LMFDB - Embedded newform 9.12.a.a.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [9,12,Mod(1,9)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(9, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 12, names="a")

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

chi := DirichletCharacter("9.1");

S:= CuspForms(chi, 12);

N := Newforms(S);

Level:	$ N $	$=$	$ 9 = 3^{2} $
Weight:	$ k $	$=$	$ 12 $
Character orbit:	$[\chi]$	$=$	9.a (trivial)

Newform invariants

comment: select newform

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

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

Self dual:	yes
Analytic conductor:	$6.91508862504$
Analytic rank:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 3)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	9.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q-78.0000 q^{2} +4036.00 q^{4} +5370.00 q^{5} -27760.0 q^{7} -155064. q^{8} +O(q^{10})$

\(q-78.0000 q^{2} +4036.00 q^{4} +5370.00 q^{5} -27760.0 q^{7} -155064. q^{8} -418860. q^{10} -637836. q^{11} +766214. q^{13} +2.16528e6 q^{14} +3.82926e6 q^{16} -3.08435e6 q^{17} -1.95114e7 q^{19} +2.16733e7 q^{20} +4.97512e7 q^{22} -1.53124e7 q^{23} -1.99912e7 q^{25} -5.97647e7 q^{26} -1.12039e8 q^{28} -1.07513e7 q^{29} -5.09374e7 q^{31} +1.88885e7 q^{32} +2.40580e8 q^{34} -1.49071e8 q^{35} +6.64741e8 q^{37} +1.52189e9 q^{38} -8.32694e8 q^{40} -8.98833e8 q^{41} -9.57947e8 q^{43} -2.57431e9 q^{44} +1.19436e9 q^{46} +1.55574e9 q^{47} -1.20671e9 q^{49} +1.55932e9 q^{50} +3.09244e9 q^{52} -3.79242e9 q^{53} -3.42518e9 q^{55} +4.30458e9 q^{56} +8.38598e8 q^{58} -5.55307e8 q^{59} +4.95042e9 q^{61} +3.97312e9 q^{62} -9.31563e9 q^{64} +4.11457e9 q^{65} +5.29240e9 q^{67} -1.24485e10 q^{68} +1.16276e10 q^{70} +1.48311e10 q^{71} +1.39710e10 q^{73} -5.18498e10 q^{74} -7.87480e10 q^{76} +1.77063e10 q^{77} +3.72054e9 q^{79} +2.05631e10 q^{80} +7.01090e10 q^{82} -8.76845e9 q^{83} -1.65630e10 q^{85} +7.47199e10 q^{86} +9.89054e10 q^{88} +2.54728e10 q^{89} -2.12701e10 q^{91} -6.18007e10 q^{92} -1.21348e11 q^{94} -1.04776e11 q^{95} -3.90925e10 q^{97} +9.41233e10 q^{98} +O(q^{100})\)

Display 100 coefficients 10 coefficients

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 $

$2$	−78.0000	−1.72357	−0.861786	−	0.507271i	$-0.830654\pi$
$2$	−78.0000	−1.72357	−0.861786	+	0.507271i	$0.830654\pi$
$3$	0	0
$3$	0	0
$4$	4036.00	1.97070
$5$	5370.00	0.768492	0.384246	−	0.923231i	$-0.374461\pi$
$5$	5370.00	0.768492	0.384246	+	0.923231i	$0.374461\pi$
$6$	0	0
$7$	−27760.0	−0.624281	−0.312141	−	0.950036i	$-0.601046\pi$
$7$	−27760.0	−0.624281	−0.312141	+	0.950036i	$0.601046\pi$
$8$	−155064.	−1.67308
$9$	0	0
$10$	−418860.	−1.32455
$11$	−637836.	−1.19412	−0.597062	−	0.802195i	$-0.703665\pi$
$11$	−637836.	−1.19412	−0.597062	+	0.802195i	$0.703665\pi$
$12$	0	0
$13$	766214.	0.572350	0.286175	−	0.958177i	$-0.407616\pi$
$13$	766214.	0.572350	0.286175	+	0.958177i	$0.407616\pi$
$14$	2.16528e6	1.07599
$15$	0	0
$16$	3.82926e6	0.912968
$17$	−3.08435e6	−0.526860	−0.263430	−	0.964679i	$-0.584854\pi$
$17$	−3.08435e6	−0.526860	−0.263430	+	0.964679i	$0.584854\pi$
$18$	0	0
$19$	−1.95114e7	−1.80777	−0.903886	−	0.427773i	$-0.859298\pi$
$19$	−1.95114e7	−1.80777	−0.903886	+	0.427773i	$0.859298\pi$
$20$	2.16733e7	1.51447
$21$	0	0
$22$	4.97512e7	2.05816
$23$	−1.53124e7	−0.496066	−0.248033	−	0.968752i	$-0.579784\pi$
$23$	−1.53124e7	−0.496066	−0.248033	+	0.968752i	$0.579784\pi$
$24$	0	0
$25$	−1.99912e7	−0.409420
$26$	−5.97647e7	−0.986487
$27$	0	0
$28$	−1.12039e8	−1.23027
$29$	−1.07513e7	−0.0973353	−0.0486677	−	0.998815i	$-0.515498\pi$
$29$	−1.07513e7	−0.0973353	−0.0486677	+	0.998815i	$0.515498\pi$
$30$	0	0
$31$	−5.09374e7	−0.319556	−0.159778	−	0.987153i	$-0.551078\pi$
$31$	−5.09374e7	−0.319556	−0.159778	+	0.987153i	$0.551078\pi$
$32$	1.88885e7	0.0995112
$33$	0	0
$34$	2.40580e8	0.908081
$35$	−1.49071e8	−0.479755
$36$	0	0
$37$	6.64741e8	1.57595	0.787976	−	0.615706i	$-0.211129\pi$
$37$	6.64741e8	1.57595	0.787976	+	0.615706i	$0.211129\pi$
$38$	1.52189e9	3.11583
$39$	0	0
$40$	−8.32694e8	−1.28575
$41$	−8.98833e8	−1.21162	−0.605812	−	0.795608i	$-0.707152\pi$
$41$	−8.98833e8	−1.21162	−0.605812	+	0.795608i	$0.707152\pi$
$42$	0	0
$43$	−9.57947e8	−0.993722	−0.496861	−	0.867830i	$-0.665514\pi$
$43$	−9.57947e8	−0.993722	−0.496861	+	0.867830i	$0.665514\pi$
$44$	−2.57431e9	−2.35326
$45$	0	0
$46$	1.19436e9	0.855005
$47$	1.55574e9	0.989462	0.494731	−	0.869046i	$-0.335267\pi$
$47$	1.55574e9	0.989462	0.494731	+	0.869046i	$0.335267\pi$
$48$	0	0
$49$	−1.20671e9	−0.610273
$50$	1.55932e9	0.705666
$51$	0	0
$52$	3.09244e9	1.12793
$53$	−3.79242e9	−1.24566	−0.622829	−	0.782358i	$-0.714017\pi$
$53$	−3.79242e9	−1.24566	−0.622829	+	0.782358i	$0.714017\pi$
$54$	0	0
$55$	−3.42518e9	−0.917674
$56$	4.30458e9	1.04447
$57$	0	0
$58$	8.38598e8	0.167765
$59$	−5.55307e8	−0.101122	−0.0505612	−	0.998721i	$-0.516101\pi$
$59$	−5.55307e8	−0.101122	−0.0505612	+	0.998721i	$0.516101\pi$
$60$	0	0
$61$	4.95042e9	0.750461	0.375230	−	0.926932i	$-0.377564\pi$
$61$	4.95042e9	0.750461	0.375230	+	0.926932i	$0.377564\pi$
$62$	3.97312e9	0.550779
$63$	0	0
$64$	−9.31563e9	−1.08448
$65$	4.11457e9	0.439846
$66$	0	0
