LMFDB - Embedded newform 243.2.a.e.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [243,2,Mod(1,243)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

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")

//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);

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

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	yes
Analytic conductor:	$1.94036476912$
Analytic rank:	$1$
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			1.2
Root			$-0.347296$ 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)

$f(q)$	$=$	$q-1.34730 q^{2} -0.184793 q^{4} -1.65270 q^{5} +2.41147 q^{7} +2.94356 q^{8} +O(q^{10})$	\(q-1.34730 q^{2} -0.184793 q^{4} -1.65270 q^{5} +2.41147 q^{7} +2.94356 q^{8} +2.22668 q^{10} -5.94356 q^{11} -3.22668 q^{13} -3.24897 q^{14} -3.59627 q^{16} -3.00000 q^{17} -6.63816 q^{19} +0.305407 q^{20} +8.00774 q^{22} +2.94356 q^{23} -2.26857 q^{25} +4.34730 q^{26} -0.445622 q^{28} +1.29086 q^{29} -0.588526 q^{31} -1.04189 q^{32} +4.04189 q^{34} -3.98545 q^{35} +0.0418891 q^{37} +8.94356 q^{38} -4.86484 q^{40} +4.90167 q^{41} -5.18479 q^{43} +1.09833 q^{44} -3.96585 q^{46} +3.73648 q^{47} -1.18479 q^{49} +3.05644 q^{50} +0.596267 q^{52} -11.6382 q^{53} +9.82295 q^{55} +7.09833 q^{56} -1.73917 q^{58} +7.34730 q^{59} +11.0496 q^{61} +0.792919 q^{62} +8.59627 q^{64} +5.33275 q^{65} +1.85710 q^{67} +0.554378 q^{68} +5.36959 q^{70} +5.51249 q^{71} +5.55438 q^{73} -0.0564370 q^{74} +1.22668 q^{76} -14.3327 q^{77} -3.78106 q^{79} +5.94356 q^{80} -6.60401 q^{82} -3.98545 q^{83} +4.95811 q^{85} +6.98545 q^{86} -17.4953 q^{88} -8.15064 q^{89} -7.78106 q^{91} -0.543948 q^{92} -5.03415 q^{94} +10.9709 q^{95} +0.260830 q^{97} +1.59627 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 3 q^{2} + 3 q^{4} - 6 q^{5} - 3 q^{7} - 6 q^{8}+O(q^{10}) $ 3 * q - 3 * q^2 + 3 * q^4 - 6 * q^5 - 3 * q^7 - 6 * q^8	$ 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}+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

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	−1.34730	−0.952682	−0.476341	−	0.879261i	$-0.658037\pi$
$2$	−1.34730	−0.952682	−0.476341	+	0.879261i	$0.658037\pi$
$3$	0	0
$3$	0	0
$4$	−0.184793	−0.0923963
$5$	−1.65270	−0.739112	−0.369556	−	0.929209i	$-0.620490\pi$
$5$	−1.65270	−0.739112	−0.369556	+	0.929209i	$0.620490\pi$
$6$	0	0
$7$	2.41147	0.911452	0.455726	−	0.890120i	$-0.349380\pi$
$7$	2.41147	0.911452	0.455726	+	0.890120i	$0.349380\pi$
$8$	2.94356	1.04071
$9$	0	0
$10$	2.22668	0.704139
$11$	−5.94356	−1.79205	−0.896026	−	0.444002i	$-0.853558\pi$
$11$	−5.94356	−1.79205	−0.896026	+	0.444002i	$0.853558\pi$
$12$	0	0
$13$	−3.22668	−0.894920	−0.447460	−	0.894304i	$-0.647671\pi$
$13$	−3.22668	−0.894920	−0.447460	+	0.894304i	$0.647671\pi$
$14$	−3.24897	−0.868324
$15$	0	0
$16$	−3.59627	−0.899067
$17$	−3.00000	−0.727607	−0.363803	−	0.931476i	$-0.618522\pi$
$17$	−3.00000	−0.727607	−0.363803	+	0.931476i	$0.618522\pi$
$18$	0	0
$19$	−6.63816	−1.52290	−0.761449	−	0.648225i	$-0.775512\pi$
$19$	−6.63816	−1.52290	−0.761449	+	0.648225i	$0.775512\pi$
$20$	0.305407	0.0682911
$21$	0	0
$22$	8.00774	1.70726
$23$	2.94356	0.613775	0.306888	−	0.951746i	$-0.400712\pi$
$23$	2.94356	0.613775	0.306888	+	0.951746i	$0.400712\pi$
$24$	0	0
$25$	−2.26857	−0.453714
$26$	4.34730	0.852575
$27$	0	0
$28$	−0.445622	−0.0842147
$29$	1.29086	0.239707	0.119853	−	0.992792i	$-0.461758\pi$
$29$	1.29086	0.239707	0.119853	+	0.992792i	$0.461758\pi$
$30$	0	0
$31$	−0.588526	−0.105702	−0.0528512	−	0.998602i	$-0.516831\pi$
$31$	−0.588526	−0.105702	−0.0528512	+	0.998602i	$0.516831\pi$
$32$	−1.04189	−0.184182
$33$	0	0
$34$	4.04189	0.693178
$35$	−3.98545	−0.673664
$36$	0	0
$37$	0.0418891	0.00688652	0.00344326	−	0.999994i	$-0.498904\pi$
$37$	0.0418891	0.00688652	0.00344326	+	0.999994i	$0.498904\pi$
$38$	8.94356	1.45084
$39$	0	0
$40$	−4.86484	−0.769198
$41$	4.90167	0.765513	0.382756	−	0.923849i	$-0.374975\pi$
$41$	4.90167	0.765513	0.382756	+	0.923849i	$0.374975\pi$
$42$	0	0
$43$	−5.18479	−0.790673	−0.395337	−	0.918536i	$-0.629372\pi$
$43$	−5.18479	−0.790673	−0.395337	+	0.918536i	$0.629372\pi$
$44$	1.09833	0.165579
$45$	0	0
$46$	−3.96585	−0.584733
$47$	3.73648	0.545022	0.272511	−	0.962153i	$-0.412146\pi$
$47$	3.73648	0.545022	0.272511	+	0.962153i	$0.412146\pi$
$48$	0	0
$49$	−1.18479	−0.169256
$50$	3.05644	0.432245
$51$	0	0
$52$	0.596267	0.0826873
$53$	−11.6382	−1.59862	−0.799312	−	0.600916i	$-0.794802\pi$
$53$	−11.6382	−1.59862	−0.799312	+	0.600916i	$0.794802\pi$
$54$	0	0
$55$	9.82295	1.32453
$56$	7.09833	0.948554
$57$	0	0
$58$	−1.73917	−0.228364
$59$	7.34730	0.956537	0.478268	−	0.878214i	$-0.341265\pi$
$59$	7.34730	0.956537	0.478268	+	0.878214i	$0.341265\pi$
$60$	0	0
$61$	11.0496	1.41476	0.707380	−	0.706833i	$-0.249877\pi$
$61$	11.0496	1.41476	0.707380	+	0.706833i	$0.249877\pi$
$62$	0.792919	0.100701
$63$	0	0
$64$	8.59627	1.07453
$65$	5.33275	0.661446
$66$	0	0
$67$	1.85710	0.226880	0.113440	−	0.993545i	$-0.463813\pi$
$67$	1.85710	0.226880	0.113440	+	0.993545i	$0.463813\pi$
$68$	0.554378	0.0672282
$69$	0	0
$70$	5.36959	0.641788
$71$	5.51249	0.654212	0.327106	−	0.944988i	$-0.393927\pi$
$71$	5.51249	0.654212	0.327106	+	0.944988i	$0.393927\pi$
