LMFDB - Embedded newform 729.2.e.e.325.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [729,2,Mod(82,729)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("729.82"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(729, base_ring=CyclotomicField(18)) chi = DirichletCharacter(H, H._module([8])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 729 = 3^{6} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	729.e (of order $9$, degree $6$, not minimal)

Newform invariants

comment:select newform

sage:traces = [6,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(19)] == traces)

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

Self dual:	no
Analytic conductor:	$5.82109430735$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	$\Q(\zeta_{18})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - x^{3} + 1 $ x^6 - x^3 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 243)
Sato-Tate group:	$\mathrm{U}(1)[D_{9}]$

Embedding invariants

Embedding label			325.1
Root			$0.939693 + 0.342020i$ of defining polynomial
Character	$\chi$	$=$	729.325
Dual form			729.2.e.e.406.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+(-0.347296 + 1.96962i) q^{4} +(-0.694593 - 3.93923i) q^{7} +(-5.36231 - 4.49951i) q^{13} +(-3.75877 - 1.36808i) q^{16} +(0.500000 + 0.866025i) q^{19} +(-3.83022 + 3.21394i) q^{25} +8.00000 q^{28} +(1.91013 - 10.8329i) q^{31} +(5.00000 - 8.66025i) q^{37} +(-4.69846 - 1.71010i) q^{43} +(-8.45723 + 3.07818i) q^{49} +(10.7246 - 8.99903i) q^{52} +(-0.173648 - 0.984808i) q^{61} +(4.00000 - 6.92820i) q^{64} +(3.83022 + 3.21394i) q^{67} +(3.50000 + 6.06218i) q^{73} +(-1.87939 + 0.684040i) q^{76} +(-9.95858 + 8.35624i) q^{79} +(-14.0000 + 24.2487i) q^{91} +(-4.69846 - 1.71010i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 3 q^{19} + 48 q^{28} + 30 q^{37} + 24 q^{64} + 21 q^{73} - 84 q^{91}+O(q^{100}) $ 6 * q + 3 * q^19 + 48 * q^28 + 30 * q^37 + 24 * q^64 + 21 * q^73 - 84 * q^91

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/729\mathbb{Z}\right)^\times$.

