LMFDB - Embedded newform 2070.2.a.h.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:	no (minimal twist has level 690)
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)

$f(q)$

$=$

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

$q-1.00000 q^{2} +1.00000 q^{4} +1.00000 q^{5} -1.00000 q^{8} -1.00000 q^{10} +6.00000 q^{13} +1.00000 q^{16} -2.00000 q^{17} +1.00000 q^{20} +1.00000 q^{23} +1.00000 q^{25} -6.00000 q^{26} -6.00000 q^{29} +8.00000 q^{31} -1.00000 q^{32} +2.00000 q^{34} +10.0000 q^{37} -1.00000 q^{40} +6.00000 q^{41} -8.00000 q^{43} -1.00000 q^{46} -8.00000 q^{47} -7.00000 q^{49} -1.00000 q^{50} +6.00000 q^{52} +6.00000 q^{53} +6.00000 q^{58} +4.00000 q^{59} -6.00000 q^{61} -8.00000 q^{62} +1.00000 q^{64} +6.00000 q^{65} +8.00000 q^{67} -2.00000 q^{68} +8.00000 q^{71} +10.0000 q^{73} -10.0000 q^{74} -8.00000 q^{79} +1.00000 q^{80} -6.00000 q^{82} +8.00000 q^{83} -2.00000 q^{85} +8.00000 q^{86} +6.00000 q^{89} +1.00000 q^{92} +8.00000 q^{94} +18.0000 q^{97} +7.00000 q^{98} +O(q^{100})$

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
$7$	0	0		−	1.00000i	$-0.5\pi$
$7$	0	0			1.00000i	$0.5\pi$
$8$	−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
$13$	6.00000	1.66410	0.832050	−	0.554700i	$-0.187167\pi$
$13$	6.00000	1.66410	0.832050	+	0.554700i	$0.187167\pi$
$14$	0	0
$15$	0	0
$16$	1.00000	0.250000
$17$	−2.00000	−0.485071	−0.242536	−	0.970143i	$-0.577979\pi$
$17$	−2.00000	−0.485071	−0.242536	+	0.970143i	$0.577979\pi$
$18$	0	0
$19$	0	0		−	1.00000i	$-0.5\pi$
$19$	0	0			1.00000i	$0.5\pi$
$20$	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
$26$	−6.00000	−1.17670
$27$	0	0
$28$	0	0
$29$	−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
$31$	8.00000	1.43684	0.718421	−	0.695608i	$-0.244865\pi$
$31$	8.00000	1.43684	0.718421	+	0.695608i	$0.244865\pi$
$32$	−1.00000	−0.176777
$33$	0	0
$34$	2.00000	0.342997
$35$	0	0
$36$	0	0
$37$	10.0000	1.64399	0.821995	−	0.569495i	$-0.192861\pi$
$37$	10.0000	1.64399	0.821995	+	0.569495i	$0.192861\pi$
$38$	0	0
$39$	0	0
$40$	−1.00000	−0.158114
$41$	6.00000	0.937043	0.468521	−	0.883452i	$-0.344787\pi$
$41$	6.00000	0.937043	0.468521	+	0.883452i	$0.344787\pi$
$42$	0	0
$43$	−8.00000	−1.21999	−0.609994	−	0.792406i	$-0.708828\pi$
$43$	−8.00000	−1.21999	−0.609994	+	0.792406i	$0.708828\pi$
$44$	0	0
$45$	0	0
$46$	−1.00000	−0.147442
$47$	−8.00000	−1.16692	−0.583460	−	0.812142i	$-0.698301\pi$
$47$	−8.00000	−1.16692	−0.583460	+	0.812142i	$0.698301\pi$
$48$	0	0
$49$	−7.00000	−1.00000
$50$	−1.00000	−0.141421
$51$	0	0
$52$	6.00000	0.832050
$53$	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
$56$	0	0
$57$	0	0
$58$	6.00000	0.787839
$59$	4.00000	0.520756	0.260378	−	0.965507i	$-0.416153\pi$
$59$	4.00000	0.520756	0.260378	+	0.965507i	$0.416153\pi$
$60$	0	0
$61$	−6.00000	−0.768221	−0.384111	−	0.923287i	$-0.625492\pi$
$61$	−6.00000	−0.768221	−0.384111	+	0.923287i	$0.625492\pi$
$62$	−8.00000	−1.01600
$63$	0	0
$64$	1.00000	0.125000
$65$	6.00000	0.744208
$66$	0	0
$67$	8.00000	0.977356	0.488678	−	0.872464i	$-0.337479\pi$
$67$	8.00000	0.977356	0.488678	+	0.872464i	$0.337479\pi$
$68$	−2.00000	−0.242536
$69$	0	0
$70$	0	0
$71$	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
$73$	10.0000	1.17041	0.585206	−	0.810885i	$-0.301014\pi$
$73$	10.0000	1.17041	0.585206	+	0.810885i	$0.301014\pi$
$74$	−10.0000	−1.16248
$75$	0	0
$76$	0	0
$77$	0	0
$78$	0	0
$79$	−8.00000	−0.900070	−0.450035	−	0.893011i	$-0.648589\pi$
$79$	−8.00000	−0.900070	−0.450035	+	0.893011i	$0.648589\pi$
$80$	1.00000	0.111803
$81$	0	0
$82$	−6.00000	−0.662589
$83$	8.00000	0.878114	0.439057	−	0.898459i	$-0.355313\pi$
$83$	8.00000	0.878114	0.439057	+	0.898459i	$0.355313\pi$
$84$	0	0
$85$	−2.00000	−0.216930
$86$	8.00000	0.862662
$87$	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
$95$	0	0
$96$	0	0
$97$	18.0000	1.82762	0.913812	−	0.406138i	$-0.133125\pi$
$97$	18.0000	1.82762	0.913812	+	0.406138i	$0.133125\pi$
$98$	7.00000	0.707107
$99$	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
