988483	+	0.151330i	$0.951645\pi$
$684$	−1.17829e7	−	1.25343e6i	−0.0368200	−	0.00391681i
$685$	3.06482e7			0.0953526
$686$		−	1.80234e8i		−	0.558296i
$687$	−208517.	+	3.93139e6i	−0.000643089	+	0.0121248i
$688$	3.12438e8			0.959397
$689$		−	1.95406e6i		−	0.00597419i
$690$	1.16600e8	+	6.18435e6i	0.354937	+	0.0188255i
$691$	5.37804e8			1.63001			0.815005	−	0.579455i	$-0.196734\pi$
$691$	5.37804e8			1.63001			0.815005	+	0.579455i	$0.196734\pi$
$692$		−	2.23204e7i		−	0.0673571i
$693$	−4.03347e7	+	3.79167e8i	−0.121193	+	1.13928i
$694$	−2.49365e8			−0.746032
$695$			9.56680e6i			0.0284979i
$696$	3.21742e6	−	6.06614e7i	0.00954289	−	0.179922i
$697$	8.09581e7			0.239090
$698$		−	4.93122e7i		−	0.145007i
$699$	−1.75998e8	−	9.33476e6i	−0.515320	−	0.0273320i
$700$	−2.78758e7			−0.0812706
$701$			8.72547e7i			0.253300i	0.991947	+	0.126650i	$0.0404225\pi$
$701$			8.72547e7i			0.253300i	−0.991947	+	0.126650i	$0.959578\pi$
$702$	−1.60039e6	−	256575.i	−0.00462609	−	0.000741658i
$703$	2.15362e8			0.619873
$704$			2.44474e8i			0.700671i
$705$	2.52816e6	−	4.76661e7i	0.00721502	−	0.136032i
$706$	−1.21240e7			−0.0344533
$707$			3.39605e8i			0.960983i
$708$	1.17452e7	+	622951.i	0.0330948	+	0.00175531i
$709$	−3.81980e8			−1.07177			−0.535886	−	0.844290i	$-0.680022\pi$
$709$	−3.81980e8			−1.07177			−0.535886	+	0.844290i	$0.680022\pi$
$710$		−	7.93853e7i		−	0.221802i
$711$	−5.73511e8	−	6.10085e7i	−1.59563	−	0.169739i
$712$	3.43002e7			0.0950292
$713$			2.71935e8i			0.750234i
$714$	−5.86258e7	+	1.10533e9i	−0.161062	+	3.03668i
$715$	−273023.			−0.000746933
$716$		−	2.54888e7i		−	0.0694402i
$717$	1.84886e8	+	9.80619e6i	0.501589	+	0.0266037i
$718$	2.28996e8			0.618663
$719$		−	6.86531e8i		−	1.84703i	−0.383564	−	0.923514i	$-0.625303\pi$
$719$		−	6.86531e8i		−	1.84703i	0.383564	−	0.923514i	$-0.374697\pi$
$720$	−9.11600e6	+	8.56951e7i	−0.0244234	+	0.229593i
$721$	4.88120e8			1.30233
$722$			2.15476e8i			0.572514i
$723$	591550.	−	1.11531e7i	0.00156523	−	0.0295108i
$724$	−2.50164e7			−0.0659188
$725$			6.73593e7i			0.176760i
$726$	−1.73694e8	−	9.21255e6i	−0.453916	−	0.0240752i
$727$	−7.14336e8			−1.85909			−0.929543	−	0.368714i	$-0.879798\pi$
$727$	−7.14336e8			−1.85909			−0.929543	+	0.368714i	$0.879798\pi$
$728$			2.61921e6i			0.00678854i
$729$	−3.68004e8	−	1.21110e8i	−0.949883	−	0.312607i
$730$	1.65317e8			0.424961
$731$			6.86816e8i			1.75828i
$732$	−95409.1	+	1.79885e6i	−0.000243252	+	0.00458629i
$733$	−3.40557e8			−0.864725			−0.432363	−	0.901700i	$-0.642320\pi$
$733$	−3.40557e8			−0.864725			−0.432363	+	0.901700i	$0.642320\pi$
$734$		−	3.73793e8i		−	0.945242i
$735$	1.17782e8	+	6.24703e6i	0.296631	+	0.0157330i
$736$	−6.98948e7			−0.175312
$737$			4.47908e7i			0.111889i
