LMFDB - Embedded newform 693.1.bl.a.142.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [693,1,Mod(142,693)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(693, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([4, 2, 3]))

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

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

chi := DirichletCharacter("693.142");

S:= CuspForms(chi, 1);

N := Newforms(S);

Level:	$ N $	$=$	$ 693 = 3^{2} \cdot 7 \cdot 11 $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	693.bl (of order $6$, degree $2$, 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:	$0.345852053755$
Analytic rank:	$0$
Dimension:	$4$
Relative dimension:	$2$ over $\Q(\zeta_{6})$
Coefficient field:	$\Q(\zeta_{12})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{2} + 1 $ x^4 - x^2 + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Projective image:	$A_{4}$
Projective field:	Galois closure of 4.0.480249.1
Artin image:	$\SL(2,3):C_2$
Artin field:	Galois closure of $\mathbb{Q}[x]/(x^{16} - \cdots)$

Embedding invariants

Embedding label			142.1
Root			$0.866025 - 0.500000i$ of defining polynomial
Character	$\chi$	$=$	693.142
Dual form			693.1.bl.a.571.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-0.866025 + 0.500000i) q^{2} -1.00000 q^{3} +(0.866025 - 0.500000i) q^{6} +1.00000i q^{7} -1.00000i q^{8} +1.00000 q^{9} +O(q^{10})$	\(q+(-0.866025 + 0.500000i) q^{2} -1.00000 q^{3} +(0.866025 - 0.500000i) q^{6} +1.00000i q^{7} -1.00000i q^{8} +1.00000 q^{9} +1.00000 q^{11} +(-0.866025 + 0.500000i) q^{13} +(-0.500000 - 0.866025i) q^{14} +(0.500000 + 0.866025i) q^{16} +(-0.866025 + 0.500000i) q^{17} +(-0.866025 + 0.500000i) q^{18} +(0.866025 + 0.500000i) q^{19} -1.00000i q^{21} +(-0.866025 + 0.500000i) q^{22} +1.00000i q^{24} -1.00000 q^{25} +(0.500000 - 0.866025i) q^{26} -1.00000 q^{27} +(-0.866025 - 0.500000i) q^{29} +(-0.500000 + 0.866025i) q^{31} -1.00000 q^{33} +(0.500000 - 0.866025i) q^{34} +(-0.500000 + 0.866025i) q^{37} -1.00000 q^{38} +(0.866025 - 0.500000i) q^{39} +(-0.866025 + 0.500000i) q^{41} +(0.500000 + 0.866025i) q^{42} +(0.866025 + 0.500000i) q^{43} +(0.500000 + 0.866025i) q^{47} +(-0.500000 - 0.866025i) q^{48} -1.00000 q^{49} +(0.866025 - 0.500000i) q^{50} +(0.866025 - 0.500000i) q^{51} +(-0.500000 - 0.866025i) q^{53} +(0.866025 - 0.500000i) q^{54} +1.00000 q^{56} +(-0.866025 - 0.500000i) q^{57} +1.00000 q^{58} +(-0.500000 + 0.866025i) q^{59} +(0.866025 - 0.500000i) q^{61} -1.00000i q^{62} +1.00000i q^{63} -1.00000 q^{64} +(0.866025 - 0.500000i) q^{66} +(0.500000 - 0.866025i) q^{67} -1.00000i q^{72} +(0.866025 - 0.500000i) q^{73} -1.00000i q^{74} +1.00000 q^{75} +1.00000i q^{77} +(-0.500000 + 0.866025i) q^{78} +(-0.866025 + 0.500000i) q^{79} +1.00000 q^{81} +(0.500000 - 0.866025i) q^{82} +(0.866025 + 0.500000i) q^{83} -1.00000 q^{86} +(0.866025 + 0.500000i) q^{87} -1.00000i q^{88} +(-0.500000 + 0.866025i) q^{89} +(-0.500000 - 0.866025i) q^{91} +(0.500000 - 0.866025i) q^{93} +(-0.866025 - 0.500000i) q^{94} +(0.500000 - 0.866025i) q^{97} +(0.866025 - 0.500000i) q^{98} +1.00000 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 4 q^{3} + 4 q^{9}+O(q^{10}) $ 4 * q - 4 * q^3 + 4 * q^9	$ 4 q - 4 q^{3} + 4 q^{9} + 4 q^{11} - 2 q^{14} + 2 q^{16} - 4 q^{25} + 2 q^{26} - 4 q^{27} - 2 q^{31} - 4 q^{33} + 2 q^{34} - 2 q^{37} - 4 q^{38} + 2 q^{42} + 2 q^{47} - 2 q^{48} - 4 q^{49} - 2 q^{53} + 4 q^{56} + 4 q^{58} - 2 q^{59} - 4 q^{64} + 2 q^{67} + 4 q^{75} - 2 q^{78} + 4 q^{81} + 2 q^{82} - 4 q^{86} - 2 q^{89} - 2 q^{91} + 2 q^{93} + 2 q^{97} + 4 q^{99}+O(q^{100}) $ 4 * q - 4 * q^3 + 4 * q^9 + 4 * q^11 - 2 * q^14 + 2 * q^16 - 4 * q^25 + 2 * q^26 - 4 * q^27 - 2 * q^31 - 4 * q^33 + 2 * q^34 - 2 * q^37 - 4 * q^38 + 2 * q^42 + 2 * q^47 - 2 * q^48 - 4 * q^49 - 2 * q^53 + 4 * q^56 + 4 * q^58 - 2 * q^59 - 4 * q^64 + 2 * q^67 + 4 * q^75 - 2 * q^78 + 4 * q^81 + 2 * q^82 - 4 * q^86 - 2 * q^89 - 2 * q^91 + 2 * q^93 + 2 * q^97 + 4 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/693\mathbb{Z}\right)^\times$.

