LMFDB - Embedded newform 109.8.b.a.108.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [109,8,Mod(108,109)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(109, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1]))

N = Newforms(chi, 8, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("109.108");

S:= CuspForms(chi, 8);

N := Newforms(S);

Level:	$ N $	$=$	$ 109 $
Weight:	$ k $	$=$	$ 8 $
Character orbit:	$[\chi]$	$=$	109.b (of order $2$, degree $1$, minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$34.0499677778$
Analytic rank:	$0$
Dimension:	$62$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			108.3
Character	$\chi$	$=$	109.108
Dual form			109.8.b.a.108.60

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-20.6762i q^{2} -63.1733 q^{3} -299.506 q^{4} -321.718 q^{5} +1306.18i q^{6} +710.246 q^{7} +3546.09i q^{8} +1803.86 q^{9} +O(q^{10})$	\(q-20.6762i q^{2} -63.1733 q^{3} -299.506 q^{4} -321.718 q^{5} +1306.18i q^{6} +710.246 q^{7} +3546.09i q^{8} +1803.86 q^{9} +6651.91i q^{10} -1192.20i q^{11} +18920.8 q^{12} -1572.49i q^{13} -14685.2i q^{14} +20324.0 q^{15} +34983.0 q^{16} +18385.9i q^{17} -37297.0i q^{18} -6835.01i q^{19} +96356.4 q^{20} -44868.6 q^{21} -24650.2 q^{22} +79967.7i q^{23} -224018. i q^{24} +25377.3 q^{25} -32513.1 q^{26} +24204.1 q^{27} -212723. q^{28} -35246.0 q^{29} -420223. i q^{30} -199862. q^{31} -269416. i q^{32} +75315.3i q^{33} +380151. q^{34} -228499. q^{35} -540267. q^{36} +348608. i q^{37} -141322. q^{38} +99339.2i q^{39} -1.14084e6i q^{40} -632802. i q^{41} +927712. i q^{42} +773281. q^{43} +357072. i q^{44} -580334. q^{45} +1.65343e6 q^{46} +216626. i q^{47} -2.20999e6 q^{48} -319093. q^{49} -524708. i q^{50} -1.16150e6i q^{51} +470969. i q^{52} -1.11561e6i q^{53} -500450. i q^{54} +383553. i q^{55} +2.51860e6i q^{56} +431790. i q^{57} +728754. i q^{58} +1.10542e6i q^{59} -6.08715e6 q^{60} -1.25024e6 q^{61} +4.13238e6i q^{62} +1.28119e6 q^{63} -1.09269e6 q^{64} +505897. i q^{65} +1.55724e6 q^{66} +241109. i q^{67} -5.50669e6i q^{68} -5.05182e6i q^{69} +4.72449e6i q^{70} -2.92360e6 q^{71} +6.39666e6i q^{72} -4.25942e6 q^{73} +7.20789e6 q^{74} -1.60317e6 q^{75} +2.04713e6i q^{76} -846757. i q^{77} +2.05396e6 q^{78} -4.77319e6i q^{79} -1.12547e7 q^{80} -5.47410e6 q^{81} -1.30840e7 q^{82} +5.49556e6 q^{83} +1.34384e7 q^{84} -5.91508e6i q^{85} -1.59885e7i q^{86} +2.22661e6 q^{87} +4.22766e6 q^{88} +8.82466e6 q^{89} +1.19991e7i q^{90} -1.11685e6i q^{91} -2.39508e7i q^{92} +1.26259e7 q^{93} +4.47900e6 q^{94} +2.19894e6i q^{95} +1.70199e7i q^{96} +1.43933e7 q^{97} +6.59764e6i q^{98} -2.15057e6i q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 62 q - 56 q^{3} - 3584 q^{4} + 194 q^{5} + 370 q^{7} + 40462 q^{9}+O(q^{10}) $ 62 * q - 56 * q^3 - 3584 * q^4 + 194 * q^5 + 370 * q^7 + 40462 * q^9	$ 62 q - 56 q^{3} - 3584 q^{4} + 194 q^{5} + 370 q^{7} + 40462 q^{9} + 6248 q^{12} - 21376 q^{15} + 227856 q^{16} - 216666 q^{20} + 42914 q^{21} - 101446 q^{22} + 788644 q^{25} - 282610 q^{26} - 235946 q^{27} - 69142 q^{28} - 213678 q^{29} + 300350 q^{31} - 401602 q^{34} - 377134 q^{35} - 2176902 q^{36} - 2421542 q^{38} + 2022550 q^{43} + 2223056 q^{45} + 2665174 q^{46} + 1060208 q^{48} + 7702792 q^{49} + 9894904 q^{60} + 278126 q^{61} - 6010300 q^{63} - 9362182 q^{64} - 23291020 q^{66} + 15373744 q^{71} - 1378934 q^{73} - 19097780 q^{74} - 41528062 q^{75} + 5705654 q^{78} + 42766270 q^{80} + 29291446 q^{81} + 14605464 q^{82} + 1191422 q^{83} - 6959552 q^{84} - 22121732 q^{87} + 6337872 q^{88} + 29157388 q^{89} + 14091936 q^{93} - 17482078 q^{94} - 22763314 q^{97}+O(q^{100}) $ 62 * q - 56 * q^3 - 3584 * q^4 + 194 * q^5 + 370 * q^7 + 40462 * q^9 + 6248 * q^12 - 21376 * q^15 + 227856 * q^16 - 216666 * q^20 + 42914 * q^21 - 101446 * q^22 + 788644 * q^25 - 282610 * q^26 - 235946 * q^27 - 69142 * q^28 - 213678 * q^29 + 300350 * q^31 - 401602 * q^34 - 377134 * q^35 - 2176902 * q^36 - 2421542 * q^38 + 2022550 * q^43 + 2223056 * q^45 + 2665174 * q^46 + 1060208 * q^48 + 7702792 * q^49 + 9894904 * q^60 + 278126 * q^61 - 6010300 * q^63 - 9362182 * q^64 - 23291020 * q^66 + 15373744 * q^71 - 1378934 * q^73 - 19097780 * q^74 - 41528062 * q^75 + 5705654 * q^78 + 42766270 * q^80 + 29291446 * q^81 + 14605464 * q^82 + 1191422 * q^83 - 6959552 * q^84 - 22121732 * q^87 + 6337872 * q^88 + 29157388 * q^89 + 14091936 * q^93 - 17482078 * q^94 - 22763314 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/109\mathbb{Z}\right)^\times$.

$n$	$6$
$\chi(n)$	$-1$

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 $

$2$		−	20.6762i		−	1.82754i	−0.406236	−	0.913768i	$-0.633159\pi$
$2$		−	20.6762i		−	1.82754i	0.406236	−	0.913768i	$-0.366841\pi$
$3$	−63.1733			−1.35086			−0.675428	−	0.737426i	$-0.736041\pi$
$3$	−63.1733			−1.35086			−0.675428	+	0.737426i	$0.736041\pi$
$4$	−299.506			−2.33989
$5$	−321.718			−1.15101			−0.575506	−	0.817797i	$-0.695195\pi$
$5$	−321.718			−1.15101			−0.575506	+	0.817797i	$0.695195\pi$
$6$			1306.18i			2.46874i
$7$	710.246			0.782647			0.391323	−	0.920253i	$-0.372017\pi$
$7$	710.246			0.782647			0.391323	+	0.920253i	$0.372017\pi$
$8$			3546.09i			2.44870i
$9$	1803.86			0.824811
$10$			6651.91i			2.10352i
$11$		−	1192.20i		−	0.270070i	−0.990841	−	0.135035i	$-0.956885\pi$
$11$		−	1192.20i		−	0.270070i	0.990841	−	0.135035i	$-0.0431146\pi$
$12$	18920.8			3.16085
$13$		−	1572.49i		−	0.198511i	−0.995062	−	0.0992557i	$-0.968354\pi$
$13$		−	1572.49i		−	0.198511i	0.995062	−	0.0992557i	$-0.0316462\pi$
$14$		−	14685.2i		−	1.43032i
$15$	20324.0			1.55485
$16$	34983.0			2.13519
$17$			18385.9i			0.907641i	0.891093	+	0.453820i	$0.149939\pi$
$17$			18385.9i			0.907641i	−0.891093	+	0.453820i	$0.850061\pi$
$18$		−	37297.0i		−	1.50737i
$19$		−	6835.01i		−	0.228613i	−0.993446	−	0.114307i	$-0.963535\pi$
$19$		−	6835.01i		−	0.228613i	0.993446	−	0.114307i	$-0.0364646\pi$
$20$	96356.4			2.69324
$21$	−44868.6			−1.05724
$22$	−24650.2			−0.493562
$23$			79967.7i			1.37046i	0.728325	+	0.685232i	$0.240299\pi$
$23$			79967.7i			1.37046i	−0.728325	+	0.685232i	$0.759701\pi$
$24$		−	224018.i		−	3.30784i
$25$	25377.3			0.324830
$26$	−32513.1			−0.362787
$27$	24204.1			0.236655
$28$	−212723.			−1.83131
$29$	−35246.0			−0.268360			−0.134180	−	0.990957i	$-0.542840\pi$
$29$	−35246.0			−0.268360			−0.134180	+	0.990957i	$0.542840\pi$
$30$		−	420223.i		−	2.84155i
$31$	−199862.			−1.20493			−0.602467	−	0.798144i	$-0.705816\pi$
$31$	−199862.			−1.20493			−0.602467	+	0.798144i	$0.705816\pi$
$32$		−	269416.i		−	1.45345i
$33$			75315.3i			0.364825i
$34$	380151.			1.65875
$35$	−228499.			−0.900836
$36$	−540267.			−1.92997
$37$			348608.i			1.13144i	0.824598	+	0.565719i	$0.191401\pi$
$37$			348608.i			1.13144i	−0.824598	+	0.565719i	$0.808599\pi$
$38$	−141322.			−0.417799
$39$			99339.2i			0.268160i
$40$		−	1.14084e6i		−	2.81848i
$41$		−	632802.i		−	1.43392i	−0.697115	−	0.716960i	$-0.745533\pi$
$41$		−	632802.i		−	1.43392i	0.697115	−	0.716960i	$-0.254467\pi$
$42$			927712.i			1.93215i
$43$	773281.			1.48319			0.741596	−	0.670846i	$-0.234069\pi$
