LMFDB - Embedded newform 729.2.e.e.649.1

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:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 243)
Sato-Tate group:	$\mathrm{U}(1)[D_{9}]$

Embedding invariants

Embedding label			649.1
Root			$-0.766044 - 0.642788i$ of defining polynomial
Character	$\chi$	$=$	729.649
Dual form			729.2.e.e.82.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(1.87939 + 0.684040i) q^{4} +(3.75877 - 1.36808i) q^{7} +O(q^{10})$	$q+(1.87939 + 0.684040i) q^{4} +(3.75877 - 1.36808i) q^{7} +(-1.21554 - 6.89365i) q^{13} +(3.06418 + 2.57115i) q^{16} +(0.500000 - 0.866025i) q^{19} +(-0.868241 + 4.92404i) q^{25} +8.00000 q^{28} +(-10.3366 - 3.76222i) q^{31} +(5.00000 + 8.66025i) q^{37} +(3.83022 + 3.21394i) q^{43} +(6.89440 - 5.78509i) q^{49} +(2.43107 - 13.7873i) q^{52} +(0.939693 - 0.342020i) q^{61} +(4.00000 + 6.92820i) q^{64} +(0.868241 + 4.92404i) q^{67} +(3.50000 - 6.06218i) q^{73} +(1.53209 - 1.28558i) q^{76} +(-2.25743 + 12.8025i) q^{79} +(-14.0000 - 24.2487i) q^{91} +(3.83022 + 3.21394i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q+O(q^{10}) $ 6 * 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

Display 100 coefficients 10 coefficients

Character values

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

$n$	$2$
$\chi(n)$	$e\left(\frac{5}{9}\right)$

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		−0.984808	−	0.173648i	$-0.944444\pi$
$2$		0			0		0.984808	+	0.173648i	$0.0555556\pi$
$3$		0			0
$3$		0			0
$4$	1.87939	+	0.684040i	0.939693	+	0.342020i
$5$		0			0		−0.642788	−	0.766044i	$-0.722222\pi$
$5$		0			0		0.642788	+	0.766044i	$0.277778\pi$
$6$		0			0
$7$	3.75877	−	1.36808i	1.42068	−	0.517086i	0.486436	−	0.873716i	$-0.338297\pi$
$7$	3.75877	−	1.36808i	1.42068	−	0.517086i	0.934246	+	0.356630i	$0.116074\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$		0			0		0.642788	−	0.766044i	$-0.277778\pi$
$11$		0			0		−0.642788	+	0.766044i	$0.722222\pi$
$12$		0			0
$13$	−1.21554	−	6.89365i	−0.337129	−	1.91196i	−0.405108	−	0.914269i	$-0.632766\pi$
$13$	−1.21554	−	6.89365i	−0.337129	−	1.91196i	0.0679785	−	0.997687i	$-0.478345\pi$
$14$		0			0
$15$		0			0
$16$	3.06418	+	2.57115i	0.766044	+	0.642788i
$17$		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		0.342020	−	0.939693i	$-0.388889\pi$
$23$		0			0		−0.342020	+	0.939693i	$0.611111\pi$
$24$		0			0
$25$	−0.868241	+	4.92404i	−0.173648	+	0.984808i
$26$		0			0
$27$		0			0
$28$	8.00000			1.51186
$29$		0			0		−0.984808	−	0.173648i	$-0.944444\pi$
$29$		0			0		0.984808	+	0.173648i	$0.0555556\pi$
$30$		0			0
$31$	−10.3366	−	3.76222i	−1.85651	−	0.675715i	−0.981455	−	0.191695i	$-0.938602\pi$
$31$	−10.3366	−	3.76222i	−1.85651	−	0.675715i	−0.875057	−	0.484020i	$-0.839176\pi$
$32$		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		0.984808	−	0.173648i	$-0.0555556\pi$
$41$		0			0		−0.984808	+	0.173648i	$0.944444\pi$
$42$		0			0
$43$	3.83022	+	3.21394i	0.584103	+	0.490121i	0.886292	−	0.463127i	$-0.153273\pi$
$43$	3.83022	+	3.21394i	0.584103	+	0.490121i	−0.302188	+	0.953248i	$0.597717\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$		0			0		−0.342020	−	0.939693i	$-0.611111\pi$
$47$		0			0		0.342020	+	0.939693i	$0.388889\pi$
$48$		0			0
$49$	6.89440	−	5.78509i	0.984914	−	0.826441i
$50$		0			0
$51$		0			0
$52$	2.43107	−	13.7873i	0.337129	−	1.91196i
$53$		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		−0.642788	−	0.766044i	$-0.722222\pi$
$59$		0			0		0.642788	+	0.766044i	$0.277778\pi$
$60$		0			0
$61$	0.939693	−	0.342020i	0.120315	−	0.0437912i	−0.281161	−	0.959661i	$-0.590719\pi$
