LMFDB - Embedded newform 39.2.j.a.4.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [39,2,Mod(4,39)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(39, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 1]))

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

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

chi := DirichletCharacter("39.4");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 39 = 3 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	39.j (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:	$0.311416567883$
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:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			4.1
Root			$0.500000 + 0.866025i$ of defining polynomial
Character	$\chi$	$=$	39.4
Dual form			39.2.j.a.10.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(0.500000 - 0.866025i) q^{3} +(-1.00000 - 1.73205i) q^{4} +3.46410i q^{5} +(-1.50000 + 0.866025i) q^{7} +(-0.500000 - 0.866025i) q^{9} +O(q^{10})$	\(q+(0.500000 - 0.866025i) q^{3} +(-1.00000 - 1.73205i) q^{4} +3.46410i q^{5} +(-1.50000 + 0.866025i) q^{7} +(-0.500000 - 0.866025i) q^{9} +(-3.00000 - 1.73205i) q^{11} -2.00000 q^{12} +(3.50000 - 0.866025i) q^{13} +(3.00000 + 1.73205i) q^{15} +(-2.00000 + 3.46410i) q^{16} +(3.00000 - 1.73205i) q^{19} +(6.00000 - 3.46410i) q^{20} +1.73205i q^{21} +(3.00000 - 5.19615i) q^{23} -7.00000 q^{25} -1.00000 q^{27} +(3.00000 + 1.73205i) q^{28} +(-3.00000 + 5.19615i) q^{29} +1.73205i q^{31} +(-3.00000 + 1.73205i) q^{33} +(-3.00000 - 5.19615i) q^{35} +(-1.00000 + 1.73205i) q^{36} +(1.00000 - 3.46410i) q^{39} +(-6.00000 - 3.46410i) q^{41} +(0.500000 + 0.866025i) q^{43} +6.92820i q^{44} +(3.00000 - 1.73205i) q^{45} +3.46410i q^{47} +(2.00000 + 3.46410i) q^{48} +(-2.00000 + 3.46410i) q^{49} +(-5.00000 - 5.19615i) q^{52} +12.0000 q^{53} +(6.00000 - 10.3923i) q^{55} -3.46410i q^{57} +(-3.00000 + 1.73205i) q^{59} -6.92820i q^{60} +(-0.500000 - 0.866025i) q^{61} +(1.50000 + 0.866025i) q^{63} +8.00000 q^{64} +(3.00000 + 12.1244i) q^{65} +(7.50000 + 4.33013i) q^{67} +(-3.00000 - 5.19615i) q^{69} +(-9.00000 + 5.19615i) q^{71} +1.73205i q^{73} +(-3.50000 + 6.06218i) q^{75} +(-6.00000 - 3.46410i) q^{76} +6.00000 q^{77} -11.0000 q^{79} +(-12.0000 - 6.92820i) q^{80} +(-0.500000 + 0.866025i) q^{81} -13.8564i q^{83} +(3.00000 - 1.73205i) q^{84} +(3.00000 + 5.19615i) q^{87} +(6.00000 + 3.46410i) q^{89} +(-4.50000 + 4.33013i) q^{91} -12.0000 q^{92} +(1.50000 + 0.866025i) q^{93} +(6.00000 + 10.3923i) q^{95} +(-4.50000 + 2.59808i) q^{97} +3.46410i q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + q^{3} - 2 q^{4} - 3 q^{7} - q^{9}+O(q^{10}) $ 2 * q + q^3 - 2 * q^4 - 3 * q^7 - q^9	$ 2 q + q^{3} - 2 q^{4} - 3 q^{7} - q^{9} - 6 q^{11} - 4 q^{12} + 7 q^{13} + 6 q^{15} - 4 q^{16} + 6 q^{19} + 12 q^{20} + 6 q^{23} - 14 q^{25} - 2 q^{27} + 6 q^{28} - 6 q^{29} - 6 q^{33} - 6 q^{35} - 2 q^{36} + 2 q^{39} - 12 q^{41} + q^{43} + 6 q^{45} + 4 q^{48} - 4 q^{49} - 10 q^{52} + 24 q^{53} + 12 q^{55} - 6 q^{59} - q^{61} + 3 q^{63} + 16 q^{64} + 6 q^{65} + 15 q^{67} - 6 q^{69} - 18 q^{71} - 7 q^{75} - 12 q^{76} + 12 q^{77} - 22 q^{79} - 24 q^{80} - q^{81} + 6 q^{84} + 6 q^{87} + 12 q^{89} - 9 q^{91} - 24 q^{92} + 3 q^{93} + 12 q^{95} - 9 q^{97}+O(q^{100}) $ 2 * q + q^3 - 2 * q^4 - 3 * q^7 - q^9 - 6 * q^11 - 4 * q^12 + 7 * q^13 + 6 * q^15 - 4 * q^16 + 6 * q^19 + 12 * q^20 + 6 * q^23 - 14 * q^25 - 2 * q^27 + 6 * q^28 - 6 * q^29 - 6 * q^33 - 6 * q^35 - 2 * q^36 + 2 * q^39 - 12 * q^41 + q^43 + 6 * q^45 + 4 * q^48 - 4 * q^49 - 10 * q^52 + 24 * q^53 + 12 * q^55 - 6 * q^59 - q^61 + 3 * q^63 + 16 * q^64 + 6 * q^65 + 15 * q^67 - 6 * q^69 - 18 * q^71 - 7 * q^75 - 12 * q^76 + 12 * q^77 - 22 * q^79 - 24 * q^80 - q^81 + 6 * q^84 + 6 * q^87 + 12 * q^89 - 9 * q^91 - 24 * q^92 + 3 * q^93 + 12 * q^95 - 9 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/39\mathbb{Z}\right)^\times$.

$n$	$14$	$28$
$\chi(n)$	$1$	$e\left(\frac{1}{6}\right)$

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 $

$2$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$2$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$3$	0.500000	−	0.866025i	0.288675	−	0.500000i
$3$	0.500000	−	0.866025i	0.288675	−	0.500000i
$4$	−1.00000	−	1.73205i	−0.500000	−	0.866025i
$5$			3.46410i			1.54919i	0.632456	+	0.774597i	$0.282047\pi$
$5$			3.46410i			1.54919i	−0.632456	+	0.774597i	$0.717953\pi$
$6$		0			0
$7$	−1.50000	+	0.866025i	−0.566947	+	0.327327i	−0.755929	−	0.654654i	$-0.772814\pi$
$7$	−1.50000	+	0.866025i	−0.566947	+	0.327327i	0.188982	+	0.981981i	$0.439481\pi$
$8$		0			0
$9$	−0.500000	−	0.866025i	−0.166667	−	0.288675i
$10$		0			0
$11$	−3.00000	−	1.73205i	−0.904534	−	0.522233i	−0.0258656	−	0.999665i	$-0.508234\pi$
$11$	−3.00000	−	1.73205i	−0.904534	−	0.522233i	−0.878668	+	0.477432i	$0.841568\pi$
$12$	−2.00000			−0.577350
$13$	3.50000	−	0.866025i	0.970725	−	0.240192i
$13$	3.50000	−	0.866025i	0.970725	−	0.240192i
$14$		0			0
$15$	3.00000	+	1.73205i	0.774597	+	0.447214i
$16$	−2.00000	+	3.46410i	−0.500000	+	0.866025i
$17$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$17$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$18$		0			0
$19$	3.00000	−	1.73205i	0.688247	−	0.397360i	−0.114708	−	0.993399i	$-0.536593\pi$
$19$	3.00000	−	1.73205i	0.688247	−	0.397360i	0.802955	+	0.596040i	$0.203260\pi$
$20$	6.00000	−	3.46410i	1.34164	−	0.774597i
$21$			1.73205i			0.377964i
$22$		0			0
$23$	3.00000	−	5.19615i	0.625543	−	1.08347i	−0.362892	−	0.931831i	$-0.618211\pi$
$23$	3.00000	−	5.19615i	0.625543	−	1.08347i	0.988436	−	0.151642i	$-0.0484560\pi$
$24$		0			0
$25$	−7.00000			−1.40000
$26$		0			0
$27$	−1.00000			−0.192450
$28$	3.00000	+	1.73205i	0.566947	+	0.327327i
$29$	−3.00000	+	5.19615i	−0.557086	+	0.964901i	0.440652	+	0.897678i	$0.354747\pi$
$29$	−3.00000	+	5.19615i	−0.557086	+	0.964901i	−0.997738	+	0.0672232i	$0.978586\pi$
$30$		0			0
$31$			1.73205i			0.311086i	0.987829	+	0.155543i	$0.0497126\pi$
$31$			1.73205i			0.311086i	−0.987829	+	0.155543i	$0.950287\pi$
$32$		0			0
$33$	−3.00000	+	1.73205i	−0.522233	+	0.301511i
$34$		0			0
$35$	−3.00000	−	5.19615i	−0.507093	−	0.878310i
$36$	−1.00000	+	1.73205i	−0.166667	+	0.288675i
$37$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$37$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$38$		0			0
$39$	1.00000	−	3.46410i	0.160128	−	0.554700i
$40$		0			0
$41$	−6.00000	−	3.46410i	−0.937043	−	0.541002i	−0.0480106	−	0.998847i	$-0.515288\pi$
$41$	−6.00000	−	3.46410i	−0.937043	−	0.541002i	−0.889032	+	0.457845i	$0.848621\pi$
$42$		0			0
$43$	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	$0.809039\pi$
$44$			6.92820i			1.04447i
$45$	3.00000	−	1.73205i	0.447214	−	0.258199i
$46$		0			0
$47$			3.46410i			0.505291i	0.967559	+	0.252646i	$0.0813007\pi$
$47$			3.46410i			0.505291i	−0.967559	+	0.252646i	$0.918699\pi$
$48$	2.00000	+	3.46410i	0.288675	+	0.500000i
$49$	−2.00000	+	3.46410i	−0.285714	+	0.494872i
$50$		0			0
$51$		0			0
$52$	−5.00000	−	5.19615i	−0.693375	−	0.720577i
$53$	12.0000			1.64833			0.824163	−	0.566352i	$-0.191646\pi$
$53$	12.0000			1.64833			0.824163	+	0.566352i	$0.191646\pi$
$54$		0			0
$55$	6.00000	−	10.3923i	0.809040	−	1.40130i
$56$		0			0
$57$		−	3.46410i		−	0.458831i
$58$		0			0
