LMFDB - Embedded newform 8085.2.a.i.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8085,2,Mod(1,8085)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8085, 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("8085.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8085 = 3 \cdot 5 \cdot 7^{2} \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8085.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:	$64.5590500342$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 1155)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	8085.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q-1.00000 q^{3} -2.00000 q^{4} -1.00000 q^{5} +1.00000 q^{9} +O(q^{10})$

$q-1.00000 q^{3} -2.00000 q^{4} -1.00000 q^{5} +1.00000 q^{9} -1.00000 q^{11} +2.00000 q^{12} +1.00000 q^{13} +1.00000 q^{15} +4.00000 q^{16} -4.00000 q^{17} +2.00000 q^{19} +2.00000 q^{20} +1.00000 q^{23} +1.00000 q^{25} -1.00000 q^{27} -9.00000 q^{29} +1.00000 q^{33} -2.00000 q^{36} -1.00000 q^{39} +5.00000 q^{41} -11.0000 q^{43} +2.00000 q^{44} -1.00000 q^{45} +9.00000 q^{47} -4.00000 q^{48} +4.00000 q^{51} -2.00000 q^{52} +13.0000 q^{53} +1.00000 q^{55} -2.00000 q^{57} -4.00000 q^{59} -2.00000 q^{60} -2.00000 q^{61} -8.00000 q^{64} -1.00000 q^{65} -10.0000 q^{67} +8.00000 q^{68} -1.00000 q^{69} -2.00000 q^{73} -1.00000 q^{75} -4.00000 q^{76} +4.00000 q^{79} -4.00000 q^{80} +1.00000 q^{81} +10.0000 q^{83} +4.00000 q^{85} +9.00000 q^{87} -4.00000 q^{89} -2.00000 q^{92} -2.00000 q^{95} -14.0000 q^{97} -1.00000 q^{99} +O(q^{100})$

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		−	1.00000i	$-0.5\pi$
$2$	0	0			1.00000i	$0.5\pi$
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	−2.00000	−1.00000
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	2.00000	0.577350
$13$	1.00000	0.277350	0.138675	−	0.990338i	$-0.455716\pi$
$13$	1.00000	0.277350	0.138675	+	0.990338i	$0.455716\pi$
$14$	0	0
$15$	1.00000	0.258199
$16$	4.00000	1.00000
$17$	−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
$19$	2.00000	0.458831	0.229416	−	0.973329i	$-0.426318\pi$
$19$	2.00000	0.458831	0.229416	+	0.973329i	$0.426318\pi$
$20$	2.00000	0.447214
$21$	0	0
$22$	0	0
$23$	1.00000	0.208514	0.104257	−	0.994550i	$-0.466753\pi$
$23$	1.00000	0.208514	0.104257	+	0.994550i	$0.466753\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	−1.00000	−0.192450
$28$	0	0
$29$	−9.00000	−1.67126	−0.835629	−	0.549294i	$-0.814897\pi$
$29$	−9.00000	−1.67126	−0.835629	+	0.549294i	$0.814897\pi$
$30$	0	0
$31$	0	0		−	1.00000i	$-0.5\pi$
$31$	0	0			1.00000i	$0.5\pi$
$32$	0	0
$33$	1.00000	0.174078
$34$	0	0
$35$	0	0
$36$	−2.00000	−0.333333
$37$	0	0		−	1.00000i	$-0.5\pi$
$37$	0	0			1.00000i	$0.5\pi$
$38$	0	0
$39$	−1.00000	−0.160128
$40$	0	0
$41$	5.00000	0.780869	0.390434	−	0.920631i	$-0.372325\pi$
$41$	5.00000	0.780869	0.390434	+	0.920631i	$0.372325\pi$
$42$	0	0
$43$	−11.0000	−1.67748	−0.838742	−	0.544529i	$-0.816708\pi$
$43$	−11.0000	−1.67748	−0.838742	+	0.544529i	$0.816708\pi$
$44$	2.00000	0.301511
$45$	−1.00000	−0.149071
$46$	0	0
$47$	9.00000	1.31278	0.656392	−	0.754420i	$-0.272082\pi$
$47$	9.00000	1.31278	0.656392	+	0.754420i	$0.272082\pi$
$48$	−4.00000	−0.577350
$49$	0	0
$50$	0	0
$51$	4.00000	0.560112
$52$	−2.00000	−0.277350
$53$	13.0000	1.78569	0.892844	−	0.450367i	$-0.148707\pi$
$53$	13.0000	1.78569	0.892844	+	0.450367i	$0.148707\pi$
$54$	0	0
$55$	1.00000	0.134840
$56$	0	0
$57$	−2.00000	−0.264906
$58$	0	0
$59$	−4.00000	−0.520756	−0.260378	−	0.965507i	$-0.583847\pi$
$59$	−4.00000	−0.520756	−0.260378	+	0.965507i	$0.583847\pi$
$60$	−2.00000	−0.258199
$61$	−2.00000	−0.256074	−0.128037	−	0.991769i	$-0.540868\pi$
$61$	−2.00000	−0.256074	−0.128037	+	0.991769i	$0.540868\pi$
$62$	0	0
$63$	0	0
$64$	−8.00000	−1.00000
$65$	−1.00000	−0.124035
$66$	0	0
$67$	−10.0000	−1.22169	−0.610847	−	0.791748i	$-0.709171\pi$
$67$	−10.0000	−1.22169	−0.610847	+	0.791748i	$0.709171\pi$
$68$	8.00000	0.970143
$69$	−1.00000	−0.120386
$70$	0	0
$71$	0	0		−	1.00000i	$-0.5\pi$
$71$	0	0			1.00000i	$0.5\pi$
$72$	0	0
$73$	−2.00000	−0.234082	−0.117041	−	0.993127i	$-0.537341\pi$
$73$	−2.00000	−0.234082	−0.117041	+	0.993127i	$0.537341\pi$
$74$	0	0
$75$	−1.00000	−0.115470
$76$	−4.00000	−0.458831
$77$	0	0
$78$	0	0
$79$	4.00000	0.450035	0.225018	−	0.974355i	$-0.427756\pi$
$79$	4.00000	0.450035	0.225018	+	0.974355i	$0.427756\pi$
