LMFDB - Embedded newform 1476.1.o.a.655.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1476,1,Mod(655,1476)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1476, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([3, 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("1476.655");

S:= CuspForms(chi, 1);

N := Newforms(S);

Level:	$ N $	$=$	$ 1476 = 2^{2} \cdot 3^{2} \cdot 41 $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	1476.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:	$0.736619958646$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{6})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x + 1 $ x^2 - x + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Projective image:	$D_{3}$
Projective field:	Galois closure of 3.1.13284.1

Embedding invariants

Embedding label			655.1
Root			$0.500000 + 0.866025i$ of defining polynomial
Character	$\chi$	$=$	1476.655
Dual form			1476.1.o.a.1147.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-0.500000 + 0.866025i) q^{2} -1.00000 q^{3} +(-0.500000 - 0.866025i) q^{4} +(0.500000 + 0.866025i) q^{5} +(0.500000 - 0.866025i) q^{6} +(-0.500000 + 0.866025i) q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q+(-0.500000 + 0.866025i) q^{2} -1.00000 q^{3} +(-0.500000 - 0.866025i) q^{4} +(0.500000 + 0.866025i) q^{5} +(0.500000 - 0.866025i) q^{6} +(-0.500000 + 0.866025i) q^{7} +1.00000 q^{8} +1.00000 q^{9} -1.00000 q^{10} +(1.00000 - 1.73205i) q^{11} +(0.500000 + 0.866025i) q^{12} +(-0.500000 - 0.866025i) q^{14} +(-0.500000 - 0.866025i) q^{15} +(-0.500000 + 0.866025i) q^{16} +(-0.500000 + 0.866025i) q^{18} +1.00000 q^{19} +(0.500000 - 0.866025i) q^{20} +(0.500000 - 0.866025i) q^{21} +(1.00000 + 1.73205i) q^{22} -1.00000 q^{24} -1.00000 q^{27} +1.00000 q^{28} +1.00000 q^{30} +(-0.500000 - 0.866025i) q^{32} +(-1.00000 + 1.73205i) q^{33} -1.00000 q^{35} +(-0.500000 - 0.866025i) q^{36} +2.00000 q^{37} +(-0.500000 + 0.866025i) q^{38} +(0.500000 + 0.866025i) q^{40} +(-0.500000 - 0.866025i) q^{41} +(0.500000 + 0.866025i) q^{42} -2.00000 q^{44} +(0.500000 + 0.866025i) q^{45} +(-0.500000 + 0.866025i) q^{47} +(0.500000 - 0.866025i) q^{48} +(0.500000 - 0.866025i) q^{54} +2.00000 q^{55} +(-0.500000 + 0.866025i) q^{56} -1.00000 q^{57} +(-0.500000 + 0.866025i) q^{60} +(-1.00000 + 1.73205i) q^{61} +(-0.500000 + 0.866025i) q^{63} +1.00000 q^{64} +(-1.00000 - 1.73205i) q^{66} +(1.00000 + 1.73205i) q^{67} +(0.500000 - 0.866025i) q^{70} +1.00000 q^{71} +1.00000 q^{72} -1.00000 q^{73} +(-1.00000 + 1.73205i) q^{74} +(-0.500000 - 0.866025i) q^{76} +(1.00000 + 1.73205i) q^{77} +(-0.500000 + 0.866025i) q^{79} -1.00000 q^{80} +1.00000 q^{81} +1.00000 q^{82} -1.00000 q^{84} +(1.00000 - 1.73205i) q^{88} -1.00000 q^{90} +(-0.500000 - 0.866025i) q^{94} +(0.500000 + 0.866025i) q^{95} +(0.500000 + 0.866025i) q^{96} +(1.00000 - 1.73205i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - q^{2} - 2 q^{3} - q^{4} + q^{5} + q^{6} - q^{7} + 2 q^{8} + 2 q^{9}+O(q^{10}) $ 2 * q - q^2 - 2 * q^3 - q^4 + q^5 + q^6 - q^7 + 2 * q^8 + 2 * q^9	$ 2 q - q^{2} - 2 q^{3} - q^{4} + q^{5} + q^{6} - q^{7} + 2 q^{8} + 2 q^{9} - 2 q^{10} + 2 q^{11} + q^{12} - q^{14} - q^{15} - q^{16} - q^{18} + 2 q^{19} + q^{20} + q^{21} + 2 q^{22} - 2 q^{24} - 2 q^{27} + 2 q^{28} + 2 q^{30} - q^{32} - 2 q^{33} - 2 q^{35} - q^{36} + 4 q^{37} - q^{38} + q^{40} - q^{41} + q^{42} - 4 q^{44} + q^{45} - q^{47} + q^{48} + q^{54} + 4 q^{55} - q^{56} - 2 q^{57} - q^{60} - 2 q^{61} - q^{63} + 2 q^{64} - 2 q^{66} + 2 q^{67} + q^{70} + 2 q^{71} + 2 q^{72} - 2 q^{73} - 2 q^{74} - q^{76} + 2 q^{77} - q^{79} - 2 q^{80} + 2 q^{81} + 2 q^{82} - 2 q^{84} + 2 q^{88} - 2 q^{90} - q^{94} + q^{95} + q^{96} + 2 q^{99}+O(q^{100}) $ 2 * q - q^2 - 2 * q^3 - q^4 + q^5 + q^6 - q^7 + 2 * q^8 + 2 * q^9 - 2 * q^10 + 2 * q^11 + q^12 - q^14 - q^15 - q^16 - q^18 + 2 * q^19 + q^20 + q^21 + 2 * q^22 - 2 * q^24 - 2 * q^27 + 2 * q^28 + 2 * q^30 - q^32 - 2 * q^33 - 2 * q^35 - q^36 + 4 * q^37 - q^38 + q^40 - q^41 + q^42 - 4 * q^44 + q^45 - q^47 + q^48 + q^54 + 4 * q^55 - q^56 - 2 * q^57 - q^60 - 2 * q^61 - q^63 + 2 * q^64 - 2 * q^66 + 2 * q^67 + q^70 + 2 * q^71 + 2 * q^72 - 2 * q^73 - 2 * q^74 - q^76 + 2 * q^77 - q^79 - 2 * q^80 + 2 * q^81 + 2 * q^82 - 2 * q^84 + 2 * q^88 - 2 * q^90 - q^94 + q^95 + q^96 + 2 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1476\mathbb{Z}\right)^\times$.

