LMFDB - Embedded newform 578.2.c.d.327.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [578,2,Mod(251,578)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("578.251"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(578, base_ring=CyclotomicField(4)) chi = DirichletCharacter(H, H._module([1])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 578 = 2 \cdot 17^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	578.c (of order $4$, degree $2$, not minimal)

Newform invariants

comment:select newform

sage:traces = [2,0,4,-2,4,4,0,0,0,4,4,-4,-4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(13)] == traces)

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

Self dual:	no
Analytic conductor:	$4.61535323683$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(i)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} + 1 $ x^2 + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 34)
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			327.1
Root			$-1.00000i$ of defining polynomial
Character	$\chi$	$=$	578.327
Dual form			578.2.c.d.251.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+1.00000i q^{2} +(2.00000 - 2.00000i) q^{3} -1.00000 q^{4} +(2.00000 - 2.00000i) q^{5} +(2.00000 + 2.00000i) q^{6} -1.00000i q^{8} -5.00000i q^{9} +(2.00000 + 2.00000i) q^{10} +(2.00000 + 2.00000i) q^{11} +(-2.00000 + 2.00000i) q^{12} -2.00000 q^{13} -8.00000i q^{15} +1.00000 q^{16} +5.00000 q^{18} +4.00000i q^{19} +(-2.00000 + 2.00000i) q^{20} +(-2.00000 + 2.00000i) q^{22} +(-4.00000 - 4.00000i) q^{23} +(-2.00000 - 2.00000i) q^{24} -3.00000i q^{25} -2.00000i q^{26} +(-4.00000 - 4.00000i) q^{27} +(2.00000 - 2.00000i) q^{29} +8.00000 q^{30} +1.00000i q^{32} +8.00000 q^{33} +5.00000i q^{36} +(-6.00000 + 6.00000i) q^{37} -4.00000 q^{38} +(-4.00000 + 4.00000i) q^{39} +(-2.00000 - 2.00000i) q^{40} +(4.00000 + 4.00000i) q^{41} -4.00000i q^{43} +(-2.00000 - 2.00000i) q^{44} +(-10.0000 - 10.0000i) q^{45} +(4.00000 - 4.00000i) q^{46} +(2.00000 - 2.00000i) q^{48} -7.00000i q^{49} +3.00000 q^{50} +2.00000 q^{52} -6.00000i q^{53} +(4.00000 - 4.00000i) q^{54} +8.00000 q^{55} +(8.00000 + 8.00000i) q^{57} +(2.00000 + 2.00000i) q^{58} +12.0000i q^{59} +8.00000i q^{60} +(6.00000 + 6.00000i) q^{61} -1.00000 q^{64} +(-4.00000 + 4.00000i) q^{65} +8.00000i q^{66} -4.00000 q^{67} -16.0000 q^{69} +(4.00000 - 4.00000i) q^{71} -5.00000 q^{72} +(-6.00000 - 6.00000i) q^{74} +(-6.00000 - 6.00000i) q^{75} -4.00000i q^{76} +(-4.00000 - 4.00000i) q^{78} +(12.0000 + 12.0000i) q^{79} +(2.00000 - 2.00000i) q^{80} -1.00000 q^{81} +(-4.00000 + 4.00000i) q^{82} +12.0000i q^{83} +4.00000 q^{86} -8.00000i q^{87} +(2.00000 - 2.00000i) q^{88} -6.00000 q^{89} +(10.0000 - 10.0000i) q^{90} +(4.00000 + 4.00000i) q^{92} +(8.00000 + 8.00000i) q^{95} +(2.00000 + 2.00000i) q^{96} +(-12.0000 + 12.0000i) q^{97} +7.00000 q^{98} +(10.0000 - 10.0000i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 4 q^{3} - 2 q^{4} + 4 q^{5} + 4 q^{6} + 4 q^{10} + 4 q^{11} - 4 q^{12} - 4 q^{13} + 2 q^{16} + 10 q^{18} - 4 q^{20} - 4 q^{22} - 8 q^{23} - 4 q^{24} - 8 q^{27} + 4 q^{29} + 16 q^{30} + 16 q^{33} - 12 q^{37}+ \cdots + 20 q^{99}+O(q^{100}) $ 2 * q + 4 * q^3 - 2 * q^4 + 4 * q^5 + 4 * q^6 + 4 * q^10 + 4 * q^11 - 4 * q^12 - 4 * q^13 + 2 * q^16 + 10 * q^18 - 4 * q^20 - 4 * q^22 - 8 * q^23 - 4 * q^24 - 8 * q^27 + 4 * q^29 + 16 * q^30 + 16 * q^33 - 12 * q^37 - 8 * q^38 - 8 * q^39 - 4 * q^40 + 8 * q^41 - 4 * q^44 - 20 * q^45 + 8 * q^46 + 4 * q^48 + 6 * q^50 + 4 * q^52 + 8 * q^54 + 16 * q^55 + 16 * q^57 + 4 * q^58 + 12 * q^61 - 2 * q^64 - 8 * q^65 - 8 * q^67 - 32 * q^69 + 8 * q^71 - 10 * q^72 - 12 * q^74 - 12 * q^75 - 8 * q^78 + 24 * q^79 + 4 * q^80 - 2 * q^81 - 8 * q^82 + 8 * q^86 + 4 * q^88 - 12 * q^89 + 20 * q^90 + 8 * q^92 + 16 * q^95 + 4 * q^96 - 24 * q^97 + 14 * q^98 + 20 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/578\mathbb{Z}\right)^\times$.

$n$	$3$
$\chi(n)$	$e\left(\frac{3}{4}\right)$

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 $

$2$			1.00000i			0.707107i
$2$			1.00000i			0.707107i
$3$	2.00000	−	2.00000i	1.15470	−	1.15470i	0.169102	−	0.985599i	$-0.445913\pi$
$3$	2.00000	−	2.00000i	1.15470	−	1.15470i	0.985599	−	0.169102i	$-0.0540867\pi$
$4$	−1.00000			−0.500000
$5$	2.00000	−	2.00000i	0.894427	−	0.894427i	−0.100509	−	0.994936i	$-0.532047\pi$
$5$	2.00000	−	2.00000i	0.894427	−	0.894427i	0.994936	+	0.100509i	$0.0320471\pi$
$6$	2.00000	+	2.00000i	0.816497	+	0.816497i
$7$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$7$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$8$		−	1.00000i		−	0.353553i
$9$		−	5.00000i		−	1.66667i
$10$	2.00000	+	2.00000i	0.632456	+	0.632456i
$11$	2.00000	+	2.00000i	0.603023	+	0.603023i	0.941113	−	0.338091i	$-0.109781\pi$
$11$	2.00000	+	2.00000i	0.603023	+	0.603023i	−0.338091	+	0.941113i	$0.609781\pi$
$12$	−2.00000	+	2.00000i	−0.577350	+	0.577350i
$13$	−2.00000			−0.554700			−0.277350	−	0.960769i	$-0.589456\pi$
$13$	−2.00000			−0.554700			−0.277350	+	0.960769i	$0.589456\pi$
$14$		0			0
$15$		−	8.00000i		−	2.06559i
$16$	1.00000			0.250000
$17$		0			0
$17$		0			0
$18$	5.00000			1.17851
$19$			4.00000i			0.917663i	0.888523	+	0.458831i	$0.151732\pi$
$19$			4.00000i			0.917663i	−0.888523	+	0.458831i	$0.848268\pi$
$20$	−2.00000	+	2.00000i	−0.447214	+	0.447214i
$21$		0			0
$22$	−2.00000	+	2.00000i	−0.426401	+	0.426401i
$23$	−4.00000	−	4.00000i	−0.834058	−	0.834058i	0.154011	−	0.988069i	$-0.450781\pi$
$23$	−4.00000	−	4.00000i	−0.834058	−	0.834058i	−0.988069	+	0.154011i	$0.950781\pi$
$24$	−2.00000	−	2.00000i	−0.408248	−	0.408248i
$25$		−	3.00000i		−	0.600000i
$26$		−	2.00000i		−	0.392232i
$27$	−4.00000	−	4.00000i	−0.769800	−	0.769800i
$28$		0			0
$29$	2.00000	−	2.00000i	0.371391	−	0.371391i	−0.496593	−	0.867984i	$-0.665416\pi$
$29$	2.00000	−	2.00000i	0.371391	−	0.371391i	0.867984	+	0.496593i	$0.165416\pi$
$30$	8.00000			1.46059
$31$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$31$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$32$			1.00000i			0.176777i
$33$	8.00000			1.39262
$34$		0			0
$35$		0			0
$36$			5.00000i			0.833333i
$37$	−6.00000	+	6.00000i	−0.986394	+	0.986394i	−0.999909	−	0.0135147i	$-0.995698\pi$
$37$	−6.00000	+	6.00000i	−0.986394	+	0.986394i	0.0135147	+	0.999909i	$0.495698\pi$
$38$	−4.00000			−0.648886
$39$	−4.00000	+	4.00000i	−0.640513	+	0.640513i
$40$	−2.00000	−	2.00000i	−0.316228	−	0.316228i
$41$	4.00000	+	4.00000i	0.624695	+	0.624695i	0.946728	−	0.322033i	$-0.104366\pi$
$41$	4.00000	+	4.00000i	0.624695	+	0.624695i	−0.322033	+	0.946728i	$0.604366\pi$
$42$		0			0
$43$		−	4.00000i		−	0.609994i	−0.952353	−	0.304997i	$-0.901344\pi$
$43$		−	4.00000i		−	0.609994i	0.952353	−	0.304997i	$-0.0986555\pi$
$44$	−2.00000	−	2.00000i	−0.301511	−	0.301511i
$45$	−10.0000	−	10.0000i	−1.49071	−	1.49071i
$46$	4.00000	−	4.00000i	0.589768	−	0.589768i
$47$		0			0			−	1.00000i	$-0.5\pi$
$47$		0			0				1.00000i	$0.5\pi$
$48$	2.00000	−	2.00000i	0.288675	−	0.288675i
$49$		−	7.00000i		−	1.00000i
$50$	3.00000			0.424264
$51$		0			0
$52$	2.00000			0.277350
$53$		−	6.00000i		−	0.824163i	−0.911147	−	0.412082i	$-0.864802\pi$
$53$		−	6.00000i		−	0.824163i	0.911147	−	0.412082i	$-0.135198\pi$
$54$	4.00000	−	4.00000i	0.544331	−	0.544331i
$55$	8.00000			1.07872
$56$		0			0
$57$	8.00000	+	8.00000i	1.05963	+	1.05963i
$58$	2.00000	+	2.00000i	0.262613	+	0.262613i
$59$			12.0000i			1.56227i	0.624364	+	0.781133i	$0.285358\pi$
