LMFDB - Embedded newform 6561.2.a.d.1.20

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6561,2,Mod(1,6561)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6561, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("6561.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6561 = 3^{8} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6561.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:	$52.3898487662$
Analytic rank:	$0$
Dimension:	$72$
Twist minimal:	no (minimal twist has level 81)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.20
Character	$\chi$	$=$	6561.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.36534 q^{2} -0.135854 q^{4} +3.81012 q^{5} +1.74698 q^{7} +2.91616 q^{8} +O(q^{10})$	\(q-1.36534 q^{2} -0.135854 q^{4} +3.81012 q^{5} +1.74698 q^{7} +2.91616 q^{8} -5.20210 q^{10} +3.06068 q^{11} +1.86816 q^{13} -2.38522 q^{14} -3.70984 q^{16} -7.00231 q^{17} +4.94221 q^{19} -0.517619 q^{20} -4.17886 q^{22} +1.90491 q^{23} +9.51701 q^{25} -2.55067 q^{26} -0.237334 q^{28} -0.226517 q^{29} -3.93934 q^{31} -0.767143 q^{32} +9.56051 q^{34} +6.65620 q^{35} -0.368499 q^{37} -6.74779 q^{38} +11.1109 q^{40} +4.33031 q^{41} -3.24717 q^{43} -0.415805 q^{44} -2.60085 q^{46} +1.74782 q^{47} -3.94806 q^{49} -12.9939 q^{50} -0.253796 q^{52} -6.06284 q^{53} +11.6616 q^{55} +5.09447 q^{56} +0.309272 q^{58} -1.99797 q^{59} +12.7894 q^{61} +5.37853 q^{62} +8.46708 q^{64} +7.11791 q^{65} -6.34476 q^{67} +0.951289 q^{68} -9.08796 q^{70} +13.9570 q^{71} +3.16797 q^{73} +0.503125 q^{74} -0.671418 q^{76} +5.34695 q^{77} -2.62141 q^{79} -14.1349 q^{80} -5.91234 q^{82} -0.224829 q^{83} -26.6796 q^{85} +4.43348 q^{86} +8.92544 q^{88} +7.11806 q^{89} +3.26363 q^{91} -0.258790 q^{92} -2.38636 q^{94} +18.8304 q^{95} +15.5794 q^{97} +5.39044 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 72 q + 9 q^{2} + 63 q^{4} + 18 q^{5} + 27 q^{8}+O(q^{10}) $ 72 * q + 9 * q^2 + 63 * q^4 + 18 * q^5 + 27 * q^8	$ 72 q + 9 q^{2} + 63 q^{4} + 18 q^{5} + 27 q^{8} + 36 q^{11} + 36 q^{14} + 45 q^{16} + 36 q^{17} + 54 q^{20} + 54 q^{23} + 36 q^{25} + 45 q^{26} + 9 q^{28} + 54 q^{29} + 63 q^{32} + 72 q^{35} + 54 q^{38} + 72 q^{41} + 90 q^{44} + 90 q^{47} + 18 q^{49} + 45 q^{50} + 45 q^{53} + 9 q^{55} + 108 q^{56} + 18 q^{58} + 108 q^{59} + 72 q^{62} + 9 q^{64} + 72 q^{65} + 108 q^{68} + 126 q^{71} + 90 q^{74} + 72 q^{77} + 144 q^{80} - 18 q^{82} + 108 q^{83} + 90 q^{86} + 108 q^{89} + 72 q^{92} + 144 q^{95} + 81 q^{98}+O(q^{100}) $ 72 * q + 9 * q^2 + 63 * q^4 + 18 * q^5 + 27 * q^8 + 36 * q^11 + 36 * q^14 + 45 * q^16 + 36 * q^17 + 54 * q^20 + 54 * q^23 + 36 * q^25 + 45 * q^26 + 9 * q^28 + 54 * q^29 + 63 * q^32 + 72 * q^35 + 54 * q^38 + 72 * q^41 + 90 * q^44 + 90 * q^47 + 18 * q^49 + 45 * q^50 + 45 * q^53 + 9 * q^55 + 108 * q^56 + 18 * q^58 + 108 * q^59 + 72 * q^62 + 9 * q^64 + 72 * q^65 + 108 * q^68 + 126 * q^71 + 90 * q^74 + 72 * q^77 + 144 * q^80 - 18 * q^82 + 108 * q^83 + 90 * q^86 + 108 * q^89 + 72 * q^92 + 144 * q^95 + 81 * q^98

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 $

$2$	−1.36534	−0.965439	−0.482720	−	0.875775i	$-0.660351\pi$
$2$	−1.36534	−0.965439	−0.482720	+	0.875775i	$0.660351\pi$
$3$	0	0
$3$	0	0
$4$	−0.135854	−0.0679268
$5$	3.81012	1.70394	0.851969	−	0.523593i	$-0.175409\pi$
$5$	3.81012	1.70394	0.851969	+	0.523593i	$0.175409\pi$
$6$	0	0
$7$	1.74698	0.660296	0.330148	−	0.943929i	$-0.392901\pi$
$7$	1.74698	0.660296	0.330148	+	0.943929i	$0.392901\pi$
$8$	2.91616	1.03102
$9$	0	0
$10$	−5.20210	−1.64505
$11$	3.06068	0.922831	0.461415	−	0.887184i	$-0.347342\pi$
$11$	3.06068	0.922831	0.461415	+	0.887184i	$0.347342\pi$
$12$	0	0
$13$	1.86816	0.518134	0.259067	−	0.965859i	$-0.416585\pi$
$13$	1.86816	0.518134	0.259067	+	0.965859i	$0.416585\pi$
$14$	−2.38522	−0.637476
$15$	0	0
$16$	−3.70984	−0.927459
$17$	−7.00231	−1.69831	−0.849154	−	0.528145i	$-0.822888\pi$
$17$	−7.00231	−1.69831	−0.849154	+	0.528145i	$0.822888\pi$
$18$	0	0
$19$	4.94221	1.13382	0.566911	−	0.823779i	$-0.308138\pi$
$19$	4.94221	1.13382	0.566911	+	0.823779i	$0.308138\pi$
$20$	−0.517619	−0.115743
$21$	0	0
$22$	−4.17886	−0.890937
$23$	1.90491	0.397202	0.198601	−	0.980080i	$-0.436360\pi$
$23$	1.90491	0.397202	0.198601	+	0.980080i	$0.436360\pi$
$24$	0	0
$25$	9.51701	1.90340
$26$	−2.55067	−0.500227
$27$	0	0
$28$	−0.237334	−0.0448518
$29$	−0.226517	−0.0420631	−0.0210315	−	0.999779i	$-0.506695\pi$
$29$	−0.226517	−0.0420631	−0.0210315	+	0.999779i	$0.506695\pi$
$30$	0	0
$31$	−3.93934	−0.707526	−0.353763	−	0.935335i	$-0.615098\pi$
$31$	−3.93934	−0.707526	−0.353763	+	0.935335i	$0.615098\pi$
$32$	−0.767143	−0.135613
$33$	0	0
$34$	9.56051	1.63961
$35$	6.65620	1.12510
$36$	0	0
$37$	−0.368499	−0.0605808	−0.0302904	−	0.999541i	$-0.509643\pi$
$37$	−0.368499	−0.0605808	−0.0302904	+	0.999541i	$0.509643\pi$
$38$	−6.74779	−1.09464
$39$	0	0
$40$	11.1109	1.75679
$41$	4.33031	0.676281	0.338140	−	0.941096i	$-0.390202\pi$
$41$	4.33031	0.676281	0.338140	+	0.941096i	$0.390202\pi$
$42$	0	0
$43$	−3.24717	−0.495188	−0.247594	−	0.968864i	$-0.579640\pi$
$43$	−3.24717	−0.495188	−0.247594	+	0.968864i	$0.579640\pi$
$44$	−0.415805	−0.0626849
$45$	0	0
$46$	−2.60085	−0.383475
$47$	1.74782	0.254945	0.127473	−	0.991842i	$-0.459314\pi$
$47$	1.74782	0.254945	0.127473	+	0.991842i	$0.459314\pi$
$48$	0	0