$103$	−8.00000	−0.788263	−0.394132	−	0.919054i	$-0.628955\pi$
$103$	−8.00000	−0.788263	−0.394132	+	0.919054i	$0.628955\pi$
$104$	−6.00000	−0.588348
$105$	0	0
$106$	−6.00000	−0.582772
$107$	0	0		−	1.00000i	$-0.5\pi$
$107$	0	0			1.00000i	$0.5\pi$
$108$	0	0
$109$	−6.00000	−0.574696	−0.287348	−	0.957826i	$-0.592774\pi$
$109$	−6.00000	−0.574696	−0.287348	+	0.957826i	$0.592774\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	6.00000	0.564433	0.282216	−	0.959351i	$-0.408930\pi$
$113$	6.00000	0.564433	0.282216	+	0.959351i	$0.408930\pi$
$114$	0	0
$115$	1.00000	0.0932505
$116$	−6.00000	−0.557086
$117$	0	0
$118$	−4.00000	−0.368230
$119$	0	0
$120$	0	0
$121$	−11.0000	−1.00000
$122$	6.00000	0.543214
$123$	0	0
$124$	8.00000	0.718421
$125$	1.00000	0.0894427
$126$	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
$130$	−6.00000	−0.526235
$131$	−4.00000	−0.349482	−0.174741	−	0.984614i	$-0.555909\pi$
$131$	−4.00000	−0.349482	−0.174741	+	0.984614i	$0.555909\pi$
$132$	0	0
$133$	0	0
$134$	−8.00000	−0.691095
$135$	0	0
$136$	2.00000	0.171499
$137$	6.00000	0.512615	0.256307	−	0.966595i	$-0.417494\pi$
$137$	6.00000	0.512615	0.256307	+	0.966595i	$0.417494\pi$
$138$	0	0
$139$	−4.00000	−0.339276	−0.169638	−	0.985506i	$-0.554260\pi$
$139$	−4.00000	−0.339276	−0.169638	+	0.985506i	$0.554260\pi$
$140$	0	0
$141$	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
$148$	10.0000	0.821995
$149$	6.00000	0.491539	0.245770	−	0.969328i	$-0.420959\pi$
$149$	6.00000	0.491539	0.245770	+	0.969328i	$0.420959\pi$
$150$	0	0
$151$	−16.0000	−1.30206	−0.651031	−	0.759051i	$-0.725663\pi$
$151$	−16.0000	−1.30206	−0.651031	+	0.759051i	$0.725663\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	8.00000	0.642575
$156$	0	0
$157$	18.0000	1.43656	0.718278	−	0.695756i	$-0.244931\pi$
$157$	18.0000	1.43656	0.718278	+	0.695756i	$0.244931\pi$
$158$	8.00000	0.636446
$159$	0	0
$160$	−1.00000	−0.0790569
$161$	0	0
$162$	0	0
$163$	12.0000	0.939913	0.469956	−	0.882690i	$-0.344270\pi$
$163$	12.0000	0.939913	0.469956	+	0.882690i	$0.344270\pi$
$164$	6.00000	0.468521
$165$	0	0
$166$	−8.00000	−0.620920
$167$	16.0000	1.23812	0.619059	−	0.785345i	$-0.287514\pi$
$167$	16.0000	1.23812	0.619059	+	0.785345i	$0.287514\pi$
$168$	0	0
$169$	23.0000	1.76923
$170$	2.00000	0.153393
$171$	0	0
$172$	−8.00000	−0.609994
$173$	18.0000	1.36851	0.684257	−	0.729241i	$-0.260127\pi$
$173$	18.0000	1.36851	0.684257	+	0.729241i	$0.260127\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	0	0
$178$	−6.00000	−0.449719
$179$	4.00000	0.298974	0.149487	−	0.988764i	$-0.452238\pi$
$179$	4.00000	0.298974	0.149487	+	0.988764i	$0.452238\pi$
$180$	0	0
$181$	2.00000	0.148659	0.0743294	−	0.997234i	$-0.476318\pi$
$181$	2.00000	0.148659	0.0743294	+	0.997234i	$0.476318\pi$
$182$	0	0
$183$	0	0
$184$	−1.00000	−0.0737210
$185$	10.0000	0.735215
$186$	0	0
$187$	0	0
$188$	−8.00000	−0.583460
$189$	0	0
$190$	0	0
$191$	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	$-0.275675\pi$
$193$	18.0000	1.29567	0.647834	+	0.761781i	$0.275675\pi$
$194$	−18.0000	−1.29232
$195$	0	0
$196$	−7.00000	−0.500000
$197$	−22.0000	−1.56744	−0.783718	−	0.621117i	$-0.786679\pi$
$197$	−22.0000	−1.56744	−0.783718	+	0.621117i	$0.786679\pi$
$198$	0	0
$199$	−16.0000	−1.13421	−0.567105	−	0.823646i	$-0.691937\pi$
$199$	−16.0000	−1.13421	−0.567105	+	0.823646i	$0.691937\pi$
$200$	−1.00000	−0.0707107
$201$	0	0
$202$	6.00000	0.422159
$203$	0	0
$204$	0	0
$205$	6.00000	0.419058
$206$	8.00000	0.557386
$207$	0	0
$208$	6.00000	0.416025
$209$	0	0
$210$	0	0
$211$	4.00000	0.275371	0.137686	−	0.990476i	$-0.456034\pi$
$211$	4.00000	0.275371	0.137686	+	0.990476i	$0.456034\pi$
$212$	6.00000	0.412082
$213$	0	0
$214$	0	0
$215$	−8.00000	−0.545595
$216$	0	0
$217$	0	0
$218$	6.00000	0.406371
$219$	0	0
$220$	0	0
$221$	−12.0000	−0.807207
$222$	0	0
$223$	0	0		−	1.00000i	$-0.5\pi$
$223$	0	0			1.00000i	$0.5\pi$
$224$	0	0
$225$	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
$229$	18.0000	1.18947	0.594737	−	0.803921i	$-0.297256\pi$
$229$	18.0000	1.18947	0.594737	+	0.803921i	$0.297256\pi$
