LMFDB - Embedded newform 729.2.e.f.568.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [729,2,Mod(82,729)]

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

lf = mfeigenbasis(mf)

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")

//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);

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

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:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 27)
Sato-Tate group:	$\mathrm{U}(1)[D_{9}]$

Embedding invariants

Embedding label			568.1
Root			$-0.173648 - 0.984808i$ of defining polynomial
Character	$\chi$	$=$	729.568
Dual form			729.2.e.f.163.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-1.53209 - 1.28558i) q^{4} +(-0.766044 + 0.642788i) q^{7} +(-4.69846 + 1.71010i) q^{13} +(0.694593 + 3.93923i) q^{16} +(3.50000 + 6.06218i) q^{19} +(4.69846 + 1.71010i) q^{25} +2.00000 q^{28} +(-3.06418 - 2.57115i) q^{31} +(-5.50000 + 9.52628i) q^{37} +(1.38919 + 7.87846i) q^{43} +(-1.04189 + 5.90885i) q^{49} +(9.39693 + 3.42020i) q^{52} +(-0.766044 + 0.642788i) q^{61} +(4.00000 - 6.92820i) q^{64} +(-4.69846 + 1.71010i) q^{67} +(3.50000 + 6.06218i) q^{73} +(2.43107 - 13.7873i) q^{76} +(-15.9748 - 5.81434i) q^{79} +(2.50000 - 4.33013i) q^{91} +(-3.29932 - 18.7113i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 21 q^{19} + 12 q^{28} - 33 q^{37} + 24 q^{64} + 21 q^{73} + 15 q^{91}+O(q^{100}) $ 6 * q + 21 * q^19 + 12 * q^28 - 33 * q^37 + 24 * q^64 + 21 * q^73 + 15 * q^91

