LMFDB - Embedded newform 1176.1.ck.a.149.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1176,1,Mod(53,1176)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1176, base_ring=CyclotomicField(42))

chi = DirichletCharacter(H, H._module([0, 21, 21, 10]))

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

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

chi := DirichletCharacter("1176.53");

S:= CuspForms(chi, 1);

N := Newforms(S);

Level:	$ N $	$=$	$ 1176 = 2^{3} \cdot 3 \cdot 7^{2} $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	1176.ck (of order $42$, degree $12$, 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:	$0.586900454856$
Analytic rank:	$0$
Dimension:	$12$
Coefficient field:	$\Q(\zeta_{21})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - x^{11} + x^{9} - x^{8} + x^{6} - x^{4} + x^{3} - x + 1 $ x^12 - x^11 + x^9 - x^8 + x^6 - x^4 + x^3 - x + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Projective image:	$D_{21}$
Projective field:	Galois closure of $\mathbb{Q}[x]/(x^{21} + \cdots)$

Embedding invariants

Embedding label			149.1
Root			$-0.733052 - 0.680173i$ of defining polynomial
Character	$\chi$	$=$	1176.149
Dual form			1176.1.ck.a.221.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-0.955573 - 0.294755i) q^{2} +(-0.365341 + 0.930874i) q^{3} +(0.826239 + 0.563320i) q^{4} +(-1.44973 + 0.218511i) q^{5} +(0.623490 - 0.781831i) q^{6} +(-0.988831 - 0.149042i) q^{7} +(-0.623490 - 0.781831i) q^{8} +(-0.733052 - 0.680173i) q^{9} +O(q^{10})$	\(q+(-0.955573 - 0.294755i) q^{2} +(-0.365341 + 0.930874i) q^{3} +(0.826239 + 0.563320i) q^{4} +(-1.44973 + 0.218511i) q^{5} +(0.623490 - 0.781831i) q^{6} +(-0.988831 - 0.149042i) q^{7} +(-0.623490 - 0.781831i) q^{8} +(-0.733052 - 0.680173i) q^{9} +(1.44973 + 0.218511i) q^{10} +(1.21135 - 1.12397i) q^{11} +(-0.826239 + 0.563320i) q^{12} +(0.900969 + 0.433884i) q^{14} +(0.326239 - 1.42935i) q^{15} +(0.365341 + 0.930874i) q^{16} +(0.500000 + 0.866025i) q^{18} +(-1.32091 - 0.636119i) q^{20} +(0.500000 - 0.866025i) q^{21} +(-1.48883 + 0.716983i) q^{22} +(0.955573 - 0.294755i) q^{24} +(1.09839 - 0.338809i) q^{25} +(0.900969 - 0.433884i) q^{27} +(-0.733052 - 0.680173i) q^{28} +(1.72188 + 0.829215i) q^{29} +(-0.733052 + 1.26968i) q^{30} +(-0.365341 - 0.632789i) q^{31} +(-0.0747301 - 0.997204i) q^{32} +(0.603718 + 1.53825i) q^{33} +1.46610 q^{35} +(-0.222521 - 0.974928i) q^{36} +(1.07473 + 0.997204i) q^{40} +(-0.733052 + 0.680173i) q^{42} +(1.63402 - 0.246289i) q^{44} +(1.21135 + 0.825886i) q^{45} -1.00000 q^{48} +(0.955573 + 0.294755i) q^{49} -1.14946 q^{50} +(0.826239 + 0.563320i) q^{53} +(-0.988831 + 0.149042i) q^{54} +(-1.51053 + 1.89415i) q^{55} +(0.500000 + 0.866025i) q^{56} +(-1.40097 - 1.29991i) q^{58} +(-0.988831 - 0.149042i) q^{59} +(1.07473 - 0.997204i) q^{60} +(0.162592 + 0.712362i) q^{62} +(0.623490 + 0.781831i) q^{63} +(-0.222521 + 0.974928i) q^{64} +(-0.123490 - 1.64786i) q^{66} +(-1.40097 - 0.432142i) q^{70} +(-0.0747301 + 0.997204i) q^{72} +(1.19158 - 0.367554i) q^{73} +(-0.0858993 + 1.14625i) q^{75} +(-1.36534 + 0.930874i) q^{77} +(0.988831 - 1.71271i) q^{79} +(-0.733052 - 1.26968i) q^{80} +(0.0747301 + 0.997204i) q^{81} +(0.425270 - 1.86323i) q^{83} +(0.900969 - 0.433884i) q^{84} +(-1.40097 + 1.29991i) q^{87} +(-1.63402 - 0.246289i) q^{88} +(-0.914101 - 1.14625i) q^{90} +(0.722521 - 0.108903i) q^{93} +(0.955573 + 0.294755i) q^{96} +0.149460 q^{97} +(-0.826239 - 0.563320i) q^{98} -1.65248 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q - q^{2} - q^{3} + q^{4} + q^{5} - 2 q^{6} + q^{7} + 2 q^{8} + q^{9}+O(q^{10}) $ 12 * q - q^2 - q^3 + q^4 + q^5 - 2 * q^6 + q^7 + 2 * q^8 + q^9	$ 12 q - q^{2} - q^{3} + q^{4} + q^{5} - 2 q^{6} + q^{7} + 2 q^{8} + q^{9} - q^{10} + q^{11} - q^{12} + 2 q^{14} - 5 q^{15} + q^{16} + 6 q^{18} - 2 q^{20} + 6 q^{21} - 5 q^{22} + q^{24} + 2 q^{27} + q^{28} - 2 q^{29} + q^{30} - q^{31} - q^{32} - q^{33} - 2 q^{35} - 2 q^{36} + 13 q^{40} + q^{42} + q^{44} + q^{45} - 12 q^{48} + q^{49} - 14 q^{50} + q^{53} + q^{54} - 9 q^{55} + 6 q^{56} - 8 q^{58} + q^{59} + 13 q^{60} - 2 q^{62} - 2 q^{63} - 2 q^{64} + 8 q^{66} - 8 q^{70} - q^{72} + 2 q^{73} - 14 q^{75} - 13 q^{77} - q^{79} + q^{80} + q^{81} + 5 q^{83} + 2 q^{84} - 8 q^{87} - q^{88} + 2 q^{90} + 8 q^{93} + q^{96} + 2 q^{97} - q^{98} - 2 q^{99}+O(q^{100}) $ 12 * q - q^2 - q^3 + q^4 + q^5 - 2 * q^6 + q^7 + 2 * q^8 + q^9 - q^10 + q^11 - q^12 + 2 * q^14 - 5 * q^15 + q^16 + 6 * q^18 - 2 * q^20 + 6 * q^21 - 5 * q^22 + q^24 + 2 * q^27 + q^28 - 2 * q^29 + q^30 - q^31 - q^32 - q^33 - 2 * q^35 - 2 * q^36 + 13 * q^40 + q^42 + q^44 + q^45 - 12 * q^48 + q^49 - 14 * q^50 + q^53 + q^54 - 9 * q^55 + 6 * q^56 - 8 * q^58 + q^59 + 13 * q^60 - 2 * q^62 - 2 * q^63 - 2 * q^64 + 8 * q^66 - 8 * q^70 - q^72 + 2 * q^73 - 14 * q^75 - 13 * q^77 - q^79 + q^80 + q^81 + 5 * q^83 + 2 * q^84 - 8 * q^87 - q^88 + 2 * q^90 + 8 * q^93 + q^96 + 2 * q^97 - q^98 - 2 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1176\mathbb{Z}\right)^\times$.

$n$	$295$	$589$	$785$	$1081$
$\chi(n)$	$1$	$-1$	$-1$	$e\left(\frac{13}{21}\right)$

