LMFDB - Embedded newform 3479.1.g.d.1206.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3479,1,Mod(851,3479)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3479, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([4, 3]))

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

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

chi := DirichletCharacter("3479.851");

S:= CuspForms(chi, 1);

N := Newforms(S);

Level:	$ N $	$=$	$ 3479 = 7^{2} \cdot 71 $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	3479.g (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:	$1.73624717895$
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:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 71)
Projective image:	$D_{7}$
Projective field:	Galois closure of 7.1.357911.1
Artin image:	$C_6\times D_7$
Artin field:	Galois closure of $\mathbb{Q}[x]/(x^{42} - \cdots)$

Embedding invariants

Embedding label			1206.3
Root			$-0.623490 + 1.07992i$ of defining polynomial
Character	$\chi$	$=$	3479.1206
Dual form			3479.1.g.d.851.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(0.900969 - 1.56052i) q^{2} +(-0.222521 - 0.385418i) q^{3} +(-1.12349 - 1.94594i) q^{4} +(0.623490 - 1.07992i) q^{5} -0.801938 q^{6} -2.24698 q^{8} +(0.400969 - 0.694498i) q^{9} +O(q^{10})$	\(q+(0.900969 - 1.56052i) q^{2} +(-0.222521 - 0.385418i) q^{3} +(-1.12349 - 1.94594i) q^{4} +(0.623490 - 1.07992i) q^{5} -0.801938 q^{6} -2.24698 q^{8} +(0.400969 - 0.694498i) q^{9} +(-1.12349 - 1.94594i) q^{10} +(-0.500000 + 0.866025i) q^{12} -0.554958 q^{15} +(-0.900969 + 1.56052i) q^{16} +(-0.722521 - 1.25144i) q^{18} +(-0.900969 + 1.56052i) q^{19} -2.80194 q^{20} +(0.500000 + 0.866025i) q^{24} +(-0.277479 - 0.480608i) q^{25} -0.801938 q^{27} -0.445042 q^{29} +(-0.500000 + 0.866025i) q^{30} +(0.500000 + 0.866025i) q^{32} -1.80194 q^{36} +(0.900969 - 1.56052i) q^{37} +(1.62349 + 2.81197i) q^{38} +(-1.40097 + 2.42655i) q^{40} +1.24698 q^{43} +(-0.500000 - 0.866025i) q^{45} +0.801938 q^{48} -1.00000 q^{50} +(-0.722521 + 1.25144i) q^{54} +0.801938 q^{57} +(-0.400969 + 0.694498i) q^{58} +(0.623490 + 1.07992i) q^{60} +1.00000 q^{71} +(-0.900969 + 1.56052i) q^{72} +(0.623490 + 1.07992i) q^{73} +(-1.62349 - 2.81197i) q^{74} +(-0.123490 + 0.213891i) q^{75} +4.04892 q^{76} +(-0.623490 + 1.07992i) q^{79} +(1.12349 + 1.94594i) q^{80} +(-0.222521 - 0.385418i) q^{81} +1.80194 q^{83} +(1.12349 - 1.94594i) q^{86} +(0.0990311 + 0.171527i) q^{87} +(-0.222521 + 0.385418i) q^{89} -1.80194 q^{90} +(1.12349 + 1.94594i) q^{95} +(0.222521 - 0.385418i) q^{96} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + q^{2} - q^{3} - 2 q^{4} - q^{5} + 4 q^{6} - 4 q^{8} - 2 q^{9}+O(q^{10}) $ 6 * q + q^2 - q^3 - 2 * q^4 - q^5 + 4 * q^6 - 4 * q^8 - 2 * q^9	$ 6 q + q^{2} - q^{3} - 2 q^{4} - q^{5} + 4 q^{6} - 4 q^{8} - 2 q^{9} - 2 q^{10} - 3 q^{12} - 4 q^{15} - q^{16} - 4 q^{18} - q^{19} - 8 q^{20} + 3 q^{24} - 2 q^{25} + 4 q^{27} - 2 q^{29} - 3 q^{30} + 3 q^{32} - 2 q^{36} + q^{37} + 5 q^{38} - 4 q^{40} - 2 q^{43} - 3 q^{45} - 4 q^{48} - 6 q^{50} - 4 q^{54} - 4 q^{57} + 2 q^{58} - q^{60} + 6 q^{71} - q^{72} - q^{73} - 5 q^{74} + 4 q^{75} + 6 q^{76} + q^{79} + 2 q^{80} - q^{81} + 2 q^{83} + 2 q^{86} + 5 q^{87} - q^{89} - 2 q^{90} + 2 q^{95} + q^{96}+O(q^{100}) $ 6 * q + q^2 - q^3 - 2 * q^4 - q^5 + 4 * q^6 - 4 * q^8 - 2 * q^9 - 2 * q^10 - 3 * q^12 - 4 * q^15 - q^16 - 4 * q^18 - q^19 - 8 * q^20 + 3 * q^24 - 2 * q^25 + 4 * q^27 - 2 * q^29 - 3 * q^30 + 3 * q^32 - 2 * q^36 + q^37 + 5 * q^38 - 4 * q^40 - 2 * q^43 - 3 * q^45 - 4 * q^48 - 6 * q^50 - 4 * q^54 - 4 * q^57 + 2 * q^58 - q^60 + 6 * q^71 - q^72 - q^73 - 5 * q^74 + 4 * q^75 + 6 * q^76 + q^79 + 2 * q^80 - q^81 + 2 * q^83 + 2 * q^86 + 5 * q^87 - q^89 - 2 * q^90 + 2 * q^95 + q^96

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/3479\mathbb{Z}\right)^\times$.