$59$			12.0000i			1.56227i	−0.624364	+	0.781133i	$0.714642\pi$
$60$			8.00000i			1.03280i
$61$	6.00000	+	6.00000i	0.768221	+	0.768221i	0.977793	−	0.209572i	$-0.0672070\pi$
$61$	6.00000	+	6.00000i	0.768221	+	0.768221i	−0.209572	+	0.977793i	$0.567207\pi$
$62$		0			0
$63$		0			0
$64$	−1.00000			−0.125000
$65$	−4.00000	+	4.00000i	−0.496139	+	0.496139i
$66$			8.00000i			0.984732i
$67$	−4.00000			−0.488678			−0.244339	−	0.969690i	$-0.578571\pi$
$67$	−4.00000			−0.488678			−0.244339	+	0.969690i	$0.578571\pi$
$68$		0			0
$69$	−16.0000			−1.92617
$70$		0			0
$71$	4.00000	−	4.00000i	0.474713	−	0.474713i	−0.428723	−	0.903436i	$-0.641036\pi$
$71$	4.00000	−	4.00000i	0.474713	−	0.474713i	0.903436	+	0.428723i	$0.141036\pi$
$72$	−5.00000			−0.589256
$73$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$73$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$74$	−6.00000	−	6.00000i	−0.697486	−	0.697486i
$75$	−6.00000	−	6.00000i	−0.692820	−	0.692820i
$76$		−	4.00000i		−	0.458831i
$77$		0			0
$78$	−4.00000	−	4.00000i	−0.452911	−	0.452911i
$79$	12.0000	+	12.0000i	1.35011	+	1.35011i	0.885537	+	0.464568i	$0.153790\pi$
$79$	12.0000	+	12.0000i	1.35011	+	1.35011i	0.464568	+	0.885537i	$0.346210\pi$
$80$	2.00000	−	2.00000i	0.223607	−	0.223607i
$81$	−1.00000			−0.111111
$82$	−4.00000	+	4.00000i	−0.441726	+	0.441726i
$83$			12.0000i			1.31717i	0.752506	+	0.658586i	$0.228845\pi$
$83$			12.0000i			1.31717i	−0.752506	+	0.658586i	$0.771155\pi$
$84$		0			0
$85$		0			0
$86$	4.00000			0.431331
$87$		−	8.00000i		−	0.857690i
$88$	2.00000	−	2.00000i	0.213201	−	0.213201i
$89$	−6.00000			−0.635999			−0.317999	−	0.948091i	$-0.603011\pi$
$89$	−6.00000			−0.635999			−0.317999	+	0.948091i	$0.603011\pi$
$90$	10.0000	−	10.0000i	1.05409	−	1.05409i
$91$		0			0
$92$	4.00000	+	4.00000i	0.417029	+	0.417029i
$93$		0			0
$94$		0			0
$95$	8.00000	+	8.00000i	0.820783	+	0.820783i
$96$	2.00000	+	2.00000i	0.204124	+	0.204124i
$97$	−12.0000	+	12.0000i	−1.21842	+	1.21842i	−0.250229	+	0.968187i	$0.580506\pi$
$97$	−12.0000	+	12.0000i	−1.21842	+	1.21842i	−0.968187	+	0.250229i	$0.919494\pi$
$98$	7.00000			0.707107
$99$	10.0000	−	10.0000i	1.00504	−	1.00504i
$100$			3.00000i			0.300000i
$101$	−6.00000			−0.597022			−0.298511	−	0.954406i	$-0.596490\pi$
$101$	−6.00000			−0.597022			−0.298511	+	0.954406i	$0.596490\pi$
$102$		0			0
$103$	8.00000			0.788263			0.394132	−	0.919054i	$-0.371045\pi$
$103$	8.00000			0.788263			0.394132	+	0.919054i	$0.371045\pi$
$104$			2.00000i			0.196116i
$105$		0			0
$106$	6.00000			0.582772
$107$	2.00000	−	2.00000i	0.193347	−	0.193347i	−0.603793	−	0.797141i	$-0.706345\pi$
$107$	2.00000	−	2.00000i	0.193347	−	0.193347i	0.797141	+	0.603793i	$0.206345\pi$
$108$	4.00000	+	4.00000i	0.384900	+	0.384900i
$109$	−6.00000	−	6.00000i	−0.574696	−	0.574696i	0.358741	−	0.933437i	$-0.383206\pi$
$109$	−6.00000	−	6.00000i	−0.574696	−	0.574696i	−0.933437	+	0.358741i	$0.883206\pi$
$110$			8.00000i			0.762770i
$111$			24.0000i			2.27798i
$112$		0			0
$113$	8.00000	+	8.00000i	0.752577	+	0.752577i	0.974959	−	0.222383i	$-0.0713835\pi$
$113$	8.00000	+	8.00000i	0.752577	+	0.752577i	−0.222383	+	0.974959i	$0.571383\pi$
$114$	−8.00000	+	8.00000i	−0.749269	+	0.749269i
$115$	−16.0000			−1.49201
$116$	−2.00000	+	2.00000i	−0.185695	+	0.185695i
$117$			10.0000i			0.924500i
$118$	−12.0000			−1.10469
$119$		0			0
$120$	−8.00000			−0.730297
$121$		−	3.00000i		−	0.272727i
$122$	−6.00000	+	6.00000i	−0.543214	+	0.543214i
$123$	16.0000			1.44267
$124$		0			0
$125$	4.00000	+	4.00000i	0.357771	+	0.357771i
$126$		0			0
$127$		−	16.0000i		−	1.41977i	−0.704317	−	0.709885i	$-0.748747\pi$
$127$		−	16.0000i		−	1.41977i	0.704317	−	0.709885i	$-0.251253\pi$
$128$		−	1.00000i		−	0.0883883i
$129$	−8.00000	−	8.00000i	−0.704361	−	0.704361i
$130$	−4.00000	−	4.00000i	−0.350823	−	0.350823i
$131$	2.00000	−	2.00000i	0.174741	−	0.174741i	−0.614318	−	0.789059i	$-0.710569\pi$
$131$	2.00000	−	2.00000i	0.174741	−	0.174741i	0.789059	+	0.614318i	$0.210569\pi$
$132$	−8.00000			−0.696311
$133$		0			0
$134$		−	4.00000i		−	0.345547i
$135$	−16.0000			−1.37706
$136$		0			0
$137$	−18.0000			−1.53784			−0.768922	−	0.639343i	$-0.779207\pi$
$137$	−18.0000			−1.53784			−0.768922	+	0.639343i	$0.779207\pi$
$138$		−	16.0000i		−	1.36201i
$139$	6.00000	−	6.00000i	0.508913	−	0.508913i	−0.405279	−	0.914193i	$-0.632826\pi$
$139$	6.00000	−	6.00000i	0.508913	−	0.508913i	0.914193	+	0.405279i	$0.132826\pi$
$140$		0			0
$141$		0			0
$142$	4.00000	+	4.00000i	0.335673	+	0.335673i
$143$	−4.00000	−	4.00000i	−0.334497	−	0.334497i
$144$		−	5.00000i		−	0.416667i
$145$		−	8.00000i		−	0.664364i
$146$		0			0
$147$	−14.0000	−	14.0000i	−1.15470	−	1.15470i
$148$	6.00000	−	6.00000i	0.493197	−	0.493197i
$149$	−6.00000			−0.491539			−0.245770	−	0.969328i	$-0.579041\pi$
$149$	−6.00000			−0.491539			−0.245770	+	0.969328i	$0.579041\pi$
$150$	6.00000	−	6.00000i	0.489898	−	0.489898i
$151$		−	8.00000i		−	0.651031i	−0.945537	−	0.325515i	$-0.894462\pi$
$151$		−	8.00000i		−	0.651031i	0.945537	−	0.325515i	$-0.105538\pi$
$152$	4.00000			0.324443
$153$		0			0
$154$		0			0
$155$		0			0
$156$	4.00000	−	4.00000i	0.320256	−	0.320256i
$157$	−14.0000			−1.11732			−0.558661	−	0.829396i	$-0.688685\pi$
$157$	−14.0000			−1.11732			−0.558661	+	0.829396i	$0.688685\pi$
$158$	−12.0000	+	12.0000i	−0.954669	+	0.954669i
$159$	−12.0000	−	12.0000i	−0.951662	−	0.951662i
$160$	2.00000	+	2.00000i	0.158114	+	0.158114i
$161$		0			0
$162$		−	1.00000i		−	0.0785674i
$163$	−6.00000	−	6.00000i	−0.469956	−	0.469956i	0.431944	−	0.901900i	$-0.357828\pi$
$163$	−6.00000	−	6.00000i	−0.469956	−	0.469956i	−0.901900	+	0.431944i	$0.857828\pi$
$164$	−4.00000	−	4.00000i	−0.312348	−	0.312348i
$165$	16.0000	−	16.0000i	1.24560	−	1.24560i
$166$	−12.0000			−0.931381
$167$	−8.00000	+	8.00000i	−0.619059	+	0.619059i	−0.945290	−	0.326231i	$-0.894221\pi$
$167$	−8.00000	+	8.00000i	−0.619059	+	0.619059i	0.326231	+	0.945290i	$0.394221\pi$
$168$		0			0
$169$	−9.00000			−0.692308
$170$		0			0
$171$	20.0000			1.52944
$172$			4.00000i			0.304997i
$173$	−2.00000	+	2.00000i	−0.152057	+	0.152057i	−0.779036	−	0.626979i	$-0.784291\pi$
$173$	−2.00000	+	2.00000i	−0.152057	+	0.152057i	0.626979	+	0.779036i	$0.284291\pi$
$174$	8.00000			0.606478
$175$		0			0
$176$	2.00000	+	2.00000i	0.150756	+	0.150756i
$177$	24.0000	+	24.0000i	1.80395	+	1.80395i
$178$		−	6.00000i		−	0.449719i
$179$			12.0000i			0.896922i	0.893802	+	0.448461i	$0.148028\pi$
$179$			12.0000i			0.896922i	−0.893802	+	0.448461i	$0.851972\pi$
$180$	10.0000	+	10.0000i	0.745356	+	0.745356i
$181$	−6.00000	−	6.00000i	−0.445976	−	0.445976i	0.448038	−	0.894015i	$-0.352123\pi$
$181$	−6.00000	−	6.00000i	−0.445976	−	0.445976i	−0.894015	+	0.448038i	$0.852123\pi$
$182$		0			0
$183$	24.0000			1.77413
$184$	−4.00000	+	4.00000i	−0.294884	+	0.294884i
$185$			24.0000i			1.76452i
$186$		0			0
$187$		0			0
$188$		0			0
$189$		0			0
$190$	−8.00000	+	8.00000i	−0.580381	+	0.580381i
$191$		0			0			−	1.00000i	$-0.5\pi$
$191$		0			0				1.00000i	$0.5\pi$
$192$	−2.00000	+	2.00000i	−0.144338	+	0.144338i
$193$	−12.0000	−	12.0000i	−0.863779	−	0.863779i	0.127996	−	0.991775i	$-0.459146\pi$
$193$	−12.0000	−	12.0000i	−0.863779	−	0.863779i	−0.991775	+	0.127996i	$0.959146\pi$
$194$	−12.0000	−	12.0000i	−0.861550	−	0.861550i
$195$			16.0000i			1.14578i
$196$			7.00000i			0.500000i
$197$	−2.00000	−	2.00000i	−0.142494	−	0.142494i	0.632261	−	0.774755i	$-0.282127\pi$
$197$	−2.00000	−	2.00000i	−0.142494	−	0.142494i	−0.774755	+	0.632261i	$0.782127\pi$
