LMFDB - Embedded newform 1320.1.dk.a.413.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1320,1,Mod(173,1320)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1320, base_ring=CyclotomicField(20))

chi = DirichletCharacter(H, H._module([0, 10, 10, 15, 6]))

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

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

chi := DirichletCharacter("1320.173");

S:= CuspForms(chi, 1);

N := Newforms(S);

Level:	$ N $	$=$	$ 1320 = 2^{3} \cdot 3 \cdot 5 \cdot 11 $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	1320.dk (of order $20$, degree $8$, 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:	$0.658765816676$
Analytic rank:	$0$
Dimension:	$16$
Relative dimension:	$2$ over $\Q(\zeta_{20})$
Coefficient field:	$\Q(\zeta_{40})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} - x^{12} + x^{8} - x^{4} + 1 $ x^16 - x^12 + x^8 - x^4 + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Projective image:	$D_{20}$
Projective field:	Galois closure of $\mathbb{Q}[x]/(x^{20} - \cdots)$

Embedding invariants

Embedding label			413.1
Root			$0.156434 + 0.987688i$ of defining polynomial
Character	$\chi$	$=$	1320.413
Dual form			1320.1.dk.a.1157.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q+(-0.453990 + 0.891007i) q^{2} +(0.156434 + 0.987688i) q^{3} +(-0.587785 - 0.809017i) q^{4} +(-0.156434 + 0.987688i) q^{5} +(-0.951057 - 0.309017i) q^{6} +(0.896802 + 0.142040i) q^{7} +(0.987688 - 0.156434i) q^{8} +(-0.951057 + 0.309017i) q^{9} +(-0.809017 - 0.587785i) q^{10} +(-0.707107 + 0.707107i) q^{11} +(0.707107 - 0.707107i) q^{12} +(-0.533698 + 0.734572i) q^{14} -1.00000 q^{15} +(-0.309017 + 0.951057i) q^{16} +(0.156434 - 0.987688i) q^{18} +(0.891007 - 0.453990i) q^{20} +0.907981i q^{21} +(-0.309017 - 0.951057i) q^{22} +(0.309017 + 0.951057i) q^{24} +(-0.951057 - 0.309017i) q^{25} +(-0.453990 - 0.891007i) q^{27} +(-0.412215 - 0.809017i) q^{28} +(0.253116 - 0.183900i) q^{29} +(0.453990 - 0.891007i) q^{30} +(0.587785 + 1.80902i) q^{31} +(-0.707107 - 0.707107i) q^{32} +(-0.809017 - 0.587785i) q^{33} +(-0.280582 + 0.863541i) q^{35} +(0.809017 + 0.587785i) q^{36} +1.00000i q^{40} +(-0.809017 - 0.412215i) q^{42} +(0.987688 + 0.156434i) q^{44} +(-0.156434 - 0.987688i) q^{45} +(-0.987688 - 0.156434i) q^{48} +(-0.166977 - 0.0542543i) q^{49} +(0.707107 - 0.707107i) q^{50} +(0.280582 - 0.550672i) q^{53} +1.00000 q^{54} +(-0.587785 - 0.809017i) q^{55} +0.907981 q^{56} +(0.0489435 + 0.309017i) q^{58} +(-1.04744 - 1.44168i) q^{59} +(0.587785 + 0.809017i) q^{60} +(-1.87869 - 0.297556i) q^{62} +(-0.896802 + 0.142040i) q^{63} +(0.951057 - 0.309017i) q^{64} +(0.891007 - 0.453990i) q^{66} +(-0.642040 - 0.642040i) q^{70} +(-0.891007 + 0.453990i) q^{72} +(0.309017 - 1.95106i) q^{73} +(0.156434 - 0.987688i) q^{75} +(-0.734572 + 0.533698i) q^{77} +(0.500000 + 1.53884i) q^{79} +(-0.891007 - 0.453990i) q^{80} +(0.809017 - 0.587785i) q^{81} +(0.734572 + 1.44168i) q^{83} +(0.734572 - 0.533698i) q^{84} +(0.221232 + 0.221232i) q^{87} +(-0.587785 + 0.809017i) q^{88} +(0.951057 + 0.309017i) q^{90} +(-1.69480 + 0.863541i) q^{93} +(0.587785 - 0.809017i) q^{96} +(-1.76007 - 0.896802i) q^{97} +(0.124147 - 0.124147i) q^{98} +(0.453990 - 0.891007i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q - 4 q^{7} - 4 q^{10} - 16 q^{15} + 4 q^{16} + 4 q^{22} - 4 q^{24} - 16 q^{28} - 4 q^{33} + 4 q^{36} - 4 q^{42} + 16 q^{54} + 16 q^{58} + 4 q^{63} - 4 q^{70} - 4 q^{73} + 8 q^{79} + 4 q^{81} + 4 q^{87}+ \cdots - 4 q^{97}+O(q^{100}) $ 16 * q - 4 * q^7 - 4 * q^10 - 16 * q^15 + 4 * q^16 + 4 * q^22 - 4 * q^24 - 16 * q^28 - 4 * q^33 + 4 * q^36 - 4 * q^42 + 16 * q^54 + 16 * q^58 + 4 * q^63 - 4 * q^70 - 4 * q^73 + 8 * q^79 + 4 * q^81 + 4 * q^87 - 4 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1320\mathbb{Z}\right)^\times$.

$n$	$661$	$881$	$991$	$1057$	$1201$
$\chi(n)$	$-1$	$-1$	$1$	$e\left(\frac{3}{4}\right)$	$e\left(\frac{9}{10}\right)$

