LMFDB - Embedded newform 6561.2.a.d.1.7

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6561,2,Mod(1,6561)]

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

lf = mfeigenbasis(mf)

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")

//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);

Level:	$ N $	$=$	$ 6561 = 3^{8} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6561.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:	$52.3898487662$
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			1.7
Character	$\chi$	$=$	6561.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.19283 q^{2} +2.80849 q^{4} +3.59965 q^{5} +1.46244 q^{7} -1.77287 q^{8} +O(q^{10})$	\(q-2.19283 q^{2} +2.80849 q^{4} +3.59965 q^{5} +1.46244 q^{7} -1.77287 q^{8} -7.89340 q^{10} -4.51171 q^{11} -3.07200 q^{13} -3.20688 q^{14} -1.72938 q^{16} +3.35908 q^{17} +0.817062 q^{19} +10.1096 q^{20} +9.89339 q^{22} +2.88835 q^{23} +7.95745 q^{25} +6.73636 q^{26} +4.10725 q^{28} -8.96778 q^{29} +0.638458 q^{31} +7.33796 q^{32} -7.36589 q^{34} +5.26427 q^{35} -3.75576 q^{37} -1.79167 q^{38} -6.38170 q^{40} +9.75366 q^{41} +0.0603845 q^{43} -12.6711 q^{44} -6.33364 q^{46} -2.19160 q^{47} -4.86126 q^{49} -17.4493 q^{50} -8.62767 q^{52} -1.75168 q^{53} -16.2405 q^{55} -2.59272 q^{56} +19.6648 q^{58} +9.87188 q^{59} -2.79506 q^{61} -1.40003 q^{62} -12.6321 q^{64} -11.0581 q^{65} -0.315669 q^{67} +9.43394 q^{68} -11.5436 q^{70} +0.0593099 q^{71} +1.32460 q^{73} +8.23573 q^{74} +2.29471 q^{76} -6.59811 q^{77} -9.13522 q^{79} -6.22515 q^{80} -21.3881 q^{82} +13.8181 q^{83} +12.0915 q^{85} -0.132413 q^{86} +7.99867 q^{88} +14.6100 q^{89} -4.49262 q^{91} +8.11188 q^{92} +4.80580 q^{94} +2.94113 q^{95} +11.2136 q^{97} +10.6599 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 72 q + 9 q^{2} + 63 q^{4} + 18 q^{5} + 27 q^{8}+O(q^{10}) $ 72 * q + 9 * q^2 + 63 * q^4 + 18 * q^5 + 27 * q^8	$ 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}+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

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 $

$2$	−2.19283	−1.55056	−0.775281	−	0.631616i	$-0.782392\pi$
$2$	−2.19283	−1.55056	−0.775281	+	0.631616i	$0.782392\pi$
$3$	0	0
$3$	0	0
$4$	2.80849	1.40424
$5$	3.59965	1.60981	0.804905	−	0.593403i	$-0.202216\pi$
$5$	3.59965	1.60981	0.804905	+	0.593403i	$0.202216\pi$
$6$	0	0
$7$	1.46244	0.552751	0.276376	−	0.961050i	$-0.410867\pi$
$7$	1.46244	0.552751	0.276376	+	0.961050i	$0.410867\pi$
$8$	−1.77287	−0.626804
$9$	0	0
$10$	−7.89340	−2.49611
$11$	−4.51171	−1.36033	−0.680165	−	0.733059i	$-0.738092\pi$
$11$	−4.51171	−1.36033	−0.680165	+	0.733059i	$0.738092\pi$
$12$	0	0
$13$	−3.07200	−0.852019	−0.426010	−	0.904719i	$-0.640081\pi$
$13$	−3.07200	−0.852019	−0.426010	+	0.904719i	$0.640081\pi$
$14$	−3.20688	−0.857075
$15$	0	0
$16$	−1.72938	−0.432344
$17$	3.35908	0.814698	0.407349	−	0.913273i	$-0.366453\pi$
$17$	3.35908	0.814698	0.407349	+	0.913273i	$0.366453\pi$
$18$	0	0
$19$	0.817062	0.187447	0.0937234	−	0.995598i	$-0.470123\pi$
$19$	0.817062	0.187447	0.0937234	+	0.995598i	$0.470123\pi$
$20$	10.1096	2.26057
$21$	0	0
$22$	9.89339	2.10928
$23$	2.88835	0.602262	0.301131	−	0.953583i	$-0.402636\pi$
$23$	2.88835	0.602262	0.301131	+	0.953583i	$0.402636\pi$
$24$	0	0
$25$	7.95745	1.59149
$26$	6.73636	1.32111
$27$	0	0
$28$	4.10725	0.776197
$29$	−8.96778	−1.66527	−0.832637	−	0.553819i	$-0.813170\pi$
$29$	−8.96778	−1.66527	−0.832637	+	0.553819i	$0.813170\pi$
$30$	0	0
$31$	0.638458	0.114670	0.0573352	−	0.998355i	$-0.481740\pi$
$31$	0.638458	0.114670	0.0573352	+	0.998355i	$0.481740\pi$
$32$	7.33796	1.29718
$33$	0	0
$34$	−7.36589	−1.26324
$35$	5.26427	0.889825
$36$	0	0
$37$	−3.75576	−0.617443	−0.308721	−	0.951152i	$-0.599901\pi$
$37$	−3.75576	−0.617443	−0.308721	+	0.951152i	$0.599901\pi$
$38$	−1.79167	−0.290648
$39$	0	0
$40$	−6.38170	−1.00904
$41$	9.75366	1.52327	0.761633	−	0.648009i	$-0.224398\pi$
$41$	9.75366	1.52327	0.761633	+	0.648009i	$0.224398\pi$
$42$	0	0
$43$	0.0603845	0.00920855	0.00460428	−	0.999989i	$-0.498534\pi$
$43$	0.0603845	0.00920855	0.00460428	+	0.999989i	$0.498534\pi$
$44$	−12.6711	−1.91023
$45$	0	0
$46$	−6.33364	−0.933845
$47$	−2.19160	−0.319678	−0.159839	−	0.987143i	$-0.551097\pi$
$47$	−2.19160	−0.319678	−0.159839	+	0.987143i	$0.551097\pi$
$48$	0	0
$49$	−4.86126	−0.694466
$50$	−17.4493	−2.46771
$51$	0	0
$52$	−8.62767	−1.19644
$53$	−1.75168	−0.240611	−0.120306	−	0.992737i	$-0.538387\pi$
$53$	−1.75168	−0.240611	−0.120306	+	0.992737i	$0.538387\pi$
$54$	0	0
$55$	−16.2405	−2.18987
$56$	−2.59272	−0.346467
$57$	0	0
$58$	19.6648	2.58211
$59$	9.87188	1.28521	0.642604	−	0.766198i	$-0.277854\pi$
$59$	9.87188	1.28521	0.642604	+	0.766198i	$0.277854\pi$
$60$	0	0
$61$	−2.79506	−0.357871	−0.178936	−	0.983861i	$-0.557265\pi$
$61$	−2.79506	−0.357871	−0.178936	+	0.983861i	$0.557265\pi$
$62$	−1.40003	−0.177804
$63$	0	0
$64$	−12.6321	−1.57902
$65$	−11.0581	−1.37159
$66$	0	0
$67$	−0.315669	−0.0385651	−0.0192826	−	0.999814i	$-0.506138\pi$
$67$	−0.315669	−0.0385651	−0.0192826	+	0.999814i	$0.506138\pi$
$68$	9.43394	1.14403
$69$	0	0
