LMFDB - Embedded newform 2142.2.a.l.1.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [2142,2,Mod(1,2142)] 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("2142.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2142, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 2142 = 2 \cdot 3^{2} \cdot 7 \cdot 17 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2142.a (trivial)

Newform invariants

comment:select newform

sage:traces = [1,1,0,1,-4,0,1,1,0,-4,4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)

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

Self dual:	yes
Analytic conductor:	$17.1039561130$
Analytic rank:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 238)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	2142.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$

$=$

$q+1.00000 q^{2} +1.00000 q^{4} -4.00000 q^{5} +1.00000 q^{7} +1.00000 q^{8} -4.00000 q^{10} +4.00000 q^{11} -4.00000 q^{13} +1.00000 q^{14} +1.00000 q^{16} +1.00000 q^{17} -6.00000 q^{19} -4.00000 q^{20} +4.00000 q^{22} +11.0000 q^{25} -4.00000 q^{26} +1.00000 q^{28} -6.00000 q^{29} +4.00000 q^{31} +1.00000 q^{32} +1.00000 q^{34} -4.00000 q^{35} -10.0000 q^{37} -6.00000 q^{38} -4.00000 q^{40} -6.00000 q^{41} +4.00000 q^{44} -4.00000 q^{47} +1.00000 q^{49} +11.0000 q^{50} -4.00000 q^{52} -14.0000 q^{53} -16.0000 q^{55} +1.00000 q^{56} -6.00000 q^{58} +6.00000 q^{59} -12.0000 q^{61} +4.00000 q^{62} +1.00000 q^{64} +16.0000 q^{65} +4.00000 q^{67} +1.00000 q^{68} -4.00000 q^{70} +8.00000 q^{71} +2.00000 q^{73} -10.0000 q^{74} -6.00000 q^{76} +4.00000 q^{77} -4.00000 q^{80} -6.00000 q^{82} -10.0000 q^{83} -4.00000 q^{85} +4.00000 q^{88} -10.0000 q^{89} -4.00000 q^{91} -4.00000 q^{94} +24.0000 q^{95} +6.00000 q^{97} +1.00000 q^{98} +O(q^{100})$

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
$5$	−4.00000	−1.78885	−0.894427	−	0.447214i	$-0.852416\pi$
$5$	−4.00000	−1.78885	−0.894427	+	0.447214i	$0.852416\pi$
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	1.00000	0.353553
$9$	0	0
$10$	−4.00000	−1.26491
$11$	4.00000	1.20605	0.603023	−	0.797724i	$-0.293963\pi$
$11$	4.00000	1.20605	0.603023	+	0.797724i	$0.293963\pi$
$12$	0	0
$13$	−4.00000	−1.10940	−0.554700	−	0.832050i	$-0.687167\pi$
$13$	−4.00000	−1.10940	−0.554700	+	0.832050i	$0.687167\pi$
$14$	1.00000	0.267261
$15$	0	0
$16$	1.00000	0.250000
$17$	1.00000	0.242536
$17$	1.00000	0.242536
$18$	0	0
$19$	−6.00000	−1.37649	−0.688247	−	0.725476i	$-0.741620\pi$
$19$	−6.00000	−1.37649	−0.688247	+	0.725476i	$0.741620\pi$
$20$	−4.00000	−0.894427
$21$	0	0
$22$	4.00000	0.852803
$23$	0	0		−	1.00000i	$-0.5\pi$
$23$	0	0			1.00000i	$0.5\pi$
$24$	0	0
$25$	11.0000	2.20000
$26$	−4.00000	−0.784465
$27$	0	0
$28$	1.00000	0.188982
$29$	−6.00000	−1.11417	−0.557086	−	0.830455i	$-0.688081\pi$
$29$	−6.00000	−1.11417	−0.557086	+	0.830455i	$0.688081\pi$
$30$	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
$35$	−4.00000	−0.676123
$36$	0	0
$37$	−10.0000	−1.64399	−0.821995	−	0.569495i	$-0.807139\pi$
$37$	−10.0000	−1.64399	−0.821995	+	0.569495i	$0.807139\pi$
$38$	−6.00000	−0.973329
$39$	0	0
$40$	−4.00000	−0.632456
$41$	−6.00000	−0.937043	−0.468521	−	0.883452i	$-0.655213\pi$
$41$	−6.00000	−0.937043	−0.468521	+	0.883452i	$0.655213\pi$
$42$	0	0
$43$	0	0		−	1.00000i	$-0.5\pi$
$43$	0	0			1.00000i	$0.5\pi$
$44$	4.00000	0.603023
$45$	0	0
$46$	0	0
$47$	−4.00000	−0.583460	−0.291730	−	0.956501i	$-0.594231\pi$
$47$	−4.00000	−0.583460	−0.291730	+	0.956501i	$0.594231\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	11.0000	1.55563
$51$	0	0
$52$	−4.00000	−0.554700
$53$	−14.0000	−1.92305	−0.961524	−	0.274721i	$-0.911414\pi$
$53$	−14.0000	−1.92305	−0.961524	+	0.274721i	$0.911414\pi$
$54$	0	0
$55$	−16.0000	−2.15744
$56$	1.00000	0.133631
$57$	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
$61$	−12.0000	−1.53644	−0.768221	−	0.640184i	$-0.778858\pi$
$61$	−12.0000	−1.53644	−0.768221	+	0.640184i	$0.778858\pi$
$62$	4.00000	0.508001
$63$	0	0
$64$	1.00000	0.125000
$65$	16.0000	1.98456
$66$	0	0
$67$	4.00000	0.488678	0.244339	−	0.969690i	$-0.421429\pi$
$67$	4.00000	0.488678	0.244339	+	0.969690i	$0.421429\pi$
$68$	1.00000	0.121268
$69$	0	0
$70$	−4.00000	−0.478091
$71$	8.00000	0.949425	0.474713	−	0.880141i	$-0.342552\pi$
$71$	8.00000	0.949425	0.474713	+	0.880141i	$0.342552\pi$
$72$	0	0
$73$	2.00000	0.234082	0.117041	−	0.993127i	$-0.462659\pi$
$73$	2.00000	0.234082	0.117041	+	0.993127i	$0.462659\pi$
$74$	−10.0000	−1.16248
$75$	0	0
$76$	−6.00000	−0.688247
