LMFDB - Embedded newform 729.2.e.e.163.1

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			163.1
Root			$-0.173648 + 0.984808i$ of defining polynomial
Character	$\chi$	$=$	729.163
Dual form			729.2.e.e.568.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+(-1.53209 + 1.28558i) q^{4} +(-3.06418 - 2.57115i) q^{7} +(6.57785 + 2.39414i) q^{13} +(0.694593 - 3.93923i) q^{16} +(0.500000 - 0.866025i) q^{19} +(4.69846 - 1.71010i) q^{25} +8.00000 q^{28} +(8.42649 - 7.07066i) q^{31} +(5.00000 + 8.66025i) q^{37} +(0.868241 - 4.92404i) q^{43} +(1.56283 + 8.86327i) q^{49} +(-13.1557 + 4.78828i) q^{52} +(-0.766044 - 0.642788i) q^{61} +(4.00000 + 6.92820i) q^{64} +(-4.69846 - 1.71010i) q^{67} +(3.50000 - 6.06218i) q^{73} +(0.347296 + 1.96962i) q^{76} +(12.2160 - 4.44626i) q^{79} +(-14.0000 - 24.2487i) q^{91} +(0.868241 - 4.92404i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $\left(\mathbb{Z}/729\mathbb{Z}\right)^\times$.

$n$	$2$
$\chi(n)$	$e\left(\frac{8}{9}\right)$

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		−0.342020	−	0.939693i	$-0.611111\pi$
$2$		0			0		0.342020	+	0.939693i	$0.388889\pi$
$3$		0			0
$3$		0			0
$4$	−1.53209	+	1.28558i	−0.766044	+	0.642788i
$5$		0			0		0.984808	−	0.173648i	$-0.0555556\pi$
$5$		0			0		−0.984808	+	0.173648i	$0.944444\pi$
$6$		0			0
$7$	−3.06418	−	2.57115i	−1.15815	−	0.971804i	−0.158272	−	0.987396i	$-0.550592\pi$
$7$	−3.06418	−	2.57115i	−1.15815	−	0.971804i	−0.999878	+	0.0155920i	$0.995037\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$		0			0		−0.984808	−	0.173648i	$-0.944444\pi$
$11$		0			0		0.984808	+	0.173648i	$0.0555556\pi$
$12$		0			0
$13$	6.57785	+	2.39414i	1.82437	+	0.664015i	0.994334	+	0.106301i	$0.0339006\pi$
$13$	6.57785	+	2.39414i	1.82437	+	0.664015i	0.830033	+	0.557714i	$0.188322\pi$
$14$		0			0
$15$		0			0
$16$	0.694593	−	3.93923i	0.173648	−	0.984808i
$17$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$17$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$18$		0			0
$19$	0.500000	−	0.866025i	0.114708	−	0.198680i	−0.802955	−	0.596040i	$-0.796740\pi$
$19$	0.500000	−	0.866025i	0.114708	−	0.198680i	0.917663	+	0.397360i	$0.130073\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$		0			0		−0.642788	−	0.766044i	$-0.722222\pi$
$23$		0			0		0.642788	+	0.766044i	$0.277778\pi$
$24$		0			0
$25$	4.69846	−	1.71010i	0.939693	−	0.342020i
$26$		0			0
$27$		0			0
$28$	8.00000			1.51186
$29$		0			0		−0.342020	−	0.939693i	$-0.611111\pi$
$29$		0			0		0.342020	+	0.939693i	$0.388889\pi$
$30$		0			0
$31$	8.42649	−	7.07066i	1.51344	−	1.26993i	0.656740	−	0.754117i	$-0.271935\pi$
$31$	8.42649	−	7.07066i	1.51344	−	1.26993i	0.856702	−	0.515812i	$-0.172510\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		0			0
$36$		0			0
$37$	5.00000	+	8.66025i	0.821995	+	1.42374i	0.904194	+	0.427121i	$0.140472\pi$
$37$	5.00000	+	8.66025i	0.821995	+	1.42374i	−0.0821995	+	0.996616i	$0.526194\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$		0			0		0.342020	−	0.939693i	$-0.388889\pi$
$41$		0			0		−0.342020	+	0.939693i	$0.611111\pi$
$42$		0			0
$43$	0.868241	−	4.92404i	0.132405	−	0.750909i	−0.844226	−	0.535988i	$-0.819939\pi$
$43$	0.868241	−	4.92404i	0.132405	−	0.750909i	0.976631	−	0.214921i	$-0.0689495\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$		0			0		0.642788	−	0.766044i	$-0.277778\pi$
$47$		0			0		−0.642788	+	0.766044i	$0.722222\pi$
$48$		0			0
$49$	1.56283	+	8.86327i	0.223262	+	1.26618i
$50$		0			0
$51$		0			0
$52$	−13.1557	+	4.78828i	−1.82437	+	0.664015i
$53$		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		0.984808	−	0.173648i	$-0.0555556\pi$
$59$		0			0		−0.984808	+	0.173648i	$0.944444\pi$
$60$		0			0
$61$	−0.766044	−	0.642788i	−0.0980819	−	0.0823005i	0.592428	−	0.805623i	$-0.298169\pi$
$61$	−0.766044	−	0.642788i	−0.0980819	−	0.0823005i	−0.690510	+	0.723323i	$0.742614\pi$
$62$		0			0
$63$		0			0
$64$	4.00000	+	6.92820i	0.500000	+	0.866025i
$65$		0			0
$66$		0			0
$67$	−4.69846	−	1.71010i	−0.574009	−	0.208922i	0.0386729	−	0.999252i	$-0.487687\pi$
$67$	−4.69846	−	1.71010i	−0.574009	−	0.208922i	−0.612682	+	0.790330i	$0.709909\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$71$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$72$		0			0
$73$	3.50000	−	6.06218i	0.409644	−	0.709524i	−0.585206	−	0.810885i	$-0.698986\pi$
