LMFDB - Embedded newform 63.10.a.d.1.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [63,10,Mod(1,63)] 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("63.1"); S:= CuspForms(chi, 10); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(63, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 10, names="a")

Level:	$ N $	$=$	$ 63 = 3^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 10 $
Character orbit:	$[\chi]$	$=$	63.a (trivial)

Newform invariants

comment:select newform

sage:traces = [2,6] 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:	$32.4472576783$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{193}) $
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 48 $ x^2 - x - 48
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 7)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-6.44622$ of defining polynomial
Character	$\chi$	$=$	63.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q-10.8924 q^{2} -393.355 q^{4} -200.782 q^{5} -2401.00 q^{7} +9861.52 q^{8} +2187.01 q^{10} -63864.3 q^{11} -164679. q^{13} +26152.8 q^{14} +93981.5 q^{16} +362910. q^{17} -436498. q^{19} +78978.6 q^{20} +695638. q^{22} -918199. q^{23} -1.91281e6 q^{25} +1.79375e6 q^{26} +944445. q^{28} +3.68643e6 q^{29} +3.47629e6 q^{31} -6.07279e6 q^{32} -3.95298e6 q^{34} +482078. q^{35} +1.88149e7 q^{37} +4.75453e6 q^{38} -1.98002e6 q^{40} -2.40714e6 q^{41} -1.25306e7 q^{43} +2.51213e7 q^{44} +1.00014e7 q^{46} +5.54509e7 q^{47} +5.76480e6 q^{49} +2.08352e7 q^{50} +6.47772e7 q^{52} +9.26889e7 q^{53} +1.28228e7 q^{55} -2.36775e7 q^{56} -4.01542e7 q^{58} +2.52600e7 q^{59} +6.93275e7 q^{61} -3.78653e7 q^{62} +1.80290e7 q^{64} +3.30646e7 q^{65} -2.33494e7 q^{67} -1.42752e8 q^{68} -5.25101e6 q^{70} +1.06194e8 q^{71} -2.10115e8 q^{73} -2.04940e8 q^{74} +1.71699e8 q^{76} +1.53338e8 q^{77} -149606. q^{79} -1.88698e7 q^{80} +2.62197e7 q^{82} -5.21565e8 q^{83} -7.28659e7 q^{85} +1.36489e8 q^{86} -6.29799e8 q^{88} -2.98587e8 q^{89} +3.95394e8 q^{91} +3.61178e8 q^{92} -6.03996e8 q^{94} +8.76410e7 q^{95} -8.95983e8 q^{97} -6.27928e7 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 6 q^{2} - 620 q^{4} + 2238 q^{5} - 4802 q^{7} - 2616 q^{8} + 43384 q^{10} - 35316 q^{11} - 26530 q^{13} - 14406 q^{14} - 752 q^{16} + 463920 q^{17} - 925426 q^{19} - 473760 q^{20} + 1177888 q^{22}+ \cdots + 34588806 q^{98}+O(q^{100}) $ 2 * q + 6 * q^2 - 620 * q^4 + 2238 * q^5 - 4802 * q^7 - 2616 * q^8 + 43384 * q^10 - 35316 * q^11 - 26530 * q^13 - 14406 * q^14 - 752 * q^16 + 463920 * q^17 - 925426 * q^19 - 473760 * q^20 + 1177888 * q^22 - 778128 * q^23 + 2081722 * q^25 + 4127424 * q^26 + 1488620 * q^28 + 10003584 * q^29 + 2467260 * q^31 - 1284576 * q^32 - 2246676 * q^34 - 5373438 * q^35 + 30735552 * q^37 - 3504660 * q^38 - 32409984 * q^40 + 19103448 * q^41 + 4065100 * q^43 + 18650976 * q^44 + 12367584 * q^46 + 82195020 * q^47 + 11529602 * q^49 + 88312626 * q^50 + 33466384 * q^52 + 55189812 * q^53 + 82445816 * q^55 + 6281016 * q^56 + 66558004 * q^58 + 7069218 * q^59 + 44316386 * q^61 - 54910200 * q^62 + 147417152 * q^64 + 369979260 * q^65 - 241921336 * q^67 - 165645816 * q^68 - 104164984 * q^70 - 206493816 * q^71 - 499153188 * q^73 - 3571524 * q^74 + 282511768 * q^76 + 84793716 * q^77 + 468535096 * q^79 - 249904128 * q^80 + 389586092 * q^82 - 444023958 * q^83 + 173475060 * q^85 + 416830608 * q^86 - 986010816 * q^88 - 636267396 * q^89 + 63698530 * q^91 + 329431488 * q^92 - 152223192 * q^94 - 1104747984 * q^95 - 1632716064 * q^97 + 34588806 * q^98

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 $

$2$	−10.8924	−0.481383	−0.240691	−	0.970602i	$-0.577374\pi$
$2$	−10.8924	−0.481383	−0.240691	+	0.970602i	$0.577374\pi$
$3$	0	0
$3$	0	0
$4$	−393.355	−0.768271
$5$	−200.782	−0.143668	−0.0718340	−	0.997417i	$-0.522885\pi$
$5$	−200.782	−0.143668	−0.0718340	+	0.997417i	$0.522885\pi$
$6$	0	0
$7$	−2401.00	−0.377964
$7$	−2401.00	−0.377964
$8$	9861.52	0.851215
$9$	0	0
$10$	2187.01	0.0691593
$11$	−63864.3	−1.31520	−0.657599	−	0.753369i	$-0.728428\pi$
$11$	−63864.3	−1.31520	−0.657599	+	0.753369i	$0.728428\pi$
$12$	0	0
$13$	−164679.	−1.59916	−0.799581	−	0.600558i	$-0.794945\pi$
$13$	−164679.	−1.59916	−0.799581	+	0.600558i	$0.794945\pi$
$14$	26152.8	0.181946
$15$	0	0
$16$	93981.5	0.358511
$17$	362910.	1.05385	0.526925	−	0.849912i	$-0.323345\pi$
$17$	362910.	1.05385	0.526925	+	0.849912i	$0.323345\pi$
$18$	0	0
$19$	−436498.	−0.768406	−0.384203	−	0.923249i	$-0.625524\pi$
$19$	−436498.	−0.768406	−0.384203	+	0.923249i	$0.625524\pi$
$20$	78978.6	0.110376
$21$	0	0
$22$	695638.	0.633113
$23$	−918199.	−0.684166	−0.342083	−	0.939670i	$-0.611132\pi$
$23$	−918199.	−0.684166	−0.342083	+	0.939670i	$0.611132\pi$
$24$	0	0
$25$	−1.91281e6	−0.979359
$26$	1.79375e6	0.769809
$27$	0	0
$28$	944445.	0.290379
$29$	3.68643e6	0.967865	0.483932	−	0.875105i	$-0.339208\pi$
$29$	3.68643e6	0.967865	0.483932	+	0.875105i	$0.339208\pi$
$30$	0	0
$31$	3.47629e6	0.676064	0.338032	−	0.941135i	$-0.390239\pi$
$31$	3.47629e6	0.676064	0.338032	+	0.941135i	$0.390239\pi$
$32$	−6.07279e6	−1.02380
$33$	0	0
$34$	−3.95298e6	−0.507305
$35$	482078.	0.0543014
$36$	0	0
$37$	1.88149e7	1.65042	0.825210	−	0.564826i	$-0.191057\pi$
$37$	1.88149e7	1.65042	0.825210	+	0.564826i	$0.191057\pi$
$38$	4.75453e6	0.369897
$39$	0	0
$40$	−1.98002e6	−0.122292
$41$	−2.40714e6	−0.133038	−0.0665188	−	0.997785i	$-0.521189\pi$
$41$	−2.40714e6	−0.133038	−0.0665188	+	0.997785i	$0.521189\pi$
$42$	0	0
$43$	−1.25306e7	−0.558938	−0.279469	−	0.960155i	$-0.590158\pi$
$43$	−1.25306e7	−0.558938	−0.279469	+	0.960155i	$0.590158\pi$
$44$	2.51213e7	1.01043
$45$	0	0
$46$	1.00014e7	0.329346
