LMFDB - Embedded newform 6561.2.a.d.1.12

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			1.12
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)

$f(q)$	$=$	$q-2.01118 q^{2} +2.04483 q^{4} +3.50138 q^{5} +2.92740 q^{7} -0.0901573 q^{8} +O(q^{10})$	\(q-2.01118 q^{2} +2.04483 q^{4} +3.50138 q^{5} +2.92740 q^{7} -0.0901573 q^{8} -7.04188 q^{10} +1.54329 q^{11} +2.12692 q^{13} -5.88751 q^{14} -3.90833 q^{16} +4.76079 q^{17} +6.16451 q^{19} +7.15971 q^{20} -3.10383 q^{22} -1.95430 q^{23} +7.25964 q^{25} -4.27760 q^{26} +5.98603 q^{28} +5.79565 q^{29} -0.921998 q^{31} +8.04066 q^{32} -9.57478 q^{34} +10.2499 q^{35} +3.40956 q^{37} -12.3979 q^{38} -0.315675 q^{40} -8.85830 q^{41} -4.41620 q^{43} +3.15577 q^{44} +3.93043 q^{46} +6.30194 q^{47} +1.56966 q^{49} -14.6004 q^{50} +4.34918 q^{52} +8.39886 q^{53} +5.40365 q^{55} -0.263926 q^{56} -11.6561 q^{58} -2.25990 q^{59} -13.8080 q^{61} +1.85430 q^{62} -8.35452 q^{64} +7.44713 q^{65} +2.01794 q^{67} +9.73499 q^{68} -20.6144 q^{70} +0.243126 q^{71} -4.64231 q^{73} -6.85723 q^{74} +12.6054 q^{76} +4.51783 q^{77} -3.17937 q^{79} -13.6846 q^{80} +17.8156 q^{82} +14.7346 q^{83} +16.6693 q^{85} +8.88176 q^{86} -0.139139 q^{88} +10.3658 q^{89} +6.22633 q^{91} -3.99620 q^{92} -12.6743 q^{94} +21.5843 q^{95} -13.2418 q^{97} -3.15687 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	−2.01118	−1.42212	−0.711058	−	0.703133i	$-0.751784\pi$
$2$	−2.01118	−1.42212	−0.711058	+	0.703133i	$0.751784\pi$
$3$	0	0
$3$	0	0
$4$	2.04483	1.02241
$5$	3.50138	1.56586	0.782932	−	0.622108i	$-0.213723\pi$
$5$	3.50138	1.56586	0.782932	+	0.622108i	$0.213723\pi$
$6$	0	0
$7$	2.92740	1.10645	0.553226	−	0.833031i	$-0.313396\pi$
$7$	2.92740	1.10645	0.553226	+	0.833031i	$0.313396\pi$
$8$	−0.0901573	−0.0318754
$9$	0	0
$10$	−7.04188	−2.22684
$11$	1.54329	0.465320	0.232660	−	0.972558i	$-0.425257\pi$
$11$	1.54329	0.465320	0.232660	+	0.972558i	$0.425257\pi$
$12$	0	0
$13$	2.12692	0.589900	0.294950	−	0.955513i	$-0.404697\pi$
$13$	2.12692	0.589900	0.294950	+	0.955513i	$0.404697\pi$
$14$	−5.88751	−1.57350
$15$	0	0
$16$	−3.90833	−0.977084
$17$	4.76079	1.15466	0.577330	−	0.816511i	$-0.304095\pi$
$17$	4.76079	1.15466	0.577330	+	0.816511i	$0.304095\pi$
$18$	0	0
$19$	6.16451	1.41424	0.707118	−	0.707096i	$-0.249995\pi$
$19$	6.16451	1.41424	0.707118	+	0.707096i	$0.249995\pi$
$20$	7.15971	1.60096
$21$	0	0
$22$	−3.10383	−0.661739
$23$	−1.95430	−0.407499	−0.203749	−	0.979023i	$-0.565313\pi$
$23$	−1.95430	−0.407499	−0.203749	+	0.979023i	$0.565313\pi$
$24$	0	0
$25$	7.25964	1.45193
$26$	−4.27760	−0.838907
$27$	0	0
$28$	5.98603	1.13125
$29$	5.79565	1.07622	0.538112	−	0.842873i	$-0.319138\pi$
$29$	5.79565	1.07622	0.538112	+	0.842873i	$0.319138\pi$
$30$	0	0
$31$	−0.921998	−0.165596	−0.0827979	−	0.996566i	$-0.526386\pi$
$31$	−0.921998	−0.165596	−0.0827979	+	0.996566i	$0.526386\pi$
$32$	8.04066	1.42140
$33$	0	0
$34$	−9.57478	−1.64206
$35$	10.2499	1.73255
$36$	0	0
$37$	3.40956	0.560528	0.280264	−	0.959923i	$-0.409578\pi$
$37$	3.40956	0.560528	0.280264	+	0.959923i	$0.409578\pi$
$38$	−12.3979	−2.01121
$39$	0	0
$40$	−0.315675	−0.0499126
$41$	−8.85830	−1.38343	−0.691717	−	0.722168i	$-0.743146\pi$
$41$	−8.85830	−1.38343	−0.691717	+	0.722168i	$0.743146\pi$
$42$	0	0
$43$	−4.41620	−0.673465	−0.336732	−	0.941600i	$-0.609322\pi$
$43$	−4.41620	−0.673465	−0.336732	+	0.941600i	$0.609322\pi$
$44$	3.15577	0.475750
$45$	0	0
$46$	3.93043	0.579511
$47$	6.30194	0.919233	0.459616	−	0.888118i	$-0.347987\pi$
$47$	6.30194	0.919233	0.459616	+	0.888118i	$0.347987\pi$
$48$	0	0
$49$	1.56966	0.224238
$50$	−14.6004	−2.06481
$51$	0	0
$52$	4.34918	0.603122
$53$	8.39886	1.15367	0.576836	−	0.816860i	$-0.304287\pi$
$53$	8.39886	1.15367	0.576836	+	0.816860i	$0.304287\pi$
$54$	0	0
$55$	5.40365	0.728628
$56$	−0.263926	−0.0352687
$57$	0	0
$58$	−11.6561	−1.53052
$59$	−2.25990	−0.294214	−0.147107	−	0.989121i	$-0.546996\pi$
$59$	−2.25990	−0.294214	−0.147107	+	0.989121i	$0.546996\pi$
$60$	0	0
$61$	−13.8080	−1.76793	−0.883964	−	0.467556i	$-0.845135\pi$
$61$	−13.8080	−1.76793	−0.883964	+	0.467556i	$0.845135\pi$
$62$	1.85430	0.235496
$63$	0	0
$64$	−8.35452	−1.04431
$65$	7.44713	0.923703
$66$	0	0
$67$	2.01794	0.246530	0.123265	−	0.992374i	$-0.460663\pi$
$67$	2.01794	0.246530	0.123265	+	0.992374i	$0.460663\pi$
$68$	9.73499	1.18054
$69$	0	0
$70$	−20.6144	−2.46389
$71$	0.243126	0.0288537	0.0144269	−	0.999896i	$-0.495408\pi$
$71$	0.243126	0.0288537	0.0144269	+	0.999896i	$0.495408\pi$
$72$	0	0
$73$	−4.64231	−0.543341	−0.271671	−	0.962390i	$-0.587576\pi$
$73$	−4.64231	−0.543341	−0.271671	+	0.962390i	$0.587576\pi$
$74$	−6.85723	−0.797136
$75$	0	0
$76$	12.6054	1.44593
$77$	4.51783	0.514855
$78$	0	0
$79$	−3.17937	−0.357707	−0.178854	−	0.983876i	$-0.557239\pi$
$79$	−3.17937	−0.357707	−0.178854	+	0.983876i	$0.557239\pi$
$80$	−13.6846	−1.52998
$81$	0	0
$82$	17.8156	1.96740
$83$	14.7346	1.61733	0.808666	−	0.588269i	$-0.200190\pi$
