LMFDB - Embedded newform 6561.2.a.d.1.42

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			1.42
Character	$\chi$	$=$	6561.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+0.661372 q^{2} -1.56259 q^{4} +2.51349 q^{5} +4.11204 q^{7} -2.35619 q^{8} +1.66235 q^{10} -2.96060 q^{11} -3.24317 q^{13} +2.71959 q^{14} +1.56685 q^{16} +3.29829 q^{17} -3.28896 q^{19} -3.92756 q^{20} -1.95805 q^{22} -2.49612 q^{23} +1.31766 q^{25} -2.14494 q^{26} -6.42542 q^{28} -5.56701 q^{29} +6.42268 q^{31} +5.74866 q^{32} +2.18139 q^{34} +10.3356 q^{35} +2.73427 q^{37} -2.17522 q^{38} -5.92228 q^{40} +9.04853 q^{41} -7.81757 q^{43} +4.62619 q^{44} -1.65086 q^{46} +10.1199 q^{47} +9.90886 q^{49} +0.871461 q^{50} +5.06773 q^{52} +0.00988863 q^{53} -7.44144 q^{55} -9.68876 q^{56} -3.68186 q^{58} +11.3730 q^{59} +9.01190 q^{61} +4.24778 q^{62} +0.668297 q^{64} -8.15168 q^{65} +5.84527 q^{67} -5.15386 q^{68} +6.83567 q^{70} +2.37345 q^{71} +4.72541 q^{73} +1.80837 q^{74} +5.13928 q^{76} -12.1741 q^{77} +9.00201 q^{79} +3.93828 q^{80} +5.98444 q^{82} +3.70134 q^{83} +8.29022 q^{85} -5.17032 q^{86} +6.97574 q^{88} +12.4961 q^{89} -13.3360 q^{91} +3.90041 q^{92} +6.69304 q^{94} -8.26678 q^{95} -7.15193 q^{97} +6.55344 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 72 q + 9 q^{2} + 63 q^{4} + 18 q^{5} + 27 q^{8} + 36 q^{11} + 36 q^{14} + 45 q^{16} + 36 q^{17} + 54 q^{20} + 54 q^{23} + 36 q^{25} + 45 q^{26} + 9 q^{28} + 54 q^{29} + 63 q^{32} + 72 q^{35} + 54 q^{38}+ \cdots + 81 q^{98}+O(q^{100}) $ 72 * q + 9 * q^2 + 63 * q^4 + 18 * q^5 + 27 * q^8 + 36 * q^11 + 36 * q^14 + 45 * q^16 + 36 * q^17 + 54 * q^20 + 54 * q^23 + 36 * q^25 + 45 * q^26 + 9 * q^28 + 54 * q^29 + 63 * q^32 + 72 * q^35 + 54 * q^38 + 72 * q^41 + 90 * q^44 + 90 * q^47 + 18 * q^49 + 45 * q^50 + 45 * q^53 + 9 * q^55 + 108 * q^56 + 18 * q^58 + 108 * q^59 + 72 * q^62 + 9 * q^64 + 72 * q^65 + 108 * q^68 + 126 * q^71 + 90 * q^74 + 72 * q^77 + 144 * q^80 - 18 * q^82 + 108 * q^83 + 90 * q^86 + 108 * q^89 + 72 * q^92 + 144 * q^95 + 81 * q^98

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0.661372	0.467661	0.233830	−	0.972277i	$-0.424874\pi$
$2$	0.661372	0.467661	0.233830	+	0.972277i	$0.424874\pi$
$3$	0	0
$3$	0	0
$4$	−1.56259	−0.781294
$5$	2.51349	1.12407	0.562035	−	0.827114i	$-0.310019\pi$
$5$	2.51349	1.12407	0.562035	+	0.827114i	$0.310019\pi$
$6$	0	0
$7$	4.11204	1.55420	0.777102	−	0.629374i	$-0.216689\pi$
$7$	4.11204	1.55420	0.777102	+	0.629374i	$0.216689\pi$
$8$	−2.35619	−0.833041
$9$	0	0
$10$	1.66235	0.525683
$11$	−2.96060	−0.892653	−0.446327	−	0.894870i	$-0.647268\pi$
$11$	−2.96060	−0.892653	−0.446327	+	0.894870i	$0.647268\pi$
$12$	0	0
$13$	−3.24317	−0.899492	−0.449746	−	0.893156i	$-0.648486\pi$
$13$	−3.24317	−0.899492	−0.449746	+	0.893156i	$0.648486\pi$
$14$	2.71959	0.726840
$15$	0	0
$16$	1.56685	0.391713
$17$	3.29829	0.799952	0.399976	−	0.916526i	$-0.369018\pi$
$17$	3.29829	0.799952	0.399976	+	0.916526i	$0.369018\pi$
$18$	0	0
$19$	−3.28896	−0.754539	−0.377269	−	0.926104i	$-0.623137\pi$
$19$	−3.28896	−0.754539	−0.377269	+	0.926104i	$0.623137\pi$
$20$	−3.92756	−0.878228
$21$	0	0
$22$	−1.95805	−0.417459
$23$	−2.49612	−0.520477	−0.260239	−	0.965544i	$-0.583801\pi$
$23$	−2.49612	−0.520477	−0.260239	+	0.965544i	$0.583801\pi$
$24$	0	0
$25$	1.31766	0.263531
$26$	−2.14494	−0.420657
$27$	0	0
$28$	−6.42542	−1.21429
$29$	−5.56701	−1.03377	−0.516884	−	0.856056i	$-0.672908\pi$
$29$	−5.56701	−1.03377	−0.516884	+	0.856056i	$0.672908\pi$
$30$	0	0
$31$	6.42268	1.15355	0.576774	−	0.816904i	$-0.304311\pi$
$31$	6.42268	1.15355	0.576774	+	0.816904i	$0.304311\pi$
$32$	5.74866	1.01623
$33$	0	0
$34$	2.18139	0.374106
$35$	10.3356	1.74703
$36$	0	0
$37$	2.73427	0.449511	0.224756	−	0.974415i	$-0.427842\pi$
$37$	2.73427	0.449511	0.224756	+	0.974415i	$0.427842\pi$
$38$	−2.17522	−0.352868
$39$	0	0
$40$	−5.92228	−0.936395
$41$	9.04853	1.41314	0.706571	−	0.707642i	$-0.250241\pi$
$41$	9.04853	1.41314	0.706571	+	0.707642i	$0.250241\pi$
$42$	0	0
$43$	−7.81757	−1.19217	−0.596084	−	0.802922i	$-0.703278\pi$
$43$	−7.81757	−1.19217	−0.596084	+	0.802922i	$0.703278\pi$
$44$	4.62619	0.697424
$45$	0	0
$46$	−1.65086	−0.243407
$47$	10.1199	1.47614	0.738072	−	0.674722i	$-0.235737\pi$
$47$	10.1199	1.47614	0.738072	+	0.674722i	$0.235737\pi$
$48$	0	0
$49$	9.90886	1.41555
$50$	0.871461	0.123243
$51$	0	0
$52$	5.06773	0.702768
$53$	0.00988863	0.00135831	0.000679154	−	1.00000i	$-0.499784\pi$
$53$	0.00988863	0.00135831	0.000679154		1.00000i	$0.499784\pi$
$54$	0	0
$55$	−7.44144	−1.00340
$56$	−9.68876	−1.29472
$57$	0	0
$58$	−3.68186	−0.483452
$59$	11.3730	1.48064	0.740321	−	0.672253i	$-0.234673\pi$
$59$	11.3730	1.48064	0.740321	+	0.672253i	$0.234673\pi$
$60$	0	0
$61$	9.01190	1.15386	0.576928	−	0.816795i	$-0.304251\pi$
$61$	9.01190	1.15386	0.576928	+	0.816795i	$0.304251\pi$
$62$	4.24778	0.539469
$63$	0	0
$64$	0.668297	0.0835371
$65$	−8.15168	−1.01109
$66$	0	0
$67$	5.84527	0.714113	0.357056	−	0.934083i	$-0.383780\pi$
$67$	5.84527	0.714113	0.357056	+	0.934083i	$0.383780\pi$