$61$	0.939693	−	0.342020i	0.120315	−	0.0437912i	0.401476	+	0.915869i	$0.368497\pi$
$62$		0			0
$63$		0			0
$64$	4.00000	+	6.92820i	0.500000	+	0.866025i
$65$		0			0
$66$		0			0
$67$	0.868241	+	4.92404i	0.106073	+	0.601567i	0.990787	+	0.135433i	$0.0432425\pi$
$67$	0.868241	+	4.92404i	0.106073	+	0.601567i	−0.884714	+	0.466134i	$0.845646\pi$
$68$		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
$76$	1.53209	−	1.28558i	0.175743	−	0.147466i
$77$		0			0
$78$		0			0
$79$	−2.25743	+	12.8025i	−0.253980	+	1.44039i	0.544696	+	0.838633i	$0.316645\pi$
$79$	−2.25743	+	12.8025i	−0.253980	+	1.44039i	−0.798677	+	0.601760i	$0.794466\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$		0			0		−0.984808	−	0.173648i	$-0.944444\pi$
$83$		0			0		0.984808	+	0.173648i	$0.0555556\pi$
$84$		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
$97$	3.83022	+	3.21394i	0.388900	+	0.326326i	0.816185	−	0.577791i	$-0.196085\pi$
$97$	3.83022	+	3.21394i	0.388900	+	0.326326i	−0.427284	+	0.904117i	$0.640530\pi$
$98$		0			0
$99$		0			0
$100$	−5.00000	+	8.66025i	−0.500000	+	0.866025i
$101$		0			0		−0.342020	−	0.939693i	$-0.611111\pi$
$101$		0			0		0.342020	+	0.939693i	$0.388889\pi$
$102$		0			0
$103$	−9.95858	+	8.35624i	−0.981248	+	0.823365i	−0.984277	−	0.176631i	$-0.943480\pi$
$103$	−9.95858	+	8.35624i	−0.981248	+	0.823365i	0.00302937	+	0.999995i	$0.499036\pi$
$104$		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
$112$	15.0351	+	5.47232i	1.42068	+	0.517086i
$113$		0			0		−0.642788	−	0.766044i	$-0.722222\pi$
$113$		0			0		0.642788	+	0.766044i	$0.277778\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$		0			0
$118$		0			0
$119$		0			0
$120$		0			0
$121$	−1.91013	−	10.8329i	−0.173648	−	0.984808i
$122$		0			0
$123$		0			0
$124$	−16.8530	−	14.1413i	−1.51344	−	1.26993i
$125$		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		0.342020	−	0.939693i	$-0.388889\pi$
$131$		0			0		−0.342020	+	0.939693i	$0.611111\pi$
$132$		0			0
$133$	0.694593	−	3.93923i	0.0602288	−	0.341575i
$134$		0			0
$135$		0			0
$136$		0			0
$137$		0			0		−0.984808	−	0.173648i	$-0.944444\pi$
$137$		0			0		0.984808	+	0.173648i	$0.0555556\pi$
$138$		0			0
$139$	15.0351	+	5.47232i	1.27526	+	0.464156i	0.888861	−	0.458176i	$-0.151497\pi$
$139$	15.0351	+	5.47232i	1.27526	+	0.464156i	0.386398	+	0.922332i	$0.373719\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		0			0
$144$		0			0
$145$		0			0
$146$		0			0
$147$		0			0
$148$	3.47296	+	19.6962i	0.285476	+	1.61901i
$149$		0			0		0.984808	−	0.173648i	$-0.0555556\pi$
$149$		0			0		−0.984808	+	0.173648i	$0.944444\pi$
$150$		0			0
$151$	−14.5548	−	12.2130i	−1.18446	−	0.993877i	−0.999939	−	0.0110477i	$-0.996483\pi$
$151$	−14.5548	−	12.2130i	−1.18446	−	0.993877i	−0.184517	−	0.982829i	$-0.559072\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		0			0
$156$		0			0
$157$	−19.1511	+	16.0697i	−1.52843	+	1.28250i	−0.719785	+	0.694197i	$0.755760\pi$
$157$	−19.1511	+	16.0697i	−1.52843	+	1.28250i	−0.808640	+	0.588304i	$0.799796\pi$
$158$		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		−0.642788	−	0.766044i	$-0.722222\pi$
$167$		0			0		0.642788	+	0.766044i	$0.277778\pi$
$168$		0			0
$169$	−33.8289	+	12.3127i	−2.60223	+	0.947133i
$170$		0			0
$171$		0			0
$172$	5.00000	+	8.66025i	0.381246	+	0.660338i
$173$		0			0		0.642788	−	0.766044i	$-0.277778\pi$
$173$		0			0		−0.642788	+	0.766044i	$0.722222\pi$
$174$		0			0
$175$	3.47296	+	19.6962i	0.262531	+	1.48889i
$176$		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		−0.984808	−	0.173648i	$-0.944444\pi$
$191$		0			0		0.984808	+	0.173648i	$0.0555556\pi$
$192$		0			0
