LMFDB - Embedded newform 117.2.l.a.88.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [117,2,Mod(4,117)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(117, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([2, 1]))

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

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

chi := DirichletCharacter("117.4");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 117 = 3^{2} \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	117.l (of order $6$, degree $2$, minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	no
Analytic conductor:	$0.934249703649$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{6})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x + 1 $ x^2 - x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			88.1
Root			$0.500000 - 0.866025i$ of defining polynomial
Character	$\chi$	$=$	117.88
Dual form			117.2.l.a.4.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q+1.73205i q^{2} -1.73205i q^{3} -1.00000 q^{4} +(1.50000 - 0.866025i) q^{5} +3.00000 q^{6} +(1.50000 - 0.866025i) q^{7} +1.73205i q^{8} -3.00000 q^{9} +(1.50000 + 2.59808i) q^{10} +3.46410i q^{11} +1.73205i q^{12} +(-1.00000 - 3.46410i) q^{13} +(1.50000 + 2.59808i) q^{14} +(-1.50000 - 2.59808i) q^{15} -5.00000 q^{16} +(-1.50000 + 2.59808i) q^{17} -5.19615i q^{18} +(-1.50000 - 0.866025i) q^{19} +(-1.50000 + 0.866025i) q^{20} +(-1.50000 - 2.59808i) q^{21} -6.00000 q^{22} +(1.50000 - 2.59808i) q^{23} +3.00000 q^{24} +(-1.00000 + 1.73205i) q^{25} +(6.00000 - 1.73205i) q^{26} +5.19615i q^{27} +(-1.50000 + 0.866025i) q^{28} -6.00000 q^{29} +(4.50000 - 2.59808i) q^{30} +(7.50000 - 4.33013i) q^{31} -5.19615i q^{32} +6.00000 q^{33} +(-4.50000 - 2.59808i) q^{34} +(1.50000 - 2.59808i) q^{35} +3.00000 q^{36} +(-4.50000 + 2.59808i) q^{37} +(1.50000 - 2.59808i) q^{38} +(-6.00000 + 1.73205i) q^{39} +(1.50000 + 2.59808i) q^{40} +(-10.5000 - 6.06218i) q^{41} +(4.50000 - 2.59808i) q^{42} +(0.500000 + 0.866025i) q^{43} -3.46410i q^{44} +(-4.50000 + 2.59808i) q^{45} +(4.50000 + 2.59808i) q^{46} +(4.50000 + 2.59808i) q^{47} +8.66025i q^{48} +(-2.00000 + 3.46410i) q^{49} +(-3.00000 - 1.73205i) q^{50} +(4.50000 + 2.59808i) q^{51} +(1.00000 + 3.46410i) q^{52} +6.00000 q^{53} -9.00000 q^{54} +(3.00000 + 5.19615i) q^{55} +(1.50000 + 2.59808i) q^{56} +(-1.50000 + 2.59808i) q^{57} -10.3923i q^{58} +3.46410i q^{59} +(1.50000 + 2.59808i) q^{60} +(2.50000 + 4.33013i) q^{61} +(7.50000 + 12.9904i) q^{62} +(-4.50000 + 2.59808i) q^{63} -1.00000 q^{64} +(-4.50000 - 4.33013i) q^{65} +10.3923i q^{66} +(10.5000 + 6.06218i) q^{67} +(1.50000 - 2.59808i) q^{68} +(-4.50000 - 2.59808i) q^{69} +(4.50000 + 2.59808i) q^{70} +(-7.50000 - 4.33013i) q^{71} -5.19615i q^{72} -6.92820i q^{73} +(-4.50000 - 7.79423i) q^{74} +(3.00000 + 1.73205i) q^{75} +(1.50000 + 0.866025i) q^{76} +(3.00000 + 5.19615i) q^{77} +(-3.00000 - 10.3923i) q^{78} +(5.50000 - 9.52628i) q^{79} +(-7.50000 + 4.33013i) q^{80} +9.00000 q^{81} +(10.5000 - 18.1865i) q^{82} +(4.50000 + 2.59808i) q^{83} +(1.50000 + 2.59808i) q^{84} +5.19615i q^{85} +(-1.50000 + 0.866025i) q^{86} +10.3923i q^{87} -6.00000 q^{88} +(13.5000 - 7.79423i) q^{89} +(-4.50000 - 7.79423i) q^{90} +(-4.50000 - 4.33013i) q^{91} +(-1.50000 + 2.59808i) q^{92} +(-7.50000 - 12.9904i) q^{93} +(-4.50000 + 7.79423i) q^{94} -3.00000 q^{95} -9.00000 q^{96} +(13.5000 - 7.79423i) q^{97} +(-6.00000 - 3.46410i) q^{98} -10.3923i q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{4} + 3 q^{5} + 6 q^{6} + 3 q^{7} - 6 q^{9} + 3 q^{10} - 2 q^{13} + 3 q^{14} - 3 q^{15} - 10 q^{16} - 3 q^{17} - 3 q^{19} - 3 q^{20} - 3 q^{21} - 12 q^{22} + 3 q^{23} + 6 q^{24} - 2 q^{25} + 12 q^{26}+ \cdots - 12 q^{98}+O(q^{100}) $ 2 * q - 2 * q^4 + 3 * q^5 + 6 * q^6 + 3 * q^7 - 6 * q^9 + 3 * q^10 - 2 * q^13 + 3 * q^14 - 3 * q^15 - 10 * q^16 - 3 * q^17 - 3 * q^19 - 3 * q^20 - 3 * q^21 - 12 * q^22 + 3 * q^23 + 6 * q^24 - 2 * q^25 + 12 * q^26 - 3 * q^28 - 12 * q^29 + 9 * q^30 + 15 * q^31 + 12 * q^33 - 9 * q^34 + 3 * q^35 + 6 * q^36 - 9 * q^37 + 3 * q^38 - 12 * q^39 + 3 * q^40 - 21 * q^41 + 9 * q^42 + q^43 - 9 * q^45 + 9 * q^46 + 9 * q^47 - 4 * q^49 - 6 * q^50 + 9 * q^51 + 2 * q^52 + 12 * q^53 - 18 * q^54 + 6 * q^55 + 3 * q^56 - 3 * q^57 + 3 * q^60 + 5 * q^61 + 15 * q^62 - 9 * q^63 - 2 * q^64 - 9 * q^65 + 21 * q^67 + 3 * q^68 - 9 * q^69 + 9 * q^70 - 15 * q^71 - 9 * q^74 + 6 * q^75 + 3 * q^76 + 6 * q^77 - 6 * q^78 + 11 * q^79 - 15 * q^80 + 18 * q^81 + 21 * q^82 + 9 * q^83 + 3 * q^84 - 3 * q^86 - 12 * q^88 + 27 * q^89 - 9 * q^90 - 9 * q^91 - 3 * q^92 - 15 * q^93 - 9 * q^94 - 6 * q^95 - 18 * q^96 + 27 * q^97 - 12 * q^98

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/117\mathbb{Z}\right)^\times$.

$n$	$28$	$92$
$\chi(n)$	$e\left(\frac{5}{6}\right)$	$e\left(\frac{2}{3}\right)$

Coefficient data

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

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

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

$2$			1.73205i			1.22474i	0.790569	+	0.612372i	$0.209785\pi$
$2$			1.73205i			1.22474i	−0.790569	+	0.612372i	$0.790215\pi$
$3$		−	1.73205i		−	1.00000i
$3$		−	1.73205i		−	1.00000i
$4$	−1.00000			−0.500000
$5$	1.50000	−	0.866025i	0.670820	−	0.387298i	−0.125567	−	0.992085i	$-0.540075\pi$
$5$	1.50000	−	0.866025i	0.670820	−	0.387298i	0.796387	+	0.604787i	$0.206742\pi$
$6$	3.00000			1.22474
$7$	1.50000	−	0.866025i	0.566947	−	0.327327i	−0.188982	−	0.981981i	$-0.560519\pi$
$7$	1.50000	−	0.866025i	0.566947	−	0.327327i	0.755929	+	0.654654i	$0.227186\pi$
$8$			1.73205i			0.612372i
$9$	−3.00000			−1.00000
$10$	1.50000	+	2.59808i	0.474342	+	0.821584i
$11$			3.46410i			1.04447i	0.852803	+	0.522233i	$0.174901\pi$
$11$			3.46410i			1.04447i	−0.852803	+	0.522233i	$0.825099\pi$
$12$			1.73205i			0.500000i
$13$	−1.00000	−	3.46410i	−0.277350	−	0.960769i
$13$	−1.00000	−	3.46410i	−0.277350	−	0.960769i
$14$	1.50000	+	2.59808i	0.400892	+	0.694365i
$15$	−1.50000	−	2.59808i	−0.387298	−	0.670820i
$16$	−5.00000			−1.25000
$17$	−1.50000	+	2.59808i	−0.363803	+	0.630126i	−0.988583	−	0.150675i	$-0.951855\pi$
$17$	−1.50000	+	2.59808i	−0.363803	+	0.630126i	0.624780	+	0.780801i	$0.285189\pi$
$18$		−	5.19615i		−	1.22474i
$19$	−1.50000	−	0.866025i	−0.344124	−	0.198680i	0.317970	−	0.948101i	$-0.396999\pi$
$19$	−1.50000	−	0.866025i	−0.344124	−	0.198680i	−0.662094	+	0.749421i	$0.730332\pi$
$20$	−1.50000	+	0.866025i	−0.335410	+	0.193649i
$21$	−1.50000	−	2.59808i	−0.327327	−	0.566947i
$22$	−6.00000			−1.27920
$23$	1.50000	−	2.59808i	0.312772	−	0.541736i	−0.666190	−	0.745782i	$-0.732076\pi$
$23$	1.50000	−	2.59808i	0.312772	−	0.541736i	0.978961	+	0.204046i	$0.0654092\pi$
$24$	3.00000			0.612372
$25$	−1.00000	+	1.73205i	−0.200000	+	0.346410i
$26$	6.00000	−	1.73205i	1.17670	−	0.339683i
$27$			5.19615i			1.00000i
$28$	−1.50000	+	0.866025i	−0.283473	+	0.163663i
$29$	−6.00000			−1.11417			−0.557086	−	0.830455i	$-0.688081\pi$