$67$	5.29240e9	0.478896	0.239448	−	0.970909i	$-0.423034\pi$
$67$	5.29240e9	0.478896	0.239448	+	0.970909i	$0.423034\pi$
$68$	−1.24485e10	−1.03828
$69$	0	0
$70$	1.16276e10	0.826893
$71$	1.48311e10	0.975556	0.487778	−	0.872968i	$-0.337808\pi$
$71$	1.48311e10	0.975556	0.487778	+	0.872968i	$0.337808\pi$
$72$	0	0
$73$	1.39710e10	0.788773	0.394386	−	0.918945i	$-0.370957\pi$
$73$	1.39710e10	0.788773	0.394386	+	0.918945i	$0.370957\pi$
$74$	−5.18498e10	−2.71627
$75$	0	0
$76$	−7.87480e10	−3.56258
$77$	1.77063e10	0.745469
$78$	0	0
$79$	3.72054e9	0.136037	0.0680185	−	0.997684i	$-0.478332\pi$
$79$	3.72054e9	0.136037	0.0680185	+	0.997684i	$0.478332\pi$
$80$	2.05631e10	0.701608
$81$	0	0
$82$	7.01090e10	2.08832
$83$	−8.76845e9	−0.244339	−0.122170	−	0.992509i	$-0.538985\pi$
$83$	−8.76845e9	−0.244339	−0.122170	+	0.992509i	$0.538985\pi$
$84$	0	0
$85$	−1.65630e10	−0.404888
$86$	7.47199e10	1.71275
$87$	0	0
$88$	9.89054e10	1.99786
$89$	2.54728e10	0.483539	0.241769	−	0.970334i	$-0.422272\pi$
$89$	2.54728e10	0.483539	0.241769	+	0.970334i	$0.422272\pi$
$90$	0	0
$91$	−2.12701e10	−0.357307
$92$	−6.18007e10	−0.977598
$93$	0	0
$94$	−1.21348e11	−1.70541
$95$	−1.04776e11	−1.38926
$96$	0	0
$97$	−3.90925e10	−0.462220	−0.231110	−	0.972928i	$-0.574236\pi$
$97$	−3.90925e10	−0.462220	−0.231110	+	0.972928i	$0.574236\pi$
$98$	9.41233e10	1.05185
$99$	0	0
$100$	−8.06846e10	−0.806846
$101$	−9.31078e9	−0.0881492	−0.0440746	−	0.999028i	$-0.514034\pi$
$101$	−9.31078e9	−0.0881492	−0.0440746	+	0.999028i	$0.514034\pi$
$102$	0	0
$103$	4.85751e10	0.412865	0.206433	−	0.978461i	$-0.433815\pi$
$103$	4.85751e10	0.412865	0.206433	+	0.978461i	$0.433815\pi$
$104$	−1.18812e11	−0.957586
$105$	0	0
$106$	2.95809e11	2.14698
$107$	−2.25596e11	−1.55496	−0.777482	−	0.628906i	$-0.783503\pi$
$107$	−2.25596e11	−1.55496	−0.777482	+	0.628906i	$0.783503\pi$
$108$	0	0
$109$	−6.94512e10	−0.432348	−0.216174	−	0.976355i	$-0.569358\pi$
$109$	−6.94512e10	−0.432348	−0.216174	+	0.976355i	$0.569358\pi$
$110$	2.67164e11	1.58168
$111$	0	0
$112$	−1.06300e11	−0.569949
$113$	3.59665e11	1.83640	0.918198	−	0.396122i	$-0.129644\pi$
$113$	3.59665e11	1.83640	0.918198	+	0.396122i	$0.129644\pi$
$114$	0	0
$115$	−8.22274e10	−0.381222
$116$	−4.33921e10	−0.191819
$117$	0	0
$118$	4.33139e10	0.174292
$119$	8.56217e10	0.328909
$120$	0	0
$121$	1.21523e11	0.425931
$122$	−3.86133e11	−1.29347
$123$	0	0
$124$	−2.05583e11	−0.629751
$125$	−3.69560e11	−1.08313
$126$	0	0
$127$	2.50273e11	0.672192	0.336096	−	0.941828i	$-0.390893\pi$
$127$	2.50273e11	0.672192	0.336096	+	0.941828i	$0.390893\pi$
$128$	6.87936e11	1.76967
$129$	0	0
$130$	−3.20936e11	−0.758107
$131$	9.78918e10	0.221694	0.110847	−	0.993837i	$-0.464644\pi$
$131$	9.78918e10	0.221694	0.110847	+	0.993837i	$0.464644\pi$
$132$	0	0
$133$	5.41637e11	1.12856
$134$	−4.12807e11	−0.825412
$135$	0	0
$136$	4.78272e11	0.881477
$137$	1.55015e11	0.274416	0.137208	−	0.990542i	$-0.456187\pi$
$137$	1.55015e11	0.274416	0.137208	+	0.990542i	$0.456187\pi$
$138$	0	0
$139$	−1.12627e12	−1.84102	−0.920512	−	0.390715i	$-0.872228\pi$
$139$	−1.12627e12	−1.84102	−0.920512	+	0.390715i	$0.872228\pi$
$140$	−6.01651e11	−0.945455
$141$	0	0
$142$	−1.15682e12	−1.68144
$143$	−4.88719e11	−0.683457
$144$	0	0
$145$	−5.77343e10	−0.0748014
$146$	−1.08974e12	−1.35951
$147$	0	0
$148$	2.68289e12	3.10573
$149$	1.38458e12	1.54452	0.772261	−	0.635306i	$-0.219126\pi$
$149$	1.38458e12	1.54452	0.772261	+	0.635306i	$0.219126\pi$
$150$	0	0
$151$	−6.98601e11	−0.724196	−0.362098	−	0.932140i	$-0.617939\pi$
$151$	−6.98601e11	−0.724196	−0.362098	+	0.932140i	$0.617939\pi$
$152$	3.02552e12	3.02454
$153$	0	0
$154$	−1.38109e12	−1.28487
$155$	−2.73534e11	−0.245576
$156$	0	0
$157$	2.13127e12	1.78316	0.891580	−	0.452863i	$-0.149597\pi$
$157$	2.13127e12	1.78316	0.891580	+	0.452863i	$0.149597\pi$
$158$	−2.90202e11	−0.234470
$159$	0	0
$160$	1.01431e11	0.0764736
$161$	4.25071e11	0.309684
$162$	0	0
$163$	−1.63564e11	−0.111342	−0.0556708	−	0.998449i	$-0.517730\pi$
$163$	−1.63564e11	−0.111342	−0.0556708	+	0.998449i	$0.517730\pi$
$164$	−3.62769e12	−2.38775
$165$	0	0
$166$	6.83939e11	0.421137
$167$	8.80943e9	0.00524816	0.00262408	−	0.999997i	$-0.499165\pi$
$167$	8.80943e9	0.00524816	0.00262408	+	0.999997i	$0.499165\pi$
$168$	0	0
$169$	−1.20508e12	−0.672416
$170$	1.29191e12	0.697853
$171$	0	0
$172$	−3.86627e12	−1.95833
$173$	7.30852e11	0.358571	0.179286	−	0.983797i	$-0.442621\pi$
$173$	7.30852e11	0.358571	0.179286	+	0.983797i	$0.442621\pi$
$174$	0	0
$175$	5.54956e11	0.255593
$176$	−2.44244e12	−1.09020
$177$	0	0
$178$	−1.98688e12	−0.833414
$179$	−3.92371e12	−1.59590	−0.797950	−	0.602724i	$-0.794082\pi$
$179$	−3.92371e12	−1.59590	−0.797950	+	0.602724i	$0.794082\pi$
$180$	0	0
$181$	2.27931e12	0.872110	0.436055	−	0.899920i	$-0.356375\pi$
$181$	2.27931e12	0.872110	0.436055	+	0.899920i	$0.356375\pi$
$182$	1.65907e12	0.615845
$183$	0	0
$184$	2.37440e12	0.829956
$185$	3.56966e12	1.21111
$186$	0	0
$187$	1.96731e12	0.629136
$188$	6.27897e12	1.94994
$189$	0	0
$190$	8.17255e12	2.39449
$191$	−3.38709e12	−0.964147	−0.482073	−	0.876131i	$-0.660116\pi$
$191$	−3.38709e12	−0.964147	−0.482073	+	0.876131i	$0.660116\pi$
$192$	0	0
$193$	−4.92921e12	−1.32499	−0.662494	−	0.749067i	$-0.730502\pi$
