LMFDB - Embedded newform 81.12.a.d.1.10

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			1.10
Root			$-35.7085$ 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)

$f(q)$	$=$	\(q+73.5092 q^{2} +3355.60 q^{4} +3419.73 q^{5} -54197.9 q^{7} +96120.6 q^{8} +251382. q^{10} -646669. q^{11} -20727.3 q^{13} -3.98405e6 q^{14} +193480. q^{16} +5.43454e6 q^{17} -1.12341e7 q^{19} +1.14752e7 q^{20} -4.75361e7 q^{22} -5.06575e7 q^{23} -3.71336e7 q^{25} -1.52365e6 q^{26} -1.81867e8 q^{28} -5.57175e7 q^{29} +2.69221e8 q^{31} -1.82632e8 q^{32} +3.99489e8 q^{34} -1.85342e8 q^{35} -2.61847e8 q^{37} -8.25806e8 q^{38} +3.28707e8 q^{40} +9.21115e8 q^{41} +1.86274e9 q^{43} -2.16996e9 q^{44} -3.72379e9 q^{46} -1.85264e9 q^{47} +9.60090e8 q^{49} -2.72966e9 q^{50} -6.95526e7 q^{52} -2.53225e9 q^{53} -2.21143e9 q^{55} -5.20954e9 q^{56} -4.09575e9 q^{58} -5.61583e9 q^{59} -3.69275e9 q^{61} +1.97902e10 q^{62} -1.38214e10 q^{64} -7.08818e7 q^{65} -2.50794e9 q^{67} +1.82362e10 q^{68} -1.36244e10 q^{70} -5.97576e9 q^{71} +1.54028e10 q^{73} -1.92481e10 q^{74} -3.76970e10 q^{76} +3.50481e10 q^{77} -3.94570e10 q^{79} +6.61650e8 q^{80} +6.77104e10 q^{82} +1.71107e8 q^{83} +1.85847e10 q^{85} +1.36928e11 q^{86} -6.21582e10 q^{88} +7.91822e10 q^{89} +1.12338e9 q^{91} -1.69986e11 q^{92} -1.36186e11 q^{94} -3.84174e10 q^{95} +3.54701e10 q^{97} +7.05754e10 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	73.5092	1.62434	0.812169	−	0.583422i	$-0.198286\pi$
$2$	73.5092	1.62434	0.812169	+	0.583422i	$0.198286\pi$
$3$	0	0
$3$	0	0
$4$	3355.60	1.63848
$5$	3419.73	0.489392	0.244696	−	0.969600i	$-0.421312\pi$
$5$	3419.73	0.489392	0.244696	+	0.969600i	$0.421312\pi$
$6$	0	0
$7$	−54197.9	−1.21883	−0.609416	−	0.792851i	$-0.708596\pi$
$7$	−54197.9	−1.21883	−0.609416	+	0.792851i	$0.708596\pi$
$8$	96120.6	1.03710
$9$	0	0
$10$	251382.	0.794938
$11$	−646669.	−1.21066	−0.605330	−	0.795975i	$-0.706959\pi$
$11$	−646669.	−1.21066	−0.605330	+	0.795975i	$0.706959\pi$
$12$	0	0
$13$	−20727.3	−0.0154830	−0.00774149	−	0.999970i	$-0.502464\pi$
$13$	−20727.3	−0.0154830	−0.00774149	+	0.999970i	$0.502464\pi$
$14$	−3.98405e6	−1.97979
$15$	0	0
$16$	193480.	0.0461293
$17$	5.43454e6	0.928312	0.464156	−	0.885753i	$-0.346358\pi$
$17$	5.43454e6	0.928312	0.464156	+	0.885753i	$0.346358\pi$
$18$	0	0
$19$	−1.12341e7	−1.04086	−0.520429	−	0.853905i	$-0.674228\pi$
$19$	−1.12341e7	−1.04086	−0.520429	+	0.853905i	$0.674228\pi$
$20$	1.14752e7	0.801857
$21$	0	0
$22$	−4.75361e7	−1.96652
$23$	−5.06575e7	−1.64112	−0.820560	−	0.571560i	$-0.806339\pi$
$23$	−5.06575e7	−1.64112	−0.820560	+	0.571560i	$0.806339\pi$
$24$	0	0
$25$	−3.71336e7	−0.760496
$26$	−1.52365e6	−0.0251496
$27$	0	0
$28$	−1.81867e8	−1.99703
$29$	−5.57175e7	−0.504432	−0.252216	−	0.967671i	$-0.581159\pi$
$29$	−5.57175e7	−0.504432	−0.252216	+	0.967671i	$0.581159\pi$
$30$	0	0
$31$	2.69221e8	1.68896	0.844479	−	0.535588i	$-0.179910\pi$
$31$	2.69221e8	1.68896	0.844479	+	0.535588i	$0.179910\pi$
$32$	−1.82632e8	−0.962173
$33$	0	0
$34$	3.99489e8	1.50789
$35$	−1.85342e8	−0.596486
$36$	0	0
$37$	−2.61847e8	−0.620780	−0.310390	−	0.950609i	$-0.600460\pi$
$37$	−2.61847e8	−0.620780	−0.310390	+	0.950609i	$0.600460\pi$
$38$	−8.25806e8	−1.69071
$39$	0	0
$40$	3.28707e8	0.507550
$41$	9.21115e8	1.24166	0.620830	−	0.783945i	$-0.286796\pi$
$41$	9.21115e8	1.24166	0.620830	+	0.783945i	$0.286796\pi$
$42$	0	0
$43$	1.86274e9	1.93230	0.966152	−	0.257974i	$-0.0830550\pi$
$43$	1.86274e9	1.93230	0.966152	+	0.257974i	$0.0830550\pi$
$44$	−2.16996e9	−1.98364
$45$	0	0
$46$	−3.72379e9	−2.66574
$47$	−1.85264e9	−1.17829	−0.589146	−	0.808026i	$-0.700536\pi$
$47$	−1.85264e9	−1.17829	−0.589146	+	0.808026i	$0.700536\pi$
$48$	0	0
$49$	9.60090e8	0.485549
$50$	−2.72966e9	−1.23530
$51$	0	0
$52$	−6.95526e7	−0.0253685
$53$	−2.53225e9	−0.831743	−0.415872	−	0.909423i	$-0.636523\pi$
$53$	−2.53225e9	−0.831743	−0.415872	+	0.909423i	$0.636523\pi$
$54$	0	0
$55$	−2.21143e9	−0.592487
$56$	−5.20954e9	−1.26405
$57$	0	0
$58$	−4.09575e9	−0.819368
$59$	−5.61583e9	−1.02265	−0.511326	−	0.859387i	$-0.670845\pi$
$59$	−5.61583e9	−1.02265	−0.511326	+	0.859387i	$0.670845\pi$
$60$	0	0
$61$	−3.69275e9	−0.559804	−0.279902	−	0.960029i	$-0.590302\pi$
$61$	−3.69275e9	−0.559804	−0.279902	+	0.960029i	$0.590302\pi$
$62$	1.97902e10	2.74344
$63$	0	0
$64$	−1.38214e10	−1.60902
$65$	−7.08818e7	−0.00757725
$66$	0	0
$67$	−2.50794e9	−0.226938	−0.113469	−	0.993542i	$-0.536196\pi$
$67$	−2.50794e9	−0.226938	−0.113469	+	0.993542i	$0.536196\pi$
$68$	1.82362e10	1.52102
$69$	0	0
$70$	−1.36244e10	−0.968895
$71$	−5.97576e9	−0.393072	−0.196536	−	0.980497i	$-0.562969\pi$
$71$	−5.97576e9	−0.393072	−0.196536	+	0.980497i	$0.562969\pi$
$72$	0	0
$73$	1.54028e10	0.869608	0.434804	−	0.900525i	$-0.356818\pi$
$73$	1.54028e10	0.869608	0.434804	+	0.900525i	$0.356818\pi$
$74$	−1.92481e10	−1.00836
$75$	0	0
$76$	−3.76970e10	−1.70542
$77$	3.50481e10	1.47559
$78$	0	0
$79$	−3.94570e10	−1.44270	−0.721349	−	0.692572i	$-0.756478\pi$
$79$	−3.94570e10	−1.44270	−0.721349	+	0.692572i	$0.756478\pi$
$80$	6.61650e8	0.0225753
$81$	0	0
$82$	6.77104e10	2.01688
$83$	1.71107e8	0.00476803	0.00238401	−	0.999997i	$-0.499241\pi$
$83$	1.71107e8	0.00476803	0.00238401	+	0.999997i	$0.499241\pi$