$80$	−4.00000	−0.447214
$81$	1.00000	0.111111
$82$	0	0
$83$	10.0000	1.09764	0.548821	−	0.835940i	$-0.315077\pi$
$83$	10.0000	1.09764	0.548821	+	0.835940i	$0.315077\pi$
$84$	0	0
$85$	4.00000	0.433861
$86$	0	0
$87$	9.00000	0.964901
$88$	0	0
$89$	−4.00000	−0.423999	−0.212000	−	0.977270i	$-0.567998\pi$
$89$	−4.00000	−0.423999	−0.212000	+	0.977270i	$0.567998\pi$
$90$	0	0
$91$	0	0
$92$	−2.00000	−0.208514
$93$	0	0
$94$	0	0
$95$	−2.00000	−0.205196
$96$	0	0
$97$	−14.0000	−1.42148	−0.710742	−	0.703452i	$-0.751641\pi$
$97$	−14.0000	−1.42148	−0.710742	+	0.703452i	$0.751641\pi$
$98$	0	0
$99$	−1.00000	−0.100504
$100$	−2.00000	−0.200000
$101$	−5.00000	−0.497519	−0.248759	−	0.968565i	$-0.580023\pi$
$101$	−5.00000	−0.497519	−0.248759	+	0.968565i	$0.580023\pi$
$102$	0	0
$103$	−12.0000	−1.18240	−0.591198	−	0.806527i	$-0.701345\pi$
$103$	−12.0000	−1.18240	−0.591198	+	0.806527i	$0.701345\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−6.00000	−0.580042	−0.290021	−	0.957020i	$-0.593662\pi$
$107$	−6.00000	−0.580042	−0.290021	+	0.957020i	$0.593662\pi$
$108$	2.00000	0.192450
$109$	−4.00000	−0.383131	−0.191565	−	0.981480i	$-0.561356\pi$
$109$	−4.00000	−0.383131	−0.191565	+	0.981480i	$0.561356\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−6.00000	−0.564433	−0.282216	−	0.959351i	$-0.591070\pi$
$113$	−6.00000	−0.564433	−0.282216	+	0.959351i	$0.591070\pi$
$114$	0	0
$115$	−1.00000	−0.0932505
$116$	18.0000	1.67126
$117$	1.00000	0.0924500
$118$	0	0
$119$	0	0
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	−5.00000	−0.450835
$124$	0	0
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	−13.0000	−1.15356	−0.576782	−	0.816898i	$-0.695692\pi$
$127$	−13.0000	−1.15356	−0.576782	+	0.816898i	$0.695692\pi$
$128$	0	0
$129$	11.0000	0.968496
$130$	0	0
$131$	−7.00000	−0.611593	−0.305796	−	0.952097i	$-0.598923\pi$
$131$	−7.00000	−0.611593	−0.305796	+	0.952097i	$0.598923\pi$
$132$	−2.00000	−0.174078
$133$	0	0
$134$	0	0
$135$	1.00000	0.0860663
$136$	0	0
$137$	−7.00000	−0.598050	−0.299025	−	0.954245i	$-0.596661\pi$
$137$	−7.00000	−0.598050	−0.299025	+	0.954245i	$0.596661\pi$
$138$	0	0
$139$	2.00000	0.169638	0.0848189	−	0.996396i	$-0.472969\pi$
$139$	2.00000	0.169638	0.0848189	+	0.996396i	$0.472969\pi$
$140$	0	0
$141$	−9.00000	−0.757937
$142$	0	0
$143$	−1.00000	−0.0836242
$144$	4.00000	0.333333
$145$	9.00000	0.747409
$146$	0	0
$147$	0	0
$148$	0	0
$149$	3.00000	0.245770	0.122885	−	0.992421i	$-0.460785\pi$
$149$	3.00000	0.245770	0.122885	+	0.992421i	$0.460785\pi$
$150$	0	0
$151$	−12.0000	−0.976546	−0.488273	−	0.872691i	$-0.662373\pi$
$151$	−12.0000	−0.976546	−0.488273	+	0.872691i	$0.662373\pi$
$152$	0	0
$153$	−4.00000	−0.323381
$154$	0	0
$155$	0	0
$156$	2.00000	0.160128
$157$	0	0		−	1.00000i	$-0.5\pi$
$157$	0	0			1.00000i	$0.5\pi$
$158$	0	0
$159$	−13.0000	−1.03097
$160$	0	0
$161$	0	0
$162$	0	0
$163$	16.0000	1.25322	0.626608	−	0.779334i	$-0.284443\pi$
$163$	16.0000	1.25322	0.626608	+	0.779334i	$0.284443\pi$
$164$	−10.0000	−0.780869
$165$	−1.00000	−0.0778499
$166$	0	0
$167$	12.0000	0.928588	0.464294	−	0.885681i	$-0.346308\pi$
$167$	12.0000	0.928588	0.464294	+	0.885681i	$0.346308\pi$
$168$	0	0
$169$	−12.0000	−0.923077
$170$	0	0
$171$	2.00000	0.152944
$172$	22.0000	1.67748
$173$	22.0000	1.67263	0.836315	−	0.548250i	$-0.184706\pi$
$173$	22.0000	1.67263	0.836315	+	0.548250i	$0.184706\pi$
$174$	0	0
$175$	0	0
$176$	−4.00000	−0.301511
$177$	4.00000	0.300658
$178$	0	0
$179$	−14.0000	−1.04641	−0.523205	−	0.852207i	$-0.675264\pi$
$179$	−14.0000	−1.04641	−0.523205	+	0.852207i	$0.675264\pi$
$180$	2.00000	0.149071
$181$	−13.0000	−0.966282	−0.483141	−	0.875542i	$-0.660504\pi$
$181$	−13.0000	−0.966282	−0.483141	+	0.875542i	$0.660504\pi$
$182$	0	0
$183$	2.00000	0.147844
$184$	0	0
$185$	0	0
$186$	0	0
$187$	4.00000	0.292509
$188$	−18.0000	−1.31278
$189$	0	0
$190$	0	0
$191$	−12.0000	−0.868290	−0.434145	−	0.900843i	$-0.642949\pi$
$191$	−12.0000	−0.868290	−0.434145	+	0.900843i	$0.642949\pi$
$192$	8.00000	0.577350
$193$	−5.00000	−0.359908	−0.179954	−	0.983675i	$-0.557595\pi$
$193$	−5.00000	−0.359908	−0.179954	+	0.983675i	$0.557595\pi$
$194$	0	0
$195$	1.00000	0.0716115
$196$	0	0
$197$	12.0000	0.854965	0.427482	−	0.904024i	$-0.359401\pi$
$197$	12.0000	0.854965	0.427482	+	0.904024i	$0.359401\pi$
$198$	0	0
$199$	17.0000	1.20510	0.602549	−	0.798082i	$-0.294152\pi$
