LMFDB - Embedded newform 3069.1.cd.b.2665.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3069,1,Mod(433,3069)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3069, base_ring=CyclotomicField(10))

chi = DirichletCharacter(H, H._module([0, 2, 5]))

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

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

chi := DirichletCharacter("3069.433");

S:= CuspForms(chi, 1);

N := Newforms(S);

Level:	$ N $	$=$	$ 3069 = 3^{2} \cdot 11 \cdot 31 $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	3069.cd (of order $10$, degree $4$, minimal)

Newform invariants

comment: select newform

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

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

Self dual:	no
Analytic conductor:	$1.53163052377$
Analytic rank:	$0$
Dimension:	$8$
Relative dimension:	$2$ over $\Q(\zeta_{10})$
Coefficient field:	$\Q(\zeta_{20})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - x^{6} + x^{4} - x^{2} + 1 $ x^8 - x^6 + x^4 - x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Projective image:	$A_{5}$
Projective field:	Galois closure of 5.1.126630009.1

Embedding invariants

Embedding label			2665.2
Root			$-0.587785 - 0.809017i$ of defining polynomial
Character	$\chi$	$=$	3069.2665
Dual form			3069.1.cd.b.433.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(1.30902 - 0.951057i) q^{2} +(0.500000 - 1.53884i) q^{4} +(0.809017 + 0.587785i) q^{5} +(-0.309017 + 0.951057i) q^{7} +(-0.309017 - 0.951057i) q^{8} +O(q^{10})$	\(q+(1.30902 - 0.951057i) q^{2} +(0.500000 - 1.53884i) q^{4} +(0.809017 + 0.587785i) q^{5} +(-0.309017 + 0.951057i) q^{7} +(-0.309017 - 0.951057i) q^{8} +1.61803 q^{10} +(0.951057 + 0.309017i) q^{11} +(-0.587785 - 0.809017i) q^{13} +(0.500000 + 1.53884i) q^{14} +(-0.951057 + 1.30902i) q^{17} +(1.30902 - 0.951057i) q^{20} +(1.53884 - 0.500000i) q^{22} -1.61803i q^{23} +(-1.53884 - 0.500000i) q^{26} +(1.30902 + 0.951057i) q^{28} +(0.587785 + 0.190983i) q^{29} +(0.587785 + 0.809017i) q^{31} +1.00000 q^{32} +2.61803i q^{34} +(-0.809017 + 0.587785i) q^{35} +(-0.951057 - 0.309017i) q^{37} +(0.309017 - 0.951057i) q^{40} +0.618034i q^{43} +(0.951057 - 1.30902i) q^{44} +(-1.53884 - 2.11803i) q^{46} +(-0.500000 - 1.53884i) q^{47} +(-1.53884 + 0.500000i) q^{52} +(0.587785 + 0.809017i) q^{55} +1.00000 q^{56} +(0.951057 - 0.309017i) q^{58} +(0.587785 - 0.809017i) q^{61} +(1.53884 + 0.500000i) q^{62} +(1.30902 - 0.951057i) q^{64} -1.00000i q^{65} -1.00000 q^{67} +(1.53884 + 2.11803i) q^{68} +(-0.500000 + 1.53884i) q^{70} +(-0.809017 - 0.587785i) q^{71} +(0.951057 + 0.309017i) q^{73} +(-1.53884 + 0.500000i) q^{74} +(-0.587785 + 0.809017i) q^{77} +(-0.587785 - 0.809017i) q^{79} +(-0.587785 + 0.809017i) q^{83} +(-1.53884 + 0.500000i) q^{85} +(0.587785 + 0.809017i) q^{86} -1.00000i q^{88} -1.00000i q^{89} +(0.951057 - 0.309017i) q^{91} +(-2.48990 - 0.809017i) q^{92} +(-2.11803 - 1.53884i) q^{94} +(-0.809017 + 0.587785i) q^{97} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 6 q^{2} + 4 q^{4} + 2 q^{5} + 2 q^{7} + 2 q^{8}+O(q^{10}) $ 8 * q + 6 * q^2 + 4 * q^4 + 2 * q^5 + 2 * q^7 + 2 * q^8	$ 8 q + 6 q^{2} + 4 q^{4} + 2 q^{5} + 2 q^{7} + 2 q^{8} + 4 q^{10} + 4 q^{14} + 6 q^{20} + 6 q^{28} + 8 q^{32} - 2 q^{35} - 2 q^{40} - 4 q^{47} + 8 q^{56} + 6 q^{64} - 8 q^{67} - 4 q^{70} - 2 q^{71} - 8 q^{94} - 2 q^{97}+O(q^{100}) $ 8 * q + 6 * q^2 + 4 * q^4 + 2 * q^5 + 2 * q^7 + 2 * q^8 + 4 * q^10 + 4 * q^14 + 6 * q^20 + 6 * q^28 + 8 * q^32 - 2 * q^35 - 2 * q^40 - 4 * q^47 + 8 * q^56 + 6 * q^64 - 8 * q^67 - 4 * q^70 - 2 * q^71 - 8 * q^94 - 2 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/3069\mathbb{Z}\right)^\times$.