$n$	$739$	$821$	$1441$
$\chi(n)$	$-1$	$e\left(\frac{2}{3}\right)$	$-1$

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 $

$2$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$2$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$3$	−1.00000			−1.00000
$3$	−1.00000			−1.00000
$4$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$5$	0.500000	+	0.866025i	0.500000	+	0.866025i	1.00000			$0$
$5$	0.500000	+	0.866025i	0.500000	+	0.866025i	−0.500000	+	0.866025i	$0.666667\pi$
$6$	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			$\pi$
$8$	1.00000			1.00000
$9$	1.00000			1.00000
$10$	−1.00000			−1.00000
$11$	1.00000	−	1.73205i	1.00000	−	1.73205i	0.500000	−	0.866025i	$-0.333333\pi$
$11$	1.00000	−	1.73205i	1.00000	−	1.73205i	0.500000	−	0.866025i	$-0.333333\pi$
$12$	0.500000	+	0.866025i	0.500000	+	0.866025i
$13$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$13$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$14$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$15$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$16$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$17$		0			0		1.00000			$0$
$17$		0			0		−1.00000			$\pi$
$18$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$19$	1.00000			1.00000			0.500000	−	0.866025i	$-0.333333\pi$
$19$	1.00000			1.00000			0.500000	+	0.866025i	$0.333333\pi$
$20$	0.500000	−	0.866025i	0.500000	−	0.866025i
$21$	0.500000	−	0.866025i	0.500000	−	0.866025i
$22$	1.00000	+	1.73205i	1.00000	+	1.73205i
$23$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$23$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$24$	−1.00000			−1.00000
$25$		0			0
$26$		0			0
$27$	−1.00000			−1.00000
$28$	1.00000			1.00000
$29$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$29$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$30$	1.00000			1.00000
$31$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$31$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$32$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$33$	−1.00000	+	1.73205i	−1.00000	+	1.73205i
$34$		0			0
$35$	−1.00000			−1.00000
$36$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$37$	2.00000			2.00000			1.00000			$0$
$37$	2.00000			2.00000			1.00000			$0$
$38$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$39$		0			0
$40$	0.500000	+	0.866025i	0.500000	+	0.866025i
$41$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$41$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$42$	0.500000	+	0.866025i	0.500000	+	0.866025i
$43$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$43$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$44$	−2.00000			−2.00000
$45$	0.500000	+	0.866025i	0.500000	+	0.866025i
$46$		0			0
$47$	−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			$\pi$
$48$	0.500000	−	0.866025i	0.500000	−	0.866025i
$49$		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
$55$	2.00000			2.00000
$56$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$57$	−1.00000			−1.00000
$58$		0			0
$59$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$59$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$60$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$61$	−1.00000	+	1.73205i	−1.00000	+	1.73205i	−0.500000	+	0.866025i	$0.666667\pi$
$61$	−1.00000	+	1.73205i	−1.00000	+	1.73205i	−0.500000	+	0.866025i	$0.666667\pi$
$62$		0			0
$63$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$64$	1.00000			1.00000
$65$		0			0
$66$	−1.00000	−	1.73205i	−1.00000	−	1.73205i
$67$	1.00000	+	1.73205i	1.00000	+	1.73205i	0.500000	+	0.866025i	$0.333333\pi$
$67$	1.00000	+	1.73205i	1.00000	+	1.73205i	0.500000	+	0.866025i	$0.333333\pi$
$68$		0			0
$69$		0			0
$70$	0.500000	−	0.866025i	0.500000	−	0.866025i
$71$	1.00000			1.00000			0.500000	−	0.866025i	$-0.333333\pi$
$71$	1.00000			1.00000			0.500000	+	0.866025i	$0.333333\pi$
$72$	1.00000			1.00000
$73$	−1.00000			−1.00000			−0.500000	−	0.866025i	$-0.666667\pi$
$73$	−1.00000			−1.00000			−0.500000	+	0.866025i	$0.666667\pi$
$74$	−1.00000	+	1.73205i	−1.00000	+	1.73205i
$75$		0			0
$76$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$77$	1.00000	+	1.73205i	1.00000	+	1.73205i
$78$		0			0
$79$	−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			$\pi$
$80$	−1.00000			−1.00000
$81$	1.00000			1.00000
$82$	1.00000			1.00000
$83$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$83$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$84$	−1.00000			−1.00000
$85$		0			0
$86$		0			0
$87$		0			0
$88$	1.00000	−	1.73205i	1.00000	−	1.73205i
$89$		0			0		1.00000			$0$
$89$		0			0		−1.00000			$\pi$
$90$	−1.00000			−1.00000
$91$		0			0
$92$		0			0
$93$		0			0
$94$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$95$	0.500000	+	0.866025i	0.500000	+	0.866025i
$96$	0.500000	+	0.866025i	0.500000	+	0.866025i
$97$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$97$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$98$		0			0
$99$	1.00000	−	1.73205i	1.00000	−	1.73205i
$100$		0			0
$101$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$101$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$102$		0			0
$103$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$103$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$104$		0			0
$105$	1.00000			1.00000
$106$		0			0
$107$		0			0		1.00000			$0$
$107$		0			0		−1.00000			$\pi$
$108$	0.500000	+	0.866025i	0.500000	+	0.866025i
$109$		0			0		1.00000			$0$
$109$		0			0		−1.00000			$\pi$
$110$	−1.00000	+	1.73205i	−1.00000	+	1.73205i
$111$	−2.00000			−2.00000
$112$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$113$	−1.00000	−	1.73205i	−1.00000	−	1.73205i	−0.500000	−	0.866025i	$-0.666667\pi$
$113$	−1.00000	−	1.73205i	−1.00000	−	1.73205i	−0.500000	−	0.866025i	$-0.666667\pi$
$114$	0.500000	−	0.866025i	0.500000	−	0.866025i
$115$		0			0
$116$		0			0
$117$		0			0
$118$		0			0
$119$		0			0
$120$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$121$	−1.50000	−	2.59808i	−1.50000	−	2.59808i
$122$	−1.00000	−	1.73205i	−1.00000	−	1.73205i
$123$	0.500000	+	0.866025i	0.500000	+	0.866025i
$124$		0			0
$125$	1.00000			1.00000
$126$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$127$		0			0		1.00000			$0$
$127$		0			0		−1.00000			$\pi$
$128$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$129$		0			0
$130$		0			0
$131$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$131$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$132$	2.00000			2.00000