Character values

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

$n$	$2$
$\chi(n)$	$e\left(\frac{1}{9}\right)$

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	$-0.388889\pi$
$2$		0			0		−0.342020	+	0.939693i	$0.611111\pi$
$3$		0			0
$3$		0			0
$4$	−1.53209	−	1.28558i	−0.766044	−	0.642788i
$5$		0			0		−0.984808	−	0.173648i	$-0.944444\pi$
$5$		0			0		0.984808	+	0.173648i	$0.0555556\pi$
$6$		0			0
$7$	−0.766044	+	0.642788i	−0.289538	+	0.242951i	−0.775974	−	0.630765i	$-0.782741\pi$
$7$	−0.766044	+	0.642788i	−0.289538	+	0.242951i	0.486436	+	0.873716i	$0.338297\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$		0			0		0.984808	−	0.173648i	$-0.0555556\pi$
$11$		0			0		−0.984808	+	0.173648i	$0.944444\pi$
$12$		0			0
$13$	−4.69846	+	1.71010i	−1.30312	+	0.474297i	−0.898011	−	0.439972i	$-0.854988\pi$
$13$	−4.69846	+	1.71010i	−1.30312	+	0.474297i	−0.405108	+	0.914269i	$0.632766\pi$
$14$		0			0
$15$		0			0
$16$	0.694593	+	3.93923i	0.173648	+	0.984808i
$17$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$17$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$18$		0			0
$19$	3.50000	+	6.06218i	0.802955	+	1.39076i	0.917663	+	0.397360i	$0.130073\pi$
$19$	3.50000	+	6.06218i	0.802955	+	1.39076i	−0.114708	+	0.993399i	$0.536593\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$		0			0		0.642788	−	0.766044i	$-0.277778\pi$
$23$		0			0		−0.642788	+	0.766044i	$0.722222\pi$
$24$		0			0
$25$	4.69846	+	1.71010i	0.939693	+	0.342020i
$26$		0			0
$27$		0			0
$28$	2.00000			0.377964
$29$		0			0		0.342020	−	0.939693i	$-0.388889\pi$
$29$		0			0		−0.342020	+	0.939693i	$0.611111\pi$
$30$		0			0
$31$	−3.06418	−	2.57115i	−0.550343	−	0.461792i	0.324714	−	0.945812i	$-0.394732\pi$
$31$	−3.06418	−	2.57115i	−0.550343	−	0.461792i	−0.875057	+	0.484020i	$0.839176\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		0			0
$36$		0			0
$37$	−5.50000	+	9.52628i	−0.904194	+	1.56611i	−0.0821995	+	0.996616i	$0.526194\pi$
$37$	−5.50000	+	9.52628i	−0.904194	+	1.56611i	−0.821995	+	0.569495i	$0.807139\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$		0			0		−0.342020	−	0.939693i	$-0.611111\pi$
$41$		0			0		0.342020	+	0.939693i	$0.388889\pi$
$42$		0			0
$43$	1.38919	+	7.87846i	0.211849	+	1.20145i	0.886292	+	0.463127i	$0.153273\pi$
$43$	1.38919	+	7.87846i	0.211849	+	1.20145i	−0.674443	+	0.738327i	$0.735616\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$		0			0		−0.642788	−	0.766044i	$-0.722222\pi$
$47$		0			0		0.642788	+	0.766044i	$0.277778\pi$
$48$		0			0
$49$	−1.04189	+	5.90885i	−0.148841	+	0.844121i
$50$		0			0
$51$		0			0
$52$	9.39693	+	3.42020i	1.30312	+	0.474297i
$53$		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	$-0.944444\pi$
$59$		0			0		0.984808	+	0.173648i	$0.0555556\pi$
$60$		0			0
$61$	−0.766044	+	0.642788i	−0.0980819	+	0.0823005i	−0.690510	−	0.723323i	$-0.742614\pi$
$61$	−0.766044	+	0.642788i	−0.0980819	+	0.0823005i	0.592428	+	0.805623i	$0.298169\pi$
$62$		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	−0.612682	−	0.790330i	$-0.709909\pi$
$67$	−4.69846	+	1.71010i	−0.574009	+	0.208922i	0.0386729	+	0.999252i	$0.487687\pi$
$68$		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
$76$	2.43107	−	13.7873i	0.278863	−	1.58151i
$77$		0			0
$78$		0			0
$79$	−15.9748	−	5.81434i	−1.79730	−	0.654165i	−0.998626	−	0.0524041i	$-0.983312\pi$
$79$	−15.9748	−	5.81434i	−1.79730	−	0.654165i	−0.798677	−	0.601760i	$-0.794466\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$		0			0		0.342020	−	0.939693i	$-0.388889\pi$
$83$		0			0		−0.342020	+	0.939693i	$0.611111\pi$
$84$		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
$91$	2.50000	−	4.33013i	0.262071	−	0.453921i
$92$		0			0
$93$		0			0
$94$		0			0
$95$		0			0
$96$		0			0
$97$	−3.29932	−	18.7113i	−0.334995	−	1.89985i	−0.427284	−	0.904117i	$-0.640530\pi$
$97$	−3.29932	−	18.7113i	−0.334995	−	1.89985i	0.0922897	−	0.995732i	$-0.470581\pi$
$98$		0			0
$99$		0			0
$100$	−5.00000	−	8.66025i	−0.500000	−	0.866025i
$101$		0			0		−0.642788	−	0.766044i	$-0.722222\pi$
$101$		0			0		0.642788	+	0.766044i	$0.277778\pi$
$102$		0			0
$103$	−2.25743	+	12.8025i	−0.222431	+	1.26147i	0.645105	+	0.764094i	$0.276813\pi$
$103$	−2.25743	+	12.8025i	−0.222431	+	1.26147i	−0.867536	+	0.497374i	$0.834298\pi$
$104$		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
$109$	2.00000			0.191565			0.0957826	−	0.995402i	$-0.469465\pi$
$109$	2.00000			0.191565			0.0957826	+	0.995402i	$0.469465\pi$
$110$		0			0
$111$		0			0
$112$	−3.06418	−	2.57115i	−0.289538	−	0.242951i
$113$		0			0		−0.984808	−	0.173648i	$-0.944444\pi$
$113$		0			0		0.984808	+	0.173648i	$0.0555556\pi$
$114$		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
$124$	1.38919	+	7.87846i	0.124753	+	0.707507i
$125$		0			0
$126$		0			0
$127$	−10.0000	−	17.3205i	−0.887357	−	1.53695i	−0.842989	−	0.537931i	$-0.819206\pi$
$127$	−10.0000	−	17.3205i	−0.887357	−	1.53695i	−0.0443678	−	0.999015i	$-0.514127\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$		0			0		0.642788	−	0.766044i	$-0.277778\pi$
$131$		0			0		−0.642788	+	0.766044i	$0.722222\pi$
$132$		0			0
$133$	−6.57785	−	2.39414i	−0.570372	−	0.207598i
$134$		0			0
$135$		0			0
$136$		0			0
$137$		0			0		0.342020	−	0.939693i	$-0.388889\pi$
$137$		0			0		−0.342020	+	0.939693i	$0.611111\pi$
$138$		0			0
$139$	17.6190	+	14.7841i	1.49443	+	1.25397i	0.888861	+	0.458176i	$0.151497\pi$
$139$	17.6190	+	14.7841i	1.49443	+	1.25397i	0.605564	+	0.795796i	$0.292947\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		0			0
