LMFDB - Embedded newform 1386.2.k.c.793.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1386,2,Mod(793,1386)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1386, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 2, 0]))

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

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

chi := DirichletCharacter("1386.793");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1386 = 2 \cdot 3^{2} \cdot 7 \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1386.k (of order $3$, degree $2$, minimal)

Newform invariants

comment: select newform

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

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

Self dual:	no
Analytic conductor:	$11.0672657201$
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 462)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			793.1
Root			$0.500000 - 0.866025i$ of defining polynomial
Character	$\chi$	$=$	1386.793
Dual form			1386.2.k.c.991.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-0.500000 + 0.866025i) q^{2} +(-0.500000 - 0.866025i) q^{4} +(-1.50000 + 2.59808i) q^{5} +(2.50000 - 0.866025i) q^{7} +1.00000 q^{8} +O(q^{10})$	\(q+(-0.500000 + 0.866025i) q^{2} +(-0.500000 - 0.866025i) q^{4} +(-1.50000 + 2.59808i) q^{5} +(2.50000 - 0.866025i) q^{7} +1.00000 q^{8} +(-1.50000 - 2.59808i) q^{10} +(-0.500000 - 0.866025i) q^{11} +2.00000 q^{13} +(-0.500000 + 2.59808i) q^{14} +(-0.500000 + 0.866025i) q^{16} +(1.50000 + 2.59808i) q^{17} +(-1.00000 + 1.73205i) q^{19} +3.00000 q^{20} +1.00000 q^{22} +(1.50000 - 2.59808i) q^{23} +(-2.00000 - 3.46410i) q^{25} +(-1.00000 + 1.73205i) q^{26} +(-2.00000 - 1.73205i) q^{28} +6.00000 q^{29} +(2.00000 + 3.46410i) q^{31} +(-0.500000 - 0.866025i) q^{32} -3.00000 q^{34} +(-1.50000 + 7.79423i) q^{35} +(-1.00000 + 1.73205i) q^{37} +(-1.00000 - 1.73205i) q^{38} +(-1.50000 + 2.59808i) q^{40} +3.00000 q^{41} +2.00000 q^{43} +(-0.500000 + 0.866025i) q^{44} +(1.50000 + 2.59808i) q^{46} +(-4.50000 + 7.79423i) q^{47} +(5.50000 - 4.33013i) q^{49} +4.00000 q^{50} +(-1.00000 - 1.73205i) q^{52} +(3.00000 + 5.19615i) q^{53} +3.00000 q^{55} +(2.50000 - 0.866025i) q^{56} +(-3.00000 + 5.19615i) q^{58} +(-6.00000 - 10.3923i) q^{59} +(-2.50000 + 4.33013i) q^{61} -4.00000 q^{62} +1.00000 q^{64} +(-3.00000 + 5.19615i) q^{65} +(-2.50000 - 4.33013i) q^{67} +(1.50000 - 2.59808i) q^{68} +(-6.00000 - 5.19615i) q^{70} -12.0000 q^{71} +(8.00000 + 13.8564i) q^{73} +(-1.00000 - 1.73205i) q^{74} +2.00000 q^{76} +(-2.00000 - 1.73205i) q^{77} +(-8.50000 + 14.7224i) q^{79} +(-1.50000 - 2.59808i) q^{80} +(-1.50000 + 2.59808i) q^{82} -9.00000 q^{83} -9.00000 q^{85} +(-1.00000 + 1.73205i) q^{86} +(-0.500000 - 0.866025i) q^{88} +(-3.00000 + 5.19615i) q^{89} +(5.00000 - 1.73205i) q^{91} -3.00000 q^{92} +(-4.50000 - 7.79423i) q^{94} +(-3.00000 - 5.19615i) q^{95} +17.0000 q^{97} +(1.00000 + 6.92820i) q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - q^{2} - q^{4} - 3 q^{5} + 5 q^{7} + 2 q^{8}+O(q^{10}) $ 2 * q - q^2 - q^4 - 3 * q^5 + 5 * q^7 + 2 * q^8	$ 2 q - q^{2} - q^{4} - 3 q^{5} + 5 q^{7} + 2 q^{8} - 3 q^{10} - q^{11} + 4 q^{13} - q^{14} - q^{16} + 3 q^{17} - 2 q^{19} + 6 q^{20} + 2 q^{22} + 3 q^{23} - 4 q^{25} - 2 q^{26} - 4 q^{28} + 12 q^{29} + 4 q^{31} - q^{32} - 6 q^{34} - 3 q^{35} - 2 q^{37} - 2 q^{38} - 3 q^{40} + 6 q^{41} + 4 q^{43} - q^{44} + 3 q^{46} - 9 q^{47} + 11 q^{49} + 8 q^{50} - 2 q^{52} + 6 q^{53} + 6 q^{55} + 5 q^{56} - 6 q^{58} - 12 q^{59} - 5 q^{61} - 8 q^{62} + 2 q^{64} - 6 q^{65} - 5 q^{67} + 3 q^{68} - 12 q^{70} - 24 q^{71} + 16 q^{73} - 2 q^{74} + 4 q^{76} - 4 q^{77} - 17 q^{79} - 3 q^{80} - 3 q^{82} - 18 q^{83} - 18 q^{85} - 2 q^{86} - q^{88} - 6 q^{89} + 10 q^{91} - 6 q^{92} - 9 q^{94} - 6 q^{95} + 34 q^{97} + 2 q^{98}+O(q^{100}) $ 2 * q - q^2 - q^4 - 3 * q^5 + 5 * q^7 + 2 * q^8 - 3 * q^10 - q^11 + 4 * q^13 - q^14 - q^16 + 3 * q^17 - 2 * q^19 + 6 * q^20 + 2 * q^22 + 3 * q^23 - 4 * q^25 - 2 * q^26 - 4 * q^28 + 12 * q^29 + 4 * q^31 - q^32 - 6 * q^34 - 3 * q^35 - 2 * q^37 - 2 * q^38 - 3 * q^40 + 6 * q^41 + 4 * q^43 - q^44 + 3 * q^46 - 9 * q^47 + 11 * q^49 + 8 * q^50 - 2 * q^52 + 6 * q^53 + 6 * q^55 + 5 * q^56 - 6 * q^58 - 12 * q^59 - 5 * q^61 - 8 * q^62 + 2 * q^64 - 6 * q^65 - 5 * q^67 + 3 * q^68 - 12 * q^70 - 24 * q^71 + 16 * q^73 - 2 * q^74 + 4 * q^76 - 4 * q^77 - 17 * q^79 - 3 * q^80 - 3 * q^82 - 18 * q^83 - 18 * q^85 - 2 * q^86 - q^88 - 6 * q^89 + 10 * q^91 - 6 * q^92 - 9 * q^94 - 6 * q^95 + 34 * q^97 + 2 * q^98

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1386\mathbb{Z}\right)^\times$.

