LMFDB - Embedded newform 189.2.e.d.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:	$4$
Relative dimension:	$2$ over $\Q(\zeta_{3})$
Coefficient field:	$\Q(\sqrt{-2}, \sqrt{-3})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 2x^{2} + 4 $ x^4 - 2*x^2 + 4
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 3 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			109.1
Root			$1.22474 + 0.707107i$ of defining polynomial
Character	$\chi$	$=$	189.109
Dual form			189.2.e.d.163.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-1.22474 - 2.12132i) q^{2} +(-2.00000 + 3.46410i) q^{4} +(-1.22474 - 2.12132i) q^{5} +(-0.500000 - 2.59808i) q^{7} +4.89898 q^{8} +O(q^{10})$	\(q+(-1.22474 - 2.12132i) q^{2} +(-2.00000 + 3.46410i) q^{4} +(-1.22474 - 2.12132i) q^{5} +(-0.500000 - 2.59808i) q^{7} +4.89898 q^{8} +(-3.00000 + 5.19615i) q^{10} +(-2.44949 + 4.24264i) q^{11} -4.00000 q^{13} +(-4.89898 + 4.24264i) q^{14} +(-2.00000 - 3.46410i) q^{16} +(1.22474 - 2.12132i) q^{17} +(0.500000 + 0.866025i) q^{19} +9.79796 q^{20} +12.0000 q^{22} +(-1.22474 - 2.12132i) q^{23} +(-0.500000 + 0.866025i) q^{25} +(4.89898 + 8.48528i) q^{26} +(10.0000 + 3.46410i) q^{28} -7.34847 q^{29} +(3.50000 - 6.06218i) q^{31} -6.00000 q^{34} +(-4.89898 + 4.24264i) q^{35} +(-4.00000 - 6.92820i) q^{37} +(1.22474 - 2.12132i) q^{38} +(-6.00000 - 10.3923i) q^{40} +7.34847 q^{41} -1.00000 q^{43} +(-9.79796 - 16.9706i) q^{44} +(-3.00000 + 5.19615i) q^{46} +(-1.22474 - 2.12132i) q^{47} +(-6.50000 + 2.59808i) q^{49} +2.44949 q^{50} +(8.00000 - 13.8564i) q^{52} +(1.22474 - 2.12132i) q^{53} +12.0000 q^{55} +(-2.44949 - 12.7279i) q^{56} +(9.00000 + 15.5885i) q^{58} +(4.89898 - 8.48528i) q^{59} +(-2.50000 - 4.33013i) q^{61} -17.1464 q^{62} -8.00000 q^{64} +(4.89898 + 8.48528i) q^{65} +(-1.00000 + 1.73205i) q^{67} +(4.89898 + 8.48528i) q^{68} +(15.0000 + 5.19615i) q^{70} +(0.500000 - 0.866025i) q^{73} +(-9.79796 + 16.9706i) q^{74} -4.00000 q^{76} +(12.2474 + 4.24264i) q^{77} +(2.00000 + 3.46410i) q^{79} +(-4.89898 + 8.48528i) q^{80} +(-9.00000 - 15.5885i) q^{82} +14.6969 q^{83} -6.00000 q^{85} +(1.22474 + 2.12132i) q^{86} +(-12.0000 + 20.7846i) q^{88} +(-1.22474 - 2.12132i) q^{89} +(2.00000 + 10.3923i) q^{91} +9.79796 q^{92} +(-3.00000 + 5.19615i) q^{94} +(1.22474 - 2.12132i) q^{95} -1.00000 q^{97} +(13.4722 + 10.6066i) q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 8 q^{4} - 2 q^{7}+O(q^{10}) $ 4 * q - 8 * q^4 - 2 * q^7	$ 4 q - 8 q^{4} - 2 q^{7} - 12 q^{10} - 16 q^{13} - 8 q^{16} + 2 q^{19} + 48 q^{22} - 2 q^{25} + 40 q^{28} + 14 q^{31} - 24 q^{34} - 16 q^{37} - 24 q^{40} - 4 q^{43} - 12 q^{46} - 26 q^{49} + 32 q^{52} + 48 q^{55} + 36 q^{58} - 10 q^{61} - 32 q^{64} - 4 q^{67} + 60 q^{70} + 2 q^{73} - 16 q^{76} + 8 q^{79} - 36 q^{82} - 24 q^{85} - 48 q^{88} + 8 q^{91} - 12 q^{94} - 4 q^{97}+O(q^{100}) $ 4 * q - 8 * q^4 - 2 * q^7 - 12 * q^10 - 16 * q^13 - 8 * q^16 + 2 * q^19 + 48 * q^22 - 2 * q^25 + 40 * q^28 + 14 * q^31 - 24 * q^34 - 16 * q^37 - 24 * q^40 - 4 * q^43 - 12 * q^46 - 26 * q^49 + 32 * q^52 + 48 * q^55 + 36 * q^58 - 10 * q^61 - 32 * q^64 - 4 * q^67 + 60 * q^70 + 2 * q^73 - 16 * q^76 + 8 * q^79 - 36 * q^82 - 24 * q^85 - 48 * q^88 + 8 * q^91 - 12 * q^94 - 4 * q^97

Display 100 coefficients 10 coefficients

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 $

$2$	−1.22474	−	2.12132i	−0.866025	−	1.50000i	−0.866025	−	0.500000i	$-0.833333\pi$
$2$	−1.22474	−	2.12132i	−0.866025	−	1.50000i		−	1.00000i	$-0.5\pi$
$3$		0			0
$3$		0			0
$4$	−2.00000	+	3.46410i	−1.00000	+	1.73205i
$5$	−1.22474	−	2.12132i	−0.547723	−	0.948683i	−0.998430	−	0.0560116i	$-0.982162\pi$
$5$	−1.22474	−	2.12132i	−0.547723	−	0.948683i	0.450708	−	0.892672i	$-0.351172\pi$
$6$		0			0
$7$	−0.500000	−	2.59808i	−0.188982	−	0.981981i
$7$	−0.500000	−	2.59808i	−0.188982	−	0.981981i
$8$	4.89898			1.73205
$9$		0			0
$10$	−3.00000	+	5.19615i	−0.948683	+	1.64317i
$11$	−2.44949	+	4.24264i	−0.738549	+	1.27920i	0.214600	+	0.976702i	$0.431155\pi$
$11$	−2.44949	+	4.24264i	−0.738549	+	1.27920i	−0.953149	+	0.302502i	$0.902178\pi$
$12$		0			0
$13$	−4.00000			−1.10940			−0.554700	−	0.832050i	$-0.687167\pi$
$13$	−4.00000			−1.10940			−0.554700	+	0.832050i	$0.687167\pi$
$14$	−4.89898	+	4.24264i	−1.30931	+	1.13389i
$15$		0			0
$16$	−2.00000	−	3.46410i	−0.500000	−	0.866025i
$17$	1.22474	−	2.12132i	0.297044	−	0.514496i	−0.678414	−	0.734680i	$-0.737332\pi$
$17$	1.22474	−	2.12132i	0.297044	−	0.514496i	0.975458	+	0.220184i	$0.0706658\pi$
$18$		0			0
$19$	0.500000	+	0.866025i	0.114708	+	0.198680i	0.917663	−	0.397360i	$-0.130073\pi$
$19$	0.500000	+	0.866025i	0.114708	+	0.198680i	−0.802955	+	0.596040i	$0.796740\pi$
$20$	9.79796			2.19089
$21$		0			0
$22$	12.0000			2.55841
$23$	−1.22474	−	2.12132i	−0.255377	−	0.442326i	0.709621	−	0.704584i	$-0.248866\pi$
$23$	−1.22474	−	2.12132i	−0.255377	−	0.442326i	−0.964998	+	0.262258i	$0.915533\pi$
$24$		0			0
$25$	−0.500000	+	0.866025i	−0.100000	+	0.173205i
$26$	4.89898	+	8.48528i	0.960769	+	1.66410i
$27$		0			0
$28$	10.0000	+	3.46410i	1.88982	+	0.654654i
$29$	−7.34847			−1.36458			−0.682288	−	0.731083i	$-0.739015\pi$
$29$	−7.34847			−1.36458			−0.682288	+	0.731083i	$0.739015\pi$
$30$		0			0
$31$	3.50000	−	6.06218i	0.628619	−	1.08880i	−0.359211	−	0.933257i	$-0.616954\pi$
$31$	3.50000	−	6.06218i	0.628619	−	1.08880i	0.987829	−	0.155543i	$-0.0497126\pi$
$32$		0			0
$33$		0			0
$34$	−6.00000			−1.02899
$35$	−4.89898	+	4.24264i	−0.828079	+	0.717137i
$36$		0			0
$37$	−4.00000	−	6.92820i	−0.657596	−	1.13899i	−0.981236	−	0.192809i	$-0.938240\pi$
$37$	−4.00000	−	6.92820i	−0.657596	−	1.13899i	0.323640	−	0.946180i	$-0.395093\pi$
$38$	1.22474	−	2.12132i	0.198680	−	0.344124i
$39$		0			0
$40$	−6.00000	−	10.3923i	−0.948683	−	1.64317i
$41$	7.34847			1.14764			0.573819	−	0.818982i	$-0.305461\pi$
$41$	7.34847			1.14764			0.573819	+	0.818982i	$0.305461\pi$
$42$		0			0
$43$	−1.00000			−0.152499			−0.0762493	−	0.997089i	$-0.524294\pi$
$43$	−1.00000			−0.152499			−0.0762493	+	0.997089i	$0.524294\pi$
$44$	−9.79796	−	16.9706i	−1.47710	−	2.55841i
$45$		0			0
$46$	−3.00000	+	5.19615i	−0.442326	+	0.766131i
$47$	−1.22474	−	2.12132i	−0.178647	−	0.309426i	0.762770	−	0.646670i	$-0.223839\pi$
$47$	−1.22474	−	2.12132i	−0.178647	−	0.309426i	−0.941417	+	0.337243i	$0.890505\pi$
$48$		0			0
$49$	−6.50000	+	2.59808i	−0.928571	+	0.371154i
$50$	2.44949			0.346410
$51$		0			0
$52$	8.00000	−	13.8564i	1.10940	−	1.92154i
$53$	1.22474	−	2.12132i	0.168232	−	0.291386i	−0.769567	−	0.638567i	$-0.779528\pi$
$53$	1.22474	−	2.12132i	0.168232	−	0.291386i	0.937798	+	0.347181i	$0.112861\pi$
$54$		0			0
$55$	12.0000			1.61808
$56$	−2.44949	−	12.7279i	−0.327327	−	1.70084i
$57$		0			0
$58$	9.00000	+	15.5885i	1.18176	+	2.04686i
$59$	4.89898	−	8.48528i	0.637793	−	1.10469i	−0.348123	−	0.937449i	$-0.613181\pi$
$59$	4.89898	−	8.48528i	0.637793	−	1.10469i	0.985916	−	0.167241i	$-0.0534857\pi$
$60$		0			0
$61$	−2.50000	−	4.33013i	−0.320092	−	0.554416i	0.660415	−	0.750901i	$-0.270381\pi$
$61$	−2.50000	−	4.33013i	−0.320092	−	0.554416i	−0.980507	+	0.196485i	$0.937047\pi$
$62$	−17.1464			−2.17760
$63$		0			0
$64$	−8.00000			−1.00000