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 $

$2$	−0.453990	+	0.891007i	−0.453990	+	0.891007i
$2$	−0.453990	+	0.891007i	−0.453990	+	0.891007i
$3$	0.156434	+	0.987688i	0.156434	+	0.987688i
$3$	0.156434	+	0.987688i	0.156434	+	0.987688i
$4$	−0.587785	−	0.809017i	−0.587785	−	0.809017i
$5$	−0.156434	+	0.987688i	−0.156434	+	0.987688i
$5$	−0.156434	+	0.987688i	−0.156434	+	0.987688i
$6$	−0.951057	−	0.309017i	−0.951057	−	0.309017i
$7$	0.896802	+	0.142040i	0.896802	+	0.142040i	0.587785	−	0.809017i	$-0.300000\pi$
$7$	0.896802	+	0.142040i	0.896802	+	0.142040i	0.309017	+	0.951057i	$0.400000\pi$
$8$	0.987688	−	0.156434i	0.987688	−	0.156434i
$9$	−0.951057	+	0.309017i	−0.951057	+	0.309017i
$10$	−0.809017	−	0.587785i	−0.809017	−	0.587785i
$11$	−0.707107	+	0.707107i	−0.707107	+	0.707107i
$11$	−0.707107	+	0.707107i	−0.707107	+	0.707107i
$12$	0.707107	−	0.707107i	0.707107	−	0.707107i
$13$		0			0		−0.891007	−	0.453990i	$-0.850000\pi$
$13$		0			0		0.891007	+	0.453990i	$0.150000\pi$
$14$	−0.533698	+	0.734572i	−0.533698	+	0.734572i
$15$	−1.00000			−1.00000
$16$	−0.309017	+	0.951057i	−0.309017	+	0.951057i
$17$		0			0		0.891007	−	0.453990i	$-0.150000\pi$
$17$		0			0		−0.891007	+	0.453990i	$0.850000\pi$
$18$	0.156434	−	0.987688i	0.156434	−	0.987688i
$19$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$19$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$20$	0.891007	−	0.453990i	0.891007	−	0.453990i
$21$			0.907981i			0.907981i
$22$	−0.309017	−	0.951057i	−0.309017	−	0.951057i
$23$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$23$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$24$	0.309017	+	0.951057i	0.309017	+	0.951057i
$25$	−0.951057	−	0.309017i	−0.951057	−	0.309017i
$26$		0			0
$27$	−0.453990	−	0.891007i	−0.453990	−	0.891007i
$28$	−0.412215	−	0.809017i	−0.412215	−	0.809017i
$29$	0.253116	−	0.183900i	0.253116	−	0.183900i	−0.453990	−	0.891007i	$-0.650000\pi$
$29$	0.253116	−	0.183900i	0.253116	−	0.183900i	0.707107	+	0.707107i	$0.250000\pi$
$30$	0.453990	−	0.891007i	0.453990	−	0.891007i
$31$	0.587785	+	1.80902i	0.587785	+	1.80902i	0.587785	+	0.809017i	$0.300000\pi$
$31$	0.587785	+	1.80902i	0.587785	+	1.80902i			1.00000i	$0.5\pi$
$32$	−0.707107	−	0.707107i	−0.707107	−	0.707107i
$33$	−0.809017	−	0.587785i	−0.809017	−	0.587785i
$34$		0			0
$35$	−0.280582	+	0.863541i	−0.280582	+	0.863541i
$36$	0.809017	+	0.587785i	0.809017	+	0.587785i
$37$		0			0		0.156434	−	0.987688i	$-0.450000\pi$
$37$		0			0		−0.156434	+	0.987688i	$0.550000\pi$
$38$		0			0
$39$		0			0
$40$			1.00000i			1.00000i
$41$		0			0		0.587785	−	0.809017i	$-0.300000\pi$
$41$		0			0		−0.587785	+	0.809017i	$0.700000\pi$
$42$	−0.809017	−	0.412215i	−0.809017	−	0.412215i
$43$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$43$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$44$	0.987688	+	0.156434i	0.987688	+	0.156434i
$45$	−0.156434	−	0.987688i	−0.156434	−	0.987688i
$46$		0			0
$47$		0			0		0.987688	−	0.156434i	$-0.0500000\pi$
$47$		0			0		−0.987688	+	0.156434i	$0.950000\pi$
$48$	−0.987688	−	0.156434i	−0.987688	−	0.156434i
$49$	−0.166977	−	0.0542543i	−0.166977	−	0.0542543i
$50$	0.707107	−	0.707107i	0.707107	−	0.707107i
$51$		0			0
$52$		0			0
$53$	0.280582	−	0.550672i	0.280582	−	0.550672i	−0.707107	−	0.707107i	$-0.750000\pi$
$53$	0.280582	−	0.550672i	0.280582	−	0.550672i	0.987688	+	0.156434i	$0.0500000\pi$
$54$	1.00000			1.00000
$55$	−0.587785	−	0.809017i	−0.587785	−	0.809017i
$56$	0.907981			0.907981
$57$		0			0
$58$	0.0489435	+	0.309017i	0.0489435	+	0.309017i
$59$	−1.04744	−	1.44168i	−1.04744	−	1.44168i	−0.891007	−	0.453990i	$-0.850000\pi$
$59$	−1.04744	−	1.44168i	−1.04744	−	1.44168i	−0.156434	−	0.987688i	$-0.550000\pi$
$60$	0.587785	+	0.809017i	0.587785	+	0.809017i
$61$		0			0		−0.951057	−	0.309017i	$-0.900000\pi$
$61$		0			0		0.951057	+	0.309017i	$0.100000\pi$
$62$	−1.87869	−	0.297556i	−1.87869	−	0.297556i
$63$	−0.896802	+	0.142040i	−0.896802	+	0.142040i
$64$	0.951057	−	0.309017i	0.951057	−	0.309017i
$65$		0			0
$66$	0.891007	−	0.453990i	0.891007	−	0.453990i
$67$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$67$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$68$		0			0
$69$		0			0
$70$	−0.642040	−	0.642040i	−0.642040	−	0.642040i
$71$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$71$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$72$	−0.891007	+	0.453990i	−0.891007	+	0.453990i
$73$	0.309017	−	1.95106i	0.309017	−	1.95106i		−	1.00000i	$-0.5\pi$
$73$	0.309017	−	1.95106i	0.309017	−	1.95106i	0.309017	−	0.951057i	$-0.400000\pi$
$74$		0			0
$75$	0.156434	−	0.987688i	0.156434	−	0.987688i
$76$		0			0
$77$	−0.734572	+	0.533698i	−0.734572	+	0.533698i
$78$		0			0
$79$	0.500000	+	1.53884i	0.500000	+	1.53884i	0.809017	+	0.587785i	$0.200000\pi$
$79$	0.500000	+	1.53884i	0.500000	+	1.53884i	−0.309017	+	0.951057i	$0.600000\pi$
$80$	−0.891007	−	0.453990i	−0.891007	−	0.453990i
$81$	0.809017	−	0.587785i	0.809017	−	0.587785i
$82$		0			0
$83$	0.734572	+	1.44168i	0.734572	+	1.44168i	0.891007	+	0.453990i	$0.150000\pi$
$83$	0.734572	+	1.44168i	0.734572	+	1.44168i	−0.156434	+	0.987688i	$0.550000\pi$
$84$	0.734572	−	0.533698i	0.734572	−	0.533698i
$85$		0			0
$86$		0			0
$87$	0.221232	+	0.221232i	0.221232	+	0.221232i
$88$	−0.587785	+	0.809017i	−0.587785	+	0.809017i
$89$		0			0			−	1.00000i	$-0.5\pi$
$89$		0			0				1.00000i	$0.5\pi$
$90$	0.951057	+	0.309017i	0.951057	+	0.309017i
$91$		0			0
$92$		0			0
$93$	−1.69480	+	0.863541i	−1.69480	+	0.863541i
$94$		0			0
$95$		0			0
$96$	0.587785	−	0.809017i	0.587785	−	0.809017i
$97$	−1.76007	−	0.896802i	−1.76007	−	0.896802i	−0.951057	−	0.309017i	$-0.900000\pi$
$97$	−1.76007	−	0.896802i	−1.76007	−	0.896802i	−0.809017	−	0.587785i	$-0.800000\pi$
$98$	0.124147	−	0.124147i	0.124147	−	0.124147i
$99$	0.453990	−	0.891007i	0.453990	−	0.891007i
$100$	0.309017	+	0.951057i	0.309017	+	0.951057i
$101$	1.87869	−	0.610425i	1.87869	−	0.610425i	0.891007	−	0.453990i	$-0.150000\pi$
$101$	1.87869	−	0.610425i	1.87869	−	0.610425i	0.987688	−	0.156434i	$-0.0500000\pi$
$102$		0			0
$103$	1.76007	+	0.278768i	1.76007	+	0.278768i	0.951057	−	0.309017i	$-0.100000\pi$
$103$	1.76007	+	0.278768i	1.76007	+	0.278768i	0.809017	+	0.587785i	$0.200000\pi$
$104$		0			0
$105$	−0.896802	−	0.142040i	−0.896802	−	0.142040i
$106$	0.363271	+	0.500000i	0.363271	+	0.500000i
$107$	0.0966818	+	0.610425i	0.0966818	+	0.610425i	0.987688	+	0.156434i	$0.0500000\pi$
$107$	0.0966818	+	0.610425i	0.0966818	+	0.610425i	−0.891007	+	0.453990i	$0.850000\pi$
$108$	−0.453990	+	0.891007i	−0.453990	+	0.891007i
$109$		0			0		1.00000			$0$
$109$		0			0		−1.00000			$\pi$
$110$	0.987688	−	0.156434i	0.987688	−	0.156434i
$111$		0			0
$112$	−0.412215	+	0.809017i	−0.412215	+	0.809017i
$113$		0			0		−0.156434	−	0.987688i	$-0.550000\pi$
$113$		0			0		0.156434	+	0.987688i	$0.450000\pi$
$114$		0			0
$115$		0			0
$116$	−0.297556	−	0.0966818i	−0.297556	−	0.0966818i
$117$		0			0
$118$	1.76007	−	0.278768i	1.76007	−	0.278768i
$119$		0			0
$120$	−0.987688	+	0.156434i	−0.987688	+	0.156434i
$121$		−	1.00000i		−	1.00000i
$122$		0			0
$123$		0			0
$124$	1.11803	−	1.53884i	1.11803	−	1.53884i
$125$	0.453990	−	0.891007i	0.453990	−	0.891007i
$126$	0.280582	−	0.863541i	0.280582	−	0.863541i
$127$	1.26007	−	0.642040i	1.26007	−	0.642040i	0.309017	−	0.951057i	$-0.400000\pi$
$127$	1.26007	−	0.642040i	1.26007	−	0.642040i	0.951057	+	0.309017i	$0.100000\pi$
$128$	−0.156434	+	0.987688i	−0.156434	+	0.987688i
$129$		0			0
$130$		0			0
$131$			1.78201i			1.78201i	0.453990	+	0.891007i	$0.350000\pi$
$131$			1.78201i			1.78201i	−0.453990	+	0.891007i	$0.650000\pi$
$132$			1.00000i			1.00000i
$133$		0			0
$134$		0			0
$135$	0.951057	−	0.309017i	0.951057	−	0.309017i
$136$		0			0
$137$		0			0		−0.453990	−	0.891007i	$-0.650000\pi$
$137$		0			0		0.453990	+	0.891007i	$0.350000\pi$
$138$		0			0
$139$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$139$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$140$	0.863541	−	0.280582i	0.863541	−	0.280582i
