LMFDB - Embedded newform 169.8.a.i.1.18

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)

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")

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);

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			1.18
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)

$f(q)$	$=$	\(q+17.7683 q^{2} +58.1116 q^{3} +187.714 q^{4} +15.5333 q^{5} +1032.55 q^{6} +1688.45 q^{7} +1061.02 q^{8} +1189.96 q^{9} +276.002 q^{10} -4665.45 q^{11} +10908.4 q^{12} +30000.9 q^{14} +902.667 q^{15} -5174.84 q^{16} +37309.4 q^{17} +21143.6 q^{18} +20164.9 q^{19} +2915.82 q^{20} +98118.4 q^{21} -82897.4 q^{22} +65513.9 q^{23} +61657.6 q^{24} -77883.7 q^{25} -57939.6 q^{27} +316945. q^{28} -59346.3 q^{29} +16038.9 q^{30} -125426. q^{31} -227759. q^{32} -271117. q^{33} +662927. q^{34} +26227.2 q^{35} +223372. q^{36} +216208. q^{37} +358297. q^{38} +16481.2 q^{40} -154839. q^{41} +1.74340e6 q^{42} -124739. q^{43} -875771. q^{44} +18484.1 q^{45} +1.16407e6 q^{46} +359489. q^{47} -300718. q^{48} +2.02731e6 q^{49} -1.38386e6 q^{50} +2.16811e6 q^{51} -1.76397e6 q^{53} -1.02949e6 q^{54} -72470.0 q^{55} +1.79148e6 q^{56} +1.17181e6 q^{57} -1.05449e6 q^{58} -784579. q^{59} +169443. q^{60} -1.82630e6 q^{61} -2.22860e6 q^{62} +2.00919e6 q^{63} -3.38452e6 q^{64} -4.81730e6 q^{66} -1.23293e6 q^{67} +7.00350e6 q^{68} +3.80712e6 q^{69} +466014. q^{70} +4.03286e6 q^{71} +1.26257e6 q^{72} +2.54855e6 q^{73} +3.84166e6 q^{74} -4.52595e6 q^{75} +3.78523e6 q^{76} -7.87737e6 q^{77} +3.36048e6 q^{79} -80382.4 q^{80} -5.96941e6 q^{81} -2.75122e6 q^{82} +2.84027e6 q^{83} +1.84182e7 q^{84} +579540. q^{85} -2.21640e6 q^{86} -3.44871e6 q^{87} -4.95014e6 q^{88} -1.27649e6 q^{89} +328431. q^{90} +1.22979e7 q^{92} -7.28868e6 q^{93} +6.38753e6 q^{94} +313228. q^{95} -1.32354e7 q^{96} -6.59083e6 q^{97} +3.60220e7 q^{98} -5.55170e6 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	17.7683	1.57051	0.785257	−	0.619170i	$-0.212531\pi$
$2$	17.7683	1.57051	0.785257	+	0.619170i	$0.212531\pi$
$3$	58.1116	1.24262	0.621310	−	0.783565i	$-0.286601\pi$
$3$	58.1116	1.24262	0.621310	+	0.783565i	$0.286601\pi$
$4$	187.714	1.46652
$5$	15.5333	0.0555737	0.0277869	−	0.999614i	$-0.491154\pi$
$5$	15.5333	0.0555737	0.0277869	+	0.999614i	$0.491154\pi$
$6$	1032.55	1.95155
$7$	1688.45	1.86056	0.930282	−	0.366846i	$-0.119562\pi$
$7$	1688.45	1.86056	0.930282	+	0.366846i	$0.119562\pi$
$8$	1061.02	0.732670
$9$	1189.96	0.544106
$10$	276.002	0.0872794
$11$	−4665.45	−1.05686	−0.528432	−	0.848975i	$-0.677220\pi$
$11$	−4665.45	−1.05686	−0.528432	+	0.848975i	$0.677220\pi$
$12$	10908.4	1.82232
$13$	0	0
$13$	0	0
$14$	30000.9	2.92204
$15$	902.667	0.0690571
$16$	−5174.84	−0.315847
$17$	37309.4	1.84182	0.920910	−	0.389774i	$-0.127447\pi$
$17$	37309.4	1.84182	0.920910	+	0.389774i	$0.127447\pi$
$18$	21143.6	0.854527
$19$	20164.9	0.674463	0.337232	−	0.941422i	$-0.390509\pi$
$19$	20164.9	0.674463	0.337232	+	0.941422i	$0.390509\pi$
$20$	2915.82	0.0814998
$21$	98118.4	2.31197
$22$	−82897.4	−1.65982
$23$	65513.9	1.12276	0.561379	−	0.827559i	$-0.310271\pi$
$23$	65513.9	1.12276	0.561379	+	0.827559i	$0.310271\pi$
$24$	61657.6	0.910431
$25$	−77883.7	−0.996912
$26$	0	0
$27$	−57939.6	−0.566503
$28$	316945.	2.72855
$29$	−59346.3	−0.451857	−0.225928	−	0.974144i	$-0.572542\pi$
$29$	−59346.3	−0.451857	−0.225928	+	0.974144i	$0.572542\pi$
$30$	16038.9	0.108455
$31$	−125426.	−0.756171	−0.378085	−	0.925771i	$-0.623417\pi$
$31$	−125426.	−0.756171	−0.378085	+	0.925771i	$0.623417\pi$
$32$	−227759.	−1.22871
$33$	−271117.	−1.31328
$34$	662927.	2.89261
$35$	26227.2	0.103398
$36$	223372.	0.797941
$37$	216208.	0.701723	0.350862	−	0.936427i	$-0.385889\pi$
$37$	216208.	0.701723	0.350862	+	0.936427i	$0.385889\pi$
$38$	358297.	1.05925
$39$	0	0
$40$	16481.2	0.0407172
$41$	−154839.	−0.350861	−0.175431	−	0.984492i	$-0.556132\pi$
$41$	−154839.	−0.350861	−0.175431	+	0.984492i	$0.556132\pi$
$42$	1.74340e6	3.63099
$43$	−124739.	−0.239256	−0.119628	−	0.992819i	$-0.538170\pi$
$43$	−124739.	−0.239256	−0.119628	+	0.992819i	$0.538170\pi$
$44$	−875771.	−1.54991
$45$	18484.1	0.0302380
$46$	1.16407e6	1.76331
$47$	359489.	0.505061	0.252530	−	0.967589i	$-0.418737\pi$
$47$	359489.	0.505061	0.252530	+	0.967589i	$0.418737\pi$
$48$	−300718.	−0.392478
$49$	2.02731e6	2.46170
$50$	−1.38386e6	−1.56566
$51$	2.16811e6	2.28868
$52$	0	0
$53$	−1.76397e6	−1.62752	−0.813761	−	0.581200i	$-0.802583\pi$
$53$	−1.76397e6	−1.62752	−0.813761	+	0.581200i	$0.802583\pi$
$54$	−1.02949e6	−0.889701
$55$	−72470.0	−0.0587339
$56$	1.79148e6	1.36318
$57$	1.17181e6	0.838102
$58$	−1.05449e6	−0.709648
$59$	−784579.	−0.497342	−0.248671	−	0.968588i	$-0.579994\pi$
$59$	−784579.	−0.497342	−0.248671	+	0.968588i	$0.579994\pi$
$60$	169443.	0.101273
$61$	−1.82630e6	−1.03019	−0.515096	−	0.857133i	$-0.672244\pi$
$61$	−1.82630e6	−1.03019	−0.515096	+	0.857133i	$0.672244\pi$
$62$	−2.22860e6	−1.18758
$63$	2.00919e6	1.01234
$64$	−3.38452e6	−1.61386
$65$	0	0
$66$	−4.81730e6	−2.06253
$67$	−1.23293e6	−0.500813	−0.250406	−	0.968141i	$-0.580564\pi$
$67$	−1.23293e6	−0.500813	−0.250406	+	0.968141i	$0.580564\pi$
$68$	7.00350e6	2.70106
$69$	3.80712e6	1.39516
$70$	466014.	0.162389
$71$	4.03286e6	1.33724	0.668619	−	0.743605i	$-0.266886\pi$
$71$	4.03286e6	1.33724	0.668619	+	0.743605i	$0.266886\pi$
$72$	1.26257e6	0.398650
$73$	2.54855e6	0.766767	0.383384	−	0.923589i	$-0.374759\pi$
$73$	2.54855e6	0.766767	0.383384	+	0.923589i	$0.374759\pi$
$74$	3.84166e6	1.10207
$75$	−4.52595e6	−1.23878
$76$	3.78523e6	0.989111
$77$	−7.87737e6	−1.96636
$78$	0	0
$79$	3.36048e6	0.766843	0.383422	−	0.923573i	$-0.374746\pi$
$79$	3.36048e6	0.766843	0.383422	+	0.923573i	$0.374746\pi$
$80$	−80382.4	−0.0175528
