LMFDB - Embedded newform 169.8.b.f.168.29

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [169,8,Mod(168,169)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("169.168"); S:= CuspForms(chi, 8); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(169, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1])) N = Newforms(chi, 8, names="a")

Level:	$ N $	$=$	$ 169 = 13^{2} $
Weight:	$ k $	$=$	$ 8 $
Character orbit:	$[\chi]$	$=$	169.b (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [42] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)

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

Self dual:	no
Analytic conductor:	$52.7930693068$
Analytic rank:	$0$
Dimension:	$42$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			168.29
Character	$\chi$	$=$	169.168
Dual form			169.8.b.f.168.14

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+8.99747i q^{2} +87.5208 q^{3} +47.0455 q^{4} -190.033i q^{5} +787.466i q^{6} +44.6495i q^{7} +1574.97i q^{8} +5472.89 q^{9} +1709.81 q^{10} +7589.80i q^{11} +4117.46 q^{12} -401.733 q^{14} -16631.8i q^{15} -8148.89 q^{16} +23353.5 q^{17} +49242.1i q^{18} -16826.8i q^{19} -8940.18i q^{20} +3907.76i q^{21} -68289.0 q^{22} -74929.6 q^{23} +137842. i q^{24} +42012.6 q^{25} +287583. q^{27} +2100.56i q^{28} -140338. q^{29} +149644. q^{30} -40335.6i q^{31} +128276. i q^{32} +664265. i q^{33} +210123. i q^{34} +8484.87 q^{35} +257475. q^{36} +235757. i q^{37} +151399. q^{38} +299295. q^{40} -400429. i q^{41} -35160.0 q^{42} +3696.88 q^{43} +357066. i q^{44} -1.04003e6i q^{45} -674177. i q^{46} -831675. i q^{47} -713197. q^{48} +821549. q^{49} +378007. i q^{50} +2.04392e6 q^{51} -1.10224e6 q^{53} +2.58752e6i q^{54} +1.44231e6 q^{55} -70321.5 q^{56} -1.47269e6i q^{57} -1.26268e6i q^{58} -236199. i q^{59} -782452. i q^{60} +2.81617e6 q^{61} +362918. q^{62} +244362. i q^{63} -2.19722e6 q^{64} -5.97671e6 q^{66} +2.44794e6i q^{67} +1.09868e6 q^{68} -6.55789e6 q^{69} +76342.3i q^{70} -1.62315e6i q^{71} +8.61961e6i q^{72} +891091. i q^{73} -2.12122e6 q^{74} +3.67698e6 q^{75} -791625. i q^{76} -338881. q^{77} +4.25155e6 q^{79} +1.54855e6i q^{80} +1.32003e7 q^{81} +3.60285e6 q^{82} +1.90485e6i q^{83} +183843. i q^{84} -4.43793e6i q^{85} +33262.6i q^{86} -1.22825e7 q^{87} -1.19537e7 q^{88} +6.79531e6i q^{89} +9.35761e6 q^{90} -3.52510e6 q^{92} -3.53020e6i q^{93} +7.48297e6 q^{94} -3.19764e6 q^{95} +1.12268e7i q^{96} -1.03105e7i q^{97} +7.39187e6i q^{98} +4.15381e7i q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 42 q - 52 q^{3} - 2818 q^{4} + 30930 q^{9} + 10334 q^{10} - 43590 q^{12} - 358 q^{14} + 226410 q^{16} - 90032 q^{17} - 548348 q^{22} + 256210 q^{23} - 733074 q^{25} - 213286 q^{27} - 124658 q^{29} - 19522 q^{30}+ \cdots - 35320062 q^{95}+O(q^{100}) $ 42 * q - 52 * q^3 - 2818 * q^4 + 30930 * q^9 + 10334 * q^10 - 43590 * q^12 - 358 * q^14 + 226410 * q^16 - 90032 * q^17 - 548348 * q^22 + 256210 * q^23 - 733074 * q^25 - 213286 * q^27 - 124658 * q^29 - 19522 * q^30 + 1697720 * q^35 + 548490 * q^36 - 451662 * q^38 - 4577654 * q^40 + 5473672 * q^42 - 1788978 * q^43 + 12604776 * q^48 - 263060 * q^49 + 2368682 * q^51 - 9100054 * q^53 + 11048060 * q^55 - 4668772 * q^56 + 11434024 * q^61 + 27187306 * q^62 - 19535848 * q^64 - 35613954 * q^66 + 24504984 * q^68 - 5544412 * q^69 + 19250156 * q^74 + 86424794 * q^75 - 18892072 * q^77 - 37226562 * q^79 + 82110898 * q^81 - 48380654 * q^82 + 18055788 * q^87 + 151356360 * q^88 - 6671120 * q^90 - 138678780 * q^92 + 57911426 * q^94 - 35320062 * q^95

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/169\mathbb{Z}\right)^\times$.

$n$	$2$
$\chi(n)$	$-1$

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 $

$2$	8.99747i	0.795272i	0.917543	+	0.397636i	$0.130169\pi$
$2$	8.99747i	0.795272i	−0.917543	+	0.397636i	$0.869831\pi$
$3$	87.5208	1.87149	0.935743	−	0.352682i	$-0.114730\pi$
$3$	87.5208	1.87149	0.935743	+	0.352682i	$0.114730\pi$
$4$	47.0455	0.367543
$5$	− 190.033i	− 0.679881i	−0.940447	−	0.339941i	$-0.889593\pi$
$5$	− 190.033i	− 0.679881i	0.940447	−	0.339941i	$-0.110407\pi$
$6$	787.466i	1.48834i
$7$	44.6495i	0.0492010i	0.999697	+	0.0246005i	$0.00783137\pi$
$7$	44.6495i	0.0492010i	−0.999697	+	0.0246005i	$0.992169\pi$
$8$	1574.97i	1.08757i
$9$	5472.89	2.50246
$10$	1709.81	0.540690
$11$	7589.80i	1.71932i	0.510869	+	0.859659i	$0.329324\pi$
$11$	7589.80i	1.71932i	−0.510869	+	0.859659i	$0.670676\pi$
$12$	4117.46	0.687852
$13$	0	0
$13$	0	0
$14$	−401.733	−0.0391281
$15$	− 16631.8i	− 1.27239i
$16$	−8148.89	−0.497369
$17$	23353.5	1.15287	0.576436	−	0.817142i	$-0.304443\pi$
$17$	23353.5	1.15287	0.576436	+	0.817142i	$0.304443\pi$
$18$	49242.1i	1.99014i
$19$	− 16826.8i	− 0.562812i	−0.959589	−	0.281406i	$-0.909199\pi$
$19$	− 16826.8i	− 0.562812i	0.959589	−	0.281406i	$-0.0908008\pi$
$20$	− 8940.18i	− 0.249886i
$21$	3907.76i	0.0920790i
$22$	−68289.0	−1.36732
$23$	−74929.6	−1.28412	−0.642060	−	0.766654i	$-0.721920\pi$
$23$	−74929.6	−1.28412	−0.642060	+	0.766654i	$0.721920\pi$
$24$	137842.i	2.03537i
$25$	42012.6	0.537761
$26$	0	0
$27$	287583.	2.81184
$28$	2100.56i	0.0180835i
$29$	−140338.	−1.06852	−0.534258	−	0.845321i	$-0.679409\pi$
$29$	−140338.	−1.06852	−0.534258	+	0.845321i	$0.679409\pi$
$30$	149644.	1.01189
$31$	− 40335.6i	− 0.243177i	−0.992581	−	0.121588i	$-0.961201\pi$
$31$	− 40335.6i	− 0.243177i	0.992581	−	0.121588i	$-0.0387988\pi$
$32$	128276.i	0.692025i
$33$	664265.i	3.21768i
$34$	210123.i	0.916847i
$35$	8484.87	0.0334508
$36$	257475.	0.919763
$37$	235757.i	0.765171i	0.923920	+	0.382586i	$0.124966\pi$
$37$	235757.i	0.765171i	−0.923920	+	0.382586i	$0.875034\pi$
$38$	151399.	0.447589
$39$	0	0
$40$	299295.	0.739417
$41$	− 400429.i	− 0.907365i	−0.891163	−	0.453682i	$-0.850110\pi$
$41$	− 400429.i	− 0.907365i	0.891163	−	0.453682i	$-0.149890\pi$
$42$	−35160.0	−0.0732278
$43$	3696.88	0.00709081	0.00354540	−	0.999994i	$-0.498871\pi$
$43$	3696.88	0.00709081	0.00354540	+	0.999994i	$0.498871\pi$
$44$	357066.i	0.631923i
$45$	− 1.04003e6i	− 1.70138i
$46$	− 674177.i	− 1.02122i
$47$	− 831675.i	− 1.16845i	−0.811591	−	0.584226i	$-0.801398\pi$
$47$	− 831675.i	− 1.16845i	0.811591	−	0.584226i	$-0.198602\pi$
$48$	−713197.	−0.930819
$49$	821549.	0.997579
$50$	378007.i	0.427666i
$51$	2.04392e6	2.15759
$52$	0	0