$47$	5.54509e7	1.65756	0.828779	−	0.559577i	$-0.189036\pi$
$47$	5.54509e7	1.65756	0.828779	+	0.559577i	$0.189036\pi$
$48$	0	0
$49$	5.76480e6	0.142857
$50$	2.08352e7	0.471447
$51$	0	0
$52$	6.47772e7	1.22859
$53$	9.26889e7	1.61356	0.806782	−	0.590849i	$-0.201207\pi$
$53$	9.26889e7	1.61356	0.806782	+	0.590849i	$0.201207\pi$
$54$	0	0
$55$	1.28228e7	0.188952
$56$	−2.36775e7	−0.321729
$57$	0	0
$58$	−4.01542e7	−0.465913
$59$	2.52600e7	0.271393	0.135696	−	0.990750i	$-0.456673\pi$
$59$	2.52600e7	0.271393	0.135696	+	0.990750i	$0.456673\pi$
$60$	0	0
$61$	6.93275e7	0.641093	0.320547	−	0.947233i	$-0.396133\pi$
$61$	6.93275e7	0.641093	0.320547	+	0.947233i	$0.396133\pi$
$62$	−3.78653e7	−0.325446
$63$	0	0
$64$	1.80290e7	0.134326
$65$	3.30646e7	0.229748
$66$	0	0
$67$	−2.33494e7	−0.141559	−0.0707796	−	0.997492i	$-0.522549\pi$
$67$	−2.33494e7	−0.141559	−0.0707796	+	0.997492i	$0.522549\pi$
$68$	−1.42752e8	−0.809643
$69$	0	0
$70$	−5.25101e6	−0.0261398
$71$	1.06194e8	0.495950	0.247975	−	0.968766i	$-0.420235\pi$
$71$	1.06194e8	0.495950	0.247975	+	0.968766i	$0.420235\pi$
$72$	0	0
$73$	−2.10115e8	−0.865974	−0.432987	−	0.901400i	$-0.642540\pi$
$73$	−2.10115e8	−0.865974	−0.432987	+	0.901400i	$0.642540\pi$
$74$	−2.04940e8	−0.794483
$75$	0	0
$76$	1.71699e8	0.590344
$77$	1.53338e8	0.497098
$78$	0	0
$79$	−149606.	−0.000432144 0	−0.000216072	−	1.00000i	$-0.500069\pi$
$79$	−149606.	−0.000432144 0	−0.000216072		1.00000i	$0.500069\pi$
$80$	−1.88698e7	−0.0515066
$81$	0	0
$82$	2.62197e7	0.0640420
$83$	−5.21565e8	−1.20630	−0.603152	−	0.797626i	$-0.706089\pi$
$83$	−5.21565e8	−1.20630	−0.603152	+	0.797626i	$0.706089\pi$
$84$	0	0
$85$	−7.28659e7	−0.151405
$86$	1.36489e8	0.269063
$87$	0	0
$88$	−6.29799e8	−1.11952
$89$	−2.98587e8	−0.504448	−0.252224	−	0.967669i	$-0.581162\pi$
$89$	−2.98587e8	−0.504448	−0.252224	+	0.967669i	$0.581162\pi$
$90$	0	0
$91$	3.95394e8	0.604426
$92$	3.61178e8	0.525625
$93$	0	0
$94$	−6.03996e8	−0.797919
$95$	8.76410e7	0.110395
$96$	0	0
$97$	−8.95983e8	−1.02761	−0.513803	−	0.857908i	$-0.671764\pi$
$97$	−8.95983e8	−1.02761	−0.513803	+	0.857908i	$0.671764\pi$
$98$	−6.27928e7	−0.0687689
$99$	0	0
$100$	7.52413e8	0.752413
$101$	4.13043e7	0.0394956	0.0197478	−	0.999805i	$-0.493714\pi$
$101$	4.13043e7	0.0394956	0.0197478	+	0.999805i	$0.493714\pi$
$102$	0	0
$103$	4.29472e8	0.375982	0.187991	−	0.982171i	$-0.439802\pi$
$103$	4.29472e8	0.375982	0.187991	+	0.982171i	$0.439802\pi$
$104$	−1.62398e9	−1.36123
$105$	0	0
$106$	−1.00961e9	−0.776742
$107$	−1.23287e9	−0.909265	−0.454632	−	0.890679i	$-0.650229\pi$
$107$	−1.23287e9	−0.909265	−0.454632	+	0.890679i	$0.650229\pi$
$108$	0	0
$109$	1.69619e9	1.15095	0.575475	−	0.817820i	$-0.304817\pi$
$109$	1.69619e9	1.15095	0.575475	+	0.817820i	$0.304817\pi$
$110$	−1.39672e8	−0.0909581
$111$	0	0
$112$	−2.25650e8	−0.135504
$113$	1.42449e9	0.821878	0.410939	−	0.911663i	$-0.365201\pi$
$113$	1.42449e9	0.821878	0.410939	+	0.911663i	$0.365201\pi$
$114$	0	0
$115$	1.84358e8	0.0982928
$116$	−1.45007e9	−0.743582
$117$	0	0
$118$	−2.75143e8	−0.130644
$119$	−8.71347e8	−0.398318
$120$	0	0
$121$	1.72070e9	0.729744
$122$	−7.55146e8	−0.308611
$123$	0	0
$124$	−1.36741e9	−0.519401
$125$	7.76211e8	0.284371
$126$	0	0
$127$	−3.12858e9	−1.06716	−0.533581	−	0.845749i	$-0.679154\pi$
$127$	−3.12858e9	−1.06716	−0.533581	+	0.845749i	$0.679154\pi$
$128$	2.91289e9	0.959133
$129$	0	0
$130$	−3.60154e8	−0.110597
$131$	−7.03123e8	−0.208598	−0.104299	−	0.994546i	$-0.533260\pi$
$131$	−7.03123e8	−0.208598	−0.104299	+	0.994546i	$0.533260\pi$
$132$	0	0
$133$	1.04803e9	0.290430
$134$	2.54332e8	0.0681442
$135$	0	0
$136$	3.57885e9	0.897053
$137$	3.07310e9	0.745306	0.372653	−	0.927971i	$-0.378448\pi$
$137$	3.07310e9	0.745306	0.372653	+	0.927971i	$0.378448\pi$
$138$	0	0
$139$	5.07806e9	1.15380	0.576901	−	0.816814i	$-0.304262\pi$
$139$	5.07806e9	1.15380	0.576901	+	0.816814i	$0.304262\pi$
$140$	−1.89628e8	−0.0417182
$141$	0	0
$142$	−1.15671e9	−0.238742
$143$	1.05171e10	2.10321
$144$	0	0
$145$	−7.40169e8	−0.139051
$146$	2.28867e9	0.416865
$147$	0	0
$148$	−7.40093e9	−1.26797
$149$	−2.13455e9	−0.354788	−0.177394	−	0.984140i	$-0.556767\pi$
$149$	−2.13455e9	−0.354788	−0.177394	+	0.984140i	$0.556767\pi$
$150$	0	0
$151$	−2.35298e9	−0.368318	−0.184159	−	0.982897i	$-0.558956\pi$
$151$	−2.35298e9	−0.368318	−0.184159	+	0.982897i	$0.558956\pi$
$152$	−4.30454e9	−0.654079
$153$	0	0
$154$	−1.67023e9	−0.239294
$155$	−6.97977e8	−0.0971288
$156$	0	0
$157$	−2.98482e9	−0.392075	−0.196038	−	0.980596i	$-0.562807\pi$
$157$	−2.98482e9	−0.392075	−0.196038	+	0.980596i	$0.562807\pi$
$158$	1.62958e6	0.000208027 0
$159$	0	0
$160$	1.21931e9	0.147087
$161$	2.20460e9	0.258591
$162$	0	0
$163$	9.20745e9	1.02163	0.510817	−	0.859690i	$-0.329343\pi$
$163$	9.20745e9	1.02163	0.510817	+	0.859690i	$0.329343\pi$
$164$	9.46860e8	0.102209
$165$	0	0
$166$	5.68111e9	0.580694
$167$	4.95501e9	0.492970	0.246485	−	0.969147i	$-0.420724\pi$
$167$	4.95501e9	0.492970	0.246485	+	0.969147i	$0.420724\pi$
$168$	0	0
$169$	1.65146e10	1.55732
$170$	7.93688e8	0.0728835
$171$	0	0
$172$	4.92897e9	0.429416
$173$	−3.77546e9	−0.320452	−0.160226	−	0.987080i	$-0.551222\pi$
$173$	−3.77546e9	−0.320452	−0.160226	+	0.987080i	$0.551222\pi$
$174$	0	0
$175$	4.59266e9	0.370163
$176$	−6.00206e9	−0.471513
$177$	0	0
$178$	3.25235e9	0.242833
$179$	1.94362e10	1.41505	0.707526	−	0.706687i	$-0.249811\pi$
$179$	1.94362e10	1.41505	0.707526	+	0.706687i	$0.249811\pi$
$180$	0	0
$181$	−9.81530e9	−0.679751	−0.339875	−	0.940470i	$-0.610385\pi$
$181$	−9.81530e9	−0.679751	−0.339875	+	0.940470i	$0.610385\pi$
$182$	−4.30680e9	−0.290960
$183$	0	0
$184$	−9.05485e9	−0.582373
$185$	−3.77770e9	−0.237113
$186$	0	0
