LMFDB - Embedded newform 8085.2.a.x.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)

$f(q)$

$=$

$q+2.00000 q^{2} -1.00000 q^{3} +2.00000 q^{4} -1.00000 q^{5} -2.00000 q^{6} +1.00000 q^{9} +O(q^{10})$

\(q+2.00000 q^{2} -1.00000 q^{3} +2.00000 q^{4} -1.00000 q^{5} -2.00000 q^{6} +1.00000 q^{9} -2.00000 q^{10} -1.00000 q^{11} -2.00000 q^{12} +5.00000 q^{13} +1.00000 q^{15} -4.00000 q^{16} +4.00000 q^{17} +2.00000 q^{18} +2.00000 q^{19} -2.00000 q^{20} -2.00000 q^{22} +3.00000 q^{23} +1.00000 q^{25} +10.0000 q^{26} -1.00000 q^{27} -3.00000 q^{29} +2.00000 q^{30} -8.00000 q^{32} +1.00000 q^{33} +8.00000 q^{34} +2.00000 q^{36} -6.00000 q^{37} +4.00000 q^{38} -5.00000 q^{39} +3.00000 q^{41} -7.00000 q^{43} -2.00000 q^{44} -1.00000 q^{45} +6.00000 q^{46} +3.00000 q^{47} +4.00000 q^{48} +2.00000 q^{50} -4.00000 q^{51} +10.0000 q^{52} -5.00000 q^{53} -2.00000 q^{54} +1.00000 q^{55} -2.00000 q^{57} -6.00000 q^{58} -4.00000 q^{59} +2.00000 q^{60} +6.00000 q^{61} -8.00000 q^{64} -5.00000 q^{65} +2.00000 q^{66} +12.0000 q^{67} +8.00000 q^{68} -3.00000 q^{69} +6.00000 q^{71} -10.0000 q^{73} -12.0000 q^{74} -1.00000 q^{75} +4.00000 q^{76} -10.0000 q^{78} -4.00000 q^{79} +4.00000 q^{80} +1.00000 q^{81} +6.00000 q^{82} +4.00000 q^{83} -4.00000 q^{85} -14.0000 q^{86} +3.00000 q^{87} +14.0000 q^{89} -2.00000 q^{90} +6.00000 q^{92} +6.00000 q^{94} -2.00000 q^{95} +8.00000 q^{96} +8.00000 q^{97} -1.00000 q^{99} +O(q^{100})\)

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 $

$2$	2.00000	1.41421	0.707107	−	0.707107i	$-0.250000\pi$
$2$	2.00000	1.41421	0.707107	+	0.707107i	$0.250000\pi$
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	2.00000	1.00000
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	−2.00000	−0.816497
$7$	0	0
$7$	0	0
$8$	0	0
$9$	1.00000	0.333333
$10$	−2.00000	−0.632456
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	−2.00000	−0.577350
$13$	5.00000	1.38675	0.693375	−	0.720577i	$-0.256123\pi$
$13$	5.00000	1.38675	0.693375	+	0.720577i	$0.256123\pi$
$14$	0	0
$15$	1.00000	0.258199
$16$	−4.00000	−1.00000
$17$	4.00000	0.970143	0.485071	−	0.874475i	$-0.338794\pi$
$17$	4.00000	0.970143	0.485071	+	0.874475i	$0.338794\pi$
$18$	2.00000	0.471405
$19$	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
$22$	−2.00000	−0.426401
$23$	3.00000	0.625543	0.312772	−	0.949828i	$-0.398743\pi$
$23$	3.00000	0.625543	0.312772	+	0.949828i	$0.398743\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	10.0000	1.96116
$27$	−1.00000	−0.192450
$28$	0	0
$29$	−3.00000	−0.557086	−0.278543	−	0.960424i	$-0.589851\pi$
$29$	−3.00000	−0.557086	−0.278543	+	0.960424i	$0.589851\pi$
$30$	2.00000	0.365148
$31$	0	0		−	1.00000i	$-0.5\pi$
$31$	0	0			1.00000i	$0.5\pi$
$32$	−8.00000	−1.41421
$33$	1.00000	0.174078
$34$	8.00000	1.37199
$35$	0	0
$36$	2.00000	0.333333
$37$	−6.00000	−0.986394	−0.493197	−	0.869918i	$-0.664172\pi$
$37$	−6.00000	−0.986394	−0.493197	+	0.869918i	$0.664172\pi$
$38$	4.00000	0.648886
$39$	−5.00000	−0.800641
$40$	0	0
$41$	3.00000	0.468521	0.234261	−	0.972174i	$-0.424733\pi$
$41$	3.00000	0.468521	0.234261	+	0.972174i	$0.424733\pi$
$42$	0	0
$43$	−7.00000	−1.06749	−0.533745	−	0.845645i	$-0.679216\pi$
$43$	−7.00000	−1.06749	−0.533745	+	0.845645i	$0.679216\pi$
$44$	−2.00000	−0.301511
$45$	−1.00000	−0.149071
$46$	6.00000	0.884652
$47$	3.00000	0.437595	0.218797	−	0.975770i	$-0.429787\pi$
$47$	3.00000	0.437595	0.218797	+	0.975770i	$0.429787\pi$
$48$	4.00000	0.577350
$49$	0	0
$50$	2.00000	0.282843
$51$	−4.00000	−0.560112
$52$	10.0000	1.38675
$53$	−5.00000	−0.686803	−0.343401	−	0.939189i	$-0.611579\pi$
$53$	−5.00000	−0.686803	−0.343401	+	0.939189i	$0.611579\pi$
$54$	−2.00000	−0.272166
$55$	1.00000	0.134840
$56$	0	0
$57$	−2.00000	−0.264906
$58$	−6.00000	−0.787839
$59$	−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
$61$	6.00000	0.768221	0.384111	−	0.923287i	$-0.374508\pi$
$61$	6.00000	0.768221	0.384111	+	0.923287i	$0.374508\pi$
$62$	0	0
$63$	0	0
$64$	−8.00000	−1.00000
$65$	−5.00000	−0.620174
$66$	2.00000	0.246183
$67$	12.0000	1.46603	0.733017	−	0.680211i	$-0.238112\pi$
$67$	12.0000	1.46603	0.733017	+	0.680211i	$0.238112\pi$
$68$	8.00000	0.970143
$69$	−3.00000	−0.361158
$70$	0	0
$71$	6.00000	0.712069	0.356034	−	0.934473i	$-0.384129\pi$
$71$	6.00000	0.712069	0.356034	+	0.934473i	$0.384129\pi$
$72$	0	0
$73$	−10.0000	−1.17041	−0.585206	−	0.810885i	$-0.698986\pi$
$73$	−10.0000	−1.17041	−0.585206	+	0.810885i	$0.698986\pi$
$74$	−12.0000	−1.39497
$75$	−1.00000	−0.115470
$76$	4.00000	0.458831
$77$	0	0
$78$	−10.0000	−1.13228
$79$	−4.00000	−0.450035	−0.225018	−	0.974355i	$-0.572244\pi$
$79$	−4.00000	−0.450035	−0.225018	+	0.974355i	$0.572244\pi$
$80$	4.00000	0.447214
