LMFDB - Embedded newform 648.2.n.c.541.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [648,2,Mod(109,648)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(648, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 3, 2]))

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

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

chi := DirichletCharacter("648.109");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 648 = 2^{3} \cdot 3^{4} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	648.n (of order $6$, degree $2$, 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:	$5.17430605098$
Analytic rank:	$0$
Dimension:	$4$
Relative dimension:	$2$ over $\Q(\zeta_{6})$
Coefficient field:	$\Q(\zeta_{12})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{2} + 1 $ x^4 - x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{25}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 24)
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			541.2
Root			$-0.866025 + 0.500000i$ of defining polynomial
Character	$\chi$	$=$	648.541
Dual form			648.2.n.c.109.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(0.366025 + 1.36603i) q^{2} +(-1.73205 + 1.00000i) q^{4} +(1.73205 - 1.00000i) q^{5} +(1.00000 - 1.73205i) q^{7} +(-2.00000 - 2.00000i) q^{8} +O(q^{10})$	\(q+(0.366025 + 1.36603i) q^{2} +(-1.73205 + 1.00000i) q^{4} +(1.73205 - 1.00000i) q^{5} +(1.00000 - 1.73205i) q^{7} +(-2.00000 - 2.00000i) q^{8} +(2.00000 + 2.00000i) q^{10} +(3.46410 - 2.00000i) q^{13} +(2.73205 + 0.732051i) q^{14} +(2.00000 - 3.46410i) q^{16} +2.00000 q^{17} -4.00000i q^{19} +(-2.00000 + 3.46410i) q^{20} +(2.00000 + 3.46410i) q^{23} +(-0.500000 + 0.866025i) q^{25} +(4.00000 + 4.00000i) q^{26} +4.00000i q^{28} +(5.19615 + 3.00000i) q^{29} +(-1.00000 - 1.73205i) q^{31} +(5.46410 + 1.46410i) q^{32} +(0.732051 + 2.73205i) q^{34} -4.00000i q^{35} -8.00000i q^{37} +(5.46410 - 1.46410i) q^{38} +(-5.46410 - 1.46410i) q^{40} +(1.00000 + 1.73205i) q^{41} +(-3.46410 - 2.00000i) q^{43} +(-4.00000 + 4.00000i) q^{46} +(-6.00000 + 10.3923i) q^{47} +(1.50000 + 2.59808i) q^{49} +(-1.36603 - 0.366025i) q^{50} +(-4.00000 + 6.92820i) q^{52} +6.00000i q^{53} +(-5.46410 + 1.46410i) q^{56} +(-2.19615 + 8.19615i) q^{58} +(3.46410 - 2.00000i) q^{59} +(2.00000 - 2.00000i) q^{62} +8.00000i q^{64} +(4.00000 - 6.92820i) q^{65} +(10.3923 - 6.00000i) q^{67} +(-3.46410 + 2.00000i) q^{68} +(5.46410 - 1.46410i) q^{70} -12.0000 q^{71} -6.00000 q^{73} +(10.9282 - 2.92820i) q^{74} +(4.00000 + 6.92820i) q^{76} +(-5.00000 + 8.66025i) q^{79} -8.00000i q^{80} +(-2.00000 + 2.00000i) q^{82} +(-13.8564 - 8.00000i) q^{83} +(3.46410 - 2.00000i) q^{85} +(1.46410 - 5.46410i) q^{86} +10.0000 q^{89} -8.00000i q^{91} +(-6.92820 - 4.00000i) q^{92} +(-16.3923 - 4.39230i) q^{94} +(-4.00000 - 6.92820i) q^{95} +(1.00000 - 1.73205i) q^{97} +(-3.00000 + 3.00000i) q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 2 q^{2} + 4 q^{7} - 8 q^{8}+O(q^{10}) $ 4 * q - 2 * q^2 + 4 * q^7 - 8 * q^8	$ 4 q - 2 q^{2} + 4 q^{7} - 8 q^{8} + 8 q^{10} + 4 q^{14} + 8 q^{16} + 8 q^{17} - 8 q^{20} + 8 q^{23} - 2 q^{25} + 16 q^{26} - 4 q^{31} + 8 q^{32} - 4 q^{34} + 8 q^{38} - 8 q^{40} + 4 q^{41} - 16 q^{46} - 24 q^{47} + 6 q^{49} - 2 q^{50} - 16 q^{52} - 8 q^{56} + 12 q^{58} + 8 q^{62} + 16 q^{65} + 8 q^{70} - 48 q^{71} - 24 q^{73} + 16 q^{74} + 16 q^{76} - 20 q^{79} - 8 q^{82} - 8 q^{86} + 40 q^{89} - 24 q^{94} - 16 q^{95} + 4 q^{97} - 12 q^{98}+O(q^{100}) $ 4 * q - 2 * q^2 + 4 * q^7 - 8 * q^8 + 8 * q^10 + 4 * q^14 + 8 * q^16 + 8 * q^17 - 8 * q^20 + 8 * q^23 - 2 * q^25 + 16 * q^26 - 4 * q^31 + 8 * q^32 - 4 * q^34 + 8 * q^38 - 8 * q^40 + 4 * q^41 - 16 * q^46 - 24 * q^47 + 6 * q^49 - 2 * q^50 - 16 * q^52 - 8 * q^56 + 12 * q^58 + 8 * q^62 + 16 * q^65 + 8 * q^70 - 48 * q^71 - 24 * q^73 + 16 * q^74 + 16 * q^76 - 20 * q^79 - 8 * q^82 - 8 * q^86 + 40 * q^89 - 24 * q^94 - 16 * q^95 + 4 * q^97 - 12 * q^98

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/648\mathbb{Z}\right)^\times$.

$n$	$325$	$487$	$569$
$\chi(n)$	$-1$	$1$	$e\left(\frac{2}{3}\right)$

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 $

$2$	0.366025	+	1.36603i	0.258819	+	0.965926i
$2$	0.366025	+	1.36603i	0.258819	+	0.965926i
$3$		0			0
$3$		0			0
$4$	−1.73205	+	1.00000i	−0.866025	+	0.500000i
$5$	1.73205	−	1.00000i	0.774597	−	0.447214i	−0.0599153	−	0.998203i	$-0.519083\pi$
$5$	1.73205	−	1.00000i	0.774597	−	0.447214i	0.834512	+	0.550990i	$0.185750\pi$
$6$		0			0
$7$	1.00000	−	1.73205i	0.377964	−	0.654654i	−0.612801	−	0.790237i	$-0.709957\pi$
$7$	1.00000	−	1.73205i	0.377964	−	0.654654i	0.990766	+	0.135583i	$0.0432908\pi$
$8$	−2.00000	−	2.00000i	−0.707107	−	0.707107i
$9$		0			0
$10$	2.00000	+	2.00000i	0.632456	+	0.632456i
$11$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$11$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$12$		0			0
$13$	3.46410	−	2.00000i	0.960769	−	0.554700i	0.0643593	−	0.997927i	$-0.479500\pi$
$13$	3.46410	−	2.00000i	0.960769	−	0.554700i	0.896410	+	0.443227i	$0.146166\pi$
$14$	2.73205	+	0.732051i	0.730171	+	0.195649i
$15$		0			0
$16$	2.00000	−	3.46410i	0.500000	−	0.866025i
$17$	2.00000			0.485071			0.242536	−	0.970143i	$-0.422021\pi$
$17$	2.00000			0.485071			0.242536	+	0.970143i	$0.422021\pi$
$18$		0			0
$19$		−	4.00000i		−	0.917663i	−0.888523	−	0.458831i	$-0.848268\pi$
$19$		−	4.00000i		−	0.917663i	0.888523	−	0.458831i	$-0.151732\pi$
$20$	−2.00000	+	3.46410i	−0.447214	+	0.774597i
$21$		0			0
$22$		0			0
$23$	2.00000	+	3.46410i	0.417029	+	0.722315i	0.995639	−	0.0932891i	$-0.0297381\pi$
$23$	2.00000	+	3.46410i	0.417029	+	0.722315i	−0.578610	+	0.815604i	$0.696405\pi$
$24$		0			0
$25$	−0.500000	+	0.866025i	−0.100000	+	0.173205i
$26$	4.00000	+	4.00000i	0.784465	+	0.784465i
$27$		0			0
$28$			4.00000i			0.755929i
$29$	5.19615	+	3.00000i	0.964901	+	0.557086i	0.897678	−	0.440652i	$-0.145253\pi$
$29$	5.19615	+	3.00000i	0.964901	+	0.557086i	0.0672232	+	0.997738i	$0.478586\pi$
$30$		0			0
$31$	−1.00000	−	1.73205i	−0.179605	−	0.311086i	0.762140	−	0.647412i	$-0.224149\pi$
$31$	−1.00000	−	1.73205i	−0.179605	−	0.311086i	−0.941745	+	0.336327i	$0.890815\pi$
$32$	5.46410	+	1.46410i	0.965926	+	0.258819i
$33$		0			0
$34$	0.732051	+	2.73205i	0.125546	+	0.468543i
$35$		−	4.00000i		−	0.676123i
$36$		0			0
$37$		−	8.00000i		−	1.31519i	−0.753371	−	0.657596i	$-0.771573\pi$
$37$		−	8.00000i		−	1.31519i	0.753371	−	0.657596i	$-0.228427\pi$
$38$	5.46410	−	1.46410i	0.886394	−	0.237509i
$39$		0			0
$40$	−5.46410	−	1.46410i	−0.863950	−	0.231495i
$41$	1.00000	+	1.73205i	0.156174	+	0.270501i	0.933486	−	0.358614i	$-0.116751\pi$
$41$	1.00000	+	1.73205i	0.156174	+	0.270501i	−0.777312	+	0.629115i	$0.783417\pi$
$42$		0			0
$43$	−3.46410	−	2.00000i	−0.528271	−	0.304997i	0.212041	−	0.977261i	$-0.431989\pi$
$43$	−3.46410	−	2.00000i	−0.528271	−	0.304997i	−0.740312	+	0.672264i	$0.765322\pi$
$44$		0			0
$45$		0			0
$46$	−4.00000	+	4.00000i	−0.589768	+	0.589768i
$47$	−6.00000	+	10.3923i	−0.875190	+	1.51587i	−0.0186297	+	0.999826i	$0.505930\pi$
$47$	−6.00000	+	10.3923i	−0.875190	+	1.51587i	−0.856560	+	0.516047i	$0.827403\pi$
$48$		0			0
$49$	1.50000	+	2.59808i	0.214286	+	0.371154i
$50$	−1.36603	−	0.366025i	−0.193185	−	0.0517638i
$51$		0			0
$52$	−4.00000	+	6.92820i	−0.554700	+	0.960769i
$53$			6.00000i			0.824163i	0.911147	+	0.412082i	$0.135198\pi$
$53$			6.00000i			0.824163i	−0.911147	+	0.412082i	$0.864802\pi$
$54$		0			0
$55$		0			0
$56$	−5.46410	+	1.46410i	−0.730171	+	0.195649i
$57$		0			0
$58$	−2.19615	+	8.19615i	−0.288369	+	1.07621i
$59$	3.46410	−	2.00000i	0.450988	−	0.260378i	−0.257260	−	0.966342i	$-0.582820\pi$
