LMFDB - Embedded newform 504.2.cc.a.293.8

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [504,2,Mod(293,504)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("504.293"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(504, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 3, 5, 3])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 504 = 2^{3} \cdot 3^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	504.cc (of order $6$, degree $2$, minimal)

Newform invariants

comment:select newform

sage:traces = [16] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)

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

Self dual:	no
Analytic conductor:	$4.02446026187$
Analytic rank:	$0$
Dimension:	$16$
Relative dimension:	$8$ over $\Q(\zeta_{6})$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} + \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} + 10x^{12} + 19x^{8} + 810x^{4} + 6561 $ x^16 + 10x^12 + 19x^8 + 810*x^4 + 6561
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{6}\cdot 3 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{6}]$

Embedding invariants

Embedding label			293.8
Root			$1.20455 - 1.24461i$ of defining polynomial
Character	$\chi$	$=$	504.293
Dual form			504.2.cc.a.461.8

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+(1.22474 - 0.707107i) q^{2} +(1.68014 - 0.420861i) q^{3} +(1.00000 - 1.73205i) q^{4} +(0.0658376 + 0.0380114i) q^{5} +(1.76015 - 1.70349i) q^{6} +(1.32288 + 2.29129i) q^{7} -2.82843i q^{8} +(2.64575 - 1.41421i) q^{9} +0.107512 q^{10} +(0.951188 - 3.33095i) q^{12} +(-2.27533 + 3.94099i) q^{13} +(3.24037 + 1.87083i) q^{14} +(0.126614 + 0.0361560i) q^{15} +(-2.00000 - 3.46410i) q^{16} +(2.24037 - 3.60288i) q^{18} -7.99179 q^{19} +(0.131675 - 0.0760228i) q^{20} +(3.18693 + 3.29294i) q^{21} +(1.25963 + 0.727248i) q^{23} +(-1.19038 - 4.75216i) q^{24} +(-2.49711 - 4.32512i) q^{25} +6.43560i q^{26} +(3.85005 - 3.48957i) q^{27} +5.29150 q^{28} +(0.180636 - 0.0452478i) q^{30} +(-4.89898 - 2.82843i) q^{32} +0.201137i q^{35} +(0.196262 - 5.99679i) q^{36} +(-9.78791 + 5.65105i) q^{38} +(-2.16427 + 7.57901i) q^{39} +(0.107512 - 0.186217i) q^{40} +(6.23164 + 1.77951i) q^{42} +(0.227946 + 0.00746017i) q^{45} +2.05697 q^{46} +(-4.81819 - 4.97846i) q^{48} +(-3.50000 + 6.06218i) q^{49} +(-6.11665 - 3.53145i) q^{50} +(4.55066 + 7.88197i) q^{52} +(2.24783 - 6.99623i) q^{54} +(6.48074 - 3.74166i) q^{56} +(-13.4273 + 3.36344i) q^{57} +(4.58621 + 2.64785i) q^{59} +(0.189238 - 0.183146i) q^{60} +(-4.13812 - 7.16744i) q^{61} +(6.74037 + 4.19135i) q^{63} -8.00000 q^{64} +(-0.299605 + 0.172977i) q^{65} +(2.42243 + 0.691749i) q^{69} +(0.142226 + 0.246342i) q^{70} +16.5762i q^{71} +(-4.00000 - 7.48331i) q^{72} +(-6.01578 - 6.21588i) q^{75} +(-7.99179 + 13.8422i) q^{76} +(2.70850 + 10.8127i) q^{78} +(6.02559 + 10.4366i) q^{79} -0.304091i q^{80} +(5.00000 - 7.48331i) q^{81} +(15.7330 - 9.08345i) q^{83} +(8.89047 - 2.22699i) q^{84} +(0.284451 - 0.152046i) q^{90} -12.0399 q^{91} +(2.51926 - 1.45450i) q^{92} +(-0.526161 - 0.303779i) q^{95} +(-9.42135 - 2.69037i) q^{96} +9.89949i q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q + 16 q^{4} - 16 q^{15} - 32 q^{16} - 16 q^{18} + 72 q^{23} + 40 q^{25} + 32 q^{30} - 40 q^{39} - 56 q^{49} - 144 q^{50} + 8 q^{57} - 16 q^{60} + 56 q^{63} - 128 q^{64} - 72 q^{65} - 64 q^{72} + 128 q^{78}+ \cdots + 144 q^{92}+O(q^{100}) $ 16 * q + 16 * q^4 - 16 * q^15 - 32 * q^16 - 16 * q^18 + 72 * q^23 + 40 * q^25 + 32 * q^30 - 40 * q^39 - 56 * q^49 - 144 * q^50 + 8 * q^57 - 16 * q^60 + 56 * q^63 - 128 * q^64 - 72 * q^65 - 64 * q^72 + 128 * q^78 + 80 * q^81 + 144 * q^92

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/504\mathbb{Z}\right)^\times$.