$68$	−5.15386	−0.624997
$69$	0	0
$70$	6.83567	0.817018
$71$	2.37345	0.281677	0.140839	−	0.990033i	$-0.455020\pi$
$71$	2.37345	0.281677	0.140839	+	0.990033i	$0.455020\pi$
$72$	0	0
$73$	4.72541	0.553067	0.276534	−	0.961004i	$-0.410814\pi$
$73$	4.72541	0.553067	0.276534	+	0.961004i	$0.410814\pi$
$74$	1.80837	0.210219
$75$	0	0
$76$	5.13928	0.589516
$77$	−12.1741	−1.38737
$78$	0	0
$79$	9.00201	1.01281	0.506403	−	0.862297i	$-0.330975\pi$
$79$	9.00201	1.01281	0.506403	+	0.862297i	$0.330975\pi$
$80$	3.93828	0.440313
$81$	0	0
$82$	5.98444	0.660871
$83$	3.70134	0.406275	0.203138	−	0.979150i	$-0.434886\pi$
$83$	3.70134	0.406275	0.203138	+	0.979150i	$0.434886\pi$
$84$	0	0
$85$	8.29022	0.899201
$86$	−5.17032	−0.557530
$87$	0	0
$88$	6.97574	0.743616
$89$	12.4961	1.32459	0.662293	−	0.749245i	$-0.269584\pi$
$89$	12.4961	1.32459	0.662293	+	0.749245i	$0.269584\pi$
$90$	0	0
$91$	−13.3360	−1.39799
$92$	3.90041	0.406646
$93$	0	0
$94$	6.69304	0.690334
$95$	−8.26678	−0.848153
$96$	0	0
$97$	−7.15193	−0.726168	−0.363084	−	0.931756i	$-0.618276\pi$
$97$	−7.15193	−0.726168	−0.363084	+	0.931756i	$0.618276\pi$
$98$	6.55344	0.661997
$99$	0	0
$100$	−2.05895	−0.205895
$101$	2.10610	0.209565	0.104783	−	0.994495i	$-0.466585\pi$
$101$	2.10610	0.209565	0.104783	+	0.994495i	$0.466585\pi$
$102$	0	0
$103$	−2.06534	−0.203504	−0.101752	−	0.994810i	$-0.532445\pi$
$103$	−2.06534	−0.203504	−0.101752	+	0.994810i	$0.532445\pi$
$104$	7.64153	0.749314
$105$	0	0
$106$	0.00654006	0.000635227 0
$107$	−14.2636	−1.37891	−0.689456	−	0.724327i	$-0.742150\pi$
$107$	−14.2636	−1.37891	−0.689456	+	0.724327i	$0.742150\pi$
$108$	0	0
$109$	13.4047	1.28394	0.641970	−	0.766730i	$-0.278117\pi$
$109$	13.4047	1.28394	0.641970	+	0.766730i	$0.278117\pi$
$110$	−4.92156	−0.469252
$111$	0	0
$112$	6.44296	0.608803
$113$	2.96033	0.278485	0.139242	−	0.990258i	$-0.455533\pi$
$113$	2.96033	0.278485	0.139242	+	0.990258i	$0.455533\pi$
$114$	0	0
$115$	−6.27399	−0.585053
$116$	8.69893	0.807676
$117$	0	0
$118$	7.52180	0.692438
$119$	13.5627	1.24329
$120$	0	0
$121$	−2.23488	−0.203171
$122$	5.96021	0.539612
$123$	0	0
$124$	−10.0360	−0.901259
$125$	−9.25555	−0.827842
$126$	0	0
$127$	12.0319	1.06766	0.533828	−	0.845593i	$-0.320753\pi$
$127$	12.0319	1.06766	0.533828	+	0.845593i	$0.320753\pi$
$128$	−11.0553	−0.977163
$129$	0	0
$130$	−5.39129	−0.472848
$131$	5.13626	0.448757	0.224379	−	0.974502i	$-0.427965\pi$
$131$	5.13626	0.448757	0.224379	+	0.974502i	$0.427965\pi$
$132$	0	0
$133$	−13.5243	−1.17271
$134$	3.86589	0.333962
$135$	0	0
$136$	−7.77140	−0.666392
$137$	−1.90105	−0.162417	−0.0812087	−	0.996697i	$-0.525878\pi$
$137$	−1.90105	−0.162417	−0.0812087	+	0.996697i	$0.525878\pi$
$138$	0	0
$139$	9.35782	0.793720	0.396860	−	0.917879i	$-0.370100\pi$
$139$	9.35782	0.793720	0.396860	+	0.917879i	$0.370100\pi$
$140$	−16.1503	−1.36495
$141$	0	0
$142$	1.56973	0.131729
$143$	9.60170	0.802935
$144$	0	0
$145$	−13.9926	−1.16203
$146$	3.12525	0.258648
$147$	0	0
$148$	−4.27253	−0.351200
$149$	0.888642	0.0728004	0.0364002	−	0.999337i	$-0.488411\pi$
$149$	0.888642	0.0728004	0.0364002	+	0.999337i	$0.488411\pi$
$150$	0	0
$151$	−8.05855	−0.655796	−0.327898	−	0.944713i	$-0.606340\pi$
$151$	−8.05855	−0.655796	−0.327898	+	0.944713i	$0.606340\pi$
$152$	7.74942	0.628561
$153$	0	0
$154$	−8.05159	−0.648816
$155$	16.1434	1.29667
$156$	0	0
$157$	−14.4705	−1.15487	−0.577437	−	0.816435i	$-0.695947\pi$
$157$	−14.4705	−1.15487	−0.577437	+	0.816435i	$0.695947\pi$
$158$	5.95368	0.473649
$159$	0	0
$160$	14.4492	1.14231
$161$	−10.2641	−0.808928
$162$	0	0
$163$	8.05495	0.630912	0.315456	−	0.948940i	$-0.397843\pi$
$163$	8.05495	0.630912	0.315456	+	0.948940i	$0.397843\pi$
$164$	−14.1391	−1.10408
$165$	0	0
$166$	2.44796	0.189999
$167$	−16.8824	−1.30640	−0.653198	−	0.757187i	$-0.726573\pi$
$167$	−16.8824	−1.30640	−0.653198	+	0.757187i	$0.726573\pi$
$168$	0	0
$169$	−2.48188	−0.190914
$170$	5.48292	0.420521
$171$	0	0
$172$	12.2156	0.931434
$173$	−6.72317	−0.511153	−0.255577	−	0.966789i	$-0.582265\pi$
$173$	−6.72317	−0.511153	−0.255577	+	0.966789i	$0.582265\pi$
$174$	0	0
$175$	5.41825	0.409582
$176$	−4.63882	−0.349664
$177$	0	0
$178$	8.26458	0.619456
$179$	15.2057	1.13652	0.568262	−	0.822848i	$-0.307616\pi$
$179$	15.2057	1.13652	0.568262	+	0.822848i	$0.307616\pi$
$180$	0	0
$181$	−5.60500	−0.416616	−0.208308	−	0.978063i	$-0.566796\pi$
$181$	−5.60500	−0.416616	−0.208308	+	0.978063i	$0.566796\pi$
$182$	−8.82007	−0.653787
$183$	0	0
$184$	5.88135	0.433579
$185$	6.87257	0.505281
$186$	0	0
$187$	−9.76489	−0.714079
$188$	−15.8133	−1.15330
$189$	0	0
$190$	−5.46741	−0.396648
$191$	4.29567	0.310824	0.155412	−	0.987850i	$-0.450329\pi$
$191$	4.29567	0.310824	0.155412	+	0.987850i	$0.450329\pi$
$192$	0	0
$193$	2.83368	0.203973	0.101986	−	0.994786i	$-0.467480\pi$
$193$	2.83368	0.203973	0.101986	+	0.994786i	$0.467480\pi$
$194$	−4.73008	−0.339600
$195$	0	0
$196$	−15.4835	−1.10596
$197$	21.2481	1.51387	0.756933	−	0.653492i	$-0.226697\pi$
$197$	21.2481	1.51387	0.756933	+	0.653492i	$0.226697\pi$
$198$	0	0