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 $

$2$	−0.955573	−	0.294755i	−0.955573	−	0.294755i
$2$	−0.955573	−	0.294755i	−0.955573	−	0.294755i
$3$	−0.365341	+	0.930874i	−0.365341	+	0.930874i
$3$	−0.365341	+	0.930874i	−0.365341	+	0.930874i
$4$	0.826239	+	0.563320i	0.826239	+	0.563320i
$5$	−1.44973	+	0.218511i	−1.44973	+	0.218511i	−0.826239	−	0.563320i	$-0.809524\pi$
$5$	−1.44973	+	0.218511i	−1.44973	+	0.218511i	−0.623490	+	0.781831i	$0.714286\pi$
$6$	0.623490	−	0.781831i	0.623490	−	0.781831i
$7$	−0.988831	−	0.149042i	−0.988831	−	0.149042i
$7$	−0.988831	−	0.149042i	−0.988831	−	0.149042i
$8$	−0.623490	−	0.781831i	−0.623490	−	0.781831i
$9$	−0.733052	−	0.680173i	−0.733052	−	0.680173i
$10$	1.44973	+	0.218511i	1.44973	+	0.218511i
$11$	1.21135	−	1.12397i	1.21135	−	1.12397i	0.222521	−	0.974928i	$-0.428571\pi$
$11$	1.21135	−	1.12397i	1.21135	−	1.12397i	0.988831	−	0.149042i	$-0.0476190\pi$
$12$	−0.826239	+	0.563320i	−0.826239	+	0.563320i
$13$		0			0		−0.222521	−	0.974928i	$-0.571429\pi$
$13$		0			0		0.222521	+	0.974928i	$0.428571\pi$
$14$	0.900969	+	0.433884i	0.900969	+	0.433884i
$15$	0.326239	−	1.42935i	0.326239	−	1.42935i
$16$	0.365341	+	0.930874i	0.365341	+	0.930874i
$17$		0			0		−0.0747301	−	0.997204i	$-0.523810\pi$
$17$		0			0		0.0747301	+	0.997204i	$0.476190\pi$
$18$	0.500000	+	0.866025i	0.500000	+	0.866025i
$19$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$19$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$20$	−1.32091	−	0.636119i	−1.32091	−	0.636119i
$21$	0.500000	−	0.866025i	0.500000	−	0.866025i
$22$	−1.48883	+	0.716983i	−1.48883	+	0.716983i
$23$		0			0		0.0747301	−	0.997204i	$-0.476190\pi$
$23$		0			0		−0.0747301	+	0.997204i	$0.523810\pi$
$24$	0.955573	−	0.294755i	0.955573	−	0.294755i
$25$	1.09839	−	0.338809i	1.09839	−	0.338809i
$26$		0			0
$27$	0.900969	−	0.433884i	0.900969	−	0.433884i
$28$	−0.733052	−	0.680173i	−0.733052	−	0.680173i
$29$	1.72188	+	0.829215i	1.72188	+	0.829215i	0.988831	+	0.149042i	$0.0476190\pi$
$29$	1.72188	+	0.829215i	1.72188	+	0.829215i	0.733052	+	0.680173i	$0.238095\pi$
$30$	−0.733052	+	1.26968i	−0.733052	+	1.26968i
$31$	−0.365341	−	0.632789i	−0.365341	−	0.632789i	0.623490	−	0.781831i	$-0.285714\pi$
$31$	−0.365341	−	0.632789i	−0.365341	−	0.632789i	−0.988831	+	0.149042i	$0.952381\pi$
$32$	−0.0747301	−	0.997204i	−0.0747301	−	0.997204i
$33$	0.603718	+	1.53825i	0.603718	+	1.53825i
$34$		0			0
$35$	1.46610			1.46610
$36$	−0.222521	−	0.974928i	−0.222521	−	0.974928i
$37$		0			0		0.826239	−	0.563320i	$-0.190476\pi$
$37$		0			0		−0.826239	+	0.563320i	$0.809524\pi$
$38$		0			0
$39$		0			0
$40$	1.07473	+	0.997204i	1.07473	+	0.997204i
$41$		0			0		−0.623490	−	0.781831i	$-0.714286\pi$
$41$		0			0		0.623490	+	0.781831i	$0.285714\pi$
$42$	−0.733052	+	0.680173i	−0.733052	+	0.680173i
$43$		0			0		0.623490	−	0.781831i	$-0.285714\pi$
$43$		0			0		−0.623490	+	0.781831i	$0.714286\pi$
$44$	1.63402	−	0.246289i	1.63402	−	0.246289i
$45$	1.21135	+	0.825886i	1.21135	+	0.825886i
$46$		0			0
$47$		0			0		−0.955573	−	0.294755i	$-0.904762\pi$
$47$		0			0		0.955573	+	0.294755i	$0.0952381\pi$
$48$	−1.00000			−1.00000
$49$	0.955573	+	0.294755i	0.955573	+	0.294755i
$50$	−1.14946			−1.14946
$51$		0			0
$52$		0			0
$53$	0.826239	+	0.563320i	0.826239	+	0.563320i	0.900969	−	0.433884i	$-0.142857\pi$
$53$	0.826239	+	0.563320i	0.826239	+	0.563320i	−0.0747301	+	0.997204i	$0.523810\pi$
$54$	−0.988831	+	0.149042i	−0.988831	+	0.149042i
$55$	−1.51053	+	1.89415i	−1.51053	+	1.89415i
$56$	0.500000	+	0.866025i	0.500000	+	0.866025i
$57$		0			0
$58$	−1.40097	−	1.29991i	−1.40097	−	1.29991i
$59$	−0.988831	−	0.149042i	−0.988831	−	0.149042i	−0.365341	−	0.930874i	$-0.619048\pi$
$59$	−0.988831	−	0.149042i	−0.988831	−	0.149042i	−0.623490	+	0.781831i	$0.714286\pi$
$60$	1.07473	−	0.997204i	1.07473	−	0.997204i
$61$		0			0		0.826239	−	0.563320i	$-0.190476\pi$
$61$		0			0		−0.826239	+	0.563320i	$0.809524\pi$
$62$	0.162592	+	0.712362i	0.162592	+	0.712362i
$63$	0.623490	+	0.781831i	0.623490	+	0.781831i
$64$	−0.222521	+	0.974928i	−0.222521	+	0.974928i
$65$		0			0
$66$	−0.123490	−	1.64786i	−0.123490	−	1.64786i
$67$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$67$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$68$		0			0
$69$		0			0
$70$	−1.40097	−	0.432142i	−1.40097	−	0.432142i
$71$		0			0		0.900969	−	0.433884i	$-0.142857\pi$
$71$		0			0		−0.900969	+	0.433884i	$0.857143\pi$
$72$	−0.0747301	+	0.997204i	−0.0747301	+	0.997204i
$73$	1.19158	−	0.367554i	1.19158	−	0.367554i	0.365341	−	0.930874i	$-0.380952\pi$
$73$	1.19158	−	0.367554i	1.19158	−	0.367554i	0.826239	+	0.563320i	$0.190476\pi$
$74$		0			0
$75$	−0.0858993	+	1.14625i	−0.0858993	+	1.14625i
$76$		0			0
$77$	−1.36534	+	0.930874i	−1.36534	+	0.930874i
$78$		0			0
$79$	0.988831	−	1.71271i	0.988831	−	1.71271i	0.365341	−	0.930874i	$-0.380952\pi$
$79$	0.988831	−	1.71271i	0.988831	−	1.71271i	0.623490	−	0.781831i	$-0.285714\pi$
$80$	−0.733052	−	1.26968i	−0.733052	−	1.26968i
$81$	0.0747301	+	0.997204i	0.0747301	+	0.997204i
$82$		0			0
$83$	0.425270	−	1.86323i	0.425270	−	1.86323i	−0.0747301	−	0.997204i	$-0.523810\pi$
$83$	0.425270	−	1.86323i	0.425270	−	1.86323i	0.500000	−	0.866025i	$-0.333333\pi$
$84$	0.900969	−	0.433884i	0.900969	−	0.433884i
$85$		0			0
$86$		0			0
$87$	−1.40097	+	1.29991i	−1.40097	+	1.29991i
$88$	−1.63402	−	0.246289i	−1.63402	−	0.246289i
$89$		0			0		−0.733052	−	0.680173i	$-0.761905\pi$
$89$		0			0		0.733052	+	0.680173i	$0.238095\pi$
$90$	−0.914101	−	1.14625i	−0.914101	−	1.14625i
$91$		0			0
$92$		0			0
$93$	0.722521	−	0.108903i	0.722521	−	0.108903i
$94$		0			0
$95$		0			0
$96$	0.955573	+	0.294755i	0.955573	+	0.294755i
$97$	0.149460			0.149460			0.0747301	−	0.997204i	$-0.476190\pi$
$97$	0.149460			0.149460			0.0747301	+	0.997204i	$0.476190\pi$
$98$	−0.826239	−	0.563320i	−0.826239	−	0.563320i
$99$	−1.65248			−1.65248
$100$	1.09839	+	0.338809i	1.09839	+	0.338809i
$101$	0.658322	−	1.67738i	0.658322	−	1.67738i	−0.0747301	−	0.997204i	$-0.523810\pi$
$101$	0.658322	−	1.67738i	0.658322	−	1.67738i	0.733052	−	0.680173i	$-0.238095\pi$
$102$		0			0
$103$	1.78181	−	0.268565i	1.78181	−	0.268565i	0.826239	−	0.563320i	$-0.190476\pi$
$103$	1.78181	−	0.268565i	1.78181	−	0.268565i	0.955573	+	0.294755i	$0.0952381\pi$
$104$		0			0
$105$	−0.535628	+	1.36476i	−0.535628	+	1.36476i
$106$	−0.623490	−	0.781831i	−0.623490	−	0.781831i
$107$	−1.44973	−	1.34515i	−1.44973	−	1.34515i	−0.826239	−	0.563320i	$-0.809524\pi$
$107$	−1.44973	−	1.34515i	−1.44973	−	1.34515i	−0.623490	−	0.781831i	$-0.714286\pi$
$108$	0.988831	+	0.149042i	0.988831	+	0.149042i
$109$		0			0		0.733052	−	0.680173i	$-0.238095\pi$
$109$		0			0		−0.733052	+	0.680173i	$0.761905\pi$
$110$	2.00173	−	1.36476i	2.00173	−	1.36476i
$111$		0			0
$112$	−0.222521	−	0.974928i	−0.222521	−	0.974928i
$113$		0			0		0.222521	−	0.974928i	$-0.428571\pi$
$113$		0			0		−0.222521	+	0.974928i	$0.571429\pi$
$114$		0			0
$115$		0			0
$116$	0.955573	+	1.65510i	0.955573	+	1.65510i
$117$		0			0
$118$	0.900969	+	0.433884i	0.900969	+	0.433884i
$119$		0			0
$120$	−1.32091	+	0.636119i	−1.32091	+	0.636119i
$121$	0.129334	−	1.72584i	0.129334	−	1.72584i
$122$		0			0
$123$		0			0
$124$	0.0546039	−	0.728639i	0.0546039	−	0.728639i
$125$	−0.197424	+	0.0950744i	−0.197424	+	0.0950744i
$126$	−0.365341	−	0.930874i	−0.365341	−	0.930874i
$127$	−0.658322	−	0.317031i	−0.658322	−	0.317031i	0.0747301	−	0.997204i	$-0.476190\pi$
$127$	−0.658322	−	0.317031i	−0.658322	−	0.317031i	−0.733052	+	0.680173i	$0.761905\pi$
$128$	0.500000	−	0.866025i	0.500000	−	0.866025i
$129$		0			0
$130$		0			0
$131$	0.722521	+	1.84095i	0.722521	+	1.84095i	0.500000	+	0.866025i	$0.333333\pi$
$131$	0.722521	+	1.84095i	0.722521	+	1.84095i	0.222521	+	0.974928i	$0.428571\pi$
$132$	−0.367711	+	1.61105i	−0.367711	+	1.61105i
$133$		0			0
$134$		0			0
$135$	−1.21135	+	0.825886i	−1.21135	+	0.825886i
$136$		0			0
$137$		0			0		−0.988831	−	0.149042i	$-0.952381\pi$
$137$		0			0		0.988831	+	0.149042i	$0.0476190\pi$
$138$		0			0
$139$		0			0		−0.623490	−	0.781831i	$-0.714286\pi$