$n$	$73$	$127$	$253$	$281$
$\chi(n)$	$-1$	$1$	$-1$	$e\left(\frac{5}{6}\right)$

Coefficient data

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

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

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

$2$	1.22474	−	0.707107i	0.866025	−	0.500000i
$2$	1.22474	−	0.707107i	0.866025	−	0.500000i
$3$	1.68014	−	0.420861i	0.970030	−	0.242984i
$3$	1.68014	−	0.420861i	0.970030	−	0.242984i
$4$	1.00000	−	1.73205i	0.500000	−	0.866025i
$5$	0.0658376	+	0.0380114i	0.0294435	+	0.0169992i	0.514649	−	0.857401i	$-0.327922\pi$
$5$	0.0658376	+	0.0380114i	0.0294435	+	0.0169992i	−0.485206	+	0.874400i	$0.661255\pi$
$6$	1.76015	−	1.70349i	0.718579	−	0.695446i
$7$	1.32288	+	2.29129i	0.500000	+	0.866025i
$7$	1.32288	+	2.29129i	0.500000	+	0.866025i
$8$		−	2.82843i		−	1.00000i
$9$	2.64575	−	1.41421i	0.881917	−	0.471405i
$10$	0.107512			0.0339984
$11$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$11$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$12$	0.951188	−	3.33095i	0.274584	−	0.961563i
$13$	−2.27533	+	3.94099i	−0.631063	+	1.09303i	0.356272	+	0.934382i	$0.384048\pi$
$13$	−2.27533	+	3.94099i	−0.631063	+	1.09303i	−0.987335	+	0.158651i	$0.949286\pi$
$14$	3.24037	+	1.87083i	0.866025	+	0.500000i
$15$	0.126614	+	0.0361560i	0.0326916	+	0.00933543i
$16$	−2.00000	−	3.46410i	−0.500000	−	0.866025i
$17$		0			0			−	1.00000i	$-0.5\pi$
$17$		0			0				1.00000i	$0.5\pi$
$18$	2.24037	−	3.60288i	0.528060	−	0.849207i
$19$	−7.99179			−1.83344			−0.916721	−	0.399527i	$-0.869174\pi$
$19$	−7.99179			−1.83344			−0.916721	+	0.399527i	$0.869174\pi$
$20$	0.131675	−	0.0760228i	0.0294435	−	0.0169992i
$21$	3.18693	+	3.29294i	0.695446	+	0.718579i
$22$		0			0
$23$	1.25963	+	0.727248i	0.262651	+	0.151642i	0.625543	−	0.780189i	$-0.284877\pi$
$23$	1.25963	+	0.727248i	0.262651	+	0.151642i	−0.362892	+	0.931831i	$0.618211\pi$
$24$	−1.19038	−	4.75216i	−0.242984	−	0.970030i
$25$	−2.49711	−	4.32512i	−0.499422	−	0.865024i
$26$			6.43560i			1.26213i
$27$	3.85005	−	3.48957i	0.740942	−	0.671569i
$28$	5.29150			1.00000
$29$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$29$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$30$	0.180636	−	0.0452478i	0.0329795	−	0.00826109i
$31$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$31$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$32$	−4.89898	−	2.82843i	−0.866025	−	0.500000i
$33$		0			0
$34$		0			0
$35$			0.201137i			0.0339984i
$36$	0.196262	−	5.99679i	0.0327103	−	0.999465i
$37$		0			0		1.00000			$0$
$37$		0			0		−1.00000			$\pi$
$38$	−9.78791	+	5.65105i	−1.58781	+	0.916721i
$39$	−2.16427	+	7.57901i	−0.346560	+	1.21361i
$40$	0.107512	−	0.186217i	0.0169992	−	0.0294435i
$41$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$41$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$42$	6.23164	+	1.77951i	0.961563	+	0.274584i
$43$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$43$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$44$		0			0
$45$	0.227946	+	0.00746017i	0.0339802	+	0.00111210i
$46$	2.05697			0.303283
$47$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$47$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$48$	−4.81819	−	4.97846i	−0.695446	−	0.718579i
$49$	−3.50000	+	6.06218i	−0.500000	+	0.866025i
$50$	−6.11665	−	3.53145i	−0.865024	−	0.499422i
$51$		0			0
$52$	4.55066	+	7.88197i	0.631063	+	1.09303i
$53$		0			0			−	1.00000i	$-0.5\pi$
$53$		0			0				1.00000i	$0.5\pi$
$54$	2.24783	−	6.99623i	0.305890	−	0.952067i
$55$		0			0
$56$	6.48074	−	3.74166i	0.866025	−	0.500000i
$57$	−13.4273	+	3.36344i	−1.77849	+	0.445498i
$58$		0			0
$59$	4.58621	+	2.64785i	0.597074	+	0.344721i	0.767890	−	0.640582i	$-0.221307\pi$
$59$	4.58621	+	2.64785i	0.597074	+	0.344721i	−0.170816	+	0.985303i	$0.554640\pi$
$60$	0.189238	−	0.183146i	0.0244305	−	0.0236441i
$61$	−4.13812	−	7.16744i	−0.529832	−	0.917696i	−0.999394	−	0.0347968i	$-0.988922\pi$
$61$	−4.13812	−	7.16744i	−0.529832	−	0.917696i	0.469562	−	0.882899i	$-0.344412\pi$
$62$		0			0
$63$	6.74037	+	4.19135i	0.849207	+	0.528060i
$64$	−8.00000			−1.00000
$65$	−0.299605	+	0.172977i	−0.0371614	+	0.0214551i
$66$		0			0
$67$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$67$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$68$		0			0
$69$	2.42243	+	0.691749i	0.291626	+	0.0832768i
$70$	0.142226	+	0.246342i	0.0169992	+	0.0294435i
$71$			16.5762i			1.96723i	0.180288	+	0.983614i	$0.442297\pi$
$71$			16.5762i			1.96723i	−0.180288	+	0.983614i	$0.557703\pi$
$72$	−4.00000	−	7.48331i	−0.471405	−	0.881917i
$73$		0			0		1.00000			$0$
$73$		0			0		−1.00000			$\pi$
$74$		0			0
$75$	−6.01578	−	6.21588i	−0.694642	−	0.717748i
$76$	−7.99179	+	13.8422i	−0.916721	+	1.58781i
$77$		0			0
$78$	2.70850	+	10.8127i	0.306677	+	1.22430i
$79$	6.02559	+	10.4366i	0.677932	+	1.17421i	0.975603	+	0.219544i	$0.0704571\pi$
$79$	6.02559	+	10.4366i	0.677932	+	1.17421i	−0.297670	+	0.954669i	$0.596210\pi$
$80$		−	0.304091i		−	0.0339984i
$81$	5.00000	−	7.48331i	0.555556	−	0.831479i
$82$		0			0
$83$	15.7330	−	9.08345i	1.72692	−	0.997039i	0.825010	−	0.565118i	$-0.191170\pi$
$83$	15.7330	−	9.08345i	1.72692	−	0.997039i	0.901912	−	0.431920i	$-0.142164\pi$
$84$	8.89047	−	2.22699i	0.970030	−	0.242984i
$85$		0			0
$86$		0			0
$87$		0			0
$88$		0			0
$89$		0			0			−	1.00000i	$-0.5\pi$
$89$		0			0				1.00000i	$0.5\pi$
$90$	0.284451	−	0.152046i	0.0299838	−	0.0160270i
$91$	−12.0399			−1.26213
$92$	2.51926	−	1.45450i	0.262651	−	0.151642i
$93$		0			0
$94$		0			0
$95$	−0.526161	−	0.303779i	−0.0539830	−	0.0311671i
$96$	−9.42135	−	2.69037i	−0.961563	−	0.274584i
$97$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$97$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$98$			9.89949i			1.00000i
$99$		0			0
$100$	−9.98844			−0.998844
$101$	9.61372	−	5.55048i	0.956600	−	0.552294i	0.0614754	−	0.998109i	$-0.480419\pi$
$101$	9.61372	−	5.55048i	0.956600	−	0.552294i	0.895125	+	0.445815i	$0.147086\pi$
$102$		0			0
$103$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$103$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$104$	11.1468	+	6.43560i	1.09303	+	0.631063i
$105$	0.0846509	+	0.337939i	0.00826109	+	0.0329795i
$106$		0			0
$107$		0			0			−	1.00000i	$-0.5\pi$
$107$		0			0				1.00000i	$0.5\pi$
$108$	−2.19407	−	10.1581i	−0.211124	−	0.977459i
$109$		0			0		1.00000			$0$
$109$		0			0		−1.00000			$\pi$
$110$		0			0
$111$		0			0
$112$	5.29150	−	9.16515i	0.500000	−	0.866025i
$113$	−18.0296	−	10.4094i	−1.69608	−	0.979234i	−0.949409	−	0.314044i	$-0.898316\pi$
$113$	−18.0296	−	10.4094i	−1.69608	−	0.979234i	−0.746674	−	0.665190i	$-0.768350\pi$
$114$	−14.0668	+	13.6139i	−1.31747	+	1.27506i
$115$	0.0552874	+	0.0957605i	0.00515557	+	0.00892972i
$116$		0			0
$117$	−0.446560	+	13.6447i	−0.0412845	+	1.26145i
$118$	7.48925			0.689442
$119$		0			0
$120$	0.102265	−	0.358119i	0.00933543	−	0.0326916i
$121$	5.50000	−	9.52628i	0.500000	−	0.866025i
$122$	−10.1363	−	5.85219i	−0.917696	−	0.529832i
