LMFDB - Embedded newform 208.2.a.d.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [208,2,Mod(1,208)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(208, 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("208.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 208 = 2^{4} \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	208.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:	$1.66088836204$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 26)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	208.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q+3.00000 q^{3} -1.00000 q^{5} -1.00000 q^{7} +6.00000 q^{9} +O(q^{10})$

$q+3.00000 q^{3} -1.00000 q^{5} -1.00000 q^{7} +6.00000 q^{9} +2.00000 q^{11} -1.00000 q^{13} -3.00000 q^{15} -3.00000 q^{17} -6.00000 q^{19} -3.00000 q^{21} +4.00000 q^{23} -4.00000 q^{25} +9.00000 q^{27} +2.00000 q^{29} -4.00000 q^{31} +6.00000 q^{33} +1.00000 q^{35} +3.00000 q^{37} -3.00000 q^{39} +5.00000 q^{43} -6.00000 q^{45} -13.0000 q^{47} -6.00000 q^{49} -9.00000 q^{51} +12.0000 q^{53} -2.00000 q^{55} -18.0000 q^{57} +10.0000 q^{59} -8.00000 q^{61} -6.00000 q^{63} +1.00000 q^{65} +2.00000 q^{67} +12.0000 q^{69} +5.00000 q^{71} -10.0000 q^{73} -12.0000 q^{75} -2.00000 q^{77} +4.00000 q^{79} +9.00000 q^{81} +3.00000 q^{85} +6.00000 q^{87} +6.00000 q^{89} +1.00000 q^{91} -12.0000 q^{93} +6.00000 q^{95} +14.0000 q^{97} +12.0000 q^{99} +O(q^{100})$

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 $

$2$	0	0
$2$	0	0
$3$	3.00000	1.73205	0.866025	−	0.500000i	$-0.166667\pi$
$3$	3.00000	1.73205	0.866025	+	0.500000i	$0.166667\pi$
$4$	0	0
$5$	−1.00000	−0.447214	−0.223607	−	0.974679i	$-0.571783\pi$
$5$	−1.00000	−0.447214	−0.223607	+	0.974679i	$0.571783\pi$
$6$	0	0
$7$	−1.00000	−0.377964	−0.188982	−	0.981981i	$-0.560519\pi$
$7$	−1.00000	−0.377964	−0.188982	+	0.981981i	$0.560519\pi$
$8$	0	0
$9$	6.00000	2.00000
$10$	0	0
$11$	2.00000	0.603023	0.301511	−	0.953463i	$-0.402509\pi$
$11$	2.00000	0.603023	0.301511	+	0.953463i	$0.402509\pi$
$12$	0	0
$13$	−1.00000	−0.277350
$13$	−1.00000	−0.277350
$14$	0	0
$15$	−3.00000	−0.774597
$16$	0	0
$17$	−3.00000	−0.727607	−0.363803	−	0.931476i	$-0.618522\pi$
$17$	−3.00000	−0.727607	−0.363803	+	0.931476i	$0.618522\pi$
$18$	0	0
$19$	−6.00000	−1.37649	−0.688247	−	0.725476i	$-0.741620\pi$
$19$	−6.00000	−1.37649	−0.688247	+	0.725476i	$0.741620\pi$
$20$	0	0
$21$	−3.00000	−0.654654
$22$	0	0
$23$	4.00000	0.834058	0.417029	−	0.908893i	$-0.363071\pi$
$23$	4.00000	0.834058	0.417029	+	0.908893i	$0.363071\pi$
$24$	0	0
$25$	−4.00000	−0.800000
$26$	0	0
$27$	9.00000	1.73205
$28$	0	0
$29$	2.00000	0.371391	0.185695	−	0.982607i	$-0.440546\pi$
$29$	2.00000	0.371391	0.185695	+	0.982607i	$0.440546\pi$
$30$	0	0
$31$	−4.00000	−0.718421	−0.359211	−	0.933257i	$-0.616954\pi$
$31$	−4.00000	−0.718421	−0.359211	+	0.933257i	$0.616954\pi$
$32$	0	0
$33$	6.00000	1.04447
$34$	0	0
$35$	1.00000	0.169031
$36$	0	0
$37$	3.00000	0.493197	0.246598	−	0.969118i	$-0.420687\pi$
$37$	3.00000	0.493197	0.246598	+	0.969118i	$0.420687\pi$
$38$	0	0
$39$	−3.00000	−0.480384
$40$	0	0
$41$	0	0		−	1.00000i	$-0.5\pi$
$41$	0	0			1.00000i	$0.5\pi$
$42$	0	0
$43$	5.00000	0.762493	0.381246	−	0.924473i	$-0.375495\pi$
$43$	5.00000	0.762493	0.381246	+	0.924473i	$0.375495\pi$
$44$	0	0
$45$	−6.00000	−0.894427
$46$	0	0
$47$	−13.0000	−1.89624	−0.948122	−	0.317905i	$-0.897021\pi$
$47$	−13.0000	−1.89624	−0.948122	+	0.317905i	$0.897021\pi$
$48$	0	0
$49$	−6.00000	−0.857143
$50$	0	0
$51$	−9.00000	−1.26025
$52$	0	0
$53$	12.0000	1.64833	0.824163	−	0.566352i	$-0.191646\pi$
$53$	12.0000	1.64833	0.824163	+	0.566352i	$0.191646\pi$
$54$	0	0
$55$	−2.00000	−0.269680
$56$	0	0
$57$	−18.0000	−2.38416
$58$	0	0
$59$	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	$0.671150\pi$
$62$	0	0
$63$	−6.00000	−0.755929
$64$	0	0
$65$	1.00000	0.124035
$66$	0	0
$67$	2.00000	0.244339	0.122169	−	0.992509i	$-0.461015\pi$
$67$	2.00000	0.244339	0.122169	+	0.992509i	$0.461015\pi$
$68$	0	0
$69$	12.0000	1.44463
$70$	0	0
$71$	5.00000	0.593391	0.296695	−	0.954972i	$-0.404115\pi$
$71$	5.00000	0.593391	0.296695	+	0.954972i	$0.404115\pi$
$72$	0	0
$73$	−10.0000	−1.17041	−0.585206	−	0.810885i	$-0.698986\pi$
$73$	−10.0000	−1.17041	−0.585206	+	0.810885i	$0.698986\pi$
$74$	0	0
$75$	−12.0000	−1.38564
$76$	0	0
$77$	−2.00000	−0.227921
$78$	0	0
$79$	4.00000	0.450035	0.225018	−	0.974355i	$-0.427756\pi$
$79$	4.00000	0.450035	0.225018	+	0.974355i	$0.427756\pi$
$80$	0	0
$81$	9.00000	1.00000
$82$	0	0
$83$	0	0		−	1.00000i	$-0.5\pi$
$83$	0	0			1.00000i	$0.5\pi$
$84$	0	0
$85$	3.00000	0.325396
$86$	0	0
$87$	6.00000	0.643268