$187$	−2.31770e10	−1.38602
$188$	−2.18119e10	−1.27345
$189$	0	0
$190$	−9.54625e8	−0.0531424
$191$	1.84579e10	1.00354	0.501768	−	0.865002i	$-0.332683\pi$
$191$	1.84579e10	1.00354	0.501768	+	0.865002i	$0.332683\pi$
$192$	0	0
$193$	4.83605e9	0.250890	0.125445	−	0.992101i	$-0.459964\pi$
$193$	4.83605e9	0.250890	0.125445	+	0.992101i	$0.459964\pi$
$194$	9.75944e9	0.494672
$195$	0	0
$196$	−2.26761e9	−0.109753
$197$	−2.70242e10	−1.27836	−0.639182	−	0.769056i	$-0.720727\pi$
$197$	−2.70242e10	−1.27836	−0.639182	+	0.769056i	$0.720727\pi$
$198$	0	0
$199$	1.65772e10	0.749330	0.374665	−	0.927160i	$-0.377758\pi$
$199$	1.65772e10	0.749330	0.374665	+	0.927160i	$0.377758\pi$
$200$	−1.88632e10	−0.833645
$201$	0	0
$202$	−4.49904e8	−0.0190125
$203$	−8.85111e9	−0.365819
$204$	0	0
$205$	4.83311e8	0.0191132
$206$	−4.67800e9	−0.180991
$207$	0	0
$208$	−1.54768e10	−0.573317
$209$	2.78766e10	1.01061
$210$	0	0
$211$	−5.44866e10	−1.89243	−0.946213	−	0.323544i	$-0.895126\pi$
$211$	−5.44866e10	−1.89243	−0.946213	+	0.323544i	$0.895126\pi$
$212$	−3.64596e10	−1.23965
$213$	0	0
$214$	1.34290e10	0.437704
$215$	2.51592e9	0.0803015
$216$	0	0
$217$	−8.34657e9	−0.255528
$218$	−1.84757e10	−0.554047
$219$	0	0
$220$	−5.04391e9	−0.145166
$221$	−5.97636e10	−1.68528
$222$	0	0
$223$	−2.05500e10	−0.556468	−0.278234	−	0.960513i	$-0.589749\pi$
$223$	−2.05500e10	−0.556468	−0.278234	+	0.960513i	$0.589749\pi$
$224$	1.45808e10	0.386958
$225$	0	0
$226$	−1.55162e10	−0.395638
$227$	3.94058e10	0.985017	0.492509	−	0.870307i	$-0.336080\pi$
$227$	3.94058e10	0.985017	0.492509	+	0.870307i	$0.336080\pi$
$228$	0	0
$229$	−1.82516e10	−0.438572	−0.219286	−	0.975661i	$-0.570373\pi$
$229$	−1.82516e10	−0.438572	−0.219286	+	0.975661i	$0.570373\pi$
$230$	−2.00811e9	−0.0473165
$231$	0	0
$232$	3.63538e10	0.823861
$233$	5.08161e10	1.12953	0.564767	−	0.825250i	$-0.308966\pi$
$233$	5.08161e10	1.12953	0.564767	+	0.825250i	$0.308966\pi$
$234$	0	0
$235$	−1.11336e10	−0.238138
$236$	−9.93612e9	−0.208503
$237$	0	0
$238$	9.49110e9	0.191743
$239$	−3.80447e10	−0.754230	−0.377115	−	0.926166i	$-0.623084\pi$
$239$	−3.80447e10	−0.754230	−0.377115	+	0.926166i	$0.623084\pi$
$240$	0	0
$241$	9.80077e10	1.87147	0.935737	−	0.352699i	$-0.114736\pi$
$241$	9.80077e10	1.87147	0.935737	+	0.352699i	$0.114736\pi$
$242$	−1.87426e10	−0.351286
$243$	0	0
$244$	−2.72703e10	−0.492533
$245$	−1.15747e9	−0.0205240
$246$	0	0
$247$	7.18819e10	1.22881
$248$	3.42815e10	0.575476
$249$	0	0
$250$	−8.45484e9	−0.136891
$251$	−7.75125e10	−1.23265	−0.616325	−	0.787492i	$-0.711379\pi$
$251$	−7.75125e10	−1.23265	−0.616325	+	0.787492i	$0.711379\pi$
$252$	0	0
$253$	5.86401e10	0.899814
$254$	3.40778e10	0.513713
$255$	0	0
$256$	−4.09593e10	−0.596036
$257$	1.07589e11	1.53840	0.769199	−	0.639009i	$-0.220655\pi$
$257$	1.07589e11	1.53840	0.769199	+	0.639009i	$0.220655\pi$
$258$	0	0
$259$	−4.51746e10	−0.623800
$260$	−1.30061e10	−0.176509
$261$	0	0
$262$	7.65873e9	0.100416
$263$	1.21615e10	0.156742	0.0783708	−	0.996924i	$-0.475028\pi$
$263$	1.21615e10	0.156742	0.0783708	+	0.996924i	$0.475028\pi$
$264$	0	0
$265$	−1.86103e10	−0.231818
$266$	−1.14156e10	−0.139808
$267$	0	0
$268$	9.18458e9	0.108756
$269$	1.23517e11	1.43827	0.719134	−	0.694872i	$-0.244539\pi$
$269$	1.23517e11	1.43827	0.719134	+	0.694872i	$0.244539\pi$
$270$	0	0
$271$	1.34305e11	1.51263	0.756313	−	0.654210i	$-0.226999\pi$
$271$	1.34305e11	1.51263	0.756313	+	0.654210i	$0.226999\pi$
$272$	3.41068e10	0.377817
$273$	0	0
$274$	−3.34736e10	−0.358777
$275$	1.22160e11	1.28805
$276$	0	0
$277$	−2.16684e10	−0.221140	−0.110570	−	0.993868i	$-0.535268\pi$
$277$	−2.16684e10	−0.221140	−0.110570	+	0.993868i	$0.535268\pi$
$278$	−5.53125e10	−0.555420
$279$	0	0
$280$	4.75402e9	0.0462222
$281$	−7.73283e9	−0.0739878	−0.0369939	−	0.999315i	$-0.511778\pi$
$281$	−7.73283e9	−0.0739878	−0.0369939	+	0.999315i	$0.511778\pi$
$282$	0	0
$283$	−7.19601e10	−0.666888	−0.333444	−	0.942770i	$-0.608211\pi$
$283$	−7.19601e10	−0.666888	−0.333444	+	0.942770i	$0.608211\pi$
$284$	−4.17720e10	−0.381024
$285$	0	0
$286$	−1.14557e11	−1.01245
$287$	5.77955e9	0.0502835
$288$	0	0
$289$	1.31159e10	0.110601
$290$	8.06225e9	0.0669368
$291$	0	0
$292$	8.26498e10	0.665303
$293$	−1.73674e11	−1.37667	−0.688335	−	0.725393i	$-0.741658\pi$
$293$	−1.73674e11	−1.37667	−0.688335	+	0.725393i	$0.741658\pi$
$294$	0	0
$295$	−5.07175e9	−0.0389905
$296$	1.85544e11	1.40486
$297$	0	0
$298$	2.32505e10	0.170789
$299$	1.51208e11	1.09409
$300$	0	0
$301$	3.00860e10	0.211259
$302$	2.56297e10	0.177302
$303$	0	0
$304$	−4.10227e10	−0.275482
$305$	−1.39197e10	−0.0921046
$306$	0	0
$307$	2.27108e11	1.45918	0.729591	−	0.683884i	$-0.239710\pi$
$307$	2.27108e11	1.45918	0.729591	+	0.683884i	$0.239710\pi$
$308$	−6.03163e10	−0.381906
$309$	0	0
$310$	7.60267e9	0.0467561
$311$	8.68962e10	0.526719	0.263360	−	0.964698i	$-0.415169\pi$
$311$	8.68962e10	0.526719	0.263360	+	0.964698i	$0.415169\pi$
$312$	0	0
$313$	−5.25289e10	−0.309349	−0.154674	−	0.987965i	$-0.549433\pi$
$313$	−5.25289e10	−0.309349	−0.154674	+	0.987965i	$0.549433\pi$
$314$	3.25120e10	0.188738
$315$	0	0
$316$	5.88484e7	0.000332004 0
$317$	−2.63784e11	−1.46718	−0.733588	−	0.679595i	$-0.762156\pi$
$317$	−2.63784e11	−1.46718	−0.733588	+	0.679595i	$0.762156\pi$
$318$	0	0
$319$	−2.35431e11	−1.27293
$320$	−3.61990e9	−0.0192984
$321$	0	0
$322$	−2.40134e10	−0.124481
$323$	−1.58410e11	−0.809786
$324$	0	0
$325$	3.14999e11	1.56615
$326$	−1.00292e11	−0.491797
$327$	0	0
$328$	−2.37381e10	−0.113244
$329$	−1.33138e11	−0.626498
$330$	0	0
$331$	−2.11350e11	−0.967780	−0.483890	−	0.875129i	$-0.660776\pi$
$331$	−2.11350e11	−0.967780	−0.483890	+	0.875129i	$0.660776\pi$
$332$	2.05160e11	0.926768
$333$	0	0