$n$	$838$	$2080$	$2729$
$\chi(n)$	$e\left(\frac{4}{5}\right)$	$-1$	$1$

Coefficient data

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

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

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

$2$	1.30902	−	0.951057i	1.30902	−	0.951057i	0.309017	−	0.951057i	$-0.400000\pi$
$2$	1.30902	−	0.951057i	1.30902	−	0.951057i	1.00000			$0$
$3$		0			0
$3$		0			0
$4$	0.500000	−	1.53884i	0.500000	−	1.53884i
$5$	0.809017	+	0.587785i	0.809017	+	0.587785i	0.913545	−	0.406737i	$-0.133333\pi$
$5$	0.809017	+	0.587785i	0.809017	+	0.587785i	−0.104528	+	0.994522i	$0.533333\pi$
$6$		0			0
$7$	−0.309017	+	0.951057i	−0.309017	+	0.951057i	0.669131	+	0.743145i	$0.266667\pi$
$7$	−0.309017	+	0.951057i	−0.309017	+	0.951057i	−0.978148	+	0.207912i	$0.933333\pi$
$8$	−0.309017	−	0.951057i	−0.309017	−	0.951057i
$9$		0			0
$10$	1.61803			1.61803
$11$	0.951057	+	0.309017i	0.951057	+	0.309017i
$11$	0.951057	+	0.309017i	0.951057	+	0.309017i
$12$		0			0
$13$	−0.587785	−	0.809017i	−0.587785	−	0.809017i	0.406737	−	0.913545i	$-0.366667\pi$
$13$	−0.587785	−	0.809017i	−0.587785	−	0.809017i	−0.994522	+	0.104528i	$0.966667\pi$
$14$	0.500000	+	1.53884i	0.500000	+	1.53884i
$15$		0			0
$16$		0			0
$17$	−0.951057	+	1.30902i	−0.951057	+	1.30902i			1.00000i	$0.5\pi$
$17$	−0.951057	+	1.30902i	−0.951057	+	1.30902i	−0.951057	+	0.309017i	$0.900000\pi$
$18$		0			0
$19$		0			0		0.951057	−	0.309017i	$-0.100000\pi$
$19$		0			0		−0.951057	+	0.309017i	$0.900000\pi$
$20$	1.30902	−	0.951057i	1.30902	−	0.951057i
$21$		0			0
$22$	1.53884	−	0.500000i	1.53884	−	0.500000i
$23$		−	1.61803i		−	1.61803i	−0.587785	−	0.809017i	$-0.700000\pi$
$23$		−	1.61803i		−	1.61803i	0.587785	−	0.809017i	$-0.300000\pi$
$24$		0			0
$25$		0			0
$26$	−1.53884	−	0.500000i	−1.53884	−	0.500000i
$27$		0			0
$28$	1.30902	+	0.951057i	1.30902	+	0.951057i
$29$	0.587785	+	0.190983i	0.587785	+	0.190983i	0.587785	−	0.809017i	$-0.300000\pi$
$29$	0.587785	+	0.190983i	0.587785	+	0.190983i			1.00000i	$0.5\pi$
$30$		0			0
$31$	0.587785	+	0.809017i	0.587785	+	0.809017i
$31$	0.587785	+	0.809017i	0.587785	+	0.809017i
$32$	1.00000			1.00000
$33$		0			0
$34$			2.61803i			2.61803i
$35$	−0.809017	+	0.587785i	−0.809017	+	0.587785i
$36$		0			0
$37$	−0.951057	−	0.309017i	−0.951057	−	0.309017i	−0.207912	−	0.978148i	$-0.566667\pi$
$37$	−0.951057	−	0.309017i	−0.951057	−	0.309017i	−0.743145	+	0.669131i	$0.766667\pi$
$38$		0			0
$39$		0			0
$40$	0.309017	−	0.951057i	0.309017	−	0.951057i
$41$		0			0		0.951057	−	0.309017i	$-0.100000\pi$
$41$		0			0		−0.951057	+	0.309017i	$0.900000\pi$
$42$		0			0
$43$			0.618034i			0.618034i	0.951057	+	0.309017i	$0.100000\pi$
$43$			0.618034i			0.618034i	−0.951057	+	0.309017i	$0.900000\pi$
$44$	0.951057	−	1.30902i	0.951057	−	1.30902i
$45$		0			0
$46$	−1.53884	−	2.11803i	−1.53884	−	2.11803i
$47$	−0.500000	−	1.53884i	−0.500000	−	1.53884i	−0.809017	−	0.587785i	$-0.800000\pi$
$47$	−0.500000	−	1.53884i	−0.500000	−	1.53884i	0.309017	−	0.951057i	$-0.400000\pi$
$48$		0			0
$49$		0			0
$50$		0			0
$51$		0			0
$52$	−1.53884	+	0.500000i	−1.53884	+	0.500000i
$53$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$53$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$54$		0			0
$55$	0.587785	+	0.809017i	0.587785	+	0.809017i
$56$	1.00000			1.00000
$57$		0			0
$58$	0.951057	−	0.309017i	0.951057	−	0.309017i
$59$		0			0		−0.951057	−	0.309017i	$-0.900000\pi$
$59$		0			0		0.951057	+	0.309017i	$0.100000\pi$
$60$		0			0
$61$	0.587785	−	0.809017i	0.587785	−	0.809017i	−0.406737	−	0.913545i	$-0.633333\pi$
$61$	0.587785	−	0.809017i	0.587785	−	0.809017i	0.994522	+	0.104528i	$0.0333333\pi$
$62$	1.53884	+	0.500000i	1.53884	+	0.500000i
$63$		0			0
$64$	1.30902	−	0.951057i	1.30902	−	0.951057i
$65$		−	1.00000i		−	1.00000i
$66$		0			0
$67$	−1.00000			−1.00000			−0.500000	−	0.866025i	$-0.666667\pi$
$67$	−1.00000			−1.00000			−0.500000	+	0.866025i	$0.666667\pi$
$68$	1.53884	+	2.11803i	1.53884	+	2.11803i
$69$		0			0
$70$	−0.500000	+	1.53884i	−0.500000	+	1.53884i
$71$	−0.809017	−	0.587785i	−0.809017	−	0.587785i	0.104528	−	0.994522i	$-0.466667\pi$
$71$	−0.809017	−	0.587785i	−0.809017	−	0.587785i	−0.913545	+	0.406737i	$0.866667\pi$
$72$		0			0
$73$	0.951057	+	0.309017i	0.951057	+	0.309017i	0.743145	−	0.669131i	$-0.233333\pi$
$73$	0.951057	+	0.309017i	0.951057	+	0.309017i	0.207912	+	0.978148i	$0.433333\pi$
$74$	−1.53884	+	0.500000i	−1.53884	+	0.500000i
$75$		0			0
$76$		0			0
$77$	−0.587785	+	0.809017i	−0.587785	+	0.809017i
$78$		0			0
$79$	−0.587785	−	0.809017i	−0.587785	−	0.809017i	0.406737	−	0.913545i	$-0.366667\pi$
$79$	−0.587785	−	0.809017i	−0.587785	−	0.809017i	−0.994522	+	0.104528i	$0.966667\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	−0.587785	+	0.809017i	−0.587785	+	0.809017i	−0.994522	−	0.104528i	$-0.966667\pi$
$83$	−0.587785	+	0.809017i	−0.587785	+	0.809017i	0.406737	+	0.913545i	$0.366667\pi$
$84$		0			0
$85$	−1.53884	+	0.500000i	−1.53884	+	0.500000i
$86$	0.587785	+	0.809017i	0.587785	+	0.809017i
$87$		0			0
$88$		−	1.00000i		−	1.00000i
$89$		−	1.00000i		−	1.00000i	−0.866025	−	0.500000i	$-0.833333\pi$
$89$		−	1.00000i		−	1.00000i	0.866025	−	0.500000i	$-0.166667\pi$
$90$		0			0
$91$	0.951057	−	0.309017i	0.951057	−	0.309017i
$92$	−2.48990	−	0.809017i	−2.48990	−	0.809017i
$93$		0			0
$94$	−2.11803	−	1.53884i	−2.11803	−	1.53884i
$95$		0			0
$96$		0			0
$97$	−0.809017	+	0.587785i	−0.809017	+	0.587785i	−0.913545	−	0.406737i	$-0.866667\pi$
$97$	−0.809017	+	0.587785i	−0.809017	+	0.587785i	0.104528	+	0.994522i	$0.466667\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	0.809017	−	0.587785i	0.809017	−	0.587785i	−0.104528	−	0.994522i	$-0.533333\pi$
$101$	0.809017	−	0.587785i	0.809017	−	0.587785i	0.913545	+	0.406737i	$0.133333\pi$
$102$		0			0
$103$	−0.500000	+	1.53884i	−0.500000	+	1.53884i	0.309017	+	0.951057i	$0.400000\pi$
$103$	−0.500000	+	1.53884i	−0.500000	+	1.53884i	−0.809017	+	0.587785i	$0.800000\pi$
$104$	−0.587785	+	0.809017i	−0.587785	+	0.809017i
$105$		0			0
$106$		0			0
$107$	−0.500000	−	1.53884i	−0.500000	−	1.53884i	−0.809017	−	0.587785i	$-0.800000\pi$
$107$	−0.500000	−	1.53884i	−0.500000	−	1.53884i	0.309017	−	0.951057i	$-0.400000\pi$
$108$		0			0
$109$	−1.61803			−1.61803			−0.809017	−	0.587785i	$-0.800000\pi$
$109$	−1.61803			−1.61803			−0.809017	+	0.587785i	$0.800000\pi$
$110$	1.53884	+	0.500000i	1.53884	+	0.500000i
$111$		0			0
$112$		0			0
$113$	−0.309017	−	0.951057i	−0.309017	−	0.951057i	−0.978148	−	0.207912i	$-0.933333\pi$
$113$	−0.309017	−	0.951057i	−0.309017	−	0.951057i	0.669131	−	0.743145i	$-0.266667\pi$
$114$		0			0
$115$	0.951057	−	1.30902i	0.951057	−	1.30902i
$116$	0.587785	−	0.809017i	0.587785	−	0.809017i
$117$		0			0
$118$		0			0
$119$	−0.951057	−	1.30902i	−0.951057	−	1.30902i
$120$		0			0
$121$	0.809017	+	0.587785i	0.809017	+	0.587785i
$122$		−	1.61803i		−	1.61803i
$123$		0			0
$124$	1.53884	−	0.500000i	1.53884	−	0.500000i
$125$	0.309017	−	0.951057i	0.309017	−	0.951057i
$126$		0			0
$127$	−0.951057	+	1.30902i	−0.951057	+	1.30902i			1.00000i	$0.5\pi$
$127$	−0.951057	+	1.30902i	−0.951057	+	1.30902i	−0.951057	+	0.309017i	$0.900000\pi$
$128$	0.500000	−	1.53884i	0.500000	−	1.53884i
$129$		0			0
$130$	−0.951057	−	1.30902i	−0.951057	−	1.30902i
$131$	−1.00000			−1.00000			−0.500000	−	0.866025i	$-0.666667\pi$
$131$	−1.00000			−1.00000			−0.500000	+	0.866025i	$0.666667\pi$
$132$		0			0
$133$		0			0
$134$	−1.30902	+	0.951057i	−1.30902	+	0.951057i
$135$		0			0
$136$	1.53884	+	0.500000i	1.53884	+	0.500000i
$137$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$137$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$138$		0			0
$139$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$139$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$140$	0.500000	+	1.53884i	0.500000	+	1.53884i
$141$		0			0
$142$	−1.61803			−1.61803
$143$	−0.309017	−	0.951057i	−0.309017	−	0.951057i
$144$		0			0
$145$	0.363271	+	0.500000i	0.363271	+	0.500000i
