LMFDB - Embedded newform 2535.2.a.m.1.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [2535,2,Mod(1,2535)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("2535.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2535, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 2535 = 3 \cdot 5 \cdot 13^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2535.a (trivial)

Newform invariants

comment:select newform

sage:traces = [1,2,1,2,-1,2,5] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)

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

Self dual:	yes
Analytic conductor:	$20.2420769124$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 195)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	2535.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$

$=$

\(q+2.00000 q^{2} +1.00000 q^{3} +2.00000 q^{4} -1.00000 q^{5} +2.00000 q^{6} +5.00000 q^{7} +1.00000 q^{9} -2.00000 q^{10} +2.00000 q^{11} +2.00000 q^{12} +10.0000 q^{14} -1.00000 q^{15} -4.00000 q^{16} +2.00000 q^{17} +2.00000 q^{18} -2.00000 q^{20} +5.00000 q^{21} +4.00000 q^{22} +6.00000 q^{23} +1.00000 q^{25} +1.00000 q^{27} +10.0000 q^{28} -4.00000 q^{29} -2.00000 q^{30} -7.00000 q^{31} -8.00000 q^{32} +2.00000 q^{33} +4.00000 q^{34} -5.00000 q^{35} +2.00000 q^{36} -2.00000 q^{37} +6.00000 q^{41} +10.0000 q^{42} +1.00000 q^{43} +4.00000 q^{44} -1.00000 q^{45} +12.0000 q^{46} -8.00000 q^{47} -4.00000 q^{48} +18.0000 q^{49} +2.00000 q^{50} +2.00000 q^{51} -4.00000 q^{53} +2.00000 q^{54} -2.00000 q^{55} -8.00000 q^{58} +12.0000 q^{59} -2.00000 q^{60} -13.0000 q^{61} -14.0000 q^{62} +5.00000 q^{63} -8.00000 q^{64} +4.00000 q^{66} -7.00000 q^{67} +4.00000 q^{68} +6.00000 q^{69} -10.0000 q^{70} +12.0000 q^{71} +15.0000 q^{73} -4.00000 q^{74} +1.00000 q^{75} +10.0000 q^{77} +3.00000 q^{79} +4.00000 q^{80} +1.00000 q^{81} +12.0000 q^{82} +8.00000 q^{83} +10.0000 q^{84} -2.00000 q^{85} +2.00000 q^{86} -4.00000 q^{87} +14.0000 q^{89} -2.00000 q^{90} +12.0000 q^{92} -7.00000 q^{93} -16.0000 q^{94} -8.00000 q^{96} -5.00000 q^{97} +36.0000 q^{98} +2.00000 q^{99} +O(q^{100})\)

Coefficient data

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

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

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

$2$	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
$7$	5.00000	1.88982	0.944911	−	0.327327i	$-0.106148\pi$
$7$	5.00000	1.88982	0.944911	+	0.327327i	$0.106148\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	−2.00000	−0.632456
$11$	2.00000	0.603023	0.301511	−	0.953463i	$-0.402509\pi$
$11$	2.00000	0.603023	0.301511	+	0.953463i	$0.402509\pi$
$12$	2.00000	0.577350
$13$	0	0
$13$	0	0
$14$	10.0000	2.67261
$15$	−1.00000	−0.258199
$16$	−4.00000	−1.00000
$17$	2.00000	0.485071	0.242536	−	0.970143i	$-0.422021\pi$
$17$	2.00000	0.485071	0.242536	+	0.970143i	$0.422021\pi$
$18$	2.00000	0.471405
$19$	0	0		−	1.00000i	$-0.5\pi$
$19$	0	0			1.00000i	$0.5\pi$
$20$	−2.00000	−0.447214
$21$	5.00000	1.09109
$22$	4.00000	0.852803
$23$	6.00000	1.25109	0.625543	−	0.780189i	$-0.284877\pi$
$23$	6.00000	1.25109	0.625543	+	0.780189i	$0.284877\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	1.00000	0.192450
$28$	10.0000	1.88982
$29$	−4.00000	−0.742781	−0.371391	−	0.928477i	$-0.621119\pi$
$29$	−4.00000	−0.742781	−0.371391	+	0.928477i	$0.621119\pi$
$30$	−2.00000	−0.365148
$31$	−7.00000	−1.25724	−0.628619	−	0.777714i	$-0.716379\pi$
$31$	−7.00000	−1.25724	−0.628619	+	0.777714i	$0.716379\pi$
$32$	−8.00000	−1.41421
$33$	2.00000	0.348155
$34$	4.00000	0.685994
$35$	−5.00000	−0.845154
$36$	2.00000	0.333333
$37$	−2.00000	−0.328798	−0.164399	−	0.986394i	$-0.552568\pi$
$37$	−2.00000	−0.328798	−0.164399	+	0.986394i	$0.552568\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	6.00000	0.937043	0.468521	−	0.883452i	$-0.344787\pi$
$41$	6.00000	0.937043	0.468521	+	0.883452i	$0.344787\pi$
$42$	10.0000	1.54303
$43$	1.00000	0.152499	0.0762493	−	0.997089i	$-0.475706\pi$
$43$	1.00000	0.152499	0.0762493	+	0.997089i	$0.475706\pi$
$44$	4.00000	0.603023
$45$	−1.00000	−0.149071
$46$	12.0000	1.76930
$47$	−8.00000	−1.16692	−0.583460	−	0.812142i	$-0.698301\pi$
$47$	−8.00000	−1.16692	−0.583460	+	0.812142i	$0.698301\pi$
$48$	−4.00000	−0.577350
$49$	18.0000	2.57143
$50$	2.00000	0.282843
$51$	2.00000	0.280056
$52$	0	0
$53$	−4.00000	−0.549442	−0.274721	−	0.961524i	$-0.588586\pi$
$53$	−4.00000	−0.549442	−0.274721	+	0.961524i	$0.588586\pi$
$54$	2.00000	0.272166
$55$	−2.00000	−0.269680
$56$	0	0
$57$	0	0
$58$	−8.00000	−1.05045
$59$	12.0000	1.56227	0.781133	−	0.624364i	$-0.214642\pi$
$59$	12.0000	1.56227	0.781133	+	0.624364i	$0.214642\pi$
$60$	−2.00000	−0.258199
$61$	−13.0000	−1.66448	−0.832240	−	0.554416i	$-0.812942\pi$
$61$	−13.0000	−1.66448	−0.832240	+	0.554416i	$0.812942\pi$
$62$	−14.0000	−1.77800
$63$	5.00000	0.629941
$64$	−8.00000	−1.00000
$65$	0	0
$66$	4.00000	0.492366
$67$	−7.00000	−0.855186	−0.427593	−	0.903971i	$-0.640638\pi$
$67$	−7.00000	−0.855186	−0.427593	+	0.903971i	$0.640638\pi$
