LMFDB - Embedded newform 192.12.a.q.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [192,12,Mod(1,192)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(192, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("192.1");

S:= CuspForms(chi, 12);

N := Newforms(S);

Level:	$ N $	$=$	$ 192 = 2^{6} \cdot 3 $
Weight:	$ k $	$=$	$ 12 $
Character orbit:	$[\chi]$	$=$	192.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:	$147.521890667$
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	$\chi$	$=$	192.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q+243.000 q^{3} +5370.00 q^{5} -27760.0 q^{7} +59049.0 q^{9} +O(q^{10})$

$q+243.000 q^{3} +5370.00 q^{5} -27760.0 q^{7} +59049.0 q^{9} -637836. q^{11} -766214. q^{13} +1.30491e6 q^{15} +3.08435e6 q^{17} +1.95114e7 q^{19} -6.74568e6 q^{21} +1.53124e7 q^{23} -1.99912e7 q^{25} +1.43489e7 q^{27} -1.07513e7 q^{29} -5.09374e7 q^{31} -1.54994e8 q^{33} -1.49071e8 q^{35} -6.64741e8 q^{37} -1.86190e8 q^{39} +8.98833e8 q^{41} +9.57947e8 q^{43} +3.17093e8 q^{45} -1.55574e9 q^{47} -1.20671e9 q^{49} +7.49498e8 q^{51} -3.79242e9 q^{53} -3.42518e9 q^{55} +4.74127e9 q^{57} -5.55307e8 q^{59} -4.95042e9 q^{61} -1.63920e9 q^{63} -4.11457e9 q^{65} -5.29240e9 q^{67} +3.72090e9 q^{69} -1.48311e10 q^{71} +1.39710e10 q^{73} -4.85787e9 q^{75} +1.77063e10 q^{77} +3.72054e9 q^{79} +3.48678e9 q^{81} -8.76845e9 q^{83} +1.65630e10 q^{85} -2.61256e9 q^{87} -2.54728e10 q^{89} +2.12701e10 q^{91} -1.23778e10 q^{93} +1.04776e11 q^{95} -3.90925e10 q^{97} -3.76636e10 q^{99} +O(q^{100})$

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 $

$2$	0	0
$2$	0	0
$3$	243.000	0.577350
$3$	243.000	0.577350
$4$	0	0
$5$	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	$0.601046\pi$
$8$	0	0
$9$	59049.0	0.333333
$10$	0	0
$11$	−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	$-0.592384\pi$
$13$	−766214.	−0.572350	−0.286175	+	0.958177i	$0.592384\pi$
$14$	0	0
$15$	1.30491e6	0.443689
$16$	0	0
$17$	3.08435e6	0.526860	0.263430	−	0.964679i	$-0.415146\pi$
$17$	3.08435e6	0.526860	0.263430	+	0.964679i	$0.415146\pi$
$18$	0	0
$19$	1.95114e7	1.80777	0.903886	−	0.427773i	$-0.140702\pi$
$19$	1.95114e7	1.80777	0.903886	+	0.427773i	$0.140702\pi$
$20$	0	0
$21$	−6.74568e6	−0.360429
$22$	0	0
$23$	1.53124e7	0.496066	0.248033	−	0.968752i	$-0.420216\pi$
$23$	1.53124e7	0.496066	0.248033	+	0.968752i	$0.420216\pi$
$24$	0	0
$25$	−1.99912e7	−0.409420
$26$	0	0
$27$	1.43489e7	0.192450
$28$	0	0
$29$	−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	$0.551078\pi$
$32$	0	0
$33$	−1.54994e8	−0.689428
$34$	0	0
$35$	−1.49071e8	−0.479755
$36$	0	0
$37$	−6.64741e8	−1.57595	−0.787976	−	0.615706i	$-0.788871\pi$
$37$	−6.64741e8	−1.57595	−0.787976	+	0.615706i	$0.788871\pi$
$38$	0	0
$39$	−1.86190e8	−0.330446
$40$	0	0
$41$	8.98833e8	1.21162	0.605812	−	0.795608i	$-0.292848\pi$
$41$	8.98833e8	1.21162	0.605812	+	0.795608i	$0.292848\pi$
$42$	0	0
$43$	9.57947e8	0.993722	0.496861	−	0.867830i	$-0.334486\pi$
$43$	9.57947e8	0.993722	0.496861	+	0.867830i	$0.334486\pi$
$44$	0	0
$45$	3.17093e8	0.256164
$46$	0	0
$47$	−1.55574e9	−0.989462	−0.494731	−	0.869046i	$-0.664733\pi$
$47$	−1.55574e9	−0.989462	−0.494731	+	0.869046i	$0.664733\pi$
$48$	0	0
$49$	−1.20671e9	−0.610273
$50$	0	0
$51$	7.49498e8	0.304183
$52$	0	0
$53$	−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
$56$	0	0
$57$	4.74127e9	1.04372
$58$	0	0
$59$	−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	$-0.622436\pi$
$61$	−4.95042e9	−0.750461	−0.375230	+	0.926932i	$0.622436\pi$
$62$	0	0
$63$	−1.63920e9	−0.208094
$64$	0	0
$65$	−4.11457e9	−0.439846
$66$	0	0
$67$	−5.29240e9	−0.478896	−0.239448	−	0.970909i	$-0.576966\pi$
$67$	−5.29240e9	−0.478896	−0.239448	+	0.970909i	$0.576966\pi$
$68$	0	0
$69$	3.72090e9	0.286404
$70$	0	0
$71$	−1.48311e10	−0.975556	−0.487778	−	0.872968i	$-0.662192\pi$
$71$	−1.48311e10	−0.975556	−0.487778	+	0.872968i	$0.662192\pi$
$72$	0	0