$49$	−3.94806	−0.564009
$50$	−12.9939	−1.83762
$51$	0	0
$52$	−0.253796	−0.0351952
$53$	−6.06284	−0.832795	−0.416398	−	0.909183i	$-0.636708\pi$
$53$	−6.06284	−0.832795	−0.416398	+	0.909183i	$0.636708\pi$
$54$	0	0
$55$	11.6616	1.57245
$56$	5.09447	0.680778
$57$	0	0
$58$	0.309272	0.0406094
$59$	−1.99797	−0.260113	−0.130057	−	0.991507i	$-0.541516\pi$
$59$	−1.99797	−0.260113	−0.130057	+	0.991507i	$0.541516\pi$
$60$	0	0
$61$	12.7894	1.63751	0.818756	−	0.574142i	$-0.194664\pi$
$61$	12.7894	1.63751	0.818756	+	0.574142i	$0.194664\pi$
$62$	5.37853	0.683074
$63$	0	0
$64$	8.46708	1.05839
$65$	7.11791	0.882868
$66$	0	0
$67$	−6.34476	−0.775136	−0.387568	−	0.921841i	$-0.626685\pi$
$67$	−6.34476	−0.775136	−0.387568	+	0.921841i	$0.626685\pi$
$68$	0.951289	0.115361
$69$	0	0
$70$	−9.08796	−1.08622
$71$	13.9570	1.65639	0.828194	−	0.560441i	$-0.189368\pi$
$71$	13.9570	1.65639	0.828194	+	0.560441i	$0.189368\pi$
$72$	0	0
$73$	3.16797	0.370782	0.185391	−	0.982665i	$-0.440645\pi$
$73$	3.16797	0.370782	0.185391	+	0.982665i	$0.440645\pi$
$74$	0.503125	0.0584871
$75$	0	0
$76$	−0.671418	−0.0770169
$77$	5.34695	0.609342
$78$	0	0
$79$	−2.62141	−0.294931	−0.147466	−	0.989067i	$-0.547112\pi$
$79$	−2.62141	−0.294931	−0.147466	+	0.989067i	$0.547112\pi$
$80$	−14.1349	−1.58033
$81$	0	0
$82$	−5.91234	−0.652908
$83$	−0.224829	−0.0246782	−0.0123391	−	0.999924i	$-0.503928\pi$
$83$	−0.224829	−0.0246782	−0.0123391	+	0.999924i	$0.503928\pi$
$84$	0	0
$85$	−26.6796	−2.89381
$86$	4.43348	0.478074
$87$	0	0
$88$	8.92544	0.951456
$89$	7.11806	0.754513	0.377256	−	0.926109i	$-0.376868\pi$
$89$	7.11806	0.754513	0.377256	+	0.926109i	$0.376868\pi$
$90$	0	0
$91$	3.26363	0.342122
$92$	−0.258790	−0.0269807
$93$	0	0
$94$	−2.38636	−0.246134
$95$	18.8304	1.93196
$96$	0	0
$97$	15.5794	1.58184	0.790922	−	0.611917i	$-0.209601\pi$
$97$	15.5794	1.58184	0.790922	+	0.611917i	$0.209601\pi$
$98$	5.39044	0.544516
$99$	0	0
$100$	−1.29292	−0.129292
$101$	8.59334	0.855070	0.427535	−	0.903999i	$-0.359382\pi$
$101$	8.59334	0.855070	0.427535	+	0.903999i	$0.359382\pi$
$102$	0	0
$103$	−1.15497	−0.113802	−0.0569011	−	0.998380i	$-0.518122\pi$
$103$	−1.15497	−0.113802	−0.0569011	+	0.998380i	$0.518122\pi$
$104$	5.44785	0.534206
$105$	0	0
$106$	8.27782	0.804013
$107$	11.0635	1.06955	0.534776	−	0.844994i	$-0.320396\pi$
$107$	11.0635	1.06955	0.534776	+	0.844994i	$0.320396\pi$
$108$	0	0
$109$	15.2032	1.45621	0.728103	−	0.685468i	$-0.240402\pi$
$109$	15.2032	1.45621	0.728103	+	0.685468i	$0.240402\pi$
$110$	−15.9220	−1.51810
$111$	0	0
$112$	−6.48101	−0.612398
$113$	2.12061	0.199491	0.0997453	−	0.995013i	$-0.468197\pi$
$113$	2.12061	0.199491	0.0997453	+	0.995013i	$0.468197\pi$
$114$	0	0
$115$	7.25795	0.676807
$116$	0.0307731	0.00285721
$117$	0	0
$118$	2.72790	0.251124
$119$	−12.2329	−1.12139
$120$	0	0
$121$	−1.63222	−0.148384
$122$	−17.4618	−1.58092
$123$	0	0
$124$	0.535174	0.0480600
$125$	17.2104	1.53934
$126$	0	0
$127$	−11.3247	−1.00491	−0.502453	−	0.864605i	$-0.667569\pi$
$127$	−11.3247	−1.00491	−0.502453	+	0.864605i	$0.667569\pi$
$128$	−10.0261	−0.886194
$129$	0	0
$130$	−9.71834	−0.852355
$131$	10.8326	0.946449	0.473224	−	0.880942i	$-0.343090\pi$
$131$	10.8326	0.946449	0.473224	+	0.880942i	$0.343090\pi$
$132$	0	0
$133$	8.63395	0.748658
$134$	8.66274	0.748347
$135$	0	0
$136$	−20.4199	−1.75099
$137$	12.6684	1.08233	0.541166	−	0.840916i	$-0.317983\pi$
$137$	12.6684	1.08233	0.541166	+	0.840916i	$0.317983\pi$
$138$	0	0
$139$	−9.71198	−0.823759	−0.411880	−	0.911238i	$-0.635128\pi$
$139$	−9.71198	−0.823759	−0.411880	+	0.911238i	$0.635128\pi$
$140$	−0.904269	−0.0764247
$141$	0	0
$142$	−19.0560	−1.59914
$143$	5.71784	0.478150
$144$	0	0
$145$	−0.863056	−0.0716729
$146$	−4.32534	−0.357968
$147$	0	0
$148$	0.0500619	0.00411506
$149$	−4.15142	−0.340098	−0.170049	−	0.985436i	$-0.554393\pi$
$149$	−4.15142	−0.340098	−0.170049	+	0.985436i	$0.554393\pi$
$150$	0	0
$151$	−3.23899	−0.263586	−0.131793	−	0.991277i	$-0.542073\pi$
$151$	−3.23899	−0.263586	−0.131793	+	0.991277i	$0.542073\pi$
$152$	14.4123	1.16899
$153$	0	0
$154$	−7.30039	−0.588282
$155$	−15.0094	−1.20558
$156$	0	0
$157$	−7.47730	−0.596753	−0.298377	−	0.954448i	$-0.596445\pi$
$157$	−7.47730	−0.596753	−0.298377	+	0.954448i	$0.596445\pi$
$158$	3.57911	0.284738
$159$	0	0
$160$	−2.92291	−0.231076
$161$	3.32785	0.262271
$162$	0	0
$163$	−9.71856	−0.761216	−0.380608	−	0.924736i	$-0.624285\pi$
$163$	−9.71856	−0.761216	−0.380608	+	0.924736i	$0.624285\pi$
$164$	−0.588288	−0.0459376
$165$	0	0
$166$	0.306968	0.0238253
$167$	0.720490	0.0557532	0.0278766	−	0.999611i	$-0.491125\pi$
$167$	0.720490	0.0557532	0.0278766	+	0.999611i	$0.491125\pi$
$168$	0	0
$169$	−9.50999	−0.731537
$170$	36.4267	2.79380
$171$	0	0
$172$	0.441139	0.0336365
$173$	−2.19370	−0.166784	−0.0833920	−	0.996517i	$-0.526575\pi$
$173$	−2.19370	−0.166784	−0.0833920	+	0.996517i	$0.526575\pi$
$174$	0	0
$175$	16.6260	1.25681
$176$	−11.3546	−0.855888
$177$	0	0
$178$	−9.71855	−0.728436
$179$	25.3953	1.89814	0.949068	−	0.315073i	$-0.102029\pi$
$179$	25.3953	1.89814	0.949068	+	0.315073i	$0.102029\pi$
$180$	0	0
$181$	12.0564	0.896145	0.448073	−	0.893997i	$-0.352111\pi$
$181$	12.0564	0.896145	0.448073	+	0.893997i	$0.352111\pi$
$182$	−4.45596	−0.330298
$183$	0	0
$184$	5.55504	0.409523
$185$	−1.40402	−0.103226
$186$	0	0
$187$	−21.4318	−1.56725
