LMFDB - Embedded newform 144.16.a.l.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [144,16,Mod(1,144)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(144, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 16, names="a")

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

chi := DirichletCharacter("144.1");

S:= CuspForms(chi, 16);

N := Newforms(S);

Level:	$ N $	$=$	$ 144 = 2^{4} \cdot 3^{2} $
Weight:	$ k $	$=$	$ 16 $
Character orbit:	$[\chi]$	$=$	144.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:	$205.478647344$
Analytic rank:	$0$
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	$\chi$	$=$	144.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q+221490. q^{5} +2.14900e6 q^{7} +O(q^{10})$

$q+221490. q^{5} +2.14900e6 q^{7} +3.71693e7 q^{11} -2.79974e8 q^{13} -2.49291e9 q^{17} +4.66978e9 q^{19} -1.84679e10 q^{23} +1.85402e10 q^{25} +1.15953e11 q^{29} +5.61870e10 q^{31} +4.75982e11 q^{35} +6.14765e11 q^{37} -5.49860e11 q^{41} +9.82884e11 q^{43} +2.07614e12 q^{47} -1.29361e11 q^{49} +1.20484e13 q^{53} +8.23263e12 q^{55} +2.30879e13 q^{59} -8.50581e12 q^{61} -6.20115e13 q^{65} +1.23310e13 q^{67} +5.89892e13 q^{71} -5.60983e12 q^{73} +7.98769e13 q^{77} -1.59919e14 q^{79} +5.76759e13 q^{83} -5.52155e14 q^{85} +3.62288e14 q^{89} -6.01665e14 q^{91} +1.03431e15 q^{95} -5.39787e14 q^{97} +O(q^{100})$

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
$3$	0	0
$3$	0	0
$4$	0	0
$5$	221490.	1.26788	0.633941	−	0.773381i	$-0.281436\pi$
$5$	221490.	1.26788	0.633941	+	0.773381i	$0.281436\pi$
$6$	0	0
$7$	2.14900e6	0.986282	0.493141	−	0.869949i	$-0.335849\pi$
$7$	2.14900e6	0.986282	0.493141	+	0.869949i	$0.335849\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	3.71693e7	0.575095	0.287547	−	0.957766i	$-0.407160\pi$
$11$	3.71693e7	0.575095	0.287547	+	0.957766i	$0.407160\pi$
$12$	0	0
$13$	−2.79974e8	−1.23749	−0.618747	−	0.785590i	$-0.712359\pi$
$13$	−2.79974e8	−1.23749	−0.618747	+	0.785590i	$0.712359\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−2.49291e9	−1.47347	−0.736733	−	0.676184i	$-0.763633\pi$
$17$	−2.49291e9	−1.47347	−0.736733	+	0.676184i	$0.763633\pi$
$18$	0	0
$19$	4.66978e9	1.19852	0.599259	−	0.800555i	$-0.295462\pi$
$19$	4.66978e9	1.19852	0.599259	+	0.800555i	$0.295462\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−1.84679e10	−1.13099	−0.565496	−	0.824751i	$-0.691315\pi$
$23$	−1.84679e10	−1.13099	−0.565496	+	0.824751i	$0.691315\pi$
$24$	0	0
$25$	1.85402e10	0.607527
$26$	0	0
$27$	0	0
$28$	0	0
$29$	1.15953e11	1.24824	0.624121	−	0.781328i	$-0.285457\pi$
$29$	1.15953e11	1.24824	0.624121	+	0.781328i	$0.285457\pi$
$30$	0	0
$31$	5.61870e10	0.366795	0.183397	−	0.983039i	$-0.441290\pi$
$31$	5.61870e10	0.366795	0.183397	+	0.983039i	$0.441290\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	4.75982e11	1.25049
$36$	0	0
$37$	6.14765e11	1.06462	0.532312	−	0.846548i	$-0.321323\pi$
$37$	6.14765e11	1.06462	0.532312	+	0.846548i	$0.321323\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−5.49860e11	−0.440933	−0.220467	−	0.975394i	$-0.570758\pi$
$41$	−5.49860e11	−0.440933	−0.220467	+	0.975394i	$0.570758\pi$
$42$	0	0
$43$	9.82884e11	0.551428	0.275714	−	0.961240i	$-0.411086\pi$
$43$	9.82884e11	0.551428	0.275714	+	0.961240i	$0.411086\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	2.07614e12	0.597756	0.298878	−	0.954291i	$-0.403388\pi$
$47$	2.07614e12	0.597756	0.298878	+	0.954291i	$0.403388\pi$
$48$	0	0
$49$	−1.29361e11	−0.0272478
$50$	0	0
$51$	0	0
$52$	0	0
$53$	1.20484e13	1.40883	0.704417	−	0.709787i	$-0.251209\pi$
$53$	1.20484e13	1.40883	0.704417	+	0.709787i	$0.251209\pi$
$54$	0	0
$55$	8.23263e12	0.729153
$56$	0	0
$57$	0	0
$58$	0	0
$59$	2.30879e13	1.20780	0.603899	−	0.797061i	$-0.293613\pi$
$59$	2.30879e13	1.20780	0.603899	+	0.797061i	$0.293613\pi$
$60$	0	0
$61$	−8.50581e12	−0.346531	−0.173265	−	0.984875i	$-0.555432\pi$
$61$	−8.50581e12	−0.346531	−0.173265	+	0.984875i	$0.555432\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−6.20115e13	−1.56900
$66$	0	0
$67$	1.23310e13	0.248564	0.124282	−	0.992247i	$-0.460337\pi$
$67$	1.23310e13	0.248564	0.124282	+	0.992247i	$0.460337\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	5.89892e13	0.769724	0.384862	−	0.922974i	$-0.374249\pi$
$71$	5.89892e13	0.769724	0.384862	+	0.922974i	$0.374249\pi$
$72$	0	0
$73$	−5.60983e12	−0.0594331	−0.0297165	−	0.999558i	$-0.509460\pi$
$73$	−5.60983e12	−0.0594331	−0.0297165	+	0.999558i	$0.509460\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	7.98769e13	0.567206
