LMFDB - Embedded newform 6480.2.a.bk.1.1

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			1.1
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)

$f(q)$	$=$	$q+1.00000 q^{5} -3.73205 q^{7} +5.46410 q^{11} +1.46410 q^{13} -7.46410 q^{17} +2.00000 q^{19} +0.267949 q^{23} +1.00000 q^{25} -8.46410 q^{29} +2.00000 q^{31} -3.73205 q^{35} -10.3923 q^{37} +3.92820 q^{41} +11.4641 q^{43} +3.73205 q^{47} +6.92820 q^{49} -6.00000 q^{53} +5.46410 q^{55} -6.39230 q^{59} -1.53590 q^{61} +1.46410 q^{65} -9.73205 q^{67} +2.53590 q^{71} -6.92820 q^{73} -20.3923 q^{77} -8.53590 q^{79} -2.80385 q^{83} -7.46410 q^{85} -3.92820 q^{89} -5.46410 q^{91} +2.00000 q^{95} +4.92820 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$7$	−3.73205	−1.41058	−0.705291	−	0.708918i	$-0.749184\pi$
$7$	−3.73205	−1.41058	−0.705291	+	0.708918i	$0.749184\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	5.46410	1.64749	0.823744	−	0.566961i	$-0.191881\pi$
$11$	5.46410	1.64749	0.823744	+	0.566961i	$0.191881\pi$
$12$	0	0
$13$	1.46410	0.406069	0.203034	−	0.979172i	$-0.434920\pi$
$13$	1.46410	0.406069	0.203034	+	0.979172i	$0.434920\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−7.46410	−1.81031	−0.905155	−	0.425081i	$-0.860246\pi$
$17$	−7.46410	−1.81031	−0.905155	+	0.425081i	$0.860246\pi$
$18$	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
$23$	0.267949	0.0558713	0.0279356	−	0.999610i	$-0.491107\pi$
$23$	0.267949	0.0558713	0.0279356	+	0.999610i	$0.491107\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	0	0
$28$	0	0
$29$	−8.46410	−1.57174	−0.785872	−	0.618389i	$-0.787786\pi$
$29$	−8.46410	−1.57174	−0.785872	+	0.618389i	$0.787786\pi$
$30$	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
$35$	−3.73205	−0.630832
$36$	0	0
$37$	−10.3923	−1.70848	−0.854242	−	0.519875i	$-0.825978\pi$
$37$	−10.3923	−1.70848	−0.854242	+	0.519875i	$0.825978\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	3.92820	0.613482	0.306741	−	0.951793i	$-0.400761\pi$
$41$	3.92820	0.613482	0.306741	+	0.951793i	$0.400761\pi$
$42$	0	0
$43$	11.4641	1.74826	0.874130	−	0.485693i	$-0.161433\pi$
$43$	11.4641	1.74826	0.874130	+	0.485693i	$0.161433\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	3.73205	0.544376	0.272188	−	0.962244i	$-0.412253\pi$
$47$	3.73205	0.544376	0.272188	+	0.962244i	$0.412253\pi$
$48$	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
$55$	5.46410	0.736779
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−6.39230	−0.832207	−0.416104	−	0.909317i	$-0.636605\pi$
$59$	−6.39230	−0.832207	−0.416104	+	0.909317i	$0.636605\pi$
$60$	0	0
$61$	−1.53590	−0.196652	−0.0983258	−	0.995154i	$-0.531349\pi$
$61$	−1.53590	−0.196652	−0.0983258	+	0.995154i	$0.531349\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	1.46410	0.181599
$66$	0	0
$67$	−9.73205	−1.18896	−0.594480	−	0.804111i	$-0.702642\pi$
$67$	−9.73205	−1.18896	−0.594480	+	0.804111i	$0.702642\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	2.53590	0.300956	0.150478	−	0.988613i	$-0.451919\pi$
$71$	2.53590	0.300956	0.150478	+	0.988613i	$0.451919\pi$
$72$	0	0
$73$	−6.92820	−0.810885	−0.405442	−	0.914121i	$-0.632883\pi$
$73$	−6.92820	−0.810885	−0.405442	+	0.914121i	$0.632883\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−20.3923	−2.32392
$78$	0	0
$79$	−8.53590	−0.960364	−0.480182	−	0.877169i	$-0.659429\pi$
$79$	−8.53590	−0.960364	−0.480182	+	0.877169i	$0.659429\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	−2.80385	−0.307762	−0.153881	−	0.988089i	$-0.549177\pi$
$83$	−2.80385	−0.307762	−0.153881	+	0.988089i	$0.549177\pi$
$84$	0	0
$85$	−7.46410	−0.809595
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−3.92820	−0.416389	−0.208194	−	0.978087i	$-0.566759\pi$
$89$	−3.92820	−0.416389	−0.208194	+	0.978087i	$0.566759\pi$
$90$	0	0
$91$	−5.46410	−0.572793
$92$	0	0
$93$	0	0
$94$	0	0
$95$	2.00000	0.205196
$96$	0	0
$97$	4.92820	0.500383	0.250192	−	0.968196i	$-0.419506\pi$
$97$	4.92820	0.500383	0.250192	+	0.968196i	$0.419506\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	12.9282	1.28640	0.643202	−	0.765696i	$-0.277605\pi$
$101$	12.9282	1.28640	0.643202	+	0.765696i	$0.277605\pi$
$102$	0	0
$103$	6.39230	0.629853	0.314926	−	0.949116i	$-0.398020\pi$