$146$	1.53884	−	0.500000i	1.53884	−	0.500000i
$147$		0			0
$148$	−0.951057	+	1.30902i	−0.951057	+	1.30902i
$149$	−0.500000	−	0.363271i	−0.500000	−	0.363271i	0.309017	−	0.951057i	$-0.400000\pi$
$149$	−0.500000	−	0.363271i	−0.500000	−	0.363271i	−0.809017	+	0.587785i	$0.800000\pi$
$150$		0			0
$151$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$151$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$152$		0			0
$153$		0			0
$154$			1.61803i			1.61803i
$155$			1.00000i			1.00000i
$156$		0			0
$157$	0.309017	+	0.951057i	0.309017	+	0.951057i	0.978148	+	0.207912i	$0.0666667\pi$
$157$	0.309017	+	0.951057i	0.309017	+	0.951057i	−0.669131	+	0.743145i	$0.733333\pi$
$158$	−1.53884	−	0.500000i	−1.53884	−	0.500000i
$159$		0			0
$160$	0.809017	+	0.587785i	0.809017	+	0.587785i
$161$	1.53884	+	0.500000i	1.53884	+	0.500000i
$162$		0			0
$163$	−0.500000	+	0.363271i	−0.500000	+	0.363271i	−0.809017	−	0.587785i	$-0.800000\pi$
$163$	−0.500000	+	0.363271i	−0.500000	+	0.363271i	0.309017	+	0.951057i	$0.400000\pi$
$164$		0			0
$165$		0			0
$166$			1.61803i			1.61803i
$167$	0.951057	+	1.30902i	0.951057	+	1.30902i	0.951057	+	0.309017i	$0.100000\pi$
$167$	0.951057	+	1.30902i	0.951057	+	1.30902i			1.00000i	$0.5\pi$
$168$		0			0
$169$		0			0
$170$	−1.53884	+	2.11803i	−1.53884	+	2.11803i
$171$		0			0
$172$	0.951057	+	0.309017i	0.951057	+	0.309017i
$173$	0.309017	+	0.951057i	0.309017	+	0.951057i	0.978148	+	0.207912i	$0.0666667\pi$
$173$	0.309017	+	0.951057i	0.309017	+	0.951057i	−0.669131	+	0.743145i	$0.733333\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$		0			0
$178$	−0.951057	−	1.30902i	−0.951057	−	1.30902i
$179$	0.951057	−	0.309017i	0.951057	−	0.309017i	0.207912	−	0.978148i	$-0.433333\pi$
$179$	0.951057	−	0.309017i	0.951057	−	0.309017i	0.743145	+	0.669131i	$0.233333\pi$
$180$		0			0
$181$	0.587785	−	0.809017i	0.587785	−	0.809017i	−0.406737	−	0.913545i	$-0.633333\pi$
$181$	0.587785	−	0.809017i	0.587785	−	0.809017i	0.994522	+	0.104528i	$0.0333333\pi$
$182$	0.951057	−	1.30902i	0.951057	−	1.30902i
$183$		0			0
$184$	−1.53884	+	0.500000i	−1.53884	+	0.500000i
$185$	−0.587785	−	0.809017i	−0.587785	−	0.809017i
$186$		0			0
$187$	−1.30902	+	0.951057i	−1.30902	+	0.951057i
$188$	−2.61803			−2.61803
$189$		0			0
$190$		0			0
$191$	0.190983	−	0.587785i	0.190983	−	0.587785i	−0.809017	−	0.587785i	$-0.800000\pi$
$191$	0.190983	−	0.587785i	0.190983	−	0.587785i	1.00000			$0$
$192$		0			0
$193$		0			0		0.587785	−	0.809017i	$-0.300000\pi$
$193$		0			0		−0.587785	+	0.809017i	$0.700000\pi$
$194$	−0.500000	+	1.53884i	−0.500000	+	1.53884i
$195$		0			0
$196$		0			0
$197$		−	1.00000i		−	1.00000i	−0.866025	−	0.500000i	$-0.833333\pi$
$197$		−	1.00000i		−	1.00000i	0.866025	−	0.500000i	$-0.166667\pi$
$198$		0			0
$199$		−	1.61803i		−	1.61803i	−0.587785	−	0.809017i	$-0.700000\pi$
$199$		−	1.61803i		−	1.61803i	0.587785	−	0.809017i	$-0.300000\pi$
$200$		0			0
$201$		0			0
$202$	0.500000	−	1.53884i	0.500000	−	1.53884i
$203$	−0.363271	+	0.500000i	−0.363271	+	0.500000i
$204$		0			0
$205$		0			0
$206$	0.809017	+	2.48990i	0.809017	+	2.48990i
$207$		0			0
$208$		0			0
$209$		0			0
$210$		0			0
$211$		0			0		−0.587785	−	0.809017i	$-0.700000\pi$
$211$		0			0		0.587785	+	0.809017i	$0.300000\pi$
$212$		0			0
$213$		0			0
$214$	−2.11803	−	1.53884i	−2.11803	−	1.53884i
$215$	−0.363271	+	0.500000i	−0.363271	+	0.500000i
$216$		0			0
$217$	−0.951057	+	0.309017i	−0.951057	+	0.309017i
$218$	−2.11803	+	1.53884i	−2.11803	+	1.53884i
$219$		0			0
$220$	1.53884	−	0.500000i	1.53884	−	0.500000i
$221$	1.61803			1.61803
$222$		0			0
$223$	−0.587785	+	0.190983i	−0.587785	+	0.190983i	−0.587785	−	0.809017i	$-0.700000\pi$
$223$	−0.587785	+	0.190983i	−0.587785	+	0.190983i			1.00000i	$0.5\pi$
$224$	−0.309017	+	0.951057i	−0.309017	+	0.951057i
$225$		0			0
$226$	−1.30902	−	0.951057i	−1.30902	−	0.951057i
$227$		0			0		−0.951057	−	0.309017i	$-0.900000\pi$
$227$		0			0		0.951057	+	0.309017i	$0.100000\pi$
$228$		0			0
$229$	0.587785	+	0.809017i	0.587785	+	0.809017i	0.994522	−	0.104528i	$-0.0333333\pi$
$229$	0.587785	+	0.809017i	0.587785	+	0.809017i	−0.406737	+	0.913545i	$0.633333\pi$
$230$		−	2.61803i		−	2.61803i
$231$		0			0
$232$		−	0.618034i		−	0.618034i
$233$		0			0		−0.587785	−	0.809017i	$-0.700000\pi$
$233$		0			0		0.587785	+	0.809017i	$0.300000\pi$
$234$		0			0
$235$	0.500000	−	1.53884i	0.500000	−	1.53884i
$236$		0			0
$237$		0			0
$238$	−2.48990	−	0.809017i	−2.48990	−	0.809017i
$239$	−0.951057	+	0.309017i	−0.951057	+	0.309017i	−0.743145	−	0.669131i	$-0.766667\pi$
$239$	−0.951057	+	0.309017i	−0.951057	+	0.309017i	−0.207912	+	0.978148i	$0.566667\pi$
$240$		0			0
$241$			1.61803i			1.61803i	0.587785	+	0.809017i	$0.300000\pi$
$241$			1.61803i			1.61803i	−0.587785	+	0.809017i	$0.700000\pi$
$242$	1.61803			1.61803
$243$		0			0
$244$	−0.951057	−	1.30902i	−0.951057	−	1.30902i
$245$		0			0
$246$		0			0
$247$		0			0
$248$	0.587785	−	0.809017i	0.587785	−	0.809017i
$249$		0			0
$250$	−0.500000	−	1.53884i	−0.500000	−	1.53884i
$251$	−0.587785	−	0.809017i	−0.587785	−	0.809017i	0.406737	−	0.913545i	$-0.366667\pi$
$251$	−0.587785	−	0.809017i	−0.587785	−	0.809017i	−0.994522	+	0.104528i	$0.966667\pi$
$252$		0			0
$253$	0.500000	−	1.53884i	0.500000	−	1.53884i
$254$			2.61803i			2.61803i
$255$		0			0
$256$	−0.309017	−	0.951057i	−0.309017	−	0.951057i
$257$	−0.190983	+	0.587785i	−0.190983	+	0.587785i	0.809017	+	0.587785i	$0.200000\pi$
$257$	−0.190983	+	0.587785i	−0.190983	+	0.587785i	−1.00000			$\pi$
$258$		0			0
$259$	0.587785	−	0.809017i	0.587785	−	0.809017i
$260$	−1.53884	−	0.500000i	−1.53884	−	0.500000i
$261$		0			0
$262$	−1.30902	+	0.951057i	−1.30902	+	0.951057i
$263$			0.618034i			0.618034i	0.951057	+	0.309017i	$0.100000\pi$
$263$			0.618034i			0.618034i	−0.951057	+	0.309017i	$0.900000\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$		0			0
$268$	−0.500000	+	1.53884i	−0.500000	+	1.53884i
$269$	−0.587785	+	0.809017i	−0.587785	+	0.809017i	−0.994522	−	0.104528i	$-0.966667\pi$
$269$	−0.587785	+	0.809017i	−0.587785	+	0.809017i	0.406737	+	0.913545i	$0.366667\pi$
$270$		0			0
$271$	−0.587785	−	0.190983i	−0.587785	−	0.190983i		−	1.00000i	$-0.5\pi$
$271$	−0.587785	−	0.190983i	−0.587785	−	0.190983i	−0.587785	+	0.809017i	$0.700000\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$		0			0
$276$		0			0
$277$	0.951057	+	1.30902i	0.951057	+	1.30902i	0.951057	+	0.309017i	$0.100000\pi$
$277$	0.951057	+	1.30902i	0.951057	+	1.30902i			1.00000i	$0.5\pi$
$278$		0			0
$279$		0			0
$280$	0.809017	+	0.587785i	0.809017	+	0.587785i
$281$	0.500000	+	0.363271i	0.500000	+	0.363271i	0.809017	−	0.587785i	$-0.200000\pi$
$281$	0.500000	+	0.363271i	0.500000	+	0.363271i	−0.309017	+	0.951057i	$0.600000\pi$
$282$		0			0
$283$	0.309017	+	0.951057i	0.309017	+	0.951057i	0.978148	+	0.207912i	$0.0666667\pi$
$283$	0.309017	+	0.951057i	0.309017	+	0.951057i	−0.669131	+	0.743145i	$0.733333\pi$
$284$	−1.30902	+	0.951057i	−1.30902	+	0.951057i
$285$		0			0
$286$	−1.30902	−	0.951057i	−1.30902	−	0.951057i
$287$		0			0
$288$		0			0
$289$	−0.500000	−	1.53884i	−0.500000	−	1.53884i
$290$	0.951057	+	0.309017i	0.951057	+	0.309017i
$291$		0			0
$292$	0.951057	−	1.30902i	0.951057	−	1.30902i
$293$	0.309017	−	0.951057i	0.309017	−	0.951057i	−0.669131	−	0.743145i	$-0.733333\pi$
$293$	0.309017	−	0.951057i	0.309017	−	0.951057i	0.978148	−	0.207912i	$-0.0666667\pi$
$294$		0			0
$295$		0			0
$296$			1.00000i			1.00000i
$297$		0			0
$298$	−1.00000			−1.00000
$299$	−1.30902	+	0.951057i	−1.30902	+	0.951057i
$300$		0			0
$301$	−0.587785	−	0.190983i	−0.587785	−	0.190983i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	0.951057	−	0.309017i	0.951057	−	0.309017i
$306$		0			0
$307$	0.618034			0.618034			0.309017	−	0.951057i	$-0.400000\pi$
$307$	0.618034			0.618034			0.309017	+	0.951057i	$0.400000\pi$
$308$	0.951057	+	1.30902i	0.951057	+	1.30902i
$309$		0			0
$310$	0.951057	+	1.30902i	0.951057	+	1.30902i