$73$	1.39710e10	0.788773	0.394386	−	0.918945i	$-0.370957\pi$
$73$	1.39710e10	0.788773	0.394386	+	0.918945i	$0.370957\pi$
$74$	0	0
$75$	−4.85787e9	−0.236379
$76$	0	0
$77$	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	$0.478332\pi$
$80$	0	0
$81$	3.48678e9	0.111111
$82$	0	0
$83$	−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
$86$	0	0
$87$	−2.61256e9	−0.0561966
$88$	0	0
$89$	−2.54728e10	−0.483539	−0.241769	−	0.970334i	$-0.577728\pi$
$89$	−2.54728e10	−0.483539	−0.241769	+	0.970334i	$0.577728\pi$
$90$	0	0
$91$	2.12701e10	0.357307
$92$	0	0
$93$	−1.23778e10	−0.184496
$94$	0	0
$95$	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	$0.574236\pi$
$98$	0	0
$99$	−3.76636e10	−0.398041
$100$	0	0
$101$	−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	$0.433815\pi$
$104$	0	0
$105$	−3.62243e10	−0.276987
$106$	0	0
$107$	−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	$-0.430642\pi$
$109$	6.94512e10	0.432348	0.216174	+	0.976355i	$0.430642\pi$
$110$	0	0
$111$	−1.61532e11	−0.909876
$112$	0	0
$113$	−3.59665e11	−1.83640	−0.918198	−	0.396122i	$-0.870356\pi$
$113$	−3.59665e11	−1.83640	−0.918198	+	0.396122i	$0.870356\pi$
$114$	0	0
$115$	8.22274e10	0.381222
$116$	0	0
$117$	−4.52442e10	−0.190783
$118$	0	0
$119$	−8.56217e10	−0.328909
$120$	0	0
$121$	1.21523e11	0.425931
$122$	0	0
$123$	2.18417e11	0.699532
$124$	0	0
$125$	−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	$0.390893\pi$
$128$	0	0
$129$	2.32781e11	0.573726
$130$	0	0
$131$	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
$134$	0	0
$135$	7.70536e10	0.147896
$136$	0	0
$137$	−1.55015e11	−0.274416	−0.137208	−	0.990542i	$-0.543813\pi$
$137$	−1.55015e11	−0.274416	−0.137208	+	0.990542i	$0.543813\pi$
$138$	0	0
$139$	1.12627e12	1.84102	0.920512	−	0.390715i	$-0.127772\pi$
$139$	1.12627e12	1.84102	0.920512	+	0.390715i	$0.127772\pi$
$140$	0	0
$141$	−3.78045e11	−0.571266
$142$	0	0
$143$	4.88719e11	0.683457
$144$	0	0
$145$	−5.77343e10	−0.0748014
$146$	0	0
$147$	−2.93230e11	−0.352341
$148$	0	0
$149$	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	$0.617939\pi$
$152$	0	0
$153$	1.82128e11	0.175620
$154$	0	0
$155$	−2.73534e11	−0.245576
$156$	0	0
$157$	−2.13127e12	−1.78316	−0.891580	−	0.452863i	$-0.850403\pi$
$157$	−2.13127e12	−1.78316	−0.891580	+	0.452863i	$0.850403\pi$
$158$	0	0
$159$	−9.21557e11	−0.719181
$160$	0	0
$161$	−4.25071e11	−0.309684
$162$	0	0
$163$	1.63564e11	0.111342	0.0556708	−	0.998449i	$-0.482270\pi$
$163$	1.63564e11	0.111342	0.0556708	+	0.998449i	$0.482270\pi$
$164$	0	0
$165$	−8.32319e11	−0.529819
$166$	0	0
$167$	−8.80943e9	−0.00524816	−0.00262408	−	0.999997i	$-0.500835\pi$
$167$	−8.80943e9	−0.00524816	−0.00262408	+	0.999997i	$0.500835\pi$
$168$	0	0
$169$	−1.20508e12	−0.672416
$170$	0	0
$171$	1.15213e12	0.602591
$172$	0	0
$173$	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
$176$	0	0
$177$	−1.34940e11	−0.0583830
$178$	0	0
$179$	−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	$-0.643625\pi$
$181$	−2.27931e12	−0.872110	−0.436055	+	0.899920i	$0.643625\pi$
$182$	0	0
$183$	−1.20295e12	−0.433279
$184$	0	0
$185$	−3.56966e12	−1.21111
$186$	0	0
$187$	−1.96731e12	−0.629136
$188$	0	0
$189$	−3.98326e11	−0.120143
$190$	0	0
$191$	3.38709e12	0.964147	0.482073	−	0.876131i	$-0.339884\pi$
$191$	3.38709e12	0.964147	0.482073	+	0.876131i	$0.339884\pi$
$192$	0	0
$193$	−4.92921e12	−1.32499	−0.662494	−	0.749067i	$-0.730502\pi$
$193$	−4.92921e12	−1.32499	−0.662494	+	0.749067i	$0.730502\pi$
$194$	0	0
$195$	−9.99840e11	−0.253945
$196$	0	0
$197$	−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	$0.641464\pi$
$200$	0	0
$201$	−1.28605e12	−0.276491
$202$	0	0
$203$	2.98455e11	0.0607646
$204$	0	0
$205$	4.82674e12	0.931123
$206$	0	0
$207$	9.04180e11	0.165355
$208$	0	0
$209$	−1.24451e13	−2.15870
$210$	0	0
$211$	−1.79494e11	−0.0295458	−0.0147729	−	0.999891i	$-0.504703\pi$
$211$	−1.79494e11	−0.0295458	−0.0147729	+	0.999891i	$0.504703\pi$