$103$	6.39230	0.629853	0.314926	+	0.949116i	$0.398020\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−5.19615	−0.502331	−0.251166	−	0.967944i	$-0.580814\pi$
$107$	−5.19615	−0.502331	−0.251166	+	0.967944i	$0.580814\pi$
$108$	0	0
$109$	3.39230	0.324924	0.162462	−	0.986715i	$-0.448057\pi$
$109$	3.39230	0.324924	0.162462	+	0.986715i	$0.448057\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	13.8564	1.30350	0.651751	−	0.758433i	$-0.274035\pi$
$113$	13.8564	1.30350	0.651751	+	0.758433i	$0.274035\pi$
$114$	0	0
$115$	0.267949	0.0249864
$116$	0	0
$117$	0	0
$118$	0	0
$119$	27.8564	2.55359
$120$	0	0
$121$	18.8564	1.71422
$122$	0	0
$123$	0	0
$124$	0	0
$125$	1.00000	0.0894427
$126$	0	0
$127$	−19.7321	−1.75094	−0.875468	−	0.483276i	$-0.839447\pi$
$127$	−19.7321	−1.75094	−0.875468	+	0.483276i	$0.839447\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	3.46410	0.302660	0.151330	−	0.988483i	$-0.451644\pi$
$131$	3.46410	0.302660	0.151330	+	0.988483i	$0.451644\pi$
$132$	0	0
$133$	−7.46410	−0.647220
$134$	0	0
$135$	0	0
$136$	0	0
$137$	5.85641	0.500347	0.250173	−	0.968201i	$-0.419512\pi$
$137$	5.85641	0.500347	0.250173	+	0.968201i	$0.419512\pi$
$138$	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
$145$	−8.46410	−0.702905
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−4.46410	−0.365713	−0.182857	−	0.983140i	$-0.558534\pi$
$149$	−4.46410	−0.365713	−0.182857	+	0.983140i	$0.558534\pi$
$150$	0	0
$151$	1.46410	0.119147	0.0595734	−	0.998224i	$-0.481026\pi$
$151$	1.46410	0.119147	0.0595734	+	0.998224i	$0.481026\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	2.00000	0.160644
$156$	0	0
$157$	−8.92820	−0.712548	−0.356274	−	0.934381i	$-0.615953\pi$
$157$	−8.92820	−0.712548	−0.356274	+	0.934381i	$0.615953\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−1.00000	−0.0788110
$162$	0	0
$163$	−21.3205	−1.66995	−0.834976	−	0.550287i	$-0.814518\pi$
$163$	−21.3205	−1.66995	−0.834976	+	0.550287i	$0.814518\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	9.58846	0.741977	0.370989	−	0.928637i	$-0.379019\pi$
$167$	9.58846	0.741977	0.370989	+	0.928637i	$0.379019\pi$
$168$	0	0
$169$	−10.8564	−0.835108
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−22.9282	−1.74320	−0.871600	−	0.490219i	$-0.836917\pi$
$173$	−22.9282	−1.74320	−0.871600	+	0.490219i	$0.836917\pi$
$174$	0	0
$175$	−3.73205	−0.282117
$176$	0	0
$177$	0	0
$178$	0	0
$179$	6.53590	0.488516	0.244258	−	0.969710i	$-0.421456\pi$
$179$	6.53590	0.488516	0.244258	+	0.969710i	$0.421456\pi$
$180$	0	0
$181$	−16.4641	−1.22377	−0.611884	−	0.790948i	$-0.709588\pi$
$181$	−16.4641	−1.22377	−0.611884	+	0.790948i	$0.709588\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−10.3923	−0.764057
$186$	0	0
$187$	−40.7846	−2.98247
$188$	0	0
$189$	0	0
$190$	0	0
$191$	3.07180	0.222267	0.111134	−	0.993805i	$-0.464552\pi$
$191$	3.07180	0.222267	0.111134	+	0.993805i	$0.464552\pi$
$192$	0	0
$193$	−10.5359	−0.758391	−0.379195	−	0.925317i	$-0.623799\pi$
$193$	−10.5359	−0.758391	−0.379195	+	0.925317i	$0.623799\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−1.07180	−0.0763624	−0.0381812	−	0.999271i	$-0.512156\pi$
$197$	−1.07180	−0.0763624	−0.0381812	+	0.999271i	$0.512156\pi$
$198$	0	0
$199$	−20.9282	−1.48356	−0.741780	−	0.670643i	$-0.766018\pi$
$199$	−20.9282	−1.48356	−0.741780	+	0.670643i	$0.766018\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	31.5885	2.21708
$204$	0	0
$205$	3.92820	0.274358
$206$	0	0
$207$	0	0
$208$	0	0
$209$	10.9282	0.755920
$210$	0	0
$211$	20.3923	1.40386	0.701932	−	0.712244i	$-0.252321\pi$
$211$	20.3923	1.40386	0.701932	+	0.712244i	$0.252321\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	11.4641	0.781845
$216$	0	0
$217$	−7.46410	−0.506696
$218$	0	0
$219$	0	0
$220$	0	0
$221$	−10.9282	−0.735111
$222$	0	0
$223$	−5.33975	−0.357576	−0.178788	−	0.983888i	$-0.557218\pi$
$223$	−5.33975	−0.357576	−0.178788	+	0.983888i	$0.557218\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−24.2487	−1.60944	−0.804722	−	0.593652i	$-0.797686\pi$
$227$	−24.2487	−1.60944	−0.804722	+	0.593652i	$0.797686\pi$
$228$	0	0
$229$	−23.2487	−1.53632	−0.768159	−	0.640259i	$-0.778827\pi$
