LMFDB - Embedded newform 8016.2.a.b.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8016,2,Mod(1,8016)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8016, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("8016.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8016 = 2^{4} \cdot 3 \cdot 167 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8016.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:	$64.0080822603$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 1002)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	8016.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q-1.00000 q^{3} -1.00000 q^{5} +1.00000 q^{7} +1.00000 q^{9} +O(q^{10})$

$q-1.00000 q^{3} -1.00000 q^{5} +1.00000 q^{7} +1.00000 q^{9} -2.00000 q^{11} -6.00000 q^{13} +1.00000 q^{15} -4.00000 q^{17} +4.00000 q^{19} -1.00000 q^{21} -8.00000 q^{23} -4.00000 q^{25} -1.00000 q^{27} -2.00000 q^{29} +5.00000 q^{31} +2.00000 q^{33} -1.00000 q^{35} +3.00000 q^{37} +6.00000 q^{39} -6.00000 q^{41} +8.00000 q^{43} -1.00000 q^{45} -3.00000 q^{47} -6.00000 q^{49} +4.00000 q^{51} -7.00000 q^{53} +2.00000 q^{55} -4.00000 q^{57} -11.0000 q^{59} +4.00000 q^{61} +1.00000 q^{63} +6.00000 q^{65} +7.00000 q^{67} +8.00000 q^{69} +4.00000 q^{71} +6.00000 q^{73} +4.00000 q^{75} -2.00000 q^{77} -4.00000 q^{79} +1.00000 q^{81} +15.0000 q^{83} +4.00000 q^{85} +2.00000 q^{87} +9.00000 q^{89} -6.00000 q^{91} -5.00000 q^{93} -4.00000 q^{95} -13.0000 q^{97} -2.00000 q^{99} +O(q^{100})$

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	0	0
$2$	0	0
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	0	0
$5$	−1.00000	−0.447214	−0.223607	−	0.974679i	$-0.571783\pi$
$5$	−1.00000	−0.447214	−0.223607	+	0.974679i	$0.571783\pi$
$6$	0	0
$7$	1.00000	0.377964	0.188982	−	0.981981i	$-0.439481\pi$
$7$	1.00000	0.377964	0.188982	+	0.981981i	$0.439481\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	−2.00000	−0.603023	−0.301511	−	0.953463i	$-0.597491\pi$
$11$	−2.00000	−0.603023	−0.301511	+	0.953463i	$0.597491\pi$
$12$	0	0
$13$	−6.00000	−1.66410	−0.832050	−	0.554700i	$-0.812833\pi$
$13$	−6.00000	−1.66410	−0.832050	+	0.554700i	$0.812833\pi$
$14$	0	0
$15$	1.00000	0.258199
$16$	0	0
$17$	−4.00000	−0.970143	−0.485071	−	0.874475i	$-0.661206\pi$
$17$	−4.00000	−0.970143	−0.485071	+	0.874475i	$0.661206\pi$
$18$	0	0
$19$	4.00000	0.917663	0.458831	−	0.888523i	$-0.348268\pi$
$19$	4.00000	0.917663	0.458831	+	0.888523i	$0.348268\pi$
$20$	0	0
$21$	−1.00000	−0.218218
$22$	0	0
$23$	−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
$25$	−4.00000	−0.800000
$26$	0	0
$27$	−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	$0.559454\pi$
$30$	0	0
$31$	5.00000	0.898027	0.449013	−	0.893525i	$-0.351776\pi$
$31$	5.00000	0.898027	0.449013	+	0.893525i	$0.351776\pi$
$32$	0	0
$33$	2.00000	0.348155
$34$	0	0
$35$	−1.00000	−0.169031
$36$	0	0
$37$	3.00000	0.493197	0.246598	−	0.969118i	$-0.420687\pi$
$37$	3.00000	0.493197	0.246598	+	0.969118i	$0.420687\pi$
$38$	0	0
$39$	6.00000	0.960769
$40$	0	0
$41$	−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
$43$	8.00000	1.21999	0.609994	−	0.792406i	$-0.291172\pi$
$43$	8.00000	1.21999	0.609994	+	0.792406i	$0.291172\pi$
$44$	0	0
$45$	−1.00000	−0.149071
$46$	0	0
$47$	−3.00000	−0.437595	−0.218797	−	0.975770i	$-0.570213\pi$
$47$	−3.00000	−0.437595	−0.218797	+	0.975770i	$0.570213\pi$
$48$	0	0
$49$	−6.00000	−0.857143
$50$	0	0
$51$	4.00000	0.560112
$52$	0	0
$53$	−7.00000	−0.961524	−0.480762	−	0.876851i	$-0.659640\pi$
$53$	−7.00000	−0.961524	−0.480762	+	0.876851i	$0.659640\pi$
$54$	0	0
$55$	2.00000	0.269680
$56$	0	0
$57$	−4.00000	−0.529813
$58$	0	0
$59$	−11.0000	−1.43208	−0.716039	−	0.698060i	$-0.754047\pi$
$59$	−11.0000	−1.43208	−0.716039	+	0.698060i	$0.754047\pi$
$60$	0	0
$61$	4.00000	0.512148	0.256074	−	0.966657i	$-0.417571\pi$
$61$	4.00000	0.512148	0.256074	+	0.966657i	$0.417571\pi$
$62$	0	0
$63$	1.00000	0.125988
$64$	0	0
$65$	6.00000	0.744208
$66$	0	0
$67$	7.00000	0.855186	0.427593	−	0.903971i	$-0.359362\pi$
$67$	7.00000	0.855186	0.427593	+	0.903971i	$0.359362\pi$
$68$	0	0
$69$	8.00000	0.963087
$70$	0	0
$71$	4.00000	0.474713	0.237356	−	0.971423i	$-0.423719\pi$
$71$	4.00000	0.474713	0.237356	+	0.971423i	$0.423719\pi$
$72$	0	0
$73$	6.00000	0.702247	0.351123	−	0.936329i	$-0.385800\pi$
$73$	6.00000	0.702247	0.351123	+	0.936329i	$0.385800\pi$