$141$		0			0
$142$		0			0
$143$		0			0
$144$		−	1.00000i		−	1.00000i
$145$	0.142040	+	0.278768i	0.142040	+	0.278768i
$146$	1.59811	+	1.16110i	1.59811	+	1.16110i
$147$	0.0274653	−	0.173409i	0.0274653	−	0.173409i
$148$		0			0
$149$	−0.550672	+	1.69480i	−0.550672	+	1.69480i	0.156434	+	0.987688i	$0.450000\pi$
$149$	−0.550672	+	1.69480i	−0.550672	+	1.69480i	−0.707107	+	0.707107i	$0.750000\pi$
$150$	0.809017	+	0.587785i	0.809017	+	0.587785i
$151$	−0.690983	+	0.951057i	−0.690983	+	0.951057i	0.309017	+	0.951057i	$0.400000\pi$
$151$	−0.690983	+	0.951057i	−0.690983	+	0.951057i	−1.00000			$1.00000\pi$
$152$		0			0
$153$		0			0
$154$	−0.142040	−	0.896802i	−0.142040	−	0.896802i
$155$	−1.87869	+	0.297556i	−1.87869	+	0.297556i
$156$		0			0
$157$		0			0		0.987688	−	0.156434i	$-0.0500000\pi$
$157$		0			0		−0.987688	+	0.156434i	$0.950000\pi$
$158$	−1.59811	−	0.253116i	−1.59811	−	0.253116i
$159$	0.587785	+	0.190983i	0.587785	+	0.190983i
$160$	0.809017	−	0.587785i	0.809017	−	0.587785i
$161$		0			0
$162$	0.156434	+	0.987688i	0.156434	+	0.987688i
$163$		0			0		0.453990	−	0.891007i	$-0.350000\pi$
$163$		0			0		−0.453990	+	0.891007i	$0.650000\pi$
$164$		0			0
$165$	0.707107	−	0.707107i	0.707107	−	0.707107i
$166$	−1.61803			−1.61803
$167$		0			0		0.453990	−	0.891007i	$-0.350000\pi$
$167$		0			0		−0.453990	+	0.891007i	$0.650000\pi$
$168$	0.142040	+	0.896802i	0.142040	+	0.896802i
$169$	0.587785	+	0.809017i	0.587785	+	0.809017i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	0.610425	−	0.0966818i	0.610425	−	0.0966818i	0.156434	−	0.987688i	$-0.450000\pi$
$173$	0.610425	−	0.0966818i	0.610425	−	0.0966818i	0.453990	+	0.891007i	$0.350000\pi$
$174$	−0.297556	+	0.0966818i	−0.297556	+	0.0966818i
$175$	−0.809017	−	0.412215i	−0.809017	−	0.412215i
$176$	−0.453990	−	0.891007i	−0.453990	−	0.891007i
$177$	1.26007	−	1.26007i	1.26007	−	1.26007i
$178$		0			0
$179$	0.183900	−	0.253116i	0.183900	−	0.253116i	−0.707107	−	0.707107i	$-0.750000\pi$
$179$	0.183900	−	0.253116i	0.183900	−	0.253116i	0.891007	+	0.453990i	$0.150000\pi$
$180$	−0.707107	+	0.707107i	−0.707107	+	0.707107i
$181$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$181$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$		0			0
$186$		−	1.90211i		−	1.90211i
$187$		0			0
$188$		0			0
$189$	−0.280582	−	0.863541i	−0.280582	−	0.863541i
$190$		0			0
$191$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$191$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$192$	0.453990	+	0.891007i	0.453990	+	0.891007i
$193$	0.809017	+	1.58779i	0.809017	+	1.58779i	0.809017	+	0.587785i	$0.200000\pi$
$193$	0.809017	+	1.58779i	0.809017	+	1.58779i			1.00000i	$0.500000\pi$
$194$	1.59811	−	1.16110i	1.59811	−	1.16110i
$195$		0			0
$196$	0.0542543	+	0.166977i	0.0542543	+	0.166977i
$197$	−0.831254	−	0.831254i	−0.831254	−	0.831254i	0.156434	−	0.987688i	$-0.450000\pi$
$197$	−0.831254	−	0.831254i	−0.831254	−	0.831254i	−0.987688	+	0.156434i	$0.950000\pi$
$198$	0.587785	+	0.809017i	0.587785	+	0.809017i
$199$			1.61803i			1.61803i	0.587785	+	0.809017i	$0.300000\pi$
$199$			1.61803i			1.61803i	−0.587785	+	0.809017i	$0.700000\pi$
$200$	−0.987688	−	0.156434i	−0.987688	−	0.156434i
$201$		0			0
$202$	−0.309017	+	1.95106i	−0.309017	+	1.95106i
$203$	0.253116	−	0.128969i	0.253116	−	0.128969i
$204$		0			0
$205$		0			0
$206$	−1.04744	+	1.44168i	−1.04744	+	1.44168i
$207$		0			0
$208$		0			0
$209$		0			0
$210$	0.533698	−	0.734572i	0.533698	−	0.734572i
$211$		0			0		0.951057	−	0.309017i	$-0.100000\pi$
$211$		0			0		−0.951057	+	0.309017i	$0.900000\pi$
$212$	−0.610425	+	0.0966818i	−0.610425	+	0.0966818i
$213$		0			0
$214$	−0.587785	−	0.190983i	−0.587785	−	0.190983i
$215$		0			0
$216$	−0.587785	−	0.809017i	−0.587785	−	0.809017i
$217$	0.270175	+	1.70582i	0.270175	+	1.70582i
$218$		0			0
$219$	1.97538			1.97538
$220$	−0.309017	+	0.951057i	−0.309017	+	0.951057i
$221$		0			0
$222$		0			0
$223$	−0.309017	−	1.95106i	−0.309017	−	1.95106i	−0.309017	−	0.951057i	$-0.600000\pi$
$223$	−0.309017	−	1.95106i	−0.309017	−	1.95106i		−	1.00000i	$-0.5\pi$
$224$	−0.533698	−	0.734572i	−0.533698	−	0.734572i
$225$	1.00000			1.00000
$226$		0			0
$227$	1.16110	+	0.183900i	1.16110	+	0.183900i	0.707107	−	0.707107i	$-0.250000\pi$
$227$	1.16110	+	0.183900i	1.16110	+	0.183900i	0.453990	+	0.891007i	$0.350000\pi$
$228$		0			0
$229$		0			0		0.951057	−	0.309017i	$-0.100000\pi$
$229$		0			0		−0.951057	+	0.309017i	$0.900000\pi$
$230$		0			0
$231$	−0.642040	−	0.642040i	−0.642040	−	0.642040i
$232$	0.221232	−	0.221232i	0.221232	−	0.221232i
$233$		0			0		−0.891007	−	0.453990i	$-0.850000\pi$
$233$		0			0		0.891007	+	0.453990i	$0.150000\pi$
$234$		0			0
$235$		0			0
$236$	−0.550672	+	1.69480i	−0.550672	+	1.69480i
$237$	−1.44168	+	0.734572i	−1.44168	+	0.734572i
$238$		0			0
$239$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$239$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$240$	0.309017	−	0.951057i	0.309017	−	0.951057i
$241$			0.618034i			0.618034i	0.951057	+	0.309017i	$0.100000\pi$
$241$			0.618034i			0.618034i	−0.951057	+	0.309017i	$0.900000\pi$
$242$	0.891007	+	0.453990i	0.891007	+	0.453990i
$243$	0.707107	+	0.707107i	0.707107	+	0.707107i
$244$		0			0
$245$	0.0797073	−	0.156434i	0.0797073	−	0.156434i
$246$		0			0
$247$		0			0
$248$	0.863541	+	1.69480i	0.863541	+	1.69480i
$249$	−1.30902	+	0.951057i	−1.30902	+	0.951057i
$250$	0.587785	+	0.809017i	0.587785	+	0.809017i
$251$	−0.610425	−	1.87869i	−0.610425	−	1.87869i	−0.453990	−	0.891007i	$-0.650000\pi$
$251$	−0.610425	−	1.87869i	−0.610425	−	1.87869i	−0.156434	−	0.987688i	$-0.550000\pi$
$252$	0.642040	+	0.642040i	0.642040	+	0.642040i
$253$		0			0
$254$			1.41421i			1.41421i
$255$		0			0
$256$	−0.809017	−	0.587785i	−0.809017	−	0.587785i
$257$		0			0		0.156434	−	0.987688i	$-0.450000\pi$
$257$		0			0		−0.156434	+	0.987688i	$0.550000\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$	−0.183900	+	0.253116i	−0.183900	+	0.253116i
$262$	−1.58779	−	0.809017i	−1.58779	−	0.809017i
$263$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$263$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$264$	−0.891007	−	0.453990i	−0.891007	−	0.453990i
$265$	0.500000	+	0.363271i	0.500000	+	0.363271i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	1.34500	+	0.437016i	1.34500	+	0.437016i	0.891007	−	0.453990i	$-0.150000\pi$
$269$	1.34500	+	0.437016i	1.34500	+	0.437016i	0.453990	+	0.891007i	$0.350000\pi$
$270$	−0.156434	+	0.987688i	−0.156434	+	0.987688i
$271$		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
$274$		0			0
$275$	0.891007	−	0.453990i	0.891007	−	0.453990i
$276$		0			0
$277$		0			0		0.453990	−	0.891007i	$-0.350000\pi$
$277$		0			0		−0.453990	+	0.891007i	$0.650000\pi$
$278$		0			0
$279$	−1.11803	−	1.53884i	−1.11803	−	1.53884i
$280$	−0.142040	+	0.896802i	−0.142040	+	0.896802i
$281$		0			0		−0.951057	−	0.309017i	$-0.900000\pi$
$281$		0			0		0.951057	+	0.309017i	$0.100000\pi$
$282$		0			0
$283$		0			0		0.987688	−	0.156434i	$-0.0500000\pi$
$283$		0			0		−0.987688	+	0.156434i	$0.950000\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$	0.891007	+	0.453990i	0.891007	+	0.453990i
$289$	0.587785	−	0.809017i	0.587785	−	0.809017i
$290$	−0.312869			−0.312869
$291$	0.610425	−	1.87869i	0.610425	−	1.87869i
$292$	−1.76007	+	0.896802i	−1.76007	+	0.896802i
$293$	0.297556	−	1.87869i	0.297556	−	1.87869i	−0.156434	−	0.987688i	$-0.550000\pi$
$293$	0.297556	−	1.87869i	0.297556	−	1.87869i	0.453990	−	0.891007i	$-0.350000\pi$
$294$	0.142040	+	0.103198i	0.142040	+	0.103198i
$295$	1.58779	−	0.809017i	1.58779	−	0.809017i
$296$		0			0
$297$	0.951057	+	0.309017i	0.951057	+	0.309017i
$298$	−1.26007	−	1.26007i	−1.26007	−	1.26007i
$299$		0			0
$300$	−0.891007	+	0.453990i	−0.891007	+	0.453990i
$301$		0			0
$302$	−0.533698	−	1.04744i	−0.533698	−	1.04744i
$303$	0.896802	+	1.76007i	0.896802	+	1.76007i
$304$		0			0
$305$		0			0
$306$		0			0
