LMFDB - Embedded newform 1530.2.a.k.1.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1530,2,Mod(1,1530)] 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("1530.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1530, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 1530 = 2 \cdot 3^{2} \cdot 5 \cdot 17 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1530.a (trivial)

Newform invariants

comment:select newform

sage:traces = [1,1,0,1,-1,0,-2,1,0,-1,2,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(13)] == traces)

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

Self dual:	yes
Analytic conductor:	$12.2171115093$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	1530.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$

$=$

$q+1.00000 q^{2} +1.00000 q^{4} -1.00000 q^{5} -2.00000 q^{7} +1.00000 q^{8} -1.00000 q^{10} +2.00000 q^{11} -2.00000 q^{14} +1.00000 q^{16} -1.00000 q^{17} +4.00000 q^{19} -1.00000 q^{20} +2.00000 q^{22} +8.00000 q^{23} +1.00000 q^{25} -2.00000 q^{28} +6.00000 q^{29} +4.00000 q^{31} +1.00000 q^{32} -1.00000 q^{34} +2.00000 q^{35} +4.00000 q^{38} -1.00000 q^{40} +4.00000 q^{41} -4.00000 q^{43} +2.00000 q^{44} +8.00000 q^{46} +8.00000 q^{47} -3.00000 q^{49} +1.00000 q^{50} -2.00000 q^{53} -2.00000 q^{55} -2.00000 q^{56} +6.00000 q^{58} +6.00000 q^{59} +2.00000 q^{61} +4.00000 q^{62} +1.00000 q^{64} -16.0000 q^{67} -1.00000 q^{68} +2.00000 q^{70} -8.00000 q^{71} +10.0000 q^{73} +4.00000 q^{76} -4.00000 q^{77} +4.00000 q^{79} -1.00000 q^{80} +4.00000 q^{82} +4.00000 q^{83} +1.00000 q^{85} -4.00000 q^{86} +2.00000 q^{88} +12.0000 q^{89} +8.00000 q^{92} +8.00000 q^{94} -4.00000 q^{95} -2.00000 q^{97} -3.00000 q^{98} +O(q^{100})$

Coefficient data

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

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

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

$2$	1.00000	0.707107
$2$	1.00000	0.707107
$3$	0	0
$3$	0	0
$4$	1.00000	0.500000
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	−2.00000	−0.755929	−0.377964	−	0.925820i	$-0.623376\pi$
$7$	−2.00000	−0.755929	−0.377964	+	0.925820i	$0.623376\pi$
$8$	1.00000	0.353553
$9$	0	0
$10$	−1.00000	−0.316228
$11$	2.00000	0.603023	0.301511	−	0.953463i	$-0.402509\pi$
$11$	2.00000	0.603023	0.301511	+	0.953463i	$0.402509\pi$
$12$	0	0
$13$	0	0		−	1.00000i	$-0.5\pi$
$13$	0	0			1.00000i	$0.5\pi$
$14$	−2.00000	−0.534522
$15$	0	0
$16$	1.00000	0.250000
$17$	−1.00000	−0.242536
$17$	−1.00000	−0.242536
$18$	0	0
$19$	4.00000	0.917663	0.458831	−	0.888523i	$-0.348268\pi$
$19$	4.00000	0.917663	0.458831	+	0.888523i	$0.348268\pi$
$20$	−1.00000	−0.223607
$21$	0	0
$22$	2.00000	0.426401
$23$	8.00000	1.66812	0.834058	−	0.551677i	$-0.186012\pi$
$23$	8.00000	1.66812	0.834058	+	0.551677i	$0.186012\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	0	0
$28$	−2.00000	−0.377964
$29$	6.00000	1.11417	0.557086	−	0.830455i	$-0.311919\pi$
$29$	6.00000	1.11417	0.557086	+	0.830455i	$0.311919\pi$
$30$	0	0
$31$	4.00000	0.718421	0.359211	−	0.933257i	$-0.383046\pi$
$31$	4.00000	0.718421	0.359211	+	0.933257i	$0.383046\pi$
$32$	1.00000	0.176777
$33$	0	0
$34$	−1.00000	−0.171499
$35$	2.00000	0.338062
$36$	0	0
$37$	0	0		−	1.00000i	$-0.5\pi$
$37$	0	0			1.00000i	$0.5\pi$
$38$	4.00000	0.648886
$39$	0	0
$40$	−1.00000	−0.158114
$41$	4.00000	0.624695	0.312348	−	0.949968i	$-0.398885\pi$
$41$	4.00000	0.624695	0.312348	+	0.949968i	$0.398885\pi$
$42$	0	0
$43$	−4.00000	−0.609994	−0.304997	−	0.952353i	$-0.598656\pi$
$43$	−4.00000	−0.609994	−0.304997	+	0.952353i	$0.598656\pi$
$44$	2.00000	0.301511
$45$	0	0
$46$	8.00000	1.17954
$47$	8.00000	1.16692	0.583460	−	0.812142i	$-0.301699\pi$
$47$	8.00000	1.16692	0.583460	+	0.812142i	$0.301699\pi$
$48$	0	0
$49$	−3.00000	−0.428571
$50$	1.00000	0.141421
$51$	0	0
$52$	0	0
$53$	−2.00000	−0.274721	−0.137361	−	0.990521i	$-0.543862\pi$
$53$	−2.00000	−0.274721	−0.137361	+	0.990521i	$0.543862\pi$
$54$	0	0
$55$	−2.00000	−0.269680
$56$	−2.00000	−0.267261
$57$	0	0
$58$	6.00000	0.787839
$59$	6.00000	0.781133	0.390567	−	0.920575i	$-0.372279\pi$
$59$	6.00000	0.781133	0.390567	+	0.920575i	$0.372279\pi$
$60$	0	0
$61$	2.00000	0.256074	0.128037	−	0.991769i	$-0.459132\pi$
$61$	2.00000	0.256074	0.128037	+	0.991769i	$0.459132\pi$
$62$	4.00000	0.508001
$63$	0	0
$64$	1.00000	0.125000
$65$	0	0
$66$	0	0
$67$	−16.0000	−1.95471	−0.977356	−	0.211604i	$-0.932131\pi$
$67$	−16.0000	−1.95471	−0.977356	+	0.211604i	$0.932131\pi$
$68$	−1.00000	−0.121268
$69$	0	0
