LMFDB - Embedded newform 756.2.be.b.431.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [756,2,Mod(107,756)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(756, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([3, 3, 2]))

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

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

chi := DirichletCharacter("756.107");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 756 = 2^{2} \cdot 3^{3} \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	756.be (of order $6$, 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:	$6.03669039281$
Analytic rank:	$0$
Dimension:	$4$
Relative dimension:	$2$ over $\Q(\zeta_{6})$
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:	$\Z[a_1, \ldots, a_{25}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			431.1
Root			$-1.22474 + 0.707107i$ of defining polynomial
Character	$\chi$	$=$	756.431
Dual form			756.2.be.b.107.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-1.22474 + 0.707107i) q^{2} +(1.00000 - 1.73205i) q^{4} +(-2.44949 + 1.41421i) q^{5} +(2.00000 - 1.73205i) q^{7} +2.82843i q^{8} +O(q^{10})$	\(q+(-1.22474 + 0.707107i) q^{2} +(1.00000 - 1.73205i) q^{4} +(-2.44949 + 1.41421i) q^{5} +(2.00000 - 1.73205i) q^{7} +2.82843i q^{8} +(2.00000 - 3.46410i) q^{10} +(1.22474 - 2.12132i) q^{11} -1.00000 q^{13} +(-1.22474 + 3.53553i) q^{14} +(-2.00000 - 3.46410i) q^{16} +(-1.22474 - 0.707107i) q^{17} +(-6.00000 + 3.46410i) q^{19} +5.65685i q^{20} +3.46410i q^{22} +(-1.22474 - 2.12132i) q^{23} +(1.50000 - 2.59808i) q^{25} +(1.22474 - 0.707107i) q^{26} +(-1.00000 - 5.19615i) q^{28} -7.07107i q^{29} +(4.50000 + 2.59808i) q^{31} +(4.89898 + 2.82843i) q^{32} +2.00000 q^{34} +(-2.44949 + 7.07107i) q^{35} +(-2.50000 - 4.33013i) q^{37} +(4.89898 - 8.48528i) q^{38} +(-4.00000 - 6.92820i) q^{40} -7.07107i q^{41} -8.66025i q^{43} +(-2.44949 - 4.24264i) q^{44} +(3.00000 + 1.73205i) q^{46} +(3.67423 + 6.36396i) q^{47} +(1.00000 - 6.92820i) q^{49} +4.24264i q^{50} +(-1.00000 + 1.73205i) q^{52} +(-8.57321 - 4.94975i) q^{53} +6.92820i q^{55} +(4.89898 + 5.65685i) q^{56} +(5.00000 + 8.66025i) q^{58} +(3.67423 - 6.36396i) q^{59} +(-2.50000 - 4.33013i) q^{61} -7.34847 q^{62} -8.00000 q^{64} +(2.44949 - 1.41421i) q^{65} +(-4.50000 - 2.59808i) q^{67} +(-2.44949 + 1.41421i) q^{68} +(-2.00000 - 10.3923i) q^{70} -14.6969 q^{71} +(2.00000 - 3.46410i) q^{73} +(6.12372 + 3.53553i) q^{74} +13.8564i q^{76} +(-1.22474 - 6.36396i) q^{77} +(13.5000 - 7.79423i) q^{79} +(9.79796 + 5.65685i) q^{80} +(5.00000 + 8.66025i) q^{82} +2.44949 q^{83} +4.00000 q^{85} +(6.12372 + 10.6066i) q^{86} +(6.00000 + 3.46410i) q^{88} +(12.2474 - 7.07107i) q^{89} +(-2.00000 + 1.73205i) q^{91} -4.89898 q^{92} +(-9.00000 - 5.19615i) q^{94} +(9.79796 - 16.9706i) q^{95} +5.00000 q^{97} +(3.67423 + 9.19239i) q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 4 q^{4} + 8 q^{7}+O(q^{10}) $ 4 * q + 4 * q^4 + 8 * q^7	$ 4 q + 4 q^{4} + 8 q^{7} + 8 q^{10} - 4 q^{13} - 8 q^{16} - 24 q^{19} + 6 q^{25} - 4 q^{28} + 18 q^{31} + 8 q^{34} - 10 q^{37} - 16 q^{40} + 12 q^{46} + 4 q^{49} - 4 q^{52} + 20 q^{58} - 10 q^{61} - 32 q^{64} - 18 q^{67} - 8 q^{70} + 8 q^{73} + 54 q^{79} + 20 q^{82} + 16 q^{85} + 24 q^{88} - 8 q^{91} - 36 q^{94} + 20 q^{97}+O(q^{100}) $ 4 * q + 4 * q^4 + 8 * q^7 + 8 * q^10 - 4 * q^13 - 8 * q^16 - 24 * q^19 + 6 * q^25 - 4 * q^28 + 18 * q^31 + 8 * q^34 - 10 * q^37 - 16 * q^40 + 12 * q^46 + 4 * q^49 - 4 * q^52 + 20 * q^58 - 10 * q^61 - 32 * q^64 - 18 * q^67 - 8 * q^70 + 8 * q^73 + 54 * q^79 + 20 * q^82 + 16 * q^85 + 24 * q^88 - 8 * q^91 - 36 * q^94 + 20 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/756\mathbb{Z}\right)^\times$.

$n$	$29$	$325$	$379$
$\chi(n)$	$-1$	$e\left(\frac{2}{3}\right)$	$-1$

Coefficient data

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

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

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

$2$	−1.22474	+	0.707107i	−0.866025	+	0.500000i
$2$	−1.22474	+	0.707107i	−0.866025	+	0.500000i
$3$		0			0
$3$		0			0
$4$	1.00000	−	1.73205i	0.500000	−	0.866025i
$5$	−2.44949	+	1.41421i	−1.09545	+	0.632456i	−0.935021	−	0.354593i	$-0.884620\pi$
$5$	−2.44949	+	1.41421i	−1.09545	+	0.632456i	−0.160424	+	0.987048i	$0.551286\pi$
$6$		0			0
$7$	2.00000	−	1.73205i	0.755929	−	0.654654i
$7$	2.00000	−	1.73205i	0.755929	−	0.654654i
$8$			2.82843i			1.00000i
$9$		0			0
$10$	2.00000	−	3.46410i	0.632456	−	1.09545i
$11$	1.22474	−	2.12132i	0.369274	−	0.639602i	−0.620178	−	0.784461i	$-0.712940\pi$
$11$	1.22474	−	2.12132i	0.369274	−	0.639602i	0.989452	+	0.144859i	$0.0462729\pi$
$12$		0			0
$13$	−1.00000			−0.277350			−0.138675	−	0.990338i	$-0.544284\pi$
$13$	−1.00000			−0.277350			−0.138675	+	0.990338i	$0.544284\pi$
$14$	−1.22474	+	3.53553i	−0.327327	+	0.944911i
$15$		0			0
$16$	−2.00000	−	3.46410i	−0.500000	−	0.866025i
$17$	−1.22474	−	0.707107i	−0.297044	−	0.171499i	0.344070	−	0.938944i	$-0.388194\pi$
$17$	−1.22474	−	0.707107i	−0.297044	−	0.171499i	−0.641114	+	0.767445i	$0.721528\pi$
$18$		0			0
$19$	−6.00000	+	3.46410i	−1.37649	+	0.794719i	−0.991736	−	0.128298i	$-0.959049\pi$
$19$	−6.00000	+	3.46410i	−1.37649	+	0.794719i	−0.384759	+	0.923017i	$0.625715\pi$
$20$			5.65685i			1.26491i
$21$		0			0
$22$			3.46410i			0.738549i
$23$	−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
$25$	1.50000	−	2.59808i	0.300000	−	0.519615i
$26$	1.22474	−	0.707107i	0.240192	−	0.138675i
$27$		0			0
$28$	−1.00000	−	5.19615i	−0.188982	−	0.981981i
$29$		−	7.07107i		−	1.31306i	−0.754298	−	0.656532i	$-0.772023\pi$
$29$		−	7.07107i		−	1.31306i	0.754298	−	0.656532i	$-0.227977\pi$
$30$		0			0
$31$	4.50000	+	2.59808i	0.808224	+	0.466628i	0.846339	−	0.532645i	$-0.178802\pi$
$31$	4.50000	+	2.59808i	0.808224	+	0.466628i	−0.0381148	+	0.999273i	$0.512135\pi$
$32$	4.89898	+	2.82843i	0.866025	+	0.500000i
$33$		0			0
$34$	2.00000			0.342997
$35$	−2.44949	+	7.07107i	−0.414039	+	1.19523i
$36$		0			0
$37$	−2.50000	−	4.33013i	−0.410997	−	0.711868i	0.584002	−	0.811752i	$-0.301486\pi$
$37$	−2.50000	−	4.33013i	−0.410997	−	0.711868i	−0.994999	+	0.0998840i	$0.968153\pi$
$38$	4.89898	−	8.48528i	0.794719	−	1.37649i
$39$		0			0
$40$	−4.00000	−	6.92820i	−0.632456	−	1.09545i
$41$		−	7.07107i		−	1.10432i	−0.833740	−	0.552158i	$-0.813805\pi$
$41$		−	7.07107i		−	1.10432i	0.833740	−	0.552158i	$-0.186195\pi$
$42$		0			0
$43$		−	8.66025i		−	1.32068i	−0.750968	−	0.660338i	$-0.770413\pi$
$43$		−	8.66025i		−	1.32068i	0.750968	−	0.660338i	$-0.229587\pi$
$44$	−2.44949	−	4.24264i	−0.369274	−	0.639602i
$45$		0			0
$46$	3.00000	+	1.73205i	0.442326	+	0.255377i
$47$	3.67423	+	6.36396i	0.535942	+	0.928279i	0.999117	+	0.0420122i	$0.0133768\pi$
$47$	3.67423	+	6.36396i	0.535942	+	0.928279i	−0.463175	+	0.886267i	$0.653290\pi$
$48$		0			0
$49$	1.00000	−	6.92820i	0.142857	−	0.989743i
$50$			4.24264i			0.600000i
$51$		0			0
$52$	−1.00000	+	1.73205i	−0.138675	+	0.240192i
$53$	−8.57321	−	4.94975i	−1.17762	−	0.679900i	−0.222158	−	0.975011i	$-0.571310\pi$
$53$	−8.57321	−	4.94975i	−1.17762	−	0.679900i	−0.955463	+	0.295110i	$0.904644\pi$
$54$		0			0
$55$			6.92820i			0.934199i
$56$	4.89898	+	5.65685i	0.654654	+	0.755929i
$57$		0			0
$58$	5.00000	+	8.66025i	0.656532	+	1.13715i
