LMFDB - Embedded newform 169.8.a.i.1.6

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [169,8,Mod(1,169)] 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("169.1"); 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([0])) N = Newforms(chi, 8, names="a")

Level:	$ N $	$=$	$ 169 = 13^{2} $
Weight:	$ k $	$=$	$ 8 $
Character orbit:	$[\chi]$	$=$	169.a (trivial)

Newform invariants

comment:select newform

sage:traces = [21,31] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(2)] == traces)

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

Self dual:	yes
Analytic conductor:	$52.7930693068$
Analytic rank:	$0$
Dimension:	$21$
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.6
Character	$\chi$	$=$	169.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q-8.99747 q^{2} +87.5208 q^{3} -47.0455 q^{4} +190.033 q^{5} -787.466 q^{6} +44.6495 q^{7} +1574.97 q^{8} +5472.89 q^{9} -1709.81 q^{10} +7589.80 q^{11} -4117.46 q^{12} -401.733 q^{14} +16631.8 q^{15} -8148.89 q^{16} -23353.5 q^{17} -49242.1 q^{18} +16826.8 q^{19} -8940.18 q^{20} +3907.76 q^{21} -68289.0 q^{22} +74929.6 q^{23} +137842. q^{24} -42012.6 q^{25} +287583. q^{27} -2100.56 q^{28} -140338. q^{29} -149644. q^{30} +40335.6 q^{31} -128276. q^{32} +664265. q^{33} +210123. q^{34} +8484.87 q^{35} -257475. q^{36} +235757. q^{37} -151399. q^{38} +299295. q^{40} +400429. q^{41} -35160.0 q^{42} -3696.88 q^{43} -357066. q^{44} +1.04003e6 q^{45} -674177. q^{46} -831675. q^{47} -713197. q^{48} -821549. q^{49} +378007. q^{50} -2.04392e6 q^{51} -1.10224e6 q^{53} -2.58752e6 q^{54} +1.44231e6 q^{55} +70321.5 q^{56} +1.47269e6 q^{57} +1.26268e6 q^{58} -236199. q^{59} -782452. q^{60} +2.81617e6 q^{61} -362918. q^{62} +244362. q^{63} +2.19722e6 q^{64} -5.97671e6 q^{66} -2.44794e6 q^{67} +1.09868e6 q^{68} +6.55789e6 q^{69} -76342.3 q^{70} +1.62315e6 q^{71} +8.61961e6 q^{72} +891091. q^{73} -2.12122e6 q^{74} -3.67698e6 q^{75} -791625. q^{76} +338881. q^{77} +4.25155e6 q^{79} -1.54855e6 q^{80} +1.32003e7 q^{81} -3.60285e6 q^{82} -1.90485e6 q^{83} -183843. q^{84} -4.43793e6 q^{85} +33262.6 q^{86} -1.22825e7 q^{87} +1.19537e7 q^{88} +6.79531e6 q^{89} -9.35761e6 q^{90} -3.52510e6 q^{92} +3.53020e6 q^{93} +7.48297e6 q^{94} +3.19764e6 q^{95} -1.12268e7 q^{96} +1.03105e7 q^{97} +7.39187e6 q^{98} +4.15381e7 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 21 q + 31 q^{2} - 26 q^{3} + 1409 q^{4} + 680 q^{5} + 1470 q^{6} + 2929 q^{7} + 4716 q^{8} + 15465 q^{9} - 5167 q^{10} + 14824 q^{11} + 21795 q^{12} - 179 q^{14} + 36398 q^{15} + 113205 q^{16} + 45016 q^{17}+ \cdots + 37605493 q^{99}+O(q^{100}) $ 21 * q + 31 * q^2 - 26 * q^3 + 1409 * q^4 + 680 * q^5 + 1470 * q^6 + 2929 * q^7 + 4716 * q^8 + 15465 * q^9 - 5167 * q^10 + 14824 * q^11 + 21795 * q^12 - 179 * q^14 + 36398 * q^15 + 113205 * q^16 + 45016 * q^17 + 72967 * q^18 + 119316 * q^19 + 131130 * q^20 + 94178 * q^21 - 274174 * q^22 - 128105 * q^23 + 316902 * q^24 + 366537 * q^25 - 106643 * q^27 + 170352 * q^28 - 62329 * q^29 + 9761 * q^30 + 467367 * q^31 + 677464 * q^32 + 587690 * q^33 + 578614 * q^34 + 848860 * q^35 - 274245 * q^36 + 697064 * q^37 + 225831 * q^38 - 2288827 * q^40 + 2351315 * q^41 + 2736836 * q^42 + 894489 * q^43 - 263025 * q^44 - 1938345 * q^45 - 3170392 * q^46 + 1070934 * q^47 + 6302388 * q^48 + 131530 * q^49 + 3970474 * q^50 - 1184341 * q^51 - 4550027 * q^53 + 7916233 * q^54 + 5524030 * q^55 + 2334386 * q^56 + 5590305 * q^57 + 7510959 * q^58 + 10746233 * q^59 + 15186203 * q^60 + 5717012 * q^61 - 13593653 * q^62 + 22347508 * q^63 + 9767924 * q^64 - 17806977 * q^66 + 11305923 * q^67 + 12252492 * q^68 + 2772206 * q^69 + 28670666 * q^70 + 23267025 * q^71 + 49774322 * q^72 + 4763727 * q^73 + 9625078 * q^74 - 43212397 * q^75 + 63359948 * q^76 + 9446036 * q^77 - 18613281 * q^79 + 57904093 * q^80 + 41055449 * q^81 + 24190327 * q^82 + 21455789 * q^83 + 60768429 * q^84 + 26216249 * q^85 + 35632347 * q^86 + 9027894 * q^87 - 75678180 * q^88 + 54037507 * q^89 + 3335560 * q^90 - 69339390 * q^92 + 20085986 * q^93 + 28955713 * q^94 + 17660031 * q^95 + 66667956 * q^96 + 10916609 * q^97 + 41738840 * q^98 + 37605493 * q^99

