LMFDB - Embedded newform 169.8.a.g.1.13

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 = [14,0] 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:	$1$
Dimension:	$14$
Coefficient field:	$\mathbb{Q}[x]/(x^{14} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{14} - 1279 x^{12} + 629380 x^{10} - 148562016 x^{8} + 16872573312 x^{6} - 790180980480 x^{4} + \cdots - 4669637050368 $ x^14 - 1279x^12 + 629380x^10 - 148562016x^8 + 16872573312x^6 - 790180980480x^4 + 10484997239808x^2 - 4669637050368
Coefficient ring:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$ 2^{13}\cdot 3^{3}\cdot 13^{6} $
Twist minimal:	no (minimal twist has level 13)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.13
Root			$16.7657$ of defining polynomial
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)

$f(q)$	$=$	\(q+16.7657 q^{2} -81.0229 q^{3} +153.090 q^{4} -223.318 q^{5} -1358.41 q^{6} +666.907 q^{7} +420.649 q^{8} +4377.70 q^{9} -3744.09 q^{10} +7590.80 q^{11} -12403.8 q^{12} +11181.2 q^{14} +18093.9 q^{15} -12543.0 q^{16} -14102.9 q^{17} +73395.4 q^{18} -8343.98 q^{19} -34187.7 q^{20} -54034.7 q^{21} +127265. q^{22} +58145.3 q^{23} -34082.2 q^{24} -28254.0 q^{25} -177497. q^{27} +102097. q^{28} -33045.0 q^{29} +303357. q^{30} -91450.2 q^{31} -264136. q^{32} -615028. q^{33} -236445. q^{34} -148932. q^{35} +670182. q^{36} -355545. q^{37} -139893. q^{38} -93938.5 q^{40} -582075. q^{41} -905932. q^{42} -104498. q^{43} +1.16207e6 q^{44} -977620. q^{45} +974848. q^{46} +398024. q^{47} +1.01627e6 q^{48} -378778. q^{49} -473700. q^{50} +1.14266e6 q^{51} +1.49881e6 q^{53} -2.97587e6 q^{54} -1.69516e6 q^{55} +280534. q^{56} +676053. q^{57} -554024. q^{58} -330628. q^{59} +2.76999e6 q^{60} -1.44273e6 q^{61} -1.53323e6 q^{62} +2.91952e6 q^{63} -2.82292e6 q^{64} -1.03114e7 q^{66} -2.56680e6 q^{67} -2.15901e6 q^{68} -4.71110e6 q^{69} -2.49696e6 q^{70} +1.29773e6 q^{71} +1.84148e6 q^{72} -2.66195e6 q^{73} -5.96097e6 q^{74} +2.28922e6 q^{75} -1.27738e6 q^{76} +5.06236e6 q^{77} +2.13906e6 q^{79} +2.80108e6 q^{80} +4.80728e6 q^{81} -9.75892e6 q^{82} -1.63158e6 q^{83} -8.27217e6 q^{84} +3.14943e6 q^{85} -1.75198e6 q^{86} +2.67740e6 q^{87} +3.19306e6 q^{88} -401206. q^{89} -1.63905e7 q^{90} +8.90145e6 q^{92} +7.40955e6 q^{93} +6.67316e6 q^{94} +1.86336e6 q^{95} +2.14010e7 q^{96} -1.10423e7 q^{97} -6.35048e6 q^{98} +3.32303e7 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 14 q - 52 q^{3} + 766 q^{4} + 6982 q^{9} - 1018 q^{10} - 38380 q^{12} - 47916 q^{14} + 1266 q^{16} - 76806 q^{17} + 251764 q^{22} - 137100 q^{23} - 39380 q^{25} - 432400 q^{27} - 443166 q^{29} - 315780 q^{30}+ \cdots - 86840772 q^{95}+O(q^{100}) $ 14 * q - 52 * q^3 + 766 * q^4 + 6982 * q^9 - 1018 * q^10 - 38380 * q^12 - 47916 * q^14 + 1266 * q^16 - 76806 * q^17 + 251764 * q^22 - 137100 * q^23 - 39380 * q^25 - 432400 * q^27 - 443166 * q^29 - 315780 * q^30 - 1319232 * q^35 + 1271350 * q^36 - 953640 * q^38 - 77182 * q^40 - 412812 * q^42 - 1765384 * q^43 - 2267260 * q^48 + 5307958 * q^49 + 38964 * q^51 - 1791210 * q^53 - 4907072 * q^55 - 14915904 * q^56 - 7717050 * q^61 + 3808284 * q^62 - 14570922 * q^64 - 31956396 * q^66 - 19139238 * q^68 - 27487680 * q^69 - 17280210 * q^74 + 20632304 * q^75 - 19599336 * q^77 - 30688440 * q^79 - 27772442 * q^81 - 74719438 * q^82 - 25539372 * q^87 + 54091072 * q^88 - 41193546 * q^90 - 47685588 * q^92 - 38277620 * q^94 - 86840772 * q^95

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 $

$2$	16.7657	1.48190	0.740948	−	0.671563i	$-0.234377\pi$
$2$	16.7657	1.48190	0.740948	+	0.671563i	$0.234377\pi$
$3$	−81.0229	−1.73254	−0.866270	−	0.499576i	$-0.833489\pi$
$3$	−81.0229	−1.73254	−0.866270	+	0.499576i	$0.833489\pi$
$4$	153.090	1.19601
$5$	−223.318	−0.798967	−0.399483	−	0.916740i	$-0.630811\pi$
$5$	−223.318	−0.798967	−0.399483	+	0.916740i	$0.630811\pi$
$6$	−1358.41	−2.56744
$7$	666.907	0.734890	0.367445	−	0.930045i	$-0.380233\pi$
$7$	666.907	0.734890	0.367445	+	0.930045i	$0.380233\pi$
$8$	420.649	0.290472
$9$	4377.70	2.00169
$10$	−3744.09	−1.18399
$11$	7590.80	1.71954	0.859771	−	0.510679i	$-0.170606\pi$
$11$	7590.80	1.71954	0.859771	+	0.510679i	$0.170606\pi$
$12$	−12403.8	−2.07214
$13$	0	0
$13$	0	0
$14$	11181.2	1.08903
$15$	18093.9	1.38424
$16$	−12543.0	−0.765564
$17$	−14102.9	−0.696204	−0.348102	−	0.937457i	$-0.613174\pi$
$17$	−14102.9	−0.696204	−0.348102	+	0.937457i	$0.613174\pi$
$18$	73395.4	2.96630
$19$	−8343.98	−0.279085	−0.139542	−	0.990216i	$-0.544563\pi$
$19$	−8343.98	−0.279085	−0.139542	+	0.990216i	$0.544563\pi$
$20$	−34187.7	−0.955576
$21$	−54034.7	−1.27323
$22$	127265.	2.54818
$23$	58145.3	0.996476	0.498238	−	0.867040i	$-0.333981\pi$
$23$	58145.3	0.996476	0.498238	+	0.867040i	$0.333981\pi$
$24$	−34082.2	−0.503255
$25$	−28254.0	−0.361652
$26$	0	0
$27$	−177497.	−1.73547
$28$	102097.	0.878939
$29$	−33045.0	−0.251601	−0.125801	−	0.992056i	$-0.540150\pi$
$29$	−33045.0	−0.251601	−0.125801	+	0.992056i	$0.540150\pi$
$30$	303357.	2.05130
$31$	−91450.2	−0.551339	−0.275669	−	0.961252i	$-0.588900\pi$
$31$	−91450.2	−0.551339	−0.275669	+	0.961252i	$0.588900\pi$
$32$	−264136.	−1.42496
$33$	−615028.	−2.97918
$34$	−236445.	−1.03170
$35$	−148932.	−0.587153
$36$	670182.	2.39405
$37$	−355545.	−1.15395	−0.576977	−	0.816760i	$-0.695768\pi$
$37$	−355545.	−1.15395	−0.576977	+	0.816760i	$0.695768\pi$
$38$	−139893.	−0.413574
$39$	0	0
$40$	−93938.5	−0.232078
$41$	−582075.	−1.31897	−0.659486	−	0.751717i	$-0.729226\pi$
$41$	−582075.	−1.31897	−0.659486	+	0.751717i	$0.729226\pi$
$42$	−905932.	−1.88679
$43$	−104498.	−0.200432	−0.100216	−	0.994966i	$-0.531953\pi$
$43$	−104498.	−0.200432	−0.100216	+	0.994966i	$0.531953\pi$
$44$	1.16207e6	2.05660
$45$	−977620.	−1.59929
$46$	974848.	1.47667
$47$	398024.	0.559199	0.279600	−	0.960117i	$-0.409798\pi$
$47$	398024.	0.559199	0.279600	+	0.960117i	$0.409798\pi$
$48$	1.01627e6	1.32637
$49$	−378778.	−0.459937
$50$	−473700.	−0.535930
$51$	1.14266e6	1.20620
$52$	0	0
$53$	1.49881e6	1.38287	0.691433	−	0.722441i	$-0.256980\pi$
