LMFDB - Embedded newform 3577.1.p.a.2481.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3577,1,Mod(656,3577)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3577, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([5, 3]))

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

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

chi := DirichletCharacter("3577.656");

S:= CuspForms(chi, 1);

N := Newforms(S);

Level:	$ N $	$=$	$ 3577 = 7^{2} \cdot 73 $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	3577.p (of order $6$, degree $2$, not minimal)

Newform invariants

comment: select newform

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

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

Self dual:	no
Analytic conductor:	$1.78515555019$
Analytic rank:	$0$
Dimension:	$6$
Relative dimension:	$3$ over $\Q(\zeta_{6})$
Coefficient field:	6.0.64827.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - x^{5} + 3x^{4} + 5x^{2} - 2x + 1 $ x^6 - x^5 + 3x^4 + 5x^2 - 2*x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{4}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 511)
Projective image:	$D_{7}$
Projective field:	Galois closure of 7.1.133432831.1
Artin image:	$C_3\times D_{14}$
Artin field:	Galois closure of $\mathbb{Q}[x]/(x^{42} - \cdots)$

Embedding invariants

Embedding label			2481.3
Root			$0.900969 - 1.56052i$ of defining polynomial
Character	$\chi$	$=$	3577.2481
Dual form			3577.1.p.a.656.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(0.900969 + 1.56052i) q^{2} +(-1.12349 + 1.94594i) q^{4} +(0.623490 + 1.07992i) q^{5} -2.24698 q^{8} +(-0.500000 - 0.866025i) q^{9} +O(q^{10})$	\(q+(0.900969 + 1.56052i) q^{2} +(-1.12349 + 1.94594i) q^{4} +(0.623490 + 1.07992i) q^{5} -2.24698 q^{8} +(-0.500000 - 0.866025i) q^{9} +(-1.12349 + 1.94594i) q^{10} +0.445042 q^{13} +(-0.900969 - 1.56052i) q^{16} +(-0.900969 + 1.56052i) q^{17} +(0.900969 - 1.56052i) q^{18} -2.80194 q^{20} +(0.222521 + 0.385418i) q^{23} +(-0.277479 + 0.480608i) q^{25} +(0.400969 + 0.694498i) q^{26} +(-0.900969 + 1.56052i) q^{31} +(0.500000 - 0.866025i) q^{32} -3.24698 q^{34} +2.24698 q^{36} +(-0.623490 - 1.07992i) q^{37} +(-1.40097 - 2.42655i) q^{40} +(0.623490 - 1.07992i) q^{45} +(-0.400969 + 0.694498i) q^{46} +(-0.222521 - 0.385418i) q^{47} -1.00000 q^{50} +(-0.500000 + 0.866025i) q^{52} +(-0.222521 + 0.385418i) q^{59} -3.24698 q^{62} +(0.277479 + 0.480608i) q^{65} +(0.900969 - 1.56052i) q^{67} +(-2.02446 - 3.50647i) q^{68} +1.24698 q^{71} +(1.12349 + 1.94594i) q^{72} +(0.500000 - 0.866025i) q^{73} +(1.12349 - 1.94594i) q^{74} +(0.900969 + 1.56052i) q^{79} +(1.12349 - 1.94594i) q^{80} +(-0.500000 + 0.866025i) q^{81} +1.80194 q^{83} -2.24698 q^{85} +2.24698 q^{90} -1.00000 q^{92} +(0.400969 - 0.694498i) q^{94} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + q^{2} - 2 q^{4} - q^{5} - 4 q^{8} - 3 q^{9}+O(q^{10}) $ 6 * q + q^2 - 2 * q^4 - q^5 - 4 * q^8 - 3 * q^9	$ 6 q + q^{2} - 2 q^{4} - q^{5} - 4 q^{8} - 3 q^{9} - 2 q^{10} + 2 q^{13} - q^{16} - q^{17} + q^{18} - 8 q^{20} + q^{23} - 2 q^{25} - 2 q^{26} - q^{31} + 3 q^{32} - 10 q^{34} + 4 q^{36} + q^{37} - 4 q^{40} - q^{45} + 2 q^{46} - q^{47} - 6 q^{50} - 3 q^{52} - q^{59} - 10 q^{62} + 2 q^{65} + q^{67} - 3 q^{68} - 2 q^{71} + 2 q^{72} + 3 q^{73} + 2 q^{74} + q^{79} + 2 q^{80} - 3 q^{81} + 2 q^{83} - 4 q^{85} + 4 q^{90} - 6 q^{92} - 2 q^{94}+O(q^{100}) $ 6 * q + q^2 - 2 * q^4 - q^5 - 4 * q^8 - 3 * q^9 - 2 * q^10 + 2 * q^13 - q^16 - q^17 + q^18 - 8 * q^20 + q^23 - 2 * q^25 - 2 * q^26 - q^31 + 3 * q^32 - 10 * q^34 + 4 * q^36 + q^37 - 4 * q^40 - q^45 + 2 * q^46 - q^47 - 6 * q^50 - 3 * q^52 - q^59 - 10 * q^62 + 2 * q^65 + q^67 - 3 * q^68 - 2 * q^71 + 2 * q^72 + 3 * q^73 + 2 * q^74 + q^79 + 2 * q^80 - 3 * q^81 + 2 * q^83 - 4 * q^85 + 4 * q^90 - 6 * q^92 - 2 * q^94

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/3577\mathbb{Z}\right)^\times$.

