LMFDB - Embedded newform 729.2.e.f.82.1

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			82.1
Root			$-0.766044 + 0.642788i$ of defining polynomial
Character	$\chi$	$=$	729.82
Dual form			729.2.e.f.649.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(1.87939 - 0.684040i) q^{4} +(0.939693 + 0.342020i) q^{7} +O(q^{10})$	$q+(1.87939 - 0.684040i) q^{4} +(0.939693 + 0.342020i) q^{7} +(0.868241 - 4.92404i) q^{13} +(3.06418 - 2.57115i) q^{16} +(3.50000 + 6.06218i) q^{19} +(-0.868241 - 4.92404i) q^{25} +2.00000 q^{28} +(3.75877 - 1.36808i) q^{31} +(-5.50000 + 9.52628i) q^{37} +(6.12836 - 5.14230i) q^{43} +(-4.59627 - 3.85673i) q^{49} +(-1.73648 - 9.84808i) q^{52} +(0.939693 + 0.342020i) q^{61} +(4.00000 - 6.92820i) q^{64} +(0.868241 - 4.92404i) q^{67} +(3.50000 + 6.06218i) q^{73} +(10.7246 + 8.99903i) q^{76} +(2.95202 + 16.7417i) q^{79} +(2.50000 - 4.33013i) q^{91} +(-14.5548 + 12.2130i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q+O(q^{10}) $ 6 * 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

Display 100 coefficients 10 coefficients

Character values

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

$n$	$2$
$\chi(n)$	$e\left(\frac{4}{9}\right)$

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	$-0.0555556\pi$
$2$		0			0		−0.984808	+	0.173648i	$0.944444\pi$
$3$		0			0
$3$		0			0
$4$	1.87939	−	0.684040i	0.939693	−	0.342020i
$5$		0			0		0.642788	−	0.766044i	$-0.277778\pi$
$5$		0			0		−0.642788	+	0.766044i	$0.722222\pi$
$6$		0			0
$7$	0.939693	+	0.342020i	0.355170	+	0.129271i	0.513442	−	0.858124i	$-0.328370\pi$
$7$	0.939693	+	0.342020i	0.355170	+	0.129271i	−0.158272	+	0.987396i	$0.550592\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$		0			0		−0.642788	−	0.766044i	$-0.722222\pi$
$11$		0			0		0.642788	+	0.766044i	$0.277778\pi$
$12$		0			0
$13$	0.868241	−	4.92404i	0.240807	−	1.36568i	−0.589226	−	0.807968i	$-0.700567\pi$
$13$	0.868241	−	4.92404i	0.240807	−	1.36568i	0.830033	−	0.557714i	$-0.188322\pi$
$14$		0			0
$15$		0			0
$16$	3.06418	−	2.57115i	0.766044	−	0.642788i
$17$		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		−0.342020	−	0.939693i	$-0.611111\pi$
$23$		0			0		0.342020	+	0.939693i	$0.388889\pi$
$24$		0			0
$25$	−0.868241	−	4.92404i	−0.173648	−	0.984808i
$26$		0			0
$27$		0			0
$28$	2.00000			0.377964
$29$		0			0		0.984808	−	0.173648i	$-0.0555556\pi$
$29$		0			0		−0.984808	+	0.173648i	$0.944444\pi$
$30$		0			0
$31$	3.75877	−	1.36808i	0.675095	−	0.245715i	0.0183550	−	0.999832i	$-0.494157\pi$
$31$	3.75877	−	1.36808i	0.675095	−	0.245715i	0.656740	+	0.754117i	$0.271935\pi$
$32$		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		−0.984808	−	0.173648i	$-0.944444\pi$
$41$		0			0		0.984808	+	0.173648i	$0.0555556\pi$
$42$		0			0
$43$	6.12836	−	5.14230i	0.934565	−	0.784194i	−0.0420659	−	0.999115i	$-0.513394\pi$
$43$	6.12836	−	5.14230i	0.934565	−	0.784194i	0.976631	+	0.214921i	$0.0689495\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$		0			0		0.342020	−	0.939693i	$-0.388889\pi$
$47$		0			0		−0.342020	+	0.939693i	$0.611111\pi$
$48$		0			0
$49$	−4.59627	−	3.85673i	−0.656610	−	0.550961i
$50$		0			0
$51$		0			0
$52$	−1.73648	−	9.84808i	−0.240807	−	1.36568i
$53$		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	$-0.277778\pi$
$59$		0			0		−0.642788	+	0.766044i	$0.722222\pi$
$60$		0			0
$61$	0.939693	+	0.342020i	0.120315	+	0.0437912i	0.401476	−	0.915869i	$-0.368497\pi$
$61$	0.939693	+	0.342020i	0.120315	+	0.0437912i	−0.281161	+	0.959661i	$0.590719\pi$
$62$		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	−0.884714	−	0.466134i	$-0.845646\pi$
$67$	0.868241	−	4.92404i	0.106073	−	0.601567i	0.990787	−	0.135433i	$-0.0432425\pi$
$68$		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
