LMFDB - Embedded newform 460.2.a.d.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [460,2,Mod(1,460)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(460, 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("460.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 460 = 2^{2} \cdot 5 \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	460.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:	$3.67311849298$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	460.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q+3.00000 q^{3} -1.00000 q^{5} +2.00000 q^{7} +6.00000 q^{9} +O(q^{10})$

$q+3.00000 q^{3} -1.00000 q^{5} +2.00000 q^{7} +6.00000 q^{9} -3.00000 q^{13} -3.00000 q^{15} +4.00000 q^{17} -4.00000 q^{19} +6.00000 q^{21} -1.00000 q^{23} +1.00000 q^{25} +9.00000 q^{27} +1.00000 q^{29} +1.00000 q^{31} -2.00000 q^{35} -8.00000 q^{37} -9.00000 q^{39} +11.0000 q^{41} -10.0000 q^{43} -6.00000 q^{45} -1.00000 q^{47} -3.00000 q^{49} +12.0000 q^{51} -6.00000 q^{53} -12.0000 q^{57} -8.00000 q^{59} -8.00000 q^{61} +12.0000 q^{63} +3.00000 q^{65} +12.0000 q^{67} -3.00000 q^{69} +13.0000 q^{71} +7.00000 q^{73} +3.00000 q^{75} -12.0000 q^{79} +9.00000 q^{81} +16.0000 q^{83} -4.00000 q^{85} +3.00000 q^{87} -6.00000 q^{89} -6.00000 q^{91} +3.00000 q^{93} +4.00000 q^{95} +2.00000 q^{97} +O(q^{100})$

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
$5$	−1.00000	−0.447214
$6$	0	0
$7$	2.00000	0.755929	0.377964	−	0.925820i	$-0.376624\pi$
$7$	2.00000	0.755929	0.377964	+	0.925820i	$0.376624\pi$
$8$	0	0
$9$	6.00000	2.00000
$10$	0	0
$11$	0	0		−	1.00000i	$-0.5\pi$
$11$	0	0			1.00000i	$0.5\pi$
$12$	0	0
$13$	−3.00000	−0.832050	−0.416025	−	0.909353i	$-0.636577\pi$
$13$	−3.00000	−0.832050	−0.416025	+	0.909353i	$0.636577\pi$
$14$	0	0
$15$	−3.00000	−0.774597
$16$	0	0
$17$	4.00000	0.970143	0.485071	−	0.874475i	$-0.338794\pi$
$17$	4.00000	0.970143	0.485071	+	0.874475i	$0.338794\pi$
$18$	0	0
$19$	−4.00000	−0.917663	−0.458831	−	0.888523i	$-0.651732\pi$
$19$	−4.00000	−0.917663	−0.458831	+	0.888523i	$0.651732\pi$
$20$	0	0
$21$	6.00000	1.30931
$22$	0	0
$23$	−1.00000	−0.208514
$23$	−1.00000	−0.208514
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	9.00000	1.73205
$28$	0	0
$29$	1.00000	0.185695	0.0928477	−	0.995680i	$-0.470403\pi$
$29$	1.00000	0.185695	0.0928477	+	0.995680i	$0.470403\pi$
$30$	0	0
$31$	1.00000	0.179605	0.0898027	−	0.995960i	$-0.471376\pi$
$31$	1.00000	0.179605	0.0898027	+	0.995960i	$0.471376\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−2.00000	−0.338062
$36$	0	0
$37$	−8.00000	−1.31519	−0.657596	−	0.753371i	$-0.728427\pi$
$37$	−8.00000	−1.31519	−0.657596	+	0.753371i	$0.728427\pi$
$38$	0	0
$39$	−9.00000	−1.44115
$40$	0	0
$41$	11.0000	1.71791	0.858956	−	0.512050i	$-0.171114\pi$
$41$	11.0000	1.71791	0.858956	+	0.512050i	$0.171114\pi$
$42$	0	0
$43$	−10.0000	−1.52499	−0.762493	−	0.646997i	$-0.776025\pi$
$43$	−10.0000	−1.52499	−0.762493	+	0.646997i	$0.776025\pi$
$44$	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
$49$	−3.00000	−0.428571
$50$	0	0
$51$	12.0000	1.68034
$52$	0	0
$53$	−6.00000	−0.824163	−0.412082	−	0.911147i	$-0.635198\pi$
$53$	−6.00000	−0.824163	−0.412082	+	0.911147i	$0.635198\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	−12.0000	−1.58944
$58$	0	0
$59$	−8.00000	−1.04151	−0.520756	−	0.853706i	$-0.674350\pi$
$59$	−8.00000	−1.04151	−0.520756	+	0.853706i	$0.674350\pi$
$60$	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
$63$	12.0000	1.51186
$64$	0	0
$65$	3.00000	0.372104
$66$	0	0
$67$	12.0000	1.46603	0.733017	−	0.680211i	$-0.238112\pi$
$67$	12.0000	1.46603	0.733017	+	0.680211i	$0.238112\pi$
$68$	0	0
$69$	−3.00000	−0.361158
$70$	0	0
$71$	13.0000	1.54282	0.771408	−	0.636341i	$-0.219553\pi$
$71$	13.0000	1.54282	0.771408	+	0.636341i	$0.219553\pi$
$72$	0	0
$73$	7.00000	0.819288	0.409644	−	0.912245i	$-0.365653\pi$
$73$	7.00000	0.819288	0.409644	+	0.912245i	$0.365653\pi$
$74$	0	0
$75$	3.00000	0.346410
$76$	0	0
$77$	0	0
$78$	0	0
$79$	−12.0000	−1.35011	−0.675053	−	0.737769i	$-0.735879\pi$
$79$	−12.0000	−1.35011	−0.675053	+	0.737769i	$0.735879\pi$
$80$	0	0
$81$	9.00000	1.00000
$82$	0	0
$83$	16.0000	1.75623	0.878114	−	0.478451i	$-0.158802\pi$
$83$	16.0000	1.75623	0.878114	+	0.478451i	$0.158802\pi$
