LMFDB - Embedded newform 24.2.d.a.13.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [24,2,Mod(13,24)] 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("24.13"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(24, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 24 = 2^{3} \cdot 3 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	24.d (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

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

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

Self dual:	no
Analytic conductor:	$0.191640964851$
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			13.1
Root			$-1.00000i$ of defining polynomial
Character	$\chi$	$=$	24.13
Dual form			24.2.d.a.13.2

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+(-1.00000 - 1.00000i) q^{2} -1.00000i q^{3} +2.00000i q^{4} +2.00000i q^{5} +(-1.00000 + 1.00000i) q^{6} -2.00000 q^{7} +(2.00000 - 2.00000i) q^{8} -1.00000 q^{9} +(2.00000 - 2.00000i) q^{10} +2.00000 q^{12} -4.00000i q^{13} +(2.00000 + 2.00000i) q^{14} +2.00000 q^{15} -4.00000 q^{16} -2.00000 q^{17} +(1.00000 + 1.00000i) q^{18} +4.00000i q^{19} -4.00000 q^{20} +2.00000i q^{21} +4.00000 q^{23} +(-2.00000 - 2.00000i) q^{24} +1.00000 q^{25} +(-4.00000 + 4.00000i) q^{26} +1.00000i q^{27} -4.00000i q^{28} -6.00000i q^{29} +(-2.00000 - 2.00000i) q^{30} +2.00000 q^{31} +(4.00000 + 4.00000i) q^{32} +(2.00000 + 2.00000i) q^{34} -4.00000i q^{35} -2.00000i q^{36} +8.00000i q^{37} +(4.00000 - 4.00000i) q^{38} -4.00000 q^{39} +(4.00000 + 4.00000i) q^{40} +2.00000 q^{41} +(2.00000 - 2.00000i) q^{42} -4.00000i q^{43} -2.00000i q^{45} +(-4.00000 - 4.00000i) q^{46} -12.0000 q^{47} +4.00000i q^{48} -3.00000 q^{49} +(-1.00000 - 1.00000i) q^{50} +2.00000i q^{51} +8.00000 q^{52} +6.00000i q^{53} +(1.00000 - 1.00000i) q^{54} +(-4.00000 + 4.00000i) q^{56} +4.00000 q^{57} +(-6.00000 + 6.00000i) q^{58} +4.00000i q^{59} +4.00000i q^{60} +(-2.00000 - 2.00000i) q^{62} +2.00000 q^{63} -8.00000i q^{64} +8.00000 q^{65} -12.0000i q^{67} -4.00000i q^{68} -4.00000i q^{69} +(-4.00000 + 4.00000i) q^{70} +12.0000 q^{71} +(-2.00000 + 2.00000i) q^{72} -6.00000 q^{73} +(8.00000 - 8.00000i) q^{74} -1.00000i q^{75} -8.00000 q^{76} +(4.00000 + 4.00000i) q^{78} +10.0000 q^{79} -8.00000i q^{80} +1.00000 q^{81} +(-2.00000 - 2.00000i) q^{82} +16.0000i q^{83} -4.00000 q^{84} -4.00000i q^{85} +(-4.00000 + 4.00000i) q^{86} -6.00000 q^{87} -10.0000 q^{89} +(-2.00000 + 2.00000i) q^{90} +8.00000i q^{91} +8.00000i q^{92} -2.00000i q^{93} +(12.0000 + 12.0000i) q^{94} -8.00000 q^{95} +(4.00000 - 4.00000i) q^{96} -2.00000 q^{97} +(3.00000 + 3.00000i) q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{2} - 2 q^{6} - 4 q^{7} + 4 q^{8} - 2 q^{9} + 4 q^{10} + 4 q^{12} + 4 q^{14} + 4 q^{15} - 8 q^{16} - 4 q^{17} + 2 q^{18} - 8 q^{20} + 8 q^{23} - 4 q^{24} + 2 q^{25} - 8 q^{26} - 4 q^{30} + 4 q^{31}+ \cdots + 6 q^{98}+O(q^{100}) $ 2 * q - 2 * q^2 - 2 * q^6 - 4 * q^7 + 4 * q^8 - 2 * q^9 + 4 * q^10 + 4 * q^12 + 4 * q^14 + 4 * q^15 - 8 * q^16 - 4 * q^17 + 2 * q^18 - 8 * q^20 + 8 * q^23 - 4 * q^24 + 2 * q^25 - 8 * q^26 - 4 * q^30 + 4 * q^31 + 8 * q^32 + 4 * q^34 + 8 * q^38 - 8 * q^39 + 8 * q^40 + 4 * q^41 + 4 * q^42 - 8 * q^46 - 24 * q^47 - 6 * q^49 - 2 * q^50 + 16 * q^52 + 2 * q^54 - 8 * q^56 + 8 * q^57 - 12 * q^58 - 4 * q^62 + 4 * q^63 + 16 * q^65 - 8 * q^70 + 24 * q^71 - 4 * q^72 - 12 * q^73 + 16 * q^74 - 16 * q^76 + 8 * q^78 + 20 * q^79 + 2 * q^81 - 4 * q^82 - 8 * q^84 - 8 * q^86 - 12 * q^87 - 20 * q^89 - 4 * q^90 + 24 * q^94 - 16 * q^95 + 8 * q^96 - 4 * q^97 + 6 * q^98

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/24\mathbb{Z}\right)^\times$.

$n$	$7$	$13$	$17$
$\chi(n)$	$1$	$-1$	$1$

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
$3$		−	1.00000i		−	0.577350i
$4$			2.00000i			1.00000i
$5$			2.00000i			0.894427i	0.894427	+	0.447214i	$0.147584\pi$
$5$			2.00000i			0.894427i	−0.894427	+	0.447214i	$0.852416\pi$
$6$	−1.00000	+	1.00000i	−0.408248	+	0.408248i
$7$	−2.00000			−0.755929			−0.377964	−	0.925820i	$-0.623376\pi$
$7$	−2.00000			−0.755929			−0.377964	+	0.925820i	$0.623376\pi$
$8$	2.00000	−	2.00000i	0.707107	−	0.707107i
$9$	−1.00000			−0.333333
$10$	2.00000	−	2.00000i	0.632456	−	0.632456i
$11$		0			0		1.00000			$0$
$11$		0			0		−1.00000			$\pi$
$12$	2.00000			0.577350
$13$		−	4.00000i		−	1.10940i	−0.832050	−	0.554700i	$-0.812833\pi$
$13$		−	4.00000i		−	1.10940i	0.832050	−	0.554700i	$-0.187167\pi$
$14$	2.00000	+	2.00000i	0.534522	+	0.534522i
$15$	2.00000			0.516398
$16$	−4.00000			−1.00000
$17$	−2.00000			−0.485071			−0.242536	−	0.970143i	$-0.577979\pi$
$17$	−2.00000			−0.485071			−0.242536	+	0.970143i	$0.577979\pi$
$18$	1.00000	+	1.00000i	0.235702	+	0.235702i
$19$			4.00000i			0.917663i	0.888523	+	0.458831i	$0.151732\pi$
$19$			4.00000i			0.917663i	−0.888523	+	0.458831i	$0.848268\pi$
$20$	−4.00000			−0.894427
$21$			2.00000i			0.436436i
$22$		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
$25$	1.00000			0.200000
$26$	−4.00000	+	4.00000i	−0.784465	+	0.784465i
$27$			1.00000i			0.192450i
$28$		−	4.00000i		−	0.755929i
$29$		−	6.00000i		−	1.11417i	−0.830455	−	0.557086i	$-0.811919\pi$
$29$		−	6.00000i		−	1.11417i	0.830455	−	0.557086i	$-0.188081\pi$
$30$	−2.00000	−	2.00000i	−0.365148	−	0.365148i
$31$	2.00000			0.359211			0.179605	−	0.983739i	$-0.442518\pi$
$31$	2.00000			0.359211			0.179605	+	0.983739i	$0.442518\pi$
$32$	4.00000	+	4.00000i	0.707107	+	0.707107i
$33$		0			0
$34$	2.00000	+	2.00000i	0.342997	+	0.342997i
$35$		−	4.00000i		−	0.676123i
$36$		−	2.00000i		−	0.333333i
$37$			8.00000i			1.31519i	0.753371	+	0.657596i	$0.228427\pi$
$37$			8.00000i			1.31519i	−0.753371	+	0.657596i	$0.771573\pi$
$38$	4.00000	−	4.00000i	0.648886	−	0.648886i
$39$	−4.00000			−0.640513
$40$	4.00000	+	4.00000i	0.632456	+	0.632456i
$41$	2.00000			0.312348			0.156174	−	0.987730i	$-0.450084\pi$
$41$	2.00000			0.312348			0.156174	+	0.987730i	$0.450084\pi$
$42$	2.00000	−	2.00000i	0.308607	−	0.308607i
$43$		−	4.00000i		−	0.609994i	−0.952353	−	0.304997i	$-0.901344\pi$
$43$		−	4.00000i		−	0.609994i	0.952353	−	0.304997i	$-0.0986555\pi$
$44$		0			0
$45$		−	2.00000i		−	0.298142i
$46$	−4.00000	−	4.00000i	−0.589768	−	0.589768i
$47$	−12.0000			−1.75038			−0.875190	−	0.483779i	$-0.839264\pi$
$47$	−12.0000			−1.75038			−0.875190	+	0.483779i	$0.839264\pi$
$48$			4.00000i			0.577350i
$49$	−3.00000			−0.428571
$50$	−1.00000	−	1.00000i	−0.141421	−	0.141421i
$51$			2.00000i			0.280056i
$52$	8.00000			1.10940
$53$			6.00000i			0.824163i	0.911147	+	0.412082i	$0.135198\pi$
$53$			6.00000i			0.824163i	−0.911147	+	0.412082i	$0.864802\pi$
$54$	1.00000	−	1.00000i	0.136083	−	0.136083i
$55$		0			0
$56$	−4.00000	+	4.00000i	−0.534522	+	0.534522i
$57$	4.00000			0.529813
$58$	−6.00000	+	6.00000i	−0.787839	+	0.787839i
$59$			4.00000i			0.520756i	0.965507	+	0.260378i	$0.0838471\pi$
$59$			4.00000i			0.520756i	−0.965507	+	0.260378i	$0.916153\pi$
$60$			4.00000i			0.516398i
$61$		0			0		1.00000			$0$
$61$		0			0		−1.00000			$\pi$
$62$	−2.00000	−	2.00000i	−0.254000	−	0.254000i
$63$	2.00000			0.251976
$64$		−	8.00000i		−	1.00000i
$65$	8.00000			0.992278
$66$		0			0
$67$		−	12.0000i		−	1.46603i	−0.680211	−	0.733017i	$-0.738112\pi$
$67$		−	12.0000i		−	1.46603i	0.680211	−	0.733017i	$-0.261888\pi$
$68$		−	4.00000i		−	0.485071i
$69$		−	4.00000i		−	0.481543i
$70$	−4.00000	+	4.00000i	−0.478091	+	0.478091i
$71$	12.0000			1.42414			0.712069	−	0.702109i	$-0.247758\pi$