$77$	4.00000	0.455842
$78$	0	0
$79$	0	0		−	1.00000i	$-0.5\pi$
$79$	0	0			1.00000i	$0.5\pi$
$80$	−4.00000	−0.447214
$81$	0	0
$82$	−6.00000	−0.662589
$83$	−10.0000	−1.09764	−0.548821	−	0.835940i	$-0.684923\pi$
$83$	−10.0000	−1.09764	−0.548821	+	0.835940i	$0.684923\pi$
$84$	0	0
$85$	−4.00000	−0.433861
$86$	0	0
$87$	0	0
$88$	4.00000	0.426401
$89$	−10.0000	−1.06000	−0.529999	−	0.847998i	$-0.677808\pi$
$89$	−10.0000	−1.06000	−0.529999	+	0.847998i	$0.677808\pi$
$90$	0	0
$91$	−4.00000	−0.419314
$92$	0	0
$93$	0	0
$94$	−4.00000	−0.412568
$95$	24.0000	2.46235
$96$	0	0
$97$	6.00000	0.609208	0.304604	−	0.952479i	$-0.401476\pi$
$97$	6.00000	0.609208	0.304604	+	0.952479i	$0.401476\pi$
$98$	1.00000	0.101015
$99$	0	0
$100$	11.0000	1.10000
$101$	−16.0000	−1.59206	−0.796030	−	0.605257i	$-0.793070\pi$
$101$	−16.0000	−1.59206	−0.796030	+	0.605257i	$0.793070\pi$
$102$	0	0
$103$	−4.00000	−0.394132	−0.197066	−	0.980390i	$-0.563141\pi$
$103$	−4.00000	−0.394132	−0.197066	+	0.980390i	$0.563141\pi$
$104$	−4.00000	−0.392232
$105$	0	0
$106$	−14.0000	−1.35980
$107$	0	0		−	1.00000i	$-0.5\pi$
$107$	0	0			1.00000i	$0.5\pi$
$108$	0	0
$109$	−2.00000	−0.191565	−0.0957826	−	0.995402i	$-0.530535\pi$
$109$	−2.00000	−0.191565	−0.0957826	+	0.995402i	$0.530535\pi$
$110$	−16.0000	−1.52554
$111$	0	0
$112$	1.00000	0.0944911
$113$	−14.0000	−1.31701	−0.658505	−	0.752577i	$-0.728811\pi$
$113$	−14.0000	−1.31701	−0.658505	+	0.752577i	$0.728811\pi$
$114$	0	0
$115$	0	0
$116$	−6.00000	−0.557086
$117$	0	0
$118$	6.00000	0.552345
$119$	1.00000	0.0916698
$120$	0	0
$121$	5.00000	0.454545
$122$	−12.0000	−1.08643
$123$	0	0
$124$	4.00000	0.359211
$125$	−24.0000	−2.14663
$126$	0	0
$127$	−8.00000	−0.709885	−0.354943	−	0.934888i	$-0.615500\pi$
$127$	−8.00000	−0.709885	−0.354943	+	0.934888i	$0.615500\pi$
$128$	1.00000	0.0883883
$129$	0	0
$130$	16.0000	1.40329
$131$	18.0000	1.57267	0.786334	−	0.617802i	$-0.211977\pi$
$131$	18.0000	1.57267	0.786334	+	0.617802i	$0.211977\pi$
$132$	0	0
$133$	−6.00000	−0.520266
$134$	4.00000	0.345547
$135$	0	0
$136$	1.00000	0.0857493
$137$	−2.00000	−0.170872	−0.0854358	−	0.996344i	$-0.527228\pi$
$137$	−2.00000	−0.170872	−0.0854358	+	0.996344i	$0.527228\pi$
$138$	0	0
$139$	10.0000	0.848189	0.424094	−	0.905618i	$-0.360592\pi$
$139$	10.0000	0.848189	0.424094	+	0.905618i	$0.360592\pi$
$140$	−4.00000	−0.338062
$141$	0	0
$142$	8.00000	0.671345
$143$	−16.0000	−1.33799
$144$	0	0
$145$	24.0000	1.99309
$146$	2.00000	0.165521
$147$	0	0
$148$	−10.0000	−0.821995
$149$	2.00000	0.163846	0.0819232	−	0.996639i	$-0.473894\pi$
$149$	2.00000	0.163846	0.0819232	+	0.996639i	$0.473894\pi$
$150$	0	0
$151$	−16.0000	−1.30206	−0.651031	−	0.759051i	$-0.725663\pi$
$151$	−16.0000	−1.30206	−0.651031	+	0.759051i	$0.725663\pi$
$152$	−6.00000	−0.486664
$153$	0	0
$154$	4.00000	0.322329
$155$	−16.0000	−1.28515
$156$	0	0
$157$	4.00000	0.319235	0.159617	−	0.987179i	$-0.448974\pi$
$157$	4.00000	0.319235	0.159617	+	0.987179i	$0.448974\pi$
$158$	0	0
$159$	0	0
$160$	−4.00000	−0.316228
$161$	0	0
$162$	0	0
$163$	4.00000	0.313304	0.156652	−	0.987654i	$-0.449930\pi$
$163$	4.00000	0.313304	0.156652	+	0.987654i	$0.449930\pi$
$164$	−6.00000	−0.468521
$165$	0	0
$166$	−10.0000	−0.776151
$167$	20.0000	1.54765	0.773823	−	0.633402i	$-0.218342\pi$
$167$	20.0000	1.54765	0.773823	+	0.633402i	$0.218342\pi$
$168$	0	0
$169$	3.00000	0.230769
$170$	−4.00000	−0.306786
$171$	0	0
$172$	0	0
$173$	8.00000	0.608229	0.304114	−	0.952636i	$-0.401639\pi$
$173$	8.00000	0.608229	0.304114	+	0.952636i	$0.401639\pi$
$174$	0	0
$175$	11.0000	0.831522
$176$	4.00000	0.301511
$177$	0	0
$178$	−10.0000	−0.749532
$179$	12.0000	0.896922	0.448461	−	0.893802i	$-0.351972\pi$
$179$	12.0000	0.896922	0.448461	+	0.893802i	$0.351972\pi$
$180$	0	0
$181$	0	0		−	1.00000i	$-0.5\pi$
$181$	0	0			1.00000i	$0.5\pi$
$182$	−4.00000	−0.296500
$183$	0	0
$184$	0	0
$185$	40.0000	2.94086
$186$	0	0
$187$	4.00000	0.292509
$188$	−4.00000	−0.291730
$189$	0	0
$190$	24.0000	1.74114
$191$	24.0000	1.73658	0.868290	−	0.496058i	$-0.165220\pi$
$191$	24.0000	1.73658	0.868290	+	0.496058i	$0.165220\pi$
$192$	0	0
$193$	−2.00000	−0.143963	−0.0719816	−	0.997406i	$-0.522932\pi$
$193$	−2.00000	−0.143963	−0.0719816	+	0.997406i	$0.522932\pi$
$194$	6.00000	0.430775
$195$	0	0
$196$	1.00000	0.0714286
$197$	−2.00000	−0.142494	−0.0712470	−	0.997459i	$-0.522698\pi$
$197$	−2.00000	−0.142494	−0.0712470	+	0.997459i	$0.522698\pi$
$198$	0	0
$199$	28.0000	1.98487	0.992434	−	0.122782i	$-0.0391815\pi$
$199$	28.0000	1.98487	0.992434	+	0.122782i	$0.0391815\pi$
