LMFDB - Embedded newform 729.2.e.f.325.1

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

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-0.347296 + 1.96962i) q^{4} +(-0.173648 - 0.984808i) q^{7} +O(q^{10})$	$q+(-0.347296 + 1.96962i) q^{4} +(-0.173648 - 0.984808i) q^{7} +(3.83022 + 3.21394i) q^{13} +(-3.75877 - 1.36808i) q^{16} +(3.50000 + 6.06218i) q^{19} +(-3.83022 + 3.21394i) q^{25} +2.00000 q^{28} +(-0.694593 + 3.93923i) q^{31} +(-5.50000 + 9.52628i) q^{37} +(-7.51754 - 2.73616i) q^{43} +(5.63816 - 2.05212i) q^{49} +(-7.66044 + 6.42788i) q^{52} +(-0.173648 - 0.984808i) q^{61} +(4.00000 - 6.92820i) q^{64} +(3.83022 + 3.21394i) q^{67} +(3.50000 + 6.06218i) q^{73} +(-13.1557 + 4.78828i) q^{76} +(13.0228 - 10.9274i) q^{79} +(2.50000 - 4.33013i) q^{91} +(17.8542 + 6.49838i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $\left(\mathbb{Z}/729\mathbb{Z}\right)^\times$.

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

Coefficient data

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

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

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

$2$		0			0		−0.642788	−	0.766044i	$-0.722222\pi$
$2$		0			0		0.642788	+	0.766044i	$0.277778\pi$
$3$		0			0
$3$		0			0
$4$	−0.347296	+	1.96962i	−0.173648	+	0.984808i
$5$		0			0		−0.342020	−	0.939693i	$-0.611111\pi$
$5$		0			0		0.342020	+	0.939693i	$0.388889\pi$
$6$		0			0
$7$	−0.173648	−	0.984808i	−0.0656328	−	0.372222i	−0.999878	−	0.0155920i	$-0.995037\pi$
$7$	−0.173648	−	0.984808i	−0.0656328	−	0.372222i	0.934246	−	0.356630i	$-0.116074\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$		0			0		0.342020	−	0.939693i	$-0.388889\pi$
$11$		0			0		−0.342020	+	0.939693i	$0.611111\pi$
$12$		0			0
$13$	3.83022	+	3.21394i	1.06231	+	0.891386i	0.994334	−	0.106301i	$-0.0339006\pi$
$13$	3.83022	+	3.21394i	1.06231	+	0.891386i	0.0679785	+	0.997687i	$0.478345\pi$
$14$		0			0
$15$		0			0
$16$	−3.75877	−	1.36808i	−0.939693	−	0.342020i
$17$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$17$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$18$		0			0
$19$	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		−0.984808	−	0.173648i	$-0.944444\pi$
$23$		0			0		0.984808	+	0.173648i	$0.0555556\pi$
$24$		0			0
$25$	−3.83022	+	3.21394i	−0.766044	+	0.642788i
$26$		0			0
$27$		0			0
$28$	2.00000			0.377964
$29$		0			0		−0.642788	−	0.766044i	$-0.722222\pi$
$29$		0			0		0.642788	+	0.766044i	$0.277778\pi$
$30$		0			0
$31$	−0.694593	+	3.93923i	−0.124753	+	0.707507i	0.856702	+	0.515812i	$0.172510\pi$
$31$	−0.694593	+	3.93923i	−0.124753	+	0.707507i	−0.981455	+	0.191695i	$0.938602\pi$
$32$		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		0.642788	−	0.766044i	$-0.277778\pi$
$41$		0			0		−0.642788	+	0.766044i	$0.722222\pi$
$42$		0			0
$43$	−7.51754	−	2.73616i	−1.14641	−	0.417261i	−0.302188	−	0.953248i	$-0.597717\pi$
$43$	−7.51754	−	2.73616i	−1.14641	−	0.417261i	−0.844226	+	0.535988i	$0.819939\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$		0			0		0.984808	−	0.173648i	$-0.0555556\pi$
$47$		0			0		−0.984808	+	0.173648i	$0.944444\pi$
$48$		0			0
$49$	5.63816	−	2.05212i	0.805451	−	0.293160i
$50$		0			0
$51$		0			0
$52$	−7.66044	+	6.42788i	−1.06231	+	0.891386i
$53$		0			0			−	1.00000i	$-0.5\pi$
$53$		0			0				1.00000i	$0.5\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$		0			0
$58$		0			0
$59$		0			0		−0.342020	−	0.939693i	$-0.611111\pi$
$59$		0			0		0.342020	+	0.939693i	$0.388889\pi$
$60$		0			0
$61$	−0.173648	−	0.984808i	−0.0222334	−	0.126092i	0.971671	−	0.236338i	$-0.0759472\pi$
$61$	−0.173648	−	0.984808i	−0.0222334	−	0.126092i	−0.993904	+	0.110246i	$0.964836\pi$
$62$		0			0
$63$		0			0
$64$	4.00000	−	6.92820i	0.500000	−	0.866025i
$65$		0			0
$66$		0			0
$67$	3.83022	+	3.21394i	0.467936	+	0.392645i	0.846041	−	0.533118i	$-0.178980\pi$
$67$	3.83022	+	3.21394i	0.467936	+	0.392645i	−0.378105	+	0.925763i	$0.623424\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$71$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$72$		0			0
$73$	3.50000	+	6.06218i	0.409644	+	0.709524i	0.994850	−	0.101361i	$-0.0323196\pi$
$73$	3.50000	+	6.06218i	0.409644	+	0.709524i	−0.585206	+	0.810885i	$0.698986\pi$
$74$		0			0
$75$		0			0
$76$	−13.1557	+	4.78828i	−1.50906	+	0.549254i