$88$	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
$91$	1.00000	0.104828
$92$	0	0
$93$	−12.0000	−1.24434
$94$	0	0
$95$	6.00000	0.615587
$96$	0	0
$97$	14.0000	1.42148	0.710742	−	0.703452i	$-0.248359\pi$
$97$	14.0000	1.42148	0.710742	+	0.703452i	$0.248359\pi$
$98$	0	0
$99$	12.0000	1.20605
$100$	0	0
$101$	4.00000	0.398015	0.199007	−	0.979998i	$-0.436228\pi$
$101$	4.00000	0.398015	0.199007	+	0.979998i	$0.436228\pi$
$102$	0	0
$103$	8.00000	0.788263	0.394132	−	0.919054i	$-0.371045\pi$
$103$	8.00000	0.788263	0.394132	+	0.919054i	$0.371045\pi$
$104$	0	0
$105$	3.00000	0.292770
$106$	0	0
$107$	4.00000	0.386695	0.193347	−	0.981130i	$-0.438066\pi$
$107$	4.00000	0.386695	0.193347	+	0.981130i	$0.438066\pi$
$108$	0	0
$109$	19.0000	1.81987	0.909935	−	0.414751i	$-0.136131\pi$
$109$	19.0000	1.81987	0.909935	+	0.414751i	$0.136131\pi$
$110$	0	0
$111$	9.00000	0.854242
$112$	0	0
$113$	2.00000	0.188144	0.0940721	−	0.995565i	$-0.470012\pi$
$113$	2.00000	0.188144	0.0940721	+	0.995565i	$0.470012\pi$
$114$	0	0
$115$	−4.00000	−0.373002
$116$	0	0
$117$	−6.00000	−0.554700
$118$	0	0
$119$	3.00000	0.275010
$120$	0	0
$121$	−7.00000	−0.636364
$122$	0	0
$123$	0	0
$124$	0	0
$125$	9.00000	0.804984
$126$	0	0
$127$	−16.0000	−1.41977	−0.709885	−	0.704317i	$-0.751253\pi$
$127$	−16.0000	−1.41977	−0.709885	+	0.704317i	$0.751253\pi$
$128$	0	0
$129$	15.0000	1.32068
$130$	0	0
$131$	1.00000	0.0873704	0.0436852	−	0.999045i	$-0.486090\pi$
$131$	1.00000	0.0873704	0.0436852	+	0.999045i	$0.486090\pi$
$132$	0	0
$133$	6.00000	0.520266
$134$	0	0
$135$	−9.00000	−0.774597
$136$	0	0
$137$	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
$139$	−7.00000	−0.593732	−0.296866	−	0.954919i	$-0.595942\pi$
$139$	−7.00000	−0.593732	−0.296866	+	0.954919i	$0.595942\pi$
$140$	0	0
$141$	−39.0000	−3.28439
$142$	0	0
$143$	−2.00000	−0.167248
$144$	0	0
$145$	−2.00000	−0.166091
$146$	0	0
$147$	−18.0000	−1.48461
$148$	0	0
$149$	−18.0000	−1.47462	−0.737309	−	0.675556i	$-0.763904\pi$
$149$	−18.0000	−1.47462	−0.737309	+	0.675556i	$0.763904\pi$
$150$	0	0
$151$	9.00000	0.732410	0.366205	−	0.930534i	$-0.380657\pi$
$151$	9.00000	0.732410	0.366205	+	0.930534i	$0.380657\pi$
$152$	0	0
$153$	−18.0000	−1.45521
$154$	0	0
$155$	4.00000	0.321288
$156$	0	0
$157$	−10.0000	−0.798087	−0.399043	−	0.916932i	$-0.630658\pi$
$157$	−10.0000	−0.798087	−0.399043	+	0.916932i	$0.630658\pi$
$158$	0	0
$159$	36.0000	2.85499
$160$	0	0
$161$	−4.00000	−0.315244
$162$	0	0
$163$	4.00000	0.313304	0.156652	−	0.987654i	$-0.449930\pi$
$163$	4.00000	0.313304	0.156652	+	0.987654i	$0.449930\pi$
$164$	0	0
$165$	−6.00000	−0.467099
$166$	0	0
$167$	0	0		−	1.00000i	$-0.5\pi$
$167$	0	0			1.00000i	$0.5\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	−36.0000	−2.75299
$172$	0	0
$173$	20.0000	1.52057	0.760286	−	0.649589i	$-0.225059\pi$
$173$	20.0000	1.52057	0.760286	+	0.649589i	$0.225059\pi$
$174$	0	0
$175$	4.00000	0.302372
$176$	0	0
$177$	30.0000	2.25494
$178$	0	0
$179$	9.00000	0.672692	0.336346	−	0.941739i	$-0.390809\pi$
$179$	9.00000	0.672692	0.336346	+	0.941739i	$0.390809\pi$
$180$	0	0
$181$	0	0		−	1.00000i	$-0.5\pi$
$181$	0	0			1.00000i	$0.5\pi$
$182$	0	0
$183$	−24.0000	−1.77413
$184$	0	0
$185$	−3.00000	−0.220564
$186$	0	0
$187$	−6.00000	−0.438763
$188$	0	0
$189$	−9.00000	−0.654654
$190$	0	0
$191$	−10.0000	−0.723575	−0.361787	−	0.932261i	$-0.617833\pi$
$191$	−10.0000	−0.723575	−0.361787	+	0.932261i	$0.617833\pi$
$192$	0	0
$193$	−16.0000	−1.15171	−0.575853	−	0.817554i	$-0.695330\pi$
$193$	−16.0000	−1.15171	−0.575853	+	0.817554i	$0.695330\pi$
$194$	0	0
$195$	3.00000	0.214834
$196$	0	0
$197$	9.00000	0.641223	0.320612	−	0.947211i	$-0.396112\pi$
$197$	9.00000	0.641223	0.320612	+	0.947211i	$0.396112\pi$
$198$	0	0
$199$	10.0000	0.708881	0.354441	−	0.935079i	$-0.384671\pi$
$199$	10.0000	0.708881	0.354441	+	0.935079i	$0.384671\pi$
$200$	0	0
$201$	6.00000	0.423207
$202$	0	0
$203$	−2.00000	−0.140372
$204$	0	0
$205$	0	0
$206$	0	0
$207$	24.0000	1.66812
$208$	0	0
$209$	−12.0000	−0.830057
$210$	0	0
$211$	−23.0000	−1.58339	−0.791693	−	0.610920i	$-0.790800\pi$
$211$	−23.0000	−1.58339	−0.791693	+	0.610920i	$0.790800\pi$
$212$	0	0
$213$	15.0000	1.02778
$214$	0	0
$215$	−5.00000	−0.340997
$216$	0	0
$217$	4.00000	0.271538
$218$	0	0
$219$	−30.0000	−2.02721
$220$	0	0
$221$	3.00000	0.201802
$222$	0	0
$223$	21.0000	1.40626	0.703132	−	0.711059i	$-0.251784\pi$
$223$	21.0000	1.40626	0.703132	+	0.711059i	$0.251784\pi$
$224$	0	0
$225$	−24.0000	−1.60000
$226$	0	0
$227$	24.0000	1.59294	0.796468	−	0.604681i	$-0.206699\pi$
$227$	24.0000	1.59294	0.796468	+	0.604681i	$0.206699\pi$
$228$	0	0