$200$	11.0000	0.777817
$201$	0	0
$202$	−16.0000	−1.12576
$203$	−6.00000	−0.421117
$204$	0	0
$205$	24.0000	1.67623
$206$	−4.00000	−0.278693
$207$	0	0
$208$	−4.00000	−0.277350
$209$	−24.0000	−1.66011
$210$	0	0
$211$	−8.00000	−0.550743	−0.275371	−	0.961338i	$-0.588801\pi$
$211$	−8.00000	−0.550743	−0.275371	+	0.961338i	$0.588801\pi$
$212$	−14.0000	−0.961524
$213$	0	0
$214$	0	0
$215$	0	0
$216$	0	0
$217$	4.00000	0.271538
$218$	−2.00000	−0.135457
$219$	0	0
$220$	−16.0000	−1.07872
$221$	−4.00000	−0.269069
$222$	0	0
$223$	24.0000	1.60716	0.803579	−	0.595198i	$-0.202926\pi$
$223$	24.0000	1.60716	0.803579	+	0.595198i	$0.202926\pi$
$224$	1.00000	0.0668153
$225$	0	0
$226$	−14.0000	−0.931266
$227$	2.00000	0.132745	0.0663723	−	0.997795i	$-0.478857\pi$
$227$	2.00000	0.132745	0.0663723	+	0.997795i	$0.478857\pi$
$228$	0	0
$229$	−28.0000	−1.85029	−0.925146	−	0.379611i	$-0.876058\pi$
$229$	−28.0000	−1.85029	−0.925146	+	0.379611i	$0.876058\pi$
$230$	0	0
$231$	0	0
$232$	−6.00000	−0.393919
$233$	−26.0000	−1.70332	−0.851658	−	0.524097i	$-0.824403\pi$
$233$	−26.0000	−1.70332	−0.851658	+	0.524097i	$0.824403\pi$
$234$	0	0
$235$	16.0000	1.04372
$236$	6.00000	0.390567
$237$	0	0
$238$	1.00000	0.0648204
$239$	8.00000	0.517477	0.258738	−	0.965947i	$-0.416693\pi$
$239$	8.00000	0.517477	0.258738	+	0.965947i	$0.416693\pi$
$240$	0	0
$241$	−2.00000	−0.128831	−0.0644157	−	0.997923i	$-0.520518\pi$
$241$	−2.00000	−0.128831	−0.0644157	+	0.997923i	$0.520518\pi$
$242$	5.00000	0.321412
$243$	0	0
$244$	−12.0000	−0.768221
$245$	−4.00000	−0.255551
$246$	0	0
$247$	24.0000	1.52708
$248$	4.00000	0.254000
$249$	0	0
$250$	−24.0000	−1.51789
$251$	−14.0000	−0.883672	−0.441836	−	0.897096i	$-0.645673\pi$
$251$	−14.0000	−0.883672	−0.441836	+	0.897096i	$0.645673\pi$
$252$	0	0
$253$	0	0
$254$	−8.00000	−0.501965
$255$	0	0
$256$	1.00000	0.0625000
$257$	6.00000	0.374270	0.187135	−	0.982334i	$-0.440080\pi$
$257$	6.00000	0.374270	0.187135	+	0.982334i	$0.440080\pi$
$258$	0	0
$259$	−10.0000	−0.621370
$260$	16.0000	0.992278
$261$	0	0
$262$	18.0000	1.11204
$263$	−24.0000	−1.47990	−0.739952	−	0.672660i	$-0.765152\pi$
$263$	−24.0000	−1.47990	−0.739952	+	0.672660i	$0.765152\pi$
$264$	0	0
$265$	56.0000	3.44005
$266$	−6.00000	−0.367884
$267$	0	0
$268$	4.00000	0.244339
$269$	24.0000	1.46331	0.731653	−	0.681677i	$-0.238749\pi$
$269$	24.0000	1.46331	0.731653	+	0.681677i	$0.238749\pi$
$270$	0	0
$271$	0	0		−	1.00000i	$-0.5\pi$
$271$	0	0			1.00000i	$0.5\pi$
$272$	1.00000	0.0606339
$273$	0	0
$274$	−2.00000	−0.120824
$275$	44.0000	2.65330
$276$	0	0
$277$	18.0000	1.08152	0.540758	−	0.841178i	$-0.318138\pi$
$277$	18.0000	1.08152	0.540758	+	0.841178i	$0.318138\pi$
$278$	10.0000	0.599760
$279$	0	0
$280$	−4.00000	−0.239046
$281$	6.00000	0.357930	0.178965	−	0.983855i	$-0.442725\pi$
$281$	6.00000	0.357930	0.178965	+	0.983855i	$0.442725\pi$
$282$	0	0
$283$	14.0000	0.832214	0.416107	−	0.909316i	$-0.363394\pi$
$283$	14.0000	0.832214	0.416107	+	0.909316i	$0.363394\pi$
$284$	8.00000	0.474713
$285$	0	0
$286$	−16.0000	−0.946100
$287$	−6.00000	−0.354169
$288$	0	0
$289$	1.00000	0.0588235
$290$	24.0000	1.40933
$291$	0	0
$292$	2.00000	0.117041
$293$	0	0		−	1.00000i	$-0.5\pi$
$293$	0	0			1.00000i	$0.5\pi$
$294$	0	0
$295$	−24.0000	−1.39733
$296$	−10.0000	−0.581238
$297$	0	0
$298$	2.00000	0.115857
$299$	0	0
$300$	0	0
$301$	0	0
$302$	−16.0000	−0.920697
$303$	0	0
$304$	−6.00000	−0.344124
$305$	48.0000	2.74847
$306$	0	0
$307$	10.0000	0.570730	0.285365	−	0.958419i	$-0.407885\pi$
$307$	10.0000	0.570730	0.285365	+	0.958419i	$0.407885\pi$
$308$	4.00000	0.227921
$309$	0	0
$310$	−16.0000	−0.908739
$311$	−8.00000	−0.453638	−0.226819	−	0.973937i	$-0.572833\pi$
$311$	−8.00000	−0.453638	−0.226819	+	0.973937i	$0.572833\pi$
$312$	0	0
$313$	6.00000	0.339140	0.169570	−	0.985518i	$-0.445762\pi$
$313$	6.00000	0.339140	0.169570	+	0.985518i	$0.445762\pi$
$314$	4.00000	0.225733
$315$	0	0
$316$	0	0
$317$	22.0000	1.23564	0.617822	−	0.786318i	$-0.288015\pi$
$317$	22.0000	1.23564	0.617822	+	0.786318i	$0.288015\pi$
$318$	0	0
$319$	−24.0000	−1.34374
$320$	−4.00000	−0.223607
$321$	0	0
$322$	0	0
$323$	−6.00000	−0.333849
$324$	0	0
$325$	−44.0000	−2.44068
$326$	4.00000	0.221540
$327$	0	0
$328$	−6.00000	−0.331295
$329$	−4.00000	−0.220527
$330$	0	0
$331$	0	0		−	1.00000i	$-0.5\pi$
$331$	0	0			1.00000i	$0.5\pi$
$332$	−10.0000	−0.548821
$333$	0	0
$334$	20.0000	1.09435
$335$	−16.0000	−0.874173
$336$	0	0
$337$	22.0000	1.19842	0.599208	−	0.800593i	$-0.295482\pi$
$337$	22.0000	1.19842	0.599208	+	0.800593i	$0.295482\pi$
