LMFDB - Embedded newform 243.2.a.f.1.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [243,2,Mod(1,243)] 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("243.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(243, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 243 = 3^{5} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	243.a (trivial)

Newform invariants

comment:select newform

sage:traces = [3,3,0,3,6,0,-3] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$1.94036476912$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	$\Q(\zeta_{18})^+$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - 3x - 1 $ x^3 - 3*x - 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$1.87939$ of defining polynomial
Character	$\chi$	$=$	243.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q-0.879385 q^{2} -1.22668 q^{4} +3.87939 q^{5} -2.18479 q^{7} +2.83750 q^{8} -3.41147 q^{10} +0.162504 q^{11} +2.41147 q^{13} +1.92127 q^{14} -0.0418891 q^{16} +3.00000 q^{17} +3.59627 q^{19} -4.75877 q^{20} -0.142903 q^{22} +2.83750 q^{23} +10.0496 q^{25} -2.12061 q^{26} +2.68004 q^{28} +6.71688 q^{29} -5.18479 q^{31} -5.63816 q^{32} -2.63816 q^{34} -8.47565 q^{35} -6.63816 q^{37} -3.16250 q^{38} +11.0077 q^{40} -5.80066 q^{41} -6.22668 q^{43} -0.199340 q^{44} -2.49525 q^{46} +7.39693 q^{47} -2.22668 q^{49} -8.83750 q^{50} -2.95811 q^{52} +1.40373 q^{53} +0.630415 q^{55} -6.19934 q^{56} -5.90673 q^{58} -5.12061 q^{59} -3.78106 q^{61} +4.55943 q^{62} +5.04189 q^{64} +9.35504 q^{65} -5.86484 q^{67} -3.68004 q^{68} +7.45336 q^{70} -15.3182 q^{71} +8.68004 q^{73} +5.83750 q^{74} -4.41147 q^{76} -0.355037 q^{77} -1.26857 q^{79} -0.162504 q^{80} +5.10101 q^{82} -8.47565 q^{83} +11.6382 q^{85} +5.47565 q^{86} +0.461104 q^{88} +7.72193 q^{89} -5.26857 q^{91} -3.48070 q^{92} -6.50475 q^{94} +13.9513 q^{95} -3.90673 q^{97} +1.95811 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 3 q^{2} + 3 q^{4} + 6 q^{5} - 3 q^{7} + 6 q^{8} + 3 q^{11} - 3 q^{13} - 3 q^{14} + 3 q^{16} + 9 q^{17} - 3 q^{19} - 3 q^{20} + 6 q^{23} + 3 q^{25} - 12 q^{26} - 12 q^{28} + 12 q^{29} - 12 q^{31} + 9 q^{34}+ \cdots + 9 q^{98}+O(q^{100}) $ 3 * q + 3 * q^2 + 3 * q^4 + 6 * q^5 - 3 * q^7 + 6 * q^8 + 3 * q^11 - 3 * q^13 - 3 * q^14 + 3 * q^16 + 9 * q^17 - 3 * q^19 - 3 * q^20 + 6 * q^23 + 3 * q^25 - 12 * q^26 - 12 * q^28 + 12 * q^29 - 12 * q^31 + 9 * q^34 - 6 * q^35 - 3 * q^37 - 12 * q^38 + 9 * q^40 - 3 * q^41 - 12 * q^43 - 15 * q^44 + 9 * q^46 - 6 * q^47 - 24 * q^50 - 12 * q^52 + 18 * q^53 + 9 * q^55 - 33 * q^56 + 9 * q^58 - 21 * q^59 + 6 * q^61 - 12 * q^62 + 12 * q^64 + 3 * q^65 + 6 * q^67 + 9 * q^68 + 9 * q^70 - 9 * q^71 + 6 * q^73 + 15 * q^74 - 3 * q^76 + 24 * q^77 + 6 * q^79 - 3 * q^80 + 18 * q^82 - 6 * q^83 + 18 * q^85 - 3 * q^86 - 36 * q^88 - 6 * q^91 + 24 * q^92 - 36 * q^94 + 3 * q^95 + 15 * q^97 + 9 * 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 $

$2$	−0.879385	−0.621819	−0.310910	−	0.950439i	$-0.600634\pi$
$2$	−0.879385	−0.621819	−0.310910	+	0.950439i	$0.600634\pi$
$3$	0	0
$3$	0	0
$4$	−1.22668	−0.613341
$5$	3.87939	1.73491	0.867457	−	0.497512i	$-0.165753\pi$
$5$	3.87939	1.73491	0.867457	+	0.497512i	$0.165753\pi$
$6$	0	0
$7$	−2.18479	−0.825774	−0.412887	−	0.910782i	$-0.635480\pi$
$7$	−2.18479	−0.825774	−0.412887	+	0.910782i	$0.635480\pi$
$8$	2.83750	1.00321
$9$	0	0
$10$	−3.41147	−1.07880
$11$	0.162504	0.0489967	0.0244984	−	0.999700i	$-0.492201\pi$
$11$	0.162504	0.0489967	0.0244984	+	0.999700i	$0.492201\pi$
$12$	0	0
$13$	2.41147	0.668823	0.334411	−	0.942427i	$-0.391463\pi$
$13$	2.41147	0.668823	0.334411	+	0.942427i	$0.391463\pi$
$14$	1.92127	0.513482
$15$	0	0
$16$	−0.0418891	−0.0104723
$17$	3.00000	0.727607	0.363803	−	0.931476i	$-0.381478\pi$
$17$	3.00000	0.727607	0.363803	+	0.931476i	$0.381478\pi$
$18$	0	0
$19$	3.59627	0.825040	0.412520	−	0.910949i	$-0.364649\pi$
$19$	3.59627	0.825040	0.412520	+	0.910949i	$0.364649\pi$
$20$	−4.75877	−1.06409
$21$	0	0
$22$	−0.142903	−0.0304671
$23$	2.83750	0.591659	0.295829	−	0.955241i	$-0.404404\pi$
$23$	2.83750	0.591659	0.295829	+	0.955241i	$0.404404\pi$
$24$	0	0
$25$	10.0496	2.00993
$26$	−2.12061	−0.415887
$27$	0	0
$28$	2.68004	0.506481
$29$	6.71688	1.24729	0.623647	−	0.781706i	$-0.285650\pi$
$29$	6.71688	1.24729	0.623647	+	0.781706i	$0.285650\pi$
$30$	0	0
$31$	−5.18479	−0.931216	−0.465608	−	0.884991i	$-0.654164\pi$
$31$	−5.18479	−0.931216	−0.465608	+	0.884991i	$0.654164\pi$
$32$	−5.63816	−0.996695
$33$	0	0
$34$	−2.63816	−0.452440
$35$	−8.47565	−1.43265
$36$	0	0
$37$	−6.63816	−1.09131	−0.545653	−	0.838011i	$-0.683718\pi$
$37$	−6.63816	−1.09131	−0.545653	+	0.838011i	$0.683718\pi$
$38$	−3.16250	−0.513026
$39$	0	0
$40$	11.0077	1.74048
$41$	−5.80066	−0.905911	−0.452955	−	0.891533i	$-0.649630\pi$
$41$	−5.80066	−0.905911	−0.452955	+	0.891533i	$0.649630\pi$
$42$	0	0
$43$	−6.22668	−0.949560	−0.474780	−	0.880105i	$-0.657472\pi$
$43$	−6.22668	−0.949560	−0.474780	+	0.880105i	$0.657472\pi$
$44$	−0.199340	−0.0300517
$45$	0	0
$46$	−2.49525	−0.367905
$47$	7.39693	1.07895	0.539476	−	0.842001i	$-0.318622\pi$
$47$	7.39693	1.07895	0.539476	+	0.842001i	$0.318622\pi$
$48$	0	0
$49$	−2.22668	−0.318097
$50$	−8.83750	−1.24981
$51$	0	0
$52$	−2.95811	−0.410216
$53$	1.40373	0.192818	0.0964088	−	0.995342i	$-0.469264\pi$
$53$	1.40373	0.192818	0.0964088	+	0.995342i	$0.469264\pi$