$n$	$2$
$\chi(n)$	$e\left(\frac{7}{9}\right)$

Coefficient data

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

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

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

$2$		0			0		−0.642788	−	0.766044i	$-0.722222\pi$
$2$		0			0		0.642788	+	0.766044i	$0.277778\pi$
$3$		0			0
$3$		0			0
$4$	−0.347296	+	1.96962i	−0.173648	+	0.984808i
$5$		0			0		−0.342020	−	0.939693i	$-0.611111\pi$
$5$		0			0		0.342020	+	0.939693i	$0.388889\pi$
$6$		0			0
$7$	−0.694593	−	3.93923i	−0.262531	−	1.48889i	−0.775974	−	0.630765i	$-0.782741\pi$
$7$	−0.694593	−	3.93923i	−0.262531	−	1.48889i	0.513442	−	0.858124i	$-0.328370\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$		0			0		0.342020	−	0.939693i	$-0.388889\pi$
$11$		0			0		−0.342020	+	0.939693i	$0.611111\pi$
$12$		0			0
$13$	−5.36231	−	4.49951i	−1.48724	−	1.24794i	−0.898011	−	0.439972i	$-0.854988\pi$
$13$	−5.36231	−	4.49951i	−1.48724	−	1.24794i	−0.589226	−	0.807968i	$-0.700567\pi$
$14$		0			0
$15$		0			0
$16$	−3.75877	−	1.36808i	−0.939693	−	0.342020i
$17$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$17$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$18$		0			0
$19$	0.500000	+	0.866025i	0.114708	+	0.198680i	0.917663	−	0.397360i	$-0.130073\pi$
$19$	0.500000	+	0.866025i	0.114708	+	0.198680i	−0.802955	+	0.596040i	$0.796740\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$		0			0		−0.984808	−	0.173648i	$-0.944444\pi$
$23$		0			0		0.984808	+	0.173648i	$0.0555556\pi$
$24$		0			0
$25$	−3.83022	+	3.21394i	−0.766044	+	0.642788i
$26$		0			0
$27$		0			0
$28$	8.00000			1.51186
$29$		0			0		−0.642788	−	0.766044i	$-0.722222\pi$
$29$		0			0		0.642788	+	0.766044i	$0.277778\pi$
$30$		0			0
$31$	1.91013	−	10.8329i	0.343069	−	1.94564i	0.0183550	−	0.999832i	$-0.494157\pi$
$31$	1.91013	−	10.8329i	0.343069	−	1.94564i	0.324714	−	0.945812i	$-0.394732\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		0			0
$36$		0			0
$37$	5.00000	−	8.66025i	0.821995	−	1.42374i	−0.0821995	−	0.996616i	$-0.526194\pi$
$37$	5.00000	−	8.66025i	0.821995	−	1.42374i	0.904194	−	0.427121i	$-0.140472\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$		0			0		0.642788	−	0.766044i	$-0.277778\pi$
$41$		0			0		−0.642788	+	0.766044i	$0.722222\pi$
$42$		0			0
$43$	−4.69846	−	1.71010i	−0.716509	−	0.260788i	−0.0420659	−	0.999115i	$-0.513394\pi$
$43$	−4.69846	−	1.71010i	−0.716509	−	0.260788i	−0.674443	+	0.738327i	$0.735616\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$		0			0		0.984808	−	0.173648i	$-0.0555556\pi$
$47$		0			0		−0.984808	+	0.173648i	$0.944444\pi$
$48$		0			0
$49$	−8.45723	+	3.07818i	−1.20818	+	0.439740i
$50$		0			0
$51$		0			0
$52$	10.7246	−	8.99903i	1.48724	−	1.24794i
$53$		0			0			−	1.00000i	$-0.5\pi$
$53$		0			0				1.00000i	$0.5\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$		0			0
$58$		0			0
$59$		0			0		−0.342020	−	0.939693i	$-0.611111\pi$
$59$		0			0		0.342020	+	0.939693i	$0.388889\pi$
$60$		0			0
$61$	−0.173648	−	0.984808i	−0.0222334	−	0.126092i	0.971671	−	0.236338i	$-0.0759472\pi$
$61$	−0.173648	−	0.984808i	−0.0222334	−	0.126092i	−0.993904	+	0.110246i	$0.964836\pi$
$62$		0			0
$63$		0			0
$64$	4.00000	−	6.92820i	0.500000	−	0.866025i
$65$		0			0
$66$		0			0
$67$	3.83022	+	3.21394i	0.467936	+	0.392645i	0.846041	−	0.533118i	$-0.178980\pi$
$67$	3.83022	+	3.21394i	0.467936	+	0.392645i	−0.378105	+	0.925763i	$0.623424\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$71$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$72$		0			0
$73$	3.50000	+	6.06218i	0.409644	+	0.709524i	0.994850	−	0.101361i	$-0.0323196\pi$
$73$	3.50000	+	6.06218i	0.409644	+	0.709524i	−0.585206	+	0.810885i	$0.698986\pi$
$74$		0			0
$75$		0			0
$76$	−1.87939	+	0.684040i	−0.215580	+	0.0784648i
$77$		0			0
$78$		0			0
$79$	−9.95858	+	8.35624i	−1.12043	+	0.940150i	−0.998626	−	0.0524041i	$-0.983312\pi$
$79$	−9.95858	+	8.35624i	−1.12043	+	0.940150i	−0.121802	+	0.992554i	$0.538867\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$		0			0		−0.642788	−	0.766044i	$-0.722222\pi$
$83$		0			0		0.642788	+	0.766044i	$0.277778\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$		0			0
$88$		0			0
$89$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$89$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$90$		0			0
$91$	−14.0000	+	24.2487i	−1.46760	+	2.54196i
$92$		0			0
$93$		0			0
$94$		0			0
$95$		0			0
$96$		0			0
$97$	−4.69846	−	1.71010i	−0.477057	−	0.173634i	0.0922897	−	0.995732i	$-0.470581\pi$
$97$	−4.69846	−	1.71010i	−0.477057	−	0.173634i	−0.569346	+	0.822098i	$0.692804\pi$
$98$		0			0
$99$		0			0
$100$	−5.00000	−	8.66025i	−0.500000	−	0.866025i
$101$		0			0		0.984808	−	0.173648i	$-0.0555556\pi$
$101$		0			0		−0.984808	+	0.173648i	$0.944444\pi$
$102$		0			0
$103$	12.2160	−	4.44626i	1.20368	−	0.438103i	0.339172	−	0.940724i	$-0.389853\pi$
$103$	12.2160	−	4.44626i	1.20368	−	0.438103i	0.864507	+	0.502621i	$0.167631\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		0			0			−	1.00000i	$-0.5\pi$
$107$		0			0				1.00000i	$0.5\pi$
$108$		0			0
$109$	−19.0000			−1.81987			−0.909935	−	0.414751i	$-0.863869\pi$
$109$	−19.0000			−1.81987			−0.909935	+	0.414751i	$0.863869\pi$
$110$		0			0
$111$		0			0
$112$	−2.77837	+	15.7569i	−0.262531	+	1.48889i
$113$		0			0		−0.342020	−	0.939693i	$-0.611111\pi$
$113$		0			0		0.342020	+	0.939693i	$0.388889\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$		0			0
$118$		0			0
$119$		0			0
$120$		0			0
$121$	−8.42649	−	7.07066i	−0.766044	−	0.642788i
$122$		0			0
$123$		0			0
$124$	20.6732	+	7.52444i	1.85651	+	0.675715i
$125$		0			0
$126$		0			0
$127$	0.500000	+	0.866025i	0.0443678	+	0.0768473i	0.887357	−	0.461084i	$-0.152539\pi$
$127$	0.500000	+	0.866025i	0.0443678	+	0.0768473i	−0.842989	+	0.537931i	$0.819206\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$		0			0		−0.984808	−	0.173648i	$-0.944444\pi$
$131$		0			0		0.984808	+	0.173648i	$0.0555556\pi$
$132$		0			0
$133$	3.06418	−	2.57115i	0.265698	−	0.222947i
$134$		0			0
$135$		0			0
$136$		0			0
$137$		0			0		−0.642788	−	0.766044i	$-0.722222\pi$
$137$		0			0		0.642788	+	0.766044i	$0.277778\pi$
$138$		0			0
$139$	−2.77837	+	15.7569i	−0.235658	+	1.33648i	0.605564	+	0.795796i	$0.292947\pi$
