LMFDB - Embedded newform 693.1.o.a.373.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [693,1,Mod(340,693)]

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, 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("693.340");

S:= CuspForms(chi, 1);

N := Newforms(S);

Level:	$ N $	$=$	$ 693 = 3^{2} \cdot 7 \cdot 11 $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	693.o (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:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Projective image:	$A_{4}$
Projective field:	Galois closure of 4.0.480249.1

Embedding invariants

Embedding label			373.2
Root			$0.866025 - 0.500000i$ of defining polynomial
Character	$\chi$	$=$	693.373
Dual form			693.1.o.a.340.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000i q^{2} +(0.500000 - 0.866025i) q^{3} +(0.866025 + 0.500000i) q^{6} +(0.866025 + 0.500000i) q^{7} +1.00000i q^{8} +(-0.500000 - 0.866025i) q^{9} +O(q^{10})$	\(q+1.00000i q^{2} +(0.500000 - 0.866025i) q^{3} +(0.866025 + 0.500000i) q^{6} +(0.866025 + 0.500000i) q^{7} +1.00000i q^{8} +(-0.500000 - 0.866025i) q^{9} +(-0.500000 + 0.866025i) q^{11} +(-0.866025 - 0.500000i) q^{13} +(-0.500000 + 0.866025i) q^{14} -1.00000 q^{16} +(0.866025 - 0.500000i) q^{17} +(0.866025 - 0.500000i) q^{18} +(-0.866025 - 0.500000i) q^{19} +(0.866025 - 0.500000i) q^{21} +(-0.866025 - 0.500000i) q^{22} +(0.866025 + 0.500000i) q^{24} +(0.500000 - 0.866025i) q^{25} +(0.500000 - 0.866025i) q^{26} -1.00000 q^{27} +(-0.866025 + 0.500000i) q^{29} +1.00000 q^{31} +(0.500000 + 0.866025i) q^{33} +(0.500000 + 0.866025i) q^{34} +(-0.500000 + 0.866025i) q^{37} +(0.500000 - 0.866025i) q^{38} +(-0.866025 + 0.500000i) q^{39} +(-0.866025 - 0.500000i) q^{41} +(0.500000 + 0.866025i) q^{42} +(0.866025 - 0.500000i) q^{43} -1.00000 q^{47} +(-0.500000 + 0.866025i) q^{48} +(0.500000 + 0.866025i) q^{49} +(0.866025 + 0.500000i) q^{50} -1.00000i q^{51} +(-0.500000 - 0.866025i) q^{53} -1.00000i q^{54} +(-0.500000 + 0.866025i) q^{56} +(-0.866025 + 0.500000i) q^{57} +(-0.500000 - 0.866025i) q^{58} +1.00000 q^{59} -1.00000i q^{61} +1.00000i q^{62} -1.00000i q^{63} -1.00000 q^{64} +(-0.866025 + 0.500000i) q^{66} -1.00000 q^{67} +(0.866025 - 0.500000i) q^{72} +(-0.866025 + 0.500000i) q^{73} +(-0.866025 - 0.500000i) q^{74} +(-0.500000 - 0.866025i) q^{75} +(-0.866025 + 0.500000i) q^{77} +(-0.500000 - 0.866025i) q^{78} +1.00000i q^{79} +(-0.500000 + 0.866025i) q^{81} +(0.500000 - 0.866025i) q^{82} +(0.866025 - 0.500000i) q^{83} +(0.500000 + 0.866025i) q^{86} +1.00000i q^{87} +(-0.866025 - 0.500000i) q^{88} +(-0.500000 + 0.866025i) q^{89} +(-0.500000 - 0.866025i) q^{91} +(0.500000 - 0.866025i) q^{93} -1.00000i q^{94} +(0.500000 + 0.866025i) q^{97} +(-0.866025 + 0.500000i) q^{98} +1.00000 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 2 q^{3} - 2 q^{9}+O(q^{10}) $ 4 * q + 2 * q^3 - 2 * q^9	$ 4 q + 2 q^{3} - 2 q^{9} - 2 q^{11} - 2 q^{14} - 4 q^{16} + 2 q^{25} + 2 q^{26} - 4 q^{27} + 4 q^{31} + 2 q^{33} + 2 q^{34} - 2 q^{37} + 2 q^{38} + 2 q^{42} - 4 q^{47} - 2 q^{48} + 2 q^{49} - 2 q^{53} - 2 q^{56} - 2 q^{58} + 4 q^{59} - 4 q^{64} - 4 q^{67} - 2 q^{75} - 2 q^{78} - 2 q^{81} + 2 q^{82} + 2 q^{86} - 2 q^{89} - 2 q^{91} + 2 q^{93} + 2 q^{97} + 4 q^{99}+O(q^{100}) $ 4 * q + 2 * q^3 - 2 * q^9 - 2 * q^11 - 2 * q^14 - 4 * q^16 + 2 * q^25 + 2 * q^26 - 4 * q^27 + 4 * q^31 + 2 * q^33 + 2 * q^34 - 2 * q^37 + 2 * q^38 + 2 * q^42 - 4 * q^47 - 2 * q^48 + 2 * q^49 - 2 * q^53 - 2 * q^56 - 2 * q^58 + 4 * q^59 - 4 * q^64 - 4 * q^67 - 2 * q^75 - 2 * q^78 - 2 * q^81 + 2 * q^82 + 2 * 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 $\left(\mathbb{Z}/693\mathbb{Z}\right)^\times$.