$59$	−3.00000	+	1.73205i	−0.390567	+	0.225494i	−0.682406	−	0.730974i	$-0.739066\pi$
$59$	−3.00000	+	1.73205i	−0.390567	+	0.225494i	0.291839	+	0.956467i	$0.405733\pi$
$60$		−	6.92820i		−	0.894427i
$61$	−0.500000	−	0.866025i	−0.0640184	−	0.110883i	0.832240	−	0.554416i	$-0.187058\pi$
$61$	−0.500000	−	0.866025i	−0.0640184	−	0.110883i	−0.896258	+	0.443533i	$0.853725\pi$
$62$		0			0
$63$	1.50000	+	0.866025i	0.188982	+	0.109109i
$64$	8.00000			1.00000
$65$	3.00000	+	12.1244i	0.372104	+	1.50384i
$66$		0			0
$67$	7.50000	+	4.33013i	0.916271	+	0.529009i	0.882443	−	0.470418i	$-0.155897\pi$
$67$	7.50000	+	4.33013i	0.916271	+	0.529009i	0.0338274	+	0.999428i	$0.489230\pi$
$68$		0			0
$69$	−3.00000	−	5.19615i	−0.361158	−	0.625543i
$70$		0			0
$71$	−9.00000	+	5.19615i	−1.06810	+	0.616670i	−0.927663	−	0.373419i	$-0.878185\pi$
$71$	−9.00000	+	5.19615i	−1.06810	+	0.616670i	−0.140441	+	0.990089i	$0.544852\pi$
$72$		0			0
$73$			1.73205i			0.202721i	0.994850	+	0.101361i	$0.0323196\pi$
$73$			1.73205i			0.202721i	−0.994850	+	0.101361i	$0.967680\pi$
$74$		0			0
$75$	−3.50000	+	6.06218i	−0.404145	+	0.700000i
$76$	−6.00000	−	3.46410i	−0.688247	−	0.397360i
$77$	6.00000			0.683763
$78$		0			0
$79$	−11.0000			−1.23760			−0.618798	−	0.785550i	$-0.712380\pi$
$79$	−11.0000			−1.23760			−0.618798	+	0.785550i	$0.712380\pi$
$80$	−12.0000	−	6.92820i	−1.34164	−	0.774597i
$81$	−0.500000	+	0.866025i	−0.0555556	+	0.0962250i
$82$		0			0
$83$		−	13.8564i		−	1.52094i	−0.649374	−	0.760469i	$-0.724969\pi$
$83$		−	13.8564i		−	1.52094i	0.649374	−	0.760469i	$-0.275031\pi$
$84$	3.00000	−	1.73205i	0.327327	−	0.188982i
$85$		0			0
$86$		0			0
$87$	3.00000	+	5.19615i	0.321634	+	0.557086i
$88$		0			0
$89$	6.00000	+	3.46410i	0.635999	+	0.367194i	0.783072	−	0.621932i	$-0.213652\pi$
$89$	6.00000	+	3.46410i	0.635999	+	0.367194i	−0.147073	+	0.989126i	$0.546985\pi$
$90$		0			0
$91$	−4.50000	+	4.33013i	−0.471728	+	0.453921i
$92$	−12.0000			−1.25109
$93$	1.50000	+	0.866025i	0.155543	+	0.0898027i
$94$		0			0
$95$	6.00000	+	10.3923i	0.615587	+	1.06623i
$96$		0			0
$97$	−4.50000	+	2.59808i	−0.456906	+	0.263795i	−0.710742	−	0.703452i	$-0.751641\pi$
$97$	−4.50000	+	2.59808i	−0.456906	+	0.263795i	0.253837	+	0.967247i	$0.418307\pi$
$98$		0			0
$99$			3.46410i			0.348155i
$100$	7.00000	+	12.1244i	0.700000	+	1.21244i
$101$	9.00000	−	15.5885i	0.895533	−	1.55111i	0.0623905	−	0.998052i	$-0.480128\pi$
$101$	9.00000	−	15.5885i	0.895533	−	1.55111i	0.833143	−	0.553058i	$-0.186539\pi$
$102$		0			0
$103$	−1.00000			−0.0985329			−0.0492665	−	0.998786i	$-0.515688\pi$
$103$	−1.00000			−0.0985329			−0.0492665	+	0.998786i	$0.515688\pi$
$104$		0			0
$105$	−6.00000			−0.585540
$106$		0			0
$107$	3.00000	−	5.19615i	0.290021	−	0.502331i	−0.683793	−	0.729676i	$-0.739671\pi$
$107$	3.00000	−	5.19615i	0.290021	−	0.502331i	0.973814	+	0.227345i	$0.0730044\pi$
$108$	1.00000	+	1.73205i	0.0962250	+	0.166667i
$109$		−	15.5885i		−	1.49310i	−0.665327	−	0.746552i	$-0.731708\pi$
$109$		−	15.5885i		−	1.49310i	0.665327	−	0.746552i	$-0.268292\pi$
$110$		0			0
$111$		0			0
$112$		−	6.92820i		−	0.654654i
$113$	−3.00000	−	5.19615i	−0.282216	−	0.488813i	0.689714	−	0.724082i	$-0.257736\pi$
$113$	−3.00000	−	5.19615i	−0.282216	−	0.488813i	−0.971930	+	0.235269i	$0.924403\pi$
$114$		0			0
$115$	18.0000	+	10.3923i	1.67851	+	0.969087i
$116$	12.0000			1.11417
$117$	−2.50000	−	2.59808i	−0.231125	−	0.240192i
$118$		0			0
$119$		0			0
$120$		0			0
$121$	0.500000	+	0.866025i	0.0454545	+	0.0787296i
$122$		0			0
$123$	−6.00000	+	3.46410i	−0.541002	+	0.312348i
$124$	3.00000	−	1.73205i	0.269408	−	0.155543i
$125$		−	6.92820i		−	0.619677i
$126$		0			0
$127$	−6.50000	+	11.2583i	−0.576782	+	0.999015i	0.419064	+	0.907957i	$0.362358\pi$
$127$	−6.50000	+	11.2583i	−0.576782	+	0.999015i	−0.995846	+	0.0910585i	$0.970975\pi$
$128$		0			0
$129$	1.00000			0.0880451
$130$		0			0
$131$	6.00000			0.524222			0.262111	−	0.965038i	$-0.415581\pi$
$131$	6.00000			0.524222			0.262111	+	0.965038i	$0.415581\pi$
$132$	6.00000	+	3.46410i	0.522233	+	0.301511i
$133$	−3.00000	+	5.19615i	−0.260133	+	0.450564i
$134$		0			0
$135$		−	3.46410i		−	0.298142i
$136$		0			0
$137$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$137$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$138$		0			0
$139$	−2.50000	−	4.33013i	−0.212047	−	0.367277i	0.740308	−	0.672268i	$-0.234680\pi$
$139$	−2.50000	−	4.33013i	−0.212047	−	0.367277i	−0.952355	+	0.304991i	$0.901346\pi$
$140$	−6.00000	+	10.3923i	−0.507093	+	0.878310i
$141$	3.00000	+	1.73205i	0.252646	+	0.145865i
$142$		0			0
$143$	−12.0000	−	3.46410i	−1.00349	−	0.289683i
$144$	4.00000			0.333333
$145$	−18.0000	−	10.3923i	−1.49482	−	0.863034i
$146$		0			0
$147$	2.00000	+	3.46410i	0.164957	+	0.285714i
$148$		0			0
$149$	−6.00000	+	3.46410i	−0.491539	+	0.283790i	−0.725213	−	0.688525i	$-0.758259\pi$
$149$	−6.00000	+	3.46410i	−0.491539	+	0.283790i	0.233674	+	0.972315i	$0.424925\pi$
$150$		0			0
$151$		−	3.46410i		−	0.281905i	−0.990016	−	0.140952i	$-0.954984\pi$
$151$		−	3.46410i		−	0.281905i	0.990016	−	0.140952i	$-0.0450164\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−6.00000			−0.481932
$156$	−7.00000	+	1.73205i	−0.560449	+	0.138675i
$157$	11.0000			0.877896			0.438948	−	0.898513i	$-0.355351\pi$
$157$	11.0000			0.877896			0.438948	+	0.898513i	$0.355351\pi$
$158$		0			0
$159$	6.00000	−	10.3923i	0.475831	−	0.824163i
$160$		0			0
$161$			10.3923i			0.819028i
$162$		0			0
$163$	−16.5000	+	9.52628i	−1.29238	+	0.746156i	−0.979076	−	0.203497i	$-0.934769\pi$
$163$	−16.5000	+	9.52628i	−1.29238	+	0.746156i	−0.313304	+	0.949653i	$0.601436\pi$
$164$			13.8564i			1.08200i
$165$	−6.00000	−	10.3923i	−0.467099	−	0.809040i
$166$		0			0
$167$	−6.00000	−	3.46410i	−0.464294	−	0.268060i	0.249554	−	0.968361i	$-0.419716\pi$
$167$	−6.00000	−	3.46410i	−0.464294	−	0.268060i	−0.713848	+	0.700301i	$0.753049\pi$
$168$		0			0
$169$	11.5000	−	6.06218i	0.884615	−	0.466321i
$170$		0			0
$171$	−3.00000	−	1.73205i	−0.229416	−	0.132453i
$172$	1.00000	−	1.73205i	0.0762493	−	0.132068i
$173$	3.00000	+	5.19615i	0.228086	+	0.395056i	0.957241	−	0.289292i	$-0.0934200\pi$
$173$	3.00000	+	5.19615i	0.228086	+	0.395056i	−0.729155	+	0.684349i	$0.760087\pi$
$174$		0			0
$175$	10.5000	−	6.06218i	0.793725	−	0.458258i
$176$	12.0000	−	6.92820i	0.904534	−	0.522233i
$177$			3.46410i			0.260378i
$178$		0			0
$179$	−6.00000	+	10.3923i	−0.448461	+	0.776757i	−0.998286	−	0.0585225i	$-0.981361\pi$
$179$	−6.00000	+	10.3923i	−0.448461	+	0.776757i	0.549825	+	0.835280i	$0.314694\pi$
$180$	−6.00000	−	3.46410i	−0.447214	−	0.258199i
$181$	14.0000			1.04061			0.520306	−	0.853980i	$-0.325818\pi$
$181$	14.0000			1.04061			0.520306	+	0.853980i	$0.325818\pi$
$182$		0			0
$183$	−1.00000			−0.0739221
$184$		0			0
$185$		0			0
$186$		0			0
$187$		0			0
$188$	6.00000	−	3.46410i	0.437595	−	0.252646i
$189$	1.50000	−	0.866025i	0.109109	−	0.0629941i
$190$		0			0
$191$	9.00000	+	15.5885i	0.651217	+	1.12794i	0.982828	+	0.184525i	$0.0590746\pi$
$191$	9.00000	+	15.5885i	0.651217	+	1.12794i	−0.331611	+	0.943416i	$0.607592\pi$
$192$	4.00000	−	6.92820i	0.288675	−	0.500000i
$193$	−13.5000	−	7.79423i	−0.971751	−	0.561041i	−0.0719816	−	0.997406i	$-0.522932\pi$
$193$	−13.5000	−	7.79423i	−0.971751	−	0.561041i	−0.899770	+	0.436365i	$0.856266\pi$
$194$		0			0
