LMFDB - Embedded newform 8670.2.a.r.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8670,2,Mod(1,8670)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8670, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("8670.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8670 = 2 \cdot 3 \cdot 5 \cdot 17^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8670.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:	$69.2302985525$
Analytic rank:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 510)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	8670.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

\(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{5} -1.00000 q^{6} +1.00000 q^{8} +1.00000 q^{9} +1.00000 q^{10} -4.00000 q^{11} -1.00000 q^{12} +2.00000 q^{13} -1.00000 q^{15} +1.00000 q^{16} +1.00000 q^{18} -4.00000 q^{19} +1.00000 q^{20} -4.00000 q^{22} -4.00000 q^{23} -1.00000 q^{24} +1.00000 q^{25} +2.00000 q^{26} -1.00000 q^{27} -2.00000 q^{29} -1.00000 q^{30} +4.00000 q^{31} +1.00000 q^{32} +4.00000 q^{33} +1.00000 q^{36} +6.00000 q^{37} -4.00000 q^{38} -2.00000 q^{39} +1.00000 q^{40} +10.0000 q^{41} -8.00000 q^{43} -4.00000 q^{44} +1.00000 q^{45} -4.00000 q^{46} -1.00000 q^{48} -7.00000 q^{49} +1.00000 q^{50} +2.00000 q^{52} +6.00000 q^{53} -1.00000 q^{54} -4.00000 q^{55} +4.00000 q^{57} -2.00000 q^{58} -8.00000 q^{59} -1.00000 q^{60} -10.0000 q^{61} +4.00000 q^{62} +1.00000 q^{64} +2.00000 q^{65} +4.00000 q^{66} -8.00000 q^{67} +4.00000 q^{69} -8.00000 q^{71} +1.00000 q^{72} +2.00000 q^{73} +6.00000 q^{74} -1.00000 q^{75} -4.00000 q^{76} -2.00000 q^{78} -4.00000 q^{79} +1.00000 q^{80} +1.00000 q^{81} +10.0000 q^{82} +4.00000 q^{83} -8.00000 q^{86} +2.00000 q^{87} -4.00000 q^{88} -14.0000 q^{89} +1.00000 q^{90} -4.00000 q^{92} -4.00000 q^{93} -4.00000 q^{95} -1.00000 q^{96} +10.0000 q^{97} -7.00000 q^{98} -4.00000 q^{99} +O(q^{100})\)

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	1.00000	0.707107
$2$	1.00000	0.707107
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	1.00000	0.500000
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	−1.00000	−0.408248
$7$	0	0		−	1.00000i	$-0.5\pi$
$7$	0	0			1.00000i	$0.5\pi$
$8$	1.00000	0.353553
$9$	1.00000	0.333333
$10$	1.00000	0.316228
$11$	−4.00000	−1.20605	−0.603023	−	0.797724i	$-0.706037\pi$
$11$	−4.00000	−1.20605	−0.603023	+	0.797724i	$0.706037\pi$
$12$	−1.00000	−0.288675
$13$	2.00000	0.554700	0.277350	−	0.960769i	$-0.410544\pi$
$13$	2.00000	0.554700	0.277350	+	0.960769i	$0.410544\pi$
$14$	0	0
$15$	−1.00000	−0.258199
$16$	1.00000	0.250000
$17$	0	0
$17$	0	0
$18$	1.00000	0.235702
$19$	−4.00000	−0.917663	−0.458831	−	0.888523i	$-0.651732\pi$
$19$	−4.00000	−0.917663	−0.458831	+	0.888523i	$0.651732\pi$
$20$	1.00000	0.223607
$21$	0	0
$22$	−4.00000	−0.852803
$23$	−4.00000	−0.834058	−0.417029	−	0.908893i	$-0.636929\pi$
$23$	−4.00000	−0.834058	−0.417029	+	0.908893i	$0.636929\pi$
$24$	−1.00000	−0.204124
$25$	1.00000	0.200000
$26$	2.00000	0.392232
$27$	−1.00000	−0.192450
$28$	0	0
$29$	−2.00000	−0.371391	−0.185695	−	0.982607i	$-0.559454\pi$
$29$	−2.00000	−0.371391	−0.185695	+	0.982607i	$0.559454\pi$
$30$	−1.00000	−0.182574
$31$	4.00000	0.718421	0.359211	−	0.933257i	$-0.383046\pi$
$31$	4.00000	0.718421	0.359211	+	0.933257i	$0.383046\pi$
$32$	1.00000	0.176777
$33$	4.00000	0.696311
$34$	0	0
$35$	0	0
$36$	1.00000	0.166667
$37$	6.00000	0.986394	0.493197	−	0.869918i	$-0.335828\pi$
$37$	6.00000	0.986394	0.493197	+	0.869918i	$0.335828\pi$
$38$	−4.00000	−0.648886
$39$	−2.00000	−0.320256
$40$	1.00000	0.158114
$41$	10.0000	1.56174	0.780869	−	0.624695i	$-0.214777\pi$
$41$	10.0000	1.56174	0.780869	+	0.624695i	$0.214777\pi$
$42$	0	0
$43$	−8.00000	−1.21999	−0.609994	−	0.792406i	$-0.708828\pi$
$43$	−8.00000	−1.21999	−0.609994	+	0.792406i	$0.708828\pi$
$44$	−4.00000	−0.603023
$45$	1.00000	0.149071
$46$	−4.00000	−0.589768
$47$	0	0		−	1.00000i	$-0.5\pi$
$47$	0	0			1.00000i	$0.5\pi$
$48$	−1.00000	−0.144338
$49$	−7.00000	−1.00000
$50$	1.00000	0.141421
$51$	0	0
$52$	2.00000	0.277350
$53$	6.00000	0.824163	0.412082	−	0.911147i	$-0.364802\pi$
$53$	6.00000	0.824163	0.412082	+	0.911147i	$0.364802\pi$
$54$	−1.00000	−0.136083
$55$	−4.00000	−0.539360
$56$	0	0
$57$	4.00000	0.529813
$58$	−2.00000	−0.262613
$59$	−8.00000	−1.04151	−0.520756	−	0.853706i	$-0.674350\pi$
$59$	−8.00000	−1.04151	−0.520756	+	0.853706i	$0.674350\pi$
$60$	−1.00000	−0.129099
$61$	−10.0000	−1.28037	−0.640184	−	0.768221i	$-0.721142\pi$
$61$	−10.0000	−1.28037	−0.640184	+	0.768221i	$0.721142\pi$
$62$	4.00000	0.508001
$63$	0	0
$64$	1.00000	0.125000
$65$	2.00000	0.248069
$66$	4.00000	0.492366
$67$	−8.00000	−0.977356	−0.488678	−	0.872464i	$-0.662521\pi$
$67$	−8.00000	−0.977356	−0.488678	+	0.872464i	$0.662521\pi$
$68$	0	0
$69$	4.00000	0.481543
$70$	0	0
$71$	−8.00000	−0.949425	−0.474713	−	0.880141i	$-0.657448\pi$
$71$	−8.00000	−0.949425	−0.474713	+	0.880141i	$0.657448\pi$
$72$	1.00000	0.117851