$43$	773281.			1.48319			0.741596	+	0.670846i	$0.234069\pi$
$44$			357072.i			0.631933i
$45$	−580334.			−0.949368
$46$	1.65343e6			2.50457
$47$			216626.i			0.304346i	0.988354	+	0.152173i	$0.0486271\pi$
$47$			216626.i			0.304346i	−0.988354	+	0.152173i	$0.951373\pi$
$48$	−2.20999e6			−2.88434
$49$	−319093.			−0.387464
$50$		−	524708.i		−	0.593639i
$51$		−	1.16150e6i		−	1.22609i
$52$			470969.i			0.464495i
$53$		−	1.11561e6i		−	1.02931i	−0.857398	−	0.514654i	$-0.827920\pi$
$53$		−	1.11561e6i		−	1.02931i	0.857398	−	0.514654i	$-0.172080\pi$
$54$		−	500450.i		−	0.432496i
$55$			383553.i			0.310853i
$56$			2.51860e6i			1.91646i
$57$			431790.i			0.308824i
$58$			728754.i			0.490437i
$59$			1.10542e6i			0.700721i	0.936615	+	0.350361i	$0.113941\pi$
$59$			1.10542e6i			0.700721i	−0.936615	+	0.350361i	$0.886059\pi$
$60$	−6.08715e6			−3.63818
$61$	−1.25024e6			−0.705245			−0.352622	−	0.935766i	$-0.614710\pi$
$61$	−1.25024e6			−0.705245			−0.352622	+	0.935766i	$0.614710\pi$
$62$			4.13238e6i			2.20206i
$63$	1.28119e6			0.645536
$64$	−1.09269e6			−0.521034
$65$			505897.i			0.228489i
$66$	1.55724e6			0.666731
$67$			241109.i			0.0979380i	0.998800	+	0.0489690i	$0.0155935\pi$
$67$			241109.i			0.0979380i	−0.998800	+	0.0489690i	$0.984406\pi$
$68$		−	5.50669e6i		−	2.12378i
$69$		−	5.05182e6i		−	1.85130i
$70$			4.72449e6i			1.64631i
$71$	−2.92360e6			−0.969425			−0.484712	−	0.874674i	$-0.661076\pi$
$71$	−2.92360e6			−0.969425			−0.484712	+	0.874674i	$0.661076\pi$
$72$			6.39666e6i			2.01971i
$73$	−4.25942e6			−1.28151			−0.640753	−	0.767747i	$-0.721378\pi$
$73$	−4.25942e6			−1.28151			−0.640753	+	0.767747i	$0.721378\pi$
$74$	7.20789e6			2.06775
$75$	−1.60317e6			−0.438799
$76$			2.04713e6i			0.534930i
$77$		−	846757.i		−	0.211369i
$78$	2.05396e6			0.490073
$79$		−	4.77319e6i		−	1.08922i	−0.838691	−	0.544608i	$-0.816678\pi$
$79$		−	4.77319e6i		−	1.08922i	0.838691	−	0.544608i	$-0.183322\pi$
$80$	−1.12547e7			−2.45763
$81$	−5.47410e6			−1.14450
$82$	−1.30840e7			−2.62054
$83$	5.49556e6			1.05497			0.527483	−	0.849565i	$-0.323136\pi$
$83$	5.49556e6			1.05497			0.527483	+	0.849565i	$0.323136\pi$
$84$	1.34384e7			2.47383
$85$		−	5.91508e6i		−	1.04471i
$86$		−	1.59885e7i		−	2.71059i
$87$	2.22661e6			0.362515
$88$	4.22766e6			0.661319
$89$	8.82466e6			1.32688			0.663442	−	0.748228i	$-0.269095\pi$
$89$	8.82466e6			1.32688			0.663442	+	0.748228i	$0.269095\pi$
$90$			1.19991e7i			1.73500i
$91$		−	1.11685e6i		−	0.155364i
$92$		−	2.39508e7i		−	3.20673i
$93$	1.26259e7			1.62769
$94$	4.47900e6			0.556203
$95$			2.19894e6i			0.263137i
$96$			1.70199e7i			1.96340i
$97$	1.43933e7			1.60125			0.800625	−	0.599166i	$-0.204501\pi$
$97$	1.43933e7			1.60125			0.800625	+	0.599166i	$0.204501\pi$
$98$			6.59764e6i			0.708105i
$99$		−	2.15057e6i		−	0.222756i
$100$	−7.60066e6			−0.760066
$101$		−	6.69846e6i		−	0.646919i	−0.946242	−	0.323460i	$-0.895154\pi$
$101$		−	6.69846e6i		−	0.646919i	0.946242	−	0.323460i	$-0.104846\pi$
$102$	−2.40154e7			−2.24073
$103$		−	1.52467e7i		−	1.37482i	−0.726269	−	0.687410i	$-0.758748\pi$
$103$		−	1.52467e7i		−	1.37482i	0.726269	−	0.687410i	$-0.241252\pi$
$104$	5.57619e6			0.486094
$105$	1.44350e7			1.21690
$106$	−2.30665e7			−1.88110
$107$			1.31900e7i			1.04089i	0.853896	+	0.520443i	$0.174233\pi$
$107$			1.31900e7i			1.04089i	−0.853896	+	0.520443i	$0.825767\pi$
$108$	−7.24928e6			−0.553747
$109$	−1.22458e7	+	5.73095e6i	−0.905723	+	0.423871i
$109$	−1.22458e7	+	5.73095e6i	−0.905723	+	0.423871i
$110$	7.93042e6			0.568096
$111$		−	2.20227e7i		−	1.52841i
$112$	2.48466e7			1.67110
$113$	−1.45983e7			−0.951762			−0.475881	−	0.879510i	$-0.657871\pi$
$113$	−1.45983e7			−0.951762			−0.475881	+	0.879510i	$0.657871\pi$
$114$	8.92778e6			0.564386
$115$		−	2.57270e7i		−	1.57742i
$116$	1.05564e7			0.627932
$117$		−	2.83655e6i		−	0.163734i
$118$	2.28559e7			1.28059
$119$			1.30585e7i			0.710362i
$120$			7.20706e7i			3.80736i
$121$	1.80658e7			0.927062
$122$			2.58503e7i			1.28886i
$123$			3.99762e7i			1.93702i
$124$	5.98597e7			2.81941
$125$	1.69699e7			0.777129
$126$		−	2.64901e7i		−	1.17974i
$127$		−	2.58314e7i		−	1.11901i	−0.828826	−	0.559507i	$-0.810990\pi$
$127$		−	2.58314e7i		−	1.11901i	0.828826	−	0.559507i	$-0.189010\pi$
$128$		−	1.18927e7i		−	0.501238i
$129$	−4.88507e7			−2.00358
$130$	1.04600e7			0.417572
$131$	3.20890e7			1.24712			0.623558	−	0.781777i	$-0.285686\pi$
$131$	3.20890e7			1.24712			0.623558	+	0.781777i	$0.285686\pi$
$132$		−	2.25574e7i		−	0.853650i
$133$		−	4.85454e6i		−	0.178923i
$134$	4.98522e6			0.178985
$135$	−7.78690e6			−0.272393
$136$	−6.51981e7			−2.22254
$137$	−3.09276e7			−1.02760			−0.513800	−	0.857910i	$-0.671763\pi$
$137$	−3.09276e7			−1.02760			−0.513800	+	0.857910i	$0.671763\pi$
$138$	−1.04453e8			−3.38331
$139$			1.48202e6i			0.0468062i	0.999726	+	0.0234031i	$0.00745011\pi$
$139$			1.48202e6i			0.0468062i	−0.999726	+	0.0234031i	$0.992550\pi$
$140$	6.84368e7			2.10786
$141$		−	1.36850e7i		−	0.411127i
$142$			6.04490e7i			1.77166i
$143$	−1.87472e6			−0.0536119
$144$	6.31045e7			1.76113
$145$	1.13393e7			0.308885
$146$			8.80687e7i			2.34200i
$147$	2.01582e7			0.523408
$148$		−	1.04410e8i		−	2.64744i
$149$			4.13477e7i			1.02400i	0.858986	+	0.512000i	$0.171095\pi$
$149$			4.13477e7i			1.02400i	−0.858986	+	0.512000i	$0.828905\pi$
$150$			3.31475e7i			0.801920i
$151$			6.07191e6i			0.143518i	0.997422	+	0.0717589i	$0.0228612\pi$
$151$			6.07191e6i			0.143518i	−0.997422	+	0.0717589i	$0.977139\pi$
$152$	2.42376e7			0.559805
$153$			3.31656e7i			0.748632i
$154$	−1.75077e7			−0.386285
$155$	6.42990e7			1.38689
$156$		−	2.97527e7i		−	0.627465i
$157$	4.49865e6			0.0927755			0.0463878	−	0.998924i	$-0.485229\pi$
$157$	4.49865e6			0.0927755			0.0463878	+	0.998924i	$0.485229\pi$
$158$	−9.86916e7			−1.99058
$159$			7.04765e7i			1.39045i
$160$			8.66761e7i			1.67294i
$161$			5.67968e7i			1.07259i
$162$			1.13184e8i			2.09161i
$163$		−	7.76537e7i		−	1.40445i	−0.711956	−	0.702224i	$-0.752191\pi$
$163$		−	7.76537e7i		−	1.40445i	0.711956	−	0.702224i	$-0.247809\pi$
$164$			1.89528e8i			3.35521i
$165$		−	2.42303e7i		−	0.419918i
$166$		−	1.13627e8i		−	1.92799i
$167$			6.20269e7i			1.03056i	0.857022	+	0.515279i	$0.172312\pi$
$167$			6.20269e7i			1.03056i	−0.857022	+	0.515279i	$0.827688\pi$
$168$		−	1.59108e8i		−	2.58887i
$169$	6.02758e7			0.960593
$170$	−1.22301e8			−1.90924
$171$		−	1.23294e7i		−	0.188563i
$172$	−2.31602e8			−3.47051
$173$	4.68725e7			0.688267			0.344133	−	0.938921i	$-0.388173\pi$
$173$	4.68725e7			0.688267			0.344133	+	0.938921i	$0.388173\pi$
$174$		−	4.60378e7i		−	0.662510i
$175$	1.80242e7			0.254227
$176$		−	4.17068e7i		−	0.576651i
$177$		−	6.98330e7i		−	0.946574i
$178$		−	1.82461e8i		−	2.42493i
$179$			2.96998e7i			0.387051i	0.981095	+	0.193525i	$0.0619922\pi$
$179$			2.96998e7i			0.387051i	−0.981095	+	0.193525i	$0.938008\pi$
$180$	1.73814e8			2.22142
$181$		−	1.40026e8i		−	1.75523i	−0.479367	−	0.877615i	$-0.659134\pi$
$181$		−	1.40026e8i		−	1.75523i	0.479367	−	0.877615i	$-0.340866\pi$
$182$	−2.30923e7			−0.283934
$183$	7.89819e7			0.952684
