LMFDB - Embedded newform 2070.2.a.j.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2070,2,Mod(1,2070)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2070, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("2070.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2070 = 2 \cdot 3^{2} \cdot 5 \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2070.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	yes
Analytic conductor:	$16.5290332184$
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	$\chi$	$=$	2070.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q+1.00000 q^{2} +1.00000 q^{4} -1.00000 q^{5} -4.00000 q^{7} +1.00000 q^{8} +O(q^{10})$

$q+1.00000 q^{2} +1.00000 q^{4} -1.00000 q^{5} -4.00000 q^{7} +1.00000 q^{8} -1.00000 q^{10} -4.00000 q^{14} +1.00000 q^{16} +4.00000 q^{17} +8.00000 q^{19} -1.00000 q^{20} -1.00000 q^{23} +1.00000 q^{25} -4.00000 q^{28} -2.00000 q^{29} +4.00000 q^{31} +1.00000 q^{32} +4.00000 q^{34} +4.00000 q^{35} -2.00000 q^{37} +8.00000 q^{38} -1.00000 q^{40} +8.00000 q^{41} -4.00000 q^{43} -1.00000 q^{46} +8.00000 q^{47} +9.00000 q^{49} +1.00000 q^{50} +6.00000 q^{53} -4.00000 q^{56} -2.00000 q^{58} +10.0000 q^{59} -8.00000 q^{61} +4.00000 q^{62} +1.00000 q^{64} +12.0000 q^{67} +4.00000 q^{68} +4.00000 q^{70} +16.0000 q^{71} -2.00000 q^{73} -2.00000 q^{74} +8.00000 q^{76} -6.00000 q^{79} -1.00000 q^{80} +8.00000 q^{82} -2.00000 q^{83} -4.00000 q^{85} -4.00000 q^{86} +6.00000 q^{89} -1.00000 q^{92} +8.00000 q^{94} -8.00000 q^{95} -10.0000 q^{97} +9.00000 q^{98} +O(q^{100})$

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	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
$7$	−4.00000	−1.51186	−0.755929	−	0.654654i	$-0.772814\pi$
$7$	−4.00000	−1.51186	−0.755929	+	0.654654i	$0.772814\pi$
$8$	1.00000	0.353553
$9$	0	0
$10$	−1.00000	−0.316228
$11$	0	0		−	1.00000i	$-0.5\pi$
$11$	0	0			1.00000i	$0.5\pi$
$12$	0	0
$13$	0	0		−	1.00000i	$-0.5\pi$
$13$	0	0			1.00000i	$0.5\pi$
$14$	−4.00000	−1.06904
$15$	0	0
$16$	1.00000	0.250000
$17$	4.00000	0.970143	0.485071	−	0.874475i	$-0.338794\pi$
$17$	4.00000	0.970143	0.485071	+	0.874475i	$0.338794\pi$
$18$	0	0
$19$	8.00000	1.83533	0.917663	−	0.397360i	$-0.130073\pi$
$19$	8.00000	1.83533	0.917663	+	0.397360i	$0.130073\pi$
$20$	−1.00000	−0.223607
$21$	0	0
$22$	0	0
$23$	−1.00000	−0.208514
$23$	−1.00000	−0.208514
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	0	0
$28$	−4.00000	−0.755929
$29$	−2.00000	−0.371391	−0.185695	−	0.982607i	$-0.559454\pi$
$29$	−2.00000	−0.371391	−0.185695	+	0.982607i	$0.559454\pi$
$30$	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
$34$	4.00000	0.685994
$35$	4.00000	0.676123
$36$	0	0
$37$	−2.00000	−0.328798	−0.164399	−	0.986394i	$-0.552568\pi$
$37$	−2.00000	−0.328798	−0.164399	+	0.986394i	$0.552568\pi$
$38$	8.00000	1.29777
$39$	0	0
$40$	−1.00000	−0.158114
$41$	8.00000	1.24939	0.624695	−	0.780869i	$-0.285223\pi$
$41$	8.00000	1.24939	0.624695	+	0.780869i	$0.285223\pi$
$42$	0	0
$43$	−4.00000	−0.609994	−0.304997	−	0.952353i	$-0.598656\pi$
$43$	−4.00000	−0.609994	−0.304997	+	0.952353i	$0.598656\pi$
$44$	0	0
$45$	0	0
$46$	−1.00000	−0.147442
$47$	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
$49$	9.00000	1.28571
$50$	1.00000	0.141421
$51$	0	0
$52$	0	0
$53$	6.00000	0.824163	0.412082	−	0.911147i	$-0.364802\pi$
$53$	6.00000	0.824163	0.412082	+	0.911147i	$0.364802\pi$
$54$	0	0
$55$	0	0
$56$	−4.00000	−0.534522
$57$	0	0
$58$	−2.00000	−0.262613
$59$	10.0000	1.30189	0.650945	−	0.759125i	$-0.274373\pi$
$59$	10.0000	1.30189	0.650945	+	0.759125i	$0.274373\pi$
$60$	0	0
$61$	−8.00000	−1.02430	−0.512148	−	0.858898i	$-0.671150\pi$
$61$	−8.00000	−1.02430	−0.512148	+	0.858898i	$0.671150\pi$
$62$	4.00000	0.508001
$63$	0	0
$64$	1.00000	0.125000
$65$	0	0
$66$	0	0
$67$	12.0000	1.46603	0.733017	−	0.680211i	$-0.238112\pi$
$67$	12.0000	1.46603	0.733017	+	0.680211i	$0.238112\pi$
$68$	4.00000	0.485071
$69$	0	0
$70$	4.00000	0.478091
$71$	16.0000	1.89885	0.949425	−	0.313993i	$-0.101667\pi$
$71$	16.0000	1.89885	0.949425	+	0.313993i	$0.101667\pi$
$72$	0	0
$73$	−2.00000	−0.234082	−0.117041	−	0.993127i	$-0.537341\pi$
$73$	−2.00000	−0.234082	−0.117041	+	0.993127i	$0.537341\pi$
$74$	−2.00000	−0.232495
$75$	0	0
$76$	8.00000	0.917663
$77$	0	0
$78$	0	0