$59$	3.46410	−	2.00000i	0.450988	−	0.260378i	0.708247	+	0.705965i	$0.249486\pi$
$60$		0			0
$61$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$61$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$62$	2.00000	−	2.00000i	0.254000	−	0.254000i
$63$		0			0
$64$			8.00000i			1.00000i
$65$	4.00000	−	6.92820i	0.496139	−	0.859338i
$66$		0			0
$67$	10.3923	−	6.00000i	1.26962	−	0.733017i	0.294706	−	0.955588i	$-0.404778\pi$
$67$	10.3923	−	6.00000i	1.26962	−	0.733017i	0.974916	+	0.222571i	$0.0714450\pi$
$68$	−3.46410	+	2.00000i	−0.420084	+	0.242536i
$69$		0			0
$70$	5.46410	−	1.46410i	0.653085	−	0.174994i
$71$	−12.0000			−1.42414			−0.712069	−	0.702109i	$-0.752242\pi$
$71$	−12.0000			−1.42414			−0.712069	+	0.702109i	$0.752242\pi$
$72$		0			0
$73$	−6.00000			−0.702247			−0.351123	−	0.936329i	$-0.614200\pi$
$73$	−6.00000			−0.702247			−0.351123	+	0.936329i	$0.614200\pi$
$74$	10.9282	−	2.92820i	1.27038	−	0.340397i
$75$		0			0
$76$	4.00000	+	6.92820i	0.458831	+	0.794719i
$77$		0			0
$78$		0			0
$79$	−5.00000	+	8.66025i	−0.562544	+	0.974355i	0.434730	+	0.900561i	$0.356844\pi$
$79$	−5.00000	+	8.66025i	−0.562544	+	0.974355i	−0.997274	+	0.0737937i	$0.976489\pi$
$80$		−	8.00000i		−	0.894427i
$81$		0			0
$82$	−2.00000	+	2.00000i	−0.220863	+	0.220863i
$83$	−13.8564	−	8.00000i	−1.52094	−	0.878114i	−0.999695	−	0.0247060i	$-0.992135\pi$
$83$	−13.8564	−	8.00000i	−1.52094	−	0.878114i	−0.521243	−	0.853408i	$-0.674532\pi$
$84$		0			0
$85$	3.46410	−	2.00000i	0.375735	−	0.216930i
$86$	1.46410	−	5.46410i	0.157878	−	0.589209i
$87$		0			0
$88$		0			0
$89$	10.0000			1.06000			0.529999	−	0.847998i	$-0.322192\pi$
$89$	10.0000			1.06000			0.529999	+	0.847998i	$0.322192\pi$
$90$		0			0
$91$		−	8.00000i		−	0.838628i
$92$	−6.92820	−	4.00000i	−0.722315	−	0.417029i
$93$		0			0
$94$	−16.3923	−	4.39230i	−1.69074	−	0.453032i
$95$	−4.00000	−	6.92820i	−0.410391	−	0.710819i
$96$		0			0
$97$	1.00000	−	1.73205i	0.101535	−	0.175863i	−0.810782	−	0.585348i	$-0.800958\pi$
$97$	1.00000	−	1.73205i	0.101535	−	0.175863i	0.912317	+	0.409484i	$0.134291\pi$
$98$	−3.00000	+	3.00000i	−0.303046	+	0.303046i
$99$		0			0
$100$		−	2.00000i		−	0.200000i
$101$	8.66025	+	5.00000i	0.861727	+	0.497519i	0.864590	−	0.502477i	$-0.167578\pi$
$101$	8.66025	+	5.00000i	0.861727	+	0.497519i	−0.00286291	+	0.999996i	$0.500911\pi$
$102$		0			0
$103$	3.00000	+	5.19615i	0.295599	+	0.511992i	0.975124	−	0.221660i	$-0.0711475\pi$
$103$	3.00000	+	5.19615i	0.295599	+	0.511992i	−0.679525	+	0.733652i	$0.737814\pi$
$104$	−10.9282	−	2.92820i	−1.07160	−	0.287134i
$105$		0			0
$106$	−8.19615	+	2.19615i	−0.796081	+	0.213309i
$107$		−	12.0000i		−	1.16008i	−0.814587	−	0.580042i	$-0.803036\pi$
$107$		−	12.0000i		−	1.16008i	0.814587	−	0.580042i	$-0.196964\pi$
$108$		0			0
$109$		−	4.00000i		−	0.383131i	−0.981480	−	0.191565i	$-0.938644\pi$
$109$		−	4.00000i		−	0.383131i	0.981480	−	0.191565i	$-0.0613564\pi$
$110$		0			0
$111$		0			0
$112$	−4.00000	−	6.92820i	−0.377964	−	0.654654i
$113$	−3.00000	−	5.19615i	−0.282216	−	0.488813i	0.689714	−	0.724082i	$-0.257736\pi$
$113$	−3.00000	−	5.19615i	−0.282216	−	0.488813i	−0.971930	+	0.235269i	$0.924403\pi$
$114$		0			0
$115$	6.92820	+	4.00000i	0.646058	+	0.373002i
$116$	−12.0000			−1.11417
$117$		0			0
$118$	4.00000	+	4.00000i	0.368230	+	0.368230i
$119$	2.00000	−	3.46410i	0.183340	−	0.317554i
$120$		0			0
$121$	−5.50000	−	9.52628i	−0.500000	−	0.866025i
$122$		0			0
$123$		0			0
$124$	3.46410	+	2.00000i	0.311086	+	0.179605i
$125$			12.0000i			1.07331i
$126$		0			0
$127$	−2.00000			−0.177471			−0.0887357	−	0.996055i	$-0.528283\pi$
$127$	−2.00000			−0.177471			−0.0887357	+	0.996055i	$0.528283\pi$
$128$	−10.9282	+	2.92820i	−0.965926	+	0.258819i
$129$		0			0
$130$	10.9282	+	2.92820i	0.958467	+	0.256820i
$131$	−17.3205	+	10.0000i	−1.51330	+	0.873704i	−0.513421	+	0.858137i	$0.671622\pi$
$131$	−17.3205	+	10.0000i	−1.51330	+	0.873704i	−0.999879	+	0.0155672i	$0.995045\pi$
$132$		0			0
$133$	−6.92820	−	4.00000i	−0.600751	−	0.346844i
$134$	12.0000	+	12.0000i	1.03664	+	1.03664i
$135$		0			0
$136$	−4.00000	−	4.00000i	−0.342997	−	0.342997i
$137$	9.00000	−	15.5885i	0.768922	−	1.33181i	−0.169226	−	0.985577i	$-0.554127\pi$
$137$	9.00000	−	15.5885i	0.768922	−	1.33181i	0.938148	−	0.346235i	$-0.112540\pi$
$138$		0			0
$139$	−3.46410	+	2.00000i	−0.293821	+	0.169638i	−0.639664	−	0.768655i	$-0.720926\pi$
$139$	−3.46410	+	2.00000i	−0.293821	+	0.169638i	0.345843	+	0.938293i	$0.387593\pi$
$140$	4.00000	+	6.92820i	0.338062	+	0.585540i
$141$		0			0
$142$	−4.39230	−	16.3923i	−0.368594	−	1.37561i
$143$		0			0
$144$		0			0
$145$	12.0000			0.996546
$146$	−2.19615	−	8.19615i	−0.181755	−	0.678318i
$147$		0			0
$148$	8.00000	+	13.8564i	0.657596	+	1.13899i
$149$	−5.19615	+	3.00000i	−0.425685	+	0.245770i	−0.697507	−	0.716578i	$-0.745707\pi$
$149$	−5.19615	+	3.00000i	−0.425685	+	0.245770i	0.271821	+	0.962348i	$0.412374\pi$
$150$		0			0
$151$	9.00000	−	15.5885i	0.732410	−	1.26857i	−0.223441	−	0.974717i	$-0.571729\pi$
$151$	9.00000	−	15.5885i	0.732410	−	1.26857i	0.955851	−	0.293853i	$-0.0949377\pi$
$152$	−8.00000	+	8.00000i	−0.648886	+	0.648886i
$153$		0			0
$154$		0			0
$155$	−3.46410	−	2.00000i	−0.278243	−	0.160644i
$156$		0			0
$157$	−6.92820	+	4.00000i	−0.552931	+	0.319235i	−0.750303	−	0.661094i	$-0.770093\pi$
$157$	−6.92820	+	4.00000i	−0.552931	+	0.319235i	0.197372	+	0.980329i	$0.436759\pi$
$158$	−13.6603	−	3.66025i	−1.08675	−	0.291194i
$159$		0			0
$160$	10.9282	−	2.92820i	0.863950	−	0.231495i
$161$	8.00000			0.630488
$162$		0			0
$163$			4.00000i			0.313304i	0.987654	+	0.156652i	$0.0500701\pi$
$163$			4.00000i			0.313304i	−0.987654	+	0.156652i	$0.949930\pi$
$164$	−3.46410	−	2.00000i	−0.270501	−	0.156174i
$165$		0			0
$166$	5.85641	−	21.8564i	0.454545	−	1.69639i
$167$	4.00000	+	6.92820i	0.309529	+	0.536120i	0.978259	−	0.207385i	$-0.0664952\pi$
$167$	4.00000	+	6.92820i	0.309529	+	0.536120i	−0.668730	+	0.743505i	$0.733162\pi$
$168$		0			0
$169$	1.50000	−	2.59808i	0.115385	−	0.199852i
$170$	4.00000	+	4.00000i	0.306786	+	0.306786i
$171$		0			0
$172$	8.00000			0.609994
$173$	−5.19615	−	3.00000i	−0.395056	−	0.228086i	0.289292	−	0.957241i	$-0.406580\pi$
$173$	−5.19615	−	3.00000i	−0.395056	−	0.228086i	−0.684349	+	0.729155i	$0.739913\pi$
$174$		0			0
$175$	1.00000	+	1.73205i	0.0755929	+	0.130931i
$176$		0			0
$177$		0			0
$178$	3.66025	+	13.6603i	0.274348	+	1.02388i
$179$			4.00000i			0.298974i	0.988764	+	0.149487i	$0.0477622\pi$
$179$			4.00000i			0.298974i	−0.988764	+	0.149487i	$0.952238\pi$
$180$		0			0
$181$			20.0000i			1.48659i	0.668965	+	0.743294i	$0.266738\pi$
$181$			20.0000i			1.48659i	−0.668965	+	0.743294i	$0.733262\pi$
$182$	10.9282	−	2.92820i	0.810052	−	0.217053i
$183$		0			0
$184$	2.92820	−	10.9282i	0.215870	−	0.805638i
$185$	−8.00000	−	13.8564i	−0.588172	−	1.01874i
$186$		0			0
$187$		0			0
$188$		−	24.0000i		−	1.75038i
$189$		0			0
$190$	8.00000	−	8.00000i	0.580381	−	0.580381i
$191$	−4.00000	+	6.92820i	−0.289430	+	0.501307i	−0.973674	−	0.227946i	$-0.926799\pi$
$191$	−4.00000	+	6.92820i	−0.289430	+	0.501307i	0.684244	+	0.729253i	$0.260132\pi$
$192$		0			0
$193$	3.00000	+	5.19615i	0.215945	+	0.374027i	0.953564	−	0.301189i	$-0.0973836\pi$
$193$	3.00000	+	5.19615i	0.215945	+	0.374027i	−0.737620	+	0.675216i	$0.764050\pi$
$194$	2.73205	+	0.732051i	0.196150	+	0.0525582i
$195$		0			0
$196$	−5.19615	−	3.00000i	−0.371154	−	0.214286i
$197$		−	2.00000i		−	0.142494i	−0.997459	−	0.0712470i	$-0.977302\pi$
$197$		−	2.00000i		−	0.142494i	0.997459	−	0.0712470i	$-0.0226979\pi$
$198$		0			0
$199$	10.0000			0.708881			0.354441	−	0.935079i	$-0.384671\pi$
$199$	10.0000			0.708881			0.354441	+	0.935079i	$0.384671\pi$
$200$	2.73205	−	0.732051i	0.193185	−	0.0517638i