$133$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$134$	−2.00000			−2.00000
$135$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$136$		0			0
$137$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$137$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$138$		0			0
$139$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$139$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$140$	0.500000	+	0.866025i	0.500000	+	0.866025i
$141$	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
$145$		0			0
$146$	0.500000	−	0.866025i	0.500000	−	0.866025i
$147$		0			0
$148$	−1.00000	−	1.73205i	−1.00000	−	1.73205i
$149$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$149$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$150$		0			0
$151$	1.00000	−	1.73205i	1.00000	−	1.73205i	0.500000	−	0.866025i	$-0.333333\pi$
$151$	1.00000	−	1.73205i	1.00000	−	1.73205i	0.500000	−	0.866025i	$-0.333333\pi$
$152$	1.00000			1.00000
$153$		0			0
$154$	−2.00000			−2.00000
$155$		0			0
$156$		0			0
$157$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$157$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$158$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$159$		0			0
$160$	0.500000	−	0.866025i	0.500000	−	0.866025i
$161$		0			0
$162$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$163$		0			0		1.00000			$0$
$163$		0			0		−1.00000			$\pi$
$164$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$165$	−2.00000			−2.00000
$166$		0			0
$167$	−0.500000	−	0.866025i	−0.500000	−	0.866025i	0.500000	−	0.866025i	$-0.333333\pi$
$167$	−0.500000	−	0.866025i	−0.500000	−	0.866025i	−1.00000			$\pi$
$168$	0.500000	−	0.866025i	0.500000	−	0.866025i
$169$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$170$		0			0
$171$	1.00000			1.00000
$172$		0			0
$173$	0.500000	−	0.866025i	0.500000	−	0.866025i	−0.500000	−	0.866025i	$-0.666667\pi$
$173$	0.500000	−	0.866025i	0.500000	−	0.866025i	1.00000			$0$
$174$		0			0
$175$		0			0
$176$	1.00000	+	1.73205i	1.00000	+	1.73205i
$177$		0			0
$178$		0			0
$179$	1.00000			1.00000			0.500000	−	0.866025i	$-0.333333\pi$
$179$	1.00000			1.00000			0.500000	+	0.866025i	$0.333333\pi$
$180$	0.500000	−	0.866025i	0.500000	−	0.866025i
$181$		0			0		1.00000			$0$
$181$		0			0		−1.00000			$\pi$
$182$		0			0
$183$	1.00000	−	1.73205i	1.00000	−	1.73205i
$184$		0			0
$185$	1.00000	+	1.73205i	1.00000	+	1.73205i
$186$		0			0
$187$		0			0
$188$	1.00000			1.00000
$189$	0.500000	−	0.866025i	0.500000	−	0.866025i
$190$	−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			$\pi$
$192$	−1.00000			−1.00000
$193$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$193$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−1.00000			−1.00000			−0.500000	−	0.866025i	$-0.666667\pi$
$197$	−1.00000			−1.00000			−0.500000	+	0.866025i	$0.666667\pi$
$198$	1.00000	+	1.73205i	1.00000	+	1.73205i
$199$	−2.00000			−2.00000			−1.00000			$\pi$
$199$	−2.00000			−2.00000			−1.00000			$\pi$
$200$		0			0
$201$	−1.00000	−	1.73205i	−1.00000	−	1.73205i
$202$		0			0
$203$		0			0
$204$		0			0
$205$	0.500000	−	0.866025i	0.500000	−	0.866025i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	1.00000	−	1.73205i	1.00000	−	1.73205i
$210$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$211$	1.00000	+	1.73205i	1.00000	+	1.73205i	0.500000	+	0.866025i	$0.333333\pi$
$211$	1.00000	+	1.73205i	1.00000	+	1.73205i	0.500000	+	0.866025i	$0.333333\pi$
$212$		0			0
$213$	−1.00000			−1.00000
$214$		0			0
$215$		0			0
$216$	−1.00000			−1.00000
$217$		0			0
$218$		0			0
$219$	1.00000			1.00000
$220$	−1.00000	−	1.73205i	−1.00000	−	1.73205i
$221$		0			0
$222$	1.00000	−	1.73205i	1.00000	−	1.73205i
$223$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$223$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$224$	1.00000			1.00000
$225$		0			0
$226$	2.00000			2.00000
$227$	1.00000	−	1.73205i	1.00000	−	1.73205i	0.500000	−	0.866025i	$-0.333333\pi$
$227$	1.00000	−	1.73205i	1.00000	−	1.73205i	0.500000	−	0.866025i	$-0.333333\pi$
$228$	0.500000	+	0.866025i	0.500000	+	0.866025i
$229$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$229$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$230$		0			0
$231$	−1.00000	−	1.73205i	−1.00000	−	1.73205i
$232$		0			0
$233$		0			0		1.00000			$0$
$233$		0			0		−1.00000			$\pi$
$234$		0			0
$235$	−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	0.500000	−	0.866025i	$-0.333333\pi$
$239$	−0.500000	−	0.866025i	−0.500000	−	0.866025i	−1.00000			$\pi$
$240$	1.00000			1.00000
$241$	0.500000	−	0.866025i	0.500000	−	0.866025i	−0.500000	−	0.866025i	$-0.666667\pi$
$241$	0.500000	−	0.866025i	0.500000	−	0.866025i	1.00000			$0$
$242$	3.00000			3.00000
$243$	−1.00000			−1.00000
$244$	2.00000			2.00000
$245$		0			0
$246$	−1.00000			−1.00000
$247$		0			0
$248$		0			0
$249$		0			0
$250$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$251$		0			0		1.00000			$0$
$251$		0			0		−1.00000			$\pi$
$252$	1.00000			1.00000
$253$		0			0
$254$		0			0
$255$		0			0
$256$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$257$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$257$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$258$		0			0
$259$	−1.00000	+	1.73205i	−1.00000	+	1.73205i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	−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			$\pi$
$264$	−1.00000	+	1.73205i	−1.00000	+	1.73205i
$265$		0			0
$266$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$267$		0			0
$268$	1.00000	−	1.73205i	1.00000	−	1.73205i
$269$	−1.00000			−1.00000			−0.500000	−	0.866025i	$-0.666667\pi$
$269$	−1.00000			−1.00000			−0.500000	+	0.866025i	$0.666667\pi$
$270$	1.00000			1.00000
$271$		0			0		1.00000			$0$
$271$		0			0		−1.00000			$\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$		0			0
$276$		0			0
$277$	0.500000	−	0.866025i	0.500000	−	0.866025i	−0.500000	−	0.866025i	$-0.666667\pi$
$277$	0.500000	−	0.866025i	0.500000	−	0.866025i	1.00000			$0$
$278$		0			0
$279$		0			0
$280$	−1.00000			−1.00000
$281$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$281$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$282$	0.500000	+	0.866025i	0.500000	+	0.866025i
$283$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$283$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$284$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$285$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$286$		0			0