$229$	−23.2487	−1.53632	−0.768159	+	0.640259i	$0.778827\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−10.3923	−0.680823	−0.340411	−	0.940277i	$-0.610566\pi$
$233$	−10.3923	−0.680823	−0.340411	+	0.940277i	$0.610566\pi$
$234$	0	0
$235$	3.73205	0.243452
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−20.7846	−1.34444	−0.672222	−	0.740349i	$-0.734660\pi$
$239$	−20.7846	−1.34444	−0.672222	+	0.740349i	$0.734660\pi$
$240$	0	0
$241$	−19.9282	−1.28369	−0.641844	−	0.766835i	$-0.721830\pi$
$241$	−19.9282	−1.28369	−0.641844	+	0.766835i	$0.721830\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	6.92820	0.442627
$246$	0	0
$247$	2.92820	0.186317
$248$	0	0
$249$	0	0
$250$	0	0
$251$	19.8564	1.25333	0.626663	−	0.779291i	$-0.284420\pi$
$251$	19.8564	1.25333	0.626663	+	0.779291i	$0.284420\pi$
$252$	0	0
$253$	1.46410	0.0920473
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−12.9282	−0.806439	−0.403220	−	0.915103i	$-0.632109\pi$
$257$	−12.9282	−0.806439	−0.403220	+	0.915103i	$0.632109\pi$
$258$	0	0
$259$	38.7846	2.40996
$260$	0	0
$261$	0	0
$262$	0	0
$263$	1.60770	0.0991347	0.0495674	−	0.998771i	$-0.484216\pi$
$263$	1.60770	0.0991347	0.0495674	+	0.998771i	$0.484216\pi$
$264$	0	0
$265$	−6.00000	−0.368577
$266$	0	0
$267$	0	0
$268$	0	0
$269$	9.39230	0.572659	0.286329	−	0.958131i	$-0.407565\pi$
$269$	9.39230	0.572659	0.286329	+	0.958131i	$0.407565\pi$
$270$	0	0
$271$	−7.85641	−0.477243	−0.238621	−	0.971113i	$-0.576695\pi$
$271$	−7.85641	−0.477243	−0.238621	+	0.971113i	$0.576695\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	5.46410	0.329498
$276$	0	0
$277$	6.39230	0.384076	0.192038	−	0.981387i	$-0.438490\pi$
$277$	6.39230	0.384076	0.192038	+	0.981387i	$0.438490\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−12.8564	−0.766949	−0.383474	−	0.923551i	$-0.625273\pi$
$281$	−12.8564	−0.766949	−0.383474	+	0.923551i	$0.625273\pi$
$282$	0	0
$283$	−18.2679	−1.08592	−0.542958	−	0.839760i	$-0.682696\pi$
$283$	−18.2679	−1.08592	−0.542958	+	0.839760i	$0.682696\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−14.6603	−0.865367
$288$	0	0
$289$	38.7128	2.27722
$290$	0	0
$291$	0	0
$292$	0	0
$293$	0.535898	0.0313075	0.0156538	−	0.999877i	$-0.495017\pi$
$293$	0.535898	0.0313075	0.0156538	+	0.999877i	$0.495017\pi$
$294$	0	0
$295$	−6.39230	−0.372174
$296$	0	0
$297$	0	0
$298$	0	0
$299$	0.392305	0.0226876
$300$	0	0
$301$	−42.7846	−2.46606
$302$	0	0
$303$	0	0
$304$	0	0
$305$	−1.53590	−0.0879453
$306$	0	0
$307$	20.1244	1.14856	0.574279	−	0.818660i	$-0.305283\pi$
$307$	20.1244	1.14856	0.574279	+	0.818660i	$0.305283\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	16.9282	0.959910	0.479955	−	0.877293i	$-0.340653\pi$
$311$	16.9282	0.959910	0.479955	+	0.877293i	$0.340653\pi$
$312$	0	0
$313$	−8.39230	−0.474361	−0.237181	−	0.971466i	$-0.576223\pi$
$313$	−8.39230	−0.474361	−0.237181	+	0.971466i	$0.576223\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−9.46410	−0.531557	−0.265778	−	0.964034i	$-0.585629\pi$
$317$	−9.46410	−0.531557	−0.265778	+	0.964034i	$0.585629\pi$
$318$	0	0
$319$	−46.2487	−2.58943
$320$	0	0
$321$	0	0
$322$	0	0
$323$	−14.9282	−0.830627
$324$	0	0
$325$	1.46410	0.0812137
$326$	0	0
$327$	0	0
$328$	0	0
$329$	−13.9282	−0.767887
$330$	0	0
$331$	−3.46410	−0.190404	−0.0952021	−	0.995458i	$-0.530350\pi$
$331$	−3.46410	−0.190404	−0.0952021	+	0.995458i	$0.530350\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	−9.73205	−0.531719
$336$	0	0
$337$	22.9282	1.24898	0.624489	−	0.781033i	$-0.285307\pi$
$337$	22.9282	1.24898	0.624489	+	0.781033i	$0.285307\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	10.9282	0.591795
$342$	0	0
$343$	0.267949	0.0144679
$344$	0	0
$345$	0	0
$346$	0	0
$347$	2.39230	0.128426	0.0642128	−	0.997936i	$-0.479546\pi$
$347$	2.39230	0.128426	0.0642128	+	0.997936i	$0.479546\pi$
$348$	0	0
$349$	−6.46410	−0.346015	−0.173008	−	0.984920i	$-0.555349\pi$
$349$	−6.46410	−0.346015	−0.173008	+	0.984920i	$0.555349\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−5.85641	−0.311705	−0.155853	−	0.987780i	$-0.549812\pi$
$353$	−5.85641	−0.311705	−0.155853	+	0.987780i	$0.549812\pi$
$354$	0	0
$355$	2.53590	0.134592
$356$	0	0
$357$	0	0
$358$	0	0
$359$	0.928203	0.0489887	0.0244943	−	0.999700i	$-0.492202\pi$
