LMFDB - Embedded newform 243.2.c.b.82.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [243,2,Mod(82,243)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("243.82"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(243, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([2])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 243 = 3^{5} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	243.c (of order $3$, degree $2$, not minimal)

Newform invariants

comment:select newform

sage:traces = [2,0,0,2,0,0,4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$1.94036476912$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{6})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x + 1 $ x^2 - x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{3}]$

Embedding invariants

Embedding label			82.1
Root			$0.500000 - 0.866025i$ of defining polynomial
Character	$\chi$	$=$	243.82
Dual form			243.2.c.b.163.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+(1.00000 - 1.73205i) q^{4} +(2.00000 + 3.46410i) q^{7} +(3.50000 - 6.06218i) q^{13} +(-2.00000 - 3.46410i) q^{16} -1.00000 q^{19} +(2.50000 + 4.33013i) q^{25} +8.00000 q^{28} +(-5.50000 + 9.52628i) q^{31} -10.0000 q^{37} +(-2.50000 - 4.33013i) q^{43} +(-4.50000 + 7.79423i) q^{49} +(-7.00000 - 12.1244i) q^{52} +(0.500000 + 0.866025i) q^{61} -8.00000 q^{64} +(-2.50000 + 4.33013i) q^{67} -7.00000 q^{73} +(-1.00000 + 1.73205i) q^{76} +(6.50000 + 11.2583i) q^{79} +28.0000 q^{91} +(-2.50000 - 4.33013i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{4} + 4 q^{7} + 7 q^{13} - 4 q^{16} - 2 q^{19} + 5 q^{25} + 16 q^{28} - 11 q^{31} - 20 q^{37} - 5 q^{43} - 9 q^{49} - 14 q^{52} + q^{61} - 16 q^{64} - 5 q^{67} - 14 q^{73} - 2 q^{76} + 13 q^{79}+ \cdots - 5 q^{97}+O(q^{100}) $ 2 * q + 2 * q^4 + 4 * q^7 + 7 * q^13 - 4 * q^16 - 2 * q^19 + 5 * q^25 + 16 * q^28 - 11 * q^31 - 20 * q^37 - 5 * q^43 - 9 * q^49 - 14 * q^52 + q^61 - 16 * q^64 - 5 * q^67 - 14 * q^73 - 2 * q^76 + 13 * q^79 + 56 * q^91 - 5 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/243\mathbb{Z}\right)^\times$.

$n$	$2$
$\chi(n)$	$e\left(\frac{1}{3}\right)$

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		0.866025	−	0.500000i	$-0.166667\pi$
$2$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$3$		0			0
$3$		0			0
$4$	1.00000	−	1.73205i	0.500000	−	0.866025i
$5$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$5$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$6$		0			0
$7$	2.00000	+	3.46410i	0.755929	+	1.30931i	0.944911	+	0.327327i	$0.106148\pi$
$7$	2.00000	+	3.46410i	0.755929	+	1.30931i	−0.188982	+	0.981981i	$0.560519\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$11$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$12$		0			0
$13$	3.50000	−	6.06218i	0.970725	−	1.68135i	0.277350	−	0.960769i	$-0.410544\pi$
$13$	3.50000	−	6.06218i	0.970725	−	1.68135i	0.693375	−	0.720577i	$-0.256123\pi$
$14$		0			0
$15$		0			0
$16$	−2.00000	−	3.46410i	−0.500000	−	0.866025i
$17$		0			0			−	1.00000i	$-0.5\pi$
$17$		0			0				1.00000i	$0.5\pi$
$18$		0			0
$19$	−1.00000			−0.229416			−0.114708	−	0.993399i	$-0.536593\pi$
$19$	−1.00000			−0.229416			−0.114708	+	0.993399i	$0.536593\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$23$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$24$		0			0
$25$	2.50000	+	4.33013i	0.500000	+	0.866025i
$26$		0			0
$27$		0			0
$28$	8.00000			1.51186
$29$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$29$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$30$		0			0
$31$	−5.50000	+	9.52628i	−0.987829	+	1.71097i	−0.359211	+	0.933257i	$0.616954\pi$
$31$	−5.50000	+	9.52628i	−0.987829	+	1.71097i	−0.628619	+	0.777714i	$0.716379\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		0			0
$36$		0			0
$37$	−10.0000			−1.64399			−0.821995	−	0.569495i	$-0.807139\pi$
$37$	−10.0000			−1.64399			−0.821995	+	0.569495i	$0.807139\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$41$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$42$		0			0
$43$	−2.50000	−	4.33013i	−0.381246	−	0.660338i	0.609994	−	0.792406i	$-0.291172\pi$
$43$	−2.50000	−	4.33013i	−0.381246	−	0.660338i	−0.991241	+	0.132068i	$0.957838\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$47$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$48$		0			0