$287$	1.00000			1.00000
$288$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$289$	1.00000			1.00000
$290$		0			0
$291$		0			0
$292$	0.500000	+	0.866025i	0.500000	+	0.866025i
$293$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$293$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$294$		0			0
$295$		0			0
$296$	2.00000			2.00000
$297$	−1.00000	+	1.73205i	−1.00000	+	1.73205i
$298$		0			0
$299$		0			0
$300$		0			0
$301$		0			0
$302$	1.00000	+	1.73205i	1.00000	+	1.73205i
$303$		0			0
$304$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$305$	−2.00000			−2.00000
$306$		0			0
$307$		0			0		1.00000			$0$
$307$		0			0		−1.00000			$\pi$
$308$	1.00000	−	1.73205i	1.00000	−	1.73205i
$309$		0			0
$310$		0			0
$311$	1.00000	+	1.73205i	1.00000	+	1.73205i	0.500000	+	0.866025i	$0.333333\pi$
$311$	1.00000	+	1.73205i	1.00000	+	1.73205i	0.500000	+	0.866025i	$0.333333\pi$
$312$		0			0
$313$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$313$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$314$		0			0
$315$	−1.00000			−1.00000
$316$	1.00000			1.00000
$317$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$317$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$318$		0			0
$319$		0			0
$320$	0.500000	+	0.866025i	0.500000	+	0.866025i
$321$		0			0
$322$		0			0
$323$		0			0
$324$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$325$		0			0
$326$		0			0
$327$		0			0
$328$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$329$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$330$	1.00000	−	1.73205i	1.00000	−	1.73205i
$331$	−0.500000	+	0.866025i	−0.500000	+	0.866025i	0.500000	+	0.866025i	$0.333333\pi$
$331$	−0.500000	+	0.866025i	−0.500000	+	0.866025i	−1.00000			$\pi$
$332$		0			0
$333$	2.00000			2.00000
$334$	1.00000			1.00000
$335$	−1.00000	+	1.73205i	−1.00000	+	1.73205i
$336$	0.500000	+	0.866025i	0.500000	+	0.866025i
$337$	0.500000	+	0.866025i	0.500000	+	0.866025i	1.00000			$0$
$337$	0.500000	+	0.866025i	0.500000	+	0.866025i	−0.500000	+	0.866025i	$0.666667\pi$
$338$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$339$	1.00000	+	1.73205i	1.00000	+	1.73205i
$340$		0			0
$341$		0			0
$342$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$343$	−1.00000			−1.00000
$344$		0			0
$345$		0			0
$346$	0.500000	+	0.866025i	0.500000	+	0.866025i
$347$	−0.500000	−	0.866025i	−0.500000	−	0.866025i	0.500000	−	0.866025i	$-0.333333\pi$
$347$	−0.500000	−	0.866025i	−0.500000	−	0.866025i	−1.00000			$\pi$
$348$		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			0
$352$	−2.00000			−2.00000
$353$	−1.00000	+	1.73205i	−1.00000	+	1.73205i	−0.500000	+	0.866025i	$0.666667\pi$
$353$	−1.00000	+	1.73205i	−1.00000	+	1.73205i	−0.500000	+	0.866025i	$0.666667\pi$
$354$		0			0
$355$	0.500000	+	0.866025i	0.500000	+	0.866025i
$356$		0			0
$357$		0			0
$358$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$359$		0			0		1.00000			$0$
$359$		0			0		−1.00000			$\pi$
$360$	0.500000	+	0.866025i	0.500000	+	0.866025i
$361$		0			0
$362$		0			0
$363$	1.50000	+	2.59808i	1.50000	+	2.59808i
$364$		0			0
$365$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$366$	1.00000	+	1.73205i	1.00000	+	1.73205i
$367$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$367$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$368$		0			0
$369$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$370$	−2.00000			−2.00000
$371$		0			0
$372$		0			0
$373$	0.500000	+	0.866025i	0.500000	+	0.866025i	1.00000			$0$
$373$	0.500000	+	0.866025i	0.500000	+	0.866025i	−0.500000	+	0.866025i	$0.666667\pi$
$374$		0			0
$375$	−1.00000			−1.00000
$376$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$377$		0			0
$378$	0.500000	+	0.866025i	0.500000	+	0.866025i
$379$		0			0		1.00000			$0$
$379$		0			0		−1.00000			$\pi$
$380$	0.500000	−	0.866025i	0.500000	−	0.866025i
$381$		0			0
$382$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$383$	−0.500000	−	0.866025i	−0.500000	−	0.866025i	0.500000	−	0.866025i	$-0.333333\pi$
$383$	−0.500000	−	0.866025i	−0.500000	−	0.866025i	−1.00000			$\pi$
$384$	0.500000	−	0.866025i	0.500000	−	0.866025i
$385$	−1.00000	+	1.73205i	−1.00000	+	1.73205i
$386$		0			0
$387$		0			0
$388$		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			0
$391$		0			0
$392$		0			0
$393$		0			0
$394$	0.500000	−	0.866025i	0.500000	−	0.866025i
$395$	−1.00000			−1.00000
$396$	−2.00000			−2.00000
$397$		0			0		1.00000			$0$
$397$		0			0		−1.00000			$\pi$
$398$	1.00000	−	1.73205i	1.00000	−	1.73205i
$399$	0.500000	−	0.866025i	0.500000	−	0.866025i
$400$		0			0
$401$	0.500000	+	0.866025i	0.500000	+	0.866025i	1.00000			$0$
$401$	0.500000	+	0.866025i	0.500000	+	0.866025i	−0.500000	+	0.866025i	$0.666667\pi$
$402$	2.00000			2.00000
$403$		0			0
$404$		0			0
$405$	0.500000	+	0.866025i	0.500000	+	0.866025i
$406$		0			0
$407$	2.00000	−	3.46410i	2.00000	−	3.46410i
$408$		0			0
$409$	−1.00000	−	1.73205i	−1.00000	−	1.73205i	−0.500000	−	0.866025i	$-0.666667\pi$
$409$	−1.00000	−	1.73205i	−1.00000	−	1.73205i	−0.500000	−	0.866025i	$-0.666667\pi$
$410$	0.500000	+	0.866025i	0.500000	+	0.866025i
$411$		0			0
$412$		0			0
$413$		0			0
$414$		0			0
$415$		0			0
$416$		0			0
$417$		0			0
$418$	1.00000	+	1.73205i	1.00000	+	1.73205i
$419$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$419$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$420$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$421$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$421$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$422$	−2.00000			−2.00000
$423$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$424$		0			0
$425$		0			0
$426$	0.500000	−	0.866025i	0.500000	−	0.866025i
$427$	−1.00000	−	1.73205i	−1.00000	−	1.73205i
$428$		0			0
$429$		0			0
$430$		0			0
$431$		0			0		1.00000			$0$
$431$		0			0		−1.00000			$\pi$
$432$	0.500000	−	0.866025i	0.500000	−	0.866025i
$433$	−1.00000			−1.00000			−0.500000	−	0.866025i	$-0.666667\pi$
$433$	−1.00000			−1.00000			−0.500000	+	0.866025i	$0.666667\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$		0			0
$438$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$439$	1.00000	−	1.73205i	1.00000	−	1.73205i	0.500000	−	0.866025i	$-0.333333\pi$
$439$	1.00000	−	1.73205i	1.00000	−	1.73205i	0.500000	−	0.866025i	$-0.333333\pi$
