LMFDB - Embedded newform 513.2.h.b.235.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [513,2,Mod(235,513)] 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("513.235"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(513, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([4, 2])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 513 = 3^{3} \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	513.h (of order $3$, degree $2$, not minimal)

Newform invariants

comment:select newform

sage:traces = [2,2,0,-2,3] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)

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

Self dual:	no
Analytic conductor:	$4.09632562369$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{6})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x + 1 $ x^2 - x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 171)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			235.1
Root			$0.500000 - 0.866025i$ of defining polynomial
Character	$\chi$	$=$	513.235
Dual form			513.2.h.b.334.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+1.00000 q^{2} -1.00000 q^{4} +(1.50000 - 2.59808i) q^{5} +(-0.500000 + 0.866025i) q^{7} -3.00000 q^{8} +(1.50000 - 2.59808i) q^{10} +(2.50000 - 4.33013i) q^{11} +2.00000 q^{13} +(-0.500000 + 0.866025i) q^{14} -1.00000 q^{16} +(-2.50000 - 4.33013i) q^{17} +(-4.00000 - 1.73205i) q^{19} +(-1.50000 + 2.59808i) q^{20} +(2.50000 - 4.33013i) q^{22} +8.00000 q^{23} +(-2.00000 - 3.46410i) q^{25} +2.00000 q^{26} +(0.500000 - 0.866025i) q^{28} +(-0.500000 - 0.866025i) q^{29} +(-1.50000 - 2.59808i) q^{31} +5.00000 q^{32} +(-2.50000 - 4.33013i) q^{34} +(1.50000 + 2.59808i) q^{35} -6.00000 q^{37} +(-4.00000 - 1.73205i) q^{38} +(-4.50000 + 7.79423i) q^{40} +(-4.50000 + 7.79423i) q^{41} +8.00000 q^{43} +(-2.50000 + 4.33013i) q^{44} +8.00000 q^{46} +(1.50000 + 2.59808i) q^{47} +(3.00000 + 5.19615i) q^{49} +(-2.00000 - 3.46410i) q^{50} -2.00000 q^{52} +(-0.500000 + 0.866025i) q^{53} +(-7.50000 - 12.9904i) q^{55} +(1.50000 - 2.59808i) q^{56} +(-0.500000 - 0.866025i) q^{58} +(2.50000 - 4.33013i) q^{59} +(6.50000 + 11.2583i) q^{61} +(-1.50000 - 2.59808i) q^{62} +7.00000 q^{64} +(3.00000 - 5.19615i) q^{65} -4.00000 q^{67} +(2.50000 + 4.33013i) q^{68} +(1.50000 + 2.59808i) q^{70} +(1.50000 + 2.59808i) q^{71} +(2.50000 + 4.33013i) q^{73} -6.00000 q^{74} +(4.00000 + 1.73205i) q^{76} +(2.50000 + 4.33013i) q^{77} +4.00000 q^{79} +(-1.50000 + 2.59808i) q^{80} +(-4.50000 + 7.79423i) q^{82} +(4.50000 - 7.79423i) q^{83} -15.0000 q^{85} +8.00000 q^{86} +(-7.50000 + 12.9904i) q^{88} +(-4.50000 + 7.79423i) q^{89} +(-1.00000 + 1.73205i) q^{91} -8.00000 q^{92} +(1.50000 + 2.59808i) q^{94} +(-10.5000 + 7.79423i) q^{95} -10.0000 q^{97} +(3.00000 + 5.19615i) q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{2} - 2 q^{4} + 3 q^{5} - q^{7} - 6 q^{8} + 3 q^{10} + 5 q^{11} + 4 q^{13} - q^{14} - 2 q^{16} - 5 q^{17} - 8 q^{19} - 3 q^{20} + 5 q^{22} + 16 q^{23} - 4 q^{25} + 4 q^{26} + q^{28} - q^{29}+ \cdots + 6 q^{98}+O(q^{100}) $ 2 * q + 2 * q^2 - 2 * q^4 + 3 * q^5 - q^7 - 6 * q^8 + 3 * q^10 + 5 * q^11 + 4 * q^13 - q^14 - 2 * q^16 - 5 * q^17 - 8 * q^19 - 3 * q^20 + 5 * q^22 + 16 * q^23 - 4 * q^25 + 4 * q^26 + q^28 - q^29 - 3 * q^31 + 10 * q^32 - 5 * q^34 + 3 * q^35 - 12 * q^37 - 8 * q^38 - 9 * q^40 - 9 * q^41 + 16 * q^43 - 5 * q^44 + 16 * q^46 + 3 * q^47 + 6 * q^49 - 4 * q^50 - 4 * q^52 - q^53 - 15 * q^55 + 3 * q^56 - q^58 + 5 * q^59 + 13 * q^61 - 3 * q^62 + 14 * q^64 + 6 * q^65 - 8 * q^67 + 5 * q^68 + 3 * q^70 + 3 * q^71 + 5 * q^73 - 12 * q^74 + 8 * q^76 + 5 * q^77 + 8 * q^79 - 3 * q^80 - 9 * q^82 + 9 * q^83 - 30 * q^85 + 16 * q^86 - 15 * q^88 - 9 * q^89 - 2 * q^91 - 16 * q^92 + 3 * q^94 - 21 * q^95 - 20 * q^97 + 6 * q^98

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/513\mathbb{Z}\right)^\times$.

$n$	$191$	$325$
$\chi(n)$	$e\left(\frac{2}{3}\right)$	$e\left(\frac{1}{3}\right)$

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 $

$2$	1.00000			0.707107			0.353553	−	0.935414i	$-0.384973\pi$
$2$	1.00000			0.707107			0.353553	+	0.935414i	$0.384973\pi$
$3$		0			0
$3$		0			0
$4$	−1.00000			−0.500000
$5$	1.50000	−	2.59808i	0.670820	−	1.16190i	−0.306851	−	0.951757i	$-0.599275\pi$
$5$	1.50000	−	2.59808i	0.670820	−	1.16190i	0.977672	−	0.210138i	$-0.0673912\pi$
$6$		0			0
$7$	−0.500000	+	0.866025i	−0.188982	+	0.327327i	−0.944911	−	0.327327i	$-0.893852\pi$
$7$	−0.500000	+	0.866025i	−0.188982	+	0.327327i	0.755929	+	0.654654i	$0.227186\pi$
$8$	−3.00000			−1.06066
$9$		0			0
$10$	1.50000	−	2.59808i	0.474342	−	0.821584i
$11$	2.50000	−	4.33013i	0.753778	−	1.30558i	−0.192201	−	0.981356i	$-0.561563\pi$
$11$	2.50000	−	4.33013i	0.753778	−	1.30558i	0.945979	−	0.324227i	$-0.105104\pi$
$12$		0			0
$13$	2.00000			0.554700			0.277350	−	0.960769i	$-0.410544\pi$
$13$	2.00000			0.554700			0.277350	+	0.960769i	$0.410544\pi$
$14$	−0.500000	+	0.866025i	−0.133631	+	0.231455i
$15$		0			0
$16$	−1.00000			−0.250000
$17$	−2.50000	−	4.33013i	−0.606339	−	1.05021i	−0.991838	−	0.127502i	$-0.959304\pi$
$17$	−2.50000	−	4.33013i	−0.606339	−	1.05021i	0.385499	−	0.922708i	$-0.374029\pi$
$18$		0			0
$19$	−4.00000	−	1.73205i	−0.917663	−	0.397360i
$19$	−4.00000	−	1.73205i	−0.917663	−	0.397360i
$20$	−1.50000	+	2.59808i	−0.335410	+	0.580948i
$21$		0			0
$22$	2.50000	−	4.33013i	0.533002	−	0.923186i
$23$	8.00000			1.66812			0.834058	−	0.551677i	$-0.186012\pi$
$23$	8.00000			1.66812			0.834058	+	0.551677i	$0.186012\pi$
$24$		0			0
$25$	−2.00000	−	3.46410i	−0.400000	−	0.692820i
$26$	2.00000			0.392232
$27$		0			0
$28$	0.500000	−	0.866025i	0.0944911	−	0.163663i
$29$	−0.500000	−	0.866025i	−0.0928477	−	0.160817i	0.815861	−	0.578249i	$-0.196264\pi$
$29$	−0.500000	−	0.866025i	−0.0928477	−	0.160817i	−0.908708	+	0.417432i	$0.862930\pi$
$30$		0			0
$31$	−1.50000	−	2.59808i	−0.269408	−	0.466628i	0.699301	−	0.714827i	$-0.253495\pi$
$31$	−1.50000	−	2.59808i	−0.269408	−	0.466628i	−0.968709	+	0.248199i	$0.920161\pi$
$32$	5.00000			0.883883
$33$		0			0
$34$	−2.50000	−	4.33013i	−0.428746	−	0.742611i
$35$	1.50000	+	2.59808i	0.253546	+	0.439155i
$36$		0			0
$37$	−6.00000			−0.986394			−0.493197	−	0.869918i	$-0.664172\pi$
$37$	−6.00000			−0.986394			−0.493197	+	0.869918i	$0.664172\pi$
$38$	−4.00000	−	1.73205i	−0.648886	−	0.280976i
$39$		0			0
$40$	−4.50000	+	7.79423i	−0.711512	+	1.23238i
$41$	−4.50000	+	7.79423i	−0.702782	+	1.21725i	0.264704	+	0.964330i	$0.414726\pi$
$41$	−4.50000	+	7.79423i	−0.702782	+	1.21725i	−0.967486	+	0.252924i	$0.918608\pi$
$42$		0			0
$43$	8.00000			1.21999			0.609994	−	0.792406i	$-0.291172\pi$
$43$	8.00000			1.21999			0.609994	+	0.792406i	$0.291172\pi$
$44$	−2.50000	+	4.33013i	−0.376889	+	0.652791i
$45$		0			0
$46$	8.00000			1.17954
$47$	1.50000	+	2.59808i	0.218797	+	0.378968i	0.954441	−	0.298401i	$-0.0964533\pi$
$47$	1.50000	+	2.59808i	0.218797	+	0.378968i	−0.735643	+	0.677369i	$0.763120\pi$
$48$		0			0
$49$	3.00000	+	5.19615i	0.428571	+	0.742307i
$50$	−2.00000	−	3.46410i	−0.282843	−	0.489898i
$51$		0			0
$52$	−2.00000			−0.277350
$53$	−0.500000	+	0.866025i	−0.0686803	+	0.118958i	−0.898321	−	0.439340i	$-0.855212\pi$
$53$	−0.500000	+	0.866025i	−0.0686803	+	0.118958i	0.829640	+	0.558298i	$0.188546\pi$
$54$		0			0
