LMFDB - Embedded newform 87.7.b.a.59.8

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			59.8
Character	$\chi$	$=$	87.59
Dual form			87.7.b.a.59.49

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q-12.4411i q^{2} +(1.67176 + 26.9482i) q^{3} -90.7821 q^{4} -69.4567i q^{5} +(335.266 - 20.7986i) q^{6} +181.456 q^{7} +333.200i q^{8} +(-723.410 + 90.1019i) q^{9} -864.121 q^{10} +2518.11i q^{11} +(-151.766 - 2446.41i) q^{12} -1777.19 q^{13} -2257.52i q^{14} +(1871.73 - 116.115i) q^{15} -1664.67 q^{16} +1548.23i q^{17} +(1120.97 + 9000.05i) q^{18} +12597.7 q^{19} +6305.43i q^{20} +(303.351 + 4889.91i) q^{21} +31328.2 q^{22} +13378.3i q^{23} +(-8979.13 + 557.031i) q^{24} +10800.8 q^{25} +22110.3i q^{26} +(-3637.45 - 19344.0i) q^{27} -16473.0 q^{28} +4528.92i q^{29} +(-1444.61 - 23286.5i) q^{30} +16974.1 q^{31} +42035.1i q^{32} +(-67858.6 + 4209.69i) q^{33} +19261.7 q^{34} -12603.3i q^{35} +(65672.7 - 8179.64i) q^{36} +95.9247 q^{37} -156730. i q^{38} +(-2971.04 - 47892.1i) q^{39} +23143.0 q^{40} +44000.1i q^{41} +(60836.1 - 3774.04i) q^{42} +122875. q^{43} -228600. i q^{44} +(6258.19 + 50245.7i) q^{45} +166441. q^{46} +138705. i q^{47} +(-2782.93 - 44859.8i) q^{48} -84722.7 q^{49} -134374. i q^{50} +(-41721.9 + 2588.27i) q^{51} +161337. q^{52} +87390.6i q^{53} +(-240661. + 45254.1i) q^{54} +174900. q^{55} +60461.1i q^{56} +(21060.4 + 339486. i) q^{57} +56345.0 q^{58} -134377. i q^{59} +(-169920. + 10541.2i) q^{60} -408931. q^{61} -211177. i q^{62} +(-131267. + 16349.5i) q^{63} +416427. q^{64} +123438. i q^{65} +(52373.3 + 844239. i) q^{66} -377861. q^{67} -140551. i q^{68} +(-360520. + 22365.3i) q^{69} -156800. q^{70} +283247. i q^{71} +(-30021.9 - 241040. i) q^{72} +232402. q^{73} -1193.41i q^{74} +(18056.3 + 291061. i) q^{75} -1.14365e6 q^{76} +456927. i q^{77} +(-595832. + 36963.1i) q^{78} +496261. q^{79} +115622. i q^{80} +(515204. - 130361. i) q^{81} +547412. q^{82} -69521.1i q^{83} +(-27538.9 - 443916. i) q^{84} +107535. q^{85} -1.52870e6i q^{86} +(-122046. + 7571.28i) q^{87} -839035. q^{88} +115565. i q^{89} +(625114. - 77859.0i) q^{90} -322482. q^{91} -1.21451e6i q^{92} +(28376.6 + 457420. i) q^{93} +1.72565e6 q^{94} -874998. i q^{95} +(-1.13277e6 + 70272.8i) q^{96} -928194. q^{97} +1.05405e6i q^{98} +(-226887. - 1.82163e6i) 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 $

$2$		−	12.4411i		−	1.55514i	−0.628794	−	0.777572i	$-0.716451\pi$
$2$		−	12.4411i		−	1.55514i	0.628794	−	0.777572i	$-0.283549\pi$
$3$	1.67176	+	26.9482i	0.0619171	+	0.998081i
$3$	1.67176	+	26.9482i	0.0619171	+	0.998081i
$4$	−90.7821			−1.41847
$5$		−	69.4567i		−	0.555654i	−0.960631	−	0.277827i	$-0.910386\pi$
$5$		−	69.4567i		−	0.555654i	0.960631	−	0.277827i	$-0.0896142\pi$
$6$	335.266	−	20.7986i	1.55216	−	0.0962900i
$7$	181.456			0.529026			0.264513	−	0.964382i	$-0.414789\pi$
$7$	181.456			0.529026			0.264513	+	0.964382i	$0.414789\pi$
$8$			333.200i			0.650781i
$9$	−723.410	+	90.1019i	−0.992333	+	0.123597i
$10$	−864.121			−0.864121
$11$			2518.11i			1.89190i	0.324318	+	0.945948i	$0.394865\pi$
$11$			2518.11i			1.89190i	−0.324318	+	0.945948i	$0.605135\pi$
$12$	−151.766	−	2446.41i	−0.0878276	−	1.41575i
$13$	−1777.19			−0.808917			−0.404459	−	0.914556i	$-0.632540\pi$
$13$	−1777.19			−0.808917			−0.404459	+	0.914556i	$0.632540\pi$
$14$		−	2257.52i		−	0.822711i
$15$	1871.73	−	116.115i	0.554588	−	0.0344045i
$16$	−1664.67			−0.406413
$17$			1548.23i			0.315129i	0.987509	+	0.157564i	$0.0503642\pi$
$17$			1548.23i			0.315129i	−0.987509	+	0.157564i	$0.949636\pi$
$18$	1120.97	+	9000.05i	0.192210	+	1.54322i
$19$	12597.7			1.83667			0.918336	−	0.395801i	$-0.129533\pi$
$19$	12597.7			1.83667			0.918336	+	0.395801i	$0.129533\pi$
$20$			6305.43i			0.788178i
$21$	303.351	+	4889.91i	0.0327558	+	0.528011i
$22$	31328.2			2.94217
$23$			13378.3i			1.09955i	0.835311	+	0.549777i	$0.185287\pi$
$23$			13378.3i			1.09955i	−0.835311	+	0.549777i	$0.814713\pi$
$24$	−8979.13	+	557.031i	−0.649532	+	0.0402945i
$25$	10800.8			0.691249
$26$			22110.3i			1.25798i
$27$	−3637.45	−	19344.0i	−0.184802	−	0.982776i
$28$	−16473.0			−0.750408
$29$			4528.92i			0.185695i
$29$			4528.92i			0.185695i
$30$	−1444.61	−	23286.5i	−0.0535039	−	0.862463i
$31$	16974.1			0.569771			0.284886	−	0.958561i	$-0.408044\pi$
$31$	16974.1			0.569771			0.284886	+	0.958561i	$0.408044\pi$
$32$			42035.1i			1.28281i
$33$	−67858.6	+	4209.69i	−1.88827	+	0.117141i
$34$	19261.7			0.490070
$35$		−	12603.3i		−	0.293955i
$36$	65672.7	−	8179.64i	1.40759	−	0.175318i
$37$	95.9247			0.00189376			0.000946880	−	1.00000i	$-0.499699\pi$
$37$	95.9247			0.00189376			0.000946880		1.00000i	$0.499699\pi$
$38$		−	156730.i		−	2.85629i
$39$	−2971.04	−	47892.1i	−0.0500858	−	0.807365i
$40$	23143.0			0.361609
$41$			44000.1i			0.638414i	0.947685	+	0.319207i	$0.103417\pi$
$41$			44000.1i			0.638414i	−0.947685	+	0.319207i	$0.896583\pi$
$42$	60836.1	−	3774.04i	0.821133	−	0.0509399i
$43$	122875.			1.54546			0.772729	−	0.634737i	$-0.218891\pi$
$43$	122875.			1.54546			0.772729	+	0.634737i	$0.218891\pi$
$44$		−	228600.i		−	2.68360i
$45$	6258.19	+	50245.7i	0.0686769	+	0.551393i
$46$	166441.			1.70996
$47$			138705.i			1.33597i	0.744173	+	0.667987i	$0.232844\pi$
$47$			138705.i			1.33597i	−0.744173	+	0.667987i	$0.767156\pi$
$48$	−2782.93	−	44859.8i	−0.0251639	−	0.405633i
$49$	−84722.7			−0.720131
$50$		−	134374.i		−	1.07499i
$51$	−41721.9	+	2588.27i	−0.314524	+	0.0195119i
$52$	161337.			1.14742
$53$			87390.6i			0.586998i	0.955959	+	0.293499i	$0.0948198\pi$
$53$			87390.6i			0.586998i	−0.955959	+	0.293499i	$0.905180\pi$
$54$	−240661.	+	45254.1i	−1.52836	+	0.287393i
$55$	174900.			1.05124
$56$			60461.1i			0.344280i
$57$	21060.4	+	339486.i	0.113721	+	1.83315i
$58$	56345.0			0.288783
$59$		−	134377.i		−	0.654287i	−0.944975	−	0.327144i	$-0.893914\pi$
$59$		−	134377.i		−	0.654287i	0.944975	−	0.327144i	$-0.106086\pi$
$60$	−169920.	+	10541.2i	−0.786666	+	0.0488017i
$61$	−408931.			−1.80161			−0.900805	−	0.434224i	$-0.857023\pi$
$61$	−408931.			−1.80161			−0.900805	+	0.434224i	$0.857023\pi$
$62$		−	211177.i		−	0.886076i
$63$	−131267.	+	16349.5i	−0.524970	+	0.0653858i
$64$	416427.			1.58854
$65$			123438.i			0.449478i
$66$	52373.3	+	844239.i	0.182171	+	2.93652i
$67$	−377861.			−1.25634			−0.628170	−	0.778076i	$-0.716196\pi$
$67$	−377861.			−1.25634			−0.628170	+	0.778076i	$0.716196\pi$
$68$		−	140551.i		−	0.447001i
$69$	−360520.	+	22365.3i	−1.09744	+	0.0680812i
$70$	−156800.			−0.457143
$71$			283247.i			0.791389i	0.918382	+	0.395694i	$0.129496\pi$
$71$			283247.i			0.791389i	−0.918382	+	0.395694i	$0.870504\pi$
$72$	−30021.9	−	241040.i	−0.0804343	−	0.645791i
$73$	232402.			0.597408			0.298704	−	0.954346i	$-0.403446\pi$
$73$	232402.			0.597408			0.298704	+	0.954346i	$0.403446\pi$
$74$		−	1193.41i		−	0.00294507i
$75$	18056.3	+	291061.i	0.0428001	+	0.689922i
$76$	−1.14365e6			−2.60526
$77$			456927.i			1.00086i
$78$	−595832.	+	36963.1i	−1.25557	+	0.0778906i
$79$	496261.			1.00653			0.503267	−	0.864131i	$-0.332131\pi$
$79$	496261.			1.00653			0.503267	+	0.864131i	$0.332131\pi$
$80$			115622.i			0.225825i
$81$	515204.	−	130361.i	0.969448	−	0.245298i
$82$	547412.			0.992825
$83$		−	69521.1i		−	0.121586i	−0.998150	−	0.0607928i	$-0.980637\pi$
$83$		−	69521.1i		−	0.121586i	0.998150	−	0.0607928i	$-0.0193629\pi$
$84$	−27538.9	−	443916.i	−0.0464631	−	0.748968i
$85$	107535.			0.175102
$86$		−	1.52870e6i		−	2.40341i
$87$	−122046.	+	7571.28i	−0.185339	+	0.0114977i
$88$	−839035.			−1.23121
$89$			115565.i			0.163929i	0.996635	+	0.0819646i	$0.0261195\pi$