$70$	2.00000	0.239046
$71$	−8.00000	−0.949425	−0.474713	−	0.880141i	$-0.657448\pi$
$71$	−8.00000	−0.949425	−0.474713	+	0.880141i	$0.657448\pi$
$72$	0	0
$73$	10.0000	1.17041	0.585206	−	0.810885i	$-0.301014\pi$
$73$	10.0000	1.17041	0.585206	+	0.810885i	$0.301014\pi$
$74$	0	0
$75$	0	0
$76$	4.00000	0.458831
$77$	−4.00000	−0.455842
$78$	0	0
$79$	4.00000	0.450035	0.225018	−	0.974355i	$-0.427756\pi$
$79$	4.00000	0.450035	0.225018	+	0.974355i	$0.427756\pi$
$80$	−1.00000	−0.111803
$81$	0	0
$82$	4.00000	0.441726
$83$	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
$85$	1.00000	0.108465
$86$	−4.00000	−0.431331
$87$	0	0
$88$	2.00000	0.213201
$89$	12.0000	1.27200	0.635999	−	0.771690i	$-0.280588\pi$
$89$	12.0000	1.27200	0.635999	+	0.771690i	$0.280588\pi$
$90$	0	0
$91$	0	0
$92$	8.00000	0.834058
$93$	0	0
$94$	8.00000	0.825137
$95$	−4.00000	−0.410391
$96$	0	0
$97$	−2.00000	−0.203069	−0.101535	−	0.994832i	$-0.532375\pi$
$97$	−2.00000	−0.203069	−0.101535	+	0.994832i	$0.532375\pi$
$98$	−3.00000	−0.303046
$99$	0	0
$100$	1.00000	0.100000
$101$	−6.00000	−0.597022	−0.298511	−	0.954406i	$-0.596490\pi$
$101$	−6.00000	−0.597022	−0.298511	+	0.954406i	$0.596490\pi$
$102$	0	0
$103$	−14.0000	−1.37946	−0.689730	−	0.724066i	$-0.742271\pi$
$103$	−14.0000	−1.37946	−0.689730	+	0.724066i	$0.742271\pi$
$104$	0	0
$105$	0	0
$106$	−2.00000	−0.194257
$107$	4.00000	0.386695	0.193347	−	0.981130i	$-0.438066\pi$
$107$	4.00000	0.386695	0.193347	+	0.981130i	$0.438066\pi$
$108$	0	0
$109$	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
$111$	0	0
$112$	−2.00000	−0.188982
$113$	−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
$115$	−8.00000	−0.746004
$116$	6.00000	0.557086
$117$	0	0
$118$	6.00000	0.552345
$119$	2.00000	0.183340
$120$	0	0
$121$	−7.00000	−0.636364
$122$	2.00000	0.181071
$123$	0	0
$124$	4.00000	0.359211
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	−14.0000	−1.24230	−0.621150	−	0.783692i	$-0.713334\pi$
$127$	−14.0000	−1.24230	−0.621150	+	0.783692i	$0.713334\pi$
$128$	1.00000	0.0883883
$129$	0	0
$130$	0	0
$131$	6.00000	0.524222	0.262111	−	0.965038i	$-0.415581\pi$
$131$	6.00000	0.524222	0.262111	+	0.965038i	$0.415581\pi$
$132$	0	0
$133$	−8.00000	−0.693688
$134$	−16.0000	−1.38219
$135$	0	0
$136$	−1.00000	−0.0857493
$137$	18.0000	1.53784	0.768922	−	0.639343i	$-0.220793\pi$
$137$	18.0000	1.53784	0.768922	+	0.639343i	$0.220793\pi$
$138$	0	0
$139$	−4.00000	−0.339276	−0.169638	−	0.985506i	$-0.554260\pi$
$139$	−4.00000	−0.339276	−0.169638	+	0.985506i	$0.554260\pi$
$140$	2.00000	0.169031
$141$	0	0
$142$	−8.00000	−0.671345
$143$	0	0
$144$	0	0
$145$	−6.00000	−0.498273
$146$	10.0000	0.827606
$147$	0	0
$148$	0	0
$149$	−2.00000	−0.163846	−0.0819232	−	0.996639i	$-0.526106\pi$
$149$	−2.00000	−0.163846	−0.0819232	+	0.996639i	$0.526106\pi$
$150$	0	0
$151$	−16.0000	−1.30206	−0.651031	−	0.759051i	$-0.725663\pi$
$151$	−16.0000	−1.30206	−0.651031	+	0.759051i	$0.725663\pi$
$152$	4.00000	0.324443
$153$	0	0
$154$	−4.00000	−0.322329
$155$	−4.00000	−0.321288
$156$	0	0
$157$	0	0		−	1.00000i	$-0.5\pi$
$157$	0	0			1.00000i	$0.5\pi$
$158$	4.00000	0.318223
$159$	0	0
$160$	−1.00000	−0.0790569
$161$	−16.0000	−1.26098
$162$	0	0
$163$	−4.00000	−0.313304	−0.156652	−	0.987654i	$-0.550070\pi$
$163$	−4.00000	−0.313304	−0.156652	+	0.987654i	$0.550070\pi$
$164$	4.00000	0.312348
$165$	0	0
$166$	4.00000	0.310460
$167$	−8.00000	−0.619059	−0.309529	−	0.950890i	$-0.600171\pi$
$167$	−8.00000	−0.619059	−0.309529	+	0.950890i	$0.600171\pi$
$168$	0	0
$169$	−13.0000	−1.00000
$170$	1.00000	0.0766965
$171$	0	0
$172$	−4.00000	−0.304997
$173$	−2.00000	−0.152057	−0.0760286	−	0.997106i	$-0.524224\pi$
$173$	−2.00000	−0.152057	−0.0760286	+	0.997106i	$0.524224\pi$
$174$	0	0
$175$	−2.00000	−0.151186
$176$	2.00000	0.150756
$177$	0	0
$178$	12.0000	0.899438
$179$	−2.00000	−0.149487	−0.0747435	−	0.997203i	$-0.523814\pi$
$179$	−2.00000	−0.149487	−0.0747435	+	0.997203i	$0.523814\pi$
$180$	0	0
$181$	26.0000	1.93256	0.966282	−	0.257485i	$-0.0828937\pi$
$181$	26.0000	1.93256	0.966282	+	0.257485i	$0.0828937\pi$
$182$	0	0
$183$	0	0
$184$	8.00000	0.589768
$185$	0	0
$186$	0	0
$187$	−2.00000	−0.146254
$188$	8.00000	0.583460
$189$	0	0
$190$	−4.00000	−0.290191
$191$	−4.00000	−0.289430	−0.144715	−	0.989473i	$-0.546227\pi$
$191$	−4.00000	−0.289430	−0.144715	+	0.989473i	$0.546227\pi$
$192$	0	0
$193$	−14.0000	−1.00774	−0.503871	−	0.863779i	$-0.668091\pi$
$193$	−14.0000	−1.00774	−0.503871	+	0.863779i	$0.668091\pi$
$194$	−2.00000	−0.143592
$195$	0	0