$n$	$589$	$3286$
$\chi(n)$	$-1$	$e\left(\frac{1}{6}\right)$

Coefficient data

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

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

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

$2$	0.900969	+	1.56052i	0.900969	+	1.56052i	0.826239	+	0.563320i	$0.190476\pi$
$2$	0.900969	+	1.56052i	0.900969	+	1.56052i	0.0747301	+	0.997204i	$0.476190\pi$
$3$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$3$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$4$	−1.12349	+	1.94594i	−1.12349	+	1.94594i
$5$	0.623490	+	1.07992i	0.623490	+	1.07992i	0.988831	+	0.149042i	$0.0476190\pi$
$5$	0.623490	+	1.07992i	0.623490	+	1.07992i	−0.365341	+	0.930874i	$0.619048\pi$
$6$		0			0
$7$		0			0
$7$		0			0
$8$	−2.24698			−2.24698
$9$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$10$	−1.12349	+	1.94594i	−1.12349	+	1.94594i
$11$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$11$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$12$		0			0
$13$	0.445042			0.445042			0.222521	−	0.974928i	$-0.428571\pi$
$13$	0.445042			0.445042			0.222521	+	0.974928i	$0.428571\pi$
$14$		0			0
$15$		0			0
$16$	−0.900969	−	1.56052i	−0.900969	−	1.56052i
$17$	−0.900969	+	1.56052i	−0.900969	+	1.56052i	−0.0747301	+	0.997204i	$0.523810\pi$
$17$	−0.900969	+	1.56052i	−0.900969	+	1.56052i	−0.826239	+	0.563320i	$0.809524\pi$
$18$	0.900969	−	1.56052i	0.900969	−	1.56052i
$19$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$19$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$20$	−2.80194			−2.80194
$21$		0			0
$22$		0			0
$23$	0.222521	+	0.385418i	0.222521	+	0.385418i	0.955573	−	0.294755i	$-0.0952381\pi$
$23$	0.222521	+	0.385418i	0.222521	+	0.385418i	−0.733052	+	0.680173i	$0.761905\pi$
$24$		0			0
$25$	−0.277479	+	0.480608i	−0.277479	+	0.480608i
$26$	0.400969	+	0.694498i	0.400969	+	0.694498i
$27$		0			0
$28$		0			0
$29$		0			0		1.00000			$0$
$29$		0			0		−1.00000			$\pi$
$30$		0			0
$31$	−0.900969	+	1.56052i	−0.900969	+	1.56052i	−0.0747301	+	0.997204i	$0.523810\pi$
$31$	−0.900969	+	1.56052i	−0.900969	+	1.56052i	−0.826239	+	0.563320i	$0.809524\pi$
$32$	0.500000	−	0.866025i	0.500000	−	0.866025i
$33$		0			0
$34$	−3.24698			−3.24698
$35$		0			0
$36$	2.24698			2.24698
$37$	−0.623490	−	1.07992i	−0.623490	−	1.07992i	−0.988831	−	0.149042i	$-0.952381\pi$
$37$	−0.623490	−	1.07992i	−0.623490	−	1.07992i	0.365341	−	0.930874i	$-0.380952\pi$
$38$		0			0
$39$		0			0
$40$	−1.40097	−	2.42655i	−1.40097	−	2.42655i
$41$		0			0		1.00000			$0$
$41$		0			0		−1.00000			$\pi$
$42$		0			0
$43$		0			0		1.00000			$0$
$43$		0			0		−1.00000			$\pi$
$44$		0			0
$45$	0.623490	−	1.07992i	0.623490	−	1.07992i
$46$	−0.400969	+	0.694498i	−0.400969	+	0.694498i
$47$	−0.222521	−	0.385418i	−0.222521	−	0.385418i	0.733052	−	0.680173i	$-0.238095\pi$
$47$	−0.222521	−	0.385418i	−0.222521	−	0.385418i	−0.955573	+	0.294755i	$0.904762\pi$
$48$		0			0
$49$		0			0
$50$	−1.00000			−1.00000
$51$		0			0
$52$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$53$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$53$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−0.222521	+	0.385418i	−0.222521	+	0.385418i	−0.955573	−	0.294755i	$-0.904762\pi$
$59$	−0.222521	+	0.385418i	−0.222521	+	0.385418i	0.733052	+	0.680173i	$0.238095\pi$
$60$		0			0
$61$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$61$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$62$	−3.24698			−3.24698
$63$		0			0
$64$		0			0
$65$	0.277479	+	0.480608i	0.277479	+	0.480608i
$66$		0			0
$67$	0.900969	−	1.56052i	0.900969	−	1.56052i	0.0747301	−	0.997204i	$-0.476190\pi$
$67$	0.900969	−	1.56052i	0.900969	−	1.56052i	0.826239	−	0.563320i	$-0.190476\pi$
$68$	−2.02446	−	3.50647i	−2.02446	−	3.50647i
$69$		0			0
$70$		0			0
$71$	1.24698			1.24698			0.623490	−	0.781831i	$-0.285714\pi$
$71$	1.24698			1.24698			0.623490	+	0.781831i	$0.285714\pi$
$72$	1.12349	+	1.94594i	1.12349	+	1.94594i
$73$	0.500000	−	0.866025i	0.500000	−	0.866025i
$73$	0.500000	−	0.866025i	0.500000	−	0.866025i
$74$	1.12349	−	1.94594i	1.12349	−	1.94594i
$75$		0			0
$76$		0			0
$77$		0			0
$78$		0			0
$79$	0.900969	+	1.56052i	0.900969	+	1.56052i	0.826239	+	0.563320i	$0.190476\pi$
$79$	0.900969	+	1.56052i	0.900969	+	1.56052i	0.0747301	+	0.997204i	$0.476190\pi$
$80$	1.12349	−	1.94594i	1.12349	−	1.94594i
$81$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$82$		0			0
$83$	1.80194			1.80194			0.900969	−	0.433884i	$-0.142857\pi$
$83$	1.80194			1.80194			0.900969	+	0.433884i	$0.142857\pi$
$84$		0			0
$85$	−2.24698			−2.24698
$86$		0			0
$87$		0			0
$88$		0			0
$89$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$89$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$90$	2.24698			2.24698
$91$		0			0
$92$	−1.00000			−1.00000
$93$		0			0
$94$	0.400969	−	0.694498i	0.400969	−	0.694498i
$95$		0			0
$96$		0			0
$97$		0			0		1.00000			$0$
$97$		0			0		−1.00000			$\pi$
$98$		0			0
$99$		0			0
$100$	−0.623490	−	1.07992i	−0.623490	−	1.07992i
$101$	1.00000	−	1.73205i	1.00000	−	1.73205i	0.500000	−	0.866025i	$-0.333333\pi$
$101$	1.00000	−	1.73205i	1.00000	−	1.73205i	0.500000	−	0.866025i	$-0.333333\pi$
$102$		0			0
$103$	0.623490	+	1.07992i	0.623490	+	1.07992i	0.988831	+	0.149042i	$0.0476190\pi$
$103$	0.623490	+	1.07992i	0.623490	+	1.07992i	−0.365341	+	0.930874i	$0.619048\pi$
$104$	−1.00000			−1.00000
$105$		0			0
$106$		0			0
$107$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$107$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$108$		0			0
$109$	0.900969	−	1.56052i	0.900969	−	1.56052i	0.0747301	−	0.997204i	$-0.476190\pi$
$109$	0.900969	−	1.56052i	0.900969	−	1.56052i	0.826239	−	0.563320i	$-0.190476\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$		0			0		1.00000			$0$
$113$		0			0		−1.00000			$\pi$
$114$		0			0
$115$	−0.277479	+	0.480608i	−0.277479	+	0.480608i
$116$		0			0
$117$	−0.222521	−	0.385418i	−0.222521	−	0.385418i
$118$	−0.801938			−0.801938
$119$		0			0
$120$		0			0
$121$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$122$		0			0
$123$		0			0
$124$	−2.02446	−	3.50647i	−2.02446	−	3.50647i
$125$	0.554958			0.554958
$126$		0			0
$127$	1.24698			1.24698			0.623490	−	0.781831i	$-0.285714\pi$
$127$	1.24698			1.24698			0.623490	+	0.781831i	$0.285714\pi$
$128$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$129$		0			0
$130$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$131$	0.623490	+	1.07992i	0.623490	+	1.07992i	0.988831	+	0.149042i	$0.0476190\pi$