$738$	−5.08996e7	−	5.41455e6i	−0.126632	−	0.0134708i
$739$	−8.56092e7			−0.212123			−0.106061	−	0.994360i	$-0.533824\pi$
$739$	−8.56092e7			−0.212123			−0.106061	+	0.994360i	$0.533824\pi$
$740$			4.60784e6i			0.0113711i
$741$	65388.1	−	1.23283e6i	0.000160710	−	0.00303004i
$742$	8.43727e8			2.06533
$743$		−	3.20134e8i		−	0.780486i	−0.920712	−	0.390243i	$-0.872391\pi$
$743$		−	3.20134e8i		−	0.780486i	0.920712	−	0.390243i	$-0.127609\pi$
$744$	−1.89826e8	−	1.00682e7i	−0.460933	−	0.0244474i
$745$	1.12180e8			0.271298
$746$		−	4.00370e8i		−	0.964374i
$747$	−4.11664e7	+	3.86985e8i	−0.0987599	+	0.928394i
$748$	3.35450e7			0.0801537
$749$			1.16789e9i			2.77943i
$750$	9.82971e6	−	1.85330e8i	0.0233001	−	0.439301i
$751$	7.03872e8			1.66178			0.830890	−	0.556436i	$-0.187832\pi$
$751$	7.03872e8			1.66178			0.830890	+	0.556436i	$0.187832\pi$
$752$			2.77964e8i			0.653635i
$753$	−1.57687e8	−	8.36354e6i	−0.369326	−	0.0195887i
$754$	−372941.			−0.000870013
$755$			1.47836e8i			0.343511i
$756$	5.83976e6	−	3.64255e7i	0.0135154	−	0.0843024i
$757$	−5.02352e8			−1.15803			−0.579016	−	0.815316i	$-0.696563\pi$
$757$	−5.02352e8			−1.15803			−0.579016	+	0.815316i	$0.696563\pi$
$758$			3.25027e8i			0.746298i
$759$	2.72716e7	−	5.14180e8i	0.0623714	−	1.17595i
$760$	−6.21710e7			−0.141627
$761$		−	6.15805e8i		−	1.39730i	−0.715464	−	0.698650i	$-0.753785\pi$
$761$		−	6.15805e8i		−	1.39730i	0.715464	−	0.698650i	$-0.246215\pi$
$762$	7.44557e8	+	3.94906e7i	1.68280	+	0.0892541i
$763$	7.05890e8			1.58914
$764$			3.75800e6i			0.00842707i
$765$	−1.88379e8	−	2.00392e7i	−0.420774	−	0.0447607i
$766$	−6.51463e8			−1.44945
$767$			1.22543e6i			0.00271582i
$768$	3.99302e6	−	7.52847e7i	0.00881492	−	0.166197i
$769$	−3.83492e8			−0.843291			−0.421646	−	0.906761i	$-0.638547\pi$
$769$	−3.83492e8			−0.843291			−0.421646	+	0.906761i	$0.638547\pi$
$770$		−	1.17887e8i		−	0.258222i
$771$	3.48339e8	+	1.84755e7i	0.760045	+	0.0403120i
$772$	−1.85093e7			−0.0402288
$773$		−	1.45254e8i		−	0.314477i	−0.987561	−	0.157239i	$-0.949741\pi$
$773$		−	1.45254e8i		−	0.314477i	0.987561	−	0.157239i	$-0.0502592\pi$
$774$	4.59349e7	−	4.31812e8i	0.0990650	−	0.931261i
$775$	2.10786e8			0.452831
$776$			6.92210e8i			1.48133i
$777$	−3.55119e7	+	6.69544e8i	−0.0757026	+	1.42730i
$778$	7.51955e8			1.59681
$779$		−	3.89884e7i		−	0.0824751i
$780$	26377.4	+	1399.03i	5.55838e−5	+	2.94811e-6i
$781$	−3.50072e8			−0.734858
$782$		−	1.49470e9i		−	3.12561i
$783$	−8.80188e7	−	1.41112e7i	−0.183354	−	0.0293954i
$784$	−6.86843e8			−1.42531
$785$			1.43723e8i			0.297111i
$786$	1.88152e6	−	3.54743e7i	0.00387473	−	0.0730543i
$787$	5.41839e8			1.11159			0.555797	−	0.831318i	$-0.312413\pi$
$787$	5.41839e8			1.11159			0.555797	+	0.831318i	$0.312413\pi$
