LMFDB - Embedded newform 216.2.v.a.59.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [216,2,Mod(11,216)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(216, base_ring=CyclotomicField(18))

chi = DirichletCharacter(H, H._module([9, 9, 13]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("216.11");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 216 = 2^{3} \cdot 3^{3} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	216.v (of order $18$, degree $6$, 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:	$1.72476868366$
Analytic rank:	$0$
Dimension:	$12$
Relative dimension:	$2$ over $\Q(\zeta_{18})$
Coefficient field:	12.0.101559956668416.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - 8x^{6} + 64 $ x^12 - 8*x^6 + 64
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{18}]$

Embedding invariants

Embedding label			59.1
Root			$0.483690 + 1.32893i$ of defining polynomial
Character	$\chi$	$=$	216.59
Dual form			216.2.v.a.11.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-0.909039 - 1.08335i) q^{2} +(1.42338 - 0.986906i) q^{3} +(-0.347296 + 1.96962i) q^{4} +(-2.36307 - 0.644886i) q^{6} +(2.44949 - 1.41421i) q^{8} +(1.05203 - 2.80949i) q^{9} +O(q^{10})$	\(q+(-0.909039 - 1.08335i) q^{2} +(1.42338 - 0.986906i) q^{3} +(-0.347296 + 1.96962i) q^{4} +(-2.36307 - 0.644886i) q^{6} +(2.44949 - 1.41421i) q^{8} +(1.05203 - 2.80949i) q^{9} +(1.85296 - 5.09097i) q^{11} +(1.44949 + 3.14626i) q^{12} +(-3.75877 - 1.36808i) q^{16} +(-0.412814 - 0.238338i) q^{17} +(-4.00000 + 1.41421i) q^{18} +(2.39076 + 4.14092i) q^{19} +(-7.19972 + 2.62048i) q^{22} +(2.09086 - 4.43038i) q^{24} +(-3.83022 + 3.21394i) q^{25} +(-1.27526 - 5.03723i) q^{27} +(1.93476 + 5.31570i) q^{32} +(-2.38684 - 9.07509i) q^{33} +(0.117060 + 0.663880i) q^{34} +(5.16824 + 3.04783i) q^{36} +(2.31277 - 6.35429i) q^{38} +(-5.51466 + 6.57212i) q^{41} +(12.3238 + 4.48548i) q^{43} +(9.38372 + 5.41770i) q^{44} +(-6.70033 + 1.76225i) q^{48} +(-6.57785 + 2.39414i) q^{49} +(6.96364 + 1.22788i) q^{50} +(-0.822809 + 0.0681621i) q^{51} +(-4.29783 + 5.96059i) q^{54} +(7.48967 + 3.53466i) q^{57} +(5.24706 + 14.4162i) q^{59} +(4.00000 - 6.92820i) q^{64} +(-7.66178 + 10.8354i) q^{66} +(8.26366 + 6.93404i) q^{67} +(0.612803 - 0.730310i) q^{68} +(-1.39627 - 8.36961i) q^{72} +(-6.22028 - 10.7738i) q^{73} +(-2.28002 + 8.35473i) q^{75} +(-8.98632 + 3.27075i) q^{76} +(-6.78645 - 5.91135i) q^{81} +12.1329 q^{82} +(-9.11103 - 10.8581i) q^{83} +(-6.34343 - 17.4284i) q^{86} +(-2.66091 - 15.0908i) q^{88} +(-15.9495 + 9.20844i) q^{89} +(8.00000 + 5.65685i) q^{96} +(17.3366 + 6.31001i) q^{97} +(8.57321 + 4.94975i) q^{98} +(-12.3536 - 10.5617i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q+O(q^{10}) $ 12 * q	$ 12 q + 18 q^{11} - 12 q^{12} - 48 q^{18} + 12 q^{22} - 30 q^{27} + 6 q^{33} - 24 q^{34} + 72 q^{38} - 18 q^{41} + 30 q^{43} - 12 q^{51} + 42 q^{57} + 36 q^{59} + 48 q^{64} - 42 q^{67} - 72 q^{68} - 24 q^{76} + 36 q^{86} + 48 q^{88} - 162 q^{89} + 96 q^{96} + 30 q^{97}+O(q^{100}) $ 12 * q + 18 * q^11 - 12 * q^12 - 48 * q^18 + 12 * q^22 - 30 * q^27 + 6 * q^33 - 24 * q^34 + 72 * q^38 - 18 * q^41 + 30 * q^43 - 12 * q^51 + 42 * q^57 + 36 * q^59 + 48 * q^64 - 42 * q^67 - 72 * q^68 - 24 * q^76 + 36 * q^86 + 48 * q^88 - 162 * q^89 + 96 * q^96 + 30 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/216\mathbb{Z}\right)^\times$.

$n$	$55$	$109$	$137$
$\chi(n)$	$-1$	$-1$	$e\left(\frac{5}{18}\right)$

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 $

$2$	−0.909039	−	1.08335i	−0.642788	−	0.766044i
$2$	−0.909039	−	1.08335i	−0.642788	−	0.766044i
$3$	1.42338	−	0.986906i	0.821790	−	0.569790i
$3$	1.42338	−	0.986906i	0.821790	−	0.569790i
$4$	−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	$0.388889\pi$
$6$	−2.36307	−	0.644886i	−0.964721	−	0.263274i
$7$		0			0		−0.173648	−	0.984808i	$-0.555556\pi$
$7$		0			0		0.173648	+	0.984808i	$0.444444\pi$
$8$	2.44949	−	1.41421i	0.866025	−	0.500000i
$9$	1.05203	−	2.80949i	0.350678	−	0.936496i
$10$		0			0
$11$	1.85296	−	5.09097i	0.558689	−	1.53498i	−0.262853	−	0.964836i	$-0.584663\pi$
$11$	1.85296	−	5.09097i	0.558689	−	1.53498i	0.821541	−	0.570149i	$-0.193114\pi$
$12$	1.44949	+	3.14626i	0.418432	+	0.908248i
$13$		0			0		−0.766044	−	0.642788i	$-0.777778\pi$
$13$		0			0		0.766044	+	0.642788i	$0.222222\pi$
$14$		0			0
$15$		0			0
$16$	−3.75877	−	1.36808i	−0.939693	−	0.342020i
$17$	−0.412814	−	0.238338i	−0.100122	−	0.0578055i	0.449103	−	0.893480i	$-0.351744\pi$
$17$	−0.412814	−	0.238338i	−0.100122	−	0.0578055i	−0.549225	+	0.835675i	$0.685077\pi$
$18$	−4.00000	+	1.41421i	−0.942809	+	0.333333i
$19$	2.39076	+	4.14092i	0.548478	+	0.949992i	0.998379	+	0.0569137i	$0.0181260\pi$
$19$	2.39076	+	4.14092i	0.548478	+	0.949992i	−0.449901	+	0.893079i	$0.648541\pi$
$20$		0			0
$21$		0			0
$22$	−7.19972	+	2.62048i	−1.53498	+	0.558689i
$23$		0			0		−0.984808	−	0.173648i	$-0.944444\pi$
$23$		0			0		0.984808	+	0.173648i	$0.0555556\pi$
$24$	2.09086	−	4.43038i	0.426796	−	0.904348i
$25$	−3.83022	+	3.21394i	−0.766044	+	0.642788i
$26$		0			0
$27$	−1.27526	−	5.03723i	−0.245423	−	0.969416i
$28$		0			0
$29$		0			0		−0.642788	−	0.766044i	$-0.722222\pi$
$29$		0			0		0.642788	+	0.766044i	$0.277778\pi$
$30$		0			0
$31$		0			0		0.173648	−	0.984808i	$-0.444444\pi$
$31$		0			0		−0.173648	+	0.984808i	$0.555556\pi$
$32$	1.93476	+	5.31570i	0.342020	+	0.939693i
$33$	−2.38684	−	9.07509i	−0.415495	−	1.57977i
$34$	0.117060	+	0.663880i	0.0200756	+	0.113855i
$35$		0			0
$36$	5.16824	+	3.04783i	0.861374	+	0.507971i
$37$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$37$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$38$	2.31277	−	6.35429i	0.375181	−	1.03080i
$39$		0			0
$40$		0			0
$41$	−5.51466	+	6.57212i	−0.861245	+	1.02639i	0.138108	+	0.990417i	$0.455898\pi$
$41$	−5.51466	+	6.57212i	−0.861245	+	1.02639i	−0.999353	+	0.0359748i	$0.988546\pi$
$42$		0			0
$43$	12.3238	+	4.48548i	1.87936	+	0.684030i	0.941562	+	0.336840i	$0.109358\pi$