$229$	−15.0000	−0.991228	−0.495614	−	0.868543i	$-0.665057\pi$
$229$	−15.0000	−0.991228	−0.495614	+	0.868543i	$0.665057\pi$
$230$	0	0
$231$	−6.00000	−0.394771
$232$	0	0
$233$	−11.0000	−0.720634	−0.360317	−	0.932830i	$-0.617331\pi$
$233$	−11.0000	−0.720634	−0.360317	+	0.932830i	$0.617331\pi$
$234$	0	0
$235$	13.0000	0.848026
$236$	0	0
$237$	12.0000	0.779484
$238$	0	0
$239$	−9.00000	−0.582162	−0.291081	−	0.956698i	$-0.594015\pi$
$239$	−9.00000	−0.582162	−0.291081	+	0.956698i	$0.594015\pi$
$240$	0	0
$241$	18.0000	1.15948	0.579741	−	0.814801i	$-0.303154\pi$
$241$	18.0000	1.15948	0.579741	+	0.814801i	$0.303154\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	6.00000	0.383326
$246$	0	0
$247$	6.00000	0.381771
$248$	0	0
$249$	0	0
$250$	0	0
$251$	0	0		−	1.00000i	$-0.5\pi$
$251$	0	0			1.00000i	$0.5\pi$
$252$	0	0
$253$	8.00000	0.502956
$254$	0	0
$255$	9.00000	0.563602
$256$	0	0
$257$	−15.0000	−0.935674	−0.467837	−	0.883815i	$-0.654967\pi$
$257$	−15.0000	−0.935674	−0.467837	+	0.883815i	$0.654967\pi$
$258$	0	0
$259$	−3.00000	−0.186411
$260$	0	0
$261$	12.0000	0.742781
$262$	0	0
$263$	−12.0000	−0.739952	−0.369976	−	0.929041i	$-0.620634\pi$
$263$	−12.0000	−0.739952	−0.369976	+	0.929041i	$0.620634\pi$
$264$	0	0
$265$	−12.0000	−0.737154
$266$	0	0
$267$	18.0000	1.10158
$268$	0	0
$269$	−24.0000	−1.46331	−0.731653	−	0.681677i	$-0.761251\pi$
$269$	−24.0000	−1.46331	−0.731653	+	0.681677i	$0.761251\pi$
$270$	0	0
$271$	−13.0000	−0.789694	−0.394847	−	0.918747i	$-0.629202\pi$
$271$	−13.0000	−0.789694	−0.394847	+	0.918747i	$0.629202\pi$
$272$	0	0
$273$	3.00000	0.181568
$274$	0	0
$275$	−8.00000	−0.482418
$276$	0	0
$277$	12.0000	0.721010	0.360505	−	0.932757i	$-0.382604\pi$
$277$	12.0000	0.721010	0.360505	+	0.932757i	$0.382604\pi$
$278$	0	0
$279$	−24.0000	−1.43684
$280$	0	0
$281$	−26.0000	−1.55103	−0.775515	−	0.631329i	$-0.782510\pi$
$281$	−26.0000	−1.55103	−0.775515	+	0.631329i	$0.782510\pi$
$282$	0	0
$283$	−4.00000	−0.237775	−0.118888	−	0.992908i	$-0.537933\pi$
$283$	−4.00000	−0.237775	−0.118888	+	0.992908i	$0.537933\pi$
$284$	0	0
$285$	18.0000	1.06623
$286$	0	0
$287$	0	0
$288$	0	0
$289$	−8.00000	−0.470588
$290$	0	0
$291$	42.0000	2.46208
$292$	0	0
$293$	7.00000	0.408944	0.204472	−	0.978872i	$-0.434452\pi$
$293$	7.00000	0.408944	0.204472	+	0.978872i	$0.434452\pi$
$294$	0	0
$295$	−10.0000	−0.582223
$296$	0	0
$297$	18.0000	1.04447
$298$	0	0
$299$	−4.00000	−0.231326
$300$	0	0
$301$	−5.00000	−0.288195
$302$	0	0
$303$	12.0000	0.689382
$304$	0	0
$305$	8.00000	0.458079
$306$	0	0
$307$	−14.0000	−0.799022	−0.399511	−	0.916728i	$-0.630820\pi$
$307$	−14.0000	−0.799022	−0.399511	+	0.916728i	$0.630820\pi$
$308$	0	0
$309$	24.0000	1.36531
$310$	0	0
$311$	−18.0000	−1.02069	−0.510343	−	0.859971i	$-0.670482\pi$
$311$	−18.0000	−1.02069	−0.510343	+	0.859971i	$0.670482\pi$
$312$	0	0
$313$	−1.00000	−0.0565233	−0.0282617	−	0.999601i	$-0.508997\pi$
$313$	−1.00000	−0.0565233	−0.0282617	+	0.999601i	$0.508997\pi$
$314$	0	0
$315$	6.00000	0.338062
$316$	0	0
$317$	−18.0000	−1.01098	−0.505490	−	0.862832i	$-0.668688\pi$
$317$	−18.0000	−1.01098	−0.505490	+	0.862832i	$0.668688\pi$
$318$	0	0
$319$	4.00000	0.223957
$320$	0	0
$321$	12.0000	0.669775
$322$	0	0
$323$	18.0000	1.00155
$324$	0	0
$325$	4.00000	0.221880
$326$	0	0
$327$	57.0000	3.15211
$328$	0	0
$329$	13.0000	0.716713
$330$	0	0
$331$	4.00000	0.219860	0.109930	−	0.993939i	$-0.464937\pi$
$331$	4.00000	0.219860	0.109930	+	0.993939i	$0.464937\pi$
$332$	0	0
$333$	18.0000	0.986394
$334$	0	0
$335$	−2.00000	−0.109272
$336$	0	0
$337$	23.0000	1.25289	0.626445	−	0.779466i	$-0.284509\pi$
$337$	23.0000	1.25289	0.626445	+	0.779466i	$0.284509\pi$
$338$	0	0
$339$	6.00000	0.325875
$340$	0	0
$341$	−8.00000	−0.433224
$342$	0	0
$343$	13.0000	0.701934
$344$	0	0
$345$	−12.0000	−0.646058
$346$	0	0
$347$	9.00000	0.483145	0.241573	−	0.970383i	$-0.422337\pi$
$347$	9.00000	0.483145	0.241573	+	0.970383i	$0.422337\pi$
$348$	0	0
$349$	7.00000	0.374701	0.187351	−	0.982293i	$-0.440010\pi$
$349$	7.00000	0.374701	0.187351	+	0.982293i	$0.440010\pi$
$350$	0	0
$351$	−9.00000	−0.480384
$352$	0	0
$353$	4.00000	0.212899	0.106449	−	0.994318i	$-0.466052\pi$
$353$	4.00000	0.212899	0.106449	+	0.994318i	$0.466052\pi$
$354$	0	0
$355$	−5.00000	−0.265372
$356$	0	0
$357$	9.00000	0.476331
$358$	0	0
$359$	−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
$363$	−21.0000	−1.10221
$364$	0	0
$365$	10.0000	0.523424
$366$	0	0
$367$	10.0000	0.521996	0.260998	−	0.965339i	$-0.415948\pi$
$367$	10.0000	0.521996	0.260998	+	0.965339i	$0.415948\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	−12.0000	−0.623009
$372$	0	0
$373$	−4.00000	−0.207112	−0.103556	−	0.994624i	$-0.533022\pi$