$84$	0	0
$85$	1.85847e10	0.454308
$86$	1.36928e11	3.13872
$87$	0	0
$88$	−6.21582e10	−1.25558
$89$	7.91822e10	1.50308	0.751541	−	0.659686i	$-0.229311\pi$
$89$	7.91822e10	1.50308	0.751541	+	0.659686i	$0.229311\pi$
$90$	0	0
$91$	1.12338e9	0.0188711
$92$	−1.69986e11	−2.68894
$93$	0	0
$94$	−1.36186e11	−1.91395
$95$	−3.84174e10	−0.509387
$96$	0	0
$97$	3.54701e10	0.419389	0.209695	−	0.977767i	$-0.432753\pi$
$97$	3.54701e10	0.419389	0.209695	+	0.977767i	$0.432753\pi$
$98$	7.05754e10	0.788697
$99$	0	0
$100$	−1.24605e11	−1.24605
$101$	8.89197e10	0.841842	0.420921	−	0.907097i	$-0.361707\pi$
$101$	8.89197e10	0.841842	0.420921	+	0.907097i	$0.361707\pi$
$102$	0	0
$103$	−1.77812e10	−0.151132	−0.0755660	−	0.997141i	$-0.524076\pi$
$103$	−1.77812e10	−0.151132	−0.0755660	+	0.997141i	$0.524076\pi$
$104$	−1.99232e9	−0.0160574
$105$	0	0
$106$	−1.86144e11	−1.35103
$107$	4.82012e10	0.332236	0.166118	−	0.986106i	$-0.446877\pi$
$107$	4.82012e10	0.332236	0.166118	+	0.986106i	$0.446877\pi$
$108$	0	0
$109$	−2.59616e11	−1.61616	−0.808082	−	0.589069i	$-0.799494\pi$
$109$	−2.59616e11	−1.61616	−0.808082	+	0.589069i	$0.799494\pi$
$110$	−1.62561e11	−0.962400
$111$	0	0
$112$	−1.04862e10	−0.0562238
$113$	−8.50226e10	−0.434113	−0.217056	−	0.976159i	$-0.569646\pi$
$113$	−8.50226e10	−0.434113	−0.217056	+	0.976159i	$0.569646\pi$
$114$	0	0
$115$	−1.73235e11	−0.803151
$116$	−1.86966e11	−0.826500
$117$	0	0
$118$	−4.12815e11	−1.66113
$119$	−2.94541e11	−1.13146
$120$	0	0
$121$	1.32869e11	0.465697
$122$	−2.71451e11	−0.909311
$123$	0	0
$124$	9.03397e11	2.76732
$125$	−2.93966e11	−0.861572
$126$	0	0
$127$	−5.11579e10	−0.137402	−0.0687008	−	0.997637i	$-0.521885\pi$
$127$	−5.11579e10	−0.137402	−0.0687008	+	0.997637i	$0.521885\pi$
$128$	−6.41969e11	−1.65143
$129$	0	0
$130$	−5.21046e9	−0.0123080
$131$	5.90022e11	1.33621	0.668107	−	0.744066i	$-0.267105\pi$
$131$	5.90022e11	1.33621	0.668107	+	0.744066i	$0.267105\pi$
$132$	0	0
$133$	6.08862e11	1.26863
$134$	−1.84357e11	−0.368623
$135$	0	0
$136$	5.22372e11	0.962755
$137$	6.58027e11	1.16488	0.582439	−	0.812874i	$-0.302099\pi$
$137$	6.58027e11	1.16488	0.582439	+	0.812874i	$0.302099\pi$
$138$	0	0
$139$	1.14950e12	1.87901	0.939506	−	0.342533i	$-0.111285\pi$
$139$	1.14950e12	1.87901	0.939506	+	0.342533i	$0.111285\pi$
$140$	−6.21935e11	−0.977328
$141$	0	0
$142$	−4.39273e11	−0.638482
$143$	1.34037e10	0.0187446
$144$	0	0
$145$	−1.90539e11	−0.246865
$146$	1.13225e12	1.41254
$147$	0	0
$148$	−8.78653e11	−1.01713
$149$	8.16344e11	0.910644	0.455322	−	0.890327i	$-0.349524\pi$
$149$	8.16344e11	0.910644	0.455322	+	0.890327i	$0.349524\pi$
$150$	0	0
$151$	−6.61749e11	−0.685994	−0.342997	−	0.939337i	$-0.611442\pi$
$151$	−6.61749e11	−0.685994	−0.342997	+	0.939337i	$0.611442\pi$
$152$	−1.07982e12	−1.07948
$153$	0	0
$154$	2.57636e12	2.39686
$155$	9.20662e11	0.826563
$156$	0	0
$157$	−1.21819e11	−0.101922	−0.0509609	−	0.998701i	$-0.516228\pi$
$157$	−1.21819e11	−0.101922	−0.0509609	+	0.998701i	$0.516228\pi$
$158$	−2.90045e12	−2.34343
$159$	0	0
$160$	−6.24554e11	−0.470880
$161$	2.74553e12	2.00025
$162$	0	0
$163$	−2.04979e12	−1.39533	−0.697666	−	0.716423i	$-0.745778\pi$
$163$	−2.04979e12	−1.39533	−0.697666	+	0.716423i	$0.745778\pi$
$164$	3.09090e12	2.03443
$165$	0	0
$166$	1.25780e10	0.00774490
$167$	5.04298e11	0.300432	0.150216	−	0.988653i	$-0.452003\pi$
$167$	5.04298e11	0.300432	0.150216	+	0.988653i	$0.452003\pi$
$168$	0	0
$169$	−1.79173e12	−0.999760
$170$	1.36614e12	0.737951
$171$	0	0
$172$	6.25061e12	3.16603
$173$	3.20516e12	1.57252	0.786260	−	0.617896i	$-0.212015\pi$
$173$	3.20516e12	1.57252	0.786260	+	0.617896i	$0.212015\pi$
$174$	0	0
$175$	2.01256e12	0.926916
$176$	−1.25118e11	−0.0558468
$177$	0	0
$178$	5.82062e12	2.44151
$179$	4.59980e11	0.187089	0.0935443	−	0.995615i	$-0.470180\pi$
$179$	4.59980e11	0.187089	0.0935443	+	0.995615i	$0.470180\pi$
$180$	0	0
$181$	3.40048e12	1.30109	0.650546	−	0.759467i	$-0.274540\pi$
$181$	3.40048e12	1.30109	0.650546	+	0.759467i	$0.274540\pi$
$182$	8.25786e10	0.0306531
$183$	0	0
$184$	−4.86923e12	−1.70201
$185$	−8.95445e11	−0.303805
$186$	0	0
$187$	−3.51435e12	−1.12387
$188$	−6.21672e12	−1.93061
$189$	0	0
$190$	−2.82403e12	−0.827418
$191$	−3.47851e12	−0.990171	−0.495086	−	0.868844i	$-0.664863\pi$
$191$	−3.47851e12	−0.990171	−0.495086	+	0.868844i	$0.664863\pi$
$192$	0	0
$193$	7.20013e12	1.93542	0.967710	−	0.252065i	$-0.0811098\pi$
$193$	7.20013e12	1.93542	0.967710	+	0.252065i	$0.0811098\pi$
$194$	2.60738e12	0.681230
$195$	0	0
$196$	3.22168e12	0.795561
$197$	−5.27753e12	−1.26726	−0.633632	−	0.773635i	$-0.718436\pi$
$197$	−5.27753e12	−1.26726	−0.633632	+	0.773635i	$0.718436\pi$
$198$	0	0
$199$	−6.85412e12	−1.55690	−0.778449	−	0.627708i	$-0.783993\pi$
$199$	−6.85412e12	−1.55690	−0.778449	+	0.627708i	$0.783993\pi$
$200$	−3.56930e12	−0.788712
$201$	0	0
$202$	6.53642e12	1.36744
$203$	3.01977e12	0.614817
$204$	0	0
$205$	3.14997e12	0.607658
$206$	−1.30708e12	−0.245490
$207$	0	0
$208$	−4.01033e9	−0.000714219 0
$209$	7.26471e12	1.26013
$210$	0	0
$211$	−6.03995e12	−0.994214	−0.497107	−	0.867689i	$-0.665604\pi$
$211$	−6.03995e12	−0.994214	−0.497107	+	0.867689i	$0.665604\pi$
$212$	−8.49722e12	−1.36279
$213$	0	0
$214$	3.54323e12	0.539664