$43$	12.3238	+	4.48548i	1.87936	+	0.684030i	0.937795	+	0.347190i	$0.112864\pi$
$44$	9.38372	+	5.41770i	1.41465	+	0.816748i
$45$		0			0
$46$		0			0
$47$		0			0		0.984808	−	0.173648i	$-0.0555556\pi$
$47$		0			0		−0.984808	+	0.173648i	$0.944444\pi$
$48$	−6.70033	+	1.76225i	−0.967110	+	0.254359i
$49$	−6.57785	+	2.39414i	−0.939693	+	0.342020i
$50$	6.96364	+	1.22788i	0.984808	+	0.173648i
$51$	−0.822809	+	0.0681621i	−0.115216	+	0.00954460i
$52$		0			0
$53$		0			0			−	1.00000i	$-0.5\pi$
$53$		0			0				1.00000i	$0.5\pi$
$54$	−4.29783	+	5.96059i	−0.584861	+	0.811134i
$55$		0			0
$56$		0			0
$57$	7.48967	+	3.53466i	0.992030	+	0.468176i
$58$		0			0
$59$	5.24706	+	14.4162i	0.683109	+	1.87683i	0.390567	+	0.920575i	$0.372279\pi$
$59$	5.24706	+	14.4162i	0.683109	+	1.87683i	0.292542	+	0.956253i	$0.405499\pi$
$60$		0			0
$61$		0			0		−0.173648	−	0.984808i	$-0.555556\pi$
$61$		0			0		0.173648	+	0.984808i	$0.444444\pi$
$62$		0			0
$63$		0			0
$64$	4.00000	−	6.92820i	0.500000	−	0.866025i
$65$		0			0
$66$	−7.66178	+	10.8354i	−0.943100	+	1.33374i
$67$	8.26366	+	6.93404i	1.00957	+	0.847127i	0.988281	−	0.152646i	$-0.0487795\pi$
$67$	8.26366	+	6.93404i	1.00957	+	0.847127i	0.0212861	+	0.999773i	$0.493224\pi$
$68$	0.612803	−	0.730310i	0.0743133	−	0.0885631i
$69$		0			0
$70$		0			0
$71$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$71$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$72$	−1.39627	−	8.36961i	−0.164552	−	0.986368i
$73$	−6.22028	−	10.7738i	−0.728028	−	1.26098i	−0.957715	−	0.287718i	$-0.907104\pi$
$73$	−6.22028	−	10.7738i	−0.728028	−	1.26098i	0.229687	−	0.973265i	$-0.426230\pi$
$74$		0			0
$75$	−2.28002	+	8.35473i	−0.263274	+	0.964721i
$76$	−8.98632	+	3.27075i	−1.03080	+	0.375181i
$77$		0			0
$78$		0			0
$79$		0			0		0.766044	−	0.642788i	$-0.222222\pi$
$79$		0			0		−0.766044	+	0.642788i	$0.777778\pi$
$80$		0			0
$81$	−6.78645	−	5.91135i	−0.754050	−	0.656817i
$82$	12.1329			1.33986
$83$	−9.11103	−	10.8581i	−1.00007	−	1.19183i	−0.981394	−	0.192006i	$-0.938501\pi$
$83$	−9.11103	−	10.8581i	−1.00007	−	1.19183i	−0.0186715	−	0.999826i	$-0.505944\pi$
$84$		0			0
$85$		0			0
$86$	−6.34343	−	17.4284i	−0.684030	−	1.87936i
$87$		0			0
$88$	−2.66091	−	15.0908i	−0.283654	−	1.60868i
$89$	−15.9495	+	9.20844i	−1.69064	+	0.976093i	−0.736644	+	0.676280i	$0.763591\pi$
$89$	−15.9495	+	9.20844i	−1.69064	+	0.976093i	−0.953998	+	0.299813i	$0.903076\pi$
$90$		0			0
$91$		0			0
$92$		0			0
$93$		0			0
$94$		0			0
$95$		0			0
$96$	8.00000	+	5.65685i	0.816497	+	0.577350i
$97$	17.3366	+	6.31001i	1.76026	+	0.640684i	0.999961	−	0.00888289i	$-0.00282755\pi$
$97$	17.3366	+	6.31001i	1.76026	+	0.640684i	0.760304	+	0.649567i	$0.225050\pi$
$98$	8.57321	+	4.94975i	0.866025	+	0.500000i
$99$	−12.3536	−	10.5617i	−1.24159	−	1.06149i
$100$	−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	$0.944444\pi$
$102$	0.821809	+	0.829428i	0.0813712	+	0.0821256i
$103$		0			0		0.939693	−	0.342020i	$-0.111111\pi$
$103$		0			0		−0.939693	+	0.342020i	$0.888889\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		−	19.0236i		−	1.83908i	−0.392992	−	0.919542i	$-0.628560\pi$
$107$		−	19.0236i		−	1.83908i	0.392992	−	0.919542i	$-0.371440\pi$
$108$	10.3643	−	0.762349i	0.997306	−	0.0733571i
$109$		0			0		1.00000			$0$
$109$		0			0		−1.00000			$\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	3.39681	+	9.33265i	0.319545	+	0.877942i	0.990631	+	0.136563i	$0.0436057\pi$
$113$	3.39681	+	9.33265i	0.319545	+	0.877942i	−0.671087	+	0.741379i	$0.734172\pi$
$114$	−2.97913	−	11.3271i	−0.279021	−	1.06088i
$115$		0			0
$116$		0			0
$117$		0			0
$118$	10.8480	−	18.7893i	0.998639	−	1.72969i
$119$		0			0
$120$		0			0
$121$	−14.0580	−	11.7961i	−1.27800	−	1.07237i
$122$		0			0
$123$	−1.36341	+	14.7971i	−0.122934	+	1.33421i
$124$		0			0
$125$		0			0
$126$		0			0
$127$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$127$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$128$	−11.1418	+	1.96460i	−0.984808	+	0.173648i
$129$	21.9682	−	5.77784i	1.93419	−	0.508710i
$130$		0			0
$131$	−22.3153	−	3.93479i	−1.94969	−	0.343784i	−0.999482	−	0.0321817i	$-0.989754\pi$
$131$	−22.3153	−	3.93479i	−1.94969	−	0.343784i	−0.950213	−	0.311602i	$-0.899134\pi$
$132$	18.7034	−	1.54940i	1.62792	−	0.134858i
$133$		0			0
$134$		−	15.2558i		−	1.31790i
$135$		0			0
$136$	−1.34824			−0.115611
$137$	6.32378	+	7.53639i	0.540277	+	0.643877i	0.965250	−	0.261329i	$-0.0841608\pi$
$137$	6.32378	+	7.53639i	0.540277	+	0.643877i	−0.424973	+	0.905206i	$0.639716\pi$
$138$		0			0
$139$	0.787687	−	4.46720i	0.0668108	−	0.378903i	−0.933008	−	0.359856i	$-0.882826\pi$
$139$	0.787687	−	4.46720i	0.0668108	−	0.378903i	0.999819	−	0.0190466i	$-0.00606310\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		0			0
$144$	−7.79796	+	9.12096i	−0.649830	+	0.760080i
$145$		0			0
$146$	−6.01737	+	16.5326i	−0.498001	+	1.36825i
$147$	−7.00000	+	9.89949i	−0.577350	+	0.816497i
$148$		0			0
$149$		0			0		0.642788	−	0.766044i	$-0.277778\pi$
$149$		0			0		−0.642788	+	0.766044i	$0.722222\pi$
$150$	11.1237	−	5.12472i	0.908248	−	0.418432i
$151$		0			0		−0.939693	−	0.342020i	$-0.888889\pi$
$151$		0			0		0.939693	+	0.342020i	$0.111111\pi$
$152$	11.7123	+	6.76209i	0.949992	+	0.548478i
$153$	−1.10390	+	0.909055i	−0.0892452	+	0.0734928i
$154$		0			0
$155$		0			0
$156$		0			0
$157$		0			0		0.939693	−	0.342020i	$-0.111111\pi$
$157$		0			0		−0.939693	+	0.342020i	$0.888889\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$	−0.234917	+	12.7258i	−0.0184568	+	0.999830i
$163$	−23.0454			−1.80506			−0.902528	−	0.430632i	$-0.858291\pi$
$163$	−23.0454			−1.80506			−0.902528	+	0.430632i	$0.858291\pi$
$164$	−11.0293	−	13.1442i	−0.861245	−	1.02639i
$165$		0			0
$166$	−3.48085	+	19.7409i	−0.270166	+	1.53219i
$167$		0			0		−0.342020	−	0.939693i	$-0.611111\pi$
$167$		0			0		0.342020	+	0.939693i	$0.388889\pi$
$168$		0			0
$169$	2.25743	+	12.8025i	0.173648	+	0.984808i
$170$		0			0
$171$	14.1490	−	2.36043i	1.08200	−	0.180507i
$172$	−13.1147	+	22.7153i	−0.999985	+	1.73202i
