LMFDB - Embedded newform 6480.2.a.bk.1.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [6480,2,Mod(1,6480)] 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("6480.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(6480, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 6480 = 2^{4} \cdot 3^{4} \cdot 5 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6480.a (trivial)

Newform invariants

comment:select newform

sage:traces = [2,0,0,0,2,0,-4,0,0,0,4,0,-4,0,0,0,-8] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(17)] == traces)

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

Self dual:	yes
Analytic conductor:	$51.7430605098$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{12})^+$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 3 $ x^2 - 3
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 360)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$1.73205$ of defining polynomial
Character	$\chi$	$=$	6480.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{5} -0.267949 q^{7} -1.46410 q^{11} -5.46410 q^{13} -0.535898 q^{17} +2.00000 q^{19} +3.73205 q^{23} +1.00000 q^{25} -1.53590 q^{29} +2.00000 q^{31} -0.267949 q^{35} +10.3923 q^{37} -9.92820 q^{41} +4.53590 q^{43} +0.267949 q^{47} -6.92820 q^{49} -6.00000 q^{53} -1.46410 q^{55} +14.3923 q^{59} -8.46410 q^{61} -5.46410 q^{65} -6.26795 q^{67} +9.46410 q^{71} +6.92820 q^{73} +0.392305 q^{77} -15.4641 q^{79} -13.1962 q^{83} -0.535898 q^{85} +9.92820 q^{89} +1.46410 q^{91} +2.00000 q^{95} -8.92820 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{5} - 4 q^{7} + 4 q^{11} - 4 q^{13} - 8 q^{17} + 4 q^{19} + 4 q^{23} + 2 q^{25} - 10 q^{29} + 4 q^{31} - 4 q^{35} - 6 q^{41} + 16 q^{43} + 4 q^{47} - 12 q^{53} + 4 q^{55} + 8 q^{59} - 10 q^{61}+ \cdots - 4 q^{97}+O(q^{100}) $ 2 * q + 2 * q^5 - 4 * q^7 + 4 * q^11 - 4 * q^13 - 8 * q^17 + 4 * q^19 + 4 * q^23 + 2 * q^25 - 10 * q^29 + 4 * q^31 - 4 * q^35 - 6 * q^41 + 16 * q^43 + 4 * q^47 - 12 * q^53 + 4 * q^55 + 8 * q^59 - 10 * q^61 - 4 * q^65 - 16 * q^67 + 12 * q^71 - 20 * q^77 - 24 * q^79 - 16 * q^83 - 8 * q^85 + 6 * q^89 - 4 * q^91 + 4 * q^95 - 4 * q^97

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
$3$	0	0
$3$	0	0
$4$	0	0
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	−0.267949	−0.101275	−0.0506376	−	0.998717i	$-0.516125\pi$
$7$	−0.267949	−0.101275	−0.0506376	+	0.998717i	$0.516125\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	−1.46410	−0.441443	−0.220722	−	0.975337i	$-0.570841\pi$
$11$	−1.46410	−0.441443	−0.220722	+	0.975337i	$0.570841\pi$
$12$	0	0
$13$	−5.46410	−1.51547	−0.757735	−	0.652563i	$-0.773694\pi$
$13$	−5.46410	−1.51547	−0.757735	+	0.652563i	$0.773694\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−0.535898	−0.129974	−0.0649872	−	0.997886i	$-0.520701\pi$
$17$	−0.535898	−0.129974	−0.0649872	+	0.997886i	$0.520701\pi$
$18$	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
$21$	0	0
$22$	0	0
$23$	3.73205	0.778186	0.389093	−	0.921198i	$-0.372788\pi$
$23$	3.73205	0.778186	0.389093	+	0.921198i	$0.372788\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	0	0
$28$	0	0
$29$	−1.53590	−0.285209	−0.142605	−	0.989780i	$-0.545548\pi$
$29$	−1.53590	−0.285209	−0.142605	+	0.989780i	$0.545548\pi$
$30$	0	0
$31$	2.00000	0.359211	0.179605	−	0.983739i	$-0.442518\pi$
$31$	2.00000	0.359211	0.179605	+	0.983739i	$0.442518\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−0.267949	−0.0452917
$36$	0	0
$37$	10.3923	1.70848	0.854242	−	0.519875i	$-0.174022\pi$
$37$	10.3923	1.70848	0.854242	+	0.519875i	$0.174022\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−9.92820	−1.55052	−0.775262	−	0.631639i	$-0.782382\pi$
$41$	−9.92820	−1.55052	−0.775262	+	0.631639i	$0.782382\pi$
$42$	0	0
$43$	4.53590	0.691718	0.345859	−	0.938286i	$-0.387588\pi$
$43$	4.53590	0.691718	0.345859	+	0.938286i	$0.387588\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	0.267949	0.0390844	0.0195422	−	0.999809i	$-0.493779\pi$
$47$	0.267949	0.0390844	0.0195422	+	0.999809i	$0.493779\pi$
$48$	0	0
$49$	−6.92820	−0.989743
$50$	0	0
$51$	0	0
$52$	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
$55$	−1.46410	−0.197419
$56$	0	0
$57$	0	0
$58$	0	0
$59$	14.3923	1.87372	0.936859	−	0.349707i	$-0.113719\pi$
$59$	14.3923	1.87372	0.936859	+	0.349707i	$0.113719\pi$
$60$	0	0
$61$	−8.46410	−1.08372	−0.541859	−	0.840470i	$-0.682279\pi$
$61$	−8.46410	−1.08372	−0.541859	+	0.840470i	$0.682279\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−5.46410	−0.677738
$66$	0	0
$67$	−6.26795	−0.765752	−0.382876	−	0.923800i	$-0.625066\pi$