$215$	6.37006e12	0.945654
$216$	0	0
$217$	−1.45912e13	−2.05856
$218$	−1.90842e13	−2.62520
$219$	0	0
$220$	−7.42068e12	−0.970776
$221$	−1.12644e11	−0.0143730
$222$	0	0
$223$	1.22695e11	0.0148988	0.00744938	−	0.999972i	$-0.497629\pi$
$223$	1.22695e11	0.0148988	0.00744938	+	0.999972i	$0.497629\pi$
$224$	9.89830e12	1.17273
$225$	0	0
$226$	−6.24994e12	−0.705147
$227$	1.17238e12	0.129100	0.0645498	−	0.997914i	$-0.479439\pi$
$227$	1.17238e12	0.129100	0.0645498	+	0.997914i	$0.479439\pi$
$228$	0	0
$229$	9.85291e12	1.03388	0.516939	−	0.856022i	$-0.327071\pi$
$229$	9.85291e12	1.03388	0.516939	+	0.856022i	$0.327071\pi$
$230$	−1.27344e13	−1.30459
$231$	0	0
$232$	−5.35560e12	−0.523148
$233$	−6.28150e12	−0.599247	−0.299623	−	0.954058i	$-0.596861\pi$
$233$	−6.28150e12	−0.599247	−0.299623	+	0.954058i	$0.596861\pi$
$234$	0	0
$235$	−6.33553e12	−0.576647
$236$	−1.88445e13	−1.67559
$237$	0	0
$238$	−2.16515e13	−1.83787
$239$	1.02518e13	0.850377	0.425188	−	0.905105i	$-0.360208\pi$
$239$	1.02518e13	0.850377	0.425188	+	0.905105i	$0.360208\pi$
$240$	0	0
$241$	−1.72778e13	−1.36897	−0.684486	−	0.729026i	$-0.739973\pi$
$241$	−1.72778e13	−1.36897	−0.684486	+	0.729026i	$0.739973\pi$
$242$	9.76708e12	0.756450
$243$	0	0
$244$	−1.23914e13	−0.917225
$245$	3.28325e12	0.237624
$246$	0	0
$247$	2.32852e11	0.0161156
$248$	2.58777e13	1.75162
$249$	0	0
$250$	−2.16092e13	−1.39949
$251$	−6.39888e12	−0.405414	−0.202707	−	0.979239i	$-0.564974\pi$
$251$	−6.39888e12	−0.405414	−0.202707	+	0.979239i	$0.564974\pi$
$252$	0	0
$253$	3.27586e13	1.98684
$254$	−3.76057e12	−0.223187
$255$	0	0
$256$	−1.88844e13	−1.07345
$257$	1.19172e13	0.663044	0.331522	−	0.943447i	$-0.392438\pi$
$257$	1.19172e13	0.663044	0.331522	+	0.943447i	$0.392438\pi$
$258$	0	0
$259$	1.41916e13	0.756626
$260$	−2.37851e11	−0.0124151
$261$	0	0
$262$	4.33720e13	2.17046
$263$	−8.44531e10	−0.00413865	−0.00206933	−	0.999998i	$-0.500659\pi$
$263$	−8.44531e10	−0.00413865	−0.00206933	+	0.999998i	$0.500659\pi$
$264$	0	0
$265$	−8.65961e12	−0.407048
$266$	4.47570e13	2.06069
$267$	0	0
$268$	−8.41566e12	−0.371832
$269$	2.99327e12	0.129571	0.0647855	−	0.997899i	$-0.479364\pi$
$269$	2.99327e12	0.129571	0.0647855	+	0.997899i	$0.479364\pi$
$270$	0	0
$271$	−2.88736e12	−0.119997	−0.0599985	−	0.998198i	$-0.519110\pi$
$271$	−2.88736e12	−0.119997	−0.0599985	+	0.998198i	$0.519110\pi$
$272$	1.05148e12	0.0428224
$273$	0	0
$274$	4.83710e13	1.89216
$275$	2.40131e13	0.920702
$276$	0	0
$277$	−1.02283e13	−0.376846	−0.188423	−	0.982088i	$-0.560338\pi$
$277$	−1.02283e13	−0.376846	−0.188423	+	0.982088i	$0.560338\pi$
$278$	8.44992e13	3.05215
$279$	0	0
$280$	−1.78152e13	−0.618617
$281$	1.03563e13	0.352631	0.176315	−	0.984334i	$-0.443582\pi$
$281$	1.03563e13	0.352631	0.176315	+	0.984334i	$0.443582\pi$
$282$	0	0
$283$	−4.33086e13	−1.41824	−0.709118	−	0.705090i	$-0.750907\pi$
$283$	−4.33086e13	−1.41824	−0.709118	+	0.705090i	$0.750907\pi$
$284$	−2.00523e13	−0.644039
$285$	0	0
$286$	9.85296e11	0.0304476
$287$	−4.99226e13	−1.51337
$288$	0	0
$289$	−4.73763e12	−0.138237
$290$	−1.40063e13	−0.400992
$291$	0	0
$292$	5.16856e13	1.42483
$293$	−5.22048e13	−1.41234	−0.706169	−	0.708044i	$-0.749578\pi$
$293$	−5.22048e13	−1.41234	−0.706169	+	0.708044i	$0.749578\pi$
$294$	0	0
$295$	−1.92046e13	−0.500477
$296$	−2.51689e13	−0.643813
$297$	0	0
$298$	6.00088e13	1.47919
$299$	1.04999e12	0.0254094
$300$	0	0
$301$	−1.00957e14	−2.35515
$302$	−4.86446e13	−1.11429
$303$	0	0
$304$	−2.17357e12	−0.0480140
$305$	−1.26282e13	−0.273963
$306$	0	0
$307$	−5.07230e13	−1.06156	−0.530780	−	0.847510i	$-0.678101\pi$
$307$	−5.07230e13	−1.06156	−0.530780	+	0.847510i	$0.678101\pi$
$308$	1.17607e14	2.41772
$309$	0	0
$310$	6.76771e13	1.34262
$311$	−8.94273e13	−1.74296	−0.871481	−	0.490429i	$-0.836840\pi$
$311$	−8.94273e13	−1.74296	−0.871481	+	0.490429i	$0.836840\pi$
$312$	0	0
$313$	9.59921e12	0.180610	0.0903050	−	0.995914i	$-0.471216\pi$
$313$	9.59921e12	0.180610	0.0903050	+	0.995914i	$0.471216\pi$
$314$	−8.95482e12	−0.165555
$315$	0	0
$316$	−1.32402e14	−2.36383
$317$	4.94097e13	0.866934	0.433467	−	0.901169i	$-0.357290\pi$
$317$	4.94097e13	0.866934	0.433467	+	0.901169i	$0.357290\pi$
$318$	0	0
$319$	3.60308e13	0.610696
$320$	−4.72655e13	−0.787443
$321$	0	0
$322$	2.01822e14	3.24908
$323$	−6.10519e13	−0.966241
$324$	0	0
$325$	7.69679e11	0.0117747
$326$	−1.50678e14	−2.26649
$327$	0	0
$328$	8.85382e13	1.28773
$329$	1.00409e14	1.43614
$330$	0	0
$331$	4.26693e13	0.590285	0.295143	−	0.955453i	$-0.404633\pi$
$331$	4.26693e13	0.590285	0.295143	+	0.955453i	$0.404633\pi$
$332$	5.74168e11	0.00781231
$333$	0	0
$334$	3.70705e13	0.488004
$335$	−8.57649e12	−0.111061
$336$	0	0
$337$	−4.66498e13	−0.584635	−0.292318	−	0.956321i	$-0.594426\pi$
$337$	−4.66498e13	−0.584635	−0.292318	+	0.956321i	$0.594426\pi$
$338$	−1.31709e14	−1.62395
$339$	0	0
$340$	6.23627e13	0.744374
$341$	−1.74097e14	−2.04475
$342$	0	0
$343$	5.51322e13	0.627028
$344$	1.79048e14	2.00400
$345$	0	0
$346$	2.35609e14	2.55430
$347$	−6.91678e13	−0.738060	−0.369030	−	0.929417i	$-0.620310\pi$
$347$	−6.91678e13	−0.738060	−0.369030	+	0.929417i	$0.620310\pi$
$348$	0	0
$349$	−4.21104e13	−0.435361	−0.217680	−	0.976020i	$-0.569849\pi$
$349$	−4.21104e13	−0.435361	−0.217680	+	0.976020i	$0.569849\pi$
