LMFDB - Embedded newform 560.2.a.f.1.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [560,2,Mod(1,560)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("560.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(560, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 560 = 2^{4} \cdot 5 \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	560.a (trivial)

Newform invariants

comment:select newform

sage:traces = [1,0,3,0,1] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)

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

Self dual:	yes
Analytic conductor:	$4.47162251319$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 280)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	560.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$

$=$

$q+3.00000 q^{3} +1.00000 q^{5} -1.00000 q^{7} +6.00000 q^{9} +5.00000 q^{11} -5.00000 q^{13} +3.00000 q^{15} -7.00000 q^{17} +2.00000 q^{19} -3.00000 q^{21} +2.00000 q^{23} +1.00000 q^{25} +9.00000 q^{27} +7.00000 q^{29} -4.00000 q^{31} +15.0000 q^{33} -1.00000 q^{35} -6.00000 q^{37} -15.0000 q^{39} -12.0000 q^{41} +2.00000 q^{43} +6.00000 q^{45} -1.00000 q^{47} +1.00000 q^{49} -21.0000 q^{51} +5.00000 q^{55} +6.00000 q^{57} +4.00000 q^{59} +4.00000 q^{61} -6.00000 q^{63} -5.00000 q^{65} -8.00000 q^{67} +6.00000 q^{69} +6.00000 q^{73} +3.00000 q^{75} -5.00000 q^{77} +3.00000 q^{79} +9.00000 q^{81} +4.00000 q^{83} -7.00000 q^{85} +21.0000 q^{87} +5.00000 q^{91} -12.0000 q^{93} +2.00000 q^{95} +13.0000 q^{97} +30.0000 q^{99} +O(q^{100})$

Coefficient data

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

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

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

$2$	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
$5$	1.00000	0.447214
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	0	0
$9$	6.00000	2.00000
$10$	0	0
$11$	5.00000	1.50756	0.753778	−	0.657129i	$-0.228229\pi$
$11$	5.00000	1.50756	0.753778	+	0.657129i	$0.228229\pi$
$12$	0	0
$13$	−5.00000	−1.38675	−0.693375	−	0.720577i	$-0.743877\pi$
$13$	−5.00000	−1.38675	−0.693375	+	0.720577i	$0.743877\pi$
$14$	0	0
$15$	3.00000	0.774597
$16$	0	0
$17$	−7.00000	−1.69775	−0.848875	−	0.528594i	$-0.822719\pi$
$17$	−7.00000	−1.69775	−0.848875	+	0.528594i	$0.822719\pi$
$18$	0	0
$19$	2.00000	0.458831	0.229416	−	0.973329i	$-0.426318\pi$
$19$	2.00000	0.458831	0.229416	+	0.973329i	$0.426318\pi$
$20$	0	0
$21$	−3.00000	−0.654654
$22$	0	0
$23$	2.00000	0.417029	0.208514	−	0.978019i	$-0.433137\pi$
$23$	2.00000	0.417029	0.208514	+	0.978019i	$0.433137\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	9.00000	1.73205
$28$	0	0
$29$	7.00000	1.29987	0.649934	−	0.759991i	$-0.274797\pi$
$29$	7.00000	1.29987	0.649934	+	0.759991i	$0.274797\pi$
$30$	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
$33$	15.0000	2.61116
$34$	0	0
$35$	−1.00000	−0.169031
$36$	0	0
$37$	−6.00000	−0.986394	−0.493197	−	0.869918i	$-0.664172\pi$
$37$	−6.00000	−0.986394	−0.493197	+	0.869918i	$0.664172\pi$
$38$	0	0
$39$	−15.0000	−2.40192
$40$	0	0
$41$	−12.0000	−1.87409	−0.937043	−	0.349215i	$-0.886448\pi$
$41$	−12.0000	−1.87409	−0.937043	+	0.349215i	$0.886448\pi$
$42$	0	0
$43$	2.00000	0.304997	0.152499	−	0.988304i	$-0.451268\pi$
$43$	2.00000	0.304997	0.152499	+	0.988304i	$0.451268\pi$
$44$	0	0
$45$	6.00000	0.894427
$46$	0	0
$47$	−1.00000	−0.145865	−0.0729325	−	0.997337i	$-0.523236\pi$
$47$	−1.00000	−0.145865	−0.0729325	+	0.997337i	$0.523236\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	−21.0000	−2.94059
$52$	0	0
$53$	0	0		−	1.00000i	$-0.5\pi$
$53$	0	0			1.00000i	$0.5\pi$
$54$	0	0
$55$	5.00000	0.674200
$56$	0	0
$57$	6.00000	0.794719
$58$	0	0
$59$	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
$61$	4.00000	0.512148	0.256074	−	0.966657i	$-0.417571\pi$
$61$	4.00000	0.512148	0.256074	+	0.966657i	$0.417571\pi$
$62$	0	0
$63$	−6.00000	−0.755929
$64$	0	0
$65$	−5.00000	−0.620174
$66$	0	0
$67$	−8.00000	−0.977356	−0.488678	−	0.872464i	$-0.662521\pi$
$67$	−8.00000	−0.977356	−0.488678	+	0.872464i	$0.662521\pi$
$68$	0	0
$69$	6.00000	0.722315
$70$	0	0
$71$	0	0		−	1.00000i	$-0.5\pi$
$71$	0	0			1.00000i	$0.5\pi$
$72$	0	0
$73$	6.00000	0.702247	0.351123	−	0.936329i	$-0.385800\pi$
$73$	6.00000	0.702247	0.351123	+	0.936329i	$0.385800\pi$
$74$	0	0