$73$	2.00000	0.234082	0.117041	−	0.993127i	$-0.462659\pi$
$73$	2.00000	0.234082	0.117041	+	0.993127i	$0.462659\pi$
$74$	6.00000	0.697486
$75$	−1.00000	−0.115470
$76$	−4.00000	−0.458831
$77$	0	0
$78$	−2.00000	−0.226455
$79$	−4.00000	−0.450035	−0.225018	−	0.974355i	$-0.572244\pi$
$79$	−4.00000	−0.450035	−0.225018	+	0.974355i	$0.572244\pi$
$80$	1.00000	0.111803
$81$	1.00000	0.111111
$82$	10.0000	1.10432
$83$	4.00000	0.439057	0.219529	−	0.975606i	$-0.429548\pi$
$83$	4.00000	0.439057	0.219529	+	0.975606i	$0.429548\pi$
$84$	0	0
$85$	0	0
$86$	−8.00000	−0.862662
$87$	2.00000	0.214423
$88$	−4.00000	−0.426401
$89$	−14.0000	−1.48400	−0.741999	−	0.670402i	$-0.766122\pi$
$89$	−14.0000	−1.48400	−0.741999	+	0.670402i	$0.766122\pi$
$90$	1.00000	0.105409
$91$	0	0
$92$	−4.00000	−0.417029
$93$	−4.00000	−0.414781
$94$	0	0
$95$	−4.00000	−0.410391
$96$	−1.00000	−0.102062
$97$	10.0000	1.01535	0.507673	−	0.861550i	$-0.330506\pi$
$97$	10.0000	1.01535	0.507673	+	0.861550i	$0.330506\pi$
$98$	−7.00000	−0.707107
$99$	−4.00000	−0.402015
$100$	1.00000	0.100000
$101$	−6.00000	−0.597022	−0.298511	−	0.954406i	$-0.596490\pi$
$101$	−6.00000	−0.597022	−0.298511	+	0.954406i	$0.596490\pi$
$102$	0	0
$103$	16.0000	1.57653	0.788263	−	0.615338i	$-0.210980\pi$
$103$	16.0000	1.57653	0.788263	+	0.615338i	$0.210980\pi$
$104$	2.00000	0.196116
$105$	0	0
$106$	6.00000	0.582772
$107$	−12.0000	−1.16008	−0.580042	−	0.814587i	$-0.696964\pi$
$107$	−12.0000	−1.16008	−0.580042	+	0.814587i	$0.696964\pi$
$108$	−1.00000	−0.0962250
$109$	14.0000	1.34096	0.670478	−	0.741929i	$-0.266089\pi$
$109$	14.0000	1.34096	0.670478	+	0.741929i	$0.266089\pi$
$110$	−4.00000	−0.381385
$111$	−6.00000	−0.569495
$112$	0	0
$113$	−18.0000	−1.69330	−0.846649	−	0.532152i	$-0.821383\pi$
$113$	−18.0000	−1.69330	−0.846649	+	0.532152i	$0.821383\pi$
$114$	4.00000	0.374634
$115$	−4.00000	−0.373002
$116$	−2.00000	−0.185695
$117$	2.00000	0.184900
$118$	−8.00000	−0.736460
$119$	0	0
$120$	−1.00000	−0.0912871
$121$	5.00000	0.454545
$122$	−10.0000	−0.905357
$123$	−10.0000	−0.901670
$124$	4.00000	0.359211
$125$	1.00000	0.0894427
$126$	0	0
$127$	−16.0000	−1.41977	−0.709885	−	0.704317i	$-0.751253\pi$
$127$	−16.0000	−1.41977	−0.709885	+	0.704317i	$0.751253\pi$
$128$	1.00000	0.0883883
$129$	8.00000	0.704361
$130$	2.00000	0.175412
$131$	−4.00000	−0.349482	−0.174741	−	0.984614i	$-0.555909\pi$
$131$	−4.00000	−0.349482	−0.174741	+	0.984614i	$0.555909\pi$
$132$	4.00000	0.348155
$133$	0	0
$134$	−8.00000	−0.691095
$135$	−1.00000	−0.0860663
$136$	0	0
$137$	−6.00000	−0.512615	−0.256307	−	0.966595i	$-0.582506\pi$
$137$	−6.00000	−0.512615	−0.256307	+	0.966595i	$0.582506\pi$
$138$	4.00000	0.340503
$139$	4.00000	0.339276	0.169638	−	0.985506i	$-0.445740\pi$
$139$	4.00000	0.339276	0.169638	+	0.985506i	$0.445740\pi$
$140$	0	0
$141$	0	0
$142$	−8.00000	−0.671345
$143$	−8.00000	−0.668994
$144$	1.00000	0.0833333
$145$	−2.00000	−0.166091
$146$	2.00000	0.165521
$147$	7.00000	0.577350
$148$	6.00000	0.493197
$149$	2.00000	0.163846	0.0819232	−	0.996639i	$-0.473894\pi$
$149$	2.00000	0.163846	0.0819232	+	0.996639i	$0.473894\pi$
$150$	−1.00000	−0.0816497
$151$	0	0		−	1.00000i	$-0.5\pi$
$151$	0	0			1.00000i	$0.5\pi$
$152$	−4.00000	−0.324443
$153$	0	0
$154$	0	0
$155$	4.00000	0.321288
$156$	−2.00000	−0.160128
$157$	−6.00000	−0.478852	−0.239426	−	0.970915i	$-0.576959\pi$
$157$	−6.00000	−0.478852	−0.239426	+	0.970915i	$0.576959\pi$
$158$	−4.00000	−0.318223
$159$	−6.00000	−0.475831
$160$	1.00000	0.0790569
$161$	0	0
$162$	1.00000	0.0785674
$163$	−12.0000	−0.939913	−0.469956	−	0.882690i	$-0.655730\pi$
$163$	−12.0000	−0.939913	−0.469956	+	0.882690i	$0.655730\pi$
$164$	10.0000	0.780869
$165$	4.00000	0.311400
$166$	4.00000	0.310460
$167$	−12.0000	−0.928588	−0.464294	−	0.885681i	$-0.653692\pi$
$167$	−12.0000	−0.928588	−0.464294	+	0.885681i	$0.653692\pi$
$168$	0	0
$169$	−9.00000	−0.692308
$170$	0	0
$171$	−4.00000	−0.305888
$172$	−8.00000	−0.609994
$173$	−2.00000	−0.152057	−0.0760286	−	0.997106i	$-0.524224\pi$
$173$	−2.00000	−0.152057	−0.0760286	+	0.997106i	$0.524224\pi$
$174$	2.00000	0.151620
$175$	0	0
$176$	−4.00000	−0.301511
$177$	8.00000	0.601317
$178$	−14.0000	−1.04934
$179$	0	0		−	1.00000i	$-0.5\pi$
$179$	0	0			1.00000i	$0.5\pi$
$180$	1.00000	0.0745356
$181$	14.0000	1.04061	0.520306	−	0.853980i	$-0.325818\pi$
$181$	14.0000	1.04061	0.520306	+	0.853980i	$0.325818\pi$
$182$	0	0
$183$	10.0000	0.739221
$184$	−4.00000	−0.294884
$185$	6.00000	0.441129
$186$	−4.00000	−0.293294
$187$	0	0
$188$	0	0
$189$	0	0
$190$	−4.00000	−0.290191
$191$	0	0		−	1.00000i	$-0.5\pi$
$191$	0	0			1.00000i	$0.5\pi$
$192$	−1.00000	−0.0721688
$193$	−6.00000	−0.431889	−0.215945	−	0.976406i	$-0.569283\pi$
$193$	−6.00000	−0.431889	−0.215945	+	0.976406i	$0.569283\pi$
$194$	10.0000	0.717958
$195$	−2.00000	−0.143223
$196$	−7.00000	−0.500000
$197$	−18.0000	−1.28245	−0.641223	−	0.767354i	$-0.721573\pi$
$197$	−18.0000	−1.28245	−0.641223	+	0.767354i	$0.721573\pi$