$65$	4.89898	+	8.48528i	0.607644	+	1.05247i
$66$		0			0
$67$	−1.00000	+	1.73205i	−0.122169	+	0.211604i	−0.920623	−	0.390453i	$-0.872318\pi$
$67$	−1.00000	+	1.73205i	−0.122169	+	0.211604i	0.798454	+	0.602056i	$0.205652\pi$
$68$	4.89898	+	8.48528i	0.594089	+	1.02899i
$69$		0			0
$70$	15.0000	+	5.19615i	1.79284	+	0.621059i
$71$		0			0			−	1.00000i	$-0.5\pi$
$71$		0			0				1.00000i	$0.5\pi$
$72$		0			0
$73$	0.500000	−	0.866025i	0.0585206	−	0.101361i	−0.835281	−	0.549823i	$-0.814695\pi$
$73$	0.500000	−	0.866025i	0.0585206	−	0.101361i	0.893801	+	0.448463i	$0.148028\pi$
$74$	−9.79796	+	16.9706i	−1.13899	+	1.97279i
$75$		0			0
$76$	−4.00000			−0.458831
$77$	12.2474	+	4.24264i	1.39573	+	0.483494i
$78$		0			0
$79$	2.00000	+	3.46410i	0.225018	+	0.389742i	0.956325	−	0.292306i	$-0.0944227\pi$
$79$	2.00000	+	3.46410i	0.225018	+	0.389742i	−0.731307	+	0.682048i	$0.761089\pi$
$80$	−4.89898	+	8.48528i	−0.547723	+	0.948683i
$81$		0			0
$82$	−9.00000	−	15.5885i	−0.993884	−	1.72146i
$83$	14.6969			1.61320			0.806599	−	0.591099i	$-0.201306\pi$
$83$	14.6969			1.61320			0.806599	+	0.591099i	$0.201306\pi$
$84$		0			0
$85$	−6.00000			−0.650791
$86$	1.22474	+	2.12132i	0.132068	+	0.228748i
$87$		0			0
$88$	−12.0000	+	20.7846i	−1.27920	+	2.21565i
$89$	−1.22474	−	2.12132i	−0.129823	−	0.224860i	0.793785	−	0.608198i	$-0.208107\pi$
$89$	−1.22474	−	2.12132i	−0.129823	−	0.224860i	−0.923608	+	0.383339i	$0.874774\pi$
$90$		0			0
$91$	2.00000	+	10.3923i	0.209657	+	1.08941i
$92$	9.79796			1.02151
$93$		0			0
$94$	−3.00000	+	5.19615i	−0.309426	+	0.535942i
$95$	1.22474	−	2.12132i	0.125656	−	0.217643i
$96$		0			0
$97$	−1.00000			−0.101535			−0.0507673	−	0.998711i	$-0.516167\pi$
$97$	−1.00000			−0.101535			−0.0507673	+	0.998711i	$0.516167\pi$
$98$	13.4722	+	10.6066i	1.36090	+	1.07143i
$99$		0			0
$100$	−2.00000	−	3.46410i	−0.200000	−	0.346410i
$101$	−2.44949	+	4.24264i	−0.243733	+	0.422159i	−0.961775	−	0.273842i	$-0.911706\pi$
$101$	−2.44949	+	4.24264i	−0.243733	+	0.422159i	0.718041	+	0.696000i	$0.245039\pi$
$102$		0			0
$103$	−1.00000	−	1.73205i	−0.0985329	−	0.170664i	0.812545	−	0.582899i	$-0.198082\pi$
$103$	−1.00000	−	1.73205i	−0.0985329	−	0.170664i	−0.911078	+	0.412235i	$0.864748\pi$
$104$	−19.5959			−1.92154
$105$		0			0
$106$	−6.00000			−0.582772
$107$	9.79796	+	16.9706i	0.947204	+	1.64061i	0.751276	+	0.659988i	$0.229439\pi$
$107$	9.79796	+	16.9706i	0.947204	+	1.64061i	0.195928	+	0.980618i	$0.437228\pi$
$108$		0			0
$109$	0.500000	−	0.866025i	0.0478913	−	0.0829502i	−0.841086	−	0.540901i	$-0.818083\pi$
$109$	0.500000	−	0.866025i	0.0478913	−	0.0829502i	0.888977	+	0.457951i	$0.151417\pi$
$110$	−14.6969	−	25.4558i	−1.40130	−	2.42712i
$111$		0			0
$112$	−8.00000	+	6.92820i	−0.755929	+	0.654654i
$113$	−14.6969			−1.38257			−0.691286	−	0.722581i	$-0.742955\pi$
$113$	−14.6969			−1.38257			−0.691286	+	0.722581i	$0.742955\pi$
$114$		0			0
$115$	−3.00000	+	5.19615i	−0.279751	+	0.484544i
$116$	14.6969	−	25.4558i	1.36458	−	2.36352i
$117$		0			0
$118$	−24.0000			−2.20938
$119$	−6.12372	−	2.12132i	−0.561361	−	0.194461i
$120$		0			0
$121$	−6.50000	−	11.2583i	−0.590909	−	1.02348i
$122$	−6.12372	+	10.6066i	−0.554416	+	0.960277i
$123$		0			0
$124$	14.0000	+	24.2487i	1.25724	+	2.17760i
$125$	−9.79796			−0.876356
$126$		0			0
$127$	11.0000			0.976092			0.488046	−	0.872818i	$-0.337710\pi$
$127$	11.0000			0.976092			0.488046	+	0.872818i	$0.337710\pi$
$128$	9.79796	+	16.9706i	0.866025	+	1.50000i
$129$		0			0
$130$	12.0000	−	20.7846i	1.05247	−	1.82293i
$131$	−1.22474	−	2.12132i	−0.107006	−	0.185341i	0.807550	−	0.589799i	$-0.200793\pi$
$131$	−1.22474	−	2.12132i	−0.107006	−	0.185341i	−0.914556	+	0.404459i	$0.867460\pi$
$132$		0			0
$133$	2.00000	−	1.73205i	0.173422	−	0.150188i
$134$	4.89898			0.423207
$135$		0			0
$136$	6.00000	−	10.3923i	0.514496	−	0.891133i
$137$	8.57321	−	14.8492i	0.732459	−	1.26866i	−0.223370	−	0.974734i	$-0.571706\pi$
$137$	8.57321	−	14.8492i	0.732459	−	1.26866i	0.955829	−	0.293923i	$-0.0949608\pi$
$138$		0			0
$139$	14.0000			1.18746			0.593732	−	0.804663i	$-0.297654\pi$
$139$	14.0000			1.18746			0.593732	+	0.804663i	$0.297654\pi$
$140$	−4.89898	−	25.4558i	−0.414039	−	2.15141i
$141$		0			0
$142$		0			0
$143$	9.79796	−	16.9706i	0.819346	−	1.41915i
$144$		0			0
$145$	9.00000	+	15.5885i	0.747409	+	1.29455i
$146$	−2.44949			−0.202721
$147$		0			0
$148$	32.0000			2.63038
$149$	−1.22474	−	2.12132i	−0.100335	−	0.173785i	0.811488	−	0.584370i	$-0.198658\pi$
$149$	−1.22474	−	2.12132i	−0.100335	−	0.173785i	−0.911823	+	0.410584i	$0.865325\pi$
$150$		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	$0.231881\pi$
$152$	2.44949	+	4.24264i	0.198680	+	0.344124i
$153$		0			0
$154$	−6.00000	−	31.1769i	−0.483494	−	2.51231i
$155$	−17.1464			−1.37723
$156$		0			0
$157$	−10.0000	+	17.3205i	−0.798087	+	1.38233i	0.122774	+	0.992435i	$0.460821\pi$
$157$	−10.0000	+	17.3205i	−0.798087	+	1.38233i	−0.920860	+	0.389892i	$0.872512\pi$
$158$	4.89898	−	8.48528i	0.389742	−	0.675053i
$159$		0			0
$160$		0			0
$161$	−4.89898	+	4.24264i	−0.386094	+	0.334367i
$162$		0			0
$163$	0.500000	+	0.866025i	0.0391630	+	0.0678323i	0.884943	−	0.465700i	$-0.154198\pi$
$163$	0.500000	+	0.866025i	0.0391630	+	0.0678323i	−0.845780	+	0.533533i	$0.820864\pi$
$164$	−14.6969	+	25.4558i	−1.14764	+	1.98777i
$165$		0			0
$166$	−18.0000	−	31.1769i	−1.39707	−	2.41980i
$167$	−14.6969			−1.13728			−0.568642	−	0.822585i	$-0.692531\pi$
$167$	−14.6969			−1.13728			−0.568642	+	0.822585i	$0.692531\pi$
$168$		0			0
$169$	3.00000			0.230769
$170$	7.34847	+	12.7279i	0.563602	+	0.976187i
$171$		0			0
$172$	2.00000	−	3.46410i	0.152499	−	0.264135i
$173$	−1.22474	−	2.12132i	−0.0931156	−	0.161281i	0.815705	−	0.578468i	$-0.196349\pi$
$173$	−1.22474	−	2.12132i	−0.0931156	−	0.161281i	−0.908821	+	0.417187i	$0.863016\pi$
$174$		0			0
$175$	2.50000	+	0.866025i	0.188982	+	0.0654654i
$176$	19.5959			1.47710
$177$		0			0
$178$	−3.00000	+	5.19615i	−0.224860	+	0.389468i
$179$	1.22474	−	2.12132i	0.0915417	−	0.158555i	−0.816618	−	0.577178i	$-0.804154\pi$
$179$	1.22474	−	2.12132i	0.0915417	−	0.158555i	0.908160	+	0.418623i	$0.137487\pi$
$180$		0			0
$181$	−7.00000			−0.520306			−0.260153	−	0.965567i	$-0.583773\pi$
$181$	−7.00000			−0.520306			−0.260153	+	0.965567i	$0.583773\pi$
$182$	19.5959	−	16.9706i	1.45255	−	1.25794i
$183$		0			0
$184$	−6.00000	−	10.3923i	−0.442326	−	0.766131i
$185$	−9.79796	+	16.9706i	−0.720360	+	1.24770i
$186$		0			0
$187$	6.00000	+	10.3923i	0.438763	+	0.759961i
$188$	9.79796			0.714590
$189$		0			0
$190$	−6.00000			−0.435286
$191$	−1.22474	−	2.12132i	−0.0886194	−	0.153493i	0.818308	−	0.574779i	$-0.194912\pi$
$191$	−1.22474	−	2.12132i	−0.0886194	−	0.153493i	−0.906928	+	0.421286i	$0.861579\pi$
$192$		0			0
$193$	−10.0000	+	17.3205i	−0.719816	+	1.24676i	0.241257	+	0.970461i	$0.422440\pi$
$193$	−10.0000	+	17.3205i	−0.719816	+	1.24676i	−0.961073	+	0.276296i	$0.910893\pi$
$194$	1.22474	+	2.12132i	0.0879316	+	0.152302i
$195$		0			0
$196$	4.00000	−	27.7128i	0.285714	−	1.97949i
$197$		0			0			−	1.00000i	$-0.5\pi$
$197$		0			0				1.00000i	$0.5\pi$
$198$		0			0
$199$	9.50000	−	16.4545i	0.673437	−	1.16643i	−0.303486	−	0.952836i	$-0.598151\pi$