$201$		0			0
$202$	−3.66025	+	13.6603i	−0.257535	+	0.961132i
$203$	10.3923	−	6.00000i	0.729397	−	0.421117i
$204$		0			0
$205$	3.46410	+	2.00000i	0.241943	+	0.139686i
$206$	−6.00000	+	6.00000i	−0.418040	+	0.418040i
$207$		0			0
$208$		−	16.0000i		−	1.10940i
$209$		0			0
$210$		0			0
$211$	−17.3205	+	10.0000i	−1.19239	+	0.688428i	−0.958849	−	0.283918i	$-0.908366\pi$
$211$	−17.3205	+	10.0000i	−1.19239	+	0.688428i	−0.233544	+	0.972346i	$0.575032\pi$
$212$	−6.00000	−	10.3923i	−0.412082	−	0.713746i
$213$		0			0
$214$	16.3923	−	4.39230i	1.12055	−	0.300252i
$215$	−8.00000			−0.545595
$216$		0			0
$217$	−4.00000			−0.271538
$218$	5.46410	−	1.46410i	0.370076	−	0.0991615i
$219$		0			0
$220$		0			0
$221$	6.92820	−	4.00000i	0.466041	−	0.269069i
$222$		0			0
$223$	−7.00000	+	12.1244i	−0.468755	+	0.811907i	−0.999362	−	0.0357107i	$-0.988630\pi$
$223$	−7.00000	+	12.1244i	−0.468755	+	0.811907i	0.530607	+	0.847618i	$0.321964\pi$
$224$	8.00000	−	8.00000i	0.534522	−	0.534522i
$225$		0			0
$226$	6.00000	−	6.00000i	0.399114	−	0.399114i
$227$	−6.92820	−	4.00000i	−0.459841	−	0.265489i	0.252136	−	0.967692i	$-0.418867\pi$
$227$	−6.92820	−	4.00000i	−0.459841	−	0.265489i	−0.711977	+	0.702202i	$0.752200\pi$
$228$		0			0
$229$	−3.46410	+	2.00000i	−0.228914	+	0.132164i	−0.610071	−	0.792347i	$-0.708859\pi$
$229$	−3.46410	+	2.00000i	−0.228914	+	0.132164i	0.381157	+	0.924510i	$0.375526\pi$
$230$	−2.92820	+	10.9282i	−0.193080	+	0.720584i
$231$		0			0
$232$	−4.39230	−	16.3923i	−0.288369	−	1.07621i
$233$	−14.0000			−0.917170			−0.458585	−	0.888650i	$-0.651644\pi$
$233$	−14.0000			−0.917170			−0.458585	+	0.888650i	$0.651644\pi$
$234$		0			0
$235$			24.0000i			1.56559i
$236$	−4.00000	+	6.92820i	−0.260378	+	0.450988i
$237$		0			0
$238$	5.46410	+	1.46410i	0.354185	+	0.0949036i
$239$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$239$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$240$		0			0
$241$	−1.00000	+	1.73205i	−0.0644157	+	0.111571i	−0.896435	−	0.443176i	$-0.853852\pi$
$241$	−1.00000	+	1.73205i	−0.0644157	+	0.111571i	0.832019	+	0.554747i	$0.187185\pi$
$242$	11.0000	−	11.0000i	0.707107	−	0.707107i
$243$		0			0
$244$		0			0
$245$	5.19615	+	3.00000i	0.331970	+	0.191663i
$246$		0			0
$247$	−8.00000	−	13.8564i	−0.509028	−	0.881662i
$248$	−1.46410	+	5.46410i	−0.0929705	+	0.346971i
$249$		0			0
$250$	−16.3923	+	4.39230i	−1.03674	+	0.277794i
$251$		0			0		1.00000			$0$
$251$		0			0		−1.00000			$\pi$
$252$		0			0
$253$		0			0
$254$	−0.732051	−	2.73205i	−0.0459330	−	0.171424i
$255$		0			0
$256$	−8.00000	−	13.8564i	−0.500000	−	0.866025i
$257$	9.00000	+	15.5885i	0.561405	+	0.972381i	0.997374	+	0.0724199i	$0.0230722\pi$
$257$	9.00000	+	15.5885i	0.561405	+	0.972381i	−0.435970	+	0.899961i	$0.643595\pi$
$258$		0			0
$259$	−13.8564	−	8.00000i	−0.860995	−	0.497096i
$260$			16.0000i			0.992278i
$261$		0			0
$262$	−20.0000	−	20.0000i	−1.23560	−	1.23560i
$263$	−8.00000	+	13.8564i	−0.493301	+	0.854423i	−0.999970	−	0.00771799i	$-0.997543\pi$
$263$	−8.00000	+	13.8564i	−0.493301	+	0.854423i	0.506669	+	0.862141i	$0.330877\pi$
$264$		0			0
$265$	6.00000	+	10.3923i	0.368577	+	0.638394i
$266$	2.92820	−	10.9282i	0.179540	−	0.670051i
$267$		0			0
$268$	−12.0000	+	20.7846i	−0.733017	+	1.26962i
$269$		−	6.00000i		−	0.365826i	−0.983129	−	0.182913i	$-0.941447\pi$
$269$		−	6.00000i		−	0.365826i	0.983129	−	0.182913i	$-0.0585527\pi$
$270$		0			0
$271$	−18.0000			−1.09342			−0.546711	−	0.837321i	$-0.684120\pi$
$271$	−18.0000			−1.09342			−0.546711	+	0.837321i	$0.684120\pi$
$272$	4.00000	−	6.92820i	0.242536	−	0.420084i
$273$		0			0
$274$	24.5885	+	6.58846i	1.48544	+	0.398023i
$275$		0			0
$276$		0			0
$277$	24.2487	+	14.0000i	1.45696	+	0.841178i	0.998861	−	0.0477206i	$-0.0151957\pi$
$277$	24.2487	+	14.0000i	1.45696	+	0.841178i	0.458103	+	0.888899i	$0.348529\pi$
$278$	−4.00000	−	4.00000i	−0.239904	−	0.239904i
$279$		0			0
$280$	−8.00000	+	8.00000i	−0.478091	+	0.478091i
$281$	−9.00000	+	15.5885i	−0.536895	+	0.929929i	0.462174	+	0.886789i	$0.347070\pi$
$281$	−9.00000	+	15.5885i	−0.536895	+	0.929929i	−0.999069	+	0.0431402i	$0.986264\pi$
$282$		0			0
$283$	3.46410	−	2.00000i	0.205919	−	0.118888i	−0.393494	−	0.919327i	$-0.628734\pi$
$283$	3.46410	−	2.00000i	0.205919	−	0.118888i	0.599414	+	0.800439i	$0.295400\pi$
$284$	20.7846	−	12.0000i	1.23334	−	0.712069i
$285$		0			0
$286$		0			0
$287$	4.00000			0.236113
$288$		0			0
$289$	−13.0000			−0.764706
$290$	4.39230	+	16.3923i	0.257925	+	0.962589i
$291$		0			0
$292$	10.3923	−	6.00000i	0.608164	−	0.351123i
$293$	5.19615	−	3.00000i	0.303562	−	0.175262i	−0.340480	−	0.940252i	$-0.610589\pi$
$293$	5.19615	−	3.00000i	0.303562	−	0.175262i	0.644042	+	0.764990i	$0.277256\pi$
$294$		0			0
$295$	4.00000	−	6.92820i	0.232889	−	0.403376i
$296$	−16.0000	+	16.0000i	−0.929981	+	0.929981i
$297$		0			0
$298$	−6.00000	−	6.00000i	−0.347571	−	0.347571i
$299$	13.8564	+	8.00000i	0.801337	+	0.462652i
$300$		0			0
$301$	−6.92820	+	4.00000i	−0.399335	+	0.230556i
$302$	24.5885	+	6.58846i	1.41491	+	0.379123i
$303$		0			0
$304$	−13.8564	−	8.00000i	−0.794719	−	0.458831i
$305$		0			0
$306$		0			0
$307$			12.0000i			0.684876i	0.939540	+	0.342438i	$0.111253\pi$
$307$			12.0000i			0.684876i	−0.939540	+	0.342438i	$0.888747\pi$
$308$		0			0
$309$		0			0
$310$	1.46410	−	5.46410i	0.0831554	−	0.310340i
$311$	−4.00000	−	6.92820i	−0.226819	−	0.392862i	0.730044	−	0.683400i	$-0.239499\pi$
$311$	−4.00000	−	6.92820i	−0.226819	−	0.392862i	−0.956864	+	0.290537i	$0.906166\pi$
$312$		0			0
$313$	−7.00000	+	12.1244i	−0.395663	+	0.685309i	−0.993186	−	0.116543i	$-0.962819\pi$
$313$	−7.00000	+	12.1244i	−0.395663	+	0.685309i	0.597522	+	0.801852i	$0.296152\pi$
$314$	−8.00000	−	8.00000i	−0.451466	−	0.451466i
$315$		0			0
$316$		−	20.0000i		−	1.12509i
$317$	19.0526	+	11.0000i	1.07010	+	0.617822i	0.928208	−	0.372061i	$-0.121349\pi$
$317$	19.0526	+	11.0000i	1.07010	+	0.617822i	0.141890	+	0.989882i	$0.454682\pi$
$318$		0			0
$319$		0			0
$320$	8.00000	+	13.8564i	0.447214	+	0.774597i
$321$		0			0
$322$	2.92820	+	10.9282i	0.163182	+	0.609005i
$323$		−	8.00000i		−	0.445132i
$324$		0			0
$325$			4.00000i			0.221880i
$326$	−5.46410	+	1.46410i	−0.302629	+	0.0810891i
$327$		0			0
$328$	1.46410	−	5.46410i	0.0808415	−	0.301705i
$329$	12.0000	+	20.7846i	0.661581	+	1.14589i
$330$		0			0
$331$	17.3205	+	10.0000i	0.952021	+	0.549650i	0.893708	−	0.448649i	$-0.148095\pi$
$331$	17.3205	+	10.0000i	0.952021	+	0.549650i	0.0583130	+	0.998298i	$0.481428\pi$
$332$	32.0000			1.75623
$333$		0			0
$334$	−8.00000	+	8.00000i	−0.437741	+	0.437741i
$335$	12.0000	−	20.7846i	0.655630	−	1.13558i
$336$		0			0
$337$	1.00000	+	1.73205i	0.0544735	+	0.0943508i	0.891976	−	0.452082i	$-0.149319\pi$
$337$	1.00000	+	1.73205i	0.0544735	+	0.0943508i	−0.837503	+	0.546433i	$0.815985\pi$
$338$	4.09808	+	1.09808i	0.222906	+	0.0597275i
$339$		0			0
$340$	−4.00000	+	6.92820i	−0.216930	+	0.375735i
$341$		0			0
$342$		0			0
$343$	20.0000			1.07990
$344$	2.92820	+	10.9282i	0.157878	+	0.589209i
$345$		0			0
$346$	2.19615	−	8.19615i	0.118066	−	0.440628i
$347$	6.92820	−	4.00000i	0.371925	−	0.214731i	−0.302374	−	0.953189i	$-0.597779\pi$
$347$	6.92820	−	4.00000i	0.371925	−	0.214731i	0.674299	+	0.738458i	$0.264446\pi$
$348$		0			0
$349$	−13.8564	−	8.00000i	−0.741716	−	0.428230i	0.0809766	−	0.996716i	$-0.474196\pi$
$349$	−13.8564	−	8.00000i	−0.741716	−	0.428230i	−0.822693	+	0.568486i	$0.807529\pi$
$350$	−2.00000	+	2.00000i	−0.106904	+	0.106904i
$351$		0			0
$352$		0			0
$353$	−3.00000	+	5.19615i	−0.159674	+	0.276563i	−0.934751	−	0.355303i	$-0.884378\pi$
$353$	−3.00000	+	5.19615i	−0.159674	+	0.276563i	0.775077	+	0.631867i	$0.217711\pi$
$354$		0			0