$74$	0	0
$75$	4.00000	0.461880
$76$	0	0
$77$	−2.00000	−0.227921
$78$	0	0
$79$	−4.00000	−0.450035	−0.225018	−	0.974355i	$-0.572244\pi$
$79$	−4.00000	−0.450035	−0.225018	+	0.974355i	$0.572244\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	15.0000	1.64646	0.823232	−	0.567705i	$-0.192169\pi$
$83$	15.0000	1.64646	0.823232	+	0.567705i	$0.192169\pi$
$84$	0	0
$85$	4.00000	0.433861
$86$	0	0
$87$	2.00000	0.214423
$88$	0	0
$89$	9.00000	0.953998	0.476999	−	0.878904i	$-0.341725\pi$
$89$	9.00000	0.953998	0.476999	+	0.878904i	$0.341725\pi$
$90$	0	0
$91$	−6.00000	−0.628971
$92$	0	0
$93$	−5.00000	−0.518476
$94$	0	0
$95$	−4.00000	−0.410391
$96$	0	0
$97$	−13.0000	−1.31995	−0.659975	−	0.751288i	$-0.729433\pi$
$97$	−13.0000	−1.31995	−0.659975	+	0.751288i	$0.729433\pi$
$98$	0	0
$99$	−2.00000	−0.201008
$100$	0	0
$101$	1.00000	0.0995037	0.0497519	−	0.998762i	$-0.484157\pi$
$101$	1.00000	0.0995037	0.0497519	+	0.998762i	$0.484157\pi$
$102$	0	0
$103$	−6.00000	−0.591198	−0.295599	−	0.955312i	$-0.595519\pi$
$103$	−6.00000	−0.591198	−0.295599	+	0.955312i	$0.595519\pi$
$104$	0	0
$105$	1.00000	0.0975900
$106$	0	0
$107$	10.0000	0.966736	0.483368	−	0.875417i	$-0.339413\pi$
$107$	10.0000	0.966736	0.483368	+	0.875417i	$0.339413\pi$
$108$	0	0
$109$	2.00000	0.191565	0.0957826	−	0.995402i	$-0.469465\pi$
$109$	2.00000	0.191565	0.0957826	+	0.995402i	$0.469465\pi$
$110$	0	0
$111$	−3.00000	−0.284747
$112$	0	0
$113$	−4.00000	−0.376288	−0.188144	−	0.982141i	$-0.560247\pi$
$113$	−4.00000	−0.376288	−0.188144	+	0.982141i	$0.560247\pi$
$114$	0	0
$115$	8.00000	0.746004
$116$	0	0
$117$	−6.00000	−0.554700
$118$	0	0
$119$	−4.00000	−0.366679
$120$	0	0
$121$	−7.00000	−0.636364
$122$	0	0
$123$	6.00000	0.541002
$124$	0	0
$125$	9.00000	0.804984
$126$	0	0
$127$	17.0000	1.50851	0.754253	−	0.656584i	$-0.227999\pi$
$127$	17.0000	1.50851	0.754253	+	0.656584i	$0.227999\pi$
$128$	0	0
$129$	−8.00000	−0.704361
$130$	0	0
$131$	12.0000	1.04844	0.524222	−	0.851581i	$-0.324356\pi$
$131$	12.0000	1.04844	0.524222	+	0.851581i	$0.324356\pi$
$132$	0	0
$133$	4.00000	0.346844
$134$	0	0
$135$	1.00000	0.0860663
$136$	0	0
$137$	9.00000	0.768922	0.384461	−	0.923141i	$-0.374387\pi$
$137$	9.00000	0.768922	0.384461	+	0.923141i	$0.374387\pi$
$138$	0	0
$139$	7.00000	0.593732	0.296866	−	0.954919i	$-0.404058\pi$
$139$	7.00000	0.593732	0.296866	+	0.954919i	$0.404058\pi$
$140$	0	0
$141$	3.00000	0.252646
$142$	0	0
$143$	12.0000	1.00349
$144$	0	0
$145$	2.00000	0.166091
$146$	0	0
$147$	6.00000	0.494872
$148$	0	0
$149$	9.00000	0.737309	0.368654	−	0.929567i	$-0.379819\pi$
$149$	9.00000	0.737309	0.368654	+	0.929567i	$0.379819\pi$
$150$	0	0
$151$	−20.0000	−1.62758	−0.813788	−	0.581161i	$-0.802599\pi$
$151$	−20.0000	−1.62758	−0.813788	+	0.581161i	$0.802599\pi$
$152$	0	0
$153$	−4.00000	−0.323381
$154$	0	0
$155$	−5.00000	−0.401610
$156$	0	0
$157$	−18.0000	−1.43656	−0.718278	−	0.695756i	$-0.755069\pi$
$157$	−18.0000	−1.43656	−0.718278	+	0.695756i	$0.755069\pi$
$158$	0	0
$159$	7.00000	0.555136
$160$	0	0
$161$	−8.00000	−0.630488
$162$	0	0
$163$	−9.00000	−0.704934	−0.352467	−	0.935824i	$-0.614657\pi$
$163$	−9.00000	−0.704934	−0.352467	+	0.935824i	$0.614657\pi$
$164$	0	0
$165$	−2.00000	−0.155700
$166$	0	0
$167$	−1.00000	−0.0773823
$167$	−1.00000	−0.0773823
$168$	0	0
$169$	23.0000	1.76923
$170$	0	0
$171$	4.00000	0.305888
$172$	0	0
$173$	−4.00000	−0.304114	−0.152057	−	0.988372i	$-0.548590\pi$
$173$	−4.00000	−0.304114	−0.152057	+	0.988372i	$0.548590\pi$
$174$	0	0
$175$	−4.00000	−0.302372
$176$	0	0
$177$	11.0000	0.826811
$178$	0	0
$179$	−2.00000	−0.149487	−0.0747435	−	0.997203i	$-0.523814\pi$
$179$	−2.00000	−0.149487	−0.0747435	+	0.997203i	$0.523814\pi$
$180$	0	0
$181$	20.0000	1.48659	0.743294	−	0.668965i	$-0.233262\pi$
$181$	20.0000	1.48659	0.743294	+	0.668965i	$0.233262\pi$
$182$	0	0
$183$	−4.00000	−0.295689
$184$	0	0
$185$	−3.00000	−0.220564
$186$	0	0
$187$	8.00000	0.585018
$188$	0	0
$189$	−1.00000	−0.0727393
$190$	0	0
$191$	−15.0000	−1.08536	−0.542681	−	0.839939i	$-0.682591\pi$
$191$	−15.0000	−1.08536	−0.542681	+	0.839939i	$0.682591\pi$
$192$	0	0
$193$	16.0000	1.15171	0.575853	−	0.817554i	$-0.304670\pi$
$193$	16.0000	1.15171	0.575853	+	0.817554i	$0.304670\pi$
$194$	0	0
$195$	−6.00000	−0.429669
$196$	0	0
$197$	6.00000	0.427482	0.213741	−	0.976890i	$-0.431435\pi$
$197$	6.00000	0.427482	0.213741	+	0.976890i	$0.431435\pi$
