LMFDB - Embedded newform 6561.2.a.d.1.56

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			1.56
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)

$f(q)$	$=$	\(q+1.73145 q^{2} +0.997916 q^{4} +1.37705 q^{5} +3.28825 q^{7} -1.73506 q^{8} +2.38429 q^{10} +0.205344 q^{11} +5.25480 q^{13} +5.69344 q^{14} -5.00000 q^{16} -5.07493 q^{17} -2.93751 q^{19} +1.37418 q^{20} +0.355543 q^{22} +3.76215 q^{23} -3.10373 q^{25} +9.09842 q^{26} +3.28140 q^{28} +6.30823 q^{29} +5.08133 q^{31} -5.18713 q^{32} -8.78699 q^{34} +4.52809 q^{35} +8.31960 q^{37} -5.08614 q^{38} -2.38926 q^{40} +2.76252 q^{41} -1.58659 q^{43} +0.204916 q^{44} +6.51398 q^{46} +2.08130 q^{47} +3.81261 q^{49} -5.37396 q^{50} +5.24385 q^{52} +9.03326 q^{53} +0.282769 q^{55} -5.70531 q^{56} +10.9224 q^{58} +14.8459 q^{59} -1.50670 q^{61} +8.79806 q^{62} +1.01875 q^{64} +7.23612 q^{65} -13.4089 q^{67} -5.06436 q^{68} +7.84015 q^{70} +4.67402 q^{71} -16.2828 q^{73} +14.4050 q^{74} -2.93138 q^{76} +0.675224 q^{77} -1.48696 q^{79} -6.88524 q^{80} +4.78316 q^{82} +14.2514 q^{83} -6.98844 q^{85} -2.74710 q^{86} -0.356284 q^{88} -5.14015 q^{89} +17.2791 q^{91} +3.75432 q^{92} +3.60367 q^{94} -4.04509 q^{95} -1.79542 q^{97} +6.60133 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	1.73145	1.22432	0.612160	−	0.790734i	$-0.290301\pi$
$2$	1.73145	1.22432	0.612160	+	0.790734i	$0.290301\pi$
$3$	0	0
$3$	0	0
$4$	0.997916	0.498958
$5$	1.37705	0.615835	0.307918	−	0.951413i	$-0.400368\pi$
$5$	1.37705	0.615835	0.307918	+	0.951413i	$0.400368\pi$
$6$	0	0
$7$	3.28825	1.24284	0.621421	−	0.783477i	$-0.286556\pi$
$7$	3.28825	1.24284	0.621421	+	0.783477i	$0.286556\pi$
$8$	−1.73506	−0.613435
$9$	0	0
$10$	2.38429	0.753979
$11$	0.205344	0.0619136	0.0309568	−	0.999521i	$-0.490145\pi$
$11$	0.205344	0.0619136	0.0309568	+	0.999521i	$0.490145\pi$
$12$	0	0
$13$	5.25480	1.45742	0.728709	−	0.684823i	$-0.240121\pi$
$13$	5.25480	1.45742	0.728709	+	0.684823i	$0.240121\pi$
$14$	5.69344	1.52164
$15$	0	0
$16$	−5.00000	−1.25000
$17$	−5.07493	−1.23085	−0.615426	−	0.788195i	$-0.711016\pi$
$17$	−5.07493	−1.23085	−0.615426	+	0.788195i	$0.711016\pi$
$18$	0	0
$19$	−2.93751	−0.673910	−0.336955	−	0.941521i	$-0.609397\pi$
$19$	−2.93751	−0.673910	−0.336955	+	0.941521i	$0.609397\pi$
$20$	1.37418	0.307276
$21$	0	0
$22$	0.355543	0.0758021
$23$	3.76215	0.784464	0.392232	−	0.919866i	$-0.371703\pi$
$23$	3.76215	0.784464	0.392232	+	0.919866i	$0.371703\pi$
$24$	0	0
$25$	−3.10373	−0.620747
$26$	9.09842	1.78435
$27$	0	0
$28$	3.28140	0.620127
$29$	6.30823	1.17141	0.585704	−	0.810525i	$-0.300818\pi$
$29$	6.30823	1.17141	0.585704	+	0.810525i	$0.300818\pi$
$30$	0	0
$31$	5.08133	0.912633	0.456317	−	0.889817i	$-0.349168\pi$
$31$	5.08133	0.912633	0.456317	+	0.889817i	$0.349168\pi$
$32$	−5.18713	−0.916963
$33$	0	0
$34$	−8.78699	−1.50696
$35$	4.52809	0.765387
$36$	0	0
$37$	8.31960	1.36773	0.683867	−	0.729606i	$-0.260297\pi$
$37$	8.31960	1.36773	0.683867	+	0.729606i	$0.260297\pi$
$38$	−5.08614	−0.825081
$39$	0	0
$40$	−2.38926	−0.377775
$41$	2.76252	0.431432	0.215716	−	0.976456i	$-0.430791\pi$
$41$	2.76252	0.431432	0.215716	+	0.976456i	$0.430791\pi$
$42$	0	0
$43$	−1.58659	−0.241953	−0.120977	−	0.992655i	$-0.538603\pi$
$43$	−1.58659	−0.241953	−0.120977	+	0.992655i	$0.538603\pi$
$44$	0.204916	0.0308923
$45$	0	0
$46$	6.51398	0.960434
$47$	2.08130	0.303589	0.151795	−	0.988412i	$-0.451495\pi$
$47$	2.08130	0.303589	0.151795	+	0.988412i	$0.451495\pi$
$48$	0	0
$49$	3.81261	0.544658
$50$	−5.37396	−0.759992
$51$	0	0
$52$	5.24385	0.727191
$53$	9.03326	1.24081	0.620407	−	0.784280i	$-0.286967\pi$
$53$	9.03326	1.24081	0.620407	+	0.784280i	$0.286967\pi$
$54$	0	0
$55$	0.282769	0.0381286
$56$	−5.70531	−0.762403
$57$	0	0
$58$	10.9224	1.43418
$59$	14.8459	1.93278	0.966389	−	0.257084i	$-0.0827618\pi$
$59$	14.8459	1.93278	0.966389	+	0.257084i	$0.0827618\pi$
$60$	0	0
$61$	−1.50670	−0.192913	−0.0964566	−	0.995337i	$-0.530751\pi$
$61$	−1.50670	−0.192913	−0.0964566	+	0.995337i	$0.530751\pi$
$62$	8.79806	1.11735
$63$	0	0
$64$	1.01875	0.127343
$65$	7.23612	0.897530
$66$	0	0
$67$	−13.4089	−1.63815	−0.819077	−	0.573683i	$-0.805514\pi$
$67$	−13.4089	−1.63815	−0.819077	+	0.573683i	$0.805514\pi$
$68$	−5.06436	−0.614144
$69$	0	0
$70$	7.84015	0.937078
$71$	4.67402	0.554704	0.277352	−	0.960768i	$-0.410543\pi$
$71$	4.67402	0.554704	0.277352	+	0.960768i	$0.410543\pi$
$72$	0	0
$73$	−16.2828	−1.90576	−0.952879	−	0.303350i	$-0.901895\pi$
$73$	−16.2828	−1.90576	−0.952879	+	0.303350i	$0.901895\pi$
$74$	14.4050	1.67454
$75$	0	0
$76$	−2.93138	−0.336253
$77$	0.675224	0.0769489
$78$	0	0
$79$	−1.48696	−0.167296	−0.0836478	−	0.996495i	$-0.526657\pi$
$79$	−1.48696	−0.167296	−0.0836478	+	0.996495i	$0.526657\pi$
$80$	−6.88524	−0.769794
$81$	0	0
$82$	4.78316	0.528211
$83$	14.2514	1.56430	0.782149	−	0.623091i	$-0.214123\pi$
$83$	14.2514	1.56430	0.782149	+	0.623091i	$0.214123\pi$
$84$	0	0
$85$	−6.98844	−0.758002
$86$	−2.74710	−0.296228
$87$	0	0