$68$	4.00000	0.485071
$69$	6.00000	0.722315
$70$	−10.0000	−1.19523
$71$	12.0000	1.42414	0.712069	−	0.702109i	$-0.247758\pi$
$71$	12.0000	1.42414	0.712069	+	0.702109i	$0.247758\pi$
$72$	0	0
$73$	15.0000	1.75562	0.877809	−	0.479012i	$-0.159005\pi$
$73$	15.0000	1.75562	0.877809	+	0.479012i	$0.159005\pi$
$74$	−4.00000	−0.464991
$75$	1.00000	0.115470
$76$	0	0
$77$	10.0000	1.13961
$78$	0	0
$79$	3.00000	0.337526	0.168763	−	0.985657i	$-0.446023\pi$
$79$	3.00000	0.337526	0.168763	+	0.985657i	$0.446023\pi$
$80$	4.00000	0.447214
$81$	1.00000	0.111111
$82$	12.0000	1.32518
$83$	8.00000	0.878114	0.439057	−	0.898459i	$-0.355313\pi$
$83$	8.00000	0.878114	0.439057	+	0.898459i	$0.355313\pi$
$84$	10.0000	1.09109
$85$	−2.00000	−0.216930
$86$	2.00000	0.215666
$87$	−4.00000	−0.428845
$88$	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
$92$	12.0000	1.25109
$93$	−7.00000	−0.725866
$94$	−16.0000	−1.65027
$95$	0	0
$96$	−8.00000	−0.816497
$97$	−5.00000	−0.507673	−0.253837	−	0.967247i	$-0.581693\pi$
$97$	−5.00000	−0.507673	−0.253837	+	0.967247i	$0.581693\pi$
$98$	36.0000	3.63655
$99$	2.00000	0.201008
$100$	2.00000	0.200000
$101$	−18.0000	−1.79107	−0.895533	−	0.444994i	$-0.853206\pi$
$101$	−18.0000	−1.79107	−0.895533	+	0.444994i	$0.853206\pi$
$102$	4.00000	0.396059
$103$	−7.00000	−0.689730	−0.344865	−	0.938652i	$-0.612075\pi$
$103$	−7.00000	−0.689730	−0.344865	+	0.938652i	$0.612075\pi$
$104$	0	0
$105$	−5.00000	−0.487950
$106$	−8.00000	−0.777029
$107$	4.00000	0.386695	0.193347	−	0.981130i	$-0.438066\pi$
$107$	4.00000	0.386695	0.193347	+	0.981130i	$0.438066\pi$
$108$	2.00000	0.192450
$109$	−11.0000	−1.05361	−0.526804	−	0.849987i	$-0.676610\pi$
$109$	−11.0000	−1.05361	−0.526804	+	0.849987i	$0.676610\pi$
$110$	−4.00000	−0.381385
$111$	−2.00000	−0.189832
$112$	−20.0000	−1.88982
$113$	−2.00000	−0.188144	−0.0940721	−	0.995565i	$-0.529988\pi$
$113$	−2.00000	−0.188144	−0.0940721	+	0.995565i	$0.529988\pi$
$114$	0	0
$115$	−6.00000	−0.559503
$116$	−8.00000	−0.742781
$117$	0	0
$118$	24.0000	2.20938
$119$	10.0000	0.916698
$120$	0	0
$121$	−7.00000	−0.636364
$122$	−26.0000	−2.35393
$123$	6.00000	0.541002
$124$	−14.0000	−1.25724
$125$	−1.00000	−0.0894427
$126$	10.0000	0.890871
$127$	−11.0000	−0.976092	−0.488046	−	0.872818i	$-0.662290\pi$
$127$	−11.0000	−0.976092	−0.488046	+	0.872818i	$0.662290\pi$
$128$	0	0
$129$	1.00000	0.0880451
$130$	0	0
$131$	−4.00000	−0.349482	−0.174741	−	0.984614i	$-0.555909\pi$
$131$	−4.00000	−0.349482	−0.174741	+	0.984614i	$0.555909\pi$
$132$	4.00000	0.348155
$133$	0	0
$134$	−14.0000	−1.20942
$135$	−1.00000	−0.0860663
$136$	0	0
$137$	−2.00000	−0.170872	−0.0854358	−	0.996344i	$-0.527228\pi$
$137$	−2.00000	−0.170872	−0.0854358	+	0.996344i	$0.527228\pi$
$138$	12.0000	1.02151
$139$	−3.00000	−0.254457	−0.127228	−	0.991873i	$-0.540608\pi$
$139$	−3.00000	−0.254457	−0.127228	+	0.991873i	$0.540608\pi$
$140$	−10.0000	−0.845154
$141$	−8.00000	−0.673722
$142$	24.0000	2.01404
$143$	0	0
$144$	−4.00000	−0.333333
$145$	4.00000	0.332182
$146$	30.0000	2.48282
$147$	18.0000	1.48461
$148$	−4.00000	−0.328798
$149$	−12.0000	−0.983078	−0.491539	−	0.870855i	$-0.663566\pi$
$149$	−12.0000	−0.983078	−0.491539	+	0.870855i	$0.663566\pi$
$150$	2.00000	0.163299
$151$	−8.00000	−0.651031	−0.325515	−	0.945537i	$-0.605538\pi$
$151$	−8.00000	−0.651031	−0.325515	+	0.945537i	$0.605538\pi$
$152$	0	0
$153$	2.00000	0.161690
$154$	20.0000	1.61165
$155$	7.00000	0.562254
$156$	0	0
$157$	−15.0000	−1.19713	−0.598565	−	0.801074i	$-0.704262\pi$
$157$	−15.0000	−1.19713	−0.598565	+	0.801074i	$0.704262\pi$
$158$	6.00000	0.477334
$159$	−4.00000	−0.317221
$160$	8.00000	0.632456
$161$	30.0000	2.36433
$162$	2.00000	0.157135
$163$	−15.0000	−1.17489	−0.587445	−	0.809264i	$-0.699866\pi$
$163$	−15.0000	−1.17489	−0.587445	+	0.809264i	$0.699866\pi$
$164$	12.0000	0.937043
$165$	−2.00000	−0.155700
$166$	16.0000	1.24184
$167$	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
$169$	0	0
$170$	−4.00000	−0.306786
$171$	0	0
$172$	2.00000	0.152499
$173$	0	0		−	1.00000i	$-0.5\pi$
$173$	0	0			1.00000i	$0.5\pi$
$174$	−8.00000	−0.606478
$175$	5.00000	0.377964
$176$	−8.00000	−0.603023
$177$	12.0000	0.901975
$178$	28.0000	2.09869
$179$	−6.00000	−0.448461	−0.224231	−	0.974536i	$-0.571987\pi$
$179$	−6.00000	−0.448461	−0.224231	+	0.974536i	$0.571987\pi$
$180$	−2.00000	−0.149071
$181$	−22.0000	−1.63525	−0.817624	−	0.575753i	$-0.804709\pi$
$181$	−22.0000	−1.63525	−0.817624	+	0.575753i	$0.804709\pi$
$182$	0	0
$183$	−13.0000	−0.960988
$184$	0	0
$185$	2.00000	0.147043
$186$	−14.0000	−1.02653
$187$	4.00000	0.292509
$188$	−16.0000	−1.16692
$189$	5.00000	0.363696
$190$	0	0
$191$	12.0000	0.868290	0.434145	−	0.900843i	$-0.357051\pi$
$191$	12.0000	0.868290	0.434145	+	0.900843i	$0.357051\pi$
$192$	−8.00000	−0.577350
$193$	−11.0000	−0.791797	−0.395899	−	0.918294i	$-0.629567\pi$
$193$	−11.0000	−0.791797	−0.395899	+	0.918294i	$0.629567\pi$
$194$	−10.0000	−0.717958
$195$	0	0
$196$	36.0000	2.57143