$334$	−5.39722e10	−0.237307
$335$	4.68813e9	0.0203375
$336$	0	0
$337$	−3.67482e11	−1.55203	−0.776017	−	0.630712i	$-0.782763\pi$
$337$	−3.67482e11	−1.55203	−0.776017	+	0.630712i	$0.782763\pi$
$338$	−1.79884e11	−0.749666
$339$	0	0
$340$	2.86621e10	0.116320
$341$	−2.22011e11	−0.889158
$342$	0	0
$343$	−1.38413e10	−0.0539949
$344$	−1.23571e11	−0.475776
$345$	0	0
$346$	4.11240e10	0.154260
$347$	−4.33866e11	−1.60647	−0.803235	−	0.595662i	$-0.796890\pi$
$347$	−4.33866e11	−1.60647	−0.803235	+	0.595662i	$0.796890\pi$
$348$	0	0
$349$	1.04086e11	0.375559	0.187780	−	0.982211i	$-0.439871\pi$
$349$	1.04086e11	0.375559	0.187780	+	0.982211i	$0.439871\pi$
$350$	−5.00253e10	−0.178190
$351$	0	0
$352$	3.87834e11	1.34649
$353$	4.46890e11	1.53184	0.765922	−	0.642934i	$-0.222283\pi$
$353$	4.46890e11	1.53184	0.765922	+	0.642934i	$0.222283\pi$
$354$	0	0
$355$	−2.13219e10	−0.0712522
$356$	1.17451e11	0.387553
$357$	0	0
$358$	−2.11708e11	−0.681182
$359$	1.96551e11	0.624526	0.312263	−	0.949996i	$-0.398913\pi$
$359$	1.96551e11	0.624526	0.312263	+	0.949996i	$0.398913\pi$
$360$	0	0
$361$	−1.32157e11	−0.409551
$362$	1.06913e11	0.327220
$363$	0	0
$364$	−1.55530e11	−0.464363
$365$	4.21874e10	0.124413
$366$	0	0
$367$	−2.25518e11	−0.648908	−0.324454	−	0.945901i	$-0.605181\pi$
$367$	−2.25518e11	−0.648908	−0.324454	+	0.945901i	$0.605181\pi$
$368$	−8.62937e10	−0.245281
$369$	0	0
$370$	4.11484e10	0.114142
$371$	−2.22546e11	−0.609870
$372$	0	0
$373$	−2.04414e11	−0.546791	−0.273395	−	0.961902i	$-0.588147\pi$
$373$	−2.04414e11	−0.546791	−0.273395	+	0.961902i	$0.588147\pi$
$374$	2.52454e11	0.667206
$375$	0	0
$376$	5.46831e11	1.41094
$377$	−6.07076e11	−1.54777
$378$	0	0
$379$	4.03306e10	0.100406	0.0502029	−	0.998739i	$-0.484013\pi$
$379$	4.03306e10	0.100406	0.0502029	+	0.998739i	$0.484013\pi$
$380$	−3.44740e10	−0.0848136
$381$	0	0
$382$	−2.01052e11	−0.483084
$383$	3.98325e11	0.945895	0.472947	−	0.881091i	$-0.343190\pi$
$383$	3.98325e11	0.945895	0.472947	+	0.881091i	$0.343190\pi$
$384$	0	0
$385$	−3.07876e10	−0.0714171
$386$	−5.26764e10	−0.120774
$387$	0	0
$388$	3.52439e11	0.789480
$389$	4.22504e11	0.935531	0.467765	−	0.883853i	$-0.345059\pi$
$389$	4.22504e11	0.935531	0.467765	+	0.883853i	$0.345059\pi$
$390$	0	0
$391$	−3.33224e11	−0.721009
$392$	5.68497e10	0.121602
$393$	0	0
$394$	2.94359e11	0.615382
$395$	3.00383e7	6.20853e−5 0
$396$	0	0
$397$	6.50999e11	1.31529	0.657647	−	0.753326i	$-0.271552\pi$
$397$	6.50999e11	1.31529	0.657647	+	0.753326i	$0.271552\pi$
$398$	−1.80567e11	−0.360714
$399$	0	0
$400$	−1.79769e11	−0.351111
$401$	3.10509e11	0.599687	0.299843	−	0.953988i	$-0.403066\pi$
$401$	3.10509e11	0.599687	0.299843	+	0.953988i	$0.403066\pi$
$402$	0	0
$403$	−5.72471e11	−1.08114
$404$	−1.62472e10	−0.0303433
$405$	0	0
$406$	9.64102e10	0.176099
$407$	−1.20160e12	−2.17063
$408$	0	0
$409$	2.99519e11	0.529261	0.264630	−	0.964350i	$-0.414750\pi$
$409$	2.99519e11	0.529261	0.264630	+	0.964350i	$0.414750\pi$
$410$	−5.26444e9	−0.00920078
$411$	0	0
$412$	−1.68935e11	−0.288856
$413$	−6.06492e10	−0.102577
$414$	0	0
$415$	1.04721e11	0.173307
$416$	1.00006e12	1.63722
$417$	0	0
$418$	−3.03645e11	−0.486488
$419$	4.35217e11	0.689832	0.344916	−	0.938634i	$-0.387907\pi$
$419$	4.35217e11	0.689832	0.344916	+	0.938634i	$0.387907\pi$
$420$	0	0
$421$	−2.07600e11	−0.322076	−0.161038	−	0.986948i	$-0.551484\pi$
$421$	−2.07600e11	−0.322076	−0.161038	+	0.986948i	$0.551484\pi$
$422$	5.93493e11	0.910981
$423$	0	0
$424$	9.14054e11	1.37349
$425$	−6.94179e11	−1.03210
$426$	0	0
$427$	−1.66455e11	−0.242310
$428$	4.84955e11	0.698562
$429$	0	0
$430$	−2.74045e10	−0.0386557
$431$	−8.61584e11	−1.20268	−0.601340	−	0.798993i	$-0.705366\pi$
$431$	−8.61584e11	−1.20268	−0.601340	+	0.798993i	$0.705366\pi$
$432$	0	0
$433$	1.45840e11	0.199379	0.0996896	−	0.995019i	$-0.468215\pi$
$433$	1.45840e11	0.199379	0.0996896	+	0.995019i	$0.468215\pi$
$434$	9.09145e10	0.123007
$435$	0	0
$436$	−6.67206e11	−0.884241
$437$	4.00792e11	0.525718
$438$	0	0
$439$	−1.07131e11	−0.137665	−0.0688324	−	0.997628i	$-0.521927\pi$
$439$	−1.07131e11	−0.137665	−0.0688324	+	0.997628i	$0.521927\pi$
$440$	1.26452e11	0.160839
$441$	0	0
$442$	6.50972e11	0.811263
$443$	1.36838e11	0.168806	0.0844031	−	0.996432i	$-0.473102\pi$
$443$	1.36838e11	0.168806	0.0844031	+	0.996432i	$0.473102\pi$
$444$	0	0
$445$	5.99510e10	0.0724731
$446$	2.23840e11	0.267874
$447$	0	0
$448$	−4.32876e10	−0.0507706
$449$	7.65671e11	0.889065	0.444532	−	0.895763i	$-0.353370\pi$
$449$	7.65671e11	0.889065	0.444532	+	0.895763i	$0.353370\pi$
$450$	0	0
$451$	1.53730e11	0.174971
$452$	−5.60331e11	−0.631425
$453$	0	0
$454$	−4.29226e11	−0.474170
$455$	−7.93880e10	−0.0868368
$456$	0	0
$457$	5.41842e11	0.581099	0.290549	−	0.956860i	$-0.406162\pi$
$457$	5.41842e11	0.581099	0.290549	+	0.956860i	$0.406162\pi$
$458$	1.98805e11	0.211121
$459$	0	0
$460$	−7.25181e10	−0.0755155
$461$	7.10838e11	0.733021	0.366510	−	0.930414i	$-0.380552\pi$
$461$	7.10838e11	0.733021	0.366510	+	0.930414i	$0.380552\pi$
$462$	0	0
$463$	7.96272e11	0.805280	0.402640	−	0.915358i	$-0.368093\pi$
$463$	7.96272e11	0.805280	0.402640	+	0.915358i	$0.368093\pi$
$464$	3.46456e11	0.346990
$465$	0	0
$466$	−5.53511e11	−0.543738
$467$	1.67673e12	1.63132	0.815658	−	0.578535i	$-0.196375\pi$
$467$	1.67673e12	1.63132	0.815658	+	0.578535i	$0.196375\pi$
$468$	0	0
$469$	5.60618e10	0.0535044
$470$	1.21272e11	0.114635
$471$	0	0
$472$	2.49102e11	0.231014
$473$	8.00257e11	0.735114
$474$	0	0
$475$	8.34938e11	0.752546
$476$	3.42749e11	0.306016
$477$	0	0
$478$	4.14400e11	0.363073
$479$	−1.17129e12	−1.01661	−0.508305	−	0.861177i	$-0.669728\pi$
$479$	−1.17129e12	−1.01661	−0.508305	+	0.861177i	$0.669728\pi$