$373$	−4.00000	−0.207112	−0.103556	+	0.994624i	$0.533022\pi$
$374$	0	0
$375$	27.0000	1.39427
$376$	0	0
$377$	−2.00000	−0.103005
$378$	0	0
$379$	−16.0000	−0.821865	−0.410932	−	0.911666i	$-0.634797\pi$
$379$	−16.0000	−0.821865	−0.410932	+	0.911666i	$0.634797\pi$
$380$	0	0
$381$	−48.0000	−2.45911
$382$	0	0
$383$	−27.0000	−1.37964	−0.689818	−	0.723983i	$-0.742309\pi$
$383$	−27.0000	−1.37964	−0.689818	+	0.723983i	$0.742309\pi$
$384$	0	0
$385$	2.00000	0.101929
$386$	0	0
$387$	30.0000	1.52499
$388$	0	0
$389$	−30.0000	−1.52106	−0.760530	−	0.649303i	$-0.775061\pi$
$389$	−30.0000	−1.52106	−0.760530	+	0.649303i	$0.775061\pi$
$390$	0	0
$391$	−12.0000	−0.606866
$392$	0	0
$393$	3.00000	0.151330
$394$	0	0
$395$	−4.00000	−0.201262
$396$	0	0
$397$	−22.0000	−1.10415	−0.552074	−	0.833795i	$-0.686163\pi$
$397$	−22.0000	−1.10415	−0.552074	+	0.833795i	$0.686163\pi$
$398$	0	0
$399$	18.0000	0.901127
$400$	0	0
$401$	24.0000	1.19850	0.599251	−	0.800561i	$-0.295465\pi$
$401$	24.0000	1.19850	0.599251	+	0.800561i	$0.295465\pi$
$402$	0	0
$403$	4.00000	0.199254
$404$	0	0
$405$	−9.00000	−0.447214
$406$	0	0
$407$	6.00000	0.297409
$408$	0	0
$409$	4.00000	0.197787	0.0988936	−	0.995098i	$-0.468470\pi$
$409$	4.00000	0.197787	0.0988936	+	0.995098i	$0.468470\pi$
$410$	0	0
$411$	36.0000	1.77575
$412$	0	0
$413$	−10.0000	−0.492068
$414$	0	0
$415$	0	0
$416$	0	0
$417$	−21.0000	−1.02837
$418$	0	0
$419$	−21.0000	−1.02592	−0.512959	−	0.858413i	$-0.671451\pi$
$419$	−21.0000	−1.02592	−0.512959	+	0.858413i	$0.671451\pi$
$420$	0	0
$421$	−5.00000	−0.243685	−0.121843	−	0.992549i	$-0.538880\pi$
$421$	−5.00000	−0.243685	−0.121843	+	0.992549i	$0.538880\pi$
$422$	0	0
$423$	−78.0000	−3.79249
$424$	0	0
$425$	12.0000	0.582086
$426$	0	0
$427$	8.00000	0.387147
$428$	0	0
$429$	−6.00000	−0.289683
$430$	0	0
$431$	−33.0000	−1.58955	−0.794777	−	0.606902i	$-0.792412\pi$
$431$	−33.0000	−1.58955	−0.794777	+	0.606902i	$0.792412\pi$
$432$	0	0
$433$	7.00000	0.336399	0.168199	−	0.985753i	$-0.446205\pi$
$433$	7.00000	0.336399	0.168199	+	0.985753i	$0.446205\pi$
$434$	0	0
$435$	−6.00000	−0.287678
$436$	0	0
$437$	−24.0000	−1.14808
$438$	0	0
$439$	22.0000	1.05000	0.525001	−	0.851101i	$-0.324065\pi$
$439$	22.0000	1.05000	0.525001	+	0.851101i	$0.324065\pi$
$440$	0	0
$441$	−36.0000	−1.71429
$442$	0	0
$443$	39.0000	1.85295	0.926473	−	0.376361i	$-0.122825\pi$
$443$	39.0000	1.85295	0.926473	+	0.376361i	$0.122825\pi$
$444$	0	0
$445$	−6.00000	−0.284427
$446$	0	0
$447$	−54.0000	−2.55411
$448$	0	0
$449$	−26.0000	−1.22702	−0.613508	−	0.789689i	$-0.710242\pi$
$449$	−26.0000	−1.22702	−0.613508	+	0.789689i	$0.710242\pi$
$450$	0	0
$451$	0	0
$452$	0	0
$453$	27.0000	1.26857
$454$	0	0
$455$	−1.00000	−0.0468807
$456$	0	0
$457$	10.0000	0.467780	0.233890	−	0.972263i	$-0.424854\pi$
$457$	10.0000	0.467780	0.233890	+	0.972263i	$0.424854\pi$
$458$	0	0
$459$	−27.0000	−1.26025
$460$	0	0
$461$	−21.0000	−0.978068	−0.489034	−	0.872265i	$-0.662651\pi$
$461$	−21.0000	−0.978068	−0.489034	+	0.872265i	$0.662651\pi$
$462$	0	0
$463$	−16.0000	−0.743583	−0.371792	−	0.928316i	$-0.621256\pi$
$463$	−16.0000	−0.743583	−0.371792	+	0.928316i	$0.621256\pi$
$464$	0	0
$465$	12.0000	0.556487
$466$	0	0
$467$	−20.0000	−0.925490	−0.462745	−	0.886492i	$-0.653135\pi$
$467$	−20.0000	−0.925490	−0.462745	+	0.886492i	$0.653135\pi$
$468$	0	0
$469$	−2.00000	−0.0923514
$470$	0	0
$471$	−30.0000	−1.38233
$472$	0	0
$473$	10.0000	0.459800
$474$	0	0
$475$	24.0000	1.10120
$476$	0	0
$477$	72.0000	3.29665
$478$	0	0
$479$	3.00000	0.137073	0.0685367	−	0.997649i	$-0.478167\pi$
$479$	3.00000	0.137073	0.0685367	+	0.997649i	$0.478167\pi$
$480$	0	0
$481$	−3.00000	−0.136788
$482$	0	0
$483$	−12.0000	−0.546019
$484$	0	0
$485$	−14.0000	−0.635707
$486$	0	0
$487$	16.0000	0.725029	0.362515	−	0.931978i	$-0.381918\pi$
$487$	16.0000	0.725029	0.362515	+	0.931978i	$0.381918\pi$
$488$	0	0
$489$	12.0000	0.542659
$490$	0	0
$491$	5.00000	0.225647	0.112823	−	0.993615i	$-0.464011\pi$
$491$	5.00000	0.225647	0.112823	+	0.993615i	$0.464011\pi$
$492$	0	0
$493$	−6.00000	−0.270226
$494$	0	0
$495$	−12.0000	−0.539360
$496$	0	0
$497$	−5.00000	−0.224281
$498$	0	0
$499$	32.0000	1.43252	0.716258	−	0.697835i	$-0.245853\pi$
$499$	32.0000	1.43252	0.716258	+	0.697835i	$0.245853\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	14.0000	0.624229	0.312115	−	0.950044i	$-0.398963\pi$
$503$	14.0000	0.624229	0.312115	+	0.950044i	$0.398963\pi$
$504$	0	0
$505$	−4.00000	−0.177998
$506$	0	0
$507$	3.00000	0.133235
$508$	0	0
$509$	34.0000	1.50702	0.753512	−	0.657434i	$-0.228358\pi$
$509$	34.0000	1.50702	0.753512	+	0.657434i	$0.228358\pi$