$n$	$155$	$199$	$442$
$\chi(n)$	$e\left(\frac{2}{3}\right)$	$e\left(\frac{1}{3}\right)$	$-1$

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 $

$2$	−0.866025	+	0.500000i	−0.866025	+	0.500000i	−0.866025	−	0.500000i	$-0.833333\pi$
$2$	−0.866025	+	0.500000i	−0.866025	+	0.500000i			1.00000i	$0.5\pi$
$3$	−1.00000			−1.00000
$3$	−1.00000			−1.00000
$4$		0			0
$5$		0			0			−	1.00000i	$-0.5\pi$
$5$		0			0				1.00000i	$0.5\pi$
$6$	0.866025	−	0.500000i	0.866025	−	0.500000i
$7$			1.00000i			1.00000i
$7$			1.00000i			1.00000i
$8$		−	1.00000i		−	1.00000i
$9$	1.00000			1.00000
$10$		0			0
$11$	1.00000			1.00000
$11$	1.00000			1.00000
$12$		0			0
$13$	−0.866025	+	0.500000i	−0.866025	+	0.500000i	−0.866025	−	0.500000i	$-0.833333\pi$
$13$	−0.866025	+	0.500000i	−0.866025	+	0.500000i			1.00000i	$0.5\pi$
$14$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$15$		0			0
$16$	0.500000	+	0.866025i	0.500000	+	0.866025i
$17$	−0.866025	+	0.500000i	−0.866025	+	0.500000i	−0.866025	−	0.500000i	$-0.833333\pi$
$17$	−0.866025	+	0.500000i	−0.866025	+	0.500000i			1.00000i	$0.5\pi$
$18$	−0.866025	+	0.500000i	−0.866025	+	0.500000i
$19$	0.866025	+	0.500000i	0.866025	+	0.500000i	0.866025	−	0.500000i	$-0.166667\pi$
$19$	0.866025	+	0.500000i	0.866025	+	0.500000i			1.00000i	$0.5\pi$
$20$		0			0
$21$		−	1.00000i		−	1.00000i
$22$	−0.866025	+	0.500000i	−0.866025	+	0.500000i
$23$		0			0			−	1.00000i	$-0.5\pi$
$23$		0			0				1.00000i	$0.5\pi$
$24$			1.00000i			1.00000i
$25$	−1.00000			−1.00000
$26$	0.500000	−	0.866025i	0.500000	−	0.866025i
$27$	−1.00000			−1.00000
$28$		0			0
$29$	−0.866025	−	0.500000i	−0.866025	−	0.500000i		−	1.00000i	$-0.5\pi$
$29$	−0.866025	−	0.500000i	−0.866025	−	0.500000i	−0.866025	+	0.500000i	$0.833333\pi$
$30$		0			0
$31$	−0.500000	+	0.866025i	−0.500000	+	0.866025i	0.500000	+	0.866025i	$0.333333\pi$
$31$	−0.500000	+	0.866025i	−0.500000	+	0.866025i	−1.00000			$\pi$
$32$		0			0
$33$	−1.00000			−1.00000
$34$	0.500000	−	0.866025i	0.500000	−	0.866025i
$35$		0			0
$36$		0			0
$37$	−0.500000	+	0.866025i	−0.500000	+	0.866025i	0.500000	+	0.866025i	$0.333333\pi$
$37$	−0.500000	+	0.866025i	−0.500000	+	0.866025i	−1.00000			$\pi$
$38$	−1.00000			−1.00000
$39$	0.866025	−	0.500000i	0.866025	−	0.500000i
$40$		0			0
$41$	−0.866025	+	0.500000i	−0.866025	+	0.500000i	−0.866025	−	0.500000i	$-0.833333\pi$
$41$	−0.866025	+	0.500000i	−0.866025	+	0.500000i			1.00000i	$0.5\pi$
$42$	0.500000	+	0.866025i	0.500000	+	0.866025i
$43$	0.866025	+	0.500000i	0.866025	+	0.500000i	0.866025	−	0.500000i	$-0.166667\pi$
$43$	0.866025	+	0.500000i	0.866025	+	0.500000i			1.00000i	$0.5\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	0.500000	+	0.866025i	0.500000	+	0.866025i	1.00000			$0$
$47$	0.500000	+	0.866025i	0.500000	+	0.866025i	−0.500000	+	0.866025i	$0.666667\pi$
$48$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$49$	−1.00000			−1.00000
$50$	0.866025	−	0.500000i	0.866025	−	0.500000i
$51$	0.866025	−	0.500000i	0.866025	−	0.500000i
$52$		0			0
$53$	−0.500000	−	0.866025i	−0.500000	−	0.866025i	0.500000	−	0.866025i	$-0.333333\pi$
$53$	−0.500000	−	0.866025i	−0.500000	−	0.866025i	−1.00000			$\pi$
$54$	0.866025	−	0.500000i	0.866025	−	0.500000i
$55$		0			0
$56$	1.00000			1.00000
$57$	−0.866025	−	0.500000i	−0.866025	−	0.500000i
$58$	1.00000			1.00000
$59$	−0.500000	+	0.866025i	−0.500000	+	0.866025i	0.500000	+	0.866025i	$0.333333\pi$
$59$	−0.500000	+	0.866025i	−0.500000	+	0.866025i	−1.00000			$\pi$
$60$		0			0
$61$	0.866025	−	0.500000i	0.866025	−	0.500000i		−	1.00000i	$-0.5\pi$
$61$	0.866025	−	0.500000i	0.866025	−	0.500000i	0.866025	+	0.500000i	$0.166667\pi$
$62$		−	1.00000i		−	1.00000i
$63$			1.00000i			1.00000i
$64$	−1.00000			−1.00000
$65$		0			0
$66$	0.866025	−	0.500000i	0.866025	−	0.500000i
$67$	0.500000	−	0.866025i	0.500000	−	0.866025i	−0.500000	−	0.866025i	$-0.666667\pi$
$67$	0.500000	−	0.866025i	0.500000	−	0.866025i	1.00000			$0$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		0			0			−	1.00000i	$-0.5\pi$
$71$		0			0				1.00000i	$0.5\pi$
$72$		−	1.00000i		−	1.00000i
$73$	0.866025	−	0.500000i	0.866025	−	0.500000i		−	1.00000i	$-0.5\pi$
$73$	0.866025	−	0.500000i	0.866025	−	0.500000i	0.866025	+	0.500000i	$0.166667\pi$
$74$		−	1.00000i		−	1.00000i
$75$	1.00000			1.00000
$76$		0			0
$77$			1.00000i			1.00000i
$78$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$79$	−0.866025	+	0.500000i	−0.866025	+	0.500000i	−0.866025	−	0.500000i	$-0.833333\pi$
$79$	−0.866025	+	0.500000i	−0.866025	+	0.500000i			1.00000i	$0.5\pi$
$80$		0			0
$81$	1.00000			1.00000
$82$	0.500000	−	0.866025i	0.500000	−	0.866025i
$83$	0.866025	+	0.500000i	0.866025	+	0.500000i	0.866025	−	0.500000i	$-0.166667\pi$
$83$	0.866025	+	0.500000i	0.866025	+	0.500000i			1.00000i	$0.5\pi$
$84$		0			0
$85$		0			0
$86$	−1.00000			−1.00000
$87$	0.866025	+	0.500000i	0.866025	+	0.500000i
$88$		−	1.00000i		−	1.00000i
$89$	−0.500000	+	0.866025i	−0.500000	+	0.866025i	0.500000	+	0.866025i	$0.333333\pi$
$89$	−0.500000	+	0.866025i	−0.500000	+	0.866025i	−1.00000			$\pi$
$90$		0			0
$91$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$92$		0			0
$93$	0.500000	−	0.866025i	0.500000	−	0.866025i
$94$	−0.866025	−	0.500000i	−0.866025	−	0.500000i
$95$		0			0
$96$		0			0
$97$	0.500000	−	0.866025i	0.500000	−	0.866025i	−0.500000	−	0.866025i	$-0.666667\pi$
$97$	0.500000	−	0.866025i	0.500000	−	0.866025i	1.00000			$0$
$98$	0.866025	−	0.500000i	0.866025	−	0.500000i
$99$	1.00000			1.00000
$100$		0			0
$101$		0			0		1.00000			$0$
$101$		0			0		−1.00000			$\pi$
$102$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$103$		0			0			−	1.00000i	$-0.5\pi$
$103$		0			0				1.00000i	$0.5\pi$