$131$	0.623490	+	1.07992i	0.623490	+	1.07992i	−0.365341	+	0.930874i	$0.619048\pi$
$132$		0			0
$133$		0			0
$134$	3.24698			3.24698
$135$		0			0
$136$	2.02446	−	3.50647i	2.02446	−	3.50647i
$137$	−0.623490	+	1.07992i	−0.623490	+	1.07992i	0.365341	+	0.930874i	$0.380952\pi$
$137$	−0.623490	+	1.07992i	−0.623490	+	1.07992i	−0.988831	+	0.149042i	$0.952381\pi$
$138$		0			0
$139$	0.445042			0.445042			0.222521	−	0.974928i	$-0.428571\pi$
$139$	0.445042			0.445042			0.222521	+	0.974928i	$0.428571\pi$
$140$		0			0
$141$		0			0
$142$	1.12349	+	1.94594i	1.12349	+	1.94594i
$143$		0			0
$144$	−0.900969	+	1.56052i	−0.900969	+	1.56052i
$145$		0			0
$146$	1.80194			1.80194
$147$		0			0
$148$	2.80194			2.80194
$149$	0.222521	+	0.385418i	0.222521	+	0.385418i	0.955573	−	0.294755i	$-0.0952381\pi$
$149$	0.222521	+	0.385418i	0.222521	+	0.385418i	−0.733052	+	0.680173i	$0.761905\pi$
$150$		0			0
$151$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$151$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$152$		0			0
$153$	1.80194			1.80194
$154$		0			0
$155$	−2.24698			−2.24698
$156$		0			0
$157$	−0.900969	+	1.56052i	−0.900969	+	1.56052i	−0.0747301	+	0.997204i	$0.523810\pi$
$157$	−0.900969	+	1.56052i	−0.900969	+	1.56052i	−0.826239	+	0.563320i	$0.809524\pi$
$158$	−1.62349	+	2.81197i	−1.62349	+	2.81197i
$159$		0			0
$160$	1.24698			1.24698
$161$		0			0
$162$	−1.80194			−1.80194
$163$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$163$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$164$		0			0
$165$		0			0
$166$	1.62349	+	2.81197i	1.62349	+	2.81197i
$167$	−1.24698			−1.24698			−0.623490	−	0.781831i	$-0.714286\pi$
$167$	−1.24698			−1.24698			−0.623490	+	0.781831i	$0.714286\pi$
$168$		0			0
$169$	−0.801938			−0.801938
$170$	−2.02446	−	3.50647i	−2.02446	−	3.50647i
$171$		0			0
$172$		0			0
$173$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$173$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$		0			0
$178$		0			0
$179$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$179$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$180$	1.40097	+	2.42655i	1.40097	+	2.42655i
$181$		0			0		1.00000			$0$
$181$		0			0		−1.00000			$\pi$
$182$		0			0
$183$		0			0
$184$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$185$	0.777479	−	1.34663i	0.777479	−	1.34663i
$186$		0			0
$187$		0			0
$188$	1.00000			1.00000
$189$		0			0
$190$		0			0
$191$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$191$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$192$		0			0
$193$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$193$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$		0			0		1.00000			$0$
$197$		0			0		−1.00000			$\pi$
$198$		0			0
$199$	0.623490	−	1.07992i	0.623490	−	1.07992i	−0.365341	−	0.930874i	$-0.619048\pi$
$199$	0.623490	−	1.07992i	0.623490	−	1.07992i	0.988831	−	0.149042i	$-0.0476190\pi$
$200$	0.623490	−	1.07992i	0.623490	−	1.07992i
$201$		0			0
$202$	3.60388			3.60388
$203$		0			0
$204$		0			0
$205$		0			0
$206$	−1.12349	+	1.94594i	−1.12349	+	1.94594i
$207$	0.222521	−	0.385418i	0.222521	−	0.385418i
$208$	−0.400969	−	0.694498i	−0.400969	−	0.694498i
$209$		0			0
$210$		0			0
$211$	−0.445042			−0.445042			−0.222521	−	0.974928i	$-0.571429\pi$
$211$	−0.445042			−0.445042			−0.222521	+	0.974928i	$0.571429\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		0			0
$216$		0			0
$217$		0			0
$218$	3.24698			3.24698
$219$		0			0
$220$		0			0
$221$	−0.400969	+	0.694498i	−0.400969	+	0.694498i
$222$		0			0
$223$		0			0		1.00000			$0$
$223$		0			0		−1.00000			$\pi$
$224$		0			0
$225$	0.554958			0.554958
$226$		0			0
$227$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$227$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$228$		0			0
$229$	−0.900969	−	1.56052i	−0.900969	−	1.56052i	−0.826239	−	0.563320i	$-0.809524\pi$
$229$	−0.900969	−	1.56052i	−0.900969	−	1.56052i	−0.0747301	−	0.997204i	$-0.523810\pi$
$230$	−1.00000			−1.00000
$231$		0			0
$232$		0			0
$233$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$233$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$234$	0.400969	−	0.694498i	0.400969	−	0.694498i
$235$	0.277479	−	0.480608i	0.277479	−	0.480608i
$236$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$237$		0			0
$238$		0			0
$239$		0			0		1.00000			$0$
$239$		0			0		−1.00000			$\pi$
$240$		0			0
$241$	−0.222521	+	0.385418i	−0.222521	+	0.385418i	−0.955573	−	0.294755i	$-0.904762\pi$
$241$	−0.222521	+	0.385418i	−0.222521	+	0.385418i	0.733052	+	0.680173i	$0.238095\pi$
$242$	0.900969	−	1.56052i	0.900969	−	1.56052i
$243$		0			0
$244$		0			0
$245$		0			0
$246$		0			0
$247$		0			0
$248$	2.02446	−	3.50647i	2.02446	−	3.50647i
$249$		0			0
$250$	0.500000	+	0.866025i	0.500000	+	0.866025i
$251$		0			0		1.00000			$0$
$251$		0			0		−1.00000			$\pi$
$252$		0			0
$253$		0			0
$254$	1.12349	+	1.94594i	1.12349	+	1.94594i
$255$		0			0
$256$	0.900969	−	1.56052i	0.900969	−	1.56052i
$257$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$257$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$258$		0			0
$259$		0			0
$260$	−1.24698			−1.24698
$261$		0			0
$262$	−1.12349	+	1.94594i	−1.12349	+	1.94594i
$263$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$263$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$		0			0
$268$	2.02446	+	3.50647i	2.02446	+	3.50647i
$269$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$269$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$270$		0			0
$271$	−0.900969	−	1.56052i	−0.900969	−	1.56052i	−0.826239	−	0.563320i	$-0.809524\pi$
$271$	−0.900969	−	1.56052i	−0.900969	−	1.56052i	−0.0747301	−	0.997204i	$-0.523810\pi$
$272$	3.24698			3.24698
$273$		0			0
$274$	−2.24698			−2.24698
$275$		0			0
$276$		0			0
$277$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$277$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$278$	0.400969	+	0.694498i	0.400969	+	0.694498i
$279$	1.80194			1.80194
$280$		0			0
$281$		0			0		1.00000			$0$
$281$		0			0		−1.00000			$\pi$
$282$		0			0
$283$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$283$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$284$	−1.40097	+	2.42655i	−1.40097	+	2.42655i
$285$		0			0
$286$		0			0
$287$		0			0
$288$	−1.00000			−1.00000
$289$	−1.12349	−	1.94594i	−1.12349	−	1.94594i
$290$		0			0
$291$		0			0
$292$	1.12349	+	1.94594i	1.12349	+	1.94594i
$293$		0			0		1.00000			$0$
$293$		0			0		−1.00000			$\pi$
$294$		0			0
$295$	−0.554958			−0.554958
$296$	1.40097	+	2.42655i	1.40097	+	2.42655i
$297$		0			0
$298$	−0.400969	+	0.694498i	−0.400969	+	0.694498i