$350$	1.47942e14	1.50563
$351$	0	0
$352$	1.18103e14	1.16486
$353$	−1.72856e14	−1.67851	−0.839256	−	0.543737i	$-0.817009\pi$
$353$	−1.72856e14	−1.67851	−0.839256	+	0.543737i	$0.817009\pi$
$354$	0	0
$355$	−2.04355e13	−0.192366
$356$	2.65704e14	2.46277
$357$	0	0
$358$	3.38127e13	0.303895
$359$	−1.10755e14	−0.980269	−0.490134	−	0.871647i	$-0.663052\pi$
$359$	−1.10755e14	−0.980269	−0.490134	+	0.871647i	$0.663052\pi$
$360$	0	0
$361$	9.71363e12	0.0833858
$362$	2.49967e14	2.11342
$363$	0	0
$364$	3.76961e12	0.0309199
$365$	5.26734e13	0.425579
$366$	0	0
$367$	−1.34942e14	−1.05800	−0.528999	−	0.848623i	$-0.677432\pi$
$367$	−1.34942e14	−1.05800	−0.528999	+	0.848623i	$0.677432\pi$
$368$	−9.80122e12	−0.0757037
$369$	0	0
$370$	−6.58234e13	−0.493482
$371$	1.37243e14	1.01375
$372$	0	0
$373$	−1.29446e14	−0.928301	−0.464151	−	0.885756i	$-0.653640\pi$
$373$	−1.29446e14	−0.928301	−0.464151	+	0.885756i	$0.653640\pi$
$374$	−2.58337e14	−1.82555
$375$	0	0
$376$	−1.78077e14	−1.22201
$377$	1.15487e12	0.00781011
$378$	0	0
$379$	1.28256e14	0.842486	0.421243	−	0.906948i	$-0.361594\pi$
$379$	1.28256e14	0.842486	0.421243	+	0.906948i	$0.361594\pi$
$380$	−1.28913e14	−0.834620
$381$	0	0
$382$	−2.55703e14	−1.60837
$383$	6.22249e13	0.385808	0.192904	−	0.981218i	$-0.438209\pi$
$383$	6.22249e13	0.385808	0.192904	+	0.981218i	$0.438209\pi$
$384$	0	0
$385$	1.19855e14	0.722142
$386$	5.29276e14	3.14378
$387$	0	0
$388$	1.19023e14	0.687160
$389$	2.17341e14	1.23714	0.618569	−	0.785730i	$-0.287713\pi$
$389$	2.17341e14	1.23714	0.618569	+	0.785730i	$0.287713\pi$
$390$	0	0
$391$	−2.75300e14	−1.52347
$392$	9.22844e13	0.503564
$393$	0	0
$394$	−3.87947e14	−2.05846
$395$	−1.34932e14	−0.706045
$396$	0	0
$397$	−7.19033e13	−0.365933	−0.182966	−	0.983119i	$-0.558570\pi$
$397$	−7.19033e13	−0.365933	−0.182966	+	0.983119i	$0.558570\pi$
$398$	−5.03841e14	−2.52893
$399$	0	0
$400$	−7.18461e12	−0.0350811
$401$	4.83974e13	0.233092	0.116546	−	0.993185i	$-0.462818\pi$
$401$	4.83974e13	0.233092	0.116546	+	0.993185i	$0.462818\pi$
$402$	0	0
$403$	−5.58022e12	−0.0261501
$404$	2.98379e14	1.37934
$405$	0	0
$406$	2.21981e14	0.998672
$407$	1.69328e14	0.751553
$408$	0	0
$409$	−7.85346e13	−0.339299	−0.169650	−	0.985504i	$-0.554264\pi$
$409$	−7.85346e13	−0.339299	−0.169650	+	0.985504i	$0.554264\pi$
$410$	2.31551e14	0.987043
$411$	0	0
$412$	−5.96666e13	−0.247626
$413$	3.04366e14	1.24644
$414$	0	0
$415$	5.85140e11	0.00233343
$416$	3.78548e12	0.0148973
$417$	0	0
$418$	5.34023e14	2.04687
$419$	−2.73347e13	−0.103404	−0.0517020	−	0.998663i	$-0.516465\pi$
$419$	−2.73347e13	−0.103404	−0.0517020	+	0.998663i	$0.516465\pi$
$420$	0	0
$421$	−3.99915e14	−1.47372	−0.736862	−	0.676044i	$-0.763693\pi$
$421$	−3.99915e14	−1.47372	−0.736862	+	0.676044i	$0.763693\pi$
$422$	−4.43992e14	−1.61494
$423$	0	0
$424$	−2.43402e14	−0.862603
$425$	−2.01804e14	−0.705977
$426$	0	0
$427$	2.00139e14	0.682306
$428$	1.61744e14	0.544361
$429$	0	0
$430$	4.68258e14	1.53606
$431$	−3.70048e14	−1.19849	−0.599243	−	0.800567i	$-0.704532\pi$
$431$	−3.70048e14	−1.19849	−0.599243	+	0.800567i	$0.704532\pi$
$432$	0	0
$433$	−5.79057e14	−1.82826	−0.914130	−	0.405422i	$-0.867125\pi$
$433$	−5.79057e14	−1.82826	−0.914130	+	0.405422i	$0.867125\pi$
$434$	−1.07259e15	−3.34379
$435$	0	0
$436$	−8.71167e14	−2.64805
$437$	5.69089e14	1.70817
$438$	0	0
$439$	8.02363e13	0.234864	0.117432	−	0.993081i	$-0.462534\pi$
$439$	8.02363e13	0.234864	0.117432	+	0.993081i	$0.462534\pi$
$440$	−2.12564e14	−0.614470
$441$	0	0
$442$	−8.28033e12	−0.0233467
$443$	1.55825e14	0.433927	0.216963	−	0.976180i	$-0.430385\pi$
$443$	1.55825e14	0.433927	0.216963	+	0.976180i	$0.430385\pi$
$444$	0	0
$445$	2.70782e14	0.735596
$446$	9.01921e12	0.0242006
$447$	0	0
$448$	7.49092e14	1.96113
$449$	−5.37095e14	−1.38898	−0.694490	−	0.719502i	$-0.744370\pi$
$449$	−5.37095e14	−1.38898	−0.694490	+	0.719502i	$0.744370\pi$
$450$	0	0
$451$	−5.95657e14	−1.50323
$452$	−2.85302e14	−0.711284
$453$	0	0
$454$	8.61805e13	0.209702
$455$	3.84165e12	0.00923538
$456$	0	0
$457$	−2.81854e14	−0.661431	−0.330716	−	0.943730i	$-0.607290\pi$
$457$	−2.81854e14	−0.661431	−0.330716	+	0.943730i	$0.607290\pi$
$458$	7.24280e14	1.67937
$459$	0	0
$460$	−5.81307e14	−1.31594
$461$	2.92217e14	0.653657	0.326828	−	0.945084i	$-0.394020\pi$
$461$	2.92217e14	0.653657	0.326828	+	0.945084i	$0.394020\pi$
$462$	0	0
$463$	5.48103e14	1.19720	0.598600	−	0.801048i	$-0.295724\pi$
$463$	5.48103e14	1.19720	0.598600	+	0.801048i	$0.295724\pi$
$464$	−1.07802e13	−0.0232691
$465$	0	0
$466$	−4.61748e14	−0.973380
$467$	7.06397e14	1.47165	0.735827	−	0.677170i	$-0.236794\pi$
$467$	7.06397e14	1.47165	0.735827	+	0.677170i	$0.236794\pi$
$468$	0	0
$469$	1.35925e14	0.276599
$470$	−4.65720e14	−0.936670
$471$	0	0
$472$	−5.39797e14	−1.06059
$473$	−1.20457e15	−2.33936
$474$	0	0
$475$	4.17160e14	0.791568
$476$	−9.88362e14	−1.85386
$477$	0	0
$478$	7.53601e14	1.38130
$479$	1.43103e14	0.259301	0.129650	−	0.991560i	$-0.458615\pi$
$479$	1.43103e14	0.259301	0.129650	+	0.991560i	$0.458615\pi$
$480$	0	0
$481$	5.42738e12	0.00961153
$482$	−1.27008e15	−2.22367
$483$	0	0
$484$	4.45855e14	0.763034
$485$	1.21298e14	0.205246
$486$	0	0
$487$	−6.88418e14	−1.13879	−0.569394	−	0.822065i	$-0.692822\pi$
