LMFDB - Embedded newform 144.16.a.f.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 1)
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)

$f(q)$

$=$

$q-52110.0 q^{5} -2.82246e6 q^{7} +O(q^{10})$

$q-52110.0 q^{5} -2.82246e6 q^{7} +2.05869e7 q^{11} -1.90073e8 q^{13} -1.64653e9 q^{17} -1.56326e9 q^{19} +9.45112e9 q^{23} -2.78021e10 q^{25} +3.69026e10 q^{29} -7.15885e10 q^{31} +1.47078e11 q^{35} -1.03365e12 q^{37} -1.64197e12 q^{41} +4.92403e11 q^{43} -3.41068e12 q^{47} +3.21870e12 q^{49} -6.79715e12 q^{53} -1.07278e12 q^{55} +9.85886e12 q^{59} +4.93184e12 q^{61} +9.90472e12 q^{65} +2.88378e13 q^{67} +1.25050e14 q^{71} -8.21715e13 q^{73} -5.81055e13 q^{77} +2.54131e13 q^{79} -2.81737e14 q^{83} +8.58006e13 q^{85} -7.15619e14 q^{89} +5.36474e14 q^{91} +8.14613e13 q^{95} +6.12786e14 q^{97} +O(q^{100})$

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
$5$	−52110.0	−0.298295	−0.149148	−	0.988815i	$-0.547653\pi$
$5$	−52110.0	−0.298295	−0.149148	+	0.988815i	$0.547653\pi$
$6$	0	0
$7$	−2.82246e6	−1.29536	−0.647682	−	0.761911i	$-0.724261\pi$
$7$	−2.82246e6	−1.29536	−0.647682	+	0.761911i	$0.724261\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	2.05869e7	0.318526	0.159263	−	0.987236i	$-0.449088\pi$
$11$	2.05869e7	0.318526	0.159263	+	0.987236i	$0.449088\pi$
$12$	0	0
$13$	−1.90073e8	−0.840129	−0.420065	−	0.907494i	$-0.637993\pi$
$13$	−1.90073e8	−0.840129	−0.420065	+	0.907494i	$0.637993\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−1.64653e9	−0.973200	−0.486600	−	0.873625i	$-0.661763\pi$
$17$	−1.64653e9	−0.973200	−0.486600	+	0.873625i	$0.661763\pi$
$18$	0	0
$19$	−1.56326e9	−0.401216	−0.200608	−	0.979672i	$-0.564292\pi$
$19$	−1.56326e9	−0.401216	−0.200608	+	0.979672i	$0.564292\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	9.45112e9	0.578794	0.289397	−	0.957209i	$-0.406545\pi$
$23$	9.45112e9	0.578794	0.289397	+	0.957209i	$0.406545\pi$
$24$	0	0
$25$	−2.78021e10	−0.911020
$26$	0	0
$27$	0	0
$28$	0	0
$29$	3.69026e10	0.397257	0.198629	−	0.980075i	$-0.436351\pi$
$29$	3.69026e10	0.397257	0.198629	+	0.980075i	$0.436351\pi$
$30$	0	0
$31$	−7.15885e10	−0.467337	−0.233669	−	0.972316i	$-0.575073\pi$
$31$	−7.15885e10	−0.467337	−0.233669	+	0.972316i	$0.575073\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	1.47078e11	0.386401
$36$	0	0
$37$	−1.03365e12	−1.79003	−0.895017	−	0.446031i	$-0.852837\pi$
$37$	−1.03365e12	−1.79003	−0.895017	+	0.446031i	$0.852837\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−1.64197e12	−1.31670	−0.658351	−	0.752711i	$-0.728746\pi$
$41$	−1.64197e12	−1.31670	−0.658351	+	0.752711i	$0.728746\pi$
$42$	0	0
$43$	4.92403e11	0.276253	0.138127	−	0.990415i	$-0.455892\pi$
$43$	4.92403e11	0.276253	0.138127	+	0.990415i	$0.455892\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−3.41068e12	−0.981991	−0.490996	−	0.871162i	$-0.663367\pi$
$47$	−3.41068e12	−0.981991	−0.490996	+	0.871162i	$0.663367\pi$
$48$	0	0
$49$	3.21870e12	0.677968
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−6.79715e12	−0.794800	−0.397400	−	0.917645i	$-0.630087\pi$
$53$	−6.79715e12	−0.794800	−0.397400	+	0.917645i	$0.630087\pi$
$54$	0	0
$55$	−1.07278e12	−0.0950147
$56$	0	0
$57$	0	0
$58$	0	0
$59$	9.85886e12	0.515747	0.257873	−	0.966179i	$-0.416978\pi$
$59$	9.85886e12	0.515747	0.257873	+	0.966179i	$0.416978\pi$
$60$	0	0
$61$	4.93184e12	0.200926	0.100463	−	0.994941i	$-0.467968\pi$
$61$	4.93184e12	0.200926	0.100463	+	0.994941i	$0.467968\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	9.90472e12	0.250606
$66$	0	0
$67$	2.88378e13	0.581302	0.290651	−	0.956829i	$-0.406128\pi$
$67$	2.88378e13	0.581302	0.290651	+	0.956829i	$0.406128\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	1.25050e14	1.63172	0.815862	−	0.578247i	$-0.196263\pi$
$71$	1.25050e14	1.63172	0.815862	+	0.578247i	$0.196263\pi$
$72$	0	0
$73$	−8.21715e13	−0.870562	−0.435281	−	0.900295i	$-0.643351\pi$
$73$	−8.21715e13	−0.870562	−0.435281	+	0.900295i	$0.643351\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−5.81055e13	−0.412607
$78$	0	0
$79$	2.54131e13	0.148886	0.0744430	−	0.997225i	$-0.476282\pi$
$79$	2.54131e13	0.148886	0.0744430	+	0.997225i	$0.476282\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	−2.81737e14	−1.13961	−0.569807	−	0.821779i	$-0.692982\pi$
$83$	−2.81737e14	−1.13961	−0.569807	+	0.821779i	$0.692982\pi$
$84$	0	0
$85$	8.58006e13	0.290301
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−7.15619e14	−1.71497	−0.857485	−	0.514509i	$-0.827974\pi$