$193$	−4.92921e12	−1.32499	−0.662494	+	0.749067i	$0.730502\pi$
$194$	3.04921e12	0.796670
$195$	0	0
$196$	−4.87028e12	−1.20267
$197$	−6.35785e12	−1.52667	−0.763337	−	0.646001i	$-0.776440\pi$
$197$	−6.35785e12	−1.52667	−0.763337	+	0.646001i	$0.776440\pi$
$198$	0	0
$199$	−3.78554e12	−0.859875	−0.429938	−	0.902859i	$-0.641464\pi$
$199$	−3.78554e12	−0.859875	−0.429938	+	0.902859i	$0.641464\pi$
$200$	3.09992e12	0.684992
$201$	0	0
$202$	7.26241e11	0.151932
$203$	2.98455e11	0.0607646
$204$	0	0
$205$	−4.82674e12	−0.931123
$206$	−3.78885e12	−0.711604
$207$	0	0
$208$	2.93404e12	0.522537
$209$	1.24451e13	2.15870
$210$	0	0
$211$	1.79494e11	0.0295458	0.0147729	−	0.999891i	$-0.495297\pi$
$211$	1.79494e11	0.0295458	0.0147729	+	0.999891i	$0.495297\pi$
$212$	−1.53062e13	−2.45482
$213$	0	0
$214$	1.75965e13	2.68009
$215$	−5.14418e12	−0.763668
$216$	0	0
$217$	1.41402e12	0.199493
$218$	5.41719e12	0.745184
$219$	0	0
$220$	−1.38240e13	−1.80846
$221$	−2.36328e12	−0.301548
$222$	0	0
$223$	2.22568e12	0.270263	0.135132	−	0.990828i	$-0.456854\pi$
$223$	2.22568e12	0.270263	0.135132	+	0.990828i	$0.456854\pi$
$224$	−5.24344e11	−0.0621230
$225$	0	0
$226$	−2.80538e13	−3.16516
$227$	4.15848e11	0.0457923	0.0228961	−	0.999738i	$-0.492711\pi$
$227$	4.15848e11	0.0457923	0.0228961	+	0.999738i	$0.492711\pi$
$228$	0	0
$229$	1.81248e13	1.90186	0.950928	−	0.309414i	$-0.100133\pi$
$229$	1.81248e13	1.90186	0.950928	+	0.309414i	$0.100133\pi$
$230$	6.41374e12	0.657064
$231$	0	0
$232$	1.66713e12	0.162850
$233$	1.87641e10	0.00179007	0.000895033	−	1.00000i	$-0.499715\pi$
$233$	1.87641e10	0.00179007	0.000895033		1.00000i	$0.499715\pi$
$234$	0	0
$235$	8.35433e12	0.760394
$236$	−2.24122e12	−0.199282
$237$	0	0
$238$	−6.67849e12	−0.566898
$239$	1.76252e13	1.46200	0.730999	−	0.682379i	$-0.239054\pi$
$239$	1.76252e13	1.46200	0.730999	+	0.682379i	$0.239054\pi$
$240$	0	0
$241$	−8.90117e11	−0.0705267	−0.0352633	−	0.999378i	$-0.511227\pi$
$241$	−8.90117e11	−0.0705267	−0.0352633	+	0.999378i	$0.511227\pi$
$242$	−9.47880e12	−0.734123
$243$	0	0
$244$	1.99799e13	1.47894
$245$	−6.48003e12	−0.468990
$246$	0	0
$247$	−1.49499e13	−1.03468
$248$	7.89856e12	0.534643
$249$	0	0
$250$	2.88257e13	1.86685
$251$	2.42280e13	1.53502	0.767508	−	0.641040i	$-0.221497\pi$
$251$	2.42280e13	1.53502	0.767508	+	0.641040i	$0.221497\pi$
$252$	0	0
$253$	9.76677e12	0.592364
$254$	−1.95213e13	−1.15857
$255$	0	0
$256$	−3.45806e13	−1.96568
$257$	−7.80492e12	−0.434246	−0.217123	−	0.976144i	$-0.569667\pi$
$257$	−7.80492e12	−0.434246	−0.217123	+	0.976144i	$0.569667\pi$
$258$	0	0
$259$	−1.84532e13	−0.983837
$260$	1.66064e13	0.866806
$261$	0	0
$262$	−7.63556e12	−0.382106
$263$	−1.65956e13	−0.813272	−0.406636	−	0.913590i	$-0.633298\pi$
$263$	−1.65956e13	−0.813272	−0.406636	+	0.913590i	$0.633298\pi$
$264$	0	0
$265$	−2.03653e13	−0.957278
$266$	−4.22477e13	−1.94515
$267$	0	0
$268$	2.13601e13	0.943762
$269$	−2.85236e13	−1.23471	−0.617357	−	0.786683i	$-0.711797\pi$
$269$	−2.85236e13	−1.23471	−0.617357	+	0.786683i	$0.711797\pi$
$270$	0	0
$271$	2.33800e13	0.971658	0.485829	−	0.874054i	$-0.338518\pi$
$271$	2.33800e13	0.971658	0.485829	+	0.874054i	$0.338518\pi$
$272$	−1.18108e13	−0.481006
$273$	0	0
$274$	−1.20911e13	−0.472976
$275$	1.27511e13	0.488898
$276$	0	0
$277$	−3.03641e13	−1.11872	−0.559361	−	0.828924i	$-0.688953\pi$
$277$	−3.03641e13	−1.11872	−0.559361	+	0.828924i	$0.688953\pi$
$278$	8.78487e13	3.17314
$279$	0	0
$280$	2.31156e13	0.802667
$281$	−1.59749e13	−0.543942	−0.271971	−	0.962306i	$-0.587675\pi$
$281$	−1.59749e13	−0.543942	−0.271971	+	0.962306i	$0.587675\pi$
$282$	0	0
$283$	2.87045e13	0.939993	0.469997	−	0.882668i	$-0.344255\pi$
$283$	2.87045e13	0.939993	0.469997	+	0.882668i	$0.344255\pi$
$284$	5.98583e13	1.92253
$285$	0	0
$286$	3.81201e13	1.17799
$287$	2.49516e13	0.756394
$288$	0	0
$289$	−2.47587e13	−0.722419
$290$	4.50327e12	0.128926
$291$	0	0
$292$	5.63870e13	1.55444
$293$	−4.81754e13	−1.30333	−0.651663	−	0.758508i	$-0.725929\pi$
$293$	−4.81754e13	−1.30333	−0.651663	+	0.758508i	$0.725929\pi$
$294$	0	0
$295$	−2.98200e12	−0.0777117
$296$	−1.03077e14	−2.63669
$297$	0	0
$298$	−1.07997e14	−2.66209
$299$	−1.17325e13	−0.283923
$300$	0	0
$301$	2.65926e13	0.620362
$302$	5.44909e13	1.24820
$303$	0	0
$304$	−7.47143e13	−1.65044
$305$	2.65838e13	0.576723
$306$	0	0
$307$	2.57350e13	0.538597	0.269298	−	0.963057i	$-0.413208\pi$
$307$	2.57350e13	0.538597	0.269298	+	0.963057i	$0.413208\pi$
$308$	7.14627e13	1.46910
$309$	0	0
$310$	2.13356e13	0.423269
$311$	−3.46043e13	−0.674446	−0.337223	−	0.941425i	$-0.609488\pi$
$311$	−3.46043e13	−0.674446	−0.337223	+	0.941425i	$0.609488\pi$
$312$	0	0
$313$	1.26066e13	0.237194	0.118597	−	0.992942i	$-0.462160\pi$
$313$	1.26066e13	0.237194	0.118597	+	0.992942i	$0.462160\pi$
$314$	−1.66239e14	−3.07341
$315$	0	0
$316$	1.50161e13	0.268089
$317$	8.18243e13	1.43568	0.717838	−	0.696210i	$-0.245132\pi$
$317$	8.18243e13	1.43568	0.717838	+	0.696210i	$0.245132\pi$
$318$	0	0
$319$	6.85754e12	0.116230
$320$	−5.00250e13	−0.833416
$321$	0	0
$322$	−3.31555e13	−0.533764
$323$	6.01801e13	0.952443
$324$	0	0
$325$	−1.53176e13	−0.234332
$326$	1.27580e13	0.191905
$327$	0	0
$328$	1.39377e14	2.02714