$307$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$307$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$308$	0.863541	+	0.280582i	0.863541	+	0.280582i
$309$			1.78201i			1.78201i
$310$	0.587785	−	1.80902i	0.587785	−	1.80902i
$311$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$311$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$312$		0			0
$313$	−0.278768	+	0.142040i	−0.278768	+	0.142040i	−0.587785	−	0.809017i	$-0.700000\pi$
$313$	−0.278768	+	0.142040i	−0.278768	+	0.142040i	0.309017	+	0.951057i	$0.400000\pi$
$314$		0			0
$315$		−	0.907981i		−	0.907981i
$316$	0.951057	−	1.30902i	0.951057	−	1.30902i
$317$		0			0		0.453990	−	0.891007i	$-0.350000\pi$
$317$		0			0		−0.453990	+	0.891007i	$0.650000\pi$
$318$	−0.437016	+	0.437016i	−0.437016	+	0.437016i
$319$	−0.0489435	+	0.309017i	−0.0489435	+	0.309017i
$320$	0.156434	+	0.987688i	0.156434	+	0.987688i
$321$	−0.587785	+	0.190983i	−0.587785	+	0.190983i
$322$		0			0
$323$		0			0
$324$	−0.951057	−	0.309017i	−0.951057	−	0.309017i
$325$		0			0
$326$		0			0
$327$		0			0
$328$		0			0
$329$		0			0
$330$	0.309017	+	0.951057i	0.309017	+	0.951057i
$331$		0			0		1.00000			$0$
$331$		0			0		−1.00000			$\pi$
$332$	0.734572	−	1.44168i	0.734572	−	1.44168i
$333$		0			0
$334$		0			0
$335$		0			0
$336$	−0.863541	−	0.280582i	−0.863541	−	0.280582i
$337$	1.39680	+	0.221232i	1.39680	+	0.221232i	0.809017	−	0.587785i	$-0.200000\pi$
$337$	1.39680	+	0.221232i	1.39680	+	0.221232i	0.587785	+	0.809017i	$0.300000\pi$
$338$	−0.987688	+	0.156434i	−0.987688	+	0.156434i
$339$		0			0
$340$		0			0
$341$	−1.69480	−	0.863541i	−1.69480	−	0.863541i
$342$		0			0
$343$	−0.951057	−	0.484587i	−0.951057	−	0.484587i
$344$		0			0
$345$		0			0
$346$	−0.190983	+	0.587785i	−0.190983	+	0.587785i
$347$	1.69480	−	0.863541i	1.69480	−	0.863541i	0.707107	−	0.707107i	$-0.250000\pi$
$347$	1.69480	−	0.863541i	1.69480	−	0.863541i	0.987688	−	0.156434i	$-0.0500000\pi$
$348$	0.0489435	−	0.309017i	0.0489435	−	0.309017i
$349$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$349$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$350$	0.734572	−	0.533698i	0.734572	−	0.533698i
$351$		0			0
$352$	1.00000			1.00000
$353$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$353$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$354$	0.550672	+	1.69480i	0.550672	+	1.69480i
$355$		0			0
$356$		0			0
$357$		0			0
$358$	0.142040	+	0.278768i	0.142040	+	0.278768i
$359$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$359$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$360$	−0.309017	−	0.951057i	−0.309017	−	0.951057i
$361$	0.309017	+	0.951057i	0.309017	+	0.951057i
$362$		0			0
$363$	0.987688	−	0.156434i	0.987688	−	0.156434i
$364$		0			0
$365$	1.87869	+	0.610425i	1.87869	+	0.610425i
$366$		0			0
$367$	0.0489435	−	0.309017i	0.0489435	−	0.309017i	−0.951057	−	0.309017i	$-0.900000\pi$
$367$	0.0489435	−	0.309017i	0.0489435	−	0.309017i	1.00000			$0$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	0.329843	−	0.453990i	0.329843	−	0.453990i
$372$	1.69480	+	0.863541i	1.69480	+	0.863541i
$373$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$373$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$374$		0			0
$375$	0.951057	+	0.309017i	0.951057	+	0.309017i
$376$		0			0
$377$		0			0
$378$	0.896802	+	0.142040i	0.896802	+	0.142040i
$379$		0			0		−0.951057	−	0.309017i	$-0.900000\pi$
$379$		0			0		0.951057	+	0.309017i	$0.100000\pi$
$380$		0			0
$381$	0.831254	+	1.14412i	0.831254	+	1.14412i
$382$		0			0
$383$		0			0		0.453990	−	0.891007i	$-0.350000\pi$
$383$		0			0		−0.453990	+	0.891007i	$0.650000\pi$
$384$	−1.00000			−1.00000
$385$	−0.412215	−	0.809017i	−0.412215	−	0.809017i
$386$	−1.78201			−1.78201
$387$		0			0
$388$	0.309017	+	1.95106i	0.309017	+	1.95106i
$389$	−0.831254	−	1.14412i	−0.831254	−	1.14412i	−0.987688	−	0.156434i	$-0.950000\pi$
$389$	−0.831254	−	1.14412i	−0.831254	−	1.14412i	0.156434	−	0.987688i	$-0.450000\pi$
$390$		0			0
$391$		0			0
$392$	−0.173409	−	0.0274653i	−0.173409	−	0.0274653i
$393$	−1.76007	+	0.278768i	−1.76007	+	0.278768i
$394$	1.11803	−	0.363271i	1.11803	−	0.363271i
$395$	−1.59811	+	0.253116i	−1.59811	+	0.253116i
$396$	−0.987688	+	0.156434i	−0.987688	+	0.156434i
$397$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$397$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$398$	−1.44168	−	0.734572i	−1.44168	−	0.734572i
$399$		0			0
$400$	0.587785	−	0.809017i	0.587785	−	0.809017i
$401$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$401$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$402$		0			0
$403$		0			0
$404$	−1.59811	−	1.16110i	−1.59811	−	1.16110i
$405$	0.453990	+	0.891007i	0.453990	+	0.891007i
$406$			0.284079i			0.284079i
$407$		0			0
$408$		0			0
$409$	−0.500000	−	1.53884i	−0.500000	−	1.53884i	−0.809017	−	0.587785i	$-0.800000\pi$
$409$	−0.500000	−	1.53884i	−0.500000	−	1.53884i	0.309017	−	0.951057i	$-0.400000\pi$
$410$		0			0
$411$		0			0
$412$	−0.809017	−	1.58779i	−0.809017	−	1.58779i
$413$	−0.734572	−	1.44168i	−0.734572	−	1.44168i
$414$		0			0
$415$	−1.53884	+	0.500000i	−1.53884	+	0.500000i
$416$		0			0
$417$		0			0
$418$		0			0
$419$			0.312869i			0.312869i	0.987688	+	0.156434i	$0.0500000\pi$
$419$			0.312869i			0.312869i	−0.987688	+	0.156434i	$0.950000\pi$
$420$	0.412215	+	0.809017i	0.412215	+	0.809017i
$421$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$421$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$422$		0			0
$423$		0			0
$424$	0.190983	−	0.587785i	0.190983	−	0.587785i
$425$		0			0
$426$		0			0
$427$		0			0
$428$	0.437016	−	0.437016i	0.437016	−	0.437016i
$429$		0			0
$430$		0			0
$431$		0			0		0.951057	−	0.309017i	$-0.100000\pi$
$431$		0			0		−0.951057	+	0.309017i	$0.900000\pi$
$432$	0.987688	−	0.156434i	0.987688	−	0.156434i
$433$	−0.896802	−	0.142040i	−0.896802	−	0.142040i	−0.309017	−	0.951057i	$-0.600000\pi$
$433$	−0.896802	−	0.142040i	−0.896802	−	0.142040i	−0.587785	+	0.809017i	$0.700000\pi$
$434$	−1.64255	−	0.533698i	−1.64255	−	0.533698i
$435$	−0.253116	+	0.183900i	−0.253116	+	0.183900i
$436$		0			0
$437$		0			0
$438$	−0.896802	+	1.76007i	−0.896802	+	1.76007i
$439$	−1.90211			−1.90211			−0.951057	−	0.309017i	$-0.900000\pi$
$439$	−1.90211			−1.90211			−0.951057	+	0.309017i	$0.900000\pi$
$440$	−0.707107	−	0.707107i	−0.707107	−	0.707107i
$441$	0.175571			0.175571
$442$		0			0
$443$	−0.0966818	−	0.610425i	−0.0966818	−	0.610425i	−0.987688	−	0.156434i	$-0.950000\pi$
$443$	−0.0966818	−	0.610425i	−0.0966818	−	0.610425i	0.891007	−	0.453990i	$-0.150000\pi$
$444$		0			0
$445$		0			0
$446$	1.87869	+	0.610425i	1.87869	+	0.610425i
$447$	−1.76007	−	0.278768i	−1.76007	−	0.278768i
$448$	0.896802	−	0.142040i	0.896802	−	0.142040i
$449$		0			0		0.951057	−	0.309017i	$-0.100000\pi$
$449$		0			0		−0.951057	+	0.309017i	$0.900000\pi$
$450$	−0.453990	+	0.891007i	−0.453990	+	0.891007i
$451$		0			0
$452$		0			0
$453$	−1.04744	−	0.533698i	−1.04744	−	0.533698i
$454$	−0.690983	+	0.951057i	−0.690983	+	0.951057i
$455$		0			0
$456$		0			0
$457$	−0.278768	+	0.142040i	−0.278768	+	0.142040i	−0.587785	−	0.809017i	$-0.700000\pi$
$457$	−0.278768	+	0.142040i	−0.278768	+	0.142040i	0.309017	+	0.951057i	$0.400000\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$		−	1.41421i		−	1.41421i	−0.707107	−	0.707107i	$-0.750000\pi$
$461$		−	1.41421i		−	1.41421i	0.707107	−	0.707107i	$-0.250000\pi$
$462$	0.863541	−	0.280582i	0.863541	−	0.280582i
$463$	−0.642040	−	0.642040i	−0.642040	−	0.642040i	0.309017	−	0.951057i	$-0.400000\pi$
$463$	−0.642040	−	0.642040i	−0.642040	−	0.642040i	−0.951057	+	0.309017i	$0.900000\pi$
$464$	0.0966818	+	0.297556i	0.0966818	+	0.297556i
$465$	−0.587785	−	1.80902i	−0.587785	−	1.80902i
$466$		0			0
$467$	−0.863541	−	1.69480i	−0.863541	−	1.69480i	−0.707107	−	0.707107i	$-0.750000\pi$
$467$	−0.863541	−	1.69480i	−0.863541	−	1.69480i	−0.156434	−	0.987688i	$-0.550000\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$		0			0
$472$	−1.26007	−	1.26007i	−1.26007	−	1.26007i
$473$		0			0