$70$	−11.5436	−1.37973
$71$	0.0593099	0.00703879	0.00351940	−	0.999994i	$-0.498880\pi$
$71$	0.0593099	0.00703879	0.00351940	+	0.999994i	$0.498880\pi$
$72$	0	0
$73$	1.32460	0.155032	0.0775161	−	0.996991i	$-0.475301\pi$
$73$	1.32460	0.155032	0.0775161	+	0.996991i	$0.475301\pi$
$74$	8.23573	0.957384
$75$	0	0
$76$	2.29471	0.263221
$77$	−6.59811	−0.751924
$78$	0	0
$79$	−9.13522	−1.02779	−0.513896	−	0.857852i	$-0.671798\pi$
$79$	−9.13522	−1.02779	−0.513896	+	0.857852i	$0.671798\pi$
$80$	−6.22515	−0.695992
$81$	0	0
$82$	−21.3881	−2.36192
$83$	13.8181	1.51674	0.758369	−	0.651826i	$-0.225997\pi$
$83$	13.8181	1.51674	0.758369	+	0.651826i	$0.225997\pi$
$84$	0	0
$85$	12.0915	1.31151
$86$	−0.132413	−0.0142784
$87$	0	0
$88$	7.99867	0.852661
$89$	14.6100	1.54866	0.774328	−	0.632785i	$-0.218088\pi$
$89$	14.6100	1.54866	0.774328	+	0.632785i	$0.218088\pi$
$90$	0	0
$91$	−4.49262	−0.470955
$92$	8.11188	0.845722
$93$	0	0
$94$	4.80580	0.495680
$95$	2.94113	0.301754
$96$	0	0
$97$	11.2136	1.13857	0.569283	−	0.822142i	$-0.307221\pi$
$97$	11.2136	1.13857	0.569283	+	0.822142i	$0.307221\pi$
$98$	10.6599	1.07681
$99$	0	0
$100$	22.3484	2.23484
$101$	11.5373	1.14800	0.574000	−	0.818855i	$-0.305391\pi$
$101$	11.5373	1.14800	0.574000	+	0.818855i	$0.305391\pi$
$102$	0	0
$103$	0.816852	0.0804868	0.0402434	−	0.999190i	$-0.487187\pi$
$103$	0.816852	0.0804868	0.0402434	+	0.999190i	$0.487187\pi$
$104$	5.44626	0.534049
$105$	0	0
$106$	3.84112	0.373083
$107$	2.50126	0.241806	0.120903	−	0.992664i	$-0.461421\pi$
$107$	2.50126	0.241806	0.120903	+	0.992664i	$0.461421\pi$
$108$	0	0
$109$	−6.91805	−0.662629	−0.331315	−	0.943520i	$-0.607492\pi$
$109$	−6.91805	−0.662629	−0.331315	+	0.943520i	$0.607492\pi$
$110$	35.6127	3.39554
$111$	0	0
$112$	−2.52911	−0.238979
$113$	3.67941	0.346130	0.173065	−	0.984910i	$-0.444633\pi$
$113$	3.67941	0.346130	0.173065	+	0.984910i	$0.444633\pi$
$114$	0	0
$115$	10.3970	0.969528
$116$	−25.1859	−2.33845
$117$	0	0
$118$	−21.6473	−1.99280
$119$	4.91247	0.450325
$120$	0	0
$121$	9.35549	0.850499
$122$	6.12909	0.554902
$123$	0	0
$124$	1.79310	0.161025
$125$	10.6458	0.952188
$126$	0	0
$127$	16.7316	1.48469	0.742345	−	0.670018i	$-0.233714\pi$
$127$	16.7316	1.48469	0.742345	+	0.670018i	$0.233714\pi$
$128$	13.0241	1.15118
$129$	0	0
$130$	24.2485	2.12674
$131$	13.9136	1.21564	0.607818	−	0.794076i	$-0.292045\pi$
$131$	13.9136	1.21564	0.607818	+	0.794076i	$0.292045\pi$
$132$	0	0
$133$	1.19491	0.103611
$134$	0.692207	0.0597976
$135$	0	0
$136$	−5.95522	−0.510656
$137$	5.06612	0.432827	0.216414	−	0.976302i	$-0.430564\pi$
$137$	5.06612	0.432827	0.216414	+	0.976302i	$0.430564\pi$
$138$	0	0
$139$	6.09899	0.517309	0.258655	−	0.965970i	$-0.416721\pi$
$139$	6.09899	0.517309	0.258655	+	0.965970i	$0.416721\pi$
$140$	14.7846	1.24953
$141$	0	0
$142$	−0.130056	−0.0109141
$143$	13.8600	1.15903
$144$	0	0
$145$	−32.2808	−2.68078
$146$	−2.90461	−0.240387
$147$	0	0
$148$	−10.5480	−0.867040
$149$	−1.47928	−0.121187	−0.0605937	−	0.998163i	$-0.519299\pi$
$149$	−1.47928	−0.121187	−0.0605937	+	0.998163i	$0.519299\pi$
$150$	0	0
$151$	14.4095	1.17263	0.586315	−	0.810083i	$-0.300578\pi$
$151$	14.4095	1.17263	0.586315	+	0.810083i	$0.300578\pi$
$152$	−1.44854	−0.117492
$153$	0	0
$154$	14.4685	1.16591
$155$	2.29822	0.184598
$156$	0	0
$157$	−17.5449	−1.40024	−0.700118	−	0.714027i	$-0.746869\pi$
$157$	−17.5449	−1.40024	−0.700118	+	0.714027i	$0.746869\pi$
$158$	20.0319	1.59366
$159$	0	0
$160$	26.4141	2.08822
$161$	4.22404	0.332901
$162$	0	0
$163$	15.2996	1.19835	0.599177	−	0.800616i	$-0.295494\pi$
$163$	15.2996	1.19835	0.599177	+	0.800616i	$0.295494\pi$
$164$	27.3930	2.13904
$165$	0	0
$166$	−30.3008	−2.35180
$167$	14.2688	1.10415	0.552077	−	0.833793i	$-0.313835\pi$
$167$	14.2688	1.10415	0.552077	+	0.833793i	$0.313835\pi$
$168$	0	0
$169$	−3.56282	−0.274063
$170$	−26.5146	−2.03358
$171$	0	0
$172$	0.169589	0.0129311
$173$	17.1902	1.30695	0.653474	−	0.756949i	$-0.273311\pi$
$173$	17.1902	1.30695	0.653474	+	0.756949i	$0.273311\pi$
$174$	0	0
$175$	11.6373	0.879698
$176$	7.80244	0.588131
$177$	0	0
$178$	−32.0372	−2.40129
$179$	−5.11273	−0.382143	−0.191072	−	0.981576i	$-0.561196\pi$
$179$	−5.11273	−0.382143	−0.191072	+	0.981576i	$0.561196\pi$
$180$	0	0
$181$	8.92675	0.663520	0.331760	−	0.943364i	$-0.392358\pi$
$181$	8.92675	0.663520	0.331760	+	0.943364i	$0.392358\pi$
$182$	9.85154	0.730244
$183$	0	0
$184$	−5.12066	−0.377500
$185$	−13.5194	−0.993966
$186$	0	0
$187$	−15.1552	−1.10826
$188$	−6.15508	−0.448905
$189$	0	0
$190$	−6.44939	−0.467888
$191$	5.31186	0.384353	0.192176	−	0.981360i	$-0.438445\pi$
$191$	5.31186	0.384353	0.192176	+	0.981360i	$0.438445\pi$
$192$	0	0
$193$	7.38026	0.531243	0.265621	−	0.964077i	$-0.414423\pi$
$193$	7.38026	0.531243	0.265621	+	0.964077i	$0.414423\pi$
$194$	−24.5894	−1.76542
$195$	0	0
$196$	−13.6528	−0.975199
$197$	24.4984	1.74544	0.872719	−	0.488224i	$-0.162355\pi$
$197$	24.4984	1.74544	0.872719	+	0.488224i	$0.162355\pi$
$198$	0	0
$199$	1.49625	0.106067	0.0530333	−	0.998593i	$-0.483111\pi$
$199$	1.49625	0.106067	0.0530333	+	0.998593i	$0.483111\pi$
$200$	−14.1075	−0.997553