$788$			1.52336e7i			0.0311333i
$789$	2.10922e8	+	1.11871e7i	0.429429	+	0.0227765i
$790$	1.78310e8			0.361655
$791$		−	1.13841e9i		−	2.30022i
$792$	3.57918e8	+	3.80743e7i	0.720456	+	0.0766401i
$793$	−187682.			−0.000376360
$794$			6.91970e8i			1.38237i
$795$	−7.64821e6	+	1.44200e8i	−0.0152215	+	0.286988i
$796$	−3.30898e7			−0.0656077
$797$			4.02944e7i			0.0795921i	0.999208	+	0.0397961i	$0.0126708\pi$
$797$			4.02944e7i			0.0795921i	−0.999208	+	0.0397961i	$0.987329\pi$
$798$	5.32315e8	+	2.82334e7i	1.04751	+	0.0555591i
$799$	−6.11035e8			−1.19791
$800$			5.41777e7i			0.105816i
$801$	5.32434e6	−	5.00515e7i	0.0103602	−	0.0973912i
$802$	3.42769e8			0.664474
$803$		−	7.29012e8i		−	1.40795i
$804$	229518.	−	4.32734e6i	0.000441619	−	0.00832632i
$805$	−2.76890e8			−0.530786
$806$			1.16703e6i			0.00222884i
$807$	−3.28933e6	−	174463.i	−0.00625874	−	0.000331957i
$808$	3.20573e8			0.607705
$809$		−	7.65099e8i		−	1.44501i	−0.691364	−	0.722507i	$-0.742990\pi$
$809$		−	7.65099e8i		−	1.44501i	0.691364	−	0.722507i	$-0.257010\pi$
$810$	1.17097e8	+	2.51979e7i	0.220338	+	0.0474144i
$811$	2.70817e8			0.507707			0.253854	−	0.967243i	$-0.418302\pi$
$811$	2.70817e8			0.507707			0.253854	+	0.967243i	$0.418302\pi$
$812$		−	8.48828e6i		−	0.0158545i
$813$	−2.00891e7	+	3.78760e8i	−0.0373842	+	0.704844i
$814$	3.85477e8			0.714702
$815$			9.35343e7i			0.172782i
$816$	1.10163e9	+	5.84293e7i	2.02752	+	0.107538i
$817$	3.30762e8			0.606526
$818$			3.04754e8i			0.556787i
$819$	3.82200e6	+	406574.i	0.00695727	+	0.000740095i
$820$	834188.			0.00151294
$821$			3.68452e8i			0.665812i	0.942960	+	0.332906i	$0.108029\pi$
$821$			3.68452e8i			0.665812i	−0.942960	+	0.332906i	$0.891971\pi$
$822$	−1.31381e7	+	2.47707e8i	−0.0236547	+	0.445987i
$823$	−1.09763e8			−0.196904			−0.0984521	−	0.995142i	$-0.531389\pi$
$823$	−1.09763e8			−0.196904			−0.0984521	+	0.995142i	$0.531389\pi$
$824$		−	4.60765e8i		−	0.823565i
$825$	−3.98558e8	−	2.11391e7i	−0.709790	−	0.0376465i
$826$	−5.29117e8			−0.938884
$827$			5.95216e8i			1.05234i	0.850378	+	0.526172i	$0.176373\pi$
$827$			5.95216e8i			1.05234i	−0.850378	+	0.526172i	$0.823627\pi$
$828$	−5.26954e6	+	4.95364e7i	−0.00928286	+	0.0872636i
$829$	1.18601e8			0.208173			0.104087	−	0.994568i	$-0.466808\pi$
$829$	1.18601e8			0.208173			0.104087	+	0.994568i	$0.466808\pi$
$830$		−	1.20317e8i		−	0.210423i
$831$	4.07028e7	−	7.67414e8i	0.0709286	−	1.33729i
$832$	2.46429e6			0.00427881
$833$		−	1.50985e9i		−	2.61216i
$834$	−7.73215e7	−	4.10105e6i	−0.133291	−	0.00706964i
$835$	1.61062e7			0.0276651
$836$		−	1.61549e7i		−	0.0276493i
$837$	−4.41580e7	+	2.75435e8i	−0.0753065	+	0.469724i
$838$	4.81442e8			0.818110
$839$		−	5.53616e8i		−	0.937395i	−0.883359	−	0.468697i	$-0.844723\pi$