$67$	−6.26795	−0.765752	−0.382876	+	0.923800i	$0.625066\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	9.46410	1.12318	0.561591	−	0.827415i	$-0.310189\pi$
$71$	9.46410	1.12318	0.561591	+	0.827415i	$0.310189\pi$
$72$	0	0
$73$	6.92820	0.810885	0.405442	−	0.914121i	$-0.367117\pi$
$73$	6.92820	0.810885	0.405442	+	0.914121i	$0.367117\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	0.392305	0.0447073
$78$	0	0
$79$	−15.4641	−1.73985	−0.869924	−	0.493186i	$-0.835832\pi$
$79$	−15.4641	−1.73985	−0.869924	+	0.493186i	$0.835832\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	−13.1962	−1.44847	−0.724233	−	0.689555i	$-0.757806\pi$
$83$	−13.1962	−1.44847	−0.724233	+	0.689555i	$0.757806\pi$
$84$	0	0
$85$	−0.535898	−0.0581263
$86$	0	0
$87$	0	0
$88$	0	0
$89$	9.92820	1.05239	0.526194	−	0.850365i	$-0.323619\pi$
$89$	9.92820	1.05239	0.526194	+	0.850365i	$0.323619\pi$
$90$	0	0
$91$	1.46410	0.153480
$92$	0	0
$93$	0	0
$94$	0	0
$95$	2.00000	0.205196
$96$	0	0
$97$	−8.92820	−0.906522	−0.453261	−	0.891378i	$-0.649739\pi$
$97$	−8.92820	−0.906522	−0.453261	+	0.891378i	$0.649739\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−0.928203	−0.0923597	−0.0461798	−	0.998933i	$-0.514705\pi$
$101$	−0.928203	−0.0923597	−0.0461798	+	0.998933i	$0.514705\pi$
$102$	0	0
$103$	−14.3923	−1.41812	−0.709058	−	0.705150i	$-0.750880\pi$
$103$	−14.3923	−1.41812	−0.709058	+	0.705150i	$0.750880\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	5.19615	0.502331	0.251166	−	0.967944i	$-0.419186\pi$
$107$	5.19615	0.502331	0.251166	+	0.967944i	$0.419186\pi$
$108$	0	0
$109$	−17.3923	−1.66588	−0.832940	−	0.553363i	$-0.813344\pi$
$109$	−17.3923	−1.66588	−0.832940	+	0.553363i	$0.813344\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−13.8564	−1.30350	−0.651751	−	0.758433i	$-0.725965\pi$
$113$	−13.8564	−1.30350	−0.651751	+	0.758433i	$0.725965\pi$
$114$	0	0
$115$	3.73205	0.348016
$116$	0	0
$117$	0	0
$118$	0	0
$119$	0.143594	0.0131632
$120$	0	0
$121$	−8.85641	−0.805128
$122$	0	0
$123$	0	0
$124$	0	0
$125$	1.00000	0.0894427
$126$	0	0
$127$	−16.2679	−1.44355	−0.721774	−	0.692129i	$-0.756673\pi$
$127$	−16.2679	−1.44355	−0.721774	+	0.692129i	$0.756673\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−3.46410	−0.302660	−0.151330	−	0.988483i	$-0.548356\pi$
$131$	−3.46410	−0.302660	−0.151330	+	0.988483i	$0.548356\pi$
$132$	0	0
$133$	−0.535898	−0.0464683
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−21.8564	−1.86732	−0.933659	−	0.358162i	$-0.883404\pi$
$137$	−21.8564	−1.86732	−0.933659	+	0.358162i	$0.883404\pi$
$138$	0	0
$139$	0	0		−	1.00000i	$-0.5\pi$
$139$	0	0			1.00000i	$0.5\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	8.00000	0.668994
$144$	0	0
$145$	−1.53590	−0.127549
$146$	0	0
$147$	0	0
$148$	0	0
$149$	2.46410	0.201867	0.100934	−	0.994893i	$-0.467817\pi$
$149$	2.46410	0.201867	0.100934	+	0.994893i	$0.467817\pi$
$150$	0	0
$151$	−5.46410	−0.444662	−0.222331	−	0.974971i	$-0.571367\pi$
$151$	−5.46410	−0.444662	−0.222331	+	0.974971i	$0.571367\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	2.00000	0.160644
$156$	0	0
$157$	4.92820	0.393313	0.196657	−	0.980472i	$-0.436992\pi$
$157$	4.92820	0.393313	0.196657	+	0.980472i	$0.436992\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−1.00000	−0.0788110
$162$	0	0
$163$	13.3205	1.04334	0.521671	−	0.853147i	$-0.325309\pi$
$163$	13.3205	1.04334	0.521671	+	0.853147i	$0.325309\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−21.5885	−1.67056	−0.835282	−	0.549821i	$-0.814696\pi$
$167$	−21.5885	−1.67056	−0.835282	+	0.549821i	$0.814696\pi$
$168$	0	0
$169$	16.8564	1.29665
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−9.07180	−0.689716	−0.344858	−	0.938655i	$-0.612073\pi$
$173$	−9.07180	−0.689716	−0.344858	+	0.938655i	$0.612073\pi$
$174$	0	0
$175$	−0.267949	−0.0202551
$176$	0	0
$177$	0	0
$178$	0	0
$179$	13.4641	1.00635	0.503177	−	0.864183i	$-0.332164\pi$
$179$	13.4641	1.00635	0.503177	+	0.864183i	$0.332164\pi$
$180$	0	0
$181$	−9.53590	−0.708798	−0.354399	−	0.935094i	$-0.615314\pi$
$181$	−9.53590	−0.708798	−0.354399	+	0.935094i	$0.615314\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	10.3923	0.764057
$186$	0	0
$187$	0.784610	0.0573763
$188$	0	0
$189$	0	0
$190$	0	0
$191$	16.9282	1.22488	0.612441	−	0.790516i	$-0.290188\pi$
$191$	16.9282	1.22488	0.612441	+	0.790516i	$0.290188\pi$
$192$	0	0
$193$	−17.4641	−1.25709	−0.628547	−	0.777772i	$-0.716350\pi$
$193$	−17.4641	−1.25709	−0.628547	+	0.777772i	$0.716350\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−14.9282	−1.06359	−0.531795	−	0.846873i	$-0.678482\pi$