$53$	1.49881e6	1.38287	0.691433	+	0.722441i	$0.256980\pi$
$54$	−2.97587e6	−2.57179
$55$	−1.69516e6	−1.37386
$56$	280534.	0.213465
$57$	676053.	0.483525
$58$	−554024.	−0.372847
$59$	−330628.	−0.209584	−0.104792	−	0.994494i	$-0.533418\pi$
$59$	−330628.	−0.209584	−0.104792	+	0.994494i	$0.533418\pi$
$60$	2.76999e6	1.65557
$61$	−1.44273e6	−0.813825	−0.406912	−	0.913467i	$-0.633395\pi$
$61$	−1.44273e6	−0.813825	−0.406912	+	0.913467i	$0.633395\pi$
$62$	−1.53323e6	−0.817027
$63$	2.91952e6	1.47102
$64$	−2.82292e6	−1.34608
$65$	0	0
$66$	−1.03114e7	−4.41483
$67$	−2.56680e6	−1.04263	−0.521316	−	0.853364i	$-0.674559\pi$
$67$	−2.56680e6	−1.04263	−0.521316	+	0.853364i	$0.674559\pi$
$68$	−2.15901e6	−0.832670
$69$	−4.71110e6	−1.72643
$70$	−2.49696e6	−0.870099
$71$	1.29773e6	0.430307	0.215154	−	0.976580i	$-0.430975\pi$
$71$	1.29773e6	0.430307	0.215154	+	0.976580i	$0.430975\pi$
$72$	1.84148e6	0.581437
$73$	−2.66195e6	−0.800885	−0.400442	−	0.916322i	$-0.631144\pi$
$73$	−2.66195e6	−0.800885	−0.400442	+	0.916322i	$0.631144\pi$
$74$	−5.96097e6	−1.71004
$75$	2.28922e6	0.626576
$76$	−1.27738e6	−0.333789
$77$	5.06236e6	1.26367
$78$	0	0
$79$	2.13906e6	0.488121	0.244060	−	0.969760i	$-0.421520\pi$
$79$	2.13906e6	0.488121	0.244060	+	0.969760i	$0.421520\pi$
$80$	2.80108e6	0.611661
$81$	4.80728e6	1.00508
$82$	−9.75892e6	−1.95458
$83$	−1.63158e6	−0.313211	−0.156605	−	0.987661i	$-0.550055\pi$
$83$	−1.63158e6	−0.313211	−0.156605	+	0.987661i	$0.550055\pi$
$84$	−8.27217e6	−1.52280
$85$	3.14943e6	0.556244
$86$	−1.75198e6	−0.297020
$87$	2.67740e6	0.435909
$88$	3.19306e6	0.499480
$89$	−401206.	−0.0603258	−0.0301629	−	0.999545i	$-0.509603\pi$
$89$	−401206.	−0.0603258	−0.0301629	+	0.999545i	$0.509603\pi$
$90$	−1.63905e7	−2.36998
$91$	0	0
$92$	8.90145e6	1.19180
$93$	7.40955e6	0.955216
$94$	6.67316e6	0.828675
$95$	1.86336e6	0.222979
$96$	2.14010e7	2.46880
$97$	−1.10423e7	−1.22846	−0.614228	−	0.789128i	$-0.710533\pi$
$97$	−1.10423e7	−1.22846	−0.614228	+	0.789128i	$0.710533\pi$
$98$	−6.35048e6	−0.681578
$99$	3.32303e7	3.44200
$100$	−4.32541e6	−0.432541
$101$	7.68319e6	0.742022	0.371011	−	0.928629i	$-0.379011\pi$
$101$	7.68319e6	0.742022	0.371011	+	0.928629i	$0.379011\pi$
$102$	1.91575e7	1.78746
$103$	−485826.	−0.0438077	−0.0219039	−	0.999760i	$-0.506973\pi$
$103$	−485826.	−0.0438077	−0.0219039	+	0.999760i	$0.506973\pi$
$104$	0	0
$105$	1.20669e7	1.01727
$106$	2.51286e7	2.04926
$107$	−69974.3	−0.00552198	−0.00276099	−	0.999996i	$-0.500879\pi$
$107$	−69974.3	−0.00552198	−0.00276099	+	0.999996i	$0.500879\pi$
$108$	−2.71730e7	−2.07565
$109$	−4.06366e6	−0.300556	−0.150278	−	0.988644i	$-0.548017\pi$
$109$	−4.06366e6	−0.300556	−0.150278	+	0.988644i	$0.548017\pi$
$110$	−2.84206e7	−2.03591
$111$	2.88073e7	1.99927
$112$	−8.36502e6	−0.562606
$113$	−2.09954e7	−1.36883	−0.684417	−	0.729091i	$-0.739943\pi$
$113$	−2.09954e7	−1.36883	−0.684417	+	0.729091i	$0.739943\pi$
$114$	1.13345e7	0.716534
$115$	−1.29849e7	−0.796152
$116$	−5.05885e6	−0.300919
$117$	0	0
$118$	−5.54322e6	−0.310581
$119$	−9.40531e6	−0.511633
$120$	7.61117e6	0.402084
$121$	3.81330e7	1.95683
$122$	−2.41884e7	−1.20600
$123$	4.71614e7	2.28517
$124$	−1.40001e7	−0.659409
$125$	2.37564e7	1.08791
$126$	4.89479e7	2.17990
$127$	9.33567e6	0.404420	0.202210	−	0.979342i	$-0.435188\pi$
$127$	9.33567e6	0.404420	0.202210	+	0.979342i	$0.435188\pi$
$128$	−1.35190e7	−0.569784
$129$	8.46671e6	0.347257
$130$	0	0
$131$	−4.07546e7	−1.58390	−0.791948	−	0.610588i	$-0.790933\pi$
$131$	−4.07546e7	−1.58390	−0.791948	+	0.610588i	$0.790933\pi$
$132$	−9.41545e7	−3.56314
$133$	−5.56466e6	−0.205096
$134$	−4.30343e7	−1.54507
$135$	3.96383e7	1.38659
$136$	−5.93236e6	−0.202228
$137$	−4.87238e7	−1.61890	−0.809449	−	0.587190i	$-0.800234\pi$
$137$	−4.87238e7	−1.61890	−0.809449	+	0.587190i	$0.800234\pi$
$138$	−7.89850e7	−2.55840
$139$	−5.32895e7	−1.68302	−0.841511	−	0.540240i	$-0.818334\pi$
$139$	−5.32895e7	−1.68302	−0.841511	+	0.540240i	$0.818334\pi$
$140$	−2.28000e7	−0.702243
$141$	−3.22490e7	−0.968835
$142$	2.17573e7	0.637670
$143$	0	0
$144$	−5.49096e7	−1.53243
$145$	7.37955e6	0.201021
$146$	−4.46296e7	−1.18683
$147$	3.06896e7	0.796858
$148$	−5.44303e7	−1.38015
$149$	−3.62391e7	−0.897480	−0.448740	−	0.893662i	$-0.648127\pi$
$149$	−3.62391e7	−0.897480	−0.448740	+	0.893662i	$0.648127\pi$
$150$	3.83805e7	0.928520
$151$	2.55676e7	0.604326	0.302163	−	0.953256i	$-0.402291\pi$
$151$	2.55676e7	0.604326	0.302163	+	0.953256i	$0.402291\pi$
$152$	−3.50989e6	−0.0810664
$153$	−6.17382e7	−1.39359
$154$	8.48741e7	1.87263
$155$	2.04225e7	0.440502
$156$	0	0
$157$	4.23295e7	0.872960	0.436480	−	0.899714i	$-0.356225\pi$
$157$	4.23295e7	0.872960	0.436480	+	0.899714i	$0.356225\pi$
$158$	3.58629e7	0.723344
$159$	−1.21438e8	−2.39587
$160$	5.89863e7	1.13849
$161$	3.87775e7	0.732301
$162$	8.05976e7	1.48943
$163$	3.44898e7	0.623784	0.311892	−	0.950118i	$-0.399037\pi$
$163$	3.44898e7	0.623784	0.311892	+	0.950118i	$0.399037\pi$
$164$	−8.91098e7	−1.57751
$165$	1.37347e8	2.38026
$166$	−2.73547e7	−0.464145
$167$	8.00957e6	0.133077	0.0665383	−	0.997784i	$-0.478805\pi$
$167$	8.00957e6	0.133077	0.0665383	+	0.997784i	$0.478805\pi$
$168$	−2.27297e7	−0.369837
$169$	0	0
$170$	5.28024e7	0.824295
$171$	−3.65275e7	−0.558642
$172$	−1.59975e7	−0.239720
$173$	3.73141e7	0.547912	0.273956	−	0.961742i	$-0.411668\pi$
$173$	3.73141e7	0.547912	0.273956	+	0.961742i	$0.411668\pi$
$174$	4.48886e7	0.645972
$175$	−1.88428e7	−0.265774
$176$	−9.52114e7	−1.31642
$177$	2.67884e7	0.363112
$178$	−6.72652e6	−0.0893965
$179$	−6.85207e7	−0.892968	−0.446484	−	0.894792i	$-0.647324\pi$
$179$	−6.85207e7	−0.892968	−0.446484	+	0.894792i	$0.647324\pi$
$180$	−1.49664e8	−1.91277
$181$	1.20860e8	1.51498	0.757489	−	0.652848i	$-0.226426\pi$
$181$	1.20860e8	1.51498	0.757489	+	0.652848i	$0.226426\pi$
$182$	0	0
$183$	1.16894e8	1.40998
$184$	2.44588e7	0.289449
$185$	7.93996e7	0.921972
$186$	1.24227e8	1.41553
$187$	−1.07052e8	−1.19715
$188$	6.09334e7	0.668810
$189$	−1.18374e8	−1.27538
$190$	3.12406e7	0.330432
$191$	−8.40445e7	−0.872755	−0.436377	−	0.899764i	$-0.643739\pi$