$79$	−6.00000	−0.675053	−0.337526	−	0.941316i	$-0.609590\pi$
$79$	−6.00000	−0.675053	−0.337526	+	0.941316i	$0.609590\pi$
$80$	−1.00000	−0.111803
$81$	0	0
$82$	8.00000	0.883452
$83$	−2.00000	−0.219529	−0.109764	−	0.993958i	$-0.535010\pi$
$83$	−2.00000	−0.219529	−0.109764	+	0.993958i	$0.535010\pi$
$84$	0	0
$85$	−4.00000	−0.433861
$86$	−4.00000	−0.431331
$87$	0	0
$88$	0	0
$89$	6.00000	0.635999	0.317999	−	0.948091i	$-0.396989\pi$
$89$	6.00000	0.635999	0.317999	+	0.948091i	$0.396989\pi$
$90$	0	0
$91$	0	0
$92$	−1.00000	−0.104257
$93$	0	0
$94$	8.00000	0.825137
$95$	−8.00000	−0.820783
$96$	0	0
$97$	−10.0000	−1.01535	−0.507673	−	0.861550i	$-0.669494\pi$
$97$	−10.0000	−1.01535	−0.507673	+	0.861550i	$0.669494\pi$
$98$	9.00000	0.909137
$99$	0	0
$100$	1.00000	0.100000
$101$	−2.00000	−0.199007	−0.0995037	−	0.995037i	$-0.531726\pi$
$101$	−2.00000	−0.199007	−0.0995037	+	0.995037i	$0.531726\pi$
$102$	0	0
$103$	−4.00000	−0.394132	−0.197066	−	0.980390i	$-0.563141\pi$
$103$	−4.00000	−0.394132	−0.197066	+	0.980390i	$0.563141\pi$
$104$	0	0
$105$	0	0
$106$	6.00000	0.582772
$107$	−2.00000	−0.193347	−0.0966736	−	0.995316i	$-0.530820\pi$
$107$	−2.00000	−0.193347	−0.0966736	+	0.995316i	$0.530820\pi$
$108$	0	0
$109$	−4.00000	−0.383131	−0.191565	−	0.981480i	$-0.561356\pi$
$109$	−4.00000	−0.383131	−0.191565	+	0.981480i	$0.561356\pi$
$110$	0	0
$111$	0	0
$112$	−4.00000	−0.377964
$113$	0	0		−	1.00000i	$-0.5\pi$
$113$	0	0			1.00000i	$0.5\pi$
$114$	0	0
$115$	1.00000	0.0932505
$116$	−2.00000	−0.185695
$117$	0	0
$118$	10.0000	0.920575
$119$	−16.0000	−1.46672
$120$	0	0
$121$	−11.0000	−1.00000
$122$	−8.00000	−0.724286
$123$	0	0
$124$	4.00000	0.359211
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	−10.0000	−0.887357	−0.443678	−	0.896186i	$-0.646327\pi$
$127$	−10.0000	−0.887357	−0.443678	+	0.896186i	$0.646327\pi$
$128$	1.00000	0.0883883
$129$	0	0
$130$	0	0
$131$	−6.00000	−0.524222	−0.262111	−	0.965038i	$-0.584419\pi$
$131$	−6.00000	−0.524222	−0.262111	+	0.965038i	$0.584419\pi$
$132$	0	0
$133$	−32.0000	−2.77475
$134$	12.0000	1.03664
$135$	0	0
$136$	4.00000	0.342997
$137$	12.0000	1.02523	0.512615	−	0.858619i	$-0.328677\pi$
$137$	12.0000	1.02523	0.512615	+	0.858619i	$0.328677\pi$
$138$	0	0
$139$	20.0000	1.69638	0.848189	−	0.529694i	$-0.177693\pi$
$139$	20.0000	1.69638	0.848189	+	0.529694i	$0.177693\pi$
$140$	4.00000	0.338062
$141$	0	0
$142$	16.0000	1.34269
$143$	0	0
$144$	0	0
$145$	2.00000	0.166091
$146$	−2.00000	−0.165521
$147$	0	0
$148$	−2.00000	−0.164399
$149$	−6.00000	−0.491539	−0.245770	−	0.969328i	$-0.579041\pi$
$149$	−6.00000	−0.491539	−0.245770	+	0.969328i	$0.579041\pi$
$150$	0	0
$151$	20.0000	1.62758	0.813788	−	0.581161i	$-0.197401\pi$
$151$	20.0000	1.62758	0.813788	+	0.581161i	$0.197401\pi$
$152$	8.00000	0.648886
$153$	0	0
$154$	0	0
$155$	−4.00000	−0.321288
$156$	0	0
$157$	10.0000	0.798087	0.399043	−	0.916932i	$-0.369342\pi$
$157$	10.0000	0.798087	0.399043	+	0.916932i	$0.369342\pi$
$158$	−6.00000	−0.477334
$159$	0	0
$160$	−1.00000	−0.0790569
$161$	4.00000	0.315244
$162$	0	0
$163$	−8.00000	−0.626608	−0.313304	−	0.949653i	$-0.601436\pi$
$163$	−8.00000	−0.626608	−0.313304	+	0.949653i	$0.601436\pi$
$164$	8.00000	0.624695
$165$	0	0
$166$	−2.00000	−0.155230
$167$	−16.0000	−1.23812	−0.619059	−	0.785345i	$-0.712486\pi$
$167$	−16.0000	−1.23812	−0.619059	+	0.785345i	$0.712486\pi$
$168$	0	0
$169$	−13.0000	−1.00000
$170$	−4.00000	−0.306786
$171$	0	0
$172$	−4.00000	−0.304997
$173$	2.00000	0.152057	0.0760286	−	0.997106i	$-0.475776\pi$
$173$	2.00000	0.152057	0.0760286	+	0.997106i	$0.475776\pi$
$174$	0	0
$175$	−4.00000	−0.302372
$176$	0	0
$177$	0	0
$178$	6.00000	0.449719
$179$	6.00000	0.448461	0.224231	−	0.974536i	$-0.428013\pi$
$179$	6.00000	0.448461	0.224231	+	0.974536i	$0.428013\pi$
$180$	0	0
$181$	−4.00000	−0.297318	−0.148659	−	0.988889i	$-0.547496\pi$
$181$	−4.00000	−0.297318	−0.148659	+	0.988889i	$0.547496\pi$
$182$	0	0
$183$	0	0
$184$	−1.00000	−0.0737210
$185$	2.00000	0.147043
$186$	0	0
$187$	0	0
$188$	8.00000	0.583460
$189$	0	0
$190$	−8.00000	−0.580381
$191$	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
$193$	−18.0000	−1.29567	−0.647834	−	0.761781i	$-0.724325\pi$
$193$	−18.0000	−1.29567	−0.647834	+	0.761781i	$0.724325\pi$
$194$	−10.0000	−0.717958
$195$	0	0
$196$	9.00000	0.642857