$195$	12.0000	+	3.46410i	0.859338	+	0.248069i
$196$	8.00000			0.571429
$197$	12.0000	+	6.92820i	0.854965	+	0.493614i	0.862323	−	0.506359i	$-0.169009\pi$
$197$	12.0000	+	6.92820i	0.854965	+	0.493614i	−0.00735824	+	0.999973i	$0.502342\pi$
$198$		0			0
$199$	−3.50000	−	6.06218i	−0.248108	−	0.429736i	0.714893	−	0.699234i	$-0.246476\pi$
$199$	−3.50000	−	6.06218i	−0.248108	−	0.429736i	−0.963001	+	0.269498i	$0.913142\pi$
$200$		0			0
$201$	7.50000	−	4.33013i	0.529009	−	0.305424i
$202$		0			0
$203$		−	10.3923i		−	0.729397i
$204$		0			0
$205$	12.0000	−	20.7846i	0.838116	−	1.45166i
$206$		0			0
$207$	−6.00000			−0.417029
$208$	−4.00000	+	13.8564i	−0.277350	+	0.960769i
$209$	−12.0000			−0.830057
$210$		0			0
$211$	−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	$0.314345\pi$
$212$	−12.0000	−	20.7846i	−0.824163	−	1.42749i
$213$			10.3923i			0.712069i
$214$		0			0
$215$	−3.00000	+	1.73205i	−0.204598	+	0.118125i
$216$		0			0
$217$	−1.50000	−	2.59808i	−0.101827	−	0.176369i
$218$		0			0
$219$	1.50000	+	0.866025i	0.101361	+	0.0585206i
$220$	−24.0000			−1.61808
$221$		0			0
$222$		0			0
$223$	15.0000	+	8.66025i	1.00447	+	0.579934i	0.909569	−	0.415553i	$-0.136412\pi$
$223$	15.0000	+	8.66025i	1.00447	+	0.579934i	0.0949052	+	0.995486i	$0.469745\pi$
$224$		0			0
$225$	3.50000	+	6.06218i	0.233333	+	0.404145i
$226$		0			0
$227$	18.0000	−	10.3923i	1.19470	−	0.689761i	0.235333	−	0.971915i	$-0.424382\pi$
$227$	18.0000	−	10.3923i	1.19470	−	0.689761i	0.959369	+	0.282153i	$0.0910487\pi$
$228$	−6.00000	+	3.46410i	−0.397360	+	0.229416i
$229$			27.7128i			1.83131i	0.401960	+	0.915657i	$0.368329\pi$
$229$			27.7128i			1.83131i	−0.401960	+	0.915657i	$0.631671\pi$
$230$		0			0
$231$	3.00000	−	5.19615i	0.197386	−	0.341882i
$232$		0			0
$233$	−18.0000			−1.17922			−0.589610	−	0.807688i	$-0.700718\pi$
$233$	−18.0000			−1.17922			−0.589610	+	0.807688i	$0.700718\pi$
$234$		0			0
$235$	−12.0000			−0.782794
$236$	6.00000	+	3.46410i	0.390567	+	0.225494i
$237$	−5.50000	+	9.52628i	−0.357263	+	0.618798i
$238$		0			0
$239$		0			0		1.00000			$0$
$239$		0			0		−1.00000			$\pi$
$240$	−12.0000	+	6.92820i	−0.774597	+	0.447214i
$241$	18.0000	−	10.3923i	1.15948	−	0.669427i	0.208302	−	0.978065i	$-0.433206\pi$
$241$	18.0000	−	10.3923i	1.15948	−	0.669427i	0.951180	+	0.308637i	$0.0998729\pi$
$242$		0			0
$243$	0.500000	+	0.866025i	0.0320750	+	0.0555556i
$244$	−1.00000	+	1.73205i	−0.0640184	+	0.110883i
$245$	−12.0000	−	6.92820i	−0.766652	−	0.442627i
$246$		0			0
$247$	9.00000	−	8.66025i	0.572656	−	0.551039i
$248$		0			0
$249$	−12.0000	−	6.92820i	−0.760469	−	0.439057i
$250$		0			0
$251$	−6.00000	−	10.3923i	−0.378717	−	0.655956i	0.612159	−	0.790735i	$-0.290301\pi$
$251$	−6.00000	−	10.3923i	−0.378717	−	0.655956i	−0.990876	+	0.134778i	$0.956968\pi$
$252$		−	3.46410i		−	0.218218i
$253$	−18.0000	+	10.3923i	−1.13165	+	0.653359i
$254$		0			0
$255$		0			0
$256$	−8.00000	−	13.8564i	−0.500000	−	0.866025i
$257$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$257$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$258$		0			0
$259$		0			0
$260$	18.0000	−	17.3205i	1.11631	−	1.07417i
$261$	6.00000			0.371391
$262$		0			0
$263$	−6.00000	+	10.3923i	−0.369976	+	0.640817i	−0.989561	−	0.144112i	$-0.953967\pi$
$263$	−6.00000	+	10.3923i	−0.369976	+	0.640817i	0.619586	+	0.784929i	$0.287301\pi$
$264$		0			0
$265$			41.5692i			2.55358i
$266$		0			0
$267$	6.00000	−	3.46410i	0.367194	−	0.212000i
$268$		−	17.3205i		−	1.05802i
$269$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$269$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$270$		0			0
$271$	−4.50000	−	2.59808i	−0.273356	−	0.157822i	0.357056	−	0.934083i	$-0.383781\pi$
$271$	−4.50000	−	2.59808i	−0.273356	−	0.157822i	−0.630412	+	0.776261i	$0.717114\pi$
$272$		0			0
$273$	1.50000	+	6.06218i	0.0907841	+	0.366900i
$274$		0			0
$275$	21.0000	+	12.1244i	1.26635	+	0.731126i
$276$	−6.00000	+	10.3923i	−0.361158	+	0.625543i
$277$	5.00000	+	8.66025i	0.300421	+	0.520344i	0.976231	−	0.216731i	$-0.0695395\pi$
$277$	5.00000	+	8.66025i	0.300421	+	0.520344i	−0.675810	+	0.737075i	$0.736206\pi$
$278$		0			0
$279$	1.50000	−	0.866025i	0.0898027	−	0.0518476i
$280$		0			0
$281$		−	24.2487i		−	1.44656i	−0.690557	−	0.723278i	$-0.742634\pi$
$281$		−	24.2487i		−	1.44656i	0.690557	−	0.723278i	$-0.257366\pi$
$282$		0			0
$283$	5.50000	−	9.52628i	0.326941	−	0.566279i	−0.654962	−	0.755662i	$-0.727315\pi$
$283$	5.50000	−	9.52628i	0.326941	−	0.566279i	0.981903	+	0.189383i	$0.0606488\pi$
$284$	18.0000	+	10.3923i	1.06810	+	0.616670i
$285$	12.0000			0.710819
$286$		0			0
$287$	12.0000			0.708338
$288$		0			0
$289$	8.50000	−	14.7224i	0.500000	−	0.866025i
$290$		0			0
$291$			5.19615i			0.304604i
$292$	3.00000	−	1.73205i	0.175562	−	0.101361i
$293$	−15.0000	+	8.66025i	−0.876309	+	0.505937i	−0.869440	−	0.494039i	$-0.835520\pi$
$293$	−15.0000	+	8.66025i	−0.876309	+	0.505937i	−0.00686959	+	0.999976i	$0.502187\pi$
$294$		0			0
$295$	−6.00000	−	10.3923i	−0.349334	−	0.605063i
$296$		0			0
$297$	3.00000	+	1.73205i	0.174078	+	0.100504i
$298$		0			0
$299$	6.00000	−	20.7846i	0.346989	−	1.20201i
$300$	14.0000			0.808290
$301$	−1.50000	−	0.866025i	−0.0864586	−	0.0499169i
$302$		0			0
$303$	−9.00000	−	15.5885i	−0.517036	−	0.895533i
$304$			13.8564i			0.794719i
$305$	3.00000	−	1.73205i	0.171780	−	0.0991769i
$306$		0			0
$307$		−	1.73205i		−	0.0988534i	−0.998778	−	0.0494267i	$-0.984261\pi$
$307$		−	1.73205i		−	0.0988534i	0.998778	−	0.0494267i	$-0.0157394\pi$
$308$	−6.00000	−	10.3923i	−0.341882	−	0.592157i
$309$	−0.500000	+	0.866025i	−0.0284440	+	0.0492665i
$310$		0			0
$311$	−24.0000			−1.36092			−0.680458	−	0.732787i	$-0.738219\pi$
$311$	−24.0000			−1.36092			−0.680458	+	0.732787i	$0.738219\pi$
$312$		0			0
$313$	13.0000			0.734803			0.367402	−	0.930062i	$-0.380247\pi$
$313$	13.0000			0.734803			0.367402	+	0.930062i	$0.380247\pi$
$314$		0			0
$315$	−3.00000	+	5.19615i	−0.169031	+	0.292770i
$316$	11.0000	+	19.0526i	0.618798	+	1.07179i
$317$			6.92820i			0.389127i	0.980890	+	0.194563i	$0.0623290\pi$
$317$			6.92820i			0.389127i	−0.980890	+	0.194563i	$0.937671\pi$
$318$		0			0
$319$	18.0000	−	10.3923i	1.00781	−	0.581857i
$320$			27.7128i			1.54919i
$321$	−3.00000	−	5.19615i	−0.167444	−	0.290021i
$322$		0			0
$323$		0			0
$324$	2.00000			0.111111
$325$	−24.5000	+	6.06218i	−1.35902	+	0.336269i
$326$		0			0
$327$	−13.5000	−	7.79423i	−0.746552	−	0.431022i
$328$		0			0
$329$	−3.00000	−	5.19615i	−0.165395	−	0.286473i
$330$		0			0
$331$	4.50000	−	2.59808i	0.247342	−	0.142803i	−0.371204	−	0.928551i	$-0.621055\pi$
$331$	4.50000	−	2.59808i	0.247342	−	0.142803i	0.618547	+	0.785748i	$0.287722\pi$
$332$	−24.0000	+	13.8564i	−1.31717	+	0.760469i
$333$		0			0
$334$		0			0
$335$	−15.0000	+	25.9808i	−0.819538	+	1.41948i
$336$	−6.00000	−	3.46410i	−0.327327	−	0.188982i
$337$	−5.00000			−0.272367			−0.136184	−	0.990684i	$-0.543484\pi$
$337$	−5.00000			−0.272367			−0.136184	+	0.990684i	$0.543484\pi$
$338$		0			0
$339$	−6.00000			−0.325875
$340$		0			0
$341$	3.00000	−	5.19615i	0.162459	−	0.281387i
$342$		0			0
$343$		−	19.0526i		−	1.02874i
$344$		0			0
$345$	18.0000	−	10.3923i	0.969087	−	0.559503i
$346$		0			0
$347$	12.0000	+	20.7846i	0.644194	+	1.11578i	0.984487	+	0.175457i	$0.0561403\pi$
$347$	12.0000	+	20.7846i	0.644194	+	1.11578i	−0.340293	+	0.940319i	$0.610526\pi$