$139$		0			0		0.623490	+	0.781831i	$0.285714\pi$
$140$	1.21135	+	0.825886i	1.21135	+	0.825886i
$141$		0			0
$142$		0			0
$143$		0			0
$144$	0.365341	−	0.930874i	0.365341	−	0.930874i
$145$	−2.67746	−	0.825886i	−2.67746	−	0.825886i
$146$	−1.24698			−1.24698
$147$	−0.623490	+	0.781831i	−0.623490	+	0.781831i
$148$		0			0
$149$	−1.19158	−	0.367554i	−1.19158	−	0.367554i	−0.365341	−	0.930874i	$-0.619048\pi$
$149$	−1.19158	−	0.367554i	−1.19158	−	0.367554i	−0.826239	+	0.563320i	$0.809524\pi$
$150$	0.419945	−	1.07000i	0.419945	−	1.07000i
$151$	−0.826239	−	0.563320i	−0.826239	−	0.563320i	0.0747301	−	0.997204i	$-0.476190\pi$
$151$	−0.826239	−	0.563320i	−0.826239	−	0.563320i	−0.900969	+	0.433884i	$0.857143\pi$
$152$		0			0
$153$		0			0
$154$	1.57906	−	0.487076i	1.57906	−	0.487076i
$155$	0.667917	+	0.837541i	0.667917	+	0.837541i
$156$		0			0
$157$		0			0		−0.988831	−	0.149042i	$-0.952381\pi$
$157$		0			0		0.988831	+	0.149042i	$0.0476190\pi$
$158$	−1.44973	+	1.34515i	−1.44973	+	1.34515i
$159$	−0.826239	+	0.563320i	−0.826239	+	0.563320i
$160$	0.326239	+	1.42935i	0.326239	+	1.42935i
$161$		0			0
$162$	0.222521	−	0.974928i	0.222521	−	0.974928i
$163$		0			0		−0.365341	−	0.930874i	$-0.619048\pi$
$163$		0			0		0.365341	+	0.930874i	$0.380952\pi$
$164$		0			0
$165$	−1.21135	−	2.09812i	−1.21135	−	2.09812i
$166$	−0.955573	+	1.65510i	−0.955573	+	1.65510i
$167$		0			0		−0.900969	−	0.433884i	$-0.857143\pi$
$167$		0			0		0.900969	+	0.433884i	$0.142857\pi$
$168$	−0.988831	+	0.149042i	−0.988831	+	0.149042i
$169$	−0.900969	+	0.433884i	−0.900969	+	0.433884i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	0.0332580	−	0.443797i	0.0332580	−	0.443797i	−0.955573	−	0.294755i	$-0.904762\pi$
$173$	0.0332580	−	0.443797i	0.0332580	−	0.443797i	0.988831	−	0.149042i	$-0.0476190\pi$
$174$	1.72188	−	0.829215i	1.72188	−	0.829215i
$175$	−1.13662	+	0.171318i	−1.13662	+	0.171318i
$176$	1.48883	+	0.716983i	1.48883	+	0.716983i
$177$	0.500000	−	0.866025i	0.500000	−	0.866025i
$178$		0			0
$179$	0.0332580	+	0.443797i	0.0332580	+	0.443797i	0.988831	+	0.149042i	$0.0476190\pi$
$179$	0.0332580	+	0.443797i	0.0332580	+	0.443797i	−0.955573	+	0.294755i	$0.904762\pi$
$180$	0.535628	+	1.36476i	0.535628	+	1.36476i
$181$		0			0		0.222521	−	0.974928i	$-0.428571\pi$
$181$		0			0		−0.222521	+	0.974928i	$0.571429\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$		0			0
$186$	−0.722521	−	0.108903i	−0.722521	−	0.108903i
$187$		0			0
$188$		0			0
$189$	−0.955573	+	0.294755i	−0.955573	+	0.294755i
$190$		0			0
$191$		0			0		0.988831	−	0.149042i	$-0.0476190\pi$
$191$		0			0		−0.988831	+	0.149042i	$0.952381\pi$
$192$	−0.826239	−	0.563320i	−0.826239	−	0.563320i
$193$	−0.722521	+	1.84095i	−0.722521	+	1.84095i	−0.222521	+	0.974928i	$0.571429\pi$
$193$	−0.722521	+	1.84095i	−0.722521	+	1.84095i	−0.500000	+	0.866025i	$0.666667\pi$
$194$	−0.142820	−	0.0440542i	−0.142820	−	0.0440542i
$195$		0			0
$196$	0.623490	+	0.781831i	0.623490	+	0.781831i
$197$	1.80194			1.80194			0.900969	−	0.433884i	$-0.142857\pi$
$197$	1.80194			1.80194			0.900969	+	0.433884i	$0.142857\pi$
$198$	1.57906	+	0.487076i	1.57906	+	0.487076i
$199$	0.455573	−	1.16078i	0.455573	−	1.16078i	−0.500000	−	0.866025i	$-0.666667\pi$
$199$	0.455573	−	1.16078i	0.455573	−	1.16078i	0.955573	−	0.294755i	$-0.0952381\pi$
$200$	−0.949729	−	0.647514i	−0.949729	−	0.647514i
$201$		0			0
$202$	−1.12349	+	1.40881i	−1.12349	+	1.40881i
$203$	−1.57906	−	1.07659i	−1.57906	−	1.07659i
$204$		0			0
$205$		0			0
$206$	−1.78181	−	0.268565i	−1.78181	−	0.268565i
$207$		0			0
$208$		0			0
$209$		0			0
$210$	0.914101	−	1.14625i	0.914101	−	1.14625i
$211$		0			0		0.222521	−	0.974928i	$-0.428571\pi$
$211$		0			0		−0.222521	+	0.974928i	$0.571429\pi$
$212$	0.365341	+	0.930874i	0.365341	+	0.930874i
$213$		0			0
$214$	0.988831	+	1.71271i	0.988831	+	1.71271i
$215$		0			0
$216$	−0.900969	−	0.433884i	−0.900969	−	0.433884i
$217$	0.266948	+	0.680173i	0.266948	+	0.680173i
$218$		0			0
$219$	−0.0931869	+	1.24349i	−0.0931869	+	1.24349i
$220$	−2.31507	+	0.714104i	−2.31507	+	0.714104i
$221$		0			0
$222$		0			0
$223$	−0.134659	+	0.0648483i	−0.134659	+	0.0648483i	−0.500000	−	0.866025i	$-0.666667\pi$
$223$	−0.134659	+	0.0648483i	−0.134659	+	0.0648483i	0.365341	+	0.930874i	$0.380952\pi$
$224$	−0.0747301	+	0.997204i	−0.0747301	+	0.997204i
$225$	−1.03563	−	0.498732i	−1.03563	−	0.498732i
$226$		0			0
$227$	0.826239	+	1.43109i	0.826239	+	1.43109i	0.900969	+	0.433884i	$0.142857\pi$
$227$	0.826239	+	1.43109i	0.826239	+	1.43109i	−0.0747301	+	0.997204i	$0.523810\pi$
$228$		0			0
$229$		0			0		−0.365341	−	0.930874i	$-0.619048\pi$
$229$		0			0		0.365341	+	0.930874i	$0.380952\pi$
$230$		0			0
$231$	−0.367711	−	1.61105i	−0.367711	−	1.61105i
$232$	−0.425270	−	1.86323i	−0.425270	−	1.86323i
$233$		0			0		0.826239	−	0.563320i	$-0.190476\pi$
$233$		0			0		−0.826239	+	0.563320i	$0.809524\pi$
$234$		0			0
$235$		0			0
$236$	−0.733052	−	0.680173i	−0.733052	−	0.680173i
$237$	1.23305	+	1.54620i	1.23305	+	1.54620i
$238$		0			0
$239$		0			0		0.623490	−	0.781831i	$-0.285714\pi$
$239$		0			0		−0.623490	+	0.781831i	$0.714286\pi$
$240$	1.44973	−	0.218511i	1.44973	−	0.218511i
$241$	1.57906	+	1.07659i	1.57906	+	1.07659i	0.955573	+	0.294755i	$0.0952381\pi$
$241$	1.57906	+	1.07659i	1.57906	+	1.07659i	0.623490	+	0.781831i	$0.285714\pi$
$242$	−0.632289	+	1.61105i	−0.632289	+	1.61105i
$243$	−0.955573	−	0.294755i	−0.955573	−	0.294755i
$244$		0			0
$245$	−1.44973	−	0.218511i	−1.44973	−	0.218511i
$246$		0			0
$247$		0			0
$248$	−0.266948	+	0.680173i	−0.266948	+	0.680173i
$249$	1.57906	+	1.07659i	1.57906	+	1.07659i
$250$	0.216677	−	0.0326588i	0.216677	−	0.0326588i
$251$	−0.455573	+	0.571270i	−0.455573	+	0.571270i	−0.955573	−	0.294755i	$-0.904762\pi$
$251$	−0.455573	+	0.571270i	−0.455573	+	0.571270i	0.500000	+	0.866025i	$0.333333\pi$
$252$	0.0747301	+	0.997204i	0.0747301	+	0.997204i
$253$		0			0
$254$	0.535628	+	0.496990i	0.535628	+	0.496990i
$255$		0			0
$256$	−0.733052	+	0.680173i	−0.733052	+	0.680173i
$257$		0			0		0.826239	−	0.563320i	$-0.190476\pi$
$257$		0			0		−0.826239	+	0.563320i	$0.809524\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$	−0.698220	−	1.77904i	−0.698220	−	1.77904i
$262$	−0.147791	−	1.97213i	−0.147791	−	1.97213i
$263$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$263$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$264$	0.826239	−	1.43109i	0.826239	−	1.43109i
$265$	−1.32091	−	0.636119i	−1.32091	−	0.636119i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−1.82624	+	0.563320i	−1.82624	+	0.563320i	−0.826239	+	0.563320i	$0.809524\pi$
$269$	−1.82624	+	0.563320i	−1.82624	+	0.563320i	−1.00000			$\pi$
$270$	1.40097	−	0.432142i	1.40097	−	0.432142i
$271$	0.0111692	−	0.149042i	0.0111692	−	0.149042i	−0.988831	−	0.149042i	$-0.952381\pi$
$271$	0.0111692	−	0.149042i	0.0111692	−	0.149042i	1.00000			$0$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	0.949729	−	1.64498i	0.949729	−	1.64498i
$276$		0			0
$277$		0			0		−0.0747301	−	0.997204i	$-0.523810\pi$
$277$		0			0		0.0747301	+	0.997204i	$0.476190\pi$
$278$		0			0
$279$	−0.162592	+	0.712362i	−0.162592	+	0.712362i
$280$	−0.914101	−	1.14625i	−0.914101	−	1.14625i
$281$		0			0		−0.222521	−	0.974928i	$-0.571429\pi$
$281$		0			0		0.222521	+	0.974928i	$0.428571\pi$
$282$		0			0
$283$		0			0		0.733052	−	0.680173i	$-0.238095\pi$
$283$		0			0		−0.733052	+	0.680173i	$0.761905\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$	−0.623490	+	0.781831i	−0.623490	+	0.781831i
$289$	−0.988831	+	0.149042i	−0.988831	+	0.149042i
$290$	2.31507	+	1.57839i	2.31507	+	1.57839i
$291$	−0.0546039	+	0.139129i	−0.0546039	+	0.139129i
$292$	1.19158	+	0.367554i	1.19158	+	0.367554i
$293$	−0.149460			−0.149460			−0.0747301	−	0.997204i	$-0.523810\pi$
$293$	−0.149460			−0.149460			−0.0747301	+	0.997204i	$0.523810\pi$
$294$	0.826239	−	0.563320i	0.826239	−	0.563320i
$295$	1.46610			1.46610
$296$		0			0
$297$	0.603718	−	1.53825i	0.603718	−	1.53825i
$298$	1.03030	+	0.702449i	1.03030	+	0.702449i