$197$	−12.0000	−0.854965	−0.427482	−	0.904024i	$-0.640599\pi$
$197$	−12.0000	−0.854965	−0.427482	+	0.904024i	$0.640599\pi$
$198$	4.00000	0.284268
$199$	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
$201$	−7.00000	−0.493742
$202$	−36.0000	−2.53295
$203$	−20.0000	−1.40372
$204$	4.00000	0.280056
$205$	−6.00000	−0.419058
$206$	−14.0000	−0.975426
$207$	6.00000	0.417029
$208$	0	0
$209$	0	0
$210$	−10.0000	−0.690066
$211$	15.0000	1.03264	0.516321	−	0.856395i	$-0.327301\pi$
$211$	15.0000	1.03264	0.516321	+	0.856395i	$0.327301\pi$
$212$	−8.00000	−0.549442
$213$	12.0000	0.822226
$214$	8.00000	0.546869
$215$	−1.00000	−0.0681994
$216$	0	0
$217$	−35.0000	−2.37595
$218$	−22.0000	−1.49003
$219$	15.0000	1.01361
$220$	−4.00000	−0.269680
$221$	0	0
$222$	−4.00000	−0.268462
$223$	−8.00000	−0.535720	−0.267860	−	0.963458i	$-0.586316\pi$
$223$	−8.00000	−0.535720	−0.267860	+	0.963458i	$0.586316\pi$
$224$	−40.0000	−2.67261
$225$	1.00000	0.0666667
$226$	−4.00000	−0.266076
$227$	10.0000	0.663723	0.331862	−	0.943328i	$-0.392323\pi$
$227$	10.0000	0.663723	0.331862	+	0.943328i	$0.392323\pi$
$228$	0	0
$229$	14.0000	0.925146	0.462573	−	0.886581i	$-0.346926\pi$
$229$	14.0000	0.925146	0.462573	+	0.886581i	$0.346926\pi$
$230$	−12.0000	−0.791257
$231$	10.0000	0.657952
$232$	0	0
$233$	−14.0000	−0.917170	−0.458585	−	0.888650i	$-0.651644\pi$
$233$	−14.0000	−0.917170	−0.458585	+	0.888650i	$0.651644\pi$
$234$	0	0
$235$	8.00000	0.521862
$236$	24.0000	1.56227
$237$	3.00000	0.194871
$238$	20.0000	1.29641
$239$	12.0000	0.776215	0.388108	−	0.921614i	$-0.373129\pi$
$239$	12.0000	0.776215	0.388108	+	0.921614i	$0.373129\pi$
$240$	4.00000	0.258199
$241$	10.0000	0.644157	0.322078	−	0.946713i	$-0.395619\pi$
$241$	10.0000	0.644157	0.322078	+	0.946713i	$0.395619\pi$
$242$	−14.0000	−0.899954
$243$	1.00000	0.0641500
$244$	−26.0000	−1.66448
$245$	−18.0000	−1.14998
$246$	12.0000	0.765092
$247$	0	0
$248$	0	0
$249$	8.00000	0.506979
$250$	−2.00000	−0.126491
$251$	0	0		−	1.00000i	$-0.5\pi$
$251$	0	0			1.00000i	$0.5\pi$
$252$	10.0000	0.629941
$253$	12.0000	0.754434
$254$	−22.0000	−1.38040
$255$	−2.00000	−0.125245
$256$	16.0000	1.00000
$257$	−22.0000	−1.37232	−0.686161	−	0.727450i	$-0.740706\pi$
$257$	−22.0000	−1.37232	−0.686161	+	0.727450i	$0.740706\pi$
$258$	2.00000	0.124515
$259$	−10.0000	−0.621370
$260$	0	0
$261$	−4.00000	−0.247594
$262$	−8.00000	−0.494242
$263$	−10.0000	−0.616626	−0.308313	−	0.951285i	$-0.599764\pi$
$263$	−10.0000	−0.616626	−0.308313	+	0.951285i	$0.599764\pi$
$264$	0	0
$265$	4.00000	0.245718
$266$	0	0
$267$	14.0000	0.856786
$268$	−14.0000	−0.855186
$269$	−6.00000	−0.365826	−0.182913	−	0.983129i	$-0.558553\pi$
$269$	−6.00000	−0.365826	−0.182913	+	0.983129i	$0.558553\pi$
$270$	−2.00000	−0.121716
$271$	29.0000	1.76162	0.880812	−	0.473466i	$-0.156997\pi$
$271$	29.0000	1.76162	0.880812	+	0.473466i	$0.156997\pi$
$272$	−8.00000	−0.485071
$273$	0	0
$274$	−4.00000	−0.241649
$275$	2.00000	0.120605
$276$	12.0000	0.722315
$277$	10.0000	0.600842	0.300421	−	0.953807i	$-0.402873\pi$
$277$	10.0000	0.600842	0.300421	+	0.953807i	$0.402873\pi$
$278$	−6.00000	−0.359856
$279$	−7.00000	−0.419079
$280$	0	0
$281$	−12.0000	−0.715860	−0.357930	−	0.933748i	$-0.616517\pi$
$281$	−12.0000	−0.715860	−0.357930	+	0.933748i	$0.616517\pi$
$282$	−16.0000	−0.952786
$283$	5.00000	0.297219	0.148610	−	0.988896i	$-0.452520\pi$
$283$	5.00000	0.297219	0.148610	+	0.988896i	$0.452520\pi$
$284$	24.0000	1.42414
$285$	0	0
$286$	0	0
$287$	30.0000	1.77084
$288$	−8.00000	−0.471405
$289$	−13.0000	−0.764706
$290$	8.00000	0.469776
$291$	−5.00000	−0.293105
$292$	30.0000	1.75562
$293$	16.0000	0.934730	0.467365	−	0.884064i	$-0.345203\pi$
$293$	16.0000	0.934730	0.467365	+	0.884064i	$0.345203\pi$
$294$	36.0000	2.09956
$295$	−12.0000	−0.698667
$296$	0	0
$297$	2.00000	0.116052
$298$	−24.0000	−1.39028
$299$	0	0
$300$	2.00000	0.115470
$301$	5.00000	0.288195
$302$	−16.0000	−0.920697
$303$	−18.0000	−1.03407
$304$	0	0
$305$	13.0000	0.744378
$306$	4.00000	0.228665
$307$	−31.0000	−1.76926	−0.884632	−	0.466290i	$-0.845590\pi$
$307$	−31.0000	−1.76926	−0.884632	+	0.466290i	$0.845590\pi$
$308$	20.0000	1.13961
$309$	−7.00000	−0.398216
$310$	14.0000	0.795147
$311$	−22.0000	−1.24751	−0.623753	−	0.781622i	$-0.714393\pi$
$311$	−22.0000	−1.24751	−0.623753	+	0.781622i	$0.714393\pi$
$312$	0	0
$313$	−31.0000	−1.75222	−0.876112	−	0.482108i	$-0.839871\pi$
$313$	−31.0000	−1.75222	−0.876112	+	0.482108i	$0.839871\pi$
$314$	−30.0000	−1.69300
$315$	−5.00000	−0.281718
$316$	6.00000	0.337526
$317$	12.0000	0.673987	0.336994	−	0.941507i	$-0.390590\pi$
$317$	12.0000	0.673987	0.336994	+	0.941507i	$0.390590\pi$
$318$	−8.00000	−0.448618
$319$	−8.00000	−0.447914
$320$	8.00000	0.447214
$321$	4.00000	0.223258
$322$	60.0000	3.34367
$323$	0	0
$324$	2.00000	0.111111
$325$	0	0
$326$	−30.0000	−1.66155
$327$	−11.0000	−0.608301
$328$	0	0
$329$	−40.0000	−2.20527
$330$	−4.00000	−0.220193
$331$	9.00000	0.494685	0.247342	−	0.968928i	$-0.420443\pi$
$331$	9.00000	0.494685	0.247342	+	0.968928i	$0.420443\pi$