$212$	0	0
$213$	−3.60395e12	−0.563237
$214$	0	0
$215$	5.14418e12	0.763668
$216$	0	0
$217$	1.41402e12	0.199493
$218$	0	0
$219$	3.39495e12	0.455398
$220$	0	0
$221$	−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	$0.456854\pi$
$224$	0	0
$225$	−1.18046e12	−0.136473
$226$	0	0
$227$	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	$-0.899867\pi$
$229$	−1.81248e13	−1.90186	−0.950928	+	0.309414i	$0.899867\pi$
$230$	0	0
$231$	4.30264e12	0.430397
$232$	0	0
$233$	−1.87641e10	−0.00179007	−0.000895033	−	1.00000i	$-0.500285\pi$
$233$	−1.87641e10	−0.00179007	−0.000895033		1.00000i	$0.500285\pi$
$234$	0	0
$235$	−8.35433e12	−0.760394
$236$	0	0
$237$	9.04092e11	0.0785410
$238$	0	0
$239$	−1.76252e13	−1.46200	−0.730999	−	0.682379i	$-0.760946\pi$
$239$	−1.76252e13	−1.46200	−0.730999	+	0.682379i	$0.760946\pi$
$240$	0	0
$241$	−8.90117e11	−0.0705267	−0.0352633	−	0.999378i	$-0.511227\pi$
$241$	−8.90117e11	−0.0705267	−0.0352633	+	0.999378i	$0.511227\pi$
$242$	0	0
$243$	8.47289e11	0.0641500
$244$	0	0
$245$	−6.48003e12	−0.468990
$246$	0	0
$247$	−1.49499e13	−1.03468
$248$	0	0
$249$	−2.13073e12	−0.141069
$250$	0	0
$251$	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
$254$	0	0
$255$	4.02480e12	0.233762
$256$	0	0
$257$	7.80492e12	0.434246	0.217123	−	0.976144i	$-0.430333\pi$
$257$	7.80492e12	0.434246	0.217123	+	0.976144i	$0.430333\pi$
$258$	0	0
$259$	1.84532e13	0.983837
$260$	0	0
$261$	−6.34851e11	−0.0324451
$262$	0	0
$263$	1.65956e13	0.813272	0.406636	−	0.913590i	$-0.366702\pi$
$263$	1.65956e13	0.813272	0.406636	+	0.913590i	$0.366702\pi$
$264$	0	0
$265$	−2.03653e13	−0.957278
$266$	0	0
$267$	−6.18988e12	−0.279171
$268$	0	0
$269$	−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	$0.338518\pi$
$272$	0	0
$273$	5.16863e12	0.206291
$274$	0	0
$275$	1.27511e13	0.488898
$276$	0	0
$277$	3.03641e13	1.11872	0.559361	−	0.828924i	$-0.311047\pi$
$277$	3.03641e13	1.11872	0.559361	+	0.828924i	$0.311047\pi$
$278$	0	0
$279$	−3.00780e12	−0.106519
$280$	0	0
$281$	1.59749e13	0.543942	0.271971	−	0.962306i	$-0.412325\pi$
$281$	1.59749e13	0.543942	0.271971	+	0.962306i	$0.412325\pi$
$282$	0	0
$283$	−2.87045e13	−0.939993	−0.469997	−	0.882668i	$-0.655745\pi$
$283$	−2.87045e13	−0.939993	−0.469997	+	0.882668i	$0.655745\pi$
$284$	0	0
$285$	2.54606e13	0.802089
$286$	0	0
$287$	−2.49516e13	−0.756394
$288$	0	0
$289$	−2.47587e13	−0.722419
$290$	0	0
$291$	−9.49948e12	−0.266863
$292$	0	0
$293$	−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
$296$	0	0
$297$	−9.15225e12	−0.229809
$298$	0	0
$299$	−1.17325e13	−0.283923
$300$	0	0
$301$	−2.65926e13	−0.620362
$302$	0	0
$303$	−2.26252e12	−0.0508930
$304$	0	0
$305$	−2.65838e13	−0.576723
$306$	0	0
$307$	−2.57350e13	−0.538597	−0.269298	−	0.963057i	$-0.586792\pi$
$307$	−2.57350e13	−0.538597	−0.269298	+	0.963057i	$0.586792\pi$
$308$	0	0
$309$	1.18037e13	0.238368
$310$	0	0
$311$	3.46043e13	0.674446	0.337223	−	0.941425i	$-0.390512\pi$
$311$	3.46043e13	0.674446	0.337223	+	0.941425i	$0.390512\pi$
$312$	0	0
$313$	1.26066e13	0.237194	0.118597	−	0.992942i	$-0.462160\pi$
$313$	1.26066e13	0.237194	0.118597	+	0.992942i	$0.462160\pi$
$314$	0	0
$315$	−8.80251e12	−0.159918
$316$	0	0
$317$	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
$320$	0	0
$321$	−5.48198e13	−0.897758
$322$	0	0
$323$	6.01801e13	0.952443
$324$	0	0
$325$	1.53176e13	0.234332
$326$	0	0
$327$	1.68766e13	0.249616
$328$	0	0
$329$	4.31874e13	0.617703
$330$	0	0
$331$	3.40115e13	0.470513	0.235256	−	0.971933i	$-0.424407\pi$
$331$	3.40115e13	0.470513	0.235256	+	0.971933i	$0.424407\pi$
$332$	0	0
$333$	−3.92523e13	−0.525317
$334$	0	0
$335$	−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	$0.622571\pi$
$338$	0	0
$339$	−8.73985e13	−1.06024
$340$	0	0
$341$	3.24897e13	0.381590
$342$	0	0
$343$	8.83888e13	1.00526
$344$	0	0
$345$	1.99813e13	0.220099
$346$	0	0
$347$	−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	$-0.235714\pi$