$78$	0	0
$79$	−1.59919e14	−0.936906	−0.468453	−	0.883488i	$-0.655188\pi$
$79$	−1.59919e14	−0.936906	−0.468453	+	0.883488i	$0.655188\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	5.76759e13	0.233297	0.116648	−	0.993173i	$-0.462785\pi$
$83$	5.76759e13	0.233297	0.116648	+	0.993173i	$0.462785\pi$
$84$	0	0
$85$	−5.52155e14	−1.86818
$86$	0	0
$87$	0	0
$88$	0	0
$89$	3.62288e14	0.868217	0.434109	−	0.900861i	$-0.357063\pi$
$89$	3.62288e14	0.868217	0.434109	+	0.900861i	$0.357063\pi$
$90$	0	0
$91$	−6.01665e14	−1.22052
$92$	0	0
$93$	0	0
$94$	0	0
$95$	1.03431e15	1.51958
$96$	0	0
$97$	−5.39787e14	−0.678319	−0.339160	−	0.940729i	$-0.610143\pi$
$97$	−5.39787e14	−0.678319	−0.339160	+	0.940729i	$0.610143\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	4.22221e14	0.391859	0.195929	−	0.980618i	$-0.437228\pi$
$101$	4.22221e14	0.391859	0.195929	+	0.980618i	$0.437228\pi$
$102$	0	0
$103$	1.45689e14	0.116720	0.0583602	−	0.998296i	$-0.481413\pi$
$103$	1.45689e14	0.116720	0.0583602	+	0.998296i	$0.481413\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−4.08696e14	−0.246049	−0.123024	−	0.992404i	$-0.539259\pi$
$107$	−4.08696e14	−0.246049	−0.123024	+	0.992404i	$0.539259\pi$
$108$	0	0
$109$	1.10134e15	0.577060	0.288530	−	0.957471i	$-0.406833\pi$
$109$	1.10134e15	0.577060	0.288530	+	0.957471i	$0.406833\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	2.13023e15	0.851802	0.425901	−	0.904770i	$-0.359957\pi$
$113$	2.13023e15	0.851802	0.425901	+	0.904770i	$0.359957\pi$
$114$	0	0
$115$	−4.09046e15	−1.43397
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−5.35727e15	−1.45325
$120$	0	0
$121$	−2.79569e15	−0.669266
$122$	0	0
$123$	0	0
$124$	0	0
$125$	−2.65286e15	−0.497610
$126$	0	0
$127$	6.25962e15	1.04236	0.521182	−	0.853445i	$-0.325491\pi$
$127$	6.25962e15	1.04236	0.521182	+	0.853445i	$0.325491\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	1.01059e16	1.33365	0.666825	−	0.745214i	$-0.267653\pi$
$131$	1.01059e16	1.33365	0.666825	+	0.745214i	$0.267653\pi$
$132$	0	0
$133$	1.00354e16	1.18208
$134$	0	0
$135$	0	0
$136$	0	0
$137$	9.93739e15	0.937277	0.468638	−	0.883390i	$-0.344745\pi$
$137$	9.93739e15	0.937277	0.468638	+	0.883390i	$0.344745\pi$
$138$	0	0
$139$	1.95078e16	1.65043	0.825216	−	0.564817i	$-0.191053\pi$
$139$	1.95078e16	1.65043	0.825216	+	0.564817i	$0.191053\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−1.04065e16	−0.711676
$144$	0	0
$145$	2.56825e16	1.58262
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−1.58935e16	−0.798589	−0.399295	−	0.916823i	$-0.630745\pi$
$149$	−1.58935e16	−0.798589	−0.399295	+	0.916823i	$0.630745\pi$
$150$	0	0
$151$	−3.10597e16	−1.41211	−0.706057	−	0.708155i	$-0.749528\pi$
$151$	−3.10597e16	−1.41211	−0.706057	+	0.708155i	$0.749528\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	1.24449e16	0.465053
$156$	0	0
$157$	−5.41372e16	−1.83759	−0.918794	−	0.394738i	$-0.870835\pi$
$157$	−5.41372e16	−1.83759	−0.918794	+	0.394738i	$0.870835\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−3.96876e16	−1.11548
$162$	0	0
$163$	5.54612e16	1.42096	0.710481	−	0.703717i	$-0.248478\pi$
$163$	5.54612e16	1.42096	0.710481	+	0.703717i	$0.248478\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−5.83552e15	−0.124654	−0.0623270	−	0.998056i	$-0.519852\pi$
$167$	−5.83552e15	−0.124654	−0.0623270	+	0.998056i	$0.519852\pi$
$168$	0	0
$169$	2.71997e16	0.531390
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−4.40958e16	−0.722855	−0.361428	−	0.932400i	$-0.617711\pi$
$173$	−4.40958e16	−0.722855	−0.361428	+	0.932400i	$0.617711\pi$
$174$	0	0
$175$	3.98430e16	0.599193
$176$	0	0
$177$	0	0
$178$	0	0
$179$	−5.10594e16	−0.648153	−0.324077	−	0.946031i	$-0.605054\pi$
$179$	−5.10594e16	−0.648153	−0.324077	+	0.946031i	$0.605054\pi$
$180$	0	0
$181$	3.16143e16	0.369228	0.184614	−	0.982811i	$-0.440897\pi$
$181$	3.16143e16	0.369228	0.184614	+	0.982811i	$0.440897\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	1.36164e17	1.34982
$186$	0	0
$187$	−9.26599e16	−0.847383
$188$	0	0
$189$	0	0
$190$	0	0
$191$	9.44291e16	0.736810	0.368405	−	0.929665i	$-0.379904\pi$
$191$	9.44291e16	0.736810	0.368405	+	0.929665i	$0.379904\pi$
$192$	0	0
$193$	4.17744e16	0.301460	0.150730	−	0.988575i	$-0.451838\pi$
$193$	4.17744e16	0.301460	0.150730	+	0.988575i	$0.451838\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	1.48383e17	0.918094	0.459047	−	0.888412i	$-0.348191\pi$
$197$	1.48383e17	0.918094	0.459047	+	0.888412i	$0.348191\pi$
