LMFDB - Embedded newform 2916.3.c.b.1457.20

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2916,3,Mod(1457,2916)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2916, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 1]))

N = Newforms(chi, 3, names="a")

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

chi := DirichletCharacter("2916.1457");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 2916 = 2^{2} \cdot 3^{6} $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	2916.c (of order $2$, degree $1$, minimal)

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:	no
Analytic conductor:	$79.4552450875$
Analytic rank:	$0$
Dimension:	$36$
Twist minimal:	no (minimal twist has level 108)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1457.20
Character	$\chi$	$=$	2916.1457
Dual form			2916.3.c.b.1457.17

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.464464i q^{5} +10.8001 q^{7} +O(q^{10})$	$q+0.464464i q^{5} +10.8001 q^{7} -15.9172i q^{11} -17.9569 q^{13} +26.1473i q^{17} -3.55728 q^{19} -4.12593i q^{23} +24.7843 q^{25} -41.6788i q^{29} +7.04870 q^{31} +5.01625i q^{35} -9.85818 q^{37} -43.5435i q^{41} -35.5753 q^{43} -16.1318i q^{47} +67.6417 q^{49} -75.6950i q^{53} +7.39296 q^{55} +28.6908i q^{59} +59.5324 q^{61} -8.34035i q^{65} -23.2357 q^{67} +37.2238i q^{71} +52.0559 q^{73} -171.907i q^{77} -120.715 q^{79} -116.867i q^{83} -12.1445 q^{85} -135.986i q^{89} -193.936 q^{91} -1.65223i q^{95} -94.4120 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 36 q+O(q^{10}) $ 36 * q	$ 36 q - 180 q^{25} + 252 q^{49} + 18 q^{61} - 90 q^{67} + 126 q^{73} - 198 q^{91}+O(q^{100}) $ 36 * q - 180 * q^25 + 252 * q^49 + 18 * q^61 - 90 * q^67 + 126 * q^73 - 198 * q^91

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2916\mathbb{Z}\right)^\times$.

$n$	$1459$	$2189$
$\chi(n)$	$1$	$-1$

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
$5$	0.464464i	0.0928927i	0.998921	+	0.0464464i	$0.0147897\pi$
$5$	0.464464i	0.0928927i	−0.998921	+	0.0464464i	$0.985210\pi$
$6$	0	0
$7$	10.8001	1.54287	0.771434	−	0.636309i	$-0.219540\pi$
$7$	10.8001	1.54287	0.771434	+	0.636309i	$0.219540\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	− 15.9172i	− 1.44702i	−0.690315	−	0.723509i	$-0.742528\pi$
$11$	− 15.9172i	− 1.44702i	0.690315	−	0.723509i	$-0.257472\pi$
$12$	0	0
$13$	−17.9569	−1.38130	−0.690651	−	0.723188i	$-0.742676\pi$
$13$	−17.9569	−1.38130	−0.690651	+	0.723188i	$0.742676\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	26.1473i	1.53808i	0.639203	+	0.769038i	$0.279264\pi$
$17$	26.1473i	1.53808i	−0.639203	+	0.769038i	$0.720736\pi$
$18$	0	0
$19$	−3.55728	−0.187225	−0.0936126	−	0.995609i	$-0.529841\pi$
$19$	−3.55728	−0.187225	−0.0936126	+	0.995609i	$0.529841\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	− 4.12593i	− 0.179388i	−0.995969	−	0.0896940i	$-0.971411\pi$
$23$	− 4.12593i	− 0.179388i	0.995969	−	0.0896940i	$-0.0285889\pi$
$24$	0	0
$25$	24.7843	0.991371
$26$	0	0
$27$	0	0
$28$	0	0
$29$	− 41.6788i	− 1.43720i	−0.695423	−	0.718600i	$-0.744783\pi$
$29$	− 41.6788i	− 1.43720i	0.695423	−	0.718600i	$-0.255217\pi$
$30$	0	0
$31$	7.04870	0.227377	0.113689	−	0.993516i	$-0.463733\pi$
$31$	7.04870	0.227377	0.113689	+	0.993516i	$0.463733\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	5.01625i	0.143321i
$36$	0	0
$37$	−9.85818	−0.266437	−0.133219	−	0.991087i	$-0.542531\pi$
$37$	−9.85818	−0.266437	−0.133219	+	0.991087i	$0.542531\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	− 43.5435i	− 1.06204i	−0.847360	−	0.531018i	$-0.821810\pi$
$41$	− 43.5435i	− 1.06204i	0.847360	−	0.531018i	$-0.178190\pi$
$42$	0	0
$43$	−35.5753	−0.827332	−0.413666	−	0.910429i	$-0.635752\pi$
$43$	−35.5753	−0.827332	−0.413666	+	0.910429i	$0.635752\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	− 16.1318i	− 0.343231i	−0.985164	−	0.171615i	$-0.945101\pi$
$47$	− 16.1318i	− 0.343231i	0.985164	−	0.171615i	$-0.0548986\pi$
$48$	0	0
$49$	67.6417	1.38044
$50$	0	0
$51$	0	0
$52$	0	0
$53$	− 75.6950i	− 1.42821i	−0.700040	−	0.714104i	$-0.746834\pi$
$53$	− 75.6950i	− 1.42821i	0.700040	−	0.714104i	$-0.253166\pi$
$54$	0	0
$55$	7.39296	0.134417
$56$	0	0
$57$	0	0
$58$	0	0
$59$	28.6908i	0.486285i	0.969991	+	0.243143i	$0.0781783\pi$
$59$	28.6908i	0.486285i	−0.969991	+	0.243143i	$0.921822\pi$
$60$	0	0
$61$	59.5324	0.975942	0.487971	−	0.872860i	$-0.337737\pi$
$61$	59.5324	0.975942	0.487971	+	0.872860i	$0.337737\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	− 8.34035i	− 0.128313i
$66$	0	0
$67$	−23.2357	−0.346801	−0.173401	−	0.984851i	$-0.555476\pi$