$349$	1.42790e14	1.47624	0.738120	+	0.674670i	$0.235714\pi$
$350$	0	0
$351$	−1.09943e13	−0.110149
$352$	0	0
$353$	1.44246e14	1.40069	0.700346	−	0.713804i	$-0.253029\pi$
$353$	1.44246e14	1.40069	0.700346	+	0.713804i	$0.253029\pi$
$354$	0	0
$355$	−7.96429e13	−0.749707
$356$	0	0
$357$	−2.08061e13	−0.189896
$358$	0	0
$359$	−1.24349e14	−1.10059	−0.550293	−	0.834972i	$-0.685484\pi$
$359$	−1.24349e14	−1.10059	−0.550293	+	0.834972i	$0.685484\pi$
$360$	0	0
$361$	2.64205e14	2.26804
$362$	0	0
$363$	2.95301e13	0.245911
$364$	0	0
$365$	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	$0.715987\pi$
$368$	0	0
$369$	5.30752e13	0.403875
$370$	0	0
$371$	1.05277e14	0.777641
$372$	0	0
$373$	3.47258e13	0.249031	0.124516	−	0.992218i	$-0.460262\pi$
$373$	3.47258e13	0.249031	0.124516	+	0.992218i	$0.460262\pi$
$374$	0	0
$375$	−8.98031e13	−0.625344
$376$	0	0
$377$	8.23777e12	0.0557099
$378$	0	0
$379$	1.46500e14	0.962327	0.481163	−	0.876631i	$-0.340214\pi$
$379$	1.46500e14	0.962327	0.481163	+	0.876631i	$0.340214\pi$
$380$	0	0
$381$	6.08164e13	0.388090
$382$	0	0
$383$	6.43419e13	0.398933	0.199467	−	0.979905i	$-0.436079\pi$
$383$	6.43419e13	0.398933	0.199467	+	0.979905i	$0.436079\pi$
$384$	0	0
$385$	9.50830e13	0.572887
$386$	0	0
$387$	5.65658e13	0.331241
$388$	0	0
$389$	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
$392$	0	0
$393$	2.37877e13	0.127995
$394$	0	0
$395$	1.99793e13	0.104543
$396$	0	0
$397$	−1.06552e14	−0.542268	−0.271134	−	0.962542i	$-0.587399\pi$
$397$	−1.06552e14	−0.542268	−0.271134	+	0.962542i	$0.587399\pi$
$398$	0	0
$399$	−1.31618e14	−0.651573
$400$	0	0
$401$	3.41445e13	0.164447	0.0822236	−	0.996614i	$-0.473798\pi$
$401$	3.41445e13	0.164447	0.0822236	+	0.996614i	$0.473798\pi$
$402$	0	0
$403$	3.90289e13	0.182898
$404$	0	0
$405$	1.87240e13	0.0853880
$406$	0	0
$407$	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	$0.463245\pi$
$410$	0	0
$411$	−3.76686e13	−0.158434
$412$	0	0
$413$	1.54153e13	0.0631288
$414$	0	0
$415$	−4.70866e13	−0.187773
$416$	0	0
$417$	2.73682e14	1.06292
$418$	0	0
$419$	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	$-0.406015\pi$
$421$	1.57928e14	0.581981	0.290991	+	0.956726i	$0.406015\pi$
$422$	0	0
$423$	−9.18650e13	−0.329821
$424$	0	0
$425$	−6.16600e13	−0.215707
$426$	0	0
$427$	1.37424e14	0.468499
$428$	0	0
$429$	1.18759e14	0.394594
$430$	0	0
$431$	5.13171e14	1.66202	0.831012	−	0.556254i	$-0.187762\pi$
$431$	5.13171e14	1.66202	0.831012	+	0.556254i	$0.187762\pi$
$432$	0	0
$433$	−7.49248e13	−0.236560	−0.118280	−	0.992980i	$-0.537738\pi$
$433$	−7.49248e13	−0.236560	−0.118280	+	0.992980i	$0.537738\pi$
$434$	0	0
$435$	−1.40294e13	−0.0431866
$436$	0	0
$437$	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	$0.681230\pi$
$440$	0	0
$441$	−7.12550e13	−0.203424
$442$	0	0
$443$	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
$446$	0	0
$447$	3.36453e14	0.891730
$448$	0	0
$449$	−8.83314e13	−0.228434	−0.114217	−	0.993456i	$-0.536436\pi$
$449$	−8.83314e13	−0.228434	−0.114217	+	0.993456i	$0.536436\pi$
$450$	0	0
$451$	−5.73308e14	−1.44683
$452$	0	0
$453$	−1.69760e14	−0.418115
$454$	0	0
$455$	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	$0.503519\pi$
$458$	0	0
$459$	4.42571e13	0.101394
$460$	0	0
$461$	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	$0.565803\pi$
$464$	0	0
$465$	−6.64687e13	−0.141784
$466$	0	0
$467$	−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
$470$	0	0
$471$	−5.17899e14	−1.02951
$472$	0	0
$473$	−6.11013e14	−1.18663
$474$	0	0
$475$	−3.90057e14	−0.740139
$476$	0	0
$477$	−2.23938e14	−0.415219
$478$	0	0
$479$	−1.50382e14	−0.272491	−0.136245	−	0.990675i	$-0.543504\pi$
$479$	−1.50382e14	−0.272491	−0.136245	+	0.990675i	$0.543504\pi$
$480$	0	0
$481$	5.09334e14	0.901996
$482$	0	0
$483$	−1.03292e14	−0.178796
$484$	0	0
$485$	−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	$0.453353\pi$
$488$	0	0
$489$	3.97462e13	0.0642831