$299$		0			0
$300$	−0.716677	+	0.898684i	−0.716677	+	0.898684i
$301$		0			0
$302$	0.623490	+	0.781831i	0.623490	+	0.781831i
$303$	1.32091	+	1.22563i	1.32091	+	1.22563i
$304$		0			0
$305$		0			0
$306$		0			0
$307$		0			0		−0.222521	−	0.974928i	$-0.571429\pi$
$307$		0			0		0.222521	+	0.974928i	$0.428571\pi$
$308$	−1.65248			−1.65248
$309$	−0.400969	+	1.75676i	−0.400969	+	1.75676i
$310$	−0.391374	−	0.997204i	−0.391374	−	0.997204i
$311$		0			0		−0.0747301	−	0.997204i	$-0.523810\pi$
$311$		0			0		0.0747301	+	0.997204i	$0.476190\pi$
$312$		0			0
$313$	0.988831	−	1.71271i	0.988831	−	1.71271i	0.365341	−	0.930874i	$-0.380952\pi$
$313$	0.988831	−	1.71271i	0.988831	−	1.71271i	0.623490	−	0.781831i	$-0.285714\pi$
$314$		0			0
$315$	−1.07473	−	0.997204i	−1.07473	−	0.997204i
$316$	1.78181	−	0.858075i	1.78181	−	0.858075i
$317$	−0.0111692	+	0.149042i	−0.0111692	+	0.149042i	0.988831	+	0.149042i	$0.0476190\pi$
$317$	−0.0111692	+	0.149042i	−0.0111692	+	0.149042i	−1.00000			$\pi$
$318$	0.955573	−	0.294755i	0.955573	−	0.294755i
$319$	3.01782	−	0.930874i	3.01782	−	0.930874i
$320$	0.109562	−	1.46200i	0.109562	−	1.46200i
$321$	1.78181	−	0.858075i	1.78181	−	0.858075i
$322$		0			0
$323$		0			0
$324$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$325$		0			0
$326$		0			0
$327$		0			0
$328$		0			0
$329$		0			0
$330$	0.539102	+	2.36196i	0.539102	+	2.36196i
$331$		0			0		0.826239	−	0.563320i	$-0.190476\pi$
$331$		0			0		−0.826239	+	0.563320i	$0.809524\pi$
$332$	1.40097	−	1.29991i	1.40097	−	1.29991i
$333$		0			0
$334$		0			0
$335$		0			0
$336$	0.988831	+	0.149042i	0.988831	+	0.149042i
$337$	−0.623490	+	0.781831i	−0.623490	+	0.781831i	−0.988831	−	0.149042i	$-0.952381\pi$
$337$	−0.623490	+	0.781831i	−0.623490	+	0.781831i	0.365341	+	0.930874i	$0.380952\pi$
$338$	0.988831	−	0.149042i	0.988831	−	0.149042i
$339$		0			0
$340$		0			0
$341$	−1.15379	−	0.355898i	−1.15379	−	0.355898i
$342$		0			0
$343$	−0.900969	−	0.433884i	−0.900969	−	0.433884i
$344$		0			0
$345$		0			0
$346$	−0.162592	+	0.414278i	−0.162592	+	0.414278i
$347$	0.367711	+	0.250701i	0.367711	+	0.250701i	0.733052	−	0.680173i	$-0.238095\pi$
$347$	0.367711	+	0.250701i	0.367711	+	0.250701i	−0.365341	+	0.930874i	$0.619048\pi$
$348$	−1.88980	+	0.284841i	−1.88980	+	0.284841i
$349$		0			0		0.623490	−	0.781831i	$-0.285714\pi$
$349$		0			0		−0.623490	+	0.781831i	$0.714286\pi$
$350$	1.13662	+	0.171318i	1.13662	+	0.171318i
$351$		0			0
$352$	−1.21135	−	1.12397i	−1.21135	−	1.12397i
$353$		0			0		−0.988831	−	0.149042i	$-0.952381\pi$
$353$		0			0		0.988831	+	0.149042i	$0.0476190\pi$
$354$	−0.733052	+	0.680173i	−0.733052	+	0.680173i
$355$		0			0
$356$		0			0
$357$		0			0
$358$	0.0990311	−	0.433884i	0.0990311	−	0.433884i
$359$		0			0		−0.365341	−	0.930874i	$-0.619048\pi$
$359$		0			0		0.365341	+	0.930874i	$0.380952\pi$
$360$	−0.109562	−	1.46200i	−0.109562	−	1.46200i
$361$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$362$		0			0
$363$	1.55929	+	0.750915i	1.55929	+	0.750915i
$364$		0			0
$365$	−1.64715	+	0.793227i	−1.64715	+	0.793227i
$366$		0			0
$367$	−0.955573	+	0.294755i	−0.955573	+	0.294755i	−0.733052	−	0.680173i	$-0.761905\pi$
$367$	−0.955573	+	0.294755i	−0.955573	+	0.294755i	−0.222521	+	0.974928i	$0.571429\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	−0.733052	−	0.680173i	−0.733052	−	0.680173i
$372$	0.658322	+	0.317031i	0.658322	+	0.317031i
$373$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$373$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$374$		0			0
$375$	−0.0163752	−	0.218511i	−0.0163752	−	0.218511i
$376$		0			0
$377$		0			0
$378$	1.00000			1.00000
$379$		0			0		−0.222521	−	0.974928i	$-0.571429\pi$
$379$		0			0		0.222521	+	0.974928i	$0.428571\pi$
$380$		0			0
$381$	0.535628	−	0.496990i	0.535628	−	0.496990i
$382$		0			0
$383$		0			0		−0.733052	−	0.680173i	$-0.761905\pi$
$383$		0			0		0.733052	+	0.680173i	$0.238095\pi$
$384$	0.623490	+	0.781831i	0.623490	+	0.781831i
$385$	1.77597	−	1.64786i	1.77597	−	1.64786i
$386$	1.23305	−	1.54620i	1.23305	−	1.54620i
$387$		0			0
$388$	0.123490	+	0.0841939i	0.123490	+	0.0841939i
$389$	0.162592	−	0.414278i	0.162592	−	0.414278i	−0.826239	−	0.563320i	$-0.809524\pi$
$389$	0.162592	−	0.414278i	0.162592	−	0.414278i	0.988831	+	0.149042i	$0.0476190\pi$
$390$		0			0
$391$		0			0
$392$	−0.365341	−	0.930874i	−0.365341	−	0.930874i
$393$	−1.97766			−1.97766
$394$	−1.72188	−	0.531130i	−1.72188	−	0.531130i
$395$	−1.05929	+	2.69903i	−1.05929	+	2.69903i
$396$	−1.36534	−	0.930874i	−1.36534	−	0.930874i
$397$		0			0		0.988831	−	0.149042i	$-0.0476190\pi$
$397$		0			0		−0.988831	+	0.149042i	$0.952381\pi$
$398$	−0.777479	+	0.974928i	−0.777479	+	0.974928i
$399$		0			0
$400$	0.716677	+	0.898684i	0.716677	+	0.898684i
$401$		0			0		−0.733052	−	0.680173i	$-0.761905\pi$
$401$		0			0		0.733052	+	0.680173i	$0.238095\pi$
$402$		0			0
$403$		0			0
$404$	1.48883	−	1.01507i	1.48883	−	1.01507i
$405$	−0.326239	−	1.42935i	−0.326239	−	1.42935i
$406$	1.19158	+	1.49419i	1.19158	+	1.49419i
$407$		0			0
$408$		0			0
$409$	−0.0747301	−	0.997204i	−0.0747301	−	0.997204i	−0.900969	−	0.433884i	$-0.857143\pi$
$409$	−0.0747301	−	0.997204i	−0.0747301	−	0.997204i	0.826239	−	0.563320i	$-0.190476\pi$
$410$		0			0
$411$		0			0
$412$	1.62349	+	0.781831i	1.62349	+	0.781831i
$413$	0.955573	+	0.294755i	0.955573	+	0.294755i
$414$		0			0
$415$	−0.209389	+	2.79410i	−0.209389	+	2.79410i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−0.400969	+	0.193096i	−0.400969	+	0.193096i	−0.623490	−	0.781831i	$-0.714286\pi$
$419$	−0.400969	+	0.193096i	−0.400969	+	0.193096i	0.222521	+	0.974928i	$0.428571\pi$
$420$	−1.21135	+	0.825886i	−1.21135	+	0.825886i
$421$		0			0		−0.900969	−	0.433884i	$-0.857143\pi$
$421$		0			0		0.900969	+	0.433884i	$0.142857\pi$
$422$		0			0
$423$		0			0
$424$	−0.0747301	−	0.997204i	−0.0747301	−	0.997204i
$425$		0			0
$426$		0			0
$427$		0			0
$428$	−0.440071	−	1.92808i	−0.440071	−	1.92808i
$429$		0			0
$430$		0			0
$431$		0			0		−0.988831	−	0.149042i	$-0.952381\pi$
$431$		0			0		0.988831	+	0.149042i	$0.0476190\pi$
$432$	0.733052	+	0.680173i	0.733052	+	0.680173i
$433$	−0.277479	−	0.347948i	−0.277479	−	0.347948i	0.623490	−	0.781831i	$-0.285714\pi$
$433$	−0.277479	−	0.347948i	−0.277479	−	0.347948i	−0.900969	+	0.433884i	$0.857143\pi$
$434$	−0.0546039	−	0.728639i	−0.0546039	−	0.728639i
$435$	1.74698	−	2.19064i	1.74698	−	2.19064i
$436$		0			0
$437$		0			0
$438$	0.455573	−	1.16078i	0.455573	−	1.16078i
$439$	1.57906	+	0.487076i	1.57906	+	0.487076i	0.955573	−	0.294755i	$-0.0952381\pi$
$439$	1.57906	+	0.487076i	1.57906	+	0.487076i	0.623490	+	0.781831i	$0.285714\pi$
$440$	2.42270			2.42270
$441$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$442$		0			0
$443$	−0.142820	−	0.0440542i	−0.142820	−	0.0440542i	0.222521	−	0.974928i	$-0.428571\pi$
$443$	−0.142820	−	0.0440542i	−0.142820	−	0.0440542i	−0.365341	+	0.930874i	$0.619048\pi$
$444$		0			0
$445$		0			0
$446$	0.147791	−	0.0222759i	0.147791	−	0.0222759i
$447$	0.777479	−	0.974928i	0.777479	−	0.974928i
$448$	0.365341	−	0.930874i	0.365341	−	0.930874i
$449$		0			0		−0.623490	−	0.781831i	$-0.714286\pi$
$449$		0			0		0.623490	+	0.781831i	$0.285714\pi$
$450$	0.842614	+	0.781831i	0.842614	+	0.781831i
$451$		0			0
$452$		0			0
$453$	0.826239	−	0.563320i	0.826239	−	0.563320i
$454$	−0.367711	−	1.61105i	−0.367711	−	1.61105i
$455$		0			0
$456$		0			0
$457$	0.698220	+	1.77904i	0.698220	+	1.77904i	0.623490	+	0.781831i	$0.285714\pi$
$457$	0.698220	+	1.77904i	0.698220	+	1.77904i	0.0747301	+	0.997204i	$0.476190\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	1.12349	+	0.541044i	1.12349	+	0.541044i	0.900969	−	0.433884i	$-0.142857\pi$
$461$	1.12349	+	0.541044i	1.12349	+	0.541044i	0.222521	+	0.974928i	$0.428571\pi$
$462$	−0.123490	+	1.64786i	−0.123490	+	1.64786i
$463$	0.400969	−	0.193096i	0.400969	−	0.193096i	−0.222521	−	0.974928i	$-0.571429\pi$
$463$	0.400969	−	0.193096i	0.400969	−	0.193096i	0.623490	+	0.781831i	$0.285714\pi$
$464$	−0.142820	+	1.90580i	−0.142820	+	1.90580i