$197$	2.00000	0.142494	0.0712470	−	0.997459i	$-0.477302\pi$
$197$	2.00000	0.142494	0.0712470	+	0.997459i	$0.477302\pi$
$198$	0	0
$199$	−18.0000	−1.27599	−0.637993	−	0.770042i	$-0.720235\pi$
$199$	−18.0000	−1.27599	−0.637993	+	0.770042i	$0.720235\pi$
$200$	1.00000	0.0707107
$201$	0	0
$202$	−2.00000	−0.140720
$203$	8.00000	0.561490
$204$	0	0
$205$	−8.00000	−0.558744
$206$	−4.00000	−0.278693
$207$	0	0
$208$	0	0
$209$	0	0
$210$	0	0
$211$	20.0000	1.37686	0.688428	−	0.725304i	$-0.258301\pi$
$211$	20.0000	1.37686	0.688428	+	0.725304i	$0.258301\pi$
$212$	6.00000	0.412082
$213$	0	0
$214$	−2.00000	−0.136717
$215$	4.00000	0.272798
$216$	0	0
$217$	−16.0000	−1.08615
$218$	−4.00000	−0.270914
$219$	0	0
$220$	0	0
$221$	0	0
$222$	0	0
$223$	6.00000	0.401790	0.200895	−	0.979613i	$-0.435615\pi$
$223$	6.00000	0.401790	0.200895	+	0.979613i	$0.435615\pi$
$224$	−4.00000	−0.267261
$225$	0	0
$226$	0	0
$227$	−22.0000	−1.46019	−0.730096	−	0.683345i	$-0.760525\pi$
$227$	−22.0000	−1.46019	−0.730096	+	0.683345i	$0.760525\pi$
$228$	0	0
$229$	−24.0000	−1.58596	−0.792982	−	0.609245i	$-0.791473\pi$
$229$	−24.0000	−1.58596	−0.792982	+	0.609245i	$0.791473\pi$
$230$	1.00000	0.0659380
$231$	0	0
$232$	−2.00000	−0.131306
$233$	2.00000	0.131024	0.0655122	−	0.997852i	$-0.479132\pi$
$233$	2.00000	0.131024	0.0655122	+	0.997852i	$0.479132\pi$
$234$	0	0
$235$	−8.00000	−0.521862
$236$	10.0000	0.650945
$237$	0	0
$238$	−16.0000	−1.03713
$239$	−12.0000	−0.776215	−0.388108	−	0.921614i	$-0.626871\pi$
$239$	−12.0000	−0.776215	−0.388108	+	0.921614i	$0.626871\pi$
$240$	0	0
$241$	2.00000	0.128831	0.0644157	−	0.997923i	$-0.479482\pi$
$241$	2.00000	0.128831	0.0644157	+	0.997923i	$0.479482\pi$
$242$	−11.0000	−0.707107
$243$	0	0
$244$	−8.00000	−0.512148
$245$	−9.00000	−0.574989
$246$	0	0
$247$	0	0
$248$	4.00000	0.254000
$249$	0	0
$250$	−1.00000	−0.0632456
$251$	20.0000	1.26239	0.631194	−	0.775625i	$-0.282565\pi$
$251$	20.0000	1.26239	0.631194	+	0.775625i	$0.282565\pi$
$252$	0	0
$253$	0	0
$254$	−10.0000	−0.627456
$255$	0	0
$256$	1.00000	0.0625000
$257$	22.0000	1.37232	0.686161	−	0.727450i	$-0.259294\pi$
$257$	22.0000	1.37232	0.686161	+	0.727450i	$0.259294\pi$
$258$	0	0
$259$	8.00000	0.497096
$260$	0	0
$261$	0	0
$262$	−6.00000	−0.370681
$263$	4.00000	0.246651	0.123325	−	0.992366i	$-0.460644\pi$
$263$	4.00000	0.246651	0.123325	+	0.992366i	$0.460644\pi$
$264$	0	0
$265$	−6.00000	−0.368577
$266$	−32.0000	−1.96205
$267$	0	0
$268$	12.0000	0.733017
$269$	−18.0000	−1.09748	−0.548740	−	0.835993i	$-0.684892\pi$
$269$	−18.0000	−1.09748	−0.548740	+	0.835993i	$0.684892\pi$
$270$	0	0
$271$	−20.0000	−1.21491	−0.607457	−	0.794353i	$-0.707810\pi$
$271$	−20.0000	−1.21491	−0.607457	+	0.794353i	$0.707810\pi$
$272$	4.00000	0.242536
$273$	0	0
$274$	12.0000	0.724947
$275$	0	0
$276$	0	0
$277$	−8.00000	−0.480673	−0.240337	−	0.970690i	$-0.577258\pi$
$277$	−8.00000	−0.480673	−0.240337	+	0.970690i	$0.577258\pi$
$278$	20.0000	1.19952
$279$	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
$283$	12.0000	0.713326	0.356663	−	0.934233i	$-0.383914\pi$
$283$	12.0000	0.713326	0.356663	+	0.934233i	$0.383914\pi$
$284$	16.0000	0.949425
$285$	0	0
$286$	0	0
$287$	−32.0000	−1.88890
$288$	0	0
$289$	−1.00000	−0.0588235
$290$	2.00000	0.117444
$291$	0	0
$292$	−2.00000	−0.117041
$293$	6.00000	0.350524	0.175262	−	0.984522i	$-0.443923\pi$
$293$	6.00000	0.350524	0.175262	+	0.984522i	$0.443923\pi$
$294$	0	0
$295$	−10.0000	−0.582223
$296$	−2.00000	−0.116248
$297$	0	0
$298$	−6.00000	−0.347571
$299$	0	0
$300$	0	0
$301$	16.0000	0.922225
$302$	20.0000	1.15087
$303$	0	0
$304$	8.00000	0.458831
$305$	8.00000	0.458079
$306$	0	0
$307$	12.0000	0.684876	0.342438	−	0.939540i	$-0.388747\pi$
$307$	12.0000	0.684876	0.342438	+	0.939540i	$0.388747\pi$
$308$	0	0
$309$	0	0
$310$	−4.00000	−0.227185
$311$	32.0000	1.81455	0.907277	−	0.420534i	$-0.138157\pi$
$311$	32.0000	1.81455	0.907277	+	0.420534i	$0.138157\pi$
$312$	0	0
$313$	−14.0000	−0.791327	−0.395663	−	0.918396i	$-0.629485\pi$
$313$	−14.0000	−0.791327	−0.395663	+	0.918396i	$0.629485\pi$
$314$	10.0000	0.564333
$315$	0	0
$316$	−6.00000	−0.337526
$317$	30.0000	1.68497	0.842484	−	0.538721i	$-0.181092\pi$
$317$	30.0000	1.68497	0.842484	+	0.538721i	$0.181092\pi$
$318$	0	0
$319$	0	0
$320$	−1.00000	−0.0559017
$321$	0	0
$322$	4.00000	0.222911
