LMFDB - Embedded newform 6561.2.a.c.1.65

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:	$1$
Dimension:	$72$
Twist minimal:	no (minimal twist has level 81)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.65
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)

$f(q)$	$=$	\(q+2.17581 q^{2} +2.73413 q^{4} +0.646494 q^{5} -1.82934 q^{7} +1.59733 q^{8} +1.40664 q^{10} -1.20984 q^{11} -5.33346 q^{13} -3.98029 q^{14} -1.99279 q^{16} +6.84728 q^{17} +6.36190 q^{19} +1.76760 q^{20} -2.63238 q^{22} -3.11666 q^{23} -4.58205 q^{25} -11.6046 q^{26} -5.00165 q^{28} -5.88635 q^{29} -2.77332 q^{31} -7.53058 q^{32} +14.8984 q^{34} -1.18266 q^{35} -1.42881 q^{37} +13.8423 q^{38} +1.03266 q^{40} +1.11540 q^{41} -9.63443 q^{43} -3.30786 q^{44} -6.78124 q^{46} +6.10234 q^{47} -3.65352 q^{49} -9.96964 q^{50} -14.5824 q^{52} -8.52271 q^{53} -0.782154 q^{55} -2.92205 q^{56} -12.8075 q^{58} +2.08811 q^{59} -3.59004 q^{61} -6.03420 q^{62} -12.3995 q^{64} -3.44805 q^{65} +1.21977 q^{67} +18.7214 q^{68} -2.57323 q^{70} +8.15028 q^{71} +6.42239 q^{73} -3.10880 q^{74} +17.3943 q^{76} +2.21321 q^{77} -14.2584 q^{79} -1.28832 q^{80} +2.42689 q^{82} +6.71946 q^{83} +4.42672 q^{85} -20.9627 q^{86} -1.93251 q^{88} -15.5808 q^{89} +9.75671 q^{91} -8.52135 q^{92} +13.2775 q^{94} +4.11293 q^{95} -3.72929 q^{97} -7.94935 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	2.17581	1.53853	0.769264	−	0.638932i	$-0.220623\pi$
$2$	2.17581	1.53853	0.769264	+	0.638932i	$0.220623\pi$
$3$	0	0
$3$	0	0
$4$	2.73413	1.36707
$5$	0.646494	0.289121	0.144560	−	0.989496i	$-0.453823\pi$
$5$	0.646494	0.289121	0.144560	+	0.989496i	$0.453823\pi$
$6$	0	0
$7$	−1.82934	−0.691425	−0.345713	−	0.938340i	$-0.612363\pi$
$7$	−1.82934	−0.691425	−0.345713	+	0.938340i	$0.612363\pi$
$8$	1.59733	0.564741
$9$	0	0
$10$	1.40664	0.444820
$11$	−1.20984	−0.364781	−0.182390	−	0.983226i	$-0.558383\pi$
$11$	−1.20984	−0.364781	−0.182390	+	0.983226i	$0.558383\pi$
$12$	0	0
$13$	−5.33346	−1.47924	−0.739618	−	0.673027i	$-0.764994\pi$
$13$	−5.33346	−1.47924	−0.739618	+	0.673027i	$0.764994\pi$
$14$	−3.98029	−1.06378
$15$	0	0
$16$	−1.99279	−0.498197
$17$	6.84728	1.66071	0.830355	−	0.557235i	$-0.188138\pi$
$17$	6.84728	1.66071	0.830355	+	0.557235i	$0.188138\pi$
$18$	0	0
$19$	6.36190	1.45952	0.729760	−	0.683704i	$-0.239632\pi$
$19$	6.36190	1.45952	0.729760	+	0.683704i	$0.239632\pi$
$20$	1.76760	0.395247
$21$	0	0
$22$	−2.63238	−0.561225
$23$	−3.11666	−0.649868	−0.324934	−	0.945737i	$-0.605342\pi$
$23$	−3.11666	−0.649868	−0.324934	+	0.945737i	$0.605342\pi$
$24$	0	0
$25$	−4.58205	−0.916409
$26$	−11.6046	−2.27584
$27$	0	0
$28$	−5.00165	−0.945224
$29$	−5.88635	−1.09307	−0.546533	−	0.837437i	$-0.684053\pi$
$29$	−5.88635	−1.09307	−0.546533	+	0.837437i	$0.684053\pi$
$30$	0	0
$31$	−2.77332	−0.498103	−0.249051	−	0.968490i	$-0.580119\pi$
$31$	−2.77332	−0.498103	−0.249051	+	0.968490i	$0.580119\pi$
$32$	−7.53058	−1.33123
$33$	0	0
$34$	14.8984	2.55505
$35$	−1.18266	−0.199905
$36$	0	0
$37$	−1.42881	−0.234894	−0.117447	−	0.993079i	$-0.537471\pi$
$37$	−1.42881	−0.234894	−0.117447	+	0.993079i	$0.537471\pi$
$38$	13.8423	2.24551
$39$	0	0
$40$	1.03266	0.163278
$41$	1.11540	0.174196	0.0870981	−	0.996200i	$-0.472241\pi$
$41$	1.11540	0.174196	0.0870981	+	0.996200i	$0.472241\pi$
$42$	0	0
$43$	−9.63443	−1.46924	−0.734619	−	0.678480i	$-0.762639\pi$
$43$	−9.63443	−1.46924	−0.734619	+	0.678480i	$0.762639\pi$
$44$	−3.30786	−0.498679
$45$	0	0
$46$	−6.78124	−0.999840
$47$	6.10234	0.890118	0.445059	−	0.895501i	$-0.353183\pi$
$47$	6.10234	0.890118	0.445059	+	0.895501i	$0.353183\pi$
$48$	0	0
$49$	−3.65352	−0.521931
$50$	−9.96964	−1.40992
$51$	0	0
$52$	−14.5824	−2.02221
$53$	−8.52271	−1.17068	−0.585342	−	0.810786i	$-0.699040\pi$
$53$	−8.52271	−1.17068	−0.585342	+	0.810786i	$0.699040\pi$
$54$	0	0
$55$	−0.782154	−0.105466
$56$	−2.92205	−0.390476
$57$	0	0
$58$	−12.8075	−1.68171
$59$	2.08811	0.271849	0.135925	−	0.990719i	$-0.456600\pi$
$59$	2.08811	0.271849	0.135925	+	0.990719i	$0.456600\pi$
$60$	0	0
$61$	−3.59004	−0.459658	−0.229829	−	0.973231i	$-0.573817\pi$
$61$	−3.59004	−0.459658	−0.229829	+	0.973231i	$0.573817\pi$
$62$	−6.03420	−0.766344
$63$	0	0
$64$	−12.3995	−1.54994
$65$	−3.44805	−0.427678
$66$	0	0
$67$	1.21977	0.149019	0.0745095	−	0.997220i	$-0.476261\pi$
$67$	1.21977	0.149019	0.0745095	+	0.997220i	$0.476261\pi$
$68$	18.7214	2.27030
$69$	0	0
$70$	−2.57323	−0.307560
$71$	8.15028	0.967260	0.483630	−	0.875273i	$-0.339318\pi$
$71$	8.15028	0.967260	0.483630	+	0.875273i	$0.339318\pi$
$72$	0	0
$73$	6.42239	0.751684	0.375842	−	0.926684i	$-0.377354\pi$
$73$	6.42239	0.751684	0.375842	+	0.926684i	$0.377354\pi$
$74$	−3.10880	−0.361391
$75$	0	0
$76$	17.3943	1.99526
$77$	2.21321	0.252219
$78$	0	0
$79$	−14.2584	−1.60419	−0.802096	−	0.597195i	$-0.796282\pi$
$79$	−14.2584	−1.60419	−0.802096	+	0.597195i	$0.796282\pi$
$80$	−1.28832	−0.144039
$81$	0	0
$82$	2.42689	0.268006