$54$	0	0
$55$	0.630415	0.0850051
$56$	−6.19934	−0.828422
$57$	0	0
$58$	−5.90673	−0.775591
$59$	−5.12061	−0.666647	−0.333324	−	0.942812i	$-0.608170\pi$
$59$	−5.12061	−0.666647	−0.333324	+	0.942812i	$0.608170\pi$
$60$	0	0
$61$	−3.78106	−0.484115	−0.242058	−	0.970262i	$-0.577822\pi$
$61$	−3.78106	−0.484115	−0.242058	+	0.970262i	$0.577822\pi$
$62$	4.55943	0.579048
$63$	0	0
$64$	5.04189	0.630236
$65$	9.35504	1.16035
$66$	0	0
$67$	−5.86484	−0.716504	−0.358252	−	0.933625i	$-0.616627\pi$
$67$	−5.86484	−0.716504	−0.358252	+	0.933625i	$0.616627\pi$
$68$	−3.68004	−0.446271
$69$	0	0
$70$	7.45336	0.890847
$71$	−15.3182	−1.81794	−0.908968	−	0.416866i	$-0.863128\pi$
$71$	−15.3182	−1.81794	−0.908968	+	0.416866i	$0.863128\pi$
$72$	0	0
$73$	8.68004	1.01592	0.507961	−	0.861380i	$-0.330399\pi$
$73$	8.68004	1.01592	0.507961	+	0.861380i	$0.330399\pi$
$74$	5.83750	0.678595
$75$	0	0
$76$	−4.41147	−0.506031
$77$	−0.355037	−0.0404602
$78$	0	0
$79$	−1.26857	−0.142725	−0.0713627	−	0.997450i	$-0.522735\pi$
$79$	−1.26857	−0.142725	−0.0713627	+	0.997450i	$0.522735\pi$
$80$	−0.162504	−0.0181685
$81$	0	0
$82$	5.10101	0.563313
$83$	−8.47565	−0.930324	−0.465162	−	0.885226i	$-0.654004\pi$
$83$	−8.47565	−0.930324	−0.465162	+	0.885226i	$0.654004\pi$
$84$	0	0
$85$	11.6382	1.26234
$86$	5.47565	0.590455
$87$	0	0
$88$	0.461104	0.0491538
$89$	7.72193	0.818523	0.409262	−	0.912417i	$-0.365786\pi$
$89$	7.72193	0.818523	0.409262	+	0.912417i	$0.365786\pi$
$90$	0	0
$91$	−5.26857	−0.552296
$92$	−3.48070	−0.362889
$93$	0	0
$94$	−6.50475	−0.670914
$95$	13.9513	1.43137
$96$	0	0
$97$	−3.90673	−0.396668	−0.198334	−	0.980134i	$-0.563553\pi$
$97$	−3.90673	−0.396668	−0.198334	+	0.980134i	$0.563553\pi$
$98$	1.95811	0.197799
$99$	0	0
$100$	−12.3277	−1.23277
$101$	−8.11381	−0.807354	−0.403677	−	0.914902i	$-0.632268\pi$
$101$	−8.11381	−0.807354	−0.403677	+	0.914902i	$0.632268\pi$
$102$	0	0
$103$	18.6459	1.83723	0.918617	−	0.395148i	$-0.129307\pi$
$103$	18.6459	1.83723	0.918617	+	0.395148i	$0.129307\pi$
$104$	6.84255	0.670967
$105$	0	0
$106$	−1.23442	−0.119898
$107$	7.59627	0.734359	0.367179	−	0.930150i	$-0.380324\pi$
$107$	7.59627	0.734359	0.367179	+	0.930150i	$0.380324\pi$
$108$	0	0
$109$	−15.6382	−1.49786	−0.748932	−	0.662647i	$-0.769433\pi$
$109$	−15.6382	−1.49786	−0.748932	+	0.662647i	$0.769433\pi$
$110$	−0.554378	−0.0528578
$111$	0	0
$112$	0.0915189	0.00864772
$113$	−2.31315	−0.217603	−0.108801	−	0.994064i	$-0.534701\pi$
$113$	−2.31315	−0.217603	−0.108801	+	0.994064i	$0.534701\pi$
$114$	0	0
$115$	11.0077	1.02648
$116$	−8.23947	−0.765016
$117$	0	0
$118$	4.50299	0.414534
$119$	−6.55438	−0.600839
$120$	0	0
$121$	−10.9736	−0.997599
$122$	3.32501	0.301032
$123$	0	0
$124$	6.36009	0.571153
$125$	19.5895	1.75213
$126$	0	0
$127$	0.0418891	0.00371705	0.00185853	−	0.999998i	$-0.499408\pi$
$127$	0.0418891	0.00371705	0.00185853	+	0.999998i	$0.499408\pi$
$128$	6.84255	0.604802
$129$	0	0
$130$	−8.22668	−0.721528
$131$	18.3482	1.60309	0.801546	−	0.597933i	$-0.204011\pi$
$131$	18.3482	1.60309	0.801546	+	0.597933i	$0.204011\pi$
$132$	0	0
$133$	−7.85710	−0.681297
$134$	5.15745	0.445536
$135$	0	0
$136$	8.51249	0.729940
$137$	14.3131	1.22285	0.611427	−	0.791301i	$-0.290596\pi$
$137$	14.3131	1.22285	0.611427	+	0.791301i	$0.290596\pi$
$138$	0	0
$139$	10.4953	0.890196	0.445098	−	0.895482i	$-0.353169\pi$
$139$	10.4953	0.890196	0.445098	+	0.895482i	$0.353169\pi$
$140$	10.3969	0.878701
$141$	0	0
$142$	13.4706	1.13043
$143$	0.391874	0.0327701
$144$	0	0
$145$	26.0574	2.16395
$146$	−7.63310	−0.631720
$147$	0	0
$148$	8.14290	0.669343
$149$	−1.27126	−0.104146	−0.0520728	−	0.998643i	$-0.516583\pi$
$149$	−1.27126	−0.104146	−0.0520728	+	0.998643i	$0.516583\pi$
$150$	0	0
$151$	7.85710	0.639401	0.319701	−	0.947519i	$-0.396418\pi$
$151$	7.85710	0.639401	0.319701	+	0.947519i	$0.396418\pi$
$152$	10.2044	0.827686
$153$	0	0
$154$	0.312214	0.0251590
$155$	−20.1138	−1.61558
$156$	0	0
$157$	−12.3523	−0.985825	−0.492912	−	0.870079i	$-0.664068\pi$
$157$	−12.3523	−0.985825	−0.492912	+	0.870079i	$0.664068\pi$
$158$	1.11556	0.0887494
$159$	0	0
$160$	−21.8726	−1.72918
$161$	−6.19934	−0.488576
$162$	0	0
$163$	−13.7469	−1.07674	−0.538371	−	0.842708i	$-0.680960\pi$
$163$	−13.7469	−1.07674	−0.538371	+	0.842708i	$0.680960\pi$
$164$	7.11556	0.555632
$165$	0	0
$166$	7.45336	0.578493
$167$	−3.71688	−0.287621	−0.143810	−	0.989605i	$-0.545936\pi$
$167$	−3.71688	−0.287621	−0.143810	+	0.989605i	$0.545936\pi$
$168$	0	0
$169$	−7.18479	−0.552676
$170$	−10.2344	−0.784944
$171$	0	0
$172$	7.63816	0.582404
$173$	−1.55943	−0.118561	−0.0592806	−	0.998241i	$-0.518881\pi$
$173$	−1.55943	−0.118561	−0.0592806	+	0.998241i	$0.518881\pi$
$174$	0	0
$175$	−21.9564	−1.65974
$176$	−0.00680713	−0.000513107 0
$177$	0	0
$178$	−6.79055	−0.508974
$179$	−12.1925	−0.911313	−0.455656	−	0.890156i	$-0.650595\pi$
$179$	−12.1925	−0.911313	−0.455656	+	0.890156i	$0.650595\pi$
$180$	0	0
$181$	−16.8726	−1.25413	−0.627064	−	0.778967i	$-0.715744\pi$
$181$	−16.8726	−1.25413	−0.627064	+	0.778967i	$0.715744\pi$
$182$	4.63310	0.343428
$183$	0	0
$184$	8.05138	0.593556
$185$	−25.7520	−1.89332
$186$	0	0
$187$	0.487511	0.0356504
$188$	−9.07367	−0.661766
$189$	0	0
$190$	−12.2686	−0.890056
$191$	−17.4757	−1.26449	−0.632247	−	0.774767i	$-0.717867\pi$
$191$	−17.4757	−1.26449	−0.632247	+	0.774767i	$0.717867\pi$