$359$	0.928203	0.0489887	0.0244943	+	0.999700i	$0.492202\pi$
$360$	0	0
$361$	−15.0000	−0.789474
$362$	0	0
$363$	0	0
$364$	0	0
$365$	−6.92820	−0.362639
$366$	0	0
$367$	−6.39230	−0.333676	−0.166838	−	0.985984i	$-0.553356\pi$
$367$	−6.39230	−0.333676	−0.166838	+	0.985984i	$0.553356\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	22.3923	1.16255
$372$	0	0
$373$	4.92820	0.255173	0.127586	−	0.991827i	$-0.459277\pi$
$373$	4.92820	0.255173	0.127586	+	0.991827i	$0.459277\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−12.3923	−0.638236
$378$	0	0
$379$	27.1769	1.39598	0.697992	−	0.716105i	$-0.254077\pi$
$379$	27.1769	1.39598	0.697992	+	0.716105i	$0.254077\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−3.46410	−0.177007	−0.0885037	−	0.996076i	$-0.528208\pi$
$383$	−3.46410	−0.177007	−0.0885037	+	0.996076i	$0.528208\pi$
$384$	0	0
$385$	−20.3923	−1.03929
$386$	0	0
$387$	0	0
$388$	0	0
$389$	30.4641	1.54459	0.772296	−	0.635263i	$-0.219108\pi$
$389$	30.4641	1.54459	0.772296	+	0.635263i	$0.219108\pi$
$390$	0	0
$391$	−2.00000	−0.101144
$392$	0	0
$393$	0	0
$394$	0	0
$395$	−8.53590	−0.429488
$396$	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
$401$	26.7846	1.33756	0.668780	−	0.743461i	$-0.266817\pi$
$401$	26.7846	1.33756	0.668780	+	0.743461i	$0.266817\pi$
$402$	0	0
$403$	2.92820	0.145864
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−56.7846	−2.81471
$408$	0	0
$409$	−8.92820	−0.441471	−0.220736	−	0.975334i	$-0.570846\pi$
$409$	−8.92820	−0.441471	−0.220736	+	0.975334i	$0.570846\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	23.8564	1.17390
$414$	0	0
$415$	−2.80385	−0.137635
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−18.2487	−0.891508	−0.445754	−	0.895156i	$-0.647064\pi$
$419$	−18.2487	−0.891508	−0.445754	+	0.895156i	$0.647064\pi$
$420$	0	0
$421$	−23.0718	−1.12445	−0.562225	−	0.826984i	$-0.690055\pi$
$421$	−23.0718	−1.12445	−0.562225	+	0.826984i	$0.690055\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	−7.46410	−0.362062
$426$	0	0
$427$	5.73205	0.277393
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−11.4641	−0.552206	−0.276103	−	0.961128i	$-0.589043\pi$
$431$	−11.4641	−0.552206	−0.276103	+	0.961128i	$0.589043\pi$
$432$	0	0
$433$	−19.4641	−0.935385	−0.467693	−	0.883891i	$-0.654915\pi$
$433$	−19.4641	−0.935385	−0.467693	+	0.883891i	$0.654915\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	0.535898	0.0256355
$438$	0	0
$439$	−34.9282	−1.66703	−0.833516	−	0.552495i	$-0.813676\pi$
$439$	−34.9282	−1.66703	−0.833516	+	0.552495i	$0.813676\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−10.5167	−0.499662	−0.249831	−	0.968290i	$-0.580375\pi$
$443$	−10.5167	−0.499662	−0.249831	+	0.968290i	$0.580375\pi$
$444$	0	0
$445$	−3.92820	−0.186215
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−31.8564	−1.50340	−0.751698	−	0.659507i	$-0.770765\pi$
$449$	−31.8564	−1.50340	−0.751698	+	0.659507i	$0.770765\pi$
$450$	0	0
$451$	21.4641	1.01071
$452$	0	0
$453$	0	0
$454$	0	0
$455$	−5.46410	−0.256161
$456$	0	0
$457$	26.3923	1.23458	0.617290	−	0.786736i	$-0.288231\pi$
$457$	26.3923	1.23458	0.617290	+	0.786736i	$0.288231\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−25.3923	−1.18264	−0.591319	−	0.806438i	$-0.701392\pi$
$461$	−25.3923	−1.18264	−0.591319	+	0.806438i	$0.701392\pi$
$462$	0	0
$463$	−3.46410	−0.160990	−0.0804952	−	0.996755i	$-0.525650\pi$
$463$	−3.46410	−0.160990	−0.0804952	+	0.996755i	$0.525650\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−9.60770	−0.444591	−0.222296	−	0.974979i	$-0.571355\pi$
$467$	−9.60770	−0.444591	−0.222296	+	0.974979i	$0.571355\pi$
$468$	0	0
$469$	36.3205	1.67713
$470$	0	0
$471$	0	0
$472$	0	0
$473$	62.6410	2.88024
$474$	0	0
$475$	2.00000	0.0917663
$476$	0	0
$477$	0	0
$478$	0	0
$479$	6.39230	0.292072	0.146036	−	0.989279i	$-0.453348\pi$
$479$	6.39230	0.292072	0.146036	+	0.989279i	$0.453348\pi$
$480$	0	0
$481$	−15.2154	−0.693762
$482$	0	0
$483$	0	0
$484$	0	0
$485$	4.92820	0.223778
$486$	0	0
$487$	8.24871	0.373785	0.186892	−	0.982380i	$-0.440158\pi$
$487$	8.24871	0.373785	0.186892	+	0.982380i	$0.440158\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	1.85641	0.0837785	0.0418892	−	0.999122i	$-0.486662\pi$