$198$	10.0000	+	10.0000i	0.710669	+	0.710669i
$199$	12.0000	−	12.0000i	0.850657	−	0.850657i	−0.139557	−	0.990214i	$-0.544568\pi$
$199$	12.0000	−	12.0000i	0.850657	−	0.850657i	0.990214	+	0.139557i	$0.0445677\pi$
$200$	−3.00000			−0.212132
$201$	−8.00000	+	8.00000i	−0.564276	+	0.564276i
$202$		−	6.00000i		−	0.422159i
$203$		0			0
$204$		0			0
$205$	16.0000			1.11749
$206$			8.00000i			0.557386i
$207$	−20.0000	+	20.0000i	−1.39010	+	1.39010i
$208$	−2.00000			−0.138675
$209$	−8.00000	+	8.00000i	−0.553372	+	0.553372i
$210$		0			0
$211$	6.00000	+	6.00000i	0.413057	+	0.413057i	0.882802	−	0.469745i	$-0.155654\pi$
$211$	6.00000	+	6.00000i	0.413057	+	0.413057i	−0.469745	+	0.882802i	$0.655654\pi$
$212$			6.00000i			0.412082i
$213$		−	16.0000i		−	1.09630i
$214$	2.00000	+	2.00000i	0.136717	+	0.136717i
$215$	−8.00000	−	8.00000i	−0.545595	−	0.545595i
$216$	−4.00000	+	4.00000i	−0.272166	+	0.272166i
$217$		0			0
$218$	6.00000	−	6.00000i	0.406371	−	0.406371i
$219$		0			0
$220$	−8.00000			−0.539360
$221$		0			0
$222$	−24.0000			−1.61077
$223$			16.0000i			1.07144i	0.844396	+	0.535720i	$0.179960\pi$
$223$			16.0000i			1.07144i	−0.844396	+	0.535720i	$0.820040\pi$
$224$		0			0
$225$	−15.0000			−1.00000
$226$	−8.00000	+	8.00000i	−0.532152	+	0.532152i
$227$	14.0000	+	14.0000i	0.929213	+	0.929213i	0.997655	−	0.0684424i	$-0.0218029\pi$
$227$	14.0000	+	14.0000i	0.929213	+	0.929213i	−0.0684424	+	0.997655i	$0.521803\pi$
$228$	−8.00000	−	8.00000i	−0.529813	−	0.529813i
$229$		−	22.0000i		−	1.45380i	−0.686743	−	0.726900i	$-0.740960\pi$
$229$		−	22.0000i		−	1.45380i	0.686743	−	0.726900i	$-0.259040\pi$
$230$		−	16.0000i		−	1.05501i
$231$		0			0
$232$	−2.00000	−	2.00000i	−0.131306	−	0.131306i
$233$	−16.0000	+	16.0000i	−1.04819	+	1.04819i	−0.0494166	+	0.998778i	$0.515736\pi$
$233$	−16.0000	+	16.0000i	−1.04819	+	1.04819i	−0.998778	+	0.0494166i	$0.984264\pi$
$234$	−10.0000			−0.653720
$235$		0			0
$236$		−	12.0000i		−	0.781133i
$237$	48.0000			3.11794
$238$		0			0
$239$		0			0			−	1.00000i	$-0.5\pi$
$239$		0			0				1.00000i	$0.5\pi$
$240$		−	8.00000i		−	0.516398i
$241$	12.0000	−	12.0000i	0.772988	−	0.772988i	−0.205640	−	0.978628i	$-0.565928\pi$
$241$	12.0000	−	12.0000i	0.772988	−	0.772988i	0.978628	+	0.205640i	$0.0659275\pi$
$242$	3.00000			0.192847
$243$	10.0000	−	10.0000i	0.641500	−	0.641500i
$244$	−6.00000	−	6.00000i	−0.384111	−	0.384111i
$245$	−14.0000	−	14.0000i	−0.894427	−	0.894427i
$246$			16.0000i			1.02012i
$247$		−	8.00000i		−	0.509028i
$248$		0			0
$249$	24.0000	+	24.0000i	1.52094	+	1.52094i
$250$	−4.00000	+	4.00000i	−0.252982	+	0.252982i
$251$	12.0000			0.757433			0.378717	−	0.925513i	$-0.376365\pi$
$251$	12.0000			0.757433			0.378717	+	0.925513i	$0.376365\pi$
$252$		0			0
$253$		−	16.0000i		−	1.00591i
$254$	16.0000			1.00393
$255$		0			0
$256$	1.00000			0.0625000
$257$		−	6.00000i		−	0.374270i	−0.982334	−	0.187135i	$-0.940080\pi$
$257$		−	6.00000i		−	0.374270i	0.982334	−	0.187135i	$-0.0599201\pi$
$258$	8.00000	−	8.00000i	0.498058	−	0.498058i
$259$		0			0
$260$	4.00000	−	4.00000i	0.248069	−	0.248069i
$261$	−10.0000	−	10.0000i	−0.618984	−	0.618984i
$262$	2.00000	+	2.00000i	0.123560	+	0.123560i
$263$			24.0000i			1.47990i	0.672660	+	0.739952i	$0.265152\pi$
$263$			24.0000i			1.47990i	−0.672660	+	0.739952i	$0.734848\pi$
$264$		−	8.00000i		−	0.492366i
$265$	−12.0000	−	12.0000i	−0.737154	−	0.737154i
$266$		0			0
$267$	−12.0000	+	12.0000i	−0.734388	+	0.734388i
$268$	4.00000			0.244339
$269$	22.0000	−	22.0000i	1.34136	−	1.34136i	0.446660	−	0.894704i	$-0.352613\pi$
$269$	22.0000	−	22.0000i	1.34136	−	1.34136i	0.894704	−	0.446660i	$-0.147387\pi$
$270$		−	16.0000i		−	0.973729i
$271$	−16.0000			−0.971931			−0.485965	−	0.873978i	$-0.661532\pi$
$271$	−16.0000			−0.971931			−0.485965	+	0.873978i	$0.661532\pi$
$272$		0			0
$273$		0			0
$274$		−	18.0000i		−	1.08742i
$275$	6.00000	−	6.00000i	0.361814	−	0.361814i
$276$	16.0000			0.963087
$277$	18.0000	−	18.0000i	1.08152	−	1.08152i	0.0851468	−	0.996368i	$-0.472864\pi$
$277$	18.0000	−	18.0000i	1.08152	−	1.08152i	0.996368	−	0.0851468i	$-0.0271359\pi$
$278$	6.00000	+	6.00000i	0.359856	+	0.359856i
$279$		0			0
$280$		0			0
$281$			18.0000i			1.07379i	0.843649	+	0.536895i	$0.180403\pi$
$281$			18.0000i			1.07379i	−0.843649	+	0.536895i	$0.819597\pi$
$282$		0			0
$283$	−6.00000	−	6.00000i	−0.356663	−	0.356663i	0.505918	−	0.862581i	$-0.331154\pi$
$283$	−6.00000	−	6.00000i	−0.356663	−	0.356663i	−0.862581	+	0.505918i	$0.831154\pi$
$284$	−4.00000	+	4.00000i	−0.237356	+	0.237356i
$285$	32.0000			1.89552
$286$	4.00000	−	4.00000i	0.236525	−	0.236525i
$287$		0			0
$288$	5.00000			0.294628
$289$		0			0
$290$	8.00000			0.469776
$291$			48.0000i			2.81381i
$292$		0			0
$293$	−6.00000			−0.350524			−0.175262	−	0.984522i	$-0.556077\pi$
$293$	−6.00000			−0.350524			−0.175262	+	0.984522i	$0.556077\pi$
$294$	14.0000	−	14.0000i	0.816497	−	0.816497i
$295$	24.0000	+	24.0000i	1.39733	+	1.39733i
$296$	6.00000	+	6.00000i	0.348743	+	0.348743i
$297$		−	16.0000i		−	0.928414i
$298$		−	6.00000i		−	0.347571i
$299$	8.00000	+	8.00000i	0.462652	+	0.462652i
$300$	6.00000	+	6.00000i	0.346410	+	0.346410i
$301$		0			0
$302$	8.00000			0.460348
$303$	−12.0000	+	12.0000i	−0.689382	+	0.689382i
$304$			4.00000i			0.229416i
$305$	24.0000			1.37424
$306$		0			0
$307$	20.0000			1.14146			0.570730	−	0.821138i	$-0.306660\pi$
$307$	20.0000			1.14146			0.570730	+	0.821138i	$0.306660\pi$
$308$		0			0
$309$	16.0000	−	16.0000i	0.910208	−	0.910208i
$310$		0			0
$311$	8.00000	−	8.00000i	0.453638	−	0.453638i	−0.442922	−	0.896560i	$-0.646058\pi$
$311$	8.00000	−	8.00000i	0.453638	−	0.453638i	0.896560	+	0.442922i	$0.146058\pi$
$312$	4.00000	+	4.00000i	0.226455	+	0.226455i
$313$	−12.0000	−	12.0000i	−0.678280	−	0.678280i	0.281331	−	0.959611i	$-0.409224\pi$
$313$	−12.0000	−	12.0000i	−0.678280	−	0.678280i	−0.959611	+	0.281331i	$0.909224\pi$
$314$		−	14.0000i		−	0.790066i
$315$		0			0
$316$	−12.0000	−	12.0000i	−0.675053	−	0.675053i
$317$	2.00000	+	2.00000i	0.112331	+	0.112331i	0.761038	−	0.648707i	$-0.224690\pi$
$317$	2.00000	+	2.00000i	0.112331	+	0.112331i	−0.648707	+	0.761038i	$0.724690\pi$
$318$	12.0000	−	12.0000i	0.672927	−	0.672927i
$319$	8.00000			0.447914
$320$	−2.00000	+	2.00000i	−0.111803	+	0.111803i
$321$		−	8.00000i		−	0.446516i
$322$		0			0
$323$		0			0
$324$	1.00000			0.0555556
$325$			6.00000i			0.332820i
$326$	6.00000	−	6.00000i	0.332309	−	0.332309i
$327$	−24.0000			−1.32720
$328$	4.00000	−	4.00000i	0.220863	−	0.220863i
$329$		0			0
$330$	16.0000	+	16.0000i	0.880771	+	0.880771i
$331$		−	4.00000i		−	0.219860i	−0.993939	−	0.109930i	$-0.964937\pi$
$331$		−	4.00000i		−	0.219860i	0.993939	−	0.109930i	$-0.0350627\pi$
$332$		−	12.0000i		−	0.658586i
$333$	30.0000	+	30.0000i	1.64399	+	1.64399i
$334$	−8.00000	−	8.00000i	−0.437741	−	0.437741i
$335$	−8.00000	+	8.00000i	−0.437087	+	0.437087i
$336$		0			0
$337$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$337$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$338$		−	9.00000i		−	0.489535i
$339$	32.0000			1.73800
$340$		0			0
$341$		0			0
$342$			20.0000i			1.08148i
$343$		0			0
$344$	−4.00000			−0.215666
$345$	−32.0000	+	32.0000i	−1.72282	+	1.72282i
$346$	−2.00000	−	2.00000i	−0.107521	−	0.107521i
$347$	10.0000	+	10.0000i	0.536828	+	0.536828i	0.922596	−	0.385768i	$-0.126063\pi$
$347$	10.0000	+	10.0000i	0.536828	+	0.536828i	−0.385768	+	0.922596i	$0.626063\pi$
$348$			8.00000i			0.428845i