$84$	0	0
$85$	−4.00000	−0.433861
$86$	0	0
$87$	3.00000	0.321634
$88$	0	0
$89$	−6.00000	−0.635999	−0.317999	−	0.948091i	$-0.603011\pi$
$89$	−6.00000	−0.635999	−0.317999	+	0.948091i	$0.603011\pi$
$90$	0	0
$91$	−6.00000	−0.628971
$92$	0	0
$93$	3.00000	0.311086
$94$	0	0
$95$	4.00000	0.410391
$96$	0	0
$97$	2.00000	0.203069	0.101535	−	0.994832i	$-0.467625\pi$
$97$	2.00000	0.203069	0.101535	+	0.994832i	$0.467625\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−14.0000	−1.39305	−0.696526	−	0.717532i	$-0.745272\pi$
$101$	−14.0000	−1.39305	−0.696526	+	0.717532i	$0.745272\pi$
$102$	0	0
$103$	−16.0000	−1.57653	−0.788263	−	0.615338i	$-0.789020\pi$
$103$	−16.0000	−1.57653	−0.788263	+	0.615338i	$0.789020\pi$
$104$	0	0
$105$	−6.00000	−0.585540
$106$	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
$109$	2.00000	0.191565	0.0957826	−	0.995402i	$-0.469465\pi$
$109$	2.00000	0.191565	0.0957826	+	0.995402i	$0.469465\pi$
$110$	0	0
$111$	−24.0000	−2.27798
$112$	0	0
$113$	−6.00000	−0.564433	−0.282216	−	0.959351i	$-0.591070\pi$
$113$	−6.00000	−0.564433	−0.282216	+	0.959351i	$0.591070\pi$
$114$	0	0
$115$	1.00000	0.0932505
$116$	0	0
$117$	−18.0000	−1.66410
$118$	0	0
$119$	8.00000	0.733359
$120$	0	0
$121$	−11.0000	−1.00000
$122$	0	0
$123$	33.0000	2.97551
$124$	0	0
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	17.0000	1.50851	0.754253	−	0.656584i	$-0.227999\pi$
$127$	17.0000	1.50851	0.754253	+	0.656584i	$0.227999\pi$
$128$	0	0
$129$	−30.0000	−2.64135
$130$	0	0
$131$	−15.0000	−1.31056	−0.655278	−	0.755388i	$-0.727449\pi$
$131$	−15.0000	−1.31056	−0.655278	+	0.755388i	$0.727449\pi$
$132$	0	0
$133$	−8.00000	−0.693688
$134$	0	0
$135$	−9.00000	−0.774597
$136$	0	0
$137$	−12.0000	−1.02523	−0.512615	−	0.858619i	$-0.671323\pi$
$137$	−12.0000	−1.02523	−0.512615	+	0.858619i	$0.671323\pi$
$138$	0	0
$139$	11.0000	0.933008	0.466504	−	0.884519i	$-0.345513\pi$
$139$	11.0000	0.933008	0.466504	+	0.884519i	$0.345513\pi$
$140$	0	0
$141$	−3.00000	−0.252646
$142$	0	0
$143$	0	0
$144$	0	0
$145$	−1.00000	−0.0830455
$146$	0	0
$147$	−9.00000	−0.742307
$148$	0	0
$149$	−6.00000	−0.491539	−0.245770	−	0.969328i	$-0.579041\pi$
$149$	−6.00000	−0.491539	−0.245770	+	0.969328i	$0.579041\pi$
$150$	0	0
$151$	5.00000	0.406894	0.203447	−	0.979086i	$-0.434786\pi$
$151$	5.00000	0.406894	0.203447	+	0.979086i	$0.434786\pi$
$152$	0	0
$153$	24.0000	1.94029
$154$	0	0
$155$	−1.00000	−0.0803219
$156$	0	0
$157$	−2.00000	−0.159617	−0.0798087	−	0.996810i	$-0.525431\pi$
$157$	−2.00000	−0.159617	−0.0798087	+	0.996810i	$0.525431\pi$
$158$	0	0
$159$	−18.0000	−1.42749
$160$	0	0
$161$	−2.00000	−0.157622
$162$	0	0
$163$	25.0000	1.95815	0.979076	−	0.203497i	$-0.0652307\pi$
$163$	25.0000	1.95815	0.979076	+	0.203497i	$0.0652307\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	8.00000	0.619059	0.309529	−	0.950890i	$-0.399829\pi$
$167$	8.00000	0.619059	0.309529	+	0.950890i	$0.399829\pi$
$168$	0	0
$169$	−4.00000	−0.307692
$170$	0	0
$171$	−24.0000	−1.83533
$172$	0	0
$173$	14.0000	1.06440	0.532200	−	0.846619i	$-0.321365\pi$
$173$	14.0000	1.06440	0.532200	+	0.846619i	$0.321365\pi$
$174$	0	0
$175$	2.00000	0.151186
$176$	0	0
$177$	−24.0000	−1.80395
$178$	0	0
$179$	3.00000	0.224231	0.112115	−	0.993695i	$-0.464237\pi$
$179$	3.00000	0.224231	0.112115	+	0.993695i	$0.464237\pi$
$180$	0	0
$181$	20.0000	1.48659	0.743294	−	0.668965i	$-0.233262\pi$
$181$	20.0000	1.48659	0.743294	+	0.668965i	$0.233262\pi$
$182$	0	0
$183$	−24.0000	−1.77413
$184$	0	0
$185$	8.00000	0.588172
$186$	0	0
$187$	0	0
$188$	0	0
$189$	18.0000	1.30931
$190$	0	0
$191$	18.0000	1.30243	0.651217	−	0.758891i	$-0.274259\pi$
$191$	18.0000	1.30243	0.651217	+	0.758891i	$0.274259\pi$
$192$	0	0
$193$	−9.00000	−0.647834	−0.323917	−	0.946085i	$-0.605000\pi$
$193$	−9.00000	−0.647834	−0.323917	+	0.946085i	$0.605000\pi$
$194$	0	0
$195$	9.00000	0.644503
$196$	0	0
$197$	23.0000	1.63868	0.819341	−	0.573306i	$-0.194340\pi$
$197$	23.0000	1.63868	0.819341	+	0.573306i	$0.194340\pi$
$198$	0	0
$199$	−12.0000	−0.850657	−0.425329	−	0.905039i	$-0.639842\pi$
$199$	−12.0000	−0.850657	−0.425329	+	0.905039i	$0.639842\pi$
$200$	0	0
$201$	36.0000	2.53924
$202$	0	0
$203$	2.00000	0.140372
$204$	0	0
$205$	−11.0000	−0.768273
$206$	0	0
$207$	−6.00000	−0.417029
