LMFDB - Embedded newform 648.2.n.c.109.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [648,2,Mod(109,648)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage: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")

magma://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:traces = [4,-2,0,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)

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			109.1
Root			$0.866025 + 0.500000i$ of defining polynomial
Character	$\chi$	$=$	648.109
Dual form			648.2.n.c.541.1

$q$ -expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

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

Character values

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

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

Coefficient data

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

Display

a_p

with

p

up to: 50 250 1000 (See

a_n

instead) (See

a_n

instead) (See

a_n

instead) Display

a_n

with

n

up to: 50 250 1000 (See only

a_p

) (See only

a_p

) (See only

a_p

)

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

$2$	−1.36603	−	0.366025i	−0.965926	−	0.258819i
$2$	−1.36603	−	0.366025i	−0.965926	−	0.258819i
$3$		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	$-0.480917\pi$
$5$	−1.73205	−	1.00000i	−0.774597	−	0.447214i	−0.834512	+	0.550990i	$0.814250\pi$
$6$		0			0
$7$	1.00000	+	1.73205i	0.377964	+	0.654654i	0.990766	−	0.135583i	$-0.0432908\pi$
$7$	1.00000	+	1.73205i	0.377964	+	0.654654i	−0.612801	+	0.790237i	$0.709957\pi$
$8$	−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	$-0.666667\pi$
$11$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$12$		0			0
$13$	−3.46410	−	2.00000i	−0.960769	−	0.554700i	−0.0643593	−	0.997927i	$-0.520500\pi$
$13$	−3.46410	−	2.00000i	−0.960769	−	0.554700i	−0.896410	+	0.443227i	$0.853834\pi$
$14$	−0.732051	−	2.73205i	−0.195649	−	0.730171i
$15$		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	−0.578610	−	0.815604i	$-0.696405\pi$
$23$	2.00000	−	3.46410i	0.417029	−	0.722315i	0.995639	+	0.0932891i	$0.0297381\pi$
$24$		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	$-0.854747\pi$
$29$	−5.19615	+	3.00000i	−0.964901	+	0.557086i	−0.0672232	+	0.997738i	$0.521414\pi$
$30$		0			0
$31$	−1.00000	+	1.73205i	−0.179605	+	0.311086i	−0.941745	−	0.336327i	$-0.890815\pi$
$31$	−1.00000	+	1.73205i	−0.179605	+	0.311086i	0.762140	+	0.647412i	$0.224149\pi$
$32$	−1.46410	−	5.46410i	−0.258819	−	0.965926i
$33$		0			0
$34$	−2.73205	−	0.732051i	−0.468543	−	0.125546i
$35$		−	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	$-0.228427\pi$
$38$	−1.46410	+	5.46410i	−0.237509	+	0.886394i
$39$		0			0
$40$	1.46410	+	5.46410i	0.231495	+	0.863950i
$41$	1.00000	−	1.73205i	0.156174	−	0.270501i	−0.777312	−	0.629115i	$-0.783417\pi$
$41$	1.00000	−	1.73205i	0.156174	−	0.270501i	0.933486	+	0.358614i	$0.116751\pi$
$42$		0			0
$43$	3.46410	−	2.00000i	0.528271	−	0.304997i	−0.212041	−	0.977261i	$-0.568011\pi$
$43$	3.46410	−	2.00000i	0.528271	−	0.304997i	0.740312	+	0.672264i	$0.234678\pi$
$44$		0			0
$45$		0			0
$46$	−4.00000	+	4.00000i	−0.589768	+	0.589768i
$47$	−6.00000	−	10.3923i	−0.875190	−	1.51587i	−0.856560	−	0.516047i	$-0.827403\pi$
$47$	−6.00000	−	10.3923i	−0.875190	−	1.51587i	−0.0186297	−	0.999826i	$-0.505930\pi$
$48$		0			0
$49$	1.50000	−	2.59808i	0.214286	−	0.371154i
$50$	0.366025	+	1.36603i	0.0517638	+	0.193185i
$51$		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
$56$	1.46410	−	5.46410i	0.195649	−	0.730171i
$57$		0			0
$58$	8.19615	−	2.19615i	1.07621	−	0.288369i
$59$	−3.46410	−	2.00000i	−0.450988	−	0.260378i	0.257260	−	0.966342i	$-0.417180\pi$
$59$	−3.46410	−	2.00000i	−0.450988	−	0.260378i	−0.708247	+	0.705965i	$0.750514\pi$
$60$		0			0
$61$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$61$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$62$	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	$-0.595222\pi$
$67$	−10.3923	−	6.00000i	−1.26962	−	0.733017i	−0.974916	+	0.222571i	$0.928555\pi$
$68$	3.46410	+	2.00000i	0.420084	+	0.242536i
$69$		0			0
$70$	−1.46410	+	5.46410i	−0.174994	+	0.653085i
$71$	−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	$0.614200\pi$
$74$	−2.92820	+	10.9282i	−0.340397	+	1.27038i
$75$		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	−0.997274	−	0.0737937i	$-0.976489\pi$
$79$	−5.00000	−	8.66025i	−0.562544	−	0.974355i	0.434730	−	0.900561i	$-0.356844\pi$
$80$		−	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	0.521243	−	0.853408i	$-0.325468\pi$
$83$	13.8564	−	8.00000i	1.52094	−	0.878114i	0.999695	−	0.0247060i	$-0.00786498\pi$
$84$		0			0
$85$	−3.46410	−	2.00000i	−0.375735	−	0.216930i
$86$	−5.46410	+	1.46410i	−0.589209	+	0.157878i
$87$		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
$94$	4.39230	+	16.3923i	0.453032	+	1.69074i