$81$	1.00000	0.111111
$82$	6.00000	0.662589
$83$	4.00000	0.439057	0.219529	−	0.975606i	$-0.429548\pi$
$83$	4.00000	0.439057	0.219529	+	0.975606i	$0.429548\pi$
$84$	0	0
$85$	−4.00000	−0.433861
$86$	−14.0000	−1.50966
$87$	3.00000	0.321634
$88$	0	0
$89$	14.0000	1.48400	0.741999	−	0.670402i	$-0.233878\pi$
$89$	14.0000	1.48400	0.741999	+	0.670402i	$0.233878\pi$
$90$	−2.00000	−0.210819
$91$	0	0
$92$	6.00000	0.625543
$93$	0	0
$94$	6.00000	0.618853
$95$	−2.00000	−0.205196
$96$	8.00000	0.816497
$97$	8.00000	0.812277	0.406138	−	0.913812i	$-0.366875\pi$
$97$	8.00000	0.812277	0.406138	+	0.913812i	$0.366875\pi$
$98$	0	0
$99$	−1.00000	−0.100504
$100$	2.00000	0.200000
$101$	1.00000	0.0995037	0.0497519	−	0.998762i	$-0.484157\pi$
$101$	1.00000	0.0995037	0.0497519	+	0.998762i	$0.484157\pi$
$102$	−8.00000	−0.792118
$103$	4.00000	0.394132	0.197066	−	0.980390i	$-0.436859\pi$
$103$	4.00000	0.394132	0.197066	+	0.980390i	$0.436859\pi$
$104$	0	0
$105$	0	0
$106$	−10.0000	−0.971286
$107$	10.0000	0.966736	0.483368	−	0.875417i	$-0.339413\pi$
$107$	10.0000	0.966736	0.483368	+	0.875417i	$0.339413\pi$
$108$	−2.00000	−0.192450
$109$	10.0000	0.957826	0.478913	−	0.877862i	$-0.341031\pi$
$109$	10.0000	0.957826	0.478913	+	0.877862i	$0.341031\pi$
$110$	2.00000	0.190693
$111$	6.00000	0.569495
$112$	0	0
$113$	6.00000	0.564433	0.282216	−	0.959351i	$-0.408930\pi$
$113$	6.00000	0.564433	0.282216	+	0.959351i	$0.408930\pi$
$114$	−4.00000	−0.374634
$115$	−3.00000	−0.279751
$116$	−6.00000	−0.557086
$117$	5.00000	0.462250
$118$	−8.00000	−0.736460
$119$	0	0
$120$	0	0
$121$	1.00000	0.0909091
$122$	12.0000	1.08643
$123$	−3.00000	−0.270501
$124$	0	0
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	19.0000	1.68598	0.842989	−	0.537931i	$-0.180794\pi$
$127$	19.0000	1.68598	0.842989	+	0.537931i	$0.180794\pi$
$128$	0	0
$129$	7.00000	0.616316
$130$	−10.0000	−0.877058
$131$	3.00000	0.262111	0.131056	−	0.991375i	$-0.458163\pi$
$131$	3.00000	0.262111	0.131056	+	0.991375i	$0.458163\pi$
$132$	2.00000	0.174078
$133$	0	0
$134$	24.0000	2.07328
$135$	1.00000	0.0860663
$136$	0	0
$137$	−13.0000	−1.11066	−0.555332	−	0.831628i	$-0.687409\pi$
$137$	−13.0000	−1.11066	−0.555332	+	0.831628i	$0.687409\pi$
$138$	−6.00000	−0.510754
$139$	16.0000	1.35710	0.678551	−	0.734553i	$-0.262608\pi$
$139$	16.0000	1.35710	0.678551	+	0.734553i	$0.262608\pi$
$140$	0	0
$141$	−3.00000	−0.252646
$142$	12.0000	1.00702
$143$	−5.00000	−0.418121
$144$	−4.00000	−0.333333
$145$	3.00000	0.249136
$146$	−20.0000	−1.65521
$147$	0	0
$148$	−12.0000	−0.986394
$149$	21.0000	1.72039	0.860194	−	0.509968i	$-0.170343\pi$
$149$	21.0000	1.72039	0.860194	+	0.509968i	$0.170343\pi$
$150$	−2.00000	−0.163299
$151$	−10.0000	−0.813788	−0.406894	−	0.913475i	$-0.633388\pi$
$151$	−10.0000	−0.813788	−0.406894	+	0.913475i	$0.633388\pi$
$152$	0	0
$153$	4.00000	0.323381
$154$	0	0
$155$	0	0
$156$	−10.0000	−0.800641
$157$	−10.0000	−0.798087	−0.399043	−	0.916932i	$-0.630658\pi$
$157$	−10.0000	−0.798087	−0.399043	+	0.916932i	$0.630658\pi$
$158$	−8.00000	−0.636446
$159$	5.00000	0.396526
$160$	8.00000	0.632456
$161$	0	0
$162$	2.00000	0.157135
$163$	14.0000	1.09656	0.548282	−	0.836293i	$-0.315282\pi$
$163$	14.0000	1.09656	0.548282	+	0.836293i	$0.315282\pi$
$164$	6.00000	0.468521
$165$	−1.00000	−0.0778499
$166$	8.00000	0.620920
$167$	22.0000	1.70241	0.851206	−	0.524832i	$-0.175872\pi$
$167$	22.0000	1.70241	0.851206	+	0.524832i	$0.175872\pi$
$168$	0	0
$169$	12.0000	0.923077
$170$	−8.00000	−0.613572
$171$	2.00000	0.152944
$172$	−14.0000	−1.06749
$173$	22.0000	1.67263	0.836315	−	0.548250i	$-0.184706\pi$
$173$	22.0000	1.67263	0.836315	+	0.548250i	$0.184706\pi$
$174$	6.00000	0.454859
$175$	0	0
$176$	4.00000	0.301511
$177$	4.00000	0.300658
$178$	28.0000	2.09869
$179$	14.0000	1.04641	0.523205	−	0.852207i	$-0.324736\pi$
$179$	14.0000	1.04641	0.523205	+	0.852207i	$0.324736\pi$
$180$	−2.00000	−0.149071
$181$	7.00000	0.520306	0.260153	−	0.965567i	$-0.416227\pi$
$181$	7.00000	0.520306	0.260153	+	0.965567i	$0.416227\pi$
$182$	0	0
$183$	−6.00000	−0.443533
$184$	0	0
$185$	6.00000	0.441129
$186$	0	0
$187$	−4.00000	−0.292509
$188$	6.00000	0.437595
$189$	0	0
$190$	−4.00000	−0.290191
$191$	−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
$193$	−25.0000	−1.79954	−0.899770	−	0.436365i	$-0.856266\pi$
$193$	−25.0000	−1.79954	−0.899770	+	0.436365i	$0.856266\pi$
$194$	16.0000	1.14873
$195$	5.00000	0.358057
$196$	0	0
$197$	−10.0000	−0.712470	−0.356235	−	0.934396i	$-0.615940\pi$
$197$	−10.0000	−0.712470	−0.356235	+	0.934396i	$0.615940\pi$
$198$	−2.00000	−0.142134
$199$	−11.0000	−0.779769	−0.389885	−	0.920864i	$-0.627485\pi$
$199$	−11.0000	−0.779769	−0.389885	+	0.920864i	$0.627485\pi$
$200$	0	0
$201$	−12.0000	−0.846415