$67$	−23.2357	−0.346801	−0.173401	+	0.984851i	$0.555476\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	37.2238i	0.524279i	0.965030	+	0.262139i	$0.0844281\pi$
$71$	37.2238i	0.524279i	−0.965030	+	0.262139i	$0.915572\pi$
$72$	0	0
$73$	52.0559	0.713095	0.356547	−	0.934277i	$-0.383954\pi$
$73$	52.0559	0.713095	0.356547	+	0.934277i	$0.383954\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	− 171.907i	− 2.23256i
$78$	0	0
$79$	−120.715	−1.52804	−0.764018	−	0.645194i	$-0.776776\pi$
$79$	−120.715	−1.52804	−0.764018	+	0.645194i	$0.776776\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	− 116.867i	− 1.40803i	−0.710185	−	0.704015i	$-0.751389\pi$
$83$	− 116.867i	− 1.40803i	0.710185	−	0.704015i	$-0.248611\pi$
$84$	0	0
$85$	−12.1445	−0.142876
$86$	0	0
$87$	0	0
$88$	0	0
$89$	− 135.986i	− 1.52793i	−0.645257	−	0.763966i	$-0.723250\pi$
$89$	− 135.986i	− 1.52793i	0.645257	−	0.763966i	$-0.276750\pi$
$90$	0	0
$91$	−193.936	−2.13117
$92$	0	0
$93$	0	0
$94$	0	0
$95$	− 1.65223i	− 0.0173919i
$96$	0	0
$97$	−94.4120	−0.973320	−0.486660	−	0.873591i	$-0.661785\pi$
$97$	−94.4120	−0.973320	−0.486660	+	0.873591i	$0.661785\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	152.431i	1.50922i	0.656173	+	0.754611i	$0.272174\pi$
$101$	152.431i	1.50922i	−0.656173	+	0.754611i	$0.727826\pi$
$102$	0	0
$103$	21.7537	0.211201	0.105601	−	0.994409i	$-0.466323\pi$
$103$	21.7537	0.211201	0.105601	+	0.994409i	$0.466323\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	49.8431i	0.465823i	0.972498	+	0.232912i	$0.0748253\pi$
$107$	49.8431i	0.465823i	−0.972498	+	0.232912i	$0.925175\pi$
$108$	0	0
$109$	171.063	1.56939	0.784695	−	0.619882i	$-0.212820\pi$
$109$	171.063	1.56939	0.784695	+	0.619882i	$0.212820\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	− 40.7050i	− 0.360222i	−0.983646	−	0.180111i	$-0.942354\pi$
$113$	− 40.7050i	− 0.360222i	0.983646	−	0.180111i	$-0.0576456\pi$
$114$	0	0
$115$	1.91634	0.0166639
$116$	0	0
$117$	0	0
$118$	0	0
$119$	282.393i	2.37305i
$120$	0	0
$121$	−132.357	−1.09386
$122$	0	0
$123$	0	0
$124$	0	0
$125$	23.1230i	0.184984i
$126$	0	0
$127$	61.7816	0.486469	0.243235	−	0.969967i	$-0.421792\pi$
$127$	61.7816	0.486469	0.243235	+	0.969967i	$0.421792\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	− 226.158i	− 1.72639i	−0.504867	−	0.863197i	$-0.668459\pi$
$131$	− 226.158i	− 1.72639i	0.504867	−	0.863197i	$-0.331541\pi$
$132$	0	0
$133$	−38.4189	−0.288864
$134$	0	0
$135$	0	0
$136$	0	0
$137$	151.921i	1.10891i	0.832214	+	0.554455i	$0.187073\pi$
$137$	151.921i	1.10891i	−0.832214	+	0.554455i	$0.812927\pi$
$138$	0	0
$139$	−169.960	−1.22273	−0.611367	−	0.791347i	$-0.709380\pi$
$139$	−169.960	−1.22273	−0.611367	+	0.791347i	$0.709380\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	285.824i	1.99877i
$144$	0	0
$145$	19.3583	0.133506
$146$	0	0
$147$	0	0
$148$	0	0
$149$	− 179.051i	− 1.20169i	−0.799367	−	0.600843i	$-0.794832\pi$
$149$	− 179.051i	− 1.20169i	0.799367	−	0.600843i	$-0.205168\pi$
$150$	0	0
$151$	120.169	0.795822	0.397911	−	0.917424i	$-0.369735\pi$
$151$	120.169	0.795822	0.397911	+	0.917424i	$0.369735\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	3.27387i	0.0211217i
$156$	0	0
$157$	32.6084	0.207697	0.103848	−	0.994593i	$-0.466884\pi$
$157$	32.6084	0.207697	0.103848	+	0.994593i	$0.466884\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	− 44.5603i	− 0.276772i
$162$	0	0
$163$	−219.827	−1.34863	−0.674316	−	0.738443i	$-0.735561\pi$
$163$	−219.827	−1.34863	−0.674316	+	0.738443i	$0.735561\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	− 121.425i	− 0.727099i	−0.931575	−	0.363549i	$-0.881565\pi$
$167$	− 121.425i	− 0.727099i	0.931575	−	0.363549i	$-0.118435\pi$
$168$	0	0
$169$	153.452	0.907997
$170$	0	0
$171$	0	0
$172$	0	0
$173$	− 77.2012i	− 0.446250i	−0.974790	−	0.223125i	$-0.928374\pi$
$173$	− 77.2012i	− 0.446250i	0.974790	−	0.223125i	$-0.0716258\pi$
$174$	0	0
$175$	267.672	1.52956
$176$	0	0
$177$	0	0
$178$	0	0
$179$	148.480i	0.829499i	0.909936	+	0.414750i	$0.136131\pi$
$179$	148.480i	0.829499i	−0.909936	+	0.414750i	$0.863869\pi$
$180$	0	0
$181$	−84.4269	−0.466447	−0.233224	−	0.972423i	$-0.574927\pi$
$181$	−84.4269	−0.466447	−0.233224	+	0.972423i	$0.574927\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	− 4.57877i	− 0.0247501i
$186$	0	0
$187$	416.191	2.22562
$188$	0	0
$189$	0	0
$190$	0	0
$191$	1.95539i	0.0102376i	0.999987	+	0.00511882i	$0.00162938\pi$
$191$	1.95539i	0.0102376i	−0.999987	+	0.00511882i	$0.998371\pi$
$192$	0	0
$193$	331.992	1.72017	0.860083	−	0.510154i	$-0.170412\pi$
