LMFDB - Embedded newform 81.10.a.b.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [81,10,Mod(1,81)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(81, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0]))

N = Newforms(chi, 10, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("81.1");

S:= CuspForms(chi, 10);

N := Newforms(S);

Level:	$ N $	$=$	$ 81 = 3^{4} $
Weight:	$ k $	$=$	$ 10 $
Character orbit:	$[\chi]$	$=$	81.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:	$41.7179027293$
Analytic rank:	$1$
Dimension:	$4$
Coefficient field:	$\mathbb{Q}[x]/(x^{4} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 1314x^{2} + 10232x + 106624 $ x^4 - x^3 - 1314x^2 + 10232x + 106624
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{2}\cdot 3^{3} $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-5.97760$ of defining polynomial
Character	$\chi$	$=$	81.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q+2.02240 q^{2} -507.910 q^{4} -643.210 q^{5} +2952.58 q^{7} -2062.66 q^{8} -1300.83 q^{10} +61533.0 q^{11} +47852.7 q^{13} +5971.30 q^{14} +255878. q^{16} -315492. q^{17} +284556. q^{19} +326693. q^{20} +124444. q^{22} -19567.0 q^{23} -1.53941e6 q^{25} +96777.2 q^{26} -1.49965e6 q^{28} -3.09788e6 q^{29} -5.43262e6 q^{31} +1.57357e6 q^{32} -638050. q^{34} -1.89913e6 q^{35} -7.66765e6 q^{37} +575485. q^{38} +1.32673e6 q^{40} +2.22462e7 q^{41} -5.71372e6 q^{43} -3.12532e7 q^{44} -39572.2 q^{46} +3.70005e7 q^{47} -3.16359e7 q^{49} -3.11329e6 q^{50} -2.43049e7 q^{52} -1.11104e8 q^{53} -3.95786e7 q^{55} -6.09018e6 q^{56} -6.26515e6 q^{58} +1.61683e8 q^{59} -1.51282e8 q^{61} -1.09869e7 q^{62} -1.27827e8 q^{64} -3.07794e7 q^{65} -2.10872e8 q^{67} +1.60241e8 q^{68} -3.84080e6 q^{70} -2.50785e8 q^{71} -2.10869e8 q^{73} -1.55070e7 q^{74} -1.44529e8 q^{76} +1.81681e8 q^{77} -1.98853e8 q^{79} -1.64584e8 q^{80} +4.49907e7 q^{82} +4.01229e8 q^{83} +2.02928e8 q^{85} -1.15554e7 q^{86} -1.26922e8 q^{88} -5.59701e8 q^{89} +1.41289e8 q^{91} +9.93826e6 q^{92} +7.48296e7 q^{94} -1.83029e8 q^{95} +8.24474e8 q^{97} -6.39802e7 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 33 q^{2} + 853 q^{4} + 570 q^{5} - 3238 q^{7} + 4791 q^{8} - 9723 q^{10} - 96690 q^{11} - 141118 q^{13} + 3036 q^{14} - 244463 q^{16} + 285156 q^{17} - 465166 q^{19} - 1041711 q^{20} - 2244480 q^{22}+ \cdots + 735501321 q^{98}+O(q^{100}) $ 4 * q + 33 * q^2 + 853 * q^4 + 570 * q^5 - 3238 * q^7 + 4791 * q^8 - 9723 * q^10 - 96690 * q^11 - 141118 * q^13 + 3036 * q^14 - 244463 * q^16 + 285156 * q^17 - 465166 * q^19 - 1041711 * q^20 - 2244480 * q^22 - 237318 * q^23 + 3417712 * q^25 - 6433737 * q^26 + 769292 * q^28 + 2806926 * q^29 - 8244448 * q^31 + 3896415 * q^32 + 1386189 * q^34 - 31668402 * q^35 - 8204014 * q^37 - 755556 * q^38 - 54180645 * q^40 + 28889292 * q^41 - 39586246 * q^43 - 89567616 * q^44 - 93870624 * q^46 - 17514588 * q^47 - 67913400 * q^49 + 233832372 * q^50 - 207295237 * q^52 - 155826504 * q^53 - 185126142 * q^55 + 133671300 * q^56 - 164356143 * q^58 + 341256108 * q^59 - 238298254 * q^61 - 729569796 * q^62 - 286454975 * q^64 + 379881528 * q^65 - 72601486 * q^67 + 762353217 * q^68 - 780989460 * q^70 - 667543266 * q^71 - 520588288 * q^73 + 183473031 * q^74 - 413336884 * q^76 + 721530228 * q^77 - 137324662 * q^79 - 1181367123 * q^80 - 275042586 * q^82 - 278546844 * q^83 - 1408511358 * q^85 + 420627720 * q^86 - 30700416 * q^88 + 237712932 * q^89 - 762950942 * q^91 - 1238743680 * q^92 - 507333516 * q^94 - 811663194 * q^95 + 700071356 * q^97 + 735501321 * 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 $

$2$	2.02240	0.0893781	0.0446891	−	0.999001i	$-0.485770\pi$
$2$	2.02240	0.0893781	0.0446891	+	0.999001i	$0.485770\pi$
$3$	0	0
$3$	0	0
$4$	−507.910	−0.992012
$5$	−643.210	−0.460244	−0.230122	−	0.973162i	$-0.573912\pi$
$5$	−643.210	−0.460244	−0.230122	+	0.973162i	$0.573912\pi$
$6$	0	0
$7$	2952.58	0.464795	0.232397	−	0.972621i	$-0.425343\pi$
$7$	2952.58	0.464795	0.232397	+	0.972621i	$0.425343\pi$
$8$	−2062.66	−0.178042
$9$	0	0
$10$	−1300.83	−0.0411357
$11$	61533.0	1.26719	0.633594	−	0.773666i	$-0.281579\pi$
$11$	61533.0	1.26719	0.633594	+	0.773666i	$0.281579\pi$
$12$	0	0
$13$	47852.7	0.464688	0.232344	−	0.972634i	$-0.425360\pi$
$13$	47852.7	0.464688	0.232344	+	0.972634i	$0.425360\pi$
$14$	5971.30	0.0415425
$15$	0	0
$16$	255878.	0.976098
$17$	−315492.	−0.916153	−0.458076	−	0.888913i	$-0.651461\pi$
$17$	−315492.	−0.916153	−0.458076	+	0.888913i	$0.651461\pi$
$18$	0	0
$19$	284556.	0.500929	0.250465	−	0.968126i	$-0.419417\pi$
$19$	284556.	0.500929	0.250465	+	0.968126i	$0.419417\pi$
$20$	326693.	0.456567
$21$	0	0
$22$	124444.	0.113259
$23$	−19567.0	−0.0145797	−0.00728985	−	0.999973i	$-0.502320\pi$
$23$	−19567.0	−0.0145797	−0.00728985	+	0.999973i	$0.502320\pi$
$24$	0	0
$25$	−1.53941e6	−0.788176
$26$	96777.2	0.0415330
$27$	0	0
$28$	−1.49965e6	−0.461082
$29$	−3.09788e6	−0.813343	−0.406672	−	0.913574i	$-0.633311\pi$
$29$	−3.09788e6	−0.813343	−0.406672	+	0.913574i	$0.633311\pi$
$30$	0	0
$31$	−5.43262e6	−1.05653	−0.528265	−	0.849080i	$-0.677157\pi$
$31$	−5.43262e6	−1.05653	−0.528265	+	0.849080i	$0.677157\pi$
$32$	1.57357e6	0.265284
$33$	0	0
$34$	−638050.	−0.0818840
$35$	−1.89913e6	−0.213919
$36$	0	0
$37$	−7.66765e6	−0.672596	−0.336298	−	0.941756i	$-0.609175\pi$
$37$	−7.66765e6	−0.672596	−0.336298	+	0.941756i	$0.609175\pi$
$38$	575485.	0.0447721
$39$	0	0
$40$	1.32673e6	0.0819428
$41$	2.22462e7	1.22950	0.614751	−	0.788721i	$-0.289257\pi$
$41$	2.22462e7	1.22950	0.614751	+	0.788721i	$0.289257\pi$
$42$	0	0
$43$	−5.71372e6	−0.254865	−0.127433	−	0.991847i	$-0.540674\pi$