$198$	0	0
$199$	−1.97555e17	−1.13316	−0.566579	−	0.824007i	$-0.691733\pi$
$199$	−1.97555e17	−1.13316	−0.566579	+	0.824007i	$0.691733\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	2.49184e17	1.23112
$204$	0	0
$205$	−1.21788e17	−0.559052
$206$	0	0
$207$	0	0
$208$	0	0
$209$	1.73573e17	0.689262
$210$	0	0
$211$	4.48570e17	1.65849	0.829244	−	0.558887i	$-0.188772\pi$
$211$	4.48570e17	1.65849	0.829244	+	0.558887i	$0.188772\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	2.17699e17	0.699146
$216$	0	0
$217$	1.20746e17	0.361763
$218$	0	0
$219$	0	0
$220$	0	0
$221$	6.97951e17	1.82341
$222$	0	0
$223$	−1.59977e17	−0.390634	−0.195317	−	0.980740i	$-0.562574\pi$
$223$	−1.59977e17	−0.390634	−0.195317	+	0.980740i	$0.562574\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	1.45803e17	0.311582	0.155791	−	0.987790i	$-0.450207\pi$
$227$	1.45803e17	0.311582	0.155791	+	0.987790i	$0.450207\pi$
$228$	0	0
$229$	6.81194e17	1.36303	0.681514	−	0.731805i	$-0.261322\pi$
$229$	6.81194e17	1.36303	0.681514	+	0.731805i	$0.261322\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	3.89035e17	0.683628	0.341814	−	0.939768i	$-0.388959\pi$
$233$	3.89035e17	0.683628	0.341814	+	0.939768i	$0.388959\pi$
$234$	0	0
$235$	4.59845e17	0.757884
$236$	0	0
$237$	0	0
$238$	0	0
$239$	6.76227e16	0.0981993	0.0490996	−	0.998794i	$-0.484365\pi$
$239$	6.76227e16	0.0981993	0.0490996	+	0.998794i	$0.484365\pi$
$240$	0	0
$241$	−9.00528e17	−1.22848	−0.614242	−	0.789118i	$-0.710538\pi$
$241$	−9.00528e17	−1.22848	−0.614242	+	0.789118i	$0.710538\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	−2.86521e16	−0.0345470
$246$	0	0
$247$	−1.30742e18	−1.48316
$248$	0	0
$249$	0	0
$250$	0	0
$251$	1.58929e18	1.59827	0.799134	−	0.601153i	$-0.205292\pi$
$251$	1.58929e18	1.59827	0.799134	+	0.601153i	$0.205292\pi$
$252$	0	0
$253$	−6.86440e17	−0.650428
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−2.29738e18	−1.93524	−0.967618	−	0.252418i	$-0.918774\pi$
$257$	−2.29738e18	−1.93524	−0.967618	+	0.252418i	$0.918774\pi$
$258$	0	0
$259$	1.32113e18	1.05002
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−5.05657e17	−0.358252	−0.179126	−	0.983826i	$-0.557327\pi$
$263$	−5.05657e17	−0.358252	−0.179126	+	0.983826i	$0.557327\pi$
$264$	0	0
$265$	2.66860e18	1.78624
$266$	0	0
$267$	0	0
$268$	0	0
$269$	2.44731e18	1.46402	0.732009	−	0.681295i	$-0.238583\pi$
$269$	2.44731e18	1.46402	0.732009	+	0.681295i	$0.238583\pi$
$270$	0	0
$271$	1.53807e18	0.870373	0.435187	−	0.900340i	$-0.356682\pi$
$271$	1.53807e18	0.870373	0.435187	+	0.900340i	$0.356682\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	6.89128e17	0.349385
$276$	0	0
$277$	−7.89959e17	−0.379321	−0.189660	−	0.981850i	$-0.560739\pi$
$277$	−7.89959e17	−0.379321	−0.189660	+	0.981850i	$0.560739\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	6.15303e17	0.265333	0.132667	−	0.991161i	$-0.457646\pi$
$281$	6.15303e17	0.265333	0.132667	+	0.991161i	$0.457646\pi$
$282$	0	0
$283$	1.26279e18	0.516337	0.258169	−	0.966100i	$-0.416881\pi$
$283$	1.26279e18	0.516337	0.258169	+	0.966100i	$0.416881\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−1.18165e18	−0.434885
$288$	0	0
$289$	3.35219e18	1.17110
$290$	0	0
$291$	0	0
$292$	0	0
$293$	3.49864e18	1.10253	0.551267	−	0.834329i	$-0.314145\pi$
$293$	3.49864e18	1.10253	0.551267	+	0.834329i	$0.314145\pi$
$294$	0	0
$295$	5.11374e18	1.53135
$296$	0	0
$297$	0	0
$298$	0	0
$299$	5.17055e18	1.39960
$300$	0	0
$301$	2.11222e18	0.543864
$302$	0	0
$303$	0	0
$304$	0	0
$305$	−1.88395e18	−0.439360
$306$	0	0
$307$	−1.28861e18	−0.286142	−0.143071	−	0.989712i	$-0.545698\pi$
$307$	−1.28861e18	−0.286142	−0.143071	+	0.989712i	$0.545698\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	8.43030e18	1.69879	0.849396	−	0.527756i	$-0.176967\pi$
$311$	8.43030e18	1.69879	0.849396	+	0.527756i	$0.176967\pi$
$312$	0	0
$313$	7.88359e18	1.51405	0.757027	−	0.653383i	$-0.226651\pi$
$313$	7.88359e18	1.51405	0.757027	+	0.653383i	$0.226651\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	8.71539e18	1.52175	0.760873	−	0.648900i	$-0.224771\pi$
$317$	8.71539e18	1.52175	0.760873	+	0.648900i	$0.224771\pi$
$318$	0	0
$319$	4.30991e18	0.717857
$320$	0	0
$321$	0	0
$322$	0	0
$323$	−1.16414e19	−1.76598
$324$	0	0
$325$	−5.19079e18	−0.751810
$326$	0	0
$327$	0	0
$328$	0	0
$329$	4.46163e18	0.589556
$330$	0	0
$331$	−1.26995e19	−1.60353	−0.801763	−	0.597643i	$-0.796104\pi$
$331$	−1.26995e19	−1.60353	−0.801763	+	0.597643i	$0.796104\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	2.73120e18	0.315150