$81$	−5.96941e6	−1.24805
$82$	−2.75122e6	−0.551033
$83$	2.84027e6	0.545237	0.272619	−	0.962122i	$-0.412110\pi$
$83$	2.84027e6	0.545237	0.272619	+	0.962122i	$0.412110\pi$
$84$	1.84182e7	3.39055
$85$	579540.	0.102357
$86$	−2.21640e6	−0.375755
$87$	−3.44871e6	−0.561487
$88$	−4.95014e6	−0.774333
$89$	−1.27649e6	−0.191935	−0.0959673	−	0.995384i	$-0.530594\pi$
$89$	−1.27649e6	−0.191935	−0.0959673	+	0.995384i	$0.530594\pi$
$90$	328431.	0.0474893
$91$	0	0
$92$	1.22979e7	1.64654
$93$	−7.28868e6	−0.939634
$94$	6.38753e6	0.793205
$95$	313228.	0.0374824
$96$	−1.32354e7	−1.52682
$97$	−6.59083e6	−0.733228	−0.366614	−	0.930373i	$-0.619483\pi$
$97$	−6.59083e6	−0.733228	−0.366614	+	0.930373i	$0.619483\pi$
$98$	3.60220e7	3.86613
$99$	−5.55170e6	−0.575047
$100$	−1.46199e7	−1.46199
$101$	−5.59140e6	−0.540003	−0.270001	−	0.962860i	$-0.587024\pi$
$101$	−5.59140e6	−0.540003	−0.270001	+	0.962860i	$0.587024\pi$
$102$	3.85237e7	3.59441
$103$	−1.18242e7	−1.06621	−0.533105	−	0.846049i	$-0.678975\pi$
$103$	−1.18242e7	−1.06621	−0.533105	+	0.846049i	$0.678975\pi$
$104$	0	0
$105$	1.52411e6	0.128485
$106$	−3.13429e7	−2.55605
$107$	1.32849e6	0.104837	0.0524185	−	0.998625i	$-0.483307\pi$
$107$	1.32849e6	0.104837	0.0524185	+	0.998625i	$0.483307\pi$
$108$	−1.08761e7	−0.830786
$109$	1.46298e7	1.08205	0.541023	−	0.841008i	$-0.318037\pi$
$109$	1.46298e7	1.08205	0.541023	+	0.841008i	$0.318037\pi$
$110$	−1.28767e6	−0.0922425
$111$	1.25642e7	0.871976
$112$	−8.73744e6	−0.587653
$113$	−3.67572e6	−0.239645	−0.119822	−	0.992795i	$-0.538232\pi$
$113$	−3.67572e6	−0.239645	−0.119822	+	0.992795i	$0.538232\pi$
$114$	2.08212e7	1.31625
$115$	1.01765e6	0.0623958
$116$	−1.11401e7	−0.662655
$117$	0	0
$118$	−1.39407e7	−0.781082
$119$	6.29950e7	3.42682
$120$	957747.	0.0505961
$121$	2.27928e6	0.116963
$122$	−3.24503e7	−1.61793
$123$	−8.99792e6	−0.435988
$124$	−2.35441e7	−1.10894
$125$	−2.42333e6	−0.110976
$126$	3.56999e7	1.58990
$127$	−2.78149e7	−1.20494	−0.602469	−	0.798142i	$-0.705816\pi$
$127$	−2.78149e7	−1.20494	−0.602469	+	0.798142i	$0.705816\pi$
$128$	−3.09841e7	−1.30588
$129$	−7.24878e6	−0.297304
$130$	0	0
$131$	−2.63591e7	−1.02443	−0.512214	−	0.858858i	$-0.671174\pi$
$131$	−2.63591e7	−1.02443	−0.512214	+	0.858858i	$0.671174\pi$
$132$	−5.08925e7	−1.92595
$133$	3.40474e7	1.25488
$134$	−2.19071e7	−0.786534
$135$	−899995.	−0.0314827
$136$	3.95860e7	1.34945
$137$	−2.86790e7	−0.952887	−0.476443	−	0.879205i	$-0.658074\pi$
$137$	−2.86790e7	−0.952887	−0.476443	+	0.879205i	$0.658074\pi$
$138$	6.76462e7	2.19112
$139$	−3.00048e7	−0.947629	−0.473815	−	0.880625i	$-0.657123\pi$
$139$	−3.00048e7	−0.947629	−0.473815	+	0.880625i	$0.657123\pi$
$140$	4.92322e6	0.151635
$141$	2.08905e7	0.627599
$142$	7.16572e7	2.10015
$143$	0	0
$144$	−6.15785e6	−0.171854
$145$	−921846.	−0.0251114
$146$	4.52836e7	1.20422
$147$	1.17810e8	3.05895
$148$	4.05853e7	1.02909
$149$	5.15415e6	0.127645	0.0638227	−	0.997961i	$-0.479671\pi$
$149$	5.15415e6	0.127645	0.0638227	+	0.997961i	$0.479671\pi$
$150$	−8.04186e7	−1.94553
$151$	2.40636e7	0.568776	0.284388	−	0.958709i	$-0.408210\pi$
$151$	2.40636e7	0.568776	0.284388	+	0.958709i	$0.408210\pi$
$152$	2.13953e7	0.494159
$153$	4.43967e7	1.00215
$154$	−1.39968e8	−3.08820
$155$	−1.94828e6	−0.0420232
$156$	0	0
$157$	−2.16509e6	−0.0446506	−0.0223253	−	0.999751i	$-0.507107\pi$
$157$	−2.16509e6	−0.0446506	−0.0223253	+	0.999751i	$0.507107\pi$
$158$	5.97102e7	1.20434
$159$	−1.02507e8	−2.02239
$160$	−3.53785e6	−0.0682841
$161$	1.10617e8	2.08896
$162$	−1.06066e8	−1.96009
$163$	3.17203e7	0.573694	0.286847	−	0.957976i	$-0.407393\pi$
$163$	3.17203e7	0.573694	0.286847	+	0.957976i	$0.407393\pi$
$164$	−2.90654e7	−0.514544
$165$	−4.21135e6	−0.0729840
$166$	5.04668e7	0.856303
$167$	−2.41873e7	−0.401864	−0.200932	−	0.979605i	$-0.564397\pi$
$167$	−2.41873e7	−0.401864	−0.200932	+	0.979605i	$0.564397\pi$
$168$	1.04106e8	1.69391
$169$	0	0
$170$	1.02975e7	0.160753
$171$	2.39954e7	0.366980
$172$	−2.34152e7	−0.350872
$173$	2.17543e7	0.319435	0.159718	−	0.987163i	$-0.448942\pi$
$173$	2.17543e7	0.319435	0.159718	+	0.987163i	$0.448942\pi$
$174$	−6.12779e7	−0.881823
$175$	−1.31503e8	−1.85482
$176$	2.41429e7	0.333807
$177$	−4.55932e7	−0.618007
$178$	−2.26812e7	−0.301436
$179$	9.71205e7	1.26568	0.632842	−	0.774281i	$-0.281888\pi$
$179$	9.71205e7	1.26568	0.632842	+	0.774281i	$0.281888\pi$
$180$	3.46972e6	0.0443445
$181$	−921679.	−0.0115533	−0.00577663	−	0.999983i	$-0.501839\pi$
$181$	−921679.	−0.0115533	−0.00577663	+	0.999983i	$0.501839\pi$
$182$	0	0
$183$	−1.06129e8	−1.28014
$184$	6.95115e7	0.822611
$185$	3.35843e6	0.0389974
$186$	−1.29508e8	−1.47571
$187$	−1.74065e8	−1.94656
$188$	6.74812e7	0.740680
$189$	−9.78280e7	−1.05401
$190$	5.56554e6	0.0588667
$191$	−5.09497e7	−0.529084	−0.264542	−	0.964374i	$-0.585221\pi$
$191$	−5.09497e7	−0.529084	−0.264542	+	0.964374i	$0.585221\pi$
$192$	−1.96680e8	−2.00542
$193$	1.90809e8	1.91051	0.955254	−	0.295785i	$-0.0955813\pi$
$193$	1.90809e8	1.91051	0.955254	+	0.295785i	$0.0955813\pi$
$194$	−1.17108e8	−1.15155
$195$	0	0
$196$	3.80555e8	3.61012
$197$	−5.53491e6	−0.0515797	−0.0257899	−	0.999667i	$-0.508210\pi$
$197$	−5.53491e6	−0.0515797	−0.0257899	+	0.999667i	$0.508210\pi$
$198$	−9.86446e7	−0.903119
$199$	−2.13170e8	−1.91752	−0.958762	−	0.284211i	$-0.908268\pi$
$199$	−2.13170e8	−1.91752	−0.958762	+	0.284211i	$0.908268\pi$
$200$	−8.26362e7	−0.730407
$201$	−7.16474e7	−0.622320
$202$	−9.93499e7	−0.848082
$203$	−1.00203e8	−0.840708
$204$	4.06985e8	3.35639
$205$	−2.40516e6	−0.0194987
$206$	−2.10097e8	−1.67450
$207$	7.79590e7	0.610900
$208$	0	0
$209$	−9.40783e7	−0.712816
$210$	2.70808e7	0.201788
$211$	−1.75767e8	−1.28810	−0.644049	−	0.764985i	$-0.722747\pi$
