LMFDB - Embedded newform 8190.2.a.bg.1.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [8190,2,Mod(1,8190)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("8190.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(8190, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 8190 = 2 \cdot 3^{2} \cdot 5 \cdot 7 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8190.a (trivial)

Newform invariants

comment:select newform

sage:traces = [1,1,0,1,-1,0,1,1,0,-1,2,0,-1,1,0,1,4,0,-4,-1,0,2,-8] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(23)] == traces)

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

Self dual:	yes
Analytic conductor:	$65.3974792554$
Analytic rank:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	8190.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$

$=$

\(q+1.00000 q^{2} +1.00000 q^{4} -1.00000 q^{5} +1.00000 q^{7} +1.00000 q^{8} -1.00000 q^{10} +2.00000 q^{11} -1.00000 q^{13} +1.00000 q^{14} +1.00000 q^{16} +4.00000 q^{17} -4.00000 q^{19} -1.00000 q^{20} +2.00000 q^{22} -8.00000 q^{23} +1.00000 q^{25} -1.00000 q^{26} +1.00000 q^{28} -4.00000 q^{29} -8.00000 q^{31} +1.00000 q^{32} +4.00000 q^{34} -1.00000 q^{35} -8.00000 q^{37} -4.00000 q^{38} -1.00000 q^{40} -6.00000 q^{41} +6.00000 q^{43} +2.00000 q^{44} -8.00000 q^{46} -2.00000 q^{47} +1.00000 q^{49} +1.00000 q^{50} -1.00000 q^{52} -2.00000 q^{53} -2.00000 q^{55} +1.00000 q^{56} -4.00000 q^{58} +10.0000 q^{61} -8.00000 q^{62} +1.00000 q^{64} +1.00000 q^{65} -16.0000 q^{67} +4.00000 q^{68} -1.00000 q^{70} -6.00000 q^{71} -2.00000 q^{73} -8.00000 q^{74} -4.00000 q^{76} +2.00000 q^{77} +4.00000 q^{79} -1.00000 q^{80} -6.00000 q^{82} +2.00000 q^{83} -4.00000 q^{85} +6.00000 q^{86} +2.00000 q^{88} +10.0000 q^{89} -1.00000 q^{91} -8.00000 q^{92} -2.00000 q^{94} +4.00000 q^{95} +2.00000 q^{97} +1.00000 q^{98} +O(q^{100})\)

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
$3$	0	0
$3$	0	0
$4$	1.00000	0.500000
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	1.00000	0.353553
$9$	0	0
$10$	−1.00000	−0.316228
$11$	2.00000	0.603023	0.301511	−	0.953463i	$-0.402509\pi$
$11$	2.00000	0.603023	0.301511	+	0.953463i	$0.402509\pi$
$12$	0	0
$13$	−1.00000	−0.277350
$13$	−1.00000	−0.277350
$14$	1.00000	0.267261
$15$	0	0
$16$	1.00000	0.250000
$17$	4.00000	0.970143	0.485071	−	0.874475i	$-0.338794\pi$
$17$	4.00000	0.970143	0.485071	+	0.874475i	$0.338794\pi$
$18$	0	0
$19$	−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
$22$	2.00000	0.426401
$23$	−8.00000	−1.66812	−0.834058	−	0.551677i	$-0.813988\pi$
$23$	−8.00000	−1.66812	−0.834058	+	0.551677i	$0.813988\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	−1.00000	−0.196116
$27$	0	0
$28$	1.00000	0.188982
$29$	−4.00000	−0.742781	−0.371391	−	0.928477i	$-0.621119\pi$
$29$	−4.00000	−0.742781	−0.371391	+	0.928477i	$0.621119\pi$
$30$	0	0
$31$	−8.00000	−1.43684	−0.718421	−	0.695608i	$-0.755135\pi$
$31$	−8.00000	−1.43684	−0.718421	+	0.695608i	$0.755135\pi$
$32$	1.00000	0.176777
$33$	0	0
$34$	4.00000	0.685994
$35$	−1.00000	−0.169031
$36$	0	0
$37$	−8.00000	−1.31519	−0.657596	−	0.753371i	$-0.728427\pi$
$37$	−8.00000	−1.31519	−0.657596	+	0.753371i	$0.728427\pi$
$38$	−4.00000	−0.648886
$39$	0	0
$40$	−1.00000	−0.158114
$41$	−6.00000	−0.937043	−0.468521	−	0.883452i	$-0.655213\pi$
$41$	−6.00000	−0.937043	−0.468521	+	0.883452i	$0.655213\pi$
$42$	0	0
$43$	6.00000	0.914991	0.457496	−	0.889212i	$-0.348747\pi$
$43$	6.00000	0.914991	0.457496	+	0.889212i	$0.348747\pi$
$44$	2.00000	0.301511
$45$	0	0
$46$	−8.00000	−1.17954
$47$	−2.00000	−0.291730	−0.145865	−	0.989305i	$-0.546597\pi$
$47$	−2.00000	−0.291730	−0.145865	+	0.989305i	$0.546597\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	1.00000	0.141421
$51$	0	0
$52$	−1.00000	−0.138675
$53$	−2.00000	−0.274721	−0.137361	−	0.990521i	$-0.543862\pi$
$53$	−2.00000	−0.274721	−0.137361	+	0.990521i	$0.543862\pi$
$54$	0	0
$55$	−2.00000	−0.269680
$56$	1.00000	0.133631
$57$	0	0
$58$	−4.00000	−0.525226
$59$	0	0		−	1.00000i	$-0.5\pi$
$59$	0	0			1.00000i	$0.5\pi$
$60$	0	0
$61$	10.0000	1.28037	0.640184	−	0.768221i	$-0.278858\pi$
$61$	10.0000	1.28037	0.640184	+	0.768221i	$0.278858\pi$
$62$	−8.00000	−1.01600
$63$	0	0
$64$	1.00000	0.125000
$65$	1.00000	0.124035
$66$	0	0
$67$	−16.0000	−1.95471	−0.977356	−	0.211604i	$-0.932131\pi$
$67$	−16.0000	−1.95471	−0.977356	+	0.211604i	$0.932131\pi$
$68$	4.00000	0.485071
$69$	0	0
$70$	−1.00000	−0.119523
$71$	−6.00000	−0.712069	−0.356034	−	0.934473i	$-0.615871\pi$
$71$	−6.00000	−0.712069	−0.356034	+	0.934473i	$0.615871\pi$
$72$	0	0
$73$	−2.00000	−0.234082	−0.117041	−	0.993127i	$-0.537341\pi$