$144$		0			0
$145$		0			0
$146$		0			0
$147$		0			0
$148$	20.6732	−	7.52444i	1.69933	−	0.618505i
$149$		0			0		−0.342020	−	0.939693i	$-0.611111\pi$
$149$		0			0		0.342020	+	0.939693i	$0.388889\pi$
$150$		0			0
$151$	−3.29932	−	18.7113i	−0.268494	−	1.52271i	−0.758896	−	0.651211i	$-0.774261\pi$
$151$	−3.29932	−	18.7113i	−0.268494	−	1.52271i	0.490402	−	0.871496i	$-0.336850\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		0			0
$156$		0			0
$157$	2.43107	−	13.7873i	0.194021	−	1.10035i	−0.719785	−	0.694197i	$-0.755760\pi$
$157$	2.43107	−	13.7873i	0.194021	−	1.10035i	0.913806	−	0.406150i	$-0.133129\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$		0			0
$163$	−25.0000			−1.95815			−0.979076	−	0.203497i	$-0.934769\pi$
$163$	−25.0000			−1.95815			−0.979076	+	0.203497i	$0.934769\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		0			0		−0.984808	−	0.173648i	$-0.944444\pi$
$167$		0			0		0.984808	+	0.173648i	$0.0555556\pi$
$168$		0			0
$169$	9.19253	−	7.71345i	0.707118	−	0.593342i
$170$		0			0
$171$		0			0
$172$	8.00000	−	13.8564i	0.609994	−	1.05654i
$173$		0			0		0.984808	−	0.173648i	$-0.0555556\pi$
$173$		0			0		−0.984808	+	0.173648i	$0.944444\pi$
$174$		0			0
$175$	−4.69846	+	1.71010i	−0.355170	+	0.129271i
$176$		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
$181$	3.50000	+	6.06218i	0.260153	+	0.450598i	0.966282	−	0.257485i	$-0.0828937\pi$
$181$	3.50000	+	6.06218i	0.260153	+	0.450598i	−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	$-0.388889\pi$
$191$		0			0		−0.342020	+	0.939693i	$0.611111\pi$
$192$		0			0
$193$	17.6190	+	14.7841i	1.26824	+	1.06418i	0.994753	+	0.102310i	$0.0326233\pi$
$193$	17.6190	+	14.7841i	1.26824	+	1.06418i	0.273492	+	0.961874i	$0.411821\pi$
$194$		0			0
$195$		0			0
$196$	9.19253	−	7.71345i	0.656610	−	0.550961i
$197$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$197$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$198$		0			0
$199$	−5.50000	+	9.52628i	−0.389885	+	0.675300i	−0.992434	−	0.122782i	$-0.960818\pi$
$199$	−5.50000	+	9.52628i	−0.389885	+	0.675300i	0.602549	+	0.798082i	$0.294152\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$		0			0
$204$		0			0
$205$		0			0
$206$		0			0
$207$		0			0
$208$	−10.0000	−	17.3205i	−0.693375	−	1.20096i
$209$		0			0
$210$		0			0
$211$	−2.25743	+	12.8025i	−0.155408	+	0.881361i	0.803005	+	0.595973i	$0.203233\pi$
$211$	−2.25743	+	12.8025i	−0.155408	+	0.881361i	−0.958412	+	0.285388i	$0.907878\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		0			0
$216$		0			0
$217$	4.00000			0.271538
$218$		0			0
$219$		0			0
$220$		0			0
$221$		0			0
$222$		0			0
$223$	−21.4492	+	17.9981i	−1.43635	+	1.20524i	−0.494509	+	0.869172i	$0.664652\pi$
$223$	−21.4492	+	17.9981i	−1.43635	+	1.20524i	−0.941838	+	0.336066i	$0.890903\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$		0			0		0.984808	−	0.173648i	$-0.0555556\pi$
$227$		0			0		−0.984808	+	0.173648i	$0.944444\pi$
$228$		0			0
$229$	20.6732	−	7.52444i	1.36613	−	0.497229i	0.448183	−	0.893942i	$-0.352071\pi$
$229$	20.6732	−	7.52444i	1.36613	−	0.497229i	0.917943	+	0.396713i	$0.129849\pi$
$230$		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		0.642788	−	0.766044i	$-0.277778\pi$
$239$		0			0		−0.642788	+	0.766044i	$0.722222\pi$
$240$		0			0
$241$	−15.9748	−	5.81434i	−1.02903	−	0.374535i	−0.228316	−	0.973587i	$-0.573322\pi$
$241$	−15.9748	−	5.81434i	−1.02903	−	0.374535i	−0.800710	+	0.599052i	$0.795544\pi$
$242$		0			0
$243$		0			0
$244$	2.00000			0.128037
$245$		0			0
$246$		0			0
$247$	−26.8116	−	22.4976i	−1.70598	−	1.43149i
$248$		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
$256$	−15.0351	+	5.47232i	−0.939693	+	0.342020i
$257$		0			0		−0.342020	−	0.939693i	$-0.611111\pi$
$257$		0			0		0.342020	+	0.939693i	$0.388889\pi$
$258$		0			0
$259$	−1.91013	−	10.8329i	−0.118690	−	0.673123i
$260$		0			0
$261$		0			0
$262$		0			0
$263$		0			0		−0.642788	−	0.766044i	$-0.722222\pi$
$263$		0			0		0.642788	+	0.766044i	$0.277778\pi$
$264$		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
$277$	19.9172	−	16.7125i	1.19671	−	1.00416i	0.196988	−	0.980406i	$-0.436884\pi$
$277$	19.9172	−	16.7125i	1.19671	−	1.00416i	0.999718	−	0.0237496i	$-0.00756043\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$		0			0		0.984808	−	0.173648i	$-0.0555556\pi$
$281$		0			0		−0.984808	+	0.173648i	$0.944444\pi$
$282$		0			0
$283$	−30.0702	+	10.9446i	−1.78749	+	0.650592i	−0.788100	+	0.615547i	$0.788935\pi$
$283$	−30.0702	+	10.9446i	−1.78749	+	0.650592i	−0.999386	+	0.0350443i	$0.988843\pi$
$284$		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	$-0.277778\pi$
$293$		0			0		−0.642788	+	0.766044i	$0.722222\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$		0			0
$298$		0			0
$299$		0			0
$300$		0			0
$301$	−6.12836	−	5.14230i	−0.353233	−	0.296397i
$302$		0			0
$303$		0			0
$304$	−21.4492	+	17.9981i	−1.23020	+	1.03226i
$305$		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		−0.342020	−	0.939693i	$-0.611111\pi$
$311$		0			0		0.342020	+	0.939693i	$0.388889\pi$
$312$		0			0
$313$	6.07769	+	34.4683i	0.343531	+	1.94826i	0.316387	+	0.948630i	$0.397530\pi$
$313$	6.07769	+	34.4683i	0.343531	+	1.94826i	0.0271446	+	0.999632i	$0.491359\pi$
$314$		0			0
$315$		0			0
$316$	17.0000	+	29.4449i	0.956325	+	1.65640i
$317$		0			0		−0.642788	−	0.766044i	$-0.722222\pi$
$317$		0			0		0.642788	+	0.766044i	$0.277778\pi$
$318$		0			0
$319$		0			0
$320$		0			0
$321$		0			0
$322$		0			0
$323$		0			0
$324$		0			0
$325$	−25.0000			−1.38675
$326$		0			0