$29$	−6.00000			−1.11417			−0.557086	+	0.830455i	$0.688081\pi$
$30$	4.50000	−	2.59808i	0.821584	−	0.474342i
$31$	7.50000	−	4.33013i	1.34704	−	0.777714i	0.359211	−	0.933257i	$-0.383046\pi$
$31$	7.50000	−	4.33013i	1.34704	−	0.777714i	0.987829	+	0.155543i	$0.0497126\pi$
$32$		−	5.19615i		−	0.918559i
$33$	6.00000			1.04447
$34$	−4.50000	−	2.59808i	−0.771744	−	0.445566i
$35$	1.50000	−	2.59808i	0.253546	−	0.439155i
$36$	3.00000			0.500000
$37$	−4.50000	+	2.59808i	−0.739795	+	0.427121i	−0.821995	−	0.569495i	$-0.807139\pi$
$37$	−4.50000	+	2.59808i	−0.739795	+	0.427121i	0.0821995	+	0.996616i	$0.473806\pi$
$38$	1.50000	−	2.59808i	0.243332	−	0.421464i
$39$	−6.00000	+	1.73205i	−0.960769	+	0.277350i
$40$	1.50000	+	2.59808i	0.237171	+	0.410792i
$41$	−10.5000	−	6.06218i	−1.63982	−	0.946753i	−0.980892	−	0.194551i	$-0.937675\pi$
$41$	−10.5000	−	6.06218i	−1.63982	−	0.946753i	−0.658932	−	0.752202i	$-0.728992\pi$
$42$	4.50000	−	2.59808i	0.694365	−	0.400892i
$43$	0.500000	+	0.866025i	0.0762493	+	0.132068i	0.901629	−	0.432511i	$-0.142372\pi$
$43$	0.500000	+	0.866025i	0.0762493	+	0.132068i	−0.825380	+	0.564578i	$0.809039\pi$
$44$		−	3.46410i		−	0.522233i
$45$	−4.50000	+	2.59808i	−0.670820	+	0.387298i
$46$	4.50000	+	2.59808i	0.663489	+	0.383065i
$47$	4.50000	+	2.59808i	0.656392	+	0.378968i	0.790901	−	0.611944i	$-0.209612\pi$
$47$	4.50000	+	2.59808i	0.656392	+	0.378968i	−0.134509	+	0.990912i	$0.542946\pi$
$48$			8.66025i			1.25000i
$49$	−2.00000	+	3.46410i	−0.285714	+	0.494872i
$50$	−3.00000	−	1.73205i	−0.424264	−	0.244949i
$51$	4.50000	+	2.59808i	0.630126	+	0.363803i
$52$	1.00000	+	3.46410i	0.138675	+	0.480384i
$53$	6.00000			0.824163			0.412082	−	0.911147i	$-0.364802\pi$
$53$	6.00000			0.824163			0.412082	+	0.911147i	$0.364802\pi$
$54$	−9.00000			−1.22474
$55$	3.00000	+	5.19615i	0.404520	+	0.700649i
$56$	1.50000	+	2.59808i	0.200446	+	0.347183i
$57$	−1.50000	+	2.59808i	−0.198680	+	0.344124i
$58$		−	10.3923i		−	1.36458i
$59$			3.46410i			0.450988i	0.974245	+	0.225494i	$0.0723995\pi$
$59$			3.46410i			0.450988i	−0.974245	+	0.225494i	$0.927600\pi$
$60$	1.50000	+	2.59808i	0.193649	+	0.335410i
$61$	2.50000	+	4.33013i	0.320092	+	0.554416i	0.980507	−	0.196485i	$-0.0629528\pi$
$61$	2.50000	+	4.33013i	0.320092	+	0.554416i	−0.660415	+	0.750901i	$0.729619\pi$
$62$	7.50000	+	12.9904i	0.952501	+	1.64978i
$63$	−4.50000	+	2.59808i	−0.566947	+	0.327327i
$64$	−1.00000			−0.125000
$65$	−4.50000	−	4.33013i	−0.558156	−	0.537086i
$66$			10.3923i			1.27920i
$67$	10.5000	+	6.06218i	1.28278	+	0.740613i	0.977356	−	0.211604i	$-0.0678686\pi$
$67$	10.5000	+	6.06218i	1.28278	+	0.740613i	0.305424	+	0.952217i	$0.401202\pi$
$68$	1.50000	−	2.59808i	0.181902	−	0.315063i
$69$	−4.50000	−	2.59808i	−0.541736	−	0.312772i
$70$	4.50000	+	2.59808i	0.537853	+	0.310530i
$71$	−7.50000	−	4.33013i	−0.890086	−	0.513892i	−0.0161155	−	0.999870i	$-0.505130\pi$
$71$	−7.50000	−	4.33013i	−0.890086	−	0.513892i	−0.873971	+	0.485979i	$0.838463\pi$
$72$		−	5.19615i		−	0.612372i
$73$		−	6.92820i		−	0.810885i	−0.914121	−	0.405442i	$-0.867117\pi$
$73$		−	6.92820i		−	0.810885i	0.914121	−	0.405442i	$-0.132883\pi$
$74$	−4.50000	−	7.79423i	−0.523114	−	0.906061i
$75$	3.00000	+	1.73205i	0.346410	+	0.200000i
$76$	1.50000	+	0.866025i	0.172062	+	0.0993399i
$77$	3.00000	+	5.19615i	0.341882	+	0.592157i
$78$	−3.00000	−	10.3923i	−0.339683	−	1.17670i
$79$	5.50000	−	9.52628i	0.618798	−	1.07179i	−0.370907	−	0.928670i	$-0.620953\pi$
$79$	5.50000	−	9.52628i	0.618798	−	1.07179i	0.989705	−	0.143120i	$-0.0457135\pi$
$80$	−7.50000	+	4.33013i	−0.838525	+	0.484123i
$81$	9.00000			1.00000
$82$	10.5000	−	18.1865i	1.15953	−	2.00837i
$83$	4.50000	+	2.59808i	0.493939	+	0.285176i	0.726207	−	0.687476i	$-0.241281\pi$
$83$	4.50000	+	2.59808i	0.493939	+	0.285176i	−0.232268	+	0.972652i	$0.574615\pi$
$84$	1.50000	+	2.59808i	0.163663	+	0.283473i
$85$			5.19615i			0.563602i
$86$	−1.50000	+	0.866025i	−0.161749	+	0.0933859i
$87$			10.3923i			1.11417i
$88$	−6.00000			−0.639602
$89$	13.5000	−	7.79423i	1.43100	−	0.826187i	0.433800	−	0.901009i	$-0.357172\pi$
$89$	13.5000	−	7.79423i	1.43100	−	0.826187i	0.997197	+	0.0748225i	$0.0238390\pi$
$90$	−4.50000	−	7.79423i	−0.474342	−	0.821584i
$91$	−4.50000	−	4.33013i	−0.471728	−	0.453921i
$92$	−1.50000	+	2.59808i	−0.156386	+	0.270868i
$93$	−7.50000	−	12.9904i	−0.777714	−	1.34704i
$94$	−4.50000	+	7.79423i	−0.464140	+	0.803913i
$95$	−3.00000			−0.307794
$96$	−9.00000			−0.918559
$97$	13.5000	−	7.79423i	1.37072	−	0.791384i	0.379699	−	0.925110i	$-0.376028\pi$
$97$	13.5000	−	7.79423i	1.37072	−	0.791384i	0.991018	+	0.133726i	$0.0426942\pi$
$98$	−6.00000	−	3.46410i	−0.606092	−	0.349927i
$99$		−	10.3923i		−	1.04447i
$100$	1.00000	−	1.73205i	0.100000	−	0.173205i
$101$	−6.00000			−0.597022			−0.298511	−	0.954406i	$-0.596490\pi$
$101$	−6.00000			−0.597022			−0.298511	+	0.954406i	$0.596490\pi$
$102$	−4.50000	+	7.79423i	−0.445566	+	0.771744i
$103$	6.50000	+	11.2583i	0.640464	+	1.10932i	0.985329	+	0.170664i	$0.0545913\pi$
$103$	6.50000	+	11.2583i	0.640464	+	1.10932i	−0.344865	+	0.938652i	$0.612075\pi$
$104$	6.00000	−	1.73205i	0.588348	−	0.169842i
$105$	−4.50000	−	2.59808i	−0.439155	−	0.253546i
$106$			10.3923i			1.00939i
$107$	−7.50000	−	12.9904i	−0.725052	−	1.25583i	−0.958952	−	0.283567i	$-0.908482\pi$
$107$	−7.50000	−	12.9904i	−0.725052	−	1.25583i	0.233900	−	0.972261i	$-0.424851\pi$
$108$		−	5.19615i		−	0.500000i
$109$		0			0		1.00000			$0$
$109$		0			0		−1.00000			$\pi$
$110$	−9.00000	+	5.19615i	−0.858116	+	0.495434i
$111$	4.50000	+	7.79423i	0.427121	+	0.739795i
$112$	−7.50000	+	4.33013i	−0.708683	+	0.409159i
$113$	6.00000			0.564433			0.282216	−	0.959351i	$-0.408930\pi$
$113$	6.00000			0.564433			0.282216	+	0.959351i	$0.408930\pi$
$114$	−4.50000	−	2.59808i	−0.421464	−	0.243332i
$115$		−	5.19615i		−	0.484544i
$116$	6.00000			0.557086
$117$	3.00000	+	10.3923i	0.277350	+	0.960769i
$118$	−6.00000			−0.552345
$119$			5.19615i			0.476331i
$120$	4.50000	−	2.59808i	0.410792	−	0.237171i
$121$	−1.00000			−0.0909091
$122$	−7.50000	+	4.33013i	−0.679018	+	0.392031i
$123$	−10.5000	+	18.1865i	−0.946753	+	1.63982i
$124$	−7.50000	+	4.33013i	−0.673520	+	0.388857i
$125$			12.1244i			1.08444i
$126$	−4.50000	−	7.79423i	−0.400892	−	0.694365i
$127$	2.50000	+	4.33013i	0.221839	+	0.384237i	0.955366	−	0.295423i	$-0.0954607\pi$
$127$	2.50000	+	4.33013i	0.221839	+	0.384237i	−0.733527	+	0.679660i	$0.762127\pi$
$128$		−	12.1244i		−	1.07165i
$129$	1.50000	−	0.866025i	0.132068	−	0.0762493i
$130$	7.50000	−	7.79423i	0.657794	−	0.683599i
$131$	−7.50000	−	12.9904i	−0.655278	−	1.13497i	−0.981824	−	0.189794i	$-0.939218\pi$
$131$	−7.50000	−	12.9904i	−0.655278	−	1.13497i	0.326546	−	0.945181i	$-0.394115\pi$
$132$	−6.00000			−0.522233
$133$	−3.00000			−0.260133
$134$	−10.5000	+	18.1865i	−0.907062	+	1.57108i
$135$	4.50000	+	7.79423i	0.387298	+	0.670820i
$136$	−4.50000	−	2.59808i	−0.385872	−	0.222783i
$137$	−4.50000	+	2.59808i	−0.384461	+	0.221969i	−0.679757	−	0.733437i	$-0.737915\pi$