$73$	−2.00000	−0.234082	−0.117041	+	0.993127i	$0.537341\pi$
$74$	−8.00000	−0.929981
$75$	0	0
$76$	−4.00000	−0.458831
$77$	2.00000	0.227921
$78$	0	0
$79$	4.00000	0.450035	0.225018	−	0.974355i	$-0.427756\pi$
$79$	4.00000	0.450035	0.225018	+	0.974355i	$0.427756\pi$
$80$	−1.00000	−0.111803
$81$	0	0
$82$	−6.00000	−0.662589
$83$	2.00000	0.219529	0.109764	−	0.993958i	$-0.464990\pi$
$83$	2.00000	0.219529	0.109764	+	0.993958i	$0.464990\pi$
$84$	0	0
$85$	−4.00000	−0.433861
$86$	6.00000	0.646997
$87$	0	0
$88$	2.00000	0.213201
$89$	10.0000	1.06000	0.529999	−	0.847998i	$-0.322192\pi$
$89$	10.0000	1.06000	0.529999	+	0.847998i	$0.322192\pi$
$90$	0	0
$91$	−1.00000	−0.104828
$92$	−8.00000	−0.834058
$93$	0	0
$94$	−2.00000	−0.206284
$95$	4.00000	0.410391
$96$	0	0
$97$	2.00000	0.203069	0.101535	−	0.994832i	$-0.467625\pi$
$97$	2.00000	0.203069	0.101535	+	0.994832i	$0.467625\pi$
$98$	1.00000	0.101015
$99$	0	0
$100$	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
$103$	12.0000	1.18240	0.591198	−	0.806527i	$-0.298655\pi$
$103$	12.0000	1.18240	0.591198	+	0.806527i	$0.298655\pi$
$104$	−1.00000	−0.0980581
$105$	0	0
$106$	−2.00000	−0.194257
$107$	−4.00000	−0.386695	−0.193347	−	0.981130i	$-0.561934\pi$
$107$	−4.00000	−0.386695	−0.193347	+	0.981130i	$0.561934\pi$
$108$	0	0
$109$	−18.0000	−1.72409	−0.862044	−	0.506834i	$-0.830816\pi$
$109$	−18.0000	−1.72409	−0.862044	+	0.506834i	$0.830816\pi$
$110$	−2.00000	−0.190693
$111$	0	0
$112$	1.00000	0.0944911
$113$	14.0000	1.31701	0.658505	−	0.752577i	$-0.271189\pi$
$113$	14.0000	1.31701	0.658505	+	0.752577i	$0.271189\pi$
$114$	0	0
$115$	8.00000	0.746004
$116$	−4.00000	−0.371391
$117$	0	0
$118$	0	0
$119$	4.00000	0.366679
$120$	0	0
$121$	−7.00000	−0.636364
$122$	10.0000	0.905357
$123$	0	0
$124$	−8.00000	−0.718421
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	−18.0000	−1.59724	−0.798621	−	0.601834i	$-0.794437\pi$
$127$	−18.0000	−1.59724	−0.798621	+	0.601834i	$0.794437\pi$
$128$	1.00000	0.0883883
$129$	0	0
$130$	1.00000	0.0877058
$131$	4.00000	0.349482	0.174741	−	0.984614i	$-0.444091\pi$
$131$	4.00000	0.349482	0.174741	+	0.984614i	$0.444091\pi$
$132$	0	0
$133$	−4.00000	−0.346844
$134$	−16.0000	−1.38219
$135$	0	0
$136$	4.00000	0.342997
$137$	6.00000	0.512615	0.256307	−	0.966595i	$-0.417494\pi$
$137$	6.00000	0.512615	0.256307	+	0.966595i	$0.417494\pi$
$138$	0	0
$139$	10.0000	0.848189	0.424094	−	0.905618i	$-0.360592\pi$
$139$	10.0000	0.848189	0.424094	+	0.905618i	$0.360592\pi$
$140$	−1.00000	−0.0845154
$141$	0	0
$142$	−6.00000	−0.503509
$143$	−2.00000	−0.167248
$144$	0	0
$145$	4.00000	0.332182
$146$	−2.00000	−0.165521
$147$	0	0
$148$	−8.00000	−0.657596
$149$	14.0000	1.14692	0.573462	−	0.819232i	$-0.305600\pi$
$149$	14.0000	1.14692	0.573462	+	0.819232i	$0.305600\pi$
$150$	0	0
$151$	−24.0000	−1.95309	−0.976546	−	0.215308i	$-0.930924\pi$
$151$	−24.0000	−1.95309	−0.976546	+	0.215308i	$0.930924\pi$
$152$	−4.00000	−0.324443
$153$	0	0
$154$	2.00000	0.161165
$155$	8.00000	0.642575
$156$	0	0
$157$	−2.00000	−0.159617	−0.0798087	−	0.996810i	$-0.525431\pi$
$157$	−2.00000	−0.159617	−0.0798087	+	0.996810i	$0.525431\pi$
$158$	4.00000	0.318223
$159$	0	0
$160$	−1.00000	−0.0790569
$161$	−8.00000	−0.630488
$162$	0	0
$163$	4.00000	0.313304	0.156652	−	0.987654i	$-0.449930\pi$
$163$	4.00000	0.313304	0.156652	+	0.987654i	$0.449930\pi$
$164$	−6.00000	−0.468521
$165$	0	0
$166$	2.00000	0.155230
$167$	−2.00000	−0.154765	−0.0773823	−	0.997001i	$-0.524656\pi$
$167$	−2.00000	−0.154765	−0.0773823	+	0.997001i	$0.524656\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	−4.00000	−0.306786
$171$	0	0
$172$	6.00000	0.457496
$173$	24.0000	1.82469	0.912343	−	0.409426i	$-0.134271\pi$
$173$	24.0000	1.82469	0.912343	+	0.409426i	$0.134271\pi$
$174$	0	0
$175$	1.00000	0.0755929
$176$	2.00000	0.150756
$177$	0	0
$178$	10.0000	0.749532
$179$	−4.00000	−0.298974	−0.149487	−	0.988764i	$-0.547762\pi$
$179$	−4.00000	−0.298974	−0.149487	+	0.988764i	$0.547762\pi$
$180$	0	0
$181$	−2.00000	−0.148659	−0.0743294	−	0.997234i	$-0.523682\pi$
$181$	−2.00000	−0.148659	−0.0743294	+	0.997234i	$0.523682\pi$
$182$	−1.00000	−0.0741249
$183$	0	0
$184$	−8.00000	−0.589768
$185$	8.00000	0.588172
$186$	0	0
$187$	8.00000	0.585018
$188$	−2.00000	−0.145865
$189$	0	0
$190$	4.00000	0.290191
$191$	16.0000	1.15772	0.578860	−	0.815427i	$-0.303498\pi$
$191$	16.0000	1.15772	0.578860	+	0.815427i	$0.303498\pi$
$192$	0	0
$193$	4.00000	0.287926	0.143963	−	0.989583i	$-0.454015\pi$
$193$	4.00000	0.287926	0.143963	+	0.989583i	$0.454015\pi$
$194$	2.00000	0.143592
$195$	0	0
$196$	1.00000	0.0714286