$208$	0	0
$209$	0	0
$210$	0	0
$211$	−16.0000	−1.10149	−0.550743	−	0.834675i	$-0.685655\pi$
$211$	−16.0000	−1.10149	−0.550743	+	0.834675i	$0.685655\pi$
$212$	0	0
$213$	39.0000	2.67224
$214$	0	0
$215$	10.0000	0.681994
$216$	0	0
$217$	2.00000	0.135769
$218$	0	0
$219$	21.0000	1.41905
$220$	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
$225$	6.00000	0.400000
$226$	0	0
$227$	14.0000	0.929213	0.464606	−	0.885517i	$-0.346196\pi$
$227$	14.0000	0.929213	0.464606	+	0.885517i	$0.346196\pi$
$228$	0	0
$229$	−6.00000	−0.396491	−0.198246	−	0.980152i	$-0.563524\pi$
$229$	−6.00000	−0.396491	−0.198246	+	0.980152i	$0.563524\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	29.0000	1.89985	0.949927	−	0.312473i	$-0.101157\pi$
$233$	29.0000	1.89985	0.949927	+	0.312473i	$0.101157\pi$
$234$	0	0
$235$	1.00000	0.0652328
$236$	0	0
$237$	−36.0000	−2.33845
$238$	0	0
$239$	21.0000	1.35838	0.679189	−	0.733964i	$-0.262332\pi$
$239$	21.0000	1.35838	0.679189	+	0.733964i	$0.262332\pi$
$240$	0	0
$241$	14.0000	0.901819	0.450910	−	0.892570i	$-0.351100\pi$
$241$	14.0000	0.901819	0.450910	+	0.892570i	$0.351100\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	3.00000	0.191663
$246$	0	0
$247$	12.0000	0.763542
$248$	0	0
$249$	48.0000	3.04188
$250$	0	0
$251$	20.0000	1.26239	0.631194	−	0.775625i	$-0.282565\pi$
$251$	20.0000	1.26239	0.631194	+	0.775625i	$0.282565\pi$
$252$	0	0
$253$	0	0
$254$	0	0
$255$	−12.0000	−0.751469
$256$	0	0
$257$	−11.0000	−0.686161	−0.343081	−	0.939306i	$-0.611470\pi$
$257$	−11.0000	−0.686161	−0.343081	+	0.939306i	$0.611470\pi$
$258$	0	0
$259$	−16.0000	−0.994192
$260$	0	0
$261$	6.00000	0.371391
$262$	0	0
$263$	−8.00000	−0.493301	−0.246651	−	0.969104i	$-0.579330\pi$
$263$	−8.00000	−0.493301	−0.246651	+	0.969104i	$0.579330\pi$
$264$	0	0
$265$	6.00000	0.368577
$266$	0	0
$267$	−18.0000	−1.10158
$268$	0	0
$269$	−27.0000	−1.64622	−0.823110	−	0.567883i	$-0.807763\pi$
$269$	−27.0000	−1.64622	−0.823110	+	0.567883i	$0.807763\pi$
$270$	0	0
$271$	−20.0000	−1.21491	−0.607457	−	0.794353i	$-0.707810\pi$
$271$	−20.0000	−1.21491	−0.607457	+	0.794353i	$0.707810\pi$
$272$	0	0
$273$	−18.0000	−1.08941
$274$	0	0
$275$	0	0
$276$	0	0
$277$	31.0000	1.86261	0.931305	−	0.364241i	$-0.118672\pi$
$277$	31.0000	1.86261	0.931305	+	0.364241i	$0.118672\pi$
$278$	0	0
$279$	6.00000	0.359211
$280$	0	0
$281$	6.00000	0.357930	0.178965	−	0.983855i	$-0.442725\pi$
$281$	6.00000	0.357930	0.178965	+	0.983855i	$0.442725\pi$
$282$	0	0
$283$	−6.00000	−0.356663	−0.178331	−	0.983970i	$-0.557070\pi$
$283$	−6.00000	−0.356663	−0.178331	+	0.983970i	$0.557070\pi$
$284$	0	0
$285$	12.0000	0.710819
$286$	0	0
$287$	22.0000	1.29862
$288$	0	0
$289$	−1.00000	−0.0588235
$290$	0	0
$291$	6.00000	0.351726
$292$	0	0
$293$	0	0		−	1.00000i	$-0.5\pi$
$293$	0	0			1.00000i	$0.5\pi$
$294$	0	0
$295$	8.00000	0.465778
$296$	0	0
$297$	0	0
$298$	0	0
$299$	3.00000	0.173494
$300$	0	0
$301$	−20.0000	−1.15278
$302$	0	0
$303$	−42.0000	−2.41284
$304$	0	0
$305$	8.00000	0.458079
$306$	0	0
$307$	12.0000	0.684876	0.342438	−	0.939540i	$-0.388747\pi$
$307$	12.0000	0.684876	0.342438	+	0.939540i	$0.388747\pi$
$308$	0	0
$309$	−48.0000	−2.73062
$310$	0	0
$311$	17.0000	0.963982	0.481991	−	0.876176i	$-0.339914\pi$
$311$	17.0000	0.963982	0.481991	+	0.876176i	$0.339914\pi$
$312$	0	0
$313$	28.0000	1.58265	0.791327	−	0.611393i	$-0.209391\pi$
$313$	28.0000	1.58265	0.791327	+	0.611393i	$0.209391\pi$
$314$	0	0
$315$	−12.0000	−0.676123
$316$	0	0
$317$	−6.00000	−0.336994	−0.168497	−	0.985702i	$-0.553891\pi$
$317$	−6.00000	−0.336994	−0.168497	+	0.985702i	$0.553891\pi$
$318$	0	0
$319$	0	0
$320$	0	0
$321$	12.0000	0.669775
$322$	0	0
$323$	−16.0000	−0.890264
$324$	0	0
$325$	−3.00000	−0.166410
$326$	0	0
$327$	6.00000	0.331801
$328$	0	0
$329$	−2.00000	−0.110264
$330$	0	0
$331$	−17.0000	−0.934405	−0.467202	−	0.884150i	$-0.654738\pi$
$331$	−17.0000	−0.934405	−0.467202	+	0.884150i	$0.654738\pi$
$332$	0	0
$333$	−48.0000	−2.63038
$334$	0	0
$335$	−12.0000	−0.655630
$336$	0	0
$337$	−18.0000	−0.980522	−0.490261	−	0.871576i	$-0.663099\pi$
$337$	−18.0000	−0.980522	−0.490261	+	0.871576i	$0.663099\pi$
$338$	0	0
$339$	−18.0000	−0.977626
$340$	0	0
$341$	0	0
$342$	0	0
$343$	−20.0000	−1.07990
$344$	0	0
$345$	3.00000	0.161515
$346$	0	0