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

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 $

$2$			1.00000i			1.00000i	0.866025	+	0.500000i	$0.166667\pi$
$2$			1.00000i			1.00000i	−0.866025	+	0.500000i	$0.833333\pi$
$3$	0.500000	−	0.866025i	0.500000	−	0.866025i
$3$	0.500000	−	0.866025i	0.500000	−	0.866025i
$4$		0			0
$5$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$5$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$6$	0.866025	+	0.500000i	0.866025	+	0.500000i
$7$	0.866025	+	0.500000i	0.866025	+	0.500000i
$7$	0.866025	+	0.500000i	0.866025	+	0.500000i
$8$			1.00000i			1.00000i
$9$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$10$		0			0
$11$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$11$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$12$		0			0
$13$	−0.866025	−	0.500000i	−0.866025	−	0.500000i		−	1.00000i	$-0.5\pi$
$13$	−0.866025	−	0.500000i	−0.866025	−	0.500000i	−0.866025	+	0.500000i	$0.833333\pi$
$14$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$15$		0			0
$16$	−1.00000			−1.00000
$17$	0.866025	−	0.500000i	0.866025	−	0.500000i		−	1.00000i	$-0.5\pi$
$17$	0.866025	−	0.500000i	0.866025	−	0.500000i	0.866025	+	0.500000i	$0.166667\pi$
$18$	0.866025	−	0.500000i	0.866025	−	0.500000i
$19$	−0.866025	−	0.500000i	−0.866025	−	0.500000i		−	1.00000i	$-0.5\pi$
$19$	−0.866025	−	0.500000i	−0.866025	−	0.500000i	−0.866025	+	0.500000i	$0.833333\pi$
$20$		0			0
$21$	0.866025	−	0.500000i	0.866025	−	0.500000i
$22$	−0.866025	−	0.500000i	−0.866025	−	0.500000i
$23$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$23$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$24$	0.866025	+	0.500000i	0.866025	+	0.500000i
$25$	0.500000	−	0.866025i	0.500000	−	0.866025i
$26$	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	−0.866025	−	0.500000i	$-0.833333\pi$
$29$	−0.866025	+	0.500000i	−0.866025	+	0.500000i			1.00000i	$0.5\pi$
$30$		0			0
$31$	1.00000			1.00000			0.500000	−	0.866025i	$-0.333333\pi$
$31$	1.00000			1.00000			0.500000	+	0.866025i	$0.333333\pi$
$32$		0			0
$33$	0.500000	+	0.866025i	0.500000	+	0.866025i
$34$	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			$\pi$
$38$	0.500000	−	0.866025i	0.500000	−	0.866025i
$39$	−0.866025	+	0.500000i	−0.866025	+	0.500000i
$40$		0			0
$41$	−0.866025	−	0.500000i	−0.866025	−	0.500000i		−	1.00000i	$-0.5\pi$
$41$	−0.866025	−	0.500000i	−0.866025	−	0.500000i	−0.866025	+	0.500000i	$0.833333\pi$
$42$	0.500000	+	0.866025i	0.500000	+	0.866025i
$43$	0.866025	−	0.500000i	0.866025	−	0.500000i		−	1.00000i	$-0.5\pi$
$43$	0.866025	−	0.500000i	0.866025	−	0.500000i	0.866025	+	0.500000i	$0.166667\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−1.00000			−1.00000			−0.500000	−	0.866025i	$-0.666667\pi$
$47$	−1.00000			−1.00000			−0.500000	+	0.866025i	$0.666667\pi$
$48$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$49$	0.500000	+	0.866025i	0.500000	+	0.866025i
$50$	0.866025	+	0.500000i	0.866025	+	0.500000i
$51$		−	1.00000i		−	1.00000i
$52$		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			$\pi$
$54$		−	1.00000i		−	1.00000i
$55$		0			0
$56$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$57$	−0.866025	+	0.500000i	−0.866025	+	0.500000i
$58$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$59$	1.00000			1.00000			0.500000	−	0.866025i	$-0.333333\pi$
$59$	1.00000			1.00000			0.500000	+	0.866025i	$0.333333\pi$
$60$		0			0
$61$		−	1.00000i		−	1.00000i	−0.866025	−	0.500000i	$-0.833333\pi$
$61$		−	1.00000i		−	1.00000i	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
$67$	−1.00000			−1.00000			−0.500000	−	0.866025i	$-0.666667\pi$
$67$	−1.00000			−1.00000			−0.500000	+	0.866025i	$0.666667\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		0			0			−	1.00000i	$-0.5\pi$
$71$		0			0				1.00000i	$0.5\pi$
$72$	0.866025	−	0.500000i	0.866025	−	0.500000i
$73$	−0.866025	+	0.500000i	−0.866025	+	0.500000i	−0.866025	−	0.500000i	$-0.833333\pi$
$73$	−0.866025	+	0.500000i	−0.866025	+	0.500000i			1.00000i	$0.5\pi$
$74$	−0.866025	−	0.500000i	−0.866025	−	0.500000i
$75$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$76$		0			0
$77$	−0.866025	+	0.500000i	−0.866025	+	0.500000i