$197$	−18.0000	−1.28245	−0.641223	−	0.767354i	$-0.721573\pi$
$197$	−18.0000	−1.28245	−0.641223	+	0.767354i	$0.721573\pi$
$198$	0	0
$199$	−14.0000	−0.992434	−0.496217	−	0.868199i	$-0.665278\pi$
$199$	−14.0000	−0.992434	−0.496217	+	0.868199i	$0.665278\pi$
$200$	1.00000	0.0707107
$201$	0	0
$202$	−6.00000	−0.422159
$203$	−4.00000	−0.280745
$204$	0	0
$205$	6.00000	0.419058
$206$	12.0000	0.836080
$207$	0	0
$208$	−1.00000	−0.0693375
$209$	−8.00000	−0.553372
$210$	0	0
$211$	−20.0000	−1.37686	−0.688428	−	0.725304i	$-0.741699\pi$
$211$	−20.0000	−1.37686	−0.688428	+	0.725304i	$0.741699\pi$
$212$	−2.00000	−0.137361
$213$	0	0
$214$	−4.00000	−0.273434
$215$	−6.00000	−0.409197
$216$	0	0
$217$	−8.00000	−0.543075
$218$	−18.0000	−1.21911
$219$	0	0
$220$	−2.00000	−0.134840
$221$	−4.00000	−0.269069
$222$	0	0
$223$	−24.0000	−1.60716	−0.803579	−	0.595198i	$-0.797074\pi$
$223$	−24.0000	−1.60716	−0.803579	+	0.595198i	$0.797074\pi$
$224$	1.00000	0.0668153
$225$	0	0
$226$	14.0000	0.931266
$227$	−18.0000	−1.19470	−0.597351	−	0.801980i	$-0.703780\pi$
$227$	−18.0000	−1.19470	−0.597351	+	0.801980i	$0.703780\pi$
$228$	0	0
$229$	−20.0000	−1.32164	−0.660819	−	0.750546i	$-0.729791\pi$
$229$	−20.0000	−1.32164	−0.660819	+	0.750546i	$0.729791\pi$
$230$	8.00000	0.527504
$231$	0	0
$232$	−4.00000	−0.262613
$233$	6.00000	0.393073	0.196537	−	0.980497i	$-0.437031\pi$
$233$	6.00000	0.393073	0.196537	+	0.980497i	$0.437031\pi$
$234$	0	0
$235$	2.00000	0.130466
$236$	0	0
$237$	0	0
$238$	4.00000	0.259281
$239$	6.00000	0.388108	0.194054	−	0.980991i	$-0.437836\pi$
$239$	6.00000	0.388108	0.194054	+	0.980991i	$0.437836\pi$
$240$	0	0
$241$	0	0		−	1.00000i	$-0.5\pi$
$241$	0	0			1.00000i	$0.5\pi$
$242$	−7.00000	−0.449977
$243$	0	0
$244$	10.0000	0.640184
$245$	−1.00000	−0.0638877
$246$	0	0
$247$	4.00000	0.254514
$248$	−8.00000	−0.508001
$249$	0	0
$250$	−1.00000	−0.0632456
$251$	4.00000	0.252478	0.126239	−	0.992000i	$-0.459709\pi$
$251$	4.00000	0.252478	0.126239	+	0.992000i	$0.459709\pi$
$252$	0	0
$253$	−16.0000	−1.00591
$254$	−18.0000	−1.12942
$255$	0	0
$256$	1.00000	0.0625000
$257$	−16.0000	−0.998053	−0.499026	−	0.866587i	$-0.666309\pi$
$257$	−16.0000	−0.998053	−0.499026	+	0.866587i	$0.666309\pi$
$258$	0	0
$259$	−8.00000	−0.497096
$260$	1.00000	0.0620174
$261$	0	0
$262$	4.00000	0.247121
$263$	0	0		−	1.00000i	$-0.5\pi$
$263$	0	0			1.00000i	$0.5\pi$
$264$	0	0
$265$	2.00000	0.122859
$266$	−4.00000	−0.245256
$267$	0	0
$268$	−16.0000	−0.977356
$269$	−10.0000	−0.609711	−0.304855	−	0.952399i	$-0.598608\pi$
$269$	−10.0000	−0.609711	−0.304855	+	0.952399i	$0.598608\pi$
$270$	0	0
$271$	8.00000	0.485965	0.242983	−	0.970031i	$-0.421874\pi$
$271$	8.00000	0.485965	0.242983	+	0.970031i	$0.421874\pi$
$272$	4.00000	0.242536
$273$	0	0
$274$	6.00000	0.362473
$275$	2.00000	0.120605
$276$	0	0
$277$	2.00000	0.120168	0.0600842	−	0.998193i	$-0.480863\pi$
$277$	2.00000	0.120168	0.0600842	+	0.998193i	$0.480863\pi$
$278$	10.0000	0.599760
$279$	0	0
$280$	−1.00000	−0.0597614
$281$	22.0000	1.31241	0.656205	−	0.754583i	$-0.272161\pi$
$281$	22.0000	1.31241	0.656205	+	0.754583i	$0.272161\pi$
$282$	0	0
$283$	0	0		−	1.00000i	$-0.5\pi$
$283$	0	0			1.00000i	$0.5\pi$
$284$	−6.00000	−0.356034
$285$	0	0
$286$	−2.00000	−0.118262
$287$	−6.00000	−0.354169
$288$	0	0
$289$	−1.00000	−0.0588235
$290$	4.00000	0.234888
$291$	0	0
$292$	−2.00000	−0.117041
$293$	−18.0000	−1.05157	−0.525786	−	0.850617i	$-0.676229\pi$
$293$	−18.0000	−1.05157	−0.525786	+	0.850617i	$0.676229\pi$
$294$	0	0
$295$	0	0
$296$	−8.00000	−0.464991
$297$	0	0
$298$	14.0000	0.810998
$299$	8.00000	0.462652
$300$	0	0
$301$	6.00000	0.345834
$302$	−24.0000	−1.38104
$303$	0	0
$304$	−4.00000	−0.229416
$305$	−10.0000	−0.572598
$306$	0	0
$307$	20.0000	1.14146	0.570730	−	0.821138i	$-0.306660\pi$
$307$	20.0000	1.14146	0.570730	+	0.821138i	$0.306660\pi$
$308$	2.00000	0.113961
$309$	0	0
$310$	8.00000	0.454369
$311$	−8.00000	−0.453638	−0.226819	−	0.973937i	$-0.572833\pi$
$311$	−8.00000	−0.453638	−0.226819	+	0.973937i	$0.572833\pi$
$312$	0	0
$313$	6.00000	0.339140	0.169570	−	0.985518i	$-0.445762\pi$
$313$	6.00000	0.339140	0.169570	+	0.985518i	$0.445762\pi$
$314$	−2.00000	−0.112867
$315$	0	0
$316$	4.00000	0.225018
$317$	−14.0000	−0.786318	−0.393159	−	0.919470i	$-0.628618\pi$
$317$	−14.0000	−0.786318	−0.393159	+	0.919470i	$0.628618\pi$
$318$	0	0
$319$	−8.00000	−0.447914
$320$	−1.00000	−0.0559017
$321$	0	0
$322$	−8.00000	−0.445823
$323$	−16.0000	−0.890264
$324$	0	0
$325$	−1.00000	−0.0554700
$326$	4.00000	0.221540
$327$	0	0
$328$	−6.00000	−0.331295
$329$	−2.00000	−0.110264
$330$	0	0
$331$	−12.0000	−0.659580	−0.329790	−	0.944054i	$-0.606978\pi$