$139$	−2.77837	+	15.7569i	−0.235658	+	1.33648i	−0.841223	+	0.540689i	$0.818164\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		0			0
$144$		0			0
$145$		0			0
$146$		0			0
$147$		0			0
$148$	15.3209	+	12.8558i	1.25937	+	1.05674i
$149$		0			0		0.642788	−	0.766044i	$-0.277778\pi$
$149$		0			0		−0.642788	+	0.766044i	$0.722222\pi$
$150$		0			0
$151$	17.8542	+	6.49838i	1.45295	+	0.528831i	0.943414	−	0.331618i	$-0.107594\pi$
$151$	17.8542	+	6.49838i	1.45295	+	0.528831i	0.509537	+	0.860449i	$0.329817\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		0			0
$156$		0			0
$157$	23.4923	−	8.55050i	1.87489	−	0.682404i	0.913806	−	0.406150i	$-0.133129\pi$
$157$	23.4923	−	8.55050i	1.87489	−	0.682404i	0.961085	−	0.276254i	$-0.0890931\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$		0			0
$163$	8.00000			0.626608			0.313304	−	0.949653i	$-0.398564\pi$
$163$	8.00000			0.626608			0.313304	+	0.949653i	$0.398564\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		0			0		−0.342020	−	0.939693i	$-0.611111\pi$
$167$		0			0		0.342020	+	0.939693i	$0.388889\pi$
$168$		0			0
$169$	6.25133	+	35.4531i	0.480872	+	2.72716i
$170$		0			0
$171$		0			0
$172$	5.00000	−	8.66025i	0.381246	−	0.660338i
$173$		0			0		0.342020	−	0.939693i	$-0.388889\pi$
$173$		0			0		−0.342020	+	0.939693i	$0.611111\pi$
$174$		0			0
$175$	15.3209	+	12.8558i	1.15815	+	0.971804i
$176$		0			0
$177$		0			0
$178$		0			0
$179$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$179$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$180$		0			0
$181$	−13.0000	−	22.5167i	−0.966282	−	1.67365i	−0.706129	−	0.708083i	$-0.749560\pi$
$181$	−13.0000	−	22.5167i	−0.966282	−	1.67365i	−0.260153	−	0.965567i	$-0.583773\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$		0			0
$186$		0			0
$187$		0			0
$188$		0			0
$189$		0			0
$190$		0			0
$191$		0			0		−0.642788	−	0.766044i	$-0.722222\pi$
$191$		0			0		0.642788	+	0.766044i	$0.277778\pi$
$192$		0			0
$193$	3.99391	−	22.6506i	0.287488	−	1.63042i	−0.408773	−	0.912636i	$-0.634043\pi$
$193$	3.99391	−	22.6506i	0.287488	−	1.63042i	0.696261	−	0.717788i	$-0.254845\pi$
$194$		0			0
$195$		0			0
$196$	−3.12567	−	17.7265i	−0.223262	−	1.26618i
$197$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$197$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$198$		0			0
$199$	−8.50000	+	14.7224i	−0.602549	+	1.04365i	0.389885	+	0.920864i	$0.372515\pi$
$199$	−8.50000	+	14.7224i	−0.602549	+	1.04365i	−0.992434	+	0.122782i	$0.960818\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$		0			0
$204$		0			0
$205$		0			0
$206$		0			0
$207$		0			0
$208$	14.0000	+	24.2487i	0.970725	+	1.68135i
$209$		0			0
$210$		0			0
$211$	−27.2511	+	9.91858i	−1.87604	+	0.682823i	−0.917630	+	0.397436i	$0.869900\pi$
$211$	−27.2511	+	9.91858i	−1.87604	+	0.682823i	−0.958412	+	0.285388i	$0.907878\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		0			0
$216$		0			0
$217$	−44.0000			−2.98691
$218$		0			0
$219$		0			0
$220$		0			0
$221$		0			0
$222$		0			0
$223$	3.99391	+	22.6506i	0.267452	+	1.51679i	0.761961	+	0.647623i	$0.224237\pi$
$223$	3.99391	+	22.6506i	0.267452	+	1.51679i	−0.494509	+	0.869172i	$0.664652\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$		0			0		0.342020	−	0.939693i	$-0.388889\pi$
$227$		0			0		−0.342020	+	0.939693i	$0.611111\pi$
$228$		0			0
$229$	−5.36231	−	4.49951i	−0.354351	−	0.297336i	0.448183	−	0.893942i	$-0.352071\pi$
$229$	−5.36231	−	4.49951i	−0.354351	−	0.297336i	−0.802535	+	0.596606i	$0.796516\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$233$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$		0			0
$238$		0			0
$239$		0			0		−0.984808	−	0.173648i	$-0.944444\pi$
$239$		0			0		0.984808	+	0.173648i	$0.0555556\pi$
$240$		0			0
$241$	10.7246	−	8.99903i	0.690834	−	0.579678i	−0.228316	−	0.973587i	$-0.573322\pi$
$241$	10.7246	−	8.99903i	0.690834	−	0.579678i	0.919150	+	0.393909i	$0.128877\pi$
$242$		0			0
$243$		0			0
$244$	2.00000			0.128037
$245$		0			0
$246$		0			0
$247$	1.21554	−	6.89365i	0.0773428	−	0.438633i
$248$		0			0
$249$		0			0
$250$		0			0
$251$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$251$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$252$		0			0
$253$		0			0
$254$		0			0
$255$		0			0
$256$	12.2567	+	10.2846i	0.766044	+	0.642788i
$257$		0			0		0.642788	−	0.766044i	$-0.277778\pi$
$257$		0			0		−0.642788	+	0.766044i	$0.722222\pi$
$258$		0			0
$259$	−37.5877	−	13.6808i	−2.33559	−	0.850084i
$260$		0			0
$261$		0			0
$262$		0			0
$263$		0			0		0.984808	−	0.173648i	$-0.0555556\pi$
$263$		0			0		−0.984808	+	0.173648i	$0.944444\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$		0			0
$268$	−7.66044	+	6.42788i	−0.467936	+	0.392645i
$269$		0			0			−	1.00000i	$-0.5\pi$
$269$		0			0				1.00000i	$0.5\pi$
$270$		0			0
$271$	29.0000			1.76162			0.880812	−	0.473466i	$-0.156997\pi$
$271$	29.0000			1.76162			0.880812	+	0.473466i	$0.156997\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$		0			0
$276$		0			0
$277$	−5.38309	−	30.5290i	−0.323439	−	1.83431i	−0.520427	−	0.853906i	$-0.674227\pi$
$277$	−5.38309	−	30.5290i	−0.323439	−	1.83431i	0.196988	−	0.980406i	$-0.436884\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$		0			0		0.342020	−	0.939693i	$-0.388889\pi$
$281$		0			0		−0.342020	+	0.939693i	$0.611111\pi$
$282$		0			0
$283$	−5.36231	−	4.49951i	−0.318756	−	0.267468i	0.469344	−	0.883016i	$-0.344491\pi$
$283$	−5.36231	−	4.49951i	−0.318756	−	0.267468i	−0.788100	+	0.615547i	$0.788935\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$	8.50000	+	14.7224i	0.500000	+	0.866025i
$290$		0			0
$291$		0			0
$292$	−13.1557	+	4.78828i	−0.769879	+	0.280213i
$293$		0			0		−0.984808	−	0.173648i	$-0.944444\pi$
$293$		0			0		0.984808	+	0.173648i	$0.0555556\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$		0			0
$298$		0			0
$299$		0			0
$300$		0			0
$301$	−3.47296	+	19.6962i	−0.200178	+	1.13527i
$302$		0			0
$303$		0			0
$304$	−0.694593	−	3.93923i	−0.0398376	−	0.225930i
$305$		0			0
$306$		0			0
$307$	8.00000	−	13.8564i	0.456584	−	0.790827i	−0.542194	−	0.840254i	$-0.682406\pi$