$193$	331.992	1.72017	0.860083	+	0.510154i	$0.170412\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	− 246.119i	− 1.24933i	−0.780892	−	0.624666i	$-0.785235\pi$
$197$	− 246.119i	− 1.24933i	0.780892	−	0.624666i	$-0.214765\pi$
$198$	0	0
$199$	276.452	1.38921	0.694604	−	0.719392i	$-0.255580\pi$
$199$	276.452	1.38921	0.694604	+	0.719392i	$0.255580\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	− 450.135i	− 2.21741i
$204$	0	0
$205$	20.2244	0.0986555
$206$	0	0
$207$	0	0
$208$	0	0
$209$	56.6219i	0.270918i
$210$	0	0
$211$	−53.2470	−0.252356	−0.126178	−	0.992008i	$-0.540271\pi$
$211$	−53.2470	−0.252356	−0.126178	+	0.992008i	$0.540271\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	− 16.5234i	− 0.0768532i
$216$	0	0
$217$	76.1265	0.350813
$218$	0	0
$219$	0	0
$220$	0	0
$221$	− 469.525i	− 2.12455i
$222$	0	0
$223$	−108.493	−0.486515	−0.243258	−	0.969962i	$-0.578216\pi$
$223$	−108.493	−0.486515	−0.243258	+	0.969962i	$0.578216\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	− 362.469i	− 1.59678i	−0.602140	−	0.798390i	$-0.705685\pi$
$227$	− 362.469i	− 1.59678i	0.602140	−	0.798390i	$-0.294315\pi$
$228$	0	0
$229$	−289.548	−1.26440	−0.632201	−	0.774804i	$-0.717849\pi$
$229$	−289.548	−1.26440	−0.632201	+	0.774804i	$0.717849\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	− 147.412i	− 0.632672i	−0.948647	−	0.316336i	$-0.897547\pi$
$233$	− 147.412i	− 0.632672i	0.948647	−	0.316336i	$-0.102453\pi$
$234$	0	0
$235$	7.49266	0.0318836
$236$	0	0
$237$	0	0
$238$	0	0
$239$	13.6322i	0.0570386i	0.999593	+	0.0285193i	$0.00907920\pi$
$239$	13.6322i	0.0570386i	−0.999593	+	0.0285193i	$0.990921\pi$
$240$	0	0
$241$	−297.063	−1.23263	−0.616313	−	0.787502i	$-0.711374\pi$
$241$	−297.063	−1.23263	−0.616313	+	0.787502i	$0.711374\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	31.4171i	0.128233i
$246$	0	0
$247$	63.8778	0.258615
$248$	0	0
$249$	0	0
$250$	0	0
$251$	− 0.211134i	0 0.000841172i	−1.00000		0.000420586i	$-0.999866\pi$
$251$	− 0.211134i	0 0.000841172i	1.00000		0.000420586i	$-0.000133877\pi$
$252$	0	0
$253$	−65.6732	−0.259578
$254$	0	0
$255$	0	0
$256$	0	0
$257$	193.916i	0.754536i	0.926104	+	0.377268i	$0.123136\pi$
$257$	193.916i	0.754536i	−0.926104	+	0.377268i	$0.876864\pi$
$258$	0	0
$259$	−106.469	−0.411078
$260$	0	0
$261$	0	0
$262$	0	0
$263$	− 408.850i	− 1.55456i	−0.629152	−	0.777282i	$-0.716598\pi$
$263$	− 408.850i	− 1.55456i	0.629152	−	0.777282i	$-0.283402\pi$
$264$	0	0
$265$	35.1576	0.132670
$266$	0	0
$267$	0	0
$268$	0	0
$269$	120.090i	0.446433i	0.974769	+	0.223216i	$0.0716557\pi$
$269$	120.090i	0.446433i	−0.974769	+	0.223216i	$0.928344\pi$
$270$	0	0
$271$	19.0289	0.0702175	0.0351088	−	0.999383i	$-0.488822\pi$
$271$	19.0289	0.0702175	0.0351088	+	0.999383i	$0.488822\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	− 394.496i	− 1.43453i
$276$	0	0
$277$	−318.369	−1.14935	−0.574673	−	0.818383i	$-0.694871\pi$
$277$	−318.369	−1.14935	−0.574673	+	0.818383i	$0.694871\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	− 240.657i	− 0.856432i	−0.903676	−	0.428216i	$-0.859142\pi$
$281$	− 240.657i	− 0.856432i	0.903676	−	0.428216i	$-0.140858\pi$
$282$	0	0
$283$	89.3386	0.315684	0.157842	−	0.987464i	$-0.449546\pi$
$283$	89.3386	0.315684	0.157842	+	0.987464i	$0.449546\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	− 470.273i	− 1.63858i
$288$	0	0
$289$	−394.681	−1.36568
$290$	0	0
$291$	0	0
$292$	0	0
$293$	− 311.499i	− 1.06314i	−0.847015	−	0.531568i	$-0.821603\pi$
$293$	− 311.499i	− 1.06314i	0.847015	−	0.531568i	$-0.178397\pi$
$294$	0	0
$295$	−13.3258	−0.0451724
$296$	0	0
$297$	0	0
$298$	0	0
$299$	74.0890i	0.247789i
$300$	0	0
$301$	−384.216	−1.27647
$302$	0	0
$303$	0	0
$304$	0	0
$305$	27.6507i	0.0906579i
$306$	0	0
$307$	593.580	1.93349	0.966743	−	0.255751i	$-0.0823226\pi$
$307$	593.580	1.93349	0.966743	+	0.255751i	$0.0823226\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	− 546.678i	− 1.75781i	−0.477000	−	0.878903i	$-0.658276\pi$
$311$	− 546.678i	− 1.75781i	0.477000	−	0.878903i	$-0.341724\pi$
$312$	0	0
$313$	156.637	0.500437	0.250219	−	0.968189i	$-0.419498\pi$
$313$	156.637	0.500437	0.250219	+	0.968189i	$0.419498\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	− 42.3174i	− 0.133493i	−0.997770	−	0.0667466i	$-0.978738\pi$
$317$	− 42.3174i	− 0.133493i	0.997770	−	0.0667466i	$-0.0212619\pi$
$318$	0	0
$319$	−663.410	−2.07965
$320$	0	0
$321$	0	0
$322$	0	0
$323$	− 93.0132i	− 0.287966i
$324$	0	0
$325$	−445.050	−1.36938
$326$	0	0
$327$	0	0
$328$	0	0
$329$	− 174.225i	− 0.529560i
$330$	0	0