$55$	−7.50000	−	12.9904i	−1.01130	−	1.75162i
$56$	1.50000	−	2.59808i	0.200446	−	0.347183i
$57$		0			0
$58$	−0.500000	−	0.866025i	−0.0656532	−	0.113715i
$59$	2.50000	−	4.33013i	0.325472	−	0.563735i	−0.656136	−	0.754643i	$-0.727810\pi$
$59$	2.50000	−	4.33013i	0.325472	−	0.563735i	0.981608	+	0.190909i	$0.0611434\pi$
$60$		0			0
$61$	6.50000	+	11.2583i	0.832240	+	1.44148i	0.896258	+	0.443533i	$0.146275\pi$
$61$	6.50000	+	11.2583i	0.832240	+	1.44148i	−0.0640184	+	0.997949i	$0.520392\pi$
$62$	−1.50000	−	2.59808i	−0.190500	−	0.329956i
$63$		0			0
$64$	7.00000			0.875000
$65$	3.00000	−	5.19615i	0.372104	−	0.644503i
$66$		0			0
$67$	−4.00000			−0.488678			−0.244339	−	0.969690i	$-0.578571\pi$
$67$	−4.00000			−0.488678			−0.244339	+	0.969690i	$0.578571\pi$
$68$	2.50000	+	4.33013i	0.303170	+	0.525105i
$69$		0			0
$70$	1.50000	+	2.59808i	0.179284	+	0.310530i
$71$	1.50000	+	2.59808i	0.178017	+	0.308335i	0.941201	−	0.337846i	$-0.109698\pi$
$71$	1.50000	+	2.59808i	0.178017	+	0.308335i	−0.763184	+	0.646181i	$0.776365\pi$
$72$		0			0
$73$	2.50000	+	4.33013i	0.292603	+	0.506803i	0.974424	−	0.224716i	$-0.0721453\pi$
$73$	2.50000	+	4.33013i	0.292603	+	0.506803i	−0.681822	+	0.731519i	$0.738812\pi$
$74$	−6.00000			−0.697486
$75$		0			0
$76$	4.00000	+	1.73205i	0.458831	+	0.198680i
$77$	2.50000	+	4.33013i	0.284901	+	0.493464i
$78$		0			0
$79$	4.00000			0.450035			0.225018	−	0.974355i	$-0.427756\pi$
$79$	4.00000			0.450035			0.225018	+	0.974355i	$0.427756\pi$
$80$	−1.50000	+	2.59808i	−0.167705	+	0.290474i
$81$		0			0
$82$	−4.50000	+	7.79423i	−0.496942	+	0.860729i
$83$	4.50000	−	7.79423i	0.493939	−	0.855528i	−0.506036	−	0.862512i	$-0.668890\pi$
$83$	4.50000	−	7.79423i	0.493939	−	0.855528i	0.999976	+	0.00698436i	$0.00222321\pi$
$84$		0			0
$85$	−15.0000			−1.62698
$86$	8.00000			0.862662
$87$		0			0
$88$	−7.50000	+	12.9904i	−0.799503	+	1.38478i
$89$	−4.50000	+	7.79423i	−0.476999	+	0.826187i	−0.999653	−	0.0263586i	$-0.991609\pi$
$89$	−4.50000	+	7.79423i	−0.476999	+	0.826187i	0.522654	+	0.852545i	$0.324942\pi$
$90$		0			0
$91$	−1.00000	+	1.73205i	−0.104828	+	0.181568i
$92$	−8.00000			−0.834058
$93$		0			0
$94$	1.50000	+	2.59808i	0.154713	+	0.267971i
$95$	−10.5000	+	7.79423i	−1.07728	+	0.799671i
$96$		0			0
$97$	−10.0000			−1.01535			−0.507673	−	0.861550i	$-0.669494\pi$
$97$	−10.0000			−1.01535			−0.507673	+	0.861550i	$0.669494\pi$
$98$	3.00000	+	5.19615i	0.303046	+	0.524891i
$99$		0			0
$100$	2.00000	+	3.46410i	0.200000	+	0.346410i
$101$	3.50000	+	6.06218i	0.348263	+	0.603209i	0.985941	−	0.167094i	$-0.0534383\pi$
$101$	3.50000	+	6.06218i	0.348263	+	0.603209i	−0.637678	+	0.770303i	$0.720105\pi$
$102$		0			0
$103$	2.50000	+	4.33013i	0.246332	+	0.426660i	0.962505	−	0.271263i	$-0.0874412\pi$
$103$	2.50000	+	4.33013i	0.246332	+	0.426660i	−0.716173	+	0.697923i	$0.754108\pi$
$104$	−6.00000			−0.588348
$105$		0			0
$106$	−0.500000	+	0.866025i	−0.0485643	+	0.0841158i
$107$	−12.0000			−1.16008			−0.580042	−	0.814587i	$-0.696964\pi$
$107$	−12.0000			−1.16008			−0.580042	+	0.814587i	$0.696964\pi$
$108$		0			0
$109$	2.50000	+	4.33013i	0.239457	+	0.414751i	0.960558	−	0.278078i	$-0.0896974\pi$
$109$	2.50000	+	4.33013i	0.239457	+	0.414751i	−0.721102	+	0.692829i	$0.756364\pi$
$110$	−7.50000	−	12.9904i	−0.715097	−	1.23858i
$111$		0			0
$112$	0.500000	−	0.866025i	0.0472456	−	0.0818317i
$113$	−8.50000	−	14.7224i	−0.799613	−	1.38497i	−0.919868	−	0.392227i	$-0.871705\pi$
$113$	−8.50000	−	14.7224i	−0.799613	−	1.38497i	0.120256	−	0.992743i	$-0.461629\pi$
$114$		0			0
$115$	12.0000	−	20.7846i	1.11901	−	1.93817i
$116$	0.500000	+	0.866025i	0.0464238	+	0.0804084i
$117$		0			0
$118$	2.50000	−	4.33013i	0.230144	−	0.398621i
$119$	5.00000			0.458349
$120$		0			0
$121$	−7.00000	−	12.1244i	−0.636364	−	1.10221i
$122$	6.50000	+	11.2583i	0.588482	+	1.01928i
$123$		0			0
$124$	1.50000	+	2.59808i	0.134704	+	0.233314i
$125$	3.00000			0.268328
$126$		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	$0.970975\pi$
$128$	−3.00000			−0.265165
$129$		0			0
$130$	3.00000	−	5.19615i	0.263117	−	0.455733i
$131$	−3.50000	+	6.06218i	−0.305796	+	0.529655i	−0.977438	−	0.211221i	$-0.932256\pi$
$131$	−3.50000	+	6.06218i	−0.305796	+	0.529655i	0.671642	+	0.740876i	$0.265589\pi$
$132$		0			0
$133$	3.50000	−	2.59808i	0.303488	−	0.225282i
$134$	−4.00000			−0.345547
$135$		0			0
$136$	7.50000	+	12.9904i	0.643120	+	1.11392i
$137$	1.50000	+	2.59808i	0.128154	+	0.221969i	0.922961	−	0.384893i	$-0.125762\pi$
$137$	1.50000	+	2.59808i	0.128154	+	0.221969i	−0.794808	+	0.606861i	$0.792428\pi$
$138$		0			0
$139$	20.0000			1.69638			0.848189	−	0.529694i	$-0.177693\pi$
$139$	20.0000			1.69638			0.848189	+	0.529694i	$0.177693\pi$
$140$	−1.50000	−	2.59808i	−0.126773	−	0.219578i
$141$		0			0
$142$	1.50000	+	2.59808i	0.125877	+	0.218026i
$143$	5.00000	−	8.66025i	0.418121	−	0.724207i
$144$		0			0
$145$	−3.00000			−0.249136
$146$	2.50000	+	4.33013i	0.206901	+	0.358364i
$147$		0			0
$148$	6.00000			0.493197
$149$	7.50000	−	12.9904i	0.614424	−	1.06421i	−0.376061	−	0.926595i	$-0.622722\pi$
$149$	7.50000	−	12.9904i	0.614424	−	1.06421i	0.990485	−	0.137619i	$-0.0439449\pi$
$150$		0			0
$151$	7.50000	−	12.9904i	0.610341	−	1.05714i	−0.380841	−	0.924640i	$-0.624366\pi$
$151$	7.50000	−	12.9904i	0.610341	−	1.05714i	0.991183	−	0.132502i	$-0.0423010\pi$
$152$	12.0000	+	5.19615i	0.973329	+	0.421464i
$153$		0			0
$154$	2.50000	+	4.33013i	0.201456	+	0.348932i
$155$	−9.00000			−0.722897
$156$		0			0
$157$	−1.50000	+	2.59808i	−0.119713	+	0.207349i	−0.919654	−	0.392730i	$-0.871531\pi$
$157$	−1.50000	+	2.59808i	−0.119713	+	0.207349i	0.799941	+	0.600079i	$0.204864\pi$
$158$	4.00000			0.318223
$159$		0			0
$160$	7.50000	−	12.9904i	0.592927	−	1.02698i
$161$	−4.00000	+	6.92820i	−0.315244	+	0.546019i
$162$		0			0
$163$	12.0000			0.939913			0.469956	−	0.882690i	$-0.344270\pi$
$163$	12.0000			0.939913			0.469956	+	0.882690i	$0.344270\pi$
$164$	4.50000	−	7.79423i	0.351391	−	0.608627i
$165$		0			0
$166$	4.50000	−	7.79423i	0.349268	−	0.604949i
$167$	4.00000			0.309529			0.154765	−	0.987951i	$-0.450538\pi$
$167$	4.00000			0.309529			0.154765	+	0.987951i	$0.450538\pi$
$168$		0			0
$169$	−9.00000			−0.692308
$170$	−15.0000			−1.15045
$171$		0			0
$172$	−8.00000			−0.609994
$173$	6.00000			0.456172			0.228086	−	0.973641i	$-0.426753\pi$
$173$	6.00000			0.456172			0.228086	+	0.973641i	$0.426753\pi$
$174$		0			0
$175$	4.00000			0.302372
$176$	−2.50000	+	4.33013i	−0.188445	+	0.326396i
$177$		0			0
$178$	−4.50000	+	7.79423i	−0.337289	+	0.584202i
$179$	12.0000			0.896922			0.448461	−	0.893802i	$-0.351972\pi$
$179$	12.0000			0.896922			0.448461	+	0.893802i	$0.351972\pi$
$180$		0			0
$181$	−3.50000	+	6.06218i	−0.260153	+	0.450598i	−0.966282	−	0.257485i	$-0.917106\pi$
$181$	−3.50000	+	6.06218i	−0.260153	+	0.450598i	0.706129	+	0.708083i	$0.250440\pi$
$182$	−1.00000	+	1.73205i	−0.0741249	+	0.128388i
$183$		0			0
$184$	−24.0000			−1.76930
$185$	−9.00000	+	15.5885i	−0.661693	+	1.14609i
$186$		0			0
$187$	−25.0000			−1.82818
$188$	−1.50000	−	2.59808i	−0.109399	−	0.189484i
$189$		0			0
$190$	−10.5000	+	7.79423i	−0.761750	+	0.565453i
$191$	−1.50000	+	2.59808i	−0.108536	+	0.187990i	−0.915177	−	0.403051i	$-0.867950\pi$
$191$	−1.50000	+	2.59808i	−0.108536	+	0.187990i	0.806641	+	0.591041i	$0.201283\pi$
$192$		0			0