$n$	$640$	$1569$
$\chi(n)$	$e\left(\frac{1}{3}\right)$	$-1$

Coefficient data

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

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

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

$2$	0.900969	−	1.56052i	0.900969	−	1.56052i	0.0747301	−	0.997204i	$-0.476190\pi$
$2$	0.900969	−	1.56052i	0.900969	−	1.56052i	0.826239	−	0.563320i	$-0.190476\pi$
$3$	−0.222521	−	0.385418i	−0.222521	−	0.385418i	0.733052	−	0.680173i	$-0.238095\pi$
$3$	−0.222521	−	0.385418i	−0.222521	−	0.385418i	−0.955573	+	0.294755i	$0.904762\pi$
$4$	−1.12349	−	1.94594i	−1.12349	−	1.94594i
$5$	0.623490	−	1.07992i	0.623490	−	1.07992i	−0.365341	−	0.930874i	$-0.619048\pi$
$5$	0.623490	−	1.07992i	0.623490	−	1.07992i	0.988831	−	0.149042i	$-0.0476190\pi$
$6$	−0.801938			−0.801938
$7$		0			0
$7$		0			0
$8$	−2.24698			−2.24698
$9$	0.400969	−	0.694498i	0.400969	−	0.694498i
$10$	−1.12349	−	1.94594i	−1.12349	−	1.94594i
$11$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$11$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$12$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$13$		0			0		1.00000			$0$
$13$		0			0		−1.00000			$\pi$
$14$		0			0
$15$	−0.554958			−0.554958
$16$	−0.900969	+	1.56052i	−0.900969	+	1.56052i
$17$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$17$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$18$	−0.722521	−	1.25144i	−0.722521	−	1.25144i
$19$	−0.900969	+	1.56052i	−0.900969	+	1.56052i	−0.0747301	+	0.997204i	$0.523810\pi$
$19$	−0.900969	+	1.56052i	−0.900969	+	1.56052i	−0.826239	+	0.563320i	$0.809524\pi$
$20$	−2.80194			−2.80194
$21$		0			0
$22$		0			0
$23$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$23$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$24$	0.500000	+	0.866025i	0.500000	+	0.866025i
$25$	−0.277479	−	0.480608i	−0.277479	−	0.480608i
$26$		0			0
$27$	−0.801938			−0.801938
$28$		0			0
$29$	−0.445042			−0.445042			−0.222521	−	0.974928i	$-0.571429\pi$
$29$	−0.445042			−0.445042			−0.222521	+	0.974928i	$0.571429\pi$
$30$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$31$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$31$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$32$	0.500000	+	0.866025i	0.500000	+	0.866025i
$33$		0			0
$34$		0			0
$35$		0			0
$36$	−1.80194			−1.80194
$37$	0.900969	−	1.56052i	0.900969	−	1.56052i	0.0747301	−	0.997204i	$-0.476190\pi$
$37$	0.900969	−	1.56052i	0.900969	−	1.56052i	0.826239	−	0.563320i	$-0.190476\pi$
$38$	1.62349	+	2.81197i	1.62349	+	2.81197i
$39$		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
$43$	1.24698			1.24698			0.623490	−	0.781831i	$-0.285714\pi$
$43$	1.24698			1.24698			0.623490	+	0.781831i	$0.285714\pi$
$44$		0			0
$45$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$46$		0			0
$47$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$47$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$48$	0.801938			0.801938
$49$		0			0
$50$	−1.00000			−1.00000
$51$		0			0
$52$		0			0
$53$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$53$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$54$	−0.722521	+	1.25144i	−0.722521	+	1.25144i
$55$		0			0
$56$		0			0
$57$	0.801938			0.801938
$58$	−0.400969	+	0.694498i	−0.400969	+	0.694498i
$59$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$59$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$60$	0.623490	+	1.07992i	0.623490	+	1.07992i
$61$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$61$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$		0			0
$66$		0			0
$67$		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
$70$		0			0
$71$	1.00000			1.00000
$71$	1.00000			1.00000
$72$	−0.900969	+	1.56052i	−0.900969	+	1.56052i
$73$	0.623490	+	1.07992i	0.623490	+	1.07992i	0.988831	+	0.149042i	$0.0476190\pi$
$73$	0.623490	+	1.07992i	0.623490	+	1.07992i	−0.365341	+	0.930874i	$0.619048\pi$
$74$	−1.62349	−	2.81197i	−1.62349	−	2.81197i
$75$	−0.123490	+	0.213891i	−0.123490	+	0.213891i
$76$	4.04892			4.04892
$77$		0			0
$78$		0			0
$79$	−0.623490	+	1.07992i	−0.623490	+	1.07992i	0.365341	+	0.930874i	$0.380952\pi$
$79$	−0.623490	+	1.07992i	−0.623490	+	1.07992i	−0.988831	+	0.149042i	$0.952381\pi$
$80$	1.12349	+	1.94594i	1.12349	+	1.94594i
$81$	−0.222521	−	0.385418i	−0.222521	−	0.385418i
$82$		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
$85$		0			0
$86$	1.12349	−	1.94594i	1.12349	−	1.94594i
$87$	0.0990311	+	0.171527i	0.0990311	+	0.171527i
$88$		0			0
$89$	−0.222521	+	0.385418i	−0.222521	+	0.385418i	−0.955573	−	0.294755i	$-0.904762\pi$
$89$	−0.222521	+	0.385418i	−0.222521	+	0.385418i	0.733052	+	0.680173i	$0.238095\pi$
$90$	−1.80194			−1.80194
$91$		0			0
$92$		0			0
$93$		0			0
$94$		0			0
$95$	1.12349	+	1.94594i	1.12349	+	1.94594i
$96$	0.222521	−	0.385418i	0.222521	−	0.385418i
$97$		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
$101$	−0.900969	−	1.56052i	−0.900969	−	1.56052i	−0.826239	−	0.563320i	$-0.809524\pi$
$101$	−0.900969	−	1.56052i	−0.900969	−	1.56052i	−0.0747301	−	0.997204i	$-0.523810\pi$
$102$		0			0
$103$	−0.222521	+	0.385418i	−0.222521	+	0.385418i	−0.955573	−	0.294755i	$-0.904762\pi$
$103$	−0.222521	+	0.385418i	−0.222521	+	0.385418i	0.733052	+	0.680173i	$0.238095\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−1.00000	+	1.73205i	−1.00000	+	1.73205i	−0.500000	+	0.866025i	$0.666667\pi$