$491$	1.85641	0.0837785	0.0418892	+	0.999122i	$0.486662\pi$
$492$	0	0
$493$	63.1769	2.84535
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−9.46410	−0.424523
$498$	0	0
$499$	−9.85641	−0.441233	−0.220617	−	0.975361i	$-0.570807\pi$
$499$	−9.85641	−0.441233	−0.220617	+	0.975361i	$0.570807\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	6.12436	0.273072	0.136536	−	0.990635i	$-0.456403\pi$
$503$	6.12436	0.273072	0.136536	+	0.990635i	$0.456403\pi$
$504$	0	0
$505$	12.9282	0.575297
$506$	0	0
$507$	0	0
$508$	0	0
$509$	41.3923	1.83468	0.917341	−	0.398103i	$-0.130331\pi$
$509$	41.3923	1.83468	0.917341	+	0.398103i	$0.130331\pi$
$510$	0	0
$511$	25.8564	1.14382
$512$	0	0
$513$	0	0
$514$	0	0
$515$	6.39230	0.281679
$516$	0	0
$517$	20.3923	0.896853
$518$	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
$523$	−21.1962	−0.926843	−0.463422	−	0.886138i	$-0.653378\pi$
$523$	−21.1962	−0.926843	−0.463422	+	0.886138i	$0.653378\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−14.9282	−0.650283
$528$	0	0
$529$	−22.9282	−0.996878
$530$	0	0
$531$	0	0
$532$	0	0
$533$	5.75129	0.249116
$534$	0	0
$535$	−5.19615	−0.224649
$536$	0	0
$537$	0	0
$538$	0	0
$539$	37.8564	1.63059
$540$	0	0
$541$	−39.3923	−1.69361	−0.846804	−	0.531905i	$-0.821476\pi$
$541$	−39.3923	−1.69361	−0.846804	+	0.531905i	$0.821476\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	3.39230	0.145310
$546$	0	0
$547$	5.98076	0.255719	0.127859	−	0.991792i	$-0.459189\pi$
$547$	5.98076	0.255719	0.127859	+	0.991792i	$0.459189\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−16.9282	−0.721166
$552$	0	0
$553$	31.8564	1.35467
$554$	0	0
$555$	0	0
$556$	0	0
$557$	36.6410	1.55253	0.776265	−	0.630407i	$-0.217112\pi$
$557$	36.6410	1.55253	0.776265	+	0.630407i	$0.217112\pi$
$558$	0	0
$559$	16.7846	0.709913
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−43.5885	−1.83703	−0.918517	−	0.395381i	$-0.870613\pi$
$563$	−43.5885	−1.83703	−0.918517	+	0.395381i	$0.870613\pi$
$564$	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
$571$	−0.784610	−0.0328349	−0.0164174	−	0.999865i	$-0.505226\pi$
$571$	−0.784610	−0.0328349	−0.0164174	+	0.999865i	$0.505226\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	0.267949	0.0111743
$576$	0	0
$577$	19.0718	0.793969	0.396985	−	0.917825i	$-0.370057\pi$
$577$	19.0718	0.793969	0.396985	+	0.917825i	$0.370057\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	10.4641	0.434124
$582$	0	0
$583$	−32.7846	−1.35780
$584$	0	0
$585$	0	0
$586$	0	0
$587$	39.5885	1.63399	0.816995	−	0.576644i	$-0.195638\pi$
$587$	39.5885	1.63399	0.816995	+	0.576644i	$0.195638\pi$
$588$	0	0
$589$	4.00000	0.164817
$590$	0	0
$591$	0	0
$592$	0	0
$593$	32.5359	1.33609	0.668045	−	0.744121i	$-0.267132\pi$
$593$	32.5359	1.33609	0.668045	+	0.744121i	$0.267132\pi$
$594$	0	0
$595$	27.8564	1.14200
$596$	0	0
$597$	0	0
$598$	0	0
$599$	22.3923	0.914925	0.457462	−	0.889229i	$-0.348758\pi$
$599$	22.3923	0.914925	0.457462	+	0.889229i	$0.348758\pi$
$600$	0	0
$601$	−10.7846	−0.439913	−0.219957	−	0.975510i	$-0.570592\pi$
$601$	−10.7846	−0.439913	−0.219957	+	0.975510i	$0.570592\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	18.8564	0.766622
$606$	0	0
$607$	−43.7321	−1.77503	−0.887515	−	0.460780i	$-0.847570\pi$
$607$	−43.7321	−1.77503	−0.887515	+	0.460780i	$0.847570\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	5.46410	0.221054
$612$	0	0
$613$	1.60770	0.0649342	0.0324671	−	0.999473i	$-0.489664\pi$
$613$	1.60770	0.0649342	0.0324671	+	0.999473i	$0.489664\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−17.0718	−0.687285	−0.343642	−	0.939101i	$-0.611661\pi$
$617$	−17.0718	−0.687285	−0.343642	+	0.939101i	$0.611661\pi$
$618$	0	0
$619$	−17.7128	−0.711938	−0.355969	−	0.934498i	$-0.615849\pi$
$619$	−17.7128	−0.711938	−0.355969	+	0.934498i	$0.615849\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	14.6603	0.587351
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	77.5692	3.09289
$630$	0	0
$631$	23.7128	0.943992	0.471996	−	0.881601i	$-0.343534\pi$
$631$	23.7128	0.943992	0.471996	+	0.881601i	$0.343534\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	−19.7321	−0.783043
$636$	0	0
$637$	10.1436	0.401904
$638$	0	0
$639$	0	0