$193$	8.50000	−	14.7224i	0.611843	−	1.05974i	−0.379086	−	0.925361i	$-0.623762\pi$
$193$	8.50000	−	14.7224i	0.611843	−	1.05974i	0.990930	−	0.134382i	$-0.0429051\pi$
$194$	−10.0000			−0.717958
$195$		0			0
$196$	−3.00000	−	5.19615i	−0.214286	−	0.371154i
$197$	−2.00000			−0.142494			−0.0712470	−	0.997459i	$-0.522698\pi$
$197$	−2.00000			−0.142494			−0.0712470	+	0.997459i	$0.522698\pi$
$198$		0			0
$199$	1.50000	−	2.59808i	0.106332	−	0.184173i	−0.807950	−	0.589252i	$-0.799423\pi$
$199$	1.50000	−	2.59808i	0.106332	−	0.184173i	0.914282	+	0.405079i	$0.132756\pi$
$200$	6.00000	+	10.3923i	0.424264	+	0.734847i
$201$		0			0
$202$	3.50000	+	6.06218i	0.246259	+	0.426533i
$203$	1.00000			0.0701862
$204$		0			0
$205$	13.5000	+	23.3827i	0.942881	+	1.63312i
$206$	2.50000	+	4.33013i	0.174183	+	0.301694i
$207$		0			0
$208$	−2.00000			−0.138675
$209$	−17.5000	+	12.9904i	−1.21050	+	0.898563i
$210$		0			0
$211$	7.50000	−	12.9904i	0.516321	−	0.894295i	−0.483499	−	0.875345i	$-0.660634\pi$
$211$	7.50000	−	12.9904i	0.516321	−	0.894295i	0.999820	−	0.0189499i	$-0.00603229\pi$
$212$	0.500000	−	0.866025i	0.0343401	−	0.0594789i
$213$		0			0
$214$	−12.0000			−0.820303
$215$	12.0000	−	20.7846i	0.818393	−	1.41750i
$216$		0			0
$217$	3.00000			0.203653
$218$	2.50000	+	4.33013i	0.169321	+	0.293273i
$219$		0			0
$220$	7.50000	+	12.9904i	0.505650	+	0.875811i
$221$	−5.00000	−	8.66025i	−0.336336	−	0.582552i
$222$		0			0
$223$	−24.0000			−1.60716			−0.803579	−	0.595198i	$-0.797074\pi$
$223$	−24.0000			−1.60716			−0.803579	+	0.595198i	$0.797074\pi$
$224$	−2.50000	+	4.33013i	−0.167038	+	0.289319i
$225$		0			0
$226$	−8.50000	−	14.7224i	−0.565412	−	0.979322i
$227$	10.5000	−	18.1865i	0.696909	−	1.20708i	−0.272623	−	0.962121i	$-0.587891\pi$
$227$	10.5000	−	18.1865i	0.696909	−	1.20708i	0.969533	−	0.244962i	$-0.0787754\pi$
$228$		0			0
$229$	8.50000	+	14.7224i	0.561696	+	0.972886i	0.997349	+	0.0727709i	$0.0231842\pi$
$229$	8.50000	+	14.7224i	0.561696	+	0.972886i	−0.435653	+	0.900115i	$0.643482\pi$
$230$	12.0000	−	20.7846i	0.791257	−	1.37050i
$231$		0			0
$232$	1.50000	+	2.59808i	0.0984798	+	0.170572i
$233$	−4.50000	−	7.79423i	−0.294805	−	0.510617i	0.680135	−	0.733087i	$-0.261921\pi$
$233$	−4.50000	−	7.79423i	−0.294805	−	0.510617i	−0.974939	+	0.222470i	$0.928588\pi$
$234$		0			0
$235$	9.00000			0.587095
$236$	−2.50000	+	4.33013i	−0.162736	+	0.281867i
$237$		0			0
$238$	5.00000			0.324102
$239$	7.50000	+	12.9904i	0.485135	+	0.840278i	0.999854	−	0.0170808i	$-0.00543724\pi$
$239$	7.50000	+	12.9904i	0.485135	+	0.840278i	−0.514719	+	0.857359i	$0.672104\pi$
$240$		0			0
$241$	0.500000	+	0.866025i	0.0322078	+	0.0557856i	0.881680	−	0.471848i	$-0.156413\pi$
$241$	0.500000	+	0.866025i	0.0322078	+	0.0557856i	−0.849472	+	0.527633i	$0.823079\pi$
$242$	−7.00000	−	12.1244i	−0.449977	−	0.779383i
$243$		0			0
$244$	−6.50000	−	11.2583i	−0.416120	−	0.720741i
$245$	18.0000			1.14998
$246$		0			0
$247$	−8.00000	−	3.46410i	−0.509028	−	0.220416i
$248$	4.50000	+	7.79423i	0.285750	+	0.494934i
$249$		0			0
$250$	3.00000			0.189737
$251$	2.50000	−	4.33013i	0.157799	−	0.273315i	−0.776276	−	0.630393i	$-0.782894\pi$
$251$	2.50000	−	4.33013i	0.157799	−	0.273315i	0.934075	+	0.357078i	$0.116227\pi$
$252$		0			0
$253$	20.0000	−	34.6410i	1.25739	−	2.17786i
$254$	−6.50000	+	11.2583i	−0.407846	+	0.706410i
$255$		0			0
$256$	−17.0000			−1.06250
$257$	10.0000			0.623783			0.311891	−	0.950118i	$-0.399037\pi$
$257$	10.0000			0.623783			0.311891	+	0.950118i	$0.399037\pi$
$258$		0			0
$259$	3.00000	−	5.19615i	0.186411	−	0.322873i
$260$	−3.00000	+	5.19615i	−0.186052	+	0.322252i
$261$		0			0
$262$	−3.50000	+	6.06218i	−0.216231	+	0.374523i
$263$	8.00000			0.493301			0.246651	−	0.969104i	$-0.420670\pi$
$263$	8.00000			0.493301			0.246651	+	0.969104i	$0.420670\pi$
$264$		0			0
$265$	1.50000	+	2.59808i	0.0921443	+	0.159599i
$266$	3.50000	−	2.59808i	0.214599	−	0.159298i
$267$		0			0
$268$	4.00000			0.244339
$269$	−12.5000	−	21.6506i	−0.762138	−	1.32006i	−0.941746	−	0.336324i	$-0.890816\pi$
$269$	−12.5000	−	21.6506i	−0.762138	−	1.32006i	0.179608	−	0.983738i	$-0.442517\pi$
$270$		0			0
$271$	−13.5000	−	23.3827i	−0.820067	−	1.42040i	−0.905632	−	0.424064i	$-0.860603\pi$
$271$	−13.5000	−	23.3827i	−0.820067	−	1.42040i	0.0855654	−	0.996333i	$-0.472730\pi$
$272$	2.50000	+	4.33013i	0.151585	+	0.262553i
$273$		0			0
$274$	1.50000	+	2.59808i	0.0906183	+	0.156956i
$275$	−20.0000			−1.20605
$276$		0			0
$277$	−3.50000	+	6.06218i	−0.210295	+	0.364241i	−0.951807	−	0.306699i	$-0.900776\pi$
$277$	−3.50000	+	6.06218i	−0.210295	+	0.364241i	0.741512	+	0.670940i	$0.234109\pi$
$278$	20.0000			1.19952
$279$		0			0
$280$	−4.50000	−	7.79423i	−0.268926	−	0.465794i
$281$	5.50000	+	9.52628i	0.328102	+	0.568290i	0.982135	−	0.188176i	$-0.0602575\pi$
$281$	5.50000	+	9.52628i	0.328102	+	0.568290i	−0.654033	+	0.756466i	$0.726924\pi$
$282$		0			0
$283$	−14.5000	+	25.1147i	−0.861936	+	1.49292i	0.00812260	+	0.999967i	$0.497414\pi$
$283$	−14.5000	+	25.1147i	−0.861936	+	1.49292i	−0.870058	+	0.492949i	$0.835919\pi$
$284$	−1.50000	−	2.59808i	−0.0890086	−	0.154167i
$285$		0			0
$286$	5.00000	−	8.66025i	0.295656	−	0.512092i
$287$	−4.50000	−	7.79423i	−0.265627	−	0.460079i
$288$		0			0
$289$	−4.00000	+	6.92820i	−0.235294	+	0.407541i
$290$	−3.00000			−0.176166
$291$		0			0
$292$	−2.50000	−	4.33013i	−0.146301	−	0.253402i
$293$	−2.50000	−	4.33013i	−0.146052	−	0.252969i	0.783713	−	0.621123i	$-0.213323\pi$
$293$	−2.50000	−	4.33013i	−0.146052	−	0.252969i	−0.929765	+	0.368154i	$0.879990\pi$
$294$		0			0
$295$	−7.50000	−	12.9904i	−0.436667	−	0.756329i
$296$	18.0000			1.04623
$297$		0			0
$298$	7.50000	−	12.9904i	0.434463	−	0.752513i
$299$	16.0000			0.925304
$300$		0			0
$301$	−4.00000	+	6.92820i	−0.230556	+	0.399335i
$302$	7.50000	−	12.9904i	0.431577	−	0.747512i
$303$		0			0
$304$	4.00000	+	1.73205i	0.229416	+	0.0993399i
$305$	39.0000			2.23313
$306$		0			0
$307$	16.5000	+	28.5788i	0.941705	+	1.63108i	0.762218	+	0.647320i	$0.224110\pi$
$307$	16.5000	+	28.5788i	0.941705	+	1.63108i	0.179486	+	0.983760i	$0.442556\pi$
$308$	−2.50000	−	4.33013i	−0.142451	−	0.246732i
$309$		0			0
$310$	−9.00000			−0.511166
$311$	−10.5000	−	18.1865i	−0.595400	−	1.03126i	−0.993490	−	0.113917i	$-0.963660\pi$
$311$	−10.5000	−	18.1865i	−0.595400	−	1.03126i	0.398090	−	0.917346i	$-0.369673\pi$
$312$		0			0
$313$	8.50000	+	14.7224i	0.480448	+	0.832161i	0.999748	−	0.0224310i	$-0.00714060\pi$
$313$	8.50000	+	14.7224i	0.480448	+	0.832161i	−0.519300	+	0.854592i	$0.673807\pi$
$314$	−1.50000	+	2.59808i	−0.0846499	+	0.146618i
$315$		0			0
$316$	−4.00000			−0.225018
$317$	−8.50000	−	14.7224i	−0.477408	−	0.826894i	0.522257	−	0.852788i	$-0.325090\pi$
$317$	−8.50000	−	14.7224i	−0.477408	−	0.826894i	−0.999665	+	0.0258939i	$0.991757\pi$
$318$		0			0
$319$	−5.00000			−0.279946
$320$	10.5000	−	18.1865i	0.586968	−	1.01666i
$321$		0			0
$322$	−4.00000	+	6.92820i	−0.222911	+	0.386094i
$323$	2.50000	+	21.6506i	0.139104	+	1.20467i
$324$		0			0
$325$	−4.00000	−	6.92820i	−0.221880	−	0.384308i
$326$	12.0000			0.664619
$327$		0			0
$328$	13.5000	−	23.3827i	0.745413	−	1.29109i
$329$	−3.00000			−0.165395
$330$		0			0
$331$	9.50000	−	16.4545i	0.522167	−	0.904420i	−0.477500	−	0.878632i	$-0.658457\pi$
$331$	9.50000	−	16.4545i	0.522167	−	0.904420i	0.999667	−	0.0257885i	$-0.00820965\pi$