$198$	0	0
$199$	8.00000	0.567105	0.283552	−	0.958957i	$-0.408487\pi$
$199$	8.00000	0.567105	0.283552	+	0.958957i	$0.408487\pi$
$200$	0	0
$201$	−7.00000	−0.493742
$202$	0	0
$203$	−2.00000	−0.140372
$204$	0	0
$205$	6.00000	0.419058
$206$	0	0
$207$	−8.00000	−0.556038
$208$	0	0
$209$	−8.00000	−0.553372
$210$	0	0
$211$	2.00000	0.137686	0.0688428	−	0.997628i	$-0.478069\pi$
$211$	2.00000	0.137686	0.0688428	+	0.997628i	$0.478069\pi$
$212$	0	0
$213$	−4.00000	−0.274075
$214$	0	0
$215$	−8.00000	−0.545595
$216$	0	0
$217$	5.00000	0.339422
$218$	0	0
$219$	−6.00000	−0.405442
$220$	0	0
$221$	24.0000	1.61441
$222$	0	0
$223$	−29.0000	−1.94198	−0.970992	−	0.239113i	$-0.923143\pi$
$223$	−29.0000	−1.94198	−0.970992	+	0.239113i	$0.923143\pi$
$224$	0	0
$225$	−4.00000	−0.266667
$226$	0	0
$227$	−1.00000	−0.0663723	−0.0331862	−	0.999449i	$-0.510565\pi$
$227$	−1.00000	−0.0663723	−0.0331862	+	0.999449i	$0.510565\pi$
$228$	0	0
$229$	−20.0000	−1.32164	−0.660819	−	0.750546i	$-0.729791\pi$
$229$	−20.0000	−1.32164	−0.660819	+	0.750546i	$0.729791\pi$
$230$	0	0
$231$	2.00000	0.131590
$232$	0	0
$233$	−9.00000	−0.589610	−0.294805	−	0.955557i	$-0.595255\pi$
$233$	−9.00000	−0.589610	−0.294805	+	0.955557i	$0.595255\pi$
$234$	0	0
$235$	3.00000	0.195698
$236$	0	0
$237$	4.00000	0.259828
$238$	0	0
$239$	−24.0000	−1.55243	−0.776215	−	0.630468i	$-0.782863\pi$
$239$	−24.0000	−1.55243	−0.776215	+	0.630468i	$0.782863\pi$
$240$	0	0
$241$	12.0000	0.772988	0.386494	−	0.922292i	$-0.373686\pi$
$241$	12.0000	0.772988	0.386494	+	0.922292i	$0.373686\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	0	0
$245$	6.00000	0.383326
$246$	0	0
$247$	−24.0000	−1.52708
$248$	0	0
$249$	−15.0000	−0.950586
$250$	0	0
$251$	−20.0000	−1.26239	−0.631194	−	0.775625i	$-0.717435\pi$
$251$	−20.0000	−1.26239	−0.631194	+	0.775625i	$0.717435\pi$
$252$	0	0
$253$	16.0000	1.00591
$254$	0	0
$255$	−4.00000	−0.250490
$256$	0	0
$257$	−22.0000	−1.37232	−0.686161	−	0.727450i	$-0.740706\pi$
$257$	−22.0000	−1.37232	−0.686161	+	0.727450i	$0.740706\pi$
$258$	0	0
$259$	3.00000	0.186411
$260$	0	0
$261$	−2.00000	−0.123797
$262$	0	0
$263$	−15.0000	−0.924940	−0.462470	−	0.886635i	$-0.653037\pi$
$263$	−15.0000	−0.924940	−0.462470	+	0.886635i	$0.653037\pi$
$264$	0	0
$265$	7.00000	0.430007
$266$	0	0
$267$	−9.00000	−0.550791
$268$	0	0
$269$	27.0000	1.64622	0.823110	−	0.567883i	$-0.192237\pi$
$269$	27.0000	1.64622	0.823110	+	0.567883i	$0.192237\pi$
$270$	0	0
$271$	30.0000	1.82237	0.911185	−	0.411997i	$-0.135169\pi$
$271$	30.0000	1.82237	0.911185	+	0.411997i	$0.135169\pi$
$272$	0	0
$273$	6.00000	0.363137
$274$	0	0
$275$	8.00000	0.482418
$276$	0	0
$277$	−9.00000	−0.540758	−0.270379	−	0.962754i	$-0.587149\pi$
$277$	−9.00000	−0.540758	−0.270379	+	0.962754i	$0.587149\pi$
$278$	0	0
$279$	5.00000	0.299342
$280$	0	0
$281$	15.0000	0.894825	0.447412	−	0.894328i	$-0.352346\pi$
$281$	15.0000	0.894825	0.447412	+	0.894328i	$0.352346\pi$
$282$	0	0
$283$	8.00000	0.475551	0.237775	−	0.971320i	$-0.423582\pi$
$283$	8.00000	0.475551	0.237775	+	0.971320i	$0.423582\pi$
$284$	0	0
$285$	4.00000	0.236940
$286$	0	0
$287$	−6.00000	−0.354169
$288$	0	0
$289$	−1.00000	−0.0588235
$290$	0	0
$291$	13.0000	0.762073
$292$	0	0
$293$	12.0000	0.701047	0.350524	−	0.936554i	$-0.386004\pi$
$293$	12.0000	0.701047	0.350524	+	0.936554i	$0.386004\pi$
$294$	0	0
$295$	11.0000	0.640445
$296$	0	0
$297$	2.00000	0.116052
$298$	0	0
$299$	48.0000	2.77591
$300$	0	0
$301$	8.00000	0.461112
$302$	0	0
$303$	−1.00000	−0.0574485
$304$	0	0
$305$	−4.00000	−0.229039
$306$	0	0
$307$	27.0000	1.54097	0.770486	−	0.637457i	$-0.220014\pi$
$307$	27.0000	1.54097	0.770486	+	0.637457i	$0.220014\pi$
$308$	0	0
$309$	6.00000	0.341328
$310$	0	0
$311$	27.0000	1.53103	0.765515	−	0.643418i	$-0.222484\pi$
$311$	27.0000	1.53103	0.765515	+	0.643418i	$0.222484\pi$
$312$	0	0
$313$	10.0000	0.565233	0.282617	−	0.959233i	$-0.408798\pi$
$313$	10.0000	0.565233	0.282617	+	0.959233i	$0.408798\pi$
$314$	0	0
$315$	−1.00000	−0.0563436
$316$	0	0
$317$	6.00000	0.336994	0.168497	−	0.985702i	$-0.446109\pi$
$317$	6.00000	0.336994	0.168497	+	0.985702i	$0.446109\pi$
$318$	0	0
$319$	4.00000	0.223957
$320$	0	0
$321$	−10.0000	−0.558146
$322$	0	0
$323$	−16.0000	−0.890264
$324$	0	0
$325$	24.0000	1.33128
$326$	0	0
$327$	−2.00000	−0.110600
$328$	0	0
$329$	−3.00000	−0.165395
$330$	0	0
