LMFDB - Embedded newform 6561.2.a.d.1.24

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [6561,2,Mod(1,6561)] 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("6561.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage: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")

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

Newform invariants

comment:select newform

sage:traces = [72,9] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(2)] == traces)

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			1.24
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)

$f(q)$	$=$	\(q-0.833012 q^{2} -1.30609 q^{4} +1.20615 q^{5} -1.31489 q^{7} +2.75401 q^{8} -1.00474 q^{10} -2.17338 q^{11} +4.55495 q^{13} +1.09532 q^{14} +0.318052 q^{16} +3.76950 q^{17} +2.66594 q^{19} -1.57535 q^{20} +1.81045 q^{22} +6.03331 q^{23} -3.54519 q^{25} -3.79433 q^{26} +1.71737 q^{28} +4.15782 q^{29} +7.45716 q^{31} -5.77297 q^{32} -3.14004 q^{34} -1.58596 q^{35} +9.76208 q^{37} -2.22077 q^{38} +3.32176 q^{40} -5.78809 q^{41} -5.05513 q^{43} +2.83863 q^{44} -5.02582 q^{46} +2.99794 q^{47} -5.27105 q^{49} +2.95319 q^{50} -5.94918 q^{52} -12.4597 q^{53} -2.62143 q^{55} -3.62124 q^{56} -3.46352 q^{58} +10.9078 q^{59} +11.7931 q^{61} -6.21191 q^{62} +4.17285 q^{64} +5.49397 q^{65} -0.830108 q^{67} -4.92330 q^{68} +1.32113 q^{70} -7.25123 q^{71} -8.12021 q^{73} -8.13193 q^{74} -3.48196 q^{76} +2.85776 q^{77} -11.4996 q^{79} +0.383620 q^{80} +4.82155 q^{82} +4.77483 q^{83} +4.54659 q^{85} +4.21099 q^{86} -5.98551 q^{88} -3.33221 q^{89} -5.98928 q^{91} -7.88004 q^{92} -2.49732 q^{94} +3.21554 q^{95} +1.36325 q^{97} +4.39085 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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}+ \cdots + 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