$107$	−1.00000	+	1.73205i	−1.00000	+	1.73205i	−0.500000	+	0.866025i	$0.666667\pi$
$108$	0.900969	+	1.56052i	0.900969	+	1.56052i
$109$	0.222521	+	0.385418i	0.222521	+	0.385418i	0.955573	−	0.294755i	$-0.0952381\pi$
$109$	0.222521	+	0.385418i	0.222521	+	0.385418i	−0.733052	+	0.680173i	$0.761905\pi$
$110$		0			0
$111$	−0.801938			−0.801938
$112$		0			0
$113$		0			0		1.00000			$0$
$113$		0			0		−1.00000			$\pi$
$114$	0.722521	−	1.25144i	0.722521	−	1.25144i
$115$		0			0
$116$	0.500000	+	0.866025i	0.500000	+	0.866025i
$117$		0			0
$118$		0			0
$119$		0			0
$120$	1.24698			1.24698
$121$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	0.554958			0.554958
$126$		0			0
$127$		0			0		1.00000			$0$
$127$		0			0		−1.00000			$\pi$
$128$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$129$	−0.277479	−	0.480608i	−0.277479	−	0.480608i
$130$		0			0
$131$	0.623490	−	1.07992i	0.623490	−	1.07992i	−0.365341	−	0.930874i	$-0.619048\pi$
$131$	0.623490	−	1.07992i	0.623490	−	1.07992i	0.988831	−	0.149042i	$-0.0476190\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$136$		0			0
$137$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$137$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$138$		0			0
$139$		0			0		1.00000			$0$
$139$		0			0		−1.00000			$\pi$
$140$		0			0
$141$		0			0
$142$	0.900969	−	1.56052i	0.900969	−	1.56052i
$143$		0			0
$144$	0.722521	+	1.25144i	0.722521	+	1.25144i
$145$	−0.277479	+	0.480608i	−0.277479	+	0.480608i
$146$	2.24698			2.24698
$147$		0			0
$148$	−4.04892			−4.04892
$149$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$149$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$150$	0.222521	+	0.385418i	0.222521	+	0.385418i
$151$	0.222521	+	0.385418i	0.222521	+	0.385418i	0.955573	−	0.294755i	$-0.0952381\pi$
$151$	0.222521	+	0.385418i	0.222521	+	0.385418i	−0.733052	+	0.680173i	$0.761905\pi$
$152$	2.02446	−	3.50647i	2.02446	−	3.50647i
$153$		0			0
$154$		0			0
$155$		0			0
$156$		0			0
$157$	−0.222521	−	0.385418i	−0.222521	−	0.385418i	0.733052	−	0.680173i	$-0.238095\pi$
$157$	−0.222521	−	0.385418i	−0.222521	−	0.385418i	−0.955573	+	0.294755i	$0.904762\pi$
$158$	1.12349	+	1.94594i	1.12349	+	1.94594i
$159$		0			0
$160$	1.24698			1.24698
$161$		0			0
$162$	−0.801938			−0.801938
$163$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$163$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$164$		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
$169$	1.00000			1.00000
$170$		0			0
$171$	0.722521	+	1.25144i	0.722521	+	1.25144i
$172$	−1.40097	−	2.42655i	−1.40097	−	2.42655i
$173$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$173$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$174$	0.356896			0.356896
$175$		0			0
$176$		0			0
$177$		0			0
$178$	0.400969	+	0.694498i	0.400969	+	0.694498i
$179$	−0.623490	−	1.07992i	−0.623490	−	1.07992i	−0.988831	−	0.149042i	$-0.952381\pi$
$179$	−0.623490	−	1.07992i	−0.623490	−	1.07992i	0.365341	−	0.930874i	$-0.380952\pi$
$180$	−1.12349	+	1.94594i	−1.12349	+	1.94594i
$181$		0			0		1.00000			$0$
$181$		0			0		−1.00000			$\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−1.12349	−	1.94594i	−1.12349	−	1.94594i
$186$		0			0
$187$		0			0
$188$		0			0
$189$		0			0
$190$	4.04892			4.04892
$191$	0.222521	−	0.385418i	0.222521	−	0.385418i	−0.733052	−	0.680173i	$-0.761905\pi$
$191$	0.222521	−	0.385418i	0.222521	−	0.385418i	0.955573	+	0.294755i	$0.0952381\pi$
$192$		0			0
$193$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$193$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$		0			0		1.00000			$0$
$197$		0			0		−1.00000			$\pi$
$198$		0			0
$199$	−0.900969	−	1.56052i	−0.900969	−	1.56052i	−0.826239	−	0.563320i	$-0.809524\pi$
$199$	−0.900969	−	1.56052i	−0.900969	−	1.56052i	−0.0747301	−	0.997204i	$-0.523810\pi$
$200$	0.623490	+	1.07992i	0.623490	+	1.07992i
$201$		0			0
$202$	−3.24698			−3.24698
$203$		0			0
$204$		0			0
$205$		0			0
$206$	0.400969	+	0.694498i	0.400969	+	0.694498i
$207$		0			0
$208$		0			0
$209$		0			0
$210$		0			0
$211$		0			0		1.00000			$0$
$211$		0			0		−1.00000			$\pi$
$212$		0			0
$213$	−0.222521	−	0.385418i	−0.222521	−	0.385418i
$214$	1.80194	+	3.12105i	1.80194	+	3.12105i
$215$	0.777479	−	1.34663i	0.777479	−	1.34663i
$216$	1.80194			1.80194
$217$		0			0
$218$	0.801938			0.801938
$219$	0.277479	−	0.480608i	0.277479	−	0.480608i
$220$		0			0
$221$		0			0
$222$	−0.722521	+	1.25144i	−0.722521	+	1.25144i
$223$	1.80194			1.80194			0.900969	−	0.433884i	$-0.142857\pi$
$223$	1.80194			1.80194			0.900969	+	0.433884i	$0.142857\pi$
$224$		0			0
$225$	−0.445042			−0.445042
$226$		0			0
$227$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$227$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$228$	−0.900969	−	1.56052i	−0.900969	−	1.56052i
$229$	0.623490	−	1.07992i	0.623490	−	1.07992i	−0.365341	−	0.930874i	$-0.619048\pi$
$229$	0.623490	−	1.07992i	0.623490	−	1.07992i	0.988831	−	0.149042i	$-0.0476190\pi$
$230$		0			0
$231$		0			0
$232$	1.00000			1.00000
$233$	0.900969	−	1.56052i	0.900969	−	1.56052i	0.0747301	−	0.997204i	$-0.476190\pi$
$233$	0.900969	−	1.56052i	0.900969	−	1.56052i	0.826239	−	0.563320i	$-0.190476\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$	0.554958			0.554958