$137$	−4.50000	+	2.59808i	−0.384461	+	0.221969i	0.295296	+	0.955406i	$0.404582\pi$
$138$	4.50000	−	7.79423i	0.383065	−	0.663489i
$139$	−16.0000			−1.35710			−0.678551	−	0.734553i	$-0.737392\pi$
$139$	−16.0000			−1.35710			−0.678551	+	0.734553i	$0.737392\pi$
$140$	−1.50000	+	2.59808i	−0.126773	+	0.219578i
$141$	4.50000	−	7.79423i	0.378968	−	0.656392i
$142$	7.50000	−	12.9904i	0.629386	−	1.09013i
$143$	12.0000	−	3.46410i	1.00349	−	0.289683i
$144$	15.0000			1.25000
$145$	−9.00000	+	5.19615i	−0.747409	+	0.431517i
$146$	12.0000			0.993127
$147$	6.00000	+	3.46410i	0.494872	+	0.285714i
$148$	4.50000	−	2.59808i	0.369898	−	0.213561i
$149$		0			0		1.00000			$0$
$149$		0			0		−1.00000			$\pi$
$150$	−3.00000	+	5.19615i	−0.244949	+	0.424264i
$151$	10.5000	+	6.06218i	0.854478	+	0.493333i	0.862159	−	0.506637i	$-0.169112\pi$
$151$	10.5000	+	6.06218i	0.854478	+	0.493333i	−0.00768132	+	0.999970i	$0.502445\pi$
$152$	1.50000	−	2.59808i	0.121666	−	0.210732i
$153$	4.50000	−	7.79423i	0.363803	−	0.630126i
$154$	−9.00000	+	5.19615i	−0.725241	+	0.418718i
$155$	7.50000	−	12.9904i	0.602414	−	1.04341i
$156$	6.00000	−	1.73205i	0.480384	−	0.138675i
$157$	−11.5000	−	19.9186i	−0.917800	−	1.58968i	−0.802749	−	0.596316i	$-0.796630\pi$
$157$	−11.5000	−	19.9186i	−0.917800	−	1.58968i	−0.115050	−	0.993360i	$-0.536703\pi$
$158$	16.5000	+	9.52628i	1.31267	+	0.757870i
$159$		−	10.3923i		−	0.824163i
$160$	−4.50000	−	7.79423i	−0.355756	−	0.616188i
$161$		−	5.19615i		−	0.409514i
$162$			15.5885i			1.22474i
$163$	10.5000	+	6.06218i	0.822423	+	0.474826i	0.851251	−	0.524758i	$-0.175844\pi$
$163$	10.5000	+	6.06218i	0.822423	+	0.474826i	−0.0288280	+	0.999584i	$0.509178\pi$
$164$	10.5000	+	6.06218i	0.819912	+	0.473377i
$165$	9.00000	−	5.19615i	0.700649	−	0.404520i
$166$	−4.50000	+	7.79423i	−0.349268	+	0.604949i
$167$	4.50000	+	2.59808i	0.348220	+	0.201045i	0.663901	−	0.747820i	$-0.268900\pi$
$167$	4.50000	+	2.59808i	0.348220	+	0.201045i	−0.315681	+	0.948865i	$0.602233\pi$
$168$	4.50000	−	2.59808i	0.347183	−	0.200446i
$169$	−11.0000	+	6.92820i	−0.846154	+	0.532939i
$170$	−9.00000			−0.690268
$171$	4.50000	+	2.59808i	0.344124	+	0.198680i
$172$	−0.500000	−	0.866025i	−0.0381246	−	0.0660338i
$173$	10.5000	+	18.1865i	0.798300	+	1.38270i	0.920722	+	0.390218i	$0.127601\pi$
$173$	10.5000	+	18.1865i	0.798300	+	1.38270i	−0.122422	+	0.992478i	$0.539066\pi$
$174$	−18.0000			−1.36458
$175$			3.46410i			0.261861i
$176$		−	17.3205i		−	1.30558i
$177$	6.00000			0.450988
$178$	13.5000	+	23.3827i	1.01187	+	1.75261i
$179$	−1.50000	−	2.59808i	−0.112115	−	0.194189i	0.804508	−	0.593942i	$-0.202429\pi$
$179$	−1.50000	−	2.59808i	−0.112115	−	0.194189i	−0.916623	+	0.399753i	$0.869096\pi$
$180$	4.50000	−	2.59808i	0.335410	−	0.193649i
$181$	−22.0000			−1.63525			−0.817624	−	0.575753i	$-0.804709\pi$
$181$	−22.0000			−1.63525			−0.817624	+	0.575753i	$0.804709\pi$
$182$	7.50000	−	7.79423i	0.555937	−	0.577747i
$183$	7.50000	−	4.33013i	0.554416	−	0.320092i
$184$	4.50000	+	2.59808i	0.331744	+	0.191533i
$185$	−4.50000	+	7.79423i	−0.330847	+	0.573043i
$186$	22.5000	−	12.9904i	1.64978	−	0.952501i
$187$	−9.00000	−	5.19615i	−0.658145	−	0.379980i
$188$	−4.50000	−	2.59808i	−0.328196	−	0.189484i
$189$	4.50000	+	7.79423i	0.327327	+	0.566947i
$190$		−	5.19615i		−	0.376969i
$191$	−1.50000	−	2.59808i	−0.108536	−	0.187990i	0.806641	−	0.591041i	$-0.201283\pi$
$191$	−1.50000	−	2.59808i	−0.108536	−	0.187990i	−0.915177	+	0.403051i	$0.867950\pi$
$192$			1.73205i			0.125000i
$193$	−4.50000	−	2.59808i	−0.323917	−	0.187014i	0.329220	−	0.944253i	$-0.393214\pi$
$193$	−4.50000	−	2.59808i	−0.323917	−	0.187014i	−0.653137	+	0.757240i	$0.726548\pi$
$194$	13.5000	+	23.3827i	0.969244	+	1.67878i
$195$	−7.50000	+	7.79423i	−0.537086	+	0.558156i
$196$	2.00000	−	3.46410i	0.142857	−	0.247436i
$197$	7.50000	−	4.33013i	0.534353	−	0.308509i	−0.208434	−	0.978036i	$-0.566837\pi$
$197$	7.50000	−	4.33013i	0.534353	−	0.308509i	0.742787	+	0.669528i	$0.233503\pi$
$198$	18.0000			1.27920
$199$	−6.50000	+	11.2583i	−0.460773	+	0.798082i	−0.999000	−	0.0447181i	$-0.985761\pi$
$199$	−6.50000	+	11.2583i	−0.460773	+	0.798082i	0.538227	+	0.842800i	$0.319094\pi$
$200$	−3.00000	−	1.73205i	−0.212132	−	0.122474i
$201$	10.5000	−	18.1865i	0.740613	−	1.28278i
$202$		−	10.3923i		−	0.731200i
$203$	−9.00000	+	5.19615i	−0.631676	+	0.364698i
$204$	−4.50000	−	2.59808i	−0.315063	−	0.181902i
$205$	−21.0000			−1.46670
$206$	−19.5000	+	11.2583i	−1.35863	+	0.784405i
$207$	−4.50000	+	7.79423i	−0.312772	+	0.541736i
$208$	5.00000	+	17.3205i	0.346688	+	1.20096i
$209$	3.00000	−	5.19615i	0.207514	−	0.359425i
$210$	4.50000	−	7.79423i	0.310530	−	0.537853i
$211$	−6.50000	+	11.2583i	−0.447478	+	0.775055i	−0.998221	−	0.0596196i	$-0.981011\pi$
$211$	−6.50000	+	11.2583i	−0.447478	+	0.775055i	0.550743	+	0.834675i	$0.314345\pi$
$212$	−6.00000			−0.412082
$213$	−7.50000	+	12.9904i	−0.513892	+	0.890086i
$214$	22.5000	−	12.9904i	1.53807	−	0.888004i
$215$	1.50000	+	0.866025i	0.102299	+	0.0590624i
$216$	−9.00000			−0.612372
$217$	7.50000	−	12.9904i	0.509133	−	0.881845i
$218$		0			0
$219$	−12.0000			−0.810885
$220$	−3.00000	−	5.19615i	−0.202260	−	0.350325i
$221$	10.5000	+	2.59808i	0.706306	+	0.174766i
$222$	−13.5000	+	7.79423i	−0.906061	+	0.523114i
$223$		−	3.46410i		−	0.231973i	−0.993251	−	0.115987i	$-0.962997\pi$
$223$		−	3.46410i		−	0.231973i	0.993251	−	0.115987i	$-0.0370030\pi$
$224$	−4.50000	−	7.79423i	−0.300669	−	0.520774i
$225$	3.00000	−	5.19615i	0.200000	−	0.346410i
$226$			10.3923i			0.691286i
$227$	−10.5000	+	6.06218i	−0.696909	+	0.402361i	−0.806195	−	0.591649i	$-0.798477\pi$
$227$	−10.5000	+	6.06218i	−0.696909	+	0.402361i	0.109286	+	0.994010i	$0.465144\pi$
$228$	1.50000	−	2.59808i	0.0993399	−	0.172062i
$229$	7.50000	−	4.33013i	0.495614	−	0.286143i	−0.231287	−	0.972886i	$-0.574293\pi$
$229$	7.50000	−	4.33013i	0.495614	−	0.286143i	0.726900	+	0.686743i	$0.240960\pi$
$230$	9.00000			0.593442
$231$	9.00000	−	5.19615i	0.592157	−	0.341882i
$232$		−	10.3923i		−	0.682288i
$233$	−18.0000			−1.17922			−0.589610	−	0.807688i	$-0.700718\pi$
$233$	−18.0000			−1.17922			−0.589610	+	0.807688i	$0.700718\pi$
$234$	−18.0000	+	5.19615i	−1.17670	+	0.339683i
$235$	9.00000			0.587095
$236$		−	3.46410i		−	0.225494i
$237$	−16.5000	−	9.52628i	−1.07179	−	0.618798i
$238$	−9.00000			−0.583383
$239$	−4.50000	+	2.59808i	−0.291081	+	0.168056i	−0.638429	−	0.769681i	$-0.720415\pi$
$239$	−4.50000	+	2.59808i	−0.291081	+	0.168056i	0.347348	+	0.937736i	$0.387082\pi$
$240$	7.50000	+	12.9904i	0.484123	+	0.838525i
$241$	−4.50000	+	2.59808i	−0.289870	+	0.167357i	−0.637883	−	0.770133i	$-0.720190\pi$
$241$	−4.50000	+	2.59808i	−0.289870	+	0.167357i	0.348013	+	0.937490i	$0.386857\pi$
$242$		−	1.73205i		−	0.111340i
$243$		−	15.5885i		−	1.00000i
$244$	−2.50000	−	4.33013i	−0.160046	−	0.277208i
$245$			6.92820i			0.442627i
$246$	−31.5000	−	18.1865i	−2.00837	−	1.15953i
$247$	−1.50000	+	6.06218i	−0.0954427	+	0.385727i
$248$	7.50000	+	12.9904i	0.476250	+	0.824890i
$249$	4.50000	−	7.79423i	0.285176	−	0.493939i
