LMFDB - Embedded newform 81.12.a.d.1.7

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [81,12,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, 12, 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, 12); N := Newforms(S);

Level:	$ N $	$=$	$ 81 = 3^{4} $
Weight:	$ k $	$=$	$ 12 $
Character orbit:	$[\chi]$	$=$	81.a (trivial)

Newform invariants

comment:select newform

sage:traces = [10,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(2)] == traces)

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

Self dual:	yes
Analytic conductor:	$62.2357976253$
Analytic rank:	$1$
Dimension:	$10$
Coefficient field:	$\mathbb{Q}[x]/(x^{10} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{10} - 4 x^{9} - 4167 x^{8} - 2152 x^{7} + 5690320 x^{6} + 20355744 x^{5} - 2749862760 x^{4} + \cdots - 910577483568 $ x^10 - 4x^9 - 4167x^8 - 2152x^7 + 5690320x^6 + 20355744x^5 - 2749862760x^4 - 9719638560x^3 + 432889632432x^2 + 840182162112*x - 910577483568
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{12}\cdot 3^{30} $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.7
Root			$16.6610$ 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)

$f(q)$	$=$	\(q+23.1974 q^{2} -1509.88 q^{4} +6661.33 q^{5} -45287.0 q^{7} -82533.6 q^{8} +154526. q^{10} +412139. q^{11} +2.42329e6 q^{13} -1.05054e6 q^{14} +1.17767e6 q^{16} -2.28802e6 q^{17} -3.36923e6 q^{19} -1.00578e7 q^{20} +9.56055e6 q^{22} -5.49956e7 q^{23} -4.45478e6 q^{25} +5.62139e7 q^{26} +6.83779e7 q^{28} +6.38654e7 q^{29} -7.68718e7 q^{31} +1.96348e8 q^{32} -5.30761e7 q^{34} -3.01671e8 q^{35} +3.81919e8 q^{37} -7.81573e7 q^{38} -5.49783e8 q^{40} -6.48281e8 q^{41} -6.82862e8 q^{43} -6.22281e8 q^{44} -1.27576e9 q^{46} -1.02700e9 q^{47} +7.35813e7 q^{49} -1.03339e8 q^{50} -3.65887e9 q^{52} -5.96226e8 q^{53} +2.74539e9 q^{55} +3.73769e9 q^{56} +1.48151e9 q^{58} -9.31757e9 q^{59} +5.51863e9 q^{61} -1.78323e9 q^{62} +2.14288e9 q^{64} +1.61423e10 q^{65} -1.99744e10 q^{67} +3.45464e9 q^{68} -6.99799e9 q^{70} -1.10915e9 q^{71} -3.51189e10 q^{73} +8.85951e9 q^{74} +5.08713e9 q^{76} -1.86645e10 q^{77} -2.72212e10 q^{79} +7.84486e9 q^{80} -1.50384e10 q^{82} +5.15112e10 q^{83} -1.52413e10 q^{85} -1.58406e10 q^{86} -3.40153e10 q^{88} -9.43745e10 q^{89} -1.09743e11 q^{91} +8.30368e10 q^{92} -2.38237e10 q^{94} -2.24435e10 q^{95} +2.78835e10 q^{97} +1.70689e9 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 10 q + 5110 q^{4} - 11836 q^{7} - 117642 q^{10} + 27110 q^{13} - 5117294 q^{16} - 24650908 q^{19} - 27880836 q^{22} - 43357796 q^{25} - 177204868 q^{28} - 56217184 q^{31} + 302628942 q^{34} - 636530326 q^{37}+ \cdots - 413983885420 q^{97}+O(q^{100}) $ 10 * q + 5110 * q^4 - 11836 * q^7 - 117642 * q^10 + 27110 * q^13 - 5117294 * q^16 - 24650908 * q^19 - 27880836 * q^22 - 43357796 * q^25 - 177204868 * q^28 - 56217184 * q^31 + 302628942 * q^34 - 636530326 * q^37 + 405215346 * q^40 + 1101619412 * q^43 - 5426695164 * q^46 - 1944391710 * q^49 - 7927944502 * q^52 - 17670840708 * q^55 - 23512516710 * q^58 - 25504032214 * q^61 - 43436567978 * q^64 - 59699336020 * q^67 - 92786466516 * q^70 - 58729185430 * q^73 - 139282283236 * q^76 - 150636181996 * q^79 - 152979717864 * q^82 - 184281748362 * q^85 - 371435708028 * q^88 - 277231971332 * q^91 - 500149068360 * q^94 - 413983885420 * q^97