$196$	−3.00000	−0.214286
$197$	18.0000	1.28245	0.641223	−	0.767354i	$-0.278427\pi$
$197$	18.0000	1.28245	0.641223	+	0.767354i	$0.278427\pi$
$198$	0	0
$199$	16.0000	1.13421	0.567105	−	0.823646i	$-0.308063\pi$
$199$	16.0000	1.13421	0.567105	+	0.823646i	$0.308063\pi$
$200$	1.00000	0.0707107
$201$	0	0
$202$	−6.00000	−0.422159
$203$	−12.0000	−0.842235
$204$	0	0
$205$	−4.00000	−0.279372
$206$	−14.0000	−0.975426
$207$	0	0
$208$	0	0
$209$	8.00000	0.553372
$210$	0	0
$211$	−20.0000	−1.37686	−0.688428	−	0.725304i	$-0.741699\pi$
$211$	−20.0000	−1.37686	−0.688428	+	0.725304i	$0.741699\pi$
$212$	−2.00000	−0.137361
$213$	0	0
$214$	4.00000	0.273434
$215$	4.00000	0.272798
$216$	0	0
$217$	−8.00000	−0.543075
$218$	10.0000	0.677285
$219$	0	0
$220$	−2.00000	−0.134840
$221$	0	0
$222$	0	0
$223$	10.0000	0.669650	0.334825	−	0.942280i	$-0.391323\pi$
$223$	10.0000	0.669650	0.334825	+	0.942280i	$0.391323\pi$
$224$	−2.00000	−0.133631
$225$	0	0
$226$	−6.00000	−0.399114
$227$	0	0		−	1.00000i	$-0.5\pi$
$227$	0	0			1.00000i	$0.5\pi$
$228$	0	0
$229$	22.0000	1.45380	0.726900	−	0.686743i	$-0.240960\pi$
$229$	22.0000	1.45380	0.726900	+	0.686743i	$0.240960\pi$
$230$	−8.00000	−0.527504
$231$	0	0
$232$	6.00000	0.393919
$233$	−10.0000	−0.655122	−0.327561	−	0.944830i	$-0.606227\pi$
$233$	−10.0000	−0.655122	−0.327561	+	0.944830i	$0.606227\pi$
$234$	0	0
$235$	−8.00000	−0.521862
$236$	6.00000	0.390567
$237$	0	0
$238$	2.00000	0.129641
$239$	12.0000	0.776215	0.388108	−	0.921614i	$-0.373129\pi$
$239$	12.0000	0.776215	0.388108	+	0.921614i	$0.373129\pi$
$240$	0	0
$241$	−2.00000	−0.128831	−0.0644157	−	0.997923i	$-0.520518\pi$
$241$	−2.00000	−0.128831	−0.0644157	+	0.997923i	$0.520518\pi$
$242$	−7.00000	−0.449977
$243$	0	0
$244$	2.00000	0.128037
$245$	3.00000	0.191663
$246$	0	0
$247$	0	0
$248$	4.00000	0.254000
$249$	0	0
$250$	−1.00000	−0.0632456
$251$	−22.0000	−1.38863	−0.694314	−	0.719672i	$-0.744292\pi$
$251$	−22.0000	−1.38863	−0.694314	+	0.719672i	$0.744292\pi$
$252$	0	0
$253$	16.0000	1.00591
$254$	−14.0000	−0.878438
$255$	0	0
$256$	1.00000	0.0625000
$257$	−6.00000	−0.374270	−0.187135	−	0.982334i	$-0.559920\pi$
$257$	−6.00000	−0.374270	−0.187135	+	0.982334i	$0.559920\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	0	0
$262$	6.00000	0.370681
$263$	8.00000	0.493301	0.246651	−	0.969104i	$-0.420670\pi$
$263$	8.00000	0.493301	0.246651	+	0.969104i	$0.420670\pi$
$264$	0	0
$265$	2.00000	0.122859
$266$	−8.00000	−0.490511
$267$	0	0
$268$	−16.0000	−0.977356
$269$	14.0000	0.853595	0.426798	−	0.904347i	$-0.359642\pi$
$269$	14.0000	0.853595	0.426798	+	0.904347i	$0.359642\pi$
$270$	0	0
$271$	−8.00000	−0.485965	−0.242983	−	0.970031i	$-0.578126\pi$
$271$	−8.00000	−0.485965	−0.242983	+	0.970031i	$0.578126\pi$
$272$	−1.00000	−0.0606339
$273$	0	0
$274$	18.0000	1.08742
$275$	2.00000	0.120605
$276$	0	0
$277$	−24.0000	−1.44202	−0.721010	−	0.692925i	$-0.756322\pi$
$277$	−24.0000	−1.44202	−0.721010	+	0.692925i	$0.756322\pi$
$278$	−4.00000	−0.239904
$279$	0	0
$280$	2.00000	0.119523
$281$	−20.0000	−1.19310	−0.596550	−	0.802576i	$-0.703462\pi$
$281$	−20.0000	−1.19310	−0.596550	+	0.802576i	$0.703462\pi$
$282$	0	0
$283$	12.0000	0.713326	0.356663	−	0.934233i	$-0.383914\pi$
$283$	12.0000	0.713326	0.356663	+	0.934233i	$0.383914\pi$
$284$	−8.00000	−0.474713
$285$	0	0
$286$	0	0
$287$	−8.00000	−0.472225
$288$	0	0
$289$	1.00000	0.0588235
$290$	−6.00000	−0.352332
$291$	0	0
$292$	10.0000	0.585206
$293$	−6.00000	−0.350524	−0.175262	−	0.984522i	$-0.556077\pi$
$293$	−6.00000	−0.350524	−0.175262	+	0.984522i	$0.556077\pi$
$294$	0	0
$295$	−6.00000	−0.349334
$296$	0	0
$297$	0	0
$298$	−2.00000	−0.115857
$299$	0	0
$300$	0	0
$301$	8.00000	0.461112
$302$	−16.0000	−0.920697
$303$	0	0
$304$	4.00000	0.229416
$305$	−2.00000	−0.114520
$306$	0	0
$307$	−28.0000	−1.59804	−0.799022	−	0.601302i	$-0.794649\pi$
$307$	−28.0000	−1.59804	−0.799022	+	0.601302i	$0.794649\pi$
$308$	−4.00000	−0.227921
$309$	0	0
$310$	−4.00000	−0.227185
$311$	−16.0000	−0.907277	−0.453638	−	0.891186i	$-0.649874\pi$
$311$	−16.0000	−0.907277	−0.453638	+	0.891186i	$0.649874\pi$
$312$	0	0
$313$	34.0000	1.92179	0.960897	−	0.276907i	$-0.0893093\pi$
$313$	34.0000	1.92179	0.960897	+	0.276907i	$0.0893093\pi$
$314$	0	0
$315$	0	0
$316$	4.00000	0.225018
$317$	−30.0000	−1.68497	−0.842484	−	0.538721i	$-0.818908\pi$
$317$	−30.0000	−1.68497	−0.842484	+	0.538721i	$0.818908\pi$
$318$	0	0
$319$	12.0000	0.671871
$320$	−1.00000	−0.0559017
$321$	0	0
$322$	−16.0000	−0.891645
$323$	−4.00000	−0.222566
$324$	0	0
$325$	0	0