$349$		−	34.0000i		−	1.81998i	−0.414632	−	0.909989i	$-0.636090\pi$
$349$		−	34.0000i		−	1.81998i	0.414632	−	0.909989i	$-0.363910\pi$
$350$		0			0
$351$	8.00000	+	8.00000i	0.427008	+	0.427008i
$352$	−2.00000	+	2.00000i	−0.106600	+	0.106600i
$353$	6.00000			0.319348			0.159674	−	0.987170i	$-0.448956\pi$
$353$	6.00000			0.319348			0.159674	+	0.987170i	$0.448956\pi$
$354$	−24.0000	+	24.0000i	−1.27559	+	1.27559i
$355$		−	16.0000i		−	0.849192i
$356$	6.00000			0.317999
$357$		0			0
$358$	−12.0000			−0.634220
$359$		−	24.0000i		−	1.26667i	−0.773877	−	0.633336i	$-0.781685\pi$
$359$		−	24.0000i		−	1.26667i	0.773877	−	0.633336i	$-0.218315\pi$
$360$	−10.0000	+	10.0000i	−0.527046	+	0.527046i
$361$	3.00000			0.157895
$362$	6.00000	−	6.00000i	0.315353	−	0.315353i
$363$	−6.00000	−	6.00000i	−0.314918	−	0.314918i
$364$		0			0
$365$		0			0
$366$			24.0000i			1.25450i
$367$	12.0000	+	12.0000i	0.626395	+	0.626395i	0.947159	−	0.320764i	$-0.103940\pi$
$367$	12.0000	+	12.0000i	0.626395	+	0.626395i	−0.320764	+	0.947159i	$0.603940\pi$
$368$	−4.00000	−	4.00000i	−0.208514	−	0.208514i
$369$	20.0000	−	20.0000i	1.04116	−	1.04116i
$370$	−24.0000			−1.24770
$371$		0			0
$372$		0			0
$373$	−22.0000			−1.13912			−0.569558	−	0.821951i	$-0.692886\pi$
$373$	−22.0000			−1.13912			−0.569558	+	0.821951i	$0.692886\pi$
$374$		0			0
$375$	16.0000			0.826236
$376$		0			0
$377$	−4.00000	+	4.00000i	−0.206010	+	0.206010i
$378$		0			0
$379$	−18.0000	+	18.0000i	−0.924598	+	0.924598i	−0.997350	−	0.0727522i	$-0.976822\pi$
$379$	−18.0000	+	18.0000i	−0.924598	+	0.924598i	0.0727522	+	0.997350i	$0.476822\pi$
$380$	−8.00000	−	8.00000i	−0.410391	−	0.410391i
$381$	−32.0000	−	32.0000i	−1.63941	−	1.63941i
$382$		0			0
$383$		0			0		1.00000			$0$
$383$		0			0		−1.00000			$\pi$
$384$	−2.00000	−	2.00000i	−0.102062	−	0.102062i
$385$		0			0
$386$	12.0000	−	12.0000i	0.610784	−	0.610784i
$387$	−20.0000			−1.01666
$388$	12.0000	−	12.0000i	0.609208	−	0.609208i
$389$			6.00000i			0.304212i	0.988364	+	0.152106i	$0.0486055\pi$
$389$			6.00000i			0.304212i	−0.988364	+	0.152106i	$0.951394\pi$
$390$	−16.0000			−0.810191
$391$		0			0
$392$	−7.00000			−0.353553
$393$		−	8.00000i		−	0.403547i
$394$	2.00000	−	2.00000i	0.100759	−	0.100759i
$395$	48.0000			2.41514
$396$	−10.0000	+	10.0000i	−0.502519	+	0.502519i
$397$	−6.00000	−	6.00000i	−0.301131	−	0.301131i	0.540325	−	0.841456i	$-0.318301\pi$
$397$	−6.00000	−	6.00000i	−0.301131	−	0.301131i	−0.841456	+	0.540325i	$0.818301\pi$
$398$	12.0000	+	12.0000i	0.601506	+	0.601506i
$399$		0			0
$400$		−	3.00000i		−	0.150000i
$401$	−20.0000	−	20.0000i	−0.998752	−	0.998752i	0.00124688	−	0.999999i	$-0.499603\pi$
$401$	−20.0000	−	20.0000i	−0.998752	−	0.998752i	−0.999999	+	0.00124688i	$0.999603\pi$
$402$	−8.00000	−	8.00000i	−0.399004	−	0.399004i
$403$		0			0
$404$	6.00000			0.298511
$405$	−2.00000	+	2.00000i	−0.0993808	+	0.0993808i
$406$		0			0
$407$	−24.0000			−1.18964
$408$		0			0
$409$	2.00000			0.0988936			0.0494468	−	0.998777i	$-0.484254\pi$
$409$	2.00000			0.0988936			0.0494468	+	0.998777i	$0.484254\pi$
$410$			16.0000i			0.790184i
$411$	−36.0000	+	36.0000i	−1.77575	+	1.77575i
$412$	−8.00000			−0.394132
$413$		0			0
$414$	−20.0000	−	20.0000i	−0.982946	−	0.982946i
$415$	24.0000	+	24.0000i	1.17811	+	1.17811i
$416$		−	2.00000i		−	0.0980581i
$417$		−	24.0000i		−	1.17529i
$418$	−8.00000	−	8.00000i	−0.391293	−	0.391293i
$419$	−10.0000	−	10.0000i	−0.488532	−	0.488532i	0.419311	−	0.907843i	$-0.362272\pi$
$419$	−10.0000	−	10.0000i	−0.488532	−	0.488532i	−0.907843	+	0.419311i	$0.862272\pi$
$420$		0			0
$421$	−26.0000			−1.26716			−0.633581	−	0.773676i	$-0.718416\pi$
$421$	−26.0000			−1.26716			−0.633581	+	0.773676i	$0.718416\pi$
$422$	−6.00000	+	6.00000i	−0.292075	+	0.292075i
$423$		0			0
$424$	−6.00000			−0.291386
$425$		0			0
$426$	16.0000			0.775203
$427$		0			0
$428$	−2.00000	+	2.00000i	−0.0966736	+	0.0966736i
$429$	−16.0000			−0.772487
$430$	8.00000	−	8.00000i	0.385794	−	0.385794i
$431$	−4.00000	−	4.00000i	−0.192673	−	0.192673i	0.604177	−	0.796850i	$-0.293502\pi$
$431$	−4.00000	−	4.00000i	−0.192673	−	0.192673i	−0.796850	+	0.604177i	$0.793502\pi$
$432$	−4.00000	−	4.00000i	−0.192450	−	0.192450i
$433$			14.0000i			0.672797i	0.941720	+	0.336399i	$0.109209\pi$
$433$			14.0000i			0.672797i	−0.941720	+	0.336399i	$0.890791\pi$
$434$		0			0
$435$	−16.0000	−	16.0000i	−0.767141	−	0.767141i
$436$	6.00000	+	6.00000i	0.287348	+	0.287348i
$437$	16.0000	−	16.0000i	0.765384	−	0.765384i
$438$		0			0
$439$	12.0000	−	12.0000i	0.572729	−	0.572729i	−0.360161	−	0.932890i	$-0.617278\pi$
$439$	12.0000	−	12.0000i	0.572729	−	0.572729i	0.932890	+	0.360161i	$0.117278\pi$
$440$		−	8.00000i		−	0.381385i
$441$	−35.0000			−1.66667
$442$		0			0
$443$	−12.0000			−0.570137			−0.285069	−	0.958507i	$-0.592016\pi$
$443$	−12.0000			−0.570137			−0.285069	+	0.958507i	$0.592016\pi$
$444$		−	24.0000i		−	1.13899i
$445$	−12.0000	+	12.0000i	−0.568855	+	0.568855i
$446$	−16.0000			−0.757622
$447$	−12.0000	+	12.0000i	−0.567581	+	0.567581i
$448$		0			0
$449$	4.00000	+	4.00000i	0.188772	+	0.188772i	0.795165	−	0.606393i	$-0.207384\pi$
$449$	4.00000	+	4.00000i	0.188772	+	0.188772i	−0.606393	+	0.795165i	$0.707384\pi$
$450$		−	15.0000i		−	0.707107i
$451$			16.0000i			0.753411i
$452$	−8.00000	−	8.00000i	−0.376288	−	0.376288i
$453$	−16.0000	−	16.0000i	−0.751746	−	0.751746i
$454$	−14.0000	+	14.0000i	−0.657053	+	0.657053i
$455$		0			0
$456$	8.00000	−	8.00000i	0.374634	−	0.374634i
$457$			10.0000i			0.467780i	0.972263	+	0.233890i	$0.0751456\pi$
$457$			10.0000i			0.467780i	−0.972263	+	0.233890i	$0.924854\pi$
$458$	22.0000			1.02799
$459$		0			0
$460$	16.0000			0.746004
$461$			18.0000i			0.838344i	0.907907	+	0.419172i	$0.137680\pi$
$461$			18.0000i			0.838344i	−0.907907	+	0.419172i	$0.862320\pi$
$462$		0			0
$463$	−32.0000			−1.48717			−0.743583	−	0.668644i	$-0.766875\pi$
$463$	−32.0000			−1.48717			−0.743583	+	0.668644i	$0.766875\pi$
$464$	2.00000	−	2.00000i	0.0928477	−	0.0928477i
$465$		0			0
$466$	−16.0000	−	16.0000i	−0.741186	−	0.741186i
$467$		−	36.0000i		−	1.66588i	−0.553362	−	0.832941i	$-0.686655\pi$
$467$		−	36.0000i		−	1.66588i	0.553362	−	0.832941i	$-0.313345\pi$
$468$		−	10.0000i		−	0.462250i
$469$		0			0
$470$		0			0
$471$	−28.0000	+	28.0000i	−1.29017	+	1.29017i
$472$	12.0000			0.552345
$473$	8.00000	−	8.00000i	0.367840	−	0.367840i
$474$			48.0000i			2.20471i
$475$	12.0000			0.550598
$476$		0			0
$477$	−30.0000			−1.37361
$478$		0			0
$479$	4.00000	−	4.00000i	0.182765	−	0.182765i	−0.609795	−	0.792559i	$-0.708748\pi$
$479$	4.00000	−	4.00000i	0.182765	−	0.182765i	0.792559	+	0.609795i	$0.208748\pi$
$480$	8.00000			0.365148
$481$	12.0000	−	12.0000i	0.547153	−	0.547153i
$482$	12.0000	+	12.0000i	0.546585	+	0.546585i
$483$		0			0
$484$			3.00000i			0.136364i
$485$			48.0000i			2.17957i
$486$	10.0000	+	10.0000i	0.453609	+	0.453609i
$487$	−12.0000	−	12.0000i	−0.543772	−	0.543772i	0.380861	−	0.924632i	$-0.375628\pi$
$487$	−12.0000	−	12.0000i	−0.543772	−	0.543772i	−0.924632	+	0.380861i	$0.875628\pi$
$488$	6.00000	−	6.00000i	0.271607	−	0.271607i
$489$	−24.0000			−1.08532
$490$	14.0000	−	14.0000i	0.632456	−	0.632456i
$491$			12.0000i			0.541552i	0.962642	+	0.270776i	$0.0872803\pi$
$491$			12.0000i			0.541552i	−0.962642	+	0.270776i	$0.912720\pi$
$492$	−16.0000			−0.721336
$493$		0			0
$494$	8.00000			0.359937
$495$		−	40.0000i		−	1.79787i
$496$		0			0
$497$		0			0
$498$	−24.0000	+	24.0000i	−1.07547	+	1.07547i