$336$	0	0
$337$	−6.64250e18	−0.733005	−0.366503	−	0.930417i	$-0.619445\pi$
$337$	−6.64250e18	−0.733005	−0.366503	+	0.930417i	$0.619445\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	2.08843e18	0.210942
$342$	0	0
$343$	−1.04805e19	−1.01316
$344$	0	0
$345$	0	0
$346$	0	0
$347$	1.87835e18	0.166458	0.0832289	−	0.996530i	$-0.473477\pi$
$347$	1.87835e18	0.166458	0.0832289	+	0.996530i	$0.473477\pi$
$348$	0	0
$349$	−1.48545e19	−1.26086	−0.630432	−	0.776245i	$-0.717122\pi$
$349$	−1.48545e19	−1.26086	−0.630432	+	0.776245i	$0.717122\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	8.23380e18	0.641638	0.320819	−	0.947140i	$-0.396042\pi$
$353$	8.23380e18	0.641638	0.320819	+	0.947140i	$0.396042\pi$
$354$	0	0
$355$	1.30655e19	0.975919
$356$	0	0
$357$	0	0
$358$	0	0
$359$	1.87417e19	1.28706	0.643532	−	0.765419i	$-0.277468\pi$
$359$	1.87417e19	1.28706	0.643532	+	0.765419i	$0.277468\pi$
$360$	0	0
$361$	6.62574e18	0.436446
$362$	0	0
$363$	0	0
$364$	0	0
$365$	−1.24252e18	−0.0753542
$366$	0	0
$367$	−8.63117e18	−0.502428	−0.251214	−	0.967932i	$-0.580830\pi$
$367$	−8.63117e18	−0.502428	−0.251214	+	0.967932i	$0.580830\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	2.58920e19	1.38951
$372$	0	0
$373$	−9.95975e18	−0.513372	−0.256686	−	0.966495i	$-0.582631\pi$
$373$	−9.95975e18	−0.513372	−0.256686	+	0.966495i	$0.582631\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−3.24640e19	−1.54469
$378$	0	0
$379$	−1.00949e19	−0.461645	−0.230823	−	0.972996i	$-0.574142\pi$
$379$	−1.00949e19	−0.461645	−0.230823	+	0.972996i	$0.574142\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−2.71764e19	−1.14869	−0.574343	−	0.818614i	$-0.694743\pi$
$383$	−2.71764e19	−1.14869	−0.574343	+	0.818614i	$0.694743\pi$
$384$	0	0
$385$	1.76919e19	0.719150
$386$	0	0
$387$	0	0
$388$	0	0
$389$	9.05285e18	0.340536	0.170268	−	0.985398i	$-0.445537\pi$
$389$	9.05285e18	0.340536	0.170268	+	0.985398i	$0.445537\pi$
$390$	0	0
$391$	4.60389e19	1.66648
$392$	0	0
$393$	0	0
$394$	0	0
$395$	−3.54204e19	−1.18789
$396$	0	0
$397$	−1.76495e19	−0.569908	−0.284954	−	0.958541i	$-0.591978\pi$
$397$	−1.76495e19	−0.569908	−0.284954	+	0.958541i	$0.591978\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−2.81135e19	−0.842040	−0.421020	−	0.907051i	$-0.638328\pi$
$401$	−2.81135e19	−0.842040	−0.421020	+	0.907051i	$0.638328\pi$
$402$	0	0
$403$	−1.57309e19	−0.453906
$404$	0	0
$405$	0	0
$406$	0	0
$407$	2.28504e19	0.612260
$408$	0	0
$409$	6.42955e19	1.66056	0.830282	−	0.557343i	$-0.188179\pi$
$409$	6.42955e19	1.66056	0.830282	+	0.557343i	$0.188179\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	4.96159e19	1.19123
$414$	0	0
$415$	1.27746e19	0.295793
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−2.34474e19	−0.505230	−0.252615	−	0.967567i	$-0.581291\pi$
$419$	−2.34474e19	−0.505230	−0.252615	+	0.967567i	$0.581291\pi$
$420$	0	0
$421$	−7.28227e19	−1.51409	−0.757044	−	0.653364i	$-0.773357\pi$
$421$	−7.28227e19	−1.51409	−0.757044	+	0.653364i	$0.773357\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	−4.62192e19	−0.895170
$426$	0	0
$427$	−1.82790e19	−0.341777
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−4.71831e19	−0.822635	−0.411317	−	0.911492i	$-0.634931\pi$
$431$	−4.71831e19	−0.822635	−0.411317	+	0.911492i	$0.634931\pi$
$432$	0	0
$433$	3.79099e19	0.638400	0.319200	−	0.947687i	$-0.396586\pi$
$433$	3.79099e19	0.638400	0.319200	+	0.947687i	$0.396586\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−8.62412e19	−1.35551
$438$	0	0
$439$	2.26832e19	0.344525	0.172262	−	0.985051i	$-0.444892\pi$
$439$	2.26832e19	0.344525	0.172262	+	0.985051i	$0.444892\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−9.94888e19	−1.41171	−0.705856	−	0.708355i	$-0.749438\pi$
$443$	−9.94888e19	−1.41171	−0.705856	+	0.708355i	$0.749438\pi$
$444$	0	0
$445$	8.02431e19	1.10080
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−1.30159e20	−1.66965	−0.834825	−	0.550515i	$-0.814432\pi$
$449$	−1.30159e20	−1.66965	−0.834825	+	0.550515i	$0.814432\pi$
$450$	0	0
$451$	−2.04379e19	−0.253579
$452$	0	0
$453$	0	0
$454$	0	0
$455$	−1.33263e20	−1.54747
$456$	0	0
$457$	−1.01910e20	−1.14510	−0.572550	−	0.819870i	$-0.694046\pi$
$457$	−1.01910e20	−1.14510	−0.572550	+	0.819870i	$0.694046\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	1.08142e20	1.13825	0.569123	−	0.822253i	$-0.307283\pi$
$461$	1.08142e20	1.13825	0.569123	+	0.822253i	$0.307283\pi$
$462$	0	0