$173$		0			0		0.342020	−	0.939693i	$-0.388889\pi$
$173$		0			0		−0.342020	+	0.939693i	$0.611111\pi$
$174$		0			0
$175$		0			0
$176$	−13.9297	+	16.6008i	−1.04999	+	1.25133i
$177$	21.6960	+	15.3414i	1.63077	+	1.15313i
$178$	24.4747	+	8.90805i	1.83445	+	0.667687i
$179$	4.92679	+	2.84448i	0.368245	+	0.212607i	0.672692	−	0.739923i	$-0.265138\pi$
$179$	4.92679	+	2.84448i	0.368245	+	0.212607i	−0.304446	+	0.952529i	$0.598471\pi$
$180$		0			0
$181$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$181$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$		0			0
$186$		0			0
$187$	−1.97830	+	1.65999i	−0.144668	+	0.121390i
$188$		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	$0.277778\pi$
$192$	−1.14396	−	13.8091i	−0.0825579	−	0.996586i
$193$	4.59777	−	26.0753i	0.330955	−	1.87694i	−0.133056	−	0.991109i	$-0.542479\pi$
$193$	4.59777	−	26.0753i	0.330955	−	1.87694i	0.464011	−	0.885830i	$-0.346410\pi$
$194$	−8.92370	−	24.5177i	−0.640684	−	1.76026i
$195$		0			0
$196$	−2.43107	−	13.7873i	−0.173648	−	0.984808i
$197$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$197$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$198$	−0.212127	+	22.9844i	−0.0150752	+	1.63343i
$199$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$199$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$200$	−4.83690	+	13.2893i	−0.342020	+	0.939693i
$201$	18.6056	+	1.71433i	1.31234	+	0.120919i
$202$		0			0
$203$		0			0
$204$	0.151505	−	1.64429i	0.0106075	−	0.115123i
$205$		0			0
$206$		0			0
$207$		0			0
$208$		0			0
$209$	25.5113	−	4.49833i	1.76465	−	0.311156i
$210$		0			0
$211$	27.2938	−	9.93411i	1.87898	−	0.683893i	0.932384	−	0.361469i	$-0.117725\pi$
$211$	27.2938	−	9.93411i	1.87898	−	0.683893i	0.946595	−	0.322424i	$-0.104498\pi$
$212$		0			0
$213$		0			0
$214$	−20.6093	+	17.2932i	−1.40882	+	1.18214i
$215$		0			0
$216$	−10.2474	−	10.5352i	−0.697251	−	0.716827i
$217$		0			0
$218$		0			0
$219$	−19.4866	−	9.19646i	−1.31678	−	0.621439i
$220$		0			0
$221$		0			0
$222$		0			0
$223$		0			0		−0.173648	−	0.984808i	$-0.555556\pi$
$223$		0			0		0.173648	+	0.984808i	$0.444444\pi$
$224$		0			0
$225$	5.00000	+	14.1421i	0.333333	+	0.942809i
$226$	7.02270	−	12.1637i	0.467143	−	0.809116i
$227$	5.85433	−	16.0847i	0.388566	−	1.06758i	−0.579082	−	0.815270i	$-0.696589\pi$
$227$	5.85433	−	16.0847i	0.388566	−	1.06758i	0.967647	−	0.252306i	$-0.0811890\pi$
$228$	−9.56304	+	13.5242i	−0.633328	+	0.895661i
$229$		0			0		−0.766044	−	0.642788i	$-0.777778\pi$
$229$		0			0		0.766044	+	0.642788i	$0.222222\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−4.28241	−	2.47245i	−0.280550	−	0.161976i	0.353122	−	0.935577i	$-0.385120\pi$
$233$	−4.28241	−	2.47245i	−0.280550	−	0.161976i	−0.633672	+	0.773602i	$0.718453\pi$
$234$		0			0
$235$		0			0
$236$	−30.2166	+	5.32801i	−1.96693	+	0.346824i
$237$		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
$241$	−9.34413	+	7.84065i	−0.601908	+	0.505061i	−0.892058	−	0.451920i	$-0.850739\pi$
$241$	−9.34413	+	7.84065i	−0.601908	+	0.505061i	0.290150	+	0.956981i	$0.406295\pi$
$242$			25.9528i			1.66831i
$243$	−15.4937	−	1.71652i	−0.993919	−	0.110115i
$244$		0			0
$245$		0			0
$246$	17.2698	−	11.9741i	1.10108	−	0.763439i
$247$		0			0
$248$		0			0
$249$	−23.6844	−	6.46350i	−1.50094	−	0.409608i
$250$		0			0
$251$	17.3006	−	9.98849i	1.09200	−	0.630468i	0.157894	−	0.987456i	$-0.449530\pi$
$251$	17.3006	−	9.98849i	1.09200	−	0.630468i	0.934109	+	0.356988i	$0.116196\pi$
$252$		0			0
$253$		0			0
$254$		0			0
$255$		0			0
$256$	12.2567	+	10.2846i	0.766044	+	0.642788i
$257$	20.2535	−	24.1372i	1.26338	−	1.50564i	0.489951	−	0.871750i	$-0.337015\pi$
$257$	20.2535	−	24.1372i	1.26338	−	1.50564i	0.773427	−	0.633885i	$-0.218541\pi$
$258$	−26.2294	−	18.5470i	−1.63297	−	1.15468i
$259$		0			0
$260$		0			0
$261$		0			0
$262$	16.0227	+	27.7521i	0.989886	+	1.71453i
$263$		0			0		0.984808	−	0.173648i	$-0.0555556\pi$
$263$		0			0		−0.984808	+	0.173648i	$0.944444\pi$
$264$	−18.6806	−	18.8538i	−1.14971	−	1.16037i
$265$		0			0
$266$		0			0
$267$	−13.6144	+	28.8478i	−0.833185	+	1.76546i
$268$	−16.5273	+	13.8681i	−1.00957	+	0.847127i
$269$		0			0			−	1.00000i	$-0.5\pi$
$269$		0			0				1.00000i	$0.5\pi$
$270$		0			0
$271$		0			0		1.00000			$0$
$271$		0			0		−1.00000			$\pi$
$272$	1.22561	+	1.46062i	0.0743133	+	0.0885631i
$273$		0			0
$274$	2.41599	−	13.7017i	0.145955	−	0.827752i
$275$	9.26480	+	25.4548i	0.558689	+	1.53498i
$276$		0			0
$277$		0			0		−0.173648	−	0.984808i	$-0.555556\pi$
$277$		0			0		0.173648	+	0.984808i	$0.444444\pi$
$278$	−5.55558	+	3.20751i	−0.333201	+	0.192374i
$279$		0			0
$280$		0			0
$281$	−0.494671	+	1.35910i	−0.0295096	+	0.0810770i	−0.953573	−	0.301163i	$-0.902625\pi$
$281$	−0.494671	+	1.35910i	−0.0295096	+	0.0810770i	0.924063	+	0.382240i	$0.124847\pi$
$282$		0			0
$283$	−8.46127	−	7.09985i	−0.502970	−	0.422042i	0.355677	−	0.934609i	$-0.384250\pi$
$283$	−8.46127	−	7.09985i	−0.502970	−	0.422042i	−0.858647	+	0.512567i	$0.828695\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$	16.9698	+	0.156618i	0.999957	+	0.00922881i
$289$	−8.38639	−	14.5257i	−0.493317	−	0.854450i
$290$		0			0
$291$	30.9040	−	8.12804i	1.81162	−	0.476474i
$292$	23.3806	−	8.50984i	1.36825	−	0.498001i
$293$		0			0		−0.984808	−	0.173648i	$-0.944444\pi$
$293$		0			0		0.984808	+	0.173648i	$0.0555556\pi$
$294$	17.0879	−	1.41557i	0.996586	−	0.0825579i
$295$		0			0
$296$		0			0
$297$	−28.0074	−	2.84151i	−1.62515	−	0.164881i
$298$		0			0
$299$		0			0
$300$	−15.6638	−	7.39232i	−0.904348	−	0.426796i
$301$		0			0
$302$		0			0
$303$		0			0
$304$	−3.32121	−	18.8355i	−0.190484	−	1.08029i
$305$		0			0
$306$	1.98832	+	0.369546i	0.113664	+	0.0211255i
$307$	−14.5237	+	25.1558i	−0.828912	+	1.43572i	0.0699810	+	0.997548i	$0.477706\pi$
$307$	−14.5237	+	25.1558i	−0.828912	+	1.43572i	−0.898893	+	0.438169i	$0.855627\pi$
$308$		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
$313$	2.07825	+	0.756419i	0.117469	+	0.0427554i	0.400086	−	0.916478i	$-0.368980\pi$