$53$	−1.10224e6	−1.01697	−0.508486	−	0.861070i	$-0.669795\pi$
$53$	−1.10224e6	−1.01697	−0.508486	+	0.861070i	$0.669795\pi$
$54$	2.58752e6i	2.23618i
$55$	1.44231e6	1.16893
$56$	−70321.5	−0.0535094
$57$	− 1.47269e6i	− 1.05330i
$58$	− 1.26268e6i	− 0.849760i
$59$	− 236199.i	− 0.149726i	−0.997194	−	0.0748629i	$-0.976148\pi$
$59$	− 236199.i	− 0.149726i	0.997194	−	0.0748629i	$-0.0238519\pi$
$60$	− 782452.i	− 0.467658i
$61$	2.81617e6	1.58856	0.794281	−	0.607550i	$-0.207848\pi$
$61$	2.81617e6	1.58856	0.794281	+	0.607550i	$0.207848\pi$
$62$	362918.	0.193392
$63$	244362.i	0.123124i
$64$	−2.19722e6	−1.04772
$65$	0	0
$66$	−5.97671e6	−2.55893
$67$	2.44794e6i	0.994351i	0.867650	+	0.497176i	$0.165630\pi$
$67$	2.44794e6i	0.994351i	−0.867650	+	0.497176i	$0.834370\pi$
$68$	1.09868e6	0.423730
$69$	−6.55789e6	−2.40321
$70$	76342.3i	0.0266025i
$71$	− 1.62315e6i	− 0.538213i	−0.963110	−	0.269106i	$-0.913272\pi$
$71$	− 1.62315e6i	− 0.538213i	0.963110	−	0.269106i	$-0.0867283\pi$
$72$	8.61961e6i	2.72160i
$73$	891091.i	0.268097i	0.990975	+	0.134049i	$0.0427978\pi$
$73$	891091.i	0.268097i	−0.990975	+	0.134049i	$0.957202\pi$
$74$	−2.12122e6	−0.608519
$75$	3.67698e6	1.00641
$76$	− 791625.i	− 0.206858i
$77$	−338881.	−0.0845921
$78$	0	0
$79$	4.25155e6	0.970180	0.485090	−	0.874464i	$-0.338787\pi$
$79$	4.25155e6	0.970180	0.485090	+	0.874464i	$0.338787\pi$
$80$	1.54855e6i	0.338152i
$81$	1.32003e7	2.75986
$82$	3.60285e6	0.721601
$83$	1.90485e6i	0.365668i	0.983144	+	0.182834i	$0.0585272\pi$
$83$	1.90485e6i	0.365668i	−0.983144	+	0.182834i	$0.941473\pi$
$84$	183843.i	0.0338430i
$85$	− 4.43793e6i	− 0.783816i
$86$	33262.6i	0.00563912i
$87$	−1.22825e7	−1.99971
$88$	−1.19537e7	−1.86987
$89$	6.79531e6i	1.02175i	0.859655	+	0.510874i	$0.170678\pi$
$89$	6.79531e6i	1.02175i	−0.859655	+	0.510874i	$0.829322\pi$
$90$	9.35761e6	1.35306
$91$	0	0
$92$	−3.52510e6	−0.471970
$93$	− 3.53020e6i	− 0.455102i
$94$	7.48297e6	0.929237
$95$	−3.19764e6	−0.382646
$96$	1.12268e7i	1.29512i
$97$	− 1.03105e7i	− 1.14704i	−0.819193	−	0.573519i	$-0.805578\pi$
$97$	− 1.03105e7i	− 1.14704i	0.819193	−	0.573519i	$-0.194422\pi$
$98$	7.39187e6i	0.793346i
$99$	4.15381e7i	4.30253i
$100$	1.97651e6	0.197651
$101$	274073.	0.0264692	0.0132346	−	0.999912i	$-0.495787\pi$
$101$	274073.	0.0264692	0.0132346	+	0.999912i	$0.495787\pi$
$102$	1.83901e7i	1.71587i
$103$	2.54223e6	0.229237	0.114619	−	0.993410i	$-0.463435\pi$
$103$	2.54223e6	0.229237	0.114619	+	0.993410i	$0.463435\pi$
$104$	0	0
$105$	742602.	0.0626028
$106$	− 9.91734e6i	− 0.808769i
$107$	−1.12719e7	−0.889514	−0.444757	−	0.895651i	$-0.646710\pi$
$107$	−1.12719e7	−0.889514	−0.444757	+	0.895651i	$0.646710\pi$
$108$	1.35295e7	1.03347
$109$	− 6.86352e6i	− 0.507638i	−0.967252	−	0.253819i	$-0.918313\pi$
$109$	− 6.86352e6i	− 0.507638i	0.967252	−	0.253819i	$-0.0816868\pi$
$110$	1.29771e7i	0.929618i
$111$	2.06336e7i	1.43201i
$112$	− 363844.i	− 0.0244710i
$113$	−1.52777e6	−0.0996054	−0.0498027	−	0.998759i	$-0.515859\pi$
$113$	−1.52777e6	−0.0996054	−0.0498027	+	0.998759i	$0.515859\pi$
$114$	1.32505e7	0.837656
$115$	1.42391e7i	0.873050i
$116$	−6.60226e6	−0.392726
$117$	0	0
$118$	2.12520e6	0.119073
$119$	1.04272e6i	0.0567225i
$120$	2.61945e7	1.38381
$121$	−3.81179e7	−1.95605
$122$	2.53384e7i	1.26334i
$123$	− 3.50458e7i	− 1.69812i
$124$	− 1.89761e6i	− 0.0893780i
$125$	− 2.28301e7i	− 1.04550i
$126$	−2.19864e6	−0.0979167
$127$	−2.68776e7	−1.16433	−0.582166	−	0.813070i	$-0.697795\pi$
$127$	−2.68776e7	−1.16433	−0.582166	+	0.813070i	$0.697795\pi$
$128$	− 3.35005e6i	− 0.141194i
$129$	323554.	0.0132703
$130$	0	0
$131$	2.01789e7	0.784237	0.392119	−	0.919915i	$-0.371742\pi$
$131$	2.01789e7	0.784237	0.392119	+	0.919915i	$0.371742\pi$
$132$	3.12507e7i	1.18264i
$133$	751308.	0.0276909
$134$	−2.20253e7	−0.790779
$135$	− 5.46502e7i	− 1.91172i
$136$	3.67810e7i	1.25383i
$137$	− 2.71531e7i	− 0.902190i	−0.892476	−	0.451095i	$-0.851034\pi$
$137$	− 2.71531e7i	− 0.902190i	0.892476	−	0.451095i	$-0.148966\pi$
$138$	− 5.90045e7i	− 1.91121i
$139$	−9.19607e6	−0.290436	−0.145218	−	0.989400i	$-0.546388\pi$
$139$	−9.19607e6	−0.290436	−0.145218	+	0.989400i	$0.546388\pi$
$140$	399175.	0.0122946
$141$	− 7.27888e7i	− 2.18674i
$142$	1.46042e7	0.428025
$143$	0	0
$144$	−4.45980e7	−1.24465
$145$	2.66687e7i	0.726464i
$146$	−8.01757e6	−0.213210
$147$	7.19026e7	1.86696
$148$	1.10913e7i	0.281233i
$149$	− 7.49979e7i	− 1.85736i	−0.370878	−	0.928682i	$-0.620943\pi$
$149$	− 7.49979e7i	− 1.85736i	0.370878	−	0.928682i	$-0.379057\pi$
$150$	3.30835e7i	0.800372i
$151$	− 1.06575e7i	− 0.251905i	−0.992036	−	0.125952i	$-0.959801\pi$
$151$	− 1.06575e7i	− 0.251905i	0.992036	−	0.125952i	$-0.0401986\pi$
$152$	2.65016e7	0.612097
$153$	1.27811e8	2.88502
$154$	− 3.04907e6i	− 0.0672737i
$155$	−7.66507e6	−0.165331
$156$	0	0
$157$	−8.03440e7	−1.65693	−0.828466	−	0.560039i	$-0.810786\pi$
$157$	−8.03440e7	−1.65693	−0.828466	+	0.560039i	$0.810786\pi$
$158$	3.82532e7i	0.771557i
$159$	−9.64685e7	−1.90325
$160$	2.43767e7	0.470495
$161$	− 3.34557e6i	− 0.0631800i
$162$	1.18769e8i	2.19484i
$163$	− 8.26127e7i	− 1.49414i	−0.664748	−	0.747068i	$-0.731461\pi$
$163$	− 8.26127e7i	− 1.49414i	0.664748	−	0.747068i	$-0.268539\pi$
$164$	− 1.88384e7i	− 0.333496i
$165$	1.26232e8	2.18764
$166$	−1.71388e7	−0.290806
$167$	1.53330e7i	0.254753i	0.991854	+	0.127377i	$0.0406557\pi$
$167$	1.53330e7i	0.254753i	−0.991854	+	0.127377i	$0.959344\pi$
$168$	−6.15460e6	−0.100142
$169$	0	0
$170$	3.99302e7	0.623347
$171$	− 9.20911e7i	− 1.40842i
$172$	173922.	0.00260618
$173$	−2.26984e6	−0.0333299	−0.0166650	−	0.999861i	$-0.505305\pi$
$173$	−2.26984e6	−0.0333299	−0.0166650	+	0.999861i	$0.505305\pi$
$174$	− 1.10511e8i	− 1.59032i
$175$	1.87584e6i	0.0264584i
$176$	− 6.18485e7i	− 0.855135i
$177$	− 2.06723e7i	− 0.280210i
$178$	−6.11406e7	−0.812567
$179$	−3.37802e7	−0.440226	−0.220113	−	0.975474i	$-0.570643\pi$
$179$	−3.37802e7	−0.440226	−0.220113	+	0.975474i	$0.570643\pi$
$180$	− 4.89286e7i	− 0.625330i
$181$	−2.19086e7	−0.274625	−0.137312	−	0.990528i	$-0.543846\pi$
$181$	−2.19086e7	−0.274625	−0.137312	+	0.990528i	$0.543846\pi$
$182$	0	0
$183$	2.46473e8	2.97297
$184$	− 1.18012e8i	− 1.39657i
$185$	4.48015e7	0.520226
$186$	3.17629e7	0.361930
$187$	1.77249e8i	1.98215i
$188$	− 3.91266e7i	− 0.429457i
$189$	1.28405e7i	0.138345i