$323$	32.0000	1.78053
$324$	0	0
$325$	0	0
$326$	−8.00000	−0.443079
$327$	0	0
$328$	8.00000	0.441726
$329$	−32.0000	−1.76422
$330$	0	0
$331$	28.0000	1.53902	0.769510	−	0.638635i	$-0.220501\pi$
$331$	28.0000	1.53902	0.769510	+	0.638635i	$0.220501\pi$
$332$	−2.00000	−0.109764
$333$	0	0
$334$	−16.0000	−0.875481
$335$	−12.0000	−0.655630
$336$	0	0
$337$	18.0000	0.980522	0.490261	−	0.871576i	$-0.336901\pi$
$337$	18.0000	0.980522	0.490261	+	0.871576i	$0.336901\pi$
$338$	−13.0000	−0.707107
$339$	0	0
$340$	−4.00000	−0.216930
$341$	0	0
$342$	0	0
$343$	−8.00000	−0.431959
$344$	−4.00000	−0.215666
$345$	0	0
$346$	2.00000	0.107521
$347$	−28.0000	−1.50312	−0.751559	−	0.659665i	$-0.770698\pi$
$347$	−28.0000	−1.50312	−0.751559	+	0.659665i	$0.770698\pi$
$348$	0	0
$349$	14.0000	0.749403	0.374701	−	0.927146i	$-0.377745\pi$
$349$	14.0000	0.749403	0.374701	+	0.927146i	$0.377745\pi$
$350$	−4.00000	−0.213809
$351$	0	0
$352$	0	0
$353$	−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
$355$	−16.0000	−0.849192
$356$	6.00000	0.317999
$357$	0	0
$358$	6.00000	0.317110
$359$	−32.0000	−1.68890	−0.844448	−	0.535638i	$-0.820071\pi$
$359$	−32.0000	−1.68890	−0.844448	+	0.535638i	$0.820071\pi$
$360$	0	0
$361$	45.0000	2.36842
$362$	−4.00000	−0.210235
$363$	0	0
$364$	0	0
$365$	2.00000	0.104685
$366$	0	0
$367$	−32.0000	−1.67039	−0.835193	−	0.549957i	$-0.814644\pi$
$367$	−32.0000	−1.67039	−0.835193	+	0.549957i	$0.814644\pi$
$368$	−1.00000	−0.0521286
$369$	0	0
$370$	2.00000	0.103975
$371$	−24.0000	−1.24602
$372$	0	0
$373$	−6.00000	−0.310668	−0.155334	−	0.987862i	$-0.549645\pi$
$373$	−6.00000	−0.310668	−0.155334	+	0.987862i	$0.549645\pi$
$374$	0	0
$375$	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	$0.532759\pi$
$380$	−8.00000	−0.410391
$381$	0	0
$382$	24.0000	1.22795
$383$	0	0		−	1.00000i	$-0.5\pi$
$383$	0	0			1.00000i	$0.5\pi$
$384$	0	0
$385$	0	0
$386$	−18.0000	−0.916176
$387$	0	0
$388$	−10.0000	−0.507673
$389$	−18.0000	−0.912636	−0.456318	−	0.889817i	$-0.650832\pi$
$389$	−18.0000	−0.912636	−0.456318	+	0.889817i	$0.650832\pi$
$390$	0	0
$391$	−4.00000	−0.202289
$392$	9.00000	0.454569
$393$	0	0
$394$	2.00000	0.100759
$395$	6.00000	0.301893
$396$	0	0
$397$	36.0000	1.80679	0.903394	−	0.428811i	$-0.141067\pi$
$397$	36.0000	1.80679	0.903394	+	0.428811i	$0.141067\pi$
$398$	−18.0000	−0.902258
$399$	0	0
$400$	1.00000	0.0500000
$401$	−2.00000	−0.0998752	−0.0499376	−	0.998752i	$-0.515902\pi$
$401$	−2.00000	−0.0998752	−0.0499376	+	0.998752i	$0.515902\pi$
$402$	0	0
$403$	0	0
$404$	−2.00000	−0.0995037
$405$	0	0
$406$	8.00000	0.397033
$407$	0	0
$408$	0	0
$409$	−18.0000	−0.890043	−0.445021	−	0.895520i	$-0.646804\pi$
$409$	−18.0000	−0.890043	−0.445021	+	0.895520i	$0.646804\pi$
$410$	−8.00000	−0.395092
$411$	0	0
$412$	−4.00000	−0.197066
$413$	−40.0000	−1.96827
$414$	0	0
$415$	2.00000	0.0981761
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−24.0000	−1.17248	−0.586238	−	0.810139i	$-0.699392\pi$
$419$	−24.0000	−1.17248	−0.586238	+	0.810139i	$0.699392\pi$
$420$	0	0
$421$	4.00000	0.194948	0.0974740	−	0.995238i	$-0.468924\pi$
$421$	4.00000	0.194948	0.0974740	+	0.995238i	$0.468924\pi$
$422$	20.0000	0.973585
$423$	0	0
$424$	6.00000	0.291386
$425$	4.00000	0.194029
$426$	0	0
$427$	32.0000	1.54859
$428$	−2.00000	−0.0966736
$429$	0	0
$430$	4.00000	0.192897
$431$	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
$433$	−2.00000	−0.0961139	−0.0480569	−	0.998845i	$-0.515303\pi$
$433$	−2.00000	−0.0961139	−0.0480569	+	0.998845i	$0.515303\pi$
$434$	−16.0000	−0.768025
$435$	0	0
$436$	−4.00000	−0.191565
$437$	−8.00000	−0.382692
$438$	0	0
$439$	0	0		−	1.00000i	$-0.5\pi$
$439$	0	0			1.00000i	$0.5\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−24.0000	−1.14027	−0.570137	−	0.821549i	$-0.693110\pi$
$443$	−24.0000	−1.14027	−0.570137	+	0.821549i	$0.693110\pi$
$444$	0	0
$445$	−6.00000	−0.284427
$446$	6.00000	0.284108
$447$	0	0
$448$	−4.00000	−0.188982
$449$	20.0000	0.943858	0.471929	−	0.881636i	$-0.343558\pi$
$449$	20.0000	0.943858	0.471929	+	0.881636i	$0.343558\pi$
$450$	0	0
$451$	0	0
$452$	0	0
$453$	0	0
$454$	−22.0000	−1.03251
$455$	0	0
$456$	0	0
$457$	10.0000	0.467780	0.233890	−	0.972263i	$-0.424854\pi$
$457$	10.0000	0.467780	0.233890	+	0.972263i	$0.424854\pi$
$458$	−24.0000	−1.12145
$459$	0	0
$460$	1.00000	0.0466252