$76$	10.7246	+	8.99903i	1.23020	+	1.03226i
$77$		0			0
$78$		0			0
$79$	2.95202	+	16.7417i	0.332128	+	1.88359i	0.453930	+	0.891038i	$0.350022\pi$
$79$	2.95202	+	16.7417i	0.332128	+	1.88359i	−0.121802	+	0.992554i	$0.538867\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$		0			0		0.984808	−	0.173648i	$-0.0555556\pi$
$83$		0			0		−0.984808	+	0.173648i	$0.944444\pi$
$84$		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
$97$	−14.5548	+	12.2130i	−1.47782	+	1.24004i	−0.569346	+	0.822098i	$0.692804\pi$
$97$	−14.5548	+	12.2130i	−1.47782	+	1.24004i	−0.908474	+	0.417941i	$0.862752\pi$
$98$		0			0
$99$		0			0
$100$	−5.00000	−	8.66025i	−0.500000	−	0.866025i
$101$		0			0		0.342020	−	0.939693i	$-0.388889\pi$
$101$		0			0		−0.342020	+	0.939693i	$0.611111\pi$
$102$		0			0
$103$	−9.95858	−	8.35624i	−0.981248	−	0.823365i	0.00302937	−	0.999995i	$-0.499036\pi$
$103$	−9.95858	−	8.35624i	−0.981248	−	0.823365i	−0.984277	+	0.176631i	$0.943480\pi$
$104$		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
$112$	3.75877	−	1.36808i	0.355170	−	0.129271i
$113$		0			0		0.642788	−	0.766044i	$-0.277778\pi$
$113$		0			0		−0.642788	+	0.766044i	$0.722222\pi$
$114$		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
$124$	6.12836	−	5.14230i	0.550343	−	0.461792i
$125$		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		−0.342020	−	0.939693i	$-0.611111\pi$
$131$		0			0		0.342020	+	0.939693i	$0.388889\pi$
$132$		0			0
$133$	1.21554	+	6.89365i	0.105400	+	0.597756i
$134$		0			0
$135$		0			0
$136$		0			0
$137$		0			0		0.984808	−	0.173648i	$-0.0555556\pi$
$137$		0			0		−0.984808	+	0.173648i	$0.944444\pi$
$138$		0			0
$139$	−21.6129	+	7.86646i	−1.83318	+	0.667225i	−0.841223	+	0.540689i	$0.818164\pi$
$139$	−21.6129	+	7.86646i	−1.83318	+	0.667225i	−0.991962	+	0.126536i	$0.959614\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		0			0
$144$		0			0
$145$		0			0
$146$		0			0
$147$		0			0
$148$	−3.82026	+	21.6658i	−0.314023	+	1.78092i
$149$		0			0		−0.984808	−	0.173648i	$-0.944444\pi$
$149$		0			0		0.984808	+	0.173648i	$0.0555556\pi$
$150$		0			0
$151$	−14.5548	+	12.2130i	−1.18446	+	0.993877i	−0.184517	+	0.982829i	$0.559072\pi$
$151$	−14.5548	+	12.2130i	−1.18446	+	0.993877i	−0.999939	+	0.0110477i	$0.996483\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		0			0
$156$		0			0
$157$	10.7246	+	8.99903i	0.855918	+	0.718201i	0.961085	−	0.276254i	$-0.0890931\pi$
$157$	10.7246	+	8.99903i	0.855918	+	0.718201i	−0.105167	+	0.994455i	$0.533538\pi$
$158$		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		0.642788	−	0.766044i	$-0.277778\pi$
$167$		0			0		−0.642788	+	0.766044i	$0.722222\pi$
$168$		0			0
$169$	−11.2763	−	4.10424i	−0.867409	−	0.315711i
$170$		0			0
$171$		0			0
$172$	8.00000	−	13.8564i	0.609994	−	1.05654i
$173$		0			0		−0.642788	−	0.766044i	$-0.722222\pi$
$173$		0			0		0.642788	+	0.766044i	$0.277778\pi$
$174$		0			0
$175$	0.868241	−	4.92404i	0.0656328	−	0.372222i
$176$		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		0.984808	−	0.173648i	$-0.0555556\pi$
$191$		0			0		−0.984808	+	0.173648i	$0.944444\pi$
$192$		0			0
$193$	−21.6129	+	7.86646i	−1.55573	+	0.566240i	−0.969754	−	0.244086i	$-0.921512\pi$
$193$	−21.6129	+	7.86646i	−1.55573	+	0.566240i	−0.585979	+	0.810326i	$0.699290\pi$
$194$		0			0
$195$		0			0
$196$	−11.2763	−	4.10424i	−0.805451	−	0.293160i
$197$		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
$211$	−9.95858	−	8.35624i	−0.685577	−	0.575267i	0.232053	−	0.972703i	$-0.425456\pi$
$211$	−9.95858	−	8.35624i	−0.685577	−	0.575267i	−0.917630	+	0.397436i	$0.869900\pi$
$212$		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
$223$	26.3114	+	9.57656i	1.76194	+	0.641294i	0.999980	−	0.00632846i	$-0.00201443\pi$