$78$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$79$			1.00000i			1.00000i	0.866025	+	0.500000i	$0.166667\pi$
$79$			1.00000i			1.00000i	−0.866025	+	0.500000i	$0.833333\pi$
$80$		0			0
$81$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$82$	0.500000	−	0.866025i	0.500000	−	0.866025i
$83$	0.866025	−	0.500000i	0.866025	−	0.500000i		−	1.00000i	$-0.5\pi$
$83$	0.866025	−	0.500000i	0.866025	−	0.500000i	0.866025	+	0.500000i	$0.166667\pi$
$84$		0			0
$85$		0			0
$86$	0.500000	+	0.866025i	0.500000	+	0.866025i
$87$			1.00000i			1.00000i
$88$	−0.866025	−	0.500000i	−0.866025	−	0.500000i
$89$	−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
$94$		−	1.00000i		−	1.00000i
$95$		0			0
$96$		0			0
$97$	0.500000	+	0.866025i	0.500000	+	0.866025i	1.00000			$0$
$97$	0.500000	+	0.866025i	0.500000	+	0.866025i	−0.500000	+	0.866025i	$0.666667\pi$
$98$	−0.866025	+	0.500000i	−0.866025	+	0.500000i
$99$	1.00000			1.00000
$100$		0			0
$101$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$101$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$102$	1.00000			1.00000
$103$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$103$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$104$	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		−	1.00000i	$-0.5\pi$
$107$	−0.866025	−	0.500000i	−0.866025	−	0.500000i	−0.866025	+	0.500000i	$0.833333\pi$
$108$		0			0
$109$	−0.866025	+	0.500000i	−0.866025	+	0.500000i	−0.866025	−	0.500000i	$-0.833333\pi$
$109$	−0.866025	+	0.500000i	−0.866025	+	0.500000i			1.00000i	$0.5\pi$
$110$		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	−0.500000	−	0.866025i	$-0.666667\pi$
$113$	0.500000	−	0.866025i	0.500000	−	0.866025i	1.00000			$0$
$114$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$115$		0			0
$116$		0			0
$117$			1.00000i			1.00000i
$118$			1.00000i			1.00000i
$119$	1.00000			1.00000
$120$		0			0
$121$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$122$	1.00000			1.00000
$123$	−0.866025	+	0.500000i	−0.866025	+	0.500000i
$124$		0			0
$125$		0			0
$126$	1.00000			1.00000
$127$		0			0		1.00000			$0$
$127$		0			0		−1.00000			$\pi$
$128$		−	1.00000i		−	1.00000i
$129$		−	1.00000i		−	1.00000i
$130$		0			0
$131$	1.73205	−	1.00000i	1.73205	−	1.00000i	0.866025	−	0.500000i	$-0.166667\pi$
$131$	1.73205	−	1.00000i	1.73205	−	1.00000i	0.866025	−	0.500000i	$-0.166667\pi$
$132$		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		−0.866025	−	0.500000i	$-0.833333\pi$
$137$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$138$		0			0
$139$	0.866025	+	0.500000i	0.866025	+	0.500000i	0.866025	−	0.500000i	$-0.166667\pi$
$139$	0.866025	+	0.500000i	0.866025	+	0.500000i			1.00000i	$0.5\pi$
$140$		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		−0.500000	−	0.866025i	$-0.666667\pi$
$149$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$150$	0.866025	−	0.500000i	0.866025	−	0.500000i
$151$	1.73205	+	1.00000i	1.73205	+	1.00000i	0.866025	+	0.500000i	$0.166667\pi$
$151$	1.73205	+	1.00000i	1.73205	+	1.00000i	0.866025	+	0.500000i	$0.166667\pi$
$152$	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
$157$	−1.00000			−1.00000			−0.500000	−	0.866025i	$-0.666667\pi$
$157$	−1.00000			−1.00000			−0.500000	+	0.866025i	$0.666667\pi$
$158$	−1.00000			−1.00000
$159$	−1.00000			−1.00000
$160$		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
$166$	0.500000	+	0.866025i	0.500000	+	0.866025i
$167$	−0.866025	−	0.500000i	−0.866025	−	0.500000i		−	1.00000i	$-0.5\pi$
$167$	−0.866025	−	0.500000i	−0.866025	−	0.500000i	−0.866025	+	0.500000i	$0.833333\pi$
$168$	0.500000	+	0.866025i	0.500000	+	0.866025i
$169$		0			0
$170$		0			0
$171$			1.00000i			1.00000i
$172$		0			0
$173$			1.00000i			1.00000i	0.866025	+	0.500000i	$0.166667\pi$
$173$			1.00000i			1.00000i	−0.866025	+	0.500000i	$0.833333\pi$
$174$	−1.00000			−1.00000
$175$	0.866025	−	0.500000i	0.866025	−	0.500000i
$176$	0.500000	−	0.866025i	0.500000	−	0.866025i
$177$	0.500000	−	0.866025i	0.500000	−	0.866025i
$178$	−0.866025	−	0.500000i	−0.866025	−	0.500000i
$179$	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
$186$	0.866025	+	0.500000i	0.866025	+	0.500000i
$187$			1.00000i			1.00000i
$188$		0			0
$189$	−0.866025	−	0.500000i	−0.866025	−	0.500000i
$190$		0			0