$95$	−4.00000	+	6.92820i	−0.410391	+	0.710819i
$96$		0			0
$97$	1.00000	+	1.73205i	0.101535	+	0.175863i	0.912317	−	0.409484i	$-0.134291\pi$
$97$	1.00000	+	1.73205i	0.101535	+	0.175863i	−0.810782	+	0.585348i	$0.800958\pi$
$98$	−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	$-0.832422\pi$
$101$	−8.66025	+	5.00000i	−0.861727	+	0.497519i	0.00286291	+	0.999996i	$0.499089\pi$
$102$		0			0
$103$	3.00000	−	5.19615i	0.295599	−	0.511992i	−0.679525	−	0.733652i	$-0.737814\pi$
$103$	3.00000	−	5.19615i	0.295599	−	0.511992i	0.975124	+	0.221660i	$0.0711475\pi$
$104$	2.92820	+	10.9282i	0.287134	+	1.07160i
$105$		0			0
$106$	2.19615	−	8.19615i	0.213309	−	0.796081i
$107$		−	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	−0.971930	−	0.235269i	$-0.924403\pi$
$113$	−3.00000	+	5.19615i	−0.282216	+	0.488813i	0.689714	+	0.724082i	$0.257736\pi$
$114$		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	$0.528283\pi$
$128$	2.92820	−	10.9282i	0.258819	−	0.965926i
$129$		0			0
$130$	−2.92820	−	10.9282i	−0.256820	−	0.958467i
$131$	17.3205	+	10.0000i	1.51330	+	0.873704i	0.999879	+	0.0155672i	$0.00495539\pi$
$131$	17.3205	+	10.0000i	1.51330	+	0.873704i	0.513421	+	0.858137i	$0.328378\pi$
$132$		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	0.938148	+	0.346235i	$0.112540\pi$
$137$	9.00000	+	15.5885i	0.768922	+	1.33181i	−0.169226	+	0.985577i	$0.554127\pi$
$138$		0			0
$139$	3.46410	+	2.00000i	0.293821	+	0.169638i	0.639664	−	0.768655i	$-0.279074\pi$
$139$	3.46410	+	2.00000i	0.293821	+	0.169638i	−0.345843	+	0.938293i	$0.612407\pi$
$140$	4.00000	−	6.92820i	0.338062	−	0.585540i
$141$		0			0
$142$	16.3923	+	4.39230i	1.37561	+	0.368594i
$143$		0			0
$144$		0			0
$145$	12.0000			0.996546
$146$	8.19615	+	2.19615i	0.678318	+	0.181755i
$147$		0			0
$148$	8.00000	−	13.8564i	0.657596	−	1.13899i
$149$	5.19615	+	3.00000i	0.425685	+	0.245770i	0.697507	−	0.716578i	$-0.254293\pi$
$149$	5.19615	+	3.00000i	0.425685	+	0.245770i	−0.271821	+	0.962348i	$0.587626\pi$
$150$		0			0
$151$	9.00000	+	15.5885i	0.732410	+	1.26857i	0.955851	+	0.293853i	$0.0949377\pi$
$151$	9.00000	+	15.5885i	0.732410	+	1.26857i	−0.223441	+	0.974717i	$0.571729\pi$
$152$	−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	$-0.229907\pi$
$157$	6.92820	+	4.00000i	0.552931	+	0.319235i	−0.197372	+	0.980329i	$0.563241\pi$
$158$	3.66025	+	13.6603i	0.291194	+	1.08675i
$159$		0			0
$160$	−2.92820	+	10.9282i	−0.231495	+	0.863950i
$161$	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
$166$	−21.8564	+	5.85641i	−1.69639	+	0.454545i
$167$	4.00000	−	6.92820i	0.309529	−	0.536120i	−0.668730	−	0.743505i	$-0.733162\pi$
$167$	4.00000	−	6.92820i	0.309529	−	0.536120i	0.978259	+	0.207385i	$0.0664952\pi$
$168$		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	$-0.593420\pi$
$173$	5.19615	−	3.00000i	0.395056	−	0.228086i	0.684349	+	0.729155i	$0.260087\pi$
$174$		0			0
$175$	1.00000	−	1.73205i	0.0755929	−	0.130931i
$176$		0			0
$177$		0			0
$178$	−13.6603	−	3.66025i	−1.02388	−	0.274348i
$179$			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	$0.733262\pi$
$182$	−2.92820	+	10.9282i	−0.217053	+	0.810052i
$183$		0			0
$184$	−10.9282	+	2.92820i	−0.805638	+	0.215870i
$185$	−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	0.684244	−	0.729253i	$-0.260132\pi$
$191$	−4.00000	−	6.92820i	−0.289430	−	0.501307i	−0.973674	+	0.227946i	$0.926799\pi$
$192$		0			0
$193$	3.00000	−	5.19615i	0.215945	−	0.374027i	−0.737620	−	0.675216i	$-0.764050\pi$
$193$	3.00000	−	5.19615i	0.215945	−	0.374027i	0.953564	+	0.301189i	$0.0973836\pi$
$194$	−0.732051	−	2.73205i	−0.0525582	−	0.196150i
$195$		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	$0.384671\pi$
$200$	−0.732051	+	2.73205i	−0.0517638	+	0.193185i
$201$		0			0
$202$	13.6603	−	3.66025i	0.961132	−	0.257535i
$203$	−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	$-0.0916343\pi$
$211$	17.3205	+	10.0000i	1.19239	+	0.688428i	0.233544	+	0.972346i	$0.424968\pi$
$212$	−6.00000	+	10.3923i	−0.412082	+	0.713746i
$213$		0			0
$214$	−4.39230	+	16.3923i	−0.300252	+	1.12055i
$215$	−8.00000			−0.545595
$216$		0			0
$217$	−4.00000			−0.271538
$218$	−1.46410	+	5.46410i	−0.0991615	+	0.370076i
$219$		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	0.530607	−	0.847618i	$-0.321964\pi$
$223$	−7.00000	−	12.1244i	−0.468755	−	0.811907i	−0.999362	+	0.0357107i	$0.988630\pi$
$224$	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	$-0.581133\pi$
$227$	6.92820	−	4.00000i	0.459841	−	0.265489i	0.711977	+	0.702202i	$0.247800\pi$
$228$		0			0
$229$	3.46410	+	2.00000i	0.228914	+	0.132164i	0.610071	−	0.792347i	$-0.291141\pi$
$229$	3.46410	+	2.00000i	0.228914	+	0.132164i	−0.381157	+	0.924510i	$0.624474\pi$
$230$	10.9282	−	2.92820i	0.720584	−	0.193080i
$231$		0			0
$232$	16.3923	+	4.39230i	1.07621	+	0.288369i
$233$	−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