$326$	−4.00000	−0.221540
$327$	0	0
$328$	4.00000	0.220863
$329$	−16.0000	−0.882109
$330$	0	0
$331$	12.0000	0.659580	0.329790	−	0.944054i	$-0.393022\pi$
$331$	12.0000	0.659580	0.329790	+	0.944054i	$0.393022\pi$
$332$	4.00000	0.219529
$333$	0	0
$334$	−8.00000	−0.437741
$335$	16.0000	0.874173
$336$	0	0
$337$	2.00000	0.108947	0.0544735	−	0.998515i	$-0.482652\pi$
$337$	2.00000	0.108947	0.0544735	+	0.998515i	$0.482652\pi$
$338$	−13.0000	−0.707107
$339$	0	0
$340$	1.00000	0.0542326
$341$	8.00000	0.433224
$342$	0	0
$343$	20.0000	1.07990
$344$	−4.00000	−0.215666
$345$	0	0
$346$	−2.00000	−0.107521
$347$	−16.0000	−0.858925	−0.429463	−	0.903085i	$-0.641297\pi$
$347$	−16.0000	−0.858925	−0.429463	+	0.903085i	$0.641297\pi$
$348$	0	0
$349$	10.0000	0.535288	0.267644	−	0.963518i	$-0.413755\pi$
$349$	10.0000	0.535288	0.267644	+	0.963518i	$0.413755\pi$
$350$	−2.00000	−0.106904
$351$	0	0
$352$	2.00000	0.106600
$353$	−18.0000	−0.958043	−0.479022	−	0.877803i	$-0.659008\pi$
$353$	−18.0000	−0.958043	−0.479022	+	0.877803i	$0.659008\pi$
$354$	0	0
$355$	8.00000	0.424596
$356$	12.0000	0.635999
$357$	0	0
$358$	−2.00000	−0.105703
$359$	16.0000	0.844448	0.422224	−	0.906492i	$-0.361250\pi$
$359$	16.0000	0.844448	0.422224	+	0.906492i	$0.361250\pi$
$360$	0	0
$361$	−3.00000	−0.157895
$362$	26.0000	1.36653
$363$	0	0
$364$	0	0
$365$	−10.0000	−0.523424
$366$	0	0
$367$	−2.00000	−0.104399	−0.0521996	−	0.998637i	$-0.516623\pi$
$367$	−2.00000	−0.104399	−0.0521996	+	0.998637i	$0.516623\pi$
$368$	8.00000	0.417029
$369$	0	0
$370$	0	0
$371$	4.00000	0.207670
$372$	0	0
$373$	0	0		−	1.00000i	$-0.5\pi$
$373$	0	0			1.00000i	$0.5\pi$
$374$	−2.00000	−0.103418
$375$	0	0
$376$	8.00000	0.412568
$377$	0	0
$378$	0	0
$379$	−4.00000	−0.205466	−0.102733	−	0.994709i	$-0.532759\pi$
$379$	−4.00000	−0.205466	−0.102733	+	0.994709i	$0.532759\pi$
$380$	−4.00000	−0.205196
$381$	0	0
$382$	−4.00000	−0.204658
$383$	−32.0000	−1.63512	−0.817562	−	0.575841i	$-0.804675\pi$
$383$	−32.0000	−1.63512	−0.817562	+	0.575841i	$0.804675\pi$
$384$	0	0
$385$	4.00000	0.203859
$386$	−14.0000	−0.712581
$387$	0	0
$388$	−2.00000	−0.101535
$389$	6.00000	0.304212	0.152106	−	0.988364i	$-0.451394\pi$
$389$	6.00000	0.304212	0.152106	+	0.988364i	$0.451394\pi$
$390$	0	0
$391$	−8.00000	−0.404577
$392$	−3.00000	−0.151523
$393$	0	0
$394$	18.0000	0.906827
$395$	−4.00000	−0.201262
$396$	0	0
$397$	−8.00000	−0.401508	−0.200754	−	0.979642i	$-0.564339\pi$
$397$	−8.00000	−0.401508	−0.200754	+	0.979642i	$0.564339\pi$
$398$	16.0000	0.802008
$399$	0	0
$400$	1.00000	0.0500000
$401$	−32.0000	−1.59800	−0.799002	−	0.601329i	$-0.794638\pi$
$401$	−32.0000	−1.59800	−0.799002	+	0.601329i	$0.794638\pi$
$402$	0	0
$403$	0	0
$404$	−6.00000	−0.298511
$405$	0	0
$406$	−12.0000	−0.595550
$407$	0	0
$408$	0	0
$409$	10.0000	0.494468	0.247234	−	0.968956i	$-0.420478\pi$
$409$	10.0000	0.494468	0.247234	+	0.968956i	$0.420478\pi$
$410$	−4.00000	−0.197546
$411$	0	0
$412$	−14.0000	−0.689730
$413$	−12.0000	−0.590481
$414$	0	0
$415$	−4.00000	−0.196352
$416$	0	0
$417$	0	0
$418$	8.00000	0.391293
$419$	−26.0000	−1.27018	−0.635092	−	0.772437i	$-0.719038\pi$
$419$	−26.0000	−1.27018	−0.635092	+	0.772437i	$0.719038\pi$
$420$	0	0
$421$	−10.0000	−0.487370	−0.243685	−	0.969854i	$-0.578356\pi$
$421$	−10.0000	−0.487370	−0.243685	+	0.969854i	$0.578356\pi$
$422$	−20.0000	−0.973585
$423$	0	0
$424$	−2.00000	−0.0971286
$425$	−1.00000	−0.0485071
$426$	0	0
$427$	−4.00000	−0.193574
$428$	4.00000	0.193347
$429$	0	0
$430$	4.00000	0.192897
$431$	20.0000	0.963366	0.481683	−	0.876346i	$-0.340026\pi$
$431$	20.0000	0.963366	0.481683	+	0.876346i	$0.340026\pi$
$432$	0	0
$433$	30.0000	1.44171	0.720854	−	0.693087i	$-0.243750\pi$
$433$	30.0000	1.44171	0.720854	+	0.693087i	$0.243750\pi$
$434$	−8.00000	−0.384012
$435$	0	0
$436$	10.0000	0.478913
$437$	32.0000	1.53077
$438$	0	0
$439$	8.00000	0.381819	0.190910	−	0.981608i	$-0.438856\pi$
$439$	8.00000	0.381819	0.190910	+	0.981608i	$0.438856\pi$
$440$	−2.00000	−0.0953463
$441$	0	0
$442$	0	0
$443$	0	0		−	1.00000i	$-0.5\pi$
$443$	0	0			1.00000i	$0.5\pi$
$444$	0	0
$445$	−12.0000	−0.568855
$446$	10.0000	0.473514
$447$	0	0
$448$	−2.00000	−0.0944911
$449$	24.0000	1.13263	0.566315	−	0.824189i	$-0.308369\pi$
$449$	24.0000	1.13263	0.566315	+	0.824189i	$0.308369\pi$
$450$	0	0
$451$	8.00000	0.376705
$452$	−6.00000	−0.282216
$453$	0	0
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−30.0000	−1.40334	−0.701670	−	0.712502i	$-0.747562\pi$
$457$	−30.0000	−1.40334	−0.701670	+	0.712502i	$0.747562\pi$