$348$	6.00000	−	10.3923i	0.321634	−	0.557086i
$349$	−16.5000	−	9.52628i	−0.883225	−	0.509930i	−0.0115044	−	0.999934i	$-0.503662\pi$
$349$	−16.5000	−	9.52628i	−0.883225	−	0.509930i	−0.871720	+	0.490004i	$0.836995\pi$
$350$		0			0
$351$	−3.50000	+	0.866025i	−0.186816	+	0.0462250i
$352$		0			0
$353$	9.00000	+	5.19615i	0.479022	+	0.276563i	0.720009	−	0.693965i	$-0.244138\pi$
$353$	9.00000	+	5.19615i	0.479022	+	0.276563i	−0.240987	+	0.970528i	$0.577471\pi$
$354$		0			0
$355$	−18.0000	−	31.1769i	−0.955341	−	1.65470i
$356$		−	13.8564i		−	0.734388i
$357$		0			0
$358$		0			0
$359$			6.92820i			0.365657i	0.983145	+	0.182828i	$0.0585252\pi$
$359$			6.92820i			0.365657i	−0.983145	+	0.182828i	$0.941475\pi$
$360$		0			0
$361$	−3.50000	+	6.06218i	−0.184211	+	0.319062i
$362$		0			0
$363$	1.00000			0.0524864
$364$	12.0000	+	3.46410i	0.628971	+	0.181568i
$365$	−6.00000			−0.314054
$366$		0			0
$367$	−8.50000	+	14.7224i	−0.443696	+	0.768505i	−0.997960	−	0.0638362i	$-0.979666\pi$
$367$	−8.50000	+	14.7224i	−0.443696	+	0.768505i	0.554264	+	0.832341i	$0.313000\pi$
$368$	12.0000	+	20.7846i	0.625543	+	1.08347i
$369$			6.92820i			0.360668i
$370$		0			0
$371$	−18.0000	+	10.3923i	−0.934513	+	0.539542i
$372$		−	3.46410i		−	0.179605i
$373$	5.50000	+	9.52628i	0.284779	+	0.493252i	0.972556	−	0.232671i	$-0.0747464\pi$
$373$	5.50000	+	9.52628i	0.284779	+	0.493252i	−0.687776	+	0.725923i	$0.741413\pi$
$374$		0			0
$375$	−6.00000	−	3.46410i	−0.309839	−	0.178885i
$376$		0			0
$377$	−6.00000	+	20.7846i	−0.309016	+	1.07046i
$378$		0			0
$379$	−19.5000	−	11.2583i	−1.00165	−	0.578302i	−0.0929123	−	0.995674i	$-0.529618\pi$
$379$	−19.5000	−	11.2583i	−1.00165	−	0.578302i	−0.908735	+	0.417373i	$0.862951\pi$
$380$	12.0000	−	20.7846i	0.615587	−	1.06623i
$381$	6.50000	+	11.2583i	0.333005	+	0.576782i
$382$		0			0
$383$	24.0000	−	13.8564i	1.22634	−	0.708029i	0.260080	−	0.965587i	$-0.416251\pi$
$383$	24.0000	−	13.8564i	1.22634	−	0.708029i	0.966263	+	0.257558i	$0.0829178\pi$
$384$		0			0
$385$			20.7846i			1.05928i
$386$		0			0
$387$	0.500000	−	0.866025i	0.0254164	−	0.0440225i
$388$	9.00000	+	5.19615i	0.456906	+	0.263795i
$389$	12.0000			0.608424			0.304212	−	0.952604i	$-0.401607\pi$
$389$	12.0000			0.608424			0.304212	+	0.952604i	$0.401607\pi$
$390$		0			0
$391$		0			0
$392$		0			0
$393$	3.00000	−	5.19615i	0.151330	−	0.262111i
$394$		0			0
$395$		−	38.1051i		−	1.91728i
$396$	6.00000	−	3.46410i	0.301511	−	0.174078i
$397$	13.5000	−	7.79423i	0.677546	−	0.391181i	−0.121384	−	0.992606i	$-0.538733\pi$
$397$	13.5000	−	7.79423i	0.677546	−	0.391181i	0.798930	+	0.601424i	$0.205400\pi$
$398$		0			0
$399$	3.00000	+	5.19615i	0.150188	+	0.260133i
$400$	14.0000	−	24.2487i	0.700000	−	1.21244i
$401$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$401$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$402$		0			0
$403$	1.50000	+	6.06218i	0.0747203	+	0.301979i
$404$	−36.0000			−1.79107
$405$	−3.00000	−	1.73205i	−0.149071	−	0.0860663i
$406$		0			0
$407$		0			0
$408$		0			0
$409$	7.50000	−	4.33013i	0.370851	−	0.214111i	−0.302979	−	0.952997i	$-0.597981\pi$
$409$	7.50000	−	4.33013i	0.370851	−	0.214111i	0.673830	+	0.738886i	$0.264648\pi$
$410$		0			0
$411$		0			0
$412$	1.00000	+	1.73205i	0.0492665	+	0.0853320i
$413$	3.00000	−	5.19615i	0.147620	−	0.255686i
$414$		0			0
$415$	48.0000			2.35623
$416$		0			0
$417$	−5.00000			−0.244851
$418$		0			0
$419$	6.00000	−	10.3923i	0.293119	−	0.507697i	−0.681426	−	0.731887i	$-0.738640\pi$
$419$	6.00000	−	10.3923i	0.293119	−	0.507697i	0.974546	+	0.224189i	$0.0719734\pi$
$420$	6.00000	+	10.3923i	0.292770	+	0.507093i
$421$		−	12.1244i		−	0.590905i	−0.955357	−	0.295452i	$-0.904530\pi$
$421$		−	12.1244i		−	0.590905i	0.955357	−	0.295452i	$-0.0954704\pi$
$422$		0			0
$423$	3.00000	−	1.73205i	0.145865	−	0.0842152i
$424$		0			0
$425$		0			0
$426$		0			0
$427$	1.50000	+	0.866025i	0.0725901	+	0.0419099i
$428$	−12.0000			−0.580042
$429$	−9.00000	+	8.66025i	−0.434524	+	0.418121i
$430$		0			0
$431$	18.0000	+	10.3923i	0.867029	+	0.500580i	0.866360	−	0.499420i	$-0.166454\pi$
$431$	18.0000	+	10.3923i	0.867029	+	0.500580i	0.000669521		1.00000i	$0.499787\pi$
$432$	2.00000	−	3.46410i	0.0962250	−	0.166667i
$433$	11.5000	+	19.9186i	0.552655	+	0.957226i	0.998082	+	0.0619079i	$0.0197185\pi$
$433$	11.5000	+	19.9186i	0.552655	+	0.957226i	−0.445427	+	0.895318i	$0.646948\pi$
$434$		0			0
$435$	−18.0000	+	10.3923i	−0.863034	+	0.498273i
$436$	−27.0000	+	15.5885i	−1.29307	+	0.746552i
$437$		−	20.7846i		−	0.994263i
$438$		0			0
$439$	−17.5000	+	30.3109i	−0.835229	+	1.44666i	0.0586141	+	0.998281i	$0.481332\pi$
$439$	−17.5000	+	30.3109i	−0.835229	+	1.44666i	−0.893843	+	0.448379i	$0.852001\pi$
$440$		0			0
$441$	4.00000			0.190476
$442$		0			0
$443$	24.0000			1.14027			0.570137	−	0.821549i	$-0.306890\pi$
$443$	24.0000			1.14027			0.570137	+	0.821549i	$0.306890\pi$
$444$		0			0
$445$	−12.0000	+	20.7846i	−0.568855	+	0.985285i
$446$		0			0
$447$			6.92820i			0.327693i
$448$	−12.0000	+	6.92820i	−0.566947	+	0.327327i
$449$	33.0000	−	19.0526i	1.55737	−	0.899146i	0.559859	−	0.828588i	$-0.310855\pi$
$449$	33.0000	−	19.0526i	1.55737	−	0.899146i	0.997508	−	0.0705577i	$-0.0224779\pi$
$450$		0			0
$451$	12.0000	+	20.7846i	0.565058	+	0.978709i
$452$	−6.00000	+	10.3923i	−0.282216	+	0.488813i
$453$	−3.00000	−	1.73205i	−0.140952	−	0.0813788i
$454$		0			0
$455$	−15.0000	−	15.5885i	−0.703211	−	0.730798i
$456$		0			0
$457$	−31.5000	−	18.1865i	−1.47351	−	0.850730i	−0.473953	−	0.880550i	$-0.657173\pi$
$457$	−31.5000	−	18.1865i	−1.47351	−	0.850730i	−0.999555	+	0.0298202i	$0.990507\pi$
$458$		0			0
$459$		0			0
$460$		−	41.5692i		−	1.93817i
$461$	−36.0000	+	20.7846i	−1.67669	+	0.968036i	−0.712938	+	0.701228i	$0.752636\pi$
$461$	−36.0000	+	20.7846i	−1.67669	+	0.968036i	−0.963750	+	0.266808i	$0.914031\pi$
$462$		0			0
$463$		−	36.3731i		−	1.69040i	−0.534450	−	0.845200i	$-0.679481\pi$
$463$		−	36.3731i		−	1.69040i	0.534450	−	0.845200i	$-0.320519\pi$
$464$	−12.0000	−	20.7846i	−0.557086	−	0.964901i
$465$	−3.00000	+	5.19615i	−0.139122	+	0.240966i
$466$		0			0
$467$	6.00000			0.277647			0.138823	−	0.990317i	$-0.455668\pi$
$467$	6.00000			0.277647			0.138823	+	0.990317i	$0.455668\pi$
$468$	−2.00000	+	6.92820i	−0.0924500	+	0.320256i
$469$	−15.0000			−0.692636
$470$		0			0
$471$	5.50000	−	9.52628i	0.253427	−	0.438948i
$472$		0			0
$473$		−	3.46410i		−	0.159280i
$474$		0			0
$475$	−21.0000	+	12.1244i	−0.963546	+	0.556304i
$476$		0			0
$477$	−6.00000	−	10.3923i	−0.274721	−	0.475831i
$478$		0			0
$479$	30.0000	+	17.3205i	1.37073	+	0.791394i	0.991021	−	0.133710i	$-0.0426889\pi$
$479$	30.0000	+	17.3205i	1.37073	+	0.791394i	0.379714	+	0.925104i	$0.376022\pi$
$480$		0			0
$481$		0			0
$482$		0			0
$483$	9.00000	+	5.19615i	0.409514	+	0.236433i
$484$	1.00000	−	1.73205i	0.0454545	−	0.0787296i
$485$	−9.00000	−	15.5885i	−0.408669	−	0.707835i
$486$		0			0
$487$	−21.0000	+	12.1244i	−0.951601	+	0.549407i	−0.893578	−	0.448908i	$-0.851813\pi$
$487$	−21.0000	+	12.1244i	−0.951601	+	0.549407i	−0.0580230	+	0.998315i	$0.518480\pi$
$488$		0			0
$489$			19.0526i			0.861586i
$490$		0			0
$491$	−3.00000	+	5.19615i	−0.135388	+	0.234499i	−0.925746	−	0.378147i	$-0.876561\pi$
$491$	−3.00000	+	5.19615i	−0.135388	+	0.234499i	0.790358	+	0.612646i	$0.209895\pi$
