LMFDB - Embedded newform 252.2.bf.c.19.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [252,2,Mod(19,252)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(252, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([3, 0, 5])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("252.19"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 252 = 2^{2} \cdot 3^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	252.bf (of order $6$, degree $2$, minimal)

Newform invariants

comment:select newform

sage:traces = [4,0,0,4,0,0,2] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)

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

Self dual:	no
Analytic conductor:	$2.01223013094$
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:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			19.1
Root			$-1.22474 + 0.707107i$ of defining polynomial
Character	$\chi$	$=$	252.19
Dual form			252.2.bf.c.199.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+(-1.22474 + 0.707107i) q^{2} +(1.00000 - 1.73205i) q^{4} +(-2.44949 - 1.41421i) q^{5} +(0.500000 + 2.59808i) q^{7} +2.82843i q^{8} +4.00000 q^{10} +(4.89898 - 2.82843i) q^{11} -5.19615i q^{13} +(-2.44949 - 2.82843i) q^{14} +(-2.00000 - 3.46410i) q^{16} +(2.44949 - 1.41421i) q^{17} +(2.50000 - 4.33013i) q^{19} +(-4.89898 + 2.82843i) q^{20} +(-4.00000 + 6.92820i) q^{22} +(2.44949 + 1.41421i) q^{23} +(1.50000 + 2.59808i) q^{25} +(3.67423 + 6.36396i) q^{26} +(5.00000 + 1.73205i) q^{28} +(-0.500000 - 0.866025i) q^{31} +(4.89898 + 2.82843i) q^{32} +(-2.00000 + 3.46410i) q^{34} +(2.44949 - 7.07107i) q^{35} +(-2.50000 + 4.33013i) q^{37} +7.07107i q^{38} +(4.00000 - 6.92820i) q^{40} -5.65685i q^{41} +5.19615i q^{43} -11.3137i q^{44} -4.00000 q^{46} +(-2.44949 + 4.24264i) q^{47} +(-6.50000 + 2.59808i) q^{49} +(-3.67423 - 2.12132i) q^{50} +(-9.00000 - 5.19615i) q^{52} +(-2.44949 - 4.24264i) q^{53} -16.0000 q^{55} +(-7.34847 + 1.41421i) q^{56} +(2.44949 + 4.24264i) q^{59} +(6.00000 + 3.46410i) q^{61} +(1.22474 + 0.707107i) q^{62} -8.00000 q^{64} +(-7.34847 + 12.7279i) q^{65} +(7.50000 - 4.33013i) q^{67} -5.65685i q^{68} +(2.00000 + 10.3923i) q^{70} -2.82843i q^{71} +(1.50000 - 0.866025i) q^{73} -7.07107i q^{74} +(-5.00000 - 8.66025i) q^{76} +(9.79796 + 11.3137i) q^{77} +(-1.50000 - 0.866025i) q^{79} +11.3137i q^{80} +(4.00000 + 6.92820i) q^{82} -14.6969 q^{83} -8.00000 q^{85} +(-3.67423 - 6.36396i) q^{86} +(8.00000 + 13.8564i) q^{88} +(-9.79796 - 5.65685i) q^{89} +(13.5000 - 2.59808i) q^{91} +(4.89898 - 2.82843i) q^{92} -6.92820i q^{94} +(-12.2474 + 7.07107i) q^{95} +(6.12372 - 7.77817i) q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 4 q^{4} + 2 q^{7} + 16 q^{10} - 8 q^{16} + 10 q^{19} - 16 q^{22} + 6 q^{25} + 20 q^{28} - 2 q^{31} - 8 q^{34} - 10 q^{37} + 16 q^{40} - 16 q^{46} - 26 q^{49} - 36 q^{52} - 64 q^{55} + 24 q^{61} - 32 q^{64}+ \cdots + 54 q^{91}+O(q^{100}) $ 4 * q + 4 * q^4 + 2 * q^7 + 16 * q^10 - 8 * q^16 + 10 * q^19 - 16 * q^22 + 6 * q^25 + 20 * q^28 - 2 * q^31 - 8 * q^34 - 10 * q^37 + 16 * q^40 - 16 * q^46 - 26 * q^49 - 36 * q^52 - 64 * q^55 + 24 * q^61 - 32 * q^64 + 30 * q^67 + 8 * q^70 + 6 * q^73 - 20 * q^76 - 6 * q^79 + 16 * q^82 - 32 * q^85 + 32 * q^88 + 54 * q^91

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/252\mathbb{Z}\right)^\times$.

$n$	$29$	$73$	$127$
$\chi(n)$	$1$	$e\left(\frac{5}{6}\right)$	$-1$

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	−0.160424	−	0.987048i	$-0.551286\pi$
$5$	−2.44949	−	1.41421i	−1.09545	−	0.632456i	−0.935021	+	0.354593i	$0.884620\pi$
$6$		0			0
$7$	0.500000	+	2.59808i	0.188982	+	0.981981i
$7$	0.500000	+	2.59808i	0.188982	+	0.981981i
$8$			2.82843i			1.00000i
$9$		0			0
$10$	4.00000			1.26491
$11$	4.89898	−	2.82843i	1.47710	−	0.852803i	0.477432	−	0.878668i	$-0.341568\pi$
$11$	4.89898	−	2.82843i	1.47710	−	0.852803i	0.999665	+	0.0258656i	$0.00823419\pi$
$12$		0			0
$13$		−	5.19615i		−	1.44115i	−0.693375	−	0.720577i	$-0.743877\pi$
$13$		−	5.19615i		−	1.44115i	0.693375	−	0.720577i	$-0.256123\pi$
$14$	−2.44949	−	2.82843i	−0.654654	−	0.755929i
$15$		0			0
$16$	−2.00000	−	3.46410i	−0.500000	−	0.866025i
$17$	2.44949	−	1.41421i	0.594089	−	0.342997i	−0.172624	−	0.984988i	$-0.555225\pi$
$17$	2.44949	−	1.41421i	0.594089	−	0.342997i	0.766712	+	0.641991i	$0.221891\pi$
$18$		0			0
$19$	2.50000	−	4.33013i	0.573539	−	0.993399i	−0.422659	−	0.906289i	$-0.638903\pi$
$19$	2.50000	−	4.33013i	0.573539	−	0.993399i	0.996199	−	0.0871106i	$-0.0277634\pi$
$20$	−4.89898	+	2.82843i	−1.09545	+	0.632456i
$21$		0			0
$22$	−4.00000	+	6.92820i	−0.852803	+	1.47710i
$23$	2.44949	+	1.41421i	0.510754	+	0.294884i	0.733144	−	0.680074i	$-0.238052\pi$
$23$	2.44949	+	1.41421i	0.510754	+	0.294884i	−0.222390	+	0.974958i	$0.571386\pi$
$24$		0			0
$25$	1.50000	+	2.59808i	0.300000	+	0.519615i
$26$	3.67423	+	6.36396i	0.720577	+	1.24808i
$27$		0			0
$28$	5.00000	+	1.73205i	0.944911	+	0.327327i
$29$		0			0			−	1.00000i	$-0.5\pi$
$29$		0			0				1.00000i	$0.5\pi$
$30$		0			0
$31$	−0.500000	−	0.866025i	−0.0898027	−	0.155543i	0.817625	−	0.575751i	$-0.195290\pi$
$31$	−0.500000	−	0.866025i	−0.0898027	−	0.155543i	−0.907428	+	0.420208i	$0.861957\pi$
$32$	4.89898	+	2.82843i	0.866025	+	0.500000i
$33$		0			0
$34$	−2.00000	+	3.46410i	−0.342997	+	0.594089i
$35$	2.44949	−	7.07107i	0.414039	−	1.19523i
$36$		0			0
$37$	−2.50000	+	4.33013i	−0.410997	+	0.711868i	−0.994999	−	0.0998840i	$-0.968153\pi$
$37$	−2.50000	+	4.33013i	−0.410997	+	0.711868i	0.584002	+	0.811752i	$0.301486\pi$
$38$			7.07107i			1.14708i
$39$		0			0
$40$	4.00000	−	6.92820i	0.632456	−	1.09545i
$41$		−	5.65685i		−	0.883452i	−0.897150	−	0.441726i	$-0.854366\pi$
$41$		−	5.65685i		−	0.883452i	0.897150	−	0.441726i	$-0.145634\pi$
$42$		0			0
$43$			5.19615i			0.792406i	0.918163	+	0.396203i	$0.129672\pi$
$43$			5.19615i			0.792406i	−0.918163	+	0.396203i	$0.870328\pi$
$44$		−	11.3137i		−	1.70561i
$45$		0			0
$46$	−4.00000			−0.589768
$47$	−2.44949	+	4.24264i	−0.357295	+	0.618853i	−0.987508	−	0.157569i	$-0.949634\pi$
$47$	−2.44949	+	4.24264i	−0.357295	+	0.618853i	0.630213	+	0.776422i	$0.282968\pi$
$48$		0			0
$49$	−6.50000	+	2.59808i	−0.928571	+	0.371154i
$50$	−3.67423	−	2.12132i	−0.519615	−	0.300000i
$51$		0			0
$52$	−9.00000	−	5.19615i	−1.24808	−	0.720577i
$53$	−2.44949	−	4.24264i	−0.336463	−	0.582772i	0.647302	−	0.762234i	$-0.275897\pi$
$53$	−2.44949	−	4.24264i	−0.336463	−	0.582772i	−0.983765	+	0.179463i	$0.942564\pi$
$54$		0			0
$55$	−16.0000			−2.15744
$56$	−7.34847	+	1.41421i	−0.981981	+	0.188982i
$57$		0			0
$58$		0			0
$59$	2.44949	+	4.24264i	0.318896	+	0.552345i	0.980258	−	0.197722i	$-0.0633545\pi$
$59$	2.44949	+	4.24264i	0.318896	+	0.552345i	−0.661362	+	0.750067i	$0.730021\pi$
$60$		0			0
$61$	6.00000	+	3.46410i	0.768221	+	0.443533i	0.832240	−	0.554416i	$-0.187058\pi$
$61$	6.00000	+	3.46410i	0.768221	+	0.443533i	−0.0640184	+	0.997949i	$0.520392\pi$
$62$	1.22474	+	0.707107i	0.155543	+	0.0898027i
$63$		0			0
$64$	−8.00000			−1.00000
$65$	−7.34847	+	12.7279i	−0.911465	+	1.57870i
$66$		0			0
$67$	7.50000	−	4.33013i	0.916271	−	0.529009i	0.0338274	−	0.999428i	$-0.489230\pi$