$331$	−20.0000	−1.09930	−0.549650	−	0.835395i	$-0.685239\pi$
$331$	−20.0000	−1.09930	−0.549650	+	0.835395i	$0.685239\pi$
$332$	0	0
$333$	3.00000	0.164399
$334$	0	0
$335$	−7.00000	−0.382451
$336$	0	0
$337$	29.0000	1.57973	0.789865	−	0.613280i	$-0.210150\pi$
$337$	29.0000	1.57973	0.789865	+	0.613280i	$0.210150\pi$
$338$	0	0
$339$	4.00000	0.217250
$340$	0	0
$341$	−10.0000	−0.541530
$342$	0	0
$343$	−13.0000	−0.701934
$344$	0	0
$345$	−8.00000	−0.430706
$346$	0	0
$347$	−3.00000	−0.161048	−0.0805242	−	0.996753i	$-0.525659\pi$
$347$	−3.00000	−0.161048	−0.0805242	+	0.996753i	$0.525659\pi$
$348$	0	0
$349$	29.0000	1.55233	0.776167	−	0.630527i	$-0.217161\pi$
$349$	29.0000	1.55233	0.776167	+	0.630527i	$0.217161\pi$
$350$	0	0
$351$	6.00000	0.320256
$352$	0	0
$353$	−21.0000	−1.11772	−0.558859	−	0.829263i	$-0.688761\pi$
$353$	−21.0000	−1.11772	−0.558859	+	0.829263i	$0.688761\pi$
$354$	0	0
$355$	−4.00000	−0.212298
$356$	0	0
$357$	4.00000	0.211702
$358$	0	0
$359$	24.0000	1.26667	0.633336	−	0.773877i	$-0.281685\pi$
$359$	24.0000	1.26667	0.633336	+	0.773877i	$0.281685\pi$
$360$	0	0
$361$	−3.00000	−0.157895
$362$	0	0
$363$	7.00000	0.367405
$364$	0	0
$365$	−6.00000	−0.314054
$366$	0	0
$367$	0	0		−	1.00000i	$-0.5\pi$
$367$	0	0			1.00000i	$0.5\pi$
$368$	0	0
$369$	−6.00000	−0.312348
$370$	0	0
$371$	−7.00000	−0.363422
$372$	0	0
$373$	13.0000	0.673114	0.336557	−	0.941663i	$-0.390737\pi$
$373$	13.0000	0.673114	0.336557	+	0.941663i	$0.390737\pi$
$374$	0	0
$375$	−9.00000	−0.464758
$376$	0	0
$377$	12.0000	0.618031
$378$	0	0
$379$	11.0000	0.565032	0.282516	−	0.959263i	$-0.408831\pi$
$379$	11.0000	0.565032	0.282516	+	0.959263i	$0.408831\pi$
$380$	0	0
$381$	−17.0000	−0.870936
$382$	0	0
$383$	−16.0000	−0.817562	−0.408781	−	0.912633i	$-0.634046\pi$
$383$	−16.0000	−0.817562	−0.408781	+	0.912633i	$0.634046\pi$
$384$	0	0
$385$	2.00000	0.101929
$386$	0	0
$387$	8.00000	0.406663
$388$	0	0
$389$	34.0000	1.72387	0.861934	−	0.507020i	$-0.169253\pi$
$389$	34.0000	1.72387	0.861934	+	0.507020i	$0.169253\pi$
$390$	0	0
$391$	32.0000	1.61831
$392$	0	0
$393$	−12.0000	−0.605320
$394$	0	0
$395$	4.00000	0.201262
$396$	0	0
$397$	−26.0000	−1.30490	−0.652451	−	0.757831i	$-0.726259\pi$
$397$	−26.0000	−1.30490	−0.652451	+	0.757831i	$0.726259\pi$
$398$	0	0
$399$	−4.00000	−0.200250
$400$	0	0
$401$	−24.0000	−1.19850	−0.599251	−	0.800561i	$-0.704535\pi$
$401$	−24.0000	−1.19850	−0.599251	+	0.800561i	$0.704535\pi$
$402$	0	0
$403$	−30.0000	−1.49441
$404$	0	0
$405$	−1.00000	−0.0496904
$406$	0	0
$407$	−6.00000	−0.297409
$408$	0	0
$409$	26.0000	1.28562	0.642809	−	0.766027i	$-0.277769\pi$
$409$	26.0000	1.28562	0.642809	+	0.766027i	$0.277769\pi$
$410$	0	0
$411$	−9.00000	−0.443937
$412$	0	0
$413$	−11.0000	−0.541275
$414$	0	0
$415$	−15.0000	−0.736321
$416$	0	0
$417$	−7.00000	−0.342791
$418$	0	0
$419$	16.0000	0.781651	0.390826	−	0.920465i	$-0.372190\pi$
$419$	16.0000	0.781651	0.390826	+	0.920465i	$0.372190\pi$
$420$	0	0
$421$	2.00000	0.0974740	0.0487370	−	0.998812i	$-0.484480\pi$
$421$	2.00000	0.0974740	0.0487370	+	0.998812i	$0.484480\pi$
$422$	0	0
$423$	−3.00000	−0.145865
$424$	0	0
$425$	16.0000	0.776114
$426$	0	0
$427$	4.00000	0.193574
$428$	0	0
$429$	−12.0000	−0.579365
$430$	0	0
$431$	39.0000	1.87856	0.939282	−	0.343146i	$-0.111493\pi$
$431$	39.0000	1.87856	0.939282	+	0.343146i	$0.111493\pi$
$432$	0	0
$433$	−11.0000	−0.528626	−0.264313	−	0.964437i	$-0.585145\pi$
$433$	−11.0000	−0.528626	−0.264313	+	0.964437i	$0.585145\pi$
$434$	0	0
$435$	−2.00000	−0.0958927
$436$	0	0
$437$	−32.0000	−1.53077
$438$	0	0
$439$	8.00000	0.381819	0.190910	−	0.981608i	$-0.438856\pi$
$439$	8.00000	0.381819	0.190910	+	0.981608i	$0.438856\pi$
$440$	0	0
$441$	−6.00000	−0.285714
$442$	0	0
$443$	−9.00000	−0.427603	−0.213801	−	0.976877i	$-0.568585\pi$
$443$	−9.00000	−0.427603	−0.213801	+	0.976877i	$0.568585\pi$
$444$	0	0
$445$	−9.00000	−0.426641
$446$	0	0
$447$	−9.00000	−0.425685
$448$	0	0
$449$	5.00000	0.235965	0.117982	−	0.993016i	$-0.462357\pi$
$449$	5.00000	0.235965	0.117982	+	0.993016i	$0.462357\pi$
$450$	0	0
$451$	12.0000	0.565058
$452$	0	0
$453$	20.0000	0.939682
$454$	0	0
$455$	6.00000	0.281284
$456$	0	0
$457$	−2.00000	−0.0935561	−0.0467780	−	0.998905i	$-0.514895\pi$
$457$	−2.00000	−0.0935561	−0.0467780	+	0.998905i	$0.514895\pi$
$458$	0	0
$459$	4.00000	0.186704
$460$	0	0
$461$	−6.00000	−0.279448	−0.139724	−	0.990190i	$-0.544622\pi$