$487$	−6.88418e14	−1.13879	−0.569394	+	0.822065i	$0.692822\pi$
$488$	−3.54949e14	−0.580574
$489$	0	0
$490$	2.41349e14	0.385982
$491$	7.63703e14	1.20775	0.603874	−	0.797080i	$-0.293623\pi$
$491$	7.63703e14	1.20775	0.603874	+	0.797080i	$0.293623\pi$
$492$	0	0
$493$	−3.02799e14	−0.468270
$494$	1.71167e13	0.0261772
$495$	0	0
$496$	5.20889e13	0.0779104
$497$	3.23874e14	0.479088
$498$	0	0
$499$	−5.22407e14	−0.755886	−0.377943	−	0.925829i	$-0.623368\pi$
$499$	−5.22407e14	−0.755886	−0.377943	+	0.925829i	$0.623368\pi$
$500$	−9.86432e14	−1.41167
$501$	0	0
$502$	−4.70377e14	−0.658530
$503$	8.62796e14	1.19477	0.597385	−	0.801955i	$-0.296207\pi$
$503$	8.62796e14	1.19477	0.597385	+	0.801955i	$0.296207\pi$
$504$	0	0
$505$	3.04081e14	0.411990
$506$	2.40806e15	3.22730
$507$	0	0
$508$	−1.71665e14	−0.225129
$509$	9.29019e14	1.20525	0.602624	−	0.798025i	$-0.294122\pi$
$509$	9.29019e14	1.20525	0.602624	+	0.798025i	$0.294122\pi$
$510$	0	0
$511$	−8.34799e14	−1.05991
$512$	−7.34233e13	−0.0922251
$513$	0	0
$514$	8.76025e14	1.07701
$515$	−6.08069e13	−0.0739627
$516$	0	0
$517$	1.19805e15	1.42651
$518$	1.04321e15	1.22902
$519$	0	0
$520$	−6.81321e12	−0.00785838
$521$	−4.49865e14	−0.513423	−0.256711	−	0.966488i	$-0.582639\pi$
$521$	−4.49865e14	−0.513423	−0.256711	+	0.966488i	$0.582639\pi$
$522$	0	0
$523$	1.36029e15	1.52010	0.760050	−	0.649864i	$-0.225174\pi$
$523$	1.36029e15	1.52010	0.760050	+	0.649864i	$0.225174\pi$
$524$	1.97988e15	2.18935
$525$	0	0
$526$	−6.20808e12	−0.00672258
$527$	1.46309e15	1.56788
$528$	0	0
$529$	1.61337e15	1.69328
$530$	−6.36561e14	−0.661185
$531$	0	0
$532$	2.04310e15	2.07862
$533$	−1.90923e13	−0.0192246
$534$	0	0
$535$	1.64835e14	0.162594
$536$	−2.41065e14	−0.235358
$537$	0	0
$538$	2.20033e14	0.210467
$539$	−6.20860e14	−0.587835
$540$	0	0
$541$	2.72396e14	0.252706	0.126353	−	0.991985i	$-0.459673\pi$
$541$	2.72396e14	0.252706	0.126353	+	0.991985i	$0.459673\pi$
$542$	−2.12248e14	−0.194916
$543$	0	0
$544$	−9.92524e14	−0.893197
$545$	−8.87816e14	−0.790938
$546$	0	0
$547$	−2.17632e15	−1.90017	−0.950086	−	0.311987i	$-0.899006\pi$
$547$	−2.17632e15	−1.90017	−0.950086	+	0.311987i	$0.899006\pi$
$548$	2.20807e15	1.90863
$549$	0	0
$550$	1.76519e15	1.49553
$551$	6.25933e14	0.525042
$552$	0	0
$553$	2.13849e15	1.75841
$554$	−7.51872e14	−0.612125
$555$	0	0
$556$	3.85728e15	3.07872
$557$	−7.42272e14	−0.586624	−0.293312	−	0.956017i	$-0.594757\pi$
$557$	−7.42272e14	−0.586624	−0.293312	+	0.956017i	$0.594757\pi$
$558$	0	0
$559$	−3.86096e13	−0.0299178
$560$	−3.58601e13	−0.0275155
$561$	0	0
$562$	7.61284e14	0.572792
$563$	−9.90305e14	−0.737858	−0.368929	−	0.929458i	$-0.620275\pi$
$563$	−9.90305e14	−0.737858	−0.368929	+	0.929458i	$0.620275\pi$
$564$	0	0
$565$	−2.90754e14	−0.212451
$566$	−3.18358e15	−2.30370
$567$	0	0
$568$	−5.74394e14	−0.407656
$569$	−3.08229e14	−0.216649	−0.108324	−	0.994116i	$-0.534549\pi$
$569$	−3.08229e14	−0.216649	−0.108324	+	0.994116i	$0.534549\pi$
$570$	0	0
$571$	−1.47514e14	−0.101703	−0.0508515	−	0.998706i	$-0.516194\pi$
$571$	−1.47514e14	−0.101703	−0.0508515	+	0.998706i	$0.516194\pi$
$572$	4.49775e13	0.0307126
$573$	0	0
$574$	−3.66977e15	−2.45823
$575$	1.88109e15	1.24807
$576$	0	0
$577$	−1.25479e15	−0.816777	−0.408389	−	0.912808i	$-0.633909\pi$
$577$	−1.25479e15	−0.816777	−0.408389	+	0.912808i	$0.633909\pi$
$578$	−3.48259e14	−0.224543
$579$	0	0
$580$	−6.39372e14	−0.404482
$581$	−9.27366e12	−0.00581142
$582$	0	0
$583$	1.63753e15	1.00696
$584$	1.48053e15	0.901873
$585$	0	0
$586$	−3.83753e15	−2.29411
$587$	−1.32072e15	−0.782170	−0.391085	−	0.920355i	$-0.627900\pi$
$587$	−1.32072e15	−0.782170	−0.391085	+	0.920355i	$0.627900\pi$
$588$	0	0
$589$	−3.02444e15	−1.75797
$590$	−1.41172e15	−0.812945
$591$	0	0
$592$	−5.06622e13	−0.0286361
$593$	3.14526e15	1.76139	0.880695	−	0.473685i	$-0.157076\pi$
$593$	3.14526e15	1.76139	0.880695	+	0.473685i	$0.157076\pi$
$594$	0	0
$595$	−1.00725e15	−0.553725
$596$	2.73932e15	1.49207
$597$	0	0
$598$	7.71842e13	0.0412736
$599$	−3.44899e15	−1.82744	−0.913722	−	0.406339i	$-0.866805\pi$
$599$	−3.44899e15	−1.82744	−0.913722	+	0.406339i	$0.866805\pi$
$600$	0	0
$601$	2.72953e15	1.41997	0.709983	−	0.704218i	$-0.248702\pi$
$601$	2.72953e15	1.41997	0.709983	+	0.704218i	$0.248702\pi$
$602$	−7.42124e15	−3.82556
$603$	0	0
$604$	−2.22057e15	−1.12398
$605$	4.54375e14	0.227908
$606$	0	0
$607$	−1.25192e15	−0.616652	−0.308326	−	0.951281i	$-0.599769\pi$
$607$	−1.25192e15	−0.616652	−0.308326	+	0.951281i	$0.599769\pi$
$608$	2.05170e15	1.00149
$609$	0	0
$610$	−9.28289e14	−0.445009
$611$	3.84003e13	0.0182435
$612$	0	0
$613$	9.07978e14	0.423684	0.211842	−	0.977304i	$-0.432054\pi$
$613$	9.07978e14	0.423684	0.211842	+	0.977304i	$0.432054\pi$
$614$	−3.72861e15	−1.72433
$615$	0	0
$616$	3.36885e15	1.53034
$617$	−3.58549e13	−0.0161428	−0.00807142	−	0.999967i	$-0.502569\pi$
$617$	−3.58549e13	−0.0161428	−0.00807142	+	0.999967i	$0.502569\pi$
$618$	0	0
$619$	1.09634e15	0.484892	0.242446	−	0.970165i	$-0.422050\pi$
$619$	1.09634e15	0.484892	0.242446	+	0.970165i	$0.422050\pi$
$620$	3.08937e15	1.35430
$621$	0	0
$622$	−6.57373e15	−2.83116
$623$	−4.29151e15	−1.83200
$624$	0	0
$625$	8.07880e14	0.338849
$626$	7.05630e14	0.293372
