LMFDB - Embedded newform 1170.2.i.d.991.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1170,2,Mod(451,1170)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1170.451"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1170, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 0, 4])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 1170 = 2 \cdot 3^{2} \cdot 5 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1170.i (of order $3$, degree $2$, minimal)

Newform invariants

comment:select newform

sage:traces = [2,-1,0,-1,2,0,-2,2,0,-1,-3] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)

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

Self dual:	no
Analytic conductor:	$9.34249703649$
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:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 390)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			991.1
Root			$0.500000 - 0.866025i$ of defining polynomial
Character	$\chi$	$=$	1170.991
Dual form			1170.2.i.d.451.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+(-0.500000 - 0.866025i) q^{2} +(-0.500000 + 0.866025i) q^{4} +1.00000 q^{5} +(-1.00000 + 1.73205i) q^{7} +1.00000 q^{8} +(-0.500000 - 0.866025i) q^{10} +(-1.50000 - 2.59808i) q^{11} +(1.00000 + 3.46410i) q^{13} +2.00000 q^{14} +(-0.500000 - 0.866025i) q^{16} +(3.00000 - 5.19615i) q^{17} +(-1.00000 + 1.73205i) q^{19} +(-0.500000 + 0.866025i) q^{20} +(-1.50000 + 2.59808i) q^{22} +(1.50000 + 2.59808i) q^{23} +1.00000 q^{25} +(2.50000 - 2.59808i) q^{26} +(-1.00000 - 1.73205i) q^{28} +(1.50000 + 2.59808i) q^{29} +5.00000 q^{31} +(-0.500000 + 0.866025i) q^{32} -6.00000 q^{34} +(-1.00000 + 1.73205i) q^{35} +(3.50000 + 6.06218i) q^{37} +2.00000 q^{38} +1.00000 q^{40} +(3.00000 + 5.19615i) q^{41} +(0.500000 - 0.866025i) q^{43} +3.00000 q^{44} +(1.50000 - 2.59808i) q^{46} +3.00000 q^{47} +(1.50000 + 2.59808i) q^{49} +(-0.500000 - 0.866025i) q^{50} +(-3.50000 - 0.866025i) q^{52} +6.00000 q^{53} +(-1.50000 - 2.59808i) q^{55} +(-1.00000 + 1.73205i) q^{56} +(1.50000 - 2.59808i) q^{58} +(-4.50000 + 7.79423i) q^{59} +(-1.00000 + 1.73205i) q^{61} +(-2.50000 - 4.33013i) q^{62} +1.00000 q^{64} +(1.00000 + 3.46410i) q^{65} +(-4.00000 - 6.92820i) q^{67} +(3.00000 + 5.19615i) q^{68} +2.00000 q^{70} +(-6.00000 + 10.3923i) q^{71} +14.0000 q^{73} +(3.50000 - 6.06218i) q^{74} +(-1.00000 - 1.73205i) q^{76} +6.00000 q^{77} +5.00000 q^{79} +(-0.500000 - 0.866025i) q^{80} +(3.00000 - 5.19615i) q^{82} +6.00000 q^{83} +(3.00000 - 5.19615i) q^{85} -1.00000 q^{86} +(-1.50000 - 2.59808i) q^{88} +(-9.00000 - 15.5885i) q^{89} +(-7.00000 - 1.73205i) q^{91} -3.00000 q^{92} +(-1.50000 - 2.59808i) q^{94} +(-1.00000 + 1.73205i) q^{95} +(-7.00000 + 12.1244i) q^{97} +(1.50000 - 2.59808i) q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - q^{2} - q^{4} + 2 q^{5} - 2 q^{7} + 2 q^{8} - q^{10} - 3 q^{11} + 2 q^{13} + 4 q^{14} - q^{16} + 6 q^{17} - 2 q^{19} - q^{20} - 3 q^{22} + 3 q^{23} + 2 q^{25} + 5 q^{26} - 2 q^{28} + 3 q^{29}+ \cdots + 3 q^{98}+O(q^{100}) $ 2 * q - q^2 - q^4 + 2 * q^5 - 2 * q^7 + 2 * q^8 - q^10 - 3 * q^11 + 2 * q^13 + 4 * q^14 - q^16 + 6 * q^17 - 2 * q^19 - q^20 - 3 * q^22 + 3 * q^23 + 2 * q^25 + 5 * q^26 - 2 * q^28 + 3 * q^29 + 10 * q^31 - q^32 - 12 * q^34 - 2 * q^35 + 7 * q^37 + 4 * q^38 + 2 * q^40 + 6 * q^41 + q^43 + 6 * q^44 + 3 * q^46 + 6 * q^47 + 3 * q^49 - q^50 - 7 * q^52 + 12 * q^53 - 3 * q^55 - 2 * q^56 + 3 * q^58 - 9 * q^59 - 2 * q^61 - 5 * q^62 + 2 * q^64 + 2 * q^65 - 8 * q^67 + 6 * q^68 + 4 * q^70 - 12 * q^71 + 28 * q^73 + 7 * q^74 - 2 * q^76 + 12 * q^77 + 10 * q^79 - q^80 + 6 * q^82 + 12 * q^83 + 6 * q^85 - 2 * q^86 - 3 * q^88 - 18 * q^89 - 14 * q^91 - 6 * q^92 - 3 * q^94 - 2 * q^95 - 14 * q^97 + 3 * q^98

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1170\mathbb{Z}\right)^\times$.

