LMFDB - Embedded newform 1386.2.k.k.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:	yes
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.k.991.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(0.500000 - 0.866025i) q^{2} +(-0.500000 - 0.866025i) q^{4} +(-1.00000 + 1.73205i) q^{5} +(2.00000 + 1.73205i) q^{7} -1.00000 q^{8} +O(q^{10})$	\(q+(0.500000 - 0.866025i) q^{2} +(-0.500000 - 0.866025i) q^{4} +(-1.00000 + 1.73205i) q^{5} +(2.00000 + 1.73205i) q^{7} -1.00000 q^{8} +(1.00000 + 1.73205i) q^{10} +(-0.500000 - 0.866025i) q^{11} +2.00000 q^{13} +(2.50000 - 0.866025i) q^{14} +(-0.500000 + 0.866025i) q^{16} +(-1.50000 - 2.59808i) q^{17} +(-3.50000 + 6.06218i) q^{19} +2.00000 q^{20} -1.00000 q^{22} +(-3.50000 + 6.06218i) q^{23} +(0.500000 + 0.866025i) q^{25} +(1.00000 - 1.73205i) q^{26} +(0.500000 - 2.59808i) q^{28} +5.00000 q^{29} +(-1.00000 - 1.73205i) q^{31} +(0.500000 + 0.866025i) q^{32} -3.00000 q^{34} +(-5.00000 + 1.73205i) q^{35} +(-1.50000 + 2.59808i) q^{37} +(3.50000 + 6.06218i) q^{38} +(1.00000 - 1.73205i) q^{40} -6.00000 q^{41} +11.0000 q^{43} +(-0.500000 + 0.866025i) q^{44} +(3.50000 + 6.06218i) q^{46} +(-3.50000 + 6.06218i) q^{47} +(1.00000 + 6.92820i) q^{49} +1.00000 q^{50} +(-1.00000 - 1.73205i) q^{52} +(-2.00000 - 3.46410i) q^{53} +2.00000 q^{55} +(-2.00000 - 1.73205i) q^{56} +(2.50000 - 4.33013i) q^{58} +(5.50000 + 9.52628i) q^{59} +(-5.00000 + 8.66025i) q^{61} -2.00000 q^{62} +1.00000 q^{64} +(-2.00000 + 3.46410i) q^{65} +(2.00000 + 3.46410i) q^{67} +(-1.50000 + 2.59808i) q^{68} +(-1.00000 + 5.19615i) q^{70} +5.00000 q^{71} +(4.00000 + 6.92820i) q^{73} +(1.50000 + 2.59808i) q^{74} +7.00000 q^{76} +(0.500000 - 2.59808i) q^{77} +(4.00000 - 6.92820i) q^{79} +(-1.00000 - 1.73205i) q^{80} +(-3.00000 + 5.19615i) q^{82} -14.0000 q^{83} +6.00000 q^{85} +(5.50000 - 9.52628i) q^{86} +(0.500000 + 0.866025i) q^{88} +(1.00000 - 1.73205i) q^{89} +(4.00000 + 3.46410i) q^{91} +7.00000 q^{92} +(3.50000 + 6.06218i) q^{94} +(-7.00000 - 12.1244i) q^{95} +15.0000 q^{97} +(6.50000 + 2.59808i) q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + q^{2} - q^{4} - 2 q^{5} + 4 q^{7} - 2 q^{8}+O(q^{10}) $ 2 * q + q^2 - q^4 - 2 * q^5 + 4 * q^7 - 2 * q^8	$ 2 q + q^{2} - q^{4} - 2 q^{5} + 4 q^{7} - 2 q^{8} + 2 q^{10} - q^{11} + 4 q^{13} + 5 q^{14} - q^{16} - 3 q^{17} - 7 q^{19} + 4 q^{20} - 2 q^{22} - 7 q^{23} + q^{25} + 2 q^{26} + q^{28} + 10 q^{29} - 2 q^{31} + q^{32} - 6 q^{34} - 10 q^{35} - 3 q^{37} + 7 q^{38} + 2 q^{40} - 12 q^{41} + 22 q^{43} - q^{44} + 7 q^{46} - 7 q^{47} + 2 q^{49} + 2 q^{50} - 2 q^{52} - 4 q^{53} + 4 q^{55} - 4 q^{56} + 5 q^{58} + 11 q^{59} - 10 q^{61} - 4 q^{62} + 2 q^{64} - 4 q^{65} + 4 q^{67} - 3 q^{68} - 2 q^{70} + 10 q^{71} + 8 q^{73} + 3 q^{74} + 14 q^{76} + q^{77} + 8 q^{79} - 2 q^{80} - 6 q^{82} - 28 q^{83} + 12 q^{85} + 11 q^{86} + q^{88} + 2 q^{89} + 8 q^{91} + 14 q^{92} + 7 q^{94} - 14 q^{95} + 30 q^{97} + 13 q^{98}+O(q^{100}) $ 2 * q + q^2 - q^4 - 2 * q^5 + 4 * q^7 - 2 * q^8 + 2 * q^10 - q^11 + 4 * q^13 + 5 * q^14 - q^16 - 3 * q^17 - 7 * q^19 + 4 * q^20 - 2 * q^22 - 7 * q^23 + q^25 + 2 * q^26 + q^28 + 10 * q^29 - 2 * q^31 + q^32 - 6 * q^34 - 10 * q^35 - 3 * q^37 + 7 * q^38 + 2 * q^40 - 12 * q^41 + 22 * q^43 - q^44 + 7 * q^46 - 7 * q^47 + 2 * q^49 + 2 * q^50 - 2 * q^52 - 4 * q^53 + 4 * q^55 - 4 * q^56 + 5 * q^58 + 11 * q^59 - 10 * q^61 - 4 * q^62 + 2 * q^64 - 4 * q^65 + 4 * q^67 - 3 * q^68 - 2 * q^70 + 10 * q^71 + 8 * q^73 + 3 * q^74 + 14 * q^76 + q^77 + 8 * q^79 - 2 * q^80 - 6 * q^82 - 28 * q^83 + 12 * q^85 + 11 * q^86 + q^88 + 2 * q^89 + 8 * q^91 + 14 * q^92 + 7 * q^94 - 14 * q^95 + 30 * q^97 + 13 * 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
$5$	−1.00000	+	1.73205i	−0.447214	+	0.774597i	−0.998203	−	0.0599153i	$-0.980917\pi$
$5$	−1.00000	+	1.73205i	−0.447214	+	0.774597i	0.550990	+	0.834512i	$0.314250\pi$
$6$		0			0
$7$	2.00000	+	1.73205i	0.755929	+	0.654654i
$7$	2.00000	+	1.73205i	0.755929	+	0.654654i
$8$	−1.00000			−0.353553
$9$		0			0
$10$	1.00000	+	1.73205i	0.316228	+	0.547723i
$11$	−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	$0.410544\pi$
$14$	2.50000	−	0.866025i	0.668153	−	0.231455i
$15$		0			0
$16$	−0.500000	+	0.866025i	−0.125000	+	0.216506i
$17$	−1.50000	−	2.59808i	−0.363803	−	0.630126i	0.624780	−	0.780801i	$-0.285189\pi$
$17$	−1.50000	−	2.59808i	−0.363803	−	0.630126i	−0.988583	+	0.150675i	$0.951855\pi$
$18$		0			0
$19$	−3.50000	+	6.06218i	−0.802955	+	1.39076i	0.114708	+	0.993399i	$0.463407\pi$
$19$	−3.50000	+	6.06218i	−0.802955	+	1.39076i	−0.917663	+	0.397360i	$0.869927\pi$
$20$	2.00000			0.447214
$21$		0			0
$22$	−1.00000			−0.213201
$23$	−3.50000	+	6.06218i	−0.729800	+	1.26405i	0.227167	+	0.973856i	$0.427054\pi$
$23$	−3.50000	+	6.06218i	−0.729800	+	1.26405i	−0.956967	+	0.290196i	$0.906280\pi$
$24$		0			0
$25$	0.500000	+	0.866025i	0.100000	+	0.173205i
$26$	1.00000	−	1.73205i	0.196116	−	0.339683i
$27$		0			0
$28$	0.500000	−	2.59808i	0.0944911	−	0.490990i
$29$	5.00000			0.928477			0.464238	−	0.885710i	$-0.346328\pi$
$29$	5.00000			0.928477			0.464238	+	0.885710i	$0.346328\pi$
$30$		0			0
$31$	−1.00000	−	1.73205i	−0.179605	−	0.311086i	0.762140	−	0.647412i	$-0.224149\pi$
$31$	−1.00000	−	1.73205i	−0.179605	−	0.311086i	−0.941745	+	0.336327i	$0.890815\pi$
$32$	0.500000	+	0.866025i	0.0883883	+	0.153093i
$33$		0			0
$34$	−3.00000			−0.514496
$35$	−5.00000	+	1.73205i	−0.845154	+	0.292770i
$36$		0			0
$37$	−1.50000	+	2.59808i	−0.246598	+	0.427121i	−0.962580	−	0.270998i	$-0.912646\pi$
$37$	−1.50000	+	2.59808i	−0.246598	+	0.427121i	0.715981	+	0.698119i	$0.245980\pi$
$38$	3.50000	+	6.06218i	0.567775	+	0.983415i
$39$		0			0
$40$	1.00000	−	1.73205i	0.158114	−	0.273861i
$41$	−6.00000			−0.937043			−0.468521	−	0.883452i	$-0.655213\pi$
$41$	−6.00000			−0.937043			−0.468521	+	0.883452i	$0.655213\pi$
$42$		0			0
$43$	11.0000			1.67748			0.838742	−	0.544529i	$-0.183292\pi$
$43$	11.0000			1.67748			0.838742	+	0.544529i	$0.183292\pi$
$44$	−0.500000	+	0.866025i	−0.0753778	+	0.130558i
$45$		0			0
$46$	3.50000	+	6.06218i	0.516047	+	0.893819i
$47$	−3.50000	+	6.06218i	−0.510527	+	0.884260i	0.489398	+	0.872060i	$0.337217\pi$
$47$	−3.50000	+	6.06218i	−0.510527	+	0.884260i	−0.999926	+	0.0121990i	$0.996117\pi$
$48$		0			0
$49$	1.00000	+	6.92820i	0.142857	+	0.989743i
$50$	1.00000			0.141421
$51$		0			0
$52$	−1.00000	−	1.73205i	−0.138675	−	0.240192i
$53$	−2.00000	−	3.46410i	−0.274721	−	0.475831i	0.695344	−	0.718677i	$-0.255252\pi$
$53$	−2.00000	−	3.46410i	−0.274721	−	0.475831i	−0.970065	+	0.242846i	$0.921919\pi$
$54$		0			0
$55$	2.00000			0.269680
$56$	−2.00000	−	1.73205i	−0.267261	−	0.231455i
$57$		0			0
$58$	2.50000	−	4.33013i	0.328266	−	0.568574i
$59$	5.50000	+	9.52628i	0.716039	+	1.24022i	0.962557	+	0.271078i	$0.0873801\pi$
$59$	5.50000	+	9.52628i	0.716039	+	1.24022i	−0.246518	+	0.969138i	$0.579287\pi$
$60$		0			0
$61$	−5.00000	+	8.66025i	−0.640184	+	1.10883i	0.345207	+	0.938527i	$0.387809\pi$
$61$	−5.00000	+	8.66025i	−0.640184	+	1.10883i	−0.985391	+	0.170305i	$0.945525\pi$
$62$	−2.00000			−0.254000
$63$		0			0
$64$	1.00000			0.125000
$65$	−2.00000	+	3.46410i	−0.248069	+	0.429669i
$66$		0			0
$67$	2.00000	+	3.46410i	0.244339	+	0.423207i	0.961946	−	0.273241i	$-0.0880957\pi$