$83$	6.71946	0.737556	0.368778	−	0.929517i	$-0.379776\pi$
$83$	6.71946	0.737556	0.368778	+	0.929517i	$0.379776\pi$
$84$	0	0
$85$	4.42672	0.480146
$86$	−20.9627	−2.26046
$87$	0	0
$88$	−1.93251	−0.206006
$89$	−15.5808	−1.65156	−0.825779	−	0.563994i	$-0.809264\pi$
$89$	−15.5808	−1.65156	−0.825779	+	0.563994i	$0.809264\pi$
$90$	0	0
$91$	9.75671	1.02278
$92$	−8.52135	−0.888412
$93$	0	0
$94$	13.2775	1.36947
$95$	4.11293	0.421977
$96$	0	0
$97$	−3.72929	−0.378652	−0.189326	−	0.981914i	$-0.560630\pi$
$97$	−3.72929	−0.378652	−0.189326	+	0.981914i	$0.560630\pi$
$98$	−7.94935	−0.803005
$99$	0	0
$100$	−12.5279	−1.25279
$101$	−4.65058	−0.462750	−0.231375	−	0.972865i	$-0.574322\pi$
$101$	−4.65058	−0.462750	−0.231375	+	0.972865i	$0.574322\pi$
$102$	0	0
$103$	5.16851	0.509268	0.254634	−	0.967037i	$-0.418045\pi$
$103$	5.16851	0.509268	0.254634	+	0.967037i	$0.418045\pi$
$104$	−8.51928	−0.835384
$105$	0	0
$106$	−18.5438	−1.80113
$107$	−12.9906	−1.25584	−0.627922	−	0.778276i	$-0.716094\pi$
$107$	−12.9906	−1.25584	−0.627922	+	0.778276i	$0.716094\pi$
$108$	0	0
$109$	1.77737	0.170241	0.0851207	−	0.996371i	$-0.472872\pi$
$109$	1.77737	0.170241	0.0851207	+	0.996371i	$0.472872\pi$
$110$	−1.70182	−0.162262
$111$	0	0
$112$	3.64549	0.344466
$113$	13.8881	1.30648	0.653242	−	0.757149i	$-0.273409\pi$
$113$	13.8881	1.30648	0.653242	+	0.757149i	$0.273409\pi$
$114$	0	0
$115$	−2.01490	−0.187890
$116$	−16.0940	−1.49429
$117$	0	0
$118$	4.54333	0.418247
$119$	−12.5260	−1.14826
$120$	0	0
$121$	−9.53629	−0.866935
$122$	−7.81123	−0.707196
$123$	0	0
$124$	−7.58262	−0.680939
$125$	−6.19473	−0.554074
$126$	0	0
$127$	−11.3955	−1.01119	−0.505595	−	0.862771i	$-0.668727\pi$
$127$	−11.3955	−1.01119	−0.505595	+	0.862771i	$0.668727\pi$
$128$	−11.9177	−1.05339
$129$	0	0
$130$	−7.50228	−0.657994
$131$	−14.5807	−1.27393	−0.636963	−	0.770895i	$-0.719810\pi$
$131$	−14.5807	−1.27393	−0.636963	+	0.770895i	$0.719810\pi$
$132$	0	0
$133$	−11.6381	−1.00915
$134$	2.65399	0.229270
$135$	0	0
$136$	10.9374	0.937870
$137$	−0.370808	−0.0316803	−0.0158401	−	0.999875i	$-0.505042\pi$
$137$	−0.370808	−0.0316803	−0.0158401	+	0.999875i	$0.505042\pi$
$138$	0	0
$139$	14.9775	1.27038	0.635188	−	0.772357i	$-0.280922\pi$
$139$	14.9775	1.27038	0.635188	+	0.772357i	$0.280922\pi$
$140$	−3.23354	−0.273284
$141$	0	0
$142$	17.7334	1.48816
$143$	6.45264	0.539596
$144$	0	0
$145$	−3.80549	−0.316028
$146$	13.9739	1.15649
$147$	0	0
$148$	−3.90654	−0.321116
$149$	−1.00026	−0.0819441	−0.0409721	−	0.999160i	$-0.513045\pi$
$149$	−1.00026	−0.0819441	−0.0409721	+	0.999160i	$0.513045\pi$
$150$	0	0
$151$	0.0196324	0.00159766	0.000798831	−	1.00000i	$-0.499746\pi$
$151$	0.0196324	0.00159766	0.000798831		1.00000i	$0.499746\pi$
$152$	10.1620	0.824250
$153$	0	0
$154$	4.81551	0.388045
$155$	−1.79293	−0.144012
$156$	0	0
$157$	−7.80457	−0.622872	−0.311436	−	0.950267i	$-0.600810\pi$
$157$	−7.80457	−0.622872	−0.311436	+	0.950267i	$0.600810\pi$
$158$	−31.0235	−2.46809
$159$	0	0
$160$	−4.86847	−0.384886
$161$	5.70142	0.449335
$162$	0	0
$163$	−15.3947	−1.20581	−0.602905	−	0.797813i	$-0.705990\pi$
$163$	−15.3947	−1.20581	−0.602905	+	0.797813i	$0.705990\pi$
$164$	3.04965	0.238138
$165$	0	0
$166$	14.6202	1.13475
$167$	3.65482	0.282818	0.141409	−	0.989951i	$-0.454837\pi$
$167$	3.65482	0.282818	0.141409	+	0.989951i	$0.454837\pi$
$168$	0	0
$169$	15.4458	1.18814
$170$	9.63169	0.738717
$171$	0	0
$172$	−26.3418	−2.00854
$173$	13.2826	1.00986	0.504928	−	0.863161i	$-0.331519\pi$
$173$	13.2826	1.00986	0.504928	+	0.863161i	$0.331519\pi$
$174$	0	0
$175$	8.38212	0.633628
$176$	2.41096	0.181733
$177$	0	0
$178$	−33.9007	−2.54097
$179$	0.417943	0.0312385	0.0156193	−	0.999878i	$-0.495028\pi$
$179$	0.417943	0.0312385	0.0156193	+	0.999878i	$0.495028\pi$
$180$	0	0
$181$	20.8452	1.54941	0.774705	−	0.632323i	$-0.217898\pi$
$181$	20.8452	1.54941	0.774705	+	0.632323i	$0.217898\pi$
$182$	21.2287	1.57358
$183$	0	0
$184$	−4.97832	−0.367007
$185$	−0.923714	−0.0679128
$186$	0	0
$187$	−8.28412	−0.605795
$188$	16.6846	1.21685
$189$	0	0
$190$	8.94893	0.649223
$191$	6.88924	0.498488	0.249244	−	0.968441i	$-0.419818\pi$
$191$	6.88924	0.498488	0.249244	+	0.968441i	$0.419818\pi$
$192$	0	0
$193$	−8.74639	−0.629579	−0.314789	−	0.949162i	$-0.601934\pi$
$193$	−8.74639	−0.629579	−0.314789	+	0.949162i	$0.601934\pi$
$194$	−8.11422	−0.582567
$195$	0	0
$196$	−9.98920	−0.713514
$197$	−8.44721	−0.601839	−0.300919	−	0.953650i	$-0.597294\pi$
$197$	−8.44721	−0.601839	−0.300919	+	0.953650i	$0.597294\pi$
$198$	0	0
$199$	12.6206	0.894647	0.447324	−	0.894372i	$-0.352377\pi$
$199$	12.6206	0.894647	0.447324	+	0.894372i	$0.352377\pi$
$200$	−7.31903	−0.517533
$201$	0	0
$202$	−10.1188	−0.711953
$203$	10.7681	0.755774
$204$	0	0
$205$	0.721099	0.0503637
$206$	11.2457	0.783523
$207$	0	0
$208$	10.6285	0.736951
$209$	−7.69688	−0.532404
$210$	0	0
$211$	−5.67591	−0.390746	−0.195373	−	0.980729i	$-0.562592\pi$
$211$	−5.67591	−0.390746	−0.195373	+	0.980729i	$0.562592\pi$
$212$	−23.3022	−1.60040