$329$	−4.31874e13	−0.617703
$330$	0	0
$331$	−3.40115e13	−0.470513	−0.235256	−	0.971933i	$-0.575593\pi$
$331$	−3.40115e13	−0.470513	−0.235256	+	0.971933i	$0.575593\pi$
$332$	−3.53895e13	−0.481520
$333$	0	0
$334$	−6.87135e11	−0.00904558
$335$	2.84202e13	0.368028
$336$	0	0
$337$	−5.99439e13	−0.751244	−0.375622	−	0.926773i	$-0.622571\pi$
$337$	−5.99439e13	−0.751244	−0.375622	+	0.926773i	$0.622571\pi$
$338$	9.39960e13	1.15896
$339$	0	0
$340$	−6.68482e13	−0.797913
$341$	3.24897e13	0.381590
$342$	0	0
$343$	8.83888e13	1.00526
$344$	1.48543e14	1.66257
$345$	0	0
$346$	−5.70064e13	−0.618024
$347$	−9.78685e13	−1.04431	−0.522157	−	0.852850i	$-0.674872\pi$
$347$	−9.78685e13	−1.04431	−0.522157	+	0.852850i	$0.674872\pi$
$348$	0	0
$349$	−1.42790e14	−1.47624	−0.738120	−	0.674670i	$-0.764286\pi$
$349$	−1.42790e14	−1.47624	−0.738120	+	0.674670i	$0.764286\pi$
$350$	−4.32866e13	−0.440534
$351$	0	0
$352$	−1.20478e13	−0.118829
$353$	−1.44246e14	−1.40069	−0.700346	−	0.713804i	$-0.746971\pi$
$353$	−1.44246e14	−1.40069	−0.700346	+	0.713804i	$0.746971\pi$
$354$	0	0
$355$	7.96429e13	0.749707
$356$	1.02808e14	0.952911
$357$	0	0
$358$	3.06050e14	2.75065
$359$	1.24349e14	1.10059	0.550293	−	0.834972i	$-0.314516\pi$
$359$	1.24349e14	1.10059	0.550293	+	0.834972i	$0.314516\pi$
$360$	0	0
$361$	2.64205e14	2.26804
$362$	−1.77786e14	−1.50314
$363$	0	0
$364$	−8.58461e13	−0.704147
$365$	7.50243e13	0.606165
$366$	0	0
$367$	−1.60110e14	−1.25532	−0.627659	−	0.778488i	$-0.715987\pi$
$367$	−1.60110e14	−1.25532	−0.627659	+	0.778488i	$0.715987\pi$
$368$	−5.86351e13	−0.452892
$369$	0	0
$370$	−2.78433e14	−2.08743
$371$	1.05277e14	0.777641
$372$	0	0
$373$	−3.47258e13	−0.249031	−0.124516	−	0.992218i	$-0.539738\pi$
$373$	−3.47258e13	−0.249031	−0.124516	+	0.992218i	$0.539738\pi$
$374$	−1.53450e14	−1.08436
$375$	0	0
$376$	−2.41239e14	−1.65545
$377$	−8.23777e12	−0.0557099
$378$	0	0
$379$	−1.46500e14	−0.962327	−0.481163	−	0.876631i	$-0.659786\pi$
$379$	−1.46500e14	−0.962327	−0.481163	+	0.876631i	$0.659786\pi$
$380$	−4.22877e14	−2.73782
$381$	0	0
$382$	2.64193e14	1.66178
$383$	−6.43419e13	−0.398933	−0.199467	−	0.979905i	$-0.563921\pi$
$383$	−6.43419e13	−0.398933	−0.199467	+	0.979905i	$0.563921\pi$
$384$	0	0
$385$	9.50830e13	0.572887
$386$	3.84478e14	2.28371
$387$	0	0
$388$	−1.57777e14	−0.910899
$389$	3.98900e13	0.227061	0.113530	−	0.993535i	$-0.463784\pi$
$389$	3.98900e13	0.227061	0.113530	+	0.993535i	$0.463784\pi$
$390$	0	0
$391$	4.72287e13	0.261357
$392$	1.87117e14	1.02103
$393$	0	0
$394$	4.95912e14	2.63133
$395$	1.99793e13	0.104543
$396$	0	0
$397$	1.06552e14	0.542268	0.271134	−	0.962542i	$-0.412601\pi$
$397$	1.06552e14	0.542268	0.271134	+	0.962542i	$0.412601\pi$
$398$	2.95272e14	1.48206
$399$	0	0
$400$	−7.65517e13	−0.373787
$401$	−3.41445e13	−0.164447	−0.0822236	−	0.996614i	$-0.526202\pi$
$401$	−3.41445e13	−0.164447	−0.0822236	+	0.996614i	$0.526202\pi$
$402$	0	0
$403$	−3.90289e13	−0.182898
$404$	−3.75783e13	−0.173716
$405$	0	0
$406$	−2.32795e13	−0.104732
$407$	−4.23996e14	−1.88188
$408$	0	0
$409$	5.33349e13	0.230427	0.115213	−	0.993341i	$-0.463245\pi$
$409$	5.33349e13	0.230427	0.115213	+	0.993341i	$0.463245\pi$
$410$	3.76485e14	1.60486
$411$	0	0
$412$	1.96049e14	0.813635
$413$	1.54153e13	0.0631288
$414$	0	0
$415$	−4.70866e13	−0.187773
$416$	1.44726e13	0.0569553
$417$	0	0
$418$	−9.70716e14	−3.72068
$419$	1.01288e14	0.383159	0.191580	−	0.981477i	$-0.438639\pi$
$419$	1.01288e14	0.383159	0.191580	+	0.981477i	$0.438639\pi$
$420$	0	0
$421$	−1.57928e14	−0.581981	−0.290991	−	0.956726i	$-0.593985\pi$
$421$	−1.57928e14	−0.581981	−0.290991	+	0.956726i	$0.593985\pi$
$422$	−1.40005e13	−0.0509244
$423$	0	0
$424$	5.88067e14	2.08408
$425$	6.16600e13	0.215707
$426$	0	0
$427$	−1.37424e14	−0.468499
$428$	−9.10504e14	−3.06437
$429$	0	0
$430$	4.01246e14	1.31624
$431$	−5.13171e14	−1.66202	−0.831012	−	0.556254i	$-0.812238\pi$
$431$	−5.13171e14	−1.66202	−0.831012	+	0.556254i	$0.812238\pi$
$432$	0	0
$433$	−7.49248e13	−0.236560	−0.118280	−	0.992980i	$-0.537738\pi$
$433$	−7.49248e13	−0.236560	−0.118280	+	0.992980i	$0.537738\pi$
$434$	−1.10294e14	−0.343841
$435$	0	0
$436$	−2.80305e14	−0.852030
$437$	2.98766e14	0.896773
$438$	0	0
$439$	−3.68335e14	−1.07817	−0.539086	−	0.842250i	$-0.681230\pi$
$439$	−3.68335e14	−1.07817	−0.539086	+	0.842250i	$0.681230\pi$
$440$	5.31122e14	1.53534
$441$	0	0
$442$	1.84335e14	0.519740
$443$	1.11248e14	0.309793	0.154896	−	0.987931i	$-0.450496\pi$
$443$	1.11248e14	0.309793	0.154896	+	0.987931i	$0.450496\pi$
$444$	0	0
$445$	1.36789e14	0.371595
$446$	−1.73603e14	−0.465818
$447$	0	0
$448$	2.58602e14	0.677022
$449$	8.83314e13	0.228434	0.114217	−	0.993456i	$-0.463564\pi$
$449$	8.83314e13	0.228434	0.114217	+	0.993456i	$0.463564\pi$
$450$	0	0
$451$	5.73308e14	1.44683
$452$	1.45161e15	3.61899
$453$	0	0
$454$	−3.24361e13	−0.0789263
$455$	−1.14220e14	−0.274588
$456$	0	0
$457$	−9.42094e12	−0.0221083	−0.0110541	−	0.999939i	$-0.503519\pi$
$457$	−9.42094e12	−0.0221083	−0.0110541	+	0.999939i	$0.503519\pi$
$458$	−1.41373e15	−3.27799
$459$	0	0
$460$	−3.31870e14	−0.751276
$461$	6.97134e14	1.55941	0.779706	−	0.626146i	$-0.215369\pi$
$461$	6.97134e14	1.55941	0.779706	+	0.626146i	$0.215369\pi$
$462$	0	0