$355$	−20.7846	+	12.0000i	−1.10313	+	0.636894i
$356$	−17.3205	+	10.0000i	−0.917985	+	0.529999i
$357$		0			0
$358$	−5.46410	+	1.46410i	−0.288787	+	0.0773802i
$359$	20.0000			1.05556			0.527780	−	0.849381i	$-0.323025\pi$
$359$	20.0000			1.05556			0.527780	+	0.849381i	$0.323025\pi$
$360$		0			0
$361$	3.00000			0.157895
$362$	−27.3205	+	7.32051i	−1.43593	+	0.384757i
$363$		0			0
$364$	8.00000	+	13.8564i	0.419314	+	0.726273i
$365$	−10.3923	+	6.00000i	−0.543958	+	0.314054i
$366$		0			0
$367$	−9.00000	+	15.5885i	−0.469796	+	0.813711i	−0.999404	−	0.0345320i	$-0.989006\pi$
$367$	−9.00000	+	15.5885i	−0.469796	+	0.813711i	0.529607	+	0.848243i	$0.322339\pi$
$368$	16.0000			0.834058
$369$		0			0
$370$	16.0000	−	16.0000i	0.831800	−	0.831800i
$371$	10.3923	+	6.00000i	0.539542	+	0.311504i
$372$		0			0
$373$	−13.8564	+	8.00000i	−0.717458	+	0.414224i	−0.813816	−	0.581122i	$-0.802614\pi$
$373$	−13.8564	+	8.00000i	−0.717458	+	0.414224i	0.0963587	+	0.995347i	$0.469280\pi$
$374$		0			0
$375$		0			0
$376$	32.7846	−	8.78461i	1.69074	−	0.453032i
$377$	24.0000			1.23606
$378$		0			0
$379$		−	4.00000i		−	0.205466i	−0.994709	−	0.102733i	$-0.967241\pi$
$379$		−	4.00000i		−	0.205466i	0.994709	−	0.102733i	$-0.0327588\pi$
$380$	13.8564	+	8.00000i	0.710819	+	0.410391i
$381$		0			0
$382$	−10.9282	−	2.92820i	−0.559136	−	0.149820i
$383$	12.0000	+	20.7846i	0.613171	+	1.06204i	0.990702	+	0.136047i	$0.0434398\pi$
$383$	12.0000	+	20.7846i	0.613171	+	1.06204i	−0.377531	+	0.925997i	$0.623227\pi$
$384$		0			0
$385$		0			0
$386$	−6.00000	+	6.00000i	−0.305392	+	0.305392i
$387$		0			0
$388$			4.00000i			0.203069i
$389$	−29.4449	−	17.0000i	−1.49291	−	0.861934i	−0.492947	−	0.870059i	$-0.664080\pi$
$389$	−29.4449	−	17.0000i	−1.49291	−	0.861934i	−0.999967	+	0.00812520i	$0.997414\pi$
$390$		0			0
$391$	4.00000	+	6.92820i	0.202289	+	0.350374i
$392$	2.19615	−	8.19615i	0.110922	−	0.413968i
$393$		0			0
$394$	2.73205	−	0.732051i	0.137639	−	0.0368802i
$395$			20.0000i			1.00631i
$396$		0			0
$397$			32.0000i			1.60603i	0.595956	+	0.803017i	$0.296773\pi$
$397$			32.0000i			1.60603i	−0.595956	+	0.803017i	$0.703227\pi$
$398$	3.66025	+	13.6603i	0.183472	+	0.684727i
$399$		0			0
$400$	2.00000	+	3.46410i	0.100000	+	0.173205i
$401$	−9.00000	−	15.5885i	−0.449439	−	0.778450i	0.548911	−	0.835881i	$-0.315043\pi$
$401$	−9.00000	−	15.5885i	−0.449439	−	0.778450i	−0.998350	+	0.0574304i	$0.981709\pi$
$402$		0			0
$403$	−6.92820	−	4.00000i	−0.345118	−	0.199254i
$404$	−20.0000			−0.995037
$405$		0			0
$406$	12.0000	+	12.0000i	0.595550	+	0.595550i
$407$		0			0
$408$		0			0
$409$	−5.00000	−	8.66025i	−0.247234	−	0.428222i	0.715523	−	0.698589i	$-0.246188\pi$
$409$	−5.00000	−	8.66025i	−0.247234	−	0.428222i	−0.962757	+	0.270367i	$0.912855\pi$
$410$	−1.46410	+	5.46410i	−0.0723068	+	0.269853i
$411$		0			0
$412$	−10.3923	−	6.00000i	−0.511992	−	0.295599i
$413$		−	8.00000i		−	0.393654i
$414$		0			0
$415$	−32.0000			−1.57082
$416$	21.8564	−	5.85641i	1.07160	−	0.287134i
$417$		0			0
$418$		0			0
$419$	20.7846	−	12.0000i	1.01539	−	0.586238i	0.102628	−	0.994720i	$-0.467275\pi$
$419$	20.7846	−	12.0000i	1.01539	−	0.586238i	0.912767	+	0.408481i	$0.133942\pi$
$420$		0			0
$421$	−17.3205	−	10.0000i	−0.844150	−	0.487370i	0.0145228	−	0.999895i	$-0.495377\pi$
$421$	−17.3205	−	10.0000i	−0.844150	−	0.487370i	−0.858673	+	0.512524i	$0.828710\pi$
$422$	−20.0000	−	20.0000i	−0.973585	−	0.973585i
$423$		0			0
$424$	12.0000	−	12.0000i	0.582772	−	0.582772i
$425$	−1.00000	+	1.73205i	−0.0485071	+	0.0840168i
$426$		0			0
$427$		0			0
$428$	12.0000	+	20.7846i	0.580042	+	1.00466i
$429$		0			0
$430$	−2.92820	−	10.9282i	−0.141210	−	0.527005i
$431$	−12.0000			−0.578020			−0.289010	−	0.957326i	$-0.593326\pi$
$431$	−12.0000			−0.578020			−0.289010	+	0.957326i	$0.593326\pi$
$432$		0			0
$433$	14.0000			0.672797			0.336399	−	0.941720i	$-0.390791\pi$
$433$	14.0000			0.672797			0.336399	+	0.941720i	$0.390791\pi$
$434$	−1.46410	−	5.46410i	−0.0702791	−	0.262285i
$435$		0			0
$436$	4.00000	+	6.92820i	0.191565	+	0.331801i
$437$	13.8564	−	8.00000i	0.662842	−	0.382692i
$438$		0			0
$439$	15.0000	−	25.9808i	0.715911	−	1.23999i	−0.246696	−	0.969093i	$-0.579345\pi$
$439$	15.0000	−	25.9808i	0.715911	−	1.23999i	0.962607	−	0.270901i	$-0.0873217\pi$
$440$		0			0
$441$		0			0
$442$	8.00000	+	8.00000i	0.380521	+	0.380521i
$443$	20.7846	+	12.0000i	0.987507	+	0.570137i	0.904528	−	0.426414i	$-0.140223\pi$
$443$	20.7846	+	12.0000i	0.987507	+	0.570137i	0.0829786	+	0.996551i	$0.473557\pi$
$444$		0			0
$445$	17.3205	−	10.0000i	0.821071	−	0.474045i
$446$	−19.1244	−	5.12436i	−0.905564	−	0.242645i
$447$		0			0
$448$	13.8564	+	8.00000i	0.654654	+	0.377964i
$449$	−30.0000			−1.41579			−0.707894	−	0.706319i	$-0.750354\pi$
$449$	−30.0000			−1.41579			−0.707894	+	0.706319i	$0.750354\pi$
$450$		0			0
$451$		0			0
$452$	10.3923	+	6.00000i	0.488813	+	0.282216i
$453$		0			0
$454$	2.92820	−	10.9282i	0.137427	−	0.512886i
$455$	−8.00000	−	13.8564i	−0.375046	−	0.649598i
$456$		0			0
$457$	11.0000	−	19.0526i	0.514558	−	0.891241i	−0.485299	−	0.874348i	$-0.661289\pi$
$457$	11.0000	−	19.0526i	0.514558	−	0.891241i	0.999857	−	0.0168929i	$-0.00537742\pi$
$458$	−4.00000	−	4.00000i	−0.186908	−	0.186908i
$459$		0			0
$460$	−16.0000			−0.746004
$461$	−25.9808	−	15.0000i	−1.21004	−	0.698620i	−0.247276	−	0.968945i	$-0.579535\pi$
$461$	−25.9808	−	15.0000i	−1.21004	−	0.698620i	−0.962769	+	0.270326i	$0.912869\pi$
$462$		0			0
$463$	13.0000	+	22.5167i	0.604161	+	1.04644i	0.992183	+	0.124788i	$0.0398251\pi$
$463$	13.0000	+	22.5167i	0.604161	+	1.04644i	−0.388022	+	0.921650i	$0.626842\pi$
$464$	20.7846	−	12.0000i	0.964901	−	0.557086i
$465$		0			0
$466$	−5.12436	−	19.1244i	−0.237381	−	0.885919i
$467$			8.00000i			0.370196i	0.982720	+	0.185098i	$0.0592602\pi$
$467$			8.00000i			0.370196i	−0.982720	+	0.185098i	$0.940740\pi$
$468$		0			0
$469$		−	24.0000i		−	1.10822i
$470$	−32.7846	+	8.78461i	−1.51224	+	0.405204i
$471$		0			0
$472$	−10.9282	−	2.92820i	−0.503011	−	0.134781i
$473$		0			0
$474$		0			0
$475$	3.46410	+	2.00000i	0.158944	+	0.0917663i
$476$			8.00000i			0.366679i
$477$		0			0
$478$		0			0
$479$	10.0000	−	17.3205i	0.456912	−	0.791394i	−0.541884	−	0.840453i	$-0.682289\pi$
$479$	10.0000	−	17.3205i	0.456912	−	0.791394i	0.998796	+	0.0490589i	$0.0156222\pi$
$480$		0			0
$481$	−16.0000	−	27.7128i	−0.729537	−	1.26360i
$482$	−2.73205	−	0.732051i	−0.124442	−	0.0333440i
$483$		0			0
$484$	19.0526	+	11.0000i	0.866025	+	0.500000i
$485$		−	4.00000i		−	0.181631i
$486$		0			0
$487$	38.0000			1.72194			0.860972	−	0.508652i	$-0.169856\pi$
$487$	38.0000			1.72194			0.860972	+	0.508652i	$0.169856\pi$
$488$		0			0
$489$		0			0
$490$	−2.19615	+	8.19615i	−0.0992121	+	0.370264i
$491$	−17.3205	+	10.0000i	−0.781664	+	0.451294i	−0.837020	−	0.547173i	$-0.815704\pi$
$491$	−17.3205	+	10.0000i	−0.781664	+	0.451294i	0.0553560	+	0.998467i	$0.482371\pi$
$492$		0			0
$493$	10.3923	+	6.00000i	0.468046	+	0.270226i
$494$	16.0000	−	16.0000i	0.719874	−	0.719874i
$495$		0			0
$496$	−8.00000			−0.359211
$497$	−12.0000	+	20.7846i	−0.538274	+	0.932317i
$498$		0			0
$499$	31.1769	−	18.0000i	1.39567	−	0.805791i	0.401735	−	0.915756i	$-0.368407\pi$
$499$	31.1769	−	18.0000i	1.39567	−	0.805791i	0.993935	+	0.109965i	$0.0350740\pi$
$500$	−12.0000	−	20.7846i	−0.536656	−	0.929516i
$501$		0			0
$502$		0			0
$503$	36.0000			1.60516			0.802580	−	0.596544i	$-0.203460\pi$
$503$	36.0000			1.60516			0.802580	+	0.596544i	$0.203460\pi$
$504$		0			0
$505$	20.0000			0.889988
$506$		0			0
$507$		0			0
$508$	3.46410	−	2.00000i	0.153695	−	0.0887357i
$509$	−5.19615	+	3.00000i	−0.230315	+	0.132973i	−0.610718	−	0.791849i	$-0.709119\pi$