$238$		0			0
$239$		0			0		1.00000			$0$
$239$		0			0		−1.00000			$\pi$
$240$	0.500000	−	0.866025i	0.500000	−	0.866025i
$241$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$241$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$242$	0.900969	+	1.56052i	0.900969	+	1.56052i
$243$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$244$		0			0
$245$		0			0
$246$		0			0
$247$		0			0
$248$		0			0
$249$	−0.400969	−	0.694498i	−0.400969	−	0.694498i
$250$	0.500000	−	0.866025i	0.500000	−	0.866025i
$251$	1.80194			1.80194			0.900969	−	0.433884i	$-0.142857\pi$
$251$	1.80194			1.80194			0.900969	+	0.433884i	$0.142857\pi$
$252$		0			0
$253$		0			0
$254$		0			0
$255$		0			0
$256$	0.900969	+	1.56052i	0.900969	+	1.56052i
$257$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$257$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$258$	−1.00000			−1.00000
$259$		0			0
$260$		0			0
$261$	−0.178448	+	0.309081i	−0.178448	+	0.309081i
$262$	−1.12349	−	1.94594i	−1.12349	−	1.94594i
$263$	−0.623490	−	1.07992i	−0.623490	−	1.07992i	−0.988831	−	0.149042i	$-0.952381\pi$
$263$	−0.623490	−	1.07992i	−0.623490	−	1.07992i	0.365341	−	0.930874i	$-0.380952\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$	0.198062			0.198062
$268$		0			0
$269$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$269$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$270$	0.900969	+	1.56052i	0.900969	+	1.56052i
$271$	−0.222521	+	0.385418i	−0.222521	+	0.385418i	−0.955573	−	0.294755i	$-0.904762\pi$
$271$	−0.222521	+	0.385418i	−0.222521	+	0.385418i	0.733052	+	0.680173i	$0.238095\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$		0			0
$276$		0			0
$277$	0.900969	+	1.56052i	0.900969	+	1.56052i	0.826239	+	0.563320i	$0.190476\pi$
$277$	0.900969	+	1.56052i	0.900969	+	1.56052i	0.0747301	+	0.997204i	$0.476190\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$		0			0		1.00000			$0$
$281$		0			0		−1.00000			$\pi$
$282$		0			0
$283$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$283$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$284$	−1.12349	−	1.94594i	−1.12349	−	1.94594i
$285$	0.500000	−	0.866025i	0.500000	−	0.866025i
$286$		0			0
$287$		0			0
$288$	0.801938			0.801938
$289$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$290$	0.500000	+	0.866025i	0.500000	+	0.866025i
$291$		0			0
$292$	1.40097	−	2.42655i	1.40097	−	2.42655i
$293$	−2.00000			−2.00000			−1.00000			$\pi$
$293$	−2.00000			−2.00000			−1.00000			$\pi$
$294$		0			0
$295$		0			0
$296$	−2.02446	+	3.50647i	−2.02446	+	3.50647i
$297$		0			0
$298$		0			0
$299$		0			0
$300$	0.554958			0.554958
$301$		0			0
$302$	0.801938			0.801938
$303$	−0.400969	+	0.694498i	−0.400969	+	0.694498i
$304$	−1.62349	−	2.81197i	−1.62349	−	2.81197i
$305$		0			0
$306$		0			0
$307$		0			0		1.00000			$0$
$307$		0			0		−1.00000			$\pi$
$308$		0			0
$309$	0.198062			0.198062
$310$		0			0
$311$	−0.900969	−	1.56052i	−0.900969	−	1.56052i	−0.826239	−	0.563320i	$-0.809524\pi$
$311$	−0.900969	−	1.56052i	−0.900969	−	1.56052i	−0.0747301	−	0.997204i	$-0.523810\pi$
$312$		0			0
$313$	−0.900969	+	1.56052i	−0.900969	+	1.56052i	−0.0747301	+	0.997204i	$0.523810\pi$
$313$	−0.900969	+	1.56052i	−0.900969	+	1.56052i	−0.826239	+	0.563320i	$0.809524\pi$
$314$	−0.801938			−0.801938
$315$		0			0
$316$	2.80194			2.80194
$317$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$317$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$318$		0			0
$319$		0			0
$320$		0			0
$321$	0.890084			0.890084
$322$		0			0
$323$		0			0
$324$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$325$		0			0
$326$		0			0
$327$	0.0990311	−	0.171527i	0.0990311	−	0.171527i
$328$		0			0
$329$		0			0
$330$		0			0
$331$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$331$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$332$	−2.02446	−	3.50647i	−2.02446	−	3.50647i
$333$	−0.722521	−	1.25144i	−0.722521	−	1.25144i
$334$	−1.12349	+	1.94594i	−1.12349	+	1.94594i
$335$		0			0
$336$		0			0
$337$		0			0		1.00000			$0$
$337$		0			0		−1.00000			$\pi$
$338$	0.900969	−	1.56052i	0.900969	−	1.56052i
$339$		0			0
$340$		0			0
$341$		0			0
$342$	2.60388			2.60388
$343$		0			0
$344$	−2.80194			−2.80194
$345$		0			0
$346$		0			0
$347$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$347$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$348$	0.222521	−	0.385418i	0.222521	−	0.385418i
$349$		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	$-0.666667\pi$
$353$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$354$		0			0
$355$	0.623490	−	1.07992i	0.623490	−	1.07992i
$356$	1.00000			1.00000
$357$		0			0
$358$	−2.24698			−2.24698
$359$	−0.623490	+	1.07992i	−0.623490	+	1.07992i	0.365341	+	0.930874i	$0.380952\pi$
$359$	−0.623490	+	1.07992i	−0.623490	+	1.07992i	−0.988831	+	0.149042i	$0.952381\pi$
$360$	1.12349	+	1.94594i	1.12349	+	1.94594i
$361$	−1.12349	−	1.94594i	−1.12349	−	1.94594i
$362$		0			0
$363$	0.445042			0.445042
$364$		0			0
$365$	1.55496			1.55496
$366$		0			0
$367$	0.623490	+	1.07992i	0.623490	+	1.07992i	0.988831	+	0.149042i	$0.0476190\pi$
$367$	0.623490	+	1.07992i	0.623490	+	1.07992i	−0.365341	+	0.930874i	$0.619048\pi$
$368$		0			0
$369$		0			0