$191$	1.00000			1.00000			0.500000	−	0.866025i	$-0.333333\pi$
$191$	1.00000			1.00000			0.500000	+	0.866025i	$0.333333\pi$
$192$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$193$		−	1.00000i		−	1.00000i	−0.866025	−	0.500000i	$-0.833333\pi$
$193$		−	1.00000i		−	1.00000i	0.866025	−	0.500000i	$-0.166667\pi$
$194$	−0.866025	+	0.500000i	−0.866025	+	0.500000i
$195$		0			0
$196$		0			0
$197$		0			0		1.00000			$0$
$197$		0			0		−1.00000			$\pi$
$198$			1.00000i			1.00000i
$199$	−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			$\pi$
$200$	0.866025	+	0.500000i	0.866025	+	0.500000i
$201$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$202$		0			0
$203$	−1.00000			−1.00000
$204$		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	0.866025	−	0.500000i	$-0.166667\pi$
$211$	0.866025	+	0.500000i	0.866025	+	0.500000i			1.00000i	$0.5\pi$
$212$		0			0
$213$		0			0
$214$	0.500000	−	0.866025i	0.500000	−	0.866025i
$215$		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
$219$			1.00000i			1.00000i
$220$		0			0
$221$	−1.00000			−1.00000
$222$	−0.866025	+	0.500000i	−0.866025	+	0.500000i
$223$	0.500000	+	0.866025i	0.500000	+	0.866025i	1.00000			$0$
$223$	0.500000	+	0.866025i	0.500000	+	0.866025i	−0.500000	+	0.866025i	$0.666667\pi$
$224$		0			0
$225$	−1.00000			−1.00000
$226$	0.866025	+	0.500000i	0.866025	+	0.500000i
$227$	−1.73205	−	1.00000i	−1.73205	−	1.00000i	−0.866025	−	0.500000i	$-0.833333\pi$
$227$	−1.73205	−	1.00000i	−1.73205	−	1.00000i	−0.866025	−	0.500000i	$-0.833333\pi$
$228$		0			0
$229$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$229$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$230$		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		−	1.00000i	$-0.5\pi$
$233$	−0.866025	−	0.500000i	−0.866025	−	0.500000i	−0.866025	+	0.500000i	$0.833333\pi$
$234$	−1.00000			−1.00000
$235$		0			0
$236$		0			0
$237$	0.866025	+	0.500000i	0.866025	+	0.500000i
$238$			1.00000i			1.00000i
$239$	0.866025	+	0.500000i	0.866025	+	0.500000i	0.866025	−	0.500000i	$-0.166667\pi$
$239$	0.866025	+	0.500000i	0.866025	+	0.500000i			1.00000i	$0.5\pi$
$240$		0			0
$241$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$241$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$242$	0.866025	−	0.500000i	0.866025	−	0.500000i
$243$	0.500000	+	0.866025i	0.500000	+	0.866025i
$244$		0			0
$245$		0			0
$246$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$247$	0.500000	+	0.866025i	0.500000	+	0.866025i
$248$			1.00000i			1.00000i
$249$		−	1.00000i		−	1.00000i
$250$		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		0.866025	−	0.500000i	$-0.166667\pi$
$257$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$258$	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
$263$	1.73205	+	1.00000i	1.73205	+	1.00000i	0.866025	+	0.500000i	$0.166667\pi$
$263$	1.73205	+	1.00000i	1.73205	+	1.00000i	0.866025	+	0.500000i	$0.166667\pi$
$264$	−0.866025	+	0.500000i	−0.866025	+	0.500000i
$265$		0			0
$266$	0.866025	−	0.500000i	0.866025	−	0.500000i
$267$	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	0.866025	−	0.500000i	$-0.166667\pi$
$271$	0.866025	+	0.500000i	0.866025	+	0.500000i			1.00000i	$0.5\pi$
$272$	−0.866025	+	0.500000i	−0.866025	+	0.500000i
$273$	−1.00000			−1.00000
$274$		0			0
$275$	0.500000	+	0.866025i	0.500000	+	0.866025i
$276$		0			0
$277$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$277$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$278$	−0.500000	+	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		−	1.00000i	$-0.5\pi$
$281$	0.866025	−	0.500000i	0.866025	−	0.500000i	0.866025	+	0.500000i	$0.166667\pi$
$282$	−0.866025	−	0.500000i	−0.866025	−	0.500000i
$283$			1.00000i			1.00000i	0.866025	+	0.500000i	$0.166667\pi$
$283$			1.00000i			1.00000i	−0.866025	+	0.500000i	$0.833333\pi$
$284$		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
$291$	1.00000			1.00000
$292$		0			0
$293$	0.866025	+	0.500000i	0.866025	+	0.500000i	0.866025	−	0.500000i	$-0.166667\pi$
$293$	0.866025	+	0.500000i	0.866025	+	0.500000i			1.00000i	$0.5\pi$
$294$			1.00000i			1.00000i
$295$		0			0
$296$	−0.866025	−	0.500000i	−0.866025	−	0.500000i
$297$	0.500000	−	0.866025i	0.500000	−	0.866025i