$238$	−1.46410	−	5.46410i	−0.0949036	−	0.354185i
$239$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$239$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$240$		0			0
$241$	−1.00000	−	1.73205i	−0.0644157	−	0.111571i	0.832019	−	0.554747i	$-0.187185\pi$
$241$	−1.00000	−	1.73205i	−0.0644157	−	0.111571i	−0.896435	+	0.443176i	$0.853852\pi$
$242$	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
$248$	5.46410	−	1.46410i	0.346971	−	0.0929705i
$249$		0			0
$250$	4.39230	−	16.3923i	0.277794	−	1.03674i
$251$		0			0		1.00000			$0$
$251$		0			0		−1.00000			$\pi$
$252$		0			0
$253$		0			0
$254$	2.73205	+	0.732051i	0.171424	+	0.0459330i
$255$		0			0
$256$	−8.00000	+	13.8564i	−0.500000	+	0.866025i
$257$	9.00000	−	15.5885i	0.561405	−	0.972381i	−0.435970	−	0.899961i	$-0.643595\pi$
$257$	9.00000	−	15.5885i	0.561405	−	0.972381i	0.997374	−	0.0724199i	$-0.0230722\pi$
$258$		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	0.506669	−	0.862141i	$-0.330877\pi$
$263$	−8.00000	−	13.8564i	−0.493301	−	0.854423i	−0.999970	+	0.00771799i	$0.997543\pi$
$264$		0			0
$265$	6.00000	−	10.3923i	0.368577	−	0.638394i
$266$	−10.9282	+	2.92820i	−0.670051	+	0.179540i
$267$		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
$274$	−6.58846	−	24.5885i	−0.398023	−	1.48544i
$275$		0			0
$276$		0			0
$277$	−24.2487	+	14.0000i	−1.45696	+	0.841178i	−0.998861	−	0.0477206i	$-0.984804\pi$
$277$	−24.2487	+	14.0000i	−1.45696	+	0.841178i	−0.458103	+	0.888899i	$0.651471\pi$
$278$	−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	−0.999069	−	0.0431402i	$-0.986264\pi$
$281$	−9.00000	−	15.5885i	−0.536895	−	0.929929i	0.462174	−	0.886789i	$-0.347070\pi$
$282$		0			0
$283$	−3.46410	−	2.00000i	−0.205919	−	0.118888i	0.393494	−	0.919327i	$-0.371266\pi$
$283$	−3.46410	−	2.00000i	−0.205919	−	0.118888i	−0.599414	+	0.800439i	$0.704600\pi$
$284$	−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
$290$	−16.3923	−	4.39230i	−0.962589	−	0.257925i
$291$		0			0
$292$	−10.3923	−	6.00000i	−0.608164	−	0.351123i
$293$	−5.19615	−	3.00000i	−0.303562	−	0.175262i	0.340480	−	0.940252i	$-0.389411\pi$
$293$	−5.19615	−	3.00000i	−0.303562	−	0.175262i	−0.644042	+	0.764990i	$0.722744\pi$
$294$		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
$302$	−6.58846	−	24.5885i	−0.379123	−	1.41491i
$303$		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
$310$	−5.46410	+	1.46410i	−0.310340	+	0.0831554i
$311$	−4.00000	+	6.92820i	−0.226819	+	0.392862i	−0.956864	−	0.290537i	$-0.906166\pi$
$311$	−4.00000	+	6.92820i	−0.226819	+	0.392862i	0.730044	+	0.683400i	$0.239499\pi$
$312$		0			0
$313$	−7.00000	−	12.1244i	−0.395663	−	0.685309i	0.597522	−	0.801852i	$-0.296152\pi$
$313$	−7.00000	−	12.1244i	−0.395663	−	0.685309i	−0.993186	+	0.116543i	$0.962819\pi$
$314$	−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	$-0.878651\pi$
$317$	−19.0526	+	11.0000i	−1.07010	+	0.617822i	−0.141890	+	0.989882i	$0.545318\pi$
$318$		0			0
$319$		0			0
$320$	8.00000	−	13.8564i	0.447214	−	0.774597i
$321$		0			0
$322$	−10.9282	−	2.92820i	−0.609005	−	0.163182i
$323$		−	8.00000i		−	0.445132i
$324$		0			0
$325$			4.00000i			0.221880i
$326$	1.46410	−	5.46410i	0.0810891	−	0.302629i
$327$		0			0
$328$	−5.46410	+	1.46410i	−0.301705	+	0.0808415i
$329$	12.0000	−	20.7846i	0.661581	−	1.14589i
$330$		0			0
$331$	−17.3205	+	10.0000i	−0.952021	+	0.549650i	−0.893708	−	0.448649i	$-0.851905\pi$
$331$	−17.3205	+	10.0000i	−0.952021	+	0.549650i	−0.0583130	+	0.998298i	$0.518572\pi$
$332$	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	−0.837503	−	0.546433i	$-0.815985\pi$
$337$	1.00000	−	1.73205i	0.0544735	−	0.0943508i	0.891976	+	0.452082i	$0.149319\pi$
$338$	−1.09808	−	4.09808i	−0.0597275	−	0.222906i
$339$		0			0
$340$	−4.00000	−	6.92820i	−0.216930	−	0.375735i
$341$		0			0
$342$		0			0
$343$	20.0000			1.07990
$344$	−10.9282	−	2.92820i	−0.589209	−	0.157878i
$345$		0			0
$346$	−8.19615	+	2.19615i	−0.440628	+	0.118066i
$347$	−6.92820	−	4.00000i	−0.371925	−	0.214731i	0.302374	−	0.953189i	$-0.402221\pi$
$347$	−6.92820	−	4.00000i	−0.371925	−	0.214731i	−0.674299	+	0.738458i	$0.735554\pi$
$348$		0			0
$349$	13.8564	−	8.00000i	0.741716	−	0.428230i	−0.0809766	−	0.996716i	$-0.525804\pi$
$349$	13.8564	−	8.00000i	0.741716	−	0.428230i	0.822693	+	0.568486i	$0.192471\pi$
$350$	−2.00000	+	2.00000i	−0.106904	+	0.106904i
$351$		0			0
$352$		0			0
$353$	−3.00000	−	5.19615i	−0.159674	−	0.276563i	0.775077	−	0.631867i	$-0.217711\pi$
$353$	−3.00000	−	5.19615i	−0.159674	−	0.276563i	−0.934751	+	0.355303i	$0.884378\pi$
$354$		0			0
$355$	20.7846	+	12.0000i	1.10313	+	0.636894i
$356$	17.3205	+	10.0000i	0.917985	+	0.529999i
$357$		0			0
$358$	1.46410	−	5.46410i	0.0773802	−	0.288787i
$359$	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
$362$	7.32051	−	27.3205i	0.384757	−	1.43593i
$363$		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	0.529607	−	0.848243i	$-0.322339\pi$