$192$	0	0
$193$	−1.99226	−0.143406	−0.0717030	−	0.997426i	$-0.522843\pi$
$193$	−1.99226	−0.143406	−0.0717030	+	0.997426i	$0.522843\pi$
$194$	3.43552	0.246656
$195$	0	0
$196$	2.73143	0.195102
$197$	21.1925	1.50991	0.754953	−	0.655779i	$-0.227660\pi$
$197$	21.1925	1.50991	0.754953	+	0.655779i	$0.227660\pi$
$198$	0	0
$199$	−3.08378	−0.218603	−0.109302	−	0.994009i	$-0.534861\pi$
$199$	−3.08378	−0.218603	−0.109302	+	0.994009i	$0.534861\pi$
$200$	28.5158	2.01637
$201$	0	0
$202$	7.13516	0.502028
$203$	−14.6750	−1.02998
$204$	0	0
$205$	−22.5030	−1.57168
$206$	−16.3969	−1.14243
$207$	0	0
$208$	−0.101014	−0.00700409
$209$	0.584407	0.0404243
$210$	0	0
$211$	1.00774	0.0693757	0.0346879	−	0.999398i	$-0.488956\pi$
$211$	1.00774	0.0693757	0.0346879	+	0.999398i	$0.488956\pi$
$212$	−1.72193	−0.118263
$213$	0	0
$214$	−6.68004	−0.456638
$215$	−24.1557	−1.64740
$216$	0	0
$217$	11.3277	0.768974
$218$	13.7520	0.931400
$219$	0	0
$220$	−0.773318	−0.0521371
$221$	7.23442	0.486640
$222$	0	0
$223$	18.2841	1.22439	0.612195	−	0.790707i	$-0.290287\pi$
$223$	18.2841	1.22439	0.612195	+	0.790707i	$0.290287\pi$
$224$	12.3182	0.823044
$225$	0	0
$226$	2.03415	0.135310
$227$	−2.64496	−0.175552	−0.0877762	−	0.996140i	$-0.527976\pi$
$227$	−2.64496	−0.175552	−0.0877762	+	0.996140i	$0.527976\pi$
$228$	0	0
$229$	−3.46286	−0.228832	−0.114416	−	0.993433i	$-0.536500\pi$
$229$	−3.46286	−0.228832	−0.114416	+	0.993433i	$0.536500\pi$
$230$	−9.68004	−0.638283
$231$	0	0
$232$	19.0591	1.25129
$233$	6.12567	0.401306	0.200653	−	0.979662i	$-0.435694\pi$
$233$	6.12567	0.401306	0.200653	+	0.979662i	$0.435694\pi$
$234$	0	0
$235$	28.6955	1.87189
$236$	6.28136	0.408882
$237$	0	0
$238$	5.76382	0.373613
$239$	−28.9513	−1.87270	−0.936352	−	0.351062i	$-0.885821\pi$
$239$	−28.9513	−1.87270	−0.936352	+	0.351062i	$0.885821\pi$
$240$	0	0
$241$	22.3259	1.43814	0.719070	−	0.694937i	$-0.244568\pi$
$241$	22.3259	1.43814	0.719070	+	0.694937i	$0.244568\pi$
$242$	9.65002	0.620326
$243$	0	0
$244$	4.63816	0.296927
$245$	−8.63816	−0.551872
$246$	0	0
$247$	8.67230	0.551805
$248$	−14.7118	−0.934202
$249$	0	0
$250$	−17.2267	−1.08951
$251$	−22.7219	−1.43420	−0.717098	−	0.696972i	$-0.754530\pi$
$251$	−22.7219	−1.43420	−0.717098	+	0.696972i	$0.754530\pi$
$252$	0	0
$253$	0.461104	0.0289894
$254$	−0.0368366	−0.00231134
$255$	0	0
$256$	−16.1010	−1.00631
$257$	−19.5895	−1.22196	−0.610978	−	0.791647i	$-0.709224\pi$
$257$	−19.5895	−1.22196	−0.610978	+	0.791647i	$0.709224\pi$
$258$	0	0
$259$	14.5030	0.901172
$260$	−11.4757	−0.711690
$261$	0	0
$262$	−16.1352	−0.996834
$263$	17.8007	1.09764	0.548818	−	0.835942i	$-0.315078\pi$
$263$	17.8007	1.09764	0.548818	+	0.835942i	$0.315078\pi$
$264$	0	0
$265$	5.44562	0.334522
$266$	6.90941	0.423643
$267$	0	0
$268$	7.19429	0.439461
$269$	22.7888	1.38946	0.694729	−	0.719272i	$-0.255524\pi$
$269$	22.7888	1.38946	0.694729	+	0.719272i	$0.255524\pi$
$270$	0	0
$271$	−3.44562	−0.209307	−0.104653	−	0.994509i	$-0.533373\pi$
$271$	−3.44562	−0.209307	−0.104653	+	0.994509i	$0.533373\pi$
$272$	−0.125667	−0.00761969
$273$	0	0
$274$	−12.5868	−0.760395
$275$	1.63310	0.0984798
$276$	0	0
$277$	−2.61350	−0.157030	−0.0785151	−	0.996913i	$-0.525018\pi$
$277$	−2.61350	−0.157030	−0.0785151	+	0.996913i	$0.525018\pi$
$278$	−9.22937	−0.553541
$279$	0	0
$280$	−24.0496	−1.43724
$281$	−13.6851	−0.816384	−0.408192	−	0.912896i	$-0.633841\pi$
$281$	−13.6851	−0.816384	−0.408192	+	0.912896i	$0.633841\pi$
$282$	0	0
$283$	22.8803	1.36009	0.680047	−	0.733169i	$-0.261959\pi$
$283$	22.8803	1.36009	0.680047	+	0.733169i	$0.261959\pi$
$284$	18.7906	1.11501
$285$	0	0
$286$	−0.344608	−0.0203771
$287$	12.6732	0.748078
$288$	0	0
$289$	−8.00000	−0.470588
$290$	−22.9145	−1.34558
$291$	0	0
$292$	−10.6477	−0.623107
$293$	24.2814	1.41853	0.709266	−	0.704941i	$-0.249026\pi$
$293$	24.2814	1.41853	0.709266	+	0.704941i	$0.249026\pi$
$294$	0	0
$295$	−19.8648	−1.15658
$296$	−18.8357	−1.09481
$297$	0	0
$298$	1.11793	0.0647597
$299$	6.84255	0.395715
$300$	0	0
$301$	13.6040	0.784122
$302$	−6.90941	−0.397592
$303$	0	0
$304$	−0.150644	−0.00864004
$305$	−14.6682	−0.839898
$306$	0	0
$307$	−16.1489	−0.921666	−0.460833	−	0.887487i	$-0.652449\pi$
$307$	−16.1489	−0.921666	−0.460833	+	0.887487i	$0.652449\pi$
$308$	0.435518	0.0248159
$309$	0	0
$310$	17.6878	1.00460
$311$	18.7101	1.06095	0.530475	−	0.847700i	$-0.322013\pi$
$311$	18.7101	1.06095	0.530475	+	0.847700i	$0.322013\pi$
$312$	0	0
$313$	2.77332	0.156757	0.0783786	−	0.996924i	$-0.475026\pi$
$313$	2.77332	0.156757	0.0783786	+	0.996924i	$0.475026\pi$
$314$	10.8625	0.613005
$315$	0	0
$316$	1.55613	0.0875393
$317$	17.4825	0.981913	0.490956	−	0.871184i	$-0.336647\pi$
$317$	17.4825	0.981913	0.490956	+	0.871184i	$0.336647\pi$
$318$	0	0
$319$	1.09152	0.0611133
$320$	19.5594	1.09341
$321$	0	0
$322$	5.45161	0.303806
$323$	10.7888	0.600305
$324$	0	0
$325$	24.2344	1.34428
$326$	12.0888	0.669538
$327$	0	0
$328$	−16.4593	−0.908816
$329$	−16.1607	−0.890971
$330$	0	0
$331$	−32.4593	−1.78413	−0.892064	−	0.451910i	$-0.850743\pi$
$331$	−32.4593	−1.78413	−0.892064	+	0.451910i	$0.850743\pi$
$332$	10.3969	0.570605
$333$	0	0
$334$	3.26857	0.178848
$335$	−22.7520	−1.24307
$336$	0	0
$337$	8.28581	0.451357	0.225678	−	0.974202i	$-0.427540\pi$
$337$	8.28581	0.451357	0.225678	+	0.974202i	$0.427540\pi$
$338$	6.31820	0.343665