$370$	−4.04892			−4.04892
$371$		0			0
$372$		0			0
$373$	−0.623490	+	1.07992i	−0.623490	+	1.07992i	0.365341	+	0.930874i	$0.380952\pi$
$373$	−0.623490	+	1.07992i	−0.623490	+	1.07992i	−0.988831	+	0.149042i	$0.952381\pi$
$374$		0			0
$375$	−0.123490	−	0.213891i	−0.123490	−	0.213891i
$376$		0			0
$377$		0			0
$378$		0			0
$379$	−0.445042			−0.445042			−0.222521	−	0.974928i	$-0.571429\pi$
$379$	−0.445042			−0.445042			−0.222521	+	0.974928i	$0.571429\pi$
$380$	2.52446	−	4.37249i	2.52446	−	4.37249i
$381$		0			0
$382$	−0.400969	−	0.694498i	−0.400969	−	0.694498i
$383$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$383$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$384$	0.445042			0.445042
$385$		0			0
$386$		0			0
$387$	0.500000	−	0.866025i	0.500000	−	0.866025i
$388$		0			0
$389$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$389$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$390$		0			0
$391$		0			0
$392$		0			0
$393$	−0.554958			−0.554958
$394$		0			0
$395$	0.777479	+	1.34663i	0.777479	+	1.34663i
$396$		0			0
$397$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$397$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$398$	−3.24698			−3.24698
$399$		0			0
$400$	1.00000			1.00000
$401$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$401$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$402$		0			0
$403$		0			0
$404$	−2.02446	+	3.50647i	−2.02446	+	3.50647i
$405$	−0.554958			−0.554958
$406$		0			0
$407$		0			0
$408$		0			0
$409$	−0.222521	−	0.385418i	−0.222521	−	0.385418i	0.733052	−	0.680173i	$-0.238095\pi$
$409$	−0.222521	−	0.385418i	−0.222521	−	0.385418i	−0.955573	+	0.294755i	$0.904762\pi$
$410$		0			0
$411$		0			0
$412$	1.00000			1.00000
$413$		0			0
$414$		0			0
$415$	1.12349	−	1.94594i	1.12349	−	1.94594i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	0.445042			0.445042			0.222521	−	0.974928i	$-0.428571\pi$
$419$	0.445042			0.445042			0.222521	+	0.974928i	$0.428571\pi$
$420$		0			0
$421$		0			0		1.00000			$0$
$421$		0			0		−1.00000			$\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$		0			0
$426$	−0.801938			−0.801938
$427$		0			0
$428$	4.49396			4.49396
$429$		0			0
$430$	−1.40097	−	2.42655i	−1.40097	−	2.42655i
$431$	0.900969	+	1.56052i	0.900969	+	1.56052i	0.826239	+	0.563320i	$0.190476\pi$
$431$	0.900969	+	1.56052i	0.900969	+	1.56052i	0.0747301	+	0.997204i	$0.476190\pi$
$432$	0.722521	−	1.25144i	0.722521	−	1.25144i
$433$		0			0		1.00000			$0$
$433$		0			0		−1.00000			$\pi$
$434$		0			0
$435$	0.246980			0.246980
$436$	0.500000	−	0.866025i	0.500000	−	0.866025i
$437$		0			0
$438$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$439$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$439$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$443$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$444$	0.900969	+	1.56052i	0.900969	+	1.56052i
$445$	0.277479	+	0.480608i	0.277479	+	0.480608i
$446$	1.62349	−	2.81197i	1.62349	−	2.81197i
$447$		0			0
$448$		0			0
$449$		0			0		1.00000			$0$
$449$		0			0		−1.00000			$\pi$
$450$	−0.400969	+	0.694498i	−0.400969	+	0.694498i
$451$		0			0
$452$		0			0
$453$	0.0990311	−	0.171527i	0.0990311	−	0.171527i
$454$		0			0
$455$		0			0
$456$	−1.80194			−1.80194
$457$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$457$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$458$	−1.12349	−	1.94594i	−1.12349	−	1.94594i
$459$		0			0
$460$		0			0
$461$		0			0		1.00000			$0$
$461$		0			0		−1.00000			$\pi$
$462$		0			0
$463$	−1.80194			−1.80194			−0.900969	−	0.433884i	$-0.857143\pi$
$463$	−1.80194			−1.80194			−0.900969	+	0.433884i	$0.857143\pi$
$464$	0.400969	−	0.694498i	0.400969	−	0.694498i
$465$		0			0
$466$	−1.62349	−	2.81197i	−1.62349	−	2.81197i
$467$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$467$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$	−0.0990311	+	0.171527i	−0.0990311	+	0.171527i
$472$		0			0
$473$		0			0
$474$	0.500000	−	0.866025i	0.500000	−	0.866025i
$475$	1.00000			1.00000
$476$		0			0
$477$		0			0
$478$		0			0
$479$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$479$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$480$	−0.277479	−	0.480608i	−0.277479	−	0.480608i
$481$		0			0
$482$		0			0
$483$		0			0
$484$	2.24698			2.24698
$485$		0			0
$486$	0.900969	+	1.56052i	0.900969	+	1.56052i
$487$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$487$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$488$		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
$496$		0			0
$497$		0			0
$498$	−1.44504			−1.44504
$499$	0.222521	−	0.385418i	0.222521	−	0.385418i	−0.733052	−	0.680173i	$-0.761905\pi$
$499$	0.222521	−	0.385418i	0.222521	−	0.385418i	0.955573	+	0.294755i	$0.0952381\pi$
$500$	−0.623490	−	1.07992i	−0.623490	−	1.07992i
$501$	0.277479	+	0.480608i	0.277479	+	0.480608i
$502$	1.62349	−	2.81197i	1.62349	−	2.81197i
$503$	0.445042			0.445042			0.222521	−	0.974928i	$-0.428571\pi$
$503$	0.445042			0.445042			0.222521	+	0.974928i	$0.428571\pi$
$504$		0			0
$505$	−2.24698			−2.24698
$506$		0			0
$507$	−0.222521	−	0.385418i	−0.222521	−	0.385418i
$508$		0			0
$509$	1.00000	−	1.73205i	1.00000	−	1.73205i	0.500000	−	0.866025i	$-0.333333\pi$
$509$	1.00000	−	1.73205i	1.00000	−	1.73205i	0.500000	−	0.866025i	$-0.333333\pi$
