LMFDB - Embedded newform 104.2.b.a.53.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [104,2,Mod(53,104)] 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("104.53"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(104, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 104 = 2^{3} \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	104.b (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

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

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

Self dual:	no
Analytic conductor:	$0.830444181021$
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:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			53.1
Root			$-1.00000i$ of defining polynomial
Character	$\chi$	$=$	104.53
Dual form			104.2.b.a.53.2

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+(-1.00000 - 1.00000i) q^{2} +1.00000i q^{3} +2.00000i q^{4} -3.00000i q^{5} +(1.00000 - 1.00000i) q^{6} +3.00000 q^{7} +(2.00000 - 2.00000i) q^{8} +2.00000 q^{9} +(-3.00000 + 3.00000i) q^{10} -2.00000 q^{12} -1.00000i q^{13} +(-3.00000 - 3.00000i) q^{14} +3.00000 q^{15} -4.00000 q^{16} -7.00000 q^{17} +(-2.00000 - 2.00000i) q^{18} +4.00000i q^{19} +6.00000 q^{20} +3.00000i q^{21} +4.00000 q^{23} +(2.00000 + 2.00000i) q^{24} -4.00000 q^{25} +(-1.00000 + 1.00000i) q^{26} +5.00000i q^{27} +6.00000i q^{28} +4.00000i q^{29} +(-3.00000 - 3.00000i) q^{30} -8.00000 q^{31} +(4.00000 + 4.00000i) q^{32} +(7.00000 + 7.00000i) q^{34} -9.00000i q^{35} +4.00000i q^{36} -7.00000i q^{37} +(4.00000 - 4.00000i) q^{38} +1.00000 q^{39} +(-6.00000 - 6.00000i) q^{40} +2.00000 q^{41} +(3.00000 - 3.00000i) q^{42} +1.00000i q^{43} -6.00000i q^{45} +(-4.00000 - 4.00000i) q^{46} -7.00000 q^{47} -4.00000i q^{48} +2.00000 q^{49} +(4.00000 + 4.00000i) q^{50} -7.00000i q^{51} +2.00000 q^{52} -4.00000i q^{53} +(5.00000 - 5.00000i) q^{54} +(6.00000 - 6.00000i) q^{56} -4.00000 q^{57} +(4.00000 - 4.00000i) q^{58} +14.0000i q^{59} +6.00000i q^{60} +10.0000i q^{61} +(8.00000 + 8.00000i) q^{62} +6.00000 q^{63} -8.00000i q^{64} -3.00000 q^{65} -2.00000i q^{67} -14.0000i q^{68} +4.00000i q^{69} +(-9.00000 + 9.00000i) q^{70} -3.00000 q^{71} +(4.00000 - 4.00000i) q^{72} +14.0000 q^{73} +(-7.00000 + 7.00000i) q^{74} -4.00000i q^{75} -8.00000 q^{76} +(-1.00000 - 1.00000i) q^{78} -10.0000 q^{79} +12.0000i q^{80} +1.00000 q^{81} +(-2.00000 - 2.00000i) q^{82} -14.0000i q^{83} -6.00000 q^{84} +21.0000i q^{85} +(1.00000 - 1.00000i) q^{86} -4.00000 q^{87} +(-6.00000 + 6.00000i) q^{90} -3.00000i q^{91} +8.00000i q^{92} -8.00000i q^{93} +(7.00000 + 7.00000i) q^{94} +12.0000 q^{95} +(-4.00000 + 4.00000i) q^{96} +8.00000 q^{97} +(-2.00000 - 2.00000i) q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{2} + 2 q^{6} + 6 q^{7} + 4 q^{8} + 4 q^{9} - 6 q^{10} - 4 q^{12} - 6 q^{14} + 6 q^{15} - 8 q^{16} - 14 q^{17} - 4 q^{18} + 12 q^{20} + 8 q^{23} + 4 q^{24} - 8 q^{25} - 2 q^{26} - 6 q^{30} - 16 q^{31}+ \cdots - 4 q^{98}+O(q^{100}) $ 2 * q - 2 * q^2 + 2 * q^6 + 6 * q^7 + 4 * q^8 + 4 * q^9 - 6 * q^10 - 4 * q^12 - 6 * q^14 + 6 * q^15 - 8 * q^16 - 14 * q^17 - 4 * q^18 + 12 * q^20 + 8 * q^23 + 4 * q^24 - 8 * q^25 - 2 * q^26 - 6 * q^30 - 16 * q^31 + 8 * q^32 + 14 * q^34 + 8 * q^38 + 2 * q^39 - 12 * q^40 + 4 * q^41 + 6 * q^42 - 8 * q^46 - 14 * q^47 + 4 * q^49 + 8 * q^50 + 4 * q^52 + 10 * q^54 + 12 * q^56 - 8 * q^57 + 8 * q^58 + 16 * q^62 + 12 * q^63 - 6 * q^65 - 18 * q^70 - 6 * q^71 + 8 * q^72 + 28 * q^73 - 14 * q^74 - 16 * q^76 - 2 * q^78 - 20 * q^79 + 2 * q^81 - 4 * q^82 - 12 * q^84 + 2 * q^86 - 8 * q^87 - 12 * q^90 + 14 * q^94 + 24 * q^95 - 8 * q^96 + 16 * q^97 - 4 * q^98

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/104\mathbb{Z}\right)^\times$.

$n$	$41$	$53$	$79$
$\chi(n)$	$1$	$-1$	$1$

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 $

$2$	−1.00000	−	1.00000i	−0.707107	−	0.707107i
$2$	−1.00000	−	1.00000i	−0.707107	−	0.707107i
$3$			1.00000i			0.577350i	0.957427	+	0.288675i	$0.0932147\pi$
$3$			1.00000i			0.577350i	−0.957427	+	0.288675i	$0.906785\pi$
$4$			2.00000i			1.00000i
$5$		−	3.00000i		−	1.34164i	−0.741620	−	0.670820i	$-0.765942\pi$
$5$		−	3.00000i		−	1.34164i	0.741620	−	0.670820i	$-0.234058\pi$
$6$	1.00000	−	1.00000i	0.408248	−	0.408248i
$7$	3.00000			1.13389			0.566947	−	0.823754i	$-0.308125\pi$
$7$	3.00000			1.13389			0.566947	+	0.823754i	$0.308125\pi$
$8$	2.00000	−	2.00000i	0.707107	−	0.707107i
$9$	2.00000			0.666667
$10$	−3.00000	+	3.00000i	−0.948683	+	0.948683i
$11$		0			0		1.00000			$0$
$11$		0			0		−1.00000			$\pi$
$12$	−2.00000			−0.577350
$13$		−	1.00000i		−	0.277350i
$13$		−	1.00000i		−	0.277350i
$14$	−3.00000	−	3.00000i	−0.801784	−	0.801784i
$15$	3.00000			0.774597
$16$	−4.00000			−1.00000
$17$	−7.00000			−1.69775			−0.848875	−	0.528594i	$-0.822719\pi$
$17$	−7.00000			−1.69775			−0.848875	+	0.528594i	$0.822719\pi$
$18$	−2.00000	−	2.00000i	−0.471405	−	0.471405i
$19$			4.00000i			0.917663i	0.888523	+	0.458831i	$0.151732\pi$
$19$			4.00000i			0.917663i	−0.888523	+	0.458831i	$0.848268\pi$
$20$	6.00000			1.34164
$21$			3.00000i			0.654654i
$22$		0			0
$23$	4.00000			0.834058			0.417029	−	0.908893i	$-0.363071\pi$
$23$	4.00000			0.834058			0.417029	+	0.908893i	$0.363071\pi$
$24$	2.00000	+	2.00000i	0.408248	+	0.408248i
$25$	−4.00000			−0.800000
$26$	−1.00000	+	1.00000i	−0.196116	+	0.196116i
$27$			5.00000i			0.962250i
$28$			6.00000i			1.13389i
$29$			4.00000i			0.742781i	0.928477	+	0.371391i	$0.121119\pi$
$29$			4.00000i			0.742781i	−0.928477	+	0.371391i	$0.878881\pi$
$30$	−3.00000	−	3.00000i	−0.547723	−	0.547723i
$31$	−8.00000			−1.43684			−0.718421	−	0.695608i	$-0.755135\pi$
$31$	−8.00000			−1.43684			−0.718421	+	0.695608i	$0.755135\pi$
$32$	4.00000	+	4.00000i	0.707107	+	0.707107i
$33$		0			0
$34$	7.00000	+	7.00000i	1.20049	+	1.20049i
$35$		−	9.00000i		−	1.52128i
$36$			4.00000i			0.666667i
$37$		−	7.00000i		−	1.15079i	−0.817875	−	0.575396i	$-0.804848\pi$
$37$		−	7.00000i		−	1.15079i	0.817875	−	0.575396i	$-0.195152\pi$
$38$	4.00000	−	4.00000i	0.648886	−	0.648886i
$39$	1.00000			0.160128
$40$	−6.00000	−	6.00000i	−0.948683	−	0.948683i
$41$	2.00000			0.312348			0.156174	−	0.987730i	$-0.450084\pi$
$41$	2.00000			0.312348			0.156174	+	0.987730i	$0.450084\pi$
$42$	3.00000	−	3.00000i	0.462910	−	0.462910i
$43$			1.00000i			0.152499i	0.997089	+	0.0762493i	$0.0242945\pi$
$43$			1.00000i			0.152499i	−0.997089	+	0.0762493i	$0.975706\pi$
$44$		0			0
$45$		−	6.00000i		−	0.894427i
$46$	−4.00000	−	4.00000i	−0.589768	−	0.589768i
$47$	−7.00000			−1.02105			−0.510527	−	0.859861i	$-0.670550\pi$
$47$	−7.00000			−1.02105			−0.510527	+	0.859861i	$0.670550\pi$
$48$		−	4.00000i		−	0.577350i
$49$	2.00000			0.285714
$50$	4.00000	+	4.00000i	0.565685	+	0.565685i
$51$		−	7.00000i		−	0.980196i
$52$	2.00000			0.277350
$53$		−	4.00000i		−	0.549442i	−0.961524	−	0.274721i	$-0.911414\pi$
$53$		−	4.00000i		−	0.549442i	0.961524	−	0.274721i	$-0.0885855\pi$
$54$	5.00000	−	5.00000i	0.680414	−	0.680414i
$55$		0			0
$56$	6.00000	−	6.00000i	0.801784	−	0.801784i
$57$	−4.00000			−0.529813
$58$	4.00000	−	4.00000i	0.525226	−	0.525226i