$213$	0	0
$214$	−28.2649	−1.93215
$215$	−6.22860	−0.424787
$216$	0	0
$217$	5.07334	0.344401
$218$	3.86722	0.261921
$219$	0	0
$220$	−2.13851	−0.144178
$221$	−36.5197	−2.45658
$222$	0	0
$223$	7.53286	0.504438	0.252219	−	0.967670i	$-0.418840\pi$
$223$	7.53286	0.504438	0.252219	+	0.967670i	$0.418840\pi$
$224$	13.7760	0.920446
$225$	0	0
$226$	30.2178	2.01006
$227$	−13.6431	−0.905527	−0.452764	−	0.891631i	$-0.649562\pi$
$227$	−13.6431	−0.905527	−0.452764	+	0.891631i	$0.649562\pi$
$228$	0	0
$229$	15.9945	1.05695	0.528474	−	0.848950i	$-0.322765\pi$
$229$	15.9945	1.05695	0.528474	+	0.848950i	$0.322765\pi$
$230$	−4.38403	−0.289074
$231$	0	0
$232$	−9.40242	−0.617299
$233$	−5.35914	−0.351089	−0.175544	−	0.984471i	$-0.556169\pi$
$233$	−5.35914	−0.351089	−0.175544	+	0.984471i	$0.556169\pi$
$234$	0	0
$235$	3.94513	0.257352
$236$	5.70918	0.371636
$237$	0	0
$238$	−27.2541	−1.76662
$239$	−16.1436	−1.04424	−0.522121	−	0.852871i	$-0.674859\pi$
$239$	−16.1436	−1.04424	−0.522121	+	0.852871i	$0.674859\pi$
$240$	0	0
$241$	−23.3880	−1.50655	−0.753276	−	0.657705i	$-0.771528\pi$
$241$	−23.3880	−1.50655	−0.753276	+	0.657705i	$0.771528\pi$
$242$	−20.7491	−1.33380
$243$	0	0
$244$	−9.81564	−0.628382
$245$	−2.36198	−0.150901
$246$	0	0
$247$	−33.9309	−2.15897
$248$	−4.42990	−0.281299
$249$	0	0
$250$	−13.4785	−0.852457
$251$	24.2873	1.53300	0.766501	−	0.642244i	$-0.221996\pi$
$251$	24.2873	1.53300	0.766501	+	0.642244i	$0.221996\pi$
$252$	0	0
$253$	3.77066	0.237059
$254$	−24.7945	−1.55574
$255$	0	0
$256$	−1.13171	−0.0707317
$257$	9.63393	0.600948	0.300474	−	0.953790i	$-0.402855\pi$
$257$	9.63393	0.600948	0.300474	+	0.953790i	$0.402855\pi$
$258$	0	0
$259$	2.61377	0.162412
$260$	−9.42742	−0.584664
$261$	0	0
$262$	−31.7249	−1.95997
$263$	6.64180	0.409551	0.204776	−	0.978809i	$-0.434354\pi$
$263$	6.64180	0.409551	0.204776	+	0.978809i	$0.434354\pi$
$264$	0	0
$265$	−5.50988	−0.338469
$266$	−25.3222	−1.55260
$267$	0	0
$268$	3.33502	0.203719
$269$	29.3664	1.79050	0.895251	−	0.445563i	$-0.146996\pi$
$269$	29.3664	1.79050	0.895251	+	0.445563i	$0.146996\pi$
$270$	0	0
$271$	−18.8791	−1.14683	−0.573413	−	0.819267i	$-0.694381\pi$
$271$	−18.8791	−1.14683	−0.573413	+	0.819267i	$0.694381\pi$
$272$	−13.6452	−0.827361
$273$	0	0
$274$	−0.806806	−0.0487410
$275$	5.54354	0.334288
$276$	0	0
$277$	14.0387	0.843504	0.421752	−	0.906711i	$-0.361415\pi$
$277$	14.0387	0.843504	0.421752	+	0.906711i	$0.361415\pi$
$278$	32.5882	1.95451
$279$	0	0
$280$	−1.88909	−0.112895
$281$	−22.0804	−1.31721	−0.658604	−	0.752489i	$-0.728853\pi$
$281$	−22.0804	−1.31721	−0.658604	+	0.752489i	$0.728853\pi$
$282$	0	0
$283$	3.95647	0.235188	0.117594	−	0.993062i	$-0.462482\pi$
$283$	3.95647	0.235188	0.117594	+	0.993062i	$0.462482\pi$
$284$	22.2839	1.32231
$285$	0	0
$286$	14.0397	0.830184
$287$	−2.04045	−0.120444
$288$	0	0
$289$	29.8853	1.75796
$290$	−8.28000	−0.486218
$291$	0	0
$292$	17.5597	1.02760
$293$	7.97726	0.466037	0.233018	−	0.972472i	$-0.425140\pi$
$293$	7.97726	0.466037	0.233018	+	0.972472i	$0.425140\pi$
$294$	0	0
$295$	1.34995	0.0785973
$296$	−2.28227	−0.132654
$297$	0	0
$298$	−2.17636	−0.126073
$299$	16.6226	0.961308
$300$	0	0
$301$	17.6246	1.01587
$302$	0.0427163	0.00245805
$303$	0	0
$304$	−12.6779	−0.727128
$305$	−2.32094	−0.132897
$306$	0	0
$307$	−2.08390	−0.118935	−0.0594673	−	0.998230i	$-0.518940\pi$
$307$	−2.08390	−0.118935	−0.0594673	+	0.998230i	$0.518940\pi$
$308$	6.05120	0.344799
$309$	0	0
$310$	−3.90107	−0.221566
$311$	−15.0144	−0.851390	−0.425695	−	0.904867i	$-0.639970\pi$
$311$	−15.0144	−0.851390	−0.425695	+	0.904867i	$0.639970\pi$
$312$	0	0
$313$	5.09090	0.287755	0.143877	−	0.989596i	$-0.454043\pi$
$313$	5.09090	0.287755	0.143877	+	0.989596i	$0.454043\pi$
$314$	−16.9812	−0.958306
$315$	0	0
$316$	−38.9843	−2.19304
$317$	−27.7081	−1.55624	−0.778121	−	0.628114i	$-0.783827\pi$
$317$	−27.7081	−1.55624	−0.778121	+	0.628114i	$0.783827\pi$
$318$	0	0
$319$	7.12154	0.398730
$320$	−8.01619	−0.448119
$321$	0	0
$322$	12.4052	0.691314
$323$	43.5617	2.42384
$324$	0	0
$325$	24.4382	1.35559
$326$	−33.4960	−1.85517
$327$	0	0
$328$	1.78166	0.0983757
$329$	−11.1633	−0.615450
$330$	0	0
$331$	27.2691	1.49885	0.749423	−	0.662092i	$-0.230331\pi$
$331$	27.2691	1.49885	0.749423	+	0.662092i	$0.230331\pi$
$332$	18.3719	1.00829
$333$	0	0
$334$	7.95217	0.435123
$335$	0.788576	0.0430845
$336$	0	0
$337$	−12.7186	−0.692824	−0.346412	−	0.938082i	$-0.612600\pi$
$337$	−12.7186	−0.692824	−0.346412	+	0.938082i	$0.612600\pi$
$338$	33.6071	1.82798
$339$	0	0
$340$	12.1032	0.656391
$341$	3.35527	0.181698
$342$	0	0
$343$	19.4889	1.05230
$344$	−15.3893	−0.829738
$345$	0	0
$346$	28.9003	1.55369
$347$	8.41312	0.451640	0.225820	−	0.974169i	$-0.427494\pi$
$347$	8.41312	0.451640	0.225820	+	0.974169i	$0.427494\pi$
$348$	0	0
$349$	30.1432	1.61353	0.806764	−	0.590873i	$-0.201217\pi$
$349$	30.1432	1.61353	0.806764	+	0.590873i	$0.201217\pi$
$350$	18.2379	0.974855
$351$	0	0