$n$	$155$	$199$	$1135$
$\chi(n)$	$1$	$e\left(\frac{1}{3}\right)$	$1$

Coefficient data

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

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

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

$2$	−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
$5$	−1.50000	+	2.59808i	−0.670820	+	1.16190i	0.306851	+	0.951757i	$0.400725\pi$
$5$	−1.50000	+	2.59808i	−0.670820	+	1.16190i	−0.977672	+	0.210138i	$0.932609\pi$
$6$		0			0
$7$	2.50000	−	0.866025i	0.944911	−	0.327327i
$7$	2.50000	−	0.866025i	0.944911	−	0.327327i
$8$	1.00000			0.353553
$9$		0			0
$10$	−1.50000	−	2.59808i	−0.474342	−	0.821584i
$11$	−0.500000	−	0.866025i	−0.150756	−	0.261116i
$11$	−0.500000	−	0.866025i	−0.150756	−	0.261116i
$12$		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	+	2.59808i	−0.133631	+	0.694365i
$15$		0			0
$16$	−0.500000	+	0.866025i	−0.125000	+	0.216506i
$17$	1.50000	+	2.59808i	0.363803	+	0.630126i	0.988583	−	0.150675i	$-0.0481447\pi$
$17$	1.50000	+	2.59808i	0.363803	+	0.630126i	−0.624780	+	0.780801i	$0.714811\pi$
$18$		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	$0.240348\pi$
$20$	3.00000			0.670820
$21$		0			0
$22$	1.00000			0.213201
$23$	1.50000	−	2.59808i	0.312772	−	0.541736i	−0.666190	−	0.745782i	$-0.732076\pi$
$23$	1.50000	−	2.59808i	0.312772	−	0.541736i	0.978961	+	0.204046i	$0.0654092\pi$
$24$		0			0
$25$	−2.00000	−	3.46410i	−0.400000	−	0.692820i
$26$	−1.00000	+	1.73205i	−0.196116	+	0.339683i
$27$		0			0
$28$	−2.00000	−	1.73205i	−0.377964	−	0.327327i
$29$	6.00000			1.11417			0.557086	−	0.830455i	$-0.311919\pi$
$29$	6.00000			1.11417			0.557086	+	0.830455i	$0.311919\pi$
$30$		0			0
$31$	2.00000	+	3.46410i	0.359211	+	0.622171i	0.987829	−	0.155543i	$-0.0497126\pi$
$31$	2.00000	+	3.46410i	0.359211	+	0.622171i	−0.628619	+	0.777714i	$0.716379\pi$
$32$	−0.500000	−	0.866025i	−0.0883883	−	0.153093i
$33$		0			0
$34$	−3.00000			−0.514496
$35$	−1.50000	+	7.79423i	−0.253546	+	1.31747i
$36$		0			0
$37$	−1.00000	+	1.73205i	−0.164399	+	0.284747i	−0.936442	−	0.350823i	$-0.885902\pi$
$37$	−1.00000	+	1.73205i	−0.164399	+	0.284747i	0.772043	+	0.635571i	$0.219235\pi$
$38$	−1.00000	−	1.73205i	−0.162221	−	0.280976i
$39$		0			0
$40$	−1.50000	+	2.59808i	−0.237171	+	0.410792i
$41$	3.00000			0.468521			0.234261	−	0.972174i	$-0.424733\pi$
$41$	3.00000			0.468521			0.234261	+	0.972174i	$0.424733\pi$
$42$		0			0
$43$	2.00000			0.304997			0.152499	−	0.988304i	$-0.451268\pi$
$43$	2.00000			0.304997			0.152499	+	0.988304i	$0.451268\pi$
$44$	−0.500000	+	0.866025i	−0.0753778	+	0.130558i
$45$		0			0
$46$	1.50000	+	2.59808i	0.221163	+	0.383065i
$47$	−4.50000	+	7.79423i	−0.656392	+	1.13691i	0.325150	+	0.945662i	$0.394585\pi$
$47$	−4.50000	+	7.79423i	−0.656392	+	1.13691i	−0.981543	+	0.191243i	$0.938748\pi$
$48$		0			0
$49$	5.50000	−	4.33013i	0.785714	−	0.618590i
$50$	4.00000			0.565685
$51$		0			0
$52$	−1.00000	−	1.73205i	−0.138675	−	0.240192i
$53$	3.00000	+	5.19615i	0.412082	+	0.713746i	0.995117	−	0.0987002i	$-0.0314685\pi$
$53$	3.00000	+	5.19615i	0.412082	+	0.713746i	−0.583036	+	0.812447i	$0.698135\pi$
$54$		0			0
$55$	3.00000			0.404520
$56$	2.50000	−	0.866025i	0.334077	−	0.115728i
$57$		0			0
$58$	−3.00000	+	5.19615i	−0.393919	+	0.682288i
$59$	−6.00000	−	10.3923i	−0.781133	−	1.35296i	−0.931282	−	0.364299i	$-0.881308\pi$
$59$	−6.00000	−	10.3923i	−0.781133	−	1.35296i	0.150148	−	0.988663i	$-0.452025\pi$
$60$		0			0
$61$	−2.50000	+	4.33013i	−0.320092	+	0.554416i	−0.980507	−	0.196485i	$-0.937047\pi$
$61$	−2.50000	+	4.33013i	−0.320092	+	0.554416i	0.660415	+	0.750901i	$0.270381\pi$
$62$	−4.00000			−0.508001
$63$		0			0
$64$	1.00000			0.125000
$65$	−3.00000	+	5.19615i	−0.372104	+	0.644503i
$66$		0			0
$67$	−2.50000	−	4.33013i	−0.305424	−	0.529009i	0.671932	−	0.740613i	$-0.265465\pi$
$67$	−2.50000	−	4.33013i	−0.305424	−	0.529009i	−0.977356	+	0.211604i	$0.932131\pi$
$68$	1.50000	−	2.59808i	0.181902	−	0.315063i
$69$		0			0
$70$	−6.00000	−	5.19615i	−0.717137	−	0.621059i
$71$	−12.0000			−1.42414			−0.712069	−	0.702109i	$-0.752242\pi$
$71$	−12.0000			−1.42414			−0.712069	+	0.702109i	$0.752242\pi$
$72$		0			0
$73$	8.00000	+	13.8564i	0.936329	+	1.62177i	0.772246	+	0.635323i	$0.219133\pi$
$73$	8.00000	+	13.8564i	0.936329	+	1.62177i	0.164083	+	0.986447i	$0.447534\pi$
$74$	−1.00000	−	1.73205i	−0.116248	−	0.201347i
$75$		0			0
$76$	2.00000			0.229416
$77$	−2.00000	−	1.73205i	−0.227921	−	0.197386i
$78$		0			0
$79$	−8.50000	+	14.7224i	−0.956325	+	1.65640i	−0.225018	+	0.974355i	$0.572244\pi$
$79$	−8.50000	+	14.7224i	−0.956325	+	1.65640i	−0.731307	+	0.682048i	$0.761089\pi$
$80$	−1.50000	−	2.59808i	−0.167705	−	0.290474i
$81$		0			0
$82$	−1.50000	+	2.59808i	−0.165647	+	0.286910i
$83$	−9.00000			−0.987878			−0.493939	−	0.869496i	$-0.664443\pi$
$83$	−9.00000			−0.987878			−0.493939	+	0.869496i	$0.664443\pi$
$84$		0			0
$85$	−9.00000			−0.976187
$86$	−1.00000	+	1.73205i	−0.107833	+	0.186772i
$87$		0			0
$88$	−0.500000	−	0.866025i	−0.0533002	−	0.0923186i
$89$	−3.00000	+	5.19615i	−0.317999	+	0.550791i	−0.980071	−	0.198650i	$-0.936344\pi$
$89$	−3.00000	+	5.19615i	−0.317999	+	0.550791i	0.662071	+	0.749441i	$0.269678\pi$
$90$		0			0
$91$	5.00000	−	1.73205i	0.524142	−	0.181568i
$92$	−3.00000			−0.312772
$93$		0			0
$94$	−4.50000	−	7.79423i	−0.464140	−	0.803913i
$95$	−3.00000	−	5.19615i	−0.307794	−	0.533114i
$96$		0			0
$97$	17.0000			1.72609			0.863044	−	0.505128i	$-0.168555\pi$
$97$	17.0000			1.72609			0.863044	+	0.505128i	$0.168555\pi$
$98$	1.00000	+	6.92820i	0.101015	+	0.699854i
$99$		0			0
$100$	−2.00000	+	3.46410i	−0.200000	+	0.346410i
$101$	6.00000	+	10.3923i	0.597022	+	1.03407i	0.993258	+	0.115924i	$0.0369830\pi$
$101$	6.00000	+	10.3923i	0.597022	+	1.03407i	−0.396236	+	0.918149i	$0.629684\pi$
$102$		0			0
$103$	−7.00000	+	12.1244i	−0.689730	+	1.19465i	0.282194	+	0.959357i	$0.408938\pi$
$103$	−7.00000	+	12.1244i	−0.689730	+	1.19465i	−0.971925	+	0.235291i	$0.924396\pi$
$104$	2.00000			0.196116
$105$		0			0
$106$	−6.00000			−0.582772
$107$	1.50000	−	2.59808i	0.145010	−	0.251166i	−0.784366	−	0.620298i	$-0.787012\pi$
$107$	1.50000	−	2.59808i	0.145010	−	0.251166i	0.929377	+	0.369132i	$0.120345\pi$
$108$		0			0
$109$	−5.50000	−	9.52628i	−0.526804	−	0.912452i	−0.999512	−	0.0312328i	$-0.990057\pi$
$109$	−5.50000	−	9.52628i	−0.526804	−	0.912452i	0.472708	−	0.881219i	$-0.343277\pi$
$110$	−1.50000	+	2.59808i	−0.143019	+	0.247717i
$111$		0			0
$112$	−0.500000	+	2.59808i	−0.0472456	+	0.245495i
$113$	6.00000			0.564433			0.282216	−	0.959351i	$-0.408930\pi$
$113$	6.00000			0.564433			0.282216	+	0.959351i	$0.408930\pi$
$114$		0			0
$115$	4.50000	+	7.79423i	0.419627	+	0.726816i
$116$	−3.00000	−	5.19615i	−0.278543	−	0.482451i
$117$		0			0
$118$	12.0000			1.10469
$119$	6.00000	+	5.19615i	0.550019	+	0.476331i
$120$		0			0
$121$	−0.500000	+	0.866025i	−0.0454545	+	0.0787296i
$122$	−2.50000	−	4.33013i	−0.226339	−	0.392031i
$123$		0			0
$124$	2.00000	−	3.46410i	0.179605	−	0.311086i
$125$	−3.00000			−0.268328
$126$		0			0
$127$	11.0000			0.976092			0.488046	−	0.872818i	$-0.337710\pi$
$127$	11.0000			0.976092			0.488046	+	0.872818i	$0.337710\pi$
$128$	−0.500000	+	0.866025i	−0.0441942	+	0.0765466i
$129$		0			0
$130$	−3.00000	−	5.19615i	−0.263117	−	0.455733i
$131$	−6.00000	+	10.3923i	−0.524222	+	0.907980i	0.475380	+	0.879781i	$0.342311\pi$
$131$	−6.00000	+	10.3923i	−0.524222	+	0.907980i	−0.999602	+	0.0281993i	$0.991023\pi$
$132$		0			0
$133$	−1.00000	+	5.19615i	−0.0867110	+	0.450564i
$134$	5.00000			0.431934
$135$		0			0
$136$	1.50000	+	2.59808i	0.128624	+	0.222783i
$137$	9.00000	+	15.5885i	0.768922	+	1.33181i	0.938148	+	0.346235i	$0.112540\pi$
$137$	9.00000	+	15.5885i	0.768922	+	1.33181i	−0.169226	+	0.985577i	$0.554127\pi$
$138$		0			0
$139$	14.0000			1.18746			0.593732	−	0.804663i	$-0.297654\pi$
$139$	14.0000			1.18746			0.593732	+	0.804663i	$0.297654\pi$
$140$	7.50000	−	2.59808i	0.633866	−	0.219578i
$141$		0			0
$142$	6.00000	−	10.3923i	0.503509	−	0.872103i
$143$	−1.00000	−	1.73205i	−0.0836242	−	0.144841i
$144$		0			0
$145$	−9.00000	+	15.5885i	−0.747409	+	1.29455i
$146$	−16.0000			−1.32417
$147$		0			0
$148$	2.00000			0.164399
$149$	3.00000	−	5.19615i	0.245770	−	0.425685i	−0.716578	−	0.697507i	$-0.754293\pi$
$149$	3.00000	−	5.19615i	0.245770	−	0.425685i	0.962348	+	0.271821i	$0.0876260\pi$
$150$		0			0
$151$	0.500000	+	0.866025i	0.0406894	+	0.0704761i	0.885653	−	0.464348i	$-0.153711\pi$