$43$	−5.71372e6	−0.254865	−0.127433	+	0.991847i	$0.540674\pi$
$44$	−3.12532e7	−1.25706
$45$	0	0
$46$	−39572.2	−0.00130311
$47$	3.70005e7	1.10603	0.553015	−	0.833171i	$-0.313477\pi$
$47$	3.70005e7	1.10603	0.553015	+	0.833171i	$0.313477\pi$
$48$	0	0
$49$	−3.16359e7	−0.783966
$50$	−3.11329e6	−0.0704457
$51$	0	0
$52$	−2.43049e7	−0.460976
$53$	−1.11104e8	−1.93414	−0.967070	−	0.254512i	$-0.918085\pi$
$53$	−1.11104e8	−1.93414	−0.967070	+	0.254512i	$0.918085\pi$
$54$	0	0
$55$	−3.95786e7	−0.583215
$56$	−6.09018e6	−0.0827531
$57$	0	0
$58$	−6.26515e6	−0.0726951
$59$	1.61683e8	1.73712	0.868559	−	0.495586i	$-0.165047\pi$
$59$	1.61683e8	1.73712	0.868559	+	0.495586i	$0.165047\pi$
$60$	0	0
$61$	−1.51282e8	−1.39895	−0.699477	−	0.714655i	$-0.746583\pi$
$61$	−1.51282e8	−1.39895	−0.699477	+	0.714655i	$0.746583\pi$
$62$	−1.09869e7	−0.0944307
$63$	0	0
$64$	−1.27827e8	−0.952388
$65$	−3.07794e7	−0.213870
$66$	0	0
$67$	−2.10872e8	−1.27845	−0.639223	−	0.769021i	$-0.720744\pi$
$67$	−2.10872e8	−1.27845	−0.639223	+	0.769021i	$0.720744\pi$
$68$	1.60241e8	0.908834
$69$	0	0
$70$	−3.84080e6	−0.0191197
$71$	−2.50785e8	−1.17122	−0.585610	−	0.810593i	$-0.699145\pi$
$71$	−2.50785e8	−1.17122	−0.585610	+	0.810593i	$0.699145\pi$
$72$	0	0
$73$	−2.10869e8	−0.869082	−0.434541	−	0.900652i	$-0.643089\pi$
$73$	−2.10869e8	−0.869082	−0.434541	+	0.900652i	$0.643089\pi$
$74$	−1.55070e7	−0.0601154
$75$	0	0
$76$	−1.44529e8	−0.496928
$77$	1.81681e8	0.588982
$78$	0	0
$79$	−1.98853e8	−0.574394	−0.287197	−	0.957872i	$-0.592723\pi$
$79$	−1.98853e8	−0.574394	−0.287197	+	0.957872i	$0.592723\pi$
$80$	−1.64584e8	−0.449243
$81$	0	0
$82$	4.49907e7	0.109891
$83$	4.01229e8	0.927984	0.463992	−	0.885839i	$-0.346417\pi$
$83$	4.01229e8	0.927984	0.463992	+	0.885839i	$0.346417\pi$
$84$	0	0
$85$	2.02928e8	0.421654
$86$	−1.15554e7	−0.0227794
$87$	0	0
$88$	−1.26922e8	−0.225613
$89$	−5.59701e8	−0.945586	−0.472793	−	0.881174i	$-0.656754\pi$
$89$	−5.59701e8	−0.945586	−0.472793	+	0.881174i	$0.656754\pi$
$90$	0	0
$91$	1.41289e8	0.215985
$92$	9.93826e6	0.0144632
$93$	0	0
$94$	7.48296e7	0.0988549
$95$	−1.83029e8	−0.230550
$96$	0	0
$97$	8.24474e8	0.945593	0.472796	−	0.881172i	$-0.343245\pi$
$97$	8.24474e8	0.945593	0.472796	+	0.881172i	$0.343245\pi$
$98$	−6.39802e7	−0.0700694
$99$	0	0
$100$	7.81879e8	0.781879
$101$	−6.32029e8	−0.604353	−0.302177	−	0.953252i	$-0.597713\pi$
$101$	−6.32029e8	−0.604353	−0.302177	+	0.953252i	$0.597713\pi$
$102$	0	0
$103$	−3.15128e8	−0.275879	−0.137940	−	0.990441i	$-0.544048\pi$
$103$	−3.15128e8	−0.275879	−0.137940	+	0.990441i	$0.544048\pi$
$104$	−9.87040e7	−0.0827341
$105$	0	0
$106$	−2.24696e8	−0.172870
$107$	2.65705e9	1.95963	0.979813	−	0.199916i	$-0.0640668\pi$
$107$	2.65705e9	1.95963	0.979813	+	0.199916i	$0.0640668\pi$
$108$	0	0
$109$	−1.16479e9	−0.790368	−0.395184	−	0.918602i	$-0.629319\pi$
$109$	−1.16479e9	−0.790368	−0.395184	+	0.918602i	$0.629319\pi$
$110$	−8.00437e7	−0.0521267
$111$	0	0
$112$	7.55502e8	0.453685
$113$	−1.83747e9	−1.06015	−0.530075	−	0.847951i	$-0.677836\pi$
$113$	−1.83747e9	−1.06015	−0.530075	+	0.847951i	$0.677836\pi$
$114$	0	0
$115$	1.25857e7	0.00671021
$116$	1.57345e9	0.806846
$117$	0	0
$118$	3.26986e8	0.155260
$119$	−9.31516e8	−0.425823
$120$	0	0
$121$	1.42836e9	0.605764
$122$	−3.05952e8	−0.125036
$123$	0	0
$124$	2.75928e9	1.04809
$125$	2.24643e9	0.822997
$126$	0	0
$127$	4.41296e9	1.50527	0.752633	−	0.658440i	$-0.228783\pi$
$127$	4.41296e9	1.50527	0.752633	+	0.658440i	$0.228783\pi$
$128$	−1.06419e9	−0.350407
$129$	0	0
$130$	−6.22481e7	−0.0191153
$131$	−4.69617e9	−1.39323	−0.696615	−	0.717445i	$-0.745312\pi$
$131$	−4.69617e9	−1.39323	−0.696615	+	0.717445i	$0.745312\pi$
$132$	0	0
$133$	8.40176e8	0.232829
$134$	−4.26467e8	−0.114265
$135$	0	0
$136$	6.50753e8	0.163114
$137$	3.17527e9	0.770085	0.385042	−	0.922899i	$-0.374187\pi$
$137$	3.17527e9	0.770085	0.385042	+	0.922899i	$0.374187\pi$
$138$	0	0
$139$	5.55423e9	1.26199	0.630997	−	0.775785i	$-0.282646\pi$
$139$	5.55423e9	1.26199	0.630997	+	0.775785i	$0.282646\pi$
$140$	9.64588e8	0.212210
$141$	0	0
$142$	−5.07186e8	−0.104681
$143$	2.94452e9	0.588847
$144$	0	0
$145$	1.99259e9	0.374336
$146$	−4.26461e8	−0.0776769
$147$	0	0
$148$	3.89448e9	0.667223
$149$	−1.99493e9	−0.331582	−0.165791	−	0.986161i	$-0.553018\pi$
$149$	−1.99493e9	−0.331582	−0.165791	+	0.986161i	$0.553018\pi$
$150$	0	0
$151$	3.50374e9	0.548448	0.274224	−	0.961666i	$-0.411579\pi$
$151$	3.50374e9	0.548448	0.274224	+	0.961666i	$0.411579\pi$
$152$	−5.86943e8	−0.0891866
$153$	0	0
$154$	3.67432e8	0.0526421
$155$	3.49432e9	0.486261
$156$	0	0
$157$	−1.31064e10	−1.72161	−0.860807	−	0.508931i	$-0.830041\pi$
$157$	−1.31064e10	−1.72161	−0.860807	+	0.508931i	$0.830041\pi$
$158$	−4.02159e8	−0.0513382
$159$	0	0
$160$	−1.01214e9	−0.122095
$161$	−5.77731e7	−0.00677656
$162$	0	0
$163$	−1.33062e10	−1.47642	−0.738210	−	0.674571i	$-0.764329\pi$
$163$	−1.33062e10	−1.47642	−0.738210	+	0.674571i	$0.764329\pi$
$164$	−1.12991e10	−1.21968
$165$	0	0
$166$	8.11443e8	0.0829415
$167$	5.12547e9	0.509928	0.254964	−	0.966950i	$-0.417936\pi$
$167$	5.12547e9	0.509928	0.254964	+	0.966950i	$0.417936\pi$
$168$	0	0
$169$	−8.31462e9	−0.784065
$170$	4.10400e8	0.0376866
$171$	0	0
$172$	2.90205e9	0.252829
$173$	1.63773e9	0.139006	0.0695030	−	0.997582i	$-0.477859\pi$
$173$	1.63773e9	0.139006	0.0695030	+	0.997582i	$0.477859\pi$
$174$	0	0
$175$	−4.54523e9	−0.366340
$176$	1.57450e10	1.23690
$177$	0	0
$178$	−1.13194e9	−0.0845147
$179$	−1.20732e10	−0.878986	−0.439493	−	0.898246i	$-0.644842\pi$
$179$	−1.20732e10	−0.878986	−0.439493	+	0.898246i	$0.644842\pi$
$180$	0	0
$181$	−2.08579e9	−0.144450	−0.0722250	−	0.997388i	$-0.523010\pi$