$311$	0.309017	+	0.951057i	0.309017	+	0.951057i	0.978148	+	0.207912i	$0.0666667\pi$
$311$	0.309017	+	0.951057i	0.309017	+	0.951057i	−0.669131	+	0.743145i	$0.733333\pi$
$312$		0			0
$313$	−0.363271	+	0.500000i	−0.363271	+	0.500000i	−0.951057	−	0.309017i	$-0.900000\pi$
$313$	−0.363271	+	0.500000i	−0.363271	+	0.500000i	0.587785	+	0.809017i	$0.300000\pi$
$314$	1.30902	+	0.951057i	1.30902	+	0.951057i
$315$		0			0
$316$	−1.53884	+	0.500000i	−1.53884	+	0.500000i
$317$	0.809017	−	0.587785i	0.809017	−	0.587785i	−0.104528	−	0.994522i	$-0.533333\pi$
$317$	0.809017	−	0.587785i	0.809017	−	0.587785i	0.913545	+	0.406737i	$0.133333\pi$
$318$		0			0
$319$	0.500000	+	0.363271i	0.500000	+	0.363271i
$320$	1.61803			1.61803
$321$		0			0
$322$	2.48990	−	0.809017i	2.48990	−	0.809017i
$323$		0			0
$324$		0			0
$325$		0			0
$326$	−0.309017	+	0.951057i	−0.309017	+	0.951057i
$327$		0			0
$328$		0			0
$329$	1.61803			1.61803
$330$		0			0
$331$		−	0.618034i		−	0.618034i	−0.951057	−	0.309017i	$-0.900000\pi$
$331$		−	0.618034i		−	0.618034i	0.951057	−	0.309017i	$-0.100000\pi$
$332$	0.951057	+	1.30902i	0.951057	+	1.30902i
$333$		0			0
$334$	2.48990	+	0.809017i	2.48990	+	0.809017i
$335$	−0.809017	−	0.587785i	−0.809017	−	0.587785i
$336$		0			0
$337$	0.951057	+	0.309017i	0.951057	+	0.309017i	0.743145	−	0.669131i	$-0.233333\pi$
$337$	0.951057	+	0.309017i	0.951057	+	0.309017i	0.207912	+	0.978148i	$0.433333\pi$
$338$		0			0
$339$		0			0
$340$			2.61803i			2.61803i
$341$	0.309017	+	0.951057i	0.309017	+	0.951057i
$342$		0			0
$343$	−0.809017	+	0.587785i	−0.809017	+	0.587785i
$344$	0.587785	−	0.190983i	0.587785	−	0.190983i
$345$		0			0
$346$	1.30902	+	0.951057i	1.30902	+	0.951057i
$347$	0.587785	−	0.809017i	0.587785	−	0.809017i	−0.406737	−	0.913545i	$-0.633333\pi$
$347$	0.587785	−	0.809017i	0.587785	−	0.809017i	0.994522	+	0.104528i	$0.0333333\pi$
$348$		0			0
$349$	0.190983	+	0.587785i	0.190983	+	0.587785i	1.00000			$0$
$349$	0.190983	+	0.587785i	0.190983	+	0.587785i	−0.809017	+	0.587785i	$0.800000\pi$
$350$		0			0
$351$		0			0
$352$	0.951057	+	0.309017i	0.951057	+	0.309017i
$353$		0			0		1.00000			$0$
$353$		0			0		−1.00000			$\pi$
$354$		0			0
$355$	−0.309017	−	0.951057i	−0.309017	−	0.951057i
$356$	−1.53884	−	0.500000i	−1.53884	−	0.500000i
$357$		0			0
$358$	0.951057	−	1.30902i	0.951057	−	1.30902i
$359$		0			0		−0.951057	−	0.309017i	$-0.900000\pi$
$359$		0			0		0.951057	+	0.309017i	$0.100000\pi$
$360$		0			0
$361$	0.809017	−	0.587785i	0.809017	−	0.587785i
$362$		−	1.61803i		−	1.61803i
$363$		0			0
$364$		−	1.61803i		−	1.61803i
$365$	0.587785	+	0.809017i	0.587785	+	0.809017i
$366$		0			0
$367$	−1.53884	−	0.500000i	−1.53884	−	0.500000i	−0.587785	−	0.809017i	$-0.700000\pi$
$367$	−1.53884	−	0.500000i	−1.53884	−	0.500000i	−0.951057	+	0.309017i	$0.900000\pi$
$368$		0			0
$369$		0			0
$370$	−1.53884	−	0.500000i	−1.53884	−	0.500000i
$371$		0			0
$372$		0			0
$373$	1.61803			1.61803			0.809017	−	0.587785i	$-0.200000\pi$
$373$	1.61803			1.61803			0.809017	+	0.587785i	$0.200000\pi$
$374$	−0.809017	+	2.48990i	−0.809017	+	2.48990i
$375$		0			0
$376$	−1.30902	+	0.951057i	−1.30902	+	0.951057i
$377$	−0.190983	−	0.587785i	−0.190983	−	0.587785i
$378$		0			0
$379$		0			0		0.587785	−	0.809017i	$-0.300000\pi$
$379$		0			0		−0.587785	+	0.809017i	$0.700000\pi$
$380$		0			0
$381$		0			0
$382$	−0.309017	−	0.951057i	−0.309017	−	0.951057i
$383$	−0.587785	−	0.809017i	−0.587785	−	0.809017i	0.406737	−	0.913545i	$-0.366667\pi$
$383$	−0.587785	−	0.809017i	−0.587785	−	0.809017i	−0.994522	+	0.104528i	$0.966667\pi$
$384$		0			0
$385$	−0.951057	+	0.309017i	−0.951057	+	0.309017i
$386$		0			0
$387$		0			0
$388$	0.500000	+	1.53884i	0.500000	+	1.53884i
$389$	0.587785	+	0.190983i	0.587785	+	0.190983i	0.587785	−	0.809017i	$-0.300000\pi$
$389$	0.587785	+	0.190983i	0.587785	+	0.190983i			1.00000i	$0.5\pi$
$390$		0			0
$391$	2.11803	+	1.53884i	2.11803	+	1.53884i
$392$		0			0
$393$		0			0
$394$	−0.951057	−	1.30902i	−0.951057	−	1.30902i
$395$		−	1.00000i		−	1.00000i
$396$		0			0
$397$		0			0			−	1.00000i	$-0.5\pi$
$397$		0			0				1.00000i	$0.5\pi$
$398$	−1.53884	−	2.11803i	−1.53884	−	2.11803i
$399$		0			0
$400$		0			0
$401$	−0.363271	+	0.500000i	−0.363271	+	0.500000i	−0.951057	−	0.309017i	$-0.900000\pi$
$401$	−0.363271	+	0.500000i	−0.363271	+	0.500000i	0.587785	+	0.809017i	$0.300000\pi$
$402$		0			0
$403$	0.309017	−	0.951057i	0.309017	−	0.951057i
$404$	−0.500000	−	1.53884i	−0.500000	−	1.53884i
$405$		0			0
$406$			1.00000i			1.00000i
$407$	−0.809017	−	0.587785i	−0.809017	−	0.587785i
$408$		0			0
$409$	−0.587785	−	0.809017i	−0.587785	−	0.809017i	0.406737	−	0.913545i	$-0.366667\pi$
$409$	−0.587785	−	0.809017i	−0.587785	−	0.809017i	−0.994522	+	0.104528i	$0.966667\pi$
$410$		0			0
$411$		0			0
$412$	2.11803	+	1.53884i	2.11803	+	1.53884i
$413$		0			0
$414$		0			0
$415$	−0.951057	+	0.309017i	−0.951057	+	0.309017i
$416$	−0.587785	−	0.809017i	−0.587785	−	0.809017i
$417$		0			0
$418$		0			0
$419$	−1.00000			−1.00000			−0.500000	−	0.866025i	$-0.666667\pi$
$419$	−1.00000			−1.00000			−0.500000	+	0.866025i	$0.666667\pi$
$420$		0			0
$421$	0.500000	+	1.53884i	0.500000	+	1.53884i	0.809017	+	0.587785i	$0.200000\pi$
$421$	0.500000	+	1.53884i	0.500000	+	1.53884i	−0.309017	+	0.951057i	$0.600000\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$		0			0
$426$		0			0
$427$	0.587785	+	0.809017i	0.587785	+	0.809017i
$428$	−2.61803			−2.61803
$429$		0			0
$430$			1.00000i			1.00000i
$431$	0.809017	−	0.587785i	0.809017	−	0.587785i	−0.104528	−	0.994522i	$-0.533333\pi$
$431$	0.809017	−	0.587785i	0.809017	−	0.587785i	0.913545	+	0.406737i	$0.133333\pi$
$432$		0			0
$433$	1.53884	+	0.500000i	1.53884	+	0.500000i	0.951057	−	0.309017i	$-0.100000\pi$
$433$	1.53884	+	0.500000i	1.53884	+	0.500000i	0.587785	+	0.809017i	$0.300000\pi$
$434$	−0.951057	+	1.30902i	−0.951057	+	1.30902i
$435$		0			0
$436$	−0.809017	+	2.48990i	−0.809017	+	2.48990i
$437$		0			0
$438$		0			0
$439$	−0.618034			−0.618034			−0.309017	−	0.951057i	$-0.600000\pi$
$439$	−0.618034			−0.618034			−0.309017	+	0.951057i	$0.600000\pi$
$440$	0.587785	−	0.809017i	0.587785	−	0.809017i
$441$		0			0
$442$	2.11803	−	1.53884i	2.11803	−	1.53884i
$443$	0.500000	+	1.53884i	0.500000	+	1.53884i	0.809017	+	0.587785i	$0.200000\pi$
$443$	0.500000	+	1.53884i	0.500000	+	1.53884i	−0.309017	+	0.951057i	$0.600000\pi$
$444$		0			0
$445$	0.587785	−	0.809017i	0.587785	−	0.809017i
$446$	−0.587785	+	0.809017i	−0.587785	+	0.809017i
$447$		0			0
$448$	0.500000	+	1.53884i	0.500000	+	1.53884i
$449$	0.363271	+	0.500000i	0.363271	+	0.500000i	0.951057	−	0.309017i	$-0.100000\pi$
$449$	0.363271	+	0.500000i	0.363271	+	0.500000i	−0.587785	+	0.809017i	$0.700000\pi$
$450$		0			0
$451$		0			0
$452$	−1.61803			−1.61803
$453$		0			0
$454$		0			0
$455$	0.951057	+	0.309017i	0.951057	+	0.309017i
$456$		0			0
$457$	0.587785	−	0.809017i	0.587785	−	0.809017i	−0.406737	−	0.913545i	$-0.633333\pi$
$457$	0.587785	−	0.809017i	0.587785	−	0.809017i	0.994522	+	0.104528i	$0.0333333\pi$
$458$	1.53884	+	0.500000i	1.53884	+	0.500000i
$459$		0			0
$460$	−1.53884	−	2.11803i	−1.53884	−	2.11803i
$461$		−	1.61803i		−	1.61803i	−0.587785	−	0.809017i	$-0.700000\pi$
$461$		−	1.61803i		−	1.61803i	0.587785	−	0.809017i	$-0.300000\pi$
$462$		0			0
$463$		−	1.61803i		−	1.61803i	−0.587785	−	0.809017i	$-0.700000\pi$
$463$		−	1.61803i		−	1.61803i	0.587785	−	0.809017i	$-0.300000\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	0.809017	+	0.587785i	0.809017	+	0.587785i	0.913545	−	0.406737i	$-0.133333\pi$
$467$	0.809017	+	0.587785i	0.809017	+	0.587785i	−0.104528	+	0.994522i	$0.533333\pi$
$468$		0			0
$469$	0.309017	−	0.951057i	0.309017	−	0.951057i
$470$	−0.809017	−	2.48990i	−0.809017	−	2.48990i
$471$		0			0
$472$		0			0
$473$	−0.190983	+	0.587785i	−0.190983	+	0.587785i
$474$		0			0
$475$		0			0
$476$	−2.48990	+	0.809017i	−2.48990	+	0.809017i
$477$		0			0
$478$	−0.951057	+	1.30902i	−0.951057	+	1.30902i
$479$	0.500000	+	0.363271i	0.500000	+	0.363271i	0.809017	−	0.587785i	$-0.200000\pi$