$201$	0	0
$202$	−25.2992	−1.78005
$203$	−13.1149	−0.920482
$204$	0	0
$205$	35.1097	2.45217
$206$	−1.79121	−0.124800
$207$	0	0
$208$	5.31264	0.368366
$209$	−3.68634	−0.254990
$210$	0	0
$211$	−21.8109	−1.50152	−0.750762	−	0.660572i	$-0.770314\pi$
$211$	−21.8109	−1.50152	−0.750762	+	0.660572i	$0.770314\pi$
$212$	−4.91956	−0.337877
$213$	0	0
$214$	−5.48482	−0.374935
$215$	0.217363	0.0148240
$216$	0	0
$217$	0.933707	0.0633842
$218$	15.1701	1.02745
$219$	0	0
$220$	−45.6114	−3.07512
$221$	−10.3191	−0.694138
$222$	0	0
$223$	24.4171	1.63509	0.817546	−	0.575864i	$-0.195334\pi$
$223$	24.4171	1.63509	0.817546	+	0.575864i	$0.195334\pi$
$224$	10.7313	0.717018
$225$	0	0
$226$	−8.06831	−0.536696
$227$	5.70903	0.378922	0.189461	−	0.981888i	$-0.439326\pi$
$227$	5.70903	0.378922	0.189461	+	0.981888i	$0.439326\pi$
$228$	0	0
$229$	25.9628	1.71567	0.857834	−	0.513928i	$-0.171810\pi$
$229$	25.9628	1.71567	0.857834	+	0.513928i	$0.171810\pi$
$230$	−22.7989	−1.50331
$231$	0	0
$232$	15.8987	1.04380
$233$	−4.64858	−0.304539	−0.152269	−	0.988339i	$-0.548658\pi$
$233$	−4.64858	−0.304539	−0.152269	+	0.988339i	$0.548658\pi$
$234$	0	0
$235$	−7.88898	−0.514620
$236$	27.7250	1.80475
$237$	0	0
$238$	−10.7722	−0.698257
$239$	−11.2407	−0.727099	−0.363549	−	0.931575i	$-0.618435\pi$
$239$	−11.2407	−0.727099	−0.363549	+	0.931575i	$0.618435\pi$
$240$	0	0
$241$	−22.0714	−1.42174	−0.710871	−	0.703322i	$-0.751699\pi$
$241$	−22.0714	−1.42174	−0.710871	+	0.703322i	$0.751699\pi$
$242$	−20.5150	−1.31875
$243$	0	0
$244$	−7.84990	−0.502538
$245$	−17.4988	−1.11796
$246$	0	0
$247$	−2.51001	−0.159708
$248$	−1.13190	−0.0718759
$249$	0	0
$250$	−23.3444	−1.47643
$251$	−25.6224	−1.61727	−0.808636	−	0.588310i	$-0.799794\pi$
$251$	−25.6224	−1.61727	−0.808636	+	0.588310i	$0.799794\pi$
$252$	0	0
$253$	−13.0314	−0.819275
$254$	−36.6895	−2.30210
$255$	0	0
$256$	−3.29539	−0.205962
$257$	−17.0823	−1.06557	−0.532783	−	0.846252i	$-0.678854\pi$
$257$	−17.0823	−1.06557	−0.532783	+	0.846252i	$0.678854\pi$
$258$	0	0
$259$	−5.49258	−0.341292
$260$	−31.0566	−1.92605
$261$	0	0
$262$	−30.5101	−1.88492
$263$	−0.257391	−0.0158714	−0.00793570	−	0.999969i	$-0.502526\pi$
$263$	−0.257391	−0.0158714	−0.00793570	+	0.999969i	$0.502526\pi$
$264$	0	0
$265$	−6.30541	−0.387339
$266$	−2.62022	−0.160656
$267$	0	0
$268$	−0.886552	−0.0541548
$269$	−6.08141	−0.370790	−0.185395	−	0.982664i	$-0.559356\pi$
$269$	−6.08141	−0.370790	−0.185395	+	0.982664i	$0.559356\pi$
$270$	0	0
$271$	−11.1074	−0.674729	−0.337365	−	0.941374i	$-0.609536\pi$
$271$	−11.1074	−0.674729	−0.337365	+	0.941374i	$0.609536\pi$
$272$	−5.80912	−0.352230
$273$	0	0
$274$	−11.1091	−0.671126
$275$	−35.9017	−2.16495
$276$	0	0
$277$	−5.15034	−0.309454	−0.154727	−	0.987957i	$-0.549450\pi$
$277$	−5.15034	−0.309454	−0.154727	+	0.987957i	$0.549450\pi$
$278$	−13.3740	−0.802120
$279$	0	0
$280$	−9.33287	−0.557746
$281$	−10.3221	−0.615767	−0.307883	−	0.951424i	$-0.599621\pi$
$281$	−10.3221	−0.615767	−0.307883	+	0.951424i	$0.599621\pi$
$282$	0	0
$283$	15.6678	0.931352	0.465676	−	0.884955i	$-0.345811\pi$
$283$	15.6678	0.931352	0.465676	+	0.884955i	$0.345811\pi$
$284$	0.166571	0.00988418
$285$	0	0
$286$	−30.3925	−1.79714
$287$	14.2642	0.841987
$288$	0	0
$289$	−5.71655	−0.336268
$290$	70.7862	4.15671
$291$	0	0
$292$	3.72011	0.217703
$293$	13.2626	0.774807	0.387403	−	0.921910i	$-0.373372\pi$
$293$	13.2626	0.774807	0.387403	+	0.921910i	$0.373372\pi$
$294$	0	0
$295$	35.5353	2.06894
$296$	6.65847	0.387016
$297$	0	0
$298$	3.24381	0.187909
$299$	−8.87300	−0.513139
$300$	0	0
$301$	0.0883089	0.00509004
$302$	−31.5976	−1.81823
$303$	0	0
$304$	−1.41301	−0.0810415
$305$	−10.0612	−0.576105
$306$	0	0
$307$	6.17842	0.352621	0.176311	−	0.984335i	$-0.443584\pi$
$307$	6.17842	0.352621	0.176311	+	0.984335i	$0.443584\pi$
$308$	−18.5307	−1.05588
$309$	0	0
$310$	−5.03960	−0.286230
$311$	−9.89383	−0.561028	−0.280514	−	0.959850i	$-0.590505\pi$
$311$	−9.89383	−0.561028	−0.280514	+	0.959850i	$0.590505\pi$
$312$	0	0
$313$	6.76487	0.382373	0.191186	−	0.981554i	$-0.438767\pi$
$313$	6.76487	0.382373	0.191186	+	0.981554i	$0.438767\pi$
$314$	38.4729	2.17115
$315$	0	0
$316$	−25.6561	−1.44327
$317$	23.7604	1.33452	0.667259	−	0.744825i	$-0.267467\pi$
$317$	23.7604	1.33452	0.667259	+	0.744825i	$0.267467\pi$
$318$	0	0
$319$	40.4600	2.26532
$320$	−45.4712	−2.54192
$321$	0	0
$322$	−9.26258	−0.516184
$323$	2.74458	0.152712
$324$	0	0
$325$	−24.4453	−1.35598
$326$	−33.5493	−1.85812
$327$	0	0
$328$	−17.2920	−0.954790
$329$	−3.20509	−0.176702
$330$	0	0
$331$	25.4108	1.39670	0.698352	−	0.715754i	$-0.253917\pi$
$331$	25.4108	1.39670	0.698352	+	0.715754i	$0.253917\pi$
$332$	38.8081	2.12987
$333$	0	0
$334$	−31.2890	−1.71206
$335$	−1.13630	−0.0620825
$336$	0	0
$337$	−10.8091	−0.588808	−0.294404	−	0.955681i	$-0.595121\pi$
$337$	−10.8091	−0.588808	−0.294404	+	0.955681i	$0.595121\pi$
$338$	7.81265	0.424952
$339$	0	0
$340$	33.9589	1.84168
$341$	−2.88053	−0.155990
$342$	0	0
$343$	−17.3464	−0.936618
$344$	−0.107054	−0.00577196
$345$	0	0
$346$	−37.6952	−2.02651
$347$	19.7251	1.05890	0.529449	−	0.848342i	$-0.322399\pi$