$83$	14.7346	1.61733	0.808666	+	0.588269i	$0.200190\pi$
$84$	0	0
$85$	16.6693	1.80804
$86$	8.88176	0.957745
$87$	0	0
$88$	−0.139139	−0.0148323
$89$	10.3658	1.09877	0.549385	−	0.835570i	$-0.314862\pi$
$89$	10.3658	1.09877	0.549385	+	0.835570i	$0.314862\pi$
$90$	0	0
$91$	6.22633	0.652697
$92$	−3.99620	−0.416632
$93$	0	0
$94$	−12.6743	−1.30726
$95$	21.5843	2.21450
$96$	0	0
$97$	−13.2418	−1.34450	−0.672252	−	0.740323i	$-0.734673\pi$
$97$	−13.2418	−1.34450	−0.672252	+	0.740323i	$0.734673\pi$
$98$	−3.15687	−0.318892
$99$	0	0
$100$	14.8447	1.48447
$101$	−4.22553	−0.420456	−0.210228	−	0.977652i	$-0.567421\pi$
$101$	−4.22553	−0.420456	−0.210228	+	0.977652i	$0.567421\pi$
$102$	0	0
$103$	−5.92061	−0.583375	−0.291688	−	0.956514i	$-0.594217\pi$
$103$	−5.92061	−0.583375	−0.291688	+	0.956514i	$0.594217\pi$
$104$	−0.191757	−0.0188033
$105$	0	0
$106$	−16.8916	−1.64066
$107$	0.826631	0.0799134	0.0399567	−	0.999201i	$-0.487278\pi$
$107$	0.826631	0.0799134	0.0399567	+	0.999201i	$0.487278\pi$
$108$	0	0
$109$	10.0673	0.964271	0.482136	−	0.876097i	$-0.339861\pi$
$109$	10.0673	0.964271	0.482136	+	0.876097i	$0.339861\pi$
$110$	−10.8677	−1.03619
$111$	0	0
$112$	−11.4413	−1.08110
$113$	6.14275	0.577861	0.288931	−	0.957350i	$-0.406700\pi$
$113$	6.14275	0.577861	0.288931	+	0.957350i	$0.406700\pi$
$114$	0	0
$115$	−6.84272	−0.638087
$116$	11.8511	1.10035
$117$	0	0
$118$	4.54506	0.418406
$119$	13.9367	1.27758
$120$	0	0
$121$	−8.61825	−0.783477
$122$	27.7702	2.51420
$123$	0	0
$124$	−1.88533	−0.169307
$125$	7.91185	0.707657
$126$	0	0
$127$	−0.644154	−0.0571594	−0.0285797	−	0.999592i	$-0.509098\pi$
$127$	−0.644154	−0.0571594	−0.0285797	+	0.999592i	$0.509098\pi$
$128$	0.721077	0.0637348
$129$	0	0
$130$	−14.9775	−1.31361
$131$	−0.774142	−0.0676371	−0.0338186	−	0.999428i	$-0.510767\pi$
$131$	−0.774142	−0.0676371	−0.0338186	+	0.999428i	$0.510767\pi$
$132$	0	0
$133$	18.0460	1.56478
$134$	−4.05843	−0.350595
$135$	0	0
$136$	−0.429220	−0.0368053
$137$	−3.19062	−0.272593	−0.136296	−	0.990668i	$-0.543520\pi$
$137$	−3.19062	−0.272593	−0.136296	+	0.990668i	$0.543520\pi$
$138$	0	0
$139$	21.5149	1.82487	0.912436	−	0.409219i	$-0.134199\pi$
$139$	21.5149	1.82487	0.912436	+	0.409219i	$0.134199\pi$
$140$	20.9593	1.77139
$141$	0	0
$142$	−0.488968	−0.0410333
$143$	3.28245	0.274493
$144$	0	0
$145$	20.2927	1.68522
$146$	9.33651	0.772695
$147$	0	0
$148$	6.97197	0.573092
$149$	11.8603	0.971634	0.485817	−	0.874061i	$-0.338522\pi$
$149$	11.8603	0.971634	0.485817	+	0.874061i	$0.338522\pi$
$150$	0	0
$151$	−16.3994	−1.33456	−0.667281	−	0.744806i	$-0.732542\pi$
$151$	−16.3994	−1.33456	−0.667281	+	0.744806i	$0.732542\pi$
$152$	−0.555776	−0.0450794
$153$	0	0
$154$	−9.08616	−0.732183
$155$	−3.22826	−0.259300
$156$	0	0
$157$	5.64589	0.450591	0.225295	−	0.974290i	$-0.427665\pi$
$157$	5.64589	0.450591	0.225295	+	0.974290i	$0.427665\pi$
$158$	6.39428	0.508701
$159$	0	0
$160$	28.1534	2.22572
$161$	−5.72100	−0.450878
$162$	0	0
$163$	−3.70178	−0.289946	−0.144973	−	0.989436i	$-0.546310\pi$
$163$	−3.70178	−0.289946	−0.144973	+	0.989436i	$0.546310\pi$
$164$	−18.1137	−1.41444
$165$	0	0
$166$	−29.6338	−2.30003
$167$	0.291497	0.0225567	0.0112783	−	0.999936i	$-0.496410\pi$
$167$	0.291497	0.0225567	0.0112783	+	0.999936i	$0.496410\pi$
$168$	0	0
$169$	−8.47623	−0.652018
$170$	−33.5249	−2.57124
$171$	0	0
$172$	−9.03038	−0.688560
$173$	−5.52232	−0.419854	−0.209927	−	0.977717i	$-0.567323\pi$
$173$	−5.52232	−0.419854	−0.209927	+	0.977717i	$0.567323\pi$
$174$	0	0
$175$	21.2519	1.60649
$176$	−6.03170	−0.454657
$177$	0	0
$178$	−20.8474	−1.56258
$179$	−13.4804	−1.00757	−0.503787	−	0.863828i	$-0.668061\pi$
$179$	−13.4804	−1.00757	−0.503787	+	0.863828i	$0.668061\pi$
$180$	0	0
$181$	−25.1469	−1.86916	−0.934578	−	0.355758i	$-0.884223\pi$
$181$	−25.1469	−1.86916	−0.934578	+	0.355758i	$0.884223\pi$
$182$	−12.5222	−0.928211
$183$	0	0
$184$	0.176194	0.0129892
$185$	11.9382	0.877711
$186$	0	0
$187$	7.34728	0.537287
$188$	12.8864	0.939837
$189$	0	0
$190$	−43.4098	−3.14927
$191$	14.2664	1.03228	0.516141	−	0.856503i	$-0.327368\pi$
$191$	14.2664	1.03228	0.516141	+	0.856503i	$0.327368\pi$
$192$	0	0
$193$	2.86893	0.206510	0.103255	−	0.994655i	$-0.467074\pi$
$193$	2.86893	0.206510	0.103255	+	0.994655i	$0.467074\pi$
$194$	26.6316	1.91204
$195$	0	0
$196$	3.20969	0.229264
$197$	−2.17258	−0.154790	−0.0773950	−	0.997001i	$-0.524660\pi$
$197$	−2.17258	−0.154790	−0.0773950	+	0.997001i	$0.524660\pi$
$198$	0	0
$199$	−14.9636	−1.06074	−0.530369	−	0.847767i	$-0.677947\pi$
$199$	−14.9636	−1.06074	−0.530369	+	0.847767i	$0.677947\pi$
$200$	−0.654510	−0.0462808
$201$	0	0
$202$	8.49828	0.597937
$203$	16.9662	1.19079
$204$	0	0
$205$	−31.0163	−2.16627
$206$	11.9074	0.829628
$207$	0	0
$208$	−8.31270	−0.576382
$209$	9.51364	0.658072
$210$	0	0
$211$	11.9984	0.826004	0.413002	−	0.910730i	$-0.364480\pi$
$211$	11.9984	0.826004	0.413002	+	0.910730i	$0.364480\pi$
$212$	17.1742	1.17953
$213$	0	0