$338$	3.00000	0.163178
$339$	0	0
$340$	−4.00000	−0.216930
$341$	16.0000	0.866449
$342$	0	0
$343$	1.00000	0.0539949
$344$	0	0
$345$	0	0
$346$	8.00000	0.430083
$347$	−12.0000	−0.644194	−0.322097	−	0.946707i	$-0.604388\pi$
$347$	−12.0000	−0.644194	−0.322097	+	0.946707i	$0.604388\pi$
$348$	0	0
$349$	20.0000	1.07058	0.535288	−	0.844670i	$-0.320203\pi$
$349$	20.0000	1.07058	0.535288	+	0.844670i	$0.320203\pi$
$350$	11.0000	0.587975
$351$	0	0
$352$	4.00000	0.213201
$353$	−26.0000	−1.38384	−0.691920	−	0.721974i	$-0.743235\pi$
$353$	−26.0000	−1.38384	−0.691920	+	0.721974i	$0.743235\pi$
$354$	0	0
$355$	−32.0000	−1.69838
$356$	−10.0000	−0.529999
$357$	0	0
$358$	12.0000	0.634220
$359$	−24.0000	−1.26667	−0.633336	−	0.773877i	$-0.718315\pi$
$359$	−24.0000	−1.26667	−0.633336	+	0.773877i	$0.718315\pi$
$360$	0	0
$361$	17.0000	0.894737
$362$	0	0
$363$	0	0
$364$	−4.00000	−0.209657
$365$	−8.00000	−0.418739
$366$	0	0
$367$	−24.0000	−1.25279	−0.626395	−	0.779506i	$-0.715470\pi$
$367$	−24.0000	−1.25279	−0.626395	+	0.779506i	$0.715470\pi$
$368$	0	0
$369$	0	0
$370$	40.0000	2.07950
$371$	−14.0000	−0.726844
$372$	0	0
$373$	−10.0000	−0.517780	−0.258890	−	0.965907i	$-0.583357\pi$
$373$	−10.0000	−0.517780	−0.258890	+	0.965907i	$0.583357\pi$
$374$	4.00000	0.206835
$375$	0	0
$376$	−4.00000	−0.206284
$377$	24.0000	1.23606
$378$	0	0
$379$	−4.00000	−0.205466	−0.102733	−	0.994709i	$-0.532759\pi$
$379$	−4.00000	−0.205466	−0.102733	+	0.994709i	$0.532759\pi$
$380$	24.0000	1.23117
$381$	0	0
$382$	24.0000	1.22795
$383$	−4.00000	−0.204390	−0.102195	−	0.994764i	$-0.532587\pi$
$383$	−4.00000	−0.204390	−0.102195	+	0.994764i	$0.532587\pi$
$384$	0	0
$385$	−16.0000	−0.815436
$386$	−2.00000	−0.101797
$387$	0	0
$388$	6.00000	0.304604
$389$	30.0000	1.52106	0.760530	−	0.649303i	$-0.224939\pi$
$389$	30.0000	1.52106	0.760530	+	0.649303i	$0.224939\pi$
$390$	0	0
$391$	0	0
$392$	1.00000	0.0505076
$393$	0	0
$394$	−2.00000	−0.100759
$395$	0	0
$396$	0	0
$397$	−16.0000	−0.803017	−0.401508	−	0.915855i	$-0.631514\pi$
$397$	−16.0000	−0.803017	−0.401508	+	0.915855i	$0.631514\pi$
$398$	28.0000	1.40351
$399$	0	0
$400$	11.0000	0.550000
$401$	18.0000	0.898877	0.449439	−	0.893311i	$-0.351624\pi$
$401$	18.0000	0.898877	0.449439	+	0.893311i	$0.351624\pi$
$402$	0	0
$403$	−16.0000	−0.797017
$404$	−16.0000	−0.796030
$405$	0	0
$406$	−6.00000	−0.297775
$407$	−40.0000	−1.98273
$408$	0	0
$409$	−26.0000	−1.28562	−0.642809	−	0.766027i	$-0.722231\pi$
$409$	−26.0000	−1.28562	−0.642809	+	0.766027i	$0.722231\pi$
$410$	24.0000	1.18528
$411$	0	0
$412$	−4.00000	−0.197066
$413$	6.00000	0.295241
$414$	0	0
$415$	40.0000	1.96352
$416$	−4.00000	−0.196116
$417$	0	0
$418$	−24.0000	−1.17388
$419$	−18.0000	−0.879358	−0.439679	−	0.898155i	$-0.644908\pi$
$419$	−18.0000	−0.879358	−0.439679	+	0.898155i	$0.644908\pi$
$420$	0	0
$421$	−18.0000	−0.877266	−0.438633	−	0.898666i	$-0.644537\pi$
$421$	−18.0000	−0.877266	−0.438633	+	0.898666i	$0.644537\pi$
$422$	−8.00000	−0.389434
$423$	0	0
$424$	−14.0000	−0.679900
$425$	11.0000	0.533578
$426$	0	0
$427$	−12.0000	−0.580721
$428$	0	0
$429$	0	0
$430$	0	0
$431$	8.00000	0.385346	0.192673	−	0.981263i	$-0.438284\pi$
$431$	8.00000	0.385346	0.192673	+	0.981263i	$0.438284\pi$
$432$	0	0
$433$	−34.0000	−1.63394	−0.816968	−	0.576683i	$-0.804347\pi$
$433$	−34.0000	−1.63394	−0.816968	+	0.576683i	$0.804347\pi$
$434$	4.00000	0.192006
$435$	0	0
$436$	−2.00000	−0.0957826
$437$	0	0
$438$	0	0
$439$	40.0000	1.90910	0.954548	−	0.298057i	$-0.0963387\pi$
$439$	40.0000	1.90910	0.954548	+	0.298057i	$0.0963387\pi$
$440$	−16.0000	−0.762770
$441$	0	0
$442$	−4.00000	−0.190261
$443$	−12.0000	−0.570137	−0.285069	−	0.958507i	$-0.592016\pi$
$443$	−12.0000	−0.570137	−0.285069	+	0.958507i	$0.592016\pi$
$444$	0	0
$445$	40.0000	1.89618
$446$	24.0000	1.13643
$447$	0	0
$448$	1.00000	0.0472456
$449$	−10.0000	−0.471929	−0.235965	−	0.971762i	$-0.575825\pi$
$449$	−10.0000	−0.471929	−0.235965	+	0.971762i	$0.575825\pi$
$450$	0	0
$451$	−24.0000	−1.13012
$452$	−14.0000	−0.658505
$453$	0	0
$454$	2.00000	0.0938647
$455$	16.0000	0.750092
$456$	0	0
$457$	22.0000	1.02912	0.514558	−	0.857455i	$-0.327956\pi$
$457$	22.0000	1.02912	0.514558	+	0.857455i	$0.327956\pi$
$458$	−28.0000	−1.30835
$459$	0	0
$460$	0	0
$461$	20.0000	0.931493	0.465746	−	0.884918i	$-0.345786\pi$
$461$	20.0000	0.931493	0.465746	+	0.884918i	$0.345786\pi$
$462$	0	0
$463$	−40.0000	−1.85896	−0.929479	−	0.368875i	$-0.879743\pi$
$463$	−40.0000	−1.85896	−0.929479	+	0.368875i	$0.879743\pi$
$464$	−6.00000	−0.278543
$465$	0	0
$466$	−26.0000	−1.20443
