LMFDB - Embedded newform 81.10.a.b.1.4

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [81,10,Mod(1,81)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage: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")

magma://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:traces = [4,33] 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:	$41.7179027293$
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			1.4
Root			$29.6018$ 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)

$f(q)$	$=$	\(q+37.6018 q^{2} +901.897 q^{4} -1776.55 q^{5} +3746.45 q^{7} +14660.8 q^{8} -66801.3 q^{10} -43634.0 q^{11} -198244. q^{13} +140873. q^{14} +89502.6 q^{16} +520288. q^{17} -129326. q^{19} -1.60226e6 q^{20} -1.64072e6 q^{22} -1.93451e6 q^{23} +1.20299e6 q^{25} -7.45433e6 q^{26} +3.37891e6 q^{28} +2.30493e6 q^{29} -8.64675e6 q^{31} -4.14088e6 q^{32} +1.95638e7 q^{34} -6.65575e6 q^{35} +2.11152e6 q^{37} -4.86290e6 q^{38} -2.60456e7 q^{40} +1.32674e7 q^{41} -2.09275e6 q^{43} -3.93534e7 q^{44} -7.27411e7 q^{46} -3.60701e7 q^{47} -2.63177e7 q^{49} +4.52346e7 q^{50} -1.78796e8 q^{52} +7.10476e7 q^{53} +7.75178e7 q^{55} +5.49261e7 q^{56} +8.66697e7 q^{58} +1.13604e8 q^{59} -1.18231e7 q^{61} -3.25134e8 q^{62} -2.01530e8 q^{64} +3.52189e8 q^{65} +1.95419e8 q^{67} +4.69246e8 q^{68} -2.50268e8 q^{70} -1.47791e7 q^{71} -3.20288e8 q^{73} +7.93968e7 q^{74} -1.16639e8 q^{76} -1.63473e8 q^{77} +2.14129e8 q^{79} -1.59005e8 q^{80} +4.98880e8 q^{82} -2.48121e8 q^{83} -9.24315e8 q^{85} -7.86914e7 q^{86} -6.39711e8 q^{88} -1.21201e8 q^{89} -7.42712e8 q^{91} -1.74473e9 q^{92} -1.35630e9 q^{94} +2.29754e8 q^{95} +8.81709e8 q^{97} -9.89593e8 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	37.6018	1.66178	0.830891	−	0.556436i	$-0.187831\pi$
$2$	37.6018	1.66178	0.830891	+	0.556436i	$0.187831\pi$
$3$	0	0
$3$	0	0
$4$	901.897	1.76152
$5$	−1776.55	−1.27119	−0.635596	−	0.772022i	$-0.719246\pi$
$5$	−1776.55	−1.27119	−0.635596	+	0.772022i	$0.719246\pi$
$6$	0	0
$7$	3746.45	0.589765	0.294883	−	0.955533i	$-0.404719\pi$
$7$	3746.45	0.589765	0.294883	+	0.955533i	$0.404719\pi$
$8$	14660.8	1.26547
$9$	0	0
$10$	−66801.3	−2.11244
$11$	−43634.0	−0.898583	−0.449291	−	0.893385i	$-0.648323\pi$
$11$	−43634.0	−0.898583	−0.449291	+	0.893385i	$0.648323\pi$
$12$	0	0
$13$	−198244.	−1.92511	−0.962554	−	0.271091i	$-0.912615\pi$
$13$	−198244.	−1.92511	−0.962554	+	0.271091i	$0.912615\pi$
$14$	140873.	0.980061
$15$	0	0
$16$	89502.6	0.341425
$17$	520288.	1.51086	0.755429	−	0.655230i	$-0.227428\pi$
$17$	520288.	1.51086	0.755429	+	0.655230i	$0.227428\pi$
$18$	0	0
$19$	−129326.	−0.227665	−0.113832	−	0.993500i	$-0.536313\pi$
$19$	−129326.	−0.227665	−0.113832	+	0.993500i	$0.536313\pi$
$20$	−1.60226e6	−2.23923
$21$	0	0
$22$	−1.64072e6	−1.49325
$23$	−1.93451e6	−1.44144	−0.720718	−	0.693228i	$-0.756188\pi$
$23$	−1.93451e6	−1.44144	−0.720718	+	0.693228i	$0.756188\pi$
$24$	0	0
$25$	1.20299e6	0.615930
$26$	−7.45433e6	−3.19911
$27$	0	0
$28$	3.37891e6	1.03888
$29$	2.30493e6	0.605156	0.302578	−	0.953125i	$-0.402153\pi$
$29$	2.30493e6	0.605156	0.302578	+	0.953125i	$0.402153\pi$
$30$	0	0
$31$	−8.64675e6	−1.68161	−0.840805	−	0.541338i	$-0.817918\pi$
$31$	−8.64675e6	−1.68161	−0.840805	+	0.541338i	$0.817918\pi$
$32$	−4.14088e6	−0.698101
$33$	0	0
$34$	1.95638e7	2.51072
$35$	−6.65575e6	−0.749705
$36$	0	0
$37$	2.11152e6	0.185219	0.0926097	−	0.995702i	$-0.470479\pi$
$37$	2.11152e6	0.185219	0.0926097	+	0.995702i	$0.470479\pi$
$38$	−4.86290e6	−0.378329
$39$	0	0
$40$	−2.60456e7	−1.60866
$41$	1.32674e7	0.733263	0.366631	−	0.930366i	$-0.380511\pi$
$41$	1.32674e7	0.733263	0.366631	+	0.930366i	$0.380511\pi$
$42$	0	0
$43$	−2.09275e6	−0.0933491	−0.0466746	−	0.998910i	$-0.514862\pi$
$43$	−2.09275e6	−0.0933491	−0.0466746	+	0.998910i	$0.514862\pi$
$44$	−3.93534e7	−1.58287
$45$	0	0
$46$	−7.27411e7	−2.39535
$47$	−3.60701e7	−1.07822	−0.539110	−	0.842236i	$-0.681239\pi$
$47$	−3.60701e7	−1.07822	−0.539110	+	0.842236i	$0.681239\pi$
$48$	0	0
$49$	−2.63177e7	−0.652177
$50$	4.52346e7	1.02354
$51$	0	0
$52$	−1.78796e8	−3.39111
$53$	7.10476e7	1.23682	0.618412	−	0.785854i	$-0.287776\pi$
$53$	7.10476e7	1.23682	0.618412	+	0.785854i	$0.287776\pi$
$54$	0	0
$55$	7.75178e7	1.14227
$56$	5.49261e7	0.746333
$57$	0	0
$58$	8.66697e7	1.00564
$59$	1.13604e8	1.22056	0.610279	−	0.792186i	$-0.291057\pi$
$59$	1.13604e8	1.22056	0.610279	+	0.792186i	$0.291057\pi$
$60$	0	0
$61$	−1.18231e7	−0.109332	−0.0546658	−	0.998505i	$-0.517409\pi$
$61$	−1.18231e7	−0.109332	−0.0546658	+	0.998505i	$0.517409\pi$
$62$	−3.25134e8	−2.79447
$63$	0	0
$64$	−2.01530e8	−1.50152
$65$	3.52189e8	2.44718
$66$	0	0
$67$	1.95419e8	1.18476	0.592379	−	0.805660i	$-0.298189\pi$
$67$	1.95419e8	1.18476	0.592379	+	0.805660i	$0.298189\pi$
$68$	4.69246e8	2.66140
$69$	0	0
$70$	−2.50268e8	−1.24585
$71$	−1.47791e7	−0.0690217	−0.0345109	−	0.999404i	$-0.510987\pi$
$71$	−1.47791e7	−0.0690217	−0.0345109	+	0.999404i	$0.510987\pi$
$72$	0	0
$73$	−3.20288e8	−1.32004	−0.660020	−	0.751248i	$-0.729452\pi$
$73$	−3.20288e8	−1.32004	−0.660020	+	0.751248i	$0.729452\pi$
$74$	7.93968e7	0.307794
$75$	0	0
$76$	−1.16639e8	−0.401035
$77$	−1.63473e8	−0.529953
$78$	0	0