$67$	7.50000	−	4.33013i	0.916271	−	0.529009i	0.882443	+	0.470418i	$0.155897\pi$
$68$		−	5.65685i		−	0.685994i
$69$		0			0
$70$	2.00000	+	10.3923i	0.239046	+	1.24212i
$71$		−	2.82843i		−	0.335673i	−0.985815	−	0.167836i	$-0.946322\pi$
$71$		−	2.82843i		−	0.335673i	0.985815	−	0.167836i	$-0.0536780\pi$
$72$		0			0
$73$	1.50000	−	0.866025i	0.175562	−	0.101361i	−0.409644	−	0.912245i	$-0.634347\pi$
$73$	1.50000	−	0.866025i	0.175562	−	0.101361i	0.585206	+	0.810885i	$0.301014\pi$
$74$		−	7.07107i		−	0.821995i
$75$		0			0
$76$	−5.00000	−	8.66025i	−0.573539	−	0.993399i
$77$	9.79796	+	11.3137i	1.11658	+	1.28932i
$78$		0			0
$79$	−1.50000	−	0.866025i	−0.168763	−	0.0974355i	0.413239	−	0.910622i	$-0.364397\pi$
$79$	−1.50000	−	0.866025i	−0.168763	−	0.0974355i	−0.582003	+	0.813187i	$0.697731\pi$
$80$			11.3137i			1.26491i
$81$		0			0
$82$	4.00000	+	6.92820i	0.441726	+	0.765092i
$83$	−14.6969			−1.61320			−0.806599	−	0.591099i	$-0.798694\pi$
$83$	−14.6969			−1.61320			−0.806599	+	0.591099i	$0.798694\pi$
$84$		0			0
$85$	−8.00000			−0.867722
$86$	−3.67423	−	6.36396i	−0.396203	−	0.686244i
$87$		0			0
$88$	8.00000	+	13.8564i	0.852803	+	1.47710i
$89$	−9.79796	−	5.65685i	−1.03858	−	0.599625i	−0.119150	−	0.992876i	$-0.538017\pi$
$89$	−9.79796	−	5.65685i	−1.03858	−	0.599625i	−0.919431	+	0.393251i	$0.871350\pi$
$90$		0			0
$91$	13.5000	−	2.59808i	1.41518	−	0.272352i
$92$	4.89898	−	2.82843i	0.510754	−	0.294884i
$93$		0			0
$94$		−	6.92820i		−	0.714590i
$95$	−12.2474	+	7.07107i	−1.25656	+	0.725476i
$96$		0			0
$97$		0			0		1.00000			$0$
$97$		0			0		−1.00000			$\pi$
$98$	6.12372	−	7.77817i	0.618590	−	0.785714i
$99$		0			0
$100$	6.00000			0.600000
$101$	9.79796	−	5.65685i	0.974933	−	0.562878i	0.0741967	−	0.997244i	$-0.476361\pi$
$101$	9.79796	−	5.65685i	0.974933	−	0.562878i	0.900737	+	0.434366i	$0.143027\pi$
$102$		0			0
$103$	−0.500000	+	0.866025i	−0.0492665	+	0.0853320i	−0.889607	−	0.456727i	$-0.849022\pi$
$103$	−0.500000	+	0.866025i	−0.0492665	+	0.0853320i	0.840341	+	0.542059i	$0.182355\pi$
$104$	14.6969			1.44115
$105$		0			0
$106$	6.00000	+	3.46410i	0.582772	+	0.336463i
$107$	2.44949	+	1.41421i	0.236801	+	0.136717i	0.613706	−	0.789535i	$-0.289678\pi$
$107$	2.44949	+	1.41421i	0.236801	+	0.136717i	−0.376905	+	0.926252i	$0.623012\pi$
$108$		0			0
$109$	−2.50000	−	4.33013i	−0.239457	−	0.414751i	0.721102	−	0.692829i	$-0.243636\pi$
$109$	−2.50000	−	4.33013i	−0.239457	−	0.414751i	−0.960558	+	0.278078i	$0.910303\pi$
$110$	19.5959	−	11.3137i	1.86840	−	1.07872i
$111$		0			0
$112$	8.00000	−	6.92820i	0.755929	−	0.654654i
$113$	14.6969			1.38257			0.691286	−	0.722581i	$-0.257045\pi$
$113$	14.6969			1.38257			0.691286	+	0.722581i	$0.257045\pi$
$114$		0			0
$115$	−4.00000	−	6.92820i	−0.373002	−	0.646058i
$116$		0			0
$117$		0			0
$118$	−6.00000	−	3.46410i	−0.552345	−	0.318896i
$119$	4.89898	+	5.65685i	0.449089	+	0.518563i
$120$		0			0
$121$	10.5000	−	18.1865i	0.954545	−	1.65332i
$122$	−9.79796			−0.887066
$123$		0			0
$124$	−2.00000			−0.179605
$125$			5.65685i			0.505964i
$126$		0			0
$127$			15.5885i			1.38325i	0.722256	+	0.691626i	$0.243105\pi$
$127$			15.5885i			1.38325i	−0.722256	+	0.691626i	$0.756895\pi$
$128$	9.79796	−	5.65685i	0.866025	−	0.500000i
$129$		0			0
$130$		−	20.7846i		−	1.82293i
$131$	−9.79796	+	16.9706i	−0.856052	+	1.48272i	0.0196143	+	0.999808i	$0.493756\pi$
$131$	−9.79796	+	16.9706i	−0.856052	+	1.48272i	−0.875666	+	0.482917i	$0.839577\pi$
$132$		0			0
$133$	12.5000	+	4.33013i	1.08389	+	0.375470i
$134$	−6.12372	+	10.6066i	−0.529009	+	0.916271i
$135$		0			0
$136$	4.00000	+	6.92820i	0.342997	+	0.594089i
$137$	4.89898	+	8.48528i	0.418548	+	0.724947i	0.995794	−	0.0916241i	$-0.0292058\pi$
$137$	4.89898	+	8.48528i	0.418548	+	0.724947i	−0.577246	+	0.816571i	$0.695872\pi$
$138$		0			0
$139$	13.0000			1.10265			0.551323	−	0.834292i	$-0.314123\pi$
$139$	13.0000			1.10265			0.551323	+	0.834292i	$0.314123\pi$
$140$	−9.79796	−	11.3137i	−0.828079	−	0.956183i
$141$		0			0
$142$	2.00000	+	3.46410i	0.167836	+	0.290701i
$143$	−14.6969	−	25.4558i	−1.22902	−	2.12872i
$144$		0			0
$145$		0			0
$146$	−1.22474	+	2.12132i	−0.101361	+	0.175562i
$147$		0			0
$148$	5.00000	+	8.66025i	0.410997	+	0.711868i
$149$	−4.89898	+	8.48528i	−0.401340	+	0.695141i	−0.993888	−	0.110394i	$-0.964789\pi$
$149$	−4.89898	+	8.48528i	−0.401340	+	0.695141i	0.592548	+	0.805535i	$0.298122\pi$
$150$		0			0
$151$	−15.0000	+	8.66025i	−1.22068	+	0.704761i	−0.965064	−	0.262016i	$-0.915613\pi$
$151$	−15.0000	+	8.66025i	−1.22068	+	0.704761i	−0.255619	+	0.966778i	$0.582279\pi$
$152$	12.2474	+	7.07107i	0.993399	+	0.573539i
$153$		0			0
$154$	−20.0000	−	6.92820i	−1.61165	−	0.558291i
$155$			2.82843i			0.227185i
$156$		0			0
$157$	6.00000	−	3.46410i	0.478852	−	0.276465i	−0.241086	−	0.970504i	$-0.577504\pi$
$157$	6.00000	−	3.46410i	0.478852	−	0.276465i	0.719938	+	0.694038i	$0.244170\pi$
$158$	2.44949			0.194871
$159$		0			0
$160$	−8.00000	−	13.8564i	−0.632456	−	1.09545i
$161$	−2.44949	+	7.07107i	−0.193047	+	0.557278i
$162$		0			0
$163$	3.00000	+	1.73205i	0.234978	+	0.135665i	0.612866	−	0.790186i	$-0.290016\pi$
$163$	3.00000	+	1.73205i	0.234978	+	0.135665i	−0.377888	+	0.925851i	$0.623350\pi$
$164$	−9.79796	−	5.65685i	−0.765092	−	0.441726i
$165$		0			0
$166$	18.0000	−	10.3923i	1.39707	−	0.806599i
$167$	14.6969			1.13728			0.568642	−	0.822585i	$-0.307469\pi$
$167$	14.6969			1.13728			0.568642	+	0.822585i	$0.307469\pi$
$168$		0			0
$169$	−14.0000			−1.07692
$170$	9.79796	−	5.65685i	0.751469	−	0.433861i
$171$		0			0
$172$	9.00000	+	5.19615i	0.686244	+	0.396203i
$173$	−17.1464	−	9.89949i	−1.30362	−	0.752645i	−0.322596	−	0.946537i	$-0.604555\pi$
$173$	−17.1464	−	9.89949i	−1.30362	−	0.752645i	−0.981023	+	0.193892i	$0.937889\pi$
$174$		0			0
$175$	−6.00000	+	5.19615i	−0.453557	+	0.392792i
$176$	−19.5959	−	11.3137i	−1.47710	−	0.852803i
$177$		0			0
$178$	16.0000			1.19925
$179$	−2.44949	+	1.41421i	−0.183083	+	0.105703i	−0.588741	−	0.808322i	$-0.700376\pi$
$179$	−2.44949	+	1.41421i	−0.183083	+	0.105703i	0.405657	+	0.914025i	$0.367043\pi$
$180$		0			0
$181$			5.19615i			0.386227i	0.981176	+	0.193113i	$0.0618586\pi$
$181$			5.19615i			0.386227i	−0.981176	+	0.193113i	$0.938141\pi$
$182$	−14.6969	+	12.7279i	−1.08941	+	0.943456i
$183$		0			0
$184$	−4.00000	+	6.92820i	−0.294884	+	0.510754i
$185$	12.2474	−	7.07107i	0.900450	−	0.519875i
$186$		0			0
$187$	8.00000	−	13.8564i	0.585018	−	1.01328i
$188$	4.89898	+	8.48528i	0.357295	+	0.618853i
$189$		0			0
$190$	10.0000	−	17.3205i	0.725476	−	1.25656i
$191$	17.1464	+	9.89949i	1.24067	+	0.716302i	0.969231	−	0.246153i	$-0.0791665\pi$
$191$	17.1464	+	9.89949i	1.24067	+	0.716302i	0.271441	+	0.962455i	$0.412500\pi$
$192$		0			0
$193$	3.50000	+	6.06218i	0.251936	+	0.436365i	0.964059	−	0.265689i	$-0.0855996\pi$
$193$	3.50000	+	6.06218i	0.251936	+	0.436365i	−0.712123	+	0.702055i	$0.752266\pi$
$194$		0			0
$195$		0			0
$196$	−2.00000	+	13.8564i	−0.142857	+	0.989743i
$197$	−14.6969			−1.04711			−0.523557	−	0.851991i	$-0.675395\pi$
$197$	−14.6969			−1.04711			−0.523557	+	0.851991i	$0.675395\pi$
$198$		0			0
$199$	−2.00000	−	3.46410i	−0.141776	−	0.245564i	0.786389	−	0.617731i	$-0.211948\pi$