$332$	16.0000	0.878114
$333$	−2.00000	−0.109599
$334$	24.0000	1.31322
$335$	7.00000	0.382451
$336$	−20.0000	−1.09109
$337$	−1.00000	−0.0544735	−0.0272367	−	0.999629i	$-0.508671\pi$
$337$	−1.00000	−0.0544735	−0.0272367	+	0.999629i	$0.508671\pi$
$338$	0	0
$339$	−2.00000	−0.108625
$340$	−4.00000	−0.216930
$341$	−14.0000	−0.758143
$342$	0	0
$343$	55.0000	2.96972
$344$	0	0
$345$	−6.00000	−0.323029
$346$	0	0
$347$	−16.0000	−0.858925	−0.429463	−	0.903085i	$-0.641297\pi$
$347$	−16.0000	−0.858925	−0.429463	+	0.903085i	$0.641297\pi$
$348$	−8.00000	−0.428845
$349$	3.00000	0.160586	0.0802932	−	0.996771i	$-0.474414\pi$
$349$	3.00000	0.160586	0.0802932	+	0.996771i	$0.474414\pi$
$350$	10.0000	0.534522
$351$	0	0
$352$	−16.0000	−0.852803
$353$	6.00000	0.319348	0.159674	−	0.987170i	$-0.448956\pi$
$353$	6.00000	0.319348	0.159674	+	0.987170i	$0.448956\pi$
$354$	24.0000	1.27559
$355$	−12.0000	−0.636894
$356$	28.0000	1.48400
$357$	10.0000	0.529256
$358$	−12.0000	−0.634220
$359$	2.00000	0.105556	0.0527780	−	0.998606i	$-0.483192\pi$
$359$	2.00000	0.105556	0.0527780	+	0.998606i	$0.483192\pi$
$360$	0	0
$361$	−19.0000	−1.00000
$362$	−44.0000	−2.31259
$363$	−7.00000	−0.367405
$364$	0	0
$365$	−15.0000	−0.785136
$366$	−26.0000	−1.35904
$367$	7.00000	0.365397	0.182699	−	0.983169i	$-0.441517\pi$
$367$	7.00000	0.365397	0.182699	+	0.983169i	$0.441517\pi$
$368$	−24.0000	−1.25109
$369$	6.00000	0.312348
$370$	4.00000	0.207950
$371$	−20.0000	−1.03835
$372$	−14.0000	−0.725866
$373$	13.0000	0.673114	0.336557	−	0.941663i	$-0.390737\pi$
$373$	13.0000	0.673114	0.336557	+	0.941663i	$0.390737\pi$
$374$	8.00000	0.413670
$375$	−1.00000	−0.0516398
$376$	0	0
$377$	0	0
$378$	10.0000	0.514344
$379$	−5.00000	−0.256833	−0.128416	−	0.991720i	$-0.540989\pi$
$379$	−5.00000	−0.256833	−0.128416	+	0.991720i	$0.540989\pi$
$380$	0	0
$381$	−11.0000	−0.563547
$382$	24.0000	1.22795
$383$	−18.0000	−0.919757	−0.459879	−	0.887982i	$-0.652107\pi$
$383$	−18.0000	−0.919757	−0.459879	+	0.887982i	$0.652107\pi$
$384$	0	0
$385$	−10.0000	−0.509647
$386$	−22.0000	−1.11977
$387$	1.00000	0.0508329
$388$	−10.0000	−0.507673
$389$	8.00000	0.405616	0.202808	−	0.979219i	$-0.434993\pi$
$389$	8.00000	0.405616	0.202808	+	0.979219i	$0.434993\pi$
$390$	0	0
$391$	12.0000	0.606866
$392$	0	0
$393$	−4.00000	−0.201773
$394$	−24.0000	−1.20910
$395$	−3.00000	−0.150946
$396$	4.00000	0.201008
$397$	−15.0000	−0.752828	−0.376414	−	0.926451i	$-0.622843\pi$
$397$	−15.0000	−0.752828	−0.376414	+	0.926451i	$0.622843\pi$
$398$	34.0000	1.70427
$399$	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	$0.369183\pi$
$402$	−14.0000	−0.698257
$403$	0	0
$404$	−36.0000	−1.79107
$405$	−1.00000	−0.0496904
$406$	−40.0000	−1.98517
$407$	−4.00000	−0.198273
$408$	0	0
$409$	−15.0000	−0.741702	−0.370851	−	0.928692i	$-0.620934\pi$
$409$	−15.0000	−0.741702	−0.370851	+	0.928692i	$0.620934\pi$
$410$	−12.0000	−0.592638
$411$	−2.00000	−0.0986527
$412$	−14.0000	−0.689730
$413$	60.0000	2.95241
$414$	12.0000	0.589768
$415$	−8.00000	−0.392705
$416$	0	0
$417$	−3.00000	−0.146911
$418$	0	0
$419$	38.0000	1.85642	0.928211	−	0.372055i	$-0.121347\pi$
$419$	38.0000	1.85642	0.928211	+	0.372055i	$0.121347\pi$
$420$	−10.0000	−0.487950
$421$	−23.0000	−1.12095	−0.560476	−	0.828171i	$-0.689382\pi$
$421$	−23.0000	−1.12095	−0.560476	+	0.828171i	$0.689382\pi$
$422$	30.0000	1.46038
$423$	−8.00000	−0.388973
$424$	0	0
$425$	2.00000	0.0970143
$426$	24.0000	1.16280
$427$	−65.0000	−3.14557
$428$	8.00000	0.386695
$429$	0	0
$430$	−2.00000	−0.0964486
$431$	−28.0000	−1.34871	−0.674356	−	0.738406i	$-0.735579\pi$
$431$	−28.0000	−1.34871	−0.674356	+	0.738406i	$0.735579\pi$
$432$	−4.00000	−0.192450
$433$	1.00000	0.0480569	0.0240285	−	0.999711i	$-0.492351\pi$
$433$	1.00000	0.0480569	0.0240285	+	0.999711i	$0.492351\pi$
$434$	−70.0000	−3.36011
$435$	4.00000	0.191785
$436$	−22.0000	−1.05361
$437$	0	0
$438$	30.0000	1.43346
$439$	15.0000	0.715911	0.357955	−	0.933739i	$-0.383474\pi$
$439$	15.0000	0.715911	0.357955	+	0.933739i	$0.383474\pi$
$440$	0	0
$441$	18.0000	0.857143
$442$	0	0
$443$	26.0000	1.23530	0.617649	−	0.786454i	$-0.288085\pi$
$443$	26.0000	1.23530	0.617649	+	0.786454i	$0.288085\pi$
$444$	−4.00000	−0.189832
$445$	−14.0000	−0.663664
$446$	−16.0000	−0.757622
$447$	−12.0000	−0.567581
$448$	−40.0000	−1.88982
$449$	18.0000	0.849473	0.424736	−	0.905317i	$-0.360367\pi$
$449$	18.0000	0.849473	0.424736	+	0.905317i	$0.360367\pi$
$450$	2.00000	0.0942809
$451$	12.0000	0.565058
$452$	−4.00000	−0.188144
$453$	−8.00000	−0.375873
$454$	20.0000	0.938647
$455$	0	0
$456$	0	0
$457$	35.0000	1.63723	0.818615	−	0.574342i	$-0.194742\pi$
$457$	35.0000	1.63723	0.818615	+	0.574342i	$0.194742\pi$
$458$	28.0000	1.30835
$459$	2.00000	0.0933520
$460$	−12.0000	−0.559503
$461$	−2.00000	−0.0931493	−0.0465746	−	0.998915i	$-0.514831\pi$
$461$	−2.00000	−0.0931493	−0.0465746	+	0.998915i	$0.514831\pi$
$462$	20.0000	0.930484
$463$	3.00000	0.139422	0.0697109	−	0.997567i	$-0.477792\pi$
$463$	3.00000	0.139422	0.0697109	+	0.997567i	$0.477792\pi$