$347$	20.0000	1.07366	0.536828	−	0.843692i	$-0.319622\pi$
$347$	20.0000	1.07366	0.536828	+	0.843692i	$0.319622\pi$
$348$	0	0
$349$	5.00000	0.267644	0.133822	−	0.991005i	$-0.457275\pi$
$349$	5.00000	0.267644	0.133822	+	0.991005i	$0.457275\pi$
$350$	0	0
$351$	−27.0000	−1.44115
$352$	0	0
$353$	3.00000	0.159674	0.0798369	−	0.996808i	$-0.474560\pi$
$353$	3.00000	0.159674	0.0798369	+	0.996808i	$0.474560\pi$
$354$	0	0
$355$	−13.0000	−0.689968
$356$	0	0
$357$	24.0000	1.27021
$358$	0	0
$359$	4.00000	0.211112	0.105556	−	0.994413i	$-0.466338\pi$
$359$	4.00000	0.211112	0.105556	+	0.994413i	$0.466338\pi$
$360$	0	0
$361$	−3.00000	−0.157895
$362$	0	0
$363$	−33.0000	−1.73205
$364$	0	0
$365$	−7.00000	−0.366397
$366$	0	0
$367$	28.0000	1.46159	0.730794	−	0.682598i	$-0.239150\pi$
$367$	28.0000	1.46159	0.730794	+	0.682598i	$0.239150\pi$
$368$	0	0
$369$	66.0000	3.43582
$370$	0	0
$371$	−12.0000	−0.623009
$372$	0	0
$373$	6.00000	0.310668	0.155334	−	0.987862i	$-0.450355\pi$
$373$	6.00000	0.310668	0.155334	+	0.987862i	$0.450355\pi$
$374$	0	0
$375$	−3.00000	−0.154919
$376$	0	0
$377$	−3.00000	−0.154508
$378$	0	0
$379$	32.0000	1.64373	0.821865	−	0.569683i	$-0.192934\pi$
$379$	32.0000	1.64373	0.821865	+	0.569683i	$0.192934\pi$
$380$	0	0
$381$	51.0000	2.61281
$382$	0	0
$383$	6.00000	0.306586	0.153293	−	0.988181i	$-0.451012\pi$
$383$	6.00000	0.306586	0.153293	+	0.988181i	$0.451012\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−60.0000	−3.04997
$388$	0	0
$389$	24.0000	1.21685	0.608424	−	0.793612i	$-0.291802\pi$
$389$	24.0000	1.21685	0.608424	+	0.793612i	$0.291802\pi$
$390$	0	0
$391$	−4.00000	−0.202289
$392$	0	0
$393$	−45.0000	−2.26995
$394$	0	0
$395$	12.0000	0.603786
$396$	0	0
$397$	−3.00000	−0.150566	−0.0752828	−	0.997162i	$-0.523986\pi$
$397$	−3.00000	−0.150566	−0.0752828	+	0.997162i	$0.523986\pi$
$398$	0	0
$399$	−24.0000	−1.20150
$400$	0	0
$401$	10.0000	0.499376	0.249688	−	0.968326i	$-0.419672\pi$
$401$	10.0000	0.499376	0.249688	+	0.968326i	$0.419672\pi$
$402$	0	0
$403$	−3.00000	−0.149441
$404$	0	0
$405$	−9.00000	−0.447214
$406$	0	0
$407$	0	0
$408$	0	0
$409$	21.0000	1.03838	0.519192	−	0.854658i	$-0.326233\pi$
$409$	21.0000	1.03838	0.519192	+	0.854658i	$0.326233\pi$
$410$	0	0
$411$	−36.0000	−1.77575
$412$	0	0
$413$	−16.0000	−0.787309
$414$	0	0
$415$	−16.0000	−0.785409
$416$	0	0
$417$	33.0000	1.61602
$418$	0	0
$419$	6.00000	0.293119	0.146560	−	0.989202i	$-0.453180\pi$
$419$	6.00000	0.293119	0.146560	+	0.989202i	$0.453180\pi$
$420$	0	0
$421$	−14.0000	−0.682318	−0.341159	−	0.940006i	$-0.610819\pi$
$421$	−14.0000	−0.682318	−0.341159	+	0.940006i	$0.610819\pi$
$422$	0	0
$423$	−6.00000	−0.291730
$424$	0	0
$425$	4.00000	0.194029
$426$	0	0
$427$	−16.0000	−0.774294
$428$	0	0
$429$	0	0
$430$	0	0
$431$	26.0000	1.25238	0.626188	−	0.779672i	$-0.284614\pi$
$431$	26.0000	1.25238	0.626188	+	0.779672i	$0.284614\pi$
$432$	0	0
$433$	−38.0000	−1.82616	−0.913082	−	0.407777i	$-0.866304\pi$
$433$	−38.0000	−1.82616	−0.913082	+	0.407777i	$0.866304\pi$
$434$	0	0
$435$	−3.00000	−0.143839
$436$	0	0
$437$	4.00000	0.191346
$438$	0	0
$439$	3.00000	0.143182	0.0715911	−	0.997434i	$-0.477192\pi$
$439$	3.00000	0.143182	0.0715911	+	0.997434i	$0.477192\pi$
$440$	0	0
$441$	−18.0000	−0.857143
$442$	0	0
$443$	−9.00000	−0.427603	−0.213801	−	0.976877i	$-0.568585\pi$
$443$	−9.00000	−0.427603	−0.213801	+	0.976877i	$0.568585\pi$
$444$	0	0
$445$	6.00000	0.284427
$446$	0	0
$447$	−18.0000	−0.851371
$448$	0	0
$449$	−34.0000	−1.60456	−0.802280	−	0.596948i	$-0.796380\pi$
$449$	−34.0000	−1.60456	−0.802280	+	0.596948i	$0.796380\pi$
$450$	0	0
$451$	0	0
$452$	0	0
$453$	15.0000	0.704761
$454$	0	0
$455$	6.00000	0.281284
$456$	0	0
$457$	16.0000	0.748448	0.374224	−	0.927338i	$-0.377909\pi$
$457$	16.0000	0.748448	0.374224	+	0.927338i	$0.377909\pi$
$458$	0	0
$459$	36.0000	1.68034
$460$	0	0
$461$	−15.0000	−0.698620	−0.349310	−	0.937007i	$-0.613584\pi$
$461$	−15.0000	−0.698620	−0.349310	+	0.937007i	$0.613584\pi$
$462$	0	0
$463$	4.00000	0.185896	0.0929479	−	0.995671i	$-0.470371\pi$
$463$	4.00000	0.185896	0.0929479	+	0.995671i	$0.470371\pi$
$464$	0	0
$465$	−3.00000	−0.139122
$466$	0	0
$467$	−10.0000	−0.462745	−0.231372	−	0.972865i	$-0.574322\pi$
$467$	−10.0000	−0.462745	−0.231372	+	0.972865i	$0.574322\pi$