$199$	9.50000	−	16.4545i	0.673437	−	1.16643i	0.976923	−	0.213591i	$-0.0685161\pi$
$200$	−2.44949	+	4.24264i	−0.173205	+	0.300000i
$201$		0			0
$202$	12.0000			0.844317
$203$	3.67423	+	19.0919i	0.257881	+	1.33999i
$204$		0			0
$205$	−9.00000	−	15.5885i	−0.628587	−	1.08875i
$206$	−2.44949	+	4.24264i	−0.170664	+	0.295599i
$207$		0			0
$208$	8.00000	+	13.8564i	0.554700	+	0.960769i
$209$	−4.89898			−0.338869
$210$		0			0
$211$	5.00000			0.344214			0.172107	−	0.985078i	$-0.444942\pi$
$211$	5.00000			0.344214			0.172107	+	0.985078i	$0.444942\pi$
$212$	4.89898	+	8.48528i	0.336463	+	0.582772i
$213$		0			0
$214$	24.0000	−	41.5692i	1.64061	−	2.84161i
$215$	1.22474	+	2.12132i	0.0835269	+	0.144673i
$216$		0			0
$217$	−17.5000	−	6.06218i	−1.18798	−	0.411527i
$218$	−2.44949			−0.165900
$219$		0			0
$220$	−24.0000	+	41.5692i	−1.61808	+	2.80260i
$221$	−4.89898	+	8.48528i	−0.329541	+	0.570782i
$222$		0			0
$223$	26.0000			1.74109			0.870544	−	0.492090i	$-0.163767\pi$
$223$	26.0000			1.74109			0.870544	+	0.492090i	$0.163767\pi$
$224$		0			0
$225$		0			0
$226$	18.0000	+	31.1769i	1.19734	+	2.07386i
$227$	8.57321	−	14.8492i	0.569024	−	0.985579i	−0.427639	−	0.903950i	$-0.640654\pi$
$227$	8.57321	−	14.8492i	0.569024	−	0.985579i	0.996663	−	0.0816290i	$-0.0260123\pi$
$228$		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	$0.759040\pi$
$230$	14.6969			0.969087
$231$		0			0
$232$	−36.0000			−2.36352
$233$	−1.22474	−	2.12132i	−0.0802357	−	0.138972i	0.823115	−	0.567874i	$-0.192234\pi$
$233$	−1.22474	−	2.12132i	−0.0802357	−	0.138972i	−0.903351	+	0.428902i	$0.858901\pi$
$234$		0			0
$235$	−3.00000	+	5.19615i	−0.195698	+	0.338960i
$236$	19.5959	+	33.9411i	1.27559	+	2.20938i
$237$		0			0
$238$	3.00000	+	15.5885i	0.194461	+	1.01045i
$239$	7.34847			0.475333			0.237666	−	0.971347i	$-0.423617\pi$
$239$	7.34847			0.475333			0.237666	+	0.971347i	$0.423617\pi$
$240$		0			0
$241$	6.50000	−	11.2583i	0.418702	−	0.725213i	−0.577107	−	0.816668i	$-0.695819\pi$
$241$	6.50000	−	11.2583i	0.418702	−	0.725213i	0.995809	+	0.0914555i	$0.0291519\pi$
$242$	−15.9217	+	27.5772i	−1.02348	+	1.77273i
$243$		0			0
$244$	20.0000			1.28037
$245$	13.4722	+	10.6066i	0.860707	+	0.677631i
$246$		0			0
$247$	−2.00000	−	3.46410i	−0.127257	−	0.220416i
$248$	17.1464	−	29.6985i	1.08880	−	1.88586i
$249$		0			0
$250$	12.0000	+	20.7846i	0.758947	+	1.31453i
$251$	−22.0454			−1.39149			−0.695747	−	0.718287i	$-0.744926\pi$
$251$	−22.0454			−1.39149			−0.695747	+	0.718287i	$0.744926\pi$
$252$		0			0
$253$	12.0000			0.754434
$254$	−13.4722	−	23.3345i	−0.845321	−	1.46414i
$255$		0			0
$256$	16.0000	−	27.7128i	1.00000	−	1.73205i
$257$	−12.2474	−	21.2132i	−0.763975	−	1.32324i	−0.940787	−	0.338999i	$-0.889912\pi$
$257$	−12.2474	−	21.2132i	−0.763975	−	1.32324i	0.176812	−	0.984245i	$-0.443422\pi$
$258$		0			0
$259$	−16.0000	+	13.8564i	−0.994192	+	0.860995i
$260$	−39.1918			−2.43057
$261$		0			0
$262$	−3.00000	+	5.19615i	−0.185341	+	0.321019i
$263$	−2.44949	+	4.24264i	−0.151042	+	0.261612i	−0.931611	−	0.363457i	$-0.881596\pi$
$263$	−2.44949	+	4.24264i	−0.151042	+	0.261612i	0.780569	+	0.625070i	$0.214930\pi$
$264$		0			0
$265$	−6.00000			−0.368577
$266$	−6.12372	−	2.12132i	−0.375470	−	0.130066i
$267$		0			0
$268$	−4.00000	−	6.92820i	−0.244339	−	0.423207i
$269$	12.2474	−	21.2132i	0.746740	−	1.29339i	−0.202637	−	0.979254i	$-0.564951\pi$
$269$	12.2474	−	21.2132i	0.746740	−	1.29339i	0.949377	−	0.314138i	$-0.101715\pi$
$270$		0			0
$271$	0.500000	+	0.866025i	0.0303728	+	0.0526073i	0.880812	−	0.473466i	$-0.156997\pi$
$271$	0.500000	+	0.866025i	0.0303728	+	0.0526073i	−0.850439	+	0.526073i	$0.823664\pi$
$272$	−9.79796			−0.594089
$273$		0			0
$274$	−42.0000			−2.53731
$275$	−2.44949	−	4.24264i	−0.147710	−	0.255841i
$276$		0			0
$277$	−11.5000	+	19.9186i	−0.690968	+	1.19679i	0.280553	+	0.959839i	$0.409482\pi$
$277$	−11.5000	+	19.9186i	−0.690968	+	1.19679i	−0.971521	+	0.236953i	$0.923851\pi$
$278$	−17.1464	−	29.6985i	−1.02837	−	1.78120i
$279$		0			0
$280$	−24.0000	+	20.7846i	−1.43427	+	1.24212i
$281$	14.6969			0.876746			0.438373	−	0.898793i	$-0.355555\pi$
$281$	14.6969			0.876746			0.438373	+	0.898793i	$0.355555\pi$
$282$		0			0
$283$	12.5000	−	21.6506i	0.743048	−	1.28700i	−0.208053	−	0.978117i	$-0.566713\pi$
$283$	12.5000	−	21.6506i	0.743048	−	1.28700i	0.951101	−	0.308879i	$-0.0999539\pi$
$284$		0			0
$285$		0			0
$286$	−48.0000			−2.83830
$287$	−3.67423	−	19.0919i	−0.216883	−	1.12696i
$288$		0			0
$289$	5.50000	+	9.52628i	0.323529	+	0.560369i
$290$	22.0454	−	38.1838i	1.29455	−	2.24223i
$291$		0			0
$292$	2.00000	+	3.46410i	0.117041	+	0.202721i
$293$	29.3939			1.71721			0.858604	−	0.512639i	$-0.171332\pi$
$293$	29.3939			1.71721			0.858604	+	0.512639i	$0.171332\pi$
$294$		0			0
$295$	−24.0000			−1.39733
$296$	−19.5959	−	33.9411i	−1.13899	−	1.97279i
$297$		0			0
$298$	−3.00000	+	5.19615i	−0.173785	+	0.301005i
$299$	4.89898	+	8.48528i	0.283315	+	0.490716i
$300$		0			0
$301$	0.500000	+	2.59808i	0.0288195	+	0.149751i
$302$	12.2474			0.704761
$303$		0			0
$304$	2.00000	−	3.46410i	0.114708	−	0.198680i
$305$	−6.12372	+	10.6066i	−0.350643	+	0.607332i
$306$		0			0
$307$	−25.0000			−1.42683			−0.713413	−	0.700744i	$-0.752851\pi$
$307$	−25.0000			−1.42683			−0.713413	+	0.700744i	$0.752851\pi$
$308$	−39.1918	+	33.9411i	−2.23316	+	1.93398i
$309$		0			0
$310$	21.0000	+	36.3731i	1.19272	+	2.06585i
$311$	−17.1464	+	29.6985i	−0.972285	+	1.68405i	−0.283666	+	0.958923i	$0.591551\pi$
$311$	−17.1464	+	29.6985i	−0.972285	+	1.68405i	−0.688619	+	0.725123i	$0.741783\pi$
$312$		0			0
$313$	−11.5000	−	19.9186i	−0.650018	−	1.12586i	−0.983118	−	0.182973i	$-0.941428\pi$
$313$	−11.5000	−	19.9186i	−0.650018	−	1.12586i	0.333099	−	0.942892i	$-0.391906\pi$
$314$	48.9898			2.76465
$315$		0			0
$316$	−16.0000			−0.900070
$317$	−12.2474	−	21.2132i	−0.687885	−	1.19145i	−0.972521	−	0.232816i	$-0.925206\pi$
$317$	−12.2474	−	21.2132i	−0.687885	−	1.19145i	0.284635	−	0.958636i	$-0.408127\pi$
$318$		0			0
$319$	18.0000	−	31.1769i	1.00781	−	1.74557i
$320$	9.79796	+	16.9706i	0.547723	+	0.948683i
$321$		0			0
$322$	15.0000	+	5.19615i	0.835917	+	0.289570i
$323$	2.44949			0.136293
$324$		0			0
$325$	2.00000	−	3.46410i	0.110940	−	0.192154i
$326$	1.22474	−	2.12132i	0.0678323	−	0.117489i
$327$		0			0
$328$	36.0000			1.98777
$329$	−4.89898	+	4.24264i	−0.270089	+	0.233904i
$330$		0			0
$331$	6.50000	+	11.2583i	0.357272	+	0.618814i	0.987504	−	0.157593i	$-0.0503735\pi$
$331$	6.50000	+	11.2583i	0.357272	+	0.618814i	−0.630232	+	0.776407i	$0.717040\pi$
$332$	−29.3939	+	50.9117i	−1.61320	+	2.79414i
$333$		0			0
$334$	18.0000	+	31.1769i	0.984916	+	1.70592i
$335$	4.89898			0.267660
$336$		0			0
$337$	5.00000			0.272367			0.136184	−	0.990684i	$-0.456516\pi$
$337$	5.00000			0.272367			0.136184	+	0.990684i	$0.456516\pi$
$338$	−3.67423	−	6.36396i	−0.199852	−	0.346154i
$339$		0			0
$340$	12.0000	−	20.7846i	0.650791	−	1.12720i
$341$	17.1464	+	29.6985i	0.928531	+	1.60826i
$342$		0			0
$343$	10.0000	+	15.5885i	0.539949	+	0.841698i
$344$	−4.89898			−0.264135
$345$		0			0
$346$	−3.00000	+	5.19615i	−0.161281	+	0.279347i