$193$	−21.6129	−	7.86646i	−1.55573	−	0.566240i	−0.585979	−	0.810326i	$-0.699290\pi$
$193$	−21.6129	−	7.86646i	−1.55573	−	0.566240i	−0.969754	+	0.244086i	$0.921512\pi$
$194$		0			0
$195$		0			0
$196$	16.9145	−	6.15636i	1.20818	−	0.439740i
$197$		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
$211$	22.2153	−	18.6408i	1.52936	−	1.28329i	0.726359	−	0.687315i	$-0.241211\pi$
$211$	22.2153	−	18.6408i	1.52936	−	1.28329i	0.803005	−	0.595973i	$-0.203233\pi$
$212$		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
$223$	−21.6129	+	7.86646i	−1.44731	+	0.526777i	−0.941838	−	0.336066i	$-0.890903\pi$
$223$	−21.6129	+	7.86646i	−1.44731	+	0.526777i	−0.505471	+	0.862844i	$0.668681\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$		0			0		0.642788	−	0.766044i	$-0.277778\pi$
$227$		0			0		−0.642788	+	0.766044i	$0.722222\pi$
$228$		0			0
$229$	−1.21554	−	6.89365i	−0.0803250	−	0.455545i	−0.998268	−	0.0588329i	$-0.981262\pi$
$229$	−1.21554	−	6.89365i	−0.0803250	−	0.455545i	0.917943	−	0.396713i	$-0.129849\pi$
$230$		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		0.342020	−	0.939693i	$-0.388889\pi$
$239$		0			0		−0.342020	+	0.939693i	$0.611111\pi$
$240$		0			0
$241$	2.43107	−	13.7873i	0.156599	−	0.888119i	−0.800710	−	0.599052i	$-0.795544\pi$
$241$	2.43107	−	13.7873i	0.156599	−	0.888119i	0.957309	−	0.289066i	$-0.0933448\pi$
$242$		0			0
$243$		0			0
$244$	2.00000			0.128037
$245$		0			0
$246$		0			0
$247$	−6.57785	−	2.39414i	−0.418538	−	0.152336i
$248$		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
$256$	2.77837	+	15.7569i	0.173648	+	0.984808i
$257$		0			0		0.984808	−	0.173648i	$-0.0555556\pi$
$257$		0			0		−0.984808	+	0.173648i	$0.944444\pi$
$258$		0			0
$259$	30.6418	+	25.7115i	1.90399	+	1.59764i
$260$		0			0
$261$		0			0
$262$		0			0
$263$		0			0		−0.342020	−	0.939693i	$-0.611111\pi$
$263$		0			0		0.342020	+	0.939693i	$0.388889\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$		0			0
$268$	−1.73648	+	9.84808i	−0.106073	+	0.601567i
$269$		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
$277$	29.1305	−	10.6026i	1.75028	−	0.637050i	0.750562	−	0.660800i	$-0.229783\pi$
$277$	29.1305	−	10.6026i	1.75028	−	0.637050i	0.999718	+	0.0237496i	$0.00756043\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$		0			0		0.642788	−	0.766044i	$-0.277778\pi$
$281$		0			0		−0.642788	+	0.766044i	$0.722222\pi$
$282$		0			0
$283$	−1.21554	−	6.89365i	−0.0722562	−	0.409785i	−0.999386	−	0.0350443i	$-0.988843\pi$
$283$	−1.21554	−	6.89365i	−0.0722562	−	0.409785i	0.927130	−	0.374741i	$-0.122268\pi$
$284$		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
$292$	10.7246	−	8.99903i	0.627611	−	0.526628i
$293$		0			0		0.342020	−	0.939693i	$-0.388889\pi$
$293$		0			0		−0.342020	+	0.939693i	$0.611111\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$		0			0
$298$		0			0
$299$		0			0
$300$		0			0
$301$	18.7939	+	6.84040i	1.08326	+	0.394274i
$302$		0			0
$303$		0			0
$304$	3.75877	−	1.36808i	0.215580	−	0.0784648i
$305$		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		0.984808	−	0.173648i	$-0.0555556\pi$
$311$		0			0		−0.984808	+	0.173648i	$0.944444\pi$
$312$		0			0
$313$	−16.8530	−	14.1413i	−0.952587	−	0.799315i	0.0271446	−	0.999632i	$-0.491359\pi$
$313$	−16.8530	−	14.1413i	−0.952587	−	0.799315i	−0.979731	+	0.200316i	$0.935803\pi$
$314$		0			0
$315$		0			0
$316$	−13.0000	+	22.5167i	−0.731307	+	1.26666i
$317$		0			0		−0.342020	−	0.939693i	$-0.611111\pi$
$317$		0			0		0.342020	+	0.939693i	$0.388889\pi$
$318$		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
$331$	29.1305	−	10.6026i	1.60116	−	0.582773i	0.621492	−	0.783420i	$-0.286527\pi$