$49$	−4.50000	+	7.79423i	−0.642857	+	1.11346i
$50$		0			0
$51$		0			0
$52$	−7.00000	−	12.1244i	−0.970725	−	1.68135i
$53$		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		−0.866025	−	0.500000i	$-0.833333\pi$
$59$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$60$		0			0
$61$	0.500000	+	0.866025i	0.0640184	+	0.110883i	0.896258	−	0.443533i	$-0.146275\pi$
$61$	0.500000	+	0.866025i	0.0640184	+	0.110883i	−0.832240	+	0.554416i	$0.812942\pi$
$62$		0			0
$63$		0			0
$64$	−8.00000			−1.00000
$65$		0			0
$66$		0			0
$67$	−2.50000	+	4.33013i	−0.305424	+	0.529009i	−0.977356	−	0.211604i	$-0.932131\pi$
$67$	−2.50000	+	4.33013i	−0.305424	+	0.529009i	0.671932	+	0.740613i	$0.265465\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		0			0			−	1.00000i	$-0.5\pi$
$71$		0			0				1.00000i	$0.5\pi$
$72$		0			0
$73$	−7.00000			−0.819288			−0.409644	−	0.912245i	$-0.634347\pi$
$73$	−7.00000			−0.819288			−0.409644	+	0.912245i	$0.634347\pi$
$74$		0			0
$75$		0			0
$76$	−1.00000	+	1.73205i	−0.114708	+	0.198680i
$77$		0			0
$78$		0			0
$79$	6.50000	+	11.2583i	0.731307	+	1.26666i	0.956325	+	0.292306i	$0.0944227\pi$
$79$	6.50000	+	11.2583i	0.731307	+	1.26666i	−0.225018	+	0.974355i	$0.572244\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$83$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$		0			0
$88$		0			0
$89$		0			0			−	1.00000i	$-0.5\pi$
$89$		0			0				1.00000i	$0.5\pi$
$90$		0			0
$91$	28.0000			2.93520
$92$		0			0
$93$		0			0
$94$		0			0
$95$		0			0
$96$		0			0
$97$	−2.50000	−	4.33013i	−0.253837	−	0.439658i	0.710742	−	0.703452i	$-0.248359\pi$
$97$	−2.50000	−	4.33013i	−0.253837	−	0.439658i	−0.964579	+	0.263795i	$0.915026\pi$
$98$		0			0
$99$		0			0
$100$	10.0000			1.00000
$101$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$101$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$102$		0			0
$103$	6.50000	−	11.2583i	0.640464	−	1.10932i	−0.344865	−	0.938652i	$-0.612075\pi$
$103$	6.50000	−	11.2583i	0.640464	−	1.10932i	0.985329	−	0.170664i	$-0.0545913\pi$
$104$		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
$112$	8.00000	−	13.8564i	0.755929	−	1.30931i
$113$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$113$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$		0			0
$118$		0			0
$119$		0			0
$120$		0			0
$121$	5.50000	−	9.52628i	0.500000	−	0.866025i
$122$		0			0
$123$		0			0
$124$	11.0000	+	19.0526i	0.987829	+	1.71097i
$125$		0			0
$126$		0			0
$127$	−1.00000			−0.0887357			−0.0443678	−	0.999015i	$-0.514127\pi$
$127$	−1.00000			−0.0887357			−0.0443678	+	0.999015i	$0.514127\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$131$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$132$		0			0
$133$	−2.00000	−	3.46410i	−0.173422	−	0.300376i
$134$		0			0
$135$		0			0
$136$		0			0
$137$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$137$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$138$		0			0
$139$	8.00000	−	13.8564i	0.678551	−	1.17529i	−0.296866	−	0.954919i	$-0.595942\pi$
$139$	8.00000	−	13.8564i	0.678551	−	1.17529i	0.975417	−	0.220366i	$-0.0707252\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		0			0
$144$		0			0
$145$		0			0
$146$		0			0
$147$		0			0
$148$	−10.0000	+	17.3205i	−0.821995	+	1.42374i
$149$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$149$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$150$		0			0
$151$	9.50000	+	16.4545i	0.773099	+	1.33905i	0.935857	+	0.352381i	$0.114628\pi$
$151$	9.50000	+	16.4545i	0.773099	+	1.33905i	−0.162758	+	0.986666i	$0.552039\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		0			0
$156$		0			0
$157$	12.5000	−	21.6506i	0.997609	−	1.72791i	0.438948	−	0.898513i	$-0.355351\pi$
$157$	12.5000	−	21.6506i	0.997609	−	1.72791i	0.558661	−	0.829396i	$-0.311315\pi$
$158$		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		−0.866025	−	0.500000i	$-0.833333\pi$
$167$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$168$		0			0
$169$	−18.0000	−	31.1769i	−1.38462	−	2.39822i
$170$		0			0
$171$		0			0
$172$	−10.0000			−0.762493
$173$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$173$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$174$		0			0
$175$	−10.0000	+	17.3205i	−0.755929	+	1.30931i
$176$		0			0
$177$		0			0
$178$		0			0