$184$	−2.83573e8			−3.35585
$185$		−	1.12153e8i		−	1.30230i
$186$		−	2.61056e8i		−	2.97467i
$187$	2.19197e7			0.245126
$188$		−	6.48807e7i		−	0.712136i
$189$	1.71909e7			0.185218
$190$	4.54658e7			0.480892
$191$	2.17926e6			0.0226304			0.0113152	−	0.999936i	$-0.496398\pi$
$191$	2.17926e6			0.0226304			0.0113152	+	0.999936i	$0.496398\pi$
$192$	6.90286e7			0.703842
$193$	−4.03163e7			−0.403673			−0.201837	−	0.979419i	$-0.564691\pi$
$193$	−4.03163e7			−0.403673			−0.201837	+	0.979419i	$0.564691\pi$
$194$		−	2.97599e8i		−	2.92634i
$195$		−	3.19592e7i		−	0.308656i
$196$	9.55703e7			0.906623
$197$	1.02135e8			0.951790			0.475895	−	0.879502i	$-0.342124\pi$
$197$	1.02135e8			0.951790			0.475895	+	0.879502i	$0.342124\pi$
$198$	−4.44656e7			−0.407095
$199$			7.10692e6i			0.0639286i	0.999489	+	0.0319643i	$0.0101763\pi$
$199$			7.10692e6i			0.0639286i	−0.999489	+	0.0319643i	$0.989824\pi$
$200$			8.99904e7i			0.795410i
$201$		−	1.52316e7i		−	0.132300i
$202$	−1.38499e8			−1.18227
$203$	−2.50334e7			−0.210031
$204$			3.47876e8i			2.86892i
$205$			2.03584e8i			1.65046i
$206$	−3.15244e8			−2.51253
$207$			1.44251e8i			1.13037i
$208$		−	5.50103e7i		−	0.423860i
$209$	−8.14872e6			−0.0617415
$210$		−	2.98462e8i		−	2.22393i
$211$	−612251.			−0.00448684			−0.00224342	−	0.999997i	$-0.500714\pi$
$211$	−612251.			−0.00448684			−0.00224342	+	0.999997i	$0.500714\pi$
$212$			3.34131e8i			2.40847i
$213$	1.84694e8			1.30955
$214$	2.72720e8			1.90226
$215$	−2.48778e8			−1.70717
$216$			8.58301e7i			0.579497i
$217$	−1.41951e8			−0.943038
$218$	1.18494e8	+	2.53197e8i	0.774640	+	1.65524i
$219$	2.69081e8			1.73113
$220$		−	1.14876e8i		−	0.727363i
$221$	2.89116e7			0.180177
$222$	−4.55346e8			−2.79323
$223$	9.59138e7			0.579181			0.289590	−	0.957151i	$-0.406481\pi$
$223$	9.59138e7			0.579181			0.289590	+	0.957151i	$0.406481\pi$
$224$		−	1.91352e8i		−	1.13754i
$225$	4.57772e7			0.267923
$226$			3.01838e8i			1.73938i
$227$	1.97524e8			1.12080			0.560402	−	0.828221i	$-0.310647\pi$
$227$	1.97524e8			1.12080			0.560402	+	0.828221i	$0.310647\pi$
$228$		−	1.29324e8i		−	0.722613i
$229$		−	1.72394e7i		−	0.0948633i	−0.998874	−	0.0474316i	$-0.984896\pi$
$229$		−	1.72394e7i		−	0.0948633i	0.998874	−	0.0474316i	$-0.0151036\pi$
$230$	−5.31938e8			−2.88279
$231$			5.34924e7i			0.285529i
$232$		−	1.24986e8i		−	0.657131i
$233$	−2.00931e8			−1.04064			−0.520320	−	0.853971i	$-0.674187\pi$
$233$	−2.00931e8			−1.04064			−0.520320	+	0.853971i	$0.674187\pi$
$234$	−5.86491e7			−0.299230
$235$		−	6.96924e7i		−	0.350306i
$236$		−	3.31080e8i		−	1.63961i
$237$			3.01538e8i			1.47137i
$238$	2.70001e8			1.29821
$239$	−2.24940e8			−1.06579			−0.532897	−	0.846180i	$-0.678897\pi$
$239$	−2.24940e8			−1.06579			−0.532897	+	0.846180i	$0.678897\pi$
$240$	7.10993e8			3.31991
$241$			3.74305e8i			1.72253i	0.508159	+	0.861263i	$0.330326\pi$
$241$			3.74305e8i			1.72253i	−0.508159	+	0.861263i	$0.669674\pi$
$242$		−	3.73533e8i		−	1.69424i
$243$	2.92882e8			1.30940
$244$	3.74455e8			1.65020
$245$	1.02658e8			0.445976
$246$	8.26556e8			3.53997
$247$	−1.07480e7			−0.0453824
$248$		−	7.08728e8i		−	2.95052i
$249$	−3.47172e8			−1.42511
$250$		−	3.50872e8i		−	1.42023i
$251$			4.04183e8i			1.61332i	0.591017	+	0.806659i	$0.298727\pi$
$251$			4.04183e8i			1.61332i	−0.591017	+	0.806659i	$0.701273\pi$
$252$	−3.83723e8			−1.51048
$253$	9.53377e7			0.370120
$254$	−5.34096e8			−2.04504
$255$			3.73675e8i			1.41125i
$256$	−3.85759e8			−1.43707
$257$			2.08670e8i			0.766820i	0.923578	+	0.383410i	$0.125250\pi$
$257$			2.08670e8i			0.766820i	−0.923578	+	0.383410i	$0.874750\pi$
$258$			1.01005e9i			3.66161i
$259$			2.47597e8i			0.885517i
$260$		−	1.51519e8i		−	0.534639i
$261$	−6.35789e7			−0.221346
$262$		−	6.63480e8i		−	2.27915i
$263$	−3.44667e8			−1.16830			−0.584149	−	0.811646i	$-0.698572\pi$
$263$	−3.44667e8			−1.16830			−0.584149	+	0.811646i	$0.698572\pi$
$264$	−2.67075e8			−0.893346
$265$			3.58911e8i			1.18475i
$266$	−1.00374e8			−0.326989
$267$	−5.57483e8			−1.79243
$268$		−	7.22135e7i		−	0.229164i
$269$			3.20751e8i			1.00470i	0.864665	+	0.502349i	$0.167531\pi$
$269$			3.20751e8i			1.00470i	−0.864665	+	0.502349i	$0.832469\pi$
$270$			1.61004e8i			0.497809i
$271$		−	4.59409e8i		−	1.40219i	−0.713068	−	0.701094i	$-0.752695\pi$
$271$		−	4.59409e8i		−	1.40219i	0.713068	−	0.701094i	$-0.247305\pi$
$272$			6.43195e8i			1.93799i
$273$			7.05553e7i			0.209875i
$274$			6.39466e8i			1.87798i
$275$		−	3.02549e7i		−	0.0877267i
$276$			1.51305e9i			4.33183i
$277$		−	1.52829e8i		−	0.432043i	−0.976389	−	0.216022i	$-0.930692\pi$
$277$		−	1.52829e8i		−	0.432043i	0.976389	−	0.216022i	$-0.0693082\pi$
$278$	3.06426e7			0.0855400
$279$	−3.60523e8			−0.993843
$280$		−	8.10278e8i		−	2.20588i
$281$	4.54301e8			1.22144			0.610719	−	0.791848i	$-0.290881\pi$
$281$	4.54301e8			1.22144			0.610719	+	0.791848i	$0.290881\pi$
$282$	−2.82953e8			−0.751350
$283$		−	7.34070e8i		−	1.92524i	−0.270855	−	0.962620i	$-0.587306\pi$
$283$		−	7.34070e8i		−	1.92524i	0.270855	−	0.962620i	$-0.412694\pi$
$284$	8.75636e8			2.26835
$285$		−	1.38915e8i		−	0.355460i
$286$			3.87622e7i			0.0979777i
$287$		−	4.49446e8i		−	1.12225i
$288$		−	4.85990e8i		−	1.19882i
$289$	7.22968e7			0.176188
$290$		−	2.34453e8i		−	0.564499i
$291$	−9.09271e8			−2.16306
$292$	1.27572e9			2.99858
$293$	7.20805e8			1.67410			0.837050	−	0.547127i	$-0.184279\pi$
$293$	7.20805e8			1.67410			0.837050	+	0.547127i	$0.184279\pi$
$294$		−	4.16795e8i		−	0.956547i
$295$		−	3.55633e8i		−	0.806539i
$296$	−1.23620e9			−2.77055
$297$		−	2.88562e7i		−	0.0639134i
$298$	8.54915e8			1.87140
$299$	1.25748e8			0.272053
$300$	4.80159e8			1.02674
$301$	5.49220e8			1.16082
$302$	1.25544e8			0.262284
$303$			4.23163e8i			0.873894i
$304$		−	2.39109e8i		−	0.488134i
$305$	4.02225e8			0.811746
$306$	6.85740e8			1.36815
$307$	−3.03795e8			−0.599235			−0.299617	−	0.954059i	$-0.596859\pi$
$307$	−3.03795e8			−0.599235			−0.299617	+	0.954059i	$0.596859\pi$
$308$			2.53609e8i			0.494580i
$309$			9.63185e8i			1.85718i
$310$		−	1.32946e9i		−	2.53460i
$311$	−1.91545e8			−0.361086			−0.180543	−	0.983567i	$-0.557785\pi$
$311$	−1.91545e8			−0.361086			−0.180543	+	0.983567i	$0.557785\pi$
$312$	−3.52266e8			−0.656643
$313$		−	7.76822e8i		−	1.43191i	−0.698145	−	0.715956i	$-0.745991\pi$
$313$		−	7.76822e8i		−	1.43191i	0.698145	−	0.715956i	$-0.254009\pi$
$314$		−	9.30151e7i		−	0.169551i
$315$	−4.12180e8			−0.743020
$316$			1.42960e9i			2.54865i
$317$		−	4.99148e8i		−	0.880080i	−0.897978	−	0.440040i	$-0.854964\pi$
$317$		−	4.99148e8i		−	0.880080i	0.897978	−	0.440040i	$-0.145036\pi$
$318$	1.45719e9			2.54109
$319$			4.20204e7i			0.0724758i
$320$	3.51537e8			0.599717
$321$		−	8.33258e8i		−	1.40609i
$322$	1.17434e9			1.96019
$323$	1.25668e8			0.207499
$324$	1.63952e9			2.67800
$325$		−	3.99056e7i		−	0.0644825i
$326$	−1.60558e9			−2.56668
$327$	7.73609e8	−	3.62043e8i	1.22350	−	0.572588i
$328$	2.24398e9			3.51123
$329$			1.53858e8i			0.238195i
$330$	−5.00990e8			−0.767416
$331$	−4.82366e8			−0.731103			−0.365551	−	0.930791i	$-0.619120\pi$
$331$	−4.82366e8			−0.731103			−0.365551	+	0.930791i	$0.619120\pi$