$75$	3.00000	0.346410
$76$	0	0
$77$	−5.00000	−0.569803
$78$	0	0
$79$	3.00000	0.337526	0.168763	−	0.985657i	$-0.446023\pi$
$79$	3.00000	0.337526	0.168763	+	0.985657i	$0.446023\pi$
$80$	0	0
$81$	9.00000	1.00000
$82$	0	0
$83$	4.00000	0.439057	0.219529	−	0.975606i	$-0.429548\pi$
$83$	4.00000	0.439057	0.219529	+	0.975606i	$0.429548\pi$
$84$	0	0
$85$	−7.00000	−0.759257
$86$	0	0
$87$	21.0000	2.25144
$88$	0	0
$89$	0	0		−	1.00000i	$-0.5\pi$
$89$	0	0			1.00000i	$0.5\pi$
$90$	0	0
$91$	5.00000	0.524142
$92$	0	0
$93$	−12.0000	−1.24434
$94$	0	0
$95$	2.00000	0.205196
$96$	0	0
$97$	13.0000	1.31995	0.659975	−	0.751288i	$-0.270567\pi$
$97$	13.0000	1.31995	0.659975	+	0.751288i	$0.270567\pi$
$98$	0	0
$99$	30.0000	3.01511
$100$	0	0
$101$	−18.0000	−1.79107	−0.895533	−	0.444994i	$-0.853206\pi$
$101$	−18.0000	−1.79107	−0.895533	+	0.444994i	$0.853206\pi$
$102$	0	0
$103$	−13.0000	−1.28093	−0.640464	−	0.767988i	$-0.721258\pi$
$103$	−13.0000	−1.28093	−0.640464	+	0.767988i	$0.721258\pi$
$104$	0	0
$105$	−3.00000	−0.292770
$106$	0	0
$107$	−18.0000	−1.74013	−0.870063	−	0.492941i	$-0.835922\pi$
$107$	−18.0000	−1.74013	−0.870063	+	0.492941i	$0.835922\pi$
$108$	0	0
$109$	5.00000	0.478913	0.239457	−	0.970907i	$-0.423031\pi$
$109$	5.00000	0.478913	0.239457	+	0.970907i	$0.423031\pi$
$110$	0	0
$111$	−18.0000	−1.70848
$112$	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
$115$	2.00000	0.186501
$116$	0	0
$117$	−30.0000	−2.77350
$118$	0	0
$119$	7.00000	0.641689
$120$	0	0
$121$	14.0000	1.27273
$122$	0	0
$123$	−36.0000	−3.24601
$124$	0	0
$125$	1.00000	0.0894427
$126$	0	0
$127$	12.0000	1.06483	0.532414	−	0.846484i	$-0.321285\pi$
$127$	12.0000	1.06483	0.532414	+	0.846484i	$0.321285\pi$
$128$	0	0
$129$	6.00000	0.528271
$130$	0	0
$131$	6.00000	0.524222	0.262111	−	0.965038i	$-0.415581\pi$
$131$	6.00000	0.524222	0.262111	+	0.965038i	$0.415581\pi$
$132$	0	0
$133$	−2.00000	−0.173422
$134$	0	0
$135$	9.00000	0.774597
$136$	0	0
$137$	−8.00000	−0.683486	−0.341743	−	0.939793i	$-0.611017\pi$
$137$	−8.00000	−0.683486	−0.341743	+	0.939793i	$0.611017\pi$
$138$	0	0
$139$	−18.0000	−1.52674	−0.763370	−	0.645961i	$-0.776457\pi$
$139$	−18.0000	−1.52674	−0.763370	+	0.645961i	$0.776457\pi$
$140$	0	0
$141$	−3.00000	−0.252646
$142$	0	0
$143$	−25.0000	−2.09061
$144$	0	0
$145$	7.00000	0.581318
$146$	0	0
$147$	3.00000	0.247436
$148$	0	0
$149$	−22.0000	−1.80231	−0.901155	−	0.433497i	$-0.857280\pi$
$149$	−22.0000	−1.80231	−0.901155	+	0.433497i	$0.857280\pi$
$150$	0	0
$151$	19.0000	1.54620	0.773099	−	0.634285i	$-0.218706\pi$
$151$	19.0000	1.54620	0.773099	+	0.634285i	$0.218706\pi$
$152$	0	0
$153$	−42.0000	−3.39550
$154$	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	$0.369342\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−2.00000	−0.157622
$162$	0	0
$163$	14.0000	1.09656	0.548282	−	0.836293i	$-0.315282\pi$
$163$	14.0000	1.09656	0.548282	+	0.836293i	$0.315282\pi$
$164$	0	0
$165$	15.0000	1.16775
$166$	0	0
$167$	3.00000	0.232147	0.116073	−	0.993241i	$-0.462969\pi$
$167$	3.00000	0.232147	0.116073	+	0.993241i	$0.462969\pi$
$168$	0	0
$169$	12.0000	0.923077
$170$	0	0
$171$	12.0000	0.917663
$172$	0	0
$173$	−7.00000	−0.532200	−0.266100	−	0.963945i	$-0.585735\pi$
$173$	−7.00000	−0.532200	−0.266100	+	0.963945i	$0.585735\pi$
$174$	0	0
$175$	−1.00000	−0.0755929
$176$	0	0
$177$	12.0000	0.901975
$178$	0	0
$179$	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
$181$	8.00000	0.594635	0.297318	−	0.954779i	$-0.403908\pi$
$181$	8.00000	0.594635	0.297318	+	0.954779i	$0.403908\pi$
$182$	0	0
$183$	12.0000	0.887066
$184$	0	0
$185$	−6.00000	−0.441129
$186$	0	0
$187$	−35.0000	−2.55945
$188$	0	0
$189$	−9.00000	−0.654654
$190$	0	0
$191$	13.0000	0.940647	0.470323	−	0.882494i	$-0.344137\pi$
$191$	13.0000	0.940647	0.470323	+	0.882494i	$0.344137\pi$
$192$	0	0
$193$	−8.00000	−0.575853	−0.287926	−	0.957653i	$-0.592966\pi$
$193$	−8.00000	−0.575853	−0.287926	+	0.957653i	$0.592966\pi$
$194$	0	0
$195$	−15.0000	−1.07417
$196$	0	0
$197$	−8.00000	−0.569976	−0.284988	−	0.958531i	$-0.591990\pi$
$197$	−8.00000	−0.569976	−0.284988	+	0.958531i	$0.591990\pi$
$198$	0	0
$199$	4.00000	0.283552	0.141776	−	0.989899i	$-0.454719\pi$
$199$	4.00000	0.283552	0.141776	+	0.989899i	$0.454719\pi$
$200$	0	0
$201$	−24.0000	−1.69283
$202$	0	0
$203$	−7.00000	−0.491304
$204$	0	0