$59$	3.67423	−	6.36396i	0.478345	−	0.828517i	−0.521347	−	0.853345i	$-0.674570\pi$
$59$	3.67423	−	6.36396i	0.478345	−	0.828517i	0.999692	+	0.0248275i	$0.00790366\pi$
$60$		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	$0.937047\pi$
$62$	−7.34847			−0.933257
$63$		0			0
$64$	−8.00000			−1.00000
$65$	2.44949	−	1.41421i	0.303822	−	0.175412i
$66$		0			0
$67$	−4.50000	−	2.59808i	−0.549762	−	0.317406i	0.199264	−	0.979946i	$-0.436145\pi$
$67$	−4.50000	−	2.59808i	−0.549762	−	0.317406i	−0.749026	+	0.662540i	$0.769478\pi$
$68$	−2.44949	+	1.41421i	−0.297044	+	0.171499i
$69$		0			0
$70$	−2.00000	−	10.3923i	−0.239046	−	1.24212i
$71$	−14.6969			−1.74421			−0.872103	−	0.489323i	$-0.837244\pi$
$71$	−14.6969			−1.74421			−0.872103	+	0.489323i	$0.837244\pi$
$72$		0			0
$73$	2.00000	−	3.46410i	0.234082	−	0.405442i	−0.724923	−	0.688830i	$-0.758125\pi$
$73$	2.00000	−	3.46410i	0.234082	−	0.405442i	0.959006	+	0.283387i	$0.0914581\pi$
$74$	6.12372	+	3.53553i	0.711868	+	0.410997i
$75$		0			0
$76$			13.8564i			1.58944i
$77$	−1.22474	−	6.36396i	−0.139573	−	0.725241i
$78$		0			0
$79$	13.5000	−	7.79423i	1.51887	−	0.876919i	0.519115	−	0.854704i	$-0.326261\pi$
$79$	13.5000	−	7.79423i	1.51887	−	0.876919i	0.999753	−	0.0222151i	$-0.00707187\pi$
$80$	9.79796	+	5.65685i	1.09545	+	0.632456i
$81$		0			0
$82$	5.00000	+	8.66025i	0.552158	+	0.956365i
$83$	2.44949			0.268866			0.134433	−	0.990923i	$-0.457079\pi$
$83$	2.44949			0.268866			0.134433	+	0.990923i	$0.457079\pi$
$84$		0			0
$85$	4.00000			0.433861
$86$	6.12372	+	10.6066i	0.660338	+	1.14374i
$87$		0			0
$88$	6.00000	+	3.46410i	0.639602	+	0.369274i
$89$	12.2474	−	7.07107i	1.29823	−	0.749532i	0.318129	−	0.948047i	$-0.396945\pi$
$89$	12.2474	−	7.07107i	1.29823	−	0.749532i	0.980098	+	0.198516i	$0.0636121\pi$
$90$		0			0
$91$	−2.00000	+	1.73205i	−0.209657	+	0.181568i
$92$	−4.89898			−0.510754
$93$		0			0
$94$	−9.00000	−	5.19615i	−0.928279	−	0.535942i
$95$	9.79796	−	16.9706i	1.00525	−	1.74114i
$96$		0			0
$97$	5.00000			0.507673			0.253837	−	0.967247i	$-0.418307\pi$
$97$	5.00000			0.507673			0.253837	+	0.967247i	$0.418307\pi$
$98$	3.67423	+	9.19239i	0.371154	+	0.928571i
$99$		0			0
$100$	−3.00000	−	5.19615i	−0.300000	−	0.519615i
$101$	−12.2474	−	7.07107i	−1.21867	−	0.703598i	−0.254034	−	0.967195i	$-0.581758\pi$
$101$	−12.2474	−	7.07107i	−1.21867	−	0.703598i	−0.964633	+	0.263598i	$0.915091\pi$
$102$		0			0
$103$	−16.5000	+	9.52628i	−1.62579	+	0.938652i	−0.640464	+	0.767988i	$0.721258\pi$
$103$	−16.5000	+	9.52628i	−1.62579	+	0.938652i	−0.985329	+	0.170664i	$0.945409\pi$
$104$		−	2.82843i		−	0.277350i
$105$		0			0
$106$	14.0000			1.35980
$107$	8.57321	+	14.8492i	0.828804	+	1.43553i	0.898977	+	0.437996i	$0.144311\pi$
$107$	8.57321	+	14.8492i	0.828804	+	1.43553i	−0.0701732	+	0.997535i	$0.522355\pi$
$108$		0			0
$109$	−2.50000	+	4.33013i	−0.239457	+	0.414751i	−0.960558	−	0.278078i	$-0.910303\pi$
$109$	−2.50000	+	4.33013i	−0.239457	+	0.414751i	0.721102	+	0.692829i	$0.243636\pi$
$110$	−4.89898	−	8.48528i	−0.467099	−	0.809040i
$111$		0			0
$112$	−10.0000	−	3.46410i	−0.944911	−	0.327327i
$113$		−	2.82843i		−	0.266076i	−0.991111	−	0.133038i	$-0.957527\pi$
$113$		−	2.82843i		−	0.266076i	0.991111	−	0.133038i	$-0.0424732\pi$
$114$		0			0
$115$	6.00000	+	3.46410i	0.559503	+	0.323029i
$116$	−12.2474	−	7.07107i	−1.13715	−	0.656532i
$117$		0			0
$118$			10.3923i			0.956689i
$119$	−3.67423	+	0.707107i	−0.336817	+	0.0648204i
$120$		0			0
$121$	2.50000	+	4.33013i	0.227273	+	0.393648i
$122$	6.12372	+	3.53553i	0.554416	+	0.320092i
$123$		0			0
$124$	9.00000	−	5.19615i	0.808224	−	0.466628i
$125$		−	5.65685i		−	0.505964i
$126$		0			0
$127$		−	5.19615i		−	0.461084i	−0.973062	−	0.230542i	$-0.925950\pi$
$127$		−	5.19615i		−	0.461084i	0.973062	−	0.230542i	$-0.0740499\pi$
$128$	9.79796	−	5.65685i	0.866025	−	0.500000i
$129$		0			0
$130$	−2.00000	+	3.46410i	−0.175412	+	0.303822i
$131$	−4.89898	−	8.48528i	−0.428026	−	0.741362i	0.568672	−	0.822564i	$-0.307457\pi$
$131$	−4.89898	−	8.48528i	−0.428026	−	0.741362i	−0.996698	+	0.0812020i	$0.974124\pi$
$132$		0			0
$133$	−6.00000	+	17.3205i	−0.520266	+	1.50188i
$134$	7.34847			0.634811
$135$		0			0
$136$	2.00000	−	3.46410i	0.171499	−	0.297044i
$137$	6.12372	+	3.53553i	0.523185	+	0.302061i	0.738237	−	0.674542i	$-0.235659\pi$
$137$	6.12372	+	3.53553i	0.523185	+	0.302061i	−0.215052	+	0.976603i	$0.568992\pi$
$138$		0			0
$139$			8.66025i			0.734553i	0.930112	+	0.367277i	$0.119710\pi$
$139$			8.66025i			0.734553i	−0.930112	+	0.367277i	$0.880290\pi$
$140$	9.79796	+	11.3137i	0.828079	+	0.956183i
$141$		0			0
$142$	18.0000	−	10.3923i	1.51053	−	0.872103i
$143$	−1.22474	+	2.12132i	−0.102418	+	0.177394i
$144$		0			0
$145$	10.0000	+	17.3205i	0.830455	+	1.43839i
$146$			5.65685i			0.468165i
$147$		0			0
$148$	−10.0000			−0.821995
$149$	−2.44949	+	1.41421i	−0.200670	+	0.115857i	−0.596968	−	0.802265i	$-0.703628\pi$
$149$	−2.44949	+	1.41421i	−0.200670	+	0.115857i	0.396298	+	0.918122i	$0.370295\pi$
$150$		0			0
$151$	−7.50000	−	4.33013i	−0.610341	−	0.352381i	0.162758	−	0.986666i	$-0.447961\pi$
$151$	−7.50000	−	4.33013i	−0.610341	−	0.352381i	−0.773099	+	0.634285i	$0.781294\pi$
$152$	−9.79796	−	16.9706i	−0.794719	−	1.37649i
$153$		0			0
$154$	6.00000	+	6.92820i	0.483494	+	0.558291i
$155$	−14.6969			−1.18049
$156$		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	$0.872512\pi$
$158$	−11.0227	+	19.0919i	−0.876919	+	1.51887i
$159$		0			0
$160$	−16.0000			−1.26491
$161$	−6.12372	−	2.12132i	−0.482617	−	0.167183i
$162$		0			0
$163$	16.5000	−	9.52628i	1.29238	−	0.746156i	0.313304	−	0.949653i	$-0.398564\pi$
$163$	16.5000	−	9.52628i	1.29238	−	0.746156i	0.979076	+	0.203497i	$0.0652307\pi$
$164$	−12.2474	−	7.07107i	−0.956365	−	0.552158i
$165$		0			0
$166$	−3.00000	+	1.73205i	−0.232845	+	0.134433i
$167$	12.2474			0.947736			0.473868	−	0.880596i	$-0.342857\pi$
$167$	12.2474			0.947736			0.473868	+	0.880596i	$0.342857\pi$
$168$		0			0
$169$	−12.0000			−0.923077
$170$	−4.89898	+	2.82843i	−0.375735	+	0.216930i
$171$		0			0
$172$	−15.0000	−	8.66025i	−1.14374	−	0.660338i
$173$	−9.79796	+	5.65685i	−0.744925	+	0.430083i	−0.823857	−	0.566797i	$-0.808182\pi$
$173$	−9.79796	+	5.65685i	−0.744925	+	0.430083i	0.0789322	+	0.996880i	$0.474849\pi$
$174$		0			0
$175$	−1.50000	−	7.79423i	−0.113389	−	0.589188i
$176$	−9.79796			−0.738549
$177$		0			0
$178$	−10.0000	+	17.3205i	−0.749532	+	1.29823i
$179$	2.44949	−	4.24264i	0.183083	−	0.317110i	−0.759846	−	0.650104i	$-0.774725\pi$
$179$	2.44949	−	4.24264i	0.183083	−	0.317110i	0.942929	+	0.332994i	$0.108059\pi$
$180$		0			0
$181$	−16.0000			−1.18927			−0.594635	−	0.803996i	$-0.702704\pi$
$181$	−16.0000			−1.18927			−0.594635	+	0.803996i	$0.702704\pi$
$182$	1.22474	−	3.53553i	0.0907841	−	0.262071i
$183$		0			0
$184$	6.00000	−	3.46410i	0.442326	−	0.255377i
$185$	12.2474	+	7.07107i	0.900450	+	0.519875i
$186$		0			0
$187$	−3.00000	+	1.73205i	−0.219382	+	0.126660i
$188$	14.6969			1.07188
$189$		0			0
$190$			27.7128i			2.01050i
$191$	7.34847	+	12.7279i	0.531717	+	0.920960i	0.999315	+	0.0370189i	$0.0117862\pi$
$191$	7.34847	+	12.7279i	0.531717	+	0.920960i	−0.467598	+	0.883941i	$0.654880\pi$
$192$		0			0
$193$	−5.50000	+	9.52628i	−0.395899	+	0.685717i	−0.993215	−	0.116289i	$-0.962900\pi$