$59$			14.0000i			1.82264i	0.411693	+	0.911322i	$0.364937\pi$
$59$			14.0000i			1.82264i	−0.411693	+	0.911322i	$0.635063\pi$
$60$			6.00000i			0.774597i
$61$			10.0000i			1.28037i	0.768221	+	0.640184i	$0.221142\pi$
$61$			10.0000i			1.28037i	−0.768221	+	0.640184i	$0.778858\pi$
$62$	8.00000	+	8.00000i	1.01600	+	1.01600i
$63$	6.00000			0.755929
$64$		−	8.00000i		−	1.00000i
$65$	−3.00000			−0.372104
$66$		0			0
$67$		−	2.00000i		−	0.244339i	−0.992509	−	0.122169i	$-0.961015\pi$
$67$		−	2.00000i		−	0.244339i	0.992509	−	0.122169i	$-0.0389851\pi$
$68$		−	14.0000i		−	1.69775i
$69$			4.00000i			0.481543i
$70$	−9.00000	+	9.00000i	−1.07571	+	1.07571i
$71$	−3.00000			−0.356034			−0.178017	−	0.984027i	$-0.556968\pi$
$71$	−3.00000			−0.356034			−0.178017	+	0.984027i	$0.556968\pi$
$72$	4.00000	−	4.00000i	0.471405	−	0.471405i
$73$	14.0000			1.63858			0.819288	−	0.573382i	$-0.194369\pi$
$73$	14.0000			1.63858			0.819288	+	0.573382i	$0.194369\pi$
$74$	−7.00000	+	7.00000i	−0.813733	+	0.813733i
$75$		−	4.00000i		−	0.461880i
$76$	−8.00000			−0.917663
$77$		0			0
$78$	−1.00000	−	1.00000i	−0.113228	−	0.113228i
$79$	−10.0000			−1.12509			−0.562544	−	0.826767i	$-0.690177\pi$
$79$	−10.0000			−1.12509			−0.562544	+	0.826767i	$0.690177\pi$
$80$			12.0000i			1.34164i
$81$	1.00000			0.111111
$82$	−2.00000	−	2.00000i	−0.220863	−	0.220863i
$83$		−	14.0000i		−	1.53670i	−0.640030	−	0.768350i	$-0.721078\pi$
$83$		−	14.0000i		−	1.53670i	0.640030	−	0.768350i	$-0.278922\pi$
$84$	−6.00000			−0.654654
$85$			21.0000i			2.27777i
$86$	1.00000	−	1.00000i	0.107833	−	0.107833i
$87$	−4.00000			−0.428845
$88$		0			0
$89$		0			0			−	1.00000i	$-0.5\pi$
$89$		0			0				1.00000i	$0.5\pi$
$90$	−6.00000	+	6.00000i	−0.632456	+	0.632456i
$91$		−	3.00000i		−	0.314485i
$92$			8.00000i			0.834058i
$93$		−	8.00000i		−	0.829561i
$94$	7.00000	+	7.00000i	0.721995	+	0.721995i
$95$	12.0000			1.23117
$96$	−4.00000	+	4.00000i	−0.408248	+	0.408248i
$97$	8.00000			0.812277			0.406138	−	0.913812i	$-0.366875\pi$
$97$	8.00000			0.812277			0.406138	+	0.913812i	$0.366875\pi$
$98$	−2.00000	−	2.00000i	−0.202031	−	0.202031i
$99$		0			0
$100$		−	8.00000i		−	0.800000i
$101$		0			0		1.00000			$0$
$101$		0			0		−1.00000			$\pi$
$102$	−7.00000	+	7.00000i	−0.693103	+	0.693103i
$103$	−6.00000			−0.591198			−0.295599	−	0.955312i	$-0.595519\pi$
$103$	−6.00000			−0.591198			−0.295599	+	0.955312i	$0.595519\pi$
$104$	−2.00000	−	2.00000i	−0.196116	−	0.196116i
$105$	9.00000			0.878310
$106$	−4.00000	+	4.00000i	−0.388514	+	0.388514i
$107$			8.00000i			0.773389i	0.922208	+	0.386695i	$0.126383\pi$
$107$			8.00000i			0.773389i	−0.922208	+	0.386695i	$0.873617\pi$
$108$	−10.0000			−0.962250
$109$		−	1.00000i		−	0.0957826i	−0.998853	−	0.0478913i	$-0.984750\pi$
$109$		−	1.00000i		−	0.0957826i	0.998853	−	0.0478913i	$-0.0152501\pi$
$110$		0			0
$111$	7.00000			0.664411
$112$	−12.0000			−1.13389
$113$	−6.00000			−0.564433			−0.282216	−	0.959351i	$-0.591070\pi$
$113$	−6.00000			−0.564433			−0.282216	+	0.959351i	$0.591070\pi$
$114$	4.00000	+	4.00000i	0.374634	+	0.374634i
$115$		−	12.0000i		−	1.11901i
$116$	−8.00000			−0.742781
$117$		−	2.00000i		−	0.184900i
$118$	14.0000	−	14.0000i	1.28880	−	1.28880i
$119$	−21.0000			−1.92507
$120$	6.00000	−	6.00000i	0.547723	−	0.547723i
$121$	11.0000			1.00000
$122$	10.0000	−	10.0000i	0.905357	−	0.905357i
$123$			2.00000i			0.180334i
$124$		−	16.0000i		−	1.43684i
$125$		−	3.00000i		−	0.268328i
$126$	−6.00000	−	6.00000i	−0.534522	−	0.534522i
$127$	8.00000			0.709885			0.354943	−	0.934888i	$-0.384500\pi$
$127$	8.00000			0.709885			0.354943	+	0.934888i	$0.384500\pi$
$128$	−8.00000	+	8.00000i	−0.707107	+	0.707107i
$129$	−1.00000			−0.0880451
$130$	3.00000	+	3.00000i	0.263117	+	0.263117i
$131$			15.0000i			1.31056i	0.755388	+	0.655278i	$0.227449\pi$
$131$			15.0000i			1.31056i	−0.755388	+	0.655278i	$0.772551\pi$
$132$		0			0
$133$			12.0000i			1.04053i
$134$	−2.00000	+	2.00000i	−0.172774	+	0.172774i
$135$	15.0000			1.29099
$136$	−14.0000	+	14.0000i	−1.20049	+	1.20049i
$137$	−12.0000			−1.02523			−0.512615	−	0.858619i	$-0.671323\pi$
$137$	−12.0000			−1.02523			−0.512615	+	0.858619i	$0.671323\pi$
$138$	4.00000	−	4.00000i	0.340503	−	0.340503i
$139$		−	11.0000i		−	0.933008i	−0.884519	−	0.466504i	$-0.845513\pi$
$139$		−	11.0000i		−	0.933008i	0.884519	−	0.466504i	$-0.154487\pi$
$140$	18.0000			1.52128
$141$		−	7.00000i		−	0.589506i
$142$	3.00000	+	3.00000i	0.251754	+	0.251754i
$143$		0			0
$144$	−8.00000			−0.666667
$145$	12.0000			0.996546
$146$	−14.0000	−	14.0000i	−1.15865	−	1.15865i
$147$			2.00000i			0.164957i
$148$	14.0000			1.15079
$149$		−	6.00000i		−	0.491539i	−0.969328	−	0.245770i	$-0.920959\pi$
$149$		−	6.00000i		−	0.491539i	0.969328	−	0.245770i	$-0.0790407\pi$
$150$	−4.00000	+	4.00000i	−0.326599	+	0.326599i
$151$	17.0000			1.38344			0.691720	−	0.722166i	$-0.256853\pi$
$151$	17.0000			1.38344			0.691720	+	0.722166i	$0.256853\pi$
$152$	8.00000	+	8.00000i	0.648886	+	0.648886i
$153$	−14.0000			−1.13183
$154$		0			0
$155$			24.0000i			1.92773i
$156$			2.00000i			0.160128i
$157$		−	2.00000i		−	0.159617i	−0.996810	−	0.0798087i	$-0.974569\pi$
$157$		−	2.00000i		−	0.159617i	0.996810	−	0.0798087i	$-0.0254309\pi$
$158$	10.0000	+	10.0000i	0.795557	+	0.795557i
$159$	4.00000			0.317221
$160$	12.0000	−	12.0000i	0.948683	−	0.948683i
$161$	12.0000			0.945732
$162$	−1.00000	−	1.00000i	−0.0785674	−	0.0785674i
$163$		−	24.0000i		−	1.87983i	−0.341415	−	0.939913i	$-0.610906\pi$
$163$		−	24.0000i		−	1.87983i	0.341415	−	0.939913i	$-0.389094\pi$
$164$			4.00000i			0.312348i
$165$		0			0
$166$	−14.0000	+	14.0000i	−1.08661	+	1.08661i
$167$	−12.0000			−0.928588			−0.464294	−	0.885681i	$-0.653692\pi$
$167$	−12.0000			−0.928588			−0.464294	+	0.885681i	$0.653692\pi$
$168$	6.00000	+	6.00000i	0.462910	+	0.462910i
$169$	−1.00000			−0.0769231
$170$	21.0000	−	21.0000i	1.61063	−	1.61063i
$171$			8.00000i			0.611775i
$172$	−2.00000			−0.152499
$173$			6.00000i			0.456172i	0.973641	+	0.228086i	$0.0732467\pi$
$173$			6.00000i			0.456172i	−0.973641	+	0.228086i	$0.926753\pi$
$174$	4.00000	+	4.00000i	0.303239	+	0.303239i
$175$	−12.0000			−0.907115
$176$		0			0
$177$	−14.0000			−1.05230
$178$		0			0
$179$		−	21.0000i		−	1.56961i	−0.619740	−	0.784807i	$-0.712762\pi$
$179$		−	21.0000i		−	1.56961i	0.619740	−	0.784807i	$-0.287238\pi$
$180$	12.0000			0.894427
$181$		−	10.0000i		−	0.743294i	−0.928374	−	0.371647i	$-0.878793\pi$
$181$		−	10.0000i		−	0.743294i	0.928374	−	0.371647i	$-0.121207\pi$
$182$	−3.00000	+	3.00000i	−0.222375	+	0.222375i
$183$	−10.0000			−0.739221
$184$	8.00000	−	8.00000i	0.589768	−	0.589768i
$185$	−21.0000			−1.54395
$186$	−8.00000	+	8.00000i	−0.586588	+	0.586588i
$187$		0			0
$188$		−	14.0000i		−	1.02105i
$189$			15.0000i			1.09109i
$190$	−12.0000	−	12.0000i	−0.870572	−	0.870572i
$191$	12.0000			0.868290			0.434145	−	0.900843i	$-0.357051\pi$
$191$	12.0000			0.868290			0.434145	+	0.900843i	$0.357051\pi$
$192$	8.00000			0.577350
$193$	−6.00000			−0.431889			−0.215945	−	0.976406i	$-0.569283\pi$
$193$	−6.00000			−0.431889			−0.215945	+	0.976406i	$0.569283\pi$