$468$	0	0
$469$	24.0000	1.10822
$470$	0	0
$471$	−6.00000	−0.276465
$472$	0	0
$473$	0	0
$474$	0	0
$475$	−4.00000	−0.183533
$476$	0	0
$477$	−36.0000	−1.64833
$478$	0	0
$479$	−4.00000	−0.182765	−0.0913823	−	0.995816i	$-0.529129\pi$
$479$	−4.00000	−0.182765	−0.0913823	+	0.995816i	$0.529129\pi$
$480$	0	0
$481$	24.0000	1.09431
$482$	0	0
$483$	−6.00000	−0.273009
$484$	0	0
$485$	−2.00000	−0.0908153
$486$	0	0
$487$	21.0000	0.951601	0.475800	−	0.879553i	$-0.342158\pi$
$487$	21.0000	0.951601	0.475800	+	0.879553i	$0.342158\pi$
$488$	0	0
$489$	75.0000	3.39162
$490$	0	0
$491$	15.0000	0.676941	0.338470	−	0.940977i	$-0.390091\pi$
$491$	15.0000	0.676941	0.338470	+	0.940977i	$0.390091\pi$
$492$	0	0
$493$	4.00000	0.180151
$494$	0	0
$495$	0	0
$496$	0	0
$497$	26.0000	1.16626
$498$	0	0
$499$	−39.0000	−1.74588	−0.872940	−	0.487828i	$-0.837789\pi$
$499$	−39.0000	−1.74588	−0.872940	+	0.487828i	$0.837789\pi$
$500$	0	0
$501$	24.0000	1.07224
$502$	0	0
$503$	−18.0000	−0.802580	−0.401290	−	0.915951i	$-0.631438\pi$
$503$	−18.0000	−0.802580	−0.401290	+	0.915951i	$0.631438\pi$
$504$	0	0
$505$	14.0000	0.622992
$506$	0	0
$507$	−12.0000	−0.532939
$508$	0	0
$509$	−39.0000	−1.72864	−0.864322	−	0.502938i	$-0.832252\pi$
$509$	−39.0000	−1.72864	−0.864322	+	0.502938i	$0.832252\pi$
$510$	0	0
$511$	14.0000	0.619324
$512$	0	0
$513$	−36.0000	−1.58944
$514$	0	0
$515$	16.0000	0.705044
$516$	0	0
$517$	0	0
$518$	0	0
$519$	42.0000	1.84360
$520$	0	0
$521$	4.00000	0.175243	0.0876216	−	0.996154i	$-0.472073\pi$
$521$	4.00000	0.175243	0.0876216	+	0.996154i	$0.472073\pi$
$522$	0	0
$523$	14.0000	0.612177	0.306089	−	0.952003i	$-0.400980\pi$
$523$	14.0000	0.612177	0.306089	+	0.952003i	$0.400980\pi$
$524$	0	0
$525$	6.00000	0.261861
$526$	0	0
$527$	4.00000	0.174243
$528$	0	0
$529$	1.00000	0.0434783
$530$	0	0
$531$	−48.0000	−2.08302
$532$	0	0
$533$	−33.0000	−1.42939
$534$	0	0
$535$	−4.00000	−0.172935
$536$	0	0
$537$	9.00000	0.388379
$538$	0	0
$539$	0	0
$540$	0	0
$541$	−11.0000	−0.472927	−0.236463	−	0.971640i	$-0.575988\pi$
$541$	−11.0000	−0.472927	−0.236463	+	0.971640i	$0.575988\pi$
$542$	0	0
$543$	60.0000	2.57485
$544$	0	0
$545$	−2.00000	−0.0856706
$546$	0	0
$547$	−19.0000	−0.812381	−0.406191	−	0.913788i	$-0.633143\pi$
$547$	−19.0000	−0.812381	−0.406191	+	0.913788i	$0.633143\pi$
$548$	0	0
$549$	−48.0000	−2.04859
$550$	0	0
$551$	−4.00000	−0.170406
$552$	0	0
$553$	−24.0000	−1.02058
$554$	0	0
$555$	24.0000	1.01874
$556$	0	0
$557$	−30.0000	−1.27114	−0.635570	−	0.772043i	$-0.719235\pi$
$557$	−30.0000	−1.27114	−0.635570	+	0.772043i	$0.719235\pi$
$558$	0	0
$559$	30.0000	1.26886
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−8.00000	−0.337160	−0.168580	−	0.985688i	$-0.553918\pi$
$563$	−8.00000	−0.337160	−0.168580	+	0.985688i	$0.553918\pi$
$564$	0	0
$565$	6.00000	0.252422
$566$	0	0
$567$	18.0000	0.755929
$568$	0	0
$569$	−30.0000	−1.25767	−0.628833	−	0.777541i	$-0.716467\pi$
$569$	−30.0000	−1.25767	−0.628833	+	0.777541i	$0.716467\pi$
$570$	0	0
$571$	−32.0000	−1.33916	−0.669579	−	0.742741i	$-0.733526\pi$
$571$	−32.0000	−1.33916	−0.669579	+	0.742741i	$0.733526\pi$
$572$	0	0
$573$	54.0000	2.25588
$574$	0	0
$575$	−1.00000	−0.0417029
$576$	0	0
$577$	−17.0000	−0.707719	−0.353860	−	0.935299i	$-0.615131\pi$
$577$	−17.0000	−0.707719	−0.353860	+	0.935299i	$0.615131\pi$
$578$	0	0
$579$	−27.0000	−1.12208
$580$	0	0
$581$	32.0000	1.32758
$582$	0	0
$583$	0	0
$584$	0	0
$585$	18.0000	0.744208
$586$	0	0
$587$	−41.0000	−1.69225	−0.846126	−	0.532984i	$-0.821071\pi$
$587$	−41.0000	−1.69225	−0.846126	+	0.532984i	$0.821071\pi$
$588$	0	0
$589$	−4.00000	−0.164817
$590$	0	0
$591$	69.0000	2.83828
$592$	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
$595$	−8.00000	−0.327968
$596$	0	0
$597$	−36.0000	−1.47338
$598$	0	0
$599$	40.0000	1.63436	0.817178	−	0.576386i	$-0.195537\pi$
$599$	40.0000	1.63436	0.817178	+	0.576386i	$0.195537\pi$
$600$	0	0
$601$	−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
$603$	72.0000	2.93207
$604$	0	0
$605$	11.0000	0.447214
$606$	0	0
$607$	8.00000	0.324710	0.162355	−	0.986732i	$-0.448091\pi$
$607$	8.00000	0.324710	0.162355	+	0.986732i	$0.448091\pi$
$608$	0	0
$609$	6.00000	0.243132
$610$	0	0
$611$	3.00000	0.121367
$612$	0	0
$613$	−28.0000	−1.13091	−0.565455	−	0.824779i	$-0.691299\pi$