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 $

$2$	23.1974	0.512595	0.256297	−	0.966598i	$-0.417497\pi$
$2$	23.1974	0.512595	0.256297	+	0.966598i	$0.417497\pi$
$3$	0	0
$3$	0	0
$4$	−1509.88	−0.737246
$5$	6661.33	0.953292	0.476646	−	0.879095i	$-0.341852\pi$
$5$	6661.33	0.953292	0.476646	+	0.879095i	$0.341852\pi$
$6$	0	0
$7$	−45287.0	−1.01844	−0.509218	−	0.860637i	$-0.670065\pi$
$7$	−45287.0	−1.01844	−0.509218	+	0.860637i	$0.670065\pi$
$8$	−82533.6	−0.890504
$9$	0	0
$10$	154526.	0.488653
$11$	412139.	0.771585	0.385793	−	0.922585i	$-0.373928\pi$
$11$	412139.	0.771585	0.385793	+	0.922585i	$0.373928\pi$
$12$	0	0
$13$	2.42329e6	1.81016	0.905078	−	0.425245i	$-0.139812\pi$
$13$	2.42329e6	1.81016	0.905078	+	0.425245i	$0.139812\pi$
$14$	−1.05054e6	−0.522045
$15$	0	0
$16$	1.17767e6	0.280779
$17$	−2.28802e6	−0.390833	−0.195416	−	0.980720i	$-0.562606\pi$
$17$	−2.28802e6	−0.390833	−0.195416	+	0.980720i	$0.562606\pi$
$18$	0	0
$19$	−3.36923e6	−0.312166	−0.156083	−	0.987744i	$-0.549887\pi$
$19$	−3.36923e6	−0.312166	−0.156083	+	0.987744i	$0.549887\pi$
$20$	−1.00578e7	−0.702811
$21$	0	0
$22$	9.56055e6	0.395511
$23$	−5.49956e7	−1.78166	−0.890830	−	0.454336i	$-0.849877\pi$
$23$	−5.49956e7	−1.78166	−0.890830	+	0.454336i	$0.849877\pi$
$24$	0	0
$25$	−4.45478e6	−0.0912340
$26$	5.62139e7	0.927877
$27$	0	0
$28$	6.83779e7	0.750839
$29$	6.38654e7	0.578198	0.289099	−	0.957299i	$-0.406644\pi$
$29$	6.38654e7	0.578198	0.289099	+	0.957299i	$0.406644\pi$
$30$	0	0
$31$	−7.68718e7	−0.482256	−0.241128	−	0.970493i	$-0.577517\pi$
$31$	−7.68718e7	−0.482256	−0.241128	+	0.970493i	$0.577517\pi$
$32$	1.96348e8	1.03443
$33$	0	0
$34$	−5.30761e7	−0.200339
$35$	−3.01671e8	−0.970867
$36$	0	0
$37$	3.81919e8	0.905443	0.452722	−	0.891652i	$-0.350453\pi$
$37$	3.81919e8	0.905443	0.452722	+	0.891652i	$0.350453\pi$
$38$	−7.81573e7	−0.160015
$39$	0	0
$40$	−5.49783e8	−0.848910
$41$	−6.48281e8	−0.873880	−0.436940	−	0.899491i	$-0.643938\pi$
$41$	−6.48281e8	−0.873880	−0.436940	+	0.899491i	$0.643938\pi$
$42$	0	0
$43$	−6.82862e8	−0.708364	−0.354182	−	0.935177i	$-0.615241\pi$
$43$	−6.82862e8	−0.708364	−0.354182	+	0.935177i	$0.615241\pi$
$44$	−6.22281e8	−0.568849
$45$	0	0
$46$	−1.27576e9	−0.913270
$47$	−1.02700e9	−0.653177	−0.326588	−	0.945167i	$-0.605899\pi$
$47$	−1.02700e9	−0.653177	−0.326588	+	0.945167i	$0.605899\pi$
$48$	0	0
$49$	7.35813e7	0.0372125
$50$	−1.03339e8	−0.0467661
$51$	0	0
$52$	−3.65887e9	−1.33453
$53$	−5.96226e8	−0.195837	−0.0979183	−	0.995194i	$-0.531218\pi$
$53$	−5.96226e8	−0.195837	−0.0979183	+	0.995194i	$0.531218\pi$
$54$	0	0
$55$	2.74539e9	0.735546
$56$	3.73769e9	0.906921
$57$	0	0
$58$	1.48151e9	0.296381
$59$	−9.31757e9	−1.69675	−0.848373	−	0.529399i	$-0.822417\pi$
$59$	−9.31757e9	−1.69675	−0.848373	+	0.529399i	$0.822417\pi$
$60$	0	0
$61$	5.51863e9	0.836598	0.418299	−	0.908309i	$-0.362626\pi$
$61$	5.51863e9	0.836598	0.418299	+	0.908309i	$0.362626\pi$
$62$	−1.78323e9	−0.247202
$63$	0	0
$64$	2.14288e9	0.249464
$65$	1.61423e10	1.72561
$66$	0	0
$67$	−1.99744e10	−1.80744	−0.903718	−	0.428128i	$-0.859173\pi$
$67$	−1.99744e10	−1.80744	−0.903718	+	0.428128i	$0.859173\pi$
$68$	3.45464e9	0.288140
$69$	0	0
$70$	−6.99799e9	−0.497662
$71$	−1.10915e9	−0.0729577	−0.0364788	−	0.999334i	$-0.511614\pi$
$71$	−1.10915e9	−0.0729577	−0.0364788	+	0.999334i	$0.511614\pi$
$72$	0	0
$73$	−3.51189e10	−1.98274	−0.991370	−	0.131093i	$-0.958151\pi$
$73$	−3.51189e10	−1.98274	−0.991370	+	0.131093i	$0.958151\pi$
$74$	8.85951e9	0.464126
$75$	0	0
$76$	5.08713e9	0.230143
$77$	−1.86645e10	−0.785811
$78$	0	0
$79$	−2.72212e10	−0.995311	−0.497656	−	0.867375i	$-0.665806\pi$
$79$	−2.72212e10	−0.995311	−0.497656	+	0.867375i	$0.665806\pi$
$80$	7.84486e9	0.267664
$81$	0	0
$82$	−1.50384e10	−0.447947
$83$	5.15112e10	1.43540	0.717699	−	0.696354i	$-0.245195\pi$
$83$	5.15112e10	1.43540	0.717699	+	0.696354i	$0.245195\pi$
$84$	0	0
$85$	−1.52413e10	−0.372578
$86$	−1.58406e10	−0.363104
$87$	0	0
$88$	−3.40153e10	−0.687100
$89$	−9.43745e10	−1.79147	−0.895736	−	0.444587i	$-0.853350\pi$
$89$	−9.43745e10	−1.79147	−0.895736	+	0.444587i	$0.853350\pi$
$90$	0	0
$91$	−1.09743e11	−1.84353
$92$	8.30368e10	1.31352
$93$	0	0
$94$	−2.38237e10	−0.334815
$95$	−2.24435e10	−0.297585
$96$	0	0
$97$	2.78835e10	0.329688	0.164844	−	0.986320i	$-0.447288\pi$
$97$	2.78835e10	0.329688	0.164844	+	0.986320i	$0.447288\pi$
$98$	1.70689e9	0.0190749
$99$	0	0
$100$	6.72619e9	0.0672619
$101$	1.28898e11	1.22033	0.610165	−	0.792274i	$-0.291103\pi$
$101$	1.28898e11	1.22033	0.610165	+	0.792274i	$0.291103\pi$
$102$	0	0
$103$	−1.19799e11	−1.01824	−0.509120	−	0.860696i	$-0.670029\pi$
$103$	−1.19799e11	−1.01824	−0.509120	+	0.860696i	$0.670029\pi$
$104$	−2.00002e11	−1.61195
$105$	0	0
$106$	−1.38309e10	−0.100385
$107$	1.47222e11	1.01475	0.507377	−	0.861724i	$-0.330615\pi$
$107$	1.47222e11	1.01475	0.507377	+	0.861724i	$0.330615\pi$
$108$	0	0
$109$	1.88136e11	1.17119	0.585594	−	0.810605i	$-0.300861\pi$
$109$	1.88136e11	1.17119	0.585594	+	0.810605i	$0.300861\pi$
$110$	6.36860e10	0.377037
$111$	0	0
$112$	−5.33332e10	−0.285955
$113$	−7.81419e10	−0.398981	−0.199491	−	0.979900i	$-0.563929\pi$
$113$	−7.81419e10	−0.398981	−0.199491	+	0.979900i	$0.563929\pi$
$114$	0	0
$115$	−3.66344e11	−1.69844
$116$	−9.64292e10	−0.426275
$117$	0	0
$118$	−2.16143e11	−0.869743
$119$	1.03617e11	0.398038
$120$	0	0
$121$	−1.15453e11	−0.404656
$122$	1.28018e11	0.428836
$123$	0	0
$124$	1.16067e11	0.355542
$125$	−3.54935e11	−1.04026
$126$	0	0
$127$	4.90730e10	0.131802	0.0659010	−	0.997826i	$-0.479008\pi$
$127$	4.90730e10	0.131802	0.0659010	+	0.997826i	$0.479008\pi$
$128$	−3.52411e11	−0.906555
$129$	0	0
$130$	3.74459e11	0.884538
$131$	−2.61853e11	−0.593015	−0.296507	−	0.955031i	$-0.595822\pi$
$131$	−2.61853e11	−0.593015	−0.296507	+	0.955031i	$0.595822\pi$
$132$	0	0
$133$	1.52582e11	0.317921
$134$	−4.63355e11	−0.926482
$135$	0	0
$136$	1.88838e11	0.348038
$137$	−2.03546e11	−0.360329	−0.180165	−	0.983636i	$-0.557663\pi$
$137$	−2.03546e11	−0.360329	−0.180165	+	0.983636i	$0.557663\pi$
$138$	0	0
$139$	8.83317e10	0.144389	0.0721947	−	0.997391i	$-0.477000\pi$
$139$	8.83317e10	0.144389	0.0721947	+	0.997391i	$0.477000\pi$
$140$	4.55488e11	0.715769
$141$	0	0
$142$	−2.57295e10	−0.0373977
$143$	9.98730e11	1.39669
$144$	0	0
$145$	4.25429e11	0.551192
$146$	−8.14668e11	−1.01634
$147$	0	0
$148$	−5.76651e11	−0.667535
$149$	1.38554e12	1.54559	0.772796	−	0.634655i	$-0.218858\pi$
$149$	1.38554e12	1.54559	0.772796	+	0.634655i	$0.218858\pi$