$331$	29.1305	−	10.6026i	1.60116	−	0.582773i	0.979663	+	0.200648i	$0.0643046\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$		0			0
$336$		0			0
$337$	−5.90404	−	33.4835i	−0.321613	−	1.82396i	−0.532476	−	0.846445i	$-0.678738\pi$
$337$	−5.90404	−	33.4835i	−0.321613	−	1.82396i	0.210863	−	0.977516i	$-0.432373\pi$
$338$		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		0.342020	−	0.939693i	$-0.388889\pi$
$347$		0			0		−0.342020	+	0.939693i	$0.611111\pi$
$348$		0			0
$349$	2.43107	−	13.7873i	0.130132	−	0.738018i	−0.847994	−	0.530006i	$-0.822190\pi$
$349$	2.43107	−	13.7873i	0.130132	−	0.738018i	0.978126	−	0.208012i	$-0.0666992\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$		0			0		−0.984808	−	0.173648i	$-0.944444\pi$
$353$		0			0		0.984808	+	0.173648i	$0.0555556\pi$
$354$		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
$364$	−9.72430	−	55.1492i	−0.509692	−	2.89061i
$365$		0			0
$366$		0			0
$367$	26.8116	+	22.4976i	1.39955	+	1.17436i	0.961298	+	0.275512i	$0.0888475\pi$
$367$	26.8116	+	22.4976i	1.39955	+	1.17436i	0.438254	+	0.898851i	$0.355597\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		0			0
$372$		0			0
$373$	−19.1511	+	16.0697i	−0.991607	+	0.832057i	−0.985800	−	0.167926i	$-0.946293\pi$
$373$	−19.1511	+	16.0697i	−0.991607	+	0.832057i	−0.00580736	+	0.999983i	$0.501849\pi$
$374$		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		−0.642788	−	0.766044i	$-0.722222\pi$
$383$		0			0		0.642788	+	0.766044i	$0.277778\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$		0			0
$388$	5.00000	+	8.66025i	0.253837	+	0.439658i
$389$		0			0		0.642788	−	0.766044i	$-0.277778\pi$
$389$		0			0		−0.642788	+	0.766044i	$0.722222\pi$
$390$		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
$400$	−15.3209	+	12.8558i	−0.766044	+	0.642788i
$401$		0			0		0.342020	−	0.939693i	$-0.388889\pi$
$401$		0			0		−0.342020	+	0.939693i	$0.611111\pi$
$402$		0			0
$403$	−13.3709	+	75.8302i	−0.666052	+	3.77737i
$404$		0			0
$405$		0			0
$406$		0			0
$407$		0			0
$408$		0			0
$409$	−35.7083	−	12.9968i	−1.76566	−	0.642649i	−0.765663	−	0.643242i	$-0.777589\pi$
$409$	−35.7083	−	12.9968i	−1.76566	−	0.642649i	−1.00000		0.000593299i	$0.999811\pi$
$410$		0			0
$411$		0			0
$412$	−24.4320	+	8.89252i	−1.20368	+	0.438103i
$413$		0			0
$414$		0			0
$415$		0			0
$416$		0			0
$417$		0			0
$418$		0			0
$419$		0			0		0.984808	−	0.173648i	$-0.0555556\pi$
$419$		0			0		−0.984808	+	0.173648i	$0.944444\pi$
$420$		0			0
$421$	−16.8530	−	14.1413i	−0.821364	−	0.689206i	0.131927	−	0.991259i	$-0.457883\pi$
$421$	−16.8530	−	14.1413i	−0.821364	−	0.689206i	−0.953291	+	0.302053i	$0.902328\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$		0			0
$426$		0			0
$427$	3.06418	−	2.57115i	0.148286	−	0.124427i
$428$		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
$436$	−35.7083	−	12.9968i	−1.71012	−	0.622432i
$437$		0			0
$438$		0			0
$439$	26.3114	−	9.57656i	1.25577	−	0.457064i	0.373425	−	0.927660i	$-0.378183\pi$
$439$	26.3114	−	9.57656i	1.25577	−	0.457064i	0.882349	+	0.470596i	$0.155961\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$		0			0		0.642788	−	0.766044i	$-0.277778\pi$
$443$		0			0		−0.642788	+	0.766044i	$0.722222\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$		0			0
$448$	24.5134	+	20.5692i	1.15815	+	0.971804i
$449$		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
$457$	7.11958	−	40.3771i	0.333040	−	1.88876i	−0.112749	−	0.993624i	$-0.535966\pi$
$457$	7.11958	−	40.3771i	0.333040	−	1.88876i	0.445788	−	0.895138i	$-0.352923\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$		0			0		−0.984808	−	0.173648i	$-0.944444\pi$