$89$	−7.15619e14	−1.71497	−0.857485	+	0.514509i	$0.827974\pi$
$90$	0	0
$91$	5.36474e14	1.08827
$92$	0	0
$93$	0	0
$94$	0	0
$95$	8.14613e13	0.119681
$96$	0	0
$97$	6.12786e14	0.770054	0.385027	−	0.922905i	$-0.374192\pi$
$97$	6.12786e14	0.770054	0.385027	+	0.922905i	$0.374192\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	8.17642e14	0.758844	0.379422	−	0.925224i	$-0.376123\pi$
$101$	8.17642e14	0.758844	0.379422	+	0.925224i	$0.376123\pi$
$102$	0	0
$103$	−7.41115e14	−0.593753	−0.296877	−	0.954916i	$-0.595945\pi$
$103$	−7.41115e14	−0.593753	−0.296877	+	0.954916i	$0.595945\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−2.51430e15	−1.51370	−0.756849	−	0.653590i	$-0.773262\pi$
$107$	−2.51430e15	−1.51370	−0.756849	+	0.653590i	$0.773262\pi$
$108$	0	0
$109$	1.26835e15	0.664572	0.332286	−	0.943179i	$-0.392180\pi$
$109$	1.26835e15	0.664572	0.332286	+	0.943179i	$0.392180\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	2.05416e15	0.821385	0.410692	−	0.911774i	$-0.365287\pi$
$113$	2.05416e15	0.821385	0.410692	+	0.911774i	$0.365287\pi$
$114$	0	0
$115$	−4.92498e14	−0.172652
$116$	0	0
$117$	0	0
$118$	0	0
$119$	4.64725e15	1.26065
$120$	0	0
$121$	−3.75343e15	−0.898541
$122$	0	0
$123$	0	0
$124$	0	0
$125$	3.03904e15	0.570048
$126$	0	0
$127$	−2.99068e15	−0.498014	−0.249007	−	0.968502i	$-0.580104\pi$
$127$	−2.99068e15	−0.498014	−0.249007	+	0.968502i	$0.580104\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−1.62623e15	−0.214608	−0.107304	−	0.994226i	$-0.534222\pi$
$131$	−1.62623e15	−0.214608	−0.107304	+	0.994226i	$0.534222\pi$
$132$	0	0
$133$	4.41222e15	0.519721
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−1.05922e16	−0.999038	−0.499519	−	0.866303i	$-0.666490\pi$
$137$	−1.05922e16	−0.999038	−0.499519	+	0.866303i	$0.666490\pi$
$138$	0	0
$139$	1.86709e16	1.57963	0.789813	−	0.613347i	$-0.210177\pi$
$139$	1.86709e16	1.57963	0.789813	+	0.613347i	$0.210177\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−3.91301e15	−0.267603
$144$	0	0
$145$	−1.92299e15	−0.118500
$146$	0	0
$147$	0	0
$148$	0	0
$149$	1.25560e16	0.630889	0.315444	−	0.948944i	$-0.397846\pi$
$149$	1.25560e16	0.630889	0.315444	+	0.948944i	$0.397846\pi$
$150$	0	0
$151$	−2.87588e16	−1.30751	−0.653753	−	0.756708i	$-0.726806\pi$
$151$	−2.87588e16	−1.30751	−0.653753	+	0.756708i	$0.726806\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	3.73048e15	0.139404
$156$	0	0
$157$	−1.45276e16	−0.493114	−0.246557	−	0.969128i	$-0.579299\pi$
$157$	−1.45276e16	−0.493114	−0.246557	+	0.969128i	$0.579299\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−2.66754e16	−0.749750
$162$	0	0
$163$	−1.67741e16	−0.429767	−0.214884	−	0.976640i	$-0.568937\pi$
$163$	−1.67741e16	−0.429767	−0.214884	+	0.976640i	$0.568937\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	6.41999e16	1.37139	0.685695	−	0.727889i	$-0.259498\pi$
$167$	6.41999e16	1.37139	0.685695	+	0.727889i	$0.259498\pi$
$168$	0	0
$169$	−1.50580e16	−0.294183
$170$	0	0
$171$	0	0
$172$	0	0
$173$	7.59860e16	1.24563	0.622814	−	0.782370i	$-0.285990\pi$
$173$	7.59860e16	1.24563	0.622814	+	0.782370i	$0.285990\pi$
$174$	0	0
$175$	7.84703e16	1.18010
$176$	0	0
$177$	0	0
$178$	0	0
$179$	9.33749e16	1.18531	0.592655	−	0.805456i	$-0.298080\pi$
$179$	9.33749e16	1.18531	0.592655	+	0.805456i	$0.298080\pi$
$180$	0	0
$181$	7.43177e16	0.867966	0.433983	−	0.900921i	$-0.357108\pi$
$181$	7.43177e16	0.867966	0.433983	+	0.900921i	$0.357108\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	5.38636e16	0.533958
$186$	0	0
$187$	−3.38968e16	−0.309990
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−9.86224e16	−0.769529	−0.384765	−	0.923015i	$-0.625717\pi$
$191$	−9.86224e16	−0.769529	−0.384765	+	0.923015i	$0.625717\pi$
$192$	0	0
$193$	−8.91178e15	−0.0643109	−0.0321554	−	0.999483i	$-0.510237\pi$
$193$	−8.91178e15	−0.0643109	−0.0321554	+	0.999483i	$0.510237\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−3.54176e16	−0.219140	−0.109570	−	0.993979i	$-0.534947\pi$
$197$	−3.54176e16	−0.219140	−0.109570	+	0.993979i	$0.534947\pi$
$198$	0	0
$199$	2.86461e17	1.64311	0.821556	−	0.570127i	$-0.193106\pi$
$199$	2.86461e17	1.64311	0.821556	+	0.570127i	$0.193106\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	−1.04156e17	−0.514593
$204$	0	0
$205$	8.55633e16	0.392766
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−3.21825e16	−0.127798
$210$	0	0
$211$	−3.75834e17	−1.38956	−0.694780	−	0.719222i	$-0.744498\pi$