$88$	−0.356284	−0.0379800
$89$	−5.14015	−0.544854	−0.272427	−	0.962176i	$-0.587826\pi$
$89$	−5.14015	−0.544854	−0.272427	+	0.962176i	$0.587826\pi$
$90$	0	0
$91$	17.2791	1.81134
$92$	3.75432	0.391415
$93$	0	0
$94$	3.60367	0.371690
$95$	−4.04509	−0.415018
$96$	0	0
$97$	−1.79542	−0.182297	−0.0911486	−	0.995837i	$-0.529054\pi$
$97$	−1.79542	−0.182297	−0.0911486	+	0.995837i	$0.529054\pi$
$98$	6.60133	0.666835
$99$	0	0
$100$	−3.09727	−0.309727
$101$	11.4751	1.14181	0.570906	−	0.821016i	$-0.306592\pi$
$101$	11.4751	1.14181	0.570906	+	0.821016i	$0.306592\pi$
$102$	0	0
$103$	0.500196	0.0492858	0.0246429	−	0.999696i	$-0.492155\pi$
$103$	0.500196	0.0492858	0.0246429	+	0.999696i	$0.492155\pi$
$104$	−9.11737	−0.894032
$105$	0	0
$106$	15.6406	1.51915
$107$	−5.44365	−0.526258	−0.263129	−	0.964761i	$-0.584754\pi$
$107$	−5.44365	−0.526258	−0.263129	+	0.964761i	$0.584754\pi$
$108$	0	0
$109$	16.7796	1.60719	0.803597	−	0.595174i	$-0.202917\pi$
$109$	16.7796	1.60719	0.803597	+	0.595174i	$0.202917\pi$
$110$	0.489601	0.0466816
$111$	0	0
$112$	−16.4412	−1.55355
$113$	8.83243	0.830885	0.415443	−	0.909619i	$-0.363627\pi$
$113$	8.83243	0.830885	0.415443	+	0.909619i	$0.363627\pi$
$114$	0	0
$115$	5.18067	0.483100
$116$	6.29509	0.584484
$117$	0	0
$118$	25.7050	2.36634
$119$	−16.6877	−1.52976
$120$	0	0
$121$	−10.9578	−0.996167
$122$	−2.60877	−0.236187
$123$	0	0
$124$	5.07074	0.455366
$125$	−11.1592	−0.998113
$126$	0	0
$127$	−6.97791	−0.619189	−0.309595	−	0.950869i	$-0.600193\pi$
$127$	−6.97791	−0.619189	−0.309595	+	0.950869i	$0.600193\pi$
$128$	12.1382	1.07287
$129$	0	0
$130$	12.5290	1.09886
$131$	−5.63970	−0.492743	−0.246372	−	0.969175i	$-0.579238\pi$
$131$	−5.63970	−0.492743	−0.246372	+	0.969175i	$0.579238\pi$
$132$	0	0
$133$	−9.65926	−0.837564
$134$	−23.2168	−2.00562
$135$	0	0
$136$	8.80530	0.755048
$137$	1.11446	0.0952150	0.0476075	−	0.998866i	$-0.484840\pi$
$137$	1.11446	0.0952150	0.0476075	+	0.998866i	$0.484840\pi$
$138$	0	0
$139$	−4.75797	−0.403566	−0.201783	−	0.979430i	$-0.564674\pi$
$139$	−4.75797	−0.403566	−0.201783	+	0.979430i	$0.564674\pi$
$140$	4.51865	0.381896
$141$	0	0
$142$	8.09283	0.679135
$143$	1.07904	0.0902341
$144$	0	0
$145$	8.68675	0.721395
$146$	−28.1929	−2.33326
$147$	0	0
$148$	8.30227	0.682442
$149$	−7.67753	−0.628968	−0.314484	−	0.949263i	$-0.601831\pi$
$149$	−7.67753	−0.628968	−0.314484	+	0.949263i	$0.601831\pi$
$150$	0	0
$151$	3.08871	0.251356	0.125678	−	0.992071i	$-0.459889\pi$
$151$	3.08871	0.251356	0.125678	+	0.992071i	$0.459889\pi$
$152$	5.09674	0.413400
$153$	0	0
$154$	1.16912	0.0942100
$155$	6.99724	0.562032
$156$	0	0
$157$	6.13079	0.489291	0.244645	−	0.969613i	$-0.421328\pi$
$157$	6.13079	0.489291	0.244645	+	0.969613i	$0.421328\pi$
$158$	−2.57459	−0.204823
$159$	0	0
$160$	−7.14293	−0.564698
$161$	12.3709	0.974965
$162$	0	0
$163$	−0.171894	−0.0134638	−0.00673188	−	0.999977i	$-0.502143\pi$
$163$	−0.171894	−0.0134638	−0.00673188	+	0.999977i	$0.502143\pi$
$164$	2.75676	0.215267
$165$	0	0
$166$	24.6756	1.91520
$167$	0.294337	0.0227765	0.0113883	−	0.999935i	$-0.496375\pi$
$167$	0.294337	0.0227765	0.0113883	+	0.999935i	$0.496375\pi$
$168$	0	0
$169$	14.6129	1.12407
$170$	−12.1001	−0.928037
$171$	0	0
$172$	−1.58329	−0.120724
$173$	2.68695	0.204285	0.102143	−	0.994770i	$-0.467430\pi$
$173$	2.68695	0.204285	0.102143	+	0.994770i	$0.467430\pi$
$174$	0	0
$175$	−10.2059	−0.771491
$176$	−1.02672	−0.0773920
$177$	0	0
$178$	−8.89990	−0.667076
$179$	2.39354	0.178902	0.0894509	−	0.995991i	$-0.471489\pi$
$179$	2.39354	0.178902	0.0894509	+	0.995991i	$0.471489\pi$
$180$	0	0
$181$	16.7455	1.24468	0.622340	−	0.782747i	$-0.286182\pi$
$181$	16.7455	1.24468	0.622340	+	0.782747i	$0.286182\pi$
$182$	29.9179	2.21766
$183$	0	0
$184$	−6.52755	−0.481218
$185$	11.4565	0.842299
$186$	0	0
$187$	−1.04211	−0.0762065
$188$	2.07697	0.151478
$189$	0	0
$190$	−7.00387	−0.508114
$191$	−14.1908	−1.02681	−0.513405	−	0.858147i	$-0.671616\pi$
$191$	−14.1908	−1.02681	−0.513405	+	0.858147i	$0.671616\pi$
$192$	0	0
$193$	−4.17589	−0.300587	−0.150294	−	0.988641i	$-0.548022\pi$
$193$	−4.17589	−0.300587	−0.150294	+	0.988641i	$0.548022\pi$
$194$	−3.10868	−0.223190
$195$	0	0
$196$	3.80466	0.271762
$197$	−20.7182	−1.47611	−0.738057	−	0.674739i	$-0.764256\pi$
$197$	−20.7182	−1.47611	−0.738057	+	0.674739i	$0.764256\pi$
$198$	0	0
$199$	2.87638	0.203901	0.101951	−	0.994789i	$-0.467492\pi$
$199$	2.87638	0.203901	0.101951	+	0.994789i	$0.467492\pi$
$200$	5.38515	0.380788
$201$	0	0
$202$	19.8685	1.39794
$203$	20.7431	1.45588
$204$	0	0
$205$	3.80412	0.265691
$206$	0.866064	0.0603416
$207$	0	0
$208$	−26.2740	−1.82177
$209$	−0.603200	−0.0417242
$210$	0	0
$211$	−10.6139	−0.730690	−0.365345	−	0.930872i	$-0.619049\pi$
$211$	−10.6139	−0.730690	−0.365345	+	0.930872i	$0.619049\pi$
$212$	9.01444	0.619114
$213$	0	0
$214$	−9.42540	−0.644307
$215$	−2.18482	−0.149003
$216$	0	0
$217$	16.7087	1.13426
$218$	29.0530	1.96772
$219$	0	0
$220$	0.282180	0.0190246