$480$	0	0
$481$	−3.09842e12	−2.63929
$482$	−1.06754e12	−0.900895
$483$	0	0
$484$	−6.76844e11	−0.560641
$485$	1.79897e11	0.147634
$486$	0	0
$487$	4.49797e11	0.362357	0.181178	−	0.983450i	$-0.442009\pi$
$487$	4.49797e11	0.362357	0.181178	+	0.983450i	$0.442009\pi$
$488$	6.83675e11	0.545708
$489$	0	0
$490$	1.26077e10	0.00987990
$491$	−2.25034e12	−1.74735	−0.873677	−	0.486506i	$-0.838271\pi$
$491$	−2.25034e12	−1.74735	−0.873677	+	0.486506i	$0.838271\pi$
$492$	0	0
$493$	1.33784e12	1.01998
$494$	−7.82970e11	−0.591526
$495$	0	0
$496$	3.26707e11	0.242376
$497$	−2.54972e11	−0.187452
$498$	0	0
$499$	1.59712e10	0.0115315	0.00576574	−	0.999983i	$-0.498165\pi$
$499$	1.59712e10	0.0115315	0.00576574	+	0.999983i	$0.498165\pi$
$500$	−3.05326e11	−0.218474
$501$	0	0
$502$	8.44300e11	0.593376
$503$	−1.73375e12	−1.20762	−0.603811	−	0.797127i	$-0.706352\pi$
$503$	−1.73375e12	−1.20762	−0.603811	+	0.797127i	$0.706352\pi$
$504$	0	0
$505$	−8.29316e9	−0.00567425
$506$	−6.38734e11	−0.433155
$507$	0	0
$508$	1.23064e12	0.819869
$509$	−9.77460e11	−0.645460	−0.322730	−	0.946491i	$-0.604601\pi$
$509$	−9.77460e11	−0.645460	−0.322730	+	0.946491i	$0.604601\pi$
$510$	0	0
$511$	5.04487e11	0.327307
$512$	−1.04525e12	−0.672212
$513$	0	0
$514$	−1.17191e12	−0.740558
$515$	−8.62303e10	−0.0540166
$516$	0	0
$517$	−3.54133e12	−2.18001
$518$	4.92062e11	0.300286
$519$	0	0
$520$	3.26067e11	0.195565
$521$	−5.89223e11	−0.350356	−0.175178	−	0.984537i	$-0.556050\pi$
$521$	−5.89223e11	−0.350356	−0.175178	+	0.984537i	$0.556050\pi$
$522$	0	0
$523$	1.43218e12	0.837027	0.418513	−	0.908211i	$-0.362551\pi$
$523$	1.43218e12	0.837027	0.418513	+	0.908211i	$0.362551\pi$
$524$	2.76577e11	0.160260
$525$	0	0
$526$	−1.32468e11	−0.0754527
$527$	1.26158e12	0.712471
$528$	0	0
$529$	−9.58063e11	−0.531916
$530$	2.02711e11	0.111593
$531$	0	0
$532$	−4.12248e11	−0.223129
$533$	3.96405e11	0.212749
$534$	0	0
$535$	2.47538e11	0.130632
$536$	−2.30260e11	−0.120497
$537$	0	0
$538$	−1.34540e12	−0.692357
$539$	−3.68165e11	−0.187885
$540$	0	0
$541$	1.17323e12	0.588839	0.294419	−	0.955676i	$-0.404874\pi$
$541$	1.17323e12	0.588839	0.294419	+	0.955676i	$0.404874\pi$
$542$	−1.46291e12	−0.728152
$543$	0	0
$544$	−2.20388e12	−1.07893
$545$	−3.40565e11	−0.165355
$546$	0	0
$547$	2.59515e11	0.123942	0.0619712	−	0.998078i	$-0.480261\pi$
$547$	2.59515e11	0.123942	0.0619712	+	0.998078i	$0.480261\pi$
$548$	−1.20882e12	−0.572597
$549$	0	0
$550$	−1.33062e12	−0.620045
$551$	−1.60912e12	−0.743714
$552$	0	0
$553$	3.59205e8	0.000163335 0
$554$	2.36022e11	0.106453
$555$	0	0
$556$	−1.99748e12	−0.886432
$557$	2.36651e12	1.04174	0.520871	−	0.853635i	$-0.325607\pi$
$557$	2.36651e12	1.04174	0.520871	+	0.853635i	$0.325607\pi$
$558$	0	0
$559$	2.06352e12	0.893832
$560$	4.53064e10	0.0194676
$561$	0	0
$562$	8.42294e10	0.0356164
$563$	2.42400e12	1.01682	0.508410	−	0.861115i	$-0.330233\pi$
$563$	2.42400e12	1.01682	0.508410	+	0.861115i	$0.330233\pi$
$564$	0	0
$565$	−2.86013e11	−0.118078
$566$	7.83821e11	0.321028
$567$	0	0
$568$	1.04724e12	0.422160
$569$	2.11854e12	0.847290	0.423645	−	0.905828i	$-0.360750\pi$
$569$	2.11854e12	0.847290	0.423645	+	0.905828i	$0.360750\pi$
$570$	0	0
$571$	−7.50992e11	−0.295646	−0.147823	−	0.989014i	$-0.547227\pi$
$571$	−7.50992e11	−0.295646	−0.147823	+	0.989014i	$0.547227\pi$
$572$	−4.13695e12	−1.61584
$573$	0	0
$574$	−6.29534e10	−0.0242056
$575$	1.75634e12	0.670045
$576$	0	0
$577$	4.35951e12	1.63737	0.818685	−	0.574243i	$-0.194704\pi$
$577$	4.35951e12	1.63737	0.818685	+	0.574243i	$0.194704\pi$
$578$	−1.42864e11	−0.0532413
$579$	0	0
$580$	2.91149e11	0.106829
$581$	1.25228e12	0.455940
$582$	0	0
$583$	−5.91951e12	−2.12216
$584$	−2.07206e12	−0.737130
$585$	0	0
$586$	1.89173e12	0.662704
$587$	2.20056e12	0.765002	0.382501	−	0.923955i	$-0.375063\pi$
$587$	2.20056e12	0.765002	0.382501	+	0.923955i	$0.375063\pi$
$588$	0	0
$589$	−1.51739e12	−0.519492
$590$	5.52438e10	0.0187693
$591$	0	0
$592$	1.76825e12	0.591693
$593$	2.83604e12	0.941815	0.470907	−	0.882183i	$-0.343927\pi$
$593$	2.83604e12	0.941815	0.470907	+	0.882183i	$0.343927\pi$
$594$	0	0
$595$	1.74951e11	0.0572256
$596$	8.39636e11	0.272573
$597$	0	0
$598$	−1.64702e12	−0.526677
$599$	8.29286e11	0.263199	0.131599	−	0.991303i	$-0.457989\pi$
$599$	8.29286e11	0.263199	0.131599	+	0.991303i	$0.457989\pi$
$600$	0	0
$601$	5.62459e12	1.75855	0.879277	−	0.476311i	$-0.158026\pi$
$601$	5.62459e12	1.75855	0.879277	+	0.476311i	$0.158026\pi$
$602$	−3.27710e11	−0.101696
$603$	0	0
$604$	9.25557e11	0.282968
$605$	−3.45485e11	−0.104841
$606$	0	0
$607$	−4.68777e11	−0.140158	−0.0700789	−	0.997541i	$-0.522325\pi$
$607$	−4.68777e11	−0.140158	−0.0700789	+	0.997541i	$0.522325\pi$
$608$	2.65076e12	0.786691
$609$	0	0
$610$	1.51620e11	0.0443375
$611$	−9.13159e12	−2.65070
$612$	0	0
$613$	5.58394e12	1.59723	0.798617	−	0.601840i	$-0.205566\pi$
$613$	5.58394e12	1.59723	0.798617	+	0.601840i	$0.205566\pi$
$614$	−2.47376e12	−0.702425
$615$	0	0
$616$	1.51215e12	0.423137
$617$	3.36717e12	0.935367	0.467683	−	0.883896i	$-0.345089\pi$
$617$	3.36717e12	0.935367	0.467683	+	0.883896i	$0.345089\pi$
$618$	0	0
$619$	−5.66928e12	−1.55210	−0.776051	−	0.630671i	$-0.782780\pi$
$619$	−5.66928e12	−1.55210	−0.776051	+	0.630671i	$0.782780\pi$
$620$	2.74552e11	0.0746212
$621$	0	0
$622$	−9.46512e11	−0.253553
$623$	7.16909e11	0.190663
$624$	0	0
$625$	3.58011e12	0.938505
$626$	5.72168e11	0.148915
$627$	0	0
$628$	1.17409e12	0.301220
$629$	6.82812e12	1.73930
$630$	0	0
$631$	−1.06685e12	−0.267899	−0.133950	−	0.990988i	$-0.542766\pi$
$631$	−1.06685e12	−0.267899	−0.133950	+	0.990988i	$0.542766\pi$
$632$	−1.47535e9	−0.000367847 0
$633$	0	0
$634$	2.87325e12	0.706273