$327$		0			0
$328$		0			0
$329$		0			0
$330$		0			0
$331$	−0.766044	+	0.642788i	−0.0421056	+	0.0353308i	−0.663598	−	0.748090i	$-0.730971\pi$
$331$	−0.766044	+	0.642788i	−0.0421056	+	0.0353308i	0.621492	+	0.783420i	$0.286527\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$		0			0
$336$		0			0
$337$	−4.69846	+	1.71010i	−0.255942	+	0.0931551i	−0.466805	−	0.884361i	$-0.654595\pi$
$337$	−4.69846	+	1.71010i	−0.255942	+	0.0931551i	0.210863	+	0.977516i	$0.432373\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$		0			0
$342$		0			0
$343$	−6.50000	−	11.2583i	−0.350967	−	0.607893i
$344$		0			0
$345$		0			0
$346$		0			0
$347$		0			0		0.642788	−	0.766044i	$-0.277778\pi$
$347$		0			0		−0.642788	+	0.766044i	$0.722222\pi$
$348$		0			0
$349$	34.7686	+	12.6547i	1.86112	+	0.677393i	0.978126	+	0.208012i	$0.0666992\pi$
$349$	34.7686	+	12.6547i	1.86112	+	0.677393i	0.882996	+	0.469381i	$0.155523\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$		0			0		0.342020	−	0.939693i	$-0.388889\pi$
$353$		0			0		−0.342020	+	0.939693i	$0.611111\pi$
$354$		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
$361$	−15.0000	+	25.9808i	−0.789474	+	1.36741i
$362$		0			0
$363$		0			0
$364$	−9.39693	+	3.42020i	−0.492533	+	0.179267i
$365$		0			0
$366$		0			0
$367$	6.07769	+	34.4683i	0.317253	+	1.79923i	0.559301	+	0.828965i	$0.311070\pi$
$367$	6.07769	+	34.4683i	0.317253	+	1.79923i	−0.242048	+	0.970264i	$0.577819\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		0			0
$372$		0			0
$373$	−2.25743	+	12.8025i	−0.116885	+	0.662888i	0.868914	+	0.494962i	$0.164818\pi$
$373$	−2.25743	+	12.8025i	−0.116885	+	0.662888i	−0.985800	+	0.167926i	$0.946293\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		0			0
$378$		0			0
$379$	29.0000			1.48963			0.744815	−	0.667271i	$-0.232538\pi$
$379$	29.0000			1.48963			0.744815	+	0.667271i	$0.232538\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$		0			0		−0.984808	−	0.173648i	$-0.944444\pi$
$383$		0			0		0.984808	+	0.173648i	$0.0555556\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$		0			0
$388$	−19.0000	+	32.9090i	−0.964579	+	1.67070i
$389$		0			0		0.984808	−	0.173648i	$-0.0555556\pi$
$389$		0			0		−0.984808	+	0.173648i	$0.944444\pi$
$390$		0			0
$391$		0			0
$392$		0			0
$393$		0			0
$394$		0			0
$395$		0			0
$396$		0			0
$397$	17.0000	+	29.4449i	0.853206	+	1.47780i	0.878300	+	0.478110i	$0.158678\pi$
$397$	17.0000	+	29.4449i	0.853206	+	1.47780i	−0.0250943	+	0.999685i	$0.507989\pi$
$398$		0			0
$399$		0			0
$400$	−3.47296	+	19.6962i	−0.173648	+	0.984808i
$401$		0			0		0.642788	−	0.766044i	$-0.277778\pi$
$401$		0			0		−0.642788	+	0.766044i	$0.722222\pi$
$402$		0			0
$403$	18.7939	+	6.84040i	0.936188	+	0.340745i
$404$		0			0
$405$		0			0
$406$		0			0
$407$		0			0
$408$		0			0
$409$	−23.7474	−	19.9264i	−1.17423	−	0.985298i	−1.00000		0.000593299i	$-0.999811\pi$
$409$	−23.7474	−	19.9264i	−1.17423	−	0.985298i	−0.174232	−	0.984705i	$-0.555744\pi$
$410$		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	$-0.611111\pi$
$419$		0			0		0.342020	+	0.939693i	$0.388889\pi$
$420$		0			0
$421$	−3.29932	−	18.7113i	−0.160799	−	0.911935i	−0.953291	−	0.302053i	$-0.902328\pi$
$421$	−3.29932	−	18.7113i	−0.160799	−	0.911935i	0.792492	−	0.609882i	$-0.208783\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$		0			0
$426$		0			0
$427$	0.173648	−	0.984808i	0.00840342	−	0.0476582i
$428$		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
$433$	2.00000			0.0961139			0.0480569	−	0.998845i	$-0.484697\pi$
$433$	2.00000			0.0961139			0.0480569	+	0.998845i	$0.484697\pi$
$434$		0			0
$435$		0			0
$436$	−3.06418	−	2.57115i	−0.146748	−	0.123136i
$437$		0			0
$438$		0			0
$439$	−21.4492	+	17.9981i	−1.02372	+	0.859000i	−0.990090	−	0.140434i	$-0.955150\pi$
$439$	−21.4492	+	17.9981i	−1.02372	+	0.859000i	−0.0336266	+	0.999434i	$0.510706\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$		0			0		0.984808	−	0.173648i	$-0.0555556\pi$
$443$		0			0		−0.984808	+	0.173648i	$0.944444\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$		0			0
$448$	1.38919	+	7.87846i	0.0656328	+	0.372222i
$449$		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
$457$	9.39693	+	3.42020i	0.439570	+	0.159990i	0.552318	−	0.833633i	$-0.313743\pi$
$457$	9.39693	+	3.42020i	0.439570	+	0.159990i	−0.112749	+	0.993624i	$0.535966\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$		0			0		0.342020	−	0.939693i	$-0.388889\pi$
$461$		0			0		−0.342020	+	0.939693i	$0.611111\pi$
$462$		0			0
$463$	17.6190	+	14.7841i	0.818825	+	0.687076i	0.952697	−	0.303923i	$-0.0982964\pi$
$463$	17.6190	+	14.7841i	0.818825	+	0.687076i	−0.133871	+	0.990999i	$0.542741\pi$
$464$		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
$469$	2.50000	−	4.33013i	0.115439	−	0.199947i
$470$		0			0
$471$		0			0
$472$		0			0
$473$		0			0
$474$		0			0
$475$	6.07769	+	34.4683i	0.278863	+	1.58151i
$476$		0			0
$477$		0			0
$478$		0			0
$479$		0			0		−0.642788	−	0.766044i	$-0.722222\pi$
$479$		0			0		0.642788	+	0.766044i	$0.277778\pi$
$480$		0			0
$481$	9.55065	−	54.1644i	0.435472	−	2.46969i
$482$		0			0
$483$		0			0
$484$	−20.6732	−	7.52444i	−0.939693	−	0.342020i
$485$		0			0
$486$		0			0
$487$	−25.0000			−1.13286			−0.566429	−	0.824110i	$-0.691675\pi$
$487$	−25.0000			−1.13286			−0.566429	+	0.824110i	$0.691675\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$		0			0		−0.984808	−	0.173648i	$-0.944444\pi$
$491$		0			0		0.984808	+	0.173648i	$0.0555556\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$		0			0