$331$	−492.072	−1.48662	−0.743311	−	0.668946i	$-0.766746\pi$
$331$	−492.072	−1.48662	−0.743311	+	0.668946i	$0.766746\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	− 10.7921i	− 0.0322153i
$336$	0	0
$337$	654.938	1.94343	0.971717	−	0.236147i	$-0.0758846\pi$
$337$	654.938	1.94343	0.971717	+	0.236147i	$0.0758846\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	− 112.196i	− 0.329019i
$342$	0	0
$343$	201.332	0.586974
$344$	0	0
$345$	0	0
$346$	0	0
$347$	130.058i	0.374807i	0.982283	+	0.187404i	$0.0600073\pi$
$347$	130.058i	0.374807i	−0.982283	+	0.187404i	$0.939993\pi$
$348$	0	0
$349$	82.6088	0.236701	0.118351	−	0.992972i	$-0.462239\pi$
$349$	82.6088	0.236701	0.118351	+	0.992972i	$0.462239\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	66.5996i	0.188668i	0.995541	+	0.0943338i	$0.0300721\pi$
$353$	66.5996i	0.188668i	−0.995541	+	0.0943338i	$0.969928\pi$
$354$	0	0
$355$	−17.2891	−0.0487017
$356$	0	0
$357$	0	0
$358$	0	0
$359$	334.423i	0.931540i	0.884906	+	0.465770i	$0.154223\pi$
$359$	334.423i	0.931540i	−0.884906	+	0.465770i	$0.845777\pi$
$360$	0	0
$361$	−348.346	−0.964947
$362$	0	0
$363$	0	0
$364$	0	0
$365$	24.1781i	0.0662413i
$366$	0	0
$367$	408.479	1.11302	0.556511	−	0.830840i	$-0.312140\pi$
$367$	408.479	1.11302	0.556511	+	0.830840i	$0.312140\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	− 817.512i	− 2.20354i
$372$	0	0
$373$	−306.535	−0.821809	−0.410905	−	0.911678i	$-0.634787\pi$
$373$	−306.535	−0.821809	−0.410905	+	0.911678i	$0.634787\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	748.424i	1.98521i
$378$	0	0
$379$	−98.9004	−0.260951	−0.130475	−	0.991452i	$-0.541650\pi$
$379$	−98.9004	−0.260951	−0.130475	+	0.991452i	$0.541650\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	− 496.756i	− 1.29701i	−0.761209	−	0.648507i	$-0.775394\pi$
$383$	− 496.756i	− 1.29701i	0.761209	−	0.648507i	$-0.224606\pi$
$384$	0	0
$385$	79.8446	0.207388
$386$	0	0
$387$	0	0
$388$	0	0
$389$	475.404i	1.22212i	0.791585	+	0.611059i	$0.209256\pi$
$389$	475.404i	1.22212i	−0.791585	+	0.611059i	$0.790744\pi$
$390$	0	0
$391$	107.882	0.275912
$392$	0	0
$393$	0	0
$394$	0	0
$395$	− 56.0677i	− 0.141944i
$396$	0	0
$397$	434.075	1.09339	0.546694	−	0.837332i	$-0.315886\pi$
$397$	434.075	1.09339	0.546694	+	0.837332i	$0.315886\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	− 245.782i	− 0.612924i	−0.951883	−	0.306462i	$-0.900855\pi$
$401$	− 245.782i	− 0.612924i	0.951883	−	0.306462i	$-0.0991452\pi$
$402$	0	0
$403$	−126.573	−0.314077
$404$	0	0
$405$	0	0
$406$	0	0
$407$	156.915i	0.385539i
$408$	0	0
$409$	349.648	0.854884	0.427442	−	0.904043i	$-0.359415\pi$
$409$	349.648	0.854884	0.427442	+	0.904043i	$0.359415\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	309.863i	0.750274i
$414$	0	0
$415$	54.2803	0.130796
$416$	0	0
$417$	0	0
$418$	0	0
$419$	− 275.697i	− 0.657989i	−0.944332	−	0.328994i	$-0.893290\pi$
$419$	− 275.697i	− 0.657989i	0.944332	−	0.328994i	$-0.106710\pi$
$420$	0	0
$421$	65.3422	0.155207	0.0776036	−	0.996984i	$-0.475273\pi$
$421$	65.3422	0.155207	0.0776036	+	0.996984i	$0.475273\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	648.042i	1.52480i
$426$	0	0
$427$	642.955	1.50575
$428$	0	0
$429$	0	0
$430$	0	0
$431$	105.508i	0.244799i	0.992481	+	0.122399i	$0.0390589\pi$
$431$	105.508i	0.244799i	−0.992481	+	0.122399i	$0.960941\pi$
$432$	0	0
$433$	599.181	1.38379	0.691895	−	0.721998i	$-0.256776\pi$
$433$	599.181	1.38379	0.691895	+	0.721998i	$0.256776\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	14.6771i	0.0335860i
$438$	0	0
$439$	−63.7414	−0.145197	−0.0725984	−	0.997361i	$-0.523129\pi$
$439$	−63.7414	−0.145197	−0.0725984	+	0.997361i	$0.523129\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	− 133.177i	− 0.300625i	−0.988638	−	0.150313i	$-0.951972\pi$
$443$	− 133.177i	− 0.300625i	0.988638	−	0.150313i	$-0.0480281\pi$
$444$	0	0
$445$	63.1605	0.141934
$446$	0	0
$447$	0	0
$448$	0	0
$449$	139.123i	0.309851i	0.987926	+	0.154925i	$0.0495137\pi$
$449$	139.123i	0.309851i	−0.987926	+	0.154925i	$0.950486\pi$
$450$	0	0
$451$	−693.090	−1.53679
$452$	0	0
$453$	0	0
$454$	0	0
$455$	− 90.0764i	− 0.197970i
$456$	0	0
$457$	−121.006	−0.264783	−0.132392	−	0.991197i	$-0.542266\pi$
$457$	−121.006	−0.264783	−0.132392	+	0.991197i	$0.542266\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	270.309i	0.586353i	0.956058	+	0.293176i	$0.0947124\pi$
$461$	270.309i	0.586353i	−0.956058	+	0.293176i	$0.905288\pi$
$462$	0	0
$463$	−18.0780	−0.0390453	−0.0195227	−	0.999809i	$-0.506215\pi$