$440$	2.00000			2.00000
$441$		0			0
$442$		0			0
$443$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$443$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$444$	1.00000	+	1.73205i	1.00000	+	1.73205i
$445$		0			0
$446$		0			0
$447$		0			0
$448$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$449$	−1.00000			−1.00000			−0.500000	−	0.866025i	$-0.666667\pi$
$449$	−1.00000			−1.00000			−0.500000	+	0.866025i	$0.666667\pi$
$450$		0			0
$451$	−2.00000			−2.00000
$452$	−1.00000	+	1.73205i	−1.00000	+	1.73205i
$453$	−1.00000	+	1.73205i	−1.00000	+	1.73205i
$454$	1.00000	+	1.73205i	1.00000	+	1.73205i
$455$		0			0
$456$	−1.00000			−1.00000
$457$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$457$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−1.00000	+	1.73205i	−1.00000	+	1.73205i	−0.500000	+	0.866025i	$0.666667\pi$
$461$	−1.00000	+	1.73205i	−1.00000	+	1.73205i	−0.500000	+	0.866025i	$0.666667\pi$
$462$	2.00000			2.00000
$463$	−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			0
$465$		0			0
$466$		0			0
$467$		0			0		1.00000			$0$
$467$		0			0		−1.00000			$\pi$
$468$		0			0
$469$	−2.00000			−2.00000
$470$	0.500000	−	0.866025i	0.500000	−	0.866025i
$471$		0			0
$472$		0			0
$473$		0			0
$474$	0.500000	+	0.866025i	0.500000	+	0.866025i
$475$		0			0
$476$		0			0
$477$		0			0
$478$	1.00000			1.00000
$479$	−0.500000	+	0.866025i	−0.500000	+	0.866025i	0.500000	+	0.866025i	$0.333333\pi$
$479$	−0.500000	+	0.866025i	−0.500000	+	0.866025i	−1.00000			$\pi$
$480$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$481$		0			0
$482$	0.500000	+	0.866025i	0.500000	+	0.866025i
$483$		0			0
$484$	−1.50000	+	2.59808i	−1.50000	+	2.59808i
$485$		0			0
$486$	0.500000	−	0.866025i	0.500000	−	0.866025i
$487$		0			0		1.00000			$0$
$487$		0			0		−1.00000			$\pi$
$488$	−1.00000	+	1.73205i	−1.00000	+	1.73205i
$489$		0			0
$490$		0			0
$491$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$491$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$492$	0.500000	−	0.866025i	0.500000	−	0.866025i
$493$		0			0
$494$		0			0
$495$	2.00000			2.00000
$496$		0			0
$497$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$498$		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			$\pi$
$500$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$501$	0.500000	+	0.866025i	0.500000	+	0.866025i
$502$		0			0
$503$	1.00000			1.00000			0.500000	−	0.866025i	$-0.333333\pi$
$503$	1.00000			1.00000			0.500000	+	0.866025i	$0.333333\pi$
$504$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$505$		0			0
$506$		0			0
$507$	0.500000	−	0.866025i	0.500000	−	0.866025i
$508$		0			0
$509$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$509$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$510$		0			0
$511$	0.500000	−	0.866025i	0.500000	−	0.866025i
$512$	1.00000			1.00000
$513$	−1.00000			−1.00000
$514$		0			0
$515$		0			0
$516$		0			0
$517$	1.00000	+	1.73205i	1.00000	+	1.73205i
$518$	−1.00000	−	1.73205i	−1.00000	−	1.73205i
$519$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$520$		0			0
$521$		0			0		1.00000			$0$
$521$		0			0		−1.00000			$\pi$
$522$		0			0
$523$		0			0		1.00000			$0$
$523$		0			0		−1.00000			$\pi$
$524$		0			0
$525$		0			0
$526$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$527$		0			0
$528$	−1.00000	−	1.73205i	−1.00000	−	1.73205i
$529$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$530$		0			0
$531$		0			0
$532$	1.00000			1.00000
$533$		0			0
$534$		0			0
$535$		0			0
$536$	1.00000	+	1.73205i	1.00000	+	1.73205i
$537$	−1.00000			−1.00000
$538$	0.500000	−	0.866025i	0.500000	−	0.866025i
$539$		0			0
$540$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$541$	−1.00000			−1.00000			−0.500000	−	0.866025i	$-0.666667\pi$
$541$	−1.00000			−1.00000			−0.500000	+	0.866025i	$0.666667\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$		0			0
$546$		0			0
$547$	−0.500000	+	0.866025i	−0.500000	+	0.866025i	0.500000	+	0.866025i	$0.333333\pi$
$547$	−0.500000	+	0.866025i	−0.500000	+	0.866025i	−1.00000			$\pi$
$548$		0			0
$549$	−1.00000	+	1.73205i	−1.00000	+	1.73205i
$550$		0			0
$551$		0			0
$552$		0			0
$553$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$554$	0.500000	+	0.866025i	0.500000	+	0.866025i
$555$	−1.00000	−	1.73205i	−1.00000	−	1.73205i
$556$		0			0
$557$		0			0		1.00000			$0$
$557$		0			0		−1.00000			$\pi$
$558$		0			0
$559$		0			0
$560$	0.500000	−	0.866025i	0.500000	−	0.866025i
$561$		0			0
$562$		0			0
$563$	−0.500000	−	0.866025i	−0.500000	−	0.866025i	0.500000	−	0.866025i	$-0.333333\pi$
$563$	−0.500000	−	0.866025i	−0.500000	−	0.866025i	−1.00000			$\pi$
$564$	−1.00000			−1.00000
$565$	1.00000	−	1.73205i	1.00000	−	1.73205i
$566$		0			0
$567$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$568$	1.00000			1.00000
$569$	0.500000	−	0.866025i	0.500000	−	0.866025i	−0.500000	−	0.866025i	$-0.666667\pi$
$569$	0.500000	−	0.866025i	0.500000	−	0.866025i	1.00000			$0$
$570$	1.00000			1.00000
$571$	−0.500000	−	0.866025i	−0.500000	−	0.866025i	0.500000	−	0.866025i	$-0.333333\pi$
$571$	−0.500000	−	0.866025i	−0.500000	−	0.866025i	−1.00000			$\pi$
$572$		0			0
$573$	0.500000	−	0.866025i	0.500000	−	0.866025i
$574$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$575$		0			0
$576$	1.00000			1.00000
$577$		0			0		1.00000			$0$
$577$		0			0		−1.00000			$\pi$
$578$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$579$		0			0
$580$		0			0
$581$		0			0
$582$		0			0
$583$		0			0
$584$	−1.00000			−1.00000
$585$		0			0
$586$		0			0
$587$	1.00000	−	1.73205i	1.00000	−	1.73205i	0.500000	−	0.866025i	$-0.333333\pi$
$587$	1.00000	−	1.73205i	1.00000	−	1.73205i	0.500000	−	0.866025i	$-0.333333\pi$
$588$		0			0
$589$		0			0
$590$		0			0
$591$	1.00000			1.00000
$592$	−1.00000	+	1.73205i	−1.00000	+	1.73205i
$593$		0			0		1.00000			$0$
$593$		0			0		−1.00000			$\pi$
$594$	−1.00000	−	1.73205i	−1.00000	−	1.73205i
$595$		0			0
$596$		0			0
$597$	2.00000			2.00000
$598$		0			0
$599$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$599$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$600$		0			0
$601$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$601$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$602$		0			0
$603$	1.00000	+	1.73205i	1.00000	+	1.73205i
$604$	−2.00000			−2.00000
$605$	1.50000	−	2.59808i	1.50000	−	2.59808i
$606$		0			0