$72$	0	0
$73$	5.55438	0.650091	0.325045	−	0.945698i	$-0.394620\pi$
$73$	5.55438	0.650091	0.325045	+	0.945698i	$0.394620\pi$
$74$	−0.0564370	−0.00656067
$75$	0	0
$76$	1.22668	0.140710
$77$	−14.3327	−1.63337
$78$	0	0
$79$	−3.78106	−0.425402	−0.212701	−	0.977117i	$-0.568226\pi$
$79$	−3.78106	−0.425402	−0.212701	+	0.977117i	$0.568226\pi$
$80$	5.94356	0.664511
$81$	0	0
$82$	−6.60401	−0.729291
$83$	−3.98545	−0.437460	−0.218730	−	0.975785i	$-0.570191\pi$
$83$	−3.98545	−0.437460	−0.218730	+	0.975785i	$0.570191\pi$
$84$	0	0
$85$	4.95811	0.537783
$86$	6.98545	0.753261
$87$	0	0
$88$	−17.4953	−1.86500
$89$	−8.15064	−0.863967	−0.431983	−	0.901882i	$-0.642186\pi$
$89$	−8.15064	−0.863967	−0.431983	+	0.901882i	$0.642186\pi$
$90$	0	0
$91$	−7.78106	−0.815677
$92$	−0.543948	−0.0567105
$93$	0	0
$94$	−5.03415	−0.519233
$95$	10.9709	1.12559
$96$	0	0
$97$	0.260830	0.0264833	0.0132416	−	0.999912i	$-0.495785\pi$
$97$	0.260830	0.0264833	0.0132416	+	0.999912i	$0.495785\pi$
$98$	1.59627	0.161247
$99$	0	0
$100$	0.419215	0.0419215
$101$	−11.0273	−1.09726	−0.548631	−	0.836065i	$-0.684851\pi$
$101$	−11.0273	−1.09726	−0.548631	+	0.836065i	$0.684851\pi$
$102$	0	0
$103$	−3.90673	−0.384941	−0.192471	−	0.981303i	$-0.561650\pi$
$103$	−3.90673	−0.384941	−0.192471	+	0.981303i	$0.561650\pi$
$104$	−9.49794	−0.931350
$105$	0	0
$106$	15.6800	1.52298
$107$	2.63816	0.255040	0.127520	−	0.991836i	$-0.459298\pi$
$107$	2.63816	0.255040	0.127520	+	0.991836i	$0.459298\pi$
$108$	0	0
$109$	−8.95811	−0.858031	−0.429016	−	0.903297i	$-0.641140\pi$
$109$	−8.95811	−0.858031	−0.429016	+	0.903297i	$0.641140\pi$
$110$	−13.2344	−1.26185
$111$	0	0
$112$	−8.67230	−0.819456
$113$	−15.9290	−1.49848	−0.749238	−	0.662301i	$-0.769580\pi$
$113$	−15.9290	−1.49848	−0.749238	+	0.662301i	$0.769580\pi$
$114$	0	0
$115$	−4.86484	−0.453648
$116$	−0.238541	−0.0221480
$117$	0	0
$118$	−9.89899	−0.911275
$119$	−7.23442	−0.663178
$120$	0	0
$121$	24.3259	2.21145
$122$	−14.8871	−1.34782
$123$	0	0
$124$	0.108755	0.00976650
$125$	12.0128	1.07446
$126$	0	0
$127$	3.59627	0.319117	0.159559	−	0.987188i	$-0.448993\pi$
$127$	3.59627	0.319117	0.159559	+	0.987188i	$0.448993\pi$
$128$	−9.49794	−0.839507
$129$	0	0
$130$	−7.18479	−0.630148
$131$	17.7074	1.54710	0.773551	−	0.633734i	$-0.218479\pi$
$131$	17.7074	1.54710	0.773551	+	0.633734i	$0.218479\pi$
$132$	0	0
$133$	−16.0077	−1.38805
$134$	−2.50206	−0.216145
$135$	0	0
$136$	−8.83069	−0.757225
$137$	3.92902	0.335678	0.167839	−	0.985814i	$-0.446321\pi$
$137$	3.92902	0.335678	0.167839	+	0.985814i	$0.446321\pi$
$138$	0	0
$139$	11.9659	1.01493	0.507465	−	0.861672i	$-0.330583\pi$
$139$	11.9659	1.01493	0.507465	+	0.861672i	$0.330583\pi$
$140$	0.736482	0.0622441
$141$	0	0
$142$	−7.42696	−0.623256
$143$	19.1780	1.60374
$144$	0	0
$145$	−2.13341	−0.177170
$146$	−7.48339	−0.619330
$147$	0	0
$148$	−0.00774079	−0.000636289 0
$149$	−20.5253	−1.68150	−0.840748	−	0.541426i	$-0.817885\pi$
$149$	−20.5253	−1.68150	−0.840748	+	0.541426i	$0.817885\pi$
$150$	0	0
$151$	16.0077	1.30269	0.651346	−	0.758781i	$-0.274205\pi$
$151$	16.0077	1.30269	0.651346	+	0.758781i	$0.274205\pi$
$152$	−19.5398	−1.58489
$153$	0	0
$154$	19.3105	1.55608
$155$	0.972659	0.0781258
$156$	0	0
$157$	−21.9736	−1.75368	−0.876842	−	0.480779i	$-0.840354\pi$
$157$	−21.9736	−1.75368	−0.876842	+	0.480779i	$0.840354\pi$
$158$	5.09421	0.405273
$159$	0	0
$160$	1.72193	0.136131
$161$	7.09833	0.559426
$162$	0	0
$163$	20.5107	1.60652	0.803262	−	0.595625i	$-0.203096\pi$
$163$	20.5107	1.60652	0.803262	+	0.595625i	$0.203096\pi$
$164$	−0.905793	−0.0707305
$165$	0	0
$166$	5.36959	0.416761
$167$	−4.29086	−0.332037	−0.166018	−	0.986123i	$-0.553091\pi$
$167$	−4.29086	−0.332037	−0.166018	+	0.986123i	$0.553091\pi$
$168$	0	0
$169$	−2.58853	−0.199117
$170$	−6.68004	−0.512336
$171$	0	0
$172$	0.958111	0.0730553
$173$	−3.79292	−0.288370	−0.144185	−	0.989551i	$-0.546056\pi$
$173$	−3.79292	−0.288370	−0.144185	+	0.989551i	$0.546056\pi$
$174$	0	0
$175$	−5.47060	−0.413538
$176$	21.3746	1.61117
$177$	0	0
$178$	10.9813	0.823086
$179$	−8.27631	−0.618601	−0.309300	−	0.950964i	$-0.600095\pi$
$179$	−8.27631	−0.618601	−0.309300	+	0.950964i	$0.600095\pi$
$180$	0	0
$181$	6.72193	0.499637	0.249819	−	0.968293i	$-0.419629\pi$
$181$	6.72193	0.499637	0.249819	+	0.968293i	$0.419629\pi$
$182$	10.4834	0.777081
$183$	0	0
$184$	8.66456	0.638760
$185$	−0.0692302	−0.00508991
$186$	0	0
$187$	17.8307	1.30391
$188$	−0.690474	−0.0503580
$189$	0	0
$190$	−14.7811	−1.07233
$191$	5.01455	0.362840	0.181420	−	0.983406i	$-0.441931\pi$
$191$	5.01455	0.362840	0.181420	+	0.983406i	$0.441931\pi$
$192$	0	0
$193$	−17.8648	−1.28594	−0.642970	−	0.765892i	$-0.722298\pi$
$193$	−17.8648	−1.28594	−0.642970	+	0.765892i	$0.722298\pi$
$194$	−0.351415	−0.0252301
$195$	0	0
$196$	0.218941	0.0156386
$197$	−0.723689	−0.0515607	−0.0257803	−	0.999668i	$-0.508207\pi$
$197$	−0.723689	−0.0515607	−0.0257803	+	0.999668i	$0.508207\pi$
$198$	0	0
$199$	−10.1925	−0.722530	−0.361265	−	0.932463i	$-0.617655\pi$
$199$	−10.1925	−0.722530	−0.361265	+	0.932463i	$0.617655\pi$
$200$	−6.67768	−0.472183
$201$	0	0
$202$	14.8571	1.04534
$203$	3.11287	0.218481
$204$	0	0
$205$	−8.10101	−0.565799