$367$	−9.00000	−	15.5885i	−0.469796	−	0.813711i	−0.999404	+	0.0345320i	$0.989006\pi$
$368$	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	$-0.197386\pi$
$373$	13.8564	+	8.00000i	0.717458	+	0.414224i	−0.0963587	+	0.995347i	$0.530720\pi$
$374$		0			0
$375$		0			0
$376$	−8.78461	+	32.7846i	−0.453032	+	1.69074i
$377$	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
$382$	2.92820	+	10.9282i	0.149820	+	0.559136i
$383$	12.0000	−	20.7846i	0.613171	−	1.06204i	−0.377531	−	0.925997i	$-0.623227\pi$
$383$	12.0000	−	20.7846i	0.613171	−	1.06204i	0.990702	−	0.136047i	$-0.0434398\pi$
$384$		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	$-0.335920\pi$
$389$	29.4449	−	17.0000i	1.49291	−	0.861934i	0.999967	+	0.00812520i	$0.00258636\pi$
$390$		0			0
$391$	4.00000	−	6.92820i	0.202289	−	0.350374i
$392$	−8.19615	+	2.19615i	−0.413968	+	0.110922i
$393$		0			0
$394$	−0.732051	+	2.73205i	−0.0368802	+	0.137639i
$395$			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	$0.703227\pi$
$398$	−13.6603	−	3.66025i	−0.684727	−	0.183472i
$399$		0			0
$400$	2.00000	−	3.46410i	0.100000	−	0.173205i
$401$	−9.00000	+	15.5885i	−0.449439	+	0.778450i	−0.998350	−	0.0574304i	$-0.981709\pi$
$401$	−9.00000	+	15.5885i	−0.449439	+	0.778450i	0.548911	+	0.835881i	$0.315043\pi$
$402$		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	−0.962757	−	0.270367i	$-0.912855\pi$
$409$	−5.00000	+	8.66025i	−0.247234	+	0.428222i	0.715523	+	0.698589i	$0.246188\pi$
$410$	5.46410	−	1.46410i	0.269853	−	0.0723068i
$411$		0			0
$412$	10.3923	−	6.00000i	0.511992	−	0.295599i
$413$		−	8.00000i		−	0.393654i
$414$		0			0
$415$	−32.0000			−1.57082
$416$	−5.85641	+	21.8564i	−0.287134	+	1.07160i
$417$		0			0
$418$		0			0
$419$	−20.7846	−	12.0000i	−1.01539	−	0.586238i	−0.102628	−	0.994720i	$-0.532725\pi$
$419$	−20.7846	−	12.0000i	−1.01539	−	0.586238i	−0.912767	+	0.408481i	$0.866058\pi$
$420$		0			0
$421$	17.3205	−	10.0000i	0.844150	−	0.487370i	−0.0145228	−	0.999895i	$-0.504623\pi$
$421$	17.3205	−	10.0000i	0.844150	−	0.487370i	0.858673	+	0.512524i	$0.171290\pi$
$422$	−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
$430$	10.9282	+	2.92820i	0.527005	+	0.141210i
$431$	−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	$0.390791\pi$
$434$	5.46410	+	1.46410i	0.262285	+	0.0702791i
$435$		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	0.962607	+	0.270901i	$0.0873217\pi$
$439$	15.0000	+	25.9808i	0.715911	+	1.23999i	−0.246696	+	0.969093i	$0.579345\pi$
$440$		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	$-0.859777\pi$
$443$	−20.7846	+	12.0000i	−0.987507	+	0.570137i	−0.0829786	+	0.996551i	$0.526443\pi$
$444$		0			0
$445$	−17.3205	−	10.0000i	−0.821071	−	0.474045i
$446$	5.12436	+	19.1244i	0.242645	+	0.905564i
$447$		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
$454$	−10.9282	+	2.92820i	−0.512886	+	0.137427i
$455$	−8.00000	+	13.8564i	−0.375046	+	0.649598i
$456$		0			0
$457$	11.0000	+	19.0526i	0.514558	+	0.891241i	0.999857	+	0.0168929i	$0.00537742\pi$
$457$	11.0000	+	19.0526i	0.514558	+	0.891241i	−0.485299	+	0.874348i	$0.661289\pi$
$458$	−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	$-0.420465\pi$
$461$	25.9808	−	15.0000i	1.21004	−	0.698620i	0.962769	+	0.270326i	$0.0871313\pi$
$462$		0			0
$463$	13.0000	−	22.5167i	0.604161	−	1.04644i	−0.388022	−	0.921650i	$-0.626842\pi$
$463$	13.0000	−	22.5167i	0.604161	−	1.04644i	0.992183	−	0.124788i	$-0.0398251\pi$
$464$	−20.7846	−	12.0000i	−0.964901	−	0.557086i
$465$		0			0
$466$	19.1244	+	5.12436i	0.885919	+	0.237381i
$467$			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
$470$	8.78461	−	32.7846i	0.405204	−	1.51224i
$471$		0			0
$472$	2.92820	+	10.9282i	0.134781	+	0.503011i
$473$		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	0.998796	−	0.0490589i	$-0.0156222\pi$
$479$	10.0000	+	17.3205i	0.456912	+	0.791394i	−0.541884	+	0.840453i	$0.682289\pi$
$480$		0			0
$481$	−16.0000	+	27.7128i	−0.729537	+	1.26360i
$482$	0.732051	+	2.73205i	0.0333440	+	0.124442i
$483$		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
$490$	8.19615	−	2.19615i	0.370264	−	0.0992121i
$491$	17.3205	+	10.0000i	0.781664	+	0.451294i	0.837020	−	0.547173i	$-0.184296\pi$
$491$	17.3205	+	10.0000i	0.781664	+	0.451294i	−0.0553560	+	0.998467i	$0.517629\pi$
$492$		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	$-0.631593\pi$
$499$	−31.1769	−	18.0000i	−1.39567	−	0.805791i	−0.993935	+	0.109965i	$0.964926\pi$
$500$	−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	$-0.290881\pi$
$509$	5.19615	+	3.00000i	0.230315	+	0.132973i	−0.380402	+	0.924821i	$0.624214\pi$
$510$		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
$518$	−21.8564	+	5.85641i	−0.960315	+	0.257316i
$519$		0			0
$520$	5.85641	−	21.8564i	0.256820	−	0.958467i
$521$	−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