$77$		0			0
$78$		0			0
$79$	13.0228	−	10.9274i	1.46517	−	1.22943i	0.544696	−	0.838633i	$-0.316645\pi$
$79$	13.0228	−	10.9274i	1.46517	−	1.22943i	0.920478	−	0.390794i	$-0.127800\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$		0			0		−0.642788	−	0.766044i	$-0.722222\pi$
$83$		0			0		0.642788	+	0.766044i	$0.277778\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$		0			0
$88$		0			0
$89$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$89$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$90$		0			0
$91$	2.50000	−	4.33013i	0.262071	−	0.453921i
$92$		0			0
$93$		0			0
$94$		0			0
$95$		0			0
$96$		0			0
$97$	17.8542	+	6.49838i	1.81282	+	0.659811i	0.996631	+	0.0820195i	$0.0261370\pi$
$97$	17.8542	+	6.49838i	1.81282	+	0.659811i	0.816185	+	0.577791i	$0.196085\pi$
$98$		0			0
$99$		0			0
$100$	−5.00000	−	8.66025i	−0.500000	−	0.866025i
$101$		0			0		0.984808	−	0.173648i	$-0.0555556\pi$
$101$		0			0		−0.984808	+	0.173648i	$0.944444\pi$
$102$		0			0
$103$	12.2160	−	4.44626i	1.20368	−	0.438103i	0.339172	−	0.940724i	$-0.389853\pi$
$103$	12.2160	−	4.44626i	1.20368	−	0.438103i	0.864507	+	0.502621i	$0.167631\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		0			0			−	1.00000i	$-0.5\pi$
$107$		0			0				1.00000i	$0.5\pi$
$108$		0			0
$109$	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
$112$	−0.694593	+	3.93923i	−0.0656328	+	0.372222i
$113$		0			0		−0.342020	−	0.939693i	$-0.611111\pi$
$113$		0			0		0.342020	+	0.939693i	$0.388889\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$		0			0
$118$		0			0
$119$		0			0
$120$		0			0
$121$	−8.42649	−	7.07066i	−0.766044	−	0.642788i
$122$		0			0
$123$		0			0
$124$	−7.51754	−	2.73616i	−0.675095	−	0.245715i
$125$		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		−0.984808	−	0.173648i	$-0.944444\pi$
$131$		0			0		0.984808	+	0.173648i	$0.0555556\pi$
$132$		0			0
$133$	5.36231	−	4.49951i	0.464971	−	0.390157i
$134$		0			0
$135$		0			0
$136$		0			0
$137$		0			0		−0.642788	−	0.766044i	$-0.722222\pi$
$137$		0			0		0.642788	+	0.766044i	$0.277778\pi$
$138$		0			0
$139$	3.99391	−	22.6506i	0.338759	−	1.92120i	−0.0476387	−	0.998865i	$-0.515170\pi$
$139$	3.99391	−	22.6506i	0.338759	−	1.92120i	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
$148$	−16.8530	−	14.1413i	−1.38531	−	1.16241i
$149$		0			0		0.642788	−	0.766044i	$-0.277778\pi$
$149$		0			0		−0.642788	+	0.766044i	$0.722222\pi$
$150$		0			0
$151$	17.8542	+	6.49838i	1.45295	+	0.528831i	0.943414	−	0.331618i	$-0.107594\pi$
$151$	17.8542	+	6.49838i	1.45295	+	0.528831i	0.509537	+	0.860449i	$0.329817\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		0			0
$156$		0			0
$157$	−13.1557	+	4.78828i	−1.04994	+	0.382147i	−0.808640	−	0.588304i	$-0.799796\pi$
$157$	−13.1557	+	4.78828i	−1.04994	+	0.382147i	−0.241299	+	0.970451i	$0.577574\pi$
$158$		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		−0.342020	−	0.939693i	$-0.611111\pi$
$167$		0			0		0.342020	+	0.939693i	$0.388889\pi$
$168$		0			0
$169$	2.08378	+	11.8177i	0.160291	+	0.909053i
$170$		0			0
$171$		0			0
$172$	8.00000	−	13.8564i	0.609994	−	1.05654i
$173$		0			0		0.342020	−	0.939693i	$-0.388889\pi$
$173$		0			0		−0.342020	+	0.939693i	$0.611111\pi$
$174$		0			0
$175$	3.83022	+	3.21394i	0.289538	+	0.242951i
$176$		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		−0.642788	−	0.766044i	$-0.722222\pi$
$191$		0			0		0.642788	+	0.766044i	$0.277778\pi$
$192$		0			0
$193$	3.99391	−	22.6506i	0.287488	−	1.63042i	−0.408773	−	0.912636i	$-0.634043\pi$
$193$	3.99391	−	22.6506i	0.287488	−	1.63042i	0.696261	−	0.717788i	$-0.254845\pi$
$194$		0			0
$195$		0			0
$196$	2.08378	+	11.8177i	0.148841	+	0.844121i
$197$		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
$211$	12.2160	−	4.44626i	0.840984	−	0.306093i	0.114625	−	0.993409i	$-0.463433\pi$
$211$	12.2160	−	4.44626i	0.840984	−	0.306093i	0.726359	+	0.687315i	$0.241211\pi$
$212$		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
$223$	−4.86215	−	27.5746i	−0.325594	−	1.84653i	−0.505471	−	0.862844i	$-0.668681\pi$
$223$	−4.86215	−	27.5746i	−0.325594	−	1.84653i	0.179877	−	0.983689i	$-0.442430\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$		0			0		0.342020	−	0.939693i	$-0.388889\pi$
$227$		0			0		−0.342020	+	0.939693i	$0.611111\pi$