$191$	−8.40445e7	−0.872755	−0.436377	+	0.899764i	$0.643739\pi$
$192$	2.28721e8	2.33213
$193$	−4.45097e7	−0.445661	−0.222830	−	0.974857i	$-0.571530\pi$
$193$	−4.45097e7	−0.445661	−0.222830	+	0.974857i	$0.571530\pi$
$194$	−1.85133e8	−1.82044
$195$	0	0
$196$	−5.79870e7	−0.550091
$197$	1.26502e8	1.17887	0.589437	−	0.807815i	$-0.299350\pi$
$197$	1.26502e8	1.17887	0.589437	+	0.807815i	$0.299350\pi$
$198$	5.57130e8	5.10068
$199$	−9.62184e7	−0.865511	−0.432755	−	0.901511i	$-0.642459\pi$
$199$	−9.62184e7	−0.865511	−0.432755	+	0.901511i	$0.642459\pi$
$200$	−1.18850e7	−0.105050
$201$	2.07970e8	1.80640
$202$	1.28814e8	1.09960
$203$	−2.20380e7	−0.184899
$204$	1.74929e8	1.44263
$205$	1.29988e8	1.05382
$206$	−8.14523e6	−0.0649185
$207$	2.54543e8	1.99464
$208$	0	0
$209$	−6.33375e7	−0.479898
$210$	2.02311e8	1.50748
$211$	−5.03533e7	−0.369011	−0.184505	−	0.982832i	$-0.559068\pi$
$211$	−5.03533e7	−0.369011	−0.184505	+	0.982832i	$0.559068\pi$
$212$	2.29452e8	1.65393
$213$	−1.05145e8	−0.745524
$214$	−1.17317e6	−0.00818300
$215$	2.33362e7	0.160139
$216$	−7.46640e7	−0.504107
$217$	−6.09888e7	−0.405173
$218$	−6.81303e7	−0.445392
$219$	2.15679e8	1.38756
$220$	−2.59512e8	−1.64315
$221$	0	0
$222$	4.82975e8	2.96271
$223$	2.45161e8	1.48042	0.740208	−	0.672378i	$-0.234727\pi$
$223$	2.45161e8	1.48042	0.740208	+	0.672378i	$0.234727\pi$
$224$	−1.76154e8	−1.04719
$225$	−1.23688e8	−0.723916
$226$	−3.52004e8	−2.02847
$227$	−1.90894e6	−0.0108318	−0.00541591	−	0.999985i	$-0.501724\pi$
$227$	−1.90894e6	−0.0108318	−0.00541591	+	0.999985i	$0.501724\pi$
$228$	1.03497e8	0.578303
$229$	−2.07407e8	−1.14130	−0.570650	−	0.821194i	$-0.693309\pi$
$229$	−2.07407e8	−1.14130	−0.570650	+	0.821194i	$0.693309\pi$
$230$	−2.17701e8	−1.17981
$231$	−4.10167e8	−2.18937
$232$	−1.39004e7	−0.0730832
$233$	−3.42955e8	−1.77620	−0.888099	−	0.459652i	$-0.847974\pi$
$233$	−3.42955e8	−1.77620	−0.888099	+	0.459652i	$0.847974\pi$
$234$	0	0
$235$	−8.88859e7	−0.446782
$236$	−5.06158e7	−0.250665
$237$	−1.73312e8	−0.845689
$238$	−1.57687e8	−0.758187
$239$	3.94963e8	1.87139	0.935693	−	0.352815i	$-0.114775\pi$
$239$	3.94963e8	1.87139	0.935693	+	0.352815i	$0.114775\pi$
$240$	−2.26952e8	−1.05973
$241$	−7.77374e7	−0.357742	−0.178871	−	0.983873i	$-0.557245\pi$
$241$	−7.77374e7	−0.357742	−0.178871	+	0.983873i	$0.557245\pi$
$242$	6.39328e8	2.89981
$243$	−1.31373e6	−0.00587334
$244$	−2.20867e8	−0.973346
$245$	8.45879e7	0.367474
$246$	7.90695e8	3.38638
$247$	0	0
$248$	−3.84684e7	−0.160149
$249$	1.32196e8	0.542650
$250$	3.98293e8	1.61218
$251$	−3.50877e8	−1.40054	−0.700272	−	0.713877i	$-0.746938\pi$
$251$	−3.50877e8	−1.40054	−0.700272	+	0.713877i	$0.746938\pi$
$252$	4.46949e8	1.75937
$253$	4.41369e8	1.71348
$254$	1.56519e8	0.599308
$255$	−2.55176e8	−0.963715
$256$	1.34678e8	0.501715
$257$	−3.59200e8	−1.31999	−0.659995	−	0.751270i	$-0.729442\pi$
$257$	−3.59200e8	−1.31999	−0.659995	+	0.751270i	$0.729442\pi$
$258$	1.41951e8	0.514598
$259$	−2.37116e8	−0.848030
$260$	0	0
$261$	−1.44661e8	−0.503629
$262$	−6.83280e8	−2.34717
$263$	−1.99512e8	−0.676277	−0.338139	−	0.941096i	$-0.609797\pi$
$263$	−1.99512e8	−0.676277	−0.338139	+	0.941096i	$0.609797\pi$
$264$	−2.58711e8	−0.865368
$265$	−3.34711e8	−1.10486
$266$	−9.32957e7	−0.303932
$267$	3.25069e7	0.104517
$268$	−3.92951e8	−1.24700
$269$	−6.44857e7	−0.201990	−0.100995	−	0.994887i	$-0.532203\pi$
$269$	−6.44857e7	−0.201990	−0.100995	+	0.994887i	$0.532203\pi$
$270$	6.64565e8	2.05478
$271$	−4.79494e7	−0.146349	−0.0731747	−	0.997319i	$-0.523313\pi$
$271$	−4.79494e7	−0.146349	−0.0731747	+	0.997319i	$0.523313\pi$
$272$	1.76892e8	0.532989
$273$	0	0
$274$	−8.16891e8	−2.39904
$275$	−2.14471e8	−0.621876
$276$	−7.21221e8	−2.06484
$277$	5.93018e8	1.67644	0.838221	−	0.545331i	$-0.183596\pi$
$277$	5.93018e8	1.67644	0.838221	+	0.545331i	$0.183596\pi$
$278$	−8.93438e8	−2.49406
$279$	−4.00342e8	−1.10361
$280$	−6.26483e7	−0.170552
$281$	1.74758e8	0.469856	0.234928	−	0.972013i	$-0.424515\pi$
$281$	1.74758e8	0.469856	0.234928	+	0.972013i	$0.424515\pi$
$282$	−5.40679e8	−1.43571
$283$	6.61117e8	1.73391	0.866954	−	0.498389i	$-0.166075\pi$
$283$	6.61117e8	1.73391	0.866954	+	0.498389i	$0.166075\pi$
$284$	1.98669e8	0.514653
$285$	−1.50975e8	−0.386321
$286$	0	0
$287$	−3.88190e8	−0.969299
$288$	−1.15631e9	−2.85233
$289$	−2.11448e8	−0.515300
$290$	1.23724e8	0.297892
$291$	8.94681e8	2.12835
$292$	−4.07518e8	−0.957869
$293$	5.36743e8	1.24661	0.623304	−	0.781980i	$-0.285790\pi$
$293$	5.36743e8	1.24661	0.623304	+	0.781980i	$0.285790\pi$
$294$	5.14534e8	1.18086
$295$	7.38352e7	0.167451
$296$	−1.49560e8	−0.335192
$297$	−1.34734e9	−2.98422
$298$	−6.07574e8	−1.32997
$299$	0	0
$300$	3.50457e8	0.749394
$301$	−6.96903e7	−0.147296
$302$	4.28660e8	0.895548
$303$	−6.22514e8	−1.28558
$304$	1.04659e8	0.213657
$305$	3.22188e8	0.650219
$306$	−1.03509e9	−2.06515
$307$	3.58135e8	0.706419	0.353209	−	0.935544i	$-0.385090\pi$
$307$	3.58135e8	0.706419	0.353209	+	0.935544i	$0.385090\pi$
$308$	7.74995e8	1.51137
$309$	3.93630e7	0.0758986
$310$	3.42398e8	0.652777
$311$	−4.95093e8	−0.933309	−0.466655	−	0.884440i	$-0.654541\pi$
$311$	−4.95093e8	−0.933309	−0.466655	+	0.884440i	$0.654541\pi$
$312$	0	0
$313$	−3.14797e8	−0.580263	−0.290131	−	0.956987i	$-0.593699\pi$
$313$	−3.14797e8	−0.580263	−0.290131	+	0.956987i	$0.593699\pi$
$314$	7.09685e8	1.29364
$315$	−6.51982e8	−1.17530
$316$	3.27468e8	0.583800
$317$	6.24177e8	1.10053	0.550263	−	0.834991i	$-0.314527\pi$
$317$	6.24177e8	1.10053	0.550263	+	0.834991i	$0.314527\pi$
$318$	−2.03599e9	−3.55043
$319$	−2.50838e8	−0.432639
$320$	6.30410e8	1.07547
$321$	5.66952e6	0.00956706
$322$	6.50133e8	1.08519
$323$	1.17674e8	0.194300
$324$	7.35946e8	1.20209
$325$	0	0
$326$	5.78247e8	0.924382
$327$	3.29250e8	0.520725
$328$	−2.44849e8	−0.383125
$329$	2.65445e8	0.410950
$330$	2.30272e9	3.52730
$331$	−5.03800e8	−0.763590	−0.381795	−	0.924247i	$-0.624694\pi$
$331$	−5.03800e8	−0.763590	−0.381795	+	0.924247i	$0.624694\pi$
$332$	−2.49779e8	−0.374604
$333$	−1.55647e9	−2.30986
$334$	1.34286e8	0.197206
$335$	5.73213e8	0.833028