$467$	−18.0000	−0.832941	−0.416470	−	0.909149i	$-0.636733\pi$
$467$	−18.0000	−0.832941	−0.416470	+	0.909149i	$0.636733\pi$
$468$	0	0
$469$	4.00000	0.184703
$470$	16.0000	0.738025
$471$	0	0
$472$	6.00000	0.276172
$473$	0	0
$474$	0	0
$475$	−66.0000	−3.02829
$476$	1.00000	0.0458349
$477$	0	0
$478$	8.00000	0.365911
$479$	−4.00000	−0.182765	−0.0913823	−	0.995816i	$-0.529129\pi$
$479$	−4.00000	−0.182765	−0.0913823	+	0.995816i	$0.529129\pi$
$480$	0	0
$481$	40.0000	1.82384
$482$	−2.00000	−0.0910975
$483$	0	0
$484$	5.00000	0.227273
$485$	−24.0000	−1.08978
$486$	0	0
$487$	−32.0000	−1.45006	−0.725029	−	0.688718i	$-0.758174\pi$
$487$	−32.0000	−1.45006	−0.725029	+	0.688718i	$0.758174\pi$
$488$	−12.0000	−0.543214
$489$	0	0
$490$	−4.00000	−0.180702
$491$	28.0000	1.26362	0.631811	−	0.775122i	$-0.282312\pi$
$491$	28.0000	1.26362	0.631811	+	0.775122i	$0.282312\pi$
$492$	0	0
$493$	−6.00000	−0.270226
$494$	24.0000	1.07981
$495$	0	0
$496$	4.00000	0.179605
$497$	8.00000	0.358849
$498$	0	0
$499$	−24.0000	−1.07439	−0.537194	−	0.843459i	$-0.680516\pi$
$499$	−24.0000	−1.07439	−0.537194	+	0.843459i	$0.680516\pi$
$500$	−24.0000	−1.07331
$501$	0	0
$502$	−14.0000	−0.624851
$503$	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
$505$	64.0000	2.84796
$506$	0	0
$507$	0	0
$508$	−8.00000	−0.354943
$509$	−12.0000	−0.531891	−0.265945	−	0.963988i	$-0.585684\pi$
$509$	−12.0000	−0.531891	−0.265945	+	0.963988i	$0.585684\pi$
$510$	0	0
$511$	2.00000	0.0884748
$512$	1.00000	0.0441942
$513$	0	0
$514$	6.00000	0.264649
$515$	16.0000	0.705044
$516$	0	0
$517$	−16.0000	−0.703679
$518$	−10.0000	−0.439375
$519$	0	0
$520$	16.0000	0.701646
$521$	−14.0000	−0.613351	−0.306676	−	0.951814i	$-0.599217\pi$
$521$	−14.0000	−0.613351	−0.306676	+	0.951814i	$0.599217\pi$
$522$	0	0
$523$	34.0000	1.48672	0.743358	−	0.668894i	$-0.233232\pi$
$523$	34.0000	1.48672	0.743358	+	0.668894i	$0.233232\pi$
$524$	18.0000	0.786334
$525$	0	0
$526$	−24.0000	−1.04645
$527$	4.00000	0.174243
$528$	0	0
$529$	−23.0000	−1.00000
$530$	56.0000	2.43248
$531$	0	0
$532$	−6.00000	−0.260133
$533$	24.0000	1.03956
$534$	0	0
$535$	0	0
$536$	4.00000	0.172774
$537$	0	0
$538$	24.0000	1.03471
$539$	4.00000	0.172292
$540$	0	0
$541$	26.0000	1.11783	0.558914	−	0.829226i	$-0.311218\pi$
$541$	26.0000	1.11783	0.558914	+	0.829226i	$0.311218\pi$
$542$	0	0
$543$	0	0
$544$	1.00000	0.0428746
$545$	8.00000	0.342682
$546$	0	0
$547$	−4.00000	−0.171028	−0.0855138	−	0.996337i	$-0.527253\pi$
$547$	−4.00000	−0.171028	−0.0855138	+	0.996337i	$0.527253\pi$
$548$	−2.00000	−0.0854358
$549$	0	0
$550$	44.0000	1.87617
$551$	36.0000	1.53365
$552$	0	0
$553$	0	0
$554$	18.0000	0.764747
$555$	0	0
$556$	10.0000	0.424094
$557$	18.0000	0.762684	0.381342	−	0.924434i	$-0.375462\pi$
$557$	18.0000	0.762684	0.381342	+	0.924434i	$0.375462\pi$
$558$	0	0
$559$	0	0
$560$	−4.00000	−0.169031
$561$	0	0
$562$	6.00000	0.253095
$563$	−14.0000	−0.590030	−0.295015	−	0.955493i	$-0.595325\pi$
$563$	−14.0000	−0.590030	−0.295015	+	0.955493i	$0.595325\pi$
$564$	0	0
$565$	56.0000	2.35594
$566$	14.0000	0.588464
$567$	0	0
$568$	8.00000	0.335673
$569$	26.0000	1.08998	0.544988	−	0.838444i	$-0.316534\pi$
$569$	26.0000	1.08998	0.544988	+	0.838444i	$0.316534\pi$
$570$	0	0
$571$	36.0000	1.50655	0.753277	−	0.657704i	$-0.228472\pi$
$571$	36.0000	1.50655	0.753277	+	0.657704i	$0.228472\pi$
$572$	−16.0000	−0.668994
$573$	0	0
$574$	−6.00000	−0.250435
$575$	0	0
$576$	0	0
$577$	−22.0000	−0.915872	−0.457936	−	0.888985i	$-0.651411\pi$
$577$	−22.0000	−0.915872	−0.457936	+	0.888985i	$0.651411\pi$
$578$	1.00000	0.0415945
$579$	0	0
$580$	24.0000	0.996546
$581$	−10.0000	−0.414870
$582$	0	0
$583$	−56.0000	−2.31928
$584$	2.00000	0.0827606
$585$	0	0
$586$	0	0
$587$	18.0000	0.742940	0.371470	−	0.928445i	$-0.378854\pi$
$587$	18.0000	0.742940	0.371470	+	0.928445i	$0.378854\pi$
$588$	0	0
$589$	−24.0000	−0.988903
$590$	−24.0000	−0.988064
$591$	0	0
$592$	−10.0000	−0.410997
$593$	6.00000	0.246390	0.123195	−	0.992382i	$-0.460686\pi$
$593$	6.00000	0.246390	0.123195	+	0.992382i	$0.460686\pi$
$594$	0	0
$595$	−4.00000	−0.163984
$596$	2.00000	0.0819232
$597$	0	0
$598$	0	0
$599$	−32.0000	−1.30748	−0.653742	−	0.756717i	$-0.726802\pi$
$599$	−32.0000	−1.30748	−0.653742	+	0.756717i	$0.726802\pi$
$600$	0	0
$601$	2.00000	0.0815817	0.0407909	−	0.999168i	$-0.487012\pi$
$601$	2.00000	0.0815817	0.0407909	+	0.999168i	$0.487012\pi$
$602$	0	0
$603$	0	0
$604$	−16.0000	−0.651031
$605$	−20.0000	−0.813116
$606$	0	0
$607$	16.0000	0.649420	0.324710	−	0.945814i	$-0.394733\pi$