$228$		0			0
$229$	−16.8530	−	14.1413i	−1.11368	−	0.934485i	−0.115408	−	0.993318i	$-0.536818\pi$
$229$	−16.8530	−	14.1413i	−1.11368	−	0.934485i	−0.998268	+	0.0588329i	$0.981262\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$233$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$		0			0
$238$		0			0
$239$		0			0		−0.984808	−	0.173648i	$-0.944444\pi$
$239$		0			0		0.984808	+	0.173648i	$0.0555556\pi$
$240$		0			0
$241$	13.0228	−	10.9274i	0.838869	−	0.703895i	−0.118440	−	0.992961i	$-0.537789\pi$
$241$	13.0228	−	10.9274i	0.838869	−	0.703895i	0.957309	+	0.289066i	$0.0933448\pi$
$242$		0			0
$243$		0			0
$244$	2.00000			0.128037
$245$		0			0
$246$		0			0
$247$	−6.07769	+	34.4683i	−0.386714	+	2.19316i
$248$		0			0
$249$		0			0
$250$		0			0
$251$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$251$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$252$		0			0
$253$		0			0
$254$		0			0
$255$		0			0
$256$	12.2567	+	10.2846i	0.766044	+	0.642788i
$257$		0			0		0.642788	−	0.766044i	$-0.277778\pi$
$257$		0			0		−0.642788	+	0.766044i	$0.722222\pi$
$258$		0			0
$259$	10.3366	+	3.76222i	0.642286	+	0.233773i
$260$		0			0
$261$		0			0
$262$		0			0
$263$		0			0		0.984808	−	0.173648i	$-0.0555556\pi$
$263$		0			0		−0.984808	+	0.173648i	$0.944444\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$		0			0
$268$	−7.66044	+	6.42788i	−0.467936	+	0.392645i
$269$		0			0			−	1.00000i	$-0.5\pi$
$269$		0			0				1.00000i	$0.5\pi$
$270$		0			0
$271$	29.0000			1.76162			0.880812	−	0.473466i	$-0.156997\pi$
$271$	29.0000			1.76162			0.880812	+	0.473466i	$0.156997\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$		0			0
$276$		0			0
$277$	4.51485	+	25.6050i	0.271271	+	1.53846i	0.750562	+	0.660800i	$0.229783\pi$
$277$	4.51485	+	25.6050i	0.271271	+	1.53846i	−0.479291	+	0.877656i	$0.659106\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$		0			0		0.342020	−	0.939693i	$-0.388889\pi$
$281$		0			0		−0.342020	+	0.939693i	$0.611111\pi$
$282$		0			0
$283$	24.5134	+	20.5692i	1.45717	+	1.22271i	0.927130	+	0.374741i	$0.122268\pi$
$283$	24.5134	+	20.5692i	1.45717	+	1.22271i	0.530042	+	0.847971i	$0.322176\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$	8.50000	+	14.7224i	0.500000	+	0.866025i
$290$		0			0
$291$		0			0
$292$	−13.1557	+	4.78828i	−0.769879	+	0.280213i
$293$		0			0		−0.984808	−	0.173648i	$-0.944444\pi$
$293$		0			0		0.984808	+	0.173648i	$0.0555556\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$		0			0
$298$		0			0
$299$		0			0
$300$		0			0
$301$	−1.38919	+	7.87846i	−0.0800713	+	0.454107i
$302$		0			0
$303$		0			0
$304$	−4.86215	−	27.5746i	−0.278863	−	1.58151i
$305$		0			0
$306$		0			0
$307$	8.00000	−	13.8564i	0.456584	−	0.790827i	−0.542194	−	0.840254i	$-0.682406\pi$
$307$	8.00000	−	13.8564i	0.456584	−	0.790827i	0.998778	+	0.0494267i	$0.0157394\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$		0			0		0.642788	−	0.766044i	$-0.277778\pi$
$311$		0			0		−0.642788	+	0.766044i	$0.722222\pi$
$312$		0			0
$313$	−32.8892	−	11.9707i	−1.85901	−	0.676624i	−0.979731	−	0.200316i	$-0.935803\pi$
$313$	−32.8892	−	11.9707i	−1.85901	−	0.676624i	−0.879279	−	0.476308i	$-0.841975\pi$
$314$		0			0
$315$		0			0
$316$	17.0000	+	29.4449i	0.956325	+	1.65640i
$317$		0			0		0.984808	−	0.173648i	$-0.0555556\pi$
$317$		0			0		−0.984808	+	0.173648i	$0.944444\pi$
$318$		0			0
$319$		0			0
$320$		0			0
$321$		0			0
$322$		0			0
$323$		0			0
$324$		0			0
$325$	−25.0000			−1.38675
$326$		0			0
$327$		0			0
$328$		0			0
$329$		0			0
$330$		0			0
$331$	−0.173648	−	0.984808i	−0.00954457	−	0.0541299i	0.979663	−	0.200648i	$-0.0643046\pi$
$331$	−0.173648	−	0.984808i	−0.00954457	−	0.0541299i	−0.989208	+	0.146518i	$0.953193\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$		0			0
$336$		0			0
$337$	3.83022	+	3.21394i	0.208645	+	0.175074i	0.741122	−	0.671370i	$-0.234294\pi$
$337$	3.83022	+	3.21394i	0.208645	+	0.175074i	−0.532476	+	0.846445i	$0.678738\pi$
$338$		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		−0.984808	−	0.173648i	$-0.944444\pi$
$347$		0			0		0.984808	+	0.173648i	$0.0555556\pi$
$348$		0			0
$349$	−28.3436	+	23.7831i	−1.51720	+	1.27308i	−0.669207	+	0.743076i	$0.733366\pi$
$349$	−28.3436	+	23.7831i	−1.51720	+	1.27308i	−0.847994	+	0.530006i	$0.822190\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$		0			0		−0.642788	−	0.766044i	$-0.722222\pi$