$607$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$607$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$608$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$609$		0			0
$610$	1.00000	−	1.73205i	1.00000	−	1.73205i
$611$		0			0
$612$		0			0
$613$	−1.00000			−1.00000			−0.500000	−	0.866025i	$-0.666667\pi$
$613$	−1.00000			−1.00000			−0.500000	+	0.866025i	$0.666667\pi$
$614$		0			0
$615$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$616$	1.00000	+	1.73205i	1.00000	+	1.73205i
$617$	0.500000	+	0.866025i	0.500000	+	0.866025i	1.00000			$0$
$617$	0.500000	+	0.866025i	0.500000	+	0.866025i	−0.500000	+	0.866025i	$0.666667\pi$
$618$		0			0
$619$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$619$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$620$		0			0
$621$		0			0
$622$	−2.00000			−2.00000
$623$		0			0
$624$		0			0
$625$	0.500000	+	0.866025i	0.500000	+	0.866025i
$626$		0			0
$627$	−1.00000	+	1.73205i	−1.00000	+	1.73205i
$628$		0			0
$629$		0			0
$630$	0.500000	−	0.866025i	0.500000	−	0.866025i
$631$		0			0		1.00000			$0$
$631$		0			0		−1.00000			$\pi$
$632$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$633$	−1.00000	−	1.73205i	−1.00000	−	1.73205i
$634$		0			0
$635$		0			0
$636$		0			0
$637$		0			0
$638$		0			0
$639$	1.00000			1.00000
$640$	−1.00000			−1.00000
$641$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$641$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$642$		0			0
$643$	−0.500000	−	0.866025i	−0.500000	−	0.866025i	0.500000	−	0.866025i	$-0.333333\pi$
$643$	−0.500000	−	0.866025i	−0.500000	−	0.866025i	−1.00000			$\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		0			0		1.00000			$0$
$647$		0			0		−1.00000			$\pi$
$648$	1.00000			1.00000
$649$		0			0
$650$		0			0
$651$		0			0
$652$		0			0
$653$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$653$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$654$		0			0
$655$		0			0
$656$	1.00000			1.00000
$657$	−1.00000			−1.00000
$658$	1.00000			1.00000
$659$	−0.500000	+	0.866025i	−0.500000	+	0.866025i	0.500000	+	0.866025i	$0.333333\pi$
$659$	−0.500000	+	0.866025i	−0.500000	+	0.866025i	−1.00000			$\pi$
$660$	1.00000	+	1.73205i	1.00000	+	1.73205i
$661$	0.500000	+	0.866025i	0.500000	+	0.866025i	1.00000			$0$
$661$	0.500000	+	0.866025i	0.500000	+	0.866025i	−0.500000	+	0.866025i	$0.666667\pi$
$662$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$663$		0			0
$664$		0			0
$665$	−1.00000			−1.00000
$666$	−1.00000	+	1.73205i	−1.00000	+	1.73205i
$667$		0			0
$668$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$669$		0			0
$670$	−1.00000	−	1.73205i	−1.00000	−	1.73205i
$671$	2.00000	+	3.46410i	2.00000	+	3.46410i
$672$	−1.00000			−1.00000
$673$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$673$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$674$	−1.00000			−1.00000
$675$		0			0
$676$	1.00000			1.00000
$677$	0.500000	−	0.866025i	0.500000	−	0.866025i	−0.500000	−	0.866025i	$-0.666667\pi$
$677$	0.500000	−	0.866025i	0.500000	−	0.866025i	1.00000			$0$
$678$	−2.00000			−2.00000
$679$		0			0
$680$		0			0
$681$	−1.00000	+	1.73205i	−1.00000	+	1.73205i
$682$		0			0
$683$	1.00000			1.00000			0.500000	−	0.866025i	$-0.333333\pi$
$683$	1.00000			1.00000			0.500000	+	0.866025i	$0.333333\pi$
$684$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$685$		0			0
$686$	0.500000	−	0.866025i	0.500000	−	0.866025i
$687$		0			0
$688$		0			0
$689$		0			0
$690$		0			0
$691$	1.00000	−	1.73205i	1.00000	−	1.73205i	0.500000	−	0.866025i	$-0.333333\pi$
$691$	1.00000	−	1.73205i	1.00000	−	1.73205i	0.500000	−	0.866025i	$-0.333333\pi$
$692$	−1.00000			−1.00000
$693$	1.00000	+	1.73205i	1.00000	+	1.73205i
$694$	1.00000			1.00000
$695$		0			0
$696$		0			0
$697$		0			0
$698$	0.500000	+	0.866025i	0.500000	+	0.866025i
$699$		0			0
$700$		0			0
$701$	−1.00000			−1.00000			−0.500000	−	0.866025i	$-0.666667\pi$
$701$	−1.00000			−1.00000			−0.500000	+	0.866025i	$0.666667\pi$
$702$		0			0
$703$	2.00000			2.00000
$704$	1.00000	−	1.73205i	1.00000	−	1.73205i
$705$	1.00000			1.00000
$706$	−1.00000	−	1.73205i	−1.00000	−	1.73205i
$707$		0			0
$708$		0			0
$709$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$709$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$710$	−1.00000			−1.00000
$711$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$712$		0			0
$713$		0			0
$714$		0			0
$715$		0			0
$716$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$717$	0.500000	+	0.866025i	0.500000	+	0.866025i
$718$		0			0
$719$	−2.00000			−2.00000			−1.00000			$\pi$
$719$	−2.00000			−2.00000			−1.00000			$\pi$
$720$	−1.00000			−1.00000
$721$		0			0
$722$		0			0
$723$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$724$		0			0
$725$		0			0
$726$	−3.00000			−3.00000
$727$	−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			0
$729$	1.00000			1.00000
$730$	1.00000			1.00000
$731$		0			0
$732$	−2.00000			−2.00000
$733$	0.500000	+	0.866025i	0.500000	+	0.866025i	1.00000			$0$
$733$	0.500000	+	0.866025i	0.500000	+	0.866025i	−0.500000	+	0.866025i	$0.666667\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	4.00000			4.00000
$738$	1.00000			1.00000
$739$		0			0		1.00000			$0$
$739$		0			0		−1.00000			$\pi$
$740$	1.00000	−	1.73205i	1.00000	−	1.73205i
$741$		0			0
$742$		0			0
$743$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$743$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$744$		0			0
$745$		0			0
$746$	−1.00000			−1.00000
$747$		0			0
$748$		0			0
$749$		0			0
$750$	0.500000	−	0.866025i	0.500000	−	0.866025i
$751$	−0.500000	−	0.866025i	−0.500000	−	0.866025i	0.500000	−	0.866025i	$-0.333333\pi$
$751$	−0.500000	−	0.866025i	−0.500000	−	0.866025i	−1.00000			$\pi$
$752$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$753$		0			0
$754$		0			0
$755$	2.00000			2.00000
$756$	−1.00000			−1.00000
$757$		0			0		1.00000			$0$
$757$		0			0		−1.00000			$\pi$
$758$		0			0
$759$		0			0
$760$	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
$763$		0			0
$764$	1.00000			1.00000
$765$		0			0
$766$	1.00000			1.00000
$767$		0			0
$768$	0.500000	+	0.866025i	0.500000	+	0.866025i
$769$	0.500000	+	0.866025i	0.500000	+	0.866025i	1.00000			$0$
$769$	0.500000	+	0.866025i	0.500000	+	0.866025i	−0.500000	+	0.866025i	$0.666667\pi$
$770$	−1.00000	−	1.73205i	−1.00000	−	1.73205i
$771$		0			0
$772$		0			0
$773$		0			0		1.00000			$0$
$773$		0			0		−1.00000			$\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$	1.00000	−	1.73205i	1.00000	−	1.73205i