$151$	0.500000	+	0.866025i	0.0406894	+	0.0704761i	−0.844963	+	0.534824i	$0.820378\pi$
$152$	−1.00000	+	1.73205i	−0.0811107	+	0.140488i
$153$		0			0
$154$	2.50000	−	0.866025i	0.201456	−	0.0697863i
$155$	−12.0000			−0.963863
$156$		0			0
$157$	−10.0000	−	17.3205i	−0.798087	−	1.38233i	−0.920860	−	0.389892i	$-0.872512\pi$
$157$	−10.0000	−	17.3205i	−0.798087	−	1.38233i	0.122774	−	0.992435i	$-0.460821\pi$
$158$	−8.50000	−	14.7224i	−0.676224	−	1.17125i
$159$		0			0
$160$	3.00000			0.237171
$161$	1.50000	−	7.79423i	0.118217	−	0.614271i
$162$		0			0
$163$	0.500000	−	0.866025i	0.0391630	−	0.0678323i	−0.845780	−	0.533533i	$-0.820864\pi$
$163$	0.500000	−	0.866025i	0.0391630	−	0.0678323i	0.884943	+	0.465700i	$0.154198\pi$
$164$	−1.50000	−	2.59808i	−0.117130	−	0.202876i
$165$		0			0
$166$	4.50000	−	7.79423i	0.349268	−	0.604949i
$167$	12.0000			0.928588			0.464294	−	0.885681i	$-0.346308\pi$
$167$	12.0000			0.928588			0.464294	+	0.885681i	$0.346308\pi$
$168$		0			0
$169$	−9.00000			−0.692308
$170$	4.50000	−	7.79423i	0.345134	−	0.597790i
$171$		0			0
$172$	−1.00000	−	1.73205i	−0.0762493	−	0.132068i
$173$	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
$175$	−8.00000	−	6.92820i	−0.604743	−	0.523723i
$176$	1.00000			0.0753778
$177$		0			0
$178$	−3.00000	−	5.19615i	−0.224860	−	0.389468i
$179$	6.00000	+	10.3923i	0.448461	+	0.776757i	0.998286	−	0.0585225i	$-0.0186389\pi$
$179$	6.00000	+	10.3923i	0.448461	+	0.776757i	−0.549825	+	0.835280i	$0.685306\pi$
$180$		0			0
$181$	−10.0000			−0.743294			−0.371647	−	0.928374i	$-0.621207\pi$
$181$	−10.0000			−0.743294			−0.371647	+	0.928374i	$0.621207\pi$
$182$	−1.00000	+	5.19615i	−0.0741249	+	0.385164i
$183$		0			0
$184$	1.50000	−	2.59808i	0.110581	−	0.191533i
$185$	−3.00000	−	5.19615i	−0.220564	−	0.382029i
$186$		0			0
$187$	1.50000	−	2.59808i	0.109691	−	0.189990i
$188$	9.00000			0.656392
$189$		0			0
$190$	6.00000			0.435286
$191$	12.0000	−	20.7846i	0.868290	−	1.50392i	0.00454614	−	0.999990i	$-0.498553\pi$
$191$	12.0000	−	20.7846i	0.868290	−	1.50392i	0.863743	−	0.503932i	$-0.168114\pi$
$192$		0			0
$193$	−1.00000	−	1.73205i	−0.0719816	−	0.124676i	0.827788	−	0.561041i	$-0.189599\pi$
$193$	−1.00000	−	1.73205i	−0.0719816	−	0.124676i	−0.899770	+	0.436365i	$0.856266\pi$
$194$	−8.50000	+	14.7224i	−0.610264	+	1.05701i
$195$		0			0
$196$	−6.50000	−	2.59808i	−0.464286	−	0.185577i
$197$	−12.0000			−0.854965			−0.427482	−	0.904024i	$-0.640599\pi$
$197$	−12.0000			−0.854965			−0.427482	+	0.904024i	$0.640599\pi$
$198$		0			0
$199$	−1.00000	−	1.73205i	−0.0708881	−	0.122782i	0.828403	−	0.560133i	$-0.189250\pi$
$199$	−1.00000	−	1.73205i	−0.0708881	−	0.122782i	−0.899291	+	0.437351i	$0.855917\pi$
$200$	−2.00000	−	3.46410i	−0.141421	−	0.244949i
$201$		0			0
$202$	−12.0000			−0.844317
$203$	15.0000	−	5.19615i	1.05279	−	0.364698i
$204$		0			0
$205$	−4.50000	+	7.79423i	−0.314294	+	0.544373i
$206$	−7.00000	−	12.1244i	−0.487713	−	0.844744i
$207$		0			0
$208$	−1.00000	+	1.73205i	−0.0693375	+	0.120096i
$209$	2.00000			0.138343
$210$		0			0
$211$	−22.0000			−1.51454			−0.757271	−	0.653101i	$-0.773468\pi$
$211$	−22.0000			−1.51454			−0.757271	+	0.653101i	$0.773468\pi$
$212$	3.00000	−	5.19615i	0.206041	−	0.356873i
$213$		0			0
$214$	1.50000	+	2.59808i	0.102538	+	0.177601i
$215$	−3.00000	+	5.19615i	−0.204598	+	0.354375i
$216$		0			0
$217$	8.00000	+	6.92820i	0.543075	+	0.470317i
$218$	11.0000			0.745014
$219$		0			0
$220$	−1.50000	−	2.59808i	−0.101130	−	0.175162i
$221$	3.00000	+	5.19615i	0.201802	+	0.349531i
$222$		0			0
$223$	20.0000			1.33930			0.669650	−	0.742677i	$-0.266444\pi$
$223$	20.0000			1.33930			0.669650	+	0.742677i	$0.266444\pi$
$224$	−2.00000	−	1.73205i	−0.133631	−	0.115728i
$225$		0			0
$226$	−3.00000	+	5.19615i	−0.199557	+	0.345643i
$227$	−10.5000	−	18.1865i	−0.696909	−	1.20708i	−0.969533	−	0.244962i	$-0.921225\pi$
$227$	−10.5000	−	18.1865i	−0.696909	−	1.20708i	0.272623	−	0.962121i	$-0.412109\pi$
$228$		0			0
$229$	5.00000	−	8.66025i	0.330409	−	0.572286i	−0.652183	−	0.758062i	$-0.726147\pi$
$229$	5.00000	−	8.66025i	0.330409	−	0.572286i	0.982592	+	0.185776i	$0.0594799\pi$
$230$	−9.00000			−0.593442
$231$		0			0
$232$	6.00000			0.393919
$233$	13.5000	−	23.3827i	0.884414	−	1.53185i	0.0380310	−	0.999277i	$-0.487891\pi$
$233$	13.5000	−	23.3827i	0.884414	−	1.53185i	0.846383	−	0.532574i	$-0.178775\pi$
$234$		0			0
$235$	−13.5000	−	23.3827i	−0.880643	−	1.52532i
$236$	−6.00000	+	10.3923i	−0.390567	+	0.676481i
$237$		0			0
$238$	−7.50000	+	2.59808i	−0.486153	+	0.168408i
$239$	−6.00000			−0.388108			−0.194054	−	0.980991i	$-0.562164\pi$
$239$	−6.00000			−0.388108			−0.194054	+	0.980991i	$0.562164\pi$
$240$		0			0
$241$	5.00000	+	8.66025i	0.322078	+	0.557856i	0.980917	−	0.194429i	$-0.0622852\pi$
$241$	5.00000	+	8.66025i	0.322078	+	0.557856i	−0.658838	+	0.752285i	$0.728952\pi$
$242$	−0.500000	−	0.866025i	−0.0321412	−	0.0556702i
$243$		0			0
$244$	5.00000			0.320092
$245$	3.00000	+	20.7846i	0.191663	+	1.32788i
$246$		0			0
$247$	−2.00000	+	3.46410i	−0.127257	+	0.220416i
$248$	2.00000	+	3.46410i	0.127000	+	0.219971i
$249$		0			0
$250$	1.50000	−	2.59808i	0.0948683	−	0.164317i
$251$	24.0000			1.51487			0.757433	−	0.652913i	$-0.226453\pi$
$251$	24.0000			1.51487			0.757433	+	0.652913i	$0.226453\pi$
$252$		0			0
$253$	−3.00000			−0.188608
$254$	−5.50000	+	9.52628i	−0.345101	+	0.597732i
$255$		0			0
$256$	−0.500000	−	0.866025i	−0.0312500	−	0.0541266i
$257$	3.00000	−	5.19615i	0.187135	−	0.324127i	−0.757159	−	0.653231i	$-0.773413\pi$
$257$	3.00000	−	5.19615i	0.187135	−	0.324127i	0.944294	+	0.329104i	$0.106747\pi$
$258$		0			0
$259$	−1.00000	+	5.19615i	−0.0621370	+	0.322873i
$260$	6.00000			0.372104
$261$		0			0
$262$	−6.00000	−	10.3923i	−0.370681	−	0.642039i
$263$	−6.00000	−	10.3923i	−0.369976	−	0.640817i	0.619586	−	0.784929i	$-0.287301\pi$
$263$	−6.00000	−	10.3923i	−0.369976	−	0.640817i	−0.989561	+	0.144112i	$0.953967\pi$
$264$		0			0
$265$	−18.0000			−1.10573
$266$	−4.00000	−	3.46410i	−0.245256	−	0.212398i
$267$		0			0
$268$	−2.50000	+	4.33013i	−0.152712	+	0.264505i
$269$	4.50000	+	7.79423i	0.274370	+	0.475223i	0.969976	−	0.243201i	$-0.0781974\pi$
$269$	4.50000	+	7.79423i	0.274370	+	0.475223i	−0.695606	+	0.718423i	$0.744864\pi$
$270$		0			0
$271$	8.00000	−	13.8564i	0.485965	−	0.841717i	−0.513905	−	0.857847i	$-0.671801\pi$
$271$	8.00000	−	13.8564i	0.485965	−	0.841717i	0.999870	+	0.0161307i	$0.00513477\pi$
$272$	−3.00000			−0.181902
$273$		0			0
$274$	−18.0000			−1.08742
$275$	−2.00000	+	3.46410i	−0.120605	+	0.208893i
$276$		0			0
$277$	5.00000	+	8.66025i	0.300421	+	0.520344i	0.976231	−	0.216731i	$-0.0695395\pi$
$277$	5.00000	+	8.66025i	0.300421	+	0.520344i	−0.675810	+	0.737075i	$0.736206\pi$
$278$	−7.00000	+	12.1244i	−0.419832	+	0.727171i
$279$		0			0
$280$	−1.50000	+	7.79423i	−0.0896421	+	0.465794i
$281$	−3.00000			−0.178965			−0.0894825	−	0.995988i	$-0.528521\pi$
$281$	−3.00000			−0.178965			−0.0894825	+	0.995988i	$0.528521\pi$
$282$		0			0
$283$	−7.00000	−	12.1244i	−0.416107	−	0.720718i	0.579437	−	0.815017i	$-0.303272\pi$
$283$	−7.00000	−	12.1244i	−0.416107	−	0.720718i	−0.995544	+	0.0942988i	$0.969939\pi$
$284$	6.00000	+	10.3923i	0.356034	+	0.616670i
$285$		0			0
$286$	2.00000			0.118262
$287$	7.50000	−	2.59808i	0.442711	−	0.153360i
$288$		0			0
$289$	4.00000	−	6.92820i	0.235294	−	0.407541i
$290$	−9.00000	−	15.5885i	−0.528498	−	0.915386i
$291$		0			0
$292$	8.00000	−	13.8564i	0.468165	−	0.810885i
$293$	24.0000			1.40209			0.701047	−	0.713115i	$-0.252716\pi$
$293$	24.0000			1.40209			0.701047	+	0.713115i	$0.252716\pi$
$294$		0			0
$295$	36.0000			2.09600
$296$	−1.00000	+	1.73205i	−0.0581238	+	0.100673i
$297$		0			0
$298$	3.00000	+	5.19615i	0.173785	+	0.301005i
$299$	3.00000	−	5.19615i	0.173494	−	0.300501i
$300$		0			0
$301$	5.00000	−	1.73205i	0.288195	−	0.0998337i
$302$	−1.00000			−0.0575435
$303$		0			0
$304$	−1.00000	−	1.73205i	−0.0573539	−	0.0993399i
$305$	−7.50000	−	12.9904i	−0.429449	−	0.743827i
$306$		0			0
$307$	−22.0000			−1.25561			−0.627803	−	0.778372i	$-0.716046\pi$
$307$	−22.0000			−1.25561			−0.627803	+	0.778372i	$0.716046\pi$
$308$	−0.500000	+	2.59808i	−0.0284901	+	0.148039i
$309$		0			0
$310$	6.00000	−	10.3923i	0.340777	−	0.590243i
$311$	10.5000	+	18.1865i	0.595400	+	1.03126i	0.993490	+	0.113917i	$0.0363399\pi$
$311$	10.5000	+	18.1865i	0.595400	+	1.03126i	−0.398090	+	0.917346i	$0.630327\pi$
$312$		0			0
$313$	−1.00000	+	1.73205i	−0.0565233	+	0.0979013i	−0.892903	−	0.450250i	$-0.851335\pi$