$205$	−12.0000	−0.838116
$206$	0	0
$207$	12.0000	0.834058
$208$	0	0
$209$	10.0000	0.691714
$210$	0	0
$211$	−5.00000	−0.344214	−0.172107	−	0.985078i	$-0.555058\pi$
$211$	−5.00000	−0.344214	−0.172107	+	0.985078i	$0.555058\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	2.00000	0.136399
$216$	0	0
$217$	4.00000	0.271538
$218$	0	0
$219$	18.0000	1.21633
$220$	0	0
$221$	35.0000	2.35435
$222$	0	0
$223$	19.0000	1.27233	0.636167	−	0.771551i	$-0.280519\pi$
$223$	19.0000	1.27233	0.636167	+	0.771551i	$0.280519\pi$
$224$	0	0
$225$	6.00000	0.400000
$226$	0	0
$227$	−9.00000	−0.597351	−0.298675	−	0.954355i	$-0.596545\pi$
$227$	−9.00000	−0.597351	−0.298675	+	0.954355i	$0.596545\pi$
$228$	0	0
$229$	4.00000	0.264327	0.132164	−	0.991228i	$-0.457808\pi$
$229$	4.00000	0.264327	0.132164	+	0.991228i	$0.457808\pi$
$230$	0	0
$231$	−15.0000	−0.986928
$232$	0	0
$233$	−24.0000	−1.57229	−0.786146	−	0.618041i	$-0.787927\pi$
$233$	−24.0000	−1.57229	−0.786146	+	0.618041i	$0.787927\pi$
$234$	0	0
$235$	−1.00000	−0.0652328
$236$	0	0
$237$	9.00000	0.584613
$238$	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
$241$	−22.0000	−1.41714	−0.708572	−	0.705638i	$-0.750660\pi$
$241$	−22.0000	−1.41714	−0.708572	+	0.705638i	$0.750660\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	1.00000	0.0638877
$246$	0	0
$247$	−10.0000	−0.636285
$248$	0	0
$249$	12.0000	0.760469
$250$	0	0
$251$	−6.00000	−0.378717	−0.189358	−	0.981908i	$-0.560641\pi$
$251$	−6.00000	−0.378717	−0.189358	+	0.981908i	$0.560641\pi$
$252$	0	0
$253$	10.0000	0.628695
$254$	0	0
$255$	−21.0000	−1.31507
$256$	0	0
$257$	18.0000	1.12281	0.561405	−	0.827541i	$-0.310261\pi$
$257$	18.0000	1.12281	0.561405	+	0.827541i	$0.310261\pi$
$258$	0	0
$259$	6.00000	0.372822
$260$	0	0
$261$	42.0000	2.59973
$262$	0	0
$263$	30.0000	1.84988	0.924940	−	0.380114i	$-0.124115\pi$
$263$	30.0000	1.84988	0.924940	+	0.380114i	$0.124115\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	0	0
$268$	0	0
$269$	26.0000	1.58525	0.792624	−	0.609711i	$-0.208714\pi$
$269$	26.0000	1.58525	0.792624	+	0.609711i	$0.208714\pi$
$270$	0	0
$271$	−12.0000	−0.728948	−0.364474	−	0.931214i	$-0.618751\pi$
$271$	−12.0000	−0.728948	−0.364474	+	0.931214i	$0.618751\pi$
$272$	0	0
$273$	15.0000	0.907841
$274$	0	0
$275$	5.00000	0.301511
$276$	0	0
$277$	2.00000	0.120168	0.0600842	−	0.998193i	$-0.480863\pi$
$277$	2.00000	0.120168	0.0600842	+	0.998193i	$0.480863\pi$
$278$	0	0
$279$	−24.0000	−1.43684
$280$	0	0
$281$	19.0000	1.13344	0.566722	−	0.823909i	$-0.308211\pi$
$281$	19.0000	1.13344	0.566722	+	0.823909i	$0.308211\pi$
$282$	0	0
$283$	25.0000	1.48610	0.743048	−	0.669238i	$-0.233379\pi$
$283$	25.0000	1.48610	0.743048	+	0.669238i	$0.233379\pi$
$284$	0	0
$285$	6.00000	0.355409
$286$	0	0
$287$	12.0000	0.708338
$288$	0	0
$289$	32.0000	1.88235
$290$	0	0
$291$	39.0000	2.28622
$292$	0	0
$293$	13.0000	0.759468	0.379734	−	0.925096i	$-0.376015\pi$
$293$	13.0000	0.759468	0.379734	+	0.925096i	$0.376015\pi$
$294$	0	0
$295$	4.00000	0.232889
$296$	0	0
$297$	45.0000	2.61116
$298$	0	0
$299$	−10.0000	−0.578315
$300$	0	0
$301$	−2.00000	−0.115278
$302$	0	0
$303$	−54.0000	−3.10222
$304$	0	0
$305$	4.00000	0.229039
$306$	0	0
$307$	17.0000	0.970241	0.485121	−	0.874447i	$-0.338776\pi$
$307$	17.0000	0.970241	0.485121	+	0.874447i	$0.338776\pi$
$308$	0	0
$309$	−39.0000	−2.21863
$310$	0	0
$311$	34.0000	1.92796	0.963982	−	0.265969i	$-0.0856919\pi$
$311$	34.0000	1.92796	0.963982	+	0.265969i	$0.0856919\pi$
$312$	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
$317$	6.00000	0.336994	0.168497	−	0.985702i	$-0.446109\pi$
$317$	6.00000	0.336994	0.168497	+	0.985702i	$0.446109\pi$
$318$	0	0
$319$	35.0000	1.95962
$320$	0	0
$321$	−54.0000	−3.01399
$322$	0	0
$323$	−14.0000	−0.778981
$324$	0	0
$325$	−5.00000	−0.277350
$326$	0	0
$327$	15.0000	0.829502
$328$	0	0
$329$	1.00000	0.0551318
$330$	0	0
$331$	12.0000	0.659580	0.329790	−	0.944054i	$-0.393022\pi$
$331$	12.0000	0.659580	0.329790	+	0.944054i	$0.393022\pi$
$332$	0	0
$333$	−36.0000	−1.97279
$334$	0	0
$335$	−8.00000	−0.437087
$336$	0	0
$337$	34.0000	1.85210	0.926049	−	0.377403i	$-0.123183\pi$
$337$	34.0000	1.85210	0.926049	+	0.377403i	$0.123183\pi$
$338$	0	0
$339$	18.0000	0.977626
$340$	0	0
$341$	−20.0000	−1.08306
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	0	0
$345$	6.00000	0.323029