$465$	−1.02366	+	0.315758i	−1.02366	+	0.315758i
$466$		0			0
$467$	0.134659	−	1.79690i	0.134659	−	1.79690i	−0.365341	−	0.930874i	$-0.619048\pi$
$467$	0.134659	−	1.79690i	0.134659	−	1.79690i	0.500000	−	0.866025i	$-0.333333\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$		0			0
$472$	0.500000	+	0.866025i	0.500000	+	0.866025i
$473$		0			0
$474$	−0.722521	−	1.84095i	−0.722521	−	1.84095i
$475$		0			0
$476$		0			0
$477$	−0.222521	−	0.974928i	−0.222521	−	0.974928i
$478$		0			0
$479$		0			0		0.733052	−	0.680173i	$-0.238095\pi$
$479$		0			0		−0.733052	+	0.680173i	$0.761905\pi$
$480$	−1.44973	−	0.218511i	−1.44973	−	0.218511i
$481$		0			0
$482$	−1.19158	−	1.49419i	−1.19158	−	1.49419i
$483$		0			0
$484$	1.07906	−	1.35310i	1.07906	−	1.35310i
$485$	−0.216677	+	0.0326588i	−0.216677	+	0.0326588i
$486$	0.826239	+	0.563320i	0.826239	+	0.563320i
$487$	0.266948	−	0.680173i	0.266948	−	0.680173i	−0.733052	−	0.680173i	$-0.761905\pi$
$487$	0.266948	−	0.680173i	0.266948	−	0.680173i	1.00000			$0$
$488$		0			0
$489$		0			0
$490$	1.32091	+	0.636119i	1.32091	+	0.636119i
$491$	1.46610			1.46610			0.733052	−	0.680173i	$-0.238095\pi$
$491$	1.46610			1.46610			0.733052	+	0.680173i	$0.238095\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$	2.39564	−	0.361085i	2.39564	−	0.361085i
$496$	0.455573	−	0.571270i	0.455573	−	0.571270i
$497$		0			0
$498$	−1.19158	−	1.49419i	−1.19158	−	1.49419i
$499$		0			0		−0.733052	−	0.680173i	$-0.761905\pi$
$499$		0			0		0.733052	+	0.680173i	$0.238095\pi$
$500$	−0.216677	−	0.0326588i	−0.216677	−	0.0326588i
$501$		0			0
$502$	0.603718	−	0.411608i	0.603718	−	0.411608i
$503$		0			0		−0.222521	−	0.974928i	$-0.571429\pi$
$503$		0			0		0.222521	+	0.974928i	$0.428571\pi$
$504$	0.222521	−	0.974928i	0.222521	−	0.974928i
$505$	−0.587862	+	2.57559i	−0.587862	+	2.57559i
$506$		0			0
$507$	−0.0747301	−	0.997204i	−0.0747301	−	0.997204i
$508$	−0.365341	−	0.632789i	−0.365341	−	0.632789i
$509$	−0.500000	+	0.866025i	−0.500000	+	0.866025i	0.500000	+	0.866025i	$0.333333\pi$
$509$	−0.500000	+	0.866025i	−0.500000	+	0.866025i	−1.00000			$\pi$
$510$		0			0
$511$	−1.23305	+	0.185853i	−1.23305	+	0.185853i
$512$	0.900969	−	0.433884i	0.900969	−	0.433884i
$513$		0			0
$514$		0			0
$515$	−2.52446	+	0.778692i	−2.52446	+	0.778692i
$516$		0			0
$517$		0			0
$518$		0			0
$519$	0.400969	+	0.193096i	0.400969	+	0.193096i
$520$		0			0
$521$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$521$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$522$	0.142820	+	1.90580i	0.142820	+	1.90580i
$523$		0			0		−0.365341	−	0.930874i	$-0.619048\pi$
$523$		0			0		0.365341	+	0.930874i	$0.380952\pi$
$524$	−0.440071	+	1.92808i	−0.440071	+	1.92808i
$525$	0.255779	−	1.12064i	0.255779	−	1.12064i
$526$		0			0
$527$		0			0
$528$	−1.21135	+	1.12397i	−1.21135	+	1.12397i
$529$	−0.988831	−	0.149042i	−0.988831	−	0.149042i
$530$	1.07473	+	0.997204i	1.07473	+	0.997204i
$531$	0.623490	+	0.781831i	0.623490	+	0.781831i
$532$		0			0
$533$		0			0
$534$		0			0
$535$	2.39564	+	1.63332i	2.39564	+	1.63332i
$536$		0			0
$537$	−0.425270	−	0.131178i	−0.425270	−	0.131178i
$538$	1.91115			1.91115
$539$	1.48883	−	0.716983i	1.48883	−	0.716983i
$540$	−1.46610			−1.46610
$541$		0			0		−0.955573	−	0.294755i	$-0.904762\pi$
$541$		0			0		0.955573	+	0.294755i	$0.0952381\pi$
$542$	−0.0546039	+	0.139129i	−0.0546039	+	0.139129i
$543$		0			0
$544$		0			0
$545$		0			0
$546$		0			0
$547$		0			0		−0.623490	−	0.781831i	$-0.714286\pi$
$547$		0			0		0.623490	+	0.781831i	$0.285714\pi$
$548$		0			0
$549$		0			0
$550$	−1.39240	+	1.29196i	−1.39240	+	1.29196i
$551$		0			0
$552$		0			0
$553$	−1.23305	+	1.54620i	−1.23305	+	1.54620i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−0.733052	−	1.26968i	−0.733052	−	1.26968i	−0.955573	−	0.294755i	$-0.904762\pi$
$557$	−0.733052	−	1.26968i	−0.733052	−	1.26968i	0.222521	−	0.974928i	$-0.428571\pi$
$558$	0.365341	−	0.632789i	0.365341	−	0.632789i
$559$		0			0
$560$	0.535628	+	1.36476i	0.535628	+	1.36476i
$561$		0			0
$562$		0			0
$563$	1.88980	−	0.582926i	1.88980	−	0.582926i	0.900969	−	0.433884i	$-0.142857\pi$
$563$	1.88980	−	0.582926i	1.88980	−	0.582926i	0.988831	−	0.149042i	$-0.0476190\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$	0.0747301	−	0.997204i	0.0747301	−	0.997204i
$568$		0			0
$569$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$569$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$570$		0			0
$571$		0			0		−0.0747301	−	0.997204i	$-0.523810\pi$
$571$		0			0		0.0747301	+	0.997204i	$0.476190\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$	0.826239	−	0.563320i	0.826239	−	0.563320i
$577$	−0.535628	+	0.496990i	−0.535628	+	0.496990i	−0.900969	−	0.433884i	$-0.857143\pi$
$577$	−0.535628	+	0.496990i	−0.535628	+	0.496990i	0.365341	+	0.930874i	$0.380952\pi$
$578$	0.988831	+	0.149042i	0.988831	+	0.149042i
$579$	−1.44973	−	1.34515i	−1.44973	−	1.34515i
$580$	−1.74698	−	2.19064i	−1.74698	−	2.19064i
$581$	−0.698220	+	1.77904i	−0.698220	+	1.77904i
$582$	0.0931869	−	0.116853i	0.0931869	−	0.116853i
$583$	1.63402	−	0.246289i	1.63402	−	0.246289i
$584$	−1.03030	−	0.702449i	−1.03030	−	0.702449i
$585$		0			0
$586$	0.142820	+	0.0440542i	0.142820	+	0.0440542i
$587$	−1.65248			−1.65248			−0.826239	−	0.563320i	$-0.809524\pi$
$587$	−1.65248			−1.65248			−0.826239	+	0.563320i	$0.809524\pi$
$588$	−0.955573	+	0.294755i	−0.955573	+	0.294755i
$589$		0			0
$590$	−1.40097	−	0.432142i	−1.40097	−	0.432142i
$591$	−0.658322	+	1.67738i	−0.658322	+	1.67738i
$592$		0			0
$593$		0			0		0.988831	−	0.149042i	$-0.0476190\pi$
$593$		0			0		−0.988831	+	0.149042i	$0.952381\pi$
$594$	−1.03030	+	1.29196i	−1.03030	+	1.29196i
$595$		0			0
$596$	−0.777479	−	0.974928i	−0.777479	−	0.974928i
$597$	0.914101	+	0.848162i	0.914101	+	0.848162i
$598$		0			0
$599$		0			0		0.733052	−	0.680173i	$-0.238095\pi$
$599$		0			0		−0.733052	+	0.680173i	$0.761905\pi$
$600$	0.949729	−	0.647514i	0.949729	−	0.647514i
$601$	−0.162592	−	0.712362i	−0.162592	−	0.712362i	−0.988831	−	0.149042i	$-0.952381\pi$
$601$	−0.162592	−	0.712362i	−0.162592	−	0.712362i	0.826239	−	0.563320i	$-0.190476\pi$
$602$		0			0
$603$		0			0
$604$	−0.365341	−	0.930874i	−0.365341	−	0.930874i
$605$	0.189617	+	2.53026i	0.189617	+	2.53026i
$606$	−0.900969	−	1.56052i	−0.900969	−	1.56052i
$607$	−0.826239	+	1.43109i	−0.826239	+	1.43109i	0.0747301	+	0.997204i	$0.476190\pi$
$607$	−0.826239	+	1.43109i	−0.826239	+	1.43109i	−0.900969	+	0.433884i	$0.857143\pi$
$608$		0			0
$609$	1.57906	−	1.07659i	1.57906	−	1.07659i
$610$		0			0
$611$		0			0
$612$		0			0
$613$		0			0		0.955573	−	0.294755i	$-0.0952381\pi$
$613$		0			0		−0.955573	+	0.294755i	$0.904762\pi$
$614$		0			0
$615$		0			0
$616$	1.57906	+	0.487076i	1.57906	+	0.487076i
$617$		0			0		−0.900969	−	0.433884i	$-0.857143\pi$
$617$		0			0		0.900969	+	0.433884i	$0.142857\pi$
$618$	0.900969	−	1.56052i	0.900969	−	1.56052i
$619$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$619$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$620$	0.0800550	+	1.06826i	0.0800550	+	1.06826i
$621$		0			0
$622$		0			0
$623$		0			0
$624$		0			0
$625$	−0.684292	+	0.466542i	−0.684292	+	0.466542i
$626$	−1.44973	+	1.34515i	−1.44973	+	1.34515i
$627$		0			0
$628$		0			0
$629$		0			0
$630$	0.733052	+	1.26968i	0.733052	+	1.26968i
$631$	0.0931869	−	0.116853i	0.0931869	−	0.116853i	−0.733052	−	0.680173i	$-0.761905\pi$
$631$	0.0931869	−	0.116853i	0.0931869	−	0.116853i	0.826239	+	0.563320i	$0.190476\pi$
$632$	−1.95557	+	0.294755i	−1.95557	+	0.294755i
$633$		0			0
$634$	0.0546039	−	0.139129i	0.0546039	−	0.139129i
$635$	1.02366	+	0.315758i	1.02366	+	0.315758i
$636$	−1.00000			−1.00000
$637$		0			0
$638$	−3.15813			−3.15813
$639$		0			0
$640$	−0.535628	+	1.36476i	−0.535628	+	1.36476i
$641$		0			0		−0.826239	−	0.563320i	$-0.809524\pi$
$641$		0			0		0.826239	+	0.563320i	$0.190476\pi$
$642$	−1.95557	+	0.294755i	−1.95557	+	0.294755i
$643$		0			0		0.623490	−	0.781831i	$-0.285714\pi$
$643$		0			0		−0.623490	+	0.781831i	$0.714286\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		0			0		−0.988831	−	0.149042i	$-0.952381\pi$