$n$	$911$	$937$	$1081$
$\chi(n)$	$1$	$1$	$e\left(\frac{1}{3}\right)$

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 $

$2$	−0.500000	−	0.866025i	−0.353553	−	0.612372i
$2$	−0.500000	−	0.866025i	−0.353553	−	0.612372i
$3$		0			0
$3$		0			0
$4$	−0.500000	+	0.866025i	−0.250000	+	0.433013i
$5$	1.00000			0.447214
$5$	1.00000			0.447214
$6$		0			0
$7$	−1.00000	+	1.73205i	−0.377964	+	0.654654i	−0.990766	−	0.135583i	$-0.956709\pi$
$7$	−1.00000	+	1.73205i	−0.377964	+	0.654654i	0.612801	+	0.790237i	$0.290043\pi$
$8$	1.00000			0.353553
$9$		0			0
$10$	−0.500000	−	0.866025i	−0.158114	−	0.273861i
$11$	−1.50000	−	2.59808i	−0.452267	−	0.783349i	0.546259	−	0.837616i	$-0.316051\pi$
$11$	−1.50000	−	2.59808i	−0.452267	−	0.783349i	−0.998526	+	0.0542666i	$0.982718\pi$
$12$		0			0
$13$	1.00000	+	3.46410i	0.277350	+	0.960769i
$13$	1.00000	+	3.46410i	0.277350	+	0.960769i
$14$	2.00000			0.534522
$15$		0			0
$16$	−0.500000	−	0.866025i	−0.125000	−	0.216506i
$17$	3.00000	−	5.19615i	0.727607	−	1.26025i	−0.230285	−	0.973123i	$-0.573966\pi$
$17$	3.00000	−	5.19615i	0.727607	−	1.26025i	0.957892	−	0.287129i	$-0.0927008\pi$
$18$		0			0
$19$	−1.00000	+	1.73205i	−0.229416	+	0.397360i	−0.957635	−	0.287984i	$-0.907015\pi$
$19$	−1.00000	+	1.73205i	−0.229416	+	0.397360i	0.728219	+	0.685344i	$0.240348\pi$
$20$	−0.500000	+	0.866025i	−0.111803	+	0.193649i
$21$		0			0
$22$	−1.50000	+	2.59808i	−0.319801	+	0.553912i
$23$	1.50000	+	2.59808i	0.312772	+	0.541736i	0.978961	−	0.204046i	$-0.0654092\pi$
$23$	1.50000	+	2.59808i	0.312772	+	0.541736i	−0.666190	+	0.745782i	$0.732076\pi$
$24$		0			0
$25$	1.00000			0.200000
$26$	2.50000	−	2.59808i	0.490290	−	0.509525i
$27$		0			0
$28$	−1.00000	−	1.73205i	−0.188982	−	0.327327i
$29$	1.50000	+	2.59808i	0.278543	+	0.482451i	0.971023	−	0.238987i	$-0.0768152\pi$
$29$	1.50000	+	2.59808i	0.278543	+	0.482451i	−0.692480	+	0.721437i	$0.743482\pi$
$30$		0			0
$31$	5.00000			0.898027			0.449013	−	0.893525i	$-0.351776\pi$
$31$	5.00000			0.898027			0.449013	+	0.893525i	$0.351776\pi$
$32$	−0.500000	+	0.866025i	−0.0883883	+	0.153093i
$33$		0			0
$34$	−6.00000			−1.02899
$35$	−1.00000	+	1.73205i	−0.169031	+	0.292770i
$36$		0			0
$37$	3.50000	+	6.06218i	0.575396	+	0.996616i	0.995998	+	0.0893706i	$0.0284856\pi$
$37$	3.50000	+	6.06218i	0.575396	+	0.996616i	−0.420602	+	0.907245i	$0.638181\pi$
$38$	2.00000			0.324443
$39$		0			0
$40$	1.00000			0.158114
$41$	3.00000	+	5.19615i	0.468521	+	0.811503i	0.999353	−	0.0359748i	$-0.0114536\pi$
$41$	3.00000	+	5.19615i	0.468521	+	0.811503i	−0.530831	+	0.847477i	$0.678120\pi$
$42$		0			0
$43$	0.500000	−	0.866025i	0.0762493	−	0.132068i	−0.825380	−	0.564578i	$-0.809039\pi$
$43$	0.500000	−	0.866025i	0.0762493	−	0.132068i	0.901629	+	0.432511i	$0.142372\pi$
$44$	3.00000			0.452267
$45$		0			0
$46$	1.50000	−	2.59808i	0.221163	−	0.383065i
$47$	3.00000			0.437595			0.218797	−	0.975770i	$-0.429787\pi$
$47$	3.00000			0.437595			0.218797	+	0.975770i	$0.429787\pi$
$48$		0			0
$49$	1.50000	+	2.59808i	0.214286	+	0.371154i
$50$	−0.500000	−	0.866025i	−0.0707107	−	0.122474i
$51$		0			0
$52$	−3.50000	−	0.866025i	−0.485363	−	0.120096i
$53$	6.00000			0.824163			0.412082	−	0.911147i	$-0.364802\pi$
$53$	6.00000			0.824163			0.412082	+	0.911147i	$0.364802\pi$
$54$		0			0
$55$	−1.50000	−	2.59808i	−0.202260	−	0.350325i
$56$	−1.00000	+	1.73205i	−0.133631	+	0.231455i
$57$		0			0
$58$	1.50000	−	2.59808i	0.196960	−	0.341144i
$59$	−4.50000	+	7.79423i	−0.585850	+	1.01472i	0.408919	+	0.912571i	$0.365906\pi$
$59$	−4.50000	+	7.79423i	−0.585850	+	1.01472i	−0.994769	+	0.102151i	$0.967427\pi$
$60$		0			0
$61$	−1.00000	+	1.73205i	−0.128037	+	0.221766i	−0.922916	−	0.385002i	$-0.874201\pi$
$61$	−1.00000	+	1.73205i	−0.128037	+	0.221766i	0.794879	+	0.606768i	$0.207534\pi$
$62$	−2.50000	−	4.33013i	−0.317500	−	0.549927i
$63$		0			0
$64$	1.00000			0.125000
$65$	1.00000	+	3.46410i	0.124035	+	0.429669i
$66$		0			0
$67$	−4.00000	−	6.92820i	−0.488678	−	0.846415i	0.511237	−	0.859440i	$-0.329187\pi$
$67$	−4.00000	−	6.92820i	−0.488678	−	0.846415i	−0.999915	+	0.0130248i	$0.995854\pi$
$68$	3.00000	+	5.19615i	0.363803	+	0.630126i
$69$		0			0
$70$	2.00000			0.239046
$71$	−6.00000	+	10.3923i	−0.712069	+	1.23334i	0.252010	+	0.967725i	$0.418908\pi$
$71$	−6.00000	+	10.3923i	−0.712069	+	1.23334i	−0.964079	+	0.265615i	$0.914425\pi$
$72$		0			0
$73$	14.0000			1.63858			0.819288	−	0.573382i	$-0.194369\pi$
$73$	14.0000			1.63858			0.819288	+	0.573382i	$0.194369\pi$
$74$	3.50000	−	6.06218i	0.406867	−	0.704714i
$75$		0			0
$76$	−1.00000	−	1.73205i	−0.114708	−	0.198680i
$77$	6.00000			0.683763
$78$		0			0
$79$	5.00000			0.562544			0.281272	−	0.959628i	$-0.409244\pi$
$79$	5.00000			0.562544			0.281272	+	0.959628i	$0.409244\pi$
$80$	−0.500000	−	0.866025i	−0.0559017	−	0.0968246i
$81$		0			0
$82$	3.00000	−	5.19615i	0.331295	−	0.573819i
$83$	6.00000			0.658586			0.329293	−	0.944228i	$-0.393190\pi$
$83$	6.00000			0.658586			0.329293	+	0.944228i	$0.393190\pi$
$84$		0			0
$85$	3.00000	−	5.19615i	0.325396	−	0.563602i
$86$	−1.00000			−0.107833
$87$		0			0
$88$	−1.50000	−	2.59808i	−0.159901	−	0.276956i
$89$	−9.00000	−	15.5885i	−0.953998	−	1.65237i	−0.736644	−	0.676280i	$-0.763591\pi$
$89$	−9.00000	−	15.5885i	−0.953998	−	1.65237i	−0.217354	−	0.976093i	$-0.569742\pi$
$90$		0			0
$91$	−7.00000	−	1.73205i	−0.733799	−	0.181568i
$92$	−3.00000			−0.312772
$93$		0			0
$94$	−1.50000	−	2.59808i	−0.154713	−	0.267971i
$95$	−1.00000	+	1.73205i	−0.102598	+	0.177705i
$96$		0			0
$97$	−7.00000	+	12.1244i	−0.710742	+	1.23104i	0.253837	+	0.967247i	$0.418307\pi$
$97$	−7.00000	+	12.1244i	−0.710742	+	1.23104i	−0.964579	+	0.263795i	$0.915026\pi$
$98$	1.50000	−	2.59808i	0.151523	−	0.262445i
$99$		0			0
$100$	−0.500000	+	0.866025i	−0.0500000	+	0.0866025i
$101$	3.00000	+	5.19615i	0.298511	+	0.517036i	0.975796	−	0.218685i	$-0.0701767\pi$
$101$	3.00000	+	5.19615i	0.298511	+	0.517036i	−0.677284	+	0.735721i	$0.736843\pi$
$102$		0			0
$103$	14.0000			1.37946			0.689730	−	0.724066i	$-0.257729\pi$
$103$	14.0000			1.37946			0.689730	+	0.724066i	$0.257729\pi$
$104$	1.00000	+	3.46410i	0.0980581	+	0.339683i
$105$		0			0
$106$	−3.00000	−	5.19615i	−0.291386	−	0.504695i
$107$	−3.00000	−	5.19615i	−0.290021	−	0.502331i	0.683793	−	0.729676i	$-0.260329\pi$
$107$	−3.00000	−	5.19615i	−0.290021	−	0.502331i	−0.973814	+	0.227345i	$0.926996\pi$
$108$		0			0
$109$	14.0000			1.34096			0.670478	−	0.741929i	$-0.266089\pi$
$109$	14.0000			1.34096			0.670478	+	0.741929i	$0.266089\pi$
$110$	−1.50000	+	2.59808i	−0.143019	+	0.247717i
$111$		0			0
$112$	2.00000			0.188982
$113$	7.50000	−	12.9904i	0.705541	−	1.22203i	−0.260955	−	0.965351i	$-0.584038\pi$
$113$	7.50000	−	12.9904i	0.705541	−	1.22203i	0.966496	−	0.256681i	$-0.0826291\pi$
$114$		0			0
$115$	1.50000	+	2.59808i	0.139876	+	0.242272i
$116$	−3.00000			−0.278543
$117$		0			0
$118$	9.00000			0.828517
$119$	6.00000	+	10.3923i	0.550019	+	0.952661i
$120$		0			0
$121$	1.00000	−	1.73205i	0.0909091	−	0.157459i
$122$	2.00000			0.181071
$123$		0			0
$124$	−2.50000	+	4.33013i	−0.224507	+	0.388857i
$125$	1.00000			0.0894427
$126$		0			0
$127$	−7.00000	−	12.1244i	−0.621150	−	1.07586i	−0.989272	−	0.146085i	$-0.953333\pi$
$127$	−7.00000	−	12.1244i	−0.621150	−	1.07586i	0.368122	−	0.929777i	$-0.380001\pi$
$128$	−0.500000	−	0.866025i	−0.0441942	−	0.0765466i
$129$		0			0
$130$	2.50000	−	2.59808i	0.219265	−	0.227866i
$131$	−9.00000			−0.786334			−0.393167	−	0.919467i	$-0.628621\pi$
$131$	−9.00000			−0.786334			−0.393167	+	0.919467i	$0.628621\pi$
$132$		0			0
$133$	−2.00000	−	3.46410i	−0.173422	−	0.300376i
$134$	−4.00000	+	6.92820i	−0.345547	+	0.598506i
$135$		0			0
$136$	3.00000	−	5.19615i	0.257248	−	0.445566i
$137$	4.50000	−	7.79423i	0.384461	−	0.665906i	−0.607233	−	0.794524i	$-0.707721\pi$
$137$	4.50000	−	7.79423i	0.384461	−	0.665906i	0.991694	+	0.128618i	$0.0410540\pi$