$510$	0	0
$511$	10.0000	0.442374
$512$	0	0
$513$	−54.0000	−2.38416
$514$	0	0
$515$	−8.00000	−0.352522
$516$	0	0
$517$	−26.0000	−1.14348
$518$	0	0
$519$	60.0000	2.63371
$520$	0	0
$521$	39.0000	1.70862	0.854311	−	0.519763i	$-0.173980\pi$
$521$	39.0000	1.70862	0.854311	+	0.519763i	$0.173980\pi$
$522$	0	0
$523$	36.0000	1.57417	0.787085	−	0.616844i	$-0.211589\pi$
$523$	36.0000	1.57417	0.787085	+	0.616844i	$0.211589\pi$
$524$	0	0
$525$	12.0000	0.523723
$526$	0	0
$527$	12.0000	0.522728
$528$	0	0
$529$	−7.00000	−0.304348
$530$	0	0
$531$	60.0000	2.60378
$532$	0	0
$533$	0	0
$534$	0	0
$535$	−4.00000	−0.172935
$536$	0	0
$537$	27.0000	1.16514
$538$	0	0
$539$	−12.0000	−0.516877
$540$	0	0
$541$	17.0000	0.730887	0.365444	−	0.930834i	$-0.380917\pi$
$541$	17.0000	0.730887	0.365444	+	0.930834i	$0.380917\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	−19.0000	−0.813871
$546$	0	0
$547$	−37.0000	−1.58201	−0.791003	−	0.611812i	$-0.790441\pi$
$547$	−37.0000	−1.58201	−0.791003	+	0.611812i	$0.790441\pi$
$548$	0	0
$549$	−48.0000	−2.04859
$550$	0	0
$551$	−12.0000	−0.511217
$552$	0	0
$553$	−4.00000	−0.170097
$554$	0	0
$555$	−9.00000	−0.382029
$556$	0	0
$557$	33.0000	1.39825	0.699127	−	0.714997i	$-0.253572\pi$
$557$	33.0000	1.39825	0.699127	+	0.714997i	$0.253572\pi$
$558$	0	0
$559$	−5.00000	−0.211477
$560$	0	0
$561$	−18.0000	−0.759961
$562$	0	0
$563$	−11.0000	−0.463595	−0.231797	−	0.972764i	$-0.574461\pi$
$563$	−11.0000	−0.463595	−0.231797	+	0.972764i	$0.574461\pi$
$564$	0	0
$565$	−2.00000	−0.0841406
$566$	0	0
$567$	−9.00000	−0.377964
$568$	0	0
$569$	31.0000	1.29959	0.649794	−	0.760111i	$-0.274855\pi$
$569$	31.0000	1.29959	0.649794	+	0.760111i	$0.274855\pi$
$570$	0	0
$571$	−33.0000	−1.38101	−0.690504	−	0.723329i	$-0.742611\pi$
$571$	−33.0000	−1.38101	−0.690504	+	0.723329i	$0.742611\pi$
$572$	0	0
$573$	−30.0000	−1.25327
$574$	0	0
$575$	−16.0000	−0.667246
$576$	0	0
$577$	18.0000	0.749350	0.374675	−	0.927156i	$-0.377754\pi$
$577$	18.0000	0.749350	0.374675	+	0.927156i	$0.377754\pi$
$578$	0	0
$579$	−48.0000	−1.99481
$580$	0	0
$581$	0	0
$582$	0	0
$583$	24.0000	0.993978
$584$	0	0
$585$	6.00000	0.248069
$586$	0	0
$587$	28.0000	1.15568	0.577842	−	0.816149i	$-0.303895\pi$
$587$	28.0000	1.15568	0.577842	+	0.816149i	$0.303895\pi$
$588$	0	0
$589$	24.0000	0.988903
$590$	0	0
$591$	27.0000	1.11063
$592$	0	0
$593$	−22.0000	−0.903432	−0.451716	−	0.892162i	$-0.649188\pi$
$593$	−22.0000	−0.903432	−0.451716	+	0.892162i	$0.649188\pi$
$594$	0	0
$595$	−3.00000	−0.122988
$596$	0	0
$597$	30.0000	1.22782
$598$	0	0
$599$	2.00000	0.0817178	0.0408589	−	0.999165i	$-0.486991\pi$
$599$	2.00000	0.0817178	0.0408589	+	0.999165i	$0.486991\pi$
$600$	0	0
$601$	−35.0000	−1.42768	−0.713840	−	0.700309i	$-0.753046\pi$
$601$	−35.0000	−1.42768	−0.713840	+	0.700309i	$0.753046\pi$
$602$	0	0
$603$	12.0000	0.488678
$604$	0	0
$605$	7.00000	0.284590
$606$	0	0
$607$	−6.00000	−0.243532	−0.121766	−	0.992559i	$-0.538856\pi$
$607$	−6.00000	−0.243532	−0.121766	+	0.992559i	$0.538856\pi$
$608$	0	0
$609$	−6.00000	−0.243132
$610$	0	0
$611$	13.0000	0.525924
$612$	0	0
$613$	26.0000	1.05013	0.525065	−	0.851062i	$-0.324041\pi$
$613$	26.0000	1.05013	0.525065	+	0.851062i	$0.324041\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	16.0000	0.644136	0.322068	−	0.946717i	$-0.395622\pi$
$617$	16.0000	0.644136	0.322068	+	0.946717i	$0.395622\pi$
$618$	0	0
$619$	−4.00000	−0.160774	−0.0803868	−	0.996764i	$-0.525616\pi$
$619$	−4.00000	−0.160774	−0.0803868	+	0.996764i	$0.525616\pi$
$620$	0	0
$621$	36.0000	1.44463
$622$	0	0
$623$	−6.00000	−0.240385
$624$	0	0
$625$	11.0000	0.440000
$626$	0	0
$627$	−36.0000	−1.43770
$628$	0	0
$629$	−9.00000	−0.358854
$630$	0	0
$631$	5.00000	0.199047	0.0995234	−	0.995035i	$-0.468268\pi$
$631$	5.00000	0.199047	0.0995234	+	0.995035i	$0.468268\pi$
$632$	0	0
$633$	−69.0000	−2.74250
$634$	0	0
$635$	16.0000	0.634941
$636$	0	0
$637$	6.00000	0.237729
$638$	0	0
$639$	30.0000	1.18678
$640$	0	0
$641$	−2.00000	−0.0789953	−0.0394976	−	0.999220i	$-0.512576\pi$
$641$	−2.00000	−0.0789953	−0.0394976	+	0.999220i	$0.512576\pi$
$642$	0	0
$643$	−14.0000	−0.552106	−0.276053	−	0.961142i	$-0.589027\pi$
$643$	−14.0000	−0.552106	−0.276053	+	0.961142i	$0.589027\pi$
$644$	0	0
$645$	−15.0000	−0.590624
$646$	0	0
$647$	38.0000	1.49393	0.746967	−	0.664861i	$-0.231509\pi$
$647$	38.0000	1.49393	0.746967	+	0.664861i	$0.231509\pi$
$648$	0	0
$649$	20.0000	0.785069
$650$	0	0
$651$	12.0000	0.470317
$652$	0	0
$653$	24.0000	0.939193	0.469596	−	0.882881i	$-0.344399\pi$
$653$	24.0000	0.939193	0.469596	+	0.882881i	$0.344399\pi$
$654$	0	0
$655$	−1.00000	−0.0390732
$656$	0	0
$657$	−60.0000	−2.34082