$492$	12.0000	+	6.92820i	0.541002	+	0.312348i
$493$		0			0
$494$		0			0
$495$	−12.0000			−0.539360
$496$	−6.00000	−	3.46410i	−0.269408	−	0.155543i
$497$	9.00000	−	15.5885i	0.403705	−	0.699238i
$498$		0			0
$499$			31.1769i			1.39567i	0.716258	+	0.697835i	$0.245853\pi$
$499$			31.1769i			1.39567i	−0.716258	+	0.697835i	$0.754147\pi$
$500$	−12.0000	+	6.92820i	−0.536656	+	0.309839i
$501$	−6.00000	+	3.46410i	−0.268060	+	0.154765i
$502$		0			0
$503$	15.0000	+	25.9808i	0.668817	+	1.15842i	0.978235	+	0.207499i	$0.0665323\pi$
$503$	15.0000	+	25.9808i	0.668817	+	1.15842i	−0.309418	+	0.950926i	$0.600134\pi$
$504$		0			0
$505$	54.0000	+	31.1769i	2.40297	+	1.38735i
$506$		0			0
$507$	0.500000	−	12.9904i	0.0222058	−	0.576923i
$508$	26.0000			1.15356
$509$	−15.0000	−	8.66025i	−0.664863	−	0.383859i	0.129264	−	0.991610i	$-0.458738\pi$
$509$	−15.0000	−	8.66025i	−0.664863	−	0.383859i	−0.794128	+	0.607751i	$0.792072\pi$
$510$		0			0
$511$	−1.50000	−	2.59808i	−0.0663561	−	0.114932i
$512$		0			0
$513$	−3.00000	+	1.73205i	−0.132453	+	0.0764719i
$514$		0			0
$515$		−	3.46410i		−	0.152647i
$516$	−1.00000	−	1.73205i	−0.0440225	−	0.0762493i
$517$	6.00000	−	10.3923i	0.263880	−	0.457053i
$518$		0			0
$519$	6.00000			0.263371
$520$		0			0
$521$	−24.0000			−1.05146			−0.525730	−	0.850652i	$-0.676208\pi$
$521$	−24.0000			−1.05146			−0.525730	+	0.850652i	$0.676208\pi$
$522$		0			0
$523$	14.0000	−	24.2487i	0.612177	−	1.06032i	−0.378695	−	0.925521i	$-0.623627\pi$
$523$	14.0000	−	24.2487i	0.612177	−	1.06032i	0.990873	−	0.134801i	$-0.0430394\pi$
$524$	−6.00000	−	10.3923i	−0.262111	−	0.453990i
$525$		−	12.1244i		−	0.529150i
$526$		0			0
$527$		0			0
$528$		−	13.8564i		−	0.603023i
$529$	−6.50000	−	11.2583i	−0.282609	−	0.489493i
$530$		0			0
$531$	3.00000	+	1.73205i	0.130189	+	0.0751646i
$532$	12.0000			0.520266
$533$	−24.0000	−	6.92820i	−1.03956	−	0.300094i
$534$		0			0
$535$	18.0000	+	10.3923i	0.778208	+	0.449299i
$536$		0			0
$537$	6.00000	+	10.3923i	0.258919	+	0.448461i
$538$		0			0
$539$	12.0000	−	6.92820i	0.516877	−	0.298419i
$540$	−6.00000	+	3.46410i	−0.258199	+	0.149071i
$541$		−	29.4449i		−	1.26593i	−0.774179	−	0.632967i	$-0.781837\pi$
$541$		−	29.4449i		−	1.26593i	0.774179	−	0.632967i	$-0.218163\pi$
$542$		0			0
$543$	7.00000	−	12.1244i	0.300399	−	0.520306i
$544$		0			0
$545$	54.0000			2.31311
$546$		0			0
$547$	−19.0000			−0.812381			−0.406191	−	0.913788i	$-0.633143\pi$
$547$	−19.0000			−0.812381			−0.406191	+	0.913788i	$0.633143\pi$
$548$		0			0
$549$	−0.500000	+	0.866025i	−0.0213395	+	0.0369611i
$550$		0			0
$551$			20.7846i			0.885454i
$552$		0			0
$553$	16.5000	−	9.52628i	0.701651	−	0.405099i
$554$		0			0
$555$		0			0
$556$	−5.00000	+	8.66025i	−0.212047	+	0.367277i
$557$	−24.0000	−	13.8564i	−1.01691	−	0.587115i	−0.103704	−	0.994608i	$-0.533070\pi$
$557$	−24.0000	−	13.8564i	−1.01691	−	0.587115i	−0.913208	+	0.407493i	$0.866403\pi$
$558$		0			0
$559$	2.50000	+	2.59808i	0.105739	+	0.109887i
$560$	24.0000			1.01419
$561$		0			0
$562$		0			0
$563$	21.0000	+	36.3731i	0.885044	+	1.53294i	0.845663	+	0.533718i	$0.179206\pi$
$563$	21.0000	+	36.3731i	0.885044	+	1.53294i	0.0393818	+	0.999224i	$0.487461\pi$
$564$		−	6.92820i		−	0.291730i
$565$	18.0000	−	10.3923i	0.757266	−	0.437208i
$566$		0			0
$567$		−	1.73205i		−	0.0727393i
$568$		0			0
$569$	3.00000	−	5.19615i	0.125767	−	0.217834i	−0.796266	−	0.604947i	$-0.793194\pi$
$569$	3.00000	−	5.19615i	0.125767	−	0.217834i	0.922032	+	0.387113i	$0.126528\pi$
$570$		0			0
$571$	−44.0000			−1.84134			−0.920671	−	0.390339i	$-0.872358\pi$
$571$	−44.0000			−1.84134			−0.920671	+	0.390339i	$0.872358\pi$
$572$	6.00000	+	24.2487i	0.250873	+	1.01389i
$573$	18.0000			0.751961
$574$		0			0
$575$	−21.0000	+	36.3731i	−0.875761	+	1.51686i
$576$	−4.00000	−	6.92820i	−0.166667	−	0.288675i
$577$		−	34.6410i		−	1.44212i	−0.692870	−	0.721062i	$-0.743654\pi$
$577$		−	34.6410i		−	1.44212i	0.692870	−	0.721062i	$-0.256346\pi$
$578$		0			0
$579$	−13.5000	+	7.79423i	−0.561041	+	0.323917i
$580$			41.5692i			1.72607i
$581$	12.0000	+	20.7846i	0.497844	+	0.862291i
$582$		0			0
$583$	−36.0000	−	20.7846i	−1.49097	−	0.860811i
$584$		0			0
$585$	9.00000	−	8.66025i	0.372104	−	0.358057i
$586$		0			0
$587$	−27.0000	−	15.5885i	−1.11441	−	0.643404i	−0.174441	−	0.984668i	$-0.555812\pi$
$587$	−27.0000	−	15.5885i	−1.11441	−	0.643404i	−0.939968	+	0.341263i	$0.889145\pi$
$588$	4.00000	−	6.92820i	0.164957	−	0.285714i
$589$	3.00000	+	5.19615i	0.123613	+	0.214104i
$590$		0			0
$591$	12.0000	−	6.92820i	0.493614	−	0.284988i
$592$		0			0
$593$			3.46410i			0.142254i	0.997467	+	0.0711268i	$0.0226595\pi$
$593$			3.46410i			0.142254i	−0.997467	+	0.0711268i	$0.977341\pi$
$594$		0			0
$595$		0			0
$596$	12.0000	+	6.92820i	0.491539	+	0.283790i
$597$	−7.00000			−0.286491
$598$		0			0
$599$	−30.0000			−1.22577			−0.612883	−	0.790173i	$-0.709990\pi$
$599$	−30.0000			−1.22577			−0.612883	+	0.790173i	$0.709990\pi$
$600$		0			0
$601$	13.0000	−	22.5167i	0.530281	−	0.918474i	−0.469095	−	0.883148i	$-0.655420\pi$
$601$	13.0000	−	22.5167i	0.530281	−	0.918474i	0.999376	−	0.0353259i	$-0.0112469\pi$
$602$		0			0
$603$		−	8.66025i		−	0.352673i
$604$	−6.00000	+	3.46410i	−0.244137	+	0.140952i
$605$	−3.00000	+	1.73205i	−0.121967	+	0.0704179i
$606$		0			0
$607$	−4.00000	−	6.92820i	−0.162355	−	0.281207i	0.773358	−	0.633970i	$-0.218576\pi$
$607$	−4.00000	−	6.92820i	−0.162355	−	0.281207i	−0.935713	+	0.352763i	$0.885242\pi$
$608$		0			0
$609$	−9.00000	−	5.19615i	−0.364698	−	0.210559i
$610$		0			0
$611$	3.00000	+	12.1244i	0.121367	+	0.490499i
$612$		0			0
$613$	7.50000	+	4.33013i	0.302922	+	0.174892i	0.643755	−	0.765232i	$-0.277376\pi$
$613$	7.50000	+	4.33013i	0.302922	+	0.174892i	−0.340833	+	0.940124i	$0.610709\pi$
$614$		0			0
$615$	−12.0000	−	20.7846i	−0.483887	−	0.838116i
$616$		0			0
$617$	−9.00000	+	5.19615i	−0.362326	+	0.209189i	−0.670101	−	0.742270i	$-0.733749\pi$
$617$	−9.00000	+	5.19615i	−0.362326	+	0.209189i	0.307774	+	0.951459i	$0.400416\pi$
$618$		0			0
$619$			25.9808i			1.04425i	0.852867	+	0.522127i	$0.174861\pi$
$619$			25.9808i			1.04425i	−0.852867	+	0.522127i	$0.825139\pi$
$620$	6.00000	+	10.3923i	0.240966	+	0.417365i
$621$	−3.00000	+	5.19615i	−0.120386	+	0.208514i
$622$		0			0
$623$	−12.0000			−0.480770
$624$	10.0000	+	10.3923i	0.400320	+	0.416025i
$625$	−11.0000			−0.440000
$626$		0			0
$627$	−6.00000	+	10.3923i	−0.239617	+	0.415029i
$628$	−11.0000	−	19.0526i	−0.438948	−	0.760280i
$629$		0			0
$630$		0			0
$631$	1.50000	−	0.866025i	0.0597141	−	0.0344759i	−0.469846	−	0.882749i	$-0.655690\pi$
$631$	1.50000	−	0.866025i	0.0597141	−	0.0344759i	0.529560	+	0.848273i	$0.322357\pi$
$632$		0			0
$633$	6.50000	+	11.2583i	0.258352	+	0.447478i
$634$		0			0
$635$	−39.0000	−	22.5167i	−1.54767	−	0.893546i
$636$	−24.0000			−0.951662
$637$	−4.00000	+	13.8564i	−0.158486	+	0.549011i
$638$		0			0
$639$	9.00000	+	5.19615i	0.356034	+	0.205557i
$640$		0			0
$641$	−9.00000	−	15.5885i	−0.355479	−	0.615707i	0.631721	−	0.775196i	$-0.282349\pi$
$641$	−9.00000	−	15.5885i	−0.355479	−	0.615707i	−0.987200	+	0.159489i	$0.949015\pi$
$642$		0			0
$643$	16.5000	−	9.52628i	0.650696	−	0.375680i	−0.138027	−	0.990429i	$-0.544076\pi$
$643$	16.5000	−	9.52628i	0.650696	−	0.375680i	0.788723	+	0.614749i	$0.210743\pi$