$211$	−1.75767e8	−1.28810	−0.644049	+	0.764985i	$0.722747\pi$
$212$	−3.31123e8	−2.38679
$213$	2.34356e8	1.66168
$214$	2.36050e7	0.164648
$215$	−1.93761e6	−0.0132963
$216$	−6.14750e7	−0.415060
$217$	−2.11774e8	−1.40690
$218$	2.59947e8	1.69937
$219$	1.48101e8	0.952801
$220$	−1.36036e7	−0.0861342
$221$	0	0
$222$	2.23245e8	1.36945
$223$	−3.74636e7	−0.226226	−0.113113	−	0.993582i	$-0.536082\pi$
$223$	−3.74636e7	−0.226226	−0.113113	+	0.993582i	$0.536082\pi$
$224$	−3.84559e8	−2.28610
$225$	−9.26785e7	−0.542426
$226$	−6.53114e7	−0.376365
$227$	2.10377e8	1.19373	0.596866	−	0.802341i	$-0.296412\pi$
$227$	2.10377e8	1.19373	0.596866	+	0.802341i	$0.296412\pi$
$228$	2.19966e8	1.22909
$229$	−3.04391e8	−1.67497	−0.837486	−	0.546459i	$-0.815975\pi$
$229$	−3.04391e8	−1.67497	−0.837486	+	0.546459i	$0.815975\pi$
$230$	1.80819e7	0.0979936
$231$	−4.57767e8	−2.44344
$232$	−6.29676e7	−0.331062
$233$	−2.22293e8	−1.15128	−0.575639	−	0.817704i	$-0.695247\pi$
$233$	−2.22293e8	−1.15128	−0.575639	+	0.817704i	$0.695247\pi$
$234$	0	0
$235$	5.58407e6	0.0280681
$236$	−1.47277e8	−0.729359
$237$	1.95283e8	0.952895
$238$	1.11932e9	5.38188
$239$	1.97086e8	0.933822	0.466911	−	0.884304i	$-0.345367\pi$
$239$	1.97086e8	0.933822	0.466911	+	0.884304i	$0.345367\pi$
$240$	−4.67115e6	−0.0218115
$241$	2.98738e8	1.37477	0.687385	−	0.726293i	$-0.258758\pi$
$241$	2.98738e8	1.37477	0.687385	+	0.726293i	$0.258758\pi$
$242$	4.04990e7	0.183692
$243$	−2.20178e8	−0.984356
$244$	−3.42822e8	−1.51079
$245$	3.14909e7	0.136806
$246$	−1.59878e8	−0.684725
$247$	0	0
$248$	−1.33079e8	−0.554024
$249$	1.65052e8	0.677523
$250$	−4.30586e7	−0.174289
$251$	−9.04440e7	−0.361012	−0.180506	−	0.983574i	$-0.557774\pi$
$251$	−9.04440e7	−0.361012	−0.180506	+	0.983574i	$0.557774\pi$
$252$	3.77152e8	1.48462
$253$	−3.05652e8	−1.18660
$254$	−4.94225e8	−1.89237
$255$	3.36780e7	0.127191
$256$	−1.17319e8	−0.437046
$257$	3.35132e8	1.23155	0.615773	−	0.787924i	$-0.288844\pi$
$257$	3.35132e8	1.23155	0.615773	+	0.787924i	$0.288844\pi$
$258$	−1.28799e8	−0.466921
$259$	3.65056e8	1.30560
$260$	0	0
$261$	−7.06198e7	−0.245858
$262$	−4.68358e8	−1.60888
$263$	4.38247e7	0.148551	0.0742753	−	0.997238i	$-0.476336\pi$
$263$	4.38247e7	0.148551	0.0742753	+	0.997238i	$0.476336\pi$
$264$	−2.87660e8	−0.962202
$265$	−2.74004e7	−0.0904475
$266$	6.04965e8	1.97081
$267$	−7.41791e7	−0.238502
$268$	−2.31438e8	−0.734450
$269$	3.55234e8	1.11271	0.556355	−	0.830945i	$-0.312200\pi$
$269$	3.55234e8	1.11271	0.556355	+	0.830945i	$0.312200\pi$
$270$	−1.59914e7	−0.0494440
$271$	−4.89900e8	−1.49526	−0.747628	−	0.664118i	$-0.768807\pi$
$271$	−4.89900e8	−1.49526	−0.747628	+	0.664118i	$0.768807\pi$
$272$	−1.93070e8	−0.581733
$273$	0	0
$274$	−5.09578e8	−1.49652
$275$	3.63363e8	1.05360
$276$	7.14650e8	2.04603
$277$	−2.84193e7	−0.0803404	−0.0401702	−	0.999193i	$-0.512790\pi$
$277$	−2.84193e7	−0.0803404	−0.0401702	+	0.999193i	$0.512790\pi$
$278$	−5.33135e8	−1.48827
$279$	−1.49251e8	−0.411437
$280$	2.78276e7	0.0757570
$281$	3.58070e8	0.962711	0.481355	−	0.876525i	$-0.340145\pi$
$281$	3.58070e8	0.962711	0.481355	+	0.876525i	$0.340145\pi$
$282$	3.71190e8	0.985653
$283$	3.66356e8	0.960840	0.480420	−	0.877039i	$-0.340484\pi$
$283$	3.66356e8	0.960840	0.480420	+	0.877039i	$0.340484\pi$
$284$	7.57024e8	1.96108
$285$	1.82022e7	0.0465765
$286$	0	0
$287$	−2.61437e8	−0.652800
$288$	−2.71024e8	−0.668550
$289$	9.81655e8	2.39230
$290$	−1.63797e7	−0.0394378
$291$	−3.83004e8	−0.911124
$292$	4.78399e8	1.12448
$293$	1.64599e7	0.0382289	0.0191145	−	0.999817i	$-0.493915\pi$
$293$	1.64599e7	0.0382289	0.0191145	+	0.999817i	$0.493915\pi$
$294$	2.09330e9	4.80413
$295$	−1.21871e7	−0.0276391
$296$	2.29401e8	0.514132
$297$	2.70314e8	0.598717
$298$	9.15807e7	0.200469
$299$	0	0
$300$	−8.49584e8	−1.81670
$301$	−2.10615e8	−0.445150
$302$	4.27570e8	0.893270
$303$	−3.24925e8	−0.671018
$304$	−1.04350e8	−0.213027
$305$	−2.83685e7	−0.0572516
$306$	7.88857e8	1.57389
$307$	−9.09885e8	−1.79474	−0.897372	−	0.441275i	$-0.854526\pi$
$307$	−9.09885e8	−1.79474	−0.897372	+	0.441275i	$0.854526\pi$
$308$	−1.47869e9	−2.88370
$309$	−6.87125e8	−1.32489
$310$	−3.46176e7	−0.0659981
$311$	9.54547e8	1.79943	0.899717	−	0.436473i	$-0.143772\pi$
$311$	9.54547e8	1.79943	0.899717	+	0.436473i	$0.143772\pi$
$312$	0	0
$313$	6.29742e8	1.16080	0.580400	−	0.814331i	$-0.302896\pi$
$313$	6.29742e8	1.16080	0.580400	+	0.814331i	$0.302896\pi$
$314$	−3.84701e7	−0.0701245
$315$	3.12094e7	0.0562598
$316$	6.30809e8	1.12459
$317$	−7.62485e8	−1.34439	−0.672193	−	0.740376i	$-0.734648\pi$
$317$	−7.62485e8	−1.34439	−0.672193	+	0.740376i	$0.734648\pi$
$318$	−1.82139e9	−3.17620
$319$	2.76878e8	0.477552
$320$	−5.25728e7	−0.0896884
$321$	7.72006e7	0.130273
$322$	1.96548e9	3.28075
$323$	7.52341e8	1.24224
$324$	−1.12054e9	−1.83029
$325$	0	0
$326$	5.63617e8	0.900995
$327$	8.50162e8	1.34457
$328$	−1.64287e8	−0.257066
$329$	6.06979e8	0.939698
$330$	−7.48287e7	−0.114622
$331$	4.97597e8	0.754188	0.377094	−	0.926175i	$-0.376923\pi$
$331$	4.97597e8	0.754188	0.377094	+	0.926175i	$0.376923\pi$
$332$	5.33158e8	0.799599
$333$	2.57279e8	0.381812
$334$	−4.29768e8	−0.631133
$335$	−1.91515e7	−0.0278320
$336$	−5.07747e8	−0.730230
$337$	1.11553e9	1.58772	0.793862	−	0.608098i	$-0.208067\pi$
$337$	1.11553e9	1.58772	0.793862	+	0.608098i	$0.208067\pi$
$338$	0	0
$339$	−2.13602e8	−0.297787
$340$	1.08788e8	0.150108
$341$	5.85167e8	0.799170
$342$	4.26359e8	0.576347
$343$	2.03250e9	2.71958
$344$	−1.32350e8	−0.175296
$345$	5.91372e7	0.0775344
$346$	3.86537e8	0.501678
$347$	2.32713e8	0.298997	0.149499	−	0.988762i	$-0.452234\pi$
$347$	2.32713e8	0.298997	0.149499	+	0.988762i	$0.452234\pi$
$348$	−6.47372e8	−0.823429
$349$	8.94950e8	1.12696	0.563481	−	0.826129i	$-0.309462\pi$