$463$	−1.69113e20	−1.72314	−0.861569	−	0.507641i	$-0.830518\pi$
$463$	−1.69113e20	−1.72314	−0.861569	+	0.507641i	$0.830518\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−4.58418e19	−0.437910	−0.218955	−	0.975735i	$-0.570265\pi$
$467$	−4.58418e19	−0.437910	−0.218955	+	0.975735i	$0.570265\pi$
$468$	0	0
$469$	2.64993e19	0.245154
$470$	0	0
$471$	0	0
$472$	0	0
$473$	3.65331e19	0.317124
$474$	0	0
$475$	8.65789e19	0.728132
$476$	0	0
$477$	0	0
$478$	0	0
$479$	1.31513e20	1.03861	0.519303	−	0.854590i	$-0.326192\pi$
$479$	1.31513e20	1.03861	0.519303	+	0.854590i	$0.326192\pi$
$480$	0	0
$481$	−1.72118e20	−1.31747
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−1.19557e20	−0.860029
$486$	0	0
$487$	−6.13671e19	−0.428024	−0.214012	−	0.976831i	$-0.568653\pi$
$487$	−6.13671e19	−0.428024	−0.214012	+	0.976831i	$0.568653\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	2.42057e20	1.58784	0.793918	−	0.608024i	$-0.208038\pi$
$491$	2.42057e20	1.58784	0.793918	+	0.608024i	$0.208038\pi$
$492$	0	0
$493$	−2.89062e20	−1.83924
$494$	0	0
$495$	0	0
$496$	0	0
$497$	1.26768e20	0.759165
$498$	0	0
$499$	−1.46049e20	−0.848680	−0.424340	−	0.905503i	$-0.639494\pi$
$499$	−1.46049e20	−0.848680	−0.424340	+	0.905503i	$0.639494\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−1.68041e20	−0.919720	−0.459860	−	0.887991i	$-0.652100\pi$
$503$	−1.68041e20	−0.919720	−0.459860	+	0.887991i	$0.652100\pi$
$504$	0	0
$505$	9.35178e19	0.496831
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−7.37709e19	−0.369404	−0.184702	−	0.982795i	$-0.559132\pi$
$509$	−7.37709e19	−0.369404	−0.184702	+	0.982795i	$0.559132\pi$
$510$	0	0
$511$	−1.20555e19	−0.0586178
$512$	0	0
$513$	0	0
$514$	0	0
$515$	3.22686e19	0.147988
$516$	0	0
$517$	7.71689e19	0.343766
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−3.22023e20	−1.35396	−0.676978	−	0.736003i	$-0.736711\pi$
$521$	−3.22023e20	−1.35396	−0.676978	+	0.736003i	$0.736711\pi$
$522$	0	0
$523$	1.19064e19	0.0486429	0.0243214	−	0.999704i	$-0.492257\pi$
$523$	1.19064e19	0.0486429	0.0243214	+	0.999704i	$0.492257\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−1.40069e20	−0.540460
$528$	0	0
$529$	7.44293e19	0.279143
$530$	0	0
$531$	0	0
$532$	0	0
$533$	1.53947e20	0.545652
$534$	0	0
$535$	−9.05220e19	−0.311961
$536$	0	0
$537$	0	0
$538$	0	0
$539$	−4.80824e18	−0.0156701
$540$	0	0
$541$	1.59033e20	0.504089	0.252045	−	0.967716i	$-0.418897\pi$
$541$	1.59033e20	0.504089	0.252045	+	0.967716i	$0.418897\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	2.43935e20	0.731645
$546$	0	0
$547$	4.25405e20	1.24136	0.620680	−	0.784064i	$-0.286857\pi$
$547$	4.25405e20	1.24136	0.620680	+	0.784064i	$0.286857\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	5.41477e20	1.49604
$552$	0	0
$553$	−3.43665e20	−0.924054
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−2.71579e20	−0.691803	−0.345901	−	0.938271i	$-0.612427\pi$
$557$	−2.71579e20	−0.691803	−0.345901	+	0.938271i	$0.612427\pi$
$558$	0	0
$559$	−2.75182e20	−0.682389
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−2.68438e20	−0.631002	−0.315501	−	0.948925i	$-0.602173\pi$
$563$	−2.68438e20	−0.631002	−0.315501	+	0.948925i	$0.602173\pi$
$564$	0	0
$565$	4.71825e20	1.07999
$566$	0	0
$567$	0	0
$568$	0	0
$569$	1.34893e20	0.292853	0.146426	−	0.989222i	$-0.453223\pi$
$569$	1.34893e20	0.292853	0.146426	+	0.989222i	$0.453223\pi$
$570$	0	0
$571$	6.81153e20	1.44037	0.720185	−	0.693782i	$-0.244057\pi$
$571$	6.81153e20	1.44037	0.720185	+	0.693782i	$0.244057\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−3.42400e20	−0.687108
$576$	0	0
$577$	−2.42927e20	−0.474961	−0.237480	−	0.971392i	$-0.576322\pi$
$577$	−2.42927e20	−0.474961	−0.237480	+	0.971392i	$0.576322\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	1.23945e20	0.230096
$582$	0	0
$583$	4.47830e20	0.810213
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−1.06241e20	−0.182602	−0.0913012	−	0.995823i	$-0.529103\pi$
$587$	−1.06241e20	−0.182602	−0.0913012	+	0.995823i	$0.529103\pi$
$588$	0	0
$589$	2.62381e20	0.439610
$590$	0	0
$591$	0	0
$592$	0	0
$593$	7.70415e20	1.22692	0.613458	−	0.789727i	$-0.289778\pi$
$593$	7.70415e20	1.22692	0.613458	+	0.789727i	$0.289778\pi$
$594$	0	0
$595$	−1.18658e21	−1.84255
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−7.59796e19	−0.112201	−0.0561004	−	0.998425i	$-0.517867\pi$
$599$	−7.59796e19	−0.112201	−0.0561004	+	0.998425i	$0.517867\pi$
$600$	0	0
$601$	7.78003e20	1.12053	0.560264	−	0.828314i	$-0.310700\pi$