$499$	18.0000	+	18.0000i	0.805791	+	0.805791i	0.983994	−	0.178203i	$-0.0570284\pi$
$499$	18.0000	+	18.0000i	0.805791	+	0.805791i	−0.178203	+	0.983994i	$0.557028\pi$
$500$	−4.00000	−	4.00000i	−0.178885	−	0.178885i
$501$			32.0000i			1.42965i
$502$			12.0000i			0.535586i
$503$	−20.0000	−	20.0000i	−0.891756	−	0.891756i	0.102932	−	0.994688i	$-0.467177\pi$
$503$	−20.0000	−	20.0000i	−0.891756	−	0.891756i	−0.994688	+	0.102932i	$0.967177\pi$
$504$		0			0
$505$	−12.0000	+	12.0000i	−0.533993	+	0.533993i
$506$	16.0000			0.711287
$507$	−18.0000	+	18.0000i	−0.799408	+	0.799408i
$508$			16.0000i			0.709885i
$509$	−18.0000			−0.797836			−0.398918	−	0.916987i	$-0.630614\pi$
$509$	−18.0000			−0.797836			−0.398918	+	0.916987i	$0.630614\pi$
$510$		0			0
$511$		0			0
$512$			1.00000i			0.0441942i
$513$	16.0000	−	16.0000i	0.706417	−	0.706417i
$514$	6.00000			0.264649
$515$	16.0000	−	16.0000i	0.705044	−	0.705044i
$516$	8.00000	+	8.00000i	0.352180	+	0.352180i
$517$		0			0
$518$		0			0
$519$			8.00000i			0.351161i
$520$	4.00000	+	4.00000i	0.175412	+	0.175412i
$521$	−4.00000	−	4.00000i	−0.175243	−	0.175243i	0.614035	−	0.789279i	$-0.289545\pi$
$521$	−4.00000	−	4.00000i	−0.175243	−	0.175243i	−0.789279	+	0.614035i	$0.789545\pi$
$522$	10.0000	−	10.0000i	0.437688	−	0.437688i
$523$	−20.0000			−0.874539			−0.437269	−	0.899331i	$-0.644054\pi$
$523$	−20.0000			−0.874539			−0.437269	+	0.899331i	$0.644054\pi$
$524$	−2.00000	+	2.00000i	−0.0873704	+	0.0873704i
$525$		0			0
$526$	−24.0000			−1.04645
$527$		0			0
$528$	8.00000			0.348155
$529$			9.00000i			0.391304i
$530$	12.0000	−	12.0000i	0.521247	−	0.521247i
$531$	60.0000			2.60378
$532$		0			0
$533$	−8.00000	−	8.00000i	−0.346518	−	0.346518i
$534$	−12.0000	−	12.0000i	−0.519291	−	0.519291i
$535$		−	8.00000i		−	0.345870i
$536$			4.00000i			0.172774i
$537$	24.0000	+	24.0000i	1.03568	+	1.03568i
$538$	22.0000	+	22.0000i	0.948487	+	0.948487i
$539$	14.0000	−	14.0000i	0.603023	−	0.603023i
$540$	16.0000			0.688530
$541$	−6.00000	+	6.00000i	−0.257960	+	0.257960i	−0.824224	−	0.566264i	$-0.808388\pi$
$541$	−6.00000	+	6.00000i	−0.257960	+	0.257960i	0.566264	+	0.824224i	$0.308388\pi$
$542$		−	16.0000i		−	0.687259i
$543$	−24.0000			−1.02994
$544$		0			0
$545$	−24.0000			−1.02805
$546$		0			0
$547$	30.0000	−	30.0000i	1.28271	−	1.28271i	0.343586	−	0.939121i	$-0.388358\pi$
$547$	30.0000	−	30.0000i	1.28271	−	1.28271i	0.939121	−	0.343586i	$-0.111642\pi$
$548$	18.0000			0.768922
$549$	30.0000	−	30.0000i	1.28037	−	1.28037i
$550$	6.00000	+	6.00000i	0.255841	+	0.255841i
$551$	8.00000	+	8.00000i	0.340811	+	0.340811i
$552$			16.0000i			0.681005i
$553$		0			0
$554$	18.0000	+	18.0000i	0.764747	+	0.764747i
$555$	48.0000	+	48.0000i	2.03749	+	2.03749i
$556$	−6.00000	+	6.00000i	−0.254457	+	0.254457i
$557$	−18.0000			−0.762684			−0.381342	−	0.924434i	$-0.624538\pi$
$557$	−18.0000			−0.762684			−0.381342	+	0.924434i	$0.624538\pi$
$558$		0			0
$559$			8.00000i			0.338364i
$560$		0			0
$561$		0			0
$562$	−18.0000			−0.759284
$563$		−	36.0000i		−	1.51722i	−0.651546	−	0.758610i	$-0.725879\pi$
$563$		−	36.0000i		−	1.51722i	0.651546	−	0.758610i	$-0.274121\pi$
$564$		0			0
$565$	32.0000			1.34625
$566$	6.00000	−	6.00000i	0.252199	−	0.252199i
$567$		0			0
$568$	−4.00000	−	4.00000i	−0.167836	−	0.167836i
$569$		−	6.00000i		−	0.251533i	−0.992060	−	0.125767i	$-0.959861\pi$
$569$		−	6.00000i		−	0.251533i	0.992060	−	0.125767i	$-0.0401390\pi$
$570$			32.0000i			1.34033i
$571$	18.0000	+	18.0000i	0.753277	+	0.753277i	0.975089	−	0.221813i	$-0.0711974\pi$
$571$	18.0000	+	18.0000i	0.753277	+	0.753277i	−0.221813	+	0.975089i	$0.571197\pi$
$572$	4.00000	+	4.00000i	0.167248	+	0.167248i
$573$		0			0
$574$		0			0
$575$	−12.0000	+	12.0000i	−0.500435	+	0.500435i
$576$			5.00000i			0.208333i
$577$	38.0000			1.58196			0.790980	−	0.611842i	$-0.209571\pi$
$577$	38.0000			1.58196			0.790980	+	0.611842i	$0.209571\pi$
$578$		0			0
$579$	−48.0000			−1.99481
$580$			8.00000i			0.332182i
$581$		0			0
$582$	−48.0000			−1.98966
$583$	12.0000	−	12.0000i	0.496989	−	0.496989i
$584$		0			0
$585$	20.0000	+	20.0000i	0.826898	+	0.826898i
$586$		−	6.00000i		−	0.247858i
$587$		−	12.0000i		−	0.495293i	−0.968850	−	0.247647i	$-0.920343\pi$
$587$		−	12.0000i		−	0.495293i	0.968850	−	0.247647i	$-0.0796572\pi$
$588$	14.0000	+	14.0000i	0.577350	+	0.577350i
$589$		0			0
$590$	−24.0000	+	24.0000i	−0.988064	+	0.988064i
$591$	−8.00000			−0.329076
$592$	−6.00000	+	6.00000i	−0.246598	+	0.246598i
$593$		−	18.0000i		−	0.739171i	−0.929197	−	0.369586i	$-0.879500\pi$
$593$		−	18.0000i		−	0.739171i	0.929197	−	0.369586i	$-0.120500\pi$
$594$	16.0000			0.656488
$595$		0			0
$596$	6.00000			0.245770
$597$		−	48.0000i		−	1.96451i
$598$	−8.00000	+	8.00000i	−0.327144	+	0.327144i
$599$	−24.0000			−0.980613			−0.490307	−	0.871550i	$-0.663115\pi$
$599$	−24.0000			−0.980613			−0.490307	+	0.871550i	$0.663115\pi$
$600$	−6.00000	+	6.00000i	−0.244949	+	0.244949i
$601$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$601$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$602$		0			0
$603$			20.0000i			0.814463i
$604$			8.00000i			0.325515i
$605$	−6.00000	−	6.00000i	−0.243935	−	0.243935i
$606$	−12.0000	−	12.0000i	−0.487467	−	0.487467i
$607$	24.0000	−	24.0000i	0.974130	−	0.974130i	−0.0255438	−	0.999674i	$-0.508132\pi$
$607$	24.0000	−	24.0000i	0.974130	−	0.974130i	0.999674	+	0.0255438i	$0.00813171\pi$
$608$	−4.00000			−0.162221
$609$		0			0
$610$			24.0000i			0.971732i
$611$		0			0
$612$		0			0
$613$	−10.0000			−0.403896			−0.201948	−	0.979396i	$-0.564727\pi$
$613$	−10.0000			−0.403896			−0.201948	+	0.979396i	$0.564727\pi$
$614$			20.0000i			0.807134i
$615$	32.0000	−	32.0000i	1.29036	−	1.29036i
$616$		0			0
$617$	8.00000	−	8.00000i	0.322068	−	0.322068i	−0.527492	−	0.849560i	$-0.676868\pi$
$617$	8.00000	−	8.00000i	0.322068	−	0.322068i	0.849560	+	0.527492i	$0.176868\pi$
$618$	16.0000	+	16.0000i	0.643614	+	0.643614i
$619$	−6.00000	−	6.00000i	−0.241160	−	0.241160i	0.576170	−	0.817330i	$-0.304547\pi$
$619$	−6.00000	−	6.00000i	−0.241160	−	0.241160i	−0.817330	+	0.576170i	$0.804547\pi$
$620$		0			0
$621$			32.0000i			1.28412i
$622$	8.00000	+	8.00000i	0.320771	+	0.320771i
$623$		0			0
$624$	−4.00000	+	4.00000i	−0.160128	+	0.160128i
$625$	31.0000			1.24000
$626$	12.0000	−	12.0000i	0.479616	−	0.479616i
$627$			32.0000i			1.27796i
$628$	14.0000			0.558661
$629$		0			0
$630$		0			0
$631$		−	8.00000i		−	0.318475i	−0.987240	−	0.159237i	$-0.949096\pi$
$631$		−	8.00000i		−	0.318475i	0.987240	−	0.159237i	$-0.0509036\pi$
$632$	12.0000	−	12.0000i	0.477334	−	0.477334i
$633$	24.0000			0.953914
$634$	−2.00000	+	2.00000i	−0.0794301	+	0.0794301i
$635$	−32.0000	−	32.0000i	−1.26988	−	1.26988i
$636$	12.0000	+	12.0000i	0.475831	+	0.475831i
$637$			14.0000i			0.554700i
$638$			8.00000i			0.316723i
$639$	−20.0000	−	20.0000i	−0.791188	−	0.791188i
$640$	−2.00000	−	2.00000i	−0.0790569	−	0.0790569i
$641$	−4.00000	+	4.00000i	−0.157991	+	0.157991i	−0.781676	−	0.623685i	$-0.785635\pi$
$641$	−4.00000	+	4.00000i	−0.157991	+	0.157991i	0.623685	+	0.781676i	$0.285635\pi$
$642$	8.00000			0.315735
$643$	−6.00000	+	6.00000i	−0.236617	+	0.236617i	−0.815448	−	0.578831i	$-0.803509\pi$
$643$	−6.00000	+	6.00000i	−0.236617	+	0.236617i	0.578831	+	0.815448i	$0.303509\pi$
$644$		0			0
$645$	−32.0000			−1.26000
$646$		0			0
$647$	−24.0000			−0.943537			−0.471769	−	0.881722i	$-0.656384\pi$
$647$	−24.0000			−0.943537			−0.471769	+	0.881722i	$0.656384\pi$
$648$			1.00000i			0.0392837i
$649$	−24.0000	+	24.0000i	−0.942082	+	0.942082i