$230$	−1.00000	−0.0659380
$231$	0	0
$232$	6.00000	0.393919
$233$	22.0000	1.44127	0.720634	−	0.693316i	$-0.243851\pi$
$233$	22.0000	1.44127	0.720634	+	0.693316i	$0.243851\pi$
$234$	0	0
$235$	−8.00000	−0.521862
$236$	4.00000	0.260378
$237$	0	0
$238$	0	0
$239$	−8.00000	−0.517477	−0.258738	−	0.965947i	$-0.583307\pi$
$239$	−8.00000	−0.517477	−0.258738	+	0.965947i	$0.583307\pi$
$240$	0	0
$241$	26.0000	1.67481	0.837404	−	0.546585i	$-0.184072\pi$
$241$	26.0000	1.67481	0.837404	+	0.546585i	$0.184072\pi$
$242$	11.0000	0.707107
$243$	0	0
$244$	−6.00000	−0.384111
$245$	−7.00000	−0.447214
$246$	0	0
$247$	0	0
$248$	−8.00000	−0.508001
$249$	0	0
$250$	−1.00000	−0.0632456
$251$	−8.00000	−0.504956	−0.252478	−	0.967603i	$-0.581245\pi$
$251$	−8.00000	−0.504956	−0.252478	+	0.967603i	$0.581245\pi$
$252$	0	0
$253$	0	0
$254$	8.00000	0.501965
$255$	0	0
$256$	1.00000	0.0625000
$257$	−18.0000	−1.12281	−0.561405	−	0.827541i	$-0.689739\pi$
$257$	−18.0000	−1.12281	−0.561405	+	0.827541i	$0.689739\pi$
$258$	0	0
$259$	0	0
$260$	6.00000	0.372104
$261$	0	0
$262$	4.00000	0.247121
$263$	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
$265$	6.00000	0.368577
$266$	0	0
$267$	0	0
$268$	8.00000	0.488678
$269$	−22.0000	−1.34136	−0.670682	−	0.741745i	$-0.733998\pi$
$269$	−22.0000	−1.34136	−0.670682	+	0.741745i	$0.733998\pi$
$270$	0	0
$271$	16.0000	0.971931	0.485965	−	0.873978i	$-0.338468\pi$
$271$	16.0000	0.971931	0.485965	+	0.873978i	$0.338468\pi$
$272$	−2.00000	−0.121268
$273$	0	0
$274$	−6.00000	−0.362473
$275$	0	0
$276$	0	0
$277$	−10.0000	−0.600842	−0.300421	−	0.953807i	$-0.597127\pi$
$277$	−10.0000	−0.600842	−0.300421	+	0.953807i	$0.597127\pi$
$278$	4.00000	0.239904
$279$	0	0
$280$	0	0
$281$	22.0000	1.31241	0.656205	−	0.754583i	$-0.272161\pi$
$281$	22.0000	1.31241	0.656205	+	0.754583i	$0.272161\pi$
$282$	0	0
$283$	24.0000	1.42665	0.713326	−	0.700832i	$-0.247188\pi$
$283$	24.0000	1.42665	0.713326	+	0.700832i	$0.247188\pi$
$284$	8.00000	0.474713
$285$	0	0
$286$	0	0
$287$	0	0
$288$	0	0
$289$	−13.0000	−0.764706
$290$	6.00000	0.352332
$291$	0	0
$292$	10.0000	0.585206
$293$	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
$295$	4.00000	0.232889
$296$	−10.0000	−0.581238
$297$	0	0
$298$	−6.00000	−0.347571
$299$	6.00000	0.346989
$300$	0	0
$301$	0	0
$302$	16.0000	0.920697
$303$	0	0
$304$	0	0
$305$	−6.00000	−0.343559
$306$	0	0
$307$	−4.00000	−0.228292	−0.114146	−	0.993464i	$-0.536413\pi$
$307$	−4.00000	−0.228292	−0.114146	+	0.993464i	$0.536413\pi$
$308$	0	0
$309$	0	0
$310$	−8.00000	−0.454369
$311$	0	0		−	1.00000i	$-0.5\pi$
$311$	0	0			1.00000i	$0.5\pi$
$312$	0	0
$313$	−6.00000	−0.339140	−0.169570	−	0.985518i	$-0.554238\pi$
$313$	−6.00000	−0.339140	−0.169570	+	0.985518i	$0.554238\pi$
$314$	−18.0000	−1.01580
$315$	0	0
$316$	−8.00000	−0.450035
$317$	2.00000	0.112331	0.0561656	−	0.998421i	$-0.482113\pi$
$317$	2.00000	0.112331	0.0561656	+	0.998421i	$0.482113\pi$
$318$	0	0
$319$	0	0
$320$	1.00000	0.0559017
$321$	0	0
$322$	0	0
$323$	0	0
$324$	0	0
$325$	6.00000	0.332820
$326$	−12.0000	−0.664619
$327$	0	0
$328$	−6.00000	−0.331295
$329$	0	0
$330$	0	0
$331$	−20.0000	−1.09930	−0.549650	−	0.835395i	$-0.685239\pi$
$331$	−20.0000	−1.09930	−0.549650	+	0.835395i	$0.685239\pi$
$332$	8.00000	0.439057
$333$	0	0
$334$	−16.0000	−0.875481
$335$	8.00000	0.437087
$336$	0	0
$337$	−22.0000	−1.19842	−0.599208	−	0.800593i	$-0.704518\pi$
$337$	−22.0000	−1.19842	−0.599208	+	0.800593i	$0.704518\pi$
$338$	−23.0000	−1.25104
$339$	0	0
$340$	−2.00000	−0.108465
$341$	0	0
$342$	0	0
$343$	0	0
$344$	8.00000	0.431331
$345$	0	0
$346$	−18.0000	−0.967686
$347$	−36.0000	−1.93258	−0.966291	−	0.257454i	$-0.917117\pi$
$347$	−36.0000	−1.93258	−0.966291	+	0.257454i	$0.917117\pi$
$348$	0	0
$349$	−26.0000	−1.39175	−0.695874	−	0.718164i	$-0.744983\pi$
$349$	−26.0000	−1.39175	−0.695874	+	0.718164i	$0.744983\pi$
$350$	0	0
$351$	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
$355$	8.00000	0.424596
$356$	6.00000	0.317999
$357$	0	0
$358$	−4.00000	−0.211407