$250$	−21.0000			−1.32816
$251$	−4.50000	+	7.79423i	−0.284037	+	0.491967i	−0.972375	−	0.233423i	$-0.925007\pi$
$251$	−4.50000	+	7.79423i	−0.284037	+	0.491967i	0.688338	+	0.725390i	$0.258341\pi$
$252$	4.50000	−	2.59808i	0.283473	−	0.163663i
$253$	9.00000	+	5.19615i	0.565825	+	0.326679i
$254$	−7.50000	+	4.33013i	−0.470592	+	0.271696i
$255$	9.00000			0.563602
$256$	19.0000			1.18750
$257$	−1.50000	+	2.59808i	−0.0935674	+	0.162064i	−0.909010	−	0.416775i	$-0.863160\pi$
$257$	−1.50000	+	2.59808i	−0.0935674	+	0.162064i	0.815442	+	0.578838i	$0.196494\pi$
$258$	1.50000	+	2.59808i	0.0933859	+	0.161749i
$259$	−4.50000	+	7.79423i	−0.279616	+	0.484310i
$260$	4.50000	+	4.33013i	0.279078	+	0.268543i
$261$	18.0000			1.11417
$262$	22.5000	−	12.9904i	1.39005	−	0.802548i
$263$	24.0000			1.47990			0.739952	−	0.672660i	$-0.234848\pi$
$263$	24.0000			1.47990			0.739952	+	0.672660i	$0.234848\pi$
$264$			10.3923i			0.639602i
$265$	9.00000	−	5.19615i	0.552866	−	0.319197i
$266$		−	5.19615i		−	0.318597i
$267$	−13.5000	−	23.3827i	−0.826187	−	1.43100i
$268$	−10.5000	−	6.06218i	−0.641390	−	0.370306i
$269$	−1.50000	+	2.59808i	−0.0914566	+	0.158408i	−0.908124	−	0.418701i	$-0.862486\pi$
$269$	−1.50000	+	2.59808i	−0.0914566	+	0.158408i	0.816668	+	0.577108i	$0.195819\pi$
$270$	−13.5000	+	7.79423i	−0.821584	+	0.474342i
$271$	13.5000	−	7.79423i	0.820067	−	0.473466i	−0.0303728	−	0.999539i	$-0.509669\pi$
$271$	13.5000	−	7.79423i	0.820067	−	0.473466i	0.850439	+	0.526073i	$0.176336\pi$
$272$	7.50000	−	12.9904i	0.454754	−	0.787658i
$273$	−7.50000	+	7.79423i	−0.453921	+	0.471728i
$274$	−4.50000	−	7.79423i	−0.271855	−	0.470867i
$275$	−6.00000	−	3.46410i	−0.361814	−	0.208893i
$276$	4.50000	+	2.59808i	0.270868	+	0.156386i
$277$	−11.5000	−	19.9186i	−0.690968	−	1.19679i	−0.971521	−	0.236953i	$-0.923851\pi$
$277$	−11.5000	−	19.9186i	−0.690968	−	1.19679i	0.280553	−	0.959839i	$-0.409482\pi$
$278$		−	27.7128i		−	1.66210i
$279$	−22.5000	+	12.9904i	−1.34704	+	0.777714i
$280$	4.50000	+	2.59808i	0.268926	+	0.155265i
$281$	−4.50000	−	2.59808i	−0.268447	−	0.154988i	0.359734	−	0.933055i	$-0.382867\pi$
$281$	−4.50000	−	2.59808i	−0.268447	−	0.154988i	−0.628182	+	0.778067i	$0.716201\pi$
$282$	13.5000	+	7.79423i	0.803913	+	0.464140i
$283$	11.5000	−	19.9186i	0.683604	−	1.18404i	−0.290269	−	0.956945i	$-0.593745\pi$
$283$	11.5000	−	19.9186i	0.683604	−	1.18404i	0.973873	−	0.227092i	$-0.0729218\pi$
$284$	7.50000	+	4.33013i	0.445043	+	0.256946i
$285$			5.19615i			0.307794i
$286$	6.00000	+	20.7846i	0.354787	+	1.22902i
$287$	−21.0000			−1.23959
$288$			15.5885i			0.918559i
$289$	4.00000	+	6.92820i	0.235294	+	0.407541i
$290$	−9.00000	−	15.5885i	−0.528498	−	0.915386i
$291$	−13.5000	−	23.3827i	−0.791384	−	1.37072i
$292$			6.92820i			0.405442i
$293$			13.8564i			0.809500i	0.914427	+	0.404750i	$0.132641\pi$
$293$			13.8564i			0.809500i	−0.914427	+	0.404750i	$0.867359\pi$
$294$	−6.00000	+	10.3923i	−0.349927	+	0.606092i
$295$	3.00000	+	5.19615i	0.174667	+	0.302532i
$296$	−4.50000	−	7.79423i	−0.261557	−	0.453030i
$297$	−18.0000			−1.04447
$298$		0			0
$299$	−10.5000	−	2.59808i	−0.607231	−	0.150251i
$300$	−3.00000	−	1.73205i	−0.173205	−	0.100000i
$301$	1.50000	+	0.866025i	0.0864586	+	0.0499169i
$302$	−10.5000	+	18.1865i	−0.604207	+	1.04652i
$303$			10.3923i			0.597022i
$304$	7.50000	+	4.33013i	0.430155	+	0.248350i
$305$	7.50000	+	4.33013i	0.429449	+	0.247942i
$306$	13.5000	+	7.79423i	0.771744	+	0.445566i
$307$		−	24.2487i		−	1.38395i	−0.721923	−	0.691974i	$-0.756741\pi$
$307$		−	24.2487i		−	1.38395i	0.721923	−	0.691974i	$-0.243259\pi$
$308$	−3.00000	−	5.19615i	−0.170941	−	0.296078i
$309$	19.5000	−	11.2583i	1.10932	−	0.640464i
$310$	22.5000	+	12.9904i	1.27791	+	0.737804i
$311$	−13.5000	−	23.3827i	−0.765515	−	1.32591i	−0.939974	−	0.341246i	$-0.889151\pi$
$311$	−13.5000	−	23.3827i	−0.765515	−	1.32591i	0.174459	−	0.984664i	$-0.444182\pi$
$312$	−3.00000	−	10.3923i	−0.169842	−	0.588348i
$313$	−9.50000	+	16.4545i	−0.536972	+	0.930062i	0.462093	+	0.886831i	$0.347098\pi$
$313$	−9.50000	+	16.4545i	−0.536972	+	0.930062i	−0.999065	+	0.0432311i	$0.986235\pi$
$314$	34.5000	−	19.9186i	1.94695	−	1.12407i
$315$	−4.50000	+	7.79423i	−0.253546	+	0.439155i
$316$	−5.50000	+	9.52628i	−0.309399	+	0.535895i
$317$	7.50000	+	4.33013i	0.421242	+	0.243204i	0.695609	−	0.718421i	$-0.255135\pi$
$317$	7.50000	+	4.33013i	0.421242	+	0.243204i	−0.274367	+	0.961625i	$0.588468\pi$
$318$	18.0000			1.00939
$319$		−	20.7846i		−	1.16371i
$320$	−1.50000	+	0.866025i	−0.0838525	+	0.0484123i
$321$	−22.5000	+	12.9904i	−1.25583	+	0.725052i
$322$	9.00000			0.501550
$323$	4.50000	−	2.59808i	0.250387	−	0.144561i
$324$	−9.00000			−0.500000
$325$	7.00000	+	1.73205i	0.388290	+	0.0960769i
$326$	−10.5000	+	18.1865i	−0.581541	+	1.00726i
$327$		0			0
$328$	10.5000	−	18.1865i	0.579766	−	1.00418i
$329$	9.00000			0.496186
$330$	9.00000	+	15.5885i	0.495434	+	0.858116i
$331$	13.5000	−	7.79423i	0.742027	−	0.428410i	−0.0807788	−	0.996732i	$-0.525741\pi$
$331$	13.5000	−	7.79423i	0.742027	−	0.428410i	0.822806	+	0.568323i	$0.192407\pi$
$332$	−4.50000	−	2.59808i	−0.246970	−	0.142588i
$333$	13.5000	−	7.79423i	0.739795	−	0.427121i
$334$	−4.50000	+	7.79423i	−0.246229	+	0.426481i
$335$	21.0000			1.14735
$336$	7.50000	+	12.9904i	0.409159	+	0.708683i
$337$	14.5000	+	25.1147i	0.789865	+	1.36809i	0.926049	+	0.377403i	$0.123183\pi$
$337$	14.5000	+	25.1147i	0.789865	+	1.36809i	−0.136184	+	0.990684i	$0.543484\pi$
$338$	−12.0000	−	19.0526i	−0.652714	−	1.03632i
$339$		−	10.3923i		−	0.564433i
$340$		−	5.19615i		−	0.281801i
$341$	15.0000	+	25.9808i	0.812296	+	1.40694i
$342$	−4.50000	+	7.79423i	−0.243332	+	0.421464i
$343$			19.0526i			1.02874i
$344$	−1.50000	+	0.866025i	−0.0808746	+	0.0466930i
$345$	−9.00000			−0.484544
$346$	−31.5000	+	18.1865i	−1.69345	+	0.977714i
$347$	−12.0000			−0.644194			−0.322097	−	0.946707i	$-0.604388\pi$
$347$	−12.0000			−0.644194			−0.322097	+	0.946707i	$0.604388\pi$
$348$		−	10.3923i		−	0.557086i
$349$			27.7128i			1.48343i	0.670714	+	0.741716i	$0.265988\pi$
$349$			27.7128i			1.48343i	−0.670714	+	0.741716i	$0.734012\pi$
$350$	−6.00000			−0.320713
$351$	18.0000	−	5.19615i	0.960769	−	0.277350i
$352$	18.0000			0.959403
$353$		−	6.92820i		−	0.368751i	−0.982856	−	0.184376i	$-0.940974\pi$
$353$		−	6.92820i		−	0.368751i	0.982856	−	0.184376i	$-0.0590263\pi$
$354$			10.3923i			0.552345i
$355$	−15.0000			−0.796117
$356$	−13.5000	+	7.79423i	−0.715499	+	0.413093i
$357$	9.00000			0.476331
$358$	4.50000	−	2.59808i	0.237832	−	0.137313i
$359$			10.3923i			0.548485i	0.961661	+	0.274242i	$0.0884271\pi$
$359$			10.3923i			0.548485i	−0.961661	+	0.274242i	$0.911573\pi$
$360$	−4.50000	−	7.79423i	−0.237171	−	0.410792i
$361$	−8.00000	−	13.8564i	−0.421053	−	0.729285i
$362$		−	38.1051i		−	2.00276i
$363$			1.73205i			0.0909091i
$364$	4.50000	+	4.33013i	0.235864	+	0.226960i
$365$	−6.00000	−	10.3923i	−0.314054	−	0.543958i
$366$	7.50000	+	12.9904i	0.392031	+	0.679018i
$367$	8.00000			0.417597			0.208798	−	0.977959i	$-0.433045\pi$