$336$	6.77758e8	0.974737
$337$	9.64189e8	1.37233	0.686164	−	0.727447i	$-0.259293\pi$
$337$	9.64189e8	1.37233	0.686164	+	0.727447i	$0.259293\pi$
$338$	0	0
$339$	1.70111e9	2.37156
$340$	4.82145e8	0.665276
$341$	−6.94180e8	−0.948051
$342$	−6.12410e8	−0.827849
$343$	−8.01836e8	−1.07289
$344$	−4.39569e7	−0.0582200
$345$	1.05207e9	1.37936
$346$	6.25598e8	0.811949
$347$	4.60468e8	0.591624	0.295812	−	0.955246i	$-0.404410\pi$
$347$	4.60468e8	0.591624	0.295812	+	0.955246i	$0.404410\pi$
$348$	4.09883e8	0.521354
$349$	−2.44565e8	−0.307967	−0.153984	−	0.988073i	$-0.549210\pi$
$349$	−2.44565e8	−0.307967	−0.153984	+	0.988073i	$0.549210\pi$
$350$	−3.15914e8	−0.393850
$351$	0	0
$352$	−2.00500e9	−2.45028
$353$	−5.42061e8	−0.655899	−0.327949	−	0.944695i	$-0.606358\pi$
$353$	−5.42061e8	−0.655899	−0.327949	+	0.944695i	$0.606358\pi$
$354$	4.49128e8	0.538094
$355$	−2.89806e8	−0.343801
$356$	−6.14206e7	−0.0721505
$357$	7.62045e8	0.886425
$358$	−1.14880e9	−1.32329
$359$	−5.16223e8	−0.588853	−0.294427	−	0.955674i	$-0.595129\pi$
$359$	−5.16223e8	−0.588853	−0.294427	+	0.955674i	$0.595129\pi$
$360$	−4.11235e8	−0.464549
$361$	−8.24250e8	−0.922112
$362$	2.02630e9	2.24504
$363$	−3.08965e9	−3.39028
$364$	0	0
$365$	5.94462e8	0.639880
$366$	1.95982e9	2.08945
$367$	1.03213e9	1.08994	0.544969	−	0.838456i	$-0.316541\pi$
$367$	1.03213e9	1.08994	0.544969	+	0.838456i	$0.316541\pi$
$368$	−7.29317e8	−0.762867
$369$	−2.54815e9	−2.64018
$370$	1.33119e9	1.36627
$371$	9.99565e8	1.01625
$372$	1.13433e9	1.14245
$373$	−1.80008e9	−1.79602	−0.898012	−	0.439971i	$-0.854989\pi$
$373$	−1.80008e9	−1.79602	−0.898012	+	0.439971i	$0.854989\pi$
$374$	−1.79481e9	−1.77405
$375$	−1.92481e9	−1.88486
$376$	1.67428e8	0.162432
$377$	0	0
$378$	−1.98463e9	−1.88998
$379$	−7.94428e8	−0.749579	−0.374790	−	0.927110i	$-0.622285\pi$
$379$	−7.94428e8	−0.749579	−0.374790	+	0.927110i	$0.622285\pi$
$380$	2.85262e8	0.266686
$381$	−7.56403e8	−0.700674
$382$	−1.40907e9	−1.29333
$383$	−1.89190e9	−1.72069	−0.860346	−	0.509710i	$-0.829753\pi$
$383$	−1.89190e9	−1.72069	−0.860346	+	0.509710i	$0.829753\pi$
$384$	1.09535e9	0.987174
$385$	−1.13052e9	−1.00963
$386$	−7.46239e8	−0.660423
$387$	−4.57460e8	−0.401204
$388$	−1.69047e9	−1.46925
$389$	9.79194e8	0.843422	0.421711	−	0.906730i	$-0.361430\pi$
$389$	9.79194e8	0.843422	0.421711	+	0.906730i	$0.361430\pi$
$390$	0	0
$391$	−8.20016e8	−0.693751
$392$	−1.59332e8	−0.133599
$393$	3.30205e9	2.74416
$394$	2.12091e9	1.74697
$395$	−4.77690e8	−0.389993
$396$	5.08721e9	4.11668
$397$	−1.17845e9	−0.945244	−0.472622	−	0.881265i	$-0.656692\pi$
$397$	−1.17845e9	−0.945244	−0.472622	+	0.881265i	$0.656692\pi$
$398$	−1.61317e9	−1.28260
$399$	4.50865e8	0.355338
$400$	3.54391e8	0.276868
$401$	9.42929e8	0.730254	0.365127	−	0.930958i	$-0.381026\pi$
$401$	9.42929e8	0.730254	0.365127	+	0.930958i	$0.381026\pi$
$402$	3.48676e9	2.67690
$403$	0	0
$404$	1.17622e9	0.887468
$405$	−1.07355e9	−0.803029
$406$	−3.69483e8	−0.274001
$407$	−2.69887e9	−1.98427
$408$	4.80657e8	0.350368
$409$	1.96849e9	1.42266	0.711331	−	0.702858i	$-0.248093\pi$
$409$	1.96849e9	1.42266	0.711331	+	0.702858i	$0.248093\pi$
$410$	2.17934e9	1.56164
$411$	3.94774e9	2.80480
$412$	−7.43751e7	−0.0523947
$413$	−2.20498e8	−0.154021
$414$	4.26760e9	2.95585
$415$	3.64362e8	0.250245
$416$	0	0
$417$	4.31767e9	2.91590
$418$	−1.06190e9	−0.711159
$419$	−1.95673e9	−1.29952	−0.649758	−	0.760141i	$-0.725130\pi$
$419$	−1.95673e9	−1.29952	−0.649758	+	0.760141i	$0.725130\pi$
$420$	1.84732e9	1.21666
$421$	2.92602e8	0.191113	0.0955563	−	0.995424i	$-0.469537\pi$
$421$	2.92602e8	0.191113	0.0955563	+	0.995424i	$0.469537\pi$
$422$	−8.44209e8	−0.546835
$423$	1.74243e9	1.11935
$424$	6.30471e8	0.401684
$425$	3.98463e8	0.251783
$426$	−1.76284e9	−1.10479
$427$	−9.62168e8	−0.598072
$428$	−1.07123e7	−0.00660437
$429$	0	0
$430$	3.91249e8	0.237309
$431$	9.77619e7	0.0588165	0.0294083	−	0.999567i	$-0.490638\pi$
$431$	9.77619e7	0.0588165	0.0294083	+	0.999567i	$0.490638\pi$
$432$	2.22635e9	1.32862
$433$	−6.53277e8	−0.386714	−0.193357	−	0.981128i	$-0.561938\pi$
$433$	−6.53277e8	−0.386714	−0.193357	+	0.981128i	$0.561938\pi$
$434$	−1.02252e9	−0.600425
$435$	−5.97912e8	−0.348277
$436$	−6.22105e8	−0.359469
$437$	−4.85163e8	−0.278101
$438$	3.61601e9	2.05623
$439$	1.57565e9	0.888862	0.444431	−	0.895813i	$-0.353406\pi$
$439$	1.57565e9	0.888862	0.444431	+	0.895813i	$0.353406\pi$
$440$	−7.13068e8	−0.399068
$441$	−1.65818e9	−0.920652
$442$	0	0
$443$	2.36999e9	1.29519	0.647594	−	0.761985i	$-0.275775\pi$
$443$	2.36999e9	1.29519	0.647594	+	0.761985i	$0.275775\pi$
$444$	4.41010e9	2.39116
$445$	8.95966e7	0.0481983
$446$	4.11030e9	2.19382
$447$	2.93619e9	1.55492
$448$	−1.88263e9	−0.989217
$449$	−2.05507e8	−0.107143	−0.0535716	−	0.998564i	$-0.517061\pi$
$449$	−2.05507e8	−0.107143	−0.0535716	+	0.998564i	$0.517061\pi$
$450$	−2.07372e9	−1.07277
$451$	−4.41841e9	−2.26803
$452$	−3.21419e9	−1.63714
$453$	−2.07156e9	−1.04702
$454$	−3.20048e7	−0.0160516
$455$	0	0
$456$	2.84381e8	0.140451
$457$	2.69717e9	1.32191	0.660954	−	0.750426i	$-0.270152\pi$
$457$	2.69717e9	1.32191	0.660954	+	0.750426i	$0.270152\pi$
$458$	−3.47733e9	−1.69129
$459$	2.50322e9	1.20824
$460$	−1.98785e9	−0.952209
$461$	1.92610e9	0.915642	0.457821	−	0.889044i	$-0.348630\pi$
$461$	1.92610e9	0.915642	0.457821	+	0.889044i	$0.348630\pi$
$462$	−6.87675e9	−3.24441
$463$	−2.34776e9	−1.09931	−0.549655	−	0.835391i	$-0.685241\pi$
$463$	−2.34776e9	−1.09931	−0.549655	+	0.835391i	$0.685241\pi$
$464$	4.14484e8	0.192617
$465$	−1.65469e9	−0.763186
$466$	−5.74989e9	−2.63214
$467$	1.71342e9	0.778493	0.389246	−	0.921134i	$-0.372736\pi$
$467$	1.71342e9	0.778493	0.389246	+	0.921134i	$0.372736\pi$
$468$	0	0
$469$	−1.71182e9	−0.766219
$470$	−1.49024e9	−0.662084
$471$	−3.42966e9	−1.51244
$472$	−1.39078e8	−0.0608783
$473$	−7.93221e8	−0.344652
$474$	−2.90571e9	−1.25322
$475$	2.35751e8	0.100931
$476$	−1.43986e9	−0.611921
$477$	6.56133e9	2.76807
$478$	6.62184e9	2.77320
$479$	1.13940e8	0.0473697	0.0236848	−	0.999719i	$-0.492460\pi$
$479$	1.13940e8	0.0473697	0.0236848	+	0.999719i	$0.492460\pi$