$138$		0			0
$139$	−7.00000	+	12.1244i	−0.593732	+	1.02837i	0.399992	+	0.916519i	$0.369013\pi$
$139$	−7.00000	+	12.1244i	−0.593732	+	1.02837i	−0.993724	+	0.111856i	$0.964321\pi$
$140$	−1.00000	−	1.73205i	−0.0845154	−	0.146385i
$141$		0			0
$142$	12.0000			1.00702
$143$	7.50000	−	7.79423i	0.627182	−	0.651786i
$144$		0			0
$145$	1.50000	+	2.59808i	0.124568	+	0.215758i
$146$	−7.00000	−	12.1244i	−0.579324	−	1.00342i
$147$		0			0
$148$	−7.00000			−0.575396
$149$	−4.50000	+	7.79423i	−0.368654	+	0.638528i	−0.989355	−	0.145519i	$-0.953515\pi$
$149$	−4.50000	+	7.79423i	−0.368654	+	0.638528i	0.620701	+	0.784047i	$0.286848\pi$
$150$		0			0
$151$	8.00000			0.651031			0.325515	−	0.945537i	$-0.394462\pi$
$151$	8.00000			0.651031			0.325515	+	0.945537i	$0.394462\pi$
$152$	−1.00000	+	1.73205i	−0.0811107	+	0.140488i
$153$		0			0
$154$	−3.00000	−	5.19615i	−0.241747	−	0.418718i
$155$	5.00000			0.401610
$156$		0			0
$157$	−13.0000			−1.03751			−0.518756	−	0.854922i	$-0.673605\pi$
$157$	−13.0000			−1.03751			−0.518756	+	0.854922i	$0.673605\pi$
$158$	−2.50000	−	4.33013i	−0.198889	−	0.344486i
$159$		0			0
$160$	−0.500000	+	0.866025i	−0.0395285	+	0.0684653i
$161$	−6.00000			−0.472866
$162$		0			0
$163$	6.50000	−	11.2583i	0.509119	−	0.881820i	−0.490825	−	0.871258i	$-0.663305\pi$
$163$	6.50000	−	11.2583i	0.509119	−	0.881820i	0.999944	−	0.0105623i	$-0.00336213\pi$
$164$	−6.00000			−0.468521
$165$		0			0
$166$	−3.00000	−	5.19615i	−0.232845	−	0.403300i
$167$	−4.50000	−	7.79423i	−0.348220	−	0.603136i	0.637713	−	0.770274i	$-0.279881\pi$
$167$	−4.50000	−	7.79423i	−0.348220	−	0.603136i	−0.985933	+	0.167139i	$0.946547\pi$
$168$		0			0
$169$	−11.0000	+	6.92820i	−0.846154	+	0.532939i
$170$	−6.00000			−0.460179
$171$		0			0
$172$	0.500000	+	0.866025i	0.0381246	+	0.0660338i
$173$	6.00000	−	10.3923i	0.456172	−	0.790112i	−0.542583	−	0.840002i	$-0.682554\pi$
$173$	6.00000	−	10.3923i	0.456172	−	0.790112i	0.998755	+	0.0498898i	$0.0158870\pi$
$174$		0			0
$175$	−1.00000	+	1.73205i	−0.0755929	+	0.130931i
$176$	−1.50000	+	2.59808i	−0.113067	+	0.195837i
$177$		0			0
$178$	−9.00000	+	15.5885i	−0.674579	+	1.16840i
$179$	−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	$0.869096\pi$
$180$		0			0
$181$	−16.0000			−1.18927			−0.594635	−	0.803996i	$-0.702704\pi$
$181$	−16.0000			−1.18927			−0.594635	+	0.803996i	$0.702704\pi$
$182$	2.00000	+	6.92820i	0.148250	+	0.513553i
$183$		0			0
$184$	1.50000	+	2.59808i	0.110581	+	0.191533i
$185$	3.50000	+	6.06218i	0.257325	+	0.445700i
$186$		0			0
$187$	−18.0000			−1.31629
$188$	−1.50000	+	2.59808i	−0.109399	+	0.189484i
$189$		0			0
$190$	2.00000			0.145095
$191$	−6.00000	+	10.3923i	−0.434145	+	0.751961i	−0.997225	−	0.0744412i	$-0.976283\pi$
$191$	−6.00000	+	10.3923i	−0.434145	+	0.751961i	0.563081	+	0.826402i	$0.309616\pi$
$192$		0			0
$193$	2.00000	+	3.46410i	0.143963	+	0.249351i	0.928986	−	0.370116i	$-0.120682\pi$
$193$	2.00000	+	3.46410i	0.143963	+	0.249351i	−0.785022	+	0.619467i	$0.787349\pi$
$194$	14.0000			1.00514
$195$		0			0
$196$	−3.00000			−0.214286
$197$	12.0000	+	20.7846i	0.854965	+	1.48084i	0.876678	+	0.481078i	$0.159755\pi$
$197$	12.0000	+	20.7846i	0.854965	+	1.48084i	−0.0217133	+	0.999764i	$0.506912\pi$
$198$		0			0
$199$	−4.00000	+	6.92820i	−0.283552	+	0.491127i	−0.972257	−	0.233915i	$-0.924846\pi$
$199$	−4.00000	+	6.92820i	−0.283552	+	0.491127i	0.688705	+	0.725042i	$0.258180\pi$
$200$	1.00000			0.0707107
$201$		0			0
$202$	3.00000	−	5.19615i	0.211079	−	0.365600i
$203$	−6.00000			−0.421117
$204$		0			0
$205$	3.00000	+	5.19615i	0.209529	+	0.362915i
$206$	−7.00000	−	12.1244i	−0.487713	−	0.844744i
$207$		0			0
$208$	2.50000	−	2.59808i	0.173344	−	0.180144i
$209$	6.00000			0.415029
$210$		0			0
$211$	−10.0000	−	17.3205i	−0.688428	−	1.19239i	−0.972346	−	0.233544i	$-0.924968\pi$
$211$	−10.0000	−	17.3205i	−0.688428	−	1.19239i	0.283918	−	0.958849i	$-0.408366\pi$
$212$	−3.00000	+	5.19615i	−0.206041	+	0.356873i
$213$		0			0
$214$	−3.00000	+	5.19615i	−0.205076	+	0.355202i
$215$	0.500000	−	0.866025i	0.0340997	−	0.0590624i
$216$		0			0
$217$	−5.00000	+	8.66025i	−0.339422	+	0.587896i
$218$	−7.00000	−	12.1244i	−0.474100	−	0.821165i
$219$		0			0
$220$	3.00000			0.202260
$221$	21.0000	+	5.19615i	1.41261	+	0.349531i
$222$		0			0
$223$	5.00000	+	8.66025i	0.334825	+	0.579934i	0.983451	−	0.181173i	$-0.0579895\pi$
$223$	5.00000	+	8.66025i	0.334825	+	0.579934i	−0.648626	+	0.761107i	$0.724656\pi$
$224$	−1.00000	−	1.73205i	−0.0668153	−	0.115728i
$225$		0			0
$226$	−15.0000			−0.997785
$227$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$227$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$228$		0			0
$229$	14.0000			0.925146			0.462573	−	0.886581i	$-0.346926\pi$
$229$	14.0000			0.925146			0.462573	+	0.886581i	$0.346926\pi$
$230$	1.50000	−	2.59808i	0.0989071	−	0.171312i
$231$		0			0
$232$	1.50000	+	2.59808i	0.0984798	+	0.170572i
$233$	−21.0000			−1.37576			−0.687878	−	0.725826i	$-0.741458\pi$
$233$	−21.0000			−1.37576			−0.687878	+	0.725826i	$0.741458\pi$
$234$		0			0
$235$	3.00000			0.195698
$236$	−4.50000	−	7.79423i	−0.292925	−	0.507361i
$237$		0			0
$238$	6.00000	−	10.3923i	0.388922	−	0.673633i
$239$	−24.0000			−1.55243			−0.776215	−	0.630468i	$-0.782863\pi$
$239$	−24.0000			−1.55243			−0.776215	+	0.630468i	$0.782863\pi$
$240$		0			0
$241$	−8.50000	+	14.7224i	−0.547533	+	0.948355i	0.450910	+	0.892570i	$0.351100\pi$
$241$	−8.50000	+	14.7224i	−0.547533	+	0.948355i	−0.998443	+	0.0557856i	$0.982234\pi$
$242$	−2.00000			−0.128565
$243$		0			0
$244$	−1.00000	−	1.73205i	−0.0640184	−	0.110883i
$245$	1.50000	+	2.59808i	0.0958315	+	0.165985i
$246$		0			0
$247$	−7.00000	−	1.73205i	−0.445399	−	0.110208i
$248$	5.00000			0.317500
$249$		0			0
$250$	−0.500000	−	0.866025i	−0.0316228	−	0.0547723i
$251$	7.50000	−	12.9904i	0.473396	−	0.819946i	−0.526140	−	0.850398i	$-0.676361\pi$
$251$	7.50000	−	12.9904i	0.473396	−	0.819946i	0.999536	+	0.0304521i	$0.00969471\pi$
$252$		0			0
$253$	4.50000	−	7.79423i	0.282913	−	0.490019i
$254$	−7.00000	+	12.1244i	−0.439219	+	0.760750i
$255$		0			0
$256$	−0.500000	+	0.866025i	−0.0312500	+	0.0541266i
$257$	−10.5000	−	18.1865i	−0.654972	−	1.13444i	−0.981901	−	0.189396i	$-0.939347\pi$
$257$	−10.5000	−	18.1865i	−0.654972	−	1.13444i	0.326929	−	0.945049i	$-0.393986\pi$
$258$		0			0
$259$	−14.0000			−0.869918
$260$	−3.50000	−	0.866025i	−0.217061	−	0.0537086i
$261$		0			0
$262$	4.50000	+	7.79423i	0.278011	+	0.481529i
$263$	−7.50000	−	12.9904i	−0.462470	−	0.801021i	0.536614	−	0.843828i	$-0.319703\pi$
$263$	−7.50000	−	12.9904i	−0.462470	−	0.801021i	−0.999083	+	0.0428069i	$0.986370\pi$
$264$		0			0
$265$	6.00000			0.368577
$266$	−2.00000	+	3.46410i	−0.122628	+	0.212398i
$267$		0			0
$268$	8.00000			0.488678
$269$	9.00000	−	15.5885i	0.548740	−	0.950445i	−0.449622	−	0.893219i	$-0.648441\pi$
$269$	9.00000	−	15.5885i	0.548740	−	0.950445i	0.998361	−	0.0572259i	$-0.0182255\pi$
$270$		0			0
$271$	−5.50000	−	9.52628i	−0.334101	−	0.578680i	0.649211	−	0.760609i	$-0.275099\pi$
$271$	−5.50000	−	9.52628i	−0.334101	−	0.578680i	−0.983312	+	0.181928i	$0.941766\pi$
$272$	−6.00000			−0.363803
$273$		0			0
$274$	−9.00000			−0.543710
$275$	−1.50000	−	2.59808i	−0.0904534	−	0.156670i
$276$		0			0
$277$	0.500000	−	0.866025i	0.0300421	−	0.0520344i	−0.850613	−	0.525792i	$-0.823769\pi$
$277$	0.500000	−	0.866025i	0.0300421	−	0.0520344i	0.880656	+	0.473757i	$0.157103\pi$
$278$	14.0000			0.839664
$279$		0			0
$280$	−1.00000	+	1.73205i	−0.0597614	+	0.103510i
$281$	−18.0000			−1.07379			−0.536895	−	0.843649i	$-0.680403\pi$
$281$	−18.0000			−1.07379			−0.536895	+	0.843649i	$0.680403\pi$
$282$		0			0
$283$	15.5000	+	26.8468i	0.921379	+	1.59588i	0.797283	+	0.603606i	$0.206270\pi$
$283$	15.5000	+	26.8468i	0.921379	+	1.59588i	0.124096	+	0.992270i	$0.460397\pi$
