LMFDB - Embedded newform 171.1.o.a.151.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [171,1,Mod(94,171)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(171, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([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("171.94");

S:= CuspForms(chi, 1);

N := Newforms(S);

Level:	$ N $	$=$	$ 171 = 3^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	171.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:	$0.0853401171602$
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 4.0.29241.1
Artin image:	$\SL(2,3):C_2$
Artin field:	Galois closure of $\mathbb{Q}[x]/(x^{16} - \cdots)$

Embedding invariants

Embedding label			151.2
Root			$-0.866025 + 0.500000i$ of defining polynomial
Character	$\chi$	$=$	171.151
Dual form			171.1.o.a.94.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(0.866025 + 0.500000i) q^{2} +1.00000i q^{3} +(-0.500000 - 0.866025i) q^{5} +(-0.500000 + 0.866025i) q^{6} +(-0.500000 + 0.866025i) q^{7} -1.00000i q^{8} -1.00000 q^{9} +O(q^{10})$	\(q+(0.866025 + 0.500000i) q^{2} +1.00000i q^{3} +(-0.500000 - 0.866025i) q^{5} +(-0.500000 + 0.866025i) q^{6} +(-0.500000 + 0.866025i) q^{7} -1.00000i q^{8} -1.00000 q^{9} -1.00000i q^{10} +(0.500000 - 0.866025i) q^{11} +(-0.866025 + 0.500000i) q^{13} +(-0.866025 + 0.500000i) q^{14} +(0.866025 - 0.500000i) q^{15} +(0.500000 - 0.866025i) q^{16} +(-0.866025 - 0.500000i) q^{18} +1.00000i q^{19} +(-0.866025 - 0.500000i) q^{21} +(0.866025 - 0.500000i) q^{22} +(0.500000 + 0.866025i) q^{23} +1.00000 q^{24} -1.00000 q^{26} -1.00000i q^{27} +(-0.866025 - 0.500000i) q^{29} +1.00000 q^{30} +(0.866025 - 0.500000i) q^{31} +(0.866025 + 0.500000i) q^{33} +1.00000 q^{35} +(-0.500000 + 0.866025i) q^{38} +(-0.500000 - 0.866025i) q^{39} +(-0.866025 + 0.500000i) q^{40} +(-0.866025 + 0.500000i) q^{41} +(-0.500000 - 0.866025i) q^{42} +(-0.500000 + 0.866025i) q^{43} +(0.500000 + 0.866025i) q^{45} +1.00000i q^{46} +(-0.500000 + 0.866025i) q^{47} +(0.866025 + 0.500000i) q^{48} +(0.500000 - 0.866025i) q^{54} -1.00000 q^{55} +(0.866025 + 0.500000i) q^{56} -1.00000 q^{57} +(-0.500000 - 0.866025i) q^{58} +(0.866025 - 0.500000i) q^{59} +(0.500000 - 0.866025i) q^{61} +1.00000 q^{62} +(0.500000 - 0.866025i) q^{63} -1.00000 q^{64} +(0.866025 + 0.500000i) q^{65} +(0.500000 + 0.866025i) q^{66} +(0.866025 - 0.500000i) q^{67} +(-0.866025 + 0.500000i) q^{69} +(0.866025 + 0.500000i) q^{70} +1.00000i q^{72} +(0.500000 + 0.866025i) q^{77} -1.00000i q^{78} +(-0.866025 - 0.500000i) q^{79} -1.00000 q^{80} +1.00000 q^{81} -1.00000 q^{82} +(0.500000 - 0.866025i) q^{83} +(-0.866025 + 0.500000i) q^{86} +(0.500000 - 0.866025i) q^{87} +(-0.866025 - 0.500000i) q^{88} +1.00000i q^{90} -1.00000i q^{91} +(0.500000 + 0.866025i) q^{93} +(-0.866025 + 0.500000i) q^{94} +(0.866025 - 0.500000i) q^{95} +(0.866025 + 0.500000i) q^{97} +(-0.500000 + 0.866025i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 2 q^{5} - 2 q^{6} - 2 q^{7} - 4 q^{9}+O(q^{10}) $ 4 * q - 2 * q^5 - 2 * q^6 - 2 * q^7 - 4 * q^9	$ 4 q - 2 q^{5} - 2 q^{6} - 2 q^{7} - 4 q^{9} + 2 q^{11} + 2 q^{16} + 2 q^{23} + 4 q^{24} - 4 q^{26} + 4 q^{30} + 4 q^{35} - 2 q^{38} - 2 q^{39} - 2 q^{42} - 2 q^{43} + 2 q^{45} - 2 q^{47} + 2 q^{54} - 4 q^{55} - 4 q^{57} - 2 q^{58} + 2 q^{61} + 4 q^{62} + 2 q^{63} - 4 q^{64} + 2 q^{66} + 2 q^{77} - 4 q^{80} + 4 q^{81} - 4 q^{82} + 2 q^{83} + 2 q^{87} + 2 q^{93} - 2 q^{99}+O(q^{100}) $ 4 * q - 2 * q^5 - 2 * q^6 - 2 * q^7 - 4 * q^9 + 2 * q^11 + 2 * q^16 + 2 * q^23 + 4 * q^24 - 4 * q^26 + 4 * q^30 + 4 * q^35 - 2 * q^38 - 2 * q^39 - 2 * q^42 - 2 * q^43 + 2 * q^45 - 2 * q^47 + 2 * q^54 - 4 * q^55 - 4 * q^57 - 2 * q^58 + 2 * q^61 + 4 * q^62 + 2 * q^63 - 4 * q^64 + 2 * q^66 + 2 * q^77 - 4 * q^80 + 4 * q^81 - 4 * q^82 + 2 * q^83 + 2 * q^87 + 2 * q^93 - 2 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/171\mathbb{Z}\right)^\times$.