$347$	19.7251	1.05890	0.529449	+	0.848342i	$0.322399\pi$
$348$	0	0
$349$	−3.09656	−0.165755	−0.0828776	−	0.996560i	$-0.526411\pi$
$349$	−3.09656	−0.165755	−0.0828776	+	0.996560i	$0.526411\pi$
$350$	−25.5186	−1.36403
$351$	0	0
$352$	−33.1067	−1.76459
$353$	−6.15817	−0.327766	−0.163883	−	0.986480i	$-0.552402\pi$
$353$	−6.15817	−0.327766	−0.163883	+	0.986480i	$0.552402\pi$
$354$	0	0
$355$	0.213495	0.0113311
$356$	41.0319	2.17469
$357$	0	0
$358$	11.2113	0.592537
$359$	5.42967	0.286567	0.143284	−	0.989682i	$-0.454234\pi$
$359$	5.42967	0.286567	0.143284	+	0.989682i	$0.454234\pi$
$360$	0	0
$361$	−18.3324	−0.964864
$362$	−19.5748	−1.02883
$363$	0	0
$364$	−12.6175	−0.661335
$365$	4.76808	0.249572
$366$	0	0
$367$	−16.5477	−0.863785	−0.431893	−	0.901925i	$-0.642154\pi$
$367$	−16.5477	−0.863785	−0.431893	+	0.901925i	$0.642154\pi$
$368$	−4.99504	−0.260384
$369$	0	0
$370$	29.6457	1.54121
$371$	−2.56172	−0.132998
$372$	0	0
$373$	−13.9760	−0.723647	−0.361824	−	0.932247i	$-0.617846\pi$
$373$	−13.9760	−0.723647	−0.361824	+	0.932247i	$0.617846\pi$
$374$	33.2327	1.71842
$375$	0	0
$376$	3.88542	0.200375
$377$	27.5490	1.41885
$378$	0	0
$379$	16.8576	0.865918	0.432959	−	0.901414i	$-0.357469\pi$
$379$	16.8576	0.865918	0.432959	+	0.901414i	$0.357469\pi$
$380$	8.26013	0.423736
$381$	0	0
$382$	−11.6480	−0.595963
$383$	−18.0175	−0.920651	−0.460325	−	0.887750i	$-0.652267\pi$
$383$	−18.0175	−0.920651	−0.460325	+	0.887750i	$0.652267\pi$
$384$	0	0
$385$	−23.7509	−1.21046
$386$	−16.1836	−0.823725
$387$	0	0
$388$	31.4932	1.59882
$389$	5.73601	0.290827	0.145414	−	0.989371i	$-0.453549\pi$
$389$	5.73601	0.290827	0.145414	+	0.989371i	$0.453549\pi$
$390$	0	0
$391$	9.70220	0.490661
$392$	8.61839	0.435294
$393$	0	0
$394$	−53.7207	−2.70641
$395$	−32.8836	−1.65455
$396$	0	0
$397$	7.03379	0.353016	0.176508	−	0.984299i	$-0.443520\pi$
$397$	7.03379	0.353016	0.176508	+	0.984299i	$0.443520\pi$
$398$	−3.28102	−0.164463
$399$	0	0
$400$	−13.7614	−0.688072
$401$	35.2637	1.76098	0.880492	−	0.474061i	$-0.157212\pi$
$401$	35.2637	1.76098	0.880492	+	0.474061i	$0.157212\pi$
$402$	0	0
$403$	−1.96134	−0.0977014
$404$	32.4022	1.61207
$405$	0	0
$406$	28.7586	1.42727
$407$	16.9449	0.839927
$408$	0	0
$409$	7.67503	0.379506	0.189753	−	0.981832i	$-0.439231\pi$
$409$	7.67503	0.379506	0.189753	+	0.981832i	$0.439231\pi$
$410$	−76.9895	−3.80224
$411$	0	0
$412$	2.29412	0.113023
$413$	14.4370	0.710401
$414$	0	0
$415$	49.7404	2.44166
$416$	−22.5422	−1.10522
$417$	0	0
$418$	8.08351	0.395377
$419$	−13.7764	−0.673021	−0.336511	−	0.941680i	$-0.609247\pi$
$419$	−13.7764	−0.673021	−0.336511	+	0.941680i	$0.609247\pi$
$420$	0	0
$421$	−17.8400	−0.869469	−0.434735	−	0.900559i	$-0.643158\pi$
$421$	−17.8400	−0.869469	−0.434735	+	0.900559i	$0.643158\pi$
$422$	47.8275	2.32821
$423$	0	0
$424$	3.10549	0.150816
$425$	26.7298	1.29658
$426$	0	0
$427$	−4.08762	−0.197814
$428$	7.02475	0.339554
$429$	0	0
$430$	−0.476639	−0.0229856
$431$	41.0726	1.97840	0.989198	−	0.146583i	$-0.0468275\pi$
$431$	41.0726	1.97840	0.989198	+	0.146583i	$0.0468275\pi$
$432$	0	0
$433$	24.6306	1.18367	0.591835	−	0.806059i	$-0.298404\pi$
$433$	24.6306	1.18367	0.591835	+	0.806059i	$0.298404\pi$
$434$	−2.04746	−0.0982811
$435$	0	0
$436$	−19.4293	−0.930493
$437$	2.35996	0.112892
$438$	0	0
$439$	4.49724	0.214642	0.107321	−	0.994224i	$-0.465773\pi$
$439$	4.49724	0.214642	0.107321	+	0.994224i	$0.465773\pi$
$440$	28.7924	1.37262
$441$	0	0
$442$	22.6280	1.07630
$443$	−18.7036	−0.888634	−0.444317	−	0.895870i	$-0.646554\pi$
$443$	−18.7036	−0.888634	−0.444317	+	0.895870i	$0.646554\pi$
$444$	0	0
$445$	52.5908	2.49304
$446$	−53.5425	−2.53531
$447$	0	0
$448$	−18.4737	−0.872803
$449$	−15.0527	−0.710382	−0.355191	−	0.934794i	$-0.615584\pi$
$449$	−15.0527	−0.710382	−0.355191	+	0.934794i	$0.615584\pi$
$450$	0	0
$451$	−44.0057	−2.07215
$452$	10.3336	0.486051
$453$	0	0
$454$	−12.5189	−0.587542
$455$	−16.1718	−0.758148
$456$	0	0
$457$	−35.7287	−1.67132	−0.835658	−	0.549250i	$-0.814913\pi$
$457$	−35.7287	−1.67132	−0.835658	+	0.549250i	$0.814913\pi$
$458$	−56.9318	−2.66025
$459$	0	0
$460$	29.1999	1.36145
$461$	18.2577	0.850344	0.425172	−	0.905112i	$-0.360214\pi$
$461$	18.2577	0.850344	0.425172	+	0.905112i	$0.360214\pi$
$462$	0	0
$463$	−1.63930	−0.0761848	−0.0380924	−	0.999274i	$-0.512128\pi$
$463$	−1.63930	−0.0761848	−0.0380924	+	0.999274i	$0.512128\pi$
$464$	15.5087	0.719972
$465$	0	0
$466$	10.1935	0.472206
$467$	−35.2825	−1.63268	−0.816339	−	0.577573i	$-0.804000\pi$
$467$	−35.2825	−1.63268	−0.816339	+	0.577573i	$0.804000\pi$
$468$	0	0
$469$	−0.461648	−0.0213169
$470$	17.2992	0.797951
$471$	0	0
$472$	−17.5016	−0.805574
$473$	−0.272437	−0.0125267
$474$	0	0
$475$	6.50173	0.298320
$476$	13.7966	0.632366
$477$	0	0
$478$	24.6488	1.12741
$479$	22.4814	1.02720	0.513600	−	0.858030i	$-0.328312\pi$
$479$	22.4814	1.02720	0.513600	+	0.858030i	$0.328312\pi$
$480$	0	0
$481$	11.5377	0.526073
$482$	48.3987	2.20450
$483$	0	0
$484$	26.2748	1.19431
$485$	40.3649	1.83287
$486$	0	0
$487$	40.2579	1.82426	0.912130	−	0.409900i	$-0.134437\pi$
$487$	40.2579	1.82426	0.912130	+	0.409900i	$0.134437\pi$