$458$	22.0000	1.02799
$459$	0	0
$460$	−8.00000	−0.373002
$461$	30.0000	1.39724	0.698620	−	0.715493i	$-0.253798\pi$
$461$	30.0000	1.39724	0.698620	+	0.715493i	$0.253798\pi$
$462$	0	0
$463$	14.0000	0.650635	0.325318	−	0.945605i	$-0.394529\pi$
$463$	14.0000	0.650635	0.325318	+	0.945605i	$0.394529\pi$
$464$	6.00000	0.278543
$465$	0	0
$466$	−10.0000	−0.463241
$467$	−20.0000	−0.925490	−0.462745	−	0.886492i	$-0.653135\pi$
$467$	−20.0000	−0.925490	−0.462745	+	0.886492i	$0.653135\pi$
$468$	0	0
$469$	32.0000	1.47762
$470$	−8.00000	−0.369012
$471$	0	0
$472$	6.00000	0.276172
$473$	−8.00000	−0.367840
$474$	0	0
$475$	4.00000	0.183533
$476$	2.00000	0.0916698
$477$	0	0
$478$	12.0000	0.548867
$479$	−24.0000	−1.09659	−0.548294	−	0.836286i	$-0.684723\pi$
$479$	−24.0000	−1.09659	−0.548294	+	0.836286i	$0.684723\pi$
$480$	0	0
$481$	0	0
$482$	−2.00000	−0.0910975
$483$	0	0
$484$	−7.00000	−0.318182
$485$	2.00000	0.0908153
$486$	0	0
$487$	18.0000	0.815658	0.407829	−	0.913058i	$-0.366286\pi$
$487$	18.0000	0.815658	0.407829	+	0.913058i	$0.366286\pi$
$488$	2.00000	0.0905357
$489$	0	0
$490$	3.00000	0.135526
$491$	−6.00000	−0.270776	−0.135388	−	0.990793i	$-0.543228\pi$
$491$	−6.00000	−0.270776	−0.135388	+	0.990793i	$0.543228\pi$
$492$	0	0
$493$	−6.00000	−0.270226
$494$	0	0
$495$	0	0
$496$	4.00000	0.179605
$497$	16.0000	0.717698
$498$	0	0
$499$	−36.0000	−1.61158	−0.805791	−	0.592200i	$-0.798259\pi$
$499$	−36.0000	−1.61158	−0.805791	+	0.592200i	$0.798259\pi$
$500$	−1.00000	−0.0447214
$501$	0	0
$502$	−22.0000	−0.981908
$503$	16.0000	0.713405	0.356702	−	0.934218i	$-0.383901\pi$
$503$	16.0000	0.713405	0.356702	+	0.934218i	$0.383901\pi$
$504$	0	0
$505$	6.00000	0.266996
$506$	16.0000	0.711287
$507$	0	0
$508$	−14.0000	−0.621150
$509$	−30.0000	−1.32973	−0.664863	−	0.746965i	$-0.731510\pi$
$509$	−30.0000	−1.32973	−0.664863	+	0.746965i	$0.731510\pi$
$510$	0	0
$511$	−20.0000	−0.884748
$512$	1.00000	0.0441942
$513$	0	0
$514$	−6.00000	−0.264649
$515$	14.0000	0.616914
$516$	0	0
$517$	16.0000	0.703679
$518$	0	0
$519$	0	0
$520$	0	0
$521$	0	0		−	1.00000i	$-0.5\pi$
$521$	0	0			1.00000i	$0.5\pi$
$522$	0	0
$523$	−24.0000	−1.04945	−0.524723	−	0.851273i	$-0.675831\pi$
$523$	−24.0000	−1.04945	−0.524723	+	0.851273i	$0.675831\pi$
$524$	6.00000	0.262111
$525$	0	0
$526$	8.00000	0.348817
$527$	−4.00000	−0.174243
$528$	0	0
$529$	41.0000	1.78261
$530$	2.00000	0.0868744
$531$	0	0
$532$	−8.00000	−0.346844
$533$	0	0
$534$	0	0
$535$	−4.00000	−0.172935
$536$	−16.0000	−0.691095
$537$	0	0
$538$	14.0000	0.603583
$539$	−6.00000	−0.258438
$540$	0	0
$541$	42.0000	1.80572	0.902861	−	0.429934i	$-0.141463\pi$
$541$	42.0000	1.80572	0.902861	+	0.429934i	$0.141463\pi$
$542$	−8.00000	−0.343629
$543$	0	0
$544$	−1.00000	−0.0428746
$545$	−10.0000	−0.428353
$546$	0	0
$547$	−24.0000	−1.02617	−0.513083	−	0.858339i	$-0.671497\pi$
$547$	−24.0000	−1.02617	−0.513083	+	0.858339i	$0.671497\pi$
$548$	18.0000	0.768922
$549$	0	0
$550$	2.00000	0.0852803
$551$	24.0000	1.02243
$552$	0	0
$553$	−8.00000	−0.340195
$554$	−24.0000	−1.01966
$555$	0	0
$556$	−4.00000	−0.169638
$557$	30.0000	1.27114	0.635570	−	0.772043i	$-0.280765\pi$
$557$	30.0000	1.27114	0.635570	+	0.772043i	$0.280765\pi$
$558$	0	0
$559$	0	0
$560$	2.00000	0.0845154
$561$	0	0
$562$	−20.0000	−0.843649
$563$	32.0000	1.34864	0.674320	−	0.738440i	$-0.264437\pi$
$563$	32.0000	1.34864	0.674320	+	0.738440i	$0.264437\pi$
$564$	0	0
$565$	6.00000	0.252422
$566$	12.0000	0.504398
$567$	0	0
$568$	−8.00000	−0.335673
$569$	44.0000	1.84458	0.922288	−	0.386503i	$-0.126317\pi$
$569$	44.0000	1.84458	0.922288	+	0.386503i	$0.126317\pi$
$570$	0	0
$571$	20.0000	0.836974	0.418487	−	0.908223i	$-0.362561\pi$
$571$	20.0000	0.836974	0.418487	+	0.908223i	$0.362561\pi$
$572$	0	0
$573$	0	0
$574$	−8.00000	−0.333914
$575$	8.00000	0.333623
$576$	0	0
$577$	2.00000	0.0832611	0.0416305	−	0.999133i	$-0.486745\pi$
$577$	2.00000	0.0832611	0.0416305	+	0.999133i	$0.486745\pi$
$578$	1.00000	0.0415945
$579$	0	0
$580$	−6.00000	−0.249136
$581$	−8.00000	−0.331896
$582$	0	0
$583$	−4.00000	−0.165663
$584$	10.0000	0.413803
$585$	0	0
$586$	−6.00000	−0.247858
$587$	−28.0000	−1.15568	−0.577842	−	0.816149i	$-0.696105\pi$
$587$	−28.0000	−1.15568	−0.577842	+	0.816149i	$0.696105\pi$
$588$	0	0
$589$	16.0000	0.659269
$590$	−6.00000	−0.247016
$591$	0	0
$592$	0	0
$593$	2.00000	0.0821302	0.0410651	−	0.999156i	$-0.486925\pi$
$593$	2.00000	0.0821302	0.0410651	+	0.999156i	$0.486925\pi$
$594$	0	0
$595$	−2.00000	−0.0819920
$596$	−2.00000	−0.0819232
$597$	0	0