$640$	0	0
$641$	7.14359	0.282155	0.141077	−	0.989999i	$-0.454943\pi$
$641$	7.14359	0.282155	0.141077	+	0.989999i	$0.454943\pi$
$642$	0	0
$643$	37.7321	1.48801	0.744003	−	0.668176i	$-0.232925\pi$
$643$	37.7321	1.48801	0.744003	+	0.668176i	$0.232925\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	28.5167	1.12111	0.560553	−	0.828119i	$-0.310589\pi$
$647$	28.5167	1.12111	0.560553	+	0.828119i	$0.310589\pi$
$648$	0	0
$649$	−34.9282	−1.37105
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−24.5359	−0.960164	−0.480082	−	0.877224i	$-0.659393\pi$
$653$	−24.5359	−0.960164	−0.480082	+	0.877224i	$0.659393\pi$
$654$	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	$-0.367332\pi$
$659$	20.7846	0.809653	0.404827	+	0.914393i	$0.367332\pi$
$660$	0	0
$661$	−33.7128	−1.31128	−0.655638	−	0.755075i	$-0.727600\pi$
$661$	−33.7128	−1.31128	−0.655638	+	0.755075i	$0.727600\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−7.46410	−0.289445
$666$	0	0
$667$	−2.26795	−0.0878153
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−8.39230	−0.323981
$672$	0	0
$673$	−47.3205	−1.82407	−0.912036	−	0.410111i	$-0.865490\pi$
$673$	−47.3205	−1.82407	−0.912036	+	0.410111i	$0.865490\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−45.1769	−1.73629	−0.868145	−	0.496311i	$-0.834688\pi$
$677$	−45.1769	−1.73629	−0.868145	+	0.496311i	$0.834688\pi$
$678$	0	0
$679$	−18.3923	−0.705832
$680$	0	0
$681$	0	0
$682$	0	0
$683$	1.60770	0.0615167	0.0307584	−	0.999527i	$-0.490208\pi$
$683$	1.60770	0.0615167	0.0307584	+	0.999527i	$0.490208\pi$
$684$	0	0
$685$	5.85641	0.223762
$686$	0	0
$687$	0	0
$688$	0	0
$689$	−8.78461	−0.334667
$690$	0	0
$691$	29.6077	1.12633	0.563165	−	0.826345i	$-0.309584\pi$
$691$	29.6077	1.12633	0.563165	+	0.826345i	$0.309584\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	0	0
$696$	0	0
$697$	−29.3205	−1.11059
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−38.1769	−1.44192	−0.720961	−	0.692976i	$-0.756299\pi$
$701$	−38.1769	−1.44192	−0.720961	+	0.692976i	$0.756299\pi$
$702$	0	0
$703$	−20.7846	−0.783906
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−48.2487	−1.81458
$708$	0	0
$709$	−5.39230	−0.202512	−0.101256	−	0.994860i	$-0.532286\pi$
$709$	−5.39230	−0.202512	−0.101256	+	0.994860i	$0.532286\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	0.535898	0.0200696
$714$	0	0
$715$	8.00000	0.299183
$716$	0	0
$717$	0	0
$718$	0	0
$719$	1.32051	0.0492466	0.0246233	−	0.999697i	$-0.492161\pi$
$719$	1.32051	0.0492466	0.0246233	+	0.999697i	$0.492161\pi$
$720$	0	0
$721$	−23.8564	−0.888459
$722$	0	0
$723$	0	0
$724$	0	0
$725$	−8.46410	−0.314349
$726$	0	0
$727$	35.9808	1.33445	0.667226	−	0.744855i	$-0.267481\pi$
$727$	35.9808	1.33445	0.667226	+	0.744855i	$0.267481\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−85.5692	−3.16489
$732$	0	0
$733$	43.8564	1.61987	0.809937	−	0.586517i	$-0.199501\pi$
$733$	43.8564	1.61987	0.809937	+	0.586517i	$0.199501\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−53.1769	−1.95880
$738$	0	0
$739$	−48.2487	−1.77486	−0.887429	−	0.460945i	$-0.847511\pi$
$739$	−48.2487	−1.77486	−0.887429	+	0.460945i	$0.847511\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	45.8372	1.68160	0.840801	−	0.541344i	$-0.182084\pi$
$743$	45.8372	1.68160	0.840801	+	0.541344i	$0.182084\pi$
$744$	0	0
$745$	−4.46410	−0.163552
$746$	0	0
$747$	0	0
$748$	0	0
$749$	19.3923	0.708579
$750$	0	0
$751$	4.39230	0.160277	0.0801387	−	0.996784i	$-0.474464\pi$
$751$	4.39230	0.160277	0.0801387	+	0.996784i	$0.474464\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	1.46410	0.0532841
$756$	0	0
$757$	0.392305	0.0142586	0.00712928	−	0.999975i	$-0.497731\pi$
$757$	0.392305	0.0142586	0.00712928	+	0.999975i	$0.497731\pi$
$758$	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
$763$	−12.6603	−0.458332
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−9.35898	−0.337933
$768$	0	0
$769$	−7.78461	−0.280720	−0.140360	−	0.990101i	$-0.544826\pi$
$769$	−7.78461	−0.280720	−0.140360	+	0.990101i	$0.544826\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	−32.7846	−1.17918	−0.589590	−	0.807703i	$-0.700711\pi$
$773$	−32.7846	−1.17918	−0.589590	+	0.807703i	$0.700711\pi$
$774$	0	0