$206$	5.26352	0.366727
$207$	0	0
$208$	11.6040	0.804593
$209$	39.4543	2.72911
$210$	0	0
$211$	−14.8648	−1.02334	−0.511669	−	0.859183i	$-0.670973\pi$
$211$	−14.8648	−1.02334	−0.511669	+	0.859183i	$0.670973\pi$
$212$	2.15064	0.147707
$213$	0	0
$214$	−3.55438	−0.242972
$215$	8.56893	0.584396
$216$	0	0
$217$	−1.41921	−0.0963426
$218$	12.0692	0.817431
$219$	0	0
$220$	−1.81521	−0.122381
$221$	9.68004	0.651150
$222$	0	0
$223$	−10.9486	−0.733174	−0.366587	−	0.930384i	$-0.619474\pi$
$223$	−10.9486	−0.733174	−0.366587	+	0.930384i	$0.619474\pi$
$224$	−2.51249	−0.167873
$225$	0	0
$226$	21.4611	1.42757
$227$	17.3327	1.15041	0.575207	−	0.818008i	$-0.304921\pi$
$227$	17.3327	1.15041	0.575207	+	0.818008i	$0.304921\pi$
$228$	0	0
$229$	1.56212	0.103228	0.0516138	−	0.998667i	$-0.483563\pi$
$229$	1.56212	0.103228	0.0516138	+	0.998667i	$0.483563\pi$
$230$	6.55438	0.432183
$231$	0	0
$232$	3.79973	0.249464
$233$	−16.7888	−1.09987	−0.549935	−	0.835207i	$-0.685348\pi$
$233$	−16.7888	−1.09987	−0.549935	+	0.835207i	$0.685348\pi$
$234$	0	0
$235$	−6.17530	−0.402832
$236$	−1.35773	−0.0883804
$237$	0	0
$238$	9.74691	0.631798
$239$	4.02910	0.260621	0.130310	−	0.991473i	$-0.458403\pi$
$239$	4.02910	0.260621	0.130310	+	0.991473i	$0.458403\pi$
$240$	0	0
$241$	−3.35235	−0.215944	−0.107972	−	0.994154i	$-0.534436\pi$
$241$	−3.35235	−0.215944	−0.107972	+	0.994154i	$0.534436\pi$
$242$	−32.7743	−2.10681
$243$	0	0
$244$	−2.04189	−0.130719
$245$	1.95811	0.125099
$246$	0	0
$247$	21.4192	1.36287
$248$	−1.73236	−0.110005
$249$	0	0
$250$	−16.1848	−1.02362
$251$	23.1506	1.46126	0.730628	−	0.682776i	$-0.239227\pi$
$251$	23.1506	1.46126	0.730628	+	0.682776i	$0.239227\pi$
$252$	0	0
$253$	−17.4953	−1.09992
$254$	−4.84524	−0.304017
$255$	0	0
$256$	−4.39599	−0.274750
$257$	−12.0128	−0.749337	−0.374669	−	0.927159i	$-0.622244\pi$
$257$	−12.0128	−0.749337	−0.374669	+	0.927159i	$0.622244\pi$
$258$	0	0
$259$	0.101014	0.00627673
$260$	−0.985452	−0.0611151
$261$	0	0
$262$	−23.8571	−1.47390
$263$	−16.9017	−1.04220	−0.521101	−	0.853495i	$-0.674478\pi$
$263$	−16.9017	−1.04220	−0.521101	+	0.853495i	$0.674478\pi$
$264$	0	0
$265$	19.2344	1.18156
$266$	21.5672	1.32237
$267$	0	0
$268$	−0.343178	−0.0209629
$269$	7.91447	0.482554	0.241277	−	0.970456i	$-0.422434\pi$
$269$	7.91447	0.482554	0.241277	+	0.970456i	$0.422434\pi$
$270$	0	0
$271$	−17.2344	−1.04692	−0.523458	−	0.852051i	$-0.675358\pi$
$271$	−17.2344	−1.04692	−0.523458	+	0.852051i	$0.675358\pi$
$272$	10.7888	0.654167
$273$	0	0
$274$	−5.29355	−0.319795
$275$	13.4834	0.813079
$276$	0	0
$277$	26.4347	1.58831	0.794153	−	0.607717i	$-0.207915\pi$
$277$	26.4347	1.58831	0.794153	+	0.607717i	$0.207915\pi$
$278$	−16.1215	−0.966906
$279$	0	0
$280$	−11.7314	−0.701087
$281$	18.9959	1.13320	0.566600	−	0.823993i	$-0.308259\pi$
$281$	18.9959	1.13320	0.566600	+	0.823993i	$0.308259\pi$
$282$	0	0
$283$	−16.5868	−0.985981	−0.492991	−	0.870035i	$-0.664096\pi$
$283$	−16.5868	−0.985981	−0.492991	+	0.870035i	$0.664096\pi$
$284$	−1.01867	−0.0604467
$285$	0	0
$286$	−25.8384	−1.52786
$287$	11.8203	0.697728
$288$	0	0
$289$	−8.00000	−0.470588
$290$	2.87433	0.168787
$291$	0	0
$292$	−1.02641	−0.0600660
$293$	−19.3577	−1.13089	−0.565445	−	0.824786i	$-0.691296\pi$
$293$	−19.3577	−1.13089	−0.565445	+	0.824786i	$0.691296\pi$
$294$	0	0
$295$	−12.1429	−0.706987
$296$	0.123303	0.00716685
$297$	0	0
$298$	27.6536	1.60193
$299$	−9.49794	−0.549280
$300$	0	0
$301$	−12.5030	−0.720661
$302$	−21.5672	−1.24105
$303$	0	0
$304$	23.8726	1.36919
$305$	−18.2618	−1.04567
$306$	0	0
$307$	20.8057	1.18744	0.593722	−	0.804670i	$-0.297658\pi$
$307$	20.8057	1.18744	0.593722	+	0.804670i	$0.297658\pi$
$308$	2.64858	0.150917
$309$	0	0
$310$	−1.31046	−0.0744291
$311$	10.6655	0.604785	0.302392	−	0.953184i	$-0.402215\pi$
$311$	10.6655	0.604785	0.302392	+	0.953184i	$0.402215\pi$
$312$	0	0
$313$	3.81521	0.215648	0.107824	−	0.994170i	$-0.465612\pi$
$313$	3.81521	0.215648	0.107824	+	0.994170i	$0.465612\pi$
$314$	29.6049	1.67070
$315$	0	0
$316$	0.698711	0.0393056
$317$	−26.3892	−1.48216	−0.741082	−	0.671414i	$-0.765687\pi$
$317$	−26.3892	−1.48216	−0.741082	+	0.671414i	$0.765687\pi$
$318$	0	0
$319$	−7.67230	−0.429567
$320$	−14.2071	−0.794200
$321$	0	0
$322$	−9.56355	−0.532956
$323$	19.9145	1.10807
$324$	0	0
$325$	7.31996	0.406038
$326$	−27.6340	−1.53051
$327$	0	0
$328$	14.4284	0.796674
$329$	9.01043	0.496761
$330$	0	0
$331$	−1.57161	−0.0863837	−0.0431919	−	0.999067i	$-0.513753\pi$
$331$	−1.57161	−0.0863837	−0.0431919	+	0.999067i	$0.513753\pi$
$332$	0.736482	0.0404197
$333$	0	0
$334$	5.78106	0.316325
$335$	−3.06923	−0.167690
$336$	0	0
$337$	−8.01548	−0.436631	−0.218316	−	0.975878i	$-0.570056\pi$
$337$	−8.01548	−0.436631	−0.218316	+	0.975878i	$0.570056\pi$
$338$	3.48751	0.189696
$339$	0	0
$340$	−0.916222	−0.0496891
$341$	3.49794	0.189424
$342$	0	0
$343$	−19.7374	−1.06572
$344$	−15.2618	−0.822859
$345$	0	0
$346$	5.11019	0.274725
$347$	−19.8452	−1.06535	−0.532674	−	0.846320i	$-0.678813\pi$
$347$	−19.8452	−1.06535	−0.532674	+	0.846320i	$0.678813\pi$
$348$	0	0
$349$	11.0933	0.593809	0.296905	−	0.954907i	$-0.404046\pi$
$349$	11.0933	0.593809	0.296905	+	0.954907i	$0.404046\pi$
$350$	7.37052	0.393971
$351$	0	0
$352$	6.19253	0.330063