$194$	−8.00000	−	8.00000i	−0.574367	−	0.574367i
$195$		−	3.00000i		−	0.214834i
$196$			4.00000i			0.285714i
$197$		−	17.0000i		−	1.21120i	−0.795769	−	0.605600i	$-0.792933\pi$
$197$		−	17.0000i		−	1.21120i	0.795769	−	0.605600i	$-0.207067\pi$
$198$		0			0
$199$	−10.0000			−0.708881			−0.354441	−	0.935079i	$-0.615329\pi$
$199$	−10.0000			−0.708881			−0.354441	+	0.935079i	$0.615329\pi$
$200$	−8.00000	+	8.00000i	−0.565685	+	0.565685i
$201$	2.00000			0.141069
$202$		0			0
$203$			12.0000i			0.842235i
$204$	14.0000			0.980196
$205$		−	6.00000i		−	0.419058i
$206$	6.00000	+	6.00000i	0.418040	+	0.418040i
$207$	8.00000			0.556038
$208$			4.00000i			0.277350i
$209$		0			0
$210$	−9.00000	−	9.00000i	−0.621059	−	0.621059i
$211$			15.0000i			1.03264i	0.856395	+	0.516321i	$0.172699\pi$
$211$			15.0000i			1.03264i	−0.856395	+	0.516321i	$0.827301\pi$
$212$	8.00000			0.549442
$213$		−	3.00000i		−	0.205557i
$214$	8.00000	−	8.00000i	0.546869	−	0.546869i
$215$	3.00000			0.204598
$216$	10.0000	+	10.0000i	0.680414	+	0.680414i
$217$	−24.0000			−1.62923
$218$	−1.00000	+	1.00000i	−0.0677285	+	0.0677285i
$219$			14.0000i			0.946032i
$220$		0			0
$221$			7.00000i			0.470871i
$222$	−7.00000	−	7.00000i	−0.469809	−	0.469809i
$223$	−1.00000			−0.0669650			−0.0334825	−	0.999439i	$-0.510660\pi$
$223$	−1.00000			−0.0669650			−0.0334825	+	0.999439i	$0.510660\pi$
$224$	12.0000	+	12.0000i	0.801784	+	0.801784i
$225$	−8.00000			−0.533333
$226$	6.00000	+	6.00000i	0.399114	+	0.399114i
$227$			18.0000i			1.19470i	0.801980	+	0.597351i	$0.203780\pi$
$227$			18.0000i			1.19470i	−0.801980	+	0.597351i	$0.796220\pi$
$228$		−	8.00000i		−	0.529813i
$229$		−	21.0000i		−	1.38772i	−0.720110	−	0.693860i	$-0.755909\pi$
$229$		−	21.0000i		−	1.38772i	0.720110	−	0.693860i	$-0.244091\pi$
$230$	−12.0000	+	12.0000i	−0.791257	+	0.791257i
$231$		0			0
$232$	8.00000	+	8.00000i	0.525226	+	0.525226i
$233$	19.0000			1.24473			0.622366	−	0.782727i	$-0.286172\pi$
$233$	19.0000			1.24473			0.622366	+	0.782727i	$0.286172\pi$
$234$	−2.00000	+	2.00000i	−0.130744	+	0.130744i
$235$			21.0000i			1.36989i
$236$	−28.0000			−1.82264
$237$		−	10.0000i		−	0.649570i
$238$	21.0000	+	21.0000i	1.36123	+	1.36123i
$239$	5.00000			0.323423			0.161712	−	0.986838i	$-0.448299\pi$
$239$	5.00000			0.323423			0.161712	+	0.986838i	$0.448299\pi$
$240$	−12.0000			−0.774597
$241$	−8.00000			−0.515325			−0.257663	−	0.966235i	$-0.582952\pi$
$241$	−8.00000			−0.515325			−0.257663	+	0.966235i	$0.582952\pi$
$242$	−11.0000	−	11.0000i	−0.707107	−	0.707107i
$243$			16.0000i			1.02640i
$244$	−20.0000			−1.28037
$245$		−	6.00000i		−	0.383326i
$246$	2.00000	−	2.00000i	0.127515	−	0.127515i
$247$	4.00000			0.254514
$248$	−16.0000	+	16.0000i	−1.01600	+	1.01600i
$249$	14.0000			0.887214
$250$	−3.00000	+	3.00000i	−0.189737	+	0.189737i
$251$		−	20.0000i		−	1.26239i	−0.775625	−	0.631194i	$-0.782565\pi$
$251$		−	20.0000i		−	1.26239i	0.775625	−	0.631194i	$-0.217435\pi$
$252$			12.0000i			0.755929i
$253$		0			0
$254$	−8.00000	−	8.00000i	−0.501965	−	0.501965i
$255$	−21.0000			−1.31507
$256$	16.0000			1.00000
$257$	−7.00000			−0.436648			−0.218324	−	0.975876i	$-0.570059\pi$
$257$	−7.00000			−0.436648			−0.218324	+	0.975876i	$0.570059\pi$
$258$	1.00000	+	1.00000i	0.0622573	+	0.0622573i
$259$		−	21.0000i		−	1.30488i
$260$		−	6.00000i		−	0.372104i
$261$			8.00000i			0.495188i
$262$	15.0000	−	15.0000i	0.926703	−	0.926703i
$263$	14.0000			0.863277			0.431638	−	0.902047i	$-0.357936\pi$
$263$	14.0000			0.863277			0.431638	+	0.902047i	$0.357936\pi$
$264$		0			0
$265$	−12.0000			−0.737154
$266$	12.0000	−	12.0000i	0.735767	−	0.735767i
$267$		0			0
$268$	4.00000			0.244339
$269$		−	6.00000i		−	0.365826i	−0.983129	−	0.182913i	$-0.941447\pi$
$269$		−	6.00000i		−	0.365826i	0.983129	−	0.182913i	$-0.0585527\pi$
$270$	−15.0000	−	15.0000i	−0.912871	−	0.912871i
$271$	−3.00000			−0.182237			−0.0911185	−	0.995840i	$-0.529044\pi$
$271$	−3.00000			−0.182237			−0.0911185	+	0.995840i	$0.529044\pi$
$272$	28.0000			1.69775
$273$	3.00000			0.181568
$274$	12.0000	+	12.0000i	0.724947	+	0.724947i
$275$		0			0
$276$	−8.00000			−0.481543
$277$			18.0000i			1.08152i	0.841178	+	0.540758i	$0.181862\pi$
$277$			18.0000i			1.08152i	−0.841178	+	0.540758i	$0.818138\pi$
$278$	−11.0000	+	11.0000i	−0.659736	+	0.659736i
$279$	−16.0000			−0.957895
$280$	−18.0000	−	18.0000i	−1.07571	−	1.07571i
$281$	−8.00000			−0.477240			−0.238620	−	0.971113i	$-0.576695\pi$
$281$	−8.00000			−0.477240			−0.238620	+	0.971113i	$0.576695\pi$
$282$	−7.00000	+	7.00000i	−0.416844	+	0.416844i
$283$		−	4.00000i		−	0.237775i	−0.992908	−	0.118888i	$-0.962067\pi$
$283$		−	4.00000i		−	0.237775i	0.992908	−	0.118888i	$-0.0379328\pi$
$284$		−	6.00000i		−	0.356034i
$285$			12.0000i			0.710819i
$286$		0			0
$287$	6.00000			0.354169
$288$	8.00000	+	8.00000i	0.471405	+	0.471405i
$289$	32.0000			1.88235
$290$	−12.0000	−	12.0000i	−0.704664	−	0.704664i
$291$			8.00000i			0.468968i
$292$			28.0000i			1.63858i
$293$			1.00000i			0.0584206i	0.999573	+	0.0292103i	$0.00929925\pi$
$293$			1.00000i			0.0584206i	−0.999573	+	0.0292103i	$0.990701\pi$
$294$	2.00000	−	2.00000i	0.116642	−	0.116642i
$295$	42.0000			2.44533
$296$	−14.0000	−	14.0000i	−0.813733	−	0.813733i
$297$		0			0
$298$	−6.00000	+	6.00000i	−0.347571	+	0.347571i
$299$		−	4.00000i		−	0.231326i
$300$	8.00000			0.461880
$301$			3.00000i			0.172917i
$302$	−17.0000	−	17.0000i	−0.978240	−	0.978240i
$303$		0			0
$304$		−	16.0000i		−	0.917663i
$305$	30.0000			1.71780
$306$	14.0000	+	14.0000i	0.800327	+	0.800327i
$307$		−	2.00000i		−	0.114146i	−0.998370	−	0.0570730i	$-0.981823\pi$
$307$		−	2.00000i		−	0.114146i	0.998370	−	0.0570730i	$-0.0181768\pi$
$308$		0			0
$309$		−	6.00000i		−	0.341328i
$310$	24.0000	−	24.0000i	1.36311	−	1.36311i
$311$	32.0000			1.81455			0.907277	−	0.420534i	$-0.138157\pi$
$311$	32.0000			1.81455			0.907277	+	0.420534i	$0.138157\pi$
$312$	2.00000	−	2.00000i	0.113228	−	0.113228i
$313$	29.0000			1.63918			0.819588	−	0.572953i	$-0.194202\pi$
$313$	29.0000			1.63918			0.819588	+	0.572953i	$0.194202\pi$
$314$	−2.00000	+	2.00000i	−0.112867	+	0.112867i
$315$		−	18.0000i		−	1.01419i
$316$		−	20.0000i		−	1.12509i
$317$			18.0000i			1.01098i	0.862832	+	0.505490i	$0.168688\pi$
$317$			18.0000i			1.01098i	−0.862832	+	0.505490i	$0.831312\pi$
$318$	−4.00000	−	4.00000i	−0.224309	−	0.224309i
$319$		0			0
$320$	−24.0000			−1.34164
$321$	−8.00000			−0.446516
$322$	−12.0000	−	12.0000i	−0.668734	−	0.668734i
$323$		−	28.0000i		−	1.55796i
$324$			2.00000i			0.111111i
$325$			4.00000i			0.221880i
$326$	−24.0000	+	24.0000i	−1.32924	+	1.32924i
$327$	1.00000			0.0553001
$328$	4.00000	−	4.00000i	0.220863	−	0.220863i
$329$	−21.0000			−1.15777
$330$		0			0
$331$		−	10.0000i		−	0.549650i	−0.961494	−	0.274825i	$-0.911380\pi$
$331$		−	10.0000i		−	0.549650i	0.961494	−	0.274825i	$-0.0886199\pi$
$332$	28.0000			1.53670
$333$		−	14.0000i		−	0.767195i
$334$	12.0000	+	12.0000i	0.656611	+	0.656611i
$335$	−6.00000			−0.327815
$336$		−	12.0000i		−	0.654654i
$337$	−17.0000			−0.926049			−0.463025	−	0.886345i	$-0.653236\pi$
$337$	−17.0000			−0.926049			−0.463025	+	0.886345i	$0.653236\pi$
