LMFDB - Embedded newform 12.18.a.a.1.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [12,18,Mod(1,12)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("12.1"); S:= CuspForms(chi, 18); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(12, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 18, names="a")

Level:	$ N $	$=$	$ 12 = 2^{2} \cdot 3 $
Weight:	$ k $	$=$	$ 18 $
Character orbit:	$[\chi]$	$=$	12.a (trivial)

Newform invariants

comment:select newform

sage:traces = [1,0,-6561] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)

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

Self dual:	yes
Analytic conductor:	$21.9866504813$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	12.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$

$=$

$q-6561.00 q^{3} -1.60893e6 q^{5} -9.41718e6 q^{7} +4.30467e7 q^{9} -1.86911e8 q^{11} -2.62544e9 q^{13} +1.05562e10 q^{15} +4.37823e10 q^{17} -9.65950e10 q^{19} +6.17861e10 q^{21} +2.90868e11 q^{23} +1.82572e12 q^{25} -2.82430e11 q^{27} +1.39862e12 q^{29} +7.64790e12 q^{31} +1.22632e12 q^{33} +1.51516e13 q^{35} -3.33695e13 q^{37} +1.72255e13 q^{39} -1.20327e13 q^{41} -7.55092e11 q^{43} -6.92592e13 q^{45} -2.80540e14 q^{47} -1.43947e14 q^{49} -2.87256e14 q^{51} +4.60570e14 q^{53} +3.00726e14 q^{55} +6.33760e14 q^{57} +1.07847e15 q^{59} -1.98078e15 q^{61} -4.05379e14 q^{63} +4.22415e15 q^{65} +4.85019e15 q^{67} -1.90838e15 q^{69} +2.70757e15 q^{71} -5.00226e15 q^{73} -1.19785e16 q^{75} +1.76017e15 q^{77} -9.77448e15 q^{79} +1.85302e15 q^{81} +1.71129e16 q^{83} -7.04427e16 q^{85} -9.17633e15 q^{87} +3.46982e16 q^{89} +2.47243e16 q^{91} -5.01779e16 q^{93} +1.55415e17 q^{95} +6.86169e16 q^{97} -8.04589e15 q^{99} +O(q^{100})$

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
$3$	−6561.00	−0.577350
$3$	−6561.00	−0.577350
$4$	0	0
$5$	−1.60893e6	−1.84201	−0.921005	−	0.389550i	$-0.872631\pi$
$5$	−1.60893e6	−1.84201	−0.921005	+	0.389550i	$0.872631\pi$
$6$	0	0
$7$	−9.41718e6	−0.617430	−0.308715	−	0.951155i	$-0.599899\pi$
$7$	−9.41718e6	−0.617430	−0.308715	+	0.951155i	$0.599899\pi$
$8$	0	0
$9$	4.30467e7	0.333333
$10$	0	0
$11$	−1.86911e8	−0.262903	−0.131452	−	0.991323i	$-0.541964\pi$
$11$	−1.86911e8	−0.262903	−0.131452	+	0.991323i	$0.541964\pi$
$12$	0	0
$13$	−2.62544e9	−0.892656	−0.446328	−	0.894869i	$-0.647268\pi$
$13$	−2.62544e9	−0.892656	−0.446328	+	0.894869i	$0.647268\pi$
$14$	0	0
$15$	1.05562e10	1.06349
$16$	0	0
$17$	4.37823e10	1.52224	0.761120	−	0.648612i	$-0.224650\pi$
$17$	4.37823e10	1.52224	0.761120	+	0.648612i	$0.224650\pi$
$18$	0	0
$19$	−9.65950e10	−1.30482	−0.652408	−	0.757868i	$-0.726241\pi$
$19$	−9.65950e10	−1.30482	−0.652408	+	0.757868i	$0.726241\pi$
$20$	0	0
$21$	6.17861e10	0.356473
$22$	0	0
$23$	2.90868e11	0.774478	0.387239	−	0.921979i	$-0.373429\pi$
$23$	2.90868e11	0.774478	0.387239	+	0.921979i	$0.373429\pi$
$24$	0	0
$25$	1.82572e12	2.39300
$26$	0	0
$27$	−2.82430e11	−0.192450
$28$	0	0
$29$	1.39862e12	0.519178	0.259589	−	0.965719i	$-0.416413\pi$
$29$	1.39862e12	0.519178	0.259589	+	0.965719i	$0.416413\pi$
$30$	0	0
$31$	7.64790e12	1.61053	0.805263	−	0.592918i	$-0.202024\pi$
$31$	7.64790e12	1.61053	0.805263	+	0.592918i	$0.202024\pi$
$32$	0	0
$33$	1.22632e12	0.151787
$34$	0	0
$35$	1.51516e13	1.13731
$36$	0	0
$37$	−3.33695e13	−1.56184	−0.780918	−	0.624634i	$-0.785248\pi$
$37$	−3.33695e13	−1.56184	−0.780918	+	0.624634i	$0.785248\pi$
$38$	0	0
$39$	1.72255e13	0.515375
$40$	0	0
$41$	−1.20327e13	−0.235343	−0.117672	−	0.993053i	$-0.537543\pi$
$41$	−1.20327e13	−0.235343	−0.117672	+	0.993053i	$0.537543\pi$
$42$	0	0
$43$	−7.55092e11	−0.00985186	−0.00492593	−	0.999988i	$-0.501568\pi$
$43$	−7.55092e11	−0.00985186	−0.00492593	+	0.999988i	$0.501568\pi$
$44$	0	0
$45$	−6.92592e13	−0.614004
$46$	0	0
$47$	−2.80540e14	−1.71856	−0.859278	−	0.511509i	$-0.829087\pi$
$47$	−2.80540e14	−1.71856	−0.859278	+	0.511509i	$0.829087\pi$
$48$	0	0
$49$	−1.43947e14	−0.618780
$50$	0	0
$51$	−2.87256e14	−0.878865
$52$	0	0
$53$	4.60570e14	1.01613	0.508067	−	0.861318i	$-0.330360\pi$
$53$	4.60570e14	1.01613	0.508067	+	0.861318i	$0.330360\pi$
$54$	0	0
$55$	3.00726e14	0.484271
$56$	0	0
$57$	6.33760e14	0.753335
$58$	0	0
$59$	1.07847e15	0.956236	0.478118	−	0.878296i	$-0.341319\pi$
$59$	1.07847e15	0.956236	0.478118	+	0.878296i	$0.341319\pi$
$60$	0	0
$61$	−1.98078e15	−1.32292	−0.661458	−	0.749982i	$-0.730062\pi$
$61$	−1.98078e15	−1.32292	−0.661458	+	0.749982i	$0.730062\pi$
$62$	0	0
$63$	−4.05379e14	−0.205810
$64$	0	0
$65$	4.22415e15	1.64428
$66$	0	0
$67$	4.85019e15	1.45923	0.729614	−	0.683860i	$-0.239700\pi$
$67$	4.85019e15	1.45923	0.729614	+	0.683860i	$0.239700\pi$
$68$	0	0
$69$	−1.90838e15	−0.447145
$70$	0	0