$839$		−	5.53616e8i		−	0.937395i	0.883359	−	0.468697i	$-0.155277\pi$
$840$	1.02516e7	−	1.93285e8i	0.0172964	−	0.326107i
$841$	−2.05111e7			−0.0344828
$842$			6.40681e8i			1.07326i
$843$	−7.02780e8	−	3.72747e7i	−1.17310	−	0.0622202i
$844$	6.65232e6			0.0110649
$845$		−	1.32349e8i		−	0.219356i
$846$	3.84167e8	+	4.08666e7i	0.634466	+	0.0674927i
$847$	4.12471e8			0.678802
$848$		−	8.40899e8i		−	1.37897i
$849$	8.58215e6	−	1.61808e8i	0.0140240	−	0.264410i
$850$	−1.15859e9			−1.88658
$851$		−	9.05400e8i		−	1.46910i
$852$	3.38212e7	+	1.79384e6i	0.0546853	+	0.00290045i
$853$	5.11960e8			0.824877			0.412438	−	0.910986i	$-0.364677\pi$
$853$	5.11960e8			0.824877			0.412438	+	0.910986i	$0.364677\pi$
$854$		−	8.10378e7i		−	0.130111i
$855$	−9.65065e6	+	9.07210e7i	−0.0154404	+	0.145147i
$856$	1.10244e9			1.75765
$857$			4.09778e8i			0.651038i	0.945535	+	0.325519i	$0.105539\pi$
$857$			4.09778e8i			0.651038i	−0.945535	+	0.325519i	$0.894461\pi$
$858$	117038.	−	2.20665e6i	0.000185296	−	0.00349359i
$859$	−8.97643e8			−1.41620			−0.708099	−	0.706113i	$-0.750447\pi$
$859$	−8.97643e8			−1.41620			−0.708099	+	0.706113i	$0.750447\pi$
$860$			7.07692e6i			0.0111263i
$861$	1.21212e8	+	6.42896e6i	0.189905	+	0.0100724i
$862$	−6.76384e8			−1.05602
$863$			4.46254e7i			0.0694305i	0.999397	+	0.0347152i	$0.0110524\pi$
$863$			4.46254e7i			0.0694305i	−0.999397	+	0.0347152i	$0.988948\pi$
$864$	−7.07944e7	−	1.13498e7i	−0.109763	−	0.0175973i
$865$	−1.71853e8			−0.265527
$866$			5.61368e8i			0.864359i
$867$	−9.39244e7	+	1.77086e9i	−0.144119	+	2.71723i
$868$	−2.65622e7			−0.0406167
$869$		−	7.86308e8i		−	1.19821i
$870$	2.75212e7	+	1.45970e6i	0.0417936	+	0.00221669i
$871$	451491.			0.000683274
$872$		−	6.66330e8i		−	1.00494i
$873$	1.01009e9	+	1.07450e8i	1.51815	+	0.161497i
$874$	−7.19830e8			−1.07819
$875$			4.40103e8i			0.656946i
$876$	−3.73561e6	+	7.04315e7i	−0.00555712	+	0.104774i
$877$	−7.36575e8			−1.09199			−0.545994	−	0.837789i	$-0.683848\pi$
$877$	−7.36575e8			−1.09199			−0.545994	+	0.837789i	$0.683848\pi$
$878$		−	7.18087e8i		−	1.06095i
$879$	6.27573e8	+	3.32858e7i	0.924055	+	0.0490109i
$880$	−1.17492e8			−0.172409
$881$			9.15425e8i			1.33874i	0.742931	+	0.669368i	$0.233435\pi$
$881$			9.15425e8i			1.33874i	−0.742931	+	0.669368i	$0.766565\pi$
$882$	−1.00980e8	+	9.49267e8i	−0.147174	+	1.38351i
$883$	6.92368e8			1.00567			0.502834	−	0.864383i	$-0.332291\pi$
$883$	6.92368e8			1.00567			0.502834	+	0.864383i	$0.332291\pi$
$884$		−	338134.i		−	0.000489476i
$885$	4.79634e6	−	9.04305e7i	0.00691958	−	0.130462i
$886$	−8.94017e7			−0.128542
$887$			9.81846e8i			1.40693i	0.710730	+	0.703465i	$0.248365\pi$
$887$			9.81846e8i			1.40693i	−0.710730	+	0.703465i	$0.751635\pi$