$284$	−6.00000	−	10.3923i	−0.356034	−	0.616670i
$285$		0			0
$286$	−10.5000	−	2.59808i	−0.620878	−	0.153627i
$287$	−12.0000			−0.708338
$288$		0			0
$289$	−9.50000	−	16.4545i	−0.558824	−	0.967911i
$290$	1.50000	−	2.59808i	0.0880830	−	0.152564i
$291$		0			0
$292$	−7.00000	+	12.1244i	−0.409644	+	0.709524i
$293$	−15.0000	+	25.9808i	−0.876309	+	1.51781i	−0.0209480	+	0.999781i	$0.506668\pi$
$293$	−15.0000	+	25.9808i	−0.876309	+	1.51781i	−0.855361	+	0.518032i	$0.826665\pi$
$294$		0			0
$295$	−4.50000	+	7.79423i	−0.262000	+	0.453798i
$296$	3.50000	+	6.06218i	0.203433	+	0.352357i
$297$		0			0
$298$	9.00000			0.521356
$299$	−7.50000	+	7.79423i	−0.433736	+	0.450752i
$300$		0			0
$301$	1.00000	+	1.73205i	0.0576390	+	0.0998337i
$302$	−4.00000	−	6.92820i	−0.230174	−	0.398673i
$303$		0			0
$304$	2.00000			0.114708
$305$	−1.00000	+	1.73205i	−0.0572598	+	0.0991769i
$306$		0			0
$307$	8.00000			0.456584			0.228292	−	0.973593i	$-0.426686\pi$
$307$	8.00000			0.456584			0.228292	+	0.973593i	$0.426686\pi$
$308$	−3.00000	+	5.19615i	−0.170941	+	0.296078i
$309$		0			0
$310$	−2.50000	−	4.33013i	−0.141990	−	0.245935i
$311$	12.0000			0.680458			0.340229	−	0.940343i	$-0.389495\pi$
$311$	12.0000			0.680458			0.340229	+	0.940343i	$0.389495\pi$
$312$		0			0
$313$	8.00000			0.452187			0.226093	−	0.974106i	$-0.427405\pi$
$313$	8.00000			0.452187			0.226093	+	0.974106i	$0.427405\pi$
$314$	6.50000	+	11.2583i	0.366816	+	0.635344i
$315$		0			0
$316$	−2.50000	+	4.33013i	−0.140636	+	0.243589i
$317$	−12.0000			−0.673987			−0.336994	−	0.941507i	$-0.609410\pi$
$317$	−12.0000			−0.673987			−0.336994	+	0.941507i	$0.609410\pi$
$318$		0			0
$319$	4.50000	−	7.79423i	0.251952	−	0.436393i
$320$	1.00000			0.0559017
$321$		0			0
$322$	3.00000	+	5.19615i	0.167183	+	0.289570i
$323$	6.00000	+	10.3923i	0.333849	+	0.578243i
$324$		0			0
$325$	1.00000	+	3.46410i	0.0554700	+	0.192154i
$326$	−13.0000			−0.720003
$327$		0			0
$328$	3.00000	+	5.19615i	0.165647	+	0.286910i
$329$	−3.00000	+	5.19615i	−0.165395	+	0.286473i
$330$		0			0
$331$	−16.0000	+	27.7128i	−0.879440	+	1.52323i	−0.0274825	+	0.999622i	$0.508749\pi$
$331$	−16.0000	+	27.7128i	−0.879440	+	1.52323i	−0.851957	+	0.523612i	$0.824584\pi$
$332$	−3.00000	+	5.19615i	−0.164646	+	0.285176i
$333$		0			0
$334$	−4.50000	+	7.79423i	−0.246229	+	0.426481i
$335$	−4.00000	−	6.92820i	−0.218543	−	0.378528i
$336$		0			0
$337$	14.0000			0.762629			0.381314	−	0.924445i	$-0.375472\pi$
$337$	14.0000			0.762629			0.381314	+	0.924445i	$0.375472\pi$
$338$	11.5000	+	6.06218i	0.625518	+	0.329739i
$339$		0			0
$340$	3.00000	+	5.19615i	0.162698	+	0.281801i
$341$	−7.50000	−	12.9904i	−0.406148	−	0.703469i
$342$		0			0
$343$	−20.0000			−1.07990
$344$	0.500000	−	0.866025i	0.0269582	−	0.0466930i
$345$		0			0
$346$	−12.0000			−0.645124
$347$	−15.0000	+	25.9808i	−0.805242	+	1.39472i	0.110885	+	0.993833i	$0.464631\pi$
$347$	−15.0000	+	25.9808i	−0.805242	+	1.39472i	−0.916127	+	0.400887i	$0.868702\pi$
$348$		0			0
$349$	−4.00000	−	6.92820i	−0.214115	−	0.370858i	0.738883	−	0.673833i	$-0.235353\pi$
$349$	−4.00000	−	6.92820i	−0.214115	−	0.370858i	−0.952998	+	0.302975i	$0.902020\pi$
$350$	2.00000			0.106904
$351$		0			0
$352$	3.00000			0.159901
$353$	15.0000	+	25.9808i	0.798369	+	1.38282i	0.920677	+	0.390324i	$0.127637\pi$
$353$	15.0000	+	25.9808i	0.798369	+	1.38282i	−0.122308	+	0.992492i	$0.539030\pi$
$354$		0			0
$355$	−6.00000	+	10.3923i	−0.318447	+	0.551566i
$356$	18.0000			0.953998
$357$		0			0
$358$	−1.50000	+	2.59808i	−0.0792775	+	0.137313i
$359$	−24.0000			−1.26667			−0.633336	−	0.773877i	$-0.718315\pi$
$359$	−24.0000			−1.26667			−0.633336	+	0.773877i	$0.718315\pi$
$360$		0			0
$361$	7.50000	+	12.9904i	0.394737	+	0.683704i
$362$	8.00000	+	13.8564i	0.420471	+	0.728277i
$363$		0			0
$364$	5.00000	−	5.19615i	0.262071	−	0.272352i
$365$	14.0000			0.732793
$366$		0			0
$367$	−10.0000	−	17.3205i	−0.521996	−	0.904123i	−0.999673	−	0.0255875i	$-0.991854\pi$
$367$	−10.0000	−	17.3205i	−0.521996	−	0.904123i	0.477677	−	0.878536i	$-0.341479\pi$
$368$	1.50000	−	2.59808i	0.0781929	−	0.135434i
$369$		0			0
$370$	3.50000	−	6.06218i	0.181956	−	0.315158i
$371$	−6.00000	+	10.3923i	−0.311504	+	0.539542i
$372$		0			0
$373$	12.5000	−	21.6506i	0.647225	−	1.12103i	−0.336557	−	0.941663i	$-0.609263\pi$
$373$	12.5000	−	21.6506i	0.647225	−	1.12103i	0.983783	−	0.179364i	$-0.0574041\pi$
$374$	9.00000	+	15.5885i	0.465379	+	0.806060i
$375$		0			0
$376$	3.00000			0.154713
$377$	−7.50000	+	7.79423i	−0.386270	+	0.401423i
$378$		0			0
$379$	−19.0000	−	32.9090i	−0.975964	−	1.69042i	−0.676715	−	0.736245i	$-0.736597\pi$
$379$	−19.0000	−	32.9090i	−0.975964	−	1.69042i	−0.299249	−	0.954175i	$-0.596736\pi$
$380$	−1.00000	−	1.73205i	−0.0512989	−	0.0888523i
$381$		0			0
$382$	12.0000			0.613973
$383$	−10.5000	+	18.1865i	−0.536525	+	0.929288i	0.462563	+	0.886586i	$0.346930\pi$
$383$	−10.5000	+	18.1865i	−0.536525	+	0.929288i	−0.999088	+	0.0427020i	$0.986403\pi$
$384$		0			0
$385$	6.00000			0.305788
$386$	2.00000	−	3.46410i	0.101797	−	0.176318i
$387$		0			0
$388$	−7.00000	−	12.1244i	−0.355371	−	0.615521i
$389$	−3.00000			−0.152106			−0.0760530	−	0.997104i	$-0.524232\pi$
$389$	−3.00000			−0.152106			−0.0760530	+	0.997104i	$0.524232\pi$
$390$		0			0
$391$	18.0000			0.910299
$392$	1.50000	+	2.59808i	0.0757614	+	0.131223i
$393$		0			0
$394$	12.0000	−	20.7846i	0.604551	−	1.04711i
$395$	5.00000			0.251577
$396$		0			0
$397$	15.5000	−	26.8468i	0.777923	−	1.34740i	−0.155214	−	0.987881i	$-0.549607\pi$
$397$	15.5000	−	26.8468i	0.777923	−	1.34740i	0.933137	−	0.359521i	$-0.117060\pi$
$398$	8.00000			0.401004
$399$		0			0
$400$	−0.500000	−	0.866025i	−0.0250000	−	0.0433013i
$401$	−6.00000	−	10.3923i	−0.299626	−	0.518967i	0.676425	−	0.736512i	$-0.263528\pi$
$401$	−6.00000	−	10.3923i	−0.299626	−	0.518967i	−0.976050	+	0.217545i	$0.930195\pi$
$402$		0			0
$403$	5.00000	+	17.3205i	0.249068	+	0.862796i
$404$	−6.00000			−0.298511
$405$		0			0
$406$	3.00000	+	5.19615i	0.148888	+	0.257881i
$407$	10.5000	−	18.1865i	0.520466	−	0.901473i
$408$		0			0
$409$	5.00000	−	8.66025i	0.247234	−	0.428222i	−0.715523	−	0.698589i	$-0.753812\pi$
$409$	5.00000	−	8.66025i	0.247234	−	0.428222i	0.962757	+	0.270367i	$0.0871450\pi$
$410$	3.00000	−	5.19615i	0.148159	−	0.256620i
$411$		0			0
$412$	−7.00000	+	12.1244i	−0.344865	+	0.597324i
$413$	−9.00000	−	15.5885i	−0.442861	−	0.767058i
$414$		0			0
$415$	6.00000			0.294528
$416$	−3.50000	−	0.866025i	−0.171602	−	0.0424604i
$417$		0			0
$418$	−3.00000	−	5.19615i	−0.146735	−	0.254152i
$419$	−6.00000	−	10.3923i	−0.293119	−	0.507697i	0.681426	−	0.731887i	$-0.261360\pi$
$419$	−6.00000	−	10.3923i	−0.293119	−	0.507697i	−0.974546	+	0.224189i	$0.928027\pi$
$420$		0			0
$421$	−16.0000			−0.779792			−0.389896	−	0.920859i	$-0.627489\pi$
$421$	−16.0000			−0.779792			−0.389896	+	0.920859i	$0.627489\pi$
$422$	−10.0000	+	17.3205i	−0.486792	+	0.843149i
$423$		0			0
$424$	6.00000			0.291386
$425$	3.00000	−	5.19615i	0.145521	−	0.252050i
$426$		0			0
$427$	−2.00000	−	3.46410i	−0.0967868	−	0.167640i
$428$	6.00000			0.290021
$429$		0			0
$430$	−1.00000			−0.0482243
$431$	6.00000	+	10.3923i	0.289010	+	0.500580i	0.973574	−	0.228373i	$-0.0733406\pi$
$431$	6.00000	+	10.3923i	0.289010	+	0.500580i	−0.684564	+	0.728953i	$0.740007\pi$
$432$		0			0
$433$	20.0000	−	34.6410i	0.961139	−	1.66474i	0.241489	−	0.970404i	$-0.422364\pi$
$433$	20.0000	−	34.6410i	0.961139	−	1.66474i	0.719650	−	0.694337i	$-0.244302\pi$
$434$	10.0000			0.480015
$435$		0			0
$436$	−7.00000	+	12.1244i	−0.335239	+	0.580651i
$437$	−6.00000			−0.287019
$438$		0			0
$439$	2.00000	+	3.46410i	0.0954548	+	0.165333i	0.909798	−	0.415051i	$-0.136236\pi$
$439$	2.00000	+	3.46410i	0.0954548	+	0.165333i	−0.814344	+	0.580383i	$0.802903\pi$
$440$	−1.50000	−	2.59808i	−0.0715097	−	0.123858i
$441$		0			0
$442$	−6.00000	−	20.7846i	−0.285391	−	0.988623i