$463$	−1.87941e14	−0.410513	−0.205256	−	0.978708i	$-0.565803\pi$
$463$	−1.87941e14	−0.410513	−0.205256	+	0.978708i	$0.565803\pi$
$464$	−4.11694e13	−0.0888640
$465$	0	0
$466$	−1.46360e12	−0.00308531
$467$	−3.20007e14	−0.666678	−0.333339	−	0.942807i	$-0.608175\pi$
$467$	−3.20007e14	−0.666678	−0.333339	+	0.942807i	$0.608175\pi$
$468$	0	0
$469$	−1.46917e14	−0.298966
$470$	−6.51638e14	−1.31059
$471$	0	0
$472$	8.61081e13	0.169185
$473$	6.11013e14	1.18663
$474$	0	0
$475$	3.90057e14	0.740139
$476$	3.45569e14	0.648181
$477$	0	0
$478$	−1.37477e15	−2.51986
$479$	1.50382e14	0.272491	0.136245	−	0.990675i	$-0.456496\pi$
$479$	1.50382e14	0.272491	0.136245	+	0.990675i	$0.456496\pi$
$480$	0	0
$481$	5.09334e14	0.901996
$482$	6.94291e13	0.121558
$483$	0	0
$484$	4.90467e14	0.839384
$485$	−2.09927e14	−0.355212
$486$	0	0
$487$	1.76546e14	0.292045	0.146022	−	0.989281i	$-0.453353\pi$
$487$	1.76546e14	0.292045	0.146022	+	0.989281i	$0.453353\pi$
$488$	−7.67632e14	−1.25558
$489$	0	0
$490$	5.05442e14	0.808338
$491$	8.60958e14	1.36155	0.680775	−	0.732492i	$-0.261643\pi$
$491$	8.60958e14	1.36155	0.680775	+	0.732492i	$0.261643\pi$
$492$	0	0
$493$	3.31607e13	0.0512821
$494$	1.16609e15	1.78334
$495$	0	0
$496$	−1.95053e14	−0.291745
$497$	−4.11711e14	−0.609021
$498$	0	0
$499$	6.01209e14	0.869907	0.434953	−	0.900453i	$-0.356765\pi$
$499$	6.01209e14	0.869907	0.434953	+	0.900453i	$0.356765\pi$
$500$	−1.49154e15	−2.13452
$501$	0	0
$502$	−1.88979e15	−2.64571
$503$	−1.09203e15	−1.51221	−0.756103	−	0.654453i	$-0.772899\pi$
$503$	−1.09203e15	−1.51221	−0.756103	+	0.654453i	$0.772899\pi$
$504$	0	0
$505$	−4.99989e13	−0.0677420
$506$	−7.61808e14	−1.02098
$507$	0	0
$508$	1.01010e15	1.32469
$509$	−8.76371e14	−1.13695	−0.568474	−	0.822702i	$-0.692466\pi$
$509$	−8.76371e14	−1.13695	−0.568474	+	0.822702i	$0.692466\pi$
$510$	0	0
$511$	−3.87835e14	−0.492416
$512$	1.28839e15	1.61832
$513$	0	0
$514$	6.08784e14	0.748455
$515$	2.60848e14	0.317284
$516$	0	0
$517$	−9.92308e14	−1.18154
$518$	1.43935e15	1.69571
$519$	0	0
$520$	−6.38022e14	−0.735897
$521$	3.71989e14	0.424544	0.212272	−	0.977211i	$-0.431914\pi$
$521$	3.71989e14	0.424544	0.212272	+	0.977211i	$0.431914\pi$
$522$	0	0
$523$	−9.73507e14	−1.08788	−0.543939	−	0.839125i	$-0.683068\pi$
$523$	−9.73507e14	−1.08788	−0.543939	+	0.839125i	$0.683068\pi$
$524$	3.95091e14	0.436893
$525$	0	0
$526$	1.29445e15	1.40173
$527$	1.57109e14	0.168361
$528$	0	0
$529$	−7.18341e14	−0.753919
$530$	1.58849e15	1.64994
$531$	0	0
$532$	2.18605e15	2.22405
$533$	−6.88699e14	−0.693473
$534$	0	0
$535$	−1.21145e15	−1.19498
$536$	−8.20661e14	−0.801230
$537$	0	0
$538$	2.22484e15	2.12812
$539$	7.69683e14	0.728741
$540$	0	0
$541$	1.74606e15	1.61985	0.809925	−	0.586533i	$-0.199508\pi$
$541$	1.74606e15	1.61985	0.809925	+	0.586533i	$0.199508\pi$
$542$	−1.82364e15	−1.67472
$543$	0	0
$544$	−5.82588e13	−0.0524285
$545$	−3.72953e14	−0.332256
$546$	0	0
$547$	1.74624e14	0.152467	0.0762333	−	0.997090i	$-0.475711\pi$
$547$	1.74624e14	0.152467	0.0762333	+	0.997090i	$0.475711\pi$
$548$	6.25639e14	0.540793
$549$	0	0
$550$	−9.94588e14	−0.842652
$551$	2.09772e14	0.175960
$552$	0	0
$553$	−1.03282e14	−0.0849254
$554$	2.36840e15	1.92820
$555$	0	0
$556$	−4.54561e15	−3.62811
$557$	−1.58365e15	−1.25157	−0.625785	−	0.779995i	$-0.715221\pi$
$557$	−1.58365e15	−1.25157	−0.625785	+	0.779995i	$0.715221\pi$
$558$	0	0
$559$	−7.33993e14	−0.568757
$560$	−5.70833e14	−0.438001
$561$	0	0
$562$	1.24604e15	0.937523
$563$	9.75798e14	0.727049	0.363525	−	0.931585i	$-0.381573\pi$
$563$	9.75798e14	0.727049	0.363525	+	0.931585i	$0.381573\pi$
$564$	0	0
$565$	1.93140e15	1.41125
$566$	−2.23895e15	−1.62015
$567$	0	0
$568$	−2.29977e15	−1.63218
$569$	−1.92427e15	−1.35254	−0.676269	−	0.736655i	$-0.736404\pi$
$569$	−1.92427e15	−1.35254	−0.676269	+	0.736655i	$0.736404\pi$
$570$	0	0
$571$	1.61132e15	1.11092	0.555461	−	0.831543i	$-0.312542\pi$
$571$	1.61132e15	1.11092	0.555461	+	0.831543i	$0.312542\pi$
$572$	−1.97247e15	−1.34689
$573$	0	0
$574$	−1.94623e15	−1.30370
$575$	3.06113e14	0.203099
$576$	0	0
$577$	−2.53250e15	−1.64848	−0.824239	−	0.566242i	$-0.808397\pi$
$577$	−2.53250e15	−1.64848	−0.824239	+	0.566242i	$0.808397\pi$
$578$	1.93118e15	1.24514
$579$	0	0
$580$	−2.33016e14	−0.147411
$581$	2.43412e14	0.152536
$582$	0	0
$583$	2.41894e15	1.48747
$584$	−2.16640e15	−1.31968
$585$	0	0
$586$	3.75768e15	2.24638
$587$	3.11508e15	1.84484	0.922421	−	0.386186i	$-0.126208\pi$
$587$	3.11508e15	1.84484	0.922421	+	0.386186i	$0.126208\pi$
$588$	0	0
$589$	9.93860e14	0.577685
$590$	2.32596e14	0.133942
$591$	0	0
$592$	2.54547e15	1.43879
$593$	2.20723e15	1.23608	0.618040	−	0.786146i	$-0.287927\pi$
$593$	2.20723e15	1.23608	0.618040	+	0.786146i	$0.287927\pi$
$594$	0	0
$595$	4.59788e14	0.252764
$596$	5.58817e15	3.04379
$597$	0	0
$598$	9.15138e14	0.489362
$599$	−3.41159e15	−1.80763	−0.903813	−	0.427927i	$-0.859244\pi$
$599$	−3.41159e15	−1.80763	−0.903813	+	0.427927i	$0.859244\pi$
$600$	0	0
$601$	−1.57545e15	−0.819588	−0.409794	−	0.912178i	$-0.634399\pi$
$601$	−1.57545e15	−0.819588	−0.409794	+	0.912178i	$0.634399\pi$
$602$	−2.07422e15	−1.06924
$603$	0	0
$604$	−2.81956e15	−1.42718
$605$	6.52579e14	0.327325
$606$	0	0
$607$	−5.05592e14	−0.249036	−0.124518	−	0.992217i	$-0.539738\pi$