$359$	−16.0000	−0.844448	−0.422224	−	0.906492i	$-0.638750\pi$
$359$	−16.0000	−0.844448	−0.422224	+	0.906492i	$0.638750\pi$
$360$	0	0
$361$	−19.0000	−1.00000
$362$	−2.00000	−0.105118
$363$	0	0
$364$	0	0
$365$	10.0000	0.523424
$366$	0	0
$367$	−24.0000	−1.25279	−0.626395	−	0.779506i	$-0.715470\pi$
$367$	−24.0000	−1.25279	−0.626395	+	0.779506i	$0.715470\pi$
$368$	1.00000	0.0521286
$369$	0	0
$370$	−10.0000	−0.519875
$371$	0	0
$372$	0	0
$373$	26.0000	1.34623	0.673114	−	0.739538i	$-0.264956\pi$
$373$	26.0000	1.34623	0.673114	+	0.739538i	$0.264956\pi$
$374$	0	0
$375$	0	0
$376$	8.00000	0.412568
$377$	−36.0000	−1.85409
$378$	0	0
$379$	−24.0000	−1.23280	−0.616399	−	0.787434i	$-0.711409\pi$
$379$	−24.0000	−1.23280	−0.616399	+	0.787434i	$0.711409\pi$
$380$	0	0
$381$	0	0
$382$	−24.0000	−1.22795
$383$	32.0000	1.63512	0.817562	−	0.575841i	$-0.195325\pi$
$383$	32.0000	1.63512	0.817562	+	0.575841i	$0.195325\pi$
$384$	0	0
$385$	0	0
$386$	−18.0000	−0.916176
$387$	0	0
$388$	18.0000	0.913812
$389$	−2.00000	−0.101404	−0.0507020	−	0.998714i	$-0.516146\pi$
$389$	−2.00000	−0.101404	−0.0507020	+	0.998714i	$0.516146\pi$
$390$	0	0
$391$	−2.00000	−0.101144
$392$	7.00000	0.353553
$393$	0	0
$394$	22.0000	1.10834
$395$	−8.00000	−0.402524
$396$	0	0
$397$	6.00000	0.301131	0.150566	−	0.988600i	$-0.451890\pi$
$397$	6.00000	0.301131	0.150566	+	0.988600i	$0.451890\pi$
$398$	16.0000	0.802008
$399$	0	0
$400$	1.00000	0.0500000
$401$	−26.0000	−1.29838	−0.649189	−	0.760627i	$-0.724892\pi$
$401$	−26.0000	−1.29838	−0.649189	+	0.760627i	$0.724892\pi$
$402$	0	0
$403$	48.0000	2.39105
$404$	−6.00000	−0.298511
$405$	0	0
$406$	0	0
$407$	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	$0.420478\pi$
$410$	−6.00000	−0.296319
$411$	0	0
$412$	−8.00000	−0.394132
$413$	0	0
$414$	0	0
$415$	8.00000	0.392705
$416$	−6.00000	−0.294174
$417$	0	0
$418$	0	0
$419$	−16.0000	−0.781651	−0.390826	−	0.920465i	$-0.627810\pi$
$419$	−16.0000	−0.781651	−0.390826	+	0.920465i	$0.627810\pi$
$420$	0	0
$421$	−22.0000	−1.07221	−0.536107	−	0.844150i	$-0.680106\pi$
$421$	−22.0000	−1.07221	−0.536107	+	0.844150i	$0.680106\pi$
$422$	−4.00000	−0.194717
$423$	0	0
$424$	−6.00000	−0.291386
$425$	−2.00000	−0.0970143
$426$	0	0
$427$	0	0
$428$	0	0
$429$	0	0
$430$	8.00000	0.385794
$431$	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
$433$	−14.0000	−0.672797	−0.336399	−	0.941720i	$-0.609209\pi$
$433$	−14.0000	−0.672797	−0.336399	+	0.941720i	$0.609209\pi$
$434$	0	0
$435$	0	0
$436$	−6.00000	−0.287348
$437$	0	0
$438$	0	0
$439$	8.00000	0.381819	0.190910	−	0.981608i	$-0.438856\pi$
$439$	8.00000	0.381819	0.190910	+	0.981608i	$0.438856\pi$
$440$	0	0
$441$	0	0
$442$	12.0000	0.570782
$443$	−20.0000	−0.950229	−0.475114	−	0.879924i	$-0.657593\pi$
$443$	−20.0000	−0.950229	−0.475114	+	0.879924i	$0.657593\pi$
$444$	0	0
$445$	6.00000	0.284427
$446$	0	0
$447$	0	0
$448$	0	0
$449$	14.0000	0.660701	0.330350	−	0.943858i	$-0.392833\pi$
$449$	14.0000	0.660701	0.330350	+	0.943858i	$0.392833\pi$
$450$	0	0
$451$	0	0
$452$	6.00000	0.282216
$453$	0	0
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−38.0000	−1.77757	−0.888783	−	0.458329i	$-0.848448\pi$
$457$	−38.0000	−1.77757	−0.888783	+	0.458329i	$0.848448\pi$
$458$	−18.0000	−0.841085
$459$	0	0
$460$	1.00000	0.0466252
$461$	−6.00000	−0.279448	−0.139724	−	0.990190i	$-0.544622\pi$
$461$	−6.00000	−0.279448	−0.139724	+	0.990190i	$0.544622\pi$
$462$	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
$466$	−22.0000	−1.01913
$467$	−8.00000	−0.370196	−0.185098	−	0.982720i	$-0.559260\pi$
$467$	−8.00000	−0.370196	−0.185098	+	0.982720i	$0.559260\pi$
$468$	0	0
$469$	0	0
$470$	8.00000	0.369012
$471$	0	0
$472$	−4.00000	−0.184115
$473$	0	0
$474$	0	0
$475$	0	0
$476$	0	0
$477$	0	0
$478$	8.00000	0.365911
$479$	0	0		−	1.00000i	$-0.5\pi$
$479$	0	0			1.00000i	$0.5\pi$
$480$	0	0
$481$	60.0000	2.73576
$482$	−26.0000	−1.18427
$483$	0	0
$484$	−11.0000	−0.500000
$485$	18.0000	0.817338
$486$	0	0
$487$	16.0000	0.725029	0.362515	−	0.931978i	$-0.381918\pi$
$487$	16.0000	0.725029	0.362515	+	0.931978i	$0.381918\pi$