$443$	36.0000			1.71041			0.855206	−	0.518289i	$-0.173431\pi$
$443$	36.0000			1.71041			0.855206	+	0.518289i	$0.173431\pi$
$444$		0			0
$445$	−9.00000	−	15.5885i	−0.426641	−	0.738964i
$446$	5.00000	−	8.66025i	0.236757	−	0.410075i
$447$		0			0
$448$	−1.00000	+	1.73205i	−0.0472456	+	0.0818317i
$449$	18.0000	−	31.1769i	0.849473	−	1.47133i	−0.0322072	−	0.999481i	$-0.510254\pi$
$449$	18.0000	−	31.1769i	0.849473	−	1.47133i	0.881680	−	0.471848i	$-0.156413\pi$
$450$		0			0
$451$	9.00000	−	15.5885i	0.423793	−	0.734032i
$452$	7.50000	+	12.9904i	0.352770	+	0.611016i
$453$		0			0
$454$		0			0
$455$	−7.00000	−	1.73205i	−0.328165	−	0.0811998i
$456$		0			0
$457$	−1.00000	−	1.73205i	−0.0467780	−	0.0810219i	0.841688	−	0.539964i	$-0.181562\pi$
$457$	−1.00000	−	1.73205i	−0.0467780	−	0.0810219i	−0.888466	+	0.458942i	$0.848229\pi$
$458$	−7.00000	−	12.1244i	−0.327089	−	0.566534i
$459$		0			0
$460$	−3.00000			−0.139876
$461$	7.50000	−	12.9904i	0.349310	−	0.605022i	−0.636817	−	0.771015i	$-0.719749\pi$
$461$	7.50000	−	12.9904i	0.349310	−	0.605022i	0.986127	+	0.165992i	$0.0530827\pi$
$462$		0			0
$463$	−34.0000			−1.58011			−0.790057	−	0.613033i	$-0.789949\pi$
$463$	−34.0000			−1.58011			−0.790057	+	0.613033i	$0.789949\pi$
$464$	1.50000	−	2.59808i	0.0696358	−	0.120613i
$465$		0			0
$466$	10.5000	+	18.1865i	0.486403	+	0.842475i
$467$	−18.0000			−0.832941			−0.416470	−	0.909149i	$-0.636733\pi$
$467$	−18.0000			−0.832941			−0.416470	+	0.909149i	$0.636733\pi$
$468$		0			0
$469$	16.0000			0.738811
$470$	−1.50000	−	2.59808i	−0.0691898	−	0.119840i
$471$		0			0
$472$	−4.50000	+	7.79423i	−0.207129	+	0.358758i
$473$	−3.00000			−0.137940
$474$		0			0
$475$	−1.00000	+	1.73205i	−0.0458831	+	0.0794719i
$476$	−12.0000			−0.550019
$477$		0			0
$478$	12.0000	+	20.7846i	0.548867	+	0.950666i
$479$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$479$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$480$		0			0
$481$	−17.5000	+	18.1865i	−0.797931	+	0.829235i
$482$	17.0000			0.774329
$483$		0			0
$484$	1.00000	+	1.73205i	0.0454545	+	0.0787296i
$485$	−7.00000	+	12.1244i	−0.317854	+	0.550539i
$486$		0			0
$487$	−1.00000	+	1.73205i	−0.0453143	+	0.0784867i	−0.887793	−	0.460243i	$-0.847762\pi$
$487$	−1.00000	+	1.73205i	−0.0453143	+	0.0784867i	0.842479	+	0.538730i	$0.181096\pi$
$488$	−1.00000	+	1.73205i	−0.0452679	+	0.0784063i
$489$		0			0
$490$	1.50000	−	2.59808i	0.0677631	−	0.117369i
$491$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$491$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$492$		0			0
$493$	18.0000			0.810679
$494$	2.00000	+	6.92820i	0.0899843	+	0.311715i
$495$		0			0
$496$	−2.50000	−	4.33013i	−0.112253	−	0.194428i
$497$	−12.0000	−	20.7846i	−0.538274	−	0.932317i
$498$		0			0
$499$	32.0000			1.43252			0.716258	−	0.697835i	$-0.245853\pi$
$499$	32.0000			1.43252			0.716258	+	0.697835i	$0.245853\pi$
$500$	−0.500000	+	0.866025i	−0.0223607	+	0.0387298i
$501$		0			0
$502$	−15.0000			−0.669483
$503$	12.0000	−	20.7846i	0.535054	−	0.926740i	−0.464107	−	0.885779i	$-0.653625\pi$
$503$	12.0000	−	20.7846i	0.535054	−	0.926740i	0.999161	−	0.0409609i	$-0.0130419\pi$
$504$		0			0
$505$	3.00000	+	5.19615i	0.133498	+	0.231226i
$506$	−9.00000			−0.400099
$507$		0			0
$508$	14.0000			0.621150
$509$	1.50000	+	2.59808i	0.0664863	+	0.115158i	0.897352	−	0.441315i	$-0.145488\pi$
$509$	1.50000	+	2.59808i	0.0664863	+	0.115158i	−0.830866	+	0.556473i	$0.812154\pi$
$510$		0			0
$511$	−14.0000	+	24.2487i	−0.619324	+	1.07270i
$512$	1.00000			0.0441942
$513$		0			0
$514$	−10.5000	+	18.1865i	−0.463135	+	0.802174i
$515$	14.0000			0.616914
$516$		0			0
$517$	−4.50000	−	7.79423i	−0.197910	−	0.342790i
$518$	7.00000	+	12.1244i	0.307562	+	0.532714i
$519$		0			0
$520$	1.00000	+	3.46410i	0.0438529	+	0.151911i
$521$	30.0000			1.31432			0.657162	−	0.753749i	$-0.271757\pi$
$521$	30.0000			1.31432			0.657162	+	0.753749i	$0.271757\pi$
$522$		0			0
$523$	−5.50000	−	9.52628i	−0.240498	−	0.416555i	0.720358	−	0.693602i	$-0.243977\pi$
$523$	−5.50000	−	9.52628i	−0.240498	−	0.416555i	−0.960856	+	0.277047i	$0.910644\pi$
$524$	4.50000	−	7.79423i	0.196583	−	0.340492i
$525$		0			0
$526$	−7.50000	+	12.9904i	−0.327016	+	0.566408i
$527$	15.0000	−	25.9808i	0.653410	−	1.13174i
$528$		0			0
$529$	7.00000	−	12.1244i	0.304348	−	0.527146i
$530$	−3.00000	−	5.19615i	−0.130312	−	0.225706i
$531$		0			0
$532$	4.00000			0.173422
$533$	−15.0000	+	15.5885i	−0.649722	+	0.675211i
$534$		0			0
$535$	−3.00000	−	5.19615i	−0.129701	−	0.224649i
$536$	−4.00000	−	6.92820i	−0.172774	−	0.299253i
$537$		0			0
$538$	−18.0000			−0.776035
$539$	4.50000	−	7.79423i	0.193829	−	0.335721i
$540$		0			0
$541$	20.0000			0.859867			0.429934	−	0.902861i	$-0.358537\pi$
$541$	20.0000			0.859867			0.429934	+	0.902861i	$0.358537\pi$
$542$	−5.50000	+	9.52628i	−0.236245	+	0.409189i
$543$		0			0
$544$	3.00000	+	5.19615i	0.128624	+	0.222783i
$545$	14.0000			0.599694
$546$		0			0
$547$	20.0000			0.855138			0.427569	−	0.903983i	$-0.359370\pi$
$547$	20.0000			0.855138			0.427569	+	0.903983i	$0.359370\pi$
$548$	4.50000	+	7.79423i	0.192230	+	0.332953i
$549$		0			0
$550$	−1.50000	+	2.59808i	−0.0639602	+	0.110782i
$551$	−6.00000			−0.255609
$552$		0			0
$553$	−5.00000	+	8.66025i	−0.212622	+	0.368271i
$554$	−1.00000			−0.0424859
$555$		0			0
$556$	−7.00000	−	12.1244i	−0.296866	−	0.514187i
$557$	−3.00000	−	5.19615i	−0.127114	−	0.220168i	0.795443	−	0.606028i	$-0.207238\pi$
$557$	−3.00000	−	5.19615i	−0.127114	−	0.220168i	−0.922557	+	0.385860i	$0.873905\pi$
$558$		0			0
$559$	3.50000	+	0.866025i	0.148034	+	0.0366290i
$560$	2.00000			0.0845154
$561$		0			0
$562$	9.00000	+	15.5885i	0.379642	+	0.657559i
$563$	18.0000	−	31.1769i	0.758610	−	1.31395i	−0.184950	−	0.982748i	$-0.559212\pi$
$563$	18.0000	−	31.1769i	0.758610	−	1.31395i	0.943560	−	0.331202i	$-0.107454\pi$
$564$		0			0
$565$	7.50000	−	12.9904i	0.315527	−	0.546509i
$566$	15.5000	−	26.8468i	0.651514	−	1.12845i
$567$		0			0
$568$	−6.00000	+	10.3923i	−0.251754	+	0.436051i
$569$	3.00000	+	5.19615i	0.125767	+	0.217834i	0.922032	−	0.387113i	$-0.126528\pi$
$569$	3.00000	+	5.19615i	0.125767	+	0.217834i	−0.796266	+	0.604947i	$0.793194\pi$
$570$		0			0
$571$	−40.0000			−1.67395			−0.836974	−	0.547243i	$-0.815677\pi$
$571$	−40.0000			−1.67395			−0.836974	+	0.547243i	$0.815677\pi$
$572$	3.00000	+	10.3923i	0.125436	+	0.434524i
$573$		0			0
$574$	6.00000	+	10.3923i	0.250435	+	0.433766i
$575$	1.50000	+	2.59808i	0.0625543	+	0.108347i
$576$		0			0
$577$	2.00000			0.0832611			0.0416305	−	0.999133i	$-0.486745\pi$
$577$	2.00000			0.0832611			0.0416305	+	0.999133i	$0.486745\pi$
$578$	−9.50000	+	16.4545i	−0.395148	+	0.684416i
$579$		0			0
$580$	−3.00000			−0.124568
$581$	−6.00000	+	10.3923i	−0.248922	+	0.431145i
$582$		0			0
$583$	−9.00000	−	15.5885i	−0.372742	−	0.645608i
$584$	14.0000			0.579324
$585$		0			0
$586$	30.0000			1.23929
$587$	9.00000	+	15.5885i	0.371470	+	0.643404i	0.989792	−	0.142520i	$-0.0455206\pi$
$587$	9.00000	+	15.5885i	0.371470	+	0.643404i	−0.618322	+	0.785925i	$0.712187\pi$
$588$		0			0
$589$	−5.00000	+	8.66025i	−0.206021	+	0.356840i
$590$	9.00000			0.370524
$591$		0			0
$592$	3.50000	−	6.06218i	0.143849	−	0.249154i
$593$	−27.0000			−1.10876			−0.554379	−	0.832265i	$-0.687044\pi$
$593$	−27.0000			−1.10876			−0.554379	+	0.832265i	$0.687044\pi$
$594$		0			0
$595$	6.00000	+	10.3923i	0.245976	+	0.426043i
$596$	−4.50000	−	7.79423i	−0.184327	−	0.319264i
$597$		0			0
$598$	10.5000	+	2.59808i	0.429377	+	0.106243i
$599$	−6.00000			−0.245153			−0.122577	−	0.992459i	$-0.539116\pi$
$599$	−6.00000			−0.245153			−0.122577	+	0.992459i	$0.539116\pi$
$600$		0			0
$601$	9.50000	+	16.4545i	0.387513	+	0.671192i	0.992114	−	0.125336i	$-0.0400009\pi$
$601$	9.50000	+	16.4545i	0.387513	+	0.671192i	−0.604601	+	0.796528i	$0.706668\pi$
$602$	1.00000	−	1.73205i	0.0407570	−	0.0705931i