$490$	0	0
$491$	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
$494$	0	0
$495$	−2.02253e14	−0.305891
$496$	0	0
$497$	4.11711e14	0.609021
$498$	0	0
$499$	−6.01209e14	−0.869907	−0.434953	−	0.900453i	$-0.643235\pi$
$499$	−6.01209e14	−0.869907	−0.434953	+	0.900453i	$0.643235\pi$
$500$	0	0
$501$	−2.14069e12	−0.00303003
$502$	0	0
$503$	1.09203e15	1.51221	0.756103	−	0.654453i	$-0.227101\pi$
$503$	1.09203e15	1.51221	0.756103	+	0.654453i	$0.227101\pi$
$504$	0	0
$505$	−4.99989e13	−0.0677420
$506$	0	0
$507$	−2.92834e14	−0.388219
$508$	0	0
$509$	−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
$512$	0	0
$513$	2.79967e14	0.347906
$514$	0	0
$515$	2.60848e14	0.317284
$516$	0	0
$517$	9.92308e14	1.18154
$518$	0	0
$519$	1.77597e14	0.207021
$520$	0	0
$521$	−3.71989e14	−0.424544	−0.212272	−	0.977211i	$-0.568086\pi$
$521$	−3.71989e14	−0.424544	−0.212272	+	0.977211i	$0.568086\pi$
$522$	0	0
$523$	9.73507e14	1.08788	0.543939	−	0.839125i	$-0.316932\pi$
$523$	9.73507e14	1.08788	0.543939	+	0.839125i	$0.316932\pi$
$524$	0	0
$525$	1.34854e14	0.147567
$526$	0	0
$527$	−1.57109e14	−0.168361
$528$	0	0
$529$	−7.18341e14	−0.753919
$530$	0	0
$531$	−3.27903e13	−0.0337074
$532$	0	0
$533$	−6.88699e14	−0.693473
$534$	0	0
$535$	−1.21145e15	−1.19498
$536$	0	0
$537$	−9.53463e14	−0.921393
$538$	0	0
$539$	7.69683e14	0.728741
$540$	0	0
$541$	−1.74606e15	−1.61985	−0.809925	−	0.586533i	$-0.800492\pi$
$541$	−1.74606e15	−1.61985	−0.809925	+	0.586533i	$0.800492\pi$
$542$	0	0
$543$	−5.53872e14	−0.503513
$544$	0	0
$545$	3.72953e14	0.332256
$546$	0	0
$547$	−1.74624e14	−0.152467	−0.0762333	−	0.997090i	$-0.524289\pi$
$547$	−1.74624e14	−0.152467	−0.0762333	+	0.997090i	$0.524289\pi$
$548$	0	0
$549$	−2.92317e14	−0.250154
$550$	0	0
$551$	−2.09772e14	−0.175960
$552$	0	0
$553$	−1.03282e14	−0.0849254
$554$	0	0
$555$	−8.67427e14	−0.699232
$556$	0	0
$557$	−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
$560$	0	0
$561$	−4.78057e14	−0.363232
$562$	0	0
$563$	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
$566$	0	0
$567$	−9.67931e13	−0.0693646
$568$	0	0
$569$	1.92427e15	1.35254	0.676269	−	0.736655i	$-0.263596\pi$
$569$	1.92427e15	1.35254	0.676269	+	0.736655i	$0.263596\pi$
$570$	0	0
$571$	−1.61132e15	−1.11092	−0.555461	−	0.831543i	$-0.687458\pi$
$571$	−1.61132e15	−1.11092	−0.555461	+	0.831543i	$0.687458\pi$
$572$	0	0
$573$	8.23063e14	0.556650
$574$	0	0
$575$	−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	$0.808397\pi$
$578$	0	0
$579$	−1.19780e15	−0.764982
$580$	0	0
$581$	2.43412e14	0.152536
$582$	0	0
$583$	2.41894e15	1.48747
$584$	0	0
$585$	−2.42961e14	−0.146615
$586$	0	0
$587$	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
$590$	0	0
$591$	−1.54496e15	−0.881425
$592$	0	0
$593$	−2.20723e15	−1.23608	−0.618040	−	0.786146i	$-0.712073\pi$
$593$	−2.20723e15	−1.23608	−0.618040	+	0.786146i	$0.712073\pi$
$594$	0	0
$595$	−4.59788e14	−0.252764
$596$	0	0
$597$	−9.19885e14	−0.496449
$598$	0	0
$599$	3.41159e15	1.80763	0.903813	−	0.427927i	$-0.140756\pi$
$599$	3.41159e15	1.80763	0.903813	+	0.427927i	$0.140756\pi$
$600$	0	0
$601$	−1.57545e15	−0.819588	−0.409794	−	0.912178i	$-0.634399\pi$
$601$	−1.57545e15	−0.819588	−0.409794	+	0.912178i	$0.634399\pi$
$602$	0	0
$603$	−3.12511e14	−0.159632
$604$	0	0
$605$	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	$0.539738\pi$
$608$	0	0
$609$	7.25246e13	0.0350825
$610$	0	0
$611$	1.19203e15	0.566319
$612$	0	0
$613$	1.98375e15	0.925665	0.462832	−	0.886446i	$-0.346833\pi$
$613$	1.98375e15	0.925665	0.462832	+	0.886446i	$0.346833\pi$
$614$	0	0
$615$	1.17290e15	0.537584
$616$	0	0
$617$	−1.93099e15	−0.869382	−0.434691	−	0.900580i	$-0.643142\pi$
$617$	−1.93099e15	−0.869382	−0.434691	+	0.900580i	$0.643142\pi$
$618$	0	0
$619$	−9.47689e14	−0.419148	−0.209574	−	0.977793i	$-0.567208\pi$
$619$	−9.47689e14	−0.419148	−0.209574	+	0.977793i	$0.567208\pi$
$620$	0	0
$621$	2.19716e14	0.0954679
$622$	0	0