$199$	9.73222	0.689899	0.344949	−	0.938621i	$-0.387896\pi$
$199$	9.73222	0.689899	0.344949	+	0.938621i	$0.387896\pi$
$200$	−3.10466	−0.219532
$201$	0	0
$202$	1.39292	0.0980053
$203$	−22.8917	−1.60669
$204$	0	0
$205$	22.7434	1.58847
$206$	−1.36596	−0.0951708
$207$	0	0
$208$	−5.08157	−0.352343
$209$	9.73727	0.673541
$210$	0	0
$211$	19.5782	1.34782	0.673909	−	0.738815i	$-0.264614\pi$
$211$	19.5782	1.34782	0.673909	+	0.738815i	$0.264614\pi$
$212$	−0.0154519	−0.00106124
$213$	0	0
$214$	−9.43353	−0.644863
$215$	−19.6494	−1.34008
$216$	0	0
$217$	26.4103	1.79285
$218$	8.86551	0.600448
$219$	0	0
$220$	11.6279	0.783953
$221$	−10.6969	−0.719550
$222$	0	0
$223$	−16.1663	−1.08257	−0.541287	−	0.840838i	$-0.682063\pi$
$223$	−16.1663	−1.08257	−0.541287	+	0.840838i	$0.682063\pi$
$224$	23.6387	1.57943
$225$	0	0
$226$	1.95788	0.130236
$227$	−21.2304	−1.40911	−0.704556	−	0.709649i	$-0.748854\pi$
$227$	−21.2304	−1.40911	−0.704556	+	0.709649i	$0.748854\pi$
$228$	0	0
$229$	−8.62949	−0.570253	−0.285126	−	0.958490i	$-0.592036\pi$
$229$	−8.62949	−0.570253	−0.285126	+	0.958490i	$0.592036\pi$
$230$	−4.14944	−0.273606
$231$	0	0
$232$	13.1170	0.861170
$233$	14.9643	0.980347	0.490174	−	0.871625i	$-0.336933\pi$
$233$	14.9643	0.980347	0.490174	+	0.871625i	$0.336933\pi$
$234$	0	0
$235$	25.4364	1.65929
$236$	−17.7714	−1.15682
$237$	0	0
$238$	8.96997	0.581437
$239$	3.61051	0.233545	0.116772	−	0.993159i	$-0.462745\pi$
$239$	3.61051	0.233545	0.116772	+	0.993159i	$0.462745\pi$
$240$	0	0
$241$	−22.8914	−1.47457	−0.737283	−	0.675584i	$-0.763892\pi$
$241$	−22.8914	−1.47457	−0.737283	+	0.675584i	$0.763892\pi$
$242$	−1.47808	−0.0950149
$243$	0	0
$244$	−14.0819	−0.901500
$245$	24.9059	1.59118
$246$	0	0
$247$	10.6666	0.678702
$248$	−15.1331	−0.960952
$249$	0	0
$250$	−6.12136	−0.387149
$251$	21.1728	1.33641	0.668207	−	0.743975i	$-0.267062\pi$
$251$	21.1728	1.33641	0.668207	+	0.743975i	$0.267062\pi$
$252$	0	0
$253$	7.39001	0.464606
$254$	7.95755	0.499301
$255$	0	0
$256$	−8.64828	−0.540517
$257$	7.78703	0.485742	0.242871	−	0.970059i	$-0.421911\pi$
$257$	7.78703	0.485742	0.242871	+	0.970059i	$0.421911\pi$
$258$	0	0
$259$	11.2434	0.698632
$260$	12.7377	0.789959
$261$	0	0
$262$	3.39698	0.209866
$263$	−1.52648	−0.0941268	−0.0470634	−	0.998892i	$-0.514986\pi$
$263$	−1.52648	−0.0941268	−0.0470634	+	0.998892i	$0.514986\pi$
$264$	0	0
$265$	0.0248550	0.00152683
$266$	−8.94460	−0.548429
$267$	0	0
$268$	−9.13374	−0.557932
$269$	29.4387	1.79491	0.897454	−	0.441109i	$-0.145415\pi$
$269$	29.4387	1.79491	0.897454	+	0.441109i	$0.145415\pi$
$270$	0	0
$271$	14.5729	0.885238	0.442619	−	0.896710i	$-0.354049\pi$
$271$	14.5729	0.885238	0.442619	+	0.896710i	$0.354049\pi$
$272$	5.16793	0.313352
$273$	0	0
$274$	−1.25730	−0.0759562
$275$	−3.90105	−0.235242
$276$	0	0
$277$	10.2266	0.614456	0.307228	−	0.951636i	$-0.400599\pi$
$277$	10.2266	0.614456	0.307228	+	0.951636i	$0.400599\pi$
$278$	6.18900	0.371192
$279$	0	0
$280$	−24.3527	−1.45535
$281$	−7.02367	−0.418997	−0.209498	−	0.977809i	$-0.567183\pi$
$281$	−7.02367	−0.418997	−0.209498	+	0.977809i	$0.567183\pi$
$282$	0	0
$283$	−21.6223	−1.28531	−0.642657	−	0.766154i	$-0.722168\pi$
$283$	−21.6223	−1.28531	−0.642657	+	0.766154i	$0.722168\pi$
$284$	−3.70873	−0.220072
$285$	0	0
$286$	6.35029	0.375501
$287$	37.2079	2.19631
$288$	0	0
$289$	−6.12131	−0.360077
$290$	−9.25434	−0.543434
$291$	0	0
$292$	−7.38386	−0.432108
$293$	10.9269	0.638356	0.319178	−	0.947695i	$-0.396593\pi$
$293$	10.9269	0.638356	0.319178	+	0.947695i	$0.396593\pi$
$294$	0	0
$295$	28.5861	1.66434
$296$	−6.44247	−0.374461
$297$	0	0
$298$	0.587723	0.0340459
$299$	8.09534	0.468165
$300$	0	0
$301$	−32.1462	−1.85287
$302$	−5.32970	−0.306690
$303$	0	0
$304$	−5.15331	−0.295563
$305$	22.6514	1.29701
$306$	0	0
$307$	4.59880	0.262468	0.131234	−	0.991351i	$-0.458106\pi$
$307$	4.59880	0.262468	0.131234	+	0.991351i	$0.458106\pi$
$308$	19.0231	1.08394
$309$	0	0
$310$	10.6768	0.606400
$311$	17.4391	0.988880	0.494440	−	0.869212i	$-0.335373\pi$
$311$	17.4391	0.988880	0.494440	+	0.869212i	$0.335373\pi$
$312$	0	0
$313$	−15.8271	−0.894599	−0.447299	−	0.894384i	$-0.647614\pi$
$313$	−15.8271	−0.894599	−0.447299	+	0.894384i	$0.647614\pi$
$314$	−9.57041	−0.540089
$315$	0	0
$316$	−14.0664	−0.791298
$317$	−19.6026	−1.10099	−0.550495	−	0.834838i	$-0.685561\pi$
$317$	−19.6026	−1.10099	−0.550495	+	0.834838i	$0.685561\pi$
$318$	0	0
$319$	16.4817	0.922795
$320$	1.67976	0.0939015
$321$	0	0
$322$	−6.78842	−0.378304
$323$	−10.8479	−0.603594
$324$	0	0
$325$	−4.27338	−0.237044
$326$	5.32731	0.295053
$327$	0	0
$328$	−21.3201	−1.17721
$329$	41.6135	2.29423
$330$	0	0
$331$	−21.8785	−1.20255	−0.601276	−	0.799042i	$-0.705341\pi$
$331$	−21.8785	−1.20255	−0.601276	+	0.799042i	$0.705341\pi$
$332$	−5.78367	−0.317420
$333$	0	0
$334$	−11.1655	−0.610950
$335$	14.6920	0.802712
$336$	0	0
$337$	−8.96292	−0.488242	−0.244121	−	0.969745i	$-0.578499\pi$
$337$	−8.96292	−0.488242	−0.244121	+	0.969745i	$0.578499\pi$
$338$	−1.64144	−0.0892828
$339$	0	0
$340$	−12.9542	−0.702540
$341$	−19.0150	−1.02972