$211$	−3.75834e17	−1.38956	−0.694780	+	0.719222i	$0.744498\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−2.56591e16	−0.0824050
$216$	0	0
$217$	2.02055e17	0.605372
$218$	0	0
$219$	0	0
$220$	0	0
$221$	3.12961e17	0.817614
$222$	0	0
$223$	2.53078e16	0.0617970	0.0308985	−	0.999523i	$-0.490163\pi$
$223$	2.53078e16	0.0617970	0.0308985	+	0.999523i	$0.490163\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	3.03692e17	0.648992	0.324496	−	0.945887i	$-0.394805\pi$
$227$	3.03692e17	0.648992	0.324496	+	0.945887i	$0.394805\pi$
$228$	0	0
$229$	1.07992e17	0.216085	0.108042	−	0.994146i	$-0.465542\pi$
$229$	1.07992e17	0.216085	0.108042	+	0.994146i	$0.465542\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	7.90506e17	1.38911	0.694554	−	0.719441i	$-0.255602\pi$
$233$	7.90506e17	1.38911	0.694554	+	0.719441i	$0.255602\pi$
$234$	0	0
$235$	1.77731e17	0.292923
$236$	0	0
$237$	0	0
$238$	0	0
$239$	3.52956e17	0.512551	0.256275	−	0.966604i	$-0.417505\pi$
$239$	3.52956e17	0.512551	0.256275	+	0.966604i	$0.417505\pi$
$240$	0	0
$241$	6.85690e16	0.0935405	0.0467703	−	0.998906i	$-0.485107\pi$
$241$	6.85690e16	0.0935405	0.0467703	+	0.998906i	$0.485107\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	−1.67726e17	−0.202235
$246$	0	0
$247$	2.97134e17	0.337073
$248$	0	0
$249$	0	0
$250$	0	0
$251$	1.58806e18	1.59703	0.798515	−	0.601975i	$-0.205619\pi$
$251$	1.58806e18	1.59703	0.798515	+	0.601975i	$0.205619\pi$
$252$	0	0
$253$	1.94569e17	0.184361
$254$	0	0
$255$	0	0
$256$	0	0
$257$	8.28562e17	0.697954	0.348977	−	0.937131i	$-0.386529\pi$
$257$	8.28562e17	0.697954	0.348977	+	0.937131i	$0.386529\pi$
$258$	0	0
$259$	2.91744e18	2.31875
$260$	0	0
$261$	0	0
$262$	0	0
$263$	1.40445e18	0.995038	0.497519	−	0.867453i	$-0.334244\pi$
$263$	1.40445e18	0.995038	0.497519	+	0.867453i	$0.334244\pi$
$264$	0	0
$265$	3.54200e17	0.237085
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−1.43582e18	−0.858930	−0.429465	−	0.903083i	$-0.641298\pi$
$269$	−1.43582e18	−0.858930	−0.429465	+	0.903083i	$0.641298\pi$
$270$	0	0
$271$	−5.09160e17	−0.288127	−0.144064	−	0.989568i	$-0.546017\pi$
$271$	−5.09160e17	−0.288127	−0.144064	+	0.989568i	$0.546017\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−5.72358e17	−0.290184
$276$	0	0
$277$	5.68946e17	0.273195	0.136598	−	0.990627i	$-0.456383\pi$
$277$	5.68946e17	0.273195	0.136598	+	0.990627i	$0.456383\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	4.06184e18	1.75156	0.875780	−	0.482710i	$-0.160348\pi$
$281$	4.06184e18	1.75156	0.875780	+	0.482710i	$0.160348\pi$
$282$	0	0
$283$	−2.78506e18	−1.13877	−0.569385	−	0.822071i	$-0.692819\pi$
$283$	−2.78506e18	−1.13877	−0.569385	+	0.822071i	$0.692819\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	4.63440e18	1.70561
$288$	0	0
$289$	−1.51369e17	−0.0528813
$290$	0	0
$291$	0	0
$292$	0	0
$293$	3.63803e18	1.14646	0.573230	−	0.819395i	$-0.305690\pi$
$293$	3.63803e18	1.14646	0.573230	+	0.819395i	$0.305690\pi$
$294$	0	0
$295$	−5.13745e17	−0.153845
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−1.79641e18	−0.486262
$300$	0	0
$301$	−1.38979e18	−0.357849
$302$	0	0
$303$	0	0
$304$	0	0
$305$	−2.56998e17	−0.0599351
$306$	0	0
$307$	9.75296e17	0.216570	0.108285	−	0.994120i	$-0.465464\pi$
$307$	9.75296e17	0.216570	0.108285	+	0.994120i	$0.465464\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	3.36692e17	0.0678468	0.0339234	−	0.999424i	$-0.489200\pi$
$311$	3.36692e17	0.0678468	0.0339234	+	0.999424i	$0.489200\pi$
$312$	0	0
$313$	3.65551e18	0.702046	0.351023	−	0.936367i	$-0.385834\pi$
$313$	3.65551e18	0.702046	0.351023	+	0.936367i	$0.385834\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	7.97380e17	0.139226	0.0696131	−	0.997574i	$-0.477824\pi$
$317$	7.97380e17	0.139226	0.0696131	+	0.997574i	$0.477824\pi$
$318$	0	0
$319$	7.59708e17	0.126537
$320$	0	0
$321$	0	0
$322$	0	0
$323$	2.57395e18	0.390464
$324$	0	0
$325$	5.28444e18	0.765375
$326$	0	0
$327$	0	0
$328$	0	0
$329$	9.62651e18	1.27204
$330$	0	0
$331$	1.01585e19	1.28269	0.641343	−	0.767255i	$-0.278378\pi$
$331$	1.01585e19	1.28269	0.641343	+	0.767255i	$0.278378\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	−1.50274e18	−0.173399
$336$	0	0
$337$	−4.81465e18	−0.531301	−0.265651	−	0.964069i	$-0.585587\pi$
$337$	−4.81465e18	−0.531301	−0.265651	+	0.964069i	$0.585587\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	−1.47378e18	−0.148859
$342$	0	0
$343$	4.31515e18	0.417148
$344$	0	0
$345$	0	0
$346$	0	0
$347$	4.50275e18	0.399031	0.199516	−	0.979895i	$-0.436063\pi$