$347$	4.89898	−	8.48528i	0.262991	−	0.455514i	−0.704044	−	0.710156i	$-0.748624\pi$
$347$	4.89898	−	8.48528i	0.262991	−	0.455514i	0.967035	+	0.254642i	$0.0819577\pi$
$348$		0			0
$349$	35.0000			1.87351			0.936754	−	0.349990i	$-0.113815\pi$
$349$	35.0000			1.87351			0.936754	+	0.349990i	$0.113815\pi$
$350$	−1.22474	−	6.36396i	−0.0654654	−	0.340168i
$351$		0			0
$352$		0			0
$353$	−2.44949	+	4.24264i	−0.130373	+	0.225813i	−0.923820	−	0.382826i	$-0.874951\pi$
$353$	−2.44949	+	4.24264i	−0.130373	+	0.225813i	0.793447	+	0.608639i	$0.208284\pi$
$354$		0			0
$355$		0			0
$356$	9.79796			0.519291
$357$		0			0
$358$	−6.00000			−0.317110
$359$	9.79796	+	16.9706i	0.517116	+	0.895672i	0.999802	+	0.0198785i	$0.00632794\pi$
$359$	9.79796	+	16.9706i	0.517116	+	0.895672i	−0.482686	+	0.875794i	$0.660339\pi$
$360$		0			0
$361$	9.00000	−	15.5885i	0.473684	−	0.820445i
$362$	8.57321	+	14.8492i	0.450598	+	0.780459i
$363$		0			0
$364$	−40.0000	−	13.8564i	−2.09657	−	0.726273i
$365$	−2.44949			−0.128212
$366$		0			0
$367$	6.50000	−	11.2583i	0.339297	−	0.587680i	−0.645003	−	0.764180i	$-0.723144\pi$
$367$	6.50000	−	11.2583i	0.339297	−	0.587680i	0.984301	+	0.176500i	$0.0564774\pi$
$368$	−4.89898	+	8.48528i	−0.255377	+	0.442326i
$369$		0			0
$370$	48.0000			2.49540
$371$	−6.12372	−	2.12132i	−0.317928	−	0.110133i
$372$		0			0
$373$	−5.50000	−	9.52628i	−0.284779	−	0.493252i	0.687776	−	0.725923i	$-0.258587\pi$
$373$	−5.50000	−	9.52628i	−0.284779	−	0.493252i	−0.972556	+	0.232671i	$0.925254\pi$
$374$	14.6969	−	25.4558i	0.759961	−	1.31629i
$375$		0			0
$376$	−6.00000	−	10.3923i	−0.309426	−	0.535942i
$377$	29.3939			1.51386
$378$		0			0
$379$	−34.0000			−1.74646			−0.873231	−	0.487306i	$-0.837980\pi$
$379$	−34.0000			−1.74646			−0.873231	+	0.487306i	$0.837980\pi$
$380$	4.89898	+	8.48528i	0.251312	+	0.435286i
$381$		0			0
$382$	−3.00000	+	5.19615i	−0.153493	+	0.265858i
$383$	9.79796	+	16.9706i	0.500652	+	0.867155i	1.00000		0.000753393i	$0.000239813\pi$
$383$	9.79796	+	16.9706i	0.500652	+	0.867155i	−0.499347	+	0.866402i	$0.666427\pi$
$384$		0			0
$385$	−6.00000	−	31.1769i	−0.305788	−	1.58892i
$386$	48.9898			2.49351
$387$		0			0
$388$	2.00000	−	3.46410i	0.101535	−	0.175863i
$389$	8.57321	−	14.8492i	0.434679	−	0.752886i	−0.562590	−	0.826736i	$-0.690195\pi$
$389$	8.57321	−	14.8492i	0.434679	−	0.752886i	0.997269	+	0.0738494i	$0.0235284\pi$
$390$		0			0
$391$	−6.00000			−0.303433
$392$	−31.8434	+	12.7279i	−1.60833	+	0.642857i
$393$		0			0
$394$		0			0
$395$	4.89898	−	8.48528i	0.246494	−	0.426941i
$396$		0			0
$397$	−8.50000	−	14.7224i	−0.426603	−	0.738898i	0.569966	−	0.821668i	$-0.306956\pi$
$397$	−8.50000	−	14.7224i	−0.426603	−	0.738898i	−0.996569	+	0.0827707i	$0.973623\pi$
$398$	−46.5403			−2.33285
$399$		0			0
$400$	4.00000			0.200000
$401$	−12.2474	−	21.2132i	−0.611608	−	1.05934i	−0.990969	−	0.134088i	$-0.957189\pi$
$401$	−12.2474	−	21.2132i	−0.611608	−	1.05934i	0.379361	−	0.925249i	$-0.376144\pi$
$402$		0			0
$403$	−14.0000	+	24.2487i	−0.697390	+	1.20791i
$404$	−9.79796	−	16.9706i	−0.487467	−	0.844317i
$405$		0			0
$406$	36.0000	−	31.1769i	1.78665	−	1.54728i
$407$	39.1918			1.94267
$408$		0			0
$409$	−10.0000	+	17.3205i	−0.494468	+	0.856444i	−0.999980	−	0.00637586i	$-0.997970\pi$
$409$	−10.0000	+	17.3205i	−0.494468	+	0.856444i	0.505511	+	0.862820i	$0.331304\pi$
$410$	−22.0454	+	38.1838i	−1.08875	+	1.88576i
$411$		0			0
$412$	8.00000			0.394132
$413$	−24.4949	−	8.48528i	−1.20532	−	0.417533i
$414$		0			0
$415$	−18.0000	−	31.1769i	−0.883585	−	1.53041i
$416$		0			0
$417$		0			0
$418$	6.00000	+	10.3923i	0.293470	+	0.508304i
$419$	7.34847			0.358996			0.179498	−	0.983758i	$-0.442553\pi$
$419$	7.34847			0.358996			0.179498	+	0.983758i	$0.442553\pi$
$420$		0			0
$421$	35.0000			1.70580			0.852898	−	0.522078i	$-0.174843\pi$
$421$	35.0000			1.70580			0.852898	+	0.522078i	$0.174843\pi$
$422$	−6.12372	−	10.6066i	−0.298098	−	0.516321i
$423$		0			0
$424$	6.00000	−	10.3923i	0.291386	−	0.504695i
$425$	1.22474	+	2.12132i	0.0594089	+	0.102899i
$426$		0			0
$427$	−10.0000	+	8.66025i	−0.483934	+	0.419099i
$428$	−78.3837			−3.78882
$429$		0			0
$430$	3.00000	−	5.19615i	0.144673	−	0.250581i
$431$	−9.79796	+	16.9706i	−0.471951	+	0.817443i	−0.999485	−	0.0320907i	$-0.989783\pi$
$431$	−9.79796	+	16.9706i	−0.471951	+	0.817443i	0.527534	+	0.849534i	$0.323117\pi$
$432$		0			0
$433$	−7.00000			−0.336399			−0.168199	−	0.985753i	$-0.553795\pi$
$433$	−7.00000			−0.336399			−0.168199	+	0.985753i	$0.553795\pi$
$434$	8.57321	+	44.5477i	0.411527	+	2.13836i
$435$		0			0
$436$	2.00000	+	3.46410i	0.0957826	+	0.165900i
$437$	1.22474	−	2.12132i	0.0585875	−	0.101477i
$438$		0			0
$439$	−7.00000	−	12.1244i	−0.334092	−	0.578664i	0.649218	−	0.760602i	$-0.275096\pi$
$439$	−7.00000	−	12.1244i	−0.334092	−	0.578664i	−0.983310	+	0.181938i	$0.941763\pi$
$440$	58.7878			2.80260
$441$		0			0
$442$	24.0000			1.14156
$443$	−1.22474	−	2.12132i	−0.0581894	−	0.100787i	0.835463	−	0.549546i	$-0.185199\pi$
$443$	−1.22474	−	2.12132i	−0.0581894	−	0.100787i	−0.893653	+	0.448759i	$0.851866\pi$
$444$		0			0
$445$	−3.00000	+	5.19615i	−0.142214	+	0.246321i
$446$	−31.8434	−	55.1543i	−1.50783	−	2.61163i
$447$		0			0
$448$	4.00000	+	20.7846i	0.188982	+	0.981981i
$449$	22.0454			1.04039			0.520194	−	0.854048i	$-0.325860\pi$
$449$	22.0454			1.04039			0.520194	+	0.854048i	$0.325860\pi$
$450$		0			0
$451$	−18.0000	+	31.1769i	−0.847587	+	1.46806i
$452$	29.3939	−	50.9117i	1.38257	−	2.39468i
$453$		0			0
$454$	−42.0000			−1.97116
$455$	19.5959	−	16.9706i	0.918671	−	0.795592i
$456$		0			0
$457$	−11.5000	−	19.9186i	−0.537947	−	0.931752i	−0.999014	−	0.0443868i	$-0.985867\pi$
$457$	−11.5000	−	19.9186i	−0.537947	−	0.931752i	0.461067	−	0.887365i	$-0.347467\pi$
$458$	8.57321	−	14.8492i	0.400600	−	0.693860i
$459$		0			0
$460$	−12.0000	−	20.7846i	−0.559503	−	0.969087i
$461$	−29.3939			−1.36901			−0.684505	−	0.729008i	$-0.739981\pi$
$461$	−29.3939			−1.36901			−0.684505	+	0.729008i	$0.739981\pi$
$462$		0			0
$463$	23.0000			1.06890			0.534450	−	0.845200i	$-0.320519\pi$
$463$	23.0000			1.06890			0.534450	+	0.845200i	$0.320519\pi$
$464$	14.6969	+	25.4558i	0.682288	+	1.18176i
$465$		0			0
$466$	−3.00000	+	5.19615i	−0.138972	+	0.240707i
$467$	−12.2474	−	21.2132i	−0.566744	−	0.981630i	−0.996885	−	0.0788681i	$-0.974869\pi$
$467$	−12.2474	−	21.2132i	−0.566744	−	0.981630i	0.430141	−	0.902762i	$-0.358464\pi$
$468$		0			0
$469$	5.00000	+	1.73205i	0.230879	+	0.0799787i
$470$	14.6969			0.677919
$471$		0			0
$472$	24.0000	−	41.5692i	1.10469	−	1.91338i
$473$	2.44949	−	4.24264i	0.112628	−	0.195077i
$474$		0			0
$475$	−1.00000			−0.0458831
$476$	19.5959	−	16.9706i	0.898177	−	0.777844i
$477$		0			0
$478$	−9.00000	−	15.5885i	−0.411650	−	0.712999i
$479$	−13.4722	+	23.3345i	−0.615560	+	1.06618i	0.374726	+	0.927136i	$0.377737\pi$
$479$	−13.4722	+	23.3345i	−0.615560	+	1.06618i	−0.990286	+	0.139046i	$0.955596\pi$
$480$		0			0
$481$	16.0000	+	27.7128i	0.729537	+	1.26360i
$482$	−31.8434			−1.45043
$483$		0			0
$484$	52.0000			2.36364
$485$	1.22474	+	2.12132i	0.0556128	+	0.0963242i
$486$		0			0
$487$	9.50000	−	16.4545i	0.430486	−	0.745624i	−0.566429	−	0.824110i	$-0.691675\pi$