$79$	2.14129e8	0.618520	0.309260	−	0.950978i	$-0.399919\pi$
$79$	2.14129e8	0.618520	0.309260	+	0.950978i	$0.399919\pi$
$80$	−1.59005e8	−0.434017
$81$	0	0
$82$	4.98880e8	1.21852
$83$	−2.48121e8	−0.573869	−0.286934	−	0.957950i	$-0.592636\pi$
$83$	−2.48121e8	−0.573869	−0.286934	+	0.957950i	$0.592636\pi$
$84$	0	0
$85$	−9.24315e8	−1.92059
$86$	−7.86914e7	−0.155126
$87$	0	0
$88$	−6.39711e8	−1.13713
$89$	−1.21201e8	−0.204763	−0.102382	−	0.994745i	$-0.532646\pi$
$89$	−1.21201e8	−0.204763	−0.102382	+	0.994745i	$0.532646\pi$
$90$	0	0
$91$	−7.42712e8	−1.13536
$92$	−1.74473e9	−2.53912
$93$	0	0
$94$	−1.35630e9	−1.79176
$95$	2.29754e8	0.289406
$96$	0	0
$97$	8.81709e8	1.01124	0.505618	−	0.862758i	$-0.331265\pi$
$97$	8.81709e8	1.01124	0.505618	+	0.862758i	$0.331265\pi$
$98$	−9.89593e8	−1.08378
$99$	0	0
$100$	1.08497e9	1.08497
$101$	−2.24065e8	−0.214253	−0.107127	−	0.994245i	$-0.534165\pi$
$101$	−2.24065e8	−0.214253	−0.107127	+	0.994245i	$0.534165\pi$
$102$	0	0
$103$	1.35906e9	1.18980	0.594898	−	0.803801i	$-0.297193\pi$
$103$	1.35906e9	1.18980	0.594898	+	0.803801i	$0.297193\pi$
$104$	−2.90642e9	−2.43617
$105$	0	0
$106$	2.67152e9	2.05533
$107$	1.00263e9	0.739460	0.369730	−	0.929139i	$-0.379450\pi$
$107$	1.00263e9	0.739460	0.369730	+	0.929139i	$0.379450\pi$
$108$	0	0
$109$	−2.26633e8	−0.153782	−0.0768909	−	0.997040i	$-0.524499\pi$
$109$	−2.26633e8	−0.153782	−0.0768909	+	0.997040i	$0.524499\pi$
$110$	2.91481e9	1.89821
$111$	0	0
$112$	3.35317e8	0.201361
$113$	4.74483e8	0.273758	0.136879	−	0.990588i	$-0.456293\pi$
$113$	4.74483e8	0.273758	0.136879	+	0.990588i	$0.456293\pi$
$114$	0	0
$115$	3.43674e9	1.83234
$116$	2.07881e9	1.06599
$117$	0	0
$118$	4.27171e9	2.02830
$119$	1.94924e9	0.891052
$120$	0	0
$121$	−4.54021e8	−0.192549
$122$	−4.44568e8	−0.181685
$123$	0	0
$124$	−7.79848e9	−2.96219
$125$	1.33265e9	0.488227
$126$	0	0
$127$	−5.65066e9	−1.92745	−0.963724	−	0.266902i	$-0.914000\pi$
$127$	−5.65066e9	−1.92745	−0.963724	+	0.266902i	$0.914000\pi$
$128$	−5.45776e9	−1.79709
$129$	0	0
$130$	1.32430e10	4.06668
$131$	9.55272e8	0.283404	0.141702	−	0.989909i	$-0.454743\pi$
$131$	9.55272e8	0.283404	0.141702	+	0.989909i	$0.454743\pi$
$132$	0	0
$133$	−4.84515e8	−0.134269
$134$	7.34810e9	1.96881
$135$	0	0
$136$	7.62785e9	1.91195
$137$	3.88798e9	0.942933	0.471467	−	0.881884i	$-0.343725\pi$
$137$	3.88798e9	0.942933	0.471467	+	0.881884i	$0.343725\pi$
$138$	0	0
$139$	−2.94404e9	−0.668925	−0.334463	−	0.942409i	$-0.608555\pi$
$139$	−2.94404e9	−0.668925	−0.334463	+	0.942409i	$0.608555\pi$
$140$	−6.00280e9	−1.32062
$141$	0	0
$142$	−5.55722e8	−0.114699
$143$	8.65018e9	1.72987
$144$	0	0
$145$	−4.09482e9	−0.769270
$146$	−1.20434e10	−2.19362
$147$	0	0
$148$	1.90437e9	0.326267
$149$	4.43832e9	0.737702	0.368851	−	0.929489i	$-0.379751\pi$
$149$	4.43832e9	0.737702	0.368851	+	0.929489i	$0.379751\pi$
$150$	0	0
$151$	−5.13155e9	−0.803253	−0.401626	−	0.915804i	$-0.631555\pi$
$151$	−5.13155e9	−0.803253	−0.401626	+	0.915804i	$0.631555\pi$
$152$	−1.89603e9	−0.288104
$153$	0	0
$154$	−6.14687e9	−0.880666
$155$	1.53613e10	2.13765
$156$	0	0
$157$	−6.01238e9	−0.789765	−0.394883	−	0.918732i	$-0.629215\pi$
$157$	−6.01238e9	−0.789765	−0.394883	+	0.918732i	$0.629215\pi$
$158$	8.05164e9	1.02785
$159$	0	0
$160$	7.35647e9	0.887420
$161$	−7.24755e9	−0.850109
$162$	0	0
$163$	9.87088e9	1.09525	0.547623	−	0.836725i	$-0.315533\pi$
$163$	9.87088e9	1.09525	0.547623	+	0.836725i	$0.315533\pi$
$164$	1.19659e10	1.29165
$165$	0	0
$166$	−9.32981e9	−0.953645
$167$	−1.10715e10	−1.10149	−0.550747	−	0.834672i	$-0.685657\pi$
$167$	−1.10715e10	−1.10149	−0.550747	+	0.834672i	$0.685657\pi$
$168$	0	0
$169$	2.86962e10	2.70604
$170$	−3.47559e10	−3.19160
$171$	0	0
$172$	−1.88745e9	−0.164436
$173$	−1.13438e10	−0.962830	−0.481415	−	0.876493i	$-0.659877\pi$
$173$	−1.13438e10	−0.962830	−0.481415	+	0.876493i	$0.659877\pi$
$174$	0	0
$175$	4.50694e9	0.363254
$176$	−3.90536e9	−0.306799
$177$	0	0
$178$	−4.55738e9	−0.340271
$179$	9.63145e9	0.701218	0.350609	−	0.936522i	$-0.385975\pi$
$179$	9.63145e9	0.701218	0.350609	+	0.936522i	$0.385975\pi$
$180$	0	0
$181$	−4.35916e9	−0.301891	−0.150945	−	0.988542i	$-0.548232\pi$
$181$	−4.35916e9	−0.301891	−0.150945	+	0.988542i	$0.548232\pi$
$182$	−2.79273e10	−1.88672
$183$	0	0
$184$	−2.83615e10	−1.82410
$185$	−3.75120e9	−0.235450
$186$	0	0
$187$	−2.27023e10	−1.35763
$188$	−3.25315e10	−1.89930
$189$	0	0
$190$	8.63917e9	0.480929
$191$	−1.33132e9	−0.0723821	−0.0361911	−	0.999345i	$-0.511522\pi$
$191$	−1.33132e9	−0.0723821	−0.0361911	+	0.999345i	$0.511522\pi$
$192$	0	0
$193$	−3.29347e10	−1.70862	−0.854309	−	0.519765i	$-0.826020\pi$
$193$	−3.29347e10	−1.70862	−0.854309	+	0.519765i	$0.826020\pi$
$194$	3.31538e10	1.68045
$195$	0	0
$196$	−2.37358e10	−1.14882
$197$	−1.72555e10	−0.816264	−0.408132	−	0.912923i	$-0.633820\pi$
$197$	−1.72555e10	−0.816264	−0.408132	+	0.912923i	$0.633820\pi$
$198$	0	0
$199$	−3.03431e10	−1.37158	−0.685789	−	0.727800i	$-0.740543\pi$
$199$	−3.03431e10	−1.37158	−0.685789	+	0.727800i	$0.740543\pi$
$200$	1.76368e10	0.779444
$201$	0	0
$202$	−8.42525e9	−0.356042
$203$	8.63533e9	0.356900
$204$	0	0
$205$	−2.35702e10	−0.932118
$206$	5.11033e10	1.97718
$207$	0	0
$208$	−1.77434e10	−0.657281
$209$	5.64302e9	0.204575