$223$	26.3114	+	9.57656i	1.76194	+	0.641294i	0.761961	+	0.647623i	$0.224237\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$		0			0		−0.642788	−	0.766044i	$-0.722222\pi$
$227$		0			0		0.642788	+	0.766044i	$0.277778\pi$
$228$		0			0
$229$	−3.82026	+	21.6658i	−0.252450	+	1.43171i	0.550085	+	0.835109i	$0.314595\pi$
$229$	−3.82026	+	21.6658i	−0.252450	+	1.43171i	−0.802535	+	0.596606i	$0.796516\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$233$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$		0			0
$238$		0			0
$239$		0			0		−0.342020	−	0.939693i	$-0.611111\pi$
$239$		0			0		0.342020	+	0.939693i	$0.388889\pi$
$240$		0			0
$241$	2.95202	+	16.7417i	0.190156	+	1.07843i	0.919150	+	0.393909i	$0.128877\pi$
$241$	2.95202	+	16.7417i	0.190156	+	1.07843i	−0.728993	+	0.684521i	$0.760011\pi$
$242$		0			0
$243$		0			0
$244$	2.00000			0.128037
$245$		0			0
$246$		0			0
$247$	32.8892	−	11.9707i	2.09269	−	0.761678i
$248$		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
$256$	2.77837	−	15.7569i	0.173648	−	0.984808i
$257$		0			0		−0.984808	−	0.173648i	$-0.944444\pi$
$257$		0			0		0.984808	+	0.173648i	$0.0555556\pi$
$258$		0			0
$259$	−8.42649	+	7.07066i	−0.523597	+	0.439350i
$260$		0			0
$261$		0			0
$262$		0			0
$263$		0			0		0.342020	−	0.939693i	$-0.388889\pi$
$263$		0			0		−0.342020	+	0.939693i	$0.611111\pi$
$264$		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
$277$	−24.4320	−	8.89252i	−1.46798	−	0.534300i	−0.520427	−	0.853906i	$-0.674227\pi$
$277$	−24.4320	−	8.89252i	−1.46798	−	0.534300i	−0.947550	+	0.319606i	$0.896449\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$		0			0		−0.642788	−	0.766044i	$-0.722222\pi$
$281$		0			0		0.642788	+	0.766044i	$0.277778\pi$
$282$		0			0
$283$	5.55674	−	31.5138i	0.330314	−	1.87330i	−0.139030	−	0.990288i	$-0.544398\pi$
$283$	5.55674	−	31.5138i	0.330314	−	1.87330i	0.469344	−	0.883016i	$-0.344491\pi$
$284$		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	$-0.611111\pi$
$293$		0			0		0.342020	+	0.939693i	$0.388889\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$		0			0
$298$		0			0
$299$		0			0
$300$		0			0
$301$	7.51754	−	2.73616i	0.433304	−	0.157710i
$302$		0			0
$303$		0			0
$304$	26.3114	+	9.57656i	1.50906	+	0.549254i
$305$		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		−0.984808	−	0.173648i	$-0.944444\pi$
$311$		0			0		0.984808	+	0.173648i	$0.0555556\pi$
$312$		0			0
$313$	26.8116	−	22.4976i	1.51548	−	1.27164i	0.663345	−	0.748314i	$-0.269136\pi$
$313$	26.8116	−	22.4976i	1.51548	−	1.27164i	0.852134	−	0.523324i	$-0.175308\pi$
$314$		0			0
$315$		0			0
$316$	17.0000	+	29.4449i	0.956325	+	1.65640i
$317$		0			0		0.342020	−	0.939693i	$-0.388889\pi$
$317$		0			0		−0.342020	+	0.939693i	$0.611111\pi$
$318$		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
$331$	0.939693	+	0.342020i	0.0516502	+	0.0187991i	0.367716	−	0.929938i	$-0.380140\pi$
$331$	0.939693	+	0.342020i	0.0516502	+	0.0187991i	−0.316066	+	0.948737i	$0.602362\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$		0			0
$336$		0			0
$337$	0.868241	−	4.92404i	0.0472961	−	0.268229i	−0.951985	−	0.306145i	$-0.900961\pi$
$337$	0.868241	−	4.92404i	0.0472961	−	0.268229i	0.999281	+	0.0379157i	$0.0120718\pi$
$338$		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		−0.342020	−	0.939693i	$-0.611111\pi$
$347$		0			0		0.342020	+	0.939693i	$0.388889\pi$
$348$		0			0
$349$	−6.42498	−	36.4379i	−0.343921	−	1.95048i	−0.308920	−	0.951088i	$-0.599967\pi$
$349$	−6.42498	−	36.4379i	−0.343921	−	1.95048i	−0.0350017	−	0.999387i	$-0.511144\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$		0			0		0.984808	−	0.173648i	$-0.0555556\pi$
$353$		0			0		−0.984808	+	0.173648i	$0.944444\pi$
$354$		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
$364$	1.73648	−	9.84808i	0.0910164	−	0.516180i