$509$	−5.19615	+	3.00000i	−0.230315	+	0.132973i	0.380402	+	0.924821i	$0.375786\pi$
$510$		0			0
$511$	−6.00000	+	10.3923i	−0.265424	+	0.459728i
$512$	16.0000	−	16.0000i	0.707107	−	0.707107i
$513$		0			0
$514$	−18.0000	+	18.0000i	−0.793946	+	0.793946i
$515$	10.3923	+	6.00000i	0.457940	+	0.264392i
$516$		0			0
$517$		0			0
$518$	5.85641	−	21.8564i	0.257316	−	0.960315i
$519$		0			0
$520$	−21.8564	+	5.85641i	−0.958467	+	0.256820i
$521$	−42.0000			−1.84005			−0.920027	−	0.391856i	$-0.871833\pi$
$521$	−42.0000			−1.84005			−0.920027	+	0.391856i	$0.871833\pi$
$522$		0			0
$523$		−	36.0000i		−	1.57417i	−0.616844	−	0.787085i	$-0.711589\pi$
$523$		−	36.0000i		−	1.57417i	0.616844	−	0.787085i	$-0.288411\pi$
$524$	20.0000	−	34.6410i	0.873704	−	1.51330i
$525$		0			0
$526$	−21.8564	−	5.85641i	−0.952985	−	0.255351i
$527$	−2.00000	−	3.46410i	−0.0871214	−	0.150899i
$528$		0			0
$529$	3.50000	−	6.06218i	0.152174	−	0.263573i
$530$	−12.0000	+	12.0000i	−0.521247	+	0.521247i
$531$		0			0
$532$	16.0000			0.693688
$533$	6.92820	+	4.00000i	0.300094	+	0.173259i
$534$		0			0
$535$	−12.0000	−	20.7846i	−0.518805	−	0.898597i
$536$	−32.7846	−	8.78461i	−1.41608	−	0.379437i
$537$		0			0
$538$	8.19615	−	2.19615i	0.353361	−	0.0946829i
$539$		0			0
$540$		0			0
$541$		−	20.0000i		−	0.859867i	−0.902861	−	0.429934i	$-0.858537\pi$
$541$		−	20.0000i		−	0.859867i	0.902861	−	0.429934i	$-0.141463\pi$
$542$	−6.58846	−	24.5885i	−0.282998	−	1.05616i
$543$		0			0
$544$	10.9282	+	2.92820i	0.468543	+	0.125546i
$545$	−4.00000	−	6.92820i	−0.171341	−	0.296772i
$546$		0			0
$547$	24.2487	+	14.0000i	1.03680	+	0.598597i	0.918925	−	0.394432i	$-0.129059\pi$
$547$	24.2487	+	14.0000i	1.03680	+	0.598597i	0.117875	+	0.993028i	$0.462392\pi$
$548$			36.0000i			1.53784i
$549$		0			0
$550$		0			0
$551$	12.0000	−	20.7846i	0.511217	−	0.885454i
$552$		0			0
$553$	10.0000	+	17.3205i	0.425243	+	0.736543i
$554$	−10.2487	+	38.2487i	−0.435426	+	1.62503i
$555$		0			0
$556$	4.00000	−	6.92820i	0.169638	−	0.293821i
$557$		−	42.0000i		−	1.77960i	−0.456354	−	0.889799i	$-0.650845\pi$
$557$		−	42.0000i		−	1.77960i	0.456354	−	0.889799i	$-0.349155\pi$
$558$		0			0
$559$	−16.0000			−0.676728
$560$	−13.8564	−	8.00000i	−0.585540	−	0.338062i
$561$		0			0
$562$	−24.5885	−	6.58846i	−1.03720	−	0.277917i
$563$	−20.7846	+	12.0000i	−0.875967	+	0.505740i	−0.869326	−	0.494238i	$-0.835447\pi$
$563$	−20.7846	+	12.0000i	−0.875967	+	0.505740i	−0.00664037	+	0.999978i	$0.502114\pi$
$564$		0			0
$565$	−10.3923	−	6.00000i	−0.437208	−	0.252422i
$566$	4.00000	+	4.00000i	0.168133	+	0.168133i
$567$		0			0
$568$	24.0000	+	24.0000i	1.00702	+	1.00702i
$569$	−15.0000	+	25.9808i	−0.628833	+	1.08917i	0.358954	+	0.933355i	$0.383134\pi$
$569$	−15.0000	+	25.9808i	−0.628833	+	1.08917i	−0.987786	+	0.155815i	$0.950200\pi$
$570$		0			0
$571$	17.3205	−	10.0000i	0.724841	−	0.418487i	−0.0916910	−	0.995788i	$-0.529227\pi$
$571$	17.3205	−	10.0000i	0.724841	−	0.418487i	0.816532	+	0.577301i	$0.195894\pi$
$572$		0			0
$573$		0			0
$574$	1.46410	+	5.46410i	0.0611104	+	0.228067i
$575$	−4.00000			−0.166812
$576$		0			0
$577$	−42.0000			−1.74848			−0.874241	−	0.485491i	$-0.838641\pi$
$577$	−42.0000			−1.74848			−0.874241	+	0.485491i	$0.838641\pi$
$578$	−4.75833	−	17.7583i	−0.197920	−	0.738649i
$579$		0			0
$580$	−20.7846	+	12.0000i	−0.863034	+	0.498273i
$581$	−27.7128	+	16.0000i	−1.14972	+	0.663792i
$582$		0			0
$583$		0			0
$584$	12.0000	+	12.0000i	0.496564	+	0.496564i
$585$		0			0
$586$	6.00000	+	6.00000i	0.247858	+	0.247858i
$587$	−24.2487	−	14.0000i	−1.00085	−	0.577842i	−0.0923513	−	0.995726i	$-0.529438\pi$
$587$	−24.2487	−	14.0000i	−1.00085	−	0.577842i	−0.908500	+	0.417885i	$0.862772\pi$
$588$		0			0
$589$	−6.92820	+	4.00000i	−0.285472	+	0.164817i
$590$	10.9282	+	2.92820i	0.449907	+	0.120552i
$591$		0			0
$592$	−27.7128	−	16.0000i	−1.13899	−	0.657596i
$593$	6.00000			0.246390			0.123195	−	0.992382i	$-0.460686\pi$
$593$	6.00000			0.246390			0.123195	+	0.992382i	$0.460686\pi$
$594$		0			0
$595$		−	8.00000i		−	0.327968i
$596$	6.00000	−	10.3923i	0.245770	−	0.425685i
$597$		0			0
$598$	−5.85641	+	21.8564i	−0.239486	+	0.893775i
$599$	−10.0000	−	17.3205i	−0.408589	−	0.707697i	0.586143	−	0.810208i	$-0.300646\pi$
$599$	−10.0000	−	17.3205i	−0.408589	−	0.707697i	−0.994732	+	0.102511i	$0.967312\pi$
$600$		0			0
$601$	−1.00000	+	1.73205i	−0.0407909	+	0.0706518i	−0.885700	−	0.464258i	$-0.846321\pi$
$601$	−1.00000	+	1.73205i	−0.0407909	+	0.0706518i	0.844909	+	0.534910i	$0.179654\pi$
$602$	−8.00000	−	8.00000i	−0.326056	−	0.326056i
$603$		0			0
$604$			36.0000i			1.46482i
$605$	−19.0526	−	11.0000i	−0.774597	−	0.447214i
$606$		0			0
$607$	1.00000	+	1.73205i	0.0405887	+	0.0703018i	0.885606	−	0.464437i	$-0.153743\pi$
$607$	1.00000	+	1.73205i	0.0405887	+	0.0703018i	−0.845017	+	0.534739i	$0.820410\pi$
$608$	5.85641	−	21.8564i	0.237509	−	0.886394i
$609$		0			0
$610$		0			0
$611$			48.0000i			1.94187i
$612$		0			0
$613$		−	16.0000i		−	0.646234i	−0.946359	−	0.323117i	$-0.895269\pi$
$613$		−	16.0000i		−	0.646234i	0.946359	−	0.323117i	$-0.104731\pi$
$614$	−16.3923	+	4.39230i	−0.661540	+	0.177259i
$615$		0			0
$616$		0			0
$617$	−1.00000	−	1.73205i	−0.0402585	−	0.0697297i	0.845194	−	0.534460i	$-0.179485\pi$
$617$	−1.00000	−	1.73205i	−0.0402585	−	0.0697297i	−0.885453	+	0.464730i	$0.846151\pi$
$618$		0			0
$619$	−31.1769	−	18.0000i	−1.25311	−	0.723481i	−0.281381	−	0.959596i	$-0.590792\pi$
$619$	−31.1769	−	18.0000i	−1.25311	−	0.723481i	−0.971725	+	0.236115i	$0.924126\pi$
$620$	8.00000			0.321288
$621$		0			0
$622$	8.00000	−	8.00000i	0.320771	−	0.320771i
$623$	10.0000	−	17.3205i	0.400642	−	0.693932i
$624$		0			0
$625$	9.50000	+	16.4545i	0.380000	+	0.658179i
$626$	−19.1244	−	5.12436i	−0.764363	−	0.204810i
$627$		0			0
$628$	8.00000	−	13.8564i	0.319235	−	0.552931i
$629$		−	16.0000i		−	0.637962i
$630$		0			0
$631$	22.0000			0.875806			0.437903	−	0.899022i	$-0.355721\pi$
$631$	22.0000			0.875806			0.437903	+	0.899022i	$0.355721\pi$
$632$	27.3205	−	7.32051i	1.08675	−	0.291194i
$633$		0			0
$634$	−8.05256	+	30.0526i	−0.319808	+	1.19354i
$635$	−3.46410	+	2.00000i	−0.137469	+	0.0793676i
$636$		0			0
$637$	10.3923	+	6.00000i	0.411758	+	0.237729i
$638$		0			0
$639$		0			0
$640$	−16.0000	+	16.0000i	−0.632456	+	0.632456i
$641$	11.0000	−	19.0526i	0.434474	−	0.752531i	−0.562779	−	0.826608i	$-0.690268\pi$
$641$	11.0000	−	19.0526i	0.434474	−	0.752531i	0.997253	+	0.0740768i	$0.0236010\pi$
$642$		0			0
$643$	3.46410	−	2.00000i	0.136611	−	0.0788723i	−0.430137	−	0.902764i	$-0.641535\pi$
$643$	3.46410	−	2.00000i	0.136611	−	0.0788723i	0.566748	+	0.823891i	$0.308201\pi$
$644$	−13.8564	+	8.00000i	−0.546019	+	0.315244i
$645$		0			0
$646$	10.9282	−	2.92820i	0.429964	−	0.115209i
$647$	12.0000			0.471769			0.235884	−	0.971781i	$-0.424201\pi$
$647$	12.0000			0.471769			0.235884	+	0.971781i	$0.424201\pi$
$648$		0			0
$649$		0			0
$650$	−5.46410	+	1.46410i	−0.214320	+	0.0574268i
$651$		0			0
$652$	−4.00000	−	6.92820i	−0.156652	−	0.271329i
$653$	5.19615	−	3.00000i	0.203341	−	0.117399i	−0.394872	−	0.918736i	$-0.629211\pi$
$653$	5.19615	−	3.00000i	0.203341	−	0.117399i	0.598213	+	0.801337i	$0.295878\pi$
$654$		0			0
$655$	−20.0000	+	34.6410i	−0.781465	+	1.35354i
$656$	8.00000			0.312348
$657$		0			0
$658$	−24.0000	+	24.0000i	−0.935617	+	0.935617i
$659$	31.1769	+	18.0000i	1.21448	+	0.701180i	0.963732	−	0.266872i	$-0.0859901\pi$
$659$	31.1769	+	18.0000i	1.21448	+	0.701180i	0.250748	+	0.968052i	$0.419323\pi$
$660$		0			0
$661$	34.6410	−	20.0000i	1.34738	−	0.777910i	0.359502	−	0.933144i	$-0.382947\pi$
$661$	34.6410	−	20.0000i	1.34738	−	0.777910i	0.987878	+	0.155235i	$0.0496133\pi$
$662$	−7.32051	+	27.3205i	−0.284520	+	1.06184i
$663$		0			0