$188$	−0.237447	−0.0173176
$189$	0	0
$190$	−25.7099	−1.86519
$191$	14.6287	1.05850	0.529248	−	0.848467i	$-0.322474\pi$
$191$	14.6287	1.05850	0.529248	+	0.848467i	$0.322474\pi$
$192$	0	0
$193$	−21.2398	−1.52887	−0.764436	−	0.644700i	$-0.776982\pi$
$193$	−21.2398	−1.52887	−0.764436	+	0.644700i	$0.776982\pi$
$194$	−21.2711	−1.52717
$195$	0	0
$196$	0.536359	0.0383113
$197$	18.0748	1.28777	0.643887	−	0.765121i	$-0.277321\pi$
$197$	18.0748	1.28777	0.643887	+	0.765121i	$0.277321\pi$
$198$	0	0
$199$	−24.4003	−1.72969	−0.864847	−	0.502035i	$-0.832585\pi$
$199$	−24.4003	−1.72969	−0.864847	+	0.502035i	$0.832585\pi$
$200$	27.7531	1.96244
$201$	0	0
$202$	−11.7328	−0.825518
$203$	−0.395720	−0.0277741
$204$	0	0
$205$	16.4990	1.15234
$206$	1.57692	0.109869
$207$	0	0
$208$	−6.93056	−0.480548
$209$	15.1265	1.04633
$210$	0	0
$211$	−15.3485	−1.05663	−0.528317	−	0.849047i	$-0.677177\pi$
$211$	−15.3485	−1.05663	−0.528317	+	0.849047i	$0.677177\pi$
$212$	0.823659	0.0565691
$213$	0	0
$214$	−15.1055	−1.03259
$215$	−12.3721	−0.843769
$216$	0	0
$217$	−6.88195	−0.467177
$218$	−20.7575	−1.40588
$219$	0	0
$220$	−1.58427	−0.106811
$221$	−13.0814	−0.879951
$222$	0	0
$223$	−14.3643	−0.961907	−0.480954	−	0.876746i	$-0.659710\pi$
$223$	−14.3643	−0.961907	−0.480954	+	0.876746i	$0.659710\pi$
$224$	−1.34018	−0.0895448
$225$	0	0
$226$	−2.89536	−0.192596
$227$	−11.2196	−0.744668	−0.372334	−	0.928099i	$-0.621442\pi$
$227$	−11.2196	−0.744668	−0.372334	+	0.928099i	$0.621442\pi$
$228$	0	0
$229$	9.59904	0.634322	0.317161	−	0.948372i	$-0.397270\pi$
$229$	9.59904	0.634322	0.317161	+	0.948372i	$0.397270\pi$
$230$	−9.90955	−0.653417
$231$	0	0
$232$	−0.660559	−0.0433678
$233$	−17.5440	−1.14935	−0.574674	−	0.818383i	$-0.694871\pi$
$233$	−17.5440	−1.14935	−0.574674	+	0.818383i	$0.694871\pi$
$234$	0	0
$235$	6.65939	0.434411
$236$	0.271431	0.0176687
$237$	0	0
$238$	16.7020	1.08263
$239$	−27.3054	−1.76624	−0.883121	−	0.469145i	$-0.844562\pi$
$239$	−27.3054	−1.76624	−0.883121	+	0.469145i	$0.844562\pi$
$240$	0	0
$241$	5.20112	0.335034	0.167517	−	0.985869i	$-0.446425\pi$
$241$	5.20112	0.335034	0.167517	+	0.985869i	$0.446425\pi$
$242$	2.22853	0.143255
$243$	0	0
$244$	−1.73748	−0.111231
$245$	−15.0426	−0.961036
$246$	0	0
$247$	9.23283	0.587471
$248$	−11.4878	−0.729473
$249$	0	0
$250$	−23.4980	−1.48614
$251$	4.76719	0.300902	0.150451	−	0.988617i	$-0.451927\pi$
$251$	4.76719	0.300902	0.150451	+	0.988617i	$0.451927\pi$
$252$	0	0
$253$	5.83034	0.366550
$254$	15.4621	0.970175
$255$	0	0
$256$	−3.24510	−0.202819
$257$	−3.03893	−0.189563	−0.0947817	−	0.995498i	$-0.530215\pi$
$257$	−3.03893	−0.189563	−0.0947817	+	0.995498i	$0.530215\pi$
$258$	0	0
$259$	−0.643760	−0.0400013
$260$	−0.966993	−0.0599704
$261$	0	0
$262$	−14.7902	−0.913739
$263$	24.9814	1.54042	0.770210	−	0.637790i	$-0.220151\pi$
$263$	24.9814	1.54042	0.770210	+	0.637790i	$0.220151\pi$
$264$	0	0
$265$	−23.1002	−1.41903
$266$	−11.7882	−0.722784
$267$	0	0
$268$	0.861959	0.0526525
$269$	−25.2930	−1.54214	−0.771069	−	0.636751i	$-0.780278\pi$
$269$	−25.2930	−1.54214	−0.771069	+	0.636751i	$0.780278\pi$
$270$	0	0
$271$	22.7090	1.37947	0.689737	−	0.724060i	$-0.257726\pi$
$271$	22.7090	1.37947	0.689737	+	0.724060i	$0.257726\pi$
$272$	25.9774	1.57511
$273$	0	0
$274$	−17.2966	−1.04493
$275$	29.1286	1.75652
$276$	0	0
$277$	−28.7820	−1.72935	−0.864673	−	0.502335i	$-0.832474\pi$
$277$	−28.7820	−1.72935	−0.864673	+	0.502335i	$0.832474\pi$
$278$	13.2601	0.795290
$279$	0	0
$280$	19.4106	1.16000
$281$	1.84502	0.110065	0.0550324	−	0.998485i	$-0.482474\pi$
$281$	1.84502	0.110065	0.0550324	+	0.998485i	$0.482474\pi$
$282$	0	0
$283$	6.00565	0.356999	0.178499	−	0.983940i	$-0.442876\pi$
$283$	6.00565	0.356999	0.178499	+	0.983940i	$0.442876\pi$
$284$	−1.89611	−0.112513
$285$	0	0
$286$	−7.80678	−0.461625
$287$	7.56496	0.446546
$288$	0	0
$289$	32.0323	1.88425
$290$	1.17836	0.0691958
$291$	0	0
$292$	−0.430380	−0.0251861
$293$	−10.6211	−0.620490	−0.310245	−	0.950657i	$-0.600411\pi$
$293$	−10.6211	−0.620490	−0.310245	+	0.950657i	$0.600411\pi$
$294$	0	0
$295$	−7.61250	−0.443217
$296$	−1.07460	−0.0624600
$297$	0	0
$298$	5.66809	0.328344
$299$	3.55868	0.205804
$300$	0	0
$301$	−5.67273	−0.326971
$302$	4.42232	0.254476
$303$	0	0
$304$	−18.3348	−1.05157
$305$	48.7291	2.79022
$306$	0	0
$307$	1.69142	0.0965343	0.0482671	−	0.998834i	$-0.484630\pi$
$307$	1.69142	0.0965343	0.0482671	+	0.998834i	$0.484630\pi$
$308$	−0.726403	−0.0413906
$309$	0	0
$310$	20.4928	1.16392
$311$	27.1572	1.53994	0.769971	−	0.638079i	$-0.220271\pi$
$311$	27.1572	1.53994	0.769971	+	0.638079i	$0.220271\pi$
$312$	0	0
$313$	26.9376	1.52260	0.761300	−	0.648399i	$-0.224561\pi$
$313$	26.9376	1.52260	0.761300	+	0.648399i	$0.224561\pi$
$314$	10.2090	0.576129
$315$	0	0
$316$	0.356128	0.0200338
$317$	−5.32644	−0.299163	−0.149581	−	0.988749i	$-0.547793\pi$
$317$	−5.32644	−0.299163	−0.149581	+	0.988749i	$0.547793\pi$
$318$	0	0
$319$	−0.693296	−0.0388171
$320$	32.2606	1.80342
$321$	0	0
$322$	−4.54363	−0.253207
$323$	−34.6069	−1.92558
$324$	0	0
$325$	17.7793	0.986217
$326$	13.2691	0.734908
$327$	0	0
$328$	12.6279	0.697258
$329$	3.05340	0.168339
$330$	0	0
$331$	14.4609	0.794844	0.397422	−	0.917636i	$-0.369905\pi$
$331$	14.4609	0.794844	0.397422	+	0.917636i	$0.369905\pi$
$332$	0.0305439	0.00167631
$333$	0	0
$334$	−0.983712	−0.0538263
