LMFDB - Embedded newform 840.1.cg.b.149.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [840,1,Mod(149,840)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(840, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 3, 3, 3, 2]))

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

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

chi := DirichletCharacter("840.149");

S:= CuspForms(chi, 1);

N := Newforms(S);

Level:	$ N $	$=$	$ 840 = 2^{3} \cdot 3 \cdot 5 \cdot 7 $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	840.cg (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:	$0.419214610612$
Analytic rank:	$0$
Dimension:	$4$
Relative dimension:	$2$ over $\Q(\zeta_{6})$
Coefficient field:	$\Q(\zeta_{12})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{2} + 1 $ x^4 - x^2 + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Projective image:	$D_{6}$
Projective field:	Galois closure of 6.0.518616000.10

Embedding invariants

Embedding label			149.2
Root			$-0.866025 - 0.500000i$ of defining polynomial
Character	$\chi$	$=$	840.149
Dual form			840.1.cg.b.389.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(0.866025 + 0.500000i) q^{2} +(-0.866025 + 0.500000i) q^{3} +(0.500000 + 0.866025i) q^{4} +1.00000i q^{5} -1.00000 q^{6} +(0.500000 + 0.866025i) q^{7} +1.00000i q^{8} +(0.500000 - 0.866025i) q^{9} +O(q^{10})$	\(q+(0.866025 + 0.500000i) q^{2} +(-0.866025 + 0.500000i) q^{3} +(0.500000 + 0.866025i) q^{4} +1.00000i q^{5} -1.00000 q^{6} +(0.500000 + 0.866025i) q^{7} +1.00000i q^{8} +(0.500000 - 0.866025i) q^{9} +(-0.500000 + 0.866025i) q^{10} +(-0.866025 - 1.50000i) q^{11} +(-0.866025 - 0.500000i) q^{12} +1.00000i q^{14} +(-0.500000 - 0.866025i) q^{15} +(-0.500000 + 0.866025i) q^{16} +(0.866025 - 0.500000i) q^{18} +(-0.866025 + 0.500000i) q^{20} +(-0.866025 - 0.500000i) q^{21} -1.73205i q^{22} +(-0.500000 - 0.866025i) q^{24} -1.00000 q^{25} +1.00000i q^{27} +(-0.500000 + 0.866025i) q^{28} +1.73205 q^{29} -1.00000i q^{30} +(-0.500000 - 0.866025i) q^{31} +(-0.866025 + 0.500000i) q^{32} +(1.50000 + 0.866025i) q^{33} +(-0.866025 + 0.500000i) q^{35} +1.00000 q^{36} -1.00000 q^{40} +(-0.500000 - 0.866025i) q^{42} +(0.866025 - 1.50000i) q^{44} +(0.866025 + 0.500000i) q^{45} -1.00000i q^{48} +(-0.500000 + 0.866025i) q^{49} +(-0.866025 - 0.500000i) q^{50} +(0.866025 - 0.500000i) q^{53} +(-0.500000 + 0.866025i) q^{54} +(1.50000 - 0.866025i) q^{55} +(-0.866025 + 0.500000i) q^{56} +(1.50000 + 0.866025i) q^{58} +(0.866025 + 1.50000i) q^{59} +(0.500000 - 0.866025i) q^{60} -1.00000i q^{62} +1.00000 q^{63} -1.00000 q^{64} +(0.866025 + 1.50000i) q^{66} -1.00000 q^{70} +(0.866025 + 0.500000i) q^{72} +(0.866025 - 0.500000i) q^{75} +(0.866025 - 1.50000i) q^{77} +(0.500000 - 0.866025i) q^{79} +(-0.866025 - 0.500000i) q^{80} +(-0.500000 - 0.866025i) q^{81} -1.00000i q^{83} -1.00000i q^{84} +(-1.50000 + 0.866025i) q^{87} +(1.50000 - 0.866025i) q^{88} +(0.500000 + 0.866025i) q^{90} +(0.866025 + 0.500000i) q^{93} +(0.500000 - 0.866025i) q^{96} -1.73205i q^{97} +(-0.866025 + 0.500000i) q^{98} -1.73205 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 2 q^{4} - 4 q^{6} + 2 q^{7} + 2 q^{9}+O(q^{10}) $ 4 * q + 2 * q^4 - 4 * q^6 + 2 * q^7 + 2 * q^9	$ 4 q + 2 q^{4} - 4 q^{6} + 2 q^{7} + 2 q^{9} - 2 q^{10} - 2 q^{15} - 2 q^{16} - 2 q^{24} - 4 q^{25} - 2 q^{28} - 2 q^{31} + 6 q^{33} + 4 q^{36} - 4 q^{40} - 2 q^{42} - 2 q^{49} - 2 q^{54} + 6 q^{55} + 6 q^{58} + 2 q^{60} + 4 q^{63} - 4 q^{64} - 4 q^{70} + 2 q^{79} - 2 q^{81} - 6 q^{87} + 6 q^{88} + 2 q^{90} + 2 q^{96}+O(q^{100}) $ 4 * q + 2 * q^4 - 4 * q^6 + 2 * q^7 + 2 * q^9 - 2 * q^10 - 2 * q^15 - 2 * q^16 - 2 * q^24 - 4 * q^25 - 2 * q^28 - 2 * q^31 + 6 * q^33 + 4 * q^36 - 4 * q^40 - 2 * q^42 - 2 * q^49 - 2 * q^54 + 6 * q^55 + 6 * q^58 + 2 * q^60 + 4 * q^63 - 4 * q^64 - 4 * q^70 + 2 * q^79 - 2 * q^81 - 6 * q^87 + 6 * q^88 + 2 * q^90 + 2 * q^96