$647$		0			0		0.988831	+	0.149042i	$0.0476190\pi$
$648$	0.733052	−	0.680173i	0.733052	−	0.680173i
$649$	−1.36534	+	0.930874i	−1.36534	+	0.930874i
$650$		0			0
$651$	−0.730682			−0.730682
$652$		0			0
$653$	0.535628	+	1.36476i	0.535628	+	1.36476i	0.900969	+	0.433884i	$0.142857\pi$
$653$	0.535628	+	1.36476i	0.535628	+	1.36476i	−0.365341	+	0.930874i	$0.619048\pi$
$654$		0			0
$655$	−1.44973	−	2.51100i	−1.44973	−	2.51100i
$656$		0			0
$657$	−1.12349	−	0.541044i	−1.12349	−	0.541044i
$658$		0			0
$659$	−1.62349	+	0.781831i	−1.62349	+	0.781831i	−0.623490	+	0.781831i	$0.714286\pi$
$659$	−1.62349	+	0.781831i	−1.62349	+	0.781831i	−1.00000			$1.00000\pi$
$660$	0.181049	−	2.41593i	0.181049	−	2.41593i
$661$		0			0		0.955573	−	0.294755i	$-0.0952381\pi$
$661$		0			0		−0.955573	+	0.294755i	$0.904762\pi$
$662$		0			0
$663$		0			0
$664$	−1.72188	+	0.829215i	−1.72188	+	0.829215i
$665$		0			0
$666$		0			0
$667$		0			0
$668$		0			0
$669$	−0.0111692	−	0.149042i	−0.0111692	−	0.149042i
$670$		0			0
$671$		0			0
$672$	−0.900969	−	0.433884i	−0.900969	−	0.433884i
$673$	−0.367711	−	1.61105i	−0.367711	−	1.61105i	−0.733052	−	0.680173i	$-0.761905\pi$
$673$	−0.367711	−	1.61105i	−0.367711	−	1.61105i	0.365341	−	0.930874i	$-0.380952\pi$
$674$	0.826239	−	0.563320i	0.826239	−	0.563320i
$675$	0.842614	−	0.781831i	0.842614	−	0.781831i
$676$	−0.988831	−	0.149042i	−0.988831	−	0.149042i
$677$	1.21135	+	1.12397i	1.21135	+	1.12397i	0.988831	+	0.149042i	$0.0476190\pi$
$677$	1.21135	+	1.12397i	1.21135	+	1.12397i	0.222521	+	0.974928i	$0.428571\pi$
$678$		0			0
$679$	−0.147791	−	0.0222759i	−0.147791	−	0.0222759i
$680$		0			0
$681$	−1.63402	+	0.246289i	−1.63402	+	0.246289i
$682$	0.997630	+	0.680173i	0.997630	+	0.680173i
$683$	0.535628	−	1.36476i	0.535628	−	1.36476i	−0.365341	−	0.930874i	$-0.619048\pi$
$683$	0.535628	−	1.36476i	0.535628	−	1.36476i	0.900969	−	0.433884i	$-0.142857\pi$
$684$		0			0
$685$		0			0
$686$	0.733052	+	0.680173i	0.733052	+	0.680173i
$687$		0			0
$688$		0			0
$689$		0			0
$690$		0			0
$691$		0			0		0.988831	−	0.149042i	$-0.0476190\pi$
$691$		0			0		−0.988831	+	0.149042i	$0.952381\pi$
$692$	0.277479	−	0.347948i	0.277479	−	0.347948i
$693$	1.63402	+	0.246289i	1.63402	+	0.246289i
$694$	−0.277479	−	0.347948i	−0.277479	−	0.347948i
$695$		0			0
$696$	1.88980	+	0.284841i	1.88980	+	0.284841i
$697$		0			0
$698$		0			0
$699$		0			0
$700$	−1.03563	−	0.498732i	−1.03563	−	0.498732i
$701$	0.162592	−	0.712362i	0.162592	−	0.712362i	−0.826239	−	0.563320i	$-0.809524\pi$
$701$	0.162592	−	0.712362i	0.162592	−	0.712362i	0.988831	−	0.149042i	$-0.0476190\pi$
$702$		0			0
$703$		0			0
$704$	0.826239	+	1.43109i	0.826239	+	1.43109i
$705$		0			0
$706$		0			0
$707$	−0.900969	+	1.56052i	−0.900969	+	1.56052i
$708$	0.900969	−	0.433884i	0.900969	−	0.433884i
$709$		0			0		0.0747301	−	0.997204i	$-0.476190\pi$
$709$		0			0		−0.0747301	+	0.997204i	$0.523810\pi$
$710$		0			0
$711$	−1.88980	+	0.582926i	−1.88980	+	0.582926i
$712$		0			0
$713$		0			0
$714$		0			0
$715$		0			0
$716$	−0.222521	+	0.385418i	−0.222521	+	0.385418i
$717$		0			0
$718$		0			0
$719$		0			0		−0.365341	−	0.930874i	$-0.619048\pi$
$719$		0			0		0.365341	+	0.930874i	$0.380952\pi$
$720$	−0.326239	+	1.42935i	−0.326239	+	1.42935i
$721$	−1.80194			−1.80194
$722$	0.222521	+	0.974928i	0.222521	+	0.974928i
$723$	−1.57906	+	1.07659i	−1.57906	+	1.07659i
$724$		0			0
$725$	2.17225	+	0.327414i	2.17225	+	0.327414i
$726$	−1.26868	−	1.17716i	−1.26868	−	1.17716i
$727$	1.19158	+	1.49419i	1.19158	+	1.49419i	0.826239	+	0.563320i	$0.190476\pi$
$727$	1.19158	+	1.49419i	1.19158	+	1.49419i	0.365341	+	0.930874i	$0.380952\pi$
$728$		0			0
$729$	0.623490	−	0.781831i	0.623490	−	0.781831i
$730$	1.80778	−	0.272479i	1.80778	−	0.272479i
$731$		0			0
$732$		0			0
$733$		0			0		−0.955573	−	0.294755i	$-0.904762\pi$
$733$		0			0		0.955573	+	0.294755i	$0.0952381\pi$
$734$	1.00000			1.00000
$735$	0.733052	−	1.26968i	0.733052	−	1.26968i
$736$		0			0
$737$		0			0
$738$		0			0
$739$		0			0		−0.826239	−	0.563320i	$-0.809524\pi$
$739$		0			0		0.826239	+	0.563320i	$0.190476\pi$
$740$		0			0
$741$		0			0
$742$	0.500000	+	0.866025i	0.500000	+	0.866025i
$743$		0			0		−0.623490	−	0.781831i	$-0.714286\pi$
$743$		0			0		0.623490	+	0.781831i	$0.285714\pi$
$744$	−0.535628	−	0.496990i	−0.535628	−	0.496990i
$745$	1.80778	+	0.272479i	1.80778	+	0.272479i
$746$		0			0
$747$	−1.57906	+	1.07659i	−1.57906	+	1.07659i
$748$		0			0
$749$	1.23305	+	1.54620i	1.23305	+	1.54620i
$750$	−0.0487597	+	0.213630i	−0.0487597	+	0.213630i
$751$	0.603718	+	1.53825i	0.603718	+	1.53825i	0.826239	+	0.563320i	$0.190476\pi$
$751$	0.603718	+	1.53825i	0.603718	+	1.53825i	−0.222521	+	0.974928i	$0.571429\pi$
$752$		0			0
$753$	−0.365341	−	0.632789i	−0.365341	−	0.632789i
$754$		0			0
$755$	1.32091	+	0.636119i	1.32091	+	0.636119i
$756$	−0.955573	−	0.294755i	−0.955573	−	0.294755i
$757$		0			0		0.900969	−	0.433884i	$-0.142857\pi$
$757$		0			0		−0.900969	+	0.433884i	$0.857143\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$		0			0		0.0747301	−	0.997204i	$-0.476190\pi$
$761$		0			0		−0.0747301	+	0.997204i	$0.523810\pi$
$762$	−0.658322	+	0.317031i	−0.658322	+	0.317031i
$763$		0			0
$764$		0			0
$765$		0			0
$766$		0			0
$767$		0			0
$768$	−0.365341	−	0.930874i	−0.365341	−	0.930874i
$769$	0.440071	−	1.92808i	0.440071	−	1.92808i	0.0747301	−	0.997204i	$-0.476190\pi$
$769$	0.440071	−	1.92808i	0.440071	−	1.92808i	0.365341	−	0.930874i	$-0.380952\pi$
$770$	−2.18278	+	1.05117i	−2.18278	+	1.05117i
$771$		0			0
$772$	−1.63402	+	1.11406i	−1.63402	+	1.11406i
$773$	−1.32091	+	1.22563i	−1.32091	+	1.22563i	−0.365341	+	0.930874i	$0.619048\pi$
$773$	−1.32091	+	1.22563i	−1.32091	+	1.22563i	−0.955573	+	0.294755i	$0.904762\pi$
$774$		0			0
$775$	−0.615683	−	0.571270i	−0.615683	−	0.571270i
$776$	−0.0931869	−	0.116853i	−0.0931869	−	0.116853i
$777$		0			0
$778$	−0.277479	+	0.347948i	−0.277479	+	0.347948i
$779$		0			0
$780$		0			0
$781$		0			0
$782$		0			0
$783$	1.91115			1.91115
$784$	0.0747301	+	0.997204i	0.0747301	+	0.997204i
$785$		0			0
$786$	1.88980	+	0.582926i	1.88980	+	0.582926i
$787$		0			0		0.365341	−	0.930874i	$-0.380952\pi$
$787$		0			0		−0.365341	+	0.930874i	$0.619048\pi$
$788$	1.48883	+	1.01507i	1.48883	+	1.01507i
$789$		0			0
$790$	1.80778	−	2.26689i	1.80778	−	2.26689i
$791$		0			0
$792$	1.03030	+	1.29196i	1.03030	+	1.29196i
$793$		0			0
$794$		0			0
$795$	1.07473	−	0.997204i	1.07473	−	0.997204i
$796$	1.03030	−	0.702449i	1.03030	−	0.702449i
$797$	0.0332580	+	0.145713i	0.0332580	+	0.145713i	0.988831	−	0.149042i	$-0.0476190\pi$
$797$	0.0332580	+	0.145713i	0.0332580	+	0.145713i	−0.955573	+	0.294755i	$0.904762\pi$
$798$		0			0
$799$		0			0
$800$	−0.419945	−	1.07000i	−0.419945	−	1.07000i
$801$		0			0
$802$		0			0
$803$	1.03030	−	1.78454i	1.03030	−	1.78454i
$804$		0			0
$805$		0			0
$806$		0			0
$807$	0.142820	−	1.90580i	0.142820	−	1.90580i
$808$	−1.72188	+	0.531130i	−1.72188	+	0.531130i
$809$		0			0		0.955573	−	0.294755i	$-0.0952381\pi$
$809$		0			0		−0.955573	+	0.294755i	$0.904762\pi$
$810$	−0.109562	+	1.46200i	−0.109562	+	1.46200i
$811$		0			0		0.900969	−	0.433884i	$-0.142857\pi$
$811$		0			0		−0.900969	+	0.433884i	$0.857143\pi$
$812$	−0.698220	−	1.77904i	−0.698220	−	1.77904i
$813$	0.134659	+	0.0648483i	0.134659	+	0.0648483i
$814$		0			0
$815$		0			0
$816$		0			0
$817$		0			0
$818$	−0.222521	+	0.974928i	−0.222521	+	0.974928i
$819$		0			0
$820$		0			0
$821$	1.63402	−	1.11406i	1.63402	−	1.11406i	0.733052	−	0.680173i	$-0.238095\pi$
$821$	1.63402	−	1.11406i	1.63402	−	1.11406i	0.900969	−	0.433884i	$-0.142857\pi$
$822$		0			0
$823$	−1.23305	−	0.185853i	−1.23305	−	0.185853i	−0.500000	−	0.866025i	$-0.666667\pi$
$823$	−1.23305	−	0.185853i	−1.23305	−	0.185853i	−0.733052	+	0.680173i	$0.761905\pi$
$824$	−1.32091	−	1.22563i	−1.32091	−	1.22563i
$825$	1.18429	+	1.48506i	1.18429	+	1.48506i
$826$	−0.826239	−	0.563320i	−0.826239	−	0.563320i
$827$	0.914101	−	1.14625i	0.914101	−	1.14625i	−0.0747301	−	0.997204i	$-0.523810\pi$