$150$	0	0
$151$	1.04439e11	0.108265	0.0541327	−	0.998534i	$-0.482761\pi$
$151$	1.04439e11	0.108265	0.0541327	+	0.998534i	$0.482761\pi$
$152$	2.78074e11	0.277985
$153$	0	0
$154$	−4.32968e11	−0.402802
$155$	−5.12069e11	−0.459731
$156$	0	0
$157$	1.38080e12	1.15527	0.577635	−	0.816295i	$-0.303976\pi$
$157$	1.38080e12	1.15527	0.577635	+	0.816295i	$0.303976\pi$
$158$	−6.31462e11	−0.510192
$159$	0	0
$160$	1.30794e12	0.986114
$161$	2.49058e12	1.81451
$162$	0	0
$163$	1.44400e12	0.982958	0.491479	−	0.870889i	$-0.336456\pi$
$163$	1.44400e12	0.982958	0.491479	+	0.870889i	$0.336456\pi$
$164$	9.78827e11	0.644265
$165$	0	0
$166$	1.19493e12	0.735777
$167$	1.37826e12	0.821091	0.410546	−	0.911840i	$-0.365338\pi$
$167$	1.37826e12	0.821091	0.410546	+	0.911840i	$0.365338\pi$
$168$	0	0
$169$	4.08015e12	2.27667
$170$	−3.53558e11	−0.190981
$171$	0	0
$172$	1.03104e12	0.522239
$173$	−3.73097e12	−1.83049	−0.915247	−	0.402892i	$-0.868005\pi$
$173$	−3.73097e12	−1.83049	−0.915247	+	0.402892i	$0.868005\pi$
$174$	0	0
$175$	2.01744e11	0.0929160
$176$	4.85365e11	0.216645
$177$	0	0
$178$	−2.18924e12	−0.918299
$179$	−3.24313e12	−1.31908	−0.659542	−	0.751668i	$-0.729250\pi$
$179$	−3.24313e12	−1.31908	−0.659542	+	0.751668i	$0.729250\pi$
$180$	0	0
$181$	−2.12149e12	−0.811724	−0.405862	−	0.913934i	$-0.633029\pi$
$181$	−2.12149e12	−0.811724	−0.405862	+	0.913934i	$0.633029\pi$
$182$	−2.54576e12	−0.944983
$183$	0	0
$184$	4.53899e12	1.58658
$185$	2.54409e12	0.863152
$186$	0	0
$187$	−9.42982e11	−0.301561
$188$	1.55064e12	0.481552
$189$	0	0
$190$	−5.20631e11	−0.152541
$191$	8.01543e11	0.228162	0.114081	−	0.993471i	$-0.463608\pi$
$191$	8.01543e11	0.228162	0.114081	+	0.993471i	$0.463608\pi$
$192$	0	0
$193$	−5.29295e12	−1.42276	−0.711382	−	0.702806i	$-0.751930\pi$
$193$	−5.29295e12	−1.42276	−0.711382	+	0.702806i	$0.751930\pi$
$194$	6.46824e11	0.168996
$195$	0	0
$196$	−1.11099e11	−0.0274348
$197$	4.34920e12	1.04435	0.522173	−	0.852839i	$-0.325121\pi$
$197$	4.34920e12	1.04435	0.522173	+	0.852839i	$0.325121\pi$
$198$	0	0
$199$	1.76962e12	0.401966	0.200983	−	0.979595i	$-0.435586\pi$
$199$	1.76962e12	0.401966	0.200983	+	0.979595i	$0.435586\pi$
$200$	3.67669e11	0.0812442
$201$	0	0
$202$	2.99009e12	0.625535
$203$	−2.89227e12	−0.588858
$204$	0	0
$205$	−4.31841e12	−0.833063
$206$	−2.77903e12	−0.521944
$207$	0	0
$208$	2.85384e12	0.508254
$209$	−1.38859e12	−0.240863
$210$	0	0
$211$	−9.86755e12	−1.62426	−0.812131	−	0.583476i	$-0.801692\pi$
$211$	−9.86755e12	−1.62426	−0.812131	+	0.583476i	$0.801692\pi$
$212$	9.00231e11	0.144380
$213$	0	0
$214$	3.41516e12	0.520157
$215$	−4.54877e12	−0.675278
$216$	0	0
$217$	3.48129e12	0.491147
$218$	4.36427e12	0.600345
$219$	0	0
$220$	−4.14522e12	−0.542279
$221$	−5.54453e12	−0.707468
$222$	0	0
$223$	−2.18303e11	−0.0265083	−0.0132542	−	0.999912i	$-0.504219\pi$
$223$	−2.18303e11	−0.0265083	−0.0132542	+	0.999912i	$0.504219\pi$
$224$	−8.89199e12	−1.05350
$225$	0	0
$226$	−1.81269e12	−0.204516
$227$	1.06005e13	1.16730	0.583651	−	0.812005i	$-0.301624\pi$
$227$	1.06005e13	1.16730	0.583651	+	0.812005i	$0.301624\pi$
$228$	0	0
$229$	8.18469e12	0.858829	0.429415	−	0.903107i	$-0.358720\pi$
$229$	8.18469e12	0.858829	0.429415	+	0.903107i	$0.358720\pi$
$230$	−8.49823e12	−0.870613
$231$	0	0
$232$	−5.27104e12	−0.514888
$233$	9.83926e12	0.938652	0.469326	−	0.883025i	$-0.344497\pi$
$233$	9.83926e12	0.938652	0.469326	+	0.883025i	$0.344497\pi$
$234$	0	0
$235$	−6.84117e12	−0.622668
$236$	1.40684e13	1.25092
$237$	0	0
$238$	2.40366e12	0.204032
$239$	−2.05688e13	−1.70616	−0.853081	−	0.521778i	$-0.825269\pi$
$239$	−2.05688e13	−1.70616	−0.853081	+	0.521778i	$0.825269\pi$
$240$	0	0
$241$	−9.35141e12	−0.740940	−0.370470	−	0.928844i	$-0.620803\pi$
$241$	−9.35141e12	−0.740940	−0.370470	+	0.928844i	$0.620803\pi$
$242$	−2.67821e12	−0.207425
$243$	0	0
$244$	−8.33247e12	−0.616779
$245$	4.90149e11	0.0354744
$246$	0	0
$247$	−8.16460e12	−0.565069
$248$	6.34451e12	0.429451
$249$	0	0
$250$	−8.23357e12	−0.533234
$251$	2.07333e12	0.131360	0.0656799	−	0.997841i	$-0.479078\pi$
$251$	2.07333e12	0.131360	0.0656799	+	0.997841i	$0.479078\pi$
$252$	0	0
$253$	−2.26658e13	−1.37470
$254$	1.13837e12	0.0675611
$255$	0	0
$256$	−1.25636e13	−0.714160
$257$	9.25340e12	0.514836	0.257418	−	0.966300i	$-0.417128\pi$
$257$	9.25340e12	0.514836	0.257418	+	0.966300i	$0.417128\pi$
$258$	0	0
$259$	−1.72959e13	−0.922136
$260$	−2.43730e13	−1.27220
$261$	0	0
$262$	−6.07431e12	−0.303976
$263$	1.55658e13	0.762810	0.381405	−	0.924408i	$-0.375440\pi$
$263$	1.55658e13	0.762810	0.381405	+	0.924408i	$0.375440\pi$
$264$	0	0
$265$	−3.97166e12	−0.186689
$266$	3.53950e12	0.162965
$267$	0	0
$268$	3.01590e13	1.33253
$269$	7.69864e12	0.333255	0.166627	−	0.986020i	$-0.446712\pi$
$269$	7.69864e12	0.333255	0.166627	+	0.986020i	$0.446712\pi$
$270$	0	0
$271$	−2.73668e12	−0.113735	−0.0568673	−	0.998382i	$-0.518111\pi$
$271$	−2.73668e12	−0.113735	−0.0568673	+	0.998382i	$0.518111\pi$
$272$	−2.69454e12	−0.109738
$273$	0	0
$274$	−4.72174e12	−0.184703
$275$	−1.83599e12	−0.0703948
$276$	0	0
$277$	1.35258e13	0.498336	0.249168	−	0.968460i	$-0.419843\pi$
$277$	1.35258e13	0.498336	0.249168	+	0.968460i	$0.419843\pi$
$278$	2.04907e12	0.0740133
$279$	0	0
$280$	2.48980e13	0.864561
$281$	5.03006e13	1.71273	0.856363	−	0.516374i	$-0.172719\pi$
$281$	5.03006e13	1.71273	0.856363	+	0.516374i	$0.172719\pi$
$282$	0	0
$283$	−2.60476e13	−0.852987	−0.426493	−	0.904491i	$-0.640251\pi$
$283$	−2.60476e13	−0.852987	−0.426493	+	0.904491i	$0.640251\pi$
$284$	1.67469e12	0.0537878
$285$	0	0
$286$	2.31679e13	0.715936
$287$	2.93587e13	0.889992
$288$	0	0
$289$	−2.90369e13	−0.847250
$290$	9.86884e12	0.282538
$291$	0	0
$292$	5.30254e13	1.46177
$293$	2.12642e13	0.575278	0.287639	−	0.957739i	$-0.407130\pi$
$293$	2.12642e13	0.575278	0.287639	+	0.957739i	$0.407130\pi$
$294$	0	0
$295$	−6.20675e13	−1.61749
$296$	−3.15211e13	−0.806300
$297$	0	0
$298$	3.21409e13	0.792262
$299$	−1.33270e14	−3.22508
$300$	0	0
$301$	3.09247e13	0.721423
$302$	2.42271e12	0.0554963
$303$	0	0
$304$	−3.96784e12	−0.0876496
$305$	3.67614e13	0.797523
$306$	0	0
$307$	2.64776e13	0.554138	0.277069	−	0.960850i	$-0.410637\pi$
$307$	2.64776e13	0.554138	0.277069	+	0.960850i	$0.410637\pi$
$308$	2.81812e13	0.579336
$309$	0	0
$310$	−1.18787e13	−0.235656
$311$	−6.02016e13	−1.17335	−0.586673	−	0.809824i	$-0.699563\pi$
$311$	−6.02016e13	−1.17335	−0.586673	+	0.809824i	$0.699563\pi$
$312$	0	0
$313$	5.56934e13	1.04788	0.523938	−	0.851756i	$-0.324462\pi$