$89$			115565.i			0.163929i	−0.996635	+	0.0819646i	$0.973881\pi$
$90$	625114.	−	77859.0i	0.857496	−	0.106802i
$91$	−322482.			−0.427938
$92$		−	1.21451e6i		−	1.55969i
$93$	28376.6	+	457420.i	0.0352786	+	0.568678i
$94$	1.72565e6			2.07763
$95$		−	874998.i		−	1.02055i
$96$	−1.13277e6	+	70272.8i	−1.28035	+	0.0794279i
$97$	−928194.			−1.01701			−0.508503	−	0.861060i	$-0.669801\pi$
$97$	−928194.			−1.01701			−0.508503	+	0.861060i	$0.669801\pi$
$98$			1.05405e6i			1.11991i
$99$	−226887.	−	1.82163e6i	−0.233832	−	1.87739i
$100$	−980516.			−0.980516
$101$			246027.i			0.238791i	0.992847	+	0.119396i	$0.0380957\pi$
$101$			246027.i			0.238791i	−0.992847	+	0.119396i	$0.961904\pi$
$102$	32201.0	+	519069.i	0.0303437	+	0.489130i
$103$	−1.07519e6			−0.983954			−0.491977	−	0.870608i	$-0.663726\pi$
$103$	−1.07519e6			−0.983954			−0.491977	+	0.870608i	$0.663726\pi$
$104$		−	592160.i		−	0.526428i
$105$	339637.	−	21069.8i	0.293391	−	0.0182009i
$106$	1.08724e6			0.912867
$107$			1.21030e6i			0.987966i	0.869472	+	0.493983i	$0.164459\pi$
$107$			1.21030e6i			0.987966i	−0.869472	+	0.493983i	$0.835541\pi$
$108$	330216.	+	1.75609e6i	0.262136	+	1.39404i
$109$	−156918.			−0.121170			−0.0605849	−	0.998163i	$-0.519297\pi$
$109$	−156918.			−0.121170			−0.0605849	+	0.998163i	$0.519297\pi$
$110$		−	2.17596e6i		−	1.63483i
$111$	160.363	+	2585.00i	0.000117256	+	0.00189013i
$112$	−302064.			−0.215003
$113$		−	2.76551e6i		−	1.91663i	−0.285705	−	0.958317i	$-0.592228\pi$
$113$		−	2.76551e6i		−	1.91663i	0.285705	−	0.958317i	$-0.407772\pi$
$114$	4.22360e6	−	262016.i	2.85081	−	0.176853i
$115$	929212.			0.610972
$116$		−	411145.i		−	0.263403i
$117$	1.28564e6	−	160128.i	0.802715	−	0.0999794i
$118$	−1.67180e6			−1.01751
$119$			280935.i			0.166711i
$120$	38689.5	+	623661.i	0.0223898	+	0.360915i
$121$	−4.56934e6			−2.57927
$122$			5.08757e6i			2.80176i
$123$	−1.18572e6	+	73557.7i	−0.637189	+	0.0395287i
$124$	−1.54094e6			−0.808204
$125$		−	1.83545e6i		−	0.939749i
$126$	203407.	+	1.63311e6i	0.101684	+	0.816403i
$127$	2.20316e6			1.07556			0.537781	−	0.843085i	$-0.319263\pi$
$127$	2.20316e6			1.07556			0.537781	+	0.843085i	$0.319263\pi$
$128$		−	2.49058e6i		−	1.18760i
$129$	205417.	+	3.31125e6i	0.0956902	+	1.54249i
$130$	1.53571e6			0.699002
$131$			1.27552e6i			0.567380i	0.958916	+	0.283690i	$0.0915586\pi$
$131$			1.27552e6i			0.567380i	−0.958916	+	0.283690i	$0.908441\pi$
$132$	6.16035e6	−	382164.i	2.67845	−	0.166161i
$133$	2.28593e6			0.971648
$134$			4.70102e6i			1.95379i
$135$	−1.34357e6	+	252646.i	−0.546083	+	0.102686i
$136$	−515869.			−0.205080
$137$			1.54899e6i			0.602401i	0.953561	+	0.301201i	$0.0973874\pi$
$137$			1.54899e6i			0.602401i	−0.953561	+	0.301201i	$0.902613\pi$
$138$	278250.	+	4.48529e6i	0.105876	+	1.70668i
$139$	14076.8			0.00524154			0.00262077	−	0.999997i	$-0.499166\pi$
$139$	14076.8			0.00524154			0.00262077	+	0.999997i	$0.499166\pi$
$140$			1.14416e6i			0.416967i
$141$	−3.73785e6	+	231881.i	−1.33341	+	0.0827197i
$142$	3.52391e6			1.23072
$143$		−	4.47517e6i		−	1.53039i
$144$	1.20424e6	−	149990.i	0.403297	−	0.0502312i
$145$	314564.			0.103182
$146$		−	2.89134e6i		−	0.929055i
$147$	−141636.	−	2.28312e6i	−0.0445884	−	0.718750i
$148$	−8708.24			−0.00268624
$149$		−	3.08067e6i		−	0.931292i	−0.884971	−	0.465646i	$-0.845822\pi$
$149$		−	3.08067e6i		−	0.931292i	0.884971	−	0.465646i	$-0.154178\pi$
$150$	3.62113e6	−	224641.i	1.07293	−	0.0665603i
$151$	−2.96200e6			−0.860308			−0.430154	−	0.902756i	$-0.641541\pi$
$151$	−2.96200e6			−0.860308			−0.430154	+	0.902756i	$0.641541\pi$
$152$			4.19756e6i			1.19527i
$153$	−139498.	−	1.12000e6i	−0.0389488	−	0.312712i
$154$	5.68469e6			1.55648
$155$		−	1.17896e6i		−	0.316596i
$156$	269717.	+	4.34774e6i	0.0710452	+	1.14522i
$157$	3.51787e6			0.909035			0.454517	−	0.890738i	$-0.349812\pi$
$157$	3.51787e6			0.909035			0.454517	+	0.890738i	$0.349812\pi$
$158$		−	6.17405e6i		−	1.56530i
$159$	−2.35502e6	+	146096.i	−0.585872	+	0.0363452i
$160$	2.91962e6			0.712799
$161$			2.42757e6i			0.581693i
$162$	−1.62184e6	−	6.40973e6i	−0.381473	−	1.50763i
$163$	579605.			0.133835			0.0669174	−	0.997759i	$-0.478684\pi$
$163$	579605.			0.133835			0.0669174	+	0.997759i	$0.478684\pi$
$164$		−	3.99442e6i		−	0.905571i
$165$	292391.	+	4.71324e6i	0.0650897	+	1.04922i
$166$	−864922.			−0.189083
$167$		−	4.56476e6i		−	0.980095i	−0.871696	−	0.490048i	$-0.836979\pi$
$167$		−	4.56476e6i		−	0.980095i	0.871696	−	0.490048i	$-0.163021\pi$
$168$	−1.62932e6	+	101077.i	−0.343620	+	0.0213168i
$169$	−1.66840e6			−0.345653
$170$		−	1.33786e6i		−	0.272309i
$171$	−9.11333e6	+	1.13508e6i	−1.82259	+	0.227006i
$172$	−1.11548e7			−2.19218
$173$			7.46793e6i			1.44232i	0.692768	+	0.721160i	$0.256391\pi$
$173$			7.46793e6i			1.44232i	−0.692768	+	0.721160i	$0.743609\pi$
$174$	94195.4	+	1.51840e6i	0.0178806	+	0.288229i
$175$	1.95986e6			0.365689
$176$		−	4.19182e6i		−	0.768891i
$177$	3.62121e6	−	224646.i	0.653032	−	0.0405116i
$178$	1.43776e6			0.254933
$179$			1.99657e6i			0.348118i	0.984735	+	0.174059i	$0.0556883\pi$
$179$			1.99657e6i			0.348118i	−0.984735	+	0.174059i	$0.944312\pi$
$180$	−568131.	−	4.56141e6i	−0.0974162	−	0.782135i
$181$	−3.64456e6			−0.614624			−0.307312	−	0.951609i	$-0.599429\pi$
$181$	−3.64456e6			−0.614624			−0.307312	+	0.951609i	$0.599429\pi$
$182$			4.01204e6i			0.665505i
$183$	−683636.	−	1.10200e7i	−0.111550	−	1.79815i
$184$	−4.45764e6			−0.715569
$185$		−	6662.61i		−	0.00105228i
$186$	5.69083e6	−	353037.i	0.884376	−	0.0548633i
$187$	−3.89861e6			−0.596191
$188$		−	1.25919e7i		−	1.89504i
$189$	−660038.	−	3.51008e6i	−0.0977650	−	0.519914i
$190$	−1.08860e7			−1.58711
$191$		−	2.13196e6i		−	0.305970i	−0.988229	−	0.152985i	$-0.951111\pi$
$191$		−	2.13196e6i		−	0.305970i	0.988229	−	0.152985i	$-0.0488886\pi$
$192$	696166.	+	1.12219e7i	0.0983579	+	1.58549i
$193$	7.79139e6			1.08378			0.541892	−	0.840448i	$-0.317708\pi$
$193$	7.79139e6			1.08378			0.541892	+	0.840448i	$0.317708\pi$
$194$			1.15478e7i			1.58159i
$195$	−3.32643e6	+	206359.i	−0.448616	+	0.0278304i
$196$	7.69131e6			1.02148
$197$		−	2.76698e6i		−	0.361916i	−0.983491	−	0.180958i	$-0.942080\pi$
$197$		−	2.76698e6i		−	0.361916i	0.983491	−	0.180958i	$-0.0579198\pi$
$198$	−2.26632e7	+	2.82273e6i	−2.91961	+	0.363642i
$199$	2.85878e6			0.362762			0.181381	−	0.983413i	$-0.441943\pi$
$199$	2.85878e6			0.362762			0.181381	+	0.983413i	$0.441943\pi$
$200$			3.59881e6i			0.449851i
$201$	−631693.	−	1.01827e7i	−0.0777890	−	1.25393i
$202$	3.06086e6			0.371355
$203$			821800.i			0.0982377i
$204$	3.78760e6	−	234968.i	0.446143	−	0.0276770i
$205$	3.05611e6			0.354737
$206$			1.33766e7i			1.53019i
$207$	−1.20541e6	−	9.67799e6i	−0.135901	−	1.09112i
$208$	2.95843e6			0.328754
$209$			3.17225e7i			3.47479i
$210$	−262132.	−	4.22548e6i	−0.0283050	−	0.456266i
$211$	1.23229e7			1.31179			0.655897	−	0.754850i	$-0.272291\pi$
$211$	1.23229e7			1.31179			0.655897	+	0.754850i	$0.272291\pi$
$212$		−	7.93350e6i		−	0.832640i
$213$	−7.63299e6	+	473521.i	−0.789870	+	0.0490005i
$214$	1.50575e7			1.53643
$215$		−	8.53447e6i		−	0.858739i
$216$	6.44541e6	−	1.21200e6i	0.639572	−	0.120265i
$217$	3.08004e6			0.301424
$218$			1.95225e6i			0.188436i
$219$	388521.	+	6.26281e6i	0.0369898	+	0.596262i
$220$	−1.58778e7			−1.49115
$221$		−	2.75150e6i		−	0.254913i
$222$	32160.3	−	1995.10i	0.00293942	−	0.000182350i
$223$	9.62889e6			0.868284			0.434142	−	0.900845i	$-0.357052\pi$
$223$	9.62889e6			0.868284			0.434142	+	0.900845i	$0.357052\pi$
$224$			7.62753e6i			0.678640i
$225$	−7.81338e6	+	973170.i	−0.685949	+	0.0854360i