$603$		0			0
$604$	−4.00000	+	6.92820i	−0.162758	+	0.281905i
$605$	1.00000	−	1.73205i	0.0406558	−	0.0704179i
$606$		0			0
$607$	11.0000	−	19.0526i	0.446476	−	0.773320i	−0.551678	−	0.834058i	$-0.686012\pi$
$607$	11.0000	−	19.0526i	0.446476	−	0.773320i	0.998154	+	0.0607380i	$0.0193454\pi$
$608$	−1.00000	−	1.73205i	−0.0405554	−	0.0702439i
$609$		0			0
$610$	2.00000			0.0809776
$611$	3.00000	+	10.3923i	0.121367	+	0.420428i
$612$		0			0
$613$	15.5000	+	26.8468i	0.626039	+	1.08433i	0.988339	+	0.152270i	$0.0486583\pi$
$613$	15.5000	+	26.8468i	0.626039	+	1.08433i	−0.362300	+	0.932062i	$0.618008\pi$
$614$	−4.00000	−	6.92820i	−0.161427	−	0.279600i
$615$		0			0
$616$	6.00000			0.241747
$617$	10.5000	−	18.1865i	0.422714	−	0.732162i	−0.573490	−	0.819213i	$-0.694411\pi$
$617$	10.5000	−	18.1865i	0.422714	−	0.732162i	0.996204	+	0.0870504i	$0.0277441\pi$
$618$		0			0
$619$	−46.0000			−1.84890			−0.924448	−	0.381308i	$-0.875474\pi$
$619$	−46.0000			−1.84890			−0.924448	+	0.381308i	$0.875474\pi$
$620$	−2.50000	+	4.33013i	−0.100402	+	0.173902i
$621$		0			0
$622$	−6.00000	−	10.3923i	−0.240578	−	0.416693i
$623$	36.0000			1.44231
$624$		0			0
$625$	1.00000			0.0400000
$626$	−4.00000	−	6.92820i	−0.159872	−	0.276907i
$627$		0			0
$628$	6.50000	−	11.2583i	0.259378	−	0.449256i
$629$	42.0000			1.67465
$630$		0			0
$631$	8.00000	−	13.8564i	0.318475	−	0.551615i	−0.661695	−	0.749773i	$-0.730163\pi$
$631$	8.00000	−	13.8564i	0.318475	−	0.551615i	0.980170	+	0.198158i	$0.0634960\pi$
$632$	5.00000			0.198889
$633$		0			0
$634$	6.00000	+	10.3923i	0.238290	+	0.412731i
$635$	−7.00000	−	12.1244i	−0.277787	−	0.481140i
$636$		0			0
$637$	−7.50000	+	7.79423i	−0.297161	+	0.308819i
$638$	−9.00000			−0.356313
$639$		0			0
$640$	−0.500000	−	0.866025i	−0.0197642	−	0.0342327i
$641$	3.00000	−	5.19615i	0.118493	−	0.205236i	−0.800678	−	0.599095i	$-0.795527\pi$
$641$	3.00000	−	5.19615i	0.118493	−	0.205236i	0.919171	+	0.393860i	$0.128860\pi$
$642$		0			0
$643$	8.00000	−	13.8564i	0.315489	−	0.546443i	−0.664052	−	0.747686i	$-0.731165\pi$
$643$	8.00000	−	13.8564i	0.315489	−	0.546443i	0.979541	+	0.201243i	$0.0644981\pi$
$644$	3.00000	−	5.19615i	0.118217	−	0.204757i
$645$		0			0
$646$	6.00000	−	10.3923i	0.236067	−	0.408880i
$647$	12.0000	+	20.7846i	0.471769	+	0.817127i	0.999478	−	0.0322975i	$-0.0102824\pi$
$647$	12.0000	+	20.7846i	0.471769	+	0.817127i	−0.527710	+	0.849425i	$0.676949\pi$
$648$		0			0
$649$	27.0000			1.05984
$650$	2.50000	−	2.59808i	0.0980581	−	0.101905i
$651$		0			0
$652$	6.50000	+	11.2583i	0.254560	+	0.440910i
$653$	3.00000	+	5.19615i	0.117399	+	0.203341i	0.918736	−	0.394872i	$-0.129211\pi$
$653$	3.00000	+	5.19615i	0.117399	+	0.203341i	−0.801337	+	0.598213i	$0.795878\pi$
$654$		0			0
$655$	−9.00000			−0.351659
$656$	3.00000	−	5.19615i	0.117130	−	0.202876i
$657$		0			0
$658$	6.00000			0.233904
$659$	7.50000	−	12.9904i	0.292159	−	0.506033i	−0.682161	−	0.731202i	$-0.738960\pi$
$659$	7.50000	−	12.9904i	0.292159	−	0.506033i	0.974320	+	0.225168i	$0.0722932\pi$
$660$		0			0
$661$	−16.0000	−	27.7128i	−0.622328	−	1.07790i	−0.989051	−	0.147573i	$-0.952854\pi$
$661$	−16.0000	−	27.7128i	−0.622328	−	1.07790i	0.366723	−	0.930330i	$-0.380480\pi$
$662$	32.0000			1.24372
$663$		0			0
$664$	6.00000			0.232845
$665$	−2.00000	−	3.46410i	−0.0775567	−	0.134332i
$666$		0			0
$667$	−4.50000	+	7.79423i	−0.174241	+	0.301794i
$668$	9.00000			0.348220
$669$		0			0
$670$	−4.00000	+	6.92820i	−0.154533	+	0.267660i
$671$	6.00000			0.231627
$672$		0			0
$673$	2.00000	+	3.46410i	0.0770943	+	0.133531i	0.901995	−	0.431746i	$-0.142102\pi$
$673$	2.00000	+	3.46410i	0.0770943	+	0.133531i	−0.824901	+	0.565278i	$0.808769\pi$
$674$	−7.00000	−	12.1244i	−0.269630	−	0.467013i
$675$		0			0
$676$	−0.500000	−	12.9904i	−0.0192308	−	0.499630i
$677$	36.0000			1.38359			0.691796	−	0.722093i	$-0.256820\pi$
$677$	36.0000			1.38359			0.691796	+	0.722093i	$0.256820\pi$
$678$		0			0
$679$	−14.0000	−	24.2487i	−0.537271	−	0.930580i
$680$	3.00000	−	5.19615i	0.115045	−	0.199263i
$681$		0			0
$682$	−7.50000	+	12.9904i	−0.287190	+	0.497427i
$683$	−6.00000	+	10.3923i	−0.229584	+	0.397650i	−0.957685	−	0.287819i	$-0.907070\pi$
$683$	−6.00000	+	10.3923i	−0.229584	+	0.397650i	0.728101	+	0.685470i	$0.240403\pi$
$684$		0			0
$685$	4.50000	−	7.79423i	0.171936	−	0.297802i
$686$	10.0000	+	17.3205i	0.381802	+	0.661300i
$687$		0			0
$688$	−1.00000			−0.0381246
$689$	6.00000	+	20.7846i	0.228582	+	0.791831i
$690$		0			0
$691$	23.0000	+	39.8372i	0.874961	+	1.51548i	0.856804	+	0.515642i	$0.172447\pi$
$691$	23.0000	+	39.8372i	0.874961	+	1.51548i	0.0181572	+	0.999835i	$0.494220\pi$
$692$	6.00000	+	10.3923i	0.228086	+	0.395056i
$693$		0			0
$694$	30.0000			1.13878
$695$	−7.00000	+	12.1244i	−0.265525	+	0.459903i
$696$		0			0
$697$	36.0000			1.36360
$698$	−4.00000	+	6.92820i	−0.151402	+	0.262236i
$699$		0			0
$700$	−1.00000	−	1.73205i	−0.0377964	−	0.0654654i
$701$	21.0000			0.793159			0.396580	−	0.918000i	$-0.370197\pi$
$701$	21.0000			0.793159			0.396580	+	0.918000i	$0.370197\pi$
$702$		0			0
$703$	−14.0000			−0.528020
$704$	−1.50000	−	2.59808i	−0.0565334	−	0.0979187i
$705$		0			0
$706$	15.0000	−	25.9808i	0.564532	−	0.977799i
$707$	−12.0000			−0.451306
$708$		0			0
$709$	−16.0000	+	27.7128i	−0.600893	+	1.04078i	0.391794	+	0.920053i	$0.371855\pi$
$709$	−16.0000	+	27.7128i	−0.600893	+	1.04078i	−0.992686	+	0.120723i	$0.961479\pi$
$710$	12.0000			0.450352
$711$		0			0
$712$	−9.00000	−	15.5885i	−0.337289	−	0.584202i
$713$	7.50000	+	12.9904i	0.280877	+	0.486494i
$714$		0			0
$715$	7.50000	−	7.79423i	0.280484	−	0.291488i
$716$	3.00000			0.112115
$717$		0			0
$718$	12.0000	+	20.7846i	0.447836	+	0.775675i
$719$	18.0000	−	31.1769i	0.671287	−	1.16270i	−0.306253	−	0.951950i	$-0.599075\pi$
$719$	18.0000	−	31.1769i	0.671287	−	1.16270i	0.977539	−	0.210752i	$-0.0675914\pi$
$720$		0			0
$721$	−14.0000	+	24.2487i	−0.521387	+	0.903069i
$722$	7.50000	−	12.9904i	0.279121	−	0.483452i
$723$		0			0
$724$	8.00000	−	13.8564i	0.297318	−	0.514969i
$725$	1.50000	+	2.59808i	0.0557086	+	0.0964901i
$726$		0			0
$727$	−4.00000			−0.148352			−0.0741759	−	0.997245i	$-0.523633\pi$
$727$	−4.00000			−0.148352			−0.0741759	+	0.997245i	$0.523633\pi$
$728$	−7.00000	−	1.73205i	−0.259437	−	0.0641941i
$729$		0			0
$730$	−7.00000	−	12.1244i	−0.259082	−	0.448743i
$731$	−3.00000	−	5.19615i	−0.110959	−	0.192187i
$732$		0			0
$733$	−22.0000			−0.812589			−0.406294	−	0.913742i	$-0.633179\pi$
$733$	−22.0000			−0.812589			−0.406294	+	0.913742i	$0.633179\pi$
$734$	−10.0000	+	17.3205i	−0.369107	+	0.639312i
$735$		0			0
$736$	−3.00000			−0.110581
$737$	−12.0000	+	20.7846i	−0.442026	+	0.765611i
$738$		0			0
$739$	8.00000	+	13.8564i	0.294285	+	0.509716i	0.974818	−	0.223001i	$-0.0715853\pi$
$739$	8.00000	+	13.8564i	0.294285	+	0.509716i	−0.680534	+	0.732717i	$0.738252\pi$
$740$	−7.00000			−0.257325
$741$		0			0
$742$	12.0000			0.440534
$743$	−4.50000	−	7.79423i	−0.165089	−	0.285943i	0.771598	−	0.636111i	$-0.219458\pi$
$743$	−4.50000	−	7.79423i	−0.165089	−	0.285943i	−0.936687	+	0.350168i	$0.886124\pi$
$744$		0			0
$745$	−4.50000	+	7.79423i	−0.164867	+	0.285558i
$746$	−25.0000			−0.915315
$747$		0			0
$748$	9.00000	−	15.5885i	0.329073	−	0.569970i
$749$	12.0000			0.438470
$750$		0			0
$751$	−20.5000	−	35.5070i	−0.748056	−	1.29567i	−0.948753	−	0.316017i	$-0.897654\pi$
$751$	−20.5000	−	35.5070i	−0.748056	−	1.29567i	0.200698	−	0.979653i	$-0.435679\pi$
$752$	−1.50000	−	2.59808i	−0.0546994	−	0.0947421i
$753$		0			0
$754$	10.5000	+	2.59808i	0.382387	+	0.0946164i
$755$	8.00000			0.291150
$756$		0			0
$757$	−19.0000	−	32.9090i	−0.690567	−	1.19610i	−0.971652	−	0.236414i	$-0.924028\pi$
$757$	−19.0000	−	32.9090i	−0.690567	−	1.19610i	0.281086	−	0.959683i	$-0.409305\pi$
$758$	−19.0000	+	32.9090i	−0.690111	+	1.19531i
$759$		0			0