$104$	0.500000	+	0.866025i	0.500000	+	0.866025i
$105$		0			0
$106$	0.866025	+	0.500000i	0.866025	+	0.500000i
$107$	0.866025	+	0.500000i	0.866025	+	0.500000i	0.866025	−	0.500000i	$-0.166667\pi$
$107$	0.866025	+	0.500000i	0.866025	+	0.500000i			1.00000i	$0.5\pi$
$108$		0			0
$109$	0.866025	−	0.500000i	0.866025	−	0.500000i		−	1.00000i	$-0.5\pi$
$109$	0.866025	−	0.500000i	0.866025	−	0.500000i	0.866025	+	0.500000i	$0.166667\pi$
$110$		0			0
$111$	0.500000	−	0.866025i	0.500000	−	0.866025i
$112$	−0.866025	+	0.500000i	−0.866025	+	0.500000i
$113$	0.500000	+	0.866025i	0.500000	+	0.866025i	1.00000			$0$
$113$	0.500000	+	0.866025i	0.500000	+	0.866025i	−0.500000	+	0.866025i	$0.666667\pi$
$114$	1.00000			1.00000
$115$		0			0
$116$		0			0
$117$	−0.866025	+	0.500000i	−0.866025	+	0.500000i
$118$		−	1.00000i		−	1.00000i
$119$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$120$		0			0
$121$	1.00000			1.00000
$122$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$123$	0.866025	−	0.500000i	0.866025	−	0.500000i
$124$		0			0
$125$		0			0
$126$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$127$		0			0		1.00000			$0$
$127$		0			0		−1.00000			$\pi$
$128$	0.866025	−	0.500000i	0.866025	−	0.500000i
$129$	−0.866025	−	0.500000i	−0.866025	−	0.500000i
$130$		0			0
$131$		−	2.00000i		−	2.00000i		−	1.00000i	$-0.5\pi$
$131$		−	2.00000i		−	2.00000i		−	1.00000i	$-0.5\pi$
$132$		0			0
$133$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$134$			1.00000i			1.00000i
$135$		0			0
$136$	0.500000	+	0.866025i	0.500000	+	0.866025i
$137$		0			0			−	1.00000i	$-0.5\pi$
$137$		0			0				1.00000i	$0.5\pi$
$138$		0			0
$139$	0.866025	−	0.500000i	0.866025	−	0.500000i		−	1.00000i	$-0.5\pi$
$139$	0.866025	−	0.500000i	0.866025	−	0.500000i	0.866025	+	0.500000i	$0.166667\pi$
$140$		0			0
$141$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$142$		0			0
$143$	−0.866025	+	0.500000i	−0.866025	+	0.500000i
$144$	0.500000	+	0.866025i	0.500000	+	0.866025i
$145$		0			0
$146$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$147$	1.00000			1.00000
$148$		0			0
$149$		0			0		1.00000			$0$
$149$		0			0		−1.00000			$\pi$
$150$	−0.866025	+	0.500000i	−0.866025	+	0.500000i
$151$			2.00000i			2.00000i			1.00000i	$0.5\pi$
$151$			2.00000i			2.00000i			1.00000i	$0.5\pi$
$152$	0.500000	−	0.866025i	0.500000	−	0.866025i
$153$	−0.866025	+	0.500000i	−0.866025	+	0.500000i
$154$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$155$		0			0
$156$		0			0
$157$	0.500000	−	0.866025i	0.500000	−	0.866025i	−0.500000	−	0.866025i	$-0.666667\pi$
$157$	0.500000	−	0.866025i	0.500000	−	0.866025i	1.00000			$0$
$158$	0.500000	−	0.866025i	0.500000	−	0.866025i
$159$	0.500000	+	0.866025i	0.500000	+	0.866025i
$160$		0			0
$161$		0			0
$162$	−0.866025	+	0.500000i	−0.866025	+	0.500000i
$163$	−0.500000	+	0.866025i	−0.500000	+	0.866025i	0.500000	+	0.866025i	$0.333333\pi$
$163$	−0.500000	+	0.866025i	−0.500000	+	0.866025i	−1.00000			$\pi$
$164$		0			0
$165$		0			0
$166$	−1.00000			−1.00000
$167$	−0.866025	+	0.500000i	−0.866025	+	0.500000i	−0.866025	−	0.500000i	$-0.833333\pi$
$167$	−0.866025	+	0.500000i	−0.866025	+	0.500000i			1.00000i	$0.5\pi$
$168$	−1.00000			−1.00000
$169$		0			0
$170$		0			0
$171$	0.866025	+	0.500000i	0.866025	+	0.500000i
$172$		0			0
$173$	−0.866025	+	0.500000i	−0.866025	+	0.500000i	−0.866025	−	0.500000i	$-0.833333\pi$
$173$	−0.866025	+	0.500000i	−0.866025	+	0.500000i			1.00000i	$0.5\pi$
$174$	−1.00000			−1.00000
$175$		−	1.00000i		−	1.00000i
$176$	0.500000	+	0.866025i	0.500000	+	0.866025i
$177$	0.500000	−	0.866025i	0.500000	−	0.866025i
$178$		−	1.00000i		−	1.00000i
$179$	0.500000	+	0.866025i	0.500000	+	0.866025i	1.00000			$0$
$179$	0.500000	+	0.866025i	0.500000	+	0.866025i	−0.500000	+	0.866025i	$0.666667\pi$
$180$		0			0
$181$		0			0			−	1.00000i	$-0.5\pi$
$181$		0			0				1.00000i	$0.5\pi$
$182$	0.866025	+	0.500000i	0.866025	+	0.500000i
$183$	−0.866025	+	0.500000i	−0.866025	+	0.500000i
$184$		0			0
$185$		0			0
$186$			1.00000i			1.00000i
$187$	−0.866025	+	0.500000i	−0.866025	+	0.500000i
$188$		0			0
$189$		−	1.00000i		−	1.00000i
$190$		0			0
$191$	−0.500000	−	0.866025i	−0.500000	−	0.866025i	0.500000	−	0.866025i	$-0.333333\pi$
$191$	−0.500000	−	0.866025i	−0.500000	−	0.866025i	−1.00000			$\pi$
$192$	1.00000			1.00000
$193$	−0.866025	−	0.500000i	−0.866025	−	0.500000i		−	1.00000i	$-0.5\pi$
$193$	−0.866025	−	0.500000i	−0.866025	−	0.500000i	−0.866025	+	0.500000i	$0.833333\pi$
$194$			1.00000i			1.00000i
$195$		0			0
$196$		0			0
$197$		0			0		1.00000			$0$
$197$		0			0		−1.00000			$\pi$
$198$	−0.866025	+	0.500000i	−0.866025	+	0.500000i
$199$	−0.500000	−	0.866025i	−0.500000	−	0.866025i	0.500000	−	0.866025i	$-0.333333\pi$
$199$	−0.500000	−	0.866025i	−0.500000	−	0.866025i	−1.00000			$\pi$
$200$			1.00000i			1.00000i
$201$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$202$		0			0
$203$	0.500000	−	0.866025i	0.500000	−	0.866025i
$204$		0			0
$205$		0			0
$206$		0			0
$207$		0			0
$208$	−0.866025	−	0.500000i	−0.866025	−	0.500000i
$209$	0.866025	+	0.500000i	0.866025	+	0.500000i
$210$		0			0
$211$	0.866025	−	0.500000i	0.866025	−	0.500000i		−	1.00000i	$-0.5\pi$
$211$	0.866025	−	0.500000i	0.866025	−	0.500000i	0.866025	+	0.500000i	$0.166667\pi$
$212$		0			0
$213$		0			0
$214$	−1.00000			−1.00000
$215$		0			0
$216$			1.00000i			1.00000i
$217$	−0.866025	−	0.500000i	−0.866025	−	0.500000i
$218$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$219$	−0.866025	+	0.500000i	−0.866025	+	0.500000i
$220$		0			0
$221$	0.500000	−	0.866025i	0.500000	−	0.866025i
$222$			1.00000i			1.00000i
$223$	0.500000	−	0.866025i	0.500000	−	0.866025i	−0.500000	−	0.866025i	$-0.666667\pi$