$658$	0	0
$659$	12.0000	0.467454	0.233727	−	0.972302i	$-0.424908\pi$
$659$	12.0000	0.467454	0.233727	+	0.972302i	$0.424908\pi$
$660$	0	0
$661$	−10.0000	−0.388955	−0.194477	−	0.980907i	$-0.562301\pi$
$661$	−10.0000	−0.388955	−0.194477	+	0.980907i	$0.562301\pi$
$662$	0	0
$663$	9.00000	0.349531
$664$	0	0
$665$	−6.00000	−0.232670
$666$	0	0
$667$	8.00000	0.309761
$668$	0	0
$669$	63.0000	2.43572
$670$	0	0
$671$	−16.0000	−0.617673
$672$	0	0
$673$	37.0000	1.42625	0.713123	−	0.701039i	$-0.247280\pi$
$673$	37.0000	1.42625	0.713123	+	0.701039i	$0.247280\pi$
$674$	0	0
$675$	−36.0000	−1.38564
$676$	0	0
$677$	−36.0000	−1.38359	−0.691796	−	0.722093i	$-0.743180\pi$
$677$	−36.0000	−1.38359	−0.691796	+	0.722093i	$0.743180\pi$
$678$	0	0
$679$	−14.0000	−0.537271
$680$	0	0
$681$	72.0000	2.75905
$682$	0	0
$683$	44.0000	1.68361	0.841807	−	0.539779i	$-0.181492\pi$
$683$	44.0000	1.68361	0.841807	+	0.539779i	$0.181492\pi$
$684$	0	0
$685$	−12.0000	−0.458496
$686$	0	0
$687$	−45.0000	−1.71686
$688$	0	0
$689$	−12.0000	−0.457164
$690$	0	0
$691$	8.00000	0.304334	0.152167	−	0.988355i	$-0.451375\pi$
$691$	8.00000	0.304334	0.152167	+	0.988355i	$0.451375\pi$
$692$	0	0
$693$	−12.0000	−0.455842
$694$	0	0
$695$	7.00000	0.265525
$696$	0	0
$697$	0	0
$698$	0	0
$699$	−33.0000	−1.24817
$700$	0	0
$701$	−12.0000	−0.453234	−0.226617	−	0.973984i	$-0.572767\pi$
$701$	−12.0000	−0.453234	−0.226617	+	0.973984i	$0.572767\pi$
$702$	0	0
$703$	−18.0000	−0.678883
$704$	0	0
$705$	39.0000	1.46882
$706$	0	0
$707$	−4.00000	−0.150435
$708$	0	0
$709$	38.0000	1.42712	0.713560	−	0.700594i	$-0.247082\pi$
$709$	38.0000	1.42712	0.713560	+	0.700594i	$0.247082\pi$
$710$	0	0
$711$	24.0000	0.900070
$712$	0	0
$713$	−16.0000	−0.599205
$714$	0	0
$715$	2.00000	0.0747958
$716$	0	0
$717$	−27.0000	−1.00833
$718$	0	0
$719$	22.0000	0.820462	0.410231	−	0.911982i	$-0.365448\pi$
$719$	22.0000	0.820462	0.410231	+	0.911982i	$0.365448\pi$
$720$	0	0
$721$	−8.00000	−0.297936
$722$	0	0
$723$	54.0000	2.00828
$724$	0	0
$725$	−8.00000	−0.297113
$726$	0	0
$727$	14.0000	0.519231	0.259616	−	0.965712i	$-0.416404\pi$
$727$	14.0000	0.519231	0.259616	+	0.965712i	$0.416404\pi$
$728$	0	0
$729$	−27.0000	−1.00000
$730$	0	0
$731$	−15.0000	−0.554795
$732$	0	0
$733$	−43.0000	−1.58824	−0.794121	−	0.607760i	$-0.792068\pi$
$733$	−43.0000	−1.58824	−0.794121	+	0.607760i	$0.792068\pi$
$734$	0	0
$735$	18.0000	0.663940
$736$	0	0
$737$	4.00000	0.147342
$738$	0	0
$739$	−12.0000	−0.441427	−0.220714	−	0.975339i	$-0.570839\pi$
$739$	−12.0000	−0.441427	−0.220714	+	0.975339i	$0.570839\pi$
$740$	0	0
$741$	18.0000	0.661247
$742$	0	0
$743$	47.0000	1.72426	0.862131	−	0.506685i	$-0.169129\pi$
$743$	47.0000	1.72426	0.862131	+	0.506685i	$0.169129\pi$
$744$	0	0
$745$	18.0000	0.659469
$746$	0	0
$747$	0	0
$748$	0	0
$749$	−4.00000	−0.146157
$750$	0	0
$751$	−24.0000	−0.875772	−0.437886	−	0.899030i	$-0.644273\pi$
$751$	−24.0000	−0.875772	−0.437886	+	0.899030i	$0.644273\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	−9.00000	−0.327544
$756$	0	0
$757$	−12.0000	−0.436147	−0.218074	−	0.975932i	$-0.569977\pi$
$757$	−12.0000	−0.436147	−0.218074	+	0.975932i	$0.569977\pi$
$758$	0	0
$759$	24.0000	0.871145
$760$	0	0
$761$	6.00000	0.217500	0.108750	−	0.994069i	$-0.465315\pi$
$761$	6.00000	0.217500	0.108750	+	0.994069i	$0.465315\pi$
$762$	0	0
$763$	−19.0000	−0.687846
$764$	0	0
$765$	18.0000	0.650791
$766$	0	0
$767$	−10.0000	−0.361079
$768$	0	0
$769$	0	0		−	1.00000i	$-0.5\pi$
$769$	0	0			1.00000i	$0.5\pi$
$770$	0	0
$771$	−45.0000	−1.62064
$772$	0	0
$773$	11.0000	0.395643	0.197821	−	0.980238i	$-0.436613\pi$
$773$	11.0000	0.395643	0.197821	+	0.980238i	$0.436613\pi$
$774$	0	0
$775$	16.0000	0.574737
$776$	0	0
$777$	−9.00000	−0.322873
$778$	0	0
$779$	0	0
$780$	0	0
$781$	10.0000	0.357828
$782$	0	0
$783$	18.0000	0.643268
$784$	0	0
$785$	10.0000	0.356915
$786$	0	0
$787$	−32.0000	−1.14068	−0.570338	−	0.821410i	$-0.693188\pi$
$787$	−32.0000	−1.14068	−0.570338	+	0.821410i	$0.693188\pi$
$788$	0	0
$789$	−36.0000	−1.28163
$790$	0	0
$791$	−2.00000	−0.0711118
$792$	0	0
$793$	8.00000	0.284088
$794$	0	0
$795$	−36.0000	−1.27679
$796$	0	0
$797$	−42.0000	−1.48772	−0.743858	−	0.668338i	$-0.767006\pi$
$797$	−42.0000	−1.48772	−0.743858	+	0.668338i	$0.767006\pi$
$798$	0	0
$799$	39.0000	1.37972
$800$	0	0
$801$	36.0000	1.27200
$802$	0	0
$803$	−20.0000	−0.705785
$804$	0	0
$805$	4.00000	0.140981
$806$	0	0
$807$	−72.0000	−2.53452
$808$	0	0
$809$	−9.00000	−0.316423	−0.158212	−	0.987405i	$-0.550573\pi$
$809$	−9.00000	−0.316423	−0.158212	+	0.987405i	$0.550573\pi$
$810$	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
$813$	−39.0000	−1.36779