$331$	−12.0000	−0.659580	−0.329790	+	0.944054i	$0.606978\pi$
$332$	2.00000	0.109764
$333$	0	0
$334$	−2.00000	−0.109435
$335$	16.0000	0.874173
$336$	0	0
$337$	−18.0000	−0.980522	−0.490261	−	0.871576i	$-0.663099\pi$
$337$	−18.0000	−0.980522	−0.490261	+	0.871576i	$0.663099\pi$
$338$	1.00000	0.0543928
$339$	0	0
$340$	−4.00000	−0.216930
$341$	−16.0000	−0.866449
$342$	0	0
$343$	1.00000	0.0539949
$344$	6.00000	0.323498
$345$	0	0
$346$	24.0000	1.29025
$347$	−4.00000	−0.214731	−0.107366	−	0.994220i	$-0.534242\pi$
$347$	−4.00000	−0.214731	−0.107366	+	0.994220i	$0.534242\pi$
$348$	0	0
$349$	8.00000	0.428230	0.214115	−	0.976808i	$-0.431313\pi$
$349$	8.00000	0.428230	0.214115	+	0.976808i	$0.431313\pi$
$350$	1.00000	0.0534522
$351$	0	0
$352$	2.00000	0.106600
$353$	−18.0000	−0.958043	−0.479022	−	0.877803i	$-0.659008\pi$
$353$	−18.0000	−0.958043	−0.479022	+	0.877803i	$0.659008\pi$
$354$	0	0
$355$	6.00000	0.318447
$356$	10.0000	0.529999
$357$	0	0
$358$	−4.00000	−0.211407
$359$	10.0000	0.527780	0.263890	−	0.964553i	$-0.414994\pi$
$359$	10.0000	0.527780	0.263890	+	0.964553i	$0.414994\pi$
$360$	0	0
$361$	−3.00000	−0.157895
$362$	−2.00000	−0.105118
$363$	0	0
$364$	−1.00000	−0.0524142
$365$	2.00000	0.104685
$366$	0	0
$367$	8.00000	0.417597	0.208798	−	0.977959i	$-0.433045\pi$
$367$	8.00000	0.417597	0.208798	+	0.977959i	$0.433045\pi$
$368$	−8.00000	−0.417029
$369$	0	0
$370$	8.00000	0.415900
$371$	−2.00000	−0.103835
$372$	0	0
$373$	−26.0000	−1.34623	−0.673114	−	0.739538i	$-0.735044\pi$
$373$	−26.0000	−1.34623	−0.673114	+	0.739538i	$0.735044\pi$
$374$	8.00000	0.413670
$375$	0	0
$376$	−2.00000	−0.103142
$377$	4.00000	0.206010
$378$	0	0
$379$	−4.00000	−0.205466	−0.102733	−	0.994709i	$-0.532759\pi$
$379$	−4.00000	−0.205466	−0.102733	+	0.994709i	$0.532759\pi$
$380$	4.00000	0.205196
$381$	0	0
$382$	16.0000	0.818631
$383$	−6.00000	−0.306586	−0.153293	−	0.988181i	$-0.548988\pi$
$383$	−6.00000	−0.306586	−0.153293	+	0.988181i	$0.548988\pi$
$384$	0	0
$385$	−2.00000	−0.101929
$386$	4.00000	0.203595
$387$	0	0
$388$	2.00000	0.101535
$389$	4.00000	0.202808	0.101404	−	0.994845i	$-0.467667\pi$
$389$	4.00000	0.202808	0.101404	+	0.994845i	$0.467667\pi$
$390$	0	0
$391$	−32.0000	−1.61831
$392$	1.00000	0.0505076
$393$	0	0
$394$	−18.0000	−0.906827
$395$	−4.00000	−0.201262
$396$	0	0
$397$	18.0000	0.903394	0.451697	−	0.892171i	$-0.350819\pi$
$397$	18.0000	0.903394	0.451697	+	0.892171i	$0.350819\pi$
$398$	−14.0000	−0.701757
$399$	0	0
$400$	1.00000	0.0500000
$401$	10.0000	0.499376	0.249688	−	0.968326i	$-0.419672\pi$
$401$	10.0000	0.499376	0.249688	+	0.968326i	$0.419672\pi$
$402$	0	0
$403$	8.00000	0.398508
$404$	−6.00000	−0.298511
$405$	0	0
$406$	−4.00000	−0.198517
$407$	−16.0000	−0.793091
$408$	0	0
$409$	12.0000	0.593362	0.296681	−	0.954977i	$-0.404120\pi$
$409$	12.0000	0.593362	0.296681	+	0.954977i	$0.404120\pi$
$410$	6.00000	0.296319
$411$	0	0
$412$	12.0000	0.591198
$413$	0	0
$414$	0	0
$415$	−2.00000	−0.0981761
$416$	−1.00000	−0.0490290
$417$	0	0
$418$	−8.00000	−0.391293
$419$	36.0000	1.75872	0.879358	−	0.476162i	$-0.157972\pi$
$419$	36.0000	1.75872	0.879358	+	0.476162i	$0.157972\pi$
$420$	0	0
$421$	−2.00000	−0.0974740	−0.0487370	−	0.998812i	$-0.515520\pi$
$421$	−2.00000	−0.0974740	−0.0487370	+	0.998812i	$0.515520\pi$
$422$	−20.0000	−0.973585
$423$	0	0
$424$	−2.00000	−0.0971286
$425$	4.00000	0.194029
$426$	0	0
$427$	10.0000	0.483934
$428$	−4.00000	−0.193347
$429$	0	0
$430$	−6.00000	−0.289346
$431$	18.0000	0.867029	0.433515	−	0.901146i	$-0.357273\pi$
$431$	18.0000	0.867029	0.433515	+	0.901146i	$0.357273\pi$
$432$	0	0
$433$	2.00000	0.0961139	0.0480569	−	0.998845i	$-0.484697\pi$
$433$	2.00000	0.0961139	0.0480569	+	0.998845i	$0.484697\pi$
$434$	−8.00000	−0.384012
$435$	0	0
$436$	−18.0000	−0.862044
$437$	32.0000	1.53077
$438$	0	0
$439$	−14.0000	−0.668184	−0.334092	−	0.942541i	$-0.608430\pi$
$439$	−14.0000	−0.668184	−0.334092	+	0.942541i	$0.608430\pi$
$440$	−2.00000	−0.0953463
$441$	0	0
$442$	−4.00000	−0.190261
$443$	−36.0000	−1.71041	−0.855206	−	0.518289i	$-0.826569\pi$
$443$	−36.0000	−1.71041	−0.855206	+	0.518289i	$0.826569\pi$
$444$	0	0
$445$	−10.0000	−0.474045
$446$	−24.0000	−1.13643
$447$	0	0
$448$	1.00000	0.0472456
$449$	−6.00000	−0.283158	−0.141579	−	0.989927i	$-0.545218\pi$
$449$	−6.00000	−0.283158	−0.141579	+	0.989927i	$0.545218\pi$
$450$	0	0
$451$	−12.0000	−0.565058
$452$	14.0000	0.658505
$453$	0	0
$454$	−18.0000	−0.844782
$455$	1.00000	0.0468807
$456$	0	0
$457$	−28.0000	−1.30978	−0.654892	−	0.755722i	$-0.727286\pi$
$457$	−28.0000	−1.30978	−0.654892	+	0.755722i	$0.727286\pi$