$510$		0			0
$511$		0			0
$512$	2.24698			2.24698
$513$	0.722521	−	1.25144i	0.722521	−	1.25144i
$514$		0			0
$515$	0.277479	+	0.480608i	0.277479	+	0.480608i
$516$	−0.623490	+	1.07992i	−0.623490	+	1.07992i
$517$		0			0
$518$		0			0
$519$		0			0
$520$		0			0
$521$	0.623490	+	1.07992i	0.623490	+	1.07992i	0.988831	+	0.149042i	$0.0476190\pi$
$521$	0.623490	+	1.07992i	0.623490	+	1.07992i	−0.365341	+	0.930874i	$0.619048\pi$
$522$	0.321552	+	0.556945i	0.321552	+	0.556945i
$523$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$523$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$524$	−2.80194			−2.80194
$525$		0			0
$526$	−2.24698			−2.24698
$527$		0			0
$528$		0			0
$529$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$530$		0			0
$531$		0			0
$532$		0			0
$533$		0			0
$534$	0.178448	−	0.309081i	0.178448	−	0.309081i
$535$	1.24698	+	2.15983i	1.24698	+	2.15983i
$536$		0			0
$537$	−0.277479	+	0.480608i	−0.277479	+	0.480608i
$538$		0			0
$539$		0			0
$540$	2.24698			2.24698
$541$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$541$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$542$	0.400969	+	0.694498i	0.400969	+	0.694498i
$543$		0			0
$544$		0			0
$545$	0.554958			0.554958
$546$		0			0
$547$	−1.80194			−1.80194			−0.900969	−	0.433884i	$-0.857143\pi$
$547$	−1.80194			−1.80194			−0.900969	+	0.433884i	$0.857143\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	0.400969	−	0.694498i	0.400969	−	0.694498i
$552$		0			0
$553$		0			0
$554$	3.24698			3.24698
$555$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$556$		0			0
$557$	0.222521	+	0.385418i	0.222521	+	0.385418i	0.955573	−	0.294755i	$-0.0952381\pi$
$557$	0.222521	+	0.385418i	0.222521	+	0.385418i	−0.733052	+	0.680173i	$0.761905\pi$
$558$		0			0
$559$		0			0
$560$		0			0
$561$		0			0
$562$		0			0
$563$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$563$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$		0			0
$568$	−2.24698			−2.24698
$569$	−0.623490	+	1.07992i	−0.623490	+	1.07992i	0.365341	+	0.930874i	$0.380952\pi$
$569$	−0.623490	+	1.07992i	−0.623490	+	1.07992i	−0.988831	+	0.149042i	$0.952381\pi$
$570$	−0.900969	−	1.56052i	−0.900969	−	1.56052i
$571$	0.900969	+	1.56052i	0.900969	+	1.56052i	0.826239	+	0.563320i	$0.190476\pi$
$571$	0.900969	+	1.56052i	0.900969	+	1.56052i	0.0747301	+	0.997204i	$0.476190\pi$
$572$		0			0
$573$	−0.198062			−0.198062
$574$		0			0
$575$		0			0
$576$		0			0
$577$	0.623490	+	1.07992i	0.623490	+	1.07992i	0.988831	+	0.149042i	$0.0476190\pi$
$577$	0.623490	+	1.07992i	0.623490	+	1.07992i	−0.365341	+	0.930874i	$0.619048\pi$
$578$	0.900969	+	1.56052i	0.900969	+	1.56052i
$579$		0			0
$580$	1.24698			1.24698
$581$		0			0
$582$		0			0
$583$		0			0
$584$	−1.40097	−	2.42655i	−1.40097	−	2.42655i
$585$		0			0
$586$	−1.80194	+	3.12105i	−1.80194	+	3.12105i
$587$	0.445042			0.445042			0.222521	−	0.974928i	$-0.428571\pi$
$587$	0.445042			0.445042			0.222521	+	0.974928i	$0.428571\pi$
$588$		0			0
$589$		0			0
$590$		0			0
$591$		0			0
$592$	1.62349	+	2.81197i	1.62349	+	2.81197i
$593$	−0.222521	+	0.385418i	−0.222521	+	0.385418i	−0.955573	−	0.294755i	$-0.904762\pi$
$593$	−0.222521	+	0.385418i	−0.222521	+	0.385418i	0.733052	+	0.680173i	$0.238095\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$	−0.400969	+	0.694498i	−0.400969	+	0.694498i
$598$		0			0
$599$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$599$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$600$	0.277479	−	0.480608i	0.277479	−	0.480608i
$601$		0			0		1.00000			$0$
$601$		0			0		−1.00000			$\pi$
$602$		0			0
$603$		0			0
$604$	0.500000	−	0.866025i	0.500000	−	0.866025i
$605$	0.623490	+	1.07992i	0.623490	+	1.07992i
$606$	0.722521	+	1.25144i	0.722521	+	1.25144i
$607$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$607$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$608$	−1.80194			−1.80194
$609$		0			0
$610$		0			0
$611$		0			0
$612$		0			0
$613$	0.222521	+	0.385418i	0.222521	+	0.385418i	0.955573	−	0.294755i	$-0.0952381\pi$
$613$	0.222521	+	0.385418i	0.222521	+	0.385418i	−0.733052	+	0.680173i	$0.761905\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	1.24698			1.24698			0.623490	−	0.781831i	$-0.285714\pi$
$617$	1.24698			1.24698			0.623490	+	0.781831i	$0.285714\pi$
$618$	0.178448	−	0.309081i	0.178448	−	0.309081i
$619$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$619$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$620$		0			0
$621$		0			0
$622$	−3.24698			−3.24698
$623$		0			0
$624$		0			0
$625$	0.623490	−	1.07992i	0.623490	−	1.07992i
$626$	1.62349	+	2.81197i	1.62349	+	2.81197i
$627$		0			0
$628$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$629$		0			0
$630$		0			0
$631$		0			0		1.00000			$0$
$631$		0			0		−1.00000			$\pi$
$632$	1.40097	−	2.42655i	1.40097	−	2.42655i
$633$		0			0
$634$		0			0
$635$		0			0
$636$		0			0
$637$		0			0
$638$		0			0
$639$	0.400969	−	0.694498i	0.400969	−	0.694498i
$640$	0.623490	+	1.07992i	0.623490	+	1.07992i
$641$	0.900969	+	1.56052i	0.900969	+	1.56052i	0.826239	+	0.563320i	$0.190476\pi$
$641$	0.900969	+	1.56052i	0.900969	+	1.56052i	0.0747301	+	0.997204i	$0.476190\pi$
$642$	0.801938	−	1.38900i	0.801938	−	1.38900i
$643$	−2.00000			−2.00000			−1.00000			$\pi$