$480$	−4.77924e9	−1.97249
$481$	0	0
$482$	−1.30332e9	−0.530137
$483$	−3.14186e9	−1.26874
$484$	5.83778e9	2.34039
$485$	2.46595e9	0.981496
$486$	−2.20257e7	−0.00870367
$487$	7.96258e8	0.312394	0.156197	−	0.987726i	$-0.450077\pi$
$487$	7.96258e8	0.312394	0.156197	+	0.987726i	$0.450077\pi$
$488$	−6.06883e8	−0.236394
$489$	−2.79446e9	−1.08073
$490$	1.41818e9	0.544558
$491$	3.65621e9	1.39395	0.696973	−	0.717097i	$-0.254530\pi$
$491$	3.65621e9	1.39395	0.696973	+	0.717097i	$0.254530\pi$
$492$	7.21993e9	2.73310
$493$	4.66030e8	0.175166
$494$	0	0
$495$	−7.42092e9	−2.75004
$496$	1.14706e9	0.422085
$497$	8.65463e8	0.316228
$498$	2.21636e9	0.804150
$499$	7.46131e8	0.268821	0.134411	−	0.990926i	$-0.457086\pi$
$499$	7.46131e8	0.268821	0.134411	+	0.990926i	$0.457086\pi$
$500$	3.63686e9	1.30116
$501$	−6.48959e8	−0.230560
$502$	−5.88270e9	−2.07546
$503$	−4.68159e8	−0.164023	−0.0820117	−	0.996631i	$-0.526134\pi$
$503$	−4.68159e8	−0.164023	−0.0820117	+	0.996631i	$0.526134\pi$
$504$	1.22809e9	0.427292
$505$	−1.71579e9	−0.592851
$506$	7.39987e9	2.53920
$507$	0	0
$508$	1.42920e9	0.483692
$509$	5.93386e8	0.199446	0.0997229	−	0.995015i	$-0.468204\pi$
$509$	5.93386e8	0.199446	0.0997229	+	0.995015i	$0.468204\pi$
$510$	−4.27821e9	−1.42812
$511$	−1.77527e9	−0.588562
$512$	3.98841e9	1.31327
$513$	1.48103e9	0.484344
$514$	−6.02225e9	−1.95609
$515$	1.08494e8	0.0350009
$516$	1.29617e9	0.415324
$517$	3.02132e9	0.961567
$518$	−3.97542e9	−1.25669
$519$	−3.02329e9	−0.949280
$520$	0	0
$521$	2.35950e9	0.730950	0.365475	−	0.930821i	$-0.380907\pi$
$521$	2.35950e9	0.730950	0.365475	+	0.930821i	$0.380907\pi$
$522$	−2.42535e9	−0.746325
$523$	5.49264e8	0.167890	0.0839451	−	0.996470i	$-0.473248\pi$
$523$	5.49264e8	0.167890	0.0839451	+	0.996470i	$0.473248\pi$
$524$	−6.23911e9	−1.89436
$525$	1.52670e9	0.460465
$526$	−3.34497e9	−1.00217
$527$	1.28971e9	0.383844
$528$	7.71430e9	2.28075
$529$	−2.39521e7	−0.00703475
$530$	−5.61167e9	−1.63729
$531$	−1.44739e9	−0.419522
$532$	−8.51893e8	−0.245298
$533$	0	0
$534$	5.45002e8	0.154883
$535$	1.56265e7	0.00441188
$536$	−1.07972e9	−0.302856
$537$	5.55174e9	1.54710
$538$	−1.08115e9	−0.299328
$539$	−2.87522e9	−0.790881
$540$	6.06822e9	1.65838
$541$	−6.19192e7	−0.0168126	−0.00840630	−	0.999965i	$-0.502676\pi$
$541$	−6.19192e7	−0.0168126	−0.00840630	+	0.999965i	$0.502676\pi$
$542$	−8.03907e8	−0.216874
$543$	−9.79240e9	−2.62476
$544$	3.72507e9	0.992062
$545$	9.07489e8	0.240134
$546$	0	0
$547$	−4.11747e9	−1.07566	−0.537830	−	0.843053i	$-0.680756\pi$
$547$	−4.11747e9	−1.07566	−0.537830	+	0.843053i	$0.680756\pi$
$548$	−7.45912e9	−1.93622
$549$	−6.31585e9	−1.62903
$550$	−3.59576e9	−0.921555
$551$	2.75727e8	0.0702181
$552$	−1.98172e9	−0.501482
$553$	1.42655e9	0.358715
$554$	9.94238e9	2.48431
$555$	−6.43319e9	−1.59735
$556$	−8.15808e9	−2.01292
$557$	4.03807e9	0.990105	0.495052	−	0.868863i	$-0.335149\pi$
$557$	4.03807e9	0.990105	0.495052	+	0.868863i	$0.335149\pi$
$558$	−6.71202e9	−1.63544
$559$	0	0
$560$	1.86806e9	0.449503
$561$	8.67366e9	2.07411
$562$	2.92994e9	0.696277
$563$	−6.53949e8	−0.154442	−0.0772209	−	0.997014i	$-0.524605\pi$
$563$	−6.53949e8	−0.154442	−0.0772209	+	0.997014i	$0.524605\pi$
$564$	−4.93700e9	−1.15874
$565$	4.68866e9	1.09365
$566$	1.10841e10	2.56947
$567$	3.20601e9	0.738626
$568$	5.45887e8	0.124992
$569$	3.78098e9	0.860423	0.430211	−	0.902728i	$-0.358439\pi$
$569$	3.78098e9	0.860423	0.430211	+	0.902728i	$0.358439\pi$
$570$	−2.53121e9	−0.572487
$571$	1.41244e9	0.317499	0.158750	−	0.987319i	$-0.449254\pi$
$571$	1.41244e9	0.317499	0.158750	+	0.987319i	$0.449254\pi$
$572$	0	0
$573$	6.80952e9	1.51208
$574$	−6.50829e9	−1.43640
$575$	−1.64284e9	−0.360377
$576$	−1.23579e10	−2.69443
$577$	2.80901e9	0.608749	0.304374	−	0.952553i	$-0.401553\pi$
$577$	2.80901e9	0.608749	0.304374	+	0.952553i	$0.401553\pi$
$578$	−3.54507e9	−0.763621
$579$	3.60631e9	0.772125
$580$	1.12973e9	0.240424
$581$	−1.08812e9	−0.230175
$582$	1.50000e10	3.15399
$583$	1.13771e10	2.37790
$584$	−1.11975e9	−0.232635
$585$	0	0
$586$	8.99889e9	1.84734
$587$	−5.36072e9	−1.09393	−0.546965	−	0.837156i	$-0.684217\pi$
$587$	−5.36072e9	−1.09393	−0.546965	+	0.837156i	$0.684217\pi$
$588$	4.69827e9	0.953054
$589$	7.63059e8	0.153870
$590$	1.23790e9	0.248144
$591$	−1.02496e10	−2.04245
$592$	4.45960e9	0.883426
$593$	2.19757e9	0.432765	0.216382	−	0.976309i	$-0.430574\pi$
$593$	2.19757e9	0.432765	0.216382	+	0.976309i	$0.430574\pi$
$594$	−2.25892e10	−4.42230
$595$	2.10038e9	0.408778
$596$	−5.54783e9	−1.07340
$597$	7.79589e9	1.49953
$598$	0	0
$599$	−9.75825e9	−1.85515	−0.927573	−	0.373642i	$-0.878109\pi$
$599$	−9.75825e9	−1.85515	−0.927573	+	0.373642i	$0.878109\pi$
$600$	9.62960e8	0.182003
$601$	3.68732e9	0.692868	0.346434	−	0.938074i	$-0.387393\pi$
$601$	3.68732e9	0.692868	0.346434	+	0.938074i	$0.387393\pi$
$602$	−1.16841e9	−0.218277
$603$	−1.12367e10	−2.08703
$604$	3.91414e9	0.722782
$605$	−8.51579e9	−1.56344
$606$	−1.04369e10	−1.90510
$607$	5.90498e9	1.07166	0.535831	−	0.844325i	$-0.319998\pi$
$607$	5.90498e9	1.07166	0.535831	+	0.844325i	$0.319998\pi$
$608$	2.20394e9	0.397684
$609$	1.78558e9	0.320345
$610$	5.40171e9	0.963557
$611$	0	0
$612$	−9.45149e9	−1.66675
$613$	8.40999e9	1.47463	0.737316	−	0.675548i	$-0.236093\pi$
$613$	8.40999e9	1.47463	0.737316	+	0.675548i	$0.236093\pi$
$614$	6.00439e9	1.04684
$615$	−1.05320e10	−1.82578
$616$	2.12948e9	0.367063
$617$	8.51257e9	1.45902	0.729512	−	0.683967i	$-0.239747\pi$
$617$	8.51257e9	1.45902	0.729512	+	0.683967i	$0.239747\pi$
$618$	6.59950e8	0.112474
$619$	8.70392e9	1.47502	0.737509	−	0.675337i	$-0.236002\pi$
$619$	8.70392e9	1.47502	0.737509	+	0.675337i	$0.236002\pi$
$620$	3.12647e9	0.526846
$621$	−1.03206e10	−1.72936
$622$	−8.30060e9	−1.38307
$623$	−2.67568e8	−0.0443328
$624$	0	0
$625$	−3.09788e9	−0.507556
$626$	−5.27780e9	−0.859889
$627$	5.13178e9	0.831442
$628$	6.48022e9	1.04407
$629$	5.01421e9	0.803388
$630$	−1.09310e10	−1.74167
$631$	−9.88656e9	−1.56654	−0.783272	−	0.621679i	$-0.786451\pi$
$631$	−9.88656e9	−1.56654	−0.783272	+	0.621679i	$0.786451\pi$
