LMFDB - Embedded newform 189.2.e.a.109.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [189,2,Mod(109,189)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(189, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 4]))

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

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

chi := DirichletCharacter("189.109");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 189 = 3^{3} \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	189.e (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:	$1.50917259820$
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			109.1
Root			$0.500000 + 0.866025i$ of defining polynomial
Character	$\chi$	$=$	189.109
Dual form			189.2.e.a.163.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q+(-0.500000 - 0.866025i) q^{2} +(0.500000 - 0.866025i) q^{4} +(2.00000 + 3.46410i) q^{5} +(2.00000 + 1.73205i) q^{7} -3.00000 q^{8} +(2.00000 - 3.46410i) q^{10} +(-1.00000 + 1.73205i) q^{11} +1.00000 q^{13} +(0.500000 - 2.59808i) q^{14} +(0.500000 + 0.866025i) q^{16} +(3.00000 - 5.19615i) q^{17} +(-2.00000 - 3.46410i) q^{19} +4.00000 q^{20} +2.00000 q^{22} +(-3.00000 - 5.19615i) q^{23} +(-5.50000 + 9.52628i) q^{25} +(-0.500000 - 0.866025i) q^{26} +(2.50000 - 0.866025i) q^{28} +2.00000 q^{29} +(-1.50000 + 2.59808i) q^{31} +(-2.50000 + 4.33013i) q^{32} -6.00000 q^{34} +(-2.00000 + 10.3923i) q^{35} +(-1.50000 - 2.59808i) q^{37} +(-2.00000 + 3.46410i) q^{38} +(-6.00000 - 10.3923i) q^{40} -2.00000 q^{41} -1.00000 q^{43} +(1.00000 + 1.73205i) q^{44} +(-3.00000 + 5.19615i) q^{46} +(-3.00000 - 5.19615i) q^{47} +(1.00000 + 6.92820i) q^{49} +11.0000 q^{50} +(0.500000 - 0.866025i) q^{52} +(3.00000 - 5.19615i) q^{53} -8.00000 q^{55} +(-6.00000 - 5.19615i) q^{56} +(-1.00000 - 1.73205i) q^{58} +(-3.00000 + 5.19615i) q^{59} +(2.50000 + 4.33013i) q^{61} +3.00000 q^{62} +7.00000 q^{64} +(2.00000 + 3.46410i) q^{65} +(-3.50000 + 6.06218i) q^{67} +(-3.00000 - 5.19615i) q^{68} +(10.0000 - 3.46410i) q^{70} +(3.00000 - 5.19615i) q^{73} +(-1.50000 + 2.59808i) q^{74} -4.00000 q^{76} +(-5.00000 + 1.73205i) q^{77} +(-5.50000 - 9.52628i) q^{79} +(-2.00000 + 3.46410i) q^{80} +(1.00000 + 1.73205i) q^{82} +6.00000 q^{83} +24.0000 q^{85} +(0.500000 + 0.866025i) q^{86} +(3.00000 - 5.19615i) q^{88} +(2.00000 + 3.46410i) q^{89} +(2.00000 + 1.73205i) q^{91} -6.00000 q^{92} +(-3.00000 + 5.19615i) q^{94} +(8.00000 - 13.8564i) q^{95} +9.00000 q^{97} +(5.50000 - 4.33013i) q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - q^{2} + q^{4} + 4 q^{5} + 4 q^{7} - 6 q^{8} + 4 q^{10} - 2 q^{11} + 2 q^{13} + q^{14} + q^{16} + 6 q^{17} - 4 q^{19} + 8 q^{20} + 4 q^{22} - 6 q^{23} - 11 q^{25} - q^{26} + 5 q^{28} + 4 q^{29}+ \cdots + 11 q^{98}+O(q^{100}) $ 2 * q - q^2 + q^4 + 4 * q^5 + 4 * q^7 - 6 * q^8 + 4 * q^10 - 2 * q^11 + 2 * q^13 + q^14 + q^16 + 6 * q^17 - 4 * q^19 + 8 * q^20 + 4 * q^22 - 6 * q^23 - 11 * q^25 - q^26 + 5 * q^28 + 4 * q^29 - 3 * q^31 - 5 * q^32 - 12 * q^34 - 4 * q^35 - 3 * q^37 - 4 * q^38 - 12 * q^40 - 4 * q^41 - 2 * q^43 + 2 * q^44 - 6 * q^46 - 6 * q^47 + 2 * q^49 + 22 * q^50 + q^52 + 6 * q^53 - 16 * q^55 - 12 * q^56 - 2 * q^58 - 6 * q^59 + 5 * q^61 + 6 * q^62 + 14 * q^64 + 4 * q^65 - 7 * q^67 - 6 * q^68 + 20 * q^70 + 6 * q^73 - 3 * q^74 - 8 * q^76 - 10 * q^77 - 11 * q^79 - 4 * q^80 + 2 * q^82 + 12 * q^83 + 48 * q^85 + q^86 + 6 * q^88 + 4 * q^89 + 4 * q^91 - 12 * q^92 - 6 * q^94 + 16 * q^95 + 18 * q^97 + 11 * q^98

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/189\mathbb{Z}\right)^\times$.

$n$	$29$	$136$
$\chi(n)$	$1$	$e\left(\frac{2}{3}\right)$

Coefficient data

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

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

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

$2$	−0.500000	−	0.866025i	−0.353553	−	0.612372i	0.633316	−	0.773893i	$-0.281693\pi$
$2$	−0.500000	−	0.866025i	−0.353553	−	0.612372i	−0.986869	+	0.161521i	$0.948360\pi$
$3$		0			0
$3$		0			0
$4$	0.500000	−	0.866025i	0.250000	−	0.433013i
$5$	2.00000	+	3.46410i	0.894427	+	1.54919i	0.834512	+	0.550990i	$0.185750\pi$
$5$	2.00000	+	3.46410i	0.894427	+	1.54919i	0.0599153	+	0.998203i	$0.480917\pi$
$6$		0			0
$7$	2.00000	+	1.73205i	0.755929	+	0.654654i
$7$	2.00000	+	1.73205i	0.755929	+	0.654654i
$8$	−3.00000			−1.06066
$9$		0			0
$10$	2.00000	−	3.46410i	0.632456	−	1.09545i
$11$	−1.00000	+	1.73205i	−0.301511	+	0.522233i	−0.976478	−	0.215615i	$-0.930824\pi$
$11$	−1.00000	+	1.73205i	−0.301511	+	0.522233i	0.674967	+	0.737848i	$0.264158\pi$
$12$		0			0
$13$	1.00000			0.277350			0.138675	−	0.990338i	$-0.455716\pi$
$13$	1.00000			0.277350			0.138675	+	0.990338i	$0.455716\pi$
$14$	0.500000	−	2.59808i	0.133631	−	0.694365i
$15$		0			0
$16$	0.500000	+	0.866025i	0.125000	+	0.216506i
$17$	3.00000	−	5.19615i	0.727607	−	1.26025i	−0.230285	−	0.973123i	$-0.573966\pi$
$17$	3.00000	−	5.19615i	0.727607	−	1.26025i	0.957892	−	0.287129i	$-0.0927008\pi$
$18$		0			0
$19$	−2.00000	−	3.46410i	−0.458831	−	0.794719i	0.540068	−	0.841621i	$-0.318398\pi$
$19$	−2.00000	−	3.46410i	−0.458831	−	0.794719i	−0.998899	+	0.0469020i	$0.985065\pi$
$20$	4.00000			0.894427
$21$		0			0
$22$	2.00000			0.426401
$23$	−3.00000	−	5.19615i	−0.625543	−	1.08347i	−0.988436	−	0.151642i	$-0.951544\pi$
$23$	−3.00000	−	5.19615i	−0.625543	−	1.08347i	0.362892	−	0.931831i	$-0.381789\pi$
$24$		0			0
$25$	−5.50000	+	9.52628i	−1.10000	+	1.90526i
$26$	−0.500000	−	0.866025i	−0.0980581	−	0.169842i
$27$		0			0
$28$	2.50000	−	0.866025i	0.472456	−	0.163663i
$29$	2.00000			0.371391			0.185695	−	0.982607i	$-0.440546\pi$
$29$	2.00000			0.371391			0.185695	+	0.982607i	$0.440546\pi$
$30$		0			0
$31$	−1.50000	+	2.59808i	−0.269408	+	0.466628i	−0.968709	−	0.248199i	$-0.920161\pi$
$31$	−1.50000	+	2.59808i	−0.269408	+	0.466628i	0.699301	+	0.714827i	$0.253495\pi$
$32$	−2.50000	+	4.33013i	−0.441942	+	0.765466i
$33$		0			0
$34$	−6.00000			−1.02899
$35$	−2.00000	+	10.3923i	−0.338062	+	1.75662i
$36$		0			0
$37$	−1.50000	−	2.59808i	−0.246598	−	0.427121i	0.715981	−	0.698119i	$-0.245980\pi$
$37$	−1.50000	−	2.59808i	−0.246598	−	0.427121i	−0.962580	+	0.270998i	$0.912646\pi$
$38$	−2.00000	+	3.46410i	−0.324443	+	0.561951i
$39$		0			0
$40$	−6.00000	−	10.3923i	−0.948683	−	1.64317i
$41$	−2.00000			−0.312348			−0.156174	−	0.987730i	$-0.549916\pi$
$41$	−2.00000			−0.312348			−0.156174	+	0.987730i	$0.549916\pi$
$42$		0			0
$43$	−1.00000			−0.152499			−0.0762493	−	0.997089i	$-0.524294\pi$
$43$	−1.00000			−0.152499			−0.0762493	+	0.997089i	$0.524294\pi$
$44$	1.00000	+	1.73205i	0.150756	+	0.261116i
$45$		0			0
$46$	−3.00000	+	5.19615i	−0.442326	+	0.766131i
$47$	−3.00000	−	5.19615i	−0.437595	−	0.757937i	0.559908	−	0.828554i	$-0.310836\pi$
$47$	−3.00000	−	5.19615i	−0.437595	−	0.757937i	−0.997503	+	0.0706177i	$0.977503\pi$
$48$		0			0
$49$	1.00000	+	6.92820i	0.142857	+	0.989743i
$50$	11.0000			1.55563
$51$		0			0
$52$	0.500000	−	0.866025i	0.0693375	−	0.120096i
$53$	3.00000	−	5.19615i	0.412082	−	0.713746i	−0.583036	−	0.812447i	$-0.698135\pi$
$53$	3.00000	−	5.19615i	0.412082	−	0.713746i	0.995117	+	0.0987002i	$0.0314685\pi$
$54$		0			0
$55$	−8.00000			−1.07872
$56$	−6.00000	−	5.19615i	−0.801784	−	0.694365i
$57$		0			0
$58$	−1.00000	−	1.73205i	−0.131306	−	0.227429i