$458$	−20.0000	−0.934539
$459$	0	0
$460$	8.00000	0.373002
$461$	18.0000	0.838344	0.419172	−	0.907907i	$-0.362320\pi$
$461$	18.0000	0.838344	0.419172	+	0.907907i	$0.362320\pi$
$462$	0	0
$463$	36.0000	1.67306	0.836531	−	0.547920i	$-0.184580\pi$
$463$	36.0000	1.67306	0.836531	+	0.547920i	$0.184580\pi$
$464$	−4.00000	−0.185695
$465$	0	0
$466$	6.00000	0.277945
$467$	0	0		−	1.00000i	$-0.5\pi$
$467$	0	0			1.00000i	$0.5\pi$
$468$	0	0
$469$	−16.0000	−0.738811
$470$	2.00000	0.0922531
$471$	0	0
$472$	0	0
$473$	12.0000	0.551761
$474$	0	0
$475$	−4.00000	−0.183533
$476$	4.00000	0.183340
$477$	0	0
$478$	6.00000	0.274434
$479$	4.00000	0.182765	0.0913823	−	0.995816i	$-0.470871\pi$
$479$	4.00000	0.182765	0.0913823	+	0.995816i	$0.470871\pi$
$480$	0	0
$481$	8.00000	0.364769
$482$	0	0
$483$	0	0
$484$	−7.00000	−0.318182
$485$	−2.00000	−0.0908153
$486$	0	0
$487$	−32.0000	−1.45006	−0.725029	−	0.688718i	$-0.758174\pi$
$487$	−32.0000	−1.45006	−0.725029	+	0.688718i	$0.758174\pi$
$488$	10.0000	0.452679
$489$	0	0
$490$	−1.00000	−0.0451754
$491$	−16.0000	−0.722070	−0.361035	−	0.932552i	$-0.617576\pi$
$491$	−16.0000	−0.722070	−0.361035	+	0.932552i	$0.617576\pi$
$492$	0	0
$493$	−16.0000	−0.720604
$494$	4.00000	0.179969
$495$	0	0
$496$	−8.00000	−0.359211
$497$	−6.00000	−0.269137
$498$	0	0
$499$	28.0000	1.25345	0.626726	−	0.779240i	$-0.284395\pi$
$499$	28.0000	1.25345	0.626726	+	0.779240i	$0.284395\pi$
$500$	−1.00000	−0.0447214
$501$	0	0
$502$	4.00000	0.178529
$503$	40.0000	1.78351	0.891756	−	0.452517i	$-0.149474\pi$
$503$	40.0000	1.78351	0.891756	+	0.452517i	$0.149474\pi$
$504$	0	0
$505$	6.00000	0.266996
$506$	−16.0000	−0.711287
$507$	0	0
$508$	−18.0000	−0.798621
$509$	6.00000	0.265945	0.132973	−	0.991120i	$-0.457548\pi$
$509$	6.00000	0.265945	0.132973	+	0.991120i	$0.457548\pi$
$510$	0	0
$511$	−2.00000	−0.0884748
$512$	1.00000	0.0441942
$513$	0	0
$514$	−16.0000	−0.705730
$515$	−12.0000	−0.528783
$516$	0	0
$517$	−4.00000	−0.175920
$518$	−8.00000	−0.351500
$519$	0	0
$520$	1.00000	0.0438529
$521$	26.0000	1.13908	0.569540	−	0.821963i	$-0.307121\pi$
$521$	26.0000	1.13908	0.569540	+	0.821963i	$0.307121\pi$
$522$	0	0
$523$	20.0000	0.874539	0.437269	−	0.899331i	$-0.355946\pi$
$523$	20.0000	0.874539	0.437269	+	0.899331i	$0.355946\pi$
$524$	4.00000	0.174741
$525$	0	0
$526$	0	0
$527$	−32.0000	−1.39394
$528$	0	0
$529$	41.0000	1.78261
$530$	2.00000	0.0868744
$531$	0	0
$532$	−4.00000	−0.173422
$533$	6.00000	0.259889
$534$	0	0
$535$	4.00000	0.172935
$536$	−16.0000	−0.691095
$537$	0	0
$538$	−10.0000	−0.431131
$539$	2.00000	0.0861461
$540$	0	0
$541$	−30.0000	−1.28980	−0.644900	−	0.764267i	$-0.723101\pi$
$541$	−30.0000	−1.28980	−0.644900	+	0.764267i	$0.723101\pi$
$542$	8.00000	0.343629
$543$	0	0
$544$	4.00000	0.171499
$545$	18.0000	0.771035
$546$	0	0
$547$	22.0000	0.940652	0.470326	−	0.882493i	$-0.344136\pi$
$547$	22.0000	0.940652	0.470326	+	0.882493i	$0.344136\pi$
$548$	6.00000	0.256307
$549$	0	0
$550$	2.00000	0.0852803
$551$	16.0000	0.681623
$552$	0	0
$553$	4.00000	0.170097
$554$	2.00000	0.0849719
$555$	0	0
$556$	10.0000	0.424094
$557$	42.0000	1.77960	0.889799	−	0.456354i	$-0.150845\pi$
$557$	42.0000	1.77960	0.889799	+	0.456354i	$0.150845\pi$
$558$	0	0
$559$	−6.00000	−0.253773
$560$	−1.00000	−0.0422577
$561$	0	0
$562$	22.0000	0.928014
$563$	4.00000	0.168580	0.0842900	−	0.996441i	$-0.473138\pi$
$563$	4.00000	0.168580	0.0842900	+	0.996441i	$0.473138\pi$
$564$	0	0
$565$	−14.0000	−0.588984
$566$	0	0
$567$	0	0
$568$	−6.00000	−0.251754
$569$	24.0000	1.00613	0.503066	−	0.864248i	$-0.332205\pi$
$569$	24.0000	1.00613	0.503066	+	0.864248i	$0.332205\pi$
$570$	0	0
$571$	−8.00000	−0.334790	−0.167395	−	0.985890i	$-0.553535\pi$
$571$	−8.00000	−0.334790	−0.167395	+	0.985890i	$0.553535\pi$
$572$	−2.00000	−0.0836242
$573$	0	0
$574$	−6.00000	−0.250435
$575$	−8.00000	−0.333623
$576$	0	0
$577$	−22.0000	−0.915872	−0.457936	−	0.888985i	$-0.651411\pi$
$577$	−22.0000	−0.915872	−0.457936	+	0.888985i	$0.651411\pi$
$578$	−1.00000	−0.0415945
$579$	0	0
$580$	4.00000	0.166091
$581$	2.00000	0.0829740
$582$	0	0
$583$	−4.00000	−0.165663
$584$	−2.00000	−0.0827606
$585$	0	0
$586$	−18.0000	−0.743573
$587$	42.0000	1.73353	0.866763	−	0.498721i	$-0.166197\pi$
$587$	42.0000	1.73353	0.866763	+	0.498721i	$0.166197\pi$
$588$	0	0
$589$	32.0000	1.31854
$590$	0	0
$591$	0	0
$592$	−8.00000	−0.328798
$593$	−14.0000	−0.574911	−0.287456	−	0.957794i	$-0.592809\pi$
$593$	−14.0000	−0.574911	−0.287456	+	0.957794i	$0.592809\pi$
$594$	0	0
$595$	−4.00000	−0.163984
$596$	14.0000	0.573462
$597$	0	0
$598$	8.00000	0.327144
$599$	−24.0000	−0.980613	−0.490307	−	0.871550i	$-0.663115\pi$