$613$	−28.0000	−1.13091	−0.565455	+	0.824779i	$0.691299\pi$
$614$	0	0
$615$	−33.0000	−1.33069
$616$	0	0
$617$	−36.0000	−1.44931	−0.724653	−	0.689114i	$-0.758000\pi$
$617$	−36.0000	−1.44931	−0.724653	+	0.689114i	$0.758000\pi$
$618$	0	0
$619$	−20.0000	−0.803868	−0.401934	−	0.915669i	$-0.631662\pi$
$619$	−20.0000	−0.803868	−0.401934	+	0.915669i	$0.631662\pi$
$620$	0	0
$621$	−9.00000	−0.361158
$622$	0	0
$623$	−12.0000	−0.480770
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−32.0000	−1.27592
$630$	0	0
$631$	−18.0000	−0.716569	−0.358284	−	0.933613i	$-0.616638\pi$
$631$	−18.0000	−0.716569	−0.358284	+	0.933613i	$0.616638\pi$
$632$	0	0
$633$	−48.0000	−1.90783
$634$	0	0
$635$	−17.0000	−0.674624
$636$	0	0
$637$	9.00000	0.356593
$638$	0	0
$639$	78.0000	3.08563
$640$	0	0
$641$	−18.0000	−0.710957	−0.355479	−	0.934684i	$-0.615682\pi$
$641$	−18.0000	−0.710957	−0.355479	+	0.934684i	$0.615682\pi$
$642$	0	0
$643$	34.0000	1.34083	0.670415	−	0.741987i	$-0.266116\pi$
$643$	34.0000	1.34083	0.670415	+	0.741987i	$0.266116\pi$
$644$	0	0
$645$	30.0000	1.18125
$646$	0	0
$647$	−13.0000	−0.511083	−0.255541	−	0.966798i	$-0.582254\pi$
$647$	−13.0000	−0.511083	−0.255541	+	0.966798i	$0.582254\pi$
$648$	0	0
$649$	0	0
$650$	0	0
$651$	6.00000	0.235159
$652$	0	0
$653$	−23.0000	−0.900060	−0.450030	−	0.893014i	$-0.648587\pi$
$653$	−23.0000	−0.900060	−0.450030	+	0.893014i	$0.648587\pi$
$654$	0	0
$655$	15.0000	0.586098
$656$	0	0
$657$	42.0000	1.63858
$658$	0	0
$659$	−4.00000	−0.155818	−0.0779089	−	0.996960i	$-0.524824\pi$
$659$	−4.00000	−0.155818	−0.0779089	+	0.996960i	$0.524824\pi$
$660$	0	0
$661$	−42.0000	−1.63361	−0.816805	−	0.576913i	$-0.804257\pi$
$661$	−42.0000	−1.63361	−0.816805	+	0.576913i	$0.804257\pi$
$662$	0	0
$663$	−36.0000	−1.39812
$664$	0	0
$665$	8.00000	0.310227
$666$	0	0
$667$	−1.00000	−0.0387202
$668$	0	0
$669$	0	0
$670$	0	0
$671$	0	0
$672$	0	0
$673$	29.0000	1.11787	0.558934	−	0.829212i	$-0.311211\pi$
$673$	29.0000	1.11787	0.558934	+	0.829212i	$0.311211\pi$
$674$	0	0
$675$	9.00000	0.346410
$676$	0	0
$677$	18.0000	0.691796	0.345898	−	0.938272i	$-0.387574\pi$
$677$	18.0000	0.691796	0.345898	+	0.938272i	$0.387574\pi$
$678$	0	0
$679$	4.00000	0.153506
$680$	0	0
$681$	42.0000	1.60944
$682$	0	0
$683$	13.0000	0.497431	0.248716	−	0.968577i	$-0.419992\pi$
$683$	13.0000	0.497431	0.248716	+	0.968577i	$0.419992\pi$
$684$	0	0
$685$	12.0000	0.458496
$686$	0	0
$687$	−18.0000	−0.686743
$688$	0	0
$689$	18.0000	0.685745
$690$	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
$693$	0	0
$694$	0	0
$695$	−11.0000	−0.417254
$696$	0	0
$697$	44.0000	1.66662
$698$	0	0
$699$	87.0000	3.29064
$700$	0	0
$701$	10.0000	0.377695	0.188847	−	0.982006i	$-0.439525\pi$
$701$	10.0000	0.377695	0.188847	+	0.982006i	$0.439525\pi$
$702$	0	0
$703$	32.0000	1.20690
$704$	0	0
$705$	3.00000	0.112987
$706$	0	0
$707$	−28.0000	−1.05305
$708$	0	0
$709$	46.0000	1.72757	0.863783	−	0.503864i	$-0.168089\pi$
$709$	46.0000	1.72757	0.863783	+	0.503864i	$0.168089\pi$
$710$	0	0
$711$	−72.0000	−2.70021
$712$	0	0
$713$	−1.00000	−0.0374503
$714$	0	0
$715$	0	0
$716$	0	0
$717$	63.0000	2.35278
$718$	0	0
$719$	−4.00000	−0.149175	−0.0745874	−	0.997214i	$-0.523764\pi$
$719$	−4.00000	−0.149175	−0.0745874	+	0.997214i	$0.523764\pi$
$720$	0	0
$721$	−32.0000	−1.19174
$722$	0	0
$723$	42.0000	1.56200
$724$	0	0
$725$	1.00000	0.0371391
$726$	0	0
$727$	34.0000	1.26099	0.630495	−	0.776193i	$-0.282852\pi$
$727$	34.0000	1.26099	0.630495	+	0.776193i	$0.282852\pi$
$728$	0	0
$729$	−27.0000	−1.00000
$730$	0	0
$731$	−40.0000	−1.47945
$732$	0	0
$733$	28.0000	1.03420	0.517102	−	0.855924i	$-0.327011\pi$
$733$	28.0000	1.03420	0.517102	+	0.855924i	$0.327011\pi$
$734$	0	0
$735$	9.00000	0.331970
$736$	0	0
$737$	0	0
$738$	0	0
$739$	−25.0000	−0.919640	−0.459820	−	0.888012i	$-0.652086\pi$
$739$	−25.0000	−0.919640	−0.459820	+	0.888012i	$0.652086\pi$
$740$	0	0
$741$	36.0000	1.32249
$742$	0	0
$743$	16.0000	0.586983	0.293492	−	0.955962i	$-0.405183\pi$
$743$	16.0000	0.586983	0.293492	+	0.955962i	$0.405183\pi$
$744$	0	0
$745$	6.00000	0.219823
$746$	0	0
$747$	96.0000	3.51246
$748$	0	0
$749$	8.00000	0.292314
$750$	0	0
$751$	6.00000	0.218943	0.109472	−	0.993990i	$-0.465084\pi$
$751$	6.00000	0.218943	0.109472	+	0.993990i	$0.465084\pi$