$197$	−14.9282	−1.06359	−0.531795	+	0.846873i	$0.678482\pi$
$198$	0	0
$199$	−7.07180	−0.501306	−0.250653	−	0.968077i	$-0.580645\pi$
$199$	−7.07180	−0.501306	−0.250653	+	0.968077i	$0.580645\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	0.411543	0.0288846
$204$	0	0
$205$	−9.92820	−0.693416
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−2.92820	−0.202548
$210$	0	0
$211$	−0.392305	−0.0270074	−0.0135037	−	0.999909i	$-0.504298\pi$
$211$	−0.392305	−0.0270074	−0.0135037	+	0.999909i	$0.504298\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	4.53590	0.309346
$216$	0	0
$217$	−0.535898	−0.0363792
$218$	0	0
$219$	0	0
$220$	0	0
$221$	2.92820	0.196972
$222$	0	0
$223$	−22.6603	−1.51744	−0.758721	−	0.651415i	$-0.774176\pi$
$223$	−22.6603	−1.51744	−0.758721	+	0.651415i	$0.774176\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	24.2487	1.60944	0.804722	−	0.593652i	$-0.202314\pi$
$227$	24.2487	1.60944	0.804722	+	0.593652i	$0.202314\pi$
$228$	0	0
$229$	25.2487	1.66848	0.834241	−	0.551400i	$-0.185906\pi$
$229$	25.2487	1.66848	0.834241	+	0.551400i	$0.185906\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	10.3923	0.680823	0.340411	−	0.940277i	$-0.389434\pi$
$233$	10.3923	0.680823	0.340411	+	0.940277i	$0.389434\pi$
$234$	0	0
$235$	0.267949	0.0174791
$236$	0	0
$237$	0	0
$238$	0	0
$239$	20.7846	1.34444	0.672222	−	0.740349i	$-0.265340\pi$
$239$	20.7846	1.34444	0.672222	+	0.740349i	$0.265340\pi$
$240$	0	0
$241$	−6.07180	−0.391119	−0.195559	−	0.980692i	$-0.562652\pi$
$241$	−6.07180	−0.391119	−0.195559	+	0.980692i	$0.562652\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	−6.92820	−0.442627
$246$	0	0
$247$	−10.9282	−0.695345
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−7.85641	−0.495892	−0.247946	−	0.968774i	$-0.579756\pi$
$251$	−7.85641	−0.495892	−0.247946	+	0.968774i	$0.579756\pi$
$252$	0	0
$253$	−5.46410	−0.343525
$254$	0	0
$255$	0	0
$256$	0	0
$257$	0.928203	0.0578997	0.0289499	−	0.999581i	$-0.490784\pi$
$257$	0.928203	0.0578997	0.0289499	+	0.999581i	$0.490784\pi$
$258$	0	0
$259$	−2.78461	−0.173027
$260$	0	0
$261$	0	0
$262$	0	0
$263$	22.3923	1.38077	0.690384	−	0.723443i	$-0.257441\pi$
$263$	22.3923	1.38077	0.690384	+	0.723443i	$0.257441\pi$
$264$	0	0
$265$	−6.00000	−0.368577
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−11.3923	−0.694601	−0.347301	−	0.937754i	$-0.612902\pi$
$269$	−11.3923	−0.694601	−0.347301	+	0.937754i	$0.612902\pi$
$270$	0	0
$271$	19.8564	1.20619	0.603095	−	0.797669i	$-0.293934\pi$
$271$	19.8564	1.20619	0.603095	+	0.797669i	$0.293934\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−1.46410	−0.0882886
$276$	0	0
$277$	−14.3923	−0.864750	−0.432375	−	0.901694i	$-0.642324\pi$
$277$	−14.3923	−0.864750	−0.432375	+	0.901694i	$0.642324\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	14.8564	0.886259	0.443129	−	0.896458i	$-0.353868\pi$
$281$	14.8564	0.886259	0.443129	+	0.896458i	$0.353868\pi$
$282$	0	0
$283$	−21.7321	−1.29184	−0.645918	−	0.763407i	$-0.723525\pi$
$283$	−21.7321	−1.29184	−0.645918	+	0.763407i	$0.723525\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	2.66025	0.157030
$288$	0	0
$289$	−16.7128	−0.983107
$290$	0	0
$291$	0	0
$292$	0	0
$293$	7.46410	0.436057	0.218029	−	0.975942i	$-0.430037\pi$
$293$	7.46410	0.436057	0.218029	+	0.975942i	$0.430037\pi$
$294$	0	0
$295$	14.3923	0.837952
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−20.3923	−1.17932
$300$	0	0
$301$	−1.21539	−0.0700539
$302$	0	0
$303$	0	0
$304$	0	0
$305$	−8.46410	−0.484653
$306$	0	0
$307$	−4.12436	−0.235389	−0.117695	−	0.993050i	$-0.537550\pi$
$307$	−4.12436	−0.235389	−0.117695	+	0.993050i	$0.537550\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	3.07180	0.174186	0.0870928	−	0.996200i	$-0.472242\pi$
$311$	3.07180	0.174186	0.0870928	+	0.996200i	$0.472242\pi$
$312$	0	0
$313$	12.3923	0.700454	0.350227	−	0.936665i	$-0.386104\pi$
$313$	12.3923	0.700454	0.350227	+	0.936665i	$0.386104\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−2.53590	−0.142430	−0.0712151	−	0.997461i	$-0.522688\pi$
$317$	−2.53590	−0.142430	−0.0712151	+	0.997461i	$0.522688\pi$
$318$	0	0
$319$	2.24871	0.125904
$320$	0	0
$321$	0	0
$322$	0	0
$323$	−1.07180	−0.0596364
$324$	0	0
$325$	−5.46410	−0.303094
$326$	0	0
$327$	0	0
$328$	0	0
$329$	−0.0717968	−0.00395828
$330$	0	0
$331$	3.46410	0.190404	0.0952021	−	0.995458i	$-0.469650\pi$
$331$	3.46410	0.190404	0.0952021	+	0.995458i	$0.469650\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	−6.26795	−0.342455
$336$	0	0