$313$	2.07825	+	0.756419i	0.117469	+	0.0427554i	−0.282617	+	0.959233i	$0.591202\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$		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
$321$	−18.7745	−	27.0779i	−1.04789	−	1.51134i
$322$		0			0
$323$		−	2.27924i		−	0.126820i
$324$	14.0000	−	11.3137i	0.777778	−	0.628539i
$325$		0			0
$326$	20.9492	+	24.9663i	1.16027	+	1.38275i
$327$		0			0
$328$	−4.21373	+	23.8972i	−0.232664	+	1.31950i
$329$		0			0
$330$		0			0
$331$	−6.08557	−	34.5130i	−0.334493	−	1.89701i	−0.432178	−	0.901788i	$-0.642255\pi$
$331$	−6.08557	−	34.5130i	−0.334493	−	1.89701i	0.0976852	−	0.995217i	$-0.468856\pi$
$332$	24.5505	−	14.1742i	1.34738	−	0.777913i
$333$		0			0
$334$		0			0
$335$		0			0
$336$		0			0
$337$	−8.49720	−	7.13000i	−0.462872	−	0.388396i	0.381314	−	0.924445i	$-0.375472\pi$
$337$	−8.49720	−	7.13000i	−0.462872	−	0.388396i	−0.844187	+	0.536050i	$0.819916\pi$
$338$	11.8175	−	14.0836i	0.642788	−	0.766044i
$339$	14.0454	+	9.93160i	0.762842	+	0.539411i
$340$		0			0
$341$		0			0
$342$	−15.4192	−	13.1826i	−0.833774	−	0.712835i
$343$		0			0
$344$	36.5304	−	6.44129i	1.96959	−	0.347291i
$345$		0			0
$346$		0			0
$347$	32.0062	+	5.64356i	1.71818	+	0.302962i	0.943987	−	0.329983i	$-0.107043\pi$
$347$	32.0062	+	5.64356i	1.71818	+	0.302962i	0.774197	+	0.632945i	$0.218154\pi$
$348$		0			0
$349$		0			0		0.766044	−	0.642788i	$-0.222222\pi$
$349$		0			0		−0.766044	+	0.642788i	$0.777778\pi$
$350$		0			0
$351$		0			0
$352$	30.6471			1.63350
$353$	23.5371	+	28.0504i	1.25275	+	1.49297i	0.798369	+	0.602168i	$0.205696\pi$
$353$	23.5371	+	28.0504i	1.25275	+	1.49297i	0.454384	+	0.890806i	$0.349859\pi$
$354$	−3.10241	−	37.4503i	−0.164891	−	1.99046i
$355$		0			0
$356$	−12.5979	−	34.6124i	−0.667687	−	1.83445i
$357$		0			0
$358$	−1.39707	−	7.92318i	−0.0738375	−	0.418753i
$359$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$359$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$360$		0			0
$361$	−1.93148	+	3.34542i	−0.101657	+	0.176075i
$362$		0			0
$363$	−31.6515	−	2.91638i	−1.66127	−	0.153070i
$364$		0			0
$365$		0			0
$366$		0			0
$367$		0			0		−0.939693	−	0.342020i	$-0.888889\pi$
$367$		0			0		0.939693	+	0.342020i	$0.111111\pi$
$368$		0			0
$369$	12.6627	+	22.4075i	0.659193	+	1.16649i
$370$		0			0
$371$		0			0
$372$		0			0
$373$		0			0		0.939693	−	0.342020i	$-0.111111\pi$
$373$		0			0		−0.939693	+	0.342020i	$0.888889\pi$
$374$	3.59670	+	0.634196i	0.185981	+	0.0327935i
$375$		0			0
$376$		0			0
$377$		0			0
$378$		0			0
$379$	−38.6105			−1.98329			−0.991643	−	0.129012i	$-0.958819\pi$
$379$	−38.6105			−1.98329			−0.991643	+	0.129012i	$0.958819\pi$
$380$		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	$0.388889\pi$
$384$	−13.9202	+	13.7923i	−0.710362	+	0.703836i
$385$		0			0
$386$	−32.4282	+	18.7224i	−1.65055	+	0.952947i
$387$	25.5669	−	29.9046i	1.29964	−	1.52014i
$388$	−18.4492	+	31.9550i	−0.936617	+	1.62227i
$389$		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
$392$	−12.7265	+	15.1669i	−0.642788	+	0.766044i
$393$	−35.6464	+	16.4224i	−1.79812	+	0.828399i
$394$		0			0
$395$		0			0
$396$	25.0929	−	20.6639i	1.26097	−	1.03840i
$397$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$397$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$398$		0			0
$399$		0			0
$400$	18.7939	−	6.84040i	0.939693	−	0.342020i
$401$	−38.6656	−	6.81778i	−1.93087	−	0.340464i	−0.931158	−	0.364615i	$-0.881200\pi$
$401$	−38.6656	−	6.81778i	−1.93087	−	0.340464i	−0.999708	+	0.0241516i	$0.992312\pi$
$402$	−15.0560	−	21.7148i	−0.750925	−	1.08303i
$403$		0			0
$404$		0			0
$405$		0			0
$406$		0			0
$407$		0			0
$408$	−1.91907	+	1.33059i	−0.0950079	+	0.0658740i
$409$	−6.71496	+	38.0824i	−0.332033	+	1.88305i	0.122726	+	0.992441i	$0.460836\pi$
$409$	−6.71496	+	38.0824i	−0.332033	+	1.88305i	−0.454759	+	0.890614i	$0.650275\pi$
$410$		0			0
$411$	16.4389	+	4.48618i	0.810869	+	0.221287i
$412$		0			0
$413$		0			0
$414$		0			0
$415$		0			0
$416$		0			0
$417$	−3.28752	−	7.13590i	−0.160991	−	0.349447i
$418$	−28.0640	−	23.5485i	−1.37266	−	1.15179i
$419$	−1.79744	+	2.14210i	−0.0878107	+	0.104649i	−0.808161	−	0.588962i	$-0.799537\pi$
$419$	−1.79744	+	2.14210i	−0.0878107	+	0.104649i	0.720350	+	0.693611i	$0.243981\pi$
$420$		0			0
$421$		0			0		−0.939693	−	0.342020i	$-0.888889\pi$
$421$		0			0		0.939693	+	0.342020i	$0.111111\pi$
$422$	−35.5732	−	20.5382i	−1.73168	−	0.999784i
$423$		0			0
$424$		0			0
$425$	2.34717	−	0.413870i	0.113855	−	0.0200756i
$426$		0			0
$427$		0			0
$428$	37.4692	+	6.60684i	1.81114	+	0.319354i
$429$		0			0
$430$		0			0
$431$		0			0			−	1.00000i	$-0.5\pi$
$431$		0			0				1.00000i	$0.5\pi$
$432$	−2.09795	+	20.6785i	−0.100938	+	0.994893i
$433$	40.0182			1.92315			0.961575	−	0.274543i	$-0.0885264\pi$
$433$	40.0182			1.92315			0.961575	+	0.274543i	$0.0885264\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$		0			0
$438$	7.75109	+	29.4708i	0.370361	+	1.40817i
$439$		0			0		−0.173648	−	0.984808i	$-0.555556\pi$
$439$		0			0		0.173648	+	0.984808i	$0.444444\pi$
$440$		0			0
$441$	−0.193805	+	20.9991i	−0.00922881	+	0.999957i
$442$		0			0
$443$	−4.00403	+	11.0010i	−0.190237	+	0.522672i	−0.997740	−	0.0671913i	$-0.978596\pi$
$443$	−4.00403	+	11.0010i	−0.190237	+	0.522672i	0.807503	+	0.589863i	$0.200818\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$		0			0
$448$		0			0
$449$	−36.6712	−	21.1721i	−1.73062	−	0.999174i	−0.885630	−	0.464391i	$-0.846273\pi$
$449$	−36.6712	−	21.1721i	−1.73062	−	0.999174i	−0.844990	−	0.534783i	$-0.820394\pi$
$450$	10.7757	−	18.2725i	0.507971	−	0.861374i
$451$	23.2400	+	40.2528i	1.09433	+	1.89543i
$452$	−19.5614	+	3.44921i	−0.920093	+	0.162237i
$453$		0			0
$454$	−22.7471	+	8.27928i	−1.06758	+	0.388566i
$455$		0			0
$456$	23.3446	−	1.93388i	1.09321	−	0.0905624i
$457$	−27.6087	+	23.1664i	−1.29148	+	1.08368i	−0.299928	+	0.953962i	$0.596963\pi$
$457$	−27.6087	+	23.1664i	−1.29148	+	1.08368i	−0.991551	+	0.129718i	$0.958593\pi$
$458$		0			0
$459$	−0.674122	+	2.38338i	−0.0314653	+	0.111247i