$198$	−4.00000	−0.284268
$199$	4.00000	0.283552	0.141776	−	0.989899i	$-0.454719\pi$
$199$	4.00000	0.283552	0.141776	+	0.989899i	$0.454719\pi$
$200$	1.00000	0.0707107
$201$	8.00000	0.564276
$202$	−6.00000	−0.422159
$203$	0	0
$204$	0	0
$205$	10.0000	0.698430
$206$	16.0000	1.11477
$207$	−4.00000	−0.278019
$208$	2.00000	0.138675
$209$	16.0000	1.10674
$210$	0	0
$211$	−12.0000	−0.826114	−0.413057	−	0.910705i	$-0.635539\pi$
$211$	−12.0000	−0.826114	−0.413057	+	0.910705i	$0.635539\pi$
$212$	6.00000	0.412082
$213$	8.00000	0.548151
$214$	−12.0000	−0.820303
$215$	−8.00000	−0.545595
$216$	−1.00000	−0.0680414
$217$	0	0
$218$	14.0000	0.948200
$219$	−2.00000	−0.135147
$220$	−4.00000	−0.269680
$221$	0	0
$222$	−6.00000	−0.402694
$223$	24.0000	1.60716	0.803579	−	0.595198i	$-0.202926\pi$
$223$	24.0000	1.60716	0.803579	+	0.595198i	$0.202926\pi$
$224$	0	0
$225$	1.00000	0.0666667
$226$	−18.0000	−1.19734
$227$	−12.0000	−0.796468	−0.398234	−	0.917284i	$-0.630377\pi$
$227$	−12.0000	−0.796468	−0.398234	+	0.917284i	$0.630377\pi$
$228$	4.00000	0.264906
$229$	6.00000	0.396491	0.198246	−	0.980152i	$-0.436476\pi$
$229$	6.00000	0.396491	0.198246	+	0.980152i	$0.436476\pi$
$230$	−4.00000	−0.263752
$231$	0	0
$232$	−2.00000	−0.131306
$233$	30.0000	1.96537	0.982683	−	0.185296i	$-0.0593245\pi$
$233$	30.0000	1.96537	0.982683	+	0.185296i	$0.0593245\pi$
$234$	2.00000	0.130744
$235$	0	0
$236$	−8.00000	−0.520756
$237$	4.00000	0.259828
$238$	0	0
$239$	8.00000	0.517477	0.258738	−	0.965947i	$-0.416693\pi$
$239$	8.00000	0.517477	0.258738	+	0.965947i	$0.416693\pi$
$240$	−1.00000	−0.0645497
$241$	−10.0000	−0.644157	−0.322078	−	0.946713i	$-0.604381\pi$
$241$	−10.0000	−0.644157	−0.322078	+	0.946713i	$0.604381\pi$
$242$	5.00000	0.321412
$243$	−1.00000	−0.0641500
$244$	−10.0000	−0.640184
$245$	−7.00000	−0.447214
$246$	−10.0000	−0.637577
$247$	−8.00000	−0.509028
$248$	4.00000	0.254000
$249$	−4.00000	−0.253490
$250$	1.00000	0.0632456
$251$	0	0		−	1.00000i	$-0.5\pi$
$251$	0	0			1.00000i	$0.5\pi$
$252$	0	0
$253$	16.0000	1.00591
$254$	−16.0000	−1.00393
$255$	0	0
$256$	1.00000	0.0625000
$257$	18.0000	1.12281	0.561405	−	0.827541i	$-0.310261\pi$
$257$	18.0000	1.12281	0.561405	+	0.827541i	$0.310261\pi$
$258$	8.00000	0.498058
$259$	0	0
$260$	2.00000	0.124035
$261$	−2.00000	−0.123797
$262$	−4.00000	−0.247121
$263$	16.0000	0.986602	0.493301	−	0.869859i	$-0.335790\pi$
$263$	16.0000	0.986602	0.493301	+	0.869859i	$0.335790\pi$
$264$	4.00000	0.246183
$265$	6.00000	0.368577
$266$	0	0
$267$	14.0000	0.856786
$268$	−8.00000	−0.488678
$269$	14.0000	0.853595	0.426798	−	0.904347i	$-0.359642\pi$
$269$	14.0000	0.853595	0.426798	+	0.904347i	$0.359642\pi$
$270$	−1.00000	−0.0608581
$271$	−8.00000	−0.485965	−0.242983	−	0.970031i	$-0.578126\pi$
$271$	−8.00000	−0.485965	−0.242983	+	0.970031i	$0.578126\pi$
$272$	0	0
$273$	0	0
$274$	−6.00000	−0.362473
$275$	−4.00000	−0.241209
$276$	4.00000	0.240772
$277$	−10.0000	−0.600842	−0.300421	−	0.953807i	$-0.597127\pi$
$277$	−10.0000	−0.600842	−0.300421	+	0.953807i	$0.597127\pi$
$278$	4.00000	0.239904
$279$	4.00000	0.239474
$280$	0	0
$281$	−30.0000	−1.78965	−0.894825	−	0.446417i	$-0.852700\pi$
$281$	−30.0000	−1.78965	−0.894825	+	0.446417i	$0.852700\pi$
$282$	0	0
$283$	−20.0000	−1.18888	−0.594438	−	0.804141i	$-0.702626\pi$
$283$	−20.0000	−1.18888	−0.594438	+	0.804141i	$0.702626\pi$
$284$	−8.00000	−0.474713
$285$	4.00000	0.236940
$286$	−8.00000	−0.473050
$287$	0	0
$288$	1.00000	0.0589256
$289$	0	0
$290$	−2.00000	−0.117444
$291$	−10.0000	−0.586210
$292$	2.00000	0.117041
$293$	6.00000	0.350524	0.175262	−	0.984522i	$-0.443923\pi$
$293$	6.00000	0.350524	0.175262	+	0.984522i	$0.443923\pi$
$294$	7.00000	0.408248
$295$	−8.00000	−0.465778
$296$	6.00000	0.348743
$297$	4.00000	0.232104
$298$	2.00000	0.115857
$299$	−8.00000	−0.462652
$300$	−1.00000	−0.0577350
$301$	0	0
$302$	0	0
$303$	6.00000	0.344691
$304$	−4.00000	−0.229416
$305$	−10.0000	−0.572598
$306$	0	0
$307$	16.0000	0.913168	0.456584	−	0.889680i	$-0.349073\pi$
$307$	16.0000	0.913168	0.456584	+	0.889680i	$0.349073\pi$
$308$	0	0
$309$	−16.0000	−0.910208
$310$	4.00000	0.227185
$311$	8.00000	0.453638	0.226819	−	0.973937i	$-0.427167\pi$
$311$	8.00000	0.453638	0.226819	+	0.973937i	$0.427167\pi$
$312$	−2.00000	−0.113228
$313$	2.00000	0.113047	0.0565233	−	0.998401i	$-0.481998\pi$
$313$	2.00000	0.113047	0.0565233	+	0.998401i	$0.481998\pi$
$314$	−6.00000	−0.338600
$315$	0	0
$316$	−4.00000	−0.225018
$317$	−26.0000	−1.46031	−0.730153	−	0.683284i	$-0.760551\pi$
$317$	−26.0000	−1.46031	−0.730153	+	0.683284i	$0.760551\pi$
$318$	−6.00000	−0.336463
$319$	8.00000	0.447914
$320$	1.00000	0.0559017
$321$	12.0000	0.669775
$322$	0	0
$323$	0	0
$324$	1.00000	0.0555556
$325$	2.00000	0.110940
$326$	−12.0000	−0.664619
$327$	−14.0000	−0.774202
$328$	10.0000	0.552158
$329$	0	0
$330$	4.00000	0.220193
$331$	20.0000	1.09930	0.549650	−	0.835395i	$-0.314761\pi$
$331$	20.0000	1.09930	0.549650	+	0.835395i	$0.314761\pi$