$623$	7.07124e14	0.301864
$624$	0	0
$625$	−1.00840e15	−0.422955
$626$	0	0
$627$	−3.02415e15	−1.24633
$628$	0	0
$629$	−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	$0.435854\pi$
$632$	0	0
$633$	−4.36170e13	−0.0170583
$634$	0	0
$635$	1.34397e15	0.516574
$636$	0	0
$637$	9.24597e14	0.349290
$638$	0	0
$639$	−8.75761e14	−0.325185
$640$	0	0
$641$	1.13144e15	0.412966	0.206483	−	0.978450i	$-0.433798\pi$
$641$	1.13144e15	0.412966	0.206483	+	0.978450i	$0.433798\pi$
$642$	0	0
$643$	8.68306e14	0.311539	0.155769	−	0.987793i	$-0.450214\pi$
$643$	8.68306e14	0.311539	0.155769	+	0.987793i	$0.450214\pi$
$644$	0	0
$645$	1.25003e15	0.440904
$646$	0	0
$647$	−1.37636e15	−0.477265	−0.238633	−	0.971110i	$-0.576699\pi$
$647$	−1.37636e15	−0.477265	−0.238633	+	0.971110i	$0.576699\pi$
$648$	0	0
$649$	3.54195e14	0.120753
$650$	0	0
$651$	3.43607e14	0.115177
$652$	0	0
$653$	−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
$656$	0	0
$657$	8.24974e14	0.262924
$658$	0	0
$659$	−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	$-0.405233\pi$
$661$	1.90332e15	0.586683	0.293342	+	0.956008i	$0.405233\pi$
$662$	0	0
$663$	−5.74276e14	−0.174099
$664$	0	0
$665$	−2.90859e15	−0.867288
$666$	0	0
$667$	−1.64627e14	−0.0482847
$668$	0	0
$669$	5.40841e14	0.156037
$670$	0	0
$671$	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	$0.258394\pi$
$674$	0	0
$675$	−2.86852e14	−0.0787930
$676$	0	0
$677$	−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
$680$	0	0
$681$	1.01051e14	0.0264382
$682$	0	0
$683$	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
$686$	0	0
$687$	−4.40432e15	−1.09804
$688$	0	0
$689$	2.90580e15	0.712952
$690$	0	0
$691$	−4.03729e15	−0.974901	−0.487450	−	0.873151i	$-0.662073\pi$
$691$	−4.03729e15	−0.974901	−0.487450	+	0.873151i	$0.662073\pi$
$692$	0	0
$693$	1.04554e15	0.248490
$694$	0	0
$695$	6.04805e15	1.41481
$696$	0	0
$697$	2.77232e15	0.638356
$698$	0	0
$699$	−4.55967e12	−0.00103350
$700$	0	0
$701$	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
$704$	0	0
$705$	−2.03010e15	−0.439013
$706$	0	0
$707$	2.58467e14	0.0550299
$708$	0	0
$709$	−8.66706e14	−0.181684	−0.0908422	−	0.995865i	$-0.528956\pi$
$709$	−8.66706e14	−0.181684	−0.0908422	+	0.995865i	$0.528956\pi$
$710$	0	0
$711$	2.19694e14	0.0453457
$712$	0	0
$713$	−7.79972e14	−0.158521
$714$	0	0
$715$	2.62442e15	0.525231
$716$	0	0
$717$	−4.28293e15	−0.844085
$718$	0	0
$719$	2.62666e13	0.00509795	0.00254897	−	0.999997i	$-0.499189\pi$
$719$	2.62666e13	0.00509795	0.00254897	+	0.999997i	$0.499189\pi$
$720$	0	0
$721$	−1.34844e15	−0.257744
$722$	0	0
$723$	−2.16298e14	−0.0407186
$724$	0	0
$725$	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	$0.673573\pi$
$728$	0	0
$729$	2.05891e14	0.0370370
$730$	0	0
$731$	2.95465e15	0.523552
$732$	0	0
$733$	1.62312e15	0.283320	0.141660	−	0.989915i	$-0.454756\pi$
$733$	1.62312e15	0.283320	0.141660	+	0.989915i	$0.454756\pi$
$734$	0	0
$735$	−1.57465e15	−0.270771
$736$	0	0
$737$	3.37568e15	0.571861
$738$	0	0
$739$	−3.64794e15	−0.608840	−0.304420	−	0.952538i	$-0.598463\pi$
$739$	−3.64794e15	−0.608840	−0.304420	+	0.952538i	$0.598463\pi$
$740$	0	0
$741$	−3.63283e15	−0.597372
$742$	0	0
$743$	6.76045e15	1.09531	0.547655	−	0.836704i	$-0.315521\pi$
$743$	6.76045e15	1.09531	0.547655	+	0.836704i	$0.315521\pi$
$744$	0	0
$745$	7.43520e15	1.18695
$746$	0	0
$747$	−5.17768e14	−0.0814465
$748$	0	0
$749$	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	$0.499356\pi$
$752$	0	0
$753$	5.88741e15	0.886242
$754$	0	0
$755$	−3.75149e15	−0.556539
$756$	0	0
$757$	−7.37364e15	−1.07809	−0.539045	−	0.842277i	$-0.681215\pi$
$757$	−7.37364e15	−1.07809	−0.539045	+	0.842277i	$0.681215\pi$
$758$	0	0
$759$	−2.37333e15	−0.342001
$760$	0	0
$761$	6.20983e15	0.881991	0.440996	−	0.897509i	$-0.354625\pi$
$761$	6.20983e15	0.881991	0.440996	+	0.897509i	$0.354625\pi$
$762$	0	0
$763$	−1.92796e15	−0.269907
$764$	0	0
$765$	9.78027e14	0.134963
$766$	0	0