$349$	8.94950e8	1.12696	0.563481	+	0.826129i	$0.309462\pi$
$350$	−2.33658e9	−2.91302
$351$	0	0
$352$	1.06260e9	1.29858
$353$	3.82992e8	0.463424	0.231712	−	0.972784i	$-0.425567\pi$
$353$	3.82992e8	0.463424	0.231712	+	0.972784i	$0.425567\pi$
$354$	−8.10115e8	−0.970589
$355$	6.26437e7	0.0743153
$356$	−2.39616e8	−0.281475
$357$	3.66074e9	4.25824
$358$	1.72567e9	1.98777
$359$	9.49411e8	1.08299	0.541494	−	0.840704i	$-0.317859\pi$
$359$	9.49411e8	1.08299	0.541494	+	0.840704i	$0.317859\pi$
$360$	1.96119e7	0.0221545
$361$	−4.87249e8	−0.545099
$362$	−1.63767e7	−0.0181446
$363$	1.32453e8	0.145341
$364$	0	0
$365$	3.95875e7	0.0426121
$366$	−1.88574e9	−2.01047
$367$	−6.86490e8	−0.724941	−0.362471	−	0.931995i	$-0.618067\pi$
$367$	−6.86490e8	−0.724941	−0.362471	+	0.931995i	$0.618067\pi$
$368$	−3.39024e8	−0.354620
$369$	−1.84252e8	−0.190906
$370$	5.96738e7	0.0612460
$371$	−2.97838e9	−3.02811
$372$	−1.36819e9	−1.37799
$373$	−7.55087e8	−0.753384	−0.376692	−	0.926339i	$-0.622938\pi$
$373$	−7.55087e8	−0.753384	−0.376692	+	0.926339i	$0.622938\pi$
$374$	−3.09285e9	−3.05709
$375$	−1.40824e8	−0.137901
$376$	3.81425e8	0.370043
$377$	0	0
$378$	−1.73824e9	−1.65535
$379$	−1.24310e9	−1.17292	−0.586459	−	0.809979i	$-0.699478\pi$
$379$	−1.24310e9	−1.17292	−0.586459	+	0.809979i	$0.699478\pi$
$380$	5.87973e7	0.0549686
$381$	−1.61637e9	−1.49728
$382$	−9.05292e8	−0.830935
$383$	−9.32249e8	−0.847883	−0.423942	−	0.905689i	$-0.639354\pi$
$383$	−9.32249e8	−0.847883	−0.423942	+	0.905689i	$0.639354\pi$
$384$	−1.80054e9	−1.62272
$385$	−1.22362e8	−0.109278
$386$	3.39037e9	3.00048
$387$	−1.48434e8	−0.130181
$388$	−1.23719e9	−1.07529
$389$	−3.80482e7	−0.0327726	−0.0163863	−	0.999866i	$-0.505216\pi$
$389$	−3.80482e7	−0.0327726	−0.0163863	+	0.999866i	$0.505216\pi$
$390$	0	0
$391$	2.44429e9	2.06792
$392$	2.15102e9	1.80361
$393$	−1.53177e9	−1.27297
$394$	−9.83462e7	−0.0810067
$395$	5.21994e7	0.0426163
$396$	−1.04213e9	−0.843315
$397$	9.57613e8	0.768110	0.384055	−	0.923310i	$-0.374527\pi$
$397$	9.57613e8	0.768110	0.384055	+	0.923310i	$0.374527\pi$
$398$	−3.78768e9	−3.01150
$399$	1.97855e9	1.55934
$400$	4.03035e8	0.314871
$401$	−4.24299e8	−0.328599	−0.164300	−	0.986410i	$-0.552536\pi$
$401$	−4.24299e8	−0.328599	−0.164300	+	0.986410i	$0.552536\pi$
$402$	−1.27306e9	−0.977363
$403$	0	0
$404$	−1.04958e9	−0.791922
$405$	−9.27248e7	−0.0693591
$406$	−1.78044e9	−1.32034
$407$	−1.00871e9	−0.741627
$408$	2.30041e9	1.67685
$409$	1.52045e9	1.09886	0.549428	−	0.835541i	$-0.314846\pi$
$409$	1.52045e9	1.09886	0.549428	+	0.835541i	$0.314846\pi$
$410$	−4.27357e7	−0.0306230
$411$	−1.66658e9	−1.18408
$412$	−2.21957e9	−1.56361
$413$	−1.32472e9	−0.925335
$414$	1.38520e9	0.959427
$415$	4.41188e7	0.0303009
$416$	0	0
$417$	−1.74363e9	−1.17754
$418$	−1.67162e9	−1.11949
$419$	1.12521e9	0.747282	0.373641	−	0.927573i	$-0.378109\pi$
$419$	1.12521e9	0.747282	0.373641	+	0.927573i	$0.378109\pi$
$420$	2.86096e8	0.188425
$421$	1.54255e9	1.00752	0.503758	−	0.863845i	$-0.331950\pi$
$421$	1.54255e9	1.00752	0.503758	+	0.863845i	$0.331950\pi$
$422$	−3.12309e9	−2.02298
$423$	4.27778e8	0.274807
$424$	−1.87161e9	−1.19244
$425$	−2.90580e9	−1.83613
$426$	4.16412e9	2.60969
$427$	−3.08361e9	−1.91674
$428$	2.49376e8	0.153745
$429$	0	0
$430$	−3.44281e7	−0.0208821
$431$	−4.93342e8	−0.296810	−0.148405	−	0.988927i	$-0.547414\pi$
$431$	−4.93342e8	−0.296810	−0.148405	+	0.988927i	$0.547414\pi$
$432$	2.99828e8	0.178928
$433$	1.11794e9	0.661777	0.330889	−	0.943670i	$-0.392652\pi$
$433$	1.11794e9	0.661777	0.330889	+	0.943670i	$0.392652\pi$
$434$	−3.76288e9	−2.20956
$435$	−5.35700e7	−0.0312039
$436$	2.74622e9	1.58684
$437$	1.32108e9	0.757259
$438$	2.63150e9	1.49639
$439$	2.00439e8	0.113072	0.0565361	−	0.998401i	$-0.481994\pi$
$439$	2.00439e8	0.113072	0.0565361	+	0.998401i	$0.481994\pi$
$440$	−7.68921e7	−0.0430326
$441$	2.41242e9	1.33942
$442$	0	0
$443$	2.99756e9	1.63815	0.819077	−	0.573683i	$-0.194486\pi$
$443$	2.99756e9	1.63815	0.819077	+	0.573683i	$0.194486\pi$
$444$	2.35848e9	1.27877
$445$	−1.98282e7	−0.0106665
$446$	−6.65665e8	−0.355291
$447$	2.99516e8	0.158615
$448$	−5.71458e9	−3.00270
$449$	1.31389e9	0.685009	0.342505	−	0.939516i	$-0.388725\pi$
$449$	1.31389e9	0.685009	0.342505	+	0.939516i	$0.388725\pi$
$450$	−1.64674e9	−0.851888
$451$	7.22392e8	0.370813
$452$	−6.89984e8	−0.351443
$453$	1.39837e9	0.706772
$454$	3.73804e9	1.87477
$455$	0	0
$456$	1.24332e9	0.614052
$457$	2.19208e8	0.107436	0.0537180	−	0.998556i	$-0.482893\pi$
$457$	2.19208e8	0.107436	0.0537180	+	0.998556i	$0.482893\pi$
$458$	−5.40852e9	−2.63057
$459$	−2.16169e9	−1.04340
$460$	1.91027e8	0.0915045
$461$	2.62315e8	0.124701	0.0623504	−	0.998054i	$-0.480140\pi$
$461$	2.62315e8	0.124701	0.0623504	+	0.998054i	$0.480140\pi$
$462$	−8.13376e9	−3.83747
$463$	1.34993e9	0.632090	0.316045	−	0.948744i	$-0.397645\pi$
$463$	1.34993e9	0.632090	0.316045	+	0.948744i	$0.397645\pi$
$464$	3.07108e8	0.142718
$465$	−1.13217e8	−0.0522190
$466$	−3.94978e9	−1.80810
$467$	2.08017e9	0.945125	0.472563	−	0.881297i	$-0.343329\pi$
$467$	2.08017e9	0.945125	0.472563	+	0.881297i	$0.343329\pi$
$468$	0	0
$469$	−2.08173e9	−0.931794
$470$	9.92197e7	0.0440814
$471$	−1.25817e8	−0.0554838
$472$	−8.32454e8	−0.364387
$473$	5.81964e8	0.252861
$474$	3.46985e9	1.49654
$475$	−1.57052e9	−0.672380
$476$	1.18251e10	5.02549
$477$	−2.09906e9	−0.885545
$478$	3.50190e9	1.46658
$479$	1.47547e9	0.613416	0.306708	−	0.951804i	$-0.400772\pi$
$479$	1.47547e9	0.613416	0.306708	+	0.951804i	$0.400772\pi$
$480$	−2.05590e8	−0.0848513
$481$	0	0
$482$	5.30807e9	2.15910
$483$	6.42812e9	2.59579
$484$	4.27853e8	0.171528
$485$	−1.02378e8	−0.0407482
$486$	−3.91220e9	−1.54594
$487$	1.36444e9	0.535308	0.267654	−	0.963515i	$-0.413752\pi$
$487$	1.36444e9	0.535308	0.267654	+	0.963515i	$0.413752\pi$