$202$	2.00000	0.140720
$203$	0	0
$204$	−8.00000	−0.560112
$205$	−3.00000	−0.209529
$206$	8.00000	0.557386
$207$	3.00000	0.208514
$208$	−20.0000	−1.38675
$209$	−2.00000	−0.138343
$210$	0	0
$211$	10.0000	0.688428	0.344214	−	0.938891i	$-0.388145\pi$
$211$	10.0000	0.688428	0.344214	+	0.938891i	$0.388145\pi$
$212$	−10.0000	−0.686803
$213$	−6.00000	−0.411113
$214$	20.0000	1.36717
$215$	7.00000	0.477396
$216$	0	0
$217$	0	0
$218$	20.0000	1.35457
$219$	10.0000	0.675737
$220$	2.00000	0.134840
$221$	20.0000	1.34535
$222$	12.0000	0.805387
$223$	4.00000	0.267860	0.133930	−	0.990991i	$-0.457240\pi$
$223$	4.00000	0.267860	0.133930	+	0.990991i	$0.457240\pi$
$224$	0	0
$225$	1.00000	0.0666667
$226$	12.0000	0.798228
$227$	0	0		−	1.00000i	$-0.5\pi$
$227$	0	0			1.00000i	$0.5\pi$
$228$	−4.00000	−0.264906
$229$	5.00000	0.330409	0.165205	−	0.986259i	$-0.447172\pi$
$229$	5.00000	0.330409	0.165205	+	0.986259i	$0.447172\pi$
$230$	−6.00000	−0.395628
$231$	0	0
$232$	0	0
$233$	18.0000	1.17922	0.589610	−	0.807688i	$-0.299282\pi$
$233$	18.0000	1.17922	0.589610	+	0.807688i	$0.299282\pi$
$234$	10.0000	0.653720
$235$	−3.00000	−0.195698
$236$	−8.00000	−0.520756
$237$	4.00000	0.259828
$238$	0	0
$239$	0	0		−	1.00000i	$-0.5\pi$
$239$	0	0			1.00000i	$0.5\pi$
$240$	−4.00000	−0.258199
$241$	26.0000	1.67481	0.837404	−	0.546585i	$-0.184072\pi$
$241$	26.0000	1.67481	0.837404	+	0.546585i	$0.184072\pi$
$242$	2.00000	0.128565
$243$	−1.00000	−0.0641500
$244$	12.0000	0.768221
$245$	0	0
$246$	−6.00000	−0.382546
$247$	10.0000	0.636285
$248$	0	0
$249$	−4.00000	−0.253490
$250$	−2.00000	−0.126491
$251$	24.0000	1.51487	0.757433	−	0.652913i	$-0.226453\pi$
$251$	24.0000	1.51487	0.757433	+	0.652913i	$0.226453\pi$
$252$	0	0
$253$	−3.00000	−0.188608
$254$	38.0000	2.38433
$255$	4.00000	0.250490
$256$	16.0000	1.00000
$257$	−13.0000	−0.810918	−0.405459	−	0.914113i	$-0.632888\pi$
$257$	−13.0000	−0.810918	−0.405459	+	0.914113i	$0.632888\pi$
$258$	14.0000	0.871602
$259$	0	0
$260$	−10.0000	−0.620174
$261$	−3.00000	−0.185695
$262$	6.00000	0.370681
$263$	18.0000	1.10993	0.554964	−	0.831875i	$-0.312732\pi$
$263$	18.0000	1.10993	0.554964	+	0.831875i	$0.312732\pi$
$264$	0	0
$265$	5.00000	0.307148
$266$	0	0
$267$	−14.0000	−0.856786
$268$	24.0000	1.46603
$269$	6.00000	0.365826	0.182913	−	0.983129i	$-0.441447\pi$
$269$	6.00000	0.365826	0.182913	+	0.983129i	$0.441447\pi$
$270$	2.00000	0.121716
$271$	−20.0000	−1.21491	−0.607457	−	0.794353i	$-0.707810\pi$
$271$	−20.0000	−1.21491	−0.607457	+	0.794353i	$0.707810\pi$
$272$	−16.0000	−0.970143
$273$	0	0
$274$	−26.0000	−1.57072
$275$	−1.00000	−0.0603023
$276$	−6.00000	−0.361158
$277$	−13.0000	−0.781094	−0.390547	−	0.920583i	$-0.627714\pi$
$277$	−13.0000	−0.781094	−0.390547	+	0.920583i	$0.627714\pi$
$278$	32.0000	1.91923
$279$	0	0
$280$	0	0
$281$	−27.0000	−1.61068	−0.805342	−	0.592810i	$-0.798019\pi$
$281$	−27.0000	−1.61068	−0.805342	+	0.592810i	$0.798019\pi$
$282$	−6.00000	−0.357295
$283$	−28.0000	−1.66443	−0.832214	−	0.554455i	$-0.812927\pi$
$283$	−28.0000	−1.66443	−0.832214	+	0.554455i	$0.812927\pi$
$284$	12.0000	0.712069
$285$	2.00000	0.118470
$286$	−10.0000	−0.591312
$287$	0	0
$288$	−8.00000	−0.471405
$289$	−1.00000	−0.0588235
$290$	6.00000	0.352332
$291$	−8.00000	−0.468968
$292$	−20.0000	−1.17041
$293$	−12.0000	−0.701047	−0.350524	−	0.936554i	$-0.613996\pi$
$293$	−12.0000	−0.701047	−0.350524	+	0.936554i	$0.613996\pi$
$294$	0	0
$295$	4.00000	0.232889
$296$	0	0
$297$	1.00000	0.0580259
$298$	42.0000	2.43299
$299$	15.0000	0.867472
$300$	−2.00000	−0.115470
$301$	0	0
$302$	−20.0000	−1.15087
$303$	−1.00000	−0.0574485
$304$	−8.00000	−0.458831
$305$	−6.00000	−0.343559
$306$	8.00000	0.457330
$307$	9.00000	0.513657	0.256829	−	0.966457i	$-0.417322\pi$
$307$	9.00000	0.513657	0.256829	+	0.966457i	$0.417322\pi$
$308$	0	0
$309$	−4.00000	−0.227552
$310$	0	0
$311$	−10.0000	−0.567048	−0.283524	−	0.958965i	$-0.591504\pi$
$311$	−10.0000	−0.567048	−0.283524	+	0.958965i	$0.591504\pi$
$312$	0	0
$313$	−16.0000	−0.904373	−0.452187	−	0.891923i	$-0.649356\pi$
$313$	−16.0000	−0.904373	−0.452187	+	0.891923i	$0.649356\pi$
$314$	−20.0000	−1.12867
$315$	0	0
$316$	−8.00000	−0.450035
$317$	13.0000	0.730153	0.365076	−	0.930978i	$-0.381043\pi$
$317$	13.0000	0.730153	0.365076	+	0.930978i	$0.381043\pi$
$318$	10.0000	0.560772
$319$	3.00000	0.167968
$320$	8.00000	0.447214
$321$	−10.0000	−0.558146
$322$	0	0
$323$	8.00000	0.445132
$324$	2.00000	0.111111
$325$	5.00000	0.277350
$326$	28.0000	1.55078
$327$	−10.0000	−0.553001
$328$	0	0
$329$	0	0
$330$	−2.00000	−0.110096
$331$	17.0000	0.934405	0.467202	−	0.884150i	$-0.345262\pi$
$331$	17.0000	0.934405	0.467202	+	0.884150i	$0.345262\pi$
$332$	8.00000	0.439057
$333$	−6.00000	−0.328798