$346$	0	0
$347$	−26.0000	−1.39575	−0.697877	−	0.716218i	$-0.745872\pi$
$347$	−26.0000	−1.39575	−0.697877	+	0.716218i	$0.745872\pi$
$348$	0	0
$349$	18.0000	0.963518	0.481759	−	0.876304i	$-0.339998\pi$
$349$	18.0000	0.963518	0.481759	+	0.876304i	$0.339998\pi$
$350$	0	0
$351$	−45.0000	−2.40192
$352$	0	0
$353$	5.00000	0.266123	0.133062	−	0.991108i	$-0.457519\pi$
$353$	5.00000	0.266123	0.133062	+	0.991108i	$0.457519\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	21.0000	1.11144
$358$	0	0
$359$	0	0		−	1.00000i	$-0.5\pi$
$359$	0	0			1.00000i	$0.5\pi$
$360$	0	0
$361$	−15.0000	−0.789474
$362$	0	0
$363$	42.0000	2.20443
$364$	0	0
$365$	6.00000	0.314054
$366$	0	0
$367$	7.00000	0.365397	0.182699	−	0.983169i	$-0.441517\pi$
$367$	7.00000	0.365397	0.182699	+	0.983169i	$0.441517\pi$
$368$	0	0
$369$	−72.0000	−3.74817
$370$	0	0
$371$	0	0
$372$	0	0
$373$	16.0000	0.828449	0.414224	−	0.910175i	$-0.364053\pi$
$373$	16.0000	0.828449	0.414224	+	0.910175i	$0.364053\pi$
$374$	0	0
$375$	3.00000	0.154919
$376$	0	0
$377$	−35.0000	−1.80259
$378$	0	0
$379$	20.0000	1.02733	0.513665	−	0.857991i	$-0.328287\pi$
$379$	20.0000	1.02733	0.513665	+	0.857991i	$0.328287\pi$
$380$	0	0
$381$	36.0000	1.84434
$382$	0	0
$383$	−12.0000	−0.613171	−0.306586	−	0.951843i	$-0.599187\pi$
$383$	−12.0000	−0.613171	−0.306586	+	0.951843i	$0.599187\pi$
$384$	0	0
$385$	−5.00000	−0.254824
$386$	0	0
$387$	12.0000	0.609994
$388$	0	0
$389$	−15.0000	−0.760530	−0.380265	−	0.924878i	$-0.624167\pi$
$389$	−15.0000	−0.760530	−0.380265	+	0.924878i	$0.624167\pi$
$390$	0	0
$391$	−14.0000	−0.708010
$392$	0	0
$393$	18.0000	0.907980
$394$	0	0
$395$	3.00000	0.150946
$396$	0	0
$397$	−23.0000	−1.15434	−0.577168	−	0.816625i	$-0.695842\pi$
$397$	−23.0000	−1.15434	−0.577168	+	0.816625i	$0.695842\pi$
$398$	0	0
$399$	−6.00000	−0.300376
$400$	0	0
$401$	−15.0000	−0.749064	−0.374532	−	0.927214i	$-0.622197\pi$
$401$	−15.0000	−0.749064	−0.374532	+	0.927214i	$0.622197\pi$
$402$	0	0
$403$	20.0000	0.996271
$404$	0	0
$405$	9.00000	0.447214
$406$	0	0
$407$	−30.0000	−1.48704
$408$	0	0
$409$	−14.0000	−0.692255	−0.346128	−	0.938187i	$-0.612504\pi$
$409$	−14.0000	−0.692255	−0.346128	+	0.938187i	$0.612504\pi$
$410$	0	0
$411$	−24.0000	−1.18383
$412$	0	0
$413$	−4.00000	−0.196827
$414$	0	0
$415$	4.00000	0.196352
$416$	0	0
$417$	−54.0000	−2.64439
$418$	0	0
$419$	24.0000	1.17248	0.586238	−	0.810139i	$-0.300608\pi$
$419$	24.0000	1.17248	0.586238	+	0.810139i	$0.300608\pi$
$420$	0	0
$421$	−3.00000	−0.146211	−0.0731055	−	0.997324i	$-0.523291\pi$
$421$	−3.00000	−0.146211	−0.0731055	+	0.997324i	$0.523291\pi$
$422$	0	0
$423$	−6.00000	−0.291730
$424$	0	0
$425$	−7.00000	−0.339550
$426$	0	0
$427$	−4.00000	−0.193574
$428$	0	0
$429$	−75.0000	−3.62103
$430$	0	0
$431$	25.0000	1.20421	0.602104	−	0.798418i	$-0.294329\pi$
$431$	25.0000	1.20421	0.602104	+	0.798418i	$0.294329\pi$
$432$	0	0
$433$	14.0000	0.672797	0.336399	−	0.941720i	$-0.390791\pi$
$433$	14.0000	0.672797	0.336399	+	0.941720i	$0.390791\pi$
$434$	0	0
$435$	21.0000	1.00687
$436$	0	0
$437$	4.00000	0.191346
$438$	0	0
$439$	26.0000	1.24091	0.620456	−	0.784241i	$-0.286947\pi$
$439$	26.0000	1.24091	0.620456	+	0.784241i	$0.286947\pi$
$440$	0	0
$441$	6.00000	0.285714
$442$	0	0
$443$	−6.00000	−0.285069	−0.142534	−	0.989790i	$-0.545525\pi$
$443$	−6.00000	−0.285069	−0.142534	+	0.989790i	$0.545525\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	−66.0000	−3.12169
$448$	0	0
$449$	−9.00000	−0.424736	−0.212368	−	0.977190i	$-0.568118\pi$
$449$	−9.00000	−0.424736	−0.212368	+	0.977190i	$0.568118\pi$
$450$	0	0
$451$	−60.0000	−2.82529
$452$	0	0
$453$	57.0000	2.67809
$454$	0	0
$455$	5.00000	0.234404
$456$	0	0
$457$	−8.00000	−0.374224	−0.187112	−	0.982339i	$-0.559913\pi$
$457$	−8.00000	−0.374224	−0.187112	+	0.982339i	$0.559913\pi$
$458$	0	0
$459$	−63.0000	−2.94059
$460$	0	0
$461$	0	0		−	1.00000i	$-0.5\pi$
$461$	0	0			1.00000i	$0.5\pi$
$462$	0	0
$463$	36.0000	1.67306	0.836531	−	0.547920i	$-0.184580\pi$
$463$	36.0000	1.67306	0.836531	+	0.547920i	$0.184580\pi$
$464$	0	0
$465$	−12.0000	−0.556487
$466$	0	0
$467$	−11.0000	−0.509019	−0.254510	−	0.967070i	$-0.581914\pi$
$467$	−11.0000	−0.509019	−0.254510	+	0.967070i	$0.581914\pi$
$468$	0	0
$469$	8.00000	0.369406
$470$	0	0
$471$	30.0000	1.38233
$472$	0	0
$473$	10.0000	0.459800
$474$	0	0