$335$	−24.1743	−1.32078
$336$	0	0
$337$	14.4878	0.789200	0.394600	−	0.918853i	$-0.370883\pi$
$337$	14.4878	0.789200	0.394600	+	0.918853i	$0.370883\pi$
$338$	12.9843	0.706255
$339$	0	0
$340$	3.62452	0.196567
$341$	−12.0571	−0.652927
$342$	0	0
$343$	−19.1260	−1.03271
$344$	−9.46926	−0.510548
$345$	0	0
$346$	2.99514	0.161020
$347$	3.54218	0.190154	0.0950772	−	0.995470i	$-0.469690\pi$
$347$	3.54218	0.190154	0.0950772	+	0.995470i	$0.469690\pi$
$348$	0	0
$349$	−29.6243	−1.58575	−0.792875	−	0.609384i	$-0.791417\pi$
$349$	−29.6243	−1.58575	−0.792875	+	0.609384i	$0.791417\pi$
$350$	−22.7001	−1.21337
$351$	0	0
$352$	−2.34798	−0.125148
$353$	−27.2686	−1.45136	−0.725681	−	0.688032i	$-0.758475\pi$
$353$	−27.2686	−1.45136	−0.725681	+	0.688032i	$0.758475\pi$
$354$	0	0
$355$	53.1778	2.82238
$356$	−0.967014	−0.0512516
$357$	0	0
$358$	−34.6732	−1.83253
$359$	1.69023	0.0892068	0.0446034	−	0.999005i	$-0.485798\pi$
$359$	1.69023	0.0892068	0.0446034	+	0.999005i	$0.485798\pi$
$360$	0	0
$361$	5.42547	0.285551
$362$	−16.4611	−0.865174
$363$	0	0
$364$	−0.443376	−0.0232392
$365$	12.0703	0.631790
$366$	0	0
$367$	5.45574	0.284787	0.142394	−	0.989810i	$-0.454520\pi$
$367$	5.45574	0.284787	0.142394	+	0.989810i	$0.454520\pi$
$368$	−7.06692	−0.368389
$369$	0	0
$370$	1.91697	0.0996584
$371$	−10.5917	−0.549892
$372$	0	0
$373$	20.9460	1.08454	0.542271	−	0.840204i	$-0.317565\pi$
$373$	20.9460	1.08454	0.542271	+	0.840204i	$0.317565\pi$
$374$	29.2617	1.51309
$375$	0	0
$376$	5.09691	0.262853
$377$	−0.423169	−0.0217943
$378$	0	0
$379$	34.0299	1.74800	0.873998	−	0.485929i	$-0.161519\pi$
$379$	34.0299	1.74800	0.873998	+	0.485929i	$0.161519\pi$
$380$	−2.55818	−0.131232
$381$	0	0
$382$	−19.9731	−1.02191
$383$	9.95142	0.508494	0.254247	−	0.967139i	$-0.418172\pi$
$383$	9.95142	0.508494	0.254247	+	0.967139i	$0.418172\pi$
$384$	0	0
$385$	20.3725	1.03828
$386$	28.9994	1.47603
$387$	0	0
$388$	−2.11651	−0.107450
$389$	17.5565	0.890149	0.445075	−	0.895493i	$-0.353177\pi$
$389$	17.5565	0.890149	0.445075	+	0.895493i	$0.353177\pi$
$390$	0	0
$391$	−13.3388	−0.674572
$392$	−11.5132	−0.581504
$393$	0	0
$394$	−24.6781	−1.24327
$395$	−9.98788	−0.502545
$396$	0	0
$397$	−16.7952	−0.842929	−0.421464	−	0.906845i	$-0.638484\pi$
$397$	−16.7952	−0.842929	−0.421464	+	0.906845i	$0.638484\pi$
$398$	33.3147	1.66992
$399$	0	0
$400$	−35.3066	−1.76533
$401$	2.62836	0.131254	0.0656269	−	0.997844i	$-0.479095\pi$
$401$	2.62836	0.131254	0.0656269	+	0.997844i	$0.479095\pi$
$402$	0	0
$403$	−7.35931	−0.366593
$404$	−1.16744	−0.0580822
$405$	0	0
$406$	0.540291	0.0268142
$407$	−1.12786	−0.0559058
$408$	0	0
$409$	−7.85503	−0.388406	−0.194203	−	0.980961i	$-0.562212\pi$
$409$	−7.85503	−0.388406	−0.194203	+	0.980961i	$0.562212\pi$
$410$	−22.5267	−1.11251
$411$	0	0
$412$	0.156906	0.00773022
$413$	−3.49041	−0.171752
$414$	0	0
$415$	−0.856626	−0.0420501
$416$	−1.43314	−0.0702657
$417$	0	0
$418$	−20.6528	−1.01016
$419$	−31.6570	−1.54655	−0.773273	−	0.634073i	$-0.781382\pi$
$419$	−31.6570	−1.54655	−0.773273	+	0.634073i	$0.781382\pi$
$420$	0	0
$421$	−0.518148	−0.0252530	−0.0126265	−	0.999920i	$-0.504019\pi$
$421$	−0.518148	−0.0252530	−0.0126265	+	0.999920i	$0.504019\pi$
$422$	20.9559	1.02012
$423$	0	0
$424$	−17.6802	−0.858627
$425$	−66.6410	−3.23257
$426$	0	0
$427$	22.3428	1.08124
$428$	−1.50302	−0.0726513
$429$	0	0
$430$	16.8921	0.814608
$431$	−11.3347	−0.545973	−0.272986	−	0.962018i	$-0.588011\pi$
$431$	−11.3347	−0.545973	−0.272986	+	0.962018i	$0.588011\pi$
$432$	0	0
$433$	−7.19613	−0.345824	−0.172912	−	0.984937i	$-0.555318\pi$
$433$	−7.19613	−0.345824	−0.172912	+	0.984937i	$0.555318\pi$
$434$	9.39618	0.451031
$435$	0	0
$436$	−2.06541	−0.0989154
$437$	9.41449	0.450356
$438$	0	0
$439$	−1.82760	−0.0872265	−0.0436132	−	0.999048i	$-0.513887\pi$
$439$	−1.82760	−0.0872265	−0.0436132	+	0.999048i	$0.513887\pi$
$440$	34.0070	1.62122
$441$	0	0
$442$	17.8605	0.849539
$443$	15.8094	0.751129	0.375564	−	0.926796i	$-0.377449\pi$
$443$	15.8094	0.751129	0.375564	+	0.926796i	$0.377449\pi$
$444$	0	0
$445$	27.1207	1.28564
$446$	19.6122	0.928663
$447$	0	0
$448$	14.7918	0.698848
$449$	−9.70627	−0.458067	−0.229034	−	0.973419i	$-0.573557\pi$
$449$	−9.70627	−0.458067	−0.229034	+	0.973419i	$0.573557\pi$
$450$	0	0
$451$	13.2537	0.624093
$452$	−0.288093	−0.0135508
$453$	0	0
$454$	15.3185	0.718932
$455$	12.4348	0.582954
$456$	0	0
$457$	2.56954	0.120198	0.0600990	−	0.998192i	$-0.480858\pi$
$457$	2.56954	0.120198	0.0600990	+	0.998192i	$0.480858\pi$
$458$	−13.1059	−0.612400
$459$	0	0
$460$	−0.986019	−0.0459734
$461$	−3.24248	−0.151017	−0.0755086	−	0.997145i	$-0.524058\pi$
$461$	−3.24248	−0.151017	−0.0755086	+	0.997145i	$0.524058\pi$
$462$	0	0
$463$	9.70706	0.451125	0.225563	−	0.974229i	$-0.427578\pi$
$463$	9.70706	0.451125	0.225563	+	0.974229i	$0.427578\pi$
$464$	0.840340	0.0390118
$465$	0	0
$466$	23.9535	1.10963
$467$	−3.17093	−0.146733	−0.0733666	−	0.997305i	$-0.523374\pi$
$467$	−3.17093	−0.146733	−0.0733666	+	0.997305i	$0.523374\pi$
$468$	0	0
$469$	−11.0842	−0.511820
$470$	−9.09231	−0.419397
$471$	0	0
$472$	−5.82640	−0.268182
$473$	−9.93854	−0.456975
$474$	0	0
$475$	47.0351	2.15812
$476$	1.66188	0.0761722
$477$	0	0
$478$	37.2811	1.70520
$479$	27.6936	1.26535	0.632677	−	0.774416i	$-0.281956\pi$
$479$	27.6936	1.26535	0.632677	+	0.774416i	$0.281956\pi$