$767$	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	$0.702516\pi$
$770$	0	0
$771$	1.89659e15	0.250712
$772$	0	0
$773$	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
$776$	0	0
$777$	4.48413e15	0.568019
$778$	0	0
$779$	1.75375e16	2.19034
$780$	0	0
$781$	9.45980e15	1.16493
$782$	0	0
$783$	−1.54269e14	−0.0187322
$784$	0	0
$785$	−1.14449e16	−1.37034
$786$	0	0
$787$	−1.35545e16	−1.60038	−0.800190	−	0.599746i	$-0.795268\pi$
$787$	−1.35545e16	−1.60038	−0.800190	+	0.599746i	$0.795268\pi$
$788$	0	0
$789$	4.03272e15	0.469543
$790$	0	0
$791$	9.98429e15	1.14643
$792$	0	0
$793$	3.79308e15	0.429526
$794$	0	0
$795$	−4.94876e15	−0.552685
$796$	0	0
$797$	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
$800$	0	0
$801$	−1.50414e15	−0.161180
$802$	0	0
$803$	−8.91121e15	−0.941892
$804$	0	0
$805$	−2.28263e15	−0.237990
$806$	0	0
$807$	−6.93123e15	−0.712863
$808$	0	0
$809$	9.45859e15	0.959643	0.479821	−	0.877366i	$-0.340701\pi$
$809$	9.45859e15	0.959643	0.479821	+	0.877366i	$0.340701\pi$
$810$	0	0
$811$	1.07996e16	1.08092	0.540461	−	0.841369i	$-0.318250\pi$
$811$	1.07996e16	1.08092	0.540461	+	0.841369i	$0.318250\pi$
$812$	0	0
$813$	5.68134e15	0.560987
$814$	0	0
$815$	8.78341e14	0.0855651
$816$	0	0
$817$	1.86909e16	1.79642
$818$	0	0
$819$	1.25598e15	0.119102
$820$	0	0
$821$	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	$0.302231\pi$
$824$	0	0
$825$	3.09852e15	0.282266
$826$	0	0
$827$	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	$-0.415344\pi$
$829$	5.92589e15	0.525659	0.262829	+	0.964842i	$0.415344\pi$
$830$	0	0
$831$	7.37848e15	0.645894
$832$	0	0
$833$	−3.72192e15	−0.321528
$834$	0	0
$835$	−4.73066e13	−0.00403317
$836$	0	0
$837$	−7.30896e14	−0.0614986
$838$	0	0
$839$	2.10209e16	1.74566	0.872831	−	0.488022i	$-0.162282\pi$
$839$	2.10209e16	1.74566	0.872831	+	0.488022i	$0.162282\pi$
$840$	0	0
$841$	−1.20849e16	−0.990526
$842$	0	0
$843$	3.88189e15	0.314045
$844$	0	0
$845$	−6.47126e15	−0.516746
$846$	0	0
$847$	−3.37348e15	−0.265901
$848$	0	0
$849$	−6.97520e15	−0.542705
$850$	0	0
$851$	−1.01788e16	−0.781775
$852$	0	0
$853$	1.98378e16	1.50409	0.752046	−	0.659110i	$-0.229067\pi$
$853$	1.98378e16	1.50409	0.752046	+	0.659110i	$0.229067\pi$
$854$	0	0
$855$	6.18693e15	0.463086
$856$	0	0
$857$	7.96639e15	0.588663	0.294332	−	0.955703i	$-0.404903\pi$
$857$	7.96639e15	0.588663	0.294332	+	0.955703i	$0.404903\pi$
$858$	0	0
$859$	1.59749e16	1.16540	0.582701	−	0.812687i	$-0.301996\pi$
$859$	1.59749e16	1.16540	0.582701	+	0.812687i	$0.301996\pi$
$860$	0	0
$861$	−6.06324e15	−0.436704
$862$	0	0
$863$	−4.02926e15	−0.286527	−0.143264	−	0.989685i	$-0.545760\pi$
$863$	−4.02926e15	−0.286527	−0.143264	+	0.989685i	$0.545760\pi$
$864$	0	0
$865$	3.92467e15	0.275559
$866$	0	0
$867$	−6.01635e15	−0.417089
$868$	0	0
$869$	−2.37310e15	−0.162445
$870$	0	0
$871$	4.05511e15	0.274096
$872$	0	0
$873$	−2.30837e15	−0.154073
$874$	0	0
$875$	1.02590e16	0.676176
$876$	0	0
$877$	−9.76958e15	−0.635884	−0.317942	−	0.948110i	$-0.602992\pi$
$877$	−9.76958e15	−0.635884	−0.317942	+	0.948110i	$0.602992\pi$
$878$	0	0
$879$	−1.17066e16	−0.752476
$880$	0	0
$881$	1.62583e16	1.03207	0.516033	−	0.856569i	$-0.327408\pi$
$881$	1.62583e16	1.03207	0.516033	+	0.856569i	$0.327408\pi$
$882$	0	0
$883$	−1.66061e16	−1.04108	−0.520540	−	0.853837i	$-0.674269\pi$
$883$	−1.66061e16	−1.04108	−0.520540	+	0.853837i	$0.674269\pi$
$884$	0	0
$885$	−7.24626e14	−0.0448669
$886$	0	0
$887$	−1.91314e16	−1.16995	−0.584975	−	0.811051i	$-0.698896\pi$
$887$	−1.91314e16	−1.16995	−0.584975	+	0.811051i	$0.698896\pi$
$888$	0	0
$889$	−6.94758e15	−0.419637
$890$	0	0
$891$	−2.22400e15	−0.132680
$892$	0	0
$893$	−3.03547e16	−1.78872
$894$	0	0
$895$	−2.10703e16	−1.22644
$896$	0	0
$897$	−2.85101e15	−0.163923
$898$	0	0
$899$	5.47641e14	0.0311041
$900$	0	0
$901$	−1.16972e16	−0.656287
$902$	0	0
$903$	−6.46201e15	−0.358166
$904$	0	0
$905$	−1.22399e16	−0.670209
$906$	0	0