$488$	4.95529	0.224315
$489$	0	0
$490$	38.3719	1.73347
$491$	7.30514	0.329676	0.164838	−	0.986321i	$-0.447290\pi$
$491$	7.30514	0.329676	0.164838	+	0.986321i	$0.447290\pi$
$492$	0	0
$493$	−30.1235	−1.35670
$494$	5.50402	0.247638
$495$	0	0
$496$	−1.10413	−0.0495771
$497$	0.0867373	0.00389070
$498$	0	0
$499$	44.0051	1.96994	0.984970	−	0.172727i	$-0.0552577\pi$
$499$	44.0051	1.96994	0.984970	+	0.172727i	$0.0552577\pi$
$500$	29.8985	1.33710
$501$	0	0
$502$	56.1855	2.50768
$503$	−32.1920	−1.43537	−0.717685	−	0.696368i	$-0.754798\pi$
$503$	−32.1920	−1.43537	−0.717685	+	0.696368i	$0.754798\pi$
$504$	0	0
$505$	41.5300	1.84806
$506$	28.5755	1.27034
$507$	0	0
$508$	46.9905	2.08487
$509$	9.07758	0.402357	0.201178	−	0.979555i	$-0.435523\pi$
$509$	9.07758	0.402357	0.201178	+	0.979555i	$0.435523\pi$
$510$	0	0
$511$	1.93714	0.0856942
$512$	−18.8220	−0.831824
$513$	0	0
$514$	37.4586	1.65223
$515$	2.94038	0.129568
$516$	0	0
$517$	9.88785	0.434867
$518$	12.0443	0.529195
$519$	0	0
$520$	19.6046	0.859718
$521$	−12.5012	−0.547687	−0.273844	−	0.961774i	$-0.588295\pi$
$521$	−12.5012	−0.547687	−0.273844	+	0.961774i	$0.588295\pi$
$522$	0	0
$523$	25.0816	1.09674	0.548370	−	0.836236i	$-0.315249\pi$
$523$	25.0816	1.09674	0.548370	+	0.836236i	$0.315249\pi$
$524$	39.0761	1.70705
$525$	0	0
$526$	0.564413	0.0246096
$527$	2.14463	0.0934217
$528$	0	0
$529$	−14.6575	−0.637281
$530$	13.8267	0.600592
$531$	0	0
$532$	3.35588	0.145496
$533$	−29.9632	−1.29785
$534$	0	0
$535$	9.00364	0.389261
$536$	0.559640	0.0241728
$537$	0	0
$538$	13.3355	0.574933
$539$	21.9326	0.944704
$540$	0	0
$541$	−31.5409	−1.35605	−0.678025	−	0.735039i	$-0.737164\pi$
$541$	−31.5409	−1.35605	−0.678025	+	0.735039i	$0.737164\pi$
$542$	24.3567	1.04621
$543$	0	0
$544$	24.6488	1.05681
$545$	−24.9025	−1.06671
$546$	0	0
$547$	−28.2864	−1.20944	−0.604720	−	0.796438i	$-0.706715\pi$
$547$	−28.2864	−1.20944	−0.604720	+	0.796438i	$0.706715\pi$
$548$	14.2281	0.607795
$549$	0	0
$550$	78.7262	3.35689
$551$	−7.32723	−0.312150
$552$	0	0
$553$	−13.3597	−0.568114
$554$	11.2938	0.479827
$555$	0	0
$556$	17.1289	0.726428
$557$	−18.8212	−0.797478	−0.398739	−	0.917064i	$-0.630552\pi$
$557$	−18.8212	−0.797478	−0.398739	+	0.917064i	$0.630552\pi$
$558$	0	0
$559$	−0.185501	−0.00784587
$560$	−9.10391	−0.384711
$561$	0	0
$562$	22.6346	0.954785
$563$	−4.58170	−0.193096	−0.0965479	−	0.995328i	$-0.530780\pi$
$563$	−4.58170	−0.193096	−0.0965479	+	0.995328i	$0.530780\pi$
$564$	0	0
$565$	13.2446	0.557204
$566$	−34.3567	−1.44412
$567$	0	0
$568$	−0.105149	−0.00441194
$569$	25.4927	1.06871	0.534354	−	0.845261i	$-0.320555\pi$
$569$	25.4927	1.06871	0.534354	+	0.845261i	$0.320555\pi$
$570$	0	0
$571$	28.8727	1.20829	0.604143	−	0.796876i	$-0.293516\pi$
$571$	28.8727	1.20829	0.604143	+	0.796876i	$0.293516\pi$
$572$	38.9255	1.62756
$573$	0	0
$574$	−31.2788	−1.30555
$575$	22.9839	0.958494
$576$	0	0
$577$	−4.71781	−0.196405	−0.0982024	−	0.995166i	$-0.531309\pi$
$577$	−4.71781	−0.196405	−0.0982024	+	0.995166i	$0.531309\pi$
$578$	12.5354	0.521404
$579$	0	0
$580$	−90.6603	−3.76446
$581$	20.2082	0.838379
$582$	0	0
$583$	7.90305	0.327311
$584$	−2.34834	−0.0971748
$585$	0	0
$586$	−29.0825	−1.20139
$587$	−47.5488	−1.96255	−0.981274	−	0.192616i	$-0.938303\pi$
$587$	−47.5488	−1.96255	−0.981274	+	0.192616i	$0.938303\pi$
$588$	0	0
$589$	0.521659	0.0214946
$590$	−77.9227	−3.20803
$591$	0	0
$592$	6.49512	0.266948
$593$	−31.6796	−1.30093	−0.650463	−	0.759538i	$-0.725425\pi$
$593$	−31.6796	−1.30093	−0.650463	+	0.759538i	$0.725425\pi$
$594$	0	0
$595$	17.6831	0.724938
$596$	−4.15454	−0.170177
$597$	0	0
$598$	19.4569	0.795654
$599$	32.3200	1.32056	0.660281	−	0.751019i	$-0.270437\pi$
$599$	32.3200	1.32056	0.660281	+	0.751019i	$0.270437\pi$
$600$	0	0
$601$	30.3351	1.23740	0.618698	−	0.785629i	$-0.287660\pi$
$601$	30.3351	1.23740	0.618698	+	0.785629i	$0.287660\pi$
$602$	−0.193646	−0.00789242
$603$	0	0
$604$	40.4689	1.64666
$605$	33.6765	1.36914
$606$	0	0
$607$	5.19753	0.210961	0.105481	−	0.994421i	$-0.466362\pi$
$607$	5.19753	0.210961	0.105481	+	0.994421i	$0.466362\pi$
$608$	5.99557	0.243152
$609$	0	0
$610$	22.0626	0.893287
$611$	6.73259	0.272371
$612$	0	0
$613$	0.0736394	0.00297427	0.00148713	−	0.999999i	$-0.499527\pi$
$613$	0.0736394	0.00297427	0.00148713	+	0.999999i	$0.499527\pi$
$614$	−13.5482	−0.546761
$615$	0	0
$616$	11.6976	0.471309
$617$	−34.0533	−1.37093	−0.685467	−	0.728104i	$-0.740402\pi$
$617$	−34.0533	−1.37093	−0.685467	+	0.728104i	$0.740402\pi$
$618$	0	0
$619$	−6.83279	−0.274633	−0.137317	−	0.990527i	$-0.543848\pi$
$619$	−6.83279	−0.274633	−0.137317	+	0.990527i	$0.543848\pi$
$620$	6.45452	0.259220
$621$	0	0
$622$	21.6955	0.869909
$623$	21.3663	0.856021
$624$	0	0
$625$	−1.46621	−0.0586483
$626$	−14.8342	−0.592893
$627$	0	0
$628$	−49.2746	−1.96627
$629$	−12.6159	−0.503029
$630$	0	0
$631$	41.2309	1.64138	0.820689	−	0.571376i	$-0.193590\pi$
$631$	41.2309	1.64138	0.820689	+	0.571376i	$0.193590\pi$
$632$	16.1956	0.644225
$633$	0	0
$634$	−52.1025	−2.06925
$635$	60.2279	2.39007
$636$	0	0
$637$	14.9338	0.591699
$638$	−88.7217	−3.51253
$639$	0	0
$640$	46.8822	1.85318