$123$		0			0
$124$		0			0
$125$		−	0.759788i		−	0.0679575i
$126$	11.2190	+	0.367172i	0.999465	+	0.0327103i
$127$	−20.4422			−1.81395			−0.906977	−	0.421180i	$-0.861616\pi$
$127$	−20.4422			−1.81395			−0.906977	+	0.421180i	$0.861616\pi$
$128$	−9.79796	+	5.65685i	−0.866025	+	0.500000i
$129$		0			0
$130$	−0.244626	+	0.423705i	−0.0214551	+	0.0371614i
$131$	19.7500	+	11.4027i	1.72557	+	0.996256i	0.906010	+	0.423256i	$0.139113\pi$
$131$	19.7500	+	11.4027i	1.72557	+	0.996256i	0.819555	+	0.573000i	$0.194221\pi$
$132$		0			0
$133$	−10.5721	−	18.3115i	−0.916721	−	1.58781i
$134$		0			0
$135$	0.386122	−	0.0833996i	0.0332321	−	0.00717790i
$136$		0			0
$137$	12.9615	−	7.48331i	1.10737	−	0.639343i	0.169226	−	0.985577i	$-0.445873\pi$
$137$	12.9615	−	7.48331i	1.10737	−	0.639343i	0.938148	+	0.346235i	$0.112540\pi$
$138$	3.45600	−	0.865698i	0.294194	−	0.0736931i
$139$	−10.7334	+	18.5908i	−0.910396	+	1.57685i	−0.0968913	+	0.995295i	$0.530890\pi$
$139$	−10.7334	+	18.5908i	−0.910396	+	1.57685i	−0.813505	+	0.581558i	$0.802443\pi$
$140$	0.348380	+	0.201137i	0.0294435	+	0.0169992i
$141$		0			0
$142$	11.7211	+	20.3016i	0.983614	+	1.70367i
$143$		0			0
$144$	−10.1905	−	6.33672i	−0.849207	−	0.528060i
$145$		0			0
$146$		0			0
$147$	−3.32916	+	11.6583i	−0.274584	+	0.961563i
$148$		0			0
$149$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$149$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$150$	−11.7631	−	3.35907i	−0.960452	−	0.274267i
$151$	7.22111	+	12.5073i	0.587646	+	1.01783i	0.994540	+	0.104357i	$0.0332784\pi$
$151$	7.22111	+	12.5073i	0.587646	+	1.01783i	−0.406894	+	0.913475i	$0.633388\pi$
$152$			22.6042i			1.83344i
$153$		0			0
$154$		0			0
$155$		0			0
$156$	10.9630	+	11.3276i	0.877740	+	0.906936i
$157$	12.1299	−	21.0096i	0.968072	−	1.67675i	0.266949	−	0.963711i	$-0.413984\pi$
$157$	12.1299	−	21.0096i	0.968072	−	1.67675i	0.701123	−	0.713040i	$-0.252682\pi$
$158$	14.7596	+	8.52148i	1.17421	+	0.677932i
$159$		0			0
$160$	−0.215025	−	0.372434i	−0.0169992	−	0.0294435i
$161$			3.84823i			0.303283i
$162$	0.832222	−	12.7007i	0.0653855	−	0.997860i
$163$		0			0		1.00000			$0$
$163$		0			0		−1.00000			$\pi$
$164$		0			0
$165$		0			0
$166$	12.8459	−	22.2498i	0.997039	−	1.72692i
$167$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$167$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$168$	9.31384	−	9.01401i	0.718579	−	0.695446i
$169$	−3.85425	−	6.67575i	−0.296481	−	0.513520i
$170$		0			0
$171$	−21.1443	+	11.3021i	−1.61694	+	0.864293i
$172$		0			0
$173$	−21.3064	+	12.3013i	−1.61990	+	0.935247i	−0.632950	+	0.774193i	$0.718156\pi$
$173$	−21.3064	+	12.3013i	−1.61990	+	0.935247i	−0.986945	+	0.161055i	$0.948510\pi$
$174$		0			0
$175$	6.60673	−	11.4432i	0.499422	−	0.865024i
$176$		0			0
$177$	8.81986	+	2.51860i	0.662942	+	0.189310i
$178$		0			0
$179$		0			0			−	1.00000i	$-0.5\pi$
$179$		0			0				1.00000i	$0.5\pi$
$180$	0.240868	−	0.387354i	0.0179532	−	0.0288717i
$181$	−21.5743			−1.60361			−0.801804	−	0.597587i	$-0.796126\pi$
$181$	−21.5743			−1.60361			−0.801804	+	0.597587i	$0.796126\pi$
$182$	−14.7458	+	8.51350i	−1.09303	+	0.631063i
$183$	−9.96913	−	10.3007i	−0.736939	−	0.761452i
$184$	2.05697	−	3.56277i	0.151642	−	0.262651i
$185$		0			0
$186$		0			0
$187$		0			0
$188$		0			0
$189$	13.0888	+	4.20530i	0.952067	+	0.305890i
$190$	−0.859217			−0.0623341
$191$	21.7038	−	12.5307i	1.57043	−	0.906691i	0.574320	−	0.818631i	$-0.305267\pi$
$191$	21.7038	−	12.5307i	1.57043	−	0.906691i	0.996115	−	0.0880597i	$-0.0280666\pi$
$192$	−13.4411	+	3.36689i	−0.970030	+	0.242984i
$193$	2.94014	−	5.09248i	0.211636	−	0.366565i	−0.740591	−	0.671957i	$-0.765454\pi$
$193$	2.94014	−	5.09248i	0.211636	−	0.366565i	0.952227	+	0.305392i	$0.0987875\pi$
$194$		0			0
$195$	−0.430579	+	0.416718i	−0.0308344	+	0.0298418i
$196$	7.00000	+	12.1244i	0.500000	+	0.866025i
$197$		0			0			−	1.00000i	$-0.5\pi$
$197$		0			0				1.00000i	$0.5\pi$
$198$		0			0
$199$		0			0		1.00000			$0$
$199$		0			0		−1.00000			$\pi$
$200$	−12.2333	+	7.06289i	−0.865024	+	0.499422i
$201$		0			0
$202$	7.84957	−	13.5958i	0.552294	−	0.956600i
$203$		0			0
$204$		0			0
$205$		0			0
$206$		0			0
$207$	4.36115	+	0.142731i	0.303121	+	0.00992047i
$208$	18.2026			1.26213
$209$		0			0
$210$	0.342635	+	0.354032i	0.0236441	+	0.0244305i
$211$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$211$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$212$		0			0
$213$	6.97626	+	27.8503i	0.478006	+	1.90827i
$214$		0			0
$215$		0			0
$216$	−9.87000	−	10.8896i	−0.671569	−	0.740942i
$217$		0			0
$218$		0			0
$219$		0			0
$220$		0			0
$221$		0			0
$222$		0			0
$223$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$223$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$224$		−	14.9666i		−	1.00000i
$225$	−12.7234	−	7.91175i	−0.848225	−	0.527450i
$226$	−29.4422			−1.95847
$227$	26.0939	−	15.0653i	1.73191	−	0.999920i	0.859612	−	0.510947i	$-0.170705\pi$
$227$	26.0939	−	15.0653i	1.73191	−	0.999920i	0.872299	−	0.488973i	$-0.162628\pi$
$228$	−7.60170	+	26.6203i	−0.503435	+	1.76297i
$229$	10.4646	−	18.1253i	0.691522	−	1.19775i	−0.279817	−	0.960053i	$-0.590274\pi$
$229$	10.4646	−	18.1253i	0.691522	−	1.19775i	0.971339	−	0.237699i	$-0.0763930\pi$
$230$	0.135426	+	0.0781881i	0.00892972	+	0.00515557i
$231$		0			0
$232$		0			0
$233$			9.77048i			0.640085i	0.947403	+	0.320043i	$0.103697\pi$
$233$			9.77048i			0.640085i	−0.947403	+	0.320043i	$0.896303\pi$
$234$	9.10132	+	17.0270i	0.594972	+	1.11309i
$235$		0			0
$236$	9.17242	−	5.29570i	0.597074	−	0.344721i
$237$	14.5162	+	14.9991i	0.942930	+	0.974295i
$238$		0			0
$239$	−25.7404	−	14.8612i	−1.66501	−	0.961292i	−0.970269	−	0.242028i	$-0.922188\pi$
$239$	−25.7404	−	14.8612i	−1.66501	−	0.961292i	−0.694737	−	0.719264i	$-0.744479\pi$
$240$	−0.127980	−	0.510916i	−0.00826109	−	0.0329795i
$241$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$241$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$242$		−	15.5563i		−	1.00000i
$243$	5.25127	−	14.6773i	0.336869	−	0.941551i
$244$	−16.5525			−1.05966
$245$	−0.460864	+	0.266080i	−0.0294435	+	0.0169992i
$246$		0			0
$247$	18.1840	−	31.4955i	1.15702	−	2.00401i
$248$		0			0
$249$	22.6108	−	21.8829i	1.43290	−	1.38677i
$250$	−0.537251	−	0.930547i	−0.0339788	−	0.0588529i
$251$		−	30.2827i		−	1.91142i	−0.294303	−	0.955712i	$-0.595087\pi$
$251$		−	30.2827i		−	1.91142i	0.294303	−	0.955712i	$-0.404913\pi$
$252$	14.0000	−	7.48331i	0.881917	−	0.471405i
$253$		0			0
$254$	−25.0365	+	14.4548i	−1.57093	+	0.906977i
$255$		0			0
$256$	−8.00000	+	13.8564i	−0.500000	+	0.866025i
$257$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$257$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$258$		0			0
$259$		0			0
$260$			0.691907i			0.0429103i
$261$		0			0
$262$	32.2516			1.99251
$263$	−0.341568	+	0.197204i	−0.0210620	+	0.0121601i	−0.510494	−	0.859881i	$-0.670537\pi$
$263$	−0.341568	+	0.197204i	−0.0210620	+	0.0121601i	0.489432	+	0.872041i	$0.337204\pi$