$332$	4.00000	0.219529
$333$	6.00000	0.328798
$334$	−12.0000	−0.656611
$335$	−8.00000	−0.437087
$336$	0	0
$337$	−6.00000	−0.326841	−0.163420	−	0.986557i	$-0.552253\pi$
$337$	−6.00000	−0.326841	−0.163420	+	0.986557i	$0.552253\pi$
$338$	−9.00000	−0.489535
$339$	18.0000	0.977626
$340$	0	0
$341$	−16.0000	−0.866449
$342$	−4.00000	−0.216295
$343$	0	0
$344$	−8.00000	−0.431331
$345$	4.00000	0.215353
$346$	−2.00000	−0.107521
$347$	−4.00000	−0.214731	−0.107366	−	0.994220i	$-0.534242\pi$
$347$	−4.00000	−0.214731	−0.107366	+	0.994220i	$0.534242\pi$
$348$	2.00000	0.107211
$349$	−34.0000	−1.81998	−0.909989	−	0.414632i	$-0.863910\pi$
$349$	−34.0000	−1.81998	−0.909989	+	0.414632i	$0.863910\pi$
$350$	0	0
$351$	−2.00000	−0.106752
$352$	−4.00000	−0.213201
$353$	−30.0000	−1.59674	−0.798369	−	0.602168i	$-0.794304\pi$
$353$	−30.0000	−1.59674	−0.798369	+	0.602168i	$0.794304\pi$
$354$	8.00000	0.425195
$355$	−8.00000	−0.424596
$356$	−14.0000	−0.741999
$357$	0	0
$358$	0	0
$359$	32.0000	1.68890	0.844448	−	0.535638i	$-0.179929\pi$
$359$	32.0000	1.68890	0.844448	+	0.535638i	$0.179929\pi$
$360$	1.00000	0.0527046
$361$	−3.00000	−0.157895
$362$	14.0000	0.735824
$363$	−5.00000	−0.262432
$364$	0	0
$365$	2.00000	0.104685
$366$	10.0000	0.522708
$367$	−8.00000	−0.417597	−0.208798	−	0.977959i	$-0.566955\pi$
$367$	−8.00000	−0.417597	−0.208798	+	0.977959i	$0.566955\pi$
$368$	−4.00000	−0.208514
$369$	10.0000	0.520579
$370$	6.00000	0.311925
$371$	0	0
$372$	−4.00000	−0.207390
$373$	18.0000	0.932005	0.466002	−	0.884783i	$-0.345694\pi$
$373$	18.0000	0.932005	0.466002	+	0.884783i	$0.345694\pi$
$374$	0	0
$375$	−1.00000	−0.0516398
$376$	0	0
$377$	−4.00000	−0.206010
$378$	0	0
$379$	−20.0000	−1.02733	−0.513665	−	0.857991i	$-0.671713\pi$
$379$	−20.0000	−1.02733	−0.513665	+	0.857991i	$0.671713\pi$
$380$	−4.00000	−0.205196
$381$	16.0000	0.819705
$382$	0	0
$383$	−24.0000	−1.22634	−0.613171	−	0.789950i	$-0.710106\pi$
$383$	−24.0000	−1.22634	−0.613171	+	0.789950i	$0.710106\pi$
$384$	−1.00000	−0.0510310
$385$	0	0
$386$	−6.00000	−0.305392
$387$	−8.00000	−0.406663
$388$	10.0000	0.507673
$389$	−30.0000	−1.52106	−0.760530	−	0.649303i	$-0.775061\pi$
$389$	−30.0000	−1.52106	−0.760530	+	0.649303i	$0.775061\pi$
$390$	−2.00000	−0.101274
$391$	0	0
$392$	−7.00000	−0.353553
$393$	4.00000	0.201773
$394$	−18.0000	−0.906827
$395$	−4.00000	−0.201262
$396$	−4.00000	−0.201008
$397$	−18.0000	−0.903394	−0.451697	−	0.892171i	$-0.649181\pi$
$397$	−18.0000	−0.903394	−0.451697	+	0.892171i	$0.649181\pi$
$398$	4.00000	0.200502
$399$	0	0
$400$	1.00000	0.0500000
$401$	−6.00000	−0.299626	−0.149813	−	0.988714i	$-0.547867\pi$
$401$	−6.00000	−0.299626	−0.149813	+	0.988714i	$0.547867\pi$
$402$	8.00000	0.399004
$403$	8.00000	0.398508
$404$	−6.00000	−0.298511
$405$	1.00000	0.0496904
$406$	0	0
$407$	−24.0000	−1.18964
$408$	0	0
$409$	10.0000	0.494468	0.247234	−	0.968956i	$-0.420478\pi$
$409$	10.0000	0.494468	0.247234	+	0.968956i	$0.420478\pi$
$410$	10.0000	0.493865
$411$	6.00000	0.295958
$412$	16.0000	0.788263
$413$	0	0
$414$	−4.00000	−0.196589
$415$	4.00000	0.196352
$416$	2.00000	0.0980581
$417$	−4.00000	−0.195881
$418$	16.0000	0.782586
$419$	−28.0000	−1.36789	−0.683945	−	0.729534i	$-0.739737\pi$
$419$	−28.0000	−1.36789	−0.683945	+	0.729534i	$0.739737\pi$
$420$	0	0
$421$	30.0000	1.46211	0.731055	−	0.682318i	$-0.239028\pi$
$421$	30.0000	1.46211	0.731055	+	0.682318i	$0.239028\pi$
$422$	−12.0000	−0.584151
$423$	0	0
$424$	6.00000	0.291386
$425$	0	0
$426$	8.00000	0.387601
$427$	0	0
$428$	−12.0000	−0.580042
$429$	8.00000	0.386244
$430$	−8.00000	−0.385794
$431$	16.0000	0.770693	0.385346	−	0.922772i	$-0.374082\pi$
$431$	16.0000	0.770693	0.385346	+	0.922772i	$0.374082\pi$
$432$	−1.00000	−0.0481125
$433$	−6.00000	−0.288342	−0.144171	−	0.989553i	$-0.546051\pi$
$433$	−6.00000	−0.288342	−0.144171	+	0.989553i	$0.546051\pi$
$434$	0	0
$435$	2.00000	0.0958927
$436$	14.0000	0.670478
$437$	16.0000	0.765384
$438$	−2.00000	−0.0955637
$439$	−20.0000	−0.954548	−0.477274	−	0.878755i	$-0.658375\pi$
$439$	−20.0000	−0.954548	−0.477274	+	0.878755i	$0.658375\pi$
$440$	−4.00000	−0.190693
$441$	−7.00000	−0.333333
$442$	0	0
$443$	−28.0000	−1.33032	−0.665160	−	0.746701i	$-0.731637\pi$
$443$	−28.0000	−1.33032	−0.665160	+	0.746701i	$0.731637\pi$
$444$	−6.00000	−0.284747
$445$	−14.0000	−0.663664
$446$	24.0000	1.13643
$447$	−2.00000	−0.0945968
$448$	0	0
$449$	26.0000	1.22702	0.613508	−	0.789689i	$-0.289758\pi$
$449$	26.0000	1.22702	0.613508	+	0.789689i	$0.289758\pi$
$450$	1.00000	0.0471405
$451$	−40.0000	−1.88353
$452$	−18.0000	−0.846649
$453$	0	0
$454$	−12.0000	−0.563188
$455$	0	0
$456$	4.00000	0.187317
$457$	2.00000	0.0935561	0.0467780	−	0.998905i	$-0.485105\pi$
$457$	2.00000	0.0935561	0.0467780	+	0.998905i	$0.485105\pi$
$458$	6.00000	0.280362
$459$	0	0
$460$	−4.00000	−0.186501
$461$	−38.0000	−1.76984	−0.884918	−	0.465746i	$-0.845786\pi$
$461$	−38.0000	−1.76984	−0.884918	+	0.465746i	$0.845786\pi$
$462$	0	0