$332$	−4.50000	+	7.79423i	−0.246970	+	0.427764i
$333$		0			0
$334$	4.00000			0.218870
$335$	−6.00000	+	10.3923i	−0.327815	+	0.567792i
$336$		0			0
$337$	2.50000	−	4.33013i	0.136184	−	0.235877i	−0.789865	−	0.613280i	$-0.789850\pi$
$337$	2.50000	−	4.33013i	0.136184	−	0.235877i	0.926049	+	0.377403i	$0.123183\pi$
$338$	−9.00000			−0.489535
$339$		0			0
$340$	15.0000			0.813489
$341$	−15.0000			−0.812296
$342$		0			0
$343$	−13.0000			−0.701934
$344$	−24.0000			−1.29399
$345$		0			0
$346$	6.00000			0.322562
$347$	0.500000	−	0.866025i	0.0268414	−	0.0464907i	−0.852293	−	0.523065i	$-0.824788\pi$
$347$	0.500000	−	0.866025i	0.0268414	−	0.0464907i	0.879134	+	0.476575i	$0.158122\pi$
$348$		0			0
$349$	−7.50000	+	12.9904i	−0.401466	+	0.695359i	−0.993903	−	0.110257i	$-0.964832\pi$
$349$	−7.50000	+	12.9904i	−0.401466	+	0.695359i	0.592437	+	0.805617i	$0.298166\pi$
$350$	4.00000			0.213809
$351$		0			0
$352$	12.5000	−	21.6506i	0.666252	−	1.15398i
$353$	−18.5000	+	32.0429i	−0.984656	+	1.70547i	−0.341199	+	0.939991i	$0.610833\pi$
$353$	−18.5000	+	32.0429i	−0.984656	+	1.70547i	−0.643457	+	0.765482i	$0.722500\pi$
$354$		0			0
$355$	9.00000			0.477670
$356$	4.50000	−	7.79423i	0.238500	−	0.413093i
$357$		0			0
$358$	12.0000			0.634220
$359$	−6.50000	−	11.2583i	−0.343057	−	0.594192i	0.641942	−	0.766753i	$-0.278129\pi$
$359$	−6.50000	−	11.2583i	−0.343057	−	0.594192i	−0.984999	+	0.172561i	$0.944796\pi$
$360$		0			0
$361$	13.0000	+	13.8564i	0.684211	+	0.729285i
$362$	−3.50000	+	6.06218i	−0.183956	+	0.318621i
$363$		0			0
$364$	1.00000	−	1.73205i	0.0524142	−	0.0907841i
$365$	15.0000			0.785136
$366$		0			0
$367$	2.50000	+	4.33013i	0.130499	+	0.226031i	0.923869	−	0.382709i	$-0.125009\pi$
$367$	2.50000	+	4.33013i	0.130499	+	0.226031i	−0.793370	+	0.608740i	$0.791675\pi$
$368$	−8.00000			−0.417029
$369$		0			0
$370$	−9.00000	+	15.5885i	−0.467888	+	0.810405i
$371$	−0.500000	−	0.866025i	−0.0259587	−	0.0449618i
$372$		0			0
$373$	2.50000	+	4.33013i	0.129445	+	0.224205i	0.923462	−	0.383691i	$-0.125347\pi$
$373$	2.50000	+	4.33013i	0.129445	+	0.224205i	−0.794017	+	0.607896i	$0.792014\pi$
$374$	−25.0000			−1.29272
$375$		0			0
$376$	−4.50000	−	7.79423i	−0.232070	−	0.401957i
$377$	−1.00000	−	1.73205i	−0.0515026	−	0.0892052i
$378$		0			0
$379$	−20.0000			−1.02733			−0.513665	−	0.857991i	$-0.671713\pi$
$379$	−20.0000			−1.02733			−0.513665	+	0.857991i	$0.671713\pi$
$380$	10.5000	−	7.79423i	0.538639	−	0.399835i
$381$		0			0
$382$	−1.50000	+	2.59808i	−0.0767467	+	0.132929i
$383$	−11.5000	+	19.9186i	−0.587623	+	1.01779i	0.406920	+	0.913464i	$0.366603\pi$
$383$	−11.5000	+	19.9186i	−0.587623	+	1.01779i	−0.994543	+	0.104328i	$0.966731\pi$
$384$		0			0
$385$	15.0000			0.764471
$386$	8.50000	−	14.7224i	0.432639	−	0.749352i
$387$		0			0
$388$	10.0000			0.507673
$389$	1.50000	+	2.59808i	0.0760530	+	0.131728i	0.901544	−	0.432688i	$-0.142435\pi$
$389$	1.50000	+	2.59808i	0.0760530	+	0.131728i	−0.825491	+	0.564416i	$0.809102\pi$
$390$		0			0
$391$	−20.0000	−	34.6410i	−1.01144	−	1.75187i
$392$	−9.00000	−	15.5885i	−0.454569	−	0.787336i
$393$		0			0
$394$	−2.00000			−0.100759
$395$	6.00000	−	10.3923i	0.301893	−	0.522894i
$396$		0			0
$397$	−3.50000	−	6.06218i	−0.175660	−	0.304252i	0.764730	−	0.644351i	$-0.222873\pi$
$397$	−3.50000	−	6.06218i	−0.175660	−	0.304252i	−0.940389	+	0.340099i	$0.889539\pi$
$398$	1.50000	−	2.59808i	0.0751882	−	0.130230i
$399$		0			0
$400$	2.00000	+	3.46410i	0.100000	+	0.173205i
$401$	−16.5000	+	28.5788i	−0.823971	+	1.42716i	0.0787327	+	0.996896i	$0.474913\pi$
$401$	−16.5000	+	28.5788i	−0.823971	+	1.42716i	−0.902703	+	0.430263i	$0.858421\pi$
$402$		0			0
$403$	−3.00000	−	5.19615i	−0.149441	−	0.258839i
$404$	−3.50000	−	6.06218i	−0.174132	−	0.301605i
$405$		0			0
$406$	1.00000			0.0496292
$407$	−15.0000	+	25.9808i	−0.743522	+	1.28782i
$408$		0			0
$409$	10.0000			0.494468			0.247234	−	0.968956i	$-0.420478\pi$
$409$	10.0000			0.494468			0.247234	+	0.968956i	$0.420478\pi$
$410$	13.5000	+	23.3827i	0.666717	+	1.15479i
$411$		0			0
$412$	−2.50000	−	4.33013i	−0.123166	−	0.213330i
$413$	2.50000	+	4.33013i	0.123017	+	0.213072i
$414$		0			0
$415$	−13.5000	−	23.3827i	−0.662689	−	1.14781i
$416$	10.0000			0.490290
$417$		0			0
$418$	−17.5000	+	12.9904i	−0.855953	+	0.635380i
$419$	−0.500000	−	0.866025i	−0.0244266	−	0.0423081i	0.853554	−	0.521005i	$-0.174443\pi$
$419$	−0.500000	−	0.866025i	−0.0244266	−	0.0423081i	−0.877980	+	0.478697i	$0.841109\pi$
$420$		0			0
$421$	26.0000			1.26716			0.633581	−	0.773676i	$-0.281584\pi$
$421$	26.0000			1.26716			0.633581	+	0.773676i	$0.281584\pi$
$422$	7.50000	−	12.9904i	0.365094	−	0.632362i
$423$		0			0
$424$	1.50000	−	2.59808i	0.0728464	−	0.126174i
$425$	−10.0000	+	17.3205i	−0.485071	+	0.840168i
$426$		0			0
$427$	−13.0000			−0.629114
$428$	12.0000			0.580042
$429$		0			0
$430$	12.0000	−	20.7846i	0.578691	−	1.00232i
$431$	0.500000	−	0.866025i	0.0240842	−	0.0417150i	−0.853732	−	0.520712i	$-0.825666\pi$
$431$	0.500000	−	0.866025i	0.0240842	−	0.0417150i	0.877816	+	0.478997i	$0.159000\pi$
$432$		0			0
$433$	−13.5000	+	23.3827i	−0.648769	+	1.12370i	0.334649	+	0.942343i	$0.391382\pi$
$433$	−13.5000	+	23.3827i	−0.648769	+	1.12370i	−0.983417	+	0.181357i	$0.941951\pi$
$434$	3.00000			0.144005
$435$		0			0
$436$	−2.50000	−	4.33013i	−0.119728	−	0.207375i
$437$	−32.0000	−	13.8564i	−1.53077	−	0.662842i
$438$		0			0
$439$	−16.0000			−0.763638			−0.381819	−	0.924237i	$-0.624702\pi$
$439$	−16.0000			−0.763638			−0.381819	+	0.924237i	$0.624702\pi$
$440$	22.5000	+	38.9711i	1.07265	+	1.85788i
$441$		0			0
$442$	−5.00000	−	8.66025i	−0.237826	−	0.411926i
$443$	−12.5000	−	21.6506i	−0.593893	−	1.02865i	−0.993702	−	0.112054i	$-0.964257\pi$
$443$	−12.5000	−	21.6506i	−0.593893	−	1.02865i	0.399809	−	0.916598i	$-0.369076\pi$
$444$		0			0
$445$	13.5000	+	23.3827i	0.639961	+	1.10845i
$446$	−24.0000			−1.13643
$447$		0			0
$448$	−3.50000	+	6.06218i	−0.165359	+	0.286411i
$449$	−30.0000			−1.41579			−0.707894	−	0.706319i	$-0.750354\pi$
$449$	−30.0000			−1.41579			−0.707894	+	0.706319i	$0.750354\pi$
$450$		0			0
$451$	22.5000	+	38.9711i	1.05948	+	1.83508i
$452$	8.50000	+	14.7224i	0.399806	+	0.692485i
$453$		0			0
$454$	10.5000	−	18.1865i	0.492789	−	0.853536i
$455$	3.00000	+	5.19615i	0.140642	+	0.243599i
$456$		0			0
$457$	−9.50000	+	16.4545i	−0.444391	+	0.769708i	−0.998010	−	0.0630623i	$-0.979913\pi$
$457$	−9.50000	+	16.4545i	−0.444391	+	0.769708i	0.553618	+	0.832771i	$0.313247\pi$
$458$	8.50000	+	14.7224i	0.397179	+	0.687934i
$459$		0			0
$460$	−12.0000	+	20.7846i	−0.559503	+	0.969087i
$461$	30.0000			1.39724			0.698620	−	0.715493i	$-0.253798\pi$
$461$	30.0000			1.39724			0.698620	+	0.715493i	$0.253798\pi$
$462$		0			0
$463$	0.500000	+	0.866025i	0.0232370	+	0.0402476i	0.877410	−	0.479741i	$-0.159269\pi$
$463$	0.500000	+	0.866025i	0.0232370	+	0.0402476i	−0.854173	+	0.519989i	$0.825936\pi$
$464$	0.500000	+	0.866025i	0.0232119	+	0.0402042i
$465$		0			0
$466$	−4.50000	−	7.79423i	−0.208458	−	0.361061i
$467$	−8.00000			−0.370196			−0.185098	−	0.982720i	$-0.559260\pi$
$467$	−8.00000			−0.370196			−0.185098	+	0.982720i	$0.559260\pi$
$468$		0			0
$469$	2.00000	−	3.46410i	0.0923514	−	0.159957i
$470$	9.00000			0.415139
$471$		0			0
$472$	−7.50000	+	12.9904i	−0.345215	+	0.597931i