$479$	0.500000	+	0.363271i	0.500000	+	0.363271i	−0.309017	+	0.951057i	$0.600000\pi$
$480$		0			0
$481$	0.309017	+	0.951057i	0.309017	+	0.951057i
$482$	1.53884	+	2.11803i	1.53884	+	2.11803i
$483$		0			0
$484$	1.30902	−	0.951057i	1.30902	−	0.951057i
$485$	−1.00000			−1.00000
$486$		0			0
$487$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$487$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$488$	−0.951057	−	0.309017i	−0.951057	−	0.309017i
$489$		0			0
$490$		0			0
$491$	0.951057	+	0.309017i	0.951057	+	0.309017i	0.743145	−	0.669131i	$-0.233333\pi$
$491$	0.951057	+	0.309017i	0.951057	+	0.309017i	0.207912	+	0.978148i	$0.433333\pi$
$492$		0			0
$493$	−0.809017	+	0.587785i	−0.809017	+	0.587785i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	0.809017	−	0.587785i	0.809017	−	0.587785i
$498$		0			0
$499$	−0.587785	−	0.190983i	−0.587785	−	0.190983i		−	1.00000i	$-0.5\pi$
$499$	−0.587785	−	0.190983i	−0.587785	−	0.190983i	−0.587785	+	0.809017i	$0.700000\pi$
$500$	−1.30902	−	0.951057i	−1.30902	−	0.951057i
$501$		0			0
$502$	−1.53884	−	0.500000i	−1.53884	−	0.500000i
$503$	−0.309017	−	0.951057i	−0.309017	−	0.951057i	−0.978148	−	0.207912i	$-0.933333\pi$
$503$	−0.309017	−	0.951057i	−0.309017	−	0.951057i	0.669131	−	0.743145i	$-0.266667\pi$
$504$		0			0
$505$	1.00000			1.00000
$506$	−0.809017	−	2.48990i	−0.809017	−	2.48990i
$507$		0			0
$508$	1.53884	+	2.11803i	1.53884	+	2.11803i
$509$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$509$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$510$		0			0
$511$	−0.587785	+	0.809017i	−0.587785	+	0.809017i
$512$		0			0
$513$		0			0
$514$	0.309017	+	0.951057i	0.309017	+	0.951057i
$515$	−1.30902	+	0.951057i	−1.30902	+	0.951057i
$516$		0			0
$517$		−	1.61803i		−	1.61803i
$518$		−	1.61803i		−	1.61803i
$519$		0			0
$520$	−0.951057	+	0.309017i	−0.951057	+	0.309017i
$521$	−0.309017	+	0.951057i	−0.309017	+	0.951057i	0.669131	+	0.743145i	$0.266667\pi$
$521$	−0.309017	+	0.951057i	−0.309017	+	0.951057i	−0.978148	+	0.207912i	$0.933333\pi$
$522$		0			0
$523$	−0.951057	+	1.30902i	−0.951057	+	1.30902i			1.00000i	$0.5\pi$
$523$	−0.951057	+	1.30902i	−0.951057	+	1.30902i	−0.951057	+	0.309017i	$0.900000\pi$
$524$	−0.500000	+	1.53884i	−0.500000	+	1.53884i
$525$		0			0
$526$	0.587785	+	0.809017i	0.587785	+	0.809017i
$527$	−1.61803			−1.61803
$528$		0			0
$529$	−1.61803			−1.61803
$530$		0			0
$531$		0			0
$532$		0			0
$533$		0			0
$534$		0			0
$535$	0.500000	−	1.53884i	0.500000	−	1.53884i
$536$	0.309017	+	0.951057i	0.309017	+	0.951057i
$537$		0			0
$538$			1.61803i			1.61803i
$539$		0			0
$540$		0			0
$541$	0.809017	−	0.587785i	0.809017	−	0.587785i	−0.104528	−	0.994522i	$-0.533333\pi$
$541$	0.809017	−	0.587785i	0.809017	−	0.587785i	0.913545	+	0.406737i	$0.133333\pi$
$542$	−0.951057	+	0.309017i	−0.951057	+	0.309017i
$543$		0			0
$544$	−0.951057	+	1.30902i	−0.951057	+	1.30902i
$545$	−1.30902	−	0.951057i	−1.30902	−	0.951057i
$546$		0			0
$547$	−0.309017	−	0.951057i	−0.309017	−	0.951057i	−0.978148	−	0.207912i	$-0.933333\pi$
$547$	−0.309017	−	0.951057i	−0.309017	−	0.951057i	0.669131	−	0.743145i	$-0.266667\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$		0			0
$552$		0			0
$553$	0.951057	−	0.309017i	0.951057	−	0.309017i
$554$	2.48990	+	0.809017i	2.48990	+	0.809017i
$555$		0			0
$556$		0			0
$557$	−0.587785	−	0.190983i	−0.587785	−	0.190983i		−	1.00000i	$-0.5\pi$
$557$	−0.587785	−	0.190983i	−0.587785	−	0.190983i	−0.587785	+	0.809017i	$0.700000\pi$
$558$		0			0
$559$	0.500000	−	0.363271i	0.500000	−	0.363271i
$560$		0			0
$561$		0			0
$562$	1.00000			1.00000
$563$	−0.809017	+	0.587785i	−0.809017	+	0.587785i	−0.913545	−	0.406737i	$-0.866667\pi$
$563$	−0.809017	+	0.587785i	−0.809017	+	0.587785i	0.104528	+	0.994522i	$0.466667\pi$
$564$		0			0
$565$	0.309017	−	0.951057i	0.309017	−	0.951057i
$566$	1.30902	+	0.951057i	1.30902	+	0.951057i
$567$		0			0
$568$	−0.309017	+	0.951057i	−0.309017	+	0.951057i
$569$	−0.951057	+	0.309017i	−0.951057	+	0.309017i	−0.743145	−	0.669131i	$-0.766667\pi$
$569$	−0.951057	+	0.309017i	−0.951057	+	0.309017i	−0.207912	+	0.978148i	$0.566667\pi$
$570$		0			0
$571$			1.00000i			1.00000i	0.866025	+	0.500000i	$0.166667\pi$
$571$			1.00000i			1.00000i	−0.866025	+	0.500000i	$0.833333\pi$
$572$	−1.61803			−1.61803
$573$		0			0
$574$		0			0
$575$		0			0
$576$		0			0
$577$	−0.809017	−	0.587785i	−0.809017	−	0.587785i	0.104528	−	0.994522i	$-0.466667\pi$
$577$	−0.809017	−	0.587785i	−0.809017	−	0.587785i	−0.913545	+	0.406737i	$0.866667\pi$
$578$	−2.11803	−	1.53884i	−2.11803	−	1.53884i
$579$		0			0
$580$	0.951057	−	0.309017i	0.951057	−	0.309017i
$581$	−0.587785	−	0.809017i	−0.587785	−	0.809017i
$582$		0			0
$583$		0			0
$584$		−	1.00000i		−	1.00000i
$585$		0			0
$586$	−0.500000	−	1.53884i	−0.500000	−	1.53884i
$587$	0.587785	+	0.190983i	0.587785	+	0.190983i	0.587785	−	0.809017i	$-0.300000\pi$
$587$	0.587785	+	0.190983i	0.587785	+	0.190983i			1.00000i	$0.5\pi$
$588$		0			0
$589$		0			0
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−1.00000			−1.00000			−0.500000	−	0.866025i	$-0.666667\pi$
$593$	−1.00000			−1.00000			−0.500000	+	0.866025i	$0.666667\pi$
$594$		0			0
$595$		−	1.61803i		−	1.61803i
$596$	−0.809017	+	0.587785i	−0.809017	+	0.587785i
$597$		0			0
$598$	−0.809017	+	2.48990i	−0.809017	+	2.48990i
$599$		0			0		0.587785	−	0.809017i	$-0.300000\pi$
$599$		0			0		−0.587785	+	0.809017i	$0.700000\pi$
$600$		0			0
$601$	−0.587785	−	0.190983i	−0.587785	−	0.190983i		−	1.00000i	$-0.5\pi$
$601$	−0.587785	−	0.190983i	−0.587785	−	0.190983i	−0.587785	+	0.809017i	$0.700000\pi$
$602$	−0.951057	+	0.309017i	−0.951057	+	0.309017i
$603$		0			0
$604$		0			0
$605$	0.309017	+	0.951057i	0.309017	+	0.951057i
$606$		0			0
$607$		0			0		−0.587785	−	0.809017i	$-0.700000\pi$
$607$		0			0		0.587785	+	0.809017i	$0.300000\pi$
$608$		0			0
$609$		0			0
$610$	0.951057	−	1.30902i	0.951057	−	1.30902i
$611$	−0.951057	+	1.30902i	−0.951057	+	1.30902i
$612$		0			0
$613$	1.53884	−	0.500000i	1.53884	−	0.500000i	0.587785	−	0.809017i	$-0.300000\pi$
$613$	1.53884	−	0.500000i	1.53884	−	0.500000i	0.951057	+	0.309017i	$0.100000\pi$
$614$	0.809017	−	0.587785i	0.809017	−	0.587785i
$615$		0			0
$616$	0.951057	+	0.309017i	0.951057	+	0.309017i
$617$	−1.61803			−1.61803			−0.809017	−	0.587785i	$-0.800000\pi$
$617$	−1.61803			−1.61803			−0.809017	+	0.587785i	$0.800000\pi$
$618$		0			0
$619$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$619$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$620$	1.53884	+	0.500000i	1.53884	+	0.500000i
$621$		0			0
$622$	1.30902	+	0.951057i	1.30902	+	0.951057i
$623$	0.951057	+	0.309017i	0.951057	+	0.309017i
$624$		0			0
$625$	0.809017	−	0.587785i	0.809017	−	0.587785i
$626$			1.00000i			1.00000i
$627$		0			0
$628$	1.61803			1.61803
$629$	1.30902	−	0.951057i	1.30902	−	0.951057i
$630$		0			0
$631$	1.53884	+	0.500000i	1.53884	+	0.500000i	0.951057	−	0.309017i	$-0.100000\pi$
$631$	1.53884	+	0.500000i	1.53884	+	0.500000i	0.587785	+	0.809017i	$0.300000\pi$
$632$	−0.587785	+	0.809017i	−0.587785	+	0.809017i
$633$		0			0
$634$	0.500000	−	1.53884i	0.500000	−	1.53884i
$635$	−1.53884	+	0.500000i	−1.53884	+	0.500000i
$636$		0			0
$637$		0			0
$638$	1.00000			1.00000
$639$		0			0
$640$	1.30902	−	0.951057i	1.30902	−	0.951057i
$641$	−1.53884	+	0.500000i	−1.53884	+	0.500000i	−0.951057	−	0.309017i	$-0.900000\pi$
$641$	−1.53884	+	0.500000i	−1.53884	+	0.500000i	−0.587785	+	0.809017i	$0.700000\pi$
$642$		0			0
$643$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$643$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$644$	1.53884	−	2.11803i	1.53884	−	2.11803i
$645$		0			0
$646$		0			0
$647$	−0.363271	−	0.500000i	−0.363271	−	0.500000i	0.587785	−	0.809017i	$-0.300000\pi$
$647$	−0.363271	−	0.500000i	−0.363271	−	0.500000i	−0.951057	+	0.309017i	$0.900000\pi$
$648$		0			0
$649$		0			0
$650$		0			0
$651$		0			0
$652$	0.309017	+	0.951057i	0.309017	+	0.951057i
