LMFDB - Embedded newform 1089.1.s.b.457.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1089,1,Mod(40,1089)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1089, base_ring=CyclotomicField(30))

chi = DirichletCharacter(H, H._module([10, 21]))

N = Newforms(chi, 1, names="a")

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

chi := DirichletCharacter("1089.40");

S:= CuspForms(chi, 1);

N := Newforms(S);

Level:	$N$	$=$	$1089 = 3^{2} \cdot 11^{2}$
Weight:	$k$	$=$	$1$
Character orbit:	$[\chi]$	$=$	1089.s (of order $30$ , degree $8$ , not minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	no
Analytic conductor:	$0.543481798757$
Analytic rank:	$0$
Dimension:	$16$
Relative dimension:	$2$ over $\Q(\zeta_{30})$
Coefficient field:	16.0.26873856000000000000.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$x^{16} + 2x^{14} - 8x^{10} - 16x^{8} - 32x^{6} + 128x^{2} + 256$ x^16 + 2x^14 - 8x^10 - 16x^8 - 32x^6 + 128*x^2 + 256
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$1$
Twist minimal:	yes
Projective image:	$S_{4}$
Projective field:	Galois closure of 4.2.107811.1

Embedding invariants

Embedding label			457.2
Root			$0.294032 - 1.38331i$ of defining polynomial
Character	$\chi$	$=$	1089.457
Dual form			1089.1.s.b.112.2

$q$ -expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(0.294032 - 1.38331i) q^{2} +(0.104528 - 0.994522i) q^{3} +(-0.913545 - 0.406737i) q^{4} +(0.978148 - 0.207912i) q^{5} +(-1.34500 - 0.437016i) q^{6} +(-1.40647 + 0.147826i) q^{7} +(-0.978148 - 0.207912i) q^{9} -1.41421i q^{10} +(-0.500000 + 0.866025i) q^{12} +(-0.209057 + 1.98904i) q^{14} +(-0.104528 - 0.994522i) q^{15} +(-0.669131 - 0.743145i) q^{16} +(-0.575212 + 1.29195i) q^{18} +(-0.978148 - 0.207912i) q^{20} +1.41421i q^{21} +(-0.309017 + 0.951057i) q^{27} +(1.34500 + 0.437016i) q^{28} +(1.40647 - 0.147826i) q^{29} +(-1.40647 - 0.147826i) q^{30} +(0.669131 - 0.743145i) q^{31} +(-1.22474 + 0.707107i) q^{32} +(-1.34500 + 0.437016i) q^{35} +(0.809017 + 0.587785i) q^{36} +(0.809017 - 0.587785i) q^{37} +(1.95630 + 0.415823i) q^{42} -1.00000 q^{45} +(-0.913545 + 0.406737i) q^{47} +(-0.809017 + 0.587785i) q^{48} +(0.978148 - 0.207912i) q^{49} +(0.309017 + 0.951057i) q^{53} +(1.22474 + 0.707107i) q^{54} +(0.209057 - 1.98904i) q^{58} +(0.913545 + 0.406737i) q^{59} +(-0.309017 + 0.951057i) q^{60} +(1.05097 - 0.946294i) q^{61} +(-0.831254 - 1.14412i) q^{62} +(1.40647 + 0.147826i) q^{63} +(0.309017 + 0.951057i) q^{64} +(0.500000 + 0.866025i) q^{67} +(0.209057 + 1.98904i) q^{70} +(-0.309017 + 0.951057i) q^{71} +(0.831254 + 1.14412i) q^{73} +(-0.575212 - 1.29195i) q^{74} +(-0.809017 - 0.587785i) q^{80} +(0.913545 + 0.406737i) q^{81} +(1.05097 - 0.946294i) q^{83} +(0.575212 - 1.29195i) q^{84} -1.41421i q^{87} +(-0.294032 + 1.38331i) q^{90} +(-0.669131 - 0.743145i) q^{93} +(0.294032 + 1.38331i) q^{94} +(0.575212 + 1.29195i) q^{96} +(-0.978148 - 0.207912i) q^{97} -1.41421i q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$16 q - 2 q^{3} - 2 q^{4} - 2 q^{5} + 2 q^{9} - 8 q^{12} + 4 q^{14} + 2 q^{15} - 2 q^{16} + 2 q^{20} + 4 q^{27} + 2 q^{31} + 4 q^{36} + 4 q^{37} - 4 q^{42} - 16 q^{45} - 2 q^{47} - 4 q^{48} - 2 q^{49} - 4 q^{53}+ \cdots + 2 q^{97}+O(q^{100})$ 16 * q - 2 * q^3 - 2 * q^4 - 2 * q^5 + 2 * q^9 - 8 * q^12 + 4 * q^14 + 2 * q^15 - 2 * q^16 + 2 * q^20 + 4 * q^27 + 2 * q^31 + 4 * q^36 + 4 * q^37 - 4 * q^42 - 16 * q^45 - 2 * q^47 - 4 * q^48 - 2 * q^49 - 4 * q^53 - 4 * q^58 + 2 * q^59 + 4 * q^60 - 4 * q^64 + 8 * q^67 - 4 * q^70 + 4 * q^71 - 4 * q^80 + 2 * q^81 - 2 * q^93 + 2 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1089\mathbb{Z}\right)^\times$ .

$n$	$244$	$848$
$\chi(n)$	$e\left(\frac{9}{10}\right)$	$e\left(\frac{2}{3}\right)$

Coefficient data

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

Display

a_p

with

p

up to: 50 250 1000 (See

a_n

instead) (See

a_n

instead) (See

a_n

instead) Display

a_n

with

n

up to: 50 250 1000 (See only

a_p

) (See only

a_p

) (See only

a_p

)