$460$		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
$463$		0			0		0.173648	−	0.984808i	$-0.444444\pi$
$463$		0			0		−0.173648	+	0.984808i	$0.555556\pi$
$464$		0			0
$465$		0			0
$466$	1.21435	+	6.88690i	0.0562535	+	0.319030i
$467$	−16.4335	+	9.48788i	−0.760452	+	0.439047i	−0.829458	−	0.558569i	$-0.811350\pi$
$467$	−16.4335	+	9.48788i	−0.760452	+	0.439047i	0.0690063	+	0.997616i	$0.478017\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$		0			0
$472$	33.2402	+	27.8918i	1.53000	+	1.28383i
$473$	45.6709	−	54.4285i	2.09995	−	2.50262i
$474$		0			0
$475$	−22.4658	−	8.17688i	−1.03080	−	0.375181i
$476$		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
$481$		0			0
$482$	16.9883	+	2.99550i	0.773798	+	0.136441i
$483$		0			0
$484$	28.1160	−	23.5921i	1.27800	−	1.07237i
$485$		0			0
$486$	12.2247	+	18.3455i	0.554526	+	0.832167i
$487$		0			0		1.00000			$0$
$487$		0			0		−1.00000			$\pi$
$488$		0			0
$489$	−32.8024	+	22.7436i	−1.48338	+	1.02850i
$490$		0			0
$491$	5.52827	+	15.1888i	0.249487	+	0.685461i	0.999705	+	0.0242702i	$0.00772622\pi$
$491$	5.52827	+	15.1888i	0.249487	+	0.685461i	−0.750218	+	0.661190i	$0.770052\pi$
$492$	−28.6711	−	7.82436i	−1.29259	−	0.352750i
$493$		0			0
$494$		0			0
$495$		0			0
$496$		0			0
$497$		0			0
$498$	14.5278	+	31.5341i	0.651007	+	1.41308i
$499$	29.1065	+	24.4232i	1.30298	+	1.09333i	0.989621	+	0.143700i	$0.0459001\pi$
$499$	29.1065	+	24.4232i	1.30298	+	1.09333i	0.313363	+	0.949633i	$0.398544\pi$
$500$		0			0
$501$		0			0
$502$	−26.5479	−	9.66266i	−1.18489	−	0.431266i
$503$		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
$507$	15.8480	+	15.9950i	0.703836	+	0.710362i
$508$		0			0
$509$		0			0		−0.984808	−	0.173648i	$-0.944444\pi$
$509$		0			0		0.984808	+	0.173648i	$0.0555556\pi$
$510$		0			0
$511$		0			0
$512$		−	22.6274i		−	1.00000i
$513$	17.8100	−	17.3236i	0.786329	−	0.764854i
$514$	−44.5602			−1.96547
$515$		0			0
$516$	3.75065	+	45.2755i	0.165113	+	1.99314i
$517$		0			0
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−33.3627	+	19.2620i	−1.46165	+	0.843882i	−0.999088	−	0.0427062i	$-0.986402\pi$
$521$	−33.3627	+	19.2620i	−1.46165	+	0.843882i	−0.462559	+	0.886588i	$0.653069\pi$
$522$		0			0
$523$	20.5227	−	35.5464i	0.897395	−	1.55433i	0.0665832	−	0.997781i	$-0.478790\pi$
$523$	20.5227	−	35.5464i	0.897395	−	1.55433i	0.830812	−	0.556553i	$-0.187876\pi$
$524$	15.5000	−	42.5860i	0.677122	−	1.86038i
$525$		0			0
$526$		0			0
$527$		0			0
$528$	−3.44389	+	37.3766i	−0.149876	+	1.62661i
$529$	21.6129	+	7.86646i	0.939693	+	0.342020i
$530$		0			0
$531$	46.0222	+	0.424748i	1.99719	+	0.0184325i
$532$		0			0
$533$		0			0
$534$	43.6282	−	11.4746i	1.88798	−	0.496556i
$535$		0			0
$536$	30.0480	+	5.29827i	1.29787	+	0.228850i
$537$	9.81993	−	0.813490i	0.423761	−	0.0351047i
$538$		0			0
$539$			37.9239i			1.63350i
$540$		0			0
$541$		0			0		1.00000			$0$
$541$		0			0		−1.00000			$\pi$
$542$		0			0
$543$		0			0
$544$	0.468240	−	2.65552i	0.0200756	−	0.113855i
$545$		0			0
$546$		0			0
$547$	−7.06614	−	40.0741i	−0.302126	−	1.71344i	−0.636732	−	0.771085i	$-0.719714\pi$
$547$	−7.06614	−	40.0741i	−0.302126	−	1.71344i	0.334606	−	0.942358i	$-0.391397\pi$
$548$	−17.0400	+	9.83806i	−0.727913	+	0.420261i
$549$		0			0
$550$	19.1544	−	33.1765i	0.816748	−	1.41465i
$551$		0			0
$552$		0			0
$553$		0			0
$554$		0			0
$555$		0			0
$556$	8.52510	+	3.10288i	0.361545	+	0.131592i
$557$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$557$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$558$		0			0
$559$		0			0
$560$		0			0
$561$	−1.17762	+	4.31519i	−0.0497192	+	0.182188i
$562$	1.92205	−	0.699571i	0.0810770	−	0.0295096i
$563$	−35.3834	−	6.23904i	−1.49123	−	0.262944i	−0.632175	−	0.774826i	$-0.717837\pi$
$563$	−35.3834	−	6.23904i	−1.49123	−	0.262944i	−0.859056	+	0.511882i	$0.828949\pi$
$564$		0			0
$565$		0			0
$566$			15.6206i			0.656581i
$567$		0			0
$568$		0			0
$569$	−22.9010	−	27.2924i	−0.960061	−	1.14416i	−0.989492	−	0.144589i	$-0.953814\pi$
$569$	−22.9010	−	27.2924i	−0.960061	−	1.14416i	0.0294311	−	0.999567i	$-0.490630\pi$
$570$		0			0
$571$	1.80910	−	10.2599i	0.0757085	−	0.429364i	−0.923269	−	0.384154i	$-0.874493\pi$
$571$	1.80910	−	10.2599i	0.0757085	−	0.429364i	0.998978	−	0.0452101i	$-0.0143957\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$	−15.2566	−	18.5267i	−0.635691	−	0.771944i
$577$	−13.7607	+	23.8343i	−0.572866	+	0.992234i	0.423403	+	0.905941i	$0.360835\pi$
$577$	−13.7607	+	23.8343i	−0.572866	+	0.992234i	−0.996270	+	0.0862925i	$0.972498\pi$
$578$	−8.11282	+	22.2898i	−0.337449	+	0.927133i
$579$	−19.1894	−	41.6526i	−0.797486	−	1.73102i
$580$		0			0
$581$		0			0
$582$	−36.8984	−	26.0911i	−1.52949	−	1.08151i
$583$		0			0
$584$	−30.4730	−	17.5936i	−1.26098	−	0.728028i
$585$		0			0
$586$		0			0
$587$	−14.0418	+	2.47595i	−0.579567	+	0.102193i	−0.455744	−	0.890111i	$-0.650627\pi$
$587$	−14.0418	+	2.47595i	−0.579567	+	0.102193i	−0.123823	+	0.992304i	$0.539516\pi$
$588$	−17.0671	−	17.2254i	−0.703836	−	0.710362i
$589$		0			0
$590$		0			0
$591$		0			0
$592$		0			0
$593$			7.03896i			0.289055i	0.989501	+	0.144528i	$0.0461663\pi$
$593$			7.03896i			0.289055i	−0.989501	+	0.144528i	$0.953834\pi$
$594$	22.3815	+	32.9249i	0.918322	+	1.35092i
$595$		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	$0.388889\pi$
$600$	6.23050	+	23.6893i	0.254359	+	0.967110i
$601$	−4.28921	−	24.3253i	−0.174960	−	0.992250i	−0.938190	−	0.346122i	$-0.887498\pi$
$601$	−4.28921	−	24.3253i	−0.174960	−	0.992250i	0.763229	−	0.646128i	$-0.223613\pi$
$602$		0			0
$603$	28.1747	−	15.9218i	1.14736	−	0.648387i
$604$		0			0
$605$		0			0
$606$		0			0
$607$		0			0		−0.766044	−	0.642788i	$-0.777778\pi$
$607$		0			0		0.766044	+	0.642788i	$0.222222\pi$
$608$	−17.3864	+	20.7203i	−0.705110	+	0.840317i
$609$		0			0
$610$		0			0
$611$		0			0
$612$	−1.40711	−	2.48997i	−0.0568790	−	0.100651i
$613$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$613$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$614$	40.4552	−	7.13334i	1.63264	−	0.287878i