$347$	4.50275e18	0.399031	0.199516	+	0.979895i	$0.436063\pi$
$348$	0	0
$349$	2.24323e19	1.90407	0.952036	−	0.305986i	$-0.0989860\pi$
$349$	2.24323e19	1.90407	0.952036	+	0.305986i	$0.0989860\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−8.02510e18	−0.625374	−0.312687	−	0.949856i	$-0.601229\pi$
$353$	−8.02510e18	−0.625374	−0.312687	+	0.949856i	$0.601229\pi$
$354$	0	0
$355$	−6.51636e18	−0.486735
$356$	0	0
$357$	0	0
$358$	0	0
$359$	1.61507e18	0.110913	0.0554567	−	0.998461i	$-0.482339\pi$
$359$	1.61507e18	0.110913	0.0554567	+	0.998461i	$0.482339\pi$
$360$	0	0
$361$	−1.27374e19	−0.839026
$362$	0	0
$363$	0	0
$364$	0	0
$365$	4.28195e18	0.259684
$366$	0	0
$367$	9.97799e18	0.580828	0.290414	−	0.956901i	$-0.406207\pi$
$367$	9.97799e18	0.580828	0.290414	+	0.956901i	$0.406207\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	1.91847e19	1.02956
$372$	0	0
$373$	−2.36866e19	−1.22092	−0.610459	−	0.792048i	$-0.709015\pi$
$373$	−2.36866e19	−1.22092	−0.610459	+	0.792048i	$0.709015\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−7.01419e18	−0.333747
$378$	0	0
$379$	−1.86851e19	−0.854480	−0.427240	−	0.904138i	$-0.640514\pi$
$379$	−1.86851e19	−0.854480	−0.427240	+	0.904138i	$0.640514\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−3.02521e19	−1.27869	−0.639343	−	0.768921i	$-0.720794\pi$
$383$	−3.02521e19	−1.27869	−0.639343	+	0.768921i	$0.720794\pi$
$384$	0	0
$385$	3.02788e18	0.123079
$386$	0	0
$387$	0	0
$388$	0	0
$389$	1.00714e18	0.0378852	0.0189426	−	0.999821i	$-0.493970\pi$
$389$	1.00714e18	0.0378852	0.0189426	+	0.999821i	$0.493970\pi$
$390$	0	0
$391$	−1.55615e19	−0.563283
$392$	0	0
$393$	0	0
$394$	0	0
$395$	−1.32428e18	−0.0444120
$396$	0	0
$397$	3.56324e19	1.15058	0.575290	−	0.817950i	$-0.304889\pi$
$397$	3.56324e19	1.15058	0.575290	+	0.817950i	$0.304889\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−3.94327e19	−1.18106	−0.590532	−	0.807014i	$-0.701082\pi$
$401$	−3.94327e19	−1.18106	−0.590532	+	0.807014i	$0.701082\pi$
$402$	0	0
$403$	1.36071e19	0.392624
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−2.12796e19	−0.570172
$408$	0	0
$409$	−5.27823e19	−1.36321	−0.681607	−	0.731719i	$-0.738718\pi$
$409$	−5.27823e19	−1.36321	−0.681607	+	0.731719i	$0.738718\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	−2.78262e19	−0.668080
$414$	0	0
$415$	1.46813e19	0.339941
$416$	0	0
$417$	0	0
$418$	0	0
$419$	8.62630e18	0.185874	0.0929372	−	0.995672i	$-0.470374\pi$
$419$	8.62630e18	0.185874	0.0929372	+	0.995672i	$0.470374\pi$
$420$	0	0
$421$	−4.29249e19	−0.892469	−0.446235	−	0.894916i	$-0.647235\pi$
$421$	−4.29249e19	−0.892469	−0.446235	+	0.894916i	$0.647235\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	4.57770e19	0.886605
$426$	0	0
$427$	−1.39199e19	−0.260272
$428$	0	0
$429$	0	0
$430$	0	0
$431$	5.04764e19	0.880053	0.440026	−	0.897985i	$-0.354969\pi$
$431$	5.04764e19	0.880053	0.440026	+	0.897985i	$0.354969\pi$
$432$	0	0
$433$	5.05734e19	0.851653	0.425827	−	0.904805i	$-0.359983\pi$
$433$	5.05734e19	0.851653	0.425827	+	0.904805i	$0.359983\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−1.47745e19	−0.232222
$438$	0	0
$439$	−2.47946e19	−0.376594	−0.188297	−	0.982112i	$-0.560297\pi$
$439$	−2.47946e19	−0.376594	−0.188297	+	0.982112i	$0.560297\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−1.30654e20	−1.85394	−0.926970	−	0.375135i	$-0.877596\pi$
$443$	−1.30654e20	−1.85394	−0.926970	+	0.375135i	$0.877596\pi$
$444$	0	0
$445$	3.72909e19	0.511567
$446$	0	0
$447$	0	0
$448$	0	0
$449$	7.78280e19	0.998363	0.499181	−	0.866498i	$-0.333634\pi$
$449$	7.78280e19	0.998363	0.499181	+	0.866498i	$0.333634\pi$
$450$	0	0
$451$	−3.38031e19	−0.419404
$452$	0	0
$453$	0	0
$454$	0	0
$455$	−2.79556e19	−0.324627
$456$	0	0
$457$	−1.18451e20	−1.33096	−0.665482	−	0.746414i	$-0.731774\pi$
$457$	−1.18451e20	−1.33096	−0.665482	+	0.746414i	$0.731774\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−1.38643e20	−1.45929	−0.729644	−	0.683827i	$-0.760314\pi$
$461$	−1.38643e20	−1.45929	−0.729644	+	0.683827i	$0.760314\pi$
$462$	0	0
$463$	−1.75645e20	−1.78969	−0.894846	−	0.446375i	$-0.852715\pi$
$463$	−1.75645e20	−1.78969	−0.894846	+	0.446375i	$0.852715\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−1.36631e20	−1.30519	−0.652593	−	0.757708i	$-0.726319\pi$
$467$	−1.36631e20	−1.30519	−0.652593	+	0.757708i	$0.726319\pi$
$468$	0	0
$469$	−8.13935e19	−0.752997
$470$	0	0
$471$	0	0
$472$	0	0
$473$	1.01370e19	0.0879938
$474$	0	0
$475$	4.34619e19	0.365516