$264$		0			0
$265$		0			0
$266$	−25.8964	−	14.9513i	−1.58781	−	0.916721i
$267$		0			0
$268$		0			0
$269$		−	11.9055i		−	0.725892i	−0.931810	−	0.362946i	$-0.881771\pi$
$269$		−	11.9055i		−	0.725892i	0.931810	−	0.362946i	$-0.118229\pi$
$270$	0.413928	−	0.375173i	0.0251909	−	0.0228323i
$271$		0			0		1.00000			$0$
$271$		0			0		−1.00000			$\pi$
$272$		0			0
$273$	−20.2288	+	5.06713i	−1.22430	+	0.306677i
$274$	10.5830	−	18.3303i	0.639343	−	1.10737i
$275$		0			0
$276$	3.62057	−	3.50402i	0.217933	−	0.210917i
$277$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$277$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$278$			30.3587i			1.82079i
$279$		0			0
$280$	0.568902			0.0339984
$281$	16.0193	−	9.24872i	0.955629	−	0.551733i	0.0608039	−	0.998150i	$-0.480634\pi$
$281$	16.0193	−	9.24872i	0.955629	−	0.551733i	0.894825	+	0.446417i	$0.147300\pi$
$282$		0			0
$283$	−12.2721	+	21.2560i	−0.729503	+	1.26354i	0.227591	+	0.973757i	$0.426915\pi$
$283$	−12.2721	+	21.2560i	−0.729503	+	1.26354i	−0.957094	+	0.289779i	$0.906418\pi$
$284$	28.7107	+	16.5762i	1.70367	+	0.983614i
$285$	−1.01187	−	0.288951i	−0.0599382	−	0.0171160i
$286$		0			0
$287$		0			0
$288$	−16.9615	−	0.555112i	−0.999465	−	0.0327103i
$289$	−17.0000			−1.00000
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−29.5379	−	17.0537i	−1.72562	−	0.996289i	−0.905848	−	0.423603i	$-0.860765\pi$
$293$	−29.5379	−	17.0537i	−1.72562	−	0.996289i	−0.819775	−	0.572686i	$-0.805902\pi$
$294$	4.16632	+	16.6326i	0.242984	+	0.970030i
$295$	0.201297	+	0.348657i	0.0117200	+	0.0202996i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−5.73215	+	3.30946i	−0.331499	+	0.191391i
$300$	−16.7820	+	4.20375i	−0.968909	+	0.242704i
$301$		0			0
$302$	17.6880	+	10.2122i	1.01783	+	0.587646i
$303$	13.8164	−	13.3716i	0.793733	−	0.768180i
$304$	15.9836	+	27.6844i	0.916721	+	1.58781i
$305$		−	0.629183i		−	0.0360269i
$306$		0			0
$307$	20.8218			1.18836			0.594180	−	0.804332i	$-0.297477\pi$
$307$	20.8218			1.18836			0.594180	+	0.804332i	$0.297477\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$311$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$312$	21.4367	+	6.12147i	1.21361	+	0.346560i
$313$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$313$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$314$		−	34.3086i		−	1.93614i
$315$	0.284451	+	0.532159i	0.0160270	+	0.0299838i
$316$	24.1024			1.35586
$317$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$317$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$318$		0			0
$319$		0			0
$320$	−0.526701	−	0.304091i	−0.0294435	−	0.0169992i
$321$		0			0
$322$	2.72111	+	4.71310i	0.151642	+	0.262651i
$323$		0			0
$324$	−7.96148	−	16.1436i	−0.442305	−	0.896865i
$325$	22.7270			1.26067
$326$		0			0
$327$		0			0
$328$		0			0
$329$		0			0
$330$		0			0
$331$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$331$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$332$		−	36.3338i		−	1.99408i
$333$		0			0
$334$		0			0
$335$		0			0
$336$	5.03321	−	17.6257i	0.274584	−	0.961563i
$337$	−13.2288	+	22.9129i	−0.720616	+	1.24814i	0.240137	+	0.970739i	$0.422808\pi$
$337$	−13.2288	+	22.9129i	−0.720616	+	1.24814i	−0.960753	+	0.277405i	$0.910526\pi$
$338$	−9.44094	−	5.45073i	−0.513520	−	0.296481i
$339$	−34.6732	−	9.90130i	−1.88319	−	0.537765i
$340$		0			0
$341$		0			0
$342$	−17.9046	+	28.7935i	−0.968168	+	1.55697i
$343$	−18.5203			−1.00000
$344$		0			0
$345$	0.133193	+	0.137623i	0.00717084	+	0.00740937i
$346$	−17.3966	+	30.1318i	−0.935247	+	1.61990i
$347$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$347$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$348$		0			0
$349$	14.3152	+	24.7947i	0.766277	+	1.32723i	0.939569	+	0.342361i	$0.111226\pi$
$349$	14.3152	+	24.7947i	0.766277	+	1.32723i	−0.173291	+	0.984871i	$0.555440\pi$
$350$		−	18.6867i		−	0.998844i
$351$	4.99223	+	23.1129i	0.266466	+	1.23368i
$352$		0			0
$353$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$353$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$354$	12.5830	−	3.15194i	0.668779	−	0.167524i
$355$	−0.630083	+	1.09133i	−0.0334413	+	0.0579220i
$356$		0			0
$357$		0			0
$358$		0			0
$359$		−	23.9044i		−	1.26163i	−0.775934	−	0.630814i	$-0.782721\pi$
$359$		−	23.9044i		−	1.26163i	0.775934	−	0.630814i	$-0.217279\pi$
$360$	0.0211006	−	0.644729i	0.00111210	−	0.0339802i
$361$	44.8687			2.36151
$362$	−26.4231	+	15.2554i	−1.38877	+	0.801804i
$363$	5.23153	−	18.3202i	0.274584	−	0.961563i
$364$	−12.0399	+	20.8537i	−0.631063	+	1.09303i
$365$		0			0
$366$	−19.4934	−	5.56653i	−1.01893	−	0.290967i
$367$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$367$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$368$		−	5.81798i		−	0.303283i
$369$		0			0
$370$		0			0
$371$		0			0
$372$		0			0
$373$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$373$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$374$		0			0
$375$	−0.319766	−	1.27655i	−0.0165126	−	0.0659209i
$376$		0			0
$377$		0			0
$378$	19.0040	−	4.10473i	0.977459	−	0.211124i
$379$		0			0		1.00000			$0$
$379$		0			0		−1.00000			$\pi$
$380$	−1.05232	+	0.607558i	−0.0539830	+	0.0311671i
$381$	−34.3458	+	8.60334i	−1.75959	+	0.440763i
$382$	17.7211	−	30.6939i	0.906691	−	1.57043i
$383$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$383$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$384$	−14.0812	+	13.6279i	−0.718579	+	0.695446i
$385$		0			0
$386$		−	8.31598i		−	0.423272i
$387$		0			0
$388$		0			0
$389$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$389$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$390$	−0.232686	+	0.814838i	−0.0117825	+	0.0412609i
$391$		0			0
$392$	17.1464	+	9.89949i	0.866025	+	0.500000i
$393$	37.9817	+	10.8461i	1.91593	+	0.547113i
$394$		0			0
$395$			0.916165i			0.0460972i
$396$		0			0
$397$	37.7318			1.89370			0.946852	−	0.321668i	$-0.104244\pi$
$397$	37.7318			1.89370			0.946852	+	0.321668i	$0.104244\pi$
$398$		0			0
$399$	−25.4693	−	26.3165i	−1.27506	−	1.31747i
$400$	−9.98844	+	17.3005i	−0.499422	+	0.865024i
$401$	4.01580	+	2.31852i	0.200540	+	0.115782i	0.596907	−	0.802310i	$-0.296396\pi$
$401$	4.01580	+	2.31852i	0.200540	+	0.115782i	−0.396368	+	0.918092i	$0.629729\pi$
$402$		0			0
$403$		0			0
$404$		−	22.2019i		−	1.10459i
$405$	0.613639	−	0.302627i	0.0304920	−	0.0150377i
$406$		0			0
$407$		0			0
$408$		0			0
$409$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$409$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$410$		0			0
$411$	18.6277	−	18.0280i	0.918836	−	0.889256i
$412$		0			0
$413$			14.0111i			0.689442i
$414$	5.44222	−	2.90899i	0.267471	−	0.142969i
$415$	1.38110			0.0677955
$416$	22.2936	−	12.8712i	1.09303	−	0.631063i
$417$	−10.2095	+	35.7525i	−0.499961	+	1.75081i
$418$		0			0
$419$	−26.4889	−	15.2934i	−1.29407	−	0.747130i	−0.314695	−	0.949193i	$-0.601902\pi$
$419$	−26.4889	−	15.2934i	−1.29407	−	0.747130i	−0.979373	+	0.202063i	$0.935236\pi$
$420$	0.669979	+	0.191319i	0.0326916	+	0.00933543i