$488$	−1.93774e9	−0.754790
$489$	1.84332e9	0.712885
$490$	5.59541e8	0.214855
$491$	9.41414e8	0.358918	0.179459	−	0.983765i	$-0.442565\pi$
$491$	9.41414e8	0.358918	0.179459	+	0.983765i	$0.442565\pi$
$492$	−1.68904e9	−0.639383
$493$	−2.21418e9	−0.832239
$494$	0	0
$495$	−8.62365e7	−0.0319575
$496$	6.49056e8	0.238834
$497$	6.80927e9	2.48802
$498$	2.93271e9	1.06406
$499$	4.44915e9	1.60297	0.801485	−	0.598015i	$-0.204044\pi$
$499$	4.44915e9	1.60297	0.801485	+	0.598015i	$0.204044\pi$
$500$	−4.54894e8	−0.162748
$501$	−1.40556e9	−0.499365
$502$	−1.60704e9	−0.566975
$503$	2.33266e9	0.817265	0.408633	−	0.912699i	$-0.366006\pi$
$503$	2.33266e9	0.817265	0.408633	+	0.912699i	$0.366006\pi$
$504$	2.13179e9	0.741714
$505$	−8.68531e7	−0.0300100
$506$	−5.43093e9	−1.86358
$507$	0	0
$508$	−5.22125e9	−1.76706
$509$	3.07289e9	1.03284	0.516422	−	0.856334i	$-0.327264\pi$
$509$	3.07289e9	1.03284	0.516422	+	0.856334i	$0.327264\pi$
$510$	5.98402e8	0.199755
$511$	4.30310e9	1.42662
$512$	1.88141e9	0.619497
$513$	−1.16835e9	−0.382085
$514$	5.95475e9	1.93416
$515$	−1.83670e8	−0.0592532
$516$	−1.36070e9	−0.436001
$517$	−1.67718e9	−0.533781
$518$	6.48644e9	2.05047
$519$	1.26418e9	0.396937
$520$	0	0
$521$	−1.59236e9	−0.493298	−0.246649	−	0.969105i	$-0.579330\pi$
$521$	−1.59236e9	−0.493298	−0.246649	+	0.969105i	$0.579330\pi$
$522$	−1.25480e9	−0.386124
$523$	−2.27098e9	−0.694156	−0.347078	−	0.937836i	$-0.612826\pi$
$523$	−2.27098e9	−0.694156	−0.347078	+	0.937836i	$0.612826\pi$
$524$	−4.94798e9	−1.50234
$525$	−7.64183e9	−2.30483
$526$	7.78693e8	0.233301
$527$	−4.67955e9	−1.39273
$528$	1.40299e9	0.414796
$529$	8.87246e8	0.260585
$530$	−4.86860e8	−0.142049
$531$	−9.33618e8	−0.270607
$532$	6.39117e9	1.84030
$533$	0	0
$534$	−1.31804e9	−0.374571
$535$	2.06358e7	0.00582618
$536$	−1.30816e9	−0.366931
$537$	5.64383e9	1.57276
$538$	6.31192e9	1.74753
$539$	−9.45833e9	−2.60168
$540$	−1.68942e8	−0.0461699
$541$	−4.79640e9	−1.30234	−0.651171	−	0.758931i	$-0.725722\pi$
$541$	−4.79640e9	−1.30234	−0.651171	+	0.758931i	$0.725722\pi$
$542$	−8.70472e9	−2.34832
$543$	−5.35603e7	−0.0143563
$544$	−8.49755e9	−2.26307
$545$	2.27250e8	0.0601333
$546$	0	0
$547$	1.04080e9	0.271902	0.135951	−	0.990716i	$-0.456591\pi$
$547$	1.04080e9	0.271902	0.135951	+	0.990716i	$0.456591\pi$
$548$	−5.38344e9	−1.39742
$549$	−2.17323e9	−0.560534
$550$	6.45636e9	1.65470
$551$	−1.19671e9	−0.304761
$552$	4.03943e9	1.02219
$553$	5.67399e9	1.42676
$554$	−5.04964e8	−0.126176
$555$	1.95164e8	0.0484590
$556$	−5.63231e9	−1.38971
$557$	3.62664e9	0.889224	0.444612	−	0.895723i	$-0.353342\pi$
$557$	3.62664e9	0.889224	0.444612	+	0.895723i	$0.353342\pi$
$558$	−2.65195e9	−0.646168
$559$	0	0
$560$	−1.35722e8	−0.0326581
$561$	−1.01152e10	−2.41883
$562$	6.36231e9	1.51195
$563$	−6.84603e9	−1.61681	−0.808405	−	0.588626i	$-0.799669\pi$
$563$	−6.84603e9	−1.61681	−0.808405	+	0.588626i	$0.799669\pi$
$564$	3.92144e9	0.920384
$565$	−5.70962e7	−0.0133179
$566$	6.50954e9	1.50901
$567$	−1.00790e10	−2.32208
$568$	4.27894e9	0.979754
$569$	4.84834e9	1.10332	0.551659	−	0.834070i	$-0.313995\pi$
$569$	4.84834e9	1.10332	0.551659	+	0.834070i	$0.313995\pi$
$570$	3.23423e8	0.0731490
$571$	−1.65781e9	−0.372655	−0.186328	−	0.982488i	$-0.559659\pi$
$571$	−1.65781e9	−0.372655	−0.186328	+	0.982488i	$0.559659\pi$
$572$	0	0
$573$	−2.96077e9	−0.657451
$574$	−4.64530e9	−1.02523
$575$	−5.10247e9	−1.11929
$576$	−4.02744e9	−0.878113
$577$	−3.67938e9	−0.797370	−0.398685	−	0.917088i	$-0.630533\pi$
$577$	−3.67938e9	−0.797370	−0.398685	+	0.917088i	$0.630533\pi$
$578$	1.74424e10	3.75715
$579$	1.10882e10	2.37404
$580$	−1.73043e8	−0.0368262
$581$	4.79564e9	1.01445
$582$	−6.80534e9	−1.43093
$583$	8.22974e9	1.72007
$584$	2.70406e9	0.561788
$585$	0	0
$586$	2.92466e8	0.0600390
$587$	−1.62623e9	−0.331855	−0.165927	−	0.986138i	$-0.553062\pi$
$587$	−1.62623e9	−0.331855	−0.165927	+	0.986138i	$0.553062\pi$
$588$	2.21147e10	4.48601
$589$	−2.52919e9	−0.510009
$590$	−2.16545e8	−0.0434077
$591$	−3.21643e8	−0.0640940
$592$	−1.11884e9	−0.221637
$593$	1.07604e7	0.00211903	0.00105952	−	0.999999i	$-0.499663\pi$
$593$	1.07604e7	0.00211903	0.00105952	+	0.999999i	$0.499663\pi$
$594$	4.80304e9	0.940294
$595$	9.78522e8	0.190441
$596$	9.67507e8	0.187194
$597$	−1.23877e10	−2.38275
$598$	0	0
$599$	−8.38421e9	−1.59393	−0.796964	−	0.604027i	$-0.793562\pi$
$599$	−8.38421e9	−1.59393	−0.796964	+	0.604027i	$0.793562\pi$
$600$	−4.80212e9	−0.907619
$601$	2.73577e9	0.514066	0.257033	−	0.966403i	$-0.417255\pi$
$601$	2.73577e9	0.514066	0.257033	+	0.966403i	$0.417255\pi$
$602$	−3.74228e9	−0.699115
$603$	−1.46713e9	−0.272495
$604$	4.51707e9	0.834118
$605$	3.54048e7	0.00650007
$606$	−5.77339e9	−1.05384
$607$	−5.18763e9	−0.941475	−0.470738	−	0.882273i	$-0.656012\pi$
$607$	−5.18763e9	−0.941475	−0.470738	+	0.882273i	$0.656012\pi$
$608$	−4.59273e9	−0.828721
$609$	−5.82297e9	−1.04468
$610$	−5.04062e8	−0.0899144
$611$	0	0
$612$	8.33389e9	1.46966
$613$	−2.00248e9	−0.351121	−0.175561	−	0.984469i	$-0.556174\pi$
$613$	−2.00248e9	−0.351121	−0.175561	+	0.984469i	$0.556174\pi$
$614$	−1.61672e10	−2.81867
$615$	−1.39768e8	−0.0242295
$616$	−8.35805e9	−1.44070
$617$	2.91244e9	0.499183	0.249591	−	0.968351i	$-0.419704\pi$
$617$	2.91244e9	0.499183	0.249591	+	0.968351i	$0.419704\pi$
$618$	−1.22091e10	−2.08076
$619$	4.29088e9	0.727158	0.363579	−	0.931563i	$-0.381555\pi$
$619$	4.29088e9	0.727158	0.363579	+	0.931563i	$0.381555\pi$
$620$	−3.65719e8	−0.0616278
$621$	−3.79585e9	−0.636046
$622$	1.69607e10	2.82604
$623$	−2.15529e9	−0.357107
$624$	0	0
$625$	6.04702e9	0.990744
$626$	1.11895e10	1.82305
$627$	−5.46704e9	−0.885760
$628$	−4.06418e8	−0.0654809
$629$	8.06660e9	1.29245
$630$	5.54539e8	0.0883568