$193$	−5.50000	+	9.52628i	−0.395899	+	0.685717i	0.597317	+	0.802005i	$0.296234\pi$
$194$	−6.12372	+	3.53553i	−0.439658	+	0.253837i
$195$		0			0
$196$	−11.0000	−	8.66025i	−0.785714	−	0.618590i
$197$			22.6274i			1.61214i	0.591822	+	0.806068i	$0.298409\pi$
$197$			22.6274i			1.61214i	−0.591822	+	0.806068i	$0.701591\pi$
$198$		0			0
$199$	7.50000	+	4.33013i	0.531661	+	0.306955i	0.741693	−	0.670740i	$-0.234023\pi$
$199$	7.50000	+	4.33013i	0.531661	+	0.306955i	−0.210032	+	0.977695i	$0.567357\pi$
$200$	7.34847	+	4.24264i	0.519615	+	0.300000i
$201$		0			0
$202$	20.0000			1.40720
$203$	−12.2474	−	14.1421i	−0.859602	−	0.992583i
$204$		0			0
$205$	10.0000	+	17.3205i	0.698430	+	1.20972i
$206$	13.4722	−	23.3345i	0.938652	−	1.62579i
$207$		0			0
$208$	2.00000	+	3.46410i	0.138675	+	0.240192i
$209$			16.9706i			1.17388i
$210$		0			0
$211$			8.66025i			0.596196i	0.954535	+	0.298098i	$0.0963523\pi$
$211$			8.66025i			0.596196i	−0.954535	+	0.298098i	$0.903648\pi$
$212$	−17.1464	+	9.89949i	−1.17762	+	0.679900i
$213$		0			0
$214$	−21.0000	−	12.1244i	−1.43553	−	0.828804i
$215$	12.2474	+	21.2132i	0.835269	+	1.44673i
$216$		0			0
$217$	13.5000	−	2.59808i	0.916440	−	0.176369i
$218$		−	7.07107i		−	0.478913i
$219$		0			0
$220$	12.0000	+	6.92820i	0.809040	+	0.467099i
$221$	1.22474	+	0.707107i	0.0823853	+	0.0475651i
$222$		0			0
$223$			6.92820i			0.463947i	0.972722	+	0.231973i	$0.0745182\pi$
$223$			6.92820i			0.463947i	−0.972722	+	0.231973i	$0.925482\pi$
$224$	14.6969	−	2.82843i	0.981981	−	0.188982i
$225$		0			0
$226$	2.00000	+	3.46410i	0.133038	+	0.230429i
$227$	−6.12372	+	10.6066i	−0.406446	+	0.703985i	−0.994489	−	0.104845i	$-0.966565\pi$
$227$	−6.12372	+	10.6066i	−0.406446	+	0.703985i	0.588043	+	0.808830i	$0.299899\pi$
$228$		0			0
$229$	−8.50000	−	14.7224i	−0.561696	−	0.972886i	−0.997349	−	0.0727709i	$-0.976816\pi$
$229$	−8.50000	−	14.7224i	−0.561696	−	0.972886i	0.435653	−	0.900115i	$-0.356518\pi$
$230$	−9.79796			−0.646058
$231$		0			0
$232$	20.0000			1.31306
$233$	−9.79796	+	5.65685i	−0.641886	+	0.370593i	−0.785341	−	0.619064i	$-0.787512\pi$
$233$	−9.79796	+	5.65685i	−0.641886	+	0.370593i	0.143455	+	0.989657i	$0.454179\pi$
$234$		0			0
$235$	−18.0000	−	10.3923i	−1.17419	−	0.677919i
$236$	−7.34847	−	12.7279i	−0.478345	−	0.828517i
$237$		0			0
$238$	4.00000	−	3.46410i	0.259281	−	0.224544i
$239$	−24.4949			−1.58444			−0.792222	−	0.610234i	$-0.791076\pi$
$239$	−24.4949			−1.58444			−0.792222	+	0.610234i	$0.791076\pi$
$240$		0			0
$241$	12.5000	−	21.6506i	0.805196	−	1.39464i	−0.110963	−	0.993825i	$-0.535394\pi$
$241$	12.5000	−	21.6506i	0.805196	−	1.39464i	0.916159	−	0.400815i	$-0.131273\pi$
$242$	−6.12372	−	3.53553i	−0.393648	−	0.227273i
$243$		0			0
$244$	−10.0000			−0.640184
$245$	7.34847	+	18.3848i	0.469476	+	1.17456i
$246$		0			0
$247$	6.00000	−	3.46410i	0.381771	−	0.220416i
$248$	−7.34847	+	12.7279i	−0.466628	+	0.808224i
$249$		0			0
$250$	4.00000	+	6.92820i	0.252982	+	0.438178i
$251$	22.0454			1.39149			0.695747	−	0.718287i	$-0.255074\pi$
$251$	22.0454			1.39149			0.695747	+	0.718287i	$0.255074\pi$
$252$		0			0
$253$	−6.00000			−0.377217
$254$	3.67423	+	6.36396i	0.230542	+	0.399310i
$255$		0			0
$256$	−8.00000	+	13.8564i	−0.500000	+	0.866025i
$257$	−13.4722	+	7.77817i	−0.840372	+	0.485189i	−0.857391	−	0.514666i	$-0.827916\pi$
$257$	−13.4722	+	7.77817i	−0.840372	+	0.485189i	0.0170184	+	0.999855i	$0.494583\pi$
$258$		0			0
$259$	−12.5000	−	4.33013i	−0.776712	−	0.269061i
$260$		−	5.65685i		−	0.350823i
$261$		0			0
$262$	12.0000	+	6.92820i	0.741362	+	0.428026i
$263$	1.22474	−	2.12132i	0.0755210	−	0.130806i	−0.825792	−	0.563975i	$-0.809271\pi$
$263$	1.22474	−	2.12132i	0.0755210	−	0.130806i	0.901313	+	0.433169i	$0.142605\pi$
$264$		0			0
$265$	28.0000			1.72003
$266$	−4.89898	−	25.4558i	−0.300376	−	1.56080i
$267$		0			0
$268$	−9.00000	+	5.19615i	−0.549762	+	0.317406i
$269$	9.79796	+	5.65685i	0.597392	+	0.344904i	0.768015	−	0.640432i	$-0.221245\pi$
$269$	9.79796	+	5.65685i	0.597392	+	0.344904i	−0.170623	+	0.985336i	$0.554578\pi$
$270$		0			0
$271$	7.50000	−	4.33013i	0.455593	−	0.263036i	−0.254597	−	0.967047i	$-0.581943\pi$
$271$	7.50000	−	4.33013i	0.455593	−	0.263036i	0.710189	+	0.704011i	$0.248609\pi$
$272$			5.65685i			0.342997i
$273$		0			0
$274$	−10.0000			−0.604122
$275$	−3.67423	−	6.36396i	−0.221565	−	0.383761i
$276$		0			0
$277$	6.50000	−	11.2583i	0.390547	−	0.676448i	−0.601975	−	0.798515i	$-0.705619\pi$
$277$	6.50000	−	11.2583i	0.390547	−	0.676448i	0.992522	+	0.122068i	$0.0389525\pi$
$278$	−6.12372	−	10.6066i	−0.367277	−	0.636142i
$279$		0			0
$280$	−20.0000	−	6.92820i	−1.19523	−	0.414039i
$281$		−	19.7990i		−	1.18111i	−0.806998	−	0.590554i	$-0.798909\pi$
$281$		−	19.7990i		−	1.18111i	0.806998	−	0.590554i	$-0.201091\pi$
$282$		0			0
$283$	13.5000	+	7.79423i	0.802492	+	0.463319i	0.844342	−	0.535805i	$-0.179992\pi$
$283$	13.5000	+	7.79423i	0.802492	+	0.463319i	−0.0418500	+	0.999124i	$0.513325\pi$
$284$	−14.6969	+	25.4558i	−0.872103	+	1.51053i
$285$		0			0
$286$		−	3.46410i		−	0.204837i
$287$	−12.2474	−	14.1421i	−0.722944	−	0.834784i
$288$		0			0
$289$	−7.50000	−	12.9904i	−0.441176	−	0.764140i
$290$	−24.4949	−	14.1421i	−1.43839	−	0.830455i
$291$		0			0
$292$	−4.00000	−	6.92820i	−0.234082	−	0.405442i
$293$			9.89949i			0.578335i	0.957279	+	0.289167i	$0.0933784\pi$
$293$			9.89949i			0.578335i	−0.957279	+	0.289167i	$0.906622\pi$
$294$		0			0
$295$			20.7846i			1.21013i
$296$	12.2474	−	7.07107i	0.711868	−	0.410997i
$297$		0			0
$298$	2.00000	−	3.46410i	0.115857	−	0.200670i
$299$	1.22474	+	2.12132i	0.0708288	+	0.122679i
$300$		0			0
$301$	−15.0000	−	17.3205i	−0.864586	−	0.998337i
$302$	12.2474			0.704761
$303$		0			0
$304$	24.0000	+	13.8564i	1.37649	+	0.794719i
$305$	12.2474	+	7.07107i	0.701287	+	0.404888i
$306$		0			0
$307$			5.19615i			0.296560i	0.988945	+	0.148280i	$0.0473737\pi$
$307$			5.19615i			0.296560i	−0.988945	+	0.148280i	$0.952626\pi$
$308$	−12.2474	−	4.24264i	−0.697863	−	0.241747i
$309$		0			0
$310$	18.0000	−	10.3923i	1.02233	−	0.590243i
$311$	7.34847	−	12.7279i	0.416693	−	0.721734i	−0.578911	−	0.815391i	$-0.696522\pi$
$311$	7.34847	−	12.7279i	0.416693	−	0.721734i	0.995605	+	0.0936564i	$0.0298555\pi$
$312$		0			0
$313$	−7.00000	−	12.1244i	−0.395663	−	0.685309i	0.597522	−	0.801852i	$-0.296152\pi$
$313$	−7.00000	−	12.1244i	−0.395663	−	0.685309i	−0.993186	+	0.116543i	$0.962819\pi$
$314$		−	28.2843i		−	1.59617i
$315$		0			0
$316$		−	31.1769i		−	1.75384i
$317$	23.2702	−	13.4350i	1.30698	−	0.754586i	0.325391	−	0.945580i	$-0.394504\pi$
$317$	23.2702	−	13.4350i	1.30698	−	0.754586i	0.981591	+	0.190993i	$0.0611709\pi$
$318$		0			0
$319$	−15.0000	−	8.66025i	−0.839839	−	0.484881i
$320$	19.5959	−	11.3137i	1.09545	−	0.632456i
$321$		0			0
$322$	9.00000	−	1.73205i	0.501550	−	0.0965234i
$323$	9.79796			0.545173
$324$		0			0
$325$	−1.50000	+	2.59808i	−0.0832050	+	0.144115i
$326$	−13.4722	+	23.3345i	−0.746156	+	1.29238i
$327$		0			0
$328$	20.0000			1.10432
$329$	18.3712	+	6.36396i	1.01284	+	0.350857i
$330$		0			0
$331$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$331$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$332$	2.44949	−	4.24264i	0.134433	−	0.232845i
$333$		0			0
$334$	−15.0000	+	8.66025i	−0.820763	+	0.473868i
$335$	14.6969			0.802980
$336$		0			0