$476$	0	0
$477$	0	0
$478$	0	0
$479$	6.41058e19	0.506269	0.253134	−	0.967431i	$-0.418539\pi$
$479$	6.41058e19	0.506269	0.253134	+	0.967431i	$0.418539\pi$
$480$	0	0
$481$	1.96470e20	1.50386
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−3.19323e19	−0.229703
$486$	0	0
$487$	2.41343e19	0.168332	0.0841662	−	0.996452i	$-0.473177\pi$
$487$	2.41343e19	0.168332	0.0841662	+	0.996452i	$0.473177\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−2.80908e19	−0.184269	−0.0921346	−	0.995747i	$-0.529369\pi$
$491$	−2.80908e19	−0.184269	−0.0921346	+	0.995747i	$0.529369\pi$
$492$	0	0
$493$	−6.07611e19	−0.386611
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−3.52948e20	−2.11368
$498$	0	0
$499$	1.71994e20	0.999443	0.499722	−	0.866186i	$-0.333436\pi$
$499$	1.71994e20	0.999443	0.499722	+	0.866186i	$0.333436\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−1.83497e20	−1.00431	−0.502155	−	0.864778i	$-0.667459\pi$
$503$	−1.83497e20	−1.00431	−0.502155	+	0.864778i	$0.667459\pi$
$504$	0	0
$505$	−4.26073e19	−0.226359
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−2.67204e20	−1.33801	−0.669004	−	0.743258i	$-0.733279\pi$
$509$	−2.67204e20	−1.33801	−0.669004	+	0.743258i	$0.733279\pi$
$510$	0	0
$511$	2.31925e20	1.12769
$512$	0	0
$513$	0	0
$514$	0	0
$515$	3.86195e19	0.177114
$516$	0	0
$517$	−7.02153e19	−0.312790
$518$	0	0
$519$	0	0
$520$	0	0
$521$	2.01468e20	0.847076	0.423538	−	0.905878i	$-0.360788\pi$
$521$	2.01468e20	0.847076	0.423538	+	0.905878i	$0.360788\pi$
$522$	0	0
$523$	−3.58989e20	−1.46662	−0.733311	−	0.679894i	$-0.762026\pi$
$523$	−3.58989e20	−1.46662	−0.733311	+	0.679894i	$0.762026\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	1.17872e20	0.454813
$528$	0	0
$529$	−1.77312e20	−0.664997
$530$	0	0
$531$	0	0
$532$	0	0
$533$	3.12095e20	1.10620
$534$	0	0
$535$	1.31020e20	0.451528
$536$	0	0
$537$	0	0
$538$	0	0
$539$	6.62628e19	0.215950
$540$	0	0
$541$	2.02328e20	0.641323	0.320662	−	0.947194i	$-0.396095\pi$
$541$	2.02328e20	0.641323	0.320662	+	0.947194i	$0.396095\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	−6.60939e19	−0.198238
$546$	0	0
$547$	−7.40963e19	−0.216218	−0.108109	−	0.994139i	$-0.534480\pi$
$547$	−7.40963e19	−0.216218	−0.108109	+	0.994139i	$0.534480\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−5.76882e19	−0.159386
$552$	0	0
$553$	−7.17273e19	−0.192862
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−2.09626e18	−0.00533987	−0.00266994	−	0.999996i	$-0.500850\pi$
$557$	−2.09626e18	−0.00533987	−0.00266994	+	0.999996i	$0.500850\pi$
$558$	0	0
$559$	−9.35927e19	−0.232088
$560$	0	0
$561$	0	0
$562$	0	0
$563$	6.87353e20	1.61572	0.807861	−	0.589373i	$-0.200625\pi$
$563$	6.87353e20	1.61572	0.807861	+	0.589373i	$0.200625\pi$
$564$	0	0
$565$	−1.07042e20	−0.245015
$566$	0	0
$567$	0	0
$568$	0	0
$569$	9.05218e19	0.196522	0.0982610	−	0.995161i	$-0.468672\pi$
$569$	9.05218e19	0.196522	0.0982610	+	0.995161i	$0.468672\pi$
$570$	0	0
$571$	−2.05774e20	−0.435130	−0.217565	−	0.976046i	$-0.569811\pi$
$571$	−2.05774e20	−0.435130	−0.217565	+	0.976046i	$0.569811\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−2.62761e20	−0.527293
$576$	0	0
$577$	5.70778e20	1.11596	0.557980	−	0.829854i	$-0.311576\pi$
$577$	5.70778e20	1.11596	0.557980	+	0.829854i	$0.311576\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	7.95190e20	1.47622
$582$	0	0
$583$	−1.39932e20	−0.253164
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−9.30363e20	−1.59907	−0.799534	−	0.600621i	$-0.794920\pi$
$587$	−9.30363e20	−1.59907	−0.799534	+	0.600621i	$0.794920\pi$
$588$	0	0
$589$	1.11911e20	0.187503
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−3.54225e20	−0.564116	−0.282058	−	0.959397i	$-0.591017\pi$
$593$	−3.54225e20	−0.564116	−0.282058	+	0.959397i	$0.591017\pi$
$594$	0	0
$595$	−2.42168e20	−0.376045
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−3.30045e20	−0.487385	−0.243693	−	0.969853i	$-0.578359\pi$
$599$	−3.30045e20	−0.487385	−0.243693	+	0.969853i	$0.578359\pi$
$600$	0	0
$601$	−3.35884e20	−0.483761	−0.241880	−	0.970306i	$-0.577764\pi$
$601$	−3.35884e20	−0.483761	−0.241880	+	0.970306i	$0.577764\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	1.95591e20	0.268030
$606$	0	0
$607$	1.33438e21	1.78387	0.891934	−	0.452165i	$-0.149348\pi$
$607$	1.33438e21	1.78387	0.891934	+	0.452165i	$0.149348\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	6.48280e20	0.825000
$612$	0	0
$613$	5.68844e18	0.00706381	0.00353191	−	0.999994i	$-0.498876\pi$