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 $

$2$	−0.833012	−0.589029	−0.294514	−	0.955647i	$-0.595158\pi$
$2$	−0.833012	−0.589029	−0.294514	+	0.955647i	$0.595158\pi$
$3$	0	0
$3$	0	0
$4$	−1.30609	−0.653045
$5$	1.20615	0.539408	0.269704	−	0.962943i	$-0.413074\pi$
$5$	1.20615	0.539408	0.269704	+	0.962943i	$0.413074\pi$
$6$	0	0
$7$	−1.31489	−0.496983	−0.248492	−	0.968634i	$-0.579935\pi$
$7$	−1.31489	−0.496983	−0.248492	+	0.968634i	$0.579935\pi$
$8$	2.75401	0.973691
$9$	0	0
$10$	−1.00474	−0.317727
$11$	−2.17338	−0.655298	−0.327649	−	0.944800i	$-0.606256\pi$
$11$	−2.17338	−0.655298	−0.327649	+	0.944800i	$0.606256\pi$
$12$	0	0
$13$	4.55495	1.26332	0.631658	−	0.775247i	$-0.282375\pi$
$13$	4.55495	1.26332	0.631658	+	0.775247i	$0.282375\pi$
$14$	1.09532	0.292738
$15$	0	0
$16$	0.318052	0.0795131
$17$	3.76950	0.914237	0.457119	−	0.889406i	$-0.348882\pi$
$17$	3.76950	0.914237	0.457119	+	0.889406i	$0.348882\pi$
$18$	0	0
$19$	2.66594	0.611610	0.305805	−	0.952094i	$-0.401074\pi$
$19$	2.66594	0.611610	0.305805	+	0.952094i	$0.401074\pi$
$20$	−1.57535	−0.352258
$21$	0	0
$22$	1.81045	0.385989
$23$	6.03331	1.25803	0.629016	−	0.777393i	$-0.283458\pi$
$23$	6.03331	1.25803	0.629016	+	0.777393i	$0.283458\pi$
$24$	0	0
$25$	−3.54519	−0.709039
$26$	−3.79433	−0.744130
$27$	0	0
$28$	1.71737	0.324553
$29$	4.15782	0.772088	0.386044	−	0.922480i	$-0.373841\pi$
$29$	4.15782	0.772088	0.386044	+	0.922480i	$0.373841\pi$
$30$	0	0
$31$	7.45716	1.33935	0.669673	−	0.742656i	$-0.266434\pi$
$31$	7.45716	1.33935	0.669673	+	0.742656i	$0.266434\pi$
$32$	−5.77297	−1.02053
$33$	0	0
$34$	−3.14004	−0.538512
$35$	−1.58596	−0.268077
$36$	0	0
$37$	9.76208	1.60488	0.802438	−	0.596736i	$-0.203536\pi$
$37$	9.76208	1.60488	0.802438	+	0.596736i	$0.203536\pi$
$38$	−2.22077	−0.360256
$39$	0	0
$40$	3.32176	0.525217
$41$	−5.78809	−0.903947	−0.451974	−	0.892031i	$-0.649280\pi$
$41$	−5.78809	−0.903947	−0.451974	+	0.892031i	$0.649280\pi$
$42$	0	0
$43$	−5.05513	−0.770900	−0.385450	−	0.922729i	$-0.625954\pi$
$43$	−5.05513	−0.770900	−0.385450	+	0.922729i	$0.625954\pi$
$44$	2.83863	0.427939
$45$	0	0
$46$	−5.02582	−0.741017
$47$	2.99794	0.437295	0.218648	−	0.975804i	$-0.429835\pi$
$47$	2.99794	0.437295	0.218648	+	0.975804i	$0.429835\pi$
$48$	0	0
$49$	−5.27105	−0.753007
$50$	2.95319	0.417644
$51$	0	0
$52$	−5.94918	−0.825003
$53$	−12.4597	−1.71148	−0.855739	−	0.517408i	$-0.826897\pi$
$53$	−12.4597	−1.71148	−0.855739	+	0.517408i	$0.826897\pi$
$54$	0	0
$55$	−2.62143	−0.353473
$56$	−3.62124	−0.483908
$57$	0	0
$58$	−3.46352	−0.454782
$59$	10.9078	1.42008	0.710040	−	0.704161i	$-0.248677\pi$
$59$	10.9078	1.42008	0.710040	+	0.704161i	$0.248677\pi$
$60$	0	0
$61$	11.7931	1.50996	0.754978	−	0.655750i	$-0.227647\pi$
$61$	11.7931	1.50996	0.754978	+	0.655750i	$0.227647\pi$
$62$	−6.21191	−0.788913
$63$	0	0
$64$	4.17285	0.521606
$65$	5.49397	0.681443
$66$	0	0
$67$	−0.830108	−0.101414	−0.0507069	−	0.998714i	$-0.516147\pi$
$67$	−0.830108	−0.101414	−0.0507069	+	0.998714i	$0.516147\pi$
$68$	−4.92330	−0.597038
$69$	0	0
$70$	1.32113	0.157905
$71$	−7.25123	−0.860563	−0.430281	−	0.902695i	$-0.641586\pi$
$71$	−7.25123	−0.860563	−0.430281	+	0.902695i	$0.641586\pi$
$72$	0	0
$73$	−8.12021	−0.950399	−0.475199	−	0.879878i	$-0.657624\pi$
$73$	−8.12021	−0.950399	−0.475199	+	0.879878i	$0.657624\pi$
$74$	−8.13193	−0.945318
$75$	0	0
$76$	−3.48196	−0.399409
$77$	2.85776	0.325672
$78$	0	0
$79$	−11.4996	−1.29380	−0.646901	−	0.762574i	$-0.723935\pi$
$79$	−11.4996	−1.29380	−0.646901	+	0.762574i	$0.723935\pi$
$80$	0.383620	0.0428900
$81$	0	0
$82$	4.82155	0.532451
$83$	4.77483	0.524106	0.262053	−	0.965053i	$-0.415600\pi$
$83$	4.77483	0.524106	0.262053	+	0.965053i	$0.415600\pi$
$84$	0	0
$85$	4.54659	0.493147
$86$	4.21099	0.454082
$87$	0	0
$88$	−5.98551	−0.638058
$89$	−3.33221	−0.353213	−0.176607	−	0.984282i	$-0.556512\pi$
$89$	−3.33221	−0.353213	−0.176607	+	0.984282i	$0.556512\pi$
$90$	0	0
$91$	−5.98928	−0.627847
$92$	−7.88004	−0.821551
$93$	0	0
$94$	−2.49732	−0.257579
$95$	3.21554	0.329907
$96$	0	0
$97$	1.36325	0.138417	0.0692087	−	0.997602i	$-0.477953\pi$
$97$	1.36325	0.138417	0.0692087	+	0.997602i	$0.477953\pi$
$98$	4.39085	0.443543
$99$	0	0
$100$	4.63034	0.463034
$101$	−2.31484	−0.230335	−0.115168	−	0.993346i	$-0.536741\pi$
$101$	−2.31484	−0.230335	−0.115168	+	0.993346i	$0.536741\pi$
$102$	0	0
$103$	4.54956	0.448281	0.224141	−	0.974557i	$-0.428042\pi$
$103$	4.54956	0.448281	0.224141	+	0.974557i	$0.428042\pi$
$104$	12.5444	1.23008
$105$	0	0
$106$	10.3791	1.00811
$107$	12.2933	1.18844	0.594219	−	0.804303i	$-0.297461\pi$
$107$	12.2933	1.18844	0.594219	+	0.804303i	$0.297461\pi$
$108$	0	0
$109$	−8.90655	−0.853093	−0.426546	−	0.904466i	$-0.640270\pi$
$109$	−8.90655	−0.853093	−0.426546	+	0.904466i	$0.640270\pi$
$110$	2.18368	0.208206
$111$	0	0
$112$	−0.418205	−0.0395167
$113$	17.3907	1.63598	0.817991	−	0.575231i	$-0.195088\pi$
$113$	17.3907	1.63598	0.817991	+	0.575231i	$0.195088\pi$
$114$	0	0
$115$	7.27709	0.678593
$116$	−5.43049	−0.504208
$117$	0	0
$118$	−9.08637	−0.836468
$119$	−4.95649	−0.454361
$120$	0	0
$121$	−6.27643	−0.570585
$122$	−9.82383	−0.889408
$123$	0	0
$124$	−9.73972	−0.874653
$125$	−10.3068	−0.921870
$126$	0	0
$127$	5.68185	0.504182	0.252091	−	0.967703i	$-0.418882\pi$
$127$	5.68185	0.504182	0.252091	+	0.967703i	$0.418882\pi$
$128$	8.06990	0.713285
$129$	0	0
$130$	−4.57655	−0.401390
$131$	19.5185	1.70534	0.852668	−	0.522452i	$-0.174983\pi$
$131$	19.5185	1.70534	0.852668	+	0.522452i	$0.174983\pi$
$132$	0	0
$133$	−3.50544	−0.303960
$134$	0.691490	0.0597356
$135$	0	0
$136$	10.3812	0.890185
$137$	−15.8542	−1.35452	−0.677259	−	0.735745i	$-0.736832\pi$
$137$	−15.8542	−1.35452	−0.677259	+	0.735745i	$0.736832\pi$
$138$	0	0
$139$	−10.4392	−0.885444	−0.442722	−	0.896659i	$-0.645987\pi$
$139$	−10.4392	−0.885444	−0.442722	+	0.896659i	$0.645987\pi$
$140$	2.07141	0.175066
$141$	0	0
$142$	6.04036	0.506896
$143$	−9.89963	−0.827849
$144$	0	0
$145$	5.01497	0.416471
$146$	6.76424	0.559812
$147$	0	0
$148$	−12.7502	−1.04806
$149$	2.91511	0.238815	0.119408	−	0.992845i	$-0.461900\pi$