$650$	−6.00000			−0.235339
$651$		0			0
$652$	6.00000	+	6.00000i	0.234978	+	0.234978i
$653$	10.0000	+	10.0000i	0.391330	+	0.391330i	0.875161	−	0.483831i	$-0.160755\pi$
$653$	10.0000	+	10.0000i	0.391330	+	0.391330i	−0.483831	+	0.875161i	$0.660755\pi$
$654$		−	24.0000i		−	0.938474i
$655$		−	8.00000i		−	0.312586i
$656$	4.00000	+	4.00000i	0.156174	+	0.156174i
$657$		0			0
$658$		0			0
$659$	12.0000			0.467454			0.233727	−	0.972302i	$-0.424908\pi$
$659$	12.0000			0.467454			0.233727	+	0.972302i	$0.424908\pi$
$660$	−16.0000	+	16.0000i	−0.622799	+	0.622799i
$661$			10.0000i			0.388955i	0.980907	+	0.194477i	$0.0623011\pi$
$661$			10.0000i			0.388955i	−0.980907	+	0.194477i	$0.937699\pi$
$662$	4.00000			0.155464
$663$		0			0
$664$	12.0000			0.465690
$665$		0			0
$666$	−30.0000	+	30.0000i	−1.16248	+	1.16248i
$667$	−16.0000			−0.619522
$668$	8.00000	−	8.00000i	0.309529	−	0.309529i
$669$	32.0000	+	32.0000i	1.23719	+	1.23719i
$670$	−8.00000	−	8.00000i	−0.309067	−	0.309067i
$671$			24.0000i			0.926510i
$672$		0			0
$673$	−12.0000	−	12.0000i	−0.462566	−	0.462566i	0.436930	−	0.899496i	$-0.356066\pi$
$673$	−12.0000	−	12.0000i	−0.462566	−	0.462566i	−0.899496	+	0.436930i	$0.856066\pi$
$674$		0			0
$675$	−12.0000	+	12.0000i	−0.461880	+	0.461880i
$676$	9.00000			0.346154
$677$	10.0000	−	10.0000i	0.384331	−	0.384331i	−0.488329	−	0.872660i	$-0.662393\pi$
$677$	10.0000	−	10.0000i	0.384331	−	0.384331i	0.872660	+	0.488329i	$0.162393\pi$
$678$			32.0000i			1.22895i
$679$		0			0
$680$		0			0
$681$	56.0000			2.14592
$682$		0			0
$683$	−14.0000	+	14.0000i	−0.535695	+	0.535695i	−0.922262	−	0.386566i	$-0.873661\pi$
$683$	−14.0000	+	14.0000i	−0.535695	+	0.535695i	0.386566	+	0.922262i	$0.373661\pi$
$684$	−20.0000			−0.764719
$685$	−36.0000	+	36.0000i	−1.37549	+	1.37549i
$686$		0			0
$687$	−44.0000	−	44.0000i	−1.67870	−	1.67870i
$688$		−	4.00000i		−	0.152499i
$689$			12.0000i			0.457164i
$690$	−32.0000	−	32.0000i	−1.21822	−	1.21822i
$691$	−6.00000	−	6.00000i	−0.228251	−	0.228251i	0.583711	−	0.811962i	$-0.301600\pi$
$691$	−6.00000	−	6.00000i	−0.228251	−	0.228251i	−0.811962	+	0.583711i	$0.801600\pi$
$692$	2.00000	−	2.00000i	0.0760286	−	0.0760286i
$693$		0			0
$694$	−10.0000	+	10.0000i	−0.379595	+	0.379595i
$695$		−	24.0000i		−	0.910372i
$696$	−8.00000			−0.303239
$697$		0			0
$698$	34.0000			1.28692
$699$			64.0000i			2.42070i
$700$		0			0
$701$	30.0000			1.13308			0.566542	−	0.824033i	$-0.308281\pi$
$701$	30.0000			1.13308			0.566542	+	0.824033i	$0.308281\pi$
$702$	−8.00000	+	8.00000i	−0.301941	+	0.301941i
$703$	−24.0000	−	24.0000i	−0.905177	−	0.905177i
$704$	−2.00000	−	2.00000i	−0.0753778	−	0.0753778i
$705$		0			0
$706$			6.00000i			0.225813i
$707$		0			0
$708$	−24.0000	−	24.0000i	−0.901975	−	0.901975i
$709$	−6.00000	+	6.00000i	−0.225335	+	0.225335i	−0.810740	−	0.585406i	$-0.800935\pi$
$709$	−6.00000	+	6.00000i	−0.225335	+	0.225335i	0.585406	+	0.810740i	$0.300935\pi$
$710$	16.0000			0.600469
$711$	60.0000	−	60.0000i	2.25018	−	2.25018i
$712$			6.00000i			0.224860i
$713$		0			0
$714$		0			0
$715$	−16.0000			−0.598366
$716$		−	12.0000i		−	0.448461i
$717$		0			0
$718$	24.0000			0.895672
$719$	−28.0000	+	28.0000i	−1.04422	+	1.04422i	−0.0452480	+	0.998976i	$0.514408\pi$
$719$	−28.0000	+	28.0000i	−1.04422	+	1.04422i	−0.998976	+	0.0452480i	$0.985592\pi$
$720$	−10.0000	−	10.0000i	−0.372678	−	0.372678i
$721$		0			0
$722$			3.00000i			0.111648i
$723$		−	48.0000i		−	1.78514i
$724$	6.00000	+	6.00000i	0.222988	+	0.222988i
$725$	−6.00000	−	6.00000i	−0.222834	−	0.222834i
$726$	6.00000	−	6.00000i	0.222681	−	0.222681i
$727$	−8.00000			−0.296704			−0.148352	−	0.988935i	$-0.547397\pi$
$727$	−8.00000			−0.296704			−0.148352	+	0.988935i	$0.547397\pi$
$728$		0			0
$729$		−	43.0000i		−	1.59259i
$730$		0			0
$731$		0			0
$732$	−24.0000			−0.887066
$733$		−	14.0000i		−	0.517102i	−0.965998	−	0.258551i	$-0.916755\pi$
$733$		−	14.0000i		−	0.517102i	0.965998	−	0.258551i	$-0.0832450\pi$
$734$	−12.0000	+	12.0000i	−0.442928	+	0.442928i
$735$	−56.0000			−2.06559
$736$	4.00000	−	4.00000i	0.147442	−	0.147442i
$737$	−8.00000	−	8.00000i	−0.294684	−	0.294684i
$738$	20.0000	+	20.0000i	0.736210	+	0.736210i
$739$			44.0000i			1.61857i	0.587419	+	0.809283i	$0.300144\pi$
$739$			44.0000i			1.61857i	−0.587419	+	0.809283i	$0.699856\pi$
$740$		−	24.0000i		−	0.882258i
$741$	−16.0000	−	16.0000i	−0.587775	−	0.587775i
$742$		0			0
$743$	8.00000	−	8.00000i	0.293492	−	0.293492i	−0.544966	−	0.838458i	$-0.683458\pi$
$743$	8.00000	−	8.00000i	0.293492	−	0.293492i	0.838458	+	0.544966i	$0.183458\pi$
$744$		0			0
$745$	−12.0000	+	12.0000i	−0.439646	+	0.439646i
$746$		−	22.0000i		−	0.805477i
$747$	60.0000			2.19529
$748$		0			0
$749$		0			0
$750$			16.0000i			0.584237i
$751$	−12.0000	+	12.0000i	−0.437886	+	0.437886i	−0.891300	−	0.453414i	$-0.850206\pi$
$751$	−12.0000	+	12.0000i	−0.437886	+	0.437886i	0.453414	+	0.891300i	$0.350206\pi$
$752$		0			0
$753$	24.0000	−	24.0000i	0.874609	−	0.874609i
$754$	−4.00000	−	4.00000i	−0.145671	−	0.145671i
$755$	−16.0000	−	16.0000i	−0.582300	−	0.582300i
$756$		0			0
$757$		−	22.0000i		−	0.799604i	−0.916602	−	0.399802i	$-0.869079\pi$
$757$		−	22.0000i		−	0.799604i	0.916602	−	0.399802i	$-0.130921\pi$
$758$	−18.0000	−	18.0000i	−0.653789	−	0.653789i
$759$	−32.0000	−	32.0000i	−1.16153	−	1.16153i
$760$	8.00000	−	8.00000i	0.290191	−	0.290191i
$761$	−30.0000			−1.08750			−0.543750	−	0.839248i	$-0.682996\pi$
$761$	−30.0000			−1.08750			−0.543750	+	0.839248i	$0.682996\pi$
$762$	32.0000	−	32.0000i	1.15924	−	1.15924i
$763$		0			0
$764$		0			0
$765$		0			0
$766$		0			0
$767$		−	24.0000i		−	0.866590i
$768$	2.00000	−	2.00000i	0.0721688	−	0.0721688i
$769$	10.0000			0.360609			0.180305	−	0.983611i	$-0.442292\pi$
$769$	10.0000			0.360609			0.180305	+	0.983611i	$0.442292\pi$
$770$		0			0
$771$	−12.0000	−	12.0000i	−0.432169	−	0.432169i
$772$	12.0000	+	12.0000i	0.431889	+	0.431889i
$773$		−	54.0000i		−	1.94225i	−0.238581	−	0.971123i	$-0.576682\pi$
$773$		−	54.0000i		−	1.94225i	0.238581	−	0.971123i	$-0.423318\pi$
$774$		−	20.0000i		−	0.718885i
$775$		0			0
$776$	12.0000	+	12.0000i	0.430775	+	0.430775i
$777$		0			0
$778$	−6.00000			−0.215110
$779$	−16.0000	+	16.0000i	−0.573259	+	0.573259i
$780$		−	16.0000i		−	0.572892i
$781$	16.0000			0.572525
$782$		0			0
$783$	−16.0000			−0.571793
$784$		−	7.00000i		−	0.250000i
$785$	−28.0000	+	28.0000i	−0.999363	+	0.999363i
$786$	8.00000			0.285351
$787$	−30.0000	+	30.0000i	−1.06938	+	1.06938i	−0.0719783	+	0.997406i	$0.522931\pi$
$787$	−30.0000	+	30.0000i	−1.06938	+	1.06938i	−0.997406	+	0.0719783i	$0.977069\pi$
$788$	2.00000	+	2.00000i	0.0712470	+	0.0712470i
$789$	48.0000	+	48.0000i	1.70885	+	1.70885i
$790$			48.0000i			1.70776i
$791$		0			0
$792$	−10.0000	−	10.0000i	−0.355335	−	0.355335i
$793$	−12.0000	−	12.0000i	−0.426132	−	0.426132i
$794$	6.00000	−	6.00000i	0.212932	−	0.212932i
$795$	−48.0000			−1.70238
$796$	−12.0000	+	12.0000i	−0.425329	+	0.425329i
$797$			18.0000i			0.637593i	0.947823	+	0.318796i	$0.103279\pi$
$797$			18.0000i			0.637593i	−0.947823	+	0.318796i	$0.896721\pi$
$798$		0			0
$799$		0			0
$800$	3.00000			0.106066
$801$			30.0000i			1.06000i
$802$	20.0000	−	20.0000i	0.706225	−	0.706225i
$803$		0			0
$804$	8.00000	−	8.00000i	0.282138	−	0.282138i
$805$		0			0
$806$		0			0
$807$		−	88.0000i		−	3.09775i
$808$			6.00000i			0.211079i
$809$	28.0000	+	28.0000i	0.984428	+	0.984428i	0.999881	−	0.0154530i	$-0.00491904\pi$