$461$	−30.0000	−1.39724	−0.698620	−	0.715493i	$-0.746202\pi$
$461$	−30.0000	−1.39724	−0.698620	+	0.715493i	$0.746202\pi$
$462$	0	0
$463$	−26.0000	−1.20832	−0.604161	−	0.796862i	$-0.706492\pi$
$463$	−26.0000	−1.20832	−0.604161	+	0.796862i	$0.706492\pi$
$464$	−2.00000	−0.0928477
$465$	0	0
$466$	2.00000	0.0926482
$467$	2.00000	0.0925490	0.0462745	−	0.998929i	$-0.485265\pi$
$467$	2.00000	0.0925490	0.0462745	+	0.998929i	$0.485265\pi$
$468$	0	0
$469$	−48.0000	−2.21643
$470$	−8.00000	−0.369012
$471$	0	0
$472$	10.0000	0.460287
$473$	0	0
$474$	0	0
$475$	8.00000	0.367065
$476$	−16.0000	−0.733359
$477$	0	0
$478$	−12.0000	−0.548867
$479$	8.00000	0.365529	0.182765	−	0.983157i	$-0.441495\pi$
$479$	8.00000	0.365529	0.182765	+	0.983157i	$0.441495\pi$
$480$	0	0
$481$	0	0
$482$	2.00000	0.0910975
$483$	0	0
$484$	−11.0000	−0.500000
$485$	10.0000	0.454077
$486$	0	0
$487$	30.0000	1.35943	0.679715	−	0.733476i	$-0.262104\pi$
$487$	30.0000	1.35943	0.679715	+	0.733476i	$0.262104\pi$
$488$	−8.00000	−0.362143
$489$	0	0
$490$	−9.00000	−0.406579
$491$	−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
$493$	−8.00000	−0.360302
$494$	0	0
$495$	0	0
$496$	4.00000	0.179605
$497$	−64.0000	−2.87079
$498$	0	0
$499$	36.0000	1.61158	0.805791	−	0.592200i	$-0.201741\pi$
$499$	36.0000	1.61158	0.805791	+	0.592200i	$0.201741\pi$
$500$	−1.00000	−0.0447214
$501$	0	0
$502$	20.0000	0.892644
$503$	24.0000	1.07011	0.535054	−	0.844818i	$-0.320291\pi$
$503$	24.0000	1.07011	0.535054	+	0.844818i	$0.320291\pi$
$504$	0	0
$505$	2.00000	0.0889988
$506$	0	0
$507$	0	0
$508$	−10.0000	−0.443678
$509$	6.00000	0.265945	0.132973	−	0.991120i	$-0.457548\pi$
$509$	6.00000	0.265945	0.132973	+	0.991120i	$0.457548\pi$
$510$	0	0
$511$	8.00000	0.353899
$512$	1.00000	0.0441942
$513$	0	0
$514$	22.0000	0.970378
$515$	4.00000	0.176261
$516$	0	0
$517$	0	0
$518$	8.00000	0.351500
$519$	0	0
$520$	0	0
$521$	−38.0000	−1.66481	−0.832405	−	0.554168i	$-0.813037\pi$
$521$	−38.0000	−1.66481	−0.832405	+	0.554168i	$0.813037\pi$
$522$	0	0
$523$	−28.0000	−1.22435	−0.612177	−	0.790721i	$-0.709706\pi$
$523$	−28.0000	−1.22435	−0.612177	+	0.790721i	$0.709706\pi$
$524$	−6.00000	−0.262111
$525$	0	0
$526$	4.00000	0.174408
$527$	16.0000	0.696971
$528$	0	0
$529$	1.00000	0.0434783
$530$	−6.00000	−0.260623
$531$	0	0
$532$	−32.0000	−1.38738
$533$	0	0
$534$	0	0
$535$	2.00000	0.0864675
$536$	12.0000	0.518321
$537$	0	0
$538$	−18.0000	−0.776035
$539$	0	0
$540$	0	0
$541$	22.0000	0.945854	0.472927	−	0.881102i	$-0.343197\pi$
$541$	22.0000	0.945854	0.472927	+	0.881102i	$0.343197\pi$
$542$	−20.0000	−0.859074
$543$	0	0
$544$	4.00000	0.171499
$545$	4.00000	0.171341
$546$	0	0
$547$	8.00000	0.342055	0.171028	−	0.985266i	$-0.445291\pi$
$547$	8.00000	0.342055	0.171028	+	0.985266i	$0.445291\pi$
$548$	12.0000	0.512615
$549$	0	0
$550$	0	0
$551$	−16.0000	−0.681623
$552$	0	0
$553$	24.0000	1.02058
$554$	−8.00000	−0.339887
$555$	0	0
$556$	20.0000	0.848189
$557$	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
$563$	34.0000	1.43293	0.716465	−	0.697623i	$-0.245759\pi$
$563$	34.0000	1.43293	0.716465	+	0.697623i	$0.245759\pi$
$564$	0	0
$565$	0	0
$566$	12.0000	0.504398
$567$	0	0
$568$	16.0000	0.671345
$569$	6.00000	0.251533	0.125767	−	0.992060i	$-0.459861\pi$
$569$	6.00000	0.251533	0.125767	+	0.992060i	$0.459861\pi$
$570$	0	0
$571$	4.00000	0.167395	0.0836974	−	0.996491i	$-0.473327\pi$
$571$	4.00000	0.167395	0.0836974	+	0.996491i	$0.473327\pi$
$572$	0	0
$573$	0	0
$574$	−32.0000	−1.33565
$575$	−1.00000	−0.0417029
$576$	0	0
$577$	−14.0000	−0.582828	−0.291414	−	0.956597i	$-0.594126\pi$
$577$	−14.0000	−0.582828	−0.291414	+	0.956597i	$0.594126\pi$
$578$	−1.00000	−0.0415945
$579$	0	0
$580$	2.00000	0.0830455
$581$	8.00000	0.331896
$582$	0	0
$583$	0	0
$584$	−2.00000	−0.0827606
$585$	0	0
$586$	6.00000	0.247858
$587$	−32.0000	−1.32078	−0.660391	−	0.750922i	$-0.729609\pi$
$587$	−32.0000	−1.32078	−0.660391	+	0.750922i	$0.729609\pi$
$588$	0	0
$589$	32.0000	1.31854
$590$	−10.0000	−0.411693
$591$	0	0
$592$	−2.00000	−0.0821995
$593$	−10.0000	−0.410651	−0.205325	−	0.978694i	$-0.565825\pi$
$593$	−10.0000	−0.410651	−0.205325	+	0.978694i	$0.565825\pi$
$594$	0	0
$595$	16.0000	0.655936
$596$	−6.00000	−0.245770
$597$	0	0
$598$	0	0
$599$	40.0000	1.63436	0.817178	−	0.576386i	$-0.195537\pi$