$342$	0	0
$343$	11.9613	0.645851
$344$	18.4197	0.993125
$345$	0	0
$346$	−4.44651	−0.239046
$347$	−19.2319	−1.03242	−0.516210	−	0.856462i	$-0.672658\pi$
$347$	−19.2319	−1.03242	−0.516210	+	0.856462i	$0.672658\pi$
$348$	0	0
$349$	21.3541	1.14306	0.571529	−	0.820582i	$-0.306350\pi$
$349$	21.3541	1.14306	0.571529	+	0.820582i	$0.306350\pi$
$350$	3.58348	0.191545
$351$	0	0
$352$	−17.0195	−0.907140
$353$	14.7122	0.783053	0.391527	−	0.920167i	$-0.371947\pi$
$353$	14.7122	0.783053	0.391527	+	0.920167i	$0.371947\pi$
$354$	0	0
$355$	5.96566	0.316624
$356$	−19.5263	−1.03489
$357$	0	0
$358$	10.0566	0.531507
$359$	−10.7096	−0.565229	−0.282615	−	0.959234i	$-0.591202\pi$
$359$	−10.7096	−0.565229	−0.282615	+	0.959234i	$0.591202\pi$
$360$	0	0
$361$	−8.18276	−0.430672
$362$	−3.70699	−0.194835
$363$	0	0
$364$	20.8387	1.09224
$365$	11.8773	0.621686
$366$	0	0
$367$	−3.96567	−0.207006	−0.103503	−	0.994629i	$-0.533005\pi$
$367$	−3.96567	−0.207006	−0.103503	+	0.994629i	$0.533005\pi$
$368$	−3.91106	−0.203878
$369$	0	0
$370$	4.54532	0.236300
$371$	0.0406624	0.00211109
$372$	0	0
$373$	−23.7229	−1.22832	−0.614162	−	0.789180i	$-0.710506\pi$
$373$	−23.7229	−1.22832	−0.614162	+	0.789180i	$0.710506\pi$
$374$	−6.45822	−0.333947
$375$	0	0
$376$	−23.8445	−1.22969
$377$	18.0547	0.929866
$378$	0	0
$379$	−26.0188	−1.33650	−0.668249	−	0.743938i	$-0.732956\pi$
$379$	−26.0188	−1.33650	−0.668249	+	0.743938i	$0.732956\pi$
$380$	12.9176	0.662657
$381$	0	0
$382$	2.84104	0.145360
$383$	29.4253	1.50356	0.751780	−	0.659414i	$-0.229195\pi$
$383$	29.4253	1.50356	0.751780	+	0.659414i	$0.229195\pi$
$384$	0	0
$385$	−30.5995	−1.55949
$386$	1.87411	0.0953899
$387$	0	0
$388$	11.1755	0.567351
$389$	26.5039	1.34380	0.671900	−	0.740642i	$-0.265478\pi$
$389$	26.5039	1.34380	0.671900	+	0.740642i	$0.265478\pi$
$390$	0	0
$391$	−8.23292	−0.416357
$392$	−23.3472	−1.17921
$393$	0	0
$394$	14.0529	0.707976
$395$	22.6265	1.13846
$396$	0	0
$397$	−28.4605	−1.42839	−0.714197	−	0.699945i	$-0.753208\pi$
$397$	−28.4605	−1.42839	−0.714197	+	0.699945i	$0.753208\pi$
$398$	6.43662	0.322638
$399$	0	0
$400$	2.06458	0.103229
$401$	14.2226	0.710241	0.355121	−	0.934820i	$-0.384440\pi$
$401$	14.2226	0.710241	0.355121	+	0.934820i	$0.384440\pi$
$402$	0	0
$403$	−20.8298	−1.03761
$404$	−3.29097	−0.163732
$405$	0	0
$406$	−15.1400	−0.751383
$407$	−8.09506	−0.401257
$408$	0	0
$409$	3.11398	0.153976	0.0769882	−	0.997032i	$-0.475470\pi$
$409$	3.11398	0.153976	0.0769882	+	0.997032i	$0.475470\pi$
$410$	15.0419	0.742865
$411$	0	0
$412$	3.22727	0.158996
$413$	46.7663	2.30122
$414$	0	0
$415$	9.30330	0.456681
$416$	−18.6439	−0.914091
$417$	0	0
$418$	6.43996	0.314989
$419$	6.85107	0.334697	0.167348	−	0.985898i	$-0.446480\pi$
$419$	6.85107	0.334697	0.167348	+	0.985898i	$0.446480\pi$
$420$	0	0
$421$	−14.9325	−0.727765	−0.363882	−	0.931445i	$-0.618549\pi$
$421$	−14.9325	−0.727765	−0.363882	+	0.931445i	$0.618549\pi$
$422$	12.9485	0.630321
$423$	0	0
$424$	−0.0232996	−0.00113153
$425$	4.34601	0.210812
$426$	0	0
$427$	37.0573	1.79333
$428$	22.2881	1.07734
$429$	0	0
$430$	−12.9956	−0.626702
$431$	−30.8667	−1.48680	−0.743399	−	0.668848i	$-0.766788\pi$
$431$	−30.8667	−1.48680	−0.743399	+	0.668848i	$0.766788\pi$
$432$	0	0
$433$	−28.7698	−1.38259	−0.691295	−	0.722573i	$-0.742959\pi$
$433$	−28.7698	−1.38259	−0.691295	+	0.722573i	$0.742959\pi$
$434$	17.4670	0.838444
$435$	0	0
$436$	−20.9461	−1.00313
$437$	8.20964	0.392720
$438$	0	0
$439$	−4.42382	−0.211137	−0.105569	−	0.994412i	$-0.533666\pi$
$439$	−4.42382	−0.211137	−0.105569	+	0.994412i	$0.533666\pi$
$440$	17.5335	0.835876
$441$	0	0
$442$	−7.07462	−0.336505
$443$	2.66758	0.126741	0.0633703	−	0.997990i	$-0.479815\pi$
$443$	2.66758	0.126741	0.0633703	+	0.997990i	$0.479815\pi$
$444$	0	0
$445$	31.4089	1.48893
$446$	−10.6919	−0.506278
$447$	0	0
$448$	2.74806	0.129834
$449$	41.6366	1.96495	0.982475	−	0.186393i	$-0.0596799\pi$
$449$	41.6366	1.96495	0.982475	+	0.186393i	$0.0596799\pi$
$450$	0	0
$451$	−26.7890	−1.26145
$452$	−4.62578	−0.217578
$453$	0	0
$454$	−14.0412	−0.658986
$455$	−33.5200	−1.57144
$456$	0	0
$457$	7.66487	0.358547	0.179274	−	0.983799i	$-0.442625\pi$
$457$	7.66487	0.358547	0.179274	+	0.983799i	$0.442625\pi$
$458$	−5.70730	−0.266685
$459$	0	0
$460$	9.80366	0.457098
$461$	−22.0430	−1.02665	−0.513324	−	0.858195i	$-0.671586\pi$
$461$	−22.0430	−1.02665	−0.513324	+	0.858195i	$0.671586\pi$
$462$	0	0
$463$	37.1596	1.72695	0.863475	−	0.504391i	$-0.168283\pi$
$463$	37.1596	1.72695	0.863475	+	0.504391i	$0.168283\pi$
$464$	−8.72268	−0.404940
$465$	0	0
$466$	9.89700	0.458470
$467$	−8.31480	−0.384763	−0.192381	−	0.981320i	$-0.561621\pi$
$467$	−8.31480	−0.384763	−0.192381	+	0.981320i	$0.561621\pi$
$468$	0	0
$469$	24.0360	1.10988
$470$	16.8229	0.775983
$471$	0	0
$472$	−26.7971	−1.23344
$473$	23.1447	1.06419
$474$	0	0
$475$	−4.33372	−0.198845
$476$	−21.1929	−0.971373
$477$	0	0
$478$	2.38789	0.109220
$479$	−2.55297	−0.116648	−0.0583240	−	0.998298i	$-0.518576\pi$
$479$	−2.55297	−0.116648	−0.0583240	+	0.998298i	$0.518576\pi$