$352$	9.11079	0.485607
$353$	23.2314	1.23648	0.618241	−	0.785989i	$-0.287846\pi$
$353$	23.2314	1.23648	0.618241	+	0.785989i	$0.287846\pi$
$354$	0	0
$355$	5.26910	0.279655
$356$	−42.5999	−2.25779
$357$	0	0
$358$	0.909363	0.0480613
$359$	4.16804	0.219981	0.109990	−	0.993933i	$-0.464918\pi$
$359$	4.16804	0.219981	0.109990	+	0.993933i	$0.464918\pi$
$360$	0	0
$361$	21.4737	1.13020
$362$	45.3551	2.38381
$363$	0	0
$364$	26.6761	1.39821
$365$	4.15203	0.217327
$366$	0	0
$367$	−11.7210	−0.611833	−0.305917	−	0.952058i	$-0.598963\pi$
$367$	−11.7210	−0.611833	−0.305917	+	0.952058i	$0.598963\pi$
$368$	6.21084	0.323762
$369$	0	0
$370$	−2.00982	−0.104486
$371$	15.5909	0.809441
$372$	0	0
$373$	36.3718	1.88326	0.941629	−	0.336653i	$-0.109295\pi$
$373$	36.3718	1.88326	0.941629	+	0.336653i	$0.109295\pi$
$374$	−18.0246	−0.932032
$375$	0	0
$376$	9.74744	0.502686
$377$	31.3946	1.61690
$378$	0	0
$379$	−10.0503	−0.516250	−0.258125	−	0.966112i	$-0.583105\pi$
$379$	−10.0503	−0.516250	−0.258125	+	0.966112i	$0.583105\pi$
$380$	11.2453	0.576871
$381$	0	0
$382$	14.9896	0.766937
$383$	−0.302610	−0.0154626	−0.00773131	−	0.999970i	$-0.502461\pi$
$383$	−0.302610	−0.0154626	−0.00773131	+	0.999970i	$0.502461\pi$
$384$	0	0
$385$	1.43083	0.0729216
$386$	−19.0304	−0.968624
$387$	0	0
$388$	−10.1964	−0.517643
$389$	7.51954	0.381256	0.190628	−	0.981662i	$-0.438948\pi$
$389$	7.51954	0.381256	0.190628	+	0.981662i	$0.438948\pi$
$390$	0	0
$391$	−21.3406	−1.07924
$392$	−5.83587	−0.294756
$393$	0	0
$394$	−18.3795	−0.925946
$395$	−9.21795	−0.463805
$396$	0	0
$397$	17.0450	0.855462	0.427731	−	0.903906i	$-0.359313\pi$
$397$	17.0450	0.855462	0.427731	+	0.903906i	$0.359313\pi$
$398$	27.4599	1.37644
$399$	0	0
$400$	9.13105	0.456552
$401$	13.5774	0.678025	0.339013	−	0.940782i	$-0.389907\pi$
$401$	13.5774	0.678025	0.339013	+	0.940782i	$0.389907\pi$
$402$	0	0
$403$	14.7914	0.736811
$404$	−12.7153	−0.632609
$405$	0	0
$406$	23.4293	1.16278
$407$	1.72863	0.0856848
$408$	0	0
$409$	−0.672351	−0.0332456	−0.0166228	−	0.999862i	$-0.505291\pi$
$409$	−0.672351	−0.0332456	−0.0166228	+	0.999862i	$0.505291\pi$
$410$	1.56897	0.0774860
$411$	0	0
$412$	14.1314	0.696203
$413$	−3.81987	−0.187963
$414$	0	0
$415$	4.34409	0.213243
$416$	40.1640	1.96920
$417$	0	0
$418$	−16.7469	−0.819118
$419$	−19.8861	−0.971498	−0.485749	−	0.874098i	$-0.661453\pi$
$419$	−19.8861	−0.971498	−0.485749	+	0.874098i	$0.661453\pi$
$420$	0	0
$421$	12.5556	0.611921	0.305960	−	0.952044i	$-0.401023\pi$
$421$	12.5556	0.611921	0.305960	+	0.952044i	$0.401023\pi$
$422$	−12.3497	−0.601173
$423$	0	0
$424$	−13.6136	−0.661133
$425$	−31.3746	−1.52189
$426$	0	0
$427$	6.56740	0.317819
$428$	−35.5179	−1.71682
$429$	0	0
$430$	−13.5522	−0.653546
$431$	−21.6026	−1.04056	−0.520281	−	0.853995i	$-0.674173\pi$
$431$	−21.6026	−1.04056	−0.520281	+	0.853995i	$0.674173\pi$
$432$	0	0
$433$	3.99939	0.192199	0.0960993	−	0.995372i	$-0.469363\pi$
$433$	3.99939	0.192199	0.0960993	+	0.995372i	$0.469363\pi$
$434$	11.0386	0.529870
$435$	0	0
$436$	4.85957	0.232731
$437$	−19.8279	−0.948495
$438$	0	0
$439$	−31.2112	−1.48963	−0.744814	−	0.667273i	$-0.767462\pi$
$439$	−31.2112	−1.48963	−0.744814	+	0.667273i	$0.767462\pi$
$440$	−1.24936	−0.0595607
$441$	0	0
$442$	−79.4598	−3.77952
$443$	1.60218	0.0761217	0.0380609	−	0.999275i	$-0.487882\pi$
$443$	1.60218	0.0761217	0.0380609	+	0.999275i	$0.487882\pi$
$444$	0	0
$445$	−10.0729	−0.477500
$446$	16.3900	0.776091
$447$	0	0
$448$	22.6829	1.07167
$449$	10.5049	0.495756	0.247878	−	0.968791i	$-0.420267\pi$
$449$	10.5049	0.495756	0.247878	+	0.968791i	$0.420267\pi$
$450$	0	0
$451$	−1.34946	−0.0635434
$452$	37.9719	1.78605
$453$	0	0
$454$	−29.6848	−1.39318
$455$	6.30765	0.295707
$456$	0	0
$457$	41.1409	1.92449	0.962244	−	0.272187i	$-0.0877468\pi$
$457$	41.1409	1.92449	0.962244	+	0.272187i	$0.0877468\pi$
$458$	34.8010	1.62614
$459$	0	0
$460$	−5.50900	−0.256858
$461$	−34.2989	−1.59746	−0.798730	−	0.601689i	$-0.794495\pi$
$461$	−34.2989	−1.59746	−0.798730	+	0.601689i	$0.794495\pi$
$462$	0	0
$463$	−33.3329	−1.54911	−0.774556	−	0.632505i	$-0.782027\pi$
$463$	−33.3329	−1.54911	−0.774556	+	0.632505i	$0.782027\pi$
$464$	11.7302	0.544563
$465$	0	0
$466$	−11.6605	−0.540160
$467$	10.3904	0.480811	0.240406	−	0.970673i	$-0.422720\pi$
$467$	10.3904	0.480811	0.240406	+	0.970673i	$0.422720\pi$
$468$	0	0
$469$	−2.23138	−0.103036
$470$	8.58383	0.395942
$471$	0	0
$472$	3.33540	0.153524
$473$	11.6561	0.535949
$474$	0	0
$475$	−29.1505	−1.33752
$476$	−34.2477	−1.56974
$477$	0	0
$478$	−35.1253	−1.60659
$479$	20.2965	0.927370	0.463685	−	0.886000i	$-0.346527\pi$
$479$	20.2965	0.927370	0.463685	+	0.886000i	$0.346527\pi$
$480$	0	0
$481$	7.62048	0.347464
$482$	−50.8877	−2.31787
$483$	0	0
$484$	−26.0735	−1.18516
$485$	−2.41096	−0.109476
$486$	0	0
$487$	−27.5716	−1.24939	−0.624694	−	0.780870i	$-0.714776\pi$
$487$	−27.5716	−1.24939	−0.624694	+	0.780870i	$0.714776\pi$
$488$	−5.73447	−0.259587
$489$	0	0
$490$	−5.13920	−0.232165