$214$	−1.66250	−0.113646
$215$	−15.4628	−1.05455
$216$	0	0
$217$	−2.69906	−0.183224
$218$	−20.2471	−1.37131
$219$	0	0
$220$	11.0495	0.744959
$221$	10.1258	0.681134
$222$	0	0
$223$	8.84697	0.592437	0.296218	−	0.955120i	$-0.404274\pi$
$223$	8.84697	0.592437	0.296218	+	0.955120i	$0.404274\pi$
$224$	23.5382	1.57271
$225$	0	0
$226$	−12.3541	−0.821786
$227$	−17.9339	−1.19031	−0.595156	−	0.803610i	$-0.702910\pi$
$227$	−17.9339	−1.19031	−0.595156	+	0.803610i	$0.702910\pi$
$228$	0	0
$229$	−6.26890	−0.414261	−0.207130	−	0.978313i	$-0.566412\pi$
$229$	−6.26890	−0.414261	−0.207130	+	0.978313i	$0.566412\pi$
$230$	13.7619	0.907434
$231$	0	0
$232$	−0.522520	−0.0343051
$233$	−2.24499	−0.147074	−0.0735371	−	0.997292i	$-0.523429\pi$
$233$	−2.24499	−0.147074	−0.0735371	+	0.997292i	$0.523429\pi$
$234$	0	0
$235$	22.0655	1.43939
$236$	−4.62111	−0.300809
$237$	0	0
$238$	−28.0292	−1.81686
$239$	−19.0857	−1.23455	−0.617276	−	0.786746i	$-0.711764\pi$
$239$	−19.0857	−1.23455	−0.617276	+	0.786746i	$0.711764\pi$
$240$	0	0
$241$	−18.5790	−1.19678	−0.598388	−	0.801206i	$-0.704192\pi$
$241$	−18.5790	−1.19678	−0.598388	+	0.801206i	$0.704192\pi$
$242$	17.3328	1.11420
$243$	0	0
$244$	−28.2349	−1.80755
$245$	5.49598	0.351125
$246$	0	0
$247$	13.1114	0.834258
$248$	0.0831249	0.00527844
$249$	0	0
$250$	−15.9121	−1.00637
$251$	11.1361	0.702906	0.351453	−	0.936206i	$-0.385688\pi$
$251$	11.1361	0.702906	0.351453	+	0.936206i	$0.385688\pi$
$252$	0	0
$253$	−3.01605	−0.189617
$254$	1.29551	0.0812874
$255$	0	0
$256$	15.2588	0.953676
$257$	−18.8438	−1.17544	−0.587722	−	0.809063i	$-0.699975\pi$
$257$	−18.8438	−1.17544	−0.587722	+	0.809063i	$0.699975\pi$
$258$	0	0
$259$	9.98114	0.620198
$260$	15.2281	0.944407
$261$	0	0
$262$	1.55694	0.0961878
$263$	−14.4209	−0.889233	−0.444617	−	0.895721i	$-0.646660\pi$
$263$	−14.4209	−0.889233	−0.444617	+	0.895721i	$0.646660\pi$
$264$	0	0
$265$	29.4076	1.80649
$266$	−36.2936	−2.22530
$267$	0	0
$268$	4.12634	0.252056
$269$	20.4921	1.24942	0.624712	−	0.780856i	$-0.285217\pi$
$269$	20.4921	1.24942	0.624712	+	0.780856i	$0.285217\pi$
$270$	0	0
$271$	−15.6506	−0.950705	−0.475353	−	0.879795i	$-0.657680\pi$
$271$	−15.6506	−0.950705	−0.475353	+	0.879795i	$0.657680\pi$
$272$	−18.6067	−1.12820
$273$	0	0
$274$	6.41689	0.387659
$275$	11.2037	0.675611
$276$	0	0
$277$	−20.2527	−1.21687	−0.608433	−	0.793606i	$-0.708201\pi$
$277$	−20.2527	−1.21687	−0.608433	+	0.793606i	$0.708201\pi$
$278$	−43.2703	−2.59518
$279$	0	0
$280$	−0.924106	−0.0552259
$281$	−13.3272	−0.795036	−0.397518	−	0.917594i	$-0.630128\pi$
$281$	−13.3272	−0.795036	−0.397518	+	0.917594i	$0.630128\pi$
$282$	0	0
$283$	−30.6109	−1.81963	−0.909815	−	0.415013i	$-0.863777\pi$
$283$	−30.6109	−1.81963	−0.909815	+	0.415013i	$0.863777\pi$
$284$	0.497150	0.0295004
$285$	0	0
$286$	−6.60159	−0.390360
$287$	−25.9318	−1.53070
$288$	0	0
$289$	5.66508	0.333240
$290$	−40.8123	−2.39658
$291$	0	0
$292$	−9.49273	−0.555520
$293$	−17.1725	−1.00323	−0.501613	−	0.865092i	$-0.667260\pi$
$293$	−17.1725	−1.00323	−0.501613	+	0.865092i	$0.667260\pi$
$294$	0	0
$295$	−7.91276	−0.460699
$296$	−0.307397	−0.0178671
$297$	0	0
$298$	−23.8532	−1.38178
$299$	−4.15662	−0.240384
$300$	0	0
$301$	−12.9280	−0.745157
$302$	32.9820	1.89790
$303$	0	0
$304$	−24.0930	−1.38183
$305$	−48.3468	−2.76833
$306$	0	0
$307$	31.1805	1.77957	0.889783	−	0.456383i	$-0.150855\pi$
$307$	31.1805	1.77957	0.889783	+	0.456383i	$0.150855\pi$
$308$	9.23819	0.526395
$309$	0	0
$310$	6.49260	0.368755
$311$	−24.2773	−1.37664	−0.688319	−	0.725408i	$-0.741651\pi$
$311$	−24.2773	−1.37664	−0.688319	+	0.725408i	$0.741651\pi$
$312$	0	0
$313$	−2.76360	−0.156208	−0.0781039	−	0.996945i	$-0.524887\pi$
$313$	−2.76360	−0.156208	−0.0781039	+	0.996945i	$0.524887\pi$
$314$	−11.3549	−0.640793
$315$	0	0
$316$	−6.50127	−0.365725
$317$	14.6665	0.823751	0.411876	−	0.911240i	$-0.364874\pi$
$317$	14.6665	0.823751	0.411876	+	0.911240i	$0.364874\pi$
$318$	0	0
$319$	8.94438	0.500789
$320$	−29.2523	−1.63525
$321$	0	0
$322$	11.5059	0.641201
$323$	29.3479	1.63296
$324$	0	0
$325$	15.4406	0.856493
$326$	7.44494	0.412337
$327$	0	0
$328$	0.798641	0.0440976
$329$	18.4483	1.01709
$330$	0	0
$331$	−8.85392	−0.486655	−0.243328	−	0.969944i	$-0.578239\pi$
$331$	−8.85392	−0.486655	−0.243328	+	0.969944i	$0.578239\pi$
$332$	30.1297	1.65358
$333$	0	0
$334$	−0.586251	−0.0320782
$335$	7.06556	0.386033
$336$	0	0
$337$	−26.6182	−1.44999	−0.724993	−	0.688757i	$-0.758157\pi$
$337$	−26.6182	−1.44999	−0.724993	+	0.688757i	$0.758157\pi$
$338$	17.0472	0.927245
$339$	0	0
$340$	34.0859	1.84857
$341$	−1.42291	−0.0770550
$342$	0	0
$343$	−15.8968	−0.858344
$344$	0.398153	0.0214670
$345$	0	0
$346$	11.1064	0.597082
$347$	−13.1520	−0.706037	−0.353019	−	0.935616i	$-0.614845\pi$
$347$	−13.1520	−0.706037	−0.353019	+	0.935616i	$0.614845\pi$
$348$	0	0
$349$	0.658011	0.0352225	0.0176113	−	0.999845i	$-0.494394\pi$
$349$	0.658011	0.0352225	0.0176113	+	0.999845i	$0.494394\pi$
$350$	−42.7412	−2.28461