$615$		0			0
$616$		0			0
$617$	−22.3237	−	3.93628i	−0.898720	−	0.158469i	−0.294843	−	0.955546i	$-0.595267\pi$
$617$	−22.3237	−	3.93628i	−0.898720	−	0.158469i	−0.603877	+	0.797077i	$0.706378\pi$
$618$		0			0
$619$	−7.60017	+	6.37730i	−0.305477	+	0.256325i	−0.782620	−	0.622500i	$-0.786117\pi$
$619$	−7.60017	+	6.37730i	−0.305477	+	0.256325i	0.477143	+	0.878826i	$0.341672\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$		0			0
$625$	4.34120	−	24.6202i	0.173648	−	0.984808i
$626$	−1.06974	−	2.93908i	−0.0427554	−	0.117469i
$627$	31.8729	−	31.5801i	1.27288	−	1.26119i
$628$		0			0
$629$		0			0
$630$		0			0
$631$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$631$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$632$		0			0
$633$	29.0454	−	41.0764i	1.15445	−	1.63264i
$634$		0			0
$635$		0			0
$636$		0			0
$637$		0			0
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−12.0281	+	2.12088i	−0.475082	+	0.0837698i	−0.406062	−	0.913845i	$-0.633098\pi$
$641$	−12.0281	+	2.12088i	−0.475082	+	0.0837698i	−0.0690201	+	0.997615i	$0.521987\pi$
$642$	−12.2681	+	44.9543i	−0.484182	+	1.77420i
$643$	−16.0112	+	5.82761i	−0.631421	+	0.229818i	−0.637850	−	0.770161i	$-0.720176\pi$
$643$	−16.0112	+	5.82761i	−0.631421	+	0.229818i	0.00642884	+	0.999979i	$0.497954\pi$
$644$		0			0
$645$		0			0
$646$	−2.46921	+	2.07192i	−0.0971499	+	0.0815184i
$647$		0			0			−	1.00000i	$-0.5\pi$
$647$		0			0				1.00000i	$0.5\pi$
$648$	−24.9833	−	4.88230i	−0.981435	−	0.191795i
$649$	83.1149			3.26255
$650$		0			0
$651$		0			0
$652$	8.00359	−	45.3906i	0.313445	−	1.77763i
$653$		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
$656$	29.7195	−	17.1586i	1.16035	−	0.669930i
$657$	−36.8129	+	6.14136i	−1.43621	+	0.239597i
$658$		0			0
$659$	−13.5543	+	37.2401i	−0.528000	+	1.45067i	0.333422	+	0.942778i	$0.391797\pi$
$659$	−13.5543	+	37.2401i	−0.528000	+	1.45067i	−0.861422	+	0.507891i	$0.830425\pi$
$660$		0			0
$661$		0			0		−0.766044	−	0.642788i	$-0.777778\pi$
$661$		0			0		0.766044	+	0.642788i	$0.222222\pi$
$662$	−31.8576	+	37.9665i	−1.23818	+	1.47561i
$663$		0			0
$664$	−37.6730	−	13.7119i	−1.46200	−	0.532124i
$665$		0			0
$666$		0			0
$667$		0			0
$668$		0			0
$669$		0			0
$670$		0			0
$671$		0			0
$672$		0			0
$673$	37.6057	−	31.5550i	1.44959	−	1.21635i	0.516720	−	0.856154i	$-0.327153\pi$
$673$	37.6057	−	31.5550i	1.44959	−	1.21635i	0.932875	−	0.360200i	$-0.117292\pi$
$674$			15.6869i			0.604237i
$675$	21.0739	+	15.1951i	0.811134	+	0.584861i
$676$	−26.0000			−1.00000
$677$		0			0		−0.642788	−	0.766044i	$-0.722222\pi$
$677$		0			0		0.642788	+	0.766044i	$0.277778\pi$
$678$	−2.00842	−	24.2443i	−0.0771328	−	0.931097i
$679$		0			0
$680$		0			0
$681$	−7.54108	−	28.6723i	−0.288975	−	1.09872i
$682$		0			0
$683$	40.4993	−	23.3823i	1.54966	−	0.894699i	0.551497	−	0.834177i	$-0.314057\pi$
$683$	40.4993	−	23.3823i	1.54966	−	0.894699i	0.998167	−	0.0605215i	$-0.0192764\pi$
$684$	−0.264767	+	28.6879i	−0.0101236	+	1.09691i
$685$		0			0
$686$		0			0
$687$		0			0
$688$	−40.1857	−	33.7198i	−1.53207	−	1.28556i
$689$		0			0
$690$		0			0
$691$	−0.897023	−	0.326490i	−0.0341244	−	0.0124203i	0.324902	−	0.945748i	$-0.394669\pi$
$691$	−0.897023	−	0.326490i	−0.0341244	−	0.0124203i	−0.359026	+	0.933328i	$0.616891\pi$
$692$		0			0
$693$		0			0
$694$	−22.9809	−	39.8042i	−0.872345	−	1.51095i
$695$		0			0
$696$		0			0
$697$	3.84291	−	1.39871i	0.145561	−	0.0529797i
$698$		0			0
$699$	−8.53558	+	0.707094i	−0.322845	+	0.0267447i
$700$		0			0
$701$		0			0			−	1.00000i	$-0.5\pi$
$701$		0			0				1.00000i	$0.5\pi$
$702$		0			0
$703$		0			0
$704$	−27.8594	−	33.2016i	−1.04999	−	1.25133i
$705$		0			0
$706$	8.99230	−	50.9979i	0.338430	−	1.91933i
$707$		0			0
$708$	−37.7516	+	37.4048i	−1.41879	+	1.40576i
$709$		0			0		−0.173648	−	0.984808i	$-0.555556\pi$
$709$		0			0		0.173648	+	0.984808i	$0.444444\pi$
$710$		0			0
$711$		0			0
$712$	−26.0454	+	45.1120i	−0.976093	+	1.69064i
$713$		0			0
$714$		0			0
$715$		0			0
$716$	−7.31359	+	8.71600i	−0.273322	+	0.325732i
$717$		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
$721$		0			0
$722$	5.38005	−	0.948648i	0.200225	−	0.0353050i
$723$	−5.56228	+	20.3820i	−0.206863	+	0.758015i
$724$		0			0
$725$		0			0
$726$	25.6130	+	36.9408i	0.950588	+	1.37100i
$727$		0			0		0.766044	−	0.642788i	$-0.222222\pi$
$727$		0			0		−0.766044	+	0.642788i	$0.777778\pi$
$728$		0			0
$729$	−23.7474	+	12.8475i	−0.879535	+	0.475834i
$730$		0			0
$731$	−4.01836	−	4.78889i	−0.148624	−	0.177124i
$732$		0			0
$733$		0			0		0.173648	−	0.984808i	$-0.444444\pi$
$733$		0			0		−0.173648	+	0.984808i	$0.555556\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	50.6132	−	29.2215i	1.86436	−	1.07639i
$738$	12.7643	−	34.0874i	0.469859	−	1.25477i
$739$	23.8429	−	41.2972i	0.877077	−	1.51914i	0.0225433	−	0.999746i	$-0.492824\pi$
$739$	23.8429	−	41.2972i	0.877077	−	1.51914i	0.854534	−	0.519396i	$-0.173843\pi$
$740$		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
$747$	−40.0908	+	14.1742i	−1.46685	+	0.518608i
$748$	−2.58249	−	4.47300i	−0.0944250	−	0.163549i
$749$		0			0
$750$		0			0
$751$		0			0		0.939693	−	0.342020i	$-0.111111\pi$
$751$		0			0		−0.939693	+	0.342020i	$0.888889\pi$
$752$		0			0
$753$	14.7676	−	31.2915i	0.538162	−	1.14032i
$754$		0			0
$755$		0			0
$756$		0			0
$757$		0			0		1.00000			$0$
$757$		0			0		−1.00000			$\pi$
$758$	35.0984	+	41.8287i	1.27483	+	1.51929i
$759$		0			0
$760$		0			0
$761$	3.86952	+	10.6314i	0.140270	+	0.385388i	0.989858	−	0.142058i	$-0.0453719\pi$
$761$	3.86952	+	10.6314i	0.140270	+	0.385388i	−0.849589	+	0.527446i	$0.823150\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$		0			0
$766$		0			0
$767$		0			0
$768$	27.5959	+	2.54270i	0.995782	+	0.0917517i
$769$	42.2020	+	35.4117i	1.52184	+	1.27698i	0.835134	+	0.550046i	$0.185390\pi$
$769$	42.2020	+	35.4117i	1.52184	+	1.27698i	0.686709	+	0.726932i	$0.259055\pi$
$770$		0			0
$771$	5.00734	−	54.3447i	0.180335	−	1.95718i
