LMFDB - Embedded newform 63.10.a.e.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [63,10,Mod(1,63)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

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")

//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);

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

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	yes
Analytic conductor:	$32.4472576783$
Analytic rank:	$1$
Dimension:	$3$
Coefficient field:	$\mathbb{Q}[x]/(x^{3} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 426x + 2016 $ x^3 - x^2 - 426*x + 2016
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2\cdot 3 $
Twist minimal:	no (minimal twist has level 7)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-22.2358$ 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)

$f(q)$	$=$	\(q-13.3607 q^{2} -333.491 q^{4} -1922.19 q^{5} +2401.00 q^{7} +11296.4 q^{8} +25681.8 q^{10} +90199.9 q^{11} -3199.89 q^{13} -32079.1 q^{14} +19819.7 q^{16} -116494. q^{17} -142449. q^{19} +641033. q^{20} -1.20514e6 q^{22} -1.27391e6 q^{23} +1.74168e6 q^{25} +42752.8 q^{26} -800712. q^{28} +1.42931e6 q^{29} +9.67494e6 q^{31} -6.04855e6 q^{32} +1.55645e6 q^{34} -4.61518e6 q^{35} -8.67744e6 q^{37} +1.90323e6 q^{38} -2.17138e7 q^{40} -1.32544e7 q^{41} -2.97554e7 q^{43} -3.00809e7 q^{44} +1.70204e7 q^{46} +1.07969e7 q^{47} +5.76480e6 q^{49} -2.32702e7 q^{50} +1.06713e6 q^{52} -7.07399e7 q^{53} -1.73381e8 q^{55} +2.71226e7 q^{56} -1.90966e7 q^{58} -6.40400e6 q^{59} +1.69190e8 q^{61} -1.29264e8 q^{62} +7.06653e7 q^{64} +6.15078e6 q^{65} -1.16276e8 q^{67} +3.88498e7 q^{68} +6.16621e7 q^{70} -1.44496e8 q^{71} +1.60155e8 q^{73} +1.15937e8 q^{74} +4.75056e7 q^{76} +2.16570e8 q^{77} -4.89322e8 q^{79} -3.80972e7 q^{80} +1.77088e8 q^{82} +8.31590e7 q^{83} +2.23924e8 q^{85} +3.97553e8 q^{86} +1.01893e9 q^{88} -2.08083e6 q^{89} -7.68292e6 q^{91} +4.24838e8 q^{92} -1.44255e8 q^{94} +2.73815e8 q^{95} -3.15885e8 q^{97} -7.70219e7 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 21 q^{2} + 1557 q^{4} - 1554 q^{5} + 7203 q^{7} - 14055 q^{8} - 97860 q^{10} + 3444 q^{11} - 19782 q^{13} - 50421 q^{14} + 482961 q^{16} - 1016694 q^{17} + 222852 q^{19} + 1922088 q^{20} - 2847048 q^{22}+ \cdots - 121060821 q^{98}+O(q^{100}) $ 3 * q - 21 * q^2 + 1557 * q^4 - 1554 * q^5 + 7203 * q^7 - 14055 * q^8 - 97860 * q^10 + 3444 * q^11 - 19782 * q^13 - 50421 * q^14 + 482961 * q^16 - 1016694 * q^17 + 222852 * q^19 + 1922088 * q^20 - 2847048 * q^22 - 1885632 * q^23 + 3073221 * q^25 + 8785056 * q^26 + 3738357 * q^28 - 4081818 * q^29 + 2869440 * q^31 - 25221951 * q^32 - 3981642 * q^34 - 3731154 * q^35 + 1395618 * q^37 - 43479870 * q^38 - 82859280 * q^40 + 14420658 * q^41 - 61631172 * q^43 - 97011984 * q^44 + 89747664 * q^46 + 10368960 * q^47 + 17294403 * q^49 - 73325055 * q^50 - 80908044 * q^52 - 67502610 * q^53 - 105823032 * q^55 - 33746055 * q^56 - 159163830 * q^58 + 42590100 * q^59 + 191746842 * q^61 + 46983468 * q^62 + 7852161 * q^64 - 364283220 * q^65 - 255175788 * q^67 - 743485806 * q^68 - 234961860 * q^70 - 296514504 * q^71 + 344213310 * q^73 + 690696462 * q^74 + 728839986 * q^76 + 8269044 * q^77 - 960412656 * q^79 + 1333333344 * q^80 + 562675302 * q^82 + 1100517180 * q^83 + 438179412 * q^85 + 880982256 * q^86 + 1206124800 * q^88 - 506816478 * q^89 - 47496582 * q^91 - 691123488 * q^92 + 1388004828 * q^94 + 2203071072 * q^95 - 647498250 * q^97 - 121060821 * 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 $

$2$	−13.3607	−0.590466	−0.295233	−	0.955425i	$-0.595397\pi$
$2$	−13.3607	−0.590466	−0.295233	+	0.955425i	$0.595397\pi$
$3$	0	0
$3$	0	0
$4$	−333.491	−0.651350
$5$	−1922.19	−1.37541	−0.687703	−	0.725992i	$-0.741381\pi$
$5$	−1922.19	−1.37541	−0.687703	+	0.725992i	$0.741381\pi$
$6$	0	0
$7$	2401.00	0.377964
$7$	2401.00	0.377964
$8$	11296.4	0.975066
$9$	0	0
$10$	25681.8	0.812131
$11$	90199.9	1.85754	0.928772	−	0.370652i	$-0.120866\pi$
$11$	90199.9	1.85754	0.928772	+	0.370652i	$0.120866\pi$
$12$	0	0
$13$	−3199.89	−0.0310734	−0.0155367	−	0.999879i	$-0.504946\pi$
$13$	−3199.89	−0.0310734	−0.0155367	+	0.999879i	$0.504946\pi$
$14$	−32079.1	−0.223175
$15$	0	0
$16$	19819.7	0.0756061
$17$	−116494.	−0.338286	−0.169143	−	0.985591i	$-0.554100\pi$
$17$	−116494.	−0.338286	−0.169143	+	0.985591i	$0.554100\pi$
$18$	0	0
$19$	−142449.	−0.250767	−0.125383	−	0.992108i	$-0.540016\pi$
$19$	−142449.	−0.250767	−0.125383	+	0.992108i	$0.540016\pi$
$20$	641033.	0.895870
$21$	0	0
$22$	−1.20514e6	−1.09682
$23$	−1.27391e6	−0.949213	−0.474606	−	0.880198i	$-0.657410\pi$
$23$	−1.27391e6	−0.949213	−0.474606	+	0.880198i	$0.657410\pi$
$24$	0	0
$25$	1.74168e6	0.891742
$26$	42752.8	0.0183478
$27$	0	0
$28$	−800712.	−0.246187
$29$	1.42931e6	0.375262	0.187631	−	0.982240i	$-0.439919\pi$
$29$	1.42931e6	0.375262	0.187631	+	0.982240i	$0.439919\pi$
$30$	0	0
$31$	9.67494e6	1.88157	0.940786	−	0.339001i	$-0.110089\pi$
$31$	9.67494e6	1.88157	0.940786	+	0.339001i	$0.110089\pi$
$32$	−6.04855e6	−1.01971
$33$	0	0
$34$	1.55645e6	0.199747
$35$	−4.61518e6	−0.519855
$36$	0	0
$37$	−8.67744e6	−0.761174	−0.380587	−	0.924745i	$-0.624278\pi$
$37$	−8.67744e6	−0.761174	−0.380587	+	0.924745i	$0.624278\pi$
$38$	1.90323e6	0.148069
$39$	0	0
$40$	−2.17138e7	−1.34111
$41$	−1.32544e7	−0.732541	−0.366271	−	0.930508i	$-0.619366\pi$
$41$	−1.32544e7	−0.732541	−0.366271	+	0.930508i	$0.619366\pi$
$42$	0	0
$43$	−2.97554e7	−1.32726	−0.663632	−	0.748060i	$-0.730986\pi$
$43$	−2.97554e7	−1.32726	−0.663632	+	0.748060i	$0.730986\pi$
$44$	−3.00809e7	−1.20991
$45$	0	0
$46$	1.70204e7	0.560478
$47$	1.07969e7	0.322745	0.161373	−	0.986894i	$-0.448408\pi$