$641$	−44.4518	−1.75574	−0.877871	−	0.478897i	$-0.841037\pi$
$641$	−44.4518	−1.75574	−0.877871	+	0.478897i	$0.841037\pi$
$642$	0	0
$643$	−45.0610	−1.77703	−0.888516	−	0.458846i	$-0.848263\pi$
$643$	−45.0610	−1.77703	−0.888516	+	0.458846i	$0.848263\pi$
$644$	11.8632	0.467474
$645$	0	0
$646$	−6.01838	−0.236790
$647$	3.51356	0.138132	0.0690661	−	0.997612i	$-0.477998\pi$
$647$	3.51356	0.138132	0.0690661	+	0.997612i	$0.477998\pi$
$648$	0	0
$649$	−44.5390	−1.74831
$650$	53.6043	2.10253
$651$	0	0
$652$	42.9686	1.68278
$653$	−33.7610	−1.32117	−0.660584	−	0.750752i	$-0.729691\pi$
$653$	−33.7610	−1.32117	−0.660584	+	0.750752i	$0.729691\pi$
$654$	0	0
$655$	50.0840	1.95694
$656$	−16.8678	−0.658575
$657$	0	0
$658$	7.02820	0.273988
$659$	−17.9517	−0.699298	−0.349649	−	0.936881i	$-0.613699\pi$
$659$	−17.9517	−0.699298	−0.349649	+	0.936881i	$0.613699\pi$
$660$	0	0
$661$	−40.7402	−1.58461	−0.792306	−	0.610124i	$-0.791120\pi$
$661$	−40.7402	−1.58461	−0.792306	+	0.610124i	$0.791120\pi$
$662$	−55.7215	−2.16568
$663$	0	0
$664$	−24.4978	−0.950698
$665$	4.30124	0.166795
$666$	0	0
$667$	−25.9020	−1.00293
$668$	40.0738	1.55050
$669$	0	0
$670$	2.49170	0.0962628
$671$	12.6105	0.486823
$672$	0	0
$673$	−8.44256	−0.325437	−0.162718	−	0.986673i	$-0.552026\pi$
$673$	−8.44256	−0.325437	−0.162718	+	0.986673i	$0.552026\pi$
$674$	23.7024	0.912983
$675$	0	0
$676$	−10.0061	−0.384851
$677$	29.0735	1.11738	0.558692	−	0.829375i	$-0.311303\pi$
$677$	29.0735	1.11738	0.558692	+	0.829375i	$0.311303\pi$
$678$	0	0
$679$	16.3992	0.629343
$680$	−21.4367	−0.822059
$681$	0	0
$682$	6.31651	0.241872
$683$	−13.8900	−0.531487	−0.265744	−	0.964044i	$-0.585617\pi$
$683$	−13.8900	−0.531487	−0.265744	+	0.964044i	$0.585617\pi$
$684$	0	0
$685$	18.2362	0.696770
$686$	38.0377	1.45228
$687$	0	0
$688$	−0.104428	−0.00398127
$689$	5.38115	0.205005
$690$	0	0
$691$	−11.3494	−0.431753	−0.215877	−	0.976421i	$-0.569261\pi$
$691$	−11.3494	−0.431753	−0.215877	+	0.976421i	$0.569261\pi$
$692$	48.2785	1.83527
$693$	0	0
$694$	−43.2537	−1.64189
$695$	21.9542	0.832770
$696$	0	0
$697$	32.7634	1.24100
$698$	6.79022	0.257014
$699$	0	0
$700$	32.6832	1.23531
$701$	−13.9226	−0.525849	−0.262925	−	0.964816i	$-0.584687\pi$
$701$	−13.9226	−0.525849	−0.262925	+	0.964816i	$0.584687\pi$
$702$	0	0
$703$	−3.06869	−0.115738
$704$	56.9924	2.14798
$705$	0	0
$706$	13.5038	0.508222
$707$	16.8726	0.634558
$708$	0	0
$709$	15.6649	0.588308	0.294154	−	0.955758i	$-0.404962\pi$
$709$	15.6649	0.588308	0.294154	+	0.955758i	$0.404962\pi$
$710$	−0.468157	−0.0175696
$711$	0	0
$712$	−25.9016	−0.970704
$713$	1.84409	0.0690616
$714$	0	0
$715$	49.8909	1.86582
$716$	−14.3590	−0.536622
$717$	0	0
$718$	−11.9063	−0.444340
$719$	50.5418	1.88489	0.942445	−	0.334360i	$-0.108520\pi$
$719$	50.5418	1.88489	0.942445	+	0.334360i	$0.108520\pi$
$720$	0	0
$721$	1.19460	0.0444892
$722$	40.1998	1.49608
$723$	0	0
$724$	25.0706	0.931743
$725$	−71.3607	−2.65027
$726$	0	0
$727$	1.18294	0.0438729	0.0219365	−	0.999759i	$-0.493017\pi$
$727$	1.18294	0.0438729	0.0219365	+	0.999759i	$0.493017\pi$
$728$	7.96483	0.295196
$729$	0	0
$730$	−10.4556	−0.386978
$731$	0.202837	0.00750219
$732$	0	0
$733$	−28.7734	−1.06277	−0.531385	−	0.847130i	$-0.678328\pi$
$733$	−28.7734	−1.06277	−0.531385	+	0.847130i	$0.678328\pi$
$734$	36.2863	1.33935
$735$	0	0
$736$	21.1946	0.781243
$737$	1.42421	0.0524613
$738$	0	0
$739$	14.1618	0.520950	0.260475	−	0.965481i	$-0.416121\pi$
$739$	14.1618	0.520950	0.260475	+	0.965481i	$0.416121\pi$
$740$	−37.9691	−1.39577
$741$	0	0
$742$	5.61742	0.206222
$743$	−5.71199	−0.209552	−0.104776	−	0.994496i	$-0.533413\pi$
$743$	−5.71199	−0.209552	−0.104776	+	0.994496i	$0.533413\pi$
$744$	0	0
$745$	−5.32489	−0.195089
$746$	30.6468	1.12206
$747$	0	0
$748$	−42.5632	−1.55626
$749$	3.65795	0.133658
$750$	0	0
$751$	11.6990	0.426902	0.213451	−	0.976954i	$-0.431530\pi$
$751$	11.6990	0.426902	0.213451	+	0.976954i	$0.431530\pi$
$752$	3.79010	0.138211
$753$	0	0
$754$	−60.4102	−2.20001
$755$	51.8691	1.88771
$756$	0	0
$757$	−0.245448	−0.00892097	−0.00446049	−	0.999990i	$-0.501420\pi$
$757$	−0.245448	−0.00892097	−0.00446049	+	0.999990i	$0.501420\pi$
$758$	−36.9659	−1.34266
$759$	0	0
$760$	−5.21425	−0.189141
$761$	12.0984	0.438567	0.219284	−	0.975661i	$-0.429628\pi$
$761$	12.0984	0.438567	0.219284	+	0.975661i	$0.429628\pi$
$762$	0	0
$763$	−10.1173	−0.366269
$764$	14.9183	0.539725
$765$	0	0
$766$	39.5092	1.42753
$767$	−30.3264	−1.09502
$768$	0	0
$769$	8.50282	0.306620	0.153310	−	0.988178i	$-0.451007\pi$
$769$	8.50282	0.306620	0.153310	+	0.988178i	$0.451007\pi$
$770$	52.0815	1.87689
$771$	0	0
$772$	20.7274	0.745994
$773$	−38.5995	−1.38833	−0.694165	−	0.719816i	$-0.744226\pi$
$773$	−38.5995	−1.38833	−0.694165	+	0.719816i	$0.744226\pi$
$774$	0	0
$775$	5.08050	0.182497
$776$	−19.8802	−0.713658
$777$	0	0
$778$	−12.5781	−0.450945
$779$	7.96934	0.285531
$780$	0	0
$781$	−0.267589	−0.00957509
$782$	−21.2752	−0.760801
$783$	0	0
$784$	8.40696	0.300248
$785$	−63.1554	−2.25411
$786$	0	0
$787$	37.8537	1.34934	0.674669	−	0.738120i	$-0.264286\pi$
$787$	37.8537	1.34934	0.674669	+	0.738120i	$0.264286\pi$