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

$2$	0.294032	−	1.38331i	0.294032	−	1.38331i	−0.544639	−	0.838671i	$-0.683333\pi$
$2$	0.294032	−	1.38331i	0.294032	−	1.38331i	0.838671	−	0.544639i	$-0.183333\pi$
$3$	0.104528	−	0.994522i	0.104528	−	0.994522i
$3$	0.104528	−	0.994522i	0.104528	−	0.994522i
$4$	−0.913545	−	0.406737i	−0.913545	−	0.406737i
$5$	0.978148	−	0.207912i	0.978148	−	0.207912i	0.309017	−	0.951057i	$-0.400000\pi$
$5$	0.978148	−	0.207912i	0.978148	−	0.207912i	0.669131	+	0.743145i	$0.266667\pi$
$6$	−1.34500	−	0.437016i	−1.34500	−	0.437016i
$7$	−1.40647	+	0.147826i	−1.40647	+	0.147826i	−0.777146	−	0.629320i	$-0.783333\pi$
$7$	−1.40647	+	0.147826i	−1.40647	+	0.147826i	−0.629320	+	0.777146i	$0.716667\pi$
$8$		0			0
$9$	−0.978148	−	0.207912i	−0.978148	−	0.207912i
$10$		−	1.41421i		−	1.41421i
$11$		0			0
$11$		0			0
$12$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$13$		0			0		0.669131	−	0.743145i	$-0.266667\pi$
$13$		0			0		−0.669131	+	0.743145i	$0.733333\pi$
$14$	−0.209057	+	1.98904i	−0.209057	+	1.98904i
$15$	−0.104528	−	0.994522i	−0.104528	−	0.994522i
$16$	−0.669131	−	0.743145i	−0.669131	−	0.743145i
$17$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$17$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$18$	−0.575212	+	1.29195i	−0.575212	+	1.29195i
$19$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$19$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$20$	−0.978148	−	0.207912i	−0.978148	−	0.207912i
$21$			1.41421i			1.41421i
$22$		0			0
$23$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$23$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$24$		0			0
$25$		0			0
$26$		0			0
$27$	−0.309017	+	0.951057i	−0.309017	+	0.951057i
$28$	1.34500	+	0.437016i	1.34500	+	0.437016i
$29$	1.40647	−	0.147826i	1.40647	−	0.147826i	0.629320	−	0.777146i	$-0.283333\pi$
$29$	1.40647	−	0.147826i	1.40647	−	0.147826i	0.777146	+	0.629320i	$0.216667\pi$
$30$	−1.40647	−	0.147826i	−1.40647	−	0.147826i
$31$	0.669131	−	0.743145i	0.669131	−	0.743145i	−0.309017	−	0.951057i	$-0.600000\pi$
$31$	0.669131	−	0.743145i	0.669131	−	0.743145i	0.978148	+	0.207912i	$0.0666667\pi$
$32$	−1.22474	+	0.707107i	−1.22474	+	0.707107i
$33$		0			0
$34$		0			0
$35$	−1.34500	+	0.437016i	−1.34500	+	0.437016i
$36$	0.809017	+	0.587785i	0.809017	+	0.587785i
$37$	0.809017	−	0.587785i	0.809017	−	0.587785i	−0.104528	−	0.994522i	$-0.533333\pi$
$37$	0.809017	−	0.587785i	0.809017	−	0.587785i	0.913545	+	0.406737i	$0.133333\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$		0			0		0.104528	−	0.994522i	$-0.466667\pi$
$41$		0			0		−0.104528	+	0.994522i	$0.533333\pi$
$42$	1.95630	+	0.415823i	1.95630	+	0.415823i
$43$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$43$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$44$		0			0
$45$	−1.00000			−1.00000
$46$		0			0
$47$	−0.913545	+	0.406737i	−0.913545	+	0.406737i	−0.809017	−	0.587785i	$-0.800000\pi$
$47$	−0.913545	+	0.406737i	−0.913545	+	0.406737i	−0.104528	+	0.994522i	$0.533333\pi$
$48$	−0.809017	+	0.587785i	−0.809017	+	0.587785i
$49$	0.978148	−	0.207912i	0.978148	−	0.207912i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	0.309017	+	0.951057i	0.309017	+	0.951057i	0.978148	+	0.207912i	$0.0666667\pi$
$53$	0.309017	+	0.951057i	0.309017	+	0.951057i	−0.669131	+	0.743145i	$0.733333\pi$
$54$	1.22474	+	0.707107i	1.22474	+	0.707107i
$55$		0			0
$56$		0			0
$57$		0			0
$58$	0.209057	−	1.98904i	0.209057	−	1.98904i
$59$	0.913545	+	0.406737i	0.913545	+	0.406737i	0.809017	−	0.587785i	$-0.200000\pi$
$59$	0.913545	+	0.406737i	0.913545	+	0.406737i	0.104528	+	0.994522i	$0.466667\pi$
$60$	−0.309017	+	0.951057i	−0.309017	+	0.951057i
$61$	1.05097	−	0.946294i	1.05097	−	0.946294i	0.0523360	−	0.998630i	$-0.483333\pi$
$61$	1.05097	−	0.946294i	1.05097	−	0.946294i	0.998630	+	0.0523360i	$0.0166667\pi$
$62$	−0.831254	−	1.14412i	−0.831254	−	1.14412i
$63$	1.40647	+	0.147826i	1.40647	+	0.147826i
$64$	0.309017	+	0.951057i	0.309017	+	0.951057i
$65$		0			0
$66$		0			0
$67$	0.500000	+	0.866025i	0.500000	+	0.866025i	1.00000			$0$
$67$	0.500000	+	0.866025i	0.500000	+	0.866025i	−0.500000	+	0.866025i	$0.666667\pi$
$68$		0			0
$69$		0			0
$70$	0.209057	+	1.98904i	0.209057	+	1.98904i
$71$	−0.309017	+	0.951057i	−0.309017	+	0.951057i	0.669131	+	0.743145i	$0.266667\pi$
$71$	−0.309017	+	0.951057i	−0.309017	+	0.951057i	−0.978148	+	0.207912i	$0.933333\pi$
$72$		0			0
$73$	0.831254	+	1.14412i	0.831254	+	1.14412i	0.987688	+	0.156434i	$0.0500000\pi$
$73$	0.831254	+	1.14412i	0.831254	+	1.14412i	−0.156434	+	0.987688i	$0.550000\pi$
$74$	−0.575212	−	1.29195i	−0.575212	−	1.29195i
$75$		0			0
$76$		0			0
$77$		0			0
$78$		0			0
$79$		0			0		−0.978148	−	0.207912i	$-0.933333\pi$
$79$		0			0		0.978148	+	0.207912i	$0.0666667\pi$
$80$	−0.809017	−	0.587785i	−0.809017	−	0.587785i
$81$	0.913545	+	0.406737i	0.913545	+	0.406737i
$82$		0			0
$83$	1.05097	−	0.946294i	1.05097	−	0.946294i	0.0523360	−	0.998630i	$-0.483333\pi$
$83$	1.05097	−	0.946294i	1.05097	−	0.946294i	0.998630	+	0.0523360i	$0.0166667\pi$
$84$	0.575212	−	1.29195i	0.575212	−	1.29195i
$85$		0			0
$86$		0			0
$87$		−	1.41421i		−	1.41421i
$88$		0			0
$89$		0			0			−	1.00000i	$-0.5\pi$
$89$		0			0				1.00000i	$0.5\pi$
$90$	−0.294032	+	1.38331i	−0.294032	+	1.38331i
$91$		0			0
$92$		0			0
$93$	−0.669131	−	0.743145i	−0.669131	−	0.743145i
$94$	0.294032	+	1.38331i	0.294032	+	1.38331i
$95$		0			0
$96$	0.575212	+	1.29195i	0.575212	+	1.29195i
$97$	−0.978148	−	0.207912i	−0.978148	−	0.207912i	−0.309017	−	0.951057i	$-0.600000\pi$
$97$	−0.978148	−	0.207912i	−0.978148	−	0.207912i	−0.669131	+	0.743145i	$0.733333\pi$
$98$		−	1.41421i		−	1.41421i
$99$		0			0
$100$		0			0
$101$	−0.294032	+	1.38331i	−0.294032	+	1.38331i	0.544639	+	0.838671i	$0.316667\pi$
$101$	−0.294032	+	1.38331i	−0.294032	+	1.38331i	−0.838671	+	0.544639i	$0.816667\pi$
$102$		0			0
$103$	−0.913545	−	0.406737i	−0.913545	−	0.406737i	−0.104528	−	0.994522i	$-0.533333\pi$
$103$	−0.913545	−	0.406737i	−0.913545	−	0.406737i	−0.809017	+	0.587785i	$0.800000\pi$
$104$		0			0
$105$	0.294032	+	1.38331i	0.294032	+	1.38331i
$106$	1.40647	−	0.147826i	1.40647	−	0.147826i
$107$	−0.831254	+	1.14412i	−0.831254	+	1.14412i	0.156434	+	0.987688i	$0.450000\pi$
$107$	−0.831254	+	1.14412i	−0.831254	+	1.14412i	−0.987688	+	0.156434i	$0.950000\pi$
$108$	0.669131	−	0.743145i	0.669131	−	0.743145i
$109$		0			0		1.00000			$0$
$109$		0			0		−1.00000			$\pi$
$110$		0			0
$111$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$112$	1.05097	+	0.946294i	1.05097	+	0.946294i
$113$	0.104528	−	0.994522i	0.104528	−	0.994522i	−0.809017	−	0.587785i	$-0.800000\pi$
$113$	0.104528	−	0.994522i	0.104528	−	0.994522i	0.913545	−	0.406737i	$-0.133333\pi$
$114$		0			0
$115$		0			0
$116$	−1.34500	−	0.437016i	−1.34500	−	0.437016i
$117$		0			0
$118$	0.831254	−	1.14412i	0.831254	−	1.14412i
$119$		0			0
$120$		0			0
$121$		0			0
$122$	−1.00000	−	1.73205i	−1.00000	−	1.73205i
$123$		0			0
$124$	−0.913545	+	0.406737i	−0.913545	+	0.406737i
$125$	−0.809017	+	0.587785i	−0.809017	+	0.587785i
$126$	0.618034	−	1.90211i	0.618034	−	1.90211i
$127$	−1.34500	−	0.437016i	−1.34500	−	0.437016i	−0.453990	−	0.891007i	$-0.650000\pi$
$127$	−1.34500	−	0.437016i	−1.34500	−	0.437016i	−0.891007	+	0.453990i	$0.850000\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$131$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$132$		0			0
$133$		0			0
$134$	1.34500	−	0.437016i	1.34500	−	0.437016i
$135$	−0.104528	+	0.994522i	−0.104528	+	0.994522i
$136$		0			0
$137$	−0.669131	−	0.743145i	−0.669131	−	0.743145i	0.309017	−	0.951057i	$-0.400000\pi$
$137$	−0.669131	−	0.743145i	−0.669131	−	0.743145i	−0.978148	+	0.207912i	$0.933333\pi$
$138$		0			0
$139$	0.575212	−	1.29195i	0.575212	−	1.29195i	−0.358368	−	0.933580i	$-0.616667\pi$
$139$	0.575212	−	1.29195i	0.575212	−	1.29195i	0.933580	−	0.358368i	$-0.116667\pi$
$140$	1.40647	+	0.147826i	1.40647	+	0.147826i