$226$	−3.44061e7			−2.98064
$227$			4.76958e6i			0.407758i	0.978996	+	0.203879i	$0.0653549\pi$
$227$			4.76958e6i			0.407758i	−0.978996	+	0.203879i	$0.934645\pi$
$228$	−1.91191e6	−	3.08193e7i	−0.161310	−	2.60027i
$229$	−4.50772e6			−0.375362			−0.187681	−	0.982230i	$-0.560097\pi$
$229$	−4.50772e6			−0.375362			−0.187681	+	0.982230i	$0.560097\pi$
$230$		−	1.15605e7i		−	0.950148i
$231$	−1.23134e7	+	763873.i	−0.998942	+	0.0619705i
$232$	−1.50904e6			−0.120847
$233$			857505.i			0.0677905i	0.999425	+	0.0338952i	$0.0107913\pi$
$233$			857505.i			0.0677905i	−0.999425	+	0.0338952i	$0.989209\pi$
$234$	−1.99218e6	−	1.59948e7i	−0.155482	−	1.24834i
$235$	9.63399e6			0.742339
$236$			1.21990e7i			0.928087i
$237$	829629.	+	1.33733e7i	0.0623217	+	1.00460i
$238$	3.49515e6			0.259260
$239$		−	2.47698e7i		−	1.81439i	−0.420715	−	0.907193i	$-0.638221\pi$
$239$		−	2.47698e7i		−	1.81439i	0.420715	−	0.907193i	$-0.361779\pi$
$240$	−3.11581e6	+	193293.i	−0.225392	+	0.0139824i
$241$	2.79142e6			0.199423			0.0997114	−	0.995016i	$-0.468208\pi$
$241$	2.79142e6			0.199423			0.0997114	+	0.995016i	$0.468208\pi$
$242$			5.68478e7i			4.01114i
$243$	4.37430e6	+	1.36659e7i	0.304853	+	0.952400i
$244$	3.71236e7			2.55553
$245$			5.88456e6i			0.400144i
$246$	915142.	+	1.47518e7i	0.0614729	+	0.990920i
$247$	−2.23886e7			−1.48572
$248$			5.65575e6i			0.370796i
$249$	1.87347e6	−	116223.i	0.121352	−	0.00752823i
$250$	−2.28351e7			−1.46144
$251$			2.61685e7i			1.65484i	0.561580	+	0.827422i	$0.310194\pi$
$251$			2.61685e7i			1.65484i	−0.561580	+	0.827422i	$0.689806\pi$
$252$	1.19167e7	−	1.48424e6i	0.744654	−	0.0927479i
$253$	−3.36880e7			−2.08024
$254$		−	2.74098e7i		−	1.67265i
$255$	179773.	+	2.89787e6i	0.0108418	+	0.174766i
$256$	−4.33430e6			−0.258344
$257$		−	2.32276e7i		−	1.36838i	−0.729305	−	0.684188i	$-0.760157\pi$
$257$		−	2.32276e7i		−	1.36838i	0.729305	−	0.684188i	$-0.239843\pi$
$258$	4.11957e7	−	2.55563e6i	2.39880	−	0.148812i
$259$	17406.1			0.00100185
$260$		−	1.12059e7i		−	0.637571i
$261$	−408065.	−	3.27627e6i	−0.0229513	−	0.184272i
$262$	1.58689e7			0.882357
$263$			2.02481e7i			1.11305i	0.830830	+	0.556527i	$0.187866\pi$
$263$			2.02481e7i			1.11305i	−0.830830	+	0.556527i	$0.812134\pi$
$264$	−1.40267e6	−	2.26105e7i	−0.0762330	−	1.22885i
$265$	6.06986e6			0.326168
$266$		−	2.84396e7i		−	1.51105i
$267$	−3.11427e6	+	193197.i	−0.163615	+	0.0101500i
$268$	3.43030e7			1.78208
$269$		−	3.66111e7i		−	1.88086i	−0.339990	−	0.940429i	$-0.610424\pi$
$269$		−	3.66111e7i		−	1.88086i	0.339990	−	0.940429i	$-0.389576\pi$
$270$	3.14320e6	+	1.67155e7i	0.159691	+	0.849237i
$271$	1.04674e7			0.525935			0.262968	−	0.964805i	$-0.415299\pi$
$271$	1.04674e7			0.525935			0.262968	+	0.964805i	$0.415299\pi$
$272$		−	2.57728e6i		−	0.128072i
$273$	−539113.	−	8.69031e6i	−0.0264967	−	0.427117i
$274$	1.92712e7			0.936820
$275$			2.71976e7i			1.30777i
$276$	3.27288e7	−	2.03037e6i	1.55669	−	0.0965712i
$277$	3.17187e7			1.49237			0.746184	−	0.665740i	$-0.231884\pi$
$277$	3.17187e7			1.49237			0.746184	+	0.665740i	$0.231884\pi$
$278$		−	175131.i		−	0.00815135i
$279$	−1.22792e7	+	1.52940e6i	−0.565403	+	0.0704218i
$280$	4.19943e6			0.191301
$281$			2.60108e7i			1.17229i	0.810206	+	0.586145i	$0.199355\pi$
$281$			2.60108e7i			1.17229i	−0.810206	+	0.586145i	$0.800645\pi$
$282$	2.88487e6	+	4.65031e7i	0.128641	+	2.07364i
$283$	2.71504e7			1.19789			0.598945	−	0.800790i	$-0.295587\pi$
$283$	2.71504e7			1.19789			0.598945	+	0.800790i	$0.295587\pi$
$284$		−	2.57137e7i		−	1.12256i
$285$	2.35796e7	−	1.46279e6i	1.01860	−	0.0631898i
$286$	−5.56762e7			−2.37997
$287$			7.98409e6i			0.337738i
$288$	−3.78745e6	−	3.04087e7i	−0.158551	−	1.27297i
$289$	2.17406e7			0.900694
$290$		−	3.91354e6i		−	0.160463i
$291$	−1.55172e6	−	2.50131e7i	−0.0629701	−	1.01505i
$292$	−2.10979e7			−0.847405
$293$		−	1.61364e6i		−	0.0641511i	−0.999485	−	0.0320755i	$-0.989788\pi$
$293$		−	1.61364e6i		−	0.0641511i	0.999485	−	0.0320755i	$-0.0102117\pi$
$294$	−2.84047e7	+	1.76212e6i	−1.11776	+	0.0693414i
$295$	−9.33338e6			−0.363557
$296$			31962.1i			0.00123242i
$297$	4.87103e7	−	9.15953e6i	1.85931	−	0.349626i
$298$	−3.83270e7			−1.44829
$299$		−	2.37758e7i		−	0.889448i
$300$	−1.63919e6	−	2.64231e7i	−0.0607107	−	0.978634i
$301$	2.22963e7			0.817587
$302$			3.68506e7i			1.33790i
$303$	−6.62998e6	+	411299.i	−0.238333	+	0.0147853i
$304$	−2.09710e7			−0.746447
$305$			2.84030e7i			1.00107i
$306$	−1.39341e7	+	1.73552e6i	−0.486313	+	0.0605710i
$307$	1.09561e7			0.378652			0.189326	−	0.981914i	$-0.439370\pi$
$307$	1.09561e7			0.378652			0.189326	+	0.981914i	$0.439370\pi$
$308$		−	4.14808e7i		−	1.41969i
$309$	−1.79747e6	−	2.89745e7i	−0.0609236	−	0.982066i
$310$	−1.46676e7			−0.492352
$311$		−	1.03736e7i		−	0.344865i	−0.985021	−	0.172432i	$-0.944837\pi$
$311$		−	1.03736e7i		−	0.344865i	0.985021	−	0.172432i	$-0.0551626\pi$
$312$	1.59576e7	−	989950.i	0.525418	−	0.0325949i
$313$	2.42037e7			0.789313			0.394656	−	0.918829i	$-0.370864\pi$
$313$	2.42037e7			0.789313			0.394656	+	0.918829i	$0.370864\pi$
$314$		−	4.37663e7i		−	1.41368i
$315$	1.13559e6	+	9.11739e6i	0.0363319	+	0.291702i
$316$	−4.50516e7			−1.42774
$317$			1.10132e7i			0.345728i	0.984946	+	0.172864i	$0.0553021\pi$
$317$			1.10132e7i			0.345728i	−0.984946	+	0.172864i	$0.944698\pi$
$318$	1.81760e6	+	2.92991e7i	0.0565221	+	0.911115i
$319$	−1.14043e7			−0.351316
$320$		−	2.89236e7i		−	0.882679i
$321$	−3.26154e7	+	2.02333e6i	−0.986070	+	0.0611720i
$322$	3.02017e7			0.904616
$323$			1.95042e7i			0.578788i
$324$	−4.67713e7	+	1.18345e7i	−1.37513	+	0.347948i
$325$	−1.91950e7			−0.559163
$326$		−	7.21095e6i		−	0.208132i
$327$	−262330.	−	4.22867e6i	−0.00750249	−	0.120937i
$328$	−1.46608e7			−0.415468
$329$			2.51688e7i			0.706765i
$330$	5.86381e7	−	3.63768e6i	1.63169	−	0.101224i
$331$	−1.27658e7			−0.352018			−0.176009	−	0.984389i	$-0.556319\pi$
$331$	−1.27658e7			−0.352018			−0.176009	+	0.984389i	$0.556319\pi$
$332$			6.31127e6i			0.172466i
$333$	−69392.9	+	8643.00i	−0.00187924	+	0.000234062i
$334$	−5.67908e7			−1.52419
$335$			2.62450e7i			0.698090i
$336$	−504979.	−	8.14007e6i	−0.0133124	−	0.214590i
$337$	−3.78399e7			−0.988691			−0.494346	−	0.869265i	$-0.664592\pi$
$337$	−3.78399e7			−0.988691			−0.494346	+	0.869265i	$0.664592\pi$
$338$			2.07568e7i			0.537540i
$339$	7.45254e7	−	4.62327e6i	1.91296	−	0.118672i
$340$	−9.76223e6			−0.248378
$341$			4.27426e7i			1.07795i
$342$	1.41217e7	+	1.13380e8i	0.353028	+	2.83439i
$343$	−3.67216e7			−0.909994
$344$			4.09418e7i			1.00575i
$345$	1.55342e6	+	2.50406e7i	0.0378296	+	0.609799i
$346$	9.29096e7			2.24302
$347$			5.07869e7i			1.21552i	0.794120	+	0.607761i	$0.207932\pi$
$347$			5.07869e7i			1.21552i	−0.794120	+	0.607761i	$0.792068\pi$
$348$	1.10796e7	−	687337.i	0.262898	−	0.0163092i
$349$	−4.56293e6			−0.107342			−0.0536708	−	0.998559i	$-0.517092\pi$
$349$	−4.56293e6			−0.107342			−0.0536708	+	0.998559i	$0.517092\pi$
$350$		−	2.43829e7i		−	0.568698i
$351$	6.46445e6	+	3.43779e7i	0.149489	+	0.794984i
$352$	−1.05849e8			−2.42695
$353$		−	4.89641e7i		−	1.11315i	−0.830798	−	0.556575i	$-0.812115\pi$
$353$		−	4.89641e7i		−	1.11315i	0.830798	−	0.556575i	$-0.187885\pi$
$354$	−2.79485e6	−	4.50520e7i	−0.0630013	−	1.01556i
$355$	1.96734e7			0.439738
$356$		−	1.04912e7i		−	0.232529i
$357$	−7.57069e6	+	469657.i	−0.166391	+	0.0103223i
$358$	2.48397e7			0.541373
$359$			1.12217e7i			0.242536i	0.992620	+	0.121268i	$0.0386961\pi$
$359$			1.12217e7i			0.242536i	−0.992620	+	0.121268i	$0.961304\pi$
$360$	−1.67419e7	+	2.08523e6i	−0.358836	+	0.0446936i
