LMFDB - Embedded newform 6561.2.a.c.1.11

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			1.11
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)

$f(q)$	$=$	\(q-2.22247 q^{2} +2.93935 q^{4} -2.94555 q^{5} +2.95569 q^{7} -2.08768 q^{8} +6.54639 q^{10} -5.30728 q^{11} -0.735254 q^{13} -6.56892 q^{14} -1.23890 q^{16} +1.45289 q^{17} +5.49777 q^{19} -8.65803 q^{20} +11.7953 q^{22} -0.450858 q^{23} +3.67629 q^{25} +1.63408 q^{26} +8.68782 q^{28} +3.55469 q^{29} -1.11262 q^{31} +6.92879 q^{32} -3.22899 q^{34} -8.70614 q^{35} -3.70407 q^{37} -12.2186 q^{38} +6.14938 q^{40} -10.2263 q^{41} -7.93111 q^{43} -15.6000 q^{44} +1.00202 q^{46} +5.52262 q^{47} +1.73609 q^{49} -8.17042 q^{50} -2.16117 q^{52} +3.66041 q^{53} +15.6329 q^{55} -6.17054 q^{56} -7.90017 q^{58} +7.16579 q^{59} +4.35500 q^{61} +2.47277 q^{62} -12.9212 q^{64} +2.16573 q^{65} +11.3165 q^{67} +4.27055 q^{68} +19.3491 q^{70} +13.4521 q^{71} -3.96916 q^{73} +8.23217 q^{74} +16.1599 q^{76} -15.6867 q^{77} +2.84265 q^{79} +3.64926 q^{80} +22.7277 q^{82} +0.351845 q^{83} -4.27955 q^{85} +17.6266 q^{86} +11.0799 q^{88} -5.68641 q^{89} -2.17318 q^{91} -1.32523 q^{92} -12.2738 q^{94} -16.1940 q^{95} -7.04954 q^{97} -3.85841 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	−2.22247	−1.57152	−0.785760	−	0.618531i	$-0.787728\pi$
$2$	−2.22247	−1.57152	−0.785760	+	0.618531i	$0.787728\pi$
$3$	0	0
$3$	0	0
$4$	2.93935	1.46968
$5$	−2.94555	−1.31729	−0.658646	−	0.752453i	$-0.728870\pi$
$5$	−2.94555	−1.31729	−0.658646	+	0.752453i	$0.728870\pi$
$6$	0	0
$7$	2.95569	1.11715	0.558573	−	0.829456i	$-0.311349\pi$
$7$	2.95569	1.11715	0.558573	+	0.829456i	$0.311349\pi$
$8$	−2.08768	−0.738107
$9$	0	0
$10$	6.54639	2.07015
$11$	−5.30728	−1.60021	−0.800103	−	0.599862i	$-0.795222\pi$
$11$	−5.30728	−1.60021	−0.800103	+	0.599862i	$0.795222\pi$
$12$	0	0
$13$	−0.735254	−0.203923	−0.101961	−	0.994788i	$-0.532512\pi$
$13$	−0.735254	−0.203923	−0.101961	+	0.994788i	$0.532512\pi$
$14$	−6.56892	−1.75562
$15$	0	0
$16$	−1.23890	−0.309726
$17$	1.45289	0.352377	0.176188	−	0.984356i	$-0.443623\pi$
$17$	1.45289	0.352377	0.176188	+	0.984356i	$0.443623\pi$
$18$	0	0
$19$	5.49777	1.26127	0.630637	−	0.776078i	$-0.282794\pi$
$19$	5.49777	1.26127	0.630637	+	0.776078i	$0.282794\pi$
$20$	−8.65803	−1.93599
$21$	0	0
$22$	11.7953	2.51476
$23$	−0.450858	−0.0940103	−0.0470052	−	0.998895i	$-0.514968\pi$
$23$	−0.450858	−0.0940103	−0.0470052	+	0.998895i	$0.514968\pi$
$24$	0	0
$25$	3.67629	0.735257
$26$	1.63408	0.320469
$27$	0	0
$28$	8.68782	1.64184
$29$	3.55469	0.660089	0.330044	−	0.943965i	$-0.392936\pi$
$29$	3.55469	0.660089	0.330044	+	0.943965i	$0.392936\pi$
$30$	0	0
$31$	−1.11262	−0.199833	−0.0999166	−	0.994996i	$-0.531858\pi$
$31$	−1.11262	−0.199833	−0.0999166	+	0.994996i	$0.531858\pi$
$32$	6.92879	1.22485
$33$	0	0
$34$	−3.22899	−0.553767
$35$	−8.70614	−1.47161
$36$	0	0
$37$	−3.70407	−0.608946	−0.304473	−	0.952521i	$-0.598480\pi$
$37$	−3.70407	−0.608946	−0.304473	+	0.952521i	$0.598480\pi$
$38$	−12.2186	−1.98212
$39$	0	0
$40$	6.14938	0.972303
$41$	−10.2263	−1.59709	−0.798543	−	0.601937i	$-0.794396\pi$
$41$	−10.2263	−1.59709	−0.798543	+	0.601937i	$0.794396\pi$
$42$	0	0
$43$	−7.93111	−1.20948	−0.604742	−	0.796422i	$-0.706724\pi$
$43$	−7.93111	−1.20948	−0.604742	+	0.796422i	$0.706724\pi$
$44$	−15.6000	−2.35179
$45$	0	0
$46$	1.00202	0.147739
$47$	5.52262	0.805557	0.402779	−	0.915297i	$-0.368044\pi$
$47$	5.52262	0.805557	0.402779	+	0.915297i	$0.368044\pi$
$48$	0	0
$49$	1.73609	0.248013
$50$	−8.17042	−1.15547
$51$	0	0
$52$	−2.16117	−0.299700
$53$	3.66041	0.502796	0.251398	−	0.967884i	$-0.419110\pi$
$53$	3.66041	0.502796	0.251398	+	0.967884i	$0.419110\pi$
$54$	0	0
$55$	15.6329	2.10794
$56$	−6.17054	−0.824573
$57$	0	0
$58$	−7.90017	−1.03734
$59$	7.16579	0.932906	0.466453	−	0.884546i	$-0.345532\pi$
$59$	7.16579	0.932906	0.466453	+	0.884546i	$0.345532\pi$
$60$	0	0
$61$	4.35500	0.557600	0.278800	−	0.960349i	$-0.410063\pi$
$61$	4.35500	0.557600	0.278800	+	0.960349i	$0.410063\pi$
$62$	2.47277	0.314042
$63$	0	0
$64$	−12.9212	−1.61515
$65$	2.16573	0.268626
$66$	0	0
$67$	11.3165	1.38253	0.691266	−	0.722600i	$-0.257053\pi$
$67$	11.3165	1.38253	0.691266	+	0.722600i	$0.257053\pi$
$68$	4.27055	0.517880
$69$	0	0
$70$	19.3491	2.31266
$71$	13.4521	1.59647	0.798234	−	0.602348i	$-0.205768\pi$
$71$	13.4521	1.59647	0.798234	+	0.602348i	$0.205768\pi$
$72$	0	0
$73$	−3.96916	−0.464554	−0.232277	−	0.972650i	$-0.574618\pi$
$73$	−3.96916	−0.464554	−0.232277	+	0.972650i	$0.574618\pi$
$74$	8.23217	0.956971
$75$	0	0
$76$	16.1599	1.85367
$77$	−15.6867	−1.78766
$78$	0	0
$79$	2.84265	0.319823	0.159911	−	0.987131i	$-0.448879\pi$
$79$	2.84265	0.319823	0.159911	+	0.987131i	$0.448879\pi$
$80$	3.64926	0.408000
$81$	0	0
$82$	22.7277	2.50986
$83$	0.351845	0.0386200	0.0193100	−	0.999814i	$-0.493853\pi$
$83$	0.351845	0.0386200	0.0193100	+	0.999814i	$0.493853\pi$
$84$	0	0
$85$	−4.27955	−0.464183