$71$	12.0000			1.42414			0.712069	+	0.702109i	$0.247758\pi$
$72$	−2.00000	+	2.00000i	−0.235702	+	0.235702i
$73$	−6.00000			−0.702247			−0.351123	−	0.936329i	$-0.614200\pi$
$73$	−6.00000			−0.702247			−0.351123	+	0.936329i	$0.614200\pi$
$74$	8.00000	−	8.00000i	0.929981	−	0.929981i
$75$		−	1.00000i		−	0.115470i
$76$	−8.00000			−0.917663
$77$		0			0
$78$	4.00000	+	4.00000i	0.452911	+	0.452911i
$79$	10.0000			1.12509			0.562544	−	0.826767i	$-0.309823\pi$
$79$	10.0000			1.12509			0.562544	+	0.826767i	$0.309823\pi$
$80$		−	8.00000i		−	0.894427i
$81$	1.00000			0.111111
$82$	−2.00000	−	2.00000i	−0.220863	−	0.220863i
$83$			16.0000i			1.75623i	0.478451	+	0.878114i	$0.341198\pi$
$83$			16.0000i			1.75623i	−0.478451	+	0.878114i	$0.658802\pi$
$84$	−4.00000			−0.436436
$85$		−	4.00000i		−	0.433861i
$86$	−4.00000	+	4.00000i	−0.431331	+	0.431331i
$87$	−6.00000			−0.643268
$88$		0			0
$89$	−10.0000			−1.06000			−0.529999	−	0.847998i	$-0.677808\pi$
$89$	−10.0000			−1.06000			−0.529999	+	0.847998i	$0.677808\pi$
$90$	−2.00000	+	2.00000i	−0.210819	+	0.210819i
$91$			8.00000i			0.838628i
$92$			8.00000i			0.834058i
$93$		−	2.00000i		−	0.207390i
$94$	12.0000	+	12.0000i	1.23771	+	1.23771i
$95$	−8.00000			−0.820783
$96$	4.00000	−	4.00000i	0.408248	−	0.408248i
$97$	−2.00000			−0.203069			−0.101535	−	0.994832i	$-0.532375\pi$
$97$	−2.00000			−0.203069			−0.101535	+	0.994832i	$0.532375\pi$
$98$	3.00000	+	3.00000i	0.303046	+	0.303046i
$99$		0			0
$100$			2.00000i			0.200000i
$101$		−	10.0000i		−	0.995037i	−0.867453	−	0.497519i	$-0.834245\pi$
$101$		−	10.0000i		−	0.995037i	0.867453	−	0.497519i	$-0.165755\pi$
$102$	2.00000	−	2.00000i	0.198030	−	0.198030i
$103$	−6.00000			−0.591198			−0.295599	−	0.955312i	$-0.595519\pi$
$103$	−6.00000			−0.591198			−0.295599	+	0.955312i	$0.595519\pi$
$104$	−8.00000	−	8.00000i	−0.784465	−	0.784465i
$105$	−4.00000			−0.390360
$106$	6.00000	−	6.00000i	0.582772	−	0.582772i
$107$		−	12.0000i		−	1.16008i	−0.814587	−	0.580042i	$-0.803036\pi$
$107$		−	12.0000i		−	1.16008i	0.814587	−	0.580042i	$-0.196964\pi$
$108$	−2.00000			−0.192450
$109$			4.00000i			0.383131i	0.981480	+	0.191565i	$0.0613564\pi$
$109$			4.00000i			0.383131i	−0.981480	+	0.191565i	$0.938644\pi$
$110$		0			0
$111$	8.00000			0.759326
$112$	8.00000			0.755929
$113$	−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
$115$			8.00000i			0.746004i
$116$	12.0000			1.11417
$117$			4.00000i			0.369800i
$118$	4.00000	−	4.00000i	0.368230	−	0.368230i
$119$	4.00000			0.366679
$120$	4.00000	−	4.00000i	0.365148	−	0.365148i
$121$	11.0000			1.00000
$122$		0			0
$123$		−	2.00000i		−	0.180334i
$124$			4.00000i			0.359211i
$125$			12.0000i			1.07331i
$126$	−2.00000	−	2.00000i	−0.178174	−	0.178174i
$127$	−2.00000			−0.177471			−0.0887357	−	0.996055i	$-0.528283\pi$
$127$	−2.00000			−0.177471			−0.0887357	+	0.996055i	$0.528283\pi$
$128$	−8.00000	+	8.00000i	−0.707107	+	0.707107i
$129$	−4.00000			−0.352180
$130$	−8.00000	−	8.00000i	−0.701646	−	0.701646i
$131$		−	20.0000i		−	1.74741i	−0.486458	−	0.873704i	$-0.661711\pi$
$131$		−	20.0000i		−	1.74741i	0.486458	−	0.873704i	$-0.338289\pi$
$132$		0			0
$133$		−	8.00000i		−	0.693688i
$134$	−12.0000	+	12.0000i	−1.03664	+	1.03664i
$135$	−2.00000			−0.172133
$136$	−4.00000	+	4.00000i	−0.342997	+	0.342997i
$137$	18.0000			1.53784			0.768922	−	0.639343i	$-0.220793\pi$
$137$	18.0000			1.53784			0.768922	+	0.639343i	$0.220793\pi$
$138$	−4.00000	+	4.00000i	−0.340503	+	0.340503i
$139$			4.00000i			0.339276i	0.985506	+	0.169638i	$0.0542598\pi$
$139$			4.00000i			0.339276i	−0.985506	+	0.169638i	$0.945740\pi$
$140$	8.00000			0.676123
$141$			12.0000i			1.01058i
$142$	−12.0000	−	12.0000i	−1.00702	−	1.00702i
$143$		0			0
$144$	4.00000			0.333333
$145$	12.0000			0.996546
$146$	6.00000	+	6.00000i	0.496564	+	0.496564i
$147$			3.00000i			0.247436i
$148$	−16.0000			−1.31519
$149$		−	6.00000i		−	0.491539i	−0.969328	−	0.245770i	$-0.920959\pi$
$149$		−	6.00000i		−	0.491539i	0.969328	−	0.245770i	$-0.0790407\pi$
$150$	−1.00000	+	1.00000i	−0.0816497	+	0.0816497i
$151$	−18.0000			−1.46482			−0.732410	−	0.680864i	$-0.761604\pi$
$151$	−18.0000			−1.46482			−0.732410	+	0.680864i	$0.761604\pi$
$152$	8.00000	+	8.00000i	0.648886	+	0.648886i
$153$	2.00000			0.161690
$154$		0			0
$155$			4.00000i			0.321288i
$156$		−	8.00000i		−	0.640513i
$157$			8.00000i			0.638470i	0.947676	+	0.319235i	$0.103426\pi$
$157$			8.00000i			0.638470i	−0.947676	+	0.319235i	$0.896574\pi$
$158$	−10.0000	−	10.0000i	−0.795557	−	0.795557i
$159$	6.00000			0.475831
$160$	−8.00000	+	8.00000i	−0.632456	+	0.632456i
$161$	−8.00000			−0.630488
$162$	−1.00000	−	1.00000i	−0.0785674	−	0.0785674i
$163$		−	4.00000i		−	0.313304i	−0.987654	−	0.156652i	$-0.949930\pi$
$163$		−	4.00000i		−	0.313304i	0.987654	−	0.156652i	$-0.0500701\pi$
$164$			4.00000i			0.312348i
$165$		0			0
$166$	16.0000	−	16.0000i	1.24184	−	1.24184i
$167$	8.00000			0.619059			0.309529	−	0.950890i	$-0.399829\pi$
$167$	8.00000			0.619059			0.309529	+	0.950890i	$0.399829\pi$
$168$	4.00000	+	4.00000i	0.308607	+	0.308607i
$169$	−3.00000			−0.230769
$170$	−4.00000	+	4.00000i	−0.306786	+	0.306786i
$171$		−	4.00000i		−	0.305888i
$172$	8.00000			0.609994
$173$			6.00000i			0.456172i	0.973641	+	0.228086i	$0.0732467\pi$
$173$			6.00000i			0.456172i	−0.973641	+	0.228086i	$0.926753\pi$
$174$	6.00000	+	6.00000i	0.454859	+	0.454859i
$175$	−2.00000			−0.151186
$176$		0			0
$177$	4.00000			0.300658
$178$	10.0000	+	10.0000i	0.749532	+	0.749532i
$179$			4.00000i			0.298974i	0.988764	+	0.149487i	$0.0477622\pi$
$179$			4.00000i			0.298974i	−0.988764	+	0.149487i	$0.952238\pi$
$180$	4.00000			0.298142
$181$		−	20.0000i		−	1.48659i	−0.668965	−	0.743294i	$-0.733262\pi$
$181$		−	20.0000i		−	1.48659i	0.668965	−	0.743294i	$-0.266738\pi$
$182$	8.00000	−	8.00000i	0.592999	−	0.592999i
$183$		0			0
$184$	8.00000	−	8.00000i	0.589768	−	0.589768i
$185$	−16.0000			−1.17634
$186$	−2.00000	+	2.00000i	−0.146647	+	0.146647i
$187$		0			0
$188$		−	24.0000i		−	1.75038i
$189$		−	2.00000i		−	0.145479i
$190$	8.00000	+	8.00000i	0.580381	+	0.580381i
$191$	−8.00000			−0.578860			−0.289430	−	0.957199i	$-0.593466\pi$
$191$	−8.00000			−0.578860			−0.289430	+	0.957199i	$0.593466\pi$
$192$	−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	$0.569283\pi$
$194$	2.00000	+	2.00000i	0.143592	+	0.143592i
$195$		−	8.00000i		−	0.572892i
$196$		−	6.00000i		−	0.428571i
$197$		−	2.00000i		−	0.142494i	−0.997459	−	0.0712470i	$-0.977302\pi$
$197$		−	2.00000i		−	0.142494i	0.997459	−	0.0712470i	$-0.0226979\pi$
$198$		0			0
$199$	10.0000			0.708881			0.354441	−	0.935079i	$-0.384671\pi$
$199$	10.0000			0.708881			0.354441	+	0.935079i	$0.384671\pi$
$200$	2.00000	−	2.00000i	0.141421	−	0.141421i
$201$	−12.0000			−0.846415
$202$	−10.0000	+	10.0000i	−0.703598	+	0.703598i
$203$			12.0000i			0.842235i
$204$	−4.00000			−0.280056
$205$			4.00000i			0.279372i
$206$	6.00000	+	6.00000i	0.418040	+	0.418040i
$207$	−4.00000			−0.278019
$208$			16.0000i			1.10940i
$209$		0			0
$210$	4.00000	+	4.00000i	0.276026	+	0.276026i
$211$			20.0000i			1.37686i	0.725304	+	0.688428i	$0.241699\pi$
$211$			20.0000i			1.37686i	−0.725304	+	0.688428i	$0.758301\pi$
$212$	−12.0000			−0.824163
$213$		−	12.0000i		−	0.822226i
$214$	−12.0000	+	12.0000i	−0.820303	+	0.820303i
$215$	8.00000			0.545595
$216$	2.00000	+	2.00000i	0.136083	+	0.136083i
$217$	−4.00000			−0.271538
$218$	4.00000	−	4.00000i	0.270914	−	0.270914i