$599$	−24.0000	−0.980613	−0.490307	+	0.871550i	$0.663115\pi$
$600$	0	0
$601$	−10.0000	−0.407909	−0.203954	−	0.978980i	$-0.565379\pi$
$601$	−10.0000	−0.407909	−0.203954	+	0.978980i	$0.565379\pi$
$602$	6.00000	0.244542
$603$	0	0
$604$	−24.0000	−0.976546
$605$	7.00000	0.284590
$606$	0	0
$607$	−8.00000	−0.324710	−0.162355	−	0.986732i	$-0.551909\pi$
$607$	−8.00000	−0.324710	−0.162355	+	0.986732i	$0.551909\pi$
$608$	−4.00000	−0.162221
$609$	0	0
$610$	−10.0000	−0.404888
$611$	2.00000	0.0809113
$612$	0	0
$613$	−20.0000	−0.807792	−0.403896	−	0.914805i	$-0.632344\pi$
$613$	−20.0000	−0.807792	−0.403896	+	0.914805i	$0.632344\pi$
$614$	20.0000	0.807134
$615$	0	0
$616$	2.00000	0.0805823
$617$	18.0000	0.724653	0.362326	−	0.932051i	$-0.381983\pi$
$617$	18.0000	0.724653	0.362326	+	0.932051i	$0.381983\pi$
$618$	0	0
$619$	−24.0000	−0.964641	−0.482321	−	0.875995i	$-0.660206\pi$
$619$	−24.0000	−0.964641	−0.482321	+	0.875995i	$0.660206\pi$
$620$	8.00000	0.321288
$621$	0	0
$622$	−8.00000	−0.320771
$623$	10.0000	0.400642
$624$	0	0
$625$	1.00000	0.0400000
$626$	6.00000	0.239808
$627$	0	0
$628$	−2.00000	−0.0798087
$629$	−32.0000	−1.27592
$630$	0	0
$631$	40.0000	1.59237	0.796187	−	0.605050i	$-0.206847\pi$
$631$	40.0000	1.59237	0.796187	+	0.605050i	$0.206847\pi$
$632$	4.00000	0.159111
$633$	0	0
$634$	−14.0000	−0.556011
$635$	18.0000	0.714308
$636$	0	0
$637$	−1.00000	−0.0396214
$638$	−8.00000	−0.316723
$639$	0	0
$640$	−1.00000	−0.0395285
$641$	0	0		−	1.00000i	$-0.5\pi$
$641$	0	0			1.00000i	$0.5\pi$
$642$	0	0
$643$	12.0000	0.473234	0.236617	−	0.971603i	$-0.423961\pi$
$643$	12.0000	0.473234	0.236617	+	0.971603i	$0.423961\pi$
$644$	−8.00000	−0.315244
$645$	0	0
$646$	−16.0000	−0.629512
$647$	−12.0000	−0.471769	−0.235884	−	0.971781i	$-0.575799\pi$
$647$	−12.0000	−0.471769	−0.235884	+	0.971781i	$0.575799\pi$
$648$	0	0
$649$	0	0
$650$	−1.00000	−0.0392232
$651$	0	0
$652$	4.00000	0.156652
$653$	−6.00000	−0.234798	−0.117399	−	0.993085i	$-0.537456\pi$
$653$	−6.00000	−0.234798	−0.117399	+	0.993085i	$0.537456\pi$
$654$	0	0
$655$	−4.00000	−0.156293
$656$	−6.00000	−0.234261
$657$	0	0
$658$	−2.00000	−0.0779681
$659$	−8.00000	−0.311636	−0.155818	−	0.987786i	$-0.549801\pi$
$659$	−8.00000	−0.311636	−0.155818	+	0.987786i	$0.549801\pi$
$660$	0	0
$661$	−32.0000	−1.24466	−0.622328	−	0.782757i	$-0.713813\pi$
$661$	−32.0000	−1.24466	−0.622328	+	0.782757i	$0.713813\pi$
$662$	−12.0000	−0.466393
$663$	0	0
$664$	2.00000	0.0776151
$665$	4.00000	0.155113
$666$	0	0
$667$	32.0000	1.23904
$668$	−2.00000	−0.0773823
$669$	0	0
$670$	16.0000	0.618134
$671$	20.0000	0.772091
$672$	0	0
$673$	−6.00000	−0.231283	−0.115642	−	0.993291i	$-0.536892\pi$
$673$	−6.00000	−0.231283	−0.115642	+	0.993291i	$0.536892\pi$
$674$	−18.0000	−0.693334
$675$	0	0
$676$	1.00000	0.0384615
$677$	24.0000	0.922395	0.461197	−	0.887298i	$-0.347420\pi$
$677$	24.0000	0.922395	0.461197	+	0.887298i	$0.347420\pi$
$678$	0	0
$679$	2.00000	0.0767530
$680$	−4.00000	−0.153393
$681$	0	0
$682$	−16.0000	−0.612672
$683$	−24.0000	−0.918334	−0.459167	−	0.888350i	$-0.651852\pi$
$683$	−24.0000	−0.918334	−0.459167	+	0.888350i	$0.651852\pi$
$684$	0	0
$685$	−6.00000	−0.229248
$686$	1.00000	0.0381802
$687$	0	0
$688$	6.00000	0.228748
$689$	2.00000	0.0761939
$690$	0	0
$691$	24.0000	0.913003	0.456502	−	0.889723i	$-0.349102\pi$
$691$	24.0000	0.913003	0.456502	+	0.889723i	$0.349102\pi$
$692$	24.0000	0.912343
$693$	0	0
$694$	−4.00000	−0.151838
$695$	−10.0000	−0.379322
$696$	0	0
$697$	−24.0000	−0.909065
$698$	8.00000	0.302804
$699$	0	0
$700$	1.00000	0.0377964
$701$	−48.0000	−1.81293	−0.906467	−	0.422276i	$-0.861231\pi$
$701$	−48.0000	−1.81293	−0.906467	+	0.422276i	$0.861231\pi$
$702$	0	0
$703$	32.0000	1.20690
$704$	2.00000	0.0753778
$705$	0	0
$706$	−18.0000	−0.677439
$707$	−6.00000	−0.225653
$708$	0	0
$709$	50.0000	1.87779	0.938895	−	0.344204i	$-0.111851\pi$
$709$	50.0000	1.87779	0.938895	+	0.344204i	$0.111851\pi$
$710$	6.00000	0.225176
$711$	0	0
$712$	10.0000	0.374766
$713$	64.0000	2.39682
$714$	0	0
$715$	2.00000	0.0747958
$716$	−4.00000	−0.149487
$717$	0	0
$718$	10.0000	0.373197
$719$	0	0		−	1.00000i	$-0.5\pi$
$719$	0	0			1.00000i	$0.5\pi$
$720$	0	0
$721$	12.0000	0.446903
$722$	−3.00000	−0.111648
$723$	0	0
$724$	−2.00000	−0.0743294
$725$	−4.00000	−0.148556
$726$	0	0
$727$	−28.0000	−1.03846	−0.519231	−	0.854634i	$-0.673782\pi$
$727$	−28.0000	−1.03846	−0.519231	+	0.854634i	$0.673782\pi$
$728$	−1.00000	−0.0370625
$729$	0	0
$730$	2.00000	0.0740233
$731$	24.0000	0.887672
$732$	0	0
$733$	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
$735$	0	0
$736$	−8.00000	−0.294884