$488$	6.00000	0.271607
$489$	0	0
$490$	7.00000	0.316228
$491$	−36.0000	−1.62466	−0.812329	−	0.583200i	$-0.801800\pi$
$491$	−36.0000	−1.62466	−0.812329	+	0.583200i	$0.801800\pi$
$492$	0	0
$493$	12.0000	0.540453
$494$	0	0
$495$	0	0
$496$	8.00000	0.359211
$497$	0	0
$498$	0	0
$499$	−20.0000	−0.895323	−0.447661	−	0.894203i	$-0.647743\pi$
$499$	−20.0000	−0.895323	−0.447661	+	0.894203i	$0.647743\pi$
$500$	1.00000	0.0447214
$501$	0	0
$502$	8.00000	0.357057
$503$	−16.0000	−0.713405	−0.356702	−	0.934218i	$-0.616099\pi$
$503$	−16.0000	−0.713405	−0.356702	+	0.934218i	$0.616099\pi$
$504$	0	0
$505$	−6.00000	−0.266996
$506$	0	0
$507$	0	0
$508$	−8.00000	−0.354943
$509$	26.0000	1.15243	0.576215	−	0.817298i	$-0.304529\pi$
$509$	26.0000	1.15243	0.576215	+	0.817298i	$0.304529\pi$
$510$	0	0
$511$	0	0
$512$	−1.00000	−0.0441942
$513$	0	0
$514$	18.0000	0.793946
$515$	−8.00000	−0.352522
$516$	0	0
$517$	0	0
$518$	0	0
$519$	0	0
$520$	−6.00000	−0.263117
$521$	14.0000	0.613351	0.306676	−	0.951814i	$-0.400783\pi$
$521$	14.0000	0.613351	0.306676	+	0.951814i	$0.400783\pi$
$522$	0	0
$523$	−16.0000	−0.699631	−0.349816	−	0.936819i	$-0.613756\pi$
$523$	−16.0000	−0.699631	−0.349816	+	0.936819i	$0.613756\pi$
$524$	−4.00000	−0.174741
$525$	0	0
$526$	−8.00000	−0.348817
$527$	−16.0000	−0.696971
$528$	0	0
$529$	1.00000	0.0434783
$530$	−6.00000	−0.260623
$531$	0	0
$532$	0	0
$533$	36.0000	1.55933
$534$	0	0
$535$	0	0
$536$	−8.00000	−0.345547
$537$	0	0
$538$	22.0000	0.948487
$539$	0	0
$540$	0	0
$541$	−10.0000	−0.429934	−0.214967	−	0.976621i	$-0.568964\pi$
$541$	−10.0000	−0.429934	−0.214967	+	0.976621i	$0.568964\pi$
$542$	−16.0000	−0.687259
$543$	0	0
$544$	2.00000	0.0857493
$545$	−6.00000	−0.257012
$546$	0	0
$547$	4.00000	0.171028	0.0855138	−	0.996337i	$-0.472747\pi$
$547$	4.00000	0.171028	0.0855138	+	0.996337i	$0.472747\pi$
$548$	6.00000	0.256307
$549$	0	0
$550$	0	0
$551$	0	0
$552$	0	0
$553$	0	0
$554$	10.0000	0.424859
$555$	0	0
$556$	−4.00000	−0.169638
$557$	−10.0000	−0.423714	−0.211857	−	0.977301i	$-0.567951\pi$
$557$	−10.0000	−0.423714	−0.211857	+	0.977301i	$0.567951\pi$
$558$	0	0
$559$	−48.0000	−2.03018
$560$	0	0
$561$	0	0
$562$	−22.0000	−0.928014
$563$	−32.0000	−1.34864	−0.674320	−	0.738440i	$-0.735563\pi$
$563$	−32.0000	−1.34864	−0.674320	+	0.738440i	$0.735563\pi$
$564$	0	0
$565$	6.00000	0.252422
$566$	−24.0000	−1.00880
$567$	0	0
$568$	−8.00000	−0.335673
$569$	14.0000	0.586911	0.293455	−	0.955973i	$-0.405195\pi$
$569$	14.0000	0.586911	0.293455	+	0.955973i	$0.405195\pi$
$570$	0	0
$571$	8.00000	0.334790	0.167395	−	0.985890i	$-0.446465\pi$
$571$	8.00000	0.334790	0.167395	+	0.985890i	$0.446465\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	1.00000	0.0417029
$576$	0	0
$577$	2.00000	0.0832611	0.0416305	−	0.999133i	$-0.486745\pi$
$577$	2.00000	0.0832611	0.0416305	+	0.999133i	$0.486745\pi$
$578$	13.0000	0.540729
$579$	0	0
$580$	−6.00000	−0.249136
$581$	0	0
$582$	0	0
$583$	0	0
$584$	−10.0000	−0.413803
$585$	0	0
$586$	−6.00000	−0.247858
$587$	−12.0000	−0.495293	−0.247647	−	0.968850i	$-0.579657\pi$
$587$	−12.0000	−0.495293	−0.247647	+	0.968850i	$0.579657\pi$
$588$	0	0
$589$	0	0
$590$	−4.00000	−0.164677
$591$	0	0
$592$	10.0000	0.410997
$593$	30.0000	1.23195	0.615976	−	0.787765i	$-0.288762\pi$
$593$	30.0000	1.23195	0.615976	+	0.787765i	$0.288762\pi$
$594$	0	0
$595$	0	0
$596$	6.00000	0.245770
$597$	0	0
$598$	−6.00000	−0.245358
$599$	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	$-0.434621\pi$
$601$	10.0000	0.407909	0.203954	+	0.978980i	$0.434621\pi$
$602$	0	0
$603$	0	0
$604$	−16.0000	−0.651031
$605$	−11.0000	−0.447214
$606$	0	0
$607$	−16.0000	−0.649420	−0.324710	−	0.945814i	$-0.605267\pi$
$607$	−16.0000	−0.649420	−0.324710	+	0.945814i	$0.605267\pi$
$608$	0	0
$609$	0	0
$610$	6.00000	0.242933
$611$	−48.0000	−1.94187
$612$	0	0
$613$	−46.0000	−1.85792	−0.928961	−	0.370177i	$-0.879297\pi$
$613$	−46.0000	−1.85792	−0.928961	+	0.370177i	$0.879297\pi$
$614$	4.00000	0.161427
$615$	0	0
$616$	0	0
$617$	−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