$664$	11.7128	+	43.7128i	0.454545	+	1.69639i
$665$	−16.0000			−0.620453
$666$		0			0
$667$			24.0000i			0.929284i
$668$	−13.8564	−	8.00000i	−0.536120	−	0.309529i
$669$		0			0
$670$	32.7846	+	8.78461i	1.26658	+	0.339379i
$671$		0			0
$672$		0			0
$673$	−7.00000	+	12.1244i	−0.269830	+	0.467360i	−0.968818	−	0.247774i	$-0.920301\pi$
$673$	−7.00000	+	12.1244i	−0.269830	+	0.467360i	0.698988	+	0.715134i	$0.253634\pi$
$674$	−2.00000	+	2.00000i	−0.0770371	+	0.0770371i
$675$		0			0
$676$			6.00000i			0.230769i
$677$	−15.5885	−	9.00000i	−0.599113	−	0.345898i	0.169580	−	0.985517i	$-0.445759\pi$
$677$	−15.5885	−	9.00000i	−0.599113	−	0.345898i	−0.768693	+	0.639618i	$0.779092\pi$
$678$		0			0
$679$	−2.00000	−	3.46410i	−0.0767530	−	0.132940i
$680$	−10.9282	−	2.92820i	−0.419077	−	0.112291i
$681$		0			0
$682$		0			0
$683$			16.0000i			0.612223i	0.951996	+	0.306111i	$0.0990280\pi$
$683$			16.0000i			0.612223i	−0.951996	+	0.306111i	$0.900972\pi$
$684$		0			0
$685$		−	36.0000i		−	1.37549i
$686$	7.32051	+	27.3205i	0.279498	+	1.04310i
$687$		0			0
$688$	−13.8564	+	8.00000i	−0.528271	+	0.304997i
$689$	12.0000	+	20.7846i	0.457164	+	0.791831i
$690$		0			0
$691$	−17.3205	−	10.0000i	−0.658903	−	0.380418i	0.132956	−	0.991122i	$-0.457553\pi$
$691$	−17.3205	−	10.0000i	−0.658903	−	0.380418i	−0.791859	+	0.610704i	$0.790887\pi$
$692$	12.0000			0.456172
$693$		0			0
$694$	8.00000	+	8.00000i	0.303676	+	0.303676i
$695$	−4.00000	+	6.92820i	−0.151729	+	0.262802i
$696$		0			0
$697$	2.00000	+	3.46410i	0.0757554	+	0.131212i
$698$	5.85641	−	21.8564i	0.221668	−	0.827277i
$699$		0			0
$700$	−3.46410	−	2.00000i	−0.130931	−	0.0755929i
$701$			50.0000i			1.88847i	0.329267	+	0.944237i	$0.393198\pi$
$701$			50.0000i			1.88847i	−0.329267	+	0.944237i	$0.606802\pi$
$702$		0			0
$703$	−32.0000			−1.20690
$704$		0			0
$705$		0			0
$706$	−8.19615	−	2.19615i	−0.308466	−	0.0826533i
$707$	17.3205	−	10.0000i	0.651405	−	0.376089i
$708$		0			0
$709$	3.46410	+	2.00000i	0.130097	+	0.0751116i	0.563636	−	0.826023i	$-0.309402\pi$
$709$	3.46410	+	2.00000i	0.130097	+	0.0751116i	−0.433539	+	0.901135i	$0.642735\pi$
$710$	−24.0000	−	24.0000i	−0.900704	−	0.900704i
$711$		0			0
$712$	−20.0000	−	20.0000i	−0.749532	−	0.749532i
$713$	4.00000	−	6.92820i	0.149801	−	0.259463i
$714$		0			0
$715$		0			0
$716$	−4.00000	−	6.92820i	−0.149487	−	0.258919i
$717$		0			0
$718$	7.32051	+	27.3205i	0.273199	+	1.01959i
$719$	20.0000			0.745874			0.372937	−	0.927857i	$-0.378351\pi$
$719$	20.0000			0.745874			0.372937	+	0.927857i	$0.378351\pi$
$720$		0			0
$721$	12.0000			0.446903
$722$	1.09808	+	4.09808i	0.0408662	+	0.152515i
$723$		0			0
$724$	−20.0000	−	34.6410i	−0.743294	−	1.28742i
$725$	−5.19615	+	3.00000i	−0.192980	+	0.111417i
$726$		0			0
$727$	21.0000	−	36.3731i	0.778847	−	1.34900i	−0.153760	−	0.988108i	$-0.549138\pi$
$727$	21.0000	−	36.3731i	0.778847	−	1.34900i	0.932607	−	0.360894i	$-0.117528\pi$
$728$	−16.0000	+	16.0000i	−0.592999	+	0.592999i
$729$		0			0
$730$	−12.0000	−	12.0000i	−0.444140	−	0.444140i
$731$	−6.92820	−	4.00000i	−0.256249	−	0.147945i
$732$		0			0
$733$	3.46410	−	2.00000i	0.127950	−	0.0738717i	−0.434659	−	0.900595i	$-0.643131\pi$
$733$	3.46410	−	2.00000i	0.127950	−	0.0738717i	0.562609	+	0.826723i	$0.309798\pi$
$734$	−24.5885	−	6.58846i	−0.907577	−	0.243184i
$735$		0			0
$736$	5.85641	+	21.8564i	0.215870	+	0.805638i
$737$		0			0
$738$		0			0
$739$		−	44.0000i		−	1.61857i	−0.587419	−	0.809283i	$-0.699856\pi$
$739$		−	44.0000i		−	1.61857i	0.587419	−	0.809283i	$-0.300144\pi$
$740$	27.7128	+	16.0000i	1.01874	+	0.588172i
$741$		0			0
$742$	−4.39230	+	16.3923i	−0.161247	+	0.601780i
$743$	−8.00000	−	13.8564i	−0.293492	−	0.508342i	0.681141	−	0.732152i	$-0.261484\pi$
$743$	−8.00000	−	13.8564i	−0.293492	−	0.508342i	−0.974633	+	0.223810i	$0.928151\pi$
$744$		0			0
$745$	−6.00000	+	10.3923i	−0.219823	+	0.380745i
$746$	−16.0000	−	16.0000i	−0.585802	−	0.585802i
$747$		0			0
$748$		0			0
$749$	−20.7846	−	12.0000i	−0.759453	−	0.438470i
$750$		0			0
$751$	−1.00000	−	1.73205i	−0.0364905	−	0.0632034i	0.847203	−	0.531269i	$-0.178285\pi$
$751$	−1.00000	−	1.73205i	−0.0364905	−	0.0632034i	−0.883694	+	0.468065i	$0.844951\pi$
$752$	24.0000	+	41.5692i	0.875190	+	1.51587i
$753$		0			0
$754$	8.78461	+	32.7846i	0.319917	+	1.19395i
$755$		−	36.0000i		−	1.31017i
$756$		0			0
$757$			12.0000i			0.436147i	0.975932	+	0.218074i	$0.0699773\pi$
$757$			12.0000i			0.436147i	−0.975932	+	0.218074i	$0.930023\pi$
$758$	5.46410	−	1.46410i	0.198465	−	0.0531786i
$759$		0			0
$760$	−5.85641	+	21.8564i	−0.212434	+	0.792815i
$761$	−19.0000	−	32.9090i	−0.688749	−	1.19295i	−0.972243	−	0.233975i	$-0.924827\pi$
$761$	−19.0000	−	32.9090i	−0.688749	−	1.19295i	0.283493	−	0.958974i	$-0.408507\pi$
$762$		0			0
$763$	−6.92820	−	4.00000i	−0.250818	−	0.144810i
$764$		−	16.0000i		−	0.578860i
$765$		0			0
$766$	−24.0000	+	24.0000i	−0.867155	+	0.867155i
$767$	8.00000	−	13.8564i	0.288863	−	0.500326i
$768$		0			0
$769$	−5.00000	−	8.66025i	−0.180305	−	0.312297i	0.761680	−	0.647954i	$-0.224375\pi$
$769$	−5.00000	−	8.66025i	−0.180305	−	0.312297i	−0.941984	+	0.335657i	$0.891042\pi$
$770$		0			0
$771$		0			0
$772$	−10.3923	−	6.00000i	−0.374027	−	0.215945i
$773$		−	34.0000i		−	1.22290i	−0.791285	−	0.611448i	$-0.790588\pi$
$773$		−	34.0000i		−	1.22290i	0.791285	−	0.611448i	$-0.209412\pi$
$774$		0			0
$775$	2.00000			0.0718421
$776$	−5.46410	+	1.46410i	−0.196150	+	0.0525582i
$777$		0			0
$778$	12.4449	−	46.4449i	0.446170	−	1.66513i
$779$	6.92820	−	4.00000i	0.248229	−	0.143315i
$780$		0			0
$781$		0			0
$782$	−8.00000	+	8.00000i	−0.286079	+	0.286079i
$783$		0			0
$784$	12.0000			0.428571
$785$	−8.00000	+	13.8564i	−0.285532	+	0.494556i
$786$		0			0
$787$	10.3923	−	6.00000i	0.370446	−	0.213877i	−0.303207	−	0.952925i	$-0.598058\pi$
$787$	10.3923	−	6.00000i	0.370446	−	0.213877i	0.673653	+	0.739048i	$0.264724\pi$
$788$	2.00000	+	3.46410i	0.0712470	+	0.123404i
$789$		0			0
$790$	−27.3205	+	7.32051i	−0.972020	+	0.260452i
$791$	−12.0000			−0.426671
$792$		0			0
$793$		0			0
$794$	−43.7128	+	11.7128i	−1.55131	+	0.415672i
$795$		0			0
$796$	−17.3205	+	10.0000i	−0.613909	+	0.354441i
$797$	−1.73205	+	1.00000i	−0.0613524	+	0.0354218i	−0.530362	−	0.847771i	$-0.677944\pi$
$797$	−1.73205	+	1.00000i	−0.0613524	+	0.0354218i	0.469010	+	0.883193i	$0.344611\pi$
$798$		0			0
$799$	−12.0000	+	20.7846i	−0.424529	+	0.735307i
$800$	−4.00000	+	4.00000i	−0.141421	+	0.141421i
$801$		0			0
$802$	18.0000	−	18.0000i	0.635602	−	0.635602i
$803$		0			0
$804$		0			0
$805$	13.8564	−	8.00000i	0.488374	−	0.281963i
$806$	2.92820	−	10.9282i	0.103142	−	0.384930i
$807$		0			0
$808$	−7.32051	−	27.3205i	−0.257535	−	0.961132i
$809$	30.0000			1.05474			0.527372	−	0.849635i	$-0.323177\pi$
$809$	30.0000			1.05474			0.527372	+	0.849635i	$0.323177\pi$
$810$		0			0
$811$		−	20.0000i		−	0.702295i	−0.936320	−	0.351147i	$-0.885792\pi$
$811$		−	20.0000i		−	0.702295i	0.936320	−	0.351147i	$-0.114208\pi$
$812$	−12.0000	+	20.7846i	−0.421117	+	0.729397i
$813$		0			0
$814$		0			0
$815$	4.00000	+	6.92820i	0.140114	+	0.242684i
$816$		0			0
$817$	−8.00000	+	13.8564i	−0.279885	+	0.484774i
$818$	10.0000	−	10.0000i	0.349642	−	0.349642i
$819$		0			0
$820$	−8.00000			−0.279372
$821$	8.66025	+	5.00000i	0.302245	+	0.174501i	0.643451	−	0.765487i	$-0.277502\pi$
$821$	8.66025	+	5.00000i	0.302245	+	0.174501i	−0.341206	+	0.939989i	$0.610835\pi$
$822$		0			0
$823$	−7.00000	−	12.1244i	−0.244005	−	0.422628i	0.717847	−	0.696201i	$-0.245128\pi$
$823$	−7.00000	−	12.1244i	−0.244005	−	0.422628i	−0.961851	+	0.273573i	$0.911795\pi$
$824$	4.39230	−	16.3923i	0.153013	−	0.571053i
$825$		0			0
$826$	10.9282	−	2.92820i	0.380241	−	0.101885i