$601$	7.78003e20	1.12053	0.560264	+	0.828314i	$0.310700\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	−6.19217e20	−0.848551
$606$	0	0
$607$	−3.21867e20	−0.430290	−0.215145	−	0.976582i	$-0.569022\pi$
$607$	−3.21867e20	−0.430290	−0.215145	+	0.976582i	$0.569022\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−5.81267e20	−0.739719
$612$	0	0
$613$	6.68867e20	0.830589	0.415294	−	0.909687i	$-0.363679\pi$
$613$	6.68867e20	0.830589	0.415294	+	0.909687i	$0.363679\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	8.47904e20	1.00279	0.501393	−	0.865220i	$-0.332821\pi$
$617$	8.47904e20	1.00279	0.501393	+	0.865220i	$0.332821\pi$
$618$	0	0
$619$	3.51027e20	0.405192	0.202596	−	0.979262i	$-0.435062\pi$
$619$	3.51027e20	0.405192	0.202596	+	0.979262i	$0.435062\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	7.78556e20	0.856307
$624$	0	0
$625$	−1.15339e21	−1.23844
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−1.53256e21	−1.56869
$630$	0	0
$631$	1.30718e21	1.30652	0.653260	−	0.757134i	$-0.273401\pi$
$631$	1.30718e21	1.30652	0.653260	+	0.757134i	$0.273401\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	1.38644e21	1.32160
$636$	0	0
$637$	3.62176e19	0.0337190
$638$	0	0
$639$	0	0
$640$	0	0
$641$	1.77975e21	1.58097	0.790486	−	0.612480i	$-0.209828\pi$
$641$	1.77975e21	1.58097	0.790486	+	0.612480i	$0.209828\pi$
$642$	0	0
$643$	5.18269e19	0.0449752	0.0224876	−	0.999747i	$-0.492841\pi$
$643$	5.18269e19	0.0449752	0.0224876	+	0.999747i	$0.492841\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	1.80268e21	1.49327	0.746634	−	0.665235i	$-0.231669\pi$
$647$	1.80268e21	1.49327	0.746634	+	0.665235i	$0.231669\pi$
$648$	0	0
$649$	8.58162e20	0.694599
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−6.01117e20	−0.464633	−0.232317	−	0.972640i	$-0.574631\pi$
$653$	−6.01117e20	−0.464633	−0.232317	+	0.972640i	$0.574631\pi$
$654$	0	0
$655$	2.23837e21	1.69091
$656$	0	0
$657$	0	0
$658$	0	0
$659$	−1.34492e21	−0.970635	−0.485318	−	0.874338i	$-0.661296\pi$
$659$	−1.34492e21	−0.970635	−0.485318	+	0.874338i	$0.661296\pi$
$660$	0	0
$661$	3.00301e20	0.211858	0.105929	−	0.994374i	$-0.466218\pi$
$661$	3.00301e20	0.211858	0.105929	+	0.994374i	$0.466218\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	2.22273e21	1.49873
$666$	0	0
$667$	−2.14142e21	−1.41175
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−3.16155e20	−0.199288
$672$	0	0
$673$	−2.48252e21	−1.53031	−0.765156	−	0.643845i	$-0.777338\pi$
$673$	−2.48252e21	−1.53031	−0.765156	+	0.643845i	$0.777338\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	1.84000e21	1.08493	0.542467	−	0.840077i	$-0.317491\pi$
$677$	1.84000e21	1.08493	0.542467	+	0.840077i	$0.317491\pi$
$678$	0	0
$679$	−1.16000e21	−0.669014
$680$	0	0
$681$	0	0
$682$	0	0
$683$	2.19144e21	1.20941	0.604705	−	0.796450i	$-0.293291\pi$
$683$	2.19144e21	1.20941	0.604705	+	0.796450i	$0.293291\pi$
$684$	0	0
$685$	2.20103e21	1.18836
$686$	0	0
$687$	0	0
$688$	0	0
$689$	−3.37324e21	−1.74342
$690$	0	0
$691$	5.25387e20	0.265701	0.132851	−	0.991136i	$-0.457587\pi$
$691$	5.25387e20	0.265701	0.132851	+	0.991136i	$0.457587\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	4.32079e21	2.09256
$696$	0	0
$697$	1.37075e21	0.649701
$698$	0	0
$699$	0	0
$700$	0	0
$701$	3.78227e21	1.71739	0.858693	−	0.512491i	$-0.171277\pi$
$701$	3.78227e21	1.71739	0.858693	+	0.512491i	$0.171277\pi$
$702$	0	0
$703$	2.87082e21	1.27597
$704$	0	0
$705$	0	0
$706$	0	0
$707$	9.07353e20	0.386483
$708$	0	0
$709$	−4.24647e21	−1.77085	−0.885424	−	0.464784i	$-0.846132\pi$
$709$	−4.24647e21	−1.77085	−0.885424	+	0.464784i	$0.846132\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−1.03766e21	−0.414842
$714$	0	0
$715$	−2.30493e21	−0.902322
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−3.17360e20	−0.119148	−0.0595738	−	0.998224i	$-0.518974\pi$
$719$	−3.17360e20	−0.119148	−0.0595738	+	0.998224i	$0.518974\pi$
$720$	0	0
$721$	3.13085e20	0.115119
$722$	0	0
$723$	0	0
$724$	0	0
$725$	2.14981e21	0.758340
$726$	0	0
$727$	−1.25112e21	−0.432307	−0.216154	−	0.976359i	$-0.569351\pi$
$727$	−1.25112e21	−0.432307	−0.216154	+	0.976359i	$0.569351\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−2.45025e21	−0.812511
$732$	0	0
$733$	4.26993e21	1.38721	0.693603	−	0.720357i	$-0.256022\pi$
$733$	4.26993e21	1.38721	0.693603	+	0.720357i	$0.256022\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	4.58335e20	0.142948
$738$	0	0
$739$	−1.73302e21	−0.529627	−0.264813	−	0.964300i	$-0.585310\pi$