$71$	2.70757e15	0.497605	0.248802	−	0.968554i	$-0.419963\pi$
$71$	2.70757e15	0.497605	0.248802	+	0.968554i	$0.419963\pi$
$72$	0	0
$73$	−5.00226e15	−0.725976	−0.362988	−	0.931794i	$-0.618243\pi$
$73$	−5.00226e15	−0.725976	−0.362988	+	0.931794i	$0.618243\pi$
$74$	0	0
$75$	−1.19785e16	−1.38160
$76$	0	0
$77$	1.76017e15	0.162324
$78$	0	0
$79$	−9.77448e15	−0.724875	−0.362438	−	0.932008i	$-0.618055\pi$
$79$	−9.77448e15	−0.724875	−0.362438	+	0.932008i	$0.618055\pi$
$80$	0	0
$81$	1.85302e15	0.111111
$82$	0	0
$83$	1.71129e16	0.833989	0.416994	−	0.908909i	$-0.363083\pi$
$83$	1.71129e16	0.833989	0.416994	+	0.908909i	$0.363083\pi$
$84$	0	0
$85$	−7.04427e16	−2.80398
$86$	0	0
$87$	−9.17633e15	−0.299747
$88$	0	0
$89$	3.46982e16	0.934311	0.467156	−	0.884175i	$-0.345279\pi$
$89$	3.46982e16	0.934311	0.467156	+	0.884175i	$0.345279\pi$
$90$	0	0
$91$	2.47243e16	0.551152
$92$	0	0
$93$	−5.01779e16	−0.929838
$94$	0	0
$95$	1.55415e17	2.40348
$96$	0	0
$97$	6.86169e16	0.888938	0.444469	−	0.895794i	$-0.353392\pi$
$97$	6.86169e16	0.888938	0.444469	+	0.895794i	$0.353392\pi$
$98$	0	0
$99$	−8.04589e15	−0.0876344
$100$	0	0
$101$	−7.25756e16	−0.666897	−0.333449	−	0.942768i	$-0.608212\pi$
$101$	−7.25756e16	−0.666897	−0.333449	+	0.942768i	$0.608212\pi$
$102$	0	0
$103$	3.78078e16	0.294080	0.147040	−	0.989131i	$-0.453025\pi$
$103$	3.78078e16	0.294080	0.147040	+	0.989131i	$0.453025\pi$
$104$	0	0
$105$	−9.94096e16	−0.656628
$106$	0	0
$107$	1.77782e17	1.00029	0.500145	−	0.865942i	$-0.333280\pi$
$107$	1.77782e17	1.00029	0.500145	+	0.865942i	$0.333280\pi$
$108$	0	0
$109$	−3.23153e16	−0.155340	−0.0776701	−	0.996979i	$-0.524748\pi$
$109$	−3.23153e16	−0.155340	−0.0776701	+	0.996979i	$0.524748\pi$
$110$	0	0
$111$	2.18937e17	0.901726
$112$	0	0
$113$	1.45944e17	0.516440	0.258220	−	0.966086i	$-0.416864\pi$
$113$	1.45944e17	0.516440	0.258220	+	0.966086i	$0.416864\pi$
$114$	0	0
$115$	−4.67986e17	−1.42660
$116$	0	0
$117$	−1.13017e17	−0.297552
$118$	0	0
$119$	−4.12306e17	−0.939876
$120$	0	0
$121$	−4.70511e17	−0.930882
$122$	0	0
$123$	7.89468e16	0.135875
$124$	0	0
$125$	−1.70993e18	−2.56593
$126$	0	0
$127$	1.38813e18	1.82012	0.910058	−	0.414480i	$-0.136037\pi$
$127$	1.38813e18	1.82012	0.910058	+	0.414480i	$0.136037\pi$
$128$	0	0
$129$	4.95416e15	0.00568797
$130$	0	0
$131$	9.07566e17	0.914265	0.457133	−	0.889399i	$-0.348876\pi$
$131$	9.07566e17	0.914265	0.457133	+	0.889399i	$0.348876\pi$
$132$	0	0
$133$	9.09653e17	0.805632
$134$	0	0
$135$	4.54409e17	0.354495
$136$	0	0
$137$	−2.35557e18	−1.62170	−0.810851	−	0.585252i	$-0.800995\pi$
$137$	−2.35557e18	−1.62170	−0.810851	+	0.585252i	$0.800995\pi$
$138$	0	0
$139$	1.88350e18	1.14641	0.573204	−	0.819413i	$-0.305700\pi$
$139$	1.88350e18	1.14641	0.573204	+	0.819413i	$0.305700\pi$
$140$	0	0
$141$	1.84063e18	0.992208
$142$	0	0
$143$	4.90723e17	0.234682
$144$	0	0
$145$	−2.25028e18	−0.956331
$146$	0	0
$147$	9.44437e17	0.357253
$148$	0	0
$149$	5.50730e18	1.85719	0.928593	−	0.371100i	$-0.121019\pi$
$149$	5.50730e18	1.85719	0.928593	+	0.371100i	$0.121019\pi$
$150$	0	0
$151$	1.34665e18	0.405463	0.202732	−	0.979234i	$-0.435018\pi$
$151$	1.34665e18	0.405463	0.202732	+	0.979234i	$0.435018\pi$
$152$	0	0
$153$	1.88468e18	0.507413
$154$	0	0
$155$	−1.23049e19	−2.96661
$156$	0	0
$157$	1.49201e18	0.322570	0.161285	−	0.986908i	$-0.448436\pi$
$157$	1.49201e18	0.322570	0.161285	+	0.986908i	$0.448436\pi$
$158$	0	0
$159$	−3.02180e18	−0.586665
$160$	0	0
$161$	−2.73916e18	−0.478186
$162$	0	0
$163$	8.75681e17	0.137642	0.0688211	−	0.997629i	$-0.478076\pi$
$163$	8.75681e17	0.137642	0.0688211	+	0.997629i	$0.478076\pi$
$164$	0	0
$165$	−1.97306e18	−0.279594
$166$	0	0
$167$	5.32202e18	0.680749	0.340374	−	0.940290i	$-0.389446\pi$
$167$	5.32202e18	0.680749	0.340374	+	0.940290i	$0.389446\pi$
$168$	0	0
$169$	−1.75747e18	−0.203166
$170$	0	0
$171$	−4.15810e18	−0.434938
$172$	0	0
$173$	−2.09639e18	−0.198646	−0.0993231	−	0.995055i	$-0.531668\pi$
$173$	−2.09639e18	−0.198646	−0.0993231	+	0.995055i	$0.531668\pi$
$174$	0	0
$175$	−1.71931e19	−1.47751
$176$	0	0
$177$	−7.07583e18	−0.552083
$178$	0	0
$179$	6.16941e18	0.437515	0.218757	−	0.975779i	$-0.429800\pi$
$179$	6.16941e18	0.437515	0.218757	+	0.975779i	$0.429800\pi$
$180$	0	0
$181$	−2.82801e18	−0.182479	−0.0912394	−	0.995829i	$-0.529083\pi$
$181$	−2.82801e18	−0.182479	−0.0912394	+	0.995829i	$0.529083\pi$
$182$	0	0
$183$	1.29959e19	0.763786
$184$	0	0
$185$	5.36892e19	2.87692
$186$	0	0
$187$	−8.18337e18	−0.400202
$188$	0	0
$189$	2.65969e18	0.118824
$190$	0	0
$191$	1.63173e19	0.666600	0.333300	−	0.942821i	$-0.391838\pi$
$191$	1.63173e19	0.666600	0.333300	+	0.942821i	$0.391838\pi$
$192$	0	0
$193$	4.37787e19	1.63691	0.818456	−	0.574569i	$-0.194830\pi$