$313$	5.56934e13	1.04788	0.523938	+	0.851756i	$0.324462\pi$
$314$	3.20310e13	0.592185
$315$	0	0
$316$	4.11008e13	0.733790
$317$	−1.06838e14	−1.87456	−0.937282	−	0.348571i	$-0.886667\pi$
$317$	−1.06838e14	−1.87456	−0.937282	+	0.348571i	$0.886667\pi$
$318$	0	0
$319$	2.63214e13	0.446129
$320$	1.42745e13	0.237812
$321$	0	0
$322$	5.77751e13	0.930108
$323$	7.70886e12	0.122005
$324$	0	0
$325$	−1.07952e13	−0.165148
$326$	3.34970e13	0.503859
$327$	0	0
$328$	5.35049e13	0.778194
$329$	4.65096e13	0.665219
$330$	0	0
$331$	−2.93553e13	−0.406100	−0.203050	−	0.979168i	$-0.565085\pi$
$331$	−2.93553e13	−0.406100	−0.203050	+	0.979168i	$0.565085\pi$
$332$	−7.77758e13	−1.05824
$333$	0	0
$334$	3.19721e13	0.420887
$335$	−1.33056e14	−1.72301
$336$	0	0
$337$	−1.67726e13	−0.210202	−0.105101	−	0.994462i	$-0.533517\pi$
$337$	−1.67726e13	−0.210202	−0.105101	+	0.994462i	$0.533517\pi$
$338$	9.46489e13	1.16701
$339$	0	0
$340$	2.30125e13	0.274682
$341$	−3.16819e13	−0.372102
$342$	0	0
$343$	8.62148e13	0.980538
$344$	5.63590e13	0.630801
$345$	0	0
$346$	−8.65489e13	−0.938302
$347$	9.80389e13	1.04613	0.523066	−	0.852292i	$-0.324788\pi$
$347$	9.80389e13	1.04613	0.523066	+	0.852292i	$0.324788\pi$
$348$	0	0
$349$	−1.15311e13	−0.119215	−0.0596074	−	0.998222i	$-0.518985\pi$
$349$	−1.15311e13	−0.119215	−0.0596074	+	0.998222i	$0.518985\pi$
$350$	4.67993e12	0.0476283
$351$	0	0
$352$	8.09225e13	0.798151
$353$	−6.16619e13	−0.598765	−0.299382	−	0.954133i	$-0.596781\pi$
$353$	−6.16619e13	−0.598765	−0.299382	+	0.954133i	$0.596781\pi$
$354$	0	0
$355$	−7.38844e12	−0.0695500
$356$	1.42494e14	1.32076
$357$	0	0
$358$	−7.52321e13	−0.676156
$359$	2.07857e14	1.83969	0.919846	−	0.392279i	$-0.128313\pi$
$359$	2.07857e14	1.83969	0.919846	+	0.392279i	$0.128313\pi$
$360$	0	0
$361$	−1.05139e14	−0.902553
$362$	−4.92130e13	−0.416086
$363$	0	0
$364$	1.65699e14	1.35914
$365$	−2.33939e14	−1.89013
$366$	0	0
$367$	1.39044e14	1.09016	0.545080	−	0.838384i	$-0.316499\pi$
$367$	1.39044e14	1.09016	0.545080	+	0.838384i	$0.316499\pi$
$368$	−6.47668e13	−0.500253
$369$	0	0
$370$	5.90162e13	0.442447
$371$	2.70013e13	0.199447
$372$	0	0
$373$	−9.20465e13	−0.660098	−0.330049	−	0.943964i	$-0.607065\pi$
$373$	−9.20465e13	−0.660098	−0.330049	+	0.943964i	$0.607065\pi$
$374$	−2.18747e13	−0.154578
$375$	0	0
$376$	8.47617e13	0.581656
$377$	1.54764e14	1.04663
$378$	0	0
$379$	−2.49260e14	−1.63733	−0.818665	−	0.574271i	$-0.805286\pi$
$379$	−2.49260e14	−1.63733	−0.818665	+	0.574271i	$0.805286\pi$
$380$	3.38871e13	0.219394
$381$	0	0
$382$	1.85937e13	0.116955
$383$	−3.01373e13	−0.186858	−0.0934288	−	0.995626i	$-0.529783\pi$
$383$	−3.01373e13	−0.186858	−0.0934288	+	0.995626i	$0.529783\pi$
$384$	0	0
$385$	−1.24331e14	−0.749107
$386$	−1.22783e14	−0.729302
$387$	0	0
$388$	−4.21008e13	−0.243061
$389$	1.50936e14	0.859150	0.429575	−	0.903031i	$-0.358663\pi$
$389$	1.50936e14	0.859150	0.429575	+	0.903031i	$0.358663\pi$
$390$	0	0
$391$	1.25831e14	0.696331
$392$	−6.07293e12	−0.0331379
$393$	0	0
$394$	1.00890e14	0.535327
$395$	−1.81330e14	−0.948823
$396$	0	0
$397$	−1.32504e14	−0.674341	−0.337171	−	0.941444i	$-0.609470\pi$
$397$	−1.32504e14	−0.674341	−0.337171	+	0.941444i	$0.609470\pi$
$398$	4.10507e13	0.206046
$399$	0	0
$400$	−5.24628e12	−0.0256166
$401$	−1.94718e14	−0.937806	−0.468903	−	0.883250i	$-0.655351\pi$
$401$	−1.94718e14	−0.937806	−0.468903	+	0.883250i	$0.655351\pi$
$402$	0	0
$403$	−1.86282e14	−0.872959
$404$	−1.94620e14	−0.899684
$405$	0	0
$406$	−6.70931e13	−0.301846
$407$	1.57404e14	0.698627
$408$	0	0
$409$	−1.31584e14	−0.568493	−0.284247	−	0.958751i	$-0.591743\pi$
$409$	−1.31584e14	−0.568493	−0.284247	+	0.958751i	$0.591743\pi$
$410$	−1.00176e14	−0.427024
$411$	0	0
$412$	1.80883e14	0.750693
$413$	4.21965e14	1.72803
$414$	0	0
$415$	3.43133e14	1.36835
$416$	4.75806e14	1.87248
$417$	0	0
$418$	−3.22117e13	−0.123465
$419$	1.82316e14	0.689679	0.344840	−	0.938662i	$-0.387933\pi$
$419$	1.82316e14	0.689679	0.344840	+	0.938662i	$0.387933\pi$
$420$	0	0
$421$	−3.56128e14	−1.31236	−0.656182	−	0.754602i	$-0.727830\pi$
$421$	−3.56128e14	−1.31236	−0.656182	+	0.754602i	$0.727830\pi$
$422$	−2.28901e14	−0.832588
$423$	0	0
$424$	4.92087e13	0.174393
$425$	1.01926e13	0.0356572
$426$	0	0
$427$	−2.49922e14	−0.852022
$428$	−2.22287e14	−0.748124
$429$	0	0
$430$	−1.05520e14	−0.346144
$431$	9.50676e13	0.307899	0.153949	−	0.988079i	$-0.450801\pi$
$431$	9.50676e13	0.307899	0.153949	+	0.988079i	$0.450801\pi$
$432$	0	0
$433$	−5.78537e14	−1.82662	−0.913309	−	0.407266i	$-0.866482\pi$
$433$	−5.78537e14	−1.82662	−0.913309	+	0.407266i	$0.866482\pi$
$434$	8.07569e13	0.251760
$435$	0	0
$436$	−2.84063e14	−0.863454
$437$	1.85293e14	0.556174
$438$	0	0
$439$	−2.01434e14	−0.589627	−0.294813	−	0.955555i	$-0.595257\pi$
$439$	−2.01434e14	−0.589627	−0.294813	+	0.955555i	$0.595257\pi$
$440$	−2.26587e14	−0.655007
$441$	0	0
$442$	−1.28619e14	−0.362645
$443$	−1.44356e14	−0.401988	−0.200994	−	0.979592i	$-0.564417\pi$
$443$	−1.44356e14	−0.401988	−0.200994	+	0.979592i	$0.564417\pi$
$444$	0	0
$445$	−6.28660e14	−1.70780
$446$	−5.06406e12	−0.0135880
$447$	0	0
$448$	−9.70446e13	−0.254064
$449$	−4.58290e14	−1.18518	−0.592592	−	0.805503i	$-0.701895\pi$
$449$	−4.58290e14	−1.18518	−0.592592	+	0.805503i	$0.701895\pi$
$450$	0	0
$451$	−2.67182e14	−0.674273
$452$	1.17985e14	0.294147
$453$	0	0
$454$	2.45903e14	0.598353
$455$	−7.31036e14	−1.75742
$456$	0	0
$457$	−2.25248e14	−0.528595	−0.264297	−	0.964441i	$-0.585140\pi$
$457$	−2.25248e14	−0.528595	−0.264297	+	0.964441i	$0.585140\pi$
$458$	1.89863e14	0.440232
$459$	0	0
$460$	5.53136e14	1.25217
$461$	3.41062e13	0.0762918	0.0381459	−	0.999272i	$-0.487855\pi$
$461$	3.41062e13	0.0762918	0.0381459	+	0.999272i	$0.487855\pi$
$462$	0	0
$463$	−1.34765e14	−0.294361	−0.147181	−	0.989110i	$-0.547020\pi$
$463$	−1.34765e14	−0.294361	−0.147181	+	0.989110i	$0.547020\pi$
$464$	7.52125e13	0.162346
$465$	0	0
$466$	2.28245e14	0.481148
$467$	−7.23813e13	−0.150794	−0.0753969	−	0.997154i	$-0.524022\pi$
$467$	−7.23813e13	−0.150794	−0.0753969	+	0.997154i	$0.524022\pi$
$468$	0	0
$469$	9.04581e14	1.84076
$470$	−1.58697e14	−0.319177
$471$	0	0
$472$	7.69013e14	1.51096
$473$	−2.81434e14	−0.546563
$474$	0	0
$475$	1.50092e13	0.0284801
$476$	−1.56450e14	−0.293452
$477$	0	0
$478$	−4.77143e14	−0.874570
$479$	3.38626e14	0.613584	0.306792	−	0.951777i	$-0.400744\pi$
$479$	3.38626e14	0.613584	0.306792	+	0.951777i	$0.400744\pi$
$480$	0	0
$481$	9.25497e14	1.63899
$482$	−2.16928e14	−0.379802