$47$	1.07969e7	0.322745	0.161373	+	0.986894i	$0.448408\pi$
$48$	0	0
$49$	5.76480e6	0.142857
$50$	−2.32702e7	−0.526544
$51$	0	0
$52$	1.06713e6	0.0202397
$53$	−7.07399e7	−1.23147	−0.615734	−	0.787954i	$-0.711140\pi$
$53$	−7.07399e7	−1.23147	−0.615734	+	0.787954i	$0.711140\pi$
$54$	0	0
$55$	−1.73381e8	−2.55488
$56$	2.71226e7	0.368540
$57$	0	0
$58$	−1.90966e7	−0.221579
$59$	−6.40400e6	−0.0688046	−0.0344023	−	0.999408i	$-0.510953\pi$
$59$	−6.40400e6	−0.0688046	−0.0344023	+	0.999408i	$0.510953\pi$
$60$	0	0
$61$	1.69190e8	1.56455	0.782275	−	0.622933i	$-0.214059\pi$
$61$	1.69190e8	1.56455	0.782275	+	0.622933i	$0.214059\pi$
$62$	−1.29264e8	−1.11100
$63$	0	0
$64$	7.06653e7	0.526498
$65$	6.15078e6	0.0427386
$66$	0	0
$67$	−1.16276e8	−0.704943	−0.352471	−	0.935823i	$-0.614659\pi$
$67$	−1.16276e8	−0.704943	−0.352471	+	0.935823i	$0.614659\pi$
$68$	3.88498e7	0.220343
$69$	0	0
$70$	6.16621e7	0.306957
$71$	−1.44496e8	−0.674826	−0.337413	−	0.941357i	$-0.609552\pi$
$71$	−1.44496e8	−0.674826	−0.337413	+	0.941357i	$0.609552\pi$
$72$	0	0
$73$	1.60155e8	0.660066	0.330033	−	0.943969i	$-0.392940\pi$
$73$	1.60155e8	0.660066	0.330033	+	0.943969i	$0.392940\pi$
$74$	1.15937e8	0.449447
$75$	0	0
$76$	4.75056e7	0.163337
$77$	2.16570e8	0.702085
$78$	0	0
$79$	−4.89322e8	−1.41343	−0.706713	−	0.707500i	$-0.749823\pi$
$79$	−4.89322e8	−1.41343	−0.706713	+	0.707500i	$0.749823\pi$
$80$	−3.80972e7	−0.103989
$81$	0	0
$82$	1.77088e8	0.432541
$83$	8.31590e7	0.192335	0.0961674	−	0.995365i	$-0.469342\pi$
$83$	8.31590e7	0.192335	0.0961674	+	0.995365i	$0.469342\pi$
$84$	0	0
$85$	2.23924e8	0.465281
$86$	3.97553e8	0.783704
$87$	0	0
$88$	1.01893e9	1.81123
$89$	−2.08083e6	−0.00351546	−0.00175773	−	0.999998i	$-0.500560\pi$
$89$	−2.08083e6	−0.00351546	−0.00175773	+	0.999998i	$0.500560\pi$
$90$	0	0
$91$	−7.68292e6	−0.0117447
$92$	4.24838e8	0.618270
$93$	0	0
$94$	−1.44255e8	−0.190570
$95$	2.73815e8	0.344906
$96$	0	0
$97$	−3.15885e8	−0.362290	−0.181145	−	0.983456i	$-0.557980\pi$
$97$	−3.15885e8	−0.362290	−0.181145	+	0.983456i	$0.557980\pi$
$98$	−7.70219e7	−0.0843523
$99$	0	0
$100$	−5.80836e8	−0.580836
$101$	5.74841e8	0.549669	0.274835	−	0.961492i	$-0.411377\pi$
$101$	5.74841e8	0.549669	0.274835	+	0.961492i	$0.411377\pi$
$102$	0	0
$103$	−1.51870e9	−1.32955	−0.664775	−	0.747044i	$-0.731473\pi$
$103$	−1.51870e9	−1.32955	−0.664775	+	0.747044i	$0.731473\pi$
$104$	−3.61471e7	−0.0302987
$105$	0	0
$106$	9.45137e8	0.727140
$107$	2.01863e8	0.148878	0.0744390	−	0.997226i	$-0.476283\pi$
$107$	2.01863e8	0.148878	0.0744390	+	0.997226i	$0.476283\pi$
$108$	0	0
$109$	−8.73952e8	−0.593019	−0.296509	−	0.955030i	$-0.595823\pi$
$109$	−8.73952e8	−0.593019	−0.296509	+	0.955030i	$0.595823\pi$
$110$	2.31650e9	1.50857
$111$	0	0
$112$	4.75871e7	0.0285764
$113$	−1.52955e9	−0.882491	−0.441245	−	0.897386i	$-0.645463\pi$
$113$	−1.52955e9	−0.882491	−0.441245	+	0.897386i	$0.645463\pi$
$114$	0	0
$115$	2.44870e9	1.30555
$116$	−4.76661e8	−0.244427
$117$	0	0
$118$	8.55621e7	0.0406268
$119$	−2.79703e8	−0.127860
$120$	0	0
$121$	5.77807e9	2.45047
$122$	−2.26050e9	−0.923814
$123$	0	0
$124$	−3.22651e9	−1.22556
$125$	4.06429e8	0.148898
$126$	0	0
$127$	−8.71958e8	−0.297426	−0.148713	−	0.988880i	$-0.547513\pi$
$127$	−8.71958e8	−0.297426	−0.148713	+	0.988880i	$0.547513\pi$
$128$	2.15272e9	0.708830
$129$	0	0
$130$	−8.21789e7	−0.0252357
$131$	−2.24404e9	−0.665747	−0.332874	−	0.942971i	$-0.608018\pi$
$131$	−2.24404e9	−0.665747	−0.332874	+	0.942971i	$0.608018\pi$
$132$	0	0
$133$	−3.42021e8	−0.0947809
$134$	1.55353e9	0.416245
$135$	0	0
$136$	−1.31596e9	−0.329852
$137$	−4.16141e9	−1.00925	−0.504624	−	0.863339i	$-0.668369\pi$
$137$	−4.16141e9	−1.00925	−0.504624	+	0.863339i	$0.668369\pi$
$138$	0	0
$139$	−6.03383e9	−1.37097	−0.685483	−	0.728089i	$-0.740409\pi$
$139$	−6.03383e9	−1.37097	−0.685483	+	0.728089i	$0.740409\pi$
$140$	1.53912e9	0.338607
$141$	0	0
$142$	1.93057e9	0.398462
$143$	−2.88629e8	−0.0577203
$144$	0	0
$145$	−2.74740e9	−0.516137
$146$	−2.13979e9	−0.389747
$147$	0	0
$148$	2.89385e9	0.495790
$149$	4.37832e9	0.727728	0.363864	−	0.931452i	$-0.381457\pi$
$149$	4.37832e9	0.727728	0.363864	+	0.931452i	$0.381457\pi$
$150$	0	0
$151$	−2.69365e9	−0.421642	−0.210821	−	0.977525i	$-0.567614\pi$
$151$	−2.69365e9	−0.421642	−0.210821	+	0.977525i	$0.567614\pi$
$152$	−1.60916e9	−0.244514
$153$	0	0
$154$	−2.89353e9	−0.414558
$155$	−1.85971e10	−2.58793
$156$	0	0
$157$	−1.33044e9	−0.174762	−0.0873810	−	0.996175i	$-0.527850\pi$
$157$	−1.33044e9	−0.174762	−0.0873810	+	0.996175i	$0.527850\pi$
$158$	6.53770e9	0.834580
$159$	0	0
$160$	1.16265e10	1.40251
$161$	−3.05866e9	−0.358769
$162$	0	0
$163$	−3.56094e9	−0.395112	−0.197556	−	0.980292i	$-0.563301\pi$
$163$	−3.56094e9	−0.395112	−0.197556	+	0.980292i	$0.563301\pi$
$164$	4.42022e9	0.477140
$165$	0	0
$166$	−1.11106e9	−0.113567
$167$	1.04285e10	1.03752	0.518762	−	0.854919i	$-0.326393\pi$
$167$	1.04285e10	1.03752	0.518762	+	0.854919i	$0.326393\pi$
$168$	0	0
$169$	−1.05943e10	−0.999034
$170$	−2.99179e9	−0.274733
$171$	0	0
$172$	9.92314e9	0.864512
$173$	−2.04717e10	−1.73759	−0.868793	−	0.495176i	$-0.835104\pi$
$173$	−2.04717e10	−1.73759	−0.868793	+	0.495176i	$0.835104\pi$
$174$	0	0
$175$	4.18178e9	0.337047
$176$	1.78773e9	0.140442
$177$	0	0
$178$	2.78014e7	0.00207576
$179$	−5.46705e9	−0.398029	−0.199014	−	0.979997i	$-0.563774\pi$
$179$	−5.46705e9	−0.398029	−0.199014	+	0.979997i	$0.563774\pi$
$180$	0	0
$181$	−2.11628e9	−0.146561	−0.0732807	−	0.997311i	$-0.523347\pi$
$181$	−2.11628e9	−0.146561	−0.0732807	+	0.997311i	$0.523347\pi$
$182$	1.02649e8	0.00693482
$183$	0	0
$184$	−1.43906e10	−0.925545
$185$	1.66797e10	1.04692