$332$	−1.64595e9			−2.46850
$333$			6.28840e8i			0.933223i
$334$	1.28248e9			1.88338
$335$		−	7.75690e7i		−	0.112728i
$336$	−1.56964e9			−2.25742
$337$		−	9.05263e8i		−	1.28846i	−0.764833	−	0.644229i	$-0.777178\pi$
$337$		−	9.05263e8i		−	1.28846i	0.764833	−	0.644229i	$-0.222822\pi$
$338$		−	1.24628e9i		−	1.75552i
$339$	9.22224e8			1.28569
$340$			1.77160e9i			2.44450i
$341$			2.38276e8i			0.325416i
$342$	−2.54926e8			−0.344605
$343$	−8.11553e8			−1.08589
$344$			2.74212e9i			3.63189i
$345$			1.62526e9i			2.13087i
$346$		−	9.69146e8i		−	1.25783i
$347$	−1.01024e9			−1.29799			−0.648995	−	0.760792i	$-0.724811\pi$
$347$	−1.01024e9			−1.29799			−0.648995	+	0.760792i	$0.724811\pi$
$348$	−6.66882e8			−0.848245
$349$	1.24639e9			1.56952			0.784758	−	0.619802i	$-0.212787\pi$
$349$	1.24639e9			1.56952			0.784758	+	0.619802i	$0.212787\pi$
$350$		−	3.72672e8i		−	0.464610i
$351$		−	3.80607e7i		−	0.0469788i
$352$	−3.21199e8			−0.392532
$353$	1.24202e9			1.50285			0.751426	−	0.659817i	$-0.229366\pi$
$353$	1.24202e9			1.50285			0.751426	+	0.659817i	$0.229366\pi$
$354$	−1.44388e9			−1.72990
$355$	9.40575e8			1.11582
$356$	−2.64304e9			−3.10476
$357$		−	8.24950e8i		−	0.959597i
$358$	6.14079e8			0.707349
$359$		−	1.10382e9i		−	1.25912i	−0.776953	−	0.629559i	$-0.783236\pi$
$359$		−	1.10382e9i		−	1.25912i	0.776953	−	0.629559i	$-0.216764\pi$
$360$		−	2.05792e9i		−	2.32471i
$361$	8.47154e8			0.947736
$362$	−2.89521e9			−3.20775
$363$	−1.14128e9			−1.25233
$364$			3.34504e8i			0.363535i
$365$	1.37033e9			1.47503
$366$		−	1.63305e9i		−	1.74106i
$367$			2.76894e8i			0.292403i	0.989255	+	0.146202i	$0.0467048\pi$
$367$			2.76894e8i			0.292403i	−0.989255	+	0.146202i	$0.953295\pi$
$368$			2.79751e9i			2.92620i
$369$		−	1.14149e9i		−	1.18271i
$370$	−2.31891e9			−2.38000
$371$		−	7.92356e8i		−	0.805585i
$372$	−3.78153e9			−3.80862
$373$	−2.54457e8			−0.253883			−0.126941	−	0.991910i	$-0.540516\pi$
$373$	−2.54457e8			−0.253883			−0.126941	+	0.991910i	$0.540516\pi$
$374$		−	4.53217e8i		−	0.447977i
$375$	−1.07204e9			−1.04979
$376$	−7.68175e8			−0.745251
$377$			5.54239e7i			0.0532724i
$378$		−	3.55443e8i		−	0.338492i
$379$			1.06294e7i			0.0100293i	0.999987	+	0.00501466i	$0.00159622\pi$
$379$			1.06294e7i			0.0100293i	−0.999987	+	0.00501466i	$0.998404\pi$
$380$		−	6.58597e8i		−	0.615711i
$381$			1.63186e9i			1.51163i
$382$		−	4.50589e7i		−	0.0413580i
$383$		−	9.89202e8i		−	0.899682i	−0.893109	−	0.449841i	$-0.851481\pi$
$383$		−	9.89202e8i		−	0.899682i	0.893109	−	0.449841i	$-0.148519\pi$
$384$			7.51299e8i			0.677101i
$385$			2.72417e8i			0.243288i
$386$			8.33588e8i			0.737728i
$387$	1.39489e9			1.22335
$388$	−4.31087e9			−3.74675
$389$			1.06126e9i			0.914107i	0.889439	+	0.457053i	$0.151095\pi$
$389$			1.06126e9i			0.914107i	−0.889439	+	0.457053i	$0.848905\pi$
$390$	−6.60795e8			−0.564080
$391$	−1.47028e9			−1.24389
$392$		−	1.13153e9i		−	0.948782i
$393$	−2.02717e9			−1.68467
$394$		−	2.11176e9i		−	1.73943i
$395$			1.53562e9i			1.25370i
$396$			6.44108e8i			0.521225i
$397$		−	1.08619e9i		−	0.871246i	−0.900129	−	0.435623i	$-0.856528\pi$
$397$		−	1.08619e9i		−	0.871246i	0.900129	−	0.435623i	$-0.143472\pi$
$398$	1.46944e8			0.116832
$399$			3.06677e8i			0.241700i
$400$	8.87776e8			0.693575
$401$	1.56857e9			1.21478			0.607390	−	0.794404i	$-0.292217\pi$
$401$	1.56857e9			1.21478			0.607390	+	0.794404i	$0.292217\pi$
$402$	−3.14932e8			−0.241783
$403$			3.14280e8i			0.239193i
$404$			2.00623e9i			1.51372i
$405$	1.76111e9			1.31733
$406$			5.17595e8i			0.383839i
$407$	4.15611e8			0.305567
$408$	4.11878e9			3.00233
$409$	9.18361e8			0.663715			0.331858	−	0.943329i	$-0.392325\pi$
$409$	9.18361e8			0.663715			0.331858	+	0.943329i	$0.392325\pi$
$410$	4.20934e9			3.01627
$411$	1.95380e9			1.38814
$412$			4.56648e9i			3.21693i
$413$			7.85121e8i			0.548417i
$414$	2.98256e9			2.06580
$415$	−1.76802e9			−1.21428
$416$	−4.23654e8			−0.288526
$417$		−	9.36242e7i		−	0.0632284i
$418$			1.68485e8i			0.112835i
$419$		−	1.41315e9i		−	0.938513i	−0.883062	−	0.469256i	$-0.844522\pi$
$419$		−	1.41315e9i		−	0.938513i	0.883062	−	0.469256i	$-0.155478\pi$
$420$	−4.32337e9			−2.84741
$421$	−1.55087e8			−0.101295			−0.0506474	−	0.998717i	$-0.516128\pi$
$421$	−1.55087e8			−0.101295			−0.0506474	+	0.998717i	$0.516128\pi$
$422$			1.26590e7i			0.00819987i
$423$			3.90763e8i			0.251028i
$424$	3.95604e9			2.52046
$425$			4.66586e8i			0.294829i
$426$		−	3.81876e9i		−	2.39326i
$427$	−8.87980e8			−0.551958
$428$		−	3.95050e9i		−	2.43556i
$429$	1.18432e8			0.0724219
$430$			5.14379e9i			3.11992i
$431$	1.07161e9			0.644715			0.322358	−	0.946618i	$-0.395525\pi$
$431$	1.07161e9			0.644715			0.322358	+	0.946618i	$0.395525\pi$
$432$	8.46733e8			0.505305
$433$	−3.18798e8			−0.188715			−0.0943577	−	0.995538i	$-0.530080\pi$
$433$	−3.18798e8			−0.188715			−0.0943577	+	0.995538i	$0.530080\pi$
$434$			2.93501e9i			1.72344i
$435$	−7.16339e8			−0.417259
$436$	3.66770e9	−	1.71645e9i	2.11929	−	0.991811i
$437$	5.46580e8			0.313306
$438$		−	5.56359e9i		−	3.16370i
$439$	−1.11974e8			−0.0631671			−0.0315836	−	0.999501i	$-0.510055\pi$
$439$	−1.11974e8			−0.0631671			−0.0315836	+	0.999501i	$0.510055\pi$
$440$	−1.36011e9			−0.761186
$441$	−5.75600e8			−0.319584
$442$		−	5.97783e8i		−	0.329280i
$443$	−4.64689e8			−0.253951			−0.126975	−	0.991906i	$-0.540527\pi$
$443$	−4.64689e8			−0.253951			−0.126975	+	0.991906i	$0.540527\pi$
$444$			6.59593e9i			3.57631i
$445$	−2.83905e9			−1.52726
$446$		−	1.98313e9i		−	1.05847i
$447$		−	2.61207e9i		−	1.38328i
$448$	−7.76077e8			−0.407786
$449$		−	1.81960e9i		−	0.948667i	−0.880345	−	0.474334i	$-0.842689\pi$
$449$		−	1.81960e9i		−	0.948667i	0.880345	−	0.474334i	$-0.157311\pi$
$450$		−	9.46500e8i		−	0.489640i
$451$	−7.54429e8			−0.387258
$452$	4.37228e9			2.22702
$453$		−	3.83582e8i		−	0.193872i
$454$		−	4.08405e9i		−	2.04831i
$455$			3.59312e8i			0.178826i
$456$	−1.53117e9			−0.756215
$457$	6.41126e8			0.314222			0.157111	−	0.987581i	$-0.449782\pi$
$457$	6.41126e8			0.314222			0.157111	+	0.987581i	$0.449782\pi$
$458$	−3.56446e8			−0.173366
$459$			4.45015e8i			0.214798i
$460$			7.70540e9i			3.69099i
$461$	1.92267e9			0.914009			0.457005	−	0.889464i	$-0.348922\pi$
$461$	1.92267e9			0.914009			0.457005	+	0.889464i	$0.348922\pi$
$462$	1.10602e9			0.521815
$463$	3.09433e8			0.144888			0.0724441	−	0.997372i	$-0.476920\pi$
$463$	3.09433e8			0.144888			0.0724441	+	0.997372i	$0.476920\pi$
$464$	−1.23301e9			−0.573000
$465$	−4.06198e9			−1.87349
$466$			4.15449e9i			1.90181i
$467$	2.61488e9			1.18807			0.594037	−	0.804438i	$-0.297533\pi$
$467$	2.61488e9			1.18807			0.594037	+	0.804438i	$0.297533\pi$
$468$			8.49563e8i			0.383120i
$469$			1.71247e8i			0.0766508i
$470$	−1.44097e9			−0.640197
$471$	−2.84194e8			−0.125326
$472$	−3.91992e9			−1.71585
$473$		−	9.21907e8i		−	0.400565i
$474$	6.23467e9			2.68899
$475$		−	1.73454e8i		−	0.0742605i
$476$		−	3.91111e9i		−	1.66217i
$477$		−	2.01240e9i		−	0.848985i
$478$			4.65090e9i			1.94778i
$479$	−2.96501e9			−1.23268			−0.616341	−	0.787479i	$-0.711386\pi$