$473$	20.0000	−	34.6410i	0.919601	−	1.59280i
$474$		0			0
$475$	2.00000	+	17.3205i	0.0917663	+	0.794719i
$476$	−5.00000			−0.229175
$477$		0			0
$478$	7.50000	+	12.9904i	0.343042	+	0.594166i
$479$	−2.50000	−	4.33013i	−0.114228	−	0.197849i	0.803243	−	0.595652i	$-0.203106\pi$
$479$	−2.50000	−	4.33013i	−0.114228	−	0.197849i	−0.917471	+	0.397803i	$0.869773\pi$
$480$		0			0
$481$	−12.0000			−0.547153
$482$	0.500000	+	0.866025i	0.0227744	+	0.0394464i
$483$		0			0
$484$	7.00000	+	12.1244i	0.318182	+	0.551107i
$485$	−15.0000	+	25.9808i	−0.681115	+	1.17973i
$486$		0			0
$487$	32.0000			1.45006			0.725029	−	0.688718i	$-0.241826\pi$
$487$	32.0000			1.45006			0.725029	+	0.688718i	$0.241826\pi$
$488$	−19.5000	−	33.7750i	−0.882724	−	1.52892i
$489$		0			0
$490$	18.0000			0.813157
$491$	−13.5000	+	23.3827i	−0.609246	+	1.05525i	0.382118	+	0.924113i	$0.375195\pi$
$491$	−13.5000	+	23.3827i	−0.609246	+	1.05525i	−0.991365	+	0.131132i	$0.958139\pi$
$492$		0			0
$493$	−2.50000	+	4.33013i	−0.112594	+	0.195019i
$494$	−8.00000	−	3.46410i	−0.359937	−	0.155857i
$495$		0			0
$496$	1.50000	+	2.59808i	0.0673520	+	0.116657i
$497$	−3.00000			−0.134568
$498$		0			0
$499$	−14.5000	+	25.1147i	−0.649109	+	1.12429i	0.334227	+	0.942493i	$0.391525\pi$
$499$	−14.5000	+	25.1147i	−0.649109	+	1.12429i	−0.983336	+	0.181797i	$0.941809\pi$
$500$	−3.00000			−0.134164
$501$		0			0
$502$	2.50000	−	4.33013i	0.111580	−	0.193263i
$503$	14.5000	−	25.1147i	0.646523	−	1.11981i	−0.337424	−	0.941353i	$-0.609556\pi$
$503$	14.5000	−	25.1147i	0.646523	−	1.11981i	0.983948	−	0.178458i	$-0.0571109\pi$
$504$		0			0
$505$	21.0000			0.934488
$506$	20.0000	−	34.6410i	0.889108	−	1.53998i
$507$		0			0
$508$	6.50000	−	11.2583i	0.288391	−	0.499508i
$509$	−18.0000			−0.797836			−0.398918	−	0.916987i	$-0.630614\pi$
$509$	−18.0000			−0.797836			−0.398918	+	0.916987i	$0.630614\pi$
$510$		0			0
$511$	−5.00000			−0.221187
$512$	−11.0000			−0.486136
$513$		0			0
$514$	10.0000			0.441081
$515$	15.0000			0.660979
$516$		0			0
$517$	15.0000			0.659699
$518$	3.00000	−	5.19615i	0.131812	−	0.228306i
$519$		0			0
$520$	−9.00000	+	15.5885i	−0.394676	+	0.683599i
$521$	42.0000			1.84005			0.920027	−	0.391856i	$-0.128167\pi$
$521$	42.0000			1.84005			0.920027	+	0.391856i	$0.128167\pi$
$522$		0			0
$523$	−14.5000	+	25.1147i	−0.634041	+	1.09819i	0.352677	+	0.935745i	$0.385272\pi$
$523$	−14.5000	+	25.1147i	−0.634041	+	1.09819i	−0.986718	+	0.162446i	$0.948062\pi$
$524$	3.50000	−	6.06218i	0.152898	−	0.264827i
$525$		0			0
$526$	8.00000			0.348817
$527$	−7.50000	+	12.9904i	−0.326705	+	0.565870i
$528$		0			0
$529$	41.0000			1.78261
$530$	1.50000	+	2.59808i	0.0651558	+	0.112853i
$531$		0			0
$532$	−3.50000	+	2.59808i	−0.151744	+	0.112641i
$533$	−9.00000	+	15.5885i	−0.389833	+	0.675211i
$534$		0			0
$535$	−18.0000	+	31.1769i	−0.778208	+	1.34790i
$536$	12.0000			0.518321
$537$		0			0
$538$	−12.5000	−	21.6506i	−0.538913	−	0.933425i
$539$	30.0000			1.29219
$540$		0			0
$541$	−1.50000	+	2.59808i	−0.0644900	+	0.111700i	−0.896468	−	0.443109i	$-0.853875\pi$
$541$	−1.50000	+	2.59808i	−0.0644900	+	0.111700i	0.831978	+	0.554809i	$0.187209\pi$
$542$	−13.5000	−	23.3827i	−0.579875	−	1.00437i
$543$		0			0
$544$	−12.5000	−	21.6506i	−0.535933	−	0.928263i
$545$	15.0000			0.642529
$546$		0			0
$547$	2.50000	+	4.33013i	0.106892	+	0.185143i	0.914510	−	0.404564i	$-0.132577\pi$
$547$	2.50000	+	4.33013i	0.106892	+	0.185143i	−0.807617	+	0.589707i	$0.799243\pi$
$548$	−1.50000	−	2.59808i	−0.0640768	−	0.110984i
$549$		0			0
$550$	−20.0000			−0.852803
$551$	0.500000	+	4.33013i	0.0213007	+	0.184470i
$552$		0			0
$553$	−2.00000	+	3.46410i	−0.0850487	+	0.147309i
$554$	−3.50000	+	6.06218i	−0.148701	+	0.257557i
$555$		0			0
$556$	−20.0000			−0.848189
$557$	−6.50000	+	11.2583i	−0.275414	+	0.477031i	−0.970239	−	0.242147i	$-0.922148\pi$
$557$	−6.50000	+	11.2583i	−0.275414	+	0.477031i	0.694826	+	0.719178i	$0.255482\pi$
$558$		0			0
$559$	16.0000			0.676728
$560$	−1.50000	−	2.59808i	−0.0633866	−	0.109789i
$561$		0			0
$562$	5.50000	+	9.52628i	0.232003	+	0.401842i
$563$	13.5000	+	23.3827i	0.568957	+	0.985463i	0.996669	+	0.0815478i	$0.0259863\pi$
$563$	13.5000	+	23.3827i	0.568957	+	0.985463i	−0.427712	+	0.903915i	$0.640680\pi$
$564$		0			0
$565$	−51.0000			−2.14559
$566$	−14.5000	+	25.1147i	−0.609480	+	1.05565i
$567$		0			0
$568$	−4.50000	−	7.79423i	−0.188816	−	0.327039i
$569$	−4.50000	+	7.79423i	−0.188650	+	0.326751i	−0.944800	−	0.327647i	$-0.893744\pi$
$569$	−4.50000	+	7.79423i	−0.188650	+	0.326751i	0.756151	+	0.654398i	$0.227078\pi$
$570$		0			0
$571$	10.5000	+	18.1865i	0.439411	+	0.761083i	0.997644	−	0.0686016i	$-0.0218537\pi$
$571$	10.5000	+	18.1865i	0.439411	+	0.761083i	−0.558233	+	0.829684i	$0.688520\pi$
$572$	−5.00000	+	8.66025i	−0.209061	+	0.362103i
$573$		0			0
$574$	−4.50000	−	7.79423i	−0.187826	−	0.325325i
$575$	−16.0000	−	27.7128i	−0.667246	−	1.15570i
$576$		0			0
$577$	−42.0000			−1.74848			−0.874241	−	0.485491i	$-0.838641\pi$
$577$	−42.0000			−1.74848			−0.874241	+	0.485491i	$0.838641\pi$
$578$	−4.00000	+	6.92820i	−0.166378	+	0.288175i
$579$		0			0
$580$	3.00000			0.124568
$581$	4.50000	+	7.79423i	0.186691	+	0.323359i
$582$		0			0
$583$	2.50000	+	4.33013i	0.103539	+	0.179336i
$584$	−7.50000	−	12.9904i	−0.310352	−	0.537546i
$585$		0			0
$586$	−2.50000	−	4.33013i	−0.103274	−	0.178876i
$587$	4.00000			0.165098			0.0825488	−	0.996587i	$-0.473694\pi$
$587$	4.00000			0.165098			0.0825488	+	0.996587i	$0.473694\pi$
$588$		0			0
$589$	1.50000	+	12.9904i	0.0618064	+	0.535259i
$590$	−7.50000	−	12.9904i	−0.308770	−	0.534806i
$591$		0			0
$592$	6.00000			0.246598
$593$	−4.50000	+	7.79423i	−0.184793	+	0.320071i	−0.943507	−	0.331353i	$-0.892495\pi$
$593$	−4.50000	+	7.79423i	−0.184793	+	0.320071i	0.758714	+	0.651424i	$0.225828\pi$
$594$		0			0
$595$	7.50000	−	12.9904i	0.307470	−	0.532554i
$596$	−7.50000	+	12.9904i	−0.307212	+	0.532107i
$597$		0			0
$598$	16.0000			0.654289
$599$	12.0000			0.490307			0.245153	−	0.969484i	$-0.421162\pi$
$599$	12.0000			0.490307			0.245153	+	0.969484i	$0.421162\pi$
$600$		0			0
$601$	−1.50000	+	2.59808i	−0.0611863	+	0.105978i	−0.894996	−	0.446074i	$-0.852822\pi$
$601$	−1.50000	+	2.59808i	−0.0611863	+	0.105978i	0.833810	+	0.552052i	$0.186155\pi$
$602$	−4.00000	+	6.92820i	−0.163028	+	0.282372i
$603$		0			0
$604$	−7.50000	+	12.9904i	−0.305171	+	0.528571i
$605$	−42.0000			−1.70754
$606$		0			0
$607$	−11.5000	−	19.9186i	−0.466771	−	0.808470i	0.532509	−	0.846424i	$-0.321249\pi$
$607$	−11.5000	−	19.9186i	−0.466771	−	0.808470i	−0.999279	+	0.0379540i	$0.987916\pi$
$608$	−20.0000	−	8.66025i	−0.811107	−	0.351220i
$609$		0			0
$610$	39.0000			1.57906
$611$	3.00000	+	5.19615i	0.121367	+	0.210214i
$612$		0			0
$613$	4.50000	+	7.79423i	0.181753	+	0.314806i	0.942478	−	0.334269i	$-0.108489\pi$
$613$	4.50000	+	7.79423i	0.181753	+	0.314806i	−0.760724	+	0.649075i	$0.775156\pi$
$614$	16.5000	+	28.5788i	0.665886	+	1.15335i
$615$		0			0
$616$	−7.50000	−	12.9904i	−0.302184	−	0.523397i
$617$	−30.0000			−1.20775			−0.603877	−	0.797077i	$-0.706378\pi$
$617$	−30.0000			−1.20775			−0.603877	+	0.797077i	$0.706378\pi$
$618$		0			0
$619$	5.50000	−	9.52628i	0.221064	−	0.382893i	−0.734068	−	0.679076i	$-0.762380\pi$
$619$	5.50000	−	9.52628i	0.221064	−	0.382893i	0.955131	+	0.296183i	$0.0957138\pi$