$480$	0	0
$481$	−0.688414	−0.0313890
$482$	−7.10129	−0.323455
$483$	0	0
$484$	0.221743	0.0100792
$485$	59.3592	2.69536
$486$	0	0
$487$	10.4889	0.475298	0.237649	−	0.971351i	$-0.423623\pi$
$487$	10.4889	0.475298	0.237649	+	0.971351i	$0.423623\pi$
$488$	37.2959	1.68831
$489$	0	0
$490$	20.5382	0.927822
$491$	5.92754	0.267506	0.133753	−	0.991015i	$-0.457297\pi$
$491$	5.92754	0.267506	0.133753	+	0.991015i	$0.457297\pi$
$492$	0	0
$493$	1.58614	0.0714361
$494$	−12.6059	−0.567168
$495$	0	0
$496$	14.6143	0.656202
$497$	24.3826	1.09371
$498$	0	0
$499$	−38.1174	−1.70637	−0.853184	−	0.521609i	$-0.825332\pi$
$499$	−38.1174	−1.70637	−0.853184	+	0.521609i	$0.825332\pi$
$500$	−2.33809	−0.104563
$501$	0	0
$502$	−6.50882	−0.290503
$503$	13.0023	0.579742	0.289871	−	0.957066i	$-0.406388\pi$
$503$	13.0023	0.579742	0.289871	+	0.957066i	$0.406388\pi$
$504$	0	0
$505$	32.7417	1.45699
$506$	−7.96038	−0.353882
$507$	0	0
$508$	1.53850	0.0682600
$509$	−27.2056	−1.20587	−0.602933	−	0.797792i	$-0.706001\pi$
$509$	−27.2056	−1.20587	−0.602933	+	0.797792i	$0.706001\pi$
$510$	0	0
$511$	5.53437	0.244826
$512$	24.4829	1.08200
$513$	0	0
$514$	4.14917	0.183012
$515$	−4.40056	−0.193912
$516$	0	0
$517$	5.34951	0.235271
$518$	0.878949	0.0386188
$519$	0	0
$520$	20.7570	0.910253
$521$	9.15349	0.401022	0.200511	−	0.979691i	$-0.435740\pi$
$521$	9.15349	0.401022	0.200511	+	0.979691i	$0.435740\pi$
$522$	0	0
$523$	−26.1539	−1.14363	−0.571815	−	0.820383i	$-0.693760\pi$
$523$	−26.1539	−1.14363	−0.571815	+	0.820383i	$0.693760\pi$
$524$	−1.47165	−0.0642893
$525$	0	0
$526$	−34.1081	−1.48718
$527$	27.5845	1.20160
$528$	0	0
$529$	−19.3713	−0.842231
$530$	31.5395	1.36999
$531$	0	0
$532$	−1.17295	−0.0508540
$533$	8.08970	0.350404
$534$	0	0
$535$	42.1534	1.82245
$536$	−18.5024	−0.799180
$537$	0	0
$538$	34.5334	1.48884
$539$	−12.0838	−0.520485
$540$	0	0
$541$	−8.73933	−0.375733	−0.187867	−	0.982195i	$-0.560157\pi$
$541$	−8.73933	−0.375733	−0.187867	+	0.982195i	$0.560157\pi$
$542$	−31.0055	−1.33180
$543$	0	0
$544$	5.37177	0.230313
$545$	57.9261	2.48128
$546$	0	0
$547$	30.3875	1.29928	0.649638	−	0.760244i	$-0.274921\pi$
$547$	30.3875	1.29928	0.649638	+	0.760244i	$0.274921\pi$
$548$	−1.72104	−0.0735194
$549$	0	0
$550$	−39.7703	−1.69581
$551$	−1.11949	−0.0476920
$552$	0	0
$553$	−4.57955	−0.194742
$554$	39.2972	1.66958
$555$	0	0
$556$	1.31941	0.0559553
$557$	26.0844	1.10523	0.552617	−	0.833436i	$-0.313629\pi$
$557$	26.0844	1.10523	0.552617	+	0.833436i	$0.313629\pi$
$558$	0	0
$559$	−6.06622	−0.256574
$560$	−24.6934	−1.04349
$561$	0	0
$562$	−2.51908	−0.106261
$563$	−43.3510	−1.82703	−0.913513	−	0.406811i	$-0.866641\pi$
$563$	−43.3510	−1.82703	−0.913513	+	0.406811i	$0.866641\pi$
$564$	0	0
$565$	8.07980	0.339920
$566$	−8.19974	−0.344661
$567$	0	0
$568$	40.7008	1.70777
$569$	−6.56294	−0.275133	−0.137566	−	0.990493i	$-0.543928\pi$
$569$	−6.56294	−0.275133	−0.137566	+	0.990493i	$0.543928\pi$
$570$	0	0
$571$	23.9258	1.00127	0.500633	−	0.865660i	$-0.333101\pi$
$571$	23.9258	1.00127	0.500633	+	0.865660i	$0.333101\pi$
$572$	−0.776789	−0.0324792
$573$	0	0
$574$	−10.3287	−0.431113
$575$	18.1291	0.756036
$576$	0	0
$577$	−31.3816	−1.30643	−0.653216	−	0.757172i	$-0.726581\pi$
$577$	−31.3816	−1.30643	−0.653216	+	0.757172i	$0.726581\pi$
$578$	−43.7349	−1.81913
$579$	0	0
$580$	0.117249	0.00486851
$581$	−0.392772	−0.0162949
$582$	0	0
$583$	−18.5564	−0.768529
$584$	9.23830	0.382284
$585$	0	0
$586$	14.5013	0.599045
$587$	44.3977	1.83249	0.916244	−	0.400620i	$-0.131205\pi$
$587$	44.3977	1.83249	0.916244	+	0.400620i	$0.131205\pi$
$588$	0	0
$589$	−19.4691	−0.802209
$590$	10.3936	0.427899
$591$	0	0
$592$	1.36707	0.0561862
$593$	19.0954	0.784156	0.392078	−	0.919932i	$-0.371756\pi$
$593$	19.0954	0.784156	0.392078	+	0.919932i	$0.371756\pi$
$594$	0	0
$595$	−46.6088	−1.91077
$596$	0.563985	0.0231017
$597$	0	0
$598$	−4.85880	−0.198691
$599$	−0.881551	−0.0360192	−0.0180096	−	0.999838i	$-0.505733\pi$
$599$	−0.881551	−0.0360192	−0.0180096	+	0.999838i	$0.505733\pi$
$600$	0	0
$601$	28.2737	1.15331	0.576654	−	0.816988i	$-0.304358\pi$
$601$	28.2737	1.15331	0.576654	+	0.816988i	$0.304358\pi$
$602$	7.74519	0.315670
$603$	0	0
$604$	0.440029	0.0179045
$605$	−6.21896	−0.252837
$606$	0	0
$607$	−37.3791	−1.51717	−0.758585	−	0.651574i	$-0.774109\pi$
$607$	−37.3791	−1.51717	−0.758585	+	0.651574i	$0.774109\pi$
$608$	−3.79138	−0.153761
$609$	0	0
$610$	−66.5316	−2.69379
$611$	3.26520	0.132096
$612$	0	0
$613$	−8.47213	−0.342186	−0.171093	−	0.985255i	$-0.554730\pi$
$613$	−8.47213	−0.342186	−0.171093	+	0.985255i	$0.554730\pi$
$614$	−2.30936	−0.0931980
$615$	0	0
$616$	15.5926	0.628242
$617$	−37.2023	−1.49771	−0.748853	−	0.662736i	$-0.769395\pi$
$617$	−37.2023	−1.49771	−0.748853	+	0.662736i	$0.769395\pi$
$618$	0	0
$619$	−38.6764	−1.55453	−0.777267	−	0.629171i	$-0.783395\pi$
$619$	−38.6764	−1.55453	−0.777267	+	0.629171i	$0.783395\pi$
$620$	2.03908	0.0818913
$621$	0	0
$622$	−37.0787	−1.48672
$623$	12.4351	0.498202
$624$	0	0
$625$	17.9885	0.719539
$626$	−36.7789	−1.46998
$627$	0	0
$628$	1.01582	0.0405355
$629$	2.58034	0.102885
$630$	0	0
$631$	−0.0681348	−0.00271241	−0.00135620	−	0.999999i	$-0.500432\pi$
$631$	−0.0681348	−0.00271241	−0.00135620	+	0.999999i	$0.500432\pi$
$632$	−7.64445	−0.304080
$633$	0	0
$634$	7.27239	0.288824
$635$	−43.1485	−1.71230