$526$	5.85641	+	21.8564i	0.255351	+	0.952985i
$527$	−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
$536$	8.78461	+	32.7846i	0.379437	+	1.41608i
$537$		0			0
$538$	−2.19615	+	8.19615i	−0.0946829	+	0.353361i
$539$		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	$-0.141463\pi$
$542$	24.5885	+	6.58846i	1.05616	+	0.282998i
$543$		0			0
$544$	−2.92820	−	10.9282i	−0.125546	−	0.468543i
$545$	−4.00000	+	6.92820i	−0.171341	+	0.296772i
$546$		0			0
$547$	−24.2487	+	14.0000i	−1.03680	+	0.598597i	−0.918925	−	0.394432i	$-0.870941\pi$
$547$	−24.2487	+	14.0000i	−1.03680	+	0.598597i	−0.117875	+	0.993028i	$0.537608\pi$
$548$			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
$554$	38.2487	−	10.2487i	1.62503	−	0.435426i
$555$		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
$562$	6.58846	+	24.5885i	0.277917	+	1.03720i
$563$	20.7846	+	12.0000i	0.875967	+	0.505740i	0.869326	−	0.494238i	$-0.164553\pi$
$563$	20.7846	+	12.0000i	0.875967	+	0.505740i	0.00664037	+	0.999978i	$0.497886\pi$
$564$		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	−0.987786	−	0.155815i	$-0.950200\pi$
$569$	−15.0000	−	25.9808i	−0.628833	−	1.08917i	0.358954	−	0.933355i	$-0.383134\pi$
$570$		0			0
$571$	−17.3205	−	10.0000i	−0.724841	−	0.418487i	0.0916910	−	0.995788i	$-0.470773\pi$
$571$	−17.3205	−	10.0000i	−0.724841	−	0.418487i	−0.816532	+	0.577301i	$0.804106\pi$
$572$		0			0
$573$		0			0
$574$	−5.46410	−	1.46410i	−0.228067	−	0.0611104i
$575$	−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	$0.838641\pi$
$578$	17.7583	+	4.75833i	0.738649	+	0.197920i
$579$		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	$-0.470562\pi$
$587$	24.2487	−	14.0000i	1.00085	−	0.577842i	0.908500	+	0.417885i	$0.137228\pi$
$588$		0			0
$589$	6.92820	+	4.00000i	0.285472	+	0.164817i
$590$	−2.92820	−	10.9282i	−0.120552	−	0.449907i
$591$		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
$598$	21.8564	−	5.85641i	0.893775	−	0.239486i
$599$	−10.0000	+	17.3205i	−0.408589	+	0.707697i	−0.994732	−	0.102511i	$-0.967312\pi$
$599$	−10.0000	+	17.3205i	−0.408589	+	0.707697i	0.586143	+	0.810208i	$0.300646\pi$
$600$		0			0
$601$	−1.00000	−	1.73205i	−0.0407909	−	0.0706518i	0.844909	−	0.534910i	$-0.179654\pi$
$601$	−1.00000	−	1.73205i	−0.0407909	−	0.0706518i	−0.885700	+	0.464258i	$0.846321\pi$
$602$	−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	−0.845017	−	0.534739i	$-0.820410\pi$
$607$	1.00000	−	1.73205i	0.0405887	−	0.0703018i	0.885606	+	0.464437i	$0.153743\pi$
$608$	−21.8564	+	5.85641i	−0.886394	+	0.237509i
$609$		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	$-0.104731\pi$
$614$	4.39230	−	16.3923i	0.177259	−	0.661540i
$615$		0			0
$616$		0			0
$617$	−1.00000	+	1.73205i	−0.0402585	+	0.0697297i	−0.885453	−	0.464730i	$-0.846151\pi$
$617$	−1.00000	+	1.73205i	−0.0402585	+	0.0697297i	0.845194	+	0.534460i	$0.179485\pi$
$618$		0			0
$619$	31.1769	−	18.0000i	1.25311	−	0.723481i	0.281381	−	0.959596i	$-0.409208\pi$
$619$	31.1769	−	18.0000i	1.25311	−	0.723481i	0.971725	+	0.236115i	$0.0758742\pi$
$620$	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
$626$	5.12436	+	19.1244i	0.204810	+	0.764363i
$627$		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	$0.355721\pi$
$632$	−7.32051	+	27.3205i	−0.291194	+	1.08675i
$633$		0			0
$634$	30.0526	−	8.05256i	1.19354	−	0.319808i
$635$	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	0.997253	−	0.0740768i	$-0.0236010\pi$
$641$	11.0000	+	19.0526i	0.434474	+	0.752531i	−0.562779	+	0.826608i	$0.690268\pi$
$642$		0			0
$643$	−3.46410	−	2.00000i	−0.136611	−	0.0788723i	0.430137	−	0.902764i	$-0.358465\pi$
$643$	−3.46410	−	2.00000i	−0.136611	−	0.0788723i	−0.566748	+	0.823891i	$0.691799\pi$
$644$	13.8564	+	8.00000i	0.546019	+	0.315244i
$645$		0			0
$646$	−2.92820	+	10.9282i	−0.115209	+	0.429964i
$647$	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
$650$	1.46410	−	5.46410i	0.0574268	−	0.214320i
$651$		0			0
$652$	−4.00000	+	6.92820i	−0.156652	+	0.271329i
$653$	−5.19615	−	3.00000i	−0.203341	−	0.117399i	0.394872	−	0.918736i	$-0.370789\pi$
$653$	−5.19615	−	3.00000i	−0.203341	−	0.117399i	−0.598213	+	0.801337i	$0.704122\pi$
$654$		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	$-0.914010\pi$
$659$	−31.1769	+	18.0000i	−1.21448	+	0.701180i	−0.250748	+	0.968052i	$0.580677\pi$
$660$		0			0
$661$	−34.6410	−	20.0000i	−1.34738	−	0.777910i	−0.359502	−	0.933144i	$-0.617053\pi$
$661$	−34.6410	−	20.0000i	−1.34738	−	0.777910i	−0.987878	+	0.155235i	$0.950387\pi$
$662$	27.3205	−	7.32051i	1.06184	−	0.284520i
$663$		0			0
$664$	−43.7128	−	11.7128i	−1.69639	−	0.454545i
$665$	−16.0000			−0.620453
$666$		0			0
$667$			24.0000i			0.929284i
$668$	13.8564	−	8.00000i	0.536120	−	0.309529i
$669$		0			0
$670$	−8.78461	−	32.7846i	−0.339379	−	1.26658i
$671$		0			0
$672$		0			0