$181$	−2.08579e9	−0.144450	−0.0722250	+	0.997388i	$0.523010\pi$
$182$	2.85743e8	0.0193043
$183$	0	0
$184$	4.03601e7	0.00259580
$185$	4.93191e9	0.309558
$186$	0	0
$187$	−1.94132e10	−1.16094
$188$	−1.87929e10	−1.09719
$189$	0	0
$190$	−3.70158e8	−0.0206061
$191$	1.41695e10	0.770376	0.385188	−	0.922838i	$-0.374137\pi$
$191$	1.41695e10	0.770376	0.385188	+	0.922838i	$0.374137\pi$
$192$	0	0
$193$	−1.50524e10	−0.780903	−0.390452	−	0.920623i	$-0.627681\pi$
$193$	−1.50524e10	−0.780903	−0.390452	+	0.920623i	$0.627681\pi$
$194$	1.66741e9	0.0845153
$195$	0	0
$196$	1.60682e10	0.777703
$197$	−3.47012e10	−1.64152	−0.820760	−	0.571272i	$-0.806450\pi$
$197$	−3.47012e10	−1.64152	−0.820760	+	0.571272i	$0.806450\pi$
$198$	0	0
$199$	−2.00127e10	−0.904621	−0.452310	−	0.891861i	$-0.649400\pi$
$199$	−2.00127e10	−0.904621	−0.452310	+	0.891861i	$0.649400\pi$
$200$	3.17527e9	0.140329
$201$	0	0
$202$	−1.27821e9	−0.0540160
$203$	−9.14676e9	−0.378038
$204$	0	0
$205$	−1.43090e10	−0.565871
$206$	−6.37313e8	−0.0246576
$207$	0	0
$208$	1.22445e10	0.453581
$209$	1.75096e10	0.634771
$210$	0	0
$211$	−3.53163e9	−0.122660	−0.0613302	−	0.998118i	$-0.519534\pi$
$211$	−3.53163e9	−0.122660	−0.0613302	+	0.998118i	$0.519534\pi$
$212$	5.64308e10	1.91869
$213$	0	0
$214$	5.37362e9	0.175148
$215$	3.67512e9	0.117300
$216$	0	0
$217$	−1.60403e10	−0.491069
$218$	−2.35567e9	−0.0706417
$219$	0	0
$220$	2.01024e10	0.578556
$221$	−1.50971e10	−0.425725
$222$	0	0
$223$	−1.90687e10	−0.516355	−0.258177	−	0.966098i	$-0.583122\pi$
$223$	−1.90687e10	−0.516355	−0.258177	+	0.966098i	$0.583122\pi$
$224$	4.64610e9	0.123303
$225$	0	0
$226$	−3.71610e9	−0.0947543
$227$	−4.24420e9	−0.106091	−0.0530456	−	0.998592i	$-0.516893\pi$
$227$	−4.24420e9	−0.106091	−0.0530456	+	0.998592i	$0.516893\pi$
$228$	0	0
$229$	−2.89589e10	−0.695861	−0.347931	−	0.937520i	$-0.613116\pi$
$229$	−2.89589e10	−0.695861	−0.347931	+	0.937520i	$0.613116\pi$
$230$	2.54532e7	0.000599746 0
$231$	0	0
$232$	6.38989e9	0.144810
$233$	4.32035e10	0.960324	0.480162	−	0.877180i	$-0.340578\pi$
$233$	4.32035e10	0.960324	0.480162	+	0.877180i	$0.340578\pi$
$234$	0	0
$235$	−2.37991e10	−0.509043
$236$	−8.21202e10	−1.72324
$237$	0	0
$238$	−1.88390e9	−0.0380593
$239$	−4.43684e10	−0.879595	−0.439798	−	0.898097i	$-0.644950\pi$
$239$	−4.43684e10	−0.879595	−0.439798	+	0.898097i	$0.644950\pi$
$240$	0	0
$241$	1.51214e10	0.288746	0.144373	−	0.989523i	$-0.453883\pi$
$241$	1.51214e10	0.288746	0.144373	+	0.989523i	$0.453883\pi$
$242$	2.88871e9	0.0541421
$243$	0	0
$244$	7.68377e10	1.38778
$245$	2.03485e10	0.360815
$246$	0	0
$247$	1.36168e10	0.232776
$248$	1.12057e10	0.188107
$249$	0	0
$250$	4.54317e9	0.0735579
$251$	5.52302e10	0.878304	0.439152	−	0.898413i	$-0.355279\pi$
$251$	5.52302e10	0.878304	0.439152	+	0.898413i	$0.355279\pi$
$252$	0	0
$253$	−1.20401e9	−0.0184752
$254$	8.92476e9	0.134538
$255$	0	0
$256$	6.32954e10	0.921069
$257$	−1.14200e11	−1.63293	−0.816463	−	0.577398i	$-0.804068\pi$
$257$	−1.14200e11	−1.63293	−0.816463	+	0.577398i	$0.804068\pi$
$258$	0	0
$259$	−2.26394e10	−0.312619
$260$	1.56331e10	0.212161
$261$	0	0
$262$	−9.49752e9	−0.124524
$263$	1.05397e11	1.35841	0.679203	−	0.733951i	$-0.262326\pi$
$263$	1.05397e11	1.35841	0.679203	+	0.733951i	$0.262326\pi$
$264$	0	0
$265$	7.14631e10	0.890175
$266$	1.69917e9	0.0208099
$267$	0	0
$268$	1.07104e11	1.26823
$269$	4.73937e10	0.551868	0.275934	−	0.961177i	$-0.411013\pi$
$269$	4.73937e10	0.551868	0.275934	+	0.961177i	$0.411013\pi$
$270$	0	0
$271$	1.35742e11	1.52880	0.764402	−	0.644740i	$-0.223035\pi$
$271$	1.35742e11	1.52880	0.764402	+	0.644740i	$0.223035\pi$
$272$	−8.07275e10	−0.894255
$273$	0	0
$274$	6.42166e9	0.0688287
$275$	−9.47242e10	−0.998766
$276$	0	0
$277$	1.37164e11	1.39985	0.699927	−	0.714215i	$-0.253216\pi$
$277$	1.37164e11	1.39985	0.699927	+	0.714215i	$0.253216\pi$
$278$	1.12329e10	0.112795
$279$	0	0
$280$	3.91727e9	0.0380866
$281$	−1.44374e10	−0.138137	−0.0690686	−	0.997612i	$-0.522003\pi$
$281$	−1.44374e10	−0.138137	−0.0690686	+	0.997612i	$0.522003\pi$
$282$	0	0
$283$	−7.59649e10	−0.704002	−0.352001	−	0.936000i	$-0.614499\pi$
$283$	−7.59649e10	−0.704002	−0.352001	+	0.936000i	$0.614499\pi$
$284$	1.27376e11	1.16186
$285$	0	0
$286$	5.95499e9	0.0526300
$287$	6.56839e10	0.571466
$288$	0	0
$289$	−1.90528e10	−0.160664
$290$	4.02981e9	0.0334575
$291$	0	0
$292$	1.07103e11	0.862139
$293$	−1.78898e11	−1.41808	−0.709042	−	0.705166i	$-0.750872\pi$
$293$	−1.78898e11	−1.41808	−0.709042	+	0.705166i	$0.750872\pi$
$294$	0	0
$295$	−1.03996e11	−0.799497
$296$	1.58158e10	0.119751
$297$	0	0
$298$	−4.03455e9	−0.0296362
$299$	−9.36333e8	−0.00677501
$300$	0	0
$301$	−1.68702e10	−0.118460
$302$	7.08594e9	0.0490192
$303$	0	0
$304$	7.28117e10	0.488956
$305$	9.73062e10	0.643859
$306$	0	0
$307$	−1.35070e11	−0.867832	−0.433916	−	0.900953i	$-0.642869\pi$
$307$	−1.35070e11	−0.867832	−0.433916	+	0.900953i	$0.642869\pi$
$308$	−9.22777e10	−0.584277
$309$	0	0
$310$	7.06689e9	0.0434611
$311$	7.02212e10	0.425644	0.212822	−	0.977091i	$-0.431735\pi$
$311$	7.02212e10	0.425644	0.212822	+	0.977091i	$0.431735\pi$
$312$	0	0
$313$	−3.05164e11	−1.79715	−0.898573	−	0.438824i	$-0.855395\pi$
$313$	−3.05164e11	−1.79715	−0.898573	+	0.438824i	$0.855395\pi$
$314$	−2.65064e10	−0.153875
$315$	0	0
$316$	1.00999e11	0.569805
$317$	−3.03284e10	−0.168688	−0.0843438	−	0.996437i	$-0.526879\pi$
$317$	−3.03284e10	−0.168688	−0.0843438	+	0.996437i	$0.526879\pi$
$318$	0	0
$319$	−1.90622e11	−1.03066
$320$	8.22198e10	0.438330
$321$	0	0
$322$	−1.16840e8	−0.000605677 0
$323$	−8.97751e10	−0.458928
$324$	0	0
$325$	−7.36647e10	−0.366256
$326$	−2.69104e10	−0.131960
$327$	0	0