$190$	− 2.87707e7i	− 0.304307i
$191$	−1.31760e8	−1.36826	−0.684128	−	0.729362i	$-0.739817\pi$
$191$	−1.31760e8	−1.36826	−0.684128	+	0.729362i	$0.739817\pi$
$192$	−1.92302e8	−1.96079
$193$	− 7.46458e6i	− 0.0747403i	−0.999301	−	0.0373701i	$-0.988102\pi$
$193$	− 7.46458e6i	− 0.0747403i	0.999301	−	0.0373701i	$-0.0118981\pi$
$194$	9.27682e7	0.912206
$195$	0	0
$196$	3.86502e7	0.366653
$197$	− 4.69866e7i	− 0.437867i	−0.975740	−	0.218933i	$-0.929742\pi$
$197$	− 4.69866e7i	− 0.437867i	0.975740	−	0.218933i	$-0.0702577\pi$
$198$	−3.73738e8	−3.42168
$199$	−1.29435e7	−0.116430	−0.0582149	−	0.998304i	$-0.518541\pi$
$199$	−1.29435e7	−0.116430	−0.0582149	+	0.998304i	$0.518541\pi$
$200$	6.61685e7i	0.584852i
$201$	2.14246e8i	1.86092i
$202$	2.46596e6i	0.0210502i
$203$	− 6.26601e6i	− 0.0525720i
$204$	9.61572e7	0.793006
$205$	−7.60945e7	−0.616900
$206$	2.28737e7i	0.182306i
$207$	−4.10081e8	−3.21346
$208$	0	0
$209$	1.27712e8	0.967653
$210$	6.68154e6i	0.0497862i
$211$	1.85248e6	0.0135758	0.00678790	−	0.999977i	$-0.497839\pi$
$211$	1.85248e6	0.0135758	0.00678790	+	0.999977i	$0.497839\pi$
$212$	−5.18553e7	−0.373781
$213$	− 1.42059e8i	− 1.00726i
$214$	− 1.01418e8i	− 0.707405i
$215$	− 702527.i	− 0.00482091i
$216$	4.52934e8i	3.05807i
$217$	1.80096e6	0.0119645
$218$	6.17543e7	0.403710
$219$	7.79890e7i	0.501740i
$220$	6.78542e7	0.429633
$221$	0	0
$222$	−1.85651e8	−1.13883
$223$	− 1.60027e8i	− 0.966330i	−0.875529	−	0.483165i	$-0.839487\pi$
$223$	− 1.60027e8i	− 0.966330i	0.875529	−	0.483165i	$-0.160513\pi$
$224$	−5.72748e6	−0.0340483
$225$	2.29930e8	1.34573
$226$	− 1.37460e7i	− 0.0792134i
$227$	2.82188e8i	1.60121i	0.599194	+	0.800604i	$0.295488\pi$
$227$	2.82188e8i	1.60121i	−0.599194	+	0.800604i	$0.704512\pi$
$228$	− 6.92837e7i	− 0.387132i
$229$	− 7.12916e6i	− 0.0392296i	−0.999808	−	0.0196148i	$-0.993756\pi$
$229$	− 7.12916e6i	− 0.0392296i	0.999808	−	0.0196148i	$-0.00624399\pi$
$230$	−1.28116e8	−0.694312
$231$	−2.96591e7	−0.158313
$232$	− 2.21027e8i	− 1.16208i
$233$	−8.80122e6	−0.0455824	−0.0227912	−	0.999740i	$-0.507255\pi$
$233$	−8.80122e6	−0.0455824	−0.0227912	+	0.999740i	$0.507255\pi$
$234$	0	0
$235$	−1.58045e8	−0.794409
$236$	− 1.11121e7i	− 0.0550307i
$237$	3.72099e8	1.81568
$238$	−9.38188e6	−0.0451098
$239$	− 1.35488e7i	− 0.0641958i	−0.999485	−	0.0320979i	$-0.989781\pi$
$239$	− 1.35488e7i	− 0.0641958i	0.999485	−	0.0320979i	$-0.0102188\pi$
$240$	1.35531e8i	0.632847i
$241$	− 2.45021e8i	− 1.12757i	−0.825921	−	0.563785i	$-0.809345\pi$
$241$	− 2.45021e8i	− 1.12757i	0.825921	−	0.563785i	$-0.190655\pi$
$242$	− 3.42965e8i	− 1.55559i
$243$	5.26357e8	2.35320
$244$	1.32488e8	0.583865
$245$	− 1.56121e8i	− 0.678235i
$246$	3.15324e8	1.35047
$247$	0	0
$248$	6.35272e7	0.264471
$249$	1.66714e8i	0.684344i
$250$	2.05413e8	0.831453
$251$	1.24387e8	0.496496	0.248248	−	0.968696i	$-0.420145\pi$
$251$	1.24387e8	0.496496	0.248248	+	0.968696i	$0.420145\pi$
$252$	1.14961e7i	0.0452533i
$253$	− 5.68701e8i	− 2.20781i
$254$	− 2.41830e8i	− 0.925960i
$255$	− 3.88411e8i	− 1.46690i
$256$	−2.51102e8	−0.935429
$257$	−1.54924e8	−0.569317	−0.284658	−	0.958629i	$-0.591880\pi$
$257$	−1.54924e8	−0.569317	−0.284658	+	0.958629i	$0.591880\pi$
$258$	2.91116e6i	0.0105535i
$259$	−1.05264e7	−0.0376472
$260$	0	0
$261$	−7.68052e8	−2.67392
$262$	1.81559e8i	0.623681i
$263$	1.82841e8	0.619766	0.309883	−	0.950775i	$-0.399710\pi$
$263$	1.82841e8	0.619766	0.309883	+	0.950775i	$0.399710\pi$
$264$	−1.04620e9	−3.49945
$265$	2.09461e8i	0.691420i
$266$	6.75987e6i	0.0220218i
$267$	5.94730e8i	1.91219i
$268$	1.15165e8i	0.365467i
$269$	−5.92306e8	−1.85530	−0.927648	−	0.373456i	$-0.878173\pi$
$269$	−5.92306e8	−1.85530	−0.927648	+	0.373456i	$0.878173\pi$
$270$	4.91714e8	1.52033
$271$	− 2.92411e8i	− 0.892487i	−0.894912	−	0.446243i	$-0.852762\pi$
$271$	− 2.92411e8i	− 0.892487i	0.894912	−	0.446243i	$-0.147238\pi$
$272$	−1.90305e8	−0.573403
$273$	0	0
$274$	2.44309e8	0.717486
$275$	3.18867e8i	0.924583i
$276$	−3.08520e8	−0.883285
$277$	5.12690e8	1.44936	0.724679	−	0.689087i	$-0.241988\pi$
$277$	5.12690e8	1.44936	0.724679	+	0.689087i	$0.241988\pi$
$278$	− 8.27414e7i	− 0.230975i
$279$	− 2.20752e8i	− 0.608541i
$280$	1.33634e7i	0.0363801i
$281$	− 2.43625e7i	− 0.0655012i	−0.999464	−	0.0327506i	$-0.989573\pi$
$281$	− 2.43625e7i	− 0.0655012i	0.999464	−	0.0327506i	$-0.0104267\pi$
$282$	6.54915e8	1.73905
$283$	1.36204e8	0.357221	0.178611	−	0.983920i	$-0.442840\pi$
$283$	1.36204e8	0.357221	0.178611	+	0.983920i	$0.442840\pi$
$284$	− 7.63618e7i	− 0.197816i
$285$	−2.79860e8	−0.716116
$286$	0	0
$287$	1.78790e7	0.0446432
$288$	7.02042e8i	1.73177i
$289$	1.35049e8	0.329115
$290$	−2.39951e8	−0.577736
$291$	− 9.02381e8i	− 2.14667i
$292$	4.19219e7i	0.0985373i
$293$	5.54711e8i	1.28834i	0.764883	+	0.644170i	$0.222797\pi$
$293$	5.54711e8i	1.28834i	−0.764883	+	0.644170i	$0.777203\pi$
$294$	6.46942e8i	1.48474i
$295$	−4.48855e7	−0.101796
$296$	−3.71309e8	−0.832176
$297$	2.18270e9i	4.83444i
$298$	6.74791e8	1.47711
$299$	0	0
$300$	1.72985e8	0.369900
$301$	165064.i	0 0.000348875i
$302$	9.58906e7	0.200333
$303$	2.39871e7	0.0495368
$304$	1.37120e8i	0.279925i
$305$	− 5.35164e8i	− 1.08003i
$306$	1.14998e9i	2.29437i
$307$	− 7.96114e8i	− 1.57033i	−0.619286	−	0.785165i	$-0.712578\pi$
$307$	− 7.96114e8i	− 1.57033i	0.619286	−	0.785165i	$-0.287422\pi$
$308$	−1.59428e7	−0.0310913
$309$	2.22498e8	0.429014
$310$	− 6.89662e7i	− 0.131483i
$311$	4.47275e8	0.843167	0.421583	−	0.906790i	$-0.361474\pi$
$311$	4.47275e8	0.843167	0.421583	+	0.906790i	$0.361474\pi$
$312$	0	0
$313$	4.58943e8	0.845967	0.422984	−	0.906137i	$-0.360983\pi$
$313$	4.58943e8	0.845967	0.422984	+	0.906137i	$0.360983\pi$
$314$	− 7.22893e8i	− 1.31771i
$315$	4.64367e7	0.0837095
$316$	2.00016e8	0.356583
$317$	9.03574e8i	1.59315i	0.604541	+	0.796574i	$0.293357\pi$
$317$	9.03574e8i	1.59315i	−0.604541	+	0.796574i	$0.706643\pi$
$318$	− 8.67973e8i	− 1.51360i
$319$	− 1.06513e9i	− 1.83712i
$320$	4.17544e8i	0.712323i
$321$	−9.86522e8	−1.66471
$322$	3.01017e7	0.0502453
$323$	− 3.92965e8i	− 0.648851i
$324$	6.21016e8	1.01437
$325$	0	0
$326$	7.43305e8	1.18824
$327$	− 6.00701e8i	− 0.950038i
$328$	6.30662e8	0.986821
$329$	3.71339e7	0.0574890
$330$	1.13577e9i	1.73977i
$331$	6.97509e8i	1.05719i	0.848875	+	0.528593i	$0.177280\pi$
$331$	6.97509e8i	1.05719i	−0.848875	+	0.528593i	$0.822720\pi$