$223$	0.500000	−	0.866025i	0.500000	−	0.866025i	1.00000			$0$
$224$		0			0
$225$	−1.00000			−1.00000
$226$	−0.866025	−	0.500000i	−0.866025	−	0.500000i
$227$		−	2.00000i		−	2.00000i		−	1.00000i	$-0.5\pi$
$227$		−	2.00000i		−	2.00000i		−	1.00000i	$-0.5\pi$
$228$		0			0
$229$		0			0			−	1.00000i	$-0.5\pi$
$229$		0			0				1.00000i	$0.5\pi$
$230$		0			0
$231$		−	1.00000i		−	1.00000i
$232$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$233$	0.866025	+	0.500000i	0.866025	+	0.500000i	0.866025	−	0.500000i	$-0.166667\pi$
$233$	0.866025	+	0.500000i	0.866025	+	0.500000i			1.00000i	$0.5\pi$
$234$	0.500000	−	0.866025i	0.500000	−	0.866025i
$235$		0			0
$236$		0			0
$237$	0.866025	−	0.500000i	0.866025	−	0.500000i
$238$	0.866025	+	0.500000i	0.866025	+	0.500000i
$239$	0.866025	−	0.500000i	0.866025	−	0.500000i		−	1.00000i	$-0.5\pi$
$239$	0.866025	−	0.500000i	0.866025	−	0.500000i	0.866025	+	0.500000i	$0.166667\pi$
$240$		0			0
$241$		0			0		1.00000			$0$
$241$		0			0		−1.00000			$\pi$
$242$	−0.866025	+	0.500000i	−0.866025	+	0.500000i
$243$	−1.00000			−1.00000
$244$		0			0
$245$		0			0
$246$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$247$	−1.00000			−1.00000
$248$	0.866025	+	0.500000i	0.866025	+	0.500000i
$249$	−0.866025	−	0.500000i	−0.866025	−	0.500000i
$250$		0			0
$251$		0			0			−	1.00000i	$-0.5\pi$
$251$		0			0				1.00000i	$0.5\pi$
$252$		0			0
$253$		0			0
$254$		0			0
$255$		0			0
$256$		0			0
$257$		0			0			−	1.00000i	$-0.5\pi$
$257$		0			0				1.00000i	$0.5\pi$
$258$	1.00000			1.00000
$259$	−0.866025	−	0.500000i	−0.866025	−	0.500000i
$260$		0			0
$261$	−0.866025	−	0.500000i	−0.866025	−	0.500000i
$262$	1.00000	+	1.73205i	1.00000	+	1.73205i
$263$			2.00000i			2.00000i			1.00000i	$0.5\pi$
$263$			2.00000i			2.00000i			1.00000i	$0.5\pi$
$264$			1.00000i			1.00000i
$265$		0			0
$266$		−	1.00000i		−	1.00000i
$267$	0.500000	−	0.866025i	0.500000	−	0.866025i
$268$		0			0
$269$	0.500000	+	0.866025i	0.500000	+	0.866025i	1.00000			$0$
$269$	0.500000	+	0.866025i	0.500000	+	0.866025i	−0.500000	+	0.866025i	$0.666667\pi$
$270$		0			0
$271$	−0.866025	−	0.500000i	−0.866025	−	0.500000i		−	1.00000i	$-0.5\pi$
$271$	−0.866025	−	0.500000i	−0.866025	−	0.500000i	−0.866025	+	0.500000i	$0.833333\pi$
$272$	−0.866025	−	0.500000i	−0.866025	−	0.500000i
$273$	0.500000	+	0.866025i	0.500000	+	0.866025i
$274$		0			0
$275$	−1.00000			−1.00000
$276$		0			0
$277$		0			0		1.00000			$0$
$277$		0			0		−1.00000			$\pi$
$278$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$279$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$280$		0			0
$281$	0.866025	+	0.500000i	0.866025	+	0.500000i	0.866025	−	0.500000i	$-0.166667\pi$
$281$	0.866025	+	0.500000i	0.866025	+	0.500000i			1.00000i	$0.5\pi$
$282$	0.866025	+	0.500000i	0.866025	+	0.500000i
$283$	0.866025	+	0.500000i	0.866025	+	0.500000i	0.866025	−	0.500000i	$-0.166667\pi$
$283$	0.866025	+	0.500000i	0.866025	+	0.500000i			1.00000i	$0.5\pi$
$284$		0			0
$285$		0			0
$286$	0.500000	−	0.866025i	0.500000	−	0.866025i
$287$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$288$		0			0
$289$		0			0
$290$		0			0
$291$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$292$		0			0
$293$	0.866025	−	0.500000i	0.866025	−	0.500000i		−	1.00000i	$-0.5\pi$
$293$	0.866025	−	0.500000i	0.866025	−	0.500000i	0.866025	+	0.500000i	$0.166667\pi$
$294$	−0.866025	+	0.500000i	−0.866025	+	0.500000i
$295$		0			0
$296$	0.866025	+	0.500000i	0.866025	+	0.500000i
$297$	−1.00000			−1.00000
$298$		0			0
$299$		0			0
$300$		0			0
$301$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$302$	−1.00000	−	1.73205i	−1.00000	−	1.73205i
$303$		0			0
$304$			1.00000i			1.00000i
$305$		0			0
$306$	0.500000	−	0.866025i	0.500000	−	0.866025i
$307$		0			0		1.00000			$0$
$307$		0			0		−1.00000			$\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	0.500000	−	0.866025i	0.500000	−	0.866025i	−0.500000	−	0.866025i	$-0.666667\pi$
$311$	0.500000	−	0.866025i	0.500000	−	0.866025i	1.00000			$0$
$312$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$313$	0.500000	+	0.866025i	0.500000	+	0.866025i	1.00000			$0$
$313$	0.500000	+	0.866025i	0.500000	+	0.866025i	−0.500000	+	0.866025i	$0.666667\pi$
$314$			1.00000i			1.00000i
$315$		0			0
$316$		0			0
$317$	−0.500000	−	0.866025i	−0.500000	−	0.866025i	0.500000	−	0.866025i	$-0.333333\pi$
$317$	−0.500000	−	0.866025i	−0.500000	−	0.866025i	−1.00000			$\pi$
$318$	−0.866025	−	0.500000i	−0.866025	−	0.500000i
$319$	−0.866025	−	0.500000i	−0.866025	−	0.500000i
$320$		0			0
$321$	−0.866025	−	0.500000i	−0.866025	−	0.500000i
$322$		0			0
$323$	−1.00000			−1.00000
$324$		0			0
$325$	0.866025	−	0.500000i	0.866025	−	0.500000i
$326$		−	1.00000i		−	1.00000i
$327$	−0.866025	+	0.500000i	−0.866025	+	0.500000i
$328$	0.500000	+	0.866025i	0.500000	+	0.866025i
$329$	−0.866025	+	0.500000i	−0.866025	+	0.500000i
$330$		0			0
$331$	0.500000	+	0.866025i	0.500000	+	0.866025i	1.00000			$0$
$331$	0.500000	+	0.866025i	0.500000	+	0.866025i	−0.500000	+	0.866025i	$0.666667\pi$
$332$		0			0
$333$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$334$	0.500000	−	0.866025i	0.500000	−	0.866025i
$335$		0			0
$336$	0.866025	−	0.500000i	0.866025	−	0.500000i
$337$	0.866025	−	0.500000i	0.866025	−	0.500000i		−	1.00000i	$-0.5\pi$
$337$	0.866025	−	0.500000i	0.866025	−	0.500000i	0.866025	+	0.500000i	$0.166667\pi$
$338$		0			0
$339$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$340$		0			0
$341$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$342$	−1.00000			−1.00000
$343$		−	1.00000i		−	1.00000i
$344$	0.500000	−	0.866025i	0.500000	−	0.866025i
$345$		0			0
$346$	0.500000	−	0.866025i	0.500000	−	0.866025i