$653$	0.309017	−	0.951057i	0.309017	−	0.951057i	−0.669131	−	0.743145i	$-0.733333\pi$
$653$	0.309017	−	0.951057i	0.309017	−	0.951057i	0.978148	−	0.207912i	$-0.0666667\pi$
$654$		0			0
$655$	−0.809017	−	0.587785i	−0.809017	−	0.587785i
$656$		0			0
$657$		0			0
$658$	2.11803	−	1.53884i	2.11803	−	1.53884i
$659$	−0.618034			−0.618034			−0.309017	−	0.951057i	$-0.600000\pi$
$659$	−0.618034			−0.618034			−0.309017	+	0.951057i	$0.600000\pi$
$660$		0			0
$661$	1.00000			1.00000			0.500000	−	0.866025i	$-0.333333\pi$
$661$	1.00000			1.00000			0.500000	+	0.866025i	$0.333333\pi$
$662$	−0.587785	−	0.809017i	−0.587785	−	0.809017i
$663$		0			0
$664$	0.951057	+	0.309017i	0.951057	+	0.309017i
$665$		0			0
$666$		0			0
$667$	0.309017	−	0.951057i	0.309017	−	0.951057i
$668$	2.48990	−	0.809017i	2.48990	−	0.809017i
$669$		0			0
$670$	−1.61803			−1.61803
$671$	0.809017	−	0.587785i	0.809017	−	0.587785i
$672$		0			0
$673$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$673$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$674$	1.53884	−	0.500000i	1.53884	−	0.500000i
$675$		0			0
$676$		0			0
$677$	0.363271	−	0.500000i	0.363271	−	0.500000i	−0.587785	−	0.809017i	$-0.700000\pi$
$677$	0.363271	−	0.500000i	0.363271	−	0.500000i	0.951057	+	0.309017i	$0.100000\pi$
$678$		0			0
$679$	−0.309017	−	0.951057i	−0.309017	−	0.951057i
$680$	0.951057	+	1.30902i	0.951057	+	1.30902i
$681$		0			0
$682$	1.30902	+	0.951057i	1.30902	+	0.951057i
$683$	−0.618034			−0.618034			−0.309017	−	0.951057i	$-0.600000\pi$
$683$	−0.618034			−0.618034			−0.309017	+	0.951057i	$0.600000\pi$
$684$		0			0
$685$		0			0
$686$	−0.500000	+	1.53884i	−0.500000	+	1.53884i
$687$		0			0
$688$		0			0
$689$		0			0
$690$		0			0
$691$		0			0		−0.587785	−	0.809017i	$-0.700000\pi$
$691$		0			0		0.587785	+	0.809017i	$0.300000\pi$
$692$	1.61803			1.61803
$693$		0			0
$694$		−	1.61803i		−	1.61803i
$695$		0			0
$696$		0			0
$697$		0			0
$698$	0.809017	+	0.587785i	0.809017	+	0.587785i
$699$		0			0
$700$		0			0
$701$	0.309017	+	0.951057i	0.309017	+	0.951057i	0.978148	+	0.207912i	$0.0666667\pi$
$701$	0.309017	+	0.951057i	0.309017	+	0.951057i	−0.669131	+	0.743145i	$0.733333\pi$
$702$		0			0
$703$		0			0
$704$	1.53884	−	0.500000i	1.53884	−	0.500000i
$705$		0			0
$706$		0			0
$707$	0.309017	+	0.951057i	0.309017	+	0.951057i
$708$		0			0
$709$	−0.951057	+	1.30902i	−0.951057	+	1.30902i			1.00000i	$0.5\pi$
$709$	−0.951057	+	1.30902i	−0.951057	+	1.30902i	−0.951057	+	0.309017i	$0.900000\pi$
$710$	−1.30902	−	0.951057i	−1.30902	−	0.951057i
$711$		0			0
$712$	−0.951057	+	0.309017i	−0.951057	+	0.309017i
$713$	1.30902	−	0.951057i	1.30902	−	0.951057i
$714$		0			0
$715$	0.309017	−	0.951057i	0.309017	−	0.951057i
$716$		−	1.61803i		−	1.61803i
$717$		0			0
$718$		0			0
$719$	0.587785	+	0.190983i	0.587785	+	0.190983i	0.587785	−	0.809017i	$-0.300000\pi$
$719$	0.587785	+	0.190983i	0.587785	+	0.190983i			1.00000i	$0.5\pi$
$720$		0			0
$721$	−1.30902	−	0.951057i	−1.30902	−	0.951057i
$722$	0.500000	−	1.53884i	0.500000	−	1.53884i
$723$		0			0
$724$	−0.951057	−	1.30902i	−0.951057	−	1.30902i
$725$		0			0
$726$		0			0
$727$	1.61803			1.61803			0.809017	−	0.587785i	$-0.200000\pi$
$727$	1.61803			1.61803			0.809017	+	0.587785i	$0.200000\pi$
$728$	−0.587785	−	0.809017i	−0.587785	−	0.809017i
$729$		0			0
$730$	1.53884	+	0.500000i	1.53884	+	0.500000i
$731$	−0.809017	−	0.587785i	−0.809017	−	0.587785i
$732$		0			0
$733$	−0.309017	+	0.951057i	−0.309017	+	0.951057i	0.669131	+	0.743145i	$0.266667\pi$
$733$	−0.309017	+	0.951057i	−0.309017	+	0.951057i	−0.978148	+	0.207912i	$0.933333\pi$
$734$	−2.48990	+	0.809017i	−2.48990	+	0.809017i
$735$		0			0
$736$		−	1.61803i		−	1.61803i
$737$	−0.951057	−	0.309017i	−0.951057	−	0.309017i
$738$		0			0
$739$	−0.587785	−	0.809017i	−0.587785	−	0.809017i	0.406737	−	0.913545i	$-0.366667\pi$
$739$	−0.587785	−	0.809017i	−0.587785	−	0.809017i	−0.994522	+	0.104528i	$0.966667\pi$
$740$	−1.53884	+	0.500000i	−1.53884	+	0.500000i
$741$		0			0
$742$		0			0
$743$	−0.587785	+	0.809017i	−0.587785	+	0.809017i	−0.994522	−	0.104528i	$-0.966667\pi$
$743$	−0.587785	+	0.809017i	−0.587785	+	0.809017i	0.406737	+	0.913545i	$0.366667\pi$
$744$		0			0
$745$	−0.190983	−	0.587785i	−0.190983	−	0.587785i
$746$	2.11803	−	1.53884i	2.11803	−	1.53884i
$747$		0			0
$748$	0.809017	+	2.48990i	0.809017	+	2.48990i
$749$	1.61803			1.61803
$750$		0			0
$751$	0.309017	+	0.951057i	0.309017	+	0.951057i	0.978148	+	0.207912i	$0.0666667\pi$
$751$	0.309017	+	0.951057i	0.309017	+	0.951057i	−0.669131	+	0.743145i	$0.733333\pi$
$752$		0			0
$753$		0			0
$754$	−0.809017	−	0.587785i	−0.809017	−	0.587785i
$755$		0			0
$756$		0			0
$757$	−0.587785	−	0.809017i	−0.587785	−	0.809017i	0.406737	−	0.913545i	$-0.366667\pi$
$757$	−0.587785	−	0.809017i	−0.587785	−	0.809017i	−0.994522	+	0.104528i	$0.966667\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	0.587785	+	0.809017i	0.587785	+	0.809017i	0.994522	−	0.104528i	$-0.0333333\pi$
$761$	0.587785	+	0.809017i	0.587785	+	0.809017i	−0.406737	+	0.913545i	$0.633333\pi$
$762$		0			0
$763$	0.500000	−	1.53884i	0.500000	−	1.53884i
$764$	−0.809017	−	0.587785i	−0.809017	−	0.587785i
$765$		0			0
$766$	−1.53884	−	0.500000i	−1.53884	−	0.500000i
$767$		0			0
$768$		0			0
$769$		0			0			−	1.00000i	$-0.5\pi$
$769$		0			0				1.00000i	$0.5\pi$
$770$	−0.951057	+	1.30902i	−0.951057	+	1.30902i
$771$		0			0
$772$		0			0
$773$	0.587785	−	0.190983i	0.587785	−	0.190983i		−	1.00000i	$-0.5\pi$
$773$	0.587785	−	0.190983i	0.587785	−	0.190983i	0.587785	+	0.809017i	$0.300000\pi$
$774$		0			0
$775$		0			0
$776$	0.809017	+	0.587785i	0.809017	+	0.587785i
$777$		0			0
$778$	0.951057	−	0.309017i	0.951057	−	0.309017i
$779$		0			0
$780$		0			0
$781$	−0.587785	−	0.809017i	−0.587785	−	0.809017i
$782$	4.23607			4.23607
$783$		0			0
$784$		0			0
$785$	−0.309017	+	0.951057i	−0.309017	+	0.951057i
$786$		0			0
$787$	−0.951057	+	1.30902i	−0.951057	+	1.30902i			1.00000i	$0.5\pi$
$787$	−0.951057	+	1.30902i	−0.951057	+	1.30902i	−0.951057	+	0.309017i	$0.900000\pi$
$788$	−1.53884	−	0.500000i	−1.53884	−	0.500000i
$789$		0			0
$790$	−0.951057	−	1.30902i	−0.951057	−	1.30902i
$791$	1.00000			1.00000
$792$		0			0
$793$	−1.00000			−1.00000
$794$		0			0
$795$		0			0
$796$	−2.48990	−	0.809017i	−2.48990	−	0.809017i
$797$	0.587785	−	0.809017i	0.587785	−	0.809017i	−0.406737	−	0.913545i	$-0.633333\pi$
$797$	0.587785	−	0.809017i	0.587785	−	0.809017i	0.994522	+	0.104528i	$0.0333333\pi$
$798$		0			0
$799$	2.48990	+	0.809017i	2.48990	+	0.809017i
$800$		0			0
$801$		0			0
$802$			1.00000i			1.00000i
$803$	0.809017	+	0.587785i	0.809017	+	0.587785i
$804$		0			0
$805$	0.951057	+	1.30902i	0.951057	+	1.30902i
$806$	−0.500000	−	1.53884i	−0.500000	−	1.53884i
$807$		0			0
$808$	−0.809017	−	0.587785i	−0.809017	−	0.587785i
$809$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$809$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$810$		0			0
$811$		0			0		0.951057	−	0.309017i	$-0.100000\pi$
$811$		0			0		−0.951057	+	0.309017i	$0.900000\pi$
$812$	0.587785	+	0.809017i	0.587785	+	0.809017i
$813$		0			0
$814$	−1.61803			−1.61803
$815$	−0.618034			−0.618034
$816$		0			0
$817$		0			0
$818$	−1.53884	−	0.500000i	−1.53884	−	0.500000i
$819$		0			0
$820$		0			0
$821$	0.951057	+	0.309017i	0.951057	+	0.309017i	0.743145	−	0.669131i	$-0.233333\pi$
$821$	0.951057	+	0.309017i	0.951057	+	0.309017i	0.207912	+	0.978148i	$0.433333\pi$
$822$		0			0
$823$	−0.363271	−	0.500000i	−0.363271	−	0.500000i	0.587785	−	0.809017i	$-0.300000\pi$
$823$	−0.363271	−	0.500000i	−0.363271	−	0.500000i	−0.951057	+	0.309017i	$0.900000\pi$
$824$	1.61803			1.61803
$825$		0			0
$826$		0			0
$827$	−0.587785	−	0.809017i	−0.587785	−	0.809017i	0.406737	−	0.913545i	$-0.366667\pi$
$827$	−0.587785	−	0.809017i	−0.587785	−	0.809017i	−0.994522	+	0.104528i	$0.966667\pi$
$828$		0			0
$829$	0.951057	+	0.309017i	0.951057	+	0.309017i	0.743145	−	0.669131i	$-0.233333\pi$
$829$	0.951057	+	0.309017i	0.951057	+	0.309017i	0.207912	+	0.978148i	$0.433333\pi$