$599$	40.0000	1.63436	0.817178	+	0.576386i	$0.195537\pi$
$600$	0	0
$601$	−2.00000	−0.0815817	−0.0407909	−	0.999168i	$-0.512988\pi$
$601$	−2.00000	−0.0815817	−0.0407909	+	0.999168i	$0.512988\pi$
$602$	16.0000	0.652111
$603$	0	0
$604$	20.0000	0.813788
$605$	11.0000	0.447214
$606$	0	0
$607$	26.0000	1.05531	0.527654	−	0.849460i	$-0.323072\pi$
$607$	26.0000	1.05531	0.527654	+	0.849460i	$0.323072\pi$
$608$	8.00000	0.324443
$609$	0	0
$610$	8.00000	0.323911
$611$	0	0
$612$	0	0
$613$	−46.0000	−1.85792	−0.928961	−	0.370177i	$-0.879297\pi$
$613$	−46.0000	−1.85792	−0.928961	+	0.370177i	$0.879297\pi$
$614$	12.0000	0.484281
$615$	0	0
$616$	0	0
$617$	−36.0000	−1.44931	−0.724653	−	0.689114i	$-0.758000\pi$
$617$	−36.0000	−1.44931	−0.724653	+	0.689114i	$0.758000\pi$
$618$	0	0
$619$	−20.0000	−0.803868	−0.401934	−	0.915669i	$-0.631662\pi$
$619$	−20.0000	−0.803868	−0.401934	+	0.915669i	$0.631662\pi$
$620$	−4.00000	−0.160644
$621$	0	0
$622$	32.0000	1.28308
$623$	−24.0000	−0.961540
$624$	0	0
$625$	1.00000	0.0400000
$626$	−14.0000	−0.559553
$627$	0	0
$628$	10.0000	0.399043
$629$	−8.00000	−0.318981
$630$	0	0
$631$	−18.0000	−0.716569	−0.358284	−	0.933613i	$-0.616638\pi$
$631$	−18.0000	−0.716569	−0.358284	+	0.933613i	$0.616638\pi$
$632$	−6.00000	−0.238667
$633$	0	0
$634$	30.0000	1.19145
$635$	10.0000	0.396838
$636$	0	0
$637$	0	0
$638$	0	0
$639$	0	0
$640$	−1.00000	−0.0395285
$641$	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
$643$	−20.0000	−0.788723	−0.394362	−	0.918955i	$-0.629034\pi$
$643$	−20.0000	−0.788723	−0.394362	+	0.918955i	$0.629034\pi$
$644$	4.00000	0.157622
$645$	0	0
$646$	32.0000	1.25902
$647$	8.00000	0.314512	0.157256	−	0.987558i	$-0.449735\pi$
$647$	8.00000	0.314512	0.157256	+	0.987558i	$0.449735\pi$
$648$	0	0
$649$	0	0
$650$	0	0
$651$	0	0
$652$	−8.00000	−0.313304
$653$	−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
$656$	8.00000	0.312348
$657$	0	0
$658$	−32.0000	−1.24749
$659$	20.0000	0.779089	0.389545	−	0.921008i	$-0.372632\pi$
$659$	20.0000	0.779089	0.389545	+	0.921008i	$0.372632\pi$
$660$	0	0
$661$	−48.0000	−1.86698	−0.933492	−	0.358599i	$-0.883255\pi$
$661$	−48.0000	−1.86698	−0.933492	+	0.358599i	$0.883255\pi$
$662$	28.0000	1.08825
$663$	0	0
$664$	−2.00000	−0.0776151
$665$	32.0000	1.24091
$666$	0	0
$667$	2.00000	0.0774403
$668$	−16.0000	−0.619059
$669$	0	0
$670$	−12.0000	−0.463600
$671$	0	0
$672$	0	0
$673$	14.0000	0.539660	0.269830	−	0.962908i	$-0.413032\pi$
$673$	14.0000	0.539660	0.269830	+	0.962908i	$0.413032\pi$
$674$	18.0000	0.693334
$675$	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
$679$	40.0000	1.53506
$680$	−4.00000	−0.153393
$681$	0	0
$682$	0	0
$683$	−8.00000	−0.306111	−0.153056	−	0.988218i	$-0.548911\pi$
$683$	−8.00000	−0.306111	−0.153056	+	0.988218i	$0.548911\pi$
$684$	0	0
$685$	−12.0000	−0.458496
$686$	−8.00000	−0.305441
$687$	0	0
$688$	−4.00000	−0.152499
$689$	0	0
$690$	0	0
$691$	20.0000	0.760836	0.380418	−	0.924815i	$-0.375780\pi$
$691$	20.0000	0.760836	0.380418	+	0.924815i	$0.375780\pi$
$692$	2.00000	0.0760286
$693$	0	0
$694$	−28.0000	−1.06287
$695$	−20.0000	−0.758643
$696$	0	0
$697$	32.0000	1.21209
$698$	14.0000	0.529908
$699$	0	0
$700$	−4.00000	−0.151186
$701$	46.0000	1.73740	0.868698	−	0.495342i	$-0.164957\pi$
$701$	46.0000	1.73740	0.868698	+	0.495342i	$0.164957\pi$
$702$	0	0
$703$	−16.0000	−0.603451
$704$	0	0
$705$	0	0
$706$	−18.0000	−0.677439
$707$	8.00000	0.300871
$708$	0	0
$709$	−20.0000	−0.751116	−0.375558	−	0.926799i	$-0.622549\pi$
$709$	−20.0000	−0.751116	−0.375558	+	0.926799i	$0.622549\pi$
$710$	−16.0000	−0.600469
$711$	0	0
$712$	6.00000	0.224860
$713$	−4.00000	−0.149801
$714$	0	0
$715$	0	0
$716$	6.00000	0.224231
$717$	0	0
$718$	−32.0000	−1.19423
$719$	−28.0000	−1.04422	−0.522112	−	0.852877i	$-0.674856\pi$
$719$	−28.0000	−1.04422	−0.522112	+	0.852877i	$0.674856\pi$
$720$	0	0
$721$	16.0000	0.595871
$722$	45.0000	1.67473
$723$	0	0
$724$	−4.00000	−0.148659
$725$	−2.00000	−0.0742781
$726$	0	0
$727$	−32.0000	−1.18681	−0.593407	−	0.804902i	$-0.702218\pi$
$727$	−32.0000	−1.18681	−0.593407	+	0.804902i	$0.702218\pi$
$728$	0	0
$729$	0	0
$730$	2.00000	0.0740233
$731$	−16.0000	−0.591781
$732$	0	0
$733$	−14.0000	−0.517102	−0.258551	−	0.965998i	$-0.583245\pi$
$733$	−14.0000	−0.517102	−0.258551	+	0.965998i	$0.583245\pi$