$67$	2.00000	+	3.46410i	0.244339	+	0.423207i	−0.717607	+	0.696449i	$0.754762\pi$
$68$	−1.50000	+	2.59808i	−0.181902	+	0.315063i
$69$		0			0
$70$	−1.00000	+	5.19615i	−0.119523	+	0.621059i
$71$	5.00000			0.593391			0.296695	−	0.954972i	$-0.404115\pi$
$71$	5.00000			0.593391			0.296695	+	0.954972i	$0.404115\pi$
$72$		0			0
$73$	4.00000	+	6.92820i	0.468165	+	0.810885i	0.999338	−	0.0363782i	$-0.0115821\pi$
$73$	4.00000	+	6.92820i	0.468165	+	0.810885i	−0.531174	+	0.847263i	$0.678249\pi$
$74$	1.50000	+	2.59808i	0.174371	+	0.302020i
$75$		0			0
$76$	7.00000			0.802955
$77$	0.500000	−	2.59808i	0.0569803	−	0.296078i
$78$		0			0
$79$	4.00000	−	6.92820i	0.450035	−	0.779484i	−0.548352	−	0.836247i	$-0.684745\pi$
$79$	4.00000	−	6.92820i	0.450035	−	0.779484i	0.998388	+	0.0567635i	$0.0180781\pi$
$80$	−1.00000	−	1.73205i	−0.111803	−	0.193649i
$81$		0			0
$82$	−3.00000	+	5.19615i	−0.331295	+	0.573819i
$83$	−14.0000			−1.53670			−0.768350	−	0.640030i	$-0.778922\pi$
$83$	−14.0000			−1.53670			−0.768350	+	0.640030i	$0.778922\pi$
$84$		0			0
$85$	6.00000			0.650791
$86$	5.50000	−	9.52628i	0.593080	−	1.02725i
$87$		0			0
$88$	0.500000	+	0.866025i	0.0533002	+	0.0923186i
$89$	1.00000	−	1.73205i	0.106000	−	0.183597i	−0.808146	−	0.588982i	$-0.799529\pi$
$89$	1.00000	−	1.73205i	0.106000	−	0.183597i	0.914146	+	0.405385i	$0.132862\pi$
$90$		0			0
$91$	4.00000	+	3.46410i	0.419314	+	0.363137i
$92$	7.00000			0.729800
$93$		0			0
$94$	3.50000	+	6.06218i	0.360997	+	0.625266i
$95$	−7.00000	−	12.1244i	−0.718185	−	1.24393i
$96$		0			0
$97$	15.0000			1.52302			0.761510	−	0.648154i	$-0.224459\pi$
$97$	15.0000			1.52302			0.761510	+	0.648154i	$0.224459\pi$
$98$	6.50000	+	2.59808i	0.656599	+	0.262445i
$99$		0			0
$100$	0.500000	−	0.866025i	0.0500000	−	0.0866025i
$101$	1.50000	+	2.59808i	0.149256	+	0.258518i	0.930953	−	0.365140i	$-0.118979\pi$
$101$	1.50000	+	2.59808i	0.149256	+	0.258518i	−0.781697	+	0.623658i	$0.785646\pi$
$102$		0			0
$103$	5.00000	−	8.66025i	0.492665	−	0.853320i	−0.507300	−	0.861770i	$-0.669356\pi$
$103$	5.00000	−	8.66025i	0.492665	−	0.853320i	0.999964	+	0.00844953i	$0.00268960\pi$
$104$	−2.00000			−0.196116
$105$		0			0
$106$	−4.00000			−0.388514
$107$	9.00000	−	15.5885i	0.870063	−	1.50699i	0.00813215	−	0.999967i	$-0.497411\pi$
$107$	9.00000	−	15.5885i	0.870063	−	1.50699i	0.861931	−	0.507026i	$-0.169255\pi$
$108$		0			0
$109$	−2.00000	−	3.46410i	−0.191565	−	0.331801i	0.754204	−	0.656640i	$-0.228023\pi$
$109$	−2.00000	−	3.46410i	−0.191565	−	0.331801i	−0.945769	+	0.324840i	$0.894690\pi$
$110$	1.00000	−	1.73205i	0.0953463	−	0.165145i
$111$		0			0
$112$	−2.50000	+	0.866025i	−0.236228	+	0.0818317i
$113$	−2.00000			−0.188144			−0.0940721	−	0.995565i	$-0.529988\pi$
$113$	−2.00000			−0.188144			−0.0940721	+	0.995565i	$0.529988\pi$
$114$		0			0
$115$	−7.00000	−	12.1244i	−0.652753	−	1.13060i
$116$	−2.50000	−	4.33013i	−0.232119	−	0.402042i
$117$		0			0
$118$	11.0000			1.01263
$119$	1.50000	−	7.79423i	0.137505	−	0.714496i
$120$		0			0
$121$	−0.500000	+	0.866025i	−0.0454545	+	0.0787296i
$122$	5.00000	+	8.66025i	0.452679	+	0.784063i
$123$		0			0
$124$	−1.00000	+	1.73205i	−0.0898027	+	0.155543i
$125$	−12.0000			−1.07331
$126$		0			0
$127$	−19.0000			−1.68598			−0.842989	−	0.537931i	$-0.819206\pi$
$127$	−19.0000			−1.68598			−0.842989	+	0.537931i	$0.819206\pi$
$128$	0.500000	−	0.866025i	0.0441942	−	0.0765466i
$129$		0			0
$130$	2.00000	+	3.46410i	0.175412	+	0.303822i
$131$	−4.00000	+	6.92820i	−0.349482	+	0.605320i	−0.986157	−	0.165812i	$-0.946976\pi$
$131$	−4.00000	+	6.92820i	−0.349482	+	0.605320i	0.636676	+	0.771132i	$0.280309\pi$
$132$		0			0
$133$	−17.5000	+	6.06218i	−1.51744	+	0.525657i
$134$	4.00000			0.345547
$135$		0			0
$136$	1.50000	+	2.59808i	0.128624	+	0.222783i
$137$	3.00000	+	5.19615i	0.256307	+	0.443937i	0.965250	−	0.261329i	$-0.0841608\pi$
$137$	3.00000	+	5.19615i	0.256307	+	0.443937i	−0.708942	+	0.705266i	$0.750827\pi$
$138$		0			0
$139$	17.0000			1.44192			0.720961	−	0.692976i	$-0.243701\pi$
$139$	17.0000			1.44192			0.720961	+	0.692976i	$0.243701\pi$
$140$	4.00000	+	3.46410i	0.338062	+	0.292770i
$141$		0			0
$142$	2.50000	−	4.33013i	0.209795	−	0.363376i
$143$	−1.00000	−	1.73205i	−0.0836242	−	0.144841i
$144$		0			0
$145$	−5.00000	+	8.66025i	−0.415227	+	0.719195i
$146$	8.00000			0.662085
$147$		0			0
$148$	3.00000			0.246598
$149$	8.50000	−	14.7224i	0.696347	−	1.20611i	−0.273377	−	0.961907i	$-0.588141\pi$
$149$	8.50000	−	14.7224i	0.696347	−	1.20611i	0.969724	−	0.244202i	$-0.0785259\pi$
$150$		0			0
$151$	2.50000	+	4.33013i	0.203447	+	0.352381i	0.949637	−	0.313353i	$-0.101452\pi$
$151$	2.50000	+	4.33013i	0.203447	+	0.352381i	−0.746190	+	0.665733i	$0.768119\pi$
$152$	3.50000	−	6.06218i	0.283887	−	0.491708i
$153$		0			0
$154$	−2.00000	−	1.73205i	−0.161165	−	0.139573i
$155$	4.00000			0.321288
$156$		0			0
$157$	−0.500000	−	0.866025i	−0.0399043	−	0.0691164i	0.845383	−	0.534160i	$-0.179372\pi$
$157$	−0.500000	−	0.866025i	−0.0399043	−	0.0691164i	−0.885288	+	0.465044i	$0.846039\pi$
$158$	−4.00000	−	6.92820i	−0.318223	−	0.551178i
$159$		0			0
$160$	−2.00000			−0.158114
$161$	−17.5000	+	6.06218i	−1.37919	+	0.477767i
$162$		0			0
$163$	−3.00000	+	5.19615i	−0.234978	+	0.406994i	−0.959266	−	0.282503i	$-0.908835\pi$
$163$	−3.00000	+	5.19615i	−0.234978	+	0.406994i	0.724288	+	0.689497i	$0.242169\pi$
$164$	3.00000	+	5.19615i	0.234261	+	0.405751i
$165$		0			0
$166$	−7.00000	+	12.1244i	−0.543305	+	0.941033i
$167$	10.0000			0.773823			0.386912	−	0.922117i	$-0.373542\pi$
$167$	10.0000			0.773823			0.386912	+	0.922117i	$0.373542\pi$
$168$		0			0
$169$	−9.00000			−0.692308
$170$	3.00000	−	5.19615i	0.230089	−	0.398527i
$171$		0			0
$172$	−5.50000	−	9.52628i	−0.419371	−	0.726372i
$173$	7.00000	−	12.1244i	0.532200	−	0.921798i	−0.467093	−	0.884208i	$-0.654699\pi$
$173$	7.00000	−	12.1244i	0.532200	−	0.921798i	0.999293	−	0.0375896i	$-0.0119679\pi$
$174$		0			0
$175$	−0.500000	+	2.59808i	−0.0377964	+	0.196396i
$176$	1.00000			0.0753778
$177$		0			0
$178$	−1.00000	−	1.73205i	−0.0749532	−	0.129823i
$179$	−4.50000	−	7.79423i	−0.336346	−	0.582568i	0.647397	−	0.762153i	$-0.275858\pi$
$179$	−4.50000	−	7.79423i	−0.336346	−	0.582568i	−0.983742	+	0.179585i	$0.942524\pi$
$180$		0			0
$181$	−10.0000			−0.743294			−0.371647	−	0.928374i	$-0.621207\pi$
$181$	−10.0000			−0.743294			−0.371647	+	0.928374i	$0.621207\pi$
$182$	5.00000	−	1.73205i	0.370625	−	0.128388i
$183$		0			0
$184$	3.50000	−	6.06218i	0.258023	−	0.446910i
$185$	−3.00000	−	5.19615i	−0.220564	−	0.382029i
$186$		0			0
$187$	−1.50000	+	2.59808i	−0.109691	+	0.189990i
$188$	7.00000			0.510527
$189$		0			0
$190$	−14.0000			−1.01567
$191$	−4.00000	+	6.92820i	−0.289430	+	0.501307i	−0.973674	−	0.227946i	$-0.926799\pi$
$191$	−4.00000	+	6.92820i	−0.289430	+	0.501307i	0.684244	+	0.729253i	$0.260132\pi$
$192$		0			0
$193$	−12.0000	−	20.7846i	−0.863779	−	1.49611i	−0.868255	−	0.496119i	$-0.834758\pi$
$193$	−12.0000	−	20.7846i	−0.863779	−	1.49611i	0.00447566	−	0.999990i	$-0.498575\pi$
$194$	7.50000	−	12.9904i	0.538469	−	0.932655i
$195$		0			0
$196$	5.50000	−	4.33013i	0.392857	−	0.309295i
$197$	−15.0000			−1.06871			−0.534353	−	0.845262i	$-0.679445\pi$
$197$	−15.0000			−1.06871			−0.534353	+	0.845262i	$0.679445\pi$
$198$		0			0
$199$	−5.00000	−	8.66025i	−0.354441	−	0.613909i	0.632581	−	0.774494i	$-0.281995\pi$