$221$	−26.6677	−1.79387
$222$	0	0
$223$	−23.6751	−1.58540	−0.792700	−	0.609612i	$-0.791325\pi$
$223$	−23.6751	−1.58540	−0.792700	+	0.609612i	$0.791325\pi$
$224$	−17.0566	−1.13964
$225$	0	0
$226$	15.2929	1.01727
$227$	−0.459014	−0.0304658	−0.0152329	−	0.999884i	$-0.504849\pi$
$227$	−0.459014	−0.0304658	−0.0152329	+	0.999884i	$0.504849\pi$
$228$	0	0
$229$	15.9932	1.05686	0.528430	−	0.848977i	$-0.322781\pi$
$229$	15.9932	1.05686	0.528430	+	0.848977i	$0.322781\pi$
$230$	8.97008	0.591469
$231$	0	0
$232$	−10.9451	−0.718583
$233$	−7.94808	−0.520696	−0.260348	−	0.965515i	$-0.583837\pi$
$233$	−7.94808	−0.520696	−0.260348	+	0.965515i	$0.583837\pi$
$234$	0	0
$235$	2.86606	0.186961
$236$	14.8150	0.964375
$237$	0	0
$238$	−28.8938	−1.87291
$239$	21.2175	1.37244	0.686222	−	0.727392i	$-0.259268\pi$
$239$	21.2175	1.37244	0.686222	+	0.727392i	$0.259268\pi$
$240$	0	0
$241$	−18.0698	−1.16398	−0.581988	−	0.813197i	$-0.697725\pi$
$241$	−18.0698	−1.16398	−0.581988	+	0.813197i	$0.697725\pi$
$242$	−18.9729	−1.21963
$243$	0	0
$244$	−1.50356	−0.0962556
$245$	5.25015	0.335420
$246$	0	0
$247$	−15.4360	−0.982169
$248$	−8.81639	−0.559841
$249$	0	0
$250$	−19.3217	−1.22201
$251$	8.07484	0.509679	0.254840	−	0.966983i	$-0.417977\pi$
$251$	8.07484	0.509679	0.254840	+	0.966983i	$0.417977\pi$
$252$	0	0
$253$	0.772537	0.0485690
$254$	−12.0819	−0.758086
$255$	0	0
$256$	18.9791	1.18619
$257$	−16.5335	−1.03133	−0.515665	−	0.856790i	$-0.672455\pi$
$257$	−16.5335	−1.03133	−0.515665	+	0.856790i	$0.672455\pi$
$258$	0	0
$259$	27.3570	1.69988
$260$	7.22104	0.447830
$261$	0	0
$262$	−9.76486	−0.603275
$263$	0.0167896	0.00103529	0.000517645	−	1.00000i	$-0.499835\pi$
$263$	0.0167896	0.00103529	0.000517645		1.00000i	$0.499835\pi$
$264$	0	0
$265$	12.4393	0.764137
$266$	−16.7245	−1.02545
$267$	0	0
$268$	−13.3809	−0.817371
$269$	−6.32675	−0.385749	−0.192874	−	0.981223i	$-0.561781\pi$
$269$	−6.32675	−0.385749	−0.192874	+	0.981223i	$0.561781\pi$
$270$	0	0
$271$	19.9750	1.21340	0.606698	−	0.794932i	$-0.292494\pi$
$271$	19.9750	1.21340	0.606698	+	0.794932i	$0.292494\pi$
$272$	25.3746	1.53856
$273$	0	0
$274$	1.92964	0.116574
$275$	−0.637334	−0.0384327
$276$	0	0
$277$	6.25577	0.375873	0.187936	−	0.982181i	$-0.439820\pi$
$277$	6.25577	0.375873	0.187936	+	0.982181i	$0.439820\pi$
$278$	−8.23819	−0.494094
$279$	0	0
$280$	−7.85649	−0.469515
$281$	5.82158	0.347286	0.173643	−	0.984809i	$-0.444446\pi$
$281$	5.82158	0.347286	0.173643	+	0.984809i	$0.444446\pi$
$282$	0	0
$283$	−8.59696	−0.511036	−0.255518	−	0.966804i	$-0.582246\pi$
$283$	−8.59696	−0.511036	−0.255518	+	0.966804i	$0.582246\pi$
$284$	4.66428	0.276774
$285$	0	0
$286$	1.86831	0.110475
$287$	9.08385	0.536203
$288$	0	0
$289$	8.75494	0.514997
$290$	15.0407	0.883218
$291$	0	0
$292$	−16.2489	−0.950894
$293$	−6.43584	−0.375986	−0.187993	−	0.982170i	$-0.560198\pi$
$293$	−6.43584	−0.375986	−0.187993	+	0.982170i	$0.560198\pi$
$294$	0	0
$295$	20.4436	1.19027
$296$	−14.4350	−0.839017
$297$	0	0
$298$	−13.2933	−0.770057
$299$	19.7694	1.14329
$300$	0	0
$301$	−5.21712	−0.300710
$302$	5.34795	0.307740
$303$	0	0
$304$	14.6875	0.842387
$305$	−2.07480	−0.118803
$306$	0	0
$307$	24.5495	1.40111	0.700557	−	0.713596i	$-0.252935\pi$
$307$	24.5495	1.40111	0.700557	+	0.713596i	$0.252935\pi$
$308$	0.673817	0.0383943
$309$	0	0
$310$	12.1154	0.688107
$311$	12.9868	0.736413	0.368206	−	0.929744i	$-0.379972\pi$
$311$	12.9868	0.736413	0.368206	+	0.929744i	$0.379972\pi$
$312$	0	0
$313$	5.22809	0.295509	0.147754	−	0.989024i	$-0.452795\pi$
$313$	5.22809	0.295509	0.147754	+	0.989024i	$0.452795\pi$
$314$	10.6152	0.599048
$315$	0	0
$316$	−1.48386	−0.0834735
$317$	30.5569	1.71624	0.858122	−	0.513446i	$-0.171631\pi$
$317$	30.5569	1.71624	0.858122	+	0.513446i	$0.171631\pi$
$318$	0	0
$319$	1.29536	0.0725262
$320$	1.40287	0.0784226
$321$	0	0
$322$	21.4196	1.19367
$323$	14.9076	0.829483
$324$	0	0
$325$	−16.3095	−0.904688
$326$	−0.297625	−0.0164839
$327$	0	0
$328$	−4.79312	−0.264656
$329$	6.84385	0.377313
$330$	0	0
$331$	3.18367	0.174990	0.0874952	−	0.996165i	$-0.472114\pi$
$331$	3.18367	0.174990	0.0874952	+	0.996165i	$0.472114\pi$
$332$	14.2217	0.780520
$333$	0	0
$334$	0.509630	0.0278857
$335$	−18.4647	−1.00883
$336$	0	0
$337$	−32.7717	−1.78519	−0.892593	−	0.450863i	$-0.851116\pi$
$337$	−32.7717	−1.78519	−0.892593	+	0.450863i	$0.851116\pi$
$338$	25.3015	1.37622
$339$	0	0
$340$	−6.97387	−0.378211
$341$	1.04342	0.0565045
$342$	0	0
$343$	−10.4810	−0.565919
$344$	2.75283	0.148423
$345$	0	0
$346$	4.65232	0.250110
$347$	22.4072	1.20288	0.601441	−	0.798917i	$-0.294594\pi$
$347$	22.4072	1.20288	0.601441	+	0.798917i	$0.294594\pi$
$348$	0	0
$349$	22.0677	1.18126	0.590629	−	0.806944i	$-0.298880\pi$
$349$	22.0677	1.18126	0.590629	+	0.806944i	$0.298880\pi$
$350$	−17.6709	−0.944551
$351$	0	0
$352$	−1.06515	−0.0567725
$353$	10.9168	0.581043	0.290522	−	0.956868i	$-0.406171\pi$
$353$	10.9168	0.581043	0.290522	+	0.956868i	$0.406171\pi$
$354$	0	0
$355$	6.43636	0.341607