$598$	0	0
$599$	0	0		−	1.00000i	$-0.5\pi$
$599$	0	0			1.00000i	$0.5\pi$
$600$	0	0
$601$	−10.0000	−0.407909	−0.203954	−	0.978980i	$-0.565379\pi$
$601$	−10.0000	−0.407909	−0.203954	+	0.978980i	$0.565379\pi$
$602$	8.00000	0.326056
$603$	0	0
$604$	−16.0000	−0.651031
$605$	7.00000	0.284590
$606$	0	0
$607$	−14.0000	−0.568242	−0.284121	−	0.958788i	$-0.591702\pi$
$607$	−14.0000	−0.568242	−0.284121	+	0.958788i	$0.591702\pi$
$608$	4.00000	0.162221
$609$	0	0
$610$	−2.00000	−0.0809776
$611$	0	0
$612$	0	0
$613$	−12.0000	−0.484675	−0.242338	−	0.970192i	$-0.577914\pi$
$613$	−12.0000	−0.484675	−0.242338	+	0.970192i	$0.577914\pi$
$614$	−28.0000	−1.12999
$615$	0	0
$616$	−4.00000	−0.161165
$617$	−2.00000	−0.0805170	−0.0402585	−	0.999189i	$-0.512818\pi$
$617$	−2.00000	−0.0805170	−0.0402585	+	0.999189i	$0.512818\pi$
$618$	0	0
$619$	−12.0000	−0.482321	−0.241160	−	0.970485i	$-0.577528\pi$
$619$	−12.0000	−0.482321	−0.241160	+	0.970485i	$0.577528\pi$
$620$	−4.00000	−0.160644
$621$	0	0
$622$	−16.0000	−0.641542
$623$	−24.0000	−0.961540
$624$	0	0
$625$	1.00000	0.0400000
$626$	34.0000	1.35891
$627$	0	0
$628$	0	0
$629$	0	0
$630$	0	0
$631$	36.0000	1.43314	0.716569	−	0.697517i	$-0.245712\pi$
$631$	36.0000	1.43314	0.716569	+	0.697517i	$0.245712\pi$
$632$	4.00000	0.159111
$633$	0	0
$634$	−30.0000	−1.19145
$635$	14.0000	0.555573
$636$	0	0
$637$	0	0
$638$	12.0000	0.475085
$639$	0	0
$640$	−1.00000	−0.0395285
$641$	20.0000	0.789953	0.394976	−	0.918691i	$-0.370753\pi$
$641$	20.0000	0.789953	0.394976	+	0.918691i	$0.370753\pi$
$642$	0	0
$643$	4.00000	0.157745	0.0788723	−	0.996885i	$-0.474868\pi$
$643$	4.00000	0.157745	0.0788723	+	0.996885i	$0.474868\pi$
$644$	−16.0000	−0.630488
$645$	0	0
$646$	−4.00000	−0.157378
$647$	0	0		−	1.00000i	$-0.5\pi$
$647$	0	0			1.00000i	$0.5\pi$
$648$	0	0
$649$	12.0000	0.471041
$650$	0	0
$651$	0	0
$652$	−4.00000	−0.156652
$653$	−14.0000	−0.547862	−0.273931	−	0.961749i	$-0.588324\pi$
$653$	−14.0000	−0.547862	−0.273931	+	0.961749i	$0.588324\pi$
$654$	0	0
$655$	−6.00000	−0.234439
$656$	4.00000	0.156174
$657$	0	0
$658$	−16.0000	−0.623745
$659$	18.0000	0.701180	0.350590	−	0.936529i	$-0.385981\pi$
$659$	18.0000	0.701180	0.350590	+	0.936529i	$0.385981\pi$
$660$	0	0
$661$	18.0000	0.700119	0.350059	−	0.936727i	$-0.386161\pi$
$661$	18.0000	0.700119	0.350059	+	0.936727i	$0.386161\pi$
$662$	12.0000	0.466393
$663$	0	0
$664$	4.00000	0.155230
$665$	8.00000	0.310227
$666$	0	0
$667$	48.0000	1.85857
$668$	−8.00000	−0.309529
$669$	0	0
$670$	16.0000	0.618134
$671$	4.00000	0.154418
$672$	0	0
$673$	42.0000	1.61898	0.809491	−	0.587133i	$-0.199743\pi$
$673$	42.0000	1.61898	0.809491	+	0.587133i	$0.199743\pi$
$674$	2.00000	0.0770371
$675$	0	0
$676$	−13.0000	−0.500000
$677$	−6.00000	−0.230599	−0.115299	−	0.993331i	$-0.536783\pi$
$677$	−6.00000	−0.230599	−0.115299	+	0.993331i	$0.536783\pi$
$678$	0	0
$679$	4.00000	0.153506
$680$	1.00000	0.0383482
$681$	0	0
$682$	8.00000	0.306336
$683$	−44.0000	−1.68361	−0.841807	−	0.539779i	$-0.818508\pi$
$683$	−44.0000	−1.68361	−0.841807	+	0.539779i	$0.818508\pi$
$684$	0	0
$685$	−18.0000	−0.687745
$686$	20.0000	0.763604
$687$	0	0
$688$	−4.00000	−0.152499
$689$	0	0
$690$	0	0
$691$	−20.0000	−0.760836	−0.380418	−	0.924815i	$-0.624220\pi$
$691$	−20.0000	−0.760836	−0.380418	+	0.924815i	$0.624220\pi$
$692$	−2.00000	−0.0760286
$693$	0	0
$694$	−16.0000	−0.607352
$695$	4.00000	0.151729
$696$	0	0
$697$	−4.00000	−0.151511
$698$	10.0000	0.378506
$699$	0	0
$700$	−2.00000	−0.0755929
$701$	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
$704$	2.00000	0.0753778
$705$	0	0
$706$	−18.0000	−0.677439
$707$	12.0000	0.451306
$708$	0	0
$709$	14.0000	0.525781	0.262891	−	0.964826i	$-0.415324\pi$
$709$	14.0000	0.525781	0.262891	+	0.964826i	$0.415324\pi$
$710$	8.00000	0.300235
$711$	0	0
$712$	12.0000	0.449719
$713$	32.0000	1.19841
$714$	0	0
$715$	0	0
$716$	−2.00000	−0.0747435
$717$	0	0
$718$	16.0000	0.597115
$719$	52.0000	1.93927	0.969636	−	0.244551i	$-0.0786406\pi$
$719$	52.0000	1.93927	0.969636	+	0.244551i	$0.0786406\pi$
$720$	0	0
$721$	28.0000	1.04277
$722$	−3.00000	−0.111648
$723$	0	0
$724$	26.0000	0.966282
$725$	6.00000	0.222834
$726$	0	0
$727$	−26.0000	−0.964287	−0.482143	−	0.876092i	$-0.660142\pi$
$727$	−26.0000	−0.964287	−0.482143	+	0.876092i	$0.660142\pi$
$728$	0	0
$729$	0	0
$730$	−10.0000	−0.370117
$731$	4.00000	0.147945
$732$	0	0
$733$	−20.0000	−0.738717	−0.369358	−	0.929287i	$-0.620423\pi$
$733$	−20.0000	−0.738717	−0.369358	+	0.929287i	$0.620423\pi$
$734$	−2.00000	−0.0738213