$339$	0	0
$340$	−14.2763	−0.774242
$341$	−0.842549	−0.0456266
$342$	0	0
$343$	20.1584	1.08845
$344$	−17.6682	−0.952605
$345$	0	0
$346$	1.37134	0.0737237
$347$	14.9632	0.803265	0.401632	−	0.915801i	$-0.368443\pi$
$347$	14.9632	0.803265	0.401632	+	0.915801i	$0.368443\pi$
$348$	0	0
$349$	33.6459	1.80102	0.900512	−	0.434832i	$-0.143192\pi$
$349$	33.6459	1.80102	0.900512	+	0.434832i	$0.143192\pi$
$350$	19.3081	1.03206
$351$	0	0
$352$	−0.916222	−0.0488348
$353$	−15.7537	−0.838486	−0.419243	−	0.907874i	$-0.637704\pi$
$353$	−15.7537	−0.838486	−0.419243	+	0.907874i	$0.637704\pi$
$354$	0	0
$355$	−59.4252	−3.15396
$356$	−9.47235	−0.502034
$357$	0	0
$358$	10.7219	0.566672
$359$	18.1257	0.956636	0.478318	−	0.878187i	$-0.341247\pi$
$359$	18.1257	0.956636	0.478318	+	0.878187i	$0.341247\pi$
$360$	0	0
$361$	−6.06687	−0.319309
$362$	14.8375	0.779841
$363$	0	0
$364$	6.46286	0.338746
$365$	33.6732	1.76254
$366$	0	0
$367$	−19.1429	−0.999251	−0.499626	−	0.866241i	$-0.666529\pi$
$367$	−19.1429	−0.999251	−0.499626	+	0.866241i	$0.666529\pi$
$368$	−0.118860	−0.00619601
$369$	0	0
$370$	22.6459	1.17730
$371$	−3.06687	−0.159224
$372$	0	0
$373$	−15.2499	−0.789610	−0.394805	−	0.918765i	$-0.629188\pi$
$373$	−15.2499	−0.789610	−0.394805	+	0.918765i	$0.629188\pi$
$374$	−0.428710	−0.0221681
$375$	0	0
$376$	20.9887	1.08241
$377$	16.1976	0.834218
$378$	0	0
$379$	9.84760	0.505837	0.252919	−	0.967488i	$-0.418610\pi$
$379$	9.84760	0.505837	0.252919	+	0.967488i	$0.418610\pi$
$380$	−17.1138	−0.877920
$381$	0	0
$382$	15.3678	0.786287
$383$	28.3901	1.45067	0.725334	−	0.688397i	$-0.241685\pi$
$383$	28.3901	1.45067	0.725334	+	0.688397i	$0.241685\pi$
$384$	0	0
$385$	−1.37733	−0.0701950
$386$	1.75196	0.0891726
$387$	0	0
$388$	4.79231	0.243293
$389$	10.8844	0.551863	0.275931	−	0.961177i	$-0.411014\pi$
$389$	10.8844	0.551863	0.275931	+	0.961177i	$0.411014\pi$
$390$	0	0
$391$	8.51249	0.430495
$392$	−6.31820	−0.319117
$393$	0	0
$394$	−18.6364	−0.938888
$395$	−4.92127	−0.247616
$396$	0	0
$397$	−18.1070	−0.908764	−0.454382	−	0.890807i	$-0.650140\pi$
$397$	−18.1070	−0.908764	−0.454382	+	0.890807i	$0.650140\pi$
$398$	2.71183	0.135932
$399$	0	0
$400$	−0.420970	−0.0210485
$401$	1.43376	0.0715987	0.0357993	−	0.999359i	$-0.488602\pi$
$401$	1.43376	0.0715987	0.0357993	+	0.999359i	$0.488602\pi$
$402$	0	0
$403$	−12.5030	−0.622818
$404$	9.95306	0.495183
$405$	0	0
$406$	12.9050	0.640463
$407$	−1.07873	−0.0534704
$408$	0	0
$409$	8.60401	0.425441	0.212720	−	0.977113i	$-0.431768\pi$
$409$	8.60401	0.425441	0.212720	+	0.977113i	$0.431768\pi$
$410$	19.7888	0.977299
$411$	0	0
$412$	−22.8726	−1.12685
$413$	11.1875	0.550500
$414$	0	0
$415$	−32.8803	−1.61403
$416$	−13.5963	−0.666612
$417$	0	0
$418$	−0.513919	−0.0251366
$419$	−12.3114	−0.601451	−0.300725	−	0.953711i	$-0.597229\pi$
$419$	−12.3114	−0.601451	−0.300725	+	0.953711i	$0.597229\pi$
$420$	0	0
$421$	11.1165	0.541785	0.270892	−	0.962610i	$-0.412681\pi$
$421$	11.1165	0.541785	0.270892	+	0.962610i	$0.412681\pi$
$422$	−0.886192	−0.0431392
$423$	0	0
$424$	3.98309	0.193436
$425$	30.1489	1.46244
$426$	0	0
$427$	8.26083	0.399770
$428$	−9.31820	−0.450412
$429$	0	0
$430$	21.2422	1.02439
$431$	−36.8958	−1.77721	−0.888604	−	0.458675i	$-0.848324\pi$
$431$	−36.8958	−1.77721	−0.888604	+	0.458675i	$0.848324\pi$
$432$	0	0
$433$	−37.9982	−1.82608	−0.913040	−	0.407871i	$-0.866271\pi$
$433$	−37.9982	−1.82608	−0.913040	+	0.407871i	$0.866271\pi$
$434$	−9.96141	−0.478163
$435$	0	0
$436$	19.1830	0.918701
$437$	10.2044	0.488142
$438$	0	0
$439$	0.202029	0.00964231	0.00482115	−	0.999988i	$-0.498465\pi$
$439$	0.202029	0.00964231	0.00482115	+	0.999988i	$0.498465\pi$
$440$	1.78880	0.0852777
$441$	0	0
$442$	−6.36184	−0.302602
$443$	21.2294	1.00864	0.504319	−	0.863517i	$-0.331744\pi$
$443$	21.2294	1.00864	0.504319	+	0.863517i	$0.331744\pi$
$444$	0	0
$445$	29.9564	1.42007
$446$	−16.0787	−0.761350
$447$	0	0
$448$	−11.0155	−0.520433
$449$	−33.2594	−1.56961	−0.784804	−	0.619744i	$-0.787236\pi$
$449$	−33.2594	−1.56961	−0.784804	+	0.619744i	$0.787236\pi$
$450$	0	0
$451$	−0.942629	−0.0443867
$452$	2.83750	0.133465
$453$	0	0
$454$	2.32594	0.109162
$455$	−20.4388	−0.958186
$456$	0	0
$457$	−0.0341483	−0.00159739	−0.000798695	−	1.00000i	$-0.500254\pi$
$457$	−0.0341483	−0.00159739	−0.000798695		1.00000i	$0.500254\pi$
$458$	3.04519	0.142292
$459$	0	0
$460$	−13.5030	−0.629580
$461$	−14.9864	−0.697986	−0.348993	−	0.937125i	$-0.613476\pi$
$461$	−14.9864	−0.697986	−0.348993	+	0.937125i	$0.613476\pi$
$462$	0	0
$463$	−30.4424	−1.41478	−0.707390	−	0.706823i	$-0.750128\pi$
$463$	−30.4424	−1.41478	−0.707390	+	0.706823i	$0.750128\pi$
$464$	−0.281364	−0.0130620
$465$	0	0
$466$	−5.38682	−0.249540
$467$	−0.510734	−0.0236339	−0.0118170	−	0.999930i	$-0.503762\pi$
$467$	−0.510734	−0.0236339	−0.0118170	+	0.999930i	$0.503762\pi$
$468$	0	0
$469$	12.8135	0.591670
$470$	−25.2344	−1.16398
$471$	0	0
$472$	−14.5297	−0.668785
$473$	−1.01186	−0.0465254
$474$	0	0
$475$	36.1411	1.65827
$476$	8.04013	0.368519
$477$	0	0
$478$	25.4593	1.16448
$479$	−15.4507	−0.705959	−0.352980	−	0.935631i	$-0.614832\pi$
$479$	−15.4507	−0.705959	−0.352980	+	0.935631i	$0.614832\pi$
$480$	0	0
$481$	−16.0077	−0.729890
$482$	−19.6331	−0.894263
$483$	0	0
$484$	13.4611	0.611868
$485$	−15.1557	−0.688185
$486$	0	0
$487$	29.5107	1.33726	0.668629	−	0.743596i	$-0.266881\pi$