$737$	−32.0000	−1.17874
$738$	0	0
$739$	−28.0000	−1.03000	−0.514998	−	0.857191i	$-0.672207\pi$
$739$	−28.0000	−1.03000	−0.514998	+	0.857191i	$0.672207\pi$
$740$	8.00000	0.294086
$741$	0	0
$742$	−2.00000	−0.0734223
$743$	−36.0000	−1.32071	−0.660356	−	0.750953i	$-0.729595\pi$
$743$	−36.0000	−1.32071	−0.660356	+	0.750953i	$0.729595\pi$
$744$	0	0
$745$	−14.0000	−0.512920
$746$	−26.0000	−0.951928
$747$	0	0
$748$	8.00000	0.292509
$749$	−4.00000	−0.146157
$750$	0	0
$751$	−8.00000	−0.291924	−0.145962	−	0.989290i	$-0.546628\pi$
$751$	−8.00000	−0.291924	−0.145962	+	0.989290i	$0.546628\pi$
$752$	−2.00000	−0.0729325
$753$	0	0
$754$	4.00000	0.145671
$755$	24.0000	0.873449
$756$	0	0
$757$	−18.0000	−0.654221	−0.327111	−	0.944986i	$-0.606075\pi$
$757$	−18.0000	−0.654221	−0.327111	+	0.944986i	$0.606075\pi$
$758$	−4.00000	−0.145287
$759$	0	0
$760$	4.00000	0.145095
$761$	−30.0000	−1.08750	−0.543750	−	0.839248i	$-0.682996\pi$
$761$	−30.0000	−1.08750	−0.543750	+	0.839248i	$0.682996\pi$
$762$	0	0
$763$	−18.0000	−0.651644
$764$	16.0000	0.578860
$765$	0	0
$766$	−6.00000	−0.216789
$767$	0	0
$768$	0	0
$769$	−4.00000	−0.144244	−0.0721218	−	0.997396i	$-0.522977\pi$
$769$	−4.00000	−0.144244	−0.0721218	+	0.997396i	$0.522977\pi$
$770$	−2.00000	−0.0720750
$771$	0	0
$772$	4.00000	0.143963
$773$	−34.0000	−1.22290	−0.611448	−	0.791285i	$-0.709412\pi$
$773$	−34.0000	−1.22290	−0.611448	+	0.791285i	$0.709412\pi$
$774$	0	0
$775$	−8.00000	−0.287368
$776$	2.00000	0.0717958
$777$	0	0
$778$	4.00000	0.143407
$779$	24.0000	0.859889
$780$	0	0
$781$	−12.0000	−0.429394
$782$	−32.0000	−1.14432
$783$	0	0
$784$	1.00000	0.0357143
$785$	2.00000	0.0713831
$786$	0	0
$787$	44.0000	1.56843	0.784215	−	0.620489i	$-0.213066\pi$
$787$	44.0000	1.56843	0.784215	+	0.620489i	$0.213066\pi$
$788$	−18.0000	−0.641223
$789$	0	0
$790$	−4.00000	−0.142314
$791$	14.0000	0.497783
$792$	0	0
$793$	−10.0000	−0.355110
$794$	18.0000	0.638796
$795$	0	0
$796$	−14.0000	−0.496217
$797$	−36.0000	−1.27519	−0.637593	−	0.770374i	$-0.720070\pi$
$797$	−36.0000	−1.27519	−0.637593	+	0.770374i	$0.720070\pi$
$798$	0	0
$799$	−8.00000	−0.283020
$800$	1.00000	0.0353553
$801$	0	0
$802$	10.0000	0.353112
$803$	−4.00000	−0.141157
$804$	0	0
$805$	8.00000	0.281963
$806$	8.00000	0.281788
$807$	0	0
$808$	−6.00000	−0.211079
$809$	48.0000	1.68759	0.843795	−	0.536666i	$-0.180316\pi$
$809$	48.0000	1.68759	0.843795	+	0.536666i	$0.180316\pi$
$810$	0	0
$811$	16.0000	0.561836	0.280918	−	0.959732i	$-0.409361\pi$
$811$	16.0000	0.561836	0.280918	+	0.959732i	$0.409361\pi$
$812$	−4.00000	−0.140372
$813$	0	0
$814$	−16.0000	−0.560800
$815$	−4.00000	−0.140114
$816$	0	0
$817$	−24.0000	−0.839654
$818$	12.0000	0.419570
$819$	0	0
$820$	6.00000	0.209529
$821$	30.0000	1.04701	0.523504	−	0.852023i	$-0.324625\pi$
$821$	30.0000	1.04701	0.523504	+	0.852023i	$0.324625\pi$
$822$	0	0
$823$	30.0000	1.04573	0.522867	−	0.852414i	$-0.324862\pi$
$823$	30.0000	1.04573	0.522867	+	0.852414i	$0.324862\pi$
$824$	12.0000	0.418040
$825$	0	0
$826$	0	0
$827$	12.0000	0.417281	0.208640	−	0.977992i	$-0.433096\pi$
$827$	12.0000	0.417281	0.208640	+	0.977992i	$0.433096\pi$
$828$	0	0
$829$	18.0000	0.625166	0.312583	−	0.949890i	$-0.398806\pi$
$829$	18.0000	0.625166	0.312583	+	0.949890i	$0.398806\pi$
$830$	−2.00000	−0.0694210
$831$	0	0
$832$	−1.00000	−0.0346688
$833$	4.00000	0.138592
$834$	0	0
$835$	2.00000	0.0692129
$836$	−8.00000	−0.276686
$837$	0	0
$838$	36.0000	1.24360
$839$	−8.00000	−0.276191	−0.138095	−	0.990419i	$-0.544098\pi$
$839$	−8.00000	−0.276191	−0.138095	+	0.990419i	$0.544098\pi$
$840$	0	0
$841$	−13.0000	−0.448276
$842$	−2.00000	−0.0689246
$843$	0	0
$844$	−20.0000	−0.688428
$845$	−1.00000	−0.0344010
$846$	0	0
$847$	−7.00000	−0.240523
$848$	−2.00000	−0.0686803
$849$	0	0
$850$	4.00000	0.137199
$851$	64.0000	2.19389
$852$	0	0
$853$	2.00000	0.0684787	0.0342393	−	0.999414i	$-0.489099\pi$
$853$	2.00000	0.0684787	0.0342393	+	0.999414i	$0.489099\pi$
$854$	10.0000	0.342193
$855$	0	0
$856$	−4.00000	−0.136717
$857$	4.00000	0.136637	0.0683187	−	0.997664i	$-0.478237\pi$
$857$	4.00000	0.136637	0.0683187	+	0.997664i	$0.478237\pi$
$858$	0	0
$859$	−10.0000	−0.341196	−0.170598	−	0.985341i	$-0.554570\pi$
$859$	−10.0000	−0.341196	−0.170598	+	0.985341i	$0.554570\pi$
$860$	−6.00000	−0.204598
$861$	0	0
$862$	18.0000	0.613082
$863$	24.0000	0.816970	0.408485	−	0.912765i	$-0.366057\pi$
$863$	24.0000	0.816970	0.408485	+	0.912765i	$0.366057\pi$
$864$	0	0
$865$	−24.0000	−0.816024
$866$	2.00000	0.0679628
$867$	0	0
$868$	−8.00000	−0.271538
$869$	8.00000	0.271381
$870$	0	0
$871$	16.0000	0.542139
$872$	−18.0000	−0.609557
$873$	0	0