$463$	−24.0000	−1.11537	−0.557687	−	0.830051i	$-0.688311\pi$
$463$	−24.0000	−1.11537	−0.557687	+	0.830051i	$0.688311\pi$
$464$	−2.00000	−0.0928477
$465$	−4.00000	−0.185496
$466$	30.0000	1.38972
$467$	−12.0000	−0.555294	−0.277647	−	0.960683i	$-0.589555\pi$
$467$	−12.0000	−0.555294	−0.277647	+	0.960683i	$0.589555\pi$
$468$	2.00000	0.0924500
$469$	0	0
$470$	0	0
$471$	6.00000	0.276465
$472$	−8.00000	−0.368230
$473$	32.0000	1.47136
$474$	4.00000	0.183726
$475$	−4.00000	−0.183533
$476$	0	0
$477$	6.00000	0.274721
$478$	8.00000	0.365911
$479$	0	0		−	1.00000i	$-0.5\pi$
$479$	0	0			1.00000i	$0.5\pi$
$480$	−1.00000	−0.0456435
$481$	12.0000	0.547153
$482$	−10.0000	−0.455488
$483$	0	0
$484$	5.00000	0.227273
$485$	10.0000	0.454077
$486$	−1.00000	−0.0453609
$487$	40.0000	1.81257	0.906287	−	0.422664i	$-0.138905\pi$
$487$	40.0000	1.81257	0.906287	+	0.422664i	$0.138905\pi$
$488$	−10.0000	−0.452679
$489$	12.0000	0.542659
$490$	−7.00000	−0.316228
$491$	24.0000	1.08310	0.541552	−	0.840667i	$-0.317837\pi$
$491$	24.0000	1.08310	0.541552	+	0.840667i	$0.317837\pi$
$492$	−10.0000	−0.450835
$493$	0	0
$494$	−8.00000	−0.359937
$495$	−4.00000	−0.179787
$496$	4.00000	0.179605
$497$	0	0
$498$	−4.00000	−0.179244
$499$	−28.0000	−1.25345	−0.626726	−	0.779240i	$-0.715605\pi$
$499$	−28.0000	−1.25345	−0.626726	+	0.779240i	$0.715605\pi$
$500$	1.00000	0.0447214
$501$	12.0000	0.536120
$502$	0	0
$503$	28.0000	1.24846	0.624229	−	0.781241i	$-0.285413\pi$
$503$	28.0000	1.24846	0.624229	+	0.781241i	$0.285413\pi$
$504$	0	0
$505$	−6.00000	−0.266996
$506$	16.0000	0.711287
$507$	9.00000	0.399704
$508$	−16.0000	−0.709885
$509$	−6.00000	−0.265945	−0.132973	−	0.991120i	$-0.542452\pi$
$509$	−6.00000	−0.265945	−0.132973	+	0.991120i	$0.542452\pi$
$510$	0	0
$511$	0	0
$512$	1.00000	0.0441942
$513$	4.00000	0.176604
$514$	18.0000	0.793946
$515$	16.0000	0.705044
$516$	8.00000	0.352180
$517$	0	0
$518$	0	0
$519$	2.00000	0.0877903
$520$	2.00000	0.0877058
$521$	−30.0000	−1.31432	−0.657162	−	0.753749i	$-0.728243\pi$
$521$	−30.0000	−1.31432	−0.657162	+	0.753749i	$0.728243\pi$
$522$	−2.00000	−0.0875376
$523$	−40.0000	−1.74908	−0.874539	−	0.484955i	$-0.838836\pi$
$523$	−40.0000	−1.74908	−0.874539	+	0.484955i	$0.838836\pi$
$524$	−4.00000	−0.174741
$525$	0	0
$526$	16.0000	0.697633
$527$	0	0
$528$	4.00000	0.174078
$529$	−7.00000	−0.304348
$530$	6.00000	0.260623
$531$	−8.00000	−0.347170
$532$	0	0
$533$	20.0000	0.866296
$534$	14.0000	0.605839
$535$	−12.0000	−0.518805
$536$	−8.00000	−0.345547
$537$	0	0
$538$	14.0000	0.603583
$539$	28.0000	1.20605
$540$	−1.00000	−0.0430331
$541$	22.0000	0.945854	0.472927	−	0.881102i	$-0.343197\pi$
$541$	22.0000	0.945854	0.472927	+	0.881102i	$0.343197\pi$
$542$	−8.00000	−0.343629
$543$	−14.0000	−0.600798
$544$	0	0
$545$	14.0000	0.599694
$546$	0	0
$547$	20.0000	0.855138	0.427569	−	0.903983i	$-0.359370\pi$
$547$	20.0000	0.855138	0.427569	+	0.903983i	$0.359370\pi$
$548$	−6.00000	−0.256307
$549$	−10.0000	−0.426790
$550$	−4.00000	−0.170561
$551$	8.00000	0.340811
$552$	4.00000	0.170251
$553$	0	0
$554$	−10.0000	−0.424859
$555$	−6.00000	−0.254686
$556$	4.00000	0.169638
$557$	30.0000	1.27114	0.635570	−	0.772043i	$-0.280765\pi$
$557$	30.0000	1.27114	0.635570	+	0.772043i	$0.280765\pi$
$558$	4.00000	0.169334
$559$	−16.0000	−0.676728
$560$	0	0
$561$	0	0
$562$	−30.0000	−1.26547
$563$	20.0000	0.842900	0.421450	−	0.906852i	$-0.361521\pi$
$563$	20.0000	0.842900	0.421450	+	0.906852i	$0.361521\pi$
$564$	0	0
$565$	−18.0000	−0.757266
$566$	−20.0000	−0.840663
$567$	0	0
$568$	−8.00000	−0.335673
$569$	26.0000	1.08998	0.544988	−	0.838444i	$-0.316534\pi$
$569$	26.0000	1.08998	0.544988	+	0.838444i	$0.316534\pi$
$570$	4.00000	0.167542
$571$	44.0000	1.84134	0.920671	−	0.390339i	$-0.127642\pi$
$571$	44.0000	1.84134	0.920671	+	0.390339i	$0.127642\pi$
$572$	−8.00000	−0.334497
$573$	0	0
$574$	0	0
$575$	−4.00000	−0.166812
$576$	1.00000	0.0416667
$577$	−6.00000	−0.249783	−0.124892	−	0.992170i	$-0.539858\pi$
$577$	−6.00000	−0.249783	−0.124892	+	0.992170i	$0.539858\pi$
$578$	0	0
$579$	6.00000	0.249351
$580$	−2.00000	−0.0830455
$581$	0	0
$582$	−10.0000	−0.414513
$583$	−24.0000	−0.993978
$584$	2.00000	0.0827606
$585$	2.00000	0.0826898
$586$	6.00000	0.247858
$587$	−12.0000	−0.495293	−0.247647	−	0.968850i	$-0.579657\pi$
$587$	−12.0000	−0.495293	−0.247647	+	0.968850i	$0.579657\pi$
$588$	7.00000	0.288675
$589$	−16.0000	−0.659269
$590$	−8.00000	−0.329355
$591$	18.0000	0.740421
$592$	6.00000	0.246598
$593$	−30.0000	−1.23195	−0.615976	−	0.787765i	$-0.711238\pi$
$593$	−30.0000	−1.23195	−0.615976	+	0.787765i	$0.711238\pi$
$594$	4.00000	0.164122
$595$	0	0
$596$	2.00000	0.0819232
$597$	−4.00000	−0.163709
$598$	−8.00000	−0.327144
$599$	0	0		−	1.00000i	$-0.5\pi$
$599$	0	0			1.00000i	$0.5\pi$
$600$	−1.00000	−0.0408248
$601$	−34.0000	−1.38689	−0.693444	−	0.720510i	$-0.743908\pi$
$601$	−34.0000	−1.38689	−0.693444	+	0.720510i	$0.743908\pi$
$602$	0	0
$603$	−8.00000	−0.325785
$604$	0	0