$607$	16.0000	0.649420	0.324710	+	0.945814i	$0.394733\pi$
$608$	−6.00000	−0.243332
$609$	0	0
$610$	48.0000	1.94346
$611$	16.0000	0.647291
$612$	0	0
$613$	−14.0000	−0.565455	−0.282727	−	0.959200i	$-0.591239\pi$
$613$	−14.0000	−0.565455	−0.282727	+	0.959200i	$0.591239\pi$
$614$	10.0000	0.403567
$615$	0	0
$616$	4.00000	0.161165
$617$	−14.0000	−0.563619	−0.281809	−	0.959470i	$-0.590935\pi$
$617$	−14.0000	−0.563619	−0.281809	+	0.959470i	$0.590935\pi$
$618$	0	0
$619$	−26.0000	−1.04503	−0.522514	−	0.852631i	$-0.675006\pi$
$619$	−26.0000	−1.04503	−0.522514	+	0.852631i	$0.675006\pi$
$620$	−16.0000	−0.642575
$621$	0	0
$622$	−8.00000	−0.320771
$623$	−10.0000	−0.400642
$624$	0	0
$625$	41.0000	1.64000
$626$	6.00000	0.239808
$627$	0	0
$628$	4.00000	0.159617
$629$	−10.0000	−0.398726
$630$	0	0
$631$	16.0000	0.636950	0.318475	−	0.947931i	$-0.396829\pi$
$631$	16.0000	0.636950	0.318475	+	0.947931i	$0.396829\pi$
$632$	0	0
$633$	0	0
$634$	22.0000	0.873732
$635$	32.0000	1.26988
$636$	0	0
$637$	−4.00000	−0.158486
$638$	−24.0000	−0.950169
$639$	0	0
$640$	−4.00000	−0.158114
$641$	18.0000	0.710957	0.355479	−	0.934684i	$-0.384318\pi$
$641$	18.0000	0.710957	0.355479	+	0.934684i	$0.384318\pi$
$642$	0	0
$643$	34.0000	1.34083	0.670415	−	0.741987i	$-0.266116\pi$
$643$	34.0000	1.34083	0.670415	+	0.741987i	$0.266116\pi$
$644$	0	0
$645$	0	0
$646$	−6.00000	−0.236067
$647$	−36.0000	−1.41531	−0.707653	−	0.706560i	$-0.750246\pi$
$647$	−36.0000	−1.41531	−0.707653	+	0.706560i	$0.750246\pi$
$648$	0	0
$649$	24.0000	0.942082
$650$	−44.0000	−1.72582
$651$	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
$655$	−72.0000	−2.81327
$656$	−6.00000	−0.234261
$657$	0	0
$658$	−4.00000	−0.155936
$659$	24.0000	0.934907	0.467454	−	0.884018i	$-0.345171\pi$
$659$	24.0000	0.934907	0.467454	+	0.884018i	$0.345171\pi$
$660$	0	0
$661$	24.0000	0.933492	0.466746	−	0.884391i	$-0.345426\pi$
$661$	24.0000	0.933492	0.466746	+	0.884391i	$0.345426\pi$
$662$	0	0
$663$	0	0
$664$	−10.0000	−0.388075
$665$	24.0000	0.930680
$666$	0	0
$667$	0	0
$668$	20.0000	0.773823
$669$	0	0
$670$	−16.0000	−0.618134
$671$	−48.0000	−1.85302
$672$	0	0
$673$	−30.0000	−1.15642	−0.578208	−	0.815890i	$-0.696248\pi$
$673$	−30.0000	−1.15642	−0.578208	+	0.815890i	$0.696248\pi$
$674$	22.0000	0.847408
$675$	0	0
$676$	3.00000	0.115385
$677$	−16.0000	−0.614930	−0.307465	−	0.951559i	$-0.599481\pi$
$677$	−16.0000	−0.614930	−0.307465	+	0.951559i	$0.599481\pi$
$678$	0	0
$679$	6.00000	0.230259
$680$	−4.00000	−0.153393
$681$	0	0
$682$	16.0000	0.612672
$683$	16.0000	0.612223	0.306111	−	0.951996i	$-0.400972\pi$
$683$	16.0000	0.612223	0.306111	+	0.951996i	$0.400972\pi$
$684$	0	0
$685$	8.00000	0.305664
$686$	1.00000	0.0381802
$687$	0	0
$688$	0	0
$689$	56.0000	2.13343
$690$	0	0
$691$	6.00000	0.228251	0.114125	−	0.993466i	$-0.463593\pi$
$691$	6.00000	0.228251	0.114125	+	0.993466i	$0.463593\pi$
$692$	8.00000	0.304114
$693$	0	0
$694$	−12.0000	−0.455514
$695$	−40.0000	−1.51729
$696$	0	0
$697$	−6.00000	−0.227266
$698$	20.0000	0.757011
$699$	0	0
$700$	11.0000	0.415761
$701$	−42.0000	−1.58632	−0.793159	−	0.609015i	$-0.791565\pi$
$701$	−42.0000	−1.58632	−0.793159	+	0.609015i	$0.791565\pi$
$702$	0	0
$703$	60.0000	2.26294
$704$	4.00000	0.150756
$705$	0	0
$706$	−26.0000	−0.978523
$707$	−16.0000	−0.601742
$708$	0	0
$709$	−2.00000	−0.0751116	−0.0375558	−	0.999295i	$-0.511957\pi$
$709$	−2.00000	−0.0751116	−0.0375558	+	0.999295i	$0.511957\pi$
$710$	−32.0000	−1.20094
$711$	0	0
$712$	−10.0000	−0.374766
$713$	0	0
$714$	0	0
$715$	64.0000	2.39346
$716$	12.0000	0.448461
$717$	0	0
$718$	−24.0000	−0.895672
$719$	−36.0000	−1.34257	−0.671287	−	0.741198i	$-0.734258\pi$
$719$	−36.0000	−1.34257	−0.671287	+	0.741198i	$0.734258\pi$
$720$	0	0
$721$	−4.00000	−0.148968
$722$	17.0000	0.632674
$723$	0	0
$724$	0	0
$725$	−66.0000	−2.45118
$726$	0	0
$727$	12.0000	0.445055	0.222528	−	0.974926i	$-0.428569\pi$
$727$	12.0000	0.445055	0.222528	+	0.974926i	$0.428569\pi$
$728$	−4.00000	−0.148250
$729$	0	0
$730$	−8.00000	−0.296093
$731$	0	0
$732$	0	0
$733$	−24.0000	−0.886460	−0.443230	−	0.896408i	$-0.646168\pi$
$733$	−24.0000	−0.886460	−0.443230	+	0.896408i	$0.646168\pi$
$734$	−24.0000	−0.885856
$735$	0	0
$736$	0	0
$737$	16.0000	0.589368
$738$	0	0
$739$	0	0		−	1.00000i	$-0.5\pi$
$739$	0	0			1.00000i	$0.5\pi$
$740$	40.0000	1.47043
$741$	0	0
$742$	−14.0000	−0.513956
$743$	48.0000	1.76095	0.880475	−	0.474093i	$-0.157224\pi$
$743$	48.0000	1.76095	0.880475	+	0.474093i	$0.157224\pi$
$744$	0	0
$745$	−8.00000	−0.293097
$746$	−10.0000	−0.366126
$747$	0	0