$613$	5.68844e18	0.00706381	0.00353191	+	0.999994i	$0.498876\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−3.98915e20	−0.471783	−0.235891	−	0.971779i	$-0.575801\pi$
$617$	−3.98915e20	−0.471783	−0.235891	+	0.971779i	$0.575801\pi$
$618$	0	0
$619$	5.40017e20	0.623343	0.311672	−	0.950190i	$-0.399111\pi$
$619$	5.40017e20	0.623343	0.311672	+	0.950190i	$0.399111\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	2.01980e21	2.22151
$624$	0	0
$625$	6.90089e20	0.740978
$626$	0	0
$627$	0	0
$628$	0	0
$629$	1.70194e21	1.74206
$630$	0	0
$631$	−9.59111e20	−0.958625	−0.479312	−	0.877644i	$-0.659114\pi$
$631$	−9.59111e20	−0.958625	−0.479312	+	0.877644i	$0.659114\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	1.55844e20	0.148555
$636$	0	0
$637$	−6.11788e20	−0.569581
$638$	0	0
$639$	0	0
$640$	0	0
$641$	9.25925e20	0.822509	0.411255	−	0.911521i	$-0.365091\pi$
$641$	9.25925e20	0.822509	0.411255	+	0.911521i	$0.365091\pi$
$642$	0	0
$643$	7.65928e20	0.664669	0.332335	−	0.943162i	$-0.392164\pi$
$643$	7.65928e20	0.664669	0.332335	+	0.943162i	$0.392164\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	1.36075e21	1.12719	0.563596	−	0.826051i	$-0.309418\pi$
$647$	1.36075e21	1.12719	0.563596	+	0.826051i	$0.309418\pi$
$648$	0	0
$649$	2.02963e20	0.164279
$650$	0	0
$651$	0	0
$652$	0	0
$653$	2.71809e20	0.210094	0.105047	−	0.994467i	$-0.466501\pi$
$653$	2.71809e20	0.210094	0.105047	+	0.994467i	$0.466501\pi$
$654$	0	0
$655$	8.47427e19	0.0640166
$656$	0	0
$657$	0	0
$658$	0	0
$659$	6.74316e20	0.486657	0.243329	−	0.969944i	$-0.421761\pi$
$659$	6.74316e20	0.486657	0.243329	+	0.969944i	$0.421761\pi$
$660$	0	0
$661$	1.26727e21	0.894042	0.447021	−	0.894524i	$-0.352485\pi$
$661$	1.26727e21	0.894042	0.447021	+	0.894524i	$0.352485\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−2.29921e20	−0.155030
$666$	0	0
$667$	3.48770e20	0.229930
$668$	0	0
$669$	0	0
$670$	0	0
$671$	1.01531e20	0.0640000
$672$	0	0
$673$	−1.13945e21	−0.702394	−0.351197	−	0.936302i	$-0.614225\pi$
$673$	−1.13945e21	−0.702394	−0.351197	+	0.936302i	$0.614225\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−1.74431e21	−1.02851	−0.514256	−	0.857637i	$-0.671932\pi$
$677$	−1.74431e21	−1.02851	−0.514256	+	0.857637i	$0.671932\pi$
$678$	0	0
$679$	−1.72956e21	−0.997500
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−1.43739e21	−0.793267	−0.396634	−	0.917977i	$-0.629822\pi$
$683$	−1.43739e21	−0.793267	−0.396634	+	0.917977i	$0.629822\pi$
$684$	0	0
$685$	5.51960e20	0.298008
$686$	0	0
$687$	0	0
$688$	0	0
$689$	1.29196e21	0.667735
$690$	0	0
$691$	1.77548e21	0.897903	0.448951	−	0.893556i	$-0.351798\pi$
$691$	1.77548e21	0.897903	0.448951	+	0.893556i	$0.351798\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−9.72941e20	−0.471195
$696$	0	0
$697$	2.70356e21	1.28141
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−1.43100e21	−0.649764	−0.324882	−	0.945755i	$-0.605325\pi$
$701$	−1.43100e21	−0.649764	−0.324882	+	0.945755i	$0.605325\pi$
$702$	0	0
$703$	1.61586e21	0.718191
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−2.30776e21	−0.982980
$708$	0	0
$709$	−2.41840e21	−1.00851	−0.504257	−	0.863554i	$-0.668234\pi$
$709$	−2.41840e21	−1.00851	−0.504257	+	0.863554i	$0.668234\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−6.76591e20	−0.270492
$714$	0	0
$715$	2.03907e20	0.0798246
$716$	0	0
$717$	0	0
$718$	0	0
$719$	4.74444e21	1.78122	0.890611	−	0.454766i	$-0.150277\pi$
$719$	4.74444e21	1.78122	0.890611	+	0.454766i	$0.150277\pi$
$720$	0	0
$721$	2.09176e21	0.769127
$722$	0	0
$723$	0	0
$724$	0	0
$725$	−1.02597e21	−0.361909
$726$	0	0
$727$	3.59265e21	1.24138	0.620692	−	0.784054i	$-0.286852\pi$
$727$	3.59265e21	1.24138	0.620692	+	0.784054i	$0.286852\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−8.10755e20	−0.268850
$732$	0	0
$733$	−2.76824e21	−0.899339	−0.449669	−	0.893195i	$-0.648458\pi$
$733$	−2.76824e21	−0.899339	−0.449669	+	0.893195i	$0.648458\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	5.93680e20	0.185160
$738$	0	0
$739$	−3.55824e21	−1.08743	−0.543716	−	0.839269i	$-0.682983\pi$
$739$	−3.55824e21	−1.08743	−0.543716	+	0.839269i	$0.682983\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−1.94092e21	−0.569628	−0.284814	−	0.958583i	$-0.591932\pi$
$743$	−1.94092e21	−0.569628	−0.284814	+	0.958583i	$0.591932\pi$
$744$	0	0
$745$	−6.54291e20	−0.188191
$746$	0	0
$747$	0	0
$748$	0	0
$749$	7.09651e21	1.96079
$750$	0	0
$751$	−4.75565e21	−1.28798	−0.643992	−	0.765032i	$-0.722723\pi$