$735$	0	0
$736$	8.00000	0.294884
$737$	−32.0000	−1.17874
$738$	0	0
$739$	4.00000	0.147142	0.0735712	−	0.997290i	$-0.476560\pi$
$739$	4.00000	0.147142	0.0735712	+	0.997290i	$0.476560\pi$
$740$	0	0
$741$	0	0
$742$	4.00000	0.146845
$743$	−8.00000	−0.293492	−0.146746	−	0.989174i	$-0.546880\pi$
$743$	−8.00000	−0.293492	−0.146746	+	0.989174i	$0.546880\pi$
$744$	0	0
$745$	2.00000	0.0732743
$746$	0	0
$747$	0	0
$748$	−2.00000	−0.0731272
$749$	−8.00000	−0.292314
$750$	0	0
$751$	12.0000	0.437886	0.218943	−	0.975738i	$-0.429739\pi$
$751$	12.0000	0.437886	0.218943	+	0.975738i	$0.429739\pi$
$752$	8.00000	0.291730
$753$	0	0
$754$	0	0
$755$	16.0000	0.582300
$756$	0	0
$757$	−4.00000	−0.145382	−0.0726912	−	0.997354i	$-0.523159\pi$
$757$	−4.00000	−0.145382	−0.0726912	+	0.997354i	$0.523159\pi$
$758$	−4.00000	−0.145287
$759$	0	0
$760$	−4.00000	−0.145095
$761$	−20.0000	−0.724999	−0.362500	−	0.931984i	$-0.618077\pi$
$761$	−20.0000	−0.724999	−0.362500	+	0.931984i	$0.618077\pi$
$762$	0	0
$763$	−20.0000	−0.724049
$764$	−4.00000	−0.144715
$765$	0	0
$766$	−32.0000	−1.15621
$767$	0	0
$768$	0	0
$769$	−2.00000	−0.0721218	−0.0360609	−	0.999350i	$-0.511481\pi$
$769$	−2.00000	−0.0721218	−0.0360609	+	0.999350i	$0.511481\pi$
$770$	4.00000	0.144150
$771$	0	0
$772$	−14.0000	−0.503871
$773$	−38.0000	−1.36677	−0.683383	−	0.730061i	$-0.739492\pi$
$773$	−38.0000	−1.36677	−0.683383	+	0.730061i	$0.739492\pi$
$774$	0	0
$775$	4.00000	0.143684
$776$	−2.00000	−0.0717958
$777$	0	0
$778$	6.00000	0.215110
$779$	16.0000	0.573259
$780$	0	0
$781$	−16.0000	−0.572525
$782$	−8.00000	−0.286079
$783$	0	0
$784$	−3.00000	−0.107143
$785$	0	0
$786$	0	0
$787$	−52.0000	−1.85360	−0.926800	−	0.375555i	$-0.877452\pi$
$787$	−52.0000	−1.85360	−0.926800	+	0.375555i	$0.877452\pi$
$788$	18.0000	0.641223
$789$	0	0
$790$	−4.00000	−0.142314
$791$	12.0000	0.426671
$792$	0	0
$793$	0	0
$794$	−8.00000	−0.283909
$795$	0	0
$796$	16.0000	0.567105
$797$	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
$799$	−8.00000	−0.283020
$800$	1.00000	0.0353553
$801$	0	0
$802$	−32.0000	−1.12996
$803$	20.0000	0.705785
$804$	0	0
$805$	16.0000	0.563926
$806$	0	0
$807$	0	0
$808$	−6.00000	−0.211079
$809$	12.0000	0.421898	0.210949	−	0.977497i	$-0.432345\pi$
$809$	12.0000	0.421898	0.210949	+	0.977497i	$0.432345\pi$
$810$	0	0
$811$	−28.0000	−0.983213	−0.491606	−	0.870817i	$-0.663590\pi$
$811$	−28.0000	−0.983213	−0.491606	+	0.870817i	$0.663590\pi$
$812$	−12.0000	−0.421117
$813$	0	0
$814$	0	0
$815$	4.00000	0.140114
$816$	0	0
$817$	−16.0000	−0.559769
$818$	10.0000	0.349642
$819$	0	0
$820$	−4.00000	−0.139686
$821$	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
$823$	−14.0000	−0.488009	−0.244005	−	0.969774i	$-0.578461\pi$
$823$	−14.0000	−0.488009	−0.244005	+	0.969774i	$0.578461\pi$
$824$	−14.0000	−0.487713
$825$	0	0
$826$	−12.0000	−0.417533
$827$	24.0000	0.834562	0.417281	−	0.908778i	$-0.362983\pi$
$827$	24.0000	0.834562	0.417281	+	0.908778i	$0.362983\pi$
$828$	0	0
$829$	−14.0000	−0.486240	−0.243120	−	0.969996i	$-0.578171\pi$
$829$	−14.0000	−0.486240	−0.243120	+	0.969996i	$0.578171\pi$
$830$	−4.00000	−0.138842
$831$	0	0
$832$	0	0
$833$	3.00000	0.103944
$834$	0	0
$835$	8.00000	0.276851
$836$	8.00000	0.276686
$837$	0	0
$838$	−26.0000	−0.898155
$839$	36.0000	1.24286	0.621429	−	0.783470i	$-0.286552\pi$
$839$	36.0000	1.24286	0.621429	+	0.783470i	$0.286552\pi$
$840$	0	0
$841$	7.00000	0.241379
$842$	−10.0000	−0.344623
$843$	0	0
$844$	−20.0000	−0.688428
$845$	13.0000	0.447214
$846$	0	0
$847$	14.0000	0.481046
$848$	−2.00000	−0.0686803
$849$	0	0
$850$	−1.00000	−0.0342997
$851$	0	0
$852$	0	0
$853$	24.0000	0.821744	0.410872	−	0.911693i	$-0.365224\pi$
$853$	24.0000	0.821744	0.410872	+	0.911693i	$0.365224\pi$
$854$	−4.00000	−0.136877
$855$	0	0
$856$	4.00000	0.136717
$857$	22.0000	0.751506	0.375753	−	0.926720i	$-0.377384\pi$
$857$	22.0000	0.751506	0.375753	+	0.926720i	$0.377384\pi$
$858$	0	0
$859$	−20.0000	−0.682391	−0.341196	−	0.939992i	$-0.610832\pi$
$859$	−20.0000	−0.682391	−0.341196	+	0.939992i	$0.610832\pi$
$860$	4.00000	0.136399
$861$	0	0
$862$	20.0000	0.681203
$863$	−32.0000	−1.08929	−0.544646	−	0.838666i	$-0.683336\pi$
$863$	−32.0000	−1.08929	−0.544646	+	0.838666i	$0.683336\pi$
$864$	0	0
$865$	2.00000	0.0680020
$866$	30.0000	1.01944
$867$	0	0
$868$	−8.00000	−0.271538
$869$	8.00000	0.271381
$870$	0	0
$871$	0	0
$872$	10.0000	0.338643
$873$	0	0
$874$	32.0000	1.08242
$875$	2.00000	0.0676123
$876$	0	0
$877$	36.0000	1.21563	0.607817	−	0.794077i	$-0.292045\pi$