$748$	4.00000	0.146254
$749$	0	0
$750$	0	0
$751$	−40.0000	−1.45962	−0.729810	−	0.683650i	$-0.760392\pi$
$751$	−40.0000	−1.45962	−0.729810	+	0.683650i	$0.760392\pi$
$752$	−4.00000	−0.145865
$753$	0	0
$754$	24.0000	0.874028
$755$	64.0000	2.32920
$756$	0	0
$757$	−30.0000	−1.09037	−0.545184	−	0.838316i	$-0.683540\pi$
$757$	−30.0000	−1.09037	−0.545184	+	0.838316i	$0.683540\pi$
$758$	−4.00000	−0.145287
$759$	0	0
$760$	24.0000	0.870572
$761$	10.0000	0.362500	0.181250	−	0.983437i	$-0.441986\pi$
$761$	10.0000	0.362500	0.181250	+	0.983437i	$0.441986\pi$
$762$	0	0
$763$	−2.00000	−0.0724049
$764$	24.0000	0.868290
$765$	0	0
$766$	−4.00000	−0.144526
$767$	−24.0000	−0.866590
$768$	0	0
$769$	−26.0000	−0.937584	−0.468792	−	0.883309i	$-0.655311\pi$
$769$	−26.0000	−0.937584	−0.468792	+	0.883309i	$0.655311\pi$
$770$	−16.0000	−0.576600
$771$	0	0
$772$	−2.00000	−0.0719816
$773$	−32.0000	−1.15096	−0.575480	−	0.817816i	$-0.695185\pi$
$773$	−32.0000	−1.15096	−0.575480	+	0.817816i	$0.695185\pi$
$774$	0	0
$775$	44.0000	1.58053
$776$	6.00000	0.215387
$777$	0	0
$778$	30.0000	1.07555
$779$	36.0000	1.28983
$780$	0	0
$781$	32.0000	1.14505
$782$	0	0
$783$	0	0
$784$	1.00000	0.0357143
$785$	−16.0000	−0.571064
$786$	0	0
$787$	−10.0000	−0.356462	−0.178231	−	0.983989i	$-0.557037\pi$
$787$	−10.0000	−0.356462	−0.178231	+	0.983989i	$0.557037\pi$
$788$	−2.00000	−0.0712470
$789$	0	0
$790$	0	0
$791$	−14.0000	−0.497783
$792$	0	0
$793$	48.0000	1.70453
$794$	−16.0000	−0.567819
$795$	0	0
$796$	28.0000	0.992434
$797$	28.0000	0.991811	0.495905	−	0.868377i	$-0.334836\pi$
$797$	28.0000	0.991811	0.495905	+	0.868377i	$0.334836\pi$
$798$	0	0
$799$	−4.00000	−0.141510
$800$	11.0000	0.388909
$801$	0	0
$802$	18.0000	0.635602
$803$	8.00000	0.282314
$804$	0	0
$805$	0	0
$806$	−16.0000	−0.563576
$807$	0	0
$808$	−16.0000	−0.562878
$809$	26.0000	0.914111	0.457056	−	0.889438i	$-0.348904\pi$
$809$	26.0000	0.914111	0.457056	+	0.889438i	$0.348904\pi$
$810$	0	0
$811$	−26.0000	−0.912983	−0.456492	−	0.889728i	$-0.650894\pi$
$811$	−26.0000	−0.912983	−0.456492	+	0.889728i	$0.650894\pi$
$812$	−6.00000	−0.210559
$813$	0	0
$814$	−40.0000	−1.40200
$815$	−16.0000	−0.560456
$816$	0	0
$817$	0	0
$818$	−26.0000	−0.909069
$819$	0	0
$820$	24.0000	0.838116
$821$	54.0000	1.88461	0.942306	−	0.334751i	$-0.108652\pi$
$821$	54.0000	1.88461	0.942306	+	0.334751i	$0.108652\pi$
$822$	0	0
$823$	32.0000	1.11545	0.557725	−	0.830026i	$-0.311674\pi$
$823$	32.0000	1.11545	0.557725	+	0.830026i	$0.311674\pi$
$824$	−4.00000	−0.139347
$825$	0	0
$826$	6.00000	0.208767
$827$	8.00000	0.278187	0.139094	−	0.990279i	$-0.455581\pi$
$827$	8.00000	0.278187	0.139094	+	0.990279i	$0.455581\pi$
$828$	0	0
$829$	16.0000	0.555703	0.277851	−	0.960624i	$-0.410378\pi$
$829$	16.0000	0.555703	0.277851	+	0.960624i	$0.410378\pi$
$830$	40.0000	1.38842
$831$	0	0
$832$	−4.00000	−0.138675
$833$	1.00000	0.0346479
$834$	0	0
$835$	−80.0000	−2.76851
$836$	−24.0000	−0.830057
$837$	0	0
$838$	−18.0000	−0.621800
$839$	−52.0000	−1.79524	−0.897620	−	0.440771i	$-0.854705\pi$
$839$	−52.0000	−1.79524	−0.897620	+	0.440771i	$0.854705\pi$
$840$	0	0
$841$	7.00000	0.241379
$842$	−18.0000	−0.620321
$843$	0	0
$844$	−8.00000	−0.275371
$845$	−12.0000	−0.412813
$846$	0	0
$847$	5.00000	0.171802
$848$	−14.0000	−0.480762
$849$	0	0
$850$	11.0000	0.377297
$851$	0	0
$852$	0	0
$853$	16.0000	0.547830	0.273915	−	0.961754i	$-0.411681\pi$
$853$	16.0000	0.547830	0.273915	+	0.961754i	$0.411681\pi$
$854$	−12.0000	−0.410632
$855$	0	0
$856$	0	0
$857$	−30.0000	−1.02478	−0.512390	−	0.858753i	$-0.671240\pi$
$857$	−30.0000	−1.02478	−0.512390	+	0.858753i	$0.671240\pi$
$858$	0	0
$859$	30.0000	1.02359	0.511793	−	0.859109i	$-0.328981\pi$
$859$	30.0000	1.02359	0.511793	+	0.859109i	$0.328981\pi$
$860$	0	0
$861$	0	0
$862$	8.00000	0.272481
$863$	−16.0000	−0.544646	−0.272323	−	0.962206i	$-0.587792\pi$
$863$	−16.0000	−0.544646	−0.272323	+	0.962206i	$0.587792\pi$
$864$	0	0
$865$	−32.0000	−1.08803
$866$	−34.0000	−1.15537
$867$	0	0
$868$	4.00000	0.135769
$869$	0	0
$870$	0	0
$871$	−16.0000	−0.542139
$872$	−2.00000	−0.0677285
$873$	0	0
$874$	0	0
$875$	−24.0000	−0.811348
$876$	0	0
$877$	−26.0000	−0.877958	−0.438979	−	0.898497i	$-0.644660\pi$
$877$	−26.0000	−0.877958	−0.438979	+	0.898497i	$0.644660\pi$
$878$	40.0000	1.34993
$879$	0	0
$880$	−16.0000	−0.539360
$881$	−50.0000	−1.68454	−0.842271	−	0.539054i	$-0.818782\pi$
$881$	−50.0000	−1.68454	−0.842271	+	0.539054i	$0.818782\pi$
$882$	0	0
$883$	−4.00000	−0.134611	−0.0673054	−	0.997732i	$-0.521440\pi$
$883$	−4.00000	−0.134611	−0.0673054	+	0.997732i	$0.521440\pi$