$827$	0.914101	−	1.14625i	0.914101	−	1.14625i	0.988831	−	0.149042i	$-0.0476190\pi$
$828$		0			0
$829$		0			0		−0.826239	−	0.563320i	$-0.809524\pi$
$829$		0			0		0.826239	+	0.563320i	$0.190476\pi$
$830$	1.02366	−	2.60825i	1.02366	−	2.60825i
$831$		0			0
$832$		0			0
$833$		0			0
$834$		0			0
$835$		0			0
$836$		0			0
$837$	−0.603718	−	0.411608i	−0.603718	−	0.411608i
$838$	0.440071	−	0.0663300i	0.440071	−	0.0663300i
$839$		0			0		0.623490	−	0.781831i	$-0.285714\pi$
$839$		0			0		−0.623490	+	0.781831i	$0.714286\pi$
$840$	1.40097	−	0.432142i	1.40097	−	0.432142i
$841$	1.65379	+	2.07379i	1.65379	+	2.07379i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	1.21135	−	0.825886i	1.21135	−	0.825886i
$846$		0			0
$847$	−0.385113	+	1.68729i	−0.385113	+	1.68729i
$848$	−0.222521	+	0.974928i	−0.222521	+	0.974928i
$849$		0			0
$850$		0			0
$851$		0			0
$852$		0			0
$853$		0			0		−0.900969	−	0.433884i	$-0.857143\pi$
$853$		0			0		0.900969	+	0.433884i	$0.142857\pi$
$854$		0			0
$855$		0			0
$856$	−0.147791	+	1.97213i	−0.147791	+	1.97213i
$857$		0			0		0.955573	−	0.294755i	$-0.0952381\pi$
$857$		0			0		−0.955573	+	0.294755i	$0.904762\pi$
$858$		0			0
$859$		0			0		0.0747301	−	0.997204i	$-0.476190\pi$
$859$		0			0		−0.0747301	+	0.997204i	$0.523810\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$863$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$864$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$865$	0.0487597	+	0.650653i	0.0487597	+	0.650653i
$866$	0.162592	+	0.414278i	0.162592	+	0.414278i
$867$	0.222521	−	0.974928i	0.222521	−	0.974928i
$868$	−0.162592	+	0.712362i	−0.162592	+	0.712362i
$869$	−0.727208	−	3.18610i	−0.727208	−	3.18610i
$870$	−2.31507	+	1.57839i	−2.31507	+	1.57839i
$871$		0			0
$872$		0			0
$873$	−0.109562	−	0.101659i	−0.109562	−	0.101659i
$874$		0			0
$875$	0.209389	−	0.0645880i	0.209389	−	0.0645880i
$876$	−0.777479	+	0.974928i	−0.777479	+	0.974928i
$877$		0			0		0.988831	−	0.149042i	$-0.0476190\pi$
$877$		0			0		−0.988831	+	0.149042i	$0.952381\pi$
$878$	−1.36534	−	0.930874i	−1.36534	−	0.930874i
$879$	0.0546039	−	0.139129i	0.0546039	−	0.139129i
$880$	−2.31507	−	0.714104i	−2.31507	−	0.714104i
$881$		0			0		1.00000			$0$
$881$		0			0		−1.00000			$\pi$
$882$	0.222521	+	0.974928i	0.222521	+	0.974928i
$883$		0			0		1.00000			$0$
$883$		0			0		−1.00000			$\pi$
$884$		0			0
$885$	−0.535628	+	1.36476i	−0.535628	+	1.36476i
$886$	0.123490	+	0.0841939i	0.123490	+	0.0841939i
$887$		0			0		0.988831	−	0.149042i	$-0.0476190\pi$
$887$		0			0		−0.988831	+	0.149042i	$0.952381\pi$
$888$		0			0
$889$	0.603718	+	0.411608i	0.603718	+	0.411608i
$890$		0			0
$891$	1.21135	+	1.12397i	1.21135	+	1.12397i
$892$	−0.147791	−	0.0222759i	−0.147791	−	0.0222759i
$893$		0			0
$894$	−1.03030	+	0.702449i	−1.03030	+	0.702449i
$895$	−0.145190	−	0.636119i	−0.145190	−	0.636119i
$896$	−0.623490	+	0.781831i	−0.623490	+	0.781831i
$897$		0			0
$898$		0			0
$899$	−0.104356	−	1.39254i	−0.104356	−	1.39254i
$900$	−0.574730	−	0.995462i	−0.574730	−	0.995462i
$901$		0			0
$902$		0			0
$903$		0			0
$904$		0			0
$905$		0			0
$906$	−0.955573	+	0.294755i	−0.955573	+	0.294755i
$907$		0			0		0.955573	−	0.294755i	$-0.0952381\pi$
$907$		0			0		−0.955573	+	0.294755i	$0.904762\pi$
$908$	−0.123490	+	1.64786i	−0.123490	+	1.64786i
$909$	−1.62349	+	0.781831i	−1.62349	+	0.781831i
$910$		0			0
$911$		0			0		−0.900969	−	0.433884i	$-0.857143\pi$
$911$		0			0		0.900969	+	0.433884i	$0.142857\pi$
$912$		0			0
$913$	−1.57906	−	2.73502i	−1.57906	−	2.73502i
$914$	−0.142820	−	1.90580i	−0.142820	−	1.90580i
$915$		0			0
$916$		0			0
$917$	−0.440071	−	1.92808i	−0.440071	−	1.92808i
$918$		0			0
$919$	−1.48883	+	1.01507i	−1.48883	+	1.01507i	−0.500000	+	0.866025i	$0.666667\pi$
$919$	−1.48883	+	1.01507i	−1.48883	+	1.01507i	−0.988831	+	0.149042i	$0.952381\pi$
$920$		0			0
$921$		0			0
$922$	−0.914101	−	0.848162i	−0.914101	−	0.848162i
$923$		0			0
$924$	0.603718	−	1.53825i	0.603718	−	1.53825i
$925$		0			0
$926$	−0.440071	+	0.0663300i	−0.440071	+	0.0663300i
$927$	−1.48883	−	1.01507i	−1.48883	−	1.01507i
$928$	0.698220	−	1.77904i	0.698220	−	1.77904i
$929$		0			0		−0.955573	−	0.294755i	$-0.904762\pi$
$929$		0			0		0.955573	+	0.294755i	$0.0952381\pi$
$930$	1.07126			1.07126
$931$		0			0
$932$		0			0
$933$		0			0
$934$	−0.658322	+	1.67738i	−0.658322	+	1.67738i
$935$		0			0
$936$		0			0
$937$	0.455573	−	0.571270i	0.455573	−	0.571270i	−0.500000	−	0.866025i	$-0.666667\pi$
$937$	0.455573	−	0.571270i	0.455573	−	0.571270i	0.955573	+	0.294755i	$0.0952381\pi$
$938$		0			0
$939$	1.23305	+	1.54620i	1.23305	+	1.54620i
$940$		0			0
$941$	0.147791	+	0.0222759i	0.147791	+	0.0222759i	0.222521	−	0.974928i	$-0.428571\pi$
$941$	0.147791	+	0.0222759i	0.147791	+	0.0222759i	−0.0747301	+	0.997204i	$0.523810\pi$
$942$		0			0
$943$		0			0
$944$	−0.222521	−	0.974928i	−0.222521	−	0.974928i
$945$	1.32091	−	0.636119i	1.32091	−	0.636119i
$946$		0			0
$947$	−0.455573	−	1.16078i	−0.455573	−	1.16078i	−0.955573	−	0.294755i	$-0.904762\pi$
$947$	−0.455573	−	1.16078i	−0.455573	−	1.16078i	0.500000	−	0.866025i	$-0.333333\pi$
$948$	0.147791	+	1.97213i	0.147791	+	1.97213i
$949$		0			0
$950$		0			0
$951$	−0.134659	−	0.0648483i	−0.134659	−	0.0648483i
$952$		0			0
$953$		0			0		0.900969	−	0.433884i	$-0.142857\pi$
$953$		0			0		−0.900969	+	0.433884i	$0.857143\pi$
$954$	−0.0747301	+	0.997204i	−0.0747301	+	0.997204i
$955$		0			0
$956$		0			0
$957$	−0.236007	+	3.14929i	−0.236007	+	3.14929i
$958$		0			0
$959$		0			0
$960$	1.32091	+	0.636119i	1.32091	+	0.636119i
$961$	0.233052	−	0.403658i	0.233052	−	0.403658i
$962$		0			0
$963$	0.147791	+	1.97213i	0.147791	+	1.97213i
$964$	0.698220	+	1.77904i	0.698220	+	1.77904i
$965$	0.645190	−	2.82676i	0.645190	−	2.82676i
$966$		0			0
$967$	0.440071	+	1.92808i	0.440071	+	1.92808i	0.365341	+	0.930874i	$0.380952\pi$
$967$	0.440071	+	1.92808i	0.440071	+	1.92808i	0.0747301	+	0.997204i	$0.476190\pi$
$968$	−1.42996	+	0.974928i	−1.42996	+	0.974928i
$969$		0			0
$970$	0.216677	+	0.0326588i	0.216677	+	0.0326588i
$971$	0.535628	+	0.496990i	0.535628	+	0.496990i	0.900969	−	0.433884i	$-0.142857\pi$
$971$	0.535628	+	0.496990i	0.535628	+	0.496990i	−0.365341	+	0.930874i	$0.619048\pi$
$972$	−0.623490	−	0.781831i	−0.623490	−	0.781831i
$973$		0			0
$974$	−0.455573	+	0.571270i	−0.455573	+	0.571270i
$975$		0			0
$976$		0			0
$977$		0			0		0.365341	−	0.930874i	$-0.380952\pi$
$977$		0			0		−0.365341	+	0.930874i	$0.619048\pi$
$978$		0			0
$979$		0			0
$980$	−1.07473	−	0.997204i	−1.07473	−	0.997204i
$981$		0			0
$982$	−1.40097	−	0.432142i	−1.40097	−	0.432142i
$983$		0			0		0.365341	−	0.930874i	$-0.380952\pi$
$983$		0			0		−0.365341	+	0.930874i	$0.619048\pi$
$984$		0			0
$985$	−2.61232	+	0.393744i	−2.61232	+	0.393744i
$986$		0			0
$987$		0			0
$988$		0			0
$989$		0			0
$990$	−2.39564	−	0.361085i	−2.39564	−	0.361085i
$991$	−1.21135	+	1.12397i	−1.21135	+	1.12397i	−0.222521	+	0.974928i	$0.571429\pi$
$991$	−1.21135	+	1.12397i	−1.21135	+	1.12397i	−0.988831	+	0.149042i	$0.952381\pi$
$992$	−0.603718	+	0.411608i	−0.603718	+	0.411608i
$993$		0			0
$994$		0			0
$995$	−0.406813	+	1.78236i	−0.406813	+	1.78236i
$996$	0.698220	+	1.77904i	0.698220	+	1.77904i
$997$		0			0		−0.0747301	−	0.997204i	$-0.523810\pi$
$997$		0			0		0.0747301	+	0.997204i	$0.476190\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1176.1.ck.a.149.1	✓	12
3.2	odd	2		1176.1.ck.b.149.1	yes	12
8.5	even	2		1176.1.ck.b.149.1	yes	12
24.5	odd	2	CM	1176.1.ck.a.149.1	✓	12
49.25	even	21	inner	1176.1.ck.a.221.1	yes	12
147.74	odd	42		1176.1.ck.b.221.1	yes	12
392.221	even	42		1176.1.ck.b.221.1	yes	12
1176.221	odd	42	inner	1176.1.ck.a.221.1	yes	12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1176.1.ck.a.149.1	✓	12	1.1	even	1	trivial
1176.1.ck.a.149.1	✓	12	24.5	odd	2	CM
1176.1.ck.a.221.1	yes	12	49.25	even	21	inner
1176.1.ck.a.221.1	yes	12	1176.221	odd	42	inner
1176.1.ck.b.149.1	yes	12	3.2	odd	2
1176.1.ck.b.149.1	yes	12	8.5	even	2
1176.1.ck.b.221.1	yes	12	147.74	odd	42
1176.1.ck.b.221.1	yes	12	392.221	even	42