$210$	0	0
$211$	2.95450e10	1.02616	0.513078	−	0.858342i	$-0.328505\pi$
$211$	2.95450e10	1.02616	0.513078	+	0.858342i	$0.328505\pi$
$212$	6.40776e10	2.17869
$213$	0	0
$214$	3.77008e10	1.22882
$215$	3.71787e9	0.118665
$216$	0	0
$217$	−3.23947e10	−0.991755
$218$	−8.52183e9	−0.255552
$219$	0	0
$220$	6.99131e10	2.01213
$221$	−1.03144e11	−2.90856
$222$	0	0
$223$	−4.85005e10	−1.31333	−0.656666	−	0.754181i	$-0.728034\pi$
$223$	−4.85005e10	−1.31333	−0.656666	+	0.754181i	$0.728034\pi$
$224$	−1.55136e10	−0.411715
$225$	0	0
$226$	1.78414e10	0.454927
$227$	4.12800e10	1.03187	0.515933	−	0.856629i	$-0.327445\pi$
$227$	4.12800e10	1.03187	0.515933	+	0.856629i	$0.327445\pi$
$228$	0	0
$229$	1.21823e10	0.292730	0.146365	−	0.989231i	$-0.453243\pi$
$229$	1.21823e10	0.292730	0.146365	+	0.989231i	$0.453243\pi$
$230$	1.29228e11	3.04495
$231$	0	0
$232$	3.37923e10	0.765810
$233$	7.14702e9	0.158863	0.0794316	−	0.996840i	$-0.474689\pi$
$233$	7.14702e9	0.158863	0.0794316	+	0.996840i	$0.474689\pi$
$234$	0	0
$235$	6.40802e10	1.37062
$236$	1.02459e11	2.15004
$237$	0	0
$238$	7.32948e10	1.48073
$239$	3.86505e9	0.0766240	0.0383120	−	0.999266i	$-0.487802\pi$
$239$	3.86505e9	0.0766240	0.0383120	+	0.999266i	$0.487802\pi$
$240$	0	0
$241$	−8.42727e10	−1.60920	−0.804600	−	0.593817i	$-0.797620\pi$
$241$	−8.42727e10	−1.60920	−0.804600	+	0.593817i	$0.797620\pi$
$242$	−1.70720e10	−0.319975
$243$	0	0
$244$	−1.06632e10	−0.192589
$245$	4.67546e10	0.829042
$246$	0	0
$247$	2.56382e10	0.438279
$248$	−1.26769e11	−2.12804
$249$	0	0
$250$	5.01101e10	0.811326
$251$	−4.52000e10	−0.718798	−0.359399	−	0.933184i	$-0.617018\pi$
$251$	−4.52000e10	−0.718798	−0.359399	+	0.933184i	$0.617018\pi$
$252$	0	0
$253$	8.44104e10	1.29525
$254$	−2.12475e11	−3.20300
$255$	0	0
$256$	−1.02038e11	−1.48486
$257$	−1.12948e11	−1.61503	−0.807513	−	0.589850i	$-0.799187\pi$
$257$	−1.12948e11	−1.61503	−0.807513	+	0.589850i	$0.799187\pi$
$258$	0	0
$259$	7.91070e9	0.109236
$260$	3.17639e11	4.31075
$261$	0	0
$262$	3.59199e10	0.470956
$263$	−1.21136e11	−1.56125	−0.780623	−	0.625003i	$-0.785098\pi$
$263$	−1.21136e11	−1.56125	−0.780623	+	0.625003i	$0.785098\pi$
$264$	0	0
$265$	−1.26219e11	−1.57224
$266$	−1.82186e10	−0.223125
$267$	0	0
$268$	1.76247e11	2.08697
$269$	−2.98343e10	−0.347401	−0.173700	−	0.984799i	$-0.555572\pi$
$269$	−2.98343e10	−0.347401	−0.173700	+	0.984799i	$0.555572\pi$
$270$	0	0
$271$	−1.06831e11	−1.20319	−0.601597	−	0.798800i	$-0.705469\pi$
$271$	−1.06831e11	−1.20319	−0.601597	+	0.798800i	$0.705469\pi$
$272$	4.65671e10	0.515845
$273$	0	0
$274$	1.46195e11	1.56695
$275$	−5.24912e10	−0.553464
$276$	0	0
$277$	7.33121e9	0.0748199	0.0374099	−	0.999300i	$-0.488089\pi$
$277$	7.33121e9	0.0748199	0.0374099	+	0.999300i	$0.488089\pi$
$278$	−1.10701e11	−1.11161
$279$	0	0
$280$	−9.75787e10	−0.948733
$281$	−7.26917e10	−0.695515	−0.347757	−	0.937585i	$-0.613057\pi$
$281$	−7.26917e10	−0.695515	−0.347757	+	0.937585i	$0.613057\pi$
$282$	0	0
$283$	−6.70808e10	−0.621669	−0.310835	−	0.950464i	$-0.600609\pi$
$283$	−6.70808e10	−0.621669	−0.310835	+	0.950464i	$0.600609\pi$
$284$	−1.33292e10	−0.121583
$285$	0	0
$286$	3.25262e11	2.87466
$287$	4.97058e10	0.432453
$288$	0	0
$289$	1.52112e11	1.28269
$290$	−1.53973e11	−1.27836
$291$	0	0
$292$	−2.88866e11	−2.32527
$293$	9.70631e10	0.769396	0.384698	−	0.923043i	$-0.374306\pi$
$293$	9.70631e10	0.769396	0.384698	+	0.923043i	$0.374306\pi$
$294$	0	0
$295$	−2.01822e11	−1.55157
$296$	3.09566e10	0.234391
$297$	0	0
$298$	1.66889e11	1.22590
$299$	3.83505e11	2.77492
$300$	0	0
$301$	−7.84041e9	−0.0550541
$302$	−1.92956e11	−1.33483
$303$	0	0
$304$	−1.15750e10	−0.0777305
$305$	2.10042e10	0.138981
$306$	0	0
$307$	−3.37348e8	−0.00216748	−0.00108374	−	0.999999i	$-0.500345\pi$
$307$	−3.37348e8	−0.00216748	−0.00108374	+	0.999999i	$0.500345\pi$
$308$	−1.47436e11	−0.933521
$309$	0	0
$310$	5.77615e11	3.55231
$311$	−2.24027e11	−1.35793	−0.678966	−	0.734170i	$-0.737572\pi$
$311$	−2.24027e11	−1.35793	−0.678966	+	0.734170i	$0.737572\pi$
$312$	0	0
$313$	6.27538e10	0.369565	0.184782	−	0.982779i	$-0.440842\pi$
$313$	6.27538e10	0.369565	0.184782	+	0.982779i	$0.440842\pi$
$314$	−2.26076e11	−1.31242
$315$	0	0
$316$	1.93122e11	1.08953
$317$	1.58022e11	0.878921	0.439461	−	0.898262i	$-0.355170\pi$
$317$	1.58022e11	0.878921	0.439461	+	0.898262i	$0.355170\pi$
$318$	0	0
$319$	−1.00574e11	−0.543783
$320$	3.58027e11	1.90872
$321$	0	0
$322$	−2.72521e11	−1.41270
$323$	−6.72869e10	−0.343969
$324$	0	0
$325$	−2.38485e11	−1.18573
$326$	3.71163e11	1.82006
$327$	0	0
$328$	1.94512e11	0.927926
$329$	−1.35135e11	−0.635896
$330$	0	0
$331$	−9.87367e10	−0.452119	−0.226059	−	0.974114i	$-0.572584\pi$
$331$	−9.87367e10	−0.452119	−0.226059	+	0.974114i	$0.572584\pi$
$332$	−2.23780e11	−1.01088
$333$	0	0
$334$	−4.16308e11	−1.83044
$335$	−3.47170e11	−1.50605
$336$	0	0
$337$	1.47765e11	0.624075	0.312037	−	0.950070i	$-0.398989\pi$
$337$	1.47765e11	0.624075	0.312037	+	0.950070i	$0.398989\pi$
$338$	1.07903e12	4.49684
$339$	0	0
$340$	−8.33637e11	−3.38316
$341$	3.77292e11	1.51107
$342$	0	0
$343$	−2.49781e11	−0.974397
$344$	−3.06815e10	−0.118131
$345$	0	0
$346$	−4.26546e11	−1.60001
$347$	−3.40507e10	−0.126079	−0.0630395	−	0.998011i	$-0.520079\pi$