$59$	−3.00000	+	5.19615i	−0.390567	+	0.676481i	−0.992524	−	0.122047i	$-0.961054\pi$
$59$	−3.00000	+	5.19615i	−0.390567	+	0.676481i	0.601958	+	0.798528i	$0.294388\pi$
$60$		0			0
$61$	2.50000	+	4.33013i	0.320092	+	0.554416i	0.980507	−	0.196485i	$-0.0629528\pi$
$61$	2.50000	+	4.33013i	0.320092	+	0.554416i	−0.660415	+	0.750901i	$0.729619\pi$
$62$	3.00000			0.381000
$63$		0			0
$64$	7.00000			0.875000
$65$	2.00000	+	3.46410i	0.248069	+	0.429669i
$66$		0			0
$67$	−3.50000	+	6.06218i	−0.427593	+	0.740613i	−0.996659	−	0.0816792i	$-0.973972\pi$
$67$	−3.50000	+	6.06218i	−0.427593	+	0.740613i	0.569066	+	0.822292i	$0.307305\pi$
$68$	−3.00000	−	5.19615i	−0.363803	−	0.630126i
$69$		0			0
$70$	10.0000	−	3.46410i	1.19523	−	0.414039i
$71$		0			0			−	1.00000i	$-0.5\pi$
$71$		0			0				1.00000i	$0.5\pi$
$72$		0			0
$73$	3.00000	−	5.19615i	0.351123	−	0.608164i	−0.635323	−	0.772246i	$-0.719133\pi$
$73$	3.00000	−	5.19615i	0.351123	−	0.608164i	0.986447	+	0.164083i	$0.0524664\pi$
$74$	−1.50000	+	2.59808i	−0.174371	+	0.302020i
$75$		0			0
$76$	−4.00000			−0.458831
$77$	−5.00000	+	1.73205i	−0.569803	+	0.197386i
$78$		0			0
$79$	−5.50000	−	9.52628i	−0.618798	−	1.07179i	−0.989705	−	0.143120i	$-0.954286\pi$
$79$	−5.50000	−	9.52628i	−0.618798	−	1.07179i	0.370907	−	0.928670i	$-0.379047\pi$
$80$	−2.00000	+	3.46410i	−0.223607	+	0.387298i
$81$		0			0
$82$	1.00000	+	1.73205i	0.110432	+	0.191273i
$83$	6.00000			0.658586			0.329293	−	0.944228i	$-0.393190\pi$
$83$	6.00000			0.658586			0.329293	+	0.944228i	$0.393190\pi$
$84$		0			0
$85$	24.0000			2.60317
$86$	0.500000	+	0.866025i	0.0539164	+	0.0933859i
$87$		0			0
$88$	3.00000	−	5.19615i	0.319801	−	0.553912i
$89$	2.00000	+	3.46410i	0.212000	+	0.367194i	0.952340	−	0.305038i	$-0.0986691\pi$
$89$	2.00000	+	3.46410i	0.212000	+	0.367194i	−0.740341	+	0.672232i	$0.765336\pi$
$90$		0			0
$91$	2.00000	+	1.73205i	0.209657	+	0.181568i
$92$	−6.00000			−0.625543
$93$		0			0
$94$	−3.00000	+	5.19615i	−0.309426	+	0.535942i
$95$	8.00000	−	13.8564i	0.820783	−	1.42164i
$96$		0			0
$97$	9.00000			0.913812			0.456906	−	0.889515i	$-0.348958\pi$
$97$	9.00000			0.913812			0.456906	+	0.889515i	$0.348958\pi$
$98$	5.50000	−	4.33013i	0.555584	−	0.437409i
$99$		0			0
$100$	5.50000	+	9.52628i	0.550000	+	0.952628i
$101$	4.00000	−	6.92820i	0.398015	−	0.689382i	−0.595466	−	0.803380i	$-0.703033\pi$
$101$	4.00000	−	6.92820i	0.398015	−	0.689382i	0.993481	+	0.113998i	$0.0363659\pi$
$102$		0			0
$103$	−8.50000	−	14.7224i	−0.837530	−	1.45064i	−0.891954	−	0.452126i	$-0.850666\pi$
$103$	−8.50000	−	14.7224i	−0.837530	−	1.45064i	0.0544240	−	0.998518i	$-0.482668\pi$
$104$	−3.00000			−0.294174
$105$		0			0
$106$	−6.00000			−0.582772
$107$	−1.00000	−	1.73205i	−0.0966736	−	0.167444i	0.813632	−	0.581380i	$-0.197487\pi$
$107$	−1.00000	−	1.73205i	−0.0966736	−	0.167444i	−0.910306	+	0.413936i	$0.864154\pi$
$108$		0			0
$109$	−4.50000	+	7.79423i	−0.431022	+	0.746552i	−0.996962	−	0.0778949i	$-0.975180\pi$
$109$	−4.50000	+	7.79423i	−0.431022	+	0.746552i	0.565940	+	0.824447i	$0.308513\pi$
$110$	4.00000	+	6.92820i	0.381385	+	0.660578i
$111$		0			0
$112$	−0.500000	+	2.59808i	−0.0472456	+	0.245495i
$113$	−16.0000			−1.50515			−0.752577	−	0.658505i	$-0.771189\pi$
$113$	−16.0000			−1.50515			−0.752577	+	0.658505i	$0.771189\pi$
$114$		0			0
$115$	12.0000	−	20.7846i	1.11901	−	1.93817i
$116$	1.00000	−	1.73205i	0.0928477	−	0.160817i
$117$		0			0
$118$	6.00000			0.552345
$119$	15.0000	−	5.19615i	1.37505	−	0.476331i
$120$		0			0
$121$	3.50000	+	6.06218i	0.318182	+	0.551107i
$122$	2.50000	−	4.33013i	0.226339	−	0.392031i
$123$		0			0
$124$	1.50000	+	2.59808i	0.134704	+	0.233314i
$125$	−24.0000			−2.14663
$126$		0			0
$127$	−9.00000			−0.798621			−0.399310	−	0.916816i	$-0.630750\pi$
$127$	−9.00000			−0.798621			−0.399310	+	0.916816i	$0.630750\pi$
$128$	1.50000	+	2.59808i	0.132583	+	0.229640i
$129$		0			0
$130$	2.00000	−	3.46410i	0.175412	−	0.303822i
$131$	2.00000	+	3.46410i	0.174741	+	0.302660i	0.940072	−	0.340977i	$-0.110758\pi$
$131$	2.00000	+	3.46410i	0.174741	+	0.302660i	−0.765331	+	0.643637i	$0.777425\pi$
$132$		0			0
$133$	2.00000	−	10.3923i	0.173422	−	0.901127i
$134$	7.00000			0.604708
$135$		0			0
$136$	−9.00000	+	15.5885i	−0.771744	+	1.33670i
$137$	−9.00000	+	15.5885i	−0.768922	+	1.33181i	0.169226	+	0.985577i	$0.445873\pi$
$137$	−9.00000	+	15.5885i	−0.768922	+	1.33181i	−0.938148	+	0.346235i	$0.887460\pi$
$138$		0			0
$139$	9.00000			0.763370			0.381685	−	0.924292i	$-0.375344\pi$
$139$	9.00000			0.763370			0.381685	+	0.924292i	$0.375344\pi$
$140$	8.00000	+	6.92820i	0.676123	+	0.585540i
$141$		0			0
$142$		0			0
$143$	−1.00000	+	1.73205i	−0.0836242	+	0.144841i
$144$		0			0
$145$	4.00000	+	6.92820i	0.332182	+	0.575356i
$146$	−6.00000			−0.496564
$147$		0			0
$148$	−3.00000			−0.246598
$149$	12.0000	+	20.7846i	0.983078	+	1.70274i	0.650183	+	0.759778i	$0.274692\pi$
$149$	12.0000	+	20.7846i	0.983078	+	1.70274i	0.332896	+	0.942964i	$0.391974\pi$
$150$		0			0
$151$	−2.50000	+	4.33013i	−0.203447	+	0.352381i	−0.949637	−	0.313353i	$-0.898548\pi$
$151$	−2.50000	+	4.33013i	−0.203447	+	0.352381i	0.746190	+	0.665733i	$0.231881\pi$
$152$	6.00000	+	10.3923i	0.486664	+	0.842927i
$153$		0			0
$154$	4.00000	+	3.46410i	0.322329	+	0.279145i
$155$	−12.0000			−0.963863
$156$		0			0
$157$	−5.00000	+	8.66025i	−0.399043	+	0.691164i	−0.993608	−	0.112884i	$-0.963991\pi$
$157$	−5.00000	+	8.66025i	−0.399043	+	0.691164i	0.594565	+	0.804048i	$0.297324\pi$
$158$	−5.50000	+	9.52628i	−0.437557	+	0.757870i
$159$		0			0
$160$	−20.0000			−1.58114
$161$	3.00000	−	15.5885i	0.236433	−	1.22854i
$162$		0			0
$163$	−9.50000	−	16.4545i	−0.744097	−	1.28881i	−0.950615	−	0.310372i	$-0.899546\pi$
$163$	−9.50000	−	16.4545i	−0.744097	−	1.28881i	0.206518	−	0.978443i	$-0.433787\pi$
$164$	−1.00000	+	1.73205i	−0.0780869	+	0.135250i
$165$		0			0
$166$	−3.00000	−	5.19615i	−0.232845	−	0.403300i
$167$	14.0000			1.08335			0.541676	−	0.840587i	$-0.317790\pi$
$167$	14.0000			1.08335			0.541676	+	0.840587i	$0.317790\pi$
$168$		0			0
$169$	−12.0000			−0.923077
$170$	−12.0000	−	20.7846i	−0.920358	−	1.59411i
$171$		0			0
$172$	−0.500000	+	0.866025i	−0.0381246	+	0.0660338i
$173$	−8.00000	−	13.8564i	−0.608229	−	1.05348i	−0.991532	−	0.129861i	$-0.958547\pi$
$173$	−8.00000	−	13.8564i	−0.608229	−	1.05348i	0.383304	−	0.923622i	$-0.374786\pi$
$174$		0			0
$175$	−27.5000	+	9.52628i	−2.07880	+	0.720119i
$176$	−2.00000			−0.150756
$177$		0			0
$178$	2.00000	−	3.46410i	0.149906	−	0.259645i
$179$	−2.00000	+	3.46410i	−0.149487	+	0.258919i	−0.931038	−	0.364922i	$-0.881096\pi$
$179$	−2.00000	+	3.46410i	−0.149487	+	0.258919i	0.781551	+	0.623841i	$0.214429\pi$
$180$		0			0
$181$	−2.00000			−0.148659			−0.0743294	−	0.997234i	$-0.523682\pi$
$181$	−2.00000			−0.148659			−0.0743294	+	0.997234i	$0.523682\pi$
$182$	0.500000	−	2.59808i	0.0370625	−	0.192582i
$183$		0			0
$184$	9.00000	+	15.5885i	0.663489	+	1.14920i
$185$	6.00000	−	10.3923i	0.441129	−	0.764057i
$186$		0			0
$187$	6.00000	+	10.3923i	0.438763	+	0.759961i
$188$	−6.00000			−0.437595
$189$		0			0
$190$	−16.0000			−1.16076
$191$	2.00000	+	3.46410i	0.144715	+	0.250654i	0.929267	−	0.369410i	$-0.120440\pi$
$191$	2.00000	+	3.46410i	0.144715	+	0.250654i	−0.784552	+	0.620063i	$0.787107\pi$
$192$		0			0