$179$		0			0			−	1.00000i	$-0.5\pi$
$179$		0			0				1.00000i	$0.5\pi$
$180$		0			0
$181$	26.0000			1.93256			0.966282	−	0.257485i	$-0.0828937\pi$
$181$	26.0000			1.93256			0.966282	+	0.257485i	$0.0828937\pi$
$182$		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		0.866025	−	0.500000i	$-0.166667\pi$
$191$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$192$		0			0
$193$	−11.5000	+	19.9186i	−0.827788	+	1.43377i	0.0719816	+	0.997406i	$0.477068\pi$
$193$	−11.5000	+	19.9186i	−0.827788	+	1.43377i	−0.899770	+	0.436365i	$0.856266\pi$
$194$		0			0
$195$		0			0
$196$	9.00000	+	15.5885i	0.642857	+	1.11346i
$197$		0			0			−	1.00000i	$-0.5\pi$
$197$		0			0				1.00000i	$0.5\pi$
$198$		0			0
$199$	17.0000			1.20510			0.602549	−	0.798082i	$-0.294152\pi$
$199$	17.0000			1.20510			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
$208$	−28.0000			−1.94145
$209$		0			0
$210$		0			0
$211$	−14.5000	+	25.1147i	−0.998221	+	1.72897i	−0.447478	+	0.894295i	$0.647678\pi$
$211$	−14.5000	+	25.1147i	−0.998221	+	1.72897i	−0.550743	+	0.834675i	$0.685655\pi$
$212$		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
$223$	−11.5000	−	19.9186i	−0.770097	−	1.33385i	−0.937509	−	0.347960i	$-0.886874\pi$
$223$	−11.5000	−	19.9186i	−0.770097	−	1.33385i	0.167412	−	0.985887i	$-0.446459\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$227$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$228$		0			0
$229$	3.50000	−	6.06218i	0.231287	−	0.400600i	−0.726900	−	0.686743i	$-0.759040\pi$
$229$	3.50000	−	6.06218i	0.231287	−	0.400600i	0.958187	+	0.286143i	$0.0923732\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		0			0			−	1.00000i	$-0.5\pi$
$233$		0			0				1.00000i	$0.5\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$		0			0
$238$		0			0
$239$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$239$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$240$		0			0
$241$	−7.00000	−	12.1244i	−0.450910	−	0.780998i	0.547533	−	0.836784i	$-0.315567\pi$
$241$	−7.00000	−	12.1244i	−0.450910	−	0.780998i	−0.998443	+	0.0557856i	$0.982234\pi$
$242$		0			0
$243$		0			0
$244$	2.00000			0.128037
$245$		0			0
$246$		0			0
$247$	−3.50000	+	6.06218i	−0.222700	+	0.385727i
$248$		0			0
$249$		0			0
$250$		0			0
$251$		0			0			−	1.00000i	$-0.5\pi$
$251$		0			0				1.00000i	$0.5\pi$
$252$		0			0
$253$		0			0
$254$		0			0
$255$		0			0
$256$	−8.00000	+	13.8564i	−0.500000	+	0.866025i
$257$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$257$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$258$		0			0
$259$	−20.0000	−	34.6410i	−1.24274	−	2.15249i
$260$		0			0
$261$		0			0
$262$		0			0
$263$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$263$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$		0			0
$268$	5.00000	+	8.66025i	0.305424	+	0.529009i
$269$		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
$277$	15.5000	+	26.8468i	0.931305	+	1.61307i	0.781094	+	0.624413i	$0.214662\pi$
$277$	15.5000	+	26.8468i	0.931305	+	1.61307i	0.150210	+	0.988654i	$0.452005\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$281$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$282$		0			0
$283$	3.50000	−	6.06218i	0.208053	−	0.360359i	−0.743048	−	0.669238i	$-0.766621\pi$
$283$	3.50000	−	6.06218i	0.208053	−	0.360359i	0.951101	+	0.308879i	$0.0999539\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$	−17.0000			−1.00000
$290$		0			0
$291$		0			0
$292$	−7.00000	+	12.1244i	−0.409644	+	0.709524i
$293$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$293$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$		0			0
$298$		0			0
$299$		0			0
$300$		0			0
$301$	10.0000	−	17.3205i	0.576390	−	0.998337i
$302$		0			0
$303$		0			0
$304$	2.00000	+	3.46410i	0.114708	+	0.198680i
$305$		0			0
$306$		0			0
$307$	−16.0000			−0.913168			−0.456584	−	0.889680i	$-0.650927\pi$
$307$	−16.0000			−0.913168			−0.456584	+	0.889680i	$0.650927\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$311$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$312$		0			0
$313$	11.0000	+	19.0526i	0.621757	+	1.07691i	0.989158	+	0.146852i	$0.0469141\pi$
$313$	11.0000	+	19.0526i	0.621757	+	1.07691i	−0.367402	+	0.930062i	$0.619753\pi$
$314$		0			0
$315$		0			0
$316$	26.0000			1.46261