$328$	−4.58865e10	−0.218903
$329$	1.09247e11	0.514077
$330$	0	0
$331$	1.58072e11	0.723819	0.361910	−	0.932213i	$-0.382125\pi$
$331$	1.58072e11	0.723819	0.361910	+	0.932213i	$0.382125\pi$
$332$	−2.03788e11	−0.920571
$333$	0	0
$334$	1.03657e10	0.0455765
$335$	1.35635e11	0.588397
$336$	0	0
$337$	4.89525e8	0.00206748	0.00103374	−	0.999999i	$-0.499671\pi$
$337$	4.89525e8	0.00206748	0.00103374	+	0.999999i	$0.499671\pi$
$338$	−1.68155e10	−0.0700783
$339$	0	0
$340$	−1.03069e11	−0.418285
$341$	−3.34285e11	−1.33882
$342$	0	0
$343$	−2.12555e11	−0.829178
$344$	1.17855e10	0.0453768
$345$	0	0
$346$	3.31213e9	0.0124241
$347$	2.55900e11	0.947517	0.473759	−	0.880655i	$-0.342897\pi$
$347$	2.55900e11	0.947517	0.473759	+	0.880655i	$0.342897\pi$
$348$	0	0
$349$	−6.31618e10	−0.227898	−0.113949	−	0.993487i	$-0.536350\pi$
$349$	−6.31618e10	−0.227898	−0.113949	+	0.993487i	$0.536350\pi$
$350$	−9.19225e9	−0.0327428
$351$	0	0
$352$	9.68265e10	0.336165
$353$	5.04911e11	1.73073	0.865364	−	0.501143i	$-0.167087\pi$
$353$	5.04911e11	1.73073	0.865364	+	0.501143i	$0.167087\pi$
$354$	0	0
$355$	1.61307e11	0.539046
$356$	2.84278e11	0.938032
$357$	0	0
$358$	−2.44167e10	−0.0785622
$359$	−3.39429e11	−1.07851	−0.539254	−	0.842143i	$-0.681294\pi$
$359$	−3.39429e11	−1.07851	−0.539254	+	0.842143i	$0.681294\pi$
$360$	0	0
$361$	−2.41716e11	−0.749070
$362$	−4.21830e9	−0.0129107
$363$	0	0
$364$	−7.17622e10	−0.214259
$365$	1.35633e11	0.399989
$366$	0	0
$367$	−1.11046e11	−0.319525	−0.159762	−	0.987155i	$-0.551073\pi$
$367$	−1.11046e11	−0.319525	−0.159762	+	0.987155i	$0.551073\pi$
$368$	−5.00676e9	−0.0142312
$369$	0	0
$370$	9.97428e9	0.0276677
$371$	−3.28044e11	−0.898978
$372$	0	0
$373$	−5.93701e11	−1.58810	−0.794051	−	0.607851i	$-0.792032\pi$
$373$	−5.93701e11	−1.58810	−0.794051	+	0.607851i	$0.792032\pi$
$374$	−3.92611e10	−0.103762
$375$	0	0
$376$	−7.63195e10	−0.196920
$377$	−1.48242e11	−0.377951
$378$	0	0
$379$	−8.56009e10	−0.213109	−0.106554	−	0.994307i	$-0.533982\pi$
$379$	−8.56009e10	−0.213109	−0.106554	+	0.994307i	$0.533982\pi$
$380$	9.29624e10	0.228708
$381$	0	0
$382$	2.86563e10	0.0688548
$383$	−3.76619e11	−0.894351	−0.447176	−	0.894446i	$-0.647570\pi$
$383$	−3.76619e11	−0.894351	−0.447176	+	0.894446i	$0.647570\pi$
$384$	0	0
$385$	−1.16859e11	−0.271075
$386$	−3.04419e10	−0.0697957
$387$	0	0
$388$	−4.18758e11	−0.938039
$389$	−2.79708e11	−0.619344	−0.309672	−	0.950843i	$-0.600219\pi$
$389$	−2.79708e11	−0.619344	−0.309672	+	0.950843i	$0.600219\pi$
$390$	0	0
$391$	6.17322e9	0.0133572
$392$	6.52541e10	0.139579
$393$	0	0
$394$	−7.01796e10	−0.146716
$395$	1.27904e11	0.264361
$396$	0	0
$397$	6.81921e11	1.37777	0.688886	−	0.724870i	$-0.258100\pi$
$397$	6.81921e11	1.37777	0.688886	+	0.724870i	$0.258100\pi$
$398$	−4.04736e10	−0.0808533
$399$	0	0
$400$	−3.93901e11	−0.769337
$401$	5.49901e11	1.06202	0.531012	−	0.847364i	$-0.321812\pi$
$401$	5.49901e11	1.06202	0.531012	+	0.847364i	$0.321812\pi$
$402$	0	0
$403$	−2.59966e11	−0.490957
$404$	3.21014e11	0.599525
$405$	0	0
$406$	−1.84984e10	−0.0337883
$407$	−4.71813e11	−0.852306
$408$	0	0
$409$	1.02139e12	1.80484	0.902418	−	0.430861i	$-0.141790\pi$
$409$	1.02139e12	1.80484	0.902418	+	0.430861i	$0.141790\pi$
$410$	−2.89385e10	−0.0505765
$411$	0	0
$412$	1.60056e11	0.273675
$413$	4.77381e11	0.807403
$414$	0	0
$415$	−2.58074e11	−0.427099
$416$	7.52996e10	0.123274
$417$	0	0
$418$	3.54113e10	0.0567347
$419$	−3.43844e11	−0.545002	−0.272501	−	0.962156i	$-0.587851\pi$
$419$	−3.43844e11	−0.545002	−0.272501	+	0.962156i	$0.587851\pi$
$420$	0	0
$421$	6.76990e11	1.05030	0.525149	−	0.851010i	$-0.324009\pi$
$421$	6.76990e11	1.05030	0.525149	+	0.851010i	$0.324009\pi$
$422$	−7.14236e9	−0.0109632
$423$	0	0
$424$	2.29170e11	0.344359
$425$	4.85670e11	0.722089
$426$	0	0
$427$	−4.46673e11	−0.650226
$428$	−1.34954e12	−1.94397
$429$	0	0
$430$	7.43255e9	0.0104841
$431$	2.07875e11	0.290171	0.145085	−	0.989419i	$-0.453654\pi$
$431$	2.07875e11	0.290171	0.145085	+	0.989419i	$0.453654\pi$
$432$	0	0
$433$	3.79352e11	0.518617	0.259309	−	0.965795i	$-0.416505\pi$
$433$	3.79352e11	0.518617	0.259309	+	0.965795i	$0.416505\pi$
$434$	−3.24398e10	−0.0438909
$435$	0	0
$436$	5.91610e11	0.784055
$437$	−5.56790e9	−0.00730340
$438$	0	0
$439$	1.39999e12	1.79901	0.899504	−	0.436912i	$-0.143928\pi$
$439$	1.39999e12	1.79901	0.899504	+	0.436912i	$0.143928\pi$
$440$	8.16374e10	0.103837
$441$	0	0
$442$	−3.05324e10	−0.0380505
$443$	2.61274e11	0.322314	0.161157	−	0.986929i	$-0.448478\pi$
$443$	2.61274e11	0.322314	0.161157	+	0.986929i	$0.448478\pi$
$444$	0	0
$445$	3.60005e11	0.435200
$446$	−3.85644e10	−0.0461508
$447$	0	0
$448$	−3.77421e11	−0.442665
$449$	6.64538e11	0.771634	0.385817	−	0.922575i	$-0.373920\pi$
$449$	6.64538e11	0.771634	0.385817	+	0.922575i	$0.373920\pi$
$450$	0	0
$451$	1.36888e12	1.55801
$452$	9.33270e11	1.05168
$453$	0	0
$454$	−8.58345e9	−0.00948223
$455$	−9.08786e10	−0.0994055
$456$	0	0
$457$	5.50957e10	0.0590874	0.0295437	−	0.999563i	$-0.490595\pi$
$457$	5.50957e10	0.0590874	0.0295437	+	0.999563i	$0.490595\pi$
$458$	−5.85664e10	−0.0621948
$459$	0	0
$460$	−6.39239e9	−0.00665661
$461$	−9.65322e11	−0.995447	−0.497723	−	0.867336i	$-0.665831\pi$
$461$	−9.65322e11	−0.995447	−0.497723	+	0.867336i	$0.665831\pi$
$462$	0	0
$463$	1.47645e12	1.49315	0.746576	−	0.665300i	$-0.231696\pi$
$463$	1.47645e12	1.49315	0.746576	+	0.665300i	$0.231696\pi$
$464$	−7.92681e11	−0.793903
$465$	0	0
$466$	8.73747e10	0.0858320
$467$	1.48210e12	1.44195	0.720976	−	0.692960i	$-0.243694\pi$
$467$	1.48210e12	1.44195	0.720976	+	0.692960i	$0.243694\pi$
$468$	0	0
$469$	−6.22618e11	−0.594215
$470$	−4.81312e10	−0.0454973
$471$	0	0
$472$	−3.33497e11	−0.309280