$353$		0			0		0.642788	+	0.766044i	$0.277778\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		0			0
$358$		0			0
$359$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$359$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$360$		0			0
$361$	−15.0000	+	25.9808i	−0.789474	+	1.36741i
$362$		0			0
$363$		0			0
$364$	7.66044	+	6.42788i	0.401516	+	0.336912i
$365$		0			0
$366$		0			0
$367$	−32.8892	−	11.9707i	−1.71680	−	0.624866i	−0.719249	−	0.694752i	$-0.755514\pi$
$367$	−32.8892	−	11.9707i	−1.71680	−	0.624866i	−0.997555	+	0.0698862i	$0.977736\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		0			0
$372$		0			0
$373$	12.2160	−	4.44626i	0.632521	−	0.230219i	−0.00580736	−	0.999983i	$-0.501849\pi$
$373$	12.2160	−	4.44626i	0.632521	−	0.230219i	0.638328	+	0.769764i	$0.279626\pi$
$374$		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		−0.342020	−	0.939693i	$-0.611111\pi$
$383$		0			0		0.342020	+	0.939693i	$0.388889\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$		0			0
$388$	−19.0000	+	32.9090i	−0.964579	+	1.67070i
$389$		0			0		0.342020	−	0.939693i	$-0.388889\pi$
$389$		0			0		−0.342020	+	0.939693i	$0.611111\pi$
$390$		0			0
$391$		0			0
$392$		0			0
$393$		0			0
$394$		0			0
$395$		0			0
$396$		0			0
$397$	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
$400$	18.7939	−	6.84040i	0.939693	−	0.342020i
$401$		0			0		−0.984808	−	0.173648i	$-0.944444\pi$
$401$		0			0		0.984808	+	0.173648i	$0.0555556\pi$
$402$		0			0
$403$	−15.3209	+	12.8558i	−0.763188	+	0.640391i
$404$		0			0
$405$		0			0
$406$		0			0
$407$		0			0
$408$		0			0
$409$	−5.38309	+	30.5290i	−0.266177	+	1.50956i	0.499486	+	0.866322i	$0.333522\pi$
$409$	−5.38309	+	30.5290i	−0.266177	+	1.50956i	−0.765663	+	0.643242i	$0.777589\pi$
$410$		0			0
$411$		0			0
$412$	4.51485	+	25.6050i	0.222431	+	1.26147i
$413$		0			0
$414$		0			0
$415$		0			0
$416$		0			0
$417$		0			0
$418$		0			0
$419$		0			0		0.642788	−	0.766044i	$-0.277778\pi$
$419$		0			0		−0.642788	+	0.766044i	$0.722222\pi$
$420$		0			0
$421$	17.8542	+	6.49838i	0.870159	+	0.316712i	0.738231	−	0.674548i	$-0.235661\pi$
$421$	17.8542	+	6.49838i	0.870159	+	0.316712i	0.131927	+	0.991259i	$0.457883\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$		0			0
$426$		0			0
$427$	−0.939693	+	0.342020i	−0.0454749	+	0.0165515i
$428$		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
$436$	−0.694593	+	3.93923i	−0.0332650	+	0.188655i
$437$		0			0
$438$		0			0
$439$	−4.86215	−	27.5746i	−0.232058	−	1.31606i	−0.848722	−	0.528839i	$-0.822628\pi$
$439$	−4.86215	−	27.5746i	−0.232058	−	1.31606i	0.616665	−	0.787226i	$-0.288483\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$		0			0		0.342020	−	0.939693i	$-0.388889\pi$
$443$		0			0		−0.342020	+	0.939693i	$0.611111\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$		0			0
$448$	−7.51754	−	2.73616i	−0.355170	−	0.129271i
$449$		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
$457$	−7.66044	+	6.42788i	−0.358340	+	0.300683i	−0.804129	−	0.594455i	$-0.797368\pi$
$457$	−7.66044	+	6.42788i	−0.358340	+	0.300683i	0.445788	+	0.895138i	$0.352923\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$		0			0		−0.642788	−	0.766044i	$-0.722222\pi$
$461$		0			0		0.642788	+	0.766044i	$0.277778\pi$
$462$		0			0
$463$	3.99391	−	22.6506i	0.185613	−	1.05266i	−0.739553	−	0.673098i	$-0.764963\pi$
$463$	3.99391	−	22.6506i	0.185613	−	1.05266i	0.925166	−	0.379563i	$-0.123926\pi$
$464$		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
$475$	−32.8892	−	11.9707i	−1.50906	−	0.549254i
$476$		0			0
$477$		0			0
$478$		0			0
$479$		0			0		0.984808	−	0.173648i	$-0.0555556\pi$
$479$		0			0		−0.984808	+	0.173648i	$0.944444\pi$
$480$		0			0
$481$	−51.6831	+	18.8111i	−2.35655	+	0.857713i
$482$		0			0
$483$		0			0
$484$	16.8530	−	14.1413i	0.766044	−	0.642788i
$485$		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		−0.342020	−	0.939693i	$-0.611111\pi$
$491$		0			0		0.342020	+	0.939693i	$0.388889\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$		0			0
$496$	8.00000	−	13.8564i	0.359211	−	0.622171i
$497$		0			0
$498$		0			0
$499$	24.5134	+	20.5692i	1.09737	+	0.920804i	0.997246	−	0.0741708i	$-0.0236310\pi$