$199$	−5.00000	−	8.66025i	−0.354441	−	0.613909i	−0.987022	+	0.160585i	$0.948662\pi$
$200$	−0.500000	−	0.866025i	−0.0353553	−	0.0612372i
$201$		0			0
$202$	3.00000			0.211079
$203$	10.0000	+	8.66025i	0.701862	+	0.607831i
$204$		0			0
$205$	6.00000	−	10.3923i	0.419058	−	0.725830i
$206$	−5.00000	−	8.66025i	−0.348367	−	0.603388i
$207$		0			0
$208$	−1.00000	+	1.73205i	−0.0693375	+	0.120096i
$209$	7.00000			0.484200
$210$		0			0
$211$	20.0000			1.37686			0.688428	−	0.725304i	$-0.258301\pi$
$211$	20.0000			1.37686			0.688428	+	0.725304i	$0.258301\pi$
$212$	−2.00000	+	3.46410i	−0.137361	+	0.237915i
$213$		0			0
$214$	−9.00000	−	15.5885i	−0.615227	−	1.06561i
$215$	−11.0000	+	19.0526i	−0.750194	+	1.29937i
$216$		0			0
$217$	1.00000	−	5.19615i	0.0678844	−	0.352738i
$218$	−4.00000			−0.270914
$219$		0			0
$220$	−1.00000	−	1.73205i	−0.0674200	−	0.116775i
$221$	−3.00000	−	5.19615i	−0.201802	−	0.349531i
$222$		0			0
$223$	−14.0000			−0.937509			−0.468755	−	0.883328i	$-0.655297\pi$
$223$	−14.0000			−0.937509			−0.468755	+	0.883328i	$0.655297\pi$
$224$	−0.500000	+	2.59808i	−0.0334077	+	0.173591i
$225$		0			0
$226$	−1.00000	+	1.73205i	−0.0665190	+	0.115214i
$227$	5.00000	+	8.66025i	0.331862	+	0.574801i	0.982877	−	0.184263i	$-0.0589899\pi$
$227$	5.00000	+	8.66025i	0.331862	+	0.574801i	−0.651015	+	0.759065i	$0.725657\pi$
$228$		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	$0.0594799\pi$
$230$	−14.0000			−0.923133
$231$		0			0
$232$	−5.00000			−0.328266
$233$	−4.50000	+	7.79423i	−0.294805	+	0.510617i	−0.974939	−	0.222470i	$-0.928588\pi$
$233$	−4.50000	+	7.79423i	−0.294805	+	0.510617i	0.680135	+	0.733087i	$0.261921\pi$
$234$		0			0
$235$	−7.00000	−	12.1244i	−0.456630	−	0.790906i
$236$	5.50000	−	9.52628i	0.358020	−	0.620108i
$237$		0			0
$238$	−6.00000	−	5.19615i	−0.388922	−	0.336817i
$239$	10.0000			0.646846			0.323423	−	0.946254i	$-0.395166\pi$
$239$	10.0000			0.646846			0.323423	+	0.946254i	$0.395166\pi$
$240$		0			0
$241$	2.00000	+	3.46410i	0.128831	+	0.223142i	0.923224	−	0.384262i	$-0.125544\pi$
$241$	2.00000	+	3.46410i	0.128831	+	0.223142i	−0.794393	+	0.607404i	$0.792211\pi$
$242$	0.500000	+	0.866025i	0.0321412	+	0.0556702i
$243$		0			0
$244$	10.0000			0.640184
$245$	−13.0000	−	5.19615i	−0.830540	−	0.331970i
$246$		0			0
$247$	−7.00000	+	12.1244i	−0.445399	+	0.771454i
$248$	1.00000	+	1.73205i	0.0635001	+	0.109985i
$249$		0			0
$250$	−6.00000	+	10.3923i	−0.379473	+	0.657267i
$251$	−5.00000			−0.315597			−0.157799	−	0.987471i	$-0.550440\pi$
$251$	−5.00000			−0.315597			−0.157799	+	0.987471i	$0.550440\pi$
$252$		0			0
$253$	7.00000			0.440086
$254$	−9.50000	+	16.4545i	−0.596083	+	1.03245i
$255$		0			0
$256$	−0.500000	−	0.866025i	−0.0312500	−	0.0541266i
$257$	−7.00000	+	12.1244i	−0.436648	+	0.756297i	−0.997429	−	0.0716680i	$-0.977168\pi$
$257$	−7.00000	+	12.1244i	−0.436648	+	0.756297i	0.560781	+	0.827964i	$0.310501\pi$
$258$		0			0
$259$	−7.50000	+	2.59808i	−0.466027	+	0.161437i
$260$	4.00000			0.248069
$261$		0			0
$262$	4.00000	+	6.92820i	0.247121	+	0.428026i
$263$	15.0000	+	25.9808i	0.924940	+	1.60204i	0.791658	+	0.610964i	$0.209218\pi$
$263$	15.0000	+	25.9808i	0.924940	+	1.60204i	0.133281	+	0.991078i	$0.457449\pi$
$264$		0			0
$265$	8.00000			0.491436
$266$	−3.50000	+	18.1865i	−0.214599	+	1.11509i
$267$		0			0
$268$	2.00000	−	3.46410i	0.122169	−	0.211604i
$269$	−10.0000	−	17.3205i	−0.609711	−	1.05605i	−0.991288	−	0.131713i	$-0.957952\pi$
$269$	−10.0000	−	17.3205i	−0.609711	−	1.05605i	0.381577	−	0.924337i	$-0.375381\pi$
$270$		0			0
$271$	−4.00000	+	6.92820i	−0.242983	+	0.420858i	−0.961563	−	0.274586i	$-0.911459\pi$
$271$	−4.00000	+	6.92820i	−0.242983	+	0.420858i	0.718580	+	0.695444i	$0.244792\pi$
$272$	3.00000			0.181902
$273$		0			0
$274$	6.00000			0.362473
$275$	0.500000	−	0.866025i	0.0301511	−	0.0522233i
$276$		0			0
$277$	−12.0000	−	20.7846i	−0.721010	−	1.24883i	−0.960595	−	0.277951i	$-0.910345\pi$
$277$	−12.0000	−	20.7846i	−0.721010	−	1.24883i	0.239585	−	0.970875i	$-0.422989\pi$
$278$	8.50000	−	14.7224i	0.509796	−	0.882993i
$279$		0			0
$280$	5.00000	−	1.73205i	0.298807	−	0.103510i
$281$	−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
$283$	−12.0000	−	20.7846i	−0.713326	−	1.23552i	−0.963602	−	0.267342i	$-0.913855\pi$
$283$	−12.0000	−	20.7846i	−0.713326	−	1.23552i	0.250276	−	0.968175i	$-0.419479\pi$
$284$	−2.50000	−	4.33013i	−0.148348	−	0.256946i
$285$		0			0
$286$	−2.00000			−0.118262
$287$	−12.0000	−	10.3923i	−0.708338	−	0.613438i
$288$		0			0
$289$	4.00000	−	6.92820i	0.235294	−	0.407541i
$290$	5.00000	+	8.66025i	0.293610	+	0.508548i
$291$		0			0
$292$	4.00000	−	6.92820i	0.234082	−	0.405442i
$293$	3.00000			0.175262			0.0876309	−	0.996153i	$-0.472070\pi$
$293$	3.00000			0.175262			0.0876309	+	0.996153i	$0.472070\pi$
$294$		0			0
$295$	−22.0000			−1.28089
$296$	1.50000	−	2.59808i	0.0871857	−	0.151010i
$297$		0			0
$298$	−8.50000	−	14.7224i	−0.492392	−	0.852848i
$299$	−7.00000	+	12.1244i	−0.404820	+	0.701170i
$300$		0			0
$301$	22.0000	+	19.0526i	1.26806	+	1.09817i
$302$	5.00000			0.287718
$303$		0			0
$304$	−3.50000	−	6.06218i	−0.200739	−	0.347690i
$305$	−10.0000	−	17.3205i	−0.572598	−	0.991769i
$306$		0			0
$307$	28.0000			1.59804			0.799022	−	0.601302i	$-0.205351\pi$
$307$	28.0000			1.59804			0.799022	+	0.601302i	$0.205351\pi$
$308$	−2.50000	+	0.866025i	−0.142451	+	0.0493464i
$309$		0			0
$310$	2.00000	−	3.46410i	0.113592	−	0.196748i
$311$	13.5000	+	23.3827i	0.765515	+	1.32591i	0.939974	+	0.341246i	$0.110849\pi$
$311$	13.5000	+	23.3827i	0.765515	+	1.32591i	−0.174459	+	0.984664i	$0.555818\pi$
$312$		0			0
$313$	−7.50000	+	12.9904i	−0.423925	+	0.734260i	−0.996319	−	0.0857188i	$-0.972681\pi$
$313$	−7.50000	+	12.9904i	−0.423925	+	0.734260i	0.572394	+	0.819979i	$0.306015\pi$
$314$	−1.00000			−0.0564333
$315$		0			0
$316$	−8.00000			−0.450035
$317$	15.0000	−	25.9808i	0.842484	−	1.45922i	−0.0453045	−	0.998973i	$-0.514426\pi$
$317$	15.0000	−	25.9808i	0.842484	−	1.45922i	0.887788	−	0.460252i	$-0.152241\pi$
$318$		0			0
$319$	−2.50000	−	4.33013i	−0.139973	−	0.242441i
$320$	−1.00000	+	1.73205i	−0.0559017	+	0.0968246i
$321$		0			0
$322$	−3.50000	+	18.1865i	−0.195047	+	1.01350i
$323$	21.0000			1.16847
$324$		0			0
$325$	1.00000	+	1.73205i	0.0554700	+	0.0960769i
$326$	3.00000	+	5.19615i	0.166155	+	0.287788i
$327$		0			0
$328$	6.00000			0.331295
$329$	−17.5000	+	6.06218i	−0.964806	+	0.334219i
$330$		0			0
$331$	7.00000	−	12.1244i	0.384755	−	0.666415i	−0.606980	−	0.794717i	$-0.707619\pi$
$331$	7.00000	−	12.1244i	0.384755	−	0.666415i	0.991735	+	0.128302i	$0.0409527\pi$
$332$	7.00000	+	12.1244i	0.384175	+	0.665410i
$333$		0			0
$334$	5.00000	−	8.66025i	0.273588	−	0.473868i
$335$	−8.00000			−0.437087
$336$		0			0
$337$	8.00000			0.435788			0.217894	−	0.975972i	$-0.430081\pi$
$337$	8.00000			0.435788			0.217894	+	0.975972i	$0.430081\pi$
$338$	−4.50000	+	7.79423i	−0.244768	+	0.423950i
$339$		0			0
$340$	−3.00000	−	5.19615i	−0.162698	−	0.281801i
$341$	−1.00000	+	1.73205i	−0.0541530	+	0.0937958i
$342$		0			0
$343$	−10.0000	+	15.5885i	−0.539949	+	0.841698i
$344$	−11.0000			−0.593080
$345$		0			0
$346$	−7.00000	−	12.1244i	−0.376322	−	0.651809i
$347$	12.0000	+	20.7846i	0.644194	+	1.11578i	0.984487	+	0.175457i	$0.0561403\pi$
$347$	12.0000	+	20.7846i	0.644194	+	1.11578i	−0.340293	+	0.940319i	$0.610526\pi$