$193$	12.5000	−	21.6506i	0.899770	−	1.55845i	0.0719816	−	0.997406i	$-0.477068\pi$
$193$	12.5000	−	21.6506i	0.899770	−	1.55845i	0.827788	−	0.561041i	$-0.189599\pi$
$194$	−4.50000	−	7.79423i	−0.323081	−	0.559593i
$195$		0			0
$196$	6.50000	+	2.59808i	0.464286	+	0.185577i
$197$	20.0000			1.42494			0.712470	−	0.701702i	$-0.247576\pi$
$197$	20.0000			1.42494			0.712470	+	0.701702i	$0.247576\pi$
$198$		0			0
$199$	4.50000	−	7.79423i	0.318997	−	0.552518i	−0.661282	−	0.750137i	$-0.729987\pi$
$199$	4.50000	−	7.79423i	0.318997	−	0.552518i	0.980279	+	0.197619i	$0.0633208\pi$
$200$	16.5000	−	28.5788i	1.16673	−	2.02083i
$201$		0			0
$202$	−8.00000			−0.562878
$203$	4.00000	+	3.46410i	0.280745	+	0.243132i
$204$		0			0
$205$	−4.00000	−	6.92820i	−0.279372	−	0.483887i
$206$	−8.50000	+	14.7224i	−0.592223	+	1.02576i
$207$		0			0
$208$	0.500000	+	0.866025i	0.0346688	+	0.0600481i
$209$	8.00000			0.553372
$210$		0			0
$211$	−5.00000			−0.344214			−0.172107	−	0.985078i	$-0.555058\pi$
$211$	−5.00000			−0.344214			−0.172107	+	0.985078i	$0.555058\pi$
$212$	−3.00000	−	5.19615i	−0.206041	−	0.356873i
$213$		0			0
$214$	−1.00000	+	1.73205i	−0.0683586	+	0.118401i
$215$	−2.00000	−	3.46410i	−0.136399	−	0.236250i
$216$		0			0
$217$	−7.50000	+	2.59808i	−0.509133	+	0.176369i
$218$	9.00000			0.609557
$219$		0			0
$220$	−4.00000	+	6.92820i	−0.269680	+	0.467099i
$221$	3.00000	−	5.19615i	0.201802	−	0.349531i
$222$		0			0
$223$	16.0000			1.07144			0.535720	−	0.844396i	$-0.320040\pi$
$223$	16.0000			1.07144			0.535720	+	0.844396i	$0.320040\pi$
$224$	−12.5000	+	4.33013i	−0.835191	+	0.289319i
$225$		0			0
$226$	8.00000	+	13.8564i	0.532152	+	0.921714i
$227$	−9.00000	+	15.5885i	−0.597351	+	1.03464i	0.395860	+	0.918311i	$0.370447\pi$
$227$	−9.00000	+	15.5885i	−0.597351	+	1.03464i	−0.993210	+	0.116331i	$0.962887\pi$
$228$		0			0
$229$	3.50000	+	6.06218i	0.231287	+	0.400600i	0.958187	−	0.286143i	$-0.0923732\pi$
$229$	3.50000	+	6.06218i	0.231287	+	0.400600i	−0.726900	+	0.686743i	$0.759040\pi$
$230$	−24.0000			−1.58251
$231$		0			0
$232$	−6.00000			−0.393919
$233$	12.0000	+	20.7846i	0.786146	+	1.36165i	0.928312	+	0.371802i	$0.121260\pi$
$233$	12.0000	+	20.7846i	0.786146	+	1.36165i	−0.142166	+	0.989843i	$0.545407\pi$
$234$		0			0
$235$	12.0000	−	20.7846i	0.782794	−	1.35584i
$236$	3.00000	+	5.19615i	0.195283	+	0.338241i
$237$		0			0
$238$	−12.0000	−	10.3923i	−0.777844	−	0.673633i
$239$	−12.0000			−0.776215			−0.388108	−	0.921614i	$-0.626871\pi$
$239$	−12.0000			−0.776215			−0.388108	+	0.921614i	$0.626871\pi$
$240$		0			0
$241$	−8.50000	+	14.7224i	−0.547533	+	0.948355i	0.450910	+	0.892570i	$0.351100\pi$
$241$	−8.50000	+	14.7224i	−0.547533	+	0.948355i	−0.998443	+	0.0557856i	$0.982234\pi$
$242$	3.50000	−	6.06218i	0.224989	−	0.389692i
$243$		0			0
$244$	5.00000			0.320092
$245$	−22.0000	+	17.3205i	−1.40553	+	1.10657i
$246$		0			0
$247$	−2.00000	−	3.46410i	−0.127257	−	0.220416i
$248$	4.50000	−	7.79423i	0.285750	−	0.494934i
$249$		0			0
$250$	12.0000	+	20.7846i	0.758947	+	1.31453i
$251$	26.0000			1.64111			0.820553	−	0.571571i	$-0.193666\pi$
$251$	26.0000			1.64111			0.820553	+	0.571571i	$0.193666\pi$
$252$		0			0
$253$	12.0000			0.754434
$254$	4.50000	+	7.79423i	0.282355	+	0.489053i
$255$		0			0
$256$	8.50000	−	14.7224i	0.531250	−	0.920152i
$257$	−5.00000	−	8.66025i	−0.311891	−	0.540212i	0.666880	−	0.745165i	$-0.267629\pi$
$257$	−5.00000	−	8.66025i	−0.311891	−	0.540212i	−0.978772	+	0.204953i	$0.934296\pi$
$258$		0			0
$259$	1.50000	−	7.79423i	0.0932055	−	0.484310i
$260$	4.00000			0.248069
$261$		0			0
$262$	2.00000	−	3.46410i	0.123560	−	0.214013i
$263$	−1.00000	+	1.73205i	−0.0616626	+	0.106803i	−0.895209	−	0.445647i	$-0.852974\pi$
$263$	−1.00000	+	1.73205i	−0.0616626	+	0.106803i	0.833546	+	0.552450i	$0.186307\pi$
$264$		0			0
$265$	24.0000			1.47431
$266$	−10.0000	+	3.46410i	−0.613139	+	0.212398i
$267$		0			0
$268$	3.50000	+	6.06218i	0.213797	+	0.370306i
$269$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$269$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$270$		0			0
$271$	5.50000	+	9.52628i	0.334101	+	0.578680i	0.983312	−	0.181928i	$-0.0582339\pi$
$271$	5.50000	+	9.52628i	0.334101	+	0.578680i	−0.649211	+	0.760609i	$0.724901\pi$
$272$	6.00000			0.363803
$273$		0			0
$274$	18.0000			1.08742
$275$	−11.0000	−	19.0526i	−0.663325	−	1.14891i
$276$		0			0
$277$	−6.50000	+	11.2583i	−0.390547	+	0.676448i	−0.992522	−	0.122068i	$-0.961047\pi$
$277$	−6.50000	+	11.2583i	−0.390547	+	0.676448i	0.601975	+	0.798515i	$0.294381\pi$
$278$	−4.50000	−	7.79423i	−0.269892	−	0.467467i
$279$		0			0
$280$	6.00000	−	31.1769i	0.358569	−	1.86318i
$281$	16.0000			0.954480			0.477240	−	0.878773i	$-0.341637\pi$
$281$	16.0000			0.954480			0.477240	+	0.878773i	$0.341637\pi$
$282$		0			0
$283$	2.50000	−	4.33013i	0.148610	−	0.257399i	−0.782104	−	0.623148i	$-0.785854\pi$
$283$	2.50000	−	4.33013i	0.148610	−	0.257399i	0.930714	+	0.365748i	$0.119187\pi$
$284$		0			0
$285$		0			0
$286$	2.00000			0.118262
$287$	−4.00000	−	3.46410i	−0.236113	−	0.204479i
$288$		0			0
$289$	−9.50000	−	16.4545i	−0.558824	−	0.967911i
$290$	4.00000	−	6.92820i	0.234888	−	0.406838i
$291$		0			0
$292$	−3.00000	−	5.19615i	−0.175562	−	0.304082i
$293$	22.0000			1.28525			0.642627	−	0.766179i	$-0.277845\pi$
$293$	22.0000			1.28525			0.642627	+	0.766179i	$0.277845\pi$
$294$		0			0
$295$	−24.0000			−1.39733
$296$	4.50000	+	7.79423i	0.261557	+	0.453030i
$297$		0			0
$298$	12.0000	−	20.7846i	0.695141	−	1.20402i
$299$	−3.00000	−	5.19615i	−0.173494	−	0.300501i
$300$		0			0
$301$	−2.00000	−	1.73205i	−0.115278	−	0.0998337i
$302$	5.00000			0.287718
$303$		0			0
$304$	2.00000	−	3.46410i	0.114708	−	0.198680i
$305$	−10.0000	+	17.3205i	−0.572598	+	0.991769i
$306$		0			0
$307$	25.0000			1.42683			0.713413	−	0.700744i	$-0.247149\pi$
$307$	25.0000			1.42683			0.713413	+	0.700744i	$0.247149\pi$
$308$	−1.00000	+	5.19615i	−0.0569803	+	0.296078i
$309$		0			0
$310$	6.00000	+	10.3923i	0.340777	+	0.590243i
$311$	−12.0000	+	20.7846i	−0.680458	+	1.17859i	0.294384	+	0.955687i	$0.404886\pi$
$311$	−12.0000	+	20.7846i	−0.680458	+	1.17859i	−0.974841	+	0.222900i	$0.928448\pi$
$312$		0			0
$313$	11.0000	+	19.0526i	0.621757	+	1.07691i	0.989158	+	0.146852i	$0.0469141\pi$
$313$	11.0000	+	19.0526i	0.621757	+	1.07691i	−0.367402	+	0.930062i	$0.619753\pi$
$314$	10.0000			0.564333
$315$		0			0
$316$	−11.0000			−0.618798
$317$	−15.0000	−	25.9808i	−0.842484	−	1.45922i	−0.887788	−	0.460252i	$-0.847759\pi$
$317$	−15.0000	−	25.9808i	−0.842484	−	1.45922i	0.0453045	−	0.998973i	$-0.485574\pi$
$318$		0			0
$319$	−2.00000	+	3.46410i	−0.111979	+	0.193952i
$320$	14.0000	+	24.2487i	0.782624	+	1.35554i
$321$		0			0
$322$	−15.0000	+	5.19615i	−0.835917	+	0.289570i
$323$	−24.0000			−1.33540
$324$		0			0
$325$	−5.50000	+	9.52628i	−0.305085	+	0.528423i
$326$	−9.50000	+	16.4545i	−0.526156	+	0.911330i
$327$		0			0
$328$	6.00000			0.331295
$329$	3.00000	−	15.5885i	0.165395	−	0.859419i
$330$		0			0
$331$	14.0000	+	24.2487i	0.769510	+	1.33283i	0.937829	+	0.347097i	$0.112833\pi$
$331$	14.0000	+	24.2487i	0.769510	+	1.33283i	−0.168320	+	0.985732i	$0.553834\pi$
$332$	3.00000	−	5.19615i	0.164646	−	0.285176i
$333$		0			0
$334$	−7.00000	−	12.1244i	−0.383023	−	0.663415i
$335$	−28.0000			−1.52980
$336$		0			0