$365$		0			0
$366$		0			0
$367$	26.8116	−	22.4976i	1.39955	−	1.17436i	0.438254	−	0.898851i	$-0.355597\pi$
$367$	26.8116	−	22.4976i	1.39955	−	1.17436i	0.961298	−	0.275512i	$-0.0888475\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		0			0
$372$		0			0
$373$	−9.95858	−	8.35624i	−0.515636	−	0.432670i	0.347472	−	0.937691i	$-0.387040\pi$
$373$	−9.95858	−	8.35624i	−0.515636	−	0.432670i	−0.863107	+	0.505021i	$0.831485\pi$
$374$		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		0.642788	−	0.766044i	$-0.277778\pi$
$383$		0			0		−0.642788	+	0.766044i	$0.722222\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$		0			0
$388$	−19.0000	+	32.9090i	−0.964579	+	1.67070i
$389$		0			0		−0.642788	−	0.766044i	$-0.722222\pi$
$389$		0			0		0.642788	+	0.766044i	$0.277778\pi$
$390$		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
$400$	−15.3209	−	12.8558i	−0.766044	−	0.642788i
$401$		0			0		−0.342020	−	0.939693i	$-0.611111\pi$
$401$		0			0		0.342020	+	0.939693i	$0.388889\pi$
$402$		0			0
$403$	−3.47296	−	19.6962i	−0.173001	−	0.981135i
$404$		0			0
$405$		0			0
$406$		0			0
$407$		0			0
$408$		0			0
$409$	29.1305	−	10.6026i	1.44041	−	0.524266i	0.500514	−	0.865729i	$-0.333144\pi$
$409$	29.1305	−	10.6026i	1.44041	−	0.524266i	0.939895	+	0.341463i	$0.110922\pi$
$410$		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	$-0.944444\pi$
$419$		0			0		0.984808	+	0.173648i	$0.0555556\pi$
$420$		0			0
$421$	−14.5548	+	12.2130i	−0.709360	+	0.595223i	−0.924419	−	0.381377i	$-0.875450\pi$
$421$	−14.5548	+	12.2130i	−0.709360	+	0.595223i	0.215060	+	0.976601i	$0.431005\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$		0			0
$426$		0			0
$427$	0.766044	+	0.642788i	0.0370715	+	0.0311067i
$428$		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
$436$	3.75877	−	1.36808i	0.180012	−	0.0655192i
$437$		0			0
$438$		0			0
$439$	26.3114	+	9.57656i	1.25577	+	0.457064i	0.882349	−	0.470596i	$-0.155961\pi$
$439$	26.3114	+	9.57656i	1.25577	+	0.457064i	0.373425	+	0.927660i	$0.378183\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$		0			0		−0.642788	−	0.766044i	$-0.722222\pi$
$443$		0			0		0.642788	+	0.766044i	$0.277778\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$		0			0
$448$	6.12836	−	5.14230i	0.289538	−	0.242951i
$449$		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
$457$	−1.73648	−	9.84808i	−0.0812292	−	0.460674i	−0.998107	−	0.0615051i	$-0.980410\pi$
$457$	−1.73648	−	9.84808i	−0.0812292	−	0.460674i	0.916878	−	0.399169i	$-0.130701\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$		0			0		0.984808	−	0.173648i	$-0.0555556\pi$
$461$		0			0		−0.984808	+	0.173648i	$0.944444\pi$
$462$		0			0
$463$	−21.6129	+	7.86646i	−1.00444	+	0.365586i	−0.791294	−	0.611435i	$-0.790592\pi$
$463$	−21.6129	+	7.86646i	−1.00444	+	0.365586i	−0.213144	+	0.977021i	$0.568370\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$467$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$468$		0			0
$469$	2.50000	−	4.33013i	0.115439	−	0.199947i
$470$		0			0
$471$		0			0
$472$		0			0
$473$		0			0
$474$		0			0
$475$	26.8116	−	22.4976i	1.23020	−	1.03226i
$476$		0			0
$477$		0			0
$478$		0			0
$479$		0			0		0.342020	−	0.939693i	$-0.388889\pi$
$479$		0			0		−0.342020	+	0.939693i	$0.611111\pi$
$480$		0			0
$481$	42.1324	+	35.3533i	1.92107	+	1.61197i
$482$		0			0
$483$		0			0
$484$	3.82026	+	21.6658i	0.173648	+	0.984808i
$485$		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		0.642788	−	0.766044i	$-0.277778\pi$
$491$		0			0		−0.642788	+	0.766044i	$0.722222\pi$
$492$		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
$499$	5.55674	−	31.5138i	0.248754	−	1.41075i	−0.562857	−	0.826555i	$-0.690298\pi$
$499$	5.55674	−	31.5138i	0.248754	−	1.41075i	0.811610	−	0.584199i	$-0.198591\pi$