$752$	0	0
$753$	60.0000	2.18652
$754$	0	0
$755$	−5.00000	−0.181969
$756$	0	0
$757$	8.00000	0.290765	0.145382	−	0.989376i	$-0.453559\pi$
$757$	8.00000	0.290765	0.145382	+	0.989376i	$0.453559\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−3.00000	−0.108750	−0.0543750	−	0.998521i	$-0.517317\pi$
$761$	−3.00000	−0.108750	−0.0543750	+	0.998521i	$0.517317\pi$
$762$	0	0
$763$	4.00000	0.144810
$764$	0	0
$765$	−24.0000	−0.867722
$766$	0	0
$767$	24.0000	0.866590
$768$	0	0
$769$	−18.0000	−0.649097	−0.324548	−	0.945869i	$-0.605212\pi$
$769$	−18.0000	−0.649097	−0.324548	+	0.945869i	$0.605212\pi$
$770$	0	0
$771$	−33.0000	−1.18847
$772$	0	0
$773$	50.0000	1.79838	0.899188	−	0.437564i	$-0.144158\pi$
$773$	50.0000	1.79838	0.899188	+	0.437564i	$0.144158\pi$
$774$	0	0
$775$	1.00000	0.0359211
$776$	0	0
$777$	−48.0000	−1.72199
$778$	0	0
$779$	−44.0000	−1.57646
$780$	0	0
$781$	0	0
$782$	0	0
$783$	9.00000	0.321634
$784$	0	0
$785$	2.00000	0.0713831
$786$	0	0
$787$	16.0000	0.570338	0.285169	−	0.958477i	$-0.407950\pi$
$787$	16.0000	0.570338	0.285169	+	0.958477i	$0.407950\pi$
$788$	0	0
$789$	−24.0000	−0.854423
$790$	0	0
$791$	−12.0000	−0.426671
$792$	0	0
$793$	24.0000	0.852265
$794$	0	0
$795$	18.0000	0.638394
$796$	0	0
$797$	−48.0000	−1.70025	−0.850124	−	0.526583i	$-0.823473\pi$
$797$	−48.0000	−1.70025	−0.850124	+	0.526583i	$0.823473\pi$
$798$	0	0
$799$	−4.00000	−0.141510
$800$	0	0
$801$	−36.0000	−1.27200
$802$	0	0
$803$	0	0
$804$	0	0
$805$	2.00000	0.0704907
$806$	0	0
$807$	−81.0000	−2.85134
$808$	0	0
$809$	34.0000	1.19538	0.597688	−	0.801729i	$-0.296086\pi$
$809$	34.0000	1.19538	0.597688	+	0.801729i	$0.296086\pi$
$810$	0	0
$811$	−11.0000	−0.386262	−0.193131	−	0.981173i	$-0.561864\pi$
$811$	−11.0000	−0.386262	−0.193131	+	0.981173i	$0.561864\pi$
$812$	0	0
$813$	−60.0000	−2.10429
$814$	0	0
$815$	−25.0000	−0.875712
$816$	0	0
$817$	40.0000	1.39942
$818$	0	0
$819$	−36.0000	−1.25794
$820$	0	0
$821$	−14.0000	−0.488603	−0.244302	−	0.969699i	$-0.578559\pi$
$821$	−14.0000	−0.488603	−0.244302	+	0.969699i	$0.578559\pi$
$822$	0	0
$823$	−37.0000	−1.28974	−0.644869	−	0.764293i	$-0.723088\pi$
$823$	−37.0000	−1.28974	−0.644869	+	0.764293i	$0.723088\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	20.0000	0.695468	0.347734	−	0.937593i	$-0.386951\pi$
$827$	20.0000	0.695468	0.347734	+	0.937593i	$0.386951\pi$
$828$	0	0
$829$	−18.0000	−0.625166	−0.312583	−	0.949890i	$-0.601194\pi$
$829$	−18.0000	−0.625166	−0.312583	+	0.949890i	$0.601194\pi$
$830$	0	0
$831$	93.0000	3.22613
$832$	0	0
$833$	−12.0000	−0.415775
$834$	0	0
$835$	−8.00000	−0.276851
$836$	0	0
$837$	9.00000	0.311086
$838$	0	0
$839$	2.00000	0.0690477	0.0345238	−	0.999404i	$-0.489009\pi$
$839$	2.00000	0.0690477	0.0345238	+	0.999404i	$0.489009\pi$
$840$	0	0
$841$	−28.0000	−0.965517
$842$	0	0
$843$	18.0000	0.619953
$844$	0	0
$845$	4.00000	0.137604
$846$	0	0
$847$	−22.0000	−0.755929
$848$	0	0
$849$	−18.0000	−0.617758
$850$	0	0
$851$	8.00000	0.274236
$852$	0	0
$853$	30.0000	1.02718	0.513590	−	0.858036i	$-0.328315\pi$
$853$	30.0000	1.02718	0.513590	+	0.858036i	$0.328315\pi$
$854$	0	0
$855$	24.0000	0.820783
$856$	0	0
$857$	33.0000	1.12726	0.563629	−	0.826028i	$-0.309405\pi$
$857$	33.0000	1.12726	0.563629	+	0.826028i	$0.309405\pi$
$858$	0	0
$859$	1.00000	0.0341196	0.0170598	−	0.999854i	$-0.494569\pi$
$859$	1.00000	0.0341196	0.0170598	+	0.999854i	$0.494569\pi$
$860$	0	0
$861$	66.0000	2.24927
$862$	0	0
$863$	21.0000	0.714848	0.357424	−	0.933942i	$-0.383655\pi$
$863$	21.0000	0.714848	0.357424	+	0.933942i	$0.383655\pi$
$864$	0	0
$865$	−14.0000	−0.476014
$866$	0	0
$867$	−3.00000	−0.101885
$868$	0	0
$869$	0	0
$870$	0	0
$871$	−36.0000	−1.21981
$872$	0	0
$873$	12.0000	0.406138
$874$	0	0
$875$	−2.00000	−0.0676123
$876$	0	0
$877$	−14.0000	−0.472746	−0.236373	−	0.971662i	$-0.575959\pi$
$877$	−14.0000	−0.472746	−0.236373	+	0.971662i	$0.575959\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−48.0000	−1.61716	−0.808581	−	0.588386i	$-0.799764\pi$
$881$	−48.0000	−1.61716	−0.808581	+	0.588386i	$0.799764\pi$
$882$	0	0
$883$	4.00000	0.134611	0.0673054	−	0.997732i	$-0.478560\pi$
$883$	4.00000	0.134611	0.0673054	+	0.997732i	$0.478560\pi$
$884$	0	0
$885$	24.0000	0.806751
$886$	0	0