$483$	0	0
$484$	1.74320e14	0.298331
$485$	1.85741e14	0.314289
$486$	0	0
$487$	−4.00482e14	−0.662482	−0.331241	−	0.943546i	$-0.607467\pi$
$487$	−4.00482e14	−0.662482	−0.331241	+	0.943546i	$0.607467\pi$
$488$	−4.55472e14	−0.744994
$489$	0	0
$490$	1.13702e13	0.0181840
$491$	6.70479e14	1.06032	0.530160	−	0.847898i	$-0.322132\pi$
$491$	6.70479e14	1.06032	0.530160	+	0.847898i	$0.322132\pi$
$492$	0	0
$493$	−1.46125e14	−0.225979
$494$	−1.89397e14	−0.289651
$495$	0	0
$496$	−9.05298e13	−0.135407
$497$	5.02302e13	0.0743027
$498$	0	0
$499$	−7.12612e14	−1.03110	−0.515549	−	0.856860i	$-0.672412\pi$
$499$	−7.12612e14	−1.03110	−0.515549	+	0.856860i	$0.672412\pi$
$500$	5.35910e14	0.766932
$501$	0	0
$502$	4.80958e13	0.0673343
$503$	1.06773e15	1.47856	0.739280	−	0.673398i	$-0.235166\pi$
$503$	1.06773e15	1.47856	0.739280	+	0.673398i	$0.235166\pi$
$504$	0	0
$505$	8.58630e14	1.16333
$506$	−5.25789e14	−0.704666
$507$	0	0
$508$	−7.40944e13	−0.0971706
$509$	5.09968e14	0.661599	0.330800	−	0.943701i	$-0.392682\pi$
$509$	5.09968e14	0.661599	0.330800	+	0.943701i	$0.392682\pi$
$510$	0	0
$511$	1.59043e15	2.01929
$512$	4.30294e14	0.540481
$513$	0	0
$514$	2.14655e14	0.263902
$515$	−7.98024e14	−0.970680
$516$	0	0
$517$	−4.23265e14	−0.503982
$518$	−4.01220e14	−0.472682
$519$	0	0
$520$	−1.33228e15	−1.53666
$521$	2.78072e14	0.317358	0.158679	−	0.987330i	$-0.449276\pi$
$521$	2.78072e14	0.317358	0.158679	+	0.987330i	$0.449276\pi$
$522$	0	0
$523$	−5.61857e14	−0.627865	−0.313932	−	0.949445i	$-0.601647\pi$
$523$	−5.61857e14	−0.627865	−0.313932	+	0.949445i	$0.601647\pi$
$524$	3.95367e14	0.437198
$525$	0	0
$526$	3.61087e14	0.391012
$527$	1.75884e14	0.188481
$528$	0	0
$529$	2.07171e15	2.17432
$530$	−9.21322e13	−0.0956961
$531$	0	0
$532$	−2.30381e14	−0.234386
$533$	−1.57097e15	−1.58186
$534$	0	0
$535$	9.80692e14	0.967357
$536$	1.64856e15	1.60953
$537$	0	0
$538$	1.78588e14	0.170825
$539$	3.03257e13	0.0287126
$540$	0	0
$541$	1.22708e15	1.13838	0.569192	−	0.822204i	$-0.307256\pi$
$541$	1.22708e15	1.13838	0.569192	+	0.822204i	$0.307256\pi$
$542$	−6.34838e13	−0.0582997
$543$	0	0
$544$	−4.49247e14	−0.404289
$545$	1.25324e15	1.11648
$546$	0	0
$547$	−4.33699e14	−0.378668	−0.189334	−	0.981913i	$-0.560633\pi$
$547$	−4.33699e14	−0.378668	−0.189334	+	0.981913i	$0.560633\pi$
$548$	3.07330e14	0.265652
$549$	0	0
$550$	−4.25902e13	−0.0360840
$551$	−2.15177e14	−0.180494
$552$	0	0
$553$	1.23277e15	1.01366
$554$	3.13762e14	0.255445
$555$	0	0
$556$	−1.33370e14	−0.106451
$557$	−1.09378e15	−0.864422	−0.432211	−	0.901773i	$-0.642266\pi$
$557$	−1.09378e15	−0.864422	−0.432211	+	0.901773i	$0.642266\pi$
$558$	0	0
$559$	−1.65477e15	−1.28225
$560$	−3.55270e14	−0.272599
$561$	0	0
$562$	1.16684e15	0.877935
$563$	9.87487e14	0.735758	0.367879	−	0.929874i	$-0.380084\pi$
$563$	9.87487e14	0.735758	0.367879	+	0.929874i	$0.380084\pi$
$564$	0	0
$565$	−5.20529e14	−0.380346
$566$	−6.04237e14	−0.437237
$567$	0	0
$568$	9.15425e13	0.0649691
$569$	−2.06963e14	−0.145471	−0.0727354	−	0.997351i	$-0.523173\pi$
$569$	−2.06963e14	−0.145471	−0.0727354	+	0.997351i	$0.523173\pi$
$570$	0	0
$571$	6.41876e14	0.442540	0.221270	−	0.975213i	$-0.428980\pi$
$571$	6.41876e14	0.442540	0.221270	+	0.975213i	$0.428980\pi$
$572$	−1.50796e15	−1.02970
$573$	0	0
$574$	6.81045e14	0.456205
$575$	2.44994e14	0.162548
$576$	0	0
$577$	1.89416e15	1.23296	0.616482	−	0.787369i	$-0.288557\pi$
$577$	1.89416e15	1.23296	0.616482	+	0.787369i	$0.288557\pi$
$578$	−6.73580e14	−0.434296
$579$	0	0
$580$	−6.42347e14	−0.406364
$581$	−2.33279e15	−1.46186
$582$	0	0
$583$	−2.45728e14	−0.151105
$584$	2.89849e15	1.76564
$585$	0	0
$586$	4.93275e14	0.294884
$587$	2.46549e15	1.46014	0.730068	−	0.683375i	$-0.239489\pi$
$587$	2.46549e15	1.46014	0.730068	+	0.683375i	$0.239489\pi$
$588$	0	0
$589$	2.58999e14	0.150544
$590$	−1.43980e15	−0.829119
$591$	0	0
$592$	4.49775e14	0.254229
$593$	−1.38178e15	−0.773817	−0.386908	−	0.922118i	$-0.626457\pi$
$593$	−1.38178e15	−0.773817	−0.386908	+	0.922118i	$0.626457\pi$
$594$	0	0
$595$	6.90230e14	0.379447
$596$	−2.09200e15	−1.13948
$597$	0	0
$598$	−3.09152e15	−1.65316
$599$	−7.33743e14	−0.388773	−0.194387	−	0.980925i	$-0.562272\pi$
$599$	−7.33743e14	−0.388773	−0.194387	+	0.980925i	$0.562272\pi$
$600$	0	0
$601$	−1.04584e15	−0.544071	−0.272035	−	0.962287i	$-0.587697\pi$
$601$	−1.04584e15	−0.544071	−0.272035	+	0.962287i	$0.587697\pi$
$602$	7.17373e14	0.369798
$603$	0	0
$604$	−1.57690e14	−0.0798183
$605$	−7.69071e14	−0.385755
$606$	0	0
$607$	−1.43172e15	−0.705214	−0.352607	−	0.935771i	$-0.614705\pi$
$607$	−1.43172e15	−0.705214	−0.352607	+	0.935771i	$0.614705\pi$
$608$	−6.61540e14	−0.322913
$609$	0	0
$610$	8.52769e14	0.408806
$611$	−2.48871e15	−1.18235
$612$	0	0
$613$	−1.57774e15	−0.736212	−0.368106	−	0.929784i	$-0.619994\pi$
$613$	−1.57774e15	−0.736212	−0.368106	+	0.929784i	$0.619994\pi$
$614$	6.14212e14	0.284049
$615$	0	0
$616$	1.54045e15	0.699767
$617$	2.07917e15	0.936097	0.468048	−	0.883703i	$-0.344957\pi$
$617$	2.07917e15	0.936097	0.468048	+	0.883703i	$0.344957\pi$
$618$	0	0
$619$	−2.06773e15	−0.914525	−0.457263	−	0.889332i	$-0.651170\pi$
$619$	−2.06773e15	−0.914525	−0.457263	+	0.889332i	$0.651170\pi$
$620$	7.73163e14	0.338935
$621$	0	0
$622$	−1.39652e15	−0.601451
$623$	4.27394e15	1.82450
$624$	0	0
$625$	−2.14682e15	−0.900442
$626$	1.29194e15	0.537136
$627$	0	0
$628$	−2.08485e15	−0.851718
$629$	−8.73837e14	−0.353877
$630$	0	0
$631$	4.82782e15	1.92127	0.960637	−	0.277807i	$-0.0896076\pi$
$631$	4.82782e15	1.92127	0.960637	+	0.277807i	$0.0896076\pi$
$632$	2.24667e15	0.886328
$633$	0	0
$634$	−2.47837e15	−0.960892
$635$	3.26892e14	0.125646
$636$	0	0
$637$	1.78308e14	0.0673605
$638$	6.10589e14	0.228684
$639$	0	0
$640$	−2.34752e15	−0.864212
$641$	3.81285e15	1.39165	0.695827	−	0.718210i	$-0.255038\pi$
$641$	3.81285e15	1.39165	0.695827	+	0.718210i	$0.255038\pi$
$642$	0	0
$643$	2.81442e15	1.00978	0.504892	−	0.863183i	$-0.331532\pi$
$643$	2.81442e15	1.00978	0.504892	+	0.863183i	$0.331532\pi$
$644$	−3.76049e15	−1.33774
$645$	0	0
$646$	1.78825e14	0.0625389
$647$	−5.11427e15	−1.77342	−0.886708	−	0.462331i	$-0.847013\pi$
$647$	−5.11427e15	−1.77342	−0.886708	+	0.462331i	$0.847013\pi$
$648$	0	0
$649$	−3.84014e15	−1.30918
$650$	−2.50421e14	−0.0846539
$651$	0	0
$652$	−2.18027e15	−0.724682
$653$	−3.49269e15	−1.15116	−0.575582	−	0.817744i	$-0.695225\pi$
$653$	−3.49269e15	−1.15116	−0.575582	+	0.817744i	$0.695225\pi$
$654$	0	0
$655$	−1.74429e15	−0.565317