$775$	2.00000	0.0718421
$776$	0	0
$777$	0	0
$778$	0	0
$779$	7.85641	0.281485
$780$	0	0
$781$	13.8564	0.495821
$782$	0	0
$783$	0	0
$784$	0	0
$785$	−8.92820	−0.318661
$786$	0	0
$787$	33.3205	1.18775	0.593874	−	0.804558i	$-0.297598\pi$
$787$	33.3205	1.18775	0.593874	+	0.804558i	$0.297598\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−51.7128	−1.83870
$792$	0	0
$793$	−2.24871	−0.0798541
$794$	0	0
$795$	0	0
$796$	0	0
$797$	28.3923	1.00571	0.502854	−	0.864372i	$-0.332284\pi$
$797$	28.3923	1.00571	0.502854	+	0.864372i	$0.332284\pi$
$798$	0	0
$799$	−27.8564	−0.985489
$800$	0	0
$801$	0	0
$802$	0	0
$803$	−37.8564	−1.33592
$804$	0	0
$805$	−1.00000	−0.0352454
$806$	0	0
$807$	0	0
$808$	0	0
$809$	35.0718	1.23306	0.616529	−	0.787332i	$-0.288538\pi$
$809$	35.0718	1.23306	0.616529	+	0.787332i	$0.288538\pi$
$810$	0	0
$811$	29.6077	1.03967	0.519833	−	0.854268i	$-0.325994\pi$
$811$	29.6077	1.03967	0.519833	+	0.854268i	$0.325994\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−21.3205	−0.746825
$816$	0	0
$817$	22.9282	0.802156
$818$	0	0
$819$	0	0
$820$	0	0
$821$	25.2487	0.881186	0.440593	−	0.897707i	$-0.354768\pi$
$821$	25.2487	0.881186	0.440593	+	0.897707i	$0.354768\pi$
$822$	0	0
$823$	−20.2679	−0.706496	−0.353248	−	0.935530i	$-0.614923\pi$
$823$	−20.2679	−0.706496	−0.353248	+	0.935530i	$0.614923\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−55.0526	−1.91437	−0.957183	−	0.289485i	$-0.906516\pi$
$827$	−55.0526	−1.91437	−0.957183	+	0.289485i	$0.906516\pi$
$828$	0	0
$829$	−9.39230	−0.326208	−0.163104	−	0.986609i	$-0.552151\pi$
$829$	−9.39230	−0.326208	−0.163104	+	0.986609i	$0.552151\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	−51.7128	−1.79174
$834$	0	0
$835$	9.58846	0.331822
$836$	0	0
$837$	0	0
$838$	0	0
$839$	8.14359	0.281148	0.140574	−	0.990070i	$-0.455105\pi$
$839$	8.14359	0.281148	0.140574	+	0.990070i	$0.455105\pi$
$840$	0	0
$841$	42.6410	1.47038
$842$	0	0
$843$	0	0
$844$	0	0
$845$	−10.8564	−0.373472
$846$	0	0
$847$	−70.3731	−2.41805
$848$	0	0
$849$	0	0
$850$	0	0
$851$	−2.78461	−0.0954552
$852$	0	0
$853$	−42.5359	−1.45640	−0.728201	−	0.685364i	$-0.759643\pi$
$853$	−42.5359	−1.45640	−0.728201	+	0.685364i	$0.759643\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	30.2487	1.03328	0.516638	−	0.856204i	$-0.327183\pi$
$857$	30.2487	1.03328	0.516638	+	0.856204i	$0.327183\pi$
$858$	0	0
$859$	−0.392305	−0.0133853	−0.00669263	−	0.999978i	$-0.502130\pi$
$859$	−0.392305	−0.0133853	−0.00669263	+	0.999978i	$0.502130\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	9.33975	0.317929	0.158964	−	0.987284i	$-0.449185\pi$
$863$	9.33975	0.317929	0.158964	+	0.987284i	$0.449185\pi$
$864$	0	0
$865$	−22.9282	−0.779582
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−46.6410	−1.58219
$870$	0	0
$871$	−14.2487	−0.482799
$872$	0	0
$873$	0	0
$874$	0	0
$875$	−3.73205	−0.126166
$876$	0	0
$877$	32.3923	1.09381	0.546905	−	0.837195i	$-0.315806\pi$
$877$	32.3923	1.09381	0.546905	+	0.837195i	$0.315806\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	31.6410	1.06601	0.533006	−	0.846111i	$-0.321062\pi$
$881$	31.6410	1.06601	0.533006	+	0.846111i	$0.321062\pi$
$882$	0	0
$883$	−1.19615	−0.0402537	−0.0201269	−	0.999797i	$-0.506407\pi$
$883$	−1.19615	−0.0402537	−0.0201269	+	0.999797i	$0.506407\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	25.3205	0.850179	0.425090	−	0.905151i	$-0.360243\pi$
$887$	25.3205	0.850179	0.425090	+	0.905151i	$0.360243\pi$
$888$	0	0
$889$	73.6410	2.46984
$890$	0	0
$891$	0	0
$892$	0	0
$893$	7.46410	0.249777
$894$	0	0
$895$	6.53590	0.218471
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−16.9282	−0.564587
$900$	0	0
$901$	44.7846	1.49199
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−16.4641	−0.547285
$906$	0	0
$907$	10.2679	0.340942	0.170471	−	0.985363i	$-0.445471\pi$
$907$	10.2679	0.340942	0.170471	+	0.985363i	$0.445471\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−19.4641	−0.644874	−0.322437	−	0.946591i	$-0.604502\pi$
$911$	−19.4641	−0.644874	−0.322437	+	0.946591i	$0.604502\pi$
$912$	0	0
$913$	−15.3205	−0.507035
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−12.9282	−0.426927
$918$	0	0
$919$	38.3923	1.26645	0.633223	−	0.773970i	$-0.281732\pi$