$337$	10.0000			0.544735			0.272367	−	0.962193i	$-0.412193\pi$
$337$	10.0000			0.544735			0.272367	+	0.962193i	$0.412193\pi$
$338$	6.00000	+	10.3923i	0.326357	+	0.565267i
$339$		0			0
$340$	12.0000	−	20.7846i	0.650791	−	1.12720i
$341$	−3.00000	−	5.19615i	−0.162459	−	0.281387i
$342$		0			0
$343$	−10.0000	+	15.5885i	−0.539949	+	0.841698i
$344$	3.00000			0.161749
$345$		0			0
$346$	−8.00000	+	13.8564i	−0.430083	+	0.744925i
$347$	−8.00000	+	13.8564i	−0.429463	+	0.743851i	−0.996826	−	0.0796169i	$-0.974630\pi$
$347$	−8.00000	+	13.8564i	−0.429463	+	0.743851i	0.567363	+	0.823468i	$0.307964\pi$
$348$		0			0
$349$	−5.00000			−0.267644			−0.133822	−	0.991005i	$-0.542725\pi$
$349$	−5.00000			−0.267644			−0.133822	+	0.991005i	$0.542725\pi$
$350$	22.0000	+	19.0526i	1.17595	+	1.01840i
$351$		0			0
$352$	−5.00000	−	8.66025i	−0.266501	−	0.461593i
$353$	14.0000	−	24.2487i	0.745145	−	1.29063i	−0.204982	−	0.978766i	$-0.565714\pi$
$353$	14.0000	−	24.2487i	0.745145	−	1.29063i	0.950127	−	0.311863i	$-0.100953\pi$
$354$		0			0
$355$		0			0
$356$	4.00000			0.212000
$357$		0			0
$358$	4.00000			0.211407
$359$	−11.0000	−	19.0526i	−0.580558	−	1.00556i	−0.995413	−	0.0956683i	$-0.969501\pi$
$359$	−11.0000	−	19.0526i	−0.580558	−	1.00556i	0.414855	−	0.909887i	$-0.363832\pi$
$360$		0			0
$361$	1.50000	−	2.59808i	0.0789474	−	0.136741i
$362$	1.00000	+	1.73205i	0.0525588	+	0.0910346i
$363$		0			0
$364$	2.50000	−	0.866025i	0.131036	−	0.0453921i
$365$	24.0000			1.25622
$366$		0			0
$367$	−6.00000	+	10.3923i	−0.313197	+	0.542474i	−0.979053	−	0.203607i	$-0.934733\pi$
$367$	−6.00000	+	10.3923i	−0.313197	+	0.542474i	0.665855	+	0.746081i	$0.268067\pi$
$368$	3.00000	−	5.19615i	0.156386	−	0.270868i
$369$		0			0
$370$	−12.0000			−0.623850
$371$	15.0000	−	5.19615i	0.778761	−	0.269771i
$372$		0			0
$373$	−13.0000	−	22.5167i	−0.673114	−	1.16587i	−0.977016	−	0.213165i	$-0.931623\pi$
$373$	−13.0000	−	22.5167i	−0.673114	−	1.16587i	0.303902	−	0.952703i	$-0.401711\pi$
$374$	6.00000	−	10.3923i	0.310253	−	0.537373i
$375$		0			0
$376$	9.00000	+	15.5885i	0.464140	+	0.803913i
$377$	2.00000			0.103005
$378$		0			0
$379$	−9.00000			−0.462299			−0.231149	−	0.972918i	$-0.574249\pi$
$379$	−9.00000			−0.462299			−0.231149	+	0.972918i	$0.574249\pi$
$380$	−8.00000	−	13.8564i	−0.410391	−	0.710819i
$381$		0			0
$382$	2.00000	−	3.46410i	0.102329	−	0.177239i
$383$	9.00000	+	15.5885i	0.459879	+	0.796533i	0.998954	−	0.0457244i	$-0.0145596\pi$
$383$	9.00000	+	15.5885i	0.459879	+	0.796533i	−0.539076	+	0.842257i	$0.681226\pi$
$384$		0			0
$385$	−16.0000	−	13.8564i	−0.815436	−	0.706188i
$386$	−25.0000			−1.27247
$387$		0			0
$388$	4.50000	−	7.79423i	0.228453	−	0.395692i
$389$	6.00000	−	10.3923i	0.304212	−	0.526911i	−0.672874	−	0.739758i	$-0.734940\pi$
$389$	6.00000	−	10.3923i	0.304212	−	0.526911i	0.977086	+	0.212847i	$0.0682735\pi$
$390$		0			0
$391$	−36.0000			−1.82060
$392$	−3.00000	−	20.7846i	−0.151523	−	1.04978i
$393$		0			0
$394$	−10.0000	−	17.3205i	−0.503793	−	0.872595i
$395$	22.0000	−	38.1051i	1.10694	−	1.91728i
$396$		0			0
$397$	16.5000	+	28.5788i	0.828111	+	1.43433i	0.899518	+	0.436884i	$0.143918\pi$
$397$	16.5000	+	28.5788i	0.828111	+	1.43433i	−0.0714068	+	0.997447i	$0.522749\pi$
$398$	−9.00000			−0.451129
$399$		0			0
$400$	−11.0000			−0.550000
$401$	−15.0000	−	25.9808i	−0.749064	−	1.29742i	−0.948272	−	0.317460i	$-0.897170\pi$
$401$	−15.0000	−	25.9808i	−0.749064	−	1.29742i	0.199207	−	0.979957i	$-0.436163\pi$
$402$		0			0
$403$	−1.50000	+	2.59808i	−0.0747203	+	0.129419i
$404$	−4.00000	−	6.92820i	−0.199007	−	0.344691i
$405$		0			0
$406$	1.00000	−	5.19615i	0.0496292	−	0.257881i
$407$	6.00000			0.297409
$408$		0			0
$409$	−2.50000	+	4.33013i	−0.123617	+	0.214111i	−0.921192	−	0.389109i	$-0.872783\pi$
$409$	−2.50000	+	4.33013i	−0.123617	+	0.214111i	0.797574	+	0.603220i	$0.206116\pi$
$410$	−4.00000	+	6.92820i	−0.197546	+	0.342160i
$411$		0			0
$412$	−17.0000			−0.837530
$413$	−15.0000	+	5.19615i	−0.738102	+	0.255686i
$414$		0			0
$415$	12.0000	+	20.7846i	0.589057	+	1.02028i
$416$	−2.50000	+	4.33013i	−0.122573	+	0.212302i
$417$		0			0
$418$	−4.00000	−	6.92820i	−0.195646	−	0.338869i
$419$	−12.0000			−0.586238			−0.293119	−	0.956076i	$-0.594693\pi$
$419$	−12.0000			−0.586238			−0.293119	+	0.956076i	$0.594693\pi$
$420$		0			0
$421$	−10.0000			−0.487370			−0.243685	−	0.969854i	$-0.578356\pi$
$421$	−10.0000			−0.487370			−0.243685	+	0.969854i	$0.578356\pi$
$422$	2.50000	+	4.33013i	0.121698	+	0.210787i
$423$		0			0
$424$	−9.00000	+	15.5885i	−0.437079	+	0.757042i
$425$	33.0000	+	57.1577i	1.60074	+	2.77255i
$426$		0			0
$427$	−2.50000	+	12.9904i	−0.120983	+	0.628649i
$428$	−2.00000			−0.0966736
$429$		0			0
$430$	−2.00000	+	3.46410i	−0.0964486	+	0.167054i
$431$	−9.00000	+	15.5885i	−0.433515	+	0.750870i	−0.997173	−	0.0751385i	$-0.976060\pi$
$431$	−9.00000	+	15.5885i	−0.433515	+	0.750870i	0.563658	+	0.826008i	$0.309393\pi$
$432$		0			0
$433$	−17.0000			−0.816968			−0.408484	−	0.912766i	$-0.633942\pi$
$433$	−17.0000			−0.816968			−0.408484	+	0.912766i	$0.633942\pi$
$434$	6.00000	+	5.19615i	0.288009	+	0.249423i
$435$		0			0
$436$	4.50000	+	7.79423i	0.215511	+	0.373276i
$437$	−12.0000	+	20.7846i	−0.574038	+	0.994263i
$438$		0			0
$439$	−12.0000	−	20.7846i	−0.572729	−	0.991995i	−0.996284	−	0.0861252i	$-0.972552\pi$
$439$	−12.0000	−	20.7846i	−0.572729	−	0.991995i	0.423556	−	0.905870i	$-0.360782\pi$
$440$	24.0000			1.14416
$441$		0			0
$442$	−6.00000			−0.285391
$443$	12.0000	+	20.7846i	0.570137	+	0.987507i	0.996551	+	0.0829786i	$0.0264433\pi$
$443$	12.0000	+	20.7846i	0.570137	+	0.987507i	−0.426414	+	0.904528i	$0.640223\pi$
$444$		0			0
$445$	−8.00000	+	13.8564i	−0.379236	+	0.656857i
$446$	−8.00000	−	13.8564i	−0.378811	−	0.656120i
$447$		0			0
$448$	14.0000	+	12.1244i	0.661438	+	0.572822i
$449$	−36.0000			−1.69895			−0.849473	−	0.527633i	$-0.823080\pi$
$449$	−36.0000			−1.69895			−0.849473	+	0.527633i	$0.823080\pi$
$450$		0			0
$451$	2.00000	−	3.46410i	0.0941763	−	0.163118i
$452$	−8.00000	+	13.8564i	−0.376288	+	0.651751i
$453$		0			0
$454$	18.0000			0.844782
$455$	−2.00000	+	10.3923i	−0.0937614	+	0.487199i
$456$		0			0
$457$	−6.50000	−	11.2583i	−0.304057	−	0.526642i	0.672994	−	0.739648i	$-0.265008\pi$
$457$	−6.50000	−	11.2583i	−0.304057	−	0.526642i	−0.977051	+	0.213006i	$0.931675\pi$
$458$	3.50000	−	6.06218i	0.163544	−	0.283267i
$459$		0			0
$460$	−12.0000	−	20.7846i	−0.559503	−	0.969087i
$461$	−2.00000			−0.0931493			−0.0465746	−	0.998915i	$-0.514831\pi$
$461$	−2.00000			−0.0931493			−0.0465746	+	0.998915i	$0.514831\pi$
$462$		0			0
$463$	−32.0000			−1.48717			−0.743583	−	0.668644i	$-0.766875\pi$
$463$	−32.0000			−1.48717			−0.743583	+	0.668644i	$0.766875\pi$
$464$	1.00000	+	1.73205i	0.0464238	+	0.0804084i
$465$		0			0
$466$	12.0000	−	20.7846i	0.555889	−	0.962828i
$467$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$467$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$468$		0			0
$469$	−17.5000	+	6.06218i	−0.808075	+	0.279925i
$470$	−24.0000			−1.10704
$471$		0			0
$472$	9.00000	−	15.5885i	0.414259	−	0.717517i
$473$	1.00000	−	1.73205i	0.0459800	−	0.0796398i
$474$		0			0
$475$	44.0000			2.01886
$476$	3.00000	−	15.5885i	0.137505	−	0.714496i
$477$		0			0
$478$	6.00000	+	10.3923i	0.274434	+	0.475333i