$141$	0.309017	+	0.951057i	0.309017	+	0.951057i
$142$	1.22474	+	0.707107i	1.22474	+	0.707107i
$143$		0			0
$144$	0.500000	+	0.866025i	0.500000	+	0.866025i
$145$	1.34500	−	0.437016i	1.34500	−	0.437016i
$146$	1.82709	−	0.813473i	1.82709	−	0.813473i
$147$	−0.104528	−	0.994522i	−0.104528	−	0.994522i
$148$	−0.978148	+	0.207912i	−0.978148	+	0.207912i
$149$		0			0		0.978148	−	0.207912i	$-0.0666667\pi$
$149$		0			0		−0.978148	+	0.207912i	$0.933333\pi$
$150$		0			0
$151$		0			0		0.913545	−	0.406737i	$-0.133333\pi$
$151$		0			0		−0.913545	+	0.406737i	$0.866667\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	0.500000	−	0.866025i	0.500000	−	0.866025i
$156$		0			0
$157$	−0.104528	+	0.994522i	−0.104528	+	0.994522i	0.809017	+	0.587785i	$0.200000\pi$
$157$	−0.104528	+	0.994522i	−0.104528	+	0.994522i	−0.913545	+	0.406737i	$0.866667\pi$
$158$		0			0
$159$	0.978148	−	0.207912i	0.978148	−	0.207912i
$160$	−1.05097	+	0.946294i	−1.05097	+	0.946294i
$161$		0			0
$162$	0.831254	−	1.14412i	0.831254	−	1.14412i
$163$	−0.309017	−	0.951057i	−0.309017	−	0.951057i	−0.978148	−	0.207912i	$-0.933333\pi$
$163$	−0.309017	−	0.951057i	−0.309017	−	0.951057i	0.669131	−	0.743145i	$-0.266667\pi$
$164$		0			0
$165$		0			0
$166$	−1.00000	−	1.73205i	−1.00000	−	1.73205i
$167$		0			0		0.669131	−	0.743145i	$-0.266667\pi$
$167$		0			0		−0.669131	+	0.743145i	$0.733333\pi$
$168$		0			0
$169$	−0.104528	−	0.994522i	−0.104528	−	0.994522i
$170$		0			0
$171$		0			0
$172$		0			0
$173$		0			0		0.913545	−	0.406737i	$-0.133333\pi$
$173$		0			0		−0.913545	+	0.406737i	$0.866667\pi$
$174$	−1.95630	−	0.415823i	−1.95630	−	0.415823i
$175$		0			0
$176$		0			0
$177$	0.500000	−	0.866025i	0.500000	−	0.866025i
$178$		0			0
$179$	0.809017	+	0.587785i	0.809017	+	0.587785i	0.913545	−	0.406737i	$-0.133333\pi$
$179$	0.809017	+	0.587785i	0.809017	+	0.587785i	−0.104528	+	0.994522i	$0.533333\pi$
$180$	0.913545	+	0.406737i	0.913545	+	0.406737i
$181$	−0.309017	+	0.951057i	−0.309017	+	0.951057i	0.669131	+	0.743145i	$0.266667\pi$
$181$	−0.309017	+	0.951057i	−0.309017	+	0.951057i	−0.978148	+	0.207912i	$0.933333\pi$
$182$		0			0
$183$	−0.831254	−	1.14412i	−0.831254	−	1.14412i
$184$		0			0
$185$	0.669131	−	0.743145i	0.669131	−	0.743145i
$186$	−1.22474	+	0.707107i	−1.22474	+	0.707107i
$187$		0			0
$188$	1.00000			1.00000
$189$	0.294032	−	1.38331i	0.294032	−	1.38331i
$190$		0			0
$191$	0.104528	+	0.994522i	0.104528	+	0.994522i	0.913545	+	0.406737i	$0.133333\pi$
$191$	0.104528	+	0.994522i	0.104528	+	0.994522i	−0.809017	+	0.587785i	$0.800000\pi$
$192$	0.978148	−	0.207912i	0.978148	−	0.207912i
$193$		0			0		0.978148	−	0.207912i	$-0.0666667\pi$
$193$		0			0		−0.978148	+	0.207912i	$0.933333\pi$
$194$	−0.575212	+	1.29195i	−0.575212	+	1.29195i
$195$		0			0
$196$	−0.978148	−	0.207912i	−0.978148	−	0.207912i
$197$			1.41421i			1.41421i	0.707107	+	0.707107i	$0.250000\pi$
$197$			1.41421i			1.41421i	−0.707107	+	0.707107i	$0.750000\pi$
$198$		0			0
$199$	−1.00000			−1.00000			−0.500000	−	0.866025i	$-0.666667\pi$
$199$	−1.00000			−1.00000			−0.500000	+	0.866025i	$0.666667\pi$
$200$		0			0
$201$	0.913545	−	0.406737i	0.913545	−	0.406737i
$202$	1.82709	+	0.813473i	1.82709	+	0.813473i
$203$	−1.95630	+	0.415823i	−1.95630	+	0.415823i
$204$		0			0
$205$		0			0
$206$	−0.831254	+	1.14412i	−0.831254	+	1.14412i
$207$		0			0
$208$		0			0
$209$		0			0
$210$	2.00000			2.00000
$211$	−1.05097	−	0.946294i	−1.05097	−	0.946294i	−0.0523360	−	0.998630i	$-0.516667\pi$
$211$	−1.05097	−	0.946294i	−1.05097	−	0.946294i	−0.998630	+	0.0523360i	$0.983333\pi$
$212$	0.104528	−	0.994522i	0.104528	−	0.994522i
$213$	0.913545	+	0.406737i	0.913545	+	0.406737i
$214$	1.33826	+	1.48629i	1.33826	+	1.48629i
$215$		0			0
$216$		0			0
$217$	−0.831254	+	1.14412i	−0.831254	+	1.14412i
$218$		0			0
$219$	1.22474	−	0.707107i	1.22474	−	0.707107i
$220$		0			0
$221$		0			0
$222$	−1.34500	+	0.437016i	−1.34500	+	0.437016i
$223$		0			0		−0.406737	−	0.913545i	$-0.633333\pi$
$223$		0			0		0.406737	+	0.913545i	$0.366667\pi$
$224$	1.61803	−	1.17557i	1.61803	−	1.17557i
$225$		0			0
$226$	−1.34500	−	0.437016i	−1.34500	−	0.437016i
$227$		0			0		−0.104528	−	0.994522i	$-0.533333\pi$
$227$		0			0		0.104528	+	0.994522i	$0.466667\pi$
$228$		0			0
$229$		0			0		−0.743145	−	0.669131i	$-0.766667\pi$
$229$		0			0		0.743145	+	0.669131i	$0.233333\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	1.34500	−	0.437016i	1.34500	−	0.437016i	0.453990	−	0.891007i	$-0.350000\pi$
$233$	1.34500	−	0.437016i	1.34500	−	0.437016i	0.891007	+	0.453990i	$0.150000\pi$
$234$		0			0
$235$	−0.809017	+	0.587785i	−0.809017	+	0.587785i
$236$	−0.669131	−	0.743145i	−0.669131	−	0.743145i
$237$		0			0
$238$		0			0
$239$	1.40647	+	0.147826i	1.40647	+	0.147826i	0.777146	−	0.629320i	$-0.216667\pi$
$239$	1.40647	+	0.147826i	1.40647	+	0.147826i	0.629320	+	0.777146i	$0.283333\pi$
$240$	−0.669131	+	0.743145i	−0.669131	+	0.743145i
$241$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$241$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$242$		0			0
$243$	0.500000	−	0.866025i	0.500000	−	0.866025i
$244$	−1.34500	+	0.437016i	−1.34500	+	0.437016i
$245$	0.913545	−	0.406737i	0.913545	−	0.406737i
$246$		0			0
$247$		0			0
$248$		0			0
$249$	−0.831254	−	1.14412i	−0.831254	−	1.14412i
$250$	0.575212	+	1.29195i	0.575212	+	1.29195i
$251$		0			0		0.951057	−	0.309017i	$-0.100000\pi$
$251$		0			0		−0.951057	+	0.309017i	$0.900000\pi$
$252$	−1.22474	−	0.707107i	−1.22474	−	0.707107i
$253$		0			0
$254$	−1.00000	+	1.73205i	−1.00000	+	1.73205i
$255$		0			0
$256$	−0.104528	+	0.994522i	−0.104528	+	0.994522i
$257$	−1.82709	−	0.813473i	−1.82709	−	0.813473i	−0.913545	−	0.406737i	$-0.866667\pi$
$257$	−1.82709	−	0.813473i	−1.82709	−	0.813473i	−0.913545	−	0.406737i	$-0.866667\pi$
$258$		0			0
$259$	−1.05097	+	0.946294i	−1.05097	+	0.946294i
$260$		0			0
$261$	−1.40647	−	0.147826i	−1.40647	−	0.147826i
$262$		0			0
$263$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$263$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$264$		0			0
$265$	0.500000	+	0.866025i	0.500000	+	0.866025i
$266$		0			0
$267$		0			0
$268$	−0.104528	−	0.994522i	−0.104528	−	0.994522i
$269$	0.309017	−	0.951057i	0.309017	−	0.951057i	−0.669131	−	0.743145i	$-0.733333\pi$
$269$	0.309017	−	0.951057i	0.309017	−	0.951057i	0.978148	−	0.207912i	$-0.0666667\pi$
$270$	1.34500	+	0.437016i	1.34500	+	0.437016i
$271$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$271$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$272$		0			0
$273$		0			0
$274$	−1.22474	+	0.707107i	−1.22474	+	0.707107i
$275$		0			0
$276$		0			0
$277$		0			0		−0.978148	−	0.207912i	$-0.933333\pi$
$277$		0			0		0.978148	+	0.207912i	$0.0666667\pi$
$278$	−1.61803	−	1.17557i	−1.61803	−	1.17557i
$279$	−0.809017	+	0.587785i	−0.809017	+	0.587785i
$280$		0			0
$281$		0			0		−0.669131	−	0.743145i	$-0.733333\pi$
$281$		0			0		0.669131	+	0.743145i	$0.266667\pi$
$282$	1.40647	−	0.147826i	1.40647	−	0.147826i
$283$	−1.40647	−	0.147826i	−1.40647	−	0.147826i	−0.629320	−	0.777146i	$-0.716667\pi$
$283$	−1.40647	−	0.147826i	−1.40647	−	0.147826i	−0.777146	+	0.629320i	$0.783333\pi$
$284$	0.669131	−	0.743145i	0.669131	−	0.743145i
$285$		0			0
$286$		0			0
$287$		0			0
$288$	1.34500	−	0.437016i	1.34500	−	0.437016i
$289$	−0.809017	−	0.587785i	−0.809017	−	0.587785i
$290$	−0.209057	−	1.98904i	−0.209057	−	1.98904i
$291$	−0.309017	+	0.951057i	−0.309017	+	0.951057i
$292$	−0.294032	−	1.38331i	−0.294032	−	1.38331i
$293$	−0.575212	+	1.29195i	−0.575212	+	1.29195i	0.358368	+	0.933580i	$0.383333\pi$
$293$	−0.575212	+	1.29195i	−0.575212	+	1.29195i	−0.933580	+	0.358368i	$0.883333\pi$
$294$	−1.40647	−	0.147826i	−1.40647	−	0.147826i
$295$	0.978148	+	0.207912i	0.978148	+	0.207912i
$296$		0			0
$297$		0			0
$298$		0			0
$299$		0			0
$300$		0			0
$301$		0			0
$302$		0			0
$303$	1.34500	+	0.437016i	1.34500	+	0.437016i
$304$		0			0
$305$	0.831254	−	1.14412i	0.831254	−	1.14412i
$306$		0			0
$307$		0			0		1.00000			$0$