$619$	−16.0000	−0.643094	−0.321547	−	0.946894i	$-0.604203\pi$
$619$	−16.0000	−0.643094	−0.321547	+	0.946894i	$0.604203\pi$
$620$	8.00000	0.321288
$621$	0	0
$622$	0	0
$623$	0	0
$624$	0	0
$625$	1.00000	0.0400000
$626$	6.00000	0.239808
$627$	0	0
$628$	18.0000	0.718278
$629$	−20.0000	−0.797452
$630$	0	0
$631$	−40.0000	−1.59237	−0.796187	−	0.605050i	$-0.793153\pi$
$631$	−40.0000	−1.59237	−0.796187	+	0.605050i	$0.793153\pi$
$632$	8.00000	0.318223
$633$	0	0
$634$	−2.00000	−0.0794301
$635$	−8.00000	−0.317470
$636$	0	0
$637$	−42.0000	−1.66410
$638$	0	0
$639$	0	0
$640$	−1.00000	−0.0395285
$641$	22.0000	0.868948	0.434474	−	0.900684i	$-0.356934\pi$
$641$	22.0000	0.868948	0.434474	+	0.900684i	$0.356934\pi$
$642$	0	0
$643$	−16.0000	−0.630978	−0.315489	−	0.948929i	$-0.602169\pi$
$643$	−16.0000	−0.630978	−0.315489	+	0.948929i	$0.602169\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	24.0000	0.943537	0.471769	−	0.881722i	$-0.343616\pi$
$647$	24.0000	0.943537	0.471769	+	0.881722i	$0.343616\pi$
$648$	0	0
$649$	0	0
$650$	−6.00000	−0.235339
$651$	0	0
$652$	12.0000	0.469956
$653$	42.0000	1.64359	0.821794	−	0.569785i	$-0.192974\pi$
$653$	42.0000	1.64359	0.821794	+	0.569785i	$0.192974\pi$
$654$	0	0
$655$	−4.00000	−0.156293
$656$	6.00000	0.234261
$657$	0	0
$658$	0	0
$659$	48.0000	1.86981	0.934907	−	0.354892i	$-0.115482\pi$
$659$	48.0000	1.86981	0.934907	+	0.354892i	$0.115482\pi$
$660$	0	0
$661$	2.00000	0.0777910	0.0388955	−	0.999243i	$-0.487616\pi$
$661$	2.00000	0.0777910	0.0388955	+	0.999243i	$0.487616\pi$
$662$	20.0000	0.777322
$663$	0	0
$664$	−8.00000	−0.310460
$665$	0	0
$666$	0	0
$667$	−6.00000	−0.232321
$668$	16.0000	0.619059
$669$	0	0
$670$	−8.00000	−0.309067
$671$	0	0
$672$	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
$676$	23.0000	0.884615
$677$	6.00000	0.230599	0.115299	−	0.993331i	$-0.463217\pi$
$677$	6.00000	0.230599	0.115299	+	0.993331i	$0.463217\pi$
$678$	0	0
$679$	0	0
$680$	2.00000	0.0766965
$681$	0	0
$682$	0	0
$683$	4.00000	0.153056	0.0765279	−	0.997067i	$-0.475617\pi$
$683$	4.00000	0.153056	0.0765279	+	0.997067i	$0.475617\pi$
$684$	0	0
$685$	6.00000	0.229248
$686$	0	0
$687$	0	0
$688$	−8.00000	−0.304997
$689$	36.0000	1.37149
$690$	0	0
$691$	−12.0000	−0.456502	−0.228251	−	0.973602i	$-0.573301\pi$
$691$	−12.0000	−0.456502	−0.228251	+	0.973602i	$0.573301\pi$
$692$	18.0000	0.684257
$693$	0	0
$694$	36.0000	1.36654
$695$	−4.00000	−0.151729
$696$	0	0
$697$	−12.0000	−0.454532
$698$	26.0000	0.984115
$699$	0	0
$700$	0	0
$701$	−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
$703$	0	0
$704$	0	0
$705$	0	0
$706$	18.0000	0.677439
$707$	0	0
$708$	0	0
$709$	26.0000	0.976450	0.488225	−	0.872718i	$-0.337644\pi$
$709$	26.0000	0.976450	0.488225	+	0.872718i	$0.337644\pi$
$710$	−8.00000	−0.300235
$711$	0	0
$712$	−6.00000	−0.224860
$713$	8.00000	0.299602
$714$	0	0
$715$	0	0
$716$	4.00000	0.149487
$717$	0	0
$718$	16.0000	0.597115
$719$	−40.0000	−1.49175	−0.745874	−	0.666087i	$-0.767968\pi$
$719$	−40.0000	−1.49175	−0.745874	+	0.666087i	$0.767968\pi$
$720$	0	0
$721$	0	0
$722$	19.0000	0.707107
$723$	0	0
$724$	2.00000	0.0743294
$725$	−6.00000	−0.222834
$726$	0	0
$727$	−16.0000	−0.593407	−0.296704	−	0.954970i	$-0.595887\pi$
$727$	−16.0000	−0.593407	−0.296704	+	0.954970i	$0.595887\pi$
$728$	0	0
$729$	0	0
$730$	−10.0000	−0.370117
$731$	16.0000	0.591781
$732$	0	0
$733$	2.00000	0.0738717	0.0369358	−	0.999318i	$-0.488240\pi$
$733$	2.00000	0.0738717	0.0369358	+	0.999318i	$0.488240\pi$
$734$	24.0000	0.885856
$735$	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	$0.523440\pi$
$740$	10.0000	0.367607
$741$	0	0
$742$	0	0
$743$	8.00000	0.293492	0.146746	−	0.989174i	$-0.453120\pi$
$743$	8.00000	0.293492	0.146746	+	0.989174i	$0.453120\pi$
$744$	0	0
$745$	6.00000	0.219823
$746$	−26.0000	−0.951928
$747$	0	0
$748$	0	0
$749$	0	0
$750$	0	0
$751$	0	0		−	1.00000i	$-0.5\pi$
$751$	0	0			1.00000i	$0.5\pi$
$752$	−8.00000	−0.291730
$753$	0	0
$754$	36.0000	1.31104
$755$	−16.0000	−0.582300
$756$	0	0
$757$	34.0000	1.23575	0.617876	−	0.786276i	$-0.287994\pi$