$353$	2.86390	0.152430	0.0762151	−	0.997091i	$-0.475716\pi$
$353$	2.86390	0.152430	0.0762151	+	0.997091i	$0.475716\pi$
$354$	0	0
$355$	−9.11051	−0.483536
$356$	1.50618	0.0798273
$357$	0	0
$358$	11.1506	0.589330
$359$	−28.7888	−1.51941	−0.759707	−	0.650265i	$-0.774658\pi$
$359$	−28.7888	−1.51941	−0.759707	+	0.650265i	$0.774658\pi$
$360$	0	0
$361$	25.0651	1.31922
$362$	−9.05644	−0.475996
$363$	0	0
$364$	1.43788	0.0753655
$365$	−9.17974	−0.480490
$366$	0	0
$367$	−10.9923	−0.573791	−0.286896	−	0.957962i	$-0.592623\pi$
$367$	−10.9923	−0.573791	−0.286896	+	0.957962i	$0.592623\pi$
$368$	−10.5858	−0.551825
$369$	0	0
$370$	0.0932736	0.00484906
$371$	−28.0651	−1.45707
$372$	0	0
$373$	33.4097	1.72989	0.864945	−	0.501867i	$-0.167353\pi$
$373$	33.4097	1.72989	0.864945	+	0.501867i	$0.167353\pi$
$374$	−24.0232	−1.24221
$375$	0	0
$376$	10.9986	0.567208
$377$	−4.16519	−0.214518
$378$	0	0
$379$	20.9394	1.07559	0.537794	−	0.843077i	$-0.319258\pi$
$379$	20.9394	1.07559	0.537794	+	0.843077i	$0.319258\pi$
$380$	−2.02734	−0.104000
$381$	0	0
$382$	−6.75608	−0.345671
$383$	4.11112	0.210068	0.105034	−	0.994469i	$-0.466505\pi$
$383$	4.11112	0.210068	0.105034	+	0.994469i	$0.466505\pi$
$384$	0	0
$385$	23.6878	1.20724
$386$	24.0692	1.22509
$387$	0	0
$388$	−0.0481994	−0.00244695
$389$	−17.0942	−0.866711	−0.433355	−	0.901223i	$-0.642670\pi$
$389$	−17.0942	−0.866711	−0.433355	+	0.901223i	$0.642670\pi$
$390$	0	0
$391$	−8.83069	−0.446587
$392$	−3.48751	−0.176146
$393$	0	0
$394$	0.975023	0.0491209
$395$	6.24897	0.314420
$396$	0	0
$397$	22.4020	1.12432	0.562162	−	0.827027i	$-0.309970\pi$
$397$	22.4020	1.12432	0.562162	+	0.827027i	$0.309970\pi$
$398$	13.7324	0.688341
$399$	0	0
$400$	8.15839	0.407919
$401$	14.5817	0.728176	0.364088	−	0.931364i	$-0.381381\pi$
$401$	14.5817	0.728176	0.364088	+	0.931364i	$0.381381\pi$
$402$	0	0
$403$	1.89899	0.0945952
$404$	2.03777	0.101383
$405$	0	0
$406$	−4.19396	−0.208143
$407$	−0.248970	−0.0123410
$408$	0	0
$409$	−17.5030	−0.865467	−0.432734	−	0.901522i	$-0.642451\pi$
$409$	−17.5030	−0.865467	−0.432734	+	0.901522i	$0.642451\pi$
$410$	10.9145	0.539027
$411$	0	0
$412$	0.721934	0.0355671
$413$	17.7178	0.871837
$414$	0	0
$415$	6.58677	0.323332
$416$	3.36184	0.164828
$417$	0	0
$418$	−53.1566	−2.59998
$419$	−18.8621	−0.921476	−0.460738	−	0.887536i	$-0.652415\pi$
$419$	−18.8621	−0.921476	−0.460738	+	0.887536i	$0.652415\pi$
$420$	0	0
$421$	−32.3337	−1.57585	−0.787924	−	0.615773i	$-0.788844\pi$
$421$	−32.3337	−1.57585	−0.787924	+	0.615773i	$0.788844\pi$
$422$	20.0273	0.974916
$423$	0	0
$424$	−34.2576	−1.66370
$425$	6.80571	0.330126
$426$	0	0
$427$	26.6459	1.28949
$428$	−0.487511	−0.0235648
$429$	0	0
$430$	−11.5449	−0.556744
$431$	−34.3164	−1.65297	−0.826483	−	0.562962i	$-0.809662\pi$
$431$	−34.3164	−1.65297	−0.826483	+	0.562962i	$0.809662\pi$
$432$	0	0
$433$	−25.0669	−1.20464	−0.602318	−	0.798256i	$-0.705756\pi$
$433$	−25.0669	−1.20464	−0.602318	+	0.798256i	$0.705756\pi$
$434$	1.91210	0.0917839
$435$	0	0
$436$	1.65539	0.0792789
$437$	−19.5398	−0.934717
$438$	0	0
$439$	−23.2080	−1.10766	−0.553829	−	0.832630i	$-0.686834\pi$
$439$	−23.2080	−1.10766	−0.553829	+	0.832630i	$0.686834\pi$
$440$	28.9145	1.37844
$441$	0	0
$442$	−13.0419	−0.620339
$443$	4.12155	0.195821	0.0979103	−	0.995195i	$-0.468784\pi$
$443$	4.12155	0.195821	0.0979103	+	0.995195i	$0.468784\pi$
$444$	0	0
$445$	13.4706	0.638568
$446$	14.7510	0.698482
$447$	0	0
$448$	20.7297	0.979385
$449$	−18.3414	−0.865585	−0.432793	−	0.901494i	$-0.642472\pi$
$449$	−18.3414	−0.865585	−0.432793	+	0.901494i	$0.642472\pi$
$450$	0	0
$451$	−29.1334	−1.37184
$452$	2.94356	0.138454
$453$	0	0
$454$	−23.3523	−1.09598
$455$	12.8598	0.602876
$456$	0	0
$457$	−19.4611	−0.910352	−0.455176	−	0.890401i	$-0.650424\pi$
$457$	−19.4611	−0.910352	−0.455176	+	0.890401i	$0.650424\pi$
$458$	−2.10464	−0.0983432
$459$	0	0
$460$	0.898986	0.0419154
$461$	−27.7493	−1.29241	−0.646206	−	0.763163i	$-0.723645\pi$
$461$	−27.7493	−1.29241	−0.646206	+	0.763163i	$0.723645\pi$
$462$	0	0
$463$	38.6860	1.79789	0.898946	−	0.438059i	$-0.144334\pi$
$463$	38.6860	1.79789	0.898946	+	0.438059i	$0.144334\pi$
$464$	−4.64227	−0.215512
$465$	0	0
$466$	22.6195	1.04783
$467$	−29.7638	−1.37731	−0.688653	−	0.725091i	$-0.741798\pi$
$467$	−29.7638	−1.37731	−0.688653	+	0.725091i	$0.741798\pi$
$468$	0	0
$469$	4.47834	0.206791
$470$	8.31996	0.383771
$471$	0	0
$472$	21.6272	0.995474
$473$	30.8161	1.41693
$474$	0	0
$475$	15.0591	0.690960
$476$	1.33687	0.0612752
$477$	0	0
$478$	−5.42839	−0.248289
$479$	37.6759	1.72146	0.860728	−	0.509064i	$-0.170008\pi$
$479$	37.6759	1.72146	0.860728	+	0.509064i	$0.170008\pi$
$480$	0	0
$481$	−0.135163	−0.00616289
$482$	4.51661	0.205726
$483$	0	0
$484$	−4.49525	−0.204330
$485$	−0.431074	−0.0195741
$486$	0	0
$487$	−0.763823	−0.0346121	−0.0173061	−	0.999850i	$-0.505509\pi$
$487$	−0.763823	−0.0346121	−0.0173061	+	0.999850i	$0.505509\pi$
$488$	32.5253	1.47235
$489$	0	0
$490$	−2.63816	−0.119180
$491$	0.497941	0.0224717	0.0112359	−	0.999937i	$-0.496423\pi$
$491$	0.497941	0.0224717	0.0112359	+	0.999937i	$0.496423\pi$
$492$	0	0
$493$	−3.87258	−0.174412
$494$	−28.8580	−1.29838
$495$	0	0
$496$	2.11650	0.0950335
$497$	13.2932	0.596283