$219$			6.00000i			0.405442i
$220$		0			0
$221$			8.00000i			0.538138i
$222$	−8.00000	−	8.00000i	−0.536925	−	0.536925i
$223$	14.0000			0.937509			0.468755	−	0.883328i	$-0.344703\pi$
$223$	14.0000			0.937509			0.468755	+	0.883328i	$0.344703\pi$
$224$	−8.00000	−	8.00000i	−0.534522	−	0.534522i
$225$	−1.00000			−0.0666667
$226$	6.00000	+	6.00000i	0.399114	+	0.399114i
$227$			8.00000i			0.530979i	0.964114	+	0.265489i	$0.0855335\pi$
$227$			8.00000i			0.530979i	−0.964114	+	0.265489i	$0.914466\pi$
$228$			8.00000i			0.529813i
$229$			4.00000i			0.264327i	0.991228	+	0.132164i	$0.0421925\pi$
$229$			4.00000i			0.264327i	−0.991228	+	0.132164i	$0.957808\pi$
$230$	8.00000	−	8.00000i	0.527504	−	0.527504i
$231$		0			0
$232$	−12.0000	−	12.0000i	−0.787839	−	0.787839i
$233$	14.0000			0.917170			0.458585	−	0.888650i	$-0.348356\pi$
$233$	14.0000			0.917170			0.458585	+	0.888650i	$0.348356\pi$
$234$	4.00000	−	4.00000i	0.261488	−	0.261488i
$235$		−	24.0000i		−	1.56559i
$236$	−8.00000			−0.520756
$237$		−	10.0000i		−	0.649570i
$238$	−4.00000	−	4.00000i	−0.259281	−	0.259281i
$239$		0			0			−	1.00000i	$-0.5\pi$
$239$		0			0				1.00000i	$0.5\pi$
$240$	−8.00000			−0.516398
$241$	2.00000			0.128831			0.0644157	−	0.997923i	$-0.479482\pi$
$241$	2.00000			0.128831			0.0644157	+	0.997923i	$0.479482\pi$
$242$	−11.0000	−	11.0000i	−0.707107	−	0.707107i
$243$		−	1.00000i		−	0.0641500i
$244$		0			0
$245$		−	6.00000i		−	0.383326i
$246$	−2.00000	+	2.00000i	−0.127515	+	0.127515i
$247$	16.0000			1.01806
$248$	4.00000	−	4.00000i	0.254000	−	0.254000i
$249$	16.0000			1.01396
$250$	12.0000	−	12.0000i	0.758947	−	0.758947i
$251$		0			0		1.00000			$0$
$251$		0			0		−1.00000			$\pi$
$252$			4.00000i			0.251976i
$253$		0			0
$254$	2.00000	+	2.00000i	0.125491	+	0.125491i
$255$	−4.00000			−0.250490
$256$	16.0000			1.00000
$257$	18.0000			1.12281			0.561405	−	0.827541i	$-0.310261\pi$
$257$	18.0000			1.12281			0.561405	+	0.827541i	$0.310261\pi$
$258$	4.00000	+	4.00000i	0.249029	+	0.249029i
$259$		−	16.0000i		−	0.994192i
$260$			16.0000i			0.992278i
$261$			6.00000i			0.371391i
$262$	−20.0000	+	20.0000i	−1.23560	+	1.23560i
$263$	−16.0000			−0.986602			−0.493301	−	0.869859i	$-0.664210\pi$
$263$	−16.0000			−0.986602			−0.493301	+	0.869859i	$0.664210\pi$
$264$		0			0
$265$	−12.0000			−0.737154
$266$	−8.00000	+	8.00000i	−0.490511	+	0.490511i
$267$			10.0000i			0.611990i
$268$	24.0000			1.46603
$269$		−	6.00000i		−	0.365826i	−0.983129	−	0.182913i	$-0.941447\pi$
$269$		−	6.00000i		−	0.365826i	0.983129	−	0.182913i	$-0.0585527\pi$
$270$	2.00000	+	2.00000i	0.121716	+	0.121716i
$271$	−18.0000			−1.09342			−0.546711	−	0.837321i	$-0.684120\pi$
$271$	−18.0000			−1.09342			−0.546711	+	0.837321i	$0.684120\pi$
$272$	8.00000			0.485071
$273$	8.00000			0.484182
$274$	−18.0000	−	18.0000i	−1.08742	−	1.08742i
$275$		0			0
$276$	8.00000			0.481543
$277$			28.0000i			1.68236i	0.540758	+	0.841178i	$0.318138\pi$
$277$			28.0000i			1.68236i	−0.540758	+	0.841178i	$0.681862\pi$
$278$	4.00000	−	4.00000i	0.239904	−	0.239904i
$279$	−2.00000			−0.119737
$280$	−8.00000	−	8.00000i	−0.478091	−	0.478091i
$281$	−18.0000			−1.07379			−0.536895	−	0.843649i	$-0.680403\pi$
$281$	−18.0000			−1.07379			−0.536895	+	0.843649i	$0.680403\pi$
$282$	12.0000	−	12.0000i	0.714590	−	0.714590i
$283$		−	4.00000i		−	0.237775i	−0.992908	−	0.118888i	$-0.962067\pi$
$283$		−	4.00000i		−	0.237775i	0.992908	−	0.118888i	$-0.0379328\pi$
$284$			24.0000i			1.42414i
$285$			8.00000i			0.473879i
$286$		0			0
$287$	−4.00000			−0.236113
$288$	−4.00000	−	4.00000i	−0.235702	−	0.235702i
$289$	−13.0000			−0.764706
$290$	−12.0000	−	12.0000i	−0.704664	−	0.704664i
$291$			2.00000i			0.117242i
$292$		−	12.0000i		−	0.702247i
$293$			6.00000i			0.350524i	0.984522	+	0.175262i	$0.0560772\pi$
$293$			6.00000i			0.350524i	−0.984522	+	0.175262i	$0.943923\pi$
$294$	3.00000	−	3.00000i	0.174964	−	0.174964i
$295$	−8.00000			−0.465778
$296$	16.0000	+	16.0000i	0.929981	+	0.929981i
$297$		0			0
$298$	−6.00000	+	6.00000i	−0.347571	+	0.347571i
$299$		−	16.0000i		−	0.925304i
$300$	2.00000			0.115470
$301$			8.00000i			0.461112i
$302$	18.0000	+	18.0000i	1.03578	+	1.03578i
$303$	−10.0000			−0.574485
$304$		−	16.0000i		−	0.917663i
$305$		0			0
$306$	−2.00000	−	2.00000i	−0.114332	−	0.114332i
$307$		−	12.0000i		−	0.684876i	−0.939540	−	0.342438i	$-0.888747\pi$
$307$		−	12.0000i		−	0.684876i	0.939540	−	0.342438i	$-0.111253\pi$
$308$		0			0
$309$			6.00000i			0.341328i
$310$	4.00000	−	4.00000i	0.227185	−	0.227185i
$311$	−8.00000			−0.453638			−0.226819	−	0.973937i	$-0.572833\pi$
$311$	−8.00000			−0.453638			−0.226819	+	0.973937i	$0.572833\pi$
$312$	−8.00000	+	8.00000i	−0.452911	+	0.452911i
$313$	14.0000			0.791327			0.395663	−	0.918396i	$-0.370515\pi$
$313$	14.0000			0.791327			0.395663	+	0.918396i	$0.370515\pi$
$314$	8.00000	−	8.00000i	0.451466	−	0.451466i
$315$			4.00000i			0.225374i
$316$			20.0000i			1.12509i
$317$		−	22.0000i		−	1.23564i	−0.786318	−	0.617822i	$-0.788015\pi$
$317$		−	22.0000i		−	1.23564i	0.786318	−	0.617822i	$-0.211985\pi$
$318$	−6.00000	−	6.00000i	−0.336463	−	0.336463i
$319$		0			0
$320$	16.0000			0.894427
$321$	−12.0000			−0.669775
$322$	8.00000	+	8.00000i	0.445823	+	0.445823i
$323$		−	8.00000i		−	0.445132i
$324$			2.00000i			0.111111i
$325$		−	4.00000i		−	0.221880i
$326$	−4.00000	+	4.00000i	−0.221540	+	0.221540i
$327$	4.00000			0.221201
$328$	4.00000	−	4.00000i	0.220863	−	0.220863i
$329$	24.0000			1.32316
$330$		0			0
$331$			20.0000i			1.09930i	0.835395	+	0.549650i	$0.185239\pi$
$331$			20.0000i			1.09930i	−0.835395	+	0.549650i	$0.814761\pi$
$332$	−32.0000			−1.75623
$333$		−	8.00000i		−	0.438397i
$334$	−8.00000	−	8.00000i	−0.437741	−	0.437741i
$335$	24.0000			1.31126
$336$		−	8.00000i		−	0.436436i
$337$	−2.00000			−0.108947			−0.0544735	−	0.998515i	$-0.517348\pi$
$337$	−2.00000			−0.108947			−0.0544735	+	0.998515i	$0.517348\pi$
$338$	3.00000	+	3.00000i	0.163178	+	0.163178i
$339$			6.00000i			0.325875i
$340$	8.00000			0.433861
$341$		0			0
$342$	−4.00000	+	4.00000i	−0.216295	+	0.216295i
$343$	20.0000			1.07990
$344$	−8.00000	−	8.00000i	−0.431331	−	0.431331i
$345$	8.00000			0.430706
$346$	6.00000	−	6.00000i	0.322562	−	0.322562i
$347$			8.00000i			0.429463i	0.976673	+	0.214731i	$0.0688876\pi$
$347$			8.00000i			0.429463i	−0.976673	+	0.214731i	$0.931112\pi$
$348$		−	12.0000i		−	0.643268i
$349$		−	16.0000i		−	0.856460i	−0.903670	−	0.428230i	$-0.859137\pi$
$349$		−	16.0000i		−	0.856460i	0.903670	−	0.428230i	$-0.140863\pi$
$350$	2.00000	+	2.00000i	0.106904	+	0.106904i
$351$	4.00000			0.213504
$352$		0			0
$353$	−6.00000			−0.319348			−0.159674	−	0.987170i	$-0.551044\pi$
$353$	−6.00000			−0.319348			−0.159674	+	0.987170i	$0.551044\pi$
$354$	−4.00000	−	4.00000i	−0.212598	−	0.212598i
$355$			24.0000i			1.27379i
$356$		−	20.0000i		−	1.06000i
$357$		−	4.00000i		−	0.211702i
$358$	4.00000	−	4.00000i	0.211407	−	0.211407i
$359$	−20.0000			−1.05556			−0.527780	−	0.849381i	$-0.676975\pi$
$359$	−20.0000			−1.05556			−0.527780	+	0.849381i	$0.676975\pi$
$360$	−4.00000	−	4.00000i	−0.210819	−	0.210819i
$361$	3.00000			0.157895
$362$	−20.0000	+	20.0000i	−1.05118	+	1.05118i
$363$		−	11.0000i		−	0.577350i
$364$	−16.0000			−0.838628
$365$		−	12.0000i		−	0.628109i
$366$		0			0
$367$	18.0000			0.939592			0.469796	−	0.882775i	$-0.344327\pi$
$367$	18.0000			0.939592			0.469796	+	0.882775i	$0.344327\pi$
$368$	−16.0000			−0.834058
$369$	−2.00000			−0.104116