$149$	2.91511	0.238815	0.119408	+	0.992845i	$0.461900\pi$
$150$	0	0
$151$	−12.7598	−1.03838	−0.519190	−	0.854659i	$-0.673766\pi$
$151$	−12.7598	−1.03838	−0.519190	+	0.854659i	$0.673766\pi$
$152$	7.34205	0.595519
$153$	0	0
$154$	−2.38055	−0.191830
$155$	8.99448	0.722454
$156$	0	0
$157$	−1.27064	−0.101408	−0.0507039	−	0.998714i	$-0.516146\pi$
$157$	−1.27064	−0.101408	−0.0507039	+	0.998714i	$0.516146\pi$
$158$	9.57927	0.762086
$159$	0	0
$160$	−6.96309	−0.550481
$161$	−7.93316	−0.625221
$162$	0	0
$163$	12.3636	0.968391	0.484195	−	0.874960i	$-0.339112\pi$
$163$	12.3636	0.968391	0.484195	+	0.874960i	$0.339112\pi$
$164$	7.55976	0.590318
$165$	0	0
$166$	−3.97749	−0.308713
$167$	−3.81354	−0.295101	−0.147550	−	0.989055i	$-0.547139\pi$
$167$	−3.81354	−0.295101	−0.147550	+	0.989055i	$0.547139\pi$
$168$	0	0
$169$	7.74759	0.595969
$170$	−3.78737	−0.290478
$171$	0	0
$172$	6.60246	0.503433
$173$	24.0701	1.83002	0.915009	−	0.403432i	$-0.132183\pi$
$173$	24.0701	1.83002	0.915009	+	0.403432i	$0.132183\pi$
$174$	0	0
$175$	4.66156	0.352380
$176$	−0.691248	−0.0521048
$177$	0	0
$178$	2.77577	0.208053
$179$	6.33921	0.473815	0.236907	−	0.971532i	$-0.423866\pi$
$179$	6.33921	0.473815	0.236907	+	0.971532i	$0.423866\pi$
$180$	0	0
$181$	6.19367	0.460372	0.230186	−	0.973147i	$-0.426067\pi$
$181$	6.19367	0.460372	0.230186	+	0.973147i	$0.426067\pi$
$182$	4.98915	0.369820
$183$	0	0
$184$	16.6158	1.22493
$185$	11.7746	0.865683
$186$	0	0
$187$	−8.19254	−0.599098
$188$	−3.91559	−0.285573
$189$	0	0
$190$	−2.67858	−0.194325
$191$	−6.75265	−0.488605	−0.244302	−	0.969699i	$-0.578559\pi$
$191$	−6.75265	−0.488605	−0.244302	+	0.969699i	$0.578559\pi$
$192$	0	0
$193$	−15.9116	−1.14534	−0.572669	−	0.819786i	$-0.694092\pi$
$193$	−15.9116	−1.14534	−0.572669	+	0.819786i	$0.694092\pi$
$194$	−1.13561	−0.0815318
$195$	0	0
$196$	6.88447	0.491748
$197$	11.5923	0.825919	0.412959	−	0.910749i	$-0.364495\pi$
$197$	11.5923	0.825919	0.412959	+	0.910749i	$0.364495\pi$
$198$	0	0
$199$	4.29982	0.304806	0.152403	−	0.988318i	$-0.451299\pi$
$199$	4.29982	0.304806	0.152403	+	0.988318i	$0.451299\pi$
$200$	−9.76351	−0.690385
$201$	0	0
$202$	1.92829	0.135674
$203$	−5.46710	−0.383715
$204$	0	0
$205$	−6.98132	−0.487597
$206$	−3.78984	−0.264051
$207$	0	0
$208$	1.44871	0.100450
$209$	−5.79410	−0.400787
$210$	0	0
$211$	14.2812	0.983161	0.491581	−	0.870832i	$-0.336419\pi$
$211$	14.2812	0.983161	0.491581	+	0.870832i	$0.336419\pi$
$212$	16.2736	1.11767
$213$	0	0
$214$	−10.2405	−0.700024
$215$	−6.09726	−0.415830
$216$	0	0
$217$	−9.80538	−0.665632
$218$	7.41927	0.502496
$219$	0	0
$220$	3.42382	0.230834
$221$	17.1699	1.15497
$222$	0	0
$223$	−20.1902	−1.35204	−0.676019	−	0.736884i	$-0.736296\pi$
$223$	−20.1902	−1.35204	−0.676019	+	0.736884i	$0.736296\pi$
$224$	7.59085	0.507185
$225$	0	0
$226$	−14.4867	−0.963641
$227$	−2.72344	−0.180761	−0.0903805	−	0.995907i	$-0.528808\pi$
$227$	−2.72344	−0.180761	−0.0903805	+	0.995907i	$0.528808\pi$
$228$	0	0
$229$	1.52656	0.100878	0.0504388	−	0.998727i	$-0.483938\pi$
$229$	1.52656	0.100878	0.0504388	+	0.998727i	$0.483938\pi$
$230$	−6.06191	−0.399710
$231$	0	0
$232$	11.4507	0.751775
$233$	6.52741	0.427625	0.213812	−	0.976875i	$-0.431412\pi$
$233$	6.52741	0.427625	0.213812	+	0.976875i	$0.431412\pi$
$234$	0	0
$235$	3.61598	0.235881
$236$	−14.2466	−0.927377
$237$	0	0
$238$	4.12882	0.267632
$239$	16.8976	1.09301	0.546507	−	0.837455i	$-0.315957\pi$
$239$	16.8976	1.09301	0.546507	+	0.837455i	$0.315957\pi$
$240$	0	0
$241$	−12.5989	−0.811569	−0.405784	−	0.913969i	$-0.633002\pi$
$241$	−12.5989	−0.811569	−0.405784	+	0.913969i	$0.633002\pi$
$242$	5.22835	0.336091
$243$	0	0
$244$	−15.4029	−0.986070
$245$	−6.35770	−0.406178
$246$	0	0
$247$	12.1433	0.772657
$248$	20.5371	1.30411
$249$	0	0
$250$	8.58571	0.543008
$251$	25.8023	1.62863	0.814313	−	0.580426i	$-0.197114\pi$
$251$	25.8023	1.62863	0.814313	+	0.580426i	$0.197114\pi$
$252$	0	0
$253$	−13.1127	−0.824385
$254$	−4.73305	−0.296978
$255$	0	0
$256$	−15.0680	−0.941752
$257$	−14.4723	−0.902757	−0.451378	−	0.892333i	$-0.649068\pi$
$257$	−14.4723	−0.902757	−0.451378	+	0.892333i	$0.649068\pi$
$258$	0	0
$259$	−12.8361	−0.797597
$260$	−7.17562	−0.445013
$261$	0	0
$262$	−16.2591	−1.00449
$263$	10.4546	0.644658	0.322329	−	0.946628i	$-0.395534\pi$
$263$	10.4546	0.644658	0.322329	+	0.946628i	$0.395534\pi$
$264$	0	0
$265$	−15.0284	−0.923185
$266$	2.92007	0.179041
$267$	0	0
$268$	1.08420	0.0662278
$269$	0.210748	0.0128495	0.00642476	−	0.999979i	$-0.497955\pi$
$269$	0.210748	0.0128495	0.00642476	+	0.999979i	$0.497955\pi$
$270$	0	0
$271$	−11.4169	−0.693529	−0.346765	−	0.937952i	$-0.612720\pi$
$271$	−11.4169	−0.693529	−0.346765	+	0.937952i	$0.612720\pi$
$272$	1.19890	0.0726938
$273$	0	0
$274$	13.2068	0.797850
$275$	7.70504	0.464632
$276$	0	0
$277$	−11.1494	−0.669905	−0.334952	−	0.942235i	$-0.608720\pi$
$277$	−11.1494	−0.669905	−0.334952	+	0.942235i	$0.608720\pi$
$278$	8.69601	0.521552
$279$	0	0
$280$	−4.36777	−0.261024
$281$	21.7032	1.29470	0.647351	−	0.762192i	$-0.275877\pi$
$281$	21.7032	1.29470	0.647351	+	0.762192i	$0.275877\pi$
$282$	0	0
$283$	−0.296522	−0.0176264	−0.00881321	−	0.999961i	$-0.502805\pi$
$283$	−0.296522	−0.0176264	−0.00881321	+	0.999961i	$0.502805\pi$
$284$	9.47076	0.561986
$285$	0	0
$286$	8.24652	0.487627
$287$	7.61072	0.449247
$288$	0	0
$289$	−2.79090	−0.164171
$290$	−4.17753	−0.245313
$291$	0	0
$292$	10.6057	0.620653
$293$	14.5066	0.847485	0.423743	−	0.905783i	$-0.360716\pi$
$293$	14.5066	0.847485	0.423743	+	0.905783i	$0.360716\pi$
$294$	0	0
$295$	13.1565	0.766003
$296$	26.8849	1.56265
$297$	0	0
$298$	−2.42833	−0.140669
$299$	27.4814	1.58929
$300$	0	0
$301$	6.64696	0.383125
$302$	10.6291	0.611636
$303$	0	0
$304$	0.847910	0.0486310
$305$	14.2243	0.814483
$306$	0	0
$307$	3.78902	0.216251	0.108126	−	0.994137i	$-0.465515\pi$
$307$	3.78902	0.216251	0.108126	+	0.994137i	$0.465515\pi$
$308$	−3.73250	−0.212679
$309$	0	0
$310$	−7.49251	−0.425546
$311$	−12.3295	−0.699143	−0.349572	−	0.936910i	$-0.613673\pi$