$367$	8.00000			0.417597			0.208798	+	0.977959i	$0.433045\pi$
$368$	−7.50000	+	12.9904i	−0.390965	+	0.677170i
$369$	31.5000	+	18.1865i	1.63982	+	0.946753i
$370$	−13.5000	−	7.79423i	−0.701832	−	0.405203i
$371$	9.00000	−	5.19615i	0.467257	−	0.269771i
$372$	7.50000	+	12.9904i	0.388857	+	0.673520i
$373$	−14.0000			−0.724893			−0.362446	−	0.932005i	$-0.618058\pi$
$373$	−14.0000			−0.724893			−0.362446	+	0.932005i	$0.618058\pi$
$374$	9.00000	−	15.5885i	0.465379	−	0.806060i
$375$	21.0000			1.08444
$376$	−4.50000	+	7.79423i	−0.232070	+	0.401957i
$377$	6.00000	+	20.7846i	0.309016	+	1.07046i
$378$	−13.5000	+	7.79423i	−0.694365	+	0.400892i
$379$	−10.5000	+	6.06218i	−0.539349	+	0.311393i	−0.744815	−	0.667271i	$-0.767462\pi$
$379$	−10.5000	+	6.06218i	−0.539349	+	0.311393i	0.205466	+	0.978664i	$0.434129\pi$
$380$	3.00000			0.153897
$381$	7.50000	−	4.33013i	0.384237	−	0.221839i
$382$	4.50000	−	2.59808i	0.230240	−	0.132929i
$383$		−	3.46410i		−	0.177007i	−0.996076	−	0.0885037i	$-0.971792\pi$
$383$		−	3.46410i		−	0.177007i	0.996076	−	0.0885037i	$-0.0282085\pi$
$384$	−21.0000			−1.07165
$385$	9.00000	+	5.19615i	0.458682	+	0.264820i
$386$	4.50000	−	7.79423i	0.229044	−	0.396716i
$387$	−1.50000	−	2.59808i	−0.0762493	−	0.132068i
$388$	−13.5000	+	7.79423i	−0.685359	+	0.395692i
$389$	−1.50000	+	2.59808i	−0.0760530	+	0.131728i	−0.901544	−	0.432688i	$-0.857565\pi$
$389$	−1.50000	+	2.59808i	−0.0760530	+	0.131728i	0.825491	+	0.564416i	$0.190898\pi$
$390$	−13.5000	−	12.9904i	−0.683599	−	0.657794i
$391$	4.50000	+	7.79423i	0.227575	+	0.394171i
$392$	−6.00000	−	3.46410i	−0.303046	−	0.174964i
$393$	−22.5000	+	12.9904i	−1.13497	+	0.655278i
$394$	7.50000	+	12.9904i	0.377845	+	0.654446i
$395$		−	19.0526i		−	0.958638i
$396$			10.3923i			0.522233i
$397$	−22.5000	−	12.9904i	−1.12924	−	0.651969i	−0.185498	−	0.982645i	$-0.559390\pi$
$397$	−22.5000	−	12.9904i	−1.12924	−	0.651969i	−0.943744	+	0.330676i	$0.892723\pi$
$398$	−19.5000	−	11.2583i	−0.977447	−	0.564329i
$399$			5.19615i			0.260133i
$400$	5.00000	−	8.66025i	0.250000	−	0.433013i
$401$	25.5000	+	14.7224i	1.27341	+	0.735203i	0.975628	−	0.219431i	$-0.0704201\pi$
$401$	25.5000	+	14.7224i	1.27341	+	0.735203i	0.297781	+	0.954634i	$0.403753\pi$
$402$	31.5000	+	18.1865i	1.57108	+	0.907062i
$403$	−22.5000	−	21.6506i	−1.12080	−	1.07849i
$404$	6.00000			0.298511
$405$	13.5000	−	7.79423i	0.670820	−	0.387298i
$406$	−9.00000	−	15.5885i	−0.446663	−	0.773642i
$407$	−9.00000	−	15.5885i	−0.446113	−	0.772691i
$408$	−4.50000	+	7.79423i	−0.222783	+	0.385872i
$409$			6.92820i			0.342578i	0.985221	+	0.171289i	$0.0547931\pi$
$409$			6.92820i			0.342578i	−0.985221	+	0.171289i	$0.945207\pi$
$410$		−	36.3731i		−	1.79634i
$411$	4.50000	+	7.79423i	0.221969	+	0.384461i
$412$	−6.50000	−	11.2583i	−0.320232	−	0.554658i
$413$	3.00000	+	5.19615i	0.147620	+	0.255686i
$414$	−13.5000	−	7.79423i	−0.663489	−	0.383065i
$415$	9.00000			0.441793
$416$	−18.0000	+	5.19615i	−0.882523	+	0.254762i
$417$			27.7128i			1.35710i
$418$	9.00000	+	5.19615i	0.440204	+	0.254152i
$419$	−4.50000	+	7.79423i	−0.219839	+	0.380773i	−0.954759	−	0.297382i	$-0.903887\pi$
$419$	−4.50000	+	7.79423i	−0.219839	+	0.380773i	0.734919	+	0.678155i	$0.237220\pi$
$420$	4.50000	+	2.59808i	0.219578	+	0.126773i
$421$	7.50000	+	4.33013i	0.365528	+	0.211037i	0.671503	−	0.741002i	$-0.265649\pi$
$421$	7.50000	+	4.33013i	0.365528	+	0.211037i	−0.305975	+	0.952039i	$0.598982\pi$
$422$	−19.5000	−	11.2583i	−0.949245	−	0.548047i
$423$	−13.5000	−	7.79423i	−0.656392	−	0.378968i
$424$			10.3923i			0.504695i
$425$	−3.00000	−	5.19615i	−0.145521	−	0.252050i
$426$	−22.5000	−	12.9904i	−1.09013	−	0.629386i
$427$	7.50000	+	4.33013i	0.362950	+	0.209550i
$428$	7.50000	+	12.9904i	0.362526	+	0.627914i
$429$	−6.00000	−	20.7846i	−0.289683	−	1.00349i
$430$	−1.50000	+	2.59808i	−0.0723364	+	0.125290i
$431$	19.5000	−	11.2583i	0.939282	−	0.542295i	0.0495468	−	0.998772i	$-0.484222\pi$
$431$	19.5000	−	11.2583i	0.939282	−	0.542295i	0.889735	+	0.456477i	$0.150889\pi$
$432$		−	25.9808i		−	1.25000i
$433$	2.50000	−	4.33013i	0.120142	−	0.208093i	−0.799681	−	0.600425i	$-0.794998\pi$
$433$	2.50000	−	4.33013i	0.120142	−	0.208093i	0.919824	+	0.392332i	$0.128332\pi$
$434$	22.5000	+	12.9904i	1.08003	+	0.623558i
$435$	9.00000	+	15.5885i	0.431517	+	0.747409i
$436$		0			0
$437$	−4.50000	+	2.59808i	−0.215264	+	0.124283i
$438$		−	20.7846i		−	0.993127i
$439$	8.00000			0.381819			0.190910	−	0.981608i	$-0.438856\pi$
$439$	8.00000			0.381819			0.190910	+	0.981608i	$0.438856\pi$
$440$	−9.00000	+	5.19615i	−0.429058	+	0.247717i
$441$	6.00000	−	10.3923i	0.285714	−	0.494872i
$442$	−4.50000	+	18.1865i	−0.214043	+	0.865045i
$443$	−10.5000	+	18.1865i	−0.498870	+	0.864068i	−0.999999	−	0.00130426i	$-0.999585\pi$
$443$	−10.5000	+	18.1865i	−0.498870	+	0.864068i	0.501129	+	0.865373i	$0.332918\pi$
$444$	−4.50000	−	7.79423i	−0.213561	−	0.369898i
$445$	13.5000	−	23.3827i	0.639961	−	1.10845i
$446$	6.00000			0.284108
$447$		0			0
$448$	−1.50000	+	0.866025i	−0.0708683	+	0.0409159i
$449$	13.5000	+	7.79423i	0.637104	+	0.367832i	0.783498	−	0.621394i	$-0.213433\pi$
$449$	13.5000	+	7.79423i	0.637104	+	0.367832i	−0.146394	+	0.989226i	$0.546767\pi$
$450$	9.00000	+	5.19615i	0.424264	+	0.244949i
$451$	21.0000	−	36.3731i	0.988851	−	1.71274i
$452$	−6.00000			−0.282216
$453$	10.5000	−	18.1865i	0.493333	−	0.854478i
$454$	−10.5000	−	18.1865i	−0.492789	−	0.853536i
$455$	−10.5000	−	2.59808i	−0.492248	−	0.121800i
$456$	−4.50000	−	2.59808i	−0.210732	−	0.121666i
$457$		0			0		1.00000			$0$
$457$		0			0		−1.00000			$\pi$
$458$	7.50000	+	12.9904i	0.350452	+	0.607001i
$459$	−13.5000	−	7.79423i	−0.630126	−	0.363803i
$460$			5.19615i			0.242272i
$461$	1.50000	−	0.866025i	0.0698620	−	0.0403348i	−0.464662	−	0.885488i	$-0.653824\pi$
$461$	1.50000	−	0.866025i	0.0698620	−	0.0403348i	0.534524	+	0.845153i	$0.320491\pi$
$462$	9.00000	+	15.5885i	0.418718	+	0.725241i
$463$	−22.5000	+	12.9904i	−1.04566	+	0.603714i	−0.921432	−	0.388539i	$-0.872980\pi$
$463$	−22.5000	+	12.9904i	−1.04566	+	0.603714i	−0.124231	+	0.992253i	$0.539647\pi$
$464$	30.0000			1.39272
$465$	−22.5000	−	12.9904i	−1.04341	−	0.602414i
$466$		−	31.1769i		−	1.44424i
$467$	−12.0000			−0.555294			−0.277647	−	0.960683i	$-0.589555\pi$
$467$	−12.0000			−0.555294			−0.277647	+	0.960683i	$0.589555\pi$
$468$	−3.00000	−	10.3923i	−0.138675	−	0.480384i
$469$	21.0000			0.969690
$470$			15.5885i			0.719042i
$471$	−34.5000	+	19.9186i	−1.58968	+	0.917800i
$472$	−6.00000			−0.276172
$473$	−3.00000	+	1.73205i	−0.137940	+	0.0796398i
$474$	16.5000	−	28.5788i	0.757870	−	1.31267i
$475$	3.00000	−	1.73205i	0.137649	−	0.0794719i
$476$		−	5.19615i		−	0.238165i
$477$	−18.0000			−0.824163
$478$	−4.50000	−	7.79423i	−0.205825	−	0.356500i
$479$			24.2487i			1.10795i	0.832533	+	0.553976i	$0.186890\pi$
$479$			24.2487i			1.10795i	−0.832533	+	0.553976i	$0.813110\pi$
$480$	−13.5000	+	7.79423i	−0.616188	+	0.355756i
$481$	13.5000	+	12.9904i	0.615547	+	0.592310i