$298$		0			0
$299$		0			0
$300$		0			0
$301$	1.00000			1.00000
$302$	−1.00000	+	1.73205i	−1.00000	+	1.73205i
$303$		0			0
$304$	0.866025	+	0.500000i	0.866025	+	0.500000i
$305$		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
$311$	−1.00000			−1.00000			−0.500000	−	0.866025i	$-0.666667\pi$
$311$	−1.00000			−1.00000			−0.500000	+	0.866025i	$0.666667\pi$
$312$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$313$	−1.00000			−1.00000			−0.500000	−	0.866025i	$-0.666667\pi$
$313$	−1.00000			−1.00000			−0.500000	+	0.866025i	$0.666667\pi$
$314$		−	1.00000i		−	1.00000i
$315$		0			0
$316$		0			0
$317$	1.00000			1.00000			0.500000	−	0.866025i	$-0.333333\pi$
$317$	1.00000			1.00000			0.500000	+	0.866025i	$0.333333\pi$
$318$		−	1.00000i		−	1.00000i
$319$		−	1.00000i		−	1.00000i
$320$		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
$326$	−0.866025	−	0.500000i	−0.866025	−	0.500000i
$327$			1.00000i			1.00000i
$328$	0.500000	−	0.866025i	0.500000	−	0.866025i
$329$	−0.866025	−	0.500000i	−0.866025	−	0.500000i
$330$		0			0
$331$	−1.00000			−1.00000			−0.500000	−	0.866025i	$-0.666667\pi$
$331$	−1.00000			−1.00000			−0.500000	+	0.866025i	$0.666667\pi$
$332$		0			0
$333$	1.00000			1.00000
$334$	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	0.866025	−	0.500000i	$-0.166667\pi$
$337$	0.866025	+	0.500000i	0.866025	+	0.500000i			1.00000i	$0.5\pi$
$338$		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
$346$	−1.00000			−1.00000
$347$		−	1.00000i		−	1.00000i	−0.866025	−	0.500000i	$-0.833333\pi$
$347$		−	1.00000i		−	1.00000i	0.866025	−	0.500000i	$-0.166667\pi$
$348$		0			0
$349$	0.866025	−	0.500000i	0.866025	−	0.500000i		−	1.00000i	$-0.5\pi$
$349$	0.866025	−	0.500000i	0.866025	−	0.500000i	0.866025	+	0.500000i	$0.166667\pi$
$350$	0.500000	+	0.866025i	0.500000	+	0.866025i
$351$	0.866025	+	0.500000i	0.866025	+	0.500000i
$352$		0			0
$353$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$353$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$354$	0.866025	+	0.500000i	0.866025	+	0.500000i
$355$		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	0.866025	−	0.500000i	$-0.166667\pi$
$359$	0.866025	+	0.500000i	0.866025	+	0.500000i			1.00000i	$0.5\pi$
$360$		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		−0.866025	−	0.500000i	$-0.833333\pi$
$367$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$368$		0			0
$369$			1.00000i			1.00000i
$370$		0			0
$371$		−	1.00000i		−	1.00000i
$372$		0			0
$373$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$373$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$374$	−1.00000			−1.00000
$375$		0			0
$376$		−	1.00000i		−	1.00000i
$377$	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
$382$			1.00000i			1.00000i
$383$	−1.00000	−	1.73205i	−1.00000	−	1.73205i	−0.500000	−	0.866025i	$-0.666667\pi$
$383$	−1.00000	−	1.73205i	−1.00000	−	1.73205i	−0.500000	−	0.866025i	$-0.666667\pi$
$384$	−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		−0.866025	−	0.500000i	$-0.833333\pi$
$389$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$390$		0			0
$391$		0			0
$392$	−0.866025	+	0.500000i	−0.866025	+	0.500000i
$393$		−	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
$399$	−1.00000			−1.00000
$400$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$401$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$401$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$402$	−0.866025	−	0.500000i	−0.866025	−	0.500000i
$403$	−0.866025	−	0.500000i	−0.866025	−	0.500000i
$404$		0			0
$405$		0			0
$406$		−	1.00000i		−	1.00000i
$407$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$408$	1.00000			1.00000
$409$		−	1.00000i		−	1.00000i	−0.866025	−	0.500000i	$-0.833333\pi$
$409$		−	1.00000i		−	1.00000i	0.866025	−	0.500000i	$-0.166667\pi$
$410$		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
$418$	0.500000	+	0.866025i	0.500000	+	0.866025i
$419$	0.500000	−	0.866025i	0.500000	−	0.866025i	−0.500000	−	0.866025i	$-0.666667\pi$
$419$	0.500000	−	0.866025i	0.500000	−	0.866025i	1.00000			$0$
$420$		0			0
$421$	0.500000	+	0.866025i	0.500000	+	0.866025i	1.00000			$0$
$421$	0.500000	+	0.866025i	0.500000	+	0.866025i	−0.500000	+	0.866025i	$0.666667\pi$