$317$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$317$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$318$		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
$331$	15.5000	+	26.8468i	0.851957	+	1.47563i	0.879440	+	0.476011i	$0.157918\pi$
$331$	15.5000	+	26.8468i	0.851957	+	1.47563i	−0.0274825	+	0.999622i	$0.508749\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$		0			0
$336$		0			0
$337$	17.0000	−	29.4449i	0.926049	−	1.60396i	0.136184	−	0.990684i	$-0.456516\pi$
$337$	17.0000	−	29.4449i	0.926049	−	1.60396i	0.789865	−	0.613280i	$-0.210150\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$		0			0
$342$		0			0
$343$	−8.00000			−0.431959
$344$		0			0
$345$		0			0
$346$		0			0
$347$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$347$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$348$		0			0
$349$	−7.00000	−	12.1244i	−0.374701	−	0.649002i	0.615581	−	0.788074i	$-0.288921\pi$
$349$	−7.00000	−	12.1244i	−0.374701	−	0.649002i	−0.990282	+	0.139072i	$0.955588\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$353$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		0			0
$358$		0			0
$359$		0			0			−	1.00000i	$-0.5\pi$
$359$		0			0				1.00000i	$0.5\pi$
$360$		0			0
$361$	−18.0000			−0.947368
$362$		0			0
$363$		0			0
$364$	28.0000	−	48.4974i	1.46760	−	2.54196i
$365$		0			0
$366$		0			0
$367$	−17.5000	−	30.3109i	−0.913493	−	1.58222i	−0.809093	−	0.587680i	$-0.800041\pi$
$367$	−17.5000	−	30.3109i	−0.913493	−	1.58222i	−0.104399	−	0.994535i	$-0.533292\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		0			0
$372$		0			0
$373$	12.5000	−	21.6506i	0.647225	−	1.12103i	−0.336557	−	0.941663i	$-0.609263\pi$
$373$	12.5000	−	21.6506i	0.647225	−	1.12103i	0.983783	−	0.179364i	$-0.0574041\pi$
$374$		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		−0.866025	−	0.500000i	$-0.833333\pi$
$383$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$		0			0
$388$	−10.0000			−0.507673
$389$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$389$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$390$		0			0
$391$		0			0
$392$		0			0
$393$		0			0
$394$		0			0
$395$		0			0
$396$		0			0
$397$	−1.00000			−0.0501886			−0.0250943	−	0.999685i	$-0.507989\pi$
$397$	−1.00000			−0.0501886			−0.0250943	+	0.999685i	$0.507989\pi$
$398$		0			0
$399$		0			0
$400$	10.0000	−	17.3205i	0.500000	−	0.866025i
$401$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$401$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$402$		0			0
$403$	38.5000	+	66.6840i	1.91782	+	3.32176i
$404$		0			0
$405$		0			0
$406$		0			0
$407$		0			0
$408$		0			0
$409$	−19.0000	+	32.9090i	−0.939490	+	1.62724i	−0.173064	+	0.984911i	$0.555367\pi$
$409$	−19.0000	+	32.9090i	−0.939490	+	1.62724i	−0.766426	+	0.642333i	$0.777967\pi$
$410$		0			0
$411$		0			0
$412$	−13.0000	−	22.5167i	−0.640464	−	1.10932i
$413$		0			0
$414$		0			0
$415$		0			0
$416$		0			0
$417$		0			0
$418$		0			0
$419$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$419$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$420$		0			0
$421$	11.0000	+	19.0526i	0.536107	+	0.928565i	0.999109	+	0.0422075i	$0.0134391\pi$
$421$	11.0000	+	19.0526i	0.536107	+	0.928565i	−0.463002	+	0.886357i	$0.653228\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$		0			0
$426$		0			0
$427$	−2.00000	+	3.46410i	−0.0967868	+	0.167640i
$428$		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
$436$	−19.0000	+	32.9090i	−0.909935	+	1.57605i
$437$		0			0
$438$		0			0
$439$	14.0000	+	24.2487i	0.668184	+	1.15733i	0.978412	+	0.206666i	$0.0662612\pi$
$439$	14.0000	+	24.2487i	0.668184	+	1.15733i	−0.310228	+	0.950662i	$0.600405\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$443$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$		0			0
$448$	−16.0000	−	27.7128i	−0.755929	−	1.30931i
$449$		0			0			−	1.00000i	$-0.5\pi$
$449$		0			0				1.00000i	$0.5\pi$
$450$		0			0
$451$		0			0
$452$		0			0
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$	−20.5000	−	35.5070i	−0.958950	−	1.66095i	−0.725059	−	0.688686i	$-0.758188\pi$
$457$	−20.5000	−	35.5070i	−0.958950	−	1.66095i	−0.233890	−	0.972263i	$-0.575146\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$461$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$462$		0			0
$463$	21.5000	−	37.2391i	0.999190	−	1.73065i	0.464739	−	0.885448i	$-0.346148\pi$