$313$	−1.00000	+	1.73205i	−0.0565233	+	0.0979013i	0.836379	+	0.548151i	$0.184668\pi$
$314$	20.0000			1.12867
$315$		0			0
$316$	17.0000			0.956325
$317$	−16.5000	+	28.5788i	−0.926732	+	1.60515i	−0.137981	+	0.990435i	$0.544061\pi$
$317$	−16.5000	+	28.5788i	−0.926732	+	1.60515i	−0.788751	+	0.614713i	$0.789272\pi$
$318$		0			0
$319$	−3.00000	−	5.19615i	−0.167968	−	0.290929i
$320$	−1.50000	+	2.59808i	−0.0838525	+	0.145237i
$321$		0			0
$322$	6.00000	+	5.19615i	0.334367	+	0.289570i
$323$	−6.00000			−0.333849
$324$		0			0
$325$	−4.00000	−	6.92820i	−0.221880	−	0.384308i
$326$	0.500000	+	0.866025i	0.0276924	+	0.0479647i
$327$		0			0
$328$	3.00000			0.165647
$329$	−4.50000	+	23.3827i	−0.248093	+	1.28913i
$330$		0			0
$331$	−5.50000	+	9.52628i	−0.302307	+	0.523612i	−0.976658	−	0.214799i	$-0.931090\pi$
$331$	−5.50000	+	9.52628i	−0.302307	+	0.523612i	0.674351	+	0.738411i	$0.264424\pi$
$332$	4.50000	+	7.79423i	0.246970	+	0.427764i
$333$		0			0
$334$	−6.00000	+	10.3923i	−0.328305	+	0.568642i
$335$	15.0000			0.819538
$336$		0			0
$337$	−22.0000			−1.19842			−0.599208	−	0.800593i	$-0.704518\pi$
$337$	−22.0000			−1.19842			−0.599208	+	0.800593i	$0.704518\pi$
$338$	4.50000	−	7.79423i	0.244768	−	0.423950i
$339$		0			0
$340$	4.50000	+	7.79423i	0.244047	+	0.422701i
$341$	2.00000	−	3.46410i	0.108306	−	0.187592i
$342$		0			0
$343$	10.0000	−	15.5885i	0.539949	−	0.841698i
$344$	2.00000			0.107833
$345$		0			0
$346$	6.00000	+	10.3923i	0.322562	+	0.558694i
$347$	−16.5000	−	28.5788i	−0.885766	−	1.53419i	−0.844833	−	0.535031i	$-0.820300\pi$
$347$	−16.5000	−	28.5788i	−0.885766	−	1.53419i	−0.0409337	−	0.999162i	$-0.513033\pi$
$348$		0			0
$349$	11.0000			0.588817			0.294408	−	0.955680i	$-0.404877\pi$
$349$	11.0000			0.588817			0.294408	+	0.955680i	$0.404877\pi$
$350$	10.0000	−	3.46410i	0.534522	−	0.185164i
$351$		0			0
$352$	−0.500000	+	0.866025i	−0.0266501	+	0.0461593i
$353$	−6.00000	−	10.3923i	−0.319348	−	0.553127i	0.661004	−	0.750382i	$-0.270130\pi$
$353$	−6.00000	−	10.3923i	−0.319348	−	0.553127i	−0.980352	+	0.197256i	$0.936797\pi$
$354$		0			0
$355$	18.0000	−	31.1769i	0.955341	−	1.65470i
$356$	6.00000			0.317999
$357$		0			0
$358$	−12.0000			−0.634220
$359$	15.0000	−	25.9808i	0.791670	−	1.37121i	−0.133263	−	0.991081i	$-0.542545\pi$
$359$	15.0000	−	25.9808i	0.791670	−	1.37121i	0.924932	−	0.380131i	$-0.124121\pi$
$360$		0			0
$361$	7.50000	+	12.9904i	0.394737	+	0.683704i
$362$	5.00000	−	8.66025i	0.262794	−	0.455173i
$363$		0			0
$364$	−4.00000	−	3.46410i	−0.209657	−	0.181568i
$365$	−48.0000			−2.51243
$366$		0			0
$367$	5.00000	+	8.66025i	0.260998	+	0.452062i	0.966507	−	0.256639i	$-0.0826151\pi$
$367$	5.00000	+	8.66025i	0.260998	+	0.452062i	−0.705509	+	0.708700i	$0.749282\pi$
$368$	1.50000	+	2.59808i	0.0781929	+	0.135434i
$369$		0			0
$370$	6.00000			0.311925
$371$	12.0000	+	10.3923i	0.623009	+	0.539542i
$372$		0			0
$373$	0.500000	−	0.866025i	0.0258890	−	0.0448411i	−0.852791	−	0.522253i	$-0.825092\pi$
$373$	0.500000	−	0.866025i	0.0258890	−	0.0448411i	0.878680	+	0.477412i	$0.158425\pi$
$374$	1.50000	+	2.59808i	0.0775632	+	0.134343i
$375$		0			0
$376$	−4.50000	+	7.79423i	−0.232070	+	0.401957i
$377$	12.0000			0.618031
$378$		0			0
$379$	29.0000			1.48963			0.744815	−	0.667271i	$-0.232538\pi$
$379$	29.0000			1.48963			0.744815	+	0.667271i	$0.232538\pi$
$380$	−3.00000	+	5.19615i	−0.153897	+	0.266557i
$381$		0			0
$382$	12.0000	+	20.7846i	0.613973	+	1.06343i
$383$	12.0000	−	20.7846i	0.613171	−	1.06204i	−0.377531	−	0.925997i	$-0.623227\pi$
$383$	12.0000	−	20.7846i	0.613171	−	1.06204i	0.990702	−	0.136047i	$-0.0434398\pi$
$384$		0			0
$385$	7.50000	−	2.59808i	0.382235	−	0.132410i
$386$	2.00000			0.101797
$387$		0			0
$388$	−8.50000	−	14.7224i	−0.431522	−	0.747418i
$389$	10.5000	+	18.1865i	0.532371	+	0.922094i	0.999286	+	0.0377914i	$0.0120322\pi$
$389$	10.5000	+	18.1865i	0.532371	+	0.922094i	−0.466915	+	0.884302i	$0.654634\pi$
$390$		0			0
$391$	9.00000			0.455150
$392$	5.50000	−	4.33013i	0.277792	−	0.218704i
$393$		0			0
$394$	6.00000	−	10.3923i	0.302276	−	0.523557i
$395$	−25.5000	−	44.1673i	−1.28304	−	2.22230i
$396$		0			0
$397$	−1.00000	+	1.73205i	−0.0501886	+	0.0869291i	−0.890028	−	0.455905i	$-0.849316\pi$
$397$	−1.00000	+	1.73205i	−0.0501886	+	0.0869291i	0.839840	+	0.542834i	$0.182649\pi$
$398$	2.00000			0.100251
$399$		0			0
$400$	4.00000			0.200000
$401$	15.0000	−	25.9808i	0.749064	−	1.29742i	−0.199207	−	0.979957i	$-0.563837\pi$
$401$	15.0000	−	25.9808i	0.749064	−	1.29742i	0.948272	−	0.317460i	$-0.102830\pi$
$402$		0			0
$403$	4.00000	+	6.92820i	0.199254	+	0.345118i
$404$	6.00000	−	10.3923i	0.298511	−	0.517036i
$405$		0			0
$406$	−3.00000	+	15.5885i	−0.148888	+	0.773642i
$407$	2.00000			0.0991363
$408$		0			0
$409$	−7.00000	−	12.1244i	−0.346128	−	0.599511i	0.639430	−	0.768849i	$-0.279170\pi$
$409$	−7.00000	−	12.1244i	−0.346128	−	0.599511i	−0.985558	+	0.169338i	$0.945837\pi$
$410$	−4.50000	−	7.79423i	−0.222239	−	0.384930i
$411$		0			0
$412$	14.0000			0.689730
$413$	−24.0000	−	20.7846i	−1.18096	−	1.02274i
$414$		0			0
$415$	13.5000	−	23.3827i	0.662689	−	1.14781i
$416$	−1.00000	−	1.73205i	−0.0490290	−	0.0849208i
$417$		0			0
$418$	−1.00000	+	1.73205i	−0.0489116	+	0.0847174i
$419$	24.0000			1.17248			0.586238	−	0.810139i	$-0.300608\pi$
$419$	24.0000			1.17248			0.586238	+	0.810139i	$0.300608\pi$
$420$		0			0
$421$	20.0000			0.974740			0.487370	−	0.873195i	$-0.337956\pi$
$421$	20.0000			0.974740			0.487370	+	0.873195i	$0.337956\pi$
$422$	11.0000	−	19.0526i	0.535472	−	0.927464i
$423$		0			0
$424$	3.00000	+	5.19615i	0.145693	+	0.252347i
$425$	6.00000	−	10.3923i	0.291043	−	0.504101i
$426$		0			0
$427$	−2.50000	+	12.9904i	−0.120983	+	0.628649i
$428$	−3.00000			−0.145010
$429$		0			0
$430$	−3.00000	−	5.19615i	−0.144673	−	0.250581i
$431$	−3.00000	−	5.19615i	−0.144505	−	0.250290i	0.784683	−	0.619897i	$-0.212826\pi$
$431$	−3.00000	−	5.19615i	−0.144505	−	0.250290i	−0.929188	+	0.369607i	$0.879492\pi$
$432$		0			0
$433$	35.0000			1.68199			0.840996	−	0.541041i	$-0.181970\pi$
$433$	35.0000			1.68199			0.840996	+	0.541041i	$0.181970\pi$
$434$	−10.0000	+	3.46410i	−0.480015	+	0.166282i
$435$		0			0
$436$	−5.50000	+	9.52628i	−0.263402	+	0.456226i
$437$	3.00000	+	5.19615i	0.143509	+	0.248566i
$438$		0			0
$439$	9.50000	−	16.4545i	0.453410	−	0.785330i	−0.545185	−	0.838316i	$-0.683541\pi$
$439$	9.50000	−	16.4545i	0.453410	−	0.785330i	0.998595	+	0.0529862i	$0.0168739\pi$
$440$	3.00000			0.143019
$441$		0			0
$442$	−6.00000			−0.285391
$443$	9.00000	−	15.5885i	0.427603	−	0.740630i	−0.569057	−	0.822298i	$-0.692691\pi$
$443$	9.00000	−	15.5885i	0.427603	−	0.740630i	0.996660	+	0.0816684i	$0.0260248\pi$
$444$		0			0
$445$	−9.00000	−	15.5885i	−0.426641	−	0.738964i
$446$	−10.0000	+	17.3205i	−0.473514	+	0.820150i
$447$		0			0
$448$	2.50000	−	0.866025i	0.118114	−	0.0409159i
$449$	−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
$451$	−1.50000	−	2.59808i	−0.0706322	−	0.122339i
$452$	−3.00000	−	5.19615i	−0.141108	−	0.244406i
$453$		0			0
$454$	21.0000			0.985579
$455$	−3.00000	+	15.5885i	−0.140642	+	0.730798i
$456$		0			0
$457$	−19.0000	+	32.9090i	−0.888783	+	1.53942i	−0.0474665	+	0.998873i	$0.515115\pi$
$457$	−19.0000	+	32.9090i	−0.888783	+	1.53942i	−0.841316	+	0.540544i	$0.818219\pi$
$458$	5.00000	+	8.66025i	0.233635	+	0.404667i
$459$		0			0
$460$	4.50000	−	7.79423i	0.209814	−	0.363408i
$461$	−6.00000			−0.279448			−0.139724	−	0.990190i	$-0.544622\pi$
$461$	−6.00000			−0.279448			−0.139724	+	0.990190i	$0.544622\pi$
$462$		0			0
$463$	−16.0000			−0.743583			−0.371792	−	0.928316i	$-0.621256\pi$
$463$	−16.0000			−0.743583			−0.371792	+	0.928316i	$0.621256\pi$
$464$	−3.00000	+	5.19615i	−0.139272	+	0.241225i
$465$		0			0
$466$	13.5000	+	23.3827i	0.625375	+	1.08318i
$467$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$467$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$468$		0			0
$469$	−10.0000	−	8.66025i	−0.461757	−	0.399893i
$470$	27.0000			1.24542
$471$		0			0
$472$	−6.00000	−	10.3923i	−0.276172	−	0.478345i
$473$	−1.00000	−	1.73205i	−0.0459800	−	0.0796398i
$474$		0			0
$475$	8.00000			0.367065
$476$	1.50000	−	7.79423i	0.0687524	−	0.357248i
$477$		0			0
$478$	3.00000	−	5.19615i	0.137217	−	0.237666i
$479$	12.0000	+	20.7846i	0.548294	+	0.949673i	0.998392	+	0.0566937i	$0.0180558\pi$
$479$	12.0000	+	20.7846i	0.548294	+	0.949673i	−0.450098	+	0.892979i	$0.648611\pi$
$480$		0			0
$481$	−2.00000	+	3.46410i	−0.0911922	+	0.157949i
$482$	−10.0000			−0.455488