$473$	−3.51582e11	−0.322962
$474$	0	0
$475$	−4.38047e11	−0.394820
$476$	4.73126e11	0.422421
$477$	0	0
$478$	−8.97304e10	−0.0786166
$479$	−1.46496e12	−1.27150	−0.635751	−	0.771894i	$-0.719310\pi$
$479$	−1.46496e12	−1.27150	−0.635751	+	0.771894i	$0.719310\pi$
$480$	0	0
$481$	−3.66918e11	−0.312548
$482$	3.05816e10	0.0258076
$483$	0	0
$484$	−7.25478e11	−0.600925
$485$	−5.30310e11	−0.435203
$486$	0	0
$487$	1.26436e12	1.01857	0.509286	−	0.860598i	$-0.329910\pi$
$487$	1.26436e12	1.01857	0.509286	+	0.860598i	$0.329910\pi$
$488$	3.12044e11	0.249073
$489$	0	0
$490$	4.11527e10	0.0322490
$491$	1.07798e12	0.837038	0.418519	−	0.908208i	$-0.362549\pi$
$491$	1.07798e12	0.837038	0.418519	+	0.908208i	$0.362549\pi$
$492$	0	0
$493$	9.77357e11	0.745147
$494$	2.75385e10	0.0208051
$495$	0	0
$496$	−1.39009e12	−1.03128
$497$	−7.40463e11	−0.544377
$498$	0	0
$499$	−5.39531e11	−0.389551	−0.194775	−	0.980848i	$-0.562398\pi$
$499$	−5.39531e11	−0.389551	−0.194775	+	0.980848i	$0.562398\pi$
$500$	−1.14098e12	−0.816422
$501$	0	0
$502$	1.11697e11	0.0785012
$503$	1.09449e10	0.00762354	0.00381177	−	0.999993i	$-0.498787\pi$
$503$	1.09449e10	0.00762354	0.00381177	+	0.999993i	$0.498787\pi$
$504$	0	0
$505$	4.06527e11	0.278150
$506$	−2.43499e9	−0.00165128
$507$	0	0
$508$	−2.24139e12	−1.49324
$509$	9.46630e11	0.625101	0.312551	−	0.949901i	$-0.398817\pi$
$509$	9.46630e11	0.625101	0.312551	+	0.949901i	$0.398817\pi$
$510$	0	0
$511$	−6.22610e11	−0.403945
$512$	6.72871e11	0.432730
$513$	0	0
$514$	−2.30957e11	−0.145948
$515$	2.02693e11	0.126972
$516$	0	0
$517$	2.27675e12	1.40155
$518$	−4.57858e10	−0.0279413
$519$	0	0
$520$	6.34874e10	0.0380779
$521$	−2.51616e12	−1.49612	−0.748062	−	0.663629i	$-0.769016\pi$
$521$	−2.51616e12	−1.49612	−0.748062	+	0.663629i	$0.769016\pi$
$522$	0	0
$523$	−2.27526e12	−1.32976	−0.664881	−	0.746949i	$-0.731518\pi$
$523$	−2.27526e12	−1.32976	−0.664881	+	0.746949i	$0.731518\pi$
$524$	2.38523e12	1.38210
$525$	0	0
$526$	2.13156e11	0.121412
$527$	1.71395e12	0.967943
$528$	0	0
$529$	−1.80077e12	−0.999787
$530$	1.44527e11	0.0795622
$531$	0	0
$532$	−4.26734e11	−0.230969
$533$	1.06454e12	0.571335
$534$	0	0
$535$	−1.70904e12	−0.901906
$536$	4.34958e11	0.227618
$537$	0	0
$538$	9.58488e10	0.0493249
$539$	−1.94665e12	−0.993432
$540$	0	0
$541$	1.83546e12	0.921205	0.460602	−	0.887607i	$-0.347633\pi$
$541$	1.83546e12	0.921205	0.460602	+	0.887607i	$0.347633\pi$
$542$	2.74524e11	0.136642
$543$	0	0
$544$	−4.96449e11	−0.243041
$545$	7.49207e11	0.363762
$546$	0	0
$547$	2.78550e12	1.33033	0.665166	−	0.746695i	$-0.268361\pi$
$547$	2.78550e12	1.33033	0.665166	+	0.746695i	$0.268361\pi$
$548$	−1.61275e12	−0.763933
$549$	0	0
$550$	−1.91570e11	−0.0892679
$551$	−8.81521e11	−0.407428
$552$	0	0
$553$	−5.87129e11	−0.266975
$554$	2.77401e11	0.125116
$555$	0	0
$556$	−2.82105e12	−1.25191
$557$	−4.09039e12	−1.80060	−0.900298	−	0.435274i	$-0.856651\pi$
$557$	−4.09039e12	−1.80060	−0.900298	+	0.435274i	$0.856651\pi$
$558$	0	0
$559$	−2.73417e11	−0.118433
$560$	−4.85947e11	−0.208806
$561$	0	0
$562$	−2.91982e10	−0.0123465
$563$	2.55335e12	1.07108	0.535540	−	0.844510i	$-0.320108\pi$
$563$	2.55335e12	1.07108	0.535540	+	0.844510i	$0.320108\pi$
$564$	0	0
$565$	1.18188e12	0.487928
$566$	−1.53631e11	−0.0629224
$567$	0	0
$568$	5.17284e11	0.208527
$569$	2.27693e11	0.0910635	0.0455318	−	0.998963i	$-0.485502\pi$
$569$	2.27693e11	0.0910635	0.0455318	+	0.998963i	$0.485502\pi$
$570$	0	0
$571$	1.76034e10	0.00693002	0.00346501	−	0.999994i	$-0.498897\pi$
$571$	1.76034e10	0.00693002	0.00346501	+	0.999994i	$0.498897\pi$
$572$	−1.49555e12	−0.584143
$573$	0	0
$574$	1.32839e11	0.0510766
$575$	3.01215e10	0.0114914
$576$	0	0
$577$	−3.85224e12	−1.44684	−0.723422	−	0.690406i	$-0.757432\pi$
$577$	−3.85224e12	−1.44684	−0.723422	+	0.690406i	$0.757432\pi$
$578$	−3.85323e10	−0.0143598
$579$	0	0
$580$	−1.01206e12	−0.371346
$581$	1.18466e12	0.431322
$582$	0	0
$583$	−6.83655e12	−2.45092
$584$	4.34952e11	0.154733
$585$	0	0
$586$	−3.61803e11	−0.126746
$587$	−1.24716e12	−0.433561	−0.216781	−	0.976220i	$-0.569556\pi$
$587$	−1.24716e12	−0.433561	−0.216781	+	0.976220i	$0.569556\pi$
$588$	0	0
$589$	−1.54588e12	−0.529247
$590$	−2.10321e11	−0.0714576
$591$	0	0
$592$	−1.96199e12	−0.656520
$593$	2.85539e12	0.948242	0.474121	−	0.880460i	$-0.342766\pi$
$593$	2.85539e12	0.948242	0.474121	+	0.880460i	$0.342766\pi$
$594$	0	0
$595$	5.99161e11	0.195982
$596$	1.01325e12	0.328933
$597$	0	0
$598$	−1.89364e9	−0.000605538 0
$599$	2.74976e12	0.872720	0.436360	−	0.899772i	$-0.356267\pi$
$599$	2.74976e12	0.872720	0.436360	+	0.899772i	$0.356267\pi$
$600$	0	0
$601$	4.71796e12	1.47509	0.737546	−	0.675297i	$-0.235985\pi$
$601$	4.71796e12	1.47509	0.737546	+	0.675297i	$0.235985\pi$
$602$	−3.41183e10	−0.0105877
$603$	0	0
$604$	−1.77958e12	−0.544066
$605$	−9.18735e11	−0.278799
$606$	0	0
$607$	2.37767e11	0.0710891	0.0355445	−	0.999368i	$-0.488683\pi$
$607$	2.37767e11	0.0710891	0.0355445	+	0.999368i	$0.488683\pi$
$608$	4.47769e11	0.132889
$609$	0	0
$610$	1.96792e11	0.0575470
$611$	1.77057e12	0.513959
$612$	0	0
$613$	2.76439e12	0.790729	0.395365	−	0.918524i	$-0.370618\pi$
$613$	2.76439e12	0.790729	0.395365	+	0.918524i	$0.370618\pi$
$614$	−2.73165e11	−0.0775652
$615$	0	0
$616$	−3.74747e11	−0.104864
$617$	−3.68615e12	−1.02398	−0.511988	−	0.858993i	$-0.671091\pi$
$617$	−3.68615e12	−1.02398	−0.511988	+	0.858993i	$0.671091\pi$
$618$	0	0
$619$	−8.43418e11	−0.230906	−0.115453	−	0.993313i	$-0.536832\pi$
$619$	−8.43418e11	−0.230906	−0.115453	+	0.993313i	$0.536832\pi$
$620$	−1.77480e12	−0.482377
$621$	0	0
$622$	1.42015e11	0.0380433
$623$	−1.65256e12	−0.439503
$624$	0	0