$337$	20.0000			1.08947			0.544735	−	0.838608i	$-0.316630\pi$
$337$	20.0000			1.08947			0.544735	+	0.838608i	$0.316630\pi$
$338$	14.6969	−	8.48528i	0.799408	−	0.461538i
$339$		0			0
$340$	4.00000	−	6.92820i	0.216930	−	0.375735i
$341$	11.0227	−	6.36396i	0.596913	−	0.344628i
$342$		0			0
$343$	−10.0000	−	15.5885i	−0.539949	−	0.841698i
$344$	24.4949			1.32068
$345$		0			0
$346$	8.00000	−	13.8564i	0.430083	−	0.744925i
$347$	14.6969	−	25.4558i	0.788973	−	1.36654i	−0.137624	−	0.990485i	$-0.543946\pi$
$347$	14.6969	−	25.4558i	0.788973	−	1.36654i	0.926597	−	0.376057i	$-0.122720\pi$
$348$		0			0
$349$	−13.0000			−0.695874			−0.347937	−	0.937518i	$-0.613118\pi$
$349$	−13.0000			−0.695874			−0.347937	+	0.937518i	$0.613118\pi$
$350$	7.34847	+	8.48528i	0.392792	+	0.453557i
$351$		0			0
$352$	12.0000	−	6.92820i	0.639602	−	0.369274i
$353$	9.79796	+	5.65685i	0.521493	+	0.301084i	0.737545	−	0.675298i	$-0.235985\pi$
$353$	9.79796	+	5.65685i	0.521493	+	0.301084i	−0.216052	+	0.976382i	$0.569318\pi$
$354$		0			0
$355$	36.0000	−	20.7846i	1.91068	−	1.10313i
$356$		−	28.2843i		−	1.49906i
$357$		0			0
$358$			6.92820i			0.366167i
$359$	−6.12372	−	10.6066i	−0.323198	−	0.559795i	0.657948	−	0.753063i	$-0.271425\pi$
$359$	−6.12372	−	10.6066i	−0.323198	−	0.559795i	−0.981146	+	0.193268i	$0.938091\pi$
$360$		0			0
$361$	14.5000	−	25.1147i	0.763158	−	1.32183i
$362$	19.5959	−	11.3137i	1.02994	−	0.594635i
$363$		0			0
$364$	1.00000	+	5.19615i	0.0524142	+	0.272352i
$365$			11.3137i			0.592187i
$366$		0			0
$367$	−3.00000	−	1.73205i	−0.156599	−	0.0904123i	0.419653	−	0.907685i	$-0.362152\pi$
$367$	−3.00000	−	1.73205i	−0.156599	−	0.0904123i	−0.576252	+	0.817272i	$0.695485\pi$
$368$	−4.89898	+	8.48528i	−0.255377	+	0.442326i
$369$		0			0
$370$	−20.0000			−1.03975
$371$	−25.7196	+	4.94975i	−1.33530	+	0.256978i
$372$		0			0
$373$	−10.0000	−	17.3205i	−0.517780	−	0.896822i	−0.999787	−	0.0206542i	$-0.993425\pi$
$373$	−10.0000	−	17.3205i	−0.517780	−	0.896822i	0.482006	−	0.876168i	$-0.339908\pi$
$374$	2.44949	−	4.24264i	0.126660	−	0.219382i
$375$		0			0
$376$	−18.0000	+	10.3923i	−0.928279	+	0.535942i
$377$			7.07107i			0.364179i
$378$		0			0
$379$			25.9808i			1.33454i	0.744815	+	0.667271i	$0.232538\pi$
$379$			25.9808i			1.33454i	−0.744815	+	0.667271i	$0.767462\pi$
$380$	−19.5959	−	33.9411i	−1.00525	−	1.74114i
$381$		0			0
$382$	−18.0000	−	10.3923i	−0.920960	−	0.531717i
$383$	13.4722	+	23.3345i	0.688397	+	1.19234i	0.972356	+	0.233502i	$0.0750186\pi$
$383$	13.4722	+	23.3345i	0.688397	+	1.19234i	−0.283959	+	0.958836i	$0.591648\pi$
$384$		0			0
$385$	12.0000	+	13.8564i	0.611577	+	0.706188i
$386$		−	15.5563i		−	0.791797i
$387$		0			0
$388$	5.00000	−	8.66025i	0.253837	−	0.439658i
$389$	9.79796	+	5.65685i	0.496776	+	0.286814i	0.727381	−	0.686234i	$-0.240737\pi$
$389$	9.79796	+	5.65685i	0.496776	+	0.286814i	−0.230605	+	0.973047i	$0.574071\pi$
$390$		0			0
$391$			3.46410i			0.175187i
$392$	19.5959	+	2.82843i	0.989743	+	0.142857i
$393$		0			0
$394$	−16.0000	−	27.7128i	−0.806068	−	1.39615i
$395$	−22.0454	+	38.1838i	−1.10922	+	1.92123i
$396$		0			0
$397$	6.50000	+	11.2583i	0.326226	+	0.565039i	0.981760	−	0.190126i	$-0.0608897\pi$
$397$	6.50000	+	11.2583i	0.326226	+	0.565039i	−0.655534	+	0.755166i	$0.727556\pi$
$398$	−12.2474			−0.613909
$399$		0			0
$400$	−12.0000			−0.600000
$401$	−13.4722	+	7.77817i	−0.672769	+	0.388424i	−0.797125	−	0.603814i	$-0.793647\pi$
$401$	−13.4722	+	7.77817i	−0.672769	+	0.388424i	0.124356	+	0.992238i	$0.460314\pi$
$402$		0			0
$403$	−4.50000	−	2.59808i	−0.224161	−	0.129419i
$404$	−24.4949	+	14.1421i	−1.21867	+	0.703598i
$405$		0			0
$406$	25.0000	+	8.66025i	1.24073	+	0.429801i
$407$	−12.2474			−0.607083
$408$		0			0
$409$	3.50000	−	6.06218i	0.173064	−	0.299755i	−0.766426	−	0.642333i	$-0.777967\pi$
$409$	3.50000	−	6.06218i	0.173064	−	0.299755i	0.939490	+	0.342578i	$0.111300\pi$
$410$	−24.4949	−	14.1421i	−1.20972	−	0.698430i
$411$		0			0
$412$			38.1051i			1.87730i
$413$	−3.67423	−	19.0919i	−0.180797	−	0.939450i
$414$		0			0
$415$	−6.00000	+	3.46410i	−0.294528	+	0.170046i
$416$	−4.89898	−	2.82843i	−0.240192	−	0.138675i
$417$		0			0
$418$	−12.0000	−	20.7846i	−0.586939	−	1.01661i
$419$	4.89898			0.239331			0.119665	−	0.992814i	$-0.461818\pi$
$419$	4.89898			0.239331			0.119665	+	0.992814i	$0.461818\pi$
$420$		0			0
$421$	−16.0000			−0.779792			−0.389896	−	0.920859i	$-0.627489\pi$
$421$	−16.0000			−0.779792			−0.389896	+	0.920859i	$0.627489\pi$
$422$	−6.12372	−	10.6066i	−0.298098	−	0.516321i
$423$		0			0
$424$	14.0000	−	24.2487i	0.679900	−	1.17762i
$425$	−3.67423	+	2.12132i	−0.178227	+	0.102899i
$426$		0			0
$427$	−12.5000	−	4.33013i	−0.604917	−	0.209550i
$428$	34.2929			1.65761
$429$		0			0
$430$	−30.0000	−	17.3205i	−1.44673	−	0.835269i
$431$	6.12372	−	10.6066i	0.294969	−	0.510902i	−0.680008	−	0.733204i	$-0.738024\pi$
$431$	6.12372	−	10.6066i	0.294969	−	0.510902i	0.974978	+	0.222302i	$0.0713572\pi$
$432$		0			0
$433$	−19.0000			−0.913082			−0.456541	−	0.889702i	$-0.650912\pi$
$433$	−19.0000			−0.913082			−0.456541	+	0.889702i	$0.650912\pi$
$434$	−14.6969	+	12.7279i	−0.705476	+	0.610960i
$435$		0			0
$436$	5.00000	+	8.66025i	0.239457	+	0.414751i
$437$	14.6969	+	8.48528i	0.703050	+	0.405906i
$438$		0			0
$439$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$439$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$440$	−19.5959			−0.934199
$441$		0			0
$442$	−2.00000			−0.0951303
$443$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$443$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$444$		0			0
$445$	−20.0000	+	34.6410i	−0.948091	+	1.64214i
$446$	−4.89898	−	8.48528i	−0.231973	−	0.401790i
$447$		0			0
$448$	−16.0000	+	13.8564i	−0.755929	+	0.654654i
$449$			31.1127i			1.46830i	0.678988	+	0.734150i	$0.262419\pi$
$449$			31.1127i			1.46830i	−0.678988	+	0.734150i	$0.737581\pi$
$450$		0			0
$451$	−15.0000	−	8.66025i	−0.706322	−	0.407795i
$452$	−4.89898	−	2.82843i	−0.230429	−	0.133038i
$453$		0			0
$454$		−	17.3205i		−	0.812892i
$455$	2.44949	−	7.07107i	0.114834	−	0.331497i
$456$		0			0
$457$	−2.50000	−	4.33013i	−0.116945	−	0.202555i	0.801611	−	0.597847i	$-0.203977\pi$
$457$	−2.50000	−	4.33013i	−0.116945	−	0.202555i	−0.918556	+	0.395292i	$0.870643\pi$
$458$	20.8207	+	12.0208i	0.972886	+	0.561696i
$459$		0			0
$460$	12.0000	−	6.92820i	0.559503	−	0.323029i
$461$			1.41421i			0.0658665i	0.999458	+	0.0329332i	$0.0104849\pi$
$461$			1.41421i			0.0658665i	−0.999458	+	0.0329332i	$0.989515\pi$
$462$		0			0
$463$		−	6.92820i		−	0.321981i	−0.986956	−	0.160990i	$-0.948531\pi$
$463$		−	6.92820i		−	0.321981i	0.986956	−	0.160990i	$-0.0514688\pi$
$464$	−24.4949	+	14.1421i	−1.13715	+	0.656532i
$465$		0			0
$466$	8.00000	−	13.8564i	0.370593	−	0.641886i
$467$	12.2474	+	21.2132i	0.566744	+	0.981630i	0.996885	+	0.0788681i	$0.0251306\pi$
$467$	12.2474	+	21.2132i	0.566744	+	0.981630i	−0.430141	+	0.902762i	$0.641536\pi$
$468$		0			0
$469$	−13.5000	+	2.59808i	−0.623372	+	0.119968i
$470$	29.3939			1.35584
$471$		0			0
$472$	18.0000	+	10.3923i	0.828517	+	0.478345i
$473$	−18.3712	−	10.6066i	−0.844707	−	0.487692i
$474$		0			0
$475$			20.7846i			0.953663i
$476$	−2.44949	+	7.07107i	−0.112272	+	0.324102i
$477$		0			0
$478$	30.0000	−	17.3205i	1.37217	−	0.792222i