$757$	34.0000	1.23575	0.617876	+	0.786276i	$0.287994\pi$
$758$	24.0000	0.871719
$759$	0	0
$760$	0	0
$761$	−42.0000	−1.52250	−0.761249	−	0.648459i	$-0.775414\pi$
$761$	−42.0000	−1.52250	−0.761249	+	0.648459i	$0.775414\pi$
$762$	0	0
$763$	0	0
$764$	24.0000	0.868290
$765$	0	0
$766$	−32.0000	−1.15621
$767$	24.0000	0.866590
$768$	0	0
$769$	−6.00000	−0.216366	−0.108183	−	0.994131i	$-0.534503\pi$
$769$	−6.00000	−0.216366	−0.108183	+	0.994131i	$0.534503\pi$
$770$	0	0
$771$	0	0
$772$	18.0000	0.647834
$773$	−2.00000	−0.0719350	−0.0359675	−	0.999353i	$-0.511451\pi$
$773$	−2.00000	−0.0719350	−0.0359675	+	0.999353i	$0.511451\pi$
$774$	0	0
$775$	8.00000	0.287368
$776$	−18.0000	−0.646162
$777$	0	0
$778$	2.00000	0.0717035
$779$	0	0
$780$	0	0
$781$	0	0
$782$	2.00000	0.0715199
$783$	0	0
$784$	−7.00000	−0.250000
$785$	18.0000	0.642448
$786$	0	0
$787$	−32.0000	−1.14068	−0.570338	−	0.821410i	$-0.693188\pi$
$787$	−32.0000	−1.14068	−0.570338	+	0.821410i	$0.693188\pi$
$788$	−22.0000	−0.783718
$789$	0	0
$790$	8.00000	0.284627
$791$	0	0
$792$	0	0
$793$	−36.0000	−1.27840
$794$	−6.00000	−0.212932
$795$	0	0
$796$	−16.0000	−0.567105
$797$	54.0000	1.91278	0.956389	−	0.292096i	$-0.0943526\pi$
$797$	54.0000	1.91278	0.956389	+	0.292096i	$0.0943526\pi$
$798$	0	0
$799$	16.0000	0.566039
$800$	−1.00000	−0.0353553
$801$	0	0
$802$	26.0000	0.918092
$803$	0	0
$804$	0	0
$805$	0	0
$806$	−48.0000	−1.69073
$807$	0	0
$808$	6.00000	0.211079
$809$	6.00000	0.210949	0.105474	−	0.994422i	$-0.466364\pi$
$809$	6.00000	0.210949	0.105474	+	0.994422i	$0.466364\pi$
$810$	0	0
$811$	28.0000	0.983213	0.491606	−	0.870817i	$-0.336410\pi$
$811$	28.0000	0.983213	0.491606	+	0.870817i	$0.336410\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	12.0000	0.420342
$816$	0	0
$817$	0	0
$818$	−10.0000	−0.349642
$819$	0	0
$820$	6.00000	0.209529
$821$	10.0000	0.349002	0.174501	−	0.984657i	$-0.444169\pi$
$821$	10.0000	0.349002	0.174501	+	0.984657i	$0.444169\pi$
$822$	0	0
$823$	56.0000	1.95204	0.976019	−	0.217687i	$-0.0698512\pi$
$823$	56.0000	1.95204	0.976019	+	0.217687i	$0.0698512\pi$
$824$	8.00000	0.278693
$825$	0	0
$826$	0	0
$827$	48.0000	1.66912	0.834562	−	0.550914i	$-0.185721\pi$
$827$	48.0000	1.66912	0.834562	+	0.550914i	$0.185721\pi$
$828$	0	0
$829$	−10.0000	−0.347314	−0.173657	−	0.984806i	$-0.555558\pi$
$829$	−10.0000	−0.347314	−0.173657	+	0.984806i	$0.555558\pi$
$830$	−8.00000	−0.277684
$831$	0	0
$832$	6.00000	0.208013
$833$	14.0000	0.485071
$834$	0	0
$835$	16.0000	0.553703
$836$	0	0
$837$	0	0
$838$	16.0000	0.552711
$839$	−24.0000	−0.828572	−0.414286	−	0.910147i	$-0.635969\pi$
$839$	−24.0000	−0.828572	−0.414286	+	0.910147i	$0.635969\pi$
$840$	0	0
$841$	7.00000	0.241379
$842$	22.0000	0.758170
$843$	0	0
$844$	4.00000	0.137686
$845$	23.0000	0.791224
$846$	0	0
$847$	0	0
$848$	6.00000	0.206041
$849$	0	0
$850$	2.00000	0.0685994
$851$	10.0000	0.342796
$852$	0	0
$853$	−50.0000	−1.71197	−0.855984	−	0.517003i	$-0.827048\pi$
$853$	−50.0000	−1.71197	−0.855984	+	0.517003i	$0.827048\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−10.0000	−0.341593	−0.170797	−	0.985306i	$-0.554634\pi$
$857$	−10.0000	−0.341593	−0.170797	+	0.985306i	$0.554634\pi$
$858$	0	0
$859$	20.0000	0.682391	0.341196	−	0.939992i	$-0.389168\pi$
$859$	20.0000	0.682391	0.341196	+	0.939992i	$0.389168\pi$
$860$	−8.00000	−0.272798
$861$	0	0
$862$	−8.00000	−0.272481
$863$	−48.0000	−1.63394	−0.816970	−	0.576681i	$-0.804348\pi$
$863$	−48.0000	−1.63394	−0.816970	+	0.576681i	$0.804348\pi$
$864$	0	0
$865$	18.0000	0.612018
$866$	14.0000	0.475739
$867$	0	0
$868$	0	0
$869$	0	0
$870$	0	0
$871$	48.0000	1.62642
$872$	6.00000	0.203186
$873$	0	0
$874$	0	0
$875$	0	0
$876$	0	0
$877$	14.0000	0.472746	0.236373	−	0.971662i	$-0.424041\pi$
$877$	14.0000	0.472746	0.236373	+	0.971662i	$0.424041\pi$
$878$	−8.00000	−0.269987
$879$	0	0
$880$	0	0
$881$	−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
$883$	36.0000	1.21150	0.605748	−	0.795656i	$-0.292874\pi$
$883$	36.0000	1.21150	0.605748	+	0.795656i	$0.292874\pi$
$884$	−12.0000	−0.403604