$656$	−7.63462e14	−0.245367
$657$	0	0
$658$	1.07890e15	0.340988
$659$	−1.05126e15	−0.329487	−0.164744	−	0.986336i	$-0.552680\pi$
$659$	−1.05126e15	−0.329487	−0.164744	+	0.986336i	$0.552680\pi$
$660$	0	0
$661$	3.55844e15	1.09686	0.548431	−	0.836196i	$-0.315225\pi$
$661$	3.55844e15	1.09686	0.548431	+	0.836196i	$0.315225\pi$
$662$	−6.80967e14	−0.208165
$663$	0	0
$664$	−4.25140e15	−1.27823
$665$	1.01640e15	0.303072
$666$	0	0
$667$	−3.51232e15	−1.03015
$668$	−2.08101e15	−0.605347
$669$	0	0
$670$	−3.08656e15	−0.883209
$671$	2.27444e15	0.645507
$672$	0	0
$673$	2.13737e15	0.596755	0.298377	−	0.954448i	$-0.403555\pi$
$673$	2.13737e15	0.596755	0.298377	+	0.954448i	$0.403555\pi$
$674$	−3.89081e14	−0.107748
$675$	0	0
$676$	−6.16054e15	−1.67846
$677$	2.11604e15	0.571855	0.285927	−	0.958251i	$-0.407698\pi$
$677$	2.11604e15	0.571855	0.285927	+	0.958251i	$0.407698\pi$
$678$	0	0
$679$	−1.26276e15	−0.335766
$680$	1.25792e15	0.331782
$681$	0	0
$682$	−7.34937e14	−0.190738
$683$	−1.01393e15	−0.261031	−0.130516	−	0.991446i	$-0.541663\pi$
$683$	−1.01393e15	−0.261031	−0.130516	+	0.991446i	$0.541663\pi$
$684$	0	0
$685$	−1.35589e15	−0.343499
$686$	1.99996e15	0.502619
$687$	0	0
$688$	−8.04187e14	−0.198894
$689$	−1.44483e15	−0.354495
$690$	0	0
$691$	6.60866e15	1.59582	0.797910	−	0.602777i	$-0.205939\pi$
$691$	6.60866e15	1.59582	0.797910	+	0.602777i	$0.205939\pi$
$692$	5.63332e15	1.34953
$693$	0	0
$694$	2.27425e15	0.536242
$695$	5.88407e14	0.137645
$696$	0	0
$697$	1.48328e15	0.341541
$698$	−2.67491e14	−0.0611089
$699$	0	0
$700$	−3.04609e14	−0.0685020
$701$	3.22703e15	0.720037	0.360018	−	0.932945i	$-0.382770\pi$
$701$	3.22703e15	0.720037	0.360018	+	0.932945i	$0.382770\pi$
$702$	0	0
$703$	−1.28677e15	−0.282648
$704$	8.83166e14	0.192483
$705$	0	0
$706$	−1.43040e15	−0.306924
$707$	−5.83738e15	−1.24283
$708$	0	0
$709$	−2.95770e15	−0.620012	−0.310006	−	0.950735i	$-0.600331\pi$
$709$	−2.95770e15	−0.620012	−0.310006	+	0.950735i	$0.600331\pi$
$710$	−1.71393e14	−0.0356510
$711$	0	0
$712$	7.78907e15	1.59531
$713$	4.22761e15	0.859217
$714$	0	0
$715$	6.65288e15	1.33145
$716$	4.89674e15	0.972490
$717$	0	0
$718$	4.82174e15	0.943017
$719$	−3.18146e15	−0.617471	−0.308736	−	0.951148i	$-0.599906\pi$
$719$	−3.18146e15	−0.617471	−0.308736	+	0.951148i	$0.599906\pi$
$720$	0	0
$721$	5.42535e15	1.03701
$722$	−2.43894e15	−0.462644
$723$	0	0
$724$	3.20319e15	0.598441
$725$	−2.84507e14	−0.0527513
$726$	0	0
$727$	4.00650e15	0.731688	0.365844	−	0.930676i	$-0.380780\pi$
$727$	4.00650e15	0.731688	0.365844	+	0.930676i	$0.380780\pi$
$728$	9.05750e15	1.64167
$729$	0	0
$730$	−5.42677e15	−0.968871
$731$	1.56240e15	0.276852
$732$	0	0
$733$	−6.90271e15	−1.20489	−0.602446	−	0.798160i	$-0.705807\pi$
$733$	−6.90271e15	−1.20489	−0.602446	+	0.798160i	$0.705807\pi$
$734$	3.22547e15	0.558811
$735$	0	0
$736$	−1.07983e16	−1.84300
$737$	−8.23224e15	−1.39459
$738$	0	0
$739$	6.84614e15	1.14262	0.571309	−	0.820735i	$-0.306436\pi$
$739$	6.84614e15	1.14262	0.571309	+	0.820735i	$0.306436\pi$
$740$	−3.84127e15	−0.636356
$741$	0	0
$742$	6.26359e14	0.102236
$743$	1.07977e16	1.74941	0.874707	−	0.484652i	$-0.161054\pi$
$743$	1.07977e16	1.74941	0.874707	+	0.484652i	$0.161054\pi$
$744$	0	0
$745$	9.22954e15	1.47340
$746$	−2.13524e15	−0.338363
$747$	0	0
$748$	1.42379e15	0.222325
$749$	−6.66722e15	−1.03346
$750$	0	0
$751$	−1.17462e15	−0.179424	−0.0897118	−	0.995968i	$-0.528595\pi$
$751$	−1.17462e15	−0.179424	−0.0897118	+	0.995968i	$0.528595\pi$
$752$	−1.20947e15	−0.183398
$753$	0	0
$754$	3.59012e15	0.536497
$755$	6.95703e14	0.103209
$756$	0	0
$757$	−3.10476e15	−0.453943	−0.226972	−	0.973901i	$-0.572882\pi$
$757$	−3.10476e15	−0.453943	−0.226972	+	0.973901i	$0.572882\pi$
$758$	−5.78218e15	−0.839287
$759$	0	0
$760$	1.85234e15	0.265001
$761$	2.40919e15	0.342180	0.171090	−	0.985255i	$-0.445271\pi$
$761$	2.40919e15	0.342180	0.171090	+	0.985255i	$0.445271\pi$
$762$	0	0
$763$	−8.52011e15	−1.19278
$764$	−1.21023e15	−0.168212
$765$	0	0
$766$	−6.99107e14	−0.0957823
$767$	−2.25791e16	−3.07138
$768$	0	0
$769$	8.82621e15	1.18353	0.591765	−	0.806110i	$-0.298431\pi$
$769$	8.82621e15	1.18353	0.591765	+	0.806110i	$0.298431\pi$
$770$	−2.88415e15	−0.383988
$771$	0	0
$772$	7.99173e15	1.04893
$773$	3.57848e15	0.466349	0.233175	−	0.972435i	$-0.425089\pi$
$773$	3.57848e15	0.466349	0.233175	+	0.972435i	$0.425089\pi$
$774$	0	0
$775$	3.42447e14	0.0439982
$776$	−2.30132e15	−0.293588
$777$	0	0
$778$	3.50131e15	0.440396
$779$	2.18420e15	0.272796
$780$	0	0
$781$	−4.57126e14	−0.0562931
$782$	2.91895e15	0.356936
$783$	0	0
$784$	8.66546e13	0.0104485
$785$	9.19798e15	1.10131
$786$	0	0
$787$	−6.37897e15	−0.753164	−0.376582	−	0.926383i	$-0.622901\pi$
$787$	−6.37897e15	−0.753164	−0.376582	+	0.926383i	$0.622901\pi$
$788$	−6.56677e15	−0.769941
$789$	0	0
$790$	−4.20638e15	−0.486362
$791$	3.53881e15	0.406337
$792$	0	0
$793$	1.33732e16	1.51437
$794$	−3.07374e15	−0.345664
$795$	0	0
$796$	−2.67192e15	−0.296348
$797$	8.33604e15	0.918204	0.459102	−	0.888384i	$-0.348171\pi$
$797$	8.33604e15	0.918204	0.459102	+	0.888384i	$0.348171\pi$
$798$	0	0
$799$	2.34979e15	0.255283
$800$	−8.74687e14	−0.0943751
$801$	0	0
$802$	−4.51696e15	−0.480714
$803$	−1.44739e16	−1.52985
$804$	0	0
$805$	1.65906e16	1.72976
$806$	−4.32127e15	−0.447475
$807$	0	0
$808$	−1.06384e16	−1.08671
$809$	−8.81902e15	−0.894754	−0.447377	−	0.894346i	$-0.647642\pi$
$809$	−8.81902e15	−0.894754	−0.447377	+	0.894346i	$0.647642\pi$
$810$	0	0
$811$	9.88676e15	0.989553	0.494777	−	0.869020i	$-0.335250\pi$
$811$	9.88676e15	0.989553	0.494777	+	0.869020i	$0.335250\pi$
$812$	4.36698e15	0.434134
$813$	0	0
$814$	3.65135e15	0.358113
$815$	9.61895e15	0.937046
$816$	0	0
$817$	2.30072e15	0.221127
$818$	−3.05241e15	−0.291407
$819$	0	0
$820$	6.52029e15	0.614173
$821$	8.35385e15	0.781626	0.390813	−	0.920470i	$-0.372194\pi$
$821$	8.35385e15	0.781626	0.390813	+	0.920470i	$0.372194\pi$
$822$	0	0
$823$	−1.65177e16	−1.52493	−0.762465	−	0.647029i	$-0.776011\pi$
$823$	−1.65177e16	−1.52493	−0.762465	+	0.647029i	$0.776011\pi$
$824$	9.88747e15	0.906746
$825$	0	0
$826$	9.78848e15	0.885778
$827$	4.87197e15	0.437950	0.218975	−	0.975730i	$-0.429729\pi$
$827$	4.87197e15	0.437950	0.218975	+	0.975730i	$0.429729\pi$
$828$	0	0
$829$	−7.09701e15	−0.629543	−0.314771	−	0.949168i	$-0.601928\pi$
$829$	−7.09701e15	−0.629543	−0.314771	+	0.949168i	$0.601928\pi$
$830$	7.95980e15	0.701411
$831$	0	0
$832$	5.19282e15	0.451569
$833$	−1.68355e14	−0.0145439
$834$	0	0
$835$	9.18107e15	0.782740