$607$	−5.05592e14	−0.249036	−0.124518	+	0.992217i	$0.539738\pi$
$608$	−3.68541e14	−0.179894
$609$	0	0
$610$	−2.07353e15	−0.994024
$611$	1.19203e15	0.566319
$612$	0	0
$613$	−1.98375e15	−0.925665	−0.462832	−	0.886446i	$-0.653167\pi$
$613$	−1.98375e15	−0.925665	−0.462832	+	0.886446i	$0.653167\pi$
$614$	−2.00733e15	−0.928310
$615$	0	0
$616$	−2.74561e15	−1.24723
$617$	1.93099e15	0.869382	0.434691	−	0.900580i	$-0.356858\pi$
$617$	1.93099e15	0.869382	0.434691	+	0.900580i	$0.356858\pi$
$618$	0	0
$619$	9.47689e14	0.419148	0.209574	−	0.977793i	$-0.432792\pi$
$619$	9.47689e14	0.419148	0.209574	+	0.977793i	$0.432792\pi$
$620$	−1.10398e15	−0.483958
$621$	0	0
$622$	2.69913e15	1.16246
$623$	−7.07124e14	−0.301864
$624$	0	0
$625$	−1.00840e15	−0.422955
$626$	−9.83314e14	−0.408821
$627$	0	0
$628$	8.60181e15	3.51408
$629$	−2.05030e15	−0.830306
$630$	0	0
$631$	1.00593e15	0.400318	0.200159	−	0.979763i	$-0.435854\pi$
$631$	1.00593e15	0.400318	0.200159	+	0.979763i	$0.435854\pi$
$632$	−5.76922e14	−0.227601
$633$	0	0
$634$	−6.38230e15	−2.47449
$635$	1.34397e15	0.516574
$636$	0	0
$637$	−9.24597e14	−0.349290
$638$	−5.34888e14	−0.200332
$639$	0	0
$640$	3.69422e15	1.35998
$641$	−1.13144e15	−0.412966	−0.206483	−	0.978450i	$-0.566202\pi$
$641$	−1.13144e15	−0.412966	−0.206483	+	0.978450i	$0.566202\pi$
$642$	0	0
$643$	−8.68306e14	−0.311539	−0.155769	−	0.987793i	$-0.549786\pi$
$643$	−8.68306e14	−0.311539	−0.155769	+	0.987793i	$0.549786\pi$
$644$	1.71559e15	0.610296
$645$	0	0
$646$	−4.69405e15	−1.64160
$647$	1.37636e15	0.477265	0.238633	−	0.971110i	$-0.423301\pi$
$647$	1.37636e15	0.477265	0.238633	+	0.971110i	$0.423301\pi$
$648$	0	0
$649$	3.54195e14	0.120753
$650$	1.19477e15	0.403888
$651$	0	0
$652$	−6.60146e14	−0.219421
$653$	−2.55276e15	−0.841370	−0.420685	−	0.907207i	$-0.638210\pi$
$653$	−2.55276e15	−0.841370	−0.420685	+	0.907207i	$0.638210\pi$
$654$	0	0
$655$	5.25679e14	0.170370
$656$	−3.44187e15	−1.10617
$657$	0	0
$658$	3.36862e15	1.06466
$659$	−9.26792e14	−0.290478	−0.145239	−	0.989397i	$-0.546395\pi$
$659$	−9.26792e14	−0.290478	−0.145239	+	0.989397i	$0.546395\pi$
$660$	0	0
$661$	−1.90332e15	−0.586683	−0.293342	−	0.956008i	$-0.594767\pi$
$661$	−1.90332e15	−0.586683	−0.293342	+	0.956008i	$0.594767\pi$
$662$	2.65289e15	0.810963
$663$	0	0
$664$	1.35967e15	0.408799
$665$	2.90859e15	0.867288
$666$	0	0
$667$	1.64627e14	0.0482847
$668$	3.55548e13	0.0103426
$669$	0	0
$670$	−2.21677e15	−0.634322
$671$	−3.15756e15	−0.896143
$672$	0	0
$673$	4.92990e15	1.37643	0.688217	−	0.725505i	$-0.258394\pi$
$673$	4.92990e15	1.37643	0.688217	+	0.725505i	$0.258394\pi$
$674$	4.67563e15	1.29482
$675$	0	0
$676$	−4.86369e15	−1.32513
$677$	−4.30293e14	−0.116286	−0.0581429	−	0.998308i	$-0.518518\pi$
$677$	−4.30293e14	−0.116286	−0.0581429	+	0.998308i	$0.518518\pi$
$678$	0	0
$679$	1.08521e15	0.288555
$680$	2.56832e15	0.677408
$681$	0	0
$682$	−2.53420e15	−0.657698
$683$	3.81244e15	0.981498	0.490749	−	0.871301i	$-0.336723\pi$
$683$	3.81244e15	0.981498	0.490749	+	0.871301i	$0.336723\pi$
$684$	0	0
$685$	8.32429e14	0.210887
$686$	−6.89433e15	−1.73264
$687$	0	0
$688$	−3.66823e15	−0.907236
$689$	−2.90580e15	−0.712952
$690$	0	0
$691$	4.03729e15	0.974901	0.487450	−	0.873151i	$-0.337927\pi$
$691$	4.03729e15	0.974901	0.487450	+	0.873151i	$0.337927\pi$
$692$	2.94972e15	0.706638
$693$	0	0
$694$	7.63374e15	1.79995
$695$	−6.04805e15	−1.41481
$696$	0	0
$697$	2.77232e15	0.638356
$698$	1.11376e16	2.54441
$699$	0	0
$700$	2.23980e15	0.503699
$701$	4.71267e15	1.05152	0.525761	−	0.850632i	$-0.323781\pi$
$701$	4.71267e15	1.05152	0.525761	+	0.850632i	$0.323781\pi$
$702$	0	0
$703$	−1.29700e16	−2.84896
$704$	5.94185e15	1.29501
$705$	0	0
$706$	1.12512e16	2.41419
$707$	2.58467e14	0.0550299
$708$	0	0
$709$	8.66706e14	0.181684	0.0908422	−	0.995865i	$-0.471044\pi$
$709$	8.66706e14	0.181684	0.0908422	+	0.995865i	$0.471044\pi$
$710$	−6.21215e15	−1.29217
$711$	0	0
$712$	−3.94991e15	−0.808997
$713$	7.79972e14	0.158521
$714$	0	0
$715$	−2.62442e15	−0.525231
$716$	−1.58361e16	−3.14505
$717$	0	0
$718$	−9.69924e15	−1.89694
$719$	−2.62666e13	−0.00509795	−0.00254897	−	0.999997i	$-0.500811\pi$
$719$	−2.62666e13	−0.00509795	−0.00254897	+	0.999997i	$0.500811\pi$
$720$	0	0
$721$	−1.34844e15	−0.257744
$722$	−2.06080e16	−3.90913
$723$	0	0
$724$	9.19929e15	1.71867
$725$	2.14931e14	0.0398511
$726$	0	0
$727$	−5.68018e15	−1.03734	−0.518672	−	0.854973i	$-0.673573\pi$
$727$	−5.68018e15	−1.03734	−0.518672	+	0.854973i	$0.673573\pi$
$728$	3.29823e15	0.597803
$729$	0	0
$730$	−5.85190e15	−1.04477
$731$	2.95465e15	0.523552
$732$	0	0
$733$	−1.62312e15	−0.283320	−0.141660	−	0.989915i	$-0.545244\pi$
$733$	−1.62312e15	−0.283320	−0.141660	+	0.989915i	$0.545244\pi$
$734$	1.24885e16	2.16363
$735$	0	0
$736$	−2.89227e14	−0.0493641
$737$	−3.37568e15	−0.571861
$738$	0	0
$739$	3.64794e15	0.608840	0.304420	−	0.952538i	$-0.401537\pi$
$739$	3.64794e15	0.608840	0.304420	+	0.952538i	$0.401537\pi$
$740$	1.44071e16	2.38673
$741$	0	0
$742$	−8.21164e15	−1.34032
$743$	−6.76045e15	−1.09531	−0.547655	−	0.836704i	$-0.684479\pi$
$743$	−6.76045e15	−1.09531	−0.547655	+	0.836704i	$0.684479\pi$
$744$	0	0
$745$	7.43520e15	1.18695
$746$	2.70861e15	0.429224