$73$	3.50000	−	6.06218i	0.409644	−	0.709524i	0.994850	+	0.101361i	$0.0323196\pi$
$74$		0			0
$75$		0			0
$76$	0.347296	+	1.96962i	0.0398376	+	0.225930i
$77$		0			0
$78$		0			0
$79$	12.2160	−	4.44626i	1.37441	−	0.500244i	0.453930	−	0.891038i	$-0.350022\pi$
$79$	12.2160	−	4.44626i	1.37441	−	0.500244i	0.920478	+	0.390794i	$0.127800\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$		0			0		−0.342020	−	0.939693i	$-0.611111\pi$
$83$		0			0		0.342020	+	0.939693i	$0.388889\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$		0			0
$88$		0			0
$89$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$89$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$90$		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
$97$	0.868241	−	4.92404i	0.0881565	−	0.499960i	−0.908474	−	0.417941i	$-0.862752\pi$
$97$	0.868241	−	4.92404i	0.0881565	−	0.499960i	0.996631	−	0.0820195i	$-0.0261370\pi$
$98$		0			0
$99$		0			0
$100$	−5.00000	+	8.66025i	−0.500000	+	0.866025i
$101$		0			0		0.642788	−	0.766044i	$-0.277778\pi$
$101$		0			0		−0.642788	+	0.766044i	$0.722222\pi$
$102$		0			0
$103$	−2.25743	−	12.8025i	−0.222431	−	1.26147i	−0.867536	−	0.497374i	$-0.834298\pi$
$103$	−2.25743	−	12.8025i	−0.222431	−	1.26147i	0.645105	−	0.764094i	$-0.276813\pi$
$104$		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
$112$	−12.2567	+	10.2846i	−1.15815	+	0.971804i
$113$		0			0		0.984808	−	0.173648i	$-0.0555556\pi$
$113$		0			0		−0.984808	+	0.173648i	$0.944444\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$		0			0
$118$		0			0
$119$		0			0
$120$		0			0
$121$	10.3366	+	3.76222i	0.939693	+	0.342020i
$122$		0			0
$123$		0			0
$124$	−3.82026	+	21.6658i	−0.343069	+	1.94564i
$125$		0			0
$126$		0			0
$127$	0.500000	−	0.866025i	0.0443678	−	0.0768473i	−0.842989	−	0.537931i	$-0.819206\pi$
$127$	0.500000	−	0.866025i	0.0443678	−	0.0768473i	0.887357	+	0.461084i	$0.152539\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$		0			0		−0.642788	−	0.766044i	$-0.722222\pi$
$131$		0			0		0.642788	+	0.766044i	$0.277778\pi$
$132$		0			0
$133$	−3.75877	+	1.36808i	−0.325927	+	0.118628i
$134$		0			0
$135$		0			0
$136$		0			0
$137$		0			0		−0.342020	−	0.939693i	$-0.611111\pi$
$137$		0			0		0.342020	+	0.939693i	$0.388889\pi$
$138$		0			0
$139$	−12.2567	+	10.2846i	−1.03960	+	0.872329i	−0.991962	−	0.126536i	$-0.959614\pi$
$139$	−12.2567	+	10.2846i	−1.03960	+	0.872329i	−0.0476387	+	0.998865i	$0.515170\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		0			0
$144$		0			0
$145$		0			0
$146$		0			0
$147$		0			0
$148$	−18.7939	−	6.84040i	−1.54485	−	0.562278i
$149$		0			0		0.342020	−	0.939693i	$-0.388889\pi$
$149$		0			0		−0.342020	+	0.939693i	$0.611111\pi$
$150$		0			0
$151$	−3.29932	+	18.7113i	−0.268494	+	1.52271i	0.490402	+	0.871496i	$0.336850\pi$
$151$	−3.29932	+	18.7113i	−0.268494	+	1.52271i	−0.758896	+	0.651211i	$0.774261\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		0			0
$156$		0			0
$157$	−4.34120	−	24.6202i	−0.346466	−	1.96491i	−0.241299	−	0.970451i	$-0.577574\pi$
$157$	−4.34120	−	24.6202i	−0.346466	−	1.96491i	−0.105167	−	0.994455i	$-0.533538\pi$
$158$		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		0.984808	−	0.173648i	$-0.0555556\pi$
$167$		0			0		−0.984808	+	0.173648i	$0.944444\pi$
$168$		0			0
$169$	27.5776	+	23.1404i	2.12135	+	1.78003i
$170$		0			0
$171$		0			0
$172$	5.00000	+	8.66025i	0.381246	+	0.660338i
$173$		0			0		−0.984808	−	0.173648i	$-0.944444\pi$
$173$		0			0		0.984808	+	0.173648i	$0.0555556\pi$
$174$		0			0
$175$	−18.7939	−	6.84040i	−1.42068	−	0.517086i
$176$		0			0
$177$		0			0
$178$		0			0
$179$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$179$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$180$		0			0
$181$	−13.0000	+	22.5167i	−0.966282	+	1.67365i	−0.260153	+	0.965567i	$0.583773\pi$
$181$	−13.0000	+	22.5167i	−0.966282	+	1.67365i	−0.706129	+	0.708083i	$0.749560\pi$
$182$		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		−0.342020	−	0.939693i	$-0.611111\pi$
$191$		0			0		0.342020	+	0.939693i	$0.388889\pi$
$192$		0			0
$193$	17.6190	−	14.7841i	1.26824	−	1.06418i	0.273492	−	0.961874i	$-0.411821\pi$
$193$	17.6190	−	14.7841i	1.26824	−	1.06418i	0.994753	−	0.102310i	$-0.0326233\pi$
$194$		0			0
$195$		0			0