$498$	0	0
$499$	8.96585	0.401367	0.200683	−	0.979656i	$-0.435684\pi$
$499$	8.96585	0.401367	0.200683	+	0.979656i	$0.435684\pi$
$500$	−2.21987	−0.0992758
$501$	0	0
$502$	−31.1908	−1.39211
$503$	−18.3618	−0.818714	−0.409357	−	0.912374i	$-0.634247\pi$
$503$	−18.3618	−0.818714	−0.409357	+	0.912374i	$0.634247\pi$
$504$	0	0
$505$	18.2249	0.810999
$506$	23.5713	1.04787
$507$	0	0
$508$	−0.664563	−0.0294852
$509$	28.3705	1.25750	0.628751	−	0.777607i	$-0.283567\pi$
$509$	28.3705	1.25750	0.628751	+	0.777607i	$0.283567\pi$
$510$	0	0
$511$	13.3942	0.592526
$512$	24.9186	1.10126
$513$	0	0
$514$	16.1848	0.713881
$515$	6.45666	0.284514
$516$	0	0
$517$	−22.2080	−0.976707
$518$	−0.136096	−0.00597973
$519$	0	0
$520$	15.6973	0.688371
$521$	32.6382	1.42990	0.714952	−	0.699174i	$-0.246449\pi$
$521$	32.6382	1.42990	0.714952	+	0.699174i	$0.246449\pi$
$522$	0	0
$523$	−22.0232	−0.963008	−0.481504	−	0.876444i	$-0.659909\pi$
$523$	−22.0232	−0.963008	−0.481504	+	0.876444i	$0.659909\pi$
$524$	−3.27219	−0.142946
$525$	0	0
$526$	22.7716	0.992887
$527$	1.76558	0.0769098
$528$	0	0
$529$	−14.3354	−0.623280
$530$	−25.9145	−1.12565
$531$	0	0
$532$	2.95811	0.128250
$533$	−15.8161	−0.685073
$534$	0	0
$535$	−4.36009	−0.188503
$536$	5.46648	0.236116
$537$	0	0
$538$	−10.6631	−0.459720
$539$	7.04189	0.303316
$540$	0	0
$541$	−15.7870	−0.678738	−0.339369	−	0.940653i	$-0.610214\pi$
$541$	−15.7870	−0.678738	−0.339369	+	0.940653i	$0.610214\pi$
$542$	23.2199	0.997379
$543$	0	0
$544$	3.12567	0.134012
$545$	14.8051	0.634181
$546$	0	0
$547$	−27.4192	−1.17236	−0.586180	−	0.810180i	$-0.699369\pi$
$547$	−27.4192	−1.17236	−0.586180	+	0.810180i	$0.699369\pi$
$548$	−0.726053	−0.0310154
$549$	0	0
$550$	−18.1661	−0.774606
$551$	−8.56893	−0.365048
$552$	0	0
$553$	−9.11793	−0.387734
$554$	−35.6154	−1.51315
$555$	0	0
$556$	−2.21120	−0.0937758
$557$	−29.4020	−1.24580	−0.622901	−	0.782301i	$-0.714046\pi$
$557$	−29.4020	−1.24580	−0.622901	+	0.782301i	$0.714046\pi$
$558$	0	0
$559$	16.7297	0.707590
$560$	14.3327	0.605669
$561$	0	0
$562$	−25.5931	−1.07958
$563$	10.3705	0.437065	0.218533	−	0.975830i	$-0.429873\pi$
$563$	10.3705	0.437065	0.218533	+	0.975830i	$0.429873\pi$
$564$	0	0
$565$	26.3259	1.10754
$566$	22.3473	0.939327
$567$	0	0
$568$	16.2264	0.680843
$569$	32.9691	1.38214	0.691069	−	0.722788i	$-0.257140\pi$
$569$	32.9691	1.38214	0.691069	+	0.722788i	$0.257140\pi$
$570$	0	0
$571$	−0.737415	−0.0308599	−0.0154299	−	0.999881i	$-0.504912\pi$
$571$	−0.737415	−0.0308599	−0.0154299	+	0.999881i	$0.504912\pi$
$572$	−3.54395	−0.148180
$573$	0	0
$574$	−15.9254	−0.664713
$575$	−6.67768	−0.278479
$576$	0	0
$577$	19.3432	0.805267	0.402634	−	0.915361i	$-0.368095\pi$
$577$	19.3432	0.805267	0.402634	+	0.915361i	$0.368095\pi$
$578$	10.7784	0.448321
$579$	0	0
$580$	0.394238	0.0163698
$581$	−9.61081	−0.398724
$582$	0	0
$583$	69.1721	2.86482
$584$	16.3497	0.676554
$585$	0	0
$586$	26.0806	1.07738
$587$	31.9121	1.31715	0.658577	−	0.752514i	$-0.271159\pi$
$587$	31.9121	1.31715	0.658577	+	0.752514i	$0.271159\pi$
$588$	0	0
$589$	3.90673	0.160974
$590$	16.3601	0.673534
$591$	0	0
$592$	−0.150644	−0.00619144
$593$	−31.6783	−1.30087	−0.650436	−	0.759561i	$-0.725414\pi$
$593$	−31.6783	−1.30087	−0.650436	+	0.759561i	$0.725414\pi$
$594$	0	0
$595$	11.9564	0.490163
$596$	3.79292	0.155364
$597$	0	0
$598$	12.7965	0.523289
$599$	12.6236	0.515787	0.257893	−	0.966173i	$-0.416972\pi$
$599$	12.6236	0.515787	0.257893	+	0.966173i	$0.416972\pi$
$600$	0	0
$601$	−8.90848	−0.363385	−0.181692	−	0.983355i	$-0.558157\pi$
$601$	−8.90848	−0.363385	−0.181692	+	0.983355i	$0.558157\pi$
$602$	16.8452	0.686561
$603$	0	0
$604$	−2.95811	−0.120364
$605$	−40.2036	−1.63451
$606$	0	0
$607$	−33.1242	−1.34447	−0.672236	−	0.740337i	$-0.734666\pi$
$607$	−33.1242	−1.34447	−0.672236	+	0.740337i	$0.734666\pi$
$608$	6.91622	0.280490
$609$	0	0
$610$	24.6040	0.996187
$611$	−12.0564	−0.487751
$612$	0	0
$613$	17.6800	0.714090	0.357045	−	0.934087i	$-0.383784\pi$
$613$	17.6800	0.714090	0.357045	+	0.934087i	$0.383784\pi$
$614$	−28.0315	−1.13126
$615$	0	0
$616$	−42.1893	−1.69986
$617$	25.7324	1.03595	0.517973	−	0.855397i	$-0.326687\pi$
$617$	25.7324	1.03595	0.517973	+	0.855397i	$0.326687\pi$
$618$	0	0
$619$	27.7948	1.11717	0.558583	−	0.829448i	$-0.311345\pi$
$619$	27.7948	1.11717	0.558583	+	0.829448i	$0.311345\pi$
$620$	−0.179740	−0.00721854
$621$	0	0
$622$	−14.3696	−0.576168
$623$	−19.6551	−0.787464
$624$	0	0
$625$	−8.51073	−0.340429
$626$	−5.14022	−0.205444
$627$	0	0
$628$	4.06056	0.162034
$629$	−0.125667	−0.00501068
$630$	0	0
$631$	26.8138	1.06744	0.533720	−	0.845661i	$-0.320794\pi$
$631$	26.8138	1.06744	0.533720	+	0.845661i	$0.320794\pi$
$632$	−11.1298	−0.442719
$633$	0	0
$634$	35.5541	1.41203
$635$	−5.94356	−0.235863
$636$	0	0
$637$	3.82295	0.151471
$638$	10.3369	0.409240
$639$	0	0
$640$	15.6973	0.620490
$641$	−12.6905	−0.501244	−0.250622	−	0.968085i	$-0.580635\pi$
$641$	−12.6905	−0.501244	−0.250622	+	0.968085i	$0.580635\pi$
$642$	0	0
$643$	15.4766	0.610337	0.305168	−	0.952298i	$-0.401287\pi$
$643$	15.4766	0.610337	0.305168	+	0.952298i	$0.401287\pi$
$644$	−1.31172	−0.0516889
$645$	0	0
$646$	−26.8307	−1.05564
$647$	−11.1506	−0.438377	−0.219189	−	0.975683i	$-0.570341\pi$