$673$	−7.00000	−	12.1244i	−0.269830	−	0.467360i	0.698988	−	0.715134i	$-0.253634\pi$
$673$	−7.00000	−	12.1244i	−0.269830	−	0.467360i	−0.968818	+	0.247774i	$0.920301\pi$
$674$	−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	$-0.554241\pi$
$677$	15.5885	−	9.00000i	0.599113	−	0.345898i	0.768693	+	0.639618i	$0.220908\pi$
$678$		0			0
$679$	−2.00000	+	3.46410i	−0.0767530	+	0.132940i
$680$	2.92820	+	10.9282i	0.112291	+	0.419077i
$681$		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
$686$	−27.3205	−	7.32051i	−1.04310	−	0.279498i
$687$		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	$-0.542447\pi$
$691$	17.3205	−	10.0000i	0.658903	−	0.380418i	0.791859	+	0.610704i	$0.209113\pi$
$692$	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
$698$	−21.8564	+	5.85641i	−0.827277	+	0.221668i
$699$		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
$706$	2.19615	+	8.19615i	0.0826533	+	0.308466i
$707$	−17.3205	−	10.0000i	−0.651405	−	0.376089i
$708$		0			0
$709$	−3.46410	+	2.00000i	−0.130097	+	0.0751116i	−0.563636	−	0.826023i	$-0.690598\pi$
$709$	−3.46410	+	2.00000i	−0.130097	+	0.0751116i	0.433539	+	0.901135i	$0.357265\pi$
$710$	−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
$718$	−27.3205	−	7.32051i	−1.01959	−	0.273199i
$719$	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
$722$	−4.09808	−	1.09808i	−0.152515	−	0.0408662i
$723$		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	0.932607	+	0.360894i	$0.117528\pi$
$727$	21.0000	+	36.3731i	0.778847	+	1.34900i	−0.153760	+	0.988108i	$0.549138\pi$
$728$	−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	$-0.356869\pi$
$733$	−3.46410	−	2.00000i	−0.127950	−	0.0738717i	−0.562609	+	0.826723i	$0.690202\pi$
$734$	6.58846	+	24.5885i	0.243184	+	0.907577i
$735$		0			0
$736$	−21.8564	−	5.85641i	−0.805638	−	0.215870i
$737$		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
$742$	16.3923	−	4.39230i	0.601780	−	0.161247i
$743$	−8.00000	+	13.8564i	−0.293492	+	0.508342i	−0.974633	−	0.223810i	$-0.928151\pi$
$743$	−8.00000	+	13.8564i	−0.293492	+	0.508342i	0.681141	+	0.732152i	$0.261484\pi$
$744$		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	−0.883694	−	0.468065i	$-0.844951\pi$
$751$	−1.00000	+	1.73205i	−0.0364905	+	0.0632034i	0.847203	+	0.531269i	$0.178285\pi$
$752$	24.0000	−	41.5692i	0.875190	−	1.51587i
$753$		0			0
$754$	−32.7846	−	8.78461i	−1.19395	−	0.319917i
$755$		−	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	$0.930023\pi$
$758$	−1.46410	+	5.46410i	−0.0531786	+	0.198465i
$759$		0			0
$760$	21.8564	−	5.85641i	0.792815	−	0.212434i
$761$	−19.0000	+	32.9090i	−0.688749	+	1.19295i	0.283493	+	0.958974i	$0.408507\pi$
$761$	−19.0000	+	32.9090i	−0.688749	+	1.19295i	−0.972243	+	0.233975i	$0.924827\pi$
$762$		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	−0.941984	−	0.335657i	$-0.891042\pi$
$769$	−5.00000	+	8.66025i	−0.180305	+	0.312297i	0.761680	+	0.647954i	$0.224375\pi$
$770$		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
$776$	1.46410	−	5.46410i	0.0525582	−	0.196150i
$777$		0			0
$778$	−46.4449	+	12.4449i	−1.66513	+	0.446170i
$779$	−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	$-0.401942\pi$
$787$	−10.3923	−	6.00000i	−0.370446	−	0.213877i	−0.673653	+	0.739048i	$0.735276\pi$
$788$	2.00000	−	3.46410i	0.0712470	−	0.123404i
$789$		0			0
$790$	7.32051	−	27.3205i	0.260452	−	0.972020i
$791$	−12.0000			−0.426671
$792$		0			0
$793$		0			0
$794$	11.7128	−	43.7128i	0.415672	−	1.55131i
$795$		0			0
$796$	17.3205	+	10.0000i	0.613909	+	0.354441i
$797$	1.73205	+	1.00000i	0.0613524	+	0.0354218i	0.530362	−	0.847771i	$-0.322056\pi$
$797$	1.73205	+	1.00000i	0.0613524	+	0.0354218i	−0.469010	+	0.883193i	$0.655389\pi$
$798$		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
$806$	−10.9282	+	2.92820i	−0.384930	+	0.103142i
$807$		0			0
$808$	27.3205	+	7.32051i	0.961132	+	0.257535i
$809$	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	$-0.722498\pi$
$821$	−8.66025	+	5.00000i	−0.302245	+	0.174501i	0.341206	+	0.939989i	$0.389165\pi$
$822$		0			0
$823$	−7.00000	+	12.1244i	−0.244005	+	0.422628i	−0.961851	−	0.273573i	$-0.911795\pi$
$823$	−7.00000	+	12.1244i	−0.244005	+	0.422628i	0.717847	+	0.696201i	$0.245128\pi$
$824$	−16.3923	+	4.39230i	−0.571053	+	0.153013i
$825$		0			0
$826$	−2.92820	+	10.9282i	−0.101885	+	0.380241i
$827$		−	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	$-0.0221285\pi$
$830$	43.7128	+	11.7128i	1.51729	+	0.406558i
$831$		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	0.985397	−	0.170272i	$-0.0544647\pi$
$839$	10.0000	+	17.3205i	0.345238	+	0.597970i	−0.640159	+	0.768243i	$0.721131\pi$
$840$		0			0
$841$	3.50000	−	6.06218i	0.120690	−	0.209041i
$842$	−27.3205	+	7.32051i	−0.941527	+	0.252281i
$843$		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
$850$	0.732051	+	2.73205i	0.0251091	+	0.0937086i
$851$	−27.7128	−	16.0000i	−0.949983	−	0.548473i