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 $

$2$	−8.99747	−0.795272	−0.397636	−	0.917543i	$-0.630169\pi$
$2$	−8.99747	−0.795272	−0.397636	+	0.917543i	$0.630169\pi$
$3$	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
$5$	190.033	0.679881	0.339941	−	0.940447i	$-0.389593\pi$
$5$	190.033	0.679881	0.339941	+	0.940447i	$0.389593\pi$
$6$	−787.466	−1.48834
$7$	44.6495	0.0492010	0.0246005	−	0.999697i	$-0.492169\pi$
$7$	44.6495	0.0492010	0.0246005	+	0.999697i	$0.492169\pi$
$8$	1574.97	1.08757
$9$	5472.89	2.50246
$10$	−1709.81	−0.540690
$11$	7589.80	1.71932	0.859659	−	0.510869i	$-0.170676\pi$
$11$	7589.80	1.71932	0.859659	+	0.510869i	$0.170676\pi$
$12$	−4117.46	−0.687852
$13$	0	0
$13$	0	0
$14$	−401.733	−0.0391281
$15$	16631.8	1.27239
$16$	−8148.89	−0.497369
$17$	−23353.5	−1.15287	−0.576436	−	0.817142i	$-0.695557\pi$
$17$	−23353.5	−1.15287	−0.576436	+	0.817142i	$0.695557\pi$
$18$	−49242.1	−1.99014
$19$	16826.8	0.562812	0.281406	−	0.959589i	$-0.409199\pi$
$19$	16826.8	0.562812	0.281406	+	0.959589i	$0.409199\pi$
$20$	−8940.18	−0.249886
$21$	3907.76	0.0920790
$22$	−68289.0	−1.36732
$23$	74929.6	1.28412	0.642060	−	0.766654i	$-0.278080\pi$
$23$	74929.6	1.28412	0.642060	+	0.766654i	$0.278080\pi$
$24$	137842.	2.03537
$25$	−42012.6	−0.537761
$26$	0	0
$27$	287583.	2.81184
$28$	−2100.56	−0.0180835
$29$	−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
$31$	40335.6	0.243177	0.121588	−	0.992581i	$-0.461201\pi$
$31$	40335.6	0.243177	0.121588	+	0.992581i	$0.461201\pi$
$32$	−128276.	−0.692025
$33$	664265.	3.21768
$34$	210123.	0.916847
$35$	8484.87	0.0334508
$36$	−257475.	−0.919763
$37$	235757.	0.765171	0.382586	−	0.923920i	$-0.375034\pi$
$37$	235757.	0.765171	0.382586	+	0.923920i	$0.375034\pi$
$38$	−151399.	−0.447589
$39$	0	0
$40$	299295.	0.739417
$41$	400429.	0.907365	0.453682	−	0.891163i	$-0.350110\pi$
$41$	400429.	0.907365	0.453682	+	0.891163i	$0.350110\pi$
$42$	−35160.0	−0.0732278
$43$	−3696.88	−0.00709081	−0.00354540	−	0.999994i	$-0.501129\pi$
$43$	−3696.88	−0.00709081	−0.00354540	+	0.999994i	$0.501129\pi$
$44$	−357066.	−0.631923
$45$	1.04003e6	1.70138
$46$	−674177.	−1.02122
$47$	−831675.	−1.16845	−0.584226	−	0.811591i	$-0.698602\pi$
$47$	−831675.	−1.16845	−0.584226	+	0.811591i	$0.698602\pi$
$48$	−713197.	−0.930819
$49$	−821549.	−0.997579
$50$	378007.	0.427666
$51$	−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	$0.669795\pi$
$54$	−2.58752e6	−2.23618
$55$	1.44231e6	1.16893
$56$	70321.5	0.0535094
$57$	1.47269e6	1.05330
$58$	1.26268e6	0.849760
$59$	−236199.	−0.149726	−0.0748629	−	0.997194i	$-0.523852\pi$
$59$	−236199.	−0.149726	−0.0748629	+	0.997194i	$0.523852\pi$
$60$	−782452.	−0.467658
$61$	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.	0.123124
$64$	2.19722e6	1.04772
$65$	0	0
$66$	−5.97671e6	−2.55893
$67$	−2.44794e6	−0.994351	−0.497176	−	0.867650i	$-0.665630\pi$
$67$	−2.44794e6	−0.994351	−0.497176	+	0.867650i	$0.665630\pi$
$68$	1.09868e6	0.423730
$69$	6.55789e6	2.40321
$70$	−76342.3	−0.0266025
$71$	1.62315e6	0.538213	0.269106	−	0.963110i	$-0.413272\pi$
$71$	1.62315e6	0.538213	0.269106	+	0.963110i	$0.413272\pi$
$72$	8.61961e6	2.72160
$73$	891091.	0.268097	0.134049	−	0.990975i	$-0.457202\pi$
$73$	891091.	0.268097	0.134049	+	0.990975i	$0.457202\pi$
$74$	−2.12122e6	−0.608519
$75$	−3.67698e6	−1.00641
$76$	−791625.	−0.206858
$77$	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	$0.338787\pi$
$80$	−1.54855e6	−0.338152
$81$	1.32003e7	2.75986
$82$	−3.60285e6	−0.721601
$83$	−1.90485e6	−0.365668	−0.182834	−	0.983144i	$-0.558527\pi$
$83$	−1.90485e6	−0.365668	−0.182834	+	0.983144i	$0.558527\pi$
$84$	−183843.	−0.0338430
$85$	−4.43793e6	−0.783816
$86$	33262.6	0.00563912
$87$	−1.22825e7	−1.99971
$88$	1.19537e7	1.86987
$89$	6.79531e6	1.02175	0.510874	−	0.859655i	$-0.329322\pi$
$89$	6.79531e6	1.02175	0.510874	+	0.859655i	$0.329322\pi$
$90$	−9.35761e6	−1.35306
$91$	0	0
$92$	−3.52510e6	−0.471970
$93$	3.53020e6	0.455102
$94$	7.48297e6	0.929237
$95$	3.19764e6	0.382646
$96$	−1.12268e7	−1.29512
$97$	1.03105e7	1.14704	0.573519	−	0.819193i	$-0.305578\pi$
$97$	1.03105e7	1.14704	0.573519	+	0.819193i	$0.305578\pi$
$98$	7.39187e6	0.793346
$99$	4.15381e7	4.30253
$100$	1.97651e6	0.197651
$101$	−274073.	−0.0264692	−0.0132346	−	0.999912i	$-0.504213\pi$
$101$	−274073.	−0.0264692	−0.0132346	+	0.999912i	$0.504213\pi$
$102$	1.83901e7	1.71587
$103$	−2.54223e6	−0.229237	−0.114619	−	0.993410i	$-0.536565\pi$
$103$	−2.54223e6	−0.229237	−0.114619	+	0.993410i	$0.536565\pi$
$104$	0	0
$105$	742602.	0.0626028
$106$	9.91734e6	0.808769
$107$	−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
$109$	6.86352e6	0.507638	0.253819	−	0.967252i	$-0.418313\pi$
$109$	6.86352e6	0.507638	0.253819	+	0.967252i	$0.418313\pi$
$110$	−1.29771e7	−0.929618
$111$	2.06336e7	1.43201
$112$	−363844.	−0.0244710
$113$	−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
$115$	1.42391e7	0.873050
$116$	6.60226e6	0.392726
$117$	0	0
$118$	2.12520e6	0.119073
$119$	−1.04272e6	−0.0567225
$120$	2.61945e7	1.38381
$121$	3.81179e7	1.95605
$122$	−2.53384e7	−1.26334
$123$	3.50458e7	1.69812
$124$	−1.89761e6	−0.0893780
$125$	−2.28301e7	−1.04550
$126$	−2.19864e6	−0.0979167
$127$	2.68776e7	1.16433	0.582166	−	0.813070i	$-0.302205\pi$
$127$	2.68776e7	1.16433	0.582166	+	0.813070i	$0.302205\pi$
$128$	−3.35005e6	−0.141194
$129$	−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	$0.371742\pi$
$132$	−3.12507e7	−1.18264
$133$	751308.	0.0276909
$134$	2.20253e7	0.790779
$135$	5.46502e7	1.91172
$136$	−3.67810e7	−1.25383
$137$	−2.71531e7	−0.902190	−0.451095	−	0.892476i	$-0.648966\pi$
$137$	−2.71531e7	−0.902190	−0.451095	+	0.892476i	$0.648966\pi$
$138$	−5.90045e7	−1.91121
$139$	−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
$141$	−7.27888e7	−2.18674
$142$	−1.46042e7	−0.428025
$143$	0	0
$144$	−4.45980e7	−1.24465
$145$	−2.66687e7	−0.726464
$146$	−8.01757e6	−0.213210
$147$	−7.19026e7	−1.86696
$148$	−1.10913e7	−0.281233
$149$	7.49979e7	1.85736	0.928682	−	0.370878i	$-0.120943\pi$
$149$	7.49979e7	1.85736	0.928682	+	0.370878i	$0.120943\pi$
$150$	3.30835e7	0.800372
$151$	−1.06575e7	−0.251905	−0.125952	−	0.992036i	$-0.540199\pi$
$151$	−1.06575e7	−0.251905	−0.125952	+	0.992036i	$0.540199\pi$
$152$	2.65016e7	0.612097
$153$	−1.27811e8	−2.88502
$154$	−3.04907e6	−0.0672737
$155$	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	$0.810786\pi$
