LMFDB - Embedded newform 273.1.s.b.107.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [273,1,Mod(74,273)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(273, base_ring=CyclotomicField(6))

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

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

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

chi := DirichletCharacter("273.74");

S:= CuspForms(chi, 1);

N := Newforms(S);

Level:	$ N $	$=$	$ 273 = 3 \cdot 7 \cdot 13 $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	273.s (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:	$0.136244748449$
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 4.0.74529.1

Embedding invariants

Embedding label			107.1
Root			$-0.866025 + 0.500000i$ of defining polynomial
Character	$\chi$	$=$	273.107
Dual form			273.1.s.b.74.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000i q^{2} +(-0.866025 - 0.500000i) q^{3} +(0.866025 + 0.500000i) q^{5} +(-0.500000 + 0.866025i) q^{6} +(-0.500000 - 0.866025i) q^{7} -1.00000i q^{8} +(0.500000 + 0.866025i) q^{9} +O(q^{10})$	\(q-1.00000i q^{2} +(-0.866025 - 0.500000i) q^{3} +(0.866025 + 0.500000i) q^{5} +(-0.500000 + 0.866025i) q^{6} +(-0.500000 - 0.866025i) q^{7} -1.00000i q^{8} +(0.500000 + 0.866025i) q^{9} +(0.500000 - 0.866025i) q^{10} -1.00000 q^{13} +(-0.866025 + 0.500000i) q^{14} +(-0.500000 - 0.866025i) q^{15} -1.00000 q^{16} +1.00000i q^{17} +(0.866025 - 0.500000i) q^{18} +1.00000i q^{21} +1.00000i q^{23} +(-0.500000 + 0.866025i) q^{24} +1.00000i q^{26} -1.00000i q^{27} +(0.866025 - 0.500000i) q^{29} +(-0.866025 + 0.500000i) q^{30} +(0.500000 + 0.866025i) q^{31} +1.00000 q^{34} -1.00000i q^{35} +1.00000 q^{37} +(0.866025 + 0.500000i) q^{39} +(0.500000 - 0.866025i) q^{40} +(-0.866025 + 0.500000i) q^{41} +1.00000 q^{42} +(-0.500000 + 0.866025i) q^{43} +1.00000i q^{45} +1.00000 q^{46} +(0.866025 + 0.500000i) q^{47} +(0.866025 + 0.500000i) q^{48} +(-0.500000 + 0.866025i) q^{49} +(0.500000 - 0.866025i) q^{51} +(-0.866025 + 0.500000i) q^{53} -1.00000 q^{54} +(-0.866025 + 0.500000i) q^{56} +(-0.500000 - 0.866025i) q^{58} -1.00000i q^{59} +(0.866025 - 0.500000i) q^{62} +(0.500000 - 0.866025i) q^{63} -1.00000 q^{64} +(-0.866025 - 0.500000i) q^{65} +(0.500000 - 0.866025i) q^{69} -1.00000 q^{70} +(-0.866025 - 0.500000i) q^{71} +(0.866025 - 0.500000i) q^{72} +(-0.500000 - 0.866025i) q^{73} -1.00000i q^{74} +(0.500000 - 0.866025i) q^{78} +(-0.500000 + 0.866025i) q^{79} +(-0.866025 - 0.500000i) q^{80} +(-0.500000 + 0.866025i) q^{81} +(0.500000 + 0.866025i) q^{82} +(-0.500000 + 0.866025i) q^{85} +(0.866025 + 0.500000i) q^{86} -1.00000 q^{87} -1.00000i q^{89} +1.00000 q^{90} +(0.500000 + 0.866025i) q^{91} -1.00000i q^{93} +(0.500000 - 0.866025i) q^{94} +(0.500000 - 0.866025i) q^{97} +(0.866025 + 0.500000i) q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 2 q^{6} - 2 q^{7} + 2 q^{9}+O(q^{10}) $ 4 * q - 2 * q^6 - 2 * q^7 + 2 * q^9	$ 4 q - 2 q^{6} - 2 q^{7} + 2 q^{9} + 2 q^{10} - 4 q^{13} - 2 q^{15} - 4 q^{16} - 2 q^{24} + 2 q^{31} + 4 q^{34} + 4 q^{37} + 2 q^{40} + 4 q^{42} - 2 q^{43} + 4 q^{46} - 2 q^{49} + 2 q^{51} - 4 q^{54} - 2 q^{58} + 2 q^{63} - 4 q^{64} + 2 q^{69} - 4 q^{70} - 2 q^{73} + 2 q^{78} - 2 q^{79} - 2 q^{81} + 2 q^{82} - 2 q^{85} - 4 q^{87} + 4 q^{90} + 2 q^{91} + 2 q^{94} + 2 q^{97}+O(q^{100}) $ 4 * q - 2 * q^6 - 2 * q^7 + 2 * q^9 + 2 * q^10 - 4 * q^13 - 2 * q^15 - 4 * q^16 - 2 * q^24 + 2 * q^31 + 4 * q^34 + 4 * q^37 + 2 * q^40 + 4 * q^42 - 2 * q^43 + 4 * q^46 - 2 * q^49 + 2 * q^51 - 4 * q^54 - 2 * q^58 + 2 * q^63 - 4 * q^64 + 2 * q^69 - 4 * q^70 - 2 * q^73 + 2 * q^78 - 2 * q^79 - 2 * q^81 + 2 * q^82 - 2 * q^85 - 4 * q^87 + 4 * q^90 + 2 * q^91 + 2 * q^94 + 2 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/273\mathbb{Z}\right)^\times$.