$311$	−12.3295	−0.699143	−0.349572	+	0.936910i	$0.613673\pi$
$312$	0	0
$313$	6.80582	0.384688	0.192344	−	0.981328i	$-0.438391\pi$
$313$	6.80582	0.384688	0.192344	+	0.981328i	$0.438391\pi$
$314$	1.05846	0.0597321
$315$	0	0
$316$	15.0195	0.844911
$317$	14.9365	0.838919	0.419459	−	0.907774i	$-0.362220\pi$
$317$	14.9365	0.838919	0.419459	+	0.907774i	$0.362220\pi$
$318$	0	0
$319$	−9.03651	−0.505948
$320$	5.03310	0.281359
$321$	0	0
$322$	6.60842	0.368273
$323$	10.0493	0.559156
$324$	0	0
$325$	−16.1482	−0.895740
$326$	−10.2990	−0.570410
$327$	0	0
$328$	−15.9405	−0.880165
$329$	−3.94198	−0.217328
$330$	0	0
$331$	−3.69381	−0.203030	−0.101515	−	0.994834i	$-0.532369\pi$
$331$	−3.69381	−0.203030	−0.101515	+	0.994834i	$0.532369\pi$
$332$	−6.23636	−0.342265
$333$	0	0
$334$	3.17673	0.173823
$335$	−1.00124	−0.0547034
$336$	0	0
$337$	18.4446	1.00474	0.502371	−	0.864652i	$-0.332461\pi$
$337$	18.4446	1.00474	0.502371	+	0.864652i	$0.332461\pi$
$338$	−6.45384	−0.351043
$339$	0	0
$340$	−5.93826	−0.322047
$341$	−16.2072	−0.877670
$342$	0	0
$343$	16.1351	0.871216
$344$	−13.9219	−0.750619
$345$	0	0
$346$	−20.0507	−1.07793
$347$	33.5960	1.80353	0.901764	−	0.432228i	$-0.142272\pi$
$347$	33.5960	1.80353	0.901764	+	0.432228i	$0.142272\pi$
$348$	0	0
$349$	−32.9376	−1.76311	−0.881555	−	0.472082i	$-0.843503\pi$
$349$	−32.9376	−1.76311	−0.881555	+	0.472082i	$0.843503\pi$
$350$	−3.88313	−0.207562
$351$	0	0
$352$	12.5468	0.668749
$353$	−13.3414	−0.710091	−0.355046	−	0.934849i	$-0.615535\pi$
$353$	−13.3414	−0.710091	−0.355046	+	0.934849i	$0.615535\pi$
$354$	0	0
$355$	−8.74610	−0.464195
$356$	4.35216	0.230664
$357$	0	0
$358$	−5.28064	−0.279090
$359$	−9.02629	−0.476390	−0.238195	−	0.971217i	$-0.576556\pi$
$359$	−9.02629	−0.476390	−0.238195	+	0.971217i	$0.576556\pi$
$360$	0	0
$361$	−11.8927	−0.625934
$362$	−5.15940	−0.271172
$363$	0	0
$364$	7.82254	0.410013
$365$	−9.79422	−0.512653
$366$	0	0
$367$	25.6039	1.33651	0.668257	−	0.743931i	$-0.267041\pi$
$367$	25.6039	1.33651	0.668257	+	0.743931i	$0.267041\pi$
$368$	1.91891	0.100030
$369$	0	0
$370$	−9.80836	−0.509912
$371$	16.3833	0.850576
$372$	0	0
$373$	−21.8677	−1.13227	−0.566133	−	0.824314i	$-0.691561\pi$
$373$	−21.8677	−1.13227	−0.566133	+	0.824314i	$0.691561\pi$
$374$	6.82449	0.352886
$375$	0	0
$376$	8.25638	0.425790
$377$	18.9387	0.975391
$378$	0	0
$379$	10.5343	0.541113	0.270556	−	0.962704i	$-0.412792\pi$
$379$	10.5343	0.541113	0.270556	+	0.962704i	$0.412792\pi$
$380$	−4.19978	−0.215444
$381$	0	0
$382$	5.62504	0.287802
$383$	−21.8686	−1.11743	−0.558716	−	0.829359i	$-0.688706\pi$
$383$	−21.8686	−1.11743	−0.558716	+	0.829359i	$0.688706\pi$
$384$	0	0
$385$	3.44690	0.175670
$386$	13.2545	0.674637
$387$	0	0
$388$	−1.78053	−0.0903928
$389$	−24.3906	−1.23665	−0.618327	−	0.785921i	$-0.712189\pi$
$389$	−24.3906	−1.23665	−0.618327	+	0.785921i	$0.712189\pi$
$390$	0	0
$391$	22.7425	1.15014
$392$	−14.5166	−0.733197
$393$	0	0
$394$	−9.65655	−0.486490
$395$	−13.8702	−0.697887
$396$	0	0
$397$	23.9984	1.20445	0.602223	−	0.798328i	$-0.294282\pi$
$397$	23.9984	1.20445	0.602223	+	0.798328i	$0.294282\pi$
$398$	−3.58180	−0.179539
$399$	0	0
$400$	−1.12756	−0.0563779
$401$	36.9137	1.84338	0.921692	−	0.387922i	$-0.126807\pi$
$401$	36.9137	1.84338	0.921692	+	0.387922i	$0.126807\pi$
$402$	0	0
$403$	33.9670	1.69202
$404$	3.02339	0.150419
$405$	0	0
$406$	4.55416	0.226019
$407$	−21.2167	−1.05167
$408$	0	0
$409$	−2.85521	−0.141181	−0.0705906	−	0.997505i	$-0.522488\pi$
$409$	−2.85521	−0.141181	−0.0705906	+	0.997505i	$0.522488\pi$
$410$	5.81553	0.287208
$411$	0	0
$412$	−5.94213	−0.292748
$413$	−14.3427	−0.705757
$414$	0	0
$415$	5.75918	0.282707
$416$	−26.2956	−1.28925
$417$	0	0
$418$	4.82656	0.236075
$419$	−34.3748	−1.67932	−0.839659	−	0.543113i	$-0.817245\pi$
$419$	−34.3748	−1.67932	−0.839659	+	0.543113i	$0.817245\pi$
$420$	0	0
$421$	6.25163	0.304686	0.152343	−	0.988328i	$-0.451318\pi$
$421$	6.25163	0.304686	0.152343	+	0.988328i	$0.451318\pi$
$422$	−11.8965	−0.579110
$423$	0	0
$424$	−34.3143	−1.66645
$425$	−13.3636	−0.648229
$426$	0	0
$427$	−15.5067	−0.750424
$428$	−16.0562	−0.776104
$429$	0	0
$430$	5.07910	0.244936
$431$	−3.01725	−0.145336	−0.0726679	−	0.997356i	$-0.523151\pi$
$431$	−3.01725	−0.145336	−0.0726679	+	0.997356i	$0.523151\pi$
$432$	0	0
$433$	30.7319	1.47688	0.738439	−	0.674320i	$-0.235563\pi$
$433$	30.7319	1.47688	0.738439	+	0.674320i	$0.235563\pi$
$434$	8.16800	0.392077
$435$	0	0
$436$	11.6328	0.557108
$437$	16.0845	0.769424
$438$	0	0
$439$	0.457630	0.0218415	0.0109207	−	0.999940i	$-0.496524\pi$
$439$	0.457630	0.0218415	0.0109207	+	0.999940i	$0.496524\pi$
$440$	−7.21945	−0.344174
$441$	0	0
$442$	−14.3027	−0.680311
$443$	34.9086	1.65856	0.829278	−	0.558836i	$-0.188752\pi$
$443$	34.9086	1.65856	0.829278	+	0.558836i	$0.188752\pi$
$444$	0	0
$445$	−4.01915	−0.190526
$446$	16.8187	0.796389
$447$	0	0
$448$	−5.48686	−0.259230
$449$	7.15619	0.337722	0.168861	−	0.985640i	$-0.445991\pi$
$449$	7.15619	0.337722	0.168861	+	0.985640i	$0.445991\pi$
$450$	0	0
$451$	12.5797	0.592355
$452$	−22.7139	−1.06837
$453$	0	0
$454$	2.26866	0.106473
$455$	−7.22400	−0.338666
$456$	0	0
$457$	−12.6824	−0.593258	−0.296629	−	0.954993i	$-0.595863\pi$
$457$	−12.6824	−0.593258	−0.296629	+	0.954993i	$0.595863\pi$
$458$	−1.27164	−0.0594198
$459$	0	0
$460$	−9.50454	−0.443152
$461$	−0.461815	−0.0215089	−0.0107544	−	0.999942i	$-0.503423\pi$
$461$	−0.461815	−0.0215089	−0.0107544	+	0.999942i	$0.503423\pi$
$462$	0	0
$463$	37.1009	1.72423	0.862113	−	0.506716i	$-0.169141\pi$
$463$	37.1009	1.72423	0.862113	+	0.506716i	$0.169141\pi$
$464$	1.32240	0.0613911
$465$	0	0
$466$	−5.43741	−0.251883
$467$	−27.5069	−1.27287	−0.636434	−	0.771331i	$-0.719591\pi$
$467$	−27.5069	−1.27287	−0.636434	+	0.771331i	$0.719591\pi$
$468$	0	0
$469$	1.09150	0.0504010
$470$	−3.01216	−0.138940
$471$	0	0
$472$	30.0404	1.38272
$473$	10.9867	0.505169
$474$	0	0
$475$	−9.45129	−0.433655
$476$	6.47362	0.296718
$477$	0	0
$478$	−14.0759	−0.643816
$479$	14.2605	0.651579	0.325789	−	0.945442i	$-0.394370\pi$
$479$	14.2605	0.651579	0.325789	+	0.945442i	$0.394370\pi$