$199$	17.0000	1.20510	0.602549	+	0.798082i	$0.294152\pi$
$200$	0	0
$201$	10.0000	0.705346
$202$	0	0
$203$	0	0
$204$	−8.00000	−0.560112
$205$	−5.00000	−0.349215
$206$	0	0
$207$	1.00000	0.0695048
$208$	4.00000	0.277350
$209$	−2.00000	−0.138343
$210$	0	0
$211$	20.0000	1.37686	0.688428	−	0.725304i	$-0.258301\pi$
$211$	20.0000	1.37686	0.688428	+	0.725304i	$0.258301\pi$
$212$	−26.0000	−1.78569
$213$	0	0
$214$	0	0
$215$	11.0000	0.750194
$216$	0	0
$217$	0	0
$218$	0	0
$219$	2.00000	0.135147
$220$	−2.00000	−0.134840
$221$	−4.00000	−0.269069
$222$	0	0
$223$	10.0000	0.669650	0.334825	−	0.942280i	$-0.391323\pi$
$223$	10.0000	0.669650	0.334825	+	0.942280i	$0.391323\pi$
$224$	0	0
$225$	1.00000	0.0666667
$226$	0	0
$227$	18.0000	1.19470	0.597351	−	0.801980i	$-0.296220\pi$
$227$	18.0000	1.19470	0.597351	+	0.801980i	$0.296220\pi$
$228$	4.00000	0.264906
$229$	29.0000	1.91637	0.958187	−	0.286143i	$-0.0923732\pi$
$229$	29.0000	1.91637	0.958187	+	0.286143i	$0.0923732\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	8.00000	0.524097	0.262049	−	0.965055i	$-0.415602\pi$
$233$	8.00000	0.524097	0.262049	+	0.965055i	$0.415602\pi$
$234$	0	0
$235$	−9.00000	−0.587095
$236$	8.00000	0.520756
$237$	−4.00000	−0.259828
$238$	0	0
$239$	−8.00000	−0.517477	−0.258738	−	0.965947i	$-0.583307\pi$
$239$	−8.00000	−0.517477	−0.258738	+	0.965947i	$0.583307\pi$
$240$	4.00000	0.258199
$241$	4.00000	0.257663	0.128831	−	0.991667i	$-0.458877\pi$
$241$	4.00000	0.257663	0.128831	+	0.991667i	$0.458877\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	4.00000	0.256074
$245$	0	0
$246$	0	0
$247$	2.00000	0.127257
$248$	0	0
$249$	−10.0000	−0.633724
$250$	0	0
$251$	10.0000	0.631194	0.315597	−	0.948893i	$-0.397795\pi$
$251$	10.0000	0.631194	0.315597	+	0.948893i	$0.397795\pi$
$252$	0	0
$253$	−1.00000	−0.0628695
$254$	0	0
$255$	−4.00000	−0.250490
$256$	16.0000	1.00000
$257$	25.0000	1.55946	0.779729	−	0.626118i	$-0.215357\pi$
$257$	25.0000	1.55946	0.779729	+	0.626118i	$0.215357\pi$
$258$	0	0
$259$	0	0
$260$	2.00000	0.124035
$261$	−9.00000	−0.557086
$262$	0	0
$263$	−20.0000	−1.23325	−0.616626	−	0.787256i	$-0.711501\pi$
$263$	−20.0000	−1.23325	−0.616626	+	0.787256i	$0.711501\pi$
$264$	0	0
$265$	−13.0000	−0.798584
$266$	0	0
$267$	4.00000	0.244796
$268$	20.0000	1.22169
$269$	4.00000	0.243884	0.121942	−	0.992537i	$-0.461088\pi$
$269$	4.00000	0.243884	0.121942	+	0.992537i	$0.461088\pi$
$270$	0	0
$271$	26.0000	1.57939	0.789694	−	0.613501i	$-0.210239\pi$
$271$	26.0000	1.57939	0.789694	+	0.613501i	$0.210239\pi$
$272$	−16.0000	−0.970143
$273$	0	0
$274$	0	0
$275$	−1.00000	−0.0603023
$276$	2.00000	0.120386
$277$	19.0000	1.14160	0.570800	−	0.821089i	$-0.306633\pi$
$277$	19.0000	1.14160	0.570800	+	0.821089i	$0.306633\pi$
$278$	0	0
$279$	0	0
$280$	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
$283$	28.0000	1.66443	0.832214	−	0.554455i	$-0.187073\pi$
$283$	28.0000	1.66443	0.832214	+	0.554455i	$0.187073\pi$
$284$	0	0
$285$	2.00000	0.118470
$286$	0	0
$287$	0	0
$288$	0	0
$289$	−1.00000	−0.0588235
$290$	0	0
$291$	14.0000	0.820695
$292$	4.00000	0.234082
$293$	22.0000	1.28525	0.642627	−	0.766179i	$-0.277845\pi$
$293$	22.0000	1.28525	0.642627	+	0.766179i	$0.277845\pi$
$294$	0	0
$295$	4.00000	0.232889
$296$	0	0
$297$	1.00000	0.0580259
$298$	0	0
$299$	1.00000	0.0578315
$300$	2.00000	0.115470
$301$	0	0
$302$	0	0
$303$	5.00000	0.287242
$304$	8.00000	0.458831
$305$	2.00000	0.114520
$306$	0	0
$307$	−23.0000	−1.31268	−0.656340	−	0.754466i	$-0.727896\pi$
$307$	−23.0000	−1.31268	−0.656340	+	0.754466i	$0.727896\pi$
$308$	0	0
$309$	12.0000	0.682656
$310$	0	0
$311$	−20.0000	−1.13410	−0.567048	−	0.823685i	$-0.691915\pi$
$311$	−20.0000	−1.13410	−0.567048	+	0.823685i	$0.691915\pi$
$312$	0	0
$313$	−6.00000	−0.339140	−0.169570	−	0.985518i	$-0.554238\pi$
$313$	−6.00000	−0.339140	−0.169570	+	0.985518i	$0.554238\pi$
$314$	0	0
$315$	0	0
$316$	−8.00000	−0.450035
$317$	−1.00000	−0.0561656	−0.0280828	−	0.999606i	$-0.508940\pi$
$317$	−1.00000	−0.0561656	−0.0280828	+	0.999606i	$0.508940\pi$
$318$	0	0
$319$	9.00000	0.503903
$320$	8.00000	0.447214
$321$	6.00000	0.334887
$322$	0	0
$323$	−8.00000	−0.445132
$324$	−2.00000	−0.111111
$325$	1.00000	0.0554700
$326$	0	0
$327$	4.00000	0.221201
$328$	0	0
$329$	0	0
$330$	0	0
$331$	25.0000	1.37412	0.687062	−	0.726599i	$-0.258900\pi$
$331$	25.0000	1.37412	0.687062	+	0.726599i	$0.258900\pi$
$332$	−20.0000	−1.09764