$199$	−2.00000	−	3.46410i	−0.141776	−	0.245564i	−0.928166	+	0.372168i	$0.878615\pi$
$200$	−7.34847	+	4.24264i	−0.519615	+	0.300000i
$201$		0			0
$202$	−8.00000	+	13.8564i	−0.562878	+	0.974933i
$203$		0			0
$204$		0			0
$205$	−8.00000	+	13.8564i	−0.558744	+	0.967773i
$206$		−	1.41421i		−	0.0985329i
$207$		0			0
$208$	−18.0000	+	10.3923i	−1.24808	+	0.720577i
$209$		−	28.2843i		−	1.95646i
$210$		0			0
$211$		−	10.3923i		−	0.715436i	−0.933830	−	0.357718i	$-0.883555\pi$
$211$		−	10.3923i		−	0.715436i	0.933830	−	0.357718i	$-0.116445\pi$
$212$	−9.79796			−0.672927
$213$		0			0
$214$	−4.00000			−0.273434
$215$	7.34847	−	12.7279i	0.501161	−	0.868037i
$216$		0			0
$217$	2.00000	−	1.73205i	0.135769	−	0.117579i
$218$	6.12372	+	3.53553i	0.414751	+	0.239457i
$219$		0			0
$220$	−16.0000	+	27.7128i	−1.07872	+	1.86840i
$221$	−7.34847	−	12.7279i	−0.494312	−	0.856173i
$222$		0			0
$223$	4.00000			0.267860			0.133930	−	0.990991i	$-0.457240\pi$
$223$	4.00000			0.267860			0.133930	+	0.990991i	$0.457240\pi$
$224$	−4.89898	+	14.1421i	−0.327327	+	0.944911i
$225$		0			0
$226$	−18.0000	+	10.3923i	−1.19734	+	0.691286i
$227$	−4.89898	−	8.48528i	−0.325157	−	0.563188i	0.656387	−	0.754424i	$-0.272084\pi$
$227$	−4.89898	−	8.48528i	−0.325157	−	0.563188i	−0.981544	+	0.191236i	$0.938750\pi$
$228$		0			0
$229$	−16.5000	−	9.52628i	−1.09035	−	0.629514i	−0.156681	−	0.987649i	$-0.550079\pi$
$229$	−16.5000	−	9.52628i	−1.09035	−	0.629514i	−0.933670	+	0.358135i	$0.883413\pi$
$230$	9.79796	+	5.65685i	0.646058	+	0.373002i
$231$		0			0
$232$		0			0
$233$	−12.2474	+	21.2132i	−0.802357	+	1.38972i	0.115704	+	0.993284i	$0.463088\pi$
$233$	−12.2474	+	21.2132i	−0.802357	+	1.38972i	−0.918061	+	0.396439i	$0.870246\pi$
$234$		0			0
$235$	12.0000	−	6.92820i	0.782794	−	0.451946i
$236$	9.79796			0.637793
$237$		0			0
$238$	−10.0000	−	3.46410i	−0.648204	−	0.224544i
$239$		−	11.3137i		−	0.731823i	−0.930650	−	0.365911i	$-0.880757\pi$
$239$		−	11.3137i		−	0.731823i	0.930650	−	0.365911i	$-0.119243\pi$
$240$		0			0
$241$	−12.0000	+	6.92820i	−0.772988	+	0.446285i	−0.833939	−	0.551856i	$-0.813920\pi$
$241$	−12.0000	+	6.92820i	−0.772988	+	0.446285i	0.0609515	+	0.998141i	$0.480586\pi$
$242$			29.6985i			1.90909i
$243$		0			0
$244$	12.0000	−	6.92820i	0.768221	−	0.443533i
$245$	19.5959	+	2.82843i	1.25194	+	0.180702i
$246$		0			0
$247$	−22.5000	−	12.9904i	−1.43164	−	0.826558i
$248$	2.44949	−	1.41421i	0.155543	−	0.0898027i
$249$		0			0
$250$	−4.00000	−	6.92820i	−0.252982	−	0.438178i
$251$	14.6969			0.927663			0.463831	−	0.885924i	$-0.346474\pi$
$251$	14.6969			0.927663			0.463831	+	0.885924i	$0.346474\pi$
$252$		0			0
$253$	16.0000			1.00591
$254$	−11.0227	−	19.0919i	−0.691626	−	1.19793i
$255$		0			0
$256$	−8.00000	+	13.8564i	−0.500000	+	0.866025i
$257$	−2.44949	−	1.41421i	−0.152795	−	0.0882162i	0.421653	−	0.906757i	$-0.361450\pi$
$257$	−2.44949	−	1.41421i	−0.152795	−	0.0882162i	−0.574448	+	0.818541i	$0.694783\pi$
$258$		0			0
$259$	−12.5000	−	4.33013i	−0.776712	−	0.269061i
$260$	14.6969	+	25.4558i	0.911465	+	1.57870i
$261$		0			0
$262$		−	27.7128i		−	1.71210i
$263$	4.89898	−	2.82843i	0.302084	−	0.174408i	−0.341295	−	0.939956i	$-0.610865\pi$
$263$	4.89898	−	2.82843i	0.302084	−	0.174408i	0.643379	+	0.765548i	$0.277532\pi$
$264$		0			0
$265$			13.8564i			0.851192i
$266$	−18.3712	+	3.53553i	−1.12641	+	0.216777i
$267$		0			0
$268$		−	17.3205i		−	1.05802i
$269$	9.79796	−	5.65685i	0.597392	−	0.344904i	−0.170623	−	0.985336i	$-0.554578\pi$
$269$	9.79796	−	5.65685i	0.597392	−	0.344904i	0.768015	+	0.640432i	$0.221245\pi$
$270$		0			0
$271$	−14.0000	+	24.2487i	−0.850439	+	1.47300i	0.0303728	+	0.999539i	$0.490331\pi$
$271$	−14.0000	+	24.2487i	−0.850439	+	1.47300i	−0.880812	+	0.473466i	$0.843003\pi$
$272$	−9.79796	−	5.65685i	−0.594089	−	0.342997i
$273$		0			0
$274$	−12.0000	−	6.92820i	−0.724947	−	0.418548i
$275$	14.6969	+	8.48528i	0.886259	+	0.511682i
$276$		0			0
$277$	−5.50000	−	9.52628i	−0.330463	−	0.572379i	0.652140	−	0.758099i	$-0.273872\pi$
$277$	−5.50000	−	9.52628i	−0.330463	−	0.572379i	−0.982603	+	0.185720i	$0.940538\pi$
$278$	−15.9217	+	9.19239i	−0.954919	+	0.551323i
$279$		0			0
$280$	20.0000	+	6.92820i	1.19523	+	0.414039i
$281$	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
$283$	14.5000	+	25.1147i	0.861936	+	1.49292i	0.870058	+	0.492949i	$0.164081\pi$
$283$	14.5000	+	25.1147i	0.861936	+	1.49292i	−0.00812260	+	0.999967i	$0.502586\pi$
$284$	−4.89898	−	2.82843i	−0.290701	−	0.167836i
$285$		0			0
$286$	36.0000	+	20.7846i	2.12872	+	1.22902i
$287$	14.6969	−	2.82843i	0.867533	−	0.166957i
$288$		0			0
$289$	−4.50000	+	7.79423i	−0.264706	+	0.458484i
$290$		0			0
$291$		0			0
$292$		−	3.46410i		−	0.202721i
$293$			11.3137i			0.660954i	0.943814	+	0.330477i	$0.107210\pi$
$293$			11.3137i			0.660954i	−0.943814	+	0.330477i	$0.892790\pi$
$294$		0			0
$295$		−	13.8564i		−	0.806751i
$296$	−12.2474	−	7.07107i	−0.711868	−	0.410997i
$297$		0			0
$298$		−	13.8564i		−	0.802680i
$299$	7.34847	−	12.7279i	0.424973	−	0.736075i
$300$		0			0
$301$	−13.5000	+	2.59808i	−0.778127	+	0.149751i
$302$	12.2474	−	21.2132i	0.704761	−	1.22068i
$303$		0			0
$304$	−20.0000			−1.14708
$305$	−9.79796	−	16.9706i	−0.561029	−	0.971732i
$306$		0			0
$307$	1.00000			0.0570730			0.0285365	−	0.999593i	$-0.490915\pi$
$307$	1.00000			0.0570730			0.0285365	+	0.999593i	$0.490915\pi$
$308$	29.3939	−	5.65685i	1.67487	−	0.322329i
$309$		0			0
$310$	−2.00000	−	3.46410i	−0.113592	−	0.196748i
$311$	9.79796	+	16.9706i	0.555591	+	0.962312i	0.997857	+	0.0654284i	$0.0208414\pi$
$311$	9.79796	+	16.9706i	0.555591	+	0.962312i	−0.442266	+	0.896884i	$0.645825\pi$
$312$		0			0
$313$	19.5000	+	11.2583i	1.10221	+	0.636358i	0.936799	−	0.349867i	$-0.113773\pi$
$313$	19.5000	+	11.2583i	1.10221	+	0.636358i	0.165406	+	0.986226i	$0.447107\pi$
$314$	−4.89898	+	8.48528i	−0.276465	+	0.478852i
$315$		0			0
$316$	−3.00000	+	1.73205i	−0.168763	+	0.0974355i
$317$	9.79796	−	16.9706i	0.550308	−	0.953162i	−0.447944	−	0.894062i	$-0.647844\pi$
$317$	9.79796	−	16.9706i	0.550308	−	0.953162i	0.998252	−	0.0591001i	$-0.0188231\pi$
$318$		0			0
$319$		0			0
$320$	19.5959	+	11.3137i	1.09545	+	0.632456i
$321$		0			0
$322$	−2.00000	−	10.3923i	−0.111456	−	0.579141i
$323$		−	14.1421i		−	0.786889i
$324$		0			0
$325$	13.5000	−	7.79423i	0.748845	−	0.432346i
$326$	−4.89898			−0.271329
$327$		0			0
$328$	16.0000			0.883452
$329$	−12.2474	−	4.24264i	−0.675224	−	0.233904i
$330$		0			0
$331$	25.5000	+	14.7224i	1.40161	+	0.809218i	0.994558	−	0.104188i	$-0.0332244\pi$
$331$	25.5000	+	14.7224i	1.40161	+	0.809218i	0.407049	+	0.913406i	$0.366558\pi$
$332$	−14.6969	+	25.4558i	−0.806599	+	1.39707i
$333$		0			0
$334$	−18.0000	+	10.3923i	−0.984916	+	0.568642i
$335$	−24.4949			−1.33830
$336$		0			0
$337$	29.0000			1.57973			0.789865	−	0.613280i	$-0.210150\pi$
$337$	29.0000			1.57973			0.789865	+	0.613280i	$0.210150\pi$
$338$	17.1464	−	9.89949i	0.932643	−	0.538462i
$339$		0			0
$340$	−8.00000	+	13.8564i	−0.433861	+	0.751469i
$341$	−4.89898	−	2.82843i	−0.265295	−	0.153168i
$342$		0			0
$343$	−10.0000	−	15.5885i	−0.539949	−	0.841698i
$344$	−14.6969			−0.792406