$299$	0.0990311	+	0.171527i	0.0990311	+	0.171527i
$300$		0			0
$301$		0			0
$302$		0			0
$303$		0			0
$304$		0			0
$305$		0			0
$306$	1.62349	+	2.81197i	1.62349	+	2.81197i
$307$	−1.24698			−1.24698			−0.623490	−	0.781831i	$-0.714286\pi$
$307$	−1.24698			−1.24698			−0.623490	+	0.781831i	$0.714286\pi$
$308$		0			0
$309$		0			0
$310$	−2.02446	−	3.50647i	−2.02446	−	3.50647i
$311$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$311$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$312$		0			0
$313$	−0.222521	−	0.385418i	−0.222521	−	0.385418i	0.733052	−	0.680173i	$-0.238095\pi$
$313$	−0.222521	−	0.385418i	−0.222521	−	0.385418i	−0.955573	+	0.294755i	$0.904762\pi$
$314$	−3.24698			−3.24698
$315$		0			0
$316$	−4.04892			−4.04892
$317$	0.900969	+	1.56052i	0.900969	+	1.56052i	0.826239	+	0.563320i	$0.190476\pi$
$317$	0.900969	+	1.56052i	0.900969	+	1.56052i	0.0747301	+	0.997204i	$0.476190\pi$
$318$		0			0
$319$		0			0
$320$		0			0
$321$		0			0
$322$		0			0
$323$		0			0
$324$	−1.12349	−	1.94594i	−1.12349	−	1.94594i
$325$	−0.123490	+	0.213891i	−0.123490	+	0.213891i
$326$		0			0
$327$		0			0
$328$		0			0
$329$		0			0
$330$		0			0
$331$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$331$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$332$	−2.02446	+	3.50647i	−2.02446	+	3.50647i
$333$	−0.623490	+	1.07992i	−0.623490	+	1.07992i
$334$	−1.12349	−	1.94594i	−1.12349	−	1.94594i
$335$	2.24698			2.24698
$336$		0			0
$337$		0			0		1.00000			$0$
$337$		0			0		−1.00000			$\pi$
$338$	−0.722521	−	1.25144i	−0.722521	−	1.25144i
$339$		0			0
$340$	2.52446	−	4.37249i	2.52446	−	4.37249i
$341$		0			0
$342$		0			0
$343$		0			0
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−0.623490	+	1.07992i	−0.623490	+	1.07992i	0.365341	+	0.930874i	$0.380952\pi$
$347$	−0.623490	+	1.07992i	−0.623490	+	1.07992i	−0.988831	+	0.149042i	$0.952381\pi$
$348$		0			0
$349$		0			0		1.00000			$0$
$349$		0			0		−1.00000			$\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$353$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$354$		0			0
$355$	0.777479	+	1.34663i	0.777479	+	1.34663i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	0.222521	+	0.385418i	0.222521	+	0.385418i	0.955573	−	0.294755i	$-0.0952381\pi$
$359$	0.222521	+	0.385418i	0.222521	+	0.385418i	−0.733052	+	0.680173i	$0.761905\pi$
$360$	−1.40097	+	2.42655i	−1.40097	+	2.42655i
$361$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$362$		0			0
$363$		0			0
$364$		0			0
$365$	1.24698			1.24698
$366$		0			0
$367$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$367$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$368$	0.400969	−	0.694498i	0.400969	−	0.694498i
$369$		0			0
$370$	2.80194			2.80194
$371$		0			0
$372$		0			0
$373$	−0.623490	−	1.07992i	−0.623490	−	1.07992i	−0.988831	−	0.149042i	$-0.952381\pi$
$373$	−0.623490	−	1.07992i	−0.623490	−	1.07992i	0.365341	−	0.930874i	$-0.380952\pi$
$374$		0			0
$375$		0			0
$376$	0.500000	+	0.866025i	0.500000	+	0.866025i
$377$		0			0
$378$		0			0
$379$		0			0		1.00000			$0$
$379$		0			0		−1.00000			$\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$383$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$		0			0
$388$		0			0
$389$	0.900969	−	1.56052i	0.900969	−	1.56052i	0.0747301	−	0.997204i	$-0.476190\pi$
$389$	0.900969	−	1.56052i	0.900969	−	1.56052i	0.826239	−	0.563320i	$-0.190476\pi$
$390$		0			0
$391$	−0.801938			−0.801938
$392$		0			0
$393$		0			0
$394$		0			0
$395$	−1.12349	+	1.94594i	−1.12349	+	1.94594i
$396$		0			0
$397$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$397$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$398$	2.24698			2.24698
$399$		0			0
$400$	1.00000			1.00000
$401$	0.222521	+	0.385418i	0.222521	+	0.385418i	0.955573	−	0.294755i	$-0.0952381\pi$
$401$	0.222521	+	0.385418i	0.222521	+	0.385418i	−0.733052	+	0.680173i	$0.761905\pi$
$402$		0			0
$403$	−0.400969	+	0.694498i	−0.400969	+	0.694498i
$404$	2.24698	+	3.89188i	2.24698	+	3.89188i
$405$	−1.24698			−1.24698
$406$		0			0
$407$		0			0
$408$		0			0
$409$	0.623490	−	1.07992i	0.623490	−	1.07992i	−0.365341	−	0.930874i	$-0.619048\pi$
$409$	0.623490	−	1.07992i	0.623490	−	1.07992i	0.988831	−	0.149042i	$-0.0476190\pi$
$410$		0			0
$411$		0			0
$412$	−2.80194			−2.80194
$413$		0			0
$414$	0.801938			0.801938
$415$	1.12349	+	1.94594i	1.12349	+	1.94594i
$416$	0.222521	−	0.385418i	0.222521	−	0.385418i
$417$		0			0
$418$		0			0
$419$		0			0		1.00000			$0$
$419$		0			0		−1.00000			$\pi$
$420$		0			0
$421$		0			0		1.00000			$0$
$421$		0			0		−1.00000			$\pi$
$422$	−0.400969	−	0.694498i	−0.400969	−	0.694498i
$423$	−0.222521	+	0.385418i	−0.222521	+	0.385418i
$424$		0			0
$425$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$426$		0			0
$427$		0			0
$428$		0			0
$429$		0			0
$430$		0			0
$431$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$431$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$432$		0			0
$433$	−1.24698			−1.24698			−0.623490	−	0.781831i	$-0.714286\pi$
$433$	−1.24698			−1.24698			−0.623490	+	0.781831i	$0.714286\pi$
$434$		0			0
$435$		0			0
$436$	2.02446	+	3.50647i	2.02446	+	3.50647i
$437$		0			0
$438$		0			0
$439$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$439$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$440$		0			0
$441$		0			0
$442$	−1.44504			−1.44504
$443$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$443$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$		0			0
$448$		0			0
$449$		0			0		1.00000			$0$
$449$		0			0		−1.00000			$\pi$
$450$	0.500000	+	0.866025i	0.500000	+	0.866025i
$451$		0			0
$452$		0			0
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$	0.222521	+	0.385418i	0.222521	+	0.385418i	0.955573	−	0.294755i	$-0.0952381\pi$
$457$	0.222521	+	0.385418i	0.222521	+	0.385418i	−0.733052	+	0.680173i	$0.761905\pi$
$458$	1.62349	−	2.81197i	1.62349	−	2.81197i
$459$		0			0
$460$	−0.623490	−	1.07992i	−0.623490	−	1.07992i
$461$		0			0		1.00000			$0$
$461$		0			0		−1.00000			$\pi$
$462$		0			0
$463$	1.24698			1.24698			0.623490	−	0.781831i	$-0.285714\pi$
$463$	1.24698			1.24698			0.623490	+	0.781831i	$0.285714\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	1.00000	+	1.73205i	1.00000	+	1.73205i	0.500000	+	0.866025i	$0.333333\pi$
$467$	1.00000	+	1.73205i	1.00000	+	1.73205i	0.500000	+	0.866025i	$0.333333\pi$
$468$	1.00000			1.00000
$469$		0			0
$470$	1.00000			1.00000
$471$		0			0
$472$	0.500000	−	0.866025i	0.500000	−	0.866025i
$473$		0			0
$474$		0			0
$475$		0			0