$351$	0	0
$352$	12.4091	0.661407
$353$	12.5117	0.665931	0.332966	−	0.942939i	$-0.391951\pi$
$353$	12.5117	0.665931	0.332966	+	0.942939i	$0.391951\pi$
$354$	0	0
$355$	0.851274	0.0451810
$356$	21.1962	1.12340
$357$	0	0
$358$	27.1115	1.43289
$359$	−17.5650	−0.927045	−0.463522	−	0.886085i	$-0.653415\pi$
$359$	−17.5650	−0.927045	−0.463522	+	0.886085i	$0.653415\pi$
$360$	0	0
$361$	19.0012	1.00006
$362$	50.5749	2.65816
$363$	0	0
$364$	12.7318	0.667327
$365$	−16.2545	−0.850798
$366$	0	0
$367$	32.6289	1.70322	0.851608	−	0.524180i	$-0.175628\pi$
$367$	32.6289	1.70322	0.851608	+	0.524180i	$0.175628\pi$
$368$	7.63804	0.398160
$369$	0	0
$370$	−24.0097	−1.24821
$371$	24.5868	1.27648
$372$	0	0
$373$	2.02585	0.104895	0.0524473	−	0.998624i	$-0.483298\pi$
$373$	2.02585	0.104895	0.0524473	+	0.998624i	$0.483298\pi$
$374$	−14.7767	−0.764084
$375$	0	0
$376$	−0.568166	−0.0293009
$377$	12.3269	0.634865
$378$	0	0
$379$	6.83696	0.351191	0.175596	−	0.984462i	$-0.443815\pi$
$379$	6.83696	0.351191	0.175596	+	0.984462i	$0.443815\pi$
$380$	44.1361	2.26413
$381$	0	0
$382$	−28.6923	−1.46803
$383$	4.31882	0.220681	0.110341	−	0.993894i	$-0.464806\pi$
$383$	4.31882	0.220681	0.110341	+	0.993894i	$0.464806\pi$
$384$	0	0
$385$	15.8186	0.806192
$386$	−5.76992	−0.293681
$387$	0	0
$388$	−27.0773	−1.37464
$389$	−15.7981	−0.800996	−0.400498	−	0.916298i	$-0.631163\pi$
$389$	−15.7981	−0.800996	−0.400498	+	0.916298i	$0.631163\pi$
$390$	0	0
$391$	−9.30398	−0.470523
$392$	−0.141517	−0.00714767
$393$	0	0
$394$	4.36944	0.220129
$395$	−11.1322	−0.560121
$396$	0	0
$397$	−19.2906	−0.968166	−0.484083	−	0.875022i	$-0.660847\pi$
$397$	−19.2906	−0.968166	−0.484083	+	0.875022i	$0.660847\pi$
$398$	30.0944	1.50849
$399$	0	0
$400$	−28.3731	−1.41865
$401$	−21.2801	−1.06268	−0.531340	−	0.847159i	$-0.678311\pi$
$401$	−21.2801	−1.06268	−0.531340	+	0.847159i	$0.678311\pi$
$402$	0	0
$403$	−1.96101	−0.0976850
$404$	−8.64048	−0.429880
$405$	0	0
$406$	−34.1219	−1.69344
$407$	5.26195	0.260825
$408$	0	0
$409$	−14.0274	−0.693613	−0.346806	−	0.937937i	$-0.612734\pi$
$409$	−14.0274	−0.693613	−0.346806	+	0.937937i	$0.612734\pi$
$410$	62.3791	3.08069
$411$	0	0
$412$	−12.1066	−0.596451
$413$	−6.61563	−0.325534
$414$	0	0
$415$	51.5913	2.53252
$416$	17.1018	0.838485
$417$	0	0
$418$	−19.1336	−0.935855
$419$	22.9131	1.11938	0.559690	−	0.828702i	$-0.310920\pi$
$419$	22.9131	1.11938	0.559690	+	0.828702i	$0.310920\pi$
$420$	0	0
$421$	−7.95284	−0.387598	−0.193799	−	0.981041i	$-0.562081\pi$
$421$	−7.95284	−0.387598	−0.193799	+	0.981041i	$0.562081\pi$
$422$	−24.1309	−1.17467
$423$	0	0
$424$	−0.757219	−0.0367738
$425$	34.5616	1.67648
$426$	0	0
$427$	−40.4214	−1.95613
$428$	1.69032	0.0817046
$429$	0	0
$430$	31.0984	1.49970
$431$	27.2781	1.31394	0.656971	−	0.753916i	$-0.271838\pi$
$431$	27.2781	1.31394	0.656971	+	0.753916i	$0.271838\pi$
$432$	0	0
$433$	25.2629	1.21406	0.607028	−	0.794680i	$-0.292361\pi$
$433$	25.2629	1.21406	0.607028	+	0.794680i	$0.292361\pi$
$434$	5.42828	0.260566
$435$	0	0
$436$	20.5859	0.985884
$437$	−12.0473	−0.576299
$438$	0	0
$439$	35.3672	1.68799	0.843993	−	0.536355i	$-0.180199\pi$
$439$	35.3672	1.68799	0.843993	+	0.536355i	$0.180199\pi$
$440$	−0.487179	−0.0232253
$441$	0	0
$442$	−20.3647	−0.968652
$443$	−31.1073	−1.47795	−0.738976	−	0.673731i	$-0.764690\pi$
$443$	−31.1073	−1.47795	−0.738976	+	0.673731i	$0.764690\pi$
$444$	0	0
$445$	36.2945	1.72052
$446$	−17.7928	−0.842514
$447$	0	0
$448$	−24.4570	−1.15548
$449$	−5.60323	−0.264433	−0.132216	−	0.991221i	$-0.542209\pi$
$449$	−5.60323	−0.264433	−0.132216	+	0.991221i	$0.542209\pi$
$450$	0	0
$451$	−13.6710	−0.643740
$452$	12.5609	0.590813
$453$	0	0
$454$	36.0681	1.69276
$455$	21.8007	1.02203
$456$	0	0
$457$	−18.7263	−0.875980	−0.437990	−	0.898980i	$-0.644309\pi$
$457$	−18.7263	−0.875980	−0.437990	+	0.898980i	$0.644309\pi$
$458$	12.6079	0.589127
$459$	0	0
$460$	−13.9922	−0.652389
$461$	−15.6814	−0.730356	−0.365178	−	0.930938i	$-0.618992\pi$
$461$	−15.6814	−0.730356	−0.365178	+	0.930938i	$0.618992\pi$
$462$	0	0
$463$	−2.97257	−0.138147	−0.0690736	−	0.997612i	$-0.522004\pi$
$463$	−2.97257	−0.138147	−0.0690736	+	0.997612i	$0.522004\pi$
$464$	−22.6513	−1.05156
$465$	0	0
$466$	4.51507	0.209157
$467$	8.16984	0.378055	0.189028	−	0.981972i	$-0.439466\pi$
$467$	8.16984	0.378055	0.189028	+	0.981972i	$0.439466\pi$
$468$	0	0
$469$	5.90731	0.272774
$470$	−44.3775	−2.04698
$471$	0	0
$472$	0.203747	0.00937820
$473$	−6.81549	−0.313377
$474$	0	0
$475$	44.7521	2.05337
$476$	28.4982	1.30621
$477$	0	0
$478$	38.3847	1.75568
$479$	−0.793040	−0.0362349	−0.0181175	−	0.999836i	$-0.505767\pi$
$479$	−0.793040	−0.0362349	−0.0181175	+	0.999836i	$0.505767\pi$
$480$	0	0
$481$	7.25185	0.330656
$482$	37.3656	1.70195
$483$	0	0
$484$	−17.6228	−0.801038
$485$	−46.3646	−2.10531
$486$	0	0
$487$	2.96218	0.134229	0.0671145	−	0.997745i	$-0.478621\pi$
$487$	2.96218	0.134229	0.0671145	+	0.997745i	$0.478621\pi$
$488$	1.24489	0.0563534