$499$	24.5134	+	20.5692i	1.09737	+	0.920804i	0.100126	+	0.994975i	$0.468075\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$503$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$		0			0
$508$	37.5877	−	13.6808i	1.66768	−	0.606988i
$509$		0			0		−0.984808	−	0.173648i	$-0.944444\pi$
$509$		0			0		0.984808	+	0.173648i	$0.0555556\pi$
$510$		0			0
$511$	5.36231	−	4.49951i	0.237215	−	0.199047i
$512$		0			0
$513$		0			0
$514$		0			0
$515$		0			0
$516$		0			0
$517$		0			0
$518$		0			0
$519$		0			0
$520$		0			0
$521$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$521$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$522$		0			0
$523$	21.5000	−	37.2391i	0.940129	−	1.62835i	0.174908	−	0.984585i	$-0.444037\pi$
$523$	21.5000	−	37.2391i	0.940129	−	1.62835i	0.765222	−	0.643767i	$-0.222629\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$		0			0
$528$		0			0
$529$	21.6129	+	7.86646i	0.939693	+	0.342020i
$530$		0			0
$531$		0			0
$532$	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
$547$	−0.173648	−	0.984808i	−0.00742466	−	0.0421073i	0.980870	−	0.194662i	$-0.0623610\pi$
$547$	−0.173648	−	0.984808i	−0.00742466	−	0.0421073i	−0.988295	+	0.152555i	$0.951250\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$		0			0
$552$		0			0
$553$	−13.0228	−	10.9274i	−0.553784	−	0.464680i
$554$		0			0
$555$		0			0
$556$	43.2259	+	15.7329i	1.83318	+	0.667225i
$557$		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		−0.984808	−	0.173648i	$-0.944444\pi$
$563$		0			0		0.984808	+	0.173648i	$0.0555556\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$		0			0
$568$		0			0
$569$		0			0		−0.642788	−	0.766044i	$-0.722222\pi$
$569$		0			0		0.642788	+	0.766044i	$0.277778\pi$
$570$		0			0
$571$	−5.38309	+	30.5290i	−0.225275	+	1.27760i	0.636882	+	0.770961i	$0.280224\pi$
$571$	−5.38309	+	30.5290i	−0.225275	+	1.27760i	−0.862158	+	0.506640i	$0.830887\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$		0			0
$577$	−5.50000	+	9.52628i	−0.228968	+	0.396584i	−0.957503	−	0.288425i	$-0.906868\pi$
$577$	−5.50000	+	9.52628i	−0.228968	+	0.396584i	0.728535	+	0.685009i	$0.240202\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$		0			0
$582$		0			0
$583$		0			0
$584$		0			0
$585$		0			0
$586$		0			0
$587$		0			0		0.984808	−	0.173648i	$-0.0555556\pi$
$587$		0			0		−0.984808	+	0.173648i	$0.944444\pi$
$588$		0			0
$589$	−26.3114	+	9.57656i	−1.08414	+	0.394595i
$590$		0			0
$591$		0			0
$592$	33.7060	−	28.2827i	1.38531	−	1.16241i
$593$		0			0			−	1.00000i	$-0.5\pi$
$593$		0			0				1.00000i	$0.5\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$		0			0
$598$		0			0
$599$		0			0		−0.342020	−	0.939693i	$-0.611111\pi$
$599$		0			0		0.342020	+	0.939693i	$0.388889\pi$
$600$		0			0
$601$	4.51485	+	25.6050i	0.184165	+	1.04445i	0.927024	+	0.375002i	$0.122358\pi$
$601$	4.51485	+	25.6050i	0.184165	+	1.04445i	−0.742859	+	0.669448i	$0.766531\pi$
$602$		0			0
$603$		0			0
$604$	−19.0000	+	32.9090i	−0.773099	+	1.33905i
$605$		0			0
$606$		0			0
$607$	−37.5362	−	31.4966i	−1.52355	−	1.27841i	−0.829557	−	0.558422i	$-0.811407\pi$
$607$	−37.5362	−	31.4966i	−1.52355	−	1.27841i	−0.693990	−	0.719985i	$-0.744149\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		0			0
$612$		0			0
$613$	−23.5000	−	40.7032i	−0.949156	−	1.64399i	−0.747208	−	0.664590i	$-0.768606\pi$
$613$	−23.5000	−	40.7032i	−0.949156	−	1.64399i	−0.201948	−	0.979396i	$-0.564727\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$		0			0		−0.984808	−	0.173648i	$-0.944444\pi$
$617$		0			0		0.984808	+	0.173648i	$0.0555556\pi$
$618$		0			0
$619$	13.0228	−	10.9274i	0.523429	−	0.439209i	−0.342396	−	0.939556i	$-0.611239\pi$
$619$	13.0228	−	10.9274i	0.523429	−	0.439209i	0.865825	+	0.500347i	$0.166794\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$		0			0
$625$	4.34120	−	24.6202i	0.173648	−	0.984808i
$626$		0			0
$627$		0			0
$628$	−4.86215	−	27.5746i	−0.194021	−	1.10035i
$629$		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
$637$	28.1908	+	10.2606i	1.11696	+	0.406540i
$638$		0			0
$639$		0			0
$640$		0			0
$641$		0			0		0.984808	−	0.173648i	$-0.0555556\pi$
$641$		0			0		−0.984808	+	0.173648i	$0.944444\pi$
$642$		0			0
$643$	37.5877	−	13.6808i	1.48231	−	0.539518i	0.530901	−	0.847434i	$-0.321854\pi$