$332$	8.96147e7i	0.134399i
$333$	1.29027e9i	1.91481i
$334$	−1.37958e8	−0.202598
$335$	4.65189e8	0.676041
$336$	− 3.18439e7i	− 0.0457972i
$337$	1.04121e9	1.48195	0.740976	−	0.671532i	$-0.234363\pi$
$337$	1.04121e9	1.48195	0.740976	+	0.671532i	$0.234363\pi$
$338$	0	0
$339$	−1.33711e8	−0.186410
$340$	− 2.08785e8i	− 0.288086i
$341$	3.06139e8	0.418098
$342$	8.28587e8	1.12007
$343$	7.34526e7i	0.0982829i
$344$	5.82246e6i	0.00771173i
$345$	1.24621e9i	1.63390i
$346$	− 2.04228e7i	− 0.0265063i
$347$	−1.03087e9	−1.32450	−0.662251	−	0.749282i	$-0.730399\pi$
$347$	−1.03087e9	−1.32450	−0.662251	+	0.749282i	$0.730399\pi$
$348$	−5.77835e8	−0.734981
$349$	1.57566e9i	1.98415i	0.125660	+	0.992073i	$0.459895\pi$
$349$	1.57566e9i	1.98415i	−0.125660	+	0.992073i	$0.540105\pi$
$350$	−1.68778e7	−0.0210416
$351$	0	0
$352$	−9.73592e8	−1.18981
$353$	− 2.54911e8i	− 0.308445i	−0.988036	−	0.154223i	$-0.950713\pi$
$353$	− 2.54911e8i	− 0.308445i	0.988036	−	0.154223i	$-0.0492873\pi$
$354$	1.85999e8	0.222843
$355$	−3.08451e8	−0.365921
$356$	3.19689e8i	0.375537i
$357$	9.12600e7i	0.106155i
$358$	− 3.03936e8i	− 0.350099i
$359$	− 1.10775e8i	− 0.126360i	−0.998002	−	0.0631800i	$-0.979876\pi$
$359$	− 1.10775e8i	− 0.126360i	0.998002	−	0.0631800i	$-0.0201242\pi$
$360$	1.63801e9	1.85036
$361$	6.10731e8	0.683242
$362$	− 1.97122e8i	− 0.218401i
$363$	−3.33611e9	−3.66073
$364$	0	0
$365$	1.69336e8	0.182274
$366$	2.21764e9i	2.36432i
$367$	−7.95882e8	−0.840461	−0.420230	−	0.907417i	$-0.638051\pi$
$367$	−7.95882e8	−0.840461	−0.420230	+	0.907417i	$0.638051\pi$
$368$	6.10593e8	0.638682
$369$	− 2.19150e9i	− 2.27065i
$370$	4.03100e8i	0.413721i
$371$	− 4.92143e7i	− 0.0500360i
$372$	− 1.66080e8i	− 0.167270i
$373$	2.92736e8	0.292075	0.146038	−	0.989279i	$-0.453348\pi$
$373$	2.92736e8	0.292075	0.146038	+	0.989279i	$0.453348\pi$
$374$	−1.59479e9	−1.57635
$375$	− 1.99810e9i	− 1.95663i
$376$	1.30986e9	1.27077
$377$	0	0
$378$	−1.15532e8	−0.110022
$379$	− 5.27578e8i	− 0.497794i	−0.968530	−	0.248897i	$-0.919932\pi$
$379$	− 5.27578e8i	− 0.497794i	0.968530	−	0.248897i	$-0.0800681\pi$
$380$	−1.50435e8	−0.140639
$381$	−2.35235e9	−2.17903
$382$	− 1.18551e9i	− 1.08814i
$383$	− 1.02879e8i	− 0.0935686i	−0.998905	−	0.0467843i	$-0.985103\pi$
$383$	− 1.02879e8i	− 0.0935686i	0.998905	−	0.0467843i	$-0.0148973\pi$
$384$	− 2.93199e8i	− 0.264243i
$385$	6.43985e7i	0.0575126i
$386$	6.71623e7	0.0594388
$387$	2.02326e7	0.0177445
$388$	− 4.85062e8i	− 0.421586i
$389$	−8.52263e8	−0.734091	−0.367046	−	0.930203i	$-0.619631\pi$
$389$	−8.52263e8	−0.734091	−0.367046	+	0.930203i	$0.619631\pi$
$390$	0	0
$391$	−1.74987e9	−1.48043
$392$	1.29391e9i	1.08494i
$393$	1.76607e9	1.46769
$394$	4.22760e8	0.348223
$395$	− 8.07933e8i	− 0.659608i
$396$	1.95418e9i	1.58136i
$397$	− 1.28264e9i	− 1.02881i	−0.857547	−	0.514406i	$-0.828012\pi$
$397$	− 1.28264e9i	− 1.02881i	0.857547	−	0.514406i	$-0.171988\pi$
$398$	− 1.16458e8i	− 0.0925934i
$399$	6.57551e7	0.0518232
$400$	−3.42356e8	−0.267466
$401$	− 5.51461e8i	− 0.427080i	−0.976934	−	0.213540i	$-0.931501\pi$
$401$	− 5.51461e8i	− 0.427080i	0.976934	−	0.213540i	$-0.0684994\pi$
$402$	−1.92767e9	−1.47993
$403$	0	0
$404$	1.28939e7	0.00972859
$405$	− 2.50849e9i	− 1.87638i
$406$	5.63782e7	0.0418090
$407$	−1.78935e9	−1.31557
$408$	3.21910e9i	2.34652i
$409$	− 1.82547e9i	− 1.31930i	−0.751573	−	0.659650i	$-0.770705\pi$
$409$	− 1.82547e9i	− 1.31930i	0.751573	−	0.659650i	$-0.229295\pi$
$410$	− 6.84658e8i	− 0.490603i
$411$	− 2.37646e9i	− 1.68844i
$412$	1.19601e8	0.0842545
$413$	1.05462e7	0.00736665
$414$	− 3.68969e9i	− 2.55558i
$415$	3.61984e8	0.248611
$416$	0	0
$417$	−8.04847e8	−0.543547
$418$	1.14909e9i	0.769547i
$419$	−2.20929e9	−1.46725	−0.733626	−	0.679554i	$-0.762173\pi$
$419$	−2.20929e9	−1.46725	−0.733626	+	0.679554i	$0.762173\pi$
$420$	3.49361e7	0.0230092
$421$	1.35671e9i	0.886133i	0.896489	+	0.443066i	$0.146109\pi$
$421$	1.35671e9i	0.886133i	−0.896489	+	0.443066i	$0.853891\pi$
$422$	1.66677e7i	0.0107965i
$423$	− 4.55166e9i	− 2.92401i
$424$	− 1.73599e9i	− 1.10603i
$425$	9.81143e8	0.619970
$426$	1.27817e9	0.801043
$427$	1.25741e8i	0.0781588i
$428$	−5.30291e8	−0.326935
$429$	0	0
$430$	6.32097e6	0.00383393
$431$	− 2.84335e9i	− 1.71064i	−0.518096	−	0.855322i	$-0.673359\pi$
$431$	− 2.84335e9i	− 1.71064i	0.518096	−	0.855322i	$-0.326641\pi$
$432$	−2.34348e9	−1.39852
$433$	7.55617e8	0.447295	0.223648	−	0.974670i	$-0.428204\pi$
$433$	7.55617e8	0.447295	0.223648	+	0.974670i	$0.428204\pi$
$434$	1.62041e7i	0.00951505i
$435$	2.33407e9i	1.35957i
$436$	− 3.22898e8i	− 0.186579i
$437$	1.26082e9i	0.722719i
$438$	−7.01704e8	−0.399020
$439$	3.94623e8	0.222616	0.111308	−	0.993786i	$-0.464496\pi$
$439$	3.94623e8	0.222616	0.111308	+	0.993786i	$0.464496\pi$
$440$	2.27159e9i	1.27129i
$441$	4.49625e9	2.49641
$442$	0	0
$443$	1.96580e9	1.07430	0.537150	−	0.843487i	$-0.319501\pi$
$443$	1.96580e9	1.07430	0.537150	+	0.843487i	$0.319501\pi$
$444$	9.70720e8i	0.526325i
$445$	1.29133e9	0.694668
$446$	1.43984e9	0.768494
$447$	− 6.56387e9i	− 3.47603i
$448$	− 9.81049e7i	− 0.0515487i
$449$	2.72494e9i	1.42068i	0.703860	+	0.710339i	$0.251458\pi$
$449$	2.72494e9i	1.42068i	−0.703860	+	0.710339i	$0.748542\pi$
$450$	2.06879e9i	1.07022i
$451$	3.03918e9	1.56005
$452$	−7.18747e7	−0.0366093
$453$	− 9.32753e8i	− 0.471436i
$454$	−2.53898e9	−1.27339
$455$	0	0
$456$	2.31944e9	1.14553
$457$	− 1.27696e9i	− 0.625850i	−0.949778	−	0.312925i	$-0.898691\pi$
$457$	− 1.27696e9i	− 0.625850i	0.949778	−	0.312925i	$-0.101309\pi$
$458$	6.41444e7	0.0311982
$459$	6.71609e9	3.24169
$460$	6.69884e8i	0.320883i
$461$	4.87491e8i	0.231746i	0.993264	+	0.115873i	$0.0369666\pi$
$461$	4.87491e8i	0.231746i	−0.993264	+	0.115873i	$0.963033\pi$
$462$	− 2.66857e8i	− 0.125902i
$463$	2.46579e9i	1.15458i	0.816541	+	0.577288i	$0.195889\pi$
$463$	2.46579e9i	1.15458i	−0.816541	+	0.577288i	$0.804111\pi$
$464$	1.14360e9	0.531447
$465$	−6.70853e8	−0.309415
$466$	− 7.91887e7i	− 0.0362504i
$467$	−2.40297e9	−1.09179	−0.545894	−	0.837854i	$-0.683810\pi$
$467$	−2.40297e9	−1.09179	−0.545894	+	0.837854i	$0.683810\pi$
$468$	0	0
$469$	−1.09300e8	−0.0489231
$470$	− 1.42201e9i	− 0.631771i
$471$	−7.03177e9	−3.10093
$472$	3.72006e8	0.162837
$473$	2.80586e7i	0.0121913i
$474$	3.34795e9i	1.44396i
$475$	− 7.06937e8i	− 0.302659i