$86$	17.6266	1.90073
$87$	0	0
$88$	11.0799	1.18112
$89$	−5.68641	−0.602758	−0.301379	−	0.953504i	$-0.597447\pi$
$89$	−5.68641	−0.602758	−0.301379	+	0.953504i	$0.597447\pi$
$90$	0	0
$91$	−2.17318	−0.227811
$92$	−1.32523	−0.138165
$93$	0	0
$94$	−12.2738	−1.26595
$95$	−16.1940	−1.66147
$96$	0	0
$97$	−7.04954	−0.715773	−0.357886	−	0.933765i	$-0.616502\pi$
$97$	−7.04954	−0.715773	−0.357886	+	0.933765i	$0.616502\pi$
$98$	−3.85841	−0.389758
$99$	0	0
$100$	10.8059	1.08059
$101$	10.6118	1.05591	0.527956	−	0.849271i	$-0.322958\pi$
$101$	10.6118	1.05591	0.527956	+	0.849271i	$0.322958\pi$
$102$	0	0
$103$	−17.8213	−1.75599	−0.877994	−	0.478672i	$-0.841118\pi$
$103$	−17.8213	−1.75599	−0.877994	+	0.478672i	$0.841118\pi$
$104$	1.53498	0.150517
$105$	0	0
$106$	−8.13513	−0.790154
$107$	−0.0498293	−0.00481718	−0.00240859	−	0.999997i	$-0.500767\pi$
$107$	−0.0498293	−0.00481718	−0.00240859	+	0.999997i	$0.500767\pi$
$108$	0	0
$109$	−0.405040	−0.0387958	−0.0193979	−	0.999812i	$-0.506175\pi$
$109$	−0.405040	−0.0387958	−0.0193979	+	0.999812i	$0.506175\pi$
$110$	−34.7436	−3.31267
$111$	0	0
$112$	−3.66181	−0.346009
$113$	−11.3758	−1.07014	−0.535071	−	0.844807i	$-0.679715\pi$
$113$	−11.3758	−1.07014	−0.535071	+	0.844807i	$0.679715\pi$
$114$	0	0
$115$	1.32803	0.123839
$116$	10.4485	0.970117
$117$	0	0
$118$	−15.9257	−1.46608
$119$	4.29428	0.393656
$120$	0	0
$121$	17.1673	1.56066
$122$	−9.67884	−0.876281
$123$	0	0
$124$	−3.27040	−0.293690
$125$	3.89907	0.348744
$126$	0	0
$127$	13.0455	1.15760	0.578800	−	0.815470i	$-0.303521\pi$
$127$	13.0455	1.15760	0.578800	+	0.815470i	$0.303521\pi$
$128$	14.8593	1.31339
$129$	0	0
$130$	−4.81326	−0.422151
$131$	8.00769	0.699635	0.349818	−	0.936818i	$-0.386244\pi$
$131$	8.00769	0.699635	0.349818	+	0.936818i	$0.386244\pi$
$132$	0	0
$133$	16.2497	1.40903
$134$	−25.1506	−2.17268
$135$	0	0
$136$	−3.03317	−0.260092
$137$	−10.3205	−0.881739	−0.440869	−	0.897571i	$-0.645330\pi$
$137$	−10.3205	−0.881739	−0.440869	+	0.897571i	$0.645330\pi$
$138$	0	0
$139$	−17.4540	−1.48043	−0.740213	−	0.672373i	$-0.765275\pi$
$139$	−17.4540	−1.48043	−0.740213	+	0.672373i	$0.765275\pi$
$140$	−25.5904	−2.16279
$141$	0	0
$142$	−29.8968	−2.50888
$143$	3.90220	0.326318
$144$	0	0
$145$	−10.4705	−0.869529
$146$	8.82131	0.730057
$147$	0	0
$148$	−10.8876	−0.894954
$149$	−8.31420	−0.681126	−0.340563	−	0.940222i	$-0.610618\pi$
$149$	−8.31420	−0.681126	−0.340563	+	0.940222i	$0.610618\pi$
$150$	0	0
$151$	−17.6006	−1.43232	−0.716158	−	0.697938i	$-0.754101\pi$
$151$	−17.6006	−1.43232	−0.716158	+	0.697938i	$0.754101\pi$
$152$	−11.4776	−0.930956
$153$	0	0
$154$	34.8631	2.80935
$155$	3.27729	0.263239
$156$	0	0
$157$	−2.15482	−0.171973	−0.0859867	−	0.996296i	$-0.527404\pi$
$157$	−2.15482	−0.171973	−0.0859867	+	0.996296i	$0.527404\pi$
$158$	−6.31769	−0.502608
$159$	0	0
$160$	−20.4091	−1.61348
$161$	−1.33260	−0.105023
$162$	0	0
$163$	9.47594	0.742213	0.371106	−	0.928590i	$-0.378979\pi$
$163$	9.47594	0.742213	0.371106	+	0.928590i	$0.378979\pi$
$164$	−30.0589	−2.34720
$165$	0	0
$166$	−0.781964	−0.0606922
$167$	−3.82156	−0.295721	−0.147860	−	0.989008i	$-0.547239\pi$
$167$	−3.82156	−0.295721	−0.147860	+	0.989008i	$0.547239\pi$
$168$	0	0
$169$	−12.4594	−0.958416
$170$	9.51116	0.729473
$171$	0	0
$172$	−23.3124	−1.77755
$173$	8.36675	0.636112	0.318056	−	0.948072i	$-0.396970\pi$
$173$	8.36675	0.636112	0.318056	+	0.948072i	$0.396970\pi$
$174$	0	0
$175$	10.8660	0.821389
$176$	6.57522	0.495626
$177$	0	0
$178$	12.6378	0.947247
$179$	5.48081	0.409655	0.204827	−	0.978798i	$-0.434337\pi$
$179$	5.48081	0.409655	0.204827	+	0.978798i	$0.434337\pi$
$180$	0	0
$181$	21.9260	1.62975	0.814874	−	0.579638i	$-0.196806\pi$
$181$	21.9260	1.62975	0.814874	+	0.579638i	$0.196806\pi$
$182$	4.82982	0.358010
$183$	0	0
$184$	0.941248	0.0693897
$185$	10.9105	0.802159
$186$	0	0
$187$	−7.71088	−0.563875
$188$	16.2329	1.18391
$189$	0	0
$190$	35.9905	2.61103
$191$	−12.7424	−0.922006	−0.461003	−	0.887399i	$-0.652510\pi$
$191$	−12.7424	−0.922006	−0.461003	+	0.887399i	$0.652510\pi$
$192$	0	0
$193$	20.3328	1.46358	0.731792	−	0.681528i	$-0.238684\pi$
$193$	20.3328	1.46358	0.731792	+	0.681528i	$0.238684\pi$
$194$	15.6674	1.12485
$195$	0	0
$196$	5.10299	0.364500
$197$	−4.01375	−0.285968	−0.142984	−	0.989725i	$-0.545670\pi$
$197$	−4.01375	−0.285968	−0.142984	+	0.989725i	$0.545670\pi$
$198$	0	0
$199$	3.84576	0.272619	0.136309	−	0.990666i	$-0.456476\pi$
$199$	3.84576	0.272619	0.136309	+	0.990666i	$0.456476\pi$
$200$	−7.67492	−0.542699
$201$	0	0
$202$	−23.5843	−1.65939
$203$	10.5065	0.737415
$204$	0	0
$205$	30.1222	2.10383
$206$	39.6073	2.75957
$207$	0	0
$208$	0.910909	0.0631602
$209$	−29.1782	−2.01830
$210$	0	0
$211$	7.12810	0.490719	0.245359	−	0.969432i	$-0.421094\pi$
$211$	7.12810	0.490719	0.245359	+	0.969432i	$0.421094\pi$
$212$	10.7592	0.738947
$213$	0	0
$214$	0.110744	0.00757030
$215$	23.3615	1.59324
$216$	0	0
$217$	−3.28857	−0.223243
$218$	0.900187	0.0609683