$647$	−11.1506	−0.438377	−0.219189	+	0.975683i	$0.570341\pi$
$648$	0	0
$649$	−43.6691	−1.71416
$650$	−9.86215	−0.386825
$651$	0	0
$652$	−3.79023	−0.148437
$653$	−44.5921	−1.74503	−0.872513	−	0.488591i	$-0.837511\pi$
$653$	−44.5921	−1.74503	−0.872513	+	0.488591i	$0.837511\pi$
$654$	0	0
$655$	−29.2651	−1.14348
$656$	−17.6277	−0.688247
$657$	0	0
$658$	−12.1397	−0.473255
$659$	14.0966	0.549124	0.274562	−	0.961569i	$-0.411467\pi$
$659$	14.0966	0.549124	0.274562	+	0.961569i	$0.411467\pi$
$660$	0	0
$661$	36.1147	1.40470	0.702350	−	0.711831i	$-0.252134\pi$
$661$	36.1147	1.40470	0.702350	+	0.711831i	$0.252134\pi$
$662$	2.11743	0.0822962
$663$	0	0
$664$	−11.7314	−0.455268
$665$	26.4561	1.02592
$666$	0	0
$667$	3.79973	0.147126
$668$	0.792919	0.0306789
$669$	0	0
$670$	4.13516	0.159755
$671$	−65.6742	−2.53532
$672$	0	0
$673$	2.24216	0.0864290	0.0432145	−	0.999066i	$-0.486240\pi$
$673$	2.24216	0.0864290	0.0432145	+	0.999066i	$0.486240\pi$
$674$	10.7992	0.415971
$675$	0	0
$676$	0.478340	0.0183977
$677$	−35.1762	−1.35193	−0.675966	−	0.736933i	$-0.736273\pi$
$677$	−35.1762	−1.35193	−0.675966	+	0.736933i	$0.736273\pi$
$678$	0	0
$679$	0.628984	0.0241382
$680$	14.5945	0.559674
$681$	0	0
$682$	−4.71276	−0.180461
$683$	17.7638	0.679714	0.339857	−	0.940477i	$-0.389621\pi$
$683$	17.7638	0.679714	0.339857	+	0.940477i	$0.389621\pi$
$684$	0	0
$685$	−6.49350	−0.248104
$686$	26.5921	1.01529
$687$	0	0
$688$	18.6459	0.710868
$689$	37.5526	1.43064
$690$	0	0
$691$	−44.0306	−1.67500	−0.837502	−	0.546434i	$-0.815985\pi$
$691$	−44.0306	−1.67500	−0.837502	+	0.546434i	$0.815985\pi$
$692$	0.700903	0.0266443
$693$	0	0
$694$	26.7374	1.01494
$695$	−19.7760	−0.750147
$696$	0	0
$697$	−14.7050	−0.556992
$698$	−14.9459	−0.565712
$699$	0	0
$700$	1.01093	0.0382094
$701$	30.1052	1.13706	0.568530	−	0.822663i	$-0.307512\pi$
$701$	30.1052	1.13706	0.568530	+	0.822663i	$0.307512\pi$
$702$	0	0
$703$	−0.278066	−0.0104875
$704$	−51.0925	−1.92562
$705$	0	0
$706$	−3.85853	−0.145218
$707$	−26.5921	−1.00010
$708$	0	0
$709$	−24.7374	−0.929033	−0.464517	−	0.885564i	$-0.653772\pi$
$709$	−24.7374	−0.929033	−0.464517	+	0.885564i	$0.653772\pi$
$710$	12.2746	0.460656
$711$	0	0
$712$	−23.9919	−0.899136
$713$	−1.73236	−0.0648775
$714$	0	0
$715$	−31.6955	−1.18535
$716$	1.52940	0.0571564
$717$	0	0
$718$	38.7870	1.44752
$719$	43.5526	1.62424	0.812119	−	0.583491i	$-0.198314\pi$
$719$	43.5526	1.62424	0.812119	+	0.583491i	$0.198314\pi$
$720$	0	0
$721$	−9.42097	−0.350855
$722$	−33.7701	−1.25679
$723$	0	0
$724$	−1.24216	−0.0461646
$725$	−2.92841	−0.108758
$726$	0	0
$727$	20.5371	0.761680	0.380840	−	0.924641i	$-0.375635\pi$
$727$	20.5371	0.761680	0.380840	+	0.924641i	$0.375635\pi$
$728$	−22.9040	−0.848880
$729$	0	0
$730$	12.3678	0.457754
$731$	15.5544	0.575299
$732$	0	0
$733$	14.0060	0.517323	0.258661	−	0.965968i	$-0.416719\pi$
$733$	14.0060	0.517323	0.258661	+	0.965968i	$0.416719\pi$
$734$	14.8098	0.546641
$735$	0	0
$736$	−3.06687	−0.113046
$737$	−11.0378	−0.406581
$738$	0	0
$739$	−41.9813	−1.54431	−0.772154	−	0.635435i	$-0.780821\pi$
$739$	−41.9813	−1.54431	−0.772154	+	0.635435i	$0.780821\pi$
$740$	0.0127932	0.000470288 0
$741$	0	0
$742$	37.8120	1.38812
$743$	−27.8621	−1.02216	−0.511082	−	0.859532i	$-0.670755\pi$
$743$	−27.8621	−1.02216	−0.511082	+	0.859532i	$0.670755\pi$
$744$	0	0
$745$	33.9222	1.24281
$746$	−45.0128	−1.64804
$747$	0	0
$748$	−3.29498	−0.120476
$749$	6.36184	0.232457
$750$	0	0
$751$	−52.9050	−1.93053	−0.965265	−	0.261273i	$-0.915858\pi$
$751$	−52.9050	−1.93053	−0.965265	+	0.261273i	$0.915858\pi$
$752$	−13.4374	−0.490011
$753$	0	0
$754$	5.61175	0.204368
$755$	−26.4561	−0.962834
$756$	0	0
$757$	−41.4858	−1.50783	−0.753913	−	0.656975i	$-0.771836\pi$
$757$	−41.4858	−1.50783	−0.753913	+	0.656975i	$0.771836\pi$
$758$	−28.2116	−1.02469
$759$	0	0
$760$	32.2935	1.17141
$761$	45.3874	1.64529	0.822647	−	0.568553i	$-0.192497\pi$
$761$	45.3874	1.64529	0.822647	+	0.568553i	$0.192497\pi$
$762$	0	0
$763$	−21.6023	−0.782054
$764$	−0.926651	−0.0335251
$765$	0	0
$766$	−5.53890	−0.200128
$767$	−23.7074	−0.856024
$768$	0	0
$769$	5.11650	0.184506	0.0922528	−	0.995736i	$-0.470593\pi$
$769$	5.11650	0.184506	0.0922528	+	0.995736i	$0.470593\pi$
$770$	−31.9145	−1.15012
$771$	0	0
$772$	3.30129	0.118816
$773$	52.6427	1.89343	0.946713	−	0.322077i	$-0.104381\pi$
$773$	52.6427	1.89343	0.946713	+	0.322077i	$0.104381\pi$
$774$	0	0
$775$	1.33511	0.0479587
$776$	0.767769	0.0275613
$777$	0	0
$778$	23.0310	0.825700
$779$	−32.5381	−1.16580
$780$	0	0
$781$	−32.7638	−1.17238
$782$	11.8976	0.425456
$783$	0	0
$784$	4.26083	0.152172
$785$	36.3158	1.29617
$786$	0	0
$787$	−20.7624	−0.740099	−0.370050	−	0.929012i	$-0.620659\pi$
$787$	−20.7624	−0.740099	−0.370050	+	0.929012i	$0.620659\pi$
$788$	0.133732	0.00476401
$789$	0	0
$790$	−8.41921	−0.299542
$791$	−38.4124	−1.36579
$792$	0	0
$793$	−35.6536	−1.26610
$794$	−30.1821	−1.07112
$795$	0	0
$796$	1.88350	0.0667590
$797$	45.5800	1.61453	0.807263	−	0.590192i	$-0.200948\pi$
$797$	45.5800	1.61453	0.807263	+	0.590192i	$0.200948\pi$
$798$	0	0
$799$	−11.2094	−0.396562
$800$	2.36360	0.0835658
$801$	0	0
$802$	−19.6459	−0.693721
$803$	−33.0128	−1.16500
$804$	0	0
$805$	−11.7314	−0.413479