$479$	−8.00000	+	13.8564i	−0.365529	+	0.633115i	−0.988861	−	0.148842i	$-0.952445\pi$
$479$	−8.00000	+	13.8564i	−0.365529	+	0.633115i	0.623332	+	0.781958i	$0.285779\pi$
$480$		0			0
$481$	−1.50000	−	2.59808i	−0.0683941	−	0.118462i
$482$	17.0000			0.774329
$483$		0			0
$484$	7.00000			0.318182
$485$	18.0000	+	31.1769i	0.817338	+	1.41567i
$486$		0			0
$487$	−8.00000	+	13.8564i	−0.362515	+	0.627894i	−0.988374	−	0.152042i	$-0.951415\pi$
$487$	−8.00000	+	13.8564i	−0.362515	+	0.627894i	0.625859	+	0.779936i	$0.284748\pi$
$488$	−7.50000	−	12.9904i	−0.339509	−	0.588047i
$489$		0			0
$490$	26.0000	+	10.3923i	1.17456	+	0.469476i
$491$	34.0000			1.53440			0.767199	−	0.641409i	$-0.221650\pi$
$491$	34.0000			1.53440			0.767199	+	0.641409i	$0.221650\pi$
$492$		0			0
$493$	6.00000	−	10.3923i	0.270226	−	0.468046i
$494$	−2.00000	+	3.46410i	−0.0899843	+	0.155857i
$495$		0			0
$496$	−3.00000			−0.134704
$497$		0			0
$498$		0			0
$499$	8.50000	+	14.7224i	0.380512	+	0.659067i	0.991136	−	0.132855i	$-0.0424144\pi$
$499$	8.50000	+	14.7224i	0.380512	+	0.659067i	−0.610623	+	0.791921i	$0.709081\pi$
$500$	−12.0000	+	20.7846i	−0.536656	+	0.929516i
$501$		0			0
$502$	−13.0000	−	22.5167i	−0.580218	−	1.00497i
$503$	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
$505$	32.0000			1.42398
$506$	−6.00000	−	10.3923i	−0.266733	−	0.461994i
$507$		0			0
$508$	−4.50000	+	7.79423i	−0.199655	+	0.345813i
$509$	−11.0000	−	19.0526i	−0.487566	−	0.844490i	0.512331	−	0.858788i	$-0.328782\pi$
$509$	−11.0000	−	19.0526i	−0.487566	−	0.844490i	−0.999898	+	0.0142980i	$0.995449\pi$
$510$		0			0
$511$	15.0000	−	5.19615i	0.663561	−	0.229864i
$512$	−11.0000			−0.486136
$513$		0			0
$514$	−5.00000	+	8.66025i	−0.220541	+	0.381987i
$515$	34.0000	−	58.8897i	1.49822	−	2.59499i
$516$		0			0
$517$	12.0000			0.527759
$518$	−7.50000	+	2.59808i	−0.329531	+	0.114153i
$519$		0			0
$520$	−6.00000	−	10.3923i	−0.263117	−	0.455733i
$521$	−9.00000	+	15.5885i	−0.394297	+	0.682943i	−0.993011	−	0.118020i	$-0.962345\pi$
$521$	−9.00000	+	15.5885i	−0.394297	+	0.682943i	0.598714	+	0.800963i	$0.295679\pi$
$522$		0			0
$523$	8.50000	+	14.7224i	0.371679	+	0.643767i	0.989824	−	0.142297i	$-0.0454489\pi$
$523$	8.50000	+	14.7224i	0.371679	+	0.643767i	−0.618145	+	0.786064i	$0.712116\pi$
$524$	4.00000			0.174741
$525$		0			0
$526$	2.00000			0.0872041
$527$	9.00000	+	15.5885i	0.392046	+	0.679044i
$528$		0			0
$529$	−6.50000	+	11.2583i	−0.282609	+	0.489493i
$530$	−12.0000	−	20.7846i	−0.521247	−	0.902826i
$531$		0			0
$532$	−8.00000	−	6.92820i	−0.346844	−	0.300376i
$533$	−2.00000			−0.0866296
$534$		0			0
$535$	4.00000	−	6.92820i	0.172935	−	0.299532i
$536$	10.5000	−	18.1865i	0.453531	−	0.785539i
$537$		0			0
$538$		0			0
$539$	−13.0000	−	5.19615i	−0.559950	−	0.223814i
$540$		0			0
$541$	5.00000	+	8.66025i	0.214967	+	0.372333i	0.953262	−	0.302144i	$-0.0977023\pi$
$541$	5.00000	+	8.66025i	0.214967	+	0.372333i	−0.738296	+	0.674477i	$0.764369\pi$
$542$	5.50000	−	9.52628i	0.236245	−	0.409189i
$543$		0			0
$544$	15.0000	+	25.9808i	0.643120	+	1.11392i
$545$	−36.0000			−1.54207
$546$		0			0
$547$	43.0000			1.83855			0.919274	−	0.393619i	$-0.128777\pi$
$547$	43.0000			1.83855			0.919274	+	0.393619i	$0.128777\pi$
$548$	9.00000	+	15.5885i	0.384461	+	0.665906i
$549$		0			0
$550$	−11.0000	+	19.0526i	−0.469042	+	0.812404i
$551$	−4.00000	−	6.92820i	−0.170406	−	0.295151i
$552$		0			0
$553$	5.50000	−	28.5788i	0.233884	−	1.21530i
$554$	13.0000			0.552317
$555$		0			0
$556$	4.50000	−	7.79423i	0.190843	−	0.330549i
$557$	14.0000	−	24.2487i	0.593199	−	1.02745i	−0.400599	−	0.916253i	$-0.631198\pi$
$557$	14.0000	−	24.2487i	0.593199	−	1.02745i	0.993798	−	0.111198i	$-0.0354686\pi$
$558$		0			0
$559$	−1.00000			−0.0422955
$560$	−10.0000	+	3.46410i	−0.422577	+	0.146385i
$561$		0			0
$562$	−8.00000	−	13.8564i	−0.337460	−	0.584497i
$563$	5.00000	−	8.66025i	0.210725	−	0.364986i	−0.741217	−	0.671266i	$-0.765751\pi$
$563$	5.00000	−	8.66025i	0.210725	−	0.364986i	0.951942	+	0.306280i	$0.0990842\pi$
$564$		0			0
$565$	−32.0000	−	55.4256i	−1.34625	−	2.33177i
$566$	−5.00000			−0.210166
$567$		0			0
$568$		0			0
$569$	−16.0000	−	27.7128i	−0.670755	−	1.16178i	−0.977690	−	0.210051i	$-0.932637\pi$
$569$	−16.0000	−	27.7128i	−0.670755	−	1.16178i	0.306935	−	0.951730i	$-0.400696\pi$
$570$		0			0
$571$	8.00000	−	13.8564i	0.334790	−	0.579873i	−0.648655	−	0.761083i	$-0.724668\pi$
$571$	8.00000	−	13.8564i	0.334790	−	0.579873i	0.983444	+	0.181210i	$0.0580014\pi$
$572$	1.00000	+	1.73205i	0.0418121	+	0.0724207i
$573$		0			0
$574$	−1.00000	+	5.19615i	−0.0417392	+	0.216883i
$575$	66.0000			2.75239
$576$		0			0
$577$	8.50000	−	14.7224i	0.353860	−	0.612903i	−0.633062	−	0.774101i	$-0.718202\pi$
$577$	8.50000	−	14.7224i	0.353860	−	0.612903i	0.986922	+	0.161198i	$0.0515357\pi$
$578$	−9.50000	+	16.4545i	−0.395148	+	0.684416i
$579$		0			0
$580$	8.00000			0.332182
$581$	12.0000	+	10.3923i	0.497844	+	0.431145i
$582$		0			0
$583$	6.00000	+	10.3923i	0.248495	+	0.430405i
$584$	−9.00000	+	15.5885i	−0.372423	+	0.645055i
$585$		0			0
$586$	−11.0000	−	19.0526i	−0.454406	−	0.787054i
$587$	−28.0000			−1.15568			−0.577842	−	0.816149i	$-0.696105\pi$
$587$	−28.0000			−1.15568			−0.577842	+	0.816149i	$0.696105\pi$
$588$		0			0
$589$	12.0000			0.494451
$590$	12.0000	+	20.7846i	0.494032	+	0.855689i
$591$		0			0
$592$	1.50000	−	2.59808i	0.0616496	−	0.106780i
$593$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$593$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$594$		0			0
$595$	48.0000	+	41.5692i	1.96781	+	1.70417i
$596$	24.0000			0.983078
$597$		0			0
$598$	−3.00000	+	5.19615i	−0.122679	+	0.212486i
$599$	−6.00000	+	10.3923i	−0.245153	+	0.424618i	−0.962175	−	0.272433i	$-0.912172\pi$
$599$	−6.00000	+	10.3923i	−0.245153	+	0.424618i	0.717021	+	0.697051i	$0.245505\pi$
$600$		0			0
$601$	−33.0000			−1.34610			−0.673049	−	0.739598i	$-0.735016\pi$
$601$	−33.0000			−1.34610			−0.673049	+	0.739598i	$0.735016\pi$
$602$	−0.500000	+	2.59808i	−0.0203785	+	0.105890i
$603$		0			0
$604$	2.50000	+	4.33013i	0.101724	+	0.176190i
$605$	−14.0000	+	24.2487i	−0.569181	+	0.985850i
$606$		0			0
$607$	8.00000	+	13.8564i	0.324710	+	0.562414i	0.981454	−	0.191700i	$-0.0614000\pi$
$607$	8.00000	+	13.8564i	0.324710	+	0.562414i	−0.656744	+	0.754114i	$0.728067\pi$
$608$	20.0000			0.811107
$609$		0			0
$610$	20.0000			0.809776
$611$	−3.00000	−	5.19615i	−0.121367	−	0.210214i
$612$		0			0
$613$	−3.50000	+	6.06218i	−0.141364	+	0.244849i	−0.928010	−	0.372554i	$-0.878482\pi$
$613$	−3.50000	+	6.06218i	−0.141364	+	0.244849i	0.786647	+	0.617403i	$0.211815\pi$
$614$	−12.5000	−	21.6506i	−0.504459	−	0.873749i
$615$		0			0
$616$	15.0000	−	5.19615i	0.604367	−	0.209359i
$617$	−18.0000			−0.724653			−0.362326	−	0.932051i	$-0.618017\pi$
$617$	−18.0000			−0.724653			−0.362326	+	0.932051i	$0.618017\pi$
$618$		0			0
$619$	20.5000	−	35.5070i	0.823965	−	1.42715i	−0.0787435	−	0.996895i	$-0.525091\pi$
$619$	20.5000	−	35.5070i	0.823965	−	1.42715i	0.902708	−	0.430254i	$-0.141576\pi$
$620$	−6.00000	+	10.3923i	−0.240966	+	0.417365i
$621$		0			0
$622$	24.0000			0.962312
$623$	−2.00000	+	10.3923i	−0.0801283	+	0.416359i
$624$		0			0
$625$	−20.5000	−	35.5070i	−0.820000	−	1.42028i
$626$	11.0000	−	19.0526i	0.439648	−	0.761493i
$627$		0			0