$463$	−18.0780	−0.0390453	−0.0195227	+	0.999809i	$0.506215\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	− 507.166i	− 1.08601i	−0.839730	−	0.543004i	$-0.817287\pi$
$467$	− 507.166i	− 1.08601i	0.839730	−	0.543004i	$-0.182713\pi$
$468$	0	0
$469$	−250.947	−0.535069
$470$	0	0
$471$	0	0
$472$	0	0
$473$	566.259i	1.19716i
$474$	0	0
$475$	−88.1645	−0.185610
$476$	0	0
$477$	0	0
$478$	0	0
$479$	614.408i	1.28269i	0.767253	+	0.641344i	$0.221623\pi$
$479$	614.408i	1.28269i	−0.767253	+	0.641344i	$0.778377\pi$
$480$	0	0
$481$	177.023	0.368030
$482$	0	0
$483$	0	0
$484$	0	0
$485$	− 43.8510i	− 0.0904144i
$486$	0	0
$487$	−218.562	−0.448793	−0.224396	−	0.974498i	$-0.572041\pi$
$487$	−218.562	−0.448793	−0.224396	+	0.974498i	$0.572041\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	200.043i	0.407420i	0.979031	+	0.203710i	$0.0653000\pi$
$491$	200.043i	0.407420i	−0.979031	+	0.203710i	$0.934700\pi$
$492$	0	0
$493$	1089.79	2.21052
$494$	0	0
$495$	0	0
$496$	0	0
$497$	402.020i	0.808894i
$498$	0	0
$499$	−133.415	−0.267365	−0.133682	−	0.991024i	$-0.542680\pi$
$499$	−133.415	−0.267365	−0.133682	+	0.991024i	$0.542680\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	− 181.207i	− 0.360253i	−0.983643	−	0.180127i	$-0.942349\pi$
$503$	− 181.207i	− 0.360253i	0.983643	−	0.180127i	$-0.0576508\pi$
$504$	0	0
$505$	−70.7988	−0.140196
$506$	0	0
$507$	0	0
$508$	0	0
$509$	− 786.365i	− 1.54492i	−0.635063	−	0.772460i	$-0.719026\pi$
$509$	− 786.365i	− 1.54492i	0.635063	−	0.772460i	$-0.280974\pi$
$510$	0	0
$511$	562.208	1.10021
$512$	0	0
$513$	0	0
$514$	0	0
$515$	10.1038i	0.0196191i
$516$	0	0
$517$	−256.774	−0.496661
$518$	0	0
$519$	0	0
$520$	0	0
$521$	− 195.750i	− 0.375721i	−0.982196	−	0.187860i	$-0.939845\pi$
$521$	− 195.750i	− 0.375721i	0.982196	−	0.187860i	$-0.0601552\pi$
$522$	0	0
$523$	159.966	0.305863	0.152931	−	0.988237i	$-0.451129\pi$
$523$	159.966	0.305863	0.152931	+	0.988237i	$0.451129\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	184.304i	0.349724i
$528$	0	0
$529$	511.977	0.967820
$530$	0	0
$531$	0	0
$532$	0	0
$533$	781.908i	1.46699i
$534$	0	0
$535$	−23.1503	−0.0432716
$536$	0	0
$537$	0	0
$538$	0	0
$539$	− 1076.67i	− 1.99753i
$540$	0	0
$541$	−34.2145	−0.0632430	−0.0316215	−	0.999500i	$-0.510067\pi$
$541$	−34.2145	−0.0632430	−0.0316215	+	0.999500i	$0.510067\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	79.4528i	0.145785i
$546$	0	0
$547$	−299.429	−0.547403	−0.273701	−	0.961815i	$-0.588248\pi$
$547$	−299.429	−0.547403	−0.273701	+	0.961815i	$0.588248\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	148.263i	0.269080i
$552$	0	0
$553$	−1303.73	−2.35756
$554$	0	0
$555$	0	0
$556$	0	0
$557$	792.216i	1.42229i	0.703045	+	0.711146i	$0.251823\pi$
$557$	792.216i	1.42229i	−0.703045	+	0.711146i	$0.748177\pi$
$558$	0	0
$559$	638.823	1.14280
$560$	0	0
$561$	0	0
$562$	0	0
$563$	979.127i	1.73912i	0.493823	+	0.869562i	$0.335599\pi$
$563$	979.127i	1.73912i	−0.493823	+	0.869562i	$0.664401\pi$
$564$	0	0
$565$	18.9060	0.0334620
$566$	0	0
$567$	0	0
$568$	0	0
$569$	560.559i	0.985166i	0.870266	+	0.492583i	$0.163947\pi$
$569$	560.559i	0.985166i	−0.870266	+	0.492583i	$0.836053\pi$
$570$	0	0
$571$	31.0978	0.0544619	0.0272310	−	0.999629i	$-0.491331\pi$
$571$	31.0978	0.0544619	0.0272310	+	0.999629i	$0.491331\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	− 102.258i	− 0.177840i
$576$	0	0
$577$	190.350	0.329896	0.164948	−	0.986302i	$-0.447254\pi$
$577$	190.350	0.329896	0.164948	+	0.986302i	$0.447254\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	− 1262.17i	− 2.17241i
$582$	0	0
$583$	−1204.85	−2.06664
$584$	0	0
$585$	0	0
$586$	0	0
$587$	− 441.419i	− 0.751991i	−0.926621	−	0.375996i	$-0.877301\pi$
$587$	− 441.419i	− 0.751991i	0.926621	−	0.375996i	$-0.122699\pi$
$588$	0	0
$589$	−25.0742	−0.0425708
$590$	0	0
$591$	0	0
$592$	0	0
$593$	− 435.642i	− 0.734640i	−0.930095	−	0.367320i	$-0.880275\pi$
$593$	− 435.642i	− 0.734640i	0.930095	−	0.367320i	$-0.119725\pi$
$594$	0	0
$595$	−131.161	−0.220439
$596$	0	0
$597$	0	0
$598$	0	0
$599$	957.372i	1.59828i	0.601143	+	0.799142i	$0.294712\pi$
$599$	957.372i	1.59828i	−0.601143	+	0.799142i	$0.705288\pi$
$600$	0	0
$601$	228.520	0.380233	0.190117	−	0.981762i	$-0.439113\pi$
$601$	228.520	0.380233	0.190117	+	0.981762i	$0.439113\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	− 61.4751i	− 0.101612i
$606$	0	0
$607$	136.369	0.224660	0.112330	−	0.993671i	$-0.464169\pi$
$607$	136.369	0.224660	0.112330	+	0.993671i	$0.464169\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	289.678i	0.474106i
$612$	0	0
$613$	71.1389	0.116050	0.0580252	−	0.998315i	$-0.481520\pi$