$347$	−3.40507e10	−0.126079	−0.0630395	+	0.998011i	$0.520079\pi$
$348$	0	0
$349$	6.70967e10	0.242095	0.121048	−	0.992647i	$-0.461375\pi$
$349$	6.70967e10	0.242095	0.121048	+	0.992647i	$0.461375\pi$
$350$	1.69469e11	0.603649
$351$	0	0
$352$	1.80683e11	0.627301
$353$	9.94198e10	0.340790	0.170395	−	0.985376i	$-0.445496\pi$
$353$	9.94198e10	0.340790	0.170395	+	0.985376i	$0.445496\pi$
$354$	0	0
$355$	2.62558e10	0.0877399
$356$	−1.09311e11	−0.360694
$357$	0	0
$358$	3.62160e11	1.16527
$359$	8.45266e10	0.268577	0.134288	−	0.990942i	$-0.457125\pi$
$359$	8.45266e10	0.268577	0.134288	+	0.990942i	$0.457125\pi$
$360$	0	0
$361$	−3.05962e11	−0.948169
$362$	−1.63912e11	−0.501676
$363$	0	0
$364$	−6.69849e11	−1.99996
$365$	5.69005e11	1.67803
$366$	0	0
$367$	2.78102e11	0.800215	0.400108	−	0.916468i	$-0.368973\pi$
$367$	2.78102e11	0.800215	0.400108	+	0.916468i	$0.368973\pi$
$368$	−1.73144e11	−0.492143
$369$	0	0
$370$	−1.41052e11	−0.391266
$371$	2.66176e11	0.729436
$372$	0	0
$373$	5.89326e11	1.57640	0.788199	−	0.615420i	$-0.211014\pi$
$373$	5.89326e11	1.57640	0.788199	+	0.615420i	$0.211014\pi$
$374$	−8.53646e11	−2.25609
$375$	0	0
$376$	−5.28818e11	−1.36446
$377$	−4.56939e11	−1.16499
$378$	0	0
$379$	4.28557e10	0.106692	0.0533461	−	0.998576i	$-0.483011\pi$
$379$	4.28557e10	0.106692	0.0533461	+	0.998576i	$0.483011\pi$
$380$	2.07214e11	0.509793
$381$	0	0
$382$	−5.00600e10	−0.120283
$383$	3.31187e11	0.786463	0.393231	−	0.919440i	$-0.371357\pi$
$383$	3.31187e11	0.786463	0.393231	+	0.919440i	$0.371357\pi$
$384$	0	0
$385$	2.90417e11	0.673672
$386$	−1.23840e12	−2.83935
$387$	0	0
$388$	7.95210e11	1.78131
$389$	5.48637e10	0.121482	0.0607410	−	0.998154i	$-0.480654\pi$
$389$	5.48637e10	0.121482	0.0607410	+	0.998154i	$0.480654\pi$
$390$	0	0
$391$	−1.00650e12	−2.17781
$392$	−3.85839e11	−0.825314
$393$	0	0
$394$	−6.48840e11	−1.35645
$395$	−3.80410e11	−0.786258
$396$	0	0
$397$	−3.64760e11	−0.736971	−0.368485	−	0.929634i	$-0.620124\pi$
$397$	−3.64760e11	−0.736971	−0.368485	+	0.929634i	$0.620124\pi$
$398$	−1.14095e12	−2.27926
$399$	0	0
$400$	1.07671e11	0.210294
$401$	−1.15136e11	−0.222362	−0.111181	−	0.993800i	$-0.535463\pi$
$401$	−1.15136e11	−0.222362	−0.111181	+	0.993800i	$0.535463\pi$
$402$	0	0
$403$	1.71417e12	3.23728
$404$	−2.02083e11	−0.377411
$405$	0	0
$406$	3.24704e11	0.593090
$407$	−9.21339e10	−0.166435
$408$	0	0
$409$	−1.11527e11	−0.197073	−0.0985363	−	0.995133i	$-0.531416\pi$
$409$	−1.11527e11	−0.197073	−0.0985363	+	0.995133i	$0.531416\pi$
$410$	−8.86282e11	−1.54898
$411$	0	0
$412$	1.22574e12	2.09584
$413$	4.25611e11	0.719843
$414$	0	0
$415$	4.40799e11	0.729498
$416$	8.20905e11	1.34392
$417$	0	0
$418$	2.12188e11	0.339960
$419$	−3.64957e11	−0.578468	−0.289234	−	0.957258i	$-0.593401\pi$
$419$	−3.64957e11	−0.578468	−0.289234	+	0.957258i	$0.593401\pi$
$420$	0	0
$421$	−7.90658e11	−1.22665	−0.613323	−	0.789832i	$-0.710168\pi$
$421$	−7.90658e11	−1.22665	−0.613323	+	0.789832i	$0.710168\pi$
$422$	1.11095e12	1.70525
$423$	0	0
$424$	1.04162e12	1.56517
$425$	6.25901e11	0.930583
$426$	0	0
$427$	−4.42945e10	−0.0644800
$428$	9.04271e11	1.30257
$429$	0	0
$430$	1.39799e11	0.197195
$431$	1.34437e12	1.87660	0.938298	−	0.345828i	$-0.112402\pi$
$431$	1.34437e12	1.87660	0.938298	+	0.345828i	$0.112402\pi$
$432$	0	0
$433$	2.92568e11	0.399974	0.199987	−	0.979799i	$-0.435910\pi$
$433$	2.92568e11	0.399974	0.199987	+	0.979799i	$0.435910\pi$
$434$	−1.21810e12	−1.64808
$435$	0	0
$436$	−2.04400e11	−0.270889
$437$	2.50183e11	0.328164
$438$	0	0
$439$	−2.79934e11	−0.359720	−0.179860	−	0.983692i	$-0.557564\pi$
$439$	−2.79934e11	−0.359720	−0.179860	+	0.983692i	$0.557564\pi$
$440$	1.13648e12	1.44552
$441$	0	0
$442$	−3.87840e12	−4.83340
$443$	−1.26158e12	−1.55632	−0.778161	−	0.628064i	$-0.783847\pi$
$443$	−1.26158e12	−1.55632	−0.778161	+	0.628064i	$0.783847\pi$
$444$	0	0
$445$	2.15319e11	0.260293
$446$	−1.82371e12	−2.18247
$447$	0	0
$448$	−7.55023e11	−0.885542
$449$	7.78344e10	0.0903780	0.0451890	−	0.998978i	$-0.485611\pi$
$449$	7.78344e10	0.0903780	0.0451890	+	0.998978i	$0.485611\pi$
$450$	0	0
$451$	−5.78911e11	−0.658897
$452$	4.27935e11	0.482230
$453$	0	0
$454$	1.55220e12	1.71474
$455$	1.31946e12	1.44326
$456$	0	0
$457$	−7.15331e11	−0.767157	−0.383579	−	0.923508i	$-0.625308\pi$
$457$	−7.15331e11	−0.767157	−0.383579	+	0.923508i	$0.625308\pi$
$458$	4.58075e11	0.486454
$459$	0	0
$460$	3.09959e12	3.22770
$461$	1.73891e11	0.179318	0.0896588	−	0.995973i	$-0.471422\pi$
$461$	1.73891e11	0.179318	0.0896588	+	0.995973i	$0.471422\pi$
$462$	0	0
$463$	1.33058e12	1.34564	0.672819	−	0.739807i	$-0.265083\pi$
$463$	1.33058e12	1.34564	0.672819	+	0.739807i	$0.265083\pi$
$464$	2.06298e11	0.206616
$465$	0	0
$466$	2.68741e11	0.263996
$467$	1.12875e11	0.109818	0.0549088	−	0.998491i	$-0.482513\pi$
$467$	1.12875e11	0.109818	0.0549088	+	0.998491i	$0.482513\pi$
$468$	0	0
$469$	7.32127e11	0.698729
$470$	2.40953e12	2.27768
$471$	0	0
$472$	1.66553e12	1.54459
$473$	9.13153e10	0.0838819
$474$	0	0
$475$	−1.55578e11	−0.140226
$476$	1.75801e12	1.56960
$477$	0	0
$478$	1.45333e11	0.127332
$479$	−4.09679e11	−0.355577	−0.177789	−	0.984069i	$-0.556894\pi$
$479$	−4.09679e11	−0.355577	−0.177789	+	0.984069i	$0.556894\pi$