$483$		0			0
$484$	1.00000			0.0454545
$485$	−25.5000	+	44.1673i	−1.15790	+	2.00553i
$486$		0			0
$487$	−4.00000	−	6.92820i	−0.181257	−	0.313947i	0.761052	−	0.648691i	$-0.224683\pi$
$487$	−4.00000	−	6.92820i	−0.181257	−	0.313947i	−0.942309	+	0.334744i	$0.891350\pi$
$488$	−2.50000	+	4.33013i	−0.113170	+	0.196016i
$489$		0			0
$490$	−19.5000	−	7.79423i	−0.880920	−	0.352107i
$491$	3.00000			0.135388			0.0676941	−	0.997706i	$-0.478436\pi$
$491$	3.00000			0.135388			0.0676941	+	0.997706i	$0.478436\pi$
$492$		0			0
$493$	9.00000	+	15.5885i	0.405340	+	0.702069i
$494$	−2.00000	−	3.46410i	−0.0899843	−	0.155857i
$495$		0			0
$496$	−4.00000			−0.179605
$497$	−30.0000	+	10.3923i	−1.34568	+	0.466159i
$498$		0			0
$499$	14.0000	−	24.2487i	0.626726	−	1.08552i	−0.361478	−	0.932381i	$-0.617728\pi$
$499$	14.0000	−	24.2487i	0.626726	−	1.08552i	0.988204	−	0.153141i	$-0.0489388\pi$
$500$	1.50000	+	2.59808i	0.0670820	+	0.116190i
$501$		0			0
$502$	−12.0000	+	20.7846i	−0.535586	+	0.927663i
$503$	6.00000			0.267527			0.133763	−	0.991013i	$-0.457294\pi$
$503$	6.00000			0.267527			0.133763	+	0.991013i	$0.457294\pi$
$504$		0			0
$505$	−36.0000			−1.60198
$506$	1.50000	−	2.59808i	0.0666831	−	0.115499i
$507$		0			0
$508$	−5.50000	−	9.52628i	−0.244023	−	0.422660i
$509$	3.00000	−	5.19615i	0.132973	−	0.230315i	−0.791849	−	0.610718i	$-0.790881\pi$
$509$	3.00000	−	5.19615i	0.132973	−	0.230315i	0.924821	+	0.380402i	$0.124214\pi$
$510$		0			0
$511$	32.0000	+	27.7128i	1.41560	+	1.22594i
$512$	1.00000			0.0441942
$513$		0			0
$514$	3.00000	+	5.19615i	0.132324	+	0.229192i
$515$	−21.0000	−	36.3731i	−0.925371	−	1.60279i
$516$		0			0
$517$	9.00000			0.395820
$518$	−4.00000	−	3.46410i	−0.175750	−	0.152204i
$519$		0			0
$520$	−3.00000	+	5.19615i	−0.131559	+	0.227866i
$521$	9.00000	+	15.5885i	0.394297	+	0.682943i	0.993011	−	0.118020i	$-0.0376547\pi$
$521$	9.00000	+	15.5885i	0.394297	+	0.682943i	−0.598714	+	0.800963i	$0.704321\pi$
$522$		0			0
$523$	−13.0000	+	22.5167i	−0.568450	+	0.984585i	0.428269	+	0.903651i	$0.359124\pi$
$523$	−13.0000	+	22.5167i	−0.568450	+	0.984585i	−0.996719	+	0.0809336i	$0.974210\pi$
$524$	12.0000			0.524222
$525$		0			0
$526$	12.0000			0.523225
$527$	−6.00000	+	10.3923i	−0.261364	+	0.452696i
$528$		0			0
$529$	7.00000	+	12.1244i	0.304348	+	0.527146i
$530$	9.00000	−	15.5885i	0.390935	−	0.677119i
$531$		0			0
$532$	5.00000	−	1.73205i	0.216777	−	0.0750939i
$533$	6.00000			0.259889
$534$		0			0
$535$	4.50000	+	7.79423i	0.194552	+	0.336974i
$536$	−2.50000	−	4.33013i	−0.107984	−	0.187033i
$537$		0			0
$538$	−9.00000			−0.388018
$539$	−6.50000	−	2.59808i	−0.279975	−	0.111907i
$540$		0			0
$541$	−11.5000	+	19.9186i	−0.494424	+	0.856367i	−0.999979	−	0.00642713i	$-0.997954\pi$
$541$	−11.5000	+	19.9186i	−0.494424	+	0.856367i	0.505556	+	0.862794i	$0.331288\pi$
$542$	8.00000	+	13.8564i	0.343629	+	0.595184i
$543$		0			0
$544$	1.50000	−	2.59808i	0.0643120	−	0.111392i
$545$	33.0000			1.41356
$546$		0			0
$547$	−46.0000			−1.96682			−0.983409	−	0.181402i	$-0.941936\pi$
$547$	−46.0000			−1.96682			−0.983409	+	0.181402i	$0.941936\pi$
$548$	9.00000	−	15.5885i	0.384461	−	0.665906i
$549$		0			0
$550$	−2.00000	−	3.46410i	−0.0852803	−	0.147710i
$551$	−6.00000	+	10.3923i	−0.255609	+	0.442727i
$552$		0			0
$553$	−8.50000	+	44.1673i	−0.361457	+	1.87818i
$554$	−10.0000			−0.424859
$555$		0			0
$556$	−7.00000	−	12.1244i	−0.296866	−	0.514187i
$557$	−18.0000	−	31.1769i	−0.762684	−	1.32101i	−0.941462	−	0.337119i	$-0.890548\pi$
$557$	−18.0000	−	31.1769i	−0.762684	−	1.32101i	0.178778	−	0.983890i	$-0.442786\pi$
$558$		0			0
$559$	4.00000			0.169182
$560$	−6.00000	−	5.19615i	−0.253546	−	0.219578i
$561$		0			0
$562$	1.50000	−	2.59808i	0.0632737	−	0.109593i
$563$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$563$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$564$		0			0
$565$	−9.00000	+	15.5885i	−0.378633	+	0.655811i
$566$	14.0000			0.588464
$567$		0			0
$568$	−12.0000			−0.503509
$569$	−3.00000	+	5.19615i	−0.125767	+	0.217834i	−0.922032	−	0.387113i	$-0.873472\pi$
$569$	−3.00000	+	5.19615i	−0.125767	+	0.217834i	0.796266	+	0.604947i	$0.206806\pi$
$570$		0			0
$571$	−19.0000	−	32.9090i	−0.795125	−	1.37720i	−0.922760	−	0.385376i	$-0.874072\pi$
$571$	−19.0000	−	32.9090i	−0.795125	−	1.37720i	0.127634	−	0.991821i	$-0.459262\pi$
$572$	−1.00000	+	1.73205i	−0.0418121	+	0.0724207i
$573$		0			0
$574$	−1.50000	+	7.79423i	−0.0626088	+	0.325325i
$575$	−12.0000			−0.500435
$576$		0			0
$577$	−8.50000	−	14.7224i	−0.353860	−	0.612903i	0.633062	−	0.774101i	$-0.281798\pi$
$577$	−8.50000	−	14.7224i	−0.353860	−	0.612903i	−0.986922	+	0.161198i	$0.948464\pi$
$578$	4.00000	+	6.92820i	0.166378	+	0.288175i
$579$		0			0
$580$	18.0000			0.747409
$581$	−22.5000	+	7.79423i	−0.933457	+	0.323359i
$582$		0			0
$583$	3.00000	−	5.19615i	0.124247	−	0.215203i
$584$	8.00000	+	13.8564i	0.331042	+	0.573382i
$585$		0			0
$586$	−12.0000	+	20.7846i	−0.495715	+	0.858604i
$587$	−36.0000			−1.48588			−0.742940	−	0.669359i	$-0.766569\pi$
$587$	−36.0000			−1.48588			−0.742940	+	0.669359i	$0.766569\pi$
$588$		0			0
$589$	−8.00000			−0.329634
$590$	−18.0000	+	31.1769i	−0.741048	+	1.28353i
$591$		0			0
$592$	−1.00000	−	1.73205i	−0.0410997	−	0.0711868i
$593$	3.00000	−	5.19615i	0.123195	−	0.213380i	−0.797831	−	0.602881i	$-0.794019\pi$
$593$	3.00000	−	5.19615i	0.123195	−	0.213380i	0.921026	+	0.389501i	$0.127353\pi$
$594$		0			0
$595$	−22.5000	+	7.79423i	−0.922410	+	0.319532i
$596$	−6.00000			−0.245770
$597$		0			0
$598$	3.00000	+	5.19615i	0.122679	+	0.212486i
$599$	13.5000	+	23.3827i	0.551595	+	0.955391i	0.998160	+	0.0606393i	$0.0193139\pi$
$599$	13.5000	+	23.3827i	0.551595	+	0.955391i	−0.446565	+	0.894751i	$0.647353\pi$
$600$		0			0
$601$	−28.0000			−1.14214			−0.571072	−	0.820900i	$-0.693472\pi$
$601$	−28.0000			−1.14214			−0.571072	+	0.820900i	$0.693472\pi$
$602$	−1.00000	+	5.19615i	−0.0407570	+	0.211779i
$603$		0			0
$604$	0.500000	−	0.866025i	0.0203447	−	0.0352381i
$605$	−1.50000	−	2.59808i	−0.0609837	−	0.105627i
$606$		0			0
$607$	0.500000	−	0.866025i	0.0202944	−	0.0351509i	−0.855700	−	0.517472i	$-0.826873\pi$
$607$	0.500000	−	0.866025i	0.0202944	−	0.0351509i	0.875994	+	0.482322i	$0.160206\pi$
$608$	2.00000			0.0811107
$609$		0			0
$610$	15.0000			0.607332
$611$	−9.00000	+	15.5885i	−0.364101	+	0.630641i
$612$		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	$0.618008\pi$
$614$	11.0000	−	19.0526i	0.443924	−	0.768899i
$615$		0			0
$616$	−2.00000	−	1.73205i	−0.0805823	−	0.0697863i
$617$	−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	$-0.237620\pi$
$619$	−5.50000	−	9.52628i	−0.221064	−	0.382893i	−0.955131	+	0.296183i	$0.904286\pi$
$620$	6.00000	+	10.3923i	0.240966	+	0.417365i
$621$		0			0
$622$	−21.0000			−0.842023
$623$	−3.00000	+	15.5885i	−0.120192	+	0.624538i
$624$		0			0
$625$	14.5000	−	25.1147i	0.580000	−	1.00459i
$626$	−1.00000	−	1.73205i	−0.0399680	−	0.0692267i
$627$		0			0
$628$	−10.0000	+	17.3205i	−0.399043	+	0.691164i
$629$	−6.00000			−0.239236
$630$		0			0
$631$	−16.0000			−0.636950			−0.318475	−	0.947931i	$-0.603171\pi$
$631$	−16.0000			−0.636950			−0.318475	+	0.947931i	$0.603171\pi$
$632$	−8.50000	+	14.7224i	−0.338112	+	0.585627i
$633$		0			0
$634$	−16.5000	−	28.5788i	−0.655299	−	1.13501i
$635$	−16.5000	+	28.5788i	−0.654783	+	1.13412i
$636$		0			0
$637$	11.0000	−	8.66025i	0.435836	−	0.343132i
$638$	6.00000			0.237542
$639$		0			0
$640$	−1.50000	−	2.59808i	−0.0592927	−	0.102698i
$641$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$641$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$642$		0			0
$643$	−28.0000			−1.10421			−0.552106	−	0.833774i	$-0.686176\pi$
$643$	−28.0000			−1.10421			−0.552106	+	0.833774i	$0.686176\pi$
$644$	−7.50000	+	2.59808i	−0.295541	+	0.102379i
$645$		0			0
$646$	3.00000	−	5.19615i	0.118033	−	0.204440i
$647$	19.5000	+	33.7750i	0.766624	+	1.32783i	0.939384	+	0.342868i	$0.111398\pi$
$647$	19.5000	+	33.7750i	0.766624	+	1.32783i	−0.172760	+	0.984964i	$0.555268\pi$
$648$		0			0
$649$	−6.00000	+	10.3923i	−0.235521	+	0.407934i
$650$	8.00000			0.313786
$651$		0			0
$652$	−1.00000			−0.0391630
$653$	−10.5000	+	18.1865i	−0.410897	+	0.711694i	−0.994988	−	0.0999939i	$-0.968118\pi$
$653$	−10.5000	+	18.1865i	−0.410897	+	0.711694i	0.584091	+	0.811688i	$0.301451\pi$
$654$		0			0
$655$	−18.0000	−	31.1769i	−0.703318	−	1.21818i
$656$	−1.50000	+	2.59808i	−0.0585652	+	0.101438i
$657$		0			0
$658$	−18.0000	−	15.5885i	−0.701713	−	0.607701i