$836$	2.09660e15	0.177575
$837$	0	0
$838$	4.22925e15	0.353526
$839$	5.33515e15	0.443053	0.221527	−	0.975154i	$-0.428896\pi$
$839$	5.33515e15	0.443053	0.221527	+	0.975154i	$0.428896\pi$
$840$	0	0
$841$	−8.12172e15	−0.665687
$842$	−8.26124e15	−0.672711
$843$	0	0
$844$	1.48988e16	1.19748
$845$	2.71792e16	2.17033
$846$	0	0
$847$	5.22852e15	0.412116
$848$	−7.02159e14	−0.0549868
$849$	0	0
$850$	2.36443e14	0.0182777
$851$	−2.10038e16	−1.61319
$852$	0	0
$853$	6.44227e15	0.488449	0.244224	−	0.969719i	$-0.421467\pi$
$853$	6.44227e15	0.488449	0.244224	+	0.969719i	$0.421467\pi$
$854$	−5.79754e15	−0.436742
$855$	0	0
$856$	−1.21507e16	−0.903642
$857$	9.11174e15	0.673297	0.336649	−	0.941630i	$-0.390707\pi$
$857$	9.11174e15	0.673297	0.336649	+	0.941630i	$0.390707\pi$
$858$	0	0
$859$	1.20662e16	0.880257	0.440129	−	0.897935i	$-0.354933\pi$
$859$	1.20662e16	0.880257	0.440129	+	0.897935i	$0.354933\pi$
$860$	6.86810e15	0.497846
$861$	0	0
$862$	2.20532e15	0.157827
$863$	−4.65821e15	−0.331253	−0.165626	−	0.986189i	$-0.552965\pi$
$863$	−4.65821e15	−0.331253	−0.165626	+	0.986189i	$0.552965\pi$
$864$	0	0
$865$	−2.48532e16	−1.74500
$866$	−1.34206e16	−0.936316
$867$	0	0
$868$	−5.25633e15	−0.362097
$869$	−1.12189e16	−0.767968
$870$	0	0
$871$	−4.84037e16	−3.27174
$872$	−1.55275e16	−1.04295
$873$	0	0
$874$	4.29831e15	0.285092
$875$	1.60739e16	1.05944
$876$	0	0
$877$	−9.04585e15	−0.588778	−0.294389	−	0.955686i	$-0.595116\pi$
$877$	−9.04585e15	−0.588778	−0.294389	+	0.955686i	$0.595116\pi$
$878$	−4.67274e15	−0.302240
$879$	0	0
$880$	3.23317e15	0.206526
$881$	7.63030e15	0.484367	0.242183	−	0.970231i	$-0.422136\pi$
$881$	7.63030e15	0.484367	0.242183	+	0.970231i	$0.422136\pi$
$882$	0	0
$883$	−2.52253e16	−1.58144	−0.790719	−	0.612180i	$-0.790293\pi$
$883$	−2.52253e16	−1.58144	−0.790719	+	0.612180i	$0.790293\pi$
$884$	8.37157e15	0.521578
$885$	0	0
$886$	−3.34867e15	−0.206057
$887$	1.05582e16	0.645670	0.322835	−	0.946455i	$-0.395364\pi$
$887$	1.05582e16	0.645670	0.322835	+	0.946455i	$0.395364\pi$
$888$	0	0
$889$	−2.22237e15	−0.134232
$890$	−1.45833e16	−0.875407
$891$	0	0
$892$	3.29611e14	0.0195432
$893$	3.46018e15	0.203900
$894$	0	0
$895$	−2.16035e16	−1.25747
$896$	1.59596e16	0.923269
$897$	0	0
$898$	−1.06311e16	−0.607519
$899$	−4.90945e15	−0.278840
$900$	0	0
$901$	1.36418e15	0.0765393
$902$	−6.19792e15	−0.345629
$903$	0	0
$904$	6.44933e15	0.355294
$905$	−1.41319e16	−0.773810
$906$	0	0
$907$	2.70513e15	0.146335	0.0731675	−	0.997320i	$-0.476689\pi$
$907$	2.70513e15	0.146335	0.0731675	+	0.997320i	$0.476689\pi$
$908$	−1.60055e16	−0.860589
$909$	0	0
$910$	−1.69581e16	−0.900845
$911$	3.00588e16	1.58716	0.793579	−	0.608467i	$-0.208215\pi$
$911$	3.00588e16	1.58716	0.793579	+	0.608467i	$0.208215\pi$
$912$	0	0
$913$	2.12298e16	1.10753
$914$	−5.22518e15	−0.270955
$915$	0	0
$916$	−1.23579e16	−0.633169
$917$	1.18585e16	0.603948
$918$	0	0
$919$	−1.29261e16	−0.650478	−0.325239	−	0.945632i	$-0.605445\pi$
$919$	−1.29261e16	−0.650478	−0.325239	+	0.945632i	$0.605445\pi$
$920$	3.02357e16	1.51247
$921$	0	0
$922$	7.91174e14	0.0391068
$923$	−2.68780e15	−0.132065
$924$	0	0
$925$	−1.70136e15	−0.0826072
$926$	−3.12619e15	−0.150888
$927$	0	0
$928$	1.25398e16	0.598105
$929$	−1.82775e16	−0.866622	−0.433311	−	0.901245i	$-0.642655\pi$
$929$	−1.82775e16	−0.866622	−0.433311	+	0.901245i	$0.642655\pi$
$930$	0	0
$931$	−2.47912e14	−0.0116165
$932$	−1.48561e16	−0.692018
$933$	0	0
$934$	−1.67906e15	−0.0772962
$935$	−6.28152e15	−0.287475
$936$	0	0
$937$	2.82874e16	1.27946	0.639728	−	0.768601i	$-0.279047\pi$
$937$	2.82874e16	1.27946	0.639728	+	0.768601i	$0.279047\pi$
$938$	2.09839e16	0.943563
$939$	0	0
$940$	1.03293e16	0.459060
$941$	−2.15088e16	−0.950327	−0.475163	−	0.879898i	$-0.657611\pi$
$941$	−2.15088e16	−0.950327	−0.475163	+	0.879898i	$0.657611\pi$
$942$	0	0
$943$	3.56526e16	1.55696
$944$	−1.09730e16	−0.476410
$945$	0	0
$946$	−6.52854e15	−0.280165
$947$	−1.70849e16	−0.728933	−0.364467	−	0.931216i	$-0.618749\pi$
$947$	−1.70849e16	−0.728933	−0.364467	+	0.931216i	$0.618749\pi$
$948$	0	0
$949$	−8.51032e16	−3.58907
$950$	3.48174e14	0.0145988
$951$	0	0
$952$	−8.55192e15	−0.354454
$953$	−7.80510e15	−0.321638	−0.160819	−	0.986984i	$-0.551414\pi$
$953$	−7.80510e15	−0.321638	−0.160819	+	0.986984i	$0.551414\pi$
$954$	0	0
$955$	5.33934e15	0.217505
$956$	3.10564e16	1.25786
$957$	0	0
$958$	7.85523e15	0.314520
$959$	9.21799e15	0.366973
$960$	0	0
$961$	−1.94992e16	−0.767429
$962$	2.14691e16	0.840140
$963$	0	0
$964$	1.41195e16	0.546256
$965$	−3.52581e16	−1.35631
$966$	0	0
$967$	−1.35068e16	−0.513697	−0.256848	−	0.966452i	$-0.582684\pi$
$967$	−1.35068e16	−0.513697	−0.256848	+	0.966452i	$0.582684\pi$
$968$	9.52875e15	0.360348
$969$	0	0
$970$	4.30871e15	0.161103
$971$	3.28320e15	0.122065	0.0610325	−	0.998136i	$-0.480561\pi$
$971$	3.28320e15	0.122065	0.0610325	+	0.998136i	$0.480561\pi$
$972$	0	0
$973$	−4.00028e15	−0.147051
$974$	−9.29015e15	−0.339585
$975$	0	0
$976$	6.49913e15	0.234899
$977$	−4.04914e16	−1.45527	−0.727634	−	0.685966i	$-0.759380\pi$
$977$	−4.04914e16	−1.45527	−0.727634	+	0.685966i	$0.759380\pi$
$978$	0	0
$979$	−3.88954e16	−1.38227
$980$	−7.40067e14	−0.0261534
$981$	0	0
$982$	1.55534e16	0.543514
$983$	2.83074e16	0.983684	0.491842	−	0.870685i	$-0.336324\pi$
$983$	2.83074e16	0.983684	0.491842	+	0.870685i	$0.336324\pi$
$984$	0	0
$985$	2.89714e16	0.995568
$986$	−3.38973e15	−0.115836
$987$	0	0
$988$	1.23276e16	0.416595
$989$	3.75544e16	1.26206
$990$	0	0
$991$	−2.80386e16	−0.931861	−0.465931	−	0.884821i	$-0.654280\pi$
$991$	−2.80386e16	−0.931861	−0.465931	+	0.884821i	$0.654280\pi$
$992$	−1.50936e16	−0.498860
$993$	0	0
$994$	1.16521e15	0.0380872
$995$	1.17881e16	0.383191
$996$	0	0
$997$	1.62120e15	0.0521209	0.0260605	−	0.999660i	$-0.491704\pi$
$997$	1.62120e15	0.0521209	0.0260605	+	0.999660i	$0.491704\pi$
$998$	−1.65307e16	−0.528536
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	81.12.a.d.1.7	yes	10
3.2	odd	2	inner	81.12.a.d.1.4	✓	10
9.2	odd	6		81.12.c.m.28.7		20
9.4	even	3		81.12.c.m.55.4		20
9.5	odd	6		81.12.c.m.55.7		20
9.7	even	3		81.12.c.m.28.4		20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
81.12.a.d.1.4	✓	10	3.2	odd	2	inner
81.12.a.d.1.7	yes	10	1.1	even	1	trivial
81.12.c.m.28.4		20	9.7	even	3
81.12.c.m.28.7		20	9.2	odd	6
81.12.c.m.55.4		20	9.4	even	3
81.12.c.m.55.7		20	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:	\( k \)	\(=\)	\( 12 \)
Character orbit:	\([\chi]\)	\(=\)	81.a (trivial)