$n$	$20$	$154$
$\chi(n)$	$e\left(\frac{2}{3}\right)$	$-1$

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

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:	\( N \)	\(=\)	\( 171 = 3^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	171.o (of order \(6\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(0.866025 + 0.500000i) q^{2} +1.00000i q^{3} +(-0.500000 - 0.866025i) q^{5} +(-0.500000 + 0.866025i) q^{6} +(-0.500000 + 0.866025i) q^{7} -1.00000i q^{8} -1.00000 q^{9} +O(q^{10})\)	\(q+(0.866025 + 0.500000i) q^{2} +1.00000i q^{3} +(-0.500000 - 0.866025i) q^{5} +(-0.500000 + 0.866025i) q^{6} +(-0.500000 + 0.866025i) q^{7} -1.00000i q^{8} -1.00000 q^{9} -1.00000i q^{10} +(0.500000 - 0.866025i) q^{11} +(-0.866025 + 0.500000i) q^{13} +(-0.866025 + 0.500000i) q^{14} +(0.866025 - 0.500000i) q^{15} +(0.500000 - 0.866025i) q^{16} +(-0.866025 - 0.500000i) q^{18} +1.00000i q^{19} +(-0.866025 - 0.500000i) q^{21} +(0.866025 - 0.500000i) q^{22} +(0.500000 + 0.866025i) q^{23} +1.00000 q^{24} -1.00000 q^{26} -1.00000i q^{27} +(-0.866025 - 0.500000i) q^{29} +1.00000 q^{30} +(0.866025 - 0.500000i) q^{31} +(0.866025 + 0.500000i) q^{33} +1.00000 q^{35} +(-0.500000 + 0.866025i) q^{38} +(-0.500000 - 0.866025i) q^{39} +(-0.866025 + 0.500000i) q^{40} +(-0.866025 + 0.500000i) q^{41} +(-0.500000 - 0.866025i) q^{42} +(-0.500000 + 0.866025i) q^{43} +(0.500000 + 0.866025i) q^{45} +1.00000i q^{46} +(-0.500000 + 0.866025i) q^{47} +(0.866025 + 0.500000i) q^{48} +(0.500000 - 0.866025i) q^{54} -1.00000 q^{55} +(0.866025 + 0.500000i) q^{56} -1.00000 q^{57} +(-0.500000 - 0.866025i) q^{58} +(0.866025 - 0.500000i) q^{59} +(0.500000 - 0.866025i) q^{61} +1.00000 q^{62} +(0.500000 - 0.866025i) q^{63} -1.00000 q^{64} +(0.866025 + 0.500000i) q^{65} +(0.500000 + 0.866025i) q^{66} +(0.866025 - 0.500000i) q^{67} +(-0.866025 + 0.500000i) q^{69} +(0.866025 + 0.500000i) q^{70} +1.00000i q^{72} +(0.500000 + 0.866025i) q^{77} -1.00000i q^{78} +(-0.866025 - 0.500000i) q^{79} -1.00000 q^{80} +1.00000 q^{81} -1.00000 q^{82} +(0.500000 - 0.866025i) q^{83} +(-0.866025 + 0.500000i) q^{86} +(0.500000 - 0.866025i) q^{87} +(-0.866025 - 0.500000i) q^{88} +1.00000i q^{90} -1.00000i q^{91} +(0.500000 + 0.866025i) q^{93} +(-0.866025 + 0.500000i) q^{94} +(0.866025 - 0.500000i) q^{95} +(0.866025 + 0.500000i) q^{97} +(-0.500000 + 0.866025i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 2 q^{5} - 2 q^{6} - 2 q^{7} - 4 q^{9}+O(q^{10}) \) 4 * q - 2 * q^5 - 2 * q^6 - 2 * q^7 - 4 * q^9	\( 4 q - 2 q^{5} - 2 q^{6} - 2 q^{7} - 4 q^{9} + 2 q^{11} + 2 q^{16} + 2 q^{23} + 4 q^{24} - 4 q^{26} + 4 q^{30} + 4 q^{35} - 2 q^{38} - 2 q^{39} - 2 q^{42} - 2 q^{43} + 2 q^{45} - 2 q^{47} + 2 q^{54} - 4 q^{55} - 4 q^{57} - 2 q^{58} + 2 q^{61} + 4 q^{62} + 2 q^{63} - 4 q^{64} + 2 q^{66} + 2 q^{77} - 4 q^{80} + 4 q^{81} - 4 q^{82} + 2 q^{83} + 2 q^{87} + 2 q^{93} - 2 q^{99}+O(q^{100}) \) 4 * q - 2 * q^5 - 2 * q^6 - 2 * q^7 - 4 * q^9 + 2 * q^11 + 2 * q^16 + 2 * q^23 + 4 * q^24 - 4 * q^26 + 4 * q^30 + 4 * q^35 - 2 * q^38 - 2 * q^39 - 2 * q^42 - 2 * q^43 + 2 * q^45 - 2 * q^47 + 2 * q^54 - 4 * q^55 - 4 * q^57 - 2 * q^58 + 2 * q^61 + 4 * q^62 + 2 * q^63 - 4 * q^64 + 2 * q^66 + 2 * q^77 - 4 * q^80 + 4 * q^81 - 4 * q^82 + 2 * q^83 + 2 * q^87 + 2 * q^93 - 2 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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