$333$	0	0
$334$	0	0
$335$	10.0000	0.546358
$336$	0	0
$337$	2.00000	0.108947	0.0544735	−	0.998515i	$-0.482652\pi$
$337$	2.00000	0.108947	0.0544735	+	0.998515i	$0.482652\pi$
$338$	0	0
$339$	6.00000	0.325875
$340$	−8.00000	−0.433861
$341$	0	0
$342$	0	0
$343$	0	0
$344$	0	0
$345$	1.00000	0.0538382
$346$	0	0
$347$	−2.00000	−0.107366	−0.0536828	−	0.998558i	$-0.517096\pi$
$347$	−2.00000	−0.107366	−0.0536828	+	0.998558i	$0.517096\pi$
$348$	−18.0000	−0.964901
$349$	14.0000	0.749403	0.374701	−	0.927146i	$-0.377745\pi$
$349$	14.0000	0.749403	0.374701	+	0.927146i	$0.377745\pi$
$350$	0	0
$351$	−1.00000	−0.0533761
$352$	0	0
$353$	11.0000	0.585471	0.292735	−	0.956193i	$-0.405434\pi$
$353$	11.0000	0.585471	0.292735	+	0.956193i	$0.405434\pi$
$354$	0	0
$355$	0	0
$356$	8.00000	0.423999
$357$	0	0
$358$	0	0
$359$	25.0000	1.31945	0.659725	−	0.751507i	$-0.270673\pi$
$359$	25.0000	1.31945	0.659725	+	0.751507i	$0.270673\pi$
$360$	0	0
$361$	−15.0000	−0.789474
$362$	0	0
$363$	−1.00000	−0.0524864
$364$	0	0
$365$	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	$0.433045\pi$
$368$	4.00000	0.208514
$369$	5.00000	0.260290
$370$	0	0
$371$	0	0
$372$	0	0
$373$	−17.0000	−0.880227	−0.440113	−	0.897942i	$-0.645062\pi$
$373$	−17.0000	−0.880227	−0.440113	+	0.897942i	$0.645062\pi$
$374$	0	0
$375$	1.00000	0.0516398
$376$	0	0
$377$	−9.00000	−0.463524
$378$	0	0
$379$	−32.0000	−1.64373	−0.821865	−	0.569683i	$-0.807066\pi$
$379$	−32.0000	−1.64373	−0.821865	+	0.569683i	$0.807066\pi$
$380$	4.00000	0.205196
$381$	13.0000	0.666010
$382$	0	0
$383$	−24.0000	−1.22634	−0.613171	−	0.789950i	$-0.710106\pi$
$383$	−24.0000	−1.22634	−0.613171	+	0.789950i	$0.710106\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−11.0000	−0.559161
$388$	28.0000	1.42148
$389$	18.0000	0.912636	0.456318	−	0.889817i	$-0.349168\pi$
$389$	18.0000	0.912636	0.456318	+	0.889817i	$0.349168\pi$
$390$	0	0
$391$	−4.00000	−0.202289
$392$	0	0
$393$	7.00000	0.353103
$394$	0	0
$395$	−4.00000	−0.201262
$396$	2.00000	0.100504
$397$	−28.0000	−1.40528	−0.702640	−	0.711546i	$-0.747995\pi$
$397$	−28.0000	−1.40528	−0.702640	+	0.711546i	$0.747995\pi$
$398$	0	0
$399$	0	0
$400$	4.00000	0.200000
$401$	16.0000	0.799002	0.399501	−	0.916733i	$-0.369183\pi$
$401$	16.0000	0.799002	0.399501	+	0.916733i	$0.369183\pi$
$402$	0	0
$403$	0	0
$404$	10.0000	0.497519
$405$	−1.00000	−0.0496904
$406$	0	0
$407$	0	0
$408$	0	0
$409$	−8.00000	−0.395575	−0.197787	−	0.980245i	$-0.563376\pi$
$409$	−8.00000	−0.395575	−0.197787	+	0.980245i	$0.563376\pi$
$410$	0	0
$411$	7.00000	0.345285
$412$	24.0000	1.18240
$413$	0	0
$414$	0	0
$415$	−10.0000	−0.490881
$416$	0	0
$417$	−2.00000	−0.0979404
$418$	0	0
$419$	0	0		−	1.00000i	$-0.5\pi$
$419$	0	0			1.00000i	$0.5\pi$
$420$	0	0
$421$	−26.0000	−1.26716	−0.633581	−	0.773676i	$-0.718416\pi$
$421$	−26.0000	−1.26716	−0.633581	+	0.773676i	$0.718416\pi$
$422$	0	0
$423$	9.00000	0.437595
$424$	0	0
$425$	−4.00000	−0.194029
$426$	0	0
$427$	0	0
$428$	12.0000	0.580042
$429$	1.00000	0.0482805
$430$	0	0
$431$	27.0000	1.30054	0.650272	−	0.759701i	$-0.274655\pi$
$431$	27.0000	1.30054	0.650272	+	0.759701i	$0.274655\pi$
$432$	−4.00000	−0.192450
$433$	32.0000	1.53782	0.768911	−	0.639356i	$-0.220799\pi$
$433$	32.0000	1.53782	0.768911	+	0.639356i	$0.220799\pi$
$434$	0	0
$435$	−9.00000	−0.431517
$436$	8.00000	0.383131
$437$	2.00000	0.0956730
$438$	0	0
$439$	−6.00000	−0.286364	−0.143182	−	0.989696i	$-0.545733\pi$
$439$	−6.00000	−0.286364	−0.143182	+	0.989696i	$0.545733\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	0	0		−	1.00000i	$-0.5\pi$
$443$	0	0			1.00000i	$0.5\pi$
$444$	0	0
$445$	4.00000	0.189618
$446$	0	0
$447$	−3.00000	−0.141895
$448$	0	0
$449$	−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
$451$	−5.00000	−0.235441
$452$	12.0000	0.564433
$453$	12.0000	0.563809
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−33.0000	−1.54367	−0.771837	−	0.635820i	$-0.780662\pi$
$457$	−33.0000	−1.54367	−0.771837	+	0.635820i	$0.780662\pi$
$458$	0	0
$459$	4.00000	0.186704
$460$	2.00000	0.0932505
$461$	37.0000	1.72326	0.861631	−	0.507535i	$-0.169443\pi$
$461$	37.0000	1.72326	0.861631	+	0.507535i	$0.169443\pi$
$462$	0	0
$463$	14.0000	0.650635	0.325318	−	0.945605i	$-0.394529\pi$
$463$	14.0000	0.650635	0.325318	+	0.945605i	$0.394529\pi$
$464$	−36.0000	−1.67126
$465$	0	0
$466$	0	0
$467$	11.0000	0.509019	0.254510	−	0.967070i	$-0.418086\pi$