$888$	6.32021e8	+	3.35218e7i	0.902595	+	0.0478727i
$889$	−1.76810e9			−2.51652
$890$			1.55615e7i			0.0220740i
$891$	1.11117e8	−	5.16370e8i	0.157090	−	0.730008i
$892$	−3.30898e7			−0.0466229
$893$			2.94267e8i			0.413225i
$894$	−4.80888e7	+	9.06669e8i	−0.0673025	+	1.26893i
$895$	−1.96248e8			−0.273739
$896$			1.18673e9i			1.64979i
$897$	−5.18294e6	−	274898.i	−0.00718122	−	0.000380885i
$898$	1.15914e9			1.60068
$899$			6.41850e7i			0.0883395i
$900$	3.83973e7	+	4.08459e6i	0.0526712	+	0.00560301i
$901$	1.84851e9			2.52724
$902$		−	6.97855e7i		−	0.0950924i
$903$	−5.45408e7	+	1.02832e9i	−0.0740727	+	1.39657i
$904$	−1.07461e9			−1.45461
$905$			1.92611e8i			0.259857i
$906$	−1.19485e9	−	6.33738e7i	−1.60668	−	0.0852168i
$907$	1.08447e9			1.45344			0.726718	−	0.686936i	$-0.241045\pi$
$907$	1.08447e9			1.45344			0.726718	+	0.686936i	$0.241045\pi$
$908$		−	5.19670e7i		−	0.0694177i
$909$	4.97617e7	−	4.67786e8i	0.0662527	−	0.622810i
$910$	−1.18830e6			−0.00157689
$911$			2.80031e8i			0.370383i	0.982702	+	0.185192i	$0.0592906\pi$
$911$			2.80031e8i			0.370383i	−0.982702	+	0.185192i	$0.940709\pi$
$912$	2.81388e7	−	5.30531e8i	0.0370955	−	0.699401i
$913$	−5.30573e8			−0.697160
$914$			1.11719e8i			0.146315i
$915$	1.38500e7	+	734591.i	0.0180795	+	0.000958920i
$916$	519285.			0.000675647
$917$			8.42407e7i			0.109248i
$918$	2.42716e8	−	1.51394e9i	0.313741	−	1.95696i
$919$	−2.37598e8			−0.306124			−0.153062	−	0.988217i	$-0.548913\pi$
$919$	−2.37598e8			−0.306124			−0.153062	+	0.988217i	$0.548913\pi$
$920$			2.61373e8i			0.335658i
$921$	−6.64243e7	+	1.25237e9i	−0.0850253	+	1.60307i
$922$	1.15756e9			1.47690
$923$			3.52872e6i			0.00448758i
$924$	5.02243e7	+	2.66384e6i	0.0636646	+	0.00337670i
$925$	−7.01805e8			−0.886730
$926$			4.74171e8i			0.597176i
$927$	−6.72356e8	−	7.15234e7i	−0.844035	−	0.0897860i
$928$	−1.64973e7			−0.0206428
$929$		−	4.60429e8i		−	0.574269i	−0.957890	−	0.287135i	$-0.907297\pi$
$929$		−	4.60429e8i		−	0.574269i	0.957890	−	0.287135i	$-0.0927027\pi$
$930$	4.56779e6	−	8.61214e7i	0.00567881	−	0.107069i
$931$	−7.27126e8			−0.901074
$932$			2.32471e7i			0.0287158i
$933$	−2.58920e8	−	1.37329e7i	−0.318802	−	0.0169089i
$934$	5.30988e8			0.651694
$935$		−	2.58276e8i		−	0.315972i
$936$	383789.	−	3.60781e6i	0.000468020	−	0.00439963i
$937$	−1.43131e8			−0.173986			−0.0869932	−	0.996209i	$-0.527726\pi$
$937$	−1.43131e8			−0.173986			−0.0869932	+	0.996209i	$0.527726\pi$
$938$			1.94946e8i			0.236214i
$939$	−6.21450e7	+	1.17169e9i	−0.0750601	+	1.41519i
$940$	−6.29607e6			−0.00758029
$941$		−	8.07757e8i		−	0.969420i	−0.874675	−	0.484710i	$-0.838925\pi$
$941$		−	8.07757e8i		−	0.969420i	0.874675	−	0.484710i	$-0.161075\pi$
$942$	−1.16161e9	−	6.16107e7i	−1.38966	−	0.0737061i