$186$	0	0
$187$	−1.05078e10	−0.628381
$188$	−3.60068e9	−0.210220
$189$	0	0
$190$	−3.65836e9	−0.203655
$191$	−1.72421e10	−0.937431	−0.468715	−	0.883349i	$-0.655283\pi$
$191$	−1.72421e10	−0.937431	−0.468715	+	0.883349i	$0.655283\pi$
$192$	0	0
$193$	2.02030e10	1.04811	0.524055	−	0.851684i	$-0.324418\pi$
$193$	2.02030e10	1.04811	0.524055	+	0.851684i	$0.324418\pi$
$194$	4.22045e9	0.213920
$195$	0	0
$196$	−1.92251e9	−0.0930500
$197$	2.22592e10	1.05296	0.526481	−	0.850187i	$-0.323511\pi$
$197$	2.22592e10	1.05296	0.526481	+	0.850187i	$0.323511\pi$
$198$	0	0
$199$	1.70588e10	0.771098	0.385549	−	0.922687i	$-0.374012\pi$
$199$	1.70588e10	0.771098	0.385549	+	0.922687i	$0.374012\pi$
$200$	1.96747e10	0.869508
$201$	0	0
$202$	−7.68029e9	−0.324561
$203$	3.43176e9	0.141836
$204$	0	0
$205$	2.54774e10	1.00754
$206$	2.02909e10	0.785054
$207$	0	0
$208$	−6.34207e7	−0.00234934
$209$	−1.28489e10	−0.465810
$210$	0	0
$211$	−3.19873e10	−1.11098	−0.555490	−	0.831523i	$-0.687469\pi$
$211$	−3.19873e10	−1.11098	−0.555490	+	0.831523i	$0.687469\pi$
$212$	2.35911e10	0.802116
$213$	0	0
$214$	−2.69704e9	−0.0879074
$215$	5.71954e10	1.82553
$216$	0	0
$217$	2.32295e10	0.711167
$218$	1.16766e10	0.350157
$219$	0	0
$220$	5.78211e10	1.66412
$221$	3.72768e8	0.0105117
$222$	0	0
$223$	−2.30967e10	−0.625428	−0.312714	−	0.949847i	$-0.601238\pi$
$223$	−2.30967e10	−0.625428	−0.312714	+	0.949847i	$0.601238\pi$
$224$	−1.45226e10	−0.385414
$225$	0	0
$226$	2.04359e10	0.521081
$227$	2.30894e10	0.577160	0.288580	−	0.957456i	$-0.406817\pi$
$227$	2.30894e10	0.577160	0.288580	+	0.957456i	$0.406817\pi$
$228$	0	0
$229$	4.25496e10	1.02244	0.511218	−	0.859451i	$-0.329195\pi$
$229$	4.25496e10	1.02244	0.511218	+	0.859451i	$0.329195\pi$
$230$	−3.27164e10	−0.770885
$231$	0	0
$232$	1.61460e10	0.365905
$233$	1.26679e10	0.281582	0.140791	−	0.990039i	$-0.455036\pi$
$233$	1.26679e10	0.281582	0.140791	+	0.990039i	$0.455036\pi$
$234$	0	0
$235$	−2.07537e10	−0.443906
$236$	2.13568e9	0.0448158
$237$	0	0
$238$	3.73703e9	0.0754971
$239$	−6.37875e10	−1.26458	−0.632289	−	0.774733i	$-0.717884\pi$
$239$	−6.37875e10	−1.26458	−0.632289	+	0.774733i	$0.717884\pi$
$240$	0	0
$241$	−2.91604e10	−0.556823	−0.278412	−	0.960462i	$-0.589808\pi$
$241$	−2.91604e10	−0.556823	−0.278412	+	0.960462i	$0.589808\pi$
$242$	−7.71992e10	−1.44692
$243$	0	0
$244$	−5.64232e10	−1.01907
$245$	−1.10810e10	−0.196487
$246$	0	0
$247$	4.55822e8	0.00779218
$248$	1.09292e11	1.83466
$249$	0	0
$250$	−5.43018e9	−0.0879194
$251$	−6.28939e10	−1.00018	−0.500088	−	0.865974i	$-0.666699\pi$
$251$	−6.28939e10	−1.00018	−0.500088	+	0.865974i	$0.666699\pi$
$252$	0	0
$253$	−1.14907e11	−1.76320
$254$	1.16500e10	0.175620
$255$	0	0
$256$	−6.49425e10	−0.945038
$257$	−1.14480e11	−1.63694	−0.818469	−	0.574551i	$-0.805177\pi$
$257$	−1.14480e11	−1.63694	−0.818469	+	0.574551i	$0.805177\pi$
$258$	0	0
$259$	−2.08345e10	−0.287697
$260$	−2.05123e9	−0.0278378
$261$	0	0
$262$	2.99820e10	0.393101
$263$	−1.40705e10	−0.181346	−0.0906728	−	0.995881i	$-0.528902\pi$
$263$	−1.40705e10	−0.181346	−0.0906728	+	0.995881i	$0.528902\pi$
$264$	0	0
$265$	1.35975e11	1.69377
$266$	4.56965e9	0.0559649
$267$	0	0
$268$	3.87770e10	0.459164
$269$	−8.39143e10	−0.977127	−0.488563	−	0.872528i	$-0.662479\pi$
$269$	−8.39143e10	−0.977127	−0.488563	+	0.872528i	$0.662479\pi$
$270$	0	0
$271$	1.98401e10	0.223451	0.111726	−	0.993739i	$-0.464362\pi$
$271$	1.98401e10	0.223451	0.111726	+	0.993739i	$0.464362\pi$
$272$	−2.30888e9	−0.0255765
$273$	0	0
$274$	5.55995e10	0.595927
$275$	1.57100e11	1.65645
$276$	0	0
$277$	−7.17911e10	−0.732675	−0.366338	−	0.930482i	$-0.619389\pi$
$277$	−7.17911e10	−0.732675	−0.366338	+	0.930482i	$0.619389\pi$
$278$	8.06163e10	0.809508
$279$	0	0
$280$	−5.21347e10	−0.506893
$281$	−1.02853e11	−0.984101	−0.492050	−	0.870567i	$-0.663752\pi$
$281$	−1.02853e11	−0.984101	−0.492050	+	0.870567i	$0.663752\pi$
$282$	0	0
$283$	−5.52883e10	−0.512382	−0.256191	−	0.966626i	$-0.582468\pi$
$283$	−5.52883e10	−0.512382	−0.256191	+	0.966626i	$0.582468\pi$
$284$	4.81880e10	0.439548
$285$	0	0
$286$	3.85630e9	0.0340819
$287$	−3.18238e10	−0.276875
$288$	0	0
$289$	−1.05017e11	−0.885562
$290$	3.67072e10	0.304762
$291$	0	0
$292$	−5.34102e10	−0.429934
$293$	−1.05721e11	−0.838028	−0.419014	−	0.907980i	$-0.637624\pi$
$293$	−1.05721e11	−0.838028	−0.419014	+	0.907980i	$0.637624\pi$
$294$	0	0
$295$	1.23097e10	0.0946343
$296$	−9.80236e10	−0.742195
$297$	0	0
$298$	−5.84975e10	−0.429699
$299$	4.07637e9	0.0294953
$300$	0	0
$301$	−7.14426e10	−0.501658
$302$	3.59891e10	0.248966
$303$	0	0
$304$	−2.82330e9	−0.0189595
$305$	−3.25214e11	−2.15189
$306$	0	0
$307$	−8.10064e10	−0.520471	−0.260236	−	0.965545i	$-0.583800\pi$
$307$	−8.10064e10	−0.520471	−0.260236	+	0.965545i	$0.583800\pi$
$308$	−7.22241e10	−0.457303
$309$	0	0
$310$	2.48470e11	1.52808
$311$	3.18435e11	1.93018	0.965091	−	0.261913i	$-0.0843534\pi$
$311$	3.18435e11	1.93018	0.965091	+	0.261913i	$0.0843534\pi$
$312$	0	0
$313$	1.28876e11	0.758965	0.379483	−	0.925199i	$-0.376102\pi$
$313$	1.28876e11	0.758965	0.379483	+	0.925199i	$0.376102\pi$
$314$	1.77757e10	0.103191
$315$	0	0
$316$	1.63185e11	0.920635
$317$	7.44722e10	0.414217	0.207108	−	0.978318i	$-0.433595\pi$
$317$	7.44722e10	0.414217	0.207108	+	0.978318i	$0.433595\pi$
$318$	0	0
$319$	1.28923e11	0.697065
$320$	−1.35832e11	−0.724148
$321$	0	0
$322$	4.08659e10	0.211841
$323$	1.65945e10	0.0848309
$324$	0	0
$325$	−5.57319e9	−0.0277095
$326$	4.75768e10	0.233301
$327$	0	0
$328$	−1.49726e11	−0.714276
$329$	2.59234e10	0.121986
$330$	0	0
$331$	2.83840e11	1.29971	0.649857	−	0.760057i	$-0.274829\pi$
$331$	2.83840e11	1.29971	0.649857	+	0.760057i	$0.274829\pi$