$751$	−4.75565e21	−1.28798	−0.643992	+	0.765032i	$0.722723\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	1.49862e21	0.390022
$756$	0	0
$757$	3.62137e21	0.923960	0.461980	−	0.886890i	$-0.347139\pi$
$757$	3.62137e21	0.923960	0.461980	+	0.886890i	$0.347139\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	3.86361e21	0.947564	0.473782	−	0.880642i	$-0.342888\pi$
$761$	3.86361e21	0.947564	0.473782	+	0.880642i	$0.342888\pi$
$762$	0	0
$763$	−3.57987e21	−0.860862
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−1.87391e21	−0.433294
$768$	0	0
$769$	−5.39327e21	−1.22294	−0.611469	−	0.791268i	$-0.709421\pi$
$769$	−5.39327e21	−1.22294	−0.611469	+	0.791268i	$0.709421\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	−6.57037e21	−1.43299	−0.716496	−	0.697591i	$-0.754255\pi$
$773$	−6.57037e21	−1.43299	−0.716496	+	0.697591i	$0.754255\pi$
$774$	0	0
$775$	1.99031e21	0.425754
$776$	0	0
$777$	0	0
$778$	0	0
$779$	2.56683e21	0.528282
$780$	0	0
$781$	2.57439e21	0.519746
$782$	0	0
$783$	0	0
$784$	0	0
$785$	7.57035e20	0.147094
$786$	0	0
$787$	3.72074e20	0.0709281	0.0354641	−	0.999371i	$-0.488709\pi$
$787$	3.72074e20	0.0709281	0.0354641	+	0.999371i	$0.488709\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−5.79778e21	−1.06399
$792$	0	0
$793$	−9.37412e20	−0.168804
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−2.61511e21	−0.453474	−0.226737	−	0.973956i	$-0.572806\pi$
$797$	−2.61511e21	−0.453474	−0.226737	+	0.973956i	$0.572806\pi$
$798$	0	0
$799$	5.61579e21	0.955674
$800$	0	0
$801$	0	0
$802$	0	0
$803$	−1.69165e21	−0.277296
$804$	0	0
$805$	1.39005e21	0.223647
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−5.34899e21	−0.829198	−0.414599	−	0.910004i	$-0.636078\pi$
$809$	−5.34899e21	−0.829198	−0.414599	+	0.910004i	$0.636078\pi$
$810$	0	0
$811$	−8.46492e21	−1.28815	−0.644075	−	0.764962i	$-0.722758\pi$
$811$	−8.46492e21	−1.28815	−0.644075	+	0.764962i	$0.722758\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	8.74100e20	0.128197
$816$	0	0
$817$	−7.69753e20	−0.110837
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−7.99397e21	−1.10966	−0.554829	−	0.831965i	$-0.687216\pi$
$821$	−7.99397e21	−1.10966	−0.554829	+	0.831965i	$0.687216\pi$
$822$	0	0
$823$	−1.96841e21	−0.268297	−0.134148	−	0.990961i	$-0.542830\pi$
$823$	−1.96841e21	−0.268297	−0.134148	+	0.990961i	$0.542830\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−1.43539e22	−1.88659	−0.943296	−	0.331954i	$-0.892292\pi$
$827$	−1.43539e22	−1.88659	−0.943296	+	0.331954i	$0.892292\pi$
$828$	0	0
$829$	−8.83327e21	−1.14015	−0.570076	−	0.821592i	$-0.693086\pi$
$829$	−8.83327e21	−1.14015	−0.570076	+	0.821592i	$0.693086\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	−5.29967e21	−0.659799
$834$	0	0
$835$	−3.34546e21	−0.409079
$836$	0	0
$837$	0	0
$838$	0	0
$839$	1.26696e22	1.49469	0.747343	−	0.664439i	$-0.231329\pi$
$839$	1.26696e22	1.49469	0.747343	+	0.664439i	$0.231329\pi$
$840$	0	0
$841$	−7.26739e21	−0.842187
$842$	0	0
$843$	0	0
$844$	0	0
$845$	7.84673e20	0.0877533
$846$	0	0
$847$	1.05939e22	1.16394
$848$	0	0
$849$	0	0
$850$	0	0
$851$	−9.76917e21	−1.03606
$852$	0	0
$853$	6.00532e21	0.625776	0.312888	−	0.949790i	$-0.398704\pi$
$853$	6.00532e21	0.625776	0.312888	+	0.949790i	$0.398704\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−1.47589e22	−1.48491	−0.742453	−	0.669898i	$-0.766338\pi$
$857$	−1.47589e22	−1.48491	−0.742453	+	0.669898i	$0.766338\pi$
$858$	0	0
$859$	−9.64956e20	−0.0954023	−0.0477012	−	0.998862i	$-0.515190\pi$
$859$	−9.64956e20	−0.0954023	−0.0477012	+	0.998862i	$0.515190\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	3.44391e21	0.328829	0.164415	−	0.986391i	$-0.447426\pi$
$863$	3.44391e21	0.328829	0.164415	+	0.986391i	$0.447426\pi$
$864$	0	0
$865$	−3.95963e21	−0.371564
$866$	0	0
$867$	0	0
$868$	0	0
$869$	5.23175e20	0.0474241
$870$	0	0
$871$	−5.48130e21	−0.488368
$872$	0	0
$873$	0	0
$874$	0	0
$875$	−8.57756e21	−0.738420
$876$	0	0
$877$	−1.09850e22	−0.929617	−0.464808	−	0.885411i	$-0.653877\pi$
$877$	−1.09850e22	−0.929617	−0.464808	+	0.885411i	$0.653877\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	7.98462e21	0.653033	0.326516	−	0.945192i	$-0.394125\pi$
$881$	7.98462e21	0.653033	0.326516	+	0.945192i	$0.394125\pi$
$882$	0	0
$883$	−5.45236e21	−0.438409	−0.219204	−	0.975679i	$-0.570346\pi$
$883$	−5.45236e21	−0.438409	−0.219204	+	0.975679i	$0.570346\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	1.67127e22	1.29903	0.649517	−	0.760347i	$-0.274971\pi$