$480$	0	0
$481$	−8.86769	−0.404332
$482$	−15.1397	−0.689596
$483$	0	0
$484$	3.49219	0.158736
$485$	−17.9763	−0.816263
$486$	0	0
$487$	16.3628	0.741468	0.370734	−	0.928739i	$-0.379106\pi$
$487$	16.3628	0.741468	0.370734	+	0.928739i	$0.379106\pi$
$488$	−21.2338	−0.961208
$489$	0	0
$490$	16.4720	0.744131
$491$	36.4848	1.64653	0.823267	−	0.567654i	$-0.192149\pi$
$491$	36.4848	1.64653	0.823267	+	0.567654i	$0.192149\pi$
$492$	0	0
$493$	−18.3616	−0.826964
$494$	7.05461	0.317402
$495$	0	0
$496$	10.0634	0.451860
$497$	9.75973	0.437784
$498$	0	0
$499$	−31.7376	−1.42077	−0.710386	−	0.703812i	$-0.751480\pi$
$499$	−31.7376	−1.42077	−0.710386	+	0.703812i	$0.751480\pi$
$500$	14.4626	0.646787
$501$	0	0
$502$	14.0031	0.624989
$503$	−6.03619	−0.269141	−0.134570	−	0.990904i	$-0.542965\pi$
$503$	−6.03619	−0.269141	−0.134570	+	0.990904i	$0.542965\pi$
$504$	0	0
$505$	5.29368	0.235566
$506$	4.88754	0.217278
$507$	0	0
$508$	−18.8009	−0.834153
$509$	−26.3035	−1.16588	−0.582940	−	0.812515i	$-0.698098\pi$
$509$	−26.3035	−1.16588	−0.582940	+	0.812515i	$0.698098\pi$
$510$	0	0
$511$	19.4311	0.859579
$512$	16.3909	0.724384
$513$	0	0
$514$	5.15013	0.227162
$515$	−5.19122	−0.228753
$516$	0	0
$517$	−29.9610	−1.31768
$518$	7.43608	0.326723
$519$	0	0
$520$	19.2069	0.842280
$521$	25.3081	1.10877	0.554384	−	0.832261i	$-0.312954\pi$
$521$	25.3081	1.10877	0.554384	+	0.832261i	$0.312954\pi$
$522$	0	0
$523$	−2.94487	−0.128770	−0.0643851	−	0.997925i	$-0.520509\pi$
$523$	−2.94487	−0.128770	−0.0643851	+	0.997925i	$0.520509\pi$
$524$	−8.02585	−0.350611
$525$	0	0
$526$	−1.00957	−0.0440194
$527$	21.1838	0.922782
$528$	0	0
$529$	−16.7694	−0.729103
$530$	0.0164384	0.000714039 0
$531$	0	0
$532$	21.1329	0.916229
$533$	−29.3459	−1.27111
$534$	0	0
$535$	−35.8514	−1.54999
$536$	−13.7726	−0.594885
$537$	0	0
$538$	19.4699	0.839407
$539$	−29.3361	−1.26360
$540$	0	0
$541$	17.6934	0.760698	0.380349	−	0.924843i	$-0.375804\pi$
$541$	17.6934	0.760698	0.380349	+	0.924843i	$0.375804\pi$
$542$	9.63808	0.413991
$543$	0	0
$544$	18.9607	0.812935
$545$	33.6927	1.44324
$546$	0	0
$547$	40.7507	1.74237	0.871186	−	0.490952i	$-0.163351\pi$
$547$	40.7507	1.74237	0.871186	+	0.490952i	$0.163351\pi$
$548$	2.97055	0.126896
$549$	0	0
$550$	−2.58004	−0.110013
$551$	18.3096	0.780017
$552$	0	0
$553$	37.0166	1.57411
$554$	6.76357	0.287357
$555$	0	0
$556$	−14.6224	−0.620128
$557$	30.2389	1.28126	0.640632	−	0.767848i	$-0.278672\pi$
$557$	30.2389	1.28126	0.640632	+	0.767848i	$0.278672\pi$
$558$	0	0
$559$	25.3537	1.07235
$560$	16.1944	0.684336
$561$	0	0
$562$	−4.64526	−0.195948
$563$	18.4790	0.778798	0.389399	−	0.921069i	$-0.372683\pi$
$563$	18.4790	0.778798	0.389399	+	0.921069i	$0.372683\pi$
$564$	0	0
$565$	7.44078	0.313036
$566$	−14.3004	−0.601091
$567$	0	0
$568$	−5.59232	−0.234648
$569$	−8.99772	−0.377204	−0.188602	−	0.982054i	$-0.560396\pi$
$569$	−8.99772	−0.377204	−0.188602	+	0.982054i	$0.560396\pi$
$570$	0	0
$571$	−16.8780	−0.706324	−0.353162	−	0.935562i	$-0.614894\pi$
$571$	−16.8780	−0.706324	−0.353162	+	0.935562i	$0.614894\pi$
$572$	−15.0035	−0.627328
$573$	0	0
$574$	24.6083	1.02713
$575$	−3.28903	−0.137162
$576$	0	0
$577$	−31.4038	−1.30736	−0.653678	−	0.756773i	$-0.726775\pi$
$577$	−31.4038	−1.30736	−0.653678	+	0.756773i	$0.726775\pi$
$578$	−4.04846	−0.168394
$579$	0	0
$580$	21.8647	0.907883
$581$	15.2201	0.631434
$582$	0	0
$583$	−0.0292762	−0.00121250
$584$	−11.1340	−0.460727
$585$	0	0
$586$	7.22674	0.298534
$587$	1.37678	0.0568256	0.0284128	−	0.999596i	$-0.490955\pi$
$587$	1.37678	0.0568256	0.0284128	+	0.999596i	$0.490955\pi$
$588$	0	0
$589$	−21.1239	−0.870396
$590$	18.9060	0.778348
$591$	0	0
$592$	4.28420	0.176079
$593$	30.9859	1.27244	0.636219	−	0.771509i	$-0.280498\pi$
$593$	30.9859	1.27244	0.636219	+	0.771509i	$0.280498\pi$
$594$	0	0
$595$	34.0897	1.39754
$596$	−1.38858	−0.0568785
$597$	0	0
$598$	5.35403	0.218942
$599$	24.2514	0.990885	0.495442	−	0.868641i	$-0.335006\pi$
$599$	24.2514	0.990885	0.495442	+	0.868641i	$0.335006\pi$
$600$	0	0
$601$	7.58637	0.309454	0.154727	−	0.987957i	$-0.450550\pi$
$601$	7.58637	0.309454	0.154727	+	0.987957i	$0.450550\pi$
$602$	−21.2606	−0.866516
$603$	0	0
$604$	12.5922	0.512369
$605$	−5.61735	−0.228378
$606$	0	0
$607$	29.9647	1.21623	0.608115	−	0.793849i	$-0.291926\pi$
$607$	29.9647	1.21623	0.608115	+	0.793849i	$0.291926\pi$
$608$	−18.9071	−0.766784
$609$	0	0
$610$	14.9810	0.606562
$611$	−32.8206	−1.32778
$612$	0	0
$613$	−12.1961	−0.492597	−0.246298	−	0.969194i	$-0.579214\pi$
$613$	−12.1961	−0.492597	−0.246298	+	0.969194i	$0.579214\pi$
$614$	3.04152	0.122746
$615$	0	0
$616$	28.6845	1.15573
$617$	15.8902	0.639714	0.319857	−	0.947466i	$-0.396365\pi$
$617$	15.8902	0.639714	0.319857	+	0.947466i	$0.396365\pi$
$618$	0	0
$619$	23.2583	0.934829	0.467414	−	0.884038i	$-0.345186\pi$
$619$	23.2583	0.934829	0.467414	+	0.884038i	$0.345186\pi$
$620$	−25.2254	−1.01308
$621$	0	0
$622$	11.5337	0.462460
$623$	51.3845	2.05868
$624$	0	0
$625$	−29.8521	−1.19408
$626$	−10.4676	−0.418369
$627$	0	0
$628$	22.6115	0.902296