$219$	0	0
$220$	45.9506	3.09799
$221$	−1.06824	−0.0718576
$222$	0	0
$223$	14.8492	0.994379	0.497189	−	0.867642i	$-0.334365\pi$
$223$	14.8492	0.994379	0.497189	+	0.867642i	$0.334365\pi$
$224$	20.4793	1.36833
$225$	0	0
$226$	25.2822	1.68175
$227$	−13.4564	−0.893132	−0.446566	−	0.894751i	$-0.647353\pi$
$227$	−13.4564	−0.893132	−0.446566	+	0.894751i	$0.647353\pi$
$228$	0	0
$229$	−3.25941	−0.215388	−0.107694	−	0.994184i	$-0.534347\pi$
$229$	−3.25941	−0.215388	−0.107694	+	0.994184i	$0.534347\pi$
$230$	−2.95149	−0.194616
$231$	0	0
$232$	−7.42106	−0.487216
$233$	−8.27305	−0.541986	−0.270993	−	0.962581i	$-0.587352\pi$
$233$	−8.27305	−0.541986	−0.270993	+	0.962581i	$0.587352\pi$
$234$	0	0
$235$	−16.2672	−1.06115
$236$	21.0628	1.37107
$237$	0	0
$238$	−9.54389	−0.618638
$239$	13.0628	0.844965	0.422482	−	0.906371i	$-0.361159\pi$
$239$	13.0628	0.844965	0.422482	+	0.906371i	$0.361159\pi$
$240$	0	0
$241$	−0.632758	−0.0407595	−0.0203798	−	0.999792i	$-0.506488\pi$
$241$	−0.632758	−0.0407595	−0.0203798	+	0.999792i	$0.506488\pi$
$242$	−38.1537	−2.45261
$243$	0	0
$244$	12.8009	0.819493
$245$	−5.11376	−0.326706
$246$	0	0
$247$	−4.04225	−0.257202
$248$	2.32281	0.147498
$249$	0	0
$250$	−8.66555	−0.548058
$251$	−13.8326	−0.873109	−0.436554	−	0.899678i	$-0.643801\pi$
$251$	−13.8326	−0.873109	−0.436554	+	0.899678i	$0.643801\pi$
$252$	0	0
$253$	2.39283	0.150436
$254$	−28.9931	−1.81919
$255$	0	0
$256$	−7.18196	−0.448872
$257$	26.2056	1.63466	0.817331	−	0.576168i	$-0.195453\pi$
$257$	26.2056	1.63466	0.817331	+	0.576168i	$0.195453\pi$
$258$	0	0
$259$	−10.9481	−0.680281
$260$	6.36584	0.394793
$261$	0	0
$262$	−17.7968	−1.09949
$263$	1.42256	0.0877187	0.0438594	−	0.999038i	$-0.486035\pi$
$263$	1.42256	0.0877187	0.0438594	+	0.999038i	$0.486035\pi$
$264$	0	0
$265$	−10.7819	−0.662328
$266$	−36.1144	−2.21431
$267$	0	0
$268$	33.2633	2.03188
$269$	5.46641	0.333293	0.166646	−	0.986017i	$-0.446706\pi$
$269$	5.46641	0.333293	0.166646	+	0.986017i	$0.446706\pi$
$270$	0	0
$271$	30.7333	1.86692	0.933458	−	0.358685i	$-0.116775\pi$
$271$	30.7333	1.86692	0.933458	+	0.358685i	$0.116775\pi$
$272$	−1.79999	−0.109140
$273$	0	0
$274$	22.9369	1.38567
$275$	−19.5111	−1.17656
$276$	0	0
$277$	4.55555	0.273717	0.136858	−	0.990591i	$-0.456299\pi$
$277$	4.55555	0.273717	0.136858	+	0.990591i	$0.456299\pi$
$278$	38.7908	2.32652
$279$	0	0
$280$	18.1757	1.08620
$281$	18.2737	1.09012	0.545059	−	0.838397i	$-0.316507\pi$
$281$	18.2737	1.09012	0.545059	+	0.838397i	$0.316507\pi$
$282$	0	0
$283$	−23.6861	−1.40799	−0.703996	−	0.710204i	$-0.748603\pi$
$283$	−23.6861	−1.40799	−0.703996	+	0.710204i	$0.748603\pi$
$284$	39.5404	2.34629
$285$	0	0
$286$	−8.67251	−0.512816
$287$	−30.2259	−1.78418
$288$	0	0
$289$	−14.8891	−0.875831
$290$	23.2704	1.36648
$291$	0	0
$292$	−11.6668	−0.682745
$293$	−4.76803	−0.278551	−0.139276	−	0.990254i	$-0.544477\pi$
$293$	−4.76803	−0.278551	−0.139276	+	0.990254i	$0.544477\pi$
$294$	0	0
$295$	−21.1072	−1.22891
$296$	7.73293	0.449467
$297$	0	0
$298$	18.4780	1.07040
$299$	0.331495	0.0191708
$300$	0	0
$301$	−23.4419	−1.35117
$302$	39.1167	2.25091
$303$	0	0
$304$	−6.81121	−0.390649
$305$	−12.8279	−0.734522
$306$	0	0
$307$	−10.7915	−0.615901	−0.307951	−	0.951402i	$-0.599643\pi$
$307$	−10.7915	−0.615901	−0.307951	+	0.951402i	$0.599643\pi$
$308$	−46.1087	−2.62729
$309$	0	0
$310$	−7.28368	−0.413685
$311$	9.68139	0.548981	0.274491	−	0.961590i	$-0.411491\pi$
$311$	9.68139	0.548981	0.274491	+	0.961590i	$0.411491\pi$
$312$	0	0
$313$	−6.38710	−0.361020	−0.180510	−	0.983573i	$-0.557775\pi$
$313$	−6.38710	−0.361020	−0.180510	+	0.983573i	$0.557775\pi$
$314$	4.78901	0.270260
$315$	0	0
$316$	8.35555	0.470036
$317$	−20.6111	−1.15763	−0.578816	−	0.815458i	$-0.696485\pi$
$317$	−20.6111	−1.15763	−0.578816	+	0.815458i	$0.696485\pi$
$318$	0	0
$319$	−18.8657	−1.05628
$320$	38.0601	2.12762
$321$	0	0
$322$	2.96165	0.165046
$323$	7.98763	0.444444
$324$	0	0
$325$	−2.70300	−0.149936
$326$	−21.0600	−1.16640
$327$	0	0
$328$	21.3494	1.17882
$329$	16.3231	0.899924
$330$	0	0
$331$	20.2976	1.11566	0.557828	−	0.829957i	$-0.311635\pi$
$331$	20.2976	1.11566	0.557828	+	0.829957i	$0.311635\pi$
$332$	1.03420	0.0567590
$333$	0	0
$334$	8.49328	0.464732
$335$	−33.3334	−1.82120
$336$	0	0
$337$	−3.65631	−0.199172	−0.0995860	−	0.995029i	$-0.531752\pi$
$337$	−3.65631	−0.199172	−0.0995860	+	0.995029i	$0.531752\pi$
$338$	27.6906	1.50617
$339$	0	0
$340$	−12.5791	−0.682199
$341$	5.90501	0.319774
$342$	0	0
$343$	−15.5585	−0.840078
$344$	16.5577	0.892729
$345$	0	0
$346$	−18.5948	−0.999664
$347$	29.1562	1.56519	0.782593	−	0.622534i	$-0.213897\pi$
$347$	29.1562	1.56519	0.782593	+	0.622534i	$0.213897\pi$
$348$	0	0
$349$	16.9320	0.906350	0.453175	−	0.891421i	$-0.350291\pi$
$349$	16.9320	0.906350	0.453175	+	0.891421i	$0.350291\pi$
$350$	−24.1492	−1.29083
$351$	0	0
$352$	−36.7730	−1.96001
$353$	−22.7485	−1.21078	−0.605389	−	0.795929i	$-0.706983\pi$
$353$	−22.7485	−1.21078	−0.605389	+	0.795929i	$0.706983\pi$