$496$	8.00000	−	13.8564i	0.359211	−	0.622171i
$497$		0			0
$498$		0			0
$499$	−30.0702	+	10.9446i	−1.34613	+	0.489950i	−0.911736	−	0.410776i	$-0.865258\pi$
$499$	−30.0702	+	10.9446i	−1.34613	+	0.489950i	−0.434389	+	0.900725i	$0.643036\pi$
$500$		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
$508$	−6.94593	+	39.3923i	−0.308176	+	1.74775i
$509$		0			0		0.642788	−	0.766044i	$-0.277778\pi$
$509$		0			0		−0.642788	+	0.766044i	$0.722222\pi$
$510$		0			0
$511$	−6.57785	−	2.39414i	−0.290987	−	0.105911i
$512$		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
$529$	−3.99391	−	22.6506i	−0.173648	−	0.984808i
$530$		0			0
$531$		0			0
$532$	7.00000	+	12.1244i	0.303488	+	0.525657i
$533$		0			0
$534$		0			0
$535$		0			0
$536$		0			0
$537$		0			0
$538$		0			0
$539$		0			0
$540$		0			0
$541$	29.0000			1.24681			0.623404	−	0.781900i	$-0.285749\pi$
$541$	29.0000			1.24681			0.623404	+	0.781900i	$0.285749\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$		0			0
$546$		0			0
$547$	−0.766044	+	0.642788i	−0.0327537	+	0.0274836i	−0.659018	−	0.752128i	$-0.729028\pi$
$547$	−0.766044	+	0.642788i	−0.0327537	+	0.0274836i	0.626264	+	0.779611i	$0.284583\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$		0			0
$552$		0			0
$553$	15.9748	−	5.81434i	0.679317	−	0.247251i
$554$		0			0
$555$		0			0
$556$	−7.98782	−	45.3012i	−0.338759	−	1.92120i
$557$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$557$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$558$		0			0
$559$	−20.0000	−	34.6410i	−0.845910	−	1.46516i
$560$		0			0
$561$		0			0
$562$		0			0
$563$		0			0		0.642788	−	0.766044i	$-0.277778\pi$
$563$		0			0		−0.642788	+	0.766044i	$0.722222\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$		0			0
$568$		0			0
$569$		0			0		0.342020	−	0.939693i	$-0.388889\pi$
$569$		0			0		−0.342020	+	0.939693i	$0.611111\pi$
$570$		0			0
$571$	−23.7474	−	19.9264i	−0.993797	−	0.833895i	−0.00768386	−	0.999970i	$-0.502446\pi$
$571$	−23.7474	−	19.9264i	−0.993797	−	0.833895i	−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		−0.642788	−	0.766044i	$-0.722222\pi$
$587$		0			0		0.642788	+	0.766044i	$0.277778\pi$
$588$		0			0
$589$	4.86215	−	27.5746i	0.200341	−	1.13619i
$590$		0			0
$591$		0			0
$592$	−41.3465	−	15.0489i	−1.69933	−	0.618505i
$593$		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	$-0.944444\pi$
$599$		0			0		0.984808	+	0.173648i	$0.0555556\pi$
$600$		0			0
$601$	19.9172	−	16.7125i	0.812438	−	0.681716i	−0.138751	−	0.990327i	$-0.544309\pi$
$601$	19.9172	−	16.7125i	0.812438	−	0.681716i	0.951188	+	0.308611i	$0.0998642\pi$
$602$		0			0
$603$		0			0
$604$	−19.0000	+	32.9090i	−0.773099	+	1.33905i
$605$		0			0
$606$		0			0
$607$	46.0449	−	16.7590i	1.86891	−	0.680226i	0.898386	−	0.439206i	$-0.144740\pi$
$607$	46.0449	−	16.7590i	1.86891	−	0.680226i	0.970520	−	0.241020i	$-0.0774820\pi$
$608$		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		0.642788	−	0.766044i	$-0.277778\pi$
$617$		0			0		−0.642788	+	0.766044i	$0.722222\pi$
$618$		0			0
$619$	−15.9748	−	5.81434i	−0.642080	−	0.233698i	0.000400419	−	1.00000i	$-0.499873\pi$
$619$	−15.9748	−	5.81434i	−0.642080	−	0.233698i	−0.642481	+	0.766302i	$0.722095\pi$
$620$		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
$628$	−21.4492	+	17.9981i	−0.855918	+	0.718201i
$629$		0			0
$630$		0			0
$631$	21.5000	−	37.2391i	0.855901	−	1.48246i	−0.0199047	−	0.999802i	$-0.506336\pi$
$631$	21.5000	−	37.2391i	0.855901	−	1.48246i	0.875806	−	0.482663i	$-0.160330\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$		0			0
$636$		0			0
$637$	−5.20945	−	29.5442i	−0.206406	−	1.17059i
$638$		0			0
$639$		0			0
$640$		0			0
$641$		0			0		−0.642788	−	0.766044i	$-0.722222\pi$
$641$		0			0		0.642788	+	0.766044i	$0.277778\pi$
$642$		0			0
$643$	−6.94593	+	39.3923i	−0.273921	+	1.55348i	0.468449	+	0.883491i	$0.344813\pi$
$643$	−6.94593	+	39.3923i	−0.273921	+	1.55348i	−0.742370	+	0.669991i	$0.766298\pi$
$644$		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
$652$	38.3022	+	32.1394i	1.50003	+	1.25868i
$653$		0			0		−0.984808	−	0.173648i	$-0.944444\pi$
$653$		0			0		0.984808	+	0.173648i	$0.0555556\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$		0			0
$658$		0			0
$659$		0			0		0.984808	−	0.173648i	$-0.0555556\pi$
$659$		0			0		−0.984808	+	0.173648i	$0.944444\pi$
$660$		0			0
$661$	46.0449	−	16.7590i	1.79094	−	0.651849i	0.791783	−	0.610802i	$-0.209153\pi$
$661$	46.0449	−	16.7590i	1.79094	−	0.651849i	0.999157	−	0.0410470i	$-0.0130693\pi$
$662$		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
$673$	34.7686	+	12.6547i	1.34023	+	0.487805i	0.909886	−	0.414859i	$-0.136169\pi$
$673$	34.7686	+	12.6547i	1.34023	+	0.487805i	0.430346	+	0.902664i	$0.358391\pi$
$674$		0			0
$675$		0			0
$676$	−24.0000			−0.923077
$677$		0			0		0.342020	−	0.939693i	$-0.388889\pi$
$677$		0			0		−0.342020	+	0.939693i	$0.611111\pi$
$678$		0			0
$679$	14.5548	+	12.2130i	0.558564	+	0.468691i
$680$		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
$688$	−30.0702	+	10.9446i	−1.14641	+	0.417261i
$689$		0			0
$690$		0			0
$691$	1.38919	+	7.87846i	0.0528471	+	0.299711i	0.999763	−	0.0217706i	$-0.00693034\pi$
$691$	1.38919	+	7.87846i	0.0528471	+	0.299711i	−0.946916	+	0.321481i	$0.895819\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$		0			0
$696$		0			0
$697$		0			0
$698$		0			0
$699$		0			0
$700$	9.39693	+	3.42020i	0.355170	+	0.129271i
$701$		0			0			−	1.00000i	$-0.5\pi$
$701$		0			0				1.00000i	$0.5\pi$
$702$		0			0
$703$	−77.0000			−2.90411
$704$		0			0