$487$	29.5107	1.33726	0.668629	+	0.743596i	$0.266881\pi$
$488$	−10.7287	−0.485667
$489$	0	0
$490$	7.59627	0.343164
$491$	2.15745	0.0973644	0.0486822	−	0.998814i	$-0.484498\pi$
$491$	2.15745	0.0973644	0.0486822	+	0.998814i	$0.484498\pi$
$492$	0	0
$493$	20.1506	0.907539
$494$	−7.62630	−0.343123
$495$	0	0
$496$	0.217186	0.00975194
$497$	33.4671	1.50120
$498$	0	0
$499$	7.49525	0.335534	0.167767	−	0.985827i	$-0.446344\pi$
$499$	7.49525	0.335534	0.167767	+	0.985827i	$0.446344\pi$
$500$	−24.0300	−1.07466
$501$	0	0
$502$	19.9813	0.891811
$503$	28.5963	1.27504	0.637522	−	0.770432i	$-0.279959\pi$
$503$	28.5963	1.27504	0.637522	+	0.770432i	$0.279959\pi$
$504$	0	0
$505$	−31.4766	−1.40069
$506$	−0.405488	−0.0180261
$507$	0	0
$508$	−0.0513845	−0.00227982
$509$	−1.69190	−0.0749923	−0.0374962	−	0.999297i	$-0.511938\pi$
$509$	−1.69190	−0.0749923	−0.0374962	+	0.999297i	$0.511938\pi$
$510$	0	0
$511$	−18.9641	−0.838922
$512$	0.473897	0.0209435
$513$	0	0
$514$	17.2267	0.759836
$515$	72.3346	3.18744
$516$	0	0
$517$	1.20203	0.0528652
$518$	−12.7537	−0.560366
$519$	0	0
$520$	26.5449	1.16407
$521$	−22.4037	−0.981525	−0.490763	−	0.871293i	$-0.663282\pi$
$521$	−22.4037	−0.981525	−0.490763	+	0.871293i	$0.663282\pi$
$522$	0	0
$523$	2.42871	0.106200	0.0531000	−	0.998589i	$-0.483090\pi$
$523$	2.42871	0.106200	0.0531000	+	0.998589i	$0.483090\pi$
$524$	−22.5074	−0.983242
$525$	0	0
$526$	−15.6536	−0.682531
$527$	−15.5544	−0.677559
$528$	0	0
$529$	−14.9486	−0.649940
$530$	−4.78880	−0.208012
$531$	0	0
$532$	9.63816	0.417867
$533$	−13.9881	−0.605894
$534$	0	0
$535$	29.4688	1.27405
$536$	−16.6415	−0.718801
$537$	0	0
$538$	−20.0401	−0.863992
$539$	−0.361844	−0.0155857
$540$	0	0
$541$	38.9394	1.67414	0.837069	−	0.547098i	$-0.184267\pi$
$541$	38.9394	1.67414	0.837069	+	0.547098i	$0.184267\pi$
$542$	3.03003	0.130151
$543$	0	0
$544$	−16.9145	−0.725202
$545$	−60.6664	−2.59866
$546$	0	0
$547$	−14.6723	−0.627342	−0.313671	−	0.949532i	$-0.601559\pi$
$547$	−14.6723	−0.627342	−0.313671	+	0.949532i	$0.601559\pi$
$548$	−17.5577	−0.750027
$549$	0	0
$550$	−1.43613	−0.0612367
$551$	24.1557	1.02907
$552$	0	0
$553$	2.77156	0.117859
$554$	2.29828	0.0976444
$555$	0	0
$556$	−12.8743	−0.545993
$557$	−11.1070	−0.470619	−0.235309	−	0.971921i	$-0.575610\pi$
$557$	−11.1070	−0.470619	−0.235309	+	0.971921i	$0.575610\pi$
$558$	0	0
$559$	−15.0155	−0.635087
$560$	0.355037	0.0150031
$561$	0	0
$562$	12.0345	0.507644
$563$	16.3081	0.687304	0.343652	−	0.939097i	$-0.388336\pi$
$563$	16.3081	0.687304	0.343652	+	0.939097i	$0.388336\pi$
$564$	0	0
$565$	−8.97359	−0.377522
$566$	−20.1206	−0.845733
$567$	0	0
$568$	−43.4653	−1.82376
$569$	36.0164	1.50989	0.754943	−	0.655790i	$-0.227664\pi$
$569$	36.0164	1.50989	0.754943	+	0.655790i	$0.227664\pi$
$570$	0	0
$571$	39.1584	1.63873	0.819364	−	0.573274i	$-0.194327\pi$
$571$	39.1584	1.63873	0.819364	+	0.573274i	$0.194327\pi$
$572$	−0.480704	−0.0200993
$573$	0	0
$574$	−11.1447	−0.465169
$575$	28.5158	1.18919
$576$	0	0
$577$	11.8057	0.491478	0.245739	−	0.969336i	$-0.420969\pi$
$577$	11.8057	0.491478	0.245739	+	0.969336i	$0.420969\pi$
$578$	7.03508	0.292621
$579$	0	0
$580$	−31.9641	−1.32724
$581$	18.5175	0.768237
$582$	0	0
$583$	0.228112	0.00944744
$584$	24.6296	1.01918
$585$	0	0
$586$	−21.3527	−0.882071
$587$	−39.9614	−1.64938	−0.824692	−	0.565582i	$-0.808651\pi$
$587$	−39.9614	−1.64938	−0.824692	+	0.565582i	$0.808651\pi$
$588$	0	0
$589$	−18.6459	−0.768291
$590$	17.4688	0.719181
$591$	0	0
$592$	0.278066	0.0114284
$593$	−29.2995	−1.20319	−0.601594	−	0.798802i	$-0.705467\pi$
$593$	−29.2995	−1.20319	−0.601594	+	0.798802i	$0.705467\pi$
$594$	0	0
$595$	−25.4270	−1.04240
$596$	1.55943	0.0638767
$597$	0	0
$598$	−6.01724	−0.246063
$599$	10.0719	0.411527	0.205764	−	0.978602i	$-0.434032\pi$
$599$	10.0719	0.411527	0.205764	+	0.978602i	$0.434032\pi$
$600$	0	0
$601$	−30.4192	−1.24083	−0.620413	−	0.784275i	$-0.713035\pi$
$601$	−30.4192	−1.24083	−0.620413	+	0.784275i	$0.713035\pi$
$602$	−11.9632	−0.487582
$603$	0	0
$604$	−9.63816	−0.392171
$605$	−42.5708	−1.73075
$606$	0	0
$607$	−23.0743	−0.936556	−0.468278	−	0.883581i	$-0.655125\pi$
$607$	−23.0743	−0.936556	−0.468278	+	0.883581i	$0.655125\pi$
$608$	−20.2763	−0.822313
$609$	0	0
$610$	12.8990	0.522265
$611$	17.8375	0.721628
$612$	0	0
$613$	0.765578	0.0309214	0.0154607	−	0.999880i	$-0.495079\pi$
$613$	0.765578	0.0309214	0.0154607	+	0.999880i	$0.495079\pi$
$614$	14.2011	0.573110
$615$	0	0
$616$	−1.00742	−0.0405900
$617$	−9.28817	−0.373928	−0.186964	−	0.982367i	$-0.559865\pi$
$617$	−9.28817	−0.373928	−0.186964	+	0.982367i	$0.559865\pi$
$618$	0	0
$619$	−35.0823	−1.41008	−0.705039	−	0.709168i	$-0.749071\pi$
$619$	−35.0823	−1.41008	−0.705039	+	0.709168i	$0.749071\pi$
$620$	24.6732	0.990901
$621$	0	0
$622$	−16.4534	−0.659720
$623$	−16.8708	−0.675915
$624$	0	0
$625$	25.7469	1.02988
$626$	−2.43882	−0.0974747
$627$	0	0
$628$	15.1524	0.604647
$629$	−19.9145	−0.794042
$630$	0	0
$631$	35.7621	1.42367	0.711833	−	0.702349i	$-0.247865\pi$
$631$	35.7621	1.42367	0.711833	+	0.702349i	$0.247865\pi$
$632$	−3.59956	−0.143183
$633$	0	0
$634$	−15.3738	−0.610572
$635$	0.162504	0.00644877
$636$	0	0
$637$	−5.36959	−0.212751
$638$	−0.959866	−0.0380014
$639$	0	0
$640$	26.5449	1.04928
$641$	2.92633	0.115583	0.0577915	−	0.998329i	$-0.481594\pi$