$636$	0	0
$637$	−7.37560	−0.292232
$638$	0.946583	0.0374756
$639$	0	0
$640$	−38.2008	−1.51002
$641$	10.8567	0.428812	0.214406	−	0.976745i	$-0.431218\pi$
$641$	10.8567	0.428812	0.214406	+	0.976745i	$0.431218\pi$
$642$	0	0
$643$	31.2753	1.23338	0.616689	−	0.787207i	$-0.288474\pi$
$643$	31.2753	1.23338	0.616689	+	0.787207i	$0.288474\pi$
$644$	−0.452100	−0.0178152
$645$	0	0
$646$	47.2501	1.85903
$647$	22.3771	0.879735	0.439868	−	0.898063i	$-0.355025\pi$
$647$	22.3771	0.879735	0.439868	+	0.898063i	$0.355025\pi$
$648$	0	0
$649$	−6.11515	−0.240041
$650$	−24.2747	−0.952133
$651$	0	0
$652$	1.32030	0.0517070
$653$	37.4686	1.46626	0.733131	−	0.680088i	$-0.238058\pi$
$653$	37.4686	1.46626	0.733131	+	0.680088i	$0.238058\pi$
$654$	0	0
$655$	41.2735	1.61269
$656$	−16.0647	−0.627223
$657$	0	0
$658$	−4.16892	−0.162521
$659$	4.62582	0.180196	0.0900982	−	0.995933i	$-0.471282\pi$
$659$	4.62582	0.180196	0.0900982	+	0.995933i	$0.471282\pi$
$660$	0	0
$661$	7.33528	0.285309	0.142655	−	0.989773i	$-0.454436\pi$
$661$	7.33528	0.285309	0.142655	+	0.989773i	$0.454436\pi$
$662$	−19.7440	−0.767373
$663$	0	0
$664$	−0.655638	−0.0254437
$665$	32.8964	1.27567
$666$	0	0
$667$	−0.431495	−0.0167075
$668$	−0.0978812	−0.00378714
$669$	0	0
$670$	33.0061	1.27514
$671$	39.1442	1.51115
$672$	0	0
$673$	12.4175	0.478661	0.239331	−	0.970938i	$-0.423072\pi$
$673$	12.4175	0.478661	0.239331	+	0.970938i	$0.423072\pi$
$674$	−19.7807	−0.761925
$675$	0	0
$676$	1.29197	0.0496910
$677$	24.7335	0.950587	0.475293	−	0.879827i	$-0.342342\pi$
$677$	24.7335	0.950587	0.475293	+	0.879827i	$0.342342\pi$
$678$	0	0
$679$	27.2168	1.04449
$680$	−77.8021	−2.98357
$681$	0	0
$682$	16.4620	0.630362
$683$	−3.99404	−0.152828	−0.0764139	−	0.997076i	$-0.524347\pi$
$683$	−3.99404	−0.152828	−0.0764139	+	0.997076i	$0.524347\pi$
$684$	0	0
$685$	48.2680	1.84423
$686$	26.1135	0.997018
$687$	0	0
$688$	12.0465	0.459267
$689$	−11.3263	−0.431499
$690$	0	0
$691$	5.33588	0.202987	0.101493	−	0.994836i	$-0.467638\pi$
$691$	5.33588	0.202987	0.101493	+	0.994836i	$0.467638\pi$
$692$	0.298022	0.0113291
$693$	0	0
$694$	−4.83628	−0.183583
$695$	−37.0038	−1.40363
$696$	0	0
$697$	−30.3222	−1.14853
$698$	40.4471	1.53095
$699$	0	0
$700$	−2.25871	−0.0853711
$701$	−18.7981	−0.709994	−0.354997	−	0.934867i	$-0.615518\pi$
$701$	−18.7981	−0.709994	−0.354997	+	0.934867i	$0.615518\pi$
$702$	0	0
$703$	−1.82120	−0.0686878
$704$	25.9151	0.976710
$705$	0	0
$706$	37.2308	1.40120
$707$	15.0124	0.564599
$708$	0	0
$709$	18.3935	0.690781	0.345390	−	0.938459i	$-0.387746\pi$
$709$	18.3935	0.690781	0.345390	+	0.938459i	$0.387746\pi$
$710$	−72.6056	−2.72484
$711$	0	0
$712$	20.7574	0.777916
$713$	−7.50411	−0.281031
$714$	0	0
$715$	21.7857	0.814737
$716$	−3.45005	−0.128934
$717$	0	0
$718$	−2.30773	−0.0861238
$719$	2.66601	0.0994255	0.0497128	−	0.998764i	$-0.484169\pi$
$719$	2.66601	0.0994255	0.0497128	+	0.998764i	$0.484169\pi$
$720$	0	0
$721$	−2.01770	−0.0751431
$722$	−7.40760	−0.275682
$723$	0	0
$724$	−1.63791	−0.0608723
$725$	−2.15576	−0.0800630
$726$	0	0
$727$	−0.424942	−0.0157602	−0.00788011	−	0.999969i	$-0.502508\pi$
$727$	−0.424942	−0.0157602	−0.00788011	+	0.999969i	$0.502508\pi$
$728$	9.51728	0.352734
$729$	0	0
$730$	−16.4801	−0.609955
$731$	22.7376	0.840982
$732$	0	0
$733$	−23.8062	−0.879303	−0.439652	−	0.898168i	$-0.644898\pi$
$733$	−23.8062	−0.879303	−0.439652	+	0.898168i	$0.644898\pi$
$734$	−7.44892	−0.274945
$735$	0	0
$736$	−1.46134	−0.0538658
$737$	−19.4193	−0.715320
$738$	0	0
$739$	25.6666	0.944163	0.472081	−	0.881555i	$-0.343503\pi$
$739$	25.6666	0.944163	0.472081	+	0.881555i	$0.343503\pi$
$740$	0.190742	0.00701181
$741$	0	0
$742$	14.4612	0.530887
$743$	8.20505	0.301014	0.150507	−	0.988609i	$-0.451909\pi$
$743$	8.20505	0.301014	0.150507	+	0.988609i	$0.451909\pi$
$744$	0	0
$745$	−15.8174	−0.579505
$746$	−28.5983	−1.04706
$747$	0	0
$748$	2.91159	0.106458
$749$	19.3278	0.706221
$750$	0	0
$751$	21.4047	0.781069	0.390535	−	0.920588i	$-0.372290\pi$
$751$	21.4047	0.781069	0.390535	+	0.920588i	$0.372290\pi$
$752$	−6.48411	−0.236451
$753$	0	0
$754$	0.577768	0.0210411
$755$	−12.3410	−0.449133
$756$	0	0
$757$	−25.0862	−0.911773	−0.455886	−	0.890038i	$-0.650678\pi$
$757$	−25.0862	−0.911773	−0.455886	+	0.890038i	$0.650678\pi$
$758$	−46.4622	−1.68758
$759$	0	0
$760$	54.9125	1.99189
$761$	47.0287	1.70479	0.852395	−	0.522899i	$-0.175149\pi$
$761$	47.0287	1.70479	0.852395	+	0.522899i	$0.175149\pi$
$762$	0	0
$763$	26.5597	0.961527
$764$	−1.98736	−0.0719002
$765$	0	0
$766$	−13.5870	−0.490920
$767$	−3.73252	−0.134774
$768$	0	0
$769$	−46.5412	−1.67832	−0.839159	−	0.543886i	$-0.816953\pi$
$769$	−46.5412	−1.67832	−0.839159	+	0.543886i	$0.816953\pi$
$770$	−27.8154	−1.00240
$771$	0	0
$772$	2.88550	0.103851
$773$	39.9281	1.43612	0.718058	−	0.695983i	$-0.245031\pi$
$773$	39.9281	1.43612	0.718058	+	0.695983i	$0.245031\pi$
$774$	0	0
$775$	−37.4908	−1.34671
$776$	45.4319	1.63091
$777$	0	0
$778$	−23.9705	−0.859385
$779$	21.4013	0.766782
$780$	0	0
$781$	42.7179	1.52857
$782$	18.2120	0.651258
$783$	0	0
$784$	14.6467	0.523095
$785$	−28.4894	−1.01683
$786$	0	0
$787$	12.7534	0.454611	0.227306	−	0.973823i	$-0.427008\pi$
$787$	12.7534	0.454611	0.227306	+	0.973823i	$0.427008\pi$
$788$	−2.45552	−0.0874743
$789$	0	0
$790$	13.6368	0.485176
$791$	3.70467	0.131723
$792$	0	0