$747$	0	0
$748$	7.94007e15	1.23984
$749$	6.26254e15	0.970734
$750$	0	0
$751$	2.64833e13	0.00404532	0.00202266	−	0.999998i	$-0.499356\pi$
$751$	2.64833e13	0.00404532	0.00202266	+	0.999998i	$0.499356\pi$
$752$	5.95734e15	0.903347
$753$	0	0
$754$	6.42546e14	0.0960200
$755$	−3.75149e15	−0.556539
$756$	0	0
$757$	7.37364e15	1.07809	0.539045	−	0.842277i	$-0.318785\pi$
$757$	7.37364e15	1.07809	0.539045	+	0.842277i	$0.318785\pi$
$758$	1.14270e16	1.65864
$759$	0	0
$760$	1.62470e16	2.32434
$761$	−6.20983e15	−0.881991	−0.440996	−	0.897509i	$-0.645375\pi$
$761$	−6.20983e15	−0.881991	−0.440996	+	0.897509i	$0.645375\pi$
$762$	0	0
$763$	1.92796e15	0.269907
$764$	−1.36703e16	−1.90005
$765$	0	0
$766$	5.01866e15	0.687591
$767$	−4.25484e14	−0.0578774
$768$	0	0
$769$	−8.86195e15	−1.18832	−0.594162	−	0.804346i	$-0.702516\pi$
$769$	−8.86195e15	−1.18832	−0.594162	+	0.804346i	$0.702516\pi$
$770$	−7.41647e15	−0.987412
$771$	0	0
$772$	−1.98943e16	−2.61116
$773$	2.95047e15	0.384507	0.192254	−	0.981345i	$-0.438420\pi$
$773$	2.95047e15	0.384507	0.192254	+	0.981345i	$0.438420\pi$
$774$	0	0
$775$	1.01830e15	0.130833
$776$	6.06184e15	0.773330
$777$	0	0
$778$	−3.11142e15	−0.391355
$779$	1.75375e16	2.19034
$780$	0	0
$781$	−9.45980e15	−1.16493
$782$	−3.68384e15	−0.450468
$783$	0	0
$784$	−4.62081e15	−0.557160
$785$	1.14449e16	1.37034
$786$	0	0
$787$	1.35545e16	1.60038	0.800190	−	0.599746i	$-0.204732\pi$
$787$	1.35545e16	1.60038	0.800190	+	0.599746i	$0.204732\pi$
$788$	−2.56603e16	−3.00862
$789$	0	0
$790$	−1.55839e15	−0.180188
$791$	−9.98429e15	−1.14643
$792$	0	0
$793$	3.79308e15	0.429526
$794$	−8.31105e15	−0.934638
$795$	0	0
$796$	−1.52784e16	−1.69456
$797$	1.59922e16	1.76152	0.880762	−	0.473558i	$-0.157031\pi$
$797$	1.59922e16	1.76152	0.880762	+	0.473558i	$0.157031\pi$
$798$	0	0
$799$	−4.79846e15	−0.521308
$800$	−3.77604e14	−0.0407419
$801$	0	0
$802$	2.66327e15	0.283437
$803$	−8.91121e15	−0.941892
$804$	0	0
$805$	2.28263e15	0.237990
$806$	3.04426e15	0.315238
$807$	0	0
$808$	1.44377e15	0.147481
$809$	−9.45859e15	−0.959643	−0.479821	−	0.877366i	$-0.659299\pi$
$809$	−9.45859e15	−0.959643	−0.479821	+	0.877366i	$0.659299\pi$
$810$	0	0
$811$	−1.07996e16	−1.08092	−0.540461	−	0.841369i	$-0.681750\pi$
$811$	−1.07996e16	−1.08092	−0.540461	+	0.841369i	$0.681750\pi$
$812$	1.20456e15	0.119749
$813$	0	0
$814$	3.30717e16	3.24356
$815$	−8.78341e14	−0.0855651
$816$	0	0
$817$	1.86909e16	1.79642
$818$	−4.16012e15	−0.397158
$819$	0	0
$820$	−1.94807e16	−1.83497
$821$	1.10411e16	1.03306	0.516529	−	0.856270i	$-0.327224\pi$
$821$	1.10411e16	1.03306	0.516529	+	0.856270i	$0.327224\pi$
$822$	0	0
$823$	1.26104e16	1.16420	0.582101	−	0.813117i	$-0.302231\pi$
$823$	1.26104e16	1.16420	0.582101	+	0.813117i	$0.302231\pi$
$824$	−7.53224e15	−0.690756
$825$	0	0
$826$	−1.20239e15	−0.108807
$827$	7.86172e14	0.0706704	0.0353352	−	0.999376i	$-0.488750\pi$
$827$	7.86172e14	0.0706704	0.0353352	+	0.999376i	$0.488750\pi$
$828$	0	0
$829$	−5.92589e15	−0.525659	−0.262829	−	0.964842i	$-0.584656\pi$
$829$	−5.92589e15	−0.525659	−0.262829	+	0.964842i	$0.584656\pi$
$830$	3.67275e15	0.323640
$831$	0	0
$832$	−7.13777e15	−0.620704
$833$	3.72192e15	0.321528
$834$	0	0
$835$	4.73066e13	0.00403317
$836$	5.02283e16	4.25416
$837$	0	0
$838$	−7.90044e15	−0.660403
$839$	−2.10209e16	−1.74566	−0.872831	−	0.488022i	$-0.837718\pi$
$839$	−2.10209e16	−1.74566	−0.872831	+	0.488022i	$0.837718\pi$
$840$	0	0
$841$	−1.20849e16	−0.990526
$842$	1.23184e16	1.00309
$843$	0	0
$844$	7.24437e14	0.0582260
$845$	−6.47126e15	−0.516746
$846$	0	0
$847$	−3.37348e15	−0.265901
$848$	−1.45222e16	−1.13725
$849$	0	0
$850$	−4.80948e15	−0.371787
$851$	−1.01788e16	−0.781775
$852$	0	0
$853$	−1.98378e16	−1.50409	−0.752046	−	0.659110i	$-0.770933\pi$
$853$	−1.98378e16	−1.50409	−0.752046	+	0.659110i	$0.770933\pi$
$854$	1.07190e16	0.807491
$855$	0	0
$856$	3.49818e16	2.60157
$857$	−7.96639e15	−0.588663	−0.294332	−	0.955703i	$-0.595097\pi$
$857$	−7.96639e15	−0.588663	−0.294332	+	0.955703i	$0.595097\pi$
$858$	0	0
$859$	−1.59749e16	−1.16540	−0.582701	−	0.812687i	$-0.698004\pi$
$859$	−1.59749e16	−1.16540	−0.582701	+	0.812687i	$0.698004\pi$
$860$	−2.07619e16	−1.50496
$861$	0	0
$862$	4.00274e16	2.86462
$863$	4.02926e15	0.286527	0.143264	−	0.989685i	$-0.454240\pi$
$863$	4.02926e15	0.286527	0.143264	+	0.989685i	$0.454240\pi$
$864$	0	0
$865$	3.92467e15	0.275559
$866$	5.84413e15	0.407729
$867$	0	0
$868$	5.70699e15	0.393142
$869$	−2.37310e15	−0.162445
$870$	0	0
$871$	4.05511e15	0.274096
$872$	1.07694e16	0.723352
$873$	0	0
$874$	−2.33037e16	−1.54565
$875$	1.02590e16	0.676176
$876$	0	0
$877$	9.76958e15	0.635884	0.317942	−	0.948110i	$-0.397008\pi$
$877$	9.76958e15	0.635884	0.317942	+	0.948110i	$0.397008\pi$
$878$	2.87301e16	1.85831
$879$	0	0
$880$	−1.31159e16	−0.837807
$881$	−1.62583e16	−1.03207	−0.516033	−	0.856569i	$-0.672592\pi$
$881$	−1.62583e16	−1.03207	−0.516033	+	0.856569i	$0.672592\pi$
$882$	0	0
$883$	1.66061e16	1.04108	0.520540	−	0.853837i	$-0.325731\pi$
$883$	1.66061e16	1.04108	0.520540	+	0.853837i	$0.325731\pi$
$884$	−9.53818e15	−0.594262
$885$	0	0
$886$	−8.67733e15	−0.533950
$887$	1.91314e16	1.16995	0.584975	−	0.811051i	$-0.301104\pi$