$643$	−2.00000			−2.00000			−1.00000			$\pi$
$644$		0			0
$645$	−0.692021			−0.692021
$646$		0			0
$647$	1.00000	+	1.73205i	1.00000	+	1.73205i	0.500000	+	0.866025i	$0.333333\pi$
$647$	1.00000	+	1.73205i	1.00000	+	1.73205i	0.500000	+	0.866025i	$0.333333\pi$
$648$	0.500000	+	0.866025i	0.500000	+	0.866025i
$649$		0			0
$650$		0			0
$651$		0			0
$652$		0			0
$653$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$653$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$654$	−0.178448	−	0.309081i	−0.178448	−	0.309081i
$655$	−0.777479	−	1.34663i	−0.777479	−	1.34663i
$656$		0			0
$657$	1.00000			1.00000
$658$		0			0
$659$	−0.445042			−0.445042			−0.222521	−	0.974928i	$-0.571429\pi$
$659$	−0.445042			−0.445042			−0.222521	+	0.974928i	$0.571429\pi$
$660$		0			0
$661$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$661$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$662$		0			0
$663$		0			0
$664$	−4.04892			−4.04892
$665$		0			0
$666$	−2.60388			−2.60388
$667$		0			0
$668$	1.40097	+	2.42655i	1.40097	+	2.42655i
$669$	−0.400969	−	0.694498i	−0.400969	−	0.694498i
$670$		0			0
$671$		0			0
$672$		0			0
$673$		0			0		1.00000			$0$
$673$		0			0		−1.00000			$\pi$
$674$		0			0
$675$	0.222521	+	0.385418i	0.222521	+	0.385418i
$676$	−1.12349	−	1.94594i	−1.12349	−	1.94594i
$677$	−0.222521	+	0.385418i	−0.222521	+	0.385418i	−0.955573	−	0.294755i	$-0.904762\pi$
$677$	−0.222521	+	0.385418i	−0.222521	+	0.385418i	0.733052	+	0.680173i	$0.238095\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$		0			0
$682$		0			0
$683$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$683$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$684$	1.62349	−	2.81197i	1.62349	−	2.81197i
$685$		0			0
$686$		0			0
$687$	−0.554958			−0.554958
$688$	−1.12349	+	1.94594i	−1.12349	+	1.94594i
$689$		0			0
$690$		0			0
$691$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$691$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$		0			0
$696$	−0.222521	−	0.385418i	−0.222521	−	0.385418i
$697$		0			0
$698$		0			0
$699$	−0.801938			−0.801938
$700$		0			0
$701$		0			0		1.00000			$0$
$701$		0			0		−1.00000			$\pi$
$702$		0			0
$703$	1.62349	+	2.81197i	1.62349	+	2.81197i
$704$		0			0
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$709$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$710$	−1.12349	−	1.94594i	−1.12349	−	1.94594i
$711$	0.500000	+	0.866025i	0.500000	+	0.866025i
$712$	0.500000	−	0.866025i	0.500000	−	0.866025i
$713$		0			0
$714$		0			0
$715$		0			0
$716$	−1.40097	+	2.42655i	−1.40097	+	2.42655i
$717$		0			0
$718$	1.12349	+	1.94594i	1.12349	+	1.94594i
$719$	−0.900969	+	1.56052i	−0.900969	+	1.56052i	−0.0747301	+	0.997204i	$0.523810\pi$
$719$	−0.900969	+	1.56052i	−0.900969	+	1.56052i	−0.826239	+	0.563320i	$0.809524\pi$
$720$	1.80194			1.80194
$721$		0			0
$722$	−4.04892			−4.04892
$723$		0			0
$724$		0			0
$725$	0.123490	+	0.213891i	0.123490	+	0.213891i
$726$	0.400969	−	0.694498i	0.400969	−	0.694498i
$727$		0			0		1.00000			$0$
$727$		0			0		−1.00000			$\pi$
$728$		0			0
$729$		0			0
$730$	1.40097	−	2.42655i	1.40097	−	2.42655i
$731$		0			0
$732$		0			0
$733$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$733$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$734$	2.24698			2.24698
$735$		0			0
$736$		0			0
$737$		0			0
$738$		0			0
$739$	−1.00000	−	1.73205i	−1.00000	−	1.73205i	−0.500000	−	0.866025i	$-0.666667\pi$
$739$	−1.00000	−	1.73205i	−1.00000	−	1.73205i	−0.500000	−	0.866025i	$-0.666667\pi$
$740$	−2.52446	+	4.37249i	−2.52446	+	4.37249i
$741$		0			0
$742$		0			0
$743$		0			0		1.00000			$0$
$743$		0			0		−1.00000			$\pi$
$744$		0			0
$745$		0			0
$746$	1.12349	+	1.94594i	1.12349	+	1.94594i
$747$	0.722521	−	1.25144i	0.722521	−	1.25144i
$748$		0			0
$749$		0			0
$750$	−0.445042			−0.445042
$751$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$751$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$752$		0			0
$753$	−0.400969	−	0.694498i	−0.400969	−	0.694498i
$754$		0			0
$755$	0.554958			0.554958
$756$		0			0
$757$		0			0		1.00000			$0$
$757$		0			0		−1.00000			$\pi$
$758$	−0.400969	+	0.694498i	−0.400969	+	0.694498i
$759$		0			0
$760$	−2.52446	−	4.37249i	−2.52446	−	4.37249i
$761$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$761$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$762$		0			0
$763$		0			0
$764$	−1.00000			−1.00000
$765$		0			0
$766$		0			0
$767$		0			0
$768$	0.400969	−	0.694498i	0.400969	−	0.694498i
$769$		0			0		1.00000			$0$
$769$		0			0		−1.00000			$\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$773$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$774$	−0.900969	−	1.56052i	−0.900969	−	1.56052i
$775$		0			0
$776$		0			0
$777$		0			0
$778$		0			0
$779$		0			0
$780$		0			0
$781$		0			0
$782$		0			0
$783$	0.356896			0.356896
$784$		0			0
$785$	−0.554958			−0.554958
$786$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$787$	−0.222521	−	0.385418i	−0.222521	−	0.385418i	0.733052	−	0.680173i	$-0.238095\pi$
$787$	−0.222521	−	0.385418i	−0.222521	−	0.385418i	−0.955573	+	0.294755i	$0.904762\pi$
$788$		0			0
$789$	−0.277479	+	0.480608i	−0.277479	+	0.480608i
$790$	2.80194			2.80194
$791$		0			0
$792$		0			0
$793$		0			0
$794$		0			0
$795$		0			0
$796$	−2.02446	+	3.50647i