$463$	21.5000	−	37.2391i	0.999190	−	1.73065i	0.534450	−	0.845200i	$-0.320519\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$		0			0			−	1.00000i	$-0.5\pi$
$467$		0			0				1.00000i	$0.5\pi$
$468$		0			0
$469$	−20.0000			−0.923514
$470$		0			0
$471$		0			0
$472$		0			0
$473$		0			0
$474$		0			0
$475$	−2.50000	−	4.33013i	−0.114708	−	0.198680i
$476$		0			0
$477$		0			0
$478$		0			0
$479$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$479$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$480$		0			0
$481$	−35.0000	+	60.6218i	−1.59586	+	2.76412i
$482$		0			0
$483$		0			0
$484$	−11.0000	−	19.0526i	−0.500000	−	0.866025i
$485$		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		−0.866025	−	0.500000i	$-0.833333\pi$
$491$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$		0			0
$496$	44.0000			1.97566
$497$		0			0
$498$		0			0
$499$	−16.0000	+	27.7128i	−0.716258	+	1.24060i	0.246214	+	0.969216i	$0.420813\pi$
$499$	−16.0000	+	27.7128i	−0.716258	+	1.24060i	−0.962472	+	0.271380i	$0.912520\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		0			0			−	1.00000i	$-0.5\pi$
$503$		0			0				1.00000i	$0.5\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$		0			0
$508$	−1.00000	+	1.73205i	−0.0443678	+	0.0768473i
$509$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$509$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$510$		0			0
$511$	−14.0000	−	24.2487i	−0.619324	−	1.07270i
$512$		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			−	1.00000i	$-0.5\pi$
$521$		0			0				1.00000i	$0.5\pi$
$522$		0			0
$523$	−43.0000			−1.88026			−0.940129	−	0.340818i	$-0.889296\pi$
$523$	−43.0000			−1.88026			−0.940129	+	0.340818i	$0.889296\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$		0			0
$528$		0			0
$529$	11.5000	+	19.9186i	0.500000	+	0.866025i
$530$		0			0
$531$		0			0
$532$	−8.00000			−0.346844
$533$		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
$547$	0.500000	+	0.866025i	0.0213785	+	0.0370286i	0.876517	−	0.481371i	$-0.159861\pi$
$547$	0.500000	+	0.866025i	0.0213785	+	0.0370286i	−0.855138	+	0.518400i	$0.826528\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$		0			0
$552$		0			0
$553$	−26.0000	+	45.0333i	−1.10563	+	1.91501i
$554$		0			0
$555$		0			0
$556$	−16.0000	−	27.7128i	−0.678551	−	1.17529i
$557$		0			0			−	1.00000i	$-0.5\pi$
$557$		0			0				1.00000i	$0.5\pi$
$558$		0			0
$559$	−35.0000			−1.48034
$560$		0			0
$561$		0			0
$562$		0			0
$563$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$563$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$		0			0
$568$		0			0
$569$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$569$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$570$		0			0
$571$	8.00000	−	13.8564i	0.334790	−	0.579873i	−0.648655	−	0.761083i	$-0.724668\pi$
$571$	8.00000	−	13.8564i	0.334790	−	0.579873i	0.983444	+	0.181210i	$0.0580014\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$		0			0
$577$	11.0000			0.457936			0.228968	−	0.973434i	$-0.426465\pi$
$577$	11.0000			0.457936			0.228968	+	0.973434i	$0.426465\pi$
$578$		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		0.866025	−	0.500000i	$-0.166667\pi$
$587$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$588$		0			0
$589$	5.50000	−	9.52628i	0.226624	−	0.392524i
$590$		0			0
$591$		0			0
$592$	20.0000	+	34.6410i	0.821995	+	1.42374i
$593$		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		−0.866025	−	0.500000i	$-0.833333\pi$
$599$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$600$		0			0
$601$	−11.5000	−	19.9186i	−0.469095	−	0.812496i	0.530281	−	0.847822i	$-0.322086\pi$
$601$	−11.5000	−	19.9186i	−0.469095	−	0.812496i	−0.999376	+	0.0353259i	$0.988753\pi$
$602$		0			0
$603$		0			0
$604$	38.0000			1.54620
$605$		0			0
$606$		0			0
$607$	−10.0000	+	17.3205i	−0.405887	+	0.703018i	−0.994424	−	0.105453i	$-0.966371\pi$
$607$	−10.0000	+	17.3205i	−0.405887	+	0.703018i	0.588537	+	0.808470i	$0.299704\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		0			0
$612$		0			0
$613$	47.0000			1.89831			0.949156	−	0.314806i	$-0.101939\pi$
$613$	47.0000			1.89831			0.949156	+	0.314806i	$0.101939\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$617$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$618$		0			0