$307$	8.00000	−	13.8564i	0.456584	−	0.790827i	0.998778	+	0.0494267i	$0.0157394\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$		0			0		0.642788	−	0.766044i	$-0.277778\pi$
$311$		0			0		−0.642788	+	0.766044i	$0.722222\pi$
$312$		0			0
$313$	20.6732	+	7.52444i	1.16852	+	0.425307i	0.852134	−	0.523324i	$-0.175308\pi$
$313$	20.6732	+	7.52444i	1.16852	+	0.425307i	0.316387	+	0.948630i	$0.397530\pi$
$314$		0			0
$315$		0			0
$316$	−13.0000	−	22.5167i	−0.731307	−	1.26666i
$317$		0			0		0.984808	−	0.173648i	$-0.0555556\pi$
$317$		0			0		−0.984808	+	0.173648i	$0.944444\pi$
$318$		0			0
$319$		0			0
$320$		0			0
$321$		0			0
$322$		0			0
$323$		0			0
$324$		0			0
$325$	35.0000			1.94145
$326$		0			0
$327$		0			0
$328$		0			0
$329$		0			0
$330$		0			0
$331$	−5.38309	−	30.5290i	−0.295882	−	1.67803i	−0.663598	−	0.748090i	$-0.730971\pi$
$331$	−5.38309	−	30.5290i	−0.295882	−	1.67803i	0.367716	−	0.929938i	$-0.380140\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$		0			0
$336$		0			0
$337$	−26.0455	−	21.8548i	−1.41879	−	1.19051i	−0.951985	−	0.306145i	$-0.900961\pi$
$337$	−26.0455	−	21.8548i	−1.41879	−	1.19051i	−0.466805	−	0.884361i	$-0.654595\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$		0			0
$342$		0			0
$343$	4.00000	+	6.92820i	0.215980	+	0.374088i
$344$		0			0
$345$		0			0
$346$		0			0
$347$		0			0		−0.984808	−	0.173648i	$-0.944444\pi$
$347$		0			0		0.984808	+	0.173648i	$0.0555556\pi$
$348$		0			0
$349$	10.7246	−	8.99903i	0.574076	−	0.481707i	−0.308920	−	0.951088i	$-0.599967\pi$
$349$	10.7246	−	8.99903i	0.574076	−	0.481707i	0.882996	+	0.469381i	$0.155523\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$		0			0		−0.642788	−	0.766044i	$-0.722222\pi$
$353$		0			0		0.642788	+	0.766044i	$0.277778\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		0			0
$358$		0			0
$359$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$359$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$360$		0			0
$361$	9.00000	−	15.5885i	0.473684	−	0.820445i
$362$		0			0
$363$		0			0
$364$	−42.8985	−	35.9961i	−2.24849	−	1.88671i
$365$		0			0
$366$		0			0
$367$	−32.8892	−	11.9707i	−1.71680	−	0.624866i	−0.719249	−	0.694752i	$-0.755514\pi$
$367$	−32.8892	−	11.9707i	−1.71680	−	0.624866i	−0.997555	+	0.0698862i	$0.977736\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		0			0
$372$		0			0
$373$	23.4923	−	8.55050i	1.21639	−	0.442728i	0.347472	−	0.937691i	$-0.387040\pi$
$373$	23.4923	−	8.55050i	1.21639	−	0.442728i	0.868914	+	0.494962i	$0.164818\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		0			0
$378$		0			0
$379$	8.00000			0.410932			0.205466	−	0.978664i	$-0.434129\pi$
$379$	8.00000			0.410932			0.205466	+	0.978664i	$0.434129\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$		0			0		−0.342020	−	0.939693i	$-0.611111\pi$
$383$		0			0		0.342020	+	0.939693i	$0.388889\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$		0			0
$388$	5.00000	−	8.66025i	0.253837	−	0.439658i
$389$		0			0		0.342020	−	0.939693i	$-0.388889\pi$
$389$		0			0		−0.342020	+	0.939693i	$0.611111\pi$
$390$		0			0
$391$		0			0
$392$		0			0
$393$		0			0
$394$		0			0
$395$		0			0
$396$		0			0
$397$	0.500000	+	0.866025i	0.0250943	+	0.0434646i	0.878300	−	0.478110i	$-0.158678\pi$
$397$	0.500000	+	0.866025i	0.0250943	+	0.0434646i	−0.853206	+	0.521575i	$0.825345\pi$
$398$		0			0
$399$		0			0
$400$	18.7939	−	6.84040i	0.939693	−	0.342020i
$401$		0			0		−0.984808	−	0.173648i	$-0.944444\pi$
$401$		0			0		0.984808	+	0.173648i	$0.0555556\pi$
$402$		0			0
$403$	−58.9854	+	49.4946i	−2.93827	+	2.46550i
$404$		0			0
$405$		0			0
$406$		0			0
$407$		0			0
$408$		0			0
$409$	6.59863	−	37.4227i	0.326281	−	1.85043i	−0.174232	−	0.984705i	$-0.555744\pi$
$409$	6.59863	−	37.4227i	0.326281	−	1.85043i	0.500514	−	0.865729i	$-0.333144\pi$
$410$		0			0
$411$		0			0
$412$	4.51485	+	25.6050i	0.222431	+	1.26147i
$413$		0			0
$414$		0			0
$415$		0			0
$416$		0			0
$417$		0			0
$418$		0			0
$419$		0			0		0.642788	−	0.766044i	$-0.277778\pi$
$419$		0			0		−0.642788	+	0.766044i	$0.722222\pi$
$420$		0			0
$421$	20.6732	+	7.52444i	1.00755	+	0.366719i	0.792492	−	0.609882i	$-0.208783\pi$
$421$	20.6732	+	7.52444i	1.00755	+	0.366719i	0.215060	+	0.976601i	$0.431005\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$		0			0
$426$		0			0
$427$	−3.75877	+	1.36808i	−0.181900	+	0.0662061i
$428$		0			0
$429$		0			0
$430$		0			0
$431$		0			0			−	1.00000i	$-0.5\pi$
$431$		0			0				1.00000i	$0.5\pi$
$432$		0			0
$433$	35.0000			1.68199			0.840996	−	0.541041i	$-0.181970\pi$
$433$	35.0000			1.68199			0.840996	+	0.541041i	$0.181970\pi$
$434$		0			0
$435$		0			0
$436$	6.59863	−	37.4227i	0.316017	−	1.79222i
$437$		0			0
$438$		0			0
$439$	−4.86215	−	27.5746i	−0.232058	−	1.31606i	−0.848722	−	0.528839i	$-0.822628\pi$
$439$	−4.86215	−	27.5746i	−0.232058	−	1.31606i	0.616665	−	0.787226i	$-0.288483\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$		0			0		0.342020	−	0.939693i	$-0.388889\pi$
$443$		0			0		−0.342020	+	0.939693i	$0.611111\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$		0			0
$448$	−30.0702	−	10.9446i	−1.42068	−	0.517086i
$449$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$449$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$450$		0			0
$451$		0			0
$452$		0			0
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$	31.4078	−	26.3543i	1.46920	−	1.23280i	0.552318	−	0.833633i	$-0.313743\pi$
$457$	31.4078	−	26.3543i	1.46920	−	1.23280i	0.916878	−	0.399169i	$-0.130701\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$		0			0		−0.642788	−	0.766044i	$-0.722222\pi$
$461$		0			0		0.642788	+	0.766044i	$0.277778\pi$
$462$		0			0
$463$	−7.46687	+	42.3467i	−0.347015	+	1.96802i	−0.133871	+	0.990999i	$0.542741\pi$
$463$	−7.46687	+	42.3467i	−0.347015	+	1.96802i	−0.213144	+	0.977021i	$0.568370\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$467$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$468$		0			0
$469$	10.0000	−	17.3205i	0.461757	−	0.799787i
$470$		0			0
$471$		0			0
$472$		0			0
$473$		0			0
$474$		0			0
$475$	−4.69846	−	1.71010i	−0.215580	−	0.0784648i
$476$		0			0
$477$		0			0
$478$		0			0
$479$		0			0		0.984808	−	0.173648i	$-0.0555556\pi$
$479$		0			0		−0.984808	+	0.173648i	$0.944444\pi$
$480$		0			0