$332$	−2.77328e10	−0.125277
$333$	0	0
$334$	−1.39332e11	−0.612623
$335$	2.23505e11	0.969583
$336$	0	0
$337$	−5.61414e9	−0.0237109	−0.0118555	−	0.999930i	$-0.503774\pi$
$337$	−5.61414e9	−0.0237109	−0.0118555	+	0.999930i	$0.503774\pi$
$338$	1.41547e11	0.589896
$339$	0	0
$340$	−7.46766e10	−0.303061
$341$	8.72679e11	3.49510
$342$	0	0
$343$	1.38413e10	0.0539949
$344$	−3.36128e11	−1.29417
$345$	0	0
$346$	2.73517e11	1.02599
$347$	−3.39846e11	−1.25834	−0.629171	−	0.777267i	$-0.716606\pi$
$347$	−3.39846e11	−1.25834	−0.629171	+	0.777267i	$0.716606\pi$
$348$	0	0
$349$	3.46718e10	0.125101	0.0625506	−	0.998042i	$-0.480077\pi$
$349$	3.46718e10	0.125101	0.0625506	+	0.998042i	$0.480077\pi$
$350$	−5.58717e10	−0.199015
$351$	0	0
$352$	−5.45578e11	−1.89415
$353$	3.46900e11	1.18910	0.594550	−	0.804059i	$-0.297330\pi$
$353$	3.46900e11	1.18910	0.594550	+	0.804059i	$0.297330\pi$
$354$	0	0
$355$	2.77748e11	0.928160
$356$	6.93939e8	0.00228979
$357$	0	0
$358$	7.30438e10	0.235023
$359$	2.51011e11	0.797569	0.398785	−	0.917045i	$-0.369432\pi$
$359$	2.51011e11	0.797569	0.398785	+	0.917045i	$0.369432\pi$
$360$	0	0
$361$	−3.02396e11	−0.937116
$362$	2.82750e10	0.0865396
$363$	0	0
$364$	2.56219e9	0.00764988
$365$	−3.07848e11	−0.907859
$366$	0	0
$367$	4.79871e11	1.38079	0.690394	−	0.723433i	$-0.257437\pi$
$367$	4.79871e11	1.38079	0.690394	+	0.723433i	$0.257437\pi$
$368$	−2.52485e10	−0.0717663
$369$	0	0
$370$	−2.22853e11	−0.618173
$371$	−1.69847e11	−0.465451
$372$	0	0
$373$	−1.84794e11	−0.494308	−0.247154	−	0.968976i	$-0.579495\pi$
$373$	−1.84794e11	−0.494308	−0.247154	+	0.968976i	$0.579495\pi$
$374$	1.40391e11	0.371038
$375$	0	0
$376$	1.21966e11	0.314698
$377$	−4.57361e9	−0.0116607
$378$	0	0
$379$	−7.08908e11	−1.76487	−0.882437	−	0.470431i	$-0.844098\pi$
$379$	−7.08908e11	−1.76487	−0.882437	+	0.470431i	$0.844098\pi$
$380$	−9.13148e10	−0.224654
$381$	0	0
$382$	2.30367e11	0.553521
$383$	−1.23789e11	−0.293959	−0.146980	−	0.989140i	$-0.546955\pi$
$383$	−1.23789e11	−0.293959	−0.146980	+	0.989140i	$0.546955\pi$
$384$	0	0
$385$	−4.16288e11	−0.965653
$386$	−2.69926e11	−0.618874
$387$	0	0
$388$	1.05345e11	0.235977
$389$	2.97971e11	0.659782	0.329891	−	0.944019i	$-0.392988\pi$
$389$	2.97971e11	0.659782	0.329891	+	0.944019i	$0.392988\pi$
$390$	0	0
$391$	1.48403e11	0.321106
$392$	6.51213e10	0.139295
$393$	0	0
$394$	−2.97400e11	−0.621738
$395$	9.40570e11	1.94404
$396$	0	0
$397$	5.33426e11	1.07775	0.538873	−	0.842387i	$-0.318850\pi$
$397$	5.33426e11	1.07775	0.538873	+	0.842387i	$0.318850\pi$
$398$	−2.27918e11	−0.455307
$399$	0	0
$400$	3.45196e10	0.0674212
$401$	−4.15640e11	−0.802726	−0.401363	−	0.915919i	$-0.631463\pi$
$401$	−4.15640e11	−0.802726	−0.401363	+	0.915919i	$0.631463\pi$
$402$	0	0
$403$	−3.09587e10	−0.0584669
$404$	−1.91704e11	−0.358027
$405$	0	0
$406$	−4.58508e10	−0.0837491
$407$	−7.82704e11	−1.41391
$408$	0	0
$409$	1.07978e12	1.90801	0.954006	−	0.299788i	$-0.0969159\pi$
$409$	1.07978e12	1.90801	0.954006	+	0.299788i	$0.0969159\pi$
$410$	−3.40397e11	−0.594919
$411$	0	0
$412$	5.06473e11	0.866002
$413$	−1.53760e10	−0.0260057
$414$	0	0
$415$	−1.59847e11	−0.264539
$416$	1.93547e10	0.0316859
$417$	0	0
$418$	1.71671e11	0.275045
$419$	2.20998e11	0.350289	0.175144	−	0.984543i	$-0.443961\pi$
$419$	2.20998e11	0.350289	0.175144	+	0.984543i	$0.443961\pi$
$420$	0	0
$421$	3.47478e11	0.539086	0.269543	−	0.962988i	$-0.413127\pi$
$421$	3.47478e11	0.539086	0.269543	+	0.962988i	$0.413127\pi$
$422$	4.27373e11	0.655996
$423$	0	0
$424$	−7.99105e11	−1.20076
$425$	−2.02896e11	−0.301664
$426$	0	0
$427$	4.06224e11	0.591344
$428$	−6.73196e10	−0.0969716
$429$	0	0
$430$	−7.64172e11	−1.07791
$431$	−1.23574e11	−0.172496	−0.0862479	−	0.996274i	$-0.527488\pi$
$431$	−1.23574e11	−0.172496	−0.0862479	+	0.996274i	$0.527488\pi$
$432$	0	0
$433$	−7.27679e11	−0.994820	−0.497410	−	0.867516i	$-0.665716\pi$
$433$	−7.27679e11	−0.994820	−0.497410	+	0.867516i	$0.665716\pi$
$434$	−3.10363e11	−0.419920
$435$	0	0
$436$	2.91455e11	0.386262
$437$	1.81468e11	0.238031
$438$	0	0
$439$	−5.85966e11	−0.752977	−0.376489	−	0.926421i	$-0.622869\pi$
$439$	−5.85966e11	−0.752977	−0.376489	+	0.926421i	$0.622869\pi$
$440$	−1.95858e12	−2.49117
$441$	0	0
$442$	−4.98045e9	−0.00620681
$443$	2.11546e11	0.260969	0.130484	−	0.991450i	$-0.458347\pi$
$443$	2.11546e11	0.260969	0.130484	+	0.991450i	$0.458347\pi$
$444$	0	0
$445$	3.99975e9	0.00483519
$446$	3.08588e11	0.369294
$447$	0	0
$448$	1.69667e11	0.198997
$449$	9.21048e11	1.06948	0.534741	−	0.845016i	$-0.320409\pi$
$449$	9.21048e11	1.06948	0.534741	+	0.845016i	$0.320409\pi$
$450$	0	0
$451$	−1.19554e12	−1.36073
$452$	5.10091e11	0.574810
$453$	0	0
$454$	−3.08491e11	−0.340794
$455$	1.47680e10	0.0161537
$456$	0	0
$457$	8.11185e11	0.869955	0.434977	−	0.900441i	$-0.356756\pi$
$457$	8.11185e11	0.869955	0.434977	+	0.900441i	$0.356756\pi$
$458$	−5.68493e11	−0.603713
$459$	0	0
$460$	−8.16618e11	−0.850372
$461$	1.90069e11	0.196001	0.0980004	−	0.995186i	$-0.468755\pi$
$461$	1.90069e11	0.196001	0.0980004	+	0.995186i	$0.468755\pi$
$462$	0	0
$463$	4.76945e11	0.482341	0.241170	−	0.970483i	$-0.422469\pi$
$463$	4.76945e11	0.482341	0.241170	+	0.970483i	$0.422469\pi$
$464$	2.83284e10	0.0283721
$465$	0	0
$466$	−1.69253e11	−0.166264
$467$	−1.06392e12	−1.03510	−0.517549	−	0.855653i	$-0.673156\pi$
$467$	−1.06392e12	−1.03510	−0.517549	+	0.855653i	$0.673156\pi$
$468$	0	0
$469$	−2.79179e11	−0.266443
$470$	2.77285e11	0.262111
$471$	0	0
$472$	−7.23420e10	−0.0670890
$473$	−2.68393e12	−2.46545
$474$	0	0
$475$	−2.48102e11	−0.223619
$476$	9.32784e10	0.0832817
$477$	0	0
$478$	8.52248e11	0.746690
$479$	8.43415e11	0.732034	0.366017	−	0.930608i	$-0.380721\pi$