$643$	37.5877	−	13.6808i	1.48231	−	0.539518i	0.951414	+	0.307916i	$0.0996315\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		0			0			−	1.00000i	$-0.5\pi$
$647$		0			0				1.00000i	$0.5\pi$
$648$		0			0
$649$		0			0
$650$		0			0
$651$		0			0
$652$	8.68241	−	49.2404i	0.340029	−	1.92840i
$653$		0			0		−0.342020	−	0.939693i	$-0.611111\pi$
$653$		0			0		0.342020	+	0.939693i	$0.388889\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$		0			0
$658$		0			0
$659$		0			0		0.342020	−	0.939693i	$-0.388889\pi$
$659$		0			0		−0.342020	+	0.939693i	$0.611111\pi$
$660$		0			0
$661$	−37.5362	−	31.4966i	−1.45999	−	1.22508i	−0.924862	−	0.380303i	$-0.875820\pi$
$661$	−37.5362	−	31.4966i	−1.45999	−	1.22508i	−0.535126	−	0.844772i	$-0.679736\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$		0			0
$668$		0			0
$669$		0			0
$670$		0			0
$671$		0			0
$672$		0			0
$673$	−28.3436	+	23.7831i	−1.09257	+	0.916773i	−0.996903	−	0.0786409i	$-0.974942\pi$
$673$	−28.3436	+	23.7831i	−1.09257	+	0.916773i	−0.0956642	+	0.995414i	$0.530497\pi$
$674$		0			0
$675$		0			0
$676$	−24.0000			−0.923077
$677$		0			0		−0.642788	−	0.766044i	$-0.722222\pi$
$677$		0			0		0.642788	+	0.766044i	$0.277778\pi$
$678$		0			0
$679$	3.29932	−	18.7113i	0.126616	−	0.718076i
$680$		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
$688$	24.5134	+	20.5692i	0.934565	+	0.784194i
$689$		0			0
$690$		0			0
$691$	−7.51754	−	2.73616i	−0.285981	−	0.104088i	0.195047	−	0.980794i	$-0.437514\pi$
$691$	−7.51754	−	2.73616i	−0.285981	−	0.104088i	−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
$700$	−7.66044	+	6.42788i	−0.289538	+	0.242951i
$701$		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
$709$	9.20335	+	52.1948i	0.345639	+	1.96022i	0.268910	+	0.963165i	$0.413337\pi$
$709$	9.20335	+	52.1948i	0.345639	+	1.96022i	0.0767291	+	0.997052i	$0.475552\pi$
$710$		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
$724$	−13.1557	+	4.78828i	−0.488928	+	0.177955i
$725$		0			0
$726$		0			0
$727$	33.7060	−	28.2827i	1.25008	−	1.04895i	0.253419	−	0.967357i	$-0.418445\pi$
$727$	33.7060	−	28.2827i	1.25008	−	1.04895i	0.996666	−	0.0815889i	$-0.0259995\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$		0			0
$732$		0			0
$733$	8.68241	−	49.2404i	0.320692	−	1.81874i	−0.217671	−	0.976022i	$-0.569846\pi$
$733$	8.68241	−	49.2404i	0.320692	−	1.81874i	0.538363	−	0.842713i	$-0.319043\pi$
$734$		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		0.642788	−	0.766044i	$-0.277778\pi$
$743$		0			0		−0.642788	+	0.766044i	$0.722222\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$	−38.5274	+	14.0228i	−1.40588	+	0.511700i	−0.929919	−	0.367764i	$-0.880123\pi$
$751$	−38.5274	+	14.0228i	−1.40588	+	0.511700i	−0.475965	+	0.879464i	$0.657901\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		0			0
$756$		0			0
$757$	29.0000			1.05402			0.527011	−	0.849858i	$-0.323312\pi$
$757$	29.0000			1.05402			0.527011	+	0.849858i	$0.323312\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$		0			0		−0.342020	−	0.939693i	$-0.611111\pi$
$761$		0			0		0.342020	+	0.939693i	$0.388889\pi$
$762$		0			0
$763$	−0.347296	−	1.96962i	−0.0125730	−	0.0713049i
$764$		0			0
$765$		0			0
$766$		0			0
$767$		0			0
$768$		0			0
$769$	−37.5362	−	31.4966i	−1.35359	−	1.13580i	−0.977905	−	0.209048i	$-0.932963\pi$
$769$	−37.5362	−	31.4966i	−1.35359	−	1.13580i	−0.375684	−	0.926748i	$-0.622592\pi$
$770$		0			0
$771$		0			0
$772$	43.2259	+	15.7329i	1.55573	+	0.566240i
$773$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$773$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$774$		0			0
$775$	−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
$787$	−5.38309	+	30.5290i	−0.191887	+	1.08824i	0.724897	+	0.688858i	$0.241887\pi$
$787$	−5.38309	+	30.5290i	−0.191887	+	1.08824i	−0.916783	+	0.399385i	$0.869224\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$	2.50000	−	4.33013i	0.0887776	−	0.153767i
$794$		0			0
$795$		0			0
$796$	−16.8530	−	14.1413i	−0.597338	−	0.501226i
$797$		0			0		0.642788	−	0.766044i	$-0.277778\pi$
$797$		0			0		−0.642788	+	0.766044i	$0.722222\pi$
$798$		0			0
$799$		0			0
$800$		0			0
$801$		0			0
$802$		0			0
$803$		0			0
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$		0			0			−	1.00000i	$-0.5\pi$
$809$		0			0				1.00000i	$0.5\pi$
$810$		0			0