$476$		0			0
$477$		0			0
$478$		0			0
$479$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$479$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$480$		0			0
$481$	−0.277479	−	0.480608i	−0.277479	−	0.480608i
$482$	−0.801938			−0.801938
$483$		0			0
$484$	2.24698			2.24698
$485$		0			0
$486$		0			0
$487$	0.222521	−	0.385418i	0.222521	−	0.385418i	−0.733052	−	0.680173i	$-0.761905\pi$
$487$	0.222521	−	0.385418i	0.222521	−	0.385418i	0.955573	+	0.294755i	$0.0952381\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$		0			0		1.00000			$0$
$491$		0			0		−1.00000			$\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$		0			0
$496$	3.24698			3.24698
$497$		0			0
$498$		0			0
$499$	0.900969	+	1.56052i	0.900969	+	1.56052i	0.826239	+	0.563320i	$0.190476\pi$
$499$	0.900969	+	1.56052i	0.900969	+	1.56052i	0.0747301	+	0.997204i	$0.476190\pi$
$500$	−0.623490	+	1.07992i	−0.623490	+	1.07992i
$501$		0			0
$502$		0			0
$503$		0			0		1.00000			$0$
$503$		0			0		−1.00000			$\pi$
$504$		0			0
$505$	2.49396			2.49396
$506$		0			0
$507$		0			0
$508$	−1.40097	+	2.42655i	−1.40097	+	2.42655i
$509$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$509$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$510$		0			0
$511$		0			0
$512$	2.24698			2.24698
$513$		0			0
$514$		0			0
$515$	−0.777479	+	1.34663i	−0.777479	+	1.34663i
$516$		0			0
$517$		0			0
$518$		0			0
$519$		0			0
$520$	−0.623490	−	1.07992i	−0.623490	−	1.07992i
$521$	1.00000	−	1.73205i	1.00000	−	1.73205i	0.500000	−	0.866025i	$-0.333333\pi$
$521$	1.00000	−	1.73205i	1.00000	−	1.73205i	0.500000	−	0.866025i	$-0.333333\pi$
$522$		0			0
$523$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$523$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$524$	−2.80194			−2.80194
$525$		0			0
$526$		0			0
$527$	−1.62349	−	2.81197i	−1.62349	−	2.81197i
$528$		0			0
$529$	0.400969	−	0.694498i	0.400969	−	0.694498i
$530$		0			0
$531$	0.445042			0.445042
$532$		0			0
$533$		0			0
$534$		0			0
$535$		0			0
$536$	−2.02446	+	3.50647i	−2.02446	+	3.50647i
$537$		0			0
$538$		0			0
$539$		0			0
$540$		0			0
$541$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$541$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$542$	1.62349	−	2.81197i	1.62349	−	2.81197i
$543$		0			0
$544$	0.900969	+	1.56052i	0.900969	+	1.56052i
$545$	2.24698			2.24698
$546$		0			0
$547$	2.00000			2.00000			1.00000			$0$
$547$	2.00000			2.00000			1.00000			$0$
$548$	−1.40097	−	2.42655i	−1.40097	−	2.42655i
$549$		0			0
$550$		0			0
$551$		0			0
$552$		0			0
$553$		0			0
$554$		0			0
$555$		0			0
$556$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$557$	0.222521	−	0.385418i	0.222521	−	0.385418i	−0.733052	−	0.680173i	$-0.761905\pi$
$557$	0.222521	−	0.385418i	0.222521	−	0.385418i	0.955573	+	0.294755i	$0.0952381\pi$
$558$	1.62349	+	2.81197i	1.62349	+	2.81197i
$559$		0			0
$560$		0			0
$561$		0			0
$562$		0			0
$563$	0.623490	−	1.07992i	0.623490	−	1.07992i	−0.365341	−	0.930874i	$-0.619048\pi$
$563$	0.623490	−	1.07992i	0.623490	−	1.07992i	0.988831	−	0.149042i	$-0.0476190\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$		0			0
$568$	−2.80194			−2.80194
$569$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$569$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$570$		0			0
$571$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$571$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−0.246980			−0.246980
$576$		0			0
$577$	0.623490	−	1.07992i	0.623490	−	1.07992i	−0.365341	−	0.930874i	$-0.619048\pi$
$577$	0.623490	−	1.07992i	0.623490	−	1.07992i	0.988831	−	0.149042i	$-0.0476190\pi$
$578$	2.02446	−	3.50647i	2.02446	−	3.50647i
$579$		0			0
$580$		0			0
$581$		0			0
$582$		0			0
$583$		0			0
$584$	−1.12349	+	1.94594i	−1.12349	+	1.94594i
$585$	0.277479	−	0.480608i	0.277479	−	0.480608i
$586$		0			0
$587$		0			0		1.00000			$0$
$587$		0			0		−1.00000			$\pi$
$588$		0			0
$589$		0			0
$590$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$591$		0			0
$592$	−1.12349	+	1.94594i	−1.12349	+	1.94594i
$593$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$593$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$594$		0			0
$595$		0			0
$596$	−1.00000			−1.00000
$597$		0			0
$598$	−0.178448	+	0.309081i	−0.178448	+	0.309081i
$599$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$599$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$600$		0			0
$601$	−1.24698			−1.24698			−0.623490	−	0.781831i	$-0.714286\pi$
$601$	−1.24698			−1.24698			−0.623490	+	0.781831i	$0.714286\pi$
$602$		0			0
$603$	−1.80194			−1.80194
$604$		0			0
$605$	0.623490	−	1.07992i	0.623490	−	1.07992i
$606$		0			0
$607$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$607$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−0.0990311	−	0.171527i	−0.0990311	−	0.171527i
$612$	−2.02446	+	3.50647i	−2.02446	+	3.50647i
$613$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$613$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$614$	−1.12349	−	1.94594i	−1.12349	−	1.94594i
$615$		0			0
$616$		0			0
$617$		0			0		1.00000			$0$
$617$		0			0		−1.00000			$\pi$
$618$		0			0
$619$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$619$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$620$	2.52446	−	4.37249i	2.52446	−	4.37249i
$621$		0			0
$622$		0			0
$623$		0			0
$624$		0			0
$625$	0.623490	+	1.07992i	0.623490	+	1.07992i
$626$	0.400969	−	0.694498i	0.400969	−	0.694498i
$627$		0			0
$628$	−2.02446	−	3.50647i	−2.02446	−	3.50647i
$629$	2.24698			2.24698
$630$		0			0
$631$		0			0		1.00000			$0$
$631$		0			0		−1.00000			$\pi$
$632$	−2.02446	−	3.50647i	−2.02446	−	3.50647i
$633$		0			0
$634$	−1.62349	+	2.81197i	−1.62349	+	2.81197i
$635$	0.777479	+	1.34663i	0.777479	+	1.34663i
$636$		0			0
$637$		0			0
$638$		0			0
$639$	−0.623490	−	1.07992i	−0.623490	−	1.07992i
$640$	0.623490	−	1.07992i	0.623490	−	1.07992i
$641$	0.900969	−	1.56052i	0.900969	−	1.56052i	0.0747301	−	0.997204i	$-0.476190\pi$
$641$	0.900969	−	1.56052i	0.900969	−	1.56052i	0.826239	−	0.563320i	$-0.190476\pi$
$642$		0			0
$643$	−1.24698			−1.24698			−0.623490	−	0.781831i	$-0.714286\pi$
$643$	−1.24698			−1.24698			−0.623490	+	0.781831i	$0.714286\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−0.900969	+	1.56052i	−0.900969	+	1.56052i	−0.0747301	+	0.997204i	$0.523810\pi$
$647$	−0.900969	+	1.56052i	−0.900969	+	1.56052i	−0.826239	+	0.563320i	$0.809524\pi$
$648$	1.12349	−	1.94594i	1.12349	−	1.94594i
$649$		0			0
$650$	−0.445042			−0.445042
$651$		0			0
$652$		0			0