$644$	18.0000	−	10.3923i	0.709299	−	0.409514i
$645$			3.46410i			0.136399i
$646$		0			0
$647$	9.00000	−	15.5885i	0.353827	−	0.612845i	−0.633090	−	0.774078i	$-0.718214\pi$
$647$	9.00000	−	15.5885i	0.353827	−	0.612845i	0.986916	+	0.161233i	$0.0515470\pi$
$648$		0			0
$649$	12.0000			0.471041
$650$		0			0
$651$	−3.00000			−0.117579
$652$	33.0000	+	19.0526i	1.29238	+	0.746156i
$653$	−18.0000	+	31.1769i	−0.704394	+	1.22005i	0.262515	+	0.964928i	$0.415448\pi$
$653$	−18.0000	+	31.1769i	−0.704394	+	1.22005i	−0.966910	+	0.255119i	$0.917885\pi$
$654$		0			0
$655$			20.7846i			0.812122i
$656$	24.0000	−	13.8564i	0.937043	−	0.541002i
$657$	1.50000	−	0.866025i	0.0585206	−	0.0337869i
$658$		0			0
$659$	−24.0000	−	41.5692i	−0.934907	−	1.61931i	−0.774799	−	0.632207i	$-0.782149\pi$
$659$	−24.0000	−	41.5692i	−0.934907	−	1.61931i	−0.160108	−	0.987099i	$-0.551184\pi$
$660$	−12.0000	+	20.7846i	−0.467099	+	0.809040i
$661$	−22.5000	−	12.9904i	−0.875149	−	0.505267i	−0.00609283	−	0.999981i	$-0.501939\pi$
$661$	−22.5000	−	12.9904i	−0.875149	−	0.505267i	−0.869056	+	0.494714i	$0.835273\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	−18.0000	−	10.3923i	−0.698010	−	0.402996i
$666$		0			0
$667$	18.0000	+	31.1769i	0.696963	+	1.20717i
$668$			13.8564i			0.536120i
$669$	15.0000	−	8.66025i	0.579934	−	0.334825i
$670$		0			0
$671$			3.46410i			0.133730i
$672$		0			0
$673$	0.500000	−	0.866025i	0.0192736	−	0.0333828i	−0.856228	−	0.516599i	$-0.827198\pi$
$673$	0.500000	−	0.866025i	0.0192736	−	0.0333828i	0.875501	+	0.483216i	$0.160531\pi$
$674$		0			0
$675$	7.00000			0.269430
$676$	−22.0000	−	13.8564i	−0.846154	−	0.532939i
$677$	48.0000			1.84479			0.922395	−	0.386248i	$-0.126229\pi$
$677$	48.0000			1.84479			0.922395	+	0.386248i	$0.126229\pi$
$678$		0			0
$679$	4.50000	−	7.79423i	0.172694	−	0.299115i
$680$		0			0
$681$		−	20.7846i		−	0.796468i
$682$		0			0
$683$	21.0000	−	12.1244i	0.803543	−	0.463926i	−0.0411658	−	0.999152i	$-0.513107\pi$
$683$	21.0000	−	12.1244i	0.803543	−	0.463926i	0.844708	+	0.535227i	$0.179774\pi$
$684$			6.92820i			0.264906i
$685$		0			0
$686$		0			0
$687$	24.0000	+	13.8564i	0.915657	+	0.528655i
$688$	−4.00000			−0.152499
$689$	42.0000	−	10.3923i	1.60007	−	0.395915i
$690$		0			0
$691$	37.5000	+	21.6506i	1.42657	+	0.823629i	0.996848	−	0.0793336i	$-0.0252792\pi$
$691$	37.5000	+	21.6506i	1.42657	+	0.823629i	0.429719	+	0.902963i	$0.358613\pi$
$692$	6.00000	−	10.3923i	0.228086	−	0.395056i
$693$	−3.00000	−	5.19615i	−0.113961	−	0.197386i
$694$		0			0
$695$	15.0000	−	8.66025i	0.568982	−	0.328502i
$696$		0			0
$697$		0			0
$698$		0			0
$699$	−9.00000	+	15.5885i	−0.340411	+	0.589610i
$700$	−21.0000	−	12.1244i	−0.793725	−	0.458258i
$701$	−12.0000			−0.453234			−0.226617	−	0.973984i	$-0.572767\pi$
$701$	−12.0000			−0.453234			−0.226617	+	0.973984i	$0.572767\pi$
$702$		0			0
$703$		0			0
$704$	−24.0000	−	13.8564i	−0.904534	−	0.522233i
$705$	−6.00000	+	10.3923i	−0.225973	+	0.391397i
$706$		0			0
$707$			31.1769i			1.17253i
$708$	6.00000	−	3.46410i	0.225494	−	0.130189i
$709$	−16.5000	+	9.52628i	−0.619671	+	0.357767i	−0.776741	−	0.629821i	$-0.783128\pi$
$709$	−16.5000	+	9.52628i	−0.619671	+	0.357767i	0.157070	+	0.987587i	$0.449795\pi$
$710$		0			0
$711$	5.50000	+	9.52628i	0.206266	+	0.357263i
$712$		0			0
$713$	9.00000	+	5.19615i	0.337053	+	0.194597i
$714$		0			0
$715$	12.0000	−	41.5692i	0.448775	−	1.55460i
$716$	24.0000			0.896922
$717$		0			0
$718$		0			0
$719$	3.00000	+	5.19615i	0.111881	+	0.193784i	0.916529	−	0.399969i	$-0.130979\pi$
$719$	3.00000	+	5.19615i	0.111881	+	0.193784i	−0.804648	+	0.593753i	$0.797646\pi$
$720$			13.8564i			0.516398i
$721$	1.50000	−	0.866025i	0.0558629	−	0.0322525i
$722$		0			0
$723$		−	20.7846i		−	0.772988i
$724$	−14.0000	−	24.2487i	−0.520306	−	0.901196i
$725$	21.0000	−	36.3731i	0.779920	−	1.35086i
$726$		0			0
$727$	−35.0000			−1.29808			−0.649039	−	0.760755i	$-0.724829\pi$
$727$	−35.0000			−1.29808			−0.649039	+	0.760755i	$0.724829\pi$
$728$		0			0
$729$	1.00000			0.0370370
$730$		0			0
$731$		0			0
$732$	1.00000	+	1.73205i	0.0369611	+	0.0640184i
$733$			39.8372i			1.47142i	0.677297	+	0.735710i	$0.263151\pi$
$733$			39.8372i			1.47142i	−0.677297	+	0.735710i	$0.736849\pi$
$734$		0			0
$735$	−12.0000	+	6.92820i	−0.442627	+	0.255551i
$736$		0			0
$737$	−15.0000	−	25.9808i	−0.552532	−	0.957014i
$738$		0			0
$739$	39.0000	+	22.5167i	1.43464	+	0.828289i	0.997470	−	0.0710909i	$-0.0226481\pi$
$739$	39.0000	+	22.5167i	1.43464	+	0.828289i	0.437168	+	0.899380i	$0.355981\pi$
$740$		0			0
$741$	−3.00000	−	12.1244i	−0.110208	−	0.445399i
$742$		0			0
$743$	−9.00000	−	5.19615i	−0.330178	−	0.190628i	0.325742	−	0.945459i	$-0.394386\pi$
$743$	−9.00000	−	5.19615i	−0.330178	−	0.190628i	−0.655920	+	0.754830i	$0.727719\pi$
$744$		0			0
$745$	−12.0000	−	20.7846i	−0.439646	−	0.761489i
$746$		0			0
$747$	−12.0000	+	6.92820i	−0.439057	+	0.253490i
$748$		0			0
$749$			10.3923i			0.379727i
$750$		0			0
$751$	−4.00000	+	6.92820i	−0.145962	+	0.252814i	−0.929731	−	0.368238i	$-0.879961\pi$
$751$	−4.00000	+	6.92820i	−0.145962	+	0.252814i	0.783769	+	0.621052i	$0.213294\pi$
$752$	−12.0000	−	6.92820i	−0.437595	−	0.252646i
$753$	−12.0000			−0.437304
$754$		0			0
$755$	12.0000			0.436725
$756$	−3.00000	−	1.73205i	−0.109109	−	0.0629941i
$757$	−17.0000	+	29.4449i	−0.617876	+	1.07019i	0.371997	+	0.928234i	$0.378673\pi$
$757$	−17.0000	+	29.4449i	−0.617876	+	1.07019i	−0.989873	+	0.141958i	$0.954660\pi$
$758$		0			0
$759$			20.7846i			0.754434i
$760$		0			0
$761$	−18.0000	+	10.3923i	−0.652499	+	0.376721i	−0.789413	−	0.613862i	$-0.789615\pi$
$761$	−18.0000	+	10.3923i	−0.652499	+	0.376721i	0.136914	+	0.990583i	$0.456282\pi$
$762$		0			0
$763$	13.5000	+	23.3827i	0.488733	+	0.846510i
$764$	18.0000	−	31.1769i	0.651217	−	1.12794i
$765$		0			0
$766$		0			0
$767$	−9.00000	+	8.66025i	−0.324971	+	0.312704i
$768$	−16.0000			−0.577350
$769$	−6.00000	−	3.46410i	−0.216366	−	0.124919i	0.387901	−	0.921701i	$-0.373200\pi$
$769$	−6.00000	−	3.46410i	−0.216366	−	0.124919i	−0.604266	+	0.796782i	$0.706534\pi$
$770$		0			0
$771$		0			0
$772$			31.1769i			1.12208i
$773$	−45.0000	+	25.9808i	−1.61854	+	0.934463i	−0.631239	+	0.775589i	$0.717453\pi$
$773$	−45.0000	+	25.9808i	−1.61854	+	0.934463i	−0.987299	+	0.158874i	$0.949213\pi$
$774$		0			0
$775$		−	12.1244i		−	0.435520i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−24.0000			−0.859889
$780$	−6.00000	−	24.2487i	−0.214834	−	0.868243i
$781$	36.0000			1.28818
$782$		0			0
$783$	3.00000	−	5.19615i	0.107211	−	0.185695i
$784$	−8.00000	−	13.8564i	−0.285714	−	0.494872i
$785$			38.1051i			1.36003i
$786$		0			0
$787$	28.5000	−	16.4545i	1.01592	−	0.586539i	0.102997	−	0.994682i	$-0.467157\pi$
$787$	28.5000	−	16.4545i	1.01592	−	0.586539i	0.912918	+	0.408143i	$0.133823\pi$
$788$		−	27.7128i		−	0.987228i
$789$	6.00000	+	10.3923i	0.213606	+	0.369976i
$790$		0			0
$791$	9.00000	+	5.19615i	0.320003	+	0.184754i
$792$		0			0
$793$	−2.50000	−	2.59808i	−0.0887776	−	0.0922604i
$794$		0			0
$795$	36.0000	+	20.7846i	1.27679	+	0.737154i
$796$	−7.00000	+	12.1244i	−0.248108	+	0.429736i
$797$	−6.00000	−	10.3923i	−0.212531	−	0.368114i	0.739975	−	0.672634i	$-0.234837\pi$
$797$	−6.00000	−	10.3923i	−0.212531	−	0.368114i	−0.952506	+	0.304520i	$0.901504\pi$
$798$		0			0
$799$		0			0
$800$		0			0