$338$	1.00000	+	1.00000i	0.0543928	+	0.0543928i
$339$		−	6.00000i		−	0.325875i
$340$	−42.0000			−2.27777
$341$		0			0
$342$	8.00000	−	8.00000i	0.432590	−	0.432590i
$343$	−15.0000			−0.809924
$344$	2.00000	+	2.00000i	0.107833	+	0.107833i
$345$	12.0000			0.646058
$346$	6.00000	−	6.00000i	0.322562	−	0.322562i
$347$		−	7.00000i		−	0.375780i	−0.982190	−	0.187890i	$-0.939835\pi$
$347$		−	7.00000i		−	0.375780i	0.982190	−	0.187890i	$-0.0601648\pi$
$348$		−	8.00000i		−	0.428845i
$349$			19.0000i			1.01705i	0.861048	+	0.508523i	$0.169808\pi$
$349$			19.0000i			1.01705i	−0.861048	+	0.508523i	$0.830192\pi$
$350$	12.0000	+	12.0000i	0.641427	+	0.641427i
$351$	5.00000			0.266880
$352$		0			0
$353$	−6.00000			−0.319348			−0.159674	−	0.987170i	$-0.551044\pi$
$353$	−6.00000			−0.319348			−0.159674	+	0.987170i	$0.551044\pi$
$354$	14.0000	+	14.0000i	0.744092	+	0.744092i
$355$			9.00000i			0.477670i
$356$		0			0
$357$		−	21.0000i		−	1.11144i
$358$	−21.0000	+	21.0000i	−1.10988	+	1.10988i
$359$		0			0			−	1.00000i	$-0.5\pi$
$359$		0			0				1.00000i	$0.5\pi$
$360$	−12.0000	−	12.0000i	−0.632456	−	0.632456i
$361$	3.00000			0.157895
$362$	−10.0000	+	10.0000i	−0.525588	+	0.525588i
$363$			11.0000i			0.577350i
$364$	6.00000			0.314485
$365$		−	42.0000i		−	2.19838i
$366$	10.0000	+	10.0000i	0.522708	+	0.522708i
$367$	−22.0000			−1.14839			−0.574195	−	0.818718i	$-0.694685\pi$
$367$	−22.0000			−1.14839			−0.574195	+	0.818718i	$0.694685\pi$
$368$	−16.0000			−0.834058
$369$	4.00000			0.208232
$370$	21.0000	+	21.0000i	1.09174	+	1.09174i
$371$		−	12.0000i		−	0.623009i
$372$	16.0000			0.829561
$373$			36.0000i			1.86401i	0.362446	+	0.932005i	$0.381942\pi$
$373$			36.0000i			1.86401i	−0.362446	+	0.932005i	$0.618058\pi$
$374$		0			0
$375$	3.00000			0.154919
$376$	−14.0000	+	14.0000i	−0.721995	+	0.721995i
$377$	4.00000			0.206010
$378$	15.0000	−	15.0000i	0.771517	−	0.771517i
$379$			4.00000i			0.205466i	0.994709	+	0.102733i	$0.0327588\pi$
$379$			4.00000i			0.205466i	−0.994709	+	0.102733i	$0.967241\pi$
$380$			24.0000i			1.23117i
$381$			8.00000i			0.409852i
$382$	−12.0000	−	12.0000i	−0.613973	−	0.613973i
$383$	−1.00000			−0.0510976			−0.0255488	−	0.999674i	$-0.508133\pi$
$383$	−1.00000			−0.0510976			−0.0255488	+	0.999674i	$0.508133\pi$
$384$	−8.00000	−	8.00000i	−0.408248	−	0.408248i
$385$		0			0
$386$	6.00000	+	6.00000i	0.305392	+	0.305392i
$387$			2.00000i			0.101666i
$388$			16.0000i			0.812277i
$389$			34.0000i			1.72387i	0.507020	+	0.861934i	$0.330747\pi$
$389$			34.0000i			1.72387i	−0.507020	+	0.861934i	$0.669253\pi$
$390$	−3.00000	+	3.00000i	−0.151911	+	0.151911i
$391$	−28.0000			−1.41602
$392$	4.00000	−	4.00000i	0.202031	−	0.202031i
$393$	−15.0000			−0.756650
$394$	−17.0000	+	17.0000i	−0.856448	+	0.856448i
$395$			30.0000i			1.50946i
$396$		0			0
$397$		−	22.0000i		−	1.10415i	−0.833795	−	0.552074i	$-0.813837\pi$
$397$		−	22.0000i		−	1.10415i	0.833795	−	0.552074i	$-0.186163\pi$
$398$	10.0000	+	10.0000i	0.501255	+	0.501255i
$399$	−12.0000			−0.600751
$400$	16.0000			0.800000
$401$	32.0000			1.59800			0.799002	−	0.601329i	$-0.205362\pi$
$401$	32.0000			1.59800			0.799002	+	0.601329i	$0.205362\pi$
$402$	−2.00000	−	2.00000i	−0.0997509	−	0.0997509i
$403$			8.00000i			0.398508i
$404$		0			0
$405$		−	3.00000i		−	0.149071i
$406$	12.0000	−	12.0000i	0.595550	−	0.595550i
$407$		0			0
$408$	−14.0000	−	14.0000i	−0.693103	−	0.693103i
$409$		0			0			−	1.00000i	$-0.5\pi$
$409$		0			0				1.00000i	$0.5\pi$
$410$	−6.00000	+	6.00000i	−0.296319	+	0.296319i
$411$		−	12.0000i		−	0.591916i
$412$		−	12.0000i		−	0.591198i
$413$			42.0000i			2.06668i
$414$	−8.00000	−	8.00000i	−0.393179	−	0.393179i
$415$	−42.0000			−2.06170
$416$	4.00000	−	4.00000i	0.196116	−	0.196116i
$417$	11.0000			0.538672
$418$		0			0
$419$		−	11.0000i		−	0.537385i	−0.963226	−	0.268693i	$-0.913408\pi$
$419$		−	11.0000i		−	0.537385i	0.963226	−	0.268693i	$-0.0865916\pi$
$420$			18.0000i			0.878310i
$421$		−	15.0000i		−	0.731055i	−0.930800	−	0.365528i	$-0.880889\pi$
$421$		−	15.0000i		−	0.731055i	0.930800	−	0.365528i	$-0.119111\pi$
$422$	15.0000	−	15.0000i	0.730189	−	0.730189i
$423$	−14.0000			−0.680703
$424$	−8.00000	−	8.00000i	−0.388514	−	0.388514i
$425$	28.0000			1.35820
$426$	−3.00000	+	3.00000i	−0.145350	+	0.145350i
$427$			30.0000i			1.45180i
$428$	−16.0000			−0.773389
$429$		0			0
$430$	−3.00000	−	3.00000i	−0.144673	−	0.144673i
$431$	−3.00000			−0.144505			−0.0722525	−	0.997386i	$-0.523019\pi$
$431$	−3.00000			−0.144505			−0.0722525	+	0.997386i	$0.523019\pi$
$432$		−	20.0000i		−	0.962250i
$433$	−1.00000			−0.0480569			−0.0240285	−	0.999711i	$-0.507649\pi$
$433$	−1.00000			−0.0480569			−0.0240285	+	0.999711i	$0.507649\pi$
$434$	24.0000	+	24.0000i	1.15204	+	1.15204i
$435$			12.0000i			0.575356i
$436$	2.00000			0.0957826
$437$			16.0000i			0.765384i
$438$	14.0000	−	14.0000i	0.668946	−	0.668946i
$439$	−10.0000			−0.477274			−0.238637	−	0.971109i	$-0.576701\pi$
$439$	−10.0000			−0.477274			−0.238637	+	0.971109i	$0.576701\pi$
$440$		0			0
$441$	4.00000			0.190476
$442$	7.00000	−	7.00000i	0.332956	−	0.332956i
$443$		−	9.00000i		−	0.427603i	−0.976877	−	0.213801i	$-0.931415\pi$
$443$		−	9.00000i		−	0.427603i	0.976877	−	0.213801i	$-0.0685846\pi$
$444$			14.0000i			0.664411i
$445$		0			0
$446$	1.00000	+	1.00000i	0.0473514	+	0.0473514i
$447$	6.00000			0.283790
$448$		−	24.0000i		−	1.13389i
$449$	−20.0000			−0.943858			−0.471929	−	0.881636i	$-0.656442\pi$
$449$	−20.0000			−0.943858			−0.471929	+	0.881636i	$0.656442\pi$
$450$	8.00000	+	8.00000i	0.377124	+	0.377124i
$451$		0			0
$452$		−	12.0000i		−	0.564433i
$453$			17.0000i			0.798730i
$454$	18.0000	−	18.0000i	0.844782	−	0.844782i
$455$	−9.00000			−0.421927
$456$	−8.00000	+	8.00000i	−0.374634	+	0.374634i
$457$	−32.0000			−1.49690			−0.748448	−	0.663193i	$-0.769201\pi$
$457$	−32.0000			−1.49690			−0.748448	+	0.663193i	$0.769201\pi$
$458$	−21.0000	+	21.0000i	−0.981266	+	0.981266i
$459$		−	35.0000i		−	1.63366i
$460$	24.0000			1.11901
$461$		−	5.00000i		−	0.232873i	−0.993198	−	0.116437i	$-0.962853\pi$
$461$		−	5.00000i		−	0.232873i	0.993198	−	0.116437i	$-0.0371472\pi$
$462$		0			0
$463$	−16.0000			−0.743583			−0.371792	−	0.928316i	$-0.621256\pi$
$463$	−16.0000			−0.743583			−0.371792	+	0.928316i	$0.621256\pi$
$464$		−	16.0000i		−	0.742781i
$465$	−24.0000			−1.11297
$466$	−19.0000	−	19.0000i	−0.880158	−	0.880158i
$467$		−	12.0000i		−	0.555294i	−0.960683	−	0.277647i	$-0.910445\pi$
$467$		−	12.0000i		−	0.555294i	0.960683	−	0.277647i	$-0.0895545\pi$
$468$	4.00000			0.184900
$469$		−	6.00000i		−	0.277054i
$470$	21.0000	−	21.0000i	0.968658	−	0.968658i
$471$	2.00000			0.0921551
$472$	28.0000	+	28.0000i	1.28880	+	1.28880i
$473$		0			0
$474$	−10.0000	+	10.0000i	−0.459315	+	0.459315i
$475$		−	16.0000i		−	0.734130i
$476$		−	42.0000i		−	1.92507i
$477$		−	8.00000i		−	0.366295i
$478$	−5.00000	−	5.00000i	−0.228695	−	0.228695i
$479$	5.00000			0.228456			0.114228	−	0.993455i	$-0.463561\pi$
$479$	5.00000			0.228456			0.114228	+	0.993455i	$0.463561\pi$
$480$	12.0000	+	12.0000i	0.547723	+	0.547723i
$481$	−7.00000			−0.319173