$613$	71.1389	0.116050	0.0580252	+	0.998315i	$0.481520\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	1017.81i	1.64961i	0.565414	+	0.824807i	$0.308716\pi$
$617$	1017.81i	1.64961i	−0.565414	+	0.824807i	$0.691284\pi$
$618$	0	0
$619$	365.315	0.590170	0.295085	−	0.955471i	$-0.404652\pi$
$619$	365.315	0.590170	0.295085	+	0.955471i	$0.404652\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	− 1468.66i	− 2.35740i
$624$	0	0
$625$	608.867	0.974187
$626$	0	0
$627$	0	0
$628$	0	0
$629$	− 257.765i	− 0.409801i
$630$	0	0
$631$	−780.428	−1.23681	−0.618406	−	0.785859i	$-0.712221\pi$
$631$	−780.428	−1.23681	−0.618406	+	0.785859i	$0.712221\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	28.6953i	0.0451895i
$636$	0	0
$637$	−1214.64	−1.90681
$638$	0	0
$639$	0	0
$640$	0	0
$641$	− 282.168i	− 0.440200i	−0.975477	−	0.220100i	$-0.929362\pi$
$641$	− 282.168i	− 0.440200i	0.975477	−	0.220100i	$-0.0706383\pi$
$642$	0	0
$643$	−41.7984	−0.0650054	−0.0325027	−	0.999472i	$-0.510348\pi$
$643$	−41.7984	−0.0650054	−0.0325027	+	0.999472i	$0.510348\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	− 383.618i	− 0.592918i	−0.955046	−	0.296459i	$-0.904194\pi$
$647$	− 383.618i	− 0.592918i	0.955046	−	0.296459i	$-0.0958058\pi$
$648$	0	0
$649$	456.677	0.703663
$650$	0	0
$651$	0	0
$652$	0	0
$653$	− 61.2602i	− 0.0938135i	−0.998899	−	0.0469068i	$-0.985064\pi$
$653$	− 61.2602i	− 0.0938135i	0.998899	−	0.0469068i	$-0.0149364\pi$
$654$	0	0
$655$	105.042	0.160369
$656$	0	0
$657$	0	0
$658$	0	0
$659$	− 512.751i	− 0.778075i	−0.921222	−	0.389037i	$-0.872808\pi$
$659$	− 512.751i	− 0.778075i	0.921222	−	0.389037i	$-0.127192\pi$
$660$	0	0
$661$	−190.961	−0.288897	−0.144448	−	0.989512i	$-0.546141\pi$
$661$	−190.961	−0.288897	−0.144448	+	0.989512i	$0.546141\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	− 17.8442i	− 0.0268333i
$666$	0	0
$667$	−171.964	−0.257817
$668$	0	0
$669$	0	0
$670$	0	0
$671$	− 947.589i	− 1.41220i
$672$	0	0
$673$	−1036.84	−1.54063	−0.770314	−	0.637665i	$-0.779900\pi$
$673$	−1036.84	−1.54063	−0.770314	+	0.637665i	$0.779900\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	− 489.621i	− 0.723221i	−0.932329	−	0.361611i	$-0.882227\pi$
$677$	− 489.621i	− 0.723221i	0.932329	−	0.361611i	$-0.117773\pi$
$678$	0	0
$679$	−1019.66	−1.50170
$680$	0	0
$681$	0	0
$682$	0	0
$683$	290.146i	0.424811i	0.977182	+	0.212406i	$0.0681298\pi$
$683$	290.146i	0.424811i	−0.977182	+	0.212406i	$0.931870\pi$
$684$	0	0
$685$	−70.5617	−0.103010
$686$	0	0
$687$	0	0
$688$	0	0
$689$	1359.25i	1.97279i
$690$	0	0
$691$	194.112	0.280914	0.140457	−	0.990087i	$-0.455143\pi$
$691$	194.112	0.280914	0.140457	+	0.990087i	$0.455143\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	− 78.9403i	− 0.113583i
$696$	0	0
$697$	1138.54	1.63349
$698$	0	0
$699$	0	0
$700$	0	0
$701$	294.780i	0.420513i	0.977646	+	0.210257i	$0.0674299\pi$
$701$	294.780i	0.420513i	−0.977646	+	0.210257i	$0.932570\pi$
$702$	0	0
$703$	35.0683	0.0498837
$704$	0	0
$705$	0	0
$706$	0	0
$707$	1646.27i	2.32853i
$708$	0	0
$709$	−974.827	−1.37493	−0.687466	−	0.726216i	$-0.741277\pi$
$709$	−974.827	−1.37493	−0.687466	+	0.726216i	$0.741277\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	− 29.0824i	− 0.0407888i
$714$	0	0
$715$	−132.755	−0.185671
$716$	0	0
$717$	0	0
$718$	0	0
$719$	663.227i	0.922430i	0.887288	+	0.461215i	$0.152586\pi$
$719$	663.227i	0.922430i	−0.887288	+	0.461215i	$0.847414\pi$
$720$	0	0
$721$	234.942	0.325856
$722$	0	0
$723$	0	0
$724$	0	0
$725$	− 1032.98i	− 1.42480i
$726$	0	0
$727$	−1029.06	−1.41549	−0.707745	−	0.706467i	$-0.750288\pi$
$727$	−1029.06	−1.41549	−0.707745	+	0.706467i	$0.750288\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	− 930.197i	− 1.27250i
$732$	0	0
$733$	99.4862	0.135725	0.0678624	−	0.997695i	$-0.478382\pi$
$733$	99.4862	0.135725	0.0678624	+	0.997695i	$0.478382\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	369.847i	0.501827i
$738$	0	0
$739$	−57.5992	−0.0779421	−0.0389710	−	0.999240i	$-0.512408\pi$
$739$	−57.5992	−0.0779421	−0.0389710	+	0.999240i	$0.512408\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	1221.77i	1.64438i	0.569217	+	0.822188i	$0.307247\pi$
$743$	1221.77i	1.64438i	−0.569217	+	0.822188i	$0.692753\pi$
$744$	0	0
$745$	83.1628	0.111628
$746$	0	0
$747$	0	0
$748$	0	0
$749$	538.309i	0.718704i
$750$	0	0
$751$	814.182	1.08413	0.542065	−	0.840337i	$-0.317643\pi$
$751$	814.182	1.08413	0.542065	+	0.840337i	$0.317643\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	55.8142i	0.0739261i
$756$	0	0
$757$	646.428	0.853933	0.426967	−	0.904267i	$-0.359582\pi$