$629$	9.01840	0.359587
$630$	0	0
$631$	12.2767	0.488726	0.244363	−	0.969684i	$-0.421421\pi$
$631$	12.2767	0.488726	0.244363	+	0.969684i	$0.421421\pi$
$632$	−21.2105	−0.843708
$633$	0	0
$634$	−12.9646	−0.514890
$635$	30.2421	1.20012
$636$	0	0
$637$	−32.1361	−1.27328
$638$	10.9005	0.431555
$639$	0	0
$640$	−27.7875	−1.09840
$641$	9.36448	0.369875	0.184937	−	0.982750i	$-0.440792\pi$
$641$	9.36448	0.369875	0.184937	+	0.982750i	$0.440792\pi$
$642$	0	0
$643$	−19.6646	−0.775495	−0.387748	−	0.921766i	$-0.626747\pi$
$643$	−19.6646	−0.775495	−0.387748	+	0.921766i	$0.626747\pi$
$644$	16.0386	0.632010
$645$	0	0
$646$	−7.17451	−0.282277
$647$	14.2593	0.560591	0.280295	−	0.959914i	$-0.409568\pi$
$647$	14.2593	0.560591	0.280295	+	0.959914i	$0.409568\pi$
$648$	0	0
$649$	−33.6709	−1.32170
$650$	−2.82629	−0.110856
$651$	0	0
$652$	−12.5866	−0.492928
$653$	−40.7891	−1.59620	−0.798100	−	0.602525i	$-0.794161\pi$
$653$	−40.7891	−1.59620	−0.798100	+	0.602525i	$0.794161\pi$
$654$	0	0
$655$	12.9100	0.504434
$656$	14.1777	0.553547
$657$	0	0
$658$	27.5220	1.07292
$659$	−2.58912	−0.100858	−0.0504289	−	0.998728i	$-0.516059\pi$
$659$	−2.58912	−0.100858	−0.0504289	+	0.998728i	$0.516059\pi$
$660$	0	0
$661$	30.0328	1.16814	0.584071	−	0.811703i	$-0.301459\pi$
$661$	30.0328	1.16814	0.584071	+	0.811703i	$0.301459\pi$
$662$	−14.4698	−0.562386
$663$	0	0
$664$	−8.72108	−0.338444
$665$	−33.9933	−1.31820
$666$	0	0
$667$	13.8959	0.538052
$668$	26.3802	1.02068
$669$	0	0
$670$	9.71690	0.375397
$671$	−26.6806	−1.02999
$672$	0	0
$673$	13.8680	0.534573	0.267286	−	0.963617i	$-0.413873\pi$
$673$	13.8680	0.534573	0.267286	+	0.963617i	$0.413873\pi$
$674$	−5.92783	−0.228331
$675$	0	0
$676$	3.87815	0.149160
$677$	−20.0377	−0.770110	−0.385055	−	0.922894i	$-0.625818\pi$
$677$	−20.0377	−0.770110	−0.385055	+	0.922894i	$0.625818\pi$
$678$	0	0
$679$	−29.4090	−1.12861
$680$	−19.5334	−0.749071
$681$	0	0
$682$	−12.5760	−0.481558
$683$	−31.9122	−1.22109	−0.610543	−	0.791983i	$-0.709049\pi$
$683$	−31.9122	−1.22109	−0.610543	+	0.791983i	$0.709049\pi$
$684$	0	0
$685$	−4.77827	−0.182568
$686$	7.91089	0.302039
$687$	0	0
$688$	−12.2490	−0.466988
$689$	−0.0320705	−0.00122179
$690$	0	0
$691$	37.4436	1.42442	0.712211	−	0.701966i	$-0.247694\pi$
$691$	37.4436	1.42442	0.712211	+	0.701966i	$0.247694\pi$
$692$	10.5055	0.399361
$693$	0	0
$694$	−12.7194	−0.482822
$695$	23.5208	0.892196
$696$	0	0
$697$	29.8446	1.13045
$698$	14.1230	0.534563
$699$	0	0
$700$	−8.46650	−0.320003
$701$	23.1847	0.875673	0.437836	−	0.899055i	$-0.355745\pi$
$701$	23.1847	0.875673	0.437836	+	0.899055i	$0.355745\pi$
$702$	0	0
$703$	−8.99289	−0.339173
$704$	−1.97856	−0.0745696
$705$	0	0
$706$	9.73026	0.366203
$707$	8.66038	0.325707
$708$	0	0
$709$	4.98008	0.187031	0.0935155	−	0.995618i	$-0.470190\pi$
$709$	4.98008	0.187031	0.0935155	+	0.995618i	$0.470190\pi$
$710$	3.94552	0.148073
$711$	0	0
$712$	−29.4433	−1.10343
$713$	−16.0318	−0.600395
$714$	0	0
$715$	24.1338	0.902554
$716$	−23.7602	−0.887959
$717$	0	0
$718$	−7.08301	−0.264335
$719$	−17.5070	−0.652901	−0.326450	−	0.945214i	$-0.605853\pi$
$719$	−17.5070	−0.652901	−0.326450	+	0.945214i	$0.605853\pi$
$720$	0	0
$721$	−8.49276	−0.316287
$722$	−5.41185	−0.201408
$723$	0	0
$724$	8.75829	0.325499
$725$	−7.33540	−0.272430
$726$	0	0
$727$	8.03026	0.297826	0.148913	−	0.988850i	$-0.452423\pi$
$727$	8.03026	0.297826	0.148913	+	0.988850i	$0.452423\pi$
$728$	31.4223	1.16459
$729$	0	0
$730$	7.85530	0.290738
$731$	−25.7846	−0.953677
$732$	0	0
$733$	−30.0782	−1.11096	−0.555482	−	0.831529i	$-0.687466\pi$
$733$	−30.0782	−1.11096	−0.555482	+	0.831529i	$0.687466\pi$
$734$	−2.62278	−0.0968086
$735$	0	0
$736$	−14.3494	−0.528925
$737$	−17.3055	−0.637455
$738$	0	0
$739$	2.41162	0.0887129	0.0443565	−	0.999016i	$-0.485876\pi$
$739$	2.41162	0.0887129	0.0443565	+	0.999016i	$0.485876\pi$
$740$	−10.7390	−0.394773
$741$	0	0
$742$	0.0268930	0.000987273 0
$743$	−19.6618	−0.721321	−0.360660	−	0.932697i	$-0.617449\pi$
$743$	−19.6618	−0.721321	−0.360660	+	0.932697i	$0.617449\pi$
$744$	0	0
$745$	2.23360	0.0818327
$746$	−15.6897	−0.574439
$747$	0	0
$748$	15.2585	0.557906
$749$	−58.6524	−2.14311
$750$	0	0
$751$	−6.86957	−0.250674	−0.125337	−	0.992114i	$-0.540001\pi$
$751$	−6.86957	−0.250674	−0.125337	+	0.992114i	$0.540001\pi$
$752$	15.8564	0.578225
$753$	0	0
$754$	11.9409	0.434861
$755$	−20.2551	−0.737160
$756$	0	0
$757$	13.0042	0.472645	0.236323	−	0.971675i	$-0.424058\pi$
$757$	13.0042	0.472645	0.236323	+	0.971675i	$0.424058\pi$
$758$	−17.2081	−0.625027
$759$	0	0
$760$	19.4781	0.706546
$761$	−17.4054	−0.630946	−0.315473	−	0.948935i	$-0.602163\pi$
$761$	−17.4054	−0.630946	−0.315473	+	0.948935i	$0.602163\pi$
$762$	0	0
$763$	55.1208	1.99551
$764$	−6.71237	−0.242845
$765$	0	0
$766$	19.4610	0.703156
$767$	−36.8846	−1.33183
$768$	0	0
$769$	−18.6296	−0.671802	−0.335901	−	0.941897i	$-0.609041\pi$
$769$	−18.6296	−0.671802	−0.335901	+	0.941897i	$0.609041\pi$
$770$	−20.2376	−0.729314
$771$	0	0
$772$	−4.42787	−0.159363
$773$	−28.9738	−1.04211	−0.521057	−	0.853522i	$-0.674462\pi$