$620$	9.00000			0.361449
$621$		0			0
$622$	−10.5000	−	18.1865i	−0.421012	−	0.729214i
$623$	−4.50000	−	7.79423i	−0.180289	−	0.312269i
$624$		0			0
$625$	14.5000	−	25.1147i	0.580000	−	1.00459i
$626$	8.50000	+	14.7224i	0.339728	+	0.588427i
$627$		0			0
$628$	1.50000	−	2.59808i	0.0598565	−	0.103675i
$629$	15.0000	+	25.9808i	0.598089	+	1.03592i
$630$		0			0
$631$	19.5000	−	33.7750i	0.776283	−	1.34456i	−0.157788	−	0.987473i	$-0.550436\pi$
$631$	19.5000	−	33.7750i	0.776283	−	1.34456i	0.934071	−	0.357088i	$-0.116230\pi$
$632$	−12.0000			−0.477334
$633$		0			0
$634$	−8.50000	−	14.7224i	−0.337578	−	0.584702i
$635$	19.5000	+	33.7750i	0.773834	+	1.34032i
$636$		0			0
$637$	6.00000	+	10.3923i	0.237729	+	0.411758i
$638$	−5.00000			−0.197952
$639$		0			0
$640$	−4.50000	+	7.79423i	−0.177878	+	0.308094i
$641$	30.0000			1.18493			0.592464	−	0.805597i	$-0.298155\pi$
$641$	30.0000			1.18493			0.592464	+	0.805597i	$0.298155\pi$
$642$		0			0
$643$	3.50000	−	6.06218i	0.138027	−	0.239069i	−0.788723	−	0.614749i	$-0.789257\pi$
$643$	3.50000	−	6.06218i	0.138027	−	0.239069i	0.926750	+	0.375680i	$0.122591\pi$
$644$	4.00000	−	6.92820i	0.157622	−	0.273009i
$645$		0			0
$646$	2.50000	+	21.6506i	0.0983612	+	0.851833i
$647$	−48.0000			−1.88707			−0.943537	−	0.331266i	$-0.892524\pi$
$647$	−48.0000			−1.88707			−0.943537	+	0.331266i	$0.892524\pi$
$648$		0			0
$649$	−12.5000	−	21.6506i	−0.490668	−	0.849862i
$650$	−4.00000	−	6.92820i	−0.156893	−	0.271746i
$651$		0			0
$652$	−12.0000			−0.469956
$653$	−4.50000	−	7.79423i	−0.176099	−	0.305012i	0.764442	−	0.644692i	$-0.223014\pi$
$653$	−4.50000	−	7.79423i	−0.176099	−	0.305012i	−0.940541	+	0.339680i	$0.889681\pi$
$654$		0			0
$655$	10.5000	+	18.1865i	0.410269	+	0.710607i
$656$	4.50000	−	7.79423i	0.175695	−	0.304314i
$657$		0			0
$658$	−3.00000			−0.116952
$659$	11.5000	+	19.9186i	0.447976	+	0.775918i	0.998254	−	0.0590638i	$-0.0188115\pi$
$659$	11.5000	+	19.9186i	0.447976	+	0.775918i	−0.550278	+	0.834982i	$0.685478\pi$
$660$		0			0
$661$	−30.0000			−1.16686			−0.583432	−	0.812162i	$-0.698291\pi$
$661$	−30.0000			−1.16686			−0.583432	+	0.812162i	$0.698291\pi$
$662$	9.50000	−	16.4545i	0.369228	−	0.639522i
$663$		0			0
$664$	−13.5000	+	23.3827i	−0.523902	+	0.907424i
$665$	−1.50000	−	12.9904i	−0.0581675	−	0.503745i
$666$		0			0
$667$	−4.00000	−	6.92820i	−0.154881	−	0.268261i
$668$	−4.00000			−0.154765
$669$		0			0
$670$	−6.00000	+	10.3923i	−0.231800	+	0.401490i
$671$	65.0000			2.50930
$672$		0			0
$673$	4.50000	−	7.79423i	0.173462	−	0.300445i	−0.766166	−	0.642643i	$-0.777838\pi$
$673$	4.50000	−	7.79423i	0.173462	−	0.300445i	0.939628	+	0.342198i	$0.111171\pi$
$674$	2.50000	−	4.33013i	0.0962964	−	0.166790i
$675$		0			0
$676$	9.00000			0.346154
$677$	−18.5000	+	32.0429i	−0.711013	+	1.23151i	0.253465	+	0.967345i	$0.418430\pi$
$677$	−18.5000	+	32.0429i	−0.711013	+	1.23151i	−0.964477	+	0.264166i	$0.914903\pi$
$678$		0			0
$679$	5.00000	−	8.66025i	0.191882	−	0.332350i
$680$	45.0000			1.72567
$681$		0			0
$682$	−15.0000			−0.574380
$683$	−36.0000			−1.37750			−0.688751	−	0.724998i	$-0.741841\pi$
$683$	−36.0000			−1.37750			−0.688751	+	0.724998i	$0.741841\pi$
$684$		0			0
$685$	9.00000			0.343872
$686$	−13.0000			−0.496342
$687$		0			0
$688$	−8.00000			−0.304997
$689$	−1.00000	+	1.73205i	−0.0380970	+	0.0659859i
$690$		0			0
$691$	9.50000	−	16.4545i	0.361397	−	0.625958i	−0.626794	−	0.779185i	$-0.715633\pi$
$691$	9.50000	−	16.4545i	0.361397	−	0.625958i	0.988191	+	0.153227i	$0.0489666\pi$
$692$	−6.00000			−0.228086
$693$		0			0
$694$	0.500000	−	0.866025i	0.0189797	−	0.0328739i
$695$	30.0000	−	51.9615i	1.13796	−	1.97101i
$696$		0			0
$697$	45.0000			1.70450
$698$	−7.50000	+	12.9904i	−0.283879	+	0.491693i
$699$		0			0
$700$	−4.00000			−0.151186
$701$	9.50000	+	16.4545i	0.358810	+	0.621477i	0.987762	−	0.155967i	$-0.0498493\pi$
$701$	9.50000	+	16.4545i	0.358810	+	0.621477i	−0.628952	+	0.777444i	$0.716516\pi$
$702$		0			0
$703$	24.0000	+	10.3923i	0.905177	+	0.391953i
$704$	17.5000	−	30.3109i	0.659556	−	1.14238i
$705$		0			0
$706$	−18.5000	+	32.0429i	−0.696257	+	1.20595i
$707$	−7.00000			−0.263262
$708$		0			0
$709$	−19.5000	−	33.7750i	−0.732338	−	1.26845i	−0.955882	−	0.293752i	$-0.905096\pi$
$709$	−19.5000	−	33.7750i	−0.732338	−	1.26845i	0.223544	−	0.974694i	$-0.428237\pi$
$710$	9.00000			0.337764
$711$		0			0
$712$	13.5000	−	23.3827i	0.505934	−	0.876303i
$713$	−12.0000	−	20.7846i	−0.449404	−	0.778390i
$714$		0			0
$715$	−15.0000	−	25.9808i	−0.560968	−	0.971625i
$716$	−12.0000			−0.448461
$717$		0			0
$718$	−6.50000	−	11.2583i	−0.242578	−	0.420157i
$719$	1.50000	+	2.59808i	0.0559406	+	0.0968919i	0.892640	−	0.450771i	$-0.148851\pi$
$719$	1.50000	+	2.59808i	0.0559406	+	0.0968919i	−0.836699	+	0.547663i	$0.815518\pi$
$720$		0			0
$721$	−5.00000			−0.186210
$722$	13.0000	+	13.8564i	0.483810	+	0.515682i
$723$		0			0
$724$	3.50000	−	6.06218i	0.130076	−	0.225299i
$725$	−2.00000	+	3.46410i	−0.0742781	+	0.128654i
$726$		0			0
$727$	−8.00000			−0.296704			−0.148352	−	0.988935i	$-0.547397\pi$
$727$	−8.00000			−0.296704			−0.148352	+	0.988935i	$0.547397\pi$
$728$	3.00000	−	5.19615i	0.111187	−	0.192582i
$729$		0			0
$730$	15.0000			0.555175
$731$	−20.0000	−	34.6410i	−0.739727	−	1.28124i
$732$		0			0
$733$	−9.50000	−	16.4545i	−0.350891	−	0.607760i	0.635515	−	0.772088i	$-0.280788\pi$
$733$	−9.50000	−	16.4545i	−0.350891	−	0.607760i	−0.986406	+	0.164328i	$0.947454\pi$
$734$	2.50000	+	4.33013i	0.0922767	+	0.159828i
$735$		0			0
$736$	40.0000			1.47442
$737$	−10.0000	+	17.3205i	−0.368355	+	0.638009i
$738$		0			0
$739$	−9.50000	−	16.4545i	−0.349463	−	0.605288i	0.636691	−	0.771119i	$-0.280303\pi$
$739$	−9.50000	−	16.4545i	−0.349463	−	0.605288i	−0.986154	+	0.165831i	$0.946969\pi$
$740$	9.00000	−	15.5885i	0.330847	−	0.573043i
$741$		0			0
$742$	−0.500000	−	0.866025i	−0.0183556	−	0.0317928i
$743$	20.5000	−	35.5070i	0.752072	−	1.30263i	−0.194745	−	0.980854i	$-0.562388\pi$
$743$	20.5000	−	35.5070i	0.752072	−	1.30263i	0.946817	−	0.321773i	$-0.104279\pi$
$744$		0			0
$745$	−22.5000	−	38.9711i	−0.824336	−	1.42779i
$746$	2.50000	+	4.33013i	0.0915315	+	0.158537i
$747$		0			0
$748$	25.0000			0.914091
$749$	6.00000	−	10.3923i	0.219235	−	0.379727i
$750$		0			0
$751$	24.0000			0.875772			0.437886	−	0.899030i	$-0.355727\pi$
$751$	24.0000			0.875772			0.437886	+	0.899030i	$0.355727\pi$
$752$	−1.50000	−	2.59808i	−0.0546994	−	0.0947421i
$753$		0			0
$754$	−1.00000	−	1.73205i	−0.0364179	−	0.0630776i
$755$	−22.5000	−	38.9711i	−0.818859	−	1.41831i
$756$		0			0
$757$	−15.5000	−	26.8468i	−0.563357	−	0.975763i	−0.997200	−	0.0747748i	$-0.976176\pi$
$757$	−15.5000	−	26.8468i	−0.563357	−	0.975763i	0.433843	−	0.900988i	$-0.357157\pi$
$758$	−20.0000			−0.726433
$759$		0			0
$760$	31.5000	−	23.3827i	1.14263	−	0.848179i
$761$	−4.50000	−	7.79423i	−0.163125	−	0.282541i	0.772863	−	0.634573i	$-0.218824\pi$
$761$	−4.50000	−	7.79423i	−0.163125	−	0.282541i	−0.935988	+	0.352032i	$0.885491\pi$
$762$		0			0
$763$	−5.00000			−0.181012
$764$	1.50000	−	2.59808i	0.0542681	−	0.0939951i
$765$		0			0
$766$	−11.5000	+	19.9186i	−0.415512	+	0.719688i
$767$	5.00000	−	8.66025i	0.180540	−	0.312704i
$768$		0			0
$769$	14.0000			0.504853			0.252426	−	0.967616i	$-0.418771\pi$
$769$	14.0000			0.504853			0.252426	+	0.967616i	$0.418771\pi$