$467$	11.0000	0.509019	0.254510	+	0.967070i	$0.418086\pi$
$468$	−2.00000	−0.0924500
$469$	0	0
$470$	0	0
$471$	0	0
$472$	0	0
$473$	11.0000	0.505781
$474$	0	0
$475$	2.00000	0.0917663
$476$	0	0
$477$	13.0000	0.595229
$478$	0	0
$479$	8.00000	0.365529	0.182765	−	0.983157i	$-0.441495\pi$
$479$	8.00000	0.365529	0.182765	+	0.983157i	$0.441495\pi$
$480$	0	0
$481$	0	0
$482$	0	0
$483$	0	0
$484$	−2.00000	−0.0909091
$485$	14.0000	0.635707
$486$	0	0
$487$	32.0000	1.45006	0.725029	−	0.688718i	$-0.241826\pi$
$487$	32.0000	1.45006	0.725029	+	0.688718i	$0.241826\pi$
$488$	0	0
$489$	−16.0000	−0.723545
$490$	0	0
$491$	35.0000	1.57953	0.789764	−	0.613411i	$-0.210203\pi$
$491$	35.0000	1.57953	0.789764	+	0.613411i	$0.210203\pi$
$492$	10.0000	0.450835
$493$	36.0000	1.62136
$494$	0	0
$495$	1.00000	0.0449467
$496$	0	0
$497$	0	0
$498$	0	0
$499$	13.0000	0.581960	0.290980	−	0.956729i	$-0.406019\pi$
$499$	13.0000	0.581960	0.290980	+	0.956729i	$0.406019\pi$
$500$	2.00000	0.0894427
$501$	−12.0000	−0.536120
$502$	0	0
$503$	42.0000	1.87269	0.936344	−	0.351085i	$-0.114187\pi$
$503$	42.0000	1.87269	0.936344	+	0.351085i	$0.114187\pi$
$504$	0	0
$505$	5.00000	0.222497
$506$	0	0
$507$	12.0000	0.532939
$508$	26.0000	1.15356
$509$	34.0000	1.50702	0.753512	−	0.657434i	$-0.228358\pi$
$509$	34.0000	1.50702	0.753512	+	0.657434i	$0.228358\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	−2.00000	−0.0883022
$514$	0	0
$515$	12.0000	0.528783
$516$	−22.0000	−0.968496
$517$	−9.00000	−0.395820
$518$	0	0
$519$	−22.0000	−0.965693
$520$	0	0
$521$	−18.0000	−0.788594	−0.394297	−	0.918983i	$-0.629012\pi$
$521$	−18.0000	−0.788594	−0.394297	+	0.918983i	$0.629012\pi$
$522$	0	0
$523$	21.0000	0.918266	0.459133	−	0.888368i	$-0.348160\pi$
$523$	21.0000	0.918266	0.459133	+	0.888368i	$0.348160\pi$
$524$	14.0000	0.611593
$525$	0	0
$526$	0	0
$527$	0	0
$528$	4.00000	0.174078
$529$	−22.0000	−0.956522
$530$	0	0
$531$	−4.00000	−0.173585
$532$	0	0
$533$	5.00000	0.216574
$534$	0	0
$535$	6.00000	0.259403
$536$	0	0
$537$	14.0000	0.604145
$538$	0	0
$539$	0	0
$540$	−2.00000	−0.0860663
$541$	−4.00000	−0.171973	−0.0859867	−	0.996296i	$-0.527404\pi$
$541$	−4.00000	−0.171973	−0.0859867	+	0.996296i	$0.527404\pi$
$542$	0	0
$543$	13.0000	0.557883
$544$	0	0
$545$	4.00000	0.171341
$546$	0	0
$547$	−37.0000	−1.58201	−0.791003	−	0.611812i	$-0.790441\pi$
$547$	−37.0000	−1.58201	−0.791003	+	0.611812i	$0.790441\pi$
$548$	14.0000	0.598050
$549$	−2.00000	−0.0853579
$550$	0	0
$551$	−18.0000	−0.766826
$552$	0	0
$553$	0	0
$554$	0	0
$555$	0	0
$556$	−4.00000	−0.169638
$557$	0	0		−	1.00000i	$-0.5\pi$
$557$	0	0			1.00000i	$0.5\pi$
$558$	0	0
$559$	−11.0000	−0.465250
$560$	0	0
$561$	−4.00000	−0.168880
$562$	0	0
$563$	−12.0000	−0.505740	−0.252870	−	0.967500i	$-0.581374\pi$
$563$	−12.0000	−0.505740	−0.252870	+	0.967500i	$0.581374\pi$
$564$	18.0000	0.757937
$565$	6.00000	0.252422
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−1.00000	−0.0419222	−0.0209611	−	0.999780i	$-0.506673\pi$
$569$	−1.00000	−0.0419222	−0.0209611	+	0.999780i	$0.506673\pi$
$570$	0	0
$571$	4.00000	0.167395	0.0836974	−	0.996491i	$-0.473327\pi$
$571$	4.00000	0.167395	0.0836974	+	0.996491i	$0.473327\pi$
$572$	2.00000	0.0836242
$573$	12.0000	0.501307
$574$	0	0
$575$	1.00000	0.0417029
$576$	−8.00000	−0.333333
$577$	−12.0000	−0.499567	−0.249783	−	0.968302i	$-0.580359\pi$
$577$	−12.0000	−0.499567	−0.249783	+	0.968302i	$0.580359\pi$
$578$	0	0
$579$	5.00000	0.207793
$580$	−18.0000	−0.747409
$581$	0	0
$582$	0	0
$583$	−13.0000	−0.538405
$584$	0	0
$585$	−1.00000	−0.0413449
$586$	0	0
$587$	7.00000	0.288921	0.144460	−	0.989511i	$-0.453855\pi$
$587$	7.00000	0.288921	0.144460	+	0.989511i	$0.453855\pi$
$588$	0	0
$589$	0	0
$590$	0	0
$591$	−12.0000	−0.493614
$592$	0	0
$593$	−16.0000	−0.657041	−0.328521	−	0.944497i	$-0.606550\pi$
$593$	−16.0000	−0.657041	−0.328521	+	0.944497i	$0.606550\pi$
$594$	0	0
$595$	0	0
$596$	−6.00000	−0.245770
$597$	−17.0000	−0.695764
$598$	0	0
$599$	−40.0000	−1.63436	−0.817178	−	0.576386i	$-0.804463\pi$
$599$	−40.0000	−1.63436	−0.817178	+	0.576386i	$0.804463\pi$
$600$	0	0
$601$	−38.0000	−1.55005	−0.775026	−	0.631929i	$-0.782263\pi$
$601$	−38.0000	−1.55005	−0.775026	+	0.631929i	$0.782263\pi$
$602$	0	0
$603$	−10.0000	−0.407231
$604$	24.0000	0.976546
$605$	−1.00000	−0.0406558
$606$	0	0
$607$	−19.0000	−0.771186	−0.385593	−	0.922669i	$-0.626003\pi$