$479$	−2.44949	+	4.24264i	−0.111920	+	0.193851i	−0.916544	−	0.399933i	$-0.869033\pi$
$479$	−2.44949	+	4.24264i	−0.111920	+	0.193851i	0.804624	+	0.593784i	$0.202367\pi$
$480$		0			0
$481$	2.50000	+	4.33013i	0.113990	+	0.197437i
$482$			35.3553i			1.61039i
$483$		0			0
$484$	10.0000			0.454545
$485$	−12.2474	+	7.07107i	−0.556128	+	0.321081i
$486$		0			0
$487$	30.0000	+	17.3205i	1.35943	+	0.784867i	0.989547	−	0.144211i	$-0.0460645\pi$
$487$	30.0000	+	17.3205i	1.35943	+	0.784867i	0.369883	+	0.929078i	$0.379398\pi$
$488$	12.2474	−	7.07107i	0.554416	−	0.320092i
$489$		0			0
$490$	−22.0000	−	17.3205i	−0.993859	−	0.782461i
$491$	12.2474			0.552720			0.276360	−	0.961054i	$-0.410872\pi$
$491$	12.2474			0.552720			0.276360	+	0.961054i	$0.410872\pi$
$492$		0			0
$493$	−5.00000	+	8.66025i	−0.225189	+	0.390038i
$494$	−4.89898	+	8.48528i	−0.220416	+	0.381771i
$495$		0			0
$496$		−	20.7846i		−	0.933257i
$497$	−29.3939	+	25.4558i	−1.31850	+	1.14185i
$498$		0			0
$499$	28.5000	−	16.4545i	1.27584	−	0.736604i	0.299755	−	0.954016i	$-0.403095\pi$
$499$	28.5000	−	16.4545i	1.27584	−	0.736604i	0.976080	+	0.217412i	$0.0697616\pi$
$500$	−9.79796	−	5.65685i	−0.438178	−	0.252982i
$501$		0			0
$502$	−27.0000	+	15.5885i	−1.20507	+	0.695747i
$503$	−36.7423			−1.63826			−0.819130	−	0.573608i	$-0.805543\pi$
$503$	−36.7423			−1.63826			−0.819130	+	0.573608i	$0.805543\pi$
$504$		0			0
$505$	40.0000			1.77998
$506$	7.34847	−	4.24264i	0.326679	−	0.188608i
$507$		0			0
$508$	−9.00000	−	5.19615i	−0.399310	−	0.230542i
$509$	15.9217	−	9.19239i	0.705716	−	0.407445i	−0.103757	−	0.994603i	$-0.533086\pi$
$509$	15.9217	−	9.19239i	0.705716	−	0.407445i	0.809473	+	0.587157i	$0.199753\pi$
$510$		0			0
$511$	−2.00000	−	10.3923i	−0.0884748	−	0.459728i
$512$		−	22.6274i		−	1.00000i
$513$		0			0
$514$	11.0000	−	19.0526i	0.485189	−	0.840372i
$515$	26.9444	−	46.6690i	1.18731	−	2.05648i
$516$		0			0
$517$	18.0000			0.791639
$518$	18.3712	−	3.53553i	0.807183	−	0.155342i
$519$		0			0
$520$	4.00000	+	6.92820i	0.175412	+	0.303822i
$521$	−30.6186	−	17.6777i	−1.34143	−	0.774473i	−0.354409	−	0.935090i	$-0.615318\pi$
$521$	−30.6186	−	17.6777i	−1.34143	−	0.774473i	−0.987017	+	0.160618i	$0.948651\pi$
$522$		0			0
$523$	−1.50000	+	0.866025i	−0.0655904	+	0.0378686i	−0.532437	−	0.846470i	$-0.678724\pi$
$523$	−1.50000	+	0.866025i	−0.0655904	+	0.0378686i	0.466846	+	0.884339i	$0.345390\pi$
$524$	−19.5959			−0.856052
$525$		0			0
$526$			3.46410i			0.151042i
$527$	−3.67423	−	6.36396i	−0.160052	−	0.277218i
$528$		0			0
$529$	8.50000	−	14.7224i	0.369565	−	0.640106i
$530$	−34.2929	+	19.7990i	−1.48959	+	0.860013i
$531$		0			0
$532$	24.0000	+	27.7128i	1.04053	+	1.20150i
$533$			7.07107i			0.306282i
$534$		0			0
$535$	−42.0000	−	24.2487i	−1.81582	−	1.04836i
$536$	7.34847	−	12.7279i	0.317406	−	0.549762i
$537$		0			0
$538$	−16.0000			−0.689809
$539$	−13.4722	−	10.6066i	−0.580288	−	0.456859i
$540$		0			0
$541$	2.00000	+	3.46410i	0.0859867	+	0.148933i	0.905811	−	0.423681i	$-0.139262\pi$
$541$	2.00000	+	3.46410i	0.0859867	+	0.148933i	−0.819825	+	0.572615i	$0.805929\pi$
$542$	−6.12372	+	10.6066i	−0.263036	+	0.455593i
$543$		0			0
$544$	−4.00000	−	6.92820i	−0.171499	−	0.297044i
$545$		−	14.1421i		−	0.605783i
$546$		0			0
$547$			12.1244i			0.518400i	0.965824	+	0.259200i	$0.0834589\pi$
$547$			12.1244i			0.518400i	−0.965824	+	0.259200i	$0.916541\pi$
$548$	12.2474	−	7.07107i	0.523185	−	0.302061i
$549$		0			0
$550$	9.00000	+	5.19615i	0.383761	+	0.221565i
$551$	24.4949	+	42.4264i	1.04352	+	1.80743i
$552$		0			0
$553$	13.5000	−	38.9711i	0.574078	−	1.65722i
$554$			18.3848i			0.781094i
$555$		0			0
$556$	15.0000	+	8.66025i	0.636142	+	0.367277i
$557$	31.8434	+	18.3848i	1.34925	+	0.778988i	0.988143	−	0.153538i	$-0.0490666\pi$
$557$	31.8434	+	18.3848i	1.34925	+	0.778988i	0.361104	+	0.932526i	$0.382400\pi$
$558$		0			0
$559$			8.66025i			0.366290i
$560$	29.3939	−	5.65685i	1.24212	−	0.239046i
$561$		0			0
$562$	14.0000	+	24.2487i	0.590554	+	1.02287i
$563$	−18.3712	+	31.8198i	−0.774253	+	1.34104i	0.160961	+	0.986961i	$0.448541\pi$
$563$	−18.3712	+	31.8198i	−0.774253	+	1.34104i	−0.935214	+	0.354084i	$0.884793\pi$
$564$		0			0
$565$	4.00000	+	6.92820i	0.168281	+	0.291472i
$566$	−22.0454			−0.926638
$567$		0			0
$568$		−	41.5692i		−	1.74421i
$569$	12.2474	−	7.07107i	0.513440	−	0.296435i	−0.220807	−	0.975318i	$-0.570869\pi$
$569$	12.2474	−	7.07107i	0.513440	−	0.296435i	0.734246	+	0.678883i	$0.237536\pi$
$570$		0			0
$571$	27.0000	+	15.5885i	1.12991	+	0.652357i	0.943913	−	0.330193i	$-0.107114\pi$
$571$	27.0000	+	15.5885i	1.12991	+	0.652357i	0.186001	+	0.982549i	$0.440447\pi$
$572$	2.44949	+	4.24264i	0.102418	+	0.177394i
$573$		0			0
$574$	25.0000	+	8.66025i	1.04348	+	0.361472i
$575$	−7.34847			−0.306452
$576$		0			0
$577$	−2.50000	+	4.33013i	−0.104076	+	0.180266i	−0.913360	−	0.407152i	$-0.866522\pi$
$577$	−2.50000	+	4.33013i	−0.104076	+	0.180266i	0.809284	+	0.587417i	$0.199855\pi$
$578$	18.3712	+	10.6066i	0.764140	+	0.441176i
$579$		0			0
$580$	40.0000			1.66091
$581$	4.89898	−	4.24264i	0.203244	−	0.176014i
$582$		0			0
$583$	−21.0000	+	12.1244i	−0.869731	+	0.502140i
$584$	9.79796	+	5.65685i	0.405442	+	0.234082i
$585$		0			0
$586$	−7.00000	−	12.1244i	−0.289167	−	0.500853i
$587$	−4.89898			−0.202203			−0.101101	−	0.994876i	$-0.532237\pi$
$587$	−4.89898			−0.202203			−0.101101	+	0.994876i	$0.532237\pi$
$588$		0			0
$589$	−36.0000			−1.48335
$590$	−14.6969	−	25.4558i	−0.605063	−	1.04800i
$591$		0			0
$592$	−10.0000	+	17.3205i	−0.410997	+	0.711868i
$593$	12.2474	−	7.07107i	0.502942	−	0.290374i	−0.226985	−	0.973898i	$-0.572887\pi$
$593$	12.2474	−	7.07107i	0.502942	−	0.290374i	0.729928	+	0.683524i	$0.239554\pi$
$594$		0			0
$595$	8.00000	−	6.92820i	0.327968	−	0.284029i
$596$			5.65685i			0.231714i
$597$		0			0
$598$	−3.00000	−	1.73205i	−0.122679	−	0.0708288i
$599$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$599$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$600$		0			0
$601$	11.0000			0.448699			0.224350	−	0.974509i	$-0.427974\pi$
$601$	11.0000			0.448699			0.224350	+	0.974509i	$0.427974\pi$
$602$	30.6186	+	10.6066i	1.24792	+	0.432293i
$603$		0			0
$604$	−15.0000	+	8.66025i	−0.610341	+	0.352381i
$605$	−12.2474	−	7.07107i	−0.497930	−	0.287480i
$606$		0			0
$607$	−12.0000	+	6.92820i	−0.487065	+	0.281207i	−0.723356	−	0.690475i	$-0.757401\pi$
$607$	−12.0000	+	6.92820i	−0.487065	+	0.281207i	0.236291	+	0.971682i	$0.424068\pi$
$608$	−39.1918			−1.58944
$609$		0			0
$610$	−20.0000			−0.809776
$611$	−3.67423	−	6.36396i	−0.148644	−	0.257458i
$612$		0			0
$613$	0.500000	−	0.866025i	0.0201948	−	0.0349784i	−0.855751	−	0.517387i	$-0.826905\pi$
$613$	0.500000	−	0.866025i	0.0201948	−	0.0349784i	0.875946	+	0.482409i	$0.160238\pi$
$614$	−3.67423	−	6.36396i	−0.148280	−	0.256829i
$615$		0			0
$616$	18.0000	−	3.46410i	0.725241	−	0.139573i
$617$		−	15.5563i		−	0.626275i	−0.949708	−	0.313138i	$-0.898620\pi$
$617$		−	15.5563i		−	0.626275i	0.949708	−	0.313138i	$-0.101380\pi$
$618$		0			0
$619$	−7.50000	−	4.33013i	−0.301450	−	0.174042i	0.341644	−	0.939829i	$-0.389016\pi$
$619$	−7.50000	−	4.33013i	−0.301450	−	0.174042i	−0.643094	+	0.765787i	$0.722350\pi$
$620$	−14.6969	+	25.4558i	−0.590243	+	1.02233i
$621$		0			0
$622$			20.7846i			0.833387i
$623$	12.2474	−	35.3553i	0.490684	−	1.41648i