$337$	9.07180	0.494172	0.247086	−	0.968994i	$-0.420527\pi$
$337$	9.07180	0.494172	0.247086	+	0.968994i	$0.420527\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	−2.92820	−0.158571
$342$	0	0
$343$	3.73205	0.201512
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−18.3923	−0.987351	−0.493675	−	0.869646i	$-0.664347\pi$
$347$	−18.3923	−0.987351	−0.493675	+	0.869646i	$0.664347\pi$
$348$	0	0
$349$	0.464102	0.0248428	0.0124214	−	0.999923i	$-0.496046\pi$
$349$	0.464102	0.0248428	0.0124214	+	0.999923i	$0.496046\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	21.8564	1.16330	0.581650	−	0.813439i	$-0.302408\pi$
$353$	21.8564	1.16330	0.581650	+	0.813439i	$0.302408\pi$
$354$	0	0
$355$	9.46410	0.502302
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−12.9282	−0.682324	−0.341162	−	0.940004i	$-0.610821\pi$
$359$	−12.9282	−0.682324	−0.341162	+	0.940004i	$0.610821\pi$
$360$	0	0
$361$	−15.0000	−0.789474
$362$	0	0
$363$	0	0
$364$	0	0
$365$	6.92820	0.362639
$366$	0	0
$367$	14.3923	0.751272	0.375636	−	0.926767i	$-0.377424\pi$
$367$	14.3923	0.751272	0.375636	+	0.926767i	$0.377424\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	1.60770	0.0834674
$372$	0	0
$373$	−8.92820	−0.462285	−0.231142	−	0.972920i	$-0.574246\pi$
$373$	−8.92820	−0.462285	−0.231142	+	0.972920i	$0.574246\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	8.39230	0.432226
$378$	0	0
$379$	−35.1769	−1.80692	−0.903458	−	0.428676i	$-0.858980\pi$
$379$	−35.1769	−1.80692	−0.903458	+	0.428676i	$0.858980\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	3.46410	0.177007	0.0885037	−	0.996076i	$-0.471792\pi$
$383$	3.46410	0.177007	0.0885037	+	0.996076i	$0.471792\pi$
$384$	0	0
$385$	0.392305	0.0199937
$386$	0	0
$387$	0	0
$388$	0	0
$389$	23.5359	1.19332	0.596659	−	0.802495i	$-0.296495\pi$
$389$	23.5359	1.19332	0.596659	+	0.802495i	$0.296495\pi$
$390$	0	0
$391$	−2.00000	−0.101144
$392$	0	0
$393$	0	0
$394$	0	0
$395$	−15.4641	−0.778083
$396$	0	0
$397$	10.0000	0.501886	0.250943	−	0.968002i	$-0.419259\pi$
$397$	10.0000	0.501886	0.250943	+	0.968002i	$0.419259\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−14.7846	−0.738308	−0.369154	−	0.929368i	$-0.620353\pi$
$401$	−14.7846	−0.738308	−0.369154	+	0.929368i	$0.620353\pi$
$402$	0	0
$403$	−10.9282	−0.544373
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−15.2154	−0.754199
$408$	0	0
$409$	4.92820	0.243684	0.121842	−	0.992550i	$-0.461120\pi$
$409$	4.92820	0.243684	0.121842	+	0.992550i	$0.461120\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	−3.85641	−0.189761
$414$	0	0
$415$	−13.1962	−0.647774
$416$	0	0
$417$	0	0
$418$	0	0
$419$	30.2487	1.47775	0.738873	−	0.673845i	$-0.235358\pi$
$419$	30.2487	1.47775	0.738873	+	0.673845i	$0.235358\pi$
$420$	0	0
$421$	−36.9282	−1.79977	−0.899885	−	0.436127i	$-0.856350\pi$
$421$	−36.9282	−1.79977	−0.899885	+	0.436127i	$0.856350\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	−0.535898	−0.0259949
$426$	0	0
$427$	2.26795	0.109754
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−4.53590	−0.218487	−0.109243	−	0.994015i	$-0.534843\pi$
$431$	−4.53590	−0.218487	−0.109243	+	0.994015i	$0.534843\pi$
$432$	0	0
$433$	−12.5359	−0.602437	−0.301218	−	0.953555i	$-0.597393\pi$
$433$	−12.5359	−0.602437	−0.301218	+	0.953555i	$0.597393\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	7.46410	0.357056
$438$	0	0
$439$	−21.0718	−1.00570	−0.502851	−	0.864373i	$-0.667716\pi$
$439$	−21.0718	−1.00570	−0.502851	+	0.864373i	$0.667716\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	34.5167	1.63994	0.819968	−	0.572409i	$-0.193991\pi$
$443$	34.5167	1.63994	0.819968	+	0.572409i	$0.193991\pi$
$444$	0	0
$445$	9.92820	0.470642
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−4.14359	−0.195548	−0.0977741	−	0.995209i	$-0.531172\pi$
$449$	−4.14359	−0.195548	−0.0977741	+	0.995209i	$0.531172\pi$
$450$	0	0
$451$	14.5359	0.684469
$452$	0	0
$453$	0	0
$454$	0	0
$455$	1.46410	0.0686381
$456$	0	0
$457$	5.60770	0.262317	0.131158	−	0.991361i	$-0.458130\pi$
$457$	5.60770	0.262317	0.131158	+	0.991361i	$0.458130\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−4.60770	−0.214602	−0.107301	−	0.994227i	$-0.534221\pi$
$461$	−4.60770	−0.214602	−0.107301	+	0.994227i	$0.534221\pi$
$462$	0	0
$463$	3.46410	0.160990	0.0804952	−	0.996755i	$-0.474350\pi$
$463$	3.46410	0.160990	0.0804952	+	0.996755i	$0.474350\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−30.3923	−1.40639	−0.703194	−	0.710998i	$-0.748243\pi$
$467$	−30.3923	−1.40639	−0.703194	+	0.710998i	$0.748243\pi$
$468$	0	0
$469$	1.67949	0.0775517