$607$	−19.0000	−0.771186	−0.385593	+	0.922669i	$0.626003\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	9.00000	0.364101
$612$	8.00000	0.323381
$613$	31.0000	1.25208	0.626039	−	0.779792i	$-0.284675\pi$
$613$	31.0000	1.25208	0.626039	+	0.779792i	$0.284675\pi$
$614$	0	0
$615$	5.00000	0.201619
$616$	0	0
$617$	−13.0000	−0.523360	−0.261680	−	0.965155i	$-0.584277\pi$
$617$	−13.0000	−0.523360	−0.261680	+	0.965155i	$0.584277\pi$
$618$	0	0
$619$	−28.0000	−1.12542	−0.562708	−	0.826656i	$-0.690240\pi$
$619$	−28.0000	−1.12542	−0.562708	+	0.826656i	$0.690240\pi$
$620$	0	0
$621$	−1.00000	−0.0401286
$622$	0	0
$623$	0	0
$624$	−4.00000	−0.160128
$625$	1.00000	0.0400000
$626$	0	0
$627$	2.00000	0.0798723
$628$	0	0
$629$	0	0
$630$	0	0
$631$	19.0000	0.756378	0.378189	−	0.925728i	$-0.376547\pi$
$631$	19.0000	0.756378	0.378189	+	0.925728i	$0.376547\pi$
$632$	0	0
$633$	−20.0000	−0.794929
$634$	0	0
$635$	13.0000	0.515889
$636$	26.0000	1.03097
$637$	0	0
$638$	0	0
$639$	0	0
$640$	0	0
$641$	38.0000	1.50091	0.750455	−	0.660922i	$-0.229834\pi$
$641$	38.0000	1.50091	0.750455	+	0.660922i	$0.229834\pi$
$642$	0	0
$643$	32.0000	1.26196	0.630978	−	0.775800i	$-0.282654\pi$
$643$	32.0000	1.26196	0.630978	+	0.775800i	$0.282654\pi$
$644$	0	0
$645$	−11.0000	−0.433125
$646$	0	0
$647$	45.0000	1.76913	0.884566	−	0.466415i	$-0.154454\pi$
$647$	45.0000	1.76913	0.884566	+	0.466415i	$0.154454\pi$
$648$	0	0
$649$	4.00000	0.157014
$650$	0	0
$651$	0	0
$652$	−32.0000	−1.25322
$653$	41.0000	1.60445	0.802227	−	0.597019i	$-0.203648\pi$
$653$	41.0000	1.60445	0.802227	+	0.597019i	$0.203648\pi$
$654$	0	0
$655$	7.00000	0.273513
$656$	20.0000	0.780869
$657$	−2.00000	−0.0780274
$658$	0	0
$659$	−7.00000	−0.272681	−0.136341	−	0.990662i	$-0.543534\pi$
$659$	−7.00000	−0.272681	−0.136341	+	0.990662i	$0.543534\pi$
$660$	2.00000	0.0778499
$661$	25.0000	0.972387	0.486194	−	0.873851i	$-0.338385\pi$
$661$	25.0000	0.972387	0.486194	+	0.873851i	$0.338385\pi$
$662$	0	0
$663$	4.00000	0.155347
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−9.00000	−0.348481
$668$	−24.0000	−0.928588
$669$	−10.0000	−0.386622
$670$	0	0
$671$	2.00000	0.0772091
$672$	0	0
$673$	23.0000	0.886585	0.443292	−	0.896377i	$-0.353810\pi$
$673$	23.0000	0.886585	0.443292	+	0.896377i	$0.353810\pi$
$674$	0	0
$675$	−1.00000	−0.0384900
$676$	24.0000	0.923077
$677$	−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
$679$	0	0
$680$	0	0
$681$	−18.0000	−0.689761
$682$	0	0
$683$	−12.0000	−0.459167	−0.229584	−	0.973289i	$-0.573736\pi$
$683$	−12.0000	−0.459167	−0.229584	+	0.973289i	$0.573736\pi$
$684$	−4.00000	−0.152944
$685$	7.00000	0.267456
$686$	0	0
$687$	−29.0000	−1.10642
$688$	−44.0000	−1.67748
$689$	13.0000	0.495261
$690$	0	0
$691$	32.0000	1.21734	0.608669	−	0.793424i	$-0.291704\pi$
$691$	32.0000	1.21734	0.608669	+	0.793424i	$0.291704\pi$
$692$	−44.0000	−1.67263
$693$	0	0
$694$	0	0
$695$	−2.00000	−0.0758643
$696$	0	0
$697$	−20.0000	−0.757554
$698$	0	0
$699$	−8.00000	−0.302588
$700$	0	0
$701$	2.00000	0.0755390	0.0377695	−	0.999286i	$-0.487975\pi$
$701$	2.00000	0.0755390	0.0377695	+	0.999286i	$0.487975\pi$
$702$	0	0
$703$	0	0
$704$	8.00000	0.301511
$705$	9.00000	0.338960
$706$	0	0
$707$	0	0
$708$	−8.00000	−0.300658
$709$	−5.00000	−0.187779	−0.0938895	−	0.995583i	$-0.529930\pi$
$709$	−5.00000	−0.187779	−0.0938895	+	0.995583i	$0.529930\pi$
$710$	0	0
$711$	4.00000	0.150012
$712$	0	0
$713$	0	0
$714$	0	0
$715$	1.00000	0.0373979
$716$	28.0000	1.04641
$717$	8.00000	0.298765
$718$	0	0
$719$	−12.0000	−0.447524	−0.223762	−	0.974644i	$-0.571834\pi$
$719$	−12.0000	−0.447524	−0.223762	+	0.974644i	$0.571834\pi$
$720$	−4.00000	−0.149071
$721$	0	0
$722$	0	0
$723$	−4.00000	−0.148762
$724$	26.0000	0.966282
$725$	−9.00000	−0.334252
$726$	0	0
$727$	26.0000	0.964287	0.482143	−	0.876092i	$-0.339858\pi$
$727$	26.0000	0.964287	0.482143	+	0.876092i	$0.339858\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	44.0000	1.62740
$732$	−4.00000	−0.147844
$733$	34.0000	1.25582	0.627909	−	0.778287i	$-0.283911\pi$
$733$	34.0000	1.25582	0.627909	+	0.778287i	$0.283911\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	10.0000	0.368355
$738$	0	0
$739$	12.0000	0.441427	0.220714	−	0.975339i	$-0.429161\pi$
$739$	12.0000	0.441427	0.220714	+	0.975339i	$0.429161\pi$
$740$	0	0
$741$	−2.00000	−0.0734718
$742$	0	0