$422$	−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
$425$		−	1.00000i		−	1.00000i
$426$		0			0
$427$	0.500000	−	0.866025i	0.500000	−	0.866025i
$428$		0			0
$429$		−	1.00000i		−	1.00000i
$430$		0			0
$431$	−0.866025	+	0.500000i	−0.866025	+	0.500000i	−0.866025	−	0.500000i	$-0.833333\pi$
$431$	−0.866025	+	0.500000i	−0.866025	+	0.500000i			1.00000i	$0.5\pi$
$432$	1.00000			1.00000
$433$		0			0			−	1.00000i	$-0.5\pi$
$433$		0			0				1.00000i	$0.5\pi$
$434$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$435$		0			0
$436$		0			0
$437$		0			0
$438$	−1.00000			−1.00000
$439$			1.00000i			1.00000i	0.866025	+	0.500000i	$0.166667\pi$
$439$			1.00000i			1.00000i	−0.866025	+	0.500000i	$0.833333\pi$
$440$		0			0
$441$	0.500000	−	0.866025i	0.500000	−	0.866025i
$442$		−	1.00000i		−	1.00000i
$443$	1.00000			1.00000			0.500000	−	0.866025i	$-0.333333\pi$
$443$	1.00000			1.00000			0.500000	+	0.866025i	$0.333333\pi$
$444$		0			0
$445$		0			0
$446$	−0.866025	+	0.500000i	−0.866025	+	0.500000i
$447$		0			0
$448$	−0.866025	−	0.500000i	−0.866025	−	0.500000i
$449$		0			0			−	1.00000i	$-0.5\pi$
$449$		0			0				1.00000i	$0.5\pi$
$450$		−	1.00000i		−	1.00000i
$451$	0.866025	−	0.500000i	0.866025	−	0.500000i
$452$		0			0
$453$	1.73205	−	1.00000i	1.73205	−	1.00000i
$454$	1.00000	−	1.73205i	1.00000	−	1.73205i
$455$		0			0
$456$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$457$		−	1.00000i		−	1.00000i	−0.866025	−	0.500000i	$-0.833333\pi$
$457$		−	1.00000i		−	1.00000i	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	−0.866025	−	0.500000i	$-0.833333\pi$
$461$	−0.866025	+	0.500000i	−0.866025	+	0.500000i			1.00000i	$0.5\pi$
$462$	−1.00000			−1.00000
$463$	0.500000	−	0.866025i	0.500000	−	0.866025i	−0.500000	−	0.866025i	$-0.666667\pi$
$463$	0.500000	−	0.866025i	0.500000	−	0.866025i	1.00000			$0$
$464$	0.866025	−	0.500000i	0.866025	−	0.500000i
$465$		0			0
$466$	0.500000	−	0.866025i	0.500000	−	0.866025i
$467$	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
$472$			1.00000i			1.00000i
$473$			1.00000i			1.00000i
$474$	−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		0.500000	−	0.866025i	$-0.333333\pi$
$479$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$480$		0			0
$481$	0.866025	−	0.500000i	0.866025	−	0.500000i
$482$		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	$0.666667\pi$
$488$	1.00000			1.00000
$489$	0.500000	+	0.866025i	0.500000	+	0.866025i
$490$		0			0
$491$	−0.866025	−	0.500000i	−0.866025	−	0.500000i		−	1.00000i	$-0.5\pi$
$491$	−0.866025	−	0.500000i	−0.866025	−	0.500000i	−0.866025	+	0.500000i	$0.833333\pi$
$492$		0			0
$493$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$494$	−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		0.866025	−	0.500000i	$-0.166667\pi$
$499$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$500$		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		0.866025	−	0.500000i	$-0.166667\pi$
$509$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$510$		0			0
$511$	−1.00000			−1.00000
$512$		−	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
$518$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$519$	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			$\pi$
$522$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$523$	−0.866025	−	0.500000i	−0.866025	−	0.500000i		−	1.00000i	$-0.5\pi$
$523$	−0.866025	−	0.500000i	−0.866025	−	0.500000i	−0.866025	+	0.500000i	$0.833333\pi$
$524$		0			0
$525$		−	1.00000i		−	1.00000i
$526$	−1.00000	+	1.73205i	−1.00000	+	1.73205i
$527$	0.866025	−	0.500000i	0.866025	−	0.500000i
$528$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$529$	0.500000	−	0.866025i	0.500000	−	0.866025i
$530$		0			0
$531$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$532$		0			0
$533$	0.500000	+	0.866025i	0.500000	+	0.866025i
$534$	−0.866025	+	0.500000i	−0.866025	+	0.500000i
$535$		0			0
$536$		−	1.00000i		−	1.00000i
$537$	1.00000			1.00000
$538$	−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		−	1.00000i	$-0.5\pi$
$541$	−0.866025	−	0.500000i	−0.866025	−	0.500000i	−0.866025	+	0.500000i	$0.833333\pi$
$542$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$543$		0			0
$544$		0			0
$545$		0			0
$546$		−	1.00000i		−	1.00000i