$481$	−65.7785	+	23.9414i	−2.99924	+	1.09163i
$482$		0			0
$483$		0			0
$484$	16.8530	−	14.1413i	0.766044	−	0.642788i
$485$		0			0
$486$		0			0
$487$	−19.0000			−0.860972			−0.430486	−	0.902597i	$-0.641658\pi$
$487$	−19.0000			−0.860972			−0.430486	+	0.902597i	$0.641658\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$		0			0		−0.342020	−	0.939693i	$-0.611111\pi$
$491$		0			0		0.342020	+	0.939693i	$0.388889\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$		0			0
$496$	−22.0000	+	38.1051i	−0.987829	+	1.71097i
$497$		0			0
$498$		0			0
$499$	24.5134	+	20.5692i	1.09737	+	0.920804i	0.997246	−	0.0741708i	$-0.0236310\pi$
$499$	24.5134	+	20.5692i	1.09737	+	0.920804i	0.100126	+	0.994975i	$0.468075\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$503$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$		0			0
$508$	−1.87939	+	0.684040i	−0.0833842	+	0.0303494i
$509$		0			0		−0.984808	−	0.173648i	$-0.944444\pi$
$509$		0			0		0.984808	+	0.173648i	$0.0555556\pi$
$510$		0			0
$511$	21.4492	−	17.9981i	0.948859	−	0.796187i
$512$		0			0
$513$		0			0
$514$		0			0
$515$		0			0
$516$		0			0
$517$		0			0
$518$		0			0
$519$		0			0
$520$		0			0
$521$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$521$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$522$		0			0
$523$	21.5000	−	37.2391i	0.940129	−	1.62835i	0.174908	−	0.984585i	$-0.444037\pi$
$523$	21.5000	−	37.2391i	0.940129	−	1.62835i	0.765222	−	0.643767i	$-0.222629\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$		0			0
$528$		0			0
$529$	21.6129	+	7.86646i	0.939693	+	0.342020i
$530$		0			0
$531$		0			0
$532$	4.00000	+	6.92820i	0.173422	+	0.300376i
$533$		0			0
$534$		0			0
$535$		0			0
$536$		0			0
$537$		0			0
$538$		0			0
$539$		0			0
$540$		0			0
$541$	−46.0000			−1.97769			−0.988847	−	0.148933i	$-0.952416\pi$
$541$	−46.0000			−1.97769			−0.988847	+	0.148933i	$0.952416\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$		0			0
$546$		0			0
$547$	−0.173648	−	0.984808i	−0.00742466	−	0.0421073i	0.980870	−	0.194662i	$-0.0623610\pi$
$547$	−0.173648	−	0.984808i	−0.00742466	−	0.0421073i	−0.988295	+	0.152555i	$0.951250\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$		0			0
$552$		0			0
$553$	39.8343	+	33.4250i	1.69393	+	1.42137i
$554$		0			0
$555$		0			0
$556$	−30.0702	−	10.9446i	−1.27526	−	0.464156i
$557$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$557$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$558$		0			0
$559$	17.5000	+	30.3109i	0.740171	+	1.28201i
$560$		0			0
$561$		0			0
$562$		0			0
$563$		0			0		−0.984808	−	0.173648i	$-0.944444\pi$
$563$		0			0		0.984808	+	0.173648i	$0.0555556\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$		0			0
$568$		0			0
$569$		0			0		−0.642788	−	0.766044i	$-0.722222\pi$
$569$		0			0		0.642788	+	0.766044i	$0.277778\pi$
$570$		0			0
$571$	−2.77837	+	15.7569i	−0.116271	+	0.659407i	0.869842	+	0.493331i	$0.164221\pi$
$571$	−2.77837	+	15.7569i	−0.116271	+	0.659407i	−0.986113	+	0.166076i	$0.946890\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$		0			0
$577$	−5.50000	+	9.52628i	−0.228968	+	0.396584i	−0.957503	−	0.288425i	$-0.906868\pi$
$577$	−5.50000	+	9.52628i	−0.228968	+	0.396584i	0.728535	+	0.685009i	$0.240202\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$		0			0
$582$		0			0
$583$		0			0
$584$		0			0
$585$		0			0
$586$		0			0
$587$		0			0		0.984808	−	0.173648i	$-0.0555556\pi$
$587$		0			0		−0.984808	+	0.173648i	$0.944444\pi$
$588$		0			0
$589$	10.3366	−	3.76222i	0.425913	−	0.155020i
$590$		0			0
$591$		0			0
$592$	−30.6418	+	25.7115i	−1.25937	+	1.05674i
$593$		0			0			−	1.00000i	$-0.5\pi$
$593$		0			0				1.00000i	$0.5\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$		0			0
$598$		0			0
$599$		0			0		−0.342020	−	0.939693i	$-0.611111\pi$
$599$		0			0		0.342020	+	0.939693i	$0.388889\pi$
$600$		0			0
$601$	3.99391	+	22.6506i	0.162915	+	0.923936i	0.951188	+	0.308611i	$0.0998642\pi$
$601$	3.99391	+	22.6506i	0.162915	+	0.923936i	−0.788273	+	0.615325i	$0.789025\pi$
$602$		0			0
$603$		0			0
$604$	−19.0000	+	32.9090i	−0.773099	+	1.33905i
$605$		0			0
$606$		0			0
$607$	15.3209	+	12.8558i	0.621856	+	0.521799i	0.898386	−	0.439206i	$-0.144740\pi$
$607$	15.3209	+	12.8558i	0.621856	+	0.521799i	−0.276531	+	0.961005i	$0.589185\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		0			0
$612$		0			0
$613$	−23.5000	−	40.7032i	−0.949156	−	1.64399i	−0.747208	−	0.664590i	$-0.768606\pi$
$613$	−23.5000	−	40.7032i	−0.949156	−	1.64399i	−0.201948	−	0.979396i	$-0.564727\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$		0			0		−0.984808	−	0.173648i	$-0.944444\pi$
$617$		0			0		0.984808	+	0.173648i	$0.0555556\pi$
$618$		0			0
$619$	13.0228	−	10.9274i	0.523429	−	0.439209i	−0.342396	−	0.939556i	$-0.611239\pi$
$619$	13.0228	−	10.9274i	0.523429	−	0.439209i	0.865825	+	0.500347i	$0.166794\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$		0			0
$625$	4.34120	−	24.6202i	0.173648	−	0.984808i
$626$		0			0
$627$		0			0
$628$	8.68241	+	49.2404i	0.346466	+	1.96491i
$629$		0			0
$630$		0			0
$631$	−22.0000	+	38.1051i	−0.875806	+	1.51694i	−0.0199047	+	0.999802i	$0.506336\pi$
$631$	−22.0000	+	38.1051i	−0.875806	+	1.51694i	−0.855901	+	0.517139i	$0.826997\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$		0			0
$636$		0			0
$637$	59.2006	+	21.5473i	2.34561	+	0.853734i
$638$		0			0
$639$		0			0
$640$		0			0
$641$		0			0		0.984808	−	0.173648i	$-0.0555556\pi$
$641$		0			0		−0.984808	+	0.173648i	$0.944444\pi$
$642$		0			0
$643$	37.5877	−	13.6808i	1.48231	−	0.539518i	0.530901	−	0.847434i	$-0.321854\pi$
$643$	37.5877	−	13.6808i	1.48231	−	0.539518i	0.951414	+	0.307916i	$0.0996315\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		0			0			−	1.00000i	$-0.5\pi$
$647$		0			0				1.00000i	$0.5\pi$
$648$		0			0
$649$		0			0
$650$		0			0
$651$		0			0
$652$	−2.77837	+	15.7569i	−0.108809	+	0.617089i
$653$		0			0		−0.342020	−	0.939693i	$-0.611111\pi$
$653$		0			0		0.342020	+	0.939693i	$0.388889\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$		0			0
$658$		0			0
$659$		0			0		0.342020	−	0.939693i	$-0.388889\pi$
$659$		0			0		−0.342020	+	0.939693i	$0.611111\pi$
$660$		0			0