$734$	−32.0000	−1.18114
$735$	0	0
$736$	−1.00000	−0.0368605
$737$	0	0
$738$	0	0
$739$	−4.00000	−0.147142	−0.0735712	−	0.997290i	$-0.523440\pi$
$739$	−4.00000	−0.147142	−0.0735712	+	0.997290i	$0.523440\pi$
$740$	2.00000	0.0735215
$741$	0	0
$742$	−24.0000	−0.881068
$743$	16.0000	0.586983	0.293492	−	0.955962i	$-0.405183\pi$
$743$	16.0000	0.586983	0.293492	+	0.955962i	$0.405183\pi$
$744$	0	0
$745$	6.00000	0.219823
$746$	−6.00000	−0.219676
$747$	0	0
$748$	0	0
$749$	8.00000	0.292314
$750$	0	0
$751$	42.0000	1.53260	0.766301	−	0.642482i	$-0.222095\pi$
$751$	42.0000	1.53260	0.766301	+	0.642482i	$0.222095\pi$
$752$	8.00000	0.291730
$753$	0	0
$754$	0	0
$755$	−20.0000	−0.727875
$756$	0	0
$757$	−10.0000	−0.363456	−0.181728	−	0.983349i	$-0.558169\pi$
$757$	−10.0000	−0.363456	−0.181728	+	0.983349i	$0.558169\pi$
$758$	−4.00000	−0.145287
$759$	0	0
$760$	−8.00000	−0.290191
$761$	48.0000	1.74000	0.869999	−	0.493053i	$-0.164119\pi$
$761$	48.0000	1.74000	0.869999	+	0.493053i	$0.164119\pi$
$762$	0	0
$763$	16.0000	0.579239
$764$	24.0000	0.868290
$765$	0	0
$766$	0	0
$767$	0	0
$768$	0	0
$769$	54.0000	1.94729	0.973645	−	0.228069i	$-0.0732413\pi$
$769$	54.0000	1.94729	0.973645	+	0.228069i	$0.0732413\pi$
$770$	0	0
$771$	0	0
$772$	−18.0000	−0.647834
$773$	−34.0000	−1.22290	−0.611448	−	0.791285i	$-0.709412\pi$
$773$	−34.0000	−1.22290	−0.611448	+	0.791285i	$0.709412\pi$
$774$	0	0
$775$	4.00000	0.143684
$776$	−10.0000	−0.358979
$777$	0	0
$778$	−18.0000	−0.645331
$779$	64.0000	2.29304
$780$	0	0
$781$	0	0
$782$	−4.00000	−0.143040
$783$	0	0
$784$	9.00000	0.321429
$785$	−10.0000	−0.356915
$786$	0	0
$787$	−20.0000	−0.712923	−0.356462	−	0.934310i	$-0.616017\pi$
$787$	−20.0000	−0.712923	−0.356462	+	0.934310i	$0.616017\pi$
$788$	2.00000	0.0712470
$789$	0	0
$790$	6.00000	0.213470
$791$	0	0
$792$	0	0
$793$	0	0
$794$	36.0000	1.27759
$795$	0	0
$796$	−18.0000	−0.637993
$797$	−30.0000	−1.06265	−0.531327	−	0.847167i	$-0.678307\pi$
$797$	−30.0000	−1.06265	−0.531327	+	0.847167i	$0.678307\pi$
$798$	0	0
$799$	32.0000	1.13208
$800$	1.00000	0.0353553
$801$	0	0
$802$	−2.00000	−0.0706225
$803$	0	0
$804$	0	0
$805$	−4.00000	−0.140981
$806$	0	0
$807$	0	0
$808$	−2.00000	−0.0703598
$809$	28.0000	0.984428	0.492214	−	0.870474i	$-0.336188\pi$
$809$	28.0000	0.984428	0.492214	+	0.870474i	$0.336188\pi$
$810$	0	0
$811$	4.00000	0.140459	0.0702295	−	0.997531i	$-0.477627\pi$
$811$	4.00000	0.140459	0.0702295	+	0.997531i	$0.477627\pi$
$812$	8.00000	0.280745
$813$	0	0
$814$	0	0
$815$	8.00000	0.280228
$816$	0	0
$817$	−32.0000	−1.11954
$818$	−18.0000	−0.629355
$819$	0	0
$820$	−8.00000	−0.279372
$821$	−50.0000	−1.74501	−0.872506	−	0.488603i	$-0.837507\pi$
$821$	−50.0000	−1.74501	−0.872506	+	0.488603i	$0.837507\pi$
$822$	0	0
$823$	38.0000	1.32460	0.662298	−	0.749240i	$-0.269581\pi$
$823$	38.0000	1.32460	0.662298	+	0.749240i	$0.269581\pi$
$824$	−4.00000	−0.139347
$825$	0	0
$826$	−40.0000	−1.39178
$827$	−34.0000	−1.18230	−0.591148	−	0.806563i	$-0.701325\pi$
$827$	−34.0000	−1.18230	−0.591148	+	0.806563i	$0.701325\pi$
$828$	0	0
$829$	30.0000	1.04194	0.520972	−	0.853574i	$-0.325570\pi$
$829$	30.0000	1.04194	0.520972	+	0.853574i	$0.325570\pi$
$830$	2.00000	0.0694210
$831$	0	0
$832$	0	0
$833$	36.0000	1.24733
$834$	0	0
$835$	16.0000	0.553703
$836$	0	0
$837$	0	0
$838$	−24.0000	−0.829066
$839$	−16.0000	−0.552381	−0.276191	−	0.961103i	$-0.589072\pi$
$839$	−16.0000	−0.552381	−0.276191	+	0.961103i	$0.589072\pi$
$840$	0	0
$841$	−25.0000	−0.862069
$842$	4.00000	0.137849
$843$	0	0
$844$	20.0000	0.688428
$845$	13.0000	0.447214
$846$	0	0
$847$	44.0000	1.51186
$848$	6.00000	0.206041
$849$	0	0
$850$	4.00000	0.137199
$851$	2.00000	0.0685591
$852$	0	0
$853$	12.0000	0.410872	0.205436	−	0.978671i	$-0.434139\pi$
$853$	12.0000	0.410872	0.205436	+	0.978671i	$0.434139\pi$
$854$	32.0000	1.09502
$855$	0	0
$856$	−2.00000	−0.0683586
$857$	42.0000	1.43469	0.717346	−	0.696717i	$-0.245357\pi$
$857$	42.0000	1.43469	0.717346	+	0.696717i	$0.245357\pi$
$858$	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
$862$	8.00000	0.272481
$863$	48.0000	1.63394	0.816970	−	0.576681i	$-0.195652\pi$
$863$	48.0000	1.63394	0.816970	+	0.576681i	$0.195652\pi$
$864$	0	0
$865$	−2.00000	−0.0680020
$866$	−2.00000	−0.0679628
$867$	0	0
$868$	−16.0000	−0.543075
$869$	0	0
$870$	0	0
$871$	0	0