$624$		0			0
$625$	15.5000	+	26.8468i	0.620000	+	1.07387i
$626$	17.1464	+	9.89949i	0.685309	+	0.395663i
$627$		0			0
$628$	20.0000	+	34.6410i	0.798087	+	1.38233i
$629$			7.07107i			0.281942i
$630$		0			0
$631$		−	25.9808i		−	1.03428i	−0.855901	−	0.517139i	$-0.826997\pi$
$631$		−	25.9808i		−	1.03428i	0.855901	−	0.517139i	$-0.173003\pi$
$632$	22.0454	+	38.1838i	0.876919	+	1.51887i
$633$		0			0
$634$	−19.0000	+	32.9090i	−0.754586	+	1.30698i
$635$	7.34847	+	12.7279i	0.291615	+	0.505092i
$636$		0			0
$637$	−1.00000	+	6.92820i	−0.0396214	+	0.274505i
$638$	24.4949			0.969762
$639$		0			0
$640$	−16.0000	+	27.7128i	−0.632456	+	1.09545i
$641$	−37.9671	−	21.9203i	−1.49961	−	0.865800i	−0.499610	−	0.866250i	$-0.666523\pi$
$641$	−37.9671	−	21.9203i	−1.49961	−	0.865800i	−1.00000		0.000450234i	$0.999857\pi$
$642$		0			0
$643$		−	12.1244i		−	0.478138i	−0.971003	−	0.239069i	$-0.923158\pi$
$643$		−	12.1244i		−	0.478138i	0.971003	−	0.239069i	$-0.0768422\pi$
$644$	−9.79796	+	8.48528i	−0.386094	+	0.334367i
$645$		0			0
$646$	−12.0000	+	6.92820i	−0.472134	+	0.272587i
$647$	9.79796	−	16.9706i	0.385198	−	0.667182i	−0.606599	−	0.795008i	$-0.707467\pi$
$647$	9.79796	−	16.9706i	0.385198	−	0.667182i	0.991797	+	0.127826i	$0.0408000\pi$
$648$		0			0
$649$	−9.00000	−	15.5885i	−0.353281	−	0.611900i
$650$		−	4.24264i		−	0.166410i
$651$		0			0
$652$		−	38.1051i		−	1.49231i
$653$	−6.12372	+	3.53553i	−0.239640	+	0.138356i	−0.615011	−	0.788518i	$-0.710849\pi$
$653$	−6.12372	+	3.53553i	−0.239640	+	0.138356i	0.375371	+	0.926875i	$0.377515\pi$
$654$		0			0
$655$	24.0000	+	13.8564i	0.937758	+	0.541415i
$656$	−24.4949	+	14.1421i	−0.956365	+	0.552158i
$657$		0			0
$658$	−27.0000	+	5.19615i	−1.05257	+	0.202567i
$659$	24.4949			0.954186			0.477093	−	0.878853i	$-0.341691\pi$
$659$	24.4949			0.954186			0.477093	+	0.878853i	$0.341691\pi$
$660$		0			0
$661$	−10.0000	+	17.3205i	−0.388955	+	0.673690i	−0.992309	−	0.123784i	$-0.960497\pi$
$661$	−10.0000	+	17.3205i	−0.388955	+	0.673690i	0.603354	+	0.797473i	$0.293830\pi$
$662$		0			0
$663$		0			0
$664$			6.92820i			0.268866i
$665$	−9.79796	−	50.9117i	−0.379949	−	1.97427i
$666$		0			0
$667$	−15.0000	+	8.66025i	−0.580802	+	0.335326i
$668$	12.2474	−	21.2132i	0.473868	−	0.820763i
$669$		0			0
$670$	−18.0000	+	10.3923i	−0.695401	+	0.401490i
$671$	−12.2474			−0.472808
$672$		0			0
$673$	20.0000			0.770943			0.385472	−	0.922720i	$-0.374039\pi$
$673$	20.0000			0.770943			0.385472	+	0.922720i	$0.374039\pi$
$674$	−24.4949	+	14.1421i	−0.943508	+	0.544735i
$675$		0			0
$676$	−12.0000	+	20.7846i	−0.461538	+	0.799408i
$677$	−24.4949	+	14.1421i	−0.941415	+	0.543526i	−0.890404	−	0.455172i	$-0.849578\pi$
$677$	−24.4949	+	14.1421i	−0.941415	+	0.543526i	−0.0510117	+	0.998698i	$0.516245\pi$
$678$		0			0
$679$	10.0000	−	8.66025i	0.383765	−	0.332350i
$680$			11.3137i			0.433861i
$681$		0			0
$682$	−9.00000	+	15.5885i	−0.344628	+	0.596913i
$683$	17.1464	−	29.6985i	0.656090	−	1.13638i	−0.325530	−	0.945532i	$-0.605543\pi$
$683$	17.1464	−	29.6985i	0.656090	−	1.13638i	0.981619	−	0.190849i	$-0.0611241\pi$
$684$		0			0
$685$	−20.0000			−0.764161
$686$	23.2702	+	12.0208i	0.888459	+	0.458957i
$687$		0			0
$688$	−30.0000	+	17.3205i	−1.14374	+	0.660338i
$689$	8.57321	+	4.94975i	0.326613	+	0.188570i
$690$		0			0
$691$	−4.50000	+	2.59808i	−0.171188	+	0.0988355i	−0.583146	−	0.812367i	$-0.698178\pi$
$691$	−4.50000	+	2.59808i	−0.171188	+	0.0988355i	0.411958	+	0.911203i	$0.364845\pi$
$692$			22.6274i			0.860165i
$693$		0			0
$694$			41.5692i			1.57795i
$695$	−12.2474	−	21.2132i	−0.464572	−	0.804663i
$696$		0			0
$697$	−5.00000	+	8.66025i	−0.189389	+	0.328031i
$698$	15.9217	−	9.19239i	0.602645	−	0.347937i
$699$		0			0
$700$	−15.0000	−	5.19615i	−0.566947	−	0.196396i
$701$		−	28.2843i		−	1.06828i	−0.845395	−	0.534141i	$-0.820635\pi$
$701$		−	28.2843i		−	1.06828i	0.845395	−	0.534141i	$-0.179365\pi$
$702$		0			0
$703$	30.0000	+	17.3205i	1.13147	+	0.653255i
$704$	−9.79796	+	16.9706i	−0.369274	+	0.639602i
$705$		0			0
$706$	−16.0000			−0.602168
$707$	−36.7423	+	7.07107i	−1.38184	+	0.265935i
$708$		0			0
$709$	−11.5000	−	19.9186i	−0.431892	−	0.748058i	0.565145	−	0.824992i	$-0.308820\pi$
$709$	−11.5000	−	19.9186i	−0.431892	−	0.748058i	−0.997036	+	0.0769337i	$0.975487\pi$
$710$	−29.3939	+	50.9117i	−1.10313	+	1.91068i
$711$		0			0
$712$	20.0000	+	34.6410i	0.749532	+	1.29823i
$713$		−	12.7279i		−	0.476664i
$714$		0			0
$715$		−	6.92820i		−	0.259100i
$716$	−4.89898	−	8.48528i	−0.183083	−	0.317110i
$717$		0			0
$718$	15.0000	+	8.66025i	0.559795	+	0.323198i
$719$	−2.44949	−	4.24264i	−0.0913506	−	0.158224i	0.816729	−	0.577021i	$-0.195785\pi$
$719$	−2.44949	−	4.24264i	−0.0913506	−	0.158224i	−0.908080	+	0.418797i	$0.862452\pi$
$720$		0			0
$721$	−16.5000	+	47.6314i	−0.614492	+	1.77389i
$722$			41.0122i			1.52632i
$723$		0			0
$724$	−16.0000	+	27.7128i	−0.594635	+	1.02994i
$725$	−18.3712	−	10.6066i	−0.682288	−	0.393919i
$726$		0			0
$727$		−	8.66025i		−	0.321191i	−0.987020	−	0.160596i	$-0.948659\pi$
$727$		−	8.66025i		−	0.321191i	0.987020	−	0.160596i	$-0.0513415\pi$
$728$	−4.89898	−	5.65685i	−0.181568	−	0.209657i
$729$		0			0
$730$	−8.00000	−	13.8564i	−0.296093	−	0.512849i
$731$	−6.12372	+	10.6066i	−0.226494	+	0.392299i
$732$		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	$0.898450\pi$
$734$	4.89898			0.180825
$735$		0			0
$736$		−	13.8564i		−	0.510754i
$737$	−11.0227	+	6.36396i	−0.406027	+	0.234420i
$738$		0			0
$739$	−28.5000	−	16.4545i	−1.04839	−	0.605288i	−0.126191	−	0.992006i	$-0.540275\pi$
$739$	−28.5000	−	16.4545i	−1.04839	−	0.605288i	−0.922198	+	0.386718i	$0.873609\pi$
$740$	24.4949	−	14.1421i	0.900450	−	0.519875i
$741$		0			0
$742$	28.0000	−	24.2487i	1.02791	−	0.890198i
$743$	12.2474			0.449315			0.224658	−	0.974438i	$-0.427874\pi$
$743$	12.2474			0.449315			0.224658	+	0.974438i	$0.427874\pi$
$744$		0			0
$745$	4.00000	−	6.92820i	0.146549	−	0.253830i
$746$	24.4949	+	14.1421i	0.896822	+	0.517780i
$747$		0			0
$748$			6.92820i			0.253320i
$749$	42.8661	+	14.8492i	1.56629	+	0.542580i
$750$		0			0
$751$	−30.0000	+	17.3205i	−1.09472	+	0.632034i	−0.934828	−	0.355101i	$-0.884447\pi$
$751$	−30.0000	+	17.3205i	−1.09472	+	0.632034i	−0.159888	+	0.987135i	$0.551113\pi$
$752$	14.6969	−	25.4558i	0.535942	−	0.928279i
$753$		0			0
$754$	−5.00000	−	8.66025i	−0.182089	−	0.315388i
$755$	24.4949			0.891461
$756$		0			0
$757$	−7.00000			−0.254419			−0.127210	−	0.991876i	$-0.540602\pi$
$757$	−7.00000			−0.254419			−0.127210	+	0.991876i	$0.540602\pi$
$758$	−18.3712	−	31.8198i	−0.667271	−	1.15575i
$759$		0			0
$760$	48.0000	+	27.7128i	1.74114	+	1.00525i
$761$	−24.4949	+	14.1421i	−0.887939	+	0.512652i	−0.873268	−	0.487240i	$-0.838004\pi$
$761$	−24.4949	+	14.1421i	−0.887939	+	0.512652i	−0.0146714	+	0.999892i	$0.504670\pi$
$762$		0			0
$763$	2.50000	+	12.9904i	0.0905061	+	0.470283i
$764$	29.3939			1.06343
$765$		0			0
$766$	−33.0000	−	19.0526i	−1.19234	−	0.688397i
$767$	−3.67423	+	6.36396i	−0.132669	+	0.229789i
$768$		0			0
$769$	8.00000			0.288487			0.144244	−	0.989542i	$-0.453925\pi$
$769$	8.00000			0.288487			0.144244	+	0.989542i	$0.453925\pi$
$770$	−24.4949	−	8.48528i	−0.882735	−	0.305788i
$771$		0			0
$772$	11.0000	+	19.0526i	0.395899	+	0.685717i