$632$	8.99792e8	0.141786
$633$	4.07976e9	0.639326
$634$	1.04648e10	1.63086
$635$	−2.08482e9	−0.323118
$636$	−1.85909e10	−2.86549
$637$	0	0
$638$	−4.20548e9	−0.641126
$639$	5.68106e9	0.861343
$640$	3.01904e9	0.455239
$641$	−6.76169e9	−1.01403	−0.507017	−	0.861936i	$-0.669252\pi$
$641$	−6.76169e9	−1.01403	−0.507017	+	0.861936i	$0.669252\pi$
$642$	9.50536e7	0.0141774
$643$	2.81532e9	0.417627	0.208814	−	0.977955i	$-0.433040\pi$
$643$	2.81532e9	0.417627	0.208814	+	0.977955i	$0.433040\pi$
$644$	5.93644e9	0.875842
$645$	−1.89077e9	−0.277447
$646$	1.97289e9	0.287932
$647$	−1.19349e10	−1.73243	−0.866213	−	0.499674i	$-0.833453\pi$
$647$	−1.19349e10	−1.73243	−0.866213	+	0.499674i	$0.833453\pi$
$648$	2.02218e9	0.291949
$649$	−2.50973e9	−0.360388
$650$	0	0
$651$	4.94149e9	0.701979
$652$	5.28004e9	0.746054
$653$	1.28240e9	0.180231	0.0901153	−	0.995931i	$-0.471276\pi$
$653$	1.28240e9	0.180231	0.0901153	+	0.995931i	$0.471276\pi$
$654$	5.52011e9	0.771659
$655$	9.10123e9	1.26548
$656$	7.30097e9	1.00976
$657$	−1.16532e10	−1.60313
$658$	4.45038e9	0.608985
$659$	−4.85245e9	−0.660484	−0.330242	−	0.943896i	$-0.607130\pi$
$659$	−4.85245e9	−0.660484	−0.330242	+	0.943896i	$0.607130\pi$
$660$	2.10264e10	2.84683
$661$	−4.76107e9	−0.641209	−0.320605	−	0.947213i	$-0.603886\pi$
$661$	−4.76107e9	−0.641209	−0.320605	+	0.947213i	$0.603886\pi$
$662$	−8.44658e9	−1.13156
$663$	0	0
$664$	−6.86325e8	−0.0909790
$665$	1.24269e9	0.163865
$666$	−2.60954e10	−3.42298
$667$	−1.92141e9	−0.250715
$668$	1.22618e9	0.159162
$669$	−1.98636e10	−2.56488
$670$	9.61034e9	1.23446
$671$	−1.09515e10	−1.39941
$672$	1.42725e10	1.81429
$673$	−1.03481e10	−1.30861	−0.654303	−	0.756233i	$-0.727038\pi$
$673$	−1.03481e10	−1.30861	−0.654303	+	0.756233i	$0.727038\pi$
$674$	1.61653e10	2.03365
$675$	5.01501e9	0.627637
$676$	0	0
$677$	−1.09230e9	−0.135295	−0.0676477	−	0.997709i	$-0.521549\pi$
$677$	−1.09230e9	−0.135295	−0.0676477	+	0.997709i	$0.521549\pi$
$678$	2.85204e10	3.51440
$679$	−7.36421e9	−0.902780
$680$	1.32480e9	0.161573
$681$	1.54668e8	0.0187666
$682$	−1.16384e10	−1.40491
$683$	−7.72912e9	−0.928234	−0.464117	−	0.885774i	$-0.653628\pi$
$683$	−7.72912e9	−0.928234	−0.464117	+	0.885774i	$0.653628\pi$
$684$	−5.59199e9	−0.668144
$685$	1.08809e10	1.29345
$686$	−1.34434e10	−1.58992
$687$	1.68047e10	1.97735
$688$	1.31072e9	0.153444
$689$	0	0
$690$	1.76388e10	2.04407
$691$	−8.59771e9	−0.991310	−0.495655	−	0.868519i	$-0.665072\pi$
$691$	−8.59771e9	−0.991310	−0.495655	+	0.868519i	$0.665072\pi$
$692$	5.71240e9	0.655311
$693$	2.21615e10	2.52949
$694$	7.72008e9	0.876725
$695$	1.19005e10	1.34468
$696$	1.12625e9	0.126620
$697$	8.20893e9	0.918273
$698$	−4.10030e9	−0.456375
$699$	2.77872e10	3.07733
$700$	−2.88465e9	−0.317870
$701$	−4.34708e9	−0.476633	−0.238317	−	0.971188i	$-0.576596\pi$
$701$	−4.34708e9	−0.476633	−0.238317	+	0.971188i	$0.576596\pi$
$702$	0	0
$703$	2.96666e9	0.322051
$704$	−2.14282e10	−2.31463
$705$	7.20179e9	0.774067
$706$	−9.08805e9	−0.971973
$707$	5.12397e9	0.545304
$708$	4.10104e9	0.434287
$709$	1.22439e10	1.29020	0.645100	−	0.764098i	$-0.276816\pi$
$709$	1.22439e10	1.29020	0.645100	+	0.764098i	$0.276816\pi$
$710$	−4.85880e9	−0.509477
$711$	9.36416e9	0.977069
$712$	−1.68767e8	−0.0175230
$713$	−5.31740e9	−0.549396
$714$	1.27762e10	1.31359
$715$	0	0
$716$	−1.04898e10	−1.06800
$717$	−3.20010e10	−3.24225
$718$	−8.65486e9	−0.872619
$719$	−2.29472e9	−0.230239	−0.115119	−	0.993352i	$-0.536725\pi$
$719$	−2.29472e9	−0.230239	−0.115119	+	0.993352i	$0.536725\pi$
$720$	1.22623e10	1.22436
$721$	−3.24001e8	−0.0321939
$722$	−1.38191e10	−1.36647
$723$	6.29851e9	0.619803
$724$	1.85024e10	1.81194
$725$	9.33655e8	0.0909921
$726$	−5.18002e10	−5.02404
$727$	5.74619e9	0.554638	0.277319	−	0.960778i	$-0.410554\pi$
$727$	5.74619e9	0.554638	0.277319	+	0.960778i	$0.410554\pi$
$728$	0	0
$729$	−1.04071e10	−0.994908
$730$	9.96659e9	0.948236
$731$	1.47372e9	0.139542
$732$	1.78953e10	1.68636
$733$	1.26955e9	0.119065	0.0595326	−	0.998226i	$-0.481039\pi$
$733$	1.26955e9	0.119065	0.0595326	+	0.998226i	$0.481039\pi$
$734$	1.73044e10	1.61518
$735$	−6.85355e9	−0.636664
$736$	−1.53582e10	−1.41994
$737$	−1.94841e10	−1.79285
$738$	−4.27216e10	−3.91247
$739$	2.57407e9	0.234620	0.117310	−	0.993095i	$-0.462573\pi$
$739$	2.57407e9	0.234620	0.117310	+	0.993095i	$0.462573\pi$
$740$	1.21553e10	1.10269
$741$	0	0
$742$	1.67584e10	1.50598
$743$	4.84335e9	0.433196	0.216598	−	0.976261i	$-0.430504\pi$
$743$	4.84335e9	0.433196	0.216598	+	0.976261i	$0.430504\pi$
$744$	3.11682e9	0.277464
$745$	8.09283e9	0.717057
$746$	−3.01797e10	−2.66152
$747$	−7.14260e9	−0.626952
$748$	−1.63886e10	−1.43181
$749$	−4.66664e7	−0.00405805
$750$	−3.22708e10	−2.79316
$751$	2.07398e10	1.78676	0.893378	−	0.449307i	$-0.148329\pi$
$751$	2.07398e10	1.78676	0.893378	+	0.449307i	$0.148329\pi$
$752$	−4.99242e9	−0.428103
$753$	2.84290e10	2.42650
$754$	0	0
$755$	−5.70971e9	−0.482836
$756$	−1.81219e10	−1.52538
$757$	2.98409e9	0.250021	0.125010	−	0.992155i	$-0.460104\pi$
$757$	2.98409e9	0.250021	0.125010	+	0.992155i	$0.460104\pi$
$758$	−1.33192e10	−1.11080
$759$	−3.57610e10	−2.96868
$760$	7.83821e8	0.0647694
$761$	2.11422e10	1.73901	0.869507	−	0.493920i	$-0.164436\pi$
$761$	2.11422e10	1.73901	0.869507	+	0.493920i	$0.164436\pi$
$762$	−1.26816e10	−1.03832
$763$	−2.71009e9	−0.220875
$764$	−1.28663e10	−1.04383
$765$	1.37873e10	1.11343
$766$	−3.17191e10	−2.54989
$767$	0	0
$768$	−1.09120e10	−0.869240
$769$	−1.37622e8	−0.0109130	−0.00545652	−	0.999985i	$-0.501737\pi$
$769$	−1.37622e8	−0.0109130	−0.00545652	+	0.999985i	$0.501737\pi$
$770$	−1.89539e10	−1.49617
$771$	2.91034e10	2.28693
$772$	−6.81399e9	−0.533017
$773$	4.08746e9	0.318292	0.159146	−	0.987255i	$-0.449126\pi$
$773$	4.08746e9	0.318292	0.159146	+	0.987255i	$0.449126\pi$
$774$	−7.66966e9	−0.594542
$775$	2.58384e9	0.199393
$776$	−4.64495e9	−0.356833
$777$	1.92118e10	1.46924
$778$	1.64169e10	1.24986
$779$	4.85683e9	0.368105
$780$	0	0
$781$	9.85077e9	0.739932
$782$	−1.37482e10	−1.02807
$783$	5.86539e9	0.436648
$784$	4.75101e9	0.352111