$874$	32.0000	1.08242
$875$	−1.00000	−0.0338062
$876$	0	0
$877$	−32.0000	−1.08056	−0.540282	−	0.841484i	$-0.681682\pi$
$877$	−32.0000	−1.08056	−0.540282	+	0.841484i	$0.681682\pi$
$878$	−14.0000	−0.472477
$879$	0	0
$880$	−2.00000	−0.0674200
$881$	54.0000	1.81931	0.909653	−	0.415369i	$-0.136347\pi$
$881$	54.0000	1.81931	0.909653	+	0.415369i	$0.136347\pi$
$882$	0	0
$883$	42.0000	1.41341	0.706706	−	0.707507i	$-0.250180\pi$
$883$	42.0000	1.41341	0.706706	+	0.707507i	$0.250180\pi$
$884$	−4.00000	−0.134535
$885$	0	0
$886$	−36.0000	−1.20944
$887$	−12.0000	−0.402921	−0.201460	−	0.979497i	$-0.564569\pi$
$887$	−12.0000	−0.402921	−0.201460	+	0.979497i	$0.564569\pi$
$888$	0	0
$889$	−18.0000	−0.603701
$890$	−10.0000	−0.335201
$891$	0	0
$892$	−24.0000	−0.803579
$893$	8.00000	0.267710
$894$	0	0
$895$	4.00000	0.133705
$896$	1.00000	0.0334077
$897$	0	0
$898$	−6.00000	−0.200223
$899$	32.0000	1.06726
$900$	0	0
$901$	−8.00000	−0.266519
$902$	−12.0000	−0.399556
$903$	0	0
$904$	14.0000	0.465633
$905$	2.00000	0.0664822
$906$	0	0
$907$	−58.0000	−1.92586	−0.962929	−	0.269754i	$-0.913058\pi$
$907$	−58.0000	−1.92586	−0.962929	+	0.269754i	$0.913058\pi$
$908$	−18.0000	−0.597351
$909$	0	0
$910$	1.00000	0.0331497
$911$	44.0000	1.45779	0.728893	−	0.684628i	$-0.240035\pi$
$911$	44.0000	1.45779	0.728893	+	0.684628i	$0.240035\pi$
$912$	0	0
$913$	4.00000	0.132381
$914$	−28.0000	−0.926158
$915$	0	0
$916$	−20.0000	−0.660819
$917$	4.00000	0.132092
$918$	0	0
$919$	16.0000	0.527791	0.263896	−	0.964551i	$-0.414993\pi$
$919$	16.0000	0.527791	0.263896	+	0.964551i	$0.414993\pi$
$920$	8.00000	0.263752
$921$	0	0
$922$	18.0000	0.592798
$923$	6.00000	0.197492
$924$	0	0
$925$	−8.00000	−0.263038
$926$	36.0000	1.18303
$927$	0	0
$928$	−4.00000	−0.131306
$929$	−34.0000	−1.11550	−0.557752	−	0.830008i	$-0.688336\pi$
$929$	−34.0000	−1.11550	−0.557752	+	0.830008i	$0.688336\pi$
$930$	0	0
$931$	−4.00000	−0.131095
$932$	6.00000	0.196537
$933$	0	0
$934$	0	0
$935$	−8.00000	−0.261628
$936$	0	0
$937$	22.0000	0.718709	0.359354	−	0.933201i	$-0.382997\pi$
$937$	22.0000	0.718709	0.359354	+	0.933201i	$0.382997\pi$
$938$	−16.0000	−0.522419
$939$	0	0
$940$	2.00000	0.0652328
$941$	42.0000	1.36916	0.684580	−	0.728937i	$-0.259985\pi$
$941$	42.0000	1.36916	0.684580	+	0.728937i	$0.259985\pi$
$942$	0	0
$943$	48.0000	1.56310
$944$	0	0
$945$	0	0
$946$	12.0000	0.390154
$947$	−44.0000	−1.42981	−0.714904	−	0.699223i	$-0.753530\pi$
$947$	−44.0000	−1.42981	−0.714904	+	0.699223i	$0.753530\pi$
$948$	0	0
$949$	2.00000	0.0649227
$950$	−4.00000	−0.129777
$951$	0	0
$952$	4.00000	0.129641
$953$	6.00000	0.194359	0.0971795	−	0.995267i	$-0.469018\pi$
$953$	6.00000	0.194359	0.0971795	+	0.995267i	$0.469018\pi$
$954$	0	0
$955$	−16.0000	−0.517748
$956$	6.00000	0.194054
$957$	0	0
$958$	4.00000	0.129234
$959$	6.00000	0.193750
$960$	0	0
$961$	33.0000	1.06452
$962$	8.00000	0.257930
$963$	0	0
$964$	0	0
$965$	−4.00000	−0.128765
$966$	0	0
$967$	−28.0000	−0.900419	−0.450210	−	0.892923i	$-0.648651\pi$
$967$	−28.0000	−0.900419	−0.450210	+	0.892923i	$0.648651\pi$
$968$	−7.00000	−0.224989
$969$	0	0
$970$	−2.00000	−0.0642161
$971$	12.0000	0.385098	0.192549	−	0.981287i	$-0.438325\pi$
$971$	12.0000	0.385098	0.192549	+	0.981287i	$0.438325\pi$
$972$	0	0
$973$	10.0000	0.320585
$974$	−32.0000	−1.02535
$975$	0	0
$976$	10.0000	0.320092
$977$	2.00000	0.0639857	0.0319928	−	0.999488i	$-0.489815\pi$
$977$	2.00000	0.0639857	0.0319928	+	0.999488i	$0.489815\pi$
$978$	0	0
$979$	20.0000	0.639203
$980$	−1.00000	−0.0319438
$981$	0	0
$982$	−16.0000	−0.510581
$983$	14.0000	0.446531	0.223265	−	0.974758i	$-0.428328\pi$
$983$	14.0000	0.446531	0.223265	+	0.974758i	$0.428328\pi$
$984$	0	0
$985$	18.0000	0.573528
$986$	−16.0000	−0.509544
$987$	0	0
$988$	4.00000	0.127257
$989$	−48.0000	−1.52631
$990$	0	0
$991$	−52.0000	−1.65183	−0.825917	−	0.563791i	$-0.809342\pi$
$991$	−52.0000	−1.65183	−0.825917	+	0.563791i	$0.809342\pi$
$992$	−8.00000	−0.254000
$993$	0	0
$994$	−6.00000	−0.190308
$995$	14.0000	0.443830
$996$	0	0
$997$	−18.0000	−0.570066	−0.285033	−	0.958518i	$-0.592005\pi$
$997$	−18.0000	−0.570066	−0.285033	+	0.958518i	$0.592005\pi$
$998$	28.0000	0.886325
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8190.2.a.bg.1.1	yes	1
3.2	odd	2		8190.2.a.t.1.1	✓	1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8190.2.a.t.1.1	✓	1	3.2	odd	2
8190.2.a.bg.1.1	yes	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 8190 = 2 \cdot 3^{2} \cdot 5 \cdot 7 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8190.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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