$605$	5.00000	0.203279
$606$	6.00000	0.243733
$607$	32.0000	1.29884	0.649420	−	0.760430i	$-0.275012\pi$
$607$	32.0000	1.29884	0.649420	+	0.760430i	$0.275012\pi$
$608$	−4.00000	−0.162221
$609$	0	0
$610$	−10.0000	−0.404888
$611$	0	0
$612$	0	0
$613$	10.0000	0.403896	0.201948	−	0.979396i	$-0.435273\pi$
$613$	10.0000	0.403896	0.201948	+	0.979396i	$0.435273\pi$
$614$	16.0000	0.645707
$615$	−10.0000	−0.403239
$616$	0	0
$617$	−2.00000	−0.0805170	−0.0402585	−	0.999189i	$-0.512818\pi$
$617$	−2.00000	−0.0805170	−0.0402585	+	0.999189i	$0.512818\pi$
$618$	−16.0000	−0.643614
$619$	−44.0000	−1.76851	−0.884255	−	0.467005i	$-0.845333\pi$
$619$	−44.0000	−1.76851	−0.884255	+	0.467005i	$0.845333\pi$
$620$	4.00000	0.160644
$621$	4.00000	0.160514
$622$	8.00000	0.320771
$623$	0	0
$624$	−2.00000	−0.0800641
$625$	1.00000	0.0400000
$626$	2.00000	0.0799361
$627$	−16.0000	−0.638978
$628$	−6.00000	−0.239426
$629$	0	0
$630$	0	0
$631$	32.0000	1.27390	0.636950	−	0.770905i	$-0.280196\pi$
$631$	32.0000	1.27390	0.636950	+	0.770905i	$0.280196\pi$
$632$	−4.00000	−0.159111
$633$	12.0000	0.476957
$634$	−26.0000	−1.03259
$635$	−16.0000	−0.634941
$636$	−6.00000	−0.237915
$637$	−14.0000	−0.554700
$638$	8.00000	0.316723
$639$	−8.00000	−0.316475
$640$	1.00000	0.0395285
$641$	42.0000	1.65890	0.829450	−	0.558581i	$-0.188654\pi$
$641$	42.0000	1.65890	0.829450	+	0.558581i	$0.188654\pi$
$642$	12.0000	0.473602
$643$	20.0000	0.788723	0.394362	−	0.918955i	$-0.370966\pi$
$643$	20.0000	0.788723	0.394362	+	0.918955i	$0.370966\pi$
$644$	0	0
$645$	8.00000	0.315000
$646$	0	0
$647$	−32.0000	−1.25805	−0.629025	−	0.777385i	$-0.716546\pi$
$647$	−32.0000	−1.25805	−0.629025	+	0.777385i	$0.716546\pi$
$648$	1.00000	0.0392837
$649$	32.0000	1.25611
$650$	2.00000	0.0784465
$651$	0	0
$652$	−12.0000	−0.469956
$653$	−34.0000	−1.33052	−0.665261	−	0.746611i	$-0.731680\pi$
$653$	−34.0000	−1.33052	−0.665261	+	0.746611i	$0.731680\pi$
$654$	−14.0000	−0.547443
$655$	−4.00000	−0.156293
$656$	10.0000	0.390434
$657$	2.00000	0.0780274
$658$	0	0
$659$	24.0000	0.934907	0.467454	−	0.884018i	$-0.345171\pi$
$659$	24.0000	0.934907	0.467454	+	0.884018i	$0.345171\pi$
$660$	4.00000	0.155700
$661$	22.0000	0.855701	0.427850	−	0.903850i	$-0.359271\pi$
$661$	22.0000	0.855701	0.427850	+	0.903850i	$0.359271\pi$
$662$	20.0000	0.777322
$663$	0	0
$664$	4.00000	0.155230
$665$	0	0
$666$	6.00000	0.232495
$667$	8.00000	0.309761
$668$	−12.0000	−0.464294
$669$	−24.0000	−0.927894
$670$	−8.00000	−0.309067
$671$	40.0000	1.54418
$672$	0	0
$673$	10.0000	0.385472	0.192736	−	0.981251i	$-0.438264\pi$
$673$	10.0000	0.385472	0.192736	+	0.981251i	$0.438264\pi$
$674$	−6.00000	−0.231111
$675$	−1.00000	−0.0384900
$676$	−9.00000	−0.346154
$677$	−18.0000	−0.691796	−0.345898	−	0.938272i	$-0.612426\pi$
$677$	−18.0000	−0.691796	−0.345898	+	0.938272i	$0.612426\pi$
$678$	18.0000	0.691286
$679$	0	0
$680$	0	0
$681$	12.0000	0.459841
$682$	−16.0000	−0.612672
$683$	−44.0000	−1.68361	−0.841807	−	0.539779i	$-0.818508\pi$
$683$	−44.0000	−1.68361	−0.841807	+	0.539779i	$0.818508\pi$
$684$	−4.00000	−0.152944
$685$	−6.00000	−0.229248
$686$	0	0
$687$	−6.00000	−0.228914
$688$	−8.00000	−0.304997
$689$	12.0000	0.457164
$690$	4.00000	0.152277
$691$	36.0000	1.36950	0.684752	−	0.728776i	$-0.259910\pi$
$691$	36.0000	1.36950	0.684752	+	0.728776i	$0.259910\pi$
$692$	−2.00000	−0.0760286
$693$	0	0
$694$	−4.00000	−0.151838
$695$	4.00000	0.151729
$696$	2.00000	0.0758098
$697$	0	0
$698$	−34.0000	−1.28692
$699$	−30.0000	−1.13470
$700$	0	0
$701$	10.0000	0.377695	0.188847	−	0.982006i	$-0.439525\pi$
$701$	10.0000	0.377695	0.188847	+	0.982006i	$0.439525\pi$
$702$	−2.00000	−0.0754851
$703$	−24.0000	−0.905177
$704$	−4.00000	−0.150756
$705$	0	0
$706$	−30.0000	−1.12906
$707$	0	0
$708$	8.00000	0.300658
$709$	−10.0000	−0.375558	−0.187779	−	0.982211i	$-0.560129\pi$
$709$	−10.0000	−0.375558	−0.187779	+	0.982211i	$0.560129\pi$
$710$	−8.00000	−0.300235
$711$	−4.00000	−0.150012
$712$	−14.0000	−0.524672
$713$	−16.0000	−0.599205
$714$	0	0
$715$	−8.00000	−0.299183
$716$	0	0
$717$	−8.00000	−0.298765
$718$	32.0000	1.19423
$719$	48.0000	1.79010	0.895049	−	0.445968i	$-0.147140\pi$
$719$	48.0000	1.79010	0.895049	+	0.445968i	$0.147140\pi$
$720$	1.00000	0.0372678
$721$	0	0
$722$	−3.00000	−0.111648
$723$	10.0000	0.371904
$724$	14.0000	0.520306
$725$	−2.00000	−0.0742781
$726$	−5.00000	−0.185567
$727$	40.0000	1.48352	0.741759	−	0.670667i	$-0.233992\pi$
$727$	40.0000	1.48352	0.741759	+	0.670667i	$0.233992\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	2.00000	0.0740233
$731$	0	0
$732$	10.0000	0.369611
$733$	26.0000	0.960332	0.480166	−	0.877178i	$-0.340576\pi$
$733$	26.0000	0.960332	0.480166	+	0.877178i	$0.340576\pi$
$734$	−8.00000	−0.295285
$735$	7.00000	0.258199
$736$	−4.00000	−0.147442
$737$	32.0000	1.17874
$738$	10.0000	0.368105
$739$	−44.0000	−1.61857	−0.809283	−	0.587419i	$-0.800144\pi$
$739$	−44.0000	−1.61857	−0.809283	+	0.587419i	$0.800144\pi$
$740$	6.00000	0.220564