$196$	−13.7888	−	11.5702i	−0.984914	−	0.826441i
$197$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$197$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$198$		0			0
$199$	−8.50000	−	14.7224i	−0.602549	−	1.04365i	−0.992434	−	0.122782i	$-0.960818\pi$
$199$	−8.50000	−	14.7224i	−0.602549	−	1.04365i	0.389885	−	0.920864i	$-0.372515\pi$
$200$		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
$211$	5.03580	+	28.5594i	0.346679	+	1.96611i	0.232053	+	0.972703i	$0.425456\pi$
$211$	5.03580	+	28.5594i	0.346679	+	1.96611i	0.114625	+	0.993409i	$0.463433\pi$
$212$		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
$223$	17.6190	+	14.7841i	1.17986	+	0.990018i	0.999980	+	0.00632846i	$0.00201443\pi$
$223$	17.6190	+	14.7841i	1.17986	+	0.990018i	0.179877	+	0.983689i	$0.442430\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$		0			0		−0.984808	−	0.173648i	$-0.944444\pi$
$227$		0			0		0.984808	+	0.173648i	$0.0555556\pi$
$228$		0			0
$229$	6.57785	+	2.39414i	0.434676	+	0.158209i	0.550085	−	0.835109i	$-0.314595\pi$
$229$	6.57785	+	2.39414i	0.434676	+	0.158209i	−0.115408	+	0.993318i	$0.536818\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$233$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$		0			0
$238$		0			0
$239$		0			0		−0.642788	−	0.766044i	$-0.722222\pi$
$239$		0			0		0.642788	+	0.766044i	$0.277778\pi$
$240$		0			0
$241$	−13.1557	+	4.78828i	−0.847433	+	0.308440i	−0.728993	−	0.684521i	$-0.760011\pi$
$241$	−13.1557	+	4.78828i	−0.847433	+	0.308440i	−0.118440	+	0.992961i	$0.537789\pi$
$242$		0			0
$243$		0			0
$244$	2.00000			0.128037
$245$		0			0
$246$		0			0
$247$	5.36231	−	4.49951i	0.341196	−	0.286297i
$248$		0			0
$249$		0			0
$250$		0			0
$251$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$251$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$252$		0			0
$253$		0			0
$254$		0			0
$255$		0			0
$256$	−15.0351	−	5.47232i	−0.939693	−	0.342020i
$257$		0			0		0.342020	−	0.939693i	$-0.388889\pi$
$257$		0			0		−0.342020	+	0.939693i	$0.611111\pi$
$258$		0			0
$259$	6.94593	−	39.3923i	0.431599	−	2.44772i
$260$		0			0
$261$		0			0
$262$		0			0
$263$		0			0		0.642788	−	0.766044i	$-0.277778\pi$
$263$		0			0		−0.642788	+	0.766044i	$0.722222\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$		0			0
$268$	9.39693	−	3.42020i	0.574009	−	0.208922i
$269$		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
$277$	−23.7474	−	19.9264i	−1.42684	−	1.19726i	−0.947550	−	0.319606i	$-0.896449\pi$
$277$	−23.7474	−	19.9264i	−1.42684	−	1.19726i	−0.479291	−	0.877656i	$-0.659106\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$		0			0		−0.984808	−	0.173648i	$-0.944444\pi$
$281$		0			0		0.984808	+	0.173648i	$0.0555556\pi$
$282$		0			0
$283$	6.57785	+	2.39414i	0.391012	+	0.142317i	0.530042	−	0.847971i	$-0.322176\pi$
$283$	6.57785	+	2.39414i	0.391012	+	0.142317i	−0.139030	+	0.990288i	$0.544398\pi$
$284$		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
$292$	2.43107	+	13.7873i	0.142268	+	0.806841i
$293$		0			0		−0.642788	−	0.766044i	$-0.722222\pi$
$293$		0			0		0.642788	+	0.766044i	$0.277778\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$		0			0
$298$		0			0
$299$		0			0
$300$		0			0
$301$	−15.3209	+	12.8558i	−0.883081	+	0.740993i
$302$		0			0
$303$		0			0
$304$	−3.06418	−	2.57115i	−0.175743	−	0.147466i
$305$		0			0
$306$		0			0
$307$	8.00000	+	13.8564i	0.456584	+	0.790827i	0.998778	−	0.0494267i	$-0.0157394\pi$
$307$	8.00000	+	13.8564i	0.456584	+	0.790827i	−0.542194	+	0.840254i	$0.682406\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$		0			0		0.342020	−	0.939693i	$-0.388889\pi$
$311$		0			0		−0.342020	+	0.939693i	$0.611111\pi$
$312$		0			0
$313$	−3.82026	+	21.6658i	−0.215934	+	1.22462i	0.663345	+	0.748314i	$0.269136\pi$
$313$	−3.82026	+	21.6658i	−0.215934	+	1.22462i	−0.879279	+	0.476308i	$0.841975\pi$
$314$		0			0
$315$		0			0
$316$	−13.0000	+	22.5167i	−0.731307	+	1.26666i
$317$		0			0		0.642788	−	0.766044i	$-0.277778\pi$
$317$		0			0		−0.642788	+	0.766044i	$0.722222\pi$
$318$		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
$331$	−23.7474	−	19.9264i	−1.30527	−	1.09525i	−0.989208	−	0.146518i	$-0.953193\pi$
$331$	−23.7474	−	19.9264i	−1.30527	−	1.09525i	−0.316066	−	0.948737i	$-0.602362\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$		0			0