$474$		−	1.61803i		−	1.61803i
$475$		0			0
$476$		0			0
$477$	−0.0966818	+	0.610425i	−0.0966818	+	0.610425i
$478$		0			0
$479$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$479$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$480$	0.707107	+	0.707107i	0.707107	+	0.707107i
$481$		0			0
$482$	−0.550672	−	0.280582i	−0.550672	−	0.280582i
$483$		0			0
$484$	−0.809017	+	0.587785i	−0.809017	+	0.587785i
$485$	1.16110	−	1.59811i	1.16110	−	1.59811i
$486$	−0.951057	+	0.309017i	−0.951057	+	0.309017i
$487$	−0.896802	+	0.142040i	−0.896802	+	0.142040i	−0.587785	−	0.809017i	$-0.700000\pi$
$487$	−0.896802	+	0.142040i	−0.896802	+	0.142040i	−0.309017	+	0.951057i	$0.600000\pi$
$488$		0			0
$489$		0			0
$490$	0.103198	+	0.142040i	0.103198	+	0.142040i
$491$	−0.831254	−	1.14412i	−0.831254	−	1.14412i	−0.987688	−	0.156434i	$-0.950000\pi$
$491$	−0.831254	−	1.14412i	−0.831254	−	1.14412i	0.156434	−	0.987688i	$-0.450000\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$	0.809017	+	0.587785i	0.809017	+	0.587785i
$496$	−1.90211			−1.90211
$497$		0			0
$498$	−0.253116	−	1.59811i	−0.253116	−	1.59811i
$499$		0			0		−0.587785	−	0.809017i	$-0.700000\pi$
$499$		0			0		0.587785	+	0.809017i	$0.300000\pi$
$500$	−0.987688	+	0.156434i	−0.987688	+	0.156434i
$501$		0			0
$502$	1.95106	+	0.309017i	1.95106	+	0.309017i
$503$		0			0		0.987688	−	0.156434i	$-0.0500000\pi$
$503$		0			0		−0.987688	+	0.156434i	$0.950000\pi$
$504$	−0.863541	+	0.280582i	−0.863541	+	0.280582i
$505$	0.309017	+	1.95106i	0.309017	+	1.95106i
$506$		0			0
$507$	−0.707107	+	0.707107i	−0.707107	+	0.707107i
$508$	−1.26007	−	0.642040i	−1.26007	−	0.642040i
$509$	−1.04744	+	1.44168i	−1.04744	+	1.44168i	−0.156434	+	0.987688i	$0.550000\pi$
$509$	−1.04744	+	1.44168i	−1.04744	+	1.44168i	−0.891007	+	0.453990i	$0.850000\pi$
$510$		0			0
$511$	0.554254	−	1.70582i	0.554254	−	1.70582i
$512$	0.891007	−	0.453990i	0.891007	−	0.453990i
$513$		0			0
$514$		0			0
$515$	−0.550672	+	1.69480i	−0.550672	+	1.69480i
$516$		0			0
$517$		0			0
$518$		0			0
$519$	0.190983	+	0.587785i	0.190983	+	0.587785i
$520$		0			0
$521$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$521$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$522$	−0.142040	−	0.278768i	−0.142040	−	0.278768i
$523$		0			0		−0.453990	−	0.891007i	$-0.650000\pi$
$523$		0			0		0.453990	+	0.891007i	$0.350000\pi$
$524$	1.44168	−	1.04744i	1.44168	−	1.04744i
$525$	0.280582	−	0.863541i	0.280582	−	0.863541i
$526$		0			0
$527$		0			0
$528$	0.809017	−	0.587785i	0.809017	−	0.587785i
$529$			1.00000i			1.00000i
$530$	−0.550672	+	0.280582i	−0.550672	+	0.280582i
$531$	1.44168	+	1.04744i	1.44168	+	1.04744i
$532$		0			0
$533$		0			0
$534$		0			0
$535$	−0.618034			−0.618034
$536$		0			0
$537$	0.278768	+	0.142040i	0.278768	+	0.142040i
$538$	−1.00000	+	1.00000i	−1.00000	+	1.00000i
$539$	0.156434	−	0.0797073i	0.156434	−	0.0797073i
$540$	−0.809017	−	0.587785i	−0.809017	−	0.587785i
$541$		0			0		0.951057	−	0.309017i	$-0.100000\pi$
$541$		0			0		−0.951057	+	0.309017i	$0.900000\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$		0			0
$546$		0			0
$547$		0			0		−0.156434	−	0.987688i	$-0.550000\pi$
$547$		0			0		0.156434	+	0.987688i	$0.450000\pi$
$548$		0			0
$549$		0			0
$550$			1.00000i			1.00000i
$551$		0			0
$552$		0			0
$553$	0.229825	+	1.45106i	0.229825	+	1.45106i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	1.87869	+	0.297556i	1.87869	+	0.297556i	0.987688	−	0.156434i	$-0.0500000\pi$
$557$	1.87869	+	0.297556i	1.87869	+	0.297556i	0.891007	+	0.453990i	$0.150000\pi$
$558$	1.87869	−	0.297556i	1.87869	−	0.297556i
$559$		0			0
$560$	−0.734572	−	0.533698i	−0.734572	−	0.533698i
$561$		0			0
$562$		0			0
$563$		0			0		0.453990	−	0.891007i	$-0.350000\pi$
$563$		0			0		−0.453990	+	0.891007i	$0.650000\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$	0.809017	−	0.412215i	0.809017	−	0.412215i
$568$		0			0
$569$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$569$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$570$		0			0
$571$		0			0			−	1.00000i	$-0.5\pi$
$571$		0			0				1.00000i	$0.5\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$	−0.809017	+	0.587785i	−0.809017	+	0.587785i
$577$	−0.412215	−	0.809017i	−0.412215	−	0.809017i	0.587785	−	0.809017i	$-0.300000\pi$
$577$	−0.412215	−	0.809017i	−0.412215	−	0.809017i	−1.00000			$\pi$
$578$	0.453990	+	0.891007i	0.453990	+	0.891007i
$579$	−1.44168	+	1.04744i	−1.44168	+	1.04744i
$580$	0.142040	−	0.278768i	0.142040	−	0.278768i
$581$	0.453990	+	1.39724i	0.453990	+	1.39724i
$582$	1.39680	+	1.39680i	1.39680	+	1.39680i
$583$	0.190983	+	0.587785i	0.190983	+	0.587785i
$584$		−	1.97538i		−	1.97538i
$585$		0			0
$586$	1.53884	+	1.11803i	1.53884	+	1.11803i
$587$	−0.183900	+	1.16110i	−0.183900	+	1.16110i	0.707107	+	0.707107i	$0.250000\pi$
$587$	−0.183900	+	1.16110i	−0.183900	+	1.16110i	−0.891007	+	0.453990i	$0.850000\pi$
$588$	−0.156434	+	0.0797073i	−0.156434	+	0.0797073i
$589$		0			0
$590$			1.78201i			1.78201i
$591$	0.690983	−	0.951057i	0.690983	−	0.951057i
$592$		0			0
$593$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$593$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$594$	−0.707107	+	0.707107i	−0.707107	+	0.707107i
$595$		0			0
$596$	1.69480	−	0.550672i	1.69480	−	0.550672i
$597$	−1.59811	+	0.253116i	−1.59811	+	0.253116i
$598$		0			0
$599$		0			0		−0.951057	−	0.309017i	$-0.900000\pi$
$599$		0			0		0.951057	+	0.309017i	$0.100000\pi$
$600$		−	1.00000i		−	1.00000i
$601$	−1.11803	−	1.53884i	−1.11803	−	1.53884i	−0.809017	−	0.587785i	$-0.800000\pi$
$601$	−1.11803	−	1.53884i	−1.11803	−	1.53884i	−0.309017	−	0.951057i	$-0.600000\pi$
$602$		0			0
$603$		0			0
$604$	1.17557			1.17557
$605$	0.987688	+	0.156434i	0.987688	+	0.156434i
$606$	−1.97538			−1.97538
$607$	0.642040	−	1.26007i	0.642040	−	1.26007i	−0.309017	−	0.951057i	$-0.600000\pi$
$607$	0.642040	−	1.26007i	0.642040	−	1.26007i	0.951057	−	0.309017i	$-0.100000\pi$
$608$		0			0
$609$	0.166977	+	0.229825i	0.166977	+	0.229825i
$610$		0			0
$611$		0			0
$612$		0			0
$613$		0			0		0.987688	−	0.156434i	$-0.0500000\pi$
$613$		0			0		−0.987688	+	0.156434i	$0.950000\pi$
$614$		0			0
$615$		0			0
$616$	−0.642040	+	0.642040i	−0.642040	+	0.642040i
$617$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$617$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$618$	−1.58779	−	0.809017i	−1.58779	−	0.809017i
$619$		0			0		0.587785	−	0.809017i	$-0.300000\pi$
$619$		0			0		−0.587785	+	0.809017i	$0.700000\pi$
$620$	1.34500	+	1.34500i	1.34500	+	1.34500i
$621$		0			0
$622$		0			0
$623$		0			0
$624$		0			0
$625$	0.809017	+	0.587785i	0.809017	+	0.587785i
$626$		−	0.312869i		−	0.312869i
$627$		0			0
$628$		0			0
$629$		0			0
$630$	0.809017	+	0.412215i	0.809017	+	0.412215i
$631$	0.951057	−	0.690983i	0.951057	−	0.690983i		−	1.00000i	$-0.5\pi$
$631$	0.951057	−	0.690983i	0.951057	−	0.690983i	0.951057	+	0.309017i	$0.100000\pi$
$632$	0.734572	+	1.44168i	0.734572	+	1.44168i
$633$		0			0
$634$		0			0
$635$	0.437016	+	1.34500i	0.437016	+	1.34500i
$636$	−0.190983	−	0.587785i	−0.190983	−	0.587785i
$637$		0			0
$638$	−0.253116	−	0.183900i	−0.253116	−	0.183900i
$639$		0			0
$640$	−0.951057	−	0.309017i	−0.951057	−	0.309017i
$641$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$641$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$642$	0.0966818	−	0.610425i	0.0966818	−	0.610425i
$643$		0			0		0.891007	−	0.453990i	$-0.150000\pi$
$643$		0			0		−0.891007	+	0.453990i	$0.850000\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		0			0		−0.891007	−	0.453990i	$-0.850000\pi$
$647$		0			0		0.891007	+	0.453990i	$0.150000\pi$
$648$	0.707107	−	0.707107i	0.707107	−	0.707107i
$649$	1.76007	+	0.278768i	1.76007	+	0.278768i
$650$		0			0
$651$	−1.64255	+	0.533698i	−1.64255	+	0.533698i
$652$		0			0
$653$	−1.16110	−	0.183900i	−1.16110	−	0.183900i	−0.453990	−	0.891007i	$-0.650000\pi$
$653$	−1.16110	−	0.183900i	−1.16110	−	0.183900i	−0.707107	+	0.707107i	$0.750000\pi$