$741$	8.00000	0.293887
$742$	0	0
$743$	−28.0000	−1.02722	−0.513610	−	0.858024i	$-0.671692\pi$
$743$	−28.0000	−1.02722	−0.513610	+	0.858024i	$0.671692\pi$
$744$	−4.00000	−0.146647
$745$	2.00000	0.0732743
$746$	18.0000	0.659027
$747$	4.00000	0.146352
$748$	0	0
$749$	0	0
$750$	−1.00000	−0.0365148
$751$	4.00000	0.145962	0.0729810	−	0.997333i	$-0.476749\pi$
$751$	4.00000	0.145962	0.0729810	+	0.997333i	$0.476749\pi$
$752$	0	0
$753$	0	0
$754$	−4.00000	−0.145671
$755$	0	0
$756$	0	0
$757$	−14.0000	−0.508839	−0.254419	−	0.967094i	$-0.581884\pi$
$757$	−14.0000	−0.508839	−0.254419	+	0.967094i	$0.581884\pi$
$758$	−20.0000	−0.726433
$759$	−16.0000	−0.580763
$760$	−4.00000	−0.145095
$761$	42.0000	1.52250	0.761249	−	0.648459i	$-0.224586\pi$
$761$	42.0000	1.52250	0.761249	+	0.648459i	$0.224586\pi$
$762$	16.0000	0.579619
$763$	0	0
$764$	0	0
$765$	0	0
$766$	−24.0000	−0.867155
$767$	−16.0000	−0.577727
$768$	−1.00000	−0.0360844
$769$	2.00000	0.0721218	0.0360609	−	0.999350i	$-0.488519\pi$
$769$	2.00000	0.0721218	0.0360609	+	0.999350i	$0.488519\pi$
$770$	0	0
$771$	−18.0000	−0.648254
$772$	−6.00000	−0.215945
$773$	14.0000	0.503545	0.251773	−	0.967786i	$-0.418987\pi$
$773$	14.0000	0.503545	0.251773	+	0.967786i	$0.418987\pi$
$774$	−8.00000	−0.287554
$775$	4.00000	0.143684
$776$	10.0000	0.358979
$777$	0	0
$778$	−30.0000	−1.07555
$779$	−40.0000	−1.43315
$780$	−2.00000	−0.0716115
$781$	32.0000	1.14505
$782$	0	0
$783$	2.00000	0.0714742
$784$	−7.00000	−0.250000
$785$	−6.00000	−0.214149
$786$	4.00000	0.142675
$787$	−4.00000	−0.142585	−0.0712923	−	0.997455i	$-0.522712\pi$
$787$	−4.00000	−0.142585	−0.0712923	+	0.997455i	$0.522712\pi$
$788$	−18.0000	−0.641223
$789$	−16.0000	−0.569615
$790$	−4.00000	−0.142314
$791$	0	0
$792$	−4.00000	−0.142134
$793$	−20.0000	−0.710221
$794$	−18.0000	−0.638796
$795$	−6.00000	−0.212798
$796$	4.00000	0.141776
$797$	6.00000	0.212531	0.106265	−	0.994338i	$-0.466111\pi$
$797$	6.00000	0.212531	0.106265	+	0.994338i	$0.466111\pi$
$798$	0	0
$799$	0	0
$800$	1.00000	0.0353553
$801$	−14.0000	−0.494666
$802$	−6.00000	−0.211867
$803$	−8.00000	−0.282314
$804$	8.00000	0.282138
$805$	0	0
$806$	8.00000	0.281788
$807$	−14.0000	−0.492823
$808$	−6.00000	−0.211079
$809$	−54.0000	−1.89854	−0.949269	−	0.314464i	$-0.898175\pi$
$809$	−54.0000	−1.89854	−0.949269	+	0.314464i	$0.898175\pi$
$810$	1.00000	0.0351364
$811$	28.0000	0.983213	0.491606	−	0.870817i	$-0.336410\pi$
$811$	28.0000	0.983213	0.491606	+	0.870817i	$0.336410\pi$
$812$	0	0
$813$	8.00000	0.280572
$814$	−24.0000	−0.841200
$815$	−12.0000	−0.420342
$816$	0	0
$817$	32.0000	1.11954
$818$	10.0000	0.349642
$819$	0	0
$820$	10.0000	0.349215
$821$	6.00000	0.209401	0.104701	−	0.994504i	$-0.466612\pi$
$821$	6.00000	0.209401	0.104701	+	0.994504i	$0.466612\pi$
$822$	6.00000	0.209274
$823$	−16.0000	−0.557725	−0.278862	−	0.960331i	$-0.589957\pi$
$823$	−16.0000	−0.557725	−0.278862	+	0.960331i	$0.589957\pi$
$824$	16.0000	0.557386
$825$	4.00000	0.139262
$826$	0	0
$827$	−36.0000	−1.25184	−0.625921	−	0.779886i	$-0.715277\pi$
$827$	−36.0000	−1.25184	−0.625921	+	0.779886i	$0.715277\pi$
$828$	−4.00000	−0.139010
$829$	54.0000	1.87550	0.937749	−	0.347314i	$-0.112906\pi$
$829$	54.0000	1.87550	0.937749	+	0.347314i	$0.112906\pi$
$830$	4.00000	0.138842
$831$	10.0000	0.346896
$832$	2.00000	0.0693375
$833$	0	0
$834$	−4.00000	−0.138509
$835$	−12.0000	−0.415277
$836$	16.0000	0.553372
$837$	−4.00000	−0.138260
$838$	−28.0000	−0.967244
$839$	−16.0000	−0.552381	−0.276191	−	0.961103i	$-0.589072\pi$
$839$	−16.0000	−0.552381	−0.276191	+	0.961103i	$0.589072\pi$
$840$	0	0
$841$	−25.0000	−0.862069
$842$	30.0000	1.03387
$843$	30.0000	1.03325
$844$	−12.0000	−0.413057
$845$	−9.00000	−0.309609
$846$	0	0
$847$	0	0
$848$	6.00000	0.206041
$849$	20.0000	0.686398
$850$	0	0
$851$	−24.0000	−0.822709
$852$	8.00000	0.274075
$853$	54.0000	1.84892	0.924462	−	0.381273i	$-0.124514\pi$
$853$	54.0000	1.84892	0.924462	+	0.381273i	$0.124514\pi$
$854$	0	0
$855$	−4.00000	−0.136797
$856$	−12.0000	−0.410152
$857$	−26.0000	−0.888143	−0.444072	−	0.895991i	$-0.646466\pi$
$857$	−26.0000	−0.888143	−0.444072	+	0.895991i	$0.646466\pi$
$858$	8.00000	0.273115
$859$	20.0000	0.682391	0.341196	−	0.939992i	$-0.389168\pi$
$859$	20.0000	0.682391	0.341196	+	0.939992i	$0.389168\pi$
$860$	−8.00000	−0.272798
$861$	0	0
$862$	16.0000	0.544962
$863$	−8.00000	−0.272323	−0.136162	−	0.990687i	$-0.543477\pi$
$863$	−8.00000	−0.272323	−0.136162	+	0.990687i	$0.543477\pi$
$864$	−1.00000	−0.0340207
$865$	−2.00000	−0.0680020
$866$	−6.00000	−0.203888
$867$	0	0
$868$	0	0
$869$	16.0000	0.542763
$870$	2.00000	0.0678064
$871$	−16.0000	−0.542139
$872$	14.0000	0.474100
$873$	10.0000	0.338449
$874$	16.0000	0.541208
$875$	0	0
$876$	−2.00000	−0.0675737
$877$	−50.0000	−1.68838	−0.844190	−	0.536044i	$-0.819918\pi$
$877$	−50.0000	−1.68838	−0.844190	+	0.536044i	$0.819918\pi$
$878$	−20.0000	−0.674967
$879$	−6.00000	−0.202375
$880$	−4.00000	−0.134840