$631$	−7.47553e9	−1.18451	−0.592255	−	0.805750i	$-0.701762\pi$
$631$	−7.47553e9	−1.18451	−0.592255	+	0.805750i	$0.701762\pi$
$632$	3.56554e9	0.561843
$633$	−1.02141e10	−1.60062
$634$	−1.35481e10	−2.11138
$635$	−4.32058e8	−0.0669629
$636$	−1.92421e10	−2.96587
$637$	0	0
$638$	4.91965e9	0.750002
$639$	4.79894e9	0.727599
$640$	−4.81287e8	−0.0725728
$641$	7.59758e9	1.13939	0.569695	−	0.821856i	$-0.307061\pi$
$641$	7.59758e9	1.13939	0.569695	+	0.821856i	$0.307061\pi$
$642$	1.37173e9	0.204595
$643$	−7.35758e9	−1.09143	−0.545716	−	0.837970i	$-0.683742\pi$
$643$	−7.35758e9	−1.09143	−0.545716	+	0.837970i	$0.683742\pi$
$644$	2.07643e10	3.06350
$645$	−1.12598e8	−0.0165223
$646$	1.33678e10	1.95096
$647$	4.42469e9	0.642271	0.321135	−	0.947033i	$-0.395936\pi$
$647$	4.42469e9	0.642271	0.321135	+	0.947033i	$0.395936\pi$
$648$	−6.33366e9	−0.914412
$649$	3.66042e9	0.525623
$650$	0	0
$651$	−1.23066e10	−1.74825
$652$	5.95434e9	0.841332
$653$	6.23572e9	0.876376	0.438188	−	0.898883i	$-0.355620\pi$
$653$	6.23572e9	0.876376	0.438188	+	0.898883i	$0.355620\pi$
$654$	1.51060e10	2.11167
$655$	−4.09445e8	−0.0569313
$656$	8.01264e8	0.110818
$657$	3.03268e9	0.417203
$658$	1.07850e10	1.47581
$659$	−6.85996e9	−0.933733	−0.466866	−	0.884328i	$-0.654617\pi$
$659$	−6.85996e9	−0.933733	−0.466866	+	0.884328i	$0.654617\pi$
$660$	−7.90530e8	−0.107032
$661$	1.12545e10	1.51572	0.757861	−	0.652416i	$-0.226245\pi$
$661$	1.12545e10	1.51572	0.757861	+	0.652416i	$0.226245\pi$
$662$	8.84147e9	1.18446
$663$	0	0
$664$	3.01358e9	0.399479
$665$	5.28869e8	0.0697384
$666$	4.57142e9	0.599642
$667$	−3.88801e9	−0.507326
$668$	−4.54029e9	−0.589340
$669$	−2.17707e9	−0.281113
$670$	−3.40290e8	−0.0437106
$671$	8.52052e9	1.08877
$672$	−2.23473e10	−2.84075
$673$	1.04170e10	1.31732	0.658660	−	0.752441i	$-0.271124\pi$
$673$	1.04170e10	1.31732	0.658660	+	0.752441i	$0.271124\pi$
$674$	1.98210e10	2.49354
$675$	4.51255e9	0.564753
$676$	0	0
$677$	4.39978e9	0.544967	0.272484	−	0.962160i	$-0.412155\pi$
$677$	4.39978e9	0.544967	0.272484	+	0.962160i	$0.412155\pi$
$678$	−3.79535e9	−0.467680
$679$	−1.11283e10	−1.36422
$680$	6.14903e8	0.0749938
$681$	1.22253e10	1.48336
$682$	1.03974e10	1.25511
$683$	7.39005e9	0.887513	0.443757	−	0.896147i	$-0.353645\pi$
$683$	7.39005e9	0.887513	0.443757	+	0.896147i	$0.353645\pi$
$684$	4.50428e9	0.538182
$685$	−4.45480e8	−0.0529555
$686$	3.61142e10	4.27114
$687$	−1.76886e10	−2.08135
$688$	6.45503e8	0.0755682
$689$	0	0
$690$	1.05077e9	0.121769
$691$	−1.58357e10	−1.82584	−0.912921	−	0.408136i	$-0.866179\pi$
$691$	−1.58357e10	−1.82584	−0.912921	+	0.408136i	$0.866179\pi$
$692$	4.08358e9	0.468457
$693$	−9.37376e9	−1.06991
$694$	4.13492e9	0.469579
$695$	−4.66074e8	−0.0526633
$696$	−3.65915e9	−0.411385
$697$	−5.77694e9	−0.646224
$698$	1.59018e10	1.76991
$699$	−1.29178e10	−1.43060
$700$	−2.46849e10	−2.72012
$701$	−1.64091e10	−1.79917	−0.899586	−	0.436744i	$-0.856131\pi$
$701$	−1.64091e10	−1.79917	−0.899586	+	0.436744i	$0.856131\pi$
$702$	0	0
$703$	4.35981e9	0.473287
$704$	1.57903e10	1.70564
$705$	3.24499e8	0.0348780
$706$	6.80514e9	0.727814
$707$	−9.44079e9	−1.00471
$708$	−8.55848e9	−0.906317
$709$	−1.50021e10	−1.58084	−0.790422	−	0.612563i	$-0.790139\pi$
$709$	−1.50021e10	−1.58084	−0.790422	+	0.612563i	$0.790139\pi$
$710$	1.11307e9	0.116713
$711$	3.99884e9	0.417244
$712$	−1.35438e9	−0.140625
$713$	−8.21712e9	−0.848997
$714$	6.50453e10	6.68763
$715$	0	0
$716$	1.82309e10	1.85614
$717$	1.14530e10	1.16039
$718$	1.68695e10	1.70085
$719$	8.80929e9	0.883873	0.441936	−	0.897046i	$-0.354292\pi$
$719$	8.80929e9	0.883873	0.441936	+	0.897046i	$0.354292\pi$
$720$	−9.56519e7	−0.00955058
$721$	−1.99646e10	−1.98375
$722$	−8.65761e9	−0.856087
$723$	1.73601e10	1.70832
$724$	−1.73012e8	−0.0169431
$725$	4.62211e9	0.450461
$726$	2.35346e9	0.228260
$727$	−3.23272e9	−0.312031	−0.156015	−	0.987755i	$-0.549865\pi$
$727$	−3.23272e9	−0.312031	−0.156015	+	0.987755i	$0.549865\pi$
$728$	0	0
$729$	2.60190e8	0.0248739
$730$	7.03405e8	0.0669230
$731$	−4.65394e9	−0.440666
$732$	−1.99220e10	−1.87734
$733$	−1.26439e10	−1.18582	−0.592909	−	0.805269i	$-0.702021\pi$
$733$	−1.26439e10	−1.18582	−0.592909	+	0.805269i	$0.702021\pi$
$734$	−1.21978e10	−1.13853
$735$	1.82999e9	0.169998
$736$	−1.49214e10	−1.37955
$737$	5.75216e9	0.529291
$738$	−3.27385e9	−0.299820
$739$	−8.81222e9	−0.803211	−0.401606	−	0.915813i	$-0.631548\pi$
$739$	−8.81222e9	−0.803211	−0.401606	+	0.915813i	$0.631548\pi$
$740$	6.30425e8	0.0571903
$741$	0	0
$742$	−5.29209e10	−4.75569
$743$	−7.85295e8	−0.0702380	−0.0351190	−	0.999383i	$-0.511181\pi$
$743$	−7.85295e8	−0.0702380	−0.0351190	+	0.999383i	$0.511181\pi$
$744$	−7.73343e9	−0.688442
$745$	8.00611e7	0.00709373
$746$	−1.34166e10	−1.18320
$747$	3.37980e9	0.296667
$748$	−3.26745e10	−2.85465
$749$	2.24308e9	0.195056
$750$	−2.50221e9	−0.216575
$751$	3.05457e8	0.0263154	0.0131577	−	0.999913i	$-0.495812\pi$
$751$	3.05457e8	0.0263154	0.0131577	+	0.999913i	$0.495812\pi$
$752$	−1.86030e9	−0.159522
$753$	−5.25585e9	−0.448601
$754$	0	0
$755$	3.73788e8	0.0316090
$756$	−1.83637e10	−1.54573
$757$	1.57774e10	1.32191	0.660953	−	0.750428i	$-0.270152\pi$
$757$	1.57774e10	1.32191	0.660953	+	0.750428i	$0.270152\pi$
$758$	−2.20878e10	−1.84208
$759$	−1.77619e10	−1.47450
$760$	3.32341e8	0.0274623
$761$	9.23635e9	0.759721	0.379860	−	0.925044i	$-0.375972\pi$
$761$	9.23635e9	0.759721	0.379860	+	0.925044i	$0.375972\pi$
$762$	−2.87202e10	−2.35150
$763$	2.47017e10	2.01322
$764$	−9.56398e9	−0.775911
$765$	6.89629e8	0.0556930
$766$	−1.65645e10	−1.33161
$767$	0	0
$768$	−6.81758e9	−0.543083
$769$	−4.33833e9	−0.344017	−0.172009	−	0.985095i	$-0.555026\pi$
$769$	−4.33833e9	−0.344017	−0.172009	+	0.985095i	$0.555026\pi$
$770$	−2.17417e9	−0.171623
$771$	1.94751e10	1.53034