$739$	−1.73302e21	−0.529627	−0.264813	+	0.964300i	$0.585310\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−2.63739e21	−0.774032	−0.387016	−	0.922073i	$-0.626494\pi$
$743$	−2.63739e21	−0.774032	−0.387016	+	0.922073i	$0.626494\pi$
$744$	0	0
$745$	−3.52026e21	−1.01252
$746$	0	0
$747$	0	0
$748$	0	0
$749$	−8.78287e20	−0.242674
$750$	0	0
$751$	−2.27078e21	−0.615001	−0.307500	−	0.951548i	$-0.599493\pi$
$751$	−2.27078e21	−0.615001	−0.307500	+	0.951548i	$0.599493\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	−6.87941e21	−1.79040
$756$	0	0
$757$	−1.72959e21	−0.441291	−0.220645	−	0.975354i	$-0.570816\pi$
$757$	−1.72959e21	−0.441291	−0.220645	+	0.975354i	$0.570816\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	2.06739e20	0.0507035	0.0253518	−	0.999679i	$-0.491929\pi$
$761$	2.06739e20	0.0507035	0.0253518	+	0.999679i	$0.491929\pi$
$762$	0	0
$763$	2.36677e21	0.569144
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−6.46402e21	−1.49464
$768$	0	0
$769$	3.01490e21	0.683636	0.341818	−	0.939766i	$-0.388957\pi$
$769$	3.01490e21	0.683636	0.341818	+	0.939766i	$0.388957\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	−8.99357e21	−1.96149	−0.980745	−	0.195293i	$-0.937434\pi$
$773$	−8.99357e21	−1.96149	−0.980745	+	0.195293i	$0.937434\pi$
$774$	0	0
$775$	1.04172e21	0.222838
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−2.56773e21	−0.528467
$780$	0	0
$781$	2.19259e21	0.442664
$782$	0	0
$783$	0	0
$784$	0	0
$785$	−1.19908e22	−2.32985
$786$	0	0
$787$	5.92807e21	1.13006	0.565032	−	0.825069i	$-0.308864\pi$
$787$	5.92807e21	1.13006	0.565032	+	0.825069i	$0.308864\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	4.57787e21	0.840117
$792$	0	0
$793$	2.38141e21	0.428830
$794$	0	0
$795$	0	0
$796$	0	0
$797$	3.83837e21	0.665593	0.332797	−	0.942999i	$-0.392008\pi$
$797$	3.83837e21	0.665593	0.332797	+	0.942999i	$0.392008\pi$
$798$	0	0
$799$	−5.17565e21	−0.880773
$800$	0	0
$801$	0	0
$802$	0	0
$803$	−2.08513e20	−0.0341796
$804$	0	0
$805$	−8.79040e21	−1.41429
$806$	0	0
$807$	0	0
$808$	0	0
$809$	6.17948e21	0.957940	0.478970	−	0.877831i	$-0.341010\pi$
$809$	6.17948e21	0.957940	0.478970	+	0.877831i	$0.341010\pi$
$810$	0	0
$811$	−1.87862e21	−0.285879	−0.142939	−	0.989731i	$-0.545655\pi$
$811$	−1.87862e21	−0.285879	−0.142939	+	0.989731i	$0.545655\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	1.22841e22	1.80161
$816$	0	0
$817$	4.58986e21	0.660897
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−3.93227e21	−0.545845	−0.272923	−	0.962036i	$-0.587990\pi$
$821$	−3.93227e21	−0.545845	−0.272923	+	0.962036i	$0.587990\pi$
$822$	0	0
$823$	7.92927e21	1.08077	0.540386	−	0.841417i	$-0.318278\pi$
$823$	7.92927e21	1.08077	0.540386	+	0.841417i	$0.318278\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−1.97038e21	−0.258976	−0.129488	−	0.991581i	$-0.541333\pi$
$827$	−1.97038e21	−0.258976	−0.129488	+	0.991581i	$0.541333\pi$
$828$	0	0
$829$	2.43930e21	0.314851	0.157426	−	0.987531i	$-0.449681\pi$
$829$	2.43930e21	0.314851	0.157426	+	0.987531i	$0.449681\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	3.22484e20	0.0401487
$834$	0	0
$835$	−1.29251e21	−0.158047
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−1.35318e22	−1.59639	−0.798196	−	0.602398i	$-0.794212\pi$
$839$	−1.35318e22	−1.59639	−0.798196	+	0.602398i	$0.794212\pi$
$840$	0	0
$841$	4.81601e21	0.558107
$842$	0	0
$843$	0	0
$844$	0	0
$845$	6.02446e21	0.673741
$846$	0	0
$847$	−6.00794e21	−0.660085
$848$	0	0
$849$	0	0
$850$	0	0
$851$	−1.13534e22	−1.20408
$852$	0	0
$853$	−7.30038e21	−0.760725	−0.380363	−	0.924837i	$-0.624201\pi$
$853$	−7.30038e21	−0.760725	−0.380363	+	0.924837i	$0.624201\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−1.16475e22	−1.17187	−0.585934	−	0.810359i	$-0.699272\pi$
$857$	−1.16475e22	−1.17187	−0.585934	+	0.810359i	$0.699272\pi$
$858$	0	0
$859$	1.05473e22	1.04278	0.521388	−	0.853320i	$-0.325414\pi$
$859$	1.05473e22	1.04278	0.521388	+	0.853320i	$0.325414\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	1.25514e22	1.19843	0.599215	−	0.800588i	$-0.295480\pi$
$863$	1.25514e22	1.19843	0.599215	+	0.800588i	$0.295480\pi$
$864$	0	0
$865$	−9.76678e21	−0.916496
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−5.94407e21	−0.538810
$870$	0	0
$871$	−3.45237e21	−0.307596
$872$	0	0
$873$	0	0
$874$	0	0
$875$	−5.70100e21	−0.490784
$876$	0	0
$877$	−1.57294e22	−1.33112	−0.665558	−	0.746346i	$-0.731807\pi$
$877$	−1.57294e22	−1.33112	−0.665558	+	0.746346i	$0.731807\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−1.80412e22	−1.47553	−0.737764	−	0.675059i	$-0.764118\pi$