$480$	0	0
$481$	−4.18595e11	−0.356567
$482$	−3.16881e12	−2.67414
$483$	0	0
$484$	−4.09480e11	−0.339179
$485$	−1.56640e12	−1.28547
$486$	0	0
$487$	2.35261e12	1.89526	0.947632	−	0.319366i	$-0.103470\pi$
$487$	2.35261e12	1.89526	0.947632	+	0.319366i	$0.103470\pi$
$488$	−1.73336e11	−0.138356
$489$	0	0
$490$	1.75806e12	1.37769
$491$	1.59827e12	1.24103	0.620517	−	0.784193i	$-0.286923\pi$
$491$	1.59827e12	1.24103	0.620517	+	0.784193i	$0.286923\pi$
$492$	0	0
$493$	1.19923e12	0.914306
$494$	9.64041e11	0.728323
$495$	0	0
$496$	−7.73907e11	−0.574145
$497$	−5.53693e10	−0.0407066
$498$	0	0
$499$	1.07185e12	0.773892	0.386946	−	0.922102i	$-0.373530\pi$
$499$	1.07185e12	0.773892	0.386946	+	0.922102i	$0.373530\pi$
$500$	1.20191e12	0.860019
$501$	0	0
$502$	−1.69960e12	−1.19449
$503$	−2.53052e12	−1.76260	−0.881300	−	0.472556i	$-0.843331\pi$
$503$	−2.53052e12	−1.76260	−0.881300	+	0.472556i	$0.843331\pi$
$504$	0	0
$505$	3.98061e11	0.272357
$506$	3.17398e12	2.15242
$507$	0	0
$508$	−5.09631e12	−3.39523
$509$	1.45460e12	0.960536	0.480268	−	0.877122i	$-0.340539\pi$
$509$	1.45460e12	0.960536	0.480268	+	0.877122i	$0.340539\pi$
$510$	0	0
$511$	−1.19994e12	−0.778514
$512$	−1.04246e12	−0.670415
$513$	0	0
$514$	−4.24705e12	−2.68382
$515$	−2.41444e12	−1.51246
$516$	0	0
$517$	1.57388e12	0.968869
$518$	2.97457e11	0.181526
$519$	0	0
$520$	5.16339e12	3.09685
$521$	1.11421e12	0.662514	0.331257	−	0.943540i	$-0.392527\pi$
$521$	1.11421e12	0.662514	0.331257	+	0.943540i	$0.392527\pi$
$522$	0	0
$523$	−1.53080e12	−0.894665	−0.447332	−	0.894368i	$-0.647626\pi$
$523$	−1.53080e12	−0.894665	−0.447332	+	0.894368i	$0.647626\pi$
$524$	8.61556e11	0.499221
$525$	0	0
$526$	−4.55492e12	−2.59445
$527$	−4.49880e12	−2.54068
$528$	0	0
$529$	1.94117e12	1.07774
$530$	−4.74607e12	−2.61272
$531$	0	0
$532$	−4.36982e11	−0.236517
$533$	−2.63019e12	−1.41161
$534$	0	0
$535$	−1.78122e12	−0.939996
$536$	2.86500e12	1.49928
$537$	0	0
$538$	−1.12182e12	−0.577304
$539$	1.14835e12	0.586035
$540$	0	0
$541$	2.80574e12	1.40819	0.704094	−	0.710107i	$-0.251354\pi$
$541$	2.80574e12	1.40819	0.704094	+	0.710107i	$0.251354\pi$
$542$	−4.01704e12	−1.99945
$543$	0	0
$544$	−2.15445e12	−1.05473
$545$	4.02625e11	0.195486
$546$	0	0
$547$	−1.95570e12	−0.934027	−0.467014	−	0.884250i	$-0.654670\pi$
$547$	−1.95570e12	−0.934027	−0.467014	+	0.884250i	$0.654670\pi$
$548$	3.50655e12	1.66099
$549$	0	0
$550$	−1.97377e12	−0.919737
$551$	−2.98089e11	−0.137773
$552$	0	0
$553$	8.02225e11	0.364782
$554$	2.75667e11	0.124334
$555$	0	0
$556$	−2.65522e12	−1.17832
$557$	−3.04443e12	−1.34016	−0.670082	−	0.742287i	$-0.733741\pi$
$557$	−3.04443e12	−1.34016	−0.670082	+	0.742287i	$0.733741\pi$
$558$	0	0
$559$	4.14876e11	0.179707
$560$	−5.95707e11	−0.255968
$561$	0	0
$562$	−2.73334e12	−1.15579
$563$	−8.23592e11	−0.345481	−0.172741	−	0.984967i	$-0.555262\pi$
$563$	−8.23592e11	−0.345481	−0.172741	+	0.984967i	$0.555262\pi$
$564$	0	0
$565$	−8.42941e11	−0.348000
$566$	−2.52236e12	−1.03308
$567$	0	0
$568$	−2.16674e11	−0.0873453
$569$	−1.29328e12	−0.517234	−0.258617	−	0.965980i	$-0.583267\pi$
$569$	−1.29328e12	−0.517234	−0.258617	+	0.965980i	$0.583267\pi$
$570$	0	0
$571$	−9.40641e10	−0.0370307	−0.0185153	−	0.999829i	$-0.505894\pi$
$571$	−9.40641e10	−0.0370307	−0.0185153	+	0.999829i	$0.505894\pi$
$572$	7.80157e12	3.04719
$573$	0	0
$574$	1.86903e12	0.718642
$575$	−2.32719e12	−0.887825
$576$	0	0
$577$	3.56834e11	0.134022	0.0670108	−	0.997752i	$-0.478654\pi$
$577$	3.56834e11	0.134022	0.0670108	+	0.997752i	$0.478654\pi$
$578$	5.71968e12	2.13155
$579$	0	0
$580$	−3.69311e12	−1.35508
$581$	−9.29575e11	−0.338448
$582$	0	0
$583$	−3.10009e12	−1.11139
$584$	−4.69568e12	−1.67048
$585$	0	0
$586$	3.64975e12	1.27857
$587$	2.13672e12	0.742806	0.371403	−	0.928472i	$-0.378877\pi$
$587$	2.13672e12	0.742806	0.371403	+	0.928472i	$0.378877\pi$
$588$	0	0
$589$	1.11825e12	0.382843
$590$	−7.58889e12	−2.57836
$591$	0	0
$592$	1.88986e11	0.0632386
$593$	−2.76976e12	−0.919804	−0.459902	−	0.887970i	$-0.652116\pi$
$593$	−2.76976e12	−0.919804	−0.459902	+	0.887970i	$0.652116\pi$
$594$	0	0
$595$	−3.46291e12	−1.13270
$596$	4.00291e12	1.29947
$597$	0	0
$598$	1.44205e13	4.61131
$599$	4.23621e12	1.34449	0.672244	−	0.740330i	$-0.265331\pi$
$599$	4.23621e12	1.34449	0.672244	+	0.740330i	$0.265331\pi$
$600$	0	0
$601$	2.63193e11	0.0822885	0.0411443	−	0.999153i	$-0.486900\pi$
$601$	2.63193e11	0.0822885	0.0411443	+	0.999153i	$0.486900\pi$
$602$	−2.94814e11	−0.0914878
$603$	0	0
$604$	−4.62813e12	−1.41494
$605$	8.06589e11	0.244767
$606$	0	0
$607$	4.78384e12	1.43030	0.715150	−	0.698971i	$-0.246358\pi$
$607$	4.78384e12	1.43030	0.715150	+	0.698971i	$0.246358\pi$
$608$	5.35525e11	0.158933
$609$	0	0
$610$	7.89796e11	0.230957
$611$	7.15068e12	2.07569
$612$	0	0
$613$	5.91300e12	1.69136	0.845679	−	0.533692i	$-0.179196\pi$
$613$	5.91300e12	1.69136	0.845679	+	0.533692i	$0.179196\pi$
$614$	−1.26849e10	−0.00360188
$615$	0	0
$616$	−2.39665e12	−0.670642
$617$	−8.52460e11	−0.236805	−0.118402	−	0.992966i	$-0.537777\pi$
$617$	−8.52460e11	−0.236805	−0.118402	+	0.992966i	$0.537777\pi$
$618$	0	0
$619$	3.92018e10	0.0107324	0.00536621	−	0.999986i	$-0.498292\pi$
$619$	3.92018e10	0.0107324	0.00536621	+	0.999986i	$0.498292\pi$