$370$	16.0000	+	16.0000i	0.831800	+	0.831800i
$371$		−	12.0000i		−	0.623009i
$372$	4.00000			0.207390
$373$			16.0000i			0.828449i	0.910175	+	0.414224i	$0.135947\pi$
$373$			16.0000i			0.828449i	−0.910175	+	0.414224i	$0.864053\pi$
$374$		0			0
$375$	12.0000			0.619677
$376$	−24.0000	+	24.0000i	−1.23771	+	1.23771i
$377$	−24.0000			−1.23606
$378$	−2.00000	+	2.00000i	−0.102869	+	0.102869i
$379$			4.00000i			0.205466i	0.994709	+	0.102733i	$0.0327588\pi$
$379$			4.00000i			0.205466i	−0.994709	+	0.102733i	$0.967241\pi$
$380$		−	16.0000i		−	0.820783i
$381$			2.00000i			0.102463i
$382$	8.00000	+	8.00000i	0.409316	+	0.409316i
$383$	24.0000			1.22634			0.613171	−	0.789950i	$-0.289894\pi$
$383$	24.0000			1.22634			0.613171	+	0.789950i	$0.289894\pi$
$384$	8.00000	+	8.00000i	0.408248	+	0.408248i
$385$		0			0
$386$	6.00000	+	6.00000i	0.305392	+	0.305392i
$387$			4.00000i			0.203331i
$388$		−	4.00000i		−	0.203069i
$389$			34.0000i			1.72387i	0.507020	+	0.861934i	$0.330747\pi$
$389$			34.0000i			1.72387i	−0.507020	+	0.861934i	$0.669253\pi$
$390$	−8.00000	+	8.00000i	−0.405096	+	0.405096i
$391$	−8.00000			−0.404577
$392$	−6.00000	+	6.00000i	−0.303046	+	0.303046i
$393$	−20.0000			−1.00887
$394$	−2.00000	+	2.00000i	−0.100759	+	0.100759i
$395$			20.0000i			1.00631i
$396$		0			0
$397$		−	32.0000i		−	1.60603i	−0.595956	−	0.803017i	$-0.703227\pi$
$397$		−	32.0000i		−	1.60603i	0.595956	−	0.803017i	$-0.296773\pi$
$398$	−10.0000	−	10.0000i	−0.501255	−	0.501255i
$399$	−8.00000			−0.400501
$400$	−4.00000			−0.200000
$401$	−18.0000			−0.898877			−0.449439	−	0.893311i	$-0.648376\pi$
$401$	−18.0000			−0.898877			−0.449439	+	0.893311i	$0.648376\pi$
$402$	12.0000	+	12.0000i	0.598506	+	0.598506i
$403$		−	8.00000i		−	0.398508i
$404$	20.0000			0.995037
$405$			2.00000i			0.0993808i
$406$	12.0000	−	12.0000i	0.595550	−	0.595550i
$407$		0			0
$408$	4.00000	+	4.00000i	0.198030	+	0.198030i
$409$	10.0000			0.494468			0.247234	−	0.968956i	$-0.420478\pi$
$409$	10.0000			0.494468			0.247234	+	0.968956i	$0.420478\pi$
$410$	4.00000	−	4.00000i	0.197546	−	0.197546i
$411$		−	18.0000i		−	0.887875i
$412$		−	12.0000i		−	0.591198i
$413$		−	8.00000i		−	0.393654i
$414$	4.00000	+	4.00000i	0.196589	+	0.196589i
$415$	−32.0000			−1.57082
$416$	16.0000	−	16.0000i	0.784465	−	0.784465i
$417$	4.00000			0.195881
$418$		0			0
$419$			24.0000i			1.17248i	0.810139	+	0.586238i	$0.199392\pi$
$419$			24.0000i			1.17248i	−0.810139	+	0.586238i	$0.800608\pi$
$420$		−	8.00000i		−	0.390360i
$421$		−	20.0000i		−	0.974740i	−0.873195	−	0.487370i	$-0.837956\pi$
$421$		−	20.0000i		−	0.974740i	0.873195	−	0.487370i	$-0.162044\pi$
$422$	20.0000	−	20.0000i	0.973585	−	0.973585i
$423$	12.0000			0.583460
$424$	12.0000	+	12.0000i	0.582772	+	0.582772i
$425$	−2.00000			−0.0970143
$426$	−12.0000	+	12.0000i	−0.581402	+	0.581402i
$427$		0			0
$428$	24.0000			1.16008
$429$		0			0
$430$	−8.00000	−	8.00000i	−0.385794	−	0.385794i
$431$	12.0000			0.578020			0.289010	−	0.957326i	$-0.406674\pi$
$431$	12.0000			0.578020			0.289010	+	0.957326i	$0.406674\pi$
$432$		−	4.00000i		−	0.192450i
$433$	14.0000			0.672797			0.336399	−	0.941720i	$-0.390791\pi$
$433$	14.0000			0.672797			0.336399	+	0.941720i	$0.390791\pi$
$434$	4.00000	+	4.00000i	0.192006	+	0.192006i
$435$		−	12.0000i		−	0.575356i
$436$	−8.00000			−0.383131
$437$			16.0000i			0.765384i
$438$	6.00000	−	6.00000i	0.286691	−	0.286691i
$439$	−30.0000			−1.43182			−0.715911	−	0.698192i	$-0.753988\pi$
$439$	−30.0000			−1.43182			−0.715911	+	0.698192i	$0.753988\pi$
$440$		0			0
$441$	3.00000			0.142857
$442$	8.00000	−	8.00000i	0.380521	−	0.380521i
$443$		−	24.0000i		−	1.14027i	−0.821549	−	0.570137i	$-0.806890\pi$
$443$		−	24.0000i		−	1.14027i	0.821549	−	0.570137i	$-0.193110\pi$
$444$			16.0000i			0.759326i
$445$		−	20.0000i		−	0.948091i
$446$	−14.0000	−	14.0000i	−0.662919	−	0.662919i
$447$	−6.00000			−0.283790
$448$			16.0000i			0.755929i
$449$	30.0000			1.41579			0.707894	−	0.706319i	$-0.249646\pi$
$449$	30.0000			1.41579			0.707894	+	0.706319i	$0.249646\pi$
$450$	1.00000	+	1.00000i	0.0471405	+	0.0471405i
$451$		0			0
$452$		−	12.0000i		−	0.564433i
$453$			18.0000i			0.845714i
$454$	8.00000	−	8.00000i	0.375459	−	0.375459i
$455$	−16.0000			−0.750092
$456$	8.00000	−	8.00000i	0.374634	−	0.374634i
$457$	−22.0000			−1.02912			−0.514558	−	0.857455i	$-0.672044\pi$
$457$	−22.0000			−1.02912			−0.514558	+	0.857455i	$0.672044\pi$
$458$	4.00000	−	4.00000i	0.186908	−	0.186908i
$459$		−	2.00000i		−	0.0933520i
$460$	−16.0000			−0.746004
$461$			30.0000i			1.39724i	0.715493	+	0.698620i	$0.246202\pi$
$461$			30.0000i			1.39724i	−0.715493	+	0.698620i	$0.753798\pi$
$462$		0			0
$463$	−26.0000			−1.20832			−0.604161	−	0.796862i	$-0.706492\pi$
$463$	−26.0000			−1.20832			−0.604161	+	0.796862i	$0.706492\pi$
$464$			24.0000i			1.11417i
$465$	4.00000			0.185496
$466$	−14.0000	−	14.0000i	−0.648537	−	0.648537i
$467$			8.00000i			0.370196i	0.982720	+	0.185098i	$0.0592602\pi$
$467$			8.00000i			0.370196i	−0.982720	+	0.185098i	$0.940740\pi$
$468$	−8.00000			−0.369800
$469$			24.0000i			1.10822i
$470$	−24.0000	+	24.0000i	−1.10704	+	1.10704i
$471$	8.00000			0.368621
$472$	8.00000	+	8.00000i	0.368230	+	0.368230i
$473$		0			0
$474$	−10.0000	+	10.0000i	−0.459315	+	0.459315i
$475$			4.00000i			0.183533i
$476$			8.00000i			0.366679i
$477$		−	6.00000i		−	0.274721i
$478$		0			0
$479$	20.0000			0.913823			0.456912	−	0.889512i	$-0.348956\pi$
$479$	20.0000			0.913823			0.456912	+	0.889512i	$0.348956\pi$
$480$	8.00000	+	8.00000i	0.365148	+	0.365148i
$481$	32.0000			1.45907
$482$	−2.00000	−	2.00000i	−0.0910975	−	0.0910975i
$483$			8.00000i			0.364013i
$484$			22.0000i			1.00000i
$485$		−	4.00000i		−	0.181631i
$486$	−1.00000	+	1.00000i	−0.0453609	+	0.0453609i
$487$	38.0000			1.72194			0.860972	−	0.508652i	$-0.169856\pi$
$487$	38.0000			1.72194			0.860972	+	0.508652i	$0.169856\pi$
$488$		0			0
$489$	−4.00000			−0.180886
$490$	−6.00000	+	6.00000i	−0.271052	+	0.271052i
$491$		−	20.0000i		−	0.902587i	−0.892375	−	0.451294i	$-0.850963\pi$
$491$		−	20.0000i		−	0.902587i	0.892375	−	0.451294i	$-0.149037\pi$
$492$	4.00000			0.180334
$493$			12.0000i			0.540453i
$494$	−16.0000	−	16.0000i	−0.719874	−	0.719874i
$495$		0			0
$496$	−8.00000			−0.359211
$497$	−24.0000			−1.07655
$498$	−16.0000	−	16.0000i	−0.716977	−	0.716977i
$499$		−	36.0000i		−	1.61158i	−0.592200	−	0.805791i	$-0.701741\pi$
$499$		−	36.0000i		−	1.61158i	0.592200	−	0.805791i	$-0.298259\pi$
$500$	−24.0000			−1.07331
$501$		−	8.00000i		−	0.357414i
$502$		0			0
$503$	−36.0000			−1.60516			−0.802580	−	0.596544i	$-0.796540\pi$
$503$	−36.0000			−1.60516			−0.802580	+	0.596544i	$0.796540\pi$
$504$	4.00000	−	4.00000i	0.178174	−	0.178174i
$505$	20.0000			0.889988
$506$		0			0
$507$			3.00000i			0.133235i
$508$		−	4.00000i		−	0.177471i
$509$		−	6.00000i		−	0.265945i	−0.991120	−	0.132973i	$-0.957548\pi$
$509$		−	6.00000i		−	0.265945i	0.991120	−	0.132973i	$-0.0424523\pi$
$510$	4.00000	+	4.00000i	0.177123	+	0.177123i
$511$	12.0000			0.530849
$512$	−16.0000	−	16.0000i	−0.707107	−	0.707107i
$513$	−4.00000			−0.176604
$514$	−18.0000	−	18.0000i	−0.793946	−	0.793946i
$515$		−	12.0000i		−	0.528783i
$516$		−	8.00000i		−	0.352180i
$517$		0			0
$518$	−16.0000	+	16.0000i	−0.703000	+	0.703000i
$519$	6.00000			0.263371
$520$	16.0000	−	16.0000i	0.701646	−	0.701646i
$521$	42.0000			1.84005			0.920027	−	0.391856i	$-0.128167\pi$
$521$	42.0000			1.84005			0.920027	+	0.391856i	$0.128167\pi$