$336$		0			0
$337$	31.9495	+	11.6287i	1.74040	+	0.633455i	0.999281	−	0.0379157i	$-0.0120718\pi$
$337$	31.9495	+	11.6287i	1.74040	+	0.633455i	0.741122	+	0.671370i	$0.234294\pi$
$338$		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		−0.642788	−	0.766044i	$-0.722222\pi$
$347$		0			0		0.642788	+	0.766044i	$0.277778\pi$
$348$		0			0
$349$	−13.1557	+	4.78828i	−0.704208	+	0.256311i	−0.669207	−	0.743076i	$-0.733366\pi$
$349$	−13.1557	+	4.78828i	−0.704208	+	0.256311i	−0.0350017	+	0.999387i	$0.511144\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$		0			0		−0.342020	−	0.939693i	$-0.611111\pi$
$353$		0			0		0.342020	+	0.939693i	$0.388889\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		0			0
$358$		0			0
$359$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$359$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$360$		0			0
$361$	9.00000	+	15.5885i	0.473684	+	0.820445i
$362$		0			0
$363$		0			0
$364$	52.6228	+	19.1531i	2.75818	+	1.00390i
$365$		0			0
$366$		0			0
$367$	6.07769	−	34.4683i	0.317253	−	1.79923i	−0.242048	−	0.970264i	$-0.577819\pi$
$367$	6.07769	−	34.4683i	0.317253	−	1.79923i	0.559301	−	0.828965i	$-0.311070\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		0			0
$372$		0			0
$373$	−4.34120	−	24.6202i	−0.224779	−	1.27479i	−0.863107	−	0.505021i	$-0.831485\pi$
$373$	−4.34120	−	24.6202i	−0.224779	−	1.27479i	0.638328	−	0.769764i	$-0.279626\pi$
$374$		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		0.984808	−	0.173648i	$-0.0555556\pi$
$383$		0			0		−0.984808	+	0.173648i	$0.944444\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$		0			0
$388$	5.00000	+	8.66025i	0.253837	+	0.439658i
$389$		0			0		−0.984808	−	0.173648i	$-0.944444\pi$
$389$		0			0		0.984808	+	0.173648i	$0.0555556\pi$
$390$		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	−0.853206	−	0.521575i	$-0.825345\pi$
$397$	0.500000	−	0.866025i	0.0250943	−	0.0434646i	0.878300	+	0.478110i	$0.158678\pi$
$398$		0			0
$399$		0			0
$400$	−3.47296	−	19.6962i	−0.173648	−	0.984808i
$401$		0			0		−0.642788	−	0.766044i	$-0.722222\pi$
$401$		0			0		0.642788	+	0.766044i	$0.277778\pi$
$402$		0			0
$403$	72.3563	−	26.3356i	3.60433	−	1.31187i
$404$		0			0
$405$		0			0
$406$		0			0
$407$		0			0
$408$		0			0
$409$	29.1097	−	24.4259i	1.43938	−	1.20778i	0.499486	−	0.866322i	$-0.333522\pi$
$409$	29.1097	−	24.4259i	1.43938	−	1.20778i	0.939895	−	0.341463i	$-0.110922\pi$
$410$		0			0
$411$		0			0
$412$	19.9172	+	16.7125i	0.981248	+	0.823365i
$413$		0			0
$414$		0			0
$415$		0			0
$416$		0			0
$417$		0			0
$418$		0			0
$419$		0			0		0.342020	−	0.939693i	$-0.388889\pi$
$419$		0			0		−0.342020	+	0.939693i	$0.611111\pi$
$420$		0			0
$421$	−3.82026	+	21.6658i	−0.186188	+	1.05593i	0.738231	+	0.674548i	$0.235661\pi$
$421$	−3.82026	+	21.6658i	−0.186188	+	1.05593i	−0.924419	+	0.381377i	$0.875450\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$		0			0
$426$		0			0
$427$	0.694593	+	3.93923i	0.0336137	+	0.190633i
$428$		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
$436$	29.1097	−	24.4259i	1.39410	−	1.16979i
$437$		0			0
$438$		0			0
$439$	−21.4492	−	17.9981i	−1.02372	−	0.859000i	−0.0336266	−	0.999434i	$-0.510706\pi$
$439$	−21.4492	−	17.9981i	−1.02372	−	0.859000i	−0.990090	+	0.140434i	$0.955150\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$		0			0		−0.984808	−	0.173648i	$-0.944444\pi$
$443$		0			0		0.984808	+	0.173648i	$0.0555556\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$		0			0
$448$	5.55674	−	31.5138i	0.262531	−	1.48889i
$449$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$449$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$450$		0			0
$451$		0			0
$452$		0			0
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$	−38.5274	+	14.0228i	−1.80224	+	0.655960i	−0.804129	+	0.594455i	$0.797368\pi$
$457$	−38.5274	+	14.0228i	−1.80224	+	0.655960i	−0.998107	+	0.0615051i	$0.980410\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$		0			0		−0.342020	−	0.939693i	$-0.611111\pi$
$461$		0			0		0.342020	+	0.939693i	$0.388889\pi$
$462$		0			0
$463$	−32.9399	+	27.6399i	−1.53085	+	1.28453i	−0.739553	+	0.673098i	$0.764963\pi$
$463$	−32.9399	+	27.6399i	−1.53085	+	1.28453i	−0.791294	+	0.611435i	$0.790592\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$467$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$468$		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