$887$	1.91314e16	1.16995	0.584975	+	0.811051i	$0.301104\pi$
$888$	0	0
$889$	−6.94758e15	−0.419637
$890$	−1.06695e16	−0.640472
$891$	0	0
$892$	8.98286e15	0.532608
$893$	−3.03547e16	−1.78872
$894$	0	0
$895$	−2.10703e16	−1.22644
$896$	−1.90971e16	−1.10477
$897$	0	0
$898$	−6.88985e15	−0.393722
$899$	5.47641e14	0.0311041
$900$	0	0
$901$	1.16972e16	0.656287
$902$	−4.47180e16	−2.49372
$903$	0	0
$904$	−5.57710e16	−3.07243
$905$	1.22399e16	0.670209
$906$	0	0
$907$	−1.07252e16	−0.580182	−0.290091	−	0.956999i	$-0.593686\pi$
$907$	−1.07252e16	−0.580182	−0.290091	+	0.956999i	$0.593686\pi$
$908$	1.67836e15	0.0902430
$909$	0	0
$910$	8.90919e15	0.473272
$911$	2.82249e16	1.49032	0.745162	−	0.666883i	$-0.232372\pi$
$911$	2.82249e16	1.49032	0.745162	+	0.666883i	$0.232372\pi$
$912$	0	0
$913$	5.59284e15	0.291771
$914$	7.34833e14	0.0381052
$915$	0	0
$916$	7.31516e16	3.74799
$917$	−2.71748e15	−0.138399
$918$	0	0
$919$	1.83551e16	0.923679	0.461839	−	0.886964i	$-0.347190\pi$
$919$	1.83551e16	0.923679	0.461839	+	0.886964i	$0.347190\pi$
$920$	1.27505e16	0.637814
$921$	0	0
$922$	−5.43764e16	−2.68776
$923$	1.13638e16	0.558359
$924$	0	0
$925$	−1.32890e16	−0.645227
$926$	1.46594e16	0.707549
$927$	0	0
$928$	−2.03075e14	−0.00968596
$929$	1.10125e16	0.522155	0.261078	−	0.965318i	$-0.415922\pi$
$929$	1.10125e16	0.522155	0.261078	+	0.965318i	$0.415922\pi$
$930$	0	0
$931$	2.35446e16	1.10323
$932$	7.57317e13	0.00352769
$933$	0	0
$934$	2.49605e16	1.14907
$935$	1.05645e16	0.483486
$936$	0	0
$937$	8.36357e15	0.378289	0.189145	−	0.981949i	$-0.439429\pi$
$937$	8.36357e15	0.378289	0.189145	+	0.981949i	$0.439429\pi$
$938$	1.14595e16	0.515289
$939$	0	0
$940$	3.37181e16	1.49851
$941$	−2.22942e16	−0.985028	−0.492514	−	0.870305i	$-0.663922\pi$
$941$	−2.22942e16	−0.985028	−0.492514	+	0.870305i	$0.663922\pi$
$942$	0	0
$943$	1.37633e16	0.601045
$944$	−2.12642e15	−0.0923214
$945$	0	0
$946$	−4.76590e16	−2.04524
$947$	3.78038e16	1.61291	0.806457	−	0.591293i	$-0.201382\pi$
$947$	3.78038e16	1.61291	0.806457	+	0.591293i	$0.201382\pi$
$948$	0	0
$949$	1.07048e16	0.451454
$950$	−3.04244e16	−1.27568
$951$	0	0
$952$	−1.32768e16	−0.550290
$953$	−4.79568e15	−0.197624	−0.0988119	−	0.995106i	$-0.531504\pi$
$953$	−4.79568e15	−0.197624	−0.0988119	+	0.995106i	$0.531504\pi$
$954$	0	0
$955$	−1.81887e16	−0.740939
$956$	7.11355e16	2.88116
$957$	0	0
$958$	−1.17298e16	−0.469657
$959$	−4.30321e15	−0.171313
$960$	0	0
$961$	−2.28139e16	−0.897884
$962$	−3.97280e16	−1.55466
$963$	0	0
$964$	−3.59251e15	−0.138987
$965$	−2.64698e16	−1.01824
$966$	0	0
$967$	3.37420e16	1.28329	0.641645	−	0.767002i	$-0.278252\pi$
$967$	3.37420e16	1.28329	0.641645	+	0.767002i	$0.278252\pi$
$968$	−1.88439e16	−0.712616
$969$	0	0
$970$	1.63743e16	0.612234
$971$	−3.58587e16	−1.33318	−0.666590	−	0.745425i	$-0.732247\pi$
$971$	−3.58587e16	−1.33318	−0.666590	+	0.745425i	$0.732247\pi$
$972$	0	0
$973$	3.12651e16	1.14932
$974$	−1.37706e16	−0.503360
$975$	0	0
$976$	1.89565e16	0.685147
$977$	1.20023e16	0.431366	0.215683	−	0.976463i	$-0.430802\pi$
$977$	1.20023e16	0.431366	0.215683	+	0.976463i	$0.430802\pi$
$978$	0	0
$979$	−1.62474e16	−0.577405
$980$	−2.61534e16	−0.924240
$981$	0	0
$982$	−6.71547e16	−2.34673
$983$	4.42687e15	0.153834	0.0769170	−	0.997038i	$-0.475492\pi$
$983$	4.42687e15	0.153834	0.0769170	+	0.997038i	$0.475492\pi$
$984$	0	0
$985$	−3.41417e16	−1.17324
$986$	−2.58653e15	−0.0883884
$987$	0	0
$988$	−6.03378e16	−2.03904
$989$	1.46684e16	0.492951
$990$	0	0
$991$	−3.79167e16	−1.26016	−0.630080	−	0.776530i	$-0.716978\pi$
$991$	−3.79167e16	−1.26016	−0.630080	+	0.776530i	$0.716978\pi$
$992$	−9.62130e14	−0.0317994
$993$	0	0
$994$	3.21135e16	1.04969
$995$	−2.03283e16	−0.660807
$996$	0	0
$997$	5.76003e13	0.00185183	0.000925915	−	1.00000i	$-0.499705\pi$
$997$	5.76003e13	0.00185183	0.000925915		1.00000i	$0.499705\pi$
$998$	−4.68943e16	−1.49935
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	9.12.a.a.1.1		1
3.2	odd	2		3.12.a.a.1.1	✓	1
4.3	odd	2		144.12.a.l.1.1		1
5.2	odd	4		225.12.b.a.199.1		2
5.3	odd	4		225.12.b.a.199.2		2
5.4	even	2		225.12.a.f.1.1		1
9.2	odd	6		81.12.c.a.28.1		2
9.4	even	3		81.12.c.e.55.1		2
9.5	odd	6		81.12.c.a.55.1		2
9.7	even	3		81.12.c.e.28.1		2
12.11	even	2		48.12.a.f.1.1		1
15.2	even	4		75.12.b.a.49.2		2
15.8	even	4		75.12.b.a.49.1		2
15.14	odd	2		75.12.a.a.1.1		1
21.20	even	2		147.12.a.c.1.1		1
24.5	odd	2		192.12.a.q.1.1		1
24.11	even	2		192.12.a.g.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3.12.a.a.1.1	✓	1	3.2	odd	2
9.12.a.a.1.1		1	1.1	even	1	trivial
48.12.a.f.1.1		1	12.11	even	2
75.12.a.a.1.1		1	15.14	odd	2
75.12.b.a.49.1		2	15.8	even	4
75.12.b.a.49.2		2	15.2	even	4
81.12.c.a.28.1		2	9.2	odd	6
81.12.c.a.55.1		2	9.5	odd	6
81.12.c.e.28.1		2	9.7	even	3
81.12.c.e.55.1		2	9.4	even	3
144.12.a.l.1.1		1	4.3	odd	2
147.12.a.c.1.1		1	21.20	even	2
192.12.a.g.1.1		1	24.11	even	2
192.12.a.q.1.1		1	24.5	odd	2
225.12.a.f.1.1		1	5.4	even	2
225.12.b.a.199.1		2	5.2	odd	4
225.12.b.a.199.2		2	5.3	odd	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 9 = 3^{2} \)
Weight:	\( k \)	\(=\)	\( 12 \)
Character orbit:	\([\chi]\)	\(=\)	9.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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