$487$	9.50000	−	16.4545i	0.430486	−	0.745624i	0.996915	+	0.0784867i	$0.0250088\pi$
$488$	−12.2474	−	21.2132i	−0.554416	−	0.960277i
$489$		0			0
$490$	6.00000	−	41.5692i	0.271052	−	1.87791i
$491$	−14.6969			−0.663264			−0.331632	−	0.943409i	$-0.607599\pi$
$491$	−14.6969			−0.663264			−0.331632	+	0.943409i	$0.607599\pi$
$492$		0			0
$493$	−9.00000	+	15.5885i	−0.405340	+	0.702069i
$494$	−4.89898	+	8.48528i	−0.220416	+	0.381771i
$495$		0			0
$496$	−28.0000			−1.25724
$497$		0			0
$498$		0			0
$499$	3.50000	+	6.06218i	0.156682	+	0.271380i	0.933670	−	0.358134i	$-0.116587\pi$
$499$	3.50000	+	6.06218i	0.156682	+	0.271380i	−0.776989	+	0.629515i	$0.783254\pi$
$500$	19.5959	−	33.9411i	0.876356	−	1.51789i
$501$		0			0
$502$	27.0000	+	46.7654i	1.20507	+	2.08724i
$503$	−22.0454			−0.982956			−0.491478	−	0.870890i	$-0.663543\pi$
$503$	−22.0454			−0.982956			−0.491478	+	0.870890i	$0.663543\pi$
$504$		0			0
$505$	12.0000			0.533993
$506$	−14.6969	−	25.4558i	−0.653359	−	1.13165i
$507$		0			0
$508$	−22.0000	+	38.1051i	−0.976092	+	1.69064i
$509$	9.79796	+	16.9706i	0.434287	+	0.752207i	0.997237	−	0.0742838i	$-0.0236671\pi$
$509$	9.79796	+	16.9706i	0.434287	+	0.752207i	−0.562950	+	0.826491i	$0.690334\pi$
$510$		0			0
$511$	−2.50000	−	0.866025i	−0.110593	−	0.0383107i
$512$	−39.1918			−1.73205
$513$		0			0
$514$	−30.0000	+	51.9615i	−1.32324	+	2.29192i
$515$	−2.44949	+	4.24264i	−0.107937	+	0.186953i
$516$		0			0
$517$	12.0000			0.527759
$518$	48.9898	+	16.9706i	2.15249	+	0.745644i
$519$		0			0
$520$	24.0000	+	41.5692i	1.05247	+	1.82293i
$521$	−9.79796	+	16.9706i	−0.429256	+	0.743494i	−0.996807	−	0.0798444i	$-0.974558\pi$
$521$	−9.79796	+	16.9706i	−0.429256	+	0.743494i	0.567551	+	0.823338i	$0.307891\pi$
$522$		0			0
$523$	−4.00000	−	6.92820i	−0.174908	−	0.302949i	0.765222	−	0.643767i	$-0.222629\pi$
$523$	−4.00000	−	6.92820i	−0.174908	−	0.302949i	−0.940129	+	0.340818i	$0.889296\pi$
$524$	9.79796			0.428026
$525$		0			0
$526$	12.0000			0.523225
$527$	−8.57321	−	14.8492i	−0.373455	−	0.646843i
$528$		0			0
$529$	8.50000	−	14.7224i	0.369565	−	0.640106i
$530$	7.34847	+	12.7279i	0.319197	+	0.552866i
$531$		0			0
$532$	2.00000	+	10.3923i	0.0867110	+	0.450564i
$533$	−29.3939			−1.27319
$534$		0			0
$535$	24.0000	−	41.5692i	1.03761	−	1.79719i
$536$	−4.89898	+	8.48528i	−0.211604	+	0.366508i
$537$		0			0
$538$	−60.0000			−2.58678
$539$	4.89898	−	33.9411i	0.211014	−	1.46195i
$540$		0			0
$541$	−17.5000	−	30.3109i	−0.752384	−	1.30317i	−0.946664	−	0.322221i	$-0.895571\pi$
$541$	−17.5000	−	30.3109i	−0.752384	−	1.30317i	0.194281	−	0.980946i	$-0.437763\pi$
$542$	1.22474	−	2.12132i	0.0526073	−	0.0911185i
$543$		0			0
$544$		0			0
$545$	−2.44949			−0.104925
$546$		0			0
$547$	−37.0000			−1.58201			−0.791003	−	0.611812i	$-0.790441\pi$
$547$	−37.0000			−1.58201			−0.791003	+	0.611812i	$0.790441\pi$
$548$	34.2929	+	59.3970i	1.46492	+	2.53731i
$549$		0			0
$550$	−6.00000	+	10.3923i	−0.255841	+	0.443129i
$551$	−3.67423	−	6.36396i	−0.156528	−	0.271114i
$552$		0			0
$553$	8.00000	−	6.92820i	0.340195	−	0.294617i
$554$	56.3383			2.39358
$555$		0			0
$556$	−28.0000	+	48.4974i	−1.18746	+	2.05675i
$557$	−20.8207	+	36.0624i	−0.882200	+	1.52801i	−0.0333100	+	0.999445i	$0.510605\pi$
$557$	−20.8207	+	36.0624i	−0.882200	+	1.52801i	−0.848890	+	0.528570i	$0.822728\pi$
$558$		0			0
$559$	4.00000			0.169182
$560$	24.4949	+	8.48528i	1.03510	+	0.358569i
$561$		0			0
$562$	−18.0000	−	31.1769i	−0.759284	−	1.31512i
$563$	−6.12372	+	10.6066i	−0.258084	+	0.447015i	−0.965729	−	0.259554i	$-0.916425\pi$
$563$	−6.12372	+	10.6066i	−0.258084	+	0.447015i	0.707644	+	0.706569i	$0.249758\pi$
$564$		0			0
$565$	18.0000	+	31.1769i	0.757266	+	1.31162i
$566$	−61.2372			−2.57399
$567$		0			0
$568$		0			0
$569$	9.79796	+	16.9706i	0.410752	+	0.711443i	0.994972	−	0.100152i	$-0.0319329\pi$
$569$	9.79796	+	16.9706i	0.410752	+	0.711443i	−0.584220	+	0.811595i	$0.698600\pi$
$570$		0			0
$571$	−14.5000	+	25.1147i	−0.606806	+	1.05102i	0.384957	+	0.922934i	$0.374216\pi$
$571$	−14.5000	+	25.1147i	−0.606806	+	1.05102i	−0.991763	+	0.128085i	$0.959117\pi$
$572$	39.1918	+	67.8823i	1.63869	+	2.83830i
$573$		0			0
$574$	−36.0000	+	31.1769i	−1.50261	+	1.30130i
$575$	2.44949			0.102151
$576$		0			0
$577$	−4.00000	+	6.92820i	−0.166522	+	0.288425i	−0.937195	−	0.348806i	$-0.886587\pi$
$577$	−4.00000	+	6.92820i	−0.166522	+	0.288425i	0.770673	+	0.637231i	$0.219920\pi$
$578$	13.4722	−	23.3345i	0.560369	−	0.970588i
$579$		0			0
$580$	−72.0000			−2.98964
$581$	−7.34847	−	38.1838i	−0.304866	−	1.58413i
$582$		0			0
$583$	6.00000	+	10.3923i	0.248495	+	0.430405i
$584$	2.44949	−	4.24264i	0.101361	−	0.175562i
$585$		0			0
$586$	−36.0000	−	62.3538i	−1.48715	−	2.57581i
$587$	−7.34847			−0.303304			−0.151652	−	0.988434i	$-0.548459\pi$
$587$	−7.34847			−0.303304			−0.151652	+	0.988434i	$0.548459\pi$
$588$		0			0
$589$	7.00000			0.288430
$590$	29.3939	+	50.9117i	1.21013	+	2.09600i
$591$		0			0
$592$	−16.0000	+	27.7128i	−0.657596	+	1.13899i
$593$	−12.2474	−	21.2132i	−0.502942	−	0.871122i	−0.999994	−	0.00340097i	$-0.998917\pi$
$593$	−12.2474	−	21.2132i	−0.502942	−	0.871122i	0.497052	−	0.867721i	$-0.334416\pi$
$594$		0			0
$595$	3.00000	+	15.5885i	0.122988	+	0.639064i
$596$	9.79796			0.401340
$597$		0			0
$598$	12.0000	−	20.7846i	0.490716	−	0.849946i
$599$	15.9217	−	27.5772i	0.650542	−	1.12677i	−0.332449	−	0.943121i	$-0.607875\pi$
$599$	15.9217	−	27.5772i	0.650542	−	1.12677i	0.982991	−	0.183651i	$-0.0587917\pi$
$600$		0			0
$601$	17.0000			0.693444			0.346722	−	0.937968i	$-0.387295\pi$
$601$	17.0000			0.693444			0.346722	+	0.937968i	$0.387295\pi$
$602$	4.89898	−	4.24264i	0.199667	−	0.172917i
$603$		0			0
$604$	−10.0000	−	17.3205i	−0.406894	−	0.704761i
$605$	−15.9217	+	27.5772i	−0.647308	+	1.12117i
$606$		0			0
$607$	−14.5000	−	25.1147i	−0.588537	−	1.01938i	−0.994424	−	0.105453i	$-0.966371\pi$
$607$	−14.5000	−	25.1147i	−0.588537	−	1.01938i	0.405887	−	0.913923i	$-0.366962\pi$
$608$		0			0
$609$		0			0
$610$	30.0000			1.21466
$611$	4.89898	+	8.48528i	0.198191	+	0.343278i
$612$		0			0
$613$	−8.50000	+	14.7224i	−0.343312	+	0.594633i	−0.985046	−	0.172294i	$-0.944882\pi$
$613$	−8.50000	+	14.7224i	−0.343312	+	0.594633i	0.641734	+	0.766927i	$0.278215\pi$
$614$	30.6186	+	53.0330i	1.23567	+	2.14024i
$615$		0			0
$616$	60.0000	+	20.7846i	2.41747	+	0.837436i
$617$	29.3939			1.18335			0.591676	−	0.806176i	$-0.298466\pi$
$617$	29.3939			1.18335			0.591676	+	0.806176i	$0.298466\pi$
$618$		0			0
$619$	−7.00000	+	12.1244i	−0.281354	+	0.487319i	−0.971718	−	0.236143i	$-0.924117\pi$
$619$	−7.00000	+	12.1244i	−0.281354	+	0.487319i	0.690365	+	0.723462i	$0.257450\pi$
$620$	34.2929	−	59.3970i	1.37723	−	2.38544i
$621$		0			0
$622$	84.0000			3.36809
$623$	−4.89898	+	4.24264i	−0.196273	+	0.169978i
$624$		0			0
$625$	14.5000	+	25.1147i	0.580000	+	1.00459i
$626$	−28.1691	+	48.7904i	−1.12586	+	1.95006i
$627$		0			0
$628$	−40.0000	−	69.2820i	−1.59617	−	2.76465i
$629$	−19.5959			−0.781340
$630$		0			0
$631$	−25.0000			−0.995234			−0.497617	−	0.867397i	$-0.665792\pi$
$631$	−25.0000			−0.995234			−0.497617	+	0.867397i	$0.665792\pi$
$632$	9.79796	+	16.9706i	0.389742	+	0.675053i