$641$	2.92633	0.115583	0.0577915	+	0.998329i	$0.481594\pi$
$642$	0	0
$643$	−20.2517	−0.798647	−0.399324	−	0.916810i	$-0.630755\pi$
$643$	−20.2517	−0.798647	−0.399324	+	0.916810i	$0.630755\pi$
$644$	7.60462	0.299664
$645$	0	0
$646$	−9.48751	−0.373281
$647$	10.7219	0.421523	0.210761	−	0.977538i	$-0.432406\pi$
$647$	10.7219	0.421523	0.210761	+	0.977538i	$0.432406\pi$
$648$	0	0
$649$	−0.832119	−0.0326635
$650$	−21.3114	−0.835902
$651$	0	0
$652$	16.8631	0.660409
$653$	35.7270	1.39811	0.699053	−	0.715070i	$-0.253605\pi$
$653$	35.7270	1.39811	0.699053	+	0.715070i	$0.253605\pi$
$654$	0	0
$655$	71.1799	2.78123
$656$	0.242984	0.00948694
$657$	0	0
$658$	14.2115	0.554023
$659$	30.8658	1.20236	0.601180	−	0.799114i	$-0.294698\pi$
$659$	30.8658	1.20236	0.601180	+	0.799114i	$0.294698\pi$
$660$	0	0
$661$	−9.84793	−0.383040	−0.191520	−	0.981489i	$-0.561342\pi$
$661$	−9.84793	−0.383040	−0.191520	+	0.981489i	$0.561342\pi$
$662$	28.5443	1.10940
$663$	0	0
$664$	−24.0496	−0.933307
$665$	−30.4807	−1.18199
$666$	0	0
$667$	19.0591	0.737972
$668$	4.55943	0.176410
$669$	0	0
$670$	20.0077	0.772966
$671$	−0.614437	−0.0237201
$672$	0	0
$673$	−19.6973	−0.759274	−0.379637	−	0.925135i	$-0.623951\pi$
$673$	−19.6973	−0.759274	−0.379637	+	0.925135i	$0.623951\pi$
$674$	−7.28642	−0.280662
$675$	0	0
$676$	8.81345	0.338979
$677$	−28.4570	−1.09369	−0.546845	−	0.837234i	$-0.684171\pi$
$677$	−28.4570	−1.09369	−0.546845	+	0.837234i	$0.684171\pi$
$678$	0	0
$679$	8.53539	0.327558
$680$	33.0232	1.26638
$681$	0	0
$682$	0.740925	0.0283715
$683$	12.5107	0.478710	0.239355	−	0.970932i	$-0.423064\pi$
$683$	12.5107	0.478710	0.239355	+	0.970932i	$0.423064\pi$
$684$	0	0
$685$	55.5262	2.12155
$686$	−17.7270	−0.676819
$687$	0	0
$688$	0.260830	0.00994405
$689$	3.38507	0.128961
$690$	0	0
$691$	42.6255	1.62155	0.810775	−	0.585358i	$-0.199046\pi$
$691$	42.6255	1.62155	0.810775	+	0.585358i	$0.199046\pi$
$692$	1.91292	0.0727185
$693$	0	0
$694$	−13.1584	−0.499485
$695$	40.7151	1.54441
$696$	0	0
$697$	−17.4020	−0.659147
$698$	−29.5877	−1.11991
$699$	0	0
$700$	26.9335	1.01799
$701$	51.7701	1.95533	0.977665	−	0.210167i	$-0.0674008\pi$
$701$	51.7701	1.95533	0.977665	+	0.210167i	$0.0674008\pi$
$702$	0	0
$703$	−23.8726	−0.900371
$704$	0.819326	0.0308795
$705$	0	0
$706$	13.8536	0.521387
$707$	17.7270	0.666692
$708$	0	0
$709$	15.1584	0.569285	0.284643	−	0.958634i	$-0.408125\pi$
$709$	15.1584	0.569285	0.284643	+	0.958634i	$0.408125\pi$
$710$	52.2576	1.96119
$711$	0	0
$712$	21.9110	0.821148
$713$	−14.7118	−0.550962
$714$	0	0
$715$	1.52023	0.0568534
$716$	14.9564	0.558945
$717$	0	0
$718$	−15.9394	−0.594855
$719$	−2.61493	−0.0975206	−0.0487603	−	0.998811i	$-0.515527\pi$
$719$	−2.61493	−0.0975206	−0.0487603	+	0.998811i	$0.515527\pi$
$720$	0	0
$721$	−40.7374	−1.51714
$722$	5.33511	0.198552
$723$	0	0
$724$	20.6973	0.769208
$725$	67.5022	2.50697
$726$	0	0
$727$	−4.09926	−0.152033	−0.0760166	−	0.997107i	$-0.524220\pi$
$727$	−4.09926	−0.152033	−0.0760166	+	0.997107i	$0.524220\pi$
$728$	−14.9495	−0.554067
$729$	0	0
$730$	−29.6117	−1.09598
$731$	−18.6800	−0.690906
$732$	0	0
$733$	−38.2080	−1.41125	−0.705623	−	0.708588i	$-0.749333\pi$
$733$	−38.2080	−1.41125	−0.705623	+	0.708588i	$0.749333\pi$
$734$	16.8340	0.621354
$735$	0	0
$736$	−15.9982	−0.589703
$737$	−0.953058	−0.0351064
$738$	0	0
$739$	−24.2094	−0.890559	−0.445279	−	0.895392i	$-0.646896\pi$
$739$	−24.2094	−0.890559	−0.445279	+	0.895392i	$0.646896\pi$
$740$	31.5895	1.16125
$741$	0	0
$742$	2.69696	0.0990084
$743$	−3.31139	−0.121483	−0.0607416	−	0.998154i	$-0.519347\pi$
$743$	−3.31139	−0.121483	−0.0607416	+	0.998154i	$0.519347\pi$
$744$	0	0
$745$	−4.93170	−0.180684
$746$	13.4105	0.490995
$747$	0	0
$748$	−0.598021	−0.0218658
$749$	−16.5963	−0.606414
$750$	0	0
$751$	13.7110	0.500322	0.250161	−	0.968204i	$-0.419516\pi$
$751$	13.7110	0.500322	0.250161	+	0.968204i	$0.419516\pi$
$752$	−0.309850	−0.0112991
$753$	0	0
$754$	−14.2439	−0.518733
$755$	30.4807	1.10931
$756$	0	0
$757$	12.3833	0.450079	0.225040	−	0.974350i	$-0.427749\pi$
$757$	12.3833	0.450079	0.225040	+	0.974350i	$0.427749\pi$
$758$	−8.65984	−0.314539
$759$	0	0
$760$	39.5868	1.43596
$761$	7.58265	0.274871	0.137435	−	0.990511i	$-0.456114\pi$
$761$	7.58265	0.274871	0.137435	+	0.990511i	$0.456114\pi$
$762$	0	0
$763$	34.1661	1.23690
$764$	21.4371	0.775566
$765$	0	0
$766$	−24.9659	−0.902053
$767$	−12.3482	−0.445869
$768$	0	0
$769$	3.21719	0.116015	0.0580073	−	0.998316i	$-0.481525\pi$
$769$	3.21719	0.116015	0.0580073	+	0.998316i	$0.481525\pi$
$770$	1.21120	0.0436486
$771$	0	0
$772$	2.44387	0.0879567
$773$	−0.184468	−0.00663486	−0.00331743	−	0.999994i	$-0.501056\pi$
$773$	−0.184468	−0.00663486	−0.00331743	+	0.999994i	$0.501056\pi$
$774$	0	0
$775$	−52.1052	−1.87168
$776$	−11.0853	−0.397940
$777$	0	0
$778$	−9.57161	−0.343159
$779$	−20.8607	−0.747413
$780$	0	0
$781$	−2.48927	−0.0890729
$782$	−7.48576	−0.267690
$783$	0	0
$784$	0.0932736	0.00333120
$785$	−47.9195	−1.71032
$786$	0	0
$787$	−0.478016	−0.0170394	−0.00851971	−	0.999964i	$-0.502712\pi$
$787$	−0.478016	−0.0170394	−0.00851971	+	0.999964i	$0.502712\pi$
$788$	−25.9965	−0.926087
$789$	0	0
$790$	4.32770	0.153973
$791$	5.05375	0.179691
$792$	0	0
$793$	−9.11793	−0.323787
$794$	15.9230	0.565087
$795$	0	0
$796$	3.78281	0.134078
$797$	14.4989	0.513576	0.256788	−	0.966468i	$-0.417336\pi$