$773$	−28.9738	−1.04211	−0.521057	+	0.853522i	$0.674462\pi$
$774$	0	0
$775$	8.46289	0.303996
$776$	16.8513	0.604928
$777$	0	0
$778$	17.5289	0.628442
$779$	−29.7602	−1.06627
$780$	0	0
$781$	−7.02683	−0.251440
$782$	−5.44502	−0.194714
$783$	0	0
$784$	15.5257	0.554490
$785$	−36.3716	−1.29816
$786$	0	0
$787$	−14.8146	−0.528082	−0.264041	−	0.964511i	$-0.585055\pi$
$787$	−14.8146	−0.528082	−0.264041	+	0.964511i	$0.585055\pi$
$788$	−33.2021	−1.18277
$789$	0	0
$790$	14.9645	0.532414
$791$	12.1730	0.432822
$792$	0	0
$793$	−29.2271	−1.03788
$794$	−18.8230	−0.668003
$795$	0	0
$796$	−15.2074	−0.539013
$797$	−17.6064	−0.623650	−0.311825	−	0.950140i	$-0.600940\pi$
$797$	−17.6064	−0.623650	−0.311825	+	0.950140i	$0.600940\pi$
$798$	0	0
$799$	33.3784	1.18084
$800$	7.57476	0.267808
$801$	0	0
$802$	9.40641	0.332152
$803$	−13.9900	−0.493697
$804$	0	0
$805$	−25.7989	−0.909291
$806$	−13.7763	−0.485248
$807$	0	0
$808$	−4.96239	−0.174576
$809$	13.3577	0.469632	0.234816	−	0.972040i	$-0.424551\pi$
$809$	13.3577	0.469632	0.234816	+	0.972040i	$0.424551\pi$
$810$	0	0
$811$	13.6700	0.480017	0.240009	−	0.970771i	$-0.422850\pi$
$811$	13.6700	0.480017	0.240009	+	0.970771i	$0.422850\pi$
$812$	35.7703	1.25529
$813$	0	0
$814$	−5.35385	−0.187652
$815$	20.2461	0.709189
$816$	0	0
$817$	25.7117	0.899537
$818$	2.05950	0.0720087
$819$	0	0
$820$	−35.5386	−1.24106
$821$	−10.0404	−0.350411	−0.175205	−	0.984532i	$-0.556059\pi$
$821$	−10.0404	−0.350411	−0.175205	+	0.984532i	$0.556059\pi$
$822$	0	0
$823$	−30.2488	−1.05441	−0.527204	−	0.849739i	$-0.676760\pi$
$823$	−30.2488	−1.05441	−0.527204	+	0.849739i	$0.676760\pi$
$824$	4.86634	0.169527
$825$	0	0
$826$	30.9299	1.07619
$827$	47.0379	1.63567	0.817835	−	0.575453i	$-0.195174\pi$
$827$	47.0379	1.63567	0.817835	+	0.575453i	$0.195174\pi$
$828$	0	0
$829$	−3.50656	−0.121788	−0.0608939	−	0.998144i	$-0.519395\pi$
$829$	−3.50656	−0.121788	−0.0608939	+	0.998144i	$0.519395\pi$
$830$	6.15294	0.213572
$831$	0	0
$832$	−2.16740	−0.0751410
$833$	32.6822	1.13237
$834$	0	0
$835$	−42.4337	−1.46848
$836$	−15.2153	−0.526233
$837$	0	0
$838$	4.53111	0.156524
$839$	−36.8146	−1.27098	−0.635491	−	0.772108i	$-0.719202\pi$
$839$	−36.8146	−1.27098	−0.635491	+	0.772108i	$0.719202\pi$
$840$	0	0
$841$	1.99156	0.0686746
$842$	−9.87592	−0.340347
$843$	0	0
$844$	−30.5926	−1.05304
$845$	−6.23819	−0.214600
$846$	0	0
$847$	−9.18990	−0.315769
$848$	0.0154940	0.000532068 0
$849$	0	0
$850$	2.87433	0.0985886
$851$	−6.82507	−0.233960
$852$	0	0
$853$	−36.9110	−1.26381	−0.631904	−	0.775047i	$-0.717726\pi$
$853$	−36.9110	−1.26381	−0.631904	+	0.775047i	$0.717726\pi$
$854$	24.5086	0.838668
$855$	0	0
$856$	33.6078	1.14869
$857$	−30.1017	−1.02826	−0.514128	−	0.857714i	$-0.671884\pi$
$857$	−30.1017	−1.02826	−0.514128	+	0.857714i	$0.671884\pi$
$858$	0	0
$859$	−19.4472	−0.663530	−0.331765	−	0.943362i	$-0.607644\pi$
$859$	−19.4472	−0.663530	−0.331765	+	0.943362i	$0.607644\pi$
$860$	30.7039	1.04700
$861$	0	0
$862$	−20.4144	−0.695317
$863$	28.7133	0.977411	0.488706	−	0.872449i	$-0.337469\pi$
$863$	28.7133	0.977411	0.488706	+	0.872449i	$0.337469\pi$
$864$	0	0
$865$	−16.8986	−0.574571
$866$	−19.0276	−0.646583
$867$	0	0
$868$	−41.2684	−1.40074
$869$	−26.6513	−0.904084
$870$	0	0
$871$	−18.9572	−0.642339
$872$	−31.5842	−1.06957
$873$	0	0
$874$	5.42962	0.183660
$875$	−38.0592	−1.28664
$876$	0	0
$877$	−19.1241	−0.645775	−0.322887	−	0.946437i	$-0.604654\pi$
$877$	−19.1241	−0.645775	−0.322887	+	0.946437i	$0.604654\pi$
$878$	−2.92579	−0.0987406
$879$	0	0
$880$	−11.6596	−0.393047
$881$	−46.6937	−1.57315	−0.786576	−	0.617493i	$-0.788148\pi$
$881$	−46.6937	−1.57315	−0.786576	+	0.617493i	$0.788148\pi$
$882$	0	0
$883$	−14.0085	−0.471423	−0.235712	−	0.971823i	$-0.575742\pi$
$883$	−14.0085	−0.471423	−0.235712	+	0.971823i	$0.575742\pi$
$884$	16.7148	0.562180
$885$	0	0
$886$	1.76426	0.0592716
$887$	10.3671	0.348092	0.174046	−	0.984738i	$-0.444316\pi$
$887$	10.3671	0.348092	0.174046	+	0.984738i	$0.444316\pi$
$888$	0	0
$889$	49.4755	1.65936
$890$	20.7730	0.696312
$891$	0	0
$892$	25.2612	0.845809
$893$	−33.2840	−1.11381
$894$	0	0
$895$	38.2193	1.27753
$896$	−45.4600	−1.51871
$897$	0	0
$898$	27.5372	0.918930
$899$	−35.7551	−1.19250
$900$	0	0
$901$	0.0326155	0.00108658
$902$	−17.7175	−0.589929
$903$	0	0
$904$	−6.97512	−0.231989
$905$	−14.0881	−0.468305
$906$	0	0
$907$	−35.9662	−1.19424	−0.597119	−	0.802153i	$-0.703688\pi$
$907$	−35.9662	−1.19424	−0.597119	+	0.802153i	$0.703688\pi$
$908$	33.1744	1.10093
$909$	0	0
$910$	−22.1692	−0.734902
$911$	6.36103	0.210750	0.105375	−	0.994433i	$-0.466396\pi$
$911$	6.36103	0.210750	0.105375	+	0.994433i	$0.466396\pi$
$912$	0	0
$913$	−10.9582	−0.362663
$914$	5.06933	0.167678
$915$	0	0
$916$	13.4843	0.445535
$917$	21.1205	0.697460
$918$	0	0
$919$	−46.5768	−1.53643	−0.768213	−	0.640195i	$-0.778854\pi$
$919$	−46.5768	−1.53643	−0.768213	+	0.640195i	$0.778854\pi$
$920$	14.7827	0.487373
$921$	0	0
$922$	−14.5787	−0.480122
$923$	−7.69750	−0.253366
$924$	0	0
$925$	3.60283	0.118460
$926$	24.5763	0.807627
$927$	0	0