$347$	−0.866025	−	0.500000i	−0.866025	−	0.500000i		−	1.00000i	$-0.5\pi$
$347$	−0.866025	−	0.500000i	−0.866025	−	0.500000i	−0.866025	+	0.500000i	$0.833333\pi$
$348$		0			0
$349$	0.866025	+	0.500000i	0.866025	+	0.500000i	0.866025	−	0.500000i	$-0.166667\pi$
$349$	0.866025	+	0.500000i	0.866025	+	0.500000i			1.00000i	$0.5\pi$
$350$	0.500000	+	0.866025i	0.500000	+	0.866025i
$351$	0.866025	−	0.500000i	0.866025	−	0.500000i
$352$		0			0
$353$		0			0			−	1.00000i	$-0.5\pi$
$353$		0			0				1.00000i	$0.5\pi$
$354$			1.00000i			1.00000i
$355$		0			0
$356$		0			0
$357$	0.500000	+	0.866025i	0.500000	+	0.866025i
$358$	−0.866025	−	0.500000i	−0.866025	−	0.500000i
$359$	−0.866025	−	0.500000i	−0.866025	−	0.500000i		−	1.00000i	$-0.5\pi$
$359$	−0.866025	−	0.500000i	−0.866025	−	0.500000i	−0.866025	+	0.500000i	$0.833333\pi$
$360$		0			0
$361$		0			0
$362$		0			0
$363$	−1.00000			−1.00000
$364$		0			0
$365$		0			0
$366$	0.500000	−	0.866025i	0.500000	−	0.866025i
$367$		0			0			−	1.00000i	$-0.5\pi$
$367$		0			0				1.00000i	$0.5\pi$
$368$		0			0
$369$	−0.866025	+	0.500000i	−0.866025	+	0.500000i
$370$		0			0
$371$	0.866025	−	0.500000i	0.866025	−	0.500000i
$372$		0			0
$373$		0			0		1.00000			$0$
$373$		0			0		−1.00000			$\pi$
$374$	0.500000	−	0.866025i	0.500000	−	0.866025i
$375$		0			0
$376$	0.866025	−	0.500000i	0.866025	−	0.500000i
$377$	1.00000			1.00000
$378$	0.500000	+	0.866025i	0.500000	+	0.866025i
$379$	2.00000			2.00000			1.00000			$0$
$379$	2.00000			2.00000			1.00000			$0$
$380$		0			0
$381$		0			0
$382$	0.866025	+	0.500000i	0.866025	+	0.500000i
$383$	2.00000			2.00000			1.00000			$0$
$383$	2.00000			2.00000			1.00000			$0$
$384$	−0.866025	+	0.500000i	−0.866025	+	0.500000i
$385$		0			0
$386$	1.00000			1.00000
$387$	0.866025	+	0.500000i	0.866025	+	0.500000i
$388$		0			0
$389$		0			0			−	1.00000i	$-0.5\pi$
$389$		0			0				1.00000i	$0.5\pi$
$390$		0			0
$391$		0			0
$392$			1.00000i			1.00000i
$393$			2.00000i			2.00000i
$394$		0			0
$395$		0			0
$396$		0			0
$397$	0.500000	−	0.866025i	0.500000	−	0.866025i	−0.500000	−	0.866025i	$-0.666667\pi$
$397$	0.500000	−	0.866025i	0.500000	−	0.866025i	1.00000			$0$
$398$	0.866025	+	0.500000i	0.866025	+	0.500000i
$399$	0.500000	−	0.866025i	0.500000	−	0.866025i
$400$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$401$		0			0			−	1.00000i	$-0.5\pi$
$401$		0			0				1.00000i	$0.5\pi$
$402$		−	1.00000i		−	1.00000i
$403$		−	1.00000i		−	1.00000i
$404$		0			0
$405$		0			0
$406$			1.00000i			1.00000i
$407$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$408$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$409$	−0.866025	−	0.500000i	−0.866025	−	0.500000i		−	1.00000i	$-0.5\pi$
$409$	−0.866025	−	0.500000i	−0.866025	−	0.500000i	−0.866025	+	0.500000i	$0.833333\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	−0.866025	−	0.500000i	−0.866025	−	0.500000i
$414$		0			0
$415$		0			0
$416$		0			0
$417$	−0.866025	+	0.500000i	−0.866025	+	0.500000i
$418$	−1.00000			−1.00000
$419$	0.500000	+	0.866025i	0.500000	+	0.866025i	1.00000			$0$
$419$	0.500000	+	0.866025i	0.500000	+	0.866025i	−0.500000	+	0.866025i	$0.666667\pi$
$420$		0			0
$421$	0.500000	−	0.866025i	0.500000	−	0.866025i	−0.500000	−	0.866025i	$-0.666667\pi$
$421$	0.500000	−	0.866025i	0.500000	−	0.866025i	1.00000			$0$
$422$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$423$	0.500000	+	0.866025i	0.500000	+	0.866025i
$424$	−0.866025	+	0.500000i	−0.866025	+	0.500000i
$425$	0.866025	−	0.500000i	0.866025	−	0.500000i
$426$		0			0
$427$	0.500000	+	0.866025i	0.500000	+	0.866025i
$428$		0			0
$429$	0.866025	−	0.500000i	0.866025	−	0.500000i
$430$		0			0
$431$	0.866025	−	0.500000i	0.866025	−	0.500000i		−	1.00000i	$-0.5\pi$
$431$	0.866025	−	0.500000i	0.866025	−	0.500000i	0.866025	+	0.500000i	$0.166667\pi$
$432$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$433$		0			0			−	1.00000i	$-0.5\pi$
$433$		0			0				1.00000i	$0.5\pi$
$434$	1.00000			1.00000
$435$		0			0
$436$		0			0
$437$		0			0
$438$	0.500000	−	0.866025i	0.500000	−	0.866025i
$439$	−0.866025	+	0.500000i	−0.866025	+	0.500000i	−0.866025	−	0.500000i	$-0.833333\pi$
$439$	−0.866025	+	0.500000i	−0.866025	+	0.500000i			1.00000i	$0.5\pi$
$440$		0			0
$441$	−1.00000			−1.00000
$442$			1.00000i			1.00000i
$443$	−0.500000	−	0.866025i	−0.500000	−	0.866025i	0.500000	−	0.866025i	$-0.333333\pi$
$443$	−0.500000	−	0.866025i	−0.500000	−	0.866025i	−1.00000			$\pi$
$444$		0			0
$445$		0			0
$446$			1.00000i			1.00000i
$447$		0			0
$448$		−	1.00000i		−	1.00000i
$449$		0			0			−	1.00000i	$-0.5\pi$
$449$		0			0				1.00000i	$0.5\pi$
$450$	0.866025	−	0.500000i	0.866025	−	0.500000i
$451$	−0.866025	+	0.500000i	−0.866025	+	0.500000i
$452$		0			0
$453$		−	2.00000i		−	2.00000i
$454$	1.00000	+	1.73205i	1.00000	+	1.73205i
$455$		0			0
$456$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$457$	0.866025	−	0.500000i	0.866025	−	0.500000i		−	1.00000i	$-0.5\pi$
$457$	0.866025	−	0.500000i	0.866025	−	0.500000i	0.866025	+	0.500000i	$0.166667\pi$
$458$		0			0
$459$	0.866025	−	0.500000i	0.866025	−	0.500000i
$460$		0			0
$461$	−0.866025	−	0.500000i	−0.866025	−	0.500000i		−	1.00000i	$-0.5\pi$
$461$	−0.866025	−	0.500000i	−0.866025	−	0.500000i	−0.866025	+	0.500000i	$0.833333\pi$
$462$	0.500000	+	0.866025i	0.500000	+	0.866025i
$463$	0.500000	+	0.866025i	0.500000	+	0.866025i	1.00000			$0$
$463$	0.500000	+	0.866025i	0.500000	+	0.866025i	−0.500000	+	0.866025i	$0.666667\pi$
$464$		−	1.00000i		−	1.00000i
$465$		0			0
$466$	−1.00000			−1.00000
$467$	0.500000	−	0.866025i	0.500000	−	0.866025i	−0.500000	−	0.866025i	$-0.666667\pi$
$467$	0.500000	−	0.866025i	0.500000	−	0.866025i	1.00000			$0$
$468$		0			0