$482$	8.00000	+	8.00000i	0.364390	+	0.364390i
$483$			12.0000i			0.546019i
$484$			22.0000i			1.00000i
$485$		−	24.0000i		−	1.08978i
$486$	16.0000	−	16.0000i	0.725775	−	0.725775i
$487$	8.00000			0.362515			0.181257	−	0.983436i	$-0.441983\pi$
$487$	8.00000			0.362515			0.181257	+	0.983436i	$0.441983\pi$
$488$	20.0000	+	20.0000i	0.905357	+	0.905357i
$489$	24.0000			1.08532
$490$	−6.00000	+	6.00000i	−0.271052	+	0.271052i
$491$		−	15.0000i		−	0.676941i	−0.940977	−	0.338470i	$-0.890091\pi$
$491$		−	15.0000i		−	0.676941i	0.940977	−	0.338470i	$-0.109909\pi$
$492$	−4.00000			−0.180334
$493$		−	28.0000i		−	1.26106i
$494$	−4.00000	−	4.00000i	−0.179969	−	0.179969i
$495$		0			0
$496$	32.0000			1.43684
$497$	−9.00000			−0.403705
$498$	−14.0000	−	14.0000i	−0.627355	−	0.627355i
$499$		−	6.00000i		−	0.268597i	−0.990941	−	0.134298i	$-0.957122\pi$
$499$		−	6.00000i		−	0.268597i	0.990941	−	0.134298i	$-0.0428781\pi$
$500$	6.00000			0.268328
$501$		−	12.0000i		−	0.536120i
$502$	−20.0000	+	20.0000i	−0.892644	+	0.892644i
$503$	4.00000			0.178351			0.0891756	−	0.996016i	$-0.471577\pi$
$503$	4.00000			0.178351			0.0891756	+	0.996016i	$0.471577\pi$
$504$	12.0000	−	12.0000i	0.534522	−	0.534522i
$505$		0			0
$506$		0			0
$507$		−	1.00000i		−	0.0444116i
$508$			16.0000i			0.709885i
$509$			14.0000i			0.620539i	0.950649	+	0.310270i	$0.100419\pi$
$509$			14.0000i			0.620539i	−0.950649	+	0.310270i	$0.899581\pi$
$510$	21.0000	+	21.0000i	0.929896	+	0.929896i
$511$	42.0000			1.85797
$512$	−16.0000	−	16.0000i	−0.707107	−	0.707107i
$513$	−20.0000			−0.883022
$514$	7.00000	+	7.00000i	0.308757	+	0.308757i
$515$			18.0000i			0.793175i
$516$		−	2.00000i		−	0.0880451i
$517$		0			0
$518$	−21.0000	+	21.0000i	−0.922687	+	0.922687i
$519$	−6.00000			−0.263371
$520$	−6.00000	+	6.00000i	−0.263117	+	0.263117i
$521$	−3.00000			−0.131432			−0.0657162	−	0.997838i	$-0.520933\pi$
$521$	−3.00000			−0.131432			−0.0657162	+	0.997838i	$0.520933\pi$
$522$	8.00000	−	8.00000i	0.350150	−	0.350150i
$523$		−	44.0000i		−	1.92399i	−0.273075	−	0.961993i	$-0.588041\pi$
$523$		−	44.0000i		−	1.92399i	0.273075	−	0.961993i	$-0.411959\pi$
$524$	−30.0000			−1.31056
$525$		−	12.0000i		−	0.523723i
$526$	−14.0000	−	14.0000i	−0.610429	−	0.610429i
$527$	56.0000			2.43940
$528$		0			0
$529$	−7.00000			−0.304348
$530$	12.0000	+	12.0000i	0.521247	+	0.521247i
$531$			28.0000i			1.21510i
$532$	−24.0000			−1.04053
$533$		−	2.00000i		−	0.0866296i
$534$		0			0
$535$	24.0000			1.03761
$536$	−4.00000	−	4.00000i	−0.172774	−	0.172774i
$537$	21.0000			0.906217
$538$	−6.00000	+	6.00000i	−0.258678	+	0.258678i
$539$		0			0
$540$			30.0000i			1.29099i
$541$			25.0000i			1.07483i	0.843317	+	0.537417i	$0.180600\pi$
$541$			25.0000i			1.07483i	−0.843317	+	0.537417i	$0.819400\pi$
$542$	3.00000	+	3.00000i	0.128861	+	0.128861i
$543$	10.0000			0.429141
$544$	−28.0000	−	28.0000i	−1.20049	−	1.20049i
$545$	−3.00000			−0.128506
$546$	−3.00000	−	3.00000i	−0.128388	−	0.128388i
$547$			13.0000i			0.555840i	0.960604	+	0.277920i	$0.0896450\pi$
$547$			13.0000i			0.555840i	−0.960604	+	0.277920i	$0.910355\pi$
$548$		−	24.0000i		−	1.02523i
$549$			20.0000i			0.853579i
$550$		0			0
$551$	−16.0000			−0.681623
$552$	8.00000	+	8.00000i	0.340503	+	0.340503i
$553$	−30.0000			−1.27573
$554$	18.0000	−	18.0000i	0.764747	−	0.764747i
$555$		−	21.0000i		−	0.891400i
$556$	22.0000			0.933008
$557$			13.0000i			0.550828i	0.961326	+	0.275414i	$0.0888149\pi$
$557$			13.0000i			0.550828i	−0.961326	+	0.275414i	$0.911185\pi$
$558$	16.0000	+	16.0000i	0.677334	+	0.677334i
$559$	1.00000			0.0422955
$560$			36.0000i			1.52128i
$561$		0			0
$562$	8.00000	+	8.00000i	0.337460	+	0.337460i
$563$			31.0000i			1.30649i	0.757145	+	0.653247i	$0.226594\pi$
$563$			31.0000i			1.30649i	−0.757145	+	0.653247i	$0.773406\pi$
$564$	14.0000			0.589506
$565$			18.0000i			0.757266i
$566$	−4.00000	+	4.00000i	−0.168133	+	0.168133i
$567$	3.00000			0.125988
$568$	−6.00000	+	6.00000i	−0.251754	+	0.251754i
$569$	25.0000			1.04805			0.524027	−	0.851701i	$-0.324429\pi$
$569$	25.0000			1.04805			0.524027	+	0.851701i	$0.324429\pi$
$570$	12.0000	−	12.0000i	0.502625	−	0.502625i
$571$		−	5.00000i		−	0.209243i	−0.994512	−	0.104622i	$-0.966637\pi$
$571$		−	5.00000i		−	0.209243i	0.994512	−	0.104622i	$-0.0333632\pi$
$572$		0			0
$573$			12.0000i			0.501307i
$574$	−6.00000	−	6.00000i	−0.250435	−	0.250435i
$575$	−16.0000			−0.667246
$576$		−	16.0000i		−	0.666667i
$577$	−2.00000			−0.0832611			−0.0416305	−	0.999133i	$-0.513255\pi$
$577$	−2.00000			−0.0832611			−0.0416305	+	0.999133i	$0.513255\pi$
$578$	−32.0000	−	32.0000i	−1.33102	−	1.33102i
$579$		−	6.00000i		−	0.249351i
$580$			24.0000i			0.996546i
$581$		−	42.0000i		−	1.74245i
$582$	8.00000	−	8.00000i	0.331611	−	0.331611i
$583$		0			0
$584$	28.0000	−	28.0000i	1.15865	−	1.15865i
$585$	−6.00000			−0.248069
$586$	1.00000	−	1.00000i	0.0413096	−	0.0413096i
$587$			28.0000i			1.15568i	0.816149	+	0.577842i	$0.196105\pi$
$587$			28.0000i			1.15568i	−0.816149	+	0.577842i	$0.803895\pi$
$588$	−4.00000			−0.164957
$589$		−	32.0000i		−	1.31854i
$590$	−42.0000	−	42.0000i	−1.72911	−	1.72911i
$591$	17.0000			0.699287
$592$			28.0000i			1.15079i
$593$	24.0000			0.985562			0.492781	−	0.870153i	$-0.335980\pi$
$593$	24.0000			0.985562			0.492781	+	0.870153i	$0.335980\pi$
$594$		0			0
$595$			63.0000i			2.58275i
$596$	12.0000			0.491539
$597$		−	10.0000i		−	0.409273i
$598$	−4.00000	+	4.00000i	−0.163572	+	0.163572i
$599$	10.0000			0.408589			0.204294	−	0.978909i	$-0.434510\pi$
$599$	10.0000			0.408589			0.204294	+	0.978909i	$0.434510\pi$
$600$	−8.00000	−	8.00000i	−0.326599	−	0.326599i
$601$	27.0000			1.10135			0.550676	−	0.834719i	$-0.314370\pi$
$601$	27.0000			1.10135			0.550676	+	0.834719i	$0.314370\pi$
$602$	3.00000	−	3.00000i	0.122271	−	0.122271i
$603$		−	4.00000i		−	0.162893i
$604$			34.0000i			1.38344i
$605$		−	33.0000i		−	1.34164i
$606$		0			0
$607$	−22.0000			−0.892952			−0.446476	−	0.894795i	$-0.647321\pi$
$607$	−22.0000			−0.892952			−0.446476	+	0.894795i	$0.647321\pi$
$608$	−16.0000	+	16.0000i	−0.648886	+	0.648886i
$609$	−12.0000			−0.486265
$610$	−30.0000	−	30.0000i	−1.21466	−	1.21466i
$611$			7.00000i			0.283190i
$612$		−	28.0000i		−	1.13183i
$613$		−	14.0000i		−	0.565455i	−0.959200	−	0.282727i	$-0.908761\pi$
$613$		−	14.0000i		−	0.565455i	0.959200	−	0.282727i	$-0.0912392\pi$
$614$	−2.00000	+	2.00000i	−0.0807134	+	0.0807134i
$615$	6.00000			0.241943
$616$		0			0
$617$	−12.0000			−0.483102			−0.241551	−	0.970388i	$-0.577656\pi$
$617$	−12.0000			−0.483102			−0.241551	+	0.970388i	$0.577656\pi$
$618$	−6.00000	+	6.00000i	−0.241355	+	0.241355i
$619$		−	6.00000i		−	0.241160i	−0.992704	−	0.120580i	$-0.961525\pi$
$619$		−	6.00000i		−	0.241160i	0.992704	−	0.120580i	$-0.0384755\pi$
$620$	−48.0000			−1.92773
$621$			20.0000i			0.802572i
$622$	−32.0000	−	32.0000i	−1.28308	−	1.28308i
$623$		0			0
$624$	−4.00000			−0.160128
$625$	−29.0000			−1.16000
$626$	−29.0000	−	29.0000i	−1.15907	−	1.15907i
$627$		0			0
$628$	4.00000			0.159617
$629$			49.0000i			1.95376i
$630$	−18.0000	+	18.0000i	−0.717137	+	0.717137i