$522$	6.00000	−	6.00000i	0.262613	−	0.262613i
$523$			36.0000i			1.57417i	0.616844	+	0.787085i	$0.288411\pi$
$523$			36.0000i			1.57417i	−0.616844	+	0.787085i	$0.711589\pi$
$524$	40.0000			1.74741
$525$			2.00000i			0.0872872i
$526$	16.0000	+	16.0000i	0.697633	+	0.697633i
$527$	−4.00000			−0.174243
$528$		0			0
$529$	−7.00000			−0.304348
$530$	12.0000	+	12.0000i	0.521247	+	0.521247i
$531$		−	4.00000i		−	0.173585i
$532$	16.0000			0.693688
$533$		−	8.00000i		−	0.346518i
$534$	10.0000	−	10.0000i	0.432742	−	0.432742i
$535$	24.0000			1.03761
$536$	−24.0000	−	24.0000i	−1.03664	−	1.03664i
$537$	4.00000			0.172613
$538$	−6.00000	+	6.00000i	−0.258678	+	0.258678i
$539$		0			0
$540$		−	4.00000i		−	0.172133i
$541$			20.0000i			0.859867i	0.902861	+	0.429934i	$0.141463\pi$
$541$			20.0000i			0.859867i	−0.902861	+	0.429934i	$0.858537\pi$
$542$	18.0000	+	18.0000i	0.773166	+	0.773166i
$543$	−20.0000			−0.858282
$544$	−8.00000	−	8.00000i	−0.342997	−	0.342997i
$545$	−8.00000			−0.342682
$546$	−8.00000	−	8.00000i	−0.342368	−	0.342368i
$547$			28.0000i			1.19719i	0.801050	+	0.598597i	$0.204275\pi$
$547$			28.0000i			1.19719i	−0.801050	+	0.598597i	$0.795725\pi$
$548$			36.0000i			1.53784i
$549$		0			0
$550$		0			0
$551$	24.0000			1.02243
$552$	−8.00000	−	8.00000i	−0.340503	−	0.340503i
$553$	−20.0000			−0.850487
$554$	28.0000	−	28.0000i	1.18961	−	1.18961i
$555$			16.0000i			0.679162i
$556$	−8.00000			−0.339276
$557$		−	42.0000i		−	1.77960i	−0.456354	−	0.889799i	$-0.650845\pi$
$557$		−	42.0000i		−	1.77960i	0.456354	−	0.889799i	$-0.349155\pi$
$558$	2.00000	+	2.00000i	0.0846668	+	0.0846668i
$559$	−16.0000			−0.676728
$560$			16.0000i			0.676123i
$561$		0			0
$562$	18.0000	+	18.0000i	0.759284	+	0.759284i
$563$		−	24.0000i		−	1.01148i	−0.862686	−	0.505740i	$-0.831220\pi$
$563$		−	24.0000i		−	1.01148i	0.862686	−	0.505740i	$-0.168780\pi$
$564$	−24.0000			−1.01058
$565$		−	12.0000i		−	0.504844i
$566$	−4.00000	+	4.00000i	−0.168133	+	0.168133i
$567$	−2.00000			−0.0839921
$568$	24.0000	−	24.0000i	1.00702	−	1.00702i
$569$	−30.0000			−1.25767			−0.628833	−	0.777541i	$-0.716467\pi$
$569$	−30.0000			−1.25767			−0.628833	+	0.777541i	$0.716467\pi$
$570$	8.00000	−	8.00000i	0.335083	−	0.335083i
$571$		−	20.0000i		−	0.836974i	−0.908223	−	0.418487i	$-0.862561\pi$
$571$		−	20.0000i		−	0.836974i	0.908223	−	0.418487i	$-0.137439\pi$
$572$		0			0
$573$			8.00000i			0.334205i
$574$	4.00000	+	4.00000i	0.166957	+	0.166957i
$575$	4.00000			0.166812
$576$			8.00000i			0.333333i
$577$	−42.0000			−1.74848			−0.874241	−	0.485491i	$-0.838641\pi$
$577$	−42.0000			−1.74848			−0.874241	+	0.485491i	$0.838641\pi$
$578$	13.0000	+	13.0000i	0.540729	+	0.540729i
$579$			6.00000i			0.249351i
$580$			24.0000i			0.996546i
$581$		−	32.0000i		−	1.32758i
$582$	2.00000	−	2.00000i	0.0829027	−	0.0829027i
$583$		0			0
$584$	−12.0000	+	12.0000i	−0.496564	+	0.496564i
$585$	−8.00000			−0.330759
$586$	6.00000	−	6.00000i	0.247858	−	0.247858i
$587$			28.0000i			1.15568i	0.816149	+	0.577842i	$0.196105\pi$
$587$			28.0000i			1.15568i	−0.816149	+	0.577842i	$0.803895\pi$
$588$	−6.00000			−0.247436
$589$			8.00000i			0.329634i
$590$	8.00000	+	8.00000i	0.329355	+	0.329355i
$591$	−2.00000			−0.0822690
$592$		−	32.0000i		−	1.31519i
$593$	−6.00000			−0.246390			−0.123195	−	0.992382i	$-0.539314\pi$
$593$	−6.00000			−0.246390			−0.123195	+	0.992382i	$0.539314\pi$
$594$		0			0
$595$			8.00000i			0.327968i
$596$	12.0000			0.491539
$597$		−	10.0000i		−	0.409273i
$598$	−16.0000	+	16.0000i	−0.654289	+	0.654289i
$599$	−20.0000			−0.817178			−0.408589	−	0.912719i	$-0.633979\pi$
$599$	−20.0000			−0.817178			−0.408589	+	0.912719i	$0.633979\pi$
$600$	−2.00000	−	2.00000i	−0.0816497	−	0.0816497i
$601$	2.00000			0.0815817			0.0407909	−	0.999168i	$-0.487012\pi$
$601$	2.00000			0.0815817			0.0407909	+	0.999168i	$0.487012\pi$
$602$	8.00000	−	8.00000i	0.326056	−	0.326056i
$603$			12.0000i			0.488678i
$604$		−	36.0000i		−	1.46482i
$605$			22.0000i			0.894427i
$606$	10.0000	+	10.0000i	0.406222	+	0.406222i
$607$	−2.00000			−0.0811775			−0.0405887	−	0.999176i	$-0.512923\pi$
$607$	−2.00000			−0.0811775			−0.0405887	+	0.999176i	$0.512923\pi$
$608$	−16.0000	+	16.0000i	−0.648886	+	0.648886i
$609$	12.0000			0.486265
$610$		0			0
$611$			48.0000i			1.94187i
$612$			4.00000i			0.161690i
$613$			16.0000i			0.646234i	0.946359	+	0.323117i	$0.104731\pi$
$613$			16.0000i			0.646234i	−0.946359	+	0.323117i	$0.895269\pi$
$614$	−12.0000	+	12.0000i	−0.484281	+	0.484281i
$615$	4.00000			0.161296
$616$		0			0
$617$	−2.00000			−0.0805170			−0.0402585	−	0.999189i	$-0.512818\pi$
$617$	−2.00000			−0.0805170			−0.0402585	+	0.999189i	$0.512818\pi$
$618$	6.00000	−	6.00000i	0.241355	−	0.241355i
$619$		−	36.0000i		−	1.44696i	−0.690344	−	0.723481i	$-0.742541\pi$
$619$		−	36.0000i		−	1.44696i	0.690344	−	0.723481i	$-0.257459\pi$
$620$	−8.00000			−0.321288
$621$			4.00000i			0.160514i
$622$	8.00000	+	8.00000i	0.320771	+	0.320771i
$623$	20.0000			0.801283
$624$	16.0000			0.640513
$625$	−19.0000			−0.760000
$626$	−14.0000	−	14.0000i	−0.559553	−	0.559553i
$627$		0			0
$628$	−16.0000			−0.638470
$629$		−	16.0000i		−	0.637962i
$630$	4.00000	−	4.00000i	0.159364	−	0.159364i
$631$	22.0000			0.875806			0.437903	−	0.899022i	$-0.355721\pi$
$631$	22.0000			0.875806			0.437903	+	0.899022i	$0.355721\pi$
$632$	20.0000	−	20.0000i	0.795557	−	0.795557i
$633$	20.0000			0.794929
$634$	−22.0000	+	22.0000i	−0.873732	+	0.873732i
$635$		−	4.00000i		−	0.158735i
$636$			12.0000i			0.475831i
$637$			12.0000i			0.475457i
$638$		0			0
$639$	−12.0000			−0.474713
$640$	−16.0000	−	16.0000i	−0.632456	−	0.632456i
$641$	22.0000			0.868948			0.434474	−	0.900684i	$-0.356934\pi$
$641$	22.0000			0.868948			0.434474	+	0.900684i	$0.356934\pi$
$642$	12.0000	+	12.0000i	0.473602	+	0.473602i
$643$		−	4.00000i		−	0.157745i	−0.996885	−	0.0788723i	$-0.974868\pi$
$643$		−	4.00000i		−	0.157745i	0.996885	−	0.0788723i	$-0.0251319\pi$
$644$		−	16.0000i		−	0.630488i
$645$		−	8.00000i		−	0.315000i
$646$	−8.00000	+	8.00000i	−0.314756	+	0.314756i
$647$	−12.0000			−0.471769			−0.235884	−	0.971781i	$-0.575799\pi$
$647$	−12.0000			−0.471769			−0.235884	+	0.971781i	$0.575799\pi$
$648$	2.00000	−	2.00000i	0.0785674	−	0.0785674i
$649$		0			0
$650$	−4.00000	+	4.00000i	−0.156893	+	0.156893i
$651$			4.00000i			0.156772i
$652$	8.00000			0.313304
$653$			6.00000i			0.234798i	0.993085	+	0.117399i	$0.0374557\pi$
$653$			6.00000i			0.234798i	−0.993085	+	0.117399i	$0.962544\pi$
$654$	−4.00000	−	4.00000i	−0.156412	−	0.156412i
$655$	40.0000			1.56293
$656$	−8.00000			−0.312348
$657$	6.00000			0.234082
$658$	−24.0000	−	24.0000i	−0.935617	−	0.935617i
$659$		−	36.0000i		−	1.40236i	−0.712984	−	0.701180i	$-0.752657\pi$
$659$		−	36.0000i		−	1.40236i	0.712984	−	0.701180i	$-0.247343\pi$
$660$		0			0
$661$		−	40.0000i		−	1.55582i	−0.628376	−	0.777910i	$-0.716280\pi$
$661$		−	40.0000i		−	1.55582i	0.628376	−	0.777910i	$-0.283720\pi$
$662$	20.0000	−	20.0000i	0.777322	−	0.777322i
$663$	8.00000			0.310694
$664$	32.0000	+	32.0000i	1.24184	+	1.24184i
$665$	16.0000			0.620453
$666$	−8.00000	+	8.00000i	−0.309994	+	0.309994i
$667$		−	24.0000i		−	0.929284i
$668$			16.0000i			0.619059i
$669$		−	14.0000i		−	0.541271i
$670$	−24.0000	−	24.0000i	−0.927201	−	0.927201i
$671$		0			0
$672$	−8.00000	+	8.00000i	−0.308607	+	0.308607i
$673$	14.0000			0.539660			0.269830	−	0.962908i	$-0.413032\pi$
$673$	14.0000			0.539660			0.269830	+	0.962908i	$0.413032\pi$