$476$	4.90555e7i	0.0208480i
$477$	−6.03241e9	−2.54493
$478$	1.21905e8	0.0510531
$479$	4.04966e9i	1.68362i	0.539772	+	0.841811i	$0.318510\pi$
$479$	4.04966e9i	1.68362i	−0.539772	+	0.841811i	$0.681490\pi$
$480$	2.13347e9	0.880525
$481$	0	0
$482$	2.20457e9	0.896725
$483$	− 2.92807e8i	− 0.118241i
$484$	−1.79328e9	−0.718934
$485$	−1.95933e9	−0.779849
$486$	4.73588e9i	1.87143i
$487$	− 8.24903e8i	− 0.323632i	−0.986821	−	0.161816i	$-0.948265\pi$
$487$	− 8.24903e8i	− 0.323632i	0.986821	−	0.161816i	$-0.0517351\pi$
$488$	4.43537e9i	1.72767i
$489$	− 7.23032e9i	− 2.79625i
$490$	1.40470e9	0.539381
$491$	−3.20090e9	−1.22036	−0.610178	−	0.792264i	$-0.708902\pi$
$491$	−3.20090e9	−1.22036	−0.610178	+	0.792264i	$0.708902\pi$
$492$	− 1.64875e9i	− 0.624133i
$493$	−3.27738e9	−1.23186
$494$	0	0
$495$	7.89360e9	2.92521
$496$	3.28690e8i	0.120949i
$497$	7.24728e7	0.0264806
$498$	−1.50000e9	−0.544239
$499$	4.61602e9i	1.66309i	0.555457	+	0.831545i	$0.312543\pi$
$499$	4.61602e9i	1.66309i	−0.555457	+	0.831545i	$0.687457\pi$
$500$	− 1.07405e9i	− 0.384265i
$501$	1.34196e9i	0.476768i
$502$	1.11917e9i	0.394849i
$503$	−2.24669e9	−0.787145	−0.393573	−	0.919294i	$-0.628761\pi$
$503$	−2.24669e9	−0.787145	−0.393573	+	0.919294i	$0.628761\pi$
$504$	−3.84862e8	−0.133905
$505$	− 5.20828e7i	− 0.0179959i
$506$	5.11687e9	1.75581
$507$	0	0
$508$	−1.26447e9	−0.427942
$509$	4.39689e9i	1.47786i	0.673783	+	0.738930i	$0.264668\pi$
$509$	4.39689e9i	1.47786i	−0.673783	+	0.738930i	$0.735332\pi$
$510$	3.49472e9	1.16659
$511$	−3.97868e7	−0.0131906
$512$	− 2.68809e9i	− 0.885114i
$513$	− 4.83910e9i	− 1.58254i
$514$	− 1.39393e9i	− 0.452762i
$515$	− 4.83107e8i	− 0.155854i
$516$	1.52218e7	0.00487743
$517$	6.31225e9	2.00894
$518$	− 9.47113e7i	− 0.0299397i
$519$	−1.98658e8	−0.0623765
$520$	0	0
$521$	−5.64598e8	−0.174907	−0.0874535	−	0.996169i	$-0.527873\pi$
$521$	−5.64598e8	−0.174907	−0.0874535	+	0.996169i	$0.527873\pi$
$522$	− 6.91052e9i	− 2.12649i
$523$	−8.96882e8	−0.274144	−0.137072	−	0.990561i	$-0.543769\pi$
$523$	−8.96882e8	−0.274144	−0.137072	+	0.990561i	$0.543769\pi$
$524$	9.49326e8	0.288241
$525$	1.64175e8i	0.0495165i
$526$	1.64510e9i	0.492882i
$527$	− 9.41977e8i	− 0.280352i
$528$	− 5.41303e9i	− 1.60037i
$529$	2.20962e9	0.648966
$530$	−1.88462e9	−0.549867
$531$	− 1.29269e9i	− 0.374683i
$532$	3.53457e7	0.0101776
$533$	0	0
$534$	−5.35107e9	−1.52071
$535$	2.14202e9i	0.604764i
$536$	−3.85543e9	−1.08142
$537$	−2.95646e9	−0.823878
$538$	− 5.32926e9i	− 1.47546i
$539$	6.23540e9i	1.71516i
$540$	− 2.57105e9i	− 0.702639i
$541$	5.09067e9i	1.38224i	0.722738	+	0.691122i	$0.242883\pi$
$541$	5.09067e9i	1.38224i	−0.722738	+	0.691122i	$0.757117\pi$
$542$	2.63096e9	0.709769
$543$	−1.91746e9	−0.513956
$544$	2.99571e9i	0.797817i
$545$	−1.30429e9	−0.345134
$546$	0	0
$547$	−6.55393e9	−1.71217	−0.856083	−	0.516838i	$-0.827109\pi$
$547$	−6.55393e9	−1.71217	−0.856083	+	0.516838i	$0.827109\pi$
$548$	− 1.27743e9i	− 0.331594i
$549$	1.54126e10	3.97532
$550$	−2.86900e9	−0.735294
$551$	2.36143e9i	0.601374i
$552$	− 1.03285e10i	− 2.61366i
$553$	1.89830e8i	0.0477338i
$554$	4.61291e9i	1.15263i
$555$	3.92106e9	0.973595
$556$	−4.32634e8	−0.106748
$557$	5.25472e9i	1.28842i	0.764850	+	0.644208i	$0.222813\pi$
$557$	5.25472e9i	1.28842i	−0.764850	+	0.644208i	$0.777187\pi$
$558$	1.98621e9	0.483955
$559$	0	0
$560$	−6.91422e7	−0.0166374
$561$	1.55129e10i	3.70957i
$562$	2.19201e8	0.0520913
$563$	8.12322e9	1.91844	0.959221	−	0.282657i	$-0.0912158\pi$
$563$	8.12322e9	1.91844	0.959221	+	0.282657i	$0.0912158\pi$
$564$	− 3.42439e9i	− 0.803722i
$565$	2.90326e8i	0.0677199i
$566$	1.22549e9i	0.284088i
$567$	5.89388e8i	0.135788i
$568$	2.55640e9	0.585343
$569$	3.04502e9	0.692943	0.346471	−	0.938061i	$-0.387380\pi$
$569$	3.04502e9	0.692943	0.346471	+	0.938061i	$0.387380\pi$
$570$	− 2.51803e9i	− 0.569507i
$571$	4.99500e8	0.112282	0.0561409	−	0.998423i	$-0.482120\pi$
$571$	4.99500e8	0.112282	0.0561409	+	0.998423i	$0.482120\pi$
$572$	0	0
$573$	−1.15318e10	−2.56067
$574$	1.60865e8i	0.0355035i
$575$	−3.14799e9	−0.690551
$576$	−1.20251e10	−2.62187
$577$	− 5.43954e9i	− 1.17882i	−0.807835	−	0.589409i	$-0.799361\pi$
$577$	− 5.43954e9i	− 1.17882i	0.807835	−	0.589409i	$-0.200639\pi$
$578$	1.21510e9i	0.261736i
$579$	− 6.53306e8i	− 0.139875i
$580$	1.25464e9i	0.267007i
$581$	−8.50507e7	−0.0179912
$582$	8.11914e9	1.70718
$583$	− 8.36575e9i	− 1.74850i
$584$	−1.40344e9	−0.291574
$585$	0	0
$586$	−4.99100e9	−1.02458
$587$	− 8.19904e9i	− 1.67313i	−0.547868	−	0.836565i	$-0.684560\pi$
$587$	− 8.19904e9i	− 1.67313i	0.547868	−	0.836565i	$-0.315440\pi$
$588$	3.38270e9	0.686187
$589$	−6.78718e8	−0.136863
$590$	− 4.03856e8i	− 0.0809552i
$591$	− 4.11230e9i	− 0.819462i
$592$	− 1.92116e9i	− 0.380572i
$593$	1.83780e9i	0.361915i	0.983491	+	0.180957i	$0.0579196\pi$
$593$	1.83780e9i	0.361915i	−0.983491	+	0.180957i	$0.942080\pi$
$594$	−1.96388e10	−3.84470
$595$	1.98152e8	0.0385645
$596$	− 3.52831e9i	− 0.682661i
$597$	−1.13282e9	−0.217897
$598$	0	0
$599$	5.00835e9	0.952139	0.476070	−	0.879408i	$-0.342061\pi$
$599$	5.00835e9	0.952139	0.476070	+	0.879408i	$0.342061\pi$
$600$	5.79112e9i	1.09454i
$601$	5.99631e9	1.12674	0.563370	−	0.826205i	$-0.309505\pi$
$601$	5.99631e9	1.12674	0.563370	+	0.826205i	$0.309505\pi$
$602$	−1.48516e6	−0.000277450 0
$603$	1.33973e10i	2.48833i
$604$	− 5.01388e8i	− 0.0925858i
$605$	7.24365e9i	1.32988i
$606$	2.15823e8i	0.0393952i
$607$	4.18428e8	0.0759382	0.0379691	−	0.999279i	$-0.487911\pi$
$607$	4.18428e8	0.0759382	0.0379691	+	0.999279i	$0.487911\pi$
$608$	2.15848e9	0.389480
$609$	− 5.48406e8i	− 0.0983879i
$610$	4.81512e9	0.858920
$611$	0	0
$612$	6.01295e9	1.06037
$613$	− 1.22737e9i	− 0.215210i	−0.994194	−	0.107605i	$-0.965682\pi$
$613$	− 1.22737e9i	− 0.215210i	0.994194	−	0.107605i	$-0.0343183\pi$
$614$	7.16301e9	1.24884
$615$	−6.65985e9	−1.15452
$616$	− 5.33727e8i	− 0.0919997i
$617$	− 3.48381e9i	− 0.597113i	−0.954392	−	0.298557i	$-0.903495\pi$
$617$	− 3.48381e9i	− 0.597113i	0.954392	−	0.298557i	$-0.0965051\pi$
$618$	2.00192e9i	0.341183i
$619$	− 2.85160e9i	− 0.483250i	−0.970370	−	0.241625i	$-0.922320\pi$
$619$	− 2.85160e9i	− 0.483250i	0.970370	−	0.241625i	$-0.0776803\pi$
$620$	−3.60607e8	−0.0607664
$621$	−2.15485e10	−3.61074
$622$	4.02435e9i	0.670546i
$623$	−3.03407e8	−0.0502710
$624$	0	0