$475$	0.868241	−	4.92404i	0.0398376	−	0.225930i
$476$		0			0
$477$		0			0
$478$		0			0
$479$		0			0		0.642788	−	0.766044i	$-0.277778\pi$
$479$		0			0		−0.642788	+	0.766044i	$0.722222\pi$
$480$		0			0
$481$	12.1554	+	68.9365i	0.554237	+	3.14324i
$482$		0			0
$483$		0			0
$484$	−20.6732	+	7.52444i	−0.939693	+	0.342020i
$485$		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		0.984808	−	0.173648i	$-0.0555556\pi$
$491$		0			0		−0.984808	+	0.173648i	$0.944444\pi$
$492$		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
$499$	−30.0702	−	10.9446i	−1.34613	−	0.489950i	−0.434389	−	0.900725i	$-0.643036\pi$
$499$	−30.0702	−	10.9446i	−1.34613	−	0.489950i	−0.911736	+	0.410776i	$0.865258\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$503$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$		0			0
$508$	0.347296	+	1.96962i	0.0154088	+	0.0873876i
$509$		0			0		−0.642788	−	0.766044i	$-0.722222\pi$
$509$		0			0		0.642788	+	0.766044i	$0.277778\pi$
$510$		0			0
$511$	−26.3114	+	9.57656i	−1.16395	+	0.423642i
$512$		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	$-0.833333\pi$
$521$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$522$		0			0
$523$	21.5000	+	37.2391i	0.940129	+	1.62835i	0.765222	+	0.643767i	$0.222629\pi$
$523$	21.5000	+	37.2391i	0.940129	+	1.62835i	0.174908	+	0.984585i	$0.444037\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$		0			0
$528$		0			0
$529$	−3.99391	+	22.6506i	−0.173648	+	0.984808i
$530$		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
$547$	−0.766044	−	0.642788i	−0.0327537	−	0.0274836i	0.626264	−	0.779611i	$-0.284583\pi$
$547$	−0.766044	−	0.642788i	−0.0327537	−	0.0274836i	−0.659018	+	0.752128i	$0.729028\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$		0			0
$552$		0			0
$553$	−48.8640	−	17.7850i	−2.07791	−	0.756297i
$554$		0			0
$555$		0			0
$556$	5.55674	−	31.5138i	0.235658	−	1.33648i
$557$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$557$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$558$		0			0
$559$	17.5000	−	30.3109i	0.740171	−	1.28201i
$560$		0			0
$561$		0			0
$562$		0			0
$563$		0			0		−0.642788	−	0.766044i	$-0.722222\pi$
$563$		0			0		0.642788	+	0.766044i	$0.277778\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$		0			0
$568$		0			0
$569$		0			0		−0.342020	−	0.939693i	$-0.611111\pi$
$569$		0			0		0.342020	+	0.939693i	$0.388889\pi$
$570$		0			0
$571$	−12.2567	+	10.2846i	−0.512927	+	0.430397i	−0.862158	−	0.506640i	$-0.830887\pi$
$571$	−12.2567	+	10.2846i	−0.512927	+	0.430397i	0.349231	+	0.937037i	$0.386443\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$		0			0
$577$	−5.50000	−	9.52628i	−0.228968	−	0.396584i	0.728535	−	0.685009i	$-0.240202\pi$
$577$	−5.50000	−	9.52628i	−0.228968	−	0.396584i	−0.957503	+	0.288425i	$0.906868\pi$
$578$		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		0.642788	−	0.766044i	$-0.277778\pi$
$587$		0			0		−0.642788	+	0.766044i	$0.722222\pi$
$588$		0			0
$589$	−1.91013	−	10.8329i	−0.0787055	−	0.446361i
$590$		0			0
$591$		0			0
$592$	37.5877	−	13.6808i	1.54485	−	0.562278i
$593$		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		0.984808	−	0.173648i	$-0.0555556\pi$
$599$		0			0		−0.984808	+	0.173648i	$0.944444\pi$
$600$		0			0
$601$	17.6190	+	14.7841i	0.718695	+	0.603057i	0.927024	−	0.375002i	$-0.122358\pi$
$601$	17.6190	+	14.7841i	0.718695	+	0.603057i	−0.208329	+	0.978059i	$0.566802\pi$
$602$		0			0
$603$		0			0
$604$	−19.0000	−	32.9090i	−0.773099	−	1.33905i
$605$		0			0
$606$		0			0
$607$	−18.7939	−	6.84040i	−0.762819	−	0.277643i	−0.0688294	−	0.997628i	$-0.521926\pi$
$607$	−18.7939	−	6.84040i	−0.762819	−	0.277643i	−0.693990	+	0.719985i	$0.744149\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		0			0
$612$		0			0
$613$	−23.5000	+	40.7032i	−0.949156	+	1.64399i	−0.201948	+	0.979396i	$0.564727\pi$
$613$	−23.5000	+	40.7032i	−0.949156	+	1.64399i	−0.747208	+	0.664590i	$0.768606\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$		0			0		−0.642788	−	0.766044i	$-0.722222\pi$
$617$		0			0		0.642788	+	0.766044i	$0.277778\pi$
$618$		0			0
$619$	−15.9748	+	5.81434i	−0.642080	+	0.233698i	−0.642481	−	0.766302i	$-0.722095\pi$