$480$	0	0
$481$	44.4658	2.02747
$482$	10.4951	0.478037
$483$	0	0
$484$	8.19759	0.372618
$485$	1.64429	0.0746634
$486$	0	0
$487$	9.07752	0.411342	0.205671	−	0.978621i	$-0.434062\pi$
$487$	9.07752	0.411342	0.205671	+	0.978621i	$0.434062\pi$
$488$	32.4785	1.47023
$489$	0	0
$490$	5.29604	0.239251
$491$	27.8829	1.25834	0.629169	−	0.777268i	$-0.283395\pi$
$491$	27.8829	1.25834	0.629169	+	0.777268i	$0.283395\pi$
$492$	0	0
$493$	15.6729	0.705871
$494$	−10.1155	−0.455117
$495$	0	0
$496$	2.37177	0.106495
$497$	9.53460	0.427685
$498$	0	0
$499$	−23.7912	−1.06504	−0.532520	−	0.846417i	$-0.678755\pi$
$499$	−23.7912	−1.06504	−0.532520	+	0.846417i	$0.678755\pi$
$500$	13.4616	0.602022
$501$	0	0
$502$	−21.4936	−0.959308
$503$	8.05599	0.359199	0.179600	−	0.983740i	$-0.442520\pi$
$503$	8.05599	0.359199	0.179600	+	0.983740i	$0.442520\pi$
$504$	0	0
$505$	−2.79205	−0.124245
$506$	10.9230	0.485587
$507$	0	0
$508$	−7.42100	−0.329254
$509$	19.3048	0.855670	0.427835	−	0.903857i	$-0.359276\pi$
$509$	19.3048	0.855670	0.427835	+	0.903857i	$0.359276\pi$
$510$	0	0
$511$	10.6772	0.472333
$512$	−3.58795	−0.158566
$513$	0	0
$514$	12.0556	0.531750
$515$	5.48747	0.241807
$516$	0	0
$517$	−6.51566	−0.286559
$518$	10.6926	0.469807
$519$	0	0
$520$	15.1305	0.663515
$521$	−2.66895	−0.116929	−0.0584644	−	0.998289i	$-0.518620\pi$
$521$	−2.66895	−0.116929	−0.0584644	+	0.998289i	$0.518620\pi$
$522$	0	0
$523$	15.3637	0.671809	0.335905	−	0.941896i	$-0.390958\pi$
$523$	15.3637	0.671809	0.335905	+	0.941896i	$0.390958\pi$
$524$	−25.4929	−1.11366
$525$	0	0
$526$	−8.70881	−0.379722
$527$	28.1097	1.22448
$528$	0	0
$529$	13.4008	0.582643
$530$	12.5188	0.543783
$531$	0	0
$532$	4.57842	0.198500
$533$	−26.3645	−1.14197
$534$	0	0
$535$	14.8276	0.641053
$536$	−2.28613	−0.0987457
$537$	0	0
$538$	−0.175556	−0.00756874
$539$	11.4560	0.493444
$540$	0	0
$541$	−18.7884	−0.807777	−0.403889	−	0.914808i	$-0.632342\pi$
$541$	−18.7884	−0.807777	−0.403889	+	0.914808i	$0.632342\pi$
$542$	9.51044	0.408509
$543$	0	0
$544$	−21.7612	−0.933003
$545$	−10.7427	−0.460165
$546$	0	0
$547$	−11.4609	−0.490031	−0.245015	−	0.969519i	$-0.578793\pi$
$547$	−11.4609	−0.490031	−0.245015	+	0.969519i	$0.578793\pi$
$548$	20.7071	0.884561
$549$	0	0
$550$	−6.41840	−0.273681
$551$	11.0845	0.472216
$552$	0	0
$553$	15.1207	0.642998
$554$	9.28762	0.394593
$555$	0	0
$556$	13.6346	0.578235
$557$	−24.9737	−1.05817	−0.529086	−	0.848568i	$-0.677465\pi$
$557$	−24.9737	−1.05817	−0.529086	+	0.848568i	$0.677465\pi$
$558$	0	0
$559$	−23.0259	−0.973891
$560$	−0.504420	−0.0213156
$561$	0	0
$562$	−18.0790	−0.762616
$563$	17.5812	0.740960	0.370480	−	0.928840i	$-0.379193\pi$
$563$	17.5812	0.740960	0.370480	+	0.928840i	$0.379193\pi$
$564$	0	0
$565$	20.9759	0.882463
$566$	0.247007	0.0103825
$567$	0	0
$568$	−19.9700	−0.837922
$569$	−15.9891	−0.670298	−0.335149	−	0.942165i	$-0.608787\pi$
$569$	−15.9891	−0.670298	−0.335149	+	0.942165i	$0.608787\pi$
$570$	0	0
$571$	10.3301	0.432303	0.216151	−	0.976360i	$-0.430650\pi$
$571$	10.3301	0.432303	0.216151	+	0.976360i	$0.430650\pi$
$572$	12.9298	0.540623
$573$	0	0
$574$	−6.33983	−0.264619
$575$	−21.3892	−0.891993
$576$	0	0
$577$	39.1473	1.62972	0.814861	−	0.579656i	$-0.196813\pi$
$577$	39.1473	1.62972	0.814861	+	0.579656i	$0.196813\pi$
$578$	2.32485	0.0967012
$579$	0	0
$580$	−6.55000	−0.271974
$581$	−6.27840	−0.260472
$582$	0	0
$583$	27.0797	1.12153
$584$	−22.3632	−0.925395
$585$	0	0
$586$	−12.0842	−0.499193
$587$	17.8017	0.734756	0.367378	−	0.930072i	$-0.380256\pi$
$587$	17.8017	0.734756	0.367378	+	0.930072i	$0.380256\pi$
$588$	0	0
$589$	19.8804	0.819156
$590$	−10.9596	−0.451198
$591$	0	0
$592$	3.10485	0.127609
$593$	13.8537	0.568905	0.284452	−	0.958690i	$-0.408188\pi$
$593$	13.8537	0.568905	0.284452	+	0.958690i	$0.408188\pi$
$594$	0	0
$595$	−5.97829	−0.245086
$596$	−3.80740	−0.155957
$597$	0	0
$598$	−22.8924	−0.936139
$599$	−3.61907	−0.147871	−0.0739356	−	0.997263i	$-0.523556\pi$
$599$	−3.61907	−0.147871	−0.0739356	+	0.997263i	$0.523556\pi$
$600$	0	0
$601$	−10.3830	−0.423531	−0.211766	−	0.977321i	$-0.567921\pi$
$601$	−10.3830	−0.423531	−0.211766	+	0.977321i	$0.567921\pi$
$602$	−5.53700	−0.225671
$603$	0	0
$604$	16.6655	0.678109
$605$	−7.57034	−0.307778
$606$	0	0
$607$	13.4329	0.545223	0.272611	−	0.962124i	$-0.412113\pi$
$607$	13.4329	0.545223	0.272611	+	0.962124i	$0.412113\pi$
$608$	−15.3904	−0.624164
$609$	0	0
$610$	−11.8491	−0.479754
$611$	13.6555	0.552442
$612$	0	0
$613$	29.0651	1.17393	0.586964	−	0.809613i	$-0.300323\pi$
$613$	29.0651	1.17393	0.586964	+	0.809613i	$0.300323\pi$
$614$	−3.15630	−0.127378
$615$	0	0
$616$	7.87032	0.317104
$617$	−12.4304	−0.500427	−0.250214	−	0.968191i	$-0.580501\pi$
$617$	−12.4304	−0.500427	−0.250214	+	0.968191i	$0.580501\pi$
$618$	0	0
$619$	34.5654	1.38930	0.694651	−	0.719347i	$-0.255559\pi$
$619$	34.5654	1.38930	0.694651	+	0.719347i	$0.255559\pi$
$620$	−11.7476	−0.471795
$621$	0	0
$622$	10.2707	0.411816
$623$	4.38150	0.175541
$624$	0	0
$625$	5.29436	0.211775
$626$	−5.66934	−0.226592
$627$	0	0
$628$	1.65956	0.0662239
$629$	36.7981	1.46724
$630$	0	0
$631$	−13.2232	−0.526409	−0.263205	−	0.964740i	$-0.584779\pi$
$631$	−13.2232	−0.526409	−0.263205	+	0.964740i	$0.584779\pi$
$632$	−31.6699	−1.25976
$633$	0	0
$634$	−12.4423	−0.494147
$635$	6.85318	0.271960
$636$	0	0
$637$	−24.0094	−0.951287
$638$	7.52753	0.298018
$639$	0	0
$640$	9.73354	0.384752
$641$	5.27288	0.208266	0.104133	−	0.994563i	$-0.466793\pi$
$641$	5.27288	0.208266	0.104133	+	0.994563i	$0.466793\pi$
$642$	0	0
$643$	4.37661	0.172597	0.0862983	−	0.996269i	$-0.472496\pi$
$643$	4.37661	0.172597	0.0862983	+	0.996269i	$0.472496\pi$
$644$	10.3614	0.408297
$645$	0	0
$646$	−8.37117	−0.329359
$647$	−5.42624	−0.213327	−0.106664	−	0.994295i	$-0.534017\pi$
$647$	−5.42624	−0.213327	−0.106664	+	0.994295i	$0.534017\pi$
$648$	0	0
$649$	−23.7069	−0.930576
$650$	13.4516	0.527617
$651$	0	0
$652$	−16.1480	−0.632403
$653$	−11.1010	−0.434416	−0.217208	−	0.976125i	$-0.569695\pi$
$653$	−11.1010	−0.434416	−0.217208	+	0.976125i	$0.569695\pi$
$654$	0	0