$356$	−5.12944	−0.271860
$357$	0	0
$358$	4.14430	0.219033
$359$	24.7892	1.30832	0.654162	−	0.756354i	$-0.273021\pi$
$359$	24.7892	1.30832	0.654162	+	0.756354i	$0.273021\pi$
$360$	0	0
$361$	−10.3711	−0.545845
$362$	28.9939	1.52389
$363$	0	0
$364$	17.2431	0.903784
$365$	−22.4222	−1.17363
$366$	0	0
$367$	−12.8486	−0.670689	−0.335345	−	0.942096i	$-0.608853\pi$
$367$	−12.8486	−0.670689	−0.335345	+	0.942096i	$0.608853\pi$
$368$	−18.8108	−0.980579
$369$	0	0
$370$	19.8364	1.03124
$371$	29.7036	1.54214
$372$	0	0
$373$	2.23067	0.115500	0.0577499	−	0.998331i	$-0.481607\pi$
$373$	2.23067	0.115500	0.0577499	+	0.998331i	$0.481607\pi$
$374$	−1.80436	−0.0933011
$375$	0	0
$376$	−3.61118	−0.186232
$377$	33.1485	1.70723
$378$	0	0
$379$	−7.35893	−0.378003	−0.189001	−	0.981977i	$-0.560525\pi$
$379$	−7.35893	−0.378003	−0.189001	+	0.981977i	$0.560525\pi$
$380$	−4.03666	−0.207076
$381$	0	0
$382$	−24.5706	−1.25714
$383$	−14.4561	−0.738675	−0.369337	−	0.929295i	$-0.620415\pi$
$383$	−14.4561	−0.738675	−0.369337	+	0.929295i	$0.620415\pi$
$384$	0	0
$385$	0.929817	0.0473879
$386$	−7.23034	−0.368015
$387$	0	0
$388$	−1.79168	−0.0909587
$389$	8.50413	0.431177	0.215588	−	0.976484i	$-0.430833\pi$
$389$	8.50413	0.431177	0.215588	+	0.976484i	$0.430833\pi$
$390$	0	0
$391$	−19.0927	−0.965558
$392$	−6.61509	−0.334112
$393$	0	0
$394$	−35.8726	−1.80723
$395$	−2.04761	−0.103027
$396$	0	0
$397$	5.49229	0.275650	0.137825	−	0.990457i	$-0.455989\pi$
$397$	5.49229	0.275650	0.137825	+	0.990457i	$0.455989\pi$
$398$	4.98030	0.249640
$399$	0	0
$400$	15.5187	0.775933
$401$	−9.40888	−0.469857	−0.234929	−	0.972013i	$-0.575486\pi$
$401$	−9.40888	−0.469857	−0.234929	+	0.972013i	$0.575486\pi$
$402$	0	0
$403$	26.7013	1.33009
$404$	11.4512	0.569716
$405$	0	0
$406$	35.9155	1.78246
$407$	1.70838	0.0846814
$408$	0	0
$409$	−15.4217	−0.762555	−0.381278	−	0.924461i	$-0.624516\pi$
$409$	−15.4217	−0.762555	−0.381278	+	0.924461i	$0.624516\pi$
$410$	6.58664	0.325291
$411$	0	0
$412$	0.499154	0.0245916
$413$	48.8172	2.40214
$414$	0	0
$415$	19.6249	0.963350
$416$	−27.2573	−1.33640
$417$	0	0
$418$	−1.04441	−0.0510838
$419$	−24.7048	−1.20691	−0.603453	−	0.797398i	$-0.706209\pi$
$419$	−24.7048	−1.20691	−0.603453	+	0.797398i	$0.706209\pi$
$420$	0	0
$421$	−0.185793	−0.00905501	−0.00452750	−	0.999990i	$-0.501441\pi$
$421$	−0.185793	−0.00905501	−0.00452750	+	0.999990i	$0.501441\pi$
$422$	−18.3774	−0.894598
$423$	0	0
$424$	−15.6732	−0.761159
$425$	15.7512	0.764047
$426$	0	0
$427$	−4.95441	−0.239761
$428$	−5.43231	−0.262581
$429$	0	0
$430$	−3.78290	−0.182428
$431$	14.5522	0.700956	0.350478	−	0.936571i	$-0.386019\pi$
$431$	14.5522	0.700956	0.350478	+	0.936571i	$0.386019\pi$
$432$	0	0
$433$	−34.6192	−1.66369	−0.831846	−	0.555006i	$-0.812716\pi$
$433$	−34.6192	−1.66369	−0.831846	+	0.555006i	$0.812716\pi$
$434$	28.9302	1.38870
$435$	0	0
$436$	16.7446	0.801923
$437$	−11.0513	−0.528658
$438$	0	0
$439$	−1.96580	−0.0938227	−0.0469114	−	0.998899i	$-0.514938\pi$
$439$	−1.96580	−0.0938227	−0.0469114	+	0.998899i	$0.514938\pi$
$440$	−0.490621	−0.0233894
$441$	0	0
$442$	−46.1738	−2.19627
$443$	−31.8086	−1.51127	−0.755635	−	0.654993i	$-0.772672\pi$
$443$	−31.8086	−1.51127	−0.755635	+	0.654993i	$0.772672\pi$
$444$	0	0
$445$	−7.07824	−0.335541
$446$	−40.9922	−1.94104
$447$	0	0
$448$	3.34990	0.158268
$449$	29.7873	1.40575	0.702875	−	0.711313i	$-0.251899\pi$
$449$	29.7873	1.40575	0.702875	+	0.711313i	$0.251899\pi$
$450$	0	0
$451$	0.567267	0.0267116
$452$	8.81403	0.414577
$453$	0	0
$454$	−0.794759	−0.0372999
$455$	23.7942	1.11549
$456$	0	0
$457$	12.7464	0.596252	0.298126	−	0.954527i	$-0.403638\pi$
$457$	12.7464	0.596252	0.298126	+	0.954527i	$0.403638\pi$
$458$	27.6914	1.29394
$459$	0	0
$460$	5.16988	0.241047
$461$	−20.6680	−0.962606	−0.481303	−	0.876554i	$-0.659836\pi$
$461$	−20.6680	−0.962606	−0.481303	+	0.876554i	$0.659836\pi$
$462$	0	0
$463$	−16.6502	−0.773801	−0.386900	−	0.922121i	$-0.626454\pi$
$463$	−16.6502	−0.773801	−0.386900	+	0.922121i	$0.626454\pi$
$464$	−31.5411	−1.46426
$465$	0	0
$466$	−13.7617	−0.637498
$467$	29.2955	1.35563	0.677817	−	0.735231i	$-0.262926\pi$
$467$	29.2955	1.35563	0.677817	+	0.735231i	$0.262926\pi$
$468$	0	0
$469$	−44.0918	−2.03597
$470$	4.96243	0.228900
$471$	0	0
$472$	−25.7586	−1.18563
$473$	−0.325798	−0.0149802
$474$	0	0
$475$	9.11723	0.418327
$476$	−16.6529	−0.763284
$477$	0	0
$478$	36.7370	1.68031
$479$	−22.3619	−1.02174	−0.510871	−	0.859657i	$-0.670677\pi$
$479$	−22.3619	−1.02174	−0.510871	+	0.859657i	$0.670677\pi$
$480$	0	0
$481$	43.7178	1.99336
$482$	−31.2869	−1.42508
$483$	0	0
$484$	−10.9350	−0.497046
$485$	−2.47238	−0.112265
$486$	0	0
$487$	3.73936	0.169447	0.0847233	−	0.996405i	$-0.472999\pi$
$487$	3.73936	0.169447	0.0847233	+	0.996405i	$0.472999\pi$
$488$	2.61421	0.118340
$489$	0	0
$490$	9.09036	0.410661
$491$	1.20669	0.0544573	0.0272286	−	0.999629i	$-0.491332\pi$
$491$	1.20669	0.0544573	0.0272286	+	0.999629i	$0.491332\pi$
$492$	0	0