$193$	4.37787e19	1.63691	0.818456	+	0.574569i	$0.194830\pi$
$194$	0	0
$195$	−2.77147e19	−0.949326
$196$	0	0
$197$	−1.33554e18	−0.0419465	−0.0209732	−	0.999780i	$-0.506676\pi$
$197$	−1.33554e18	−0.0419465	−0.0209732	+	0.999780i	$0.506676\pi$
$198$	0	0
$199$	−3.17824e19	−0.916084	−0.458042	−	0.888931i	$-0.651449\pi$
$199$	−3.17824e19	−0.916084	−0.458042	+	0.888931i	$0.651449\pi$
$200$	0	0
$201$	−3.18221e19	−0.842485
$202$	0	0
$203$	−1.31710e19	−0.320556
$204$	0	0
$205$	1.93598e19	0.433505
$206$	0	0
$207$	1.25209e19	0.258159
$208$	0	0
$209$	1.80546e19	0.343040
$210$	0	0
$211$	−1.86614e19	−0.326998	−0.163499	−	0.986544i	$-0.552278\pi$
$211$	−1.86614e19	−0.326998	−0.163499	+	0.986544i	$0.552278\pi$
$212$	0	0
$213$	−1.77644e19	−0.287292
$214$	0	0
$215$	1.21489e18	0.0181472
$216$	0	0
$217$	−7.20217e19	−0.994387
$218$	0	0
$219$	3.28199e19	0.419143
$220$	0	0
$221$	−1.14948e20	−1.35884
$222$	0	0
$223$	−6.69382e19	−0.732965	−0.366482	−	0.930425i	$-0.619438\pi$
$223$	−6.69382e19	−0.732965	−0.366482	+	0.930425i	$0.619438\pi$
$224$	0	0
$225$	7.85911e19	0.797668
$226$	0	0
$227$	−5.15149e19	−0.484968	−0.242484	−	0.970155i	$-0.577962\pi$
$227$	−5.15149e19	−0.484968	−0.242484	+	0.970155i	$0.577962\pi$
$228$	0	0
$229$	−9.34272e19	−0.816341	−0.408171	−	0.912906i	$-0.633833\pi$
$229$	−9.34272e19	−0.816341	−0.408171	+	0.912906i	$0.633833\pi$
$230$	0	0
$231$	−1.15485e19	−0.0937180
$232$	0	0
$233$	5.71959e19	0.431360	0.215680	−	0.976464i	$-0.430803\pi$
$233$	5.71959e19	0.431360	0.215680	+	0.976464i	$0.430803\pi$
$234$	0	0
$235$	4.51370e20	3.16560
$236$	0	0
$237$	6.41303e19	0.418507
$238$	0	0
$239$	−2.66748e20	−1.62076	−0.810380	−	0.585905i	$-0.800739\pi$
$239$	−2.66748e20	−1.62076	−0.810380	+	0.585905i	$0.800739\pi$
$240$	0	0
$241$	4.62650e19	0.261883	0.130942	−	0.991390i	$-0.458200\pi$
$241$	4.62650e19	0.261883	0.130942	+	0.991390i	$0.458200\pi$
$242$	0	0
$243$	−1.21577e19	−0.0641500
$244$	0	0
$245$	2.31601e20	1.13980
$246$	0	0
$247$	2.53605e20	1.16475
$248$	0	0
$249$	−1.12278e20	−0.481504
$250$	0	0
$251$	−1.11918e20	−0.448407	−0.224204	−	0.974542i	$-0.571978\pi$
$251$	−1.11918e20	−0.448407	−0.224204	+	0.974542i	$0.571978\pi$
$252$	0	0
$253$	−5.43663e19	−0.203613
$254$	0	0
$255$	4.62174e20	1.61888
$256$	0	0
$257$	−5.37780e20	−1.76268	−0.881339	−	0.472484i	$-0.843358\pi$
$257$	−5.37780e20	−1.76268	−0.881339	+	0.472484i	$0.843358\pi$
$258$	0	0
$259$	3.14247e20	0.964324
$260$	0	0
$261$	6.02059e19	0.173059
$262$	0	0
$263$	3.89151e20	1.04832	0.524160	−	0.851620i	$-0.324379\pi$
$263$	3.89151e20	1.04832	0.524160	+	0.851620i	$0.324379\pi$
$264$	0	0
$265$	−7.41025e20	−1.87173
$266$	0	0
$267$	−2.27655e20	−0.539425
$268$	0	0
$269$	−1.47304e20	−0.327581	−0.163791	−	0.986495i	$-0.552372\pi$
$269$	−1.47304e20	−0.327581	−0.163791	+	0.986495i	$0.552372\pi$
$270$	0	0
$271$	−9.40590e20	−1.96409	−0.982045	−	0.188644i	$-0.939591\pi$
$271$	−9.40590e20	−1.96409	−0.982045	+	0.188644i	$0.939591\pi$
$272$	0	0
$273$	−1.62216e20	−0.318208
$274$	0	0
$275$	−3.41246e20	−0.629128
$276$	0	0
$277$	3.36947e19	0.0584095	0.0292048	−	0.999573i	$-0.490703\pi$
$277$	3.36947e19	0.0584095	0.0292048	+	0.999573i	$0.490703\pi$
$278$	0	0
$279$	3.29217e20	0.536842
$280$	0	0
$281$	5.26371e20	0.807772	0.403886	−	0.914809i	$-0.367659\pi$
$281$	5.26371e20	0.807772	0.403886	+	0.914809i	$0.367659\pi$
$282$	0	0
$283$	9.08901e20	1.31320	0.656601	−	0.754238i	$-0.271993\pi$
$283$	9.08901e20	1.31320	0.656601	+	0.754238i	$0.271993\pi$
$284$	0	0
$285$	−1.01967e21	−1.38765
$286$	0	0
$287$	1.13314e20	0.145308
$288$	0	0
$289$	1.08965e21	1.31721
$290$	0	0
$291$	−4.50196e20	−0.513229
$292$	0	0
$293$	8.93755e20	0.961266	0.480633	−	0.876922i	$-0.340407\pi$
$293$	8.93755e20	0.961266	0.480633	+	0.876922i	$0.340407\pi$
$294$	0	0
$295$	−1.73518e21	−1.76140
$296$	0	0
$297$	5.27891e19	0.0505958
$298$	0	0
$299$	−7.63657e20	−0.691343
$300$	0	0
$301$	7.11084e18	0.00608283
$302$	0	0
$303$	4.76169e20	0.385033
$304$	0	0
$305$	3.18693e21	2.43683
$306$	0	0
$307$	1.27904e20	0.0925141	0.0462571	−	0.998930i	$-0.485271\pi$
$307$	1.27904e20	0.0925141	0.0462571	+	0.998930i	$0.485271\pi$
$308$	0	0
$309$	−2.48057e20	−0.169787
$310$	0	0
$311$	2.60139e21	1.68555	0.842776	−	0.538265i	$-0.180920\pi$
$311$	2.60139e21	1.68555	0.842776	+	0.538265i	$0.180920\pi$
$312$	0	0
$313$	2.01146e20	0.123420	0.0617098	−	0.998094i	$-0.480345\pi$
$313$	2.01146e20	0.123420	0.0617098	+	0.998094i	$0.480345\pi$
$314$	0	0
$315$	6.52226e20	0.379104
$316$	0	0
$317$	8.78958e20	0.484132	0.242066	−	0.970260i	$-0.422175\pi$
$317$	8.78958e20	0.484132	0.242066	+	0.970260i	$0.422175\pi$
$318$	0	0
$319$	−2.61416e20	−0.136493
$320$	0	0
$321$	−1.16643e21	−0.577518
$322$	0	0