$943$	−1.63911e8			−0.195467
$944$			5.27344e8i			0.626871i
$945$	−2.80454e8	−	4.49625e7i	−0.332327	−	0.0532789i
$946$	5.92033e8			0.699314
$947$			3.69806e8i			0.435436i	0.976012	+	0.217718i	$0.0698613\pi$
$947$			3.69806e8i			0.435436i	−0.976012	+	0.217718i	$0.930139\pi$
$948$	−4.02921e6	+	7.59670e7i	−0.00472928	+	0.0891661i
$949$	−7.34844e6			−0.00859798
$950$			5.57964e8i			0.650782i
$951$	2.59733e6	+	137760.i	0.00301986	+	0.000160170i
$952$	−2.47773e9			−2.87173
$953$		−	3.86658e8i		−	0.446733i	−0.974735	−	0.223367i	$-0.928295\pi$
$953$		−	3.86658e8i		−	0.446733i	0.974735	−	0.223367i	$-0.0717047\pi$
$954$	−1.16218e9	−	1.23630e8i	−1.33853	−	0.142390i
$955$	2.89343e7			0.0332202
$956$		−	2.44211e7i		−	0.0279506i
$957$	6.43693e6	−	1.21362e8i	0.00734418	−	0.138468i
$958$	1.55008e9			1.76302
$959$		−	5.88229e8i		−	0.666946i
$960$	−1.81853e8	−	9.64529e6i	−0.205545	−	0.0109019i
$961$	−6.86651e8			−0.773688
$962$		−	3.88560e6i		−	0.00436449i
$963$	1.71128e8	−	1.60869e9i	0.191621	−	1.80134i
$964$	−1.47318e6			−0.00164447
$965$			1.42510e8i			0.158585i
$966$	1.18696e8	−	2.23790e9i	0.131675	−	2.48261i
$967$	−1.01671e9			−1.12440			−0.562198	−	0.827003i	$-0.690044\pi$
$967$	−1.01671e9			−1.12440			−0.562198	+	0.827003i	$0.690044\pi$
$968$		−	3.89355e8i		−	0.429259i
$969$	1.16624e9	+	6.18561e7i	1.28179	+	0.0679847i
$970$	−3.14045e8			−0.344094
$971$			1.20538e9i			1.31664i	0.752739	+	0.658319i	$0.228732\pi$
$971$			1.20538e9i			1.31664i	−0.752739	+	0.658319i	$0.771268\pi$
$972$	−1.33813e7	+	4.93183e7i	−0.0145713	+	0.0537043i
$973$	1.83615e8			0.199329
$974$			1.31577e9i			1.42398i
$975$	−213082.	+	4.01746e6i	−0.000229897	+	0.00433449i
$976$	−8.07662e7			−0.0868721
$977$		−	1.51726e9i		−	1.62696i	−0.581591	−	0.813481i	$-0.697570\pi$
$977$		−	1.51726e9i		−	1.62696i	0.581591	−	0.813481i	$-0.302430\pi$
$978$	−7.55970e8	−	4.00959e7i	−0.808142	−	0.0428630i
$979$	6.86228e7			0.0731342
$980$		−	1.55574e7i		−	0.0165295i
$981$	−9.72321e8	−	1.03433e8i	−1.02992	−	0.109560i
$982$	−1.69168e9			−1.78642
$983$			6.17688e8i			0.650292i	0.945664	+	0.325146i	$0.105413\pi$
$983$			6.17688e8i			0.650292i	−0.945664	+	0.325146i	$0.894587\pi$
$984$	6.06867e6	−	1.14419e8i	0.00636955	−	0.120092i
$985$	1.17289e8			0.122730
$986$		−	3.52796e8i		−	0.368038i
$987$	−9.14853e8	−	4.85229e7i	−0.951481	−	0.0504656i
$988$	−162841.			−0.000168847
$989$		−	1.39055e9i		−	1.43747i
$990$	−1.72737e7	+	1.62382e8i	−0.0178025	+	0.167352i
$991$	8.78695e8			0.902853			0.451427	−	0.892308i	$-0.350915\pi$
$991$	8.78695e8			0.902853			0.451427	+	0.892308i	$0.350915\pi$
$992$			5.16246e7i			0.0528837i
$993$	4.16516e7	−	7.85302e8i	0.0425387	−	0.802027i
$994$	−1.52364e9			−1.55140
$995$			2.54771e8i			0.258631i