$500$		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
$508$	−30.6418	−	25.7115i	−1.35951	−	1.14076i
$509$		0			0		−0.342020	−	0.939693i	$-0.611111\pi$
$509$		0			0		0.342020	+	0.939693i	$0.388889\pi$
$510$		0			0
$511$	1.21554	+	6.89365i	0.0537722	+	0.304957i
$512$		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
$529$	−17.6190	+	14.7841i	−0.766044	+	0.642788i
$530$		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
$547$	0.939693	+	0.342020i	0.0401784	+	0.0146237i	0.362031	−	0.932166i	$-0.382083\pi$
$547$	0.939693	+	0.342020i	0.0401784	+	0.0146237i	−0.321853	+	0.946790i	$0.604306\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$		0			0
$552$		0			0
$553$	−2.95202	+	16.7417i	−0.125533	+	0.711931i
$554$		0			0
$555$		0			0
$556$	−35.2380	+	29.5682i	−1.49443	+	1.25397i
$557$		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		−0.342020	−	0.939693i	$-0.611111\pi$
$563$		0			0		0.342020	+	0.939693i	$0.388889\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$		0			0
$568$		0			0
$569$		0			0		0.984808	−	0.173648i	$-0.0555556\pi$
$569$		0			0		−0.984808	+	0.173648i	$0.944444\pi$
$570$		0			0
$571$	29.1305	−	10.6026i	1.21907	−	0.443706i	0.349231	−	0.937037i	$-0.386443\pi$
$571$	29.1305	−	10.6026i	1.21907	−	0.443706i	0.869842	+	0.493331i	$0.164221\pi$
$572$		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		0.342020	−	0.939693i	$-0.388889\pi$
$587$		0			0		−0.342020	+	0.939693i	$0.611111\pi$
$588$		0			0
$589$	21.4492	+	17.9981i	0.883801	+	0.741597i
$590$		0			0
$591$		0			0
$592$	7.64052	+	43.3315i	0.314023	+	1.78092i
$593$		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	$-0.277778\pi$
$599$		0			0		−0.642788	+	0.766044i	$0.722222\pi$
$600$		0			0
$601$	−24.4320	−	8.89252i	−0.996602	−	0.362734i	−0.208329	−	0.978059i	$-0.566802\pi$
$601$	−24.4320	−	8.89252i	−0.996602	−	0.362734i	−0.788273	+	0.615325i	$0.789025\pi$
$602$		0			0
$603$		0			0
$604$	−19.0000	+	32.9090i	−0.773099	+	1.33905i
$605$		0			0
$606$		0			0
$607$	−8.50876	+	48.2556i	−0.345360	+	1.95863i	−0.0688294	+	0.997628i	$0.521926\pi$
$607$	−8.50876	+	48.2556i	−0.345360	+	1.95863i	−0.276531	+	0.961005i	$0.589185\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		0			0
$612$		0			0
$613$	−23.5000	−	40.7032i	−0.949156	−	1.64399i	−0.747208	−	0.664590i	$-0.768606\pi$
$613$	−23.5000	−	40.7032i	−0.949156	−	1.64399i	−0.201948	−	0.979396i	$-0.564727\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$		0			0		−0.342020	−	0.939693i	$-0.611111\pi$
$617$		0			0		0.342020	+	0.939693i	$0.388889\pi$
$618$		0			0
$619$	2.95202	+	16.7417i	0.118652	+	0.672907i	0.984877	+	0.173254i	$0.0554281\pi$
$619$	2.95202	+	16.7417i	0.118652	+	0.672907i	−0.866226	+	0.499653i	$0.833461\pi$
$620$		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
$628$	26.3114	+	9.57656i	1.04994	+	0.382147i
$629$		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
$637$	−22.9813	+	19.2836i	−0.910554	+	0.764045i
$638$		0			0
$639$		0			0
$640$		0			0
$641$		0			0		0.342020	−	0.939693i	$-0.388889\pi$
$641$		0			0		−0.342020	+	0.939693i	$0.611111\pi$
$642$		0			0
$643$	−30.6418	−	25.7115i	−1.20839	−	1.01396i	−0.999350	−	0.0360565i	$-0.988520\pi$
$643$	−30.6418	−	25.7115i	−1.20839	−	1.01396i	−0.209044	−	0.977906i	$-0.567035\pi$
$644$		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
$652$	−46.9846	+	17.1010i	−1.84006	+	0.669727i
$653$		0			0		0.642788	−	0.766044i	$-0.277778\pi$
$653$		0			0		−0.642788	+	0.766044i	$0.722222\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$		0			0
$658$		0			0
$659$		0			0		−0.642788	−	0.766044i	$-0.722222\pi$
$659$		0			0		0.642788	+	0.766044i	$0.277778\pi$
$660$		0			0
$661$	−8.50876	+	48.2556i	−0.330952	+	1.87692i	0.133078	+	0.991106i	$0.457514\pi$