$323$	−4.22915e21	−1.98624
$324$	0	0
$325$	−4.79331e21	−2.13613
$326$	0	0
$327$	2.12021e20	0.0896857
$328$	0	0
$329$	2.64190e21	1.06109
$330$	0	0
$331$	3.13804e21	1.19707	0.598537	−	0.801095i	$-0.295749\pi$
$331$	3.13804e21	1.19707	0.598537	+	0.801095i	$0.295749\pi$
$332$	0	0
$333$	−1.43645e21	−0.520612
$334$	0	0
$335$	−7.80362e21	−2.68791
$336$	0	0
$337$	1.85359e21	0.606960	0.303480	−	0.952838i	$-0.401851\pi$
$337$	1.85359e21	0.606960	0.303480	+	0.952838i	$0.401851\pi$
$338$	0	0
$339$	−9.57539e20	−0.298167
$340$	0	0
$341$	−1.42947e21	−0.423412
$342$	0	0
$343$	3.54630e21	0.999483
$344$	0	0
$345$	3.07046e21	0.823646
$346$	0	0
$347$	−2.69836e21	−0.689126	−0.344563	−	0.938763i	$-0.611973\pi$
$347$	−2.69836e21	−0.689126	−0.344563	+	0.938763i	$0.611973\pi$
$348$	0	0
$349$	−2.39727e21	−0.583043	−0.291521	−	0.956564i	$-0.594161\pi$
$349$	−2.39727e21	−0.583043	−0.291521	+	0.956564i	$0.594161\pi$
$350$	0	0
$351$	7.41503e20	0.171792
$352$	0	0
$353$	1.35894e21	0.299995	0.149998	−	0.988686i	$-0.452073\pi$
$353$	1.35894e21	0.299995	0.149998	+	0.988686i	$0.452073\pi$
$354$	0	0
$355$	−4.35630e21	−0.916593
$356$	0	0
$357$	2.70514e21	0.542638
$358$	0	0
$359$	−1.79483e21	−0.343336	−0.171668	−	0.985155i	$-0.554916\pi$
$359$	−1.79483e21	−0.343336	−0.171668	+	0.985155i	$0.554916\pi$
$360$	0	0
$361$	3.85020e21	0.702542
$362$	0	0
$363$	3.08703e21	0.537445
$364$	0	0
$365$	8.04829e21	1.33726
$366$	0	0
$367$	4.79484e21	0.760523	0.380261	−	0.924879i	$-0.375834\pi$
$367$	4.79484e21	0.760523	0.380261	+	0.924879i	$0.375834\pi$
$368$	0	0
$369$	−5.17970e20	−0.0784477
$370$	0	0
$371$	−4.33727e21	−0.627392
$372$	0	0
$373$	1.09326e20	0.0151077	0.00755387	−	0.999971i	$-0.497596\pi$
$373$	1.09326e20	0.0151077	0.00755387	+	0.999971i	$0.497596\pi$
$374$	0	0
$375$	1.12189e22	1.48144
$376$	0	0
$377$	−3.67199e21	−0.463447
$378$	0	0
$379$	1.01476e22	1.22442	0.612212	−	0.790693i	$-0.290280\pi$
$379$	1.01476e22	1.22442	0.612212	+	0.790693i	$0.290280\pi$
$380$	0	0
$381$	−9.10753e21	−1.05084
$382$	0	0
$383$	−6.24544e21	−0.689245	−0.344622	−	0.938741i	$-0.611993\pi$
$383$	−6.24544e21	−0.689245	−0.344622	+	0.938741i	$0.611993\pi$
$384$	0	0
$385$	−2.83199e21	−0.299003
$386$	0	0
$387$	−3.25043e19	−0.00328395
$388$	0	0
$389$	−1.15546e22	−1.11733	−0.558666	−	0.829393i	$-0.688687\pi$
$389$	−1.15546e22	−1.11733	−0.558666	+	0.829393i	$0.688687\pi$
$390$	0	0
$391$	1.27349e22	1.17894
$392$	0	0
$393$	−5.95454e21	−0.527851
$394$	0	0
$395$	1.57264e22	1.33523
$396$	0	0
$397$	−3.20032e21	−0.260300	−0.130150	−	0.991494i	$-0.541546\pi$
$397$	−3.20032e21	−0.260300	−0.130150	+	0.991494i	$0.541546\pi$
$398$	0	0
$399$	−5.96823e21	−0.465132
$400$	0	0
$401$	−8.03454e20	−0.0600114	−0.0300057	−	0.999550i	$-0.509553\pi$
$401$	−8.03454e20	−0.0600114	−0.0300057	+	0.999550i	$0.509553\pi$
$402$	0	0
$403$	−2.00791e22	−1.43765
$404$	0	0
$405$	−2.98138e21	−0.204668
$406$	0	0
$407$	6.23711e21	0.410612
$408$	0	0
$409$	−1.89342e22	−1.19564	−0.597818	−	0.801632i	$-0.703966\pi$
$409$	−1.89342e22	−1.19564	−0.597818	+	0.801632i	$0.703966\pi$
$410$	0	0
$411$	1.54549e22	0.936290
$412$	0	0
$413$	−1.01561e22	−0.590409
$414$	0	0
$415$	−2.75335e22	−1.53622
$416$	0	0
$417$	−1.23576e22	−0.661879
$418$	0	0
$419$	−8.50946e21	−0.437605	−0.218803	−	0.975769i	$-0.570215\pi$
$419$	−8.50946e21	−0.437605	−0.218803	+	0.975769i	$0.570215\pi$
$420$	0	0
$421$	−1.54172e22	−0.761393	−0.380697	−	0.924700i	$-0.624316\pi$
$421$	−1.54172e22	−0.761393	−0.380697	+	0.924700i	$0.624316\pi$
$422$	0	0
$423$	−1.20763e22	−0.572852
$424$	0	0
$425$	7.99341e22	3.64272
$426$	0	0
$427$	1.86534e22	0.816808
$428$	0	0
$429$	−3.21963e21	−0.135494
$430$	0	0
$431$	7.64274e21	0.309167	0.154583	−	0.987980i	$-0.450597\pi$
$431$	7.64274e21	0.309167	0.154583	+	0.987980i	$0.450597\pi$
$432$	0	0
$433$	3.80213e22	1.47870	0.739350	−	0.673321i	$-0.235133\pi$
$433$	3.80213e22	1.47870	0.739350	+	0.673321i	$0.235133\pi$
$434$	0	0
$435$	1.47641e22	0.552138
$436$	0	0
$437$	−2.80964e22	−1.01055
$438$	0	0
$439$	4.81443e22	1.66570	0.832850	−	0.553499i	$-0.186708\pi$
$439$	4.81443e22	1.66570	0.832850	+	0.553499i	$0.186708\pi$
$440$	0	0
$441$	−6.19645e21	−0.206260
$442$	0	0
$443$	2.62576e22	0.841054	0.420527	−	0.907280i	$-0.361845\pi$
$443$	2.62576e22	0.841054	0.420527	+	0.907280i	$0.361845\pi$
$444$	0	0
$445$	−5.58269e22	−1.72101
$446$	0	0
$447$	−3.61334e22	−1.07225
$448$	0	0
$449$	3.54890e22	1.01391	0.506955	−	0.861972i	$-0.330771\pi$
$449$	3.54890e22	1.01391	0.506955	+	0.861972i	$0.330771\pi$
$450$	0	0
$451$	2.24904e21	0.0618725
$452$	0	0
$453$	−8.83539e21	−0.234094
$454$	0	0
$455$	−3.97796e22	−1.01523
$456$	0	0
$457$	2.62153e22	0.644566	0.322283	−	0.946643i	$-0.395550\pi$