$489$	0	0
$490$	−11.0534	−0.499341
$491$	−3.34674	−0.151036	−0.0755182	−	0.997144i	$-0.524061\pi$
$491$	−3.34674	−0.151036	−0.0755182	+	0.997144i	$0.524061\pi$
$492$	0	0
$493$	27.5918	1.24267
$494$	−26.3693	−1.18641
$495$	0	0
$496$	3.60348	0.161801
$497$	0.711726	0.0319253
$498$	0	0
$499$	−19.1225	−0.856041	−0.428020	−	0.903769i	$-0.640789\pi$
$499$	−19.1225	−0.856041	−0.428020	+	0.903769i	$0.640789\pi$
$500$	16.1784	0.723519
$501$	0	0
$502$	−22.3967	−0.999614
$503$	5.09519	0.227183	0.113592	−	0.993528i	$-0.463764\pi$
$503$	5.09519	0.227183	0.113592	+	0.993528i	$0.463764\pi$
$504$	0	0
$505$	−14.7952	−0.658376
$506$	6.06581	0.269658
$507$	0	0
$508$	−1.31718	−0.0584406
$509$	29.0891	1.28935	0.644677	−	0.764455i	$-0.276992\pi$
$509$	29.0891	1.28935	0.644677	+	0.764455i	$0.276992\pi$
$510$	0	0
$511$	−13.5899	−0.601182
$512$	−32.1303	−1.41997
$513$	0	0
$514$	37.8982	1.67162
$515$	−20.7303	−0.913486
$516$	0	0
$517$	9.72574	0.427738
$518$	−20.0738	−0.881994
$519$	0	0
$520$	−0.671414	−0.0294434
$521$	15.1073	0.661864	0.330932	−	0.943655i	$-0.392637\pi$
$521$	15.1073	0.661864	0.330932	+	0.943655i	$0.392637\pi$
$522$	0	0
$523$	35.9846	1.57350	0.786748	−	0.617274i	$-0.211763\pi$
$523$	35.9846	1.57350	0.786748	+	0.617274i	$0.211763\pi$
$524$	−1.58299	−0.0691531
$525$	0	0
$526$	29.0030	1.26459
$527$	−4.38943	−0.191207
$528$	0	0
$529$	−19.1807	−0.833945
$530$	−59.1438	−2.56904
$531$	0	0
$532$	36.9009	1.59986
$533$	−18.8409	−0.816089
$534$	0	0
$535$	2.89435	0.125133
$536$	−0.181932	−0.00785826
$537$	0	0
$538$	−41.2131	−1.77682
$539$	2.42245	0.104342
$540$	0	0
$541$	9.43138	0.405487	0.202743	−	0.979232i	$-0.435014\pi$
$541$	9.43138	0.405487	0.202743	+	0.979232i	$0.435014\pi$
$542$	31.4761	1.35201
$543$	0	0
$544$	38.2799	1.64124
$545$	35.2494	1.50992
$546$	0	0
$547$	26.4130	1.12934	0.564668	−	0.825318i	$-0.309004\pi$
$547$	26.4130	1.12934	0.564668	+	0.825318i	$0.309004\pi$
$548$	−6.52427	−0.278703
$549$	0	0
$550$	−22.5327	−0.960798
$551$	35.7273	1.52203
$552$	0	0
$553$	−9.30729	−0.395786
$554$	40.7317	1.73052
$555$	0	0
$556$	43.9943	1.86578
$557$	22.0469	0.934156	0.467078	−	0.884216i	$-0.345307\pi$
$557$	22.0469	0.934156	0.467078	+	0.884216i	$0.345307\pi$
$558$	0	0
$559$	−9.39289	−0.397277
$560$	−40.0601	−1.69285
$561$	0	0
$562$	26.8034	1.13063
$563$	−7.78991	−0.328306	−0.164153	−	0.986435i	$-0.552489\pi$
$563$	−7.78991	−0.328306	−0.164153	+	0.986435i	$0.552489\pi$
$564$	0	0
$565$	21.5081	0.904851
$566$	61.5640	2.58773
$567$	0	0
$568$	−0.0219196	−0.000919724 0
$569$	−29.0051	−1.21596	−0.607978	−	0.793954i	$-0.708019\pi$
$569$	−29.0051	−1.21596	−0.607978	+	0.793954i	$0.708019\pi$
$570$	0	0
$571$	15.0929	0.631617	0.315808	−	0.948823i	$-0.397724\pi$
$571$	15.0929	0.631617	0.315808	+	0.948823i	$0.397724\pi$
$572$	6.71205	0.280645
$573$	0	0
$574$	52.1534	2.17684
$575$	−14.1875	−0.591659
$576$	0	0
$577$	−15.3768	−0.640144	−0.320072	−	0.947393i	$-0.603707\pi$
$577$	−15.3768	−0.640144	−0.320072	+	0.947393i	$0.603707\pi$
$578$	−11.3935	−0.473906
$579$	0	0
$580$	41.4952	1.72299
$581$	43.1340	1.78950
$582$	0	0
$583$	12.9619	0.536827
$584$	0.418538	0.0173192
$585$	0	0
$586$	34.5369	1.42670
$587$	34.5313	1.42526	0.712631	−	0.701540i	$-0.247504\pi$
$587$	34.5313	1.42526	0.712631	+	0.701540i	$0.247504\pi$
$588$	0	0
$589$	−5.68366	−0.234191
$590$	15.9140	0.655167
$591$	0	0
$592$	−13.3257	−0.547683
$593$	−19.1870	−0.787918	−0.393959	−	0.919128i	$-0.628895\pi$
$593$	−19.1870	−0.787918	−0.393959	+	0.919128i	$0.628895\pi$
$594$	0	0
$595$	48.7977	2.00051
$596$	24.2523	0.993412
$597$	0	0
$598$	8.35970	0.341853
$599$	−31.3284	−1.28004	−0.640022	−	0.768356i	$-0.721075\pi$
$599$	−31.3284	−1.28004	−0.640022	+	0.768356i	$0.721075\pi$
$600$	0	0
$601$	−38.0073	−1.55035	−0.775175	−	0.631746i	$-0.782339\pi$
$601$	−38.0073	−1.55035	−0.775175	+	0.631746i	$0.782339\pi$
$602$	26.0005	1.05970
$603$	0	0
$604$	−33.5339	−1.36448
$605$	−30.1757	−1.22682
$606$	0	0
$607$	−14.6587	−0.594977	−0.297488	−	0.954725i	$-0.596149\pi$
$607$	−14.6587	−0.594977	−0.297488	+	0.954725i	$0.596149\pi$
$608$	49.5667	2.01020
$609$	0	0
$610$	97.2340	3.93689
$611$	13.4037	0.542256
$612$	0	0
$613$	27.9502	1.12890	0.564449	−	0.825468i	$-0.309088\pi$
$613$	27.9502	1.12890	0.564449	+	0.825468i	$0.309088\pi$
$614$	−62.7095	−2.53075
$615$	0	0
$616$	−0.407316	−0.0164112
$617$	26.7404	1.07653	0.538263	−	0.842777i	$-0.319081\pi$
$617$	26.7404	1.07653	0.538263	+	0.842777i	$0.319081\pi$
$618$	0	0
$619$	−4.59234	−0.184582	−0.0922909	−	0.995732i	$-0.529419\pi$
$619$	−4.59234	−0.184582	−0.0922909	+	0.995732i	$0.529419\pi$
$620$	−6.60124	−0.265112
$621$	0	0
$622$	48.8259	1.95774
$623$	30.3447	1.21574
$624$	0	0
$625$	−8.59584	−0.343833
$626$	5.55808	0.222146
$627$	0	0
$628$	11.5449	0.460691
$629$	16.2322	0.647220
$630$	0	0
$631$	−29.1189	−1.15920	−0.579602	−	0.814899i	$-0.696792\pi$
$631$	−29.1189	−1.15920	−0.579602	+	0.814899i	$0.696792\pi$