$627$	0	0
$628$	−4.08776e14	−0.166996
$629$	−1.42302e15	−0.576278
$630$	0	0
$631$	1.97162e15	0.784623	0.392312	−	0.919832i	$-0.371675\pi$
$631$	1.97162e15	0.784623	0.392312	+	0.919832i	$0.371675\pi$
$632$	−3.79263e15	−1.49623
$633$	0	0
$634$	3.63207e15	1.40819
$635$	−1.74946e14	−0.0672433
$636$	0	0
$637$	−1.99001e13	−0.00751775
$638$	2.64859e15	0.991976
$639$	0	0
$640$	−2.19536e15	−0.808195
$641$	−7.96369e14	−0.290667	−0.145333	−	0.989383i	$-0.546425\pi$
$641$	−7.96369e14	−0.290667	−0.145333	+	0.989383i	$0.546425\pi$
$642$	0	0
$643$	1.07156e15	0.384466	0.192233	−	0.981349i	$-0.438427\pi$
$643$	1.07156e15	0.384466	0.192233	+	0.981349i	$0.438427\pi$
$644$	9.21291e15	3.27736
$645$	0	0
$646$	−4.48788e15	−1.56950
$647$	−1.57918e15	−0.547592	−0.273796	−	0.961788i	$-0.588279\pi$
$647$	−1.57918e15	−0.547592	−0.273796	+	0.961788i	$0.588279\pi$
$648$	0	0
$649$	3.63158e15	1.23808
$650$	5.65785e13	0.0191262
$651$	0	0
$652$	−6.87827e15	−2.28622
$653$	−1.56213e15	−0.514867	−0.257434	−	0.966296i	$-0.582877\pi$
$653$	−1.56213e15	−0.514867	−0.257434	+	0.966296i	$0.582877\pi$
$654$	0	0
$655$	2.01771e15	0.653932
$656$	1.78218e14	0.0572769
$657$	0	0
$658$	7.38101e15	2.33278
$659$	−4.33161e15	−1.35762	−0.678812	−	0.734312i	$-0.737505\pi$
$659$	−4.33161e15	−1.35762	−0.678812	+	0.734312i	$0.737505\pi$
$660$	0	0
$661$	−4.69271e15	−1.44649	−0.723246	−	0.690591i	$-0.757351\pi$
$661$	−4.69271e15	−1.44649	−0.723246	+	0.690591i	$0.757351\pi$
$662$	3.13659e15	0.958823
$663$	0	0
$664$	1.64469e13	0.00494494
$665$	2.08214e15	0.620857
$666$	0	0
$667$	2.82251e15	0.827834
$668$	1.69222e15	0.492251
$669$	0	0
$670$	−6.30451e14	−0.180401
$671$	2.38799e15	0.677732
$672$	0	0
$673$	3.37880e15	0.943365	0.471683	−	0.881768i	$-0.343647\pi$
$673$	3.37880e15	0.943365	0.471683	+	0.881768i	$0.343647\pi$
$674$	−3.42919e15	−0.949646
$675$	0	0
$676$	−6.01233e15	−1.63808
$677$	1.31683e15	0.355871	0.177936	−	0.984042i	$-0.443058\pi$
$677$	1.31683e15	0.355871	0.177936	+	0.984042i	$0.443058\pi$
$678$	0	0
$679$	−1.92240e15	−0.511165
$680$	1.78637e15	0.471164
$681$	0	0
$682$	−1.27977e16	−3.32137
$683$	6.27377e14	0.161516	0.0807579	−	0.996734i	$-0.474266\pi$
$683$	6.27377e14	0.161516	0.0807579	+	0.996734i	$0.474266\pi$
$684$	0	0
$685$	2.25027e15	0.570082
$686$	4.05272e15	1.01851
$687$	0	0
$688$	3.60403e14	0.0891357
$689$	5.24868e13	0.0128779
$690$	0	0
$691$	−3.92619e15	−0.948073	−0.474036	−	0.880505i	$-0.657203\pi$
$691$	−3.92619e15	−0.948073	−0.474036	+	0.880505i	$0.657203\pi$
$692$	1.07552e16	2.57654
$693$	0	0
$694$	−5.08447e15	−1.19886
$695$	3.93100e15	0.919573
$696$	0	0
$697$	5.00584e15	1.15265
$698$	−3.09550e15	−0.707174
$699$	0	0
$700$	6.75336e15	1.51873
$701$	7.61339e14	0.169875	0.0849375	−	0.996386i	$-0.472931\pi$
$701$	7.61339e14	0.169875	0.0849375	+	0.996386i	$0.472931\pi$
$702$	0	0
$703$	2.94160e15	0.646144
$704$	8.93788e15	1.94798
$705$	0	0
$706$	−1.27065e16	−2.72647
$707$	−4.81926e15	−1.02606
$708$	0	0
$709$	−5.48123e15	−1.14901	−0.574505	−	0.818501i	$-0.694805\pi$
$709$	−5.48123e15	−1.14901	−0.574505	+	0.818501i	$0.694805\pi$
$710$	−1.50220e15	−0.312468
$711$	0	0
$712$	7.61105e15	1.55885
$713$	−1.36380e16	−2.77179
$714$	0	0
$715$	4.58371e13	0.00917347
$716$	1.54351e15	0.306540
$717$	0	0
$718$	−8.14153e15	−1.59229
$719$	4.97029e15	0.964656	0.482328	−	0.875991i	$-0.339791\pi$
$719$	4.97029e15	0.964656	0.482328	+	0.875991i	$0.339791\pi$
$720$	0	0
$721$	9.63704e14	0.184204
$722$	7.14041e14	0.135447
$723$	0	0
$724$	1.14107e16	2.13181
$725$	2.06899e15	0.383618
$726$	0	0
$727$	2.72096e15	0.496915	0.248458	−	0.968643i	$-0.420076\pi$
$727$	2.72096e15	0.496915	0.248458	+	0.968643i	$0.420076\pi$
$728$	1.07980e14	0.0195713
$729$	0	0
$730$	3.87198e15	0.691285
$731$	1.01231e16	1.79378
$732$	0	0
$733$	−5.10336e15	−0.890808	−0.445404	−	0.895330i	$-0.646940\pi$
$733$	−5.10336e15	−0.890808	−0.445404	+	0.895330i	$0.646940\pi$
$734$	−9.91949e15	−1.71855
$735$	0	0
$736$	9.25170e15	1.57904
$737$	1.62181e15	0.274744
$738$	0	0
$739$	−4.27429e14	−0.0713378	−0.0356689	−	0.999364i	$-0.511356\pi$
$739$	−4.27429e14	−0.0713378	−0.0356689	+	0.999364i	$0.511356\pi$
$740$	−3.00476e15	−0.497777
$741$	0	0
$742$	1.00886e16	1.64668
$743$	−2.17073e15	−0.351695	−0.175848	−	0.984417i	$-0.556267\pi$
$743$	−2.17073e15	−0.351695	−0.175848	+	0.984417i	$0.556267\pi$
$744$	0	0
$745$	2.79168e15	0.445662
$746$	−9.51544e15	−1.50788
$747$	0	0
$748$	−1.17928e16	−1.84144
$749$	−2.61240e15	−0.404940
$750$	0	0
$751$	4.42784e15	0.676351	0.338175	−	0.941083i	$-0.390190\pi$
$751$	4.42784e15	0.676351	0.338175	+	0.941083i	$0.390190\pi$
$752$	−3.58449e14	−0.0543538
$753$	0	0
$754$	8.48939e13	0.0126863
$755$	−2.26300e15	−0.335720
$756$	0	0
$757$	1.23673e15	0.180821	0.0904104	−	0.995905i	$-0.471182\pi$
$757$	1.23673e15	0.180821	0.0904104	+	0.995905i	$0.471182\pi$
$758$	9.42801e15	1.36848
$759$	0	0
$760$	−3.69271e15	−0.528287
$761$	3.99580e15	0.567528	0.283764	−	0.958894i	$-0.408417\pi$
$761$	3.99580e15	0.567528	0.283764	+	0.958894i	$0.408417\pi$
$762$	0	0
$763$	1.40706e16	1.96983
$764$	−1.16725e16	−1.62237
$765$	0	0
$766$	4.57410e15	0.626683
$767$	1.16401e14	0.0158337
$768$	0	0
$769$	5.59039e15	0.749630	0.374815	−	0.927100i	$-0.377706\pi$