$620$	1.38544e13	3.76551
$621$	0	0
$622$	−8.42381e12	−2.25659
$623$	−4.54074e11	−0.120762
$624$	0	0
$625$	−4.71710e12	−1.23656
$626$	2.35966e12	0.614136
$627$	0	0
$628$	−5.42255e12	−1.39118
$629$	1.09860e12	0.279840
$630$	0	0
$631$	3.34191e12	0.839195	0.419598	−	0.907710i	$-0.362171\pi$
$631$	3.34191e12	0.839195	0.419598	+	0.907710i	$0.362171\pi$
$632$	3.13931e12	0.782722
$633$	0	0
$634$	5.94190e12	1.46057
$635$	1.00387e13	2.45016
$636$	0	0
$637$	5.21732e12	1.25551
$638$	−3.78175e12	−0.903649
$639$	0	0
$640$	9.69597e12	2.28445
$641$	2.50519e12	0.586111	0.293055	−	0.956095i	$-0.405328\pi$
$641$	2.50519e12	0.586111	0.293055	+	0.956095i	$0.405328\pi$
$642$	0	0
$643$	−3.97832e12	−0.917804	−0.458902	−	0.888487i	$-0.651757\pi$
$643$	−3.97832e12	−0.917804	−0.458902	+	0.888487i	$0.651757\pi$
$644$	−6.53654e12	−1.49748
$645$	0	0
$646$	−2.53011e12	−0.571601
$647$	2.26899e12	0.509053	0.254527	−	0.967066i	$-0.418080\pi$
$647$	2.26899e12	0.509053	0.254527	+	0.967066i	$0.418080\pi$
$648$	0	0
$649$	−4.95699e12	−1.09677
$650$	−8.96748e12	−1.97043
$651$	0	0
$652$	8.90252e12	1.92930
$653$	−4.06675e12	−0.875262	−0.437631	−	0.899155i	$-0.644182\pi$
$653$	−4.06675e12	−0.875262	−0.437631	+	0.899155i	$0.644182\pi$
$654$	0	0
$655$	−1.69708e12	−0.360261
$656$	1.18747e12	0.250355
$657$	0	0
$658$	−5.08132e12	−1.05672
$659$	−3.23943e12	−0.669090	−0.334545	−	0.942380i	$-0.608583\pi$
$659$	−3.23943e12	−0.669090	−0.334545	+	0.942380i	$0.608583\pi$
$660$	0	0
$661$	−2.00532e12	−0.408579	−0.204290	−	0.978910i	$-0.565488\pi$
$661$	−2.00532e12	−0.408579	−0.204290	+	0.978910i	$0.565488\pi$
$662$	−3.71268e12	−0.751323
$663$	0	0
$664$	−3.63766e12	−0.726217
$665$	8.60763e11	0.170681
$666$	0	0
$667$	−4.45892e12	−0.872295
$668$	−9.98535e12	−1.94030
$669$	0	0
$670$	−1.30542e13	−2.50273
$671$	5.15887e11	0.0982434
$672$	0	0
$673$	2.46033e11	0.0462302	0.0231151	−	0.999733i	$-0.492642\pi$
$673$	2.46033e11	0.0462302	0.0231151	+	0.999733i	$0.492642\pi$
$674$	5.55623e12	1.03708
$675$	0	0
$676$	2.58810e13	4.76673
$677$	1.99607e12	0.365196	0.182598	−	0.983188i	$-0.441549\pi$
$677$	1.99607e12	0.365196	0.182598	+	0.983188i	$0.441549\pi$
$678$	0	0
$679$	3.30328e12	0.596391
$680$	−1.35512e13	−2.43046
$681$	0	0
$682$	1.41869e13	2.51106
$683$	8.20056e12	1.44195	0.720975	−	0.692961i	$-0.243694\pi$
$683$	8.20056e12	1.44195	0.720975	+	0.692961i	$0.243694\pi$
$684$	0	0
$685$	−6.90717e12	−1.19865
$686$	−9.39222e12	−1.61923
$687$	0	0
$688$	−1.87307e11	−0.0318718
$689$	−1.40848e13	−2.38102
$690$	0	0
$691$	2.86091e12	0.477367	0.238684	−	0.971097i	$-0.423284\pi$
$691$	2.86091e12	0.477367	0.238684	+	0.971097i	$0.423284\pi$
$692$	−1.02309e13	−1.69604
$693$	0	0
$694$	−1.28037e12	−0.209516
$695$	5.23023e12	0.850333
$696$	0	0
$697$	6.90289e12	1.10786
$698$	2.52296e12	0.402310
$699$	0	0
$700$	4.06480e12	0.639879
$701$	−4.34132e12	−0.679032	−0.339516	−	0.940600i	$-0.610263\pi$
$701$	−4.34132e12	−0.679032	−0.339516	+	0.940600i	$0.610263\pi$
$702$	0	0
$703$	−2.73074e11	−0.0421679
$704$	8.79356e12	1.34924
$705$	0	0
$706$	3.73837e12	0.566319
$707$	−8.39449e11	−0.126359
$708$	0	0
$709$	1.24606e13	1.85196	0.925979	−	0.377574i	$-0.123242\pi$
$709$	1.24606e13	1.85196	0.925979	+	0.377574i	$0.123242\pi$
$710$	9.87265e11	0.145805
$711$	0	0
$712$	−1.77691e12	−0.259122
$713$	1.67272e13	2.42394
$714$	0	0
$715$	−1.53674e13	−2.19900
$716$	8.68657e12	1.23521
$717$	0	0
$718$	3.17835e12	0.446316
$719$	−7.62397e12	−1.06390	−0.531951	−	0.846775i	$-0.678541\pi$
$719$	−7.62397e12	−1.06390	−0.531951	+	0.846775i	$0.678541\pi$
$720$	0	0
$721$	5.09167e12	0.701700
$722$	−1.15047e13	−1.57565
$723$	0	0
$724$	−3.93151e12	−0.531785
$725$	2.77281e12	0.372734
$726$	0	0
$727$	8.45265e11	0.112225	0.0561123	−	0.998424i	$-0.482130\pi$
$727$	8.45265e11	0.112225	0.0561123	+	0.998424i	$0.482130\pi$
$728$	−1.08888e13	−1.43677
$729$	0	0
$730$	2.13956e13	2.78851
$731$	−1.08884e12	−0.141037
$732$	0	0
$733$	−1.25819e13	−1.60982	−0.804909	−	0.593398i	$-0.797786\pi$
$733$	−1.25819e13	−1.60982	−0.804909	+	0.593398i	$0.797786\pi$
$734$	1.04571e13	1.32978
$735$	0	0
$736$	8.01058e12	1.00627
$737$	−8.52690e12	−1.06460
$738$	0	0
$739$	−1.20238e13	−1.48301	−0.741503	−	0.670949i	$-0.765887\pi$
$739$	−1.20238e13	−1.48301	−0.741503	+	0.670949i	$0.765887\pi$
$740$	−3.38320e12	−0.414748
$741$	0	0
$742$	1.00087e13	1.21216
$743$	−1.24304e13	−1.49635	−0.748177	−	0.663499i	$-0.769071\pi$
$743$	−1.24304e13	−1.49635	−0.748177	+	0.663499i	$0.769071\pi$
$744$	0	0
$745$	−7.88489e12	−0.937761
$746$	2.21597e13	2.61963
$747$	0	0
$748$	−2.04751e13	−2.39149
$749$	3.75631e12	0.436108
$750$	0	0
$751$	3.63549e12	0.417046	0.208523	−	0.978017i	$-0.433134\pi$
$751$	3.63549e12	0.417046	0.208523	+	0.978017i	$0.433134\pi$
$752$	−3.22837e12	−0.368131
$753$	0	0
$754$	−1.71818e13	−1.93596
$755$	9.11643e12	1.02109
$756$	0	0
$757$	−1.25427e13	−1.38822	−0.694111	−	0.719868i	$-0.744202\pi$
$757$	−1.25427e13	−1.38822	−0.694111	+	0.719868i	$0.744202\pi$
$758$	1.61145e12	0.177299
$759$	0	0
$760$	3.36838e12	0.366235
$761$	1.69116e13	1.82791	0.913953	−	0.405821i	$-0.133014\pi$
$761$	1.69116e13	1.82791	0.913953	+	0.405821i	$0.133014\pi$
$762$	0	0
$763$	−8.49072e11	−0.0906951