$661$	−37.5362	−	31.4966i	−1.45999	−	1.22508i	−0.924862	−	0.380303i	$-0.875820\pi$
$661$	−37.5362	−	31.4966i	−1.45999	−	1.22508i	−0.535126	−	0.844772i	$-0.679736\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$		0			0
$668$		0			0
$669$		0			0
$670$		0			0
$671$		0			0
$672$		0			0
$673$	−9.95858	+	8.35624i	−0.383875	+	0.322109i	−0.814221	−	0.580554i	$-0.802836\pi$
$673$	−9.95858	+	8.35624i	−0.383875	+	0.322109i	0.430346	+	0.902664i	$0.358391\pi$
$674$		0			0
$675$		0			0
$676$	−72.0000			−2.76923
$677$		0			0		−0.642788	−	0.766044i	$-0.722222\pi$
$677$		0			0		0.642788	+	0.766044i	$0.277778\pi$
$678$		0			0
$679$	−3.47296	+	19.6962i	−0.133280	+	0.755869i
$680$		0			0
$681$		0			0
$682$		0			0
$683$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$683$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$		0			0
$688$	15.3209	+	12.8558i	0.584103	+	0.490121i
$689$		0			0
$690$		0			0
$691$	46.0449	+	16.7590i	1.75163	+	0.637542i	0.999763	−	0.0217706i	$-0.00693034\pi$
$691$	46.0449	+	16.7590i	1.75163	+	0.637542i	0.751869	+	0.659313i	$0.229153\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$		0			0
$696$		0			0
$697$		0			0
$698$		0			0
$699$		0			0
$700$	−30.6418	+	25.7115i	−1.15815	+	0.971804i
$701$		0			0			−	1.00000i	$-0.5\pi$
$701$		0			0				1.00000i	$0.5\pi$
$702$		0			0
$703$	10.0000			0.377157
$704$		0			0
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$	−5.38309	−	30.5290i	−0.202166	−	1.14654i	−0.901837	−	0.432077i	$-0.857781\pi$
$709$	−5.38309	−	30.5290i	−0.202166	−	1.14654i	0.699671	−	0.714466i	$-0.253330\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$		0			0
$714$		0			0
$715$		0			0
$716$		0			0
$717$		0			0
$718$		0			0
$719$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$719$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$720$		0			0
$721$	−26.0000	−	45.0333i	−0.968291	−	1.67713i
$722$		0			0
$723$		0			0
$724$	48.8640	−	17.7850i	1.81602	−	0.660976i
$725$		0			0
$726$		0			0
$727$	33.7060	−	28.2827i	1.25008	−	1.04895i	0.253419	−	0.967357i	$-0.418445\pi$
$727$	33.7060	−	28.2827i	1.25008	−	1.04895i	0.996666	−	0.0815889i	$-0.0259995\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$		0			0
$732$		0			0
$733$	−7.46687	+	42.3467i	−0.275795	+	1.56411i	0.460629	+	0.887593i	$0.347624\pi$
$733$	−7.46687	+	42.3467i	−0.275795	+	1.56411i	−0.736425	+	0.676520i	$0.763487\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		0			0
$738$		0			0
$739$	18.5000	−	32.0429i	0.680534	−	1.17872i	−0.294285	−	0.955718i	$-0.595081\pi$
$739$	18.5000	−	32.0429i	0.680534	−	1.17872i	0.974818	−	0.223001i	$-0.0715853\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$		0			0		0.642788	−	0.766044i	$-0.277778\pi$
$743$		0			0		−0.642788	+	0.766044i	$0.722222\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$	48.8640	−	17.7850i	1.78307	−	0.648986i	0.783452	−	0.621452i	$-0.213457\pi$
$751$	48.8640	−	17.7850i	1.78307	−	0.648986i	0.999621	−	0.0275338i	$-0.00876539\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		0			0
$756$		0			0
$757$	29.0000			1.05402			0.527011	−	0.849858i	$-0.323312\pi$
$757$	29.0000			1.05402			0.527011	+	0.849858i	$0.323312\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$		0			0		−0.342020	−	0.939693i	$-0.611111\pi$
$761$		0			0		0.342020	+	0.939693i	$0.388889\pi$
$762$		0			0
$763$	13.1973	+	74.8454i	0.477773	+	2.70959i
$764$		0			0
$765$		0			0
$766$		0			0
$767$		0			0
$768$		0			0
$769$	36.0041	+	30.2110i	1.29834	+	1.08944i	0.990429	+	0.138022i	$0.0440745\pi$
$769$	36.0041	+	30.2110i	1.29834	+	1.08944i	0.307912	+	0.951415i	$0.400370\pi$
$770$		0			0
$771$		0			0
$772$	43.2259	+	15.7329i	1.55573	+	0.566240i
$773$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$773$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$774$		0			0
$775$	27.5000	+	47.6314i	0.987829	+	1.71097i
$776$		0			0
$777$		0			0
$778$		0			0
$779$		0			0
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$	36.0000			1.28571
$785$		0			0
$786$		0			0
$787$	−5.38309	+	30.5290i	−0.191887	+	1.08824i	0.724897	+	0.688858i	$0.241887\pi$
$787$	−5.38309	+	30.5290i	−0.191887	+	1.08824i	−0.916783	+	0.399385i	$0.869224\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$	−3.50000	+	6.06218i	−0.124289	+	0.215274i
$794$		0			0
$795$		0			0
$796$	−26.0455	−	21.8548i	−0.923159	−	0.774622i
$797$		0			0		0.642788	−	0.766044i	$-0.277778\pi$
$797$		0			0		−0.642788	+	0.766044i	$0.722222\pi$
$798$		0			0
$799$		0			0
$800$		0			0
$801$		0			0
$802$		0			0
$803$		0			0
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$		0			0			−	1.00000i	$-0.5\pi$
$809$		0			0				1.00000i	$0.5\pi$
$810$		0			0
$811$	−19.0000			−0.667180			−0.333590	−	0.942718i	$-0.608260\pi$
$811$	−19.0000			−0.667180			−0.333590	+	0.942718i	$0.608260\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$		0			0
$816$		0			0
$817$	−0.868241	−	4.92404i	−0.0303759	−	0.172270i
$818$		0			0
$819$		0			0
$820$		0			0
$821$		0			0		0.342020	−	0.939693i	$-0.388889\pi$
$821$		0			0		−0.342020	+	0.939693i	$0.611111\pi$
$822$		0			0
$823$	36.0041	+	30.2110i	1.25502	+	1.05309i	0.996194	+	0.0871670i	$0.0277814\pi$
$823$	36.0041	+	30.2110i	1.25502	+	1.05309i	0.258830	+	0.965923i	$0.416663\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$827$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$828$		0			0
$829$	−26.5000	−	45.8993i	−0.920383	−	1.59415i	−0.798823	−	0.601566i	$-0.794544\pi$
$829$	−26.5000	−	45.8993i	−0.920383	−	1.59415i	−0.121560	−	0.992584i	$-0.538790\pi$
$830$		0			0
$831$		0			0
$832$	−52.6228	+	19.1531i	−1.82437	+	0.664015i
$833$		0			0
$834$		0			0
$835$		0			0
$836$		0			0
$837$		0			0
$838$		0			0
$839$		0			0		−0.642788	−	0.766044i	$-0.722222\pi$
$839$		0			0		0.642788	+	0.766044i	$0.277778\pi$
$840$		0			0
$841$	−5.03580	+	28.5594i	−0.173648	+	0.984808i
$842$		0			0
$843$		0			0
$844$	−10.0716	−	57.1188i	−0.346679	−	1.96611i
$845$		0			0
$846$		0			0
$847$	−22.0000	+	38.1051i	−0.755929	+	1.30931i
$848$		0			0
$849$		0			0
$850$		0			0
$851$		0			0
$852$		0			0
$853$	−32.8892	−	11.9707i	−1.12611	−	0.409869i	−0.289229	−	0.957260i	$-0.593399\pi$