$881$	−1.80412e22	−1.47553	−0.737764	+	0.675059i	$0.764118\pi$
$882$	0	0
$883$	−2.41866e22	−1.94478	−0.972388	−	0.233371i	$-0.925024\pi$
$883$	−2.41866e22	−1.94478	−0.972388	+	0.233371i	$0.925024\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	7.05392e21	0.548282	0.274141	−	0.961690i	$-0.411607\pi$
$887$	7.05392e21	0.548282	0.274141	+	0.961690i	$0.411607\pi$
$888$	0	0
$889$	1.34519e22	1.02807
$890$	0	0
$891$	0	0
$892$	0	0
$893$	9.69514e21	0.716421
$894$	0	0
$895$	−1.13091e22	−0.821782
$896$	0	0
$897$	0	0
$898$	0	0
$899$	6.51508e21	0.457849
$900$	0	0
$901$	−3.00356e22	−2.07587
$902$	0	0
$903$	0	0
$904$	0	0
$905$	7.00226e21	0.468138
$906$	0	0
$907$	−2.48498e22	−1.63406	−0.817030	−	0.576596i	$-0.804381\pi$
$907$	−2.48498e22	−1.63406	−0.817030	+	0.576596i	$0.804381\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−2.83848e21	−0.180592	−0.0902958	−	0.995915i	$-0.528781\pi$
$911$	−2.83848e21	−0.180592	−0.0902958	+	0.995915i	$0.528781\pi$
$912$	0	0
$913$	2.14377e21	0.134168
$914$	0	0
$915$	0	0
$916$	0	0
$917$	2.17177e22	1.31536
$918$	0	0
$919$	1.87762e22	1.11877	0.559387	−	0.828907i	$-0.311037\pi$
$919$	1.87762e22	1.11877	0.559387	+	0.828907i	$0.311037\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	−1.65155e22	−0.952528
$924$	0	0
$925$	1.13979e22	0.646787
$926$	0	0
$927$	0	0
$928$	0	0
$929$	2.61729e21	0.143792	0.0718958	−	0.997412i	$-0.477095\pi$
$929$	2.61729e21	0.143792	0.0718958	+	0.997412i	$0.477095\pi$
$930$	0	0
$931$	−6.04085e20	−0.0326570
$932$	0	0
$933$	0	0
$934$	0	0
$935$	−2.05232e22	−1.07438
$936$	0	0
$937$	1.24366e22	0.640701	0.320350	−	0.947299i	$-0.396199\pi$
$937$	1.24366e22	0.640701	0.320350	+	0.947299i	$0.396199\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−1.71266e22	−0.854574	−0.427287	−	0.904116i	$-0.640531\pi$
$941$	−1.71266e22	−0.854574	−0.427287	+	0.904116i	$0.640531\pi$
$942$	0	0
$943$	1.01548e22	0.498692
$944$	0	0
$945$	0	0
$946$	0	0
$947$	1.26035e21	0.0599607	0.0299804	−	0.999550i	$-0.490456\pi$
$947$	1.26035e21	0.0599607	0.0299804	+	0.999550i	$0.490456\pi$
$948$	0	0
$949$	1.57061e21	0.0735480
$950$	0	0
$951$	0	0
$952$	0	0
$953$	2.19887e22	0.997706	0.498853	−	0.866687i	$-0.333755\pi$
$953$	2.19887e22	0.997706	0.498853	+	0.866687i	$0.333755\pi$
$954$	0	0
$955$	2.09151e22	0.934189
$956$	0	0
$957$	0	0
$958$	0	0
$959$	2.13554e22	0.924419
$960$	0	0
$961$	−2.03083e22	−0.865461
$962$	0	0
$963$	0	0
$964$	0	0
$965$	9.25260e21	0.382216
$966$	0	0
$967$	−1.21130e21	−0.0492666	−0.0246333	−	0.999697i	$-0.507842\pi$
$967$	−1.21130e21	−0.0492666	−0.0246333	+	0.999697i	$0.507842\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	1.26825e22	0.500103	0.250051	−	0.968233i	$-0.419552\pi$
$971$	1.26825e22	0.500103	0.250051	+	0.968233i	$0.419552\pi$
$972$	0	0
$973$	4.19223e22	1.62779
$974$	0	0
$975$	0	0
$976$	0	0
$977$	3.29367e21	0.124014	0.0620070	−	0.998076i	$-0.480250\pi$
$977$	3.29367e21	0.124014	0.0620070	+	0.998076i	$0.480250\pi$
$978$	0	0
$979$	1.34660e22	0.499307
$980$	0	0
$981$	0	0
$982$	0	0
$983$	3.74166e22	1.34559	0.672796	−	0.739828i	$-0.265093\pi$
$983$	3.74166e22	1.34559	0.672796	+	0.739828i	$0.265093\pi$
$984$	0	0
$985$	3.28653e22	1.16404
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−1.81518e22	−0.623661
$990$	0	0
$991$	4.40305e22	1.49005	0.745026	−	0.667035i	$-0.232437\pi$
$991$	4.40305e22	1.49005	0.745026	+	0.667035i	$0.232437\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	−4.37565e22	−1.43671
$996$	0	0
$997$	2.62067e22	0.847614	0.423807	−	0.905753i	$-0.360694\pi$
$997$	2.62067e22	0.847614	0.423807	+	0.905753i	$0.360694\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	144.16.a.l.1.1		1
3.2	odd	2		48.16.a.a.1.1		1
4.3	odd	2		9.16.a.c.1.1		1
12.11	even	2		3.16.a.b.1.1	✓	1
60.23	odd	4		75.16.b.b.49.2		2
60.47	odd	4		75.16.b.b.49.1		2
60.59	even	2		75.16.a.a.1.1		1
84.83	odd	2		147.16.a.b.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3.16.a.b.1.1	✓	1	12.11	even	2
9.16.a.c.1.1		1	4.3	odd	2
48.16.a.a.1.1		1	3.2	odd	2
75.16.a.a.1.1		1	60.59	even	2
75.16.b.b.49.1		2	60.47	odd	4
75.16.b.b.49.2		2	60.23	odd	4
144.16.a.l.1.1		1	1.1	even	1	trivial
147.16.a.b.1.1		1	84.83	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 144 = 2^{4} \cdot 3^{2} \)
Weight:	\( k \)	\(=\)	\( 16 \)
Character orbit:	\([\chi]\)	\(=\)	144.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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