$158$	−3.82532e7	−0.771557
$159$	−9.64685e7	−1.90325
$160$	−2.43767e7	−0.470495
$161$	3.34557e6	0.0631800
$162$	−1.18769e8	−2.19484
$163$	−8.26127e7	−1.49414	−0.747068	−	0.664748i	$-0.768539\pi$
$163$	−8.26127e7	−1.49414	−0.747068	+	0.664748i	$0.768539\pi$
$164$	−1.88384e7	−0.333496
$165$	1.26232e8	2.18764
$166$	1.71388e7	0.290806
$167$	1.53330e7	0.254753	0.127377	−	0.991854i	$-0.459344\pi$
$167$	1.53330e7	0.254753	0.127377	+	0.991854i	$0.459344\pi$
$168$	6.15460e6	0.100142
$169$	0	0
$170$	3.99302e7	0.623347
$171$	9.20911e7	1.40842
$172$	173922.	0.00260618
$173$	2.26984e6	0.0333299	0.0166650	−	0.999861i	$-0.494695\pi$
$173$	2.26984e6	0.0333299	0.0166650	+	0.999861i	$0.494695\pi$
$174$	1.10511e8	1.59032
$175$	−1.87584e6	−0.0264584
$176$	−6.18485e7	−0.855135
$177$	−2.06723e7	−0.280210
$178$	−6.11406e7	−0.812567
$179$	3.37802e7	0.440226	0.220113	−	0.975474i	$-0.429357\pi$
$179$	3.37802e7	0.440226	0.220113	+	0.975474i	$0.429357\pi$
$180$	−4.89286e7	−0.625330
$181$	2.19086e7	0.274625	0.137312	−	0.990528i	$-0.456154\pi$
$181$	2.19086e7	0.274625	0.137312	+	0.990528i	$0.456154\pi$
$182$	0	0
$183$	2.46473e8	2.97297
$184$	1.18012e8	1.39657
$185$	4.48015e7	0.520226
$186$	−3.17629e7	−0.361930
$187$	−1.77249e8	−1.98215
$188$	3.91266e7	0.429457
$189$	1.28405e7	0.138345
$190$	−2.87707e7	−0.304307
$191$	−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
$193$	−7.46458e6	−0.0747403	−0.0373701	−	0.999301i	$-0.511898\pi$
$193$	−7.46458e6	−0.0747403	−0.0373701	+	0.999301i	$0.511898\pi$
$194$	−9.27682e7	−0.912206
$195$	0	0
$196$	3.86502e7	0.366653
$197$	4.69866e7	0.437867	0.218933	−	0.975740i	$-0.429742\pi$
$197$	4.69866e7	0.437867	0.218933	+	0.975740i	$0.429742\pi$
$198$	−3.73738e8	−3.42168
$199$	1.29435e7	0.116430	0.0582149	−	0.998304i	$-0.481459\pi$
$199$	1.29435e7	0.116430	0.0582149	+	0.998304i	$0.481459\pi$
$200$	−6.61685e7	−0.584852
$201$	−2.14246e8	−1.86092
$202$	2.46596e6	0.0210502
$203$	−6.26601e6	−0.0525720
$204$	9.61572e7	0.793006
$205$	7.60945e7	0.616900
$206$	2.28737e7	0.182306
$207$	4.10081e8	3.21346
$208$	0	0
$209$	1.27712e8	0.967653
$210$	−6.68154e6	−0.0497862
$211$	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
$213$	1.42059e8	1.00726
$214$	1.01418e8	0.707405
$215$	−702527.	−0.00482091
$216$	4.52934e8	3.05807
$217$	1.80096e6	0.0119645
$218$	−6.17543e7	−0.403710
$219$	7.79890e7	0.501740
$220$	−6.78542e7	−0.429633
$221$	0	0
$222$	−1.85651e8	−1.13883
$223$	1.60027e8	0.966330	0.483165	−	0.875529i	$-0.339487\pi$
$223$	1.60027e8	0.966330	0.483165	+	0.875529i	$0.339487\pi$
$224$	−5.72748e6	−0.0340483
$225$	−2.29930e8	−1.34573
$226$	1.37460e7	0.0792134
$227$	−2.82188e8	−1.60121	−0.800604	−	0.599194i	$-0.795488\pi$
$227$	−2.82188e8	−1.60121	−0.800604	+	0.599194i	$0.795488\pi$
$228$	−6.92837e7	−0.387132
$229$	−7.12916e6	−0.0392296	−0.0196148	−	0.999808i	$-0.506244\pi$
$229$	−7.12916e6	−0.0392296	−0.0196148	+	0.999808i	$0.506244\pi$
$230$	−1.28116e8	−0.694312
$231$	2.96591e7	0.158313
$232$	−2.21027e8	−1.16208
$233$	8.80122e6	0.0455824	0.0227912	−	0.999740i	$-0.492745\pi$
$233$	8.80122e6	0.0455824	0.0227912	+	0.999740i	$0.492745\pi$
$234$	0	0
$235$	−1.58045e8	−0.794409
$236$	1.11121e7	0.0550307
$237$	3.72099e8	1.81568
$238$	9.38188e6	0.0451098
$239$	1.35488e7	0.0641958	0.0320979	−	0.999485i	$-0.489781\pi$
$239$	1.35488e7	0.0641958	0.0320979	+	0.999485i	$0.489781\pi$
$240$	−1.35531e8	−0.632847
$241$	−2.45021e8	−1.12757	−0.563785	−	0.825921i	$-0.690655\pi$
$241$	−2.45021e8	−1.12757	−0.563785	+	0.825921i	$0.690655\pi$
$242$	−3.42965e8	−1.55559
$243$	5.26357e8	2.35320
$244$	−1.32488e8	−0.583865
$245$	−1.56121e8	−0.678235
$246$	−3.15324e8	−1.35047
$247$	0	0
$248$	6.35272e7	0.264471
$249$	−1.66714e8	−0.684344
$250$	2.05413e8	0.831453
$251$	−1.24387e8	−0.496496	−0.248248	−	0.968696i	$-0.579855\pi$
$251$	−1.24387e8	−0.496496	−0.248248	+	0.968696i	$0.579855\pi$
$252$	−1.14961e7	−0.0452533
$253$	5.68701e8	2.20781
$254$	−2.41830e8	−0.925960
$255$	−3.88411e8	−1.46690
$256$	−2.51102e8	−0.935429
$257$	1.54924e8	0.569317	0.284658	−	0.958629i	$-0.408120\pi$
$257$	1.54924e8	0.569317	0.284658	+	0.958629i	$0.408120\pi$
$258$	2.91116e6	0.0105535
$259$	1.05264e7	0.0376472
$260$	0	0
$261$	−7.68052e8	−2.67392
$262$	−1.81559e8	−0.623681
$263$	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
$265$	−2.09461e8	−0.691420
$266$	−6.75987e6	−0.0220218
$267$	5.94730e8	1.91219
$268$	1.15165e8	0.365467
$269$	−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
$271$	−2.92411e8	−0.892487	−0.446243	−	0.894912i	$-0.647238\pi$
$271$	−2.92411e8	−0.892487	−0.446243	+	0.894912i	$0.647238\pi$
$272$	1.90305e8	0.573403
$273$	0	0
$274$	2.44309e8	0.717486
$275$	−3.18867e8	−0.924583
$276$	−3.08520e8	−0.883285
$277$	−5.12690e8	−1.44936	−0.724679	−	0.689087i	$-0.758012\pi$
$277$	−5.12690e8	−1.44936	−0.724679	+	0.689087i	$0.758012\pi$
$278$	8.27414e7	0.230975
$279$	2.20752e8	0.608541
$280$	1.33634e7	0.0363801
$281$	−2.43625e7	−0.0655012	−0.0327506	−	0.999464i	$-0.510427\pi$
$281$	−2.43625e7	−0.0655012	−0.0327506	+	0.999464i	$0.510427\pi$
$282$	6.54915e8	1.73905
$283$	−1.36204e8	−0.357221	−0.178611	−	0.983920i	$-0.557160\pi$
$283$	−1.36204e8	−0.357221	−0.178611	+	0.983920i	$0.557160\pi$
$284$	−7.63618e7	−0.197816
$285$	2.79860e8	0.716116
$286$	0	0
$287$	1.78790e7	0.0446432
$288$	−7.02042e8	−1.73177
$289$	1.35049e8	0.329115
$290$	2.39951e8	0.577736
$291$	9.02381e8	2.14667
$292$	−4.19219e7	−0.0985373
$293$	5.54711e8	1.28834	0.644170	−	0.764883i	$-0.277203\pi$
$293$	5.54711e8	1.28834	0.644170	+	0.764883i	$0.277203\pi$
$294$	6.46942e8	1.48474
$295$	−4.48855e7	−0.101796
$296$	3.71309e8	0.832176
$297$	2.18270e9	4.83444
$298$	−6.74791e8	−1.47711
$299$	0	0
$300$	1.72985e8	0.369900
$301$	−165064.	−0.000348875 0
$302$	9.58906e7	0.200333
$303$	−2.39871e7	−0.0495368
$304$	−1.37120e8	−0.279925
$305$	5.35164e8	1.08003
$306$	1.14998e9	2.29437
$307$	−7.96114e8	−1.57033	−0.785165	−	0.619286i	$-0.787422\pi$
$307$	−7.96114e8	−1.57033	−0.785165	+	0.619286i	$0.787422\pi$
$308$	−1.59428e7	−0.0310913
$309$	−2.22498e8	−0.429014
$310$	−6.89662e7	−0.131483
$311$	−4.47275e8	−0.843167	−0.421583	−	0.906790i	$-0.638526\pi$
$311$	−4.47275e8	−0.843167	−0.421583	+	0.906790i	$0.638526\pi$
$312$	0	0
$313$	4.58943e8	0.845967	0.422984	−	0.906137i	$-0.360983\pi$
$313$	4.58943e8	0.845967	0.422984	+	0.906137i	$0.360983\pi$