$625$	−1.05622e9	−0.173051
$626$	4.12933e9i	0.672774i
$627$	1.11775e10	1.81095
$628$	−3.77983e9	−0.608994
$629$	5.50576e9i	0.882145i
$630$	4.17813e8i	0.0665717i
$631$	7.87572e9i	1.24792i	0.781455	+	0.623961i	$0.214478\pi$
$631$	7.87572e9i	1.24792i	−0.781455	+	0.623961i	$0.785522\pi$
$632$	6.69605e9i	1.05514i
$633$	1.62131e8	0.0254069
$634$	−8.12988e9	−1.26699
$635$	5.10761e9i	0.791608i
$636$	−4.53841e9	−0.699527
$637$	0	0
$638$	9.58352e9	1.46101
$639$	− 8.88330e9i	− 1.34686i
$640$	−6.36619e8	−0.0959953
$641$	2.81246e9	0.421777	0.210889	−	0.977510i	$-0.432364\pi$
$641$	2.81246e9	0.421777	0.210889	+	0.977510i	$0.432364\pi$
$642$	− 8.87621e9i	− 1.32390i
$643$	9.76948e9i	1.44922i	0.689161	+	0.724608i	$0.257979\pi$
$643$	9.76948e9i	1.44922i	−0.689161	+	0.724608i	$0.742021\pi$
$644$	− 1.57394e8i	− 0.0232214i
$645$	− 6.14857e7i	− 0.00902226i
$646$	3.53569e9	0.516013
$647$	7.55684e9	1.09692	0.548460	−	0.836177i	$-0.315214\pi$
$647$	7.55684e9	1.09692	0.548460	+	0.836177i	$0.315214\pi$
$648$	2.07901e10i	3.00153i
$649$	1.79271e9	0.257426
$650$	0	0
$651$	1.57622e8	0.0223915
$652$	− 3.88656e9i	− 0.549159i
$653$	−1.11232e10	−1.56326	−0.781632	−	0.623740i	$-0.785613\pi$
$653$	−1.11232e10	−1.56326	−0.781632	+	0.623740i	$0.785613\pi$
$654$	5.40479e9	0.755538
$655$	− 3.83464e9i	− 0.533188i
$656$	3.26305e9i	0.451295i
$657$	4.87684e9i	0.670903i
$658$	3.34111e8i	0.0457194i
$659$	6.76599e9	0.920943	0.460471	−	0.887675i	$-0.347680\pi$
$659$	6.76599e9	0.920943	0.460471	+	0.887675i	$0.347680\pi$
$660$	5.93866e9	0.804052
$661$	− 3.13975e9i	− 0.422854i	−0.977394	−	0.211427i	$-0.932189\pi$
$661$	− 3.13975e9i	− 0.422854i	0.977394	−	0.211427i	$-0.0678110\pi$
$662$	−6.27582e9	−0.840751
$663$	0	0
$664$	−3.00008e9	−0.397689
$665$	− 1.42773e8i	− 0.0188265i
$666$	−1.16092e10	−1.52280
$667$	1.05154e10	1.37210
$668$	7.21350e8i	0.0936329i
$669$	− 1.40057e10i	− 1.80847i
$670$	4.18553e9i	0.537636i
$671$	2.13742e10i	2.73124i
$672$	−5.01273e8	−0.0637210
$673$	8.83621e9	1.11741	0.558706	−	0.829366i	$-0.311298\pi$
$673$	8.83621e9	1.11741	0.558706	+	0.829366i	$0.311298\pi$
$674$	9.36826e9i	1.17855i
$675$	1.20821e10	1.51210
$676$	0	0
$677$	9.80819e9	1.21487	0.607433	−	0.794371i	$-0.292199\pi$
$677$	9.80819e9	1.21487	0.607433	+	0.794371i	$0.292199\pi$
$678$	− 1.20306e9i	− 0.148247i
$679$	4.60358e8	0.0564354
$680$	6.98960e9	0.852454
$681$	2.46973e10i	2.99664i
$682$	2.75448e9i	0.332501i
$683$	2.38753e9i	0.286732i	0.989670	+	0.143366i	$0.0457927\pi$
$683$	2.38753e9i	0.286732i	−0.989670	+	0.143366i	$0.954207\pi$
$684$	− 4.33248e9i	− 0.517654i
$685$	−5.15998e9	−0.613382
$686$	−6.60888e8	−0.0781616
$687$	− 6.23949e8i	− 0.0734177i
$688$	−3.01255e7	−0.00352675
$689$	0	0
$690$	−1.12128e10	−1.29939
$691$	1.48851e10i	1.71624i	0.513445	+	0.858122i	$0.328369\pi$
$691$	1.48851e10i	1.71624i	−0.513445	+	0.858122i	$0.671631\pi$
$692$	−1.06786e8	−0.0122502
$693$	−1.85466e9	−0.211689
$694$	− 9.27526e9i	− 1.05334i
$695$	1.74755e9i	0.197462i
$696$	− 1.93445e10i	− 2.17483i
$697$	− 9.35143e9i	− 1.04608i
$698$	−1.41770e10	−1.57794
$699$	−7.70289e8	−0.0853068
$700$	8.82500e7i	0.00972460i
$701$	1.19283e10	1.30787	0.653935	−	0.756551i	$-0.273117\pi$
$701$	1.19283e10	1.30787	0.653935	+	0.756551i	$0.273117\pi$
$702$	0	0
$703$	3.96703e9	0.430648
$704$	− 1.66765e10i	− 1.80136i
$705$	−1.38322e10	−1.48673
$706$	2.29356e9	0.245298
$707$	1.22372e7i	0.00130231i
$708$	− 9.72541e8i	− 0.102989i
$709$	− 5.88988e9i	− 0.620647i	−0.950631	−	0.310324i	$-0.899563\pi$
$709$	− 5.88988e9i	− 0.620647i	0.950631	−	0.310324i	$-0.100437\pi$
$710$	− 2.77528e9i	− 0.291006i
$711$	2.32683e10	2.42784
$712$	−1.07024e10	−1.11122
$713$	3.02233e9i	0.312268i
$714$	−8.21109e8	−0.0844223
$715$	0	0
$716$	−1.58921e9	−0.161802
$717$	− 1.18580e9i	− 0.120142i
$718$	9.96691e8	0.100490
$719$	−7.36634e9	−0.739095	−0.369548	−	0.929212i	$-0.620487\pi$
$719$	−7.36634e9	−0.739095	−0.369548	+	0.929212i	$0.620487\pi$
$720$	8.47506e9i	0.846212i
$721$	1.13509e8i	0.0112787i
$722$	5.49503e9i	0.543363i
$723$	− 2.14444e10i	− 2.11023i
$724$	−1.03070e9	−0.100936
$725$	−5.89595e9	−0.574607
$726$	− 3.00166e10i	− 2.91127i
$727$	−1.60000e10	−1.54437	−0.772183	−	0.635401i	$-0.780835\pi$
$727$	−1.60000e10	−1.54437	−0.772183	+	0.635401i	$0.780835\pi$
$728$	0	0
$729$	1.71981e10	1.64412
$730$	1.52360e9i	0.144958i
$731$	8.63352e7	0.00817480
$732$	1.15955e10	1.09270
$733$	− 1.72398e10i	− 1.61685i	−0.588600	−	0.808424i	$-0.700321\pi$
$733$	− 1.72398e10i	− 1.61685i	0.588600	−	0.808424i	$-0.299679\pi$
$734$	− 7.16093e9i	− 0.668395i
$735$	− 1.36638e10i	− 1.26931i
$736$	− 9.61169e9i	− 0.888643i
$737$	−1.85794e10	−1.70961
$738$	1.97180e10	1.80578
$739$	− 1.45339e10i	− 1.32473i	−0.749181	−	0.662366i	$-0.769553\pi$
$739$	− 1.45339e10i	− 1.32473i	0.749181	−	0.662366i	$-0.230447\pi$
$740$	2.10771e9	0.191205
$741$	0	0
$742$	4.42804e8	0.0397922
$743$	5.50310e9i	0.492206i	0.969244	+	0.246103i	$0.0791501\pi$
$743$	5.50310e9i	0.492206i	−0.969244	+	0.246103i	$0.920850\pi$
$744$	5.55995e9	0.494955
$745$	−1.42520e10	−1.26279
$746$	2.63388e9i	0.232279i
$747$	1.04250e10i	0.915072i
$748$	8.33876e9i	0.728527i
$749$	− 5.03283e8i	− 0.0437649i
$750$	1.79779e10	1.55605
$751$	1.82503e10	1.57228	0.786141	−	0.618047i	$-0.212076\pi$
$751$	1.82503e10	1.57228	0.786141	+	0.618047i	$0.212076\pi$
$752$	6.77722e9i	0.581152i
$753$	1.08864e10	0.929186
$754$	0	0
$755$	−2.02527e9	−0.171265
$756$	6.04086e8i	0.0508479i
$757$	2.26365e9	0.189659	0.0948297	−	0.995494i	$-0.469769\pi$
$757$	2.26365e9	0.189659	0.0948297	+	0.995494i	$0.469769\pi$
$758$	4.74687e9	0.395881
$759$	− 4.97731e10i	− 4.13189i
$760$	− 5.03618e9i	− 0.416153i
$761$	− 5.92208e9i	− 0.487111i	−0.969887	−	0.243555i	$-0.921686\pi$
$761$	− 5.92208e9i	− 0.487111i	0.969887	−	0.243555i	$-0.0783138\pi$
$762$	− 2.11652e10i	− 1.73292i
$763$	3.06453e8	0.0249763
$764$	−6.19873e9	−0.502893
$765$	− 2.42883e10i	− 1.96147i
$766$	9.25649e8	0.0744125
$767$	0	0
$768$	−2.19767e10	−1.75064
$769$	1.44521e10i	1.14601i	0.819551	+	0.573007i	$0.194223\pi$
$769$	1.44521e10i	1.14601i	−0.819551	+	0.573007i	$0.805777\pi$
$770$	−5.79423e8	−0.0457381
$771$	−1.35591e10	−1.06547
$772$	− 3.51175e8i	− 0.0274703i
$773$	2.39438e10i	1.86451i	0.361796	+	0.932257i	$0.382164\pi$
$773$	2.39438e10i	1.86451i	−0.361796	+	0.932257i	$0.617836\pi$