$345$		0			0
$346$	28.0000			1.50529
$347$	−2.44949	+	1.41421i	−0.131495	+	0.0759190i	−0.564305	−	0.825567i	$-0.690856\pi$
$347$	−2.44949	+	1.41421i	−0.131495	+	0.0759190i	0.432809	+	0.901486i	$0.357522\pi$
$348$		0			0
$349$		−	20.7846i		−	1.11257i	−0.830990	−	0.556287i	$-0.812225\pi$
$349$		−	20.7846i		−	1.11257i	0.830990	−	0.556287i	$-0.187775\pi$
$350$	3.67423	−	10.6066i	0.196396	−	0.566947i
$351$		0			0
$352$	32.0000			1.70561
$353$	−26.9444	+	15.5563i	−1.43411	+	0.827981i	−0.997431	−	0.0716387i	$-0.977177\pi$
$353$	−26.9444	+	15.5563i	−1.43411	+	0.827981i	−0.436674	+	0.899620i	$0.643844\pi$
$354$		0			0
$355$	−4.00000	+	6.92820i	−0.212298	+	0.367711i
$356$	−19.5959	+	11.3137i	−1.03858	+	0.599625i
$357$		0			0
$358$	2.00000	−	3.46410i	0.105703	−	0.183083i
$359$	−19.5959	−	11.3137i	−1.03423	−	0.597115i	−0.116039	−	0.993245i	$-0.537020\pi$
$359$	−19.5959	−	11.3137i	−1.03423	−	0.597115i	−0.918194	+	0.396130i	$0.870353\pi$
$360$		0			0
$361$	−3.00000	−	5.19615i	−0.157895	−	0.273482i
$362$	−3.67423	−	6.36396i	−0.193113	−	0.334482i
$363$		0			0
$364$	9.00000	−	25.9808i	0.471728	−	1.36176i
$365$	−4.89898			−0.256424
$366$		0			0
$367$	5.50000	+	9.52628i	0.287098	+	0.497268i	0.973116	−	0.230317i	$-0.0739762\pi$
$367$	5.50000	+	9.52628i	0.287098	+	0.497268i	−0.686018	+	0.727585i	$0.740643\pi$
$368$		−	11.3137i		−	0.589768i
$369$		0			0
$370$	−10.0000	+	17.3205i	−0.519875	+	0.900450i
$371$	9.79796	−	8.48528i	0.508685	−	0.440534i
$372$		0			0
$373$	0.500000	−	0.866025i	0.0258890	−	0.0448411i	−0.852791	−	0.522253i	$-0.825092\pi$
$373$	0.500000	−	0.866025i	0.0258890	−	0.0448411i	0.878680	+	0.477412i	$0.158425\pi$
$374$			22.6274i			1.17004i
$375$		0			0
$376$	−12.0000	−	6.92820i	−0.618853	−	0.357295i
$377$		0			0
$378$		0			0
$379$		−	36.3731i		−	1.86836i	−0.356803	−	0.934179i	$-0.616133\pi$
$379$		−	36.3731i		−	1.86836i	0.356803	−	0.934179i	$-0.383867\pi$
$380$			28.2843i			1.45095i
$381$		0			0
$382$	−28.0000			−1.43260
$383$	−17.1464	+	29.6985i	−0.876142	+	1.51752i	−0.0205998	+	0.999788i	$0.506558\pi$
$383$	−17.1464	+	29.6985i	−0.876142	+	1.51752i	−0.855542	+	0.517734i	$0.826776\pi$
$384$		0			0
$385$	−8.00000	−	41.5692i	−0.407718	−	2.11856i
$386$	−8.57321	−	4.94975i	−0.436365	−	0.251936i
$387$		0			0
$388$		0			0
$389$	12.2474	+	21.2132i	0.620970	+	1.07555i	0.989305	+	0.145859i	$0.0465946\pi$
$389$	12.2474	+	21.2132i	0.620970	+	1.07555i	−0.368335	+	0.929693i	$0.620072\pi$
$390$		0			0
$391$	8.00000			0.404577
$392$	−7.34847	−	18.3848i	−0.371154	−	0.928571i
$393$		0			0
$394$	18.0000	−	10.3923i	0.906827	−	0.523557i
$395$	2.44949	+	4.24264i	0.123247	+	0.213470i
$396$		0			0
$397$	−25.5000	−	14.7224i	−1.27981	−	0.738898i	−0.302995	−	0.952992i	$-0.597987\pi$
$397$	−25.5000	−	14.7224i	−1.27981	−	0.738898i	−0.976813	+	0.214094i	$0.931320\pi$
$398$	4.89898	+	2.82843i	0.245564	+	0.141776i
$399$		0			0
$400$	6.00000	−	10.3923i	0.300000	−	0.519615i
$401$	2.44949	−	4.24264i	0.122322	−	0.211867i	−0.798361	−	0.602179i	$-0.794299\pi$
$401$	2.44949	−	4.24264i	0.122322	−	0.211867i	0.920683	+	0.390312i	$0.127633\pi$
$402$		0			0
$403$	−4.50000	+	2.59808i	−0.224161	+	0.129419i
$404$		−	22.6274i		−	1.12576i
$405$		0			0
$406$		0			0
$407$			28.2843i			1.40200i
$408$		0			0
$409$	−25.5000	+	14.7224i	−1.26089	+	0.727977i	−0.973247	−	0.229759i	$-0.926206\pi$
$409$	−25.5000	+	14.7224i	−1.26089	+	0.727977i	−0.287646	+	0.957737i	$0.592873\pi$
$410$		−	22.6274i		−	1.11749i
$411$		0			0
$412$	1.00000	+	1.73205i	0.0492665	+	0.0853320i
$413$	−9.79796	+	8.48528i	−0.482126	+	0.417533i
$414$		0			0
$415$	36.0000	+	20.7846i	1.76717	+	1.02028i
$416$	14.6969	−	25.4558i	0.720577	−	1.24808i
$417$		0			0
$418$	20.0000	+	34.6410i	0.978232	+	1.69435i
$419$		0			0			−	1.00000i	$-0.5\pi$
$419$		0			0				1.00000i	$0.5\pi$
$420$		0			0
$421$	23.0000			1.12095			0.560476	−	0.828171i	$-0.310618\pi$
$421$	23.0000			1.12095			0.560476	+	0.828171i	$0.310618\pi$
$422$	7.34847	+	12.7279i	0.357718	+	0.619586i
$423$		0			0
$424$	12.0000	−	6.92820i	0.582772	−	0.336463i
$425$	7.34847	+	4.24264i	0.356453	+	0.205798i
$426$		0			0
$427$	−6.00000	+	17.3205i	−0.290360	+	0.838198i
$428$	4.89898	−	2.82843i	0.236801	−	0.136717i
$429$		0			0
$430$			20.7846i			1.00232i
$431$	12.2474	−	7.07107i	0.589939	−	0.340601i	−0.175134	−	0.984545i	$-0.556036\pi$
$431$	12.2474	−	7.07107i	0.589939	−	0.340601i	0.765073	+	0.643943i	$0.222703\pi$
$432$		0			0
$433$		−	5.19615i		−	0.249711i	−0.992175	−	0.124856i	$-0.960153\pi$
$433$		−	5.19615i		−	0.249711i	0.992175	−	0.124856i	$-0.0398468\pi$
$434$	−1.22474	+	3.53553i	−0.0587896	+	0.169711i
$435$		0			0
$436$	−10.0000			−0.478913
$437$	12.2474	−	7.07107i	0.585875	−	0.338255i
$438$		0			0
$439$	−2.00000	+	3.46410i	−0.0954548	+	0.165333i	−0.909798	−	0.415051i	$-0.863764\pi$
$439$	−2.00000	+	3.46410i	−0.0954548	+	0.165333i	0.814344	+	0.580383i	$0.197097\pi$
$440$		−	45.2548i		−	2.15744i
$441$		0			0
$442$	18.0000	+	10.3923i	0.856173	+	0.494312i
$443$	−26.9444	−	15.5563i	−1.28017	−	0.739104i	−0.303288	−	0.952899i	$-0.598084\pi$
$443$	−26.9444	−	15.5563i	−1.28017	−	0.739104i	−0.976879	+	0.213795i	$0.931418\pi$
$444$		0			0
$445$	16.0000	+	27.7128i	0.758473	+	1.31371i
$446$	−4.89898	+	2.82843i	−0.231973	+	0.133930i
$447$		0			0
$448$	−4.00000	−	20.7846i	−0.188982	−	0.981981i
$449$	14.6969			0.693591			0.346796	−	0.937941i	$-0.387270\pi$
$449$	14.6969			0.693591			0.346796	+	0.937941i	$0.387270\pi$
$450$		0			0
$451$	−16.0000	−	27.7128i	−0.753411	−	1.30495i
$452$	14.6969	−	25.4558i	0.691286	−	1.19734i
$453$		0			0
$454$	12.0000	+	6.92820i	0.563188	+	0.325157i
$455$	−36.7423	−	12.7279i	−1.72251	−	0.596694i
$456$		0			0
$457$	3.50000	−	6.06218i	0.163723	−	0.283577i	−0.772478	−	0.635042i	$-0.780983\pi$
$457$	3.50000	−	6.06218i	0.163723	−	0.283577i	0.936201	+	0.351465i	$0.114316\pi$
$458$	26.9444			1.25903
$459$		0			0
$460$	−16.0000			−0.746004
$461$			11.3137i			0.526932i	0.964669	+	0.263466i	$0.0848657\pi$
$461$			11.3137i			0.526932i	−0.964669	+	0.263466i	$0.915134\pi$
$462$		0			0
$463$		−	5.19615i		−	0.241486i	−0.992684	−	0.120743i	$-0.961472\pi$
$463$		−	5.19615i		−	0.241486i	0.992684	−	0.120743i	$-0.0385276\pi$
$464$		0			0
$465$		0			0
$466$		−	34.6410i		−	1.60471i
$467$	−2.44949	+	4.24264i	−0.113349	+	0.196326i	−0.917119	−	0.398615i	$-0.869491\pi$
$467$	−2.44949	+	4.24264i	−0.113349	+	0.196326i	0.803770	+	0.594941i	$0.202824\pi$
$468$		0			0
$469$	15.0000	+	17.3205i	0.692636	+	0.799787i
$470$	−9.79796	+	16.9706i	−0.451946	+	0.782794i
$471$		0			0
$472$	−12.0000	+	6.92820i	−0.552345	+	0.318896i
$473$	14.6969	+	25.4558i	0.675766	+	1.17046i
$474$		0			0
$475$	15.0000			0.688247
$476$	14.6969	−	2.82843i	0.673633	−	0.129641i
$477$		0			0
$478$	8.00000	+	13.8564i	0.365911	+	0.633777i
$479$	17.1464	+	29.6985i	0.783440	+	1.35696i	0.929926	+	0.367746i	$0.119870\pi$
$479$	17.1464	+	29.6985i	0.783440	+	1.35696i	−0.146486	+	0.989213i	$0.546796\pi$
$480$		0			0
$481$	22.5000	+	12.9904i	1.02591	+	0.592310i
$482$	9.79796	−	16.9706i	0.446285	−	0.772988i
$483$		0			0
$484$	−21.0000	−	36.3731i	−0.954545	−	1.65332i
$485$		0			0
$486$		0			0