$547$	−0.866025	+	0.500000i	−0.866025	+	0.500000i	−0.866025	−	0.500000i	$-0.833333\pi$
$547$	−0.866025	+	0.500000i	−0.866025	+	0.500000i			1.00000i	$0.5\pi$
$548$		0			0
$549$	−0.866025	+	0.500000i	−0.866025	+	0.500000i
$550$	−0.866025	+	0.500000i	−0.866025	+	0.500000i
$551$	1.00000			1.00000
$552$		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	−0.866025	−	0.500000i	$-0.833333\pi$
$557$	−0.866025	+	0.500000i	−0.866025	+	0.500000i			1.00000i	$0.5\pi$
$558$	0.866025	−	0.500000i	0.866025	−	0.500000i
$559$	−1.00000			−1.00000
$560$		0			0
$561$	0.866025	+	0.500000i	0.866025	+	0.500000i
$562$	0.500000	+	0.866025i	0.500000	+	0.866025i
$563$			1.00000i			1.00000i	0.866025	+	0.500000i	$0.166667\pi$
$563$			1.00000i			1.00000i	−0.866025	+	0.500000i	$0.833333\pi$
$564$		0			0
$565$		0			0
$566$	−1.00000			−1.00000
$567$	−0.866025	+	0.500000i	−0.866025	+	0.500000i
$568$		0			0
$569$			1.00000i			1.00000i	0.866025	+	0.500000i	$0.166667\pi$
$569$			1.00000i			1.00000i	−0.866025	+	0.500000i	$0.833333\pi$
$570$		0			0
$571$			1.00000i			1.00000i	0.866025	+	0.500000i	$0.166667\pi$
$571$			1.00000i			1.00000i	−0.866025	+	0.500000i	$0.833333\pi$
$572$		0			0
$573$	0.500000	−	0.866025i	0.500000	−	0.866025i
$574$	0.866025	−	0.500000i	0.866025	−	0.500000i
$575$		0			0
$576$	0.500000	+	0.866025i	0.500000	+	0.866025i
$577$	−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
$581$	1.00000			1.00000
$582$			1.00000i			1.00000i
$583$	1.00000			1.00000
$584$	−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
$592$	0.500000	−	0.866025i	0.500000	−	0.866025i
$593$	−0.866025	−	0.500000i	−0.866025	−	0.500000i		−	1.00000i	$-0.5\pi$
$593$	−0.866025	−	0.500000i	−0.866025	−	0.500000i	−0.866025	+	0.500000i	$0.833333\pi$
$594$	0.866025	+	0.500000i	0.866025	+	0.500000i
$595$		0			0
$596$		0			0
$597$	−1.00000			−1.00000
$598$		0			0
$599$	1.00000			1.00000			0.500000	−	0.866025i	$-0.333333\pi$
$599$	1.00000			1.00000			0.500000	+	0.866025i	$0.333333\pi$
$600$	0.866025	−	0.500000i	0.866025	−	0.500000i
$601$	0.866025	−	0.500000i	0.866025	−	0.500000i		−	1.00000i	$-0.5\pi$
$601$	0.866025	−	0.500000i	0.866025	−	0.500000i	0.866025	+	0.500000i	$0.166667\pi$
$602$			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		−0.500000	−	0.866025i	$-0.666667\pi$
$607$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$608$		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		−	1.00000i	$-0.5\pi$
$613$	0.866025	−	0.500000i	0.866025	−	0.500000i	0.866025	+	0.500000i	$0.166667\pi$
$614$		0			0
$615$		0			0
$616$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$617$	−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		−0.866025	−	0.500000i	$-0.833333\pi$
$619$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$620$		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
$625$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$626$		−	1.00000i		−	1.00000i
$627$		−	1.00000i		−	1.00000i
$628$		0			0
$629$			1.00000i			1.00000i
$630$		0			0
$631$		0			0			−	1.00000i	$-0.5\pi$
$631$		0			0				1.00000i	$0.5\pi$
$632$	−1.00000			−1.00000
$633$	0.866025	−	0.500000i	0.866025	−	0.500000i
$634$			1.00000i			1.00000i
$635$		0			0
$636$		0			0
$637$		−	1.00000i		−	1.00000i
$638$	1.00000			1.00000
$639$		0			0
$640$		0			0
$641$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$641$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$642$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$643$	−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
$646$		−	1.00000i		−	1.00000i
$647$	0.500000	+	0.866025i	0.500000	+	0.866025i	1.00000			$0$
$647$	0.500000	+	0.866025i	0.500000	+	0.866025i	−0.500000	+	0.866025i	$0.666667\pi$
$648$	−0.866025	−	0.500000i	−0.866025	−	0.500000i
$649$	−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		0.866025	−	0.500000i	$-0.166667\pi$
$653$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$654$	−1.00000			−1.00000
$655$		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	−0.866025	−	0.500000i	$-0.833333\pi$
$659$	−0.866025	+	0.500000i	−0.866025	+	0.500000i			1.00000i	$0.5\pi$
$660$		0			0
$661$	−1.00000			−1.00000			−0.500000	−	0.866025i	$-0.666667\pi$