$314$	7.22893e8	1.31771
$315$	4.64367e7	0.0837095
$316$	−2.00016e8	−0.356583
$317$	−9.03574e8	−1.59315	−0.796574	−	0.604541i	$-0.793357\pi$
$317$	−9.03574e8	−1.59315	−0.796574	+	0.604541i	$0.793357\pi$
$318$	8.67973e8	1.51360
$319$	−1.06513e9	−1.83712
$320$	4.17544e8	0.712323
$321$	−9.86522e8	−1.66471
$322$	−3.01017e7	−0.0502453
$323$	−3.92965e8	−0.648851
$324$	−6.21016e8	−1.01437
$325$	0	0
$326$	7.43305e8	1.18824
$327$	6.00701e8	0.950038
$328$	6.30662e8	0.986821
$329$	−3.71339e7	−0.0574890
$330$	−1.13577e9	−1.73977
$331$	−6.97509e8	−1.05719	−0.528593	−	0.848875i	$-0.677280\pi$
$331$	−6.97509e8	−1.05719	−0.528593	+	0.848875i	$0.677280\pi$
$332$	8.96147e7	0.134399
$333$	1.29027e9	1.91481
$334$	−1.37958e8	−0.202598
$335$	−4.65189e8	−0.676041
$336$	−3.18439e7	−0.0457972
$337$	−1.04121e9	−1.48195	−0.740976	−	0.671532i	$-0.765637\pi$
$337$	−1.04121e9	−1.48195	−0.740976	+	0.671532i	$0.765637\pi$
$338$	0	0
$339$	−1.33711e8	−0.186410
$340$	2.08785e8	0.288086
$341$	3.06139e8	0.418098
$342$	−8.28587e8	−1.12007
$343$	−7.34526e7	−0.0982829
$344$	−5.82246e6	−0.00771173
$345$	1.24621e9	1.63390
$346$	−2.04228e7	−0.0265063
$347$	−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
$349$	1.57566e9	1.98415	0.992073	−	0.125660i	$-0.0401047\pi$
$349$	1.57566e9	1.98415	0.992073	+	0.125660i	$0.0401047\pi$
$350$	1.68778e7	0.0210416
$351$	0	0
$352$	−9.73592e8	−1.18981
$353$	2.54911e8	0.308445	0.154223	−	0.988036i	$-0.450713\pi$
$353$	2.54911e8	0.308445	0.154223	+	0.988036i	$0.450713\pi$
$354$	1.85999e8	0.222843
$355$	3.08451e8	0.365921
$356$	−3.19689e8	−0.375537
$357$	−9.12600e7	−0.106155
$358$	−3.03936e8	−0.350099
$359$	−1.10775e8	−0.126360	−0.0631800	−	0.998002i	$-0.520124\pi$
$359$	−1.10775e8	−0.126360	−0.0631800	+	0.998002i	$0.520124\pi$
$360$	1.63801e9	1.85036
$361$	−6.10731e8	−0.683242
$362$	−1.97122e8	−0.218401
$363$	3.33611e9	3.66073
$364$	0	0
$365$	1.69336e8	0.182274
$366$	−2.21764e9	−2.36432
$367$	−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
$369$	2.19150e9	2.27065
$370$	−4.03100e8	−0.413721
$371$	−4.92143e7	−0.0500360
$372$	−1.66080e8	−0.167270
$373$	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
$375$	−1.99810e9	−1.95663
$376$	−1.30986e9	−1.27077
$377$	0	0
$378$	−1.15532e8	−0.110022
$379$	5.27578e8	0.497794	0.248897	−	0.968530i	$-0.419932\pi$
$379$	5.27578e8	0.497794	0.248897	+	0.968530i	$0.419932\pi$
$380$	−1.50435e8	−0.140639
$381$	2.35235e9	2.17903
$382$	1.18551e9	1.08814
$383$	1.02879e8	0.0935686	0.0467843	−	0.998905i	$-0.485103\pi$
$383$	1.02879e8	0.0935686	0.0467843	+	0.998905i	$0.485103\pi$
$384$	−2.93199e8	−0.264243
$385$	6.43985e7	0.0575126
$386$	6.71623e7	0.0594388
$387$	−2.02326e7	−0.0177445
$388$	−4.85062e8	−0.421586
$389$	8.52263e8	0.734091	0.367046	−	0.930203i	$-0.380369\pi$
$389$	8.52263e8	0.734091	0.367046	+	0.930203i	$0.380369\pi$
$390$	0	0
$391$	−1.74987e9	−1.48043
$392$	−1.29391e9	−1.08494
$393$	1.76607e9	1.46769
$394$	−4.22760e8	−0.348223
$395$	8.07933e8	0.659608
$396$	−1.95418e9	−1.58136
$397$	−1.28264e9	−1.02881	−0.514406	−	0.857547i	$-0.671988\pi$
$397$	−1.28264e9	−1.02881	−0.514406	+	0.857547i	$0.671988\pi$
$398$	−1.16458e8	−0.0925934
$399$	6.57551e7	0.0518232
$400$	3.42356e8	0.267466
$401$	−5.51461e8	−0.427080	−0.213540	−	0.976934i	$-0.568499\pi$
$401$	−5.51461e8	−0.427080	−0.213540	+	0.976934i	$0.568499\pi$
$402$	1.92767e9	1.47993
$403$	0	0
$404$	1.28939e7	0.00972859
$405$	2.50849e9	1.87638
$406$	5.63782e7	0.0418090
$407$	1.78935e9	1.31557
$408$	−3.21910e9	−2.34652
$409$	1.82547e9	1.31930	0.659650	−	0.751573i	$-0.270705\pi$
$409$	1.82547e9	1.31930	0.659650	+	0.751573i	$0.270705\pi$
$410$	−6.84658e8	−0.490603
$411$	−2.37646e9	−1.68844
$412$	1.19601e8	0.0842545
$413$	−1.05462e7	−0.00736665
$414$	−3.68969e9	−2.55558
$415$	−3.61984e8	−0.248611
$416$	0	0
$417$	−8.04847e8	−0.543547
$418$	−1.14909e9	−0.769547
$419$	−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
$421$	−1.35671e9	−0.886133	−0.443066	−	0.896489i	$-0.646109\pi$
$421$	−1.35671e9	−0.886133	−0.443066	+	0.896489i	$0.646109\pi$
$422$	−1.66677e7	−0.0107965
$423$	−4.55166e9	−2.92401
$424$	−1.73599e9	−1.10603
$425$	9.81143e8	0.619970
$426$	−1.27817e9	−0.801043
$427$	1.25741e8	0.0781588
$428$	5.30291e8	0.326935
$429$	0	0
$430$	6.32097e6	0.00383393
$431$	2.84335e9	1.71064	0.855322	−	0.518096i	$-0.173359\pi$
$431$	2.84335e9	1.71064	0.855322	+	0.518096i	$0.173359\pi$
$432$	−2.34348e9	−1.39852
$433$	−7.55617e8	−0.447295	−0.223648	−	0.974670i	$-0.571796\pi$
$433$	−7.55617e8	−0.447295	−0.223648	+	0.974670i	$0.571796\pi$
$434$	−1.62041e7	−0.00951505
$435$	−2.33407e9	−1.35957
$436$	−3.22898e8	−0.186579
$437$	1.26082e9	0.722719
$438$	−7.01704e8	−0.399020
$439$	−3.94623e8	−0.222616	−0.111308	−	0.993786i	$-0.535504\pi$
$439$	−3.94623e8	−0.222616	−0.111308	+	0.993786i	$0.535504\pi$
$440$	2.27159e9	1.27129
$441$	−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	$0.319501\pi$
$444$	−9.70720e8	−0.526325
$445$	1.29133e9	0.694668
$446$	−1.43984e9	−0.768494
$447$	6.56387e9	3.47603
$448$	9.81049e7	0.0515487
$449$	2.72494e9	1.42068	0.710339	−	0.703860i	$-0.248542\pi$
$449$	2.72494e9	1.42068	0.710339	+	0.703860i	$0.248542\pi$
$450$	2.06879e9	1.07022
$451$	3.03918e9	1.56005
$452$	7.18747e7	0.0366093
$453$	−9.32753e8	−0.471436
$454$	2.53898e9	1.27339
$455$	0	0
$456$	2.31944e9	1.14553
$457$	1.27696e9	0.625850	0.312925	−	0.949778i	$-0.398691\pi$
$457$	1.27696e9	0.625850	0.312925	+	0.949778i	$0.398691\pi$
$458$	6.41444e7	0.0311982
$459$	−6.71609e9	−3.24169
$460$	−6.69884e8	−0.320883
$461$	−4.87491e8	−0.231746	−0.115873	−	0.993264i	$-0.536967\pi$
$461$	−4.87491e8	−0.231746	−0.115873	+	0.993264i	$0.536967\pi$
$462$	−2.66857e8	−0.125902
$463$	2.46579e9	1.15458	0.577288	−	0.816541i	$-0.304111\pi$
$463$	2.46579e9	1.15458	0.577288	+	0.816541i	$0.304111\pi$
$464$	1.14360e9	0.531447
$465$	6.70853e8	0.309415
$466$	−7.91887e7	−0.0362504
$467$	2.40297e9	1.09179	0.545894	−	0.837854i	$-0.316190\pi$
$467$	2.40297e9	1.09179	0.545894	+	0.837854i	$0.316190\pi$
$468$	0	0
$469$	−1.09300e8	−0.0489231
$470$	1.42201e9	0.631771
$471$	−7.03177e9	−3.10093
$472$	−3.72006e8	−0.162837
$473$	−2.80586e7	−0.0121913
$474$	−3.34795e9	−1.44396
$475$	−7.06937e8	−0.302659
$476$	4.90555e7	0.0208480
$477$	−6.03241e9	−2.54493
$478$	−1.21905e8	−0.0510531
$479$	4.04966e9	1.68362	0.841811	−	0.539772i	$-0.181490\pi$