$479$	−2.96501e9			−1.23268			−0.616341	+	0.787479i	$0.711386\pi$
$480$		−	5.47561e9i		−	2.25989i
$481$	5.48181e8			0.224603
$482$	7.73921e9			3.14798
$483$		−	3.58804e9i		−	1.44891i
$484$	−5.41082e9			−2.16922
$485$	−4.63058e9			−1.84306
$486$		−	6.05570e9i		−	2.39297i
$487$		−	3.23736e9i		−	1.27010i	−0.772469	−	0.635052i	$-0.780979\pi$
$487$		−	3.23736e9i		−	1.27010i	0.772469	−	0.635052i	$-0.219021\pi$
$488$		−	4.43348e9i		−	1.72693i
$489$			4.90564e9i			1.89721i
$490$		−	2.12258e9i		−	0.815037i
$491$		−	4.32967e9i		−	1.65070i	−0.564619	−	0.825352i	$-0.690977\pi$
$491$		−	4.32967e9i		−	1.65070i	0.564619	−	0.825352i	$-0.309023\pi$
$492$		−	1.19731e10i		−	4.53241i
$493$		−	6.48030e8i		−	0.243574i
$494$			2.22227e8i			0.0829379i
$495$			6.91876e8i			0.256395i
$496$	−6.99176e9			−2.57277
$497$	−2.07648e9			−0.758717
$498$			7.17821e9i			2.60444i
$499$	3.42763e9			1.23493			0.617465	−	0.786599i	$-0.288160\pi$
$499$	3.42763e9			1.23493			0.617465	+	0.786599i	$0.288160\pi$
$500$	−5.08257e9			−1.81840
$501$		−	3.91844e9i		−	1.39214i
$502$	8.35698e9			2.94840
$503$			1.18414e9i			0.414871i	0.978249	+	0.207436i	$0.0665118\pi$
$503$			1.18414e9i			0.414871i	−0.978249	+	0.207436i	$0.933488\pi$
$504$			4.54320e9i			1.58072i
$505$			2.15501e9i			0.744612i
$506$		−	1.97122e9i		−	0.676409i
$507$	−3.80782e9			−1.29762
$508$			7.73667e9i			2.61837i
$509$	−6.09231e8			−0.204771			−0.102386	−	0.994745i	$-0.532648\pi$
$509$	−6.09231e8			−0.204771			−0.102386	+	0.994745i	$0.532648\pi$
$510$	7.72618e9			2.57911
$511$	−3.02524e9			−1.00297
$512$			6.45378e9i			2.12505i
$513$		−	1.65435e8i		−	0.0541026i
$514$	4.31450e9			1.40139
$515$			4.90514e9i			1.58244i
$516$	1.46311e10			4.68815
$517$	2.58262e8			0.0821946
$518$	5.11938e9			1.61831
$519$	−2.96109e9			−0.929749
$520$	−1.79396e9			−0.559501
$521$			4.49910e8i			0.139378i	0.997569	+	0.0696890i	$0.0222007\pi$
$521$			4.49910e8i			0.139378i	−0.997569	+	0.0696890i	$0.977799\pi$
$522$			1.31457e9i			0.404518i
$523$	−7.70909e8			−0.235639			−0.117820	−	0.993035i	$-0.537590\pi$
$523$	−7.70909e8			−0.235639			−0.117820	+	0.993035i	$0.537590\pi$
$524$	−9.61085e9			−2.91812
$525$	−1.13865e9			−0.343424
$526$			7.12640e9i			2.13511i
$527$		−	3.67464e9i		−	1.09365i
$528$			2.63476e9i			0.778972i
$529$	−2.99001e9			−0.878170
$530$	7.42091e9			2.16517
$531$			1.99403e9i			0.577963i
$532$			1.45396e9i			0.418661i
$533$	−9.95074e8			−0.284649
$534$			1.15266e10i			3.27573i
$535$		−	4.24347e9i		−	1.19807i
$536$	−8.54994e8			−0.239820
$537$		−	1.87623e9i		−	0.522850i
$538$	6.63193e9			1.83612
$539$			3.80424e8i			0.104642i
$540$	2.33222e9			0.637370
$541$	−9.07223e8			−0.246334			−0.123167	−	0.992386i	$-0.539305\pi$
$541$	−9.07223e8			−0.246334			−0.123167	+	0.992386i	$0.539305\pi$
$542$	−9.49883e9			−2.56255
$543$			8.84590e9i			2.37106i
$544$	4.95347e9			1.31921
$545$	3.93970e9	−	1.84375e9i	1.04250	−	0.487881i
$546$	1.45882e9			0.383554
$547$			5.82296e9i			1.52121i	0.649216	+	0.760604i	$0.275097\pi$
$547$			5.82296e9i			1.52121i	−0.649216	+	0.760604i	$0.724903\pi$
$548$	9.26300e9			2.40447
$549$	−2.25526e9			−0.581694
$550$	−6.25558e8			−0.160324
$551$			2.40907e8i			0.0613506i
$552$	1.79142e10			4.53327
$553$		−	3.39014e9i		−	0.852472i
$554$	−3.15993e9			−0.789575
$555$			7.08509e9i			1.75922i
$556$		−	4.43875e8i		−	0.109521i
$557$	2.50502e9			0.614212			0.307106	−	0.951675i	$-0.400639\pi$
$557$	2.50502e9			0.614212			0.307106	+	0.951675i	$0.400639\pi$
$558$			7.45424e9i			1.81628i
$559$		−	1.21597e9i		−	0.294431i
$560$	−7.99358e9			−1.92346
$561$	−1.38474e9			−0.331130
$562$		−	9.39322e9i		−	2.23222i
$563$			2.33572e9i			0.551621i	0.961212	+	0.275811i	$0.0889463\pi$
$563$			2.33572e9i			0.551621i	−0.961212	+	0.275811i	$0.911054\pi$
$564$			4.09872e9i			0.961993i
$565$	4.69654e9			1.09549
$566$	−1.51778e10			−3.51845
$567$	−3.88796e9			−0.895738
$568$		−	1.03674e10i		−	2.37383i
$569$			7.86238e9i			1.78921i	0.446858	+	0.894605i	$0.352543\pi$
$569$			7.86238e9i			1.78921i	−0.446858	+	0.894605i	$0.647457\pi$
$570$	−2.87223e9			−0.649616
$571$	7.16766e8			0.161121			0.0805603	−	0.996750i	$-0.474329\pi$
$571$	7.16766e8			0.161121			0.0805603	+	0.996750i	$0.474329\pi$
$572$	5.61491e8			0.125446
$573$	−1.37671e8			−0.0305705
$574$	−9.29283e9			−2.05096
$575$			2.02937e9i			0.445168i
$576$	−1.97106e9			−0.429755
$577$		−	3.37867e9i		−	0.732202i	−0.930575	−	0.366101i	$-0.880693\pi$
$577$		−	3.37867e9i		−	0.732202i	0.930575	−	0.366101i	$-0.119307\pi$
$578$		−	1.49482e9i		−	0.321990i
$579$	2.54691e9			0.545304
$580$	−3.39618e9			−0.722757
$581$	3.90320e9			0.825666
$582$			1.88003e10i			3.95306i
$583$	−1.33003e9			−0.277985
$584$		−	1.51043e10i		−	3.13802i
$585$			9.12568e8i			0.188460i
$586$		−	1.49035e10i		−	3.05948i
$587$		−	1.79201e9i		−	0.365684i	−0.983142	−	0.182842i	$-0.941470\pi$
$587$		−	1.79201e9i		−	0.365684i	0.983142	−	0.182842i	$-0.0585297\pi$
$588$	−6.03749e9			−1.22472
$589$			1.36606e9i			0.275464i
$590$	−7.35315e9			−1.47398
$591$	−6.45218e9			−1.28573
$592$			1.21954e10i			2.41584i
$593$	−1.16484e9			−0.229391			−0.114695	−	0.993401i	$-0.536589\pi$
$593$	−1.16484e9			−0.229391			−0.114695	+	0.993401i	$0.536589\pi$
$594$	−5.96637e8			−0.116804
$595$		−	4.20116e9i		−	0.817636i
$596$		−	1.23839e10i		−	2.39605i
$597$		−	4.48967e8i		−	0.0863584i
$598$		−	2.60000e9i		−	0.497186i
$599$		−	2.27529e9i		−	0.432556i	−0.976332	−	0.216278i	$-0.930608\pi$
$599$		−	2.27529e9i		−	0.432556i	0.976332	−	0.216278i	$-0.0693918\pi$
$600$		−	5.68499e9i		−	1.07448i
$601$		−	4.98984e9i		−	0.937618i	−0.883300	−	0.468809i	$-0.844683\pi$
$601$		−	4.98984e9i		−	0.937618i	0.883300	−	0.468809i	$-0.155317\pi$
$602$		−	1.13558e10i		−	2.12143i
$603$			4.34927e8i			0.0807803i
$604$		−	1.81857e9i		−	0.335816i
$605$	−5.81210e9			−1.06706
$606$	8.74942e9			1.59707
$607$		−	6.09016e8i		−	0.110527i	−0.998472	−	0.0552635i	$-0.982400\pi$
$607$		−	6.09016e8i		−	0.110527i	0.998472	−	0.0552635i	$-0.0175999\pi$
$608$	−1.84146e9			−0.332277
$609$	1.58144e9			0.283721
$610$		−	8.31650e9i		−	1.48350i
$611$	3.40641e8			0.0604161
$612$		−	9.93330e9i		−	1.75172i
$613$			6.05221e9i			1.06121i	0.847618	+	0.530606i	$0.178036\pi$
$613$			6.05221e9i			1.06121i	−0.847618	+	0.530606i	$0.821964\pi$
$614$			6.28134e9i			1.09512i
$615$		−	1.28611e10i		−	2.22953i
$616$	3.00268e9			0.517579
$617$			5.73505e9i			0.982967i	0.870887	+	0.491484i	$0.163545\pi$
$617$			5.73505e9i			0.982967i	−0.870887	+	0.491484i	$0.836455\pi$
$618$	1.99150e10			3.39407
$619$	8.48492e9			1.43791			0.718953	−	0.695059i	$-0.244622\pi$
$619$	8.48492e9			1.43791			0.718953	+	0.695059i	$0.244622\pi$
$620$	−1.92579e10			−3.24518
$621$			1.93555e9i			0.324327i
$622$			3.96043e9i			0.659897i
$623$	6.26768e9			1.03848
$624$			3.47518e9i			0.572574i
$625$	−7.44211e9			−1.21932
$626$	−1.60617e10			−2.61687
$627$	5.14781e8			0.0834039
$628$	−1.34737e9			−0.217084
$629$	−6.40947e9			−1.02694
$630$			8.52233e9i			1.35790i
$631$			8.25110e9i			1.30740i	0.756753	+	0.653701i	$0.226785\pi$
$631$			8.25110e9i			1.30740i	−0.756753	+	0.653701i	$0.773215\pi$