$354$	0	0
$355$	−39.6238	−2.10301
$356$	−16.7144	−0.885860
$357$	0	0
$358$	−12.1809	−0.643781
$359$	−15.3153	−0.808312	−0.404156	−	0.914690i	$-0.632435\pi$
$359$	−15.3153	−0.808312	−0.404156	+	0.914690i	$0.632435\pi$
$360$	0	0
$361$	11.2254	0.590812
$362$	−48.7298	−2.56118
$363$	0	0
$364$	−6.38775	−0.334809
$365$	11.6914	0.611954
$366$	0	0
$367$	14.0369	0.732720	0.366360	−	0.930473i	$-0.380604\pi$
$367$	14.0369	0.732720	0.366360	+	0.930473i	$0.380604\pi$
$368$	0.558570	0.0291175
$369$	0	0
$370$	−24.2483	−1.26061
$371$	10.8190	0.561696
$372$	0	0
$373$	−27.2465	−1.41077	−0.705385	−	0.708824i	$-0.749226\pi$
$373$	−27.2465	−1.41077	−0.705385	+	0.708824i	$0.749226\pi$
$374$	17.1372	0.886142
$375$	0	0
$376$	−11.5295	−0.594588
$377$	−2.61360	−0.134607
$378$	0	0
$379$	8.59704	0.441600	0.220800	−	0.975319i	$-0.429133\pi$
$379$	8.59704	0.441600	0.220800	+	0.975319i	$0.429133\pi$
$380$	−47.5998	−2.44182
$381$	0	0
$382$	28.3195	1.44895
$383$	−19.2886	−0.985599	−0.492800	−	0.870143i	$-0.664026\pi$
$383$	−19.2886	−0.985599	−0.492800	+	0.870143i	$0.664026\pi$
$384$	0	0
$385$	46.2059	2.35487
$386$	−45.1889	−2.30005
$387$	0	0
$388$	−20.7211	−1.05195
$389$	−31.1422	−1.57897	−0.789486	−	0.613768i	$-0.789653\pi$
$389$	−31.1422	−1.57897	−0.789486	+	0.613768i	$0.789653\pi$
$390$	0	0
$391$	−0.655045	−0.0331270
$392$	−3.62441	−0.183061
$393$	0	0
$394$	8.92042	0.449405
$395$	−8.37317	−0.421300
$396$	0	0
$397$	10.6635	0.535184	0.267592	−	0.963532i	$-0.413772\pi$
$397$	10.6635	0.535184	0.267592	+	0.963532i	$0.413772\pi$
$398$	−8.54707	−0.428426
$399$	0	0
$400$	−4.55457	−0.227728
$401$	−8.10074	−0.404532	−0.202266	−	0.979331i	$-0.564830\pi$
$401$	−8.10074	−0.404532	−0.202266	+	0.979331i	$0.564830\pi$
$402$	0	0
$403$	0.818061	0.0407505
$404$	31.1918	1.55185
$405$	0	0
$406$	−23.3504	−1.15886
$407$	19.6586	0.974439
$408$	0	0
$409$	−37.4818	−1.85336	−0.926679	−	0.375854i	$-0.877349\pi$
$409$	−37.4818	−1.85336	−0.926679	+	0.375854i	$0.877349\pi$
$410$	−66.9457	−3.30621
$411$	0	0
$412$	−52.3832	−2.58074
$413$	21.1798	1.04219
$414$	0	0
$415$	−1.03638	−0.0508738
$416$	−5.09442	−0.249774
$417$	0	0
$418$	64.8476	3.17180
$419$	−12.8958	−0.629999	−0.314999	−	0.949092i	$-0.602004\pi$
$419$	−12.8958	−0.629999	−0.314999	+	0.949092i	$0.602004\pi$
$420$	0	0
$421$	23.1329	1.12743	0.563715	−	0.825969i	$-0.309371\pi$
$421$	23.1329	1.12743	0.563715	+	0.825969i	$0.309371\pi$
$422$	−15.8420	−0.771175
$423$	0	0
$424$	−7.64177	−0.371117
$425$	5.34122	0.259087
$426$	0	0
$427$	12.8720	0.622921
$428$	−0.146466	−0.00707970
$429$	0	0
$430$	−51.9202	−2.50381
$431$	−22.6998	−1.09341	−0.546706	−	0.837324i	$-0.684119\pi$
$431$	−22.6998	−1.09341	−0.546706	+	0.837324i	$0.684119\pi$
$432$	0	0
$433$	−18.2110	−0.875165	−0.437583	−	0.899178i	$-0.644165\pi$
$433$	−18.2110	−0.875165	−0.437583	+	0.899178i	$0.644165\pi$
$434$	7.30874	0.350831
$435$	0	0
$436$	−1.19055	−0.0570172
$437$	−2.47871	−0.118573
$438$	0	0
$439$	27.4963	1.31232	0.656162	−	0.754620i	$-0.272179\pi$
$439$	27.4963	1.31232	0.656162	+	0.754620i	$0.272179\pi$
$440$	−32.6365	−1.55589
$441$	0	0
$442$	2.37413	0.112926
$443$	12.2619	0.582579	0.291290	−	0.956635i	$-0.405916\pi$
$443$	12.2619	0.582579	0.291290	+	0.956635i	$0.405916\pi$
$444$	0	0
$445$	16.7496	0.794008
$446$	−33.0019	−1.56269
$447$	0	0
$448$	−38.1910	−1.80436
$449$	−6.93959	−0.327500	−0.163750	−	0.986502i	$-0.552359\pi$
$449$	−6.93959	−0.327500	−0.163750	+	0.986502i	$0.552359\pi$
$450$	0	0
$451$	54.2741	2.55567
$452$	−33.4374	−1.57276
$453$	0	0
$454$	29.9064	1.40358
$455$	6.40122	0.300094
$456$	0	0
$457$	29.8648	1.39702	0.698509	−	0.715602i	$-0.253847\pi$
$457$	29.8648	1.39702	0.698509	+	0.715602i	$0.253847\pi$
$458$	7.24392	0.338486
$459$	0	0
$460$	3.90354	0.182003
$461$	0.820444	0.0382119	0.0191059	−	0.999817i	$-0.493918\pi$
$461$	0.820444	0.0382119	0.0191059	+	0.999817i	$0.493918\pi$
$462$	0	0
$463$	−28.6193	−1.33005	−0.665026	−	0.746820i	$-0.731580\pi$
$463$	−28.6193	−1.33005	−0.665026	+	0.746820i	$0.731580\pi$
$464$	−4.40392	−0.204447
$465$	0	0
$466$	18.3866	0.851742
$467$	12.4929	0.578102	0.289051	−	0.957314i	$-0.406660\pi$
$467$	12.4929	0.578102	0.289051	+	0.957314i	$0.406660\pi$
$468$	0	0
$469$	33.4481	1.54449
$470$	36.1532	1.66763
$471$	0	0
$472$	−14.9599	−0.688585
$473$	42.0927	1.93542
$474$	0	0
$475$	20.2114	0.927361
$476$	12.6224	0.578547
$477$	0	0
$478$	−29.0317	−1.32788
$479$	4.53590	0.207251	0.103625	−	0.994616i	$-0.466956\pi$
$479$	4.53590	0.207251	0.103625	+	0.994616i	$0.466956\pi$
$480$	0	0
$481$	2.72343	0.124178
$482$	1.40628	0.0640544
$483$	0	0
$484$	50.4607	2.29367
$485$	20.7648	0.942881
$486$	0	0
$487$	−28.2887	−1.28188	−0.640941	−	0.767590i	$-0.721456\pi$
$487$	−28.2887	−1.28188	−0.640941	+	0.767590i	$0.721456\pi$
$488$	−9.09186	−0.411569
$489$	0	0
$490$	11.3651	0.513425
$491$	−19.5841	−0.883820	−0.441910	−	0.897059i	$-0.645699\pi$
$491$	−19.5841	−0.883820	−0.441910	+	0.897059i	$0.645699\pi$