$469$	0.866025	+	0.500000i	0.866025	+	0.500000i
$470$		0			0
$471$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$472$	0.866025	+	0.500000i	0.866025	+	0.500000i
$473$	0.866025	+	0.500000i	0.866025	+	0.500000i
$474$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$475$	−0.866025	−	0.500000i	−0.866025	−	0.500000i
$476$		0			0
$477$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$478$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$479$		0			0		1.00000			$0$
$479$		0			0		−1.00000			$\pi$
$480$		0			0
$481$		−	1.00000i		−	1.00000i
$482$		0			0
$483$		0			0
$484$		0			0
$485$		0			0
$486$	0.866025	−	0.500000i	0.866025	−	0.500000i
$487$	0.500000	+	0.866025i	0.500000	+	0.866025i	1.00000			$0$
$487$	0.500000	+	0.866025i	0.500000	+	0.866025i	−0.500000	+	0.866025i	$0.666667\pi$
$488$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$489$	0.500000	−	0.866025i	0.500000	−	0.866025i
$490$		0			0
$491$	−0.866025	+	0.500000i	−0.866025	+	0.500000i	−0.866025	−	0.500000i	$-0.833333\pi$
$491$	−0.866025	+	0.500000i	−0.866025	+	0.500000i			1.00000i	$0.5\pi$
$492$		0			0
$493$	1.00000			1.00000
$494$	0.866025	−	0.500000i	0.866025	−	0.500000i
$495$		0			0
$496$	−1.00000			−1.00000
$497$		0			0
$498$	1.00000			1.00000
$499$		0			0			−	1.00000i	$-0.5\pi$
$499$		0			0				1.00000i	$0.5\pi$
$500$		0			0
$501$	0.866025	−	0.500000i	0.866025	−	0.500000i
$502$		0			0
$503$		0			0		1.00000			$0$
$503$		0			0		−1.00000			$\pi$
$504$	1.00000			1.00000
$505$		0			0
$506$		0			0
$507$		0			0
$508$		0			0
$509$		0			0			−	1.00000i	$-0.5\pi$
$509$		0			0				1.00000i	$0.5\pi$
$510$		0			0
$511$	0.500000	+	0.866025i	0.500000	+	0.866025i
$512$			1.00000i			1.00000i
$513$	−0.866025	−	0.500000i	−0.866025	−	0.500000i
$514$		0			0
$515$		0			0
$516$		0			0
$517$	0.500000	+	0.866025i	0.500000	+	0.866025i
$518$	1.00000			1.00000
$519$	0.866025	−	0.500000i	0.866025	−	0.500000i
$520$		0			0
$521$	−0.500000	−	0.866025i	−0.500000	−	0.866025i	0.500000	−	0.866025i	$-0.333333\pi$
$521$	−0.500000	−	0.866025i	−0.500000	−	0.866025i	−1.00000			$\pi$
$522$	1.00000			1.00000
$523$	0.866025	+	0.500000i	0.866025	+	0.500000i	0.866025	−	0.500000i	$-0.166667\pi$
$523$	0.866025	+	0.500000i	0.866025	+	0.500000i			1.00000i	$0.5\pi$
$524$		0			0
$525$			1.00000i			1.00000i
$526$	−1.00000	−	1.73205i	−1.00000	−	1.73205i
$527$		−	1.00000i		−	1.00000i
$528$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$529$	−1.00000			−1.00000
$530$		0			0
$531$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$532$		0			0
$533$	0.500000	−	0.866025i	0.500000	−	0.866025i
$534$			1.00000i			1.00000i
$535$		0			0
$536$	−0.866025	−	0.500000i	−0.866025	−	0.500000i
$537$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$538$	−0.866025	−	0.500000i	−0.866025	−	0.500000i
$539$	−1.00000			−1.00000
$540$		0			0
$541$	0.866025	+	0.500000i	0.866025	+	0.500000i	0.866025	−	0.500000i	$-0.166667\pi$
$541$	0.866025	+	0.500000i	0.866025	+	0.500000i			1.00000i	$0.5\pi$
$542$	1.00000			1.00000
$543$		0			0
$544$		0			0
$545$		0			0
$546$	−0.866025	−	0.500000i	−0.866025	−	0.500000i
$547$	−0.866025	−	0.500000i	−0.866025	−	0.500000i		−	1.00000i	$-0.5\pi$
$547$	−0.866025	−	0.500000i	−0.866025	−	0.500000i	−0.866025	+	0.500000i	$0.833333\pi$
$548$		0			0
$549$	0.866025	−	0.500000i	0.866025	−	0.500000i
$550$	0.866025	−	0.500000i	0.866025	−	0.500000i
$551$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$552$		0			0
$553$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	0.866025	−	0.500000i	0.866025	−	0.500000i		−	1.00000i	$-0.5\pi$
$557$	0.866025	−	0.500000i	0.866025	−	0.500000i	0.866025	+	0.500000i	$0.166667\pi$
$558$		−	1.00000i		−	1.00000i
$559$	−1.00000			−1.00000
$560$		0			0
$561$	0.866025	−	0.500000i	0.866025	−	0.500000i
$562$	−1.00000			−1.00000
$563$	0.866025	+	0.500000i	0.866025	+	0.500000i	0.866025	−	0.500000i	$-0.166667\pi$
$563$	0.866025	+	0.500000i	0.866025	+	0.500000i			1.00000i	$0.5\pi$
$564$		0			0
$565$		0			0
$566$	−1.00000			−1.00000
$567$			1.00000i			1.00000i
$568$		0			0
$569$	−0.866025	+	0.500000i	−0.866025	+	0.500000i	−0.866025	−	0.500000i	$-0.833333\pi$
$569$	−0.866025	+	0.500000i	−0.866025	+	0.500000i			1.00000i	$0.5\pi$
$570$		0			0
$571$	0.866025	+	0.500000i	0.866025	+	0.500000i	0.866025	−	0.500000i	$-0.166667\pi$
$571$	0.866025	+	0.500000i	0.866025	+	0.500000i			1.00000i	$0.5\pi$
$572$		0			0
$573$	0.500000	+	0.866025i	0.500000	+	0.866025i
$574$	0.866025	+	0.500000i	0.866025	+	0.500000i
$575$		0			0
$576$	−1.00000			−1.00000
$577$	−0.500000	−	0.866025i	−0.500000	−	0.866025i	0.500000	−	0.866025i	$-0.333333\pi$
$577$	−0.500000	−	0.866025i	−0.500000	−	0.866025i	−1.00000			$\pi$
$578$		0			0
$579$	0.866025	+	0.500000i	0.866025	+	0.500000i
$580$		0			0
$581$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$582$		−	1.00000i		−	1.00000i
$583$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$584$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$585$		0			0
$586$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$587$	−0.500000	+	0.866025i	−0.500000	+	0.866025i	0.500000	+	0.866025i	$0.333333\pi$
$587$	−0.500000	+	0.866025i	−0.500000	+	0.866025i	−1.00000			$\pi$
$588$		0			0
$589$	−0.866025	+	0.500000i	−0.866025	+	0.500000i
$590$		0			0
$591$		0			0
$592$	−1.00000			−1.00000
$593$	0.866025	+	0.500000i	0.866025	+	0.500000i	0.866025	−	0.500000i	$-0.166667\pi$
$593$	0.866025	+	0.500000i	0.866025	+	0.500000i			1.00000i	$0.5\pi$
$594$	0.866025	−	0.500000i	0.866025	−	0.500000i
$595$		0			0
$596$		0			0
$597$	0.500000	+	0.866025i	0.500000	+	0.866025i
$598$		0			0
$599$	−0.500000	+	0.866025i	−0.500000	+	0.866025i	0.500000	+	0.866025i	$0.333333\pi$