$633$		0			0
$634$	−30.0000	+	51.9615i	−1.19145	+	2.06366i
$635$	−13.4722	−	23.3345i	−0.534628	−	0.926002i
$636$		0			0
$637$	26.0000	−	10.3923i	1.03016	−	0.411758i
$638$	−88.1816			−3.49114
$639$		0			0
$640$	24.0000	−	41.5692i	0.948683	−	1.64317i
$641$	−13.4722	+	23.3345i	−0.532120	+	0.921658i	0.467177	+	0.884164i	$0.345271\pi$
$641$	−13.4722	+	23.3345i	−0.532120	+	0.921658i	−0.999297	+	0.0374946i	$0.988062\pi$
$642$		0			0
$643$	5.00000			0.197181			0.0985904	−	0.995128i	$-0.468567\pi$
$643$	5.00000			0.197181			0.0985904	+	0.995128i	$0.468567\pi$
$644$	−4.89898	−	25.4558i	−0.193047	−	1.00310i
$645$		0			0
$646$	−3.00000	−	5.19615i	−0.118033	−	0.204440i
$647$	−9.79796	+	16.9706i	−0.385198	+	0.667182i	−0.991797	−	0.127826i	$-0.959200\pi$
$647$	−9.79796	+	16.9706i	−0.385198	+	0.667182i	0.606599	+	0.795008i	$0.292533\pi$
$648$		0			0
$649$	24.0000	+	41.5692i	0.942082	+	1.63173i
$650$	−9.79796			−0.384308
$651$		0			0
$652$	−4.00000			−0.156652
$653$	−1.22474	−	2.12132i	−0.0479280	−	0.0830137i	0.841066	−	0.540932i	$-0.181928\pi$
$653$	−1.22474	−	2.12132i	−0.0479280	−	0.0830137i	−0.888994	+	0.457919i	$0.848595\pi$
$654$		0			0
$655$	−3.00000	+	5.19615i	−0.117220	+	0.203030i
$656$	−14.6969	−	25.4558i	−0.573819	−	0.993884i
$657$		0			0
$658$	15.0000	+	5.19615i	0.584761	+	0.202567i
$659$	−7.34847			−0.286256			−0.143128	−	0.989704i	$-0.545716\pi$
$659$	−7.34847			−0.286256			−0.143128	+	0.989704i	$0.545716\pi$
$660$		0			0
$661$	−5.50000	+	9.52628i	−0.213925	+	0.370529i	−0.952940	−	0.303160i	$-0.901958\pi$
$661$	−5.50000	+	9.52628i	−0.213925	+	0.370529i	0.739014	+	0.673690i	$0.235292\pi$
$662$	15.9217	−	27.5772i	0.618814	−	1.07182i
$663$		0			0
$664$	72.0000			2.79414
$665$	−6.12372	−	2.12132i	−0.237468	−	0.0822613i
$666$		0			0
$667$	9.00000	+	15.5885i	0.348481	+	0.603587i
$668$	29.3939	−	50.9117i	1.13728	−	1.96983i
$669$		0			0
$670$	−6.00000	−	10.3923i	−0.231800	−	0.401490i
$671$	24.4949			0.945615
$672$		0			0
$673$	17.0000			0.655302			0.327651	−	0.944799i	$-0.393743\pi$
$673$	17.0000			0.655302			0.327651	+	0.944799i	$0.393743\pi$
$674$	−6.12372	−	10.6066i	−0.235877	−	0.408551i
$675$		0			0
$676$	−6.00000	+	10.3923i	−0.230769	+	0.399704i
$677$	9.79796	+	16.9706i	0.376566	+	0.652232i	0.990560	−	0.137079i	$-0.0437714\pi$
$677$	9.79796	+	16.9706i	0.376566	+	0.652232i	−0.613994	+	0.789311i	$0.710438\pi$
$678$		0			0
$679$	0.500000	+	2.59808i	0.0191882	+	0.0997050i
$680$	−29.3939			−1.12720
$681$		0			0
$682$	42.0000	−	72.7461i	1.60826	−	2.78559i
$683$	12.2474	−	21.2132i	0.468636	−	0.811701i	−0.530722	−	0.847546i	$-0.678079\pi$
$683$	12.2474	−	21.2132i	0.468636	−	0.811701i	0.999357	+	0.0358455i	$0.0114124\pi$
$684$		0			0
$685$	−42.0000			−1.60474
$686$	20.8207	−	40.3051i	0.794937	−	1.53886i
$687$		0			0
$688$	2.00000	+	3.46410i	0.0762493	+	0.132068i
$689$	−4.89898	+	8.48528i	−0.186636	+	0.323263i
$690$		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	$-0.451375\pi$
$692$	9.79796			0.372463
$693$		0			0
$694$	−24.0000			−0.911028
$695$	−17.1464	−	29.6985i	−0.650401	−	1.12653i
$696$		0			0
$697$	9.00000	−	15.5885i	0.340899	−	0.590455i
$698$	−42.8661	−	74.2462i	−1.62250	−	2.81026i
$699$		0			0
$700$	−8.00000	+	6.92820i	−0.302372	+	0.261861i
$701$	22.0454			0.832644			0.416322	−	0.909217i	$-0.363319\pi$
$701$	22.0454			0.832644			0.416322	+	0.909217i	$0.363319\pi$
$702$		0			0
$703$	4.00000	−	6.92820i	0.150863	−	0.261302i
$704$	19.5959	−	33.9411i	0.738549	−	1.27920i
$705$		0			0
$706$	12.0000			0.451626
$707$	12.2474	+	4.24264i	0.460613	+	0.159561i
$708$		0			0
$709$	24.5000	+	42.4352i	0.920117	+	1.59369i	0.799232	+	0.601023i	$0.205240\pi$
$709$	24.5000	+	42.4352i	0.920117	+	1.59369i	0.120885	+	0.992667i	$0.461427\pi$
$710$		0			0
$711$		0			0
$712$	−6.00000	−	10.3923i	−0.224860	−	0.389468i
$713$	−17.1464			−0.642139
$714$		0			0
$715$	−48.0000			−1.79510
$716$	4.89898	+	8.48528i	0.183083	+	0.317110i
$717$		0			0
$718$	24.0000	−	41.5692i	0.895672	−	1.55135i
$719$	9.79796	+	16.9706i	0.365402	+	0.632895i	0.988841	−	0.148977i	$-0.0475981\pi$
$719$	9.79796	+	16.9706i	0.365402	+	0.632895i	−0.623438	+	0.781872i	$0.714265\pi$
$720$		0			0
$721$	−4.00000	+	3.46410i	−0.148968	+	0.129010i
$722$	−44.0908			−1.64089
$723$		0			0
$724$	14.0000	−	24.2487i	0.520306	−	0.901196i
$725$	3.67423	−	6.36396i	0.136458	−	0.236352i
$726$		0			0
$727$	−19.0000			−0.704671			−0.352335	−	0.935874i	$-0.614612\pi$
$727$	−19.0000			−0.704671			−0.352335	+	0.935874i	$0.614612\pi$
$728$	9.79796	+	50.9117i	0.363137	+	1.88691i
$729$		0			0
$730$	3.00000	+	5.19615i	0.111035	+	0.192318i
$731$	−1.22474	+	2.12132i	−0.0452988	+	0.0784599i
$732$		0			0
$733$	−5.50000	−	9.52628i	−0.203147	−	0.351861i	0.746394	−	0.665505i	$-0.231784\pi$
$733$	−5.50000	−	9.52628i	−0.203147	−	0.351861i	−0.949541	+	0.313644i	$0.898450\pi$
$734$	−31.8434			−1.17536
$735$		0			0
$736$		0			0
$737$	−4.89898	−	8.48528i	−0.180456	−	0.312559i
$738$		0			0
$739$	9.50000	−	16.4545i	0.349463	−	0.605288i	−0.636691	−	0.771119i	$-0.719697\pi$
$739$	9.50000	−	16.4545i	0.349463	−	0.605288i	0.986154	+	0.165831i	$0.0530307\pi$
$740$	−39.1918	−	67.8823i	−1.44072	−	2.49540i
$741$		0			0
$742$	3.00000	+	15.5885i	0.110133	+	0.572270i
$743$	7.34847			0.269589			0.134795	−	0.990874i	$-0.456963\pi$
$743$	7.34847			0.269589			0.134795	+	0.990874i	$0.456963\pi$
$744$		0			0
$745$	−3.00000	+	5.19615i	−0.109911	+	0.190372i
$746$	−13.4722	+	23.3345i	−0.493252	+	0.854338i
$747$		0			0
$748$	−48.0000			−1.75505
$749$	39.1918	−	33.9411i	1.43204	−	1.24018i
$750$		0			0
$751$	21.5000	+	37.2391i	0.784546	+	1.35887i	0.929270	+	0.369402i	$0.120437\pi$
$751$	21.5000	+	37.2391i	0.784546	+	1.35887i	−0.144724	+	0.989472i	$0.546229\pi$
$752$	−4.89898	+	8.48528i	−0.178647	+	0.309426i
$753$		0			0
$754$	−36.0000	−	62.3538i	−1.31104	−	2.27079i
$755$	12.2474			0.445730
$756$		0			0
$757$	47.0000			1.70824			0.854122	−	0.520073i	$-0.174095\pi$
$757$	47.0000			1.70824			0.854122	+	0.520073i	$0.174095\pi$
$758$	41.6413	+	72.1249i	1.51248	+	2.61969i
$759$		0			0
$760$	6.00000	−	10.3923i	0.217643	−	0.376969i
$761$	9.79796	+	16.9706i	0.355176	+	0.615182i	0.987148	−	0.159809i	$-0.0510877\pi$
$761$	9.79796	+	16.9706i	0.355176	+	0.615182i	−0.631972	+	0.774991i	$0.717754\pi$
$762$		0			0
$763$	−2.50000	−	0.866025i	−0.0905061	−	0.0313522i
$764$	9.79796			0.354478
$765$		0			0
$766$	24.0000	−	41.5692i	0.867155	−	1.50196i
$767$	−19.5959	+	33.9411i	−0.707568	+	1.22554i
$768$		0			0
$769$	5.00000			0.180305			0.0901523	−	0.995928i	$-0.471265\pi$
$769$	5.00000			0.180305			0.0901523	+	0.995928i	$0.471265\pi$
$770$	−58.7878	+	50.9117i	−2.11856	+	1.83473i
$771$		0			0
$772$	−40.0000	−	69.2820i	−1.43963	−	2.49351i
$773$	23.2702	−	40.3051i	0.836969	−	1.44967i	−0.0554478	−	0.998462i	$-0.517659\pi$
$773$	23.2702	−	40.3051i	0.836969	−	1.44967i	0.892417	−	0.451212i	$-0.149008\pi$
$774$		0			0
$775$	3.50000	+	6.06218i	0.125724	+	0.217760i
$776$	−4.89898			−0.175863
$777$		0			0
$778$	−42.0000			−1.50577
$779$	3.67423	+	6.36396i	0.131643	+	0.228013i
$780$		0			0
$781$		0			0
$782$	7.34847	+	12.7279i	0.262781	+	0.455150i