$334$	44.0000	2.40757
$335$	−12.0000	−0.655630
$336$	0	0
$337$	−18.0000	−0.980522	−0.490261	−	0.871576i	$-0.663099\pi$
$337$	−18.0000	−0.980522	−0.490261	+	0.871576i	$0.663099\pi$
$338$	24.0000	1.30543
$339$	−6.00000	−0.325875
$340$	−8.00000	−0.433861
$341$	0	0
$342$	4.00000	0.216295
$343$	0	0
$344$	0	0
$345$	3.00000	0.161515
$346$	44.0000	2.36545
$347$	6.00000	0.322097	0.161048	−	0.986947i	$-0.448512\pi$
$347$	6.00000	0.322097	0.161048	+	0.986947i	$0.448512\pi$
$348$	6.00000	0.321634
$349$	−28.0000	−1.49881	−0.749403	−	0.662114i	$-0.769659\pi$
$349$	−28.0000	−1.49881	−0.749403	+	0.662114i	$0.769659\pi$
$350$	0	0
$351$	−5.00000	−0.266880
$352$	8.00000	0.426401
$353$	−23.0000	−1.22417	−0.612083	−	0.790793i	$-0.709668\pi$
$353$	−23.0000	−1.22417	−0.612083	+	0.790793i	$0.709668\pi$
$354$	8.00000	0.425195
$355$	−6.00000	−0.318447
$356$	28.0000	1.48400
$357$	0	0
$358$	28.0000	1.47985
$359$	19.0000	1.00278	0.501391	−	0.865221i	$-0.332822\pi$
$359$	19.0000	1.00278	0.501391	+	0.865221i	$0.332822\pi$
$360$	0	0
$361$	−15.0000	−0.789474
$362$	14.0000	0.735824
$363$	−1.00000	−0.0524864
$364$	0	0
$365$	10.0000	0.523424
$366$	−12.0000	−0.627250
$367$	−8.00000	−0.417597	−0.208798	−	0.977959i	$-0.566955\pi$
$367$	−8.00000	−0.417597	−0.208798	+	0.977959i	$0.566955\pi$
$368$	−12.0000	−0.625543
$369$	3.00000	0.156174
$370$	12.0000	0.623850
$371$	0	0
$372$	0	0
$373$	31.0000	1.60512	0.802560	−	0.596572i	$-0.203471\pi$
$373$	31.0000	1.60512	0.802560	+	0.596572i	$0.203471\pi$
$374$	−8.00000	−0.413670
$375$	1.00000	0.0516398
$376$	0	0
$377$	−15.0000	−0.772539
$378$	0	0
$379$	−16.0000	−0.821865	−0.410932	−	0.911666i	$-0.634797\pi$
$379$	−16.0000	−0.821865	−0.410932	+	0.911666i	$0.634797\pi$
$380$	−4.00000	−0.205196
$381$	−19.0000	−0.973399
$382$	−24.0000	−1.22795
$383$	−8.00000	−0.408781	−0.204390	−	0.978889i	$-0.565521\pi$
$383$	−8.00000	−0.408781	−0.204390	+	0.978889i	$0.565521\pi$
$384$	0	0
$385$	0	0
$386$	−50.0000	−2.54493
$387$	−7.00000	−0.355830
$388$	16.0000	0.812277
$389$	−6.00000	−0.304212	−0.152106	−	0.988364i	$-0.548606\pi$
$389$	−6.00000	−0.304212	−0.152106	+	0.988364i	$0.548606\pi$
$390$	10.0000	0.506370
$391$	12.0000	0.606866
$392$	0	0
$393$	−3.00000	−0.151330
$394$	−20.0000	−1.00759
$395$	4.00000	0.201262
$396$	−2.00000	−0.100504
$397$	−4.00000	−0.200754	−0.100377	−	0.994949i	$-0.532005\pi$
$397$	−4.00000	−0.200754	−0.100377	+	0.994949i	$0.532005\pi$
$398$	−22.0000	−1.10276
$399$	0	0
$400$	−4.00000	−0.200000
$401$	28.0000	1.39825	0.699127	−	0.714998i	$-0.253572\pi$
$401$	28.0000	1.39825	0.699127	+	0.714998i	$0.253572\pi$
$402$	−24.0000	−1.19701
$403$	0	0
$404$	2.00000	0.0995037
$405$	−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	$0.277769\pi$
$410$	−6.00000	−0.296319
$411$	13.0000	0.641243
$412$	8.00000	0.394132
$413$	0	0
$414$	6.00000	0.294884
$415$	−4.00000	−0.196352
$416$	−40.0000	−1.96116
$417$	−16.0000	−0.783523
$418$	−4.00000	−0.195646
$419$	30.0000	1.46560	0.732798	−	0.680446i	$-0.238214\pi$
$419$	30.0000	1.46560	0.732798	+	0.680446i	$0.238214\pi$
$420$	0	0
$421$	−6.00000	−0.292422	−0.146211	−	0.989253i	$-0.546708\pi$
$421$	−6.00000	−0.292422	−0.146211	+	0.989253i	$0.546708\pi$
$422$	20.0000	0.973585
$423$	3.00000	0.145865
$424$	0	0
$425$	4.00000	0.194029
$426$	−12.0000	−0.581402
$427$	0	0
$428$	20.0000	0.966736
$429$	5.00000	0.241402
$430$	14.0000	0.675140
$431$	−31.0000	−1.49322	−0.746609	−	0.665263i	$-0.768319\pi$
$431$	−31.0000	−1.49322	−0.746609	+	0.665263i	$0.768319\pi$
$432$	4.00000	0.192450
$433$	−12.0000	−0.576683	−0.288342	−	0.957528i	$-0.593104\pi$
$433$	−12.0000	−0.576683	−0.288342	+	0.957528i	$0.593104\pi$
$434$	0	0
$435$	−3.00000	−0.143839
$436$	20.0000	0.957826
$437$	6.00000	0.287019
$438$	20.0000	0.955637
$439$	6.00000	0.286364	0.143182	−	0.989696i	$-0.454267\pi$
$439$	6.00000	0.286364	0.143182	+	0.989696i	$0.454267\pi$
$440$	0	0
$441$	0	0
$442$	40.0000	1.90261
$443$	12.0000	0.570137	0.285069	−	0.958507i	$-0.407984\pi$
$443$	12.0000	0.570137	0.285069	+	0.958507i	$0.407984\pi$
$444$	12.0000	0.569495
$445$	−14.0000	−0.663664
$446$	8.00000	0.378811
$447$	−21.0000	−0.993266
$448$	0	0
$449$	20.0000	0.943858	0.471929	−	0.881636i	$-0.343558\pi$
$449$	20.0000	0.943858	0.471929	+	0.881636i	$0.343558\pi$
$450$	2.00000	0.0942809
$451$	−3.00000	−0.141264
$452$	12.0000	0.564433
$453$	10.0000	0.469841
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−37.0000	−1.73079	−0.865393	−	0.501093i	$-0.832931\pi$
$457$	−37.0000	−1.73079	−0.865393	+	0.501093i	$0.832931\pi$
$458$	10.0000	0.467269
$459$	−4.00000	−0.186704
$460$	−6.00000	−0.279751
$461$	−37.0000	−1.72326	−0.861631	−	0.507535i	$-0.830557\pi$
$461$	−37.0000	−1.72326	−0.861631	+	0.507535i	$0.830557\pi$