$772$	3.58176e10	2.80179
$773$	1.34479e9	0.104719	0.0523597	−	0.998628i	$-0.483326\pi$
$773$	1.34479e9	0.104719	0.0523597	+	0.998628i	$0.483326\pi$
$774$	−2.63743e9	−0.204450
$775$	9.76861e9	0.753836
$776$	−6.99300e9	−0.537214
$777$	2.12140e10	1.62237
$778$	−6.76054e8	−0.0514699
$779$	−3.12230e9	−0.236643
$780$	0	0
$781$	−1.88151e10	−1.41328
$782$	4.34309e10	3.24770
$783$	3.43850e9	0.255978
$784$	−1.04910e10	−0.777519
$785$	−3.36311e7	−0.00248140
$786$	−2.72170e10	−1.99923
$787$	−6.66337e9	−0.487284	−0.243642	−	0.969865i	$-0.578342\pi$
$787$	−6.66337e9	−0.487284	−0.243642	+	0.969865i	$0.578342\pi$
$788$	−1.03898e9	−0.0756425
$789$	2.54673e9	0.184592
$790$	9.27498e8	0.0669296
$791$	−6.20626e9	−0.445874
$792$	−5.89047e9	−0.421320
$793$	0	0
$794$	1.70152e10	1.20633
$795$	−1.59228e9	−0.112392
$796$	−4.00150e10	−2.81208
$797$	1.75682e10	1.22920	0.614601	−	0.788838i	$-0.289317\pi$
$797$	1.75682e10	1.22920	0.614601	+	0.788838i	$0.289317\pi$
$798$	3.51555e10	2.44897
$799$	1.34123e10	0.930232
$800$	1.77387e10	1.22492
$801$	−1.51898e9	−0.104433
$802$	−7.53909e9	−0.516070
$803$	−1.18902e10	−0.810369
$804$	−1.34492e10	−0.912643
$805$	1.71825e9	0.116091
$806$	0	0
$807$	2.06432e10	1.38268
$808$	−5.93259e9	−0.395644
$809$	−2.39984e10	−1.59354	−0.796770	−	0.604282i	$-0.793460\pi$
$809$	−2.39984e10	−1.59354	−0.796770	+	0.604282i	$0.793460\pi$
$810$	−1.64757e9	−0.108929
$811$	1.29949e10	0.855462	0.427731	−	0.903906i	$-0.359313\pi$
$811$	1.29949e10	0.855462	0.427731	+	0.903906i	$0.359313\pi$
$812$	−1.88095e10	−1.23291
$813$	−2.84689e10	−1.85804
$814$	−1.79231e10	−1.16474
$815$	4.92722e8	0.0318823
$816$	−1.12196e10	−0.722874
$817$	−2.51535e9	−0.161369
$818$	2.70159e10	1.72577
$819$	0	0
$820$	−4.51482e8	−0.0285951
$821$	−3.50357e9	−0.220958	−0.110479	−	0.993878i	$-0.535238\pi$
$821$	−3.50357e9	−0.220958	−0.110479	+	0.993878i	$0.535238\pi$
$822$	−2.96124e10	−1.85961
$823$	−8.24920e8	−0.0515837	−0.0257918	−	0.999667i	$-0.508211\pi$
$823$	−8.24920e8	−0.0515837	−0.0257918	+	0.999667i	$0.508211\pi$
$824$	−1.25457e10	−0.781180
$825$	2.11156e10	1.30923
$826$	−2.35381e10	−1.45325
$827$	−2.02050e10	−1.24220	−0.621098	−	0.783733i	$-0.713313\pi$
$827$	−2.02050e10	−1.24220	−0.621098	+	0.783733i	$0.713313\pi$
$828$	1.46340e10	0.895894
$829$	−2.88093e10	−1.75627	−0.878135	−	0.478412i	$-0.841212\pi$
$829$	−2.88093e10	−1.75627	−0.878135	+	0.478412i	$0.841212\pi$
$830$	7.83918e8	0.0475880
$831$	−1.65149e9	−0.0998327
$832$	0	0
$833$	7.56379e10	4.53400
$834$	−3.09813e10	−1.84935
$835$	−3.75709e8	−0.0223331
$836$	−1.76598e10	−1.04536
$837$	7.26710e9	0.428373
$838$	1.99931e10	1.17362
$839$	2.61892e10	1.53093	0.765466	−	0.643476i	$-0.222509\pi$
$839$	2.61892e10	1.53093	0.765466	+	0.643476i	$0.222509\pi$
$840$	1.61711e9	0.0941372
$841$	−1.37279e10	−0.795825
$842$	2.74085e10	1.58232
$843$	2.08080e10	1.19628
$844$	−3.29939e10	−1.88902
$845$	0	0
$846$	7.60091e9	0.431588
$847$	3.84844e9	0.217617
$848$	9.12828e9	0.514048
$849$	2.12896e10	1.19396
$850$	−5.16312e10	−2.88367
$851$	1.41646e10	0.787865
$852$	4.39919e10	2.43688
$853$	2.18805e9	0.120708	0.0603539	−	0.998177i	$-0.480777\pi$
$853$	2.18805e9	0.120708	0.0603539	+	0.998177i	$0.480777\pi$
$854$	−5.47907e10	−3.01026
$855$	3.72729e8	0.0203944
$856$	1.40955e9	0.0768109
$857$	1.81674e10	0.985959	0.492979	−	0.870041i	$-0.335908\pi$
$857$	1.81674e10	0.985959	0.492979	+	0.870041i	$0.335908\pi$
$858$	0	0
$859$	−1.48867e10	−0.801351	−0.400675	−	0.916220i	$-0.631225\pi$
$859$	−1.48867e10	−0.801351	−0.400675	+	0.916220i	$0.631225\pi$
$860$	−3.63717e8	−0.0194993
$861$	−1.51925e10	−0.811183
$862$	−8.76587e9	−0.466144
$863$	2.25805e10	1.19590	0.597951	−	0.801533i	$-0.295982\pi$
$863$	2.25805e10	1.19590	0.597951	+	0.801533i	$0.295982\pi$
$864$	1.31962e10	0.696069
$865$	3.37916e8	0.0177522
$866$	1.98640e10	1.03933
$867$	5.70456e10	2.97273
$868$	−3.97530e10	−2.06325
$869$	−1.56782e10	−0.810449
$870$	−9.51850e8	−0.0490062
$871$	0	0
$872$	1.55225e10	0.792783
$873$	−7.84283e9	−0.398954
$874$	2.34734e10	1.18929
$875$	−4.09167e9	−0.206478
$876$	2.78006e10	1.39730
$877$	2.06345e10	1.03299	0.516494	−	0.856291i	$-0.327237\pi$
$877$	2.06345e10	1.03299	0.516494	+	0.856291i	$0.327237\pi$
$878$	3.56147e9	0.177582
$879$	9.56514e8	0.0475040
$880$	3.75020e8	0.0185509
$881$	−2.26396e9	−0.111546	−0.0557729	−	0.998443i	$-0.517762\pi$
$881$	−2.26396e9	−0.111546	−0.0557729	+	0.998443i	$0.517762\pi$
$882$	4.28647e10	2.10359
$883$	−1.01551e10	−0.496387	−0.248193	−	0.968711i	$-0.579837\pi$
$883$	−1.01551e10	−0.496387	−0.248193	+	0.968711i	$0.579837\pi$
$884$	0	0
$885$	−7.08214e8	−0.0343450
$886$	5.32617e10	2.57275
$887$	7.46410e9	0.359124	0.179562	−	0.983747i	$-0.442532\pi$
$887$	7.46410e9	0.359124	0.179562	+	0.983747i	$0.442532\pi$
$888$	1.33309e10	0.638871
$889$	−4.69640e10	−2.24186
$890$	−3.52314e8	−0.0167519
$891$	2.78500e10	1.31902
$892$	−7.03244e9	−0.331764
$893$	7.24907e9	0.340645
$894$	5.32190e9	0.249107
$895$	1.50860e9	0.0703388
$896$	−5.23151e10	−2.42968
$897$	0	0
$898$	2.33456e10	1.07582
$899$	7.44355e9	0.341681
$900$	−1.73971e10	−0.795476
$901$	−6.58129e10	−2.99760
$902$	1.28357e10	0.582367
$903$	−1.22392e10	−0.553153
$904$	−3.90001e9	−0.175580
$905$	−1.43168e7	−0.000642058 0
$906$	2.48468e10	1.11000
$907$	7.98911e9	0.355527	0.177764	−	0.984073i	$-0.443114\pi$
$907$	7.98911e9	0.355527	0.177764	+	0.984073i	$0.443114\pi$
$908$	3.94906e10	1.75063
$909$	−6.65355e9	−0.293819
$910$	0	0
$911$	−1.26514e10	−0.554401	−0.277200	−	0.960812i	$-0.589407\pi$
$911$	−1.26514e10	−0.554401	−0.277200	+	0.960812i	$0.589407\pi$
$912$	−6.06395e9	−0.264712
$913$	−1.32511e10	−0.576242
$914$	3.89496e9	0.168730
$915$	−1.64854e9	−0.0711420
$916$	−5.71384e10	−2.45637