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$	40.6004	−	34.0677i	1.52478	−	1.27944i	0.699671	−	0.714466i	$-0.253330\pi$
$709$	40.6004	−	34.0677i	1.52478	−	1.27944i	0.825108	−	0.564975i	$-0.191114\pi$
$710$		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
$721$	−6.50000	−	11.2583i	−0.242073	−	0.419282i
$722$		0			0
$723$		0			0
$724$	2.43107	−	13.7873i	0.0903502	−	0.512401i
$725$		0			0
$726$		0			0
$727$	−41.3465	−	15.0489i	−1.53346	−	0.558132i	−0.568991	−	0.822344i	$-0.692666\pi$
$727$	−41.3465	−	15.0489i	−1.53346	−	0.558132i	−0.964465	+	0.264211i	$0.914888\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$		0			0
$732$		0			0
$733$	38.3022	+	32.1394i	1.41472	+	1.18710i	0.954096	+	0.299503i	$0.0968207\pi$
$733$	38.3022	+	32.1394i	1.41472	+	1.18710i	0.460629	+	0.887593i	$0.347624\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		0			0
$738$		0			0
$739$	8.00000	−	13.8564i	0.294285	−	0.509716i	−0.680534	−	0.732717i	$-0.738252\pi$
$739$	8.00000	−	13.8564i	0.294285	−	0.509716i	0.974818	+	0.223001i	$0.0715853\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$		0			0		−0.342020	−	0.939693i	$-0.611111\pi$
$743$		0			0		0.342020	+	0.939693i	$0.388889\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$	7.11958	−	40.3771i	0.259797	−	1.47338i	−0.523655	−	0.851930i	$-0.675432\pi$
$751$	7.11958	−	40.3771i	0.259797	−	1.47338i	0.783452	−	0.621452i	$-0.213457\pi$
$752$		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	$-0.944444\pi$
$761$		0			0		0.984808	+	0.173648i	$0.0555556\pi$
$762$		0			0
$763$	−1.53209	+	1.28558i	−0.0554653	+	0.0465409i
$764$		0			0
$765$		0			0
$766$		0			0
$767$		0			0
$768$		0			0
$769$	46.0449	−	16.7590i	1.66042	−	0.604345i	0.669994	−	0.742367i	$-0.266297\pi$
$769$	46.0449	−	16.7590i	1.66042	−	0.604345i	0.990429	+	0.138022i	$0.0440745\pi$
$770$		0			0
$771$		0			0
$772$	−7.98782	−	45.3012i	−0.287488	−	1.63042i
$773$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$773$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$774$		0			0
$775$	−10.0000	−	17.3205i	−0.359211	−	0.622171i
$776$		0			0
$777$		0			0
$778$		0			0
$779$		0			0
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$	−24.0000			−0.857143
$785$		0			0
$786$		0			0
$787$	−23.7474	−	19.9264i	−0.846503	−	0.710300i	0.112514	−	0.993650i	$-0.464110\pi$
$787$	−23.7474	−	19.9264i	−0.846503	−	0.710300i	−0.959017	+	0.283350i	$0.908554\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$	2.50000	−	4.33013i	0.0887776	−	0.153767i
$794$		0			0
$795$		0			0
$796$	20.6732	−	7.52444i	0.732743	−	0.266697i
$797$		0			0		−0.342020	−	0.939693i	$-0.611111\pi$
$797$		0			0		0.342020	+	0.939693i	$0.388889\pi$
$798$		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
$811$	56.0000			1.96643			0.983213	−	0.182462i	$-0.0584065\pi$
$811$	56.0000			1.96643			0.983213	+	0.182462i	$0.0584065\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$		0			0
$816$		0			0
$817$	−42.8985	+	35.9961i	−1.50083	+	1.25934i
$818$		0			0
$819$		0			0
$820$		0			0
$821$		0			0		0.984808	−	0.173648i	$-0.0555556\pi$
$821$		0			0		−0.984808	+	0.173648i	$0.944444\pi$
$822$		0			0
$823$	−4.69846	+	1.71010i	−0.163778	+	0.0596104i	−0.422608	−	0.906313i	$-0.638885\pi$
$823$	−4.69846	+	1.71010i	−0.163778	+	0.0596104i	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
$829$	3.50000	+	6.06218i	0.121560	+	0.210548i	0.920383	−	0.391018i	$-0.127877\pi$
$829$	3.50000	+	6.06218i	0.121560	+	0.210548i	−0.798823	+	0.601566i	$0.794544\pi$
$830$		0			0
$831$		0			0
$832$	−6.94593	+	39.3923i	−0.240807	+	1.36568i
$833$		0			0
$834$		0			0
$835$		0			0
$836$		0			0
$837$		0			0
$838$		0			0
$839$		0			0		0.342020	−	0.939693i	$-0.388889\pi$
$839$		0			0		−0.342020	+	0.939693i	$0.611111\pi$
$840$		0			0
$841$	−22.2153	−	18.6408i	−0.766044	−	0.642788i
$842$		0			0
$843$		0			0
$844$	19.9172	−	16.7125i	0.685577	−	0.575267i
$845$		0			0
$846$		0			0
$847$	−5.50000	+	9.52628i	−0.188982	+	0.327327i
$848$		0			0
$849$		0			0
$850$		0			0
$851$		0			0
$852$		0			0
$853$	6.07769	+	34.4683i	0.208096	+	1.18017i	0.892493	+	0.451061i	$0.148954\pi$
$853$	6.07769	+	34.4683i	0.208096	+	1.18017i	−0.684397	+	0.729110i	$0.739934\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$		0			0		−0.642788	−	0.766044i	$-0.722222\pi$
$857$		0			0		0.642788	+	0.766044i	$0.277778\pi$
$858$		0			0
$859$	−2.25743	+	12.8025i	−0.0770224	+	0.436816i	0.921772	+	0.387732i	$0.126741\pi$
$859$	−2.25743	+	12.8025i	−0.0770224	+	0.436816i	−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
$868$	−6.12836	−	5.14230i	−0.208010	−	0.174541i
$869$		0			0
$870$		0			0
$871$	19.1511	−	16.0697i	0.648911	−	0.544501i
$872$		0			0
$873$		0			0
$874$		0			0
$875$		0			0
$876$		0			0
$877$	−55.4419	+	20.1792i	−1.87214	+	0.681403i	−0.906064	+	0.423141i	$0.860927\pi$
$877$	−55.4419	+	20.1792i	−1.87214	+	0.681403i	−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
$883$	−23.5000	−	40.7032i	−0.790838	−	1.36977i	−0.925449	−	0.378873i	$-0.876312\pi$
$883$	−23.5000	−	40.7032i	−0.790838	−	1.36977i	0.134611	−	0.990899i	$-0.457022\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$		0			0		0.642788	−	0.766044i	$-0.277778\pi$
$887$		0			0		−0.642788	+	0.766044i	$0.722222\pi$
$888$		0			0
$889$	18.7939	+	6.84040i	0.630326	+	0.229420i
$890$		0			0
$891$		0			0
$892$	56.0000			1.87502
$893$		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
$907$	−3.29932	−	18.7113i	−0.109552	−	0.621300i	−0.989304	−	0.145868i	$-0.953403\pi$