$361$	1.11657e8			2.37337
$362$			4.53425e7i			0.955828i
$363$	−7.63885e6	−	1.23135e8i	−0.159701	−	2.57432i
$364$	2.92756e7			0.607018
$365$		−	1.61419e7i		−	0.331952i
$366$	−1.37101e8	+	8.50521e6i	−2.79639	+	0.173477i
$367$	−1.95464e7			−0.395428			−0.197714	−	0.980260i	$-0.563352\pi$
$367$	−1.95464e7			−0.395428			−0.197714	+	0.980260i	$0.563352\pi$
$368$		−	2.22704e7i		−	0.446873i
$369$	−3.96450e6	−	3.18302e7i	−0.0789058	−	0.633519i
$370$	−82890.5			−0.00163644
$371$			1.58575e7i			0.310537i
$372$	−2.57609e6	−	4.15256e7i	−0.0500416	−	0.806653i
$373$	−4.51381e7			−0.869794			−0.434897	−	0.900480i	$-0.643215\pi$
$373$	−4.51381e7			−0.869794			−0.434897	+	0.900480i	$0.643215\pi$
$374$			4.85032e7i			0.927162i
$375$	4.94620e7	−	3.06843e6i	0.937946	−	0.0581865i
$376$	−4.62164e7			−0.869426
$377$		−	8.04876e6i		−	0.150212i
$378$	−4.36694e7	+	8.21163e6i	−0.808541	+	0.152039i
$379$	−8.91289e7			−1.63720			−0.818598	−	0.574367i	$-0.805248\pi$
$379$	−8.91289e7			−1.63720			−0.818598	+	0.574367i	$0.805248\pi$
$380$			7.94341e7i			1.44763i
$381$	3.68316e6	+	5.93712e7i	0.0665956	+	1.07350i
$382$	−2.65240e7			−0.475827
$383$		−	3.37292e7i		−	0.600357i	−0.953883	−	0.300179i	$-0.902954\pi$
$383$		−	3.37292e7i		−	0.600357i	0.953883	−	0.300179i	$-0.0970463\pi$
$384$	6.71165e7	−	4.16365e6i	1.18532	−	0.0735327i
$385$	3.17366e7			0.556133
$386$		−	9.69338e7i		−	1.68544i
$387$	−8.88888e7	+	1.10712e7i	−1.53361	+	0.191013i
$388$	8.42634e7			1.44259
$389$		−	7.00448e7i		−	1.18995i	−0.803746	−	0.594973i	$-0.797163\pi$
$389$		−	7.00448e7i		−	1.18995i	0.803746	−	0.594973i	$-0.202837\pi$
$390$	2.56734e6	+	4.13846e7i	0.0432802	+	0.697661i
$391$	−2.07126e7			−0.346501
$392$		−	2.82296e7i		−	0.468648i
$393$	−3.43730e7	+	2.13237e6i	−0.566291	+	0.0351305i
$394$	−3.44245e7			−0.562832
$395$		−	3.44686e7i		−	0.559285i
$396$	2.05973e7	+	1.65371e8i	0.331684	+	2.66302i
$397$	6.64593e7			1.06215			0.531073	−	0.847326i	$-0.321789\pi$
$397$	6.64593e7			1.06215			0.531073	+	0.847326i	$0.321789\pi$
$398$		−	3.55665e7i		−	0.564146i
$399$	3.82154e6	+	6.16018e7i	0.0601616	+	0.969783i
$400$	−1.79797e7			−0.280932
$401$		−	6.89250e7i		−	1.06892i	−0.845195	−	0.534458i	$-0.820516\pi$
$401$		−	6.89250e7i		−	1.06892i	0.845195	−	0.534458i	$-0.179484\pi$
$402$	−1.26684e8	+	7.85899e6i	−1.95004	+	0.120973i
$403$	−3.01661e7			−0.460898
$404$		−	2.23348e7i		−	0.338718i
$405$	−9.05447e6	−	3.57844e7i	−0.136301	−	0.538677i
$406$	1.02241e7			0.152774
$407$			241549.i			0.00358280i
$408$	−862410.	−	1.39017e7i	−0.0126979	−	0.204686i
$409$	1.09209e7			0.159620			0.0798099	−	0.996810i	$-0.474569\pi$
$409$	1.09209e7			0.159620			0.0798099	+	0.996810i	$0.474569\pi$
$410$		−	3.80214e7i		−	0.551667i
$411$	−4.17424e7	+	2.58954e6i	−0.601245	+	0.0372989i
$412$	9.76083e7			1.39571
$413$		−	2.43835e7i		−	0.346135i
$414$	−1.20405e8	+	1.49967e7i	−1.69685	+	0.211346i
$415$	−4.82871e6			−0.0675595
$416$		−	7.47045e7i		−	1.03769i
$417$	23533.0	+	379344.i	0.000324541	+	0.00523149i
$418$	3.94665e8			5.40380
$419$		−	1.22554e8i		−	1.66604i	−0.553239	−	0.833022i	$-0.686609\pi$
$419$		−	1.22554e8i		−	1.66604i	0.553239	−	0.833022i	$-0.313391\pi$
$420$	−3.08330e7	+	1.91276e6i	−0.416167	+	0.0258174i
$421$	1.43997e8			1.92978			0.964890	−	0.262654i	$-0.0845977\pi$
$421$	1.43997e8			1.92978			0.964890	+	0.262654i	$0.0845977\pi$
$422$		−	1.53311e8i		−	2.04003i
$423$	−1.24976e7	−	1.00341e8i	−0.165122	−	1.32573i
$424$	−2.91185e7			−0.382007
$425$			1.67220e7i			0.217832i
$426$	5.89115e6	+	9.49631e7i	0.0762028	+	1.22836i
$427$	−7.42030e7			−0.953099
$428$		−	1.09874e8i		−	1.40140i
$429$	1.20598e8	−	7.48142e6i	1.52745	−	0.0947572i
$430$	−1.06179e8			−1.33546
$431$		−	2.77292e7i		−	0.346343i	−0.984892	−	0.173171i	$-0.944599\pi$
$431$		−	2.77292e7i		−	0.346343i	0.984892	−	0.173171i	$-0.0554015\pi$
$432$	6.05515e6	+	3.22013e7i	0.0751058	+	0.399413i
$433$	−1.18052e8			−1.45416			−0.727078	−	0.686555i	$-0.759122\pi$
$433$	−1.18052e8			−1.45416			−0.727078	+	0.686555i	$0.759122\pi$
$434$		−	3.83193e7i		−	0.468757i
$435$	525876.	+	8.47694e6i	0.00638875	+	0.102984i
$436$	1.42454e7			0.171876
$437$			1.68536e8i			2.01952i
$438$	7.79165e7	−	4.83364e6i	0.927272	−	0.0575244i
$439$	8.42198e7			0.995452			0.497726	−	0.867334i	$-0.334168\pi$
$439$	8.42198e7			0.995452			0.497726	+	0.867334i	$0.334168\pi$
$440$			5.82766e7i			0.684127i
$441$	6.12893e7	−	7.63368e6i	0.714610	−	0.0890058i
$442$	−3.42318e7			−0.396426
$443$			4.29148e7i			0.493624i	0.969063	+	0.246812i	$0.0793830\pi$
$443$			4.29148e7i			0.493624i	−0.969063	+	0.246812i	$0.920617\pi$
$444$	−14558.1	−	234671.i	−0.000166324	−	0.00268109i
$445$	8.02677e6			0.0910879
$446$		−	1.19794e8i		−	1.35031i
$447$	8.30184e7	−	5.15014e6i	0.929505	−	0.0576629i
$448$	7.55631e7			0.840380
$449$			9.39085e6i			0.103745i	0.998654	+	0.0518724i	$0.0165189\pi$
$449$			9.39085e6i			0.103745i	−0.998654	+	0.0518724i	$0.983481\pi$
$450$	1.21073e7	+	9.72074e7i	0.132865	+	1.06675i
$451$	−1.10797e8			−1.20781
$452$			2.51058e8i			2.71869i
$453$	−4.95175e6	−	7.98205e7i	−0.0532678	−	0.858657i
$454$	5.93390e7			0.634122
$455$			2.23985e7i			0.237786i
$456$	−1.13117e8	+	7.01733e6i	−1.19298	+	0.0740077i
$457$	1.53491e8			1.60818			0.804089	−	0.594509i	$-0.202654\pi$
$457$	1.53491e8			1.60818			0.804089	+	0.594509i	$0.202654\pi$
$458$			5.60812e7i			0.583742i
$459$	2.99489e7	−	5.63161e6i	0.309701	−	0.0582364i
$460$	−8.43558e7			−0.866645
$461$		−	1.89066e8i		−	1.92979i	−0.262628	−	0.964897i	$-0.584589\pi$
$461$		−	1.89066e8i		−	1.92979i	0.262628	−	0.964897i	$-0.415411\pi$
$462$	9.50345e6	+	1.53192e8i	0.0963730	+	1.55350i
$463$	−2.81801e7			−0.283923			−0.141961	−	0.989872i	$-0.545341\pi$
$463$	−2.81801e7			−0.283923			−0.141961	+	0.989872i	$0.545341\pi$
$464$		−	7.53915e6i		−	0.0754689i
$465$	3.17709e7	−	1.97095e6i	0.315988	−	0.0196027i
$466$	1.06683e7			0.105424
$467$			8.85333e7i			0.869273i	0.900606	+	0.434636i	$0.143123\pi$
$467$			8.85333e7i			0.869273i	−0.900606	+	0.434636i	$0.856877\pi$
$468$	−1.16713e8	+	1.45368e7i	−1.13863	+	0.141818i
$469$	−6.85651e7			−0.664637
$470$		−	1.19858e8i		−	1.15444i
$471$	5.88104e6	+	9.48002e7i	0.0562848	+	0.907290i
$472$	4.47743e7			0.425798
$473$			3.09412e8i			2.92385i
$474$	1.66379e8	−	1.03215e7i	1.56230	−	0.0969191i
$475$	1.36065e8			1.26960
$476$		−	2.55039e7i		−	0.236475i
$477$	−7.87406e6	−	6.32192e7i	−0.0725510	−	0.582498i
$478$	−3.08165e8			−2.82163
$479$			2.48095e7i			0.225741i	0.993610	+	0.112871i	$0.0360045\pi$
$479$			2.48095e7i			0.225741i	−0.993610	+	0.112871i	$0.963995\pi$
$480$	4.88092e6	+	7.86786e7i	0.0441344	+	0.711431i
$481$	−170476.			−0.00153190
$482$		−	3.47285e7i		−	0.310131i
$483$	−6.54186e7	+	4.05832e6i	−0.580577	+	0.0360168i
$484$	4.14814e8			3.65862
$485$			6.44693e7i			0.565103i
$486$	1.70019e8	−	5.44213e7i	1.48112	−	0.474089i
$487$	−1.39665e8			−1.20920			−0.604602	−	0.796527i	$-0.706668\pi$
$487$	−1.39665e8			−1.20920			−0.604602	+	0.796527i	$0.706668\pi$
$488$		−	1.36256e8i		−	1.17245i
$489$	968961.	+	1.56193e7i	0.00828667	+	0.133578i
$490$	7.32107e7			0.622281
$491$			1.49811e8i			1.26561i	0.774312	+	0.632804i	$0.218096\pi$
$491$			1.49811e8i			1.26561i	−0.774312	+	0.632804i	$0.781904\pi$
$492$	1.07643e8	−	6.67772e6i	0.903834	−	0.0560703i
$493$	−7.01180e6			−0.0585179
$494$			2.78540e8i			2.31050i
$495$	−1.26524e8	+	1.57588e7i	−1.04318	+	0.129930i
$496$	−2.82562e7			−0.231562
$497$			5.13968e7i			0.418665i
$498$	−1.44594e6	−	2.33081e7i	−0.0117075	−	0.188720i
$499$	−9.40390e7			−0.756844			−0.378422	−	0.925633i	$-0.623533\pi$