$852$		0			0
$853$	20.7846	−	12.0000i	0.711651	−	0.410872i	−0.100021	−	0.994985i	$-0.531891\pi$
$853$	20.7846	−	12.0000i	0.711651	−	0.410872i	0.811672	+	0.584113i	$0.198558\pi$
$854$		0			0
$855$		0			0
$856$	−24.0000	+	24.0000i	−0.820303	+	0.820303i
$857$	9.00000	+	15.5885i	0.307434	+	0.532492i	0.977800	−	0.209539i	$-0.0671963\pi$
$857$	9.00000	+	15.5885i	0.307434	+	0.532492i	−0.670366	+	0.742030i	$0.733863\pi$
$858$		0			0
$859$	38.1051	+	22.0000i	1.30013	+	0.750630i	0.980426	−	0.196887i	$-0.0630833\pi$
$859$	38.1051	+	22.0000i	1.30013	+	0.750630i	0.319704	+	0.947518i	$0.396417\pi$
$860$	−13.8564	−	8.00000i	−0.472500	−	0.272798i
$861$		0			0
$862$	16.3923	+	4.39230i	0.558324	+	0.149602i
$863$	−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
$866$	−19.1244	−	5.12436i	−0.649872	−	0.174133i
$867$		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	$-0.515016\pi$
$877$	−27.7128	−	16.0000i	−0.935795	−	0.540282i	−0.888640	+	0.458606i	$0.848349\pi$
$878$	−10.9808	−	40.9808i	−0.370583	−	1.38303i
$879$		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
$886$	32.7846	−	8.78461i	1.10142	−	0.295125i
$887$	24.0000	−	41.5692i	0.805841	−	1.39576i	−0.109881	−	0.993945i	$-0.535047\pi$
$887$	24.0000	−	41.5692i	0.805841	−	1.39576i	0.915722	−	0.401813i	$-0.131620\pi$
$888$		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
$896$	21.8564	−	5.85641i	0.730171	−	0.195649i
$897$		0			0
$898$	40.9808	+	10.9808i	1.36755	+	0.366433i
$899$		−	12.0000i		−	0.400222i
$900$		0			0
$901$			12.0000i			0.399778i
$902$		0			0
$903$		0			0
$904$	16.3923	−	4.39230i	0.545200	−	0.146086i
$905$	20.0000	−	34.6410i	0.664822	−	1.15151i
$906$		0			0
$907$	−24.2487	+	14.0000i	−0.805165	+	0.464862i	−0.845274	−	0.534333i	$-0.820563\pi$
$907$	−24.2487	+	14.0000i	−0.805165	+	0.464862i	0.0401089	+	0.999195i	$0.487230\pi$
$908$	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	−0.922740	−	0.385422i	$-0.874056\pi$
$911$	−24.0000	−	41.5692i	−0.795155	−	1.37725i	0.127585	−	0.991828i	$-0.459277\pi$
$912$		0			0
$913$		0			0
$914$	−8.05256	−	30.0526i	−0.266355	−	0.994050i
$915$		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	$0.335240\pi$
$920$	21.8564	+	5.85641i	0.720584	+	0.193080i
$921$		0			0
$922$	−40.9808	+	10.9808i	−1.34963	+	0.361632i
$923$	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	−0.905516	−	0.424313i	$-0.860516\pi$
$929$	−25.0000	−	43.3013i	−0.820223	−	1.42067i	0.0852924	−	0.996356i	$-0.472818\pi$
$930$		0			0
$931$	−10.3923	−	6.00000i	−0.340594	−	0.196642i
$932$	−24.2487	−	14.0000i	−0.794293	−	0.458585i
$933$		0			0
$934$	2.92820	−	10.9282i	0.0958137	−	0.357582i
$935$		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	$0.286847\pi$
$938$	−8.78461	+	32.7846i	−0.286828	+	1.07046i
$939$		0			0
$940$	−24.0000	+	41.5692i	−0.782794	+	1.35584i
$941$	−25.9808	−	15.0000i	−0.846949	−	0.488986i	0.0126715	−	0.999920i	$-0.495966\pi$
$941$	−25.9808	−	15.0000i	−0.846949	−	0.488986i	−0.859620	+	0.510934i	$0.829300\pi$
$942$		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	$-0.729129\pi$
$947$	−10.3923	+	6.00000i	−0.337705	+	0.194974i	0.321552	+	0.946892i	$0.395796\pi$
$948$		0			0
$949$	20.7846	+	12.0000i	0.674697	+	0.389536i
$950$	5.46410	−	1.46410i	0.177279	−	0.0475017i
$951$		0			0
$952$	2.92820	−	10.9282i	0.0949036	−	0.354185i
$953$	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
$958$	−7.32051	−	27.3205i	−0.236515	−	0.882686i
$959$	−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	−0.633165	−	0.774017i	$-0.718244\pi$
$967$	11.0000	−	19.0526i	0.353736	−	0.612689i	0.986901	+	0.161328i	$0.0515777\pi$
$968$	30.0526	−	8.05256i	0.965926	−	0.258819i
$969$		0			0
$970$	−1.46410	+	5.46410i	−0.0470095	+	0.175442i
$971$		0			0		1.00000			$0$
$971$		0			0		−1.00000			$\pi$
$972$		0			0
$973$			8.00000i			0.256468i
$974$	−51.9090	−	13.9090i	−1.66327	−	0.445672i
$975$		0			0
$976$		0			0
$977$	−1.00000	+	1.73205i	−0.0319928	+	0.0554132i	−0.881579	−	0.472037i	$-0.843519\pi$
$977$	−1.00000	+	1.73205i	−0.0319928	+	0.0554132i	0.849586	+	0.527451i	$0.176852\pi$
$978$		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	0.709779	−	0.704425i	$-0.248795\pi$
$983$	−8.00000	−	13.8564i	−0.255160	−	0.441951i	−0.964939	+	0.262474i	$0.915462\pi$
$984$		0			0
$985$	−2.00000	+	3.46410i	−0.0637253	+	0.110375i
$986$	16.3923	−	4.39230i	0.522037	−	0.139879i
$987$		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	$0.489887\pi$
$992$	10.9282	+	2.92820i	0.346971	+	0.0929705i
$993$		0			0
$994$	8.78461	+	32.7846i	0.278631	+	1.03986i
$995$	−17.3205	−	10.0000i	−0.549097	−	0.317021i
$996$		0			0
$997$	−41.5692	+	24.0000i	−1.31651	+	0.760088i	−0.983165	−	0.182717i	$-0.941511\pi$
$997$	−41.5692	+	24.0000i	−1.31651	+	0.760088i	−0.333345	+	0.942805i	$0.608177\pi$
$998$	36.0000	+	36.0000i	1.13956	+	1.13956i
$999$		0			0