$599$	−0.500000	+	0.866025i	−0.500000	+	0.866025i	−1.00000			$\pi$
$600$		−	1.00000i		−	1.00000i
$601$	0.866025	+	0.500000i	0.866025	+	0.500000i	0.866025	−	0.500000i	$-0.166667\pi$
$601$	0.866025	+	0.500000i	0.866025	+	0.500000i			1.00000i	$0.5\pi$
$602$		−	1.00000i		−	1.00000i
$603$	0.500000	−	0.866025i	0.500000	−	0.866025i
$604$		0			0
$605$		0			0
$606$		0			0
$607$		0			0		1.00000			$0$
$607$		0			0		−1.00000			$\pi$
$608$		0			0
$609$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$610$		0			0
$611$	−0.866025	−	0.500000i	−0.866025	−	0.500000i
$612$		0			0
$613$	−0.866025	+	0.500000i	−0.866025	+	0.500000i	−0.866025	−	0.500000i	$-0.833333\pi$
$613$	−0.866025	+	0.500000i	−0.866025	+	0.500000i			1.00000i	$0.5\pi$
$614$		0			0
$615$		0			0
$616$	1.00000			1.00000
$617$	−0.500000	−	0.866025i	−0.500000	−	0.866025i	0.500000	−	0.866025i	$-0.333333\pi$
$617$	−0.500000	−	0.866025i	−0.500000	−	0.866025i	−1.00000			$\pi$
$618$		0			0
$619$		0			0			−	1.00000i	$-0.5\pi$
$619$		0			0				1.00000i	$0.5\pi$
$620$		0			0
$621$		0			0
$622$			1.00000i			1.00000i
$623$	−0.866025	−	0.500000i	−0.866025	−	0.500000i
$624$	0.866025	+	0.500000i	0.866025	+	0.500000i
$625$	1.00000			1.00000
$626$	−0.866025	−	0.500000i	−0.866025	−	0.500000i
$627$	−0.866025	−	0.500000i	−0.866025	−	0.500000i
$628$		0			0
$629$		−	1.00000i		−	1.00000i
$630$		0			0
$631$		0			0			−	1.00000i	$-0.5\pi$
$631$		0			0				1.00000i	$0.5\pi$
$632$	0.500000	+	0.866025i	0.500000	+	0.866025i
$633$	−0.866025	+	0.500000i	−0.866025	+	0.500000i
$634$	0.866025	+	0.500000i	0.866025	+	0.500000i
$635$		0			0
$636$		0			0
$637$	0.866025	−	0.500000i	0.866025	−	0.500000i
$638$	1.00000			1.00000
$639$		0			0
$640$		0			0
$641$		0			0			−	1.00000i	$-0.5\pi$
$641$		0			0				1.00000i	$0.5\pi$
$642$	1.00000			1.00000
$643$	−0.500000	−	0.866025i	−0.500000	−	0.866025i	0.500000	−	0.866025i	$-0.333333\pi$
$643$	−0.500000	−	0.866025i	−0.500000	−	0.866025i	−1.00000			$\pi$
$644$		0			0
$645$		0			0
$646$	0.866025	−	0.500000i	0.866025	−	0.500000i
$647$	0.500000	+	0.866025i	0.500000	+	0.866025i	1.00000			$0$
$647$	0.500000	+	0.866025i	0.500000	+	0.866025i	−0.500000	+	0.866025i	$0.666667\pi$
$648$		−	1.00000i		−	1.00000i
$649$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$650$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$651$	0.866025	+	0.500000i	0.866025	+	0.500000i
$652$		0			0
$653$		0			0			−	1.00000i	$-0.5\pi$
$653$		0			0				1.00000i	$0.5\pi$
$654$	0.500000	−	0.866025i	0.500000	−	0.866025i
$655$		0			0
$656$	−0.866025	−	0.500000i	−0.866025	−	0.500000i
$657$	0.866025	−	0.500000i	0.866025	−	0.500000i
$658$	0.500000	−	0.866025i	0.500000	−	0.866025i
$659$	−0.866025	−	0.500000i	−0.866025	−	0.500000i		−	1.00000i	$-0.5\pi$
$659$	−0.866025	−	0.500000i	−0.866025	−	0.500000i	−0.866025	+	0.500000i	$0.833333\pi$
$660$		0			0
$661$	0.500000	−	0.866025i	0.500000	−	0.866025i	−0.500000	−	0.866025i	$-0.666667\pi$
$661$	0.500000	−	0.866025i	0.500000	−	0.866025i	1.00000			$0$
$662$	−0.866025	−	0.500000i	−0.866025	−	0.500000i
$663$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$664$	0.500000	−	0.866025i	0.500000	−	0.866025i
$665$		0			0
$666$		−	1.00000i		−	1.00000i
$667$		0			0
$668$		0			0
$669$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$670$		0			0
$671$	0.866025	−	0.500000i	0.866025	−	0.500000i
$672$		0			0
$673$	−0.866025	−	0.500000i	−0.866025	−	0.500000i		−	1.00000i	$-0.5\pi$
$673$	−0.866025	−	0.500000i	−0.866025	−	0.500000i	−0.866025	+	0.500000i	$0.833333\pi$
$674$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$675$	1.00000			1.00000
$676$		0			0
$677$	−0.866025	+	0.500000i	−0.866025	+	0.500000i	−0.866025	−	0.500000i	$-0.833333\pi$
$677$	−0.866025	+	0.500000i	−0.866025	+	0.500000i			1.00000i	$0.5\pi$
$678$	0.866025	+	0.500000i	0.866025	+	0.500000i
$679$	0.866025	+	0.500000i	0.866025	+	0.500000i
$680$		0			0
$681$			2.00000i			2.00000i
$682$		−	1.00000i		−	1.00000i
$683$	0.500000	+	0.866025i	0.500000	+	0.866025i	1.00000			$0$
$683$	0.500000	+	0.866025i	0.500000	+	0.866025i	−0.500000	+	0.866025i	$0.666667\pi$
$684$		0			0
$685$		0			0
$686$	0.500000	+	0.866025i	0.500000	+	0.866025i
$687$		0			0
$688$			1.00000i			1.00000i
$689$	0.866025	+	0.500000i	0.866025	+	0.500000i
$690$		0			0
$691$	−0.500000	−	0.866025i	−0.500000	−	0.866025i	0.500000	−	0.866025i	$-0.333333\pi$
$691$	−0.500000	−	0.866025i	−0.500000	−	0.866025i	−1.00000			$\pi$
$692$		0			0
$693$			1.00000i			1.00000i
$694$	1.00000			1.00000
$695$		0			0
$696$	0.500000	−	0.866025i	0.500000	−	0.866025i
$697$	0.500000	−	0.866025i	0.500000	−	0.866025i
$698$	−1.00000			−1.00000
$699$	−0.866025	−	0.500000i	−0.866025	−	0.500000i
$700$		0			0
$701$		0			0		1.00000			$0$
$701$		0			0		−1.00000			$\pi$
$702$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$703$	−0.866025	+	0.500000i	−0.866025	+	0.500000i
$704$	−1.00000			−1.00000
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$	−0.500000	−	0.866025i	−0.500000	−	0.866025i	0.500000	−	0.866025i	$-0.333333\pi$
$709$	−0.500000	−	0.866025i	−0.500000	−	0.866025i	−1.00000			$\pi$
$710$		0			0
$711$	−0.866025	+	0.500000i	−0.866025	+	0.500000i
$712$	0.866025	+	0.500000i	0.866025	+	0.500000i
$713$		0			0
$714$	−0.866025	−	0.500000i	−0.866025	−	0.500000i
$715$		0			0
$716$		0			0
$717$	−0.866025	+	0.500000i	−0.866025	+	0.500000i
$718$	1.00000			1.00000
$719$	−0.500000	+	0.866025i	−0.500000	+	0.866025i	0.500000	+	0.866025i	$0.333333\pi$
$719$	−0.500000	+	0.866025i	−0.500000	+	0.866025i	−1.00000			$\pi$
$720$		0			0
$721$		0			0
$722$		0			0
$723$		0			0
$724$		0			0
$725$	0.866025	+	0.500000i	0.866025	+	0.500000i
$726$	0.866025	−	0.500000i	0.866025	−	0.500000i