$809$	28.0000	+	28.0000i	0.984428	+	0.984428i	−0.0154530	+	0.999881i	$0.504919\pi$
$810$	−2.00000	−	2.00000i	−0.0702728	−	0.0702728i
$811$	30.0000	−	30.0000i	1.05344	−	1.05344i	0.0549536	−	0.998489i	$-0.482499\pi$
$811$	30.0000	−	30.0000i	1.05344	−	1.05344i	0.998489	−	0.0549536i	$-0.0175011\pi$
$812$		0			0
$813$	−32.0000	+	32.0000i	−1.12229	+	1.12229i
$814$		−	24.0000i		−	0.841200i
$815$	−24.0000			−0.840683
$816$		0			0
$817$	16.0000			0.559769
$818$			2.00000i			0.0699284i
$819$		0			0
$820$	−16.0000			−0.558744
$821$	−34.0000	+	34.0000i	−1.18661	+	1.18661i	−0.208609	+	0.977999i	$0.566894\pi$
$821$	−34.0000	+	34.0000i	−1.18661	+	1.18661i	−0.977999	+	0.208609i	$0.933106\pi$
$822$	−36.0000	−	36.0000i	−1.25564	−	1.25564i
$823$	−12.0000	−	12.0000i	−0.418294	−	0.418294i	0.466322	−	0.884615i	$-0.345579\pi$
$823$	−12.0000	−	12.0000i	−0.418294	−	0.418294i	−0.884615	+	0.466322i	$0.845579\pi$
$824$		−	8.00000i		−	0.278693i
$825$		−	24.0000i		−	0.835573i
$826$		0			0
$827$	26.0000	+	26.0000i	0.904109	+	0.904109i	0.995789	−	0.0916799i	$-0.0292237\pi$
$827$	26.0000	+	26.0000i	0.904109	+	0.904109i	−0.0916799	+	0.995789i	$0.529224\pi$
$828$	20.0000	−	20.0000i	0.695048	−	0.695048i
$829$	34.0000			1.18087			0.590434	−	0.807086i	$-0.298956\pi$
$829$	34.0000			1.18087			0.590434	+	0.807086i	$0.298956\pi$
$830$	−24.0000	+	24.0000i	−0.833052	+	0.833052i
$831$		−	72.0000i		−	2.49765i
$832$	2.00000			0.0693375
$833$		0			0
$834$	24.0000			0.831052
$835$			32.0000i			1.10741i
$836$	8.00000	−	8.00000i	0.276686	−	0.276686i
$837$		0			0
$838$	10.0000	−	10.0000i	0.345444	−	0.345444i
$839$	32.0000	+	32.0000i	1.10476	+	1.10476i	0.993828	+	0.110935i	$0.0353845\pi$
$839$	32.0000	+	32.0000i	1.10476	+	1.10476i	0.110935	+	0.993828i	$0.464615\pi$
$840$		0			0
$841$			21.0000i			0.724138i
$842$		−	26.0000i		−	0.896019i
$843$	36.0000	+	36.0000i	1.23991	+	1.23991i
$844$	−6.00000	−	6.00000i	−0.206529	−	0.206529i
$845$	−18.0000	+	18.0000i	−0.619219	+	0.619219i
$846$		0			0
$847$		0			0
$848$		−	6.00000i		−	0.206041i
$849$	−24.0000			−0.823678
$850$		0			0
$851$	48.0000			1.64542
$852$			16.0000i			0.548151i
$853$	−18.0000	+	18.0000i	−0.616308	+	0.616308i	−0.944582	−	0.328274i	$-0.893533\pi$
$853$	−18.0000	+	18.0000i	−0.616308	+	0.616308i	0.328274	+	0.944582i	$0.393533\pi$
$854$		0			0
$855$	40.0000	−	40.0000i	1.36797	−	1.36797i
$856$	−2.00000	−	2.00000i	−0.0683586	−	0.0683586i
$857$	−20.0000	−	20.0000i	−0.683187	−	0.683187i	0.277530	−	0.960717i	$-0.410484\pi$
$857$	−20.0000	−	20.0000i	−0.683187	−	0.683187i	−0.960717	+	0.277530i	$0.910484\pi$
$858$		−	16.0000i		−	0.546231i
$859$		−	4.00000i		−	0.136478i	−0.997669	−	0.0682391i	$-0.978262\pi$
$859$		−	4.00000i		−	0.136478i	0.997669	−	0.0682391i	$-0.0217381\pi$
$860$	8.00000	+	8.00000i	0.272798	+	0.272798i
$861$		0			0
$862$	4.00000	−	4.00000i	0.136241	−	0.136241i
$863$	48.0000			1.63394			0.816970	−	0.576681i	$-0.195652\pi$
$863$	48.0000			1.63394			0.816970	+	0.576681i	$0.195652\pi$
$864$	4.00000	−	4.00000i	0.136083	−	0.136083i
$865$			8.00000i			0.272008i
$866$	−14.0000			−0.475739
$867$		0			0
$868$		0			0
$869$			48.0000i			1.62829i
$870$	16.0000	−	16.0000i	0.542451	−	0.542451i
$871$	8.00000			0.271070
$872$	−6.00000	+	6.00000i	−0.203186	+	0.203186i
$873$	60.0000	+	60.0000i	2.03069	+	2.03069i
$874$	16.0000	+	16.0000i	0.541208	+	0.541208i
$875$		0			0
$876$		0			0
$877$	−6.00000	−	6.00000i	−0.202606	−	0.202606i	0.598510	−	0.801115i	$-0.295760\pi$
$877$	−6.00000	−	6.00000i	−0.202606	−	0.202606i	−0.801115	+	0.598510i	$0.795760\pi$
$878$	12.0000	+	12.0000i	0.404980	+	0.404980i
$879$	−12.0000	+	12.0000i	−0.404750	+	0.404750i
$880$	8.00000			0.269680
$881$	−32.0000	+	32.0000i	−1.07811	+	1.07811i	−0.0814282	+	0.996679i	$0.525948\pi$
$881$	−32.0000	+	32.0000i	−1.07811	+	1.07811i	−0.996679	+	0.0814282i	$0.974052\pi$
$882$		−	35.0000i		−	1.17851i
$883$	44.0000			1.48072			0.740359	−	0.672212i	$-0.234656\pi$
$883$	44.0000			1.48072			0.740359	+	0.672212i	$0.234656\pi$
$884$		0			0
$885$	96.0000			3.22700
$886$		−	12.0000i		−	0.403148i
$887$	28.0000	−	28.0000i	0.940148	−	0.940148i	−0.0581593	−	0.998307i	$-0.518523\pi$
$887$	28.0000	−	28.0000i	0.940148	−	0.940148i	0.998307	+	0.0581593i	$0.0185231\pi$
$888$	24.0000			0.805387
$889$		0			0
$890$	−12.0000	−	12.0000i	−0.402241	−	0.402241i
$891$	−2.00000	−	2.00000i	−0.0670025	−	0.0670025i
$892$		−	16.0000i		−	0.535720i
$893$		0			0
$894$	−12.0000	−	12.0000i	−0.401340	−	0.401340i
$895$	24.0000	+	24.0000i	0.802232	+	0.802232i
$896$		0			0
$897$	32.0000			1.06845
$898$	−4.00000	+	4.00000i	−0.133482	+	0.133482i
$899$		0			0
$900$	15.0000			0.500000
$901$		0			0
$902$	−16.0000			−0.532742
$903$		0			0
$904$	8.00000	−	8.00000i	0.266076	−	0.266076i
$905$	−24.0000			−0.797787
$906$	16.0000	−	16.0000i	0.531564	−	0.531564i
$907$	−30.0000	−	30.0000i	−0.996134	−	0.996134i	0.00385890	−	0.999993i	$-0.498772\pi$
$907$	−30.0000	−	30.0000i	−0.996134	−	0.996134i	−0.999993	+	0.00385890i	$0.998772\pi$
$908$	−14.0000	−	14.0000i	−0.464606	−	0.464606i
$909$			30.0000i			0.995037i
$910$		0			0
$911$	40.0000	+	40.0000i	1.32526	+	1.32526i	0.909454	+	0.415806i	$0.136500\pi$
$911$	40.0000	+	40.0000i	1.32526	+	1.32526i	0.415806	+	0.909454i	$0.363500\pi$
$912$	8.00000	+	8.00000i	0.264906	+	0.264906i
$913$	−24.0000	+	24.0000i	−0.794284	+	0.794284i
$914$	−10.0000			−0.330771
$915$	48.0000	−	48.0000i	1.58683	−	1.58683i
$916$			22.0000i			0.726900i
$917$		0			0
$918$		0			0
$919$	−40.0000			−1.31948			−0.659739	−	0.751495i	$-0.729333\pi$
$919$	−40.0000			−1.31948			−0.659739	+	0.751495i	$0.729333\pi$
$920$			16.0000i			0.527504i
$921$	40.0000	−	40.0000i	1.31804	−	1.31804i
$922$	−18.0000			−0.592798
$923$	−8.00000	+	8.00000i	−0.263323	+	0.263323i
$924$		0			0
$925$	18.0000	+	18.0000i	0.591836	+	0.591836i
$926$		−	32.0000i		−	1.05159i
$927$		−	40.0000i		−	1.31377i
$928$	2.00000	+	2.00000i	0.0656532	+	0.0656532i
$929$	−4.00000	−	4.00000i	−0.131236	−	0.131236i	0.638438	−	0.769673i	$-0.279581\pi$
$929$	−4.00000	−	4.00000i	−0.131236	−	0.131236i	−0.769673	+	0.638438i	$0.779581\pi$
$930$		0			0
$931$	28.0000			0.917663
$932$	16.0000	−	16.0000i	0.524097	−	0.524097i
$933$		−	32.0000i		−	1.04763i
$934$	36.0000			1.17796
$935$		0			0
$936$	10.0000			0.326860
$937$		−	2.00000i		−	0.0653372i	−0.999466	−	0.0326686i	$-0.989599\pi$
$937$		−	2.00000i		−	0.0653372i	0.999466	−	0.0326686i	$-0.0104006\pi$
$938$		0			0
$939$	−48.0000			−1.56642
$940$		0			0
$941$	26.0000	+	26.0000i	0.847576	+	0.847576i	0.989830	−	0.142254i	$-0.0454351\pi$
$941$	26.0000	+	26.0000i	0.847576	+	0.847576i	−0.142254	+	0.989830i	$0.545435\pi$
$942$	−28.0000	−	28.0000i	−0.912289	−	0.912289i
$943$		−	32.0000i		−	1.04206i
$944$			12.0000i			0.390567i
$945$		0			0
$946$	8.00000	+	8.00000i	0.260102	+	0.260102i
$947$	2.00000	−	2.00000i	0.0649913	−	0.0649913i	−0.673864	−	0.738855i	$-0.735367\pi$
$947$	2.00000	−	2.00000i	0.0649913	−	0.0649913i	0.738855	+	0.673864i	$0.235367\pi$
$948$	−48.0000			−1.55897
$949$		0			0
$950$			12.0000i			0.389331i
$951$	8.00000			0.259418
$952$		0			0
$953$	6.00000			0.194359			0.0971795	−	0.995267i	$-0.469018\pi$
$953$	6.00000			0.194359			0.0971795	+	0.995267i	$0.469018\pi$
$954$		−	30.0000i		−	0.971286i
$955$		0			0
$956$		0			0
$957$	16.0000	−	16.0000i	0.517207	−	0.517207i
$958$	4.00000	+	4.00000i	0.129234	+	0.129234i
$959$		0			0
$960$			8.00000i			0.258199i
$961$			31.0000i			1.00000i
$962$	12.0000	+	12.0000i	0.386896	+	0.386896i