$801$		−	6.92820i		−	0.244796i
$802$		0			0
$803$	3.00000	−	5.19615i	0.105868	−	0.183368i
$804$	−15.0000	−	8.66025i	−0.529009	−	0.305424i
$805$	−36.0000			−1.26883
$806$		0			0
$807$		0			0
$808$		0			0
$809$	−6.00000	+	10.3923i	−0.210949	+	0.365374i	−0.952012	−	0.306062i	$-0.900989\pi$
$809$	−6.00000	+	10.3923i	−0.210949	+	0.365374i	0.741063	+	0.671436i	$0.234322\pi$
$810$		0			0
$811$		−	25.9808i		−	0.912308i	−0.889901	−	0.456154i	$-0.849227\pi$
$811$		−	25.9808i		−	0.912308i	0.889901	−	0.456154i	$-0.150773\pi$
$812$	−18.0000	+	10.3923i	−0.631676	+	0.364698i
$813$	−4.50000	+	2.59808i	−0.157822	+	0.0911185i
$814$		0			0
$815$	−33.0000	−	57.1577i	−1.15594	−	2.00215i
$816$		0			0
$817$	3.00000	+	1.73205i	0.104957	+	0.0605968i
$818$		0			0
$819$	6.00000	+	1.73205i	0.209657	+	0.0605228i
$820$	−48.0000			−1.67623
$821$	21.0000	+	12.1244i	0.732905	+	0.423143i	0.819484	−	0.573102i	$-0.194260\pi$
$821$	21.0000	+	12.1244i	0.732905	+	0.423143i	−0.0865789	+	0.996245i	$0.527593\pi$
$822$		0			0
$823$	−16.0000	−	27.7128i	−0.557725	−	0.966008i	−0.997686	−	0.0679910i	$-0.978341\pi$
$823$	−16.0000	−	27.7128i	−0.557725	−	0.966008i	0.439961	−	0.898017i	$-0.354992\pi$
$824$		0			0
$825$	21.0000	−	12.1244i	0.731126	−	0.422116i
$826$		0			0
$827$			48.4974i			1.68642i	0.537584	+	0.843210i	$0.319337\pi$
$827$			48.4974i			1.68642i	−0.537584	+	0.843210i	$0.680663\pi$
$828$	6.00000	+	10.3923i	0.208514	+	0.361158i
$829$	−15.5000	+	26.8468i	−0.538337	+	0.932427i	0.460657	+	0.887578i	$0.347614\pi$
$829$	−15.5000	+	26.8468i	−0.538337	+	0.932427i	−0.998994	+	0.0448490i	$0.985719\pi$
$830$		0			0
$831$	10.0000			0.346896
$832$	28.0000	−	6.92820i	0.970725	−	0.240192i
$833$		0			0
$834$		0			0
$835$	12.0000	−	20.7846i	0.415277	−	0.719281i
$836$	12.0000	+	20.7846i	0.415029	+	0.718851i
$837$		−	1.73205i		−	0.0598684i
$838$		0			0
$839$	27.0000	−	15.5885i	0.932144	−	0.538173i	0.0446547	−	0.999002i	$-0.485781\pi$
$839$	27.0000	−	15.5885i	0.932144	−	0.538173i	0.887489	+	0.460829i	$0.152448\pi$
$840$		0			0
$841$	−3.50000	−	6.06218i	−0.120690	−	0.209041i
$842$		0			0
$843$	−21.0000	−	12.1244i	−0.723278	−	0.417585i
$844$	26.0000			0.894957
$845$	21.0000	+	39.8372i	0.722422	+	1.37044i
$846$		0			0
$847$	−1.50000	−	0.866025i	−0.0515406	−	0.0297570i
$848$	−24.0000	+	41.5692i	−0.824163	+	1.42749i
$849$	−5.50000	−	9.52628i	−0.188760	−	0.326941i
$850$		0			0
$851$		0			0
$852$	18.0000	−	10.3923i	0.616670	−	0.356034i
$853$		−	25.9808i		−	0.889564i	−0.895639	−	0.444782i	$-0.853281\pi$
$853$		−	25.9808i		−	0.889564i	0.895639	−	0.444782i	$-0.146719\pi$
$854$		0			0
$855$	6.00000	−	10.3923i	0.205196	−	0.355409i
$856$		0			0
$857$	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
$859$	13.0000			0.443554			0.221777	−	0.975097i	$-0.428814\pi$
$859$	13.0000			0.443554			0.221777	+	0.975097i	$0.428814\pi$
$860$	6.00000	+	3.46410i	0.204598	+	0.118125i
$861$	6.00000	−	10.3923i	0.204479	−	0.354169i
$862$		0			0
$863$		−	17.3205i		−	0.589597i	−0.955559	−	0.294798i	$-0.904747\pi$
$863$		−	17.3205i		−	0.589597i	0.955559	−	0.294798i	$-0.0952525\pi$
$864$		0			0
$865$	−18.0000	+	10.3923i	−0.612018	+	0.353349i
$866$		0			0
$867$	−8.50000	−	14.7224i	−0.288675	−	0.500000i
$868$	−3.00000	+	5.19615i	−0.101827	+	0.176369i
$869$	33.0000	+	19.0526i	1.11945	+	0.646314i
$870$		0			0
$871$	30.0000	+	8.66025i	1.01651	+	0.293442i
$872$		0			0
$873$	4.50000	+	2.59808i	0.152302	+	0.0879316i
$874$		0			0
$875$	6.00000	+	10.3923i	0.202837	+	0.351324i
$876$		−	3.46410i		−	0.117041i
$877$	36.0000	−	20.7846i	1.21563	−	0.701846i	0.251653	−	0.967818i	$-0.419026\pi$
$877$	36.0000	−	20.7846i	1.21563	−	0.701846i	0.963981	+	0.265971i	$0.0856926\pi$
$878$		0			0
$879$			17.3205i			0.584206i
$880$	24.0000	+	41.5692i	0.809040	+	1.40130i
$881$	6.00000	−	10.3923i	0.202145	−	0.350126i	−0.747074	−	0.664741i	$-0.768542\pi$
$881$	6.00000	−	10.3923i	0.202145	−	0.350126i	0.949219	+	0.314615i	$0.101875\pi$
$882$		0			0
$883$	−5.00000			−0.168263			−0.0841317	−	0.996455i	$-0.526812\pi$
$883$	−5.00000			−0.168263			−0.0841317	+	0.996455i	$0.526812\pi$
$884$		0			0
$885$	−12.0000			−0.403376
$886$		0			0
$887$	21.0000	−	36.3731i	0.705111	−	1.22129i	−0.261540	−	0.965193i	$-0.584230\pi$
$887$	21.0000	−	36.3731i	0.705111	−	1.22129i	0.966651	−	0.256096i	$-0.0824362\pi$
$888$		0			0
$889$		−	22.5167i		−	0.755185i
$890$		0			0
$891$	3.00000	−	1.73205i	0.100504	−	0.0580259i
$892$		−	34.6410i		−	1.15987i
$893$	6.00000	+	10.3923i	0.200782	+	0.347765i
$894$		0			0
$895$	−36.0000	−	20.7846i	−1.20335	−	0.694753i
$896$		0			0
$897$	−15.0000	−	15.5885i	−0.500835	−	0.520483i
$898$		0			0
$899$	−9.00000	−	5.19615i	−0.300167	−	0.173301i
$900$	7.00000	−	12.1244i	0.233333	−	0.404145i
$901$		0			0
$902$		0			0
$903$	−1.50000	+	0.866025i	−0.0499169	+	0.0288195i
$904$		0			0
$905$			48.4974i			1.61211i
$906$		0			0
$907$	22.0000	−	38.1051i	0.730498	−	1.26526i	−0.226173	−	0.974087i	$-0.572621\pi$
$907$	22.0000	−	38.1051i	0.730498	−	1.26526i	0.956671	−	0.291172i	$-0.0940453\pi$
$908$	−36.0000	−	20.7846i	−1.19470	−	0.689761i
$909$	−18.0000			−0.597022
$910$		0			0
$911$	−12.0000			−0.397578			−0.198789	−	0.980042i	$-0.563701\pi$
$911$	−12.0000			−0.397578			−0.198789	+	0.980042i	$0.563701\pi$
$912$	12.0000	+	6.92820i	0.397360	+	0.229416i
$913$	−24.0000	+	41.5692i	−0.794284	+	1.37574i
$914$		0			0
$915$		−	3.46410i		−	0.114520i
$916$	48.0000	−	27.7128i	1.58596	−	0.915657i
$917$	−9.00000	+	5.19615i	−0.297206	+	0.171592i
$918$		0			0
$919$	16.0000	+	27.7128i	0.527791	+	0.914161i	0.999475	+	0.0323936i	$0.0103130\pi$
$919$	16.0000	+	27.7128i	0.527791	+	0.914161i	−0.471684	+	0.881768i	$0.656354\pi$
$920$		0			0
$921$	−1.50000	−	0.866025i	−0.0494267	−	0.0285365i
$922$		0			0
$923$	−27.0000	+	25.9808i	−0.888716	+	0.855167i
$924$	−12.0000			−0.394771
$925$		0			0
$926$		0			0
$927$	0.500000	+	0.866025i	0.0164222	+	0.0284440i
$928$		0			0
$929$	−18.0000	+	10.3923i	−0.590561	+	0.340960i	−0.765319	−	0.643651i	$-0.777419\pi$
$929$	−18.0000	+	10.3923i	−0.590561	+	0.340960i	0.174758	+	0.984611i	$0.444086\pi$
$930$		0			0
$931$			13.8564i			0.454125i
$932$	18.0000	+	31.1769i	0.589610	+	1.02123i
$933$	−12.0000	+	20.7846i	−0.392862	+	0.680458i
$934$		0			0
$935$		0			0
$936$		0			0
$937$	−14.0000			−0.457360			−0.228680	−	0.973502i	$-0.573441\pi$
$937$	−14.0000			−0.457360			−0.228680	+	0.973502i	$0.573441\pi$
$938$		0			0
$939$	6.50000	−	11.2583i	0.212119	−	0.367402i
$940$	12.0000	+	20.7846i	0.391397	+	0.677919i
$941$		−	10.3923i		−	0.338779i	−0.985549	−	0.169390i	$-0.945820\pi$
$941$		−	10.3923i		−	0.338779i	0.985549	−	0.169390i	$-0.0541797\pi$
$942$		0			0
$943$	−36.0000	+	20.7846i	−1.17232	+	0.676840i
$944$		−	13.8564i		−	0.450988i
$945$	3.00000	+	5.19615i	0.0975900	+	0.169031i
$946$		0			0
$947$	−42.0000	−	24.2487i	−1.36482	−	0.787977i	−0.374556	−	0.927204i	$-0.622205\pi$
$947$	−42.0000	−	24.2487i	−1.36482	−	0.787977i	−0.990260	+	0.139227i	$0.955538\pi$
$948$	22.0000			0.714527
$949$	1.50000	+	6.06218i	0.0486921	+	0.196787i
$950$		0			0
$951$	6.00000	+	3.46410i	0.194563	+	0.112331i
$952$		0			0
$953$	−3.00000	−	5.19615i	−0.0971795	−	0.168320i	0.813337	−	0.581793i	$-0.197649\pi$
$953$	−3.00000	−	5.19615i	−0.0971795	−	0.168320i	−0.910516	+	0.413473i	$0.864315\pi$
$954$		0			0