$479$	8.43415e11	0.732034	0.366017	+	0.930608i	$0.380721\pi$
$480$	0	0
$481$	2.77668e10	0.0236523
$482$	3.89605e11	0.328785
$483$	0	0
$484$	−1.92694e12	−1.59611
$485$	6.07191e11	0.498296
$486$	0	0
$487$	1.15202e12	0.928065	0.464032	−	0.885818i	$-0.346402\pi$
$487$	1.15202e12	0.928065	0.464032	+	0.885818i	$0.346402\pi$
$488$	1.91123e12	1.52554
$489$	0	0
$490$	1.48051e11	0.116019
$491$	9.68703e11	0.752184	0.376092	−	0.926582i	$-0.377268\pi$
$491$	9.68703e11	0.752184	0.376092	+	0.926582i	$0.377268\pi$
$492$	0	0
$493$	−1.66506e11	−0.126946
$494$	−6.09011e9	−0.00460102
$495$	0	0
$496$	1.91754e11	0.142258
$497$	−3.46934e11	−0.255060
$498$	0	0
$499$	2.62821e12	1.89761	0.948805	−	0.315863i	$-0.102294\pi$
$499$	2.62821e12	1.89761	0.948805	+	0.315863i	$0.102294\pi$
$500$	−1.35540e11	−0.0969848
$501$	0	0
$502$	8.40308e11	0.590570
$503$	−3.95070e11	−0.275181	−0.137590	−	0.990489i	$-0.543936\pi$
$503$	−3.95070e11	−0.275181	−0.137590	+	0.990489i	$0.543936\pi$
$504$	0	0
$505$	−1.10495e12	−0.756019
$506$	1.53524e12	1.04111
$507$	0	0
$508$	2.90790e11	0.193728
$509$	−6.64157e11	−0.438572	−0.219286	−	0.975661i	$-0.570373\pi$
$509$	−6.64157e11	−0.438572	−0.219286	+	0.975661i	$0.570373\pi$
$510$	0	0
$511$	3.84532e11	0.249482
$512$	−2.34512e11	−0.150817
$513$	0	0
$514$	1.52954e12	0.966556
$515$	2.91923e12	1.82867
$516$	0	0
$517$	9.73882e11	0.599513
$518$	2.78364e11	0.169875
$519$	0	0
$520$	6.94815e10	0.0416730
$521$	−5.22766e11	−0.310841	−0.155420	−	0.987848i	$-0.549673\pi$
$521$	−5.22766e11	−0.310841	−0.155420	+	0.987848i	$0.549673\pi$
$522$	0	0
$523$	−3.09135e12	−1.80672	−0.903360	−	0.428882i	$-0.858907\pi$
$523$	−3.09135e12	−1.80672	−0.903360	+	0.428882i	$0.858907\pi$
$524$	7.48366e11	0.433634
$525$	0	0
$526$	1.87991e11	0.107078
$527$	−1.12708e12	−0.636510
$528$	0	0
$529$	−1.78305e11	−0.0989947
$530$	−1.81673e12	−1.00011
$531$	0	0
$532$	1.14061e11	0.0617355
$533$	4.24125e10	0.0227626
$534$	0	0
$535$	−3.88019e11	−0.204768
$536$	−1.31350e12	−0.687366
$537$	0	0
$538$	1.12116e12	0.576960
$539$	5.19984e11	0.265363
$540$	0	0
$541$	−1.47490e12	−0.740243	−0.370121	−	0.928983i	$-0.620684\pi$
$541$	−1.47490e12	−0.740243	−0.370121	+	0.928983i	$0.620684\pi$
$542$	−2.65079e11	−0.131940
$543$	0	0
$544$	7.04621e11	0.344954
$545$	1.67990e12	0.815642
$546$	0	0
$547$	2.12294e12	1.01390	0.506949	−	0.861976i	$-0.330773\pi$
$547$	2.12294e12	1.01390	0.506949	+	0.861976i	$0.330773\pi$
$548$	1.38779e12	0.657374
$549$	0	0
$550$	−2.09897e12	−0.978078
$551$	−2.03604e11	−0.0941031
$552$	0	0
$553$	−1.17486e12	−0.534225
$554$	9.59181e11	0.432620
$555$	0	0
$556$	2.01223e12	0.892978
$557$	3.52857e12	1.55328	0.776641	−	0.629943i	$-0.216922\pi$
$557$	3.52857e12	1.55328	0.776641	+	0.629943i	$0.216922\pi$
$558$	0	0
$559$	9.52137e10	0.0412426
$560$	−9.14713e10	−0.0393042
$561$	0	0
$562$	1.37419e12	0.581078
$563$	3.35280e12	1.40644	0.703218	−	0.710974i	$-0.251746\pi$
$563$	3.35280e12	1.40644	0.703218	+	0.710974i	$0.251746\pi$
$564$	0	0
$565$	2.94008e12	1.21378
$566$	7.38691e11	0.302544
$567$	0	0
$568$	−1.63228e12	−0.658000
$569$	−2.99364e12	−1.19727	−0.598637	−	0.801020i	$-0.704291\pi$
$569$	−2.99364e12	−1.19727	−0.598637	+	0.801020i	$0.704291\pi$
$570$	0	0
$571$	4.67267e12	1.83951	0.919756	−	0.392490i	$-0.128386\pi$
$571$	4.67267e12	1.83951	0.919756	+	0.392490i	$0.128386\pi$
$572$	9.62553e10	0.0375961
$573$	0	0
$574$	4.25189e11	0.163485
$575$	−2.21875e12	−0.846453
$576$	0	0
$577$	−3.62799e12	−1.36262	−0.681310	−	0.731995i	$-0.738589\pi$
$577$	−3.62799e12	−1.36262	−0.681310	+	0.731995i	$0.738589\pi$
$578$	1.40310e12	0.522895
$579$	0	0
$580$	9.16232e11	0.336186
$581$	1.99665e11	0.0726957
$582$	0	0
$583$	−6.38073e12	−2.28751
$584$	1.80917e12	0.643608
$585$	0	0
$586$	1.41252e12	0.494827
$587$	−3.97200e12	−1.38082	−0.690411	−	0.723417i	$-0.742570\pi$
$587$	−3.97200e12	−1.38082	−0.690411	+	0.723417i	$0.742570\pi$
$588$	0	0
$589$	−1.37819e12	−0.471835
$590$	−1.64466e11	−0.0558783
$591$	0	0
$592$	−1.71984e11	−0.0575494
$593$	−2.67436e12	−0.888123	−0.444061	−	0.895996i	$-0.646463\pi$
$593$	−2.67436e12	−0.888123	−0.444061	+	0.895996i	$0.646463\pi$
$594$	0	0
$595$	5.37641e11	0.175860
$596$	−1.46013e12	−0.474005
$597$	0	0
$598$	−5.44632e10	−0.0174160
$599$	−4.84522e12	−1.53777	−0.768887	−	0.639384i	$-0.779189\pi$
$599$	−4.84522e12	−1.53777	−0.768887	+	0.639384i	$0.779189\pi$
$600$	0	0
$601$	−4.64764e12	−1.45311	−0.726553	−	0.687110i	$-0.758879\pi$
$601$	−4.64764e12	−1.45311	−0.726553	+	0.687110i	$0.758879\pi$
$602$	9.54525e11	0.296212
$603$	0	0
$604$	8.98307e11	0.274637
$605$	−1.11065e13	−3.37039
$606$	0	0
$607$	−6.52447e12	−1.95073	−0.975363	−	0.220604i	$-0.929197\pi$
$607$	−6.52447e12	−1.95073	−0.975363	+	0.220604i	$0.929197\pi$
$608$	8.61613e11	0.255709
$609$	0	0
$610$	4.34510e12	1.27062
$611$	−3.45489e10	−0.0100288
$612$	0	0
$613$	−9.54536e11	−0.273036	−0.136518	−	0.990638i	$-0.543591\pi$
$613$	−9.54536e11	−0.273036	−0.136518	+	0.990638i	$0.543591\pi$
$614$	1.08230e12	0.307321
$615$	0	0
$616$	2.44645e12	0.684580
$617$	4.50764e12	1.25218	0.626088	−	0.779752i	$-0.284655\pi$
$617$	4.50764e12	1.25218	0.626088	+	0.779752i	$0.284655\pi$
$618$	0	0
$619$	3.64346e12	0.997484	0.498742	−	0.866750i	$-0.333796\pi$
$619$	3.64346e12	0.997484	0.498742	+	0.866750i	$0.333796\pi$
$620$	6.20195e12	1.68564
$621$	0	0
$622$	−4.25452e12	−1.13971
$623$	−4.99608e9	−0.00132872
$624$	0	0
$625$	−4.18296e12	−1.09654
$626$	−1.72187e12	−0.448143
$627$	0	0
$628$	4.43690e11	0.113831
$629$	1.01087e12	0.257495
$630$	0	0
$631$	3.61498e12	0.907766	0.453883	−	0.891061i	$-0.350038\pi$
$631$	3.61498e12	0.907766	0.453883	+	0.891061i	$0.350038\pi$