$661$	−8.50876	+	48.2556i	−0.330952	+	1.87692i	−0.464031	+	0.885819i	$0.653597\pi$
$662$		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
$673$	−6.42498	−	36.4379i	−0.247665	−	1.40458i	−0.814221	−	0.580554i	$-0.802836\pi$
$673$	−6.42498	−	36.4379i	−0.247665	−	1.40458i	0.566557	−	0.824023i	$-0.308275\pi$
$674$		0			0
$675$		0			0
$676$	−24.0000			−0.923077
$677$		0			0		0.984808	−	0.173648i	$-0.0555556\pi$
$677$		0			0		−0.984808	+	0.173648i	$0.944444\pi$
$678$		0			0
$679$	−17.8542	+	6.49838i	−0.685180	+	0.249385i
$680$		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
$688$	5.55674	−	31.5138i	0.211849	−	1.20145i
$689$		0			0
$690$		0			0
$691$	6.12836	−	5.14230i	0.233134	−	0.195622i	−0.518735	−	0.854935i	$-0.673597\pi$
$691$	6.12836	−	5.14230i	0.233134	−	0.195622i	0.751869	+	0.659313i	$0.229153\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$		0			0
$696$		0			0
$697$		0			0
$698$		0			0
$699$		0			0
$700$	−1.73648	−	9.84808i	−0.0656328	−	0.372222i
$701$		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
$709$	−49.8037	−	18.1271i	−1.87042	−	0.680776i	−0.968581	−	0.248700i	$-0.919997\pi$
$709$	−49.8037	−	18.1271i	−1.87042	−	0.680776i	−0.901837	−	0.432077i	$-0.857781\pi$
$710$		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
$724$	10.7246	+	8.99903i	0.398577	+	0.334446i
$725$		0			0
$726$		0			0
$727$	7.64052	+	43.3315i	0.283371	+	1.60708i	0.711046	+	0.703145i	$0.248222\pi$
$727$	7.64052	+	43.3315i	0.283371	+	1.60708i	−0.427675	+	0.903933i	$0.640667\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$		0			0
$732$		0			0
$733$	−46.9846	+	17.1010i	−1.73542	+	0.631640i	−0.998992	−	0.0448796i	$-0.985710\pi$
$733$	−46.9846	+	17.1010i	−1.73542	+	0.631640i	−0.736425	+	0.676520i	$0.763487\pi$
$734$		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		−0.984808	−	0.173648i	$-0.944444\pi$
$743$		0			0		0.984808	+	0.173648i	$0.0555556\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$	31.4078	+	26.3543i	1.14609	+	0.961682i	0.999621	−	0.0275338i	$-0.00876539\pi$
$751$	31.4078	+	26.3543i	1.14609	+	0.961682i	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		0.642788	−	0.766044i	$-0.277778\pi$
$761$		0			0		−0.642788	+	0.766044i	$0.722222\pi$
$762$		0			0
$763$	1.87939	+	0.684040i	0.0680383	+	0.0247639i
$764$		0			0
$765$		0			0
$766$		0			0
$767$		0			0
$768$		0			0
$769$	−8.50876	+	48.2556i	−0.306834	+	1.74014i	0.307912	+	0.951415i	$0.400370\pi$
$769$	−8.50876	+	48.2556i	−0.306834	+	1.74014i	−0.614745	+	0.788726i	$0.710741\pi$
$770$		0			0
$771$		0			0
$772$	−35.2380	+	29.5682i	−1.26824	+	1.06418i
$773$		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
$787$	29.1305	−	10.6026i	1.03839	−	0.377943i	0.234120	−	0.972208i	$-0.424779\pi$
$787$	29.1305	−	10.6026i	1.03839	−	0.377943i	0.804269	+	0.594265i	$0.202557\pi$
$788$		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
$796$	−3.82026	+	21.6658i	−0.135406	+	0.767923i
$797$		0			0		−0.984808	−	0.173648i	$-0.944444\pi$
$797$		0			0		0.984808	+	0.173648i	$0.0555556\pi$
$798$		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
$817$	52.6228	+	19.1531i	1.84104	+	0.670083i
$818$		0			0
$819$		0			0
$820$		0			0
$821$		0			0		−0.642788	−	0.766044i	$-0.722222\pi$
$821$		0			0		0.642788	+	0.766044i	$0.277778\pi$
$822$		0			0
$823$	0.868241	−	4.92404i	0.0302650	−	0.171641i	−0.965929	−	0.258808i	$-0.916670\pi$
$823$	0.868241	−	4.92404i	0.0302650	−	0.171641i	0.996194	+	0.0871670i	$0.0277814\pi$
$824$		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
$832$	−30.6418	−	25.7115i	−1.06231	−	0.891386i
$833$		0			0
$834$		0			0
$835$		0			0
$836$		0			0
$837$		0			0
$838$		0			0
$839$		0			0		0.984808	−	0.173648i	$-0.0555556\pi$
$839$		0			0		−0.984808	+	0.173648i	$0.944444\pi$