$814$	0	0
$815$	−4.00000	−0.140114
$816$	0	0
$817$	−30.0000	−1.04957
$818$	0	0
$819$	6.00000	0.209657
$820$	0	0
$821$	−25.0000	−0.872506	−0.436253	−	0.899824i	$-0.643695\pi$
$821$	−25.0000	−0.872506	−0.436253	+	0.899824i	$0.643695\pi$
$822$	0	0
$823$	−54.0000	−1.88232	−0.941161	−	0.337959i	$-0.890263\pi$
$823$	−54.0000	−1.88232	−0.941161	+	0.337959i	$0.890263\pi$
$824$	0	0
$825$	−24.0000	−0.835573
$826$	0	0
$827$	−30.0000	−1.04320	−0.521601	−	0.853189i	$-0.674665\pi$
$827$	−30.0000	−1.04320	−0.521601	+	0.853189i	$0.674665\pi$
$828$	0	0
$829$	−38.0000	−1.31979	−0.659897	−	0.751356i	$-0.729400\pi$
$829$	−38.0000	−1.31979	−0.659897	+	0.751356i	$0.729400\pi$
$830$	0	0
$831$	36.0000	1.24883
$832$	0	0
$833$	18.0000	0.623663
$834$	0	0
$835$	0	0
$836$	0	0
$837$	−36.0000	−1.24434
$838$	0	0
$839$	−56.0000	−1.93333	−0.966667	−	0.256036i	$-0.917584\pi$
$839$	−56.0000	−1.93333	−0.966667	+	0.256036i	$0.917584\pi$
$840$	0	0
$841$	−25.0000	−0.862069
$842$	0	0
$843$	−78.0000	−2.68646
$844$	0	0
$845$	−1.00000	−0.0344010
$846$	0	0
$847$	7.00000	0.240523
$848$	0	0
$849$	−12.0000	−0.411839
$850$	0	0
$851$	12.0000	0.411355
$852$	0	0
$853$	49.0000	1.67773	0.838864	−	0.544341i	$-0.183220\pi$
$853$	49.0000	1.67773	0.838864	+	0.544341i	$0.183220\pi$
$854$	0	0
$855$	36.0000	1.23117
$856$	0	0
$857$	46.0000	1.57133	0.785665	−	0.618652i	$-0.212321\pi$
$857$	46.0000	1.57133	0.785665	+	0.618652i	$0.212321\pi$
$858$	0	0
$859$	−20.0000	−0.682391	−0.341196	−	0.939992i	$-0.610832\pi$
$859$	−20.0000	−0.682391	−0.341196	+	0.939992i	$0.610832\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	11.0000	0.374444	0.187222	−	0.982318i	$-0.440052\pi$
$863$	11.0000	0.374444	0.187222	+	0.982318i	$0.440052\pi$
$864$	0	0
$865$	−20.0000	−0.680020
$866$	0	0
$867$	−24.0000	−0.815083
$868$	0	0
$869$	8.00000	0.271381
$870$	0	0
$871$	−2.00000	−0.0677674
$872$	0	0
$873$	84.0000	2.84297
$874$	0	0
$875$	−9.00000	−0.304256
$876$	0	0
$877$	−39.0000	−1.31694	−0.658468	−	0.752609i	$-0.728795\pi$
$877$	−39.0000	−1.31694	−0.658468	+	0.752609i	$0.728795\pi$
$878$	0	0
$879$	21.0000	0.708312
$880$	0	0
$881$	21.0000	0.707508	0.353754	−	0.935339i	$-0.384905\pi$
$881$	21.0000	0.707508	0.353754	+	0.935339i	$0.384905\pi$
$882$	0	0
$883$	47.0000	1.58168	0.790838	−	0.612026i	$-0.209645\pi$
$883$	47.0000	1.58168	0.790838	+	0.612026i	$0.209645\pi$
$884$	0	0
$885$	−30.0000	−1.00844
$886$	0	0
$887$	8.00000	0.268614	0.134307	−	0.990940i	$-0.457119\pi$
$887$	8.00000	0.268614	0.134307	+	0.990940i	$0.457119\pi$
$888$	0	0
$889$	16.0000	0.536623
$890$	0	0
$891$	18.0000	0.603023
$892$	0	0
$893$	78.0000	2.61017
$894$	0	0
$895$	−9.00000	−0.300837
$896$	0	0
$897$	−12.0000	−0.400668
$898$	0	0
$899$	−8.00000	−0.266815
$900$	0	0
$901$	−36.0000	−1.19933
$902$	0	0
$903$	−15.0000	−0.499169
$904$	0	0
$905$	0	0
$906$	0	0
$907$	9.00000	0.298840	0.149420	−	0.988774i	$-0.452259\pi$
$907$	9.00000	0.298840	0.149420	+	0.988774i	$0.452259\pi$
$908$	0	0
$909$	24.0000	0.796030
$910$	0	0
$911$	54.0000	1.78910	0.894550	−	0.446968i	$-0.147496\pi$
$911$	54.0000	1.78910	0.894550	+	0.446968i	$0.147496\pi$
$912$	0	0
$913$	0	0
$914$	0	0
$915$	24.0000	0.793416
$916$	0	0
$917$	−1.00000	−0.0330229
$918$	0	0
$919$	−24.0000	−0.791687	−0.395843	−	0.918318i	$-0.629548\pi$
$919$	−24.0000	−0.791687	−0.395843	+	0.918318i	$0.629548\pi$
$920$	0	0
$921$	−42.0000	−1.38395
$922$	0	0
$923$	−5.00000	−0.164577
$924$	0	0
$925$	−12.0000	−0.394558
$926$	0	0
$927$	48.0000	1.57653
$928$	0	0
$929$	−36.0000	−1.18112	−0.590561	−	0.806993i	$-0.701093\pi$
$929$	−36.0000	−1.18112	−0.590561	+	0.806993i	$0.701093\pi$
$930$	0	0
$931$	36.0000	1.17985
$932$	0	0
$933$	−54.0000	−1.76788
$934$	0	0
$935$	6.00000	0.196221
$936$	0	0
$937$	−42.0000	−1.37208	−0.686040	−	0.727564i	$-0.740653\pi$
$937$	−42.0000	−1.37208	−0.686040	+	0.727564i	$0.740653\pi$
$938$	0	0
$939$	−3.00000	−0.0979013
$940$	0	0
$941$	25.0000	0.814977	0.407488	−	0.913210i	$-0.366405\pi$
$941$	25.0000	0.814977	0.407488	+	0.913210i	$0.366405\pi$
$942$	0	0
$943$	0	0
$944$	0	0
$945$	9.00000	0.292770
$946$	0	0
$947$	18.0000	0.584921	0.292461	−	0.956278i	$-0.405526\pi$
$947$	18.0000	0.584921	0.292461	+	0.956278i	$0.405526\pi$
$948$	0	0
$949$	10.0000	0.324614
$950$	0	0
$951$	−54.0000	−1.75107
$952$	0	0
$953$	23.0000	0.745043	0.372522	−	0.928024i	$-0.378493\pi$
$953$	23.0000	0.745043	0.372522	+	0.928024i	$0.378493\pi$
$954$	0	0
$955$	10.0000	0.323592
$956$	0	0
$957$	12.0000	0.387905
$958$	0	0
$959$	−12.0000	−0.387500
$960$	0	0
$961$	−15.0000	−0.483871
$962$	0	0
$963$	24.0000	0.773389
$964$	0	0
$965$	16.0000	0.515058
$966$	0	0