$499$	−9.40390e7			−0.756844			−0.378422	+	0.925633i	$0.623533\pi$
$500$			1.66626e8i			1.33301i
$501$	1.23012e8	−	7.63119e6i	0.978215	−	0.0606846i
$502$	3.25566e8			2.57352
$503$		−	9.09331e7i		−	0.714526i	−0.934004	−	0.357263i	$-0.883710\pi$
$503$		−	9.09331e7i		−	0.714526i	0.934004	−	0.357263i	$-0.116290\pi$
$504$	−5.44766e6	−	4.37382e7i	−0.0425519	−	0.341640i
$505$	1.70882e7			0.132685
$506$			4.19118e8i			3.23508i
$507$	−2.78917e6	−	4.49604e7i	−0.0214018	−	0.344990i
$508$	−2.00008e8			−1.52565
$509$		−	1.93046e8i		−	1.46389i	−0.681366	−	0.731943i	$-0.738614\pi$
$509$		−	1.93046e8i		−	1.46389i	0.681366	−	0.731943i	$-0.261386\pi$
$510$	3.60528e7	−	2.23658e6i	0.271787	−	0.0168606i
$511$	4.21707e7			0.316044
$512$		−	1.05473e8i		−	0.785836i
$513$	−4.58237e7	−	2.43690e8i	−0.339420	−	1.80504i
$514$	−2.88978e8			−2.12802
$515$			7.46794e7i			0.546738i
$516$	−1.86482e7	−	3.00602e8i	−0.135734	−	2.18798i
$517$	−3.49275e8			−2.52752
$518$		−	216552.i		−	0.00155802i
$519$	−2.01247e8	+	1.24846e7i	−1.43955	+	0.0893043i
$520$	−4.11295e7			−0.292512
$521$		−	6.40335e7i		−	0.452787i	−0.974036	−	0.226394i	$-0.927306\pi$
$521$		−	6.40335e7i		−	0.452787i	0.974036	−	0.226394i	$-0.0726935\pi$
$522$	−4.07606e7	+	5.07679e6i	−0.286569	+	0.0356926i
$523$	−1.19881e8			−0.838004			−0.419002	−	0.907985i	$-0.637620\pi$
$523$	−1.19881e8			−0.838004			−0.419002	+	0.907985i	$0.637620\pi$
$524$		−	1.15794e8i		−	0.804811i
$525$	3.27642e6	+	5.28148e7i	0.0226424	+	0.364987i
$526$	2.51909e8			1.73096
$527$			2.62797e7i			0.179551i
$528$	1.12962e8	−	7.00773e6i	0.767416	−	0.0476075i
$529$	−3.09425e7			−0.209020
$530$		−	7.55160e7i		−	0.507238i
$531$	1.21076e7	+	9.72096e7i	0.0808677	+	0.649270i
$532$	−2.07522e8			−1.37825
$533$		−	7.81966e7i		−	0.516424i
$534$	2.40359e6	+	3.87451e7i	0.0157847	+	0.254444i
$535$	8.40635e7			0.548967
$536$		−	1.25903e8i		−	0.817602i
$537$	−5.38041e7	+	3.33780e6i	−0.347450	+	0.0215545i
$538$	−4.55484e8			−2.92500
$539$		−	2.13342e8i		−	1.36241i
$540$	1.21972e8	−	2.29357e7i	0.774603	−	0.145657i
$541$	5.07848e7			0.320732			0.160366	−	0.987058i	$-0.448733\pi$
$541$	5.07848e7			0.320732			0.160366	+	0.987058i	$0.448733\pi$
$542$		−	1.30227e8i		−	0.817905i
$543$	−6.09284e6	−	9.82143e7i	−0.0380557	−	0.613444i
$544$	−6.50800e7			−0.404250
$545$			1.08990e7i			0.0673285i
$546$	−1.08117e8	+	6.70718e6i	−0.664228	+	0.0412062i
$547$	−2.11421e8			−1.29177			−0.645886	−	0.763434i	$-0.723512\pi$
$547$	−2.11421e8			−1.29177			−0.645886	+	0.763434i	$0.723512\pi$
$548$		−	1.40620e8i		−	0.854488i
$549$	2.95825e8	−	3.68455e7i	1.78780	−	0.222673i
$550$	3.38369e8			2.03377
$551$			5.70542e7i			0.341061i
$552$	−7.45211e6	−	1.20125e8i	−0.0443060	−	0.714196i
$553$	9.00494e7			0.532483
$554$		−	3.94617e8i		−	2.32085i
$555$	179545.	−	11138.3i	0.00105026	−	6.51538e-5i
$556$	−1.27792e6			−0.00743497
$557$			1.12071e8i			0.648529i	0.945967	+	0.324264i	$0.105117\pi$
$557$			1.12071e8i			0.648529i	−0.945967	+	0.324264i	$0.894883\pi$
$558$	1.90274e7	+	1.52767e8i	0.109516	+	0.879282i
$559$	−2.18372e8			−1.25015
$560$			2.09804e7i			0.119467i
$561$	−6.51755e6	−	1.05061e8i	−0.0369144	−	0.595047i
$562$	3.23605e8			1.82308
$563$			2.15957e8i			1.21016i	0.796166	+	0.605079i	$0.206858\pi$
$563$			2.15957e8i			1.21016i	−0.796166	+	0.605079i	$0.793142\pi$
$564$	3.39329e8	−	2.10507e7i	1.89140	−	0.117335i
$565$	−1.92083e8			−1.06499
$566$		−	3.37782e8i		−	1.86289i
$567$	9.34869e7	−	2.36548e7i	0.512863	−	0.129769i
$568$	−9.43778e7			−0.515021
$569$			3.33571e7i			0.181072i	0.995893	+	0.0905360i	$0.0288580\pi$
$569$			3.33571e7i			0.181072i	−0.995893	+	0.0905360i	$0.971142\pi$
$570$	−1.81988e7	−	2.93357e8i	−0.0982691	−	1.58406i
$571$	2.38581e8			1.28153			0.640764	−	0.767738i	$-0.278618\pi$
$571$	2.38581e8			1.28153			0.640764	+	0.767738i	$0.278618\pi$
$572$			4.06265e8i			2.17081i
$573$	5.74525e7	−	3.56413e6i	0.305383	−	0.0189448i
$574$	9.93312e7			0.525230
$575$			1.44496e8i			0.760066i
$576$	−3.01247e8	+	3.75208e7i	−1.57636	+	0.196338i
$577$	−9.08855e6			−0.0473116			−0.0236558	−	0.999720i	$-0.507531\pi$
$577$	−9.08855e6			−0.0473116			−0.0236558	+	0.999720i	$0.507531\pi$
$578$		−	2.70477e8i		−	1.40071i
$579$	1.30253e7	+	2.09964e8i	0.0671048	+	1.08170i
$580$	−2.85568e7			−0.146361
$581$		−	1.26150e7i		−	0.0643220i
$582$	−3.11192e8	+	1.93052e7i	−1.57856	+	0.0979275i
$583$	−2.20059e8			−1.11054
$584$			7.74362e7i			0.388782i
$585$	−1.11220e7	−	8.92962e7i	−0.0555539	−	0.446032i
$586$	−2.00755e7			−0.0997641
$587$			3.22743e8i			1.59567i	0.602876	+	0.797835i	$0.294021\pi$
$587$			3.22743e8i			1.59567i	−0.602876	+	0.797835i	$0.705979\pi$
$588$	1.28580e7	+	2.07267e8i	0.0632474	+	1.01952i
$589$	2.13835e8			1.04648
$590$			1.16118e8i			0.565383i
$591$	7.45652e7	−	4.62574e6i	0.361222	−	0.0224088i
$592$	−159683.			−0.000769648
$593$			1.74681e8i			0.837687i	0.908058	+	0.418844i	$0.137564\pi$
$593$			1.74681e8i			0.837687i	−0.908058	+	0.418844i	$0.862436\pi$
$594$	−1.13955e8	−	6.06012e8i	−0.543718	−	2.89149i
$595$	1.95128e7			0.0926338
$596$			2.79669e8i			1.32101i
$597$	4.77920e6	+	7.70389e7i	0.0224612	+	0.362066i
$598$	−2.95798e8			−1.38322
$599$		−	1.00065e8i		−	0.465590i	−0.972526	−	0.232795i	$-0.925213\pi$
$599$		−	1.00065e8i		−	0.465590i	0.972526	−	0.232795i	$-0.0747871\pi$
$600$	−9.69815e7	+	6.01636e6i	−0.448988	+	0.0278535i
$601$	−3.30731e8			−1.52353			−0.761766	−	0.647852i	$-0.775667\pi$
$601$	−3.30731e8			−1.52353			−0.761766	+	0.647852i	$0.775667\pi$
$602$		−	2.77392e8i		−	1.27147i
$603$	2.73348e8	−	3.40460e7i	1.24671	−	0.155279i
$604$	2.68896e8			1.22032
$605$			3.17371e8i			1.43318i
$606$	5.11703e6	+	8.24846e7i	0.0229932	+	0.370642i
$607$	−4.30119e8			−1.92319			−0.961597	−	0.274466i	$-0.911499\pi$
$607$	−4.30119e8			−1.92319			−0.961597	+	0.274466i	$0.911499\pi$
$608$			5.29548e8i			2.35610i
$609$	−2.21460e7	+	1.37385e6i	−0.0980492	+	0.00608259i
$610$	3.53366e8			1.55681
$611$		−	2.46505e8i		−	1.08069i
$612$	1.26639e7	+	1.01676e8i	0.0552478	+	0.443573i
$613$	−2.57719e8			−1.11883			−0.559415	−	0.828887i	$-0.688974\pi$
$613$	−2.57719e8			−1.11883			−0.559415	+	0.828887i	$0.688974\pi$
$614$		−	1.36306e8i		−	0.588859i
$615$	5.10908e6	+	8.23565e7i	0.0219643	+	0.354057i
$616$	−1.52248e8			−0.651342
$617$			1.30121e8i			0.553976i	0.960873	+	0.276988i	$0.0893363\pi$
$617$			1.30121e8i			0.553976i	−0.960873	+	0.276988i	$0.910664\pi$
$618$	−3.60476e8	+	2.23625e7i	−1.52725	+	0.0947449i
$619$	1.65054e8			0.695912			0.347956	−	0.937511i	$-0.386876\pi$
$619$	1.65054e8			0.695912			0.347956	+	0.937511i	$0.386876\pi$
$620$			1.07029e8i			0.449082i
$621$	2.58789e8	−	4.86629e7i	1.08062	−	0.203200i
$622$	−1.29060e8			−0.536314
$623$			2.09700e7i			0.0867229i
$624$	4.94579e6	+	7.97244e7i	0.0203555	+	0.328123i
$625$	4.12778e7			0.169074
$626$		−	3.01122e8i		−	1.22749i
$627$	−8.54865e8	+	5.30325e7i	−3.46813	+	0.215149i
$628$	−3.19359e8			−1.28944
$629$			148513.i			0.000596778i
$630$	1.13431e8	−	1.41280e7i	0.453638	−	0.0565013i
$631$	8.17589e7			0.325422			0.162711	−	0.986674i	$-0.447976\pi$
$631$	8.17589e7			0.325422			0.162711	+	0.986674i	$0.447976\pi$
$632$			1.65354e8i			0.655033i
$633$	2.06010e7	+	3.32080e8i	0.0812225	+	1.30928i
$634$	1.37016e8			0.537656
$635$		−	1.53024e8i		−	0.597640i
$636$	2.13793e8	−	1.32629e7i	0.831042	−	0.0515546i
$637$	1.50568e8			0.582527
$638$			1.41883e8i			0.546347i
$639$	−2.55211e7	−	2.04904e8i	−0.0978130	−	0.785321i
$640$	−1.72987e8			−0.659894
$641$			1.23360e8i			0.468384i	0.972190	+	0.234192i	$0.0752444\pi$
$641$			1.23360e8i			0.468384i	−0.972190	+	0.234192i	$0.924756\pi$