$632$	−5.52757e12	−1.37818
$633$	0	0
$634$	−9.95003e11	−0.244581
$635$	1.67607e12	0.409081
$636$	0	0
$637$	−1.84467e10	−0.00443906
$638$	−1.72251e12	−0.411593
$639$	0	0
$640$	−4.13793e12	−0.974929
$641$	2.66304e12	0.623041	0.311521	−	0.950239i	$-0.399162\pi$
$641$	2.66304e12	0.623041	0.311521	+	0.950239i	$0.399162\pi$
$642$	0	0
$643$	4.09899e12	0.945644	0.472822	−	0.881158i	$-0.343235\pi$
$643$	4.09899e12	0.945644	0.472822	+	0.881158i	$0.343235\pi$
$644$	1.02004e12	0.233684
$645$	0	0
$646$	−2.21715e11	−0.0500898
$647$	−6.11325e12	−1.37152	−0.685761	−	0.727827i	$-0.740531\pi$
$647$	−6.11325e12	−1.37152	−0.685761	+	0.727827i	$0.740531\pi$
$648$	0	0
$649$	−5.77640e11	−0.127807
$650$	7.44619e10	0.0163615
$651$	0	0
$652$	1.18754e12	0.257356
$653$	−7.66270e12	−1.64920	−0.824598	−	0.565719i	$-0.808599\pi$
$653$	−7.66270e12	−1.64920	−0.824598	+	0.565719i	$0.808599\pi$
$654$	0	0
$655$	4.31346e12	0.915673
$656$	−2.62698e11	−0.0553846
$657$	0	0
$658$	−3.46356e11	−0.0720287
$659$	8.20110e9	0.00169390	0.000846950	−	1.00000i	$-0.499730\pi$
$659$	8.20110e9	0.00169390	0.000846950		1.00000i	$0.499730\pi$
$660$	0	0
$661$	3.20922e12	0.653872	0.326936	−	0.945046i	$-0.393984\pi$
$661$	3.20922e12	0.653872	0.326936	+	0.945046i	$0.393984\pi$
$662$	−3.79231e12	−0.767437
$663$	0	0
$664$	9.39395e11	0.187539
$665$	6.57429e11	0.130362
$666$	0	0
$667$	−1.82081e12	−0.356203
$668$	−3.47781e12	−0.675791
$669$	0	0
$670$	−2.98618e12	−0.572506
$671$	1.52609e13	2.90622
$672$	0	0
$673$	−4.91229e12	−0.923031	−0.461515	−	0.887132i	$-0.652694\pi$
$673$	−4.91229e12	−0.923031	−0.461515	+	0.887132i	$0.652694\pi$
$674$	7.50089e10	0.0140005
$675$	0	0
$676$	3.53309e12	0.650721
$677$	3.11910e12	0.570664	0.285332	−	0.958429i	$-0.407896\pi$
$677$	3.11910e12	0.570664	0.285332	+	0.958429i	$0.407896\pi$
$678$	0	0
$679$	−7.58440e11	−0.136933
$680$	2.52953e12	0.453680
$681$	0	0
$682$	−1.16596e13	−2.06374
$683$	−2.91806e12	−0.513099	−0.256550	−	0.966531i	$-0.582586\pi$
$683$	−2.91806e12	−0.513099	−0.256550	+	0.966531i	$0.582586\pi$
$684$	0	0
$685$	7.99902e12	1.38813
$686$	−1.84930e11	−0.0318822
$687$	0	0
$688$	−5.89742e11	−0.100349
$689$	2.26360e11	0.0382660
$690$	0	0
$691$	4.74697e12	0.792073	0.396037	−	0.918235i	$-0.370385\pi$
$691$	4.74697e12	0.792073	0.396037	+	0.918235i	$0.370385\pi$
$692$	6.82712e12	1.13178
$693$	0	0
$694$	4.54058e12	0.743009
$695$	1.15982e13	1.88563
$696$	0	0
$697$	1.54406e12	0.247809
$698$	−4.63240e11	−0.0738680
$699$	0	0
$700$	−1.39459e12	−0.219535
$701$	2.24423e11	0.0351023	0.0175512	−	0.999846i	$-0.494413\pi$
$701$	2.24423e11	0.0351023	0.0175512	+	0.999846i	$0.494413\pi$
$702$	0	0
$703$	1.23610e12	0.190877
$704$	6.37400e12	0.977992
$705$	0	0
$706$	−4.63484e12	−0.702123
$707$	1.38019e12	0.207755
$708$	0	0
$709$	4.25463e12	0.632344	0.316172	−	0.948702i	$-0.397602\pi$
$709$	4.25463e12	0.632344	0.316172	+	0.948702i	$0.397602\pi$
$710$	−3.71091e12	−0.548047
$711$	0	0
$712$	−2.35059e10	−0.00342781
$713$	−1.23250e13	−1.78601
$714$	0	0
$715$	5.54800e11	0.0793888
$716$	1.82321e12	0.259256
$717$	0	0
$718$	−3.35369e12	−0.470938
$719$	4.28931e12	0.598559	0.299280	−	0.954165i	$-0.403254\pi$
$719$	4.28931e12	0.598559	0.299280	+	0.954165i	$0.403254\pi$
$720$	0	0
$721$	−3.64640e12	−0.502523
$722$	4.04023e12	0.553335
$723$	0	0
$724$	7.05761e11	0.0954627
$725$	2.48940e12	0.334637
$726$	0	0
$727$	1.28935e13	1.71185	0.855926	−	0.517098i	$-0.172988\pi$
$727$	1.28935e13	1.71185	0.855926	+	0.517098i	$0.172988\pi$
$728$	−8.67892e10	−0.0114518
$729$	0	0
$730$	4.11307e12	0.536060
$731$	3.46633e12	0.448995
$732$	0	0
$733$	2.71447e12	0.347310	0.173655	−	0.984807i	$-0.444442\pi$
$733$	2.71447e12	0.347310	0.173655	+	0.984807i	$0.444442\pi$
$734$	−6.41143e12	−0.815309
$735$	0	0
$736$	7.70531e12	0.967921
$737$	−1.04881e13	−1.30946
$738$	0	0
$739$	−9.62579e12	−1.18723	−0.593617	−	0.804747i	$-0.702301\pi$
$739$	−9.62579e12	−1.18723	−0.593617	+	0.804747i	$0.702301\pi$
$740$	−5.56252e12	−0.681913
$741$	0	0
$742$	2.26927e12	0.274833
$743$	1.24362e13	1.49706	0.748529	−	0.663103i	$-0.230761\pi$
$743$	1.24362e13	1.49706	0.748529	+	0.663103i	$0.230761\pi$
$744$	0	0
$745$	−8.41595e12	−1.00092
$746$	2.46898e12	0.291872
$747$	0	0
$748$	3.50425e12	0.409296
$749$	4.84674e11	0.0562706
$750$	0	0
$751$	4.26959e12	0.489786	0.244893	−	0.969550i	$-0.421247\pi$
$751$	4.26959e12	0.489786	0.244893	+	0.969550i	$0.421247\pi$
$752$	2.13992e11	0.0244015
$753$	0	0
$754$	6.11068e10	0.00688523
$755$	5.17770e12	0.579930
$756$	0	0
$757$	9.76840e12	1.08116	0.540582	−	0.841291i	$-0.318204\pi$
$757$	9.76840e12	1.08116	0.540582	+	0.841291i	$0.318204\pi$
$758$	9.47153e12	1.04210
$759$	0	0
$760$	3.09311e12	0.336306
$761$	−1.23336e13	−1.33309	−0.666543	−	0.745466i	$-0.732227\pi$
$761$	−1.23336e13	−1.33309	−0.666543	+	0.745466i	$0.732227\pi$
$762$	0	0
$763$	−2.09836e12	−0.224140
$764$	5.75008e12	0.610595
$765$	0	0
$766$	1.65391e12	0.173573
$767$	2.04921e10	0.00213800
$768$	0	0
$769$	−1.68919e13	−1.74185	−0.870926	−	0.491415i	$-0.836480\pi$
$769$	−1.68919e13	−1.74185	−0.870926	+	0.491415i	$0.836480\pi$
$770$	5.56191e12	0.570185
$771$	0	0
$772$	−6.73751e12	−0.682687
$773$	−1.47186e13	−1.48272	−0.741359	−	0.671109i	$-0.765818\pi$
$773$	−1.47186e13	−1.48272	−0.741359	+	0.671109i	$0.765818\pi$
$774$	0	0
$775$	1.68507e13	1.67788
$776$	−3.56835e12	−0.353257
$777$	0	0
$778$	−3.98111e12	−0.389579
$779$	1.88808e12	0.183697
$780$	0	0
$781$	−1.30335e13	−1.25352
$782$	−1.98278e12	−0.189602
$783$	0	0
$784$	1.14257e11	0.0108009
$785$	2.55736e12	0.240369
$786$	0	0
$787$	−1.82466e13	−1.69549	−0.847744	−	0.530406i	$-0.822040\pi$