$493$	−32.0138	−1.44183
$494$	−26.7266	−1.20249
$495$	0	0
$496$	−25.4066	−1.14079
$497$	15.3694	0.689410
$498$	0	0
$499$	3.33790	0.149425	0.0747124	−	0.997205i	$-0.476196\pi$
$499$	3.33790	0.149425	0.0747124	+	0.997205i	$0.476196\pi$
$500$	−11.1360	−0.498017
$501$	0	0
$502$	13.9812	0.624010
$503$	−24.4252	−1.08907	−0.544534	−	0.838739i	$-0.683293\pi$
$503$	−24.4252	−1.08907	−0.544534	+	0.838739i	$0.683293\pi$
$504$	0	0
$505$	15.8017	0.703168
$506$	1.33761	0.0594640
$507$	0	0
$508$	−6.96337	−0.308950
$509$	−30.9390	−1.37135	−0.685674	−	0.727909i	$-0.740493\pi$
$509$	−30.9390	−1.37135	−0.685674	+	0.727909i	$0.740493\pi$
$510$	0	0
$511$	−53.5420	−2.36856
$512$	8.58505	0.379409
$513$	0	0
$514$	−28.6269	−1.26268
$515$	0.688795	0.0303519
$516$	0	0
$517$	0.427383	0.0187963
$518$	47.3672	2.08119
$519$	0	0
$520$	−12.5551	−0.550577
$521$	−14.2527	−0.624421	−0.312210	−	0.950013i	$-0.601069\pi$
$521$	−14.2527	−0.624421	−0.312210	+	0.950013i	$0.601069\pi$
$522$	0	0
$523$	1.35308	0.0591662	0.0295831	−	0.999562i	$-0.490582\pi$
$523$	1.35308	0.0591662	0.0295831	+	0.999562i	$0.490582\pi$
$524$	−5.62795	−0.245858
$525$	0	0
$526$	0.0290703	0.00126753
$527$	−25.7874	−1.12332
$528$	0	0
$529$	−8.84619	−0.384617
$530$	21.5379	0.935548
$531$	0	0
$532$	−9.63913	−0.417909
$533$	14.5165	0.628778
$534$	0	0
$535$	−7.49618	−0.324088
$536$	23.2652	1.00490
$537$	0	0
$538$	−10.9544	−0.472280
$539$	0.782897	0.0337218
$540$	0	0
$541$	−25.0121	−1.07535	−0.537676	−	0.843151i	$-0.680698\pi$
$541$	−25.0121	−1.07535	−0.537676	+	0.843151i	$0.680698\pi$
$542$	34.5857	1.48558
$543$	0	0
$544$	26.3243	1.12865
$545$	23.1063	0.989767
$546$	0	0
$547$	28.3991	1.21426	0.607128	−	0.794604i	$-0.292321\pi$
$547$	28.3991	1.21426	0.607128	+	0.794604i	$0.292321\pi$
$548$	1.11214	0.0475083
$549$	0	0
$550$	−1.10351	−0.0470539
$551$	−18.5305	−0.789424
$552$	0	0
$553$	−4.88949	−0.207922
$554$	10.8315	0.460188
$555$	0	0
$556$	−4.74806	−0.201363
$557$	−36.8866	−1.56294	−0.781468	−	0.623945i	$-0.785529\pi$
$557$	−36.8866	−1.56294	−0.781468	+	0.623945i	$0.785529\pi$
$558$	0	0
$559$	−8.33722	−0.352627
$560$	−22.6404	−0.956732
$561$	0	0
$562$	10.0798	0.425189
$563$	−22.1570	−0.933806	−0.466903	−	0.884309i	$-0.654630\pi$
$563$	−22.1570	−0.933806	−0.466903	+	0.884309i	$0.654630\pi$
$564$	0	0
$565$	12.1627	0.511689
$566$	−14.8852	−0.625671
$567$	0	0
$568$	−8.10969	−0.340275
$569$	−6.40315	−0.268434	−0.134217	−	0.990952i	$-0.542852\pi$
$569$	−6.40315	−0.268434	−0.134217	+	0.990952i	$0.542852\pi$
$570$	0	0
$571$	−5.52381	−0.231164	−0.115582	−	0.993298i	$-0.536873\pi$
$571$	−5.52381	−0.231164	−0.115582	+	0.993298i	$0.536873\pi$
$572$	1.07679	0.0450230
$573$	0	0
$574$	15.7282	0.656483
$575$	−11.6767	−0.486953
$576$	0	0
$577$	−39.1190	−1.62855	−0.814273	−	0.580482i	$-0.802864\pi$
$577$	−39.1190	−1.62855	−0.814273	+	0.580482i	$0.802864\pi$
$578$	15.1587	0.630520
$579$	0	0
$580$	8.66865	0.359946
$581$	46.8623	1.94418
$582$	0	0
$583$	1.85493	0.0768233
$584$	28.2516	1.16906
$585$	0	0
$586$	−11.1433	−0.460327
$587$	−13.9082	−0.574051	−0.287026	−	0.957923i	$-0.592666\pi$
$587$	−13.9082	−0.574051	−0.287026	+	0.957923i	$0.592666\pi$
$588$	0	0
$589$	−14.9264	−0.615033
$590$	35.3971	1.45727
$591$	0	0
$592$	−41.5980	−1.70967
$593$	17.3598	0.712881	0.356441	−	0.934318i	$-0.383990\pi$
$593$	17.3598	0.712881	0.356441	+	0.934318i	$0.383990\pi$
$594$	0	0
$595$	−22.9797	−0.942078
$596$	−7.66153	−0.313829
$597$	0	0
$598$	34.2296	1.39975
$599$	−41.5732	−1.69864	−0.849318	−	0.527882i	$-0.822986\pi$
$599$	−41.5732	−1.69864	−0.849318	+	0.527882i	$0.822986\pi$
$600$	0	0
$601$	−9.23455	−0.376685	−0.188343	−	0.982103i	$-0.560312\pi$
$601$	−9.23455	−0.376685	−0.188343	+	0.982103i	$0.560312\pi$
$602$	−9.03317	−0.368165
$603$	0	0
$604$	3.08228	0.125416
$605$	−15.0895	−0.613475
$606$	0	0
$607$	−13.4897	−0.547531	−0.273766	−	0.961796i	$-0.588269\pi$
$607$	−13.4897	−0.547531	−0.273766	+	0.961796i	$0.588269\pi$
$608$	15.2372	0.617950
$609$	0	0
$610$	−3.59241	−0.145453
$611$	10.9368	0.442456
$612$	0	0
$613$	−3.88721	−0.157003	−0.0785015	−	0.996914i	$-0.525014\pi$
$613$	−3.88721	−0.157003	−0.0785015	+	0.996914i	$0.525014\pi$
$614$	42.5062	1.71541
$615$	0	0
$616$	−1.17155	−0.0472032
$617$	−24.0144	−0.966783	−0.483392	−	0.875404i	$-0.660595\pi$
$617$	−24.0144	−0.966783	−0.483392	+	0.875404i	$0.660595\pi$
$618$	0	0
$619$	−10.1787	−0.409115	−0.204557	−	0.978855i	$-0.565575\pi$
$619$	−10.1787	−0.409115	−0.204557	+	0.978855i	$0.565575\pi$
$620$	6.98266	0.280430
$621$	0	0
$622$	22.4860	0.901605
$623$	−16.9021	−0.677168
$624$	0	0
$625$	0.151831	0.00607326
$626$	9.05217	0.361797
$627$	0	0
$628$	6.11802	0.244136
$629$	−42.2214	−1.68348
$630$	0	0
$631$	−49.2350	−1.96001	−0.980007	−	0.198965i	$-0.936242\pi$
$631$	−49.2350	−1.96001	−0.980007	+	0.198965i	$0.936242\pi$
$632$	2.57995	0.102625
$633$	0	0
$634$	52.9076	2.10123
$635$	−9.60893	−0.381319
$636$	0	0
$637$	20.0345	0.793795
$638$	2.24285	0.0887952