$827$		−	12.0000i		−	0.417281i	−0.977992	−	0.208640i	$-0.933096\pi$
$827$		−	12.0000i		−	0.417281i	0.977992	−	0.208640i	$-0.0669038\pi$
$828$		0			0
$829$		−	4.00000i		−	0.138926i	−0.997585	−	0.0694629i	$-0.977871\pi$
$829$		−	4.00000i		−	0.138926i	0.997585	−	0.0694629i	$-0.0221285\pi$
$830$	−11.7128	−	43.7128i	−0.406558	−	1.51729i
$831$		0			0
$832$	16.0000	+	27.7128i	0.554700	+	0.960769i
$833$	3.00000	+	5.19615i	0.103944	+	0.180036i
$834$		0			0
$835$	13.8564	+	8.00000i	0.479521	+	0.276851i
$836$		0			0
$837$		0			0
$838$	24.0000	+	24.0000i	0.829066	+	0.829066i
$839$	10.0000	−	17.3205i	0.345238	−	0.597970i	−0.640159	−	0.768243i	$-0.721131\pi$
$839$	10.0000	−	17.3205i	0.345238	−	0.597970i	0.985397	+	0.170272i	$0.0544647\pi$
$840$		0			0
$841$	3.50000	+	6.06218i	0.120690	+	0.209041i
$842$	7.32051	−	27.3205i	0.252281	−	0.941527i
$843$		0			0
$844$	20.0000	−	34.6410i	0.688428	−	1.19239i
$845$		−	6.00000i		−	0.206406i
$846$		0			0
$847$	−22.0000			−0.755929
$848$	20.7846	+	12.0000i	0.713746	+	0.412082i
$849$		0			0
$850$	−2.73205	−	0.732051i	−0.0937086	−	0.0251091i
$851$	27.7128	−	16.0000i	0.949983	−	0.548473i
$852$		0			0
$853$	−20.7846	−	12.0000i	−0.711651	−	0.410872i	0.100021	−	0.994985i	$-0.468109\pi$
$853$	−20.7846	−	12.0000i	−0.711651	−	0.410872i	−0.811672	+	0.584113i	$0.801442\pi$
$854$		0			0
$855$		0			0
$856$	−24.0000	+	24.0000i	−0.820303	+	0.820303i
$857$	9.00000	−	15.5885i	0.307434	−	0.532492i	−0.670366	−	0.742030i	$-0.733863\pi$
$857$	9.00000	−	15.5885i	0.307434	−	0.532492i	0.977800	+	0.209539i	$0.0671963\pi$
$858$		0			0
$859$	−38.1051	+	22.0000i	−1.30013	+	0.750630i	−0.980426	−	0.196887i	$-0.936917\pi$
$859$	−38.1051	+	22.0000i	−1.30013	+	0.750630i	−0.319704	+	0.947518i	$0.603583\pi$
$860$	13.8564	−	8.00000i	0.472500	−	0.272798i
$861$		0			0
$862$	−4.39230	−	16.3923i	−0.149602	−	0.558324i
$863$	−24.0000			−0.816970			−0.408485	−	0.912765i	$-0.633943\pi$
$863$	−24.0000			−0.816970			−0.408485	+	0.912765i	$0.633943\pi$
$864$		0			0
$865$	−12.0000			−0.408012
$866$	5.12436	+	19.1244i	0.174133	+	0.649872i
$867$		0			0
$868$	6.92820	−	4.00000i	0.235159	−	0.135769i
$869$		0			0
$870$		0			0
$871$	24.0000	−	41.5692i	0.813209	−	1.40852i
$872$	−8.00000	+	8.00000i	−0.270914	+	0.270914i
$873$		0			0
$874$	16.0000	+	16.0000i	0.541208	+	0.541208i
$875$	20.7846	+	12.0000i	0.702648	+	0.405674i
$876$		0			0
$877$	27.7128	−	16.0000i	0.935795	−	0.540282i	0.0471555	−	0.998888i	$-0.484984\pi$
$877$	27.7128	−	16.0000i	0.935795	−	0.540282i	0.888640	+	0.458606i	$0.151651\pi$
$878$	40.9808	+	10.9808i	1.38303	+	0.370583i
$879$		0			0
$880$		0			0
$881$	−2.00000			−0.0673817			−0.0336909	−	0.999432i	$-0.510726\pi$
$881$	−2.00000			−0.0673817			−0.0336909	+	0.999432i	$0.510726\pi$
$882$		0			0
$883$			4.00000i			0.134611i	0.997732	+	0.0673054i	$0.0214402\pi$
$883$			4.00000i			0.134611i	−0.997732	+	0.0673054i	$0.978560\pi$
$884$	−8.00000	+	13.8564i	−0.269069	+	0.466041i
$885$		0			0
$886$	−8.78461	+	32.7846i	−0.295125	+	1.10142i
$887$	24.0000	+	41.5692i	0.805841	+	1.39576i	0.915722	+	0.401813i	$0.131620\pi$
$887$	24.0000	+	41.5692i	0.805841	+	1.39576i	−0.109881	+	0.993945i	$0.535047\pi$
$888$		0			0
$889$	−2.00000	+	3.46410i	−0.0670778	+	0.116182i
$890$	20.0000	+	20.0000i	0.670402	+	0.670402i
$891$		0			0
$892$		−	28.0000i		−	0.937509i
$893$	41.5692	+	24.0000i	1.39106	+	0.803129i
$894$		0			0
$895$	4.00000	+	6.92820i	0.133705	+	0.231584i
$896$	−5.85641	+	21.8564i	−0.195649	+	0.730171i
$897$		0			0
$898$	−10.9808	−	40.9808i	−0.366433	−	1.36755i
$899$		−	12.0000i		−	0.400222i
$900$		0			0
$901$			12.0000i			0.399778i
$902$		0			0
$903$		0			0
$904$	−4.39230	+	16.3923i	−0.146086	+	0.545200i
$905$	20.0000	+	34.6410i	0.664822	+	1.15151i
$906$		0			0
$907$	24.2487	+	14.0000i	0.805165	+	0.464862i	0.845274	−	0.534333i	$-0.179437\pi$
$907$	24.2487	+	14.0000i	0.805165	+	0.464862i	−0.0401089	+	0.999195i	$0.512770\pi$
$908$	16.0000			0.530979
$909$		0			0
$910$	16.0000	−	16.0000i	0.530395	−	0.530395i
$911$	−24.0000	+	41.5692i	−0.795155	+	1.37725i	0.127585	+	0.991828i	$0.459277\pi$
$911$	−24.0000	+	41.5692i	−0.795155	+	1.37725i	−0.922740	+	0.385422i	$0.874056\pi$
$912$		0			0
$913$		0			0
$914$	30.0526	+	8.05256i	0.994050	+	0.266355i
$915$		0			0
$916$	4.00000	−	6.92820i	0.132164	−	0.228914i
$917$			40.0000i			1.32092i
$918$		0			0
$919$	30.0000			0.989609			0.494804	−	0.869004i	$-0.335240\pi$
$919$	30.0000			0.989609			0.494804	+	0.869004i	$0.335240\pi$
$920$	−5.85641	−	21.8564i	−0.193080	−	0.720584i
$921$		0			0
$922$	10.9808	−	40.9808i	0.361632	−	1.34963i
$923$	−41.5692	+	24.0000i	−1.36827	+	0.789970i
$924$		0			0
$925$	6.92820	+	4.00000i	0.227798	+	0.131519i
$926$	−26.0000	+	26.0000i	−0.854413	+	0.854413i
$927$		0			0
$928$	24.0000	+	24.0000i	0.787839	+	0.787839i
$929$	−25.0000	+	43.3013i	−0.820223	+	1.42067i	0.0852924	+	0.996356i	$0.472818\pi$
$929$	−25.0000	+	43.3013i	−0.820223	+	1.42067i	−0.905516	+	0.424313i	$0.860516\pi$
$930$		0			0
$931$	10.3923	−	6.00000i	0.340594	−	0.196642i
$932$	24.2487	−	14.0000i	0.794293	−	0.458585i
$933$		0			0
$934$	−10.9282	+	2.92820i	−0.357582	+	0.0958137i
$935$		0			0
$936$		0			0
$937$	38.0000			1.24141			0.620703	−	0.784046i	$-0.286847\pi$
$937$	38.0000			1.24141			0.620703	+	0.784046i	$0.286847\pi$
$938$	32.7846	−	8.78461i	1.07046	−	0.286828i
$939$		0			0
$940$	−24.0000	−	41.5692i	−0.782794	−	1.35584i
$941$	25.9808	−	15.0000i	0.846949	−	0.488986i	−0.0126715	−	0.999920i	$-0.504034\pi$
$941$	25.9808	−	15.0000i	0.846949	−	0.488986i	0.859620	+	0.510934i	$0.170700\pi$
$942$		0			0
$943$	−4.00000	+	6.92820i	−0.130258	+	0.225613i
$944$		−	16.0000i		−	0.520756i
$945$		0			0
$946$		0			0
$947$	10.3923	+	6.00000i	0.337705	+	0.194974i	0.659256	−	0.751918i	$-0.270871\pi$
$947$	10.3923	+	6.00000i	0.337705	+	0.194974i	−0.321552	+	0.946892i	$0.604204\pi$
$948$		0			0
$949$	−20.7846	+	12.0000i	−0.674697	+	0.389536i
$950$	−1.46410	+	5.46410i	−0.0475017	+	0.177279i
$951$		0			0
$952$	−10.9282	+	2.92820i	−0.354185	+	0.0949036i
$953$	6.00000			0.194359			0.0971795	−	0.995267i	$-0.469018\pi$
$953$	6.00000			0.194359			0.0971795	+	0.995267i	$0.469018\pi$
$954$		0			0
$955$			16.0000i			0.517748i
$956$		0			0
$957$		0			0
$958$	27.3205	+	7.32051i	0.882686	+	0.236515i
$959$	−18.0000	−	31.1769i	−0.581250	−	1.00676i
$960$		0			0
$961$	13.5000	−	23.3827i	0.435484	−	0.754280i
$962$	32.0000	−	32.0000i	1.03172	−	1.03172i
$963$		0			0
$964$		−	4.00000i		−	0.128831i
$965$	10.3923	+	6.00000i	0.334540	+	0.193147i
$966$		0			0
$967$	11.0000	+	19.0526i	0.353736	+	0.612689i	0.986901	−	0.161328i	$-0.0515777\pi$
$967$	11.0000	+	19.0526i	0.353736	+	0.612689i	−0.633165	+	0.774017i	$0.718244\pi$
$968$	−8.05256	+	30.0526i	−0.258819	+	0.965926i
$969$		0			0
$970$	5.46410	−	1.46410i	0.175442	−	0.0470095i
$971$		0			0		1.00000			$0$
$971$		0			0		−1.00000			$\pi$
$972$		0			0
$973$			8.00000i			0.256468i
$974$	13.9090	+	51.9090i	0.445672	+	1.66327i
$975$		0			0
$976$		0			0
$977$	−1.00000	−	1.73205i	−0.0319928	−	0.0554132i	0.849586	−	0.527451i	$-0.176852\pi$
$977$	−1.00000	−	1.73205i	−0.0319928	−	0.0554132i	−0.881579	+	0.472037i	$0.843519\pi$
$978$		0			0
$979$		0			0
$980$	−12.0000			−0.383326
$981$		0			0
$982$	−20.0000	−	20.0000i	−0.638226	−	0.638226i
$983$	−8.00000	+	13.8564i	−0.255160	+	0.441951i	−0.964939	−	0.262474i	$-0.915462\pi$
$983$	−8.00000	+	13.8564i	−0.255160	+	0.441951i	0.709779	+	0.704425i	$0.248795\pi$
$984$		0			0
$985$	−2.00000	−	3.46410i	−0.0637253	−	0.110375i
$986$	−4.39230	+	16.3923i	−0.139879	+	0.522037i
$987$		0			0
$988$	27.7128	+	16.0000i	0.881662	+	0.509028i
$989$		−	16.0000i		−	0.508770i
$990$		0			0
$991$	2.00000			0.0635321			0.0317660	−	0.999495i	$-0.489887\pi$