$487$	−19.5000	+	11.2583i	−0.883629	+	0.510164i	−0.871853	−	0.489767i	$-0.837082\pi$
$487$	−19.5000	+	11.2583i	−0.883629	+	0.510164i	−0.0117760	+	0.999931i	$0.503749\pi$
$488$	−9.79796	+	16.9706i	−0.443533	+	0.768221i
$489$		0			0
$490$	−26.0000	+	10.3923i	−1.17456	+	0.469476i
$491$			5.65685i			0.255290i	0.991820	+	0.127645i	$0.0407419\pi$
$491$			5.65685i			0.255290i	−0.991820	+	0.127645i	$0.959258\pi$
$492$		0			0
$493$		0			0
$494$	36.7423			1.65312
$495$		0			0
$496$	−2.00000	+	3.46410i	−0.0898027	+	0.155543i
$497$	7.34847	−	1.41421i	0.329624	−	0.0634361i
$498$		0			0
$499$	−28.5000	−	16.4545i	−1.27584	−	0.736604i	−0.299755	−	0.954016i	$-0.596905\pi$
$499$	−28.5000	−	16.4545i	−1.27584	−	0.736604i	−0.976080	+	0.217412i	$0.930238\pi$
$500$	9.79796	+	5.65685i	0.438178	+	0.252982i
$501$		0			0
$502$	−18.0000	+	10.3923i	−0.803379	+	0.463831i
$503$	−29.3939			−1.31061			−0.655304	−	0.755365i	$-0.727460\pi$
$503$	−29.3939			−1.31061			−0.655304	+	0.755365i	$0.727460\pi$
$504$		0			0
$505$	−32.0000			−1.42398
$506$	−19.5959	+	11.3137i	−0.871145	+	0.502956i
$507$		0			0
$508$	27.0000	+	15.5885i	1.19793	+	0.691626i
$509$	26.9444	+	15.5563i	1.19429	+	0.689523i	0.959276	−	0.282469i	$-0.0911535\pi$
$509$	26.9444	+	15.5563i	1.19429	+	0.689523i	0.235013	+	0.971992i	$0.424487\pi$
$510$		0			0
$511$	3.00000	+	3.46410i	0.132712	+	0.153243i
$512$		−	22.6274i		−	1.00000i
$513$		0			0
$514$	4.00000			0.176432
$515$	2.44949	−	1.41421i	0.107937	−	0.0623177i
$516$		0			0
$517$			27.7128i			1.21881i
$518$	18.3712	−	3.53553i	0.807183	−	0.155342i
$519$		0			0
$520$	−36.0000	−	20.7846i	−1.57870	−	0.911465i
$521$	9.79796	−	5.65685i	0.429256	−	0.247831i	−0.269773	−	0.962924i	$-0.586949\pi$
$521$	9.79796	−	5.65685i	0.429256	−	0.247831i	0.699030	+	0.715093i	$0.253615\pi$
$522$		0			0
$523$	−15.5000	+	26.8468i	−0.677768	+	1.17393i	0.297884	+	0.954602i	$0.403719\pi$
$523$	−15.5000	+	26.8468i	−0.677768	+	1.17393i	−0.975652	+	0.219326i	$0.929614\pi$
$524$	19.5959	+	33.9411i	0.856052	+	1.48272i
$525$		0			0
$526$	−4.00000	+	6.92820i	−0.174408	+	0.302084i
$527$	−2.44949	−	1.41421i	−0.106701	−	0.0616041i
$528$		0			0
$529$	−7.50000	−	12.9904i	−0.326087	−	0.564799i
$530$	−9.79796	−	16.9706i	−0.425596	−	0.737154i
$531$		0			0
$532$	20.0000	−	17.3205i	0.867110	−	0.750939i
$533$	−29.3939			−1.27319
$534$		0			0
$535$	−4.00000	−	6.92820i	−0.172935	−	0.299532i
$536$	12.2474	+	21.2132i	0.529009	+	0.916271i
$537$		0			0
$538$	−8.00000	+	13.8564i	−0.344904	+	0.597392i
$539$	−24.4949	+	31.1127i	−1.05507	+	1.34012i
$540$		0			0
$541$	−11.5000	+	19.9186i	−0.494424	+	0.856367i	−0.999979	−	0.00642713i	$-0.997954\pi$
$541$	−11.5000	+	19.9186i	−0.494424	+	0.856367i	0.505556	+	0.862794i	$0.331288\pi$
$542$		−	39.5980i		−	1.70088i
$543$		0			0
$544$	16.0000			0.685994
$545$			14.1421i			0.605783i
$546$		0			0
$547$			10.3923i			0.444343i	0.975008	+	0.222171i	$0.0713145\pi$
$547$			10.3923i			0.444343i	−0.975008	+	0.222171i	$0.928686\pi$
$548$	19.5959			0.837096
$549$		0			0
$550$	−24.0000			−1.02336
$551$		0			0
$552$		0			0
$553$	1.50000	−	4.33013i	0.0637865	−	0.184136i
$554$	13.4722	+	7.77817i	0.572379	+	0.330463i
$555$		0			0
$556$	13.0000	−	22.5167i	0.551323	−	0.954919i
$557$	−9.79796	−	16.9706i	−0.415153	−	0.719066i	0.580292	−	0.814409i	$-0.302939\pi$
$557$	−9.79796	−	16.9706i	−0.415153	−	0.719066i	−0.995444	+	0.0953429i	$0.969605\pi$
$558$		0			0
$559$	27.0000			1.14198
$560$	−29.3939	+	5.65685i	−1.24212	+	0.239046i
$561$		0			0
$562$	−18.0000	+	10.3923i	−0.759284	+	0.438373i
$563$	−12.2474	−	21.2132i	−0.516168	−	0.894030i	−0.999824	−	0.0187714i	$-0.994025\pi$
$563$	−12.2474	−	21.2132i	−0.516168	−	0.894030i	0.483655	−	0.875259i	$-0.339309\pi$
$564$		0			0
$565$	−36.0000	−	20.7846i	−1.51453	−	0.874415i
$566$	−35.5176	−	20.5061i	−1.49292	−	0.861936i
$567$		0			0
$568$	8.00000			0.335673
$569$	9.79796	−	16.9706i	0.410752	−	0.711443i	−0.584220	−	0.811595i	$-0.698600\pi$
$569$	9.79796	−	16.9706i	0.410752	−	0.711443i	0.994972	+	0.100152i	$0.0319329\pi$
$570$		0			0
$571$	−10.5000	+	6.06218i	−0.439411	+	0.253694i	−0.703348	−	0.710846i	$-0.748312\pi$
$571$	−10.5000	+	6.06218i	−0.439411	+	0.253694i	0.263937	+	0.964540i	$0.414979\pi$
$572$	−58.7878			−2.45804
$573$		0			0
$574$	−16.0000	+	13.8564i	−0.667827	+	0.578355i
$575$			8.48528i			0.353861i
$576$		0			0
$577$	−7.50000	+	4.33013i	−0.312229	+	0.180266i	−0.647924	−	0.761705i	$-0.724362\pi$
$577$	−7.50000	+	4.33013i	−0.312229	+	0.180266i	0.335694	+	0.941971i	$0.391029\pi$
$578$		−	12.7279i		−	0.529412i
$579$		0			0
$580$		0			0
$581$	−7.34847	−	38.1838i	−0.304866	−	1.58413i
$582$		0			0
$583$	−24.0000	−	13.8564i	−0.993978	−	0.573874i
$584$	2.44949	+	4.24264i	0.101361	+	0.175562i
$585$		0			0
$586$	−8.00000	−	13.8564i	−0.330477	−	0.572403i
$587$		0			0			−	1.00000i	$-0.5\pi$
$587$		0			0				1.00000i	$0.5\pi$
$588$		0			0
$589$	−5.00000			−0.206021
$590$	9.79796	+	16.9706i	0.403376	+	0.698667i
$591$		0			0
$592$	20.0000			0.821995
$593$	34.2929	+	19.7990i	1.40824	+	0.813047i	0.995218	−	0.0976747i	$-0.0311405\pi$
$593$	34.2929	+	19.7990i	1.40824	+	0.813047i	0.413020	+	0.910722i	$0.364474\pi$
$594$		0			0
$595$	−4.00000	−	20.7846i	−0.163984	−	0.852086i
$596$	9.79796	+	16.9706i	0.401340	+	0.695141i
$597$		0			0
$598$			20.7846i			0.849946i
$599$	19.5959	−	11.3137i	0.800668	−	0.462266i	−0.0430370	−	0.999073i	$-0.513703\pi$
$599$	19.5959	−	11.3137i	0.800668	−	0.462266i	0.843705	+	0.536808i	$0.180370\pi$
$600$		0			0
$601$			25.9808i			1.05978i	0.848067	+	0.529889i	$0.177766\pi$
$601$			25.9808i			1.05978i	−0.848067	+	0.529889i	$0.822234\pi$
$602$	14.6969	−	12.7279i	0.599002	−	0.518751i
$603$		0			0
$604$			34.6410i			1.40952i
$605$	−51.4393	+	29.6985i	−2.09130	+	1.20742i
$606$		0			0
$607$	23.5000	−	40.7032i	0.953836	−	1.65209i	0.216825	−	0.976210i	$-0.430430\pi$
$607$	23.5000	−	40.7032i	0.953836	−	1.65209i	0.737011	−	0.675881i	$-0.236237\pi$
$608$	24.4949	−	14.1421i	0.993399	−	0.573539i
$609$		0			0
$610$	24.0000	+	13.8564i	0.971732	+	0.561029i
$611$	22.0454	+	12.7279i	0.891862	+	0.514917i
$612$		0			0
$613$	−7.00000	−	12.1244i	−0.282727	−	0.489698i	0.689328	−	0.724449i	$-0.257906\pi$
$613$	−7.00000	−	12.1244i	−0.282727	−	0.489698i	−0.972056	+	0.234751i	$0.924572\pi$
$614$	−1.22474	+	0.707107i	−0.0494267	+	0.0285365i
$615$		0			0
$616$	−32.0000	+	27.7128i	−1.28932	+	1.11658i
$617$	−29.3939			−1.18335			−0.591676	−	0.806176i	$-0.701534\pi$
$617$	−29.3939			−1.18335			−0.591676	+	0.806176i	$0.701534\pi$
$618$		0			0
$619$	14.5000	+	25.1147i	0.582804	+	1.00945i	0.995145	+	0.0984169i	$0.0313779\pi$
$619$	14.5000	+	25.1147i	0.582804	+	1.00945i	−0.412341	+	0.911030i	$0.635289\pi$
$620$	4.89898	+	2.82843i	0.196748	+	0.113592i
$621$		0			0
$622$	−24.0000	−	13.8564i	−0.962312	−	0.555591i
$623$	9.79796	−	28.2843i	0.392547	−	1.13319i
$624$		0			0
$625$	15.5000	−	26.8468i	0.620000	−	1.07387i
$626$	−31.8434			−1.27272
$627$		0			0
$628$		−	13.8564i		−	0.552931i
$629$			14.1421i			0.563884i
$630$		0			0
$631$		−	10.3923i		−	0.413711i	−0.978371	−	0.206856i	$-0.933677\pi$
$631$		−	10.3923i		−	0.413711i	0.978371	−	0.206856i	$-0.0663230\pi$