$642$	2.51726e7	+	4.05773e8i	0.0951312	+	1.53348i
$643$	2.69271e8			1.01288			0.506439	−	0.862276i	$-0.330962\pi$
$643$	2.69271e8			1.01288			0.506439	+	0.862276i	$0.330962\pi$
$644$		−	2.20380e8i		−	0.825114i
$645$	2.29989e8	−	1.42676e7i	0.857092	−	0.0531707i
$646$	2.42654e8			0.900098
$647$			2.24427e8i			0.828635i	0.910133	+	0.414317i	$0.135980\pi$
$647$			2.24427e8i			0.828635i	−0.910133	+	0.414317i	$0.864020\pi$
$648$	4.34364e7	+	1.71666e8i	0.159635	+	0.630898i
$649$	3.38376e8			1.23784
$650$			2.38808e8i			0.869578i
$651$	5.14910e6	+	8.30016e7i	0.0186633	+	0.300846i
$652$	−5.26177e7			−0.189841
$653$			2.40027e8i			0.862027i	0.902345	+	0.431014i	$0.141844\pi$
$653$			2.40027e8i			0.862027i	−0.902345	+	0.431014i	$0.858156\pi$
$654$	−5.26095e7	+	3.26369e6i	−0.188075	+	0.0116674i
$655$	8.85935e7			0.315267
$656$		−	7.32455e7i		−	0.259460i
$657$	−1.68122e8	+	2.09399e7i	−0.592827	+	0.0738376i
$658$	3.13129e8			1.09912
$659$		−	3.65877e8i		−	1.27843i	−0.769026	−	0.639217i	$-0.779258\pi$
$659$		−	3.65877e8i		−	1.27843i	0.769026	−	0.639217i	$-0.220742\pi$
$660$	−2.65439e7	−	4.27878e8i	−0.0923278	−	1.48829i
$661$	2.63071e8			0.910895			0.455448	−	0.890263i	$-0.349479\pi$
$661$	2.63071e8			0.910895			0.455448	+	0.890263i	$0.349479\pi$
$662$			1.58821e8i			0.547438i
$663$	7.41478e7	−	4.59985e6i	0.254424	−	0.0157835i
$664$	2.31644e7			0.0791256
$665$		−	1.58774e8i		−	0.539900i
$666$	107529.	+	863327.i	0.000364001	+	0.00292249i
$667$	−6.05892e7			−0.204182
$668$			4.14398e8i			1.39024i
$669$	1.60972e7	+	2.59481e8i	0.0537616	+	0.866618i
$670$	3.26517e8			1.08563
$671$		−	1.02974e9i		−	3.40846i
$672$	−2.05548e8	+	1.27514e7i	−0.677338	+	0.0420195i
$673$	4.21032e8			1.38124			0.690621	−	0.723217i	$-0.257337\pi$
$673$	4.21032e8			1.38124			0.690621	+	0.723217i	$0.257337\pi$
$674$			4.70772e8i			1.53756i
$675$	−3.92873e7	−	2.08930e8i	−0.127744	−	0.679343i
$676$	1.51461e8			0.490299
$677$		−	3.54830e8i		−	1.14355i	−0.820412	−	0.571773i	$-0.806256\pi$
$677$		−	3.54830e8i		−	1.14355i	0.820412	−	0.571773i	$-0.193744\pi$
$678$	−5.75188e7	−	9.27182e8i	−0.184553	−	2.97492i
$679$	−1.68426e8			−0.538023
$680$			3.58306e7i			0.113953i
$681$	−1.28532e8	+	7.97360e6i	−0.406976	+	0.0252472i
$682$	5.31767e8			1.67636
$683$			2.32799e8i			0.730666i	0.930877	+	0.365333i	$0.119045\pi$
$683$			2.32799e8i			0.730666i	−0.930877	+	0.365333i	$0.880955\pi$
$684$	8.27327e8	−	1.03045e8i	2.58529	−	0.322002i
$685$	1.07587e8			0.334726
$686$			4.56858e8i			1.41517i
$687$	−7.53583e6	−	1.21475e8i	−0.0232413	−	0.374642i
$688$	−2.04545e8			−0.628093
$689$		−	1.55310e8i		−	0.474833i
$690$	3.11533e8	−	1.93263e7i	0.948325	−	0.0588304i
$691$	−1.84243e8			−0.558414			−0.279207	−	0.960231i	$-0.590072\pi$
$691$	−1.84243e8			−0.558414			−0.279207	+	0.960231i	$0.590072\pi$
$692$		−	6.77954e8i		−	2.04589i
$693$	−4.11700e7	−	3.30546e8i	−0.123703	−	0.993189i
$694$	6.31847e8			1.89031
$695$		−	977728.i		−	0.00291248i
$696$	−2.52275e6	−	4.06658e7i	−0.00748249	−	0.120615i
$697$	−6.81222e7			−0.201183
$698$			5.67681e7i			0.166931i
$699$	−2.31082e7	+	1.43354e6i	−0.0676604	+	0.00419739i
$700$	−1.77920e8			−0.518718
$701$			4.14586e8i			1.20354i	0.798669	+	0.601770i	$0.205538\pi$
$701$			4.14586e8i			1.20354i	−0.798669	+	0.601770i	$0.794462\pi$
$702$	4.27701e8	−	8.04252e7i	1.23631	−	0.232477i
$703$	1.20843e6			0.00347822
$704$			1.04861e9i			3.00536i
$705$	1.61057e7	+	2.59619e8i	0.0459635	+	0.740915i
$706$	−6.09169e8			−1.73111
$707$			4.46431e7i			0.126327i
$708$	−3.28741e8	+	2.03938e7i	−0.926306	+	0.0574644i
$709$	6.25237e8			1.75431			0.877154	−	0.480210i	$-0.159439\pi$
$709$	6.25237e8			1.75431			0.877154	+	0.480210i	$0.159439\pi$
$710$		−	2.44760e8i		−	0.683856i
$711$	−3.59000e8	+	4.47140e7i	−0.998817	+	0.124404i
$712$	−3.85062e7			−0.106682
$713$			2.27084e8i			0.626495i
$714$	5.84306e6	+	9.41881e7i	0.0160526	+	0.258762i
$715$	−3.10831e8			−0.850366
$716$		−	1.81253e8i		−	0.493795i
$717$	6.67503e8	−	4.14093e7i	1.81090	−	0.112342i
$718$	1.39611e8			0.377178
$719$		−	3.34031e8i		−	0.898669i	−0.893364	−	0.449335i	$-0.851661\pi$
$719$		−	3.34031e8i		−	0.898669i	0.893364	−	0.449335i	$-0.148339\pi$
$720$	−1.04178e7	−	8.36424e7i	−0.0279112	−	0.224093i
$721$	−1.95100e8			−0.520537
$722$		−	1.38914e9i		−	3.69092i
$723$	4.66660e6	+	7.52239e7i	0.0123477	+	0.199040i
$724$	3.30861e8			0.871825
$725$			4.89158e7i			0.128362i
$726$	−1.53195e9	+	9.50360e7i	−4.00344	+	0.248358i
$727$	3.34132e8			0.869590			0.434795	−	0.900529i	$-0.356821\pi$
$727$	3.34132e8			0.869590			0.434795	+	0.900529i	$0.356821\pi$
$728$		−	1.07451e8i		−	0.278494i
$729$	−3.60958e8	+	1.40726e8i	−0.931697	+	0.363238i
$730$	−2.00823e8			−0.516233
$731$			1.90238e8i			0.487018i
$732$	6.20619e7	+	1.00041e9i	0.158231	+	2.55063i
$733$	3.95042e8			1.00307			0.501535	−	0.865137i	$-0.332769\pi$
$733$	3.95042e8			1.00307			0.501535	+	0.865137i	$0.332769\pi$
$734$			2.43179e8i			0.614948i
$735$	−1.58578e8	+	9.83759e6i	−0.399376	+	0.0247757i
$736$	−5.62358e8			−1.41052
$737$		−	9.51496e8i		−	2.37687i
$738$	−3.96004e8	+	4.93229e7i	−0.985213	+	0.122710i
$739$	−4.55994e8			−1.12986			−0.564931	−	0.825138i	$-0.691097\pi$
$739$	−4.55994e8			−1.12986			−0.564931	+	0.825138i	$0.691097\pi$
$740$			604846.i			0.00149262i
$741$	−3.74284e7	−	6.03332e8i	−0.0919912	−	1.48287i
$742$	1.97286e8			0.482930
$743$		−	5.92896e8i		−	1.44548i	−0.691120	−	0.722740i	$-0.742883\pi$
$743$		−	5.92896e8i		−	1.44548i	0.691120	−	0.722740i	$-0.257117\pi$
$744$	−1.52412e8	+	9.45507e6i	−0.370085	+	0.0229586i
$745$	−2.13973e8			−0.517476
$746$			5.61569e8i			1.35265i
$747$	6.26398e6	+	5.02923e7i	0.0150276	+	0.120653i
$748$	3.53924e8			0.845679
$749$			2.19616e8i			0.522660i
$750$	−3.81748e7	−	6.15364e8i	−0.0904884	−	1.45864i
$751$	6.62342e8			1.56373			0.781867	−	0.623446i	$-0.214268\pi$
$751$	6.62342e8			1.56373			0.781867	+	0.623446i	$0.214268\pi$
$752$		−	2.30897e8i		−	0.542957i
$753$	−7.05193e8	+	4.37475e7i	−1.65167	+	0.102463i
$754$	−1.00136e8			−0.233601
$755$			2.05731e8i			0.478033i
$756$	5.99196e7	+	3.18652e8i	0.138677	+	0.737483i
$757$	5.72807e8			1.32045			0.660223	−	0.751069i	$-0.270462\pi$
$757$	5.72807e8			1.32045			0.660223	+	0.751069i	$0.270462\pi$
$758$			1.10887e9i			2.54607i
$759$	−5.63184e7	−	9.07832e8i	−0.128803	−	2.07625i
$760$	2.91549e8			0.664157
$761$		−	2.30418e8i		−	0.522833i	−0.965226	−	0.261417i	$-0.915810\pi$
$761$		−	2.30418e8i		−	0.522833i	0.965226	−	0.261417i	$-0.0841896\pi$
$762$	7.38646e8	−	4.58227e7i	1.66944	−	0.103566i
$763$	−2.84738e7			−0.0641020
$764$			1.93544e8i			0.434009i
$765$	−7.77918e7	+	9.68909e6i	−0.173760	+	0.0216421i
$766$	−4.19630e8			−0.933641
$767$			2.38813e8i			0.529264i
$768$	−7.24592e6	−	1.16802e8i	−0.0159959	−	0.257849i
$769$	−8.48282e7			−0.186535			−0.0932677	−	0.995641i	$-0.529731\pi$
$769$	−8.48282e7			−0.186535			−0.0932677	+	0.995641i	$0.529731\pi$
$770$		−	3.94840e8i		−	0.864867i
$771$	6.25943e8	−	3.88311e7i	1.36575	−	0.0847259i
$772$	−7.07318e8			−1.53732
$773$		−	3.16831e8i		−	0.685945i	−0.939345	−	0.342973i	$-0.888566\pi$
$773$		−	3.16831e8i		−	0.685945i	0.939345	−	0.342973i	$-0.111434\pi$
$774$	1.37739e8	+	1.10588e9i	0.297053	+	2.38498i
$775$	1.83333e8			0.393854
$776$		−	3.09274e8i		−	0.661848i
$777$	29098.9	+	469063.i	6.20316e−5	+	0.000999927i
$778$	−8.71437e8			−1.85053
$779$			5.54302e8i			1.17256i
$780$	3.01980e8	−	1.87337e7i	0.636348	−	0.0394766i
$781$	−7.13248e8			−1.49723
$782$			2.57689e8i			0.538859i
$783$	8.76074e7	−	1.64738e7i	0.182497	−	0.0343168i