$307$		0			0		−1.00000			$\pi$
$308$		0			0
$309$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$310$	−1.05097	−	0.946294i	−1.05097	−	0.946294i
$311$	−0.104528	+	0.994522i	−0.104528	+	0.994522i	0.809017	+	0.587785i	$0.200000\pi$
$311$	−0.104528	+	0.994522i	−0.104528	+	0.994522i	−0.913545	+	0.406737i	$0.866667\pi$
$312$		0			0
$313$		0			0		0.743145	−	0.669131i	$-0.233333\pi$
$313$		0			0		−0.743145	+	0.669131i	$0.766667\pi$
$314$	1.34500	+	0.437016i	1.34500	+	0.437016i
$315$	1.40647	−	0.147826i	1.40647	−	0.147826i
$316$		0			0
$317$		0			0		0.207912	−	0.978148i	$-0.433333\pi$
$317$		0			0		−0.207912	+	0.978148i	$0.566667\pi$
$318$		−	1.41421i		−	1.41421i
$319$		0			0
$320$	0.500000	+	0.866025i	0.500000	+	0.866025i
$321$	1.05097	+	0.946294i	1.05097	+	0.946294i
$322$		0			0
$323$		0			0
$324$	−0.669131	−	0.743145i	−0.669131	−	0.743145i
$325$		0			0
$326$	−1.40647	+	0.147826i	−1.40647	+	0.147826i
$327$		0			0
$328$		0			0
$329$	1.22474	−	0.707107i	1.22474	−	0.707107i
$330$		0			0
$331$	−0.500000	+	0.866025i	−0.500000	+	0.866025i	0.500000	+	0.866025i	$0.333333\pi$
$331$	−0.500000	+	0.866025i	−0.500000	+	0.866025i	−1.00000			$\pi$
$332$	−1.34500	+	0.437016i	−1.34500	+	0.437016i
$333$	−0.913545	+	0.406737i	−0.913545	+	0.406737i
$334$		0			0
$335$	0.669131	+	0.743145i	0.669131	+	0.743145i
$336$	1.05097	−	0.946294i	1.05097	−	0.946294i
$337$	−0.575212	+	1.29195i	−0.575212	+	1.29195i	0.358368	+	0.933580i	$0.383333\pi$
$337$	−0.575212	+	1.29195i	−0.575212	+	1.29195i	−0.933580	+	0.358368i	$0.883333\pi$
$338$	−1.40647	−	0.147826i	−1.40647	−	0.147826i
$339$	−0.978148	−	0.207912i	−0.978148	−	0.207912i
$340$		0			0
$341$		0			0
$342$		0			0
$343$		0			0
$344$		0			0
$345$		0			0
$346$		0			0
$347$	0.294032	+	1.38331i	0.294032	+	1.38331i	0.838671	+	0.544639i	$0.183333\pi$
$347$	0.294032	+	1.38331i	0.294032	+	1.38331i	−0.544639	+	0.838671i	$0.683333\pi$
$348$	−0.575212	+	1.29195i	−0.575212	+	1.29195i
$349$		0			0		0.913545	−	0.406737i	$-0.133333\pi$
$349$		0			0		−0.913545	+	0.406737i	$0.866667\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$353$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$354$	−1.05097	−	0.946294i	−1.05097	−	0.946294i
$355$	−0.104528	+	0.994522i	−0.104528	+	0.994522i
$356$		0			0
$357$		0			0
$358$	1.05097	−	0.946294i	1.05097	−	0.946294i
$359$	0.831254	+	1.14412i	0.831254	+	1.14412i	0.987688	+	0.156434i	$0.0500000\pi$
$359$	0.831254	+	1.14412i	0.831254	+	1.14412i	−0.156434	+	0.987688i	$0.550000\pi$
$360$		0			0
$361$	0.309017	+	0.951057i	0.309017	+	0.951057i
$362$	1.22474	+	0.707107i	1.22474	+	0.707107i
$363$		0			0
$364$		0			0
$365$	1.05097	+	0.946294i	1.05097	+	0.946294i
$366$	−1.82709	+	0.813473i	−1.82709	+	0.813473i
$367$	0.104528	+	0.994522i	0.104528	+	0.994522i	0.913545	+	0.406737i	$0.133333\pi$
$367$	0.104528	+	0.994522i	0.104528	+	0.994522i	−0.809017	+	0.587785i	$0.800000\pi$
$368$		0			0
$369$		0			0
$370$	−0.831254	−	1.14412i	−0.831254	−	1.14412i
$371$	−0.575212	−	1.29195i	−0.575212	−	1.29195i
$372$	0.309017	+	0.951057i	0.309017	+	0.951057i
$373$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$373$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$374$		0			0
$375$	0.500000	+	0.866025i	0.500000	+	0.866025i
$376$		0			0
$377$		0			0
$378$	−1.82709	−	0.813473i	−1.82709	−	0.813473i
$379$		0			0		−0.951057	−	0.309017i	$-0.900000\pi$
$379$		0			0		0.951057	+	0.309017i	$0.100000\pi$
$380$		0			0
$381$	−0.575212	+	1.29195i	−0.575212	+	1.29195i
$382$	1.40647	+	0.147826i	1.40647	+	0.147826i
$383$	0.669131	−	0.743145i	0.669131	−	0.743145i	−0.309017	−	0.951057i	$-0.600000\pi$
$383$	0.669131	−	0.743145i	0.669131	−	0.743145i	0.978148	+	0.207912i	$0.0666667\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$		0			0
$388$	0.809017	+	0.587785i	0.809017	+	0.587785i
$389$	0.104528	+	0.994522i	0.104528	+	0.994522i	0.913545	+	0.406737i	$0.133333\pi$
$389$	0.104528	+	0.994522i	0.104528	+	0.994522i	−0.809017	+	0.587785i	$0.800000\pi$
$390$		0			0
$391$		0			0
$392$		0			0
$393$		0			0
$394$	1.95630	+	0.415823i	1.95630	+	0.415823i
$395$		0			0
$396$		0			0
$397$	−1.00000			−1.00000			−0.500000	−	0.866025i	$-0.666667\pi$
$397$	−1.00000			−1.00000			−0.500000	+	0.866025i	$0.666667\pi$
$398$	−0.294032	+	1.38331i	−0.294032	+	1.38331i
$399$		0			0
$400$		0			0
$401$	0.978148	−	0.207912i	0.978148	−	0.207912i	0.309017	−	0.951057i	$-0.400000\pi$
$401$	0.978148	−	0.207912i	0.978148	−	0.207912i	0.669131	+	0.743145i	$0.266667\pi$
$402$	−0.294032	−	1.38331i	−0.294032	−	1.38331i
$403$		0			0
$404$	0.831254	−	1.14412i	0.831254	−	1.14412i
$405$	0.978148	+	0.207912i	0.978148	+	0.207912i
$406$			2.82843i			2.82843i
$407$		0			0
$408$		0			0
$409$		0			0		0.669131	−	0.743145i	$-0.266667\pi$
$409$		0			0		−0.669131	+	0.743145i	$0.733333\pi$
$410$		0			0
$411$	−0.809017	+	0.587785i	−0.809017	+	0.587785i
$412$	0.669131	+	0.743145i	0.669131	+	0.743145i
$413$	−1.34500	−	0.437016i	−1.34500	−	0.437016i
$414$		0			0
$415$	0.831254	−	1.14412i	0.831254	−	1.14412i
$416$		0			0
$417$	−1.22474	−	0.707107i	−1.22474	−	0.707107i
$418$		0			0
$419$	0.500000	+	0.866025i	0.500000	+	0.866025i	1.00000			$0$
$419$	0.500000	+	0.866025i	0.500000	+	0.866025i	−0.500000	+	0.866025i	$0.666667\pi$
$420$	0.294032	−	1.38331i	0.294032	−	1.38331i
$421$	0.913545	−	0.406737i	0.913545	−	0.406737i	0.104528	−	0.994522i	$-0.466667\pi$
$421$	0.913545	−	0.406737i	0.913545	−	0.406737i	0.809017	+	0.587785i	$0.200000\pi$
$422$	−1.61803	+	1.17557i	−1.61803	+	1.17557i
$423$	0.978148	−	0.207912i	0.978148	−	0.207912i
$424$		0			0
$425$		0			0
$426$	0.831254	−	1.14412i	0.831254	−	1.14412i
$427$	−1.33826	+	1.48629i	−1.33826	+	1.48629i
$428$	1.22474	−	0.707107i	1.22474	−	0.707107i
$429$		0			0
$430$		0			0
$431$	−1.34500	+	0.437016i	−1.34500	+	0.437016i	−0.891007	−	0.453990i	$-0.850000\pi$
$431$	−1.34500	+	0.437016i	−1.34500	+	0.437016i	−0.453990	+	0.891007i	$0.650000\pi$
$432$	0.913545	−	0.406737i	0.913545	−	0.406737i
$433$		0			0		−0.587785	−	0.809017i	$-0.700000\pi$
$433$		0			0		0.587785	+	0.809017i	$0.300000\pi$
$434$	1.33826	+	1.48629i	1.33826	+	1.48629i
$435$	−0.294032	−	1.38331i	−0.294032	−	1.38331i
$436$		0			0
$437$		0			0
$438$	−0.618034	−	1.90211i	−0.618034	−	1.90211i
$439$	−1.22474	−	0.707107i	−1.22474	−	0.707107i	−0.258819	−	0.965926i	$-0.583333\pi$
$439$	−1.22474	−	0.707107i	−1.22474	−	0.707107i	−0.965926	+	0.258819i	$0.916667\pi$
$440$		0			0
$441$	−1.00000			−1.00000
$442$		0			0
$443$	0.913545	−	0.406737i	0.913545	−	0.406737i	0.104528	−	0.994522i	$-0.466667\pi$
$443$	0.913545	−	0.406737i	0.913545	−	0.406737i	0.809017	+	0.587785i	$0.200000\pi$
$444$	0.104528	+	0.994522i	0.104528	+	0.994522i
$445$		0			0
$446$		0			0
$447$		0			0
$448$	−0.575212	−	1.29195i	−0.575212	−	1.29195i
$449$	0.309017	+	0.951057i	0.309017	+	0.951057i	0.978148	+	0.207912i	$0.0666667\pi$
$449$	0.309017	+	0.951057i	0.309017	+	0.951057i	−0.669131	+	0.743145i	$0.733333\pi$
$450$		0			0
$451$		0			0
$452$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$	1.05097	−	0.946294i	1.05097	−	0.946294i	0.0523360	−	0.998630i	$-0.483333\pi$
$457$	1.05097	−	0.946294i	1.05097	−	0.946294i	0.998630	+	0.0523360i	$0.0166667\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−1.22474	−	0.707107i	−1.22474	−	0.707107i	−0.258819	−	0.965926i	$-0.583333\pi$
$461$	−1.22474	−	0.707107i	−1.22474	−	0.707107i	−0.965926	+	0.258819i	$0.916667\pi$
$462$		0			0
$463$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$463$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$464$	−1.05097	−	0.946294i	−1.05097	−	0.946294i
$465$	−0.809017	−	0.587785i	−0.809017	−	0.587785i
$466$	−0.209057	−	1.98904i	−0.209057	−	1.98904i
$467$	0.309017	−	0.951057i	0.309017	−	0.951057i	−0.669131	−	0.743145i	$-0.733333\pi$
$467$	0.309017	−	0.951057i	0.309017	−	0.951057i	0.978148	−	0.207912i	$-0.0666667\pi$
$468$		0			0
$469$	−0.831254	−	1.14412i	−0.831254	−	1.14412i