$491$	10.2173	0.461099	0.230550	−	0.973061i	$-0.425948\pi$
$491$	10.2173	0.461099	0.230550	+	0.973061i	$0.425948\pi$
$492$	0	0
$493$	−40.3055	−1.81527
$494$	−73.8271	−3.32164
$495$	0	0
$496$	5.52663	0.248153
$497$	−14.9096	−0.668788
$498$	0	0
$499$	−8.74363	−0.391419	−0.195709	−	0.980662i	$-0.562701\pi$
$499$	−8.74363	−0.391419	−0.195709	+	0.980662i	$0.562701\pi$
$500$	−16.9372	−0.757455
$501$	0	0
$502$	52.8445	2.35856
$503$	13.0837	0.583374	0.291687	−	0.956514i	$-0.405783\pi$
$503$	13.0837	0.583374	0.291687	+	0.956514i	$0.405783\pi$
$504$	0	0
$505$	−3.00657	−0.133791
$506$	8.20422	0.364722
$507$	0	0
$508$	−31.1569	−1.38236
$509$	−30.4211	−1.34839	−0.674195	−	0.738554i	$-0.735509\pi$
$509$	−30.4211	−1.34839	−0.674195	+	0.738554i	$0.735509\pi$
$510$	0	0
$511$	−11.7487	−0.519733
$512$	21.3731	0.944567
$513$	0	0
$514$	20.9616	0.924574
$515$	3.34141	0.147240
$516$	0	0
$517$	−7.38286	−0.324698
$518$	5.68706	0.249875
$519$	0	0
$520$	−5.50766	−0.241527
$521$	10.7817	0.472354	0.236177	−	0.971710i	$-0.424105\pi$
$521$	10.7817	0.472354	0.236177	+	0.971710i	$0.424105\pi$
$522$	0	0
$523$	12.3314	0.539217	0.269608	−	0.962970i	$-0.413106\pi$
$523$	12.3314	0.539217	0.269608	+	0.962970i	$0.413106\pi$
$524$	−39.8657	−1.74154
$525$	0	0
$526$	14.4513	0.630106
$527$	−18.9897	−0.827204
$528$	0	0
$529$	−13.2864	−0.577671
$530$	−11.9884	−0.520744
$531$	0	0
$532$	−31.8200	−1.37957
$533$	−5.94894	−0.257677
$534$	0	0
$535$	−8.39831	−0.363091
$536$	1.94838	0.0841571
$537$	0	0
$538$	63.8956	2.75474
$539$	4.42017	0.190390
$540$	0	0
$541$	−15.9856	−0.687274	−0.343637	−	0.939103i	$-0.611659\pi$
$541$	−15.9856	−0.687274	−0.343637	+	0.939103i	$0.611659\pi$
$542$	−41.0773	−1.76442
$543$	0	0
$544$	−51.5640	−2.21079
$545$	1.14906	0.0492203
$546$	0	0
$547$	1.23661	0.0528736	0.0264368	−	0.999650i	$-0.491584\pi$
$547$	1.23661	0.0528736	0.0264368	+	0.999650i	$0.491584\pi$
$548$	−1.01384	−0.0433090
$549$	0	0
$550$	12.0617	0.514312
$551$	−37.4483	−1.59535
$552$	0	0
$553$	26.0834	1.10918
$554$	30.5455	1.29775
$555$	0	0
$556$	40.9505	1.73669
$557$	13.1442	0.556936	0.278468	−	0.960446i	$-0.410173\pi$
$557$	13.1442	0.556936	0.278468	+	0.960446i	$0.410173\pi$
$558$	0	0
$559$	51.3849	2.17335
$560$	2.35678	0.0995923
$561$	0	0
$562$	−48.0428	−2.02656
$563$	6.08891	0.256617	0.128308	−	0.991734i	$-0.459045\pi$
$563$	6.08891	0.256617	0.128308	+	0.991734i	$0.459045\pi$
$564$	0	0
$565$	8.97858	0.377732
$566$	8.60852	0.361843
$567$	0	0
$568$	13.0187	0.546251
$569$	16.5777	0.694974	0.347487	−	0.937685i	$-0.387035\pi$
$569$	16.5777	0.694974	0.347487	+	0.937685i	$0.387035\pi$
$570$	0	0
$571$	34.8473	1.45831	0.729157	−	0.684347i	$-0.239913\pi$
$571$	34.8473	1.45831	0.729157	+	0.684347i	$0.239913\pi$
$572$	17.6424	0.737664
$573$	0	0
$574$	−4.43961	−0.185306
$575$	14.2807	0.595545
$576$	0	0
$577$	31.0211	1.29142	0.645712	−	0.763581i	$-0.276561\pi$
$577$	31.0211	1.29142	0.645712	+	0.763581i	$0.276561\pi$
$578$	65.0245	2.70466
$579$	0	0
$580$	−10.4047	−0.432032
$581$	−12.2922	−0.509965
$582$	0	0
$583$	10.3111	0.427043
$584$	10.2587	0.424506
$585$	0	0
$586$	17.3570	0.717010
$587$	−44.5443	−1.83854	−0.919271	−	0.393626i	$-0.871221\pi$
$587$	−44.5443	−1.83854	−0.919271	+	0.393626i	$0.871221\pi$
$588$	0	0
$589$	−17.6436	−0.726990
$590$	2.93723	0.120924
$591$	0	0
$592$	2.84731	0.117024
$593$	−19.8110	−0.813539	−0.406770	−	0.913531i	$-0.633345\pi$
$593$	−19.8110	−0.813539	−0.406770	+	0.913531i	$0.633345\pi$
$594$	0	0
$595$	−8.09798	−0.331985
$596$	−2.73483	−0.112023
$597$	0	0
$598$	36.1675	1.47900
$599$	−27.9283	−1.14112	−0.570560	−	0.821256i	$-0.693274\pi$
$599$	−27.9283	−1.14112	−0.570560	+	0.821256i	$0.693274\pi$
$600$	0	0
$601$	−10.1448	−0.413813	−0.206907	−	0.978361i	$-0.566340\pi$
$601$	−10.1448	−0.413813	−0.206907	+	0.978361i	$0.566340\pi$
$602$	38.3478	1.56294
$603$	0	0
$604$	0.0536776	0.00218411
$605$	−6.16515	−0.250649
$606$	0	0
$607$	10.8164	0.439025	0.219512	−	0.975610i	$-0.429553\pi$
$607$	10.8164	0.439025	0.219512	+	0.975610i	$0.429553\pi$
$608$	−47.9087	−1.94296
$609$	0	0
$610$	−5.04991	−0.204465
$611$	−32.5466	−1.31669
$612$	0	0
$613$	5.40255	0.218207	0.109103	−	0.994030i	$-0.465202\pi$
$613$	5.40255	0.218207	0.109103	+	0.994030i	$0.465202\pi$
$614$	−4.53417	−0.182984
$615$	0	0
$616$	3.53522	0.142438
$617$	−25.0403	−1.00808	−0.504042	−	0.863679i	$-0.668154\pi$
$617$	−25.0403	−1.00808	−0.504042	+	0.863679i	$0.668154\pi$
$618$	0	0
$619$	25.4851	1.02433	0.512166	−	0.858887i	$-0.328843\pi$
$619$	25.4851	1.02433	0.512166	+	0.858887i	$0.328843\pi$
$620$	−4.90211	−0.196874
$621$	0	0
$622$	−32.6685	−1.30989
$623$	28.5025	1.14193
$624$	0	0
$625$	18.9054	0.756215
$626$	11.0768	0.442718
$627$	0	0
$628$	−21.3387	−0.851507
$629$	−9.78343	−0.390091
$630$	0	0
$631$	19.9755	0.795213	0.397607	−	0.917556i	$-0.369841\pi$
$631$	19.9755	0.795213	0.397607	+	0.917556i	$0.369841\pi$
$632$	−22.7753	−0.905953
$633$	0	0
$634$	−60.2874	−2.39432