$653$	0.900969	+	1.56052i	0.900969	+	1.56052i	0.826239	+	0.563320i	$0.190476\pi$
$653$	0.900969	+	1.56052i	0.900969	+	1.56052i	0.0747301	+	0.997204i	$0.476190\pi$
$654$		0			0
$655$	−0.777479	+	1.34663i	−0.777479	+	1.34663i
$656$		0			0
$657$	−1.00000			−1.00000
$658$		0			0
$659$	−1.80194			−1.80194			−0.900969	−	0.433884i	$-0.857143\pi$
$659$	−1.80194			−1.80194			−0.900969	+	0.433884i	$0.857143\pi$
$660$		0			0
$661$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$661$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$662$		0			0
$663$		0			0
$664$	−4.04892			−4.04892
$665$		0			0
$666$	−2.24698			−2.24698
$667$		0			0
$668$	1.40097	−	2.42655i	1.40097	−	2.42655i
$669$		0			0
$670$	2.02446	+	3.50647i	2.02446	+	3.50647i
$671$		0			0
$672$		0			0
$673$	−0.445042			−0.445042			−0.222521	−	0.974928i	$-0.571429\pi$
$673$	−0.445042			−0.445042			−0.222521	+	0.974928i	$0.571429\pi$
$674$		0			0
$675$		0			0
$676$	0.900969	−	1.56052i	0.900969	−	1.56052i
$677$	−0.222521	−	0.385418i	−0.222521	−	0.385418i	0.733052	−	0.680173i	$-0.238095\pi$
$677$	−0.222521	−	0.385418i	−0.222521	−	0.385418i	−0.955573	+	0.294755i	$0.904762\pi$
$678$		0			0
$679$		0			0
$680$	5.04892			5.04892
$681$		0			0
$682$		0			0
$683$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$683$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$684$		0			0
$685$	−1.55496			−1.55496
$686$		0			0
$687$		0			0
$688$		0			0
$689$		0			0
$690$		0			0
$691$	−0.900969	−	1.56052i	−0.900969	−	1.56052i	−0.826239	−	0.563320i	$-0.809524\pi$
$691$	−0.900969	−	1.56052i	−0.900969	−	1.56052i	−0.0747301	−	0.997204i	$-0.523810\pi$
$692$		0			0
$693$		0			0
$694$	−2.24698			−2.24698
$695$	0.277479	+	0.480608i	0.277479	+	0.480608i
$696$		0			0
$697$		0			0
$698$		0			0
$699$		0			0
$700$		0			0
$701$		0			0		1.00000			$0$
$701$		0			0		−1.00000			$\pi$
$702$		0			0
$703$		0			0
$704$		0			0
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$709$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$710$	−1.40097	+	2.42655i	−1.40097	+	2.42655i
$711$	0.900969	−	1.56052i	0.900969	−	1.56052i
$712$		0			0
$713$	−0.801938			−0.801938
$714$		0			0
$715$		0			0
$716$		0			0
$717$		0			0
$718$	−0.400969	+	0.694498i	−0.400969	+	0.694498i
$719$	−0.222521	−	0.385418i	−0.222521	−	0.385418i	0.733052	−	0.680173i	$-0.238095\pi$
$719$	−0.222521	−	0.385418i	−0.222521	−	0.385418i	−0.955573	+	0.294755i	$0.904762\pi$
$720$	−2.24698			−2.24698
$721$		0			0
$722$	−1.80194			−1.80194
$723$		0			0
$724$		0			0
$725$		0			0
$726$		0			0
$727$		0			0		1.00000			$0$
$727$		0			0		−1.00000			$\pi$
$728$		0			0
$729$	1.00000			1.00000
$730$	1.12349	+	1.94594i	1.12349	+	1.94594i
$731$		0			0
$732$		0			0
$733$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$733$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$734$		0			0
$735$		0			0
$736$	0.445042			0.445042
$737$		0			0
$738$		0			0
$739$	−0.623490	+	1.07992i	−0.623490	+	1.07992i	0.365341	+	0.930874i	$0.380952\pi$
$739$	−0.623490	+	1.07992i	−0.623490	+	1.07992i	−0.988831	+	0.149042i	$0.952381\pi$
$740$	1.74698	+	3.02586i	1.74698	+	3.02586i
$741$		0			0
$742$		0			0
$743$		0			0		1.00000			$0$
$743$		0			0		−1.00000			$\pi$
$744$		0			0
$745$	−0.277479	+	0.480608i	−0.277479	+	0.480608i
$746$	1.12349	−	1.94594i	1.12349	−	1.94594i
$747$	−0.900969	−	1.56052i	−0.900969	−	1.56052i
$748$		0			0
$749$		0			0
$750$		0			0
$751$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$751$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$752$	−0.400969	+	0.694498i	−0.400969	+	0.694498i
$753$		0			0
$754$		0			0
$755$		0			0
$756$		0			0
$757$	−0.445042			−0.445042			−0.222521	−	0.974928i	$-0.571429\pi$
$757$	−0.445042			−0.445042			−0.222521	+	0.974928i	$0.571429\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−0.900969	−	1.56052i	−0.900969	−	1.56052i	−0.826239	−	0.563320i	$-0.809524\pi$
$761$	−0.900969	−	1.56052i	−0.900969	−	1.56052i	−0.0747301	−	0.997204i	$-0.523810\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$	1.12349	+	1.94594i	1.12349	+	1.94594i
$766$		0			0
$767$	−0.0990311	+	0.171527i	−0.0990311	+	0.171527i
$768$		0			0
$769$	−2.00000			−2.00000			−1.00000			$\pi$
$769$	−2.00000			−2.00000			−1.00000			$\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	1.00000	−	1.73205i	1.00000	−	1.73205i	0.500000	−	0.866025i	$-0.333333\pi$
$773$	1.00000	−	1.73205i	1.00000	−	1.73205i	0.500000	−	0.866025i	$-0.333333\pi$
$774$		0			0
$775$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$776$		0			0
$777$		0			0
$778$	3.24698			3.24698
$779$		0			0
$780$		0			0
$781$		0			0
$782$	−0.722521	−	1.25144i	−0.722521	−	1.25144i
$783$		0			0
$784$		0			0
$785$	−2.24698			−2.24698
$786$		0			0
$787$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$787$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$788$		0			0
$789$		0			0
$790$	−4.04892			−4.04892
$791$		0			0
$792$		0			0
$793$		0			0
$794$		0			0
$795$		0			0
$796$	1.40097	+	2.42655i	1.40097	+	2.42655i
$797$		0			0		1.00000			$0$
$797$		0			0		−1.00000			$\pi$
$798$		0			0
$799$	0.801938			0.801938
$800$	0.277479	+	0.480608i	0.277479	+	0.480608i
$801$		0			0
$802$	−0.400969	+	0.694498i	−0.400969	+	0.694498i
$803$		0			0
$804$		0			0
$805$		0			0
$806$	−1.44504			−1.44504
$807$		0			0
$808$	−2.24698	+	3.89188i	−2.24698	+	3.89188i
$809$	−0.623490	+	1.07992i	−0.623490	+	1.07992i	0.365341	+	0.930874i	$0.380952\pi$
$809$	−0.623490	+	1.07992i	−0.623490	+	1.07992i	−0.988831	+	0.149042i	$0.952381\pi$
$810$	−1.12349	−	1.94594i	−1.12349	−	1.94594i
$811$		0			0		1.00000			$0$
$811$		0			0		−1.00000			$\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$		0			0
$816$		0			0
$817$		0			0
$818$	2.24698			2.24698
$819$		0			0
$820$		0			0
$821$	−0.623490	−	1.07992i	−0.623490	−	1.07992i	−0.988831	−	0.149042i	$-0.952381\pi$
$821$	−0.623490	−	1.07992i	−0.623490	−	1.07992i	0.365341	−	0.930874i	$-0.380952\pi$
$822$		0			0
$823$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$823$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$824$	−1.40097	−	2.42655i	−1.40097	−	2.42655i
$825$		0			0
$826$		0			0
$827$	−0.445042			−0.445042			−0.222521	−	0.974928i	$-0.571429\pi$
$827$	−0.445042			−0.445042			−0.222521	+	0.974928i	$0.571429\pi$
$828$	0.500000	+	0.866025i	0.500000	+	0.866025i
$829$	−0.222521	+	0.385418i	−0.222521	+	0.385418i	−0.955573	−	0.294755i	$-0.904762\pi$
$829$	−0.222521	+	0.385418i	−0.222521	+	0.385418i	0.733052	+	0.680173i	$0.238095\pi$
$830$	−2.02446	+	3.50647i	−2.02446	+	3.50647i
$831$		0			0