$470$	0.575212	+	1.29195i	0.575212	+	1.29195i
$471$	0.978148	+	0.207912i	0.978148	+	0.207912i
$472$		0			0
$473$		0			0
$474$		0			0
$475$		0			0
$476$		0			0
$477$	−0.104528	−	0.994522i	−0.104528	−	0.994522i
$478$	0.618034	−	1.90211i	0.618034	−	1.90211i
$479$	−1.05097	+	0.946294i	−1.05097	+	0.946294i	−0.998630	−	0.0523360i	$-0.983333\pi$
$479$	−1.05097	+	0.946294i	−1.05097	+	0.946294i	−0.0523360	+	0.998630i	$0.516667\pi$
$480$	0.831254	+	1.14412i	0.831254	+	1.14412i
$481$		0			0
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−1.00000			−1.00000
$486$	−1.05097	−	0.946294i	−1.05097	−	0.946294i
$487$	0.809017	+	0.587785i	0.809017	+	0.587785i	0.913545	−	0.406737i	$-0.133333\pi$
$487$	0.809017	+	0.587785i	0.809017	+	0.587785i	−0.104528	+	0.994522i	$0.533333\pi$
$488$		0			0
$489$	−0.978148	+	0.207912i	−0.978148	+	0.207912i
$490$	−0.294032	−	1.38331i	−0.294032	−	1.38331i
$491$	0.575212	−	1.29195i	0.575212	−	1.29195i	−0.358368	−	0.933580i	$-0.616667\pi$
$491$	0.575212	−	1.29195i	0.575212	−	1.29195i	0.933580	−	0.358368i	$-0.116667\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$		0			0
$496$	−1.00000			−1.00000
$497$	0.294032	−	1.38331i	0.294032	−	1.38331i
$498$	−1.82709	+	0.813473i	−1.82709	+	0.813473i
$499$	0.913545	+	0.406737i	0.913545	+	0.406737i	0.809017	−	0.587785i	$-0.200000\pi$
$499$	0.913545	+	0.406737i	0.913545	+	0.406737i	0.104528	+	0.994522i	$0.466667\pi$
$500$	0.978148	−	0.207912i	0.978148	−	0.207912i
$501$		0			0
$502$		0			0
$503$	−0.831254	+	1.14412i	−0.831254	+	1.14412i	0.156434	+	0.987688i	$0.450000\pi$
$503$	−0.831254	+	1.14412i	−0.831254	+	1.14412i	−0.987688	+	0.156434i	$0.950000\pi$
$504$		0			0
$505$			1.41421i			1.41421i
$506$		0			0
$507$	−1.00000			−1.00000
$508$	1.05097	+	0.946294i	1.05097	+	0.946294i
$509$		0			0		−0.994522	−	0.104528i	$-0.966667\pi$
$509$		0			0		0.994522	+	0.104528i	$0.0333333\pi$
$510$		0			0
$511$	−1.33826	−	1.48629i	−1.33826	−	1.48629i
$512$	1.34500	+	0.437016i	1.34500	+	0.437016i
$513$		0			0
$514$	−1.66251	+	2.28825i	−1.66251	+	2.28825i
$515$	−0.978148	−	0.207912i	−0.978148	−	0.207912i
$516$		0			0
$517$		0			0
$518$	1.00000	+	1.73205i	1.00000	+	1.73205i
$519$		0			0
$520$		0			0
$521$	0.809017	−	0.587785i	0.809017	−	0.587785i	−0.104528	−	0.994522i	$-0.533333\pi$
$521$	0.809017	−	0.587785i	0.809017	−	0.587785i	0.913545	+	0.406737i	$0.133333\pi$
$522$	−0.618034	+	1.90211i	−0.618034	+	1.90211i
$523$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$523$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$		0			0
$528$		0			0
$529$	0.500000	−	0.866025i	0.500000	−	0.866025i
$530$	1.34500	−	0.437016i	1.34500	−	0.437016i
$531$	−0.809017	−	0.587785i	−0.809017	−	0.587785i
$532$		0			0
$533$		0			0
$534$		0			0
$535$	−0.575212	+	1.29195i	−0.575212	+	1.29195i
$536$		0			0
$537$	0.669131	−	0.743145i	0.669131	−	0.743145i
$538$	−1.22474	−	0.707107i	−1.22474	−	0.707107i
$539$		0			0
$540$	0.500000	−	0.866025i	0.500000	−	0.866025i
$541$	−1.34500	+	0.437016i	−1.34500	+	0.437016i	−0.891007	−	0.453990i	$-0.850000\pi$
$541$	−1.34500	+	0.437016i	−1.34500	+	0.437016i	−0.453990	+	0.891007i	$0.650000\pi$
$542$		0			0
$543$	0.913545	+	0.406737i	0.913545	+	0.406737i
$544$		0			0
$545$		0			0
$546$		0			0
$547$	−0.575212	−	1.29195i	−0.575212	−	1.29195i	−0.933580	−	0.358368i	$-0.883333\pi$
$547$	−0.575212	−	1.29195i	−0.575212	−	1.29195i	0.358368	−	0.933580i	$-0.383333\pi$
$548$	0.309017	+	0.951057i	0.309017	+	0.951057i
$549$	−1.22474	+	0.707107i	−1.22474	+	0.707107i
$550$		0			0
$551$		0			0
$552$		0			0
$553$		0			0
$554$		0			0
$555$	−0.669131	−	0.743145i	−0.669131	−	0.743145i
$556$	−1.05097	+	0.946294i	−1.05097	+	0.946294i
$557$	−0.831254	−	1.14412i	−0.831254	−	1.14412i	−0.987688	−	0.156434i	$-0.950000\pi$
$557$	−0.831254	−	1.14412i	−0.831254	−	1.14412i	0.156434	−	0.987688i	$-0.450000\pi$
$558$	0.575212	+	1.29195i	0.575212	+	1.29195i
$559$		0			0
$560$	1.22474	+	0.707107i	1.22474	+	0.707107i
$561$		0			0
$562$		0			0
$563$		0			0		0.669131	−	0.743145i	$-0.266667\pi$
$563$		0			0		−0.669131	+	0.743145i	$0.733333\pi$
$564$	0.104528	−	0.994522i	0.104528	−	0.994522i
$565$	−0.104528	−	0.994522i	−0.104528	−	0.994522i
$566$	−0.618034	+	1.90211i	−0.618034	+	1.90211i
$567$	−1.34500	−	0.437016i	−1.34500	−	0.437016i
$568$		0			0
$569$	0.575212	+	1.29195i	0.575212	+	1.29195i	0.933580	+	0.358368i	$0.116667\pi$
$569$	0.575212	+	1.29195i	0.575212	+	1.29195i	−0.358368	+	0.933580i	$0.616667\pi$
$570$		0			0
$571$	1.22474	−	0.707107i	1.22474	−	0.707107i	0.258819	−	0.965926i	$-0.416667\pi$
$571$	1.22474	−	0.707107i	1.22474	−	0.707107i	0.965926	+	0.258819i	$0.0833333\pi$
$572$		0			0
$573$	1.00000			1.00000
$574$		0			0
$575$		0			0
$576$	−0.104528	−	0.994522i	−0.104528	−	0.994522i
$577$	0.309017	−	0.951057i	0.309017	−	0.951057i	−0.669131	−	0.743145i	$-0.733333\pi$
$577$	0.309017	−	0.951057i	0.309017	−	0.951057i	0.978148	−	0.207912i	$-0.0666667\pi$
$578$	−1.05097	+	0.946294i	−1.05097	+	0.946294i
$579$		0			0
$580$	−1.40647	−	0.147826i	−1.40647	−	0.147826i
$581$	−1.33826	+	1.48629i	−1.33826	+	1.48629i
$582$	1.22474	+	0.707107i	1.22474	+	0.707107i
$583$		0			0
$584$		0			0
$585$		0			0
$586$	1.61803	+	1.17557i	1.61803	+	1.17557i
$587$	−0.104528	−	0.994522i	−0.104528	−	0.994522i	−0.913545	−	0.406737i	$-0.866667\pi$
$587$	−0.104528	−	0.994522i	−0.104528	−	0.994522i	0.809017	−	0.587785i	$-0.200000\pi$
$588$	−0.309017	+	0.951057i	−0.309017	+	0.951057i
$589$		0			0
$590$	0.575212	−	1.29195i	0.575212	−	1.29195i
$591$	1.40647	+	0.147826i	1.40647	+	0.147826i
$592$	−0.978148	−	0.207912i	−0.978148	−	0.207912i
$593$		−	1.41421i		−	1.41421i	−0.707107	−	0.707107i	$-0.750000\pi$
$593$		−	1.41421i		−	1.41421i	0.707107	−	0.707107i	$-0.250000\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$	−0.104528	+	0.994522i	−0.104528	+	0.994522i
$598$		0			0
$599$		0			0		−0.207912	−	0.978148i	$-0.566667\pi$
$599$		0			0		0.207912	+	0.978148i	$0.433333\pi$
$600$		0			0
$601$		0			0		−0.104528	−	0.994522i	$-0.533333\pi$
$601$		0			0		0.104528	+	0.994522i	$0.466667\pi$
$602$		0			0
$603$	−0.309017	−	0.951057i	−0.309017	−	0.951057i
$604$		0			0
$605$		0			0
$606$	1.00000	−	1.73205i	1.00000	−	1.73205i
$607$		0			0		0.669131	−	0.743145i	$-0.266667\pi$
$607$		0			0		−0.669131	+	0.743145i	$0.733333\pi$
$608$		0			0
$609$	0.209057	+	1.98904i	0.209057	+	1.98904i
$610$	−1.33826	−	1.48629i	−1.33826	−	1.48629i
$611$		0			0
$612$		0			0
$613$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$613$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	0.500000	+	0.866025i	0.500000	+	0.866025i	1.00000			$0$
$617$	0.500000	+	0.866025i	0.500000	+	0.866025i	−0.500000	+	0.866025i	$0.666667\pi$
$618$	1.05097	+	0.946294i	1.05097	+	0.946294i
$619$	0.913545	−	0.406737i	0.913545	−	0.406737i	0.104528	−	0.994522i	$-0.466667\pi$
$619$	0.913545	−	0.406737i	0.913545	−	0.406737i	0.809017	+	0.587785i	$0.200000\pi$
$620$	−0.809017	+	0.587785i	−0.809017	+	0.587785i
$621$		0			0
$622$	1.34500	+	0.437016i	1.34500	+	0.437016i
$623$		0			0
$624$		0			0
$625$	−0.669131	+	0.743145i	−0.669131	+	0.743145i
$626$		0			0
$627$		0			0
$628$	0.500000	−	0.866025i	0.500000	−	0.866025i
$629$		0			0
$630$	0.209057	−	1.98904i	0.209057	−	1.98904i
$631$	−0.809017	+	0.587785i	−0.809017	+	0.587785i	−0.913545	−	0.406737i	$-0.866667\pi$
$631$	−0.809017	+	0.587785i	−0.809017	+	0.587785i	0.104528	+	0.994522i	$0.466667\pi$
$632$		0			0
$633$	−1.05097	+	0.946294i	−1.05097	+	0.946294i
$634$		0			0
$635$	−1.40647	−	0.147826i	−1.40647	−	0.147826i
$636$	−0.978148	−	0.207912i	−0.978148	−	0.207912i
$637$		0			0
$638$		0			0
$639$	0.500000	−	0.866025i	0.500000	−	0.866025i
$640$		0			0
$641$	−1.82709	+	0.813473i	−1.82709	+	0.813473i	−0.913545	+	0.406737i	$0.866667\pi$
$641$	−1.82709	+	0.813473i	−1.82709	+	0.813473i	−0.913545	+	0.406737i	$0.866667\pi$
$642$	1.61803	−	1.17557i	1.61803	−	1.17557i
$643$		0			0		−0.207912	−	0.978148i	$-0.566667\pi$