$877$	36.0000	1.21563	0.607817	+	0.794077i	$0.292045\pi$
$878$	8.00000	0.269987
$879$	0	0
$880$	−2.00000	−0.0674200
$881$	16.0000	0.539054	0.269527	−	0.962993i	$-0.413133\pi$
$881$	16.0000	0.539054	0.269527	+	0.962993i	$0.413133\pi$
$882$	0	0
$883$	−56.0000	−1.88455	−0.942275	−	0.334840i	$-0.891318\pi$
$883$	−56.0000	−1.88455	−0.942275	+	0.334840i	$0.891318\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−24.0000	−0.805841	−0.402921	−	0.915235i	$-0.632005\pi$
$887$	−24.0000	−0.805841	−0.402921	+	0.915235i	$0.632005\pi$
$888$	0	0
$889$	28.0000	0.939090
$890$	−12.0000	−0.402241
$891$	0	0
$892$	10.0000	0.334825
$893$	32.0000	1.07084
$894$	0	0
$895$	2.00000	0.0668526
$896$	−2.00000	−0.0668153
$897$	0	0
$898$	24.0000	0.800890
$899$	24.0000	0.800445
$900$	0	0
$901$	2.00000	0.0666297
$902$	8.00000	0.266371
$903$	0	0
$904$	−6.00000	−0.199557
$905$	−26.0000	−0.864269
$906$	0	0
$907$	−16.0000	−0.531271	−0.265636	−	0.964073i	$-0.585582\pi$
$907$	−16.0000	−0.531271	−0.265636	+	0.964073i	$0.585582\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	28.0000	0.927681	0.463841	−	0.885919i	$-0.346471\pi$
$911$	28.0000	0.927681	0.463841	+	0.885919i	$0.346471\pi$
$912$	0	0
$913$	8.00000	0.264761
$914$	−30.0000	−0.992312
$915$	0	0
$916$	22.0000	0.726900
$917$	−12.0000	−0.396275
$918$	0	0
$919$	−20.0000	−0.659739	−0.329870	−	0.944027i	$-0.607005\pi$
$919$	−20.0000	−0.659739	−0.329870	+	0.944027i	$0.607005\pi$
$920$	−8.00000	−0.263752
$921$	0	0
$922$	30.0000	0.987997
$923$	0	0
$924$	0	0
$925$	0	0
$926$	14.0000	0.460069
$927$	0	0
$928$	6.00000	0.196960
$929$	32.0000	1.04989	0.524943	−	0.851137i	$-0.324087\pi$
$929$	32.0000	1.04989	0.524943	+	0.851137i	$0.324087\pi$
$930$	0	0
$931$	−12.0000	−0.393284
$932$	−10.0000	−0.327561
$933$	0	0
$934$	−20.0000	−0.654420
$935$	2.00000	0.0654070
$936$	0	0
$937$	34.0000	1.11073	0.555366	−	0.831606i	$-0.312578\pi$
$937$	34.0000	1.11073	0.555366	+	0.831606i	$0.312578\pi$
$938$	32.0000	1.04484
$939$	0	0
$940$	−8.00000	−0.260931
$941$	−30.0000	−0.977972	−0.488986	−	0.872292i	$-0.662633\pi$
$941$	−30.0000	−0.977972	−0.488986	+	0.872292i	$0.662633\pi$
$942$	0	0
$943$	32.0000	1.04206
$944$	6.00000	0.195283
$945$	0	0
$946$	−8.00000	−0.260102
$947$	−24.0000	−0.779895	−0.389948	−	0.920837i	$-0.627507\pi$
$947$	−24.0000	−0.779895	−0.389948	+	0.920837i	$0.627507\pi$
$948$	0	0
$949$	0	0
$950$	4.00000	0.129777
$951$	0	0
$952$	2.00000	0.0648204
$953$	42.0000	1.36051	0.680257	−	0.732974i	$-0.261868\pi$
$953$	42.0000	1.36051	0.680257	+	0.732974i	$0.261868\pi$
$954$	0	0
$955$	4.00000	0.129437
$956$	12.0000	0.388108
$957$	0	0
$958$	−24.0000	−0.775405
$959$	−36.0000	−1.16250
$960$	0	0
$961$	−15.0000	−0.483871
$962$	0	0
$963$	0	0
$964$	−2.00000	−0.0644157
$965$	14.0000	0.450676
$966$	0	0
$967$	58.0000	1.86515	0.932577	−	0.360971i	$-0.117555\pi$
$967$	58.0000	1.86515	0.932577	+	0.360971i	$0.117555\pi$
$968$	−7.00000	−0.224989
$969$	0	0
$970$	2.00000	0.0642161
$971$	58.0000	1.86131	0.930654	−	0.365900i	$-0.119239\pi$
$971$	58.0000	1.86131	0.930654	+	0.365900i	$0.119239\pi$
$972$	0	0
$973$	8.00000	0.256468
$974$	18.0000	0.576757
$975$	0	0
$976$	2.00000	0.0640184
$977$	−42.0000	−1.34370	−0.671850	−	0.740688i	$-0.734500\pi$
$977$	−42.0000	−1.34370	−0.671850	+	0.740688i	$0.734500\pi$
$978$	0	0
$979$	24.0000	0.767043
$980$	3.00000	0.0958315
$981$	0	0
$982$	−6.00000	−0.191468
$983$	24.0000	0.765481	0.382741	−	0.923856i	$-0.374980\pi$
$983$	24.0000	0.765481	0.382741	+	0.923856i	$0.374980\pi$
$984$	0	0
$985$	−18.0000	−0.573528
$986$	−6.00000	−0.191079
$987$	0	0
$988$	0	0
$989$	−32.0000	−1.01754
$990$	0	0
$991$	0	0		−	1.00000i	$-0.5\pi$
$991$	0	0			1.00000i	$0.5\pi$
$992$	4.00000	0.127000
$993$	0	0
$994$	16.0000	0.507489
$995$	−16.0000	−0.507234
$996$	0	0
$997$	52.0000	1.64686	0.823428	−	0.567420i	$-0.192059\pi$
$997$	52.0000	1.64686	0.823428	+	0.567420i	$0.192059\pi$
$998$	−36.0000	−1.13956
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1530.2.a.k.1.1	yes	1
3.2	odd	2		1530.2.a.e.1.1	✓	1
5.4	even	2		7650.2.a.bb.1.1		1
15.14	odd	2		7650.2.a.cf.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1530.2.a.e.1.1	✓	1	3.2	odd	2
1530.2.a.k.1.1	yes	1	1.1	even	1	trivial
7650.2.a.bb.1.1		1	5.4	even	2
7650.2.a.cf.1.1		1	15.14	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:	\( N \)	\(=\)	\( 1530 = 2 \cdot 3^{2} \cdot 5 \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1530.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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