$482$	−4.50000	−	7.79423i	−0.204969	−	0.355017i
$483$	−9.00000			−0.409514
$484$	1.00000			0.0454545
$485$	13.5000	−	23.3827i	0.613003	−	1.06175i
$486$	27.0000			1.22474
$487$	−7.50000	−	4.33013i	−0.339857	−	0.196217i	0.320352	−	0.947299i	$-0.396199\pi$
$487$	−7.50000	−	4.33013i	−0.339857	−	0.196217i	−0.660209	+	0.751082i	$0.729532\pi$
$488$	−7.50000	+	4.33013i	−0.339509	+	0.196016i
$489$	10.5000	−	18.1865i	0.474826	−	0.822423i
$490$	−12.0000			−0.542105
$491$	1.50000	−	2.59808i	0.0676941	−	0.117250i	−0.830192	−	0.557478i	$-0.811769\pi$
$491$	1.50000	−	2.59808i	0.0676941	−	0.117250i	0.897886	+	0.440228i	$0.145102\pi$
$492$	10.5000	−	18.1865i	0.473377	−	0.819912i
$493$	9.00000	−	15.5885i	0.405340	−	0.702069i
$494$	−10.5000	−	2.59808i	−0.472417	−	0.116893i
$495$	−9.00000	−	15.5885i	−0.404520	−	0.700649i
$496$	−37.5000	+	21.6506i	−1.68380	+	0.972142i
$497$	−15.0000			−0.672842
$498$	13.5000	+	7.79423i	0.604949	+	0.349268i
$499$	−22.5000	+	12.9904i	−1.00724	+	0.581529i	−0.910382	−	0.413769i	$-0.864212\pi$
$499$	−22.5000	+	12.9904i	−1.00724	+	0.581529i	−0.0968564	+	0.995298i	$0.530879\pi$
$500$		−	12.1244i		−	0.542218i
$501$	4.50000	−	7.79423i	0.201045	−	0.348220i
$502$	−13.5000	−	7.79423i	−0.602534	−	0.347873i
$503$	−10.5000	+	18.1865i	−0.468172	+	0.810897i	−0.999338	−	0.0363700i	$-0.988421\pi$
$503$	−10.5000	+	18.1865i	−0.468172	+	0.810897i	0.531167	+	0.847267i	$0.321754\pi$
$504$	−4.50000	−	7.79423i	−0.200446	−	0.347183i
$505$	−9.00000	+	5.19615i	−0.400495	+	0.231226i
$506$	−9.00000	+	15.5885i	−0.400099	+	0.692991i
$507$	12.0000	+	19.0526i	0.532939	+	0.846154i
$508$	−2.50000	−	4.33013i	−0.110920	−	0.192118i
$509$	1.50000	+	0.866025i	0.0664863	+	0.0383859i	0.532875	−	0.846194i	$-0.321112\pi$
$509$	1.50000	+	0.866025i	0.0664863	+	0.0383859i	−0.466388	+	0.884580i	$0.654445\pi$
$510$			15.5885i			0.690268i
$511$	−6.00000	−	10.3923i	−0.265424	−	0.459728i
$512$			8.66025i			0.382733i
$513$	4.50000	−	7.79423i	0.198680	−	0.344124i
$514$	−4.50000	−	2.59808i	−0.198486	−	0.114596i
$515$	19.5000	+	11.2583i	0.859273	+	0.496101i
$516$	−1.50000	+	0.866025i	−0.0660338	+	0.0381246i
$517$	−9.00000	+	15.5885i	−0.395820	+	0.685580i
$518$	−13.5000	−	7.79423i	−0.593156	−	0.342459i
$519$	31.5000	−	18.1865i	1.38270	−	0.798300i
$520$	7.50000	−	7.79423i	0.328897	−	0.341800i
$521$	6.00000			0.262865			0.131432	−	0.991325i	$-0.458042\pi$
$521$	6.00000			0.262865			0.131432	+	0.991325i	$0.458042\pi$
$522$			31.1769i			1.36458i
$523$	12.5000	+	21.6506i	0.546587	+	0.946716i	0.998505	+	0.0546569i	$0.0174065\pi$
$523$	12.5000	+	21.6506i	0.546587	+	0.946716i	−0.451918	+	0.892059i	$0.649260\pi$
$524$	7.50000	+	12.9904i	0.327639	+	0.567487i
$525$	6.00000			0.261861
$526$			41.5692i			1.81250i
$527$			25.9808i			1.13174i
$528$	−30.0000			−1.30558
$529$	7.00000	+	12.1244i	0.304348	+	0.527146i
$530$	9.00000	+	15.5885i	0.390935	+	0.677119i
$531$		−	10.3923i		−	0.450988i
$532$	3.00000			0.130066
$533$	−10.5000	+	42.4352i	−0.454805	+	1.83807i
$534$	40.5000	−	23.3827i	1.75261	−	1.01187i
$535$	−22.5000	−	12.9904i	−0.972760	−	0.561623i
$536$	−10.5000	+	18.1865i	−0.453531	+	0.785539i
$537$	−4.50000	+	2.59808i	−0.194189	+	0.112115i
$538$	−4.50000	−	2.59808i	−0.194009	−	0.112011i
$539$	−12.0000	−	6.92820i	−0.516877	−	0.298419i
$540$	−4.50000	−	7.79423i	−0.193649	−	0.335410i
$541$			13.8564i			0.595733i	0.954607	+	0.297867i	$0.0962751\pi$
$541$			13.8564i			0.595733i	−0.954607	+	0.297867i	$0.903725\pi$
$542$	13.5000	+	23.3827i	0.579875	+	1.00437i
$543$			38.1051i			1.63525i
$544$	13.5000	+	7.79423i	0.578808	+	0.334175i
$545$		0			0
$546$	−13.5000	−	12.9904i	−0.577747	−	0.555937i
$547$	−8.50000	+	14.7224i	−0.363434	+	0.629486i	−0.988524	−	0.151067i	$-0.951729\pi$
$547$	−8.50000	+	14.7224i	−0.363434	+	0.629486i	0.625090	+	0.780553i	$0.285062\pi$
$548$	4.50000	−	2.59808i	0.192230	−	0.110984i
$549$	−7.50000	−	12.9904i	−0.320092	−	0.554416i
$550$	6.00000	−	10.3923i	0.255841	−	0.443129i
$551$	9.00000	+	5.19615i	0.383413	+	0.221364i
$552$	4.50000	−	7.79423i	0.191533	−	0.331744i
$553$		−	19.0526i		−	0.810197i
$554$	34.5000	−	19.9186i	1.46576	−	0.846260i
$555$	13.5000	+	7.79423i	0.573043	+	0.330847i
$556$	16.0000			0.678551
$557$	−16.5000	+	9.52628i	−0.699127	+	0.403641i	−0.807022	−	0.590521i	$-0.798922\pi$
$557$	−16.5000	+	9.52628i	−0.699127	+	0.403641i	0.107895	+	0.994162i	$0.465589\pi$
$558$	−22.5000	−	38.9711i	−0.952501	−	1.64978i
$559$	2.50000	−	2.59808i	0.105739	−	0.109887i
$560$	−7.50000	+	12.9904i	−0.316933	+	0.548944i
$561$	−9.00000	+	15.5885i	−0.379980	+	0.658145i
$562$	4.50000	−	7.79423i	0.189821	−	0.328780i
$563$	24.0000			1.01148			0.505740	−	0.862686i	$-0.331220\pi$
$563$	24.0000			1.01148			0.505740	+	0.862686i	$0.331220\pi$
$564$	−4.50000	+	7.79423i	−0.189484	+	0.328196i
$565$	9.00000	−	5.19615i	0.378633	−	0.218604i
$566$	34.5000	+	19.9186i	1.45014	+	0.837241i
$567$	13.5000	−	7.79423i	0.566947	−	0.327327i
$568$	7.50000	−	12.9904i	0.314693	−	0.545064i
$569$	−18.0000			−0.754599			−0.377300	−	0.926091i	$-0.623147\pi$
$569$	−18.0000			−0.754599			−0.377300	+	0.926091i	$0.623147\pi$
$570$	−9.00000			−0.376969
$571$	2.50000	+	4.33013i	0.104622	+	0.181210i	0.913584	−	0.406651i	$-0.133303\pi$
$571$	2.50000	+	4.33013i	0.104622	+	0.181210i	−0.808962	+	0.587861i	$0.799970\pi$
$572$	−12.0000	+	3.46410i	−0.501745	+	0.144841i
$573$	−4.50000	+	2.59808i	−0.187990	+	0.108536i
$574$		−	36.3731i		−	1.51818i
$575$	3.00000	+	5.19615i	0.125109	+	0.216695i
$576$	3.00000			0.125000
$577$		−	6.92820i		−	0.288425i	−0.989547	−	0.144212i	$-0.953935\pi$
$577$		−	6.92820i		−	0.288425i	0.989547	−	0.144212i	$-0.0460649\pi$
$578$	−12.0000	+	6.92820i	−0.499134	+	0.288175i
$579$	−4.50000	+	7.79423i	−0.187014	+	0.323917i
$580$	9.00000	−	5.19615i	0.373705	−	0.215758i
$581$	9.00000			0.373383
$582$	40.5000	−	23.3827i	1.67878	−	0.969244i
$583$			20.7846i			0.860811i
$584$	12.0000			0.496564
$585$	13.5000	+	12.9904i	0.558156	+	0.537086i
$586$	−24.0000			−0.991431
$587$		−	10.3923i		−	0.428936i	−0.976731	−	0.214468i	$-0.931198\pi$
$587$		−	10.3923i		−	0.428936i	0.976731	−	0.214468i	$-0.0688018\pi$
$588$	−6.00000	−	3.46410i	−0.247436	−	0.142857i
$589$	−15.0000			−0.618064
$590$	−9.00000	+	5.19615i	−0.370524	+	0.213922i
$591$	−7.50000	−	12.9904i	−0.308509	−	0.534353i
$592$	22.5000	−	12.9904i	0.924744	−	0.533901i
$593$		−	27.7128i		−	1.13803i	−0.822328	−	0.569014i	$-0.807325\pi$
$593$		−	27.7128i		−	1.13803i	0.822328	−	0.569014i	$-0.192675\pi$
$594$		−	31.1769i		−	1.27920i
$595$	4.50000	+	7.79423i	0.184482	+	0.319532i
$596$		0			0
$597$	19.5000	+	11.2583i	0.798082	+	0.460773i
$598$	4.50000	−	18.1865i	0.184019	−	0.743703i
$599$	−13.5000	−	23.3827i	−0.551595	−	0.955391i	−0.998160	−	0.0606393i	$-0.980686\pi$
$599$	−13.5000	−	23.3827i	−0.551595	−	0.955391i	0.446565	−	0.894751i	$-0.352647\pi$
$600$	−3.00000	+	5.19615i	−0.122474	+	0.212132i
$601$	22.0000			0.897399			0.448699	−	0.893683i	$-0.351887\pi$
$601$	22.0000			0.897399			0.448699	+	0.893683i	$0.351887\pi$