$797$	14.4989	0.513576	0.256788	+	0.966468i	$0.417336\pi$
$798$	0	0
$799$	22.1908	0.785053
$800$	−56.6614	−2.00328
$801$	0	0
$802$	−1.26083	−0.0445215
$803$	1.41054	0.0497769
$804$	0	0
$805$	−24.0496	−0.847638
$806$	10.9949	0.387281
$807$	0	0
$808$	−23.0229	−0.809943
$809$	−14.8743	−0.522954	−0.261477	−	0.965210i	$-0.584210\pi$
$809$	−14.8743	−0.522954	−0.261477	+	0.965210i	$0.584210\pi$
$810$	0	0
$811$	21.5963	0.758347	0.379174	−	0.925325i	$-0.376208\pi$
$811$	21.5963	0.758347	0.379174	+	0.925325i	$0.376208\pi$
$812$	18.0015	0.631730
$813$	0	0
$814$	0.948615	0.0332490
$815$	−53.3296	−1.86805
$816$	0	0
$817$	−22.3928	−0.783425
$818$	−7.56624	−0.264547
$819$	0	0
$820$	27.6040	0.963974
$821$	−3.28136	−0.114520	−0.0572602	−	0.998359i	$-0.518236\pi$
$821$	−3.28136	−0.114520	−0.0572602	+	0.998359i	$0.518236\pi$
$822$	0	0
$823$	13.7314	0.478648	0.239324	−	0.970940i	$-0.423074\pi$
$823$	13.7314	0.478648	0.239324	+	0.970940i	$0.423074\pi$
$824$	52.9077	1.84313
$825$	0	0
$826$	−9.83811	−0.342311
$827$	20.2327	0.703559	0.351779	−	0.936083i	$-0.385577\pi$
$827$	20.2327	0.703559	0.351779	+	0.936083i	$0.385577\pi$
$828$	0	0
$829$	25.5276	0.886612	0.443306	−	0.896370i	$-0.353806\pi$
$829$	25.5276	0.886612	0.443306	+	0.896370i	$0.353806\pi$
$830$	28.9145	1.00364
$831$	0	0
$832$	12.1584	0.421516
$833$	−6.68004	−0.231450
$834$	0	0
$835$	−14.4192	−0.498998
$836$	−0.716881	−0.0247939
$837$	0	0
$838$	10.8265	0.373994
$839$	17.3800	0.600025	0.300012	−	0.953935i	$-0.403009\pi$
$839$	17.3800	0.600025	0.300012	+	0.953935i	$0.403009\pi$
$840$	0	0
$841$	16.1165	0.555741
$842$	−9.77568	−0.336892
$843$	0	0
$844$	−1.23618	−0.0425510
$845$	−27.8726	−0.958846
$846$	0	0
$847$	23.9750	0.823792
$848$	−0.0588011	−0.00201924
$849$	0	0
$850$	−26.5125	−0.909371
$851$	−18.8357	−0.645681
$852$	0	0
$853$	13.3027	0.455476	0.227738	−	0.973722i	$-0.426867\pi$
$853$	13.3027	0.455476	0.227738	+	0.973722i	$0.426867\pi$
$854$	−7.26445	−0.248584
$855$	0	0
$856$	21.5544	0.736713
$857$	−19.9213	−0.680498	−0.340249	−	0.940335i	$-0.610511\pi$
$857$	−19.9213	−0.680498	−0.340249	+	0.940335i	$0.610511\pi$
$858$	0	0
$859$	26.3446	0.898866	0.449433	−	0.893314i	$-0.351626\pi$
$859$	26.3446	0.898866	0.449433	+	0.893314i	$0.351626\pi$
$860$	29.6313	1.01042
$861$	0	0
$862$	32.4456	1.10510
$863$	38.2995	1.30373	0.651866	−	0.758334i	$-0.273987\pi$
$863$	38.2995	1.30373	0.651866	+	0.758334i	$0.273987\pi$
$864$	0	0
$865$	−6.04963	−0.205694
$866$	33.4151	1.13549
$867$	0	0
$868$	−13.8955	−0.471643
$869$	−0.206148	−0.00699308
$870$	0	0
$871$	−14.1429	−0.479214
$872$	−44.3732	−1.50267
$873$	0	0
$874$	−8.97359	−0.303536
$875$	−42.7989	−1.44687
$876$	0	0
$877$	−27.0574	−0.913662	−0.456831	−	0.889553i	$-0.651016\pi$
$877$	−27.0574	−0.913662	−0.456831	+	0.889553i	$0.651016\pi$
$878$	−0.177661	−0.00599577
$879$	0	0
$880$	−0.0264075	−0.000890196 0
$881$	30.5776	1.03019	0.515093	−	0.857134i	$-0.327757\pi$
$881$	30.5776	1.03019	0.515093	+	0.857134i	$0.327757\pi$
$882$	0	0
$883$	44.1052	1.48426	0.742130	−	0.670256i	$-0.233816\pi$
$883$	44.1052	1.48426	0.742130	+	0.670256i	$0.233816\pi$
$884$	−8.87433	−0.298476
$885$	0	0
$886$	−18.6688	−0.627190
$887$	−7.88444	−0.264734	−0.132367	−	0.991201i	$-0.542258\pi$
$887$	−7.88444	−0.264734	−0.132367	+	0.991201i	$0.542258\pi$
$888$	0	0
$889$	−0.0915189	−0.00306945
$890$	−26.3432	−0.883025
$891$	0	0
$892$	−22.4287	−0.750969
$893$	26.6013	0.890179
$894$	0	0
$895$	−47.2995	−1.58105
$896$	−14.9495	−0.499429
$897$	0	0
$898$	29.2478	0.976013
$899$	−34.8256	−1.16150
$900$	0	0
$901$	4.21120	0.140295
$902$	0.828934	0.0276005
$903$	0	0
$904$	−6.56355	−0.218300
$905$	−65.4552	−2.17581
$906$	0	0
$907$	−12.9067	−0.428561	−0.214280	−	0.976772i	$-0.568741\pi$
$907$	−12.9067	−0.428561	−0.214280	+	0.976772i	$0.568741\pi$
$908$	3.24453	0.107673
$909$	0	0
$910$	17.9736	0.595819
$911$	21.1857	0.701914	0.350957	−	0.936391i	$-0.385856\pi$
$911$	21.1857	0.701914	0.350957	+	0.936391i	$0.385856\pi$
$912$	0	0
$913$	−1.37733	−0.0455828
$914$	0.0300295	0.000993287 0
$915$	0	0
$916$	4.24783	0.140352
$917$	−40.0871	−1.32379
$918$	0	0
$919$	31.4688	1.03806	0.519031	−	0.854756i	$-0.326293\pi$
$919$	31.4688	1.03806	0.519031	+	0.854756i	$0.326293\pi$
$920$	31.2344	1.02977
$921$	0	0
$922$	13.1788	0.434021
$923$	−36.9394	−1.21588
$924$	0	0
$925$	−66.7110	−2.19344
$926$	26.7706	0.879737
$927$	0	0
$928$	−37.8708	−1.24317
$929$	−2.32676	−0.0763386	−0.0381693	−	0.999271i	$-0.512153\pi$
$929$	−2.32676	−0.0763386	−0.0381693	+	0.999271i	$0.512153\pi$
$930$	0	0
$931$	−8.00774	−0.262443
$932$	−7.51424	−0.246137
$933$	0	0
$934$	0.449132	0.0146960
$935$	1.89124	0.0618503
$936$	0	0
$937$	−10.9982	−0.359297	−0.179649	−	0.983731i	$-0.557496\pi$
$937$	−10.9982	−0.359297	−0.179649	+	0.983731i	$0.557496\pi$
$938$	−11.2680	−0.367912
$939$	0	0
$940$	−35.2003	−1.14811
$941$	−24.1037	−0.785758	−0.392879	−	0.919590i	$-0.628521\pi$
$941$	−24.1037	−0.785758	−0.392879	+	0.919590i	$0.628521\pi$
$942$	0	0
$943$	−16.4593	−0.535990
$944$	0.214498	0.00698131
$945$	0	0
$946$	0.889814	0.0289304
$947$	−11.9195	−0.387332	−0.193666	−	0.981067i	$-0.562038\pi$
$947$	−11.9195	−0.387332	−0.193666	+	0.981067i	$0.562038\pi$
$948$	0	0
$949$	20.9317	0.679472
$950$	−31.7820	−1.03114
$951$	0	0
$952$	−18.5980	−0.602765
$953$	−36.8289	−1.19301	−0.596503	−	0.802611i	$-0.703444\pi$