$774$	1.82042e8i	0.0141117i
$775$	− 1.69460e9i	− 0.130771i
$776$	1.62387e10	1.24748
$777$	−9.21282e8	−0.0704562
$778$	− 7.66821e9i	− 0.583802i
$779$	−6.73793e9	−0.510676
$780$	0	0
$781$	1.23194e10	0.925358
$782$	− 1.57444e10i	− 1.17734i
$783$	−4.03588e10	−3.00450
$784$	−6.69472e9	−0.496165
$785$	1.52680e10i	1.12652i
$786$	1.58902e10i	1.16721i
$787$	− 5.23881e9i	− 0.383108i	−0.981482	−	0.191554i	$-0.938647\pi$
$787$	− 5.23881e9i	− 0.383108i	0.981482	−	0.191554i	$-0.0613527\pi$
$788$	− 2.21051e9i	− 0.160935i
$789$	1.60024e10	1.15988
$790$	7.26935e9	0.524567
$791$	− 6.82141e7i	− 0.00490068i
$792$	−6.54212e10	−4.67929
$793$	0	0
$794$	1.15405e10	0.818185
$795$	1.83322e10i	1.29398i
$796$	−6.08932e8	−0.0427930
$797$	1.63612e10	1.14475	0.572376	−	0.819991i	$-0.306022\pi$
$797$	1.63612e10	1.14475	0.572376	+	0.819991i	$0.306022\pi$
$798$	5.91629e8i	0.0412135i
$799$	− 1.94225e10i	− 1.34708i
$800$	5.38922e9i	0.372144i
$801$	3.71899e10i	2.55689i
$802$	4.96175e9	0.339645
$803$	−6.76321e9	−0.460944
$804$	1.00793e10i	0.683967i
$805$	−6.35767e8	−0.0429549
$806$	0	0
$807$	−5.18391e10	−3.47216
$808$	4.31656e8i	0.0287871i
$809$	−1.34948e10	−0.896080	−0.448040	−	0.894014i	$-0.647878\pi$
$809$	−1.34948e10	−0.896080	−0.448040	+	0.894014i	$0.647878\pi$
$810$	2.25701e10	1.49223
$811$	2.97924e10i	1.96125i	0.195893	+	0.980625i	$0.437239\pi$
$811$	2.97924e10i	1.96125i	−0.195893	+	0.980625i	$0.562761\pi$
$812$	− 2.94788e8i	− 0.0193225i
$813$	− 2.55921e10i	− 1.67028i
$814$	− 1.60996e10i	− 1.04624i
$815$	−1.56991e10	−1.01583
$816$	−1.66557e10	−1.07312
$817$	− 6.22066e7i	− 0.00399079i
$818$	1.64246e10	1.04920
$819$	0	0
$820$	−3.57991e9	−0.226738
$821$	− 3.02544e10i	− 1.90804i	−0.299741	−	0.954021i	$-0.596900\pi$
$821$	− 3.02544e10i	− 1.90804i	0.299741	−	0.954021i	$-0.403100\pi$
$822$	2.13822e10	1.34277
$823$	−2.21462e10	−1.38484	−0.692420	−	0.721495i	$-0.743455\pi$
$823$	−2.21462e10	−1.38484	−0.692420	+	0.721495i	$0.743455\pi$
$824$	4.00393e9i	0.249311i
$825$	2.79075e10i	1.73034i
$826$	9.48890e7i	0.00585849i
$827$	1.02462e10i	0.629935i	0.949103	+	0.314967i	$0.101994\pi$
$827$	1.02462e10i	0.629935i	−0.949103	+	0.314967i	$0.898006\pi$
$828$	−1.92925e10	−1.18109
$829$	−1.42575e10	−0.869164	−0.434582	−	0.900632i	$-0.643104\pi$
$829$	−1.42575e10	−0.869164	−0.434582	+	0.900632i	$0.643104\pi$
$830$	3.25694e9i	0.197713i
$831$	4.48710e10	2.71245
$832$	0	0
$833$	1.91861e10	1.15008
$834$	− 7.24159e9i	− 0.432268i
$835$	2.91377e9	0.173202
$836$	6.00828e9	0.355654
$837$	− 1.15998e10i	− 0.683774i
$838$	− 1.98781e10i	− 1.16686i
$839$	− 3.22825e10i	− 1.88712i	−0.331195	−	0.943562i	$-0.607452\pi$
$839$	− 3.22825e10i	− 1.88712i	0.331195	−	0.943562i	$-0.392548\pi$
$840$	1.16957e9i	0.0680848i
$841$	2.44477e9	0.141727
$842$	−1.22069e10	−0.704716
$843$	− 2.13222e9i	− 0.122585i
$844$	8.71510e7	0.00498969
$845$	0	0
$846$	4.09534e10	2.32538
$847$	− 1.70195e9i	− 0.0962397i
$848$	8.98200e9	0.505810
$849$	1.19207e10	0.668535
$850$	8.82780e9i	0.493045i
$851$	− 1.76652e10i	− 0.982572i
$852$	− 6.68325e9i	− 0.370211i
$853$	− 1.14608e10i	− 0.632254i	−0.948717	−	0.316127i	$-0.897617\pi$
$853$	− 1.14608e10i	− 0.632254i	0.948717	−	0.316127i	$-0.102383\pi$
$854$	−1.13135e9	−0.0621575
$855$	−1.75003e10	−0.957557
$856$	− 1.77528e10i	− 0.967407i
$857$	5.50942e9	0.299001	0.149501	−	0.988762i	$-0.452233\pi$
$857$	5.50942e9	0.299001	0.149501	+	0.988762i	$0.452233\pi$
$858$	0	0
$859$	1.22212e10	0.657866	0.328933	−	0.944353i	$-0.393311\pi$
$859$	1.22212e10	0.657866	0.328933	+	0.944353i	$0.393311\pi$
$860$	− 3.30508e7i	− 0.00177189i
$861$	1.56478e9	0.0835492
$862$	2.55830e10	1.36043
$863$	1.56748e10i	0.830167i	0.909783	+	0.415083i	$0.136248\pi$
$863$	1.56748e10i	0.830167i	−0.909783	+	0.415083i	$0.863752\pi$
$864$	3.68901e10i	1.94586i
$865$	4.31344e8i	0.0226604i
$866$	6.79864e9i	0.355721i
$867$	1.18196e10	0.615934
$868$	8.47273e7	0.00439748
$869$	3.22684e10i	1.66805i
$870$	−2.10007e10	−1.08123
$871$	0	0
$872$	1.08098e10	0.552091
$873$	− 5.64280e10i	− 2.87042i
$874$	−1.13442e10	−0.574758
$875$	1.01935e9	0.0514394
$876$	3.66903e9i	0.184411i
$877$	− 4.79534e9i	− 0.240061i	−0.992770	−	0.120030i	$-0.961701\pi$
$877$	− 4.79534e9i	− 0.240061i	0.992770	−	0.120030i	$-0.0382992\pi$
$878$	3.55061e9i	0.177040i
$879$	4.85487e10i	2.41111i
$880$	−1.17532e10	−0.581390
$881$	−1.67020e10	−0.822910	−0.411455	−	0.911430i	$-0.634979\pi$
$881$	−1.67020e10	−0.822910	−0.411455	+	0.911430i	$0.634979\pi$
$882$	4.04548e10i	1.98532i
$883$	−2.90550e10	−1.42023	−0.710115	−	0.704085i	$-0.751357\pi$
$883$	−2.90550e10	−1.42023	−0.710115	+	0.704085i	$0.751357\pi$
$884$	0	0
$885$	−3.92842e9	−0.190509
$886$	1.76872e10i	0.854360i
$887$	1.87692e10	0.903054	0.451527	−	0.892258i	$-0.350879\pi$
$887$	1.87692e10	0.903054	0.451527	+	0.892258i	$0.350879\pi$
$888$	−3.24973e10	−1.55741
$889$	− 1.20007e9i	− 0.0572863i
$890$	1.16187e10i	0.552449i
$891$	1.00188e11i	4.74507i
$892$	− 7.52854e9i	− 0.355168i
$893$	−1.39944e10	−0.657619
$894$	5.90582e10	2.76439
$895$	6.41933e9i	0.299302i
$896$	1.49578e8	0.00694689
$897$	0	0
$898$	−2.45176e10	−1.12982
$899$	5.66059e9i	0.259838i
$900$	1.08172e10	0.494613
$901$	−2.57411e10	−1.17244
$902$	2.73449e10i	1.24066i
$903$	1.44465e7i	0 0.000652914i
$904$	− 2.40618e9i	− 0.108328i
$905$	4.16335e9i	0.186712i
$906$	8.39242e9	0.374920
$907$	−1.56436e10	−0.696162	−0.348081	−	0.937464i	$-0.613167\pi$
$907$	−1.56436e10	−0.696162	−0.348081	+	0.937464i	$0.613167\pi$
$908$	1.32757e10i	0.588513i
$909$	1.49997e9	0.0662383
$910$	0	0
$911$	2.04341e10	0.895450	0.447725	−	0.894171i	$-0.352234\pi$
$911$	2.04341e10	0.895450	0.447725	+	0.894171i	$0.352234\pi$
$912$	1.20008e10i	0.523877i
$913$	−1.44574e10	−0.628700
$914$	1.14894e10	0.497721
$915$	− 4.68380e10i	− 2.02127i
$916$	− 3.35395e8i	− 0.0144186i
$917$	9.00977e8i	0.0385852i
$918$	6.04278e10i	2.57803i
$919$	−1.70846e10	−0.726108	−0.363054	−	0.931768i	$-0.618266\pi$
$919$	−1.70846e10	−0.726108	−0.363054	+	0.931768i	$0.618266\pi$
$920$	−2.24261e10	−0.949501
$921$	− 6.96765e10i	− 2.93885i
$922$	−4.38618e9	−0.184301
$923$	0	0
$924$	−1.39533e9	−0.0581869
$925$	9.90477e9i	0.411479i
$926$	−2.21859e10	−0.918201
$927$	1.39133e10	0.573657
$928$	− 1.80020e10i	− 0.739440i
$929$	− 2.58067e9i	− 0.105603i	−0.998605	−	0.0528017i	$-0.983185\pi$