$457$	2.62153e22	0.644566	0.322283	+	0.946643i	$0.395550\pi$
$458$	0	0
$459$	−1.23654e22	−0.292955
$460$	0	0
$461$	3.34342e21	0.0763366	0.0381683	−	0.999271i	$-0.487848\pi$
$461$	3.34342e21	0.0763366	0.0381683	+	0.999271i	$0.487848\pi$
$462$	0	0
$463$	3.74210e22	0.823525	0.411762	−	0.911291i	$-0.364913\pi$
$463$	3.74210e22	0.823525	0.411762	+	0.911291i	$0.364913\pi$
$464$	0	0
$465$	8.07327e22	1.71277
$466$	0	0
$467$	1.77250e22	0.362570	0.181285	−	0.983431i	$-0.441974\pi$
$467$	1.77250e22	0.362570	0.181285	+	0.983431i	$0.441974\pi$
$468$	0	0
$469$	−4.56751e22	−0.900971
$470$	0	0
$471$	−9.78906e21	−0.186236
$472$	0	0
$473$	1.41135e20	0.00259009
$474$	0	0
$475$	−1.76355e23	−3.12243
$476$	0	0
$477$	1.98260e22	0.338711
$478$	0	0
$479$	−1.19637e23	−1.97249	−0.986243	−	0.165303i	$-0.947140\pi$
$479$	−1.19637e23	−1.97249	−0.986243	+	0.165303i	$0.947140\pi$
$480$	0	0
$481$	8.76098e22	1.39418
$482$	0	0
$483$	1.79716e22	0.276081
$484$	0	0
$485$	−1.10400e23	−1.63743
$486$	0	0
$487$	1.47191e22	0.210807	0.105404	−	0.994430i	$-0.466387\pi$
$487$	1.47191e22	0.210807	0.105404	+	0.994430i	$0.466387\pi$
$488$	0	0
$489$	−5.74534e21	−0.0794677
$490$	0	0
$491$	−7.72668e21	−0.103229	−0.0516144	−	0.998667i	$-0.516437\pi$
$491$	−7.72668e21	−0.103229	−0.0516144	+	0.998667i	$0.516437\pi$
$492$	0	0
$493$	6.12347e22	0.790312
$494$	0	0
$495$	1.29453e22	0.161424
$496$	0	0
$497$	−2.54977e22	−0.307236
$498$	0	0
$499$	1.53892e22	0.179210	0.0896048	−	0.995977i	$-0.471440\pi$
$499$	1.53892e22	0.179210	0.0896048	+	0.995977i	$0.471440\pi$
$500$	0	0
$501$	−3.49178e22	−0.393030
$502$	0	0
$503$	4.93475e22	0.536954	0.268477	−	0.963286i	$-0.413480\pi$
$503$	4.93475e22	0.536954	0.268477	+	0.963286i	$0.413480\pi$
$504$	0	0
$505$	1.16769e23	1.22843
$506$	0	0
$507$	1.15307e22	0.117298
$508$	0	0
$509$	−5.05418e22	−0.497221	−0.248610	−	0.968604i	$-0.579974\pi$
$509$	−5.05418e22	−0.497221	−0.248610	+	0.968604i	$0.579974\pi$
$510$	0	0
$511$	4.71072e22	0.448239
$512$	0	0
$513$	2.72813e22	0.251112
$514$	0	0
$515$	−6.08302e22	−0.541699
$516$	0	0
$517$	5.24359e22	0.451814
$518$	0	0
$519$	1.37544e22	0.114688
$520$	0	0
$521$	−8.91172e22	−0.719185	−0.359593	−	0.933109i	$-0.617084\pi$
$521$	−8.91172e22	−0.719185	−0.359593	+	0.933109i	$0.617084\pi$
$522$	0	0
$523$	1.27422e23	0.995363	0.497682	−	0.867360i	$-0.334185\pi$
$523$	1.27422e23	0.995363	0.497682	+	0.867360i	$0.334185\pi$
$524$	0	0
$525$	1.12804e23	0.853042
$526$	0	0
$527$	3.34843e23	2.45161
$528$	0	0
$529$	−5.64459e22	−0.400183
$530$	0	0
$531$	4.64245e22	0.318745
$532$	0	0
$533$	3.15913e22	0.210080
$534$	0	0
$535$	−2.86039e23	−1.84254
$536$	0	0
$537$	−4.04775e22	−0.252599
$538$	0	0
$539$	2.69052e22	0.162679
$540$	0	0
$541$	1.53264e22	0.0897973	0.0448986	−	0.998992i	$-0.485704\pi$
$541$	1.53264e22	0.0897973	0.0448986	+	0.998992i	$0.485704\pi$
$542$	0	0
$543$	1.85546e22	0.105354
$544$	0	0
$545$	5.19931e22	0.286138
$546$	0	0
$547$	−2.48673e23	−1.32659	−0.663293	−	0.748359i	$-0.730842\pi$
$547$	−2.48673e23	−1.32659	−0.663293	+	0.748359i	$0.730842\pi$
$548$	0	0
$549$	−8.52660e22	−0.440972
$550$	0	0
$551$	−1.35099e23	−0.677431
$552$	0	0
$553$	9.20481e22	0.447560
$554$	0	0
$555$	−3.52255e23	−1.66099
$556$	0	0
$557$	−2.81655e23	−1.28810	−0.644049	−	0.764984i	$-0.722747\pi$
$557$	−2.81655e23	−1.28810	−0.644049	+	0.764984i	$0.722747\pi$
$558$	0	0
$559$	1.98245e21	0.00879432
$560$	0	0
$561$	5.36911e22	0.231056
$562$	0	0
$563$	4.36726e23	1.82342	0.911712	−	0.410830i	$-0.134761\pi$
$563$	4.36726e23	1.82342	0.911712	+	0.410830i	$0.134761\pi$
$564$	0	0
$565$	−2.34814e23	−0.951288
$566$	0	0
$567$	−1.74502e22	−0.0686033
$568$	0	0
$569$	−1.54119e23	−0.588035	−0.294017	−	0.955800i	$-0.594992\pi$
$569$	−1.54119e23	−0.588035	−0.294017	+	0.955800i	$0.594992\pi$
$570$	0	0
$571$	3.09335e23	1.14557	0.572786	−	0.819705i	$-0.305863\pi$
$571$	3.09335e23	1.14557	0.572786	+	0.819705i	$0.305863\pi$
$572$	0	0
$573$	−1.07058e23	−0.384862
$574$	0	0
$575$	5.31042e23	1.85333
$576$	0	0
$577$	7.27005e21	0.0246345	0.0123172	−	0.999924i	$-0.496079\pi$
$577$	7.27005e21	0.0246345	0.0123172	+	0.999924i	$0.496079\pi$
$578$	0	0
$579$	−2.87232e23	−0.945072
$580$	0	0
$581$	−1.61156e23	−0.514930
$582$	0	0
$583$	−8.60854e22	−0.267145
$584$	0	0
$585$	1.81836e23	0.548094
$586$	0	0
$587$	−1.01339e23	−0.296723	−0.148361	−	0.988933i	$-0.547400\pi$
$587$	−1.01339e23	−0.296723	−0.148361	+	0.988933i	$0.547400\pi$
$588$	0	0
$589$	−7.38749e23	−2.10144
$590$	0	0
$591$	8.76251e21	0.0242178
$592$	0	0
$593$	9.59327e22	0.257633	0.128817	−	0.991668i	$-0.458882\pi$
$593$	9.59327e22	0.257633	0.128817	+	0.991668i	$0.458882\pi$
$594$	0	0
$595$	6.63372e23	1.73126
$596$	0	0
$597$	2.08524e23	0.528901
$598$	0	0