$953$	−36.8289	−1.19301	−0.596503	+	0.802611i	$0.703444\pi$
$954$	0	0
$955$	−67.7948	−2.19379
$956$	35.5140	1.14861
$957$	0	0
$958$	13.5871	0.438979
$959$	−31.2713	−1.00980
$960$	0	0
$961$	−4.11793	−0.132836
$962$	14.0770	0.453860
$963$	0	0
$964$	−27.3868	−0.882070
$965$	−7.72874	−0.248797
$966$	0	0
$967$	53.5604	1.72239	0.861193	−	0.508279i	$-0.169718\pi$
$967$	53.5604	1.72239	0.861193	+	0.508279i	$0.169718\pi$
$968$	−31.1375	−1.00080
$969$	0	0
$970$	13.3277	0.427927
$971$	53.2327	1.70832	0.854159	−	0.520012i	$-0.174073\pi$
$971$	53.2327	1.70832	0.854159	+	0.520012i	$0.174073\pi$
$972$	0	0
$973$	−22.9299	−0.735100
$974$	−25.9513	−0.831533
$975$	0	0
$976$	0.158385	0.00506978
$977$	13.4570	0.430527	0.215264	−	0.976556i	$-0.430939\pi$
$977$	13.4570	0.430527	0.215264	+	0.976556i	$0.430939\pi$
$978$	0	0
$979$	1.25484	0.0401050
$980$	10.5963	0.338485
$981$	0	0
$982$	−1.89723	−0.0605431
$983$	−10.2412	−0.326644	−0.163322	−	0.986573i	$-0.552221\pi$
$983$	−10.2412	−0.326644	−0.163322	+	0.986573i	$0.552221\pi$
$984$	0	0
$985$	82.2140	2.61956
$986$	−17.7202	−0.564325
$987$	0	0
$988$	−10.6382	−0.338445
$989$	−17.6682	−0.561816
$990$	0	0
$991$	2.00000	0.0635321	0.0317660	−	0.999495i	$-0.489887\pi$
$991$	2.00000	0.0635321	0.0317660	+	0.999495i	$0.489887\pi$
$992$	29.2327	0.928138
$993$	0	0
$994$	−29.4305	−0.933478
$995$	−11.9632	−0.379258
$996$	0	0
$997$	38.5016	1.21936	0.609678	−	0.792649i	$-0.291299\pi$
$997$	38.5016	1.21936	0.609678	+	0.792649i	$0.291299\pi$
$998$	−6.59121	−0.208641
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	243.2.a.f.1.1	yes	3
3.2	odd	2		243.2.a.e.1.3	✓	3
4.3	odd	2		3888.2.a.bk.1.3		3
5.4	even	2		6075.2.a.bq.1.3		3
9.2	odd	6		243.2.c.f.82.1		6
9.4	even	3		243.2.c.e.163.3		6
9.5	odd	6		243.2.c.f.163.1		6
9.7	even	3		243.2.c.e.82.3		6
12.11	even	2		3888.2.a.bd.1.1		3
15.14	odd	2		6075.2.a.bv.1.1		3
27.2	odd	18		729.2.e.a.568.1		6
27.4	even	9		729.2.e.b.406.1		6
27.5	odd	18		729.2.e.h.649.1		6
27.7	even	9		729.2.e.b.325.1		6
27.11	odd	18		729.2.e.h.82.1		6
27.13	even	9		729.2.e.i.163.1		6
27.14	odd	18		729.2.e.a.163.1		6
27.16	even	9		729.2.e.c.82.1		6
27.20	odd	18		729.2.e.g.325.1		6
27.22	even	9		729.2.e.c.649.1		6
27.23	odd	18		729.2.e.g.406.1		6
27.25	even	9		729.2.e.i.568.1		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
243.2.a.e.1.3	✓	3	3.2	odd	2
243.2.a.f.1.1	yes	3	1.1	even	1	trivial
243.2.c.e.82.3		6	9.7	even	3
243.2.c.e.163.3		6	9.4	even	3
243.2.c.f.82.1		6	9.2	odd	6
243.2.c.f.163.1		6	9.5	odd	6
729.2.e.a.163.1		6	27.14	odd	18
729.2.e.a.568.1		6	27.2	odd	18
729.2.e.b.325.1		6	27.7	even	9
729.2.e.b.406.1		6	27.4	even	9
729.2.e.c.82.1		6	27.16	even	9
729.2.e.c.649.1		6	27.22	even	9
729.2.e.g.325.1		6	27.20	odd	18
729.2.e.g.406.1		6	27.23	odd	18
729.2.e.h.82.1		6	27.11	odd	18
729.2.e.h.649.1		6	27.5	odd	18
729.2.e.i.163.1		6	27.13	even	9
729.2.e.i.568.1		6	27.25	even	9
3888.2.a.bd.1.1		3	12.11	even	2
3888.2.a.bk.1.3		3	4.3	odd	2
6075.2.a.bq.1.3		3	5.4	even	2
6075.2.a.bv.1.1		3	15.14	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:	\( N \)	\(=\)	\( 243 = 3^{5} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	243.a (trivial)

\(f(q)\)	\(=\)	\(q-0.879385 q^{2} -1.22668 q^{4} +3.87939 q^{5} -2.18479 q^{7} +2.83750 q^{8} -3.41147 q^{10} +0.162504 q^{11} +2.41147 q^{13} +1.92127 q^{14} -0.0418891 q^{16} +3.00000 q^{17} +3.59627 q^{19} -4.75877 q^{20} -0.142903 q^{22} +2.83750 q^{23} +10.0496 q^{25} -2.12061 q^{26} +2.68004 q^{28} +6.71688 q^{29} -5.18479 q^{31} -5.63816 q^{32} -2.63816 q^{34} -8.47565 q^{35} -6.63816 q^{37} -3.16250 q^{38} +11.0077 q^{40} -5.80066 q^{41} -6.22668 q^{43} -0.199340 q^{44} -2.49525 q^{46} +7.39693 q^{47} -2.22668 q^{49} -8.83750 q^{50} -2.95811 q^{52} +1.40373 q^{53} +0.630415 q^{55} -6.19934 q^{56} -5.90673 q^{58} -5.12061 q^{59} -3.78106 q^{61} +4.55943 q^{62} +5.04189 q^{64} +9.35504 q^{65} -5.86484 q^{67} -3.68004 q^{68} +7.45336 q^{70} -15.3182 q^{71} +8.68004 q^{73} +5.83750 q^{74} -4.41147 q^{76} -0.355037 q^{77} -1.26857 q^{79} -0.162504 q^{80} +5.10101 q^{82} -8.47565 q^{83} +11.6382 q^{85} +5.47565 q^{86} +0.461104 q^{88} +7.72193 q^{89} -5.26857 q^{91} -3.48070 q^{92} -6.50475 q^{94} +13.9513 q^{95} -3.90673 q^{97} +1.95811 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 3 q^{2} + 3 q^{4} + 6 q^{5} - 3 q^{7} + 6 q^{8} + 3 q^{11} - 3 q^{13} - 3 q^{14} + 3 q^{16} + 9 q^{17} - 3 q^{19} - 3 q^{20} + 6 q^{23} + 3 q^{25} - 12 q^{26} - 12 q^{28} + 12 q^{29} - 12 q^{31} + 9 q^{34}+ \cdots + 9 q^{98}+O(q^{100}) \) 3 * q + 3 * q^2 + 3 * q^4 + 6 * q^5 - 3 * q^7 + 6 * q^8 + 3 * q^11 - 3 * q^13 - 3 * q^14 + 3 * q^16 + 9 * q^17 - 3 * q^19 - 3 * q^20 + 6 * q^23 + 3 * q^25 - 12 * q^26 - 12 * q^28 + 12 * q^29 - 12 * q^31 + 9 * q^34 - 6 * q^35 - 3 * q^37 - 12 * q^38 + 9 * q^40 - 3 * q^41 - 12 * q^43 - 15 * q^44 + 9 * q^46 - 6 * q^47 - 24 * q^50 - 12 * q^52 + 18 * q^53 + 9 * q^55 - 33 * q^56 + 9 * q^58 - 21 * q^59 + 6 * q^61 - 12 * q^62 + 12 * q^64 + 3 * q^65 + 6 * q^67 + 9 * q^68 + 9 * q^70 - 9 * q^71 + 6 * q^73 + 15 * q^74 - 3 * q^76 + 24 * q^77 + 6 * q^79 - 3 * q^80 + 18 * q^82 - 6 * q^83 + 18 * q^85 - 3 * q^86 - 36 * q^88 - 6 * q^91 + 24 * q^92 - 36 * q^94 + 3 * q^95 + 15 * q^97 + 9 * q^98

Introduction

L-functions

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