$625$	1.56172e12	0.409397
$626$	−6.17162e11	−0.160626
$627$	0	0
$628$	6.65688e12	1.70786
$629$	2.41908e12	0.616201
$630$	0	0
$631$	−2.98482e12	−0.749526	−0.374763	−	0.927121i	$-0.622276\pi$
$631$	−2.98482e12	−0.749526	−0.374763	+	0.927121i	$0.622276\pi$
$632$	4.10166e11	0.102266
$633$	0	0
$634$	−6.13361e10	−0.0150770
$635$	−2.83846e12	−0.692789
$636$	0	0
$637$	−1.51386e12	−0.364300
$638$	−3.85513e11	−0.0921184
$639$	0	0
$640$	6.84495e11	0.161273
$641$	4.62829e12	1.08283	0.541414	−	0.840756i	$-0.317889\pi$
$641$	4.62829e12	1.08283	0.541414	+	0.840756i	$0.317889\pi$
$642$	0	0
$643$	1.98315e12	0.457516	0.228758	−	0.973483i	$-0.426534\pi$
$643$	1.98315e12	0.457516	0.228758	+	0.973483i	$0.426534\pi$
$644$	2.93436e10	0.00672243
$645$	0	0
$646$	−1.81561e11	−0.0410181
$647$	1.06389e12	0.238687	0.119343	−	0.992853i	$-0.461921\pi$
$647$	1.06389e12	0.238687	0.119343	+	0.992853i	$0.461921\pi$
$648$	0	0
$649$	9.94881e12	2.20125
$650$	−1.48979e11	−0.0327353
$651$	0	0
$652$	6.75835e12	1.46463
$653$	−2.71397e11	−0.0584111	−0.0292056	−	0.999573i	$-0.509298\pi$
$653$	−2.71397e11	−0.0584111	−0.0292056	+	0.999573i	$0.509298\pi$
$654$	0	0
$655$	3.02062e12	0.641226
$656$	5.69233e12	1.20012
$657$	0	0
$658$	2.20941e11	0.0459472
$659$	−1.58636e12	−0.327655	−0.163828	−	0.986489i	$-0.552384\pi$
$659$	−1.58636e12	−0.327655	−0.163828	+	0.986489i	$0.552384\pi$
$660$	0	0
$661$	1.39250e12	0.283719	0.141859	−	0.989887i	$-0.454692\pi$
$661$	1.39250e12	0.283719	0.141859	+	0.989887i	$0.454692\pi$
$662$	3.19685e11	0.0646936
$663$	0	0
$664$	−8.27599e11	−0.165220
$665$	−5.40409e11	−0.107158
$666$	0	0
$667$	6.06162e10	0.0118583
$668$	−2.60327e12	−0.505855
$669$	0	0
$670$	2.74308e11	0.0525898
$671$	−9.30884e12	−1.77274
$672$	0	0
$673$	3.64372e12	0.684664	0.342332	−	0.939579i	$-0.388783\pi$
$673$	3.64372e12	0.684664	0.342332	+	0.939579i	$0.388783\pi$
$674$	9.90014e8	0.000184787 0
$675$	0	0
$676$	4.22308e12	0.777802
$677$	4.82076e12	0.881995	0.440998	−	0.897508i	$-0.354625\pi$
$677$	4.82076e12	0.881995	0.440998	+	0.897508i	$0.354625\pi$
$678$	0	0
$679$	2.43433e12	0.439506
$680$	−4.18571e11	−0.0750722
$681$	0	0
$682$	−6.76057e11	−0.119661
$683$	4.75897e12	0.836797	0.418398	−	0.908264i	$-0.362592\pi$
$683$	4.75897e12	0.836797	0.418398	+	0.908264i	$0.362592\pi$
$684$	0	0
$685$	−2.04237e12	−0.354427
$686$	−4.29870e11	−0.0741104
$687$	0	0
$688$	−1.46202e12	−0.248774
$689$	−5.31662e12	−0.898771
$690$	0	0
$691$	−7.29490e11	−0.121722	−0.0608608	−	0.998146i	$-0.519385\pi$
$691$	−7.29490e11	−0.121722	−0.0608608	+	0.998146i	$0.519385\pi$
$692$	−8.31817e11	−0.137896
$693$	0	0
$694$	5.17531e11	0.0846874
$695$	−3.57254e12	−0.580825
$696$	0	0
$697$	−7.01851e12	−1.12641
$698$	−1.27738e11	−0.0203691
$699$	0	0
$700$	2.30857e12	0.363413
$701$	5.66331e12	0.885807	0.442903	−	0.896569i	$-0.353948\pi$
$701$	5.66331e12	0.885807	0.442903	+	0.896569i	$0.353948\pi$
$702$	0	0
$703$	−2.18188e12	−0.336923
$704$	−7.86560e12	−1.20685
$705$	0	0
$706$	1.02113e12	0.154689
$707$	−1.86612e12	−0.280900
$708$	0	0
$709$	2.49414e12	0.370691	0.185346	−	0.982673i	$-0.440660\pi$
$709$	2.49414e12	0.370691	0.185346	+	0.982673i	$0.440660\pi$
$710$	3.26227e11	0.0481790
$711$	0	0
$712$	1.15447e12	0.168354
$713$	1.06300e11	0.0154039
$714$	0	0
$715$	−1.89395e12	−0.271013
$716$	6.13207e12	0.871964
$717$	0	0
$718$	−6.86459e11	−0.0963951
$719$	1.09356e13	1.52603	0.763013	−	0.646383i	$-0.223719\pi$
$719$	1.09356e13	1.52603	0.763013	+	0.646383i	$0.223719\pi$
$720$	0	0
$721$	−9.30441e11	−0.128227
$722$	−4.88845e11	−0.0669505
$723$	0	0
$724$	1.05939e12	0.143296
$725$	4.76890e12	0.641058
$726$	0	0
$727$	−4.77124e12	−0.633470	−0.316735	−	0.948514i	$-0.602587\pi$
$727$	−4.77124e12	−0.633470	−0.316735	+	0.948514i	$0.602587\pi$
$728$	−2.91432e11	−0.0384544
$729$	0	0
$730$	2.74304e11	0.0357503
$731$	1.80263e12	0.233496
$732$	0	0
$733$	1.27001e13	1.62495	0.812474	−	0.582998i	$-0.198120\pi$
$733$	1.27001e13	1.62495	0.812474	+	0.582998i	$0.198120\pi$
$734$	−2.24579e11	−0.0285585
$735$	0	0
$736$	−3.07900e10	−0.00386776
$737$	−1.29756e13	−1.62003
$738$	0	0
$739$	−1.38078e13	−1.70304	−0.851521	−	0.524321i	$-0.824319\pi$
$739$	−1.38078e13	−1.70304	−0.851521	+	0.524321i	$0.824319\pi$
$740$	−2.50497e12	−0.307085
$741$	0	0
$742$	−6.63434e11	−0.0803490
$743$	1.44320e12	0.173731	0.0868654	−	0.996220i	$-0.472315\pi$
$743$	1.44320e12	0.173731	0.0868654	+	0.996220i	$0.472315\pi$
$744$	0	0
$745$	1.28316e12	0.152608
$746$	−1.20070e12	−0.141942
$747$	0	0
$748$	9.86013e12	1.15166
$749$	7.84517e12	0.910824
$750$	0	0
$751$	−1.61064e13	−1.84765	−0.923824	−	0.382819i	$-0.874953\pi$
$751$	−1.61064e13	−1.84765	−0.923824	+	0.382819i	$0.874953\pi$
$752$	9.46762e12	1.07959
$753$	0	0
$754$	−2.99804e11	−0.0337806
$755$	−2.25364e12	−0.252420
$756$	0	0
$757$	−5.19966e11	−0.0575498	−0.0287749	−	0.999586i	$-0.509161\pi$
$757$	−5.19966e11	−0.0575498	−0.0287749	+	0.999586i	$0.509161\pi$
$758$	−1.73119e11	−0.0190473
$759$	0	0
$760$	3.77528e11	0.0410476
$761$	−1.30992e13	−1.41584	−0.707922	−	0.706291i	$-0.750367\pi$
$761$	−1.30992e13	−1.41584	−0.707922	+	0.706291i	$0.750367\pi$
$762$	0	0
$763$	−3.43915e12	−0.367359
$764$	−7.19681e12	−0.764222
$765$	0	0
$766$	−7.61674e11	−0.0799354
$767$	7.73695e12	0.807218
$768$	0	0
$769$	−1.48441e12	−0.153068	−0.0765340	−	0.997067i	$-0.524385\pi$
$769$	−1.48441e12	−0.153068	−0.0765340	+	0.997067i	$0.524385\pi$
$770$	−2.36336e11	−0.0242282
$771$	0	0
$772$	7.64525e12	0.774665
$773$	−1.60264e13	−1.61446	−0.807232	−	0.590234i	$-0.799035\pi$
$773$	−1.60264e13	−1.61446	−0.807232	+	0.590234i	$0.799035\pi$
$774$	0	0
$775$	8.36301e12	0.832731
$776$	−1.70061e12	−0.168355