$806$	−2.55850	−0.0901192
$807$	0	0
$808$	−32.4597	−1.14193
$809$	4.21120	0.148058	0.0740290	−	0.997256i	$-0.476414\pi$
$809$	4.21120	0.148058	0.0740290	+	0.997256i	$0.476414\pi$
$810$	0	0
$811$	11.3618	0.398968	0.199484	−	0.979901i	$-0.436073\pi$
$811$	11.3618	0.398968	0.199484	+	0.979901i	$0.436073\pi$
$812$	−0.575236	−0.0201868
$813$	0	0
$814$	0.335437	0.0117571
$815$	−33.8982	−1.18740
$816$	0	0
$817$	34.4175	1.20411
$818$	23.5817	0.824515
$819$	0	0
$820$	1.49701	0.0522778
$821$	−1.64227	−0.0573158	−0.0286579	−	0.999589i	$-0.509123\pi$
$821$	−1.64227	−0.0573158	−0.0286579	+	0.999589i	$0.509123\pi$
$822$	0	0
$823$	11.2189	0.391068	0.195534	−	0.980697i	$-0.437356\pi$
$823$	11.2189	0.391068	0.195534	+	0.980697i	$0.437356\pi$
$824$	−11.4997	−0.400611
$825$	0	0
$826$	−23.8711	−0.830583
$827$	9.61318	0.334283	0.167141	−	0.985933i	$-0.446546\pi$
$827$	9.61318	0.334283	0.167141	+	0.985933i	$0.446546\pi$
$828$	0	0
$829$	33.4938	1.16329	0.581644	−	0.813443i	$-0.302410\pi$
$829$	33.4938	1.16329	0.581644	+	0.813443i	$0.302410\pi$
$830$	−8.87433	−0.308033
$831$	0	0
$832$	−27.7374	−0.961622
$833$	3.55438	0.123152
$834$	0	0
$835$	7.09152	0.245412
$836$	−7.29086	−0.252160
$837$	0	0
$838$	25.4129	0.877874
$839$	31.9941	1.10456	0.552280	−	0.833659i	$-0.313758\pi$
$839$	31.9941	1.10456	0.552280	+	0.833659i	$0.313758\pi$
$840$	0	0
$841$	−27.3337	−0.942541
$842$	43.5631	1.50128
$843$	0	0
$844$	2.74691	0.0945526
$845$	4.27807	0.147170
$846$	0	0
$847$	58.6614	2.01563
$848$	41.8539	1.43727
$849$	0	0
$850$	−9.16931	−0.314505
$851$	0.123303	0.00422678
$852$	0	0
$853$	35.2422	1.20667	0.603334	−	0.797488i	$-0.293838\pi$
$853$	35.2422	1.20667	0.603334	+	0.797488i	$0.293838\pi$
$854$	−35.8999	−1.22847
$855$	0	0
$856$	7.76558	0.265422
$857$	21.2490	0.725851	0.362925	−	0.931818i	$-0.381778\pi$
$857$	21.2490	0.725851	0.362925	+	0.931818i	$0.381778\pi$
$858$	0	0
$859$	51.8384	1.76870	0.884352	−	0.466820i	$-0.154601\pi$
$859$	51.8384	1.76870	0.884352	+	0.466820i	$0.154601\pi$
$860$	−1.58347	−0.0539960
$861$	0	0
$862$	46.2344	1.57475
$863$	22.6783	0.771978	0.385989	−	0.922503i	$-0.373860\pi$
$863$	22.6783	0.771978	0.385989	+	0.922503i	$0.373860\pi$
$864$	0	0
$865$	6.26857	0.213138
$866$	33.7725	1.14764
$867$	0	0
$868$	0.262260	0.00890170
$869$	22.4730	0.762343
$870$	0	0
$871$	−5.99226	−0.203040
$872$	−26.3688	−0.892959
$873$	0	0
$874$	26.3259	0.890488
$875$	28.9685	0.979315
$876$	0	0
$877$	1.13341	0.0382725	0.0191362	−	0.999817i	$-0.493908\pi$
$877$	1.13341	0.0382725	0.0191362	+	0.999817i	$0.493908\pi$
$878$	31.2681	1.05525
$879$	0	0
$880$	−35.3259	−1.19084
$881$	30.8289	1.03865	0.519327	−	0.854576i	$-0.326183\pi$
$881$	30.8289	1.03865	0.519327	+	0.854576i	$0.326183\pi$
$882$	0	0
$883$	−9.33511	−0.314152	−0.157076	−	0.987587i	$-0.550207\pi$
$883$	−9.33511	−0.314152	−0.157076	+	0.987587i	$0.550207\pi$
$884$	−1.78880	−0.0601639
$885$	0	0
$886$	−5.55295	−0.186555
$887$	14.0942	0.473237	0.236619	−	0.971603i	$-0.423961\pi$
$887$	14.0942	0.473237	0.236619	+	0.971603i	$0.423961\pi$
$888$	0	0
$889$	8.67230	0.290860
$890$	−18.1489	−0.608352
$891$	0	0
$892$	2.02322	0.0677425
$893$	−24.8033	−0.830012
$894$	0	0
$895$	13.6783	0.457215
$896$	−22.9040	−0.765170
$897$	0	0
$898$	24.7113	0.824628
$899$	−0.759704	−0.0253376
$900$	0	0
$901$	34.9145	1.16317
$902$	39.2513	1.30693
$903$	0	0
$904$	−46.8881	−1.55947
$905$	−11.1094	−0.369288
$906$	0	0
$907$	−8.73917	−0.290179	−0.145090	−	0.989419i	$-0.546347\pi$
$907$	−8.73917	−0.290179	−0.145090	+	0.989419i	$0.546347\pi$
$908$	−3.20296	−0.106294
$909$	0	0
$910$	−17.3259	−0.574349
$911$	20.6509	0.684196	0.342098	−	0.939664i	$-0.388862\pi$
$911$	20.6509	0.684196	0.342098	+	0.939664i	$0.388862\pi$
$912$	0	0
$913$	23.6878	0.783951
$914$	26.2199	0.867276
$915$	0	0
$916$	−0.288668	−0.00953785
$917$	42.7009	1.41011
$918$	0	0
$919$	−2.36009	−0.0778522	−0.0389261	−	0.999242i	$-0.512394\pi$
$919$	−2.36009	−0.0778522	−0.0389261	+	0.999242i	$0.512394\pi$
$920$	−14.3200	−0.472115
$921$	0	0
$922$	37.3865	1.23126
$923$	−17.7870	−0.585468
$924$	0	0
$925$	−0.0950283	−0.00312451
$926$	−52.1215	−1.71282
$927$	0	0
$928$	−1.34493	−0.0441496
$929$	26.8203	0.879944	0.439972	−	0.898011i	$-0.354988\pi$
$929$	26.8203	0.879944	0.439972	+	0.898011i	$0.354988\pi$
$930$	0	0
$931$	7.86484	0.257760
$932$	3.10244	0.101624
$933$	0	0
$934$	40.1007	1.31213
$935$	−29.4688	−0.963734
$936$	0	0
$937$	1.93313	0.0631527	0.0315764	−	0.999501i	$-0.489947\pi$
$937$	1.93313	0.0631527	0.0315764	+	0.999501i	$0.489947\pi$
$938$	−6.03365	−0.197006
$939$	0	0
$940$	1.14115	0.0372202
$941$	−11.9103	−0.388266	−0.194133	−	0.980975i	$-0.562189\pi$
$941$	−11.9103	−0.388266	−0.194133	+	0.980975i	$0.562189\pi$
$942$	0	0
$943$	14.4284	0.469853
$944$	−26.4228	−0.859990
$945$	0	0
$946$	−41.5185	−1.34988
$947$	0.315836	0.0102633	0.00513165	−	0.999987i	$-0.498367\pi$
$947$	0.315836	0.0102633	0.00513165	+	0.999987i	$0.498367\pi$
$948$	0	0
$949$	−17.9222	−0.581779
$950$	−20.2891	−0.658265
$951$	0	0
$952$	−21.2950	−0.690174
$953$	−3.25133	−0.105321	−0.0526605	−	0.998612i	$-0.516770\pi$
$953$	−3.25133	−0.105321	−0.0526605	+	0.998612i	$0.516770\pi$
$954$	0	0
$955$	−8.28756	−0.268179
$956$	−0.744547	−0.0240804