$917$	−4.45060e10	−1.90601
$918$	−3.84097e10	−1.63867
$919$	−2.37637e10	−1.00997	−0.504987	−	0.863127i	$-0.668503\pi$
$919$	−2.37637e10	−1.00997	−0.504987	+	0.863127i	$0.668503\pi$
$920$	1.07975e9	0.0457156
$921$	−5.28749e10	−2.23019
$922$	4.66090e9	0.195844
$923$	0	0
$924$	−8.59293e10	−3.58335
$925$	−1.68391e10	−0.699556
$926$	2.39861e10	0.992707
$927$	−1.40704e10	−0.580131
$928$	1.35166e10	0.555202
$929$	−9.87669e8	−0.0404163	−0.0202081	−	0.999796i	$-0.506433\pi$
$929$	−9.87669e8	−0.0404163	−0.0202081	+	0.999796i	$0.506433\pi$
$930$	−2.01169e9	−0.0820106
$931$	4.08805e10	1.66032
$932$	−4.17276e10	−1.68837
$933$	5.54703e10	2.23602
$934$	3.69611e10	1.48433
$935$	−2.70382e9	−0.108177
$936$	0	0
$937$	−2.77444e10	−1.10176	−0.550880	−	0.834584i	$-0.685708\pi$
$937$	−2.77444e10	−1.10176	−0.550880	+	0.834584i	$0.685708\pi$
$938$	−3.69889e10	−1.46340
$939$	3.65953e10	1.44243
$940$	1.04821e9	0.0411623
$941$	−4.22093e10	−1.65137	−0.825684	−	0.564132i	$-0.809211\pi$
$941$	−4.22093e10	−1.65137	−0.825684	+	0.564132i	$0.809211\pi$
$942$	−2.23556e9	−0.0871381
$943$	−1.01441e10	−0.393932
$944$	4.06007e9	0.157084
$945$	−1.51959e9	−0.0585755
$946$	1.03405e10	0.397122
$947$	5.20285e10	1.99075	0.995375	−	0.0960668i	$-0.0306263\pi$
$947$	5.20285e10	1.99075	0.995375	+	0.0960668i	$0.0306263\pi$
$948$	3.66574e10	1.39744
$949$	0	0
$950$	−2.79055e10	−1.05598
$951$	−4.43093e10	−1.67056
$952$	6.68390e10	2.51073
$953$	−5.60369e9	−0.209724	−0.104862	−	0.994487i	$-0.533440\pi$
$953$	−5.60369e9	−0.209724	−0.104862	+	0.994487i	$0.533440\pi$
$954$	−3.72968e10	−1.39076
$955$	−7.91419e8	−0.0294032
$956$	3.69959e10	1.36946
$957$	1.60898e10	0.593416
$958$	2.62166e10	0.963379
$959$	−4.84229e10	−1.77291
$960$	−3.05509e9	−0.111449
$961$	−1.17811e10	−0.428205
$962$	0	0
$963$	1.58085e9	0.0570425
$964$	5.60772e10	2.01612
$965$	2.96390e9	0.106174
$966$	1.14217e11	4.07672
$967$	2.36277e10	0.840289	0.420144	−	0.907457i	$-0.361979\pi$
$967$	2.36277e10	0.840289	0.420144	+	0.907457i	$0.361979\pi$
$968$	2.41836e9	0.0856953
$969$	4.37197e10	1.54363
$970$	−1.81908e9	−0.0639957
$971$	−4.26827e10	−1.49618	−0.748092	−	0.663595i	$-0.769030\pi$
$971$	−4.26827e10	−1.49618	−0.748092	+	0.663595i	$0.769030\pi$
$972$	−4.13305e10	−1.44357
$973$	−5.06615e10	−1.76312
$974$	2.42439e10	0.840709
$975$	0	0
$976$	9.45081e9	0.325383
$977$	−3.81913e10	−1.31019	−0.655094	−	0.755548i	$-0.727371\pi$
$977$	−3.81913e10	−1.31019	−0.655094	+	0.755548i	$0.727371\pi$
$978$	3.27527e10	1.11960
$979$	5.95542e9	0.202849
$980$	5.91129e9	0.200628
$981$	1.74089e10	0.588748
$982$	1.67274e10	0.563686
$983$	3.15628e10	1.05983	0.529917	−	0.848050i	$-0.322223\pi$
$983$	3.15628e10	1.05983	0.529917	+	0.848050i	$0.322223\pi$
$984$	−9.54697e9	−0.319435
$985$	−8.59756e7	−0.00286648
$986$	−3.93423e10	−1.30704
$987$	3.52725e10	1.16769
$988$	0	0
$989$	−8.17213e9	−0.268626
$990$	−1.53228e9	−0.0501897
$991$	5.55554e10	1.81329	0.906647	−	0.421890i	$-0.138633\pi$
$991$	5.55554e10	1.81329	0.906647	+	0.421890i	$0.138633\pi$
$992$	2.85668e10	0.929117
$993$	2.89162e10	0.937170
$994$	1.20989e11	3.90746
$995$	−3.31124e9	−0.106564
$996$	3.09827e10	0.993599
$997$	−3.64736e9	−0.116559	−0.0582795	−	0.998300i	$-0.518561\pi$
$997$	−3.64736e9	−0.116559	−0.0582795	+	0.998300i	$0.518561\pi$
$998$	7.90541e10	2.51749
$999$	−1.25270e10	−0.397528

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	169.8.a.i.1.18	yes	21
13.5	odd	4		169.8.b.f.168.6		42
13.8	odd	4		169.8.b.f.168.37		42
13.12	even	2		169.8.a.h.1.4	✓	21

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
169.8.a.h.1.4	✓	21	13.12	even	2
169.8.a.i.1.18	yes	21	1.1	even	1	trivial
169.8.b.f.168.6		42	13.5	odd	4
169.8.b.f.168.37		42	13.8	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)

\(f(q)\)	\(=\)	\(q+17.7683 q^{2} +58.1116 q^{3} +187.714 q^{4} +15.5333 q^{5} +1032.55 q^{6} +1688.45 q^{7} +1061.02 q^{8} +1189.96 q^{9} +276.002 q^{10} -4665.45 q^{11} +10908.4 q^{12} +30000.9 q^{14} +902.667 q^{15} -5174.84 q^{16} +37309.4 q^{17} +21143.6 q^{18} +20164.9 q^{19} +2915.82 q^{20} +98118.4 q^{21} -82897.4 q^{22} +65513.9 q^{23} +61657.6 q^{24} -77883.7 q^{25} -57939.6 q^{27} +316945. q^{28} -59346.3 q^{29} +16038.9 q^{30} -125426. q^{31} -227759. q^{32} -271117. q^{33} +662927. q^{34} +26227.2 q^{35} +223372. q^{36} +216208. q^{37} +358297. q^{38} +16481.2 q^{40} -154839. q^{41} +1.74340e6 q^{42} -124739. q^{43} -875771. q^{44} +18484.1 q^{45} +1.16407e6 q^{46} +359489. q^{47} -300718. q^{48} +2.02731e6 q^{49} -1.38386e6 q^{50} +2.16811e6 q^{51} -1.76397e6 q^{53} -1.02949e6 q^{54} -72470.0 q^{55} +1.79148e6 q^{56} +1.17181e6 q^{57} -1.05449e6 q^{58} -784579. q^{59} +169443. q^{60} -1.82630e6 q^{61} -2.22860e6 q^{62} +2.00919e6 q^{63} -3.38452e6 q^{64} -4.81730e6 q^{66} -1.23293e6 q^{67} +7.00350e6 q^{68} +3.80712e6 q^{69} +466014. q^{70} +4.03286e6 q^{71} +1.26257e6 q^{72} +2.54855e6 q^{73} +3.84166e6 q^{74} -4.52595e6 q^{75} +3.78523e6 q^{76} -7.87737e6 q^{77} +3.36048e6 q^{79} -80382.4 q^{80} -5.96941e6 q^{81} -2.75122e6 q^{82} +2.84027e6 q^{83} +1.84182e7 q^{84} +579540. q^{85} -2.21640e6 q^{86} -3.44871e6 q^{87} -4.95014e6 q^{88} -1.27649e6 q^{89} +328431. q^{90} +1.22979e7 q^{92} -7.28868e6 q^{93} +6.38753e6 q^{94} +313228. q^{95} -1.32354e7 q^{96} -6.59083e6 q^{97} +3.60220e7 q^{98} -5.55170e6 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 21 q + 31 q^{2} - 26 q^{3} + 1409 q^{4} + 680 q^{5} + 1470 q^{6} + 2929 q^{7} + 4716 q^{8} + 15465 q^{9} - 5167 q^{10} + 14824 q^{11} + 21795 q^{12} - 179 q^{14} + 36398 q^{15} + 113205 q^{16} + 45016 q^{17}+ \cdots + 37605493 q^{99}+O(q^{100}) \) 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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