$492$	0	0
$493$	5.16455	0.232600
$494$	8.98377	0.404199
$495$	0	0
$496$	1.37844	0.0618936
$497$	39.7601	1.78349
$498$	0	0
$499$	−29.5992	−1.32504	−0.662521	−	0.749043i	$-0.730514\pi$
$499$	−29.5992	−1.32504	−0.662521	+	0.749043i	$0.730514\pi$
$500$	11.4608	0.512540
$501$	0	0
$502$	30.7426	1.37211
$503$	−23.7955	−1.06099	−0.530495	−	0.847688i	$-0.677994\pi$
$503$	−23.7955	−1.06099	−0.530495	+	0.847688i	$0.677994\pi$
$504$	0	0
$505$	−31.2576	−1.39095
$506$	−5.31798	−0.236413
$507$	0	0
$508$	38.3453	1.70130
$509$	−12.4397	−0.551382	−0.275691	−	0.961246i	$-0.588907\pi$
$509$	−12.4397	−0.551382	−0.275691	+	0.961246i	$0.588907\pi$
$510$	0	0
$511$	−11.7316	−0.518975
$512$	−13.7570	−0.607979
$513$	0	0
$514$	−58.2411	−2.56891
$515$	52.4937	2.31315
$516$	0	0
$517$	−29.3101	−1.28906
$518$	24.3317	1.06908
$519$	0	0
$520$	−4.52136	−0.198275
$521$	−36.0689	−1.58021	−0.790104	−	0.612973i	$-0.789973\pi$
$521$	−36.0689	−1.58021	−0.790104	+	0.612973i	$0.789973\pi$
$522$	0	0
$523$	−19.5962	−0.856882	−0.428441	−	0.903570i	$-0.640937\pi$
$523$	−19.5962	−0.856882	−0.428441	+	0.903570i	$0.640937\pi$
$524$	23.5374	1.02824
$525$	0	0
$526$	−3.16159	−0.137852
$527$	−1.61652	−0.0704166
$528$	0	0
$529$	−22.7967	−0.991162
$530$	23.9625	1.04086
$531$	0	0
$532$	47.7636	2.07081
$533$	7.51896	0.325682
$534$	0	0
$535$	0.146775	0.00634563
$536$	−23.6253	−1.02046
$537$	0	0
$538$	−12.1489	−0.523777
$539$	−9.21394	−0.396873
$540$	0	0
$541$	−19.6333	−0.844102	−0.422051	−	0.906572i	$-0.638690\pi$
$541$	−19.6333	−0.844102	−0.422051	+	0.906572i	$0.638690\pi$
$542$	−68.3038	−2.93390
$543$	0	0
$544$	10.0667	0.431608
$545$	1.19307	0.0511053
$546$	0	0
$547$	−30.5891	−1.30790	−0.653948	−	0.756539i	$-0.726889\pi$
$547$	−30.5891	−1.30790	−0.653948	+	0.756539i	$0.726889\pi$
$548$	−30.3356	−1.29587
$549$	0	0
$550$	43.3627	1.84899
$551$	19.5428	0.832553
$552$	0	0
$553$	8.40198	0.357289
$554$	−10.1246	−0.430152
$555$	0	0
$556$	−51.3034	−2.17575
$557$	−33.4495	−1.41730	−0.708651	−	0.705559i	$-0.750696\pi$
$557$	−33.4495	−1.41730	−0.708651	+	0.705559i	$0.750696\pi$
$558$	0	0
$559$	5.83138	0.246641
$560$	10.7861	0.455795
$561$	0	0
$562$	−40.6127	−1.71314
$563$	0.209922	0.00884715	0.00442357	−	0.999990i	$-0.498592\pi$
$563$	0.209922	0.00884715	0.00442357	+	0.999990i	$0.498592\pi$
$564$	0	0
$565$	33.5079	1.40969
$566$	52.6415	2.21269
$567$	0	0
$568$	−28.0837	−1.17836
$569$	−40.9265	−1.71573	−0.857865	−	0.513876i	$-0.828209\pi$
$569$	−40.9265	−1.71573	−0.857865	+	0.513876i	$0.828209\pi$
$570$	0	0
$571$	28.1098	1.17636	0.588180	−	0.808730i	$-0.299845\pi$
$571$	28.1098	1.17636	0.588180	+	0.808730i	$0.299845\pi$
$572$	11.4699	0.479583
$573$	0	0
$574$	67.1760	2.80387
$575$	−1.65748	−0.0691218
$576$	0	0
$577$	13.6316	0.567490	0.283745	−	0.958900i	$-0.408423\pi$
$577$	13.6316	0.567490	0.283745	+	0.958900i	$0.408423\pi$
$578$	33.0906	1.37639
$579$	0	0
$580$	−30.7766	−1.27793
$581$	1.03994	0.0431442
$582$	0	0
$583$	−19.4268	−0.804577
$584$	8.28634	0.342891
$585$	0	0
$586$	10.5968	0.437749
$587$	28.1027	1.15992	0.579961	−	0.814644i	$-0.303068\pi$
$587$	28.1027	1.15992	0.579961	+	0.814644i	$0.303068\pi$
$588$	0	0
$589$	−6.11695	−0.252044
$590$	46.9100	1.93126
$591$	0	0
$592$	4.58899	0.188606
$593$	4.24975	0.174516	0.0872581	−	0.996186i	$-0.472190\pi$
$593$	4.24975	0.174516	0.0872581	+	0.996186i	$0.472190\pi$
$594$	0	0
$595$	−12.6490	−0.518560
$596$	−24.4384	−1.00103
$597$	0	0
$598$	−0.736736	−0.0301274
$599$	21.2246	0.867213	0.433606	−	0.901102i	$-0.357241\pi$
$599$	21.2246	0.867213	0.433606	+	0.901102i	$0.357241\pi$
$600$	0	0
$601$	−39.0425	−1.59258	−0.796289	−	0.604916i	$-0.793207\pi$
$601$	−39.0425	−1.59258	−0.796289	+	0.604916i	$0.793207\pi$
$602$	52.0988	2.12339
$603$	0	0
$604$	−51.7344	−2.10504
$605$	−50.5671	−2.05584
$606$	0	0
$607$	44.6975	1.81422	0.907108	−	0.420898i	$-0.138285\pi$
$607$	44.6975	1.81422	0.907108	+	0.420898i	$0.138285\pi$
$608$	38.0929	1.54487
$609$	0	0
$610$	28.5095	1.15432
$611$	−4.06053	−0.164271
$612$	0	0
$613$	−45.8643	−1.85244	−0.926221	−	0.376982i	$-0.876962\pi$
$613$	−45.8643	−1.85244	−0.926221	+	0.376982i	$0.876962\pi$
$614$	23.9837	0.967901
$615$	0	0
$616$	32.7488	1.31949
$617$	−21.7123	−0.874105	−0.437053	−	0.899436i	$-0.643978\pi$
$617$	−21.7123	−0.874105	−0.437053	+	0.899436i	$0.643978\pi$
$618$	0	0
$619$	−29.4617	−1.18417	−0.592083	−	0.805877i	$-0.701694\pi$
$619$	−29.4617	−1.18417	−0.592083	+	0.805877i	$0.701694\pi$
$620$	9.63313	0.386876
$621$	0	0
$622$	−21.5166	−0.862735
$623$	−16.8072	−0.673368
$624$	0	0
$625$	−29.8664	−1.19465
$626$	14.1951	0.567351
$627$	0	0
$628$	−6.33378	−0.252745
$629$	−5.38160	−0.214578
$630$	0	0
$631$	−30.2369	−1.20371	−0.601857	−	0.798604i	$-0.705572\pi$
$631$	−30.2369	−1.20371	−0.601857	+	0.798604i	$0.705572\pi$
$632$	−5.93455	−0.236064
$633$	0	0
$634$	45.8074	1.81924
$635$	−38.4262	−1.52490
$636$	0	0
$637$	−1.27647	−0.0505755
$638$	41.9284	1.65996