$788$	68.8034	2.45102
$789$	0	0
$790$	72.1079	2.56549
$791$	5.38093	0.191324
$792$	0	0
$793$	8.58644	0.304913
$794$	−15.4239	−0.547373
$795$	0	0
$796$	4.20221	0.148943
$797$	−25.3928	−0.899458	−0.449729	−	0.893165i	$-0.648479\pi$
$797$	−25.3928	−0.899458	−0.449729	+	0.893165i	$0.648479\pi$
$798$	0	0
$799$	−7.36177	−0.260441
$800$	58.3915	2.06445
$801$	0	0
$802$	−77.3271	−2.73052
$803$	−5.97619	−0.210895
$804$	0	0
$805$	15.2050	0.535908
$806$	4.30088	0.151492
$807$	0	0
$808$	−20.4541	−0.719571
$809$	28.9046	1.01623	0.508115	−	0.861289i	$-0.330342\pi$
$809$	28.9046	1.01623	0.508115	+	0.861289i	$0.330342\pi$
$810$	0	0
$811$	19.9649	0.701062	0.350531	−	0.936551i	$-0.386001\pi$
$811$	19.9649	0.701062	0.350531	+	0.936551i	$0.386001\pi$
$812$	−36.8329	−1.29258
$813$	0	0
$814$	−37.1572	−1.30236
$815$	55.0730	1.92912
$816$	0	0
$817$	0.0493379	0.00172611
$818$	−16.8300	−0.588447
$819$	0	0
$820$	98.6052	3.44344
$821$	4.50331	0.157166	0.0785832	−	0.996908i	$-0.474960\pi$
$821$	4.50331	0.157166	0.0785832	+	0.996908i	$0.474960\pi$
$822$	0	0
$823$	20.8243	0.725889	0.362944	−	0.931811i	$-0.381771\pi$
$823$	20.8243	0.725889	0.362944	+	0.931811i	$0.381771\pi$
$824$	−1.44817	−0.0504495
$825$	0	0
$826$	−31.6579	−1.10152
$827$	−47.6294	−1.65624	−0.828119	−	0.560553i	$-0.810589\pi$
$827$	−47.6294	−1.65624	−0.828119	+	0.560553i	$0.810589\pi$
$828$	0	0
$829$	−35.7398	−1.24129	−0.620647	−	0.784090i	$-0.713130\pi$
$829$	−35.7398	−1.24129	−0.620647	+	0.784090i	$0.713130\pi$
$830$	−109.072	−3.78595
$831$	0	0
$832$	38.8059	1.34535
$833$	−16.3294	−0.565780
$834$	0	0
$835$	51.3627	1.77748
$836$	−10.3530	−0.358067
$837$	0	0
$838$	30.2093	1.04356
$839$	−15.3417	−0.529655	−0.264828	−	0.964296i	$-0.585315\pi$
$839$	−15.3417	−0.529655	−0.264828	+	0.964296i	$0.585315\pi$
$840$	0	0
$841$	51.4210	1.77314
$842$	39.1201	1.34817
$843$	0	0
$844$	−61.2556	−2.10851
$845$	−12.8249	−0.441190
$846$	0	0
$847$	13.6819	0.470114
$848$	3.02931	0.104027
$849$	0	0
$850$	−58.6137	−2.01043
$851$	−10.8479	−0.371862
$852$	0	0
$853$	1.96446	0.0672617	0.0336309	−	0.999434i	$-0.489293\pi$
$853$	1.96446	0.0672617	0.0336309	+	0.999434i	$0.489293\pi$
$854$	8.96344	0.306723
$855$	0	0
$856$	−4.43441	−0.151565
$857$	18.6525	0.637156	0.318578	−	0.947897i	$-0.396795\pi$
$857$	18.6525	0.637156	0.318578	+	0.947897i	$0.396795\pi$
$858$	0	0
$859$	8.34193	0.284623	0.142311	−	0.989822i	$-0.454547\pi$
$859$	8.34193	0.284623	0.142311	+	0.989822i	$0.454547\pi$
$860$	0.610461	0.0208165
$861$	0	0
$862$	−90.0650	−3.06763
$863$	−13.5866	−0.462492	−0.231246	−	0.972895i	$-0.574280\pi$
$863$	−13.5866	−0.462492	−0.231246	+	0.972895i	$0.574280\pi$
$864$	0	0
$865$	61.8787	2.10394
$866$	−54.0106	−1.83535
$867$	0	0
$868$	2.62230	0.0890068
$869$	41.2154	1.39814
$870$	0	0
$871$	0.969735	0.0328582
$872$	12.2648	0.415339
$873$	0	0
$874$	−5.17498	−0.175046
$875$	15.5688	0.526323
$876$	0	0
$877$	54.3712	1.83598	0.917992	−	0.396599i	$-0.129810\pi$
$877$	54.3712	1.83598	0.917992	+	0.396599i	$0.129810\pi$
$878$	−9.86167	−0.332815
$879$	0	0
$880$	28.0860	0.946780
$881$	46.7694	1.57570	0.787851	−	0.615865i	$-0.211193\pi$
$881$	46.7694	1.57570	0.787851	+	0.615865i	$0.211193\pi$
$882$	0	0
$883$	−3.75942	−0.126515	−0.0632573	−	0.997997i	$-0.520149\pi$
$883$	−3.75942	−0.126515	−0.0632573	+	0.997997i	$0.520149\pi$
$884$	−28.9811	−0.974739
$885$	0	0
$886$	41.0137	1.37788
$887$	26.6430	0.894583	0.447291	−	0.894388i	$-0.352389\pi$
$887$	26.6430	0.894583	0.447291	+	0.894388i	$0.352389\pi$
$888$	0	0
$889$	24.4690	0.820664
$890$	−115.322	−3.86562
$891$	0	0
$892$	68.5751	2.29607
$893$	−1.79067	−0.0599225
$894$	0	0
$895$	−18.4040	−0.615178
$896$	19.0470	0.636316
$897$	0	0
$898$	33.0080	1.10149
$899$	−5.72555	−0.190958
$900$	0	0
$901$	−5.88403	−0.196025
$902$	96.4968	3.21299
$903$	0	0
$904$	−6.52312	−0.216956
$905$	32.1331	1.06814
$906$	0	0
$907$	29.9123	0.993223	0.496612	−	0.867973i	$-0.334577\pi$
$907$	29.9123	0.993223	0.496612	+	0.867973i	$0.334577\pi$
$908$	16.0337	0.532098
$909$	0	0
$910$	35.4620	1.17556
$911$	7.51929	0.249125	0.124563	−	0.992212i	$-0.460247\pi$
$911$	7.51929	0.249125	0.124563	+	0.992212i	$0.460247\pi$
$912$	0	0
$913$	−62.3434	−2.06326
$914$	78.3467	2.59148
$915$	0	0
$916$	72.9160	2.40921
$917$	20.3478	0.671944
$918$	0	0
$919$	23.6445	0.779959	0.389980	−	0.920824i	$-0.372482\pi$
$919$	23.6445	0.779959	0.389980	+	0.920824i	$0.372482\pi$
$920$	−18.4326	−0.607704
$921$	0	0
$922$	−40.0359	−1.31851
$923$	−0.182200	−0.00599719
$924$	0	0
$925$	−29.8863	−0.982655
$926$	3.59470	0.118129
$927$	0	0
$928$	−65.8052	−2.16016
$929$	−22.6402	−0.742802	−0.371401	−	0.928473i	$-0.621122\pi$
$929$	−22.6402	−0.742802	−0.371401	+	0.928473i	$0.621122\pi$
$930$	0	0
$931$	−3.97195	−0.130175
$932$	−13.0555	−0.427647
$933$	0	0
$934$	77.3683	2.53157
$935$	−54.5534	−1.78409
$936$	0	0
$937$	36.5849	1.19518	0.597588	−	0.801803i	$-0.296126\pi$
$937$	36.5849	1.19518	0.597588	+	0.801803i	$0.296126\pi$
$938$	1.01231	0.0330532
$939$	0	0
$940$	−22.1561	−0.722652
$941$	−21.7036	−0.707517	−0.353758	−	0.935337i	$-0.615097\pi$