$784$	1.41035e8			0.292671
$785$		−	2.44340e8i		−	0.505109i
$786$	2.65291e7	+	4.27639e8i	0.0546330	+	0.880664i
$787$	−9.13763e7			−0.187460			−0.0937302	−	0.995598i	$-0.529879\pi$
$787$	−9.13763e7			−0.187460			−0.0937302	+	0.995598i	$0.529879\pi$
$788$			2.51193e8i			0.513367i
$789$	−5.45649e8	+	3.38499e7i	−1.11092	+	0.0689171i
$790$	−4.28829e8			−0.869768
$791$		−	5.01818e8i		−	1.01395i
$792$	6.06967e8	−	7.55987e7i	1.22177	−	0.152173i
$793$	7.26749e8			1.45735
$794$		−	8.26830e8i		−	1.65179i
$795$	1.01474e7	+	1.63572e8i	0.0201954	+	0.325542i
$796$	−2.59526e8			−0.514567
$797$			3.72562e8i			0.735907i	0.929844	+	0.367954i	$0.119941\pi$
$797$			3.72562e8i			0.735907i	−0.929844	+	0.367954i	$0.880059\pi$
$798$	7.66397e8	−	4.75443e7i	1.50815	−	0.0935599i
$799$	−2.14747e8			−0.421004
$800$			4.54012e8i			0.886741i
$801$	−1.04126e7	−	8.36009e7i	−0.0202611	−	0.162672i
$802$	−8.57505e8			−1.66232
$803$			5.85214e8i			1.13023i
$804$	5.73464e7	+	9.24403e8i	0.110341	+	1.77866i
$805$	1.68611e8			0.323220
$806$			3.75301e8i			0.716762i
$807$	9.86603e8	−	6.12050e7i	1.87725	−	0.116457i
$808$	−8.19762e7			−0.155401
$809$			6.86413e8i			1.29640i	0.761469	+	0.648201i	$0.224478\pi$
$809$			6.86413e8i			1.29640i	−0.761469	+	0.648201i	$0.775522\pi$
$810$	−4.45199e8	+	1.12648e8i	−0.837720	+	0.211967i
$811$	5.05310e8			0.947317			0.473659	−	0.880709i	$-0.342933\pi$
$811$	5.05310e8			0.947317			0.473659	+	0.880709i	$0.342933\pi$
$812$		−	7.46047e7i		−	0.139347i
$813$	1.74991e7	+	2.82078e8i	0.0325644	+	0.524926i
$814$	3.00515e6			0.00557177
$815$		−	4.02575e7i		−	0.0743659i
$816$	6.94531e7	−	4.30860e6i	0.127827	−	0.00792987i
$817$	1.54794e9			2.83850
$818$		−	1.35868e8i		−	0.248232i
$819$	2.33287e8	−	2.90562e7i	0.424657	−	0.0528917i
$820$	−2.77440e8			−0.503184
$821$		−	4.49841e8i		−	0.812885i	−0.913676	−	0.406443i	$-0.866769\pi$
$821$		−	4.49841e8i		−	0.812885i	0.913676	−	0.406443i	$-0.133231\pi$
$822$	3.22168e7	+	5.19323e8i	0.0580052	+	0.935022i
$823$	−3.26006e8			−0.584825			−0.292412	−	0.956292i	$-0.594458\pi$
$823$	−3.26006e8			−0.584825			−0.292412	+	0.956292i	$0.594458\pi$
$824$		−	3.58254e8i		−	0.640338i
$825$	−7.32925e8	+	4.54678e7i	−1.30526	+	0.0809734i
$826$	−3.03358e8			−0.538289
$827$		−	3.87873e7i		−	0.0685761i	−0.999412	−	0.0342881i	$-0.989084\pi$
$827$		−	3.87873e7i		−	0.0685761i	0.999412	−	0.0342881i	$-0.0109164\pi$
$828$	1.09430e8	+	8.78588e8i	0.192772	+	1.54773i
$829$	−7.61916e8			−1.33735			−0.668673	−	0.743557i	$-0.733137\pi$
$829$	−7.61916e8			−1.33735			−0.668673	+	0.743557i	$0.733137\pi$
$830$			6.00747e7i			0.105065i
$831$	5.30261e7	+	8.54762e8i	0.0924031	+	1.48950i
$832$	−7.40070e8			−1.28500
$833$		−	1.31170e8i		−	0.226934i
$834$	4.71947e6	−	292778.i	0.00813571	−	0.000504708i
$835$	−3.17053e8			−0.544594
$836$		−	2.87984e9i		−	4.92889i
$837$	−6.17424e7	−	3.28346e8i	−0.105295	−	0.559958i
$838$	−1.52472e9			−2.59094
$839$		−	7.10119e8i		−	1.20239i	−0.799102	−	0.601195i	$-0.794692\pi$
$839$		−	7.10119e8i		−	1.20239i	0.799102	−	0.601195i	$-0.205308\pi$
$840$	7.02045e6	+	1.13167e8i	0.0118448	+	0.190934i
$841$	−2.05111e7			−0.0344828
$842$		−	1.79149e9i		−	3.00108i
$843$	−7.00945e8	+	4.34839e7i	−1.17004	+	0.0725848i
$844$	−1.11870e9			−1.86074
$845$			1.15882e8i			0.192063i
$846$	−1.24835e9	+	1.55484e8i	−2.06170	+	0.256788i
$847$	−8.29134e8			−1.36450
$848$		−	1.45476e8i		−	0.238564i
$849$	4.53890e7	+	7.31655e8i	0.0741699	+	1.19559i
$850$	2.08041e8			0.338760
$851$			1.28331e6i			0.00208229i
$852$	6.92939e8	−	4.29872e7i	1.12041	−	0.0695058i
$853$	5.67642e8			0.914592			0.457296	−	0.889315i	$-0.348818\pi$
$853$	5.67642e8			0.914592			0.457296	+	0.889315i	$0.348818\pi$
$854$			9.23170e8i			1.48220i
$855$	7.88390e7	+	6.32982e8i	0.126137	+	1.01273i
$856$	−4.03272e8			−0.642949
$857$		−	3.46035e8i		−	0.549766i	−0.961478	−	0.274883i	$-0.911361\pi$
$857$		−	3.46035e8i		−	0.549766i	0.961478	−	0.274883i	$-0.0886391\pi$
$858$	−9.30774e7	−	1.50037e9i	−0.147361	−	2.37541i
$859$	6.92055e8			1.09185			0.545923	−	0.837835i	$-0.316179\pi$
$859$	6.92055e8			1.09185			0.545923	+	0.837835i	$0.316179\pi$
$860$			7.74777e8i			1.21810i
$861$	−2.15157e8	+	1.33475e7i	−0.337090	+	0.0209117i
$862$	−3.44984e8			−0.538613
$863$		−	2.52626e8i		−	0.393048i	−0.980499	−	0.196524i	$-0.937035\pi$
$863$		−	2.52626e8i		−	0.393048i	0.980499	−	0.196524i	$-0.0629654\pi$
$864$	8.13127e8	−	1.52901e8i	1.26072	−	0.237066i
$865$	5.18698e8			0.801431
$866$			1.46871e9i			2.26142i
$867$	3.63450e7	+	5.85869e8i	0.0557684	+	0.898966i
$868$	−2.79613e8			−0.427561
$869$			1.24964e9i			1.90426i
$870$	1.05463e8	−	6.54251e6i	0.160155	−	0.00993542i
$871$	6.71531e8			1.01628
$872$		−	5.22852e7i		−	0.0788550i
$873$	6.71465e8	−	8.36321e7i	1.00921	−	0.125698i
$874$	2.09678e9			3.14064
$875$		−	3.33053e8i		−	0.497152i
$876$	−3.52707e7	−	5.68551e8i	−0.0524689	−	0.845779i
$877$	−8.49449e8			−1.25933			−0.629663	−	0.776868i	$-0.716807\pi$
$877$	−8.49449e8			−1.25933			−0.629663	+	0.776868i	$0.716807\pi$
$878$		−	1.04779e9i		−	1.54807i
$879$	4.34847e7	−	2.69762e6i	0.0640280	−	0.00397205i
$880$	−2.91150e8			−0.427237
$881$		−	1.67121e8i		−	0.244402i	−0.992505	−	0.122201i	$-0.961005\pi$
$881$		−	1.67121e8i		−	0.244402i	0.992505	−	0.122201i	$-0.0389952\pi$
$882$	−9.49717e7	−	7.62509e8i	−0.138417	−	1.11132i
$883$	−8.20935e8			−1.19241			−0.596206	−	0.802831i	$-0.703326\pi$
$883$	−8.20935e8			−1.19241			−0.596206	+	0.802831i	$0.703326\pi$
$884$			2.49786e8i			0.361586i
$885$	−1.56032e7	−	2.51518e8i	−0.0225104	−	0.362860i
$886$	5.33909e8			0.767656
$887$		−	1.25905e9i		−	1.80414i	−0.431588	−	0.902071i	$-0.642046\pi$
$887$		−	1.25905e9i		−	1.80414i	0.431588	−	0.902071i	$-0.357954\pi$
$888$	−861320.	+	53433.0i	−0.00123006	+	7.63081e-5i
$889$	3.99777e8			0.569000
$890$		−	9.98622e7i		−	0.141655i
$891$	3.28265e8	+	1.29734e9i	0.464078	+	1.83409i
$892$	−8.74131e8			−1.23163
$893$			1.74737e9i			2.45375i
$894$	−6.40736e7	−	1.03284e9i	−0.0896741	−	1.44551i
$895$	1.38676e8			0.193433
$896$		−	4.51930e8i		−	0.628271i
$897$	6.40714e8	−	3.97474e7i	0.887742	−	0.0550721i
$898$	1.16833e8			0.161338
$899$			7.68742e7i			0.105804i
$900$	7.09315e8	−	8.83464e7i	0.972998	−	0.121188i
$901$	−1.35300e8			−0.184980
$902$			1.37845e9i			1.87832i
$903$	3.72742e7	+	6.00846e8i	0.0506226	+	0.816019i
$904$	9.21466e8			1.24731
$905$			2.53139e8i			0.341518i
$906$	−9.93058e8	+	6.16055e7i	−1.33533	+	0.0828390i
$907$	3.02587e8			0.405535			0.202767	−	0.979227i	$-0.435006\pi$
$907$	3.02587e8			0.405535			0.202767	+	0.979227i	$0.435006\pi$
$908$		−	4.32992e8i		−	0.578393i
$909$	−2.21675e7	−	1.77979e8i	−0.0295138	−	0.236960i
$910$	2.78663e8			0.369791
$911$		−	5.71426e8i		−	0.755796i	−0.925847	−	0.377898i	$-0.876647\pi$
$911$		−	5.71426e8i		−	0.755796i	0.925847	−	0.377898i	$-0.123353\pi$
$912$	−3.50586e7	−	5.65131e8i	−0.0462178	−	0.745015i
$913$	1.75062e8			0.230027
$914$		−	1.90960e9i		−	2.50095i
$915$	−7.65410e8	+	4.74831e7i	−0.999151	+	0.0619835i
$916$	4.09220e8			0.532440
$917$			2.31451e8i			0.300159i
$918$	−7.00636e7	−	3.72598e8i	−0.0905659	−	0.481629i
$919$	6.47368e8			0.834075			0.417037	−	0.908889i	$-0.363068\pi$
$919$	6.47368e8			0.834075			0.417037	+	0.908889i	$0.363068\pi$
$920$			3.09613e8i			0.397609i
$921$	1.83160e7	+	2.95247e8i	0.0234451	+	0.377926i
$922$	−2.35220e9			−3.00111
$923$		−	5.03384e8i		−	0.640168i
$924$	1.11783e9	−	6.93460e7i	1.41697	−	0.0879033i
$925$	1.03606e6			0.00130906
$926$			3.50593e8i			0.441540i
$927$	7.77806e8	−	9.68770e7i	0.976410	−	0.121613i
$928$	−1.90374e8			−0.238212