$840$		0			0
$841$	27.2511	−	9.91858i	0.939693	−	0.342020i
$842$		0			0
$843$		0			0
$844$	−24.4320	−	8.89252i	−0.840984	−	0.306093i
$845$		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
$853$	26.8116	−	22.4976i	0.918010	−	0.770302i	−0.0556158	−	0.998452i	$-0.517712\pi$
$853$	26.8116	−	22.4976i	0.918010	−	0.770302i	0.973626	+	0.228150i	$0.0732677\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$		0			0		0.342020	−	0.939693i	$-0.388889\pi$
$857$		0			0		−0.342020	+	0.939693i	$0.611111\pi$
$858$		0			0
$859$	−9.95858	−	8.35624i	−0.339782	−	0.285111i	0.456889	−	0.889524i	$-0.348964\pi$
$859$	−9.95858	−	8.35624i	−0.339782	−	0.285111i	−0.796672	+	0.604412i	$0.793408\pi$
$860$		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
$868$	7.51754	−	2.73616i	0.255162	−	0.0928714i
$869$		0			0
$870$		0			0
$871$	−23.4923	−	8.55050i	−0.796007	−	0.289723i
$872$		0			0
$873$		0			0
$874$		0			0
$875$		0			0
$876$		0			0
$877$	10.2452	−	58.1037i	0.345957	−	1.96202i	0.0865807	−	0.996245i	$-0.472406\pi$
$877$	10.2452	−	58.1037i	0.345957	−	1.96202i	0.259377	−	0.965776i	$-0.416483\pi$
$878$		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		−0.342020	−	0.939693i	$-0.611111\pi$
$887$		0			0		0.342020	+	0.939693i	$0.388889\pi$
$888$		0			0
$889$	−3.47296	−	19.6962i	−0.116479	−	0.660588i
$890$		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
$907$	−14.5548	+	12.2130i	−0.483286	+	0.405525i	−0.851613	−	0.524171i	$-0.824375\pi$
$907$	−14.5548	+	12.2130i	−0.483286	+	0.405525i	0.368327	+	0.929696i	$0.379931\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$		0			0		0.342020	−	0.939693i	$-0.388889\pi$
$911$		0			0		−0.342020	+	0.939693i	$0.611111\pi$
$912$		0			0
$913$		0			0
$914$		0			0
$915$		0			0
$916$	7.64052	+	43.3315i	0.252450	+	1.43171i
$917$		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
$925$	51.6831	+	18.8111i	1.69933	+	0.618505i
$926$		0			0
$927$		0			0
$928$		0			0
$929$		0			0		−0.642788	−	0.766044i	$-0.722222\pi$
$929$		0			0		0.642788	+	0.766044i	$0.277778\pi$
$930$		0			0
$931$	7.29322	−	41.3619i	0.239026	−	1.35558i
$932$		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		−0.342020	−	0.939693i	$-0.611111\pi$
$941$		0			0		0.342020	+	0.939693i	$0.388889\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$		0			0
$946$		0			0
$947$		0			0		0.984808	−	0.173648i	$-0.0555556\pi$
$947$		0			0		−0.984808	+	0.173648i	$0.944444\pi$
$948$		0			0
$949$	32.8892	−	11.9707i	1.06763	−	0.388586i
$950$		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
$961$	−11.4907	+	9.64181i	−0.370667	+	0.311026i
$962$		0			0
$963$		0			0
$964$	17.0000	+	29.4449i	0.547533	+	0.948355i
$965$		0			0
$966$		0			0
$967$	31.4078	+	26.3543i	1.01001	+	0.847497i	0.988339	−	0.152268i	$-0.0486578\pi$
$967$	31.4078	+	26.3543i	1.01001	+	0.847497i	0.0216683	+	0.999765i	$0.493102\pi$
$968$		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
$976$	3.75877	−	1.36808i	0.120315	−	0.0437912i
$977$		0			0		0.642788	−	0.766044i	$-0.277778\pi$
$977$		0			0		−0.642788	+	0.766044i	$0.722222\pi$
$978$		0			0
$979$		0			0
$980$		0			0
$981$		0			0
$982$		0			0
$983$		0			0		−0.642788	−	0.766044i	$-0.722222\pi$
$983$		0			0		0.642788	+	0.766044i	$0.277778\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$	53.6231	−	44.9951i	1.70598	−	1.43149i
$989$		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
$997$	−1.73648	−	9.84808i	−0.0549949	−	0.311892i	0.944885	−	0.327403i	$-0.106173\pi$
$997$	−1.73648	−	9.84808i	−0.0549949	−	0.311892i	−0.999880	+	0.0155113i	$0.995062\pi$
$998$		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.82.1		6
3.2	odd	2	CM	729.2.e.f.82.1		6
9.2	odd	6	inner	729.2.e.f.325.1		6
9.4	even	3	inner	729.2.e.f.568.1		6
9.5	odd	6	inner	729.2.e.f.568.1		6