$639$	0	0
$640$	16.7149	0.660713
$641$	34.0792	1.34605	0.673024	−	0.739621i	$-0.264995\pi$
$641$	34.0792	1.34605	0.673024	+	0.739621i	$0.264995\pi$
$642$	0	0
$643$	−19.8762	−0.783842	−0.391921	−	0.919999i	$-0.628189\pi$
$643$	−19.8762	−0.783842	−0.391921	+	0.919999i	$0.628189\pi$
$644$	12.3451	0.486467
$645$	0	0
$646$	25.8118	1.01555
$647$	37.1636	1.46105	0.730525	−	0.682886i	$-0.239275\pi$
$647$	37.1636	1.46105	0.730525	+	0.682886i	$0.239275\pi$
$648$	0	0
$649$	3.04853	0.119665
$650$	−28.2391	−1.10763
$651$	0	0
$652$	−0.171536	−0.00671785
$653$	5.50755	0.215527	0.107763	−	0.994177i	$-0.465631\pi$
$653$	5.50755	0.215527	0.107763	+	0.994177i	$0.465631\pi$
$654$	0	0
$655$	−7.76615	−0.303449
$656$	−13.8126	−0.539290
$657$	0	0
$658$	11.8498	0.461952
$659$	−38.0834	−1.48352	−0.741759	−	0.670666i	$-0.766008\pi$
$659$	−38.0834	−1.48352	−0.741759	+	0.670666i	$0.766008\pi$
$660$	0	0
$661$	−28.1777	−1.09598	−0.547992	−	0.836483i	$-0.684608\pi$
$661$	−28.1777	−1.09598	−0.547992	+	0.836483i	$0.684608\pi$
$662$	5.51237	0.214244
$663$	0	0
$664$	−24.7271	−0.959596
$665$	−13.3013	−0.515802
$666$	0	0
$667$	23.7325	0.918928
$668$	0.293724	0.0113645
$669$	0	0
$670$	−31.9707	−1.23513
$671$	−0.309392	−0.0119440
$672$	0	0
$673$	24.8452	0.957714	0.478857	−	0.877893i	$-0.341051\pi$
$673$	24.8452	0.957714	0.478857	+	0.877893i	$0.341051\pi$
$674$	−56.7425	−2.18564
$675$	0	0
$676$	14.5825	0.560864
$677$	−35.2849	−1.35611	−0.678055	−	0.735011i	$-0.737177\pi$
$677$	−35.2849	−1.35611	−0.678055	+	0.735011i	$0.737177\pi$
$678$	0	0
$679$	−5.90379	−0.226567
$680$	12.1253	0.464985
$681$	0	0
$682$	1.80663	0.0691795
$683$	39.5379	1.51288	0.756438	−	0.654065i	$-0.226938\pi$
$683$	39.5379	1.51288	0.756438	+	0.654065i	$0.226938\pi$
$684$	0	0
$685$	1.53467	0.0586368
$686$	−18.1472	−0.692865
$687$	0	0
$688$	7.93295	0.302441
$689$	47.4680	1.80839
$690$	0	0
$691$	−16.9909	−0.646366	−0.323183	−	0.946337i	$-0.604753\pi$
$691$	−16.9909	−0.646366	−0.323183	+	0.946337i	$0.604753\pi$
$692$	2.68136	0.101930
$693$	0	0
$694$	38.7969	1.47271
$695$	−6.55197	−0.248530
$696$	0	0
$697$	−14.0196	−0.531030
$698$	38.2091	1.44624
$699$	0	0
$700$	−10.1846	−0.384942
$701$	32.7483	1.23688	0.618442	−	0.785830i	$-0.287764\pi$
$701$	32.7483	1.23688	0.618442	+	0.785830i	$0.287764\pi$
$702$	0	0
$703$	−24.4389	−0.921730
$704$	0.209194	0.00788430
$705$	0	0
$706$	18.9019	0.711383
$707$	37.7329	1.41909
$708$	0	0
$709$	32.8193	1.23255	0.616277	−	0.787529i	$-0.288640\pi$
$709$	32.8193	1.23255	0.616277	+	0.787529i	$0.288640\pi$
$710$	11.1442	0.418236
$711$	0	0
$712$	8.91845	0.334233
$713$	19.1167	0.715928
$714$	0	0
$715$	1.48590	0.0555693
$716$	2.38856	0.0892645
$717$	0	0
$718$	42.9213	1.60181
$719$	−16.8406	−0.628049	−0.314025	−	0.949415i	$-0.601677\pi$
$719$	−16.8406	−0.628049	−0.314025	+	0.949415i	$0.601677\pi$
$720$	0	0
$721$	1.64477	0.0612545
$722$	−17.9570	−0.668289
$723$	0	0
$724$	16.7106	0.621044
$725$	−19.5791	−0.727148
$726$	0	0
$727$	−8.40284	−0.311644	−0.155822	−	0.987785i	$-0.549803\pi$
$727$	−8.40284	−0.311644	−0.155822	+	0.987785i	$0.549803\pi$
$728$	−29.9802	−1.11114
$729$	0	0
$730$	−38.8230	−1.43690
$731$	8.05185	0.297808
$732$	0	0
$733$	12.3466	0.456032	0.228016	−	0.973657i	$-0.426776\pi$
$733$	12.3466	0.456032	0.228016	+	0.973657i	$0.426776\pi$
$734$	−22.2466	−0.821138
$735$	0	0
$736$	−19.5148	−0.719324
$737$	−2.75344	−0.101424
$738$	0	0
$739$	−39.7974	−1.46397	−0.731985	−	0.681321i	$-0.761406\pi$
$739$	−39.7974	−1.46397	−0.731985	+	0.681321i	$0.761406\pi$
$740$	11.4326	0.420272
$741$	0	0
$742$	51.4304	1.88807
$743$	−19.8382	−0.727794	−0.363897	−	0.931439i	$-0.618554\pi$
$743$	−19.8382	−0.727794	−0.363897	+	0.931439i	$0.618554\pi$
$744$	0	0
$745$	−10.5723	−0.387341
$746$	3.86229	0.141409
$747$	0	0
$748$	−1.03994	−0.0380239
$749$	−17.9001	−0.654055
$750$	0	0
$751$	38.3777	1.40042	0.700212	−	0.713935i	$-0.253089\pi$
$751$	38.3777	1.40042	0.700212	+	0.713935i	$0.253089\pi$
$752$	−10.4065	−0.379486
$753$	0	0
$754$	57.3949	2.09020
$755$	4.25331	0.154794
$756$	0	0
$757$	−14.1489	−0.514249	−0.257125	−	0.966378i	$-0.582775\pi$
$757$	−14.1489	−0.514249	−0.257125	+	0.966378i	$0.582775\pi$
$758$	−12.7416	−0.462796
$759$	0	0
$760$	7.01846	0.254586
$761$	−9.47907	−0.343616	−0.171808	−	0.985130i	$-0.554961\pi$
$761$	−9.47907	−0.343616	−0.171808	+	0.985130i	$0.554961\pi$
$762$	0	0
$763$	55.1756	1.99749
$764$	−14.1612	−0.512335
$765$	0	0
$766$	−25.0301	−0.904374
$767$	78.0125	2.81687
$768$	0	0
$769$	−12.5213	−0.451529	−0.225764	−	0.974182i	$-0.572488\pi$
$769$	−12.5213	−0.451529	−0.225764	+	0.974182i	$0.572488\pi$
$770$	1.60993	0.0580179
$771$	0	0
$772$	−4.16719	−0.149980
$773$	−35.9038	−1.29137	−0.645684	−	0.763604i	$-0.723428\pi$
$773$	−35.9038	−1.29137	−0.645684	+	0.763604i	$0.723428\pi$
$774$	0	0
$775$	−15.7711	−0.566514
$776$	3.11516	0.111828
$777$	0	0
$778$	14.7245	0.527898
$779$	−8.11490	−0.290747
$780$	0	0
$781$	0.959784	0.0343438
$782$	−33.0580	−1.18215