$764$	−1.20071e12	−0.127502
$765$	0	0
$766$	1.24532e13	1.30693
$767$	−2.25213e13	−2.34971
$768$	0	0
$769$	1.48482e13	1.53111	0.765555	−	0.643371i	$-0.222465\pi$
$769$	1.48482e13	1.53111	0.765555	+	0.643371i	$0.222465\pi$
$770$	1.09202e13	1.11950
$771$	0	0
$772$	−2.97037e13	−3.00976
$773$	1.70164e13	1.71419	0.857096	−	0.515157i	$-0.172266\pi$
$773$	1.70164e13	1.71419	0.857096	+	0.515157i	$0.172266\pi$
$774$	0	0
$775$	−1.04019e13	−1.03575
$776$	1.29266e13	1.27969
$777$	0	0
$778$	2.06298e12	0.201877
$779$	−1.71583e12	−0.166938
$780$	0	0
$781$	6.44872e11	0.0620217
$782$	−3.78463e13	−3.61904
$783$	0	0
$784$	−2.35550e12	−0.222670
$785$	1.06813e13	1.00394
$786$	0	0
$787$	4.72894e12	0.439418	0.219709	−	0.975565i	$-0.429489\pi$
$787$	4.72894e12	0.439418	0.219709	+	0.975565i	$0.429489\pi$
$788$	−1.55627e13	−1.43786
$789$	0	0
$790$	−1.43041e13	−1.30659
$791$	1.77763e12	0.161453
$792$	0	0
$793$	2.34385e12	0.210475
$794$	−1.37156e13	−1.22468
$795$	0	0
$796$	−2.73663e13	−2.41606
$797$	6.55471e12	0.575428	0.287714	−	0.957716i	$-0.407105\pi$
$797$	6.55471e12	0.575428	0.287714	+	0.957716i	$0.407105\pi$
$798$	0	0
$799$	−1.87668e13	−1.62904
$800$	−4.98143e12	−0.429981
$801$	0	0
$802$	−4.32931e12	−0.369517
$803$	1.39754e13	1.18617
$804$	0	0
$805$	1.28756e13	1.08065
$806$	6.44558e13	5.37965
$807$	0	0
$808$	−3.28498e12	−0.271132
$809$	2.70428e11	0.0221964	0.0110982	−	0.999938i	$-0.496467\pi$
$809$	2.70428e11	0.0221964	0.0110982	+	0.999938i	$0.496467\pi$
$810$	0	0
$811$	7.91887e12	0.642790	0.321395	−	0.946945i	$-0.395848\pi$
$811$	7.91887e12	0.642790	0.321395	+	0.946945i	$0.395848\pi$
$812$	7.78818e12	0.628686
$813$	0	0
$814$	−3.46440e12	−0.276579
$815$	−1.75361e13	−1.39227
$816$	0	0
$817$	2.70648e11	0.0212523
$818$	−4.19363e12	−0.327492
$819$	0	0
$820$	−2.12579e13	−1.64194
$821$	1.06544e13	0.818438	0.409219	−	0.912436i	$-0.365801\pi$
$821$	1.06544e13	0.818438	0.409219	+	0.912436i	$0.365801\pi$
$822$	0	0
$823$	−1.91289e13	−1.45342	−0.726711	−	0.686944i	$-0.758952\pi$
$823$	−1.91289e13	−1.45342	−0.726711	+	0.686944i	$0.758952\pi$
$824$	1.99250e13	1.50566
$825$	0	0
$826$	1.60038e13	1.19622
$827$	1.54130e13	1.14581	0.572905	−	0.819622i	$-0.305816\pi$
$827$	1.54130e13	1.14581	0.572905	+	0.819622i	$0.305816\pi$
$828$	0	0
$829$	1.56528e13	1.15105	0.575527	−	0.817783i	$-0.304797\pi$
$829$	1.56528e13	1.15105	0.575527	+	0.817783i	$0.304797\pi$
$830$	1.65748e13	1.21227
$831$	0	0
$832$	3.99521e13	2.89058
$833$	−1.36928e13	−0.985347
$834$	0	0
$835$	1.96690e13	1.40021
$836$	5.08942e12	0.360363
$837$	0	0
$838$	−1.37231e13	−0.961287
$839$	−1.84616e13	−1.28630	−0.643148	−	0.765742i	$-0.722372\pi$
$839$	−1.84616e13	−1.28630	−0.643148	+	0.765742i	$0.722372\pi$
$840$	0	0
$841$	−9.19442e12	−0.633786
$842$	−2.97302e13	−2.03842
$843$	0	0
$844$	2.66466e13	1.80759
$845$	−5.09801e13	−3.43989
$846$	0	0
$847$	−1.70097e12	−0.113559
$848$	6.35894e12	0.422283
$849$	0	0
$850$	2.35350e13	1.54643
$851$	−4.08475e12	−0.266982
$852$	0	0
$853$	−1.29767e13	−0.839254	−0.419627	−	0.907697i	$-0.637839\pi$
$853$	−1.29767e13	−0.839254	−0.419627	+	0.907697i	$0.637839\pi$
$854$	−1.66556e12	−0.107152
$855$	0	0
$856$	1.46994e13	0.935768
$857$	8.52387e12	0.539788	0.269894	−	0.962890i	$-0.413011\pi$
$857$	8.52387e12	0.539788	0.269894	+	0.962890i	$0.413011\pi$
$858$	0	0
$859$	−2.33722e13	−1.46464	−0.732318	−	0.680962i	$-0.761562\pi$
$859$	−2.33722e13	−1.46464	−0.732318	+	0.680962i	$0.761562\pi$
$860$	3.35314e12	0.209030
$861$	0	0
$862$	5.05507e13	3.11849
$863$	3.99924e12	0.245430	0.122715	−	0.992442i	$-0.460840\pi$
$863$	3.99924e12	0.245430	0.122715	+	0.992442i	$0.460840\pi$
$864$	0	0
$865$	2.01527e13	1.22394
$866$	1.10011e13	0.664669
$867$	0	0
$868$	−2.92166e13	−1.74699
$869$	−9.34331e12	−0.555791
$870$	0	0
$871$	−3.87406e13	−2.28078
$872$	−3.32263e12	−0.194607
$873$	0	0
$874$	9.40733e12	0.545337
$875$	4.99272e12	0.287939
$876$	0	0
$877$	5.99001e12	0.341924	0.170962	−	0.985278i	$-0.445313\pi$
$877$	5.99001e12	0.341924	0.170962	+	0.985278i	$0.445313\pi$
$878$	−1.05260e13	−0.597776
$879$	0	0
$880$	6.93805e12	0.390001
$881$	1.24937e13	0.698712	0.349356	−	0.936990i	$-0.386400\pi$
$881$	1.24937e13	0.698712	0.349356	+	0.936990i	$0.386400\pi$
$882$	0	0
$883$	−2.61286e13	−1.44642	−0.723208	−	0.690630i	$-0.757333\pi$
$883$	−2.61286e13	−1.44642	−0.723208	+	0.690630i	$0.757333\pi$
$884$	−9.30252e13	−5.12349
$885$	0	0
$886$	−4.74379e13	−2.58627
$887$	−7.12923e12	−0.386711	−0.193355	−	0.981129i	$-0.561937\pi$
$887$	−7.12923e12	−0.386711	−0.193355	+	0.981129i	$0.561937\pi$
$888$	0	0
$889$	−2.11699e13	−1.13674
$890$	8.09640e12	0.432550
$891$	0	0
$892$	−4.37425e13	−2.31346
$893$	4.66481e12	0.245472
$894$	0	0
$895$	−1.71107e13	−0.891383
$896$	−2.04473e13	−1.05986
$897$	0	0
$898$	2.92671e12	0.150189
$899$	−1.99302e13	−1.01764
$900$	0	0
$901$	3.69652e13	1.86867
$902$	−2.17681e13	−1.09494
$903$	0	0
$904$	6.95631e12	0.346435
$905$	7.74425e12	0.383761
$906$	0	0
$907$	2.43209e13	1.19329	0.596645	−	0.802505i	$-0.296500\pi$
$907$	2.43209e13	1.19329	0.596645	+	0.802505i	$0.296500\pi$
$908$	3.72303e13	1.81765
$909$	0	0
$910$	4.96142e13	2.39839
$911$	−2.47192e13	−1.18906	−0.594528	−	0.804075i	$-0.702661\pi$