$464$	16.0000	0.742781
$465$	7.00000	0.324617
$466$	−28.0000	−1.29707
$467$	−4.00000	−0.185098	−0.0925490	−	0.995708i	$-0.529501\pi$
$467$	−4.00000	−0.185098	−0.0925490	+	0.995708i	$0.529501\pi$
$468$	0	0
$469$	−35.0000	−1.61615
$470$	16.0000	0.738025
$471$	−15.0000	−0.691164
$472$	0	0
$473$	2.00000	0.0919601
$474$	6.00000	0.275589
$475$	0	0
$476$	20.0000	0.916698
$477$	−4.00000	−0.183147
$478$	24.0000	1.09773
$479$	42.0000	1.91903	0.959514	−	0.281659i	$-0.0908848\pi$
$479$	42.0000	1.91903	0.959514	+	0.281659i	$0.0908848\pi$
$480$	8.00000	0.365148
$481$	0	0
$482$	20.0000	0.910975
$483$	30.0000	1.36505
$484$	−14.0000	−0.636364
$485$	5.00000	0.227038
$486$	2.00000	0.0907218
$487$	−28.0000	−1.26880	−0.634401	−	0.773004i	$-0.718753\pi$
$487$	−28.0000	−1.26880	−0.634401	+	0.773004i	$0.718753\pi$
$488$	0	0
$489$	−15.0000	−0.678323
$490$	−36.0000	−1.62631
$491$	−24.0000	−1.08310	−0.541552	−	0.840667i	$-0.682163\pi$
$491$	−24.0000	−1.08310	−0.541552	+	0.840667i	$0.682163\pi$
$492$	12.0000	0.541002
$493$	−8.00000	−0.360302
$494$	0	0
$495$	−2.00000	−0.0898933
$496$	28.0000	1.25724
$497$	60.0000	2.69137
$498$	16.0000	0.716977
$499$	4.00000	0.179065	0.0895323	−	0.995984i	$-0.471463\pi$
$499$	4.00000	0.179065	0.0895323	+	0.995984i	$0.471463\pi$
$500$	−2.00000	−0.0894427
$501$	12.0000	0.536120
$502$	0	0
$503$	−6.00000	−0.267527	−0.133763	−	0.991013i	$-0.542706\pi$
$503$	−6.00000	−0.267527	−0.133763	+	0.991013i	$0.542706\pi$
$504$	0	0
$505$	18.0000	0.800989
$506$	24.0000	1.06693
$507$	0	0
$508$	−22.0000	−0.976092
$509$	−30.0000	−1.32973	−0.664863	−	0.746965i	$-0.731510\pi$
$509$	−30.0000	−1.32973	−0.664863	+	0.746965i	$0.731510\pi$
$510$	−4.00000	−0.177123
$511$	75.0000	3.31780
$512$	32.0000	1.41421
$513$	0	0
$514$	−44.0000	−1.94076
$515$	7.00000	0.308457
$516$	2.00000	0.0880451
$517$	−16.0000	−0.703679
$518$	−20.0000	−0.878750
$519$	0	0
$520$	0	0
$521$	−30.0000	−1.31432	−0.657162	−	0.753749i	$-0.728243\pi$
$521$	−30.0000	−1.31432	−0.657162	+	0.753749i	$0.728243\pi$
$522$	−8.00000	−0.350150
$523$	0	0		−	1.00000i	$-0.5\pi$
$523$	0	0			1.00000i	$0.5\pi$
$524$	−8.00000	−0.349482
$525$	5.00000	0.218218
$526$	−20.0000	−0.872041
$527$	−14.0000	−0.609850
$528$	−8.00000	−0.348155
$529$	13.0000	0.565217
$530$	8.00000	0.347498
$531$	12.0000	0.520756
$532$	0	0
$533$	0	0
$534$	28.0000	1.21168
$535$	−4.00000	−0.172935
$536$	0	0
$537$	−6.00000	−0.258919
$538$	−12.0000	−0.517357
$539$	36.0000	1.55063
$540$	−2.00000	−0.0860663
$541$	29.0000	1.24681	0.623404	−	0.781900i	$-0.285749\pi$
$541$	29.0000	1.24681	0.623404	+	0.781900i	$0.285749\pi$
$542$	58.0000	2.49131
$543$	−22.0000	−0.944110
$544$	−16.0000	−0.685994
$545$	11.0000	0.471188
$546$	0	0
$547$	−9.00000	−0.384812	−0.192406	−	0.981315i	$-0.561629\pi$
$547$	−9.00000	−0.384812	−0.192406	+	0.981315i	$0.561629\pi$
$548$	−4.00000	−0.170872
$549$	−13.0000	−0.554826
$550$	4.00000	0.170561
$551$	0	0
$552$	0	0
$553$	15.0000	0.637865
$554$	20.0000	0.849719
$555$	2.00000	0.0848953
$556$	−6.00000	−0.254457
$557$	−20.0000	−0.847427	−0.423714	−	0.905796i	$-0.639274\pi$
$557$	−20.0000	−0.847427	−0.423714	+	0.905796i	$0.639274\pi$
$558$	−14.0000	−0.592667
$559$	0	0
$560$	20.0000	0.845154
$561$	4.00000	0.168880
$562$	−24.0000	−1.01238
$563$	26.0000	1.09577	0.547885	−	0.836554i	$-0.315433\pi$
$563$	26.0000	1.09577	0.547885	+	0.836554i	$0.315433\pi$
$564$	−16.0000	−0.673722
$565$	2.00000	0.0841406
$566$	10.0000	0.420331
$567$	5.00000	0.209980
$568$	0	0
$569$	−20.0000	−0.838444	−0.419222	−	0.907884i	$-0.637697\pi$
$569$	−20.0000	−0.838444	−0.419222	+	0.907884i	$0.637697\pi$
$570$	0	0
$571$	12.0000	0.502184	0.251092	−	0.967963i	$-0.419210\pi$
$571$	12.0000	0.502184	0.251092	+	0.967963i	$0.419210\pi$
$572$	0	0
$573$	12.0000	0.501307
$574$	60.0000	2.50435
$575$	6.00000	0.250217
$576$	−8.00000	−0.333333
$577$	2.00000	0.0832611	0.0416305	−	0.999133i	$-0.486745\pi$
$577$	2.00000	0.0832611	0.0416305	+	0.999133i	$0.486745\pi$
$578$	−26.0000	−1.08146
$579$	−11.0000	−0.457144
$580$	8.00000	0.332182
$581$	40.0000	1.65948
$582$	−10.0000	−0.414513
$583$	−8.00000	−0.331326
$584$	0	0
$585$	0	0
$586$	32.0000	1.32191
$587$	28.0000	1.15568	0.577842	−	0.816149i	$-0.303895\pi$
$587$	28.0000	1.15568	0.577842	+	0.816149i	$0.303895\pi$
$588$	36.0000	1.48461
$589$	0	0
$590$	−24.0000	−0.988064
$591$	−12.0000	−0.493614
$592$	8.00000	0.328798
$593$	−10.0000	−0.410651	−0.205325	−	0.978694i	$-0.565825\pi$
$593$	−10.0000	−0.410651	−0.205325	+	0.978694i	$0.565825\pi$
$594$	4.00000	0.164122
$595$	−10.0000	−0.409960
$596$	−24.0000	−0.983078
$597$	17.0000	0.695764
$598$	0	0
$599$	−16.0000	−0.653742	−0.326871	−	0.945069i	$-0.605994\pi$
$599$	−16.0000	−0.653742	−0.326871	+	0.945069i	$0.605994\pi$
$600$	0	0
$601$	22.0000	0.897399	0.448699	−	0.893683i	$-0.351887\pi$
$601$	22.0000	0.897399	0.448699	+	0.893683i	$0.351887\pi$
$602$	10.0000	0.407570
$603$	−7.00000	−0.285062
$604$	−16.0000	−0.651031
$605$	7.00000	0.284590
$606$	−36.0000	−1.46240
$607$	16.0000	0.649420	0.324710	−	0.945814i	$-0.394733\pi$