$773$	−34.2929	−	19.7990i	−1.23343	−	0.712120i	−0.265685	−	0.964060i	$-0.585598\pi$
$773$	−34.2929	−	19.7990i	−1.23343	−	0.712120i	−0.967743	+	0.251940i	$0.918932\pi$
$774$		0			0
$775$	13.5000	−	7.79423i	0.484934	−	0.279977i
$776$			14.1421i			0.507673i
$777$		0			0
$778$	−16.0000			−0.573628
$779$	24.4949	+	42.4264i	0.877621	+	1.52008i
$780$		0			0
$781$	−18.0000	+	31.1769i	−0.644091	+	1.11560i
$782$	−2.44949	−	4.24264i	−0.0875936	−	0.151717i
$783$		0			0
$784$	−26.0000	+	10.3923i	−0.928571	+	0.371154i
$785$		−	56.5685i		−	2.01902i
$786$		0			0
$787$	22.5000	+	12.9904i	0.802038	+	0.463057i	0.844183	−	0.536054i	$-0.180086\pi$
$787$	22.5000	+	12.9904i	0.802038	+	0.463057i	−0.0421450	+	0.999112i	$0.513419\pi$
$788$	39.1918	+	22.6274i	1.39615	+	0.806068i
$789$		0			0
$790$		−	62.3538i		−	2.21845i
$791$	−4.89898	−	5.65685i	−0.174188	−	0.201135i
$792$		0			0
$793$	2.50000	+	4.33013i	0.0887776	+	0.153767i
$794$	−15.9217	−	9.19239i	−0.565039	−	0.326226i
$795$		0			0
$796$	15.0000	−	8.66025i	0.531661	−	0.306955i
$797$		−	41.0122i		−	1.45273i	−0.687311	−	0.726363i	$-0.741209\pi$
$797$		−	41.0122i		−	1.45273i	0.687311	−	0.726363i	$-0.258791\pi$
$798$		0			0
$799$		−	10.3923i		−	0.367653i
$800$	14.6969	−	8.48528i	0.519615	−	0.300000i
$801$		0			0
$802$	11.0000	−	19.0526i	0.388424	−	0.672769i
$803$	−4.89898	−	8.48528i	−0.172881	−	0.299439i
$804$		0			0
$805$	18.0000	−	3.46410i	0.634417	−	0.122094i
$806$	7.34847			0.258839
$807$		0			0
$808$	20.0000	−	34.6410i	0.703598	−	1.21867i
$809$	20.8207	+	12.0208i	0.732016	+	0.422629i	0.819159	−	0.573566i	$-0.194441\pi$
$809$	20.8207	+	12.0208i	0.732016	+	0.422629i	−0.0871435	+	0.996196i	$0.527774\pi$
$810$		0			0
$811$		−	51.9615i		−	1.82462i	−0.409505	−	0.912308i	$-0.634299\pi$
$811$		−	51.9615i		−	1.82462i	0.409505	−	0.912308i	$-0.365701\pi$
$812$	−36.7423	+	7.07107i	−1.28940	+	0.248146i
$813$		0			0
$814$	15.0000	−	8.66025i	0.525750	−	0.303542i
$815$	−26.9444	+	46.6690i	−0.943821	+	1.63475i
$816$		0			0
$817$	30.0000	+	51.9615i	1.04957	+	1.81790i
$818$			9.89949i			0.346128i
$819$		0			0
$820$	40.0000			1.39686
$821$	23.2702	−	13.4350i	0.812134	−	0.468886i	−0.0355625	−	0.999367i	$-0.511322\pi$
$821$	23.2702	−	13.4350i	0.812134	−	0.468886i	0.847696	+	0.530482i	$0.177989\pi$
$822$		0			0
$823$	22.5000	+	12.9904i	0.784301	+	0.452816i	0.837952	−	0.545743i	$-0.183753\pi$
$823$	22.5000	+	12.9904i	0.784301	+	0.452816i	−0.0536516	+	0.998560i	$0.517086\pi$
$824$	−26.9444	−	46.6690i	−0.938652	−	1.62579i
$825$		0			0
$826$	18.0000	+	20.7846i	0.626300	+	0.723189i
$827$	−29.3939			−1.02213			−0.511063	−	0.859543i	$-0.670748\pi$
$827$	−29.3939			−1.02213			−0.511063	+	0.859543i	$0.670748\pi$
$828$		0			0
$829$	−4.00000	+	6.92820i	−0.138926	+	0.240626i	−0.927090	−	0.374838i	$-0.877698\pi$
$829$	−4.00000	+	6.92820i	−0.138926	+	0.240626i	0.788165	+	0.615465i	$0.211032\pi$
$830$	4.89898	−	8.48528i	0.170046	−	0.294528i
$831$		0			0
$832$	8.00000			0.277350
$833$	−6.12372	+	7.77817i	−0.212174	+	0.269498i
$834$		0			0
$835$	−30.0000	+	17.3205i	−1.03819	+	0.599401i
$836$	29.3939	+	16.9706i	1.01661	+	0.586939i
$837$		0			0
$838$	−6.00000	+	3.46410i	−0.207267	+	0.119665i
$839$	2.44949			0.0845658			0.0422829	−	0.999106i	$-0.486537\pi$
$839$	2.44949			0.0845658			0.0422829	+	0.999106i	$0.486537\pi$
$840$		0			0
$841$	−21.0000			−0.724138
$842$	19.5959	−	11.3137i	0.675320	−	0.389896i
$843$		0			0
$844$	15.0000	+	8.66025i	0.516321	+	0.298098i
$845$	29.3939	−	16.9706i	1.01118	−	0.583805i
$846$		0			0
$847$	12.5000	+	4.33013i	0.429505	+	0.148785i
$848$			39.5980i			1.35980i
$849$		0			0
$850$	3.00000	−	5.19615i	0.102899	−	0.178227i
$851$	−6.12372	+	10.6066i	−0.209919	+	0.363590i
$852$		0			0
$853$	−40.0000			−1.36957			−0.684787	−	0.728743i	$-0.740105\pi$
$853$	−40.0000			−1.36957			−0.684787	+	0.728743i	$0.740105\pi$
$854$	18.3712	−	3.53553i	0.628649	−	0.120983i
$855$		0			0
$856$	−42.0000	+	24.2487i	−1.43553	+	0.828804i
$857$	17.1464	+	9.89949i	0.585711	+	0.338160i	0.763400	−	0.645926i	$-0.223529\pi$
$857$	17.1464	+	9.89949i	0.585711	+	0.338160i	−0.177689	+	0.984087i	$0.556862\pi$
$858$		0			0
$859$	−7.50000	+	4.33013i	−0.255897	+	0.147742i	−0.622461	−	0.782651i	$-0.713867\pi$
$859$	−7.50000	+	4.33013i	−0.255897	+	0.147742i	0.366565	+	0.930393i	$0.380534\pi$
$860$	48.9898			1.67054
$861$		0			0
$862$			17.3205i			0.589939i
$863$	4.89898	+	8.48528i	0.166763	+	0.288842i	0.937280	−	0.348577i	$-0.113335\pi$
$863$	4.89898	+	8.48528i	0.166763	+	0.288842i	−0.770517	+	0.637420i	$0.780002\pi$
$864$		0			0
$865$	16.0000	−	27.7128i	0.544016	−	0.942264i
$866$	23.2702	−	13.4350i	0.790752	−	0.456541i
$867$		0			0
$868$	9.00000	−	25.9808i	0.305480	−	0.881845i
$869$		−	38.1838i		−	1.29530i
$870$		0			0
$871$	4.50000	+	2.59808i	0.152477	+	0.0880325i
$872$	−12.2474	−	7.07107i	−0.414751	−	0.239457i
$873$		0			0
$874$	−24.0000			−0.811812
$875$	−9.79796	−	11.3137i	−0.331231	−	0.382473i
$876$		0			0
$877$	27.5000	+	47.6314i	0.928609	+	1.60840i	0.785652	+	0.618669i	$0.212328\pi$
$877$	27.5000	+	47.6314i	0.928609	+	1.60840i	0.142957	+	0.989729i	$0.454339\pi$
$878$		0			0
$879$		0			0
$880$	24.0000	−	13.8564i	0.809040	−	0.467099i
$881$		−	7.07107i		−	0.238230i	−0.992880	−	0.119115i	$-0.961994\pi$
$881$		−	7.07107i		−	0.238230i	0.992880	−	0.119115i	$-0.0380058\pi$
$882$		0			0
$883$			10.3923i			0.349729i	0.984593	+	0.174864i	$0.0559487\pi$
$883$			10.3923i			0.349729i	−0.984593	+	0.174864i	$0.944051\pi$
$884$	2.44949	−	1.41421i	0.0823853	−	0.0475651i
$885$		0			0
$886$		0			0
$887$	20.8207	+	36.0624i	0.699089	+	1.21086i	0.968783	+	0.247912i	$0.0797444\pi$
$887$	20.8207	+	36.0624i	0.699089	+	1.21086i	−0.269693	+	0.962946i	$0.586922\pi$
$888$		0			0
$889$	−9.00000	−	10.3923i	−0.301850	−	0.348547i
$890$		−	56.5685i		−	1.89618i
$891$		0			0
$892$	12.0000	+	6.92820i	0.401790	+	0.231973i
$893$	−44.0908	−	25.4558i	−1.47544	−	0.851847i
$894$		0			0
$895$			13.8564i			0.463169i
$896$	9.79796	−	28.2843i	0.327327	−	0.944911i
$897$		0			0
$898$	−22.0000	−	38.1051i	−0.734150	−	1.27158i
$899$	18.3712	−	31.8198i	0.612713	−	1.06125i
$900$		0			0
$901$	7.00000	+	12.1244i	0.233204	+	0.403921i
$902$	24.4949			0.815591
$903$		0			0
$904$	8.00000			0.266076
$905$	39.1918	−	22.6274i	1.30278	−	0.752161i
$906$		0			0
$907$	−25.5000	−	14.7224i	−0.846714	−	0.488850i	0.0128270	−	0.999918i	$-0.495917\pi$
$907$	−25.5000	−	14.7224i	−0.846714	−	0.488850i	−0.859541	+	0.511067i	$0.829250\pi$
$908$	12.2474	+	21.2132i	0.406446	+	0.703985i
$909$		0			0
$910$	2.00000	+	10.3923i	0.0662994	+	0.344502i
$911$	−26.9444			−0.892707			−0.446354	−	0.894857i	$-0.647278\pi$
$911$	−26.9444			−0.892707			−0.446354	+	0.894857i	$0.647278\pi$
$912$		0			0
$913$	3.00000	−	5.19615i	0.0992855	−	0.171968i
$914$	6.12372	+	3.53553i	0.202555	+	0.116945i
$915$		0			0
$916$	−34.0000			−1.12339
$917$	−24.4949	−	8.48528i	−0.808893	−	0.280209i
$918$		0			0
$919$	7.50000	−	4.33013i	0.247402	−	0.142838i	−0.371172	−	0.928564i	$-0.621044\pi$
$919$	7.50000	−	4.33013i	0.247402	−	0.142838i	0.618574	+	0.785726i	$0.287711\pi$
$920$	−9.79796	+	16.9706i	−0.323029	+	0.559503i
$921$		0			0
$922$	−1.00000	−	1.73205i	−0.0329332	−	0.0570421i
$923$	14.6969			0.483756
$924$		0			0
$925$	−15.0000			−0.493197
$926$	4.89898	+	8.48528i	0.160990	+	0.278844i