$674$	2.00000	+	2.00000i	0.0770371	+	0.0770371i
$675$			1.00000i			0.0384900i
$676$		−	6.00000i		−	0.230769i
$677$			18.0000i			0.691796i	0.938272	+	0.345898i	$0.112426\pi$
$677$			18.0000i			0.691796i	−0.938272	+	0.345898i	$0.887574\pi$
$678$	6.00000	−	6.00000i	0.230429	−	0.230429i
$679$	4.00000			0.153506
$680$	−8.00000	−	8.00000i	−0.306786	−	0.306786i
$681$	8.00000			0.306561
$682$		0			0
$683$			16.0000i			0.612223i	0.951996	+	0.306111i	$0.0990280\pi$
$683$			16.0000i			0.612223i	−0.951996	+	0.306111i	$0.900972\pi$
$684$	8.00000			0.305888
$685$			36.0000i			1.37549i
$686$	−20.0000	−	20.0000i	−0.763604	−	0.763604i
$687$	4.00000			0.152610
$688$			16.0000i			0.609994i
$689$	24.0000			0.914327
$690$	−8.00000	−	8.00000i	−0.304555	−	0.304555i
$691$		−	20.0000i		−	0.760836i	−0.924815	−	0.380418i	$-0.875780\pi$
$691$		−	20.0000i		−	0.760836i	0.924815	−	0.380418i	$-0.124220\pi$
$692$	−12.0000			−0.456172
$693$		0			0
$694$	8.00000	−	8.00000i	0.303676	−	0.303676i
$695$	−8.00000			−0.303457
$696$	−12.0000	+	12.0000i	−0.454859	+	0.454859i
$697$	−4.00000			−0.151511
$698$	−16.0000	+	16.0000i	−0.605609	+	0.605609i
$699$		−	14.0000i		−	0.529529i
$700$		−	4.00000i		−	0.151186i
$701$			50.0000i			1.88847i	0.329267	+	0.944237i	$0.393198\pi$
$701$			50.0000i			1.88847i	−0.329267	+	0.944237i	$0.606802\pi$
$702$	−4.00000	−	4.00000i	−0.150970	−	0.150970i
$703$	−32.0000			−1.20690
$704$		0			0
$705$	−24.0000			−0.903892
$706$	6.00000	+	6.00000i	0.225813	+	0.225813i
$707$			20.0000i			0.752177i
$708$			8.00000i			0.300658i
$709$			4.00000i			0.150223i	0.997175	+	0.0751116i	$0.0239313\pi$
$709$			4.00000i			0.150223i	−0.997175	+	0.0751116i	$0.976069\pi$
$710$	24.0000	−	24.0000i	0.900704	−	0.900704i
$711$	−10.0000			−0.375029
$712$	−20.0000	+	20.0000i	−0.749532	+	0.749532i
$713$	8.00000			0.299602
$714$	−4.00000	+	4.00000i	−0.149696	+	0.149696i
$715$		0			0
$716$	−8.00000			−0.298974
$717$		0			0
$718$	20.0000	+	20.0000i	0.746393	+	0.746393i
$719$	−20.0000			−0.745874			−0.372937	−	0.927857i	$-0.621649\pi$
$719$	−20.0000			−0.745874			−0.372937	+	0.927857i	$0.621649\pi$
$720$			8.00000i			0.298142i
$721$	12.0000			0.446903
$722$	−3.00000	−	3.00000i	−0.111648	−	0.111648i
$723$		−	2.00000i		−	0.0743808i
$724$	40.0000			1.48659
$725$		−	6.00000i		−	0.222834i
$726$	−11.0000	+	11.0000i	−0.408248	+	0.408248i
$727$	−42.0000			−1.55769			−0.778847	−	0.627214i	$-0.784195\pi$
$727$	−42.0000			−1.55769			−0.778847	+	0.627214i	$0.784195\pi$
$728$	16.0000	+	16.0000i	0.592999	+	0.592999i
$729$	−1.00000			−0.0370370
$730$	−12.0000	+	12.0000i	−0.444140	+	0.444140i
$731$			8.00000i			0.295891i
$732$		0			0
$733$		−	4.00000i		−	0.147743i	−0.997268	−	0.0738717i	$-0.976464\pi$
$733$		−	4.00000i		−	0.147743i	0.997268	−	0.0738717i	$-0.0235355\pi$
$734$	−18.0000	−	18.0000i	−0.664392	−	0.664392i
$735$	−6.00000			−0.221313
$736$	16.0000	+	16.0000i	0.589768	+	0.589768i
$737$		0			0
$738$	2.00000	+	2.00000i	0.0736210	+	0.0736210i
$739$			44.0000i			1.61857i	0.587419	+	0.809283i	$0.300144\pi$
$739$			44.0000i			1.61857i	−0.587419	+	0.809283i	$0.699856\pi$
$740$		−	32.0000i		−	1.17634i
$741$		−	16.0000i		−	0.587775i
$742$	−12.0000	+	12.0000i	−0.440534	+	0.440534i
$743$	−16.0000			−0.586983			−0.293492	−	0.955962i	$-0.594817\pi$
$743$	−16.0000			−0.586983			−0.293492	+	0.955962i	$0.594817\pi$
$744$	−4.00000	−	4.00000i	−0.146647	−	0.146647i
$745$	12.0000			0.439646
$746$	16.0000	−	16.0000i	0.585802	−	0.585802i
$747$		−	16.0000i		−	0.585409i
$748$		0			0
$749$			24.0000i			0.876941i
$750$	−12.0000	−	12.0000i	−0.438178	−	0.438178i
$751$	2.00000			0.0729810			0.0364905	−	0.999334i	$-0.488382\pi$
$751$	2.00000			0.0729810			0.0364905	+	0.999334i	$0.488382\pi$
$752$	48.0000			1.75038
$753$		0			0
$754$	24.0000	+	24.0000i	0.874028	+	0.874028i
$755$		−	36.0000i		−	1.31017i
$756$	4.00000			0.145479
$757$		−	12.0000i		−	0.436147i	−0.975932	−	0.218074i	$-0.930023\pi$
$757$		−	12.0000i		−	0.436147i	0.975932	−	0.218074i	$-0.0699773\pi$
$758$	4.00000	−	4.00000i	0.145287	−	0.145287i
$759$		0			0
$760$	−16.0000	+	16.0000i	−0.580381	+	0.580381i
$761$	−38.0000			−1.37750			−0.688749	−	0.724999i	$-0.741840\pi$
$761$	−38.0000			−1.37750			−0.688749	+	0.724999i	$0.741840\pi$
$762$	2.00000	−	2.00000i	0.0724524	−	0.0724524i
$763$		−	8.00000i		−	0.289619i
$764$		−	16.0000i		−	0.578860i
$765$			4.00000i			0.144620i
$766$	−24.0000	−	24.0000i	−0.867155	−	0.867155i
$767$	16.0000			0.577727
$768$		−	16.0000i		−	0.577350i
$769$	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
$771$		−	18.0000i		−	0.648254i
$772$		−	12.0000i		−	0.431889i
$773$		−	34.0000i		−	1.22290i	−0.791285	−	0.611448i	$-0.790588\pi$
$773$		−	34.0000i		−	1.22290i	0.791285	−	0.611448i	$-0.209412\pi$
$774$	4.00000	−	4.00000i	0.143777	−	0.143777i
$775$	2.00000			0.0718421
$776$	−4.00000	+	4.00000i	−0.143592	+	0.143592i
$777$	−16.0000			−0.573997
$778$	34.0000	−	34.0000i	1.21896	−	1.21896i
$779$			8.00000i			0.286630i
$780$	16.0000			0.572892
$781$		0			0
$782$	8.00000	+	8.00000i	0.286079	+	0.286079i
$783$	6.00000			0.214423
$784$	12.0000			0.428571
$785$	−16.0000			−0.571064
$786$	20.0000	+	20.0000i	0.713376	+	0.713376i
$787$		−	12.0000i		−	0.427754i	−0.976861	−	0.213877i	$-0.931391\pi$
$787$		−	12.0000i		−	0.427754i	0.976861	−	0.213877i	$-0.0686091\pi$
$788$	4.00000			0.142494
$789$			16.0000i			0.569615i
$790$	20.0000	−	20.0000i	0.711568	−	0.711568i
$791$	12.0000			0.426671
$792$		0			0
$793$		0			0
$794$	−32.0000	+	32.0000i	−1.13564	+	1.13564i
$795$			12.0000i			0.425596i
$796$			20.0000i			0.708881i
$797$		−	2.00000i		−	0.0708436i	−0.999372	−	0.0354218i	$-0.988723\pi$
$797$		−	2.00000i		−	0.0708436i	0.999372	−	0.0354218i	$-0.0112775\pi$
$798$	8.00000	+	8.00000i	0.283197	+	0.283197i
$799$	24.0000			0.849059
$800$	4.00000	+	4.00000i	0.141421	+	0.141421i
$801$	10.0000			0.353333
$802$	18.0000	+	18.0000i	0.635602	+	0.635602i
$803$		0			0
$804$		−	24.0000i		−	0.846415i
$805$		−	16.0000i		−	0.563926i
$806$	−8.00000	+	8.00000i	−0.281788	+	0.281788i
$807$	−6.00000			−0.211210
$808$	−20.0000	−	20.0000i	−0.703598	−	0.703598i
$809$	−30.0000			−1.05474			−0.527372	−	0.849635i	$-0.676823\pi$
$809$	−30.0000			−1.05474			−0.527372	+	0.849635i	$0.676823\pi$
$810$	2.00000	−	2.00000i	0.0702728	−	0.0702728i
$811$			20.0000i			0.702295i	0.936320	+	0.351147i	$0.114208\pi$
$811$			20.0000i			0.702295i	−0.936320	+	0.351147i	$0.885792\pi$
$812$	−24.0000			−0.842235
$813$			18.0000i			0.631288i
$814$		0			0
$815$	8.00000			0.280228
$816$		−	8.00000i		−	0.280056i
$817$	16.0000			0.559769
$818$	−10.0000	−	10.0000i	−0.349642	−	0.349642i
$819$		−	8.00000i		−	0.279543i
$820$	−8.00000			−0.279372
$821$		−	10.0000i		−	0.349002i	−0.984657	−	0.174501i	$-0.944169\pi$
$821$		−	10.0000i		−	0.349002i	0.984657	−	0.174501i	$-0.0558313\pi$
$822$	−18.0000	+	18.0000i	−0.627822	+	0.627822i
$823$	14.0000			0.488009			0.244005	−	0.969774i	$-0.421539\pi$
$823$	14.0000			0.488009			0.244005	+	0.969774i	$0.421539\pi$
$824$	−12.0000	+	12.0000i	−0.418040	+	0.418040i
$825$		0			0
$826$	−8.00000	+	8.00000i	−0.278356	+	0.278356i
$827$		−	12.0000i		−	0.417281i	−0.977992	−	0.208640i	$-0.933096\pi$
$827$		−	12.0000i		−	0.417281i	0.977992	−	0.208640i	$-0.0669038\pi$
$828$		−	8.00000i		−	0.278019i
$829$			4.00000i			0.138926i	0.997585	+	0.0694629i	$0.0221285\pi$
$829$			4.00000i			0.138926i	−0.997585	+	0.0694629i	$0.977871\pi$
$830$	32.0000	+	32.0000i	1.11074	+	1.11074i