$461$		0			0		0.984808	+	0.173648i	$0.0555556\pi$
$462$		0			0
$463$	40.4068	+	14.7069i	1.87786	+	0.683486i	0.952697	+	0.303923i	$0.0982964\pi$
$463$	40.4068	+	14.7069i	1.87786	+	0.683486i	0.925166	+	0.379563i	$0.123926\pi$
$464$		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
$475$	3.83022	+	3.21394i	0.175743	+	0.147466i
$476$		0			0
$477$		0			0
$478$		0			0
$479$		0			0		−0.342020	−	0.939693i	$-0.611111\pi$
$479$		0			0		0.342020	+	0.939693i	$0.388889\pi$
$480$		0			0
$481$	53.6231	−	44.9951i	2.44500	−	2.05160i
$482$		0			0
$483$		0			0
$484$	3.82026	−	21.6658i	0.173648	−	0.984808i
$485$		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		−0.642788	−	0.766044i	$-0.722222\pi$
$491$		0			0		0.642788	+	0.766044i	$0.277778\pi$
$492$		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
$499$	5.55674	+	31.5138i	0.248754	+	1.41075i	0.811610	+	0.584199i	$0.198591\pi$
$499$	5.55674	+	31.5138i	0.248754	+	1.41075i	−0.562857	+	0.826555i	$0.690298\pi$
$500$		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
$508$	1.53209	−	1.28558i	0.0679755	−	0.0570382i
$509$		0			0		0.342020	−	0.939693i	$-0.388889\pi$
$509$		0			0		−0.342020	+	0.939693i	$0.611111\pi$
$510$		0			0
$511$	4.86215	−	27.5746i	0.215089	−	1.21983i
$512$		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
$529$	−17.6190	−	14.7841i	−0.766044	−	0.642788i
$530$		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
$547$	0.939693	−	0.342020i	0.0401784	−	0.0146237i	−0.321853	−	0.946790i	$-0.604306\pi$
$547$	0.939693	−	0.342020i	0.0401784	−	0.0146237i	0.362031	+	0.932166i	$0.382083\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$		0			0
$552$		0			0
$553$	9.02971	+	51.2100i	0.383982	+	2.17767i
$554$		0			0
$555$		0			0
$556$	24.5134	+	20.5692i	1.03960	+	0.872329i
$557$		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		0.342020	−	0.939693i	$-0.388889\pi$
$563$		0			0		−0.342020	+	0.939693i	$0.611111\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$		0			0
$568$		0			0
$569$		0			0		−0.984808	−	0.173648i	$-0.944444\pi$
$569$		0			0		0.984808	+	0.173648i	$0.0555556\pi$
$570$		0			0
$571$	15.0351	+	5.47232i	0.629199	+	0.229010i	0.636882	−	0.770961i	$-0.280224\pi$
$571$	15.0351	+	5.47232i	0.629199	+	0.229010i	−0.00768386	+	0.999970i	$0.502446\pi$
$572$		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		−0.342020	−	0.939693i	$-0.611111\pi$
$587$		0			0		0.342020	+	0.939693i	$0.388889\pi$
$588$		0			0
$589$	−8.42649	+	7.07066i	−0.347207	+	0.291342i
$590$		0			0
$591$		0			0
$592$	−6.94593	+	39.3923i	−0.285476	+	1.61901i
$593$		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		−0.642788	−	0.766044i	$-0.722222\pi$
$599$		0			0		0.642788	+	0.766044i	$0.277778\pi$
$600$		0			0
$601$	−21.6129	+	7.86646i	−0.881610	+	0.320880i	−0.742859	−	0.669448i	$-0.766531\pi$
$601$	−21.6129	+	7.86646i	−0.881610	+	0.320880i	−0.138751	+	0.990327i	$0.544309\pi$
$602$		0			0
$603$		0			0
$604$	−19.0000	−	32.9090i	−0.773099	−	1.33905i
$605$		0			0
$606$		0			0
$607$	3.47296	+	19.6962i	0.140963	+	0.799442i	0.970520	+	0.241020i	$0.0774820\pi$
$607$	3.47296	+	19.6962i	0.140963	+	0.799442i	−0.829557	+	0.558422i	$0.811407\pi$
$608$		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		0.342020	−	0.939693i	$-0.388889\pi$
$617$		0			0		−0.342020	+	0.939693i	$0.611111\pi$
$618$		0			0
$619$	2.95202	−	16.7417i	0.118652	−	0.672907i	−0.866226	−	0.499653i	$-0.833461\pi$
$619$	2.95202	−	16.7417i	0.118652	−	0.672907i	0.984877	−	0.173254i	$-0.0554281\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$		0			0