$907$	−3.29932	−	18.7113i	−0.109552	−	0.621300i	0.879752	−	0.475433i	$-0.157708\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$		0			0		−0.642788	−	0.766044i	$-0.722222\pi$
$911$		0			0		0.642788	+	0.766044i	$0.277778\pi$
$912$		0			0
$913$		0			0
$914$		0			0
$915$		0			0
$916$	−41.3465	−	15.0489i	−1.36613	−	0.497229i
$917$		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
$925$	−42.1324	+	35.3533i	−1.38531	+	1.16241i
$926$		0			0
$927$		0			0
$928$		0			0
$929$		0			0		0.984808	−	0.173648i	$-0.0555556\pi$
$929$		0			0		−0.984808	+	0.173648i	$0.944444\pi$
$930$		0			0
$931$	−39.4671	+	14.3648i	−1.29348	+	0.470789i
$932$		0			0
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$	30.5000	+	52.8275i	0.996392	+	1.72580i	0.571700	+	0.820463i	$0.306284\pi$
$937$	30.5000	+	52.8275i	0.996392	+	1.72580i	0.424691	+	0.905338i	$0.360383\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$		0			0		0.642788	−	0.766044i	$-0.277778\pi$
$941$		0			0		−0.642788	+	0.766044i	$0.722222\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$		0			0
$946$		0			0
$947$		0			0		0.342020	−	0.939693i	$-0.388889\pi$
$947$		0			0		−0.342020	+	0.939693i	$0.611111\pi$
$948$		0			0
$949$	−26.8116	−	22.4976i	−0.870340	−	0.730302i
$950$		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
$961$	−2.60472	−	14.7721i	−0.0840233	−	0.476520i
$962$		0			0
$963$		0			0
$964$	17.0000	+	29.4449i	0.547533	+	0.948355i
$965$		0			0
$966$		0			0
$967$	7.11958	−	40.3771i	0.228950	−	1.29844i	−0.626038	−	0.779793i	$-0.715324\pi$
$967$	7.11958	−	40.3771i	0.228950	−	1.29844i	0.854988	−	0.518648i	$-0.173564\pi$
$968$		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
$973$	−23.0000			−0.737346
$974$		0			0
$975$		0			0
$976$	−3.06418	−	2.57115i	−0.0980819	−	0.0823005i
$977$		0			0		−0.984808	−	0.173648i	$-0.944444\pi$
$977$		0			0		0.984808	+	0.173648i	$0.0555556\pi$
$978$		0			0
$979$		0			0
$980$		0			0
$981$		0			0
$982$		0			0
$983$		0			0		0.984808	−	0.173648i	$-0.0555556\pi$
$983$		0			0		−0.984808	+	0.173648i	$0.944444\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$	12.1554	+	68.9365i	0.386714	+	2.19316i
$989$		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
$997$	9.39693	+	3.42020i	0.297604	+	0.108319i	0.486507	−	0.873677i	$-0.338271\pi$
$997$	9.39693	+	3.42020i	0.297604	+	0.108319i	−0.188903	+	0.981996i	$0.560493\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	729.2.e.f.568.1		6
3.2	odd	2	CM	729.2.e.f.568.1		6
9.2	odd	6	inner	729.2.e.f.82.1		6
9.4	even	3	inner	729.2.e.f.325.1		6
9.5	odd	6	inner	729.2.e.f.325.1		6
9.7	even	3	inner	729.2.e.f.82.1		6
27.2	odd	18	inner	729.2.e.f.406.1		6
27.4	even	9		81.2.c.a.28.1		2
27.5	odd	18		81.2.c.a.55.1		2
27.7	even	9	inner	729.2.e.f.163.1		6
27.11	odd	18	inner	729.2.e.f.649.1		6
27.13	even	9		27.2.a.a.1.1	✓	1
27.14	odd	18		27.2.a.a.1.1	✓	1
27.16	even	9	inner	729.2.e.f.649.1		6
27.20	odd	18	inner	729.2.e.f.163.1		6
27.22	even	9		81.2.c.a.55.1		2
27.23	odd	18		81.2.c.a.28.1		2
27.25	even	9	inner	729.2.e.f.406.1		6
108.23	even	18		1296.2.i.i.433.1		2
108.31	odd	18		1296.2.i.i.433.1		2
108.59	even	18		1296.2.i.i.865.1		2
108.67	odd	18		432.2.a.e.1.1		1
108.95	even	18		432.2.a.e.1.1		1
108.103	odd	18		1296.2.i.i.865.1		2
135.13	odd	36		675.2.b.f.649.2		2
135.14	odd	18		675.2.a.e.1.1		1
135.67	odd	36		675.2.b.f.649.1		2
135.68	even	36		675.2.b.f.649.2		2
135.94	even	18		675.2.a.e.1.1		1
135.122	even	36		675.2.b.f.649.1		2
189.13	odd	18		1323.2.a.i.1.1		1
189.41	even	18		1323.2.a.i.1.1		1
216.13	even	18		1728.2.a.n.1.1		1
216.67	odd	18		1728.2.a.o.1.1		1
216.149	odd	18		1728.2.a.n.1.1		1
216.203	even	18		1728.2.a.o.1.1		1
297.175	odd	18		3267.2.a.f.1.1		1
297.230	even	18		3267.2.a.f.1.1		1
351.311	odd	18		4563.2.a.e.1.1		1
351.337	even	18		4563.2.a.e.1.1		1
459.67	even	18		7803.2.a.k.1.1		1
459.203	odd	18		7803.2.a.k.1.1		1
513.94	odd	18		9747.2.a.f.1.1		1
513.284	even	18		9747.2.a.f.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
27.2.a.a.1.1	✓	1	27.13	even	9
27.2.a.a.1.1	✓	1	27.14	odd	18
81.2.c.a.28.1		2	27.4	even	9
81.2.c.a.28.1		2	27.23	odd	18
81.2.c.a.55.1		2	27.5	odd	18
81.2.c.a.55.1		2	27.22	even	9
432.2.a.e.1.1		1	108.67	odd	18
432.2.a.e.1.1		1	108.95	even	18
675.2.a.e.1.1		1	135.14	odd	18
675.2.a.e.1.1		1	135.94	even	18
675.2.b.f.649.1		2	135.67	odd	36
675.2.b.f.649.1		2	135.122	even	36
675.2.b.f.649.2		2	135.13	odd	36
675.2.b.f.649.2		2	135.68	even	36
729.2.e.f.82.1		6	9.2	odd	6	inner
729.2.e.f.82.1		6	9.7	even	3	inner
729.2.e.f.163.1		6	27.7	even	9	inner
729.2.e.f.163.1		6	27.20	odd	18	inner
729.2.e.f.325.1		6	9.4	even	3	inner
729.2.e.f.325.1		6	9.5	odd	6	inner
729.2.e.f.406.1		6	27.2	odd	18	inner
729.2.e.f.406.1		6	27.25	even	9	inner
729.2.e.f.568.1		6	1.1	even	1	trivial
729.2.e.f.568.1		6	3.2	odd	2	CM
729.2.e.f.649.1		6	27.11	odd	18	inner
729.2.e.f.649.1		6	27.16	even	9	inner
1296.2.i.i.433.1		2	108.23	even	18
1296.2.i.i.433.1		2	108.31	odd	18
1296.2.i.i.865.1		2	108.59	even	18
1296.2.i.i.865.1		2	108.103	odd	18
1323.2.a.i.1.1		1	189.13	odd	18
1323.2.a.i.1.1		1	189.41	even	18
1728.2.a.n.1.1		1	216.13	even	18
1728.2.a.n.1.1		1	216.149	odd	18
1728.2.a.o.1.1		1	216.67	odd	18
1728.2.a.o.1.1		1	216.203	even	18
3267.2.a.f.1.1		1	297.175	odd	18
3267.2.a.f.1.1		1	297.230	even	18
4563.2.a.e.1.1		1	351.311	odd	18
4563.2.a.e.1.1		1	351.337	even	18
7803.2.a.k.1.1		1	459.67	even	18
7803.2.a.k.1.1		1	459.203	odd	18
9747.2.a.f.1.1		1	513.94	odd	18
9747.2.a.f.1.1		1	513.284	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 $\F_{q}$