$760$	−1.00000	+	1.73205i	−0.0362738	+	0.0628281i
$761$	−6.00000	+	10.3923i	−0.217500	+	0.376721i	−0.954043	−	0.299670i	$-0.903123\pi$
$761$	−6.00000	+	10.3923i	−0.217500	+	0.376721i	0.736543	+	0.676391i	$0.236457\pi$
$762$		0			0
$763$	−14.0000	+	24.2487i	−0.506834	+	0.877862i
$764$	−6.00000	−	10.3923i	−0.217072	−	0.375980i
$765$		0			0
$766$	21.0000			0.758761
$767$	−31.5000	−	7.79423i	−1.13740	−	0.281433i
$768$		0			0
$769$	6.50000	+	11.2583i	0.234396	+	0.405986i	0.959097	−	0.283078i	$-0.0913554\pi$
$769$	6.50000	+	11.2583i	0.234396	+	0.405986i	−0.724701	+	0.689063i	$0.758022\pi$
$770$	−3.00000	−	5.19615i	−0.108112	−	0.187256i
$771$		0			0
$772$	−4.00000			−0.143963
$773$	−24.0000	+	41.5692i	−0.863220	+	1.49514i	0.00558380	+	0.999984i	$0.498223\pi$
$773$	−24.0000	+	41.5692i	−0.863220	+	1.49514i	−0.868804	+	0.495156i	$0.835111\pi$
$774$		0			0
$775$	5.00000			0.179605
$776$	−7.00000	+	12.1244i	−0.251285	+	0.435239i
$777$		0			0
$778$	1.50000	+	2.59808i	0.0537776	+	0.0931455i
$779$	−12.0000			−0.429945
$780$		0			0
$781$	36.0000			1.28818
$782$	−9.00000	−	15.5885i	−0.321839	−	0.557442i
$783$		0			0
$784$	1.50000	−	2.59808i	0.0535714	−	0.0927884i
$785$	−13.0000			−0.463990
$786$		0			0
$787$	−17.5000	+	30.3109i	−0.623808	+	1.08047i	0.364963	+	0.931022i	$0.381082\pi$
$787$	−17.5000	+	30.3109i	−0.623808	+	1.08047i	−0.988770	+	0.149444i	$0.952252\pi$
$788$	−24.0000			−0.854965
$789$		0			0
$790$	−2.50000	−	4.33013i	−0.0889460	−	0.154059i
$791$	15.0000	+	25.9808i	0.533339	+	0.923770i
$792$		0			0
$793$	−7.00000	−	1.73205i	−0.248577	−	0.0615069i
$794$	−31.0000			−1.10015
$795$		0			0
$796$	−4.00000	−	6.92820i	−0.141776	−	0.245564i
$797$	−12.0000	+	20.7846i	−0.425062	+	0.736229i	−0.996426	−	0.0844678i	$-0.973081\pi$
$797$	−12.0000	+	20.7846i	−0.425062	+	0.736229i	0.571364	+	0.820696i	$0.306414\pi$
$798$		0			0
$799$	9.00000	−	15.5885i	0.318397	−	0.551480i
$800$	−0.500000	+	0.866025i	−0.0176777	+	0.0306186i
$801$		0			0
$802$	−6.00000	+	10.3923i	−0.211867	+	0.366965i
$803$	−21.0000	−	36.3731i	−0.741074	−	1.28358i
$804$		0			0
$805$	−6.00000			−0.211472
$806$	12.5000	−	12.9904i	0.440294	−	0.457567i
$807$		0			0
$808$	3.00000	+	5.19615i	0.105540	+	0.182800i
$809$	15.0000	+	25.9808i	0.527372	+	0.913435i	0.999491	+	0.0319002i	$0.0101559\pi$
$809$	15.0000	+	25.9808i	0.527372	+	0.913435i	−0.472119	+	0.881535i	$0.656511\pi$
$810$		0			0
$811$	32.0000			1.12367			0.561836	−	0.827249i	$-0.310095\pi$
$811$	32.0000			1.12367			0.561836	+	0.827249i	$0.310095\pi$
$812$	3.00000	−	5.19615i	0.105279	−	0.182349i
$813$		0			0
$814$	−21.0000			−0.736050
$815$	6.50000	−	11.2583i	0.227685	−	0.394362i
$816$		0			0
$817$	1.00000	+	1.73205i	0.0349856	+	0.0605968i
$818$	−10.0000			−0.349642
$819$		0			0
$820$	−6.00000			−0.209529
$821$	13.5000	+	23.3827i	0.471153	+	0.816061i	0.999456	−	0.0329950i	$-0.0105045\pi$
$821$	13.5000	+	23.3827i	0.471153	+	0.816061i	−0.528302	+	0.849056i	$0.677171\pi$
$822$		0			0
$823$	−7.00000	+	12.1244i	−0.244005	+	0.422628i	−0.961851	−	0.273573i	$-0.911795\pi$
$823$	−7.00000	+	12.1244i	−0.244005	+	0.422628i	0.717847	+	0.696201i	$0.245128\pi$
$824$	14.0000			0.487713
$825$		0			0
$826$	−9.00000	+	15.5885i	−0.313150	+	0.542392i
$827$	−6.00000			−0.208640			−0.104320	−	0.994544i	$-0.533267\pi$
$827$	−6.00000			−0.208640			−0.104320	+	0.994544i	$0.533267\pi$
$828$		0			0
$829$	−13.0000	−	22.5167i	−0.451509	−	0.782036i	0.546971	−	0.837151i	$-0.315781\pi$
$829$	−13.0000	−	22.5167i	−0.451509	−	0.782036i	−0.998480	+	0.0551154i	$0.982447\pi$
$830$	−3.00000	−	5.19615i	−0.104132	−	0.180361i
$831$		0			0
$832$	1.00000	+	3.46410i	0.0346688	+	0.120096i
$833$	18.0000			0.623663
$834$		0			0
$835$	−4.50000	−	7.79423i	−0.155729	−	0.269730i
$836$	−3.00000	+	5.19615i	−0.103757	+	0.179713i
$837$		0			0
$838$	−6.00000	+	10.3923i	−0.207267	+	0.358996i
$839$	21.0000	−	36.3731i	0.725001	−	1.25574i	−0.233973	−	0.972243i	$-0.575173\pi$
$839$	21.0000	−	36.3731i	0.725001	−	1.25574i	0.958974	−	0.283495i	$-0.0914938\pi$
$840$		0			0
$841$	10.0000	−	17.3205i	0.344828	−	0.597259i
$842$	8.00000	+	13.8564i	0.275698	+	0.477523i
$843$		0			0
$844$	20.0000			0.688428
$845$	−11.0000	+	6.92820i	−0.378412	+	0.238337i
$846$		0			0
$847$	2.00000	+	3.46410i	0.0687208	+	0.119028i
$848$	−3.00000	−	5.19615i	−0.103020	−	0.178437i
$849$		0			0
$850$	−6.00000			−0.205798
$851$	−10.5000	+	18.1865i	−0.359935	+	0.623426i
$852$		0			0
$853$	−19.0000			−0.650548			−0.325274	−	0.945620i	$-0.605456\pi$
$853$	−19.0000			−0.650548			−0.325274	+	0.945620i	$0.605456\pi$
$854$	−2.00000	+	3.46410i	−0.0684386	+	0.118539i
$855$		0			0
$856$	−3.00000	−	5.19615i	−0.102538	−	0.177601i
$857$	−3.00000			−0.102478			−0.0512390	−	0.998686i	$-0.516317\pi$
$857$	−3.00000			−0.102478			−0.0512390	+	0.998686i	$0.516317\pi$
$858$		0			0
$859$	−46.0000			−1.56950			−0.784750	−	0.619813i	$-0.787209\pi$
$859$	−46.0000			−1.56950			−0.784750	+	0.619813i	$0.787209\pi$
$860$	0.500000	+	0.866025i	0.0170499	+	0.0295312i
$861$		0			0
$862$	6.00000	−	10.3923i	0.204361	−	0.353963i
$863$	45.0000			1.53182			0.765909	−	0.642949i	$-0.222289\pi$
$863$	45.0000			1.53182			0.765909	+	0.642949i	$0.222289\pi$
$864$		0			0
$865$	6.00000	−	10.3923i	0.204006	−	0.353349i
$866$	−40.0000			−1.35926
$867$		0			0
$868$	−5.00000	−	8.66025i	−0.169711	−	0.293948i
$869$	−7.50000	−	12.9904i	−0.254420	−	0.440668i
$870$		0			0
$871$	20.0000	−	20.7846i	0.677674	−	0.704260i
$872$	14.0000			0.474100
$873$		0			0
$874$	3.00000	+	5.19615i	0.101477	+	0.175762i
$875$	−1.00000	+	1.73205i	−0.0338062	+	0.0585540i
$876$		0			0
$877$	−11.5000	+	19.9186i	−0.388327	+	0.672603i	−0.992225	−	0.124459i	$-0.960280\pi$
$877$	−11.5000	+	19.9186i	−0.388327	+	0.672603i	0.603897	+	0.797062i	$0.293614\pi$
$878$	2.00000	−	3.46410i	0.0674967	−	0.116908i
$879$		0			0
$880$	−1.50000	+	2.59808i	−0.0505650	+	0.0875811i
$881$	3.00000	+	5.19615i	0.101073	+	0.175063i	0.912127	−	0.409908i	$-0.134439\pi$
$881$	3.00000	+	5.19615i	0.101073	+	0.175063i	−0.811054	+	0.584971i	$0.801106\pi$
$882$		0			0
$883$	−19.0000			−0.639401			−0.319700	−	0.947519i	$-0.603582\pi$
$883$	−19.0000			−0.639401			−0.319700	+	0.947519i	$0.603582\pi$
$884$	−15.0000	+	15.5885i	−0.504505	+	0.524297i
$885$		0			0
$886$	−18.0000	−	31.1769i	−0.604722	−	1.04741i
$887$	−28.5000	−	49.3634i	−0.956936	−	1.65746i	−0.729873	−	0.683582i	$-0.760421\pi$
$887$	−28.5000	−	49.3634i	−0.956936	−	1.65746i	−0.227063	−	0.973880i	$-0.572912\pi$
$888$		0			0
$889$	28.0000			0.939090
$890$	−9.00000	+	15.5885i	−0.301681	+	0.522526i
$891$		0			0
$892$	−10.0000			−0.334825
$893$	−3.00000	+	5.19615i	−0.100391	+	0.173883i
$894$		0			0
$895$	−1.50000	−	2.59808i	−0.0501395	−	0.0868441i
$896$	2.00000			0.0668153
$897$		0			0
$898$	−36.0000			−1.20134
$899$	7.50000	+	12.9904i	0.250139	+	0.433253i
$900$		0			0
$901$	18.0000	−	31.1769i	0.599667	−	1.03865i
$902$	−18.0000			−0.599334
$903$		0			0
$904$	7.50000	−	12.9904i	0.249446	−	0.432054i
$905$	−16.0000			−0.531858
$906$		0			0
$907$	−8.50000	−	14.7224i	−0.282238	−	0.488850i	0.689698	−	0.724097i	$-0.257743\pi$
$907$	−8.50000	−	14.7224i	−0.282238	−	0.488850i	−0.971936	+	0.235247i	$0.924410\pi$
$908$		0			0
$909$		0			0
$910$	2.00000	+	6.92820i	0.0662994	+	0.229668i
$911$	6.00000			0.198789			0.0993944	−	0.995048i	$-0.468309\pi$
$911$	6.00000			0.198789			0.0993944	+	0.995048i	$0.468309\pi$
$912$		0			0
$913$	−9.00000	−	15.5885i	−0.297857	−	0.515903i
$914$	−1.00000	+	1.73205i	−0.0330771	+	0.0572911i
$915$		0			0
$916$	−7.00000	+	12.1244i	−0.231287	+	0.400600i
$917$	9.00000	−	15.5885i	0.297206	−	0.514776i
$918$		0			0
$919$	8.00000	−	13.8564i	0.263896	−	0.457081i	−0.703378	−	0.710816i	$-0.748326\pi$
$919$	8.00000	−	13.8564i	0.263896	−	0.457081i	0.967274	+	0.253735i	$0.0816592\pi$
$920$	1.50000	+	2.59808i	0.0494535	+	0.0856560i
$921$		0			0