$628$	5.00000	+	8.66025i	0.199522	+	0.345582i
$629$	−18.0000			−0.717707
$630$		0			0
$631$	5.00000			0.199047			0.0995234	−	0.995035i	$-0.468268\pi$
$631$	5.00000			0.199047			0.0995234	+	0.995035i	$0.468268\pi$
$632$	16.5000	+	28.5788i	0.656335	+	1.13681i
$633$		0			0
$634$	−15.0000	+	25.9808i	−0.595726	+	1.03183i
$635$	−18.0000	−	31.1769i	−0.714308	−	1.23722i
$636$		0			0
$637$	1.00000	+	6.92820i	0.0396214	+	0.274505i
$638$	4.00000			0.158362
$639$		0			0
$640$	−6.00000	+	10.3923i	−0.237171	+	0.410792i
$641$	−3.00000	+	5.19615i	−0.118493	+	0.205236i	−0.919171	−	0.393860i	$-0.871140\pi$
$641$	−3.00000	+	5.19615i	−0.118493	+	0.205236i	0.800678	+	0.599095i	$0.204473\pi$
$642$		0			0
$643$	−25.0000			−0.985904			−0.492952	−	0.870057i	$-0.664082\pi$
$643$	−25.0000			−0.985904			−0.492952	+	0.870057i	$0.664082\pi$
$644$	−12.0000	−	10.3923i	−0.472866	−	0.409514i
$645$		0			0
$646$	12.0000	+	20.7846i	0.472134	+	0.817760i
$647$	−14.0000	+	24.2487i	−0.550397	+	0.953315i	0.447849	+	0.894109i	$0.352190\pi$
$647$	−14.0000	+	24.2487i	−0.550397	+	0.953315i	−0.998246	+	0.0592060i	$0.981143\pi$
$648$		0			0
$649$	−6.00000	−	10.3923i	−0.235521	−	0.407934i
$650$	11.0000			0.431455
$651$		0			0
$652$	−19.0000			−0.744097
$653$	−3.00000	−	5.19615i	−0.117399	−	0.203341i	0.801337	−	0.598213i	$-0.204122\pi$
$653$	−3.00000	−	5.19615i	−0.117399	−	0.203341i	−0.918736	+	0.394872i	$0.870789\pi$
$654$		0			0
$655$	−8.00000	+	13.8564i	−0.312586	+	0.541415i
$656$	−1.00000	−	1.73205i	−0.0390434	−	0.0676252i
$657$		0			0
$658$	−15.0000	+	5.19615i	−0.584761	+	0.202567i
$659$	12.0000			0.467454			0.233727	−	0.972302i	$-0.424908\pi$
$659$	12.0000			0.467454			0.233727	+	0.972302i	$0.424908\pi$
$660$		0			0
$661$	7.00000	−	12.1244i	0.272268	−	0.471583i	−0.697174	−	0.716902i	$-0.745559\pi$
$661$	7.00000	−	12.1244i	0.272268	−	0.471583i	0.969442	+	0.245319i	$0.0788928\pi$
$662$	14.0000	−	24.2487i	0.544125	−	0.942453i
$663$		0			0
$664$	−18.0000			−0.698535
$665$	40.0000	−	13.8564i	1.55113	−	0.537328i
$666$		0			0
$667$	−6.00000	−	10.3923i	−0.232321	−	0.402392i
$668$	7.00000	−	12.1244i	0.270838	−	0.469105i
$669$		0			0
$670$	14.0000	+	24.2487i	0.540867	+	0.936809i
$671$	−10.0000			−0.386046
$672$		0			0
$673$	22.0000			0.848038			0.424019	−	0.905653i	$-0.360619\pi$
$673$	22.0000			0.848038			0.424019	+	0.905653i	$0.360619\pi$
$674$	−5.00000	−	8.66025i	−0.192593	−	0.333581i
$675$		0			0
$676$	−6.00000	+	10.3923i	−0.230769	+	0.399704i
$677$	−6.00000	−	10.3923i	−0.230599	−	0.399409i	0.727386	−	0.686229i	$-0.240735\pi$
$677$	−6.00000	−	10.3923i	−0.230599	−	0.399409i	−0.957984	+	0.286820i	$0.907402\pi$
$678$		0			0
$679$	18.0000	+	15.5885i	0.690777	+	0.598230i
$680$	−72.0000			−2.76107
$681$		0			0
$682$	−3.00000	+	5.19615i	−0.114876	+	0.198971i
$683$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$683$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$684$		0			0
$685$	−72.0000			−2.75098
$686$	18.5000	+	0.866025i	0.706333	+	0.0330650i
$687$		0			0
$688$	−0.500000	−	0.866025i	−0.0190623	−	0.0330169i
$689$	3.00000	−	5.19615i	0.114291	−	0.197958i
$690$		0			0
$691$	−20.5000	−	35.5070i	−0.779857	−	1.35075i	−0.932024	−	0.362397i	$-0.881959\pi$
$691$	−20.5000	−	35.5070i	−0.779857	−	1.35075i	0.152167	−	0.988355i	$-0.451375\pi$
$692$	−16.0000			−0.608229
$693$		0			0
$694$	16.0000			0.607352
$695$	18.0000	+	31.1769i	0.682779	+	1.18261i
$696$		0			0
$697$	−6.00000	+	10.3923i	−0.227266	+	0.393637i
$698$	2.50000	+	4.33013i	0.0946264	+	0.163898i
$699$		0			0
$700$	−5.50000	+	28.5788i	−0.207880	+	1.08018i
$701$	24.0000			0.906467			0.453234	−	0.891392i	$-0.350270\pi$
$701$	24.0000			0.906467			0.453234	+	0.891392i	$0.350270\pi$
$702$		0			0
$703$	−6.00000	+	10.3923i	−0.226294	+	0.391953i
$704$	−7.00000	+	12.1244i	−0.263822	+	0.456954i
$705$		0			0
$706$	−28.0000			−1.05379
$707$	20.0000	−	6.92820i	0.752177	−	0.260562i
$708$		0			0
$709$	−10.5000	−	18.1865i	−0.394336	−	0.683010i	0.598680	−	0.800988i	$-0.295692\pi$
$709$	−10.5000	−	18.1865i	−0.394336	−	0.683010i	−0.993016	+	0.117978i	$0.962359\pi$
$710$		0			0
$711$		0			0
$712$	−6.00000	−	10.3923i	−0.224860	−	0.389468i
$713$	18.0000			0.674105
$714$		0			0
$715$	−8.00000			−0.299183
$716$	2.00000	+	3.46410i	0.0747435	+	0.129460i
$717$		0			0
$718$	−11.0000	+	19.0526i	−0.410516	+	0.711035i
$719$	24.0000	+	41.5692i	0.895049	+	1.55027i	0.833744	+	0.552151i	$0.186193\pi$
$719$	24.0000	+	41.5692i	0.895049	+	1.55027i	0.0613050	+	0.998119i	$0.480474\pi$
$720$		0			0
$721$	8.50000	−	44.1673i	0.316557	−	1.64488i
$722$	−3.00000			−0.111648
$723$		0			0
$724$	−1.00000	+	1.73205i	−0.0371647	+	0.0643712i
$725$	−11.0000	+	19.0526i	−0.408530	+	0.707594i
$726$		0			0
$727$	41.0000			1.52061			0.760303	−	0.649569i	$-0.225051\pi$
$727$	41.0000			1.52061			0.760303	+	0.649569i	$0.225051\pi$
$728$	−6.00000	−	5.19615i	−0.222375	−	0.192582i
$729$		0			0
$730$	−12.0000	−	20.7846i	−0.444140	−	0.769273i
$731$	−3.00000	+	5.19615i	−0.110959	+	0.192187i
$732$		0			0
$733$	−10.5000	−	18.1865i	−0.387826	−	0.671735i	0.604331	−	0.796734i	$-0.293441\pi$
$733$	−10.5000	−	18.1865i	−0.387826	−	0.671735i	−0.992157	+	0.124999i	$0.960107\pi$
$734$	12.0000			0.442928
$735$		0			0
$736$	30.0000			1.10581
$737$	−7.00000	−	12.1244i	−0.257848	−	0.446606i
$738$		0			0
$739$	4.50000	−	7.79423i	0.165535	−	0.286715i	−0.771310	−	0.636460i	$-0.780398\pi$
$739$	4.50000	−	7.79423i	0.165535	−	0.286715i	0.936845	+	0.349744i	$0.113732\pi$
$740$	−6.00000	−	10.3923i	−0.220564	−	0.382029i
$741$		0			0
$742$	−12.0000	−	10.3923i	−0.440534	−	0.381514i
$743$	−42.0000			−1.54083			−0.770415	−	0.637542i	$-0.779951\pi$
$743$	−42.0000			−1.54083			−0.770415	+	0.637542i	$0.779951\pi$
$744$		0			0
$745$	−48.0000	+	83.1384i	−1.75858	+	3.04596i
$746$	−13.0000	+	22.5167i	−0.475964	+	0.824394i
$747$		0			0
$748$	12.0000			0.438763
$749$	1.00000	−	5.19615i	0.0365392	−	0.189863i
$750$		0			0
$751$	4.00000	+	6.92820i	0.145962	+	0.252814i	0.929731	−	0.368238i	$-0.120039\pi$
$751$	4.00000	+	6.92820i	0.145962	+	0.252814i	−0.783769	+	0.621052i	$0.786706\pi$
$752$	3.00000	−	5.19615i	0.109399	−	0.189484i
$753$		0			0
$754$	−1.00000	−	1.73205i	−0.0364179	−	0.0630776i
$755$	−20.0000			−0.727875
$756$		0			0
$757$	17.0000			0.617876			0.308938	−	0.951082i	$-0.400027\pi$
$757$	17.0000			0.617876			0.308938	+	0.951082i	$0.400027\pi$
$758$	4.50000	+	7.79423i	0.163447	+	0.283099i
$759$		0			0
$760$	−24.0000	+	41.5692i	−0.870572	+	1.50787i
$761$	24.0000	+	41.5692i	0.869999	+	1.50688i	0.861996	+	0.506915i	$0.169214\pi$
$761$	24.0000	+	41.5692i	0.869999	+	1.50688i	0.00800331	+	0.999968i	$0.497452\pi$
$762$		0			0
$763$	−22.5000	+	7.79423i	−0.814555	+	0.282170i
$764$	4.00000			0.144715
$765$		0			0
$766$	9.00000	−	15.5885i	0.325183	−	0.563234i
$767$	−3.00000	+	5.19615i	−0.108324	+	0.187622i
$768$		0			0
$769$	−10.0000			−0.360609			−0.180305	−	0.983611i	$-0.557708\pi$
$769$	−10.0000			−0.360609			−0.180305	+	0.983611i	$0.557708\pi$
$770$	−4.00000	+	20.7846i	−0.144150	+	0.749025i
$771$		0			0
$772$	−12.5000	−	21.6506i	−0.449885	−	0.779223i
$773$	−8.00000	+	13.8564i	−0.287740	+	0.498380i	−0.973270	−	0.229664i	$-0.926237\pi$
$773$	−8.00000	+	13.8564i	−0.287740	+	0.498380i	0.685530	+	0.728044i	$0.259571\pi$
$774$		0			0