$632$	1.69262e10			2.66716
$633$	3.86779e7			0.00606108
$634$	−1.03205e10			−1.60838
$635$			8.31043e9i			1.28800i
$636$		−	2.11081e10i		−	3.25349i
$637$			5.01770e8i			0.0769160i
$638$	8.68823e8			0.132452
$639$	−5.27377e9			−0.799592
$640$			3.82608e9i			0.576932i
$641$		−	3.30198e9i		−	0.495189i	−0.968864	−	0.247595i	$-0.920360\pi$
$641$		−	3.30198e9i		−	0.495189i	0.968864	−	0.247595i	$-0.0796401\pi$
$642$	−1.72286e10			−2.56967
$643$			1.14589e10i			1.69983i	0.526921	+	0.849915i	$0.323347\pi$
$643$			1.14589e10i			1.69983i	−0.526921	+	0.849915i	$0.676653\pi$
$644$		−	1.70110e10i		−	2.50974i
$645$	1.57161e10			2.30615
$646$		−	2.59834e9i		−	0.379212i
$647$	1.01261e10			1.46986			0.734932	−	0.678141i	$-0.237214\pi$
$647$	1.01261e10			1.46986			0.734932	+	0.678141i	$0.237214\pi$
$648$		−	1.94117e10i		−	2.80253i
$649$	1.31788e9			0.189244
$650$	−8.25096e8			−0.117844
$651$	8.96751e9			1.27391
$652$			2.32577e10i			3.28625i
$653$	−5.19367e9			−0.729925			−0.364962	−	0.931022i	$-0.618918\pi$
$653$	−5.19367e9			−0.729925			−0.364962	+	0.931022i	$0.618918\pi$
$654$	−7.48567e9	−	1.59953e10i	−1.04643	−	2.23599i
$655$	−1.03236e10			−1.43545
$656$		−	2.21373e10i		−	3.06170i
$657$	−7.68340e9			−1.05700
$658$	3.18119e9			0.435311
$659$	−9.96842e9			−1.35684			−0.678418	−	0.734676i	$-0.737334\pi$
$659$	−9.96842e9			−1.35684			−0.678418	+	0.734676i	$0.737334\pi$
$660$			7.25711e9i			0.982562i
$661$	1.42723e10			1.92215			0.961076	−	0.276284i	$-0.0891029\pi$
$661$	1.42723e10			1.92215			0.961076	+	0.276284i	$0.0891029\pi$
$662$			9.97350e9i			1.33612i
$663$	−1.82644e9			−0.243393
$664$			1.94878e10i			2.58329i
$665$			1.56179e9i			0.205943i
$666$	1.30020e10			1.70550
$667$		−	2.81854e9i		−	0.367777i
$668$		−	1.85774e10i		−	2.41139i
$669$	−6.05919e9			−0.782389
$670$	−1.60383e9			−0.206014
$671$			1.49054e9i			0.190465i
$672$			1.20883e10i			1.53665i
$673$			8.14194e9i			1.02962i	0.857305	+	0.514808i	$0.172137\pi$
$673$			8.14194e9i			1.02962i	−0.857305	+	0.514808i	$0.827863\pi$
$674$	−1.87174e10			−2.35470
$675$	6.14237e8			0.0768728
$676$	−1.80530e10			−2.24768
$677$			1.50170e10i			1.86005i	0.367499	+	0.930024i	$0.380214\pi$
$677$			1.50170e10i			1.86005i	−0.367499	+	0.930024i	$0.619786\pi$
$678$		−	1.90681e10i		−	2.34965i
$679$	1.02228e10			1.25321
$680$	2.09754e10			2.55817
$681$	−1.24782e10			−1.51404
$682$	4.92664e9			0.594710
$683$	5.56336e9			0.668136			0.334068	−	0.942549i	$-0.391579\pi$
$683$	5.56336e9			0.668136			0.334068	+	0.942549i	$0.391579\pi$
$684$			3.69273e9i			0.441216i
$685$	9.94996e9			1.18278
$686$			1.67798e10i			1.98451i
$687$			1.08907e9i			0.128147i
$688$	2.70517e10			3.16690
$689$	−1.75428e9			−0.204329
$690$	3.36043e10			3.89424
$691$		−	4.95102e9i		−	0.570850i	−0.958401	−	0.285425i	$-0.907865\pi$
$691$		−	4.95102e9i		−	0.570850i	0.958401	−	0.285425i	$-0.0921347\pi$
$692$	−1.40386e10			−1.61047
$693$		−	1.52743e9i		−	0.174339i
$694$			2.08879e10i			2.37213i
$695$		−	4.76793e8i		−	0.0538745i
$696$			7.89575e9i			0.887690i
$697$	1.16347e10			1.30148
$698$		−	2.57707e10i		−	2.86835i
$699$	1.26934e10			1.40575
$700$	−5.39834e9			−0.594864
$701$		−	7.65635e9i		−	0.839477i	−0.907645	−	0.419739i	$-0.862122\pi$
$701$		−	7.65635e9i		−	0.839477i	0.907645	−	0.419739i	$-0.137878\pi$
$702$	−7.86951e8			−0.0858554
$703$	2.38274e9			0.258662
$704$			1.30270e9i			0.140715i
$705$			4.40269e9i			0.473213i
$706$		−	2.56802e10i		−	2.74652i
$707$		−	4.75755e9i		−	0.506309i
$708$			2.09154e10i			2.21488i
$709$		−	4.86121e9i		−	0.512251i	−0.966644	−	0.256125i	$-0.917554\pi$
$709$		−	4.86121e9i		−	0.512251i	0.966644	−	0.256125i	$-0.0824460\pi$
$710$		−	1.94475e10i		−	2.03920i
$711$		−	8.61018e9i		−	0.898397i
$712$			3.12931e10i			3.24914i
$713$		−	1.59825e10i		−	1.65132i
$714$	−1.70568e10			−1.75370
$715$	6.03132e8			0.0617080
$716$		−	8.89526e9i		−	0.905656i
$717$	1.42102e10			1.43973
$718$	−2.28227e10			−2.30108
$719$			5.34059e8i			0.0535843i	0.999641	+	0.0267922i	$0.00852923\pi$
$719$			5.34059e8i			0.0535843i	−0.999641	+	0.0267922i	$0.991471\pi$
$720$	−2.03018e10			−2.02708
$721$		−	1.08289e10i		−	1.07600i
$722$		−	1.75159e10i		−	1.73202i
$723$		−	2.36461e10i		−	2.32688i
$724$			4.19386e10i			4.10704i
$725$	−8.94451e8			−0.0871713
$726$			2.35973e10i			2.28867i
$727$	−2.15917e9			−0.208409			−0.104204	−	0.994556i	$-0.533230\pi$
$727$	−2.15917e9			−0.208409			−0.104204	+	0.994556i	$0.533230\pi$
$728$	3.96046e9			0.380440
$729$	−6.53047e9			−0.624307
$730$		−	2.83333e10i		−	2.69567i
$731$			1.42175e10i			1.34621i
$732$	−2.36555e10			−2.22918
$733$			4.06414e9i			0.381158i	0.981672	+	0.190579i	$0.0610365\pi$
$733$			4.06414e9i			0.381158i	−0.981672	+	0.190579i	$0.938964\pi$
$734$	5.72512e9			0.534378
$735$	−6.48524e9			−0.602449
$736$	2.15446e10			1.99190
$737$	2.87450e8			0.0264501
$738$	−2.36016e10			−2.16145
$739$			8.50335e9i			0.775058i	0.921857	+	0.387529i	$0.126671\pi$
$739$			8.50335e9i			0.775058i	−0.921857	+	0.387529i	$0.873329\pi$
$740$			3.35906e10i			3.04724i
$741$	6.78984e8			0.0613050
$742$	−1.63829e10			−1.47224
$743$	−3.27922e9			−0.293299			−0.146649	−	0.989189i	$-0.546849\pi$
$743$	−3.27922e9			−0.293299			−0.146649	+	0.989189i	$0.546849\pi$
$744$			4.47726e10i			3.98573i
$745$		−	1.33023e10i		−	1.17864i
$746$			5.26120e9i			0.463980i
$747$	9.91323e9			0.870148
$748$	−6.56509e9			−0.573568
$749$			9.36818e9i			0.814646i
$750$			2.21658e10i			1.91853i
$751$	1.77854e10			1.53223			0.766116	−	0.642703i	$-0.222187\pi$
$751$	1.77854e10			1.53223			0.766116	+	0.642703i	$0.222187\pi$
$752$			7.57822e9i			0.649838i
$753$		−	2.55336e10i		−	2.17936i
$754$	1.14596e9			0.0973573
$755$		−	1.95344e9i		−	0.165191i
$756$	−5.14877e9			−0.433389
$757$			2.13373e10i			1.78774i	0.448329	+	0.893869i	$0.352019\pi$
$757$			2.13373e10i			1.78774i	−0.448329	+	0.893869i	$0.647981\pi$
$758$	2.19776e8			0.0183290
$759$	−6.02280e9			−0.499979
$760$	−7.79766e9			−0.644342
$761$			1.76107e10i			1.44854i	0.689517	+	0.724269i	$0.257823\pi$
$761$			1.76107e10i			1.44854i	−0.689517	+	0.724269i	$0.742177\pi$
$762$	3.37406e10			2.76255
$763$	−8.69755e9	+	4.07038e9i	−0.708861	+	0.331741i
$764$	−6.52702e8			−0.0529527
$765$		−	1.06700e10i		−	0.861685i
$766$	−2.04529e10			−1.64420
$767$	1.73826e9			0.139101
$768$	2.43697e10			1.94127
$769$			5.75937e9i			0.456702i	0.973579	+	0.228351i	$0.0733334\pi$
$769$			5.75937e9i			0.456702i	−0.973579	+	0.228351i	$0.926667\pi$
$770$	5.63255e9			0.444619
$771$		−	1.31823e10i		−	1.03586i
$772$	1.20750e10			0.944551
$773$		−	9.53924e9i		−	0.742823i	−0.928468	−	0.371412i	$-0.878874\pi$
$773$		−	9.53924e9i		−	0.742823i	0.928468	−	0.371412i	$-0.121126\pi$
$774$		−	2.88411e10i		−	2.23572i
$775$	−5.07196e9			−0.391399
$776$			5.10399e10i			3.92097i
$777$		−	1.56415e10i		−	1.19621i
$778$	2.19428e10			1.67056
$779$	−4.32521e9			−0.327813
$780$			9.57196e9i			0.722221i
$781$			3.48553e9i			0.261812i
$782$			3.03998e10i			2.27325i
$783$	−8.53099e8			−0.0635087
$784$	−1.11628e10			−0.827311
$785$	−1.44730e9			−0.106786