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

\(f(q)\)	\(=\)	\(q+(-1.53209 - 1.28558i) q^{4} +(-0.766044 + 0.642788i) q^{7} +(-4.69846 + 1.71010i) q^{13} +(0.694593 + 3.93923i) q^{16} +(3.50000 + 6.06218i) q^{19} +(4.69846 + 1.71010i) q^{25} +2.00000 q^{28} +(-3.06418 - 2.57115i) q^{31} +(-5.50000 + 9.52628i) q^{37} +(1.38919 + 7.87846i) q^{43} +(-1.04189 + 5.90885i) q^{49} +(9.39693 + 3.42020i) q^{52} +(-0.766044 + 0.642788i) q^{61} +(4.00000 - 6.92820i) q^{64} +(-4.69846 + 1.71010i) q^{67} +(3.50000 + 6.06218i) q^{73} +(2.43107 - 13.7873i) q^{76} +(-15.9748 - 5.81434i) q^{79} +(2.50000 - 4.33013i) q^{91} +(-3.29932 - 18.7113i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 21 q^{19} + 12 q^{28} - 33 q^{37} + 24 q^{64} + 21 q^{73} + 15 q^{91}+O(q^{100}) \) 6 * q + 21 * q^19 + 12 * q^28 - 33 * q^37 + 24 * q^64 + 21 * q^73 + 15 * q^91

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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