$475$	2.00000	0.0917663
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−34.0000	−1.55350	−0.776750	−	0.629809i	$-0.783133\pi$
$479$	−34.0000	−1.55350	−0.776750	+	0.629809i	$0.783133\pi$
$480$	0	0
$481$	30.0000	1.36788
$482$	0	0
$483$	−6.00000	−0.273009
$484$	0	0
$485$	13.0000	0.590300
$486$	0	0
$487$	−22.0000	−0.996915	−0.498458	−	0.866914i	$-0.666100\pi$
$487$	−22.0000	−0.996915	−0.498458	+	0.866914i	$0.666100\pi$
$488$	0	0
$489$	42.0000	1.89931
$490$	0	0
$491$	−9.00000	−0.406164	−0.203082	−	0.979162i	$-0.565096\pi$
$491$	−9.00000	−0.406164	−0.203082	+	0.979162i	$0.565096\pi$
$492$	0	0
$493$	−49.0000	−2.20685
$494$	0	0
$495$	30.0000	1.34840
$496$	0	0
$497$	0	0
$498$	0	0
$499$	1.00000	0.0447661	0.0223831	−	0.999749i	$-0.492875\pi$
$499$	1.00000	0.0447661	0.0223831	+	0.999749i	$0.492875\pi$
$500$	0	0
$501$	9.00000	0.402090
$502$	0	0
$503$	−3.00000	−0.133763	−0.0668817	−	0.997761i	$-0.521305\pi$
$503$	−3.00000	−0.133763	−0.0668817	+	0.997761i	$0.521305\pi$
$504$	0	0
$505$	−18.0000	−0.800989
$506$	0	0
$507$	36.0000	1.59882
$508$	0	0
$509$	−14.0000	−0.620539	−0.310270	−	0.950649i	$-0.600419\pi$
$509$	−14.0000	−0.620539	−0.310270	+	0.950649i	$0.600419\pi$
$510$	0	0
$511$	−6.00000	−0.265424
$512$	0	0
$513$	18.0000	0.794719
$514$	0	0
$515$	−13.0000	−0.572848
$516$	0	0
$517$	−5.00000	−0.219900
$518$	0	0
$519$	−21.0000	−0.921798
$520$	0	0
$521$	22.0000	0.963837	0.481919	−	0.876216i	$-0.339940\pi$
$521$	22.0000	0.963837	0.481919	+	0.876216i	$0.339940\pi$
$522$	0	0
$523$	44.0000	1.92399	0.961993	−	0.273075i	$-0.0880406\pi$
$523$	44.0000	1.92399	0.961993	+	0.273075i	$0.0880406\pi$
$524$	0	0
$525$	−3.00000	−0.130931
$526$	0	0
$527$	28.0000	1.21970
$528$	0	0
$529$	−19.0000	−0.826087
$530$	0	0
$531$	24.0000	1.04151
$532$	0	0
$533$	60.0000	2.59889
$534$	0	0
$535$	−18.0000	−0.778208
$536$	0	0
$537$	12.0000	0.517838
$538$	0	0
$539$	5.00000	0.215365
$540$	0	0
$541$	7.00000	0.300954	0.150477	−	0.988614i	$-0.451919\pi$
$541$	7.00000	0.300954	0.150477	+	0.988614i	$0.451919\pi$
$542$	0	0
$543$	24.0000	1.02994
$544$	0	0
$545$	5.00000	0.214176
$546$	0	0
$547$	28.0000	1.19719	0.598597	−	0.801050i	$-0.295725\pi$
$547$	28.0000	1.19719	0.598597	+	0.801050i	$0.295725\pi$
$548$	0	0
$549$	24.0000	1.02430
$550$	0	0
$551$	14.0000	0.596420
$552$	0	0
$553$	−3.00000	−0.127573
$554$	0	0
$555$	−18.0000	−0.764057
$556$	0	0
$557$	−16.0000	−0.677942	−0.338971	−	0.940797i	$-0.610079\pi$
$557$	−16.0000	−0.677942	−0.338971	+	0.940797i	$0.610079\pi$
$558$	0	0
$559$	−10.0000	−0.422955
$560$	0	0
$561$	−105.000	−4.43310
$562$	0	0
$563$	−4.00000	−0.168580	−0.0842900	−	0.996441i	$-0.526862\pi$
$563$	−4.00000	−0.168580	−0.0842900	+	0.996441i	$0.526862\pi$
$564$	0	0
$565$	6.00000	0.252422
$566$	0	0
$567$	−9.00000	−0.377964
$568$	0	0
$569$	−22.0000	−0.922288	−0.461144	−	0.887325i	$-0.652561\pi$
$569$	−22.0000	−0.922288	−0.461144	+	0.887325i	$0.652561\pi$
$570$	0	0
$571$	−4.00000	−0.167395	−0.0836974	−	0.996491i	$-0.526673\pi$
$571$	−4.00000	−0.167395	−0.0836974	+	0.996491i	$0.526673\pi$
$572$	0	0
$573$	39.0000	1.62925
$574$	0	0
$575$	2.00000	0.0834058
$576$	0	0
$577$	−37.0000	−1.54033	−0.770165	−	0.637845i	$-0.779826\pi$
$577$	−37.0000	−1.54033	−0.770165	+	0.637845i	$0.779826\pi$
$578$	0	0
$579$	−24.0000	−0.997406
$580$	0	0
$581$	−4.00000	−0.165948
$582$	0	0
$583$	0	0
$584$	0	0
$585$	−30.0000	−1.24035
$586$	0	0
$587$	0	0		−	1.00000i	$-0.5\pi$
$587$	0	0			1.00000i	$0.5\pi$
$588$	0	0
$589$	−8.00000	−0.329634
$590$	0	0
$591$	−24.0000	−0.987228
$592$	0	0
$593$	27.0000	1.10876	0.554379	−	0.832265i	$-0.312956\pi$
$593$	27.0000	1.10876	0.554379	+	0.832265i	$0.312956\pi$
$594$	0	0
$595$	7.00000	0.286972
$596$	0	0
$597$	12.0000	0.491127
$598$	0	0
$599$	−15.0000	−0.612883	−0.306442	−	0.951889i	$-0.599138\pi$
$599$	−15.0000	−0.612883	−0.306442	+	0.951889i	$0.599138\pi$
$600$	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
$603$	−48.0000	−1.95471
$604$	0	0
$605$	14.0000	0.569181
$606$	0	0
$607$	−3.00000	−0.121766	−0.0608831	−	0.998145i	$-0.519392\pi$
$607$	−3.00000	−0.121766	−0.0608831	+	0.998145i	$0.519392\pi$
$608$	0	0
$609$	−21.0000	−0.850963
$610$	0	0
$611$	5.00000	0.202278
$612$	0	0
$613$	−30.0000	−1.21169	−0.605844	−	0.795583i	$-0.707165\pi$
$613$	−30.0000	−1.21169	−0.605844	+	0.795583i	$0.707165\pi$
$614$	0	0