$785$	−9.45295e9	−0.697466
$786$	5.53613e10	4.06656
$787$	1.83896e10	1.34481	0.672404	−	0.740184i	$-0.265262\pi$
$787$	1.83896e10	1.34481	0.672404	+	0.740184i	$0.265262\pi$
$788$	1.93662e10	1.40995
$789$	1.61651e10	1.17168
$790$	−8.00882e9	−0.577928
$791$	−1.40020e10	−1.00594
$792$	1.39783e10	0.999805
$793$	0	0
$794$	−1.97576e10	−1.40075
$795$	2.71192e10	1.91422
$796$	−1.47301e10	−1.03516
$797$	−1.07541e10	−0.752436	−0.376218	−	0.926531i	$-0.622776\pi$
$797$	−1.07541e10	−0.752436	−0.376218	+	0.926531i	$0.622776\pi$
$798$	7.55908e9	0.526573
$799$	−5.61328e9	−0.389317
$800$	7.46290e9	0.515339
$801$	−1.75636e9	−0.120754
$802$	1.58089e10	1.08216
$803$	−2.02063e10	−1.37716
$804$	3.18380e10	2.16048
$805$	−8.65972e9	−0.585084
$806$	0	0
$807$	5.22482e9	0.349956
$808$	3.23192e9	0.215537
$809$	7.93391e9	0.526826	0.263413	−	0.964683i	$-0.415152\pi$
$809$	7.93391e9	0.526826	0.263413	+	0.964683i	$0.415152\pi$
$810$	−1.79989e10	−1.19000
$811$	2.50497e10	1.64903	0.824517	−	0.565837i	$-0.191447\pi$
$811$	2.50497e10	1.64903	0.824517	+	0.565837i	$0.191447\pi$
$812$	−3.37379e9	−0.221142
$813$	3.88500e9	0.253556
$814$	−4.52485e10	−2.94049
$815$	−7.70220e9	−0.498383
$816$	−1.43323e10	−0.923424
$817$	8.71928e8	0.0559375
$818$	3.30032e10	2.10824
$819$	0	0
$820$	1.98998e10	1.26038
$821$	−5.53755e9	−0.349234	−0.174617	−	0.984636i	$-0.555869\pi$
$821$	−5.53755e9	−0.349234	−0.174617	+	0.984636i	$0.555869\pi$
$822$	6.61868e10	4.15643
$823$	2.94660e10	1.84256	0.921282	−	0.388896i	$-0.127144\pi$
$823$	2.94660e10	1.84256	0.921282	+	0.388896i	$0.127144\pi$
$824$	−2.04362e8	−0.0127249
$825$	1.73770e10	1.07742
$826$	−3.69682e9	−0.228243
$827$	2.47966e10	1.52448	0.762241	−	0.647294i	$-0.224099\pi$
$827$	2.47966e10	1.52448	0.762241	+	0.647294i	$0.224099\pi$
$828$	3.89679e10	2.38562
$829$	−1.08636e10	−0.662266	−0.331133	−	0.943584i	$-0.607431\pi$
$829$	−1.08636e10	−0.662266	−0.331133	+	0.943584i	$0.607431\pi$
$830$	6.10880e9	0.370837
$831$	−4.80480e10	−2.90450
$832$	0	0
$833$	5.34185e9	0.320210
$834$	7.23889e10	4.32106
$835$	−1.78868e9	−0.106324
$836$	−9.69632e9	−0.573965
$837$	1.62321e10	0.956834
$838$	−3.28060e10	−1.92575
$839$	−1.43527e10	−0.839007	−0.419504	−	0.907754i	$-0.637796\pi$
$839$	−1.43527e10	−0.839007	−0.419504	+	0.907754i	$0.637796\pi$
$840$	5.07594e9	0.295488
$841$	−1.61579e10	−0.936697
$842$	4.90568e9	0.283209
$843$	−1.41594e10	−0.814044
$844$	−7.70857e9	−0.441342
$845$	0	0
$846$	2.92131e10	1.65875
$847$	2.54312e10	1.43805
$848$	−1.87995e10	−1.05867
$849$	−5.35656e10	−3.00406
$850$	6.68053e9	0.373117
$851$	−2.06733e10	−1.14989
$852$	−1.60967e10	−0.891658
$853$	2.42734e10	1.33909	0.669543	−	0.742774i	$-0.266490\pi$
$853$	2.42734e10	1.33909	0.669543	+	0.742774i	$0.266490\pi$
$854$	−1.61314e10	−0.886280
$855$	8.15725e9	0.446336
$856$	−2.94346e7	−0.00160398
$857$	−9.63070e9	−0.522667	−0.261334	−	0.965249i	$-0.584162\pi$
$857$	−9.63070e9	−0.522667	−0.261334	+	0.965249i	$0.584162\pi$
$858$	0	0
$859$	2.19480e10	1.18146	0.590729	−	0.806870i	$-0.298840\pi$
$859$	2.19480e10	1.18146	0.590729	+	0.806870i	$0.298840\pi$
$860$	3.57254e9	0.191528
$861$	3.14523e10	1.67935
$862$	1.63905e9	0.0871599
$863$	−1.90983e10	−1.01148	−0.505740	−	0.862686i	$-0.668780\pi$
$863$	−1.90983e10	−1.01148	−0.505740	+	0.862686i	$0.668780\pi$
$864$	4.68833e10	2.47298
$865$	−8.33290e9	−0.437764
$866$	−1.09527e10	−0.573070
$867$	1.71321e10	0.892778
$868$	−9.33676e9	−0.484593
$869$	1.62371e10	0.839345
$870$	−1.00244e10	−0.516110
$871$	0	0
$872$	−1.70938e9	−0.0873031
$873$	−4.83401e10	−2.45899
$874$	−8.13412e9	−0.412117
$875$	1.58433e10	0.799498
$876$	3.30182e10	1.65955
$877$	−2.78380e10	−1.39360	−0.696802	−	0.717263i	$-0.745394\pi$
$877$	−2.78380e10	−1.39360	−0.696802	+	0.717263i	$0.745394\pi$
$878$	2.64169e10	1.31720
$879$	−4.34885e10	−2.15980
$880$	2.12624e10	1.05178
$881$	2.24473e10	1.10598	0.552992	−	0.833186i	$-0.313486\pi$
$881$	2.24473e10	1.10598	0.552992	+	0.833186i	$0.313486\pi$
$882$	−2.78005e10	−1.36431
$883$	−1.41508e10	−0.691700	−0.345850	−	0.938290i	$-0.612409\pi$
$883$	−1.41508e10	−0.691700	−0.345850	+	0.938290i	$0.612409\pi$
$884$	0	0
$885$	−5.98234e9	−0.290115
$886$	3.97346e10	1.91933
$887$	3.27385e10	1.57516	0.787582	−	0.616210i	$-0.211333\pi$
$887$	3.27385e10	1.57516	0.787582	+	0.616210i	$0.211333\pi$
$888$	1.21178e10	0.580733
$889$	6.22603e9	0.297204
$890$	1.50215e9	0.0714248
$891$	3.64911e10	1.72828
$892$	3.75316e10	1.77060
$893$	−3.32110e9	−0.156064
$894$	4.92274e10	2.30423
$895$	1.53019e10	0.713452
$896$	−9.01594e9	−0.418729
$897$	0	0
$898$	−3.44548e9	−0.158775
$899$	3.02197e9	0.138718
$900$	−1.89353e10	−0.865814
$901$	−2.11375e10	−0.962756
$902$	−7.40780e10	−3.36098
$903$	5.64651e9	0.255195
$904$	−8.83171e9	−0.397608
$905$	−2.69902e10	−1.21042
$906$	−3.47313e10	−1.55157
$907$	4.64948e9	0.206909	0.103454	−	0.994634i	$-0.467010\pi$
$907$	4.64948e9	0.206909	0.103454	+	0.994634i	$0.467010\pi$
$908$	−2.92239e8	−0.0129550
$909$	3.36347e10	1.48530
$910$	0	0
$911$	−1.91286e9	−0.0838242	−0.0419121	−	0.999121i	$-0.513345\pi$
$911$	−1.91286e9	−0.0838242	−0.0419121	+	0.999121i	$0.513345\pi$
$912$	−8.47974e9	−0.370170
$913$	−1.23850e10	−0.538579
$914$	4.52200e10	1.95893
$915$	−2.61046e10	−1.12653
$916$	−3.17519e10	−1.36501
$917$	−2.71795e10	−1.16399
$918$	4.19683e10	1.79049
$919$	−4.51305e10	−1.91808	−0.959039	−	0.283275i	$-0.908579\pi$
$919$	−4.51305e10	−1.91808	−0.959039	+	0.283275i	$0.908579\pi$
$920$	−5.46208e9	−0.231260
$921$	−2.90171e10	−1.22390
$922$	3.22925e10	1.35689
$923$	0	0
$924$	−6.27923e10	−2.61851
$925$	1.00456e10	0.417330
$926$	−3.93619e10	−1.62906
$927$	−2.12680e9	−0.0876897
$928$	8.72837e9	0.358522
$929$	−6.56214e9	−0.268529	−0.134264	−	0.990946i	$-0.542867\pi$
$929$	−6.56214e9	−0.268529	−0.134264	+	0.990946i	$0.542867\pi$
$930$	−2.77420e10	−1.13096
$931$	3.16051e9	0.128361
$932$	−5.25029e10	−2.12436
$933$	4.01139e10	1.61700
$934$	2.87267e10	1.15364
$935$	2.39067e10	0.956485
$936$	0	0
$937$	−1.55199e10	−0.616314	−0.308157	−	0.951336i	$-0.599712\pi$