$991$	2.00000			0.0635321			0.0317660	+	0.999495i	$0.489887\pi$
$992$	−2.92820	−	10.9282i	−0.0929705	−	0.346971i
$993$		0			0
$994$	−32.7846	−	8.78461i	−1.03986	−	0.278631i
$995$	17.3205	−	10.0000i	0.549097	−	0.317021i
$996$		0			0
$997$	41.5692	+	24.0000i	1.31651	+	0.760088i	0.983165	−	0.182717i	$-0.0584893\pi$
$997$	41.5692	+	24.0000i	1.31651	+	0.760088i	0.333345	+	0.942805i	$0.391823\pi$
$998$	36.0000	+	36.0000i	1.13956	+	1.13956i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	648.2.n.c.541.2		4
3.2	odd	2		648.2.n.k.541.1		4
4.3	odd	2		2592.2.r.g.2161.2		4
8.3	odd	2		2592.2.r.g.2161.1		4
8.5	even	2	inner	648.2.n.c.541.1		4
9.2	odd	6		24.2.d.a.13.2	yes	2
9.4	even	3	inner	648.2.n.c.109.1		4
9.5	odd	6		648.2.n.k.109.2		4
9.7	even	3		72.2.d.b.37.1		2
12.11	even	2		2592.2.r.f.2161.1		4
24.5	odd	2		648.2.n.k.541.2		4
24.11	even	2		2592.2.r.f.2161.2		4
36.7	odd	6		288.2.d.b.145.2		2
36.11	even	6		96.2.d.a.49.1		2
36.23	even	6		2592.2.r.f.433.2		4
36.31	odd	6		2592.2.r.g.433.1		4
45.2	even	12		600.2.d.b.349.1		2
45.7	odd	12		1800.2.d.i.1549.2		2
45.29	odd	6		600.2.k.b.301.1		2
45.34	even	6		1800.2.k.a.901.2		2
45.38	even	12		600.2.d.c.349.2		2
45.43	odd	12		1800.2.d.b.1549.1		2
63.20	even	6		1176.2.c.a.589.2		2
72.5	odd	6		648.2.n.k.109.1		4
72.11	even	6		96.2.d.a.49.2		2
72.13	even	6	inner	648.2.n.c.109.2		4
72.29	odd	6		24.2.d.a.13.1	✓	2
72.43	odd	6		288.2.d.b.145.1		2
72.59	even	6		2592.2.r.f.433.1		4
72.61	even	6		72.2.d.b.37.2		2
72.67	odd	6		2592.2.r.g.433.2		4
144.11	even	12		768.2.a.d.1.1		1
144.29	odd	12		768.2.a.a.1.1		1
144.43	odd	12		2304.2.a.b.1.1		1
144.61	even	12		2304.2.a.o.1.1		1
144.83	even	12		768.2.a.e.1.1		1
144.101	odd	12		768.2.a.h.1.1		1
144.115	odd	12		2304.2.a.l.1.1		1
144.133	even	12		2304.2.a.e.1.1		1
180.7	even	12		7200.2.d.g.2449.2		2
180.43	even	12		7200.2.d.d.2449.1		2
180.47	odd	12		2400.2.d.b.49.2		2
180.79	odd	6		7200.2.k.d.3601.1		2
180.83	odd	12		2400.2.d.c.49.1		2
180.119	even	6		2400.2.k.a.1201.2		2
252.83	odd	6		4704.2.c.a.2353.2		2
360.29	odd	6		600.2.k.b.301.2		2
360.43	even	12		7200.2.d.g.2449.1		2
360.83	odd	12		2400.2.d.b.49.1		2
360.133	odd	12		1800.2.d.i.1549.1		2
360.173	even	12		600.2.d.b.349.2		2
360.187	even	12		7200.2.d.d.2449.2		2
360.227	odd	12		2400.2.d.c.49.2		2
360.259	odd	6		7200.2.k.d.3601.2		2
360.277	odd	12		1800.2.d.b.1549.2		2
360.299	even	6		2400.2.k.a.1201.1		2
360.317	even	12		600.2.d.c.349.1		2
360.349	even	6		1800.2.k.a.901.1		2
504.83	odd	6		4704.2.c.a.2353.1		2
504.461	even	6		1176.2.c.a.589.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
24.2.d.a.13.1	✓	2	72.29	odd	6
24.2.d.a.13.2	yes	2	9.2	odd	6
72.2.d.b.37.1		2	9.7	even	3
72.2.d.b.37.2		2	72.61	even	6
96.2.d.a.49.1		2	36.11	even	6
96.2.d.a.49.2		2	72.11	even	6
288.2.d.b.145.1		2	72.43	odd	6
288.2.d.b.145.2		2	36.7	odd	6
600.2.d.b.349.1		2	45.2	even	12
600.2.d.b.349.2		2	360.173	even	12
600.2.d.c.349.1		2	360.317	even	12
600.2.d.c.349.2		2	45.38	even	12
600.2.k.b.301.1		2	45.29	odd	6
600.2.k.b.301.2		2	360.29	odd	6
648.2.n.c.109.1		4	9.4	even	3	inner
648.2.n.c.109.2		4	72.13	even	6	inner
648.2.n.c.541.1		4	8.5	even	2	inner
648.2.n.c.541.2		4	1.1	even	1	trivial
648.2.n.k.109.1		4	72.5	odd	6
648.2.n.k.109.2		4	9.5	odd	6
648.2.n.k.541.1		4	3.2	odd	2
648.2.n.k.541.2		4	24.5	odd	2
768.2.a.a.1.1		1	144.29	odd	12
768.2.a.d.1.1		1	144.11	even	12
768.2.a.e.1.1		1	144.83	even	12
768.2.a.h.1.1		1	144.101	odd	12
1176.2.c.a.589.1		2	504.461	even	6
1176.2.c.a.589.2		2	63.20	even	6
1800.2.d.b.1549.1		2	45.43	odd	12
1800.2.d.b.1549.2		2	360.277	odd	12
1800.2.d.i.1549.1		2	360.133	odd	12
1800.2.d.i.1549.2		2	45.7	odd	12
1800.2.k.a.901.1		2	360.349	even	6
1800.2.k.a.901.2		2	45.34	even	6
2304.2.a.b.1.1		1	144.43	odd	12
2304.2.a.e.1.1		1	144.133	even	12
2304.2.a.l.1.1		1	144.115	odd	12
2304.2.a.o.1.1		1	144.61	even	12
2400.2.d.b.49.1		2	360.83	odd	12
2400.2.d.b.49.2		2	180.47	odd	12
2400.2.d.c.49.1		2	180.83	odd	12
2400.2.d.c.49.2		2	360.227	odd	12
2400.2.k.a.1201.1		2	360.299	even	6
2400.2.k.a.1201.2		2	180.119	even	6
2592.2.r.f.433.1		4	72.59	even	6
2592.2.r.f.433.2		4	36.23	even	6
2592.2.r.f.2161.1		4	12.11	even	2
2592.2.r.f.2161.2		4	24.11	even	2
2592.2.r.g.433.1		4	36.31	odd	6
2592.2.r.g.433.2		4	72.67	odd	6
2592.2.r.g.2161.1		4	8.3	odd	2
2592.2.r.g.2161.2		4	4.3	odd	2
4704.2.c.a.2353.1		2	504.83	odd	6
4704.2.c.a.2353.2		2	252.83	odd	6
7200.2.d.d.2449.1		2	180.43	even	12
7200.2.d.d.2449.2		2	360.187	even	12
7200.2.d.g.2449.1		2	360.43	even	12
7200.2.d.g.2449.2		2	180.7	even	12
7200.2.k.d.3601.1		2	180.79	odd	6
7200.2.k.d.3601.2		2	360.259	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 648 = 2^{3} \cdot 3^{4} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	648.n (of order \(6\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(0.366025 + 1.36603i) q^{2} +(-1.73205 + 1.00000i) q^{4} +(1.73205 - 1.00000i) q^{5} +(1.00000 - 1.73205i) q^{7} +(-2.00000 - 2.00000i) q^{8} +O(q^{10})\)	\(q+(0.366025 + 1.36603i) q^{2} +(-1.73205 + 1.00000i) q^{4} +(1.73205 - 1.00000i) q^{5} +(1.00000 - 1.73205i) q^{7} +(-2.00000 - 2.00000i) q^{8} +(2.00000 + 2.00000i) q^{10} +(3.46410 - 2.00000i) q^{13} +(2.73205 + 0.732051i) q^{14} +(2.00000 - 3.46410i) q^{16} +2.00000 q^{17} -4.00000i q^{19} +(-2.00000 + 3.46410i) q^{20} +(2.00000 + 3.46410i) q^{23} +(-0.500000 + 0.866025i) q^{25} +(4.00000 + 4.00000i) q^{26} +4.00000i q^{28} +(5.19615 + 3.00000i) q^{29} +(-1.00000 - 1.73205i) q^{31} +(5.46410 + 1.46410i) q^{32} +(0.732051 + 2.73205i) q^{34} -4.00000i q^{35} -8.00000i q^{37} +(5.46410 - 1.46410i) q^{38} +(-5.46410 - 1.46410i) q^{40} +(1.00000 + 1.73205i) q^{41} +(-3.46410 - 2.00000i) q^{43} +(-4.00000 + 4.00000i) q^{46} +(-6.00000 + 10.3923i) q^{47} +(1.50000 + 2.59808i) q^{49} +(-1.36603 - 0.366025i) q^{50} +(-4.00000 + 6.92820i) q^{52} +6.00000i q^{53} +(-5.46410 + 1.46410i) q^{56} +(-2.19615 + 8.19615i) q^{58} +(3.46410 - 2.00000i) q^{59} +(2.00000 - 2.00000i) q^{62} +8.00000i q^{64} +(4.00000 - 6.92820i) q^{65} +(10.3923 - 6.00000i) q^{67} +(-3.46410 + 2.00000i) q^{68} +(5.46410 - 1.46410i) q^{70} -12.0000 q^{71} -6.00000 q^{73} +(10.9282 - 2.92820i) q^{74} +(4.00000 + 6.92820i) q^{76} +(-5.00000 + 8.66025i) q^{79} -8.00000i q^{80} +(-2.00000 + 2.00000i) q^{82} +(-13.8564 - 8.00000i) q^{83} +(3.46410 - 2.00000i) q^{85} +(1.46410 - 5.46410i) q^{86} +10.0000 q^{89} -8.00000i q^{91} +(-6.92820 - 4.00000i) q^{92} +(-16.3923 - 4.39230i) q^{94} +(-4.00000 - 6.92820i) q^{95} +(1.00000 - 1.73205i) q^{97} +(-3.00000 + 3.00000i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 2 q^{2} + 4 q^{7} - 8 q^{8}+O(q^{10}) \) 4 * q - 2 * q^2 + 4 * q^7 - 8 * q^8	\( 4 q - 2 q^{2} + 4 q^{7} - 8 q^{8} + 8 q^{10} + 4 q^{14} + 8 q^{16} + 8 q^{17} - 8 q^{20} + 8 q^{23} - 2 q^{25} + 16 q^{26} - 4 q^{31} + 8 q^{32} - 4 q^{34} + 8 q^{38} - 8 q^{40} + 4 q^{41} - 16 q^{46} - 24 q^{47} + 6 q^{49} - 2 q^{50} - 16 q^{52} - 8 q^{56} + 12 q^{58} + 8 q^{62} + 16 q^{65} + 8 q^{70} - 48 q^{71} - 24 q^{73} + 16 q^{74} + 16 q^{76} - 20 q^{79} - 8 q^{82} - 8 q^{86} + 40 q^{89} - 24 q^{94} - 16 q^{95} + 4 q^{97} - 12 q^{98}+O(q^{100}) \) 4 * q - 2 * q^2 + 4 * q^7 - 8 * q^8 + 8 * q^10 + 4 * q^14 + 8 * q^16 + 8 * q^17 - 8 * q^20 + 8 * q^23 - 2 * q^25 + 16 * q^26 - 4 * q^31 + 8 * q^32 - 4 * q^34 + 8 * q^38 - 8 * q^40 + 4 * q^41 - 16 * q^46 - 24 * q^47 + 6 * q^49 - 2 * q^50 - 16 * q^52 - 8 * q^56 + 12 * q^58 + 8 * q^62 + 16 * q^65 + 8 * q^70 - 48 * q^71 - 24 * q^73 + 16 * q^74 + 16 * q^76 - 20 * q^79 - 8 * q^82 - 8 * q^86 + 40 * q^89 - 24 * q^94 - 16 * q^95 + 4 * q^97 - 12 * q^98

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