$619$	−8.50000	−	14.7224i	−0.341644	−	0.591744i	0.643094	−	0.765787i	$-0.277650\pi$
$619$	−8.50000	−	14.7224i	−0.341644	−	0.591744i	−0.984738	+	0.174042i	$0.944317\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$		0			0
$625$	−12.5000	+	21.6506i	−0.500000	+	0.866025i
$626$		0			0
$627$		0			0
$628$	−25.0000	−	43.3013i	−0.997609	−	1.72791i
$629$		0			0
$630$		0			0
$631$	44.0000			1.75161			0.875806	−	0.482663i	$-0.160330\pi$
$631$	44.0000			1.75161			0.875806	+	0.482663i	$0.160330\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$		0			0
$636$		0			0
$637$	31.5000	+	54.5596i	1.24808	+	2.16173i
$638$		0			0
$639$		0			0
$640$		0			0
$641$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$641$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$642$		0			0
$643$	20.0000	−	34.6410i	0.788723	−	1.36611i	−0.138027	−	0.990429i	$-0.544076\pi$
$643$	20.0000	−	34.6410i	0.788723	−	1.36611i	0.926750	−	0.375680i	$-0.122591\pi$
$644$		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
$652$	8.00000	−	13.8564i	0.313304	−	0.542659i
$653$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$653$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$		0			0
$658$		0			0
$659$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$659$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$660$		0			0
$661$	24.5000	−	42.4352i	0.952940	−	1.65054i	0.213925	−	0.976850i	$-0.431375\pi$
$661$	24.5000	−	42.4352i	0.952940	−	1.65054i	0.739014	−	0.673690i	$-0.235292\pi$
$662$		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
$673$	6.50000	+	11.2583i	0.250557	+	0.433977i	0.963679	−	0.267063i	$-0.0860531\pi$
$673$	6.50000	+	11.2583i	0.250557	+	0.433977i	−0.713123	+	0.701039i	$0.752720\pi$
$674$		0			0
$675$		0			0
$676$	−72.0000			−2.76923
$677$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$677$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$678$		0			0
$679$	10.0000	−	17.3205i	0.383765	−	0.664700i
$680$		0			0
$681$		0			0
$682$		0			0
$683$		0			0			−	1.00000i	$-0.5\pi$
$683$		0			0				1.00000i	$0.5\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$		0			0
$688$	−10.0000	+	17.3205i	−0.381246	+	0.660338i
$689$		0			0
$690$		0			0
$691$	24.5000	+	42.4352i	0.932024	+	1.61431i	0.779857	+	0.625958i	$0.215292\pi$
$691$	24.5000	+	42.4352i	0.932024	+	1.61431i	0.152167	+	0.988355i	$0.451375\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$		0			0
$696$		0			0
$697$		0			0
$698$		0			0
$699$		0			0
$700$	20.0000	+	34.6410i	0.755929	+	1.30931i
$701$		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
$709$	15.5000	+	26.8468i	0.582115	+	1.00825i	0.995228	+	0.0975728i	$0.0311079\pi$
$709$	15.5000	+	26.8468i	0.582115	+	1.00825i	−0.413114	+	0.910679i	$0.635559\pi$
$710$		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			−	1.00000i	$-0.5\pi$
$719$		0			0				1.00000i	$0.5\pi$
$720$		0			0
$721$	52.0000			1.93658
$722$		0			0
$723$		0			0
$724$	26.0000	−	45.0333i	0.966282	−	1.67365i
$725$		0			0
$726$		0			0
$727$	−22.0000	−	38.1051i	−0.815935	−	1.41324i	−0.908655	−	0.417548i	$-0.862889\pi$
$727$	−22.0000	−	38.1051i	−0.815935	−	1.41324i	0.0927199	−	0.995692i	$-0.470444\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$		0			0
$732$		0			0
$733$	21.5000	−	37.2391i	0.794121	−	1.37546i	−0.129275	−	0.991609i	$-0.541265\pi$
$733$	21.5000	−	37.2391i	0.794121	−	1.37546i	0.923396	−	0.383849i	$-0.125402\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		0			0
$738$		0			0
$739$	−37.0000			−1.36107			−0.680534	−	0.732717i	$-0.738252\pi$
$739$	−37.0000			−1.36107			−0.680534	+	0.732717i	$0.738252\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$743$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$	26.0000	−	45.0333i	0.948753	−	1.64329i	0.200698	−	0.979653i	$-0.435679\pi$
$751$	26.0000	−	45.0333i	0.948753	−	1.64329i	0.748056	−	0.663636i	$-0.230988\pi$
$752$		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		−0.866025	−	0.500000i	$-0.833333\pi$
$761$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$762$		0			0
$763$	−38.0000	−	65.8179i	−1.37569	−	2.38277i
$764$		0			0
$765$		0			0
$766$		0			0
$767$		0			0
$768$		0			0
$769$	−23.5000	+	40.7032i	−0.847432	+	1.46779i	0.0360609	+	0.999350i	$0.488519\pi$
$769$	−23.5000	+	40.7032i	−0.847432	+	1.46779i	−0.883493	+	0.468445i	$0.844814\pi$