$625$	−23.4923	−	8.55050i	−0.939693	−	0.342020i
$626$		0			0
$627$		0			0
$628$	−46.9846	+	17.1010i	−1.87489	+	0.682404i
$629$		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
$637$	−48.2608	−	40.4956i	−1.91216	−	1.60449i
$638$		0			0
$639$		0			0
$640$		0			0
$641$		0			0		−0.342020	−	0.939693i	$-0.611111\pi$
$641$		0			0		0.342020	+	0.939693i	$0.388889\pi$
$642$		0			0
$643$	−30.6418	+	25.7115i	−1.20839	+	1.01396i	−0.209044	+	0.977906i	$0.567035\pi$
$643$	−30.6418	+	25.7115i	−1.20839	+	1.01396i	−0.999350	+	0.0360565i	$0.988520\pi$
$644$		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
$652$	15.0351	+	5.47232i	0.588819	+	0.214313i
$653$		0			0		−0.642788	−	0.766044i	$-0.722222\pi$
$653$		0			0		0.642788	+	0.766044i	$0.277778\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$		0			0
$658$		0			0
$659$		0			0		0.642788	−	0.766044i	$-0.277778\pi$
$659$		0			0		−0.642788	+	0.766044i	$0.722222\pi$
$660$		0			0
$661$	−8.50876	−	48.2556i	−0.330952	−	1.87692i	−0.464031	−	0.885819i	$-0.653597\pi$
$661$	−8.50876	−	48.2556i	−0.330952	−	1.87692i	0.133078	−	0.991106i	$-0.457514\pi$
$662$		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
$673$	−2.25743	+	12.8025i	−0.0870174	+	0.493500i	0.909886	+	0.414859i	$0.136169\pi$
$673$	−2.25743	+	12.8025i	−0.0870174	+	0.493500i	−0.996903	+	0.0786409i	$0.974942\pi$
$674$		0			0
$675$		0			0
$676$	−72.0000			−2.76923
$677$		0			0		−0.984808	−	0.173648i	$-0.944444\pi$
$677$		0			0		0.984808	+	0.173648i	$0.0555556\pi$
$678$		0			0
$679$	18.7939	+	6.84040i	0.721242	+	0.262511i
$680$		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
$688$	3.47296	+	19.6962i	0.132405	+	0.750909i
$689$		0			0
$690$		0			0
$691$	−37.5362	−	31.4966i	−1.42794	−	1.19819i	−0.946916	−	0.321481i	$-0.895819\pi$
$691$	−37.5362	−	31.4966i	−1.42794	−	1.19819i	−0.481028	−	0.876705i	$-0.659736\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$		0			0
$696$		0			0
$697$		0			0
$698$		0			0
$699$		0			0
$700$	−6.94593	+	39.3923i	−0.262531	+	1.48889i
$701$		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
$709$	29.1305	−	10.6026i	1.09402	−	0.398190i	0.268910	−	0.963165i	$-0.413337\pi$
$709$	29.1305	−	10.6026i	1.09402	−	0.398190i	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	$-0.166667\pi$
$719$		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
$724$	−39.8343	+	33.4250i	−1.48043	+	1.24223i
$725$		0			0
$726$		0			0
$727$	7.64052	−	43.3315i	0.283371	−	1.60708i	−0.427675	−	0.903933i	$-0.640667\pi$
$727$	7.64052	−	43.3315i	0.283371	−	1.60708i	0.711046	−	0.703145i	$-0.248222\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$		0			0
$732$		0			0
$733$	40.4068	+	14.7069i	1.49246	+	0.543210i	0.954096	−	0.299503i	$-0.0968207\pi$
$733$	40.4068	+	14.7069i	1.49246	+	0.543210i	0.538363	+	0.842713i	$0.319043\pi$
$734$		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		0.984808	−	0.173648i	$-0.0555556\pi$
$743$		0			0		−0.984808	+	0.173648i	$0.944444\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$	−39.8343	+	33.4250i	−1.45357	+	1.21969i	−0.523655	+	0.851930i	$0.675432\pi$
$751$	−39.8343	+	33.4250i	−1.45357	+	1.21969i	−0.929919	+	0.367764i	$0.880123\pi$
$752$		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		−0.642788	−	0.766044i	$-0.722222\pi$
$761$		0			0		0.642788	+	0.766044i	$0.277778\pi$
$762$		0			0
$763$	−71.4166	+	25.9935i	−2.58546	+	0.941029i
$764$		0			0
$765$		0			0
$766$		0			0
$767$		0			0
$768$		0			0
$769$	8.16146	+	46.2860i	0.294310	+	1.66911i	0.669994	+	0.742367i	$0.266297\pi$
$769$	8.16146	+	46.2860i	0.294310	+	1.66911i	−0.375684	+	0.926748i	$0.622592\pi$
$770$		0			0