$462$	0	0
$463$	−6.00000	−0.278844	−0.139422	−	0.990233i	$-0.544524\pi$
$463$	−6.00000	−0.278844	−0.139422	+	0.990233i	$0.544524\pi$
$464$	12.0000	0.557086
$465$	0	0
$466$	36.0000	1.66767
$467$	−15.0000	−0.694117	−0.347059	−	0.937843i	$-0.612820\pi$
$467$	−15.0000	−0.694117	−0.347059	+	0.937843i	$0.612820\pi$
$468$	10.0000	0.462250
$469$	0	0
$470$	−6.00000	−0.276759
$471$	10.0000	0.460776
$472$	0	0
$473$	7.00000	0.321860
$474$	8.00000	0.367452
$475$	2.00000	0.0917663
$476$	0	0
$477$	−5.00000	−0.228934
$478$	0	0
$479$	−16.0000	−0.731059	−0.365529	−	0.930800i	$-0.619112\pi$
$479$	−16.0000	−0.731059	−0.365529	+	0.930800i	$0.619112\pi$
$480$	−8.00000	−0.365148
$481$	−30.0000	−1.36788
$482$	52.0000	2.36854
$483$	0	0
$484$	2.00000	0.0909091
$485$	−8.00000	−0.363261
$486$	−2.00000	−0.0907218
$487$	−2.00000	−0.0906287	−0.0453143	−	0.998973i	$-0.514429\pi$
$487$	−2.00000	−0.0906287	−0.0453143	+	0.998973i	$0.514429\pi$
$488$	0	0
$489$	−14.0000	−0.633102
$490$	0	0
$491$	−31.0000	−1.39901	−0.699505	−	0.714628i	$-0.746596\pi$
$491$	−31.0000	−1.39901	−0.699505	+	0.714628i	$0.746596\pi$
$492$	−6.00000	−0.270501
$493$	−12.0000	−0.540453
$494$	20.0000	0.899843
$495$	1.00000	0.0449467
$496$	0	0
$497$	0	0
$498$	−8.00000	−0.358489
$499$	−19.0000	−0.850557	−0.425278	−	0.905063i	$-0.639824\pi$
$499$	−19.0000	−0.850557	−0.425278	+	0.905063i	$0.639824\pi$
$500$	−2.00000	−0.0894427
$501$	−22.0000	−0.982888
$502$	48.0000	2.14234
$503$	24.0000	1.07011	0.535054	−	0.844818i	$-0.320291\pi$
$503$	24.0000	1.07011	0.535054	+	0.844818i	$0.320291\pi$
$504$	0	0
$505$	−1.00000	−0.0444994
$506$	−6.00000	−0.266733
$507$	−12.0000	−0.532939
$508$	38.0000	1.68598
$509$	26.0000	1.15243	0.576215	−	0.817298i	$-0.304529\pi$
$509$	26.0000	1.15243	0.576215	+	0.817298i	$0.304529\pi$
$510$	8.00000	0.354246
$511$	0	0
$512$	32.0000	1.41421
$513$	−2.00000	−0.0883022
$514$	−26.0000	−1.14681
$515$	−4.00000	−0.176261
$516$	14.0000	0.616316
$517$	−3.00000	−0.131940
$518$	0	0
$519$	−22.0000	−0.965693
$520$	0	0
$521$	24.0000	1.05146	0.525730	−	0.850652i	$-0.323792\pi$
$521$	24.0000	1.05146	0.525730	+	0.850652i	$0.323792\pi$
$522$	−6.00000	−0.262613
$523$	41.0000	1.79280	0.896402	−	0.443241i	$-0.146171\pi$
$523$	41.0000	1.79280	0.896402	+	0.443241i	$0.146171\pi$
$524$	6.00000	0.262111
$525$	0	0
$526$	36.0000	1.56967
$527$	0	0
$528$	−4.00000	−0.174078
$529$	−14.0000	−0.608696
$530$	10.0000	0.434372
$531$	−4.00000	−0.173585
$532$	0	0
$533$	15.0000	0.649722
$534$	−28.0000	−1.21168
$535$	−10.0000	−0.432338
$536$	0	0
$537$	−14.0000	−0.604145
$538$	12.0000	0.517357
$539$	0	0
$540$	2.00000	0.0860663
$541$	−32.0000	−1.37579	−0.687894	−	0.725811i	$-0.741464\pi$
$541$	−32.0000	−1.37579	−0.687894	+	0.725811i	$0.741464\pi$
$542$	−40.0000	−1.71815
$543$	−7.00000	−0.300399
$544$	−32.0000	−1.37199
$545$	−10.0000	−0.428353
$546$	0	0
$547$	43.0000	1.83855	0.919274	−	0.393619i	$-0.128777\pi$
$547$	43.0000	1.83855	0.919274	+	0.393619i	$0.128777\pi$
$548$	−26.0000	−1.11066
$549$	6.00000	0.256074
$550$	−2.00000	−0.0852803
$551$	−6.00000	−0.255609
$552$	0	0
$553$	0	0
$554$	−26.0000	−1.10463
$555$	−6.00000	−0.254686
$556$	32.0000	1.35710
$557$	−42.0000	−1.77960	−0.889799	−	0.456354i	$-0.849155\pi$
$557$	−42.0000	−1.77960	−0.889799	+	0.456354i	$0.849155\pi$
$558$	0	0
$559$	−35.0000	−1.48034
$560$	0	0
$561$	4.00000	0.168880
$562$	−54.0000	−2.27785
$563$	0	0		−	1.00000i	$-0.5\pi$
$563$	0	0			1.00000i	$0.5\pi$
$564$	−6.00000	−0.252646
$565$	−6.00000	−0.252422
$566$	−56.0000	−2.35386
$567$	0	0
$568$	0	0
$569$	−19.0000	−0.796521	−0.398261	−	0.917272i	$-0.630386\pi$
$569$	−19.0000	−0.796521	−0.398261	+	0.917272i	$0.630386\pi$
$570$	4.00000	0.167542
$571$	−18.0000	−0.753277	−0.376638	−	0.926360i	$-0.622920\pi$
$571$	−18.0000	−0.753277	−0.376638	+	0.926360i	$0.622920\pi$
$572$	−10.0000	−0.418121
$573$	12.0000	0.501307
$574$	0	0
$575$	3.00000	0.125109
$576$	−8.00000	−0.333333
$577$	−28.0000	−1.16566	−0.582828	−	0.812596i	$-0.698054\pi$
$577$	−28.0000	−1.16566	−0.582828	+	0.812596i	$0.698054\pi$
$578$	−2.00000	−0.0831890
$579$	25.0000	1.03896
$580$	6.00000	0.249136
$581$	0	0
$582$	−16.0000	−0.663221
$583$	5.00000	0.207079
$584$	0	0
$585$	−5.00000	−0.206725
$586$	−24.0000	−0.991431
$587$	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
$589$	0	0
$590$	8.00000	0.329355
$591$	10.0000	0.411345
$592$	24.0000	0.986394
$593$	28.0000	1.14982	0.574911	−	0.818216i	$-0.305037\pi$
$593$	28.0000	1.14982	0.574911	+	0.818216i	$0.305037\pi$
$594$	2.00000	0.0820610
$595$	0	0
$596$	42.0000	1.72039
$597$	11.0000	0.450200
$598$	30.0000	1.22679
$599$	6.00000	0.245153	0.122577	−	0.992459i	$-0.460884\pi$
$599$	6.00000	0.245153	0.122577	+	0.992459i	$0.460884\pi$