$632$	2.44949	−	4.24264i	0.0974355	−	0.168763i
$633$		0			0
$634$			27.7128i			1.10062i
$635$	22.0454	−	38.1838i	0.874845	−	1.51528i
$636$		0			0
$637$	13.5000	+	33.7750i	0.534889	+	1.33821i
$638$		0			0
$639$		0			0
$640$	−32.0000			−1.26491
$641$	−2.44949	−	4.24264i	−0.0967490	−	0.167574i	0.813588	−	0.581442i	$-0.197511\pi$
$641$	−2.44949	−	4.24264i	−0.0967490	−	0.167574i	−0.910337	+	0.413867i	$0.864178\pi$
$642$		0			0
$643$	−29.0000			−1.14365			−0.571824	−	0.820376i	$-0.693764\pi$
$643$	−29.0000			−1.14365			−0.571824	+	0.820376i	$0.693764\pi$
$644$	9.79796	+	11.3137i	0.386094	+	0.445823i
$645$		0			0
$646$	10.0000	+	17.3205i	0.393445	+	0.681466i
$647$	9.79796	+	16.9706i	0.385198	+	0.667182i	0.991797	−	0.127826i	$-0.0408000\pi$
$647$	9.79796	+	16.9706i	0.385198	+	0.667182i	−0.606599	+	0.795008i	$0.707467\pi$
$648$		0			0
$649$	24.0000	+	13.8564i	0.942082	+	0.543912i
$650$	−11.0227	+	19.0919i	−0.432346	+	0.748845i
$651$		0			0
$652$	6.00000	−	3.46410i	0.234978	−	0.135665i
$653$	9.79796	−	16.9706i	0.383424	−	0.664109i	−0.608125	−	0.793841i	$-0.708078\pi$
$653$	9.79796	−	16.9706i	0.383424	−	0.664109i	0.991549	+	0.129732i	$0.0414116\pi$
$654$		0			0
$655$	48.0000	−	27.7128i	1.87552	−	1.08283i
$656$	−19.5959	+	11.3137i	−0.765092	+	0.441726i
$657$		0			0
$658$	18.0000	−	3.46410i	0.701713	−	0.135045i
$659$			14.1421i			0.550899i	0.961315	+	0.275450i	$0.0888267\pi$
$659$			14.1421i			0.550899i	−0.961315	+	0.275450i	$0.911173\pi$
$660$		0			0
$661$	28.5000	−	16.4545i	1.10852	−	0.640005i	0.170075	−	0.985431i	$-0.445599\pi$
$661$	28.5000	−	16.4545i	1.10852	−	0.640005i	0.938446	+	0.345426i	$0.112266\pi$
$662$	−41.6413			−1.61844
$663$		0			0
$664$		−	41.5692i		−	1.61320i
$665$	−24.4949	−	28.2843i	−0.949871	−	1.09682i
$666$		0			0
$667$		0			0
$668$	14.6969	−	25.4558i	0.568642	−	0.984916i
$669$		0			0
$670$	30.0000	−	17.3205i	1.15900	−	0.669150i
$671$	39.1918			1.51298
$672$		0			0
$673$	−19.0000			−0.732396			−0.366198	−	0.930537i	$-0.619341\pi$
$673$	−19.0000			−0.732396			−0.366198	+	0.930537i	$0.619341\pi$
$674$	−35.5176	+	20.5061i	−1.36809	+	0.789865i
$675$		0			0
$676$	−14.0000	+	24.2487i	−0.538462	+	0.932643i
$677$	4.89898	+	2.82843i	0.188283	+	0.108705i	0.591179	−	0.806541i	$-0.298663\pi$
$677$	4.89898	+	2.82843i	0.188283	+	0.108705i	−0.402895	+	0.915246i	$0.631996\pi$
$678$		0			0
$679$		0			0
$680$		−	22.6274i		−	0.867722i
$681$		0			0
$682$	8.00000			0.306336
$683$	−31.8434	+	18.3848i	−1.21845	+	0.703474i	−0.964587	−	0.263766i	$-0.915035\pi$
$683$	−31.8434	+	18.3848i	−1.21845	+	0.703474i	−0.253866	+	0.967239i	$0.581702\pi$
$684$		0			0
$685$		−	27.7128i		−	1.05885i
$686$	23.2702	+	12.0208i	0.888459	+	0.458957i
$687$		0			0
$688$	18.0000	−	10.3923i	0.686244	−	0.396203i
$689$	−22.0454	+	12.7279i	−0.839863	+	0.484895i
$690$		0			0
$691$	2.50000	−	4.33013i	0.0951045	−	0.164726i	−0.814548	−	0.580097i	$-0.803015\pi$
$691$	2.50000	−	4.33013i	0.0951045	−	0.164726i	0.909652	+	0.415371i	$0.136348\pi$
$692$	−34.2929	+	19.7990i	−1.30362	+	0.752645i
$693$		0			0
$694$	2.00000	−	3.46410i	0.0759190	−	0.131495i
$695$	−31.8434	−	18.3848i	−1.20789	−	0.697374i
$696$		0			0
$697$	−8.00000	−	13.8564i	−0.303022	−	0.524849i
$698$	14.6969	+	25.4558i	0.556287	+	0.963518i
$699$		0			0
$700$	3.00000	+	15.5885i	0.113389	+	0.589188i
$701$	44.0908			1.66529			0.832644	−	0.553809i	$-0.186826\pi$
$701$	44.0908			1.66529			0.832644	+	0.553809i	$0.186826\pi$
$702$		0			0
$703$	12.5000	+	21.6506i	0.471446	+	0.816569i
$704$	−39.1918	+	22.6274i	−1.47710	+	0.852803i
$705$		0			0
$706$	22.0000	−	38.1051i	0.827981	−	1.43411i
$707$	19.5959	+	22.6274i	0.736980	+	0.850992i
$708$		0			0
$709$	−1.00000	+	1.73205i	−0.0375558	+	0.0650485i	−0.884192	−	0.467123i	$-0.845291\pi$
$709$	−1.00000	+	1.73205i	−0.0375558	+	0.0650485i	0.846637	+	0.532172i	$0.178624\pi$
$710$		−	11.3137i		−	0.424596i
$711$		0			0
$712$	16.0000	−	27.7128i	0.599625	−	1.03858i
$713$		−	2.82843i		−	0.105925i
$714$		0			0
$715$			83.1384i			3.10920i
$716$			5.65685i			0.211407i
$717$		0			0
$718$	32.0000			1.19423
$719$	19.5959	−	33.9411i	0.730804	−	1.26579i	−0.225735	−	0.974189i	$-0.572478\pi$
$719$	19.5959	−	33.9411i	0.730804	−	1.26579i	0.956540	−	0.291602i	$-0.0941882\pi$
$720$		0			0
$721$	−2.50000	−	0.866025i	−0.0931049	−	0.0322525i
$722$	7.34847	+	4.24264i	0.273482	+	0.157895i
$723$		0			0
$724$	9.00000	+	5.19615i	0.334482	+	0.193113i
$725$		0			0
$726$		0			0
$727$	−17.0000			−0.630495			−0.315248	−	0.949009i	$-0.602088\pi$
$727$	−17.0000			−0.630495			−0.315248	+	0.949009i	$0.602088\pi$
$728$	7.34847	+	38.1838i	0.272352	+	1.41518i
$729$		0			0
$730$	6.00000	−	3.46410i	0.222070	−	0.128212i
$731$	7.34847	+	12.7279i	0.271793	+	0.470759i
$732$		0			0
$733$	−7.50000	−	4.33013i	−0.277019	−	0.159937i	0.355054	−	0.934846i	$-0.384462\pi$
$733$	−7.50000	−	4.33013i	−0.277019	−	0.159937i	−0.632073	+	0.774909i	$0.717796\pi$
$734$	−13.4722	−	7.77817i	−0.497268	−	0.287098i
$735$		0			0
$736$	8.00000	+	13.8564i	0.294884	+	0.510754i
$737$	24.4949	−	42.4264i	0.902281	−	1.56280i
$738$		0			0
$739$	−28.5000	+	16.4545i	−1.04839	+	0.605288i	−0.922198	−	0.386718i	$-0.873609\pi$
$739$	−28.5000	+	16.4545i	−1.04839	+	0.605288i	−0.126191	+	0.992006i	$0.540275\pi$
$740$		−	28.2843i		−	1.03975i
$741$		0			0
$742$	−6.00000	+	17.3205i	−0.220267	+	0.635856i
$743$		−	45.2548i		−	1.66024i	−0.557586	−	0.830119i	$-0.688272\pi$
$743$		−	45.2548i		−	1.66024i	0.557586	−	0.830119i	$-0.311728\pi$
$744$		0			0
$745$	24.0000	−	13.8564i	0.879292	−	0.507659i
$746$			1.41421i			0.0517780i
$747$		0			0
$748$	−16.0000	−	27.7128i	−0.585018	−	1.01328i
$749$	−2.44949	+	7.07107i	−0.0895024	+	0.258371i
$750$		0			0
$751$	7.50000	+	4.33013i	0.273679	+	0.158009i	0.630558	−	0.776142i	$-0.282826\pi$
$751$	7.50000	+	4.33013i	0.273679	+	0.158009i	−0.356879	+	0.934150i	$0.616159\pi$
$752$	19.5959			0.714590
$753$		0			0
$754$		0			0
$755$	48.9898			1.78292
$756$		0			0
$757$	2.00000			0.0726912			0.0363456	−	0.999339i	$-0.488428\pi$
$757$	2.00000			0.0726912			0.0363456	+	0.999339i	$0.488428\pi$
$758$	25.7196	+	44.5477i	0.934179	+	1.61805i
$759$		0			0
$760$	−20.0000	−	34.6410i	−0.725476	−	1.25656i
$761$	−17.1464	−	9.89949i	−0.621558	−	0.358856i	0.155918	−	0.987770i	$-0.450167\pi$
$761$	−17.1464	−	9.89949i	−0.621558	−	0.358856i	−0.777475	+	0.628914i	$0.783500\pi$
$762$		0			0
$763$	10.0000	−	8.66025i	0.362024	−	0.313522i
$764$	34.2929	−	19.7990i	1.24067	−	0.716302i
$765$		0			0
$766$		−	48.4974i		−	1.75228i
$767$	22.0454	−	12.7279i	0.796014	−	0.459579i
$768$		0			0
$769$			15.5885i			0.562134i	0.959688	+	0.281067i	$0.0906883\pi$
$769$			15.5885i			0.562134i	−0.959688	+	0.281067i	$0.909312\pi$
$770$	39.1918	+	45.2548i	1.41238	+	1.63087i
$771$		0			0
$772$	14.0000			0.503871
$773$	9.79796	−	5.65685i	0.352408	−	0.203463i	−0.313337	−	0.949642i	$-0.601447\pi$
$773$	9.79796	−	5.65685i	0.352408	−	0.203463i	0.665745	+	0.746179i	$0.268114\pi$
$774$		0			0
$775$	1.50000	−	2.59808i	0.0538816	−	0.0933257i
$776$		0			0
$777$		0			0
$778$	−30.0000	−	17.3205i	−1.07555	−	0.620970i
$779$	−24.4949	−	14.1421i	−0.877621	−	0.506695i