$907$	1.07252e16	0.580182	0.290091	−	0.956999i	$-0.406314\pi$
$907$	1.07252e16	0.580182	0.290091	+	0.956999i	$0.406314\pi$
$908$	0	0
$909$	−5.49792e14	−0.0293831
$910$	0	0
$911$	−2.82249e16	−1.49032	−0.745162	−	0.666883i	$-0.767628\pi$
$911$	−2.82249e16	−1.49032	−0.745162	+	0.666883i	$0.767628\pi$
$912$	0	0
$913$	5.59284e15	0.291771
$914$	0	0
$915$	−6.45985e15	−0.332971
$916$	0	0
$917$	−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	$0.347190\pi$
$920$	0	0
$921$	−6.25361e15	−0.310959
$922$	0	0
$923$	1.13638e16	0.558359
$924$	0	0
$925$	1.32890e16	0.645227
$926$	0	0
$927$	2.86831e15	0.137622
$928$	0	0
$929$	−1.10125e16	−0.522155	−0.261078	−	0.965318i	$-0.584078\pi$
$929$	−1.10125e16	−0.522155	−0.261078	+	0.965318i	$0.584078\pi$
$930$	0	0
$931$	−2.35446e16	−1.10323
$932$	0	0
$933$	8.40884e15	0.389392
$934$	0	0
$935$	−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	$0.439429\pi$
$938$	0	0
$939$	3.06340e15	0.136944
$940$	0	0
$941$	−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
$944$	0	0
$945$	−2.13901e15	−0.0923289
$946$	0	0
$947$	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
$950$	0	0
$951$	1.98833e16	0.828888
$952$	0	0
$953$	4.79568e15	0.197624	0.0988119	−	0.995106i	$-0.468496\pi$
$953$	4.79568e15	0.197624	0.0988119	+	0.995106i	$0.468496\pi$
$954$	0	0
$955$	1.81887e16	0.740939
$956$	0	0
$957$	1.66638e15	0.0671057
$958$	0	0
$959$	4.30321e15	0.171313
$960$	0	0
$961$	−2.28139e16	−0.897884
$962$	0	0
$963$	−1.33212e16	−0.518321
$964$	0	0
$965$	−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	$0.278252\pi$
$968$	0	0
$969$	1.46238e16	0.549893
$970$	0	0
$971$	−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
$974$	0	0
$975$	3.72217e15	0.135291
$976$	0	0
$977$	−1.20023e16	−0.431366	−0.215683	−	0.976463i	$-0.569198\pi$
$977$	−1.20023e16	−0.431366	−0.215683	+	0.976463i	$0.569198\pi$
$978$	0	0
$979$	1.62474e16	0.577405
$980$	0	0
$981$	4.10102e15	0.144116
$982$	0	0
$983$	−4.42687e15	−0.153834	−0.0769170	−	0.997038i	$-0.524508\pi$
$983$	−4.42687e15	−0.153834	−0.0769170	+	0.997038i	$0.524508\pi$
$984$	0	0
$985$	−3.41417e16	−1.17324
$986$	0	0
$987$	1.04945e16	0.356631
$988$	0	0
$989$	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	$0.716978\pi$
$992$	0	0
$993$	8.26479e15	0.271651
$994$	0	0
$995$	−2.03283e16	−0.660807
$996$	0	0
$997$	−5.76003e13	−0.00185183	−0.000925915	−	1.00000i	$-0.500295\pi$
$997$	−5.76003e13	−0.00185183	−0.000925915		1.00000i	$0.500295\pi$
$998$	0	0
$999$	−9.53830e15	−0.303292

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	192.12.a.q.1.1		1
4.3	odd	2		192.12.a.g.1.1		1
8.3	odd	2		48.12.a.f.1.1		1
8.5	even	2		3.12.a.a.1.1	✓	1
24.5	odd	2		9.12.a.a.1.1		1
24.11	even	2		144.12.a.l.1.1		1
40.13	odd	4		75.12.b.a.49.1		2
40.29	even	2		75.12.a.a.1.1		1
40.37	odd	4		75.12.b.a.49.2		2
56.13	odd	2		147.12.a.c.1.1		1
72.5	odd	6		81.12.c.e.55.1		2
72.13	even	6		81.12.c.a.55.1		2
72.29	odd	6		81.12.c.e.28.1		2
72.61	even	6		81.12.c.a.28.1		2
120.29	odd	2		225.12.a.f.1.1		1
120.53	even	4		225.12.b.a.199.2		2
120.77	even	4		225.12.b.a.199.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3.12.a.a.1.1	✓	1	8.5	even	2
9.12.a.a.1.1		1	24.5	odd	2
48.12.a.f.1.1		1	8.3	odd	2
75.12.a.a.1.1		1	40.29	even	2
75.12.b.a.49.1		2	40.13	odd	4
75.12.b.a.49.2		2	40.37	odd	4
81.12.c.a.28.1		2	72.61	even	6
81.12.c.a.55.1		2	72.13	even	6
81.12.c.e.28.1		2	72.29	odd	6
81.12.c.e.55.1		2	72.5	odd	6
144.12.a.l.1.1		1	24.11	even	2
147.12.a.c.1.1		1	56.13	odd	2
192.12.a.g.1.1		1	4.3	odd	2
192.12.a.q.1.1		1	1.1	even	1	trivial
225.12.a.f.1.1		1	120.29	odd	2
225.12.b.a.199.1		2	120.77	even	4
225.12.b.a.199.2		2	120.53	even	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:	\( N \)	\(=\)	\( 192 = 2^{6} \cdot 3 \)
Weight:	\( k \)	\(=\)	\( 12 \)
Character orbit:	\([\chi]\)	\(=\)	192.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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