$811$	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
$817$	−9.72430	−	55.1492i	−0.340210	−	1.92943i
$818$		0			0
$819$		0			0
$820$		0			0
$821$		0			0		0.342020	−	0.939693i	$-0.388889\pi$
$821$		0			0		−0.342020	+	0.939693i	$0.611111\pi$
$822$		0			0
$823$	3.83022	+	3.21394i	0.133513	+	0.112031i	0.707099	−	0.707115i	$-0.250004\pi$
$823$	3.83022	+	3.21394i	0.133513	+	0.112031i	−0.573586	+	0.819146i	$0.694448\pi$
$824$		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
$832$	37.5877	−	13.6808i	1.30312	−	0.474297i
$833$		0			0
$834$		0			0
$835$		0			0
$836$		0			0
$837$		0			0
$838$		0			0
$839$		0			0		−0.642788	−	0.766044i	$-0.722222\pi$
$839$		0			0		0.642788	+	0.766044i	$0.277778\pi$
$840$		0			0
$841$	−5.03580	+	28.5594i	−0.173648	+	0.984808i
$842$		0			0
$843$		0			0
$844$	4.51485	+	25.6050i	0.155408	+	0.881361i
$845$		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
$853$	−32.8892	−	11.9707i	−1.12611	−	0.409869i	−0.289229	−	0.957260i	$-0.593399\pi$
$853$	−32.8892	−	11.9707i	−1.12611	−	0.409869i	−0.836877	+	0.547391i	$0.815621\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$		0			0		0.984808	−	0.173648i	$-0.0555556\pi$
$857$		0			0		−0.984808	+	0.173648i	$0.944444\pi$
$858$		0			0
$859$	12.2160	−	4.44626i	0.416805	−	0.151704i	−0.125101	−	0.992144i	$-0.539925\pi$
$859$	12.2160	−	4.44626i	0.416805	−	0.151704i	0.541905	+	0.840440i	$0.317703\pi$
$860$		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
$868$	−1.38919	+	7.87846i	−0.0471520	+	0.267412i
$869$		0			0
$870$		0			0
$871$	4.34120	+	24.6202i	0.147096	+	0.834223i
$872$		0			0
$873$		0			0
$874$		0			0
$875$		0			0
$876$		0			0
$877$	45.1966	+	37.9245i	1.52618	+	1.28062i	0.819483	+	0.573103i	$0.194261\pi$
$877$	45.1966	+	37.9245i	1.52618	+	1.28062i	0.706698	+	0.707515i	$0.250184\pi$
$878$		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		−0.984808	−	0.173648i	$-0.944444\pi$
$887$		0			0		0.984808	+	0.173648i	$0.0555556\pi$
$888$		0			0
$889$	−15.3209	+	12.8558i	−0.513846	+	0.431168i
$890$		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
$907$	17.8542	+	6.49838i	0.592838	+	0.215775i	0.620977	−	0.783829i	$-0.286736\pi$
$907$	17.8542	+	6.49838i	0.592838	+	0.215775i	−0.0281394	+	0.999604i	$0.508958\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$		0			0		0.984808	−	0.173648i	$-0.0555556\pi$
$911$		0			0		−0.984808	+	0.173648i	$0.944444\pi$
$912$		0			0
$913$		0			0
$914$		0			0
$915$		0			0
$916$	33.7060	−	28.2827i	1.11368	−	0.934485i
$917$		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
$925$	−9.55065	−	54.1644i	−0.314023	−	1.78092i
$926$		0			0
$927$		0			0
$928$		0			0
$929$		0			0		0.342020	−	0.939693i	$-0.388889\pi$
$929$		0			0		−0.342020	+	0.939693i	$0.611111\pi$
$930$		0			0
$931$	32.1739	+	26.9971i	1.05446	+	0.884794i
$932$		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		−0.984808	−	0.173648i	$-0.944444\pi$
$941$		0			0		0.984808	+	0.173648i	$0.0555556\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$		0			0
$946$		0			0
$947$		0			0		−0.642788	−	0.766044i	$-0.722222\pi$
$947$		0			0		0.642788	+	0.766044i	$0.277778\pi$
$948$		0			0
$949$	−6.07769	+	34.4683i	−0.197290	+	1.11889i
$950$		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
$961$	14.0954	+	5.13030i	0.454690	+	0.165494i
$962$		0			0
$963$		0			0
$964$	17.0000	+	29.4449i	0.547533	+	0.948355i
$965$		0			0
$966$		0			0
$967$	−38.5274	+	14.0228i	−1.23896	+	0.450944i	−0.876656	−	0.481117i	$-0.840231\pi$
$967$	−38.5274	+	14.0228i	−1.23896	+	0.450944i	−0.362301	+	0.932061i	$0.618009\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$		0			0			−	1.00000i	$-0.5\pi$
$971$		0			0				1.00000i	$0.5\pi$
$972$		0			0
$973$	−23.0000			−0.737346
$974$		0			0
$975$		0			0
$976$	−0.694593	+	3.93923i	−0.0222334	+	0.126092i
$977$		0			0		−0.342020	−	0.939693i	$-0.611111\pi$
$977$		0			0		0.342020	+	0.939693i	$0.388889\pi$
$978$		0			0
$979$		0			0
$980$		0			0
$981$		0			0
$982$		0			0
$983$		0			0		0.342020	−	0.939693i	$-0.388889\pi$
$983$		0			0		−0.342020	+	0.939693i	$0.611111\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$	−65.7785	−	23.9414i	−2.09269	−	0.761678i
$989$		0			0
$990$		0			0
$991$	30.5000	+	52.8275i	0.968864	+	1.67812i	0.698853	+	0.715265i	$0.253694\pi$