$602$	−1.50000	+	2.59808i	−0.0611354	+	0.105890i
$603$	−31.5000	−	18.1865i	−1.28278	−	0.740613i
$604$	−10.5000	−	6.06218i	−0.427239	−	0.246667i
$605$	−1.50000	+	0.866025i	−0.0609837	+	0.0352089i
$606$	−18.0000			−0.731200
$607$	8.00000			0.324710			0.162355	−	0.986732i	$-0.448091\pi$
$607$	8.00000			0.324710			0.162355	+	0.986732i	$0.448091\pi$
$608$	−4.50000	+	7.79423i	−0.182499	+	0.316098i
$609$	9.00000	+	15.5885i	0.364698	+	0.631676i
$610$	−7.50000	+	12.9904i	−0.303666	+	0.525965i
$611$	4.50000	−	18.1865i	0.182051	−	0.735748i
$612$	−4.50000	+	7.79423i	−0.181902	+	0.315063i
$613$	−10.5000	+	6.06218i	−0.424091	+	0.244849i	−0.696826	−	0.717240i	$-0.745405\pi$
$613$	−10.5000	+	6.06218i	−0.424091	+	0.244849i	0.272735	+	0.962089i	$0.412072\pi$
$614$	42.0000			1.69498
$615$			36.3731i			1.46670i
$616$	−9.00000	+	5.19615i	−0.362620	+	0.209359i
$617$		−	27.7128i		−	1.11568i	−0.829950	−	0.557838i	$-0.811631\pi$
$617$		−	27.7128i		−	1.11568i	0.829950	−	0.557838i	$-0.188369\pi$
$618$	19.5000	+	33.7750i	0.784405	+	1.35863i
$619$	22.5000	+	12.9904i	0.904351	+	0.522127i	0.878609	−	0.477541i	$-0.158472\pi$
$619$	22.5000	+	12.9904i	0.904351	+	0.522127i	0.0257420	+	0.999669i	$0.491805\pi$
$620$	−7.50000	+	12.9904i	−0.301207	+	0.521706i
$621$	13.5000	+	7.79423i	0.541736	+	0.312772i
$622$	40.5000	−	23.3827i	1.62390	−	0.937560i
$623$	13.5000	−	23.3827i	0.540866	−	0.936808i
$624$	30.0000	−	8.66025i	1.20096	−	0.346688i
$625$	5.50000	+	9.52628i	0.220000	+	0.381051i
$626$	−28.5000	−	16.4545i	−1.13909	−	0.657653i
$627$	−9.00000	−	5.19615i	−0.359425	−	0.207514i
$628$	11.5000	+	19.9186i	0.458900	+	0.794838i
$629$		−	15.5885i		−	0.621552i
$630$	−13.5000	−	7.79423i	−0.537853	−	0.310530i
$631$	−7.50000	−	4.33013i	−0.298570	−	0.172380i	0.343230	−	0.939251i	$-0.388479\pi$
$631$	−7.50000	−	4.33013i	−0.298570	−	0.172380i	−0.641800	+	0.766872i	$0.721812\pi$
$632$	16.5000	+	9.52628i	0.656335	+	0.378935i
$633$	19.5000	+	11.2583i	0.775055	+	0.447478i
$634$	−7.50000	+	12.9904i	−0.297863	+	0.515914i
$635$	7.50000	+	4.33013i	0.297628	+	0.171836i
$636$			10.3923i			0.412082i
$637$	14.0000	+	3.46410i	0.554700	+	0.137253i
$638$	36.0000			1.42525
$639$	22.5000	+	12.9904i	0.890086	+	0.513892i
$640$	−10.5000	−	18.1865i	−0.415049	−	0.718886i
$641$	−19.5000	−	33.7750i	−0.770204	−	1.33403i	−0.937451	−	0.348117i	$-0.886821\pi$
$641$	−19.5000	−	33.7750i	−0.770204	−	1.33403i	0.167247	−	0.985915i	$-0.446512\pi$
$642$	−22.5000	−	38.9711i	−0.888004	−	1.53807i
$643$		−	3.46410i		−	0.136611i	−0.997664	−	0.0683054i	$-0.978241\pi$
$643$		−	3.46410i		−	0.136611i	0.997664	−	0.0683054i	$-0.0217592\pi$
$644$			5.19615i			0.204757i
$645$	1.50000	−	2.59808i	0.0590624	−	0.102299i
$646$	4.50000	+	7.79423i	0.177050	+	0.306660i
$647$	4.50000	+	7.79423i	0.176913	+	0.306423i	0.940822	−	0.338902i	$-0.110055\pi$
$647$	4.50000	+	7.79423i	0.176913	+	0.306423i	−0.763908	+	0.645325i	$0.776722\pi$
$648$			15.5885i			0.612372i
$649$	−12.0000			−0.471041
$650$	−3.00000	+	12.1244i	−0.117670	+	0.475556i
$651$	−22.5000	−	12.9904i	−0.881845	−	0.509133i
$652$	−10.5000	−	6.06218i	−0.411212	−	0.237413i
$653$	22.5000	−	38.9711i	0.880493	−	1.52506i	0.0296993	−	0.999559i	$-0.490545\pi$
$653$	22.5000	−	38.9711i	0.880493	−	1.52506i	0.850794	−	0.525500i	$-0.176122\pi$
$654$		0			0
$655$	−22.5000	−	12.9904i	−0.879148	−	0.507576i
$656$	52.5000	+	30.3109i	2.04978	+	1.18344i
$657$			20.7846i			0.810885i
$658$			15.5885i			0.607701i
$659$	−19.5000	−	33.7750i	−0.759612	−	1.31569i	−0.943049	−	0.332655i	$-0.892055\pi$
$659$	−19.5000	−	33.7750i	−0.759612	−	1.31569i	0.183436	−	0.983032i	$-0.441278\pi$
$660$	−9.00000	+	5.19615i	−0.350325	+	0.202260i
$661$	−40.5000	−	23.3827i	−1.57527	−	0.909481i	−0.995506	−	0.0946945i	$-0.969813\pi$
$661$	−40.5000	−	23.3827i	−1.57527	−	0.909481i	−0.579761	−	0.814787i	$-0.696854\pi$
$662$	13.5000	+	23.3827i	0.524692	+	0.908794i
$663$	4.50000	−	18.1865i	0.174766	−	0.706306i
$664$	−4.50000	+	7.79423i	−0.174634	+	0.302475i
$665$	−4.50000	+	2.59808i	−0.174503	+	0.100749i
$666$	13.5000	+	23.3827i	0.523114	+	0.906061i
$667$	−9.00000	+	15.5885i	−0.348481	+	0.603587i
$668$	−4.50000	−	2.59808i	−0.174110	−	0.100523i
$669$	−6.00000			−0.231973
$670$			36.3731i			1.40521i
$671$	−15.0000	+	8.66025i	−0.579069	+	0.334325i
$672$	−13.5000	+	7.79423i	−0.520774	+	0.300669i
$673$	−34.0000			−1.31060			−0.655302	−	0.755367i	$-0.727459\pi$
$673$	−34.0000			−1.31060			−0.655302	+	0.755367i	$0.727459\pi$
$674$	−43.5000	+	25.1147i	−1.67556	+	0.967384i
$675$	−9.00000	−	5.19615i	−0.346410	−	0.200000i
$676$	11.0000	−	6.92820i	0.423077	−	0.266469i
$677$	10.5000	−	18.1865i	0.403548	−	0.698965i	−0.590603	−	0.806962i	$-0.701110\pi$
$677$	10.5000	−	18.1865i	0.403548	−	0.698965i	0.994151	+	0.107997i	$0.0344436\pi$
$678$	18.0000			0.691286
$679$	13.5000	−	23.3827i	0.518082	−	0.897345i
$680$	−9.00000			−0.345134
$681$	10.5000	+	18.1865i	0.402361	+	0.696909i
$682$	−45.0000	+	25.9808i	−1.72314	+	0.994855i
$683$	−19.5000	−	11.2583i	−0.746147	−	0.430788i	0.0781532	−	0.996941i	$-0.475098\pi$
$683$	−19.5000	−	11.2583i	−0.746147	−	0.430788i	−0.824300	+	0.566153i	$0.808431\pi$
$684$	−4.50000	−	2.59808i	−0.172062	−	0.0993399i
$685$	−4.50000	+	7.79423i	−0.171936	+	0.297802i
$686$	−33.0000			−1.25995
$687$	−7.50000	−	12.9904i	−0.286143	−	0.495614i
$688$	−2.50000	−	4.33013i	−0.0953116	−	0.165085i
$689$	−6.00000	−	20.7846i	−0.228582	−	0.791831i
$690$		−	15.5885i		−	0.593442i
$691$			17.3205i			0.658903i	0.944172	+	0.329452i	$0.106864\pi$
$691$			17.3205i			0.658903i	−0.944172	+	0.329452i	$0.893136\pi$
$692$	−10.5000	−	18.1865i	−0.399150	−	0.691348i
$693$	−9.00000	−	15.5885i	−0.341882	−	0.592157i
$694$		−	20.7846i		−	0.788973i
$695$	−24.0000	+	13.8564i	−0.910372	+	0.525603i
$696$	−18.0000			−0.682288
$697$	31.5000	−	18.1865i	1.19315	−	0.688864i
$698$	−48.0000			−1.81683
$699$			31.1769i			1.17922i
$700$		−	3.46410i		−	0.130931i
$701$	−30.0000			−1.13308			−0.566542	−	0.824033i	$-0.691719\pi$
$701$	−30.0000			−1.13308			−0.566542	+	0.824033i	$0.691719\pi$
$702$	9.00000	+	31.1769i	0.339683	+	1.17670i
$703$	9.00000			0.339441
$704$		−	3.46410i		−	0.130558i
$705$		−	15.5885i		−	0.587095i
$706$	12.0000			0.451626
$707$	−9.00000	+	5.19615i	−0.338480	+	0.195421i
$708$	−6.00000			−0.225494
$709$	−10.5000	+	6.06218i	−0.394336	+	0.227670i	−0.684037	−	0.729447i	$-0.739777\pi$
$709$	−10.5000	+	6.06218i	−0.394336	+	0.227670i	0.289701	+	0.957117i	$0.406444\pi$
$710$		−	25.9808i		−	0.975041i
$711$	−16.5000	+	28.5788i	−0.618798	+	1.07179i
$712$	13.5000	+	23.3827i	0.505934	+	0.876303i
$713$		−	25.9808i		−	0.972987i
$714$			15.5885i			0.583383i
$715$	15.0000	−	15.5885i	0.560968	−	0.582975i
$716$	1.50000	+	2.59808i	0.0560576	+	0.0970947i
$717$	4.50000	+	7.79423i	0.168056	+	0.291081i
$718$	−18.0000			−0.671754
$719$	19.5000	−	33.7750i	0.727227	−	1.25959i	−0.230823	−	0.972996i	$-0.574142\pi$
$719$	19.5000	−	33.7750i	0.727227	−	1.25959i	0.958051	−	0.286599i	$-0.0925247\pi$