$654$		0			0
$655$	−1.76007	−	0.278768i	−1.76007	−	0.278768i
$656$		0			0
$657$	0.309017	+	1.95106i	0.309017	+	1.95106i
$658$		0			0
$659$	−0.312869			−0.312869			−0.156434	−	0.987688i	$-0.550000\pi$
$659$	−0.312869			−0.312869			−0.156434	+	0.987688i	$0.550000\pi$
$660$	−0.987688	−	0.156434i	−0.987688	−	0.156434i
$661$		0			0		1.00000			$0$
$661$		0			0		−1.00000			$\pi$
$662$		0			0
$663$		0			0
$664$	0.951057	+	1.30902i	0.951057	+	1.30902i
$665$		0			0
$666$		0			0
$667$		0			0
$668$		0			0
$669$	1.87869	−	0.610425i	1.87869	−	0.610425i
$670$		0			0
$671$		0			0
$672$	0.642040	−	0.642040i	0.642040	−	0.642040i
$673$	0.278768	+	0.142040i	0.278768	+	0.142040i	0.587785	−	0.809017i	$-0.300000\pi$
$673$	0.278768	+	0.142040i	0.278768	+	0.142040i	−0.309017	+	0.951057i	$0.600000\pi$
$674$	−0.831254	+	1.14412i	−0.831254	+	1.14412i
$675$	0.156434	+	0.987688i	0.156434	+	0.987688i
$676$	0.309017	−	0.951057i	0.309017	−	0.951057i
$677$	0.550672	−	0.280582i	0.550672	−	0.280582i	−0.156434	−	0.987688i	$-0.550000\pi$
$677$	0.550672	−	0.280582i	0.550672	−	0.280582i	0.707107	+	0.707107i	$0.250000\pi$
$678$		0			0
$679$	−1.45106	−	1.05425i	−1.45106	−	1.05425i
$680$		0			0
$681$			1.17557i			1.17557i
$682$	1.53884	−	1.11803i	1.53884	−	1.11803i
$683$	−1.34500	−	1.34500i	−1.34500	−	1.34500i	−0.891007	−	0.453990i	$-0.850000\pi$
$683$	−1.34500	−	1.34500i	−1.34500	−	1.34500i	−0.453990	−	0.891007i	$-0.650000\pi$
$684$		0			0
$685$		0			0
$686$	0.863541	−	0.627399i	0.863541	−	0.627399i
$687$		0			0
$688$		0			0
$689$		0			0
$690$		0			0
$691$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$691$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$692$	−0.437016	−	0.437016i	−0.437016	−	0.437016i
$693$	0.533698	−	0.734572i	0.533698	−	0.734572i
$694$			1.90211i			1.90211i
$695$		0			0
$696$	0.253116	+	0.183900i	0.253116	+	0.183900i
$697$		0			0
$698$		0			0
$699$		0			0
$700$	0.142040	+	0.896802i	0.142040	+	0.896802i
$701$	0.831254	−	1.14412i	0.831254	−	1.14412i	−0.156434	−	0.987688i	$-0.550000\pi$
$701$	0.831254	−	1.14412i	0.831254	−	1.14412i	0.987688	−	0.156434i	$-0.0500000\pi$
$702$		0			0
$703$		0			0
$704$	−0.453990	+	0.891007i	−0.453990	+	0.891007i
$705$		0			0
$706$		0			0
$707$	1.77152	−	0.280582i	1.77152	−	0.280582i
$708$	−1.76007	−	0.278768i	−1.76007	−	0.278768i
$709$		0			0		−0.951057	−	0.309017i	$-0.900000\pi$
$709$		0			0		0.951057	+	0.309017i	$0.100000\pi$
$710$		0			0
$711$	−0.951057	−	1.30902i	−0.951057	−	1.30902i
$712$		0			0
$713$		0			0
$714$		0			0
$715$		0			0
$716$	−0.312869			−0.312869
$717$		0			0
$718$		0			0
$719$		0			0		−0.587785	−	0.809017i	$-0.700000\pi$
$719$		0			0		0.587785	+	0.809017i	$0.300000\pi$
$720$	0.987688	+	0.156434i	0.987688	+	0.156434i
$721$	1.53884	+	0.500000i	1.53884	+	0.500000i
$722$	−0.987688	−	0.156434i	−0.987688	−	0.156434i
$723$	−0.610425	+	0.0966818i	−0.610425	+	0.0966818i
$724$		0			0
$725$	−0.297556	+	0.0966818i	−0.297556	+	0.0966818i
$726$	−0.309017	+	0.951057i	−0.309017	+	0.951057i
$727$	−1.00000	+	1.00000i	−1.00000	+	1.00000i			1.00000i	$0.5\pi$
$727$	−1.00000	+	1.00000i	−1.00000	+	1.00000i	−1.00000			$\pi$
$728$		0			0
$729$	−0.587785	+	0.809017i	−0.587785	+	0.809017i
$730$	−1.39680	+	1.39680i	−1.39680	+	1.39680i
$731$		0			0
$732$		0			0
$733$		0			0		0.156434	−	0.987688i	$-0.450000\pi$
$733$		0			0		−0.156434	+	0.987688i	$0.550000\pi$
$734$	0.253116	+	0.183900i	0.253116	+	0.183900i
$735$	0.166977	+	0.0542543i	0.166977	+	0.0542543i
$736$		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			0
$741$		0			0
$742$	0.254763	+	0.500000i	0.254763	+	0.500000i
$743$		0			0		−0.453990	−	0.891007i	$-0.650000\pi$
$743$		0			0		0.453990	+	0.891007i	$0.350000\pi$
$744$	−1.53884	+	1.11803i	−1.53884	+	1.11803i
$745$	−1.58779	−	0.809017i	−1.58779	−	0.809017i
$746$		0			0
$747$	−1.14412	−	1.14412i	−1.14412	−	1.14412i
$748$		0			0
$749$			0.561163i			0.561163i
$750$	−0.707107	+	0.707107i	−0.707107	+	0.707107i
$751$		0			0		0.587785	−	0.809017i	$-0.300000\pi$
$751$		0			0		−0.587785	+	0.809017i	$0.700000\pi$
$752$		0			0
$753$	1.76007	−	0.896802i	1.76007	−	0.896802i
$754$		0			0
$755$	−0.831254	−	0.831254i	−0.831254	−	0.831254i
$756$	−0.533698	+	0.734572i	−0.533698	+	0.734572i
$757$		0			0		−0.891007	−	0.453990i	$-0.850000\pi$
$757$		0			0		0.891007	+	0.453990i	$0.150000\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$		0			0		0.951057	−	0.309017i	$-0.100000\pi$
$761$		0			0		−0.951057	+	0.309017i	$0.900000\pi$
$762$	−1.39680	+	0.221232i	−1.39680	+	0.221232i
$763$		0			0
$764$		0			0
$765$		0			0
$766$		0			0
$767$		0			0
$768$	0.453990	−	0.891007i	0.453990	−	0.891007i
$769$	−1.17557			−1.17557			−0.587785	−	0.809017i	$-0.700000\pi$
$769$	−1.17557			−1.17557			−0.587785	+	0.809017i	$0.700000\pi$
$770$	0.907981			0.907981
$771$		0			0
$772$	0.809017	−	1.58779i	0.809017	−	1.58779i
$773$	−0.183900	−	1.16110i	−0.183900	−	1.16110i	−0.891007	−	0.453990i	$-0.850000\pi$
$773$	−0.183900	−	1.16110i	−0.183900	−	1.16110i	0.707107	−	0.707107i	$-0.250000\pi$
$774$		0			0
$775$		−	1.90211i		−	1.90211i
$776$	−1.87869	−	0.610425i	−1.87869	−	0.610425i
$777$		0			0
$778$	1.39680	−	0.221232i	1.39680	−	0.221232i
$779$		0			0
$780$		0			0
$781$		0			0
$782$		0			0
$783$	−0.278768	−	0.142040i	−0.278768	−	0.142040i
$784$	0.103198	−	0.142040i	0.103198	−	0.142040i
$785$		0			0
$786$	0.550672	−	1.69480i	0.550672	−	1.69480i
$787$		0			0		0.891007	−	0.453990i	$-0.150000\pi$
$787$		0			0		−0.891007	+	0.453990i	$0.850000\pi$
$788$	−0.183900	+	1.16110i	−0.183900	+	1.16110i
$789$		0			0
$790$	0.500000	−	1.53884i	0.500000	−	1.53884i
$791$		0			0
$792$	0.309017	−	0.951057i	0.309017	−	0.951057i
$793$		0			0
$794$		0			0
$795$	−0.280582	+	0.550672i	−0.280582	+	0.550672i
$796$	1.30902	−	0.951057i	1.30902	−	0.951057i
$797$	0.863541	+	1.69480i	0.863541	+	1.69480i	0.707107	+	0.707107i	$0.250000\pi$
$797$	0.863541	+	1.69480i	0.863541	+	1.69480i	0.156434	+	0.987688i	$0.450000\pi$
$798$		0			0
$799$		0			0
$800$	0.453990	+	0.891007i	0.453990	+	0.891007i
$801$		0			0
$802$		0			0
$803$	1.16110	+	1.59811i	1.16110	+	1.59811i
$804$		0			0
$805$		0			0
$806$		0			0
$807$	−0.221232	+	1.39680i	−0.221232	+	1.39680i
$808$	1.76007	−	0.896802i	1.76007	−	0.896802i
$809$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$809$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$810$	−1.00000			−1.00000
$811$		0			0		0.587785	−	0.809017i	$-0.300000\pi$
$811$		0			0		−0.587785	+	0.809017i	$0.700000\pi$
$812$	−0.253116	−	0.128969i	−0.253116	−	0.128969i
$813$		0			0
$814$		0			0
$815$		0			0
$816$		0			0
$817$		0			0
$818$	1.59811	+	0.253116i	1.59811	+	0.253116i
$819$		0			0
$820$		0			0
$821$	1.04744	+	1.44168i	1.04744	+	1.44168i	0.891007	+	0.453990i	$0.150000\pi$
$821$	1.04744	+	1.44168i	1.04744	+	1.44168i	0.156434	+	0.987688i	$0.450000\pi$
$822$		0			0
$823$	0.142040	−	0.278768i	0.142040	−	0.278768i	−0.809017	−	0.587785i	$-0.800000\pi$
$823$	0.142040	−	0.278768i	0.142040	−	0.278768i	0.951057	+	0.309017i	$0.100000\pi$
$824$	1.78201			1.78201
$825$	0.587785	+	0.809017i	0.587785	+	0.809017i
$826$	1.61803			1.61803
$827$	−0.863541	+	1.69480i	−0.863541	+	1.69480i	−0.156434	+	0.987688i	$0.550000\pi$
$827$	−0.863541	+	1.69480i	−0.863541	+	1.69480i	−0.707107	+	0.707107i	$0.750000\pi$
$828$		0			0
$829$		0			0		−0.587785	−	0.809017i	$-0.700000\pi$
$829$		0			0		0.587785	+	0.809017i	$0.300000\pi$
$830$	0.253116	−	1.59811i	0.253116	−	1.59811i
$831$		0			0
$832$		0			0
$833$		0			0
$834$		0			0
$835$		0			0
$836$		0			0
$837$	1.34500	−	1.34500i	1.34500	−	1.34500i
$838$	−0.278768	−	0.142040i	−0.278768	−	0.142040i
$839$		0			0		0.587785	−	0.809017i	$-0.300000\pi$
$839$		0			0		−0.587785	+	0.809017i	$0.700000\pi$
$840$	−0.907981			−0.907981
$841$	−0.278768	+	0.857960i	−0.278768	+	0.857960i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−0.891007	+	0.453990i	−0.891007	+	0.453990i
$846$		0			0
$847$	0.142040	−	0.896802i	0.142040	−	0.896802i