$659$	33.0000			1.28550			0.642749	−	0.766077i	$-0.277794\pi$
$659$	33.0000			1.28550			0.642749	+	0.766077i	$0.277794\pi$
$660$		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	$-0.380480\pi$
$662$	−5.50000	−	9.52628i	−0.213764	−	0.370249i
$663$		0			0
$664$	−9.00000			−0.349268
$665$	−12.0000	−	10.3923i	−0.465340	−	0.402996i
$666$		0			0
$667$	9.00000	−	15.5885i	0.348481	−	0.603587i
$668$	−6.00000	−	10.3923i	−0.232147	−	0.402090i
$669$		0			0
$670$	−7.50000	+	12.9904i	−0.289750	+	0.501862i
$671$	5.00000			0.193023
$672$		0			0
$673$	−34.0000			−1.31060			−0.655302	−	0.755367i	$-0.727459\pi$
$673$	−34.0000			−1.31060			−0.655302	+	0.755367i	$0.727459\pi$
$674$	11.0000	−	19.0526i	0.423704	−	0.733877i
$675$		0			0
$676$	4.50000	+	7.79423i	0.173077	+	0.299778i
$677$	−15.0000	+	25.9808i	−0.576497	+	0.998522i	0.419380	+	0.907811i	$0.362247\pi$
$677$	−15.0000	+	25.9808i	−0.576497	+	0.998522i	−0.995877	+	0.0907112i	$0.971086\pi$
$678$		0			0
$679$	42.5000	−	14.7224i	1.63100	−	0.564995i
$680$	−9.00000			−0.345134
$681$		0			0
$682$	2.00000	+	3.46410i	0.0765840	+	0.132647i
$683$	6.00000	+	10.3923i	0.229584	+	0.397650i	0.957685	−	0.287819i	$-0.0929302\pi$
$683$	6.00000	+	10.3923i	0.229584	+	0.397650i	−0.728101	+	0.685470i	$0.759597\pi$
$684$		0			0
$685$	−54.0000			−2.06323
$686$	8.50000	+	16.4545i	0.324532	+	0.628235i
$687$		0			0
$688$	−1.00000	+	1.73205i	−0.0381246	+	0.0660338i
$689$	6.00000	+	10.3923i	0.228582	+	0.395915i
$690$		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	$0.0489666\pi$
$692$	−12.0000			−0.456172
$693$		0			0
$694$	33.0000			1.25266
$695$	−21.0000	+	36.3731i	−0.796575	+	1.37971i
$696$		0			0
$697$	4.50000	+	7.79423i	0.170450	+	0.295227i
$698$	−5.50000	+	9.52628i	−0.208178	+	0.360575i
$699$		0			0
$700$	−2.00000	+	10.3923i	−0.0755929	+	0.392792i
$701$	30.0000			1.13308			0.566542	−	0.824033i	$-0.308281\pi$
$701$	30.0000			1.13308			0.566542	+	0.824033i	$0.308281\pi$
$702$		0			0
$703$	−2.00000	−	3.46410i	−0.0754314	−	0.130651i
$704$	−0.500000	−	0.866025i	−0.0188445	−	0.0326396i
$705$		0			0
$706$	12.0000			0.451626
$707$	24.0000	+	20.7846i	0.902613	+	0.781686i
$708$		0			0
$709$	14.0000	−	24.2487i	0.525781	−	0.910679i	−0.473768	−	0.880650i	$-0.657106\pi$
$709$	14.0000	−	24.2487i	0.525781	−	0.910679i	0.999549	−	0.0300298i	$-0.00956021\pi$
$710$	18.0000	+	31.1769i	0.675528	+	1.17005i
$711$		0			0
$712$	−3.00000	+	5.19615i	−0.112430	+	0.194734i
$713$	12.0000			0.449404
$714$		0			0
$715$	6.00000			0.224387
$716$	6.00000	−	10.3923i	0.224231	−	0.388379i
$717$		0			0
$718$	15.0000	+	25.9808i	0.559795	+	0.969593i
$719$	−10.5000	+	18.1865i	−0.391584	+	0.678243i	−0.992659	−	0.120950i	$-0.961406\pi$
$719$	−10.5000	+	18.1865i	−0.391584	+	0.678243i	0.601075	+	0.799193i	$0.294739\pi$
$720$		0			0
$721$	−7.00000	+	36.3731i	−0.260694	+	1.35460i
$722$	−15.0000			−0.558242
$723$		0			0
$724$	5.00000	+	8.66025i	0.185824	+	0.321856i
$725$	−12.0000	−	20.7846i	−0.445669	−	0.771921i
$726$		0			0
$727$	14.0000			0.519231			0.259616	−	0.965712i	$-0.416404\pi$
$727$	14.0000			0.519231			0.259616	+	0.965712i	$0.416404\pi$
$728$	5.00000	−	1.73205i	0.185312	−	0.0641941i
$729$		0			0
$730$	24.0000	−	41.5692i	0.888280	−	1.53855i
$731$	3.00000	+	5.19615i	0.110959	+	0.192187i
$732$		0			0
$733$	24.5000	−	42.4352i	0.904928	−	1.56738i	0.0839145	−	0.996473i	$-0.473258\pi$
$733$	24.5000	−	42.4352i	0.904928	−	1.56738i	0.821014	−	0.570909i	$-0.193409\pi$
$734$	−10.0000			−0.369107
$735$		0			0
$736$	−3.00000			−0.110581
$737$	−2.50000	+	4.33013i	−0.0920887	+	0.159502i
$738$		0			0
$739$	−19.0000	−	32.9090i	−0.698926	−	1.21058i	−0.968839	−	0.247691i	$-0.920328\pi$
$739$	−19.0000	−	32.9090i	−0.698926	−	1.21058i	0.269913	−	0.962885i	$-0.413005\pi$
$740$	−3.00000	+	5.19615i	−0.110282	+	0.191014i
$741$		0			0
$742$	−15.0000	+	5.19615i	−0.550667	+	0.190757i
$743$	30.0000			1.10059			0.550297	−	0.834969i	$-0.314515\pi$
$743$	30.0000			1.10059			0.550297	+	0.834969i	$0.314515\pi$
$744$		0			0
$745$	9.00000	+	15.5885i	0.329734	+	0.571117i
$746$	0.500000	+	0.866025i	0.0183063	+	0.0317074i
$747$		0			0
$748$	−3.00000			−0.109691
$749$	1.50000	−	7.79423i	0.0548088	−	0.284795i
$750$		0			0
$751$	−1.00000	+	1.73205i	−0.0364905	+	0.0632034i	−0.883694	−	0.468065i	$-0.844951\pi$
$751$	−1.00000	+	1.73205i	−0.0364905	+	0.0632034i	0.847203	+	0.531269i	$0.178285\pi$
$752$	−4.50000	−	7.79423i	−0.164098	−	0.284226i
$753$		0			0
$754$	−6.00000	+	10.3923i	−0.218507	+	0.378465i
$755$	−3.00000			−0.109181
$756$		0			0
$757$	14.0000			0.508839			0.254419	−	0.967094i	$-0.418116\pi$
$757$	14.0000			0.508839			0.254419	+	0.967094i	$0.418116\pi$
$758$	−14.5000	+	25.1147i	−0.526664	+	0.912208i
$759$		0			0
$760$	−3.00000	−	5.19615i	−0.108821	−	0.188484i
$761$	7.50000	−	12.9904i	0.271875	−	0.470901i	−0.697467	−	0.716617i	$-0.745690\pi$
$761$	7.50000	−	12.9904i	0.271875	−	0.470901i	0.969342	+	0.245716i	$0.0790230\pi$
$762$		0			0
$763$	−22.0000	−	19.0526i	−0.796453	−	0.689749i
$764$	−24.0000			−0.868290
$765$		0			0
$766$	12.0000	+	20.7846i	0.433578	+	0.750978i
$767$	−12.0000	−	20.7846i	−0.433295	−	0.750489i
$768$		0			0
$769$	−46.0000			−1.65880			−0.829401	−	0.558653i	$-0.811318\pi$
$769$	−46.0000			−1.65880			−0.829401	+	0.558653i	$0.811318\pi$
$770$	−1.50000	+	7.79423i	−0.0540562	+	0.280885i
$771$		0			0
$772$	−1.00000	+	1.73205i	−0.0359908	+	0.0623379i
$773$	−13.5000	−	23.3827i	−0.485561	−	0.841017i	0.514301	−	0.857610i	$-0.328051\pi$
$773$	−13.5000	−	23.3827i	−0.485561	−	0.841017i	−0.999862	+	0.0165929i	$0.994718\pi$
$774$		0			0
$775$	8.00000	−	13.8564i	0.287368	−	0.497737i
$776$	17.0000			0.610264
$777$		0			0
$778$	−21.0000			−0.752886
$779$	−3.00000	+	5.19615i	−0.107486	+	0.186171i
$780$		0			0
$781$	6.00000	+	10.3923i	0.214697	+	0.371866i
$782$	−4.50000	+	7.79423i	−0.160920	+	0.278721i
$783$		0			0
$784$	1.00000	+	6.92820i	0.0357143	+	0.247436i
$785$	60.0000			2.14149
$786$		0			0
$787$	11.0000	+	19.0526i	0.392108	+	0.679150i	0.992727	−	0.120384i	$-0.0384127\pi$
$787$	11.0000	+	19.0526i	0.392108	+	0.679150i	−0.600620	+	0.799535i	$0.705079\pi$
$788$	6.00000	+	10.3923i	0.213741	+	0.370211i
$789$		0			0
$790$	51.0000			1.81450
$791$	15.0000	−	5.19615i	0.533339	−	0.184754i
$792$		0			0
$793$	−5.00000	+	8.66025i	−0.177555	+	0.307535i
$794$	−1.00000	−	1.73205i	−0.0354887	−	0.0614682i
$795$		0			0
$796$	−1.00000	+	1.73205i	−0.0354441	+	0.0613909i
$797$	27.0000			0.956389			0.478195	−	0.878254i	$-0.341291\pi$
$797$	27.0000			0.956389			0.478195	+	0.878254i	$0.341291\pi$
$798$		0			0
$799$	−27.0000			−0.955191
$800$	−2.00000	+	3.46410i	−0.0707107	+	0.122474i
$801$		0			0
$802$	15.0000	+	25.9808i	0.529668	+	0.917413i
$803$	8.00000	−	13.8564i	0.282314	−	0.488982i
$804$		0			0
$805$	18.0000	+	15.5885i	0.634417	+	0.549421i
$806$	−8.00000			−0.281788
$807$		0			0
$808$	6.00000	+	10.3923i	0.211079	+	0.365600i
$809$	−4.50000	−	7.79423i	−0.158212	−	0.274030i	0.776012	−	0.630718i	$-0.217239\pi$
$809$	−4.50000	−	7.79423i	−0.158212	−	0.274030i	−0.934224	+	0.356687i	$0.883906\pi$
$810$		0			0
$811$	26.0000			0.912983			0.456492	−	0.889728i	$-0.349106\pi$
$811$	26.0000			0.912983			0.456492	+	0.889728i	$0.349106\pi$
$812$	−12.0000	−	10.3923i	−0.421117	−	0.364698i
$813$		0			0
$814$	−1.00000	+	1.73205i	−0.0350500	+	0.0607083i
$815$	1.50000	+	2.59808i	0.0525427	+	0.0910066i
$816$		0			0
$817$	−2.00000	+	3.46410i	−0.0699711	+	0.121194i
$818$	14.0000			0.489499
$819$		0			0
$820$	9.00000			0.314294
$821$	−12.0000	+	20.7846i	−0.418803	+	0.725388i	−0.995819	−	0.0913446i	$-0.970884\pi$
$821$	−12.0000	+	20.7846i	−0.418803	+	0.725388i	0.577016	+	0.816733i	$0.304217\pi$
$822$		0			0
$823$	−7.00000	−	12.1244i	−0.244005	−	0.422628i	0.717847	−	0.696201i	$-0.245128\pi$
$823$	−7.00000	−	12.1244i	−0.244005	−	0.422628i	−0.961851	+	0.273573i	$0.911795\pi$
$824$	−7.00000	+	12.1244i	−0.243857	+	0.422372i
$825$		0			0
$826$	30.0000	−	10.3923i	1.04383	−	0.361595i
$827$	−33.0000			−1.14752			−0.573761	−	0.819023i	$-0.694516\pi$
$827$	−33.0000			−1.14752			−0.573761	+	0.819023i	$0.694516\pi$
$828$		0			0
$829$	17.0000	+	29.4449i	0.590434	+	1.02266i	0.994174	+	0.107788i	$0.0343769\pi$
$829$	17.0000	+	29.4449i	0.590434	+	1.02266i	−0.403739	+	0.914874i	$0.632290\pi$
$830$	13.5000	+	23.3827i	0.468592	+	0.811625i
$831$		0			0
$832$	2.00000			0.0693375
$833$	19.5000	+	7.79423i	0.675635	+	0.270054i
$834$		0			0
$835$	−18.0000	+	31.1769i	−0.622916	+	1.07892i