$479$	4.04966e9	1.68362	0.841811	+	0.539772i	$0.181490\pi$
$480$	−2.13347e9	−0.880525
$481$	0	0
$482$	2.20457e9	0.896725
$483$	2.92807e8	0.118241
$484$	−1.79328e9	−0.718934
$485$	1.95933e9	0.779849
$486$	−4.73588e9	−1.87143
$487$	8.24903e8	0.323632	0.161816	−	0.986821i	$-0.448265\pi$
$487$	8.24903e8	0.323632	0.161816	+	0.986821i	$0.448265\pi$
$488$	4.43537e9	1.72767
$489$	−7.23032e9	−2.79625
$490$	1.40470e9	0.539381
$491$	3.20090e9	1.22036	0.610178	−	0.792264i	$-0.291098\pi$
$491$	3.20090e9	1.22036	0.610178	+	0.792264i	$0.291098\pi$
$492$	−1.64875e9	−0.624133
$493$	3.27738e9	1.23186
$494$	0	0
$495$	7.89360e9	2.92521
$496$	−3.28690e8	−0.120949
$497$	7.24728e7	0.0264806
$498$	1.50000e9	0.544239
$499$	−4.61602e9	−1.66309	−0.831545	−	0.555457i	$-0.812543\pi$
$499$	−4.61602e9	−1.66309	−0.831545	+	0.555457i	$0.812543\pi$
$500$	1.07405e9	0.384265
$501$	1.34196e9	0.476768
$502$	1.11917e9	0.394849
$503$	−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
$505$	−5.20828e7	−0.0179959
$506$	−5.11687e9	−1.75581
$507$	0	0
$508$	−1.26447e9	−0.427942
$509$	−4.39689e9	−1.47786	−0.738930	−	0.673783i	$-0.764668\pi$
$509$	−4.39689e9	−1.47786	−0.738930	+	0.673783i	$0.764668\pi$
$510$	3.49472e9	1.16659
$511$	3.97868e7	0.0131906
$512$	2.68809e9	0.885114
$513$	4.83910e9	1.58254
$514$	−1.39393e9	−0.452762
$515$	−4.83107e8	−0.155854
$516$	1.52218e7	0.00487743
$517$	−6.31225e9	−2.00894
$518$	−9.47113e7	−0.0299397
$519$	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	$0.527873\pi$
$522$	6.91052e9	2.12649
$523$	−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
$525$	−1.64175e8	−0.0495165
$526$	−1.64510e9	−0.492882
$527$	−9.41977e8	−0.280352
$528$	−5.41303e9	−1.60037
$529$	2.20962e9	0.648966
$530$	1.88462e9	0.549867
$531$	−1.29269e9	−0.374683
$532$	−3.53457e7	−0.0101776
$533$	0	0
$534$	−5.35107e9	−1.52071
$535$	−2.14202e9	−0.604764
$536$	−3.85543e9	−1.08142
$537$	2.95646e9	0.823878
$538$	5.32926e9	1.47546
$539$	−6.23540e9	−1.71516
$540$	−2.57105e9	−0.702639
$541$	5.09067e9	1.38224	0.691122	−	0.722738i	$-0.257117\pi$
$541$	5.09067e9	1.38224	0.691122	+	0.722738i	$0.257117\pi$
$542$	2.63096e9	0.709769
$543$	1.91746e9	0.513956
$544$	2.99571e9	0.797817
$545$	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	$0.827109\pi$
$548$	1.27743e9	0.331594
$549$	1.54126e10	3.97532
$550$	2.86900e9	0.735294
$551$	−2.36143e9	−0.601374
$552$	1.03285e10	2.61366
$553$	1.89830e8	0.0477338
$554$	4.61291e9	1.15263
$555$	3.92106e9	0.973595
$556$	4.32634e8	0.106748
$557$	5.25472e9	1.28842	0.644208	−	0.764850i	$-0.277187\pi$
$557$	5.25472e9	1.28842	0.644208	+	0.764850i	$0.277187\pi$
$558$	−1.98621e9	−0.483955
$559$	0	0
$560$	−6.91422e7	−0.0166374
$561$	−1.55129e10	−3.70957
$562$	2.19201e8	0.0520913
$563$	−8.12322e9	−1.91844	−0.959221	−	0.282657i	$-0.908784\pi$
$563$	−8.12322e9	−1.91844	−0.959221	+	0.282657i	$0.908784\pi$
$564$	3.42439e9	0.803722
$565$	−2.90326e8	−0.0677199
$566$	1.22549e9	0.284088
$567$	5.89388e8	0.135788
$568$	2.55640e9	0.585343
$569$	−3.04502e9	−0.692943	−0.346471	−	0.938061i	$-0.612620\pi$
$569$	−3.04502e9	−0.692943	−0.346471	+	0.938061i	$0.612620\pi$
$570$	−2.51803e9	−0.569507
$571$	−4.99500e8	−0.112282	−0.0561409	−	0.998423i	$-0.517880\pi$
$571$	−4.99500e8	−0.112282	−0.0561409	+	0.998423i	$0.517880\pi$
$572$	0	0
$573$	−1.15318e10	−2.56067
$574$	−1.60865e8	−0.0355035
$575$	−3.14799e9	−0.690551
$576$	1.20251e10	2.62187
$577$	5.43954e9	1.17882	0.589409	−	0.807835i	$-0.299361\pi$
$577$	5.43954e9	1.17882	0.589409	+	0.807835i	$0.299361\pi$
$578$	−1.21510e9	−0.261736
$579$	−6.53306e8	−0.139875
$580$	1.25464e9	0.267007
$581$	−8.50507e7	−0.0179912
$582$	−8.11914e9	−1.70718
$583$	−8.36575e9	−1.74850
$584$	1.40344e9	0.291574
$585$	0	0
$586$	−4.99100e9	−1.02458
$587$	8.19904e9	1.67313	0.836565	−	0.547868i	$-0.184560\pi$
$587$	8.19904e9	1.67313	0.836565	+	0.547868i	$0.184560\pi$
$588$	3.38270e9	0.686187
$589$	6.78718e8	0.136863
$590$	4.03856e8	0.0809552
$591$	4.11230e9	0.819462
$592$	−1.92116e9	−0.380572
$593$	1.83780e9	0.361915	0.180957	−	0.983491i	$-0.442080\pi$
$593$	1.83780e9	0.361915	0.180957	+	0.983491i	$0.442080\pi$
$594$	−1.96388e10	−3.84470
$595$	−1.98152e8	−0.0385645
$596$	−3.52831e9	−0.682661
$597$	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	$0.342061\pi$
$600$	−5.79112e9	−1.09454
$601$	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
$603$	−1.33973e10	−2.48833
$604$	5.01388e8	0.0925858
$605$	7.24365e9	1.32988
$606$	2.15823e8	0.0393952
$607$	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
$609$	−5.48406e8	−0.0983879
$610$	−4.81512e9	−0.858920
$611$	0	0
$612$	6.01295e9	1.06037
$613$	1.22737e9	0.215210	0.107605	−	0.994194i	$-0.465682\pi$
$613$	1.22737e9	0.215210	0.107605	+	0.994194i	$0.465682\pi$
$614$	7.16301e9	1.24884
$615$	6.65985e9	1.15452
$616$	5.33727e8	0.0919997
$617$	3.48381e9	0.597113	0.298557	−	0.954392i	$-0.403495\pi$
$617$	3.48381e9	0.597113	0.298557	+	0.954392i	$0.403495\pi$
$618$	2.00192e9	0.341183
$619$	−2.85160e9	−0.483250	−0.241625	−	0.970370i	$-0.577680\pi$
$619$	−2.85160e9	−0.483250	−0.241625	+	0.970370i	$0.577680\pi$
$620$	−3.60607e8	−0.0607664
$621$	2.15485e10	3.61074
$622$	4.02435e9	0.670546
$623$	3.03407e8	0.0502710
$624$	0	0
$625$	−1.05622e9	−0.173051
$626$	−4.12933e9	−0.672774
$627$	1.11775e10	1.81095
$628$	3.77983e9	0.608994
$629$	−5.50576e9	−0.882145
$630$	−4.17813e8	−0.0665717
$631$	7.87572e9	1.24792	0.623961	−	0.781455i	$-0.285522\pi$
$631$	7.87572e9	1.24792	0.623961	+	0.781455i	$0.285522\pi$
$632$	6.69605e9	1.05514
$633$	1.62131e8	0.0254069
$634$	8.12988e9	1.26699
$635$	5.10761e9	0.791608
$636$	4.53841e9	0.699527
$637$	0	0
$638$	9.58352e9	1.46101
$639$	8.88330e9	1.34686
$640$	−6.36619e8	−0.0959953
$641$	−2.81246e9	−0.421777	−0.210889	−	0.977510i	$-0.567636\pi$
$641$	−2.81246e9	−0.421777	−0.210889	+	0.977510i	$0.567636\pi$
$642$	8.87621e9	1.32390
$643$	−9.76948e9	−1.44922	−0.724608	−	0.689161i	$-0.757979\pi$
$643$	−9.76948e9	−1.44922	−0.724608	+	0.689161i	$0.757979\pi$
$644$	−1.57394e8	−0.0232214
$645$	−6.14857e7	−0.00902226
$646$	3.53569e9	0.516013
$647$	−7.55684e9	−1.09692	−0.548460	−	0.836177i	$-0.684786\pi$
$647$	−7.55684e9	−1.09692	−0.548460	+	0.836177i	$0.684786\pi$
$648$	2.07901e10	3.00153
$649$	−1.79271e9	−0.257426
$650$	0	0
$651$	1.57622e8	0.0223915
$652$	3.88656e9	0.549159
$653$	−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