$769$	5.59039e15	0.749630	0.374815	+	0.927100i	$0.377706\pi$
$770$	8.81045e15	1.17300
$771$	0	0
$772$	2.41608e16	3.17114
$773$	−8.68388e15	−1.13169	−0.565844	−	0.824512i	$-0.691450\pi$
$773$	−8.68388e15	−1.13169	−0.565844	+	0.824512i	$0.691450\pi$
$774$	0	0
$775$	−9.99713e15	−1.28445
$776$	3.40941e15	0.434950
$777$	0	0
$778$	1.59765e16	2.00953
$779$	−1.03479e16	−1.29239
$780$	0	0
$781$	3.86434e15	0.475877
$782$	−2.02371e16	−2.47464
$783$	0	0
$784$	1.85758e14	0.0223980
$785$	−4.16588e14	−0.0498797
$786$	0	0
$787$	−6.71593e14	−0.0792949	−0.0396475	−	0.999214i	$-0.512623\pi$
$787$	−6.71593e14	−0.0792949	−0.0396475	+	0.999214i	$0.512623\pi$
$788$	−1.77093e16	−2.07638
$789$	0	0
$790$	−9.91877e15	−1.14686
$791$	4.60805e15	0.529110
$792$	0	0
$793$	7.65408e13	0.00866743
$794$	−5.28556e15	−0.594399
$795$	0	0
$796$	−2.29997e16	−2.55094
$797$	1.61907e15	0.178339	0.0891693	−	0.996016i	$-0.471579\pi$
$797$	1.61907e15	0.178339	0.0891693	+	0.996016i	$0.471579\pi$
$798$	0	0
$799$	−1.00683e16	−1.09382
$800$	6.78180e15	0.731728
$801$	0	0
$802$	3.55765e15	0.378621
$803$	−9.96050e15	−1.05280
$804$	0	0
$805$	9.38897e15	0.978906
$806$	−4.10198e14	−0.0424767
$807$	0	0
$808$	8.54702e15	0.873076
$809$	8.86198e15	0.899112	0.449556	−	0.893252i	$-0.351582\pi$
$809$	8.86198e15	0.899112	0.449556	+	0.893252i	$0.351582\pi$
$810$	0	0
$811$	−3.62466e15	−0.362787	−0.181394	−	0.983411i	$-0.558061\pi$
$811$	−3.62466e15	−0.362787	−0.181394	+	0.983411i	$0.558061\pi$
$812$	1.01332e16	1.00736
$813$	0	0
$814$	1.24472e16	1.22078
$815$	−7.00973e15	−0.682864
$816$	0	0
$817$	−2.09261e16	−2.01125
$818$	−5.77301e15	−0.551137
$819$	0	0
$820$	1.05700e16	0.995634
$821$	6.76268e15	0.632749	0.316375	−	0.948634i	$-0.397534\pi$
$821$	6.76268e15	0.632749	0.316375	+	0.948634i	$0.397534\pi$
$822$	0	0
$823$	9.56478e15	0.883031	0.441516	−	0.897254i	$-0.354441\pi$
$823$	9.56478e15	0.883031	0.441516	+	0.897254i	$0.354441\pi$
$824$	−1.70914e15	−0.156739
$825$	0	0
$826$	2.23737e16	2.02464
$827$	1.21184e16	1.08934	0.544671	−	0.838650i	$-0.316655\pi$
$827$	1.21184e16	1.08934	0.544671	+	0.838650i	$0.316655\pi$
$828$	0	0
$829$	−9.24590e15	−0.820162	−0.410081	−	0.912049i	$-0.634499\pi$
$829$	−9.24590e15	−0.820162	−0.410081	+	0.912049i	$0.634499\pi$
$830$	4.30132e13	0.00379029
$831$	0	0
$832$	2.86481e14	0.0249125
$833$	5.21765e15	0.450741
$834$	0	0
$835$	1.72456e15	0.147029
$836$	2.43775e16	2.06469
$837$	0	0
$838$	−2.00935e15	−0.167963
$839$	2.00044e16	1.66125	0.830623	−	0.556836i	$-0.187985\pi$
$839$	2.00044e16	1.66125	0.830623	+	0.556836i	$0.187985\pi$
$840$	0	0
$841$	−9.09607e15	−0.745548
$842$	−2.93974e16	−2.39383
$843$	0	0
$844$	−2.02677e16	−1.62900
$845$	−6.12723e15	−0.489274
$846$	0	0
$847$	−7.20122e15	−0.567606
$848$	−4.89940e14	−0.0383677
$849$	0	0
$850$	−1.48345e16	−1.14675
$851$	1.32645e16	1.01878
$852$	0	0
$853$	−2.97052e14	−0.0225223	−0.0112612	−	0.999937i	$-0.503585\pi$
$853$	−2.97052e14	−0.0225223	−0.0112612	+	0.999937i	$0.503585\pi$
$854$	1.47121e16	1.10830
$855$	0	0
$856$	4.63313e15	0.344563
$857$	9.93107e15	0.733840	0.366920	−	0.930252i	$-0.380412\pi$
$857$	9.93107e15	0.733840	0.366920	+	0.930252i	$0.380412\pi$
$858$	0	0
$859$	−9.05689e15	−0.660719	−0.330359	−	0.943855i	$-0.607170\pi$
$859$	−9.05689e15	−0.660719	−0.330359	+	0.943855i	$0.607170\pi$
$860$	2.13754e16	1.54943
$861$	0	0
$862$	−2.72019e16	−1.94675
$863$	−9.05054e15	−0.643599	−0.321799	−	0.946808i	$-0.604288\pi$
$863$	−9.05054e15	−0.643599	−0.321799	+	0.946808i	$0.604288\pi$
$864$	0	0
$865$	1.09608e16	0.769578
$866$	−4.25660e16	−2.96971
$867$	0	0
$868$	−4.89622e16	−3.37290
$869$	2.55156e16	1.74662
$870$	0	0
$871$	5.19829e13	0.00351367
$872$	−2.49545e16	−1.67613
$873$	0	0
$874$	4.18333e16	2.77465
$875$	1.59323e16	1.05011
$876$	0	0
$877$	2.38703e16	1.55367	0.776836	−	0.629703i	$-0.216823\pi$
$877$	2.38703e16	1.55367	0.776836	+	0.629703i	$0.216823\pi$
$878$	5.89810e15	0.381498
$879$	0	0
$880$	−4.27868e14	−0.0273310
$881$	2.23350e16	1.41781	0.708907	−	0.705302i	$-0.249189\pi$
$881$	2.23350e16	1.41781	0.708907	+	0.705302i	$0.249189\pi$
$882$	0	0
$883$	−1.13889e15	−0.0714003	−0.0357001	−	0.999363i	$-0.511366\pi$
$883$	−1.13889e15	−0.0714003	−0.0357001	+	0.999363i	$0.511366\pi$
$884$	−3.77987e14	−0.0235499
$885$	0	0
$886$	1.14546e16	0.704844
$887$	1.68491e16	1.03038	0.515191	−	0.857076i	$-0.327721\pi$
$887$	1.68491e16	1.03038	0.515191	+	0.857076i	$0.327721\pi$
$888$	0	0
$889$	2.77265e15	0.167469
$890$	1.99050e16	1.19486
$891$	0	0
$892$	4.11716e14	0.0244113
$893$	2.08127e16	1.22644
$894$	0	0
$895$	1.57301e15	0.0915596
$896$	3.47934e16	2.01281
$897$	0	0
$898$	−3.94814e16	−2.25618
$899$	−1.50003e16	−0.851965
$900$	0	0
$901$	−1.37616e16	−0.772118
$902$	−4.37862e16	−2.44175
$903$	0	0
$904$	−8.17242e15	−0.450220
$905$	1.16287e16	0.636744
$906$	0	0
$907$	−9.39202e15	−0.508065	−0.254032	−	0.967196i	$-0.581757\pi$
$907$	−9.39202e15	−0.508065	−0.254032	+	0.967196i	$0.581757\pi$
$908$	3.93403e15	0.211527
$909$	0	0
$910$	2.82396e14	0.0150014
$911$	−2.93244e16	−1.54838	−0.774192	−	0.632951i	$-0.781843\pi$
$911$	−2.93244e16	−1.54838	−0.774192	+	0.632951i	$0.781843\pi$
$912$	0	0
$913$	−1.10650e14	−0.00577246
$914$	−2.07188e16	−1.07439
$915$	0	0