$632$	0.286644	0.0114021
$633$	0	0
$634$	−29.4969	−1.17147
$635$	−2.25543	−0.0895039
$636$	0	0
$637$	3.33854	0.132278
$638$	−17.9887	−0.712180
$639$	0	0
$640$	2.52476	0.0998001
$641$	50.5222	1.99551	0.997754	−	0.0669917i	$-0.0213401\pi$
$641$	50.5222	1.99551	0.997754	+	0.0669917i	$0.0213401\pi$
$642$	0	0
$643$	15.2058	0.599658	0.299829	−	0.953993i	$-0.403070\pi$
$643$	15.2058	0.599658	0.299829	+	0.953993i	$0.403070\pi$
$644$	−11.6985	−0.460984
$645$	0	0
$646$	−59.0238	−2.32226
$647$	8.72115	0.342864	0.171432	−	0.985196i	$-0.445161\pi$
$647$	8.72115	0.342864	0.171432	+	0.985196i	$0.445161\pi$
$648$	0	0
$649$	−3.48769	−0.136904
$650$	−31.0538	−1.21803
$651$	0	0
$652$	−7.56951	−0.296445
$653$	13.7428	0.537798	0.268899	−	0.963168i	$-0.413340\pi$
$653$	13.7428	0.537798	0.268899	+	0.963168i	$0.413340\pi$
$654$	0	0
$655$	−2.71056	−0.105910
$656$	34.6212	1.35173
$657$	0	0
$658$	−37.1028	−1.44642
$659$	−43.5790	−1.69760	−0.848799	−	0.528716i	$-0.822674\pi$
$659$	−43.5790	−1.69760	−0.848799	+	0.528716i	$0.822674\pi$
$660$	0	0
$661$	30.1020	1.17083	0.585416	−	0.810733i	$-0.300931\pi$
$661$	30.1020	1.17083	0.585416	+	0.810733i	$0.300931\pi$
$662$	17.8068	0.692081
$663$	0	0
$664$	−1.32843	−0.0515531
$665$	63.1857	2.45024
$666$	0	0
$667$	−11.3264	−0.438560
$668$	0.596060	0.0230623
$669$	0	0
$670$	−14.2101	−0.548983
$671$	−21.3097	−0.822652
$672$	0	0
$673$	22.3208	0.860402	0.430201	−	0.902733i	$-0.358443\pi$
$673$	22.3208	0.860402	0.430201	+	0.902733i	$0.358443\pi$
$674$	53.5339	2.06205
$675$	0	0
$676$	−17.3324	−0.666632
$677$	−33.4434	−1.28533	−0.642667	−	0.766145i	$-0.722172\pi$
$677$	−33.4434	−1.28533	−0.642667	+	0.766145i	$0.722172\pi$
$678$	0	0
$679$	−38.7641	−1.48763
$680$	−1.50286	−0.0576320
$681$	0	0
$682$	2.86173	0.109581
$683$	−0.629152	−0.0240738	−0.0120369	−	0.999928i	$-0.503832\pi$
$683$	−0.629152	−0.0240738	−0.0120369	+	0.999928i	$0.503832\pi$
$684$	0	0
$685$	−11.1716	−0.426843
$686$	31.9712	1.22067
$687$	0	0
$688$	17.2600	0.658031
$689$	17.8637	0.680552
$690$	0	0
$691$	44.2013	1.68150	0.840749	−	0.541425i	$-0.182115\pi$
$691$	44.2013	1.68150	0.840749	+	0.541425i	$0.182115\pi$
$692$	−11.2922	−0.429265
$693$	0	0
$694$	26.4510	1.00407
$695$	75.3319	2.85750
$696$	0	0
$697$	−42.1725	−1.59740
$698$	−1.32338	−0.0500905
$699$	0	0
$700$	43.4564	1.64250
$701$	−42.7153	−1.61333	−0.806667	−	0.591006i	$-0.798731\pi$
$701$	−42.7153	−1.61333	−0.806667	+	0.591006i	$0.798731\pi$
$702$	0	0
$703$	21.0183	0.792719
$704$	−12.8935	−0.485941
$705$	0	0
$706$	−25.1633	−0.947031
$707$	−12.3698	−0.465214
$708$	0	0
$709$	−0.308366	−0.0115809	−0.00579047	−	0.999983i	$-0.501843\pi$
$709$	−0.308366	−0.0115809	−0.00579047	+	0.999983i	$0.501843\pi$
$710$	−1.71206	−0.0642526
$711$	0	0
$712$	−0.934550	−0.0350237
$713$	1.80186	0.0674801
$714$	0	0
$715$	11.4931	0.429818
$716$	−27.5652	−1.03016
$717$	0	0
$718$	35.3263	1.31837
$719$	7.44518	0.277658	0.138829	−	0.990316i	$-0.455666\pi$
$719$	7.44518	0.277658	0.138829	+	0.990316i	$0.455666\pi$
$720$	0	0
$721$	−17.3320	−0.645477
$722$	−38.2147	−1.42220
$723$	0	0
$724$	−51.4211	−1.91105
$725$	42.0743	1.56260
$726$	0	0
$727$	−29.4244	−1.09129	−0.545645	−	0.838016i	$-0.683715\pi$
$727$	−29.4244	−1.09129	−0.545645	+	0.838016i	$0.683715\pi$
$728$	−0.561349	−0.0208050
$729$	0	0
$730$	32.6906	1.20993
$731$	−21.0246	−0.777623
$732$	0	0
$733$	31.4253	1.16072	0.580360	−	0.814360i	$-0.302912\pi$
$733$	31.4253	1.16072	0.580360	+	0.814360i	$0.302912\pi$
$734$	−65.6225	−2.42217
$735$	0	0
$736$	−15.7138	−0.579219
$737$	3.11427	0.114716
$738$	0	0
$739$	42.7277	1.57176	0.785882	−	0.618377i	$-0.212209\pi$
$739$	42.7277	1.57176	0.785882	+	0.618377i	$0.212209\pi$
$740$	24.4115	0.897384
$741$	0	0
$742$	−49.4484	−1.81531
$743$	−18.5531	−0.680649	−0.340324	−	0.940308i	$-0.610537\pi$
$743$	−18.5531	−0.680649	−0.340324	+	0.940308i	$0.610537\pi$
$744$	0	0
$745$	41.5274	1.52145
$746$	−4.07435	−0.149172
$747$	0	0
$748$	15.0239	0.549329
$749$	2.41988	0.0884204
$750$	0	0
$751$	51.6610	1.88514	0.942568	−	0.334015i	$-0.108404\pi$
$751$	51.6610	1.88514	0.942568	+	0.334015i	$0.108404\pi$
$752$	−24.6301	−0.898167
$753$	0	0
$754$	−24.7915	−0.902852
$755$	−57.4204	−2.08974
$756$	0	0
$757$	−25.4960	−0.926666	−0.463333	−	0.886184i	$-0.653347\pi$
$757$	−25.4960	−0.926666	−0.463333	+	0.886184i	$0.653347\pi$
$758$	−13.7503	−0.499434
$759$	0	0
$760$	−1.94598	−0.0705881
$761$	12.4691	0.452004	0.226002	−	0.974127i	$-0.427434\pi$
$761$	12.4691	0.452004	0.226002	+	0.974127i	$0.427434\pi$
$762$	0	0
$763$	29.4710	1.06692
$764$	29.1724	1.05542
$765$	0	0
$766$	−8.68590	−0.313834
$767$	−4.80662	−0.173557
$768$	0	0
$769$	11.5370	0.416036	0.208018	−	0.978125i	$-0.433299\pi$
$769$	11.5370	0.416036	0.208018	+	0.978125i	$0.433299\pi$
$770$	−31.8141	−1.14650
$771$	0	0
$772$	5.86646	0.211139
$773$	−3.50915	−0.126215	−0.0631077	−	0.998007i	$-0.520101\pi$
$773$	−3.50915	−0.126215	−0.0631077	+	0.998007i	$0.520101\pi$
$774$	0	0