$881$	2.00000	0.0673817	0.0336909	−	0.999432i	$-0.489274\pi$
$881$	2.00000	0.0673817	0.0336909	+	0.999432i	$0.489274\pi$
$882$	−7.00000	−0.235702
$883$	0	0		−	1.00000i	$-0.5\pi$
$883$	0	0			1.00000i	$0.5\pi$
$884$	0	0
$885$	8.00000	0.268917
$886$	−28.0000	−0.940678
$887$	−12.0000	−0.402921	−0.201460	−	0.979497i	$-0.564569\pi$
$887$	−12.0000	−0.402921	−0.201460	+	0.979497i	$0.564569\pi$
$888$	−6.00000	−0.201347
$889$	0	0
$890$	−14.0000	−0.469281
$891$	−4.00000	−0.134005
$892$	24.0000	0.803579
$893$	0	0
$894$	−2.00000	−0.0668900
$895$	0	0
$896$	0	0
$897$	8.00000	0.267112
$898$	26.0000	0.867631
$899$	−8.00000	−0.266815
$900$	1.00000	0.0333333
$901$	0	0
$902$	−40.0000	−1.33185
$903$	0	0
$904$	−18.0000	−0.598671
$905$	14.0000	0.465376
$906$	0	0
$907$	36.0000	1.19536	0.597680	−	0.801735i	$-0.296089\pi$
$907$	36.0000	1.19536	0.597680	+	0.801735i	$0.296089\pi$
$908$	−12.0000	−0.398234
$909$	−6.00000	−0.199007
$910$	0	0
$911$	−48.0000	−1.59031	−0.795155	−	0.606406i	$-0.792611\pi$
$911$	−48.0000	−1.59031	−0.795155	+	0.606406i	$0.792611\pi$
$912$	4.00000	0.132453
$913$	−16.0000	−0.529523
$914$	2.00000	0.0661541
$915$	10.0000	0.330590
$916$	6.00000	0.198246
$917$	0	0
$918$	0	0
$919$	16.0000	0.527791	0.263896	−	0.964551i	$-0.414993\pi$
$919$	16.0000	0.527791	0.263896	+	0.964551i	$0.414993\pi$
$920$	−4.00000	−0.131876
$921$	−16.0000	−0.527218
$922$	−38.0000	−1.25146
$923$	−16.0000	−0.526646
$924$	0	0
$925$	6.00000	0.197279
$926$	−24.0000	−0.788689
$927$	16.0000	0.525509
$928$	−2.00000	−0.0656532
$929$	50.0000	1.64045	0.820223	−	0.572043i	$-0.193849\pi$
$929$	50.0000	1.64045	0.820223	+	0.572043i	$0.193849\pi$
$930$	−4.00000	−0.131165
$931$	28.0000	0.917663
$932$	30.0000	0.982683
$933$	−8.00000	−0.261908
$934$	−12.0000	−0.392652
$935$	0	0
$936$	2.00000	0.0653720
$937$	2.00000	0.0653372	0.0326686	−	0.999466i	$-0.489599\pi$
$937$	2.00000	0.0653372	0.0326686	+	0.999466i	$0.489599\pi$
$938$	0	0
$939$	−2.00000	−0.0652675
$940$	0	0
$941$	30.0000	0.977972	0.488986	−	0.872292i	$-0.337367\pi$
$941$	30.0000	0.977972	0.488986	+	0.872292i	$0.337367\pi$
$942$	6.00000	0.195491
$943$	−40.0000	−1.30258
$944$	−8.00000	−0.260378
$945$	0	0
$946$	32.0000	1.04041
$947$	28.0000	0.909878	0.454939	−	0.890523i	$-0.349661\pi$
$947$	28.0000	0.909878	0.454939	+	0.890523i	$0.349661\pi$
$948$	4.00000	0.129914
$949$	4.00000	0.129845
$950$	−4.00000	−0.129777
$951$	26.0000	0.843108
$952$	0	0
$953$	−6.00000	−0.194359	−0.0971795	−	0.995267i	$-0.530982\pi$
$953$	−6.00000	−0.194359	−0.0971795	+	0.995267i	$0.530982\pi$
$954$	6.00000	0.194257
$955$	0	0
$956$	8.00000	0.258738
$957$	−8.00000	−0.258603
$958$	0	0
$959$	0	0
$960$	−1.00000	−0.0322749
$961$	−15.0000	−0.483871
$962$	12.0000	0.386896
$963$	−12.0000	−0.386695
$964$	−10.0000	−0.322078
$965$	−6.00000	−0.193147
$966$	0	0
$967$	24.0000	0.771788	0.385894	−	0.922543i	$-0.373893\pi$
$967$	24.0000	0.771788	0.385894	+	0.922543i	$0.373893\pi$
$968$	5.00000	0.160706
$969$	0	0
$970$	10.0000	0.321081
$971$	32.0000	1.02693	0.513464	−	0.858111i	$-0.328362\pi$
$971$	32.0000	1.02693	0.513464	+	0.858111i	$0.328362\pi$
$972$	−1.00000	−0.0320750
$973$	0	0
$974$	40.0000	1.28168
$975$	−2.00000	−0.0640513
$976$	−10.0000	−0.320092
$977$	18.0000	0.575871	0.287936	−	0.957650i	$-0.407031\pi$
$977$	18.0000	0.575871	0.287936	+	0.957650i	$0.407031\pi$
$978$	12.0000	0.383718
$979$	56.0000	1.78977
$980$	−7.00000	−0.223607
$981$	14.0000	0.446986
$982$	24.0000	0.765871
$983$	44.0000	1.40338	0.701691	−	0.712481i	$-0.252429\pi$
$983$	44.0000	1.40338	0.701691	+	0.712481i	$0.252429\pi$
$984$	−10.0000	−0.318788
$985$	−18.0000	−0.573528
$986$	0	0
$987$	0	0
$988$	−8.00000	−0.254514
$989$	32.0000	1.01754
$990$	−4.00000	−0.127128
$991$	44.0000	1.39771	0.698853	−	0.715265i	$-0.253694\pi$
$991$	44.0000	1.39771	0.698853	+	0.715265i	$0.253694\pi$
$992$	4.00000	0.127000
$993$	−20.0000	−0.634681
$994$	0	0
$995$	4.00000	0.126809
$996$	−4.00000	−0.126745
$997$	−10.0000	−0.316703	−0.158352	−	0.987383i	$-0.550618\pi$
$997$	−10.0000	−0.316703	−0.158352	+	0.987383i	$0.550618\pi$
$998$	−28.0000	−0.886325
$999$	−6.00000	−0.189832

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8670.2.a.r.1.1		1
17.16	even	2		510.2.a.f.1.1	✓	1
51.50	odd	2		1530.2.a.f.1.1		1
68.67	odd	2		4080.2.a.c.1.1		1
85.33	odd	4		2550.2.d.s.2449.1		2
85.67	odd	4		2550.2.d.s.2449.2		2
85.84	even	2		2550.2.a.d.1.1		1
255.254	odd	2		7650.2.a.bw.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
510.2.a.f.1.1	✓	1	17.16	even	2
1530.2.a.f.1.1		1	51.50	odd	2
2550.2.a.d.1.1		1	85.84	even	2
2550.2.d.s.2449.1		2	85.33	odd	4
2550.2.d.s.2449.2		2	85.67	odd	4
4080.2.a.c.1.1		1	68.67	odd	2
7650.2.a.bw.1.1		1	255.254	odd	2
8670.2.a.r.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:	\( N \)	\(=\)	\( 8670 = 2 \cdot 3 \cdot 5 \cdot 17^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8670.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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