$772$	49.7614	+	18.1117i	1.79095	+	0.651854i
$773$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$773$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$774$	−55.6385	−	0.513499i	−1.99988	−	0.0184573i
$775$		0			0
$776$	51.3895	−	9.06136i	1.84478	−	0.325284i
$777$		0			0
$778$		0			0
$779$	−40.3988	−	7.12341i	−1.44744	−	0.255222i
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$	28.0000			1.00000
$785$		0			0
$786$	50.1952	+	23.6890i	1.79040	+	0.844958i
$787$	−8.16935	+	46.3307i	−0.291206	+	1.65151i	0.391030	+	0.920378i	$0.372119\pi$
$787$	−8.16935	+	46.3307i	−0.291206	+	1.65151i	−0.682236	+	0.731132i	$0.738992\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		0			0
$792$	−45.1967	−	8.40019i	−1.60599	−	0.298488i
$793$		0			0
$794$		0			0
$795$		0			0
$796$		0			0
$797$		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
$800$	−24.4949	−	14.1421i	−0.866025	−	0.500000i
$801$	9.09162	+	54.4975i	0.321236	+	1.92557i
$802$	27.7625	+	48.0860i	0.980327	+	1.69798i
$803$	−66.3752	+	11.7037i	−2.34233	+	0.413016i
$804$	−9.83822	+	36.0505i	−0.346967	+	1.27140i
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$			39.4773i			1.38795i	0.720000	+	0.693974i	$0.244142\pi$
$809$			39.4773i			1.38795i	−0.720000	+	0.693974i	$0.755858\pi$
$810$		0			0
$811$	−35.1832			−1.23545			−0.617725	−	0.786394i	$-0.711946\pi$
$811$	−35.1832			−1.23545			−0.617725	+	0.786394i	$0.711946\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$		0			0
$816$	3.18600	+	0.869463i	0.111532	+	0.0304373i
$817$	10.8892	+	61.7554i	0.380963	+	2.16055i
$818$	47.3608	−	27.3438i	1.65593	−	0.956052i
$819$		0			0
$820$		0			0
$821$		0			0		0.342020	−	0.939693i	$-0.388889\pi$
$821$		0			0		−0.342020	+	0.939693i	$0.611111\pi$
$822$	−10.0835	−	21.8872i	−0.351701	−	0.763403i
$823$		0			0		−0.766044	−	0.642788i	$-0.777778\pi$
$823$		0			0		0.766044	+	0.642788i	$0.222222\pi$
$824$		0			0
$825$	38.3089	+	27.0885i	1.33374	+	0.943100i
$826$		0			0
$827$	17.1464	+	9.89949i	0.596240	+	0.344239i	0.767561	−	0.640976i	$-0.221470\pi$
$827$	17.1464	+	9.89949i	0.596240	+	0.344239i	−0.171321	+	0.985215i	$0.554804\pi$
$828$		0			0
$829$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$829$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	3.28604	+	0.579418i	0.113855	+	0.0200756i
$834$	−4.74220	+	10.0484i	−0.164209	+	0.347946i
$835$		0			0
$836$			51.8097i			1.79187i
$837$		0			0
$838$	3.95459			0.136609
$839$		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
$843$	0.637196	+	2.42271i	0.0219462	+	0.0834425i
$844$	10.0874	+	57.2083i	0.347221	+	1.96919i
$845$		0			0
$846$		0			0
$847$		0			0
$848$		0			0
$849$	−19.0505	−	1.75532i	−0.653812	−	0.0602425i
$850$	−2.58204	−	2.16659i	−0.0885631	−	0.0743133i
$851$		0			0
$852$		0			0
$853$		0			0		−0.939693	−	0.342020i	$-0.888889\pi$
$853$		0			0		0.939693	+	0.342020i	$0.111111\pi$
$854$		0			0
$855$		0			0
$856$	−26.9035	−	46.5982i	−0.919542	−	1.59269i
$857$	−22.2837	+	3.92921i	−0.761195	+	0.134219i	−0.540754	−	0.841181i	$-0.681861\pi$
$857$	−22.2837	+	3.92921i	−0.761195	+	0.134219i	−0.220441	+	0.975400i	$0.570750\pi$
$858$		0			0
$859$	1.61178	−	0.586641i	0.0549933	−	0.0200159i	−0.314377	−	0.949298i	$-0.601796\pi$
$859$	1.61178	−	0.586641i	0.0549933	−	0.0200159i	0.369370	+	0.929282i	$0.379573\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		0			0			−	1.00000i	$-0.5\pi$
$863$		0			0				1.00000i	$0.5\pi$
$864$	24.3091	−	16.5247i	0.827014	−	0.562182i
$865$		0			0
$866$	−36.3781	−	43.3537i	−1.23618	−	1.47322i
$867$	−26.2725	−	12.3990i	−0.892261	−	0.421091i
$868$		0			0
$869$		0			0
$870$		0			0
$871$		0			0
$872$		0			0
$873$	35.9666	−	42.0686i	1.21728	−	1.42381i
$874$		0			0
$875$		0			0
$876$	24.8811	−	35.1872i	0.840655	−	1.18887i
$877$		0			0		−0.766044	−	0.642788i	$-0.777778\pi$
$877$		0			0		0.766044	+	0.642788i	$0.222222\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	37.9949	+	21.9364i	1.28008	+	0.739055i	0.976863	−	0.213866i	$-0.0686057\pi$
$881$	37.9949	+	21.9364i	1.28008	+	0.739055i	0.303218	+	0.952921i	$0.401939\pi$
$882$	22.9256	−	18.8790i	0.771944	−	0.635691i
$883$	18.3238	+	31.7377i	0.616644	+	1.06806i	0.990094	+	0.140408i	$0.0448416\pi$
$883$	18.3238	+	31.7377i	0.616644	+	1.06806i	−0.373450	+	0.927650i	$0.621825\pi$
$884$		0			0
$885$		0			0
$886$	15.5577	−	5.66255i	0.522672	−	0.190237i
$887$		0			0		−0.984808	−	0.173648i	$-0.944444\pi$
$887$		0			0		0.984808	+	0.173648i	$0.0555556\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$	−42.6695	+	23.5961i	−1.42948	+	0.790499i
$892$		0			0
$893$		0			0
$894$		0			0
$895$		0			0
$896$		0			0
$897$		0			0
$898$	10.3987	+	58.9740i	0.347009	+	1.96799i
$899$		0			0
$900$	−29.5911	+	4.93657i	−0.986368	+	0.164552i
$901$		0			0
$902$	22.4819	−	61.7684i	0.748564	−	2.05666i
$903$		0			0
$904$	21.5188	+	18.0564i	0.715705	+	0.600548i
$905$		0			0
$906$		0			0
$907$	−53.3352	−	19.4124i	−1.77097	−	0.644579i	−0.999970	−	0.00773827i	$-0.997537\pi$
$907$	−53.3352	−	19.4124i	−1.77097	−	0.644579i	−0.770996	−	0.636841i	$-0.780241\pi$
$908$	29.6474	+	17.1169i	0.983883	+	0.568045i
$909$		0			0
$910$		0			0
$911$		0			0		0.984808	−	0.173648i	$-0.0555556\pi$
$911$		0			0		−0.984808	+	0.173648i	$0.944444\pi$
$912$	−23.3162	−	23.5324i	−0.772078	−	0.779236i
$913$	−72.1606	+	26.2643i	−2.38817	+	0.869222i
$914$	50.1947	+	8.85068i	1.66029	+	0.292755i
$915$		0			0
$916$		0			0
$917$		0			0
$918$	3.19484	−	1.43628i	0.105445	−	0.0474042i
$919$		0			0		1.00000			$0$
$919$		0			0		−1.00000			$\pi$
$920$		0			0
$921$	4.15362	+	50.1398i	0.136866	+	1.65216i
$922$		0			0
$923$		0			0
$924$		0			0
$925$		0			0
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−9.67379	+	26.5785i	−0.317387	+	0.872013i	0.673725	+	0.738982i	$0.264693\pi$
$929$	−9.67379	+	26.5785i	−0.317387	+	0.872013i	−0.991112	+	0.133031i	$0.957529\pi$
$930$		0			0
$931$	−25.6400	−	21.5145i	−0.840317	−	0.705110i
$932$	6.35704	−	7.57603i	0.208232	−	0.248161i
$933$		0			0