$757$	646.428	0.853933	0.426967	+	0.904267i	$0.359582\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	206.637i	0.271534i	0.990741	+	0.135767i	$0.0433499\pi$
$761$	206.637i	0.271534i	−0.990741	+	0.135767i	$0.956650\pi$
$762$	0	0
$763$	1847.50	2.42136
$764$	0	0
$765$	0	0
$766$	0	0
$767$	− 515.199i	− 0.671707i
$768$	0	0
$769$	161.135	0.209539	0.104769	−	0.994497i	$-0.466590\pi$
$769$	161.135	0.209539	0.104769	+	0.994497i	$0.466590\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	888.067i	1.14886i	0.818555	+	0.574429i	$0.194776\pi$
$773$	888.067i	1.14886i	−0.818555	+	0.574429i	$0.805224\pi$
$774$	0	0
$775$	174.697	0.225415
$776$	0	0
$777$	0	0
$778$	0	0
$779$	154.896i	0.198840i
$780$	0	0
$781$	592.499	0.758641
$782$	0	0
$783$	0	0
$784$	0	0
$785$	15.1454i	0.0192935i
$786$	0	0
$787$	1193.88	1.51701	0.758503	−	0.651669i	$-0.225931\pi$
$787$	1193.88	1.51701	0.758503	+	0.651669i	$0.225931\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	− 439.618i	− 0.555775i
$792$	0	0
$793$	−1069.02	−1.34807
$794$	0	0
$795$	0	0
$796$	0	0
$797$	484.960i	0.608481i	0.952595	+	0.304241i	$0.0984027\pi$
$797$	484.960i	0.608481i	−0.952595	+	0.304241i	$0.901597\pi$
$798$	0	0
$799$	421.804	0.527915
$800$	0	0
$801$	0	0
$802$	0	0
$803$	− 828.584i	− 1.03186i
$804$	0	0
$805$	20.6967	0.0257101
$806$	0	0
$807$	0	0
$808$	0	0
$809$	1260.59i	1.55821i	0.626893	+	0.779105i	$0.284326\pi$
$809$	1260.59i	1.55821i	−0.626893	+	0.779105i	$0.715674\pi$
$810$	0	0
$811$	740.450	0.913009	0.456504	−	0.889721i	$-0.349101\pi$
$811$	740.450	0.913009	0.456504	+	0.889721i	$0.349101\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	− 102.102i	− 0.125278i
$816$	0	0
$817$	126.551	0.154897
$818$	0	0
$819$	0	0
$820$	0	0
$821$	933.869i	1.13748i	0.822518	+	0.568739i	$0.192569\pi$
$821$	933.869i	1.13748i	−0.822518	+	0.568739i	$0.807431\pi$
$822$	0	0
$823$	−43.6448	−0.0530313	−0.0265157	−	0.999648i	$-0.508441\pi$
$823$	−43.6448	−0.0530313	−0.0265157	+	0.999648i	$0.508441\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	− 716.248i	− 0.866080i	−0.901375	−	0.433040i	$-0.857441\pi$
$827$	− 716.248i	− 0.866080i	0.901375	−	0.433040i	$-0.142559\pi$
$828$	0	0
$829$	303.737	0.366390	0.183195	−	0.983077i	$-0.441356\pi$
$829$	303.737	0.366390	0.183195	+	0.983077i	$0.441356\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	1768.65i	2.12323i
$834$	0	0
$835$	56.3977	0.0675422
$836$	0	0
$837$	0	0
$838$	0	0
$839$	− 555.133i	− 0.661661i	−0.943690	−	0.330830i	$-0.892671\pi$
$839$	− 555.133i	− 0.661661i	0.943690	−	0.330830i	$-0.107329\pi$
$840$	0	0
$841$	−896.124	−1.06555
$842$	0	0
$843$	0	0
$844$	0	0
$845$	71.2727i	0.0843464i
$846$	0	0
$847$	−1429.47	−1.68768
$848$	0	0
$849$	0	0
$850$	0	0
$851$	40.6741i	0.0477957i
$852$	0	0
$853$	260.830	0.305780	0.152890	−	0.988243i	$-0.451142\pi$
$853$	260.830	0.305780	0.152890	+	0.988243i	$0.451142\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	360.846i	0.421057i	0.977588	+	0.210529i	$0.0675185\pi$
$857$	360.846i	0.421057i	−0.977588	+	0.210529i	$0.932482\pi$
$858$	0	0
$859$	−573.451	−0.667580	−0.333790	−	0.942647i	$-0.608328\pi$
$859$	−573.451	−0.667580	−0.333790	+	0.942647i	$0.608328\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	654.390i	0.758273i	0.925341	+	0.379137i	$0.123779\pi$
$863$	654.390i	0.758273i	−0.925341	+	0.379137i	$0.876221\pi$
$864$	0	0
$865$	35.8572	0.0414534
$866$	0	0
$867$	0	0
$868$	0	0
$869$	1921.44i	2.21110i
$870$	0	0
$871$	417.242	0.479037
$872$	0	0
$873$	0	0
$874$	0	0
$875$	249.730i	0.285406i
$876$	0	0
$877$	805.155	0.918079	0.459039	−	0.888416i	$-0.348194\pi$
$877$	805.155	0.918079	0.459039	+	0.888416i	$0.348194\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	− 1204.96i	− 1.36772i	−0.729614	−	0.683860i	$-0.760300\pi$
$881$	− 1204.96i	− 1.36772i	0.729614	−	0.683860i	$-0.239700\pi$
$882$	0	0
$883$	756.767	0.857041	0.428520	−	0.903532i	$-0.359035\pi$
$883$	756.767	0.857041	0.428520	+	0.903532i	$0.359035\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	612.327i	0.690335i	0.938541	+	0.345167i	$0.112178\pi$
$887$	612.327i	0.690335i	−0.938541	+	0.345167i	$0.887822\pi$
$888$	0	0
$889$	667.246	0.750558
$890$	0	0
$891$	0	0
$892$	0	0
$893$	57.3854i	0.0642614i
$894$	0	0
$895$	−68.9637	−0.0770545
$896$	0	0
$897$	0	0
$898$	0	0
$899$	− 293.781i	− 0.326787i
$900$	0	0
$901$	1979.22	2.19669
$902$	0	0
$903$	0	0
$904$	0	0
$905$	− 39.2132i	− 0.0433296i
$906$	0	0
$907$	1609.04	1.77403	0.887013	−	0.461744i	$-0.152776\pi$