$780$		0			0
$781$	−8.00000	−	13.8564i	−0.286263	−	0.495821i
$782$	−9.79796	+	5.65685i	−0.350374	+	0.202289i
$783$		0			0
$784$	22.0000	+	17.3205i	0.785714	+	0.618590i
$785$	−19.5959			−0.699408
$786$		0			0
$787$	−8.00000	−	13.8564i	−0.285169	−	0.493928i	0.687481	−	0.726202i	$-0.258716\pi$
$787$	−8.00000	−	13.8564i	−0.285169	−	0.493928i	−0.972650	+	0.232275i	$0.925383\pi$
$788$	−14.6969	+	25.4558i	−0.523557	+	0.906827i
$789$		0			0
$790$	−6.00000	−	3.46410i	−0.213470	−	0.123247i
$791$	7.34847	+	38.1838i	0.261281	+	1.35766i
$792$		0			0
$793$	18.0000	−	31.1769i	0.639199	−	1.10712i
$794$	41.6413			1.47780
$795$		0			0
$796$	−8.00000			−0.283552
$797$		−	31.1127i		−	1.10207i	−0.834483	−	0.551034i	$-0.814233\pi$
$797$		−	31.1127i		−	1.10207i	0.834483	−	0.551034i	$-0.185767\pi$
$798$		0			0
$799$			13.8564i			0.490204i
$800$			16.9706i			0.600000i
$801$		0			0
$802$			6.92820i			0.244643i
$803$	4.89898	−	8.48528i	0.172881	−	0.299439i
$804$		0			0
$805$	16.0000	−	13.8564i	0.563926	−	0.488374i
$806$	3.67423	−	6.36396i	0.129419	−	0.224161i
$807$		0			0
$808$	16.0000	+	27.7128i	0.562878	+	0.974933i
$809$	−2.44949	−	4.24264i	−0.0861195	−	0.149163i	0.819748	−	0.572724i	$-0.194113\pi$
$809$	−2.44949	−	4.24264i	−0.0861195	−	0.149163i	−0.905868	+	0.423561i	$0.860780\pi$
$810$		0			0
$811$	−8.00000			−0.280918			−0.140459	−	0.990086i	$-0.544858\pi$
$811$	−8.00000			−0.280918			−0.140459	+	0.990086i	$0.544858\pi$
$812$		0			0
$813$		0			0
$814$	−20.0000	−	34.6410i	−0.701000	−	1.21417i
$815$	−4.89898	−	8.48528i	−0.171604	−	0.297226i
$816$		0			0
$817$	22.5000	+	12.9904i	0.787175	+	0.454476i
$818$	20.8207	−	36.0624i	0.727977	−	1.26089i
$819$		0			0
$820$	16.0000	+	27.7128i	0.558744	+	0.967773i
$821$	17.1464	−	29.6985i	0.598414	−	1.03648i	−0.394641	−	0.918835i	$-0.629131\pi$
$821$	17.1464	−	29.6985i	0.598414	−	1.03648i	0.993055	−	0.117649i	$-0.0375357\pi$
$822$		0			0
$823$	39.0000	−	22.5167i	1.35945	−	0.784881i	0.369904	−	0.929070i	$-0.379390\pi$
$823$	39.0000	−	22.5167i	1.35945	−	0.784881i	0.989550	+	0.144188i	$0.0460571\pi$
$824$	−2.44949	−	1.41421i	−0.0853320	−	0.0492665i
$825$		0			0
$826$	6.00000	−	17.3205i	0.208767	−	0.602658i
$827$		−	2.82843i		−	0.0983540i	−0.998790	−	0.0491770i	$-0.984340\pi$
$827$		−	2.82843i		−	0.0983540i	0.998790	−	0.0491770i	$-0.0156598\pi$
$828$		0			0
$829$	28.5000	−	16.4545i	0.989846	−	0.571488i	0.0846177	−	0.996413i	$-0.473033\pi$
$829$	28.5000	−	16.4545i	0.989846	−	0.571488i	0.905228	+	0.424926i	$0.139700\pi$
$830$	−58.7878			−2.04055
$831$		0			0
$832$			41.5692i			1.44115i
$833$	−12.2474	+	15.5563i	−0.424349	+	0.538996i
$834$		0			0
$835$	−36.0000	−	20.7846i	−1.24583	−	0.719281i
$836$	−48.9898	−	28.2843i	−1.69435	−	0.978232i
$837$		0			0
$838$		0			0
$839$	29.3939			1.01479			0.507395	−	0.861714i	$-0.330609\pi$
$839$	29.3939			1.01479			0.507395	+	0.861714i	$0.330609\pi$
$840$		0			0
$841$	−29.0000			−1.00000
$842$	−28.1691	+	16.2635i	−0.970772	+	0.560476i
$843$		0			0
$844$	−18.0000	−	10.3923i	−0.619586	−	0.357718i
$845$	34.2929	+	19.7990i	1.17971	+	0.681106i
$846$		0			0
$847$	52.5000	+	18.1865i	1.80392	+	0.624897i
$848$	−9.79796	+	16.9706i	−0.336463	+	0.582772i
$849$		0			0
$850$	−12.0000			−0.411597
$851$	−12.2474	+	7.07107i	−0.419837	+	0.242393i
$852$		0			0
$853$		−	15.5885i		−	0.533739i	−0.963733	−	0.266869i	$-0.914011\pi$
$853$		−	15.5885i		−	0.533739i	0.963733	−	0.266869i	$-0.0859892\pi$
$854$	−4.89898	−	25.4558i	−0.167640	−	0.871081i
$855$		0			0
$856$	−4.00000	+	6.92820i	−0.136717	+	0.236801i
$857$	−4.89898	+	2.82843i	−0.167346	+	0.0966172i	−0.581334	−	0.813665i	$-0.697469\pi$
$857$	−4.89898	+	2.82843i	−0.167346	+	0.0966172i	0.413988	+	0.910282i	$0.364136\pi$
$858$		0			0
$859$	4.00000	−	6.92820i	0.136478	−	0.236387i	−0.789683	−	0.613515i	$-0.789755\pi$
$859$	4.00000	−	6.92820i	0.136478	−	0.236387i	0.926161	+	0.377128i	$0.123088\pi$
$860$	−14.6969	−	25.4558i	−0.501161	−	0.868037i
$861$		0			0
$862$	−10.0000	+	17.3205i	−0.340601	+	0.589939i
$863$	24.4949	+	14.1421i	0.833816	+	0.481404i	0.855157	−	0.518368i	$-0.173460\pi$
$863$	24.4949	+	14.1421i	0.833816	+	0.481404i	−0.0213414	+	0.999772i	$0.506794\pi$
$864$		0			0
$865$	28.0000	+	48.4974i	0.952029	+	1.64896i
$866$	3.67423	+	6.36396i	0.124856	+	0.216256i
$867$		0			0
$868$	−1.00000	−	5.19615i	−0.0339422	−	0.176369i
$869$	−9.79796			−0.332373
$870$		0			0
$871$	−22.5000	−	38.9711i	−0.762383	−	1.32049i
$872$	12.2474	−	7.07107i	0.414751	−	0.239457i
$873$		0			0
$874$	−10.0000	+	17.3205i	−0.338255	+	0.585875i
$875$	−14.6969	+	2.82843i	−0.496847	+	0.0956183i
$876$		0			0
$877$	−19.0000	+	32.9090i	−0.641584	+	1.11126i	0.343495	+	0.939155i	$0.388389\pi$
$877$	−19.0000	+	32.9090i	−0.641584	+	1.11126i	−0.985079	+	0.172102i	$0.944944\pi$
$878$		−	5.65685i		−	0.190910i
$879$		0			0
$880$	32.0000	+	55.4256i	1.07872	+	1.86840i
$881$			53.7401i			1.81055i	0.424826	+	0.905275i	$0.360335\pi$
$881$			53.7401i			1.81055i	−0.424826	+	0.905275i	$0.639665\pi$
$882$		0			0
$883$		−	15.5885i		−	0.524593i	−0.964987	−	0.262297i	$-0.915520\pi$
$883$		−	15.5885i		−	0.524593i	0.964987	−	0.262297i	$-0.0844799\pi$
$884$	−29.3939			−0.988623
$885$		0			0
$886$	44.0000			1.47821
$887$	19.5959	−	33.9411i	0.657967	−	1.13963i	−0.323175	−	0.946339i	$-0.604750\pi$
$887$	19.5959	−	33.9411i	0.657967	−	1.13963i	0.981141	−	0.193292i	$-0.0619165\pi$
$888$		0			0
$889$	−40.5000	+	7.79423i	−1.35833	+	0.261410i
$890$	−39.1918	−	22.6274i	−1.31371	−	0.758473i
$891$		0			0
$892$	4.00000	−	6.92820i	0.133930	−	0.231973i
$893$	12.2474	+	21.2132i	0.409845	+	0.709873i
$894$		0			0
$895$	8.00000			0.267411
$896$	19.5959	+	22.6274i	0.654654	+	0.755929i
$897$		0			0
$898$	−18.0000	+	10.3923i	−0.600668	+	0.346796i
$899$		0			0
$900$		0			0
$901$	−12.0000	−	6.92820i	−0.399778	−	0.230812i
$902$	39.1918	+	22.6274i	1.30495	+	0.753411i
$903$		0			0
$904$			41.5692i			1.38257i
$905$	7.34847	−	12.7279i	0.244271	−	0.423090i
$906$		0			0
$907$	−19.5000	+	11.2583i	−0.647487	+	0.373827i	−0.787493	−	0.616324i	$-0.788621\pi$
$907$	−19.5000	+	11.2583i	−0.647487	+	0.373827i	0.140006	+	0.990151i	$0.455288\pi$
$908$	−19.5959			−0.650313
$909$		0			0
$910$	54.0000	−	10.3923i	1.79008	−	0.344502i
$911$		−	2.82843i		−	0.0937100i	−0.998902	−	0.0468550i	$-0.985080\pi$
$911$		−	2.82843i		−	0.0937100i	0.998902	−	0.0468550i	$-0.0149199\pi$
$912$		0			0
$913$	−72.0000	+	41.5692i	−2.38285	+	1.37574i
$914$			9.89949i			0.327446i
$915$		0			0
$916$	−33.0000	+	19.0526i	−1.09035	+	0.629514i
$917$	−48.9898	−	16.9706i	−1.61779	−	0.560417i
$918$		0			0
$919$	43.5000	+	25.1147i	1.43493	+	0.828459i	0.997491	−	0.0707883i	$-0.0225515\pi$
$919$	43.5000	+	25.1147i	1.43493	+	0.828459i	0.437441	+	0.899247i	$0.355885\pi$
$920$	19.5959	−	11.3137i	0.646058	−	0.373002i
$921$		0			0
$922$	−8.00000	−	13.8564i	−0.263466	−	0.456336i
$923$	−14.6969			−0.483756
$924$		0			0
$925$	−15.0000			−0.493197
$926$	3.67423	+	6.36396i	0.120743	+	0.209133i
$927$		0			0
$928$		0			0
$929$	12.2474	+	7.07107i	0.401826	+	0.231994i	0.687271	−	0.726401i	$-0.258808\pi$
$929$	12.2474	+	7.07107i	0.401826	+	0.231994i	−0.285446	+	0.958395i	$0.592142\pi$