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:	\( N \)	\(=\)	\( 1176 = 2^{3} \cdot 3 \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	1176.ck (of order \(42\), degree \(12\), minimal)

\(f(q)\)	\(=\)	\(q+(-0.955573 - 0.294755i) q^{2} +(-0.365341 + 0.930874i) q^{3} +(0.826239 + 0.563320i) q^{4} +(-1.44973 + 0.218511i) q^{5} +(0.623490 - 0.781831i) q^{6} +(-0.988831 - 0.149042i) q^{7} +(-0.623490 - 0.781831i) q^{8} +(-0.733052 - 0.680173i) q^{9} +O(q^{10})\)	\(q+(-0.955573 - 0.294755i) q^{2} +(-0.365341 + 0.930874i) q^{3} +(0.826239 + 0.563320i) q^{4} +(-1.44973 + 0.218511i) q^{5} +(0.623490 - 0.781831i) q^{6} +(-0.988831 - 0.149042i) q^{7} +(-0.623490 - 0.781831i) q^{8} +(-0.733052 - 0.680173i) q^{9} +(1.44973 + 0.218511i) q^{10} +(1.21135 - 1.12397i) q^{11} +(-0.826239 + 0.563320i) q^{12} +(0.900969 + 0.433884i) q^{14} +(0.326239 - 1.42935i) q^{15} +(0.365341 + 0.930874i) q^{16} +(0.500000 + 0.866025i) q^{18} +(-1.32091 - 0.636119i) q^{20} +(0.500000 - 0.866025i) q^{21} +(-1.48883 + 0.716983i) q^{22} +(0.955573 - 0.294755i) q^{24} +(1.09839 - 0.338809i) q^{25} +(0.900969 - 0.433884i) q^{27} +(-0.733052 - 0.680173i) q^{28} +(1.72188 + 0.829215i) q^{29} +(-0.733052 + 1.26968i) q^{30} +(-0.365341 - 0.632789i) q^{31} +(-0.0747301 - 0.997204i) q^{32} +(0.603718 + 1.53825i) q^{33} +1.46610 q^{35} +(-0.222521 - 0.974928i) q^{36} +(1.07473 + 0.997204i) q^{40} +(-0.733052 + 0.680173i) q^{42} +(1.63402 - 0.246289i) q^{44} +(1.21135 + 0.825886i) q^{45} -1.00000 q^{48} +(0.955573 + 0.294755i) q^{49} -1.14946 q^{50} +(0.826239 + 0.563320i) q^{53} +(-0.988831 + 0.149042i) q^{54} +(-1.51053 + 1.89415i) q^{55} +(0.500000 + 0.866025i) q^{56} +(-1.40097 - 1.29991i) q^{58} +(-0.988831 - 0.149042i) q^{59} +(1.07473 - 0.997204i) q^{60} +(0.162592 + 0.712362i) q^{62} +(0.623490 + 0.781831i) q^{63} +(-0.222521 + 0.974928i) q^{64} +(-0.123490 - 1.64786i) q^{66} +(-1.40097 - 0.432142i) q^{70} +(-0.0747301 + 0.997204i) q^{72} +(1.19158 - 0.367554i) q^{73} +(-0.0858993 + 1.14625i) q^{75} +(-1.36534 + 0.930874i) q^{77} +(0.988831 - 1.71271i) q^{79} +(-0.733052 - 1.26968i) q^{80} +(0.0747301 + 0.997204i) q^{81} +(0.425270 - 1.86323i) q^{83} +(0.900969 - 0.433884i) q^{84} +(-1.40097 + 1.29991i) q^{87} +(-1.63402 - 0.246289i) q^{88} +(-0.914101 - 1.14625i) q^{90} +(0.722521 - 0.108903i) q^{93} +(0.955573 + 0.294755i) q^{96} +0.149460 q^{97} +(-0.826239 - 0.563320i) q^{98} -1.65248 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - q^{2} - q^{3} + q^{4} + q^{5} - 2 q^{6} + q^{7} + 2 q^{8} + q^{9}+O(q^{10}) \) 12 * q - q^2 - q^3 + q^4 + q^5 - 2 * q^6 + q^7 + 2 * q^8 + q^9	\( 12 q - q^{2} - q^{3} + q^{4} + q^{5} - 2 q^{6} + q^{7} + 2 q^{8} + q^{9} - q^{10} + q^{11} - q^{12} + 2 q^{14} - 5 q^{15} + q^{16} + 6 q^{18} - 2 q^{20} + 6 q^{21} - 5 q^{22} + q^{24} + 2 q^{27} + q^{28} - 2 q^{29} + q^{30} - q^{31} - q^{32} - q^{33} - 2 q^{35} - 2 q^{36} + 13 q^{40} + q^{42} + q^{44} + q^{45} - 12 q^{48} + q^{49} - 14 q^{50} + q^{53} + q^{54} - 9 q^{55} + 6 q^{56} - 8 q^{58} + q^{59} + 13 q^{60} - 2 q^{62} - 2 q^{63} - 2 q^{64} + 8 q^{66} - 8 q^{70} - q^{72} + 2 q^{73} - 14 q^{75} - 13 q^{77} - q^{79} + q^{80} + q^{81} + 5 q^{83} + 2 q^{84} - 8 q^{87} - q^{88} + 2 q^{90} + 8 q^{93} + q^{96} + 2 q^{97} - q^{98} - 2 q^{99}+O(q^{100}) \) 12 * q - q^2 - q^3 + q^4 + q^5 - 2 * q^6 + q^7 + 2 * q^8 + q^9 - q^10 + q^11 - q^12 + 2 * q^14 - 5 * q^15 + q^16 + 6 * q^18 - 2 * q^20 + 6 * q^21 - 5 * q^22 + q^24 + 2 * q^27 + q^28 - 2 * q^29 + q^30 - q^31 - q^32 - q^33 - 2 * q^35 - 2 * q^36 + 13 * q^40 + q^42 + q^44 + q^45 - 12 * q^48 + q^49 - 14 * q^50 + q^53 + q^54 - 9 * q^55 + 6 * q^56 - 8 * q^58 + q^59 + 13 * q^60 - 2 * q^62 - 2 * q^63 - 2 * q^64 + 8 * q^66 - 8 * q^70 - q^72 + 2 * q^73 - 14 * q^75 - 13 * q^77 - q^79 + q^80 + q^81 + 5 * q^83 + 2 * q^84 - 8 * q^87 - q^88 + 2 * q^90 + 8 * q^93 + q^96 + 2 * q^97 - q^98 - 2 * q^99

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