$607$	16.0000	0.649420	0.324710	+	0.945814i	$0.394733\pi$
$608$	0	0
$609$	−20.0000	−0.810441
$610$	26.0000	1.05271
$611$	0	0
$612$	4.00000	0.161690
$613$	−15.0000	−0.605844	−0.302922	−	0.953015i	$-0.597962\pi$
$613$	−15.0000	−0.605844	−0.302922	+	0.953015i	$0.597962\pi$
$614$	−62.0000	−2.50212
$615$	−6.00000	−0.241943
$616$	0	0
$617$	6.00000	0.241551	0.120775	−	0.992680i	$-0.461462\pi$
$617$	6.00000	0.241551	0.120775	+	0.992680i	$0.461462\pi$
$618$	−14.0000	−0.563163
$619$	37.0000	1.48716	0.743578	−	0.668649i	$-0.233127\pi$
$619$	37.0000	1.48716	0.743578	+	0.668649i	$0.233127\pi$
$620$	14.0000	0.562254
$621$	6.00000	0.240772
$622$	−44.0000	−1.76424
$623$	70.0000	2.80449
$624$	0	0
$625$	1.00000	0.0400000
$626$	−62.0000	−2.47802
$627$	0	0
$628$	−30.0000	−1.19713
$629$	−4.00000	−0.159490
$630$	−10.0000	−0.398410
$631$	7.00000	0.278666	0.139333	−	0.990246i	$-0.455504\pi$
$631$	7.00000	0.278666	0.139333	+	0.990246i	$0.455504\pi$
$632$	0	0
$633$	15.0000	0.596196
$634$	24.0000	0.953162
$635$	11.0000	0.436522
$636$	−8.00000	−0.317221
$637$	0	0
$638$	−16.0000	−0.633446
$639$	12.0000	0.474713
$640$	0	0
$641$	2.00000	0.0789953	0.0394976	−	0.999220i	$-0.487424\pi$
$641$	2.00000	0.0789953	0.0394976	+	0.999220i	$0.487424\pi$
$642$	8.00000	0.315735
$643$	−19.0000	−0.749287	−0.374643	−	0.927169i	$-0.622235\pi$
$643$	−19.0000	−0.749287	−0.374643	+	0.927169i	$0.622235\pi$
$644$	60.0000	2.36433
$645$	−1.00000	−0.0393750
$646$	0	0
$647$	38.0000	1.49393	0.746967	−	0.664861i	$-0.231509\pi$
$647$	38.0000	1.49393	0.746967	+	0.664861i	$0.231509\pi$
$648$	0	0
$649$	24.0000	0.942082
$650$	0	0
$651$	−35.0000	−1.37176
$652$	−30.0000	−1.17489
$653$	42.0000	1.64359	0.821794	−	0.569785i	$-0.192974\pi$
$653$	42.0000	1.64359	0.821794	+	0.569785i	$0.192974\pi$
$654$	−22.0000	−0.860268
$655$	4.00000	0.156293
$656$	−24.0000	−0.937043
$657$	15.0000	0.585206
$658$	−80.0000	−3.11872
$659$	−24.0000	−0.934907	−0.467454	−	0.884018i	$-0.654829\pi$
$659$	−24.0000	−0.934907	−0.467454	+	0.884018i	$0.654829\pi$
$660$	−4.00000	−0.155700
$661$	35.0000	1.36134	0.680671	−	0.732589i	$-0.261688\pi$
$661$	35.0000	1.36134	0.680671	+	0.732589i	$0.261688\pi$
$662$	18.0000	0.699590
$663$	0	0
$664$	0	0
$665$	0	0
$666$	−4.00000	−0.154997
$667$	−24.0000	−0.929284
$668$	24.0000	0.928588
$669$	−8.00000	−0.309298
$670$	14.0000	0.540867
$671$	−26.0000	−1.00372
$672$	−40.0000	−1.54303
$673$	33.0000	1.27206	0.636028	−	0.771666i	$-0.280576\pi$
$673$	33.0000	1.27206	0.636028	+	0.771666i	$0.280576\pi$
$674$	−2.00000	−0.0770371
$675$	1.00000	0.0384900
$676$	0	0
$677$	−12.0000	−0.461197	−0.230599	−	0.973049i	$-0.574068\pi$
$677$	−12.0000	−0.461197	−0.230599	+	0.973049i	$0.574068\pi$
$678$	−4.00000	−0.153619
$679$	−25.0000	−0.959412
$680$	0	0
$681$	10.0000	0.383201
$682$	−28.0000	−1.07218
$683$	−20.0000	−0.765279	−0.382639	−	0.923898i	$-0.624985\pi$
$683$	−20.0000	−0.765279	−0.382639	+	0.923898i	$0.624985\pi$
$684$	0	0
$685$	2.00000	0.0764161
$686$	110.000	4.19982
$687$	14.0000	0.534133
$688$	−4.00000	−0.152499
$689$	0	0
$690$	−12.0000	−0.456832
$691$	37.0000	1.40755	0.703773	−	0.710425i	$-0.251497\pi$
$691$	37.0000	1.40755	0.703773	+	0.710425i	$0.251497\pi$
$692$	0	0
$693$	10.0000	0.379869
$694$	−32.0000	−1.21470
$695$	3.00000	0.113796
$696$	0	0
$697$	12.0000	0.454532
$698$	6.00000	0.227103
$699$	−14.0000	−0.529529
$700$	10.0000	0.377964
$701$	40.0000	1.51078	0.755390	−	0.655276i	$-0.227448\pi$
$701$	40.0000	1.51078	0.755390	+	0.655276i	$0.227448\pi$
$702$	0	0
$703$	0	0
$704$	−16.0000	−0.603023
$705$	8.00000	0.301297
$706$	12.0000	0.451626
$707$	−90.0000	−3.38480
$708$	24.0000	0.901975
$709$	−23.0000	−0.863783	−0.431892	−	0.901926i	$-0.642154\pi$
$709$	−23.0000	−0.863783	−0.431892	+	0.901926i	$0.642154\pi$
$710$	−24.0000	−0.900704
$711$	3.00000	0.112509
$712$	0	0
$713$	−42.0000	−1.57291
$714$	20.0000	0.748481
$715$	0	0
$716$	−12.0000	−0.448461
$717$	12.0000	0.448148
$718$	4.00000	0.149279
$719$	40.0000	1.49175	0.745874	−	0.666087i	$-0.232032\pi$
$719$	40.0000	1.49175	0.745874	+	0.666087i	$0.232032\pi$
$720$	4.00000	0.149071
$721$	−35.0000	−1.30347
$722$	−38.0000	−1.41421
$723$	10.0000	0.371904
$724$	−44.0000	−1.63525
$725$	−4.00000	−0.148556
$726$	−14.0000	−0.519589
$727$	9.00000	0.333792	0.166896	−	0.985975i	$-0.446626\pi$
$727$	9.00000	0.333792	0.166896	+	0.985975i	$0.446626\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	−30.0000	−1.11035
$731$	2.00000	0.0739727
$732$	−26.0000	−0.960988
$733$	7.00000	0.258551	0.129275	−	0.991609i	$-0.458735\pi$
$733$	7.00000	0.258551	0.129275	+	0.991609i	$0.458735\pi$
$734$	14.0000	0.516749
$735$	−18.0000	−0.663940
$736$	−48.0000	−1.76930
$737$	−14.0000	−0.515697
$738$	12.0000	0.441726
$739$	−36.0000	−1.32428	−0.662141	−	0.749380i	$-0.730352\pi$
$739$	−36.0000	−1.32428	−0.662141	+	0.749380i	$0.730352\pi$
$740$	4.00000	0.147043
$741$	0	0
$742$	−40.0000	−1.46845
$743$	24.0000	0.880475	0.440237	−	0.897881i	$-0.354894\pi$
$743$	24.0000	0.880475	0.440237	+	0.897881i	$0.354894\pi$