$643$		0			0		0.207912	+	0.978148i	$0.433333\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−0.618034	−	1.90211i	−0.618034	−	1.90211i	−0.309017	−	0.951057i	$-0.600000\pi$
$647$	−0.618034	−	1.90211i	−0.618034	−	1.90211i	−0.309017	−	0.951057i	$-0.600000\pi$
$648$		0			0
$649$		0			0
$650$		0			0
$651$	1.05097	+	0.946294i	1.05097	+	0.946294i
$652$	−0.104528	+	0.994522i	−0.104528	+	0.994522i
$653$	0.913545	+	0.406737i	0.913545	+	0.406737i	0.809017	−	0.587785i	$-0.200000\pi$
$653$	0.913545	+	0.406737i	0.913545	+	0.406737i	0.104528	+	0.994522i	$0.466667\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$	−0.575212	−	1.29195i	−0.575212	−	1.29195i
$658$	−0.618034	−	1.90211i	−0.618034	−	1.90211i
$659$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$659$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$660$		0			0
$661$	−0.500000	−	0.866025i	−0.500000	−	0.866025i	0.500000	−	0.866025i	$-0.333333\pi$
$661$	−0.500000	−	0.866025i	−0.500000	−	0.866025i	−1.00000			$\pi$
$662$	1.05097	+	0.946294i	1.05097	+	0.946294i
$663$		0			0
$664$		0			0
$665$		0			0
$666$	0.294032	+	1.38331i	0.294032	+	1.38331i
$667$		0			0
$668$		0			0
$669$		0			0
$670$	1.22474	−	0.707107i	1.22474	−	0.707107i
$671$		0			0
$672$	−1.00000	−	1.73205i	−1.00000	−	1.73205i
$673$		0			0		−0.978148	−	0.207912i	$-0.933333\pi$
$673$		0			0		0.978148	+	0.207912i	$0.0666667\pi$
$674$	1.61803	+	1.17557i	1.61803	+	1.17557i
$675$		0			0
$676$	−0.309017	+	0.951057i	−0.309017	+	0.951057i
$677$	−1.05097	+	0.946294i	−1.05097	+	0.946294i	−0.998630	−	0.0523360i	$-0.983333\pi$
$677$	−1.05097	+	0.946294i	−1.05097	+	0.946294i	−0.0523360	+	0.998630i	$0.516667\pi$
$678$	−0.575212	+	1.29195i	−0.575212	+	1.29195i
$679$	1.40647	+	0.147826i	1.40647	+	0.147826i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	1.00000			1.00000			0.500000	−	0.866025i	$-0.333333\pi$
$683$	1.00000			1.00000			0.500000	+	0.866025i	$0.333333\pi$
$684$		0			0
$685$	−0.809017	−	0.587785i	−0.809017	−	0.587785i
$686$		0			0
$687$		0			0
$688$		0			0
$689$		0			0
$690$		0			0
$691$	0.978148	+	0.207912i	0.978148	+	0.207912i	0.669131	−	0.743145i	$-0.266667\pi$
$691$	0.978148	+	0.207912i	0.978148	+	0.207912i	0.309017	+	0.951057i	$0.400000\pi$
$692$		0			0
$693$		0			0
$694$	2.00000			2.00000
$695$	0.294032	−	1.38331i	0.294032	−	1.38331i
$696$		0			0
$697$		0			0
$698$		0			0
$699$	−0.294032	−	1.38331i	−0.294032	−	1.38331i
$700$		0			0
$701$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$701$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$702$		0			0
$703$		0			0
$704$		0			0
$705$	0.500000	+	0.866025i	0.500000	+	0.866025i
$706$		0			0
$707$	0.209057	−	1.98904i	0.209057	−	1.98904i
$708$	−0.809017	+	0.587785i	−0.809017	+	0.587785i
$709$	0.669131	+	0.743145i	0.669131	+	0.743145i	0.978148	−	0.207912i	$-0.0666667\pi$
$709$	0.669131	+	0.743145i	0.669131	+	0.743145i	−0.309017	+	0.951057i	$0.600000\pi$
$710$	1.34500	+	0.437016i	1.34500	+	0.437016i
$711$		0			0
$712$		0			0
$713$		0			0
$714$		0			0
$715$		0			0
$716$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$717$	0.294032	−	1.38331i	0.294032	−	1.38331i
$718$	1.82709	−	0.813473i	1.82709	−	0.813473i
$719$	0.809017	−	0.587785i	0.809017	−	0.587785i	−0.104528	−	0.994522i	$-0.533333\pi$
$719$	0.809017	−	0.587785i	0.809017	−	0.587785i	0.913545	+	0.406737i	$0.133333\pi$
$720$	0.669131	+	0.743145i	0.669131	+	0.743145i
$721$	1.34500	+	0.437016i	1.34500	+	0.437016i
$722$	1.40647	−	0.147826i	1.40647	−	0.147826i
$723$		0			0
$724$	0.669131	−	0.743145i	0.669131	−	0.743145i
$725$		0			0
$726$		0			0
$727$	0.500000	−	0.866025i	0.500000	−	0.866025i	−0.500000	−	0.866025i	$-0.666667\pi$
$727$	0.500000	−	0.866025i	0.500000	−	0.866025i	1.00000			$0$
$728$		0			0
$729$	−0.809017	−	0.587785i	−0.809017	−	0.587785i
$730$	1.61803	−	1.17557i	1.61803	−	1.17557i
$731$		0			0
$732$	0.294032	+	1.38331i	0.294032	+	1.38331i
$733$	0.575212	−	1.29195i	0.575212	−	1.29195i	−0.358368	−	0.933580i	$-0.616667\pi$
$733$	0.575212	−	1.29195i	0.575212	−	1.29195i	0.933580	−	0.358368i	$-0.116667\pi$
$734$	1.40647	+	0.147826i	1.40647	+	0.147826i
$735$	−0.309017	−	0.951057i	−0.309017	−	0.951057i
$736$		0			0
$737$		0			0
$738$		0			0
$739$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$739$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$740$	−0.913545	+	0.406737i	−0.913545	+	0.406737i
$741$		0			0
$742$	−1.95630	+	0.415823i	−1.95630	+	0.415823i
$743$	0.294032	+	1.38331i	0.294032	+	1.38331i	0.838671	+	0.544639i	$0.183333\pi$
$743$	0.294032	+	1.38331i	0.294032	+	1.38331i	−0.544639	+	0.838671i	$0.683333\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$	−1.22474	+	0.707107i	−1.22474	+	0.707107i
$748$		0			0
$749$	1.00000	−	1.73205i	1.00000	−	1.73205i
$750$	1.34500	−	0.437016i	1.34500	−	0.437016i
$751$	0.104528	−	0.994522i	0.104528	−	0.994522i	−0.809017	−	0.587785i	$-0.800000\pi$
$751$	0.104528	−	0.994522i	0.104528	−	0.994522i	0.913545	−	0.406737i	$-0.133333\pi$
$752$	0.913545	+	0.406737i	0.913545	+	0.406737i
$753$		0			0
$754$		0			0
$755$		0			0
$756$	−0.831254	+	1.14412i	−0.831254	+	1.14412i
$757$	−0.309017	−	0.951057i	−0.309017	−	0.951057i	−0.978148	−	0.207912i	$-0.933333\pi$
$757$	−0.309017	−	0.951057i	−0.309017	−	0.951057i	0.669131	−	0.743145i	$-0.266667\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−1.05097	−	0.946294i	−1.05097	−	0.946294i	−0.0523360	−	0.998630i	$-0.516667\pi$
$761$	−1.05097	−	0.946294i	−1.05097	−	0.946294i	−0.998630	+	0.0523360i	$0.983333\pi$
$762$	1.61803	+	1.17557i	1.61803	+	1.17557i
$763$		0			0
$764$	0.309017	−	0.951057i	0.309017	−	0.951057i
$765$		0			0
$766$	−0.831254	−	1.14412i	−0.831254	−	1.14412i
$767$		0			0
$768$	0.978148	+	0.207912i	0.978148	+	0.207912i
$769$	−1.22474	+	0.707107i	−1.22474	+	0.707107i	−0.965926	−	0.258819i	$-0.916667\pi$
$769$	−1.22474	+	0.707107i	−1.22474	+	0.707107i	−0.258819	+	0.965926i	$0.583333\pi$
$770$		0			0
$771$	−1.00000	+	1.73205i	−1.00000	+	1.73205i
$772$		0			0
$773$		0			0		0.587785	−	0.809017i	$-0.300000\pi$
$773$		0			0		−0.587785	+	0.809017i	$0.700000\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$	0.831254	+	1.14412i	0.831254	+	1.14412i
$778$	1.40647	+	0.147826i	1.40647	+	0.147826i
$779$		0			0
$780$		0			0
$781$		0			0
$782$		0			0
$783$	−0.294032	+	1.38331i	−0.294032	+	1.38331i
$784$	−0.809017	−	0.587785i	−0.809017	−	0.587785i
$785$	0.104528	+	0.994522i	0.104528	+	0.994522i
$786$		0			0
$787$		0			0		0.978148	−	0.207912i	$-0.0666667\pi$
$787$		0			0		−0.978148	+	0.207912i	$0.933333\pi$
$788$	0.575212	−	1.29195i	0.575212	−	1.29195i
$789$		0			0
$790$		0			0
$791$			1.41421i			1.41421i
$792$		0			0
$793$		0			0
$794$	−0.294032	+	1.38331i	−0.294032	+	1.38331i
$795$	0.913545	−	0.406737i	0.913545	−	0.406737i
$796$	0.913545	+	0.406737i	0.913545	+	0.406737i
$797$	0.978148	−	0.207912i	0.978148	−	0.207912i	0.309017	−	0.951057i	$-0.400000\pi$
$797$	0.978148	−	0.207912i	0.978148	−	0.207912i	0.669131	+	0.743145i	$0.266667\pi$
$798$		0			0
$799$		0			0
$800$		0			0
$801$		0			0
$802$		−	1.41421i		−	1.41421i
$803$		0			0
$804$	−1.00000			−1.00000
$805$		0			0
$806$		0			0
$807$	−0.913545	−	0.406737i	−0.913545	−	0.406737i
$808$		0			0
$809$	−1.34500	−	0.437016i	−1.34500	−	0.437016i	−0.453990	−	0.891007i	$-0.650000\pi$
$809$	−1.34500	−	0.437016i	−1.34500	−	0.437016i	−0.891007	+	0.453990i	$0.850000\pi$
$810$	0.575212	−	1.29195i	0.575212	−	1.29195i
$811$	0.831254	−	1.14412i	0.831254	−	1.14412i	−0.156434	−	0.987688i	$-0.550000\pi$
$811$	0.831254	−	1.14412i	0.831254	−	1.14412i	0.987688	−	0.156434i	$-0.0500000\pi$
$812$	1.95630	+	0.415823i	1.95630	+	0.415823i
$813$		0			0
$814$		0			0
$815$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$816$		0			0
$817$		0			0
$818$		0			0
$819$		0			0
$820$		0			0
$821$		0			0		−0.104528	−	0.994522i	$-0.533333\pi$
$821$		0			0		0.104528	+	0.994522i	$0.466667\pi$
$822$	0.575212	+	1.29195i	0.575212	+	1.29195i