$461$	−6.00000	−0.279448	−0.139724	+	0.990190i	$0.544622\pi$
$462$	0	0
$463$	22.0000	1.02243	0.511213	−	0.859454i	$-0.329196\pi$
$463$	22.0000	1.02243	0.511213	+	0.859454i	$0.329196\pi$
$464$	0	0
$465$	5.00000	0.231869
$466$	0	0
$467$	38.0000	1.75843	0.879215	−	0.476425i	$-0.158068\pi$
$467$	38.0000	1.75843	0.879215	+	0.476425i	$0.158068\pi$
$468$	0	0
$469$	7.00000	0.323230
$470$	0	0
$471$	18.0000	0.829396
$472$	0	0
$473$	−16.0000	−0.735681
$474$	0	0
$475$	−16.0000	−0.734130
$476$	0	0
$477$	−7.00000	−0.320508
$478$	0	0
$479$	−24.0000	−1.09659	−0.548294	−	0.836286i	$-0.684723\pi$
$479$	−24.0000	−1.09659	−0.548294	+	0.836286i	$0.684723\pi$
$480$	0	0
$481$	−18.0000	−0.820729
$482$	0	0
$483$	8.00000	0.364013
$484$	0	0
$485$	13.0000	0.590300
$486$	0	0
$487$	16.0000	0.725029	0.362515	−	0.931978i	$-0.381918\pi$
$487$	16.0000	0.725029	0.362515	+	0.931978i	$0.381918\pi$
$488$	0	0
$489$	9.00000	0.406994
$490$	0	0
$491$	−8.00000	−0.361035	−0.180517	−	0.983572i	$-0.557777\pi$
$491$	−8.00000	−0.361035	−0.180517	+	0.983572i	$0.557777\pi$
$492$	0	0
$493$	8.00000	0.360302
$494$	0	0
$495$	2.00000	0.0898933
$496$	0	0
$497$	4.00000	0.179425
$498$	0	0
$499$	24.0000	1.07439	0.537194	−	0.843459i	$-0.319484\pi$
$499$	24.0000	1.07439	0.537194	+	0.843459i	$0.319484\pi$
$500$	0	0
$501$	1.00000	0.0446767
$502$	0	0
$503$	12.0000	0.535054	0.267527	−	0.963550i	$-0.413794\pi$
$503$	12.0000	0.535054	0.267527	+	0.963550i	$0.413794\pi$
$504$	0	0
$505$	−1.00000	−0.0444994
$506$	0	0
$507$	−23.0000	−1.02147
$508$	0	0
$509$	−16.0000	−0.709188	−0.354594	−	0.935020i	$-0.615381\pi$
$509$	−16.0000	−0.709188	−0.354594	+	0.935020i	$0.615381\pi$
$510$	0	0
$511$	6.00000	0.265424
$512$	0	0
$513$	−4.00000	−0.176604
$514$	0	0
$515$	6.00000	0.264392
$516$	0	0
$517$	6.00000	0.263880
$518$	0	0
$519$	4.00000	0.175581
$520$	0	0
$521$	−14.0000	−0.613351	−0.306676	−	0.951814i	$-0.599217\pi$
$521$	−14.0000	−0.613351	−0.306676	+	0.951814i	$0.599217\pi$
$522$	0	0
$523$	10.0000	0.437269	0.218635	−	0.975807i	$-0.429840\pi$
$523$	10.0000	0.437269	0.218635	+	0.975807i	$0.429840\pi$
$524$	0	0
$525$	4.00000	0.174574
$526$	0	0
$527$	−20.0000	−0.871214
$528$	0	0
$529$	41.0000	1.78261
$530$	0	0
$531$	−11.0000	−0.477359
$532$	0	0
$533$	36.0000	1.55933
$534$	0	0
$535$	−10.0000	−0.432338
$536$	0	0
$537$	2.00000	0.0863064
$538$	0	0
$539$	12.0000	0.516877
$540$	0	0
$541$	−3.00000	−0.128980	−0.0644900	−	0.997918i	$-0.520542\pi$
$541$	−3.00000	−0.128980	−0.0644900	+	0.997918i	$0.520542\pi$
$542$	0	0
$543$	−20.0000	−0.858282
$544$	0	0
$545$	−2.00000	−0.0856706
$546$	0	0
$547$	17.0000	0.726868	0.363434	−	0.931620i	$-0.381604\pi$
$547$	17.0000	0.726868	0.363434	+	0.931620i	$0.381604\pi$
$548$	0	0
$549$	4.00000	0.170716
$550$	0	0
$551$	−8.00000	−0.340811
$552$	0	0
$553$	−4.00000	−0.170097
$554$	0	0
$555$	3.00000	0.127343
$556$	0	0
$557$	−38.0000	−1.61011	−0.805056	−	0.593199i	$-0.797865\pi$
$557$	−38.0000	−1.61011	−0.805056	+	0.593199i	$0.797865\pi$
$558$	0	0
$559$	−48.0000	−2.03018
$560$	0	0
$561$	−8.00000	−0.337760
$562$	0	0
$563$	24.0000	1.01148	0.505740	−	0.862686i	$-0.331220\pi$
$563$	24.0000	1.01148	0.505740	+	0.862686i	$0.331220\pi$
$564$	0	0
$565$	4.00000	0.168281
$566$	0	0
$567$	1.00000	0.0419961
$568$	0	0
$569$	−4.00000	−0.167689	−0.0838444	−	0.996479i	$-0.526720\pi$
$569$	−4.00000	−0.167689	−0.0838444	+	0.996479i	$0.526720\pi$
$570$	0	0
$571$	31.0000	1.29731	0.648655	−	0.761083i	$-0.275332\pi$
$571$	31.0000	1.29731	0.648655	+	0.761083i	$0.275332\pi$
$572$	0	0
$573$	15.0000	0.626634
$574$	0	0
$575$	32.0000	1.33449
$576$	0	0
$577$	31.0000	1.29055	0.645273	−	0.763952i	$-0.276743\pi$
$577$	31.0000	1.29055	0.645273	+	0.763952i	$0.276743\pi$
$578$	0	0
$579$	−16.0000	−0.664937
$580$	0	0
$581$	15.0000	0.622305
$582$	0	0
$583$	14.0000	0.579821
$584$	0	0
$585$	6.00000	0.248069
$586$	0	0
$587$	21.0000	0.866763	0.433381	−	0.901211i	$-0.357320\pi$
$587$	21.0000	0.866763	0.433381	+	0.901211i	$0.357320\pi$
$588$	0	0
$589$	20.0000	0.824086
$590$	0	0
$591$	−6.00000	−0.246807
$592$	0	0
$593$	22.0000	0.903432	0.451716	−	0.892162i	$-0.350812\pi$
$593$	22.0000	0.903432	0.451716	+	0.892162i	$0.350812\pi$
$594$	0	0
$595$	4.00000	0.163984
$596$	0	0
$597$	−8.00000	−0.327418
$598$	0	0
$599$	−19.0000	−0.776319	−0.388159	−	0.921592i	$-0.626889\pi$
$599$	−19.0000	−0.776319	−0.388159	+	0.921592i	$0.626889\pi$