$836$	−1.00000	−	1.73205i	−0.0345857	−	0.0599042i
$837$		0			0
$838$	−12.0000	+	20.7846i	−0.414533	+	0.717992i
$839$	−33.0000			−1.13929			−0.569643	−	0.821892i	$-0.692919\pi$
$839$	−33.0000			−1.13929			−0.569643	+	0.821892i	$0.692919\pi$
$840$		0			0
$841$	7.00000			0.241379
$842$	−10.0000	+	17.3205i	−0.344623	+	0.596904i
$843$		0			0
$844$	11.0000	+	19.0526i	0.378636	+	0.655816i
$845$	13.5000	−	23.3827i	0.464414	−	0.804389i
$846$		0			0
$847$	−0.500000	+	2.59808i	−0.0171802	+	0.0892710i
$848$	−6.00000			−0.206041
$849$		0			0
$850$	6.00000	+	10.3923i	0.205798	+	0.356453i
$851$	3.00000	+	5.19615i	0.102839	+	0.178122i
$852$		0			0
$853$	17.0000			0.582069			0.291034	−	0.956713i	$-0.406001\pi$
$853$	17.0000			0.582069			0.291034	+	0.956713i	$0.406001\pi$
$854$	−10.0000	−	8.66025i	−0.342193	−	0.296348i
$855$		0			0
$856$	1.50000	−	2.59808i	0.0512689	−	0.0888004i
$857$	−10.5000	−	18.1865i	−0.358673	−	0.621240i	0.629066	−	0.777352i	$-0.283437\pi$
$857$	−10.5000	−	18.1865i	−0.358673	−	0.621240i	−0.987739	+	0.156112i	$0.950104\pi$
$858$		0			0
$859$	−17.5000	+	30.3109i	−0.597092	+	1.03419i	0.396156	+	0.918183i	$0.370344\pi$
$859$	−17.5000	+	30.3109i	−0.597092	+	1.03419i	−0.993248	+	0.116011i	$0.962989\pi$
$860$	6.00000			0.204598
$861$		0			0
$862$	6.00000			0.204361
$863$	19.5000	−	33.7750i	0.663788	−	1.14971i	−0.315825	−	0.948818i	$-0.602281\pi$
$863$	19.5000	−	33.7750i	0.663788	−	1.14971i	0.979612	−	0.200897i	$-0.0643855\pi$
$864$		0			0
$865$	18.0000	+	31.1769i	0.612018	+	1.06005i
$866$	−17.5000	+	30.3109i	−0.594674	+	1.03001i
$867$		0			0
$868$	2.00000	−	10.3923i	0.0678844	−	0.352738i
$869$	17.0000			0.576686
$870$		0			0
$871$	−5.00000	−	8.66025i	−0.169419	−	0.293442i
$872$	−5.50000	−	9.52628i	−0.186254	−	0.322601i
$873$		0			0
$874$	−6.00000			−0.202953
$875$	−7.50000	+	2.59808i	−0.253546	+	0.0878310i
$876$		0			0
$877$	18.5000	−	32.0429i	0.624701	−	1.08201i	−0.363898	−	0.931439i	$-0.618554\pi$
$877$	18.5000	−	32.0429i	0.624701	−	1.08201i	0.988599	−	0.150574i	$-0.0481123\pi$
$878$	9.50000	+	16.4545i	0.320609	+	0.555312i
$879$		0			0
$880$	−1.50000	+	2.59808i	−0.0505650	+	0.0875811i
$881$	−6.00000			−0.202145			−0.101073	−	0.994879i	$-0.532227\pi$
$881$	−6.00000			−0.202145			−0.101073	+	0.994879i	$0.532227\pi$
$882$		0			0
$883$	−13.0000			−0.437485			−0.218742	−	0.975783i	$-0.570195\pi$
$883$	−13.0000			−0.437485			−0.218742	+	0.975783i	$0.570195\pi$
$884$	3.00000	−	5.19615i	0.100901	−	0.174766i
$885$		0			0
$886$	9.00000	+	15.5885i	0.302361	+	0.523704i
$887$	6.00000	−	10.3923i	0.201460	−	0.348939i	−0.747539	−	0.664218i	$-0.768765\pi$
$887$	6.00000	−	10.3923i	0.201460	−	0.348939i	0.948999	+	0.315279i	$0.102098\pi$
$888$		0			0
$889$	27.5000	−	9.52628i	0.922320	−	0.319501i
$890$	18.0000			0.603361
$891$		0			0
$892$	−10.0000	−	17.3205i	−0.334825	−	0.579934i
$893$	−9.00000	−	15.5885i	−0.301174	−	0.521648i
$894$		0			0
$895$	−36.0000			−1.20335
$896$	−0.500000	+	2.59808i	−0.0167038	+	0.0867956i
$897$		0			0
$898$	15.0000	−	25.9808i	0.500556	−	0.866989i
$899$	12.0000	+	20.7846i	0.400222	+	0.693206i
$900$		0			0
$901$	−9.00000	+	15.5885i	−0.299833	+	0.519327i
$902$	3.00000			0.0998891
$903$		0			0
$904$	6.00000			0.199557
$905$	15.0000	−	25.9808i	0.498617	−	0.863630i
$906$		0			0
$907$	−23.5000	−	40.7032i	−0.780305	−	1.35153i	−0.931764	−	0.363064i	$-0.881731\pi$
$907$	−23.5000	−	40.7032i	−0.780305	−	1.35153i	0.151460	−	0.988463i	$-0.451603\pi$
$908$	−10.5000	+	18.1865i	−0.348455	+	0.603541i
$909$		0			0
$910$	−12.0000	−	10.3923i	−0.397796	−	0.344502i
$911$	27.0000			0.894550			0.447275	−	0.894397i	$-0.352395\pi$
$911$	27.0000			0.894550			0.447275	+	0.894397i	$0.352395\pi$
$912$		0			0
$913$	4.50000	+	7.79423i	0.148928	+	0.257951i
$914$	−19.0000	−	32.9090i	−0.628464	−	1.08853i
$915$		0			0
$916$	−10.0000			−0.330409
$917$	−6.00000	+	31.1769i	−0.198137	+	1.02955i
$918$		0			0
$919$	3.50000	−	6.06218i	0.115454	−	0.199973i	−0.802507	−	0.596643i	$-0.796501\pi$
$919$	3.50000	−	6.06218i	0.115454	−	0.199973i	0.917961	+	0.396670i	$0.129834\pi$
$920$	4.50000	+	7.79423i	0.148361	+	0.256968i
$921$		0			0
$922$	3.00000	−	5.19615i	0.0987997	−	0.171126i
$923$	−24.0000			−0.789970
$924$		0			0
$925$	8.00000			0.263038
$926$	8.00000	−	13.8564i	0.262896	−	0.455350i
$927$		0			0
$928$	−3.00000	−	5.19615i	−0.0984798	−	0.170572i
$929$	9.00000	−	15.5885i	0.295280	−	0.511441i	−0.679770	−	0.733426i	$-0.737920\pi$
$929$	9.00000	−	15.5885i	0.295280	−	0.511441i	0.975050	+	0.221985i	$0.0712536\pi$
$930$		0			0
$931$	2.00000	+	13.8564i	0.0655474	+	0.454125i
$932$	−27.0000			−0.884414
$933$		0			0
$934$		0			0
$935$	4.50000	+	7.79423i	0.147166	+	0.254899i
$936$		0			0
$937$	−34.0000			−1.11073			−0.555366	−	0.831606i	$-0.687422\pi$
$937$	−34.0000			−1.11073			−0.555366	+	0.831606i	$0.687422\pi$
$938$	12.5000	−	4.33013i	0.408139	−	0.141384i
$939$		0			0
$940$	−13.5000	+	23.3827i	−0.440321	+	0.762659i
$941$	−9.00000	−	15.5885i	−0.293392	−	0.508169i	0.681218	−	0.732081i	$-0.261451\pi$
$941$	−9.00000	−	15.5885i	−0.293392	−	0.508169i	−0.974609	+	0.223912i	$0.928117\pi$
$942$		0			0
$943$	4.50000	−	7.79423i	0.146540	−	0.253815i
$944$	12.0000			0.390567
$945$		0			0
$946$	2.00000			0.0650256
$947$	9.00000	−	15.5885i	0.292461	−	0.506557i	−0.681930	−	0.731417i	$-0.738859\pi$
$947$	9.00000	−	15.5885i	0.292461	−	0.506557i	0.974391	+	0.224860i	$0.0721926\pi$
$948$		0			0
$949$	16.0000	+	27.7128i	0.519382	+	0.899596i
$950$	−4.00000	+	6.92820i	−0.129777	+	0.224781i
$951$		0			0
$952$	6.00000	+	5.19615i	0.194461	+	0.168408i
$953$	−27.0000			−0.874616			−0.437308	−	0.899312i	$-0.644068\pi$
$953$	−27.0000			−0.874616			−0.437308	+	0.899312i	$0.644068\pi$
$954$		0			0
$955$	36.0000	+	62.3538i	1.16493	+	2.01772i
$956$	3.00000	+	5.19615i	0.0970269	+	0.168056i
$957$		0			0
$958$	−24.0000			−0.775405
$959$	36.0000	+	31.1769i	1.16250	+	1.00676i
$960$		0			0
$961$	7.50000	−	12.9904i	0.241935	−	0.419045i
$962$	−2.00000	−	3.46410i	−0.0644826	−	0.111687i
$963$		0			0
$964$	5.00000	−	8.66025i	0.161039	−	0.278928i
$965$	6.00000			0.193147
$966$		0			0
$967$	−43.0000			−1.38279			−0.691393	−	0.722478i	$-0.743003\pi$
$967$	−43.0000			−1.38279			−0.691393	+	0.722478i	$0.743003\pi$
$968$	−0.500000	+	0.866025i	−0.0160706	+	0.0278351i
$969$		0			0
$970$	−25.5000	−	44.1673i	−0.818756	−	1.41813i
$971$	−21.0000	+	36.3731i	−0.673922	+	1.16727i	0.302861	+	0.953035i	$0.402058\pi$
$971$	−21.0000	+	36.3731i	−0.673922	+	1.16727i	−0.976783	+	0.214232i	$0.931275\pi$
$972$		0			0
$973$	35.0000	−	12.1244i	1.12205	−	0.388689i
$974$	8.00000			0.256337
$975$		0			0
$976$	−2.50000	−	4.33013i	−0.0800230	−	0.138604i
$977$	−18.0000	−	31.1769i	−0.575871	−	0.997438i	−0.995946	−	0.0899487i	$-0.971330\pi$
$977$	−18.0000	−	31.1769i	−0.575871	−	0.997438i	0.420075	−	0.907489i	$-0.362004\pi$
$978$		0			0
$979$	6.00000			0.191761
$980$	16.5000	−	12.9904i	0.527073	−	0.414963i
$981$		0			0
$982$	−1.50000	+	2.59808i	−0.0478669	+	0.0829079i
$983$	−22.5000	−	38.9711i	−0.717639	−	1.24299i	−0.961933	−	0.273285i	$-0.911890\pi$
$983$	−22.5000	−	38.9711i	−0.717639	−	1.24299i	0.244294	−	0.969701i	$-0.421444\pi$
$984$		0			0
$985$	18.0000	−	31.1769i	0.573528	−	0.993379i
$986$	−18.0000			−0.573237
$987$		0			0
$988$	4.00000			0.127257
$989$	3.00000	−	5.19615i	0.0953945	−	0.165228i
$990$		0			0
$991$	11.0000	+	19.0526i	0.349427	+	0.605224i	0.986148	−	0.165870i	$-0.0530431\pi$
$991$	11.0000	+	19.0526i	0.349427	+	0.605224i	−0.636721	+	0.771094i	$0.719710\pi$
$992$	2.00000	−	3.46410i	0.0635001	−	0.109985i
$993$		0			0
$994$	6.00000	−	31.1769i	0.190308	−	0.988872i
$995$	6.00000			0.190213
$996$		0			0
$997$	−13.0000	−	22.5167i	−0.411714	−	0.713110i	0.583363	−	0.812211i	$-0.301736\pi$
$997$	−13.0000	−	22.5167i	−0.411714	−	0.713110i	−0.995077	+	0.0991016i	$0.968403\pi$
$998$	14.0000	+	24.2487i	0.443162	+	0.767580i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1386.2.k.c.793.1		2
3.2	odd	2		462.2.i.c.331.1	yes	2
7.2	even	3		9702.2.a.cf.1.1		1
7.4	even	3	inner	1386.2.k.c.991.1		2
7.5	odd	6		9702.2.a.bd.1.1		1
21.2	odd	6		3234.2.a.h.1.1		1
21.5	even	6		3234.2.a.g.1.1		1
21.11	odd	6		462.2.i.c.67.1	✓	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
462.2.i.c.67.1	✓	2	21.11	odd	6
462.2.i.c.331.1	yes	2	3.2	odd	2
1386.2.k.c.793.1		2	1.1	even	1	trivial
1386.2.k.c.991.1		2	7.4	even	3	inner
3234.2.a.g.1.1		1	21.5	even	6
3234.2.a.h.1.1		1	21.2	odd	6
9702.2.a.bd.1.1		1	7.5	odd	6
9702.2.a.cf.1.1		1	7.2	even	3