$911$	−2.47192e13	−1.18906	−0.594528	+	0.804075i	$0.702661\pi$
$912$	0	0
$913$	1.08265e13	0.515669
$914$	−2.68978e13	−1.27485
$915$	0	0
$916$	1.09871e13	0.515650
$917$	3.57888e12	0.167142
$918$	0	0
$919$	2.68306e13	1.24082	0.620411	−	0.784276i	$-0.286966\pi$
$919$	2.68306e13	1.24082	0.620411	+	0.784276i	$0.286966\pi$
$920$	5.03855e13	2.31878
$921$	0	0
$922$	6.53862e12	0.297987
$923$	2.92987e12	0.132874
$924$	0	0
$925$	2.54013e12	0.114082
$926$	5.00324e13	2.23616
$927$	0	0
$928$	−9.54446e12	−0.422460
$929$	−1.69995e13	−0.748801	−0.374401	−	0.927267i	$-0.622151\pi$
$929$	−1.69995e13	−0.748801	−0.374401	+	0.927267i	$0.622151\pi$
$930$	0	0
$931$	3.40357e12	0.148478
$932$	6.44587e12	0.279840
$933$	0	0
$934$	4.24430e12	0.182493
$935$	4.03316e13	1.72581
$936$	0	0
$937$	−1.71842e13	−0.728285	−0.364143	−	0.931343i	$-0.618638\pi$
$937$	−1.71842e13	−0.728285	−0.364143	+	0.931343i	$0.618638\pi$
$938$	2.75293e13	1.16113
$939$	0	0
$940$	5.77937e13	2.41438
$941$	−1.71372e13	−0.712504	−0.356252	−	0.934390i	$-0.615945\pi$
$941$	−1.71372e13	−0.712504	−0.356252	+	0.934390i	$0.615945\pi$
$942$	0	0
$943$	−2.56660e13	−1.05695
$944$	1.01678e13	0.416730
$945$	0	0
$946$	3.43362e12	0.139393
$947$	−3.70024e13	−1.49505	−0.747523	−	0.664236i	$-0.768757\pi$
$947$	−3.70024e13	−1.49505	−0.747523	+	0.664236i	$0.768757\pi$
$948$	0	0
$949$	6.34951e13	2.54122
$950$	−5.85002e12	−0.233024
$951$	0	0
$952$	2.85774e13	1.12760
$953$	−4.41880e13	−1.73535	−0.867673	−	0.497136i	$-0.834385\pi$
$953$	−4.41880e13	−1.73535	−0.867673	+	0.497136i	$0.834385\pi$
$954$	0	0
$955$	2.36515e12	0.0920116
$956$	3.48588e12	0.134975
$957$	0	0
$958$	−1.54047e13	−0.590891
$959$	1.45661e13	0.556109
$960$	0	0
$961$	4.83267e13	1.82781
$962$	−1.57399e13	−0.592537
$963$	0	0
$964$	−7.60052e13	−2.83463
$965$	5.85099e13	2.17198
$966$	0	0
$967$	−1.34914e13	−0.496179	−0.248090	−	0.968737i	$-0.579803\pi$
$967$	−1.34914e13	−0.496179	−0.248090	+	0.968737i	$0.579803\pi$
$968$	−6.65632e12	−0.243666
$969$	0	0
$970$	−5.88993e13	−2.13618
$971$	−8.24021e12	−0.297476	−0.148738	−	0.988877i	$-0.547521\pi$
$971$	−8.24021e12	−0.297476	−0.148738	+	0.988877i	$0.547521\pi$
$972$	0	0
$973$	−1.10297e13	−0.394509
$974$	8.84624e13	3.14951
$975$	0	0
$976$	−1.05819e12	−0.0373286
$977$	−2.85028e12	−0.100083	−0.0500416	−	0.998747i	$-0.515935\pi$
$977$	−2.85028e12	−0.100083	−0.0500416	+	0.998747i	$0.515935\pi$
$978$	0	0
$979$	5.28849e12	0.183996
$980$	4.21678e13	1.46037
$981$	0	0
$982$	6.00979e13	2.06233
$983$	−5.75121e13	−1.96457	−0.982287	−	0.187384i	$-0.939999\pi$
$983$	−5.75121e13	−1.96457	−0.982287	+	0.187384i	$0.939999\pi$
$984$	0	0
$985$	3.06552e13	1.03763
$986$	4.50932e13	1.51938
$987$	0	0
$988$	2.31230e13	0.772035
$989$	4.04845e12	0.134557
$990$	0	0
$991$	2.76543e13	0.910818	0.455409	−	0.890282i	$-0.349493\pi$
$991$	2.76543e13	0.910818	0.455409	+	0.890282i	$0.349493\pi$
$992$	3.58052e13	1.17393
$993$	0	0
$994$	−2.08199e12	−0.0676455
$995$	5.39059e13	1.74354
$996$	0	0
$997$	−2.64166e13	−0.846736	−0.423368	−	0.905958i	$-0.639152\pi$
$997$	−2.64166e13	−0.846736	−0.423368	+	0.905958i	$0.639152\pi$
$998$	4.03034e13	1.28604
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	81.10.a.b.1.4	yes	4
3.2	odd	2		81.10.a.a.1.1	✓	4
9.2	odd	6		81.10.c.k.28.4		8
9.4	even	3		81.10.c.i.55.1		8
9.5	odd	6		81.10.c.k.55.4		8
9.7	even	3		81.10.c.i.28.1		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
81.10.a.a.1.1	✓	4	3.2	odd	2
81.10.a.b.1.4	yes	4	1.1	even	1	trivial
81.10.c.i.28.1		8	9.7	even	3
81.10.c.i.55.1		8	9.4	even	3
81.10.c.k.28.4		8	9.2	odd	6
81.10.c.k.55.4		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)

\(f(q)\)	\(=\)	\(q+37.6018 q^{2} +901.897 q^{4} -1776.55 q^{5} +3746.45 q^{7} +14660.8 q^{8} -66801.3 q^{10} -43634.0 q^{11} -198244. q^{13} +140873. q^{14} +89502.6 q^{16} +520288. q^{17} -129326. q^{19} -1.60226e6 q^{20} -1.64072e6 q^{22} -1.93451e6 q^{23} +1.20299e6 q^{25} -7.45433e6 q^{26} +3.37891e6 q^{28} +2.30493e6 q^{29} -8.64675e6 q^{31} -4.14088e6 q^{32} +1.95638e7 q^{34} -6.65575e6 q^{35} +2.11152e6 q^{37} -4.86290e6 q^{38} -2.60456e7 q^{40} +1.32674e7 q^{41} -2.09275e6 q^{43} -3.93534e7 q^{44} -7.27411e7 q^{46} -3.60701e7 q^{47} -2.63177e7 q^{49} +4.52346e7 q^{50} -1.78796e8 q^{52} +7.10476e7 q^{53} +7.75178e7 q^{55} +5.49261e7 q^{56} +8.66697e7 q^{58} +1.13604e8 q^{59} -1.18231e7 q^{61} -3.25134e8 q^{62} -2.01530e8 q^{64} +3.52189e8 q^{65} +1.95419e8 q^{67} +4.69246e8 q^{68} -2.50268e8 q^{70} -1.47791e7 q^{71} -3.20288e8 q^{73} +7.93968e7 q^{74} -1.16639e8 q^{76} -1.63473e8 q^{77} +2.14129e8 q^{79} -1.59005e8 q^{80} +4.98880e8 q^{82} -2.48121e8 q^{83} -9.24315e8 q^{85} -7.86914e7 q^{86} -6.39711e8 q^{88} -1.21201e8 q^{89} -7.42712e8 q^{91} -1.74473e9 q^{92} -1.35630e9 q^{94} +2.29754e8 q^{95} +8.81709e8 q^{97} -9.89593e8 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 33 q^{2} + 853 q^{4} + 570 q^{5} - 3238 q^{7} + 4791 q^{8} - 9723 q^{10} - 96690 q^{11} - 141118 q^{13} + 3036 q^{14} - 244463 q^{16} + 285156 q^{17} - 465166 q^{19} - 1041711 q^{20} - 2244480 q^{22}+ \cdots + 735501321 q^{98}+O(q^{100}) \) 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

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