$421$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$421$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$		0			0
$426$	28.2373	+	29.1765i	1.36810	+	1.41361i
$427$	10.9484	−	18.9633i	0.529832	−	0.917696i
$428$		0			0
$429$		0			0
$430$		0			0
$431$			37.4166i			1.80229i	0.433515	+	0.901146i	$0.357273\pi$
$431$			37.4166i			1.80229i	−0.433515	+	0.901146i	$0.642727\pi$
$432$	−19.7883	−	6.35781i	−0.952067	−	0.305890i
$433$		0			0		1.00000			$0$
$433$		0			0		−1.00000			$\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−10.0667	−	5.81201i	−0.481555	−	0.278026i
$438$		0			0
$439$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$439$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$440$		0			0
$441$	−0.686915	+	20.9888i	−0.0327103	+	0.999465i
$442$		0			0
$443$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$443$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$		0			0
$448$	−10.5830	−	18.3303i	−0.500000	−	0.866025i
$449$			40.9474i			1.93243i	0.257743	+	0.966213i	$0.417021\pi$
$449$			40.9474i			1.93243i	−0.257743	+	0.966213i	$0.582979\pi$
$450$	−21.1773	−	0.693087i	−0.998310	−	0.0326725i
$451$		0			0
$452$	−36.0592	+	20.8188i	−1.69608	+	0.979234i
$453$	17.3963	+	17.9750i	0.817351	+	0.844539i
$454$	21.3056	−	36.9023i	0.999920	−	1.73191i
$455$	−0.792679	−	0.457654i	−0.0371614	−	0.0214551i
$456$	9.51324	+	37.9783i	0.445498	+	1.77849i
$457$	17.0483	+	29.5285i	0.797486	+	1.38129i	0.921249	+	0.388974i	$0.127170\pi$
$457$	17.0483	+	29.5285i	0.797486	+	1.38129i	−0.123763	+	0.992312i	$0.539496\pi$
$458$		−	29.5985i		−	1.38304i
$459$		0			0
$460$	0.221149			0.0103111
$461$	−29.1895	+	16.8526i	−1.35949	+	0.784903i	−0.989555	−	0.144154i	$-0.953954\pi$
$461$	−29.1895	+	16.8526i	−1.35949	+	0.784903i	−0.369937	+	0.929057i	$0.620621\pi$
$462$		0			0
$463$	−8.08256	+	13.9994i	−0.375628	+	0.650608i	−0.990421	−	0.138081i	$-0.955906\pi$
$463$	−8.08256	+	13.9994i	−0.375628	+	0.650608i	0.614792	+	0.788689i	$0.289240\pi$
$464$		0			0
$465$		0			0
$466$	6.90877	+	11.9663i	0.320043	+	0.554330i
$467$			41.6295i			1.92638i	0.268814	+	0.963192i	$0.413368\pi$
$467$			41.6295i			1.92638i	−0.268814	+	0.963192i	$0.586632\pi$
$468$	23.1867	+	14.4181i	1.07181	+	0.666479i
$469$		0			0
$470$		0			0
$471$	11.5378	−	40.4042i	0.531635	−	1.86173i
$472$	7.48925	−	12.9718i	0.344721	−	0.597074i
$473$		0			0
$474$	28.3846	+	8.10553i	1.30375	+	0.372299i
$475$	19.9564	+	34.5655i	0.915662	+	1.58597i
$476$		0			0
$477$		0			0
$478$	−42.0338			−1.92258
$479$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$479$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$480$	−0.518015	−	0.535246i	−0.0236441	−	0.0244305i
$481$		0			0
$482$		0			0
$483$	1.61957	+	6.46557i	0.0736931	+	0.294194i
$484$	−11.0000	−	19.0526i	−0.500000	−	0.866025i
$485$		0			0
$486$	−3.94698	−	21.6892i	−0.179039	−	0.983842i
$487$	−38.4422			−1.74198			−0.870992	−	0.491298i	$-0.836523\pi$
$487$	−38.4422			−1.74198			−0.870992	+	0.491298i	$0.836523\pi$
$488$	−20.2726	+	11.7044i	−0.917696	+	0.529832i
$489$		0			0
$490$	−0.376293	+	0.651759i	−0.0169992	+	0.0294435i
$491$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$491$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$492$		0			0
$493$		0			0
$494$		−	51.4320i		−	2.31404i
$495$		0			0
$496$		0			0
$497$	−37.9807	+	21.9282i	−1.70367	+	0.983614i
$498$	12.2189	−	42.7892i	0.547542	−	1.91743i
$499$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$499$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$500$	−1.31599	−	0.759788i	−0.0588529	−	0.0339788i
$501$		0			0
$502$	−21.4131	−	37.0885i	−0.955712	−	1.65534i
$503$		0			0			−	1.00000i	$-0.5\pi$
$503$		0			0				1.00000i	$0.5\pi$
$504$	11.8549	−	19.0646i	0.528060	−	0.849207i
$505$	0.843926			0.0375542
$506$		0			0
$507$	−9.28525	−	9.59411i	−0.412372	−	0.426089i
$508$	−20.4422	+	35.4070i	−0.906977	+	1.57093i
$509$	−32.4532	−	18.7369i	−1.43846	−	0.830497i	−0.440719	−	0.897645i	$-0.645277\pi$
$509$	−32.4532	−	18.7369i	−1.43846	−	0.830497i	−0.997743	+	0.0671482i	$0.978610\pi$
$510$		0			0
$511$		0			0
$512$			22.6274i			1.00000i
$513$	−30.7688	+	27.8879i	−1.35848	+	1.23128i
$514$		0			0
$515$		0			0
$516$		0			0
$517$		0			0
$518$		0			0
$519$	−30.6206	+	29.6349i	−1.34410	+	1.30083i
$520$	0.489252	+	0.847410i	0.0214551	+	0.0371614i
$521$		0			0			−	1.00000i	$-0.5\pi$
$521$		0			0				1.00000i	$0.5\pi$
$522$		0			0
$523$	−20.6067			−0.901069			−0.450535	−	0.892759i	$-0.648767\pi$
$523$	−20.6067			−0.901069			−0.450535	+	0.892759i	$0.648767\pi$
$524$	39.5000	−	22.8053i	1.72557	−	0.996256i
$525$	6.28425	−	22.0067i	0.274267	−	0.960452i
$526$	−0.278889	+	0.483050i	−0.0121601	+	0.0210620i
$527$		0			0
$528$		0			0
$529$	−10.4422	−	18.0865i	−0.454010	−	0.786368i
$530$		0			0
$531$	15.8786	+	0.519671i	0.689073	+	0.0225518i
$532$	−42.2886			−1.83344
$533$		0			0
$534$		0			0
$535$		0			0
$536$		0			0
$537$		0			0
$538$	−8.41847	−	14.5812i	−0.362946	−	0.628641i
$539$		0			0
$540$	0.241669	−	0.752182i	0.0103998	−	0.0323688i
$541$		0			0		1.00000			$0$
$541$		0			0		−1.00000			$\pi$
$542$		0			0
$543$	−36.2479	+	9.07981i	−1.55555	+	0.389652i
$544$		0			0
$545$		0			0
$546$	−21.1921	+	20.5098i	−0.906936	+	0.877740i
$547$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$547$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$548$		−	29.9333i		−	1.27869i
$549$	−21.0847	−	13.1111i	−0.899874	−	0.559567i
$550$		0			0
$551$		0			0
$552$	1.95656	−	6.85166i	0.0832768	−	0.291626i
$553$	−15.9422	+	27.6127i	−0.677932	+	1.17421i
$554$		0			0
$555$		0			0
$556$	21.4668	+	37.1816i	0.910396	+	1.57685i
$557$		0			0			−	1.00000i	$-0.5\pi$
$557$		0			0				1.00000i	$0.5\pi$
$558$		0			0
$559$		0			0
$560$	0.696760	−	0.402275i	0.0294435	−	0.0169992i
$561$		0			0
$562$	13.0797	−	22.6547i	0.551733	−	0.955629i
$563$	26.6206	+	15.3694i	1.12192	+	0.647743i	0.941891	−	0.335918i	$-0.109046\pi$
$563$	26.6206	+	15.3694i	1.12192	+	0.647743i	0.180032	+	0.983661i	$0.442380\pi$
$564$		0			0
$565$	−0.791351	−	1.37066i	−0.0332924	−	0.0576641i
$566$			34.7108i			1.45901i
$567$	23.7608	+	1.55694i	0.997860	+	0.0653855i
$568$	46.8844			1.96723
$569$	2.44949	−	1.41421i	0.102688	−	0.0592869i	−0.447777	−	0.894146i	$-0.647784\pi$
$569$	2.44949	−	1.41421i	0.102688	−	0.0592869i	0.550464	+	0.834859i	$0.314451\pi$
$570$	−1.44361	+	0.361611i	−0.0604660	+	0.0151462i
$571$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$571$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$572$		0			0
$573$	31.1918	−	30.1877i	1.30306	−	1.26111i
$574$		0			0
$575$		−	7.26407i		−	0.302933i
$576$	−21.1660	+	11.3137i	−0.881917	+	0.471405i
$577$		0			0		1.00000			$0$
$577$		0			0		−1.00000			$\pi$
$578$	−20.8207	+	12.0208i	−0.866025	+	0.500000i
$579$	2.79663	−	9.79348i	0.116224	−	0.407003i
$580$		0			0
$581$	41.6256	+	24.0326i	1.72692	+	0.997039i
$582$		0			0
$583$		0			0
$584$		0			0
$585$	−0.548053	+	0.881359i	−0.0226592	+	0.0364397i