$599$	2.26334e23	0.557983	0.278992	−	0.960294i	$-0.410000\pi$
$599$	2.26334e23	0.557983	0.278992	+	0.960294i	$0.410000\pi$
$600$	0	0
$601$	−3.91277e23	−0.937672	−0.468836	−	0.883285i	$-0.655326\pi$
$601$	−3.91277e23	−0.937672	−0.468836	+	0.883285i	$0.655326\pi$
$602$	0	0
$603$	2.08785e23	0.486409
$604$	0	0
$605$	7.57020e23	1.71469
$606$	0	0
$607$	4.37018e22	0.0962488	0.0481244	−	0.998841i	$-0.484676\pi$
$607$	4.37018e22	0.0962488	0.0481244	+	0.998841i	$0.484676\pi$
$608$	0	0
$609$	8.64152e22	0.185073
$610$	0	0
$611$	7.36543e23	1.53408
$612$	0	0
$613$	6.50419e23	1.31759	0.658794	−	0.752324i	$-0.271067\pi$
$613$	6.50419e23	1.31759	0.658794	+	0.752324i	$0.271067\pi$
$614$	0	0
$615$	−1.27020e23	−0.250284
$616$	0	0
$617$	−3.90594e23	−0.748689	−0.374345	−	0.927290i	$-0.622132\pi$
$617$	−3.90594e23	−0.748689	−0.374345	+	0.927290i	$0.622132\pi$
$618$	0	0
$619$	−5.06296e23	−0.944136	−0.472068	−	0.881562i	$-0.656492\pi$
$619$	−5.06296e23	−0.944136	−0.472068	+	0.881562i	$0.656492\pi$
$620$	0	0
$621$	−8.21497e22	−0.149048
$622$	0	0
$623$	−3.26759e23	−0.576872
$624$	0	0
$625$	1.35825e24	2.33346
$626$	0	0
$627$	−1.18456e23	−0.198054
$628$	0	0
$629$	−1.46099e24	−2.37749
$630$	0	0
$631$	−6.06894e23	−0.961310	−0.480655	−	0.876910i	$-0.659601\pi$
$631$	−6.06894e23	−0.961310	−0.480655	+	0.876910i	$0.659601\pi$
$632$	0	0
$633$	1.22438e23	0.188792
$634$	0	0
$635$	−2.23341e24	−3.35267
$636$	0	0
$637$	3.77925e23	0.552358
$638$	0	0
$639$	1.16552e23	0.165868
$640$	0	0
$641$	1.00024e24	1.38616	0.693078	−	0.720863i	$-0.256254\pi$
$641$	1.00024e24	1.38616	0.693078	+	0.720863i	$0.256254\pi$
$642$	0	0
$643$	3.28132e23	0.442848	0.221424	−	0.975178i	$-0.428930\pi$
$643$	3.28132e23	0.442848	0.221424	+	0.975178i	$0.428930\pi$
$644$	0	0
$645$	−7.97090e21	−0.0104773
$646$	0	0
$647$	−2.10311e22	−0.0269262	−0.0134631	−	0.999909i	$-0.504286\pi$
$647$	−2.10311e22	−0.0269262	−0.0134631	+	0.999909i	$0.504286\pi$
$648$	0	0
$649$	−2.01577e23	−0.251398
$650$	0	0
$651$	4.72534e23	0.574110
$652$	0	0
$653$	−2.68112e23	−0.317362	−0.158681	−	0.987330i	$-0.550724\pi$
$653$	−2.68112e23	−0.317362	−0.158681	+	0.987330i	$0.550724\pi$
$654$	0	0
$655$	−1.46021e24	−1.68409
$656$	0	0
$657$	−2.15331e23	−0.241992
$658$	0	0
$659$	−4.53989e23	−0.497186	−0.248593	−	0.968608i	$-0.579968\pi$
$659$	−4.53989e23	−0.497186	−0.248593	+	0.968608i	$0.579968\pi$
$660$	0	0
$661$	6.05738e23	0.646506	0.323253	−	0.946313i	$-0.395224\pi$
$661$	6.05738e23	0.646506	0.323253	+	0.946313i	$0.395224\pi$
$662$	0	0
$663$	7.54173e23	0.784524
$664$	0	0
$665$	−1.46357e24	−1.48398
$666$	0	0
$667$	4.06813e23	0.402092
$668$	0	0
$669$	4.39182e23	0.423177
$670$	0	0
$671$	3.70228e23	0.347799
$672$	0	0
$673$	−1.78163e24	−1.63189	−0.815944	−	0.578131i	$-0.803782\pi$
$673$	−1.78163e24	−1.63189	−0.815944	+	0.578131i	$0.803782\pi$
$674$	0	0
$675$	−5.15636e23	−0.460534
$676$	0	0
$677$	1.99427e24	1.73692	0.868460	−	0.495759i	$-0.165110\pi$
$677$	1.99427e24	1.73692	0.868460	+	0.495759i	$0.165110\pi$
$678$	0	0
$679$	−6.46178e23	−0.548857
$680$	0	0
$681$	3.37989e23	0.279996
$682$	0	0
$683$	2.03049e24	1.64069	0.820344	−	0.571871i	$-0.193782\pi$
$683$	2.03049e24	1.64069	0.820344	+	0.571871i	$0.193782\pi$
$684$	0	0
$685$	3.78995e24	2.98719
$686$	0	0
$687$	6.12976e23	0.471315
$688$	0	0
$689$	−1.20920e24	−0.907058
$690$	0	0
$691$	6.21944e23	0.455185	0.227592	−	0.973757i	$-0.426915\pi$
$691$	6.21944e23	0.455185	0.227592	+	0.973757i	$0.426915\pi$
$692$	0	0
$693$	7.57696e22	0.0541081
$694$	0	0
$695$	−3.03041e24	−2.11169
$696$	0	0
$697$	−5.26821e23	−0.358248
$698$	0	0
$699$	−3.75262e23	−0.249046
$700$	0	0
$701$	2.26386e24	1.46638	0.733189	−	0.680025i	$-0.238031\pi$
$701$	2.26386e24	1.46638	0.733189	+	0.680025i	$0.238031\pi$
$702$	0	0
$703$	3.22333e24	2.03791
$704$	0	0
$705$	−2.96144e24	−1.82766
$706$	0	0
$707$	6.83458e23	0.411762
$708$	0	0
$709$	−2.85033e24	−1.67650	−0.838248	−	0.545289i	$-0.816420\pi$
$709$	−2.85033e24	−1.67650	−0.838248	+	0.545289i	$0.816420\pi$
$710$	0	0
$711$	−4.20759e23	−0.241625
$712$	0	0
$713$	2.22453e24	1.24732
$714$	0	0
$715$	−7.89539e23	−0.432287
$716$	0	0
$717$	1.75013e24	0.935746
$718$	0	0
$719$	−8.46689e23	−0.442108	−0.221054	−	0.975262i	$-0.570950\pi$
$719$	−8.46689e23	−0.442108	−0.221054	+	0.975262i	$0.570950\pi$
$720$	0	0
$721$	−3.56043e23	−0.181574
$722$	0	0
$723$	−3.03545e23	−0.151198
$724$	0	0
$725$	2.55348e24	1.24239
$726$	0	0
$727$	−1.24491e23	−0.0591693	−0.0295847	−	0.999562i	$-0.509418\pi$
$727$	−1.24491e23	−0.0591693	−0.0295847	+	0.999562i	$0.509418\pi$
$728$	0	0
$729$	7.97664e22	0.0370370
$730$	0	0
$731$	−3.30597e22	−0.0149969
$732$	0	0
$733$	−1.27764e24	−0.566270	−0.283135	−	0.959080i	$-0.591374\pi$