$770$	15.0000			0.540562
$771$		0			0
$772$	−8.50000	+	14.7224i	−0.305922	+	0.529872i
$773$	−8.50000	+	14.7224i	−0.305724	+	0.529529i	−0.977422	−	0.211296i	$-0.932232\pi$
$773$	−8.50000	+	14.7224i	−0.305724	+	0.529529i	0.671698	+	0.740825i	$0.265565\pi$
$774$		0			0
$775$	−6.00000	+	10.3923i	−0.215526	+	0.373303i
$776$	30.0000			1.07694
$777$		0			0
$778$	1.50000	+	2.59808i	0.0537776	+	0.0931455i
$779$	31.5000	−	23.3827i	1.12860	−	0.837772i
$780$		0			0
$781$	15.0000			0.536742
$782$	−20.0000	−	34.6410i	−0.715199	−	1.23876i
$783$		0			0
$784$	−3.00000	−	5.19615i	−0.107143	−	0.185577i
$785$	4.50000	+	7.79423i	0.160612	+	0.278188i
$786$		0			0
$787$	−3.50000	−	6.06218i	−0.124762	−	0.216093i	0.796878	−	0.604140i	$-0.206483\pi$
$787$	−3.50000	−	6.06218i	−0.124762	−	0.216093i	−0.921640	+	0.388047i	$0.873150\pi$
$788$	2.00000			0.0712470
$789$		0			0
$790$	6.00000	−	10.3923i	0.213470	−	0.369742i
$791$	17.0000			0.604450
$792$		0			0
$793$	13.0000	+	22.5167i	0.461644	+	0.799590i
$794$	−3.50000	−	6.06218i	−0.124210	−	0.215139i
$795$		0			0
$796$	−1.50000	+	2.59808i	−0.0531661	+	0.0920864i
$797$	5.50000	+	9.52628i	0.194820	+	0.337438i	0.946841	−	0.321700i	$-0.104254\pi$
$797$	5.50000	+	9.52628i	0.194820	+	0.337438i	−0.752022	+	0.659139i	$0.770921\pi$
$798$		0			0
$799$	7.50000	−	12.9904i	0.265331	−	0.459567i
$800$	−10.0000	−	17.3205i	−0.353553	−	0.612372i
$801$		0			0
$802$	−16.5000	+	28.5788i	−0.582635	+	1.00915i
$803$	25.0000			0.882231
$804$		0			0
$805$	12.0000	+	20.7846i	0.422944	+	0.732561i
$806$	−3.00000	−	5.19615i	−0.105670	−	0.183027i
$807$		0			0
$808$	−10.5000	−	18.1865i	−0.369389	−	0.639800i
$809$	2.00000			0.0703163			0.0351581	−	0.999382i	$-0.488807\pi$
$809$	2.00000			0.0703163			0.0351581	+	0.999382i	$0.488807\pi$
$810$		0			0
$811$	−12.5000	+	21.6506i	−0.438934	+	0.760257i	−0.997608	−	0.0691313i	$-0.977977\pi$
$811$	−12.5000	+	21.6506i	−0.438934	+	0.760257i	0.558673	+	0.829388i	$0.311311\pi$
$812$	−1.00000			−0.0350931
$813$		0			0
$814$	−15.0000	+	25.9808i	−0.525750	+	0.910625i
$815$	18.0000	−	31.1769i	0.630512	−	1.09208i
$816$		0			0
$817$	−32.0000	−	13.8564i	−1.11954	−	0.484774i
$818$	10.0000			0.349642
$819$		0			0
$820$	−13.5000	−	23.3827i	−0.471440	−	0.816559i
$821$	7.50000	+	12.9904i	0.261752	+	0.453367i	0.966708	−	0.255884i	$-0.0823665\pi$
$821$	7.50000	+	12.9904i	0.261752	+	0.453367i	−0.704956	+	0.709251i	$0.749033\pi$
$822$		0			0
$823$	40.0000			1.39431			0.697156	−	0.716919i	$-0.254448\pi$
$823$	40.0000			1.39431			0.697156	+	0.716919i	$0.254448\pi$
$824$	−7.50000	−	12.9904i	−0.261275	−	0.452541i
$825$		0			0
$826$	2.50000	+	4.33013i	0.0869861	+	0.150664i
$827$	4.50000	−	7.79423i	0.156480	−	0.271032i	−0.777117	−	0.629356i	$-0.783319\pi$
$827$	4.50000	−	7.79423i	0.156480	−	0.271032i	0.933597	+	0.358325i	$0.116652\pi$
$828$		0			0
$829$	−38.0000			−1.31979			−0.659897	−	0.751356i	$-0.729400\pi$
$829$	−38.0000			−1.31979			−0.659897	+	0.751356i	$0.729400\pi$
$830$	−13.5000	−	23.3827i	−0.468592	−	0.811625i
$831$		0			0
$832$	14.0000			0.485363
$833$	15.0000	−	25.9808i	0.519719	−	0.900180i
$834$		0			0
$835$	6.00000	−	10.3923i	0.207639	−	0.359641i
$836$	17.5000	−	12.9904i	0.605250	−	0.449282i
$837$		0			0
$838$	−0.500000	−	0.866025i	−0.0172722	−	0.0299164i
$839$	24.0000			0.828572			0.414286	−	0.910147i	$-0.364031\pi$
$839$	24.0000			0.828572			0.414286	+	0.910147i	$0.364031\pi$
$840$		0			0
$841$	14.0000	−	24.2487i	0.482759	−	0.836162i
$842$	26.0000			0.896019
$843$		0			0
$844$	−7.50000	+	12.9904i	−0.258161	+	0.447147i
$845$	−13.5000	+	23.3827i	−0.464414	+	0.804389i
$846$		0			0
$847$	14.0000			0.481046
$848$	0.500000	−	0.866025i	0.0171701	−	0.0297394i
$849$		0			0
$850$	−10.0000	+	17.3205i	−0.342997	+	0.594089i
$851$	−48.0000			−1.64542
$852$		0			0
$853$	−38.0000			−1.30110			−0.650548	−	0.759465i	$-0.725461\pi$
$853$	−38.0000			−1.30110			−0.650548	+	0.759465i	$0.725461\pi$
$854$	−13.0000			−0.444851
$855$		0			0
$856$	36.0000			1.23045
$857$	−14.0000			−0.478231			−0.239115	−	0.970991i	$-0.576857\pi$
$857$	−14.0000			−0.478231			−0.239115	+	0.970991i	$0.576857\pi$
$858$		0			0
$859$	−12.0000			−0.409435			−0.204717	−	0.978821i	$-0.565628\pi$
$859$	−12.0000			−0.409435			−0.204717	+	0.978821i	$0.565628\pi$
$860$	−12.0000	+	20.7846i	−0.409197	+	0.708749i
$861$		0			0
$862$	0.500000	−	0.866025i	0.0170301	−	0.0294969i
$863$	16.0000			0.544646			0.272323	−	0.962206i	$-0.412208\pi$
$863$	16.0000			0.544646			0.272323	+	0.962206i	$0.412208\pi$
$864$		0			0
$865$	9.00000	−	15.5885i	0.306009	−	0.530023i
$866$	−13.5000	+	23.3827i	−0.458749	+	0.794576i
$867$		0			0
$868$	−3.00000			−0.101827
$869$	10.0000	−	17.3205i	0.339227	−	0.587558i
$870$		0			0
$871$	−8.00000			−0.271070
$872$	−7.50000	−	12.9904i	−0.253982	−	0.439910i
$873$		0			0
$874$	−32.0000	−	13.8564i	−1.08242	−	0.468700i
$875$	−1.50000	+	2.59808i	−0.0507093	+	0.0878310i
$876$		0			0
$877$	−19.5000	+	33.7750i	−0.658468	+	1.14050i	0.322544	+	0.946554i	$0.395462\pi$
$877$	−19.5000	+	33.7750i	−0.658468	+	1.14050i	−0.981012	+	0.193946i	$0.937871\pi$
$878$	−16.0000			−0.539974
$879$		0			0
$880$	7.50000	+	12.9904i	0.252825	+	0.437906i
$881$	2.00000			0.0673817			0.0336909	−	0.999432i	$-0.489274\pi$
$881$	2.00000			0.0673817			0.0336909	+	0.999432i	$0.489274\pi$
$882$		0			0
$883$	−14.5000	+	25.1147i	−0.487964	+	0.845178i	−0.999904	−	0.0138428i	$-0.995594\pi$
$883$	−14.5000	+	25.1147i	−0.487964	+	0.845178i	0.511940	+	0.859021i	$0.328927\pi$
$884$	5.00000	+	8.66025i	0.168168	+	0.291276i
$885$		0			0
$886$	−12.5000	−	21.6506i	−0.419946	−	0.727367i
$887$	24.0000			0.805841			0.402921	−	0.915235i	$-0.367995\pi$
$887$	24.0000			0.805841			0.402921	+	0.915235i	$0.367995\pi$
$888$		0			0
$889$	−6.50000	−	11.2583i	−0.218003	−	0.377592i
$890$	13.5000	+	23.3827i	0.452521	+	0.783789i
$891$		0			0
$892$	24.0000			0.803579
$893$	−1.50000	−	12.9904i	−0.0501956	−	0.434707i
$894$		0			0
$895$	18.0000	−	31.1769i	0.601674	−	1.04213i
$896$	1.50000	−	2.59808i	0.0501115	−	0.0867956i
$897$		0			0
$898$	−30.0000			−1.00111
$899$	−1.50000	+	2.59808i	−0.0500278	+	0.0866507i
$900$		0			0
$901$	5.00000			0.166574
$902$	22.5000	+	38.9711i	0.749168	+	1.29760i
$903$		0			0
$904$	25.5000	+	44.1673i	0.848117	+	1.46898i
$905$	10.5000	+	18.1865i	0.349032	+	0.604541i
$906$		0			0
$907$	16.0000			0.531271			0.265636	−	0.964073i	$-0.414418\pi$
$907$	16.0000			0.531271			0.265636	+	0.964073i	$0.414418\pi$
$908$	−10.5000	+	18.1865i	−0.348455	+	0.603541i
$909$		0			0
$910$	3.00000	+	5.19615i	0.0994490	+	0.172251i
$911$	22.5000	−	38.9711i	0.745458	−	1.29117i	−0.204522	−	0.978862i	$-0.565564\pi$
$911$	22.5000	−	38.9711i	0.745458	−	1.29117i	0.949980	−	0.312310i	$-0.101103\pi$
$912$		0			0
$913$	−22.5000	−	38.9711i	−0.744641	−	1.28976i
$914$	−9.50000	+	16.4545i	−0.314232	+	0.544266i
$915$		0			0
$916$	−8.50000	−	14.7224i	−0.280848	−	0.486443i
$917$	−3.50000	−	6.06218i	−0.115580	−	0.200191i
$918$		0			0
$919$	20.0000			0.659739			0.329870	−	0.944027i	$-0.392995\pi$
$919$	20.0000			0.659739			0.329870	+	0.944027i	$0.392995\pi$
$920$	−36.0000	+	62.3538i	−1.18688	+	2.05574i
$921$		0			0
$922$	30.0000			0.987997
$923$	3.00000	+	5.19615i	0.0987462	+	0.171033i
$924$		0			0
$925$	12.0000	+	20.7846i	0.394558	+	0.683394i