$771$		0			0
$772$	−35.2380	−	29.5682i	−1.26824	−	1.06418i
$773$		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
$787$	29.1305	+	10.6026i	1.03839	+	0.377943i	0.804269	−	0.594265i	$-0.202557\pi$
$787$	29.1305	+	10.6026i	1.03839	+	0.377943i	0.234120	+	0.972208i	$0.424779\pi$
$788$		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
$796$	−5.90404	−	33.4835i	−0.209263	−	1.18679i
$797$		0			0		0.984808	−	0.173648i	$-0.0555556\pi$
$797$		0			0		−0.984808	+	0.173648i	$0.944444\pi$
$798$		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
$817$	4.69846	−	1.71010i	0.164378	−	0.0598289i
$818$		0			0
$819$		0			0
$820$		0			0
$821$		0			0		0.642788	−	0.766044i	$-0.277778\pi$
$821$		0			0		−0.642788	+	0.766044i	$0.722222\pi$
$822$		0			0
$823$	8.16146	+	46.2860i	0.284491	+	1.61343i	0.707099	+	0.707115i	$0.250004\pi$
$823$	8.16146	+	46.2860i	0.284491	+	1.61343i	−0.422608	+	0.906313i	$0.638885\pi$
$824$		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
$832$	42.8985	−	35.9961i	1.48724	−	1.24794i
$833$		0			0
$834$		0			0
$835$		0			0
$836$		0			0
$837$		0			0
$838$		0			0
$839$		0			0		−0.984808	−	0.173648i	$-0.944444\pi$
$839$		0			0		0.984808	+	0.173648i	$0.0555556\pi$
$840$		0			0
$841$	27.2511	+	9.91858i	0.939693	+	0.342020i
$842$		0			0
$843$		0			0
$844$	54.5022	−	19.8372i	1.87604	−	0.682823i
$845$		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
$853$	26.8116	+	22.4976i	0.918010	+	0.770302i	0.973626	−	0.228150i	$-0.0732677\pi$
$853$	26.8116	+	22.4976i	0.918010	+	0.770302i	−0.0556158	+	0.998452i	$0.517712\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$		0			0		−0.342020	−	0.939693i	$-0.611111\pi$
$857$		0			0		0.342020	+	0.939693i	$0.388889\pi$
$858$		0			0
$859$	42.8985	−	35.9961i	1.46368	−	1.22817i	0.541905	−	0.840440i	$-0.317703\pi$
$859$	42.8985	−	35.9961i	1.46368	−	1.22817i	0.921772	−	0.387732i	$-0.126741\pi$
$860$		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
$868$	−82.6930	−	30.0978i	−2.80678	−	1.02158i
$869$		0			0
$870$		0			0
$871$	32.8892	−	11.9707i	1.11441	−	0.405612i
$872$		0			0
$873$		0			0
$874$		0			0
$875$		0			0
$876$		0			0
$877$	−5.90404	−	33.4835i	−0.199365	−	1.13066i	−0.906064	−	0.423141i	$-0.860927\pi$
$877$	−5.90404	−	33.4835i	−0.199365	−	1.13066i	0.706698	−	0.707515i	$-0.250184\pi$
$878$		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		0.342020	−	0.939693i	$-0.388889\pi$
$887$		0			0		−0.342020	+	0.939693i	$0.611111\pi$
$888$		0			0
$889$	0.694593	−	3.93923i	0.0232959	−	0.132118i
$890$		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
$907$	45.1966	+	37.9245i	1.50073	+	1.25926i	0.879752	+	0.475433i	$0.157708\pi$
$907$	45.1966	+	37.9245i	1.50073	+	1.25926i	0.620977	+	0.783829i	$0.286736\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$		0			0		−0.342020	−	0.939693i	$-0.611111\pi$
$911$		0			0		0.342020	+	0.939693i	$0.388889\pi$
$912$		0			0
$913$		0			0
$914$		0			0
$915$		0			0
$916$	2.43107	−	13.7873i	0.0803250	−	0.455545i
$917$		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
$925$	−46.9846	+	17.1010i	−1.54485	+	0.562278i
$926$		0			0
$927$		0			0
$928$		0			0
$929$		0			0		0.642788	−	0.766044i	$-0.277778\pi$
$929$		0			0		−0.642788	+	0.766044i	$0.722222\pi$
$930$		0			0
$931$	−1.56283	−	8.86327i	−0.0512198	−	0.290482i
$932$		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		0.342020	−	0.939693i	$-0.388889\pi$
$941$		0			0		−0.342020	+	0.939693i	$0.611111\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$		0			0
$946$		0			0
$947$		0			0		−0.984808	−	0.173648i	$-0.944444\pi$
$947$		0			0		0.984808	+	0.173648i	$0.0555556\pi$