$872$	−4.00000	−0.135457
$873$	0	0
$874$	−8.00000	−0.270604
$875$	4.00000	0.135225
$876$	0	0
$877$	−44.0000	−1.48577	−0.742887	−	0.669417i	$-0.766544\pi$
$877$	−44.0000	−1.48577	−0.742887	+	0.669417i	$0.766544\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−18.0000	−0.606435	−0.303218	−	0.952921i	$-0.598061\pi$
$881$	−18.0000	−0.606435	−0.303218	+	0.952921i	$0.598061\pi$
$882$	0	0
$883$	−20.0000	−0.673054	−0.336527	−	0.941674i	$-0.609252\pi$
$883$	−20.0000	−0.673054	−0.336527	+	0.941674i	$0.609252\pi$
$884$	0	0
$885$	0	0
$886$	−24.0000	−0.806296
$887$	−48.0000	−1.61168	−0.805841	−	0.592132i	$-0.798286\pi$
$887$	−48.0000	−1.61168	−0.805841	+	0.592132i	$0.798286\pi$
$888$	0	0
$889$	40.0000	1.34156
$890$	−6.00000	−0.201120
$891$	0	0
$892$	6.00000	0.200895
$893$	64.0000	2.14168
$894$	0	0
$895$	−6.00000	−0.200558
$896$	−4.00000	−0.133631
$897$	0	0
$898$	20.0000	0.667409
$899$	−8.00000	−0.266815
$900$	0	0
$901$	24.0000	0.799556
$902$	0	0
$903$	0	0
$904$	0	0
$905$	4.00000	0.132964
$906$	0	0
$907$	28.0000	0.929725	0.464862	−	0.885383i	$-0.346104\pi$
$907$	28.0000	0.929725	0.464862	+	0.885383i	$0.346104\pi$
$908$	−22.0000	−0.730096
$909$	0	0
$910$	0	0
$911$	8.00000	0.265052	0.132526	−	0.991180i	$-0.457691\pi$
$911$	8.00000	0.265052	0.132526	+	0.991180i	$0.457691\pi$
$912$	0	0
$913$	0	0
$914$	10.0000	0.330771
$915$	0	0
$916$	−24.0000	−0.792982
$917$	24.0000	0.792550
$918$	0	0
$919$	34.0000	1.12156	0.560778	−	0.827966i	$-0.310502\pi$
$919$	34.0000	1.12156	0.560778	+	0.827966i	$0.310502\pi$
$920$	1.00000	0.0329690
$921$	0	0
$922$	−30.0000	−0.987997
$923$	0	0
$924$	0	0
$925$	−2.00000	−0.0657596
$926$	−26.0000	−0.854413
$927$	0	0
$928$	−2.00000	−0.0656532
$929$	−52.0000	−1.70606	−0.853032	−	0.521858i	$-0.825239\pi$
$929$	−52.0000	−1.70606	−0.853032	+	0.521858i	$0.825239\pi$
$930$	0	0
$931$	72.0000	2.35970
$932$	2.00000	0.0655122
$933$	0	0
$934$	2.00000	0.0654420
$935$	0	0
$936$	0	0
$937$	42.0000	1.37208	0.686040	−	0.727564i	$-0.259347\pi$
$937$	42.0000	1.37208	0.686040	+	0.727564i	$0.259347\pi$
$938$	−48.0000	−1.56726
$939$	0	0
$940$	−8.00000	−0.260931
$941$	46.0000	1.49956	0.749779	−	0.661689i	$-0.230160\pi$
$941$	46.0000	1.49956	0.749779	+	0.661689i	$0.230160\pi$
$942$	0	0
$943$	−8.00000	−0.260516
$944$	10.0000	0.325472
$945$	0	0
$946$	0	0
$947$	36.0000	1.16984	0.584921	−	0.811090i	$-0.301125\pi$
$947$	36.0000	1.16984	0.584921	+	0.811090i	$0.301125\pi$
$948$	0	0
$949$	0	0
$950$	8.00000	0.259554
$951$	0	0
$952$	−16.0000	−0.518563
$953$	−48.0000	−1.55487	−0.777436	−	0.628962i	$-0.783480\pi$
$953$	−48.0000	−1.55487	−0.777436	+	0.628962i	$0.783480\pi$
$954$	0	0
$955$	−24.0000	−0.776622
$956$	−12.0000	−0.388108
$957$	0	0
$958$	8.00000	0.258468
$959$	−48.0000	−1.55000
$960$	0	0
$961$	−15.0000	−0.483871
$962$	0	0
$963$	0	0
$964$	2.00000	0.0644157
$965$	18.0000	0.579441
$966$	0	0
$967$	58.0000	1.86515	0.932577	−	0.360971i	$-0.117555\pi$
$967$	58.0000	1.86515	0.932577	+	0.360971i	$0.117555\pi$
$968$	−11.0000	−0.353553
$969$	0	0
$970$	10.0000	0.321081
$971$	0	0		−	1.00000i	$-0.5\pi$
$971$	0	0			1.00000i	$0.5\pi$
$972$	0	0
$973$	−80.0000	−2.56468
$974$	30.0000	0.961262
$975$	0	0
$976$	−8.00000	−0.256074
$977$	−32.0000	−1.02377	−0.511885	−	0.859054i	$-0.671053\pi$
$977$	−32.0000	−1.02377	−0.511885	+	0.859054i	$0.671053\pi$
$978$	0	0
$979$	0	0
$980$	−9.00000	−0.287494
$981$	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
$985$	−2.00000	−0.0637253
$986$	−8.00000	−0.254772
$987$	0	0
$988$	0	0
$989$	4.00000	0.127193
$990$	0	0
$991$	32.0000	1.01651	0.508257	−	0.861206i	$-0.330290\pi$
$991$	32.0000	1.01651	0.508257	+	0.861206i	$0.330290\pi$
$992$	4.00000	0.127000
$993$	0	0
$994$	−64.0000	−2.02996
$995$	18.0000	0.570638
$996$	0	0
$997$	−28.0000	−0.886769	−0.443384	−	0.896332i	$-0.646222\pi$
$997$	−28.0000	−0.886769	−0.443384	+	0.896332i	$0.646222\pi$
$998$	36.0000	1.13956
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2070.2.a.j.1.1	yes	1
3.2	odd	2		2070.2.a.d.1.1	✓	1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2070.2.a.d.1.1	✓	1	3.2	odd	2
2070.2.a.j.1.1	yes	1	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 2070 = 2 \cdot 3^{2} \cdot 5 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2070.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more