$655$	3.83464e9	0.533188
$656$	−3.26305e9	−0.451295
$657$	4.87684e9	0.670903
$658$	3.34111e8	0.0457194
$659$	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
$661$	−3.13975e9	−0.422854	−0.211427	−	0.977394i	$-0.567811\pi$
$661$	−3.13975e9	−0.422854	−0.211427	+	0.977394i	$0.567811\pi$
$662$	6.27582e9	0.840751
$663$	0	0
$664$	−3.00008e9	−0.397689
$665$	1.42773e8	0.0188265
$666$	−1.16092e10	−1.52280
$667$	−1.05154e10	−1.37210
$668$	−7.21350e8	−0.0936329
$669$	1.40057e10	1.80847
$670$	4.18553e9	0.537636
$671$	2.13742e10	2.73124
$672$	−5.01273e8	−0.0637210
$673$	−8.83621e9	−1.11741	−0.558706	−	0.829366i	$-0.688702\pi$
$673$	−8.83621e9	−1.11741	−0.558706	+	0.829366i	$0.688702\pi$
$674$	9.36826e9	1.17855
$675$	−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	$0.292199\pi$
$678$	1.20306e9	0.148247
$679$	4.60358e8	0.0564354
$680$	−6.98960e9	−0.852454
$681$	−2.46973e10	−2.99664
$682$	−2.75448e9	−0.332501
$683$	2.38753e9	0.286732	0.143366	−	0.989670i	$-0.454207\pi$
$683$	2.38753e9	0.286732	0.143366	+	0.989670i	$0.454207\pi$
$684$	−4.33248e9	−0.517654
$685$	−5.15998e9	−0.613382
$686$	6.60888e8	0.0781616
$687$	−6.23949e8	−0.0734177
$688$	3.01255e7	0.00352675
$689$	0	0
$690$	−1.12128e10	−1.29939
$691$	−1.48851e10	−1.71624	−0.858122	−	0.513445i	$-0.828369\pi$
$691$	−1.48851e10	−1.71624	−0.858122	+	0.513445i	$0.828369\pi$
$692$	−1.06786e8	−0.0122502
$693$	1.85466e9	0.211689
$694$	9.27526e9	1.05334
$695$	−1.74755e9	−0.197462
$696$	−1.93445e10	−2.17483
$697$	−9.35143e9	−1.04608
$698$	−1.41770e10	−1.57794
$699$	7.70289e8	0.0853068
$700$	8.82500e7	0.00972460
$701$	−1.19283e10	−1.30787	−0.653935	−	0.756551i	$-0.726883\pi$
$701$	−1.19283e10	−1.30787	−0.653935	+	0.756551i	$0.726883\pi$
$702$	0	0
$703$	3.96703e9	0.430648
$704$	1.66765e10	1.80136
$705$	−1.38322e10	−1.48673
$706$	−2.29356e9	−0.245298
$707$	−1.22372e7	−0.00130231
$708$	9.72541e8	0.102989
$709$	−5.88988e9	−0.620647	−0.310324	−	0.950631i	$-0.600437\pi$
$709$	−5.88988e9	−0.620647	−0.310324	+	0.950631i	$0.600437\pi$
$710$	−2.77528e9	−0.291006
$711$	2.32683e10	2.42784
$712$	1.07024e10	1.11122
$713$	3.02233e9	0.312268
$714$	8.21109e8	0.0844223
$715$	0	0
$716$	−1.58921e9	−0.161802
$717$	1.18580e9	0.120142
$718$	9.96691e8	0.100490
$719$	7.36634e9	0.739095	0.369548	−	0.929212i	$-0.379513\pi$
$719$	7.36634e9	0.739095	0.369548	+	0.929212i	$0.379513\pi$
$720$	−8.47506e9	−0.846212
$721$	−1.13509e8	−0.0112787
$722$	5.49503e9	0.543363
$723$	−2.14444e10	−2.11023
$724$	−1.03070e9	−0.100936
$725$	5.89595e9	0.574607
$726$	−3.00166e10	−2.91127
$727$	1.60000e10	1.54437	0.772183	−	0.635401i	$-0.219165\pi$
$727$	1.60000e10	1.54437	0.772183	+	0.635401i	$0.219165\pi$
$728$	0	0
$729$	1.71981e10	1.64412
$730$	−1.52360e9	−0.144958
$731$	8.63352e7	0.00817480
$732$	−1.15955e10	−1.09270
$733$	1.72398e10	1.61685	0.808424	−	0.588600i	$-0.200321\pi$
$733$	1.72398e10	1.61685	0.808424	+	0.588600i	$0.200321\pi$
$734$	7.16093e9	0.668395
$735$	−1.36638e10	−1.26931
$736$	−9.61169e9	−0.888643
$737$	−1.85794e10	−1.70961
$738$	−1.97180e10	−1.80578
$739$	−1.45339e10	−1.32473	−0.662366	−	0.749181i	$-0.730447\pi$
$739$	−1.45339e10	−1.32473	−0.662366	+	0.749181i	$0.730447\pi$
$740$	−2.10771e9	−0.191205
$741$	0	0
$742$	4.42804e8	0.0397922
$743$	−5.50310e9	−0.492206	−0.246103	−	0.969244i	$-0.579150\pi$
$743$	−5.50310e9	−0.492206	−0.246103	+	0.969244i	$0.579150\pi$
$744$	5.55995e9	0.494955
$745$	1.42520e10	1.26279
$746$	−2.63388e9	−0.232279
$747$	−1.04250e10	−0.915072
$748$	8.33876e9	0.728527
$749$	−5.03283e8	−0.0437649
$750$	1.79779e10	1.55605
$751$	−1.82503e10	−1.57228	−0.786141	−	0.618047i	$-0.787924\pi$
$751$	−1.82503e10	−1.57228	−0.786141	+	0.618047i	$0.787924\pi$
$752$	6.77722e9	0.581152
$753$	−1.08864e10	−0.929186
$754$	0	0
$755$	−2.02527e9	−0.171265
$756$	−6.04086e8	−0.0508479
$757$	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
$759$	4.97731e10	4.13189
$760$	5.03618e9	0.416153
$761$	−5.92208e9	−0.487111	−0.243555	−	0.969887i	$-0.578314\pi$
$761$	−5.92208e9	−0.487111	−0.243555	+	0.969887i	$0.578314\pi$
$762$	−2.11652e10	−1.73292
$763$	3.06453e8	0.0249763
$764$	6.19873e9	0.502893
$765$	−2.42883e10	−1.96147
$766$	−9.25649e8	−0.0744125
$767$	0	0
$768$	−2.19767e10	−1.75064
$769$	−1.44521e10	−1.14601	−0.573007	−	0.819551i	$-0.694223\pi$
$769$	−1.44521e10	−1.14601	−0.573007	+	0.819551i	$0.694223\pi$
$770$	−5.79423e8	−0.0457381
$771$	1.35591e10	1.06547
$772$	3.51175e8	0.0274703
$773$	−2.39438e10	−1.86451	−0.932257	−	0.361796i	$-0.882164\pi$
$773$	−2.39438e10	−1.86451	−0.932257	+	0.361796i	$0.882164\pi$
$774$	1.82042e8	0.0141117
$775$	−1.69460e9	−0.130771
$776$	1.62387e10	1.24748
$777$	9.21282e8	0.0704562
$778$	−7.66821e9	−0.583802
$779$	6.73793e9	0.510676
$780$	0	0
$781$	1.23194e10	0.925358
$782$	1.57444e10	1.17734
$783$	−4.03588e10	−3.00450
$784$	6.69472e9	0.496165
$785$	−1.52680e10	−1.12652
$786$	−1.58902e10	−1.16721
$787$	−5.23881e9	−0.383108	−0.191554	−	0.981482i	$-0.561353\pi$
$787$	−5.23881e9	−0.383108	−0.191554	+	0.981482i	$0.561353\pi$
$788$	−2.21051e9	−0.160935
$789$	1.60024e10	1.15988
$790$	−7.26935e9	−0.524567
$791$	−6.82141e7	−0.00490068
$792$	6.54212e10	4.67929
$793$	0	0
$794$	1.15405e10	0.818185
$795$	−1.83322e10	−1.29398
$796$	−6.08932e8	−0.0427930
$797$	−1.63612e10	−1.14475	−0.572376	−	0.819991i	$-0.693978\pi$
$797$	−1.63612e10	−1.14475	−0.572376	+	0.819991i	$0.693978\pi$
$798$	−5.91629e8	−0.0412135
$799$	1.94225e10	1.34708
$800$	5.38922e9	0.372144
$801$	3.71899e10	2.55689
$802$	4.96175e9	0.339645
$803$	6.76321e9	0.460944
$804$	1.00793e10	0.683967
$805$	6.35767e8	0.0429549
$806$	0	0
$807$	−5.18391e10	−3.47216
$808$	−4.31656e8	−0.0287871
$809$	−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
$811$	−2.97924e10	−1.96125	−0.980625	−	0.195893i	$-0.937239\pi$
$811$	−2.97924e10	−1.96125	−0.980625	+	0.195893i	$0.937239\pi$
$812$	2.94788e8	0.0193225
$813$	−2.55921e10	−1.67028
$814$	−1.60996e10	−1.04624
$815$	−1.56991e10	−1.01583
$816$	1.66557e10	1.07312
$817$	−6.22066e7	−0.00399079
$818$	−1.64246e10	−1.04920
$819$	0	0
$820$	−3.57991e9	−0.226738
$821$	3.02544e10	1.90804	0.954021	−	0.299741i	$-0.0969003\pi$
$821$	3.02544e10	1.90804	0.954021	+	0.299741i	$0.0969003\pi$
$822$	2.13822e10	1.34277
$823$	2.21462e10	1.38484	0.692420	−	0.721495i	$-0.256545\pi$
$823$	2.21462e10	1.38484	0.692420	+	0.721495i	$0.256545\pi$
$824$	−4.00393e9	−0.249311