$470$	0	0
$471$	0	0
$472$	0	0
$473$	−6.64102	−0.305354
$474$	0	0
$475$	2.00000	0.0917663
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−14.3923	−0.657601	−0.328801	−	0.944399i	$-0.606644\pi$
$479$	−14.3923	−0.657601	−0.328801	+	0.944399i	$0.606644\pi$
$480$	0	0
$481$	−56.7846	−2.58916
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−8.92820	−0.405409
$486$	0	0
$487$	−40.2487	−1.82384	−0.911922	−	0.410364i	$-0.865401\pi$
$487$	−40.2487	−1.82384	−0.911922	+	0.410364i	$0.865401\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−25.8564	−1.16688	−0.583442	−	0.812155i	$-0.698294\pi$
$491$	−25.8564	−1.16688	−0.583442	+	0.812155i	$0.698294\pi$
$492$	0	0
$493$	0.823085	0.0370699
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−2.53590	−0.113751
$498$	0	0
$499$	17.8564	0.799363	0.399681	−	0.916654i	$-0.369121\pi$
$499$	17.8564	0.799363	0.399681	+	0.916654i	$0.369121\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−18.1244	−0.808125	−0.404063	−	0.914731i	$-0.632402\pi$
$503$	−18.1244	−0.808125	−0.404063	+	0.914731i	$0.632402\pi$
$504$	0	0
$505$	−0.928203	−0.0413045
$506$	0	0
$507$	0	0
$508$	0	0
$509$	20.6077	0.913420	0.456710	−	0.889616i	$-0.349028\pi$
$509$	20.6077	0.913420	0.456710	+	0.889616i	$0.349028\pi$
$510$	0	0
$511$	−1.85641	−0.0821226
$512$	0	0
$513$	0	0
$514$	0	0
$515$	−14.3923	−0.634201
$516$	0	0
$517$	−0.392305	−0.0172535
$518$	0	0
$519$	0	0
$520$	0	0
$521$	13.0000	0.569540	0.284770	−	0.958596i	$-0.408083\pi$
$521$	13.0000	0.569540	0.284770	+	0.958596i	$0.408083\pi$
$522$	0	0
$523$	−10.8038	−0.472419	−0.236210	−	0.971702i	$-0.575905\pi$
$523$	−10.8038	−0.472419	−0.236210	+	0.971702i	$0.575905\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−1.07180	−0.0466882
$528$	0	0
$529$	−9.07180	−0.394426
$530$	0	0
$531$	0	0
$532$	0	0
$533$	54.2487	2.34977
$534$	0	0
$535$	5.19615	0.224649
$536$	0	0
$537$	0	0
$538$	0	0
$539$	10.1436	0.436916
$540$	0	0
$541$	−18.6077	−0.800007	−0.400004	−	0.916514i	$-0.630991\pi$
$541$	−18.6077	−0.800007	−0.400004	+	0.916514i	$0.630991\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	−17.3923	−0.745004
$546$	0	0
$547$	−45.9808	−1.96600	−0.982998	−	0.183618i	$-0.941219\pi$
$547$	−45.9808	−1.96600	−0.982998	+	0.183618i	$0.941219\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−3.07180	−0.130863
$552$	0	0
$553$	4.14359	0.176204
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−32.6410	−1.38304	−0.691522	−	0.722355i	$-0.743060\pi$
$557$	−32.6410	−1.38304	−0.691522	+	0.722355i	$0.743060\pi$
$558$	0	0
$559$	−24.7846	−1.04828
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−12.4115	−0.523084	−0.261542	−	0.965192i	$-0.584231\pi$
$563$	−12.4115	−0.523084	−0.261542	+	0.965192i	$0.584231\pi$
$564$	0	0
$565$	−13.8564	−0.582943
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−10.0000	−0.419222	−0.209611	−	0.977785i	$-0.567220\pi$
$569$	−10.0000	−0.419222	−0.209611	+	0.977785i	$0.567220\pi$
$570$	0	0
$571$	40.7846	1.70678	0.853391	−	0.521271i	$-0.174542\pi$
$571$	40.7846	1.70678	0.853391	+	0.521271i	$0.174542\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	3.73205	0.155637
$576$	0	0
$577$	32.9282	1.37082	0.685410	−	0.728158i	$-0.259623\pi$
$577$	32.9282	1.37082	0.685410	+	0.728158i	$0.259623\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	3.53590	0.146694
$582$	0	0
$583$	8.78461	0.363821
$584$	0	0
$585$	0	0
$586$	0	0
$587$	8.41154	0.347182	0.173591	−	0.984818i	$-0.444463\pi$
$587$	8.41154	0.347182	0.173591	+	0.984818i	$0.444463\pi$
$588$	0	0
$589$	4.00000	0.164817
$590$	0	0
$591$	0	0
$592$	0	0
$593$	39.4641	1.62060	0.810298	−	0.586018i	$-0.199305\pi$
$593$	39.4641	1.62060	0.810298	+	0.586018i	$0.199305\pi$
$594$	0	0
$595$	0.143594	0.00588676
$596$	0	0
$597$	0	0
$598$	0	0
$599$	1.60770	0.0656886	0.0328443	−	0.999460i	$-0.489543\pi$
$599$	1.60770	0.0656886	0.0328443	+	0.999460i	$0.489543\pi$
$600$	0	0
$601$	30.7846	1.25573	0.627865	−	0.778322i	$-0.283929\pi$
$601$	30.7846	1.25573	0.627865	+	0.778322i	$0.283929\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	−8.85641	−0.360064
$606$	0	0
$607$	−40.2679	−1.63443	−0.817213	−	0.576336i	$-0.804482\pi$
$607$	−40.2679	−1.63443	−0.817213	+	0.576336i	$0.804482\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−1.46410	−0.0592312
$612$	0	0
$613$	22.3923	0.904417	0.452208	−	0.891912i	$-0.350636\pi$
$613$	22.3923	0.904417	0.452208	+	0.891912i	$0.350636\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−30.9282	−1.24512	−0.622561	−	0.782571i	$-0.713908\pi$
$617$	−30.9282	−1.24512	−0.622561	+	0.782571i	$0.713908\pi$