$600$	0	0
$601$	27.0000	1.10135	0.550676	−	0.834719i	$-0.314370\pi$
$601$	27.0000	1.10135	0.550676	+	0.834719i	$0.314370\pi$
$602$	0	0
$603$	7.00000	0.285062
$604$	0	0
$605$	7.00000	0.284590
$606$	0	0
$607$	38.0000	1.54237	0.771186	−	0.636610i	$-0.219664\pi$
$607$	38.0000	1.54237	0.771186	+	0.636610i	$0.219664\pi$
$608$	0	0
$609$	2.00000	0.0810441
$610$	0	0
$611$	18.0000	0.728202
$612$	0	0
$613$	−4.00000	−0.161558	−0.0807792	−	0.996732i	$-0.525741\pi$
$613$	−4.00000	−0.161558	−0.0807792	+	0.996732i	$0.525741\pi$
$614$	0	0
$615$	−6.00000	−0.241943
$616$	0	0
$617$	−26.0000	−1.04672	−0.523360	−	0.852111i	$-0.675322\pi$
$617$	−26.0000	−1.04672	−0.523360	+	0.852111i	$0.675322\pi$
$618$	0	0
$619$	32.0000	1.28619	0.643094	−	0.765787i	$-0.277650\pi$
$619$	32.0000	1.28619	0.643094	+	0.765787i	$0.277650\pi$
$620$	0	0
$621$	8.00000	0.321029
$622$	0	0
$623$	9.00000	0.360577
$624$	0	0
$625$	11.0000	0.440000
$626$	0	0
$627$	8.00000	0.319489
$628$	0	0
$629$	−12.0000	−0.478471
$630$	0	0
$631$	35.0000	1.39333	0.696664	−	0.717398i	$-0.254667\pi$
$631$	35.0000	1.39333	0.696664	+	0.717398i	$0.254667\pi$
$632$	0	0
$633$	−2.00000	−0.0794929
$634$	0	0
$635$	−17.0000	−0.674624
$636$	0	0
$637$	36.0000	1.42637
$638$	0	0
$639$	4.00000	0.158238
$640$	0	0
$641$	−10.0000	−0.394976	−0.197488	−	0.980305i	$-0.563278\pi$
$641$	−10.0000	−0.394976	−0.197488	+	0.980305i	$0.563278\pi$
$642$	0	0
$643$	−40.0000	−1.57745	−0.788723	−	0.614749i	$-0.789257\pi$
$643$	−40.0000	−1.57745	−0.788723	+	0.614749i	$0.789257\pi$
$644$	0	0
$645$	8.00000	0.315000
$646$	0	0
$647$	44.0000	1.72982	0.864909	−	0.501928i	$-0.167376\pi$
$647$	44.0000	1.72982	0.864909	+	0.501928i	$0.167376\pi$
$648$	0	0
$649$	22.0000	0.863576
$650$	0	0
$651$	−5.00000	−0.195965
$652$	0	0
$653$	−36.0000	−1.40879	−0.704394	−	0.709809i	$-0.748781\pi$
$653$	−36.0000	−1.40879	−0.704394	+	0.709809i	$0.748781\pi$
$654$	0	0
$655$	−12.0000	−0.468879
$656$	0	0
$657$	6.00000	0.234082
$658$	0	0
$659$	−9.00000	−0.350590	−0.175295	−	0.984516i	$-0.556088\pi$
$659$	−9.00000	−0.350590	−0.175295	+	0.984516i	$0.556088\pi$
$660$	0	0
$661$	−14.0000	−0.544537	−0.272268	−	0.962221i	$-0.587774\pi$
$661$	−14.0000	−0.544537	−0.272268	+	0.962221i	$0.587774\pi$
$662$	0	0
$663$	−24.0000	−0.932083
$664$	0	0
$665$	−4.00000	−0.155113
$666$	0	0
$667$	16.0000	0.619522
$668$	0	0
$669$	29.0000	1.12120
$670$	0	0
$671$	−8.00000	−0.308837
$672$	0	0
$673$	−8.00000	−0.308377	−0.154189	−	0.988041i	$-0.549276\pi$
$673$	−8.00000	−0.308377	−0.154189	+	0.988041i	$0.549276\pi$
$674$	0	0
$675$	4.00000	0.153960
$676$	0	0
$677$	−22.0000	−0.845529	−0.422764	−	0.906240i	$-0.638940\pi$
$677$	−22.0000	−0.845529	−0.422764	+	0.906240i	$0.638940\pi$
$678$	0	0
$679$	−13.0000	−0.498894
$680$	0	0
$681$	1.00000	0.0383201
$682$	0	0
$683$	−16.0000	−0.612223	−0.306111	−	0.951996i	$-0.599028\pi$
$683$	−16.0000	−0.612223	−0.306111	+	0.951996i	$0.599028\pi$
$684$	0	0
$685$	−9.00000	−0.343872
$686$	0	0
$687$	20.0000	0.763048
$688$	0	0
$689$	42.0000	1.60007
$690$	0	0
$691$	20.0000	0.760836	0.380418	−	0.924815i	$-0.375780\pi$
$691$	20.0000	0.760836	0.380418	+	0.924815i	$0.375780\pi$
$692$	0	0
$693$	−2.00000	−0.0759737
$694$	0	0
$695$	−7.00000	−0.265525
$696$	0	0
$697$	24.0000	0.909065
$698$	0	0
$699$	9.00000	0.340411
$700$	0	0
$701$	0	0		−	1.00000i	$-0.5\pi$
$701$	0	0			1.00000i	$0.5\pi$
$702$	0	0
$703$	12.0000	0.452589
$704$	0	0
$705$	−3.00000	−0.112987
$706$	0	0
$707$	1.00000	0.0376089
$708$	0	0
$709$	35.0000	1.31445	0.657226	−	0.753693i	$-0.271730\pi$
$709$	35.0000	1.31445	0.657226	+	0.753693i	$0.271730\pi$
$710$	0	0
$711$	−4.00000	−0.150012
$712$	0	0
$713$	−40.0000	−1.49801
$714$	0	0
$715$	−12.0000	−0.448775
$716$	0	0
$717$	24.0000	0.896296
$718$	0	0
$719$	−4.00000	−0.149175	−0.0745874	−	0.997214i	$-0.523764\pi$
$719$	−4.00000	−0.149175	−0.0745874	+	0.997214i	$0.523764\pi$
$720$	0	0
$721$	−6.00000	−0.223452
$722$	0	0
$723$	−12.0000	−0.446285
$724$	0	0
$725$	8.00000	0.297113
$726$	0	0
$727$	−28.0000	−1.03846	−0.519231	−	0.854634i	$-0.673782\pi$
$727$	−28.0000	−1.03846	−0.519231	+	0.854634i	$0.673782\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−32.0000	−1.18356
$732$	0	0
$733$	−34.0000	−1.25582	−0.627909	−	0.778287i	$-0.716089\pi$
$733$	−34.0000	−1.25582	−0.627909	+	0.778287i	$0.716089\pi$
$734$	0	0
$735$	−6.00000	−0.221313
$736$	0	0
$737$	−14.0000	−0.515697