$775$	−6.69337	−0.240433
$776$	1.19385	0.0428566
$777$	0	0
$778$	31.7728	1.13911
$779$	−54.6071	−1.95650
$780$	0	0
$781$	0.375214	0.0134262
$782$	18.7119	0.669138
$783$	0	0
$784$	−6.13477	−0.219099
$785$	19.7684	0.705564
$786$	0	0
$787$	13.5767	0.483956	0.241978	−	0.970282i	$-0.422204\pi$
$787$	13.5767	0.483956	0.241978	+	0.970282i	$0.422204\pi$
$788$	−4.44256	−0.158260
$789$	0	0
$790$	22.3888	0.796557
$791$	17.9823	0.639376
$792$	0	0
$793$	−29.3684	−1.04290
$794$	38.7967	1.37685
$795$	0	0
$796$	−30.5979	−1.08451
$797$	50.8659	1.80176	0.900882	−	0.434064i	$-0.142921\pi$
$797$	50.8659	1.80176	0.900882	+	0.434064i	$0.142921\pi$
$798$	0	0
$799$	30.0022	1.06140
$800$	58.3723	2.06377
$801$	0	0
$802$	42.7981	1.51125
$803$	−7.16445	−0.252828
$804$	0	0
$805$	−20.0314	−0.706013
$806$	3.94394	0.138919
$807$	0	0
$808$	0.380962	0.0134022
$809$	2.81064	0.0988167	0.0494083	−	0.998779i	$-0.484266\pi$
$809$	2.81064	0.0988167	0.0494083	+	0.998779i	$0.484266\pi$
$810$	0	0
$811$	5.05065	0.177352	0.0886761	−	0.996061i	$-0.471736\pi$
$811$	5.05065	0.177352	0.0886761	+	0.996061i	$0.471736\pi$
$812$	34.6929	1.21748
$813$	0	0
$814$	−10.5827	−0.370924
$815$	−12.9613	−0.454016
$816$	0	0
$817$	−27.2237	−0.952437
$818$	28.2117	0.986398
$819$	0	0
$820$	−63.4229	−2.21482
$821$	36.9822	1.29069	0.645343	−	0.763893i	$-0.276714\pi$
$821$	36.9822	1.29069	0.645343	+	0.763893i	$0.276714\pi$
$822$	0	0
$823$	1.55764	0.0542961	0.0271480	−	0.999631i	$-0.491357\pi$
$823$	1.55764	0.0542961	0.0271480	+	0.999631i	$0.491357\pi$
$824$	0.533787	0.0185953
$825$	0	0
$826$	13.3052	0.462947
$827$	−6.71507	−0.233506	−0.116753	−	0.993161i	$-0.537249\pi$
$827$	−6.71507	−0.233506	−0.116753	+	0.993161i	$0.537249\pi$
$828$	0	0
$829$	39.8367	1.38358	0.691792	−	0.722097i	$-0.256821\pi$
$829$	39.8367	1.38358	0.691792	+	0.722097i	$0.256821\pi$
$830$	−103.759	−3.60154
$831$	0	0
$832$	−17.7694	−0.616042
$833$	7.47283	0.258918
$834$	0	0
$835$	1.02064	0.0353207
$836$	19.4538	0.672822
$837$	0	0
$838$	−46.0824	−1.59189
$839$	34.7604	1.20006	0.600032	−	0.799976i	$-0.295155\pi$
$839$	34.7604	1.20006	0.600032	+	0.799976i	$0.295155\pi$
$840$	0	0
$841$	4.58951	0.158259
$842$	15.9946	0.551209
$843$	0	0
$844$	24.5347	0.844518
$845$	−29.6785	−1.02097
$846$	0	0
$847$	−25.2290	−0.866880
$848$	−32.8255	−1.12723
$849$	0	0
$850$	−69.5094	−2.38415
$851$	−6.66329	−0.228415
$852$	0	0
$853$	−48.0264	−1.64439	−0.822196	−	0.569205i	$-0.807251\pi$
$853$	−48.0264	−1.64439	−0.822196	+	0.569205i	$0.807251\pi$
$854$	81.2945	2.78184
$855$	0	0
$856$	−0.0745268	−0.00254727
$857$	−13.8407	−0.472789	−0.236394	−	0.971657i	$-0.575966\pi$
$857$	−13.8407	−0.472789	−0.236394	+	0.971657i	$0.575966\pi$
$858$	0	0
$859$	−47.7273	−1.62843	−0.814217	−	0.580561i	$-0.802833\pi$
$859$	−47.7273	−1.62843	−0.814217	+	0.580561i	$0.802833\pi$
$860$	−31.6187	−1.07819
$861$	0	0
$862$	−54.8611	−1.86858
$863$	14.4190	0.490827	0.245414	−	0.969418i	$-0.421076\pi$
$863$	14.4190	0.490827	0.245414	+	0.969418i	$0.421076\pi$
$864$	0	0
$865$	−19.3357	−0.657435
$866$	−50.8081	−1.72653
$867$	0	0
$868$	−5.51911	−0.187331
$869$	−4.90670	−0.166448
$870$	0	0
$871$	4.29198	0.145428
$872$	−0.907640	−0.0307366
$873$	0	0
$874$	24.2292	0.819564
$875$	23.1611	0.782989
$876$	0	0
$877$	39.7659	1.34280	0.671399	−	0.741096i	$-0.265694\pi$
$877$	39.7659	1.34280	0.671399	+	0.741096i	$0.265694\pi$
$878$	−71.1297	−2.40051
$879$	0	0
$880$	−21.1193	−0.711930
$881$	−58.5622	−1.97301	−0.986505	−	0.163734i	$-0.947646\pi$
$881$	−58.5622	−1.97301	−0.986505	+	0.163734i	$0.947646\pi$
$882$	0	0
$883$	−27.4967	−0.925337	−0.462668	−	0.886531i	$-0.653108\pi$
$883$	−27.4967	−0.925337	−0.462668	+	0.886531i	$0.653108\pi$
$884$	20.7055	0.696401
$885$	0	0
$886$	62.5623	2.10182
$887$	−7.26812	−0.244040	−0.122020	−	0.992528i	$-0.538937\pi$
$887$	−7.26812	−0.244040	−0.122020	+	0.992528i	$0.538937\pi$
$888$	0	0
$889$	−1.88570	−0.0632442
$890$	−72.9945	−2.44678
$891$	0	0
$892$	18.0905	0.605716
$893$	38.8484	1.30001
$894$	0	0
$895$	−47.2001	−1.57772
$896$	2.11088	0.0705196
$897$	0	0
$898$	11.2691	0.376054
$899$	−5.34357	−0.178218
$900$	0	0
$901$	39.9852	1.33210
$902$	27.4947	0.915473
$903$	0	0
$904$	−0.553814	−0.0184196
$905$	−88.0489	−2.92684
$906$	0	0
$907$	51.7640	1.71880	0.859398	−	0.511308i	$-0.170839\pi$
$907$	51.7640	1.71880	0.859398	+	0.511308i	$0.170839\pi$
$908$	−36.6716	−1.21699
$909$	0	0
$910$	−43.8451	−1.45345
$911$	−18.7860	−0.622408	−0.311204	−	0.950343i	$-0.600732\pi$
$911$	−18.7860	−0.622408	−0.311204	+	0.950343i	$0.600732\pi$
$912$	0	0
$913$	22.7398	0.752577
$914$	37.6619	1.24575
$915$	0	0
$916$	−12.8188	−0.423546
$917$	−2.26622	−0.0748373
$918$	0	0
$919$	34.0031	1.12166	0.560830	−	0.827931i	$-0.310482\pi$
$919$	34.0031	1.12166	0.560830	+	0.827931i	$0.310482\pi$
$920$	0.616922	0.0203393
$921$	0	0
$922$	31.5381	1.03865
$923$	0.517108	0.0170208
$924$	0	0
$925$	24.7522	0.813847
$926$	5.97836	0.196461
$927$	0	0
$928$	46.6008	1.52975