$830$	−0.951057	+	1.30902i	−0.951057	+	1.30902i
$831$		0			0
$832$	−1.53884	−	0.500000i	−1.53884	−	0.500000i
$833$		0			0
$834$		0			0
$835$			1.61803i			1.61803i
$836$		0			0
$837$		0			0
$838$	−1.30902	+	0.951057i	−1.30902	+	0.951057i
$839$	−0.309017	−	0.951057i	−0.309017	−	0.951057i	−0.978148	−	0.207912i	$-0.933333\pi$
$839$	−0.309017	−	0.951057i	−0.309017	−	0.951057i	0.669131	−	0.743145i	$-0.266667\pi$
$840$		0			0
$841$	−0.500000	−	0.363271i	−0.500000	−	0.363271i
$842$	2.11803	+	1.53884i	2.11803	+	1.53884i
$843$		0			0
$844$		0			0
$845$		0			0
$846$		0			0
$847$	−0.809017	+	0.587785i	−0.809017	+	0.587785i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	−0.500000	+	1.53884i	−0.500000	+	1.53884i
$852$		0			0
$853$		0			0		0.587785	−	0.809017i	$-0.300000\pi$
$853$		0			0		−0.587785	+	0.809017i	$0.700000\pi$
$854$	1.53884	+	0.500000i	1.53884	+	0.500000i
$855$		0			0
$856$	−1.30902	+	0.951057i	−1.30902	+	0.951057i
$857$	1.61803			1.61803			0.809017	−	0.587785i	$-0.200000\pi$
$857$	1.61803			1.61803			0.809017	+	0.587785i	$0.200000\pi$
$858$		0			0
$859$			1.00000i			1.00000i	0.866025	+	0.500000i	$0.166667\pi$
$859$			1.00000i			1.00000i	−0.866025	+	0.500000i	$0.833333\pi$
$860$	0.587785	+	0.809017i	0.587785	+	0.809017i
$861$		0			0
$862$	0.500000	−	1.53884i	0.500000	−	1.53884i
$863$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$863$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$864$		0			0
$865$	−0.309017	+	0.951057i	−0.309017	+	0.951057i
$866$	2.48990	−	0.809017i	2.48990	−	0.809017i
$867$		0			0
$868$			1.61803i			1.61803i
$869$	−0.309017	−	0.951057i	−0.309017	−	0.951057i
$870$		0			0
$871$	0.587785	+	0.809017i	0.587785	+	0.809017i
$872$	0.500000	+	1.53884i	0.500000	+	1.53884i
$873$		0			0
$874$		0			0
$875$	0.809017	+	0.587785i	0.809017	+	0.587785i
$876$		0			0
$877$	−0.500000	−	1.53884i	−0.500000	−	1.53884i	−0.809017	−	0.587785i	$-0.800000\pi$
$877$	−0.500000	−	1.53884i	−0.500000	−	1.53884i	0.309017	−	0.951057i	$-0.400000\pi$
$878$	−0.809017	+	0.587785i	−0.809017	+	0.587785i
$879$		0			0
$880$		0			0
$881$		0			0		1.00000			$0$
$881$		0			0		−1.00000			$\pi$
$882$		0			0
$883$	−1.53884	+	0.500000i	−1.53884	+	0.500000i	−0.951057	−	0.309017i	$-0.900000\pi$
$883$	−1.53884	+	0.500000i	−1.53884	+	0.500000i	−0.587785	+	0.809017i	$0.700000\pi$
$884$	0.809017	−	2.48990i	0.809017	−	2.48990i
$885$		0			0
$886$	2.11803	+	1.53884i	2.11803	+	1.53884i
$887$	−0.309017	+	0.951057i	−0.309017	+	0.951057i	0.669131	+	0.743145i	$0.266667\pi$
$887$	−0.309017	+	0.951057i	−0.309017	+	0.951057i	−0.978148	+	0.207912i	$0.933333\pi$
$888$		0			0
$889$	−0.951057	−	1.30902i	−0.951057	−	1.30902i
$890$		−	1.61803i		−	1.61803i
$891$		0			0
$892$			1.00000i			1.00000i
$893$		0			0
$894$		0			0
$895$	0.951057	+	0.309017i	0.951057	+	0.309017i
$896$	1.30902	+	0.951057i	1.30902	+	0.951057i
$897$		0			0
$898$	0.951057	+	0.309017i	0.951057	+	0.309017i
$899$	0.190983	+	0.587785i	0.190983	+	0.587785i
$900$		0			0
$901$		0			0
$902$		0			0
$903$		0			0
$904$	−0.809017	+	0.587785i	−0.809017	+	0.587785i
$905$	0.951057	−	0.309017i	0.951057	−	0.309017i
$906$		0			0
$907$		0			0		0.587785	−	0.809017i	$-0.300000\pi$
$907$		0			0		−0.587785	+	0.809017i	$0.700000\pi$
$908$		0			0
$909$		0			0
$910$	1.53884	−	0.500000i	1.53884	−	0.500000i
$911$	−0.587785	−	0.809017i	−0.587785	−	0.809017i	0.406737	−	0.913545i	$-0.366667\pi$
$911$	−0.587785	−	0.809017i	−0.587785	−	0.809017i	−0.994522	+	0.104528i	$0.966667\pi$
$912$		0			0
$913$	−0.809017	+	0.587785i	−0.809017	+	0.587785i
$914$		−	1.61803i		−	1.61803i
$915$		0			0
$916$	1.53884	−	0.500000i	1.53884	−	0.500000i
$917$	0.309017	−	0.951057i	0.309017	−	0.951057i
$918$		0			0
$919$	−1.30902	−	0.951057i	−1.30902	−	0.951057i	−0.309017	−	0.951057i	$-0.600000\pi$
$919$	−1.30902	−	0.951057i	−1.30902	−	0.951057i	−1.00000			$\pi$
$920$	−1.53884	−	0.500000i	−1.53884	−	0.500000i
$921$		0			0
$922$	−1.53884	−	2.11803i	−1.53884	−	2.11803i
$923$			1.00000i			1.00000i
$924$		0			0
$925$		0			0
$926$	−1.53884	−	2.11803i	−1.53884	−	2.11803i
$927$		0			0
$928$	0.587785	+	0.190983i	0.587785	+	0.190983i
$929$	0.363271	−	0.500000i	0.363271	−	0.500000i	−0.587785	−	0.809017i	$-0.700000\pi$
$929$	0.363271	−	0.500000i	0.363271	−	0.500000i	0.951057	+	0.309017i	$0.100000\pi$
$930$		0			0
$931$		0			0
$932$		0			0
$933$		0			0
$934$	1.61803			1.61803
$935$	−1.61803			−1.61803
$936$		0			0
$937$	0.500000	−	0.363271i	0.500000	−	0.363271i	−0.309017	−	0.951057i	$-0.600000\pi$
$937$	0.500000	−	0.363271i	0.500000	−	0.363271i	0.809017	+	0.587785i	$0.200000\pi$
$938$	−0.500000	−	1.53884i	−0.500000	−	1.53884i
$939$		0			0
$940$	−2.11803	−	1.53884i	−2.11803	−	1.53884i
$941$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$941$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$		0			0
$946$	0.309017	+	0.951057i	0.309017	+	0.951057i
$947$		0			0		1.00000			$0$
$947$		0			0		−1.00000			$\pi$
$948$		0			0
$949$	−0.309017	−	0.951057i	−0.309017	−	0.951057i
$950$		0			0
$951$		0			0
$952$	−0.951057	+	1.30902i	−0.951057	+	1.30902i
$953$	1.53884	+	0.500000i	1.53884	+	0.500000i	0.951057	−	0.309017i	$-0.100000\pi$
$953$	1.53884	+	0.500000i	1.53884	+	0.500000i	0.587785	+	0.809017i	$0.300000\pi$
$954$		0			0
$955$	0.500000	−	0.363271i	0.500000	−	0.363271i
$956$			1.61803i			1.61803i
$957$		0			0
$958$	1.00000			1.00000
$959$		0			0
$960$		0			0
$961$	−0.309017	+	0.951057i	−0.309017	+	0.951057i
$962$	1.30902	+	0.951057i	1.30902	+	0.951057i
$963$		0			0
$964$	2.48990	+	0.809017i	2.48990	+	0.809017i
$965$		0			0
$966$		0			0
$967$		0			0		1.00000			$0$
$967$		0			0		−1.00000			$\pi$
$968$	0.309017	−	0.951057i	0.309017	−	0.951057i
$969$		0			0
$970$	−1.30902	+	0.951057i	−1.30902	+	0.951057i
$971$	0.190983	+	0.587785i	0.190983	+	0.587785i	1.00000			$0$
$971$	0.190983	+	0.587785i	0.190983	+	0.587785i	−0.809017	+	0.587785i	$0.800000\pi$
$972$		0			0
$973$		0			0
$974$		0			0
$975$		0			0
$976$		0			0
$977$	0.500000	−	0.363271i	0.500000	−	0.363271i	−0.309017	−	0.951057i	$-0.600000\pi$
$977$	0.500000	−	0.363271i	0.500000	−	0.363271i	0.809017	+	0.587785i	$0.200000\pi$
$978$		0			0
$979$	0.309017	−	0.951057i	0.309017	−	0.951057i
$980$		0			0
$981$		0			0
$982$	1.53884	−	0.500000i	1.53884	−	0.500000i
$983$	0.587785	+	0.190983i	0.587785	+	0.190983i	0.587785	−	0.809017i	$-0.300000\pi$
$983$	0.587785	+	0.190983i	0.587785	+	0.190983i			1.00000i	$0.5\pi$
$984$		0			0
$985$	0.587785	−	0.809017i	0.587785	−	0.809017i
$986$	−0.500000	+	1.53884i	−0.500000	+	1.53884i
$987$		0			0
$988$		0			0
$989$	1.00000			1.00000
$990$		0			0
$991$		0			0		1.00000			$0$
$991$		0			0		−1.00000			$\pi$
$992$	0.587785	+	0.809017i	0.587785	+	0.809017i
$993$		0			0
$994$	0.500000	−	1.53884i	0.500000	−	1.53884i
$995$	0.951057	−	1.30902i	0.951057	−	1.30902i
$996$		0			0
$997$	−0.190983	+	0.587785i	−0.190983	+	0.587785i	0.809017	+	0.587785i	$0.200000\pi$
$997$	−0.190983	+	0.587785i	−0.190983	+	0.587785i	−1.00000			$\pi$
$998$	−0.951057	+	0.309017i	−0.951057	+	0.309017i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3069.1.cd.b.2665.2	yes	8
3.2	odd	2		3069.1.cd.a.2665.1	yes	8
11.4	even	5	inner	3069.1.cd.b.433.1	yes	8
31.30	odd	2	inner	3069.1.cd.b.2665.1	yes	8
33.26	odd	10		3069.1.cd.a.433.2	yes	8
93.92	even	2		3069.1.cd.a.2665.2	yes	8
341.92	odd	10	inner	3069.1.cd.b.433.2	yes	8
1023.92	even	10		3069.1.cd.a.433.1	✓	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3069.1.cd.a.433.1	✓	8	1023.92	even	10
3069.1.cd.a.433.2	yes	8	33.26	odd	10
3069.1.cd.a.2665.1	yes	8	3.2	odd	2
3069.1.cd.a.2665.2	yes	8	93.92	even	2
3069.1.cd.b.433.1	yes	8	11.4	even	5	inner
3069.1.cd.b.433.2	yes	8	341.92	odd	10	inner
3069.1.cd.b.2665.1	yes	8	31.30	odd	2	inner
3069.1.cd.b.2665.2	yes	8	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 3069 = 3^{2} \cdot 11 \cdot 31 \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	3069.cd (of order \(10\), degree \(4\), minimal)