$733$	−1.27764e24	−0.566270	−0.283135	+	0.959080i	$0.591374\pi$
$734$	0	0
$735$	−1.51953e24	−0.658064
$736$	0	0
$737$	−9.06552e23	−0.383636
$738$	0	0
$739$	−3.05886e24	−1.26497	−0.632487	−	0.774571i	$-0.717966\pi$
$739$	−3.05886e24	−1.26497	−0.632487	+	0.774571i	$0.717966\pi$
$740$	0	0
$741$	−1.66390e24	−0.672469
$742$	0	0
$743$	3.19834e24	1.26334	0.631669	−	0.775238i	$-0.282370\pi$
$743$	3.19834e24	1.26334	0.631669	+	0.775238i	$0.282370\pi$
$744$	0	0
$745$	−8.86086e24	−3.42096
$746$	0	0
$747$	7.36655e23	0.277996
$748$	0	0
$749$	−1.67421e24	−0.617609
$750$	0	0
$751$	3.06697e24	1.10604	0.553019	−	0.833169i	$-0.313476\pi$
$751$	3.06697e24	1.10604	0.553019	+	0.833169i	$0.313476\pi$
$752$	0	0
$753$	7.34293e23	0.258888
$754$	0	0
$755$	−2.16667e24	−0.746868
$756$	0	0
$757$	−8.30657e22	−0.0279967	−0.0139983	−	0.999902i	$-0.504456\pi$
$757$	−8.30657e22	−0.0279967	−0.0139983	+	0.999902i	$0.504456\pi$
$758$	0	0
$759$	3.56697e23	0.117556
$760$	0	0
$761$	2.43328e24	0.784193	0.392097	−	0.919924i	$-0.371750\pi$
$761$	2.43328e24	0.784193	0.392097	+	0.919924i	$0.371750\pi$
$762$	0	0
$763$	3.04320e23	0.0959116
$764$	0	0
$765$	−3.03233e24	−0.934660
$766$	0	0
$767$	−2.83146e24	−0.853590
$768$	0	0
$769$	2.88150e24	0.849660	0.424830	−	0.905273i	$-0.360334\pi$
$769$	2.88150e24	0.849660	0.424830	+	0.905273i	$0.360334\pi$
$770$	0	0
$771$	3.52838e24	1.01768
$772$	0	0
$773$	3.45544e24	0.974940	0.487470	−	0.873140i	$-0.337920\pi$
$773$	3.45544e24	0.974940	0.487470	+	0.873140i	$0.337920\pi$
$774$	0	0
$775$	1.39629e25	3.85399
$776$	0	0
$777$	−2.06177e24	−0.556753
$778$	0	0
$779$	1.16230e24	0.307079
$780$	0	0
$781$	−5.06074e23	−0.130822
$782$	0	0
$783$	−3.95011e23	−0.0999158
$784$	0	0
$785$	−2.40054e24	−0.594177
$786$	0	0
$787$	−3.45711e24	−0.837389	−0.418695	−	0.908127i	$-0.637512\pi$
$787$	−3.45711e24	−0.837389	−0.418695	+	0.908127i	$0.637512\pi$
$788$	0	0
$789$	−2.55322e24	−0.605248
$790$	0	0
$791$	−1.37438e24	−0.318865
$792$	0	0
$793$	5.20042e24	1.18091
$794$	0	0
$795$	4.86187e24	1.08064
$796$	0	0
$797$	2.98439e24	0.649321	0.324661	−	0.945831i	$-0.394750\pi$
$797$	2.98439e24	0.649321	0.324661	+	0.945831i	$0.394750\pi$
$798$	0	0
$799$	−1.22827e25	−2.61605
$800$	0	0
$801$	1.49364e24	0.311437
$802$	0	0
$803$	9.34976e23	0.190861
$804$	0	0
$805$	4.40711e24	0.880824
$806$	0	0
$807$	9.66458e23	0.189129
$808$	0	0
$809$	−5.98989e24	−1.14777	−0.573887	−	0.818934i	$-0.694565\pi$
$809$	−5.98989e24	−1.14777	−0.573887	+	0.818934i	$0.694565\pi$
$810$	0	0
$811$	1.96570e24	0.368842	0.184421	−	0.982847i	$-0.440959\pi$
$811$	1.96570e24	0.368842	0.184421	+	0.982847i	$0.440959\pi$
$812$	0	0
$813$	6.17121e24	1.13397
$814$	0	0
$815$	−1.40891e24	−0.253538
$816$	0	0
$817$	7.29381e22	0.0128549
$818$	0	0
$819$	1.06430e24	0.183717
$820$	0	0
$821$	−5.15579e24	−0.871723	−0.435861	−	0.900014i	$-0.643556\pi$
$821$	−5.15579e24	−0.871723	−0.435861	+	0.900014i	$0.643556\pi$
$822$	0	0
$823$	5.50003e24	0.910891	0.455446	−	0.890264i	$-0.349480\pi$
$823$	5.50003e24	0.910891	0.455446	+	0.890264i	$0.349480\pi$
$824$	0	0
$825$	2.23891e24	0.363227
$826$	0	0
$827$	−2.32226e23	−0.0369074	−0.0184537	−	0.999830i	$-0.505874\pi$
$827$	−2.32226e23	−0.0369074	−0.0184537	+	0.999830i	$0.505874\pi$
$828$	0	0
$829$	−3.24844e24	−0.505779	−0.252890	−	0.967495i	$-0.581381\pi$
$829$	−3.24844e24	−0.505779	−0.252890	+	0.967495i	$0.581381\pi$
$830$	0	0
$831$	−2.21071e23	−0.0337228
$832$	0	0
$833$	−6.30234e24	−0.941931
$834$	0	0
$835$	−8.56276e24	−1.25395
$836$	0	0
$837$	−2.15999e24	−0.309946
$838$	0	0
$839$	−9.83039e24	−1.38227	−0.691137	−	0.722724i	$-0.742890\pi$
$839$	−9.83039e24	−1.38227	−0.691137	+	0.722724i	$0.742890\pi$
$840$	0	0
$841$	−5.30102e24	−0.730455
$842$	0	0
$843$	−3.45352e24	−0.466367
$844$	0	0
$845$	2.82764e24	0.374233
$846$	0	0
$847$	4.43089e24	0.574754
$848$	0	0
$849$	−5.96330e24	−0.758178
$850$	0	0
$851$	−9.70612e24	−1.20961
$852$	0	0
$853$	1.01253e25	1.23692	0.618462	−	0.785815i	$-0.287756\pi$
$853$	1.01253e25	1.23692	0.618462	+	0.785815i	$0.287756\pi$
$854$	0	0
$855$	6.69009e24	0.801161
$856$	0	0
$857$	6.93649e24	0.814335	0.407167	−	0.913354i	$-0.366517\pi$
$857$	6.93649e24	0.814335	0.407167	+	0.913354i	$0.366517\pi$
$858$	0	0
$859$	8.33551e24	0.959379	0.479690	−	0.877438i	$-0.340749\pi$
$859$	8.33551e24	0.959379	0.479690	+	0.877438i	$0.340749\pi$
$860$	0	0
$861$	−7.43456e23	−0.0838936
$862$	0	0
$863$	−5.64763e24	−0.624849	−0.312424	−	0.949943i	$-0.601141\pi$
$863$	−5.64763e24	−0.624849	−0.312424	+	0.949943i	$0.601141\pi$
$864$	0	0
$865$	3.37294e24	0.365908
$866$	0	0
$867$	−7.14920e24	−0.760492
$868$	0	0
$869$	1.82695e24	0.190572
$870$	0	0
$871$	−1.27339e25	−1.30259
$872$	0	0
$873$	2.95373e24	0.296313
$874$	0	0