$738$	0	0
$739$	−11.0000	−0.404642	−0.202321	−	0.979319i	$-0.564848\pi$
$739$	−11.0000	−0.404642	−0.202321	+	0.979319i	$0.564848\pi$
$740$	0	0
$741$	24.0000	0.881662
$742$	0	0
$743$	−21.0000	−0.770415	−0.385208	−	0.922830i	$-0.625870\pi$
$743$	−21.0000	−0.770415	−0.385208	+	0.922830i	$0.625870\pi$
$744$	0	0
$745$	−9.00000	−0.329734
$746$	0	0
$747$	15.0000	0.548821
$748$	0	0
$749$	10.0000	0.365392
$750$	0	0
$751$	−4.00000	−0.145962	−0.0729810	−	0.997333i	$-0.523251\pi$
$751$	−4.00000	−0.145962	−0.0729810	+	0.997333i	$0.523251\pi$
$752$	0	0
$753$	20.0000	0.728841
$754$	0	0
$755$	20.0000	0.727875
$756$	0	0
$757$	−38.0000	−1.38113	−0.690567	−	0.723269i	$-0.742639\pi$
$757$	−38.0000	−1.38113	−0.690567	+	0.723269i	$0.742639\pi$
$758$	0	0
$759$	−16.0000	−0.580763
$760$	0	0
$761$	27.0000	0.978749	0.489375	−	0.872074i	$-0.337225\pi$
$761$	27.0000	0.978749	0.489375	+	0.872074i	$0.337225\pi$
$762$	0	0
$763$	2.00000	0.0724049
$764$	0	0
$765$	4.00000	0.144620
$766$	0	0
$767$	66.0000	2.38312
$768$	0	0
$769$	18.0000	0.649097	0.324548	−	0.945869i	$-0.394788\pi$
$769$	18.0000	0.649097	0.324548	+	0.945869i	$0.394788\pi$
$770$	0	0
$771$	22.0000	0.792311
$772$	0	0
$773$	−45.0000	−1.61854	−0.809269	−	0.587439i	$-0.800136\pi$
$773$	−45.0000	−1.61854	−0.809269	+	0.587439i	$0.800136\pi$
$774$	0	0
$775$	−20.0000	−0.718421
$776$	0	0
$777$	−3.00000	−0.107624
$778$	0	0
$779$	−24.0000	−0.859889
$780$	0	0
$781$	−8.00000	−0.286263
$782$	0	0
$783$	2.00000	0.0714742
$784$	0	0
$785$	18.0000	0.642448
$786$	0	0
$787$	−4.00000	−0.142585	−0.0712923	−	0.997455i	$-0.522712\pi$
$787$	−4.00000	−0.142585	−0.0712923	+	0.997455i	$0.522712\pi$
$788$	0	0
$789$	15.0000	0.534014
$790$	0	0
$791$	−4.00000	−0.142224
$792$	0	0
$793$	−24.0000	−0.852265
$794$	0	0
$795$	−7.00000	−0.248264
$796$	0	0
$797$	54.0000	1.91278	0.956389	−	0.292096i	$-0.0943526\pi$
$797$	54.0000	1.91278	0.956389	+	0.292096i	$0.0943526\pi$
$798$	0	0
$799$	12.0000	0.424529
$800$	0	0
$801$	9.00000	0.317999
$802$	0	0
$803$	−12.0000	−0.423471
$804$	0	0
$805$	8.00000	0.281963
$806$	0	0
$807$	−27.0000	−0.950445
$808$	0	0
$809$	49.0000	1.72275	0.861374	−	0.507971i	$-0.169604\pi$
$809$	49.0000	1.72275	0.861374	+	0.507971i	$0.169604\pi$
$810$	0	0
$811$	−23.0000	−0.807639	−0.403820	−	0.914839i	$-0.632318\pi$
$811$	−23.0000	−0.807639	−0.403820	+	0.914839i	$0.632318\pi$
$812$	0	0
$813$	−30.0000	−1.05215
$814$	0	0
$815$	9.00000	0.315256
$816$	0	0
$817$	32.0000	1.11954
$818$	0	0
$819$	−6.00000	−0.209657
$820$	0	0
$821$	10.0000	0.349002	0.174501	−	0.984657i	$-0.444169\pi$
$821$	10.0000	0.349002	0.174501	+	0.984657i	$0.444169\pi$
$822$	0	0
$823$	−26.0000	−0.906303	−0.453152	−	0.891434i	$-0.649700\pi$
$823$	−26.0000	−0.906303	−0.453152	+	0.891434i	$0.649700\pi$
$824$	0	0
$825$	−8.00000	−0.278524
$826$	0	0
$827$	0	0		−	1.00000i	$-0.5\pi$
$827$	0	0			1.00000i	$0.5\pi$
$828$	0	0
$829$	−31.0000	−1.07667	−0.538337	−	0.842729i	$-0.680947\pi$
$829$	−31.0000	−1.07667	−0.538337	+	0.842729i	$0.680947\pi$
$830$	0	0
$831$	9.00000	0.312207
$832$	0	0
$833$	24.0000	0.831551
$834$	0	0
$835$	1.00000	0.0346064
$836$	0	0
$837$	−5.00000	−0.172825
$838$	0	0
$839$	25.0000	0.863096	0.431548	−	0.902090i	$-0.357968\pi$
$839$	25.0000	0.863096	0.431548	+	0.902090i	$0.357968\pi$
$840$	0	0
$841$	−25.0000	−0.862069
$842$	0	0
$843$	−15.0000	−0.516627
$844$	0	0
$845$	−23.0000	−0.791224
$846$	0	0
$847$	−7.00000	−0.240523
$848$	0	0
$849$	−8.00000	−0.274559
$850$	0	0
$851$	−24.0000	−0.822709
$852$	0	0
$853$	−52.0000	−1.78045	−0.890223	−	0.455525i	$-0.849452\pi$
$853$	−52.0000	−1.78045	−0.890223	+	0.455525i	$0.849452\pi$
$854$	0	0
$855$	−4.00000	−0.136797
$856$	0	0
$857$	27.0000	0.922302	0.461151	−	0.887322i	$-0.347437\pi$
$857$	27.0000	0.922302	0.461151	+	0.887322i	$0.347437\pi$
$858$	0	0
$859$	50.0000	1.70598	0.852989	−	0.521929i	$-0.174787\pi$
$859$	50.0000	1.70598	0.852989	+	0.521929i	$0.174787\pi$
$860$	0	0
$861$	6.00000	0.204479
$862$	0	0
$863$	−12.0000	−0.408485	−0.204242	−	0.978920i	$-0.565473\pi$
$863$	−12.0000	−0.408485	−0.204242	+	0.978920i	$0.565473\pi$
$864$	0	0
$865$	4.00000	0.136004
$866$	0	0
$867$	1.00000	0.0339618
$868$	0	0
$869$	8.00000	0.271381
$870$	0	0
$871$	−42.0000	−1.42312
$872$	0	0
$873$	−13.0000	−0.439983
$874$	0	0
$875$	9.00000	0.304256
$876$	0	0
$877$	26.0000	0.877958	0.438979	−	0.898497i	$-0.355340\pi$