$618$	0	0
$619$	37.7128	1.51581	0.757903	−	0.652367i	$-0.226224\pi$
$619$	37.7128	1.51581	0.757903	+	0.652367i	$0.226224\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−2.66025	−0.106581
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−5.56922	−0.222059
$630$	0	0
$631$	−31.7128	−1.26247	−0.631234	−	0.775593i	$-0.717451\pi$
$631$	−31.7128	−1.26247	−0.631234	+	0.775593i	$0.717451\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	−16.2679	−0.645574
$636$	0	0
$637$	37.8564	1.49993
$638$	0	0
$639$	0	0
$640$	0	0
$641$	34.8564	1.37675	0.688373	−	0.725357i	$-0.258325\pi$
$641$	34.8564	1.37675	0.688373	+	0.725357i	$0.258325\pi$
$642$	0	0
$643$	34.2679	1.35140	0.675698	−	0.737179i	$-0.263842\pi$
$643$	34.2679	1.35140	0.675698	+	0.737179i	$0.263842\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−16.5167	−0.649337	−0.324668	−	0.945828i	$-0.605253\pi$
$647$	−16.5167	−0.649337	−0.324668	+	0.945828i	$0.605253\pi$
$648$	0	0
$649$	−21.0718	−0.827140
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−31.4641	−1.23129	−0.615643	−	0.788025i	$-0.711104\pi$
$653$	−31.4641	−1.23129	−0.615643	+	0.788025i	$0.711104\pi$
$654$	0	0
$655$	−3.46410	−0.135354
$656$	0	0
$657$	0	0
$658$	0	0
$659$	−20.7846	−0.809653	−0.404827	−	0.914393i	$-0.632668\pi$
$659$	−20.7846	−0.809653	−0.404827	+	0.914393i	$0.632668\pi$
$660$	0	0
$661$	21.7128	0.844531	0.422265	−	0.906472i	$-0.361235\pi$
$661$	21.7128	0.844531	0.422265	+	0.906472i	$0.361235\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−0.535898	−0.0207812
$666$	0	0
$667$	−5.73205	−0.221946
$668$	0	0
$669$	0	0
$670$	0	0
$671$	12.3923	0.478400
$672$	0	0
$673$	−12.6795	−0.488758	−0.244379	−	0.969680i	$-0.578584\pi$
$673$	−12.6795	−0.488758	−0.244379	+	0.969680i	$0.578584\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	17.1769	0.660162	0.330081	−	0.943953i	$-0.392924\pi$
$677$	17.1769	0.660162	0.330081	+	0.943953i	$0.392924\pi$
$678$	0	0
$679$	2.39230	0.0918082
$680$	0	0
$681$	0	0
$682$	0	0
$683$	22.3923	0.856818	0.428409	−	0.903585i	$-0.359074\pi$
$683$	22.3923	0.856818	0.428409	+	0.903585i	$0.359074\pi$
$684$	0	0
$685$	−21.8564	−0.835090
$686$	0	0
$687$	0	0
$688$	0	0
$689$	32.7846	1.24899
$690$	0	0
$691$	50.3923	1.91701	0.958507	−	0.285070i	$-0.0920167\pi$
$691$	50.3923	1.91701	0.958507	+	0.285070i	$0.0920167\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	0	0
$696$	0	0
$697$	5.32051	0.201529
$698$	0	0
$699$	0	0
$700$	0	0
$701$	24.1769	0.913149	0.456575	−	0.889685i	$-0.349076\pi$
$701$	24.1769	0.913149	0.456575	+	0.889685i	$0.349076\pi$
$702$	0	0
$703$	20.7846	0.783906
$704$	0	0
$705$	0	0
$706$	0	0
$707$	0.248711	0.00935375
$708$	0	0
$709$	15.3923	0.578070	0.289035	−	0.957319i	$-0.406666\pi$
$709$	15.3923	0.578070	0.289035	+	0.957319i	$0.406666\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	7.46410	0.279533
$714$	0	0
$715$	8.00000	0.299183
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−33.3205	−1.24265	−0.621323	−	0.783555i	$-0.713404\pi$
$719$	−33.3205	−1.24265	−0.621323	+	0.783555i	$0.713404\pi$
$720$	0	0
$721$	3.85641	0.143620
$722$	0	0
$723$	0	0
$724$	0	0
$725$	−1.53590	−0.0570418
$726$	0	0
$727$	−15.9808	−0.592694	−0.296347	−	0.955080i	$-0.595768\pi$
$727$	−15.9808	−0.592694	−0.296347	+	0.955080i	$0.595768\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−2.43078	−0.0899057
$732$	0	0
$733$	16.1436	0.596277	0.298139	−	0.954523i	$-0.403634\pi$
$733$	16.1436	0.596277	0.298139	+	0.954523i	$0.403634\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	9.17691	0.338036
$738$	0	0
$739$	0.248711	0.00914899	0.00457450	−	0.999990i	$-0.498544\pi$
$739$	0.248711	0.00914899	0.00457450	+	0.999990i	$0.498544\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−33.8372	−1.24137	−0.620683	−	0.784062i	$-0.713144\pi$
$743$	−33.8372	−1.24137	−0.620683	+	0.784062i	$0.713144\pi$
$744$	0	0
$745$	2.46410	0.0902777
$746$	0	0
$747$	0	0
$748$	0	0
$749$	−1.39230	−0.0508737
$750$	0	0
$751$	−16.3923	−0.598164	−0.299082	−	0.954227i	$-0.596680\pi$
$751$	−16.3923	−0.598164	−0.299082	+	0.954227i	$0.596680\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	−5.46410	−0.198859
$756$	0	0
$757$	−20.3923	−0.741171	−0.370585	−	0.928798i	$-0.620843\pi$
$757$	−20.3923	−0.741171	−0.370585	+	0.928798i	$0.620843\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	21.0000	0.761249	0.380625	−	0.924730i	$-0.375709\pi$
$761$	21.0000	0.761249	0.380625	+	0.924730i	$0.375709\pi$
$762$	0	0
$763$	4.66025	0.168713
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−78.6410	−2.83956