$655$	23.5423	0.919873
$656$	−1.84091	−0.0718756
$657$	0	0
$658$	3.28372	0.128013
$659$	7.43530	0.289638	0.144819	−	0.989458i	$-0.453740\pi$
$659$	7.43530	0.289638	0.144819	+	0.989458i	$0.453740\pi$
$660$	0	0
$661$	−3.83162	−0.149033	−0.0745164	−	0.997220i	$-0.523741\pi$
$661$	−3.83162	−0.149033	−0.0745164	+	0.997220i	$0.523741\pi$
$662$	3.07699	0.119591
$663$	0	0
$664$	13.1500	0.510317
$665$	−4.22810	−0.163958
$666$	0	0
$667$	25.0854	0.971311
$668$	4.98083	0.192714
$669$	0	0
$670$	0.834043	0.0322219
$671$	−25.6309	−0.989472
$672$	0	0
$673$	42.0913	1.62250	0.811250	−	0.584700i	$-0.198788\pi$
$673$	42.0913	1.62250	0.811250	+	0.584700i	$0.198788\pi$
$674$	−15.3646	−0.591822
$675$	0	0
$676$	−10.1191	−0.389194
$677$	2.49239	0.0957902	0.0478951	−	0.998852i	$-0.484749\pi$
$677$	2.49239	0.0957902	0.0478951	+	0.998852i	$0.484749\pi$
$678$	0	0
$679$	−1.79253	−0.0687911
$680$	12.5214	0.480173
$681$	0	0
$682$	13.5008	0.516973
$683$	15.3269	0.586468	0.293234	−	0.956041i	$-0.405269\pi$
$683$	15.3269	0.586468	0.293234	+	0.956041i	$0.405269\pi$
$684$	0	0
$685$	−19.1226	−0.730638
$686$	−13.4408	−0.513171
$687$	0	0
$688$	−1.60780	−0.0612967
$689$	−56.7536	−2.16214
$690$	0	0
$691$	37.1091	1.41170	0.705848	−	0.708363i	$-0.250566\pi$
$691$	37.1091	1.41170	0.705848	+	0.708363i	$0.250566\pi$
$692$	−31.4378	−1.19508
$693$	0	0
$694$	−27.9859	−1.06233
$695$	−12.5913	−0.477616
$696$	0	0
$697$	−21.8182	−0.826422
$698$	27.4374	1.03852
$699$	0	0
$700$	−6.08841	−0.230120
$701$	−6.70977	−0.253425	−0.126712	−	0.991940i	$-0.540443\pi$
$701$	−6.70977	−0.253425	−0.126712	+	0.991940i	$0.540443\pi$
$702$	0	0
$703$	26.0252	0.981558
$704$	−9.06918	−0.341808
$705$	0	0
$706$	11.1136	0.418264
$707$	3.04377	0.114473
$708$	0	0
$709$	−23.7811	−0.893117	−0.446558	−	0.894755i	$-0.647350\pi$
$709$	−23.7811	−0.893117	−0.446558	+	0.894755i	$0.647350\pi$
$710$	7.28561	0.273424
$711$	0	0
$712$	−9.17694	−0.343921
$713$	44.9913	1.68494
$714$	0	0
$715$	−11.9405	−0.446548
$716$	−8.27958	−0.309422
$717$	0	0
$718$	7.51901	0.280607
$719$	42.6841	1.59185	0.795924	−	0.605396i	$-0.206985\pi$
$719$	42.6841	1.59185	0.795924	+	0.605396i	$0.206985\pi$
$720$	0	0
$721$	−5.98219	−0.222788
$722$	9.90680	0.368693
$723$	0	0
$724$	−8.08949	−0.300643
$725$	−14.7403	−0.547440
$726$	0	0
$727$	19.0558	0.706740	0.353370	−	0.935484i	$-0.385036\pi$
$727$	19.0558	0.706740	0.353370	+	0.935484i	$0.385036\pi$
$728$	−16.4946	−0.611329
$729$	0	0
$730$	8.15871	0.301967
$731$	−19.0553	−0.704785
$732$	0	0
$733$	−17.9632	−0.663488	−0.331744	−	0.943369i	$-0.607637\pi$
$733$	−17.9632	−0.663488	−0.331744	+	0.943369i	$0.607637\pi$
$734$	−21.3284	−0.787245
$735$	0	0
$736$	−34.8301	−1.28385
$737$	1.80414	0.0664562
$738$	0	0
$739$	12.4736	0.458849	0.229424	−	0.973327i	$-0.426316\pi$
$739$	12.4736	0.458849	0.229424	+	0.973327i	$0.426316\pi$
$740$	−15.3786	−0.565330
$741$	0	0
$742$	−13.6475	−0.501014
$743$	25.9170	0.950802	0.475401	−	0.879769i	$-0.342303\pi$
$743$	25.9170	0.950802	0.475401	+	0.879769i	$0.342303\pi$
$744$	0	0
$745$	3.51608	0.128819
$746$	18.2161	0.666937
$747$	0	0
$748$	10.7002	0.391238
$749$	−16.1644	−0.590634
$750$	0	0
$751$	10.4960	0.383003	0.191501	−	0.981492i	$-0.438664\pi$
$751$	10.4960	0.383003	0.191501	+	0.981492i	$0.438664\pi$
$752$	0.953503	0.0347707
$753$	0	0
$754$	−15.7762	−0.574534
$755$	−15.3903	−0.560111
$756$	0	0
$757$	0.172834	0.00628176	0.00314088	−	0.999995i	$-0.499000\pi$
$757$	0.172834	0.00628176	0.00314088	+	0.999995i	$0.499000\pi$
$758$	−8.77523	−0.318731
$759$	0	0
$760$	8.85564	0.321228
$761$	−1.87314	−0.0679014	−0.0339507	−	0.999424i	$-0.510809\pi$
$761$	−1.87314	−0.0679014	−0.0339507	+	0.999424i	$0.510809\pi$
$762$	0	0
$763$	11.7112	0.423973
$764$	8.81957	0.319081
$765$	0	0
$766$	18.2168	0.658200
$767$	49.6847	1.79401
$768$	0	0
$769$	−27.9100	−1.00646	−0.503230	−	0.864153i	$-0.667855\pi$
$769$	−27.9100	−1.00646	−0.503230	+	0.864153i	$0.667855\pi$
$770$	−2.87131	−0.103475
$771$	0	0
$772$	20.7819	0.747958
$773$	−0.0545862	−0.00196333	−0.000981664	−	1.00000i	$-0.500312\pi$
$773$	−0.0545862	−0.00196333	−0.000981664		1.00000i	$0.500312\pi$
$774$	0	0
$775$	−26.4371	−0.949648
$776$	3.75442	0.134776
$777$	0	0
$778$	20.3177	0.728425
$779$	−15.4307	−0.552863
$780$	0	0
$781$	15.7597	0.563925
$782$	−18.9448	−0.677465
$783$	0	0
$784$	−1.67647	−0.0598740
$785$	−1.53258	−0.0547002
$786$	0	0
$787$	−33.9196	−1.20910	−0.604551	−	0.796567i	$-0.706647\pi$
$787$	−33.9196	−1.20910	−0.604551	+	0.796567i	$0.706647\pi$
$788$	−15.1406	−0.539362
$789$	0	0
$790$	11.5541	0.411076
$791$	−22.8670	−0.813056
$792$	0	0
$793$	53.7172	1.90755
$794$	−19.9910	−0.709453
$795$	0	0
$796$	−5.61595	−0.199052
$797$	−14.5628	−0.515841	−0.257920	−	0.966166i	$-0.583037\pi$
$797$	−14.5628	−0.515841	−0.257920	+	0.966166i	$0.583037\pi$
$798$	0	0
$799$	11.3007	0.399791
$800$	20.4663	0.723593
$801$	0	0
$802$	−30.7496	−1.08581
$803$	17.6483	0.622794
$804$	0	0
$805$	−9.56861	−0.337249
$806$	−28.2949	−0.996647
$807$	0	0
$808$	−6.37511	−0.224275
$809$	−28.0189	−0.985092	−0.492546	−	0.870286i	$-0.663934\pi$
$809$	−28.0189	−0.985092	−0.492546	+	0.870286i	$0.663934\pi$
$810$	0	0
$811$	29.1924	1.02508	0.512542	−	0.858662i	$-0.328704\pi$
$811$	29.1924	1.02508	0.512542	+	0.858662i	$0.328704\pi$
$812$	7.14052	0.250583
$813$	0	0
$814$	17.6738	0.619465
$815$	14.9124	0.522358
$816$	0	0
$817$	−13.4767	−0.471490
$818$	2.37843	0.0831597
$819$	0	0
$820$	9.11824	0.318423
$821$	29.8559	1.04198	0.520989	−	0.853563i	$-0.325563\pi$
$821$	29.8559	1.04198	0.520989	+	0.853563i	$0.325563\pi$
$822$	0	0
$823$	−40.1919	−1.40100	−0.700502	−	0.713651i	$-0.747040\pi$
$823$	−40.1919	−1.40100	−0.700502	+	0.713651i	$0.747040\pi$
$824$	12.5295	0.436488
$825$	0	0
$826$	11.9476	0.415711
$827$	42.5136	1.47834	0.739172	−	0.673517i	$-0.235217\pi$
$827$	42.5136	1.47834	0.739172	+	0.673517i	$0.235217\pi$
$828$	0	0
$829$	−6.28502	−0.218288	−0.109144	−	0.994026i	$-0.534811\pi$
$829$	−6.28502	−0.218288	−0.109144	+	0.994026i	$0.534811\pi$
$830$	−4.79747	−0.166523
$831$	0	0
$832$	19.0071	0.658954