$991$	30.5000	+	52.8275i	0.968864	+	1.67812i	0.270011	+	0.962857i	$0.412973\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$		0			0
$996$		0			0
$997$	−7.66044	+	6.42788i	−0.242609	+	0.203573i	−0.755982	−	0.654593i	$-0.772840\pi$
$997$	−7.66044	+	6.42788i	−0.242609	+	0.203573i	0.513373	+	0.858166i	$0.328396\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	729.2.e.f.325.1		6
3.2	odd	2	CM	729.2.e.f.325.1		6
9.2	odd	6	inner	729.2.e.f.568.1		6
9.4	even	3	inner	729.2.e.f.82.1		6
9.5	odd	6	inner	729.2.e.f.82.1		6
9.7	even	3	inner	729.2.e.f.568.1		6
27.2	odd	18	inner	729.2.e.f.163.1		6
27.4	even	9		27.2.a.a.1.1	✓	1
27.5	odd	18		81.2.c.a.28.1		2
27.7	even	9	inner	729.2.e.f.649.1		6
27.11	odd	18	inner	729.2.e.f.406.1		6
27.13	even	9		81.2.c.a.55.1		2
27.14	odd	18		81.2.c.a.55.1		2
27.16	even	9	inner	729.2.e.f.406.1		6
27.20	odd	18	inner	729.2.e.f.649.1		6
27.22	even	9		81.2.c.a.28.1		2
27.23	odd	18		27.2.a.a.1.1	✓	1
27.25	even	9	inner	729.2.e.f.163.1		6
108.23	even	18		432.2.a.e.1.1		1
108.31	odd	18		432.2.a.e.1.1		1
108.59	even	18		1296.2.i.i.433.1		2
108.67	odd	18		1296.2.i.i.865.1		2
108.95	even	18		1296.2.i.i.865.1		2
108.103	odd	18		1296.2.i.i.433.1		2
135.4	even	18		675.2.a.e.1.1		1
135.23	even	36		675.2.b.f.649.2		2
135.58	odd	36		675.2.b.f.649.2		2
135.77	even	36		675.2.b.f.649.1		2
135.104	odd	18		675.2.a.e.1.1		1
135.112	odd	36		675.2.b.f.649.1		2
189.104	even	18		1323.2.a.i.1.1		1
189.139	odd	18		1323.2.a.i.1.1		1
216.77	odd	18		1728.2.a.n.1.1		1
216.85	even	18		1728.2.a.n.1.1		1
216.131	even	18		1728.2.a.o.1.1		1
216.139	odd	18		1728.2.a.o.1.1		1
297.131	even	18		3267.2.a.f.1.1		1
297.274	odd	18		3267.2.a.f.1.1		1
351.77	odd	18		4563.2.a.e.1.1		1
351.220	even	18		4563.2.a.e.1.1		1
459.50	odd	18		7803.2.a.k.1.1		1
459.220	even	18		7803.2.a.k.1.1		1
513.436	odd	18		9747.2.a.f.1.1		1
513.455	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	27.4	even	9
27.2.a.a.1.1	✓	1	27.23	odd	18
81.2.c.a.28.1		2	27.5	odd	18
81.2.c.a.28.1		2	27.22	even	9
81.2.c.a.55.1		2	27.13	even	9
81.2.c.a.55.1		2	27.14	odd	18
432.2.a.e.1.1		1	108.23	even	18
432.2.a.e.1.1		1	108.31	odd	18
675.2.a.e.1.1		1	135.4	even	18
675.2.a.e.1.1		1	135.104	odd	18
675.2.b.f.649.1		2	135.77	even	36
675.2.b.f.649.1		2	135.112	odd	36
675.2.b.f.649.2		2	135.23	even	36
675.2.b.f.649.2		2	135.58	odd	36
729.2.e.f.82.1		6	9.4	even	3	inner
729.2.e.f.82.1		6	9.5	odd	6	inner
729.2.e.f.163.1		6	27.2	odd	18	inner
729.2.e.f.163.1		6	27.25	even	9	inner
729.2.e.f.325.1		6	1.1	even	1	trivial
729.2.e.f.325.1		6	3.2	odd	2	CM
729.2.e.f.406.1		6	27.11	odd	18	inner
729.2.e.f.406.1		6	27.16	even	9	inner
729.2.e.f.568.1		6	9.2	odd	6	inner
729.2.e.f.568.1		6	9.7	even	3	inner
729.2.e.f.649.1		6	27.7	even	9	inner
729.2.e.f.649.1		6	27.20	odd	18	inner
1296.2.i.i.433.1		2	108.59	even	18
1296.2.i.i.433.1		2	108.103	odd	18
1296.2.i.i.865.1		2	108.67	odd	18
1296.2.i.i.865.1		2	108.95	even	18
1323.2.a.i.1.1		1	189.104	even	18
1323.2.a.i.1.1		1	189.139	odd	18
1728.2.a.n.1.1		1	216.77	odd	18
1728.2.a.n.1.1		1	216.85	even	18
1728.2.a.o.1.1		1	216.131	even	18
1728.2.a.o.1.1		1	216.139	odd	18
3267.2.a.f.1.1		1	297.131	even	18
3267.2.a.f.1.1		1	297.274	odd	18
4563.2.a.e.1.1		1	351.77	odd	18
4563.2.a.e.1.1		1	351.220	even	18
7803.2.a.k.1.1		1	459.50	odd	18
7803.2.a.k.1.1		1	459.220	even	18
9747.2.a.f.1.1		1	513.436	odd	18
9747.2.a.f.1.1		1	513.455	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)

\(f(q)\)	\(=\)	\(q+(-0.347296 + 1.96962i) q^{4} +(-0.173648 - 0.984808i) q^{7} +O(q^{10})\)	\(q+(-0.347296 + 1.96962i) q^{4} +(-0.173648 - 0.984808i) q^{7} +(3.83022 + 3.21394i) q^{13} +(-3.75877 - 1.36808i) q^{16} +(3.50000 + 6.06218i) q^{19} +(-3.83022 + 3.21394i) q^{25} +2.00000 q^{28} +(-0.694593 + 3.93923i) q^{31} +(-5.50000 + 9.52628i) q^{37} +(-7.51754 - 2.73616i) q^{43} +(5.63816 - 2.05212i) q^{49} +(-7.66044 + 6.42788i) q^{52} +(-0.173648 - 0.984808i) q^{61} +(4.00000 - 6.92820i) q^{64} +(3.83022 + 3.21394i) q^{67} +(3.50000 + 6.06218i) q^{73} +(-13.1557 + 4.78828i) q^{76} +(13.0228 - 10.9274i) q^{79} +(2.50000 - 4.33013i) q^{91} +(17.8542 + 6.49838i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q+O(q^{10}) \) 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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