$929$		−	7.92864e8i		−	0.988898i	−0.869207	−	0.494449i	$-0.835370\pi$
$929$		−	7.92864e8i		−	0.988898i	0.869207	−	0.494449i	$-0.164630\pi$
$930$	−2.45208e7	−	3.95267e8i	−0.0304850	−	0.491407i
$931$	−1.06731e9			−1.32265
$932$		−	7.78461e7i		−	0.0961588i
$933$	2.79550e8	−	1.73422e7i	0.344203	−	0.0213530i
$934$	1.10146e9			1.35184
$935$			2.70785e8i			0.331276i
$936$	5.33547e7	+	4.28374e8i	0.0650647	+	0.522391i
$937$	4.96170e8			0.603131			0.301566	−	0.953445i	$-0.402491\pi$
$937$	4.96170e8			0.603131			0.301566	+	0.953445i	$0.402491\pi$
$938$			8.53028e8i			1.03361i
$939$	4.04629e7	+	6.52247e8i	0.0488720	+	0.787798i
$940$	−8.74593e8			−1.05299
$941$		−	6.23846e7i		−	0.0748701i	−0.999299	−	0.0374351i	$-0.988081\pi$
$941$		−	6.23846e7i		−	0.0748701i	0.999299	−	0.0374351i	$-0.0119187\pi$
$942$	1.17942e9	−	7.31668e7i	1.41097	−	0.0875309i
$943$	−5.88646e8			−0.701971
$944$			2.23693e8i			0.265911i
$945$	−2.43799e8	+	4.58441e7i	−0.288892	+	0.0543235i
$946$	3.84945e9			4.54700
$947$			9.89708e8i			1.16535i	0.812705	+	0.582676i	$0.197994\pi$
$947$			9.89708e8i			1.16535i	−0.812705	+	0.582676i	$0.802006\pi$
$948$	−7.53155e7	−	1.21406e9i	−0.0884014	−	1.42500i
$949$	−4.13022e8			−0.483254
$950$		−	1.69281e9i		−	1.97441i
$951$	−2.96785e8	+	1.84114e7i	−0.345064	+	0.0214065i
$952$	−9.36075e7			−0.108493
$953$		−	5.56553e8i		−	0.643025i	−0.946905	−	0.321512i	$-0.895809\pi$
$953$		−	5.56553e8i		−	0.643025i	0.946905	−	0.321512i	$-0.104191\pi$
$954$	−7.86520e8	+	9.79623e7i	−0.905867	+	0.112827i
$955$	−1.48079e8			−0.170013
$956$			2.24866e9i			2.57365i
$957$	−1.90654e7	−	3.07327e8i	−0.0217525	−	0.350642i
$958$	3.08658e8			0.351060
$959$			2.81073e8i			0.318686i
$960$	7.79440e8	−	4.83534e7i	0.880986	−	0.0546529i
$961$	−5.99385e8			−0.675360
$962$			2.12092e6i			0.00238232i
$963$	−1.09050e8	−	8.75544e8i	−0.122109	−	0.980390i
$964$	−2.53411e8			−0.282875
$965$		−	5.41164e8i		−	0.602209i
$966$	5.04901e7	+	8.13882e8i	0.0560112	+	0.902880i
$967$	2.49433e7			0.0275851			0.0137925	−	0.999905i	$-0.495610\pi$
$967$	2.49433e7			0.0275851			0.0137925	+	0.999905i	$0.495610\pi$
$968$		−	1.52250e9i		−	1.67854i
$969$	−5.25602e8	+	3.26063e7i	−0.577678	+	0.0358369i
$970$	8.02072e8			0.878816
$971$			3.92813e8i			0.429070i	0.976716	+	0.214535i	$0.0688236\pi$
$971$			3.92813e8i			0.429070i	−0.976716	+	0.214535i	$0.931176\pi$
$972$	−3.97108e8	−	1.24062e9i	−0.432424	−	1.35095i
$973$	2.55432e6			0.00277291
$974$			1.73759e9i			1.88049i
$975$	−3.20895e7	−	5.17271e8i	−0.0346218	−	0.558090i
$976$	6.80734e8			0.732197
$977$			5.78361e8i			0.620177i	0.950708	+	0.310088i	$0.100359\pi$
$977$			5.78361e8i			0.620177i	−0.950708	+	0.310088i	$0.899641\pi$
$978$	1.94322e8	−	1.20550e7i	0.207733	−	0.0128870i
$979$	−2.91006e8			−0.310137
$980$		−	5.34213e8i		−	0.567592i
$981$	1.13516e8	−	1.41387e7i	0.120241	−	0.0149762i
$982$	1.86382e9			1.96820
$983$			9.74961e8i			1.02642i	0.858262	+	0.513212i	$0.171545\pi$
$983$			9.74961e8i			1.02642i	−0.858262	+	0.513212i	$0.828455\pi$
$984$	−2.45094e7	−	3.95083e8i	−0.0257246	−	0.414670i
$985$	−1.92186e8			−0.201100
$986$			8.72349e7i			0.0910037i
$987$	−6.78254e8	+	4.20763e7i	−0.705409	+	0.0437609i
$988$	2.03248e9			2.10744
$989$			1.64385e9i			1.69931i
$990$	1.96058e8	+	1.57411e9i	0.202059	+	1.62229i
$991$	7.41030e8			0.761404			0.380702	−	0.924698i	$-0.375682\pi$
$991$	7.41030e8			0.761404			0.380702	+	0.924698i	$0.375682\pi$
$992$			7.13507e8i			0.730909i
$993$	−2.13414e7	−	3.44016e8i	−0.0217959	−	0.351342i
$994$	6.39435e8			0.651085
$995$		−	1.98561e8i		−	0.201570i
$996$	−1.70077e8	+	1.05509e7i	−0.172135	+	0.0106786i
$997$	−6.52687e8			−0.658596			−0.329298	−	0.944226i	$-0.606812\pi$
$997$	−6.52687e8			−0.658596			−0.329298	+	0.944226i	$0.606812\pi$
$998$			1.16995e9i			1.17700i
$999$	−348922.	−	1.85556e6i	−0.000349970	−	0.00186114i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	87.7.b.a.59.8	✓	56
3.2	odd	2	inner	87.7.b.a.59.49	yes	56

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
87.7.b.a.59.8	✓	56	1.1	even	1	trivial
87.7.b.a.59.49	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-12.4411i q^{2} +(1.67176 + 26.9482i) q^{3} -90.7821 q^{4} -69.4567i q^{5} +(335.266 - 20.7986i) q^{6} +181.456 q^{7} +333.200i q^{8} +(-723.410 + 90.1019i) q^{9} -864.121 q^{10} +2518.11i q^{11} +(-151.766 - 2446.41i) q^{12} -1777.19 q^{13} -2257.52i q^{14} +(1871.73 - 116.115i) q^{15} -1664.67 q^{16} +1548.23i q^{17} +(1120.97 + 9000.05i) q^{18} +12597.7 q^{19} +6305.43i q^{20} +(303.351 + 4889.91i) q^{21} +31328.2 q^{22} +13378.3i q^{23} +(-8979.13 + 557.031i) q^{24} +10800.8 q^{25} +22110.3i q^{26} +(-3637.45 - 19344.0i) q^{27} -16473.0 q^{28} +4528.92i q^{29} +(-1444.61 - 23286.5i) q^{30} +16974.1 q^{31} +42035.1i q^{32} +(-67858.6 + 4209.69i) q^{33} +19261.7 q^{34} -12603.3i q^{35} +(65672.7 - 8179.64i) q^{36} +95.9247 q^{37} -156730. i q^{38} +(-2971.04 - 47892.1i) q^{39} +23143.0 q^{40} +44000.1i q^{41} +(60836.1 - 3774.04i) q^{42} +122875. q^{43} -228600. i q^{44} +(6258.19 + 50245.7i) q^{45} +166441. q^{46} +138705. i q^{47} +(-2782.93 - 44859.8i) q^{48} -84722.7 q^{49} -134374. i q^{50} +(-41721.9 + 2588.27i) q^{51} +161337. q^{52} +87390.6i q^{53} +(-240661. + 45254.1i) q^{54} +174900. q^{55} +60461.1i q^{56} +(21060.4 + 339486. i) q^{57} +56345.0 q^{58} -134377. i q^{59} +(-169920. + 10541.2i) q^{60} -408931. q^{61} -211177. i q^{62} +(-131267. + 16349.5i) q^{63} +416427. q^{64} +123438. i q^{65} +(52373.3 + 844239. i) q^{66} -377861. q^{67} -140551. i q^{68} +(-360520. + 22365.3i) q^{69} -156800. q^{70} +283247. i q^{71} +(-30021.9 - 241040. i) q^{72} +232402. q^{73} -1193.41i q^{74} +(18056.3 + 291061. i) q^{75} -1.14365e6 q^{76} +456927. i q^{77} +(-595832. + 36963.1i) q^{78} +496261. q^{79} +115622. i q^{80} +(515204. - 130361. i) q^{81} +547412. q^{82} -69521.1i q^{83} +(-27538.9 - 443916. i) q^{84} +107535. q^{85} -1.52870e6i q^{86} +(-122046. + 7571.28i) q^{87} -839035. q^{88} +115565. i q^{89} +(625114. - 77859.0i) q^{90} -322482. q^{91} -1.21451e6i q^{92} +(28376.6 + 457420. i) q^{93} +1.72565e6 q^{94} -874998. i q^{95} +(-1.13277e6 + 70272.8i) q^{96} -928194. q^{97} +1.05405e6i q^{98} +(-226887. - 1.82163e6i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 56 q + 52 q^{3} - 1924 q^{4} - 160 q^{6} + 160 q^{7} - 1060 q^{9} - 3588 q^{10} - 2166 q^{12} - 1400 q^{13} - 6240 q^{15} + 56588 q^{16} - 5978 q^{18} + 25000 q^{19} + 7520 q^{21} + 20970 q^{22} + 1238 q^{24}+ \cdots + 4793544 q^{99}+O(q^{100}) \) 56 * q + 52 * q^3 - 1924 * q^4 - 160 * q^6 + 160 * q^7 - 1060 * q^9 - 3588 * q^10 - 2166 * q^12 - 1400 * q^13 - 6240 * q^15 + 56588 * q^16 - 5978 * q^18 + 25000 * q^19 + 7520 * q^21 + 20970 * q^22 + 1238 * q^24 - 213304 * q^25 + 25492 * q^27 + 7870 * q^28 + 105092 * q^30 - 78656 * q^31 + 18952 * q^33 - 96642 * q^34 - 40224 * q^36 + 52120 * q^37 - 112096 * q^39 + 343440 * q^40 - 341758 * q^42 - 286760 * q^43 + 68312 * q^45 + 255516 * q^46 + 386086 * q^48 + 1051872 * q^49 - 256656 * q^51 + 58606 * q^52 + 450124 * q^54 - 409632 * q^55 - 1141736 * q^57 + 817806 * q^60 + 309976 * q^61 + 567632 * q^63 - 2844730 * q^64 + 958426 * q^66 + 748696 * q^67 - 1323704 * q^69 - 2123748 * q^70 - 1026242 * q^72 - 333728 * q^73 - 331316 * q^75 + 2846728 * q^76 + 292446 * q^78 - 661184 * q^79 + 1614852 * q^81 + 376908 * q^82 - 1670584 * q^84 + 3342000 * q^85 - 927168 * q^88 + 835882 * q^90 + 2248016 * q^91 + 3960776 * q^93 - 1368294 * q^94 - 5811456 * q^96 - 3057296 * q^97 + 4793544 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more