$927$		0			0
$928$	20.0000	−	34.6410i	0.656532	−	1.13715i
$929$	12.2474	−	7.07107i	0.401826	−	0.231994i	−0.285446	−	0.958395i	$-0.592142\pi$
$929$	12.2474	−	7.07107i	0.401826	−	0.231994i	0.687271	+	0.726401i	$0.258808\pi$
$930$		0			0
$931$	18.0000	+	45.0333i	0.589926	+	1.47591i
$932$			22.6274i			0.741186i
$933$		0			0
$934$	−30.0000	−	17.3205i	−0.981630	−	0.566744i
$935$	4.89898	−	8.48528i	0.160214	−	0.277498i
$936$		0			0
$937$	35.0000			1.14340			0.571700	−	0.820463i	$-0.306284\pi$
$937$	35.0000			1.14340			0.571700	+	0.820463i	$0.306284\pi$
$938$	14.6969	−	12.7279i	0.479872	−	0.415581i
$939$		0			0
$940$	−36.0000	+	20.7846i	−1.17419	+	0.677919i
$941$	6.12372	+	3.53553i	0.199628	+	0.115255i	0.596482	−	0.802627i	$-0.296565\pi$
$941$	6.12372	+	3.53553i	0.199628	+	0.115255i	−0.396854	+	0.917882i	$0.629898\pi$
$942$		0			0
$943$	−15.0000	+	8.66025i	−0.488467	+	0.282017i
$944$	−29.3939			−0.956689
$945$		0			0
$946$	30.0000			0.975384
$947$	−3.67423	−	6.36396i	−0.119397	−	0.206801i	0.800132	−	0.599824i	$-0.204763\pi$
$947$	−3.67423	−	6.36396i	−0.119397	−	0.206801i	−0.919529	+	0.393023i	$0.871429\pi$
$948$		0			0
$949$	−2.00000	+	3.46410i	−0.0649227	+	0.112449i
$950$	−14.6969	−	25.4558i	−0.476832	−	0.825897i
$951$		0			0
$952$	−2.00000	−	10.3923i	−0.0648204	−	0.336817i
$953$			35.3553i			1.14527i	0.819810	+	0.572636i	$0.194079\pi$
$953$			35.3553i			1.14527i	−0.819810	+	0.572636i	$0.805921\pi$
$954$		0			0
$955$	−36.0000	−	20.7846i	−1.16493	−	0.672574i
$956$	−24.4949	+	42.4264i	−0.792222	+	1.37217i
$957$		0			0
$958$		−	6.92820i		−	0.223840i
$959$	18.3712	−	3.53553i	0.593236	−	0.114168i
$960$		0			0
$961$	−2.00000	−	3.46410i	−0.0645161	−	0.111745i
$962$	−6.12372	−	3.53553i	−0.197437	−	0.113990i
$963$		0			0
$964$	−25.0000	−	43.3013i	−0.805196	−	1.39464i
$965$		−	31.1127i		−	1.00155i
$966$		0			0
$967$		−	22.5167i		−	0.724087i	−0.932161	−	0.362043i	$-0.882079\pi$
$967$		−	22.5167i		−	0.724087i	0.932161	−	0.362043i	$-0.117921\pi$
$968$	−12.2474	+	7.07107i	−0.393648	+	0.227273i
$969$		0			0
$970$	10.0000	−	17.3205i	0.321081	−	0.556128i
$971$	−24.4949	−	42.4264i	−0.786079	−	1.36153i	−0.928353	−	0.371701i	$-0.878775\pi$
$971$	−24.4949	−	42.4264i	−0.786079	−	1.36153i	0.142274	−	0.989827i	$-0.454559\pi$
$972$		0			0
$973$	15.0000	+	17.3205i	0.480878	+	0.555270i
$974$	−48.9898			−1.56973
$975$		0			0
$976$	−10.0000	+	17.3205i	−0.320092	+	0.554416i
$977$	−1.22474	−	0.707107i	−0.0391831	−	0.0226224i	0.480281	−	0.877115i	$-0.340535\pi$
$977$	−1.22474	−	0.707107i	−0.0391831	−	0.0226224i	−0.519464	+	0.854493i	$0.673868\pi$
$978$		0			0
$979$		−	34.6410i		−	1.10713i
$980$	39.1918	+	5.65685i	1.25194	+	0.180702i
$981$		0			0
$982$	−15.0000	+	8.66025i	−0.478669	+	0.276360i
$983$	12.2474	−	21.2132i	0.390633	−	0.676596i	−0.601900	−	0.798571i	$-0.705590\pi$
$983$	12.2474	−	21.2132i	0.390633	−	0.676596i	0.992533	+	0.121975i	$0.0389228\pi$
$984$		0			0
$985$	−32.0000	−	55.4256i	−1.01960	−	1.76601i
$986$		−	14.1421i		−	0.450377i
$987$		0			0
$988$		−	13.8564i		−	0.440831i
$989$	−18.3712	+	10.6066i	−0.584169	+	0.337270i
$990$		0			0
$991$	34.5000	+	19.9186i	1.09593	+	0.632735i	0.935149	−	0.354256i	$-0.115266\pi$
$991$	34.5000	+	19.9186i	1.09593	+	0.632735i	0.160780	+	0.986990i	$0.448599\pi$
$992$	14.6969	+	25.4558i	0.466628	+	0.808224i
$993$		0			0
$994$	18.0000	−	51.9615i	0.570925	−	1.64812i
$995$	−24.4949			−0.776540
$996$		0			0
$997$	12.5000	−	21.6506i	0.395879	−	0.685682i	−0.597334	−	0.801993i	$-0.703773\pi$
$997$	12.5000	−	21.6506i	0.395879	−	0.685682i	0.993213	+	0.116310i	$0.0371066\pi$
$998$	−23.2702	+	40.3051i	−0.736604	+	1.27584i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	756.2.be.b.431.1	yes	4
3.2	odd	2	inner	756.2.be.b.431.2	yes	4
4.3	odd	2		756.2.be.a.431.1	yes	4
7.2	even	3		756.2.be.a.107.2	yes	4
12.11	even	2		756.2.be.a.431.2	yes	4
21.2	odd	6		756.2.be.a.107.1	✓	4
28.23	odd	6	inner	756.2.be.b.107.2	yes	4
84.23	even	6	inner	756.2.be.b.107.1	yes	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
756.2.be.a.107.1	✓	4	21.2	odd	6
756.2.be.a.107.2	yes	4	7.2	even	3
756.2.be.a.431.1	yes	4	4.3	odd	2
756.2.be.a.431.2	yes	4	12.11	even	2
756.2.be.b.107.1	yes	4	84.23	even	6	inner
756.2.be.b.107.2	yes	4	28.23	odd	6	inner
756.2.be.b.431.1	yes	4	1.1	even	1	trivial
756.2.be.b.431.2	yes	4	3.2	odd	2	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 756 = 2^{2} \cdot 3^{3} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	756.be (of order \(6\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(-1.22474 + 0.707107i) q^{2} +(1.00000 - 1.73205i) q^{4} +(-2.44949 + 1.41421i) q^{5} +(2.00000 - 1.73205i) q^{7} +2.82843i q^{8} +O(q^{10})\)	\(q+(-1.22474 + 0.707107i) q^{2} +(1.00000 - 1.73205i) q^{4} +(-2.44949 + 1.41421i) q^{5} +(2.00000 - 1.73205i) q^{7} +2.82843i q^{8} +(2.00000 - 3.46410i) q^{10} +(1.22474 - 2.12132i) q^{11} -1.00000 q^{13} +(-1.22474 + 3.53553i) q^{14} +(-2.00000 - 3.46410i) q^{16} +(-1.22474 - 0.707107i) q^{17} +(-6.00000 + 3.46410i) q^{19} +5.65685i q^{20} +3.46410i q^{22} +(-1.22474 - 2.12132i) q^{23} +(1.50000 - 2.59808i) q^{25} +(1.22474 - 0.707107i) q^{26} +(-1.00000 - 5.19615i) q^{28} -7.07107i q^{29} +(4.50000 + 2.59808i) q^{31} +(4.89898 + 2.82843i) q^{32} +2.00000 q^{34} +(-2.44949 + 7.07107i) q^{35} +(-2.50000 - 4.33013i) q^{37} +(4.89898 - 8.48528i) q^{38} +(-4.00000 - 6.92820i) q^{40} -7.07107i q^{41} -8.66025i q^{43} +(-2.44949 - 4.24264i) q^{44} +(3.00000 + 1.73205i) q^{46} +(3.67423 + 6.36396i) q^{47} +(1.00000 - 6.92820i) q^{49} +4.24264i q^{50} +(-1.00000 + 1.73205i) q^{52} +(-8.57321 - 4.94975i) q^{53} +6.92820i q^{55} +(4.89898 + 5.65685i) q^{56} +(5.00000 + 8.66025i) q^{58} +(3.67423 - 6.36396i) q^{59} +(-2.50000 - 4.33013i) q^{61} -7.34847 q^{62} -8.00000 q^{64} +(2.44949 - 1.41421i) q^{65} +(-4.50000 - 2.59808i) q^{67} +(-2.44949 + 1.41421i) q^{68} +(-2.00000 - 10.3923i) q^{70} -14.6969 q^{71} +(2.00000 - 3.46410i) q^{73} +(6.12372 + 3.53553i) q^{74} +13.8564i q^{76} +(-1.22474 - 6.36396i) q^{77} +(13.5000 - 7.79423i) q^{79} +(9.79796 + 5.65685i) q^{80} +(5.00000 + 8.66025i) q^{82} +2.44949 q^{83} +4.00000 q^{85} +(6.12372 + 10.6066i) q^{86} +(6.00000 + 3.46410i) q^{88} +(12.2474 - 7.07107i) q^{89} +(-2.00000 + 1.73205i) q^{91} -4.89898 q^{92} +(-9.00000 - 5.19615i) q^{94} +(9.79796 - 16.9706i) q^{95} +5.00000 q^{97} +(3.67423 + 9.19239i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 4 q^{4} + 8 q^{7}+O(q^{10}) \) 4 * q + 4 * q^4 + 8 * q^7	\( 4 q + 4 q^{4} + 8 q^{7} + 8 q^{10} - 4 q^{13} - 8 q^{16} - 24 q^{19} + 6 q^{25} - 4 q^{28} + 18 q^{31} + 8 q^{34} - 10 q^{37} - 16 q^{40} + 12 q^{46} + 4 q^{49} - 4 q^{52} + 20 q^{58} - 10 q^{61} - 32 q^{64} - 18 q^{67} - 8 q^{70} + 8 q^{73} + 54 q^{79} + 20 q^{82} + 16 q^{85} + 24 q^{88} - 8 q^{91} - 36 q^{94} + 20 q^{97}+O(q^{100}) \) 4 * q + 4 * q^4 + 8 * q^7 + 8 * q^10 - 4 * q^13 - 8 * q^16 - 24 * q^19 + 6 * q^25 - 4 * q^28 + 18 * q^31 + 8 * q^34 - 10 * q^37 - 16 * q^40 + 12 * q^46 + 4 * q^49 - 4 * q^52 + 20 * q^58 - 10 * q^61 - 32 * q^64 - 18 * q^67 - 8 * q^70 + 8 * q^73 + 54 * q^79 + 20 * q^82 + 16 * q^85 + 24 * q^88 - 8 * q^91 - 36 * q^94 + 20 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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