$639$	0	0
$640$	−43.7689	−1.73012
$641$	−0.612492	−0.0241920	−0.0120960	−	0.999927i	$-0.503850\pi$
$641$	−0.612492	−0.0241920	−0.0120960	+	0.999927i	$0.503850\pi$
$642$	0	0
$643$	18.6672	0.736164	0.368082	−	0.929793i	$-0.380014\pi$
$643$	18.6672	0.736164	0.368082	+	0.929793i	$0.380014\pi$
$644$	−3.91697	−0.154350
$645$	0	0
$646$	−17.7522	−0.698452
$647$	−44.8197	−1.76204	−0.881022	−	0.473075i	$-0.843144\pi$
$647$	−44.8197	−1.76204	−0.881022	+	0.473075i	$0.843144\pi$
$648$	0	0
$649$	−38.0309	−1.49284
$650$	6.00733	0.235627
$651$	0	0
$652$	27.8531	1.09081
$653$	49.4906	1.93672	0.968359	−	0.249560i	$-0.0802860\pi$
$653$	49.4906	1.93672	0.968359	+	0.249560i	$0.0802860\pi$
$654$	0	0
$655$	−23.5871	−0.921623
$656$	12.6695	0.494659
$657$	0	0
$658$	−36.2776	−1.41425
$659$	11.3799	0.443299	0.221649	−	0.975126i	$-0.428856\pi$
$659$	11.3799	0.443299	0.221649	+	0.975126i	$0.428856\pi$
$660$	0	0
$661$	2.64285	0.102795	0.0513974	−	0.998678i	$-0.483632\pi$
$661$	2.64285	0.102795	0.0513974	+	0.998678i	$0.483632\pi$
$662$	−45.1107	−1.75328
$663$	0	0
$664$	−0.734541	−0.0285057
$665$	−47.8643	−1.85610
$666$	0	0
$667$	−1.60266	−0.0620552
$668$	−11.2329	−0.434614
$669$	0	0
$670$	74.0824	2.86205
$671$	−23.1132	−0.892276
$672$	0	0
$673$	−2.34401	−0.0903551	−0.0451776	−	0.998979i	$-0.514385\pi$
$673$	−2.34401	−0.0903551	−0.0451776	+	0.998979i	$0.514385\pi$
$674$	8.12603	0.313003
$675$	0	0
$676$	−36.6226	−1.40856
$677$	9.34103	0.359005	0.179503	−	0.983758i	$-0.442551\pi$
$677$	9.34103	0.359005	0.179503	+	0.983758i	$0.442551\pi$
$678$	0	0
$679$	−20.8362	−0.799622
$680$	8.93435	0.342617
$681$	0	0
$682$	−13.1237	−0.502532
$683$	−29.3064	−1.12138	−0.560689	−	0.828026i	$-0.689464\pi$
$683$	−29.3064	−1.12138	−0.560689	+	0.828026i	$0.689464\pi$
$684$	0	0
$685$	30.3996	1.16151
$686$	34.5782	1.32020
$687$	0	0
$688$	9.82589	0.374609
$689$	−2.69133	−0.102531
$690$	0	0
$691$	−13.6151	−0.517941	−0.258971	−	0.965885i	$-0.583383\pi$
$691$	−13.6151	−0.517941	−0.258971	+	0.965885i	$0.583383\pi$
$692$	24.5928	0.934880
$693$	0	0
$694$	−64.7986	−2.45972
$695$	51.4116	1.95015
$696$	0	0
$697$	−14.8577	−0.562776
$698$	−37.6308	−1.42435
$699$	0	0
$700$	31.9389	1.20718
$701$	−41.6203	−1.57198	−0.785989	−	0.618241i	$-0.787846\pi$
$701$	−41.6203	−1.57198	−0.785989	+	0.618241i	$0.787846\pi$
$702$	0	0
$703$	−20.3641	−0.768047
$704$	68.5764	2.58457
$705$	0	0
$706$	50.5577	1.90276
$707$	31.3652	1.17961
$708$	0	0
$709$	−11.5435	−0.433524	−0.216762	−	0.976224i	$-0.569550\pi$
$709$	−11.5435	−0.433524	−0.216762	+	0.976224i	$0.569550\pi$
$710$	88.0626	3.30493
$711$	0	0
$712$	11.8714	0.444900
$713$	0.501635	0.0187864
$714$	0	0
$715$	−11.4941	−0.429856
$716$	16.1100	0.602060
$717$	0	0
$718$	34.0378	1.27028
$719$	−31.8602	−1.18818	−0.594092	−	0.804397i	$-0.702488\pi$
$719$	−31.8602	−1.18818	−0.594092	+	0.804397i	$0.702488\pi$
$720$	0	0
$721$	−52.6743	−1.96169
$722$	−24.9481	−0.928474
$723$	0	0
$724$	64.4483	2.39520
$725$	13.0680	0.485335
$726$	0	0
$727$	−30.1927	−1.11979	−0.559893	−	0.828565i	$-0.689158\pi$
$727$	−30.1927	−1.11979	−0.559893	+	0.828565i	$0.689158\pi$
$728$	4.53691	0.168149
$729$	0	0
$730$	−25.9836	−0.961698
$731$	−11.5230	−0.426194
$732$	0	0
$733$	−15.9963	−0.590838	−0.295419	−	0.955368i	$-0.595459\pi$
$733$	−15.9963	−0.590838	−0.295419	+	0.955368i	$0.595459\pi$
$734$	−31.1965	−1.15148
$735$	0	0
$736$	−3.12390	−0.115148
$737$	−60.0600	−2.21234
$738$	0	0
$739$	−14.8174	−0.545066	−0.272533	−	0.962146i	$-0.587861\pi$
$739$	−14.8174	−0.545066	−0.272533	+	0.962146i	$0.587861\pi$
$740$	32.0700	1.17892
$741$	0	0
$742$	−24.0449	−0.882716
$743$	−1.03928	−0.0381273	−0.0190637	−	0.999818i	$-0.506069\pi$
$743$	−1.03928	−0.0381273	−0.0190637	+	0.999818i	$0.506069\pi$
$744$	0	0
$745$	24.4899	0.897241
$746$	60.5544	2.21706
$747$	0	0
$748$	−22.6650	−0.828715
$749$	−0.147280	−0.00538149
$750$	0	0
$751$	−33.0125	−1.20464	−0.602322	−	0.798253i	$-0.705758\pi$
$751$	−33.0125	−1.20464	−0.602322	+	0.798253i	$0.705758\pi$
$752$	−6.84200	−0.249502
$753$	0	0
$754$	5.80863	0.211538
$755$	51.8435	1.88678
$756$	0	0
$757$	−33.1877	−1.20623	−0.603114	−	0.797655i	$-0.706074\pi$
$757$	−33.1877	−1.20623	−0.603114	+	0.797655i	$0.706074\pi$
$758$	−19.1066	−0.693984
$759$	0	0
$760$	33.8079	1.22634
$761$	37.3624	1.35439	0.677193	−	0.735805i	$-0.263196\pi$
$761$	37.3624	1.35439	0.677193	+	0.735805i	$0.263196\pi$
$762$	0	0
$763$	−1.19717	−0.0433405
$764$	−37.4544	−1.35505
$765$	0	0
$766$	42.8682	1.54889
$767$	−5.26867	−0.190241
$768$	0	0
$769$	−42.7640	−1.54211	−0.771055	−	0.636769i	$-0.780271\pi$
$769$	−42.7640	−1.54211	−0.771055	+	0.636769i	$0.780271\pi$
$770$	−102.691	−3.70073
$771$	0	0
$772$	59.7652	2.15100
$773$	−24.6670	−0.887211	−0.443605	−	0.896222i	$-0.646301\pi$
$773$	−24.6670	−0.887211	−0.443605	+	0.896222i	$0.646301\pi$
$774$	0	0
$775$	−4.09033	−0.146929
$776$	14.7172	0.528317
$777$	0	0
$778$	69.2125	2.48139
$779$	−56.2221	−2.01436