$635$	6.28163e11	0.153317
$636$	0	0
$637$	−9.49340e11	−0.228452
$638$	2.56442e12	0.612768
$639$	0	0
$640$	−5.84856e11	−0.137797
$641$	7.83632e12	1.83337	0.916686	−	0.399607i	$-0.130853\pi$
$641$	7.83632e12	1.83337	0.916686	+	0.399607i	$0.130853\pi$
$642$	0	0
$643$	7.05971e12	1.62869	0.814343	−	0.580383i	$-0.197097\pi$
$643$	7.05971e12	1.62869	0.814343	+	0.580383i	$0.197097\pi$
$644$	−8.67188e11	−0.198668
$645$	0	0
$646$	1.72547e12	0.389817
$647$	−2.77314e12	−0.622161	−0.311081	−	0.950384i	$-0.600691\pi$
$647$	−2.77314e12	−0.622161	−0.311081	+	0.950384i	$0.600691\pi$
$648$	0	0
$649$	−1.61321e12	−0.356935
$650$	−3.43111e12	−0.753919
$651$	0	0
$652$	−3.62179e12	−0.784891
$653$	−3.09564e12	−0.666257	−0.333128	−	0.942882i	$-0.608104\pi$
$653$	−3.09564e12	−0.666257	−0.333128	+	0.942882i	$0.608104\pi$
$654$	0	0
$655$	1.41175e11	0.0299689
$656$	−2.26227e11	−0.0476954
$657$	0	0
$658$	1.45019e12	0.301585
$659$	4.80261e12	0.991958	0.495979	−	0.868335i	$-0.334809\pi$
$659$	4.80261e12	0.991958	0.495979	+	0.868335i	$0.334809\pi$
$660$	0	0
$661$	−5.13389e12	−1.04602	−0.523010	−	0.852327i	$-0.675191\pi$
$661$	−5.13389e12	−1.04602	−0.523010	+	0.852327i	$0.675191\pi$
$662$	2.30212e12	0.465872
$663$	0	0
$664$	−5.14342e12	−1.02682
$665$	−2.10426e11	−0.0417256
$666$	0	0
$667$	−3.38488e12	−0.662181
$668$	−1.94908e12	−0.378735
$669$	0	0
$670$	−5.10652e10	−0.00979014
$671$	−4.42755e12	−0.843164
$672$	0	0
$673$	4.43802e12	0.833915	0.416957	−	0.908926i	$-0.363096\pi$
$673$	4.43802e12	0.833915	0.416957	+	0.908926i	$0.363096\pi$
$674$	4.00277e12	0.747122
$675$	0	0
$676$	−6.49609e12	−1.19644
$677$	4.70675e12	0.861136	0.430568	−	0.902558i	$-0.358313\pi$
$677$	4.70675e12	0.861136	0.430568	+	0.902558i	$0.358313\pi$
$678$	0	0
$679$	2.15126e12	0.388399
$680$	−7.18569e11	−0.128878
$681$	0	0
$682$	2.41824e12	0.428025
$683$	−6.12667e11	−0.107729	−0.0538644	−	0.998548i	$-0.517154\pi$
$683$	−6.12667e11	−0.107729	−0.0538644	+	0.998548i	$0.517154\pi$
$684$	0	0
$685$	−6.17025e11	−0.107077
$686$	1.50765e11	0.0259922
$687$	0	0
$688$	−1.17764e12	−0.200385
$689$	−1.52639e13	−2.58035
$690$	0	0
$691$	2.69919e12	0.450383	0.225191	−	0.974315i	$-0.427699\pi$
$691$	2.69919e12	0.450383	0.225191	+	0.974315i	$0.427699\pi$
$692$	1.48510e12	0.246194
$693$	0	0
$694$	4.72586e12	0.773327
$695$	−1.01958e12	−0.165764
$696$	0	0
$697$	−8.73576e11	−0.140202
$698$	−1.13375e12	−0.180788
$699$	0	0
$700$	−1.80654e12	−0.284386
$701$	5.78506e12	0.904850	0.452425	−	0.891802i	$-0.350559\pi$
$701$	5.78506e12	0.904850	0.452425	+	0.891802i	$0.350559\pi$
$702$	0	0
$703$	−8.21267e12	−1.26819
$704$	−1.15141e12	−0.176666
$705$	0	0
$706$	−4.86772e12	−0.737403
$707$	−9.91715e10	−0.0149279
$708$	0	0
$709$	−3.72656e12	−0.553860	−0.276930	−	0.960890i	$-0.589317\pi$
$709$	−3.72656e12	−0.553860	−0.276930	+	0.960890i	$0.589317\pi$
$710$	2.32248e11	0.0342996
$711$	0	0
$712$	−2.94453e12	−0.429394
$713$	−3.19192e12	−0.462541
$714$	0	0
$715$	−2.11164e12	−0.302165
$716$	−7.64532e12	−1.08714
$717$	0	0
$718$	−2.14092e12	−0.300636
$719$	−1.33255e13	−1.85953	−0.929765	−	0.368154i	$-0.879990\pi$
$719$	−1.33255e13	−1.85953	−0.929765	+	0.368154i	$0.879990\pi$
$720$	0	0
$721$	−1.03116e12	−0.142108
$722$	1.43952e12	0.197151
$723$	0	0
$724$	3.86089e12	0.522233
$725$	−7.05144e12	−0.947888
$726$	0	0
$727$	−2.39852e12	−0.318448	−0.159224	−	0.987242i	$-0.550899\pi$
$727$	−2.39852e12	−0.318448	−0.159224	+	0.987242i	$0.550899\pi$
$728$	3.89918e12	0.514497
$729$	0	0
$730$	−4.59524e11	−0.0598901
$731$	−4.54748e12	−0.589037
$732$	0	0
$733$	7.93815e12	1.01567	0.507834	−	0.861455i	$-0.330446\pi$
$733$	7.93815e12	1.01567	0.507834	+	0.861455i	$0.330446\pi$
$734$	2.45644e12	0.312373
$735$	0	0
$736$	5.57603e12	0.700447
$737$	1.49119e12	0.186178
$738$	0	0
$739$	−2.20371e12	−0.271803	−0.135902	−	0.990722i	$-0.543393\pi$
$739$	−2.20371e12	−0.271803	−0.135902	+	0.990722i	$0.543393\pi$
$740$	1.48598e12	0.182167
$741$	0	0
$742$	2.42407e12	0.293581
$743$	1.09079e12	0.131308	0.0656540	−	0.997842i	$-0.479087\pi$
$743$	1.09079e12	0.131308	0.0656540	+	0.997842i	$0.479087\pi$
$744$	0	0
$745$	4.28580e11	0.0509716
$746$	2.22657e12	0.263215
$747$	0	0
$748$	9.11678e12	1.06484
$749$	2.96012e12	0.343670
$750$	0	0
$751$	−3.09647e11	−0.0355212	−0.0177606	−	0.999842i	$-0.505654\pi$
$751$	−3.09647e11	−0.0355212	−0.0177606	+	0.999842i	$0.505654\pi$
$752$	5.21136e12	0.594252
$753$	0	0
$754$	6.61254e12	0.745071
$755$	4.72437e11	0.0529155
$756$	0	0
$757$	1.44340e12	0.159755	0.0798777	−	0.996805i	$-0.474547\pi$
$757$	1.44340e12	0.159755	0.0798777	+	0.996805i	$0.474547\pi$
$758$	−4.39299e11	−0.0483336
$759$	0	0
$760$	8.64274e11	0.0939702
$761$	2.27862e12	0.246287	0.123143	−	0.992389i	$-0.460703\pi$
$761$	2.27862e12	0.246287	0.123143	+	0.992389i	$0.460703\pi$
$762$	0	0
$763$	−4.07256e12	−0.435018
$764$	−7.26051e12	−0.770987
$765$	0	0
$766$	−4.33873e12	−0.455337
$767$	−4.15978e12	−0.434001
$768$	0	0
$769$	1.09547e13	1.12962	0.564809	−	0.825221i	$-0.308950\pi$
$769$	1.09547e13	1.12962	0.564809	+	0.825221i	$0.308950\pi$
$770$	3.35352e11	0.0343789
$771$	0	0
$772$	−1.90228e12	−0.192751
$773$	−7.52114e12	−0.757663	−0.378831	−	0.925466i	$-0.623674\pi$
$773$	−7.52114e12	−0.757663	−0.378831	+	0.925466i	$0.623674\pi$
$774$	0	0
$775$	−6.64948e12	−0.662110
$776$	−8.83576e12	−0.874714
$777$	0	0
$778$	−4.60211e12	−0.450348
$779$	1.05071e12	0.102227
$780$	0	0
$781$	−6.78201e12	−0.652272
$782$	3.62962e12	0.347081
$783$	0	0
$784$	5.41785e11	0.0512158
$785$	5.99298e11	0.0563287
$786$	0	0
$787$	1.85303e12	0.172185	0.0860926	−	0.996287i	$-0.472562\pi$
$787$	1.85303e12	0.172185	0.0860926	+	0.996287i	$0.472562\pi$
$788$	1.06301e13	0.982129
$789$	0	0