$661$	−1.00000			−1.00000			−0.500000	+	0.866025i	$0.666667\pi$
$662$		−	1.00000i		−	1.00000i
$663$	−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
$669$	1.00000			1.00000
$670$		0			0
$671$	0.866025	+	0.500000i	0.866025	+	0.500000i
$672$		0			0
$673$	−0.866025	+	0.500000i	−0.866025	+	0.500000i	−0.866025	−	0.500000i	$-0.833333\pi$
$673$	−0.866025	+	0.500000i	−0.866025	+	0.500000i			1.00000i	$0.5\pi$
$674$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$675$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$676$		0			0
$677$			1.00000i			1.00000i	0.866025	+	0.500000i	$0.166667\pi$
$677$			1.00000i			1.00000i	−0.866025	+	0.500000i	$0.833333\pi$
$678$	0.866025	−	0.500000i	0.866025	−	0.500000i
$679$			1.00000i			1.00000i
$680$		0			0
$681$	−1.73205	+	1.00000i	−1.73205	+	1.00000i
$682$	−0.866025	−	0.500000i	−0.866025	−	0.500000i
$683$	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
$686$	−1.00000			−1.00000
$687$		0			0
$688$	−0.866025	+	0.500000i	−0.866025	+	0.500000i
$689$			1.00000i			1.00000i
$690$		0			0
$691$	1.00000			1.00000			0.500000	−	0.866025i	$-0.333333\pi$
$691$	1.00000			1.00000			0.500000	+	0.866025i	$0.333333\pi$
$692$		0			0
$693$	0.866025	+	0.500000i	0.866025	+	0.500000i
$694$	1.00000			1.00000
$695$		0			0
$696$	−1.00000			−1.00000
$697$	−1.00000			−1.00000
$698$	0.500000	+	0.866025i	0.500000	+	0.866025i
$699$	−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
$704$	0.500000	−	0.866025i	0.500000	−	0.866025i
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$	1.00000			1.00000			0.500000	−	0.866025i	$-0.333333\pi$
$709$	1.00000			1.00000			0.500000	+	0.866025i	$0.333333\pi$
$710$		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
$718$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$719$	−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
$725$			1.00000i			1.00000i
$726$		−	1.00000i		−	1.00000i
$727$	−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
$731$	0.500000	−	0.866025i	0.500000	−	0.866025i
$732$		0			0
$733$	−1.73205	+	1.00000i	−1.73205	+	1.00000i	−0.866025	+	0.500000i	$0.833333\pi$
$733$	−1.73205	+	1.00000i	−1.73205	+	1.00000i	−0.866025	+	0.500000i	$0.833333\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	0.500000	−	0.866025i	0.500000	−	0.866025i
$738$	−1.00000			−1.00000
$739$	0.866025	−	0.500000i	0.866025	−	0.500000i		−	1.00000i	$-0.5\pi$
$739$	0.866025	−	0.500000i	0.866025	−	0.500000i	0.866025	+	0.500000i	$0.166667\pi$
$740$		0			0
$741$	1.00000			1.00000
$742$	1.00000			1.00000
$743$	−0.866025	−	0.500000i	−0.866025	−	0.500000i		−	1.00000i	$-0.5\pi$
$743$	−0.866025	−	0.500000i	−0.866025	−	0.500000i	−0.866025	+	0.500000i	$0.833333\pi$
$744$	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		0.866025	−	0.500000i	$-0.166667\pi$
$751$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$752$	1.00000			1.00000
$753$		0			0
$754$			1.00000i			1.00000i
$755$		0			0
$756$		0			0
$757$	−2.00000			−2.00000			−1.00000			$\pi$
$757$	−2.00000			−2.00000			−1.00000			$\pi$
$758$			2.00000i			2.00000i
$759$		0			0
$760$		0			0
$761$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$761$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$762$		0			0
$763$	−1.00000			−1.00000
$764$		0			0
$765$		0			0
$766$	1.73205	−	1.00000i	1.73205	−	1.00000i
$767$	−0.866025	−	0.500000i	−0.866025	−	0.500000i
$768$		0			0
$769$	−0.866025	−	0.500000i	−0.866025	−	0.500000i		−	1.00000i	$-0.5\pi$
$769$	−0.866025	−	0.500000i	−0.866025	−	0.500000i	−0.866025	+	0.500000i	$0.833333\pi$
$770$		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			$\pi$
$774$	0.500000	−	0.866025i	0.500000	−	0.866025i
$775$	0.500000	−	0.866025i	0.500000	−	0.866025i
$776$	−0.866025	+	0.500000i	−0.866025	+	0.500000i
$777$			1.00000i			1.00000i
$778$		0			0
$779$	0.500000	+	0.866025i	0.500000	+	0.866025i
$780$		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
$786$	2.00000			2.00000
$787$		−	1.00000i		−	1.00000i	−0.866025	−	0.500000i	$-0.833333\pi$
$787$		−	1.00000i		−	1.00000i	0.866025	−	0.500000i	$-0.166667\pi$
$788$		0			0
$789$	1.73205	−	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.o (of order \(6\), degree \(2\), minimal)

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