$631$	−23.0000			−0.915616			−0.457808	−	0.889051i	$-0.651365\pi$
$631$	−23.0000			−0.915616			−0.457808	+	0.889051i	$0.651365\pi$
$632$	−20.0000	+	20.0000i	−0.795557	+	0.795557i
$633$	−15.0000			−0.596196
$634$	18.0000	−	18.0000i	0.714871	−	0.714871i
$635$		−	24.0000i		−	0.952411i
$636$			8.00000i			0.317221i
$637$		−	2.00000i		−	0.0792429i
$638$		0			0
$639$	−6.00000			−0.237356
$640$	24.0000	+	24.0000i	0.948683	+	0.948683i
$641$	2.00000			0.0789953			0.0394976	−	0.999220i	$-0.487424\pi$
$641$	2.00000			0.0789953			0.0394976	+	0.999220i	$0.487424\pi$
$642$	8.00000	+	8.00000i	0.315735	+	0.315735i
$643$			26.0000i			1.02534i	0.858586	+	0.512670i	$0.171344\pi$
$643$			26.0000i			1.02534i	−0.858586	+	0.512670i	$0.828656\pi$
$644$			24.0000i			0.945732i
$645$			3.00000i			0.118125i
$646$	−28.0000	+	28.0000i	−1.10165	+	1.10165i
$647$	18.0000			0.707653			0.353827	−	0.935311i	$-0.384880\pi$
$647$	18.0000			0.707653			0.353827	+	0.935311i	$0.384880\pi$
$648$	2.00000	−	2.00000i	0.0785674	−	0.0785674i
$649$		0			0
$650$	4.00000	−	4.00000i	0.156893	−	0.156893i
$651$		−	24.0000i		−	0.940634i
$652$	48.0000			1.87983
$653$		−	44.0000i		−	1.72185i	−0.508729	−	0.860927i	$-0.669885\pi$
$653$		−	44.0000i		−	1.72185i	0.508729	−	0.860927i	$-0.330115\pi$
$654$	−1.00000	−	1.00000i	−0.0391031	−	0.0391031i
$655$	45.0000			1.75830
$656$	−8.00000			−0.312348
$657$	28.0000			1.09238
$658$	21.0000	+	21.0000i	0.818665	+	0.818665i
$659$			4.00000i			0.155818i	0.996960	+	0.0779089i	$0.0248243\pi$
$659$			4.00000i			0.155818i	−0.996960	+	0.0779089i	$0.975176\pi$
$660$		0			0
$661$		−	30.0000i		−	1.16686i	−0.812162	−	0.583432i	$-0.801709\pi$
$661$		−	30.0000i		−	1.16686i	0.812162	−	0.583432i	$-0.198291\pi$
$662$	−10.0000	+	10.0000i	−0.388661	+	0.388661i
$663$	−7.00000			−0.271857
$664$	−28.0000	−	28.0000i	−1.08661	−	1.08661i
$665$	36.0000			1.39602
$666$	−14.0000	+	14.0000i	−0.542489	+	0.542489i
$667$			16.0000i			0.619522i
$668$		−	24.0000i		−	0.928588i
$669$		−	1.00000i		−	0.0386622i
$670$	6.00000	+	6.00000i	0.231800	+	0.231800i
$671$		0			0
$672$	−12.0000	+	12.0000i	−0.462910	+	0.462910i
$673$	29.0000			1.11787			0.558934	−	0.829212i	$-0.311211\pi$
$673$	29.0000			1.11787			0.558934	+	0.829212i	$0.311211\pi$
$674$	17.0000	+	17.0000i	0.654816	+	0.654816i
$675$		−	20.0000i		−	0.769800i
$676$		−	2.00000i		−	0.0769231i
$677$		−	22.0000i		−	0.845529i	−0.906240	−	0.422764i	$-0.861060\pi$
$677$		−	22.0000i		−	0.845529i	0.906240	−	0.422764i	$-0.138940\pi$
$678$	−6.00000	+	6.00000i	−0.230429	+	0.230429i
$679$	24.0000			0.921035
$680$	42.0000	+	42.0000i	1.61063	+	1.61063i
$681$	−18.0000			−0.689761
$682$		0			0
$683$			26.0000i			0.994862i	0.867503	+	0.497431i	$0.165723\pi$
$683$			26.0000i			0.994862i	−0.867503	+	0.497431i	$0.834277\pi$
$684$	−16.0000			−0.611775
$685$			36.0000i			1.37549i
$686$	15.0000	+	15.0000i	0.572703	+	0.572703i
$687$	21.0000			0.801200
$688$		−	4.00000i		−	0.152499i
$689$	−4.00000			−0.152388
$690$	−12.0000	−	12.0000i	−0.456832	−	0.456832i
$691$			20.0000i			0.760836i	0.924815	+	0.380418i	$0.124220\pi$
$691$			20.0000i			0.760836i	−0.924815	+	0.380418i	$0.875780\pi$
$692$	−12.0000			−0.456172
$693$		0			0
$694$	−7.00000	+	7.00000i	−0.265716	+	0.265716i
$695$	−33.0000			−1.25176
$696$	−8.00000	+	8.00000i	−0.303239	+	0.303239i
$697$	−14.0000			−0.530288
$698$	19.0000	−	19.0000i	0.719161	−	0.719161i
$699$			19.0000i			0.718646i
$700$		−	24.0000i		−	0.907115i
$701$		0			0		1.00000			$0$
$701$		0			0		−1.00000			$\pi$
$702$	−5.00000	−	5.00000i	−0.188713	−	0.188713i
$703$	28.0000			1.05604
$704$		0			0
$705$	−21.0000			−0.790906
$706$	6.00000	+	6.00000i	0.225813	+	0.225813i
$707$		0			0
$708$		−	28.0000i		−	1.05230i
$709$		−	6.00000i		−	0.225335i	−0.993633	−	0.112667i	$-0.964061\pi$
$709$		−	6.00000i		−	0.225335i	0.993633	−	0.112667i	$-0.0359394\pi$
$710$	9.00000	−	9.00000i	0.337764	−	0.337764i
$711$	−20.0000			−0.750059
$712$		0			0
$713$	−32.0000			−1.19841
$714$	−21.0000	+	21.0000i	−0.785905	+	0.785905i
$715$		0			0
$716$	42.0000			1.56961
$717$			5.00000i			0.186728i
$718$		0			0
$719$	−30.0000			−1.11881			−0.559406	−	0.828894i	$-0.688971\pi$
$719$	−30.0000			−1.11881			−0.559406	+	0.828894i	$0.688971\pi$
$720$			24.0000i			0.894427i
$721$	−18.0000			−0.670355
$722$	−3.00000	−	3.00000i	−0.111648	−	0.111648i
$723$		−	8.00000i		−	0.297523i
$724$	20.0000			0.743294
$725$		−	16.0000i		−	0.594225i
$726$	11.0000	−	11.0000i	0.408248	−	0.408248i
$727$	8.00000			0.296704			0.148352	−	0.988935i	$-0.452603\pi$
$727$	8.00000			0.296704			0.148352	+	0.988935i	$0.452603\pi$
$728$	−6.00000	−	6.00000i	−0.222375	−	0.222375i
$729$	−13.0000			−0.481481
$730$	−42.0000	+	42.0000i	−1.55449	+	1.55449i
$731$		−	7.00000i		−	0.258904i
$732$		−	20.0000i		−	0.739221i
$733$			1.00000i			0.0369358i	0.999829	+	0.0184679i	$0.00587886\pi$
$733$			1.00000i			0.0369358i	−0.999829	+	0.0184679i	$0.994121\pi$
$734$	22.0000	+	22.0000i	0.812035	+	0.812035i
$735$	6.00000			0.221313
$736$	16.0000	+	16.0000i	0.589768	+	0.589768i
$737$		0			0
$738$	−4.00000	−	4.00000i	−0.147242	−	0.147242i
$739$			34.0000i			1.25071i	0.780340	+	0.625355i	$0.215046\pi$
$739$			34.0000i			1.25071i	−0.780340	+	0.625355i	$0.784954\pi$
$740$		−	42.0000i		−	1.54395i
$741$			4.00000i			0.146944i
$742$	−12.0000	+	12.0000i	−0.440534	+	0.440534i
$743$	−21.0000			−0.770415			−0.385208	−	0.922830i	$-0.625870\pi$
$743$	−21.0000			−0.770415			−0.385208	+	0.922830i	$0.625870\pi$
$744$	−16.0000	−	16.0000i	−0.586588	−	0.586588i
$745$	−18.0000			−0.659469
$746$	36.0000	−	36.0000i	1.31805	−	1.31805i
$747$		−	28.0000i		−	1.02447i
$748$		0			0
$749$			24.0000i			0.876941i
$750$	−3.00000	−	3.00000i	−0.109545	−	0.109545i
$751$	2.00000			0.0729810			0.0364905	−	0.999334i	$-0.488382\pi$
$751$	2.00000			0.0729810			0.0364905	+	0.999334i	$0.488382\pi$
$752$	28.0000			1.02105
$753$	20.0000			0.728841
$754$	−4.00000	−	4.00000i	−0.145671	−	0.145671i
$755$		−	51.0000i		−	1.85608i
$756$	−30.0000			−1.09109
$757$		−	32.0000i		−	1.16306i	−0.813525	−	0.581530i	$-0.802454\pi$
$757$		−	32.0000i		−	1.16306i	0.813525	−	0.581530i	$-0.197546\pi$
$758$	4.00000	−	4.00000i	0.145287	−	0.145287i
$759$		0			0
$760$	24.0000	−	24.0000i	0.870572	−	0.870572i
$761$	22.0000			0.797499			0.398750	−	0.917060i	$-0.369444\pi$
$761$	22.0000			0.797499			0.398750	+	0.917060i	$0.369444\pi$
$762$	8.00000	−	8.00000i	0.289809	−	0.289809i
$763$		−	3.00000i		−	0.108607i
$764$			24.0000i			0.868290i
$765$			42.0000i			1.51851i
$766$	1.00000	+	1.00000i	0.0361315	+	0.0361315i
$767$	14.0000			0.505511
$768$			16.0000i			0.577350i
$769$	−20.0000			−0.721218			−0.360609	−	0.932717i	$-0.617431\pi$
$769$	−20.0000			−0.721218			−0.360609	+	0.932717i	$0.617431\pi$
$770$		0			0
$771$		−	7.00000i		−	0.252099i
$772$		−	12.0000i		−	0.431889i
$773$			21.0000i			0.755318i	0.925945	+	0.377659i	$0.123271\pi$
$773$			21.0000i			0.755318i	−0.925945	+	0.377659i	$0.876729\pi$
$774$	2.00000	−	2.00000i	0.0718885	−	0.0718885i
$775$	32.0000			1.14947
$776$	16.0000	−	16.0000i	0.574367	−	0.574367i
$777$	21.0000			0.753371
$778$	34.0000	−	34.0000i	1.21896	−	1.21896i