$600$	0	0
$601$	16.0000	0.652654	0.326327	−	0.945257i	$-0.394189\pi$
$601$	16.0000	0.652654	0.326327	+	0.945257i	$0.394189\pi$
$602$	0	0
$603$	12.0000	0.488678
$604$	−20.0000	−0.813788
$605$	−1.00000	−0.0406558
$606$	−2.00000	−0.0812444
$607$	21.0000	0.852364	0.426182	−	0.904638i	$-0.359858\pi$
$607$	21.0000	0.852364	0.426182	+	0.904638i	$0.359858\pi$
$608$	−16.0000	−0.648886
$609$	0	0
$610$	−12.0000	−0.485866
$611$	15.0000	0.606835
$612$	8.00000	0.323381
$613$	39.0000	1.57520	0.787598	−	0.616190i	$-0.211325\pi$
$613$	39.0000	1.57520	0.787598	+	0.616190i	$0.211325\pi$
$614$	18.0000	0.726421
$615$	3.00000	0.120972
$616$	0	0
$617$	−3.00000	−0.120775	−0.0603877	−	0.998175i	$-0.519234\pi$
$617$	−3.00000	−0.120775	−0.0603877	+	0.998175i	$0.519234\pi$
$618$	−8.00000	−0.321807
$619$	24.0000	0.964641	0.482321	−	0.875995i	$-0.339794\pi$
$619$	24.0000	0.964641	0.482321	+	0.875995i	$0.339794\pi$
$620$	0	0
$621$	−3.00000	−0.120386
$622$	−20.0000	−0.801927
$623$	0	0
$624$	20.0000	0.800641
$625$	1.00000	0.0400000
$626$	−32.0000	−1.27898
$627$	2.00000	0.0798723
$628$	−20.0000	−0.798087
$629$	−24.0000	−0.956943
$630$	0	0
$631$	−21.0000	−0.835997	−0.417998	−	0.908448i	$-0.637268\pi$
$631$	−21.0000	−0.835997	−0.417998	+	0.908448i	$0.637268\pi$
$632$	0	0
$633$	−10.0000	−0.397464
$634$	26.0000	1.03259
$635$	−19.0000	−0.753992
$636$	10.0000	0.396526
$637$	0	0
$638$	6.00000	0.237542
$639$	6.00000	0.237356
$640$	0	0
$641$	24.0000	0.947943	0.473972	−	0.880540i	$-0.342820\pi$
$641$	24.0000	0.947943	0.473972	+	0.880540i	$0.342820\pi$
$642$	−20.0000	−0.789337
$643$	16.0000	0.630978	0.315489	−	0.948929i	$-0.397831\pi$
$643$	16.0000	0.630978	0.315489	+	0.948929i	$0.397831\pi$
$644$	0	0
$645$	−7.00000	−0.275625
$646$	16.0000	0.629512
$647$	−9.00000	−0.353827	−0.176913	−	0.984226i	$-0.556611\pi$
$647$	−9.00000	−0.353827	−0.176913	+	0.984226i	$0.556611\pi$
$648$	0	0
$649$	4.00000	0.157014
$650$	10.0000	0.392232
$651$	0	0
$652$	28.0000	1.09656
$653$	−25.0000	−0.978326	−0.489163	−	0.872192i	$-0.662698\pi$
$653$	−25.0000	−0.978326	−0.489163	+	0.872192i	$0.662698\pi$
$654$	−20.0000	−0.782062
$655$	−3.00000	−0.117220
$656$	−12.0000	−0.468521
$657$	−10.0000	−0.390137
$658$	0	0
$659$	−29.0000	−1.12968	−0.564840	−	0.825201i	$-0.691062\pi$
$659$	−29.0000	−1.12968	−0.564840	+	0.825201i	$0.691062\pi$
$660$	−2.00000	−0.0778499
$661$	37.0000	1.43913	0.719567	−	0.694423i	$-0.244340\pi$
$661$	37.0000	1.43913	0.719567	+	0.694423i	$0.244340\pi$
$662$	34.0000	1.32145
$663$	−20.0000	−0.776736
$664$	0	0
$665$	0	0
$666$	−12.0000	−0.464991
$667$	−9.00000	−0.348481
$668$	44.0000	1.70241
$669$	−4.00000	−0.154649
$670$	−24.0000	−0.927201
$671$	−6.00000	−0.231627
$672$	0	0
$673$	−41.0000	−1.58043	−0.790217	−	0.612827i	$-0.790032\pi$
$673$	−41.0000	−1.58043	−0.790217	+	0.612827i	$0.790032\pi$
$674$	−36.0000	−1.38667
$675$	−1.00000	−0.0384900
$676$	24.0000	0.923077
$677$	48.0000	1.84479	0.922395	−	0.386248i	$-0.126229\pi$
$677$	48.0000	1.84479	0.922395	+	0.386248i	$0.126229\pi$
$678$	−12.0000	−0.460857
$679$	0	0
$680$	0	0
$681$	0	0
$682$	0	0
$683$	52.0000	1.98972	0.994862	−	0.101237i	$-0.0322800\pi$
$683$	52.0000	1.98972	0.994862	+	0.101237i	$0.0322800\pi$
$684$	4.00000	0.152944
$685$	13.0000	0.496704
$686$	0	0
$687$	−5.00000	−0.190762
$688$	28.0000	1.06749
$689$	−25.0000	−0.952424
$690$	6.00000	0.228416
$691$	−36.0000	−1.36950	−0.684752	−	0.728776i	$-0.740090\pi$
$691$	−36.0000	−1.36950	−0.684752	+	0.728776i	$0.740090\pi$
$692$	44.0000	1.67263
$693$	0	0
$694$	12.0000	0.455514
$695$	−16.0000	−0.606915
$696$	0	0
$697$	12.0000	0.454532
$698$	−56.0000	−2.11963
$699$	−18.0000	−0.680823
$700$	0	0
$701$	−50.0000	−1.88847	−0.944237	−	0.329267i	$-0.893198\pi$
$701$	−50.0000	−1.88847	−0.944237	+	0.329267i	$0.893198\pi$
$702$	−10.0000	−0.377426
$703$	−12.0000	−0.452589
$704$	8.00000	0.301511
$705$	3.00000	0.112987
$706$	−46.0000	−1.73123
$707$	0	0
$708$	8.00000	0.300658
$709$	7.00000	0.262891	0.131445	−	0.991323i	$-0.458038\pi$
$709$	7.00000	0.262891	0.131445	+	0.991323i	$0.458038\pi$
$710$	−12.0000	−0.450352
$711$	−4.00000	−0.150012
$712$	0	0
$713$	0	0
$714$	0	0
$715$	5.00000	0.186989
$716$	28.0000	1.04641
$717$	0	0
$718$	38.0000	1.41815
$719$	6.00000	0.223762	0.111881	−	0.993722i	$-0.464312\pi$
$719$	6.00000	0.223762	0.111881	+	0.993722i	$0.464312\pi$
$720$	4.00000	0.149071
$721$	0	0
$722$	−30.0000	−1.11648
$723$	−26.0000	−0.966950
$724$	14.0000	0.520306
$725$	−3.00000	−0.111417
$726$	−2.00000	−0.0742270
$727$	12.0000	0.445055	0.222528	−	0.974926i	$-0.428569\pi$
$727$	12.0000	0.445055	0.222528	+	0.974926i	$0.428569\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	20.0000	0.740233
$731$	−28.0000	−1.03562
$732$	−12.0000	−0.443533