$793$	23.8926	0.848450
$794$	22.9312	0.813797
$795$	0	0
$796$	3.31488	0.117493
$797$	15.6817	0.555474	0.277737	−	0.960657i	$-0.410416\pi$
$797$	15.6817	0.555474	0.277737	+	0.960657i	$0.410416\pi$
$798$	0	0
$799$	−12.2387	−0.432976
$800$	−7.30091	−0.258126
$801$	0	0
$802$	−3.58859	−0.126718
$803$	9.69614	0.342169
$804$	0	0
$805$	12.6795	0.446893
$806$	10.0479	0.353924
$807$	0	0
$808$	25.0596	0.881593
$809$	−21.4680	−0.754776	−0.377388	−	0.926055i	$-0.623178\pi$
$809$	−21.4680	−0.754776	−0.377388	+	0.926055i	$0.623178\pi$
$810$	0	0
$811$	−27.3610	−0.960774	−0.480387	−	0.877057i	$-0.659504\pi$
$811$	−27.3610	−0.960774	−0.480387	+	0.877057i	$0.659504\pi$
$812$	0.0537600	0.00188661
$813$	0	0
$814$	1.53991	0.0539737
$815$	−37.0289	−1.29707
$816$	0	0
$817$	−16.0482	−0.561455
$818$	10.7248	0.374983
$819$	0	0
$820$	−2.24145	−0.0782748
$821$	20.7419	0.723896	0.361948	−	0.932198i	$-0.382112\pi$
$821$	20.7419	0.723896	0.361948	+	0.932198i	$0.382112\pi$
$822$	0	0
$823$	−39.0938	−1.36272	−0.681362	−	0.731946i	$-0.738612\pi$
$823$	−39.0938	−1.36272	−0.681362	+	0.731946i	$0.738612\pi$
$824$	−3.36807	−0.117332
$825$	0	0
$826$	4.76559	0.165816
$827$	−30.0876	−1.04625	−0.523123	−	0.852257i	$-0.675233\pi$
$827$	−30.0876	−1.04625	−0.523123	+	0.852257i	$0.675233\pi$
$828$	0	0
$829$	−28.2463	−0.981036	−0.490518	−	0.871431i	$-0.663192\pi$
$829$	−28.2463	−0.981036	−0.490518	+	0.871431i	$0.663192\pi$
$830$	1.16958	0.0405969
$831$	0	0
$832$	15.8178	0.548385
$833$	27.6455	0.957861
$834$	0	0
$835$	2.74515	0.0949999
$836$	−2.05500	−0.0710735
$837$	0	0
$838$	43.2225	1.49310
$839$	−19.7857	−0.683077	−0.341538	−	0.939868i	$-0.610948\pi$
$839$	−19.7857	−0.683077	−0.341538	+	0.939868i	$0.610948\pi$
$840$	0	0
$841$	−28.9487	−0.998231
$842$	0.707447	0.0243802
$843$	0	0
$844$	2.08515	0.0717738
$845$	−36.2342	−1.24649
$846$	0	0
$847$	−2.85146	−0.0979772
$848$	22.4921	0.772384
$849$	0	0
$850$	90.9875	3.12085
$851$	−0.701959	−0.0240628
$852$	0	0
$853$	45.5113	1.55828	0.779138	−	0.626852i	$-0.215657\pi$
$853$	45.5113	1.55828	0.779138	+	0.626852i	$0.215657\pi$
$854$	−30.5054	−1.04387
$855$	0	0
$856$	32.2630	1.10273
$857$	−30.4161	−1.03899	−0.519496	−	0.854473i	$-0.673880\pi$
$857$	−30.4161	−1.03899	−0.519496	+	0.854473i	$0.673880\pi$
$858$	0	0
$859$	27.2144	0.928544	0.464272	−	0.885693i	$-0.346316\pi$
$859$	27.2144	0.928544	0.464272	+	0.885693i	$0.346316\pi$
$860$	1.68079	0.0573146
$861$	0	0
$862$	15.4757	0.527103
$863$	10.4437	0.355509	0.177754	−	0.984075i	$-0.443117\pi$
$863$	10.4437	0.355509	0.177754	+	0.984075i	$0.443117\pi$
$864$	0	0
$865$	−8.35826	−0.284189
$866$	9.82514	0.333872
$867$	0	0
$868$	0.934938	0.0317338
$869$	−8.02330	−0.272172
$870$	0	0
$871$	−11.8530	−0.401624
$872$	44.3351	1.50138
$873$	0	0
$874$	−12.8540	−0.434792
$875$	30.0662	1.01642
$876$	0	0
$877$	−41.6266	−1.40563	−0.702816	−	0.711372i	$-0.748074\pi$
$877$	−41.6266	−1.40563	−0.702816	+	0.711372i	$0.748074\pi$
$878$	2.49529	0.0842119
$879$	0	0
$880$	−43.2625	−1.45838
$881$	−45.8465	−1.54461	−0.772303	−	0.635254i	$-0.780895\pi$
$881$	−45.8465	−1.54461	−0.772303	+	0.635254i	$0.780895\pi$
$882$	0	0
$883$	5.33137	0.179415	0.0897075	−	0.995968i	$-0.471407\pi$
$883$	5.33137	0.179415	0.0897075	+	0.995968i	$0.471407\pi$
$884$	1.77716	0.0597723
$885$	0	0
$886$	−21.5852	−0.725169
$887$	36.7253	1.23312	0.616558	−	0.787310i	$-0.288527\pi$
$887$	36.7253	1.23312	0.616558	+	0.787310i	$0.288527\pi$
$888$	0	0
$889$	−19.7840	−0.663535
$890$	−37.0288	−1.24121
$891$	0	0
$892$	1.95145	0.0653393
$893$	8.63808	0.289062
$894$	0	0
$895$	96.7592	3.23430
$896$	−17.5155	−0.585150
$897$	0	0
$898$	13.2523	0.442236
$899$	0.892326	0.0297608
$900$	0	0
$901$	42.4539	1.41434
$902$	−18.0958	−0.602524
$903$	0	0
$904$	6.18405	0.205679
$905$	45.9363	1.52698
$906$	0	0
$907$	46.0713	1.52977	0.764886	−	0.644165i	$-0.222795\pi$
$907$	46.0713	1.52977	0.764886	+	0.644165i	$0.222795\pi$
$908$	1.52422	0.0505829
$909$	0	0
$910$	−16.9777	−0.562807
$911$	−25.6634	−0.850265	−0.425133	−	0.905131i	$-0.639773\pi$
$911$	−25.6634	−0.850265	−0.425133	+	0.905131i	$0.639773\pi$
$912$	0	0
$913$	−0.688131	−0.0227738
$914$	−3.50829	−0.116044
$915$	0	0
$916$	−1.30406	−0.0430875
$917$	18.9243	0.624937
$918$	0	0
$919$	8.98535	0.296399	0.148200	−	0.988957i	$-0.452652\pi$
$919$	8.98535	0.296399	0.148200	+	0.988957i	$0.452652\pi$
$920$	21.1654	0.697801
$921$	0	0
$922$	4.42708	0.145798
$923$	26.0738	0.858231
$924$	0	0
$925$	−3.50701	−0.115310
$926$	−13.2534	−0.435534
$927$	0	0
$928$	0.173771	0.00570430
$929$	−29.3066	−0.961518	−0.480759	−	0.876853i	$-0.659639\pi$
$929$	−29.3066	−0.961518	−0.480759	+	0.876853i	$0.659639\pi$
$930$	0	0
$931$	−19.5122	−0.639485
$932$	2.38342	0.0780715
$933$	0	0
$934$	4.32939	0.141662
$935$	−81.6579	−2.67050
$936$	0	0
$937$	−34.0522	−1.11244	−0.556218	−	0.831036i	$-0.687748\pi$
$937$	−34.0522	−1.11244	−0.556218	+	0.831036i	$0.687748\pi$
$938$	15.1336	0.494131
$939$	0	0
$940$	−0.904702	−0.0295081
$941$	−27.4807	−0.895844	−0.447922	−	0.894073i	$-0.647836\pi$
$941$	−27.4807	−0.895844	−0.447922	+	0.894073i	$0.647836\pi$
$942$	0	0
$943$	8.24887	0.268620
$944$	7.41214	0.241245
$945$	0	0
$946$	13.5695	0.441181
$947$	17.6186	0.572528	0.286264	−	0.958151i	$-0.407587\pi$
$947$	17.6186	0.572528	0.286264	+	0.958151i	$0.407587\pi$
$948$	0	0
$949$	5.91826	0.192115