$926$	0.500000	+	0.866025i	0.0164310	+	0.0284594i
$927$		0			0
$928$	−2.50000	−	4.33013i	−0.0820665	−	0.142143i
$929$	−50.0000			−1.64045			−0.820223	−	0.572043i	$-0.806151\pi$
$929$	−50.0000			−1.64045			−0.820223	+	0.572043i	$0.806151\pi$
$930$		0			0
$931$	−3.00000	−	25.9808i	−0.0983210	−	0.851485i
$932$	4.50000	+	7.79423i	0.147402	+	0.255308i
$933$		0			0
$934$	−8.00000			−0.261768
$935$	−37.5000	+	64.9519i	−1.22638	+	2.12415i
$936$		0			0
$937$	4.50000	−	7.79423i	0.147009	−	0.254626i	−0.783112	−	0.621881i	$-0.786369\pi$
$937$	4.50000	−	7.79423i	0.147009	−	0.254626i	0.930121	+	0.367254i	$0.119702\pi$
$938$	2.00000	−	3.46410i	0.0653023	−	0.113107i
$939$		0			0
$940$	−9.00000			−0.293548
$941$	−18.0000			−0.586783			−0.293392	−	0.955992i	$-0.594784\pi$
$941$	−18.0000			−0.586783			−0.293392	+	0.955992i	$0.594784\pi$
$942$		0			0
$943$	−36.0000	+	62.3538i	−1.17232	+	2.03052i
$944$	−2.50000	+	4.33013i	−0.0813681	+	0.140934i
$945$		0			0
$946$	20.0000	−	34.6410i	0.650256	−	1.12628i
$947$	−28.0000			−0.909878			−0.454939	−	0.890523i	$-0.650339\pi$
$947$	−28.0000			−0.909878			−0.454939	+	0.890523i	$0.650339\pi$
$948$		0			0
$949$	5.00000	+	8.66025i	0.162307	+	0.281124i
$950$	2.00000	+	17.3205i	0.0648886	+	0.561951i
$951$		0			0
$952$	−15.0000			−0.486153
$953$	11.5000	+	19.9186i	0.372522	+	0.645226i	0.989953	−	0.141399i	$-0.0451599\pi$
$953$	11.5000	+	19.9186i	0.372522	+	0.645226i	−0.617431	+	0.786625i	$0.711827\pi$
$954$		0			0
$955$	4.50000	+	7.79423i	0.145617	+	0.252215i
$956$	−7.50000	−	12.9904i	−0.242567	−	0.420139i
$957$		0			0
$958$	−2.50000	−	4.33013i	−0.0807713	−	0.139900i
$959$	−3.00000			−0.0968751
$960$		0			0
$961$	11.0000	−	19.0526i	0.354839	−	0.614599i
$962$	−12.0000			−0.386896
$963$		0			0
$964$	−0.500000	−	0.866025i	−0.0161039	−	0.0278928i
$965$	−25.5000	−	44.1673i	−0.820874	−	1.42180i
$966$		0			0
$967$	−18.5000	+	32.0429i	−0.594920	+	1.03043i	0.398638	+	0.917108i	$0.369483\pi$
$967$	−18.5000	+	32.0429i	−0.594920	+	1.03043i	−0.993558	+	0.113323i	$0.963850\pi$
$968$	21.0000	+	36.3731i	0.674966	+	1.16907i
$969$		0			0
$970$	−15.0000	+	25.9808i	−0.481621	+	0.834192i
$971$	−10.5000	−	18.1865i	−0.336961	−	0.583634i	0.646899	−	0.762576i	$-0.276066\pi$
$971$	−10.5000	−	18.1865i	−0.336961	−	0.583634i	−0.983860	+	0.178942i	$0.942732\pi$
$972$		0			0
$973$	−10.0000	+	17.3205i	−0.320585	+	0.555270i
$974$	32.0000			1.02535
$975$		0			0
$976$	−6.50000	−	11.2583i	−0.208060	−	0.360370i
$977$	−10.5000	−	18.1865i	−0.335925	−	0.581839i	0.647737	−	0.761864i	$-0.275715\pi$
$977$	−10.5000	−	18.1865i	−0.335925	−	0.581839i	−0.983662	+	0.180025i	$0.942382\pi$
$978$		0			0
$979$	22.5000	+	38.9711i	0.719103	+	1.24552i
$980$	−18.0000			−0.574989
$981$		0			0
$982$	−13.5000	+	23.3827i	−0.430802	+	0.746171i
$983$	−8.00000			−0.255160			−0.127580	−	0.991828i	$-0.540721\pi$
$983$	−8.00000			−0.255160			−0.127580	+	0.991828i	$0.540721\pi$
$984$		0			0
$985$	−3.00000	+	5.19615i	−0.0955879	+	0.165563i
$986$	−2.50000	+	4.33013i	−0.0796162	+	0.137899i
$987$		0			0
$988$	8.00000	+	3.46410i	0.254514	+	0.110208i
$989$	64.0000			2.03508
$990$		0			0
$991$	−11.5000	−	19.9186i	−0.365310	−	0.632735i	0.623516	−	0.781810i	$-0.285704\pi$
$991$	−11.5000	−	19.9186i	−0.365310	−	0.632735i	−0.988826	+	0.149076i	$0.952370\pi$
$992$	−7.50000	−	12.9904i	−0.238125	−	0.412445i
$993$		0			0
$994$	−3.00000			−0.0951542
$995$	−4.50000	−	7.79423i	−0.142660	−	0.247094i
$996$		0			0
$997$	2.50000	+	4.33013i	0.0791758	+	0.137136i	0.902895	−	0.429862i	$-0.141438\pi$
$997$	2.50000	+	4.33013i	0.0791758	+	0.137136i	−0.823719	+	0.566999i	$0.808104\pi$
$998$	−14.5000	+	25.1147i	−0.458989	+	0.794993i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	513.2.h.b.235.1		2
3.2	odd	2		171.2.h.a.7.1	yes	2
9.4	even	3		513.2.g.a.64.1		2
9.5	odd	6		171.2.g.a.121.1	yes	2
19.11	even	3		513.2.g.a.505.1		2
57.11	odd	6		171.2.g.a.106.1	✓	2
171.49	even	3	inner	513.2.h.b.334.1		2
171.68	odd	6		171.2.h.a.49.1	yes	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
171.2.g.a.106.1	✓	2	57.11	odd	6
171.2.g.a.121.1	yes	2	9.5	odd	6
171.2.h.a.7.1	yes	2	3.2	odd	2
171.2.h.a.49.1	yes	2	171.68	odd	6
513.2.g.a.64.1		2	9.4	even	3
513.2.g.a.505.1		2	19.11	even	3
513.2.h.b.235.1		2	1.1	even	1	trivial
513.2.h.b.334.1		2	171.49	even	3	inner

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:	\( N \)	\(=\)	\( 513 = 3^{3} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	513.h (of order \(3\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} -1.00000 q^{4} +(1.50000 - 2.59808i) q^{5} +(-0.500000 + 0.866025i) q^{7} -3.00000 q^{8} +(1.50000 - 2.59808i) q^{10} +(2.50000 - 4.33013i) q^{11} +2.00000 q^{13} +(-0.500000 + 0.866025i) q^{14} -1.00000 q^{16} +(-2.50000 - 4.33013i) q^{17} +(-4.00000 - 1.73205i) q^{19} +(-1.50000 + 2.59808i) q^{20} +(2.50000 - 4.33013i) q^{22} +8.00000 q^{23} +(-2.00000 - 3.46410i) q^{25} +2.00000 q^{26} +(0.500000 - 0.866025i) q^{28} +(-0.500000 - 0.866025i) q^{29} +(-1.50000 - 2.59808i) q^{31} +5.00000 q^{32} +(-2.50000 - 4.33013i) q^{34} +(1.50000 + 2.59808i) q^{35} -6.00000 q^{37} +(-4.00000 - 1.73205i) q^{38} +(-4.50000 + 7.79423i) q^{40} +(-4.50000 + 7.79423i) q^{41} +8.00000 q^{43} +(-2.50000 + 4.33013i) q^{44} +8.00000 q^{46} +(1.50000 + 2.59808i) q^{47} +(3.00000 + 5.19615i) q^{49} +(-2.00000 - 3.46410i) q^{50} -2.00000 q^{52} +(-0.500000 + 0.866025i) q^{53} +(-7.50000 - 12.9904i) q^{55} +(1.50000 - 2.59808i) q^{56} +(-0.500000 - 0.866025i) q^{58} +(2.50000 - 4.33013i) q^{59} +(6.50000 + 11.2583i) q^{61} +(-1.50000 - 2.59808i) q^{62} +7.00000 q^{64} +(3.00000 - 5.19615i) q^{65} -4.00000 q^{67} +(2.50000 + 4.33013i) q^{68} +(1.50000 + 2.59808i) q^{70} +(1.50000 + 2.59808i) q^{71} +(2.50000 + 4.33013i) q^{73} -6.00000 q^{74} +(4.00000 + 1.73205i) q^{76} +(2.50000 + 4.33013i) q^{77} +4.00000 q^{79} +(-1.50000 + 2.59808i) q^{80} +(-4.50000 + 7.79423i) q^{82} +(4.50000 - 7.79423i) q^{83} -15.0000 q^{85} +8.00000 q^{86} +(-7.50000 + 12.9904i) q^{88} +(-4.50000 + 7.79423i) q^{89} +(-1.00000 + 1.73205i) q^{91} -8.00000 q^{92} +(1.50000 + 2.59808i) q^{94} +(-10.5000 + 7.79423i) q^{95} -10.0000 q^{97} +(3.00000 + 5.19615i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{2} - 2 q^{4} + 3 q^{5} - q^{7} - 6 q^{8} + 3 q^{10} + 5 q^{11} + 4 q^{13} - q^{14} - 2 q^{16} - 5 q^{17} - 8 q^{19} - 3 q^{20} + 5 q^{22} + 16 q^{23} - 4 q^{25} + 4 q^{26} + q^{28} - q^{29}+ \cdots + 6 q^{98}+O(q^{100}) \) 2 * q + 2 * q^2 - 2 * q^4 + 3 * q^5 - q^7 - 6 * q^8 + 3 * q^10 + 5 * q^11 + 4 * q^13 - q^14 - 2 * q^16 - 5 * q^17 - 8 * q^19 - 3 * q^20 + 5 * q^22 + 16 * q^23 - 4 * q^25 + 4 * q^26 + q^28 - q^29 - 3 * q^31 + 10 * q^32 - 5 * q^34 + 3 * q^35 - 12 * q^37 - 8 * q^38 - 9 * q^40 - 9 * q^41 + 16 * q^43 - 5 * q^44 + 16 * q^46 + 3 * q^47 + 6 * q^49 - 4 * q^50 - 4 * q^52 - q^53 - 15 * q^55 + 3 * q^56 - q^58 + 5 * q^59 + 13 * q^61 - 3 * q^62 + 14 * q^64 + 6 * q^65 - 8 * q^67 + 5 * q^68 + 3 * q^70 + 3 * q^71 + 5 * q^73 - 12 * q^74 + 8 * q^76 + 5 * q^77 + 8 * q^79 - 3 * q^80 - 9 * q^82 + 9 * q^83 - 30 * q^85 + 16 * q^86 - 15 * q^88 - 9 * q^89 - 2 * q^91 - 16 * q^92 + 3 * q^94 - 21 * q^95 - 20 * q^97 + 6 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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