$586$	−48.2352			−1.99258
$587$	−39.6742	+	22.9059i	−1.63753	+	0.945428i	−0.655849	+	0.754892i	$0.727689\pi$
$587$	−39.6742	+	22.9059i	−1.63753	+	0.945428i	−0.981680	+	0.190536i	$0.938978\pi$
$588$	16.8637	+	17.4246i	0.695446	+	0.718579i
$589$		0			0
$590$	0.493075	+	0.284677i	0.0202996	+	0.0117200i
$591$		0			0
$592$		0			0
$593$		0			0			−	1.00000i	$-0.5\pi$
$593$		0			0				1.00000i	$0.5\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$		0			0
$598$	−4.68028	+	8.10648i	−0.191391	+	0.331499i
$599$	−9.79796	−	5.65685i	−0.400334	−	0.231133i	0.286294	−	0.958142i	$-0.407577\pi$
$599$	−9.79796	−	5.65685i	−0.400334	−	0.231133i	−0.686628	+	0.727009i	$0.740910\pi$
$600$	−17.5812	+	17.0152i	−0.717748	+	0.694642i
$601$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$601$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$602$		0			0
$603$		0			0
$604$	28.8844			1.17529
$605$	0.724214	−	0.418125i	0.0294435	−	0.0169992i
$606$	7.46641	−	26.1465i	0.303302	−	1.06213i
$607$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$607$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$608$	39.1516	+	22.6042i	1.58781	+	0.916721i
$609$		0			0
$610$	−0.444899	−	0.770588i	−0.0180135	−	0.0312002i
$611$		0			0
$612$		0			0
$613$		0			0		1.00000			$0$
$613$		0			0		−1.00000			$\pi$
$614$	25.5013	−	14.7232i	1.02915	−	0.594180i
$615$		0			0
$616$		0			0
$617$	12.2474	+	7.07107i	0.493064	+	0.284670i	0.725845	−	0.687859i	$-0.241449\pi$
$617$	12.2474	+	7.07107i	0.493064	+	0.284670i	−0.232781	+	0.972529i	$0.574782\pi$
$618$		0			0
$619$	11.7032	+	20.2706i	0.470393	+	0.814744i	0.999427	−	0.0338565i	$-0.0107789\pi$
$619$	11.7032	+	20.2706i	0.470393	+	0.814744i	−0.529034	+	0.848601i	$0.677446\pi$
$620$		0			0
$621$	7.38742	−	1.59563i	0.296447	−	0.0640305i
$622$		0			0
$623$		0			0
$624$	30.5830	−	7.66079i	1.22430	−	0.306677i
$625$	−12.4567	+	21.5756i	−0.498267	+	0.863023i
$626$		0			0
$627$		0			0
$628$	−24.2598	−	42.0192i	−0.968072	−	1.67675i
$629$		0			0
$630$	0.724674	+	0.450622i	0.0288717	+	0.0179532i
$631$	47.9141			1.90743			0.953716	−	0.300709i	$-0.0972234\pi$
$631$	47.9141			1.90743			0.953716	+	0.300709i	$0.0972234\pi$
$632$	29.5193	−	17.0430i	1.17421	−	0.677932i
$633$		0			0
$634$		0			0
$635$	−1.34587	−	0.777037i	−0.0534091	−	0.0308358i
$636$		0			0
$637$	−15.9273	−	27.5869i	−0.631063	−	1.09303i
$638$		0			0
$639$	23.4422	+	43.8564i	0.927360	+	1.73493i
$640$	−0.860099			−0.0339984
$641$	32.7265	−	18.8947i	1.29262	−	0.746295i	0.313503	−	0.949587i	$-0.398498\pi$
$641$	32.7265	−	18.8947i	1.29262	−	0.746295i	0.979118	+	0.203292i	$0.0651642\pi$
$642$		0			0
$643$	−7.63202	+	13.2190i	−0.300978	+	0.521308i	−0.976358	−	0.216161i	$-0.930646\pi$
$643$	−7.63202	+	13.2190i	−0.300978	+	0.521308i	0.675380	+	0.737470i	$0.263980\pi$
$644$	6.66533	+	3.84823i	0.262651	+	0.151642i
$645$		0			0
$646$		0			0
$647$		0			0			−	1.00000i	$-0.5\pi$
$647$		0			0				1.00000i	$0.5\pi$
$648$	−21.1660	−	14.1421i	−0.831479	−	0.555556i
$649$		0			0
$650$	27.8348	−	16.0704i	1.09177	−	0.630333i
$651$		0			0
$652$		0			0
$653$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$653$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$654$		0			0
$655$	0.866862	+	1.50145i	0.0338711	+	0.0586665i
$656$		0			0
$657$		0			0
$658$		0			0
$659$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$659$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$660$		0			0
$661$	−10.2496	+	17.7528i	−0.398664	+	0.690506i	−0.993561	−	0.113296i	$-0.963859\pi$
$661$	−10.2496	+	17.7528i	−0.398664	+	0.690506i	0.594898	+	0.803801i	$0.297193\pi$
$662$		0			0
$663$		0			0
$664$	−25.6919	−	44.4997i	−0.997039	−	1.72692i
$665$		−	1.60745i		−	0.0623341i
$666$		0			0
$667$		0			0
$668$		0			0
$669$		0			0
$670$		0			0
$671$		0			0
$672$	−6.29888	−	25.1461i	−0.242984	−	0.970030i
$673$	−12.9422	−	22.4166i	−0.498886	−	0.864096i	0.501113	−	0.865382i	$-0.332924\pi$
$673$	−12.9422	−	22.4166i	−0.498886	−	0.864096i	−0.999999	+	0.00128586i	$0.999591\pi$
$674$			37.4166i			1.44123i
$675$	−24.7068	−	7.93808i	−0.950966	−	0.305537i
$676$	−15.4170			−0.592961
$677$	−8.18524	+	4.72575i	−0.314584	+	0.181625i	−0.648976	−	0.760809i	$-0.724802\pi$
$677$	−8.18524	+	4.72575i	−0.314584	+	0.181625i	0.334392	+	0.942434i	$0.391469\pi$
$678$	−49.4671	+	12.3911i	−1.89977	+	0.475877i
$679$		0			0
$680$		0			0
$681$	37.5010	−	36.2938i	1.43704	−	1.39078i
$682$		0			0
$683$		0			0			−	1.00000i	$-0.5\pi$
$683$		0			0				1.00000i	$0.5\pi$
$684$	−1.56848	+	47.9251i	−0.0599724	+	1.83246i
$685$	1.13780			0.0434733
$686$	−22.6826	+	13.0958i	−0.866025	+	0.500000i
$687$	9.95384	−	34.8572i	0.379763	−	1.32988i
$688$		0			0
$689$		0			0
$690$	0.260441	+	0.0743716i	0.00991482	+	0.00283128i
$691$	−11.0560	−	19.1495i	−0.420588	−	0.728480i	0.575409	−	0.817866i	$-0.304843\pi$
$691$	−11.0560	−	19.1495i	−0.420588	−	0.728480i	−0.995997	+	0.0893857i	$0.971510\pi$
$692$			49.2050i			1.87049i
$693$		0			0
$694$		0			0
$695$	−1.41333	+	0.815984i	−0.0536105	+	0.0309520i
$696$		0			0
$697$		0			0
$698$	35.0650	+	20.2448i	1.32723	+	0.766277i
$699$	4.11202	+	16.4158i	0.155531	+	0.620902i
$700$	−13.2135	−	22.8864i	−0.499422	−	0.865024i
$701$		0			0			−	1.00000i	$-0.5\pi$
$701$		0			0				1.00000i	$0.5\pi$
$702$	22.4575	+	24.7774i	0.847604	+	0.935162i
$703$		0			0
$704$		0			0
$705$		0			0
$706$		0			0
$707$	25.4355	+	14.6852i	0.956600	+	0.552294i
$708$	13.1822	−	12.7578i	0.495418	−	0.479469i
$709$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$709$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$710$			1.78214i			0.0668826i
$711$	30.7019	+	19.0913i	1.15141	+	0.715978i
$712$		0			0
$713$		0			0
$714$		0			0
$715$		0			0
$716$		0			0
$717$	−49.5020	−	14.1358i	−1.84869	−	0.527911i
$718$	−16.9030	−	29.2768i	−0.630814	−	1.09260i
$719$		0			0			−	1.00000i	$-0.5\pi$
$719$		0			0				1.00000i	$0.5\pi$
$720$	−0.430050	−	0.804549i	−0.0160270	−	0.0299838i
$721$		0			0
$722$	54.9527	−	31.7270i	2.04513	−	1.18076i
$723$		0			0
$724$	−21.5743	+	37.3679i	−0.801804	+	1.38877i
$725$		0			0
$726$	−6.54707	−	26.1369i	−0.242984	−	0.970030i
$727$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$727$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$728$			34.0540i			1.26213i
$729$	2.64575	−	26.8701i	0.0979908	−	0.995187i
$730$		0			0
$731$		0			0
$732$	−27.8105	+	6.96630i	−1.02791	+	0.257482i
$733$	3.56922	−	6.18207i	0.131832	−	0.228340i	−0.792551	−	0.609806i	$-0.791247\pi$
$733$	3.56922	−	6.18207i	0.131832	−	0.228340i	0.924383	+	0.381466i	$0.124581\pi$
$734$		0			0
$735$	−0.662333	+	0.641011i	−0.0244305	+	0.0236441i
$736$	−4.11393	−	7.12554i	−0.151642	−	0.262651i
$737$		0			0
$738$		0			0
$739$		0			0		1.00000			$0$
$739$		0			0		−1.00000			$\pi$
$740$		0			0
$741$	17.2964	−	60.5699i	0.635398	−	2.22509i
$742$		0			0
$743$	6.48074	+	3.74166i	0.237755	+	0.137268i	0.614145	−	0.789193i	$-0.289501\pi$