$875$	1.61028e25	1.58428
$876$	0	0
$877$	1.63199e24	0.157478	0.0787390	−	0.996895i	$-0.474911\pi$
$877$	1.63199e24	0.157478	0.0787390	+	0.996895i	$0.474911\pi$
$878$	0	0
$879$	−5.86393e24	−0.554987
$880$	0	0
$881$	1.08087e25	1.00341	0.501705	−	0.865039i	$-0.332706\pi$
$881$	1.08087e25	1.00341	0.501705	+	0.865039i	$0.332706\pi$
$882$	0	0
$883$	2.02700e24	0.184581	0.0922906	−	0.995732i	$-0.470581\pi$
$883$	2.02700e24	0.184581	0.0922906	+	0.995732i	$0.470581\pi$
$884$	0	0
$885$	1.13845e25	1.01694
$886$	0	0
$887$	−1.06464e25	−0.932939	−0.466469	−	0.884537i	$-0.654474\pi$
$887$	−1.06464e25	−0.932939	−0.466469	+	0.884537i	$0.654474\pi$
$888$	0	0
$889$	−1.30723e25	−1.12379
$890$	0	0
$891$	−3.46349e23	−0.0292115
$892$	0	0
$893$	2.70988e25	2.24240
$894$	0	0
$895$	−9.92614e24	−0.805907
$896$	0	0
$897$	5.01035e24	0.399147
$898$	0	0
$899$	1.06965e25	0.836149
$900$	0	0
$901$	2.01648e25	1.54680
$902$	0	0
$903$	−4.66543e22	−0.00351193
$904$	0	0
$905$	4.55007e24	0.336128
$906$	0	0
$907$	−5.94517e23	−0.0431025	−0.0215512	−	0.999768i	$-0.506861\pi$
$907$	−5.94517e23	−0.0431025	−0.0215512	+	0.999768i	$0.506861\pi$
$908$	0	0
$909$	−3.12414e24	−0.222299
$910$	0	0
$911$	2.36699e25	1.65307	0.826533	−	0.562888i	$-0.190310\pi$
$911$	2.36699e25	1.65307	0.826533	+	0.562888i	$0.190310\pi$
$912$	0	0
$913$	−3.19858e24	−0.219258
$914$	0	0
$915$	−2.09095e25	−1.40690
$916$	0	0
$917$	−8.54672e24	−0.564495
$918$	0	0
$919$	9.36789e24	0.607380	0.303690	−	0.952771i	$-0.401781\pi$
$919$	9.36789e24	0.607380	0.303690	+	0.952771i	$0.401781\pi$
$920$	0	0
$921$	−8.39178e23	−0.0534130
$922$	0	0
$923$	−7.10858e24	−0.444190
$924$	0	0
$925$	−6.09233e25	−3.73748
$926$	0	0
$927$	1.62750e24	0.0980267
$928$	0	0
$929$	−2.69527e25	−1.59393	−0.796965	−	0.604025i	$-0.793563\pi$
$929$	−2.69527e25	−1.59393	−0.796965	+	0.604025i	$0.793563\pi$
$930$	0	0
$931$	1.39046e25	0.807394
$932$	0	0
$933$	−1.70677e25	−0.973154
$934$	0	0
$935$	1.31665e25	0.737176
$936$	0	0
$937$	2.62664e25	1.44416	0.722078	−	0.691811i	$-0.243187\pi$
$937$	2.62664e25	1.44416	0.722078	+	0.691811i	$0.243187\pi$
$938$	0	0
$939$	−1.31972e24	−0.0712563
$940$	0	0
$941$	−3.04508e25	−1.61468	−0.807342	−	0.590084i	$-0.799094\pi$
$941$	−3.04508e25	−1.61468	−0.807342	+	0.590084i	$0.799094\pi$
$942$	0	0
$943$	−3.49994e24	−0.182268
$944$	0	0
$945$	−4.27926e24	−0.218876
$946$	0	0
$947$	2.64356e25	1.32805	0.664025	−	0.747711i	$-0.268847\pi$
$947$	2.64356e25	1.32805	0.664025	+	0.747711i	$0.268847\pi$
$948$	0	0
$949$	1.31332e25	0.648047
$950$	0	0
$951$	−5.76684e24	−0.279514
$952$	0	0
$953$	1.17983e25	0.561734	0.280867	−	0.959747i	$-0.409378\pi$
$953$	1.17983e25	0.561734	0.280867	+	0.959747i	$0.409378\pi$
$954$	0	0
$955$	−2.62534e25	−1.22788
$956$	0	0
$957$	1.71515e24	0.0788045
$958$	0	0
$959$	2.21828e25	1.00129
$960$	0	0
$961$	3.59402e25	1.59379
$962$	0	0
$963$	7.65294e24	0.333430
$964$	0	0
$965$	−7.04368e25	−3.01521
$966$	0	0
$967$	−4.43798e25	−1.86664	−0.933318	−	0.359050i	$-0.883101\pi$
$967$	−4.43798e25	−1.86664	−0.933318	+	0.359050i	$0.883101\pi$
$968$	0	0
$969$	2.77475e25	1.14676
$970$	0	0
$971$	−1.50548e25	−0.611382	−0.305691	−	0.952131i	$-0.598887\pi$
$971$	−1.50548e25	−0.611382	−0.305691	+	0.952131i	$0.598887\pi$
$972$	0	0
$973$	−1.77372e25	−0.707826
$974$	0	0
$975$	3.14489e25	1.23329
$976$	0	0
$977$	2.53187e25	0.975748	0.487874	−	0.872914i	$-0.337773\pi$
$977$	2.53187e25	0.975748	0.487874	+	0.872914i	$0.337773\pi$
$978$	0	0
$979$	−6.48546e24	−0.245634
$980$	0	0
$981$	−1.39107e24	−0.0517800
$982$	0	0
$983$	2.47514e24	0.0905514	0.0452757	−	0.998975i	$-0.485583\pi$
$983$	2.47514e24	0.0905514	0.0452757	+	0.998975i	$0.485583\pi$
$984$	0	0
$985$	2.14880e24	0.0772658
$986$	0	0
$987$	−1.73335e25	−0.612619
$988$	0	0
$989$	−2.19632e23	−0.00763005
$990$	0	0
$991$	2.06982e25	0.706818	0.353409	−	0.935469i	$-0.385022\pi$
$991$	2.06982e25	0.706818	0.353409	+	0.935469i	$0.385022\pi$
$992$	0	0
$993$	−2.05887e25	−0.691131
$994$	0	0
$995$	5.11356e25	1.68744
$996$	0	0
$997$	3.73751e25	1.21248	0.606239	−	0.795283i	$-0.292678\pi$
$997$	3.73751e25	1.21248	0.606239	+	0.795283i	$0.292678\pi$
$998$	0	0
$999$	9.42454e24	0.300575

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	12.18.a.a.1.1	✓	1
3.2	odd	2		36.18.a.c.1.1		1
4.3	odd	2		48.18.a.c.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
12.18.a.a.1.1	✓	1	1.1	even	1	trivial
36.18.a.c.1.1		1	3.2	odd	2
48.18.a.c.1.1		1	4.3	odd	2

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:	\( N \)	\(=\)	\( 12 = 2^{2} \cdot 3 \)
Weight:	\( k \)	\(=\)	\( 18 \)
Character orbit:	\([\chi]\)	\(=\)	12.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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