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:	\( N \)	\(=\)	\( 3479 = 7^{2} \cdot 71 \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	3479.g (of order \(6\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(0.900969 - 1.56052i) q^{2} +(-0.222521 - 0.385418i) q^{3} +(-1.12349 - 1.94594i) q^{4} +(0.623490 - 1.07992i) q^{5} -0.801938 q^{6} -2.24698 q^{8} +(0.400969 - 0.694498i) q^{9} +O(q^{10})\)	\(q+(0.900969 - 1.56052i) q^{2} +(-0.222521 - 0.385418i) q^{3} +(-1.12349 - 1.94594i) q^{4} +(0.623490 - 1.07992i) q^{5} -0.801938 q^{6} -2.24698 q^{8} +(0.400969 - 0.694498i) q^{9} +(-1.12349 - 1.94594i) q^{10} +(-0.500000 + 0.866025i) q^{12} -0.554958 q^{15} +(-0.900969 + 1.56052i) q^{16} +(-0.722521 - 1.25144i) q^{18} +(-0.900969 + 1.56052i) q^{19} -2.80194 q^{20} +(0.500000 + 0.866025i) q^{24} +(-0.277479 - 0.480608i) q^{25} -0.801938 q^{27} -0.445042 q^{29} +(-0.500000 + 0.866025i) q^{30} +(0.500000 + 0.866025i) q^{32} -1.80194 q^{36} +(0.900969 - 1.56052i) q^{37} +(1.62349 + 2.81197i) q^{38} +(-1.40097 + 2.42655i) q^{40} +1.24698 q^{43} +(-0.500000 - 0.866025i) q^{45} +0.801938 q^{48} -1.00000 q^{50} +(-0.722521 + 1.25144i) q^{54} +0.801938 q^{57} +(-0.400969 + 0.694498i) q^{58} +(0.623490 + 1.07992i) q^{60} +1.00000 q^{71} +(-0.900969 + 1.56052i) q^{72} +(0.623490 + 1.07992i) q^{73} +(-1.62349 - 2.81197i) q^{74} +(-0.123490 + 0.213891i) q^{75} +4.04892 q^{76} +(-0.623490 + 1.07992i) q^{79} +(1.12349 + 1.94594i) q^{80} +(-0.222521 - 0.385418i) q^{81} +1.80194 q^{83} +(1.12349 - 1.94594i) q^{86} +(0.0990311 + 0.171527i) q^{87} +(-0.222521 + 0.385418i) q^{89} -1.80194 q^{90} +(1.12349 + 1.94594i) q^{95} +(0.222521 - 0.385418i) q^{96} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + q^{2} - q^{3} - 2 q^{4} - q^{5} + 4 q^{6} - 4 q^{8} - 2 q^{9}+O(q^{10}) \) 6 * q + q^2 - q^3 - 2 * q^4 - q^5 + 4 * q^6 - 4 * q^8 - 2 * q^9	\( 6 q + q^{2} - q^{3} - 2 q^{4} - q^{5} + 4 q^{6} - 4 q^{8} - 2 q^{9} - 2 q^{10} - 3 q^{12} - 4 q^{15} - q^{16} - 4 q^{18} - q^{19} - 8 q^{20} + 3 q^{24} - 2 q^{25} + 4 q^{27} - 2 q^{29} - 3 q^{30} + 3 q^{32} - 2 q^{36} + q^{37} + 5 q^{38} - 4 q^{40} - 2 q^{43} - 3 q^{45} - 4 q^{48} - 6 q^{50} - 4 q^{54} - 4 q^{57} + 2 q^{58} - q^{60} + 6 q^{71} - q^{72} - q^{73} - 5 q^{74} + 4 q^{75} + 6 q^{76} + q^{79} + 2 q^{80} - q^{81} + 2 q^{83} + 2 q^{86} + 5 q^{87} - q^{89} - 2 q^{90} + 2 q^{95} + q^{96}+O(q^{100}) \) 6 * q + q^2 - q^3 - 2 * q^4 - q^5 + 4 * q^6 - 4 * q^8 - 2 * q^9 - 2 * q^10 - 3 * q^12 - 4 * q^15 - q^16 - 4 * q^18 - q^19 - 8 * q^20 + 3 * q^24 - 2 * q^25 + 4 * q^27 - 2 * q^29 - 3 * q^30 + 3 * q^32 - 2 * q^36 + q^37 + 5 * q^38 - 4 * q^40 - 2 * q^43 - 3 * q^45 - 4 * q^48 - 6 * q^50 - 4 * q^54 - 4 * q^57 + 2 * q^58 - q^60 + 6 * q^71 - q^72 - q^73 - 5 * q^74 + 4 * q^75 + 6 * q^76 + q^79 + 2 * q^80 - q^81 + 2 * q^83 + 2 * q^86 + 5 * q^87 - q^89 - 2 * q^90 + 2 * q^95 + q^96

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