$779$			8.00000i			0.286630i
$780$	6.00000			0.214834
$781$		0			0
$782$	28.0000	+	28.0000i	1.00128	+	1.00128i
$783$	−20.0000			−0.714742
$784$	−8.00000			−0.285714
$785$	−6.00000			−0.214149
$786$	15.0000	+	15.0000i	0.535032	+	0.535032i
$787$		−	22.0000i		−	0.784215i	−0.919919	−	0.392108i	$-0.871746\pi$
$787$		−	22.0000i		−	0.784215i	0.919919	−	0.392108i	$-0.128254\pi$
$788$	34.0000			1.21120
$789$			14.0000i			0.498413i
$790$	30.0000	−	30.0000i	1.06735	−	1.06735i
$791$	−18.0000			−0.640006
$792$		0			0
$793$	10.0000			0.355110
$794$	−22.0000	+	22.0000i	−0.780751	+	0.780751i
$795$		−	12.0000i		−	0.425596i
$796$		−	20.0000i		−	0.708881i
$797$		−	12.0000i		−	0.425062i	−0.977154	−	0.212531i	$-0.931829\pi$
$797$		−	12.0000i		−	0.425062i	0.977154	−	0.212531i	$-0.0681706\pi$
$798$	12.0000	+	12.0000i	0.424795	+	0.424795i
$799$	49.0000			1.73350
$800$	−16.0000	−	16.0000i	−0.565685	−	0.565685i
$801$		0			0
$802$	−32.0000	−	32.0000i	−1.12996	−	1.12996i
$803$		0			0
$804$			4.00000i			0.141069i
$805$		−	36.0000i		−	1.26883i
$806$	8.00000	−	8.00000i	0.281788	−	0.281788i
$807$	6.00000			0.211210
$808$		0			0
$809$	25.0000			0.878953			0.439477	−	0.898254i	$-0.355164\pi$
$809$	25.0000			0.878953			0.439477	+	0.898254i	$0.355164\pi$
$810$	−3.00000	+	3.00000i	−0.105409	+	0.105409i
$811$		−	40.0000i		−	1.40459i	−0.711886	−	0.702295i	$-0.752159\pi$
$811$		−	40.0000i		−	1.40459i	0.711886	−	0.702295i	$-0.247841\pi$
$812$	−24.0000			−0.842235
$813$		−	3.00000i		−	0.105215i
$814$		0			0
$815$	−72.0000			−2.52205
$816$			28.0000i			0.980196i
$817$	−4.00000			−0.139942
$818$		0			0
$819$		−	6.00000i		−	0.209657i
$820$	12.0000			0.419058
$821$			45.0000i			1.57051i	0.619172	+	0.785255i	$0.287468\pi$
$821$			45.0000i			1.57051i	−0.619172	+	0.785255i	$0.712532\pi$
$822$	−12.0000	+	12.0000i	−0.418548	+	0.418548i
$823$	−46.0000			−1.60346			−0.801730	−	0.597687i	$-0.796087\pi$
$823$	−46.0000			−1.60346			−0.801730	+	0.597687i	$0.796087\pi$
$824$	−12.0000	+	12.0000i	−0.418040	+	0.418040i
$825$		0			0
$826$	42.0000	−	42.0000i	1.46137	−	1.46137i
$827$			8.00000i			0.278187i	0.990279	+	0.139094i	$0.0444189\pi$
$827$			8.00000i			0.278187i	−0.990279	+	0.139094i	$0.955581\pi$
$828$			16.0000i			0.556038i
$829$			14.0000i			0.486240i	0.969996	+	0.243120i	$0.0781709\pi$
$829$			14.0000i			0.486240i	−0.969996	+	0.243120i	$0.921829\pi$
$830$	42.0000	+	42.0000i	1.45784	+	1.45784i
$831$	−18.0000			−0.624413
$832$	−8.00000			−0.277350
$833$	−14.0000			−0.485071
$834$	−11.0000	−	11.0000i	−0.380899	−	0.380899i
$835$			36.0000i			1.24583i
$836$		0			0
$837$		−	40.0000i		−	1.38260i
$838$	−11.0000	+	11.0000i	−0.379989	+	0.379989i
$839$	−40.0000			−1.38095			−0.690477	−	0.723355i	$-0.742599\pi$
$839$	−40.0000			−1.38095			−0.690477	+	0.723355i	$0.742599\pi$
$840$	18.0000	−	18.0000i	0.621059	−	0.621059i
$841$	13.0000			0.448276
$842$	−15.0000	+	15.0000i	−0.516934	+	0.516934i
$843$		−	8.00000i		−	0.275535i
$844$	−30.0000			−1.03264
$845$			3.00000i			0.103203i
$846$	14.0000	+	14.0000i	0.481330	+	0.481330i
$847$	33.0000			1.13389
$848$			16.0000i			0.549442i
$849$	4.00000			0.137280
$850$	−28.0000	−	28.0000i	−0.960392	−	0.960392i
$851$		−	28.0000i		−	0.959828i
$852$	6.00000			0.205557
$853$		−	9.00000i		−	0.308154i	−0.988059	−	0.154077i	$-0.950760\pi$
$853$		−	9.00000i		−	0.308154i	0.988059	−	0.154077i	$-0.0492404\pi$
$854$	30.0000	−	30.0000i	1.02658	−	1.02658i
$855$	24.0000			0.820783
$856$	16.0000	+	16.0000i	0.546869	+	0.546869i
$857$	−42.0000			−1.43469			−0.717346	−	0.696717i	$-0.754643\pi$
$857$	−42.0000			−1.43469			−0.717346	+	0.696717i	$0.754643\pi$
$858$		0			0
$859$		−	36.0000i		−	1.22830i	−0.789188	−	0.614152i	$-0.789498\pi$
$859$		−	36.0000i		−	1.22830i	0.789188	−	0.614152i	$-0.210502\pi$
$860$			6.00000i			0.204598i
$861$			6.00000i			0.204479i
$862$	3.00000	+	3.00000i	0.102180	+	0.102180i
$863$	−51.0000			−1.73606			−0.868030	−	0.496512i	$-0.834614\pi$
$863$	−51.0000			−1.73606			−0.868030	+	0.496512i	$0.834614\pi$
$864$	−20.0000	+	20.0000i	−0.680414	+	0.680414i
$865$	18.0000			0.612018
$866$	1.00000	+	1.00000i	0.0339814	+	0.0339814i
$867$			32.0000i			1.08678i
$868$		−	48.0000i		−	1.62923i
$869$		0			0
$870$	12.0000	−	12.0000i	0.406838	−	0.406838i
$871$	−2.00000			−0.0677674
$872$	−2.00000	−	2.00000i	−0.0677285	−	0.0677285i
$873$	16.0000			0.541518
$874$	16.0000	−	16.0000i	0.541208	−	0.541208i
$875$		−	9.00000i		−	0.304256i
$876$	−28.0000			−0.946032
$877$		−	7.00000i		−	0.236373i	−0.992991	−	0.118187i	$-0.962292\pi$
$877$		−	7.00000i		−	0.236373i	0.992991	−	0.118187i	$-0.0377081\pi$
$878$	10.0000	+	10.0000i	0.337484	+	0.337484i
$879$	−1.00000			−0.0337292
$880$		0			0
$881$	−3.00000			−0.101073			−0.0505363	−	0.998722i	$-0.516093\pi$
$881$	−3.00000			−0.101073			−0.0505363	+	0.998722i	$0.516093\pi$
$882$	−4.00000	−	4.00000i	−0.134687	−	0.134687i
$883$			41.0000i			1.37976i	0.723924	+	0.689880i	$0.242337\pi$
$883$			41.0000i			1.37976i	−0.723924	+	0.689880i	$0.757663\pi$
$884$	−14.0000			−0.470871
$885$			42.0000i			1.41181i
$886$	−9.00000	+	9.00000i	−0.302361	+	0.302361i
$887$	18.0000			0.604381			0.302190	−	0.953248i	$-0.402282\pi$
$887$	18.0000			0.604381			0.302190	+	0.953248i	$0.402282\pi$
$888$	14.0000	−	14.0000i	0.469809	−	0.469809i
$889$	24.0000			0.804934
$890$		0			0
$891$		0			0
$892$		−	2.00000i		−	0.0669650i
$893$		−	28.0000i		−	0.936984i
$894$	−6.00000	−	6.00000i	−0.200670	−	0.200670i
$895$	−63.0000			−2.10586
$896$	−24.0000	+	24.0000i	−0.801784	+	0.801784i
$897$	4.00000			0.133556
$898$	20.0000	+	20.0000i	0.667409	+	0.667409i
$899$		−	32.0000i		−	1.06726i
$900$		−	16.0000i		−	0.533333i
$901$			28.0000i			0.932815i
$902$		0			0
$903$	−3.00000			−0.0998337
$904$	−12.0000	+	12.0000i	−0.399114	+	0.399114i
$905$	−30.0000			−0.997234
$906$	17.0000	−	17.0000i	0.564787	−	0.564787i
$907$			13.0000i			0.431658i	0.976431	+	0.215829i	$0.0692454\pi$
$907$			13.0000i			0.431658i	−0.976431	+	0.215829i	$0.930755\pi$
$908$	−36.0000			−1.19470
$909$		0			0
$910$	9.00000	+	9.00000i	0.298347	+	0.298347i
$911$	−8.00000			−0.265052			−0.132526	−	0.991180i	$-0.542309\pi$
$911$	−8.00000			−0.265052			−0.132526	+	0.991180i	$0.542309\pi$
$912$	16.0000			0.529813
$913$		0			0
$914$	32.0000	+	32.0000i	1.05847	+	1.05847i
$915$			30.0000i			0.991769i
$916$	42.0000			1.38772
$917$			45.0000i			1.48603i
$918$	−35.0000	+	35.0000i	−1.15517	+	1.15517i
$919$	60.0000			1.97922			0.989609	−	0.143787i	$-0.0459280\pi$
$919$	60.0000			1.97922			0.989609	+	0.143787i	$0.0459280\pi$
$920$	−24.0000	−	24.0000i	−0.791257	−	0.791257i
$921$	2.00000			0.0659022
$922$	−5.00000	+	5.00000i	−0.164666	+	0.164666i
$923$			3.00000i			0.0987462i
$924$		0			0
$925$			28.0000i			0.920634i
$926$	16.0000	+	16.0000i	0.525793	+	0.525793i
$927$	−12.0000			−0.394132
$928$	−16.0000	+	16.0000i	−0.525226	+	0.525226i
$929$		0			0			−	1.00000i	$-0.5\pi$
$929$		0			0				1.00000i	$0.5\pi$
$930$	24.0000	+	24.0000i	0.786991	+	0.786991i
$931$			8.00000i			0.262189i
$932$			38.0000i			1.24473i
$933$			32.0000i			1.04763i
$934$	−12.0000	+	12.0000i	−0.392652	+	0.392652i