$615$	−36.0000	−1.45166
$616$	0	0
$617$	6.00000	0.241551	0.120775	−	0.992680i	$-0.461462\pi$
$617$	6.00000	0.241551	0.120775	+	0.992680i	$0.461462\pi$
$618$	0	0
$619$	−26.0000	−1.04503	−0.522514	−	0.852631i	$-0.675006\pi$
$619$	−26.0000	−1.04503	−0.522514	+	0.852631i	$0.675006\pi$
$620$	0	0
$621$	18.0000	0.722315
$622$	0	0
$623$	0	0
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	30.0000	1.19808
$628$	0	0
$629$	42.0000	1.67465
$630$	0	0
$631$	−7.00000	−0.278666	−0.139333	−	0.990246i	$-0.544496\pi$
$631$	−7.00000	−0.278666	−0.139333	+	0.990246i	$0.544496\pi$
$632$	0	0
$633$	−15.0000	−0.596196
$634$	0	0
$635$	12.0000	0.476205
$636$	0	0
$637$	−5.00000	−0.198107
$638$	0	0
$639$	0	0
$640$	0	0
$641$	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
$643$	−21.0000	−0.828159	−0.414080	−	0.910241i	$-0.635896\pi$
$643$	−21.0000	−0.828159	−0.414080	+	0.910241i	$0.635896\pi$
$644$	0	0
$645$	6.00000	0.236250
$646$	0	0
$647$	−24.0000	−0.943537	−0.471769	−	0.881722i	$-0.656384\pi$
$647$	−24.0000	−0.943537	−0.471769	+	0.881722i	$0.656384\pi$
$648$	0	0
$649$	20.0000	0.785069
$650$	0	0
$651$	12.0000	0.470317
$652$	0	0
$653$	38.0000	1.48705	0.743527	−	0.668705i	$-0.233151\pi$
$653$	38.0000	1.48705	0.743527	+	0.668705i	$0.233151\pi$
$654$	0	0
$655$	6.00000	0.234439
$656$	0	0
$657$	36.0000	1.40449
$658$	0	0
$659$	−15.0000	−0.584317	−0.292159	−	0.956370i	$-0.594373\pi$
$659$	−15.0000	−0.584317	−0.292159	+	0.956370i	$0.594373\pi$
$660$	0	0
$661$	0	0		−	1.00000i	$-0.5\pi$
$661$	0	0			1.00000i	$0.5\pi$
$662$	0	0
$663$	105.000	4.07786
$664$	0	0
$665$	−2.00000	−0.0775567
$666$	0	0
$667$	14.0000	0.542082
$668$	0	0
$669$	57.0000	2.20375
$670$	0	0
$671$	20.0000	0.772091
$672$	0	0
$673$	0	0		−	1.00000i	$-0.5\pi$
$673$	0	0			1.00000i	$0.5\pi$
$674$	0	0
$675$	9.00000	0.346410
$676$	0	0
$677$	−29.0000	−1.11456	−0.557280	−	0.830324i	$-0.688155\pi$
$677$	−29.0000	−1.11456	−0.557280	+	0.830324i	$0.688155\pi$
$678$	0	0
$679$	−13.0000	−0.498894
$680$	0	0
$681$	−27.0000	−1.03464
$682$	0	0
$683$	−12.0000	−0.459167	−0.229584	−	0.973289i	$-0.573736\pi$
$683$	−12.0000	−0.459167	−0.229584	+	0.973289i	$0.573736\pi$
$684$	0	0
$685$	−8.00000	−0.305664
$686$	0	0
$687$	12.0000	0.457829
$688$	0	0
$689$	0	0
$690$	0	0
$691$	−44.0000	−1.67384	−0.836919	−	0.547326i	$-0.815646\pi$
$691$	−44.0000	−1.67384	−0.836919	+	0.547326i	$0.815646\pi$
$692$	0	0
$693$	−30.0000	−1.13961
$694$	0	0
$695$	−18.0000	−0.682779
$696$	0	0
$697$	84.0000	3.18173
$698$	0	0
$699$	−72.0000	−2.72329
$700$	0	0
$701$	35.0000	1.32193	0.660966	−	0.750416i	$-0.270147\pi$
$701$	35.0000	1.32193	0.660966	+	0.750416i	$0.270147\pi$
$702$	0	0
$703$	−12.0000	−0.452589
$704$	0	0
$705$	−3.00000	−0.112987
$706$	0	0
$707$	18.0000	0.676960
$708$	0	0
$709$	−25.0000	−0.938895	−0.469447	−	0.882960i	$-0.655547\pi$
$709$	−25.0000	−0.938895	−0.469447	+	0.882960i	$0.655547\pi$
$710$	0	0
$711$	18.0000	0.675053
$712$	0	0
$713$	−8.00000	−0.299602
$714$	0	0
$715$	−25.0000	−0.934947
$716$	0	0
$717$	−27.0000	−1.00833
$718$	0	0
$719$	14.0000	0.522112	0.261056	−	0.965324i	$-0.415929\pi$
$719$	14.0000	0.522112	0.261056	+	0.965324i	$0.415929\pi$
$720$	0	0
$721$	13.0000	0.484145
$722$	0	0
$723$	−66.0000	−2.45457
$724$	0	0
$725$	7.00000	0.259973
$726$	0	0
$727$	40.0000	1.48352	0.741759	−	0.670667i	$-0.233992\pi$
$727$	40.0000	1.48352	0.741759	+	0.670667i	$0.233992\pi$
$728$	0	0
$729$	−27.0000	−1.00000
$730$	0	0
$731$	−14.0000	−0.517809
$732$	0	0
$733$	−33.0000	−1.21888	−0.609441	−	0.792831i	$-0.708606\pi$
$733$	−33.0000	−1.21888	−0.609441	+	0.792831i	$0.708606\pi$
$734$	0	0
$735$	3.00000	0.110657
$736$	0	0
$737$	−40.0000	−1.47342
$738$	0	0
$739$	−51.0000	−1.87607	−0.938033	−	0.346547i	$-0.887354\pi$
$739$	−51.0000	−1.87607	−0.938033	+	0.346547i	$0.887354\pi$
$740$	0	0
$741$	−30.0000	−1.10208
$742$	0	0
$743$	−24.0000	−0.880475	−0.440237	−	0.897881i	$-0.645106\pi$
$743$	−24.0000	−0.880475	−0.440237	+	0.897881i	$0.645106\pi$
$744$	0	0
$745$	−22.0000	−0.806018
$746$	0	0
$747$	24.0000	0.878114
$748$	0	0
$749$	18.0000	0.657706
$750$	0	0
$751$	3.00000	0.109472	0.0547358	−	0.998501i	$-0.482568\pi$
$751$	3.00000	0.109472	0.0547358	+	0.998501i	$0.482568\pi$
$752$	0	0
$753$	−18.0000	−0.655956
$754$	0	0
$755$	19.0000	0.691481
$756$	0	0
$757$	−40.0000	−1.45382	−0.726912	−	0.686730i	$-0.759045\pi$