$853$	−32.8892	−	11.9707i	−1.12611	−	0.409869i	−0.836877	+	0.547391i	$0.815621\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$		0			0		0.984808	−	0.173648i	$-0.0555556\pi$
$857$		0			0		−0.984808	+	0.173648i	$0.944444\pi$
$858$		0			0
$859$	−52.6228	+	19.1531i	−1.79547	+	0.653496i	−0.796672	+	0.604412i	$0.793408\pi$
$859$	−52.6228	+	19.1531i	−1.79547	+	0.653496i	−0.998795	+	0.0490840i	$0.984370\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		0			0			−	1.00000i	$-0.5\pi$
$863$		0			0				1.00000i	$0.5\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$		0			0
$868$	15.2810	−	86.6631i	0.518672	−	2.94154i
$869$		0			0
$870$		0			0
$871$	−6.07769	−	34.4683i	−0.205935	−	1.16791i
$872$		0			0
$873$		0			0
$874$		0			0
$875$		0			0
$876$		0			0
$877$	−26.0455	−	21.8548i	−0.879494	−	0.737983i	0.0865807	−	0.996245i	$-0.472406\pi$
$877$	−26.0455	−	21.8548i	−0.879494	−	0.737983i	−0.966075	+	0.258261i	$0.916850\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$881$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$882$		0			0
$883$	27.5000	+	47.6314i	0.925449	+	1.60292i	0.790838	+	0.612026i	$0.209645\pi$
$883$	27.5000	+	47.6314i	0.925449	+	1.60292i	0.134611	+	0.990899i	$0.457022\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$		0			0		−0.984808	−	0.173648i	$-0.944444\pi$
$887$		0			0		0.984808	+	0.173648i	$0.0555556\pi$
$888$		0			0
$889$	3.06418	−	2.57115i	0.102769	−	0.0862336i
$890$		0			0
$891$		0			0
$892$	−46.0000			−1.54019
$893$		0			0
$894$		0			0
$895$		0			0
$896$		0			0
$897$		0			0
$898$		0			0
$899$		0			0
$900$		0			0
$901$		0			0
$902$		0			0
$903$		0			0
$904$		0			0
$905$		0			0
$906$		0			0
$907$	−55.4419	−	20.1792i	−1.84092	−	0.670039i	−0.989304	−	0.145868i	$-0.953403\pi$
$907$	−55.4419	−	20.1792i	−1.84092	−	0.670039i	−0.851613	−	0.524171i	$-0.824375\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$		0			0		0.984808	−	0.173648i	$-0.0555556\pi$
$911$		0			0		−0.984808	+	0.173648i	$0.944444\pi$
$912$		0			0
$913$		0			0
$914$		0			0
$915$		0			0
$916$	10.7246	−	8.99903i	0.354351	−	0.297336i
$917$		0			0
$918$		0			0
$919$	−52.0000			−1.71532			−0.857661	−	0.514216i	$-0.828083\pi$
$919$	−52.0000			−1.71532			−0.857661	+	0.514216i	$0.828083\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$		0			0
$924$		0			0
$925$	8.68241	+	49.2404i	0.285476	+	1.61901i
$926$		0			0
$927$		0			0
$928$		0			0
$929$		0			0		0.342020	−	0.939693i	$-0.388889\pi$
$929$		0			0		−0.342020	+	0.939693i	$0.611111\pi$
$930$		0			0
$931$	−6.89440	−	5.78509i	−0.225955	−	0.189599i
$932$		0			0
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$	−13.0000	−	22.5167i	−0.424691	−	0.735587i	0.571700	−	0.820463i	$-0.306284\pi$
$937$	−13.0000	−	22.5167i	−0.424691	−	0.735587i	−0.996392	+	0.0848755i	$0.972951\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$		0			0		−0.984808	−	0.173648i	$-0.944444\pi$
$941$		0			0		0.984808	+	0.173648i	$0.0555556\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$		0			0
$946$		0			0
$947$		0			0		−0.642788	−	0.766044i	$-0.722222\pi$
$947$		0			0		0.642788	+	0.766044i	$0.277778\pi$
$948$		0			0
$949$	8.50876	−	48.2556i	0.276206	−	1.56644i
$950$		0			0
$951$		0			0
$952$		0			0
$953$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$953$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	−84.5723	−	30.7818i	−2.72814	−	0.992962i
$962$		0			0
$963$		0			0
$964$	14.0000	+	24.2487i	0.450910	+	0.780998i
$965$		0			0
$966$		0			0
$967$	−38.5274	+	14.0228i	−1.23896	+	0.450944i	−0.876656	−	0.481117i	$-0.840231\pi$
$967$	−38.5274	+	14.0228i	−1.23896	+	0.450944i	−0.362301	+	0.932061i	$0.618009\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$		0			0			−	1.00000i	$-0.5\pi$
$971$		0			0				1.00000i	$0.5\pi$
$972$		0			0
$973$	64.0000			2.05175
$974$		0			0
$975$		0			0
$976$	−0.694593	+	3.93923i	−0.0222334	+	0.126092i
$977$		0			0		−0.342020	−	0.939693i	$-0.611111\pi$
$977$		0			0		0.342020	+	0.939693i	$0.388889\pi$
$978$		0			0
$979$		0			0
$980$		0			0
$981$		0			0
$982$		0			0
$983$		0			0		0.342020	−	0.939693i	$-0.388889\pi$
$983$		0			0		−0.342020	+	0.939693i	$0.611111\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$	13.1557	+	4.78828i	0.418538	+	0.152336i
$989$		0			0
$990$		0			0
$991$	30.5000	+	52.8275i	0.968864	+	1.67812i	0.698853	+	0.715265i	$0.253694\pi$
$991$	30.5000	+	52.8275i	0.968864	+	1.67812i	0.270011	+	0.962857i	$0.412973\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$		0			0
$996$		0			0
$997$	−7.66044	+	6.42788i	−0.242609	+	0.203573i	−0.755982	−	0.654593i	$-0.772840\pi$
$997$	−7.66044	+	6.42788i	−0.242609	+	0.203573i	0.513373	+	0.858166i	$0.328396\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	729.2.e.e.325.1		6
3.2	odd	2	CM	729.2.e.e.325.1		6
9.2	odd	6	inner	729.2.e.e.568.1		6
9.4	even	3	inner	729.2.e.e.82.1		6
9.5	odd	6	inner	729.2.e.e.82.1		6
9.7	even	3	inner	729.2.e.e.568.1		6
27.2	odd	18	inner	729.2.e.e.163.1		6
27.4	even	9		243.2.a.a.1.1	✓	1
27.5	odd	18		243.2.c.b.82.1		2
27.7	even	9	inner	729.2.e.e.649.1		6
27.11	odd	18	inner	729.2.e.e.406.1		6
27.13	even	9		243.2.c.b.163.1		2
27.14	odd	18		243.2.c.b.163.1		2
27.16	even	9	inner	729.2.e.e.406.1		6
27.20	odd	18	inner	729.2.e.e.649.1		6
27.22	even	9		243.2.c.b.82.1		2
27.23	odd	18		243.2.a.a.1.1	✓	1
27.25	even	9	inner	729.2.e.e.163.1		6
108.23	even	18		3888.2.a.n.1.1		1
108.31	odd	18		3888.2.a.n.1.1		1
135.4	even	18		6075.2.a.w.1.1		1
135.104	odd	18		6075.2.a.w.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
243.2.a.a.1.1	✓	1	27.4	even	9
243.2.a.a.1.1	✓	1	27.23	odd	18
243.2.c.b.82.1		2	27.5	odd	18
243.2.c.b.82.1		2	27.22	even	9
243.2.c.b.163.1		2	27.13	even	9
243.2.c.b.163.1		2	27.14	odd	18
729.2.e.e.82.1		6	9.4	even	3	inner
729.2.e.e.82.1		6	9.5	odd	6	inner
729.2.e.e.163.1		6	27.2	odd	18	inner
729.2.e.e.163.1		6	27.25	even	9	inner
729.2.e.e.325.1		6	1.1	even	1	trivial
729.2.e.e.325.1		6	3.2	odd	2	CM
729.2.e.e.406.1		6	27.11	odd	18	inner
729.2.e.e.406.1		6	27.16	even	9	inner
729.2.e.e.568.1		6	9.2	odd	6	inner
729.2.e.e.568.1		6	9.7	even	3	inner
729.2.e.e.649.1		6	27.7	even	9	inner
729.2.e.e.649.1		6	27.20	odd	18	inner
3888.2.a.n.1.1		1	108.23	even	18
3888.2.a.n.1.1		1	108.31	odd	18
6075.2.a.w.1.1		1	135.4	even	18
6075.2.a.w.1.1		1	135.104	odd	18