$768$	0	0
$769$	33.7846	1.21830	0.609152	−	0.793053i	$-0.291510\pi$
$769$	33.7846	1.21830	0.609152	+	0.793053i	$0.291510\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	8.78461	0.315960	0.157980	−	0.987442i	$-0.449502\pi$
$773$	8.78461	0.315960	0.157980	+	0.987442i	$0.449502\pi$
$774$	0	0
$775$	2.00000	0.0718421
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−19.8564	−0.711430
$780$	0	0
$781$	−13.8564	−0.495821
$782$	0	0
$783$	0	0
$784$	0	0
$785$	4.92820	0.175895
$786$	0	0
$787$	−1.32051	−0.0470710	−0.0235355	−	0.999723i	$-0.507492\pi$
$787$	−1.32051	−0.0470710	−0.0235355	+	0.999723i	$0.507492\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	3.71281	0.132012
$792$	0	0
$793$	46.2487	1.64234
$794$	0	0
$795$	0	0
$796$	0	0
$797$	7.60770	0.269478	0.134739	−	0.990881i	$-0.456980\pi$
$797$	7.60770	0.269478	0.134739	+	0.990881i	$0.456980\pi$
$798$	0	0
$799$	−0.143594	−0.00507997
$800$	0	0
$801$	0	0
$802$	0	0
$803$	−10.1436	−0.357960
$804$	0	0
$805$	−1.00000	−0.0352454
$806$	0	0
$807$	0	0
$808$	0	0
$809$	48.9282	1.72022	0.860112	−	0.510105i	$-0.170394\pi$
$809$	48.9282	1.72022	0.860112	+	0.510105i	$0.170394\pi$
$810$	0	0
$811$	50.3923	1.76951	0.884757	−	0.466053i	$-0.154325\pi$
$811$	50.3923	1.76951	0.884757	+	0.466053i	$0.154325\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	13.3205	0.466597
$816$	0	0
$817$	9.07180	0.317382
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−23.2487	−0.811386	−0.405693	−	0.914009i	$-0.632970\pi$
$821$	−23.2487	−0.811386	−0.405693	+	0.914009i	$0.632970\pi$
$822$	0	0
$823$	−23.7321	−0.827247	−0.413624	−	0.910448i	$-0.635737\pi$
$823$	−23.7321	−0.827247	−0.413624	+	0.910448i	$0.635737\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−16.9474	−0.589320	−0.294660	−	0.955602i	$-0.595206\pi$
$827$	−16.9474	−0.589320	−0.294660	+	0.955602i	$0.595206\pi$
$828$	0	0
$829$	11.3923	0.395671	0.197836	−	0.980235i	$-0.436609\pi$
$829$	11.3923	0.395671	0.197836	+	0.980235i	$0.436609\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	3.71281	0.128641
$834$	0	0
$835$	−21.5885	−0.747099
$836$	0	0
$837$	0	0
$838$	0	0
$839$	35.8564	1.23790	0.618950	−	0.785430i	$-0.287558\pi$
$839$	35.8564	1.23790	0.618950	+	0.785430i	$0.287558\pi$
$840$	0	0
$841$	−26.6410	−0.918656
$842$	0	0
$843$	0	0
$844$	0	0
$845$	16.8564	0.579878
$846$	0	0
$847$	2.37307	0.0815395
$848$	0	0
$849$	0	0
$850$	0	0
$851$	38.7846	1.32952
$852$	0	0
$853$	−49.4641	−1.69362	−0.846809	−	0.531897i	$-0.821479\pi$
$853$	−49.4641	−1.69362	−0.846809	+	0.531897i	$0.821479\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−18.2487	−0.623364	−0.311682	−	0.950186i	$-0.600892\pi$
$857$	−18.2487	−0.623364	−0.311682	+	0.950186i	$0.600892\pi$
$858$	0	0
$859$	20.3923	0.695776	0.347888	−	0.937536i	$-0.386899\pi$
$859$	20.3923	0.695776	0.347888	+	0.937536i	$0.386899\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	26.6603	0.907526	0.453763	−	0.891123i	$-0.350081\pi$
$863$	26.6603	0.907526	0.453763	+	0.891123i	$0.350081\pi$
$864$	0	0
$865$	−9.07180	−0.308450
$866$	0	0
$867$	0	0
$868$	0	0
$869$	22.6410	0.768044
$870$	0	0
$871$	34.2487	1.16047
$872$	0	0
$873$	0	0
$874$	0	0
$875$	−0.267949	−0.00905834
$876$	0	0
$877$	11.6077	0.391964	0.195982	−	0.980607i	$-0.437211\pi$
$877$	11.6077	0.391964	0.195982	+	0.980607i	$0.437211\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−37.6410	−1.26816	−0.634079	−	0.773268i	$-0.718621\pi$
$881$	−37.6410	−1.26816	−0.634079	+	0.773268i	$0.718621\pi$
$882$	0	0
$883$	9.19615	0.309475	0.154738	−	0.987956i	$-0.450547\pi$
$883$	9.19615	0.309475	0.154738	+	0.987956i	$0.450547\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−9.32051	−0.312952	−0.156476	−	0.987682i	$-0.550013\pi$
$887$	−9.32051	−0.312952	−0.156476	+	0.987682i	$0.550013\pi$
$888$	0	0
$889$	4.35898	0.146196
$890$	0	0
$891$	0	0
$892$	0	0
$893$	0.535898	0.0179332
$894$	0	0
$895$	13.4641	0.450055
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−3.07180	−0.102450
$900$	0	0
$901$	3.21539	0.107120
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−9.53590	−0.316984
$906$	0	0
$907$	13.7321	0.455965	0.227983	−	0.973665i	$-0.426787\pi$
$907$	13.7321	0.455965	0.227983	+	0.973665i	$0.426787\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−12.5359	−0.415333	−0.207666	−	0.978200i	$-0.566587\pi$
$911$	−12.5359	−0.415333	−0.207666	+	0.978200i	$0.566587\pi$
$912$	0	0
$913$	19.3205	0.639415
$914$	0	0
$915$	0	0
$916$	0	0
$917$	0.928203	0.0306520
$918$	0	0
$919$	17.6077	0.580824	0.290412	−	0.956902i	$-0.406208\pi$