$922$	−15.0000			−0.493999
$923$	−42.0000	−	10.3923i	−1.38245	−	0.342067i
$924$		0			0
$925$	3.50000	+	6.06218i	0.115079	+	0.199323i
$926$	17.0000	+	29.4449i	0.558655	+	0.967618i
$927$		0			0
$928$	−3.00000			−0.0984798
$929$	−6.00000	+	10.3923i	−0.196854	+	0.340960i	−0.947507	−	0.319736i	$-0.896406\pi$
$929$	−6.00000	+	10.3923i	−0.196854	+	0.340960i	0.750653	+	0.660697i	$0.229739\pi$
$930$		0			0
$931$	−6.00000			−0.196642
$932$	10.5000	−	18.1865i	0.343939	−	0.595720i
$933$		0			0
$934$	9.00000	+	15.5885i	0.294489	+	0.510070i
$935$	−18.0000			−0.588663
$936$		0			0
$937$	2.00000			0.0653372			0.0326686	−	0.999466i	$-0.489599\pi$
$937$	2.00000			0.0653372			0.0326686	+	0.999466i	$0.489599\pi$
$938$	−8.00000	−	13.8564i	−0.261209	−	0.452428i
$939$		0			0
$940$	−1.50000	+	2.59808i	−0.0489246	+	0.0847399i
$941$	−42.0000			−1.36916			−0.684580	−	0.728937i	$-0.740015\pi$
$941$	−42.0000			−1.36916			−0.684580	+	0.728937i	$0.740015\pi$
$942$		0			0
$943$	−9.00000	+	15.5885i	−0.293080	+	0.507630i
$944$	9.00000			0.292925
$945$		0			0
$946$	1.50000	+	2.59808i	0.0487692	+	0.0844707i
$947$	−6.00000	−	10.3923i	−0.194974	−	0.337705i	0.751918	−	0.659256i	$-0.229129\pi$
$947$	−6.00000	−	10.3923i	−0.194974	−	0.337705i	−0.946892	+	0.321552i	$0.895796\pi$
$948$		0			0
$949$	14.0000	+	48.4974i	0.454459	+	1.57429i
$950$	2.00000			0.0648886
$951$		0			0
$952$	6.00000	+	10.3923i	0.194461	+	0.336817i
$953$	4.50000	−	7.79423i	0.145769	−	0.252480i	−0.783890	−	0.620899i	$-0.786768\pi$
$953$	4.50000	−	7.79423i	0.145769	−	0.252480i	0.929660	+	0.368419i	$0.120101\pi$
$954$		0			0
$955$	−6.00000	+	10.3923i	−0.194155	+	0.336287i
$956$	12.0000	−	20.7846i	0.388108	−	0.672222i
$957$		0			0
$958$		0			0
$959$	9.00000	+	15.5885i	0.290625	+	0.503378i
$960$		0			0
$961$	−6.00000			−0.193548
$962$	24.5000	+	6.06218i	0.789912	+	0.195452i
$963$		0			0
$964$	−8.50000	−	14.7224i	−0.273767	−	0.474178i
$965$	2.00000	+	3.46410i	0.0643823	+	0.111513i
$966$		0			0
$967$	32.0000			1.02905			0.514525	−	0.857475i	$-0.327968\pi$
$967$	32.0000			1.02905			0.514525	+	0.857475i	$0.327968\pi$
$968$	1.00000	−	1.73205i	0.0321412	−	0.0556702i
$969$		0			0
$970$	14.0000			0.449513
$971$	−24.0000	+	41.5692i	−0.770197	+	1.33402i	0.167258	+	0.985913i	$0.446509\pi$
$971$	−24.0000	+	41.5692i	−0.770197	+	1.33402i	−0.937455	+	0.348107i	$0.886825\pi$
$972$		0			0
$973$	−14.0000	−	24.2487i	−0.448819	−	0.777378i
$974$	2.00000			0.0640841
$975$		0			0
$976$	2.00000			0.0640184
$977$	−28.5000	−	49.3634i	−0.911796	−	1.57928i	−0.811526	−	0.584316i	$-0.801363\pi$
$977$	−28.5000	−	49.3634i	−0.911796	−	1.57928i	−0.100270	−	0.994960i	$-0.531971\pi$
$978$		0			0
$979$	−27.0000	+	46.7654i	−0.862924	+	1.49463i
$980$	−3.00000			−0.0958315
$981$		0			0
$982$		0			0
$983$	−9.00000			−0.287055			−0.143528	−	0.989646i	$-0.545845\pi$
$983$	−9.00000			−0.287055			−0.143528	+	0.989646i	$0.545845\pi$
$984$		0			0
$985$	12.0000	+	20.7846i	0.382352	+	0.662253i
$986$	−9.00000	−	15.5885i	−0.286618	−	0.496438i
$987$		0			0
$988$	5.00000	−	5.19615i	0.159071	−	0.165312i
$989$	3.00000			0.0953945
$990$		0			0
$991$	12.5000	+	21.6506i	0.397076	+	0.687755i	0.993364	−	0.115015i	$-0.0366917\pi$
$991$	12.5000	+	21.6506i	0.397076	+	0.687755i	−0.596288	+	0.802771i	$0.703358\pi$
$992$	−2.50000	+	4.33013i	−0.0793751	+	0.137482i
$993$		0			0
$994$	−12.0000	+	20.7846i	−0.380617	+	0.659248i
$995$	−4.00000	+	6.92820i	−0.126809	+	0.219639i
$996$		0			0
$997$	11.0000	−	19.0526i	0.348373	−	0.603401i	−0.637587	−	0.770378i	$-0.720067\pi$
$997$	11.0000	−	19.0526i	0.348373	−	0.603401i	0.985961	+	0.166978i	$0.0534008\pi$
$998$	−16.0000	−	27.7128i	−0.506471	−	0.877234i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1170.2.i.d.991.1		2
3.2	odd	2		390.2.i.c.211.1	yes	2
13.9	even	3	inner	1170.2.i.d.451.1		2
15.2	even	4		1950.2.z.k.1849.1		4
15.8	even	4		1950.2.z.k.1849.2		4
15.14	odd	2		1950.2.i.n.601.1		2
39.2	even	12		5070.2.b.j.1351.2		2
39.11	even	12		5070.2.b.j.1351.1		2
39.23	odd	6		5070.2.a.v.1.1		1
39.29	odd	6		5070.2.a.j.1.1		1
39.35	odd	6		390.2.i.c.61.1	✓	2
195.74	odd	6		1950.2.i.n.451.1		2
195.113	even	12		1950.2.z.k.1699.1		4
195.152	even	12		1950.2.z.k.1699.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
390.2.i.c.61.1	✓	2	39.35	odd	6
390.2.i.c.211.1	yes	2	3.2	odd	2
1170.2.i.d.451.1		2	13.9	even	3	inner
1170.2.i.d.991.1		2	1.1	even	1	trivial
1950.2.i.n.451.1		2	195.74	odd	6
1950.2.i.n.601.1		2	15.14	odd	2
1950.2.z.k.1699.1		4	195.113	even	12
1950.2.z.k.1699.2		4	195.152	even	12
1950.2.z.k.1849.1		4	15.2	even	4
1950.2.z.k.1849.2		4	15.8	even	4
5070.2.a.j.1.1		1	39.29	odd	6
5070.2.a.v.1.1		1	39.23	odd	6
5070.2.b.j.1351.1		2	39.11	even	12
5070.2.b.j.1351.2		2	39.2	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 1170 = 2 \cdot 3^{2} \cdot 5 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1170.i (of order \(3\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(-0.500000 - 0.866025i) q^{2} +(-0.500000 + 0.866025i) q^{4} +1.00000 q^{5} +(-1.00000 + 1.73205i) q^{7} +1.00000 q^{8} +(-0.500000 - 0.866025i) q^{10} +(-1.50000 - 2.59808i) q^{11} +(1.00000 + 3.46410i) q^{13} +2.00000 q^{14} +(-0.500000 - 0.866025i) q^{16} +(3.00000 - 5.19615i) q^{17} +(-1.00000 + 1.73205i) q^{19} +(-0.500000 + 0.866025i) q^{20} +(-1.50000 + 2.59808i) q^{22} +(1.50000 + 2.59808i) q^{23} +1.00000 q^{25} +(2.50000 - 2.59808i) q^{26} +(-1.00000 - 1.73205i) q^{28} +(1.50000 + 2.59808i) q^{29} +5.00000 q^{31} +(-0.500000 + 0.866025i) q^{32} -6.00000 q^{34} +(-1.00000 + 1.73205i) q^{35} +(3.50000 + 6.06218i) q^{37} +2.00000 q^{38} +1.00000 q^{40} +(3.00000 + 5.19615i) q^{41} +(0.500000 - 0.866025i) q^{43} +3.00000 q^{44} +(1.50000 - 2.59808i) q^{46} +3.00000 q^{47} +(1.50000 + 2.59808i) q^{49} +(-0.500000 - 0.866025i) q^{50} +(-3.50000 - 0.866025i) q^{52} +6.00000 q^{53} +(-1.50000 - 2.59808i) q^{55} +(-1.00000 + 1.73205i) q^{56} +(1.50000 - 2.59808i) q^{58} +(-4.50000 + 7.79423i) q^{59} +(-1.00000 + 1.73205i) q^{61} +(-2.50000 - 4.33013i) q^{62} +1.00000 q^{64} +(1.00000 + 3.46410i) q^{65} +(-4.00000 - 6.92820i) q^{67} +(3.00000 + 5.19615i) q^{68} +2.00000 q^{70} +(-6.00000 + 10.3923i) q^{71} +14.0000 q^{73} +(3.50000 - 6.06218i) q^{74} +(-1.00000 - 1.73205i) q^{76} +6.00000 q^{77} +5.00000 q^{79} +(-0.500000 - 0.866025i) q^{80} +(3.00000 - 5.19615i) q^{82} +6.00000 q^{83} +(3.00000 - 5.19615i) q^{85} -1.00000 q^{86} +(-1.50000 - 2.59808i) q^{88} +(-9.00000 - 15.5885i) q^{89} +(-7.00000 - 1.73205i) q^{91} -3.00000 q^{92} +(-1.50000 - 2.59808i) q^{94} +(-1.00000 + 1.73205i) q^{95} +(-7.00000 + 12.1244i) q^{97} +(1.50000 - 2.59808i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - q^{2} - q^{4} + 2 q^{5} - 2 q^{7} + 2 q^{8} - q^{10} - 3 q^{11} + 2 q^{13} + 4 q^{14} - q^{16} + 6 q^{17} - 2 q^{19} - q^{20} - 3 q^{22} + 3 q^{23} + 2 q^{25} + 5 q^{26} - 2 q^{28} + 3 q^{29}+ \cdots + 3 q^{98}+O(q^{100}) \) 2 * q - q^2 - q^4 + 2 * q^5 - 2 * q^7 + 2 * q^8 - q^10 - 3 * q^11 + 2 * q^13 + 4 * q^14 - q^16 + 6 * q^17 - 2 * q^19 - q^20 - 3 * q^22 + 3 * q^23 + 2 * q^25 + 5 * q^26 - 2 * q^28 + 3 * q^29 + 10 * q^31 - q^32 - 12 * q^34 - 2 * q^35 + 7 * q^37 + 4 * q^38 + 2 * q^40 + 6 * q^41 + q^43 + 6 * q^44 + 3 * q^46 + 6 * q^47 + 3 * q^49 - q^50 - 7 * q^52 + 12 * q^53 - 3 * q^55 - 2 * q^56 + 3 * q^58 - 9 * q^59 - 2 * q^61 - 5 * q^62 + 2 * q^64 + 2 * q^65 - 8 * q^67 + 6 * q^68 + 4 * q^70 - 12 * q^71 + 28 * q^73 + 7 * q^74 - 2 * q^76 + 12 * q^77 + 10 * q^79 - q^80 + 6 * q^82 + 12 * q^83 + 6 * q^85 - 2 * q^86 - 3 * q^88 - 18 * q^89 - 14 * q^91 - 6 * q^92 - 3 * q^94 - 2 * q^95 - 14 * q^97 + 3 * q^98

Introduction

L-functions

Modular forms

Varieties

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Representations

Groups

Database

Properties

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