$743$	6.48074	+	3.74166i	0.237755	+	0.137268i	−0.376389	+	0.926462i	$0.622834\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$	28.7797	−	46.2824i	1.05299	−	1.69338i
$748$		0			0
$749$		0			0
$750$	−1.29429	−	1.33734i	−0.0472608	−	0.0488328i
$751$	22.2211	−	38.4881i	0.810860	−	1.40445i	−0.101403	−	0.994845i	$-0.532333\pi$
$751$	22.2211	−	38.4881i	0.810860	−	1.40445i	0.912263	−	0.409605i	$-0.134333\pi$
$752$		0			0
$753$	−12.7448	−	50.8792i	−0.464446	−	1.85414i
$754$		0			0
$755$			1.09794i			0.0399580i
$756$	20.3725	−	18.4651i	0.740942	−	0.671569i
$757$		0			0		1.00000			$0$
$757$		0			0		−1.00000			$\pi$
$758$		0			0
$759$		0			0
$760$	−0.859217	+	1.48821i	−0.0311671	+	0.0539830i
$761$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$761$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$762$	−35.9814	+	34.8231i	−1.30347	+	1.26151i
$763$		0			0
$764$		−	50.1229i		−	1.81338i
$765$		0			0
$766$		0			0
$767$	−20.8703	+	12.0495i	−0.753582	+	0.435081i
$768$	−7.60951	+	26.6476i	−0.274584	+	0.961563i
$769$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$769$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$770$		0			0
$771$		0			0
$772$	−5.88029	−	10.1850i	−0.211636	−	0.366565i
$773$			12.3078i			0.442680i	0.975197	+	0.221340i	$0.0710431\pi$
$773$			12.3078i			0.442680i	−0.975197	+	0.221340i	$0.928957\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$		0			0
$778$		0			0
$779$		0			0
$780$	0.291197	+	1.16250i	0.0104265	+	0.0416243i
$781$		0			0
$782$		0			0
$783$		0			0
$784$	28.0000			1.00000
$785$	1.59721	−	0.922149i	0.0570069	−	0.0329129i
$786$	54.1873	−	13.5735i	1.93280	−	0.484149i
$787$	22.6105	−	39.1626i	0.805978	−	1.39600i	−0.109650	−	0.993970i	$-0.534973\pi$
$787$	22.6105	−	39.1626i	0.805978	−	1.39600i	0.915629	−	0.402025i	$-0.131694\pi$
$788$		0			0
$789$	−0.490887	+	0.475084i	−0.0174760	+	0.0169134i
$790$	0.647826	+	1.12207i	0.0230486	+	0.0399214i
$791$		−	55.0814i		−	1.95847i
$792$		0			0
$793$	37.6624			1.33743
$794$	46.2118	−	26.6804i	1.64000	−	0.946852i
$795$		0			0
$796$		0			0
$797$	30.2347	+	17.4560i	1.07097	+	0.618323i	0.928445	−	0.371469i	$-0.121146\pi$
$797$	30.2347	+	17.4560i	1.07097	+	0.618323i	0.142521	+	0.989792i	$0.454479\pi$
$798$	−49.8020	−	14.2215i	−1.76297	−	0.503435i
$799$		0			0
$800$			28.2516i			0.998844i
$801$		0			0
$802$	6.55778			0.231563
$803$		0			0
$804$		0			0
$805$	−0.146277	+	0.253359i	−0.00515557	+	0.00892972i
$806$		0			0
$807$	−5.01057	−	20.0029i	−0.176380	−	0.704137i
$808$	−15.6991	−	27.1917i	−0.552294	−	0.956600i
$809$		−	31.1127i		−	1.09386i	−0.837177	−	0.546932i	$-0.815796\pi$
$809$		−	31.1127i		−	1.09386i	0.837177	−	0.546932i	$-0.184204\pi$
$810$	0.537562	−	0.804549i	0.0188880	−	0.0282690i
$811$	48.4452			1.70114			0.850570	−	0.525861i	$-0.176257\pi$
$811$	48.4452			1.70114			0.850570	+	0.525861i	$0.176257\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$		0			0
$816$		0			0
$817$		0			0
$818$		0			0
$819$	−31.8546	+	17.0270i	−1.11309	+	0.594972i
$820$		0			0
$821$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$821$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$822$	10.0664	−	35.2515i	0.351107	−	1.22954i
$823$	−13.2288	+	22.9129i	−0.461125	+	0.798693i	−0.999017	−	0.0443211i	$-0.985888\pi$
$823$	−13.2288	+	22.9129i	−0.461125	+	0.798693i	0.537892	+	0.843014i	$0.319221\pi$
$824$		0			0
$825$		0			0
$826$	9.90735	+	17.1600i	0.344721	+	0.597074i
$827$		0			0			−	1.00000i	$-0.5\pi$
$827$		0			0				1.00000i	$0.5\pi$
$828$	4.60837	−	7.41100i	0.160152	−	0.257550i
$829$	−1.32653			−0.0460723			−0.0230361	−	0.999735i	$-0.507333\pi$
$829$	−1.32653			−0.0460723			−0.0230361	+	0.999735i	$0.507333\pi$
$830$	1.69149	−	0.976584i	0.0587126	−	0.0338977i
$831$		0			0
$832$	18.2026	−	31.5279i	0.631063	−	1.09303i
$833$		0			0
$834$	12.7768	+	51.0069i	0.442424	+	1.76622i
$835$		0			0
$836$		0			0
$837$		0			0
$838$	−43.2562			−1.49426
$839$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$839$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$840$	0.955836	−	0.239429i	0.0329795	−	0.00826109i
$841$	14.5000	−	25.1147i	0.500000	−	0.866025i
$842$		0			0
$843$	23.0222	−	22.2811i	0.792926	−	0.767400i
$844$		0			0
$845$		−	0.586021i		−	0.0201597i
$846$		0			0
$847$	29.1033			1.00000
$848$		0			0
$849$	−11.6731	+	40.8779i	−0.400620	+	1.40293i
$850$		0			0
$851$		0			0
$852$	55.2144	+	15.7670i	1.89161	+	0.540170i
$853$	11.1097	+	19.2426i	0.380389	+	0.658853i	0.991118	−	0.132987i	$-0.0424567\pi$
$853$	11.1097	+	19.2426i	0.380389	+	0.658853i	−0.610729	+	0.791840i	$0.709123\pi$
$854$		−	30.9669i		−	1.05966i
$855$	−1.82170	−	0.0596201i	−0.0623008	−	0.00203897i
$856$		0			0
$857$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$857$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$858$		0			0
$859$	−27.1612	+	47.0446i	−0.926728	+	1.60514i	−0.137969	+	0.990437i	$0.544057\pi$
$859$	−27.1612	+	47.0446i	−0.926728	+	1.60514i	−0.788759	+	0.614703i	$0.789276\pi$
$860$		0			0
$861$		0			0
$862$	26.4575	+	45.8258i	0.901146	+	1.56083i
$863$			42.0320i			1.43079i	0.698722	+	0.715393i	$0.253752\pi$
$863$			42.0320i			1.43079i	−0.698722	+	0.715393i	$0.746248\pi$
$864$	−28.7313	+	6.20577i	−0.977459	+	0.211124i
$865$	−1.87035			−0.0635938
$866$		0			0
$867$	−28.5624	+	7.15464i	−0.970030	+	0.242984i
$868$		0			0
$869$		0			0
$870$		0			0
$871$		0			0
$872$		0			0
$873$		0			0
$874$	−16.4388			−0.556052
$875$	1.74089	−	1.00511i	0.0588529	−	0.0339788i
$876$		0			0
$877$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$877$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$878$		0			0
$879$	−56.8051	−	16.2213i	−1.91599	−	0.547131i
$880$		0			0
$881$		0			0			−	1.00000i	$-0.5\pi$
$881$		0			0				1.00000i	$0.5\pi$
$882$	14.0000	+	26.1916i	0.471405	+	0.881917i
$883$		0			0		1.00000			$0$
$883$		0			0		−1.00000			$\pi$
$884$		0			0
$885$	0.484943	+	0.501074i	0.0163012	+	0.0168434i
$886$		0			0
$887$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$887$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$888$		0			0
$889$	−27.0425	−	46.8390i	−0.906977	−	1.57093i
$890$		0			0
$891$		0			0
$892$		0			0
$893$		0			0
$894$		0			0
$895$		0			0
$896$	−25.9230	−	14.9666i	−0.866025	−	0.500000i
$897$	−8.23799	+	7.97279i	−0.275059	+	0.266204i
$898$	28.9542	+	50.1501i	0.966213	+	1.67353i
$899$		0			0
$900$	−26.4269	+	14.1258i	−0.880898	+	0.470860i
$901$		0			0
$902$		0			0
$903$		0			0
$904$	−29.4422	+	50.9954i	−0.979234	+	1.69608i
$905$	−1.42040	−	0.820071i	−0.0472158	−	0.0272601i
$906$	34.0163	+	9.71372i	1.13012	+	0.322717i
$907$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$907$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$908$		−	60.2612i		−	1.99984i
$909$	17.5859	−	28.2810i	0.583289	−	0.938023i
$910$	−1.29444			−0.0429103
$911$	45.1826	−	26.0862i	1.49697	−	0.864274i	0.496972	−	0.867766i	$-0.334445\pi$
$911$	45.1826	−	26.0862i	1.49697	−	0.864274i	0.999994	+	0.00349271i	$0.00111177\pi$
$912$	38.5060	+	39.7868i	1.27506	+	1.31747i
$913$		0			0