$787$	−1.82466e13	−1.69549	−0.847744	+	0.530406i	$0.822040\pi$
$788$	−7.42326e12	−0.685846
$789$	0	0
$790$	−1.25667e13	−1.14789
$791$	−3.67245e12	−0.333550
$792$	0	0
$793$	−5.41388e11	−0.0486160
$794$	−7.12695e12	−0.636373
$795$	0	0
$796$	−5.68895e12	−0.502254
$797$	1.21558e13	1.06714	0.533568	−	0.845757i	$-0.320851\pi$
$797$	1.21558e13	1.06714	0.533568	+	0.845757i	$0.320851\pi$
$798$	0	0
$799$	−1.25778e12	−0.109180
$800$	−1.05347e13	−0.909318
$801$	0	0
$802$	5.55325e12	0.473982
$803$	1.44460e13	1.22610
$804$	0	0
$805$	5.87932e12	0.493453
$806$	4.13631e11	0.0345227
$807$	0	0
$808$	6.49362e12	0.535964
$809$	2.61893e12	0.214959	0.107479	−	0.994207i	$-0.465722\pi$
$809$	2.61893e12	0.214959	0.107479	+	0.994207i	$0.465722\pi$
$810$	0	0
$811$	1.16994e13	0.949662	0.474831	−	0.880077i	$-0.342509\pi$
$811$	1.16994e13	0.949662	0.474831	+	0.880077i	$0.342509\pi$
$812$	−1.14446e12	−0.0923845
$813$	0	0
$814$	1.04575e13	0.834868
$815$	6.84480e12	0.543440
$816$	0	0
$817$	4.23863e12	0.332833
$818$	−1.44267e13	−1.12662
$819$	0	0
$820$	−8.49649e12	−0.656262
$821$	−5.17673e12	−0.397659	−0.198830	−	0.980034i	$-0.563714\pi$
$821$	−5.17673e12	−0.397659	−0.198830	+	0.980034i	$0.563714\pi$
$822$	0	0
$823$	−7.84017e12	−0.595698	−0.297849	−	0.954613i	$-0.596269\pi$
$823$	−7.84017e12	−0.595698	−0.297849	+	0.954613i	$0.596269\pi$
$824$	−1.71558e13	−1.29640
$825$	0	0
$826$	2.05435e11	0.0153555
$827$	1.43263e13	1.06502	0.532512	−	0.846422i	$-0.321248\pi$
$827$	1.43263e13	1.06502	0.532512	+	0.846422i	$0.321248\pi$
$828$	0	0
$829$	4.95546e12	0.364408	0.182204	−	0.983261i	$-0.441677\pi$
$829$	4.95546e12	0.364408	0.182204	+	0.983261i	$0.441677\pi$
$830$	2.13568e12	0.156201
$831$	0	0
$832$	−2.26121e11	−0.0163601
$833$	−6.71566e11	−0.0483266
$834$	0	0
$835$	−2.00456e13	−1.42702
$836$	4.28500e12	0.303405
$837$	0	0
$838$	−2.95270e12	−0.206834
$839$	−7.38235e12	−0.514358	−0.257179	−	0.966364i	$-0.582793\pi$
$839$	−7.38235e12	−0.514358	−0.257179	+	0.966364i	$0.582793\pi$
$840$	0	0
$841$	−1.24642e13	−0.859179
$842$	−4.64256e12	−0.318312
$843$	0	0
$844$	1.06675e13	0.723636
$845$	2.03642e13	1.37408
$846$	0	0
$847$	1.38732e13	0.926189
$848$	−1.40204e12	−0.0931065
$849$	0	0
$850$	2.71084e12	0.178123
$851$	1.10543e13	0.722516
$852$	0	0
$853$	2.13356e13	1.37985	0.689927	−	0.723879i	$-0.257643\pi$
$853$	2.13356e13	1.37985	0.689927	+	0.723879i	$0.257643\pi$
$854$	−5.42745e12	−0.349169
$855$	0	0
$856$	2.28032e12	0.145166
$857$	−1.65008e13	−1.04494	−0.522468	−	0.852659i	$-0.674989\pi$
$857$	−1.65008e13	−1.04494	−0.522468	+	0.852659i	$0.674989\pi$
$858$	0	0
$859$	−1.94817e13	−1.22083	−0.610417	−	0.792080i	$-0.708998\pi$
$859$	−1.94817e13	−1.22083	−0.610417	+	0.792080i	$0.708998\pi$
$860$	−1.90742e13	−1.18906
$861$	0	0
$862$	1.65103e12	0.101853
$863$	−1.61059e13	−0.988411	−0.494205	−	0.869345i	$-0.664541\pi$
$863$	−1.61059e13	−0.988411	−0.494205	+	0.869345i	$0.664541\pi$
$864$	0	0
$865$	3.93504e13	2.38989
$866$	9.72232e12	0.587408
$867$	0	0
$868$	−7.74684e12	−0.463219
$869$	−4.41368e13	−2.62550
$870$	0	0
$871$	3.72070e11	0.0219050
$872$	−9.87249e12	−0.578232
$873$	0	0
$874$	−2.42454e12	−0.140549
$875$	9.75836e11	0.0562782
$876$	0	0
$877$	−1.84919e13	−1.05556	−0.527780	−	0.849381i	$-0.676975\pi$
$877$	−1.84919e13	−1.05556	−0.527780	+	0.849381i	$0.676975\pi$
$878$	7.82892e12	0.444607
$879$	0	0
$880$	−3.43636e12	−0.193164
$881$	2.83078e13	1.58312	0.791562	−	0.611089i	$-0.209268\pi$
$881$	2.83078e13	1.58312	0.791562	+	0.611089i	$0.209268\pi$
$882$	0	0
$883$	−7.32328e12	−0.405399	−0.202700	−	0.979241i	$-0.564971\pi$
$883$	−7.32328e12	−0.405399	−0.202700	+	0.979241i	$0.564971\pi$
$884$	−1.24315e11	−0.00684681
$885$	0	0
$886$	−2.82641e12	−0.154093
$887$	−1.51372e13	−0.821087	−0.410544	−	0.911841i	$-0.634661\pi$
$887$	−1.51372e13	−0.821087	−0.410544	+	0.911841i	$0.634661\pi$
$888$	0	0
$889$	−2.09357e12	−0.112416
$890$	−5.34396e10	−0.00285501
$891$	0	0
$892$	7.70253e12	0.407372
$893$	−1.53802e12	−0.0809337
$894$	0	0
$895$	1.05087e13	0.547451
$896$	5.16867e12	0.267913
$897$	0	0
$898$	−1.23059e13	−0.631493
$899$	1.38285e13	0.706082
$900$	0	0
$901$	8.24080e12	0.416589
$902$	1.59733e13	0.803463
$903$	0	0
$904$	−1.72783e13	−0.860487
$905$	4.06789e12	0.201582
$906$	0	0
$907$	−2.24337e13	−1.10070	−0.550349	−	0.834935i	$-0.685505\pi$
$907$	−2.24337e13	−1.10070	−0.550349	+	0.834935i	$0.685505\pi$
$908$	−7.70011e12	−0.375933
$909$	0	0
$910$	−1.97312e11	−0.00953820
$911$	6.34178e12	0.305055	0.152528	−	0.988299i	$-0.451259\pi$
$911$	6.34178e12	0.305055	0.152528	+	0.988299i	$0.451259\pi$
$912$	0	0
$913$	7.50094e12	0.357270
$914$	−1.08380e13	−0.513679
$915$	0	0
$916$	−1.41899e13	−0.665963
$917$	−5.38793e12	−0.251629
$918$	0	0
$919$	9.04471e12	0.418288	0.209144	−	0.977885i	$-0.432932\pi$
$919$	9.04471e12	0.418288	0.209144	+	0.977885i	$0.432932\pi$
$920$	2.76614e13	1.27300
$921$	0	0
$922$	−2.53946e12	−0.115732
$923$	4.62369e11	0.0209692
$924$	0	0
$925$	−1.51134e13	−0.678771
$926$	−6.37233e12	−0.284806
$927$	0	0
$928$	−8.64522e12	−0.382658
$929$	−8.50641e12	−0.374693	−0.187346	−	0.982294i	$-0.559989\pi$
$929$	−8.50641e12	−0.374693	−0.187346	+	0.982294i	$0.559989\pi$
$930$	0	0
$931$	−8.21193e11	−0.0358238
$932$	−4.22464e12	−0.183408
$933$	0	0
$934$	1.42147e13	0.611191
$935$	2.01979e13	0.864280
$936$	0	0
$937$	−3.00276e13	−1.27260	−0.636300	−	0.771442i	$-0.719536\pi$
$937$	−3.00276e13	−1.27260	−0.636300	+	0.771442i	$0.719536\pi$
$938$	3.73003e12	0.157326
$939$	0	0
$940$	6.92118e12	0.289138
$941$	3.15982e13	1.31374	0.656870	−	0.754004i	$-0.271880\pi$
$941$	3.15982e13	1.31374	0.656870	+	0.754004i	$0.271880\pi$
$942$	0	0
$943$	1.68849e13	0.695338