$743$	12.0000	0.440237	0.220119	−	0.975473i	$-0.429356\pi$
$743$	12.0000	0.440237	0.220119	+	0.975473i	$0.429356\pi$
$744$	0	0
$745$	−3.00000	−0.109911
$746$	0	0
$747$	10.0000	0.365881
$748$	−8.00000	−0.292509
$749$	0	0
$750$	0	0
$751$	−4.00000	−0.145962	−0.0729810	−	0.997333i	$-0.523251\pi$
$751$	−4.00000	−0.145962	−0.0729810	+	0.997333i	$0.523251\pi$
$752$	36.0000	1.31278
$753$	−10.0000	−0.364420
$754$	0	0
$755$	12.0000	0.436725
$756$	0	0
$757$	−26.0000	−0.944986	−0.472493	−	0.881334i	$-0.656646\pi$
$757$	−26.0000	−0.944986	−0.472493	+	0.881334i	$0.656646\pi$
$758$	0	0
$759$	1.00000	0.0362977
$760$	0	0
$761$	−23.0000	−0.833749	−0.416875	−	0.908964i	$-0.636875\pi$
$761$	−23.0000	−0.833749	−0.416875	+	0.908964i	$0.636875\pi$
$762$	0	0
$763$	0	0
$764$	24.0000	0.868290
$765$	4.00000	0.144620
$766$	0	0
$767$	−4.00000	−0.144432
$768$	−16.0000	−0.577350
$769$	22.0000	0.793340	0.396670	−	0.917961i	$-0.370166\pi$
$769$	22.0000	0.793340	0.396670	+	0.917961i	$0.370166\pi$
$770$	0	0
$771$	−25.0000	−0.900353
$772$	10.0000	0.359908
$773$	−26.0000	−0.935155	−0.467578	−	0.883952i	$-0.654873\pi$
$773$	−26.0000	−0.935155	−0.467578	+	0.883952i	$0.654873\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	0	0
$778$	0	0
$779$	10.0000	0.358287
$780$	−2.00000	−0.0716115
$781$	0	0
$782$	0	0
$783$	9.00000	0.321634
$784$	0	0
$785$	0	0
$786$	0	0
$787$	−36.0000	−1.28326	−0.641631	−	0.767014i	$-0.721742\pi$
$787$	−36.0000	−1.28326	−0.641631	+	0.767014i	$0.721742\pi$
$788$	−24.0000	−0.854965
$789$	20.0000	0.712019
$790$	0	0
$791$	0	0
$792$	0	0
$793$	−2.00000	−0.0710221
$794$	0	0
$795$	13.0000	0.461062
$796$	−34.0000	−1.20510
$797$	35.0000	1.23976	0.619882	−	0.784695i	$-0.287181\pi$
$797$	35.0000	1.23976	0.619882	+	0.784695i	$0.287181\pi$
$798$	0	0
$799$	−36.0000	−1.27359
$800$	0	0
$801$	−4.00000	−0.141333
$802$	0	0
$803$	2.00000	0.0705785
$804$	−20.0000	−0.705346
$805$	0	0
$806$	0	0
$807$	−4.00000	−0.140807
$808$	0	0
$809$	−27.0000	−0.949269	−0.474635	−	0.880183i	$-0.657420\pi$
$809$	−27.0000	−0.949269	−0.474635	+	0.880183i	$0.657420\pi$
$810$	0	0
$811$	46.0000	1.61528	0.807639	−	0.589677i	$-0.200745\pi$
$811$	46.0000	1.61528	0.807639	+	0.589677i	$0.200745\pi$
$812$	0	0
$813$	−26.0000	−0.911860
$814$	0	0
$815$	−16.0000	−0.560456
$816$	16.0000	0.560112
$817$	−22.0000	−0.769683
$818$	0	0
$819$	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	$0.466612\pi$
$822$	0	0
$823$	−4.00000	−0.139431	−0.0697156	−	0.997567i	$-0.522209\pi$
$823$	−4.00000	−0.139431	−0.0697156	+	0.997567i	$0.522209\pi$
$824$	0	0
$825$	1.00000	0.0348155
$826$	0	0
$827$	2.00000	0.0695468	0.0347734	−	0.999395i	$-0.488929\pi$
$827$	2.00000	0.0695468	0.0347734	+	0.999395i	$0.488929\pi$
$828$	−2.00000	−0.0695048
$829$	−27.0000	−0.937749	−0.468874	−	0.883265i	$-0.655340\pi$
$829$	−27.0000	−0.937749	−0.468874	+	0.883265i	$0.655340\pi$
$830$	0	0
$831$	−19.0000	−0.659103
$832$	−8.00000	−0.277350
$833$	0	0
$834$	0	0
$835$	−12.0000	−0.415277
$836$	4.00000	0.138343
$837$	0	0
$838$	0	0
$839$	−42.0000	−1.45000	−0.725001	−	0.688748i	$-0.758161\pi$
$839$	−42.0000	−1.45000	−0.725001	+	0.688748i	$0.758161\pi$
$840$	0	0
$841$	52.0000	1.79310
$842$	0	0
$843$	−15.0000	−0.516627
$844$	−40.0000	−1.37686
$845$	12.0000	0.412813
$846$	0	0
$847$	0	0
$848$	52.0000	1.78569
$849$	−28.0000	−0.960958
$850$	0	0
$851$	0	0
$852$	0	0
$853$	5.00000	0.171197	0.0855984	−	0.996330i	$-0.472720\pi$
$853$	5.00000	0.171197	0.0855984	+	0.996330i	$0.472720\pi$
$854$	0	0
$855$	−2.00000	−0.0683986
$856$	0	0
$857$	18.0000	0.614868	0.307434	−	0.951569i	$-0.400530\pi$
$857$	18.0000	0.614868	0.307434	+	0.951569i	$0.400530\pi$
$858$	0	0
$859$	−1.00000	−0.0341196	−0.0170598	−	0.999854i	$-0.505431\pi$
$859$	−1.00000	−0.0341196	−0.0170598	+	0.999854i	$0.505431\pi$
$860$	−22.0000	−0.750194
$861$	0	0
$862$	0	0
$863$	−9.00000	−0.306364	−0.153182	−	0.988198i	$-0.548952\pi$
$863$	−9.00000	−0.306364	−0.153182	+	0.988198i	$0.548952\pi$
$864$	0	0
$865$	−22.0000	−0.748022
$866$	0	0
$867$	1.00000	0.0339618
$868$	0	0
$869$	−4.00000	−0.135691
$870$	0	0
$871$	−10.0000	−0.338837
$872$	0	0
$873$	−14.0000	−0.473828
$874$	0	0
$875$	0	0
$876$	−4.00000	−0.135147
$877$	59.0000	1.99229	0.996144	−	0.0877308i	$-0.0279615\pi$
$877$	59.0000	1.99229	0.996144	+	0.0877308i	$0.0279615\pi$
$878$	0	0
$879$	−22.0000	−0.742042
$880$	4.00000	0.134840
$881$	−24.0000	−0.808581	−0.404290	−	0.914631i	$-0.632481\pi$