$929$	− 2.58067e9i	− 0.105603i	0.998605	−	0.0528017i	$-0.0168151\pi$
$930$	− 6.03598e9i	− 0.246069i
$931$	− 1.38240e10i	− 0.561450i
$932$	−4.14058e8	−0.0167535
$933$	3.91459e10	1.57798
$934$	− 2.16206e10i	− 0.868268i
$935$	3.36830e10	1.34763
$936$	0	0
$937$	−5.01652e9	−0.199211	−0.0996057	−	0.995027i	$-0.531758\pi$
$937$	−5.01652e9	−0.199211	−0.0996057	+	0.995027i	$0.531758\pi$
$938$	− 9.83420e8i	− 0.0389071i
$939$	4.01671e10	1.58322
$940$	−7.43532e9	−0.291980
$941$	2.55661e10i	1.00023i	0.865958	+	0.500116i	$0.166709\pi$
$941$	2.55661e10i	1.00023i	−0.865958	+	0.500116i	$0.833291\pi$
$942$	− 6.32681e10i	− 2.46608i
$943$	3.00040e10i	1.16517i
$944$	1.92476e9i	0.0744689i
$945$	2.44011e9	0.0940584
$946$	−2.52456e8	−0.00969543
$947$	3.77249e10i	1.44345i	0.692178	+	0.721727i	$0.256651\pi$
$947$	3.77249e10i	1.44345i	−0.692178	+	0.721727i	$0.743349\pi$
$948$	1.75056e10	0.667341
$949$	0	0
$950$	6.36065e9	0.240696
$951$	7.90815e10i	2.98156i
$952$	−1.64226e9	−0.0616895
$953$	−1.31880e10	−0.493578	−0.246789	−	0.969069i	$-0.579375\pi$
$953$	−1.31880e10	−0.493578	−0.246789	+	0.969069i	$0.579375\pi$
$954$	− 5.42764e10i	− 2.02391i
$955$	2.50387e10i	0.930252i
$956$	− 6.37408e8i	− 0.0235947i
$957$	− 9.32214e10i	− 3.43814i
$958$	−3.64367e10	−1.33894
$959$	1.21237e9	0.0443886
$960$	3.65437e10i	1.33310i
$961$	2.58857e10	0.940865
$962$	0	0
$963$	−6.16896e10	−2.22597
$964$	− 1.15271e10i	− 0.414431i
$965$	−1.41851e9	−0.0508145
$966$	2.63452e9	0.0940333
$967$	2.35452e10i	0.837354i	0.908135	+	0.418677i	$0.137506\pi$
$967$	2.35452e10i	0.837354i	−0.908135	+	0.418677i	$0.862494\pi$
$968$	− 6.00345e10i	− 2.12734i
$969$	− 3.43926e10i	− 1.21432i
$970$	− 1.76290e10i	− 0.620192i
$971$	−2.74435e9	−0.0961994	−0.0480997	−	0.998843i	$-0.515317\pi$
$971$	−2.74435e9	−0.0961994	−0.0480997	+	0.998843i	$0.515317\pi$
$972$	2.47627e10	0.864902
$973$	− 4.10600e8i	− 0.0142897i
$974$	7.42204e9	0.257375
$975$	0	0
$976$	−2.29487e10	−0.790102
$977$	− 2.58483e9i	− 0.0886750i	−0.999017	−	0.0443375i	$-0.985882\pi$
$977$	− 2.58483e9i	− 0.0886750i	0.999017	−	0.0443375i	$-0.0141177\pi$
$978$	6.50546e10	2.22378
$979$	−5.15750e10	−1.75671
$980$	− 7.34480e9i	− 0.249281i
$981$	− 3.75633e10i	− 1.27035i
$982$	− 2.88000e10i	− 0.970515i
$983$	4.00140e8i	0.0134361i	0.999977	+	0.00671807i	$0.00213845\pi$
$983$	4.00140e8i	0.0134361i	−0.999977	+	0.00671807i	$0.997862\pi$
$984$	5.51960e10	1.84682
$985$	−8.92898e9	−0.297697
$986$	− 2.94881e10i	− 0.979665i
$987$	3.24999e9	0.107590
$988$	0	0
$989$	−2.77006e8	−0.00910545
$990$	7.10224e10i	2.32633i
$991$	3.08196e10	1.00593	0.502967	−	0.864305i	$-0.332242\pi$
$991$	3.08196e10	1.00593	0.502967	+	0.864305i	$0.332242\pi$
$992$	5.17410e9	0.168284
$993$	6.10465e10i	1.97851i
$994$	6.52072e8i	0.0210593i
$995$	2.45968e9i	0.0791585i
$996$	7.84315e9i	0.251526i
$997$	−5.01181e8	−0.0160163	−0.00800814	−	0.999968i	$-0.502549\pi$
$997$	−5.01181e8	−0.0160163	−0.00800814	+	0.999968i	$0.502549\pi$
$998$	−4.15325e10	−1.32261
$999$	6.77998e10i	2.15154i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	169.8.b.f.168.29		42
13.5	odd	4		169.8.a.h.1.16	✓	21
13.8	odd	4		169.8.a.i.1.6	yes	21
13.12	even	2	inner	169.8.b.f.168.14		42

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
169.8.a.h.1.16	✓	21	13.5	odd	4
169.8.a.i.1.6	yes	21	13.8	odd	4
169.8.b.f.168.14		42	13.12	even	2	inner
169.8.b.f.168.29		42	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 169 = 13^{2} \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	169.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+8.99747i q^{2} +87.5208 q^{3} +47.0455 q^{4} -190.033i q^{5} +787.466i q^{6} +44.6495i q^{7} +1574.97i q^{8} +5472.89 q^{9} +1709.81 q^{10} +7589.80i q^{11} +4117.46 q^{12} -401.733 q^{14} -16631.8i q^{15} -8148.89 q^{16} +23353.5 q^{17} +49242.1i q^{18} -16826.8i q^{19} -8940.18i q^{20} +3907.76i q^{21} -68289.0 q^{22} -74929.6 q^{23} +137842. i q^{24} +42012.6 q^{25} +287583. q^{27} +2100.56i q^{28} -140338. q^{29} +149644. q^{30} -40335.6i q^{31} +128276. i q^{32} +664265. i q^{33} +210123. i q^{34} +8484.87 q^{35} +257475. q^{36} +235757. i q^{37} +151399. q^{38} +299295. q^{40} -400429. i q^{41} -35160.0 q^{42} +3696.88 q^{43} +357066. i q^{44} -1.04003e6i q^{45} -674177. i q^{46} -831675. i q^{47} -713197. q^{48} +821549. q^{49} +378007. i q^{50} +2.04392e6 q^{51} -1.10224e6 q^{53} +2.58752e6i q^{54} +1.44231e6 q^{55} -70321.5 q^{56} -1.47269e6i q^{57} -1.26268e6i q^{58} -236199. i q^{59} -782452. i q^{60} +2.81617e6 q^{61} +362918. q^{62} +244362. i q^{63} -2.19722e6 q^{64} -5.97671e6 q^{66} +2.44794e6i q^{67} +1.09868e6 q^{68} -6.55789e6 q^{69} +76342.3i q^{70} -1.62315e6i q^{71} +8.61961e6i q^{72} +891091. i q^{73} -2.12122e6 q^{74} +3.67698e6 q^{75} -791625. i q^{76} -338881. q^{77} +4.25155e6 q^{79} +1.54855e6i q^{80} +1.32003e7 q^{81} +3.60285e6 q^{82} +1.90485e6i q^{83} +183843. i q^{84} -4.43793e6i q^{85} +33262.6i q^{86} -1.22825e7 q^{87} -1.19537e7 q^{88} +6.79531e6i q^{89} +9.35761e6 q^{90} -3.52510e6 q^{92} -3.53020e6i q^{93} +7.48297e6 q^{94} -3.19764e6 q^{95} +1.12268e7i q^{96} -1.03105e7i q^{97} +7.39187e6i q^{98} +4.15381e7i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 42 q - 52 q^{3} - 2818 q^{4} + 30930 q^{9} + 10334 q^{10} - 43590 q^{12} - 358 q^{14} + 226410 q^{16} - 90032 q^{17} - 548348 q^{22} + 256210 q^{23} - 733074 q^{25} - 213286 q^{27} - 124658 q^{29} - 19522 q^{30}+ \cdots - 35320062 q^{95}+O(q^{100}) \) 42 * q - 52 * q^3 - 2818 * q^4 + 30930 * q^9 + 10334 * q^10 - 43590 * q^12 - 358 * q^14 + 226410 * q^16 - 90032 * q^17 - 548348 * q^22 + 256210 * q^23 - 733074 * q^25 - 213286 * q^27 - 124658 * q^29 - 19522 * q^30 + 1697720 * q^35 + 548490 * q^36 - 451662 * q^38 - 4577654 * q^40 + 5473672 * q^42 - 1788978 * q^43 + 12604776 * q^48 - 263060 * q^49 + 2368682 * q^51 - 9100054 * q^53 + 11048060 * q^55 - 4668772 * q^56 + 11434024 * q^61 + 27187306 * q^62 - 19535848 * q^64 - 35613954 * q^66 + 24504984 * q^68 - 5544412 * q^69 + 19250156 * q^74 + 86424794 * q^75 - 18892072 * q^77 - 37226562 * q^79 + 82110898 * q^81 - 48380654 * q^82 + 18055788 * q^87 + 151356360 * q^88 - 6671120 * q^90 - 138678780 * q^92 + 57911426 * q^94 - 35320062 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more