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 1386 = 2 \cdot 3^{2} \cdot 7 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1386.k (of order \(3\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(-0.500000 + 0.866025i) q^{2} +(-0.500000 - 0.866025i) q^{4} +(-1.50000 + 2.59808i) q^{5} +(2.50000 - 0.866025i) q^{7} +1.00000 q^{8} +O(q^{10})\)	\(q+(-0.500000 + 0.866025i) q^{2} +(-0.500000 - 0.866025i) q^{4} +(-1.50000 + 2.59808i) q^{5} +(2.50000 - 0.866025i) q^{7} +1.00000 q^{8} +(-1.50000 - 2.59808i) q^{10} +(-0.500000 - 0.866025i) q^{11} +2.00000 q^{13} +(-0.500000 + 2.59808i) q^{14} +(-0.500000 + 0.866025i) q^{16} +(1.50000 + 2.59808i) q^{17} +(-1.00000 + 1.73205i) q^{19} +3.00000 q^{20} +1.00000 q^{22} +(1.50000 - 2.59808i) q^{23} +(-2.00000 - 3.46410i) q^{25} +(-1.00000 + 1.73205i) q^{26} +(-2.00000 - 1.73205i) q^{28} +6.00000 q^{29} +(2.00000 + 3.46410i) q^{31} +(-0.500000 - 0.866025i) q^{32} -3.00000 q^{34} +(-1.50000 + 7.79423i) q^{35} +(-1.00000 + 1.73205i) q^{37} +(-1.00000 - 1.73205i) q^{38} +(-1.50000 + 2.59808i) q^{40} +3.00000 q^{41} +2.00000 q^{43} +(-0.500000 + 0.866025i) q^{44} +(1.50000 + 2.59808i) q^{46} +(-4.50000 + 7.79423i) q^{47} +(5.50000 - 4.33013i) q^{49} +4.00000 q^{50} +(-1.00000 - 1.73205i) q^{52} +(3.00000 + 5.19615i) q^{53} +3.00000 q^{55} +(2.50000 - 0.866025i) q^{56} +(-3.00000 + 5.19615i) q^{58} +(-6.00000 - 10.3923i) q^{59} +(-2.50000 + 4.33013i) q^{61} -4.00000 q^{62} +1.00000 q^{64} +(-3.00000 + 5.19615i) q^{65} +(-2.50000 - 4.33013i) q^{67} +(1.50000 - 2.59808i) q^{68} +(-6.00000 - 5.19615i) q^{70} -12.0000 q^{71} +(8.00000 + 13.8564i) q^{73} +(-1.00000 - 1.73205i) q^{74} +2.00000 q^{76} +(-2.00000 - 1.73205i) q^{77} +(-8.50000 + 14.7224i) q^{79} +(-1.50000 - 2.59808i) q^{80} +(-1.50000 + 2.59808i) q^{82} -9.00000 q^{83} -9.00000 q^{85} +(-1.00000 + 1.73205i) q^{86} +(-0.500000 - 0.866025i) q^{88} +(-3.00000 + 5.19615i) q^{89} +(5.00000 - 1.73205i) q^{91} -3.00000 q^{92} +(-4.50000 - 7.79423i) q^{94} +(-3.00000 - 5.19615i) q^{95} +17.0000 q^{97} +(1.00000 + 6.92820i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - q^{2} - q^{4} - 3 q^{5} + 5 q^{7} + 2 q^{8}+O(q^{10}) \) 2 * q - q^2 - q^4 - 3 * q^5 + 5 * q^7 + 2 * q^8	\( 2 q - q^{2} - q^{4} - 3 q^{5} + 5 q^{7} + 2 q^{8} - 3 q^{10} - q^{11} + 4 q^{13} - q^{14} - q^{16} + 3 q^{17} - 2 q^{19} + 6 q^{20} + 2 q^{22} + 3 q^{23} - 4 q^{25} - 2 q^{26} - 4 q^{28} + 12 q^{29} + 4 q^{31} - q^{32} - 6 q^{34} - 3 q^{35} - 2 q^{37} - 2 q^{38} - 3 q^{40} + 6 q^{41} + 4 q^{43} - q^{44} + 3 q^{46} - 9 q^{47} + 11 q^{49} + 8 q^{50} - 2 q^{52} + 6 q^{53} + 6 q^{55} + 5 q^{56} - 6 q^{58} - 12 q^{59} - 5 q^{61} - 8 q^{62} + 2 q^{64} - 6 q^{65} - 5 q^{67} + 3 q^{68} - 12 q^{70} - 24 q^{71} + 16 q^{73} - 2 q^{74} + 4 q^{76} - 4 q^{77} - 17 q^{79} - 3 q^{80} - 3 q^{82} - 18 q^{83} - 18 q^{85} - 2 q^{86} - q^{88} - 6 q^{89} + 10 q^{91} - 6 q^{92} - 9 q^{94} - 6 q^{95} + 34 q^{97} + 2 q^{98}+O(q^{100}) \) 2 * q - q^2 - q^4 - 3 * q^5 + 5 * q^7 + 2 * q^8 - 3 * q^10 - q^11 + 4 * q^13 - q^14 - q^16 + 3 * q^17 - 2 * q^19 + 6 * q^20 + 2 * q^22 + 3 * q^23 - 4 * q^25 - 2 * q^26 - 4 * q^28 + 12 * q^29 + 4 * q^31 - q^32 - 6 * q^34 - 3 * q^35 - 2 * q^37 - 2 * q^38 - 3 * q^40 + 6 * q^41 + 4 * q^43 - q^44 + 3 * q^46 - 9 * q^47 + 11 * q^49 + 8 * q^50 - 2 * q^52 + 6 * q^53 + 6 * q^55 + 5 * q^56 - 6 * q^58 - 12 * q^59 - 5 * q^61 - 8 * q^62 + 2 * q^64 - 6 * q^65 - 5 * q^67 + 3 * q^68 - 12 * q^70 - 24 * q^71 + 16 * q^73 - 2 * q^74 + 4 * q^76 - 4 * q^77 - 17 * q^79 - 3 * q^80 - 3 * q^82 - 18 * q^83 - 18 * q^85 - 2 * q^86 - q^88 - 6 * q^89 + 10 * q^91 - 6 * q^92 - 9 * q^94 - 6 * q^95 + 34 * q^97 + 2 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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