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 729 = 3^{6} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	729.e (of order \(9\), degree \(6\), not minimal)

\(f(q)\)	\(=\)	\(q+(-0.347296 + 1.96962i) q^{4} +(-0.694593 - 3.93923i) q^{7} +(-5.36231 - 4.49951i) q^{13} +(-3.75877 - 1.36808i) q^{16} +(0.500000 + 0.866025i) q^{19} +(-3.83022 + 3.21394i) q^{25} +8.00000 q^{28} +(1.91013 - 10.8329i) q^{31} +(5.00000 - 8.66025i) q^{37} +(-4.69846 - 1.71010i) q^{43} +(-8.45723 + 3.07818i) q^{49} +(10.7246 - 8.99903i) q^{52} +(-0.173648 - 0.984808i) q^{61} +(4.00000 - 6.92820i) q^{64} +(3.83022 + 3.21394i) q^{67} +(3.50000 + 6.06218i) q^{73} +(-1.87939 + 0.684040i) q^{76} +(-9.95858 + 8.35624i) q^{79} +(-14.0000 + 24.2487i) q^{91} +(-4.69846 - 1.71010i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 3 q^{19} + 48 q^{28} + 30 q^{37} + 24 q^{64} + 21 q^{73} - 84 q^{91}+O(q^{100}) \) 6 * q + 3 * q^19 + 48 * q^28 + 30 * q^37 + 24 * q^64 + 21 * q^73 - 84 * q^91

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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