$744$	0	0
$745$	12.0000	0.439646
$746$	26.0000	0.951928
$747$	8.00000	0.292705
$748$	8.00000	0.292509
$749$	20.0000	0.730784
$750$	−2.00000	−0.0730297
$751$	−28.0000	−1.02173	−0.510867	−	0.859660i	$-0.670676\pi$
$751$	−28.0000	−1.02173	−0.510867	+	0.859660i	$0.670676\pi$
$752$	32.0000	1.16692
$753$	0	0
$754$	0	0
$755$	8.00000	0.291150
$756$	10.0000	0.363696
$757$	2.00000	0.0726912	0.0363456	−	0.999339i	$-0.488428\pi$
$757$	2.00000	0.0726912	0.0363456	+	0.999339i	$0.488428\pi$
$758$	−10.0000	−0.363216
$759$	12.0000	0.435572
$760$	0	0
$761$	−36.0000	−1.30500	−0.652499	−	0.757789i	$-0.726280\pi$
$761$	−36.0000	−1.30500	−0.652499	+	0.757789i	$0.726280\pi$
$762$	−22.0000	−0.796976
$763$	−55.0000	−1.99113
$764$	24.0000	0.868290
$765$	−2.00000	−0.0723102
$766$	−36.0000	−1.30073
$767$	0	0
$768$	16.0000	0.577350
$769$	−34.0000	−1.22607	−0.613036	−	0.790055i	$-0.710052\pi$
$769$	−34.0000	−1.22607	−0.613036	+	0.790055i	$0.710052\pi$
$770$	−20.0000	−0.720750
$771$	−22.0000	−0.792311
$772$	−22.0000	−0.791797
$773$	46.0000	1.65451	0.827253	−	0.561830i	$-0.189903\pi$
$773$	46.0000	1.65451	0.827253	+	0.561830i	$0.189903\pi$
$774$	2.00000	0.0718885
$775$	−7.00000	−0.251447
$776$	0	0
$777$	−10.0000	−0.358748
$778$	16.0000	0.573628
$779$	0	0
$780$	0	0
$781$	24.0000	0.858788
$782$	24.0000	0.858238
$783$	−4.00000	−0.142948
$784$	−72.0000	−2.57143
$785$	15.0000	0.535373
$786$	−8.00000	−0.285351
$787$	17.0000	0.605985	0.302992	−	0.952993i	$-0.402014\pi$
$787$	17.0000	0.605985	0.302992	+	0.952993i	$0.402014\pi$
$788$	−24.0000	−0.854965
$789$	−10.0000	−0.356009
$790$	−6.00000	−0.213470
$791$	−10.0000	−0.355559
$792$	0	0
$793$	0	0
$794$	−30.0000	−1.06466
$795$	4.00000	0.141865
$796$	34.0000	1.20510
$797$	30.0000	1.06265	0.531327	−	0.847167i	$-0.321693\pi$
$797$	30.0000	1.06265	0.531327	+	0.847167i	$0.321693\pi$
$798$	0	0
$799$	−16.0000	−0.566039
$800$	−8.00000	−0.282843
$801$	14.0000	0.494666
$802$	32.0000	1.12996
$803$	30.0000	1.05868
$804$	−14.0000	−0.493742
$805$	−30.0000	−1.05736
$806$	0	0
$807$	−6.00000	−0.211210
$808$	0	0
$809$	4.00000	0.140633	0.0703163	−	0.997525i	$-0.477599\pi$
$809$	4.00000	0.140633	0.0703163	+	0.997525i	$0.477599\pi$
$810$	−2.00000	−0.0702728
$811$	45.0000	1.58016	0.790082	−	0.613001i	$-0.210038\pi$
$811$	45.0000	1.58016	0.790082	+	0.613001i	$0.210038\pi$
$812$	−40.0000	−1.40372
$813$	29.0000	1.01707
$814$	−8.00000	−0.280400
$815$	15.0000	0.525427
$816$	−8.00000	−0.280056
$817$	0	0
$818$	−30.0000	−1.04893
$819$	0	0
$820$	−12.0000	−0.419058
$821$	−22.0000	−0.767805	−0.383903	−	0.923374i	$-0.625420\pi$
$821$	−22.0000	−0.767805	−0.383903	+	0.923374i	$0.625420\pi$
$822$	−4.00000	−0.139516
$823$	20.0000	0.697156	0.348578	−	0.937280i	$-0.386665\pi$
$823$	20.0000	0.697156	0.348578	+	0.937280i	$0.386665\pi$
$824$	0	0
$825$	2.00000	0.0696311
$826$	120.000	4.17533
$827$	46.0000	1.59958	0.799788	−	0.600282i	$-0.204945\pi$
$827$	46.0000	1.59958	0.799788	+	0.600282i	$0.204945\pi$
$828$	12.0000	0.417029
$829$	−11.0000	−0.382046	−0.191023	−	0.981586i	$-0.561180\pi$
$829$	−11.0000	−0.382046	−0.191023	+	0.981586i	$0.561180\pi$
$830$	−16.0000	−0.555368
$831$	10.0000	0.346896
$832$	0	0
$833$	36.0000	1.24733
$834$	−6.00000	−0.207763
$835$	−12.0000	−0.415277
$836$	0	0
$837$	−7.00000	−0.241955
$838$	76.0000	2.62538
$839$	−34.0000	−1.17381	−0.586905	−	0.809656i	$-0.699654\pi$
$839$	−34.0000	−1.17381	−0.586905	+	0.809656i	$0.699654\pi$
$840$	0	0
$841$	−13.0000	−0.448276
$842$	−46.0000	−1.58526
$843$	−12.0000	−0.413302
$844$	30.0000	1.03264
$845$	0	0
$846$	−16.0000	−0.550091
$847$	−35.0000	−1.20261
$848$	16.0000	0.549442
$849$	5.00000	0.171600
$850$	4.00000	0.137199
$851$	−12.0000	−0.411355
$852$	24.0000	0.822226
$853$	9.00000	0.308154	0.154077	−	0.988059i	$-0.450760\pi$
$853$	9.00000	0.308154	0.154077	+	0.988059i	$0.450760\pi$
$854$	−130.000	−4.44851
$855$	0	0
$856$	0	0
$857$	−12.0000	−0.409912	−0.204956	−	0.978771i	$-0.565705\pi$
$857$	−12.0000	−0.409912	−0.204956	+	0.978771i	$0.565705\pi$
$858$	0	0
$859$	43.0000	1.46714	0.733571	−	0.679613i	$-0.237852\pi$
$859$	43.0000	1.46714	0.733571	+	0.679613i	$0.237852\pi$
$860$	−2.00000	−0.0681994
$861$	30.0000	1.02240
$862$	−56.0000	−1.90737
$863$	6.00000	0.204242	0.102121	−	0.994772i	$-0.467437\pi$
$863$	6.00000	0.204242	0.102121	+	0.994772i	$0.467437\pi$
$864$	−8.00000	−0.272166
$865$	0	0
$866$	2.00000	0.0679628
$867$	−13.0000	−0.441503
$868$	−70.0000	−2.37595
$869$	6.00000	0.203536
$870$	8.00000	0.271225
$871$	0	0
$872$	0	0
$873$	−5.00000	−0.169224
$874$	0	0
$875$	−5.00000	−0.169031
$876$	30.0000	1.01361
$877$	6.00000	0.202606	0.101303	−	0.994856i	$-0.467699\pi$
$877$	6.00000	0.202606	0.101303	+	0.994856i	$0.467699\pi$
$878$	30.0000	1.01245
$879$	16.0000	0.539667
$880$	8.00000	0.269680
$881$	20.0000	0.673817	0.336909	−	0.941537i	$-0.390619\pi$
$881$	20.0000	0.673817	0.336909	+	0.941537i	$0.390619\pi$
$882$	36.0000	1.21218
$883$	−25.0000	−0.841317	−0.420658	−	0.907219i	$-0.638201\pi$