$635$	−7.36715	−0.292356
$636$	0	0
$637$	19.4859	0.772059
$638$	15.4951	0.613456
$639$	0	0
$640$	−7.70475	−0.304557
$641$	−8.92956	−0.352696	−0.176348	−	0.984328i	$-0.556428\pi$
$641$	−8.92956	−0.352696	−0.176348	+	0.984328i	$0.556428\pi$
$642$	0	0
$643$	13.3999	0.528441	0.264221	−	0.964462i	$-0.414885\pi$
$643$	13.3999	0.528441	0.264221	+	0.964462i	$0.414885\pi$
$644$	15.5884	0.614271
$645$	0	0
$646$	94.7818	3.72914
$647$	−48.7223	−1.91547	−0.957736	−	0.287649i	$-0.907126\pi$
$647$	−48.7223	−1.91547	−0.957736	+	0.287649i	$0.907126\pi$
$648$	0	0
$649$	−2.52628	−0.0991653
$650$	53.1727	2.08560
$651$	0	0
$652$	−42.0912	−1.64842
$653$	4.80493	0.188031	0.0940157	−	0.995571i	$-0.470030\pi$
$653$	4.80493	0.188031	0.0940157	+	0.995571i	$0.470030\pi$
$654$	0	0
$655$	−9.42636	−0.368318
$656$	−2.22276	−0.0867840
$657$	0	0
$658$	−24.2891	−0.946887
$659$	−25.2929	−0.985270	−0.492635	−	0.870236i	$-0.663966\pi$
$659$	−25.2929	−0.985270	−0.492635	+	0.870236i	$0.663966\pi$
$660$	0	0
$661$	−14.1363	−0.549840	−0.274920	−	0.961467i	$-0.588651\pi$
$661$	−14.1363	−0.549840	−0.274920	+	0.961467i	$0.588651\pi$
$662$	59.3323	2.30602
$663$	0	0
$664$	10.7332	0.416528
$665$	−7.52394	−0.291766
$666$	0	0
$667$	18.3457	0.710349
$668$	9.99275	0.386631
$669$	0	0
$670$	1.71579	0.0662867
$671$	4.34338	0.167674
$672$	0	0
$673$	2.66735	0.102819	0.0514094	−	0.998678i	$-0.483629\pi$
$673$	2.66735	0.102819	0.0514094	+	0.998678i	$0.483629\pi$
$674$	−27.6731	−1.06593
$675$	0	0
$676$	42.2308	1.62426
$677$	−14.7625	−0.567371	−0.283685	−	0.958917i	$-0.591557\pi$
$677$	−14.7625	−0.567371	−0.283685	+	0.958917i	$0.591557\pi$
$678$	0	0
$679$	6.82214	0.261810
$680$	7.07093	0.271158
$681$	0	0
$682$	7.30042	0.279548
$683$	−20.9795	−0.802758	−0.401379	−	0.915912i	$-0.631469\pi$
$683$	−20.9795	−0.802758	−0.401379	+	0.915912i	$0.631469\pi$
$684$	0	0
$685$	−0.239725	−0.00915942
$686$	42.4041	1.61899
$687$	0	0
$688$	19.1994	0.731970
$689$	45.4555	1.73172
$690$	0	0
$691$	34.8338	1.32514	0.662570	−	0.749000i	$-0.269466\pi$
$691$	34.8338	1.32514	0.662570	+	0.749000i	$0.269466\pi$
$692$	36.3163	1.38054
$693$	0	0
$694$	18.3053	0.694861
$695$	9.68287	0.367292
$696$	0	0
$697$	7.63746	0.289289
$698$	65.5858	2.48246
$699$	0	0
$700$	22.9178	0.866212
$701$	−43.6106	−1.64715	−0.823576	−	0.567207i	$-0.808024\pi$
$701$	−43.6106	−1.64715	−0.823576	+	0.567207i	$0.808024\pi$
$702$	0	0
$703$	−9.08991	−0.342833
$704$	15.0014	0.565387
$705$	0	0
$706$	50.5470	1.90236
$707$	8.50748	0.319957
$708$	0	0
$709$	18.2673	0.686045	0.343022	−	0.939327i	$-0.388549\pi$
$709$	18.2673	0.686045	0.343022	+	0.939327i	$0.388549\pi$
$710$	11.4645	0.430257
$711$	0	0
$712$	−24.8876	−0.932702
$713$	8.64348	0.323701
$714$	0	0
$715$	4.17159	0.156009
$716$	1.14271	0.0427051
$717$	0	0
$718$	9.06885	0.338446
$719$	−18.1403	−0.676518	−0.338259	−	0.941053i	$-0.609838\pi$
$719$	−18.1403	−0.676518	−0.338259	+	0.941053i	$0.609838\pi$
$720$	0	0
$721$	−9.45495	−0.352121
$722$	46.7227	1.73884
$723$	0	0
$724$	56.9934	2.11814
$725$	26.9715	1.00170
$726$	0	0
$727$	20.2539	0.751176	0.375588	−	0.926787i	$-0.377441\pi$
$727$	20.2539	0.751176	0.375588	+	0.926787i	$0.377441\pi$
$728$	15.5847	0.577606
$729$	0	0
$730$	9.03402	0.334364
$731$	−65.9697	−2.43998
$732$	0	0
$733$	−1.76799	−0.0653021	−0.0326511	−	0.999467i	$-0.510395\pi$
$733$	−1.76799	−0.0653021	−0.0326511	+	0.999467i	$0.510395\pi$
$734$	−25.5027	−0.941322
$735$	0	0
$736$	23.4702	0.865124
$737$	−1.47573	−0.0543593
$738$	0	0
$739$	−9.11841	−0.335426	−0.167713	−	0.985836i	$-0.553638\pi$
$739$	−9.11841	−0.335426	−0.167713	+	0.985836i	$0.553638\pi$
$740$	−2.52555	−0.0928412
$741$	0	0
$742$	33.9228	1.24535
$743$	−19.9215	−0.730850	−0.365425	−	0.930841i	$-0.619076\pi$
$743$	−19.9215	−0.730850	−0.365425	+	0.930841i	$0.619076\pi$
$744$	0	0
$745$	−0.646659	−0.0236917
$746$	79.1379	2.89744
$747$	0	0
$748$	−22.6499	−0.828161
$749$	23.7641	0.868323
$750$	0	0
$751$	−38.5558	−1.40692	−0.703460	−	0.710735i	$-0.748363\pi$
$751$	−38.5558	−1.40692	−0.703460	+	0.710735i	$0.748363\pi$
$752$	−12.1607	−0.443454
$753$	0	0
$754$	68.3085	2.48765
$755$	0.0126922	0.000461917 0
$756$	0	0
$757$	−50.9458	−1.85166	−0.925829	−	0.377943i	$-0.876632\pi$
$757$	−50.9458	−1.85166	−0.925829	+	0.377943i	$0.876632\pi$
$758$	−21.8675	−0.794265
$759$	0	0
$760$	6.56969	0.238308
$761$	38.9404	1.41159	0.705794	−	0.708417i	$-0.250591\pi$
$761$	38.9404	1.41159	0.705794	+	0.708417i	$0.250591\pi$
$762$	0	0
$763$	−3.25142	−0.117709
$764$	18.8361	0.681466
$765$	0	0
$766$	−0.658420	−0.0237897
$767$	−11.1369	−0.402129
$768$	0	0
$769$	34.1515	1.23153	0.615767	−	0.787928i	$-0.288846\pi$
$769$	34.1515	1.23153	0.615767	+	0.787928i	$0.288846\pi$
$770$	3.11320	0.112192
$771$	0	0
$772$	−23.9138	−0.860676
$773$	−3.02782	−0.108903	−0.0544516	−	0.998516i	$-0.517341\pi$
$773$	−3.02782	−0.108903	−0.0544516	+	0.998516i	$0.517341\pi$
$774$	0	0
$775$	12.7075	0.456466
$776$	−5.95690	−0.213840
$777$	0	0