$887$	1.67127e22	1.29903	0.649517	+	0.760347i	$0.274971\pi$
$888$	0	0
$889$	8.44105e21	0.645109
$890$	0	0
$891$	0	0
$892$	0	0
$893$	5.33178e21	0.393991
$894$	0	0
$895$	−4.86576e21	−0.353572
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−2.64180e21	−0.185653
$900$	0	0
$901$	1.11917e22	0.773500
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−3.87269e21	−0.258910
$906$	0	0
$907$	1.53384e22	1.00862	0.504309	−	0.863523i	$-0.331747\pi$
$907$	1.53384e22	1.00862	0.504309	+	0.863523i	$0.331747\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−1.49134e22	−0.948829	−0.474415	−	0.880302i	$-0.657340\pi$
$911$	−1.49134e22	−0.948829	−0.474415	+	0.880302i	$0.657340\pi$
$912$	0	0
$913$	−5.80007e21	−0.362997
$914$	0	0
$915$	0	0
$916$	0	0
$917$	4.58995e21	0.277996
$918$	0	0
$919$	5.86667e21	0.349563	0.174782	−	0.984607i	$-0.444078\pi$
$919$	5.86667e21	0.349563	0.174782	+	0.984607i	$0.444078\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	−2.37687e22	−1.37086
$924$	0	0
$925$	2.87377e22	1.63076
$926$	0	0
$927$	0	0
$928$	0	0
$929$	1.67946e22	0.922684	0.461342	−	0.887222i	$-0.347368\pi$
$929$	1.67946e22	0.922684	0.461342	+	0.887222i	$0.347368\pi$
$930$	0	0
$931$	−5.03165e21	−0.272012
$932$	0	0
$933$	0	0
$934$	0	0
$935$	1.76636e21	0.0924683
$936$	0	0
$937$	5.04466e21	0.259887	0.129944	−	0.991521i	$-0.458520\pi$
$937$	5.04466e21	0.259887	0.129944	+	0.991521i	$0.458520\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−1.65425e22	−0.825430	−0.412715	−	0.910860i	$-0.635419\pi$
$941$	−1.65425e22	−0.825430	−0.412715	+	0.910860i	$0.635419\pi$
$942$	0	0
$943$	−1.55185e22	−0.762100
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−9.81583e21	−0.466984	−0.233492	−	0.972359i	$-0.575015\pi$
$947$	−9.81583e21	−0.466984	−0.233492	+	0.972359i	$0.575015\pi$
$948$	0	0
$949$	1.56186e22	0.731384
$950$	0	0
$951$	0	0
$952$	0	0
$953$	5.97914e21	0.271295	0.135648	−	0.990757i	$-0.456689\pi$
$953$	5.97914e21	0.271295	0.135648	+	0.990757i	$0.456689\pi$
$954$	0	0
$955$	5.13921e21	0.229547
$956$	0	0
$957$	0	0
$958$	0	0
$959$	2.98960e22	1.29412
$960$	0	0
$961$	−1.83404e22	−0.781596
$962$	0	0
$963$	0	0
$964$	0	0
$965$	4.64393e20	0.0191836
$966$	0	0
$967$	1.44757e22	0.588764	0.294382	−	0.955688i	$-0.404886\pi$
$967$	1.44757e22	0.588764	0.294382	+	0.955688i	$0.404886\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	1.77921e21	0.0701590	0.0350795	−	0.999385i	$-0.488832\pi$
$971$	1.77921e21	0.0701590	0.0350795	+	0.999385i	$0.488832\pi$
$972$	0	0
$973$	−5.26978e22	−2.04619
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−1.04088e22	−0.391913	−0.195957	−	0.980613i	$-0.562781\pi$
$977$	−1.04088e22	−0.391913	−0.195957	+	0.980613i	$0.562781\pi$
$978$	0	0
$979$	−1.47323e22	−0.546262
$980$	0	0
$981$	0	0
$982$	0	0
$983$	3.26461e22	1.17403	0.587017	−	0.809575i	$-0.300302\pi$
$983$	3.26461e22	1.17403	0.587017	+	0.809575i	$0.300302\pi$
$984$	0	0
$985$	1.84561e21	0.0653684
$986$	0	0
$987$	0	0
$988$	0	0
$989$	4.65376e21	0.159894
$990$	0	0
$991$	7.47327e21	0.252906	0.126453	−	0.991973i	$-0.459641\pi$
$991$	7.47327e21	0.252906	0.126453	+	0.991973i	$0.459641\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	−1.49275e22	−0.490132
$996$	0	0
$997$	−3.10809e22	−1.00526	−0.502632	−	0.864500i	$-0.667635\pi$
$997$	−3.10809e22	−1.00526	−0.502632	+	0.864500i	$0.667635\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	144.16.a.f.1.1		1
3.2	odd	2		16.16.a.d.1.1		1
4.3	odd	2		9.16.a.a.1.1		1
12.11	even	2		1.16.a.a.1.1	✓	1
24.5	odd	2		64.16.a.c.1.1		1
24.11	even	2		64.16.a.i.1.1		1
60.23	odd	4		25.16.b.a.24.1		2
60.47	odd	4		25.16.b.a.24.2		2
60.59	even	2		25.16.a.a.1.1		1
84.11	even	6		49.16.c.c.30.1		2
84.23	even	6		49.16.c.c.18.1		2
84.47	odd	6		49.16.c.b.18.1		2
84.59	odd	6		49.16.c.b.30.1		2
84.83	odd	2		49.16.a.a.1.1		1
132.131	odd	2		121.16.a.a.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1.16.a.a.1.1	✓	1	12.11	even	2
9.16.a.a.1.1		1	4.3	odd	2
16.16.a.d.1.1		1	3.2	odd	2
25.16.a.a.1.1		1	60.59	even	2
25.16.b.a.24.1		2	60.23	odd	4
25.16.b.a.24.2		2	60.47	odd	4
49.16.a.a.1.1		1	84.83	odd	2
49.16.c.b.18.1		2	84.47	odd	6
49.16.c.b.30.1		2	84.59	odd	6
49.16.c.c.18.1		2	84.23	even	6
49.16.c.c.30.1		2	84.11	even	6
64.16.a.c.1.1		1	24.5	odd	2
64.16.a.i.1.1		1	24.11	even	2
121.16.a.a.1.1		1	132.131	odd	2
144.16.a.f.1.1		1	1.1	even	1	trivial

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

Related objects

Downloads

Learn more