$783$	0	0
$784$	−19.0630	−0.680822
$785$	8.44241	0.301322
$786$	0	0
$787$	39.9512	1.42411	0.712053	−	0.702126i	$-0.247766\pi$
$787$	39.9512	1.42411	0.712053	+	0.702126i	$0.247766\pi$
$788$	−20.6751	−0.736519
$789$	0	0
$790$	−3.54534	−0.126137
$791$	29.0433	1.03266
$792$	0	0
$793$	−7.91740	−0.281155
$794$	9.50962	0.337484
$795$	0	0
$796$	2.87039	0.101738
$797$	34.0861	1.20739	0.603696	−	0.797214i	$-0.293694\pi$
$797$	34.0861	1.20739	0.603696	+	0.797214i	$0.293694\pi$
$798$	0	0
$799$	−10.5625	−0.373673
$800$	16.0995	0.569202
$801$	0	0
$802$	−16.2910	−0.575255
$803$	−3.34358	−0.117992
$804$	0	0
$805$	17.0354	0.600418
$806$	46.2320	1.62845
$807$	0	0
$808$	−19.9099	−0.700427
$809$	−7.57622	−0.266366	−0.133183	−	0.991091i	$-0.542520\pi$
$809$	−7.57622	−0.266366	−0.133183	+	0.991091i	$0.542520\pi$
$810$	0	0
$811$	20.5558	0.721810	0.360905	−	0.932602i	$-0.382468\pi$
$811$	20.5558	0.721810	0.360905	+	0.932602i	$0.382468\pi$
$812$	20.6998	0.726422
$813$	0	0
$814$	2.95798	0.103677
$815$	−0.236706	−0.00829146
$816$	0	0
$817$	4.66062	0.163055
$818$	−26.7019	−0.933612
$819$	0	0
$820$	3.79620	0.132569
$821$	30.4750	1.06359	0.531793	−	0.846874i	$-0.321518\pi$
$821$	30.4750	1.06359	0.531793	+	0.846874i	$0.321518\pi$
$822$	0	0
$823$	−22.3183	−0.777968	−0.388984	−	0.921245i	$-0.627174\pi$
$823$	−22.3183	−0.777968	−0.388984	+	0.921245i	$0.627174\pi$
$824$	−0.867869	−0.0302336
$825$	0	0
$826$	84.5246	2.94099
$827$	35.5735	1.23701	0.618506	−	0.785780i	$-0.287738\pi$
$827$	35.5735	1.23701	0.618506	+	0.785780i	$0.287738\pi$
$828$	0	0
$829$	−17.5603	−0.609894	−0.304947	−	0.952369i	$-0.598639\pi$
$829$	−17.5603	−0.609894	−0.304947	+	0.952369i	$0.598639\pi$
$830$	33.9796	1.17945
$831$	0	0
$832$	5.35331	0.185593
$833$	−19.3487	−0.670393
$834$	0	0
$835$	0.405317	0.0140266
$836$	−0.601943	−0.0208186
$837$	0	0
$838$	−42.7751	−1.47764
$839$	38.3315	1.32335	0.661676	−	0.749790i	$-0.269845\pi$
$839$	38.3315	1.32335	0.661676	+	0.749790i	$0.269845\pi$
$840$	0	0
$841$	10.7938	0.372199
$842$	−0.321692	−0.0110862
$843$	0	0
$844$	−10.5918	−0.364584
$845$	20.1227	0.692242
$846$	0	0
$847$	−36.0321	−1.23808
$848$	−45.1663	−1.55102
$849$	0	0
$850$	27.2725	0.935438
$851$	31.2996	1.07294
$852$	0	0
$853$	8.39467	0.287428	0.143714	−	0.989619i	$-0.454095\pi$
$853$	8.39467	0.287428	0.143714	+	0.989619i	$0.454095\pi$
$854$	−8.57831	−0.293544
$855$	0	0
$856$	9.44504	0.322825
$857$	−48.1216	−1.64380	−0.821900	−	0.569631i	$-0.807086\pi$
$857$	−48.1216	−1.64380	−0.821900	+	0.569631i	$0.807086\pi$
$858$	0	0
$859$	51.0803	1.74284	0.871419	−	0.490539i	$-0.163200\pi$
$859$	51.0803	1.74284	0.871419	+	0.490539i	$0.163200\pi$
$860$	−2.18026	−0.0743464
$861$	0	0
$862$	25.1965	0.858195
$863$	−36.7771	−1.25191	−0.625953	−	0.779861i	$-0.715290\pi$
$863$	−36.7771	−1.25191	−0.625953	+	0.779861i	$0.715290\pi$
$864$	0	0
$865$	3.70007	0.125806
$866$	−59.9414	−2.03689
$867$	0	0
$868$	16.6739	0.565948
$869$	−0.305338	−0.0103579
$870$	0	0
$871$	−70.4609	−2.38748
$872$	−29.1136	−0.985909
$873$	0	0
$874$	−19.1349	−0.647246
$875$	−36.6944	−1.24050
$876$	0	0
$877$	−4.05984	−0.137091	−0.0685456	−	0.997648i	$-0.521836\pi$
$877$	−4.05984	−0.137091	−0.0685456	+	0.997648i	$0.521836\pi$
$878$	−3.40369	−0.114869
$879$	0	0
$880$	−1.41385	−0.0476607
$881$	40.6754	1.37039	0.685194	−	0.728361i	$-0.259717\pi$
$881$	40.6754	1.37039	0.685194	+	0.728361i	$0.259717\pi$
$882$	0	0
$883$	26.1833	0.881139	0.440570	−	0.897718i	$-0.354776\pi$
$883$	26.1833	0.881139	0.440570	+	0.897718i	$0.354776\pi$
$884$	−26.6122	−0.895064
$885$	0	0
$886$	−55.0749	−1.85028
$887$	−21.7392	−0.729932	−0.364966	−	0.931021i	$-0.618919\pi$
$887$	−21.7392	−0.729932	−0.364966	+	0.931021i	$0.618919\pi$
$888$	0	0
$889$	−22.9451	−0.769555
$890$	−12.2556	−0.410809
$891$	0	0
$892$	−23.6257	−0.791048
$893$	−6.11383	−0.204592
$894$	0	0
$895$	3.29603	0.110174
$896$	39.9133	1.33341
$897$	0	0
$898$	51.5752	1.72109
$899$	32.0542	1.06907
$900$	0	0
$901$	−45.8432	−1.52726
$902$	0.982194	0.0327035
$903$	0	0
$904$	−15.3248	−0.509694
$905$	23.0593	0.766519
$906$	0	0
$907$	−52.7124	−1.75029	−0.875143	−	0.483864i	$-0.839233\pi$
$907$	−52.7124	−1.75029	−0.875143	+	0.483864i	$0.839233\pi$
$908$	−0.458058	−0.0152012
$909$	0	0
$910$	41.1984	1.36571
$911$	−43.0525	−1.42639	−0.713196	−	0.700964i	$-0.752753\pi$
$911$	−43.0525	−1.42639	−0.713196	+	0.700964i	$0.752753\pi$
$912$	0	0
$913$	2.92645	0.0968514
$914$	22.0698	0.730003
$915$	0	0
$916$	15.9599	0.527329
$917$	−18.5448	−0.612402
$918$	0	0
$919$	50.4271	1.66344	0.831719	−	0.555197i	$-0.187357\pi$
$919$	50.4271	1.66344	0.831719	+	0.555197i	$0.187357\pi$
$920$	−8.98877	−0.296351
$921$	0	0
$922$	−35.7856	−1.17854
$923$	24.5610	0.808436
$924$	0	0
$925$	−25.8218	−0.849017
$926$	−28.8290	−0.947380
$927$	0	0
$928$	−32.7216	−1.07414
$929$	15.8902	0.521339	0.260670	−	0.965428i	$-0.416057\pi$
$929$	15.8902	0.521339	0.260670	+	0.965428i	$0.416057\pi$
$930$	0	0
$931$	−11.1995	−0.367050