$790$	−3.27191e8	−2.98868e−5 0
$791$	−3.42021e12	−0.310641
$792$	0	0
$793$	−1.14168e13	−1.02521
$794$	−7.09097e12	−0.633159
$795$	0	0
$796$	−6.52073e12	−0.575688
$797$	1.84378e13	1.61863	0.809314	−	0.587377i	$-0.199839\pi$
$797$	1.84378e13	1.61863	0.809314	+	0.587377i	$0.199839\pi$
$798$	0	0
$799$	2.01237e13	1.74682
$800$	1.16161e13	1.00266
$801$	0	0
$802$	−3.38220e12	−0.288679
$803$	1.34189e13	1.13893
$804$	0	0
$805$	−4.42644e11	−0.0371512
$806$	6.23560e12	0.520440
$807$	0	0
$808$	4.07323e11	0.0336192
$809$	1.03419e13	0.848856	0.424428	−	0.905462i	$-0.360475\pi$
$809$	1.03419e13	0.848856	0.424428	+	0.905462i	$0.360475\pi$
$810$	0	0
$811$	−2.31795e13	−1.88153	−0.940764	−	0.339062i	$-0.889890\pi$
$811$	−2.31795e13	−1.88153	−0.940764	+	0.339062i	$0.889890\pi$
$812$	3.48163e12	0.281048
$813$	0	0
$814$	1.30884e13	1.04490
$815$	−1.84869e12	−0.146776
$816$	0	0
$817$	5.46958e12	0.429492
$818$	−3.26250e12	−0.254777
$819$	0	0
$820$	−1.90113e11	−0.0146842
$821$	−1.10059e13	−0.845436	−0.422718	−	0.906261i	$-0.638924\pi$
$821$	−1.10059e13	−0.845436	−0.422718	+	0.906261i	$0.638924\pi$
$822$	0	0
$823$	2.30119e13	1.74845	0.874224	−	0.485523i	$-0.161371\pi$
$823$	2.30119e13	1.74845	0.874224	+	0.485523i	$0.161371\pi$
$824$	4.23525e12	0.320041
$825$	0	0
$826$	6.60618e11	0.0493787
$827$	−6.58898e12	−0.489827	−0.244914	−	0.969545i	$-0.578760\pi$
$827$	−6.58898e12	−0.489827	−0.244914	+	0.969545i	$0.578760\pi$
$828$	0	0
$829$	−1.32427e13	−0.973829	−0.486915	−	0.873450i	$-0.661878\pi$
$829$	−1.32427e13	−0.973829	−0.486915	+	0.873450i	$0.661878\pi$
$830$	−1.14067e12	−0.0834271
$831$	0	0
$832$	−2.96899e12	−0.214810
$833$	2.09210e12	0.150550
$834$	0	0
$835$	−9.94878e11	−0.0708240
$836$	−1.09654e13	−0.776419
$837$	0	0
$838$	−4.74058e12	−0.332073
$839$	5.05558e12	0.352243	0.176122	−	0.984368i	$-0.443645\pi$
$839$	5.05558e12	0.352243	0.176122	+	0.984368i	$0.443645\pi$
$840$	0	0
$841$	−9.17397e11	−0.0632376
$842$	2.26127e12	0.155042
$843$	0	0
$844$	2.14326e13	1.45390
$845$	−3.31584e12	−0.223737
$846$	0	0
$847$	−4.13139e12	−0.275817
$848$	8.71104e12	0.578480
$849$	0	0
$850$	7.56130e12	0.496834
$851$	−1.72758e13	−1.12916
$852$	0	0
$853$	1.03049e13	0.666456	0.333228	−	0.942846i	$-0.391862\pi$
$853$	1.03049e13	0.666456	0.333228	+	0.942846i	$0.391862\pi$
$854$	1.81310e12	0.116644
$855$	0	0
$856$	−1.21580e13	−0.773979
$857$	1.66788e13	1.05621	0.528105	−	0.849179i	$-0.322903\pi$
$857$	1.66788e13	1.05621	0.528105	+	0.849179i	$0.322903\pi$
$858$	0	0
$859$	−2.36677e13	−1.48316	−0.741579	−	0.670865i	$-0.765923\pi$
$859$	−2.36677e13	−1.48316	−0.741579	+	0.670865i	$0.765923\pi$
$860$	−9.89649e11	−0.0616933
$861$	0	0
$862$	9.38476e12	0.578949
$863$	−1.26428e13	−0.775880	−0.387940	−	0.921685i	$-0.626813\pi$
$863$	−1.26428e13	−0.775880	−0.387940	+	0.921685i	$0.626813\pi$
$864$	0	0
$865$	7.58046e11	0.0460387
$866$	−1.58855e12	−0.0959776
$867$	0	0
$868$	3.28316e12	0.196315
$869$	9.55450e9	0.000568354 0
$870$	0	0
$871$	3.84514e12	0.226376
$872$	1.67271e13	0.979705
$873$	0	0
$874$	−4.36561e12	−0.253071
$875$	−1.86368e12	−0.107482
$876$	0	0
$877$	2.21629e13	1.26511	0.632555	−	0.774515i	$-0.282006\pi$
$877$	2.21629e13	1.26511	0.632555	+	0.774515i	$0.282006\pi$
$878$	1.16691e12	0.0662695
$879$	0	0
$880$	1.20511e12	0.0677413
$881$	1.87132e13	1.04654	0.523272	−	0.852166i	$-0.324711\pi$
$881$	1.87132e13	1.04654	0.523272	+	0.852166i	$0.324711\pi$
$882$	0	0
$883$	−1.66483e13	−0.921611	−0.460805	−	0.887501i	$-0.652439\pi$
$883$	−1.66483e13	−0.921611	−0.460805	+	0.887501i	$0.652439\pi$
$884$	2.35083e13	1.29475
$885$	0	0
$886$	−1.49050e12	−0.0812604
$887$	1.58859e13	0.861698	0.430849	−	0.902424i	$-0.358214\pi$
$887$	1.58859e13	0.861698	0.430849	+	0.902424i	$0.358214\pi$
$888$	0	0
$889$	7.51171e12	0.403349
$890$	−6.53013e11	−0.0348873
$891$	0	0
$892$	8.08344e12	0.427518
$893$	−2.42042e13	−1.27368
$894$	0	0
$895$	−3.90244e12	−0.203298
$896$	−6.99384e12	−0.362518
$897$	0	0
$898$	−8.34003e12	−0.427980
$899$	1.28151e13	0.654339
$900$	0	0
$901$	3.36377e13	1.70046
$902$	−1.67450e12	−0.0842278
$903$	0	0
$904$	1.40477e13	0.699595
$905$	1.97074e12	0.0976585
$906$	0	0
$907$	−3.65370e13	−1.79267	−0.896333	−	0.443381i	$-0.853779\pi$
$907$	−3.65370e13	−1.79267	−0.896333	+	0.443381i	$0.853779\pi$
$908$	−1.55005e13	−0.756760
$909$	0	0
$910$	8.64729e11	0.0418017
$911$	−2.09138e13	−1.00600	−0.503002	−	0.864286i	$-0.667771\pi$
$911$	−2.09138e13	−1.00600	−0.503002	+	0.864286i	$0.667771\pi$
$912$	0	0
$913$	3.33093e13	1.58653
$914$	−5.90199e12	−0.279731
$915$	0	0
$916$	7.17935e12	0.336942
$917$	1.68820e12	0.0788427
$918$	0	0
$919$	−1.22511e12	−0.0566571	−0.0283286	−	0.999599i	$-0.509018\pi$
$919$	−1.22511e12	−0.0566571	−0.0283286	+	0.999599i	$0.509018\pi$
$920$	1.81805e12	0.0836683
$921$	0	0
$922$	−7.74276e12	−0.352863
$923$	−1.74879e13	−0.793105
$924$	0	0
$925$	−3.59894e13	−1.61635
$926$	−8.67335e12	−0.387648
$927$	0	0
$928$	−2.23869e13	−0.990896
$929$	−9.53162e12	−0.419852	−0.209926	−	0.977717i	$-0.567322\pi$
$929$	−9.53162e12	−0.419852	−0.209926	+	0.977717i	$0.567322\pi$
$930$	0	0
$931$	−2.51632e12	−0.109772
$932$	−1.99887e13	−0.867788
$933$	0	0
$934$	−1.82637e13	−0.785287
$935$	4.65353e12	0.199127
$936$	0	0
$937$	−9.70345e12	−0.411242	−0.205621	−	0.978632i	$-0.565921\pi$
$937$	−9.70345e12	−0.411242	−0.205621	+	0.978632i	$0.565921\pi$
$938$	−6.10650e11	−0.0257561
$939$	0	0
$940$	4.37944e12	0.182954
$941$	2.29625e13	0.954698	0.477349	−	0.878714i	$-0.341598\pi$
$941$	2.29625e13	0.954698	0.477349	+	0.878714i	$0.341598\pi$
$942$	0	0
$943$	2.21024e12	0.0910198
$944$	2.37397e12	0.0972973
$945$	0	0
$946$	−8.71676e12	−0.353871