$780$	0	0
$781$	−71.3940	−2.55468
$782$	1.45582	0.0520598
$783$	0	0
$784$	−2.15085	−0.0768162
$785$	6.34714	0.226539
$786$	0	0
$787$	34.1431	1.21707	0.608535	−	0.793527i	$-0.291757\pi$
$787$	34.1431	1.21707	0.608535	+	0.793527i	$0.291757\pi$
$788$	−11.7978	−0.420281
$789$	0	0
$790$	18.6091	0.662082
$791$	−33.6232	−1.19550
$792$	0	0
$793$	−3.20203	−0.113707
$794$	−23.6992	−0.841052
$795$	0	0
$796$	11.3041	0.400662
$797$	−16.9399	−0.600040	−0.300020	−	0.953933i	$-0.596993\pi$
$797$	−16.9399	−0.600040	−0.300020	+	0.953933i	$0.596993\pi$
$798$	0	0
$799$	8.02374	0.283860
$800$	25.4722	0.900578
$801$	0	0
$802$	18.0036	0.635730
$803$	21.0654	0.743383
$804$	0	0
$805$	3.92523	0.138346
$806$	−1.81811	−0.0640403
$807$	0	0
$808$	−22.1541	−0.779377
$809$	−18.5465	−0.652062	−0.326031	−	0.945359i	$-0.605711\pi$
$809$	−18.5465	−0.652062	−0.326031	+	0.945359i	$0.605711\pi$
$810$	0	0
$811$	24.9560	0.876325	0.438163	−	0.898896i	$-0.355629\pi$
$811$	24.9560	0.876325	0.438163	+	0.898896i	$0.355629\pi$
$812$	30.8825	1.08376
$813$	0	0
$814$	−43.6905	−1.53135
$815$	−27.9119	−0.977711
$816$	0	0
$817$	−43.6034	−1.52549
$818$	83.3021	2.91259
$819$	0	0
$820$	88.5400	3.09195
$821$	1.35107	0.0471527	0.0235763	−	0.999722i	$-0.492495\pi$
$821$	1.35107	0.0471527	0.0235763	+	0.999722i	$0.492495\pi$
$822$	0	0
$823$	43.8887	1.52986	0.764932	−	0.644111i	$-0.222772\pi$
$823$	43.8887	1.52986	0.764932	+	0.644111i	$0.222772\pi$
$824$	37.2053	1.29611
$825$	0	0
$826$	−47.0714	−1.63783
$827$	56.3672	1.96008	0.980041	−	0.198798i	$-0.0637037\pi$
$827$	56.3672	1.96008	0.980041	+	0.198798i	$0.0637037\pi$
$828$	0	0
$829$	14.6746	0.509671	0.254836	−	0.966984i	$-0.417979\pi$
$829$	14.6746	0.509671	0.254836	+	0.966984i	$0.417979\pi$
$830$	2.30332	0.0799493
$831$	0	0
$832$	9.50035	0.329365
$833$	2.52235	0.0873941
$834$	0	0
$835$	11.2566	0.389551
$836$	−85.7651	−2.96625
$837$	0	0
$838$	28.6604	0.990056
$839$	0.948206	0.0327357	0.0163679	−	0.999866i	$-0.494790\pi$
$839$	0.948206	0.0327357	0.0163679	+	0.999866i	$0.494790\pi$
$840$	0	0
$841$	−16.3642	−0.564283
$842$	−51.4122	−1.77178
$843$	0	0
$844$	20.9520	0.721199
$845$	36.6998	1.26251
$846$	0	0
$847$	50.7411	1.74348
$848$	−4.53489	−0.155729
$849$	0	0
$850$	−11.8707	−0.407161
$851$	1.67001	0.0572472
$852$	0	0
$853$	3.19586	0.109424	0.0547120	−	0.998502i	$-0.482576\pi$
$853$	3.19586	0.109424	0.0547120	+	0.998502i	$0.482576\pi$
$854$	−28.6076	−0.978933
$855$	0	0
$856$	0.104028	0.00355560
$857$	−29.6860	−1.01406	−0.507028	−	0.861930i	$-0.669256\pi$
$857$	−29.6860	−1.01406	−0.507028	+	0.861930i	$0.669256\pi$
$858$	0	0
$859$	−13.8351	−0.472046	−0.236023	−	0.971747i	$-0.575844\pi$
$859$	−13.8351	−0.472046	−0.236023	+	0.971747i	$0.575844\pi$
$860$	68.6678	2.34155
$861$	0	0
$862$	50.4496	1.71832
$863$	57.9543	1.97279	0.986394	−	0.164396i	$-0.0525674\pi$
$863$	57.9543	1.97279	0.986394	+	0.164396i	$0.0525674\pi$
$864$	0	0
$865$	−24.6447	−0.837945
$866$	40.4733	1.37534
$867$	0	0
$868$	−9.66628	−0.328095
$869$	−15.0867	−0.511783
$870$	0	0
$871$	−8.32051	−0.281930
$872$	0.845594	0.0286354
$873$	0	0
$874$	5.50885	0.186340
$875$	11.5244	0.389597
$876$	0	0
$877$	13.5677	0.458148	0.229074	−	0.973409i	$-0.426430\pi$
$877$	13.5677	0.458148	0.229074	+	0.973409i	$0.426430\pi$
$878$	−61.1095	−2.06235
$879$	0	0
$880$	−19.3677	−0.652883
$881$	3.11738	0.105027	0.0525135	−	0.998620i	$-0.483277\pi$
$881$	3.11738	0.105027	0.0525135	+	0.998620i	$0.483277\pi$
$882$	0	0
$883$	−5.87731	−0.197787	−0.0988936	−	0.995098i	$-0.531530\pi$
$883$	−5.87731	−0.197787	−0.0988936	+	0.995098i	$0.531530\pi$
$884$	−3.13994	−0.105607
$885$	0	0
$886$	−27.2516	−0.915535
$887$	−4.68146	−0.157188	−0.0785941	−	0.996907i	$-0.525043\pi$
$887$	−4.68146	−0.157188	−0.0785941	+	0.996907i	$0.525043\pi$
$888$	0	0
$889$	38.5584	1.29321
$890$	−37.2255	−1.24780
$891$	0	0
$892$	43.6472	1.46142
$893$	30.3621	1.01603
$894$	0	0
$895$	−16.1440	−0.539635
$896$	43.9195	1.46725
$897$	0	0
$898$	15.4230	0.514672
$899$	−3.95503	−0.131908
$900$	0	0
$901$	5.31816	0.177173
$902$	−120.622	−4.01629
$903$	0	0
$904$	23.7490	0.789879
$905$	−64.5842	−2.14685
$906$	0	0
$907$	0.933292	0.0309895	0.0154947	−	0.999880i	$-0.495068\pi$
$907$	0.933292	0.0309895	0.0154947	+	0.999880i	$0.495068\pi$
$908$	−39.5531	−1.31262
$909$	0	0
$910$	−14.2265	−0.471604
$911$	−45.4417	−1.50555	−0.752776	−	0.658277i	$-0.771286\pi$
$911$	−45.4417	−1.50555	−0.752776	+	0.658277i	$0.771286\pi$
$912$	0	0
$913$	−1.86734	−0.0618000
$914$	−66.3735	−2.19544
$915$	0	0
$916$	−9.58056	−0.316550
$917$	23.6682	0.781594
$918$	0	0
$919$	40.2641	1.32819	0.664095	−	0.747648i	$-0.268817\pi$
$919$	40.2641	1.32819	0.664095	+	0.747648i	$0.268817\pi$
$920$	−2.77250	−0.0914065
$921$	0	0
$922$	−1.82341	−0.0600508
$923$	−9.89069	−0.325556
$924$	0	0
$925$	−13.6172	−0.447732
$926$	63.6055	2.09021
$927$	0	0
$928$	24.6297	0.808508
$929$	13.0969	0.429696	0.214848	−	0.976648i	$-0.431074\pi$