$914$	41.7596	+	24.1099i	1.38129	+	0.797486i
$915$	−0.264799	−	1.05712i	−0.00875398	−	0.0349472i
$916$	−20.9293	−	36.2506i	−0.691522	−	1.19775i
$917$			60.3372i			1.99251i
$918$		0			0
$919$	−43.8002			−1.44484			−0.722418	−	0.691457i	$-0.756969\pi$
$919$	−43.8002			−1.44484			−0.722418	+	0.691457i	$0.756969\pi$
$920$	0.270852	−	0.156376i	0.00892972	−	0.00515557i
$921$	34.9835	−	8.76307i	1.15275	−	0.288753i
$922$	−23.8331	+	41.2802i	−0.784903	+	1.35949i
$923$	−65.3264	−	37.7162i	−2.15024	−	1.24144i
$924$		0			0
$925$		0			0
$926$			22.8609i			0.751257i
$927$		0			0
$928$		0			0
$929$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$929$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$930$		0			0
$931$	27.9713	−	48.4477i	0.916721	−	1.58781i
$932$	16.9230	+	9.77048i	0.554330	+	0.320043i
$933$		0			0
$934$	29.4365	+	50.9855i	0.963192	+	1.66830i
$935$		0			0
$936$	38.5930	+	1.26306i	1.26145	+	0.0412845i
$937$		0			0		1.00000			$0$
$937$		0			0		−1.00000			$\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	51.9902	+	30.0166i	1.69483	+	0.978513i	0.950510	+	0.310693i	$0.100561\pi$
$941$	51.9902	+	30.0166i	1.69483	+	0.978513i	0.744323	+	0.667820i	$0.232772\pi$
$942$	−14.4392	−	57.6433i	−0.470453	−	1.87812i
$943$		0			0
$944$		−	21.1828i		−	0.689442i
$945$	0.701884	+	0.774389i	0.0228323	+	0.0251909i
$946$		0			0
$947$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$947$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$948$	40.4954	−	10.1438i	1.31523	−	0.329454i
$949$		0			0
$950$	48.8830	+	28.2226i	1.58597	+	0.915662i
$951$		0			0
$952$		0			0
$953$		−	29.9333i		−	0.969633i	−0.874616	−	0.484817i	$-0.838886\pi$
$953$		−	29.9333i		−	0.969633i	0.874616	−	0.484817i	$-0.161114\pi$
$954$		0			0
$955$	1.90524			0.0616521
$956$	−51.4807	+	29.7224i	−1.66501	+	0.961292i
$957$		0			0
$958$		0			0
$959$	34.2929	+	19.7990i	1.10737	+	0.639343i
$960$	−1.01291	−	0.289248i	−0.0326916	−	0.00933543i
$961$	−15.5000	−	26.8468i	−0.500000	−	0.866025i
$962$		0			0
$963$		0			0
$964$		0			0
$965$	0.387144	−	0.223518i	0.0124626	−	0.00719529i
$966$	6.55541	+	6.77347i	0.210917	+	0.217933i
$967$	−4.77889	+	8.27728i	−0.153679	+	0.266179i	−0.932577	−	0.360971i	$-0.882445\pi$
$967$	−4.77889	+	8.27728i	−0.153679	+	0.266179i	0.778898	+	0.627150i	$0.215779\pi$
$968$	−26.9444	−	15.5563i	−0.866025	−	0.500000i
$969$		0			0
$970$		0			0
$971$			31.0429i			0.996214i	0.867116	+	0.498107i	$0.165971\pi$
$971$			31.0429i			0.996214i	−0.867116	+	0.498107i	$0.834029\pi$
$972$	−20.1706	−	23.7728i	−0.646973	−	0.762513i
$973$	−56.7959			−1.82079
$974$	−47.0819	+	27.1828i	−1.50860	+	0.870992i
$975$	38.1846	−	9.56492i	1.22288	−	0.306322i
$976$	−16.5525	+	28.6697i	−0.529832	+	0.917696i
$977$	−51.8459	−	29.9333i	−1.65870	−	0.957650i	−0.973317	−	0.229465i	$-0.926302\pi$
$977$	−51.8459	−	29.9333i	−1.65870	−	0.957650i	−0.685381	−	0.728184i	$-0.740364\pi$
$978$		0			0
$979$		0			0
$980$			1.06432i			0.0339984i
$981$		0			0
$982$		0			0
$983$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$983$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$	−36.3679	−	62.9911i	−1.15702	−	2.00401i
$989$		0			0
$990$		0			0
$991$	−37.0405			−1.17663			−0.588315	−	0.808632i	$-0.700209\pi$
$991$	−37.0405			−1.17663			−0.588315	+	0.808632i	$0.700209\pi$
$992$		0			0
$993$		0			0
$994$	−31.0111	+	53.7129i	−0.983614	+	1.70367i
$995$		0			0
$996$	−15.2915	−	61.0460i	−0.484530	−	1.93432i
$997$	10.1421	+	17.5666i	0.321203	+	0.556341i	0.980737	−	0.195335i	$-0.0625793\pi$
$997$	10.1421	+	17.5666i	0.321203	+	0.556341i	−0.659533	+	0.751675i	$0.729246\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	504.2.cc.a.293.8	yes	16
7.6	odd	2	inner	504.2.cc.a.293.5	✓	16
8.5	even	2	inner	504.2.cc.a.293.5	✓	16
9.2	odd	6	inner	504.2.cc.a.461.8	yes	16
56.13	odd	2	CM	504.2.cc.a.293.8	yes	16
63.20	even	6	inner	504.2.cc.a.461.5	yes	16
72.29	odd	6	inner	504.2.cc.a.461.5	yes	16
504.461	even	6	inner	504.2.cc.a.461.8	yes	16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
504.2.cc.a.293.5	✓	16	7.6	odd	2	inner
504.2.cc.a.293.5	✓	16	8.5	even	2	inner
504.2.cc.a.293.8	yes	16	1.1	even	1	trivial
504.2.cc.a.293.8	yes	16	56.13	odd	2	CM
504.2.cc.a.461.5	yes	16	63.20	even	6	inner
504.2.cc.a.461.5	yes	16	72.29	odd	6	inner
504.2.cc.a.461.8	yes	16	9.2	odd	6	inner
504.2.cc.a.461.8	yes	16	504.461	even	6	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 504 = 2^{3} \cdot 3^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	504.cc (of order \(6\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(1.22474 - 0.707107i) q^{2} +(1.68014 - 0.420861i) q^{3} +(1.00000 - 1.73205i) q^{4} +(0.0658376 + 0.0380114i) q^{5} +(1.76015 - 1.70349i) q^{6} +(1.32288 + 2.29129i) q^{7} -2.82843i q^{8} +(2.64575 - 1.41421i) q^{9} +0.107512 q^{10} +(0.951188 - 3.33095i) q^{12} +(-2.27533 + 3.94099i) q^{13} +(3.24037 + 1.87083i) q^{14} +(0.126614 + 0.0361560i) q^{15} +(-2.00000 - 3.46410i) q^{16} +(2.24037 - 3.60288i) q^{18} -7.99179 q^{19} +(0.131675 - 0.0760228i) q^{20} +(3.18693 + 3.29294i) q^{21} +(1.25963 + 0.727248i) q^{23} +(-1.19038 - 4.75216i) q^{24} +(-2.49711 - 4.32512i) q^{25} +6.43560i q^{26} +(3.85005 - 3.48957i) q^{27} +5.29150 q^{28} +(0.180636 - 0.0452478i) q^{30} +(-4.89898 - 2.82843i) q^{32} +0.201137i q^{35} +(0.196262 - 5.99679i) q^{36} +(-9.78791 + 5.65105i) q^{38} +(-2.16427 + 7.57901i) q^{39} +(0.107512 - 0.186217i) q^{40} +(6.23164 + 1.77951i) q^{42} +(0.227946 + 0.00746017i) q^{45} +2.05697 q^{46} +(-4.81819 - 4.97846i) q^{48} +(-3.50000 + 6.06218i) q^{49} +(-6.11665 - 3.53145i) q^{50} +(4.55066 + 7.88197i) q^{52} +(2.24783 - 6.99623i) q^{54} +(6.48074 - 3.74166i) q^{56} +(-13.4273 + 3.36344i) q^{57} +(4.58621 + 2.64785i) q^{59} +(0.189238 - 0.183146i) q^{60} +(-4.13812 - 7.16744i) q^{61} +(6.74037 + 4.19135i) q^{63} -8.00000 q^{64} +(-0.299605 + 0.172977i) q^{65} +(2.42243 + 0.691749i) q^{69} +(0.142226 + 0.246342i) q^{70} +16.5762i q^{71} +(-4.00000 - 7.48331i) q^{72} +(-6.01578 - 6.21588i) q^{75} +(-7.99179 + 13.8422i) q^{76} +(2.70850 + 10.8127i) q^{78} +(6.02559 + 10.4366i) q^{79} -0.304091i q^{80} +(5.00000 - 7.48331i) q^{81} +(15.7330 - 9.08345i) q^{83} +(8.89047 - 2.22699i) q^{84} +(0.284451 - 0.152046i) q^{90} -12.0399 q^{91} +(2.51926 - 1.45450i) q^{92} +(-0.526161 - 0.303779i) q^{95} +(-9.42135 - 2.69037i) q^{96} +9.89949i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q + 16 q^{4} - 16 q^{15} - 32 q^{16} - 16 q^{18} + 72 q^{23} + 40 q^{25} + 32 q^{30} - 40 q^{39} - 56 q^{49} - 144 q^{50} + 8 q^{57} - 16 q^{60} + 56 q^{63} - 128 q^{64} - 72 q^{65} - 64 q^{72} + 128 q^{78}+ \cdots + 144 q^{92}+O(q^{100}) \) 16 * q + 16 * q^4 - 16 * q^15 - 32 * q^16 - 16 * q^18 + 72 * q^23 + 40 * q^25 + 32 * q^30 - 40 * q^39 - 56 * q^49 - 144 * q^50 + 8 * q^57 - 16 * q^60 + 56 * q^63 - 128 * q^64 - 72 * q^65 - 64 * q^72 + 128 * q^78 + 80 * q^81 + 144 * q^92

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more