$778$	16.3611	0.586573
$779$	7.09606	0.254243
$780$	0	0
$781$	−9.86053	−0.352838
$782$	−46.4331	−1.66044
$783$	0	0
$784$	7.28069	0.260025
$785$	−5.04560	−0.180085
$786$	0	0
$787$	2.27120	0.0809594	0.0404797	−	0.999180i	$-0.487111\pi$
$787$	2.27120	0.0809594	0.0404797	+	0.999180i	$0.487111\pi$
$788$	−23.0958	−0.822753
$789$	0	0
$790$	−20.0565	−0.713577
$791$	−25.4061	−0.903336
$792$	0	0
$793$	19.1473	0.679942
$794$	37.0865	1.31615
$795$	0	0
$796$	34.5062	1.22304
$797$	12.5831	0.445716	0.222858	−	0.974851i	$-0.428461\pi$
$797$	12.5831	0.445716	0.222858	+	0.974851i	$0.428461\pi$
$798$	0	0
$799$	41.7844	1.47823
$800$	34.5054	1.21995
$801$	0	0
$802$	29.5419	1.04316
$803$	−7.77007	−0.274200
$804$	0	0
$805$	3.68593	0.129912
$806$	32.1832	1.13360
$807$	0	0
$808$	−7.42850	−0.261334
$809$	−13.7132	−0.482129	−0.241065	−	0.970509i	$-0.577497\pi$
$809$	−13.7132	−0.482129	−0.241065	+	0.970509i	$0.577497\pi$
$810$	0	0
$811$	1.72288	0.0604985	0.0302493	−	0.999542i	$-0.490370\pi$
$811$	1.72288	0.0604985	0.0302493	+	0.999542i	$0.490370\pi$
$812$	29.4415	1.03319
$813$	0	0
$814$	3.76116	0.131828
$815$	−9.95260	−0.348624
$816$	0	0
$817$	−61.2933	−2.14438
$818$	−1.46290	−0.0511493
$819$	0	0
$820$	1.97158	0.0688506
$821$	−51.7042	−1.80449	−0.902244	−	0.431225i	$-0.858081\pi$
$821$	−51.7042	−1.80449	−0.902244	+	0.431225i	$0.858081\pi$
$822$	0	0
$823$	20.3847	0.710566	0.355283	−	0.934759i	$-0.384384\pi$
$823$	20.3847	0.710566	0.355283	+	0.934759i	$0.384384\pi$
$824$	8.25580	0.287604
$825$	0	0
$826$	−8.31129	−0.289187
$827$	−42.8995	−1.49176	−0.745881	−	0.666079i	$-0.767971\pi$
$827$	−42.8995	−1.49176	−0.745881	+	0.666079i	$0.767971\pi$
$828$	0	0
$829$	14.2775	0.495880	0.247940	−	0.968775i	$-0.420246\pi$
$829$	14.2775	0.495880	0.247940	+	0.968775i	$0.420246\pi$
$830$	9.45189	0.328080
$831$	0	0
$832$	66.1322	2.29272
$833$	−25.0167	−0.866776
$834$	0	0
$835$	2.36282	0.0817686
$836$	−21.0443	−0.727832
$837$	0	0
$838$	−43.2682	−1.49468
$839$	30.0586	1.03774	0.518869	−	0.854854i	$-0.326353\pi$
$839$	30.0586	1.03774	0.518869	+	0.854854i	$0.326353\pi$
$840$	0	0
$841$	5.64907	0.194795
$842$	27.3185	0.941456
$843$	0	0
$844$	−15.5187	−0.534175
$845$	9.98561	0.343515
$846$	0	0
$847$	17.4451	0.599421
$848$	16.9840	0.583231
$849$	0	0
$850$	−68.2650	−2.34147
$851$	4.45310	0.152650
$852$	0	0
$853$	36.5392	1.25108	0.625539	−	0.780193i	$-0.284879\pi$
$853$	36.5392	1.25108	0.625539	+	0.780193i	$0.284879\pi$
$854$	14.2894	0.488973
$855$	0	0
$856$	−20.7502	−0.709226
$857$	−46.1597	−1.57678	−0.788392	−	0.615173i	$-0.789086\pi$
$857$	−46.1597	−1.57678	−0.788392	+	0.615173i	$0.789086\pi$
$858$	0	0
$859$	−0.308649	−0.0105310	−0.00526549	−	0.999986i	$-0.501676\pi$
$859$	−0.308649	−0.0105310	−0.00526549	+	0.999986i	$0.501676\pi$
$860$	−17.0298	−0.580712
$861$	0	0
$862$	−47.0031	−1.60093
$863$	−32.6353	−1.11092	−0.555459	−	0.831544i	$-0.687457\pi$
$863$	−32.6353	−1.11092	−0.555459	+	0.831544i	$0.687457\pi$
$864$	0	0
$865$	8.58711	0.291970
$866$	8.70190	0.295703
$867$	0	0
$868$	13.8712	0.470818
$869$	17.2504	0.585178
$870$	0	0
$871$	−6.50562	−0.220434
$872$	2.83905	0.0961422
$873$	0	0
$874$	−43.1416	−1.45929
$875$	11.3323	0.383101
$876$	0	0
$877$	6.95284	0.234781	0.117390	−	0.993086i	$-0.462547\pi$
$877$	6.95284	0.234781	0.117390	+	0.993086i	$0.462547\pi$
$878$	−67.9094	−2.29183
$879$	0	0
$880$	1.55867	0.0525427
$881$	37.2923	1.25641	0.628204	−	0.778049i	$-0.283790\pi$
$881$	37.2923	1.25641	0.628204	+	0.778049i	$0.283790\pi$
$882$	0	0
$883$	6.18813	0.208247	0.104124	−	0.994564i	$-0.466796\pi$
$883$	6.18813	0.208247	0.104124	+	0.994564i	$0.466796\pi$
$884$	−99.8497	−3.35831
$885$	0	0
$886$	3.48603	0.117115
$887$	−18.6235	−0.625316	−0.312658	−	0.949866i	$-0.601219\pi$
$887$	−18.6235	−0.625316	−0.312658	+	0.949866i	$0.601219\pi$
$888$	0	0
$889$	20.8463	0.699163
$890$	−21.9166	−0.734646
$891$	0	0
$892$	20.5958	0.689599
$893$	38.8225	1.29914
$894$	0	0
$895$	0.270198	0.00903171
$896$	21.8016	0.728340
$897$	0	0
$898$	22.8566	0.762734
$899$	16.3247	0.544460
$900$	0	0
$901$	−58.3574	−1.94417
$902$	−2.93615	−0.0977632
$903$	0	0
$904$	22.1839	0.737824
$905$	13.4763	0.447967
$906$	0	0
$907$	26.1827	0.869381	0.434691	−	0.900580i	$-0.356858\pi$
$907$	26.1827	0.869381	0.434691	+	0.900580i	$0.356858\pi$
$908$	−37.3021	−1.23791
$909$	0	0
$910$	13.7242	0.454954
$911$	7.96133	0.263771	0.131885	−	0.991265i	$-0.457897\pi$
$911$	7.96133	0.263771	0.131885	+	0.991265i	$0.457897\pi$
$912$	0	0
$913$	−8.12947	−0.269046
$914$	89.5146	2.96088
$915$	0	0
$916$	43.7311	1.44492
$917$	26.6731	0.880824
$918$	0	0
$919$	55.0648	1.81642	0.908210	−	0.418515i	$-0.137449\pi$
$919$	55.0648	1.81642	0.908210	+	0.418515i	$0.137449\pi$
$920$	−3.21845	−0.106109
$921$	0	0
$922$	−74.6278	−2.45774
$923$	−43.4692	−1.43081
$924$	0	0
$925$	6.54685	0.215259
$926$	−72.5260	−2.38335
$927$	0	0
$928$	44.3276	1.45512
$929$	−43.8587	−1.43896	−0.719478	−	0.694515i	$-0.755619\pi$