$929$	13.0969	0.429696	0.214848	+	0.976648i	$0.431074\pi$
$930$	0	0
$931$	9.54464	0.312813
$932$	−24.3174	−0.796544
$933$	0	0
$934$	−27.7650	−0.908499
$935$	22.7128	0.742788
$936$	0	0
$937$	−31.2625	−1.02130	−0.510650	−	0.859789i	$-0.670595\pi$
$937$	−31.2625	−1.02130	−0.510650	+	0.859789i	$0.670595\pi$
$938$	−74.3373	−2.42720
$939$	0	0
$940$	−47.8150	−1.55955
$941$	47.0000	1.53215	0.766077	−	0.642748i	$-0.222206\pi$
$941$	47.0000	1.53215	0.766077	+	0.642748i	$0.222206\pi$
$942$	0	0
$943$	4.61063	0.150143
$944$	−8.87772	−0.288945
$945$	0	0
$946$	−93.5495	−3.04156
$947$	30.6781	0.996903	0.498452	−	0.866917i	$-0.333902\pi$
$947$	30.6781	0.996903	0.498452	+	0.866917i	$0.333902\pi$
$948$	0	0
$949$	2.91834	0.0947332
$950$	−44.9191	−1.45737
$951$	0	0
$952$	−8.96509	−0.290560
$953$	6.76806	0.219239	0.109620	−	0.993974i	$-0.465037\pi$
$953$	6.76806	0.219239	0.109620	+	0.993974i	$0.465037\pi$
$954$	0	0
$955$	37.5334	1.21455
$956$	38.3963	1.24183
$957$	0	0
$958$	−10.0809	−0.325699
$959$	−30.5042	−0.985030
$960$	0	0
$961$	−29.7621	−0.960067
$962$	−6.05274	−0.195148
$963$	0	0
$964$	−1.85990	−0.0599033
$965$	−59.8912	−1.92797
$966$	0	0
$967$	19.3202	0.621296	0.310648	−	0.950525i	$-0.399454\pi$
$967$	19.3202	0.621296	0.310648	+	0.950525i	$0.399454\pi$
$968$	−35.8398	−1.15193
$969$	0	0
$970$	−46.1491	−1.48176
$971$	−42.4827	−1.36333	−0.681667	−	0.731662i	$-0.738745\pi$
$971$	−42.4827	−1.36333	−0.681667	+	0.731662i	$0.738745\pi$
$972$	0	0
$973$	−51.5885	−1.65385
$974$	62.8706	2.01450
$975$	0	0
$976$	−5.39543	−0.172703
$977$	38.0053	1.21590	0.607949	−	0.793976i	$-0.291992\pi$
$977$	38.0053	1.21590	0.607949	+	0.793976i	$0.291992\pi$
$978$	0	0
$979$	30.1794	0.964537
$980$	−15.0311	−0.480152
$981$	0	0
$982$	43.5251	1.38894
$983$	−5.53652	−0.176588	−0.0882938	−	0.996094i	$-0.528141\pi$
$983$	−5.53652	−0.176588	−0.0882938	+	0.996094i	$0.528141\pi$
$984$	0	0
$985$	11.8227	0.376703
$986$	−11.4780	−0.365535
$987$	0	0
$988$	−11.8816	−0.378004
$989$	3.57580	0.113704
$990$	0	0
$991$	22.1328	0.703070	0.351535	−	0.936175i	$-0.385660\pi$
$991$	22.1328	0.703070	0.351535	+	0.936175i	$0.385660\pi$
$992$	−7.70914	−0.244765
$993$	0	0
$994$	−88.3656	−2.80279
$995$	−11.3279	−0.359118
$996$	0	0
$997$	20.4917	0.648979	0.324490	−	0.945889i	$-0.394807\pi$
$997$	20.4917	0.648979	0.324490	+	0.945889i	$0.394807\pi$
$998$	65.7832	2.08233
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6561.2.a.c.1.11		72
3.2	odd	2		6561.2.a.d.1.62		72
81.2	odd	54		729.2.g.a.433.2		144
81.13	even	27		81.2.g.a.7.2	✓	144
81.14	odd	54		729.2.g.b.55.7		144
81.25	even	27		81.2.g.a.58.2	yes	144
81.29	odd	54		729.2.g.b.676.7		144
81.40	even	27		729.2.g.d.298.7		144
81.41	odd	54		729.2.g.a.298.2		144
81.52	even	27		729.2.g.c.676.2		144
81.56	odd	54		243.2.g.a.226.7		144
81.67	even	27		729.2.g.c.55.2		144
81.68	odd	54		243.2.g.a.100.7		144
81.79	even	27		729.2.g.d.433.7		144

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
81.2.g.a.7.2	✓	144	81.13	even	27
81.2.g.a.58.2	yes	144	81.25	even	27
243.2.g.a.100.7		144	81.68	odd	54
243.2.g.a.226.7		144	81.56	odd	54
729.2.g.a.298.2		144	81.41	odd	54
729.2.g.a.433.2		144	81.2	odd	54
729.2.g.b.55.7		144	81.14	odd	54
729.2.g.b.676.7		144	81.29	odd	54
729.2.g.c.55.2		144	81.67	even	27
729.2.g.c.676.2		144	81.52	even	27
729.2.g.d.298.7		144	81.40	even	27
729.2.g.d.433.7		144	81.79	even	27
6561.2.a.c.1.11		72	1.1	even	1	trivial
6561.2.a.d.1.62		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)

\(f(q)\)	\(=\)	\(q-2.22247 q^{2} +2.93935 q^{4} -2.94555 q^{5} +2.95569 q^{7} -2.08768 q^{8} +6.54639 q^{10} -5.30728 q^{11} -0.735254 q^{13} -6.56892 q^{14} -1.23890 q^{16} +1.45289 q^{17} +5.49777 q^{19} -8.65803 q^{20} +11.7953 q^{22} -0.450858 q^{23} +3.67629 q^{25} +1.63408 q^{26} +8.68782 q^{28} +3.55469 q^{29} -1.11262 q^{31} +6.92879 q^{32} -3.22899 q^{34} -8.70614 q^{35} -3.70407 q^{37} -12.2186 q^{38} +6.14938 q^{40} -10.2263 q^{41} -7.93111 q^{43} -15.6000 q^{44} +1.00202 q^{46} +5.52262 q^{47} +1.73609 q^{49} -8.17042 q^{50} -2.16117 q^{52} +3.66041 q^{53} +15.6329 q^{55} -6.17054 q^{56} -7.90017 q^{58} +7.16579 q^{59} +4.35500 q^{61} +2.47277 q^{62} -12.9212 q^{64} +2.16573 q^{65} +11.3165 q^{67} +4.27055 q^{68} +19.3491 q^{70} +13.4521 q^{71} -3.96916 q^{73} +8.23217 q^{74} +16.1599 q^{76} -15.6867 q^{77} +2.84265 q^{79} +3.64926 q^{80} +22.7277 q^{82} +0.351845 q^{83} -4.27955 q^{85} +17.6266 q^{86} +11.0799 q^{88} -5.68641 q^{89} -2.17318 q^{91} -1.32523 q^{92} -12.2738 q^{94} -16.1940 q^{95} -7.04954 q^{97} -3.85841 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 72 q - 9 q^{2} + 63 q^{4} - 18 q^{5} - 27 q^{8} - 36 q^{11} - 36 q^{14} + 45 q^{16} - 36 q^{17} - 54 q^{20} - 54 q^{23} + 36 q^{25} - 45 q^{26} + 9 q^{28} - 54 q^{29} - 63 q^{32} - 72 q^{35} - 54 q^{38}+ \cdots - 81 q^{98}+O(q^{100}) \) 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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