$825$	−2.79075e10	−1.73034
$826$	9.48890e7	0.00585849
$827$	1.02462e10	0.629935	0.314967	−	0.949103i	$-0.398006\pi$
$827$	1.02462e10	0.629935	0.314967	+	0.949103i	$0.398006\pi$
$828$	−1.92925e10	−1.18109
$829$	1.42575e10	0.869164	0.434582	−	0.900632i	$-0.356896\pi$
$829$	1.42575e10	0.869164	0.434582	+	0.900632i	$0.356896\pi$
$830$	3.25694e9	0.197713
$831$	−4.48710e10	−2.71245
$832$	0	0
$833$	1.91861e10	1.15008
$834$	7.24159e9	0.432268
$835$	2.91377e9	0.173202
$836$	−6.00828e9	−0.355654
$837$	1.15998e10	0.683774
$838$	1.98781e10	1.16686
$839$	−3.22825e10	−1.88712	−0.943562	−	0.331195i	$-0.892548\pi$
$839$	−3.22825e10	−1.88712	−0.943562	+	0.331195i	$0.892548\pi$
$840$	1.16957e9	0.0680848
$841$	2.44477e9	0.141727
$842$	1.22069e10	0.704716
$843$	−2.13222e9	−0.122585
$844$	−8.71510e7	−0.00498969
$845$	0	0
$846$	4.09534e10	2.32538
$847$	1.70195e9	0.0962397
$848$	8.98200e9	0.505810
$849$	−1.19207e10	−0.668535
$850$	−8.82780e9	−0.493045
$851$	1.76652e10	0.982572
$852$	−6.68325e9	−0.370211
$853$	−1.14608e10	−0.632254	−0.316127	−	0.948717i	$-0.602383\pi$
$853$	−1.14608e10	−0.632254	−0.316127	+	0.948717i	$0.602383\pi$
$854$	−1.13135e9	−0.0621575
$855$	1.75003e10	0.957557
$856$	−1.77528e10	−0.967407
$857$	−5.50942e9	−0.299001	−0.149501	−	0.988762i	$-0.547767\pi$
$857$	−5.50942e9	−0.299001	−0.149501	+	0.988762i	$0.547767\pi$
$858$	0	0
$859$	1.22212e10	0.657866	0.328933	−	0.944353i	$-0.393311\pi$
$859$	1.22212e10	0.657866	0.328933	+	0.944353i	$0.393311\pi$
$860$	3.30508e7	0.00177189
$861$	1.56478e9	0.0835492
$862$	−2.55830e10	−1.36043
$863$	−1.56748e10	−0.830167	−0.415083	−	0.909783i	$-0.636248\pi$
$863$	−1.56748e10	−0.830167	−0.415083	+	0.909783i	$0.636248\pi$
$864$	−3.68901e10	−1.94586
$865$	4.31344e8	0.0226604
$866$	6.79864e9	0.355721
$867$	1.18196e10	0.615934
$868$	−8.47273e7	−0.00439748
$869$	3.22684e10	1.66805
$870$	2.10007e10	1.08123
$871$	0	0
$872$	1.08098e10	0.552091
$873$	5.64280e10	2.87042
$874$	−1.13442e10	−0.574758
$875$	−1.01935e9	−0.0514394
$876$	−3.66903e9	−0.184411
$877$	4.79534e9	0.240061	0.120030	−	0.992770i	$-0.461701\pi$
$877$	4.79534e9	0.240061	0.120030	+	0.992770i	$0.461701\pi$
$878$	3.55061e9	0.177040
$879$	4.85487e10	2.41111
$880$	−1.17532e10	−0.581390
$881$	1.67020e10	0.822910	0.411455	−	0.911430i	$-0.365021\pi$
$881$	1.67020e10	0.822910	0.411455	+	0.911430i	$0.365021\pi$
$882$	4.04548e10	1.98532
$883$	2.90550e10	1.42023	0.710115	−	0.704085i	$-0.248643\pi$
$883$	2.90550e10	1.42023	0.710115	+	0.704085i	$0.248643\pi$
$884$	0	0
$885$	−3.92842e9	−0.190509
$886$	−1.76872e10	−0.854360
$887$	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
$889$	1.20007e9	0.0572863
$890$	−1.16187e10	−0.552449
$891$	1.00188e11	4.74507
$892$	−7.52854e9	−0.355168
$893$	−1.39944e10	−0.657619
$894$	−5.90582e10	−2.76439
$895$	6.41933e9	0.299302
$896$	−1.49578e8	−0.00694689
$897$	0	0
$898$	−2.45176e10	−1.12982
$899$	−5.66059e9	−0.259838
$900$	1.08172e10	0.494613
$901$	2.57411e10	1.17244
$902$	−2.73449e10	−1.24066
$903$	−1.44465e7	−0.000652914 0
$904$	−2.40618e9	−0.108328
$905$	4.16335e9	0.186712
$906$	8.39242e9	0.374920
$907$	1.56436e10	0.696162	0.348081	−	0.937464i	$-0.386833\pi$
$907$	1.56436e10	0.696162	0.348081	+	0.937464i	$0.386833\pi$
$908$	1.32757e10	0.588513
$909$	−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	$0.352234\pi$
$912$	−1.20008e10	−0.523877
$913$	−1.44574e10	−0.628700
$914$	−1.14894e10	−0.497721
$915$	4.68380e10	2.02127
$916$	3.35395e8	0.0144186
$917$	9.00977e8	0.0385852
$918$	6.04278e10	2.57803
$919$	−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
$921$	−6.96765e10	−2.93885
$922$	4.38618e9	0.184301
$923$	0	0
$924$	−1.39533e9	−0.0581869
$925$	−9.90477e9	−0.411479
$926$	−2.21859e10	−0.918201
$927$	−1.39133e10	−0.573657
$928$	1.80020e10	0.739440
$929$	2.58067e9	0.105603	0.0528017	−	0.998605i	$-0.483185\pi$
$929$	2.58067e9	0.105603	0.0528017	+	0.998605i	$0.483185\pi$
$930$	−6.03598e9	−0.246069
$931$	−1.38240e10	−0.561450
$932$	−4.14058e8	−0.0167535
$933$	−3.91459e10	−1.57798
$934$	−2.16206e10	−0.868268
$935$	−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	$0.531758\pi$
$938$	9.83420e8	0.0389071
$939$	4.01671e10	1.58322
$940$	7.43532e9	0.291980
$941$	−2.55661e10	−1.00023	−0.500116	−	0.865958i	$-0.666709\pi$
$941$	−2.55661e10	−1.00023	−0.500116	+	0.865958i	$0.666709\pi$
$942$	6.32681e10	2.46608
$943$	3.00040e10	1.16517
$944$	1.92476e9	0.0744689
$945$	2.44011e9	0.0940584
$946$	2.52456e8	0.00969543
$947$	3.77249e10	1.44345	0.721727	−	0.692178i	$-0.243349\pi$
$947$	3.77249e10	1.44345	0.721727	+	0.692178i	$0.243349\pi$
$948$	−1.75056e10	−0.667341
$949$	0	0
$950$	6.36065e9	0.240696
$951$	−7.90815e10	−2.98156
$952$	−1.64226e9	−0.0616895
$953$	1.31880e10	0.493578	0.246789	−	0.969069i	$-0.420625\pi$
$953$	1.31880e10	0.493578	0.246789	+	0.969069i	$0.420625\pi$
$954$	5.42764e10	2.02391
$955$	−2.50387e10	−0.930252
$956$	−6.37408e8	−0.0235947
$957$	−9.32214e10	−3.43814
$958$	−3.64367e10	−1.33894
$959$	−1.21237e9	−0.0443886
$960$	3.65437e10	1.33310
$961$	−2.58857e10	−0.940865
$962$	0	0
$963$	−6.16896e10	−2.22597
$964$	1.15271e10	0.414431
$965$	−1.41851e9	−0.0508145
$966$	−2.63452e9	−0.0940333
$967$	−2.35452e10	−0.837354	−0.418677	−	0.908135i	$-0.637506\pi$
$967$	−2.35452e10	−0.837354	−0.418677	+	0.908135i	$0.637506\pi$
$968$	6.00345e10	2.12734
$969$	−3.43926e10	−1.21432
$970$	−1.76290e10	−0.620192
$971$	−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
$973$	−4.10600e8	−0.0142897
$974$	−7.42204e9	−0.257375
$975$	0	0
$976$	−2.29487e10	−0.790102
$977$	2.58483e9	0.0886750	0.0443375	−	0.999017i	$-0.485882\pi$
$977$	2.58483e9	0.0886750	0.0443375	+	0.999017i	$0.485882\pi$
$978$	6.50546e10	2.22378
$979$	5.15750e10	1.75671
$980$	7.34480e9	0.249281
$981$	3.75633e10	1.27035
$982$	−2.88000e10	−0.970515
$983$	4.00140e8	0.0134361	0.00671807	−	0.999977i	$-0.497862\pi$
$983$	4.00140e8	0.0134361	0.00671807	+	0.999977i	$0.497862\pi$
$984$	5.51960e10	1.84682
$985$	8.92898e9	0.297697
$986$	−2.94881e10	−0.979665
$987$	−3.24999e9	−0.107590
$988$	0	0
$989$	−2.77006e8	−0.00910545
$990$	−7.10224e10	−2.32633
$991$	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
$993$	−6.10465e10	−1.97851
$994$	−6.52072e8	−0.0210593
$995$	2.45968e9	0.0791585
$996$	7.84315e9	0.251526
$997$	−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
$999$	6.77998e10	2.15154