$770$		0			0
$771$		0			0
$772$	23.0000	+	39.8372i	0.827788	+	1.43377i
$773$		0			0			−	1.00000i	$-0.5\pi$
$773$		0			0				1.00000i	$0.5\pi$
$774$		0			0
$775$	−55.0000			−1.97566
$776$		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
$787$	15.5000	−	26.8468i	0.552515	−	0.956985i	−0.445577	−	0.895244i	$-0.647001\pi$
$787$	15.5000	−	26.8468i	0.552515	−	0.956985i	0.998092	−	0.0617409i	$-0.0196653\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$	7.00000			0.248577
$794$		0			0
$795$		0			0
$796$	17.0000	−	29.4449i	0.602549	−	1.04365i
$797$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$797$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$798$		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
$817$	2.50000	+	4.33013i	0.0874639	+	0.151492i
$818$		0			0
$819$		0			0
$820$		0			0
$821$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$821$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$822$		0			0
$823$	−23.5000	+	40.7032i	−0.819159	+	1.41882i	0.0871445	+	0.996196i	$0.472226\pi$
$823$	−23.5000	+	40.7032i	−0.819159	+	1.41882i	−0.906303	+	0.422628i	$0.861108\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		0			0			−	1.00000i	$-0.5\pi$
$827$		0			0				1.00000i	$0.5\pi$
$828$		0			0
$829$	53.0000			1.84077			0.920383	−	0.391018i	$-0.127877\pi$
$829$	53.0000			1.84077			0.920383	+	0.391018i	$0.127877\pi$
$830$		0			0
$831$		0			0
$832$	−28.0000	+	48.4974i	−0.970725	+	1.68135i
$833$		0			0
$834$		0			0
$835$		0			0
$836$		0			0
$837$		0			0
$838$		0			0
$839$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$839$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$840$		0			0
$841$	14.5000	−	25.1147i	0.500000	−	0.866025i
$842$		0			0
$843$		0			0
$844$	29.0000	+	50.2295i	0.998221	+	1.72897i
$845$		0			0
$846$		0			0
$847$	44.0000			1.51186
$848$		0			0
$849$		0			0
$850$		0			0
$851$		0			0
$852$		0			0
$853$	−17.5000	−	30.3109i	−0.599189	−	1.03783i	−0.992941	−	0.118609i	$-0.962157\pi$
$853$	−17.5000	−	30.3109i	−0.599189	−	1.03783i	0.393753	−	0.919216i	$-0.371177\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$857$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$858$		0			0
$859$	−28.0000	+	48.4974i	−0.955348	+	1.65471i	−0.221777	+	0.975097i	$0.571186\pi$
$859$	−28.0000	+	48.4974i	−0.955348	+	1.65471i	−0.733571	+	0.679613i	$0.762148\pi$
$860$		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
$868$	−44.0000	+	76.2102i	−1.49346	+	2.58674i
$869$		0			0
$870$		0			0
$871$	17.5000	+	30.3109i	0.592965	+	1.02705i
$872$		0			0
$873$		0			0
$874$		0			0
$875$		0			0
$876$		0			0
$877$	17.0000	−	29.4449i	0.574049	−	0.994282i	−0.422095	−	0.906552i	$-0.638705\pi$
$877$	17.0000	−	29.4449i	0.574049	−	0.994282i	0.996144	−	0.0877308i	$-0.0279615\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$		0			0			−	1.00000i	$-0.5\pi$
$881$		0			0				1.00000i	$0.5\pi$
$882$		0			0
$883$	−55.0000			−1.85090			−0.925449	−	0.378873i	$-0.876312\pi$
$883$	−55.0000			−1.85090			−0.925449	+	0.378873i	$0.876312\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$887$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$888$		0			0
$889$	−2.00000	−	3.46410i	−0.0670778	−	0.116182i
$890$		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
$907$	−29.5000	−	51.0955i	−0.979531	−	1.69660i	−0.664089	−	0.747653i	$-0.731180\pi$
$907$	−29.5000	−	51.0955i	−0.979531	−	1.69660i	−0.315442	−	0.948945i	$-0.602153\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$911$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$912$		0			0
$913$		0			0
$914$		0			0
$915$		0			0
$916$	−7.00000	−	12.1244i	−0.231287	−	0.400600i
$917$		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
$925$	−25.0000	−	43.3013i	−0.821995	−	1.42374i
$926$		0			0
$927$		0			0
$928$		0			0
$929$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$929$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$930$		0			0
$931$	4.50000	−	7.79423i	0.147482	−	0.255446i
$932$		0			0
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$	26.0000			0.849383			0.424691	−	0.905338i	$-0.360383\pi$
$937$	26.0000			0.849383			0.424691	+	0.905338i	$0.360383\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$941$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$		0			0
$946$		0			0