$848$	0.437016	+	0.437016i	0.437016	+	0.437016i
$849$		0			0
$850$		0			0
$851$		0			0
$852$		0			0
$853$		0			0		−0.453990	−	0.891007i	$-0.650000\pi$
$853$		0			0		0.453990	+	0.891007i	$0.350000\pi$
$854$		0			0
$855$		0			0
$856$	0.190983	+	0.587785i	0.190983	+	0.587785i
$857$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$857$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$858$		0			0
$859$		0			0			−	1.00000i	$-0.5\pi$
$859$		0			0				1.00000i	$0.5\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		0			0		0.891007	−	0.453990i	$-0.150000\pi$
$863$		0			0		−0.891007	+	0.453990i	$0.850000\pi$
$864$	−0.309017	+	0.951057i	−0.309017	+	0.951057i
$865$			0.618034i			0.618034i
$866$	0.533698	−	0.734572i	0.533698	−	0.734572i
$867$	0.891007	+	0.453990i	0.891007	+	0.453990i
$868$	1.22123	−	1.22123i	1.22123	−	1.22123i
$869$	−1.44168	−	0.734572i	−1.44168	−	0.734572i
$870$	−0.0489435	−	0.309017i	−0.0489435	−	0.309017i
$871$		0			0
$872$		0			0
$873$	1.95106	+	0.309017i	1.95106	+	0.309017i
$874$		0			0
$875$	0.533698	−	0.734572i	0.533698	−	0.734572i
$876$	−1.16110	−	1.59811i	−1.16110	−	1.59811i
$877$		0			0		−0.156434	−	0.987688i	$-0.550000\pi$
$877$		0			0		0.156434	+	0.987688i	$0.450000\pi$
$878$	0.863541	−	1.69480i	0.863541	−	1.69480i
$879$	1.90211			1.90211
$880$	0.951057	−	0.309017i	0.951057	−	0.309017i
$881$		0			0		1.00000			$0$
$881$		0			0		−1.00000			$\pi$
$882$	−0.0797073	+	0.156434i	−0.0797073	+	0.156434i
$883$		0			0		−0.156434	−	0.987688i	$-0.550000\pi$
$883$		0			0		0.156434	+	0.987688i	$0.450000\pi$
$884$		0			0
$885$	1.04744	+	1.44168i	1.04744	+	1.44168i
$886$	0.587785	+	0.190983i	0.587785	+	0.190983i
$887$		0			0		−0.987688	−	0.156434i	$-0.950000\pi$
$887$		0			0		0.987688	+	0.156434i	$0.0500000\pi$
$888$		0			0
$889$	1.22123	−	0.396802i	1.22123	−	0.396802i
$890$		0			0
$891$	−0.156434	+	0.987688i	−0.156434	+	0.987688i
$892$	−1.39680	+	1.39680i	−1.39680	+	1.39680i
$893$		0			0
$894$	1.04744	−	1.44168i	1.04744	−	1.44168i
$895$	0.221232	+	0.221232i	0.221232	+	0.221232i
$896$	−0.280582	+	0.863541i	−0.280582	+	0.863541i
$897$		0			0
$898$		0			0
$899$	0.481456	+	0.349798i	0.481456	+	0.349798i
$900$	−0.587785	−	0.809017i	−0.587785	−	0.809017i
$901$		0			0
$902$		0			0
$903$		0			0
$904$		0			0
$905$		0			0
$906$	0.951057	−	0.690983i	0.951057	−	0.690983i
$907$		0			0		−0.453990	−	0.891007i	$-0.650000\pi$
$907$		0			0		0.453990	+	0.891007i	$0.350000\pi$
$908$	−0.533698	−	1.04744i	−0.533698	−	1.04744i
$909$	−1.59811	+	1.16110i	−1.59811	+	1.16110i
$910$		0			0
$911$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$911$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$912$		0			0
$913$	−1.53884	−	0.500000i	−1.53884	−	0.500000i
$914$		−	0.312869i		−	0.312869i
$915$		0			0
$916$		0			0
$917$	−0.253116	+	1.59811i	−0.253116	+	1.59811i
$918$		0			0
$919$	0.500000	−	1.53884i	0.500000	−	1.53884i	−0.309017	−	0.951057i	$-0.600000\pi$
$919$	0.500000	−	1.53884i	0.500000	−	1.53884i	0.809017	−	0.587785i	$-0.200000\pi$
$920$		0			0
$921$		0			0
$922$	1.26007	+	0.642040i	1.26007	+	0.642040i
$923$		0			0
$924$	−0.142040	+	0.896802i	−0.142040	+	0.896802i
$925$		0			0
$926$	0.863541	−	0.280582i	0.863541	−	0.280582i
$927$	−1.76007	+	0.278768i	−1.76007	+	0.278768i
$928$	−0.309017	−	0.0489435i	−0.309017	−	0.0489435i
$929$		0			0		−0.951057	−	0.309017i	$-0.900000\pi$
$929$		0			0		0.951057	+	0.309017i	$0.100000\pi$
$930$	1.87869	+	0.297556i	1.87869	+	0.297556i
$931$		0			0
$932$		0			0
$933$		0			0
$934$	1.90211			1.90211
$935$		0			0
$936$		0			0
$937$	0.809017	−	1.58779i	0.809017	−	1.58779i		−	1.00000i	$-0.5\pi$
$937$	0.809017	−	1.58779i	0.809017	−	1.58779i	0.809017	−	0.587785i	$-0.200000\pi$
$938$		0			0
$939$	−0.183900	−	0.253116i	−0.183900	−	0.253116i
$940$		0			0
$941$	−1.34500	−	0.437016i	−1.34500	−	0.437016i	−0.453990	−	0.891007i	$-0.650000\pi$
$941$	−1.34500	−	0.437016i	−1.34500	−	0.437016i	−0.891007	+	0.453990i	$0.850000\pi$
$942$		0			0
$943$		0			0
$944$	1.69480	−	0.550672i	1.69480	−	0.550672i
$945$	0.896802	−	0.142040i	0.896802	−	0.142040i
$946$		0			0
$947$	0.437016	−	0.437016i	0.437016	−	0.437016i	−0.453990	−	0.891007i	$-0.650000\pi$
$947$	0.437016	−	0.437016i	0.437016	−	0.437016i	0.891007	+	0.453990i	$0.150000\pi$
$948$	1.44168	+	0.734572i	1.44168	+	0.734572i
$949$		0			0
$950$		0			0
$951$		0			0
$952$		0			0
$953$		0			0		0.156434	−	0.987688i	$-0.450000\pi$
$953$		0			0		−0.156434	+	0.987688i	$0.550000\pi$
$954$	−0.500000	−	0.363271i	−0.500000	−	0.363271i
$955$		0			0
$956$		0			0
$957$	−0.312869			−0.312869
$958$		0			0
$959$		0			0
$960$	−0.951057	+	0.309017i	−0.951057	+	0.309017i
$961$	−2.11803	+	1.53884i	−2.11803	+	1.53884i
$962$		0			0
$963$	−0.280582	−	0.550672i	−0.280582	−	0.550672i
$964$	0.500000	−	0.363271i	0.500000	−	0.363271i
$965$	−1.69480	+	0.550672i	−1.69480	+	0.550672i
$966$		0			0
$967$	1.39680	+	1.39680i	1.39680	+	1.39680i	0.809017	+	0.587785i	$0.200000\pi$
$967$	1.39680	+	1.39680i	1.39680	+	1.39680i	0.587785	+	0.809017i	$0.300000\pi$
$968$	−0.156434	−	0.987688i	−0.156434	−	0.987688i
$969$		0			0
$970$	0.896802	+	1.76007i	0.896802	+	1.76007i
$971$	−1.14412	−	0.831254i	−1.14412	−	0.831254i	−0.156434	−	0.987688i	$-0.550000\pi$
$971$	−1.14412	−	0.831254i	−1.14412	−	0.831254i	−0.987688	+	0.156434i	$0.950000\pi$
$972$	0.156434	−	0.987688i	0.156434	−	0.987688i
$973$		0			0
$974$	0.280582	−	0.863541i	0.280582	−	0.863541i
$975$		0			0
$976$		0			0
$977$		0			0		−0.891007	−	0.453990i	$-0.850000\pi$
$977$		0			0		0.891007	+	0.453990i	$0.150000\pi$
$978$		0			0
$979$		0			0
$980$	−0.173409	+	0.0274653i	−0.173409	+	0.0274653i
$981$		0			0
$982$	1.39680	−	0.221232i	1.39680	−	0.221232i
$983$		0			0		−0.987688	−	0.156434i	$-0.950000\pi$
$983$		0			0		0.987688	+	0.156434i	$0.0500000\pi$
$984$		0			0
$985$	0.951057	−	0.690983i	0.951057	−	0.690983i
$986$		0			0
$987$		0			0
$988$		0			0
$989$		0			0
$990$	−0.891007	+	0.453990i	−0.891007	+	0.453990i
$991$	1.61803			1.61803			0.809017	−	0.587785i	$-0.200000\pi$
$991$	1.61803			1.61803			0.809017	+	0.587785i	$0.200000\pi$
$992$	0.863541	−	1.69480i	0.863541	−	1.69480i
$993$		0			0
$994$		0			0
$995$	−1.59811	−	0.253116i	−1.59811	−	0.253116i
$996$	1.53884	+	0.500000i	1.53884	+	0.500000i
$997$		0			0		−0.987688	−	0.156434i	$-0.950000\pi$
$997$		0			0		0.987688	+	0.156434i	$0.0500000\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1320.1.dk.a.413.1	✓	16
3.2	odd	2	inner	1320.1.dk.a.413.2	yes	16
5.2	odd	4		1320.1.dk.b.677.1	yes	16
8.5	even	2	inner	1320.1.dk.a.413.2	yes	16
11.2	odd	10		1320.1.dk.b.893.1	yes	16
15.2	even	4		1320.1.dk.b.677.2	yes	16
24.5	odd	2	CM	1320.1.dk.a.413.1	✓	16
33.2	even	10		1320.1.dk.b.893.2	yes	16
40.37	odd	4		1320.1.dk.b.677.2	yes	16
55.2	even	20	inner	1320.1.dk.a.1157.1	yes	16
88.13	odd	10		1320.1.dk.b.893.2	yes	16
120.77	even	4		1320.1.dk.b.677.1	yes	16
165.2	odd	20	inner	1320.1.dk.a.1157.2	yes	16
264.101	even	10		1320.1.dk.b.893.1	yes	16
440.277	even	20	inner	1320.1.dk.a.1157.2	yes	16
1320.1157	odd	20	inner	1320.1.dk.a.1157.1	yes	16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1320.1.dk.a.413.1	✓	16	1.1	even	1	trivial
1320.1.dk.a.413.1	✓	16	24.5	odd	2	CM
1320.1.dk.a.413.2	yes	16	3.2	odd	2	inner
1320.1.dk.a.413.2	yes	16	8.5	even	2	inner
1320.1.dk.a.1157.1	yes	16	55.2	even	20	inner
1320.1.dk.a.1157.1	yes	16	1320.1157	odd	20	inner
1320.1.dk.a.1157.2	yes	16	165.2	odd	20	inner
1320.1.dk.a.1157.2	yes	16	440.277	even	20	inner
1320.1.dk.b.677.1	yes	16	5.2	odd	4
1320.1.dk.b.677.1	yes	16	120.77	even	4
1320.1.dk.b.677.2	yes	16	15.2	even	4
1320.1.dk.b.677.2	yes	16	40.37	odd	4
1320.1.dk.b.893.1	yes	16	11.2	odd	10
1320.1.dk.b.893.1	yes	16	264.101	even	10
1320.1.dk.b.893.2	yes	16	33.2	even	10
1320.1.dk.b.893.2	yes	16	88.13	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 1320 = 2^{3} \cdot 3 \cdot 5 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	1320.dk (of order \(20\), degree \(8\), minimal)