Twists

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

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

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.a (trivial)

\(f(q)\)	\(=\)	\(q-8.99747 q^{2} +87.5208 q^{3} -47.0455 q^{4} +190.033 q^{5} -787.466 q^{6} +44.6495 q^{7} +1574.97 q^{8} +5472.89 q^{9} -1709.81 q^{10} +7589.80 q^{11} -4117.46 q^{12} -401.733 q^{14} +16631.8 q^{15} -8148.89 q^{16} -23353.5 q^{17} -49242.1 q^{18} +16826.8 q^{19} -8940.18 q^{20} +3907.76 q^{21} -68289.0 q^{22} +74929.6 q^{23} +137842. q^{24} -42012.6 q^{25} +287583. q^{27} -2100.56 q^{28} -140338. q^{29} -149644. q^{30} +40335.6 q^{31} -128276. q^{32} +664265. q^{33} +210123. q^{34} +8484.87 q^{35} -257475. q^{36} +235757. q^{37} -151399. q^{38} +299295. q^{40} +400429. q^{41} -35160.0 q^{42} -3696.88 q^{43} -357066. q^{44} +1.04003e6 q^{45} -674177. q^{46} -831675. q^{47} -713197. q^{48} -821549. q^{49} +378007. q^{50} -2.04392e6 q^{51} -1.10224e6 q^{53} -2.58752e6 q^{54} +1.44231e6 q^{55} +70321.5 q^{56} +1.47269e6 q^{57} +1.26268e6 q^{58} -236199. q^{59} -782452. q^{60} +2.81617e6 q^{61} -362918. q^{62} +244362. q^{63} +2.19722e6 q^{64} -5.97671e6 q^{66} -2.44794e6 q^{67} +1.09868e6 q^{68} +6.55789e6 q^{69} -76342.3 q^{70} +1.62315e6 q^{71} +8.61961e6 q^{72} +891091. q^{73} -2.12122e6 q^{74} -3.67698e6 q^{75} -791625. q^{76} +338881. q^{77} +4.25155e6 q^{79} -1.54855e6 q^{80} +1.32003e7 q^{81} -3.60285e6 q^{82} -1.90485e6 q^{83} -183843. q^{84} -4.43793e6 q^{85} +33262.6 q^{86} -1.22825e7 q^{87} +1.19537e7 q^{88} +6.79531e6 q^{89} -9.35761e6 q^{90} -3.52510e6 q^{92} +3.53020e6 q^{93} +7.48297e6 q^{94} +3.19764e6 q^{95} -1.12268e7 q^{96} +1.03105e7 q^{97} +7.39187e6 q^{98} +4.15381e7 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 21 q + 31 q^{2} - 26 q^{3} + 1409 q^{4} + 680 q^{5} + 1470 q^{6} + 2929 q^{7} + 4716 q^{8} + 15465 q^{9} - 5167 q^{10} + 14824 q^{11} + 21795 q^{12} - 179 q^{14} + 36398 q^{15} + 113205 q^{16} + 45016 q^{17}+ \cdots + 37605493 q^{99}+O(q^{100}) \) 21 * q + 31 * q^2 - 26 * q^3 + 1409 * q^4 + 680 * q^5 + 1470 * q^6 + 2929 * q^7 + 4716 * q^8 + 15465 * q^9 - 5167 * q^10 + 14824 * q^11 + 21795 * q^12 - 179 * q^14 + 36398 * q^15 + 113205 * q^16 + 45016 * q^17 + 72967 * q^18 + 119316 * q^19 + 131130 * q^20 + 94178 * q^21 - 274174 * q^22 - 128105 * q^23 + 316902 * q^24 + 366537 * q^25 - 106643 * q^27 + 170352 * q^28 - 62329 * q^29 + 9761 * q^30 + 467367 * q^31 + 677464 * q^32 + 587690 * q^33 + 578614 * q^34 + 848860 * q^35 - 274245 * q^36 + 697064 * q^37 + 225831 * q^38 - 2288827 * q^40 + 2351315 * q^41 + 2736836 * q^42 + 894489 * q^43 - 263025 * q^44 - 1938345 * q^45 - 3170392 * q^46 + 1070934 * q^47 + 6302388 * q^48 + 131530 * q^49 + 3970474 * q^50 - 1184341 * q^51 - 4550027 * q^53 + 7916233 * q^54 + 5524030 * q^55 + 2334386 * q^56 + 5590305 * q^57 + 7510959 * q^58 + 10746233 * q^59 + 15186203 * q^60 + 5717012 * q^61 - 13593653 * q^62 + 22347508 * q^63 + 9767924 * q^64 - 17806977 * q^66 + 11305923 * q^67 + 12252492 * q^68 + 2772206 * q^69 + 28670666 * q^70 + 23267025 * q^71 + 49774322 * q^72 + 4763727 * q^73 + 9625078 * q^74 - 43212397 * q^75 + 63359948 * q^76 + 9446036 * q^77 - 18613281 * q^79 + 57904093 * q^80 + 41055449 * q^81 + 24190327 * q^82 + 21455789 * q^83 + 60768429 * q^84 + 26216249 * q^85 + 35632347 * q^86 + 9027894 * q^87 - 75678180 * q^88 + 54037507 * q^89 + 3335560 * q^90 - 69339390 * q^92 + 20085986 * q^93 + 28955713 * q^94 + 17660031 * q^95 + 66667956 * q^96 + 10916609 * q^97 + 41738840 * q^98 + 37605493 * q^99

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