$967$	−23.0000	−0.739630	−0.369815	−	0.929105i	$-0.620579\pi$
$967$	−23.0000	−0.739630	−0.369815	+	0.929105i	$0.620579\pi$
$968$	0	0
$969$	54.0000	1.73473
$970$	0	0
$971$	15.0000	0.481373	0.240686	−	0.970603i	$-0.422627\pi$
$971$	15.0000	0.481373	0.240686	+	0.970603i	$0.422627\pi$
$972$	0	0
$973$	7.00000	0.224410
$974$	0	0
$975$	12.0000	0.384308
$976$	0	0
$977$	−30.0000	−0.959785	−0.479893	−	0.877327i	$-0.659324\pi$
$977$	−30.0000	−0.959785	−0.479893	+	0.877327i	$0.659324\pi$
$978$	0	0
$979$	12.0000	0.383522
$980$	0	0
$981$	114.000	3.63974
$982$	0	0
$983$	31.0000	0.988746	0.494373	−	0.869250i	$-0.335398\pi$
$983$	31.0000	0.988746	0.494373	+	0.869250i	$0.335398\pi$
$984$	0	0
$985$	−9.00000	−0.286764
$986$	0	0
$987$	39.0000	1.24138
$988$	0	0
$989$	20.0000	0.635963
$990$	0	0
$991$	30.0000	0.952981	0.476491	−	0.879180i	$-0.341909\pi$
$991$	30.0000	0.952981	0.476491	+	0.879180i	$0.341909\pi$
$992$	0	0
$993$	12.0000	0.380808
$994$	0	0
$995$	−10.0000	−0.317021
$996$	0	0
$997$	−10.0000	−0.316703	−0.158352	−	0.987383i	$-0.550618\pi$
$997$	−10.0000	−0.316703	−0.158352	+	0.987383i	$0.550618\pi$
$998$	0	0
$999$	27.0000	0.854242

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	208.2.a.d.1.1		1
3.2	odd	2		1872.2.a.m.1.1		1
4.3	odd	2		26.2.a.b.1.1	✓	1
5.4	even	2		5200.2.a.c.1.1		1
8.3	odd	2		832.2.a.j.1.1		1
8.5	even	2		832.2.a.a.1.1		1
12.11	even	2		234.2.a.b.1.1		1
13.5	odd	4		2704.2.f.j.337.2		2
13.8	odd	4		2704.2.f.j.337.1		2
13.12	even	2		2704.2.a.n.1.1		1
16.3	odd	4		3328.2.b.g.1665.1		2
16.5	even	4		3328.2.b.k.1665.1		2
16.11	odd	4		3328.2.b.g.1665.2		2
16.13	even	4		3328.2.b.k.1665.2		2
20.3	even	4		650.2.b.a.599.1		2
20.7	even	4		650.2.b.a.599.2		2
20.19	odd	2		650.2.a.g.1.1		1
24.5	odd	2		7488.2.a.v.1.1		1
24.11	even	2		7488.2.a.w.1.1		1
28.3	even	6		1274.2.f.a.79.1		2
28.11	odd	6		1274.2.f.l.79.1		2
28.19	even	6		1274.2.f.a.1145.1		2
28.23	odd	6		1274.2.f.l.1145.1		2
28.27	even	2		1274.2.a.o.1.1		1
36.7	odd	6		2106.2.e.h.1405.1		2
36.11	even	6		2106.2.e.t.1405.1		2
36.23	even	6		2106.2.e.t.703.1		2
36.31	odd	6		2106.2.e.h.703.1		2
44.43	even	2		3146.2.a.a.1.1		1
52.3	odd	6		338.2.c.c.191.1		2
52.7	even	12		338.2.e.d.23.2		4
52.11	even	12		338.2.e.d.147.1		4
52.15	even	12		338.2.e.d.147.2		4
52.19	even	12		338.2.e.d.23.1		4
52.23	odd	6		338.2.c.g.191.1		2
52.31	even	4		338.2.b.a.337.1		2
52.35	odd	6		338.2.c.c.315.1		2
52.43	odd	6		338.2.c.g.315.1		2
52.47	even	4		338.2.b.a.337.2		2
52.51	odd	2		338.2.a.a.1.1		1
60.23	odd	4		5850.2.e.v.5149.2		2
60.47	odd	4		5850.2.e.v.5149.1		2
60.59	even	2		5850.2.a.bn.1.1		1
68.67	odd	2		7514.2.a.i.1.1		1
76.75	even	2		9386.2.a.f.1.1		1
156.47	odd	4		3042.2.b.f.1351.1		2
156.83	odd	4		3042.2.b.f.1351.2		2
156.155	even	2		3042.2.a.l.1.1		1
260.259	odd	2		8450.2.a.y.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
26.2.a.b.1.1	✓	1	4.3	odd	2
208.2.a.d.1.1		1	1.1	even	1	trivial
234.2.a.b.1.1		1	12.11	even	2
338.2.a.a.1.1		1	52.51	odd	2
338.2.b.a.337.1		2	52.31	even	4
338.2.b.a.337.2		2	52.47	even	4
338.2.c.c.191.1		2	52.3	odd	6
338.2.c.c.315.1		2	52.35	odd	6
338.2.c.g.191.1		2	52.23	odd	6
338.2.c.g.315.1		2	52.43	odd	6
338.2.e.d.23.1		4	52.19	even	12
338.2.e.d.23.2		4	52.7	even	12
338.2.e.d.147.1		4	52.11	even	12
338.2.e.d.147.2		4	52.15	even	12
650.2.a.g.1.1		1	20.19	odd	2
650.2.b.a.599.1		2	20.3	even	4
650.2.b.a.599.2		2	20.7	even	4
832.2.a.a.1.1		1	8.5	even	2
832.2.a.j.1.1		1	8.3	odd	2
1274.2.a.o.1.1		1	28.27	even	2
1274.2.f.a.79.1		2	28.3	even	6
1274.2.f.a.1145.1		2	28.19	even	6
1274.2.f.l.79.1		2	28.11	odd	6
1274.2.f.l.1145.1		2	28.23	odd	6
1872.2.a.m.1.1		1	3.2	odd	2
2106.2.e.h.703.1		2	36.31	odd	6
2106.2.e.h.1405.1		2	36.7	odd	6
2106.2.e.t.703.1		2	36.23	even	6
2106.2.e.t.1405.1		2	36.11	even	6
2704.2.a.n.1.1		1	13.12	even	2
2704.2.f.j.337.1		2	13.8	odd	4
2704.2.f.j.337.2		2	13.5	odd	4
3042.2.a.l.1.1		1	156.155	even	2
3042.2.b.f.1351.1		2	156.47	odd	4
3042.2.b.f.1351.2		2	156.83	odd	4
3146.2.a.a.1.1		1	44.43	even	2
3328.2.b.g.1665.1		2	16.3	odd	4
3328.2.b.g.1665.2		2	16.11	odd	4
3328.2.b.k.1665.1		2	16.5	even	4
3328.2.b.k.1665.2		2	16.13	even	4
5200.2.a.c.1.1		1	5.4	even	2
5850.2.a.bn.1.1		1	60.59	even	2
5850.2.e.v.5149.1		2	60.47	odd	4
5850.2.e.v.5149.2		2	60.23	odd	4
7488.2.a.v.1.1		1	24.5	odd	2
7488.2.a.w.1.1		1	24.11	even	2
7514.2.a.i.1.1		1	68.67	odd	2
8450.2.a.y.1.1		1	260.259	odd	2
9386.2.a.f.1.1		1	76.75	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:	\( N \)	\(=\)	\( 208 = 2^{4} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	208.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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