$833$	−19.8692	−0.688427
$834$	0	0
$835$	−4.59972	−0.159180
$836$	7.56762	0.261732
$837$	0	0
$838$	28.6346	0.989167
$839$	−16.5622	−0.571790	−0.285895	−	0.958261i	$-0.592291\pi$
$839$	−16.5622	−0.571790	−0.285895	+	0.958261i	$0.592291\pi$
$840$	0	0
$841$	−11.7125	−0.403880
$842$	−5.20768	−0.179469
$843$	0	0
$844$	−18.6526	−0.642049
$845$	9.34479	0.321470
$846$	0	0
$847$	8.25285	0.283571
$848$	−3.96285	−0.136085
$849$	0	0
$850$	11.1320	0.381826
$851$	58.8976	2.01898
$852$	0	0
$853$	−52.1207	−1.78458	−0.892289	−	0.451464i	$-0.850902\pi$
$853$	−52.1207	−1.78458	−0.892289	+	0.451464i	$0.850902\pi$
$854$	12.9173	0.442021
$855$	0	0
$856$	33.8559	1.15717
$857$	38.8823	1.32820	0.664098	−	0.747646i	$-0.268816\pi$
$857$	38.8823	1.32820	0.664098	+	0.747646i	$0.268816\pi$
$858$	0	0
$859$	−21.7038	−0.740523	−0.370261	−	0.928928i	$-0.620732\pi$
$859$	−21.7038	−0.740523	−0.370261	+	0.928928i	$0.620732\pi$
$860$	7.96358	0.271556
$861$	0	0
$862$	2.51341	0.0856069
$863$	−43.8809	−1.49372	−0.746861	−	0.664980i	$-0.768440\pi$
$863$	−43.8809	−1.49372	−0.746861	+	0.664980i	$0.768440\pi$
$864$	0	0
$865$	29.0323	0.987127
$866$	−25.6000	−0.869924
$867$	0	0
$868$	12.8067	0.434688
$869$	24.9929	0.847825
$870$	0	0
$871$	−3.78110	−0.128118
$872$	−24.5288	−0.830649
$873$	0	0
$874$	−13.3986	−0.453213
$875$	13.5524	0.458154
$876$	0	0
$877$	−25.1209	−0.848272	−0.424136	−	0.905598i	$-0.639422\pi$
$877$	−25.1209	−0.848272	−0.424136	+	0.905598i	$0.639422\pi$
$878$	−0.381211	−0.0128653
$879$	0	0
$880$	−0.833751	−0.0281057
$881$	−2.67121	−0.0899953	−0.0449976	−	0.998987i	$-0.514328\pi$
$881$	−2.67121	−0.0899953	−0.0449976	+	0.998987i	$0.514328\pi$
$882$	0	0
$883$	39.3118	1.32295	0.661473	−	0.749969i	$-0.269931\pi$
$883$	39.3118	1.32295	0.661473	+	0.749969i	$0.269931\pi$
$884$	−22.4254	−0.754248
$885$	0	0
$886$	−29.0793	−0.976937
$887$	−33.4197	−1.12212	−0.561062	−	0.827774i	$-0.689607\pi$
$887$	−33.4197	−1.12212	−0.561062	+	0.827774i	$0.689607\pi$
$888$	0	0
$889$	−7.47103	−0.250570
$890$	3.34800	0.112225
$891$	0	0
$892$	26.3703	0.882942
$893$	7.99235	0.267454
$894$	0	0
$895$	7.64606	0.255579
$896$	−10.6111	−0.354491
$897$	0	0
$898$	−5.96120	−0.198928
$899$	31.0055	1.03409
$900$	0	0
$901$	−46.9670	−1.56470
$902$	−10.4790	−0.348914
$903$	0	0
$904$	47.8943	1.59294
$905$	7.47051	0.248328
$906$	0	0
$907$	−12.3034	−0.408526	−0.204263	−	0.978916i	$-0.565480\pi$
$907$	−12.3034	−0.408526	−0.204263	+	0.978916i	$0.565480\pi$
$908$	3.55706	0.118045
$909$	0	0
$910$	6.01768	0.199484
$911$	21.8531	0.724027	0.362013	−	0.932173i	$-0.382089\pi$
$911$	21.8531	0.724027	0.362013	+	0.932173i	$0.382089\pi$
$912$	0	0
$913$	−10.3775	−0.343446
$914$	10.5646	0.349446
$915$	0	0
$916$	−1.99382	−0.0658777
$917$	−25.6647	−0.847524
$918$	0	0
$919$	−55.8698	−1.84297	−0.921487	−	0.388410i	$-0.873024\pi$
$919$	−55.8698	−1.84297	−0.921487	+	0.388410i	$0.873024\pi$
$920$	20.0412	0.660739
$921$	0	0
$922$	0.384698	0.0126694
$923$	−33.0290	−1.08716
$924$	0	0
$925$	−34.6085	−1.13792
$926$	−30.9055	−1.01562
$927$	0	0
$928$	−24.0030	−0.787936
$929$	−42.9044	−1.40765	−0.703823	−	0.710375i	$-0.748525\pi$
$929$	−42.9044	−1.40765	−0.703823	+	0.710375i	$0.748525\pi$
$930$	0	0
$931$	−14.0523	−0.460547
$932$	−8.52539	−0.279258
$933$	0	0
$934$	22.9136	0.749756
$935$	−9.88146	−0.323158
$936$	0	0
$937$	−13.6439	−0.445728	−0.222864	−	0.974850i	$-0.571541\pi$
$937$	−13.6439	−0.445728	−0.222864	+	0.974850i	$0.571541\pi$
$938$	−0.909236	−0.0296876
$939$	0	0
$940$	−4.72280	−0.154041
$941$	−44.4550	−1.44919	−0.724595	−	0.689175i	$-0.757973\pi$
$941$	−44.4550	−1.44919	−0.724595	+	0.689175i	$0.757973\pi$
$942$	0	0
$943$	−34.9213	−1.13719
$944$	3.46927	0.112915
$945$	0	0
$946$	−9.15206	−0.297559
$947$	4.33524	0.140876	0.0704381	−	0.997516i	$-0.477560\pi$
$947$	4.33524	0.140876	0.0704381	+	0.997516i	$0.477560\pi$
$948$	0	0
$949$	−36.9872	−1.20065
$950$	7.87304	0.255435
$951$	0	0
$952$	−13.6502	−0.442407
$953$	−36.8566	−1.19390	−0.596952	−	0.802277i	$-0.703622\pi$
$953$	−36.8566	−1.19390	−0.596952	+	0.802277i	$0.703622\pi$
$954$	0	0
$955$	−8.14473	−0.263557
$956$	−22.0698	−0.713787
$957$	0	0
$958$	−11.8792	−0.383798
$959$	20.8466	0.673173
$960$	0	0
$961$	24.6092	0.793845
$962$	−37.0406	−1.19424
$963$	0	0
$964$	16.4553	0.529991
$965$	−19.1918	−0.617805
$966$	0	0
$967$	−10.2338	−0.329095	−0.164548	−	0.986369i	$-0.552616\pi$
$967$	−10.2338	−0.329095	−0.164548	+	0.986369i	$0.552616\pi$
$968$	−17.2854	−0.555573
$969$	0	0
$970$	−1.36972	−0.0439789
$971$	43.9486	1.41038	0.705188	−	0.709020i	$-0.250862\pi$
$971$	43.9486	1.41038	0.705188	+	0.709020i	$0.250862\pi$
$972$	0	0
$973$	13.7265	0.440051
$974$	−7.56169	−0.242292
$975$	0	0
$976$	3.75084	0.120061
$977$	−59.0076	−1.88782	−0.943911	−	0.330200i	$-0.892884\pi$
$977$	−59.0076	−1.88782	−0.943911	+	0.330200i	$0.892884\pi$
$978$	0	0
$979$	7.24214	0.231460
$980$	8.30373	0.265253
$981$	0	0
$982$	−23.2268	−0.741198
$983$	26.7887	0.854428	0.427214	−	0.904151i	$-0.359495\pi$
$983$	26.7887	0.854428	0.427214	+	0.904151i	$0.359495\pi$
$984$	0	0
$985$	13.9821	0.445507
$986$	−13.0557	−0.415779
$987$	0	0
$988$	−15.8602	−0.504580
$989$	−30.4991	−0.969816
$990$	0	0
$991$	−9.15461	−0.290806	−0.145403	−	0.989373i	$-0.546448\pi$
$991$	−9.15461	−0.290806	−0.145403	+	0.989373i	$0.546448\pi$
$992$	−43.0500	−1.36684
$993$	0	0
$994$	−7.94244	−0.251919
$995$	5.18624	0.164415
$996$	0	0
$997$	42.3276	1.34053	0.670265	−	0.742122i	$-0.266181\pi$
$997$	42.3276	1.34053	0.670265	+	0.742122i	$0.266181\pi$
$998$	19.8184	0.627339
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6561.2.a.d.1.24		72
3.2	odd	2		6561.2.a.c.1.49		72
81.5	odd	54		81.2.g.a.25.6	yes	144
81.11	odd	54		729.2.g.d.514.3		144
81.16	even	27		243.2.g.a.10.3		144
81.22	even	27		729.2.g.a.217.6		144
81.32	odd	54		729.2.g.c.703.3		144
81.38	odd	54		729.2.g.c.28.3		144
81.43	even	27		729.2.g.b.28.6		144
81.49	even	27		729.2.g.b.703.6		144
81.59	odd	54		729.2.g.d.217.3		144
81.65	odd	54		81.2.g.a.13.6	✓	144
81.70	even	27		729.2.g.a.514.6		144
81.76	even	27		243.2.g.a.73.3		144