$996$	5.12599e7	+	2.71877e6i	0.0518799	+	0.00275166i
$997$	1.83961e9			1.85626			0.928131	−	0.372253i	$-0.121415\pi$
$997$	1.83961e9			1.85626			0.928131	+	0.372253i	$0.121415\pi$
$998$			1.49559e9i			1.50460i
$999$	1.47023e8	−	9.17053e8i	0.147465	−	0.919810i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	87.7.b.a.59.16	✓	56
3.2	odd	2	inner	87.7.b.a.59.41	yes	56

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
87.7.b.a.59.16	✓	56	1.1	even	1	trivial
87.7.b.a.59.41	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-8.21957i q^{2} +(1.43004 - 26.9621i) q^{3} -3.56134 q^{4} +27.4201i q^{5} +(-221.617 - 11.7543i) q^{6} +526.273 q^{7} -496.780i q^{8} +(-724.910 - 77.1139i) q^{9} +225.382 q^{10} -993.883i q^{11} +(-5.09287 + 96.0213i) q^{12} -10.0183 q^{13} -4325.74i q^{14} +(739.304 + 39.2119i) q^{15} -4311.24 q^{16} -9477.19i q^{17} +(-633.843 + 5958.45i) q^{18} -4564.09 q^{19} -97.6524i q^{20} +(752.593 - 14189.4i) q^{21} -8169.29 q^{22} +19187.9i q^{23} +(-13394.2 - 710.416i) q^{24} +14873.1 q^{25} +82.3464i q^{26} +(-3115.80 + 19434.8i) q^{27} -1874.24 q^{28} +4528.92i q^{29} +(322.305 - 6076.76i) q^{30} +14172.3 q^{31} +3642.66i q^{32} +(-26797.2 - 1421.29i) q^{33} -77898.4 q^{34} +14430.5i q^{35} +(2581.65 + 274.629i) q^{36} -47186.1 q^{37} +37514.9i q^{38} +(-14.3266 + 270.115i) q^{39} +13621.8 q^{40} +8542.42i q^{41} +(-116631. - 6185.99i) q^{42} -72470.5 q^{43} +3539.56i q^{44} +(2114.47 - 19877.1i) q^{45} +157716. q^{46} -64474.3i q^{47} +(-6165.26 + 116240. i) q^{48} +159314. q^{49} -122251. i q^{50} +(-255525. - 13552.8i) q^{51} +35.6787 q^{52} +195048. i q^{53} +(159746. + 25610.6i) q^{54} +27252.4 q^{55} -261442. i q^{56} +(-6526.84 + 123058. i) q^{57} +37225.8 q^{58} -122318. i q^{59} +(-2632.91 - 139.647i) q^{60} +18733.9 q^{61} -116490. i q^{62} +(-381501. - 40583.0i) q^{63} -245978. q^{64} -274.704i q^{65} +(-11682.4 + 220261. i) q^{66} -45066.5 q^{67} +33751.5i q^{68} +(517345. + 27439.5i) q^{69} +118612. q^{70} -352226. i q^{71} +(-38308.6 + 360121. i) q^{72} +733499. q^{73} +387849. i q^{74} +(21269.2 - 401011. i) q^{75} +16254.3 q^{76} -523054. i q^{77} +(2220.23 + 117.759i) q^{78} +791148. q^{79} -118215. i q^{80} +(519548. + 111801. i) q^{81} +70215.0 q^{82} -533839. i q^{83} +(-2680.24 + 50533.4i) q^{84} +259866. q^{85} +595676. i q^{86} +(122109. + 6476.55i) q^{87} -493741. q^{88} +69045.2i q^{89} +(-163381. - 17380.1i) q^{90} -5272.38 q^{91} -68334.5i q^{92} +(20266.9 - 382114. i) q^{93} -529951. q^{94} -125148. i q^{95} +(98213.7 + 5209.15i) q^{96} -1.39339e6 q^{97} -1.30950e6i q^{98} +(-76642.2 + 720476. 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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