$775$	−16.5000	−	28.5788i	−0.592697	−	1.02658i
$776$	−27.0000			−0.969244
$777$		0			0
$778$	−12.0000			−0.430221
$779$	4.00000	+	6.92820i	0.143315	+	0.248229i
$780$		0			0
$781$		0			0
$782$	18.0000	+	31.1769i	0.643679	+	1.11488i
$783$		0			0
$784$	−5.50000	+	4.33013i	−0.196429	+	0.154647i
$785$	−40.0000			−1.42766
$786$		0			0
$787$	23.5000	−	40.7032i	0.837685	−	1.45091i	−0.0541413	−	0.998533i	$-0.517242\pi$
$787$	23.5000	−	40.7032i	0.837685	−	1.45091i	0.891826	−	0.452379i	$-0.149425\pi$
$788$	10.0000	−	17.3205i	0.356235	−	0.617018i
$789$		0			0
$790$	−44.0000			−1.56545
$791$	−32.0000	−	27.7128i	−1.13779	−	0.985354i
$792$		0			0
$793$	2.50000	+	4.33013i	0.0887776	+	0.153767i
$794$	16.5000	−	28.5788i	0.585563	−	1.01423i
$795$		0			0
$796$	−4.50000	−	7.79423i	−0.159498	−	0.276259i
$797$	14.0000			0.495905			0.247953	−	0.968772i	$-0.420242\pi$
$797$	14.0000			0.495905			0.247953	+	0.968772i	$0.420242\pi$
$798$		0			0
$799$	−36.0000			−1.27359
$800$	−27.5000	−	47.6314i	−0.972272	−	1.68402i
$801$		0			0
$802$	−15.0000	+	25.9808i	−0.529668	+	0.917413i
$803$	6.00000	+	10.3923i	0.211735	+	0.366736i
$804$		0			0
$805$	60.0000	−	20.7846i	2.11472	−	0.732561i
$806$	3.00000			0.105670
$807$		0			0
$808$	−12.0000	+	20.7846i	−0.422159	+	0.731200i
$809$	15.0000	−	25.9808i	0.527372	−	0.913435i	−0.472119	−	0.881535i	$-0.656511\pi$
$809$	15.0000	−	25.9808i	0.527372	−	0.913435i	0.999491	−	0.0319002i	$-0.0101559\pi$
$810$		0			0
$811$	32.0000			1.12367			0.561836	−	0.827249i	$-0.310095\pi$
$811$	32.0000			1.12367			0.561836	+	0.827249i	$0.310095\pi$
$812$	5.00000	−	1.73205i	0.175466	−	0.0607831i
$813$		0			0
$814$	−3.00000	−	5.19615i	−0.105150	−	0.182125i
$815$	38.0000	−	65.8179i	1.33108	−	2.30550i
$816$		0			0
$817$	2.00000	+	3.46410i	0.0699711	+	0.121194i
$818$	5.00000			0.174821
$819$		0			0
$820$	−8.00000			−0.279372
$821$	−11.0000	−	19.0526i	−0.383903	−	0.664939i	0.607714	−	0.794156i	$-0.292087\pi$
$821$	−11.0000	−	19.0526i	−0.383903	−	0.664939i	−0.991616	+	0.129217i	$0.958754\pi$
$822$		0			0
$823$	−22.5000	+	38.9711i	−0.784301	+	1.35845i	0.145115	+	0.989415i	$0.453645\pi$
$823$	−22.5000	+	38.9711i	−0.784301	+	1.35845i	−0.929416	+	0.369034i	$0.879689\pi$
$824$	25.5000	+	44.1673i	0.888335	+	1.53864i
$825$		0			0
$826$	12.0000	+	10.3923i	0.417533	+	0.361595i
$827$	−12.0000			−0.417281			−0.208640	−	0.977992i	$-0.566904\pi$
$827$	−12.0000			−0.417281			−0.208640	+	0.977992i	$0.566904\pi$
$828$		0			0
$829$	5.00000	−	8.66025i	0.173657	−	0.300783i	−0.766039	−	0.642795i	$-0.777775\pi$
$829$	5.00000	−	8.66025i	0.173657	−	0.300783i	0.939696	+	0.342012i	$0.111108\pi$
$830$	12.0000	−	20.7846i	0.416526	−	0.721444i
$831$		0			0
$832$	7.00000			0.242681
$833$	39.0000	+	15.5885i	1.35127	+	0.540108i
$834$		0			0
$835$	28.0000	+	48.4974i	0.968980	+	1.67832i
$836$	4.00000	−	6.92820i	0.138343	−	0.239617i
$837$		0			0
$838$	6.00000	+	10.3923i	0.207267	+	0.358996i
$839$	26.0000			0.897620			0.448810	−	0.893627i	$-0.351848\pi$
$839$	26.0000			0.897620			0.448810	+	0.893627i	$0.351848\pi$
$840$		0			0
$841$	−25.0000			−0.862069
$842$	5.00000	+	8.66025i	0.172311	+	0.298452i
$843$		0			0
$844$	−2.50000	+	4.33013i	−0.0860535	+	0.149049i
$845$	−24.0000	−	41.5692i	−0.825625	−	1.43002i
$846$		0			0
$847$	−3.50000	+	18.1865i	−0.120261	+	0.624897i
$848$	6.00000			0.206041
$849$		0			0
$850$	33.0000	−	57.1577i	1.13189	−	1.96049i
$851$	−9.00000	+	15.5885i	−0.308516	+	0.534365i
$852$		0			0
$853$	26.0000			0.890223			0.445112	−	0.895475i	$-0.353164\pi$
$853$	26.0000			0.890223			0.445112	+	0.895475i	$0.353164\pi$
$854$	12.5000	−	4.33013i	0.427741	−	0.148174i
$855$		0			0
$856$	3.00000	+	5.19615i	0.102538	+	0.177601i
$857$	8.00000	−	13.8564i	0.273275	−	0.473326i	−0.696424	−	0.717631i	$-0.745227\pi$
$857$	8.00000	−	13.8564i	0.273275	−	0.473326i	0.969698	+	0.244305i	$0.0785598\pi$
$858$		0			0
$859$	12.5000	+	21.6506i	0.426494	+	0.738710i	0.996559	−	0.0828900i	$-0.0264150\pi$
$859$	12.5000	+	21.6506i	0.426494	+	0.738710i	−0.570064	+	0.821600i	$0.693082\pi$
$860$	−4.00000			−0.136399
$861$		0			0
$862$	18.0000			0.613082
$863$	−18.0000	−	31.1769i	−0.612727	−	1.06127i	−0.990779	−	0.135490i	$-0.956739\pi$
$863$	−18.0000	−	31.1769i	−0.612727	−	1.06127i	0.378052	−	0.925785i	$-0.376594\pi$
$864$		0			0
$865$	32.0000	−	55.4256i	1.08803	−	1.88453i
$866$	8.50000	+	14.7224i	0.288842	+	0.500289i
$867$		0			0
$868$	−1.50000	+	7.79423i	−0.0509133	+	0.264553i
$869$	22.0000			0.746299
$870$		0			0
$871$	−3.50000	+	6.06218i	−0.118593	+	0.205409i
$872$	13.5000	−	23.3827i	0.457168	−	0.791838i
$873$		0			0
$874$	24.0000			0.811812
$875$	−48.0000	−	41.5692i	−1.62270	−	1.40530i
$876$		0			0
$877$	−3.50000	−	6.06218i	−0.118187	−	0.204705i	0.800862	−	0.598848i	$-0.204375\pi$
$877$	−3.50000	−	6.06218i	−0.118187	−	0.204705i	−0.919049	+	0.394143i	$0.871041\pi$
$878$	−12.0000	+	20.7846i	−0.404980	+	0.701447i
$879$		0			0
$880$	−4.00000	−	6.92820i	−0.134840	−	0.233550i
$881$	6.00000			0.202145			0.101073	−	0.994879i	$-0.467773\pi$
$881$	6.00000			0.202145			0.101073	+	0.994879i	$0.467773\pi$
$882$		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
$886$	12.0000	−	20.7846i	0.403148	−	0.698273i
$887$	−11.0000	−	19.0526i	−0.369344	−	0.639722i	0.620119	−	0.784508i	$-0.287084\pi$
$887$	−11.0000	−	19.0526i	−0.369344	−	0.639722i	−0.989463	+	0.144785i	$0.953751\pi$
$888$		0			0
$889$	−18.0000	−	15.5885i	−0.603701	−	0.522820i
$890$	16.0000			0.536321
$891$		0			0
$892$	8.00000	−	13.8564i	0.267860	−	0.463947i
$893$	−12.0000	+	20.7846i	−0.401565	+	0.695530i
$894$		0			0
$895$	−16.0000			−0.534821
$896$	−1.50000	+	7.79423i	−0.0501115	+	0.260387i
$897$		0			0
$898$	18.0000	+	31.1769i	0.600668	+	1.04039i
$899$	−3.00000	+	5.19615i	−0.100056	+	0.173301i
$900$		0			0
$901$	−18.0000	−	31.1769i	−0.599667	−	1.03865i
$902$	−4.00000			−0.133185
$903$		0			0
$904$	48.0000			1.59646
$905$	−4.00000	−	6.92820i	−0.132964	−	0.230301i
$906$		0			0
$907$	6.50000	−	11.2583i	0.215829	−	0.373827i	−0.737700	−	0.675129i	$-0.764088\pi$
$907$	6.50000	−	11.2583i	0.215829	−	0.373827i	0.953529	+	0.301302i	$0.0974213\pi$
$908$	9.00000	+	15.5885i	0.298675	+	0.517321i
$909$		0			0
$910$	10.0000	−	3.46410i	0.331497	−	0.114834i
$911$	−18.0000			−0.596367			−0.298183	−	0.954509i	$-0.596381\pi$
$911$	−18.0000			−0.596367			−0.298183	+	0.954509i	$0.596381\pi$
$912$		0			0
$913$	−6.00000	+	10.3923i	−0.198571	+	0.343935i
$914$	−6.50000	+	11.2583i	−0.215001	+	0.372392i
$915$		0			0
$916$	7.00000			0.231287
$917$	−2.00000	+	10.3923i	−0.0660458	+	0.343184i
$918$		0			0
$919$	−14.5000	−	25.1147i	−0.478311	−	0.828459i	0.521380	−	0.853325i	$-0.325417\pi$
$919$	−14.5000	−	25.1147i	−0.478311	−	0.828459i	−0.999691	+	0.0248659i	$0.992084\pi$
$920$	−36.0000	+	62.3538i	−1.18688	+	2.05574i
$921$		0			0
$922$	1.00000	+	1.73205i	0.0329332	+	0.0570421i
$923$		0			0
$924$		0			0
$925$	33.0000			1.08503
$926$	16.0000	+	27.7128i	0.525793	+	0.910700i
$927$		0			0
$928$	−5.00000	+	8.66025i	−0.164133	+	0.284287i
$929$	−2.00000	−	3.46410i	−0.0656179	−	0.113653i	0.831350	−	0.555749i	$-0.187569\pi$
$929$	−2.00000	−	3.46410i	−0.0656179	−	0.113653i	−0.896968	+	0.442096i	$0.854235\pi$
$930$		0			0
$931$	22.0000	−	17.3205i	0.721021	−	0.567657i
$932$	24.0000			0.786146
$933$		0			0