$619$	−15.9748	+	5.81434i	−0.642080	+	0.233698i	0.000400419		1.00000i	$0.499873\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$		0			0
$625$	19.1511	−	16.0697i	0.766044	−	0.642788i
$626$		0			0
$627$		0			0
$628$	38.3022	+	32.1394i	1.52843	+	1.28250i
$629$		0			0
$630$		0			0
$631$	−22.0000	−	38.1051i	−0.875806	−	1.51694i	−0.855901	−	0.517139i	$-0.826997\pi$
$631$	−22.0000	−	38.1051i	−0.875806	−	1.51694i	−0.0199047	−	0.999802i	$-0.506336\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$		0			0
$636$		0			0
$637$	−10.9398	+	62.0429i	−0.433452	+	2.45823i
$638$		0			0
$639$		0			0
$640$		0			0
$641$		0			0		0.642788	−	0.766044i	$-0.277778\pi$
$641$		0			0		−0.642788	+	0.766044i	$0.722222\pi$
$642$		0			0
$643$	−6.94593	−	39.3923i	−0.273921	−	1.55348i	−0.742370	−	0.669991i	$-0.766298\pi$
$643$	−6.94593	−	39.3923i	−0.273921	−	1.55348i	0.468449	−	0.883491i	$-0.344813\pi$
$644$		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
$652$	−12.2567	+	10.2846i	−0.480010	+	0.402776i
$653$		0			0		0.984808	−	0.173648i	$-0.0555556\pi$
$653$		0			0		−0.984808	+	0.173648i	$0.944444\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$		0			0
$658$		0			0
$659$		0			0		−0.984808	−	0.173648i	$-0.944444\pi$
$659$		0			0		0.984808	+	0.173648i	$0.0555556\pi$
$660$		0			0
$661$	46.0449	+	16.7590i	1.79094	+	0.651849i	0.999157	+	0.0410470i	$0.0130693\pi$
$661$	46.0449	+	16.7590i	1.79094	+	0.651849i	0.791783	+	0.610802i	$0.209153\pi$
$662$		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
$673$	12.2160	−	4.44626i	0.470892	−	0.171391i	−0.0956642	−	0.995414i	$-0.530497\pi$
$673$	12.2160	−	4.44626i	0.470892	−	0.171391i	0.566557	+	0.824023i	$0.308275\pi$
$674$		0			0
$675$		0			0
$676$	−72.0000			−2.76923
$677$		0			0		−0.342020	−	0.939693i	$-0.611111\pi$
$677$		0			0		0.342020	+	0.939693i	$0.388889\pi$
$678$		0			0
$679$	−15.3209	+	12.8558i	−0.587962	+	0.493358i
$680$		0			0
$681$		0			0
$682$		0			0
$683$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$683$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$		0			0
$688$	−18.7939	−	6.84040i	−0.716509	−	0.260788i
$689$		0			0
$690$		0			0
$691$	−8.50876	+	48.2556i	−0.323689	+	1.83573i	0.195047	+	0.980794i	$0.437514\pi$
$691$	−8.50876	+	48.2556i	−0.323689	+	1.83573i	−0.518735	+	0.854935i	$0.673597\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$		0			0
$696$		0			0
$697$		0			0
$698$		0			0
$699$		0			0
$700$	37.5877	−	13.6808i	1.42068	−	0.517086i
$701$		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
$709$	−23.7474	−	19.9264i	−0.891851	−	0.748352i	0.0767291	−	0.997052i	$-0.475552\pi$
$709$	−23.7474	−	19.9264i	−0.891851	−	0.748352i	−0.968581	+	0.248700i	$0.919997\pi$
$710$		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	$-0.166667\pi$
$719$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$720$		0			0
$721$	−26.0000	+	45.0333i	−0.968291	+	1.67713i
$722$		0			0
$723$		0			0
$724$	−9.02971	−	51.2100i	−0.335586	−	1.90320i
$725$		0			0
$726$		0			0
$727$	−41.3465	+	15.0489i	−1.53346	+	0.558132i	−0.964465	−	0.264211i	$-0.914888\pi$
$727$	−41.3465	+	15.0489i	−1.53346	+	0.558132i	−0.568991	+	0.822344i	$0.692666\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$		0			0
$732$		0			0
$733$	−32.9399	+	27.6399i	−1.21666	+	1.02090i	−0.217671	+	0.976022i	$0.569846\pi$
$733$	−32.9399	+	27.6399i	−1.21666	+	1.02090i	−0.998992	+	0.0448796i	$0.985710\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		0			0
$738$		0			0
$739$	18.5000	+	32.0429i	0.680534	+	1.17872i	0.974818	+	0.223001i	$0.0715853\pi$
$739$	18.5000	+	32.0429i	0.680534	+	1.17872i	−0.294285	+	0.955718i	$0.595081\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$		0			0		0.342020	−	0.939693i	$-0.388889\pi$
$743$		0			0		−0.342020	+	0.939693i	$0.611111\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$	−9.02971	−	51.2100i	−0.329499	−	1.86868i	−0.475965	−	0.879464i	$-0.657901\pi$
$751$	−9.02971	−	51.2100i	−0.329499	−	1.86868i	0.146467	−	0.989216i	$-0.453210\pi$
$752$		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		0.984808	−	0.173648i	$-0.0555556\pi$
$761$		0			0		−0.984808	+	0.173648i	$0.944444\pi$
$762$		0			0
$763$	58.2194	+	48.8519i	2.10768	+	1.76856i
$764$		0			0
$765$		0			0
$766$		0			0
$767$		0			0
$768$		0			0