$786$			4.19142e10i			3.07880i
$787$		−	1.46854e10i		−	1.07392i	−0.843606	−	0.536962i	$-0.819572\pi$
$787$		−	1.46854e10i		−	1.07392i	0.843606	−	0.536962i	$-0.180428\pi$
$788$	−3.05899e10			−2.22708
$789$	2.17737e10			1.57820
$790$	3.17508e10			2.29119
$791$	−1.03684e10			−0.744894
$792$	7.62611e9			0.545463
$793$			1.96599e9i			0.139999i
$794$	−2.24584e10			−1.59223
$795$		−	2.26736e10i		−	1.60042i
$796$		−	2.12856e9i		−	0.149586i
$797$	1.72852e10			1.20940			0.604700	−	0.796453i	$-0.293293\pi$
$797$	1.72852e10			1.20940			0.604700	+	0.796453i	$0.293293\pi$
$798$	6.34092e9			0.441715
$799$	−3.98286e9			−0.276237
$800$		−	6.83708e9i		−	0.472123i
$801$	1.59185e10			1.09443
$802$		−	3.24320e10i		−	2.22005i
$803$			5.07809e9i			0.346096i
$804$			4.56196e9i			0.309568i
$805$		−	1.82725e10i		−	1.23456i
$806$	6.49812e9			0.437134
$807$		−	2.02629e10i		−	1.35720i
$808$	2.37533e10			1.58411
$809$	7.65048e9			0.508006			0.254003	−	0.967203i	$-0.418253\pi$
$809$	7.65048e9			0.508006			0.254003	+	0.967203i	$0.418253\pi$
$810$		−	3.64132e10i		−	2.40747i
$811$	−1.29201e10			−0.850539			−0.425269	−	0.905067i	$-0.639821\pi$
$811$	−1.29201e10			−0.850539			−0.425269	+	0.905067i	$0.639821\pi$
$812$	7.49764e9			0.491449
$813$			2.90223e10i			1.89415i
$814$		−	8.59326e9i		−	0.558435i
$815$			2.49826e10i			1.61654i
$816$		−	4.06327e10i		−	2.61794i
$817$		−	5.28538e9i		−	0.339078i
$818$		−	1.89882e10i		−	1.21296i
$819$		−	2.01465e9i		−	0.128146i
$820$		−	6.09746e10i		−	3.86189i
$821$			1.11158e10i			0.701037i	0.936556	+	0.350519i	$0.113995\pi$
$821$			1.11158e10i			0.701037i	−0.936556	+	0.350519i	$0.886005\pi$
$822$		−	4.03972e10i		−	2.53688i
$823$	−1.22591e10			−0.766582			−0.383291	−	0.923628i	$-0.625209\pi$
$823$	−1.22591e10			−0.766582			−0.383291	+	0.923628i	$0.625209\pi$
$824$	5.40662e10			3.36652
$825$			1.91130e9i			0.118506i
$826$	1.62333e10			1.00225
$827$	2.37041e10			1.45731			0.728657	−	0.684878i	$-0.240145\pi$
$827$	2.37041e10			1.45731			0.728657	+	0.684878i	$0.240145\pi$
$828$		−	4.32039e10i		−	2.64495i
$829$	2.35886e10			1.43801			0.719004	−	0.695006i	$-0.244598\pi$
$829$	2.35886e10			1.43801			0.719004	+	0.695006i	$0.244598\pi$
$830$			3.65560e10i			2.21914i
$831$			9.65472e9i			0.583628i
$832$			1.71824e9i			0.103431i
$833$		−	5.86682e9i		−	0.351678i
$834$	−1.93579e9			−0.115552
$835$		−	1.99552e10i		−	1.18619i
$836$	2.44059e9			0.144468
$837$	−4.83748e9			−0.285154
$838$	−2.92187e10			−1.71517
$839$			1.72204e10i			1.00664i	0.864099	+	0.503322i	$0.167889\pi$
$839$			1.72204e10i			1.00664i	−0.864099	+	0.503322i	$0.832111\pi$
$840$			5.11879e10i			2.97982i
$841$	−1.60076e10			−0.927983
$842$			3.20660e9i			0.185120i
$843$	−2.86997e10			−1.64999
$844$	1.83373e8			0.0104987
$845$	−1.93918e10			−1.10565
$846$	8.07950e9			0.458763
$847$	1.28312e10			0.725562
$848$		−	3.90273e10i		−	2.19777i
$849$			4.63736e10i			2.60072i
$850$	9.64723e9			0.538811
$851$	−2.78774e10			−1.55060
$852$	−5.53168e10			−3.06421
$853$		−	2.68869e10i		−	1.48327i	−0.670805	−	0.741634i	$-0.734051\pi$
$853$		−	2.68869e10i		−	1.48327i	0.670805	−	0.741634i	$-0.265949\pi$
$854$			1.83601e10i			1.00872i
$855$			3.96659e9i			0.217038i
$856$	−4.67731e10			−2.54881
$857$	2.38254e10			1.29303			0.646513	−	0.762903i	$-0.276227\pi$
$857$	2.38254e10			1.29303			0.646513	+	0.762903i	$0.276227\pi$
$858$		−	2.44873e9i		−	0.132354i
$859$			2.38199e10i			1.28223i	0.767447	+	0.641113i	$0.221527\pi$
$859$			2.38199e10i			1.28223i	−0.767447	+	0.641113i	$0.778473\pi$
$860$	7.45105e10			3.99460
$861$			2.83929e10i			1.51600i
$862$		−	2.21569e10i		−	1.17824i
$863$	1.29134e9			0.0683915			0.0341958	−	0.999415i	$-0.489113\pi$
$863$	1.29134e9			0.0683915			0.0341958	+	0.999415i	$0.489113\pi$
$864$		−	6.52099e9i		−	0.343966i
$865$	−1.50797e10			−0.792204
$866$			6.59153e9i			0.344884i
$867$	−4.56723e9			−0.238005
$868$	4.25151e10			2.20661
$869$	−5.69061e9			−0.294164
$870$			1.48112e10i			0.762557i
$871$	3.79140e8			0.0194418
$872$	−2.03225e10	−	4.34248e10i	−1.03793	−	2.21784i
$873$	2.59635e10			1.32073
$874$		−	1.13012e10i		−	0.572578i
$875$	1.20528e10			0.608218
$876$	−8.05915e10			−4.05065
$877$	2.88657e10			1.44505			0.722526	−	0.691344i	$-0.242981\pi$
$877$	2.88657e10			1.44505			0.722526	+	0.691344i	$0.242981\pi$
$878$			2.31520e9i			0.115440i
$879$	−4.55356e10			−2.26147
$880$			1.34178e10i			0.663732i
$881$	−2.59652e10			−1.27931			−0.639656	−	0.768661i	$-0.720923\pi$
$881$	−2.59652e10			−1.27931			−0.639656	+	0.768661i	$0.720923\pi$
$882$			1.19012e10i			0.584052i
$883$		−	3.26545e10i		−	1.59617i	−0.602542	−	0.798087i	$-0.705845\pi$
$883$		−	3.26545e10i		−	1.59617i	0.602542	−	0.798087i	$-0.294155\pi$
$884$	−8.65920e9			−0.421594
$885$			2.24665e10i			1.08952i
$886$			9.60801e9i			0.464104i
$887$	−2.15178e9			−0.103530			−0.0517648	−	0.998659i	$-0.516485\pi$
$887$	−2.15178e9			−0.103530			−0.0517648	+	0.998659i	$0.516485\pi$
$888$	7.80945e10			3.74261
$889$		−	1.83467e10i		−	0.875793i
$890$			5.87008e10i			2.79112i
$891$			6.52623e9i			0.309094i
$892$	−2.87267e10			−1.35522
$893$	1.48064e9			0.0695776
$894$	−5.40077e10			−2.52799
$895$		−	9.55495e9i		−	0.445500i
$896$		−	8.44672e9i		−	0.392293i
$897$	−7.94393e9			−0.367504
$898$	−3.76225e10			−1.73372
$899$	7.04433e9			0.323356
$900$	−1.37105e10			−0.626911
$901$	2.05114e10			0.934242
$902$			1.55987e10i			0.707728i
$903$	−3.46960e10			−1.56810
$904$		−	5.17670e10i		−	2.33058i
$905$			4.50489e10i			2.02029i
$906$	−7.93103e9			−0.354308
$907$	2.29029e10			1.01921			0.509606	−	0.860408i	$-0.329791\pi$
$907$	2.29029e10			1.01921			0.509606	+	0.860408i	$0.329791\pi$
$908$	−5.91596e10			−2.62256
$909$		−	1.20831e10i		−	0.533586i
$910$	7.42920e9			0.326812
$911$		−	2.79357e10i		−	1.22418i	−0.790789	−	0.612089i	$-0.790329\pi$
$911$		−	2.79357e10i		−	1.22418i	0.790789	−	0.612089i	$-0.209671\pi$
$912$			1.51053e10i			0.659398i
$913$		−	6.55182e9i		−	0.284914i
$914$		−	1.32561e10i		−	0.574252i
$915$	−2.54099e10			−1.09655
$916$			5.16330e9i			0.221970i
$917$	2.27911e10			0.976052
$918$	9.20123e9			0.392551
$919$		−	2.30486e10i		−	0.979579i	−0.871841	−	0.489789i	$-0.837074\pi$
$919$		−	2.30486e10i		−	0.979579i	0.871841	−	0.489789i	$-0.162926\pi$
$920$	9.12305e10			3.86262
$921$	1.91917e10			0.809479
$922$		−	3.97535e10i		−	1.67039i
$923$			4.59733e9i			0.192442i
$924$		−	1.60213e10i		−	0.668107i
$925$			8.84674e9i			0.367525i
$926$		−	6.39790e9i		−	0.264788i
$927$		−	2.75030e10i		−	1.13397i
$928$			9.49586e9i			0.390047i
$929$		−	4.57955e9i		−	0.187399i	−0.995601	−	0.0936996i	$-0.970131\pi$
$929$		−	4.57955e9i		−	0.187399i	0.995601	−	0.0936996i	$-0.0298693\pi$
$930$			8.39864e10i			3.42388i
$931$			2.18101e9i			0.0885794i
$932$	6.01799e10			2.43498
$933$	1.21005e10			0.487775
$934$		−	5.40659e10i		−	2.17125i
$935$	−7.05197e9			−0.282143
$936$	1.00587e10			0.400936
$937$		−	3.59305e10i		−	1.42684i	−0.700738	−	0.713419i	$-0.747146\pi$
$937$		−	3.59305e10i		−	1.42684i	0.700738	−	0.713419i	$-0.252854\pi$
$938$	3.54073e9			0.140082
$939$			4.90744e10i			1.93431i