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
81.2.g.a.13.6	✓	144	81.65	odd	54
81.2.g.a.25.6	yes	144	81.5	odd	54
243.2.g.a.10.3		144	81.16	even	27
243.2.g.a.73.3		144	81.76	even	27
729.2.g.a.217.6		144	81.22	even	27
729.2.g.a.514.6		144	81.70	even	27
729.2.g.b.28.6		144	81.43	even	27
729.2.g.b.703.6		144	81.49	even	27
729.2.g.c.28.3		144	81.38	odd	54
729.2.g.c.703.3		144	81.32	odd	54
729.2.g.d.217.3		144	81.59	odd	54
729.2.g.d.514.3		144	81.11	odd	54
6561.2.a.c.1.49		72	3.2	odd	2
6561.2.a.d.1.24		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 $\F_{q}$
Belyi maps

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

\(f(q)\)	\(=\)	\(q-0.833012 q^{2} -1.30609 q^{4} +1.20615 q^{5} -1.31489 q^{7} +2.75401 q^{8} -1.00474 q^{10} -2.17338 q^{11} +4.55495 q^{13} +1.09532 q^{14} +0.318052 q^{16} +3.76950 q^{17} +2.66594 q^{19} -1.57535 q^{20} +1.81045 q^{22} +6.03331 q^{23} -3.54519 q^{25} -3.79433 q^{26} +1.71737 q^{28} +4.15782 q^{29} +7.45716 q^{31} -5.77297 q^{32} -3.14004 q^{34} -1.58596 q^{35} +9.76208 q^{37} -2.22077 q^{38} +3.32176 q^{40} -5.78809 q^{41} -5.05513 q^{43} +2.83863 q^{44} -5.02582 q^{46} +2.99794 q^{47} -5.27105 q^{49} +2.95319 q^{50} -5.94918 q^{52} -12.4597 q^{53} -2.62143 q^{55} -3.62124 q^{56} -3.46352 q^{58} +10.9078 q^{59} +11.7931 q^{61} -6.21191 q^{62} +4.17285 q^{64} +5.49397 q^{65} -0.830108 q^{67} -4.92330 q^{68} +1.32113 q^{70} -7.25123 q^{71} -8.12021 q^{73} -8.13193 q^{74} -3.48196 q^{76} +2.85776 q^{77} -11.4996 q^{79} +0.383620 q^{80} +4.82155 q^{82} +4.77483 q^{83} +4.54659 q^{85} +4.21099 q^{86} -5.98551 q^{88} -3.33221 q^{89} -5.98928 q^{91} -7.88004 q^{92} -2.49732 q^{94} +3.21554 q^{95} +1.36325 q^{97} +4.39085 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 72 q + 9 q^{2} + 63 q^{4} + 18 q^{5} + 27 q^{8} + 36 q^{11} + 36 q^{14} + 45 q^{16} + 36 q^{17} + 54 q^{20} + 54 q^{23} + 36 q^{25} + 45 q^{26} + 9 q^{28} + 54 q^{29} + 63 q^{32} + 72 q^{35} + 54 q^{38}+ \cdots + 81 q^{98}+O(q^{100}) \) 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

Introduction

L-functions

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