$937$	−1.55199e10	−0.616314	−0.308157	+	0.951336i	$0.599712\pi$
$938$	−2.86999e10	−1.13546
$939$	2.55057e10	1.00533
$940$	−1.36075e10	−0.534357
$941$	−1.49766e10	−0.585933	−0.292967	−	0.956123i	$-0.594642\pi$
$941$	−1.49766e10	−0.585933	−0.292967	+	0.956123i	$0.594642\pi$
$942$	−5.75007e10	−2.24128
$943$	−3.38449e10	−1.31432
$944$	4.14707e9	0.160450
$945$	2.64351e10	1.01899
$946$	−1.32989e10	−0.510738
$947$	1.19869e10	0.458651	0.229325	−	0.973350i	$-0.426348\pi$
$947$	1.19869e10	0.458651	0.229325	+	0.973350i	$0.426348\pi$
$948$	−2.65324e10	−1.01146
$949$	0	0
$950$	3.95254e9	0.149570
$951$	−5.05726e10	−1.90671
$952$	−3.95633e9	−0.148615
$953$	−6.04429e9	−0.226214	−0.113107	−	0.993583i	$-0.536080\pi$
$953$	−6.04429e9	−0.226214	−0.113107	+	0.993583i	$0.536080\pi$
$954$	1.10006e11	4.10200
$955$	1.87686e10	0.697302
$956$	6.04648e10	2.23820
$957$	2.03236e10	0.749565
$958$	1.91028e9	0.0701969
$959$	−3.24943e10	−1.18971
$960$	−5.10776e10	−1.86329
$961$	−1.91495e10	−0.696025
$962$	0	0
$963$	−3.06327e8	−0.0110533
$964$	−1.19008e10	−0.427865
$965$	9.93983e9	0.356068
$966$	−5.26757e10	−1.88014
$967$	4.09241e9	0.145541	0.0727706	−	0.997349i	$-0.476816\pi$
$967$	4.09241e9	0.145541	0.0727706	+	0.997349i	$0.476816\pi$
$968$	1.60406e10	0.568404
$969$	−9.53430e9	−0.336632
$970$	4.13435e10	1.45447
$971$	4.94435e10	1.73317	0.866587	−	0.499026i	$-0.166309\pi$
$971$	4.94435e10	1.73317	0.866587	+	0.499026i	$0.166309\pi$
$972$	−2.01119e8	−0.00702460
$973$	−3.55392e10	−1.23684
$974$	1.33498e10	0.462935
$975$	0	0
$976$	1.80962e10	0.623035
$977$	−4.89243e10	−1.67839	−0.839197	−	0.543828i	$-0.816974\pi$
$977$	−4.89243e10	−1.67839	−0.839197	+	0.543828i	$0.816974\pi$
$978$	−4.68512e10	−1.60153
$979$	−3.04548e9	−0.103733
$980$	1.29495e10	0.439504
$981$	−1.77895e10	−0.601620
$982$	6.12991e10	2.06568
$983$	−1.74370e10	−0.585511	−0.292755	−	0.956187i	$-0.594572\pi$
$983$	−1.74370e10	−0.585511	−0.292755	+	0.956187i	$0.594572\pi$
$984$	1.98384e10	0.663779
$985$	−2.82503e10	−0.941881
$986$	7.81333e9	0.259577
$987$	−2.15071e10	−0.711987
$988$	0	0
$989$	−6.07605e9	−0.199726
$990$	−1.24417e11	−4.07528
$991$	−4.84851e10	−1.58252	−0.791262	−	0.611477i	$-0.790576\pi$
$991$	−4.84851e10	−1.58252	−0.791262	+	0.611477i	$0.790576\pi$
$992$	2.41553e10	0.785635
$993$	4.08193e10	1.32295
$994$	1.45101e10	0.468618
$995$	2.14873e10	0.691515
$996$	2.02378e10	0.649017
$997$	2.19592e10	0.701752	0.350876	−	0.936422i	$-0.385884\pi$
$997$	2.19592e10	0.701752	0.350876	+	0.936422i	$0.385884\pi$
$998$	1.25094e10	0.398365
$999$	6.31082e10	2.00266

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	169.8.a.g.1.13		14
13.5	odd	4		169.8.b.d.168.2		14
13.6	odd	12		13.8.e.a.10.1	yes	14
13.8	odd	4		169.8.b.d.168.13		14
13.11	odd	12		13.8.e.a.4.1	✓	14
13.12	even	2	inner	169.8.a.g.1.2		14
39.11	even	12		117.8.q.b.82.7		14
39.32	even	12		117.8.q.b.10.7		14
52.11	even	12		208.8.w.a.17.1		14
52.19	even	12		208.8.w.a.49.1		14

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
13.8.e.a.4.1	✓	14	13.11	odd	12
13.8.e.a.10.1	yes	14	13.6	odd	12
117.8.q.b.10.7		14	39.32	even	12
117.8.q.b.82.7		14	39.11	even	12
169.8.a.g.1.2		14	13.12	even	2	inner
169.8.a.g.1.13		14	1.1	even	1	trivial
169.8.b.d.168.2		14	13.5	odd	4
169.8.b.d.168.13		14	13.8	odd	4
208.8.w.a.17.1		14	52.11	even	12
208.8.w.a.49.1		14	52.19	even	12

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)

\(f(q)\)	\(=\)	\(q+16.7657 q^{2} -81.0229 q^{3} +153.090 q^{4} -223.318 q^{5} -1358.41 q^{6} +666.907 q^{7} +420.649 q^{8} +4377.70 q^{9} -3744.09 q^{10} +7590.80 q^{11} -12403.8 q^{12} +11181.2 q^{14} +18093.9 q^{15} -12543.0 q^{16} -14102.9 q^{17} +73395.4 q^{18} -8343.98 q^{19} -34187.7 q^{20} -54034.7 q^{21} +127265. q^{22} +58145.3 q^{23} -34082.2 q^{24} -28254.0 q^{25} -177497. q^{27} +102097. q^{28} -33045.0 q^{29} +303357. q^{30} -91450.2 q^{31} -264136. q^{32} -615028. q^{33} -236445. q^{34} -148932. q^{35} +670182. q^{36} -355545. q^{37} -139893. q^{38} -93938.5 q^{40} -582075. q^{41} -905932. q^{42} -104498. q^{43} +1.16207e6 q^{44} -977620. q^{45} +974848. q^{46} +398024. q^{47} +1.01627e6 q^{48} -378778. q^{49} -473700. q^{50} +1.14266e6 q^{51} +1.49881e6 q^{53} -2.97587e6 q^{54} -1.69516e6 q^{55} +280534. q^{56} +676053. q^{57} -554024. q^{58} -330628. q^{59} +2.76999e6 q^{60} -1.44273e6 q^{61} -1.53323e6 q^{62} +2.91952e6 q^{63} -2.82292e6 q^{64} -1.03114e7 q^{66} -2.56680e6 q^{67} -2.15901e6 q^{68} -4.71110e6 q^{69} -2.49696e6 q^{70} +1.29773e6 q^{71} +1.84148e6 q^{72} -2.66195e6 q^{73} -5.96097e6 q^{74} +2.28922e6 q^{75} -1.27738e6 q^{76} +5.06236e6 q^{77} +2.13906e6 q^{79} +2.80108e6 q^{80} +4.80728e6 q^{81} -9.75892e6 q^{82} -1.63158e6 q^{83} -8.27217e6 q^{84} +3.14943e6 q^{85} -1.75198e6 q^{86} +2.67740e6 q^{87} +3.19306e6 q^{88} -401206. q^{89} -1.63905e7 q^{90} +8.90145e6 q^{92} +7.40955e6 q^{93} +6.67316e6 q^{94} +1.86336e6 q^{95} +2.14010e7 q^{96} -1.10423e7 q^{97} -6.35048e6 q^{98} +3.32303e7 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 14 q - 52 q^{3} + 766 q^{4} + 6982 q^{9} - 1018 q^{10} - 38380 q^{12} - 47916 q^{14} + 1266 q^{16} - 76806 q^{17} + 251764 q^{22} - 137100 q^{23} - 39380 q^{25} - 432400 q^{27} - 443166 q^{29} - 315780 q^{30}+ \cdots - 86840772 q^{95}+O(q^{100}) \) 14 * q - 52 * q^3 + 766 * q^4 + 6982 * q^9 - 1018 * q^10 - 38380 * q^12 - 47916 * q^14 + 1266 * q^16 - 76806 * q^17 + 251764 * q^22 - 137100 * q^23 - 39380 * q^25 - 432400 * q^27 - 443166 * q^29 - 315780 * q^30 - 1319232 * q^35 + 1271350 * q^36 - 953640 * q^38 - 77182 * q^40 - 412812 * q^42 - 1765384 * q^43 - 2267260 * q^48 + 5307958 * q^49 + 38964 * q^51 - 1791210 * q^53 - 4907072 * q^55 - 14915904 * q^56 - 7717050 * q^61 + 3808284 * q^62 - 14570922 * q^64 - 31956396 * q^66 - 19139238 * q^68 - 27487680 * q^69 - 17280210 * q^74 + 20632304 * q^75 - 19599336 * q^77 - 30688440 * q^79 - 27772442 * q^81 - 74719438 * q^82 - 25539372 * q^87 + 54091072 * q^88 - 41193546 * q^90 - 47685588 * q^92 - 38277620 * q^94 - 86840772 * q^95

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