$831$	28.0000			0.971309
$832$	−32.0000			−1.10940
$833$	6.00000			0.207888
$834$	−4.00000	−	4.00000i	−0.138509	−	0.138509i
$835$			16.0000i			0.553703i
$836$		0			0
$837$			2.00000i			0.0691301i
$838$	24.0000	−	24.0000i	0.829066	−	0.829066i
$839$	20.0000			0.690477			0.345238	−	0.938515i	$-0.387798\pi$
$839$	20.0000			0.690477			0.345238	+	0.938515i	$0.387798\pi$
$840$	−8.00000	+	8.00000i	−0.276026	+	0.276026i
$841$	−7.00000			−0.241379
$842$	−20.0000	+	20.0000i	−0.689246	+	0.689246i
$843$			18.0000i			0.619953i
$844$	−40.0000			−1.37686
$845$		−	6.00000i		−	0.206406i
$846$	−12.0000	−	12.0000i	−0.412568	−	0.412568i
$847$	−22.0000			−0.755929
$848$		−	24.0000i		−	0.824163i
$849$	−4.00000			−0.137280
$850$	2.00000	+	2.00000i	0.0685994	+	0.0685994i
$851$			32.0000i			1.09695i
$852$	24.0000			0.822226
$853$		−	24.0000i		−	0.821744i	−0.911693	−	0.410872i	$-0.865224\pi$
$853$		−	24.0000i		−	0.821744i	0.911693	−	0.410872i	$-0.134776\pi$
$854$		0			0
$855$	8.00000			0.273594
$856$	−24.0000	−	24.0000i	−0.820303	−	0.820303i
$857$	18.0000			0.614868			0.307434	−	0.951569i	$-0.400530\pi$
$857$	18.0000			0.614868			0.307434	+	0.951569i	$0.400530\pi$
$858$		0			0
$859$			44.0000i			1.50126i	0.660722	+	0.750630i	$0.270250\pi$
$859$			44.0000i			1.50126i	−0.660722	+	0.750630i	$0.729750\pi$
$860$			16.0000i			0.545595i
$861$			4.00000i			0.136320i
$862$	−12.0000	−	12.0000i	−0.408722	−	0.408722i
$863$	24.0000			0.816970			0.408485	−	0.912765i	$-0.366057\pi$
$863$	24.0000			0.816970			0.408485	+	0.912765i	$0.366057\pi$
$864$	−4.00000	+	4.00000i	−0.136083	+	0.136083i
$865$	−12.0000			−0.408012
$866$	−14.0000	−	14.0000i	−0.475739	−	0.475739i
$867$			13.0000i			0.441503i
$868$		−	8.00000i		−	0.271538i
$869$		0			0
$870$	−12.0000	+	12.0000i	−0.406838	+	0.406838i
$871$	−48.0000			−1.62642
$872$	8.00000	+	8.00000i	0.270914	+	0.270914i
$873$	2.00000			0.0676897
$874$	16.0000	−	16.0000i	0.541208	−	0.541208i
$875$		−	24.0000i		−	0.811348i
$876$	−12.0000			−0.405442
$877$		−	32.0000i		−	1.08056i	−0.841484	−	0.540282i	$-0.818318\pi$
$877$		−	32.0000i		−	1.08056i	0.841484	−	0.540282i	$-0.181682\pi$
$878$	30.0000	+	30.0000i	1.01245	+	1.01245i
$879$	6.00000			0.202375
$880$		0			0
$881$	2.00000			0.0673817			0.0336909	−	0.999432i	$-0.489274\pi$
$881$	2.00000			0.0673817			0.0336909	+	0.999432i	$0.489274\pi$
$882$	−3.00000	−	3.00000i	−0.101015	−	0.101015i
$883$		−	4.00000i		−	0.134611i	−0.997732	−	0.0673054i	$-0.978560\pi$
$883$		−	4.00000i		−	0.134611i	0.997732	−	0.0673054i	$-0.0214402\pi$
$884$	−16.0000			−0.538138
$885$			8.00000i			0.268917i
$886$	−24.0000	+	24.0000i	−0.806296	+	0.806296i
$887$	48.0000			1.61168			0.805841	−	0.592132i	$-0.201714\pi$
$887$	48.0000			1.61168			0.805841	+	0.592132i	$0.201714\pi$
$888$	16.0000	−	16.0000i	0.536925	−	0.536925i
$889$	4.00000			0.134156
$890$	−20.0000	+	20.0000i	−0.670402	+	0.670402i
$891$		0			0
$892$			28.0000i			0.937509i
$893$		−	48.0000i		−	1.60626i
$894$	6.00000	+	6.00000i	0.200670	+	0.200670i
$895$	−8.00000			−0.267411
$896$	16.0000	−	16.0000i	0.534522	−	0.534522i
$897$	−16.0000			−0.534224
$898$	−30.0000	−	30.0000i	−1.00111	−	1.00111i
$899$		−	12.0000i		−	0.400222i
$900$		−	2.00000i		−	0.0666667i
$901$		−	12.0000i		−	0.399778i
$902$		0			0
$903$	8.00000			0.266223
$904$	−12.0000	+	12.0000i	−0.399114	+	0.399114i
$905$	40.0000			1.32964
$906$	18.0000	−	18.0000i	0.598010	−	0.598010i
$907$			28.0000i			0.929725i	0.885383	+	0.464862i	$0.153896\pi$
$907$			28.0000i			0.929725i	−0.885383	+	0.464862i	$0.846104\pi$
$908$	−16.0000			−0.530979
$909$			10.0000i			0.331679i
$910$	16.0000	+	16.0000i	0.530395	+	0.530395i
$911$	−48.0000			−1.59031			−0.795155	−	0.606406i	$-0.792611\pi$
$911$	−48.0000			−1.59031			−0.795155	+	0.606406i	$0.792611\pi$
$912$	−16.0000			−0.529813
$913$		0			0
$914$	22.0000	+	22.0000i	0.727695	+	0.727695i
$915$		0			0
$916$	−8.00000			−0.264327
$917$			40.0000i			1.32092i
$918$	−2.00000	+	2.00000i	−0.0660098	+	0.0660098i
$919$	30.0000			0.989609			0.494804	−	0.869004i	$-0.335240\pi$
$919$	30.0000			0.989609			0.494804	+	0.869004i	$0.335240\pi$
$920$	16.0000	+	16.0000i	0.527504	+	0.527504i
$921$	−12.0000			−0.395413
$922$	30.0000	−	30.0000i	0.987997	−	0.987997i
$923$		−	48.0000i		−	1.57994i
$924$		0			0
$925$			8.00000i			0.263038i
$926$	26.0000	+	26.0000i	0.854413	+	0.854413i
$927$	6.00000			0.197066
$928$	24.0000	−	24.0000i	0.787839	−	0.787839i
$929$	−50.0000			−1.64045			−0.820223	−	0.572043i	$-0.806151\pi$
$929$	−50.0000			−1.64045			−0.820223	+	0.572043i	$0.806151\pi$
$930$	−4.00000	−	4.00000i	−0.131165	−	0.131165i
$931$		−	12.0000i		−	0.393284i
$932$			28.0000i			0.917170i
$933$			8.00000i			0.261908i
$934$	8.00000	−	8.00000i	0.261768	−	0.261768i
$935$		0			0
$936$	8.00000	+	8.00000i	0.261488	+	0.261488i
$937$	38.0000			1.24141			0.620703	−	0.784046i	$-0.286847\pi$
$937$	38.0000			1.24141			0.620703	+	0.784046i	$0.286847\pi$
$938$	24.0000	−	24.0000i	0.783628	−	0.783628i
$939$		−	14.0000i		−	0.456873i
$940$	48.0000			1.56559
$941$			30.0000i			0.977972i	0.872292	+	0.488986i	$0.162633\pi$
$941$			30.0000i			0.977972i	−0.872292	+	0.488986i	$0.837367\pi$
$942$	−8.00000	−	8.00000i	−0.260654	−	0.260654i
$943$	8.00000			0.260516
$944$		−	16.0000i		−	0.520756i
$945$	4.00000			0.130120
$946$		0			0
$947$		−	12.0000i		−	0.389948i	−0.980808	−	0.194974i	$-0.937538\pi$
$947$		−	12.0000i		−	0.389948i	0.980808	−	0.194974i	$-0.0624622\pi$
$948$	20.0000			0.649570
$949$			24.0000i			0.779073i
$950$	4.00000	−	4.00000i	0.129777	−	0.129777i
$951$	−22.0000			−0.713399
$952$	8.00000	−	8.00000i	0.259281	−	0.259281i
$953$	−6.00000			−0.194359			−0.0971795	−	0.995267i	$-0.530982\pi$
$953$	−6.00000			−0.194359			−0.0971795	+	0.995267i	$0.530982\pi$
$954$	−6.00000	+	6.00000i	−0.194257	+	0.194257i
$955$		−	16.0000i		−	0.517748i
$956$		0			0
$957$		0			0
$958$	−20.0000	−	20.0000i	−0.646171	−	0.646171i
$959$	−36.0000			−1.16250
$960$		−	16.0000i		−	0.516398i
$961$	−27.0000			−0.870968
$962$	−32.0000	−	32.0000i	−1.03172	−	1.03172i
$963$			12.0000i			0.386695i
$964$			4.00000i			0.128831i
$965$		−	12.0000i		−	0.386294i
$966$	8.00000	−	8.00000i	0.257396	−	0.257396i
$967$	−22.0000			−0.707472			−0.353736	−	0.935345i	$-0.615089\pi$
$967$	−22.0000			−0.707472			−0.353736	+	0.935345i	$0.615089\pi$
$968$	22.0000	−	22.0000i	0.707107	−	0.707107i
$969$	−8.00000			−0.256997
$970$	−4.00000	+	4.00000i	−0.128432	+	0.128432i
$971$		0			0		1.00000			$0$
$971$		0			0		−1.00000			$\pi$
$972$	2.00000			0.0641500
$973$		−	8.00000i		−	0.256468i
$974$	−38.0000	−	38.0000i	−1.21760	−	1.21760i
$975$	−4.00000			−0.128103
$976$		0			0
$977$	−2.00000			−0.0639857			−0.0319928	−	0.999488i	$-0.510185\pi$
$977$	−2.00000			−0.0639857			−0.0319928	+	0.999488i	$0.510185\pi$
$978$	4.00000	+	4.00000i	0.127906	+	0.127906i
$979$		0			0
$980$	12.0000			0.383326
$981$		−	4.00000i		−	0.127710i
$982$	−20.0000	+	20.0000i	−0.638226	+	0.638226i
$983$	−16.0000			−0.510321			−0.255160	−	0.966899i	$-0.582128\pi$
$983$	−16.0000			−0.510321			−0.255160	+	0.966899i	$0.582128\pi$
$984$	−4.00000	−	4.00000i	−0.127515	−	0.127515i
$985$	4.00000			0.127451
$986$	12.0000	−	12.0000i	0.382158	−	0.382158i
$987$		−	24.0000i		−	0.763928i
$988$			32.0000i			1.01806i
$989$		−	16.0000i		−	0.508770i
$990$		0			0
$991$	2.00000			0.0635321			0.0317660	−	0.999495i	$-0.489887\pi$
$991$	2.00000			0.0635321			0.0317660	+	0.999495i	$0.489887\pi$
$992$	8.00000	+	8.00000i	0.254000	+	0.254000i