$955$	−54.0000	+	31.1769i	−1.74740	+	1.00886i
$956$		0			0
$957$		−	20.7846i		−	0.671871i
$958$		0			0
$959$		0			0
$960$	24.0000	+	13.8564i	0.774597	+	0.447214i
$961$	28.0000			0.903226
$962$		0			0
$963$	−6.00000			−0.193347
$964$	−36.0000	−	20.7846i	−1.15948	−	0.669427i
$965$	27.0000	−	46.7654i	0.869161	−	1.50543i
$966$		0			0
$967$		−	24.2487i		−	0.779786i	−0.920860	−	0.389893i	$-0.872512\pi$
$967$		−	24.2487i		−	0.779786i	0.920860	−	0.389893i	$-0.127488\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−18.0000	−	31.1769i	−0.577647	−	1.00051i	−0.995748	−	0.0921142i	$-0.970638\pi$
$971$	−18.0000	−	31.1769i	−0.577647	−	1.00051i	0.418101	−	0.908401i	$-0.362696\pi$
$972$	1.00000	−	1.73205i	0.0320750	−	0.0555556i
$973$	7.50000	+	4.33013i	0.240439	+	0.138817i
$974$		0			0
$975$	−7.00000	+	24.2487i	−0.224179	+	0.776580i
$976$	4.00000			0.128037
$977$	−24.0000	−	13.8564i	−0.767828	−	0.443306i	0.0642712	−	0.997932i	$-0.479528\pi$
$977$	−24.0000	−	13.8564i	−0.767828	−	0.443306i	−0.832099	+	0.554627i	$0.812861\pi$
$978$		0			0
$979$	−12.0000	−	20.7846i	−0.383522	−	0.664279i
$980$			27.7128i			0.885253i
$981$	−13.5000	+	7.79423i	−0.431022	+	0.248851i
$982$		0			0
$983$		−	10.3923i		−	0.331463i	−0.986171	−	0.165732i	$-0.947001\pi$
$983$		−	10.3923i		−	0.331463i	0.986171	−	0.165732i	$-0.0529985\pi$
$984$		0			0
$985$	−24.0000	+	41.5692i	−0.764704	+	1.32451i
$986$		0			0
$987$	−6.00000			−0.190982
$988$	−24.0000	−	6.92820i	−0.763542	−	0.220416i
$989$	6.00000			0.190789
$990$		0			0
$991$	8.00000	−	13.8564i	0.254128	−	0.440163i	−0.710530	−	0.703667i	$-0.751545\pi$
$991$	8.00000	−	13.8564i	0.254128	−	0.440163i	0.964658	+	0.263504i	$0.0848781\pi$
$992$		0			0
$993$		−	5.19615i		−	0.164895i
$994$		0			0
$995$	21.0000	−	12.1244i	0.665745	−	0.384368i
$996$			27.7128i			0.878114i
$997$	−17.5000	−	30.3109i	−0.554231	−	0.959955i	−0.997963	−	0.0637961i	$-0.979679\pi$
$997$	−17.5000	−	30.3109i	−0.554231	−	0.959955i	0.443732	−	0.896159i	$-0.353654\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	39.2.j.a.4.1	✓	2
3.2	odd	2		117.2.q.a.82.1		2
4.3	odd	2		624.2.bv.b.433.1		2
5.2	odd	4		975.2.w.d.199.1		4
5.3	odd	4		975.2.w.d.199.2		4
5.4	even	2		975.2.bc.c.901.1		2
12.11	even	2		1872.2.by.f.433.1		2
13.2	odd	12		507.2.e.f.484.2		4
13.3	even	3		507.2.j.b.361.1		2
13.4	even	6		507.2.b.c.337.1		2
13.5	odd	4		507.2.e.f.22.2		4
13.6	odd	12		507.2.a.e.1.2		2
13.7	odd	12		507.2.a.e.1.1		2
13.8	odd	4		507.2.e.f.22.1		4
13.9	even	3		507.2.b.c.337.2		2
13.10	even	6	inner	39.2.j.a.10.1	yes	2
13.11	odd	12		507.2.e.f.484.1		4
13.12	even	2		507.2.j.b.316.1		2
39.17	odd	6		1521.2.b.f.1351.2		2
39.20	even	12		1521.2.a.h.1.2		2
39.23	odd	6		117.2.q.a.10.1		2
39.32	even	12		1521.2.a.h.1.1		2
39.35	odd	6		1521.2.b.f.1351.1		2
52.7	even	12		8112.2.a.bu.1.1		2
52.19	even	12		8112.2.a.bu.1.2		2
52.23	odd	6		624.2.bv.b.49.1		2
65.23	odd	12		975.2.w.d.49.1		4
65.49	even	6		975.2.bc.c.751.1		2
65.62	odd	12		975.2.w.d.49.2		4
156.23	even	6		1872.2.by.f.1297.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
39.2.j.a.4.1	✓	2	1.1	even	1	trivial
39.2.j.a.10.1	yes	2	13.10	even	6	inner
117.2.q.a.10.1		2	39.23	odd	6
117.2.q.a.82.1		2	3.2	odd	2
507.2.a.e.1.1		2	13.7	odd	12
507.2.a.e.1.2		2	13.6	odd	12
507.2.b.c.337.1		2	13.4	even	6
507.2.b.c.337.2		2	13.9	even	3
507.2.e.f.22.1		4	13.8	odd	4
507.2.e.f.22.2		4	13.5	odd	4
507.2.e.f.484.1		4	13.11	odd	12
507.2.e.f.484.2		4	13.2	odd	12
507.2.j.b.316.1		2	13.12	even	2
507.2.j.b.361.1		2	13.3	even	3
624.2.bv.b.49.1		2	52.23	odd	6
624.2.bv.b.433.1		2	4.3	odd	2
975.2.w.d.49.1		4	65.23	odd	12
975.2.w.d.49.2		4	65.62	odd	12
975.2.w.d.199.1		4	5.2	odd	4
975.2.w.d.199.2		4	5.3	odd	4
975.2.bc.c.751.1		2	65.49	even	6
975.2.bc.c.901.1		2	5.4	even	2
1521.2.a.h.1.1		2	39.32	even	12
1521.2.a.h.1.2		2	39.20	even	12
1521.2.b.f.1351.1		2	39.35	odd	6
1521.2.b.f.1351.2		2	39.17	odd	6
1872.2.by.f.433.1		2	12.11	even	2
1872.2.by.f.1297.1		2	156.23	even	6
8112.2.a.bu.1.1		2	52.7	even	12
8112.2.a.bu.1.2		2	52.19	even	12

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:	\( N \)	\(=\)	\( 39 = 3 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	39.j (of order \(6\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(0.500000 - 0.866025i) q^{3} +(-1.00000 - 1.73205i) q^{4} +3.46410i q^{5} +(-1.50000 + 0.866025i) q^{7} +(-0.500000 - 0.866025i) q^{9} +O(q^{10})\)	\(q+(0.500000 - 0.866025i) q^{3} +(-1.00000 - 1.73205i) q^{4} +3.46410i q^{5} +(-1.50000 + 0.866025i) q^{7} +(-0.500000 - 0.866025i) q^{9} +(-3.00000 - 1.73205i) q^{11} -2.00000 q^{12} +(3.50000 - 0.866025i) q^{13} +(3.00000 + 1.73205i) q^{15} +(-2.00000 + 3.46410i) q^{16} +(3.00000 - 1.73205i) q^{19} +(6.00000 - 3.46410i) q^{20} +1.73205i q^{21} +(3.00000 - 5.19615i) q^{23} -7.00000 q^{25} -1.00000 q^{27} +(3.00000 + 1.73205i) q^{28} +(-3.00000 + 5.19615i) q^{29} +1.73205i q^{31} +(-3.00000 + 1.73205i) q^{33} +(-3.00000 - 5.19615i) q^{35} +(-1.00000 + 1.73205i) q^{36} +(1.00000 - 3.46410i) q^{39} +(-6.00000 - 3.46410i) q^{41} +(0.500000 + 0.866025i) q^{43} +6.92820i q^{44} +(3.00000 - 1.73205i) q^{45} +3.46410i q^{47} +(2.00000 + 3.46410i) q^{48} +(-2.00000 + 3.46410i) q^{49} +(-5.00000 - 5.19615i) q^{52} +12.0000 q^{53} +(6.00000 - 10.3923i) q^{55} -3.46410i q^{57} +(-3.00000 + 1.73205i) q^{59} -6.92820i q^{60} +(-0.500000 - 0.866025i) q^{61} +(1.50000 + 0.866025i) q^{63} +8.00000 q^{64} +(3.00000 + 12.1244i) q^{65} +(7.50000 + 4.33013i) q^{67} +(-3.00000 - 5.19615i) q^{69} +(-9.00000 + 5.19615i) q^{71} +1.73205i q^{73} +(-3.50000 + 6.06218i) q^{75} +(-6.00000 - 3.46410i) q^{76} +6.00000 q^{77} -11.0000 q^{79} +(-12.0000 - 6.92820i) q^{80} +(-0.500000 + 0.866025i) q^{81} -13.8564i q^{83} +(3.00000 - 1.73205i) q^{84} +(3.00000 + 5.19615i) q^{87} +(6.00000 + 3.46410i) q^{89} +(-4.50000 + 4.33013i) q^{91} -12.0000 q^{92} +(1.50000 + 0.866025i) q^{93} +(6.00000 + 10.3923i) q^{95} +(-4.50000 + 2.59808i) q^{97} +3.46410i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + q^{3} - 2 q^{4} - 3 q^{7} - q^{9}+O(q^{10}) \) 2 * q + q^3 - 2 * q^4 - 3 * q^7 - q^9	\( 2 q + q^{3} - 2 q^{4} - 3 q^{7} - q^{9} - 6 q^{11} - 4 q^{12} + 7 q^{13} + 6 q^{15} - 4 q^{16} + 6 q^{19} + 12 q^{20} + 6 q^{23} - 14 q^{25} - 2 q^{27} + 6 q^{28} - 6 q^{29} - 6 q^{33} - 6 q^{35} - 2 q^{36} + 2 q^{39} - 12 q^{41} + q^{43} + 6 q^{45} + 4 q^{48} - 4 q^{49} - 10 q^{52} + 24 q^{53} + 12 q^{55} - 6 q^{59} - q^{61} + 3 q^{63} + 16 q^{64} + 6 q^{65} + 15 q^{67} - 6 q^{69} - 18 q^{71} - 7 q^{75} - 12 q^{76} + 12 q^{77} - 22 q^{79} - 24 q^{80} - q^{81} + 6 q^{84} + 6 q^{87} + 12 q^{89} - 9 q^{91} - 24 q^{92} + 3 q^{93} + 12 q^{95} - 9 q^{97}+O(q^{100}) \) 2 * q + q^3 - 2 * q^4 - 3 * q^7 - q^9 - 6 * q^11 - 4 * q^12 + 7 * q^13 + 6 * q^15 - 4 * q^16 + 6 * q^19 + 12 * q^20 + 6 * q^23 - 14 * q^25 - 2 * q^27 + 6 * q^28 - 6 * q^29 - 6 * q^33 - 6 * q^35 - 2 * q^36 + 2 * q^39 - 12 * q^41 + q^43 + 6 * q^45 + 4 * q^48 - 4 * q^49 - 10 * q^52 + 24 * q^53 + 12 * q^55 - 6 * q^59 - q^61 + 3 * q^63 + 16 * q^64 + 6 * q^65 + 15 * q^67 - 6 * q^69 - 18 * q^71 - 7 * q^75 - 12 * q^76 + 12 * q^77 - 22 * q^79 - 24 * q^80 - q^81 + 6 * q^84 + 6 * q^87 + 12 * q^89 - 9 * q^91 - 24 * q^92 + 3 * q^93 + 12 * q^95 - 9 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more