$727$	−0.500000	+	0.866025i	−0.500000	+	0.866025i	0.500000	+	0.866025i	$0.333333\pi$
$727$	−0.500000	+	0.866025i	−0.500000	+	0.866025i	−1.00000			$\pi$
$728$	−0.866025	+	0.500000i	−0.866025	+	0.500000i
$729$	1.00000			1.00000
$730$		0			0
$731$	−1.00000			−1.00000
$732$		0			0
$733$			2.00000i			2.00000i			1.00000i	$0.5\pi$
$733$			2.00000i			2.00000i			1.00000i	$0.5\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	0.500000	−	0.866025i	0.500000	−	0.866025i
$738$	0.500000	−	0.866025i	0.500000	−	0.866025i
$739$	−0.866025	+	0.500000i	−0.866025	+	0.500000i	−0.866025	−	0.500000i	$-0.833333\pi$
$739$	−0.866025	+	0.500000i	−0.866025	+	0.500000i			1.00000i	$0.5\pi$
$740$		0			0
$741$	1.00000			1.00000
$742$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$743$	−0.866025	+	0.500000i	−0.866025	+	0.500000i	−0.866025	−	0.500000i	$-0.833333\pi$
$743$	−0.866025	+	0.500000i	−0.866025	+	0.500000i			1.00000i	$0.5\pi$
$744$	−0.866025	−	0.500000i	−0.866025	−	0.500000i
$745$		0			0
$746$		0			0
$747$	0.866025	+	0.500000i	0.866025	+	0.500000i
$748$		0			0
$749$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$750$		0			0
$751$		0			0			−	1.00000i	$-0.5\pi$
$751$		0			0				1.00000i	$0.5\pi$
$752$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$753$		0			0
$754$	−0.866025	+	0.500000i	−0.866025	+	0.500000i
$755$		0			0
$756$		0			0
$757$	−2.00000			−2.00000			−1.00000			$\pi$
$757$	−2.00000			−2.00000			−1.00000			$\pi$
$758$	−1.73205	+	1.00000i	−1.73205	+	1.00000i
$759$		0			0
$760$		0			0
$761$		0			0		1.00000			$0$
$761$		0			0		−1.00000			$\pi$
$762$		0			0
$763$	0.500000	+	0.866025i	0.500000	+	0.866025i
$764$		0			0
$765$		0			0
$766$	−1.73205	+	1.00000i	−1.73205	+	1.00000i
$767$		−	1.00000i		−	1.00000i
$768$		0			0
$769$	−0.866025	+	0.500000i	−0.866025	+	0.500000i	−0.866025	−	0.500000i	$-0.833333\pi$
$769$	−0.866025	+	0.500000i	−0.866025	+	0.500000i			1.00000i	$0.5\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−0.500000	−	0.866025i	−0.500000	−	0.866025i	0.500000	−	0.866025i	$-0.333333\pi$
$773$	−0.500000	−	0.866025i	−0.500000	−	0.866025i	−1.00000			$\pi$
$774$	−1.00000			−1.00000
$775$	0.500000	−	0.866025i	0.500000	−	0.866025i
$776$	−0.866025	−	0.500000i	−0.866025	−	0.500000i
$777$	0.866025	+	0.500000i	0.866025	+	0.500000i
$778$		0			0
$779$	−1.00000			−1.00000
$780$		0			0
$781$		0			0
$782$		0			0
$783$	0.866025	+	0.500000i	0.866025	+	0.500000i
$784$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$785$		0			0
$786$	−1.00000	−	1.73205i	−1.00000	−	1.73205i
$787$	−0.866025	−	0.500000i	−0.866025	−	0.500000i		−	1.00000i</

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:	\( N \)	\(=\)	\( 693 = 3^{2} \cdot 7 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	693.bl (of order \(6\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(-0.866025 + 0.500000i) q^{2} -1.00000 q^{3} +(0.866025 - 0.500000i) q^{6} +1.00000i q^{7} -1.00000i q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q+(-0.866025 + 0.500000i) q^{2} -1.00000 q^{3} +(0.866025 - 0.500000i) q^{6} +1.00000i q^{7} -1.00000i q^{8} +1.00000 q^{9} +1.00000 q^{11} +(-0.866025 + 0.500000i) q^{13} +(-0.500000 - 0.866025i) q^{14} +(0.500000 + 0.866025i) q^{16} +(-0.866025 + 0.500000i) q^{17} +(-0.866025 + 0.500000i) q^{18} +(0.866025 + 0.500000i) q^{19} -1.00000i q^{21} +(-0.866025 + 0.500000i) q^{22} +1.00000i q^{24} -1.00000 q^{25} +(0.500000 - 0.866025i) q^{26} -1.00000 q^{27} +(-0.866025 - 0.500000i) q^{29} +(-0.500000 + 0.866025i) q^{31} -1.00000 q^{33} +(0.500000 - 0.866025i) q^{34} +(-0.500000 + 0.866025i) q^{37} -1.00000 q^{38} +(0.866025 - 0.500000i) q^{39} +(-0.866025 + 0.500000i) q^{41} +(0.500000 + 0.866025i) q^{42} +(0.866025 + 0.500000i) q^{43} +(0.500000 + 0.866025i) q^{47} +(-0.500000 - 0.866025i) q^{48} -1.00000 q^{49} +(0.866025 - 0.500000i) q^{50} +(0.866025 - 0.500000i) q^{51} +(-0.500000 - 0.866025i) q^{53} +(0.866025 - 0.500000i) q^{54} +1.00000 q^{56} +(-0.866025 - 0.500000i) q^{57} +1.00000 q^{58} +(-0.500000 + 0.866025i) q^{59} +(0.866025 - 0.500000i) q^{61} -1.00000i q^{62} +1.00000i q^{63} -1.00000 q^{64} +(0.866025 - 0.500000i) q^{66} +(0.500000 - 0.866025i) q^{67} -1.00000i q^{72} +(0.866025 - 0.500000i) q^{73} -1.00000i q^{74} +1.00000 q^{75} +1.00000i q^{77} +(-0.500000 + 0.866025i) q^{78} +(-0.866025 + 0.500000i) q^{79} +1.00000 q^{81} +(0.500000 - 0.866025i) q^{82} +(0.866025 + 0.500000i) q^{83} -1.00000 q^{86} +(0.866025 + 0.500000i) q^{87} -1.00000i q^{88} +(-0.500000 + 0.866025i) q^{89} +(-0.500000 - 0.866025i) q^{91} +(0.500000 - 0.866025i) q^{93} +(-0.866025 - 0.500000i) q^{94} +(0.500000 - 0.866025i) q^{97} +(0.866025 - 0.500000i) q^{98} +1.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{3} + 4 q^{9}+O(q^{10}) \) 4 * q - 4 * q^3 + 4 * q^9	\( 4 q - 4 q^{3} + 4 q^{9} + 4 q^{11} - 2 q^{14} + 2 q^{16} - 4 q^{25} + 2 q^{26} - 4 q^{27} - 2 q^{31} - 4 q^{33} + 2 q^{34} - 2 q^{37} - 4 q^{38} + 2 q^{42} + 2 q^{47} - 2 q^{48} - 4 q^{49} - 2 q^{53} + 4 q^{56} + 4 q^{58} - 2 q^{59} - 4 q^{64} + 2 q^{67} + 4 q^{75} - 2 q^{78} + 4 q^{81} + 2 q^{82} - 4 q^{86} - 2 q^{89} - 2 q^{91} + 2 q^{93} + 2 q^{97} + 4 q^{99}+O(q^{100}) \) 4 * q - 4 * q^3 + 4 * q^9 + 4 * q^11 - 2 * q^14 + 2 * q^16 - 4 * q^25 + 2 * q^26 - 4 * q^27 - 2 * q^31 - 4 * q^33 + 2 * q^34 - 2 * q^37 - 4 * q^38 + 2 * q^42 + 2 * q^47 - 2 * q^48 - 4 * q^49 - 2 * q^53 + 4 * q^56 + 4 * q^58 - 2 * q^59 - 4 * q^64 + 2 * q^67 + 4 * q^75 - 2 * q^78 + 4 * q^81 + 2 * q^82 - 4 * q^86 - 2 * q^89 - 2 * q^91 + 2 * q^93 + 2 * q^97 + 4 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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