$733$	−38.0000	−1.40356	−0.701781	−	0.712393i	$-0.747612\pi$
$733$	−38.0000	−1.40356	−0.701781	+	0.712393i	$0.747612\pi$
$734$	−16.0000	−0.590571
$735$	0	0
$736$	−24.0000	−0.884652
$737$	−12.0000	−0.442026
$738$	6.00000	0.220863
$739$	12.0000	0.441427	0.220714	−	0.975339i	$-0.429161\pi$
$739$	12.0000	0.441427	0.220714	+	0.975339i	$0.429161\pi$
$740$	12.0000	0.441129
$741$	−10.0000	−0.367359
$742$	0	0
$743$	−16.0000	−0.586983	−0.293492	−	0.955962i	$-0.594817\pi$
$743$	−16.0000	−0.586983	−0.293492	+	0.955962i	$0.594817\pi$
$744$	0	0
$745$	−21.0000	−0.769380
$746$	62.0000	2.26998
$747$	4.00000	0.146352
$748$	−8.00000	−0.292509
$749$	0	0
$750$	2.00000	0.0730297
$751$	16.0000	0.583848	0.291924	−	0.956441i	$-0.405705\pi$
$751$	16.0000	0.583848	0.291924	+	0.956441i	$0.405705\pi$
$752$	−12.0000	−0.437595
$753$	−24.0000	−0.874609
$754$	−30.0000	−1.09254
$755$	10.0000	0.363937
$756$	0	0
$757$	20.0000	0.726912	0.363456	−	0.931611i	$-0.381597\pi$
$757$	20.0000	0.726912	0.363456	+	0.931611i	$0.381597\pi$
$758$	−32.0000	−1.16229
$759$	3.00000	0.108893
$760$	0	0
$761$	−13.0000	−0.471250	−0.235625	−	0.971844i	$-0.575714\pi$
$761$	−13.0000	−0.471250	−0.235625	+	0.971844i	$0.575714\pi$
$762$	−38.0000	−1.37659
$763$	0	0
$764$	−24.0000	−0.868290
$765$	−4.00000	−0.144620
$766$	−16.0000	−0.578103
$767$	−20.0000	−0.722158
$768$	−16.0000	−0.577350
$769$	−38.0000	−1.37032	−0.685158	−	0.728395i	$-0.740267\pi$
$769$	−38.0000	−1.37032	−0.685158	+	0.728395i	$0.740267\pi$
$770$	0	0
$771$	13.0000	0.468184
$772$	−50.0000	−1.79954
$773$	42.0000	1.51064	0.755318	−	0.655359i	$-0.227483\pi$
$773$	42.0000	1.51064	0.755318	+	0.655359i	$0.227483\pi$
$774$	−14.0000	−0.503220
$775$	0	0
$776$	0	0
$777$	0	0
$778$	−12.0000	−0.430221
$779$	6.00000	0.214972
$780$	10.0000	0.358057
$781$	−6.00000	−0.214697
$782$	24.0000	0.858238
$783$	3.00000	0.107211
$784$	0	0
$785$	10.0000	0.356915
$786$	−6.00000	−0.214013
$787$	−32.0000	−1.14068	−0.570338	−	0.821410i	$-0.693188\pi$
$787$	−32.0000	−1.14068	−0.570338	+	0.821410i	$0.693188\pi$
$788$	−20.0000	−0.712470
$789$	−18.0000	−0.640817
$790$	8.00000	0.284627
$791$	0	0
$792$	0	0
$793$	30.0000	1.06533
$794$	−8.00000	−0.283909
$795$	−5.00000	−0.177332
$796$	−22.0000	−0.779769
$797$	45.0000	1.59398	0.796991	−	0.603991i	$-0.206424\pi$
$797$	45.0000	1.59398	0.796991	+	0.603991i	$0.206424\pi$
$798$	0	0
$799$	12.0000	0.424529
$800$	−8.00000	−0.282843
$801$	14.0000	0.494666
$802$	56.0000	1.97743
$803$	10.0000	0.352892
$804$	−24.0000	−0.846415
$805$	0	0
$806$	0	0
$807$	−6.00000	−0.211210
$808$	0	0
$809$	15.0000	0.527372	0.263686	−	0.964609i	$-0.415062\pi$
$809$	15.0000	0.527372	0.263686	+	0.964609i	$0.415062\pi$
$810$	−2.00000	−0.0702728
$811$	30.0000	1.05344	0.526721	−	0.850038i	$-0.323421\pi$
$811$	30.0000	1.05344	0.526721	+	0.850038i	$0.323421\pi$
$812$	0	0
$813$	20.0000	0.701431
$814$	12.0000	0.420600
$815$	−14.0000	−0.490399
$816$	16.0000	0.560112
$817$	−14.0000	−0.489798
$818$	52.0000	1.81814
$819$	0	0
$820$	−6.00000	−0.209529
$821$	38.0000	1.32621	0.663105	−	0.748527i	$-0.269238\pi$
$821$	38.0000	1.32621	0.663105	+	0.748527i	$0.269238\pi$
$822$	26.0000	0.906854
$823$	16.0000	0.557725	0.278862	−	0.960331i	$-0.410043\pi$
$823$	16.0000	0.557725	0.278862	+	0.960331i	$0.410043\pi$
$824$	0	0
$825$	1.00000	0.0348155
$826$	0	0
$827$	20.0000	0.695468	0.347734	−	0.937593i	$-0.386951\pi$
$827$	20.0000	0.695468	0.347734	+	0.937593i	$0.386951\pi$
$828$	6.00000	0.208514
$829$	25.0000	0.868286	0.434143	−	0.900844i	$-0.357051\pi$
$829$	25.0000	0.868286	0.434143	+	0.900844i	$0.357051\pi$
$830$	−8.00000	−0.277684
$831$	13.0000	0.450965
$832$	−40.0000	−1.38675
$833$	0	0
$834$	−32.0000	−1.10807
$835$	−22.0000	−0.761341
$836$	−4.00000	−0.138343
$837$	0	0
$838$	60.0000	2.07267
$839$	−36.0000	−1.24286	−0.621429	−	0.783470i	$-0.713448\pi$
$839$	−36.0000	−1.24286	−0.621429	+	0.783470i	$0.713448\pi$
$840$	0	0
$841$	−20.0000	−0.689655
$842$	−12.0000	−0.413547
$843$	27.0000	0.929929
$844$	20.0000	0.688428
$845$	−12.0000	−0.412813
$846$	6.00000	0.206284
$847$	0	0
$848$	20.0000	0.686803
$849$	28.0000	0.960958
$850$	8.00000	0.274398
$851$	−18.0000	−0.617032
$852$	−12.0000	−0.411113
$853$	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
$857$	−16.0000	−0.546550	−0.273275	−	0.961936i	$-0.588107\pi$
$857$	−16.0000	−0.546550	−0.273275	+	0.961936i	$0.588107\pi$
$858$	10.0000	0.341394
$859$	35.0000	1.19418	0.597092	−	0.802173i	$-0.296323\pi$
$859$	35.0000	1.19418	0.597092	+	0.802173i	$0.296323\pi$
$860$	14.0000	0.477396
$861$	0	0
$862$	−62.0000	−2.11173
$863$	21.0000	0.714848	0.357424	−	0.933942i	$-0.383655\pi$
$863$	21.0000	0.714848	0.357424	+	0.933942i	$0.383655\pi$
$864$	8.00000	0.272166