$832$		0			0
$833$		0			0
$834$		0			0
$835$	−0.777479	−	1.34663i	−0.777479	−	1.34663i
$836$		0			0
$837$		0			0
$838$		0			0
$839$		0			0		1.00000			$0$
$839$		0			0		−1.00000			$\pi$
$840$		0			0
$841$	1.00000			1.00000
$842$		0			0
$843$		0			0
$844$	0.500000	−	0.866025i	0.500000	−	0.866025i
$845$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$846$	−0.801938			−0.801938
$847$		0			0
$848$		0			0
$849$		0			0
$850$	0.900969	−	1.56052i	0.900969	−	1.56052i
$851$	0.277479	−	0.480608i	0.277479	−	0.480608i
$852$		0			0
$853$		0			0		1.00000			$0$
$853$		0			0		−1.00000			$\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$857$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$858$		0			0
$859$	1.00000	+	1.73205i	1.00000	+	1.73205i	0.500000	+	0.866025i	$0.333333\pi$
$859$	1.00000	+	1.73205i	1.00000	+	1.73205i	0.500000	+	0.866025i	$0.333333\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$863$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$864$		0			0
$865$		0			0
$866$	−1.12349	−	1.94594i	−1.12349	−	1.94594i
$867$		0			0
$868$		0			0
$869$		0			0
$870$		0			0
$871$	0.400969	−	0.694498i	0.400969	−	0.694498i
$872$	−2.02446	+	3.50647i	−2.02446	+	3.50647i
$873$		0			0
$874$		0			0
$875$		0			0
$876$		0			0
$877$	0.222521	+	0.385418i	0.222521	+	0.385418i	0.955573	−	0.294755i	$-0.0952381\pi$
$877$	0.222521	+	0.385418i	0.222521	+	0.385418i	−0.733052	+	0.680173i	$0.761905\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	0.445042			0.445042			0.222521	−	0.974928i	$-0.428571\pi$
$881$	0.445042			0.445042			0.222521	+	0.974928i	$0.428571\pi$
$882$		0			0
$883$		0			0		1.00000			$0$
$883$		0			0		−1.00000			$\pi$
$884$	−0.900969	−	1.56052i	−0.900969	−	1.56052i
$885$		0			0
$886$		0			0
$887$	−0.900969	−	1.56052i	−0.900969	−	1.56052i	−0.826239	−	0.563320i	$-0.809524\pi$
$887$	−0.900969	−	1.56052i	−0.900969	−	1.56052i	−0.0747301	−	0.997204i	$-0.523810\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$		0			0
$892$		0			0
$893$		0			0
$894$		0			0
$895$		0			0
$896$		0			0
$897$		0			0
$898$		0			0
$899$		0			0
$900$	−0.623490	+	1.07992i	−0.623490	+	1.07992i
$901$		0			0
$902$		0			0
$903$		0			0
$904$		0			0
$905$		0			0
$906$		0			0
$907$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$907$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$908$		0			0
$909$	−2.00000			−2.00000
$910$		0			0
$911$	2.00000			2.00000			1.00000			$0$
$911$	2.00000			2.00000			1.00000			$0$
$912$		0			0
$913$		0			0
$914$	−0.400969	+	0.694498i	−0.400969	+	0.694498i
$915$		0			0
$916$	4.04892			4.04892
$917$		0			0
$918$		0			0
$919$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$919$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$920$	0.623490	−	1.07992i	0.623490	−	1.07992i
$921$		0			0
$922$		0			0
$923$	0.554958			0.554958
$924$		0			0
$925$	0.692021			0.692021
$926$	1.12349	+	1.94594i	1.12349	+	1.94594i
$927$	0.623490	−	1.07992i	0.623490	−	1.07992i
$928$		0			0
$929$	0.623490	+	1.07992i	0.623490	+	1.07992i	0.988831	+	0.149042i	$0.0476190\pi$
$929$	0.623490	+	1.07992i	0.623490	+	1.07992i	−0.365341	+	0.930874i	$0.619048\pi$
$930$		0			0
$931$		0			0
$932$		0			0
$933$		0			0
$934$	−1.80194	+	3.12105i	−1.80194	+	3.12105i
$935$		0			0
$936$	0.500000	+	0.866025i	0.500000	+	0.866025i
$937$		0			0		1.00000			$0$
$937$		0			0		−1.00000			$\pi$
$938$		0			0
$939$		0			0
$940$	0.623490	+	1.07992i	0.623490	+	1.07992i
$941$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$941$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$942$		0			0
$943$		0			0
$944$	0.801938			0.801938
$945$		0			0
$946$		0			0
$947$	−0.623490	−	1.07992i	−0.623490	−	1.07992i	−0.988831	−	0.149042i	$-0.952381\pi$
$947$	−0.623490	−	1.07992i	−0.623490	−	1.07992i	0.365341	−	0.930874i	$-0.380952\pi$
$948$		0			0
$949$	0.222521	−	0.385418i	0.222521	−	0.385418i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	−1.80194			−1.80194			−0.900969	−	0.433884i	$-0.857143\pi$
$953$	−1.80194			−1.80194			−0.900969	+	0.433884i	$0.857143\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	−1.12349	−	1.94594i	−1.12349	−	1.94594i
$962$	0.500000	−	0.866025i	0.500000	−	0.866025i
$963$		0			0
$964$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$965$		0			0
$966$		0			0
$967$	−1.80194			−1.80194			−0.900969	−	0.433884i	$-0.857143\pi$
$967$	−1.80194			−1.80194			−0.900969	+	0.433884i	$0.857143\pi$
$968$	1.12349	+	1.94594i	1.12349	+	1.94594i
$969$		0			0
$970$		0			0
$971$	0.623490	+	1.07992i	0.623490	+	1.07992i	0.988831	+	0.149042i	$0.0476190\pi$
$971$	0.623490	+	1.07992i	0.623490	+	1.07992i	−0.365341	+	0.930874i	$0.619048\pi$
$972$		0			0
$973$		0			0
$974$	0.801938			0.801938
$975$		0			0
$976$		0			0
$977$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$977$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$978$		0			0
$979$		0			0
$980$		0			0
$981$	−1.80194			−1.80194
$982$		0			0
$983$	−0.222521	+	0.385418i	−0.222521	+	0.385418i	−0.955573	−	0.294755i	$-0.904762\pi$
$983$	−0.222521	+	0.385418i	−0.222521	+	0.385418i	0.733052	+	0.680173i	$0.238095\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$		0			0
$989$		0			0
$990$		0			0
$991$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$991$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$992$	0.900969	+	1.56052i	0.900969	+	1.56052i
$993$		0			0
$994$		0			0
$995$	1.55496			1.55496
$996$		0			0
$997$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$997$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$998$	−1.62349	+	2.81197i	−1.62349	+	2.81197i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3577.1.p.a.2481.3		6
7.2	even	3	inner	3577.1.p.a.656.3		6
7.3	odd	6		511.1.c.a.510.1	✓	3
7.4	even	3		511.1.c.b.510.1	yes	3
7.5	odd	6		3577.1.p.b.656.3		6
7.6	odd	2		3577.1.p.b.2481.3		6
73.72	even	2		3577.1.p.b.2481.3		6
511.72	even	6		3577.1.p.b.656.3		6
511.145	odd	6	inner	3577.1.p.a.656.3		6
511.291	even	6		511.1.c.a.510.1	✓	3
511.437	odd	6		511.1.c.b.510.1	yes	3
511.510	odd	2	CM	3577.1.p.a.2481.3		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
511.1.c.a.510.1	✓	3	7.3	odd	6
511.1.c.a.510.1	✓	3	511.291	even	6
511.1.c.b.510.1	yes	3	7.4	even	3
511.1.c.b.510.1	yes	3	511.437	odd	6
3577.1.p.a.656.3		6	7.2	even	3	inner
3577.1.p.a.656.3		6	511.145	odd	6	inner
3577.1.p.a.2481.3		6	1.1	even	1	trivial
3577.1.p.a.2481.3		6	511.510	odd	2	CM
3577.1.p.b.656.3		6	7.5	odd	6
3577.1.p.b.656.3		6	511.72	even	6
3577.1.p.b.2481.3		6	7.6	odd	2
3577.1.p.b.2481.3		6	73.72	even	2