$348$		0			0
$349$	8.00000			0.428230			0.214115	−	0.976808i	$-0.431313\pi$
$349$	8.00000			0.428230			0.214115	+	0.976808i	$0.431313\pi$
$350$	2.00000	+	1.73205i	0.106904	+	0.0925820i
$351$		0			0
$352$	0.500000	−	0.866025i	0.0266501	−	0.0461593i
$353$	5.00000	+	8.66025i	0.266123	+	0.460939i	0.967857	−	0.251500i	$-0.0809239\pi$
$353$	5.00000	+	8.66025i	0.266123	+	0.460939i	−0.701734	+	0.712439i	$0.747591\pi$
$354$		0			0
$355$	−5.00000	+	8.66025i	−0.265372	+	0.459639i
$356$	−2.00000			−0.106000
$357$		0			0
$358$	−9.00000			−0.475665
$359$	−9.00000	+	15.5885i	−0.475002	+	0.822727i	−0.999590	−	0.0286287i	$-0.990886\pi$
$359$	−9.00000	+	15.5885i	−0.475002	+	0.822727i	0.524588	+	0.851356i	$0.324219\pi$
$360$		0			0
$361$	−15.0000	−	25.9808i	−0.789474	−	1.36741i
$362$	−5.00000	+	8.66025i	−0.262794	+	0.455173i
$363$		0			0
$364$	1.00000	−	5.19615i	0.0524142	−	0.272352i
$365$	−16.0000			−0.837478
$366$		0			0
$367$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$367$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$368$	−3.50000	−	6.06218i	−0.182450	−	0.316013i
$369$		0			0
$370$	−6.00000			−0.311925
$371$	2.00000	−	10.3923i	0.103835	−	0.539542i
$372$		0			0
$373$	11.0000	−	19.0526i	0.569558	−	0.986504i	−0.427051	−	0.904227i	$-0.640448\pi$
$373$	11.0000	−	19.0526i	0.569558	−	0.986504i	0.996610	−	0.0822766i	$-0.0262191\pi$
$374$	1.50000	+	2.59808i	0.0775632	+	0.134343i
$375$		0			0
$376$	3.50000	−	6.06218i	0.180499	−	0.312633i
$377$	10.0000			0.515026
$378$		0			0
$379$	−16.0000			−0.821865			−0.410932	−	0.911666i	$-0.634797\pi$
$379$	−16.0000			−0.821865			−0.410932	+	0.911666i	$0.634797\pi$
$380$	−7.00000	+	12.1244i	−0.359092	+	0.621966i
$381$		0			0
$382$	4.00000	+	6.92820i	0.204658	+	0.354478i
$383$	15.5000	−	26.8468i	0.792013	−	1.37181i	−0.132706	−	0.991155i	$-0.542367\pi$
$383$	15.5000	−	26.8468i	0.792013	−	1.37181i	0.924719	−	0.380651i	$-0.124300\pi$
$384$		0			0
$385$	4.00000	+	3.46410i	0.203859	+	0.176547i
$386$	−24.0000			−1.22157
$387$		0			0
$388$	−7.50000	−	12.9904i	−0.380755	−	0.659487i
$389$	−3.00000	−	5.19615i	−0.152106	−	0.263455i	0.779895	−	0.625910i	$-0.215272\pi$
$389$	−3.00000	−	5.19615i	−0.152106	−	0.263455i	−0.932002	+	0.362454i	$0.881939\pi$
$390$		0			0
$391$	21.0000			1.06202
$392$	−1.00000	−	6.92820i	−0.0505076	−	0.349927i
$393$		0			0
$394$	−7.50000	+	12.9904i	−0.377845	+	0.654446i
$395$	8.00000	+	13.8564i	0.402524	+	0.697191i
$396$		0			0
$397$	1.50000	−	2.59808i	0.0752828	−	0.130394i	−0.825926	−	0.563778i	$-0.809347\pi$
$397$	1.50000	−	2.59808i	0.0752828	−	0.130394i	0.901209	+	0.433384i	$0.142681\pi$
$398$	−10.0000			−0.501255
$399$		0			0
$400$	−1.00000			−0.0500000
$401$	16.0000	−	27.7128i	0.799002	−	1.38391i	−0.121265	−	0.992620i	$-0.538695\pi$
$401$	16.0000	−	27.7128i	0.799002	−	1.38391i	0.920267	−	0.391292i	$-0.127972\pi$
$402$		0			0
$403$	−2.00000	−	3.46410i	−0.0996271	−	0.172559i
$404$	1.50000	−	2.59808i	0.0746278	−	0.129259i
$405$		0			0
$406$	12.5000	−	4.33013i	0.620365	−	0.214901i
$407$	3.00000			0.148704
$408$		0			0
$409$	−16.0000	−	27.7128i	−0.791149	−	1.37031i	−0.925256	−	0.379344i	$-0.876150\pi$
$409$	−16.0000	−	27.7128i	−0.791149	−	1.37031i	0.134107	−	0.990967i	$-0.457183\pi$
$410$	−6.00000	−	10.3923i	−0.296319	−	0.513239i
$411$		0			0
$412$	−10.0000			−0.492665
$413$	−5.50000	+	28.5788i	−0.270637	+	1.40627i
$414$		0			0
$415$	14.0000	−	24.2487i	0.687233	−	1.19032i
$416$	1.00000	+	1.73205i	0.0490290	+	0.0849208i
$417$		0			0
$418$	3.50000	−	6.06218i	0.171191	−	0.296511i
$419$	−3.00000			−0.146560			−0.0732798	−	0.997311i	$-0.523347\pi$
$419$	−3.00000			−0.146560			−0.0732798	+	0.997311i	$0.523347\pi$
$420$		0			0
$421$	−17.0000			−0.828529			−0.414265	−	0.910156i	$-0.635961\pi$
$421$	−17.0000			−0.828529			−0.414265	+	0.910156i	$0.635961\pi$
$422$	10.0000	−	17.3205i	0.486792	−	0.843149i
$423$		0			0
$424$	2.00000	+	3.46410i	0.0971286	+	0.168232i
$425$	1.50000	−	2.59808i	0.0727607	−	0.126025i
$426$		0			0
$427$	−25.0000	+	8.66025i	−1.20983	+	0.419099i
$428$	−18.0000			−0.870063
$429$		0			0
$430$	11.0000	+	19.0526i	0.530467	+	0.918796i
$431$	20.0000	+	34.6410i	0.963366	+	1.66860i	0.713942	+	0.700205i	$0.246908\pi$
$431$	20.0000	+	34.6410i	0.963366	+	1.66860i	0.249424	+	0.968394i	$0.419759\pi$
$432$		0			0
$433$	5.00000			0.240285			0.120142	−	0.992757i	$-0.461665\pi$
$433$	5.00000			0.240285			0.120142	+	0.992757i	$0.461665\pi$
$434$	−4.00000	−	3.46410i	−0.192006	−	0.166282i
$435$		0			0
$436$	−2.00000	+	3.46410i	−0.0957826	+	0.165900i
$437$	−24.5000	−	42.4352i	−1.17199	−	2.02995i
$438$		0			0
$439$	−2.50000	+	4.33013i	−0.119318	+	0.206666i	−0.919498	−	0.393095i	$-0.871404\pi$
$439$	−2.50000	+	4.33013i	−0.119318	+	0.206666i	0.800179	+	0.599761i	$0.204738\pi$
$440$	−2.00000			−0.0953463
$441$		0			0
$442$	−6.00000			−0.285391
$443$	20.5000	−	35.5070i	0.973984	−	1.68699i	0.290738	−	0.956803i	$-0.406099\pi$
$443$	20.5000	−	35.5070i	0.973984	−	1.68699i	0.683247	−	0.730188i	$-0.260567\pi$
$444$		0			0
$445$	2.00000	+	3.46410i	0.0948091	+	0.164214i
$446$	−7.00000	+	12.1244i	−0.331460	+	0.574105i
$447$		0			0
$448$	2.00000	+	1.73205i	0.0944911	+	0.0818317i
$449$	36.0000			1.69895			0.849473	−	0.527633i	$-0.176920\pi$
$449$	36.0000			1.69895			0.849473	+	0.527633i	$0.176920\pi$
$450$		0			0
$451$	3.00000	+	5.19615i	0.141264	+	0.244677i
$452$	1.00000	+	1.73205i	0.0470360	+	0.0814688i
$453$		0			0
$454$	10.0000			0.469323
$455$	−10.0000	+	3.46410i	−0.468807	+	0.162400i
$456$		0			0
$457$	−20.0000	+	34.6410i	−0.935561	+	1.62044i	−0.161929	+	0.986802i	$0.551772\pi$
$457$	−20.0000	+	34.6410i	−0.935561	+	1.62044i	−0.773631	+	0.633636i	$0.781562\pi$
$458$	−5.00000	−	8.66025i	−0.233635	−	0.404667i
$459$		0			0
$460$	−7.00000	+	12.1244i	−0.326377	+	0.565301i
$461$	−13.0000			−0.605470			−0.302735	−	0.953075i	$-0.597900\pi$
$461$	−13.0000			−0.605470			−0.302735	+	0.953075i	$0.597900\pi$
$462$		0			0
$463$	−24.0000			−1.11537			−0.557687	−	0.830051i	$-0.688311\pi$
$463$	−24.0000			−1.11537			−0.557687	+	0.830051i	$0.688311\pi$
$464$	−2.50000	+	4.33013i	−0.116060	+	0.201021i
$465$		0			0
$466$	4.50000	+	7.79423i	0.208458	+	0.361061i
$467$	6.50000	−	11.2583i	0.300784	−	0.520973i	−0.675530	−	0.737333i	$-0.736085\pi$
$467$	6.50000	−	11.2583i	0.300784	−	0.520973i	0.976314	+	0.216359i	$0.0694183\pi$
$468$		0			0
$469$	−2.00000	+	10.3923i	−0.0923514	+	0.479872i
$470$	−14.0000			−0.645772
$471$		0			0
$472$	−5.50000	−	9.52628i	−0.253158	−	0.438483i
$473$	−5.50000	−	9.52628i	−0.252890	−	0.438019i
$474$		0			0
$475$	−7.00000			−0.321182
$476$	−7.50000	+	2.59808i	−0.343762	+	0.119083i
$477$		0			0
$478$	5.00000	−	8.66025i	0.228695	−	0.396111i
$479$	9.00000	+	15.5885i	0.411220	+	0.712255i	0.995023	−	0.0996406i	$-0.0317693\pi$
$479$	9.00000	+	15.5885i	0.411220	+	0.712255i	−0.583803	+	0.811895i	$0.698436\pi$
$480$		0			0
$481$	−3.00000	+	5.19615i	−0.136788	+	0.236924i
$482$	4.00000			0.182195
$483$		0			0
$484$	1.00000			0.0454545
$485$	−15.0000	+	25.9808i	−0.681115	+	1.17973i
$486$		0			0
$487$	−9.00000	−	15.5885i	−0.407829	−	0.706380i	0.586817	−	0.809719i	$-0.300381\pi$
$487$	−9.00000	−	15.5885i	−0.407829	−	0.706380i	−0.994646	+	0.103339i	$0.967047\pi$
$488$	5.00000	−	8.66025i	0.226339	−	0.392031i