$963$	−10.0000	−	10.0000i	−0.322245	−	0.322245i
$964$	−12.0000	+	12.0000i	−0.386494	+	0.386494i
$965$	−48.0000			−1.54517
$966$		0			0
$967$		−	8.00000i		−	0.257263i	−0.991692	−	0.128631i	$-0.958942\pi$
$967$		−	8.00000i		−	0.257263i	0.991692	−	0.128631i	$-0.0410584\pi$
$968$	−3.00000			−0.0964237
$969$		0			0
$970$	−48.0000			−1.54119
$971$		−	12.0000i		−	0.385098i	−0.981287	−	0.192549i	$-0.938325\pi$
$971$		−	12.0000i		−	0.385098i	0.981287	−	0.192549i	$-0.0616755\pi$
$972$	−10.0000	+	10.0000i	−0.320750	+	0.320750i
$973$		0			0
$974$	12.0000	−	12.0000i	0.384505	−	0.384505i
$975$	12.0000	+	12.0000i	0.384308	+	0.384308i
$976$	6.00000	+	6.00000i	0.192055	+	0.192055i
$977$			18.0000i			0.575871i	0.957650	+	0.287936i	$0.0929689\pi$
$977$			18.0000i			0.575871i	−0.957650	+	0.287936i	$0.907031\pi$
$978$		−	24.0000i		−	0.767435i
$979$	−12.0000	−	12.0000i	−0.383522	−	0.383522i
$980$	14.0000	+	14.0000i	0.447214	+	0.447214i
$981$	−30.0000	+	30.0000i	−0.957826	+	0.957826i
$982$	−12.0000			−0.382935
$983$	4.00000	−	4.00000i	0.127580	−	0.127580i	−0.640433	−	0.768014i	$-0.721245\pi$
$983$	4.00000	−	4.00000i	0.127580	−	0.127580i	0.768014	+	0.640433i	$0.221245\pi$
$984$		−	16.0000i		−	0.510061i
$985$	−8.00000			−0.254901
$986$		0			0
$987$		0			0
$988$			8.00000i			0.254514i
$989$	−16.0000	+	16.0000i	−0.508770	+	0.508770i
$990$	40.0000			1.27128
$991$	36.0000	−	36.0000i	1.14358	−	1.14358i	0.155787	−	0.987791i	$-0.450209\pi$
$991$	36.0000	−	36.0000i	1.14358	−	1.14358i	0.987791	−	0.155787i	$-0.0497914\pi$
$992$		0			0
$993$	−8.00000	−	8.00000i	−0.253872	−	0.253872i
$994$		0			0
$995$		−	48.0000i		−	1.52170i
$996$	−24.0000	−	24.0000i	−0.760469	−	0.760469i
$997$	42.0000	+	42.0000i	1.33015	+	1.33015i	0.905229	+	0.424925i	$0.139699\pi$
$997$	42.0000	+	42.0000i	1.33015	+	1.33015i	0.424925	+	0.905229i	$0.360301\pi$
$998$	−18.0000	+	18.0000i	−0.569780	+	0.569780i
$999$	48.0000			1.51865

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	578.2.c.d.327.1		2
17.2	even	8		34.2.b.a.33.1	✓	2
17.3	odd	16		578.2.d.g.155.2		8
17.4	even	4		578.2.c.a.251.1		2
17.5	odd	16		578.2.d.g.423.2		8
17.6	odd	16		578.2.d.g.399.2		8
17.7	odd	16		578.2.d.g.179.1		8
17.8	even	8		578.2.a.d.1.1		2
17.9	even	8		578.2.a.d.1.2		2
17.10	odd	16		578.2.d.g.179.2		8
17.11	odd	16		578.2.d.g.399.1		8
17.12	odd	16		578.2.d.g.423.1		8
17.13	even	4	inner	578.2.c.d.251.1		2
17.14	odd	16		578.2.d.g.155.1		8
17.15	even	8		34.2.b.a.33.2	yes	2
17.16	even	2		578.2.c.a.327.1		2
51.2	odd	8		306.2.b.d.271.1		2
51.8	odd	8		5202.2.a.u.1.1		2
51.26	odd	8		5202.2.a.u.1.2		2
51.32	odd	8		306.2.b.d.271.2		2
68.15	odd	8		272.2.b.a.33.1		2
68.19	odd	8		272.2.b.a.33.2		2
68.43	odd	8		4624.2.a.s.1.1		2
68.59	odd	8		4624.2.a.s.1.2		2
85.2	odd	8		850.2.d.i.849.1		4
85.19	even	8		850.2.b.f.101.2		2
85.32	odd	8		850.2.d.i.849.2		4
85.49	even	8		850.2.b.f.101.1		2
85.53	odd	8		850.2.d.i.849.4		4
85.83	odd	8		850.2.d.i.849.3		4
119.83	odd	8		1666.2.b.c.883.1		2
119.104	odd	8		1666.2.b.c.883.2		2
136.19	odd	8		1088.2.b.a.577.1		2
136.53	even	8		1088.2.b.b.577.2		2
136.83	odd	8		1088.2.b.a.577.2		2
136.117	even	8		1088.2.b.b.577.1		2
204.83	even	8		2448.2.c.n.577.2		2
204.155	even	8		2448.2.c.n.577.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
34.2.b.a.33.1	✓	2	17.2	even	8
34.2.b.a.33.2	yes	2	17.15	even	8
272.2.b.a.33.1		2	68.15	odd	8
272.2.b.a.33.2		2	68.19	odd	8
306.2.b.d.271.1		2	51.2	odd	8
306.2.b.d.271.2		2	51.32	odd	8
578.2.a.d.1.1		2	17.8	even	8
578.2.a.d.1.2		2	17.9	even	8
578.2.c.a.251.1		2	17.4	even	4
578.2.c.a.327.1		2	17.16	even	2
578.2.c.d.251.1		2	17.13	even	4	inner
578.2.c.d.327.1		2	1.1	even	1	trivial
578.2.d.g.155.1		8	17.14	odd	16
578.2.d.g.155.2		8	17.3	odd	16
578.2.d.g.179.1		8	17.7	odd	16
578.2.d.g.179.2		8	17.10	odd	16
578.2.d.g.399.1		8	17.11	odd	16
578.2.d.g.399.2		8	17.6	odd	16
578.2.d.g.423.1		8	17.12	odd	16
578.2.d.g.423.2		8	17.5	odd	16
850.2.b.f.101.1		2	85.49	even	8
850.2.b.f.101.2		2	85.19	even	8
850.2.d.i.849.1		4	85.2	odd	8
850.2.d.i.849.2		4	85.32	odd	8
850.2.d.i.849.3		4	85.83	odd	8
850.2.d.i.849.4		4	85.53	odd	8
1088.2.b.a.577.1		2	136.19	odd	8
1088.2.b.a.577.2		2	136.83	odd	8
1088.2.b.b.577.1		2	136.117	even	8
1088.2.b.b.577.2		2	136.53	even	8
1666.2.b.c.883.1		2	119.83	odd	8
1666.2.b.c.883.2		2	119.104	odd	8
2448.2.c.n.577.1		2	204.155	even	8
2448.2.c.n.577.2		2	204.83	even	8
4624.2.a.s.1.1		2	68.43	odd	8
4624.2.a.s.1.2		2	68.59	odd	8
5202.2.a.u.1.1		2	51.8	odd	8
5202.2.a.u.1.2		2	51.26	odd	8

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 578 = 2 \cdot 17^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	578.c (of order \(4\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+1.00000i q^{2} +(2.00000 - 2.00000i) q^{3} -1.00000 q^{4} +(2.00000 - 2.00000i) q^{5} +(2.00000 + 2.00000i) q^{6} -1.00000i q^{8} -5.00000i q^{9} +(2.00000 + 2.00000i) q^{10} +(2.00000 + 2.00000i) q^{11} +(-2.00000 + 2.00000i) q^{12} -2.00000 q^{13} -8.00000i q^{15} +1.00000 q^{16} +5.00000 q^{18} +4.00000i q^{19} +(-2.00000 + 2.00000i) q^{20} +(-2.00000 + 2.00000i) q^{22} +(-4.00000 - 4.00000i) q^{23} +(-2.00000 - 2.00000i) q^{24} -3.00000i q^{25} -2.00000i q^{26} +(-4.00000 - 4.00000i) q^{27} +(2.00000 - 2.00000i) q^{29} +8.00000 q^{30} +1.00000i q^{32} +8.00000 q^{33} +5.00000i q^{36} +(-6.00000 + 6.00000i) q^{37} -4.00000 q^{38} +(-4.00000 + 4.00000i) q^{39} +(-2.00000 - 2.00000i) q^{40} +(4.00000 + 4.00000i) q^{41} -4.00000i q^{43} +(-2.00000 - 2.00000i) q^{44} +(-10.0000 - 10.0000i) q^{45} +(4.00000 - 4.00000i) q^{46} +(2.00000 - 2.00000i) q^{48} -7.00000i q^{49} +3.00000 q^{50} +2.00000 q^{52} -6.00000i q^{53} +(4.00000 - 4.00000i) q^{54} +8.00000 q^{55} +(8.00000 + 8.00000i) q^{57} +(2.00000 + 2.00000i) q^{58} +12.0000i q^{59} +8.00000i q^{60} +(6.00000 + 6.00000i) q^{61} -1.00000 q^{64} +(-4.00000 + 4.00000i) q^{65} +8.00000i q^{66} -4.00000 q^{67} -16.0000 q^{69} +(4.00000 - 4.00000i) q^{71} -5.00000 q^{72} +(-6.00000 - 6.00000i) q^{74} +(-6.00000 - 6.00000i) q^{75} -4.00000i q^{76} +(-4.00000 - 4.00000i) q^{78} +(12.0000 + 12.0000i) q^{79} +(2.00000 - 2.00000i) q^{80} -1.00000 q^{81} +(-4.00000 + 4.00000i) q^{82} +12.0000i q^{83} +4.00000 q^{86} -8.00000i q^{87} +(2.00000 - 2.00000i) q^{88} -6.00000 q^{89} +(10.0000 - 10.0000i) q^{90} +(4.00000 + 4.00000i) q^{92} +(8.00000 + 8.00000i) q^{95} +(2.00000 + 2.00000i) q^{96} +(-12.0000 + 12.0000i) q^{97} +7.00000 q^{98} +(10.0000 - 10.0000i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 4 q^{3} - 2 q^{4} + 4 q^{5} + 4 q^{6} + 4 q^{10} + 4 q^{11} - 4 q^{12} - 4 q^{13} + 2 q^{16} + 10 q^{18} - 4 q^{20} - 4 q^{22} - 8 q^{23} - 4 q^{24} - 8 q^{27} + 4 q^{29} + 16 q^{30} + 16 q^{33} - 12 q^{37}+ \cdots + 20 q^{99}+O(q^{100}) \) 2 * q + 4 * q^3 - 2 * q^4 + 4 * q^5 + 4 * q^6 + 4 * q^10 + 4 * q^11 - 4 * q^12 - 4 * q^13 + 2 * q^16 + 10 * q^18 - 4 * q^20 - 4 * q^22 - 8 * q^23 - 4 * q^24 - 8 * q^27 + 4 * q^29 + 16 * q^30 + 16 * q^33 - 12 * q^37 - 8 * q^38 - 8 * q^39 - 4 * q^40 + 8 * q^41 - 4 * q^44 - 20 * q^45 + 8 * q^46 + 4 * q^48 + 6 * q^50 + 4 * q^52 + 8 * q^54 + 16 * q^55 + 16 * q^57 + 4 * q^58 + 12 * q^61 - 2 * q^64 - 8 * q^65 - 8 * q^67 - 32 * q^69 + 8 * q^71 - 10 * q^72 - 12 * q^74 - 12 * q^75 - 8 * q^78 + 24 * q^79 + 4 * q^80 - 2 * q^81 - 8 * q^82 + 8 * q^86 + 4 * q^88 - 12 * q^89 + 20 * q^90 + 8 * q^92 + 16 * q^95 + 4 * q^96 - 24 * q^97 + 14 * q^98 + 20 * q^99

Introduction

L-functions

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