$934$		0			0
$935$	−24.0000	+	41.5692i	−0.784884	+	1.35946i
$936$		0			0
$937$	−21.0000			−0.686040			−0.343020	−	0.939328i	$-0.611450\pi$
$937$	−21.0000			−0.686040			−0.343020	+	0.939328i	$0.611450\pi$
$938$	14.0000	+	12.1244i	0.457116	+	0.395874i
$939$		0			0
$940$	−12.0000	−	20.7846i	−0.391397	−	0.677919i
$941$	−23.0000	+	39.8372i	−0.749779	+	1.29865i	0.198150	+	0.980172i	$0.436507\pi$
$941$	−23.0000	+	39.8372i	−0.749779	+	1.29865i	−0.947929	+	0.318483i	$0.896827\pi$
$942$		0			0
$943$	6.00000	+	10.3923i	0.195387	+	0.338420i
$944$	−6.00000			−0.195283
$945$		0			0
$946$	−2.00000			−0.0650256
$947$	13.0000	+	22.5167i	0.422443	+	0.731693i	0.996178	−	0.0873481i	$-0.0278392\pi$
$947$	13.0000	+	22.5167i	0.422443	+	0.731693i	−0.573735	+	0.819041i	$0.694506\pi$
$948$		0			0
$949$	3.00000	−	5.19615i	0.0973841	−	0.168674i
$950$	−22.0000	−	38.1051i	−0.713774	−	1.23629i
$951$		0			0
$952$	−45.0000	+	15.5885i	−1.45846	+	0.505225i
$953$	−14.0000			−0.453504			−0.226752	−	0.973952i	$-0.572811\pi$
$953$	−14.0000			−0.453504			−0.226752	+	0.973952i	$0.572811\pi$
$954$		0			0
$955$	−8.00000	+	13.8564i	−0.258874	+	0.448383i
$956$	−6.00000	+	10.3923i	−0.194054	+	0.336111i
$957$		0			0
$958$	16.0000			0.516937
$959$	−45.0000	+	15.5885i	−1.45313	+	0.503378i
$960$		0			0
$961$	11.0000	+	19.0526i	0.354839	+	0.614599i
$962$	−1.50000	+	2.59808i	−0.0483619	+	0.0837653i
$963$		0			0
$964$	8.50000	+	14.7224i	0.273767	+	0.474178i
$965$	100.000			3.21911
$966$		0			0
$967$	−17.0000			−0.546683			−0.273342	−	0.961917i	$-0.588129\pi$
$967$	−17.0000			−0.546683			−0.273342	+	0.961917i	$0.588129\pi$
$968$	−10.5000	−	18.1865i	−0.337483	−	0.584537i
$969$		0			0
$970$	18.0000	−	31.1769i	0.577945	−	1.00103i
$971$	6.00000	+	10.3923i	0.192549	+	0.333505i	0.946094	−	0.323891i	$-0.104991\pi$
$971$	6.00000	+	10.3923i	0.192549	+	0.333505i	−0.753545	+	0.657396i	$0.771658\pi$
$972$		0			0
$973$	18.0000	+	15.5885i	0.577054	+	0.499743i
$974$	16.0000			0.512673
$975$		0			0
$976$	−2.50000	+	4.33013i	−0.0800230	+	0.138604i
$977$	−3.00000	+	5.19615i	−0.0959785	+	0.166240i	−0.910017	−	0.414572i	$-0.863931\pi$
$977$	−3.00000	+	5.19615i	−0.0959785	+	0.166240i	0.814038	+	0.580812i	$0.197265\pi$
$978$		0			0
$979$	−8.00000			−0.255681
$980$	4.00000	+	27.7128i	0.127775	+	0.885253i
$981$		0			0
$982$	−17.0000	−	29.4449i	−0.542492	−	0.939623i
$983$	18.0000	−	31.1769i	0.574111	−	0.994389i	−0.422027	−	0.906583i	$-0.638681\pi$
$983$	18.0000	−	31.1769i	0.574111	−	0.994389i	0.996138	−	0.0878058i	$-0.0279855\pi$
$984$		0			0
$985$	40.0000	+	69.2820i	1.27451	+	2.20751i
$986$	−12.0000			−0.382158
$987$		0			0
$988$	−4.00000			−0.127257
$989$	3.00000	+	5.19615i	0.0953945	+	0.165228i
$990$		0			0
$991$	27.5000	−	47.6314i	0.873566	−	1.51306i	0.0152841	−	0.999883i	$-0.495135\pi$
$991$	27.5000	−	47.6314i	0.873566	−	1.51306i	0.858282	−	0.513178i	$-0.171532\pi$
$992$	−7.50000	−	12.9904i	−0.238125	−	0.412445i
$993$		0			0
$994$		0			0
$995$	36.0000			1.14128
$996$		0			0
$997$	14.5000	−	25.1147i	0.459220	−	0.795392i	−0.539700	−	0.841857i	$-0.681462\pi$
$997$	14.5000	−	25.1147i	0.459220	−	0.795392i	0.998920	+	0.0464655i	$0.0147958\pi$
$998$	8.50000	−	14.7224i	0.269063	−	0.466030i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	189.2.e.a.109.1	✓	2
3.2	odd	2		189.2.e.c.109.1	yes	2
7.2	even	3	inner	189.2.e.a.163.1	yes	2
7.3	odd	6		1323.2.a.p.1.1		1
7.4	even	3		1323.2.a.m.1.1		1
9.2	odd	6		567.2.g.e.109.1		2
9.4	even	3		567.2.h.e.298.1		2
9.5	odd	6		567.2.h.b.298.1		2
9.7	even	3		567.2.g.b.109.1		2
21.2	odd	6		189.2.e.c.163.1	yes	2
21.11	odd	6		1323.2.a.g.1.1		1
21.17	even	6		1323.2.a.d.1.1		1
63.2	odd	6		567.2.h.b.352.1		2
63.16	even	3		567.2.h.e.352.1		2
63.23	odd	6		567.2.g.e.541.1		2
63.58	even	3		567.2.g.b.541.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
189.2.e.a.109.1	✓	2	1.1	even	1	trivial
189.2.e.a.163.1	yes	2	7.2	even	3	inner
189.2.e.c.109.1	yes	2	3.2	odd	2
189.2.e.c.163.1	yes	2	21.2	odd	6
567.2.g.b.109.1		2	9.7	even	3
567.2.g.b.541.1		2	63.58	even	3
567.2.g.e.109.1		2	9.2	odd	6
567.2.g.e.541.1		2	63.23	odd	6
567.2.h.b.298.1		2	9.5	odd	6
567.2.h.b.352.1		2	63.2	odd	6
567.2.h.e.298.1		2	9.4	even	3
567.2.h.e.352.1		2	63.16	even	3
1323.2.a.d.1.1		1	21.17	even	6
1323.2.a.g.1.1		1	21.11	odd	6
1323.2.a.m.1.1		1	7.4	even	3
1323.2.a.p.1.1		1	7.3	odd	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:	\( N \)	\(=\)	\( 189 = 3^{3} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	189.e (of order \(3\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(-0.500000 - 0.866025i) q^{2} +(0.500000 - 0.866025i) q^{4} +(2.00000 + 3.46410i) q^{5} +(2.00000 + 1.73205i) q^{7} -3.00000 q^{8} +(2.00000 - 3.46410i) q^{10} +(-1.00000 + 1.73205i) q^{11} +1.00000 q^{13} +(0.500000 - 2.59808i) q^{14} +(0.500000 + 0.866025i) q^{16} +(3.00000 - 5.19615i) q^{17} +(-2.00000 - 3.46410i) q^{19} +4.00000 q^{20} +2.00000 q^{22} +(-3.00000 - 5.19615i) q^{23} +(-5.50000 + 9.52628i) q^{25} +(-0.500000 - 0.866025i) q^{26} +(2.50000 - 0.866025i) q^{28} +2.00000 q^{29} +(-1.50000 + 2.59808i) q^{31} +(-2.50000 + 4.33013i) q^{32} -6.00000 q^{34} +(-2.00000 + 10.3923i) q^{35} +(-1.50000 - 2.59808i) q^{37} +(-2.00000 + 3.46410i) q^{38} +(-6.00000 - 10.3923i) q^{40} -2.00000 q^{41} -1.00000 q^{43} +(1.00000 + 1.73205i) q^{44} +(-3.00000 + 5.19615i) q^{46} +(-3.00000 - 5.19615i) q^{47} +(1.00000 + 6.92820i) q^{49} +11.0000 q^{50} +(0.500000 - 0.866025i) q^{52} +(3.00000 - 5.19615i) q^{53} -8.00000 q^{55} +(-6.00000 - 5.19615i) q^{56} +(-1.00000 - 1.73205i) q^{58} +(-3.00000 + 5.19615i) q^{59} +(2.50000 + 4.33013i) q^{61} +3.00000 q^{62} +7.00000 q^{64} +(2.00000 + 3.46410i) q^{65} +(-3.50000 + 6.06218i) q^{67} +(-3.00000 - 5.19615i) q^{68} +(10.0000 - 3.46410i) q^{70} +(3.00000 - 5.19615i) q^{73} +(-1.50000 + 2.59808i) q^{74} -4.00000 q^{76} +(-5.00000 + 1.73205i) q^{77} +(-5.50000 - 9.52628i) q^{79} +(-2.00000 + 3.46410i) q^{80} +(1.00000 + 1.73205i) q^{82} +6.00000 q^{83} +24.0000 q^{85} +(0.500000 + 0.866025i) q^{86} +(3.00000 - 5.19615i) q^{88} +(2.00000 + 3.46410i) q^{89} +(2.00000 + 1.73205i) q^{91} -6.00000 q^{92} +(-3.00000 + 5.19615i) q^{94} +(8.00000 - 13.8564i) q^{95} +9.00000 q^{97} +(5.50000 - 4.33013i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - q^{2} + q^{4} + 4 q^{5} + 4 q^{7} - 6 q^{8} + 4 q^{10} - 2 q^{11} + 2 q^{13} + q^{14} + q^{16} + 6 q^{17} - 4 q^{19} + 8 q^{20} + 4 q^{22} - 6 q^{23} - 11 q^{25} - q^{26} + 5 q^{28} + 4 q^{29}+ \cdots + 11 q^{98}+O(q^{100}) \) 2 * q - q^2 + q^4 + 4 * q^5 + 4 * q^7 - 6 * q^8 + 4 * q^10 - 2 * q^11 + 2 * q^13 + q^14 + q^16 + 6 * q^17 - 4 * q^19 + 8 * q^20 + 4 * q^22 - 6 * q^23 - 11 * q^25 - q^26 + 5 * q^28 + 4 * q^29 - 3 * q^31 - 5 * q^32 - 12 * q^34 - 4 * q^35 - 3 * q^37 - 4 * q^38 - 12 * q^40 - 4 * q^41 - 2 * q^43 + 2 * q^44 - 6 * q^46 - 6 * q^47 + 2 * q^49 + 22 * q^50 + q^52 + 6 * q^53 - 16 * q^55 - 12 * q^56 - 2 * q^58 - 6 * q^59 + 5 * q^61 + 6 * q^62 + 14 * q^64 + 4 * q^65 - 7 * q^67 - 6 * q^68 + 20 * q^70 + 6 * q^73 - 3 * q^74 - 8 * q^76 - 10 * q^77 - 11 * q^79 - 4 * q^80 + 2 * q^82 + 12 * q^83 + 48 * q^85 + q^86 + 6 * q^88 + 4 * q^89 + 4 * q^91 - 12 * q^92 - 6 * q^94 + 16 * q^95 + 18 * q^97 + 11 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more