$769$	−44.1656	−	16.0749i	−1.59265	−	0.579677i	−0.614745	−	0.788726i	$-0.710741\pi$
$769$	−44.1656	−	16.0749i	−1.59265	−	0.579677i	−0.977905	+	0.209048i	$0.932963\pi$
$770$		0			0
$771$		0			0
$772$	−7.98782	+	45.3012i	−0.287488	+	1.63042i
$773$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$773$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$774$		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
$787$	−23.7474	+	19.9264i	−0.846503	+	0.710300i	−0.959017	−	0.283350i	$-0.908554\pi$
$787$	−23.7474	+	19.9264i	−0.846503	+	0.710300i	0.112514	+	0.993650i	$0.464110\pi$
$788$		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
$796$	31.9495	+	11.6287i	1.13242	+	0.412168i
$797$		0			0		0.342020	−	0.939693i	$-0.388889\pi$
$797$		0			0		−0.342020	+	0.939693i	$0.611111\pi$
$798$		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
$817$	−3.83022	−	3.21394i	−0.134003	−	0.112441i
$818$		0			0
$819$		0			0
$820$		0			0
$821$		0			0		−0.984808	−	0.173648i	$-0.944444\pi$
$821$		0			0		0.984808	+	0.173648i	$0.0555556\pi$
$822$		0			0
$823$	−44.1656	−	16.0749i	−1.53951	−	0.560337i	−0.573586	−	0.819146i	$-0.694448\pi$
$823$	−44.1656	−	16.0749i	−1.53951	−	0.560337i	−0.965929	+	0.258808i	$0.916670\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$827$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$828$		0			0
$829$	−26.5000	+	45.8993i	−0.920383	+	1.59415i	−0.121560	+	0.992584i	$0.538790\pi$
$829$	−26.5000	+	45.8993i	−0.920383	+	1.59415i	−0.798823	+	0.601566i	$0.794544\pi$
$830$		0			0
$831$		0			0
$832$	9.72430	+	55.1492i	0.337129	+	1.91196i
$833$		0			0
$834$		0			0
$835$		0			0
$836$		0			0
$837$		0			0
$838$		0			0
$839$		0			0		−0.342020	−	0.939693i	$-0.611111\pi$
$839$		0			0		0.342020	+	0.939693i	$0.388889\pi$
$840$		0			0
$841$	−22.2153	+	18.6408i	−0.766044	+	0.642788i
$842$		0			0
$843$		0			0
$844$	−44.4306	−	37.2817i	−1.52936	−	1.28329i
$845$		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
$853$	6.07769	−	34.4683i	0.208096	−	1.18017i	−0.684397	−	0.729110i	$-0.739934\pi$
$853$	6.07769	−	34.4683i	0.208096	−	1.18017i	0.892493	−	0.451061i	$-0.148954\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$		0			0		0.642788	−	0.766044i	$-0.277778\pi$
$857$		0			0		−0.642788	+	0.766044i	$0.722222\pi$
$858$		0			0
$859$	9.72430	+	55.1492i	0.331789	+	1.88167i	0.456889	+	0.889524i	$0.348964\pi$
$859$	9.72430	+	55.1492i	0.331789	+	1.88167i	−0.125101	+	0.992144i	$0.539925\pi$
$860$		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
$868$	67.4119	−	56.5653i	2.28811	−	1.91995i
$869$		0			0
$870$		0			0
$871$	−26.8116	−	22.4976i	−0.908475	−	0.762301i
$872$		0			0
$873$		0			0
$874$		0			0
$875$		0			0
$876$		0			0
$877$	31.9495	+	11.6287i	1.07886	+	0.392673i	0.819483	−	0.573103i	$-0.194261\pi$
$877$	31.9495	+	11.6287i	1.07886	+	0.392673i	0.259377	+	0.965776i	$0.416483\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$881$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$882$		0			0
$883$	27.5000	−	47.6314i	0.925449	−	1.60292i	0.134611	−	0.990899i	$-0.457022\pi$
$883$	27.5000	−	47.6314i	0.925449	−	1.60292i	0.790838	−	0.612026i	$-0.209645\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$		0			0		−0.642788	−	0.766044i	$-0.722222\pi$
$887$		0			0		0.642788	+	0.766044i	$0.277778\pi$
$888$		0			0
$889$	−3.75877	+	1.36808i	−0.126065	+	0.0458839i
$890$		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
$907$	10.2452	−	58.1037i	0.340188	−	1.92930i	−0.0281394	−	0.999604i	$-0.508958\pi$
$907$	10.2452	−	58.1037i	0.340188	−	1.92930i	0.368327	−	0.929696i	$-0.379931\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$		0			0		0.642788	−	0.766044i	$-0.277778\pi$
$911$		0			0		−0.642788	+	0.766044i	$0.722222\pi$
$912$		0			0
$913$		0			0
$914$		0			0
$915$		0			0
$916$	−13.1557	+	4.78828i	−0.434676	+	0.158209i
$917$		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
$925$	38.3022	+	32.1394i	1.25937	+	1.05674i
$926$		0			0
$927$		0			0
$928$		0			0
$929$		0			0		−0.984808	−	0.173648i	$-0.944444\pi$
$929$		0			0		0.984808	+	0.173648i	$0.0555556\pi$
$930$		0			0
$931$	8.45723	+	3.07818i	0.277175	+	0.100883i
$932$		0			0
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$	−13.0000	+	22.5167i	−0.424691	+	0.735587i	−0.996392	−	0.0848755i	$-0.972951\pi$