$489$		0			0
$490$	−11.0000	+	8.66025i	−0.496929	+	0.391230i
$491$	26.0000			1.17336			0.586682	−	0.809818i	$-0.300434\pi$
$491$	26.0000			1.17336			0.586682	+	0.809818i	$0.300434\pi$
$492$		0			0
$493$	−7.50000	−	12.9904i	−0.337783	−	0.585057i
$494$	7.00000	+	12.1244i	0.314945	+	0.545501i
$495$		0			0
$496$	2.00000			0.0898027
$497$	10.0000	+	8.66025i	0.448561	+	0.388465i
$498$		0			0
$499$	20.0000	−	34.6410i	0.895323	−	1.55074i	0.0619186	−	0.998081i	$-0.480278\pi$
$499$	20.0000	−	34.6410i	0.895323	−	1.55074i	0.833404	−	0.552664i	$-0.186389\pi$
$500$	6.00000	+	10.3923i	0.268328	+	0.464758i
$501$		0			0
$502$	−2.50000	+	4.33013i	−0.111580	+	0.193263i
$503$	30.0000			1.33763			0.668817	−	0.743427i	$-0.266801\pi$
$503$	30.0000			1.33763			0.668817	+	0.743427i	$0.266801\pi$
$504$		0			0
$505$	−6.00000			−0.266996
$506$	3.50000	−	6.06218i	0.155594	−	0.269497i
$507$		0			0
$508$	9.50000	+	16.4545i	0.421494	+	0.730050i
$509$	2.00000	−	3.46410i	0.0886484	−	0.153544i	−0.818292	−	0.574803i	$-0.805079\pi$
$509$	2.00000	−	3.46410i	0.0886484	−	0.153544i	0.906940	+	0.421260i	$0.138412\pi$
$510$		0			0
$511$	−4.00000	+	20.7846i	−0.176950	+	0.919457i
$512$	−1.00000			−0.0441942
$513$		0			0
$514$	7.00000	+	12.1244i	0.308757	+	0.534782i
$515$	10.0000	+	17.3205i	0.440653	+	0.763233i
$516$		0			0
$517$	7.00000			0.307860
$518$	−1.50000	+	7.79423i	−0.0659062	+	0.342459i
$519$		0			0
$520$	2.00000	−	3.46410i	0.0877058	−	0.151911i
$521$	−9.00000	−	15.5885i	−0.394297	−	0.682943i	0.598714	−	0.800963i	$-0.295679\pi$
$521$	−9.00000	−	15.5885i	−0.394297	−	0.682943i	−0.993011	+	0.118020i	$0.962345\pi$
$522$		0			0
$523$	10.0000	−	17.3205i	0.437269	−	0.757373i	−0.560208	−	0.828352i	$-0.689279\pi$
$523$	10.0000	−	17.3205i	0.437269	−	0.757373i	0.997478	+	0.0709788i	$0.0226123\pi$
$524$	8.00000			0.349482
$525$		0			0
$526$	30.0000			1.30806
$527$	−3.00000	+	5.19615i	−0.130682	+	0.226348i
$528$		0			0
$529$	−13.0000	−	22.5167i	−0.565217	−	0.978985i
$530$	4.00000	−	6.92820i	0.173749	−	0.300942i
$531$		0			0
$532$	14.0000	+	12.1244i	0.606977	+	0.525657i
$533$	−12.0000			−0.519778
$534$		0			0
$535$	18.0000	+	31.1769i	0.778208	+	1.34790i
$536$	−2.00000	−	3.46410i	−0.0863868	−	0.149626i
$537$		0			0
$538$	−20.0000			−0.862261
$539$	5.50000	−	4.33013i	0.236902	−	0.186512i
$540$		0			0
$541$	−8.00000	+	13.8564i	−0.343947	+	0.595733i	−0.985162	−	0.171628i	$-0.945097\pi$
$541$	−8.00000	+	13.8564i	−0.343947	+	0.595733i	0.641215	+	0.767361i	$0.278431\pi$
$542$	4.00000	+	6.92820i	0.171815	+	0.297592i
$543$		0			0
$544$	1.50000	−	2.59808i	0.0643120	−	0.111392i
$545$	8.00000			0.342682
$546$		0			0
$547$	−11.0000			−0.470326			−0.235163	−	0.971956i	$-0.575562\pi$
$547$	−11.0000			−0.470326			−0.235163	+	0.971956i	$0.575562\pi$
$548$	3.00000	−	5.19615i	0.128154	−	0.221969i
$549$		0			0
$550$	−0.500000	−	0.866025i	−0.0213201	−	0.0369274i
$551$	−17.5000	+	30.3109i	−0.745525	+	1.29129i
$552$		0			0
$553$	20.0000	−	6.92820i	0.850487	−	0.294617i
$554$	−24.0000			−1.01966
$555$		0			0
$556$	−8.50000	−	14.7224i	−0.360480	−	0.624370i
$557$	4.50000	+	7.79423i	0.190671	+	0.330252i	0.945473	−	0.325701i	$-0.105600\pi$
$557$	4.50000	+	7.79423i	0.190671	+	0.330252i	−0.754802	+	0.655953i	$0.772267\pi$
$558$		0			0
$559$	22.0000			0.930501
$560$	1.00000	−	5.19615i	0.0422577	−	0.219578i
$561$		0			0
$562$	−1.50000	+	2.59808i	−0.0632737	+	0.109593i
$563$	−21.0000	−	36.3731i	−0.885044	−	1.53294i	−0.845663	−	0.533718i	$-0.820794\pi$
$563$	−21.0000	−	36.3731i	−0.885044	−	1.53294i	−0.0393818	−	0.999224i	$-0.512539\pi$
$564$		0			0
$565$	2.00000	−	3.46410i	0.0841406	−	0.145736i
$566$	−24.0000			−1.00880
$567$		0			0
$568$	−5.00000			−0.209795
$569$	6.50000	−	11.2583i	0.272494	−	0.471974i	−0.697006	−	0.717066i	$-0.745485\pi$
$569$	6.50000	−	11.2583i	0.272494	−	0.471974i	0.969500	+	0.245092i	$0.0788181\pi$
$570$		0			0
$571$	21.5000	+	37.2391i	0.899747	+	1.55841i	0.827817	+	0.560998i	$0.189582\pi$
$571$	21.5000	+	37.2391i	0.899747	+	1.55841i	0.0719297	+	0.997410i	$0.477084\pi$
$572$	−1.00000	+	1.73205i	−0.0418121	+	0.0724207i
$573$		0			0
$574$	−15.0000	+	5.19615i	−0.626088	+	0.216883i
$575$	−7.00000			−0.291920
$576$		0			0
$577$	9.00000	+	15.5885i	0.374675	+	0.648956i	0.990278	−	0.139100i	$-0.0444210\pi$
$577$	9.00000	+	15.5885i	0.374675	+	0.648956i	−0.615603	+	0.788056i	$0.711088\pi$
$578$	−4.00000	−	6.92820i	−0.166378	−	0.288175i
$579$		0			0
$580$	10.0000			0.415227
$581$	−28.0000	−	24.2487i	−1.16164	−	1.00601i
$582$		0			0
$583$	−2.00000	+	3.46410i	−0.0828315	+	0.143468i
$584$	−4.00000	−	6.92820i	−0.165521	−	0.286691i
$585$		0			0
$586$	1.50000	−	2.59808i	0.0619644	−	0.107326i
$587$	12.0000			0.495293			0.247647	−	0.968850i	$-0.420343\pi$
$587$	12.0000			0.495293			0.247647	+	0.968850i	$0.420343\pi$
$588$		0			0
$589$	14.0000			0.576860
$590$	−11.0000	+	19.0526i	−0.452863	+	0.784381i
$591$		0			0
$592$	−1.50000	−	2.59808i	−0.0616496	−	0.106780i
$593$	−10.5000	+	18.1865i	−0.431183	+	0.746831i	−0.996976	−	0.0777165i	$-0.975237\pi$
$593$	−10.5000	+	18.1865i	−0.431183	+	0.746831i	0.565792	+	0.824548i	$0.308570\pi$
$594$		0			0
$595$	12.0000	+	10.3923i	0.491952	+	0.426043i
$596$	−17.0000			−0.696347
$597$		0			0
$598$	7.00000	+	12.1244i	0.286251	+	0.495802i
$599$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$599$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$600$		0			0
$601$		0			0			−	1.00000i	$-0.5\pi$
$601$		0			0				1.00000i	$0.5\pi$
$602$	27.5000	−	9.52628i	1.12082	−	0.388262i
$603$		0			0
$604$	2.50000	−	4.33013i	0.101724	−	0.176190i
$605$	−1.00000	−	1.73205i	−0.0406558	−	0.0704179i
$606$		0			0
$607$	−18.0000	+	31.1769i	−0.730597	+	1.26543i	0.226031	+	0.974120i	$0.427425\pi$
$607$	−18.0000	+	31.1769i	−0.730597	+	1.26543i	−0.956628	+	0.291312i	$0.905908\pi$
$608$	−7.00000			−0.283887
$609$		0			0
$610$	−20.0000			−0.809776
$611$	−7.00000	+	12.1244i	−0.283190	+	0.490499i
$612$		0			0
$613$	−1.00000	−	1.73205i	−0.0403896	−	0.0699569i	0.845124	−	0.534570i	$-0.179527\pi$
$613$	−1.00000	−	1.73205i	−0.0403896	−	0.0699569i	−0.885514	+	0.464614i	$0.846193\pi$
$614$	14.0000	−	24.2487i	0.564994	−	0.978598i
$615$		0			0
$616$	−0.500000	+	2.59808i	−0.0201456	+	0.104679i
$617$	20.0000			0.805170			0.402585	−	0.915383i	$-0.368112\pi$
$617$	20.0000			0.805170			0.402585	+	0.915383i	$0.368112\pi$
$618$		0			0
$619$	13.0000	+	22.5167i	0.522514	+	0.905021i	0.999657	+	0.0261952i	$0.00833914\pi$
$619$	13.0000	+	22.5167i	0.522514	+	0.905021i	−0.477143	+	0.878826i	$0.658328\pi$
$620$	−2.00000	−	3.46410i	−0.0803219	−	0.139122i
$621$		0			0
$622$	27.0000			1.08260
$623$	5.00000	−	1.73205i	0.200321	−	0.0693932i
$624$		0			0
$625$	9.50000	−	16.4545i	0.380000	−	0.658179i
$626$	7.50000	+	12.9904i	0.299760	+	0.519200i
$627$		0			0
$628$	−0.500000	+	0.866025i	−0.0199522	+	0.0345582i
$629$	9.00000			0.358854
$630$		0			0
$631$	−10.0000			−0.398094			−0.199047	−	0.979990i	$-0.563785\pi$
$631$	−10.0000			−0.398094			−0.199047	+	0.979990i	$0.563785\pi$
$632$	−4.00000	+	6.92820i	−0.159111	+	0.275589i
$633$		0			0
$634$	−15.0000	−	25.9808i	−0.595726	−	1.03183i
$635$	19.0000	−	32.9090i	0.753992	−	1.30595i
$636$		0			0
$637$	2.00000	+	13.8564i	0.0792429	+	0.549011i
$638$	−5.00000			−0.197952
$639$		0			0
$640$	1.00000	+	1.73205i	0.0395285	+	0.0684653i