9.7	even	3	inner	729.2.e.f.325.1		6
27.2	odd	18	inner	729.2.e.f.649.1		6
27.4	even	9		81.2.c.a.55.1		2
27.5	odd	18		27.2.a.a.1.1	✓	1
27.7	even	9	inner	729.2.e.f.406.1		6
27.11	odd	18	inner	729.2.e.f.163.1		6
27.13	even	9		81.2.c.a.28.1		2
27.14	odd	18		81.2.c.a.28.1		2
27.16	even	9	inner	729.2.e.f.163.1		6
27.20	odd	18	inner	729.2.e.f.406.1		6
27.22	even	9		27.2.a.a.1.1	✓	1
27.23	odd	18		81.2.c.a.55.1		2
27.25	even	9	inner	729.2.e.f.649.1		6
108.23	even	18		1296.2.i.i.865.1		2
108.31	odd	18		1296.2.i.i.865.1		2
108.59	even	18		432.2.a.e.1.1		1
108.67	odd	18		1296.2.i.i.433.1		2
108.95	even	18		1296.2.i.i.433.1		2
108.103	odd	18		432.2.a.e.1.1		1
135.22	odd	36		675.2.b.f.649.1		2
135.32	even	36		675.2.b.f.649.1		2
135.49	even	18		675.2.a.e.1.1		1
135.59	odd	18		675.2.a.e.1.1		1
135.103	odd	36		675.2.b.f.649.2		2
135.113	even	36		675.2.b.f.649.2		2
189.76	odd	18		1323.2.a.i.1.1		1
189.167	even	18		1323.2.a.i.1.1		1
216.5	odd	18		1728.2.a.n.1.1		1
216.59	even	18		1728.2.a.o.1.1		1
216.157	even	18		1728.2.a.n.1.1		1
216.211	odd	18		1728.2.a.o.1.1		1
297.32	even	18		3267.2.a.f.1.1		1
297.76	odd	18		3267.2.a.f.1.1		1
351.103	even	18		4563.2.a.e.1.1		1
351.194	odd	18		4563.2.a.e.1.1		1
459.356	odd	18		7803.2.a.k.1.1		1
459.373	even	18		7803.2.a.k.1.1		1
513.113	even	18		9747.2.a.f.1.1		1
513.265	odd	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	27.5	odd	18
27.2.a.a.1.1	✓	1	27.22	even	9
81.2.c.a.28.1		2	27.13	even	9
81.2.c.a.28.1		2	27.14	odd	18
81.2.c.a.55.1		2	27.4	even	9
81.2.c.a.55.1		2	27.23	odd	18
432.2.a.e.1.1		1	108.59	even	18
432.2.a.e.1.1		1	108.103	odd	18
675.2.a.e.1.1		1	135.49	even	18
675.2.a.e.1.1		1	135.59	odd	18
675.2.b.f.649.1		2	135.22	odd	36
675.2.b.f.649.1		2	135.32	even	36
675.2.b.f.649.2		2	135.103	odd	36
675.2.b.f.649.2		2	135.113	even	36
729.2.e.f.82.1		6	1.1	even	1	trivial
729.2.e.f.82.1		6	3.2	odd	2	CM
729.2.e.f.163.1		6	27.11	odd	18	inner
729.2.e.f.163.1		6	27.16	even	9	inner
729.2.e.f.325.1		6	9.2	odd	6	inner
729.2.e.f.325.1		6	9.7	even	3	inner
729.2.e.f.406.1		6	27.7	even	9	inner
729.2.e.f.406.1		6	27.20	odd	18	inner
729.2.e.f.568.1		6	9.4	even	3	inner
729.2.e.f.568.1		6	9.5	odd	6	inner
729.2.e.f.649.1		6	27.2	odd	18	inner
729.2.e.f.649.1		6	27.25	even	9	inner
1296.2.i.i.433.1		2	108.67	odd	18
1296.2.i.i.433.1		2	108.95	even	18
1296.2.i.i.865.1		2	108.23	even	18
1296.2.i.i.865.1		2	108.31	odd	18
1323.2.a.i.1.1		1	189.76	odd	18
1323.2.a.i.1.1		1	189.167	even	18
1728.2.a.n.1.1		1	216.5	odd	18
1728.2.a.n.1.1		1	216.157	even	18
1728.2.a.o.1.1		1	216.59	even	18
1728.2.a.o.1.1		1	216.211	odd	18
3267.2.a.f.1.1		1	297.32	even	18
3267.2.a.f.1.1		1	297.76	odd	18
4563.2.a.e.1.1		1	351.103	even	18
4563.2.a.e.1.1		1	351.194	odd	18
7803.2.a.k.1.1		1	459.356	odd	18
7803.2.a.k.1.1		1	459.373	even	18
9747.2.a.f.1.1		1	513.113	even	18
9747.2.a.f.1.1		1	513.265	odd	18

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q+(1.87939 - 0.684040i) q^{4} +(0.939693 + 0.342020i) q^{7} +O(q^{10})\)	\(q+(1.87939 - 0.684040i) q^{4} +(0.939693 + 0.342020i) q^{7} +(0.868241 - 4.92404i) q^{13} +(3.06418 - 2.57115i) q^{16} +(3.50000 + 6.06218i) q^{19} +(-0.868241 - 4.92404i) q^{25} +2.00000 q^{28} +(3.75877 - 1.36808i) q^{31} +(-5.50000 + 9.52628i) q^{37} +(6.12836 - 5.14230i) q^{43} +(-4.59627 - 3.85673i) q^{49} +(-1.73648 - 9.84808i) q^{52} +(0.939693 + 0.342020i) q^{61} +(4.00000 - 6.92820i) q^{64} +(0.868241 - 4.92404i) q^{67} +(3.50000 + 6.06218i) q^{73} +(10.7246 + 8.99903i) q^{76} +(2.95202 + 16.7417i) q^{79} +(2.50000 - 4.33013i) q^{91} +(-14.5548 + 12.2130i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q+O(q^{10}) \) 6 * 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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