$928$	−32.0028	−1.05054
$929$	44.9047	1.47328	0.736638	−	0.676287i	$-0.236412\pi$
$929$	44.9047	1.47328	0.736638	+	0.676287i	$0.236412\pi$
$930$	0	0
$931$	−32.5898	−1.06809
$932$	−23.3831	−0.765939
$933$	0	0
$934$	−5.49917	−0.179938
$935$	−24.5440	−0.802675
$936$	0	0
$937$	51.4265	1.68003	0.840016	−	0.542562i	$-0.182546\pi$
$937$	51.4265	1.68003	0.840016	+	0.542562i	$0.182546\pi$
$938$	15.8967	0.519046
$939$	0	0
$940$	−39.7466	−1.29639
$941$	−10.4269	−0.339908	−0.169954	−	0.985452i	$-0.554362\pi$
$941$	−10.4269	−0.339908	−0.169954	+	0.985452i	$0.554362\pi$
$942$	0	0
$943$	−22.5862	−0.735509
$944$	17.8199	0.579987
$945$	0	0
$946$	15.3072	0.497681
$947$	−45.1474	−1.46709	−0.733547	−	0.679639i	$-0.762137\pi$
$947$	−45.1474	−1.46709	−0.733547	+	0.679639i	$0.762137\pi$
$948$	0	0
$949$	−15.3253	−0.497480
$950$	−2.86620	−0.0929918
$951$	0	0
$952$	−31.9563	−1.03571
$953$	−29.2978	−0.949050	−0.474525	−	0.880242i	$-0.657380\pi$
$953$	−29.2978	−0.949050	−0.474525	+	0.880242i	$0.657380\pi$
$954$	0	0
$955$	10.7972	0.349388
$956$	−5.64174	−0.182467
$957$	0	0
$958$	−1.68846	−0.0545517
$959$	−7.81718	−0.252430
$960$	0	0
$961$	10.2508	0.330671
$962$	−5.86484	−0.189090
$963$	0	0
$964$	35.7698	1.15207
$965$	7.12244	0.229279
$966$	0	0
$967$	46.8864	1.50776	0.753882	−	0.657010i	$-0.228179\pi$
$967$	46.8864	1.50776	0.753882	+	0.657010i	$0.228179\pi$
$968$	5.26580	0.169249
$969$	0	0
$970$	−11.8890	−0.381734
$971$	−45.6691	−1.46559	−0.732795	−	0.680449i	$-0.761785\pi$
$971$	−45.6691	−1.46559	−0.732795	+	0.680449i	$0.761785\pi$
$972$	0	0
$973$	38.4797	1.23360
$974$	10.8219	0.346756
$975$	0	0
$976$	14.1203	0.451980
$977$	1.23972	0.0396622	0.0198311	−	0.999803i	$-0.493687\pi$
$977$	1.23972	0.0396622	0.0198311	+	0.999803i	$0.493687\pi$
$978$	0	0
$979$	−36.9959	−1.18239
$980$	−38.9176	−1.24318
$981$	0	0
$982$	24.1300	0.770019
$983$	36.5854	1.16689	0.583447	−	0.812151i	$-0.301704\pi$
$983$	36.5854	1.16689	0.583447	+	0.812151i	$0.301704\pi$
$984$	0	0
$985$	53.4071	1.70169
$986$	−12.1438	−0.386738
$987$	0	0
$988$	−16.6675	−0.530265
$989$	19.5136	0.620497
$990$	0	0
$991$	−41.5663	−1.32040	−0.660199	−	0.751091i	$-0.729528\pi$
$991$	−41.5663	−1.32040	−0.660199	+	0.751091i	$0.729528\pi$
$992$	36.9218	1.17227
$993$	0	0
$994$	6.45481	0.204734
$995$	24.4619	0.775494
$996$	0	0
$997$	21.1634	0.670251	0.335126	−	0.942173i	$-0.391221\pi$
$997$	21.1634	0.670251	0.335126	+	0.942173i	$0.391221\pi$
$998$	−20.9904	−0.664439
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6561.2.a.d.1.42		72
3.2	odd	2		6561.2.a.c.1.31		72
81.4	even	27		243.2.g.a.208.5		144
81.7	even	27		729.2.g.b.352.4		144
81.20	odd	54		81.2.g.a.76.4	yes	144
81.23	odd	54		729.2.g.c.379.5		144
81.31	even	27		729.2.g.a.622.4		144
81.34	even	27		729.2.g.a.109.4		144
81.47	odd	54		729.2.g.d.109.5		144
81.50	odd	54		729.2.g.d.622.5		144
81.58	even	27		729.2.g.b.379.4		144
81.61	even	27		243.2.g.a.118.5		144
81.74	odd	54		729.2.g.c.352.5		144
81.77	odd	54		81.2.g.a.16.4	✓	144

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
81.2.g.a.16.4	✓	144	81.77	odd	54
81.2.g.a.76.4	yes	144	81.20	odd	54
243.2.g.a.118.5		144	81.61	even	27
243.2.g.a.208.5		144	81.4	even	27
729.2.g.a.109.4		144	81.34	even	27
729.2.g.a.622.4		144	81.31	even	27
729.2.g.b.352.4		144	81.7	even	27
729.2.g.b.379.4		144	81.58	even	27
729.2.g.c.352.5		144	81.74	odd	54
729.2.g.c.379.5		144	81.23	odd	54
729.2.g.d.109.5		144	81.47	odd	54
729.2.g.d.622.5		144	81.50	odd	54
6561.2.a.c.1.31		72	3.2	odd	2
6561.2.a.d.1.42		72	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q+0.661372 q^{2} -1.56259 q^{4} +2.51349 q^{5} +4.11204 q^{7} -2.35619 q^{8} +1.66235 q^{10} -2.96060 q^{11} -3.24317 q^{13} +2.71959 q^{14} +1.56685 q^{16} +3.29829 q^{17} -3.28896 q^{19} -3.92756 q^{20} -1.95805 q^{22} -2.49612 q^{23} +1.31766 q^{25} -2.14494 q^{26} -6.42542 q^{28} -5.56701 q^{29} +6.42268 q^{31} +5.74866 q^{32} +2.18139 q^{34} +10.3356 q^{35} +2.73427 q^{37} -2.17522 q^{38} -5.92228 q^{40} +9.04853 q^{41} -7.81757 q^{43} +4.62619 q^{44} -1.65086 q^{46} +10.1199 q^{47} +9.90886 q^{49} +0.871461 q^{50} +5.06773 q^{52} +0.00988863 q^{53} -7.44144 q^{55} -9.68876 q^{56} -3.68186 q^{58} +11.3730 q^{59} +9.01190 q^{61} +4.24778 q^{62} +0.668297 q^{64} -8.15168 q^{65} +5.84527 q^{67} -5.15386 q^{68} +6.83567 q^{70} +2.37345 q^{71} +4.72541 q^{73} +1.80837 q^{74} +5.13928 q^{76} -12.1741 q^{77} +9.00201 q^{79} +3.93828 q^{80} +5.98444 q^{82} +3.70134 q^{83} +8.29022 q^{85} -5.17032 q^{86} +6.97574 q^{88} +12.4961 q^{89} -13.3360 q^{91} +3.90041 q^{92} +6.69304 q^{94} -8.26678 q^{95} -7.15193 q^{97} +6.55344 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 72 q + 9 q^{2} + 63 q^{4} + 18 q^{5} + 27 q^{8} + 36 q^{11} + 36 q^{14} + 45 q^{16} + 36 q^{17} + 54 q^{20} + 54 q^{23} + 36 q^{25} + 45 q^{26} + 9 q^{28} + 54 q^{29} + 63 q^{32} + 72 q^{35} + 54 q^{38}+ \cdots + 81 q^{98}+O(q^{100}) \) 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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