$757$	−40.0000	−1.45382	−0.726912	+	0.686730i	$0.759045\pi$
$758$	0	0
$759$	30.0000	1.08893
$760$	0	0
$761$	18.0000	0.652499	0.326250	−	0.945284i	$-0.394215\pi$
$761$	18.0000	0.652499	0.326250	+	0.945284i	$0.394215\pi$
$762$	0	0
$763$	−5.00000	−0.181012
$764$	0	0
$765$	−42.0000	−1.51851
$766$	0	0
$767$	−20.0000	−0.722158
$768$	0	0
$769$	34.0000	1.22607	0.613036	−	0.790055i	$-0.289948\pi$
$769$	34.0000	1.22607	0.613036	+	0.790055i	$0.289948\pi$
$770$	0	0
$771$	54.0000	1.94476
$772$	0	0
$773$	−21.0000	−0.755318	−0.377659	−	0.925945i	$-0.623271\pi$
$773$	−21.0000	−0.755318	−0.377659	+	0.925945i	$0.623271\pi$
$774$	0	0
$775$	−4.00000	−0.143684
$776$	0	0
$777$	18.0000	0.645746
$778$	0	0
$779$	−24.0000	−0.859889
$780$	0	0
$781$	0	0
$782$	0	0
$783$	63.0000	2.25144
$784$	0	0
$785$	10.0000	0.356915
$786$	0	0
$787$	−17.0000	−0.605985	−0.302992	−	0.952993i	$-0.597986\pi$
$787$	−17.0000	−0.605985	−0.302992	+	0.952993i	$0.597986\pi$
$788$	0	0
$789$	90.0000	3.20408
$790$	0	0
$791$	−6.00000	−0.213335
$792$	0	0
$793$	−20.0000	−0.710221
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−39.0000	−1.38145	−0.690725	−	0.723117i	$-0.742709\pi$
$797$	−39.0000	−1.38145	−0.690725	+	0.723117i	$0.742709\pi$
$798$	0	0
$799$	7.00000	0.247642
$800$	0	0
$801$	0	0
$802$	0	0
$803$	30.0000	1.05868
$804$	0	0
$805$	−2.00000	−0.0704907
$806$	0	0
$807$	78.0000	2.74573
$808$	0	0
$809$	−39.0000	−1.37117	−0.685583	−	0.727994i	$-0.740453\pi$
$809$	−39.0000	−1.37117	−0.685583	+	0.727994i	$0.740453\pi$
$810$	0	0
$811$	10.0000	0.351147	0.175574	−	0.984466i	$-0.443822\pi$
$811$	10.0000	0.351147	0.175574	+	0.984466i	$0.443822\pi$
$812$	0	0
$813$	−36.0000	−1.26258
$814$	0	0
$815$	14.0000	0.490399
$816$	0	0
$817$	4.00000	0.139942
$818$	0	0
$819$	30.0000	1.04828
$820$	0	0
$821$	1.00000	0.0349002	0.0174501	−	0.999848i	$-0.494445\pi$
$821$	1.00000	0.0349002	0.0174501	+	0.999848i	$0.494445\pi$
$822$	0	0
$823$	−28.0000	−0.976019	−0.488009	−	0.872838i	$-0.662277\pi$
$823$	−28.0000	−0.976019	−0.488009	+	0.872838i	$0.662277\pi$
$824$	0	0
$825$	15.0000	0.522233
$826$	0	0
$827$	−6.00000	−0.208640	−0.104320	−	0.994544i	$-0.533267\pi$
$827$	−6.00000	−0.208640	−0.104320	+	0.994544i	$0.533267\pi$
$828$	0	0
$829$	−20.0000	−0.694629	−0.347314	−	0.937749i	$-0.612906\pi$
$829$	−20.0000	−0.694629	−0.347314	+	0.937749i	$0.612906\pi$
$830$	0	0
$831$	6.00000	0.208138
$832$	0	0
$833$	−7.00000	−0.242536
$834$	0	0
$835$	3.00000	0.103819
$836$	0	0
$837$	−36.0000	−1.24434
$838$	0	0
$839$	34.0000	1.17381	0.586905	−	0.809656i	$-0.300346\pi$
$839$	34.0000	1.17381	0.586905	+	0.809656i	$0.300346\pi$
$840$	0	0
$841$	20.0000	0.689655
$842$	0	0
$843$	57.0000	1.96318
$844$	0	0
$845$	12.0000	0.412813
$846$	0	0
$847$	−14.0000	−0.481046
$848$	0	0
$849$	75.0000	2.57399
$850$	0	0
$851$	−12.0000	−0.411355
$852$	0	0
$853$	−22.0000	−0.753266	−0.376633	−	0.926363i	$-0.622918\pi$
$853$	−22.0000	−0.753266	−0.376633	+	0.926363i	$0.622918\pi$
$854$	0	0
$855$	12.0000	0.410391
$856$	0	0
$857$	6.00000	0.204956	0.102478	−	0.994735i	$-0.467323\pi$
$857$	6.00000	0.204956	0.102478	+	0.994735i	$0.467323\pi$
$858$	0	0
$859$	4.00000	0.136478	0.0682391	−	0.997669i	$-0.478262\pi$
$859$	4.00000	0.136478	0.0682391	+	0.997669i	$0.478262\pi$
$860$	0	0
$861$	36.0000	1.22688
$862$	0	0
$863$	−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
$865$	−7.00000	−0.238007
$866$	0	0
$867$	96.0000	3.26033
$868$	0	0
$869$	15.0000	0.508840
$870$	0	0
$871$	40.0000	1.35535
$872$	0	0
$873$	78.0000	2.63990
$874$	0	0
$875$	−1.00000	−0.0338062
$876$	0	0
$877$	22.0000	0.742887	0.371444	−	0.928456i	$-0.378863\pi$
$877$	22.0000	0.742887	0.371444	+	0.928456i	$0.378863\pi$
$878$	0	0
$879$	39.0000	1.31544
$880$	0	0
$881$	0	0		−	1.00000i	$-0.5\pi$
$881$	0	0			1.00000i	$0.5\pi$
$882$	0	0
$883$	40.0000	1.34611	0.673054	−	0.739594i	$-0.264982\pi$
$883$	40.0000	1.34611	0.673054	+	0.739594i	$0.264982\pi$
$884$	0	0
$885$	12.0000	0.403376
$886$	0	0
$887$	−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
$889$	−12.0000	−0.402467
$890$	0	0
$891$	45.0000	1.50756
$892$	0	0
$893$	−2.00000	−0.0669274
$894$	0	0
$895$	4.00000	0.133705
$896$	0	0
$897$	−30.0000	−1.00167
$898$	0	0
$899$	−28.0000	−0.933852
$900$	0	0
$901$	0	0
$902$	0	0
$903$	−6.00000	−0.199667
$904$	0	0
$905$	8.00000	0.265929
$906$	0	0