$778$	0.500000	+	0.866025i	0.500000	+	0.866025i
$779$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$780$		0			0
$781$	1.00000	−	1.73205i	1.00000	−	1.73205i
$782$		0			0
$783$		0			0
$784$		0			0
$785$		0			0
$786$		0			0
$787$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$787$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$788$	0.500000	+	0.866025i	0.500000	+	0.866025i
$789$	0.500000	−	0.866025i	0.500000	−	0.866025i
$790$	0.500000	−	0.866025i	0.500000	−	0.866025i
$791$	2.00000			2.00000
$792$	1.00000	−	1.73205i	1.00000	−	1.73205i
$793$		0			0
$794$		0			0
$795$		0			0
$796$	1.00000	+	1.73205i	1.00000	+	1.73205i
$797$	0.500000	+	0.866025i	0.500000	+	0.866025i	1.00000			$0$
$797$	0.500000	+	0.866025i	0.500000	+	0.866025i	−0.500000	+	0.866025i	$0.666667\pi$
$798$	0.500000	+	0.866025i	0.500000	+	0.866025i
$799$		0			0
$800$		0			0
$801$		0			0
$802$	−1.00000			−1.00000
$803$	−1.00000	+	1.73205i	−1.00000	+	1.73205i
$804$	−1.00000	+	1.73205i	−1.00000	+	1.73205i
$805$		0			0
$806$		0			0
$807$	1.00000			1.00000
$808$		0			0
$809$		0			0		1.00000			$0$
$809$		0			0		−1.00000			$\pi$
$810$	−1.00000			−1.00000
$811$		0			0		1.00000			$0$
$811$		0			0		−1.00000			$\pi$
$812$		0			0
$813$		0			0
$814$	2.00000	+	3.46410i	2.00000	+	3.46410i
$815$		0			0
$816$		0			0
$817$		0			0
$818$	2.00000			2.00000
$819$		0			0
$820$	−1.00000			−1.00000
$821$	0.500000	−	0.866025i	0.500000	−	0.866025i	−0.500000	−	0.866025i	$-0.666667\pi$
$821$	0.500000	−	0.866025i	0.500000	−	0.866025i	1.00000			$0$
$822$		0			0
$823$	−0.500000	−	0.866025i	−0.500000	−	0.866025i	0.500000	−	0.866025i	$-0.333333\pi$
$823$	−0.500000	−	0.866025i	−0.500000	−	0.866025i	−1.00000			$\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−2.00000			−2.00000			−1.00000			$\pi$
$827$	−2.00000			−2.00000			−1.00000			$\pi$
$828$		0			0
$829$	2.00000			2.00000			1.00000			$0$
$829$	2.00000			2.00000			1.00000			$0$
$830$		0			0
$831$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$832$		0			0
$833$		0			0
$834$		0			0
$835$	0.500000	−	0.866025i	0.500000	−	0.866025i
$836$	−2.00000			−2.00000
$837$		0			0
$838$		0			0
$839$	−0.500000	+	0.866025i	−0.500000	+	0.866025i	0.500000	+	0.866025i	$0.333333\pi$
$839$	−0.500000	+	0.866025i	−0.500000	+	0.866025i	−1.00000			$\pi$
$840$	1.00000			1.00000
$841$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$842$		0			0
$843$		0			0
$844$	1.00000	−	1.73205i	1.00000	−	1.73205i
$845$	−1.00000			−1.00000
$846$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$847$	3.00000			3.00000
$848$		0			0
$849$		0			0
$850$		0			0
$851$		0			0
$852$	0.500000	+	0.866025i	0.500000	+	0.866025i
$853$	0.500000	−	0.866025i	0.500000	−	0.866025i	−0.500000	−	0.866025i	$-0.666667\pi$
$853$	0.500000	−	0.866025i	0.500000	−	0.866025i	1.00000			$0$
$854$	2.00000			2.00000
$855$	0.500000	+	0.866025i	0.500000	+	0.866025i
$856$		0			0
$857$	0.500000	−	0.866025i	0.500000	−	0.866025i	−0.500000	−	0.866025i	$-0.666667\pi$
$857$	0.500000	−	0.866025i	0.500000	−	0.866025i	1.00000			$0$
$858$		0			0
$859$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$859$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$860$		0			0
$861$	−1.00000			−1.00000
$862$		0			0
$863$		0			0		1.00000			$0$
$863$		0			0		−1.00000			$\pi$
$864$	0.500000	+	0.866025i	0.500000	+	0.866025i
$865$	1.00000			1.00000
$866$	0.500000	−	0.866025i	0.500000	−	0.866025i
$867$	−1.00000			−1.00000
$868$		0			0
$869$	1.00000	+	1.73205i	1.00000	+	1.73205i
$870$		0			0
$871$		0			0
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$876$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$877$	−1.00000	−	1.73205i	−1.00000	−	1.73205i	−0.500000	−	0.866025i	$-0.666667\pi$
$877$	−1.00000	−	1.73205i	−1.00000	−	1.73205i	−0.500000	−	0.866025i	$-0.666667\pi$
$878$	1.00000	+	1.73205i	1.00000	+	1.73205i
$879$		0			0
$880$	−1.00000	+	1.73205i	−1.00000	+	1.73205i
$881$	−1.00000			−1.00000			−0.500000	−	0.866025i	$-0.666667\pi$
$881$	−1.00000			−1.00000			−0.500000	+	0.866025i	$0.666667\pi$
$882$		0			0
$883$	1.00000			1.00000			0.500000	−	0.866025i	$-0.333333\pi$
$883$	1.00000			1.00000			0.500000	+	0.866025i	$0.333333\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−0.500000	−	0.866025i	−0.500000	−	0.866025i	0.500000	−	0.866025i	$-0.333333\pi$
$887$	−0.500000	−	0.866025i	−0.500000	−	0.866025i	−1.00000			$\pi$
$888$	−2.00000			−2.00000
$889$		0			0
$890$		0			0
$891$	1.00000	−	1.73205i	1.00000	−	1.73205i
$892$		0			0
$893$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$894$		0			0
$895$	0.500000	+	0.866025i	0.500000	+	0.866025i
$896$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$897$		0			0
$898$	0.500000	−	0.866025i	0.500000	−	0.866025i
$899$		0			0
$900$		0			0
$901$		0			0
$902$	1.00000	−	1.73205i	1.00000	−	1.73205i
$903$		0			0
$904$	−1.00000	−	1.73205i	−1.00000	−	1.73205i
$905$		0			0
$906$	−1.00000	−	1.73205i	−1.00000	−	1.73205i
$907$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$907$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$908$	−2.00000			−2.00000
$909$		0			0
$910$		0			0
$911$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$911$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$912$	0.500000	−	0.866025i	0.500000	−	0.866025i
$913$		0			0
$914$		0			0
$915$	2.00000			2.00000
$916$		0			0
$917$		0			0
$918$		0			0
$919$	−2.00000			−2.00000			−1.00000			$\pi$
$919$	−2.00000			−2.00000			−1.00000			$\pi$
$920$		0			0
$921$		0			0
$922$	−1.00000	−	1.73205i	−1.00000	−	1.73205i
$923$		0			0
$924$	−1.00000	+	1.73205i	−1.00000	+	1.73205i
$925$		0			0
$926$	1.00000			1.00000
$927$		0			0
$928$		0			0
$929$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$929$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$930$		0			0
$931$		0			0
$932$		0			0
$933$	−1.00000	−	1.73205i	−1.00000	−	1.73205i
$934$		0			0
$935$		0			0
$936$		0			0
$937$		0			0		1.00000			$0$
$937$		0			0		−1.00000			$\pi$
$938$	1.00000	−	1.73205i	1.00000	−	1.73205i
$939$		0			0
$940$	0.500000	+	0.866025i	0.500000	+	0.866025i
$941$	0.500000	+	0.866025i	0.500000	+	0.866025i	1.00000			$0$
$941$	0.500000	+	0.866025i	0.500000	+	0.866025i	−0.500000	+	0.866025i	$0.666667\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$	1.00000			1.00000
$946$		0			0
$947$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$947$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$948$	−1.00000			−1.00000