$823$		0			0		−0.743145	−	0.669131i	$-0.766667\pi$
$823$		0			0		0.743145	+	0.669131i	$0.233333\pi$
$824$		0			0
$825$		0			0
$826$	−1.00000	+	1.73205i	−1.00000	+	1.73205i
$827$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$827$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$828$		0			0
$829$	−0.809017	+	0.587785i	−0.809017	+	0.587785i	−0.913545	−	0.406737i	$-0.866667\pi$
$829$	−0.809017	+	0.587785i	−0.809017	+	0.587785i	0.104528	+	0.994522i	$0.466667\pi$
$830$	−1.33826	−	1.48629i	−1.33826	−	1.48629i
$831$		0			0
$832$		0			0
$833$		0			0
$834$	−1.33826	+	1.48629i	−1.33826	+	1.48629i
$835$		0			0
$836$		0			0
$837$	0.500000	+	0.866025i	0.500000	+	0.866025i
$838$	1.34500	−	0.437016i	1.34500	−	0.437016i
$839$		0			0		−0.406737	−	0.913545i	$-0.633333\pi$
$839$		0			0		0.406737	+	0.913545i	$0.366667\pi$
$840$		0			0
$841$	0.978148	−	0.207912i	0.978148	−	0.207912i
$842$	−0.294032	−	1.38331i	−0.294032	−	1.38331i
$843$		0			0
$844$	0.575212	+	1.29195i	0.575212	+	1.29195i
$845$	−0.309017	−	0.951057i	−0.309017	−	0.951057i
$846$		−	1.41421i		−	1.41421i
$847$		0			0
$848$	0.500000	−	0.866025i	0.500000	−	0.866025i
$849$	−0.294032	+	1.38331i	−0.294032	+	1.38331i
$850$		0			0
$851$		0			0
$852$	−0.669131	−	0.743145i	−0.669131	−	0.743145i
$853$	−1.05097	+	0.946294i	−1.05097	+	0.946294i	−0.998630	−	0.0523360i	$-0.983333\pi$
$853$	−1.05097	+	0.946294i	−1.05097	+	0.946294i	−0.0523360	+	0.998630i	$0.516667\pi$
$854$	1.66251	+	2.28825i	1.66251	+	2.28825i
$855$		0			0
$856$		0			0
$857$	1.22474	+	0.707107i	1.22474	+	0.707107i	0.965926	−	0.258819i	$-0.0833333\pi$
$857$	1.22474	+	0.707107i	1.22474	+	0.707107i	0.258819	+	0.965926i	$0.416667\pi$
$858$		0			0
$859$	−0.500000	−	0.866025i	−0.500000	−	0.866025i	0.500000	−	0.866025i	$-0.333333\pi$
$859$	−0.500000	−	0.866025i	−0.500000	−	0.866025i	−1.00000			$\pi$
$860$		0			0
$861$		0			0
$862$	0.209057	+	1.98904i	0.209057	+	1.98904i
$863$		0			0		−0.951057	−	0.309017i	$-0.900000\pi$
$863$		0			0		0.951057	+	0.309017i	$0.100000\pi$
$864$	−0.294032	−	1.38331i	−0.294032	−	1.38331i
$865$		0			0
$866$		0			0
$867$	−0.669131	+	0.743145i	−0.669131	+	0.743145i
$868$	1.22474	−	0.707107i	1.22474	−	0.707107i
$869$		0			0
$870$	−2.00000			−2.00000
$871$		0			0
$872$		0			0
$873$	0.913545	+	0.406737i	0.913545	+	0.406737i
$874$		0			0
$875$	1.05097	−	0.946294i	1.05097	−	0.946294i
$876$	−1.40647	+	0.147826i	−1.40647	+	0.147826i
$877$	1.40647	+	0.147826i	1.40647	+	0.147826i	0.777146	−	0.629320i	$-0.216667\pi$
$877$	1.40647	+	0.147826i	1.40647	+	0.147826i	0.629320	+	0.777146i	$0.283333\pi$
$878$	−1.33826	+	1.48629i	−1.33826	+	1.48629i
$879$	1.22474	+	0.707107i	1.22474	+	0.707107i
$880$		0			0
$881$	−1.00000			−1.00000			−0.500000	−	0.866025i	$-0.666667\pi$
$881$	−1.00000			−1.00000			−0.500000	+	0.866025i	$0.666667\pi$
$882$	−0.294032	+	1.38331i	−0.294032	+	1.38331i
$883$	−0.809017	−	0.587785i	−0.809017	−	0.587785i	0.104528	−	0.994522i	$-0.466667\pi$
$883$	−0.809017	−	0.587785i	−0.809017	−	0.587785i	−0.913545	+	0.406737i	$0.866667\pi$
$884$		0			0
$885$	0.309017	−	0.951057i	0.309017	−	0.951057i
$886$	−0.294032	−	1.38331i	−0.294032	−	1.38331i
$887$		0			0		−0.913545	−	0.406737i	$-0.866667\pi$
$887$		0			0		0.913545	+	0.406737i	$0.133333\pi$
$888$		0			0
$889$	1.95630	+	0.415823i	1.95630	+	0.415823i
$890$		0			0
$891$		0			0
$892$		0			0
$893$		0			0
$894$		0			0
$895$	0.913545	+	0.406737i	0.913545	+	0.406737i
$896$		0			0
$897$		0			0
$898$	1.40647	−	0.147826i	1.40647	−	0.147826i
$899$	0.831254	−	1.14412i	0.831254	−	1.14412i
$900$		0			0
$901$		0			0
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−0.104528	+	0.994522i	−0.104528	+	0.994522i
$906$		0			0
$907$		0			0		0.743145	−	0.669131i	$-0.233333\pi$
$907$		0			0		−0.743145	+	0.669131i	$0.766667\pi$
$908$		0			0
$909$	0.575212	−	1.29195i	0.575212	−	1.29195i
$910$		0			0
$911$	0.978148	+	0.207912i	0.978148	+	0.207912i	0.669131	−	0.743145i	$-0.266667\pi$
$911$	0.978148	+	0.207912i	0.978148	+	0.207912i	0.309017	+	0.951057i	$0.400000\pi$
$912$		0			0
$913$		0			0
$914$	−1.00000	−	1.73205i	−1.00000	−	1.73205i
$915$	−1.05097	−	0.946294i	−1.05097	−	0.946294i
$916$		0			0
$917$		0			0
$918$		0			0
$919$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$919$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$920$		0			0
$921$		0			0
$922$	−1.33826	+	1.48629i	−1.33826	+	1.48629i
$923$		0			0
$924$		0			0
$925$		0			0
$926$		0			0
$927$	0.809017	+	0.587785i	0.809017	+	0.587785i
$928$	−1.61803	+	1.17557i	−1.61803	+	1.17557i
$929$	−0.669131	−	0.743145i	−0.669131	−	0.743145i	0.309017	−	0.951057i	$-0.400000\pi$
$929$	−0.669131	−	0.743145i	−0.669131	−	0.743145i	−0.978148	+	0.207912i	$0.933333\pi$
$930$	−1.05097	+	0.946294i	−1.05097	+	0.946294i
$931$		0			0
$932$	−1.40647	−	0.147826i	−1.40647	−	0.147826i
$933$	0.978148	+	0.207912i	0.978148	+	0.207912i
$934$	−1.22474	−	0.707107i	−1.22474	−	0.707107i
$935$		0			0
$936$		0			0
$937$	−1.34500	+	0.437016i	−1.34500	+	0.437016i	−0.891007	−	0.453990i	$-0.850000\pi$
$937$	−1.34500	+	0.437016i	−1.34500	+	0.437016i	−0.453990	+	0.891007i	$0.650000\pi$
$938$	−1.82709	+	0.813473i	−1.82709	+	0.813473i
$939$		0			0
$940$	0.978148	−	0.207912i	0.978148	−	0.207912i
$941$	−0.294032	−	1.38331i	−0.294032	−	1.38331i	−0.838671	−	0.544639i	$-0.816667\pi$
$941$	−0.294032	−	1.38331i	−0.294032	−	1.38331i	0.544639	−	0.838671i	$-0.316667\pi$
$942$	0.575212	−	1.29195i	0.575212	−	1.29195i
$943$		0			0
$944$	−0.309017	−	0.951057i	−0.309017	−	0.951057i
$945$		−	1.41421i		−	1.41421i
$946$		0			0
$947$	−0.500000	+	0.866025i	−0.500000	+	0.866025i	0.500000	+	0.866025i	$0.333333\pi$
$947$	−0.500000	+	0.866025i	−0.500000	+	0.866025i	−1.00000			$\pi$
$948$		0			0
$949$		0			0
$950$		0			0
$951$		0			0
$952$		0			0
$953$	−0.831254	−	1.14412i	−0.831254	−	1.14412i	−0.987688	−	0.156434i	$-0.950000\pi$
$953$	−0.831254	−	1.14412i	−0.831254	−	1.14412i	0.156434	−	0.987688i	$-0.450000\pi$
$954$	−1.40647	−	0.147826i	−1.40647	−	0.147826i
$955$	0.309017	+	0.951057i	0.309017	+	0.951057i
$956$	−1.22474	−	0.707107i	−1.22474	−	0.707107i
$957$		0			0
$958$	1.00000	+	1.73205i	1.00000	+	1.73205i
$959$	1.05097	+	0.946294i	1.05097	+	0.946294i
$960$	0.913545	−	0.406737i	0.913545	−	0.406737i
$961$		0			0
$962$		0			0
$963$	1.05097	−	0.946294i	1.05097	−	0.946294i
$964$		0			0
$965$		0			0
$966$		0			0
$967$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$967$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$968$		0			0
$969$		0			0
$970$	−0.294032	+	1.38331i	−0.294032	+	1.38331i
$971$		0			0		0.587785	−	0.809017i	$-0.300000\pi$
$971$		0			0		−0.587785	+	0.809017i	$0.700000\pi$
$972$	−0.809017	+	0.587785i	−0.809017	+	0.587785i
$973$	−0.618034	+	1.90211i	−0.618034	+	1.90211i
$974$	1.05097	−	0.946294i	1.05097	−	0.946294i
$975$		0			0
$976$	−1.40647	−	0.147826i	−1.40647	−	0.147826i
$977$		0			0		−0.743145	−	0.669131i	$-0.766667\pi$
$977$		0			0		0.743145	+	0.669131i	$0.233333\pi$
$978$			1.41421i			1.41421i
$979$		0			0
$980$	−1.00000			−1.00000
$981$		0			0
$982$	−1.61803	−	1.17557i	−1.61803	−	1.17557i
$983$	−0.104528	−	0.994522i	−0.104528	−	0.994522i	−0.913545	−	0.406737i	$-0.866667\pi$
$983$	−0.104528	−	0.994522i	−0.104528	−	0.994522i	0.809017	−	0.587785i	$-0.200000\pi$
$984$		0			0
$985$	0.294032	+	1.38331i	0.294032	+	1.38331i
$986$		0			0
$987$	−0.575212	−	1.29195i	−0.575212	−	1.29195i
$988$		0			0
$989$		0			0
$990$		0			0
$991$		0			0			−	1.00000i	$-0.5\pi$
$991$		0			0				1.00000i	$0.5\pi$
$992$	−0.294032	+	1.38331i	−0.294032	+	1.38331i
$993$	0.809017	+	0.587785i	0.809017	+	0.587785i
$994$	−1.82709	−	0.813473i	−1.82709	−	0.813473i
$995$	−0.978148	+	0.207912i	−0.978148	+	0.207912i
$996$	0.294032	+	1.38331i	0.294032	+	1.38331i
$997$		0			0		−0.104528	−	0.994522i	$-0.533333\pi$
$997$		0			0		0.104528	+	0.994522i	$0.466667\pi$
$998$	0.831254	−	1.14412i	0.831254	−	1.14412i
$999$	0.309017	+	0.951057i	0.309017	+	0.951057i