$865$	−22.0000	−0.748022
$866$	−24.0000	−0.815553
$867$	1.00000	0.0339618
$868$	0	0
$869$	4.00000	0.135691
$870$	−6.00000	−0.203419
$871$	60.0000	2.03302
$872$	0	0
$873$	8.00000	0.270759
$874$	12.0000	0.405906
$875$	0	0
$876$	20.0000	0.675737
$877$	−33.0000	−1.11433	−0.557165	−	0.830402i	$-0.688111\pi$
$877$	−33.0000	−1.11433	−0.557165	+	0.830402i	$0.688111\pi$
$878$	12.0000	0.404980
$879$	12.0000	0.404750
$880$	−4.00000	−0.134840
$881$	−36.0000	−1.21287	−0.606435	−	0.795133i	$-0.707401\pi$
$881$	−36.0000	−1.21287	−0.606435	+	0.795133i	$0.707401\pi$
$882$	0	0
$883$	−6.00000	−0.201916	−0.100958	−	0.994891i	$-0.532191\pi$
$883$	−6.00000	−0.201916	−0.100958	+	0.994891i	$0.532191\pi$
$884$	40.0000	1.34535
$885$	−4.00000	−0.134459
$886$	24.0000	0.806296
$887$	−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
$889$	0	0
$890$	−28.0000	−0.938562
$891$	−1.00000	−0.0335013
$892$	8.00000	0.267860
$893$	6.00000	0.200782
$894$	−42.0000	−1.40469
$895$	−14.0000	−0.467968
$896$	0	0
$897$	−15.0000	−0.500835
$898$	40.0000	1.33482
$899$	0	0
$900$	2.00000	0.0666667
$901$	−20.0000	−0.666297
$902$	−6.00000	−0.199778
$903$	0	0
$904$	0	0
$905$	−7.00000	−0.232688
$906$	20.0000	0.664455
$907$	30.0000	0.996134	0.498067	−	0.867139i	$-0.334043\pi$
$907$	30.0000	0.996134	0.498067	+	0.867139i	$0.334043\pi$
$908$	0	0
$909$	1.00000	0.0331679
$910$	0	0
$911$	32.0000	1.06021	0.530104	−	0.847933i	$-0.322153\pi$
$911$	32.0000	1.06021	0.530104	+	0.847933i	$0.322153\pi$
$912$	8.00000	0.264906
$913$	−4.00000	−0.132381
$914$	−74.0000	−2.44770
$915$	6.00000	0.198354
$916$	10.0000	0.330409
$917$	0	0
$918$	−8.00000	−0.264039
$919$	10.0000	0.329870	0.164935	−	0.986304i	$-0.447259\pi$
$919$	10.0000	0.329870	0.164935	+	0.986304i	$0.447259\pi$
$920$	0	0
$921$	−9.00000	−0.296560
$922$	−74.0000	−2.43706
$923$	30.0000	0.987462
$924$	0	0
$925$	−6.00000	−0.197279
$926$	−12.0000	−0.394344
$927$	4.00000	0.131377
$928$	24.0000	0.787839
$929$	−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
$931$	0	0
$932$	36.0000	1.17922
$933$	10.0000	0.327385
$934$	−30.0000	−0.981630
$935$	4.00000	0.130814
$936$	0	0
$937$	43.0000	1.40475	0.702374	−	0.711808i	$-0.252123\pi$
$937$	43.0000	1.40475	0.702374	+	0.711808i	$0.252123\pi$
$938$	0	0
$939$	16.0000	0.522140
$940$	−6.00000	−0.195698
$941$	−46.0000	−1.49956	−0.749779	−	0.661689i	$-0.769840\pi$
$941$	−46.0000	−1.49956	−0.749779	+	0.661689i	$0.769840\pi$
$942$	20.0000	0.651635
$943$	9.00000	0.293080
$944$	16.0000	0.520756
$945$	0	0
$946$	14.0000	0.455179
$947$	−36.0000	−1.16984	−0.584921	−	0.811090i	$-0.698875\pi$
$947$	−36.0000	−1.16984	−0.584921	+	0.811090i	$0.698875\pi$
$948$	8.00000	0.259828
$949$	−50.0000	−1.62307
$950$	4.00000	0.129777
$951$	−13.0000	−0.421554
$952$	0	0
$953$	36.0000	1.16615	0.583077	−	0.812417i	$-0.301849\pi$
$953$	36.0000	1.16615	0.583077	+	0.812417i	$0.301849\pi$
$954$	−10.0000	−0.323762
$955$	12.0000	0.388311
$956$	0	0
$957$	−3.00000	−0.0969762
$958$	−32.0000	−1.03387
$959$	0	0
$960$	−8.00000	−0.258199
$961$	−31.0000	−1.00000
$962$	−60.0000	−1.93448
$963$	10.0000	0.322245
$964$	52.0000	1.67481
$965$	25.0000	0.804778
$966$	0	0
$967$	−40.0000	−1.28631	−0.643157	−	0.765735i	$-0.722376\pi$
$967$	−40.0000	−1.28631	−0.643157	+	0.765735i	$0.722376\pi$
$968$	0	0
$969$	−8.00000	−0.256997
$970$	−16.0000	−0.513729
$971$	−48.0000	−1.54039	−0.770197	−	0.637806i	$-0.779842\pi$
$971$	−48.0000	−1.54039	−0.770197	+	0.637806i	$0.779842\pi$
$972$	−2.00000	−0.0641500
$973$	0	0
$974$	−4.00000	−0.128168
$975$	−5.00000	−0.160128
$976$	−24.0000	−0.768221
$977$	10.0000	0.319928	0.159964	−	0.987123i	$-0.448862\pi$
$977$	10.0000	0.319928	0.159964	+	0.987123i	$0.448862\pi$
$978$	−28.0000	−0.895341
$979$	−14.0000	−0.447442
$980$	0	0
$981$	10.0000	0.319275
$982$	−62.0000	−1.97850
$983$	5.00000	0.159475	0.0797376	−	0.996816i	$-0.474592\pi$
$983$	5.00000	0.159475	0.0797376	+	0.996816i	$0.474592\pi$
$984$	0	0
$985$	10.0000	0.318626
$986$	−24.0000	−0.764316
$987$	0	0
$988$	20.0000	0.636285
$989$	−21.0000	−0.667761
$990$	2.00000	0.0635642
$991$	47.0000	1.49300	0.746502	−	0.665383i	$-0.231732\pi$
$991$	47.0000	1.49300	0.746502	+	0.665383i	$0.231732\pi$
$992$	0	0
$993$	−17.0000	−0.539479
$994$	0	0
$995$	11.0000	0.348723
$996$	−8.00000	−0.253490
$997$	−39.0000	−1.23514	−0.617571	−	0.786515i	$-0.711883\pi$
$997$	−39.0000	−1.23514	−0.617571	+	0.786515i	$0.711883\pi$
$998$	−38.0000	−1.20287
$999$	6.00000	0.189832

Twists

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

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

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

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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