$n$	$92$	$106$	$157$
$\chi(n)$	$-1$	$e\left(\frac{1}{3}\right)$	$e\left(\frac{1}{3}\right)$

Coefficient data

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

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

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

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

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:	\( N \)	\(=\)	\( 273 = 3 \cdot 7 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	273.s (of order \(6\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q-1.00000i q^{2} +(-0.866025 - 0.500000i) q^{3} +(0.866025 + 0.500000i) q^{5} +(-0.500000 + 0.866025i) q^{6} +(-0.500000 - 0.866025i) q^{7} -1.00000i q^{8} +(0.500000 + 0.866025i) q^{9} +O(q^{10})\)	\(q-1.00000i q^{2} +(-0.866025 - 0.500000i) q^{3} +(0.866025 + 0.500000i) q^{5} +(-0.500000 + 0.866025i) q^{6} +(-0.500000 - 0.866025i) q^{7} -1.00000i q^{8} +(0.500000 + 0.866025i) q^{9} +(0.500000 - 0.866025i) q^{10} -1.00000 q^{13} +(-0.866025 + 0.500000i) q^{14} +(-0.500000 - 0.866025i) q^{15} -1.00000 q^{16} +1.00000i q^{17} +(0.866025 - 0.500000i) q^{18} +1.00000i q^{21} +1.00000i q^{23} +(-0.500000 + 0.866025i) q^{24} +1.00000i q^{26} -1.00000i q^{27} +(0.866025 - 0.500000i) q^{29} +(-0.866025 + 0.500000i) q^{30} +(0.500000 + 0.866025i) q^{31} +1.00000 q^{34} -1.00000i q^{35} +1.00000 q^{37} +(0.866025 + 0.500000i) q^{39} +(0.500000 - 0.866025i) q^{40} +(-0.866025 + 0.500000i) q^{41} +1.00000 q^{42} +(-0.500000 + 0.866025i) q^{43} +1.00000i q^{45} +1.00000 q^{46} +(0.866025 + 0.500000i) q^{47} +(0.866025 + 0.500000i) q^{48} +(-0.500000 + 0.866025i) q^{49} +(0.500000 - 0.866025i) q^{51} +(-0.866025 + 0.500000i) q^{53} -1.00000 q^{54} +(-0.866025 + 0.500000i) q^{56} +(-0.500000 - 0.866025i) q^{58} -1.00000i q^{59} +(0.866025 - 0.500000i) q^{62} +(0.500000 - 0.866025i) q^{63} -1.00000 q^{64} +(-0.866025 - 0.500000i) q^{65} +(0.500000 - 0.866025i) q^{69} -1.00000 q^{70} +(-0.866025 - 0.500000i) q^{71} +(0.866025 - 0.500000i) q^{72} +(-0.500000 - 0.866025i) q^{73} -1.00000i q^{74} +(0.500000 - 0.866025i) q^{78} +(-0.500000 + 0.866025i) q^{79} +(-0.866025 - 0.500000i) q^{80} +(-0.500000 + 0.866025i) q^{81} +(0.500000 + 0.866025i) q^{82} +(-0.500000 + 0.866025i) q^{85} +(0.866025 + 0.500000i) q^{86} -1.00000 q^{87} -1.00000i q^{89} +1.00000 q^{90} +(0.500000 + 0.866025i) q^{91} -1.00000i q^{93} +(0.500000 - 0.866025i) q^{94} +(0.500000 - 0.866025i) q^{97} +(0.866025 + 0.500000i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 2 q^{6} - 2 q^{7} + 2 q^{9}+O(q^{10}) \) 4 * q - 2 * q^6 - 2 * q^7 + 2 * q^9	\( 4 q - 2 q^{6} - 2 q^{7} + 2 q^{9} + 2 q^{10} - 4 q^{13} - 2 q^{15} - 4 q^{16} - 2 q^{24} + 2 q^{31} + 4 q^{34} + 4 q^{37} + 2 q^{40} + 4 q^{42} - 2 q^{43} + 4 q^{46} - 2 q^{49} + 2 q^{51} - 4 q^{54} - 2 q^{58} + 2 q^{63} - 4 q^{64} + 2 q^{69} - 4 q^{70} - 2 q^{73} + 2 q^{78} - 2 q^{79} - 2 q^{81} + 2 q^{82} - 2 q^{85} - 4 q^{87} + 4 q^{90} + 2 q^{91} + 2 q^{94} + 2 q^{97}+O(q^{100}) \) 4 * q - 2 * q^6 - 2 * q^7 + 2 * q^9 + 2 * q^10 - 4 * q^13 - 2 * q^15 - 4 * q^16 - 2 * q^24 + 2 * q^31 + 4 * q^34 + 4 * q^37 + 2 * q^40 + 4 * q^42 - 2 * q^43 + 4 * q^46 - 2 * q^49 + 2 * q^51 - 4 * q^54 - 2 * q^58 + 2 * q^63 - 4 * q^64 + 2 * q^69 - 4 * q^70 - 2 * q^73 + 2 * q^78 - 2 * q^79 - 2 * q^81 + 2 * q^82 - 2 * q^85 - 4 * q^87 + 4 * q^90 + 2 * q^91 + 2 * q^94 + 2 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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