$935$		0			0
$936$	−4.00000	−	4.00000i	−0.130744	−	0.130744i
$937$	−22.0000			−0.718709			−0.359354	−	0.933201i	$-0.617003\pi$
$937$	−22.0000			−0.718709			−0.359354	+	0.933201i	$0.617003\pi$
$938$	−6.00000	+	6.00000i	−0.195907	+	0.195907i
$939$			29.0000i			0.946379i
$940$	−42.0000			−1.36989
$941$			25.0000i			0.814977i	0.913210	+	0.407488i	$0.133595\pi$
$941$			25.0000i			0.814977i	−0.913210	+	0.407488i	$0.866405\pi$
$942$	−2.00000	−	2.00000i	−0.0651635	−	0.0651635i
$943$	8.00000			0.260516
$944$		−	56.0000i		−	1.82264i
$945$	45.0000			1.46385
$946$		0			0
$947$		−	32.0000i		−	1.03986i	−0.854209	−	0.519930i	$-0.825958\pi$
$947$		−	32.0000i		−	1.03986i	0.854209	−	0.519930i	$-0.174042\pi$
$948$	20.0000			0.649570
$949$		−	14.0000i		−	0.454459i
$950$	−16.0000	+	16.0000i	−0.519109	+	0.519109i
$951$	−18.0000			−0.583690
$952$	−42.0000	+	42.0000i	−1.36123	+	1.36123i
$953$	9.00000			0.291539			0.145769	−	0.989319i	$-0.453434\pi$
$953$	9.00000			0.291539			0.145769	+	0.989319i	$0.453434\pi$
$954$	−8.00000	+	8.00000i	−0.259010	+	0.259010i
$955$		−	36.0000i		−	1.16493i
$956$			10.0000i			0.323423i
$957$		0			0
$958$	−5.00000	−	5.00000i	−0.161543	−	0.161543i
$959$	−36.0000			−1.16250
$960$		−	24.0000i		−	0.774597i
$961$	33.0000			1.06452
$962$	7.00000	+	7.00000i	0.225689	+	0.225689i
$963$			16.0000i			0.515593i
$964$		−	16.0000i		−	0.515325i
$965$			18.0000i			0.579441i
$966$	12.0000	−	12.0000i	0.386094	−	0.386094i
$967$	−7.00000			−0.225105			−0.112552	−	0.993646i	$-0.535903\pi$
$967$	−7.00000			−0.225105			−0.112552	+	0.993646i	$0.535903\pi$
$968$	22.0000	−	22.0000i	0.707107	−	0.707107i
$969$	28.0000			0.899490
$970$	−24.0000	+	24.0000i	−0.770594	+	0.770594i
$971$			15.0000i			0.481373i	0.970603	+	0.240686i	$0.0773725\pi$
$971$			15.0000i			0.481373i	−0.970603	+	0.240686i	$0.922627\pi$
$972$	−32.0000			−1.02640
$973$		−	33.0000i		−	1.05793i
$974$	−8.00000	−	8.00000i	−0.256337	−	0.256337i
$975$	−4.00000			−0.128103
$976$		−	40.0000i		−	1.28037i
$977$	18.0000			0.575871			0.287936	−	0.957650i	$-0.407031\pi$
$977$	18.0000			0.575871			0.287936	+	0.957650i	$0.407031\pi$
$978$	−24.0000	−	24.0000i	−0.767435	−	0.767435i
$979$		0			0
$980$	12.0000			0.383326
$981$		−	2.00000i		−	0.0638551i
$982$	−15.0000	+	15.0000i	−0.478669	+	0.478669i
$983$	59.0000			1.88181			0.940904	−	0.338674i	$-0.109978\pi$
$983$	59.0000			1.88181			0.940904	+	0.338674i	$0.109978\pi$
$984$	4.00000	+	4.00000i	0.127515	+	0.127515i
$985$	−51.0000			−1.62500
$986$	−28.0000	+	28.0000i	−0.891702	+	0.891702i
$987$		−	21.0000i		−	0.668437i
$988$			8.00000i			0.254514i
$989$			4.00000i			0.127193i
$990$		0			0
$991$	−8.00000			−0.254128			−0.127064	−	0.991894i	$-0.540555\pi$
$991$	−8.00000			−0.254128			−0.127064	+	0.991894i	$0.540555\pi$
$992$	−32.0000	−	32.0000i	−1.01600	−	1.01600i
$993$	10.0000			0.317340
$994$	9.00000	+	9.00000i	0.285463	+	0.285463i
$995$			30.0000i			0.951064i
$996$			28.0000i			0.887214i
$997$		−	52.0000i		−	1.64686i	−0.567420	−	0.823428i	$-0.692059\pi$
$997$		−	52.0000i		−	1.64686i	0.567420	−	0.823428i	$-0.307941\pi$
$998$	−6.00000	+	6.00000i	−0.189927	+	0.189927i
$999$	35.0000			1.10735

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	104.2.b.a.53.1	✓	2
3.2	odd	2		936.2.g.a.469.2		2
4.3	odd	2		416.2.b.a.209.1		2
8.3	odd	2		416.2.b.a.209.2		2
8.5	even	2	inner	104.2.b.a.53.2	yes	2
12.11	even	2		3744.2.g.a.1873.2		2
16.3	odd	4		3328.2.a.g.1.1		1
16.5	even	4		3328.2.a.j.1.1		1
16.11	odd	4		3328.2.a.f.1.1		1
16.13	even	4		3328.2.a.c.1.1		1
24.5	odd	2		936.2.g.a.469.1		2
24.11	even	2		3744.2.g.a.1873.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
104.2.b.a.53.1	✓	2	1.1	even	1	trivial
104.2.b.a.53.2	yes	2	8.5	even	2	inner
416.2.b.a.209.1		2	4.3	odd	2
416.2.b.a.209.2		2	8.3	odd	2
936.2.g.a.469.1		2	24.5	odd	2
936.2.g.a.469.2		2	3.2	odd	2
3328.2.a.c.1.1		1	16.13	even	4
3328.2.a.f.1.1		1	16.11	odd	4
3328.2.a.g.1.1		1	16.3	odd	4
3328.2.a.j.1.1		1	16.5	even	4
3744.2.g.a.1873.1		2	24.11	even	2
3744.2.g.a.1873.2		2	12.11	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 104 = 2^{3} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	104.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+(-1.00000 - 1.00000i) q^{2} +1.00000i q^{3} +2.00000i q^{4} -3.00000i q^{5} +(1.00000 - 1.00000i) q^{6} +3.00000 q^{7} +(2.00000 - 2.00000i) q^{8} +2.00000 q^{9} +(-3.00000 + 3.00000i) q^{10} -2.00000 q^{12} -1.00000i q^{13} +(-3.00000 - 3.00000i) q^{14} +3.00000 q^{15} -4.00000 q^{16} -7.00000 q^{17} +(-2.00000 - 2.00000i) q^{18} +4.00000i q^{19} +6.00000 q^{20} +3.00000i q^{21} +4.00000 q^{23} +(2.00000 + 2.00000i) q^{24} -4.00000 q^{25} +(-1.00000 + 1.00000i) q^{26} +5.00000i q^{27} +6.00000i q^{28} +4.00000i q^{29} +(-3.00000 - 3.00000i) q^{30} -8.00000 q^{31} +(4.00000 + 4.00000i) q^{32} +(7.00000 + 7.00000i) q^{34} -9.00000i q^{35} +4.00000i q^{36} -7.00000i q^{37} +(4.00000 - 4.00000i) q^{38} +1.00000 q^{39} +(-6.00000 - 6.00000i) q^{40} +2.00000 q^{41} +(3.00000 - 3.00000i) q^{42} +1.00000i q^{43} -6.00000i q^{45} +(-4.00000 - 4.00000i) q^{46} -7.00000 q^{47} -4.00000i q^{48} +2.00000 q^{49} +(4.00000 + 4.00000i) q^{50} -7.00000i q^{51} +2.00000 q^{52} -4.00000i q^{53} +(5.00000 - 5.00000i) q^{54} +(6.00000 - 6.00000i) q^{56} -4.00000 q^{57} +(4.00000 - 4.00000i) q^{58} +14.0000i q^{59} +6.00000i q^{60} +10.0000i q^{61} +(8.00000 + 8.00000i) q^{62} +6.00000 q^{63} -8.00000i q^{64} -3.00000 q^{65} -2.00000i q^{67} -14.0000i q^{68} +4.00000i q^{69} +(-9.00000 + 9.00000i) q^{70} -3.00000 q^{71} +(4.00000 - 4.00000i) q^{72} +14.0000 q^{73} +(-7.00000 + 7.00000i) q^{74} -4.00000i q^{75} -8.00000 q^{76} +(-1.00000 - 1.00000i) q^{78} -10.0000 q^{79} +12.0000i q^{80} +1.00000 q^{81} +(-2.00000 - 2.00000i) q^{82} -14.0000i q^{83} -6.00000 q^{84} +21.0000i q^{85} +(1.00000 - 1.00000i) q^{86} -4.00000 q^{87} +(-6.00000 + 6.00000i) q^{90} -3.00000i q^{91} +8.00000i q^{92} -8.00000i q^{93} +(7.00000 + 7.00000i) q^{94} +12.0000 q^{95} +(-4.00000 + 4.00000i) q^{96} +8.00000 q^{97} +(-2.00000 - 2.00000i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{2} + 2 q^{6} + 6 q^{7} + 4 q^{8} + 4 q^{9} - 6 q^{10} - 4 q^{12} - 6 q^{14} + 6 q^{15} - 8 q^{16} - 14 q^{17} - 4 q^{18} + 12 q^{20} + 8 q^{23} + 4 q^{24} - 8 q^{25} - 2 q^{26} - 6 q^{30} - 16 q^{31}+ \cdots - 4 q^{98}+O(q^{100}) \) 2 * q - 2 * q^2 + 2 * q^6 + 6 * q^7 + 4 * q^8 + 4 * q^9 - 6 * q^10 - 4 * q^12 - 6 * q^14 + 6 * q^15 - 8 * q^16 - 14 * q^17 - 4 * q^18 + 12 * q^20 + 8 * q^23 + 4 * q^24 - 8 * q^25 - 2 * q^26 - 6 * q^30 - 16 * q^31 + 8 * q^32 + 14 * q^34 + 8 * q^38 + 2 * q^39 - 12 * q^40 + 4 * q^41 + 6 * q^42 - 8 * q^46 - 14 * q^47 + 4 * q^49 + 8 * q^50 + 4 * q^52 + 10 * q^54 + 12 * q^56 - 8 * q^57 + 8 * q^58 + 16 * q^62 + 12 * q^63 - 6 * q^65 - 18 * q^70 - 6 * q^71 + 8 * q^72 + 28 * q^73 - 14 * q^74 - 16 * q^76 - 2 * q^78 - 20 * q^79 + 2 * q^81 - 4 * q^82 - 12 * q^84 + 2 * q^86 - 8 * q^87 - 12 * q^90 + 14 * q^94 + 24 * q^95 - 8 * q^96 + 16 * q^97 - 4 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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