$641$	−6.00000	−	10.3923i	−0.236986	−	0.410471i	0.722862	−	0.690992i	$-0.242826\pi$
$641$	−6.00000	−	10.3923i	−0.236986	−	0.410471i	−0.959848	+	0.280521i	$0.909493\pi$
$642$		0			0
$643$	40.0000			1.57745			0.788723	−	0.614749i	$-0.210743\pi$
$643$	40.0000			1.57745			0.788723	+	0.614749i	$0.210743\pi$
$644$	14.0000	+	12.1244i	0.551677	+	0.477767i
$645$		0			0
$646$	10.5000	−	18.1865i	0.413117	−	0.715540i
$647$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$647$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$648$		0			0
$649$	5.50000	−	9.52628i	0.215894	−	0.373939i
$650$	2.00000			0.0784465
$651$		0			0
$652$	6.00000			0.234978
$653$	−5.00000	+	8.66025i	−0.195665	+	0.338902i	−0.947118	−	0.320884i	$-0.896020\pi$
$653$	−5.00000	+	8.66025i	−0.195665	+	0.338902i	0.751453	+	0.659786i	$0.229353\pi$
$654$		0			0
$655$	−8.00000	−	13.8564i	−0.312586	−	0.541415i
$656$	3.00000	−	5.19615i	0.117130	−	0.202876i
$657$		0			0
$658$	−3.50000	+	18.1865i	−0.136444	+	0.708985i
$659$	20.0000			0.779089			0.389545	−	0.921008i	$-0.372632\pi$
$659$	20.0000			0.779089			0.389545	+	0.921008i	$0.372632\pi$
$660$		0			0
$661$	5.50000	+	9.52628i	0.213925	+	0.370529i	0.952940	−	0.303160i	$-0.0980418\pi$
$661$	5.50000	+	9.52628i	0.213925	+	0.370529i	−0.739014	+	0.673690i	$0.764708\pi$
$662$	−7.00000	−	12.1244i	−0.272063	−	0.471226i
$663$		0			0
$664$	14.0000			0.543305
$665$	7.00000	−	36.3731i	0.271448	−	1.41049i
$666$		0			0
$667$	−17.5000	+	30.3109i	−0.677603	+	1.17364i
$668$	−5.00000	−	8.66025i	−0.193456	−	0.335075i
$669$		0			0
$670$	−4.00000	+	6.92820i	−0.154533	+	0.267660i
$671$	10.0000			0.386046
$672$		0			0
$673$	−10.0000			−0.385472			−0.192736	−	0.981251i	$-0.561736\pi$
$673$	−10.0000			−0.385472			−0.192736	+	0.981251i	$0.561736\pi$
$674$	4.00000	−	6.92820i	0.154074	−	0.266864i
$675$		0			0
$676$	4.50000	+	7.79423i	0.173077	+	0.299778i
$677$	−10.5000	+	18.1865i	−0.403548	+	0.698965i	−0.994151	−	0.107997i	$-0.965556\pi$
$677$	−10.5000	+	18.1865i	−0.403548	+	0.698965i	0.590603	+	0.806962i	$0.298890\pi$
$678$		0			0
$679$	30.0000	+	25.9808i	1.15129	+	0.997050i
$680$	−6.00000			−0.230089
$681$		0			0
$682$	1.00000	+	1.73205i	0.0382920	+	0.0663237i
$683$	19.5000	+	33.7750i	0.746147	+	1.29236i	0.949657	+	0.313291i	$0.101432\pi$
$683$	19.5000	+	33.7750i	0.746147	+	1.29236i	−0.203510	+	0.979073i	$0.565235\pi$
$684$		0			0
$685$	−12.0000			−0.458496
$686$	8.50000	+	16.4545i	0.324532	+	0.628235i
$687$		0			0
$688$	−5.50000	+	9.52628i	−0.209686	+	0.363186i
$689$	−4.00000	−	6.92820i	−0.152388	−	0.263944i
$690$		0			0
$691$	−9.00000	+	15.5885i	−0.342376	+	0.593013i	−0.984873	−	0.173275i	$-0.944565\pi$
$691$	−9.00000	+	15.5885i	−0.342376	+	0.593013i	0.642497	+	0.766288i	$0.277898\pi$
$692$	−14.0000			−0.532200
$693$		0			0
$694$	24.0000			0.911028
$695$	−17.0000	+	29.4449i	−0.644847	+	1.11691i
$696$		0			0
$697$	9.00000	+	15.5885i	0.340899	+	0.590455i
$698$	4.00000	−	6.92820i	0.151402	−	0.262236i
$699$		0			0
$700$	2.50000	−	0.866025i	0.0944911	−	0.0327327i
$701$	27.0000			1.01978			0.509888	−	0.860241i	$-0.329687\pi$
$701$	27.0000			1.01978			0.509888	+	0.860241i	$0.329687\pi$
$702$		0			0
$703$	−10.5000	−	18.1865i	−0.396015	−	0.685918i
$704$	−0.500000	−	0.866025i	−0.0188445	−	0.0326396i
$705$		0			0
$706$	10.0000			0.376355
$707$	−1.50000	+	7.79423i	−0.0564133	+	0.293132i
$708$		0			0
$709$	15.5000	−	26.8468i	0.582115	−	1.00825i	−0.413114	−	0.910679i	$-0.635559\pi$
$709$	15.5000	−	26.8468i	0.582115	−	1.00825i	0.995228	−	0.0975728i	$-0.0311079\pi$
$710$	5.00000	+	8.66025i	0.187647	+	0.325014i
$711$		0			0
$712$	−1.00000	+	1.73205i	−0.0374766	+	0.0649113i
$713$	14.0000			0.524304
$714$		0			0
$715$	4.00000			0.149592
$716$	−4.50000	+	7.79423i	−0.168173	+	0.291284i
$717$		0			0
$718$	9.00000	+	15.5885i	0.335877	+	0.581756i
$719$	4.50000	−	7.79423i	0.167822	−	0.290676i	−0.769832	−	0.638247i	$-0.779660\pi$
$719$	4.50000	−	7.79423i	0.167822	−	0.290676i	0.937654	+	0.347571i	$0.112993\pi$
$720$		0			0
$721$	25.0000	−	8.66025i	0.931049	−	0.322525i
$722$	−30.0000			−1.11648
$723$		0			0
$724$	5.00000	+	8.66025i	0.185824	+	0.321856i
$725$	2.50000	+	4.33013i	0.0928477	+	0.160817i
$726$		0			0
$727$	−14.0000			−0.519231			−0.259616	−	0.965712i	$-0.583596\pi$
$727$	−14.0000			−0.519231			−0.259616	+	0.965712i	$0.583596\pi$
$728$	−4.00000	−	3.46410i	−0.148250	−	0.128388i
$729$		0			0
$730$	−8.00000	+	13.8564i	−0.296093	+	0.512849i
$731$	−16.5000	−	28.5788i	−0.610275	−	1.05703i
$732$		0			0
$733$	−11.0000	+	19.0526i	−0.406294	+	0.703722i	−0.994471	−	0.105010i	$-0.966513\pi$
$733$	−11.0000	+	19.0526i	−0.406294	+	0.703722i	0.588177	+	0.808732i	$0.299846\pi$
$734$		0			0
$735$		0			0
$736$	−7.00000			−0.258023
$737$	2.00000	−	3.46410i	0.0736709	−	0.127602i
$738$		0			0
$739$	−10.0000	−	17.3205i	−0.367856	−	0.637145i	0.621374	−	0.783514i	$-0.286575\pi$
$739$	−10.0000	−	17.3205i	−0.367856	−	0.637145i	−0.989230	+	0.146369i	$0.953241\pi$
$740$	−3.00000	+	5.19615i	−0.110282	+	0.191014i
$741$		0			0
$742$	−8.00000	−	6.92820i	−0.293689	−	0.254342i
$743$	4.00000			0.146746			0.0733729	−	0.997305i	$-0.476624\pi$
$743$	4.00000			0.146746			0.0733729	+	0.997305i	$0.476624\pi$
$744$		0			0
$745$	17.0000	+	29.4449i	0.622832	+	1.07878i
$746$	−11.0000	−	19.0526i	−0.402739	−	0.697564i
$747$		0			0
$748$	3.00000			0.109691
$749$	45.0000	−	15.5885i	1.64426	−	0.569590i
$750$		0			0
$751$	20.0000	−	34.6410i	0.729810	−	1.26407i	−0.227153	−	0.973859i	$-0.572942\pi$
$751$	20.0000	−	34.6410i	0.729810	−	1.26407i	0.956963	−	0.290209i	$-0.0937250\pi$
$752$	−3.50000	−	6.06218i	−0.127632	−	0.221065i
$753$		0			0
$754$	5.00000	−	8.66025i	0.182089	−	0.315388i
$755$	−10.0000			−0.363937
$756$		0			0
$757$	15.0000			0.545184			0.272592	−	0.962130i	$-0.412119\pi$
$757$	15.0000			0.545184			0.272592	+	0.962130i	$0.412119\pi$
$758$	−8.00000	+	13.8564i	−0.290573	+	0.503287i
$759$		0			0
$760$	7.00000	+	12.1244i	0.253917	+	0.439797i
$761$	−15.0000	+	25.9808i	−0.543750	+	0.941802i	0.454935	+	0.890525i	$0.349663\pi$
$761$	−15.0000	+	25.9808i	−0.543750	+	0.941802i	−0.998684	+	0.0512772i	$0.983671\pi$
$762$		0			0
$763$	2.00000	−	10.3923i	0.0724049	−	0.376227i
$764$	8.00000			0.289430
$765$		0			0
$766$	−15.5000	−	26.8468i	−0.560038	−	0.970014i
$767$	11.0000	+	19.0526i	0.397187	+	0.687948i
$768$		0			0
$769$	28.0000			1.00971			0.504853	−	0.863205i	$-0.331547\pi$
$769$	28.0000			1.00971			0.504853	+	0.863205i	$0.331547\pi$
$770$	5.00000	−	1.73205i	0.180187	−	0.0624188i
$771$		0			0
$772$	−12.0000	+	20.7846i	−0.431889	+	0.748054i
$773$	2.00000	+	3.46410i	0.0719350	+	0.124595i	0.899749	−	0.436407i	$-0.143749\pi$
$773$	2.00000	+	3.46410i	0.0719350	+	0.124595i	−0.827814	+	0.561002i	$0.810416\pi$
$774$		0			0
$775$	1.00000	−	1.73205i	0.0359211	−	0.0622171i
$776$	−15.0000			−0.538469
$777$		0			0
$778$	−6.00000			−0.215110
$779$	21.0000	−	36.3731i	0.752403	−	1.30320i
$780$		0			0
$781$	−2.50000	−	4.33013i	−0.0894570	−	0.154944i
$782$	10.5000	−	18.1865i	0.375479	−	0.650349i
$783$		0			0
$784$	−6.50000	−	2.59808i	−0.232143	−	0.0927884i
$785$	2.00000			0.0713831
$786$		0			0
$787$	18.5000	+	32.0429i	0.659454	+	1.14221i	0.980757	+	0.195231i	$0.0625457\pi$
$787$	18.5000	+	32.0429i	0.659454	+	1.14221i	−0.321303	+	0.946976i	$0.604121\pi$