$881$	−24.0000	−0.808581	−0.404290	+	0.914631i	$0.632481\pi$
$882$	0	0
$883$	−44.0000	−1.48072	−0.740359	−	0.672212i	$-0.765344\pi$
$883$	−44.0000	−1.48072	−0.740359	+	0.672212i	$0.765344\pi$
$884$	8.00000	0.269069
$885$	−4.00000	−0.134459
$886$	0	0
$887$	12.0000	0.402921	0.201460	−	0.979497i	$-0.435431\pi$
$887$	12.0000	0.402921	0.201460	+	0.979497i	$0.435431\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	−1.00000	−0.0335013
$892$	−20.0000	−0.669650
$893$	18.0000	0.602347
$894$	0	0
$895$	14.0000	0.467968
$896$	0	0
$897$	−1.00000	−0.0333890
$898$	0	0
$899$	0	0
$900$	−2.00000	−0.0666667
$901$	−52.0000	−1.73237
$902$	0	0
$903$	0	0
$904$	0	0
$905$	13.0000	0.432135
$906$	0	0
$907$	−8.00000	−0.265636	−0.132818	−	0.991140i	$-0.542403\pi$
$907$	−8.00000	−0.265636	−0.132818	+	0.991140i	$0.542403\pi$
$908$	−36.0000	−1.19470
$909$	−5.00000	−0.165840
$910$	0	0
$911$	20.0000	0.662630	0.331315	−	0.943520i	$-0.392508\pi$
$911$	20.0000	0.662630	0.331315	+	0.943520i	$0.392508\pi$
$912$	−8.00000	−0.264906
$913$	−10.0000	−0.330952
$914$	0	0
$915$	−2.00000	−0.0661180
$916$	−58.0000	−1.91637
$917$	0	0
$918$	0	0
$919$	−44.0000	−1.45143	−0.725713	−	0.687998i	$-0.758490\pi$
$919$	−44.0000	−1.45143	−0.725713	+	0.687998i	$0.758490\pi$
$920$	0	0
$921$	23.0000	0.757876
$922$	0	0
$923$	0	0
$924$	0	0
$925$	0	0
$926$	0	0
$927$	−12.0000	−0.394132
$928$	0	0
$929$	−2.00000	−0.0656179	−0.0328089	−	0.999462i	$-0.510445\pi$
$929$	−2.00000	−0.0656179	−0.0328089	+	0.999462i	$0.510445\pi$
$930$	0	0
$931$	0	0
$932$	−16.0000	−0.524097
$933$	20.0000	0.654771
$934$	0	0
$935$	−4.00000	−0.130814
$936$	0	0
$937$	11.0000	0.359354	0.179677	−	0.983726i	$-0.442495\pi$
$937$	11.0000	0.359354	0.179677	+	0.983726i	$0.442495\pi$
$938$	0	0
$939$	6.00000	0.195803
$940$	18.0000	0.587095
$941$	−18.0000	−0.586783	−0.293392	−	0.955992i	$-0.594784\pi$
$941$	−18.0000	−0.586783	−0.293392	+	0.955992i	$0.594784\pi$
$942$	0	0
$943$	5.00000	0.162822
$944$	−16.0000	−0.520756
$945$	0	0
$946$	0	0
$947$	44.0000	1.42981	0.714904	−	0.699223i	$-0.246470\pi$
$947$	44.0000	1.42981	0.714904	+	0.699223i	$0.246470\pi$
$948$	8.00000	0.259828
$949$	−2.00000	−0.0649227
$950$	0	0
$951$	1.00000	0.0324272
$952$	0	0
$953$	46.0000	1.49009	0.745043	−	0.667016i	$-0.232429\pi$
$953$	46.0000	1.49009	0.745043	+	0.667016i	$0.232429\pi$
$954$	0	0
$955$	12.0000	0.388311
$956$	16.0000	0.517477
$957$	−9.00000	−0.290929
$958$	0	0
$959$	0	0
$960$	−8.00000	−0.258199
$961$	−31.0000	−1.00000
$962$	0	0
$963$	−6.00000	−0.193347
$964$	−8.00000	−0.257663
$965$	5.00000	0.160956
$966$	0	0
$967$	52.0000	1.67221	0.836104	−	0.548572i	$-0.184828\pi$
$967$	52.0000	1.67221	0.836104	+	0.548572i	$0.184828\pi$
$968$	0	0
$969$	8.00000	0.256997
$970$	0	0
$971$	12.0000	0.385098	0.192549	−	0.981287i	$-0.438325\pi$
$971$	12.0000	0.385098	0.192549	+	0.981287i	$0.438325\pi$
$972$	2.00000	0.0641500
$973$	0	0
$974$	0	0
$975$	−1.00000	−0.0320256
$976$	−8.00000	−0.256074
$977$	18.0000	0.575871	0.287936	−	0.957650i	$-0.407031\pi$
$977$	18.0000	0.575871	0.287936	+	0.957650i	$0.407031\pi$
$978$	0	0
$979$	4.00000	0.127841
$980$	0	0
$981$	−4.00000	−0.127710
$982$	0	0
$983$	−21.0000	−0.669796	−0.334898	−	0.942254i	$-0.608702\pi$
$983$	−21.0000	−0.669796	−0.334898	+	0.942254i	$0.608702\pi$
$984$	0	0
$985$	−12.0000	−0.382352
$986$	0	0
$987$	0	0
$988$	−4.00000	−0.127257
$989$	−11.0000	−0.349780
$990$	0	0
$991$	−13.0000	−0.412959	−0.206479	−	0.978451i	$-0.566201\pi$
$991$	−13.0000	−0.412959	−0.206479	+	0.978451i	$0.566201\pi$
$992$	0	0
$993$	−25.0000	−0.793351
$994$	0	0
$995$	−17.0000	−0.538936
$996$	20.0000	0.633724
$997$	−7.00000	−0.221692	−0.110846	−	0.993838i	$-0.535356\pi$
$997$	−7.00000	−0.221692	−0.110846	+	0.993838i	$0.535356\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8085.2.a.i.1.1		1
7.3	odd	6		1155.2.q.b.331.1	✓	2
7.5	odd	6		1155.2.q.b.991.1	yes	2
7.6	odd	2		8085.2.a.o.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1155.2.q.b.331.1	✓	2	7.3	odd	6
1155.2.q.b.991.1	yes	2	7.5	odd	6
8085.2.a.i.1.1		1	1.1	even	1	trivial
8085.2.a.o.1.1		1	7.6	odd	2

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Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 8085 = 3 \cdot 5 \cdot 7^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8085.a (trivial)

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