$877$	26.0000	0.877958	0.438979	+	0.898497i	$0.355340\pi$
$878$	0	0
$879$	−12.0000	−0.404750
$880$	0	0
$881$	−28.0000	−0.943344	−0.471672	−	0.881774i	$-0.656349\pi$
$881$	−28.0000	−0.943344	−0.471672	+	0.881774i	$0.656349\pi$
$882$	0	0
$883$	−6.00000	−0.201916	−0.100958	−	0.994891i	$-0.532191\pi$
$883$	−6.00000	−0.201916	−0.100958	+	0.994891i	$0.532191\pi$
$884$	0	0
$885$	−11.0000	−0.369761
$886$	0	0
$887$	−46.0000	−1.54453	−0.772264	−	0.635301i	$-0.780876\pi$
$887$	−46.0000	−1.54453	−0.772264	+	0.635301i	$0.780876\pi$
$888$	0	0
$889$	17.0000	0.570162
$890$	0	0
$891$	−2.00000	−0.0670025
$892$	0	0
$893$	−12.0000	−0.401565
$894$	0	0
$895$	2.00000	0.0668526
$896$	0	0
$897$	−48.0000	−1.60267
$898$	0	0
$899$	−10.0000	−0.333519
$900$	0	0
$901$	28.0000	0.932815
$902$	0	0
$903$	−8.00000	−0.266223
$904$	0	0
$905$	−20.0000	−0.664822
$906$	0	0
$907$	34.0000	1.12895	0.564476	−	0.825450i	$-0.309078\pi$
$907$	34.0000	1.12895	0.564476	+	0.825450i	$0.309078\pi$
$908$	0	0
$909$	1.00000	0.0331679
$910$	0	0
$911$	1.00000	0.0331315	0.0165657	−	0.999863i	$-0.494727\pi$
$911$	1.00000	0.0331315	0.0165657	+	0.999863i	$0.494727\pi$
$912$	0	0
$913$	−30.0000	−0.992855
$914$	0	0
$915$	4.00000	0.132236
$916$	0	0
$917$	12.0000	0.396275
$918$	0	0
$919$	11.0000	0.362857	0.181428	−	0.983404i	$-0.441928\pi$
$919$	11.0000	0.362857	0.181428	+	0.983404i	$0.441928\pi$
$920$	0	0
$921$	−27.0000	−0.889680
$922$	0	0
$923$	−24.0000	−0.789970
$924$	0	0
$925$	−12.0000	−0.394558
$926$	0	0
$927$	−6.00000	−0.197066
$928$	0	0
$929$	−33.0000	−1.08269	−0.541347	−	0.840799i	$-0.682086\pi$
$929$	−33.0000	−1.08269	−0.541347	+	0.840799i	$0.682086\pi$
$930$	0	0
$931$	−24.0000	−0.786568
$932$	0	0
$933$	−27.0000	−0.883940
$934$	0	0
$935$	−8.00000	−0.261628
$936$	0	0
$937$	−14.0000	−0.457360	−0.228680	−	0.973502i	$-0.573441\pi$
$937$	−14.0000	−0.457360	−0.228680	+	0.973502i	$0.573441\pi$
$938$	0	0
$939$	−10.0000	−0.326338
$940$	0	0
$941$	−6.00000	−0.195594	−0.0977972	−	0.995206i	$-0.531180\pi$
$941$	−6.00000	−0.195594	−0.0977972	+	0.995206i	$0.531180\pi$
$942$	0	0
$943$	48.0000	1.56310
$944$	0	0
$945$	1.00000	0.0325300
$946$	0	0
$947$	−58.0000	−1.88475	−0.942373	−	0.334563i	$-0.891411\pi$
$947$	−58.0000	−1.88475	−0.942373	+	0.334563i	$0.891411\pi$
$948$	0	0
$949$	−36.0000	−1.16861
$950$	0	0
$951$	−6.00000	−0.194563
$952$	0	0
$953$	26.0000	0.842223	0.421111	−	0.907009i	$-0.361640\pi$
$953$	26.0000	0.842223	0.421111	+	0.907009i	$0.361640\pi$
$954$	0	0
$955$	15.0000	0.485389
$956$	0	0
$957$	−4.00000	−0.129302
$958$	0	0
$959$	9.00000	0.290625
$960$	0	0
$961$	−6.00000	−0.193548
$962$	0	0
$963$	10.0000	0.322245
$964$	0	0
$965$	−16.0000	−0.515058
$966$	0	0
$967$	24.0000	0.771788	0.385894	−	0.922543i	$-0.373893\pi$
$967$	24.0000	0.771788	0.385894	+	0.922543i	$0.373893\pi$
$968$	0	0
$969$	16.0000	0.513994
$970$	0	0
$971$	47.0000	1.50830	0.754151	−	0.656701i	$-0.228049\pi$
$971$	47.0000	1.50830	0.754151	+	0.656701i	$0.228049\pi$
$972$	0	0
$973$	7.00000	0.224410
$974$	0	0
$975$	−24.0000	−0.768615
$976$	0	0
$977$	−12.0000	−0.383914	−0.191957	−	0.981403i	$-0.561483\pi$
$977$	−12.0000	−0.383914	−0.191957	+	0.981403i	$0.561483\pi$
$978$	0	0
$979$	−18.0000	−0.575282
$980$	0	0
$981$	2.00000	0.0638551
$982$	0	0
$983$	24.0000	0.765481	0.382741	−	0.923856i	$-0.374980\pi$
$983$	24.0000	0.765481	0.382741	+	0.923856i	$0.374980\pi$
$984$	0	0
$985$	−6.00000	−0.191176
$986$	0	0
$987$	3.00000	0.0954911
$988$	0	0
$989$	−64.0000	−2.03508
$990$	0	0
$991$	−20.0000	−0.635321	−0.317660	−	0.948205i	$-0.602897\pi$
$991$	−20.0000	−0.635321	−0.317660	+	0.948205i	$0.602897\pi$
$992$	0	0
$993$	20.0000	0.634681
$994$	0	0
$995$	−8.00000	−0.253617
$996$	0	0
$997$	−24.0000	−0.760088	−0.380044	−	0.924968i	$-0.624091\pi$
$997$	−24.0000	−0.760088	−0.380044	+	0.924968i	$0.624091\pi$
$998$	0	0
$999$	−3.00000	−0.0949158

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8016.2.a.b.1.1		1
4.3	odd	2		1002.2.a.c.1.1	✓	1
12.11	even	2		3006.2.a.f.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1002.2.a.c.1.1	✓	1	4.3	odd	2
3006.2.a.f.1.1		1	12.11	even	2
8016.2.a.b.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:	\( N \)	\(=\)	\( 8016 = 2^{4} \cdot 3 \cdot 167 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8016.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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