$948$		0			0
$949$	−46.0449	−	16.7590i	−1.49468	−	0.544020i
$950$		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
$961$	68.9440	+	57.8509i	2.22400	+	1.86616i
$962$		0			0
$963$		0			0
$964$	14.0000	−	24.2487i	0.450910	−	0.780998i
$965$		0			0
$966$		0			0
$967$	31.4078	−	26.3543i	1.01001	−	0.847497i	0.0216683	−	0.999765i	$-0.493102\pi$
$967$	31.4078	−	26.3543i	1.01001	−	0.847497i	0.988339	+	0.152268i	$0.0486578\pi$
$968$		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
$976$	3.75877	+	1.36808i	0.120315	+	0.0437912i
$977$		0			0		−0.642788	−	0.766044i	$-0.722222\pi$
$977$		0			0		0.642788	+	0.766044i	$0.277778\pi$
$978$		0			0
$979$		0			0
$980$		0			0
$981$		0			0
$982$		0			0
$983$		0			0		0.642788	−	0.766044i	$-0.277778\pi$
$983$		0			0		−0.642788	+	0.766044i	$0.722222\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$	−10.7246	−	8.99903i	−0.341196	−	0.286297i
$989$		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
$997$	−1.73648	+	9.84808i	−0.0549949	+	0.311892i	−0.999880	−	0.0155113i	$-0.995062\pi$
$997$	−1.73648	+	9.84808i	−0.0549949	+	0.311892i	0.944885	+	0.327403i	$0.106173\pi$
$998$		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.649.1		6
3.2	odd	2	CM	729.2.e.e.649.1		6
9.2	odd	6	inner	729.2.e.e.163.1		6
9.4	even	3	inner	729.2.e.e.406.1		6
9.5	odd	6	inner	729.2.e.e.406.1		6
9.7	even	3	inner	729.2.e.e.163.1		6
27.2	odd	18		243.2.c.b.163.1		2
27.4	even	9	inner	729.2.e.e.325.1		6
27.5	odd	18	inner	729.2.e.e.568.1		6
27.7	even	9		243.2.c.b.82.1		2
27.11	odd	18		243.2.a.a.1.1	✓	1
27.13	even	9	inner	729.2.e.e.82.1		6
27.14	odd	18	inner	729.2.e.e.82.1		6
27.16	even	9		243.2.a.a.1.1	✓	1
27.20	odd	18		243.2.c.b.82.1		2
27.22	even	9	inner	729.2.e.e.568.1		6
27.23	odd	18	inner	729.2.e.e.325.1		6
27.25	even	9		243.2.c.b.163.1		2
108.11	even	18		3888.2.a.n.1.1		1
108.43	odd	18		3888.2.a.n.1.1		1
135.119	odd	18		6075.2.a.w.1.1		1
135.124	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	27.11	odd	18
243.2.a.a.1.1	✓	1	27.16	even	9
243.2.c.b.82.1		2	27.7	even	9
243.2.c.b.82.1		2	27.20	odd	18
243.2.c.b.163.1		2	27.2	odd	18
243.2.c.b.163.1		2	27.25	even	9
729.2.e.e.82.1		6	27.13	even	9	inner
729.2.e.e.82.1		6	27.14	odd	18	inner
729.2.e.e.163.1		6	9.2	odd	6	inner
729.2.e.e.163.1		6	9.7	even	3	inner
729.2.e.e.325.1		6	27.4	even	9	inner
729.2.e.e.325.1		6	27.23	odd	18	inner
729.2.e.e.406.1		6	9.4	even	3	inner
729.2.e.e.406.1		6	9.5	odd	6	inner
729.2.e.e.568.1		6	27.5	odd	18	inner
729.2.e.e.568.1		6	27.22	even	9	inner
729.2.e.e.649.1		6	1.1	even	1	trivial
729.2.e.e.649.1		6	3.2	odd	2	CM
3888.2.a.n.1.1		1	108.11	even	18
3888.2.a.n.1.1		1	108.43	odd	18
6075.2.a.w.1.1		1	135.119	odd	18
6075.2.a.w.1.1		1	135.124	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)

\(f(q)\)	\(=\)	\(q+(1.87939 + 0.684040i) q^{4} +(3.75877 - 1.36808i) q^{7} +O(q^{10})\)	\(q+(1.87939 + 0.684040i) q^{4} +(3.75877 - 1.36808i) q^{7} +(-1.21554 - 6.89365i) q^{13} +(3.06418 + 2.57115i) q^{16} +(0.500000 - 0.866025i) q^{19} +(-0.868241 + 4.92404i) q^{25} +8.00000 q^{28} +(-10.3366 - 3.76222i) q^{31} +(5.00000 + 8.66025i) q^{37} +(3.83022 + 3.21394i) q^{43} +(6.89440 - 5.78509i) q^{49} +(2.43107 - 13.7873i) q^{52} +(0.939693 - 0.342020i) q^{61} +(4.00000 + 6.92820i) q^{64} +(0.868241 + 4.92404i) q^{67} +(3.50000 - 6.06218i) q^{73} +(1.53209 - 1.28558i) q^{76} +(-2.25743 + 12.8025i) q^{79} +(-14.0000 - 24.2487i) q^{91} +(3.83022 + 3.21394i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q+O(q^{10}) \) 6 * 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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