$941$	−21.7036	−0.707517	−0.353758	+	0.935337i	$0.615097\pi$
$942$	0	0
$943$	28.1720	0.917405
$944$	−17.0722	−0.555653
$945$	0	0
$946$	0.597408	0.0194234
$947$	−44.8926	−1.45881	−0.729406	−	0.684081i	$-0.760204\pi$
$947$	−44.8926	−1.45881	−0.729406	+	0.684081i	$0.760204\pi$
$948$	0	0
$949$	−4.06916	−0.132090
$950$	−14.2572	−0.462563
$951$	0	0
$952$	−8.70916	−0.282266
$953$	42.6264	1.38080	0.690402	−	0.723426i	$-0.257434\pi$
$953$	42.6264	1.38080	0.690402	+	0.723426i	$0.257434\pi$
$954$	0	0
$955$	19.1208	0.618735
$956$	−31.5693	−1.02102
$957$	0	0
$958$	−49.2977	−1.59274
$959$	7.40890	0.239246
$960$	0	0
$961$	−30.5924	−0.986851
$962$	−25.3001	−0.815709
$963$	0	0
$964$	−61.9872	−1.99647
$965$	26.5663	0.855201
$966$	0	0
$967$	−57.5467	−1.85058	−0.925288	−	0.379265i	$-0.876177\pi$
$967$	−57.5467	−1.85058	−0.925288	+	0.379265i	$0.876177\pi$
$968$	−16.5861	−0.533097
$969$	0	0
$970$	−88.5132	−2.84199
$971$	39.3544	1.26294	0.631471	−	0.775400i	$-0.282452\pi$
$971$	39.3544	1.26294	0.631471	+	0.775400i	$0.282452\pi$
$972$	0	0
$973$	8.91941	0.285943
$974$	−88.2786	−2.82863
$975$	0	0
$976$	4.83372	0.154724
$977$	−10.5831	−0.338583	−0.169291	−	0.985566i	$-0.554148\pi$
$977$	−10.5831	−0.338583	−0.169291	+	0.985566i	$0.554148\pi$
$978$	0	0
$979$	−65.9160	−2.10668
$980$	−49.1452	−1.56989
$981$	0	0
$982$	−16.0189	−0.511184
$983$	−20.5667	−0.655977	−0.327988	−	0.944682i	$-0.606371\pi$
$983$	−20.5667	−0.655977	−0.327988	+	0.944682i	$0.606371\pi$
$984$	0	0
$985$	88.1855	2.80982
$986$	66.0556	2.10364
$987$	0	0
$988$	−7.04934	−0.224269
$989$	0.174411	0.00554596
$990$	0	0
$991$	54.1180	1.71912	0.859558	−	0.511039i	$-0.170739\pi$
$991$	54.1180	1.71912	0.859558	+	0.511039i	$0.170739\pi$
$992$	4.68498	0.148748
$993$	0	0
$994$	−0.190200	−0.00603277
$995$	5.38598	0.170747
$996$	0	0
$997$	−22.9651	−0.727312	−0.363656	−	0.931533i	$-0.618472\pi$
$997$	−22.9651	−0.727312	−0.363656	+	0.931533i	$0.618472\pi$
$998$	−96.4956	−3.05451
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6561.2.a.d.1.7		72
3.2	odd	2		6561.2.a.c.1.66		72
81.2	odd	54		729.2.g.c.433.2		144
81.13	even	27		729.2.g.a.541.2		144
81.14	odd	54		81.2.g.a.34.7	yes	144
81.25	even	27		729.2.g.a.190.2		144
81.29	odd	54		81.2.g.a.31.7	✓	144
81.40	even	27		729.2.g.b.298.7		144
81.41	odd	54		729.2.g.c.298.2		144
81.52	even	27		243.2.g.a.145.2		144
81.56	odd	54		729.2.g.d.190.7		144
81.67	even	27		243.2.g.a.181.2		144
81.68	odd	54		729.2.g.d.541.7		144
81.79	even	27		729.2.g.b.433.7		144

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
81.2.g.a.31.7	✓	144	81.29	odd	54
81.2.g.a.34.7	yes	144	81.14	odd	54
243.2.g.a.145.2		144	81.52	even	27
243.2.g.a.181.2		144	81.67	even	27
729.2.g.a.190.2		144	81.25	even	27
729.2.g.a.541.2		144	81.13	even	27
729.2.g.b.298.7		144	81.40	even	27
729.2.g.b.433.7		144	81.79	even	27
729.2.g.c.298.2		144	81.41	odd	54
729.2.g.c.433.2		144	81.2	odd	54
729.2.g.d.190.7		144	81.56	odd	54
729.2.g.d.541.7		144	81.68	odd	54
6561.2.a.c.1.66		72	3.2	odd	2
6561.2.a.d.1.7		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 $\F_{q}$

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

\(f(q)\)	\(=\)	\(q-2.19283 q^{2} +2.80849 q^{4} +3.59965 q^{5} +1.46244 q^{7} -1.77287 q^{8} +O(q^{10})\)	\(q-2.19283 q^{2} +2.80849 q^{4} +3.59965 q^{5} +1.46244 q^{7} -1.77287 q^{8} -7.89340 q^{10} -4.51171 q^{11} -3.07200 q^{13} -3.20688 q^{14} -1.72938 q^{16} +3.35908 q^{17} +0.817062 q^{19} +10.1096 q^{20} +9.89339 q^{22} +2.88835 q^{23} +7.95745 q^{25} +6.73636 q^{26} +4.10725 q^{28} -8.96778 q^{29} +0.638458 q^{31} +7.33796 q^{32} -7.36589 q^{34} +5.26427 q^{35} -3.75576 q^{37} -1.79167 q^{38} -6.38170 q^{40} +9.75366 q^{41} +0.0603845 q^{43} -12.6711 q^{44} -6.33364 q^{46} -2.19160 q^{47} -4.86126 q^{49} -17.4493 q^{50} -8.62767 q^{52} -1.75168 q^{53} -16.2405 q^{55} -2.59272 q^{56} +19.6648 q^{58} +9.87188 q^{59} -2.79506 q^{61} -1.40003 q^{62} -12.6321 q^{64} -11.0581 q^{65} -0.315669 q^{67} +9.43394 q^{68} -11.5436 q^{70} +0.0593099 q^{71} +1.32460 q^{73} +8.23573 q^{74} +2.29471 q^{76} -6.59811 q^{77} -9.13522 q^{79} -6.22515 q^{80} -21.3881 q^{82} +13.8181 q^{83} +12.0915 q^{85} -0.132413 q^{86} +7.99867 q^{88} +14.6100 q^{89} -4.49262 q^{91} +8.11188 q^{92} +4.80580 q^{94} +2.94113 q^{95} +11.2136 q^{97} +10.6599 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 72 q + 9 q^{2} + 63 q^{4} + 18 q^{5} + 27 q^{8}+O(q^{10}) \) 72 * q + 9 * q^2 + 63 * q^4 + 18 * q^5 + 27 * q^8	\( 72 q + 9 q^{2} + 63 q^{4} + 18 q^{5} + 27 q^{8} + 36 q^{11} + 36 q^{14} + 45 q^{16} + 36 q^{17} + 54 q^{20} + 54 q^{23} + 36 q^{25} + 45 q^{26} + 9 q^{28} + 54 q^{29} + 63 q^{32} + 72 q^{35} + 54 q^{38} + 72 q^{41} + 90 q^{44} + 90 q^{47} + 18 q^{49} + 45 q^{50} + 45 q^{53} + 9 q^{55} + 108 q^{56} + 18 q^{58} + 108 q^{59} + 72 q^{62} + 9 q^{64} + 72 q^{65} + 108 q^{68} + 126 q^{71} + 90 q^{74} + 72 q^{77} + 144 q^{80} - 18 q^{82} + 108 q^{83} + 90 q^{86} + 108 q^{89} + 72 q^{92} + 144 q^{95} + 81 q^{98}+O(q^{100}) \) 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

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