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

)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1089.1.s.b.457.2		16
3.2	odd	2		3267.1.w.b.1909.1		16
9.4	even	3	inner	1089.1.s.b.94.2		16
9.5	odd	6		3267.1.w.b.820.1		16
11.2	odd	10	inner	1089.1.s.b.475.2		16
11.3	even	5		1089.1.h.a.241.1	✓	4
11.4	even	5	inner	1089.1.s.b.844.2		16
11.5	even	5	inner	1089.1.s.b.403.2		16
11.6	odd	10	inner	1089.1.s.b.403.1		16
11.7	odd	10	inner	1089.1.s.b.844.1		16
11.8	odd	10		1089.1.h.a.241.2	yes	4
11.9	even	5	inner	1089.1.s.b.475.1		16
11.10	odd	2	inner	1089.1.s.b.457.1		16
33.2	even	10		3267.1.w.b.3016.1		16
33.5	odd	10		3267.1.w.b.1855.1		16
33.8	even	10		3267.1.h.a.1693.1		4
33.14	odd	10		3267.1.h.a.1693.2		4
33.17	even	10		3267.1.w.b.1855.2		16
33.20	odd	10		3267.1.w.b.3016.2		16
33.26	odd	10		3267.1.w.b.1207.1		16
33.29	even	10		3267.1.w.b.1207.2		16
33.32	even	2		3267.1.w.b.1909.2		16
99.4	even	15	inner	1089.1.s.b.481.1		16
99.5	odd	30		3267.1.w.b.766.2		16
99.13	odd	30	inner	1089.1.s.b.112.2		16
99.14	odd	30		3267.1.h.a.604.1		4
99.31	even	15	inner	1089.1.s.b.112.1		16
99.32	even	6		3267.1.w.b.820.2		16
99.40	odd	30	inner	1089.1.s.b.481.2		16
99.41	even	30		3267.1.h.a.604.2		4
99.49	even	15	inner	1089.1.s.b.40.1		16
99.50	even	30		3267.1.w.b.766.1		16
99.58	even	15		1089.1.h.a.967.2	yes	4
99.59	odd	30		3267.1.w.b.118.2		16
99.68	even	30		3267.1.w.b.1927.1		16
99.76	odd	6	inner	1089.1.s.b.94.1		16
99.85	odd	30		1089.1.h.a.967.1	yes	4
99.86	odd	30		3267.1.w.b.1927.2		16
99.94	odd	30	inner	1089.1.s.b.40.2		16
99.95	even	30		3267.1.w.b.118.1		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1089.1.h.a.241.1	✓	4	11.3	even	5
1089.1.h.a.241.2	yes	4	11.8	odd	10
1089.1.h.a.967.1	yes	4	99.85	odd	30
1089.1.h.a.967.2	yes	4	99.58	even	15
1089.1.s.b.40.1		16	99.49	even	15	inner
1089.1.s.b.40.2		16	99.94	odd	30	inner
1089.1.s.b.94.1		16	99.76	odd	6	inner
1089.1.s.b.94.2		16	9.4	even	3	inner
1089.1.s.b.112.1		16	99.31	even	15	inner
1089.1.s.b.112.2		16	99.13	odd	30	inner
1089.1.s.b.403.1		16	11.6	odd	10	inner
1089.1.s.b.403.2		16	11.5	even	5	inner
1089.1.s.b.457.1		16	11.10	odd	2	inner
1089.1.s.b.457.2		16	1.1	even	1	trivial
1089.1.s.b.475.1		16	11.9	even	5	inner
1089.1.s.b.475.2		16	11.2	odd	10	inner
1089.1.s.b.481.1		16	99.4	even	15	inner
1089.1.s.b.481.2		16	99.40	odd	30	inner
1089.1.s.b.844.1		16	11.7	odd	10	inner
1089.1.s.b.844.2		16	11.4	even	5	inner
3267.1.h.a.604.1		4	99.14	odd	30
3267.1.h.a.604.2		4	99.41	even	30
3267.1.h.a.1693.1		4	33.8	even	10
3267.1.h.a.1693.2		4	33.14	odd	10
3267.1.w.b.118.1		16	99.95	even	30
3267.1.w.b.118.2		16	99.59	odd	30
3267.1.w.b.766.1		16	99.50	even	30
3267.1.w.b.766.2		16	99.5	odd	30
3267.1.w.b.820.1		16	9.5	odd	6
3267.1.w.b.820.2		16	99.32	even	6
3267.1.w.b.1207.1		16	33.26	odd	10
3267.1.w.b.1207.2		16	33.29	even	10
3267.1.w.b.1855.1		16	33.5	odd	10
3267.1.w.b.1855.2		16	33.17	even	10
3267.1.w.b.1909.1		16	3.2	odd	2
3267.1.w.b.1909.2		16	33.32	even	2
3267.1.w.b.1927.1		16	99.68	even	30
3267.1.w.b.1927.2		16	99.86	odd	30
3267.1.w.b.3016.1		16	33.2	even	10
3267.1.w.b.3016.2		16	33.20	odd	10

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 $\F_{q}$

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

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Newspace parameters

Newform invariants

Embedding invariants

$q$ -expansion

Character values

Coefficient data

Twists

This project is supported by grants from the US National Science Foundation, the UK Engineering and Physical Sciences Research Council, and the Simons Foundation.

Label	1089.1.s.b.457.2

Level	$1089$
Weight	$1$
Character	1089.457
Analytic conductor	$0.543$
Analytic rank	$0$
Dimension	$16$
Projective image	$S_{4}$
CM/RM	no
Inner twists	$16$