$884$	−4.00000	−0.134535
$885$	0	0
$886$	−12.0000	−0.403148
$887$	12.0000	0.402921	0.201460	−	0.979497i	$-0.435431\pi$
$887$	12.0000	0.402921	0.201460	+	0.979497i	$0.435431\pi$
$888$	0	0
$889$	−8.00000	−0.268311
$890$	40.0000	1.34080
$891$	0	0
$892$	24.0000	0.803579
$893$	24.0000	0.803129
$894$	0	0
$895$	−48.0000	−1.60446
$896$	1.00000	0.0334077
$897$	0	0
$898$	−10.0000	−0.333704
$899$	−24.0000	−0.800445
$900$	0	0
$901$	−14.0000	−0.466408
$902$	−24.0000	−0.799113
$903$	0	0
$904$	−14.0000	−0.465633
$905$	0	0
$906$	0	0
$907$	−8.00000	−0.265636	−0.132818	−	0.991140i	$-0.542403\pi$
$907$	−8.00000	−0.265636	−0.132818	+	0.991140i	$0.542403\pi$
$908$	2.00000	0.0663723
$909$	0	0
$910$	16.0000	0.530395
$911$	24.0000	0.795155	0.397578	−	0.917568i	$-0.369851\pi$
$911$	24.0000	0.795155	0.397578	+	0.917568i	$0.369851\pi$
$912$	0	0
$913$	−40.0000	−1.32381
$914$	22.0000	0.727695
$915$	0	0
$916$	−28.0000	−0.925146
$917$	18.0000	0.594412
$918$	0	0
$919$	−8.00000	−0.263896	−0.131948	−	0.991257i	$-0.542123\pi$
$919$	−8.00000	−0.263896	−0.131948	+	0.991257i	$0.542123\pi$
$920$	0	0
$921$	0	0
$922$	20.0000	0.658665
$923$	−32.0000	−1.05329
$924$	0	0
$925$	−110.000	−3.61678
$926$	−40.0000	−1.31448
$927$	0	0
$928$	−6.00000	−0.196960
$929$	50.0000	1.64045	0.820223	−	0.572043i	$-0.193849\pi$
$929$	50.0000	1.64045	0.820223	+	0.572043i	$0.193849\pi$
$930$	0	0
$931$	−6.00000	−0.196642
$932$	−26.0000	−0.851658
$933$	0	0
$934$	−18.0000	−0.588978
$935$	−16.0000	−0.523256
$936$	0	0
$937$	10.0000	0.326686	0.163343	−	0.986569i	$-0.447772\pi$
$937$	10.0000	0.326686	0.163343	+	0.986569i	$0.447772\pi$
$938$	4.00000	0.130605
$939$	0	0
$940$	16.0000	0.521862
$941$	12.0000	0.391189	0.195594	−	0.980685i	$-0.437336\pi$
$941$	12.0000	0.391189	0.195594	+	0.980685i	$0.437336\pi$
$942$	0	0
$943$	0	0
$944$	6.00000	0.195283
$945$	0	0
$946$	0	0
$947$	4.00000	0.129983	0.0649913	−	0.997886i	$-0.479298\pi$
$947$	4.00000	0.129983	0.0649913	+	0.997886i	$0.479298\pi$
$948$	0	0
$949$	−8.00000	−0.259691
$950$	−66.0000	−2.14132
$951$	0	0
$952$	1.00000	0.0324102
$953$	−26.0000	−0.842223	−0.421111	−	0.907009i	$-0.638360\pi$
$953$	−26.0000	−0.842223	−0.421111	+	0.907009i	$0.638360\pi$
$954$	0	0
$955$	−96.0000	−3.10649
$956$	8.00000	0.258738
$957$	0	0
$958$	−4.00000	−0.129234
$959$	−2.00000	−0.0645834
$960$	0	0
$961$	−15.0000	−0.483871
$962$	40.0000	1.28965
$963$	0	0
$964$	−2.00000	−0.0644157
$965$	8.00000	0.257529
$966$	0	0
$967$	32.0000	1.02905	0.514525	−	0.857475i	$-0.327968\pi$
$967$	32.0000	1.02905	0.514525	+	0.857475i	$0.327968\pi$
$968$	5.00000	0.160706
$969$	0	0
$970$	−24.0000	−0.770594
$971$	−10.0000	−0.320915	−0.160458	−	0.987043i	$-0.551297\pi$
$971$	−10.0000	−0.320915	−0.160458	+	0.987043i	$0.551297\pi$
$972$	0	0
$973$	10.0000	0.320585
$974$	−32.0000	−1.02535
$975$	0	0
$976$	−12.0000	−0.384111
$977$	54.0000	1.72761	0.863807	−	0.503824i	$-0.168074\pi$
$977$	54.0000	1.72761	0.863807	+	0.503824i	$0.168074\pi$
$978$	0	0
$979$	−40.0000	−1.27841
$980$	−4.00000	−0.127775
$981$	0	0
$982$	28.0000	0.893516
$983$	28.0000	0.893061	0.446531	−	0.894768i	$-0.352659\pi$
$983$	28.0000	0.893061	0.446531	+	0.894768i	$0.352659\pi$
$984$	0	0
$985$	8.00000	0.254901
$986$	−6.00000	−0.191079
$987$	0	0
$988$	24.0000	0.763542
$989$	0	0
$990$	0	0
$991$	−24.0000	−0.762385	−0.381193	−	0.924496i	$-0.624487\pi$
$991$	−24.0000	−0.762385	−0.381193	+	0.924496i	$0.624487\pi$
$992$	4.00000	0.127000
$993$	0	0
$994$	8.00000	0.253745
$995$	−112.000	−3.55064
$996$	0	0
$997$	−28.0000	−0.886769	−0.443384	−	0.896332i	$-0.646222\pi$
$997$	−28.0000	−0.886769	−0.443384	+	0.896332i	$0.646222\pi$
$998$	−24.0000	−0.759707
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2142.2.a.l.1.1		1
3.2	odd	2		238.2.a.b.1.1	✓	1
12.11	even	2		1904.2.a.b.1.1		1
15.14	odd	2		5950.2.a.k.1.1		1
21.20	even	2		1666.2.a.b.1.1		1
24.5	odd	2		7616.2.a.a.1.1		1
24.11	even	2		7616.2.a.i.1.1		1
51.50	odd	2		4046.2.a.b.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
238.2.a.b.1.1	✓	1	3.2	odd	2
1666.2.a.b.1.1		1	21.20	even	2
1904.2.a.b.1.1		1	12.11	even	2
2142.2.a.l.1.1		1	1.1	even	1	trivial
4046.2.a.b.1.1		1	51.50	odd	2
5950.2.a.k.1.1		1	15.14	odd	2
7616.2.a.a.1.1		1	24.5	odd	2
7616.2.a.i.1.1		1	24.11	even	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:	\( N \)	\(=\)	\( 2142 = 2 \cdot 3^{2} \cdot 7 \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2142.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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