$783$		0			0
$784$	22.0000	+	17.3205i	0.785714	+	0.618590i
$785$	48.9898			1.74852
$786$		0			0
$787$	3.50000	−	6.06218i	0.124762	−	0.216093i	−0.796878	−	0.604140i	$-0.793517\pi$
$787$	3.50000	−	6.06218i	0.124762	−	0.216093i	0.921640	+	0.388047i	$0.126850\pi$
$788$		0			0
$789$		0			0
$790$	−24.0000			−0.853882
$791$	7.34847	+	38.1838i	0.261281	+	1.35766i
$792$		0			0
$793$	10.0000	+	17.3205i	0.355110	+	0.615069i
$794$	−20.8207	+	36.0624i	−0.738898	+	1.27981i
$795$		0			0
$796$	38.0000	+	65.8179i	1.34687	+	2.33285i
$797$	−14.6969			−0.520592			−0.260296	−	0.965529i	$-0.583820\pi$
$797$	−14.6969			−0.520592			−0.260296	+	0.965529i	$0.583820\pi$
$798$		0			0
$799$	−6.00000			−0.212265
$800$		0			0
$801$		0			0
$802$	−30.0000	+	51.9615i	−1.05934	+	1.83483i
$803$	2.44949	+	4.24264i	0.0864406	+	0.149720i
$804$		0			0
$805$	15.0000	+	5.19615i	0.528681	+	0.183140i
$806$	68.5857			2.41583
$807$		0			0
$808$	−12.0000	+	20.7846i	−0.422159	+	0.731200i
$809$	12.2474	−	21.2132i	0.430597	−	0.745817i	−0.566327	−	0.824180i	$-0.691636\pi$
$809$	12.2474	−	21.2132i	0.430597	−	0.745817i	0.996925	+	0.0783638i	$0.0249696\pi$
$810$		0			0
$811$	2.00000			0.0702295			0.0351147	−	0.999383i	$-0.488820\pi$
$811$	2.00000			0.0702295			0.0351147	+	0.999383i	$0.488820\pi$
$812$	−73.4847	−	25.4558i	−2.57881	−	0.893325i
$813$		0			0
$814$	−48.0000	−	83.1384i	−1.68240	−	2.91400i
$815$	1.22474	−	2.12132i	0.0429009	−	0.0743066i
$816$		0			0
$817$	−0.500000	−	0.866025i	−0.0174928	−	0.0302984i
$818$	48.9898			1.71289
$819$		0			0
$820$	72.0000			2.51435
$821$	9.79796	+	16.9706i	0.341951	+	0.592277i	0.984795	−	0.173721i	$-0.0555789\pi$
$821$	9.79796	+	16.9706i	0.341951	+	0.592277i	−0.642844	+	0.765997i	$0.722246\pi$
$822$		0			0
$823$	12.5000	−	21.6506i	0.435723	−	0.754694i	−0.561632	−	0.827387i	$-0.689826\pi$
$823$	12.5000	−	21.6506i	0.435723	−	0.754694i	0.997354	+	0.0726937i	$0.0231595\pi$
$824$	−4.89898	−	8.48528i	−0.170664	−	0.295599i
$825$		0			0
$826$	12.0000	+	62.3538i	0.417533	+	2.16957i
$827$	44.0908			1.53319			0.766594	−	0.642132i	$-0.221950\pi$
$827$	44.0908			1.53319			0.766594	+	0.642132i	$0.221950\pi$
$828$		0			0
$829$	27.5000	−	47.6314i	0.955114	−	1.65431i	0.221009	−	0.975272i	$-0.429065\pi$
$829$	27.5000	−	47.6314i	0.955114	−	1.65431i	0.734106	−	0.679035i	$-0.237602\pi$
$830$	−44.0908	+	76.3675i	−1.53041	+	2.65076i
$831$		0			0
$832$	32.0000			1.10940
$833$	−2.44949	+	16.9706i	−0.0848698	+	0.587995i
$834$		0			0
$835$	18.0000	+	31.1769i	0.622916	+	1.07892i
$836$	9.79796	−	16.9706i	0.338869	−	0.586939i
$837$		0			0
$838$	−9.00000	−	15.5885i	−0.310900	−	0.538494i
$839$	14.6969			0.507395			0.253697	−	0.967284i	$-0.418353\pi$
$839$	14.6969			0.507395			0.253697	+	0.967284i	$0.418353\pi$
$840$		0			0
$841$	25.0000			0.862069
$842$	−42.8661	−	74.2462i	−1.47726	−	2.55869i
$843$		0			0
$844$	−10.0000	+	17.3205i	−0.344214	+	0.596196i
$845$	−3.67423	−	6.36396i	−0.126398	−	0.218927i
$846$		0			0
$847$	−26.0000	+	22.5167i	−0.893371	+	0.773682i
$848$	−9.79796			−0.336463
$849$		0			0
$850$	3.00000	−	5.19615i	0.102899	−	0.178227i
$851$	−9.79796	+	16.9706i	−0.335870	+	0.581743i
$852$		0			0
$853$	−19.0000			−0.650548			−0.325274	−	0.945620i	$-0.605456\pi$
$853$	−19.0000			−0.650548			−0.325274	+	0.945620i	$0.605456\pi$
$854$	30.6186	+	10.6066i	1.04775	+	0.362950i
$855$		0			0
$856$	48.0000	+	83.1384i	1.64061	+	2.84161i
$857$	19.5959	−	33.9411i	0.669384	−	1.15941i	−0.308693	−	0.951162i	$-0.599892\pi$
$857$	19.5959	−	33.9411i	0.669384	−	1.15941i	0.978077	−	0.208245i	$-0.0667751\pi$
$858$		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	$0.693082\pi$
$860$	−9.79796			−0.334108
$861$		0			0
$862$	48.0000			1.63489
$863$	−1.22474	−	2.12132i	−0.0416908	−	0.0722106i	0.844427	−	0.535671i	$-0.179941\pi$
$863$	−1.22474	−	2.12132i	−0.0416908	−	0.0722106i	−0.886118	+	0.463460i	$0.846608\pi$
$864$		0			0
$865$	−3.00000	+	5.19615i	−0.102003	+	0.176674i
$866$	8.57321	+	14.8492i	0.291330	+	0.504598i
$867$		0			0
$868$	56.0000	−	48.4974i	1.90076	−	1.64611i
$869$	−19.5959			−0.664746
$870$		0			0
$871$	4.00000	−	6.92820i	0.135535	−	0.234753i
$872$	2.44949	−	4.24264i	0.0829502	−	0.143674i
$873$		0			0
$874$	−6.00000			−0.202953
$875$	4.89898	+	25.4558i	0.165616	+	0.860565i
$876$		0			0
$877$	−23.5000	−	40.7032i	−0.793539	−	1.37445i	−0.923763	−	0.382965i	$-0.874903\pi$
$877$	−23.5000	−	40.7032i	−0.793539	−	1.37445i	0.130224	−	0.991485i	$-0.458430\pi$
$878$	−17.1464	+	29.6985i	−0.578664	+	1.00228i
$879$		0			0
$880$	−24.0000	−	41.5692i	−0.809040	−	1.40130i
$881$	−22.0454			−0.742729			−0.371364	−	0.928487i	$-0.621110\pi$
$881$	−22.0454			−0.742729			−0.371364	+	0.928487i	$0.621110\pi$
$882$		0			0
$883$	−7.00000			−0.235569			−0.117784	−	0.993039i	$-0.537579\pi$
$883$	−7.00000			−0.235569			−0.117784	+	0.993039i	$0.537579\pi$
$884$	−19.5959	−	33.9411i	−0.659082	−	1.14156i
$885$		0			0
$886$	−3.00000	+	5.19615i	−0.100787	+	0.174568i
$887$	9.79796	+	16.9706i	0.328983	+	0.569816i	0.982310	−	0.187260i	$-0.0599607\pi$
$887$	9.79796	+	16.9706i	0.328983	+	0.569816i	−0.653327	+	0.757076i	$0.726627\pi$
$888$		0			0
$889$	−5.50000	−	28.5788i	−0.184464	−	0.958503i
$890$	14.6969			0.492642
$891$		0			0
$892$	−52.0000	+	90.0666i	−1.74109	+	3.01565i
$893$	1.22474	−	2.12132i	0.0409845	−	0.0709873i
$894$		0			0
$895$	−6.00000			−0.200558
$896$	39.1918	−	33.9411i	1.30931	−	1.13389i
$897$		0			0
$898$	−27.0000	−	46.7654i	−0.901002	−	1.56058i
$899$	−25.7196	+	44.5477i	−0.857798	+	1.48575i
$900$		0			0
$901$	−3.00000	−	5.19615i	−0.0999445	−	0.173109i
$902$	88.1816			2.93613
$903$		0			0
$904$	−72.0000			−2.39468
$905$	8.57321	+	14.8492i	0.284983	+	0.493606i
$906$		0			0
$907$	−16.0000	+	27.7128i	−0.531271	+	0.920189i	0.468063	+	0.883695i	$0.344952\pi$
$907$	−16.0000	+	27.7128i	−0.531271	+	0.920189i	−0.999334	+	0.0364935i	$0.988381\pi$
$908$	34.2929	+	59.3970i	1.13805	+	1.97116i
$909$		0			0
$910$	−60.0000	−	20.7846i	−1.98898	−	0.689003i
$911$	−7.34847			−0.243466			−0.121733	−	0.992563i	$-0.538845\pi$
$911$	−7.34847			−0.243466			−0.121733	+	0.992563i	$0.538845\pi$
$912$		0			0
$913$	−36.0000	+	62.3538i	−1.19143	+	2.06361i
$914$	−28.1691	+	48.7904i	−0.931752	+	1.61384i
$915$		0			0
$916$	−28.0000			−0.925146
$917$	−4.89898	+	4.24264i	−0.161779	+	0.140104i
$918$		0			0
$919$	0.500000	+	0.866025i	0.0164935	+	0.0285675i	0.874154	−	0.485648i	$-0.161416\pi$
$919$	0.500000	+	0.866025i	0.0164935	+	0.0285675i	−0.857661	+	0.514216i	$0.828083\pi$
$920$	−14.6969	+	25.4558i	−0.484544	+	0.839254i
$921$		0			0
$922$	36.0000	+	62.3538i	1.18560	+	2.05351i
$923$		0			0
$924$		0			0
$925$	8.00000			0.263038
$926$	−28.1691	−	48.7904i	−0.925695	−	1.60335i
$927$		0			0
$928$		0			0
$929$	−23.2702	−	40.3051i	−0.763469	−	1.32237i	−0.941052	−	0.338261i	$-0.890161\pi$
$929$	−23.2702	−	40.3051i	−0.763469	−	1.32237i	0.177583	−	0.984106i	$-0.443172\pi$
$930$		0			0
$931$	−5.50000	−	4.33013i	−0.180255	−	0.141914i
$932$	9.79796			0.320943
$933$		0			0
$934$	−30.0000	+	51.9615i	−0.981630	+	1.70023i
$935$	14.6969	−	25.4558i	0.480641	−	0.832495i
$936$		0			0
$937$	−16.0000			−0.522697			−0.261349	−	0.965244i	$-0.584167\pi$