$916$	3.30624e16	1.69399
$917$	−3.19780e16	−1.62862
$918$	0	0
$919$	−1.40482e16	−0.706942	−0.353471	−	0.935445i	$-0.614999\pi$
$919$	−1.40482e16	−0.706942	−0.353471	+	0.935445i	$0.614999\pi$
$920$	−1.66514e16	−0.832950
$921$	0	0
$922$	2.14806e16	1.06176
$923$	1.23861e14	0.00608593
$924$	0	0
$925$	9.72331e15	0.472101
$926$	4.02906e16	1.94466
$927$	0	0
$928$	1.01758e16	0.485351
$929$	−1.97725e16	−0.937508	−0.468754	−	0.883329i	$-0.655297\pi$
$929$	−1.97725e16	−0.937508	−0.468754	+	0.883329i	$0.655297\pi$
$930$	0	0
$931$	−1.07857e16	−0.505388
$932$	−2.10782e16	−0.981852
$933$	0	0
$934$	5.19266e16	2.39046
$935$	−1.20181e16	−0.550013
$936$	0	0
$937$	3.06345e16	1.38561	0.692807	−	0.721123i	$-0.256374\pi$
$937$	3.06345e16	1.38561	0.692807	+	0.721123i	$0.256374\pi$
$938$	9.99176e15	0.449290
$939$	0	0
$940$	−2.12595e16	−0.944822
$941$	−6.83559e15	−0.302018	−0.151009	−	0.988532i	$-0.548252\pi$
$941$	−6.83559e15	−0.302018	−0.151009	+	0.988532i	$0.548252\pi$
$942$	0	0
$943$	−4.66614e16	−2.03771
$944$	−1.08655e15	−0.0471742
$945$	0	0
$946$	−8.85473e16	−3.79992
$947$	−2.14054e16	−0.913269	−0.456634	−	0.889654i	$-0.650945\pi$
$947$	−2.14054e16	−0.913269	−0.456634	+	0.889654i	$0.650945\pi$
$948$	0	0
$949$	−3.19258e14	−0.0134641
$950$	3.06651e16	1.28577
$951$	0	0
$952$	−2.83115e16	−1.17344
$953$	6.24777e15	0.257463	0.128731	−	0.991680i	$-0.458910\pi$
$953$	6.24777e15	0.257463	0.128731	+	0.991680i	$0.458910\pi$
$954$	0	0
$955$	−1.18956e16	−0.484582
$956$	3.44009e16	1.39332
$957$	0	0
$958$	1.05194e16	0.421192
$959$	−3.56637e16	−1.41979
$960$	0	0
$961$	4.70713e16	1.85258
$962$	3.98962e14	0.0156124
$963$	0	0
$964$	−5.79774e16	−2.24303
$965$	2.46225e16	0.947179
$966$	0	0
$967$	−5.67065e14	−0.0215669	−0.0107834	−	0.999942i	$-0.503433\pi$
$967$	−5.67065e14	−0.0215669	−0.0107834	+	0.999942i	$0.503433\pi$
$968$	1.27714e16	0.482976
$969$	0	0
$970$	8.91652e15	0.333389
$971$	3.46020e16	1.28646	0.643228	−	0.765675i	$-0.277595\pi$
$971$	3.46020e16	1.28646	0.643228	+	0.765675i	$0.277595\pi$
$972$	0	0
$973$	−6.23008e16	−2.29020
$974$	−5.06050e16	−1.84978
$975$	0	0
$976$	−7.14474e14	−0.0258233
$977$	−4.24356e16	−1.52514	−0.762570	−	0.646905i	$-0.776063\pi$
$977$	−4.24356e16	−1.52514	−0.762570	+	0.646905i	$0.776063\pi$
$978$	0	0
$979$	−5.12047e16	−1.81972
$980$	1.10173e16	0.389341
$981$	0	0
$982$	5.61392e16	1.96179
$983$	4.46161e16	1.55041	0.775207	−	0.631708i	$-0.217646\pi$
$983$	4.46161e16	1.55041	0.775207	+	0.631708i	$0.217646\pi$
$984$	0	0
$985$	−1.80477e16	−0.620188
$986$	−2.22585e16	−0.760630
$987$	0	0
$988$	7.81357e14	0.0264050
$989$	−9.43616e16	−3.17114
$990$	0	0
$991$	1.04382e15	0.0346911	0.0173456	−	0.999850i	$-0.494478\pi$
$991$	1.04382e15	0.0346911	0.0173456	+	0.999850i	$0.494478\pi$
$992$	−4.91684e16	−1.62507
$993$	0	0
$994$	2.38077e16	0.778202
$995$	−2.34393e16	−0.761933
$996$	0	0
$997$	1.78108e15	0.0572612	0.0286306	−	0.999590i	$-0.490885\pi$
$997$	1.78108e15	0.0572612	0.0286306	+	0.999590i	$0.490885\pi$
$998$	−3.84017e16	−1.22782
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	81.12.a.d.1.10	yes	10
3.2	odd	2	inner	81.12.a.d.1.1	✓	10
9.2	odd	6		81.12.c.m.28.10		20
9.4	even	3		81.12.c.m.55.1		20
9.5	odd	6		81.12.c.m.55.10		20
9.7	even	3		81.12.c.m.28.1		20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
81.12.a.d.1.1	✓	10	3.2	odd	2	inner
81.12.a.d.1.10	yes	10	1.1	even	1	trivial
81.12.c.m.28.1		20	9.7	even	3
81.12.c.m.28.10		20	9.2	odd	6
81.12.c.m.55.1		20	9.4	even	3
81.12.c.m.55.10		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)

\(f(q)\)	\(=\)	\(q+73.5092 q^{2} +3355.60 q^{4} +3419.73 q^{5} -54197.9 q^{7} +96120.6 q^{8} +251382. q^{10} -646669. q^{11} -20727.3 q^{13} -3.98405e6 q^{14} +193480. q^{16} +5.43454e6 q^{17} -1.12341e7 q^{19} +1.14752e7 q^{20} -4.75361e7 q^{22} -5.06575e7 q^{23} -3.71336e7 q^{25} -1.52365e6 q^{26} -1.81867e8 q^{28} -5.57175e7 q^{29} +2.69221e8 q^{31} -1.82632e8 q^{32} +3.99489e8 q^{34} -1.85342e8 q^{35} -2.61847e8 q^{37} -8.25806e8 q^{38} +3.28707e8 q^{40} +9.21115e8 q^{41} +1.86274e9 q^{43} -2.16996e9 q^{44} -3.72379e9 q^{46} -1.85264e9 q^{47} +9.60090e8 q^{49} -2.72966e9 q^{50} -6.95526e7 q^{52} -2.53225e9 q^{53} -2.21143e9 q^{55} -5.20954e9 q^{56} -4.09575e9 q^{58} -5.61583e9 q^{59} -3.69275e9 q^{61} +1.97902e10 q^{62} -1.38214e10 q^{64} -7.08818e7 q^{65} -2.50794e9 q^{67} +1.82362e10 q^{68} -1.36244e10 q^{70} -5.97576e9 q^{71} +1.54028e10 q^{73} -1.92481e10 q^{74} -3.76970e10 q^{76} +3.50481e10 q^{77} -3.94570e10 q^{79} +6.61650e8 q^{80} +6.77104e10 q^{82} +1.71107e8 q^{83} +1.85847e10 q^{85} +1.36928e11 q^{86} -6.21582e10 q^{88} +7.91822e10 q^{89} +1.12338e9 q^{91} -1.69986e11 q^{92} -1.36186e11 q^{94} -3.84174e10 q^{95} +3.54701e10 q^{97} +7.05754e10 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q + 5110 q^{4} - 11836 q^{7} - 117642 q^{10} + 27110 q^{13} - 5117294 q^{16} - 24650908 q^{19} - 27880836 q^{22} - 43357796 q^{25} - 177204868 q^{28} - 56217184 q^{31} + 302628942 q^{34} - 636530326 q^{37}+ \cdots - 413983885420 q^{97}+O(q^{100}) \) 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

Representations

Groups

Database

Properties

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