$944$	−1.26925e11	−0.00520205
$945$	0	0
$946$	3.58592e13	1.45576
$947$	8.84885e12	0.357529	0.178765	−	0.983892i	$-0.442790\pi$
$947$	8.84885e12	0.357529	0.178765	+	0.983892i	$0.442790\pi$
$948$	0	0
$949$	−5.12477e11	−0.0205105
$950$	3.31482e12	0.132040
$951$	0	0
$952$	−3.15963e12	−0.124672
$953$	4.90260e12	0.192534	0.0962672	−	0.995356i	$-0.469310\pi$
$953$	4.90260e12	0.192534	0.0962672	+	0.995356i	$0.469310\pi$
$954$	0	0
$955$	3.31425e13	1.28935
$956$	2.12726e13	0.823682
$957$	0	0
$958$	−1.12686e13	−0.432241
$959$	−9.99155e12	−0.381460
$960$	0	0
$961$	6.71649e13	2.54031
$962$	−3.70985e11	−0.0139659
$963$	0	0
$964$	9.72475e12	0.362687
$965$	−3.88339e13	−1.44158
$966$	0	0
$967$	1.55633e13	0.572378	0.286189	−	0.958173i	$-0.407612\pi$
$967$	1.55633e13	0.572378	0.286189	+	0.958173i	$0.407612\pi$
$968$	6.52713e13	2.38937
$969$	0	0
$970$	−8.11251e12	−0.294227
$971$	2.49197e13	0.899613	0.449806	−	0.893126i	$-0.351493\pi$
$971$	2.49197e13	0.899613	0.449806	+	0.893126i	$0.351493\pi$
$972$	0	0
$973$	−1.44872e13	−0.518176
$974$	−1.53918e13	−0.547991
$975$	0	0
$976$	3.35329e12	0.118290
$977$	−4.47898e12	−0.157273	−0.0786364	−	0.996903i	$-0.525057\pi$
$977$	−4.47898e12	−0.157273	−0.0786364	+	0.996903i	$0.525057\pi$
$978$	0	0
$979$	−1.87691e11	−0.00653012
$980$	3.69543e12	0.127981
$981$	0	0
$982$	−1.29426e13	−0.444139
$983$	1.43233e13	0.489275	0.244637	−	0.969615i	$-0.421331\pi$
$983$	1.43233e13	0.489275	0.244637	+	0.969615i	$0.421331\pi$
$984$	0	0
$985$	−4.27865e13	−1.44825
$986$	2.22464e12	0.0749572
$987$	0	0
$988$	−1.52013e11	−0.00507543
$989$	3.79057e13	1.25986
$990$	0	0
$991$	−2.28575e13	−0.752829	−0.376415	−	0.926451i	$-0.622843\pi$
$991$	−2.28575e13	−0.752829	−0.376415	+	0.926451i	$0.622843\pi$
$992$	−5.85194e13	−1.91866
$993$	0	0
$994$	4.63529e12	0.150605
$995$	−3.27902e13	−1.06057
$996$	0	0
$997$	−2.67132e13	−0.856243	−0.428121	−	0.903721i	$-0.640824\pi$
$997$	−2.67132e13	−0.856243	−0.428121	+	0.903721i	$0.640824\pi$
$998$	−3.51147e13	−1.12047
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	63.10.a.e.1.2		3
3.2	odd	2		7.10.a.b.1.2	✓	3
12.11	even	2		112.10.a.h.1.1		3
15.2	even	4		175.10.b.d.99.4		6
15.8	even	4		175.10.b.d.99.3		6
15.14	odd	2		175.10.a.d.1.2		3
21.2	odd	6		49.10.c.d.18.2		6
21.5	even	6		49.10.c.e.18.2		6
21.11	odd	6		49.10.c.d.30.2		6
21.17	even	6		49.10.c.e.30.2		6
21.20	even	2		49.10.a.c.1.2		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
7.10.a.b.1.2	✓	3	3.2	odd	2
49.10.a.c.1.2		3	21.20	even	2
49.10.c.d.18.2		6	21.2	odd	6
49.10.c.d.30.2		6	21.11	odd	6
49.10.c.e.18.2		6	21.5	even	6
49.10.c.e.30.2		6	21.17	even	6
63.10.a.e.1.2		3	1.1	even	1	trivial
112.10.a.h.1.1		3	12.11	even	2
175.10.a.d.1.2		3	15.14	odd	2
175.10.b.d.99.3		6	15.8	even	4
175.10.b.d.99.4		6	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 $\F_{q}$

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

\(f(q)\)	\(=\)	\(q-13.3607 q^{2} -333.491 q^{4} -1922.19 q^{5} +2401.00 q^{7} +11296.4 q^{8} +25681.8 q^{10} +90199.9 q^{11} -3199.89 q^{13} -32079.1 q^{14} +19819.7 q^{16} -116494. q^{17} -142449. q^{19} +641033. q^{20} -1.20514e6 q^{22} -1.27391e6 q^{23} +1.74168e6 q^{25} +42752.8 q^{26} -800712. q^{28} +1.42931e6 q^{29} +9.67494e6 q^{31} -6.04855e6 q^{32} +1.55645e6 q^{34} -4.61518e6 q^{35} -8.67744e6 q^{37} +1.90323e6 q^{38} -2.17138e7 q^{40} -1.32544e7 q^{41} -2.97554e7 q^{43} -3.00809e7 q^{44} +1.70204e7 q^{46} +1.07969e7 q^{47} +5.76480e6 q^{49} -2.32702e7 q^{50} +1.06713e6 q^{52} -7.07399e7 q^{53} -1.73381e8 q^{55} +2.71226e7 q^{56} -1.90966e7 q^{58} -6.40400e6 q^{59} +1.69190e8 q^{61} -1.29264e8 q^{62} +7.06653e7 q^{64} +6.15078e6 q^{65} -1.16276e8 q^{67} +3.88498e7 q^{68} +6.16621e7 q^{70} -1.44496e8 q^{71} +1.60155e8 q^{73} +1.15937e8 q^{74} +4.75056e7 q^{76} +2.16570e8 q^{77} -4.89322e8 q^{79} -3.80972e7 q^{80} +1.77088e8 q^{82} +8.31590e7 q^{83} +2.23924e8 q^{85} +3.97553e8 q^{86} +1.01893e9 q^{88} -2.08083e6 q^{89} -7.68292e6 q^{91} +4.24838e8 q^{92} -1.44255e8 q^{94} +2.73815e8 q^{95} -3.15885e8 q^{97} -7.70219e7 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 21 q^{2} + 1557 q^{4} - 1554 q^{5} + 7203 q^{7} - 14055 q^{8} - 97860 q^{10} + 3444 q^{11} - 19782 q^{13} - 50421 q^{14} + 482961 q^{16} - 1016694 q^{17} + 222852 q^{19} + 1922088 q^{20} - 2847048 q^{22}+ \cdots - 121060821 q^{98}+O(q^{100}) \) 3 * q - 21 * q^2 + 1557 * q^4 - 1554 * q^5 + 7203 * q^7 - 14055 * q^8 - 97860 * q^10 + 3444 * q^11 - 19782 * q^13 - 50421 * q^14 + 482961 * q^16 - 1016694 * q^17 + 222852 * q^19 + 1922088 * q^20 - 2847048 * q^22 - 1885632 * q^23 + 3073221 * q^25 + 8785056 * q^26 + 3738357 * q^28 - 4081818 * q^29 + 2869440 * q^31 - 25221951 * q^32 - 3981642 * q^34 - 3731154 * q^35 + 1395618 * q^37 - 43479870 * q^38 - 82859280 * q^40 + 14420658 * q^41 - 61631172 * q^43 - 97011984 * q^44 + 89747664 * q^46 + 10368960 * q^47 + 17294403 * q^49 - 73325055 * q^50 - 80908044 * q^52 - 67502610 * q^53 - 105823032 * q^55 - 33746055 * q^56 - 159163830 * q^58 + 42590100 * q^59 + 191746842 * q^61 + 46983468 * q^62 + 7852161 * q^64 - 364283220 * q^65 - 255175788 * q^67 - 743485806 * q^68 - 234961860 * q^70 - 296514504 * q^71 + 344213310 * q^73 + 690696462 * q^74 + 728839986 * q^76 + 8269044 * q^77 - 960412656 * q^79 + 1333333344 * q^80 + 562675302 * q^82 + 1100517180 * q^83 + 438179412 * q^85 + 880982256 * q^86 + 1206124800 * q^88 - 506816478 * q^89 - 47496582 * q^91 - 691123488 * q^92 + 1388004828 * q^94 + 2203071072 * q^95 - 647498250 * q^97 - 121060821 * q^98

Introduction

L-functions

Modular forms

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