$929$	34.1640	1.12088	0.560442	−	0.828193i	$-0.310631\pi$
$929$	34.1640	1.12088	0.560442	+	0.828193i	$0.310631\pi$
$930$	0	0
$931$	9.67620	0.317125
$932$	−4.59062	−0.150371
$933$	0	0
$934$	−16.4310	−0.537638
$935$	25.7256	0.841317
$936$	0	0
$937$	−34.8635	−1.13894	−0.569470	−	0.822012i	$-0.692851\pi$
$937$	−34.8635	−1.13894	−0.569470	+	0.822012i	$0.692851\pi$
$938$	−11.8806	−0.387916
$939$	0	0
$940$	45.1201	1.47166
$941$	50.3625	1.64177	0.820886	−	0.571093i	$-0.193480\pi$
$941$	50.3625	1.64177	0.820886	+	0.571093i	$0.193480\pi$
$942$	0	0
$943$	17.3117	0.563748
$944$	8.83245	0.287472
$945$	0	0
$946$	13.7072	0.445658
$947$	−53.2727	−1.73113	−0.865565	−	0.500797i	$-0.833040\pi$
$947$	−53.2727	−1.73113	−0.865565	+	0.500797i	$0.833040\pi$
$948$	0	0
$949$	−9.87381	−0.320517
$950$	−90.0043	−2.92013
$951$	0	0
$952$	−1.25650	−0.0407233
$953$	41.8153	1.35453	0.677266	−	0.735739i	$-0.263165\pi$
$953$	41.8153	1.35453	0.677266	+	0.735739i	$0.263165\pi$
$954$	0	0
$955$	49.9522	1.61641
$956$	−39.0270	−1.26222
$957$	0	0
$958$	1.59494	0.0515302
$959$	−9.34021	−0.301611
$960$	0	0
$961$	−30.1499	−0.972578
$962$	−14.5847	−0.470231
$963$	0	0
$964$	−37.9908	−1.22360
$965$	10.0452	0.323366
$966$	0	0
$967$	14.4622	0.465074	0.232537	−	0.972588i	$-0.425297\pi$
$967$	14.4622	0.465074	0.232537	+	0.972588i	$0.425297\pi$
$968$	0.776998	0.0249737
$969$	0	0
$970$	93.2474	2.99399
$971$	2.64134	0.0847647	0.0423824	−	0.999101i	$-0.486505\pi$
$971$	2.64134	0.0847647	0.0423824	+	0.999101i	$0.486505\pi$
$972$	0	0
$973$	62.9828	2.01914
$974$	−5.95746	−0.190889
$975$	0	0
$976$	53.9661	1.72741
$977$	−45.1106	−1.44322	−0.721608	−	0.692301i	$-0.756597\pi$
$977$	−45.1106	−1.44322	−0.721608	+	0.692301i	$0.756597\pi$
$978$	0	0
$979$	15.9974	0.511280
$980$	11.2383	0.358996
$981$	0	0
$982$	6.73089	0.214791
$983$	−42.9729	−1.37062	−0.685311	−	0.728250i	$-0.740334\pi$
$983$	−42.9729	−1.37062	−0.685311	+	0.728250i	$0.740334\pi$
$984$	0	0
$985$	−7.60703	−0.242380
$986$	−55.4920	−1.76723
$987$	0	0
$988$	26.8105	0.852957
$989$	8.63056	0.274436
$990$	0	0
$991$	25.4062	0.807054	0.403527	−	0.914968i	$-0.367784\pi$
$991$	25.4062	0.807054	0.403527	+	0.914968i	$0.367784\pi$
$992$	−7.41347	−0.235378
$993$	0	0
$994$	−1.43141	−0.0454014
$995$	−52.3931	−1.66097
$996$	0	0
$997$	−31.0292	−0.982704	−0.491352	−	0.870961i	$-0.663497\pi$
$997$	−31.0292	−0.982704	−0.491352	+	0.870961i	$0.663497\pi$
$998$	38.4587	1.21739
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6561.2.a.d.1.12		72
3.2	odd	2		6561.2.a.c.1.61		72
81.5	odd	54		729.2.g.c.460.7		144
81.11	odd	54		81.2.g.a.40.2	✓	144
81.16	even	27		729.2.g.b.271.2		144
81.22	even	27		243.2.g.a.235.7		144
81.32	odd	54		729.2.g.d.703.2		144
81.38	odd	54		729.2.g.d.28.2		144
81.43	even	27		729.2.g.a.28.7		144
81.49	even	27		729.2.g.a.703.7		144
81.59	odd	54		81.2.g.a.79.2	yes	144
81.65	odd	54		729.2.g.c.271.7		144
81.70	even	27		243.2.g.a.91.7		144
81.76	even	27		729.2.g.b.460.2		144

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
81.2.g.a.40.2	✓	144	81.11	odd	54
81.2.g.a.79.2	yes	144	81.59	odd	54
243.2.g.a.91.7		144	81.70	even	27
243.2.g.a.235.7		144	81.22	even	27
729.2.g.a.28.7		144	81.43	even	27
729.2.g.a.703.7		144	81.49	even	27
729.2.g.b.271.2		144	81.16	even	27
729.2.g.b.460.2		144	81.76	even	27
729.2.g.c.271.7		144	81.65	odd	54
729.2.g.c.460.7		144	81.5	odd	54
729.2.g.d.28.2		144	81.38	odd	54
729.2.g.d.703.2		144	81.32	odd	54
6561.2.a.c.1.61		72	3.2	odd	2
6561.2.a.d.1.12		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)

\(f(q)\)	\(=\)	\(q-2.01118 q^{2} +2.04483 q^{4} +3.50138 q^{5} +2.92740 q^{7} -0.0901573 q^{8} +O(q^{10})\)	\(q-2.01118 q^{2} +2.04483 q^{4} +3.50138 q^{5} +2.92740 q^{7} -0.0901573 q^{8} -7.04188 q^{10} +1.54329 q^{11} +2.12692 q^{13} -5.88751 q^{14} -3.90833 q^{16} +4.76079 q^{17} +6.16451 q^{19} +7.15971 q^{20} -3.10383 q^{22} -1.95430 q^{23} +7.25964 q^{25} -4.27760 q^{26} +5.98603 q^{28} +5.79565 q^{29} -0.921998 q^{31} +8.04066 q^{32} -9.57478 q^{34} +10.2499 q^{35} +3.40956 q^{37} -12.3979 q^{38} -0.315675 q^{40} -8.85830 q^{41} -4.41620 q^{43} +3.15577 q^{44} +3.93043 q^{46} +6.30194 q^{47} +1.56966 q^{49} -14.6004 q^{50} +4.34918 q^{52} +8.39886 q^{53} +5.40365 q^{55} -0.263926 q^{56} -11.6561 q^{58} -2.25990 q^{59} -13.8080 q^{61} +1.85430 q^{62} -8.35452 q^{64} +7.44713 q^{65} +2.01794 q^{67} +9.73499 q^{68} -20.6144 q^{70} +0.243126 q^{71} -4.64231 q^{73} -6.85723 q^{74} +12.6054 q^{76} +4.51783 q^{77} -3.17937 q^{79} -13.6846 q^{80} +17.8156 q^{82} +14.7346 q^{83} +16.6693 q^{85} +8.88176 q^{86} -0.139139 q^{88} +10.3658 q^{89} +6.22633 q^{91} -3.99620 q^{92} -12.6743 q^{94} +21.5843 q^{95} -13.2418 q^{97} -3.15687 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 72 q + 9 q^{2} + 63 q^{4} + 18 q^{5} + 27 q^{8}+O(q^{10}) \) 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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