$907$	18.0000	0.597680	0.298840	−	0.954303i	$-0.403400\pi$
$907$	18.0000	0.597680	0.298840	+	0.954303i	$0.403400\pi$
$908$	0	0
$909$	−108.000	−3.58213
$910$	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
$913$	20.0000	0.661903
$914$	0	0
$915$	12.0000	0.396708
$916$	0	0
$917$	−6.00000	−0.198137
$918$	0	0
$919$	55.0000	1.81428	0.907141	−	0.420826i	$-0.138260\pi$
$919$	55.0000	1.81428	0.907141	+	0.420826i	$0.138260\pi$
$920$	0	0
$921$	51.0000	1.68051
$922$	0	0
$923$	0	0
$924$	0	0
$925$	−6.00000	−0.197279
$926$	0	0
$927$	−78.0000	−2.56186
$928$	0	0
$929$	−32.0000	−1.04989	−0.524943	−	0.851137i	$-0.675913\pi$
$929$	−32.0000	−1.04989	−0.524943	+	0.851137i	$0.675913\pi$
$930$	0	0
$931$	2.00000	0.0655474
$932$	0	0
$933$	102.000	3.33933
$934$	0	0
$935$	−35.0000	−1.14462
$936$	0	0
$937$	13.0000	0.424691	0.212346	−	0.977195i	$-0.431890\pi$
$937$	13.0000	0.424691	0.212346	+	0.977195i	$0.431890\pi$
$938$	0	0
$939$	−3.00000	−0.0979013
$940$	0	0
$941$	28.0000	0.912774	0.456387	−	0.889781i	$-0.349143\pi$
$941$	28.0000	0.912774	0.456387	+	0.889781i	$0.349143\pi$
$942$	0	0
$943$	−24.0000	−0.781548
$944$	0	0
$945$	−9.00000	−0.292770
$946$	0	0
$947$	−40.0000	−1.29983	−0.649913	−	0.760009i	$-0.725195\pi$
$947$	−40.0000	−1.29983	−0.649913	+	0.760009i	$0.725195\pi$
$948$	0	0
$949$	−30.0000	−0.973841
$950$	0	0
$951$	18.0000	0.583690
$952$	0	0
$953$	0	0		−	1.00000i	$-0.5\pi$
$953$	0	0			1.00000i	$0.5\pi$
$954$	0	0
$955$	13.0000	0.420670
$956$	0	0
$957$	105.000	3.39417
$958$	0	0
$959$	8.00000	0.258333
$960$	0	0
$961$	−15.0000	−0.483871
$962$	0	0
$963$	−108.000	−3.48025
$964$	0	0
$965$	−8.00000	−0.257529
$966$	0	0
$967$	−6.00000	−0.192947	−0.0964735	−	0.995336i	$-0.530756\pi$
$967$	−6.00000	−0.192947	−0.0964735	+	0.995336i	$0.530756\pi$
$968$	0	0
$969$	−42.0000	−1.34923
$970$	0	0
$971$	16.0000	0.513464	0.256732	−	0.966483i	$-0.417354\pi$
$971$	16.0000	0.513464	0.256732	+	0.966483i	$0.417354\pi$
$972$	0	0
$973$	18.0000	0.577054
$974$	0	0
$975$	−15.0000	−0.480384
$976$	0	0
$977$	26.0000	0.831814	0.415907	−	0.909407i	$-0.363464\pi$
$977$	26.0000	0.831814	0.415907	+	0.909407i	$0.363464\pi$
$978$	0	0
$979$	0	0
$980$	0	0
$981$	30.0000	0.957826
$982$	0	0
$983$	−3.00000	−0.0956851	−0.0478426	−	0.998855i	$-0.515235\pi$
$983$	−3.00000	−0.0956851	−0.0478426	+	0.998855i	$0.515235\pi$
$984$	0	0
$985$	−8.00000	−0.254901
$986$	0	0
$987$	3.00000	0.0954911
$988$	0	0
$989$	4.00000	0.127193
$990$	0	0
$991$	−32.0000	−1.01651	−0.508257	−	0.861206i	$-0.669710\pi$
$991$	−32.0000	−1.01651	−0.508257	+	0.861206i	$0.669710\pi$
$992$	0	0
$993$	36.0000	1.14243
$994$	0	0
$995$	4.00000	0.126809
$996$	0	0
$997$	−3.00000	−0.0950110	−0.0475055	−	0.998871i	$-0.515127\pi$
$997$	−3.00000	−0.0950110	−0.0475055	+	0.998871i	$0.515127\pi$
$998$	0	0
$999$	−54.0000	−1.70848

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	560.2.a.f.1.1		1
3.2	odd	2		5040.2.a.a.1.1		1
4.3	odd	2		280.2.a.a.1.1	✓	1
5.2	odd	4		2800.2.g.b.449.1		2
5.3	odd	4		2800.2.g.b.449.2		2
5.4	even	2		2800.2.a.c.1.1		1
7.6	odd	2		3920.2.a.c.1.1		1
8.3	odd	2		2240.2.a.z.1.1		1
8.5	even	2		2240.2.a.a.1.1		1
12.11	even	2		2520.2.a.i.1.1		1
20.3	even	4		1400.2.g.a.449.1		2
20.7	even	4		1400.2.g.a.449.2		2
20.19	odd	2		1400.2.a.n.1.1		1
28.3	even	6		1960.2.q.a.961.1		2
28.11	odd	6		1960.2.q.o.961.1		2
28.19	even	6		1960.2.q.a.361.1		2
28.23	odd	6		1960.2.q.o.361.1		2
28.27	even	2		1960.2.a.o.1.1		1
140.139	even	2		9800.2.a.a.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
280.2.a.a.1.1	✓	1	4.3	odd	2
560.2.a.f.1.1		1	1.1	even	1	trivial
1400.2.a.n.1.1		1	20.19	odd	2
1400.2.g.a.449.1		2	20.3	even	4
1400.2.g.a.449.2		2	20.7	even	4
1960.2.a.o.1.1		1	28.27	even	2
1960.2.q.a.361.1		2	28.19	even	6
1960.2.q.a.961.1		2	28.3	even	6
1960.2.q.o.361.1		2	28.23	odd	6
1960.2.q.o.961.1		2	28.11	odd	6
2240.2.a.a.1.1		1	8.5	even	2
2240.2.a.z.1.1		1	8.3	odd	2
2520.2.a.i.1.1		1	12.11	even	2
2800.2.a.c.1.1		1	5.4	even	2
2800.2.g.b.449.1		2	5.2	odd	4
2800.2.g.b.449.2		2	5.3	odd	4
3920.2.a.c.1.1		1	7.6	odd	2
5040.2.a.a.1.1		1	3.2	odd	2
9800.2.a.a.1.1		1	140.139	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 560 = 2^{4} \cdot 5 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	560.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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