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

)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	648.2.n.c.109.1		4
3.2	odd	2		648.2.n.k.109.2		4
4.3	odd	2		2592.2.r.g.433.1		4
8.3	odd	2		2592.2.r.g.433.2		4
8.5	even	2	inner	648.2.n.c.109.2		4
9.2	odd	6		648.2.n.k.541.1		4
9.4	even	3		72.2.d.b.37.1		2
9.5	odd	6		24.2.d.a.13.2	yes	2
9.7	even	3	inner	648.2.n.c.541.2		4
12.11	even	2		2592.2.r.f.433.2		4
24.5	odd	2		648.2.n.k.109.1		4
24.11	even	2		2592.2.r.f.433.1		4
36.7	odd	6		2592.2.r.g.2161.2		4
36.11	even	6		2592.2.r.f.2161.1		4
36.23	even	6		96.2.d.a.49.1		2
36.31	odd	6		288.2.d.b.145.2		2
45.4	even	6		1800.2.k.a.901.2		2
45.13	odd	12		1800.2.d.b.1549.1		2
45.14	odd	6		600.2.k.b.301.1		2
45.22	odd	12		1800.2.d.i.1549.2		2
45.23	even	12		600.2.d.c.349.2		2
45.32	even	12		600.2.d.b.349.1		2
63.41	even	6		1176.2.c.a.589.2		2
72.5	odd	6		24.2.d.a.13.1	✓	2
72.11	even	6		2592.2.r.f.2161.2		4
72.13	even	6		72.2.d.b.37.2		2
72.29	odd	6		648.2.n.k.541.2		4
72.43	odd	6		2592.2.r.g.2161.1		4
72.59	even	6		96.2.d.a.49.2		2
72.61	even	6	inner	648.2.n.c.541.1		4
72.67	odd	6		288.2.d.b.145.1		2
144.5	odd	12		768.2.a.h.1.1		1
144.13	even	12		2304.2.a.o.1.1		1
144.59	even	12		768.2.a.d.1.1		1
144.67	odd	12		2304.2.a.l.1.1		1
144.77	odd	12		768.2.a.a.1.1		1
144.85	even	12		2304.2.a.e.1.1		1
144.131	even	12		768.2.a.e.1.1		1
144.139	odd	12		2304.2.a.b.1.1		1
180.23	odd	12		2400.2.d.c.49.1		2
180.59	even	6		2400.2.k.a.1201.2		2
180.67	even	12		7200.2.d.g.2449.2		2
180.103	even	12		7200.2.d.d.2449.1		2
180.139	odd	6		7200.2.k.d.3601.1		2
180.167	odd	12		2400.2.d.b.49.2		2
252.167	odd	6		4704.2.c.a.2353.2		2
360.13	odd	12		1800.2.d.i.1549.1		2
360.59	even	6		2400.2.k.a.1201.1		2
360.67	even	12		7200.2.d.d.2449.2		2
360.77	even	12		600.2.d.c.349.1		2
360.139	odd	6		7200.2.k.d.3601.2		2
360.149	odd	6		600.2.k.b.301.2		2
360.157	odd	12		1800.2.d.b.1549.2		2
360.203	odd	12		2400.2.d.b.49.1		2
360.229	even	6		1800.2.k.a.901.1		2
360.283	even	12		7200.2.d.g.2449.1		2
360.293	even	12		600.2.d.b.349.2		2
360.347	odd	12		2400.2.d.c.49.2		2
504.293	even	6		1176.2.c.a.589.1		2
504.419	odd	6		4704.2.c.a.2353.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
24.2.d.a.13.1	✓	2	72.5	odd	6
24.2.d.a.13.2	yes	2	9.5	odd	6
72.2.d.b.37.1		2	9.4	even	3
72.2.d.b.37.2		2	72.13	even	6
96.2.d.a.49.1		2	36.23	even	6
96.2.d.a.49.2		2	72.59	even	6
288.2.d.b.145.1		2	72.67	odd	6
288.2.d.b.145.2		2	36.31	odd	6
600.2.d.b.349.1		2	45.32	even	12
600.2.d.b.349.2		2	360.293	even	12
600.2.d.c.349.1		2	360.77	even	12
600.2.d.c.349.2		2	45.23	even	12
600.2.k.b.301.1		2	45.14	odd	6
600.2.k.b.301.2		2	360.149	odd	6
648.2.n.c.109.1		4	1.1	even	1	trivial
648.2.n.c.109.2		4	8.5	even	2	inner
648.2.n.c.541.1		4	72.61	even	6	inner
648.2.n.c.541.2		4	9.7	even	3	inner
648.2.n.k.109.1		4	24.5	odd	2
648.2.n.k.109.2		4	3.2	odd	2
648.2.n.k.541.1		4	9.2	odd	6
648.2.n.k.541.2		4	72.29	odd	6
768.2.a.a.1.1		1	144.77	odd	12
768.2.a.d.1.1		1	144.59	even	12
768.2.a.e.1.1		1	144.131	even	12
768.2.a.h.1.1		1	144.5	odd	12
1176.2.c.a.589.1		2	504.293	even	6
1176.2.c.a.589.2		2	63.41	even	6
1800.2.d.b.1549.1		2	45.13	odd	12
1800.2.d.b.1549.2		2	360.157	odd	12
1800.2.d.i.1549.1		2	360.13	odd	12
1800.2.d.i.1549.2		2	45.22	odd	12
1800.2.k.a.901.1		2	360.229	even	6
1800.2.k.a.901.2		2	45.4	even	6
2304.2.a.b.1.1		1	144.139	odd	12
2304.2.a.e.1.1		1	144.85	even	12
2304.2.a.l.1.1		1	144.67	odd	12
2304.2.a.o.1.1		1	144.13	even	12
2400.2.d.b.49.1		2	360.203	odd	12
2400.2.d.b.49.2		2	180.167	odd	12
2400.2.d.c.49.1		2	180.23	odd	12
2400.2.d.c.49.2		2	360.347	odd	12
2400.2.k.a.1201.1		2	360.59	even	6
2400.2.k.a.1201.2		2	180.59	even	6
2592.2.r.f.433.1		4	24.11	even	2
2592.2.r.f.433.2		4	12.11	even	2
2592.2.r.f.2161.1		4	36.11	even	6
2592.2.r.f.2161.2		4	72.11	even	6
2592.2.r.g.433.1		4	4.3	odd	2
2592.2.r.g.433.2		4	8.3	odd	2
2592.2.r.g.2161.1		4	72.43	odd	6
2592.2.r.g.2161.2		4	36.7	odd	6
4704.2.c.a.2353.1		2	504.419	odd	6
4704.2.c.a.2353.2		2	252.167	odd	6
7200.2.d.d.2449.1		2	180.103	even	12
7200.2.d.d.2449.2		2	360.67	even	12
7200.2.d.g.2449.1		2	360.283	even	12
7200.2.d.g.2449.2		2	180.67	even	12
7200.2.k.d.3601.1		2	180.139	odd	6
7200.2.k.d.3601.2		2	360.139	odd	6

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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Newspace parameters

Newform invariants

Embedding invariants

$q$ -expansion

Character values

Coefficient data

Twists

This project is supported by grants from the US National Science Foundation, the UK Engineering and Physical Sciences Research Council, and the Simons Foundation.

Label	648.2.n.c.109.1

Level	$648$
Weight	$2$
Character	648.109
Analytic conductor	$5.174$
Analytic rank	$0$
Dimension	$4$
Inner twists	$4$