\(f(q)\)	\(=\)	\(q+(1.30902 - 0.951057i) q^{2} +(0.500000 - 1.53884i) q^{4} +(0.809017 + 0.587785i) q^{5} +(-0.309017 + 0.951057i) q^{7} +(-0.309017 - 0.951057i) q^{8} +O(q^{10})\)	\(q+(1.30902 - 0.951057i) q^{2} +(0.500000 - 1.53884i) q^{4} +(0.809017 + 0.587785i) q^{5} +(-0.309017 + 0.951057i) q^{7} +(-0.309017 - 0.951057i) q^{8} +1.61803 q^{10} +(0.951057 + 0.309017i) q^{11} +(-0.587785 - 0.809017i) q^{13} +(0.500000 + 1.53884i) q^{14} +(-0.951057 + 1.30902i) q^{17} +(1.30902 - 0.951057i) q^{20} +(1.53884 - 0.500000i) q^{22} -1.61803i q^{23} +(-1.53884 - 0.500000i) q^{26} +(1.30902 + 0.951057i) q^{28} +(0.587785 + 0.190983i) q^{29} +(0.587785 + 0.809017i) q^{31} +1.00000 q^{32} +2.61803i q^{34} +(-0.809017 + 0.587785i) q^{35} +(-0.951057 - 0.309017i) q^{37} +(0.309017 - 0.951057i) q^{40} +0.618034i q^{43} +(0.951057 - 1.30902i) q^{44} +(-1.53884 - 2.11803i) q^{46} +(-0.500000 - 1.53884i) q^{47} +(-1.53884 + 0.500000i) q^{52} +(0.587785 + 0.809017i) q^{55} +1.00000 q^{56} +(0.951057 - 0.309017i) q^{58} +(0.587785 - 0.809017i) q^{61} +(1.53884 + 0.500000i) q^{62} +(1.30902 - 0.951057i) q^{64} -1.00000i q^{65} -1.00000 q^{67} +(1.53884 + 2.11803i) q^{68} +(-0.500000 + 1.53884i) q^{70} +(-0.809017 - 0.587785i) q^{71} +(0.951057 + 0.309017i) q^{73} +(-1.53884 + 0.500000i) q^{74} +(-0.587785 + 0.809017i) q^{77} +(-0.587785 - 0.809017i) q^{79} +(-0.587785 + 0.809017i) q^{83} +(-1.53884 + 0.500000i) q^{85} +(0.587785 + 0.809017i) q^{86} -1.00000i q^{88} -1.00000i q^{89} +(0.951057 - 0.309017i) q^{91} +(-2.48990 - 0.809017i) q^{92} +(-2.11803 - 1.53884i) q^{94} +(-0.809017 + 0.587785i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 6 q^{2} + 4 q^{4} + 2 q^{5} + 2 q^{7} + 2 q^{8}+O(q^{10}) \) 8 * q + 6 * q^2 + 4 * q^4 + 2 * q^5 + 2 * q^7 + 2 * q^8	\( 8 q + 6 q^{2} + 4 q^{4} + 2 q^{5} + 2 q^{7} + 2 q^{8} + 4 q^{10} + 4 q^{14} + 6 q^{20} + 6 q^{28} + 8 q^{32} - 2 q^{35} - 2 q^{40} - 4 q^{47} + 8 q^{56} + 6 q^{64} - 8 q^{67} - 4 q^{70} - 2 q^{71} - 8 q^{94} - 2 q^{97}+O(q^{100}) \) 8 * q + 6 * q^2 + 4 * q^4 + 2 * q^5 + 2 * q^7 + 2 * q^8 + 4 * q^10 + 4 * q^14 + 6 * q^20 + 6 * q^28 + 8 * q^32 - 2 * q^35 - 2 * q^40 - 4 * q^47 + 8 * q^56 + 6 * q^64 - 8 * q^67 - 4 * q^70 - 2 * q^71 - 8 * q^94 - 2 * q^97

Introduction

L-functions

Modular forms

Varieties

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Representations

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Database

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