$919$	38.3923	1.26645	0.633223	+	0.773970i	$0.281732\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	3.71281	0.122209
$924$	0	0
$925$	−10.3923	−0.341697
$926$	0	0
$927$	0	0
$928$	0	0
$929$	44.6410	1.46462	0.732312	−	0.680969i	$-0.238441\pi$
$929$	44.6410	1.46462	0.732312	+	0.680969i	$0.238441\pi$
$930$	0	0
$931$	13.8564	0.454125
$932$	0	0
$933$	0	0
$934$	0	0
$935$	−40.7846	−1.33380
$936$	0	0
$937$	43.7128	1.42804	0.714018	−	0.700128i	$-0.246874\pi$
$937$	43.7128	1.42804	0.714018	+	0.700128i	$0.246874\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	2.60770	0.0850084	0.0425042	−	0.999096i	$-0.486466\pi$
$941$	2.60770	0.0850084	0.0425042	+	0.999096i	$0.486466\pi$
$942$	0	0
$943$	1.05256	0.0342760
$944$	0	0
$945$	0	0
$946$	0	0
$947$	30.5167	0.991658	0.495829	−	0.868420i	$-0.334864\pi$
$947$	30.5167	0.991658	0.495829	+	0.868420i	$0.334864\pi$
$948$	0	0
$949$	−10.1436	−0.329275
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−19.6077	−0.635156	−0.317578	−	0.948232i	$-0.602869\pi$
$953$	−19.6077	−0.635156	−0.317578	+	0.948232i	$0.602869\pi$
$954$	0	0
$955$	3.07180	0.0994010
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−21.8564	−0.705780
$960$	0	0
$961$	−27.0000	−0.870968
$962$	0	0
$963$	0	0
$964$	0	0
$965$	−10.5359	−0.339163
$966$	0	0
$967$	−38.6603	−1.24323	−0.621615	−	0.783323i	$-0.713523\pi$
$967$	−38.6603	−1.24323	−0.621615	+	0.783323i	$0.713523\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−29.7128	−0.953530	−0.476765	−	0.879031i	$-0.658191\pi$
$971$	−29.7128	−0.953530	−0.476765	+	0.879031i	$0.658191\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	0	0
$976$	0	0
$977$	45.1769	1.44534	0.722669	−	0.691195i	$-0.242915\pi$
$977$	45.1769	1.44534	0.722669	+	0.691195i	$0.242915\pi$
$978$	0	0
$979$	−21.4641	−0.685996
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−4.51666	−0.144059	−0.0720295	−	0.997402i	$-0.522948\pi$
$983$	−4.51666	−0.144059	−0.0720295	+	0.997402i	$0.522948\pi$
$984$	0	0
$985$	−1.07180	−0.0341503
$986$	0	0
$987$	0	0
$988$	0	0
$989$	3.07180	0.0976775
$990$	0	0
$991$	−53.0333	−1.68466	−0.842329	−	0.538963i	$-0.818816\pi$
$991$	−53.0333	−1.68466	−0.842329	+	0.538963i	$0.818816\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	−20.9282	−0.663469
$996$	0	0
$997$	−35.5692	−1.12649	−0.563244	−	0.826290i	$-0.690447\pi$
$997$	−35.5692	−1.12649	−0.563244	+	0.826290i	$0.690447\pi$
$998$	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.1.1		2
3.2	odd	2		6480.2.a.ba.1.1		2
4.3	odd	2		3240.2.a.p.1.2		2
9.2	odd	6		720.2.q.h.481.2		4
9.4	even	3		2160.2.q.h.721.2		4
9.5	odd	6		720.2.q.h.241.2		4
9.7	even	3		2160.2.q.h.1441.2		4
12.11	even	2		3240.2.a.k.1.2		2
36.7	odd	6		1080.2.q.b.361.1		4
36.11	even	6		360.2.q.b.121.1	✓	4
36.23	even	6		360.2.q.b.241.1	yes	4
36.31	odd	6		1080.2.q.b.721.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
360.2.q.b.121.1	✓	4	36.11	even	6
360.2.q.b.241.1	yes	4	36.23	even	6
720.2.q.h.241.2		4	9.5	odd	6
720.2.q.h.481.2		4	9.2	odd	6
1080.2.q.b.361.1		4	36.7	odd	6
1080.2.q.b.721.1		4	36.31	odd	6
2160.2.q.h.721.2		4	9.4	even	3
2160.2.q.h.1441.2		4	9.7	even	3
3240.2.a.k.1.2		2	12.11	even	2
3240.2.a.p.1.2		2	4.3	odd	2
6480.2.a.ba.1.1		2	3.2	odd	2
6480.2.a.bk.1.1		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)

\(f(q)\)	\(=\)	\(q+1.00000 q^{5} -3.73205 q^{7} +5.46410 q^{11} +1.46410 q^{13} -7.46410 q^{17} +2.00000 q^{19} +0.267949 q^{23} +1.00000 q^{25} -8.46410 q^{29} +2.00000 q^{31} -3.73205 q^{35} -10.3923 q^{37} +3.92820 q^{41} +11.4641 q^{43} +3.73205 q^{47} +6.92820 q^{49} -6.00000 q^{53} +5.46410 q^{55} -6.39230 q^{59} -1.53590 q^{61} +1.46410 q^{65} -9.73205 q^{67} +2.53590 q^{71} -6.92820 q^{73} -20.3923 q^{77} -8.53590 q^{79} -2.80385 q^{83} -7.46410 q^{85} -3.92820 q^{89} -5.46410 q^{91} +2.00000 q^{95} +4.92820 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{5} - 4 q^{7} + 4 q^{11} - 4 q^{13} - 8 q^{17} + 4 q^{19} + 4 q^{23} + 2 q^{25} - 10 q^{29} + 4 q^{31} - 4 q^{35} - 6 q^{41} + 16 q^{43} + 4 q^{47} - 12 q^{53} + 4 q^{55} + 8 q^{59} - 10 q^{61}+ \cdots - 4 q^{97}+O(q^{100}) \) 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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