\(f(q)\)	\(=\)	\(q+23.1974 q^{2} -1509.88 q^{4} +6661.33 q^{5} -45287.0 q^{7} -82533.6 q^{8} +154526. q^{10} +412139. q^{11} +2.42329e6 q^{13} -1.05054e6 q^{14} +1.17767e6 q^{16} -2.28802e6 q^{17} -3.36923e6 q^{19} -1.00578e7 q^{20} +9.56055e6 q^{22} -5.49956e7 q^{23} -4.45478e6 q^{25} +5.62139e7 q^{26} +6.83779e7 q^{28} +6.38654e7 q^{29} -7.68718e7 q^{31} +1.96348e8 q^{32} -5.30761e7 q^{34} -3.01671e8 q^{35} +3.81919e8 q^{37} -7.81573e7 q^{38} -5.49783e8 q^{40} -6.48281e8 q^{41} -6.82862e8 q^{43} -6.22281e8 q^{44} -1.27576e9 q^{46} -1.02700e9 q^{47} +7.35813e7 q^{49} -1.03339e8 q^{50} -3.65887e9 q^{52} -5.96226e8 q^{53} +2.74539e9 q^{55} +3.73769e9 q^{56} +1.48151e9 q^{58} -9.31757e9 q^{59} +5.51863e9 q^{61} -1.78323e9 q^{62} +2.14288e9 q^{64} +1.61423e10 q^{65} -1.99744e10 q^{67} +3.45464e9 q^{68} -6.99799e9 q^{70} -1.10915e9 q^{71} -3.51189e10 q^{73} +8.85951e9 q^{74} +5.08713e9 q^{76} -1.86645e10 q^{77} -2.72212e10 q^{79} +7.84486e9 q^{80} -1.50384e10 q^{82} +5.15112e10 q^{83} -1.52413e10 q^{85} -1.58406e10 q^{86} -3.40153e10 q^{88} -9.43745e10 q^{89} -1.09743e11 q^{91} +8.30368e10 q^{92} -2.38237e10 q^{94} -2.24435e10 q^{95} +2.78835e10 q^{97} +1.70689e9 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q + 5110 q^{4} - 11836 q^{7} - 117642 q^{10} + 27110 q^{13} - 5117294 q^{16} - 24650908 q^{19} - 27880836 q^{22} - 43357796 q^{25} - 177204868 q^{28} - 56217184 q^{31} + 302628942 q^{34} - 636530326 q^{37}+ \cdots - 413983885420 q^{97}+O(q^{100}) \) 10 * q + 5110 * q^4 - 11836 * q^7 - 117642 * q^10 + 27110 * q^13 - 5117294 * q^16 - 24650908 * q^19 - 27880836 * q^22 - 43357796 * q^25 - 177204868 * q^28 - 56217184 * q^31 + 302628942 * q^34 - 636530326 * q^37 + 405215346 * q^40 + 1101619412 * q^43 - 5426695164 * q^46 - 1944391710 * q^49 - 7927944502 * q^52 - 17670840708 * q^55 - 23512516710 * q^58 - 25504032214 * q^61 - 43436567978 * q^64 - 59699336020 * q^67 - 92786466516 * q^70 - 58729185430 * q^73 - 139282283236 * q^76 - 150636181996 * q^79 - 152979717864 * q^82 - 184281748362 * q^85 - 371435708028 * q^88 - 277231971332 * q^91 - 500149068360 * q^94 - 413983885420 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

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Groups

Database

Properties

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