$947$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$947$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$948$		0			0
$949$	−24.5000	+	42.4352i	−0.795304	+	1.37751i
$950$		0			0
$951$		0			0
$952$		0			0
$953$		0			0			−	1.00000i	$-0.5\pi$
$953$		0			0				1.00000i	$0.5\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	−45.0000	−	77.9423i	−1.45161	−	2.51427i
$962$		0			0
$963$		0			0
$964$	−28.0000			−0.901819
$965$		0			0
$966$		0			0
$967$	−20.5000	+	35.5070i	−0.659236	+	1.14183i	0.321578	+	0.946883i	$0.395787\pi$
$967$	−20.5000	+	35.5070i	−0.659236	+	1.14183i	−0.980814	+	0.194946i	$0.937547\pi$
$968$		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
$976$	2.00000	−	3.46410i	0.0640184	−	0.110883i
$977$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$977$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$978$		0			0
$979$		0			0
$980$		0			0
$981$		0			0
$982$		0			0
$983$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$983$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$	7.00000	+	12.1244i	0.222700	+	0.385727i
$989$		0			0
$990$		0			0
$991$	−61.0000			−1.93773			−0.968864	−	0.247592i	$-0.920361\pi$
$991$	−61.0000			−1.93773			−0.968864	+	0.247592i	$0.920361\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$		0			0
$996$		0			0
$997$	5.00000	+	8.66025i	0.158352	+	0.274273i	0.934274	−	0.356555i	$-0.116049\pi$
$997$	5.00000	+	8.66025i	0.158352	+	0.274273i	−0.775923	+	0.630828i	$0.782715\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	243.2.c.b.82.1		2
3.2	odd	2	CM	243.2.c.b.82.1		2
9.2	odd	6	inner	243.2.c.b.163.1		2
9.4	even	3		243.2.a.a.1.1	✓	1
9.5	odd	6		243.2.a.a.1.1	✓	1
9.7	even	3	inner	243.2.c.b.163.1		2
27.2	odd	18		729.2.e.e.82.1		6
27.4	even	9		729.2.e.e.649.1		6
27.5	odd	18		729.2.e.e.163.1		6
27.7	even	9		729.2.e.e.568.1		6
27.11	odd	18		729.2.e.e.325.1		6
27.13	even	9		729.2.e.e.406.1		6
27.14	odd	18		729.2.e.e.406.1		6
27.16	even	9		729.2.e.e.325.1		6
27.20	odd	18		729.2.e.e.568.1		6
27.22	even	9		729.2.e.e.163.1		6
27.23	odd	18		729.2.e.e.649.1		6
27.25	even	9		729.2.e.e.82.1		6
36.23	even	6		3888.2.a.n.1.1		1
36.31	odd	6		3888.2.a.n.1.1		1
45.4	even	6		6075.2.a.w.1.1		1
45.14	odd	6		6075.2.a.w.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
243.2.a.a.1.1	✓	1	9.4	even	3
243.2.a.a.1.1	✓	1	9.5	odd	6
243.2.c.b.82.1		2	1.1	even	1	trivial
243.2.c.b.82.1		2	3.2	odd	2	CM
243.2.c.b.163.1		2	9.2	odd	6	inner
243.2.c.b.163.1		2	9.7	even	3	inner
729.2.e.e.82.1		6	27.2	odd	18
729.2.e.e.82.1		6	27.25	even	9
729.2.e.e.163.1		6	27.5	odd	18
729.2.e.e.163.1		6	27.22	even	9
729.2.e.e.325.1		6	27.11	odd	18
729.2.e.e.325.1		6	27.16	even	9
729.2.e.e.406.1		6	27.13	even	9
729.2.e.e.406.1		6	27.14	odd	18
729.2.e.e.568.1		6	27.7	even	9
729.2.e.e.568.1		6	27.20	odd	18
729.2.e.e.649.1		6	27.4	even	9
729.2.e.e.649.1		6	27.23	odd	18
3888.2.a.n.1.1		1	36.23	even	6
3888.2.a.n.1.1		1	36.31	odd	6
6075.2.a.w.1.1		1	45.4	even	6
6075.2.a.w.1.1		1	45.14	odd	6

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 243 = 3^{5} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	243.c (of order \(3\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(1.00000 - 1.73205i) q^{4} +(2.00000 + 3.46410i) q^{7} +(3.50000 - 6.06218i) q^{13} +(-2.00000 - 3.46410i) q^{16} -1.00000 q^{19} +(2.50000 + 4.33013i) q^{25} +8.00000 q^{28} +(-5.50000 + 9.52628i) q^{31} -10.0000 q^{37} +(-2.50000 - 4.33013i) q^{43} +(-4.50000 + 7.79423i) q^{49} +(-7.00000 - 12.1244i) q^{52} +(0.500000 + 0.866025i) q^{61} -8.00000 q^{64} +(-2.50000 + 4.33013i) q^{67} -7.00000 q^{73} +(-1.00000 + 1.73205i) q^{76} +(6.50000 + 11.2583i) q^{79} +28.0000 q^{91} +(-2.50000 - 4.33013i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{4} + 4 q^{7} + 7 q^{13} - 4 q^{16} - 2 q^{19} + 5 q^{25} + 16 q^{28} - 11 q^{31} - 20 q^{37} - 5 q^{43} - 9 q^{49} - 14 q^{52} + q^{61} - 16 q^{64} - 5 q^{67} - 14 q^{73} - 2 q^{76} + 13 q^{79}+ \cdots - 5 q^{97}+O(q^{100}) \) 2 * q + 2 * q^4 + 4 * q^7 + 7 * q^13 - 4 * q^16 - 2 * q^19 + 5 * q^25 + 16 * q^28 - 11 * q^31 - 20 * q^37 - 5 * q^43 - 9 * q^49 - 14 * q^52 + q^61 - 16 * q^64 - 5 * q^67 - 14 * q^73 - 2 * q^76 + 13 * q^79 + 56 * q^91 - 5 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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