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:	\( N \)	\(=\)	\( 3577 = 7^{2} \cdot 73 \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	3577.p (of order \(6\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(0.900969 + 1.56052i) q^{2} +(-1.12349 + 1.94594i) q^{4} +(0.623490 + 1.07992i) q^{5} -2.24698 q^{8} +(-0.500000 - 0.866025i) q^{9} +O(q^{10})\)	\(q+(0.900969 + 1.56052i) q^{2} +(-1.12349 + 1.94594i) q^{4} +(0.623490 + 1.07992i) q^{5} -2.24698 q^{8} +(-0.500000 - 0.866025i) q^{9} +(-1.12349 + 1.94594i) q^{10} +0.445042 q^{13} +(-0.900969 - 1.56052i) q^{16} +(-0.900969 + 1.56052i) q^{17} +(0.900969 - 1.56052i) q^{18} -2.80194 q^{20} +(0.222521 + 0.385418i) q^{23} +(-0.277479 + 0.480608i) q^{25} +(0.400969 + 0.694498i) q^{26} +(-0.900969 + 1.56052i) q^{31} +(0.500000 - 0.866025i) q^{32} -3.24698 q^{34} +2.24698 q^{36} +(-0.623490 - 1.07992i) q^{37} +(-1.40097 - 2.42655i) q^{40} +(0.623490 - 1.07992i) q^{45} +(-0.400969 + 0.694498i) q^{46} +(-0.222521 - 0.385418i) q^{47} -1.00000 q^{50} +(-0.500000 + 0.866025i) q^{52} +(-0.222521 + 0.385418i) q^{59} -3.24698 q^{62} +(0.277479 + 0.480608i) q^{65} +(0.900969 - 1.56052i) q^{67} +(-2.02446 - 3.50647i) q^{68} +1.24698 q^{71} +(1.12349 + 1.94594i) q^{72} +(0.500000 - 0.866025i) q^{73} +(1.12349 - 1.94594i) q^{74} +(0.900969 + 1.56052i) q^{79} +(1.12349 - 1.94594i) q^{80} +(-0.500000 + 0.866025i) q^{81} +1.80194 q^{83} -2.24698 q^{85} +2.24698 q^{90} -1.00000 q^{92} +(0.400969 - 0.694498i) q^{94} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + q^{2} - 2 q^{4} - q^{5} - 4 q^{8} - 3 q^{9}+O(q^{10}) \) 6 * q + q^2 - 2 * q^4 - q^5 - 4 * q^8 - 3 * q^9	\( 6 q + q^{2} - 2 q^{4} - q^{5} - 4 q^{8} - 3 q^{9} - 2 q^{10} + 2 q^{13} - q^{16} - q^{17} + q^{18} - 8 q^{20} + q^{23} - 2 q^{25} - 2 q^{26} - q^{31} + 3 q^{32} - 10 q^{34} + 4 q^{36} + q^{37} - 4 q^{40} - q^{45} + 2 q^{46} - q^{47} - 6 q^{50} - 3 q^{52} - q^{59} - 10 q^{62} + 2 q^{65} + q^{67} - 3 q^{68} - 2 q^{71} + 2 q^{72} + 3 q^{73} + 2 q^{74} + q^{79} + 2 q^{80} - 3 q^{81} + 2 q^{83} - 4 q^{85} + 4 q^{90} - 6 q^{92} - 2 q^{94}+O(q^{100}) \) 6 * q + q^2 - 2 * q^4 - q^5 - 4 * q^8 - 3 * q^9 - 2 * q^10 + 2 * q^13 - q^16 - q^17 + q^18 - 8 * q^20 + q^23 - 2 * q^25 - 2 * q^26 - q^31 + 3 * q^32 - 10 * q^34 + 4 * q^36 + q^37 - 4 * q^40 - q^45 + 2 * q^46 - q^47 - 6 * q^50 - 3 * q^52 - q^59 - 10 * q^62 + 2 * q^65 + q^67 - 3 * q^68 - 2 * q^71 + 2 * q^72 + 3 * q^73 + 2 * q^74 + q^79 + 2 * q^80 - 3 * q^81 + 2 * q^83 - 4 * q^85 + 4 * q^90 - 6 * q^92 - 2 * q^94

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