$788$	7.50000	+	12.9904i	0.267176	+	0.462763i
$789$		0			0
$790$	16.0000			0.569254
$791$	−4.00000	−	3.46410i	−0.142224	−	0.123169i
$792$		0			0
$793$	−10.0000	+	17.3205i	−0.355110	+	0.615069i
$794$	−1.50000	−	2.59808i	−0.0532330	−	0.0922023i
$795$		0			0
$796$	−5.00000	+	8.66025i	−0.177220	+	0.306955i
$797$	50.0000			1.77109			0.885545	−	0.464553i	$-0.153785\pi$
$797$	50.0000			1.77109			0.885545	+	0.464553i	$0.153785\pi$
$798$		0			0
$799$	21.0000			0.742927
$800$	−0.500000	+	0.866025i	−0.0176777	+	0.0306186i
$801$		0			0
$802$	−16.0000	−	27.7128i	−0.564980	−	0.978573i
$803$	4.00000	−	6.92820i	0.141157	−	0.244491i
$804$		0			0
$805$	7.00000	−	36.3731i	0.246718	−	1.28198i
$806$	−4.00000			−0.140894
$807$		0			0
$808$	−1.50000	−	2.59808i	−0.0527698	−	0.0914000i
$809$	−7.00000	−	12.1244i	−0.246107	−	0.426270i	0.716335	−	0.697756i	$-0.245818\pi$
$809$	−7.00000	−	12.1244i	−0.246107	−	0.426270i	−0.962442	+	0.271487i	$0.912485\pi$
$810$		0			0
$811$	−28.0000			−0.983213			−0.491606	−	0.870817i	$-0.663590\pi$
$811$	−28.0000			−0.983213			−0.491606	+	0.870817i	$0.663590\pi$
$812$	2.50000	−	12.9904i	0.0877328	−	0.455873i
$813$		0			0
$814$	1.50000	−	2.59808i	0.0525750	−	0.0910625i
$815$	−6.00000	−	10.3923i	−0.210171	−	0.364027i
$816$		0			0
$817$	−38.5000	+	66.6840i	−1.34694	+	2.33298i
$818$	−32.0000			−1.11885
$819$		0			0
$820$	−12.0000			−0.419058
$821$	25.0000	−	43.3013i	0.872506	−	1.51122i	0.0131101	−	0.999914i	$-0.495827\pi$
$821$	25.0000	−	43.3013i	0.872506	−	1.51122i	0.859396	−	0.511311i	$-0.170840\pi$
$822$		0			0
$823$	2.00000	+	3.46410i	0.0697156	+	0.120751i	0.898776	−	0.438408i	$-0.144457\pi$
$823$	2.00000	+	3.46410i	0.0697156	+	0.120751i	−0.829060	+	0.559159i	$0.811124\pi$
$824$	−5.00000	+	8.66025i	−0.174183	+	0.301694i
$825$		0			0
$826$	22.0000	+	19.0526i	0.765478	+	0.662923i
$827$	10.0000			0.347734			0.173867	−	0.984769i	$-0.444374\pi$
$827$	10.0000			0.347734			0.173867	+	0.984769i	$0.444374\pi$
$828$		0			0
$829$	8.50000	+	14.7224i	0.295217	+	0.511331i	0.975035	−	0.222049i	$-0.0712747\pi$
$829$	8.50000	+	14.7224i	0.295217	+	0.511331i	−0.679818	+	0.733381i	$0.737941\pi$
$830$	−14.0000	−	24.2487i	−0.485947	−	0.841685i
$831$		0			0
$832$	2.00000			0.0693375
$833$	16.5000	−	12.9904i	0.571691	−	0.450090i
$834$		0			0
$835$	−10.0000	+	17.3205i	−0.346064	+	0.599401i
$836$	−3.50000	−	6.06218i	−0.121050	−	0.209665i
$837$		0			0
$838$	−1.50000	+	2.59808i	−0.0518166	+	0.0897491i
$839$	−24.0000			−0.828572			−0.414286	−	0.910147i	$-0.635969\pi$
$839$	−24.0000			−0.828572			−0.414286	+	0.910147i	$0.635969\pi$
$840$		0			0
$841$	−4.00000			−0.137931
$842$	−8.50000	+	14.7224i	−0.292929	+	0.507369i
$843$		0			0
$844$	−10.0000	−	17.3205i	−0.344214	−	0.596196i
$845$	9.00000	−	15.5885i	0.309609	−	0.536259i
$846$		0			0
$847$	−2.50000	+	0.866025i	−0.0859010	+	0.0297570i
$848$	4.00000			0.137361
$849$		0			0
$850$	−1.50000	−	2.59808i	−0.0514496	−	0.0891133i
$851$	−10.5000	−	18.1865i	−0.359935	−	0.623426i
$852$		0			0
$853$	46.0000			1.57501			0.787505	−	0.616308i	$-0.211372\pi$
$853$	46.0000			1.57501			0.787505	+	0.616308i	$0.211372\pi$
$854$	−5.00000	+	25.9808i	−0.171096	+	0.889043i
$855$		0			0
$856$	−9.00000	+	15.5885i	−0.307614	+	0.532803i
$857$	10.5000	+	18.1865i	0.358673	+	0.621240i	0.987739	−	0.156112i	$-0.0498959\pi$
$857$	10.5000	+	18.1865i	0.358673	+	0.621240i	−0.629066	+	0.777352i	$0.716563\pi$
$858$		0			0
$859$	11.0000	−	19.0526i	0.375315	−	0.650065i	−0.615059	−	0.788481i	$-0.710868\pi$
$859$	11.0000	−	19.0526i	0.375315	−	0.650065i	0.990374	+	0.138416i	$0.0442012\pi$
$860$	22.0000			0.750194
$861$		0			0
$862$	40.0000			1.36241
$863$	−14.0000	+	24.2487i	−0.476566	+	0.825436i	−0.999639	−	0.0268516i	$-0.991452\pi$
$863$	−14.0000	+	24.2487i	−0.476566	+	0.825436i	0.523074	+	0.852287i	$0.324785\pi$
$864$		0			0
$865$	14.0000	+	24.2487i	0.476014	+	0.824481i
$866$	2.50000	−	4.33013i	0.0849535	−	0.147144i
$867$		0			0
$868$	−5.00000	+	1.73205i	−0.169711	+	0.0587896i
$869$	−8.00000			−0.271381
$870$		0			0
$871$	4.00000	+	6.92820i	0.135535	+	0.234753i
$872$	2.00000	+	3.46410i	0.0677285	+	0.117309i
$873$		0			0
$874$	−49.0000			−1.65745
$875$	−24.0000	−	20.7846i	−0.811348	−	0.702648i
$876$		0			0
$877$	−13.0000	+	22.5167i	−0.438979	+	0.760334i	−0.997611	−	0.0690819i	$-0.977993\pi$
$877$	−13.0000	+	22.5167i	−0.438979	+	0.760334i	0.558632	+	0.829416i	$0.311326\pi$
$878$	2.50000	+	4.33013i	0.0843709	+	0.146135i
$879$		0			0
$880$	−1.00000	+	1.73205i	−0.0337100	+	0.0583874i
$881$	−36.0000			−1.21287			−0.606435	−	0.795133i	$-0.707401\pi$
$881$	−36.0000			−1.21287			−0.606435	+	0.795133i	$0.707401\pi$
$882$		0			0
$883$	8.00000			0.269221			0.134611	−	0.990899i	$-0.457022\pi$
$883$	8.00000			0.269221			0.134611	+	0.990899i	$0.457022\pi$
$884$	−3.00000	+	5.19615i	−0.100901	+	0.174766i
$885$		0			0
$886$	−20.5000	−	35.5070i	−0.688711	−	1.19288i
$887$	−10.0000	+	17.3205i	−0.335767	+	0.581566i	−0.983632	−	0.180190i	$-0.942329\pi$
$887$	−10.0000	+	17.3205i	−0.335767	+	0.581566i	0.647865	+	0.761755i	$0.275662\pi$
$888$		0			0
$889$	−38.0000	−	32.9090i	−1.27448	−	1.10373i
$890$	4.00000			0.134080
$891$		0			0
$892$	7.00000	+	12.1244i	0.234377	+	0.405953i
$893$	−24.5000	−	42.4352i	−0.819861	−	1.42004i
$894$		0			0
$895$	18.0000			0.601674
$896$	2.50000	−	0.866025i	0.0835191	−	0.0289319i
$897$		0			0
$898$	18.0000	−	31.1769i	0.600668	−	1.04039i
$899$	−5.00000	−	8.66025i	−0.166759	−	0.288836i
$900$		0			0
$901$	−6.00000	+	10.3923i	−0.199889	+	0.346218i
$902$	6.00000			0.199778
$903$		0			0
$904$	2.00000			0.0665190
$905$	10.0000	−	17.3205i	0.332411	−	0.575753i
$906$		0			0
$907$	12.0000	+	20.7846i	0.398453	+	0.690142i	0.993535	−	0.113523i	$-0.0362137\pi$
$907$	12.0000	+	20.7846i	0.398453	+	0.690142i	−0.595082	+	0.803665i	$0.702880\pi$
$908$	5.00000	−	8.66025i	0.165931	−	0.287401i
$909$		0			0
$910$	−2.00000	+	10.3923i	−0.0662994	+	0.344502i
$911$	−13.0000			−0.430709			−0.215355	−	0.976536i	$-0.569091\pi$
$911$	−13.0000			−0.430709			−0.215355	+	0.976536i	$0.569091\pi$
$912$		0			0
$913$	7.00000	+	12.1244i	0.231666	+	0.401258i
$914$	20.0000	+	34.6410i	0.661541	+	1.14582i
$915$		0			0
$916$	−10.0000			−0.330409
$917$	−20.0000	+	6.92820i	−0.660458	+	0.228789i
$918$		0			0
$919$	−21.5000	+	37.2391i	−0.709220	+	1.22840i	0.255927	+	0.966696i	$0.417619\pi$
$919$	−21.5000	+	37.2391i	−0.709220	+	1.22840i	−0.965147	+	0.261708i	$0.915714\pi$
$920$	7.00000	+	12.1244i	0.230783	+	0.399728i
$921$		0			0
$922$	−6.50000	+	11.2583i	−0.214066	+	0.370773i
$923$	10.0000			0.329154
$924$		0			0
$925$	−3.00000			−0.0986394
$926$	−12.0000	+	20.7846i	−0.394344	+	0.683025i
$927$		0			0
$928$	2.50000	+	4.33013i	0.0820665	+	0.142143i
$929$	2.00000	−	3.46410i	0.0656179	−	0.113653i	−0.831350	−	0.555749i	$-0.812431\pi$
$929$	2.00000	−	3.46410i	0.0656179	−	0.113653i	0.896968	+	0.442096i	$0.145765\pi$
$930$		0			0
$931$	−45.5000	−	18.1865i	−1.49120	−	0.596040i
$932$	9.00000			0.294805
$933$		0			0
$934$	−6.50000	−	11.2583i	−0.212686	−	0.368384i
$935$	−3.00000	−	5.19615i	−0.0981105	−	0.169932i
$936$		0			0
$937$	26.0000			0.849383			0.424691	−	0.905338i	$-0.360383\pi$
$937$	26.0000			0.849383			0.424691	+	0.905338i	$0.360383\pi$
$938$	8.00000	+	6.92820i	0.261209	+	0.226214i
$939$		0			0