$885$	0	0
$886$	20.0000	0.671913
$887$	−16.0000	−0.537227	−0.268614	−	0.963248i	$-0.586566\pi$
$887$	−16.0000	−0.537227	−0.268614	+	0.963248i	$0.586566\pi$
$888$	0	0
$889$	0	0
$890$	−6.00000	−0.201120
$891$	0	0
$892$	0	0
$893$	0	0
$894$	0	0
$895$	4.00000	0.133705
$896$	0	0
$897$	0	0
$898$	−14.0000	−0.467186
$899$	−48.0000	−1.60089
$900$	0	0
$901$	−12.0000	−0.399778
$902$	0	0
$903$	0	0
$904$	−6.00000	−0.199557
$905$	2.00000	0.0664822
$906$	0	0
$907$	8.00000	0.265636	0.132818	−	0.991140i	$-0.457597\pi$
$907$	8.00000	0.265636	0.132818	+	0.991140i	$0.457597\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−40.0000	−1.32526	−0.662630	−	0.748947i	$-0.730560\pi$
$911$	−40.0000	−1.32526	−0.662630	+	0.748947i	$0.730560\pi$
$912$	0	0
$913$	0	0
$914$	38.0000	1.25693
$915$	0	0
$916$	18.0000	0.594737
$917$	0	0
$918$	0	0
$919$	−16.0000	−0.527791	−0.263896	−	0.964551i	$-0.585007\pi$
$919$	−16.0000	−0.527791	−0.263896	+	0.964551i	$0.585007\pi$
$920$	−1.00000	−0.0329690
$921$	0	0
$922$	6.00000	0.197599
$923$	48.0000	1.57994
$924$	0	0
$925$	10.0000	0.328798
$926$	40.0000	1.31448
$927$	0	0
$928$	6.00000	0.196960
$929$	14.0000	0.459325	0.229663	−	0.973270i	$-0.426238\pi$
$929$	14.0000	0.459325	0.229663	+	0.973270i	$0.426238\pi$
$930$	0	0
$931$	0	0
$932$	22.0000	0.720634
$933$	0	0
$934$	8.00000	0.261768
$935$	0	0
$936$	0	0
$937$	2.00000	0.0653372	0.0326686	−	0.999466i	$-0.489599\pi$
$937$	2.00000	0.0653372	0.0326686	+	0.999466i	$0.489599\pi$
$938$	0	0
$939$	0	0
$940$	−8.00000	−0.260931
$941$	−18.0000	−0.586783	−0.293392	−	0.955992i	$-0.594784\pi$
$941$	−18.0000	−0.586783	−0.293392	+	0.955992i	$0.594784\pi$
$942$	0	0
$943$	6.00000	0.195387
$944$	4.00000	0.130189
$945$	0	0
$946$	0	0
$947$	−44.0000	−1.42981	−0.714904	−	0.699223i	$-0.753530\pi$
$947$	−44.0000	−1.42981	−0.714904	+	0.699223i	$0.753530\pi$
$948$	0	0
$949$	60.0000	1.94768
$950$	0	0
$951$	0	0
$952$	0	0
$953$	6.00000	0.194359	0.0971795	−	0.995267i	$-0.469018\pi$
$953$	6.00000	0.194359	0.0971795	+	0.995267i	$0.469018\pi$
$954$	0	0
$955$	24.0000	0.776622
$956$	−8.00000	−0.258738
$957$	0	0
$958$	0	0
$959$	0	0
$960$	0	0
$961$	33.0000	1.06452
$962$	−60.0000	−1.93448
$963$	0	0
$964$	26.0000	0.837404
$965$	18.0000	0.579441
$966$	0	0
$967$	−56.0000	−1.80084	−0.900419	−	0.435023i	$-0.856740\pi$
$967$	−56.0000	−1.80084	−0.900419	+	0.435023i	$0.856740\pi$
$968$	11.0000	0.353553
$969$	0	0
$970$	−18.0000	−0.577945
$971$	32.0000	1.02693	0.513464	−	0.858111i	$-0.328362\pi$
$971$	32.0000	1.02693	0.513464	+	0.858111i	$0.328362\pi$
$972$	0	0
$973$	0	0
$974$	−16.0000	−0.512673
$975$	0	0
$976$	−6.00000	−0.192055
$977$	−50.0000	−1.59964	−0.799821	−	0.600239i	$-0.795072\pi$
$977$	−50.0000	−1.59964	−0.799821	+	0.600239i	$0.795072\pi$
$978$	0	0
$979$	0	0
$980$	−7.00000	−0.223607
$981$	0	0
$982$	36.0000	1.14881
$983$	48.0000	1.53096	0.765481	−	0.643458i	$-0.222501\pi$
$983$	48.0000	1.53096	0.765481	+	0.643458i	$0.222501\pi$
$984$	0	0
$985$	−22.0000	−0.700978
$986$	−12.0000	−0.382158
$987$	0	0
$988$	0	0
$989$	−8.00000	−0.254385
$990$	0	0
$991$	40.0000	1.27064	0.635321	−	0.772248i	$-0.280868\pi$
$991$	40.0000	1.27064	0.635321	+	0.772248i	$0.280868\pi$
$992$	−8.00000	−0.254000
$993$	0	0
$994$	0	0
$995$	−16.0000	−0.507234
$996$	0	0
$997$	38.0000	1.20347	0.601736	−	0.798695i	$-0.294476\pi$
$997$	38.0000	1.20347	0.601736	+	0.798695i	$0.294476\pi$
$998$	20.0000	0.633089
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2070.2.a.h.1.1		1
3.2	odd	2		690.2.a.h.1.1	✓	1
12.11	even	2		5520.2.a.x.1.1		1
15.2	even	4		3450.2.d.e.2899.2		2
15.8	even	4		3450.2.d.e.2899.1		2
15.14	odd	2		3450.2.a.j.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
690.2.a.h.1.1	✓	1	3.2	odd	2
2070.2.a.h.1.1		1	1.1	even	1	trivial
3450.2.a.j.1.1		1	15.14	odd	2
3450.2.d.e.2899.1		2	15.8	even	4
3450.2.d.e.2899.2		2	15.2	even	4
5520.2.a.x.1.1		1	12.11	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 $\F_{q}$

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

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