$777$	0	0
$778$	−5.65681e11	−0.0553558
$779$	6.33030e12	0.615894
$780$	0	0
$781$	−1.54315e13	−1.48415
$782$	1.24847e10	0.00119384
$783$	0	0
$784$	−8.09493e12	−0.765228
$785$	8.43019e12	0.792362
$786$	0	0
$787$	−5.17974e12	−0.481307	−0.240653	−	0.970611i	$-0.577362\pi$
$787$	−5.17974e12	−0.481307	−0.240653	+	0.970611i	$0.577362\pi$
$788$	1.76251e13	1.62841
$789$	0	0
$790$	2.58673e11	0.0236281
$791$	−5.42529e12	−0.492752
$792$	0	0
$793$	−7.23926e12	−0.650077
$794$	1.37912e12	0.123143
$795$	0	0
$796$	1.01646e13	0.897394
$797$	−6.86643e11	−0.0602793	−0.0301397	−	0.999546i	$-0.509595\pi$
$797$	−6.86643e11	−0.0602793	−0.0301397	+	0.999546i	$0.509595\pi$
$798$	0	0
$799$	−1.16733e13	−1.01329
$800$	−2.42236e12	−0.209091
$801$	0	0
$802$	1.11212e12	0.0949218
$803$	−1.29754e13	−1.10129
$804$	0	0
$805$	3.71603e10	0.00311887
$806$	−5.25754e11	−0.0438808
$807$	0	0
$808$	1.30366e12	0.107600
$809$	−1.35129e13	−1.10912	−0.554562	−	0.832142i	$-0.687114\pi$
$809$	−1.35129e13	−1.10912	−0.554562	+	0.832142i	$0.687114\pi$
$810$	0	0
$811$	−1.40287e13	−1.13874	−0.569371	−	0.822081i	$-0.692813\pi$
$811$	−1.40287e13	−1.13874	−0.569371	+	0.822081i	$0.692813\pi$
$812$	4.64573e12	0.375018
$813$	0	0
$814$	−9.54194e11	−0.0761775
$815$	8.55869e12	0.679513
$816$	0	0
$817$	−1.62587e12	−0.127669
$818$	2.06566e12	0.161313
$819$	0	0
$820$	7.26769e12	0.561350
$821$	1.31763e13	1.01216	0.506078	−	0.862487i	$-0.331095\pi$
$821$	1.31763e13	1.01216	0.506078	+	0.862487i	$0.331095\pi$
$822$	0	0
$823$	−1.45311e13	−1.10408	−0.552039	−	0.833818i	$-0.686150\pi$
$823$	−1.45311e13	−1.10408	−0.552039	+	0.833818i	$0.686150\pi$
$824$	6.50002e11	0.0491182
$825$	0	0
$826$	9.65455e11	0.0721642
$827$	1.47807e13	1.09881	0.549403	−	0.835557i	$-0.314855\pi$
$827$	1.47807e13	1.09881	0.549403	+	0.835557i	$0.314855\pi$
$828$	0	0
$829$	−1.06764e13	−0.785108	−0.392554	−	0.919729i	$-0.628408\pi$
$829$	−1.06764e13	−0.785108	−0.392554	+	0.919729i	$0.628408\pi$
$830$	−5.21928e11	−0.0381733
$831$	0	0
$832$	−6.11689e12	−0.442563
$833$	9.98085e12	0.718233
$834$	0	0
$835$	−3.29675e12	−0.234691
$836$	−8.89329e12	−0.629701
$837$	0	0
$838$	−6.95389e11	−0.0487113
$839$	−2.13448e13	−1.48718	−0.743589	−	0.668637i	$-0.766878\pi$
$839$	−2.13448e13	−1.48718	−0.743589	+	0.668637i	$0.766878\pi$
$840$	0	0
$841$	−4.91027e12	−0.338472
$842$	1.36914e12	0.0938737
$843$	0	0
$844$	1.79375e12	0.121680
$845$	5.34805e12	0.360861
$846$	0	0
$847$	4.21735e12	0.281556
$848$	−2.84291e13	−1.88791
$849$	0	0
$850$	9.82217e11	0.0645390
$851$	1.50033e11	0.00980625
$852$	0	0
$853$	1.07554e13	0.695595	0.347798	−	0.937570i	$-0.386930\pi$
$853$	1.07554e13	0.695595	0.347798	+	0.937570i	$0.386930\pi$
$854$	−9.03350e11	−0.0581160
$855$	0	0
$856$	−5.48060e12	−0.348896
$857$	2.41413e13	1.52879	0.764394	−	0.644749i	$-0.223038\pi$
$857$	2.41413e13	1.52879	0.764394	+	0.644749i	$0.223038\pi$
$858$	0	0
$859$	−9.49185e12	−0.594815	−0.297407	−	0.954751i	$-0.596122\pi$
$859$	−9.49185e12	−0.594815	−0.297407	+	0.954751i	$0.596122\pi$
$860$	−1.86663e12	−0.116363
$861$	0	0
$862$	4.20405e11	0.0259349
$863$	5.95369e12	0.365374	0.182687	−	0.983171i	$-0.441520\pi$
$863$	5.95369e12	0.365374	0.182687	+	0.983171i	$0.441520\pi$
$864$	0	0
$865$	−1.05340e12	−0.0639766
$866$	7.67200e11	0.0463530
$867$	0	0
$868$	8.14701e12	0.487146
$869$	−1.22360e13	−0.727864
$870$	0	0
$871$	−1.00908e13	−0.594079
$872$	2.40258e12	0.140719
$873$	0	0
$874$	−1.12605e10	−0.000652764 0
$875$	6.63278e12	0.382524
$876$	0	0
$877$	1.67021e13	0.953397	0.476699	−	0.879067i	$-0.341833\pi$
$877$	1.67021e13	0.953397	0.476699	+	0.879067i	$0.341833\pi$
$878$	2.83133e12	0.160792
$879$	0	0
$880$	−1.01273e13	−0.569275
$881$	−1.55578e11	−0.00870075	−0.00435037	−	0.999991i	$-0.501385\pi$
$881$	−1.55578e11	−0.00870075	−0.00435037	+	0.999991i	$0.501385\pi$
$882$	0	0
$883$	5.33298e12	0.295221	0.147610	−	0.989046i	$-0.452842\pi$
$883$	5.33298e12	0.295221	0.147610	+	0.989046i	$0.452842\pi$
$884$	7.66799e12	0.422324
$885$	0	0
$886$	5.28399e11	0.0288078
$887$	−2.96127e13	−1.60628	−0.803140	−	0.595791i	$-0.796839\pi$
$887$	−2.96127e13	−1.60628	−0.803140	+	0.595791i	$0.796839\pi$
$888$	0	0
$889$	1.30296e13	0.699640
$890$	7.28073e11	0.0388974
$891$	0	0
$892$	9.68516e12	0.512230
$893$	1.05287e13	0.554043
$894$	0	0
$895$	7.76557e12	0.404548
$896$	−3.14210e12	−0.162867
$897$	0	0
$898$	1.34396e12	0.0689672
$899$	1.68296e13	0.859321
$900$	0	0
$901$	3.50524e13	1.77197
$902$	2.76841e12	0.139252
$903$	0	0
$904$	3.79008e12	0.188752
$905$	1.34160e12	0.0664822
$906$	0	0
$907$	2.00109e13	0.981823	0.490911	−	0.871210i	$-0.336664\pi$
$907$	2.00109e13	0.981823	0.490911	+	0.871210i	$0.336664\pi$
$908$	2.15567e12	0.105244
$909$	0	0
$910$	−1.83793e11	−0.00888468
$911$	−1.93475e13	−0.930661	−0.465330	−	0.885137i	$-0.654064\pi$
$911$	−1.93475e13	−0.930661	−0.465330	+	0.885137i	$0.654064\pi$
$912$	0	0
$913$	2.46888e13	1.17593
$914$	1.11425e11	0.00528112
$915$	0	0
$916$	1.47085e13	0.690302
$917$	−1.38658e13	−0.647566
$918$	0	0
$919$	−3.35697e10	−0.00155249	−0.000776243	−	1.00000i	$-0.500247\pi$
$919$	−3.35697e10	−0.00155249	−0.000776243		1.00000i	$0.500247\pi$
$920$	−2.59600e10	−0.00119470
$921$	0	0
$922$	−1.95226e12	−0.0889712
$923$	−1.20007e13	−0.544252
$924$	0	0
$925$	1.18036e13	0.530124
$926$	2.98597e12	0.133455
$927$	0	0
$928$	−4.87474e12	−0.215767
$929$	−9.02370e12	−0.397479	−0.198739	−	0.980052i	$-0.563685\pi$
$929$	−9.02370e12	−0.397479	−0.198739	+	0.980052i	$0.563685\pi$
$930$	0	0
$931$	−9.00217e12	−0.392712
$932$	−2.19435e13	−0.952652
$933$	0	0
$934$	2.99739e12	0.128879
$935$	1.24867e13	0.534314
$936$	0	0
$937$	1.76854e13	0.749525	0.374763	−	0.927121i	$-0.377724\pi$