$930$		0			0
$931$	−5.00000	+	34.6410i	−0.163868	+	1.13531i
$932$	24.4949	+	42.4264i	0.802357	+	1.38972i
$933$		0			0
$934$		−	6.92820i		−	0.226698i
$935$	−39.1918	+	22.6274i	−1.28171	+	0.739996i
$936$		0			0
$937$		−	36.3731i		−	1.18826i	−0.804370	−	0.594128i	$-0.797497\pi$
$937$		−	36.3731i		−	1.18826i	0.804370	−	0.594128i	$-0.202503\pi$
$938$	−30.6186	−	10.6066i	−0.999733	−	0.346318i
$939$		0			0
$940$		−	27.7128i		−	0.903892i
$941$	−41.6413	+	24.0416i	−1.35747	+	0.783735i	−0.989282	−	0.146017i	$-0.953354\pi$
$941$	−41.6413	+	24.0416i	−1.35747	+	0.783735i	−0.368186	+	0.929752i	$0.620021\pi$
$942$		0			0
$943$	8.00000	−	13.8564i	0.260516	−	0.451227i
$944$	9.79796	−	16.9706i	0.318896	−	0.552345i
$945$		0			0
$946$	−36.0000	−	20.7846i	−1.17046	−	0.675766i
$947$	−4.89898	−	2.82843i	−0.159195	−	0.0919115i	0.418286	−	0.908315i	$-0.362631\pi$
$947$	−4.89898	−	2.82843i	−0.159195	−	0.0919115i	−0.577481	+	0.816404i	$0.695964\pi$
$948$		0			0
$949$	−4.50000	−	7.79423i	−0.146076	−	0.253011i
$950$	−18.3712	+	10.6066i	−0.596040	+	0.344124i
$951$		0			0
$952$	−16.0000	+	13.8564i	−0.518563	+	0.449089i
$953$	−44.0908			−1.42824			−0.714121	−	0.700022i	$-0.753173\pi$
$953$	−44.0908			−1.42824			−0.714121	+	0.700022i	$0.753173\pi$
$954$		0			0
$955$	−28.0000	−	48.4974i	−0.906059	−	1.56934i
$956$	−19.5959	−	11.3137i	−0.633777	−	0.365911i
$957$		0			0
$958$	−42.0000	−	24.2487i	−1.35696	−	0.783440i
$959$	−19.5959	+	16.9706i	−0.632785	+	0.548008i
$960$		0			0
$961$	15.0000	−	25.9808i	0.483871	−	0.838089i
$962$	−36.7423			−1.18462
$963$		0			0
$964$			27.7128i			0.892570i
$965$		−	19.7990i		−	0.637352i
$966$		0			0
$967$		−	25.9808i		−	0.835485i	−0.908565	−	0.417742i	$-0.862821\pi$
$967$		−	25.9808i		−	0.835485i	0.908565	−	0.417742i	$-0.137179\pi$
$968$	51.4393	+	29.6985i	1.65332	+	0.954545i
$969$		0			0
$970$		0			0
$971$	−17.1464	+	29.6985i	−0.550255	+	0.953070i	0.448001	+	0.894033i	$0.352136\pi$
$971$	−17.1464	+	29.6985i	−0.550255	+	0.953070i	−0.998256	+	0.0590366i	$0.981197\pi$
$972$		0			0
$973$	6.50000	+	33.7750i	0.208380	+	1.08278i
$974$	15.9217	−	27.5772i	0.510164	−	0.883629i
$975$		0			0
$976$		−	27.7128i		−	0.887066i
$977$	−9.79796	−	16.9706i	−0.313464	−	0.542936i	0.665645	−	0.746268i	$-0.268156\pi$
$977$	−9.79796	−	16.9706i	−0.313464	−	0.542936i	−0.979110	+	0.203332i	$0.934823\pi$
$978$		0			0
$979$	−64.0000			−2.04545
$980$	24.4949	−	31.1127i	0.782461	−	0.993859i
$981$		0			0
$982$	−4.00000	−	6.92820i	−0.127645	−	0.221088i
$983$	−12.2474	−	21.2132i	−0.390633	−	0.676596i	0.601900	−	0.798571i	$-0.294410\pi$
$983$	−12.2474	−	21.2132i	−0.390633	−	0.676596i	−0.992533	+	0.121975i	$0.961077\pi$
$984$		0			0
$985$	36.0000	+	20.7846i	1.14706	+	0.662253i
$986$		0			0
$987$		0			0
$988$	−45.0000	+	25.9808i	−1.43164	+	0.826558i
$989$	−7.34847	+	12.7279i	−0.233668	+	0.404724i
$990$		0			0
$991$	−19.5000	+	11.2583i	−0.619438	+	0.357633i	−0.776650	−	0.629932i	$-0.783083\pi$
$991$	−19.5000	+	11.2583i	−0.619438	+	0.357633i	0.157212	+	0.987565i	$0.449749\pi$
$992$		−	5.65685i		−	0.179605i
$993$		0			0
$994$	−8.00000	+	6.92820i	−0.253745	+	0.219749i
$995$			11.3137i			0.358669i
$996$		0			0
$997$	−16.5000	+	9.52628i	−0.522560	+	0.301700i	−0.737982	−	0.674821i	$-0.764221\pi$
$997$	−16.5000	+	9.52628i	−0.522560	+	0.301700i	0.215421	+	0.976521i	$0.430888\pi$
$998$	46.5403			1.47321
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	252.2.bf.c.19.1	yes	4
3.2	odd	2	inner	252.2.bf.c.19.2	yes	4
4.3	odd	2		252.2.bf.b.19.2	yes	4
7.2	even	3		1764.2.b.c.1567.4		4
7.3	odd	6		252.2.bf.b.199.1	yes	4
7.5	odd	6		1764.2.b.d.1567.4		4
12.11	even	2		252.2.bf.b.19.1	✓	4
21.2	odd	6		1764.2.b.c.1567.1		4
21.5	even	6		1764.2.b.d.1567.1		4
21.17	even	6		252.2.bf.b.199.2	yes	4
28.3	even	6	inner	252.2.bf.c.199.1	yes	4
28.19	even	6		1764.2.b.c.1567.3		4
28.23	odd	6		1764.2.b.d.1567.3		4
84.23	even	6		1764.2.b.d.1567.2		4
84.47	odd	6		1764.2.b.c.1567.2		4
84.59	odd	6	inner	252.2.bf.c.199.2	yes	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
252.2.bf.b.19.1	✓	4	12.11	even	2
252.2.bf.b.19.2	yes	4	4.3	odd	2
252.2.bf.b.199.1	yes	4	7.3	odd	6
252.2.bf.b.199.2	yes	4	21.17	even	6
252.2.bf.c.19.1	yes	4	1.1	even	1	trivial
252.2.bf.c.19.2	yes	4	3.2	odd	2	inner
252.2.bf.c.199.1	yes	4	28.3	even	6	inner
252.2.bf.c.199.2	yes	4	84.59	odd	6	inner
1764.2.b.c.1567.1		4	21.2	odd	6
1764.2.b.c.1567.2		4	84.47	odd	6
1764.2.b.c.1567.3		4	28.19	even	6
1764.2.b.c.1567.4		4	7.2	even	3
1764.2.b.d.1567.1		4	21.5	even	6
1764.2.b.d.1567.2		4	84.23	even	6
1764.2.b.d.1567.3		4	28.23	odd	6
1764.2.b.d.1567.4		4	7.5	odd	6

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 252 = 2^{2} \cdot 3^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	252.bf (of order \(6\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(-1.22474 + 0.707107i) q^{2} +(1.00000 - 1.73205i) q^{4} +(-2.44949 - 1.41421i) q^{5} +(0.500000 + 2.59808i) q^{7} +2.82843i q^{8} +4.00000 q^{10} +(4.89898 - 2.82843i) q^{11} -5.19615i q^{13} +(-2.44949 - 2.82843i) q^{14} +(-2.00000 - 3.46410i) q^{16} +(2.44949 - 1.41421i) q^{17} +(2.50000 - 4.33013i) q^{19} +(-4.89898 + 2.82843i) q^{20} +(-4.00000 + 6.92820i) q^{22} +(2.44949 + 1.41421i) q^{23} +(1.50000 + 2.59808i) q^{25} +(3.67423 + 6.36396i) q^{26} +(5.00000 + 1.73205i) q^{28} +(-0.500000 - 0.866025i) q^{31} +(4.89898 + 2.82843i) q^{32} +(-2.00000 + 3.46410i) q^{34} +(2.44949 - 7.07107i) q^{35} +(-2.50000 + 4.33013i) q^{37} +7.07107i q^{38} +(4.00000 - 6.92820i) q^{40} -5.65685i q^{41} +5.19615i q^{43} -11.3137i q^{44} -4.00000 q^{46} +(-2.44949 + 4.24264i) q^{47} +(-6.50000 + 2.59808i) q^{49} +(-3.67423 - 2.12132i) q^{50} +(-9.00000 - 5.19615i) q^{52} +(-2.44949 - 4.24264i) q^{53} -16.0000 q^{55} +(-7.34847 + 1.41421i) q^{56} +(2.44949 + 4.24264i) q^{59} +(6.00000 + 3.46410i) q^{61} +(1.22474 + 0.707107i) q^{62} -8.00000 q^{64} +(-7.34847 + 12.7279i) q^{65} +(7.50000 - 4.33013i) q^{67} -5.65685i q^{68} +(2.00000 + 10.3923i) q^{70} -2.82843i q^{71} +(1.50000 - 0.866025i) q^{73} -7.07107i q^{74} +(-5.00000 - 8.66025i) q^{76} +(9.79796 + 11.3137i) q^{77} +(-1.50000 - 0.866025i) q^{79} +11.3137i q^{80} +(4.00000 + 6.92820i) q^{82} -14.6969 q^{83} -8.00000 q^{85} +(-3.67423 - 6.36396i) q^{86} +(8.00000 + 13.8564i) q^{88} +(-9.79796 - 5.65685i) q^{89} +(13.5000 - 2.59808i) q^{91} +(4.89898 - 2.82843i) q^{92} -6.92820i q^{94} +(-12.2474 + 7.07107i) q^{95} +(6.12372 - 7.77817i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 4 q^{4} + 2 q^{7} + 16 q^{10} - 8 q^{16} + 10 q^{19} - 16 q^{22} + 6 q^{25} + 20 q^{28} - 2 q^{31} - 8 q^{34} - 10 q^{37} + 16 q^{40} - 16 q^{46} - 26 q^{49} - 36 q^{52} - 64 q^{55} + 24 q^{61} - 32 q^{64}+ \cdots + 54 q^{91}+O(q^{100}) \) 4 * q + 4 * q^4 + 2 * q^7 + 16 * q^10 - 8 * q^16 + 10 * q^19 - 16 * q^22 + 6 * q^25 + 20 * q^28 - 2 * q^31 - 8 * q^34 - 10 * q^37 + 16 * q^40 - 16 * q^46 - 26 * q^49 - 36 * q^52 - 64 * q^55 + 24 * q^61 - 32 * q^64 + 30 * q^67 + 8 * q^70 + 6 * q^73 - 20 * q^76 - 6 * q^79 + 16 * q^82 - 32 * q^85 + 32 * q^88 + 54 * q^91

Introduction

L-functions

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