$887$	3.00000	0.100730	0.0503651	−	0.998731i	$-0.483962\pi$
$887$	3.00000	0.100730	0.0503651	+	0.998731i	$0.483962\pi$
$888$	0	0
$889$	34.0000	1.14032
$890$	0	0
$891$	0	0
$892$	0	0
$893$	4.00000	0.133855
$894$	0	0
$895$	−3.00000	−0.100279
$896$	0	0
$897$	9.00000	0.300501
$898$	0	0
$899$	1.00000	0.0333519
$900$	0	0
$901$	−24.0000	−0.799556
$902$	0	0
$903$	−60.0000	−1.99667
$904$	0	0
$905$	−20.0000	−0.664822
$906$	0	0
$907$	4.00000	0.132818	0.0664089	−	0.997792i	$-0.478846\pi$
$907$	4.00000	0.132818	0.0664089	+	0.997792i	$0.478846\pi$
$908$	0	0
$909$	−84.0000	−2.78610
$910$	0	0
$911$	32.0000	1.06021	0.530104	−	0.847933i	$-0.322153\pi$
$911$	32.0000	1.06021	0.530104	+	0.847933i	$0.322153\pi$
$912$	0	0
$913$	0	0
$914$	0	0
$915$	24.0000	0.793416
$916$	0	0
$917$	−30.0000	−0.990687
$918$	0	0
$919$	34.0000	1.12156	0.560778	−	0.827966i	$-0.310502\pi$
$919$	34.0000	1.12156	0.560778	+	0.827966i	$0.310502\pi$
$920$	0	0
$921$	36.0000	1.18624
$922$	0	0
$923$	−39.0000	−1.28370
$924$	0	0
$925$	−8.00000	−0.263038
$926$	0	0
$927$	−96.0000	−3.15305
$928$	0	0
$929$	−37.0000	−1.21393	−0.606965	−	0.794728i	$-0.707613\pi$
$929$	−37.0000	−1.21393	−0.606965	+	0.794728i	$0.707613\pi$
$930$	0	0
$931$	12.0000	0.393284
$932$	0	0
$933$	51.0000	1.66967
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−12.0000	−0.392023	−0.196011	−	0.980602i	$-0.562799\pi$
$937$	−12.0000	−0.392023	−0.196011	+	0.980602i	$0.562799\pi$
$938$	0	0
$939$	84.0000	2.74124
$940$	0	0
$941$	−8.00000	−0.260793	−0.130396	−	0.991462i	$-0.541625\pi$
$941$	−8.00000	−0.260793	−0.130396	+	0.991462i	$0.541625\pi$
$942$	0	0
$943$	−11.0000	−0.358209
$944$	0	0
$945$	−18.0000	−0.585540
$946$	0	0
$947$	−57.0000	−1.85225	−0.926126	−	0.377215i	$-0.876882\pi$
$947$	−57.0000	−1.85225	−0.926126	+	0.377215i	$0.876882\pi$
$948$	0	0
$949$	−21.0000	−0.681689
$950$	0	0
$951$	−18.0000	−0.583690
$952$	0	0
$953$	−6.00000	−0.194359	−0.0971795	−	0.995267i	$-0.530982\pi$
$953$	−6.00000	−0.194359	−0.0971795	+	0.995267i	$0.530982\pi$
$954$	0	0
$955$	−18.0000	−0.582466
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−24.0000	−0.775000
$960$	0	0
$961$	−30.0000	−0.967742
$962$	0	0
$963$	24.0000	0.773389
$964$	0	0
$965$	9.00000	0.289720
$966$	0	0
$967$	−47.0000	−1.51142	−0.755709	−	0.654907i	$-0.772708\pi$
$967$	−47.0000	−1.51142	−0.755709	+	0.654907i	$0.772708\pi$
$968$	0	0
$969$	−48.0000	−1.54198
$970$	0	0
$971$	6.00000	0.192549	0.0962746	−	0.995355i	$-0.469307\pi$
$971$	6.00000	0.192549	0.0962746	+	0.995355i	$0.469307\pi$
$972$	0	0
$973$	22.0000	0.705288
$974$	0	0
$975$	−9.00000	−0.288231
$976$	0	0
$977$	−32.0000	−1.02377	−0.511885	−	0.859054i	$-0.671053\pi$
$977$	−32.0000	−1.02377	−0.511885	+	0.859054i	$0.671053\pi$
$978$	0	0
$979$	0	0
$980$	0	0
$981$	12.0000	0.383131
$982$	0	0
$983$	−18.0000	−0.574111	−0.287055	−	0.957914i	$-0.592676\pi$
$983$	−18.0000	−0.574111	−0.287055	+	0.957914i	$0.592676\pi$
$984$	0	0
$985$	−23.0000	−0.732841
$986$	0	0
$987$	−6.00000	−0.190982
$988$	0	0
$989$	10.0000	0.317982
$990$	0	0
$991$	56.0000	1.77890	0.889449	−	0.457034i	$-0.151088\pi$
$991$	56.0000	1.77890	0.889449	+	0.457034i	$0.151088\pi$
$992$	0	0
$993$	−51.0000	−1.61844
$994$	0	0
$995$	12.0000	0.380426
$996$	0	0
$997$	26.0000	0.823428	0.411714	−	0.911313i	$-0.364930\pi$
$997$	26.0000	0.823428	0.411714	+	0.911313i	$0.364930\pi$
$998$	0	0
$999$	−72.0000	−2.27798

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	460.2.a.d.1.1	✓	1
3.2	odd	2		4140.2.a.k.1.1		1
4.3	odd	2		1840.2.a.a.1.1		1
5.2	odd	4		2300.2.c.a.1749.1		2
5.3	odd	4		2300.2.c.a.1749.2		2
5.4	even	2		2300.2.a.a.1.1		1
8.3	odd	2		7360.2.a.bb.1.1		1
8.5	even	2		7360.2.a.b.1.1		1
20.19	odd	2		9200.2.a.bk.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
460.2.a.d.1.1	✓	1	1.1	even	1	trivial
1840.2.a.a.1.1		1	4.3	odd	2
2300.2.a.a.1.1		1	5.4	even	2
2300.2.c.a.1749.1		2	5.2	odd	4
2300.2.c.a.1749.2		2	5.3	odd	4
4140.2.a.k.1.1		1	3.2	odd	2
7360.2.a.b.1.1		1	8.5	even	2
7360.2.a.bb.1.1		1	8.3	odd	2
9200.2.a.bk.1.1		1	20.19	odd	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:	\( N \)	\(=\)	\( 460 = 2^{2} \cdot 5 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	460.a (trivial)

Introduction

L-functions

Modular forms

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