$949$		0			0
$950$		0			0
$951$		0			0
$952$		0			0
$953$	−1.00000			−1.00000			−0.500000	−	0.866025i	$-0.666667\pi$
$953$	−1.00000			−1.00000			−0.500000	+	0.866025i	$0.666667\pi$
$954$		0			0
$955$	−1.00000			−1.00000
$956$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$957$		0			0
$958$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$959$		0			0
$960$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$961$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$962$		0			0
$963$		0			0
$964$	−1.00000			−1.00000
$965$		0			0
$966$		0			0
$967$	−0.500000	−	0.866025i	−0.500000	−	0.866025i	0.500000	−	0.866025i	$-0.333333\pi$
$967$	−0.500000	−	0.866025i	−0.500000	−	0.866025i	−1.00000			$\pi$
$968$	−1.50000	−	2.59808i	−1.50000	−	2.59808i
$969$		0			0
$970$		0			0
$971$	−2.00000			−2.00000			−1.00000			$\pi$
$971$	−2.00000			−2.00000			−1.00000			$\pi$
$972$	0.500000	+	0.866025i	0.500000	+	0.866025i
$973$		0			0
$974$		0			0
$975$		0			0
$976$	−1.00000	−	1.73205i	−1.00000	−	1.73205i
$977$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$977$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$978$		0			0
$979$		0			0
$980$		0			0
$981$		0			0
$982$		0			0
$983$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$983$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$984$	0.500000	+	0.866025i	0.500000	+	0.866025i
$985$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$986$		0			0
$987$	0.500000	+	0.866025i	0.500000	+	0.866025i
$988$		0			0
$989$		0			0
$990$	−1.00000	+	1.73205i	−1.00000	+	1.73205i
$991$	1.00000			1.00000			0.500000	−	0.866025i	$-0.333333\pi$
$991$	1.00000			1.00000			0.500000	+	0.866025i	$0.333333\pi$
$992$		0			0
$993$	0.500000	−	0.866025i	0.500000	−	0.866025i
$994$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$995$	−1.00000	−	1.73205i	−1.00000	−	1.73205i
$996$		0			0
$997$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$997$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$998$	1.00000			1.00000
$999$	−2.00000			−2.00000

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1476.1.o.a.655.1	✓	2
4.3	odd	2		1476.1.o.b.655.1	yes	2
9.4	even	3	inner	1476.1.o.a.1147.1	yes	2
36.31	odd	6		1476.1.o.b.1147.1	yes	2
41.40	even	2		1476.1.o.b.655.1	yes	2
164.163	odd	2	CM	1476.1.o.a.655.1	✓	2
369.40	even	6		1476.1.o.b.1147.1	yes	2
1476.1147	odd	6	inner	1476.1.o.a.1147.1	yes	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1476.1.o.a.655.1	✓	2	1.1	even	1	trivial
1476.1.o.a.655.1	✓	2	164.163	odd	2	CM
1476.1.o.a.1147.1	yes	2	9.4	even	3	inner
1476.1.o.a.1147.1	yes	2	1476.1147	odd	6	inner
1476.1.o.b.655.1	yes	2	4.3	odd	2
1476.1.o.b.655.1	yes	2	41.40	even	2
1476.1.o.b.1147.1	yes	2	36.31	odd	6
1476.1.o.b.1147.1	yes	2	369.40	even	6

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:	\( N \)	\(=\)	\( 1476 = 2^{2} \cdot 3^{2} \cdot 41 \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	1476.o (of order \(6\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(-0.500000 + 0.866025i) q^{2} -1.00000 q^{3} +(-0.500000 - 0.866025i) q^{4} +(0.500000 + 0.866025i) q^{5} +(0.500000 - 0.866025i) q^{6} +(-0.500000 + 0.866025i) q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q+(-0.500000 + 0.866025i) q^{2} -1.00000 q^{3} +(-0.500000 - 0.866025i) q^{4} +(0.500000 + 0.866025i) q^{5} +(0.500000 - 0.866025i) q^{6} +(-0.500000 + 0.866025i) q^{7} +1.00000 q^{8} +1.00000 q^{9} -1.00000 q^{10} +(1.00000 - 1.73205i) q^{11} +(0.500000 + 0.866025i) q^{12} +(-0.500000 - 0.866025i) q^{14} +(-0.500000 - 0.866025i) q^{15} +(-0.500000 + 0.866025i) q^{16} +(-0.500000 + 0.866025i) q^{18} +1.00000 q^{19} +(0.500000 - 0.866025i) q^{20} +(0.500000 - 0.866025i) q^{21} +(1.00000 + 1.73205i) q^{22} -1.00000 q^{24} -1.00000 q^{27} +1.00000 q^{28} +1.00000 q^{30} +(-0.500000 - 0.866025i) q^{32} +(-1.00000 + 1.73205i) q^{33} -1.00000 q^{35} +(-0.500000 - 0.866025i) q^{36} +2.00000 q^{37} +(-0.500000 + 0.866025i) q^{38} +(0.500000 + 0.866025i) q^{40} +(-0.500000 - 0.866025i) q^{41} +(0.500000 + 0.866025i) q^{42} -2.00000 q^{44} +(0.500000 + 0.866025i) q^{45} +(-0.500000 + 0.866025i) q^{47} +(0.500000 - 0.866025i) q^{48} +(0.500000 - 0.866025i) q^{54} +2.00000 q^{55} +(-0.500000 + 0.866025i) q^{56} -1.00000 q^{57} +(-0.500000 + 0.866025i) q^{60} +(-1.00000 + 1.73205i) q^{61} +(-0.500000 + 0.866025i) q^{63} +1.00000 q^{64} +(-1.00000 - 1.73205i) q^{66} +(1.00000 + 1.73205i) q^{67} +(0.500000 - 0.866025i) q^{70} +1.00000 q^{71} +1.00000 q^{72} -1.00000 q^{73} +(-1.00000 + 1.73205i) q^{74} +(-0.500000 - 0.866025i) q^{76} +(1.00000 + 1.73205i) q^{77} +(-0.500000 + 0.866025i) q^{79} -1.00000 q^{80} +1.00000 q^{81} +1.00000 q^{82} -1.00000 q^{84} +(1.00000 - 1.73205i) q^{88} -1.00000 q^{90} +(-0.500000 - 0.866025i) q^{94} +(0.500000 + 0.866025i) q^{95} +(0.500000 + 0.866025i) q^{96} +(1.00000 - 1.73205i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - q^{2} - 2 q^{3} - q^{4} + q^{5} + q^{6} - q^{7} + 2 q^{8} + 2 q^{9}+O(q^{10}) \) 2 * q - q^2 - 2 * q^3 - q^4 + q^5 + q^6 - q^7 + 2 * q^8 + 2 * q^9	\( 2 q - q^{2} - 2 q^{3} - q^{4} + q^{5} + q^{6} - q^{7} + 2 q^{8} + 2 q^{9} - 2 q^{10} + 2 q^{11} + q^{12} - q^{14} - q^{15} - q^{16} - q^{18} + 2 q^{19} + q^{20} + q^{21} + 2 q^{22} - 2 q^{24} - 2 q^{27} + 2 q^{28} + 2 q^{30} - q^{32} - 2 q^{33} - 2 q^{35} - q^{36} + 4 q^{37} - q^{38} + q^{40} - q^{41} + q^{42} - 4 q^{44} + q^{45} - q^{47} + q^{48} + q^{54} + 4 q^{55} - q^{56} - 2 q^{57} - q^{60} - 2 q^{61} - q^{63} + 2 q^{64} - 2 q^{66} + 2 q^{67} + q^{70} + 2 q^{71} + 2 q^{72} - 2 q^{73} - 2 q^{74} - q^{76} + 2 q^{77} - q^{79} - 2 q^{80} + 2 q^{81} + 2 q^{82} - 2 q^{84} + 2 q^{88} - 2 q^{90} - q^{94} + q^{95} + q^{96} + 2 q^{99}+O(q^{100}) \) 2 * q - q^2 - 2 * q^3 - q^4 + q^5 + q^6 - q^7 + 2 * q^8 + 2 * q^9 - 2 * q^10 + 2 * q^11 + q^12 - q^14 - q^15 - q^16 - q^18 + 2 * q^19 + q^20 + q^21 + 2 * q^22 - 2 * q^24 - 2 * q^27 + 2 * q^28 + 2 * q^30 - q^32 - 2 * q^33 - 2 * q^35 - q^36 + 4 * q^37 - q^38 + q^40 - q^41 + q^42 - 4 * q^44 + q^45 - q^47 + q^48 + q^54 + 4 * q^55 - q^56 - 2 * q^57 - q^60 - 2 * q^61 - q^63 + 2 * q^64 - 2 * q^66 + 2 * q^67 + q^70 + 2 * q^71 + 2 * q^72 - 2 * q^73 - 2 * q^74 - q^76 + 2 * q^77 - q^79 - 2 * q^80 + 2 * q^81 + 2 * q^82 - 2 * q^84 + 2 * q^88 - 2 * q^90 - q^94 + q^95 + q^96 + 2 * q^99

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