$957$	0	0
$958$	−50.7606	−1.64000
$959$	9.47472	0.305955
$960$	0	0
$961$	−30.6536	−0.988827
$962$	0.182104	0.00587127
$963$	0	0
$964$	0.619489	0.0199524
$965$	29.5253	0.950452
$966$	0	0
$967$	10.9676	0.352694	0.176347	−	0.984328i	$-0.443572\pi$
$967$	10.9676	0.352694	0.176347	+	0.984328i	$0.443572\pi$
$968$	71.6049	2.30147
$969$	0	0
$970$	0.580785	0.0186479
$971$	−23.3868	−0.750519	−0.375259	−	0.926920i	$-0.622446\pi$
$971$	−23.3868	−0.750519	−0.375259	+	0.926920i	$0.622446\pi$
$972$	0	0
$973$	28.8553	0.925060
$974$	1.02910	0.0329744
$975$	0	0
$976$	−39.7374	−1.27196
$977$	50.1762	1.60528	0.802640	−	0.596464i	$-0.203428\pi$
$977$	50.1762	1.60528	0.802640	+	0.596464i	$0.203428\pi$
$978$	0	0
$979$	48.4439	1.54827
$980$	−0.361844	−0.0115587
$981$	0	0
$982$	−0.670874	−0.0214084
$983$	14.6946	0.468685	0.234342	−	0.972154i	$-0.424706\pi$
$983$	14.6946	0.468685	0.234342	+	0.972154i	$0.424706\pi$
$984$	0	0
$985$	1.19604	0.0381091
$986$	5.21751	0.166159
$987$	0	0
$988$	−3.95811	−0.125924
$989$	−15.2618	−0.485296
$990$	0	0
$991$	2.00000	0.0635321	0.0317660	−	0.999495i	$-0.489887\pi$
$991$	2.00000	0.0635321	0.0317660	+	0.999495i	$0.489887\pi$
$992$	0.613179	0.0194684
$993$	0	0
$994$	−17.9099	−0.568068
$995$	16.8452	0.534030
$996$	0	0
$997$	−45.8863	−1.45323	−0.726617	−	0.687043i	$-0.758908\pi$
$997$	−45.8863	−1.45323	−0.726617	+	0.687043i	$0.758908\pi$
$998$	−12.0797	−0.382375
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	243.2.a.e.1.2	✓	3
3.2	odd	2		243.2.a.f.1.2	yes	3
4.3	odd	2		3888.2.a.bd.1.2		3
5.4	even	2		6075.2.a.bv.1.2		3
9.2	odd	6		243.2.c.e.82.2		6
9.4	even	3		243.2.c.f.163.2		6
9.5	odd	6		243.2.c.e.163.2		6
9.7	even	3		243.2.c.f.82.2		6
12.11	even	2		3888.2.a.bk.1.2		3
15.14	odd	2		6075.2.a.bq.1.2		3
27.2	odd	18		729.2.e.b.568.1		6
27.4	even	9		729.2.e.h.406.1		6
27.5	odd	18		729.2.e.i.649.1		6
27.7	even	9		729.2.e.h.325.1		6
27.11	odd	18		729.2.e.i.82.1		6
27.13	even	9		729.2.e.g.163.1		6
27.14	odd	18		729.2.e.b.163.1		6
27.16	even	9		729.2.e.a.82.1		6
27.20	odd	18		729.2.e.c.325.1		6
27.22	even	9		729.2.e.a.649.1		6
27.23	odd	18		729.2.e.c.406.1		6
27.25	even	9		729.2.e.g.568.1		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
243.2.a.e.1.2	✓	3	1.1	even	1	trivial
243.2.a.f.1.2	yes	3	3.2	odd	2
243.2.c.e.82.2		6	9.2	odd	6
243.2.c.e.163.2		6	9.5	odd	6
243.2.c.f.82.2		6	9.7	even	3
243.2.c.f.163.2		6	9.4	even	3
729.2.e.a.82.1		6	27.16	even	9
729.2.e.a.649.1		6	27.22	even	9
729.2.e.b.163.1		6	27.14	odd	18
729.2.e.b.568.1		6	27.2	odd	18
729.2.e.c.325.1		6	27.20	odd	18
729.2.e.c.406.1		6	27.23	odd	18
729.2.e.g.163.1		6	27.13	even	9
729.2.e.g.568.1		6	27.25	even	9
729.2.e.h.325.1		6	27.7	even	9
729.2.e.h.406.1		6	27.4	even	9
729.2.e.i.82.1		6	27.11	odd	18
729.2.e.i.649.1		6	27.5	odd	18
3888.2.a.bd.1.2		3	4.3	odd	2
3888.2.a.bk.1.2		3	12.11	even	2
6075.2.a.bq.1.2		3	15.14	odd	2
6075.2.a.bv.1.2		3	5.4	even	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 $\F_{q}$

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

\(f(q)\)	\(=\)	\(q-1.34730 q^{2} -0.184793 q^{4} -1.65270 q^{5} +2.41147 q^{7} +2.94356 q^{8} +O(q^{10})\)	\(q-1.34730 q^{2} -0.184793 q^{4} -1.65270 q^{5} +2.41147 q^{7} +2.94356 q^{8} +2.22668 q^{10} -5.94356 q^{11} -3.22668 q^{13} -3.24897 q^{14} -3.59627 q^{16} -3.00000 q^{17} -6.63816 q^{19} +0.305407 q^{20} +8.00774 q^{22} +2.94356 q^{23} -2.26857 q^{25} +4.34730 q^{26} -0.445622 q^{28} +1.29086 q^{29} -0.588526 q^{31} -1.04189 q^{32} +4.04189 q^{34} -3.98545 q^{35} +0.0418891 q^{37} +8.94356 q^{38} -4.86484 q^{40} +4.90167 q^{41} -5.18479 q^{43} +1.09833 q^{44} -3.96585 q^{46} +3.73648 q^{47} -1.18479 q^{49} +3.05644 q^{50} +0.596267 q^{52} -11.6382 q^{53} +9.82295 q^{55} +7.09833 q^{56} -1.73917 q^{58} +7.34730 q^{59} +11.0496 q^{61} +0.792919 q^{62} +8.59627 q^{64} +5.33275 q^{65} +1.85710 q^{67} +0.554378 q^{68} +5.36959 q^{70} +5.51249 q^{71} +5.55438 q^{73} -0.0564370 q^{74} +1.22668 q^{76} -14.3327 q^{77} -3.78106 q^{79} +5.94356 q^{80} -6.60401 q^{82} -3.98545 q^{83} +4.95811 q^{85} +6.98545 q^{86} -17.4953 q^{88} -8.15064 q^{89} -7.78106 q^{91} -0.543948 q^{92} -5.03415 q^{94} +10.9709 q^{95} +0.260830 q^{97} +1.59627 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 3 q^{2} + 3 q^{4} - 6 q^{5} - 3 q^{7} - 6 q^{8}+O(q^{10}) \) 3 * q - 3 * q^2 + 3 * q^4 - 6 * q^5 - 3 * q^7 - 6 * q^8	\( 3 q - 3 q^{2} + 3 q^{4} - 6 q^{5} - 3 q^{7} - 6 q^{8} - 3 q^{11} - 3 q^{13} + 3 q^{14} + 3 q^{16} - 9 q^{17} - 3 q^{19} + 3 q^{20} - 6 q^{23} + 3 q^{25} + 12 q^{26} - 12 q^{28} - 12 q^{29} - 12 q^{31} + 9 q^{34} + 6 q^{35} - 3 q^{37} + 12 q^{38} + 9 q^{40} + 3 q^{41} - 12 q^{43} + 15 q^{44} + 9 q^{46} + 6 q^{47} + 24 q^{50} - 12 q^{52} - 18 q^{53} + 9 q^{55} + 33 q^{56} + 9 q^{58} + 21 q^{59} + 6 q^{61} + 12 q^{62} + 12 q^{64} - 3 q^{65} + 6 q^{67} - 9 q^{68} + 9 q^{70} + 9 q^{71} + 6 q^{73} - 15 q^{74} - 3 q^{76} - 24 q^{77} + 6 q^{79} + 3 q^{80} + 18 q^{82} + 6 q^{83} + 18 q^{85} + 3 q^{86} - 36 q^{88} - 6 q^{91} - 24 q^{92} - 36 q^{94} - 3 q^{95} + 15 q^{97} - 9 q^{98}+O(q^{100}) \) 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

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more