$720$	22.5000	−	12.9904i	0.838525	−	0.484123i
$721$	19.5000	+	11.2583i	0.726218	+	0.419282i
$722$	24.0000	−	13.8564i	0.893188	−	0.515682i
$723$	4.50000	+	7.79423i	0.167357	+	0.289870i
$724$	22.0000			0.817624
$725$	6.00000	−	10.3923i	0.222834	−	0.385961i
$726$	−3.00000			−0.111340
$727$	−0.500000	+	0.866025i	−0.0185440	+	0.0321191i	−0.875148	−	0.483854i	$-0.839236\pi$
$727$	−0.500000	+	0.866025i	−0.0185440	+	0.0321191i	0.856605	+	0.515974i	$0.172570\pi$
$728$	7.50000	−	7.79423i	0.277968	−	0.288873i
$729$	−27.0000			−1.00000
$730$	18.0000	−	10.3923i	0.666210	−	0.384636i
$731$	−3.00000			−0.110959
$732$	−7.50000	+	4.33013i	−0.277208	+	0.160046i
$733$	−16.5000	+	9.52628i	−0.609441	+	0.351861i	−0.772747	−	0.634714i	$-0.781118\pi$
$733$	−16.5000	+	9.52628i	−0.609441	+	0.351861i	0.163305	+	0.986576i	$0.447784\pi$
$734$			13.8564i			0.511449i
$735$	12.0000			0.442627
$736$	−13.5000	−	7.79423i	−0.497617	−	0.287299i
$737$	−21.0000	+	36.3731i	−0.773545	+	1.33982i
$738$	−31.5000	+	54.5596i	−1.15953	+	2.00837i
$739$	−28.5000	+	16.4545i	−1.04839	+	0.605288i	−0.922198	−	0.386718i	$-0.873609\pi$
$739$	−28.5000	+	16.4545i	−1.04839	+	0.605288i	−0.126191	+	0.992006i	$0.540275\pi$
$740$	4.50000	−	7.79423i	0.165423	−	0.286522i
$741$	10.5000	+	2.59808i	0.385727	+	0.0954427i
$742$	9.00000	+	15.5885i	0.330400	+	0.572270i
$743$	10.5000	+	6.06218i	0.385208	+	0.222400i	0.680082	−	0.733136i	$-0.261944\pi$
$743$	10.5000	+	6.06218i	0.385208	+	0.222400i	−0.294874	+	0.955536i	$0.595278\pi$
$744$	22.5000	−	12.9904i	0.824890	−	0.476250i
$745$		0			0
$746$		−	24.2487i		−	0.887808i
$747$	−13.5000	−	7.79423i	−0.493939	−	0.285176i
$748$	9.00000	+	5.19615i	0.329073	+	0.189990i
$749$	−22.5000	−	12.9904i	−0.822132	−	0.474658i
$750$			36.3731i			1.32816i
$751$	9.50000	−	16.4545i	0.346660	−	0.600433i	−0.638994	−	0.769212i	$-0.720649\pi$
$751$	9.50000	−	16.4545i	0.346660	−	0.600433i	0.985654	+	0.168779i	$0.0539825\pi$
$752$	−22.5000	−	12.9904i	−0.820491	−	0.473710i
$753$	13.5000	+	7.79423i	0.491967	+	0.284037i
$754$	−36.0000	+	10.3923i	−1.31104	+	0.378465i
$755$	21.0000			0.764268
$756$	−4.50000	−	7.79423i	−0.163663	−	0.283473i
$757$	−3.50000	−	6.06218i	−0.127210	−	0.220334i	0.795385	−	0.606105i	$-0.207269\pi$
$757$	−3.50000	−	6.06218i	−0.127210	−	0.220334i	−0.922595	+	0.385771i	$0.873935\pi$
$758$	−10.5000	−	18.1865i	−0.381377	−	0.660565i
$759$	9.00000	−	15.5885i	0.326679	−	0.565825i
$760$		−	5.19615i		−	0.188484i
$761$			13.8564i			0.502294i	0.967949	+	0.251147i	$0.0808078\pi$
$761$			13.8564i			0.502294i	−0.967949	+	0.251147i	$0.919192\pi$
$762$	7.50000	+	12.9904i	0.271696	+	0.470592i
$763$		0			0
$764$	1.50000	+	2.59808i	0.0542681	+	0.0939951i
$765$		−	15.5885i		−	0.563602i
$766$	6.00000			0.216789
$767$	12.0000	−	3.46410i	0.433295	−	0.125081i
$768$		−	32.9090i		−	1.18750i
$769$	19.5000	+	11.2583i	0.703188	+	0.405986i	0.808534	−	0.588450i	$-0.200262\pi$
$769$	19.5000	+	11.2583i	0.703188	+	0.405986i	−0.105346	+	0.994436i	$0.533595\pi$
$770$	−9.00000	+	15.5885i	−0.324337	+	0.561769i
\(77

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 117 = 3^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	117.l (of order \(6\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+1.73205i q^{2} -1.73205i q^{3} -1.00000 q^{4} +(1.50000 - 0.866025i) q^{5} +3.00000 q^{6} +(1.50000 - 0.866025i) q^{7} +1.73205i q^{8} -3.00000 q^{9} +(1.50000 + 2.59808i) q^{10} +3.46410i q^{11} +1.73205i q^{12} +(-1.00000 - 3.46410i) q^{13} +(1.50000 + 2.59808i) q^{14} +(-1.50000 - 2.59808i) q^{15} -5.00000 q^{16} +(-1.50000 + 2.59808i) q^{17} -5.19615i q^{18} +(-1.50000 - 0.866025i) q^{19} +(-1.50000 + 0.866025i) q^{20} +(-1.50000 - 2.59808i) q^{21} -6.00000 q^{22} +(1.50000 - 2.59808i) q^{23} +3.00000 q^{24} +(-1.00000 + 1.73205i) q^{25} +(6.00000 - 1.73205i) q^{26} +5.19615i q^{27} +(-1.50000 + 0.866025i) q^{28} -6.00000 q^{29} +(4.50000 - 2.59808i) q^{30} +(7.50000 - 4.33013i) q^{31} -5.19615i q^{32} +6.00000 q^{33} +(-4.50000 - 2.59808i) q^{34} +(1.50000 - 2.59808i) q^{35} +3.00000 q^{36} +(-4.50000 + 2.59808i) q^{37} +(1.50000 - 2.59808i) q^{38} +(-6.00000 + 1.73205i) q^{39} +(1.50000 + 2.59808i) q^{40} +(-10.5000 - 6.06218i) q^{41} +(4.50000 - 2.59808i) q^{42} +(0.500000 + 0.866025i) q^{43} -3.46410i q^{44} +(-4.50000 + 2.59808i) q^{45} +(4.50000 + 2.59808i) q^{46} +(4.50000 + 2.59808i) q^{47} +8.66025i q^{48} +(-2.00000 + 3.46410i) q^{49} +(-3.00000 - 1.73205i) q^{50} +(4.50000 + 2.59808i) q^{51} +(1.00000 + 3.46410i) q^{52} +6.00000 q^{53} -9.00000 q^{54} +(3.00000 + 5.19615i) q^{55} +(1.50000 + 2.59808i) q^{56} +(-1.50000 + 2.59808i) q^{57} -10.3923i q^{58} +3.46410i q^{59} +(1.50000 + 2.59808i) q^{60} +(2.50000 + 4.33013i) q^{61} +(7.50000 + 12.9904i) q^{62} +(-4.50000 + 2.59808i) q^{63} -1.00000 q^{64} +(-4.50000 - 4.33013i) q^{65} +10.3923i q^{66} +(10.5000 + 6.06218i) q^{67} +(1.50000 - 2.59808i) q^{68} +(-4.50000 - 2.59808i) q^{69} +(4.50000 + 2.59808i) q^{70} +(-7.50000 - 4.33013i) q^{71} -5.19615i q^{72} -6.92820i q^{73} +(-4.50000 - 7.79423i) q^{74} +(3.00000 + 1.73205i) q^{75} +(1.50000 + 0.866025i) q^{76} +(3.00000 + 5.19615i) q^{77} +(-3.00000 - 10.3923i) q^{78} +(5.50000 - 9.52628i) q^{79} +(-7.50000 + 4.33013i) q^{80} +9.00000 q^{81} +(10.5000 - 18.1865i) q^{82} +(4.50000 + 2.59808i) q^{83} +(1.50000 + 2.59808i) q^{84} +5.19615i q^{85} +(-1.50000 + 0.866025i) q^{86} +10.3923i q^{87} -6.00000 q^{88} +(13.5000 - 7.79423i) q^{89} +(-4.50000 - 7.79423i) q^{90} +(-4.50000 - 4.33013i) q^{91} +(-1.50000 + 2.59808i) q^{92} +(-7.50000 - 12.9904i) q^{93} +(-4.50000 + 7.79423i) q^{94} -3.00000 q^{95} -9.00000 q^{96} +(13.5000 - 7.79423i) q^{97} +(-6.00000 - 3.46410i) q^{98} -10.3923i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{4} + 3 q^{5} + 6 q^{6} + 3 q^{7} - 6 q^{9} + 3 q^{10} - 2 q^{13} + 3 q^{14} - 3 q^{15} - 10 q^{16} - 3 q^{17} - 3 q^{19} - 3 q^{20} - 3 q^{21} - 12 q^{22} + 3 q^{23} + 6 q^{24} - 2 q^{25} + 12 q^{26}+ \cdots - 12 q^{98}+O(q^{100}) \) 2 * q - 2 * q^4 + 3 * q^5 + 6 * q^6 + 3 * q^7 - 6 * q^9 + 3 * q^10 - 2 * q^13 + 3 * q^14 - 3 * q^15 - 10 * q^16 - 3 * q^17 - 3 * q^19 - 3 * q^20 - 3 * q^21 - 12 * q^22 + 3 * q^23 + 6 * q^24 - 2 * q^25 + 12 * q^26 - 3 * q^28 - 12 * q^29 + 9 * q^30 + 15 * q^31 + 12 * q^33 - 9 * q^34 + 3 * q^35 + 6 * q^36 - 9 * q^37 + 3 * q^38 - 12 * q^39 + 3 * q^40 - 21 * q^41 + 9 * q^42 + q^43 - 9 * q^45 + 9 * q^46 + 9 * q^47 - 4 * q^49 - 6 * q^50 + 9 * q^51 + 2 * q^52 + 12 * q^53 - 18 * q^54 + 6 * q^55 + 3 * q^56 - 3 * q^57 + 3 * q^60 + 5 * q^61 + 15 * q^62 - 9 * q^63 - 2 * q^64 - 9 * q^65 + 21 * q^67 + 3 * q^68 - 9 * q^69 + 9 * q^70 - 15 * q^71 - 9 * q^74 + 6 * q^75 + 3 * q^76 + 6 * q^77 - 6 * q^78 + 11 * q^79 - 15 * q^80 + 18 * q^81 + 21 * q^82 + 9 * q^83 + 3 * q^84 - 3 * q^86 - 12 * q^88 + 27 * q^89 - 9 * q^90 - 9 * q^91 - 3 * q^92 - 15 * q^93 - 9 * q^94 - 6 * q^95 - 18 * q^96 + 27 * q^97 - 12 * q^98

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