$932$	−7.93152	−0.259806
$933$	0	0
$934$	50.7236	1.65973
$935$	−1.43504	−0.0469307
$936$	0	0
$937$	22.9252	0.748935	0.374468	−	0.927240i	$-0.377826\pi$
$937$	22.9252	0.748935	0.374468	+	0.927240i	$0.377826\pi$
$938$	−76.3426	−2.49268
$939$	0	0
$940$	2.86008	0.0932857
$941$	−14.1397	−0.460940	−0.230470	−	0.973079i	$-0.574026\pi$
$941$	−14.1397	−0.460940	−0.230470	+	0.973079i	$0.574026\pi$
$942$	0	0
$943$	10.3930	0.338443
$944$	−74.2297	−2.41597
$945$	0	0
$946$	−0.564102	−0.0183405
$947$	57.0898	1.85517	0.927585	−	0.373611i	$-0.121881\pi$
$947$	57.0898	1.85517	0.927585	+	0.373611i	$0.121881\pi$
$948$	0	0
$949$	−85.5629	−2.77749
$950$	15.7860	0.512166
$951$	0	0
$952$	28.9540	0.938406
$953$	17.7295	0.574316	0.287158	−	0.957883i	$-0.407290\pi$
$953$	17.7295	0.574316	0.287158	+	0.957883i	$0.407290\pi$
$954$	0	0
$955$	−19.5414	−0.632345
$956$	21.1733	0.684792
$957$	0	0
$958$	−38.7185	−1.25094
$959$	3.66464	0.118337
$960$	0	0
$961$	−5.18011	−0.167100
$962$	75.6952	2.44051
$963$	0	0
$964$	−18.0321	−0.580776
$965$	−5.75041	−0.185112
$966$	0	0
$967$	−22.1182	−0.711273	−0.355636	−	0.934624i	$-0.615736\pi$
$967$	−22.1182	−0.711273	−0.355636	+	0.934624i	$0.615736\pi$
$968$	19.0125	0.611084
$969$	0	0
$970$	−4.28081	−0.137448
$971$	5.09901	0.163635	0.0818176	−	0.996647i	$-0.473928\pi$
$971$	5.09901	0.163635	0.0818176	+	0.996647i	$0.473928\pi$
$972$	0	0
$973$	−15.6454	−0.501569
$974$	6.47451	0.207457
$975$	0	0
$976$	7.53349	0.241141
$977$	−23.2327	−0.743279	−0.371640	−	0.928377i	$-0.621204\pi$
$977$	−23.2327	−0.743279	−0.371640	+	0.928377i	$0.621204\pi$
$978$	0	0
$979$	−1.05550	−0.0337339
$980$	5.23921	0.167360
$981$	0	0
$982$	2.08933	0.0666731
$983$	16.8541	0.537563	0.268781	−	0.963201i	$-0.413379\pi$
$983$	16.8541	0.537563	0.268781	+	0.963201i	$0.413379\pi$
$984$	0	0
$985$	−28.5300	−0.909043
$986$	−55.4303	−1.76526
$987$	0	0
$988$	−15.4038	−0.490061
$989$	−5.96901	−0.189803
$990$	0	0
$991$	−0.413097	−0.0131225	−0.00656123	−	0.999978i	$-0.502089\pi$
$991$	−0.413097	−0.0131225	−0.00656123	+	0.999978i	$0.502089\pi$
$992$	−26.3575	−0.836851
$993$	0	0
$994$	26.6113	0.844058
$995$	3.96092	0.125570
$996$	0	0
$997$	5.84183	0.185013	0.0925063	−	0.995712i	$-0.470512\pi$
$997$	5.84183	0.185013	0.0925063	+	0.995712i	$0.470512\pi$
$998$	5.77940	0.182944
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6561.2.a.d.1.56		72
3.2	odd	2		6561.2.a.c.1.17		72
81.2	odd	54		729.2.g.c.433.7		144
81.13	even	27		729.2.g.a.541.7		144
81.14	odd	54		81.2.g.a.34.2	yes	144
81.25	even	27		729.2.g.a.190.7		144
81.29	odd	54		81.2.g.a.31.2	✓	144
81.40	even	27		729.2.g.b.298.2		144
81.41	odd	54		729.2.g.c.298.7		144
81.52	even	27		243.2.g.a.145.7		144
81.56	odd	54		729.2.g.d.190.2		144
81.67	even	27		243.2.g.a.181.7		144
81.68	odd	54		729.2.g.d.541.2		144
81.79	even	27		729.2.g.b.433.2		144

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
81.2.g.a.31.2	✓	144	81.29	odd	54
81.2.g.a.34.2	yes	144	81.14	odd	54
243.2.g.a.145.7		144	81.52	even	27
243.2.g.a.181.7		144	81.67	even	27
729.2.g.a.190.7		144	81.25	even	27
729.2.g.a.541.7		144	81.13	even	27
729.2.g.b.298.2		144	81.40	even	27
729.2.g.b.433.2		144	81.79	even	27
729.2.g.c.298.7		144	81.41	odd	54
729.2.g.c.433.7		144	81.2	odd	54
729.2.g.d.190.2		144	81.56	odd	54
729.2.g.d.541.2		144	81.68	odd	54
6561.2.a.c.1.17		72	3.2	odd	2
6561.2.a.d.1.56		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)

\(f(q)\)	\(=\)	\(q+1.73145 q^{2} +0.997916 q^{4} +1.37705 q^{5} +3.28825 q^{7} -1.73506 q^{8} +2.38429 q^{10} +0.205344 q^{11} +5.25480 q^{13} +5.69344 q^{14} -5.00000 q^{16} -5.07493 q^{17} -2.93751 q^{19} +1.37418 q^{20} +0.355543 q^{22} +3.76215 q^{23} -3.10373 q^{25} +9.09842 q^{26} +3.28140 q^{28} +6.30823 q^{29} +5.08133 q^{31} -5.18713 q^{32} -8.78699 q^{34} +4.52809 q^{35} +8.31960 q^{37} -5.08614 q^{38} -2.38926 q^{40} +2.76252 q^{41} -1.58659 q^{43} +0.204916 q^{44} +6.51398 q^{46} +2.08130 q^{47} +3.81261 q^{49} -5.37396 q^{50} +5.24385 q^{52} +9.03326 q^{53} +0.282769 q^{55} -5.70531 q^{56} +10.9224 q^{58} +14.8459 q^{59} -1.50670 q^{61} +8.79806 q^{62} +1.01875 q^{64} +7.23612 q^{65} -13.4089 q^{67} -5.06436 q^{68} +7.84015 q^{70} +4.67402 q^{71} -16.2828 q^{73} +14.4050 q^{74} -2.93138 q^{76} +0.675224 q^{77} -1.48696 q^{79} -6.88524 q^{80} +4.78316 q^{82} +14.2514 q^{83} -6.98844 q^{85} -2.74710 q^{86} -0.356284 q^{88} -5.14015 q^{89} +17.2791 q^{91} +3.75432 q^{92} +3.60367 q^{94} -4.04509 q^{95} -1.79542 q^{97} +6.60133 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 72 q + 9 q^{2} + 63 q^{4} + 18 q^{5} + 27 q^{8} + 36 q^{11} + 36 q^{14} + 45 q^{16} + 36 q^{17} + 54 q^{20} + 54 q^{23} + 36 q^{25} + 45 q^{26} + 9 q^{28} + 54 q^{29} + 63 q^{32} + 72 q^{35} + 54 q^{38}+ \cdots + 81 q^{98}+O(q^{100}) \) 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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Database

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