$937$	−16.0000			−0.522697			−0.261349	+	0.965244i	$0.584167\pi$
$938$	−2.44949	−	12.7279i	−0.0799787	−	0.415581i
$939$		0			0
$940$	−12.0000	−	20.7846i	−0.391397	−	0.677919i
$941$	4.89898	−	8.48528i	0.159702	−	0.276612i	−0.775059	−	0.631889i	$-0.782280\pi$
$941$	4.89898	−	8.48528i	0.159702	−	0.276612i	0.934761	+	0.355276i	$0.115613\pi$
$942$		0			0
$943$	−9.00000	−	15.5885i	−0.293080	−	0.507630i
$944$	−39.1918			−1.27559
$945$		0			0
$946$	−12.0000			−0.390154
$947$	−23.2702	−	40.3051i	−0.756178	−	1.30974i	−0.944786	−	0.327687i	$-0.893731\pi$
$947$	−23.2702	−	40.3051i	−0.756178	−	1.30974i	0.188608	−	0.982052i	$-0.439602\pi$
$948$		0			0
$949$	−2.00000	+	3.46410i	−0.0649227	+	0.112449i
$950$	1.22474	+	2.12132i	0.0397360	+	0.0688247i
$951$		0			0
$952$	−30.0000	−	10.3923i	−0.972306	−	0.336817i
$953$	−22.0454			−0.714121			−0.357060	−	0.934081i	$-0.616221\pi$
$953$	−22.0454			−0.714121			−0.357060	+	0.934081i	$0.616221\pi$
$954$		0			0
$955$	−3.00000	+	5.19615i	−0.0970777	+	0.168144i
$956$	−14.6969	+	25.4558i	−0.475333	+	0.823301i
$957$		0			0
$958$	66.0000			2.13236
$959$	−42.8661	−	14.8492i	−1.38422	−	0.479507i
$960$		0			0
$961$	−9.00000	−	15.5885i	−0.290323	−	0.502853i
$962$	39.1918	−	67.8823i	1.26360	−	2.18861i
$963$		0			0
$964$	26.0000	+	45.0333i	0.837404	+	1.45043i
$965$	48.9898			1.57704
$966$		0			0
$967$	−22.0000			−0.707472			−0.353736	−	0.935345i	$-0.615089\pi$
$967$	−22.0000			−0.707472			−0.353736	+	0.935345i	$0.615089\pi$
$968$	−31.8434	−	55.1543i	−1.02348	−	1.77273i
$969$		0			0
$970$	3.00000	−	5.19615i	0.0963242	−	0.166838i
$971$	20.8207	+	36.0624i	0.668167	+	1.15730i	0.978416	+	0.206644i	$0.0662541\pi$
$971$	20.8207	+	36.0624i	0.668167	+	1.15730i	−0.310249	+	0.950655i	$0.600413\pi$
$972$		0			0
$973$	−7.00000	−	36.3731i	−0.224410	−	1.16607i
$974$	−46.5403			−1.49125
$975$		0			0
$976$	−10.0000	+	17.3205i	−0.320092	+	0.554416i
$977$	4.89898	−	8.48528i	0.156732	−	0.271468i	−0.776956	−	0.629555i	$-0.783237\pi$
$977$	4.89898	−	8.48528i	0.156732	−	0.271468i	0.933688	+	0.358086i	$0.116571\pi$
$978$		0			0
$979$	12.0000			0.383522
$980$	−63.6867	+	25.4558i	−2.03440	+	0.813157i
$981$		0			0
$982$	18.0000	+	31.1769i	0.574403	+	0.994895i
$983$	19.5959	−	33.9411i	0.625013	−	1.08255i	−0.363526	−	0.931584i	$-0.618427\pi$
$983$	19.5959	−	33.9411i	0.625013	−	1.08255i	0.988538	−	0.150970i	$-0.0482396\pi$
$984$		0			0
$985$		0			0
$986$	44.0908			1.40414
$987$		0			0
$988$	16.0000			0.509028
$989$	1.22474	+	2.12132i	0.0389446	+	0.0674541i
$990$		0			0
$991$	5.00000	−	8.66025i	0.158830	−	0.275102i	−0.775617	−	0.631204i	$-0.782561\pi$
$991$	5.00000	−	8.66025i	0.158830	−	0.275102i	0.934447	+	0.356102i	$0.115894\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	−46.5403			−1.47543
$996$		0			0
$997$	−20.5000	+	35.5070i	−0.649242	+	1.12452i	0.334063	+	0.942551i	$0.391580\pi$
$997$	−20.5000	+	35.5070i	−0.649242	+	1.12452i	−0.983304	+	0.181968i	$0.941753\pi$
$998$	8.57321	−	14.8492i	0.271380	−	0.470045i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	189.2.e.d.109.1	✓	4
3.2	odd	2	inner	189.2.e.d.109.2	yes	4
7.2	even	3	inner	189.2.e.d.163.1	yes	4
7.3	odd	6		1323.2.a.u.1.2		2
7.4	even	3		1323.2.a.v.1.2		2
9.2	odd	6		567.2.g.g.109.2		4
9.4	even	3		567.2.h.g.298.2		4
9.5	odd	6		567.2.h.g.298.1		4
9.7	even	3		567.2.g.g.109.1		4
21.2	odd	6	inner	189.2.e.d.163.2	yes	4
21.11	odd	6		1323.2.a.v.1.1		2
21.17	even	6		1323.2.a.u.1.1		2
63.2	odd	6		567.2.h.g.352.1		4
63.16	even	3		567.2.h.g.352.2		4
63.23	odd	6		567.2.g.g.541.2		4
63.58	even	3		567.2.g.g.541.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
189.2.e.d.109.1	✓	4	1.1	even	1	trivial
189.2.e.d.109.2	yes	4	3.2	odd	2	inner
189.2.e.d.163.1	yes	4	7.2	even	3	inner
189.2.e.d.163.2	yes	4	21.2	odd	6	inner
567.2.g.g.109.1		4	9.7	even	3
567.2.g.g.109.2		4	9.2	odd	6
567.2.g.g.541.1		4	63.58	even	3
567.2.g.g.541.2		4	63.23	odd	6
567.2.h.g.298.1		4	9.5	odd	6
567.2.h.g.298.2		4	9.4	even	3
567.2.h.g.352.1		4	63.2	odd	6
567.2.h.g.352.2		4	63.16	even	3
1323.2.a.u.1.1		2	21.17	even	6
1323.2.a.u.1.2		2	7.3	odd	6
1323.2.a.v.1.1		2	21.11	odd	6
1323.2.a.v.1.2		2	7.4	even	3

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 189 = 3^{3} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	189.e (of order \(3\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(-1.22474 - 2.12132i) q^{2} +(-2.00000 + 3.46410i) q^{4} +(-1.22474 - 2.12132i) q^{5} +(-0.500000 - 2.59808i) q^{7} +4.89898 q^{8} +O(q^{10})\)	\(q+(-1.22474 - 2.12132i) q^{2} +(-2.00000 + 3.46410i) q^{4} +(-1.22474 - 2.12132i) q^{5} +(-0.500000 - 2.59808i) q^{7} +4.89898 q^{8} +(-3.00000 + 5.19615i) q^{10} +(-2.44949 + 4.24264i) q^{11} -4.00000 q^{13} +(-4.89898 + 4.24264i) q^{14} +(-2.00000 - 3.46410i) q^{16} +(1.22474 - 2.12132i) q^{17} +(0.500000 + 0.866025i) q^{19} +9.79796 q^{20} +12.0000 q^{22} +(-1.22474 - 2.12132i) q^{23} +(-0.500000 + 0.866025i) q^{25} +(4.89898 + 8.48528i) q^{26} +(10.0000 + 3.46410i) q^{28} -7.34847 q^{29} +(3.50000 - 6.06218i) q^{31} -6.00000 q^{34} +(-4.89898 + 4.24264i) q^{35} +(-4.00000 - 6.92820i) q^{37} +(1.22474 - 2.12132i) q^{38} +(-6.00000 - 10.3923i) q^{40} +7.34847 q^{41} -1.00000 q^{43} +(-9.79796 - 16.9706i) q^{44} +(-3.00000 + 5.19615i) q^{46} +(-1.22474 - 2.12132i) q^{47} +(-6.50000 + 2.59808i) q^{49} +2.44949 q^{50} +(8.00000 - 13.8564i) q^{52} +(1.22474 - 2.12132i) q^{53} +12.0000 q^{55} +(-2.44949 - 12.7279i) q^{56} +(9.00000 + 15.5885i) q^{58} +(4.89898 - 8.48528i) q^{59} +(-2.50000 - 4.33013i) q^{61} -17.1464 q^{62} -8.00000 q^{64} +(4.89898 + 8.48528i) q^{65} +(-1.00000 + 1.73205i) q^{67} +(4.89898 + 8.48528i) q^{68} +(15.0000 + 5.19615i) q^{70} +(0.500000 - 0.866025i) q^{73} +(-9.79796 + 16.9706i) q^{74} -4.00000 q^{76} +(12.2474 + 4.24264i) q^{77} +(2.00000 + 3.46410i) q^{79} +(-4.89898 + 8.48528i) q^{80} +(-9.00000 - 15.5885i) q^{82} +14.6969 q^{83} -6.00000 q^{85} +(1.22474 + 2.12132i) q^{86} +(-12.0000 + 20.7846i) q^{88} +(-1.22474 - 2.12132i) q^{89} +(2.00000 + 10.3923i) q^{91} +9.79796 q^{92} +(-3.00000 + 5.19615i) q^{94} +(1.22474 - 2.12132i) q^{95} -1.00000 q^{97} +(13.4722 + 10.6066i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 8 q^{4} - 2 q^{7}+O(q^{10}) \) 4 * q - 8 * q^4 - 2 * q^7	\( 4 q - 8 q^{4} - 2 q^{7} - 12 q^{10} - 16 q^{13} - 8 q^{16} + 2 q^{19} + 48 q^{22} - 2 q^{25} + 40 q^{28} + 14 q^{31} - 24 q^{34} - 16 q^{37} - 24 q^{40} - 4 q^{43} - 12 q^{46} - 26 q^{49} + 32 q^{52} + 48 q^{55} + 36 q^{58} - 10 q^{61} - 32 q^{64} - 4 q^{67} + 60 q^{70} + 2 q^{73} - 16 q^{76} + 8 q^{79} - 36 q^{82} - 24 q^{85} - 48 q^{88} + 8 q^{91} - 12 q^{94} - 4 q^{97}+O(q^{100}) \) 4 * q - 8 * q^4 - 2 * q^7 - 12 * q^10 - 16 * q^13 - 8 * q^16 + 2 * q^19 + 48 * q^22 - 2 * q^25 + 40 * q^28 + 14 * q^31 - 24 * q^34 - 16 * q^37 - 24 * q^40 - 4 * q^43 - 12 * q^46 - 26 * q^49 + 32 * q^52 + 48 * q^55 + 36 * q^58 - 10 * q^61 - 32 * q^64 - 4 * q^67 + 60 * q^70 + 2 * q^73 - 16 * q^76 + 8 * q^79 - 36 * q^82 - 24 * q^85 - 48 * q^88 + 8 * q^91 - 12 * q^94 - 4 * q^97

Introduction

L-functions

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