$937$	1.76854e13	0.749525	0.374763	+	0.927121i	$0.377724\pi$
$938$	−1.25918e12	−0.0531099
$939$	0	0
$940$	1.20878e13	0.504977
$941$	−6.18651e12	−0.257213	−0.128606	−	0.991696i	$-0.541050\pi$
$941$	−6.18651e12	−0.257213	−0.128606	+	0.991696i	$0.541050\pi$
$942$	0	0
$943$	−4.35292e11	−0.0179258
$944$	4.13711e13	1.69560
$945$	0	0
$946$	−7.11038e11	−0.0288657
$947$	−1.51685e13	−0.612870	−0.306435	−	0.951892i	$-0.599136\pi$
$947$	−1.51685e13	−0.612870	−0.306435	+	0.951892i	$0.599136\pi$
$948$	0	0
$949$	−1.00907e13	−0.403852
$950$	−8.85905e11	−0.0352883
$951$	0	0
$952$	1.92140e12	0.0758145
$953$	1.86178e13	0.731158	0.365579	−	0.930780i	$-0.380871\pi$
$953$	1.86178e13	0.731158	0.365579	+	0.930780i	$0.380871\pi$
$954$	0	0
$955$	−9.11394e12	−0.354561
$956$	2.25351e13	0.872569
$957$	0	0
$958$	−2.96274e12	−0.113645
$959$	9.37526e12	0.357931
$960$	0	0
$961$	3.07373e12	0.116255
$962$	−7.42054e11	−0.0279349
$963$	0	0
$964$	−7.68033e12	−0.286440
$965$	9.68184e12	0.359406
$966$	0	0
$967$	4.57262e13	1.68169	0.840845	−	0.541276i	$-0.182059\pi$
$967$	4.57262e13	1.68169	0.840845	+	0.541276i	$0.182059\pi$
$968$	−2.94622e12	−0.107852
$969$	0	0
$970$	−1.07250e12	−0.0388976
$971$	3.98751e13	1.43951	0.719756	−	0.694228i	$-0.244254\pi$
$971$	3.98751e13	1.43951	0.719756	+	0.694228i	$0.244254\pi$
$972$	0	0
$973$	1.63993e13	0.586568
$974$	2.55704e12	0.0910380
$975$	0	0
$976$	−3.87098e13	−1.36552
$977$	−1.94826e12	−0.0684104	−0.0342052	−	0.999415i	$-0.510890\pi$
$977$	−1.94826e12	−0.0684104	−0.0342052	+	0.999415i	$0.510890\pi$
$978$	0	0
$979$	−3.44401e13	−1.19823
$980$	−1.03352e13	−0.357933
$981$	0	0
$982$	2.18011e12	0.0748129
$983$	−4.09293e13	−1.39812	−0.699058	−	0.715065i	$-0.746397\pi$
$983$	−4.09293e13	−1.39812	−0.699058	+	0.715065i	$0.746397\pi$
$984$	0	0
$985$	2.23202e13	0.755500
$986$	1.97660e12	0.0665999
$987$	0	0
$988$	−6.91610e12	−0.230916
$989$	1.11800e11	0.00371586
$990$	0	0
$991$	−5.11076e13	−1.68327	−0.841636	−	0.540045i	$-0.818407\pi$
$991$	−5.11076e13	−1.68327	−0.841636	+	0.540045i	$0.818407\pi$
$992$	−8.54861e12	−0.280281
$993$	0	0
$994$	−1.49751e12	−0.0486554
$995$	1.28724e13	0.416346
$996$	0	0
$997$	3.28420e13	1.05269	0.526347	−	0.850270i	$-0.323561\pi$
$997$	3.28420e13	1.05269	0.526347	+	0.850270i	$0.323561\pi$
$998$	−1.09115e12	−0.0348173
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	81.10.a.b.1.2	yes	4
3.2	odd	2		81.10.a.a.1.3	✓	4
9.2	odd	6		81.10.c.k.28.2		8
9.4	even	3		81.10.c.i.55.3		8
9.5	odd	6		81.10.c.k.55.2		8
9.7	even	3		81.10.c.i.28.3		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
81.10.a.a.1.3	✓	4	3.2	odd	2
81.10.a.b.1.2	yes	4	1.1	even	1	trivial
81.10.c.i.28.3		8	9.7	even	3
81.10.c.i.55.3		8	9.4	even	3
81.10.c.k.28.2		8	9.2	odd	6
81.10.c.k.55.2		8	9.5	odd	6

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:	\( N \)	\(=\)	\( 81 = 3^{4} \)
Weight:	\( k \)	\(=\)	\( 10 \)
Character orbit:	\([\chi]\)	\(=\)	81.a (trivial)

\(f(q)\)	\(=\)	\(q+2.02240 q^{2} -507.910 q^{4} -643.210 q^{5} +2952.58 q^{7} -2062.66 q^{8} -1300.83 q^{10} +61533.0 q^{11} +47852.7 q^{13} +5971.30 q^{14} +255878. q^{16} -315492. q^{17} +284556. q^{19} +326693. q^{20} +124444. q^{22} -19567.0 q^{23} -1.53941e6 q^{25} +96777.2 q^{26} -1.49965e6 q^{28} -3.09788e6 q^{29} -5.43262e6 q^{31} +1.57357e6 q^{32} -638050. q^{34} -1.89913e6 q^{35} -7.66765e6 q^{37} +575485. q^{38} +1.32673e6 q^{40} +2.22462e7 q^{41} -5.71372e6 q^{43} -3.12532e7 q^{44} -39572.2 q^{46} +3.70005e7 q^{47} -3.16359e7 q^{49} -3.11329e6 q^{50} -2.43049e7 q^{52} -1.11104e8 q^{53} -3.95786e7 q^{55} -6.09018e6 q^{56} -6.26515e6 q^{58} +1.61683e8 q^{59} -1.51282e8 q^{61} -1.09869e7 q^{62} -1.27827e8 q^{64} -3.07794e7 q^{65} -2.10872e8 q^{67} +1.60241e8 q^{68} -3.84080e6 q^{70} -2.50785e8 q^{71} -2.10869e8 q^{73} -1.55070e7 q^{74} -1.44529e8 q^{76} +1.81681e8 q^{77} -1.98853e8 q^{79} -1.64584e8 q^{80} +4.49907e7 q^{82} +4.01229e8 q^{83} +2.02928e8 q^{85} -1.15554e7 q^{86} -1.26922e8 q^{88} -5.59701e8 q^{89} +1.41289e8 q^{91} +9.93826e6 q^{92} +7.48296e7 q^{94} -1.83029e8 q^{95} +8.24474e8 q^{97} -6.39802e7 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 33 q^{2} + 853 q^{4} + 570 q^{5} - 3238 q^{7} + 4791 q^{8} - 9723 q^{10} - 96690 q^{11} - 141118 q^{13} + 3036 q^{14} - 244463 q^{16} + 285156 q^{17} - 465166 q^{19} - 1041711 q^{20} - 2244480 q^{22}+ \cdots + 735501321 q^{98}+O(q^{100}) \) 4 * q + 33 * q^2 + 853 * q^4 + 570 * q^5 - 3238 * q^7 + 4791 * q^8 - 9723 * q^10 - 96690 * q^11 - 141118 * q^13 + 3036 * q^14 - 244463 * q^16 + 285156 * q^17 - 465166 * q^19 - 1041711 * q^20 - 2244480 * q^22 - 237318 * q^23 + 3417712 * q^25 - 6433737 * q^26 + 769292 * q^28 + 2806926 * q^29 - 8244448 * q^31 + 3896415 * q^32 + 1386189 * q^34 - 31668402 * q^35 - 8204014 * q^37 - 755556 * q^38 - 54180645 * q^40 + 28889292 * q^41 - 39586246 * q^43 - 89567616 * q^44 - 93870624 * q^46 - 17514588 * q^47 - 67913400 * q^49 + 233832372 * q^50 - 207295237 * q^52 - 155826504 * q^53 - 185126142 * q^55 + 133671300 * q^56 - 164356143 * q^58 + 341256108 * q^59 - 238298254 * q^61 - 729569796 * q^62 - 286454975 * q^64 + 379881528 * q^65 - 72601486 * q^67 + 762353217 * q^68 - 780989460 * q^70 - 667543266 * q^71 - 520588288 * q^73 + 183473031 * q^74 - 413336884 * q^76 + 721530228 * q^77 - 137324662 * q^79 - 1181367123 * q^80 - 275042586 * q^82 - 278546844 * q^83 - 1408511358 * q^85 + 420627720 * q^86 - 30700416 * q^88 + 237712932 * q^89 - 762950942 * q^91 - 1238743680 * q^92 - 507333516 * q^94 - 811663194 * q^95 + 700071356 * q^97 + 735501321 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more