$993$	20.0000			0.634681
$994$	24.0000	+	24.0000i	0.761234	+	0.761234i
$995$			20.0000i			0.634043i
$996$			32.0000i			1.01396i
$997$			48.0000i			1.52018i	0.649821	+	0.760088i	$0.274844\pi$
$997$			48.0000i			1.52018i	−0.649821	+	0.760088i	$0.725156\pi$
$998$	−36.0000	+	36.0000i	−1.13956	+	1.13956i
$999$	−8.00000			−0.253109

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	24.2.d.a.13.1	✓	2
3.2	odd	2		72.2.d.b.37.2		2
4.3	odd	2		96.2.d.a.49.2		2
5.2	odd	4		600.2.d.c.349.1		2
5.3	odd	4		600.2.d.b.349.2		2
5.4	even	2		600.2.k.b.301.2		2
7.6	odd	2		1176.2.c.a.589.1		2
8.3	odd	2		96.2.d.a.49.1		2
8.5	even	2	inner	24.2.d.a.13.2	yes	2
9.2	odd	6		648.2.n.c.109.2		4
9.4	even	3		648.2.n.k.541.2		4
9.5	odd	6		648.2.n.c.541.1		4
9.7	even	3		648.2.n.k.109.1		4
12.11	even	2		288.2.d.b.145.1		2
15.2	even	4		1800.2.d.b.1549.2		2
15.8	even	4		1800.2.d.i.1549.1		2
15.14	odd	2		1800.2.k.a.901.1		2
16.3	odd	4		768.2.a.d.1.1		1
16.5	even	4		768.2.a.a.1.1		1
16.11	odd	4		768.2.a.e.1.1		1
16.13	even	4		768.2.a.h.1.1		1
20.3	even	4		2400.2.d.b.49.1		2
20.7	even	4		2400.2.d.c.49.2		2
20.19	odd	2		2400.2.k.a.1201.1		2
24.5	odd	2		72.2.d.b.37.1		2
24.11	even	2		288.2.d.b.145.2		2
28.27	even	2		4704.2.c.a.2353.1		2
36.7	odd	6		2592.2.r.f.433.1		4
36.11	even	6		2592.2.r.g.433.2		4
36.23	even	6		2592.2.r.g.2161.1		4
36.31	odd	6		2592.2.r.f.2161.2		4
40.3	even	4		2400.2.d.c.49.1		2
40.13	odd	4		600.2.d.c.349.2		2
40.19	odd	2		2400.2.k.a.1201.2		2
40.27	even	4		2400.2.d.b.49.2		2
40.29	even	2		600.2.k.b.301.1		2
40.37	odd	4		600.2.d.b.349.1		2
48.5	odd	4		2304.2.a.o.1.1		1
48.11	even	4		2304.2.a.l.1.1		1
48.29	odd	4		2304.2.a.e.1.1		1
48.35	even	4		2304.2.a.b.1.1		1
56.13	odd	2		1176.2.c.a.589.2		2
56.27	even	2		4704.2.c.a.2353.2		2
60.23	odd	4		7200.2.d.g.2449.1		2
60.47	odd	4		7200.2.d.d.2449.2		2
60.59	even	2		7200.2.k.d.3601.2		2
72.5	odd	6		648.2.n.c.541.2		4
72.11	even	6		2592.2.r.g.433.1		4
72.13	even	6		648.2.n.k.541.1		4
72.29	odd	6		648.2.n.c.109.1		4
72.43	odd	6		2592.2.r.f.433.2		4
72.59	even	6		2592.2.r.g.2161.2		4
72.61	even	6		648.2.n.k.109.2		4
72.67	odd	6		2592.2.r.f.2161.1		4
120.29	odd	2		1800.2.k.a.901.2		2
120.53	even	4		1800.2.d.b.1549.1		2
120.59	even	2		7200.2.k.d.3601.1		2
120.77	even	4		1800.2.d.i.1549.2		2
120.83	odd	4		7200.2.d.d.2449.1		2
120.107	odd	4		7200.2.d.g.2449.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
24.2.d.a.13.1	✓	2	1.1	even	1	trivial
24.2.d.a.13.2	yes	2	8.5	even	2	inner
72.2.d.b.37.1		2	24.5	odd	2
72.2.d.b.37.2		2	3.2	odd	2
96.2.d.a.49.1		2	8.3	odd	2
96.2.d.a.49.2		2	4.3	odd	2
288.2.d.b.145.1		2	12.11	even	2
288.2.d.b.145.2		2	24.11	even	2
600.2.d.b.349.1		2	40.37	odd	4
600.2.d.b.349.2		2	5.3	odd	4
600.2.d.c.349.1		2	5.2	odd	4
600.2.d.c.349.2		2	40.13	odd	4
600.2.k.b.301.1		2	40.29	even	2
600.2.k.b.301.2		2	5.4	even	2
648.2.n.c.109.1		4	72.29	odd	6
648.2.n.c.109.2		4	9.2	odd	6
648.2.n.c.541.1		4	9.5	odd	6
648.2.n.c.541.2		4	72.5	odd	6
648.2.n.k.109.1		4	9.7	even	3
648.2.n.k.109.2		4	72.61	even	6
648.2.n.k.541.1		4	72.13	even	6
648.2.n.k.541.2		4	9.4	even	3
768.2.a.a.1.1		1	16.5	even	4
768.2.a.d.1.1		1	16.3	odd	4
768.2.a.e.1.1		1	16.11	odd	4
768.2.a.h.1.1		1	16.13	even	4
1176.2.c.a.589.1		2	7.6	odd	2
1176.2.c.a.589.2		2	56.13	odd	2
1800.2.d.b.1549.1		2	120.53	even	4
1800.2.d.b.1549.2		2	15.2	even	4
1800.2.d.i.1549.1		2	15.8	even	4
1800.2.d.i.1549.2		2	120.77	even	4
1800.2.k.a.901.1		2	15.14	odd	2
1800.2.k.a.901.2		2	120.29	odd	2
2304.2.a.b.1.1		1	48.35	even	4
2304.2.a.e.1.1		1	48.29	odd	4
2304.2.a.l.1.1		1	48.11	even	4
2304.2.a.o.1.1		1	48.5	odd	4
2400.2.d.b.49.1		2	20.3	even	4
2400.2.d.b.49.2		2	40.27	even	4
2400.2.d.c.49.1		2	40.3	even	4
2400.2.d.c.49.2		2	20.7	even	4
2400.2.k.a.1201.1		2	20.19	odd	2
2400.2.k.a.1201.2		2	40.19	odd	2
2592.2.r.f.433.1		4	36.7	odd	6
2592.2.r.f.433.2		4	72.43	odd	6
2592.2.r.f.2161.1		4	72.67	odd	6
2592.2.r.f.2161.2		4	36.31	odd	6
2592.2.r.g.433.1		4	72.11	even	6
2592.2.r.g.433.2		4	36.11	even	6
2592.2.r.g.2161.1		4	36.23	even	6
2592.2.r.g.2161.2		4	72.59	even	6
4704.2.c.a.2353.1		2	28.27	even	2
4704.2.c.a.2353.2		2	56.27	even	2
7200.2.d.d.2449.1		2	120.83	odd	4
7200.2.d.d.2449.2		2	60.47	odd	4
7200.2.d.g.2449.1		2	60.23	odd	4
7200.2.d.g.2449.2		2	120.107	odd	4
7200.2.k.d.3601.1		2	120.59	even	2
7200.2.k.d.3601.2		2	60.59	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:	\( N \)	\(=\)	\( 24 = 2^{3} \cdot 3 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	24.d (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+(-1.00000 - 1.00000i) q^{2} -1.00000i q^{3} +2.00000i q^{4} +2.00000i q^{5} +(-1.00000 + 1.00000i) q^{6} -2.00000 q^{7} +(2.00000 - 2.00000i) q^{8} -1.00000 q^{9} +(2.00000 - 2.00000i) q^{10} +2.00000 q^{12} -4.00000i q^{13} +(2.00000 + 2.00000i) q^{14} +2.00000 q^{15} -4.00000 q^{16} -2.00000 q^{17} +(1.00000 + 1.00000i) q^{18} +4.00000i q^{19} -4.00000 q^{20} +2.00000i q^{21} +4.00000 q^{23} +(-2.00000 - 2.00000i) q^{24} +1.00000 q^{25} +(-4.00000 + 4.00000i) q^{26} +1.00000i q^{27} -4.00000i q^{28} -6.00000i q^{29} +(-2.00000 - 2.00000i) q^{30} +2.00000 q^{31} +(4.00000 + 4.00000i) q^{32} +(2.00000 + 2.00000i) q^{34} -4.00000i q^{35} -2.00000i q^{36} +8.00000i q^{37} +(4.00000 - 4.00000i) q^{38} -4.00000 q^{39} +(4.00000 + 4.00000i) q^{40} +2.00000 q^{41} +(2.00000 - 2.00000i) q^{42} -4.00000i q^{43} -2.00000i q^{45} +(-4.00000 - 4.00000i) q^{46} -12.0000 q^{47} +4.00000i q^{48} -3.00000 q^{49} +(-1.00000 - 1.00000i) q^{50} +2.00000i q^{51} +8.00000 q^{52} +6.00000i q^{53} +(1.00000 - 1.00000i) q^{54} +(-4.00000 + 4.00000i) q^{56} +4.00000 q^{57} +(-6.00000 + 6.00000i) q^{58} +4.00000i q^{59} +4.00000i q^{60} +(-2.00000 - 2.00000i) q^{62} +2.00000 q^{63} -8.00000i q^{64} +8.00000 q^{65} -12.0000i q^{67} -4.00000i q^{68} -4.00000i q^{69} +(-4.00000 + 4.00000i) q^{70} +12.0000 q^{71} +(-2.00000 + 2.00000i) q^{72} -6.00000 q^{73} +(8.00000 - 8.00000i) q^{74} -1.00000i q^{75} -8.00000 q^{76} +(4.00000 + 4.00000i) q^{78} +10.0000 q^{79} -8.00000i q^{80} +1.00000 q^{81} +(-2.00000 - 2.00000i) q^{82} +16.0000i q^{83} -4.00000 q^{84} -4.00000i q^{85} +(-4.00000 + 4.00000i) q^{86} -6.00000 q^{87} -10.0000 q^{89} +(-2.00000 + 2.00000i) q^{90} +8.00000i q^{91} +8.00000i q^{92} -2.00000i q^{93} +(12.0000 + 12.0000i) q^{94} -8.00000 q^{95} +(4.00000 - 4.00000i) q^{96} -2.00000 q^{97} +(3.00000 + 3.00000i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{2} - 2 q^{6} - 4 q^{7} + 4 q^{8} - 2 q^{9} + 4 q^{10} + 4 q^{12} + 4 q^{14} + 4 q^{15} - 8 q^{16} - 4 q^{17} + 2 q^{18} - 8 q^{20} + 8 q^{23} - 4 q^{24} + 2 q^{25} - 8 q^{26} - 4 q^{30} + 4 q^{31}+ \cdots + 6 q^{98}+O(q^{100}) \) 2 * q - 2 * q^2 - 2 * q^6 - 4 * q^7 + 4 * q^8 - 2 * q^9 + 4 * q^10 + 4 * q^12 + 4 * q^14 + 4 * q^15 - 8 * q^16 - 4 * q^17 + 2 * q^18 - 8 * q^20 + 8 * q^23 - 4 * q^24 + 2 * q^25 - 8 * q^26 - 4 * q^30 + 4 * q^31 + 8 * q^32 + 4 * q^34 + 8 * q^38 - 8 * q^39 + 8 * q^40 + 4 * q^41 + 4 * q^42 - 8 * q^46 - 24 * q^47 - 6 * q^49 - 2 * q^50 + 16 * q^52 + 2 * q^54 - 8 * q^56 + 8 * q^57 - 12 * q^58 - 4 * q^62 + 4 * q^63 + 16 * q^65 - 8 * q^70 + 24 * q^71 - 4 * q^72 - 12 * q^73 + 16 * q^74 - 16 * q^76 + 8 * q^78 + 20 * q^79 + 2 * q^81 - 4 * q^82 - 8 * q^84 - 8 * q^86 - 12 * q^87 - 20 * q^89 - 4 * q^90 + 24 * q^94 - 16 * q^95 + 8 * q^96 - 4 * q^97 + 6 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

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