$950$	−64.2188	−2.08353
$951$	0	0
$952$	−35.6731	−1.15617
$953$	−10.9295	−0.354042	−0.177021	−	0.984207i	$-0.556646\pi$
$953$	−10.9295	−0.354042	−0.177021	+	0.984207i	$0.556646\pi$
$954$	0	0
$955$	55.7371	1.80361
$956$	3.70954	0.119975
$957$	0	0
$958$	−37.8111	−1.22162
$959$	22.1314	0.714660
$960$	0	0
$961$	−15.4816	−0.499406
$962$	0.939917	0.0303042
$963$	0	0
$964$	−0.706591	−0.0227578
$965$	−80.9260	−2.60510
$966$	0	0
$967$	30.3436	0.975786	0.487893	−	0.872903i	$-0.337766\pi$
$967$	30.3436	0.975786	0.487893	+	0.872903i	$0.337766\pi$
$968$	−4.75982	−0.152986
$969$	0	0
$970$	−81.0454	−2.60221
$971$	−33.6807	−1.08087	−0.540433	−	0.841387i	$-0.681739\pi$
$971$	−33.6807	−1.08087	−0.540433	+	0.841387i	$0.681739\pi$
$972$	0	0
$973$	−16.9666	−0.543925
$974$	−14.3209	−0.458872
$975$	0	0
$976$	−47.4465	−1.51873
$977$	−6.15723	−0.196987	−0.0984936	−	0.995138i	$-0.531402\pi$
$977$	−6.15723	−0.196987	−0.0984936	+	0.995138i	$0.531402\pi$
$978$	0	0
$979$	21.7861	0.696287
$980$	2.04359	0.0652801
$981$	0	0
$982$	−8.09309	−0.258261
$983$	11.0363	0.352002	0.176001	−	0.984390i	$-0.443684\pi$
$983$	11.0363	0.352002	0.176001	+	0.984390i	$0.443684\pi$
$984$	0	0
$985$	68.8670	2.19429
$986$	−2.16562	−0.0689672
$987$	0	0
$988$	−1.25431	−0.0399050
$989$	−6.18557	−0.196690
$990$	0	0
$991$	10.8461	0.344537	0.172269	−	0.985050i	$-0.444890\pi$
$991$	10.8461	0.344537	0.172269	+	0.985050i	$0.444890\pi$
$992$	3.02204	0.0959498
$993$	0	0
$994$	−33.2904	−1.05591
$995$	−92.9683	−2.94729
$996$	0	0
$997$	−19.3962	−0.614282	−0.307141	−	0.951664i	$-0.599372\pi$
$997$	−19.3962	−0.614282	−0.307141	+	0.951664i	$0.599372\pi$
$998$	52.0431	1.64740
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6561.2.a.d.1.20		72
3.2	odd	2		6561.2.a.c.1.53		72
81.4	even	27		243.2.g.a.208.2		144
81.7	even	27		729.2.g.b.352.7		144
81.20	odd	54		81.2.g.a.76.7	yes	144
81.23	odd	54		729.2.g.c.379.2		144
81.31	even	27		729.2.g.a.622.7		144
81.34	even	27		729.2.g.a.109.7		144
81.47	odd	54		729.2.g.d.109.2		144
81.50	odd	54		729.2.g.d.622.2		144
81.58	even	27		729.2.g.b.379.7		144
81.61	even	27		243.2.g.a.118.2		144
81.74	odd	54		729.2.g.c.352.2		144
81.77	odd	54		81.2.g.a.16.7	✓	144

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
81.2.g.a.16.7	✓	144	81.77	odd	54
81.2.g.a.76.7	yes	144	81.20	odd	54
243.2.g.a.118.2		144	81.61	even	27
243.2.g.a.208.2		144	81.4	even	27
729.2.g.a.109.7		144	81.34	even	27
729.2.g.a.622.7		144	81.31	even	27
729.2.g.b.352.7		144	81.7	even	27
729.2.g.b.379.7		144	81.58	even	27
729.2.g.c.352.2		144	81.74	odd	54
729.2.g.c.379.2		144	81.23	odd	54
729.2.g.d.109.2		144	81.47	odd	54
729.2.g.d.622.2		144	81.50	odd	54
6561.2.a.c.1.53		72	3.2	odd	2
6561.2.a.d.1.20		72	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 6561 = 3^{8} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6561.a (trivial)

\(f(q)\)	\(=\)	\(q-1.36534 q^{2} -0.135854 q^{4} +3.81012 q^{5} +1.74698 q^{7} +2.91616 q^{8} +O(q^{10})\)	\(q-1.36534 q^{2} -0.135854 q^{4} +3.81012 q^{5} +1.74698 q^{7} +2.91616 q^{8} -5.20210 q^{10} +3.06068 q^{11} +1.86816 q^{13} -2.38522 q^{14} -3.70984 q^{16} -7.00231 q^{17} +4.94221 q^{19} -0.517619 q^{20} -4.17886 q^{22} +1.90491 q^{23} +9.51701 q^{25} -2.55067 q^{26} -0.237334 q^{28} -0.226517 q^{29} -3.93934 q^{31} -0.767143 q^{32} +9.56051 q^{34} +6.65620 q^{35} -0.368499 q^{37} -6.74779 q^{38} +11.1109 q^{40} +4.33031 q^{41} -3.24717 q^{43} -0.415805 q^{44} -2.60085 q^{46} +1.74782 q^{47} -3.94806 q^{49} -12.9939 q^{50} -0.253796 q^{52} -6.06284 q^{53} +11.6616 q^{55} +5.09447 q^{56} +0.309272 q^{58} -1.99797 q^{59} +12.7894 q^{61} +5.37853 q^{62} +8.46708 q^{64} +7.11791 q^{65} -6.34476 q^{67} +0.951289 q^{68} -9.08796 q^{70} +13.9570 q^{71} +3.16797 q^{73} +0.503125 q^{74} -0.671418 q^{76} +5.34695 q^{77} -2.62141 q^{79} -14.1349 q^{80} -5.91234 q^{82} -0.224829 q^{83} -26.6796 q^{85} +4.43348 q^{86} +8.92544 q^{88} +7.11806 q^{89} +3.26363 q^{91} -0.258790 q^{92} -2.38636 q^{94} +18.8304 q^{95} +15.5794 q^{97} +5.39044 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 72 q + 9 q^{2} + 63 q^{4} + 18 q^{5} + 27 q^{8}+O(q^{10}) \) 72 * q + 9 * q^2 + 63 * q^4 + 18 * q^5 + 27 * q^8	\( 72 q + 9 q^{2} + 63 q^{4} + 18 q^{5} + 27 q^{8} + 36 q^{11} + 36 q^{14} + 45 q^{16} + 36 q^{17} + 54 q^{20} + 54 q^{23} + 36 q^{25} + 45 q^{26} + 9 q^{28} + 54 q^{29} + 63 q^{32} + 72 q^{35} + 54 q^{38} + 72 q^{41} + 90 q^{44} + 90 q^{47} + 18 q^{49} + 45 q^{50} + 45 q^{53} + 9 q^{55} + 108 q^{56} + 18 q^{58} + 108 q^{59} + 72 q^{62} + 9 q^{64} + 72 q^{65} + 108 q^{68} + 126 q^{71} + 90 q^{74} + 72 q^{77} + 144 q^{80} - 18 q^{82} + 108 q^{83} + 90 q^{86} + 108 q^{89} + 72 q^{92} + 144 q^{95} + 81 q^{98}+O(q^{100}) \) 72 * q + 9 * q^2 + 63 * q^4 + 18 * q^5 + 27 * q^8 + 36 * q^11 + 36 * q^14 + 45 * q^16 + 36 * q^17 + 54 * q^20 + 54 * q^23 + 36 * q^25 + 45 * q^26 + 9 * q^28 + 54 * q^29 + 63 * q^32 + 72 * q^35 + 54 * q^38 + 72 * q^41 + 90 * q^44 + 90 * q^47 + 18 * q^49 + 45 * q^50 + 45 * q^53 + 9 * q^55 + 108 * q^56 + 18 * q^58 + 108 * q^59 + 72 * q^62 + 9 * q^64 + 72 * q^65 + 108 * q^68 + 126 * q^71 + 90 * q^74 + 72 * q^77 + 144 * q^80 - 18 * q^82 + 108 * q^83 + 90 * q^86 + 108 * q^89 + 72 * q^92 + 144 * q^95 + 81 * q^98

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