$907$	1609.04	1.77403	0.887013	+	0.461744i	$0.152776\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	− 1378.49i	− 1.51316i	−0.653903	−	0.756578i	$-0.726870\pi$
$911$	− 1378.49i	− 1.51316i	0.653903	−	0.756578i	$-0.273130\pi$
$912$	0	0
$913$	−1860.19	−2.03744
$914$	0	0
$915$	0	0
$916$	0	0
$917$	− 2442.52i	− 2.66360i
$918$	0	0
$919$	−1619.94	−1.76272	−0.881358	−	0.472449i	$-0.843370\pi$
$919$	−1619.94	−1.76272	−0.881358	+	0.472449i	$0.843370\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	− 668.426i	− 0.724188i
$924$	0	0
$925$	−244.328	−0.264138
$926$	0	0
$927$	0	0
$928$	0	0
$929$	98.5435i	0.106075i	0.998593	+	0.0530374i	$0.0168903\pi$
$929$	98.5435i	0.106075i	−0.998593	+	0.0530374i	$0.983110\pi$
$930$	0	0
$931$	−240.620	−0.258454
$932$	0	0
$933$	0	0
$934$	0	0
$935$	193.306i	0.206744i
$936$	0	0
$937$	−561.249	−0.598985	−0.299493	−	0.954099i	$-0.596817\pi$
$937$	−561.249	−0.598985	−0.299493	+	0.954099i	$0.596817\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	742.445i	0.788996i	0.918897	+	0.394498i	$0.129081\pi$
$941$	742.445i	0.788996i	−0.918897	+	0.394498i	$0.870919\pi$
$942$	0	0
$943$	−179.657	−0.190517
$944$	0	0
$945$	0	0
$946$	0	0
$947$	417.413i	0.440774i	0.975413	+	0.220387i	$0.0707320\pi$
$947$	417.413i	0.440774i	−0.975413	+	0.220387i	$0.929268\pi$
$948$	0	0
$949$	−934.765	−0.985000
$950$	0	0
$951$	0	0
$952$	0	0
$953$	547.590i	0.574596i	0.957841	+	0.287298i	$0.0927571\pi$
$953$	547.590i	0.574596i	−0.957841	+	0.287298i	$0.907243\pi$
$954$	0	0
$955$	−0.908208	−0.000951003 0
$956$	0	0
$957$	0	0
$958$	0	0
$959$	1640.76i	1.71090i
$960$	0	0
$961$	−911.316	−0.948300
$962$	0	0
$963$	0	0
$964$	0	0
$965$	154.198i	0.159791i
$966$	0	0
$967$	−359.559	−0.371829	−0.185915	−	0.982566i	$-0.559525\pi$
$967$	−359.559	−0.371829	−0.185915	+	0.982566i	$0.559525\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	842.963i	0.868139i	0.900879	+	0.434069i	$0.142923\pi$
$971$	842.963i	0.868139i	−0.900879	+	0.434069i	$0.857077\pi$
$972$	0	0
$973$	−1835.58	−1.88652
$974$	0	0
$975$	0	0
$976$	0	0
$977$	1137.63i	1.16441i	0.813043	+	0.582204i	$0.197809\pi$
$977$	1137.63i	1.16441i	−0.813043	+	0.582204i	$0.802191\pi$
$978$	0	0
$979$	−2164.51	−2.21094
$980$	0	0
$981$	0	0
$982$	0	0
$983$	− 955.512i	− 0.972037i	−0.873949	−	0.486019i	$-0.838449\pi$
$983$	− 955.512i	− 0.972037i	0.873949	−	0.486019i	$-0.161551\pi$
$984$	0	0
$985$	114.313	0.116054
$986$	0	0
$987$	0	0
$988$	0	0
$989$	146.781i	0.148414i
$990$	0	0
$991$	−1469.81	−1.48315	−0.741577	−	0.670868i	$-0.765922\pi$
$991$	−1469.81	−1.48315	−0.741577	+	0.670868i	$0.765922\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	128.402i	0.129047i
$996$	0	0
$997$	1227.20	1.23089	0.615446	−	0.788179i	$-0.288976\pi$
$997$	1227.20	1.23089	0.615446	+	0.788179i	$0.288976\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2916.3.c.b.1457.20		36
3.2	odd	2	inner	2916.3.c.b.1457.17		36
27.5	odd	18		108.3.k.a.29.2	✓	36
27.11	odd	18		324.3.k.a.233.4		36
27.16	even	9		108.3.k.a.41.2	yes	36
27.22	even	9		324.3.k.a.89.4		36
108.43	odd	18		432.3.bc.b.257.5		36
108.59	even	18		432.3.bc.b.353.5		36

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
108.3.k.a.29.2	✓	36	27.5	odd	18
108.3.k.a.41.2	yes	36	27.16	even	9
324.3.k.a.89.4		36	27.22	even	9
324.3.k.a.233.4		36	27.11	odd	18
432.3.bc.b.257.5		36	108.43	odd	18
432.3.bc.b.353.5		36	108.59	even	18
2916.3.c.b.1457.17		36	3.2	odd	2	inner
2916.3.c.b.1457.20		36	1.1	even	1	trivial

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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:	\( N \)	\(=\)	\( 2916 = 2^{2} \cdot 3^{6} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	2916.c (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+0.464464i q^{5} +10.8001 q^{7} +O(q^{10})\)	\(q+0.464464i q^{5} +10.8001 q^{7} -15.9172i q^{11} -17.9569 q^{13} +26.1473i q^{17} -3.55728 q^{19} -4.12593i q^{23} +24.7843 q^{25} -41.6788i q^{29} +7.04870 q^{31} +5.01625i q^{35} -9.85818 q^{37} -43.5435i q^{41} -35.5753 q^{43} -16.1318i q^{47} +67.6417 q^{49} -75.6950i q^{53} +7.39296 q^{55} +28.6908i q^{59} +59.5324 q^{61} -8.34035i q^{65} -23.2357 q^{67} +37.2238i q^{71} +52.0559 q^{73} -171.907i q^{77} -120.715 q^{79} -116.867i q^{83} -12.1445 q^{85} -135.986i q^{89} -193.936 q^{91} -1.65223i q^{95} -94.4120 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 36 q+O(q^{10}) \) 36 * q	\( 36 q - 180 q^{25} + 252 q^{49} + 18 q^{61} - 90 q^{67} + 126 q^{73} - 198 q^{91}+O(q^{100}) \) 36 * q - 180 * q^25 + 252 * q^49 + 18 * q^61 - 90 * q^67 + 126 * q^73 - 198 * q^91

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