$940$	−7.00000	+	12.1244i	−0.228315	+	0.395453i
$941$	7.50000	+	12.9904i	0.244493	+	0.423474i	0.961989	−	0.273088i	$-0.0880451\pi$
$941$	7.50000	+	12.9904i	0.244493	+	0.423474i	−0.717496	+	0.696563i	$0.754712\pi$
$942$		0			0
$943$	21.0000	−	36.3731i	0.683854	−	1.18447i
$944$	−11.0000			−0.358020
$945$		0			0
$946$	−11.0000			−0.357641
$947$	−13.5000	+	23.3827i	−0.438691	+	0.759835i	−0.997589	−	0.0694014i	$-0.977891\pi$
$947$	−13.5000	+	23.3827i	−0.438691	+	0.759835i	0.558898	+	0.829237i	$0.311224\pi$
$948$		0			0
$949$	8.00000	+	13.8564i	0.259691	+	0.449798i
$950$	−3.50000	+	6.06218i	−0.113555	+	0.196683i
$951$		0			0
$952$	−1.50000	+	7.79423i	−0.0486153	+	0.252612i
$953$	10.0000			0.323932			0.161966	−	0.986796i	$-0.448217\pi$
$953$	10.0000			0.323932			0.161966	+	0.986796i	$0.448217\pi$
$954$		0			0
$955$	−8.00000	−	13.8564i	−0.258874	−	0.448383i
$956$	−5.00000	−	8.66025i	−0.161712	−	0.280093i
$957$		0			0
$958$	18.0000			0.581554
$959$	−3.00000	+	15.5885i	−0.0968751	+	0.503378i
$960$		0			0
$961$	13.5000	−	23.3827i	0.435484	−	0.754280i
$962$	3.00000	+	5.19615i	0.0967239	+	0.167531i
$963$		0			0
$964$	2.00000	−	3.46410i	0.0644157	−	0.111571i
$965$	48.0000			1.54517
$966$		0			0
$967$	−3.00000			−0.0964735			−0.0482367	−	0.998836i	$-0.515360\pi$
$967$	−3.00000			−0.0964735			−0.0482367	+	0.998836i	$0.515360\pi$
$968$	0.500000	−	0.866025i	0.0160706	−	0.0278351i
$969$		0			0
$970$	15.0000	+	25.9808i	0.481621	+	0.834192i
$971$	−18.0000	+	31.1769i	−0.577647	+	1.00051i	0.418101	+	0.908401i	$0.362696\pi$
$971$	−18.0000	+	31.1769i	−0.577647	+	1.00051i	−0.995748	+	0.0921142i	$0.970638\pi$
$972$		0			0
$973$	34.0000	+	29.4449i	1.08999	+	0.943959i
$974$	−18.0000			−0.576757
$975$		0			0
$976$	−5.00000	−	8.66025i	−0.160046	−	0.277208i
$977$	−22.0000	−	38.1051i	−0.703842	−	1.21909i	−0.967108	−	0.254367i	$-0.918133\pi$
$977$	−22.0000	−	38.1051i	−0.703842	−	1.21909i	0.263265	−	0.964723i	$-0.415201\pi$
$978$		0			0
$979$	−2.00000			−0.0639203
$980$	2.00000	+	13.8564i	0.0638877	+	0.442627i
$981$		0			0
$982$	13.0000	−	22.5167i	0.414847	−	0.718536i
$983$	24.5000	+	42.4352i	0.781429	+	1.35347i	0.931110	+	0.364740i	$0.118842\pi$
$983$	24.5000	+	42.4352i	0.781429	+	1.35347i	−0.149681	+	0.988734i	$0.547825\pi$
$984$		0			0
$985$	15.0000	−	25.9808i	0.477940	−	0.827816i
$986$	−15.0000			−0.477697
$987$		0			0
$988$	14.0000			0.445399
$989$	−38.5000	+	66.6840i	−1.22423	+	2.12043i
$990$		0			0
$991$	−4.00000	−	6.92820i	−0.127064	−	0.220082i	0.795474	−	0.605988i	$-0.207222\pi$
$991$	−4.00000	−	6.92820i	−0.127064	−	0.220082i	−0.922538	+	0.385906i	$0.873889\pi$
$992$	1.00000	−	1.73205i	0.0317500	−	0.0549927i
$993$		0			0
$994$	12.5000	−	4.33013i	0.396476	−	0.137343i
$995$	20.0000			0.634043
$996$		0			0
$997$	−15.0000	−	25.9808i	−0.475055	−	0.822819i	0.524537	−	0.851388i	$-0.324238\pi$
$997$	−15.0000	−	25.9808i	−0.475055	−	0.822819i	−0.999592	+	0.0285686i	$0.990905\pi$
$998$	−20.0000	−	34.6410i	−0.633089	−	1.09654i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1386.2.k.k.793.1	yes	2
3.2	odd	2		1386.2.k.i.793.1	✓	2
7.2	even	3		9702.2.a.w.1.1		1
7.4	even	3	inner	1386.2.k.k.991.1	yes	2
7.5	odd	6		9702.2.a.j.1.1		1
21.2	odd	6		9702.2.a.bg.1.1		1
21.5	even	6		9702.2.a.bx.1.1		1
21.11	odd	6		1386.2.k.i.991.1	yes	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1386.2.k.i.793.1	✓	2	3.2	odd	2
1386.2.k.i.991.1	yes	2	21.11	odd	6
1386.2.k.k.793.1	yes	2	1.1	even	1	trivial
1386.2.k.k.991.1	yes	2	7.4	even	3	inner
9702.2.a.j.1.1		1	7.5	odd	6
9702.2.a.w.1.1		1	7.2	even	3
9702.2.a.bg.1.1		1	21.2	odd	6
9702.2.a.bx.1.1		1	21.5	even	6

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)

\(f(q)\)	\(=\)	\(q+(0.500000 - 0.866025i) q^{2} +(-0.500000 - 0.866025i) q^{4} +(-1.00000 + 1.73205i) q^{5} +(2.00000 + 1.73205i) q^{7} -1.00000 q^{8} +O(q^{10})\)	\(q+(0.500000 - 0.866025i) q^{2} +(-0.500000 - 0.866025i) q^{4} +(-1.00000 + 1.73205i) q^{5} +(2.00000 + 1.73205i) q^{7} -1.00000 q^{8} +(1.00000 + 1.73205i) q^{10} +(-0.500000 - 0.866025i) q^{11} +2.00000 q^{13} +(2.50000 - 0.866025i) q^{14} +(-0.500000 + 0.866025i) q^{16} +(-1.50000 - 2.59808i) q^{17} +(-3.50000 + 6.06218i) q^{19} +2.00000 q^{20} -1.00000 q^{22} +(-3.50000 + 6.06218i) q^{23} +(0.500000 + 0.866025i) q^{25} +(1.00000 - 1.73205i) q^{26} +(0.500000 - 2.59808i) q^{28} +5.00000 q^{29} +(-1.00000 - 1.73205i) q^{31} +(0.500000 + 0.866025i) q^{32} -3.00000 q^{34} +(-5.00000 + 1.73205i) q^{35} +(-1.50000 + 2.59808i) q^{37} +(3.50000 + 6.06218i) q^{38} +(1.00000 - 1.73205i) q^{40} -6.00000 q^{41} +11.0000 q^{43} +(-0.500000 + 0.866025i) q^{44} +(3.50000 + 6.06218i) q^{46} +(-3.50000 + 6.06218i) q^{47} +(1.00000 + 6.92820i) q^{49} +1.00000 q^{50} +(-1.00000 - 1.73205i) q^{52} +(-2.00000 - 3.46410i) q^{53} +2.00000 q^{55} +(-2.00000 - 1.73205i) q^{56} +(2.50000 - 4.33013i) q^{58} +(5.50000 + 9.52628i) q^{59} +(-5.00000 + 8.66025i) q^{61} -2.00000 q^{62} +1.00000 q^{64} +(-2.00000 + 3.46410i) q^{65} +(2.00000 + 3.46410i) q^{67} +(-1.50000 + 2.59808i) q^{68} +(-1.00000 + 5.19615i) q^{70} +5.00000 q^{71} +(4.00000 + 6.92820i) q^{73} +(1.50000 + 2.59808i) q^{74} +7.00000 q^{76} +(0.500000 - 2.59808i) q^{77} +(4.00000 - 6.92820i) q^{79} +(-1.00000 - 1.73205i) q^{80} +(-3.00000 + 5.19615i) q^{82} -14.0000 q^{83} +6.00000 q^{85} +(5.50000 - 9.52628i) q^{86} +(0.500000 + 0.866025i) q^{88} +(1.00000 - 1.73205i) q^{89} +(4.00000 + 3.46410i) q^{91} +7.00000 q^{92} +(3.50000 + 6.06218i) q^{94} +(-7.00000 - 12.1244i) q^{95} +15.0000 q^{97} +(6.50000 + 2.59808i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + q^{2} - q^{4} - 2 q^{5} + 4 q^{7} - 2 q^{8}+O(q^{10}) \) 2 * q + q^2 - q^4 - 2 * q^5 + 4 * q^7 - 2 * q^8	\( 2 q + q^{2} - q^{4} - 2 q^{5} + 4 q^{7} - 2 q^{8} + 2 q^{10} - q^{11} + 4 q^{13} + 5 q^{14} - q^{16} - 3 q^{17} - 7 q^{19} + 4 q^{20} - 2 q^{22} - 7 q^{23} + q^{25} + 2 q^{26} + q^{28} + 10 q^{29} - 2 q^{31} + q^{32} - 6 q^{34} - 10 q^{35} - 3 q^{37} + 7 q^{38} + 2 q^{40} - 12 q^{41} + 22 q^{43} - q^{44} + 7 q^{46} - 7 q^{47} + 2 q^{49} + 2 q^{50} - 2 q^{52} - 4 q^{53} + 4 q^{55} - 4 q^{56} + 5 q^{58} + 11 q^{59} - 10 q^{61} - 4 q^{62} + 2 q^{64} - 4 q^{65} + 4 q^{67} - 3 q^{68} - 2 q^{70} + 10 q^{71} + 8 q^{73} + 3 q^{74} + 14 q^{76} + q^{77} + 8 q^{79} - 2 q^{80} - 6 q^{82} - 28 q^{83} + 12 q^{85} + 11 q^{86} + q^{88} + 2 q^{89} + 8 q^{91} + 14 q^{92} + 7 q^{94} - 14 q^{95} + 30 q^{97} + 13 q^{98}+O(q^{100}) \) 2 * q + q^2 - q^4 - 2 * q^5 + 4 * q^7 - 2 * q^8 + 2 * q^10 - q^11 + 4 * q^13 + 5 * q^14 - q^16 - 3 * q^17 - 7 * q^19 + 4 * q^20 - 2 * q^22 - 7 * q^23 + q^25 + 2 * q^26 + q^28 + 10 * q^29 - 2 * q^31 + q^32 - 6 * q^34 - 10 * q^35 - 3 * q^37 + 7 * q^38 + 2 * q^40 - 12 * q^41 + 22 * q^43 - q^44 + 7 * q^46 - 7 * q^47 + 2 * q^49 + 2 * q^50 - 2 * q^52 - 4 * q^53 + 4 * q^55 - 4 * q^56 + 5 * q^58 + 11 * q^59 - 10 * q^61 - 4 * q^62 + 2 * q^64 - 4 * q^65 + 4 * q^67 - 3 * q^68 - 2 * q^70 + 10 * q^71 + 8 * q^73 + 3 * q^74 + 14 * q^76 + q^77 + 8 * q^79 - 2 * q^80 - 6 * q^82 - 28 * q^83 + 12 * q^85 + 11 * q^86 + q^88 + 2 * q^89 + 8 * q^91 + 14 * q^92 + 7 * q^94 - 14 * q^95 + 30 * q^97 + 13 * q^98

Introduction

L-functions

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Database

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