$940$			2.08733e10i			0.819678i
$941$		−	7.82990e9i		−	0.306332i	−0.988200	−	0.153166i	$-0.951053\pi$
$941$		−	7.82990e9i		−	0.306332i	0.988200	−	0.153166i	$-0.0489469\pi$
$942$			5.87606e9i			0.229038i
$943$	5.06038e10			1.96513
$944$			3.86709e10i			1.49618i
$945$	−5.53062e9			−0.213188
$946$	−1.90616e10			−0.732048
$947$	8.24206e9			0.315363			0.157682	−	0.987490i	$-0.449598\pi$
$947$	8.24206e9			0.315363			0.157682	+	0.987490i	$0.449598\pi$
$948$		−	9.03124e10i		−	3.44285i
$949$			6.69788e9i			0.254393i
$950$	−3.58638e9			−0.135714
$951$			3.15328e10i			1.18886i
$952$	−4.63067e10			−1.73946
$953$	4.83691e10			1.81027			0.905135	−	0.425125i	$-0.139770\pi$
$953$	4.83691e10			1.81027			0.905135	+	0.425125i	$0.139770\pi$
$954$	−4.16088e10			−1.55155
$955$	−7.01108e8			−0.0260479
$956$	6.73708e10			2.49384
$957$		−	2.65457e9i		−	0.0979043i
$958$			6.13051e10i			2.25277i
$959$	−2.19662e10			−0.804248
$960$	−2.22077e10			−0.810131
$961$	1.24320e10			0.451867
$962$		−	1.13343e10i		−	0.410471i
$963$			2.37930e10i			0.858534i
$964$		−	1.12107e11i		−	4.03052i
$965$	1.29705e10			0.464633
$966$	−7.41871e10			−2.64794
$967$			5.26375e10i			1.87199i	0.352019	+	0.935993i	$0.385495\pi$
$967$			5.26375e10i			1.87199i	−0.352019	+	0.935993i	$0.614505\pi$
$968$			6.40631e10i			2.27010i
$969$	−7.93885e9			−0.280301
$970$			9.57428e10i			3.36826i
$971$			2.48086e9i			0.0869632i	0.999054	+	0.0434816i	$0.0138450\pi$
$971$			2.48086e9i			0.0869632i	−0.999054	+	0.0434816i	$0.986155\pi$
$972$	−8.77199e10			−3.06384
$973$			1.05260e9i			0.0366327i
$974$	−6.69363e10			−2.32116
$975$			2.52096e9i			0.0871065i
$976$	−4.37373e10			−1.50583
$977$	−2.66236e10			−0.913348			−0.456674	−	0.889634i	$-0.650959\pi$
$977$	−2.66236e10			−0.913348			−0.456674	+	0.889634i	$0.650959\pi$
$978$	1.01430e11			3.46721
$979$		−	1.05208e10i		−	0.358351i
$980$	−3.07467e10			−1.04353
$981$	−2.20898e10	+	1.03378e10i	−0.747050	+	0.349613i
$982$	−8.95211e10			−3.01672
$983$		−	4.64890e10i		−	1.56104i	−0.625133	−	0.780518i	$-0.714955\pi$
$983$		−	4.64890e10i		−	1.56104i	0.625133	−	0.780518i	$-0.285045\pi$
$984$	−1.41759e11			−4.74317
$985$	−3.28585e10			−1.09552
$986$	−1.33988e10			−0.445141
$987$		−	9.71969e9i		−	0.321768i
$988$	3.21908e9			0.106190
$989$			6.18375e10i			2.03266i
$990$	1.43054e10			0.468572
$991$		−	1.47310e9i		−	0.0480811i	−0.999711	−	0.0240405i	$-0.992347\pi$
$991$		−	1.47310e9i		−	0.0480811i	0.999711	−	0.0240405i	$-0.00765308\pi$
$992$			5.38460e10i			1.75131i
$993$	3.04726e10			0.987614
$994$			4.29337e10i			1.38658i
$995$		−	2.28642e9i		−	0.0735827i
$996$	1.03980e11			3.33459
$997$	1.91466e10			0.611869			0.305934	−	0.952053i	$-0.401031\pi$
$997$	1.91466e10			0.611869			0.305934	+	0.952053i	$0.401031\pi$
$998$		−	7.08704e10i		−	2.25688i
$999$			8.43775e9i			0.267761i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	109.8.b.a.108.3	✓	62
109.108	even	2	inner	109.8.b.a.108.60	yes	62

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
109.8.b.a.108.3	✓	62	1.1	even	1	trivial
109.8.b.a.108.60	yes	62	109.108	even	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 109 \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	109.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-20.6762i q^{2} -63.1733 q^{3} -299.506 q^{4} -321.718 q^{5} +1306.18i q^{6} +710.246 q^{7} +3546.09i q^{8} +1803.86 q^{9} +O(q^{10})\)	\(q-20.6762i q^{2} -63.1733 q^{3} -299.506 q^{4} -321.718 q^{5} +1306.18i q^{6} +710.246 q^{7} +3546.09i q^{8} +1803.86 q^{9} +6651.91i q^{10} -1192.20i q^{11} +18920.8 q^{12} -1572.49i q^{13} -14685.2i q^{14} +20324.0 q^{15} +34983.0 q^{16} +18385.9i q^{17} -37297.0i q^{18} -6835.01i q^{19} +96356.4 q^{20} -44868.6 q^{21} -24650.2 q^{22} +79967.7i q^{23} -224018. i q^{24} +25377.3 q^{25} -32513.1 q^{26} +24204.1 q^{27} -212723. q^{28} -35246.0 q^{29} -420223. i q^{30} -199862. q^{31} -269416. i q^{32} +75315.3i q^{33} +380151. q^{34} -228499. q^{35} -540267. q^{36} +348608. i q^{37} -141322. q^{38} +99339.2i q^{39} -1.14084e6i q^{40} -632802. i q^{41} +927712. i q^{42} +773281. q^{43} +357072. i q^{44} -580334. q^{45} +1.65343e6 q^{46} +216626. i q^{47} -2.20999e6 q^{48} -319093. q^{49} -524708. i q^{50} -1.16150e6i q^{51} +470969. i q^{52} -1.11561e6i q^{53} -500450. i q^{54} +383553. i q^{55} +2.51860e6i q^{56} +431790. i q^{57} +728754. i q^{58} +1.10542e6i q^{59} -6.08715e6 q^{60} -1.25024e6 q^{61} +4.13238e6i q^{62} +1.28119e6 q^{63} -1.09269e6 q^{64} +505897. i q^{65} +1.55724e6 q^{66} +241109. i q^{67} -5.50669e6i q^{68} -5.05182e6i q^{69} +4.72449e6i q^{70} -2.92360e6 q^{71} +6.39666e6i q^{72} -4.25942e6 q^{73} +7.20789e6 q^{74} -1.60317e6 q^{75} +2.04713e6i q^{76} -846757. i q^{77} +2.05396e6 q^{78} -4.77319e6i q^{79} -1.12547e7 q^{80} -5.47410e6 q^{81} -1.30840e7 q^{82} +5.49556e6 q^{83} +1.34384e7 q^{84} -5.91508e6i q^{85} -1.59885e7i q^{86} +2.22661e6 q^{87} +4.22766e6 q^{88} +8.82466e6 q^{89} +1.19991e7i q^{90} -1.11685e6i q^{91} -2.39508e7i q^{92} +1.26259e7 q^{93} +4.47900e6 q^{94} +2.19894e6i q^{95} +1.70199e7i q^{96} +1.43933e7 q^{97} +6.59764e6i q^{98} -2.15057e6i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 62 q - 56 q^{3} - 3584 q^{4} + 194 q^{5} + 370 q^{7} + 40462 q^{9}+O(q^{10}) \) 62 * q - 56 * q^3 - 3584 * q^4 + 194 * q^5 + 370 * q^7 + 40462 * q^9	\( 62 q - 56 q^{3} - 3584 q^{4} + 194 q^{5} + 370 q^{7} + 40462 q^{9} + 6248 q^{12} - 21376 q^{15} + 227856 q^{16} - 216666 q^{20} + 42914 q^{21} - 101446 q^{22} + 788644 q^{25} - 282610 q^{26} - 235946 q^{27} - 69142 q^{28} - 213678 q^{29} + 300350 q^{31} - 401602 q^{34} - 377134 q^{35} - 2176902 q^{36} - 2421542 q^{38} + 2022550 q^{43} + 2223056 q^{45} + 2665174 q^{46} + 1060208 q^{48} + 7702792 q^{49} + 9894904 q^{60} + 278126 q^{61} - 6010300 q^{63} - 9362182 q^{64} - 23291020 q^{66} + 15373744 q^{71} - 1378934 q^{73} - 19097780 q^{74} - 41528062 q^{75} + 5705654 q^{78} + 42766270 q^{80} + 29291446 q^{81} + 14605464 q^{82} + 1191422 q^{83} - 6959552 q^{84} - 22121732 q^{87} + 6337872 q^{88} + 29157388 q^{89} + 14091936 q^{93} - 17482078 q^{94} - 22763314 q^{97}+O(q^{100}) \) 62 * q - 56 * q^3 - 3584 * q^4 + 194 * q^5 + 370 * q^7 + 40462 * q^9 + 6248 * q^12 - 21376 * q^15 + 227856 * q^16 - 216666 * q^20 + 42914 * q^21 - 101446 * q^22 + 788644 * q^25 - 282610 * q^26 - 235946 * q^27 - 69142 * q^28 - 213678 * q^29 + 300350 * q^31 - 401602 * q^34 - 377134 * q^35 - 2176902 * q^36 - 2421542 * q^38 + 2022550 * q^43 + 2223056 * q^45 + 2665174 * q^46 + 1060208 * q^48 + 7702792 * q^49 + 9894904 * q^60 + 278126 * q^61 - 6010300 * q^63 - 9362182 * q^64 - 23291020 * q^66 + 15373744 * q^71 - 1378934 * q^73 - 19097780 * q^74 - 41528062 * q^75 + 5705654 * q^78 + 42766270 * q^80 + 29291446 * q^81 + 14605464 * q^82 + 1191422 * q^83 - 6959552 * q^84 - 22121732 * q^87 + 6337872 * q^88 + 29157388 * q^89 + 14091936 * q^93 - 17482078 * q^94 - 22763314 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

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Groups

Database

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