$919$	17.6077	0.580824	0.290412	+	0.956902i	$0.406208\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	−51.7128	−1.70215
$924$	0	0
$925$	10.3923	0.341697
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−24.6410	−0.808446	−0.404223	−	0.914661i	$-0.632458\pi$
$929$	−24.6410	−0.808446	−0.404223	+	0.914661i	$0.632458\pi$
$930$	0	0
$931$	−13.8564	−0.454125
$932$	0	0
$933$	0	0
$934$	0	0
$935$	0.784610	0.0256595
$936$	0	0
$937$	−11.7128	−0.382641	−0.191320	−	0.981528i	$-0.561277\pi$
$937$	−11.7128	−0.382641	−0.191320	+	0.981528i	$0.561277\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	23.3923	0.762567	0.381284	−	0.924458i	$-0.375482\pi$
$941$	23.3923	0.762567	0.381284	+	0.924458i	$0.375482\pi$
$942$	0	0
$943$	−37.0526	−1.20660
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−14.5167	−0.471728	−0.235864	−	0.971786i	$-0.575792\pi$
$947$	−14.5167	−0.471728	−0.235864	+	0.971786i	$0.575792\pi$
$948$	0	0
$949$	−37.8564	−1.22887
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−40.3923	−1.30844	−0.654218	−	0.756306i	$-0.727002\pi$
$953$	−40.3923	−1.30844	−0.654218	+	0.756306i	$0.727002\pi$
$954$	0	0
$955$	16.9282	0.547784
$956$	0	0
$957$	0	0
$958$	0	0
$959$	5.85641	0.189113
$960$	0	0
$961$	−27.0000	−0.870968
$962$	0	0
$963$	0	0
$964$	0	0
$965$	−17.4641	−0.562189
$966$	0	0
$967$	−21.3397	−0.686240	−0.343120	−	0.939292i	$-0.611484\pi$
$967$	−21.3397	−0.686240	−0.343120	+	0.939292i	$0.611484\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	25.7128	0.825163	0.412582	−	0.910921i	$-0.364627\pi$
$971$	25.7128	0.825163	0.412582	+	0.910921i	$0.364627\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−17.1769	−0.549538	−0.274769	−	0.961510i	$-0.588601\pi$
$977$	−17.1769	−0.549538	−0.274769	+	0.961510i	$0.588601\pi$
$978$	0	0
$979$	−14.5359	−0.464569
$980$	0	0
$981$	0	0
$982$	0	0
$983$	40.5167	1.29228	0.646140	−	0.763219i	$-0.276382\pi$
$983$	40.5167	1.29228	0.646140	+	0.763219i	$0.276382\pi$
$984$	0	0
$985$	−14.9282	−0.475652
$986$	0	0
$987$	0	0
$988$	0	0
$989$	16.9282	0.538286
$990$	0	0
$991$	37.0333	1.17640	0.588201	−	0.808715i	$-0.299836\pi$
$991$	37.0333	1.17640	0.588201	+	0.808715i	$0.299836\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	−7.07180	−0.224191
$996$	0	0
$997$	47.5692	1.50653	0.753266	−	0.657716i	$-0.228477\pi$
$997$	47.5692	1.50653	0.753266	+	0.657716i	$0.228477\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6480.2.a.bk.1.2		2
3.2	odd	2		6480.2.a.ba.1.2		2
4.3	odd	2		3240.2.a.p.1.1		2
9.2	odd	6		720.2.q.h.481.1		4
9.4	even	3		2160.2.q.h.721.1		4
9.5	odd	6		720.2.q.h.241.1		4
9.7	even	3		2160.2.q.h.1441.1		4
12.11	even	2		3240.2.a.k.1.1		2
36.7	odd	6		1080.2.q.b.361.2		4
36.11	even	6		360.2.q.b.121.2	✓	4
36.23	even	6		360.2.q.b.241.2	yes	4
36.31	odd	6		1080.2.q.b.721.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
360.2.q.b.121.2	✓	4	36.11	even	6
360.2.q.b.241.2	yes	4	36.23	even	6
720.2.q.h.241.1		4	9.5	odd	6
720.2.q.h.481.1		4	9.2	odd	6
1080.2.q.b.361.2		4	36.7	odd	6
1080.2.q.b.721.2		4	36.31	odd	6
2160.2.q.h.721.1		4	9.4	even	3
2160.2.q.h.1441.1		4	9.7	even	3
3240.2.a.k.1.1		2	12.11	even	2
3240.2.a.p.1.1		2	4.3	odd	2
6480.2.a.ba.1.2		2	3.2	odd	2
6480.2.a.bk.1.2		2	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 6480 = 2^{4} \cdot 3^{4} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6480.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{5} -0.267949 q^{7} -1.46410 q^{11} -5.46410 q^{13} -0.535898 q^{17} +2.00000 q^{19} +3.73205 q^{23} +1.00000 q^{25} -1.53590 q^{29} +2.00000 q^{31} -0.267949 q^{35} +10.3923 q^{37} -9.92820 q^{41} +4.53590 q^{43} +0.267949 q^{47} -6.92820 q^{49} -6.00000 q^{53} -1.46410 q^{55} +14.3923 q^{59} -8.46410 q^{61} -5.46410 q^{65} -6.26795 q^{67} +9.46410 q^{71} +6.92820 q^{73} +0.392305 q^{77} -15.4641 q^{79} -13.1962 q^{83} -0.535898 q^{85} +9.92820 q^{89} +1.46410 q^{91} +2.00000 q^{95} -8.92820 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{5} - 4 q^{7} + 4 q^{11} - 4 q^{13} - 8 q^{17} + 4 q^{19} + 4 q^{23} + 2 q^{25} - 10 q^{29} + 4 q^{31} - 4 q^{35} - 6 q^{41} + 16 q^{43} + 4 q^{47} - 12 q^{53} + 4 q^{55} + 8 q^{59} - 10 q^{61}+ \cdots - 4 q^{97}+O(q^{100}) \) 2 * q + 2 * q^5 - 4 * q^7 + 4 * q^11 - 4 * q^13 - 8 * q^17 + 4 * q^19 + 4 * q^23 + 2 * q^25 - 10 * q^29 + 4 * q^31 - 4 * q^35 - 6 * q^41 + 16 * q^43 + 4 * q^47 - 12 * q^53 + 4 * q^55 + 8 * q^59 - 10 * q^61 - 4 * q^65 - 16 * q^67 + 12 * q^71 - 20 * q^77 - 24 * q^79 - 16 * q^83 - 8 * q^85 + 6 * q^89 - 4 * q^91 + 4 * q^95 - 4 * q^97

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