$947$	−1.40763e13	−0.568738	−0.284369	−	0.958715i	$-0.591784\pi$
$947$	−1.40763e13	−0.568738	−0.284369	+	0.958715i	$0.591784\pi$
$948$	0	0
$949$	3.46015e13	1.38483
$950$	−9.09452e12	−0.362263
$951$	0	0
$952$	−8.59281e12	−0.339054
$953$	−9.86435e12	−0.387392	−0.193696	−	0.981062i	$-0.562047\pi$
$953$	−9.86435e12	−0.387392	−0.193696	+	0.981062i	$0.562047\pi$
$954$	0	0
$955$	−3.70602e12	−0.144176
$956$	1.49651e13	0.579453
$957$	0	0
$958$	1.27582e13	0.489378
$959$	−7.37852e12	−0.281699
$960$	0	0
$961$	−1.43550e13	−0.542937
$962$	3.37493e13	1.27051
$963$	0	0
$964$	−3.85518e13	−1.43780
$965$	−9.70993e11	−0.0360449
$966$	0	0
$967$	−2.37375e12	−0.0873002	−0.0436501	−	0.999047i	$-0.513899\pi$
$967$	−2.37375e12	−0.0873002	−0.0436501	+	0.999047i	$0.513899\pi$
$968$	1.69687e13	0.621168
$969$	0	0
$970$	−1.95952e12	−0.0710686
$971$	−4.45895e13	−1.60970	−0.804852	−	0.593476i	$-0.797755\pi$
$971$	−4.45895e13	−1.60970	−0.804852	+	0.593476i	$0.797755\pi$
$972$	0	0
$973$	−1.21924e13	−0.436096
$974$	−4.89939e12	−0.174432
$975$	0	0
$976$	6.51550e12	0.229839
$977$	−8.90599e11	−0.0312721	−0.0156360	−	0.999878i	$-0.504977\pi$
$977$	−8.90599e11	−0.0312721	−0.0156360	+	0.999878i	$0.504977\pi$
$978$	0	0
$979$	1.90691e13	0.663449
$980$	4.55296e11	0.0157680
$981$	0	0
$982$	2.45117e13	0.841146
$983$	−8.23794e12	−0.281402	−0.140701	−	0.990052i	$-0.544936\pi$
$983$	−8.23794e12	−0.281402	−0.140701	+	0.990052i	$0.544936\pi$
$984$	0	0
$985$	5.42597e12	0.183660
$986$	−1.45724e13	−0.491003
$987$	0	0
$988$	−2.82751e13	−0.944056
$989$	1.15056e13	0.382407
$990$	0	0
$991$	8.90224e12	0.293203	0.146601	−	0.989196i	$-0.453167\pi$
$991$	8.90224e12	0.293203	0.146601	+	0.989196i	$0.453167\pi$
$992$	−2.11108e13	−0.692152
$993$	0	0
$994$	2.77727e12	0.0902359
$995$	−3.32841e12	−0.107655
$996$	0	0
$997$	1.45450e13	0.466216	0.233108	−	0.972451i	$-0.425110\pi$
$997$	1.45450e13	0.466216	0.233108	+	0.972451i	$0.425110\pi$
$998$	−1.73965e11	−0.00555106
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	63.10.a.d.1.1		2
3.2	odd	2		7.10.a.a.1.2	✓	2
12.11	even	2		112.10.a.e.1.2		2
15.2	even	4		175.10.b.b.99.3		4
15.8	even	4		175.10.b.b.99.2		4
15.14	odd	2		175.10.a.b.1.1		2
21.2	odd	6		49.10.c.c.18.1		4
21.5	even	6		49.10.c.b.18.1		4
21.11	odd	6		49.10.c.c.30.1		4
21.17	even	6		49.10.c.b.30.1		4
21.20	even	2		49.10.a.b.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
7.10.a.a.1.2	✓	2	3.2	odd	2
49.10.a.b.1.2		2	21.20	even	2
49.10.c.b.18.1		4	21.5	even	6
49.10.c.b.30.1		4	21.17	even	6
49.10.c.c.18.1		4	21.2	odd	6
49.10.c.c.30.1		4	21.11	odd	6
63.10.a.d.1.1		2	1.1	even	1	trivial
112.10.a.e.1.2		2	12.11	even	2
175.10.a.b.1.1		2	15.14	odd	2
175.10.b.b.99.2		4	15.8	even	4
175.10.b.b.99.3		4	15.2	even	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:	\( N \)	\(=\)	\( 63 = 3^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 10 \)
Character orbit:	\([\chi]\)	\(=\)	63.a (trivial)

\(f(q)\)	\(=\)	\(q-10.8924 q^{2} -393.355 q^{4} -200.782 q^{5} -2401.00 q^{7} +9861.52 q^{8} +2187.01 q^{10} -63864.3 q^{11} -164679. q^{13} +26152.8 q^{14} +93981.5 q^{16} +362910. q^{17} -436498. q^{19} +78978.6 q^{20} +695638. q^{22} -918199. q^{23} -1.91281e6 q^{25} +1.79375e6 q^{26} +944445. q^{28} +3.68643e6 q^{29} +3.47629e6 q^{31} -6.07279e6 q^{32} -3.95298e6 q^{34} +482078. q^{35} +1.88149e7 q^{37} +4.75453e6 q^{38} -1.98002e6 q^{40} -2.40714e6 q^{41} -1.25306e7 q^{43} +2.51213e7 q^{44} +1.00014e7 q^{46} +5.54509e7 q^{47} +5.76480e6 q^{49} +2.08352e7 q^{50} +6.47772e7 q^{52} +9.26889e7 q^{53} +1.28228e7 q^{55} -2.36775e7 q^{56} -4.01542e7 q^{58} +2.52600e7 q^{59} +6.93275e7 q^{61} -3.78653e7 q^{62} +1.80290e7 q^{64} +3.30646e7 q^{65} -2.33494e7 q^{67} -1.42752e8 q^{68} -5.25101e6 q^{70} +1.06194e8 q^{71} -2.10115e8 q^{73} -2.04940e8 q^{74} +1.71699e8 q^{76} +1.53338e8 q^{77} -149606. q^{79} -1.88698e7 q^{80} +2.62197e7 q^{82} -5.21565e8 q^{83} -7.28659e7 q^{85} +1.36489e8 q^{86} -6.29799e8 q^{88} -2.98587e8 q^{89} +3.95394e8 q^{91} +3.61178e8 q^{92} -6.03996e8 q^{94} +8.76410e7 q^{95} -8.95983e8 q^{97} -6.27928e7 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 6 q^{2} - 620 q^{4} + 2238 q^{5} - 4802 q^{7} - 2616 q^{8} + 43384 q^{10} - 35316 q^{11} - 26530 q^{13} - 14406 q^{14} - 752 q^{16} + 463920 q^{17} - 925426 q^{19} - 473760 q^{20} + 1177888 q^{22}+ \cdots + 34588806 q^{98}+O(q^{100}) \) 2 * q + 6 * q^2 - 620 * q^4 + 2238 * q^5 - 4802 * q^7 - 2616 * q^8 + 43384 * q^10 - 35316 * q^11 - 26530 * q^13 - 14406 * q^14 - 752 * q^16 + 463920 * q^17 - 925426 * q^19 - 473760 * q^20 + 1177888 * q^22 - 778128 * q^23 + 2081722 * q^25 + 4127424 * q^26 + 1488620 * q^28 + 10003584 * q^29 + 2467260 * q^31 - 1284576 * q^32 - 2246676 * q^34 - 5373438 * q^35 + 30735552 * q^37 - 3504660 * q^38 - 32409984 * q^40 + 19103448 * q^41 + 4065100 * q^43 + 18650976 * q^44 + 12367584 * q^46 + 82195020 * q^47 + 11529602 * q^49 + 88312626 * q^50 + 33466384 * q^52 + 55189812 * q^53 + 82445816 * q^55 + 6281016 * q^56 + 66558004 * q^58 + 7069218 * q^59 + 44316386 * q^61 - 54910200 * q^62 + 147417152 * q^64 + 369979260 * q^65 - 241921336 * q^67 - 165645816 * q^68 - 104164984 * q^70 - 206493816 * q^71 - 499153188 * q^73 - 3571524 * q^74 + 282511768 * q^76 + 84793716 * q^77 + 468535096 * q^79 - 249904128 * q^80 + 389586092 * q^82 - 444023958 * q^83 + 173475060 * q^85 + 416830608 * q^86 - 986010816 * q^88 - 636267396 * q^89 + 63698530 * q^91 + 329431488 * q^92 - 152223192 * q^94 - 1104747984 * q^95 - 1632716064 * q^97 + 34588806 * q^98

Introduction

L-functions

Modular forms

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Database

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