$883$	−25.0000	−0.841317	−0.420658	+	0.907219i	$0.638201\pi$
$884$	0	0
$885$	−12.0000	−0.403376
$886$	52.0000	1.74697
$887$	−44.0000	−1.47738	−0.738688	−	0.674048i	$-0.764554\pi$
$887$	−44.0000	−1.47738	−0.738688	+	0.674048i	$0.764554\pi$
$888$	0	0
$889$	−55.0000	−1.84464
$890$	−28.0000	−0.938562
$891$	2.00000	0.0670025
$892$	−16.0000	−0.535720
$893$	0	0
$894$	−24.0000	−0.802680
$895$	6.00000	0.200558
$896$	0	0
$897$	0	0
$898$	36.0000	1.20134
$899$	28.0000	0.933852
$900$	2.00000	0.0666667
$901$	−8.00000	−0.266519
$902$	24.0000	0.799113
$903$	5.00000	0.166390
$904$	0	0
$905$	22.0000	0.731305
$906$	−16.0000	−0.531564
$907$	20.0000	0.664089	0.332045	−	0.943264i	$-0.392262\pi$
$907$	20.0000	0.664089	0.332045	+	0.943264i	$0.392262\pi$
$908$	20.0000	0.663723
$909$	−18.0000	−0.597022
$910$	0	0
$911$	0	0		−	1.00000i	$-0.5\pi$
$911$	0	0			1.00000i	$0.5\pi$
$912$	0	0
$913$	16.0000	0.529523
$914$	70.0000	2.31539
$915$	13.0000	0.429767
$916$	28.0000	0.925146
$917$	−20.0000	−0.660458
$918$	4.00000	0.132020
$919$	8.00000	0.263896	0.131948	−	0.991257i	$-0.457877\pi$
$919$	8.00000	0.263896	0.131948	+	0.991257i	$0.457877\pi$
$920$	0	0
$921$	−31.0000	−1.02148
$922$	−4.00000	−0.131733
$923$	0	0
$924$	20.0000	0.657952
$925$	−2.00000	−0.0657596
$926$	6.00000	0.197172
$927$	−7.00000	−0.229910
$928$	32.0000	1.05045
$929$	52.0000	1.70606	0.853032	−	0.521858i	$-0.174761\pi$
$929$	52.0000	1.70606	0.853032	+	0.521858i	$0.174761\pi$
$930$	14.0000	0.459078
$931$	0	0
$932$	−28.0000	−0.917170
$933$	−22.0000	−0.720248
$934$	−8.00000	−0.261768
$935$	−4.00000	−0.130814
$936$	0	0
$937$	−30.0000	−0.980057	−0.490029	−	0.871706i	$-0.663014\pi$
$937$	−30.0000	−0.980057	−0.490029	+	0.871706i	$0.663014\pi$
$938$	−70.0000	−2.28558
$939$	−31.0000	−1.01165
$940$	16.0000	0.521862
$941$	0	0		−	1.00000i	$-0.5\pi$
$941$	0	0			1.00000i	$0.5\pi$
$942$	−30.0000	−0.977453
$943$	36.0000	1.17232
$944$	−48.0000	−1.56227
$945$	−5.00000	−0.162650
$946$	4.00000	0.130051
$947$	18.0000	0.584921	0.292461	−	0.956278i	$-0.405526\pi$
$947$	18.0000	0.584921	0.292461	+	0.956278i	$0.405526\pi$
$948$	6.00000	0.194871
$949$	0	0
$950$	0	0
$951$	12.0000	0.389127
$952$	0	0
$953$	6.00000	0.194359	0.0971795	−	0.995267i	$-0.469018\pi$
$953$	6.00000	0.194359	0.0971795	+	0.995267i	$0.469018\pi$
$954$	−8.00000	−0.259010
$955$	−12.0000	−0.388311
$956$	24.0000	0.776215
$957$	−8.00000	−0.258603
$958$	84.0000	2.71392
$959$	−10.0000	−0.322917
$960$	8.00000	0.258199
$961$	18.0000	0.580645
$962$	0	0
$963$	4.00000	0.128898
$964$	20.0000	0.644157
$965$	11.0000	0.354103
$966$	60.0000	1.93047
$967$	−56.0000	−1.80084	−0.900419	−	0.435023i	$-0.856740\pi$
$967$	−56.0000	−1.80084	−0.900419	+	0.435023i	$0.856740\pi$
$968$	0	0
$969$	0	0
$970$	10.0000	0.321081
$971$	20.0000	0.641831	0.320915	−	0.947108i	$-0.396010\pi$
$971$	20.0000	0.641831	0.320915	+	0.947108i	$0.396010\pi$
$972$	2.00000	0.0641500
$973$	−15.0000	−0.480878
$974$	−56.0000	−1.79436
$975$	0	0
$976$	52.0000	1.66448
$977$	60.0000	1.91957	0.959785	−	0.280736i	$-0.0905785\pi$
$977$	60.0000	1.91957	0.959785	+	0.280736i	$0.0905785\pi$
$978$	−30.0000	−0.959294
$979$	28.0000	0.894884
$980$	−36.0000	−1.14998
$981$	−11.0000	−0.351203
$982$	−48.0000	−1.53174
$983$	38.0000	1.21201	0.606006	−	0.795460i	$-0.292771\pi$
$983$	38.0000	1.21201	0.606006	+	0.795460i	$0.292771\pi$
$984$	0	0
$985$	12.0000	0.382352
$986$	−16.0000	−0.509544
$987$	−40.0000	−1.27321
$988$	0	0
$989$	6.00000	0.190789
$990$	−4.00000	−0.127128
$991$	0	0		−	1.00000i	$-0.5\pi$
$991$	0	0			1.00000i	$0.5\pi$
$992$	56.0000	1.77800
$993$	9.00000	0.285606
$994$	120.000	3.80617
$995$	−17.0000	−0.538936
$996$	16.0000	0.506979
$997$	29.0000	0.918439	0.459220	−	0.888323i	$-0.348129\pi$
$997$	29.0000	0.918439	0.459220	+	0.888323i	$0.348129\pi$
$998$	8.00000	0.253236
$999$	−2.00000	−0.0632772

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2535.2.a.m.1.1		1
3.2	odd	2		7605.2.a.a.1.1		1
13.3	even	3		195.2.i.a.61.1	yes	2
13.9	even	3		195.2.i.a.16.1	✓	2
13.12	even	2		2535.2.a.c.1.1		1
39.29	odd	6		585.2.j.b.451.1		2
39.35	odd	6		585.2.j.b.406.1		2
39.38	odd	2		7605.2.a.s.1.1		1
65.3	odd	12		975.2.bb.f.724.2		4
65.9	even	6		975.2.i.i.601.1		2
65.22	odd	12		975.2.bb.f.874.2		4
65.29	even	6		975.2.i.i.451.1		2
65.42	odd	12		975.2.bb.f.724.1		4
65.48	odd	12		975.2.bb.f.874.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
195.2.i.a.16.1	✓	2	13.9	even	3
195.2.i.a.61.1	yes	2	13.3	even	3
585.2.j.b.406.1		2	39.35	odd	6
585.2.j.b.451.1		2	39.29	odd	6
975.2.i.i.451.1		2	65.29	even	6
975.2.i.i.601.1		2	65.9	even	6
975.2.bb.f.724.1		4	65.42	odd	12
975.2.bb.f.724.2		4	65.3	odd	12
975.2.bb.f.874.1		4	65.48	odd	12
975.2.bb.f.874.2		4	65.22	odd	12
2535.2.a.c.1.1		1	13.12	even	2
2535.2.a.m.1.1		1	1.1	even	1	trivial
7605.2.a.a.1.1		1	3.2	odd	2
7605.2.a.s.1.1		1	39.38	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 2535 = 3 \cdot 5 \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2535.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more