$937$	−13.0000	+	22.5167i	−0.424691	+	0.735587i	0.571700	+	0.820463i	$0.306284\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$		0			0		−0.642788	−	0.766044i	$-0.722222\pi$
$941$		0			0		0.642788	+	0.766044i	$0.277778\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$		0			0
$946$		0			0
$947$		0			0		−0.342020	−	0.939693i	$-0.611111\pi$
$947$		0			0		0.342020	+	0.939693i	$0.388889\pi$
$948$		0			0
$949$	37.5362	−	31.4966i	1.21848	−	1.02242i
$950$		0			0
$951$		0			0
$952$		0			0
$953$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$953$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	15.6283	−	88.6327i	0.504140	−	2.85912i
$962$		0			0
$963$		0			0
$964$	14.0000	−	24.2487i	0.450910	−	0.780998i
$965$		0			0
$966$		0			0
$967$	7.11958	+	40.3771i	0.228950	+	1.29844i	0.854988	+	0.518648i	$0.173564\pi$
$967$	7.11958	+	40.3771i	0.228950	+	1.29844i	−0.626038	+	0.779793i	$0.715324\pi$
$968$		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
$976$	−3.06418	+	2.57115i	−0.0980819	+	0.0823005i
$977$		0			0		0.984808	−	0.173648i	$-0.0555556\pi$
$977$		0			0		−0.984808	+	0.173648i	$0.944444\pi$
$978$		0			0
$979$		0			0
$980$		0			0
$981$		0			0
$982$		0			0
$983$		0			0		−0.984808	−	0.173648i	$-0.944444\pi$
$983$		0			0		0.984808	+	0.173648i	$0.0555556\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$	−2.43107	+	13.7873i	−0.0773428	+	0.438633i
$989$		0			0
$990$		0			0
$991$	30.5000	−	52.8275i	0.968864	−	1.67812i	0.270011	−	0.962857i	$-0.412973\pi$
$991$	30.5000	−	52.8275i	0.968864	−	1.67812i	0.698853	−	0.715265i	$-0.253694\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$		0			0
$996$		0			0
$997$	9.39693	−	3.42020i	0.297604	−	0.108319i	−0.188903	−	0.981996i	$-0.560493\pi$
$997$	9.39693	−	3.42020i	0.297604	−	0.108319i	0.486507	+	0.873677i	$0.338271\pi$
$998$		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.163.1		6
3.2	odd	2	CM	729.2.e.e.163.1		6
9.2	odd	6	inner	729.2.e.e.406.1		6
9.4	even	3	inner	729.2.e.e.649.1		6
9.5	odd	6	inner	729.2.e.e.649.1		6
9.7	even	3	inner	729.2.e.e.406.1		6
27.2	odd	18		243.2.a.a.1.1	✓	1
27.4	even	9	inner	729.2.e.e.568.1		6
27.5	odd	18	inner	729.2.e.e.82.1		6
27.7	even	9		243.2.c.b.163.1		2
27.11	odd	18		243.2.c.b.82.1		2
27.13	even	9	inner	729.2.e.e.325.1		6
27.14	odd	18	inner	729.2.e.e.325.1		6
27.16	even	9		243.2.c.b.82.1		2
27.20	odd	18		243.2.c.b.163.1		2
27.22	even	9	inner	729.2.e.e.82.1		6
27.23	odd	18	inner	729.2.e.e.568.1		6
27.25	even	9		243.2.a.a.1.1	✓	1
108.79	odd	18		3888.2.a.n.1.1		1
108.83	even	18		3888.2.a.n.1.1		1
135.29	odd	18		6075.2.a.w.1.1		1
135.79	even	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	27.2	odd	18
243.2.a.a.1.1	✓	1	27.25	even	9
243.2.c.b.82.1		2	27.11	odd	18
243.2.c.b.82.1		2	27.16	even	9
243.2.c.b.163.1		2	27.7	even	9
243.2.c.b.163.1		2	27.20	odd	18
729.2.e.e.82.1		6	27.5	odd	18	inner
729.2.e.e.82.1		6	27.22	even	9	inner
729.2.e.e.163.1		6	1.1	even	1	trivial
729.2.e.e.163.1		6	3.2	odd	2	CM
729.2.e.e.325.1		6	27.13	even	9	inner
729.2.e.e.325.1		6	27.14	odd	18	inner
729.2.e.e.406.1		6	9.2	odd	6	inner
729.2.e.e.406.1		6	9.7	even	3	inner
729.2.e.e.568.1		6	27.4	even	9	inner
729.2.e.e.568.1		6	27.23	odd	18	inner
729.2.e.e.649.1		6	9.4	even	3	inner
729.2.e.e.649.1		6	9.5	odd	6	inner
3888.2.a.n.1.1		1	108.79	odd	18
3888.2.a.n.1.1		1	108.83	even	18
6075.2.a.w.1.1		1	135.29	odd	18
6075.2.a.w.1.1		1	135.79	even	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)

\(f(q)\)	\(=\)	\(q+(-1.53209 + 1.28558i) q^{4} +(-3.06418 - 2.57115i) q^{7} +(6.57785 + 2.39414i) q^{13} +(0.694593 - 3.93923i) q^{16} +(0.500000 - 0.866025i) q^{19} +(4.69846 - 1.71010i) q^{25} +8.00000 q^{28} +(8.42649 - 7.07066i) q^{31} +(5.00000 + 8.66025i) q^{37} +(0.868241 - 4.92404i) q^{43} +(1.56283 + 8.86327i) q^{49} +(-13.1557 + 4.78828i) q^{52} +(-0.766044 - 0.642788i) q^{61} +(4.00000 + 6.92820i) q^{64} +(-4.69846 - 1.71010i) q^{67} +(3.50000 - 6.06218i) q^{73} +(0.347296 + 1.96962i) q^{76} +(12.2160 - 4.44626i) q^{79} +(-14.0000 - 24.2487i) q^{91} +(0.868241 - 4.92404i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 3 q^{19} + 48 q^{28} + 30 q^{37} + 24 q^{64} + 21 q^{73} - 84 q^{91}+O(q^{100}) \) 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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