$929$	−43.8587	−1.43896	−0.719478	+	0.694515i	$0.755619\pi$
$930$	0	0
$931$	−23.2433	−0.761768
$932$	−14.6526	−0.479962
$933$	0	0
$934$	22.6075	0.739741
$935$	−5.35563	−0.175148
$936$	0	0
$937$	33.2464	1.08611	0.543056	−	0.839696i	$-0.317267\pi$
$937$	33.2464	1.08611	0.543056	+	0.839696i	$0.317267\pi$
$938$	−4.85505	−0.158523
$939$	0	0
$940$	10.7865	0.351817
$941$	11.2079	0.365366	0.182683	−	0.983172i	$-0.441522\pi$
$941$	11.2079	0.365366	0.182683	+	0.983172i	$0.441522\pi$
$942$	0	0
$943$	−3.47632	−0.113205
$944$	−4.16117	−0.135434
$945$	0	0
$946$	25.3615	0.824572
$947$	4.73110	0.153740	0.0768700	−	0.997041i	$-0.475507\pi$
$947$	4.73110	0.153740	0.0768700	+	0.997041i	$0.475507\pi$
$948$	0	0
$949$	−34.2536	−1.11192
$950$	−63.4258	−2.05781
$951$	0	0
$952$	−20.0081	−0.648467
$953$	−1.88947	−0.0612061	−0.0306030	−	0.999532i	$-0.509743\pi$
$953$	−1.88947	−0.0612061	−0.0306030	+	0.999532i	$0.509743\pi$
$954$	0	0
$955$	4.45385	0.144123
$956$	−44.1387	−1.42755
$957$	0	0
$958$	44.1612	1.42678
$959$	0.678334	0.0219045
$960$	0	0
$961$	−23.3087	−0.751894
$962$	16.5807	0.534583
$963$	0	0
$964$	−63.9458	−2.05956
$965$	−5.65449	−0.182024
$966$	0	0
$967$	−55.8457	−1.79588	−0.897939	−	0.440120i	$-0.854936\pi$
$967$	−55.8457	−1.79588	−0.897939	+	0.440120i	$0.854936\pi$
$968$	−15.2326	−0.489593
$969$	0	0
$970$	−5.24579	−0.168432
$971$	−6.12547	−0.196576	−0.0982879	−	0.995158i	$-0.531337\pi$
$971$	−6.12547	−0.196576	−0.0982879	+	0.995158i	$0.531337\pi$
$972$	0	0
$973$	−27.3990	−0.878370
$974$	−59.9904	−1.92222
$975$	0	0
$976$	7.15419	0.229000
$977$	28.5472	0.913305	0.456652	−	0.889645i	$-0.349048\pi$
$977$	28.5472	0.913305	0.456652	+	0.889645i	$0.349048\pi$
$978$	0	0
$979$	18.8502	0.602456
$980$	−6.45795	−0.206292
$981$	0	0
$982$	22.2308	0.709414
$983$	−22.0774	−0.704160	−0.352080	−	0.935970i	$-0.614526\pi$
$983$	−22.0774	−0.704160	−0.352080	+	0.935970i	$0.614526\pi$
$984$	0	0
$985$	−5.46107	−0.174004
$986$	−87.6969	−2.79284
$987$	0	0
$988$	−92.7716	−2.95146
$989$	30.0272	0.954810
$990$	0	0
$991$	−35.5812	−1.13028	−0.565138	−	0.824997i	$-0.691177\pi$
$991$	−35.5812	−1.13028	−0.565138	+	0.824997i	$0.691177\pi$
$992$	20.8847	0.663089
$993$	0	0
$994$	−32.4404	−1.02895
$995$	8.15911	0.258661
$996$	0	0
$997$	36.8282	1.16636	0.583180	−	0.812343i	$-0.301808\pi$
$997$	36.8282	1.16636	0.583180	+	0.812343i	$0.301808\pi$
$998$	−19.0244	−0.602208
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6561.2.a.c.1.65		72
3.2	odd	2		6561.2.a.d.1.8		72
81.5	odd	54		243.2.g.a.73.1		144
81.11	odd	54		729.2.g.a.514.8		144
81.16	even	27		81.2.g.a.13.8	✓	144
81.22	even	27		729.2.g.d.217.1		144
81.32	odd	54		729.2.g.b.703.8		144
81.38	odd	54		729.2.g.b.28.8		144
81.43	even	27		729.2.g.c.28.1		144
81.49	even	27		729.2.g.c.703.1		144
81.59	odd	54		729.2.g.a.217.8		144
81.65	odd	54		243.2.g.a.10.1		144
81.70	even	27		729.2.g.d.514.1		144
81.76	even	27		81.2.g.a.25.8	yes	144

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
81.2.g.a.13.8	✓	144	81.16	even	27
81.2.g.a.25.8	yes	144	81.76	even	27
243.2.g.a.10.1		144	81.65	odd	54
243.2.g.a.73.1		144	81.5	odd	54
729.2.g.a.217.8		144	81.59	odd	54
729.2.g.a.514.8		144	81.11	odd	54
729.2.g.b.28.8		144	81.38	odd	54
729.2.g.b.703.8		144	81.32	odd	54
729.2.g.c.28.1		144	81.43	even	27
729.2.g.c.703.1		144	81.49	even	27
729.2.g.d.217.1		144	81.22	even	27
729.2.g.d.514.1		144	81.70	even	27
6561.2.a.c.1.65		72	1.1	even	1	trivial
6561.2.a.d.1.8		72	3.2	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q+2.17581 q^{2} +2.73413 q^{4} +0.646494 q^{5} -1.82934 q^{7} +1.59733 q^{8} +1.40664 q^{10} -1.20984 q^{11} -5.33346 q^{13} -3.98029 q^{14} -1.99279 q^{16} +6.84728 q^{17} +6.36190 q^{19} +1.76760 q^{20} -2.63238 q^{22} -3.11666 q^{23} -4.58205 q^{25} -11.6046 q^{26} -5.00165 q^{28} -5.88635 q^{29} -2.77332 q^{31} -7.53058 q^{32} +14.8984 q^{34} -1.18266 q^{35} -1.42881 q^{37} +13.8423 q^{38} +1.03266 q^{40} +1.11540 q^{41} -9.63443 q^{43} -3.30786 q^{44} -6.78124 q^{46} +6.10234 q^{47} -3.65352 q^{49} -9.96964 q^{50} -14.5824 q^{52} -8.52271 q^{53} -0.782154 q^{55} -2.92205 q^{56} -12.8075 q^{58} +2.08811 q^{59} -3.59004 q^{61} -6.03420 q^{62} -12.3995 q^{64} -3.44805 q^{65} +1.21977 q^{67} +18.7214 q^{68} -2.57323 q^{70} +8.15028 q^{71} +6.42239 q^{73} -3.10880 q^{74} +17.3943 q^{76} +2.21321 q^{77} -14.2584 q^{79} -1.28832 q^{80} +2.42689 q^{82} +6.71946 q^{83} +4.42672 q^{85} -20.9627 q^{86} -1.93251 q^{88} -15.5808 q^{89} +9.75671 q^{91} -8.52135 q^{92} +13.2775 q^{94} +4.11293 q^{95} -3.72929 q^{97} -7.94935 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 72 q - 9 q^{2} + 63 q^{4} - 18 q^{5} - 27 q^{8} - 36 q^{11} - 36 q^{14} + 45 q^{16} - 36 q^{17} - 54 q^{20} - 54 q^{23} + 36 q^{25} - 45 q^{26} + 9 q^{28} - 54 q^{29} - 63 q^{32} - 72 q^{35} - 54 q^{38}+ \cdots - 81 q^{98}+O(q^{100}) \) 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

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