$934$	25.2174	+	9.17838i	0.825138	+	0.300326i
$935$		0			0
$936$		0			0
$937$	−30.5454	−	52.9062i	−0.997875	−	1.72837i	−0.555366	−	0.831606i	$-0.687422\pi$
$937$	−30.5454	−	52.9062i	−0.997875	−	1.72837i	−0.442509	−	0.896764i	$-0.645912\pi$
$938$		0			0
$939$	3.70465	−	0.974359i	0.120897	−	0.0317970i
$940$		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
$944$		−	61.3655i		−	1.99728i
$945$		0			0
$946$	−100.482			−3.26694
$947$	−14.0666	−	16.7639i	−0.457102	−	0.544753i	0.487435	−	0.873160i	$-0.337933\pi$
$947$	−14.0666	−	16.7639i	−0.457102	−	0.544753i	−0.944536	+	0.328407i	$0.893488\pi$
$948$		0			0
$949$		0			0
$950$	11.5639	+	31.7714i	0.375181	+	1.03080i
$951$		0			0
$952$		0			0
$953$	29.0149	−	16.7518i	0.939884	−	0.542643i	0.0499603	−	0.998751i	$-0.484091\pi$
$953$	29.0149	−	16.7518i	0.939884	−	0.542643i	0.889924	+	0.456109i	$0.150757\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	−29.1305	−	10.6026i	−0.939693	−	0.342020i
$962$		0			0
$963$	−53.4467	−	20.0135i	−1.72230	−	0.644926i
$964$	−12.1979	−	21.1274i	−0.392868	−	0.680467i
$965$		0			0
$966$		0			0
$967$		0			0		0.939693	−	0.342020i	$-0.111111\pi$
$967$		0			0		−0.939693	+	0.342020i	$0.888889\pi$
$968$	−51.1171	−	9.01332i	−1.64297	−	0.289699i
$969$	−2.24939	−	3.24423i	−0.0722609	−	0.104220i
$970$		0			0
$971$			31.1127i			0.998454i	0.866471	+	0.499227i	$0.166383\pi$
$971$			31.1127i			0.998454i	−0.866471	+	0.499227i	$0.833617\pi$
$972$	8.76178	−	29.9204i	0.281034	−	0.959698i
$973$		0			0
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−19.2970	−	53.0180i	−0.617365	−	1.69620i	−0.713344	−	0.700814i	$-0.752820\pi$
$977$	−19.2970	−	53.0180i	−0.617365	−	1.69620i	0.0959785	−	0.995383i	$-0.469402\pi$
$978$	54.4580	+	14.8617i	1.74138	+	0.475223i
$979$	17.3261	+	98.2612i	0.553745	+	3.14044i
$980$		0			0
$981$		0			0
$982$	11.4294	−	19.7963i	0.364726	−	0.631724i
$983$		0			0		0.342020	−	0.939693i	$-0.388889\pi$
$983$		0			0		−0.342020	+	0.939693i	$0.611111\pi$
$984$	17.5866	+	38.1735i	0.560640	+	1.21693i
$985$		0			0
$986$		0			0
$987$		0			0
$988$		0			0
$989$		0			0
$990$		0			0
$991$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$991$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$992$		0			0
$993$	−42.7232	−	43.1193i	−1.35578	−	1.36835i
$994$		0			0
$995$		0			0
$996$	20.9561	−	44.4044i	0.664020	−	1.40701i
$997$		0			0		0.766044	−	0.642788i	$-0.222222\pi$
$997$		0			0		−0.766044	+	0.642788i	$0.777778\pi$
$998$		−	53.7342i		−	1.70093i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	216.2.v.a.59.1	yes	12
3.2	odd	2		648.2.v.a.611.2		12
4.3	odd	2		864.2.bh.a.815.1		12
8.3	odd	2	CM	216.2.v.a.59.1	yes	12
8.5	even	2		864.2.bh.a.815.1		12
24.11	even	2		648.2.v.a.611.2		12
27.11	odd	18	inner	216.2.v.a.11.1	✓	12
27.16	even	9		648.2.v.a.35.2		12
108.11	even	18		864.2.bh.a.335.1		12
216.11	even	18	inner	216.2.v.a.11.1	✓	12
216.43	odd	18		648.2.v.a.35.2		12
216.173	odd	18		864.2.bh.a.335.1		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
216.2.v.a.11.1	✓	12	27.11	odd	18	inner
216.2.v.a.11.1	✓	12	216.11	even	18	inner
216.2.v.a.59.1	yes	12	1.1	even	1	trivial
216.2.v.a.59.1	yes	12	8.3	odd	2	CM
648.2.v.a.35.2		12	27.16	even	9
648.2.v.a.35.2		12	216.43	odd	18
648.2.v.a.611.2		12	3.2	odd	2
648.2.v.a.611.2		12	24.11	even	2
864.2.bh.a.335.1		12	108.11	even	18
864.2.bh.a.335.1		12	216.173	odd	18
864.2.bh.a.815.1		12	4.3	odd	2
864.2.bh.a.815.1		12	8.5	even	2

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:	\( N \)	\(=\)	\( 216 = 2^{3} \cdot 3^{3} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	216.v (of order \(18\), degree \(6\), minimal)

\(f(q)\)	\(=\)	\(q+(-0.909039 - 1.08335i) q^{2} +(1.42338 - 0.986906i) q^{3} +(-0.347296 + 1.96962i) q^{4} +(-2.36307 - 0.644886i) q^{6} +(2.44949 - 1.41421i) q^{8} +(1.05203 - 2.80949i) q^{9} +O(q^{10})\)	\(q+(-0.909039 - 1.08335i) q^{2} +(1.42338 - 0.986906i) q^{3} +(-0.347296 + 1.96962i) q^{4} +(-2.36307 - 0.644886i) q^{6} +(2.44949 - 1.41421i) q^{8} +(1.05203 - 2.80949i) q^{9} +(1.85296 - 5.09097i) q^{11} +(1.44949 + 3.14626i) q^{12} +(-3.75877 - 1.36808i) q^{16} +(-0.412814 - 0.238338i) q^{17} +(-4.00000 + 1.41421i) q^{18} +(2.39076 + 4.14092i) q^{19} +(-7.19972 + 2.62048i) q^{22} +(2.09086 - 4.43038i) q^{24} +(-3.83022 + 3.21394i) q^{25} +(-1.27526 - 5.03723i) q^{27} +(1.93476 + 5.31570i) q^{32} +(-2.38684 - 9.07509i) q^{33} +(0.117060 + 0.663880i) q^{34} +(5.16824 + 3.04783i) q^{36} +(2.31277 - 6.35429i) q^{38} +(-5.51466 + 6.57212i) q^{41} +(12.3238 + 4.48548i) q^{43} +(9.38372 + 5.41770i) q^{44} +(-6.70033 + 1.76225i) q^{48} +(-6.57785 + 2.39414i) q^{49} +(6.96364 + 1.22788i) q^{50} +(-0.822809 + 0.0681621i) q^{51} +(-4.29783 + 5.96059i) q^{54} +(7.48967 + 3.53466i) q^{57} +(5.24706 + 14.4162i) q^{59} +(4.00000 - 6.92820i) q^{64} +(-7.66178 + 10.8354i) q^{66} +(8.26366 + 6.93404i) q^{67} +(0.612803 - 0.730310i) q^{68} +(-1.39627 - 8.36961i) q^{72} +(-6.22028 - 10.7738i) q^{73} +(-2.28002 + 8.35473i) q^{75} +(-8.98632 + 3.27075i) q^{76} +(-6.78645 - 5.91135i) q^{81} +12.1329 q^{82} +(-9.11103 - 10.8581i) q^{83} +(-6.34343 - 17.4284i) q^{86} +(-2.66091 - 15.0908i) q^{88} +(-15.9495 + 9.20844i) q^{89} +(8.00000 + 5.65685i) q^{96} +(17.3366 + 6.31001i) q^{97} +(8.57321 + 4.94975i) q^{98} +(-12.3536 - 10.5617i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q+O(q^{10}) \) 12 * q	\( 12 q + 18 q^{11} - 12 q^{12} - 48 q^{18} + 12 q^{22} - 30 q^{27} + 6 q^{33} - 24 q^{34} + 72 q^{38} - 18 q^{41} + 30 q^{43} - 12 q^{51} + 42 q^{57} + 36 q^{59} + 48 q^{64} - 42 q^{67} - 72 q^{68} - 24 q^{76} + 36 q^{86} + 48 q^{88} - 162 q^{89} + 96 q^{96} + 30 q^{97}+O(q^{100}) \) 12 * q + 18 * q^11 - 12 * q^12 - 48 * q^18 + 12 * q^22 - 30 * q^27 + 6 * q^33 - 24 * q^34 + 72 * q^38 - 18 * q^41 + 30 * q^43 - 12 * q^51 + 42 * q^57 + 36 * q^59 + 48 * q^64 - 42 * q^67 - 72 * q^68 - 24 * q^76 + 36 * q^86 + 48 * q^88 - 162 * q^89 + 96 * q^96 + 30 * q^97

Introduction

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