\(f(q)\)	\(=\)	\(q+(-0.453990 + 0.891007i) q^{2} +(0.156434 + 0.987688i) q^{3} +(-0.587785 - 0.809017i) q^{4} +(-0.156434 + 0.987688i) q^{5} +(-0.951057 - 0.309017i) q^{6} +(0.896802 + 0.142040i) q^{7} +(0.987688 - 0.156434i) q^{8} +(-0.951057 + 0.309017i) q^{9} +(-0.809017 - 0.587785i) q^{10} +(-0.707107 + 0.707107i) q^{11} +(0.707107 - 0.707107i) q^{12} +(-0.533698 + 0.734572i) q^{14} -1.00000 q^{15} +(-0.309017 + 0.951057i) q^{16} +(0.156434 - 0.987688i) q^{18} +(0.891007 - 0.453990i) q^{20} +0.907981i q^{21} +(-0.309017 - 0.951057i) q^{22} +(0.309017 + 0.951057i) q^{24} +(-0.951057 - 0.309017i) q^{25} +(-0.453990 - 0.891007i) q^{27} +(-0.412215 - 0.809017i) q^{28} +(0.253116 - 0.183900i) q^{29} +(0.453990 - 0.891007i) q^{30} +(0.587785 + 1.80902i) q^{31} +(-0.707107 - 0.707107i) q^{32} +(-0.809017 - 0.587785i) q^{33} +(-0.280582 + 0.863541i) q^{35} +(0.809017 + 0.587785i) q^{36} +1.00000i q^{40} +(-0.809017 - 0.412215i) q^{42} +(0.987688 + 0.156434i) q^{44} +(-0.156434 - 0.987688i) q^{45} +(-0.987688 - 0.156434i) q^{48} +(-0.166977 - 0.0542543i) q^{49} +(0.707107 - 0.707107i) q^{50} +(0.280582 - 0.550672i) q^{53} +1.00000 q^{54} +(-0.587785 - 0.809017i) q^{55} +0.907981 q^{56} +(0.0489435 + 0.309017i) q^{58} +(-1.04744 - 1.44168i) q^{59} +(0.587785 + 0.809017i) q^{60} +(-1.87869 - 0.297556i) q^{62} +(-0.896802 + 0.142040i) q^{63} +(0.951057 - 0.309017i) q^{64} +(0.891007 - 0.453990i) q^{66} +(-0.642040 - 0.642040i) q^{70} +(-0.891007 + 0.453990i) q^{72} +(0.309017 - 1.95106i) q^{73} +(0.156434 - 0.987688i) q^{75} +(-0.734572 + 0.533698i) q^{77} +(0.500000 + 1.53884i) q^{79} +(-0.891007 - 0.453990i) q^{80} +(0.809017 - 0.587785i) q^{81} +(0.734572 + 1.44168i) q^{83} +(0.734572 - 0.533698i) q^{84} +(0.221232 + 0.221232i) q^{87} +(-0.587785 + 0.809017i) q^{88} +(0.951057 + 0.309017i) q^{90} +(-1.69480 + 0.863541i) q^{93} +(0.587785 - 0.809017i) q^{96} +(-1.76007 - 0.896802i) q^{97} +(0.124147 - 0.124147i) q^{98} +(0.453990 - 0.891007i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 4 q^{7} - 4 q^{10} - 16 q^{15} + 4 q^{16} + 4 q^{22} - 4 q^{24} - 16 q^{28} - 4 q^{33} + 4 q^{36} - 4 q^{42} + 16 q^{54} + 16 q^{58} + 4 q^{63} - 4 q^{70} - 4 q^{73} + 8 q^{79} + 4 q^{81} + 4 q^{87}+ \cdots - 4 q^{97}+O(q^{100}) \) 16 * q - 4 * q^7 - 4 * q^10 - 16 * q^15 + 4 * q^16 + 4 * q^22 - 4 * q^24 - 16 * q^28 - 4 * q^33 + 4 * q^36 - 4 * q^42 + 16 * q^54 + 16 * q^58 + 4 * q^63 - 4 * q^70 - 4 * q^73 + 8 * q^79 + 4 * q^81 + 4 * q^87 - 4 * q^97

Introduction

L-functions

Modular forms

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Database

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