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/840\mathbb{Z}\right)^\times$.

$n$	$241$	$281$	$337$	$421$	$631$
$\chi(n)$	$e\left(\frac{1}{3}\right)$	$-1$	$-1$	$-1$	$1$

Coefficient data

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

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

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

$2$	0.866025	+	0.500000i	0.866025	+	0.500000i
$2$	0.866025	+	0.500000i	0.866025	+	0.500000i
$3$	−0.866025	+	0.500000i	−0.866025	+	0.500000i
$3$	−0.866025	+	0.500000i	−0.866025	+	0.500000i
$4$	0.500000	+	0.866025i	0.500000	+	0.866025i
$5$			1.00000i			1.00000i
$5$			1.00000i			1.00000i
$6$	−1.00000			−1.00000
$7$	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
$11$	−0.866025	−	1.50000i	−0.866025	−	1.50000i	−0.866025	−	0.500000i	$-0.833333\pi$
$11$	−0.866025	−	1.50000i	−0.866025	−	1.50000i		−	1.00000i	$-0.5\pi$
$12$	−0.866025	−	0.500000i	−0.866025	−	0.500000i
$13$		0			0			−	1.00000i	$-0.5\pi$
$13$		0			0				1.00000i	$0.5\pi$
$14$			1.00000i			1.00000i
$15$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$16$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$17$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$17$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$18$	0.866025	−	0.500000i	0.866025	−	0.500000i
$19$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$19$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$20$	−0.866025	+	0.500000i	−0.866025	+	0.500000i
$21$	−0.866025	−	0.500000i	−0.866025	−	0.500000i
$22$		−	1.73205i		−	1.73205i
$23$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$23$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$24$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$25$	−1.00000			−1.00000
$26$		0			0
$27$			1.00000i			1.00000i
$28$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$29$	1.73205			1.73205			0.866025	−	0.500000i	$-0.166667\pi$
$29$	1.73205			1.73205			0.866025	+	0.500000i	$0.166667\pi$
$30$		−	1.00000i		−	1.00000i
$31$	−0.500000	−	0.866025i	−0.500000	−	0.866025i	0.500000	−	0.866025i	$-0.333333\pi$
$31$	−0.500000	−	0.866025i	−0.500000	−	0.866025i	−1.00000			$\pi$
$32$	−0.866025	+	0.500000i	−0.866025	+	0.500000i
$33$	1.50000	+	0.866025i	1.50000	+	0.866025i
$34$		0			0
$35$	−0.866025	+	0.500000i	−0.866025	+	0.500000i
$36$	1.00000			1.00000
$37$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$37$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$38$		0			0
$39$		0			0
$40$	−1.00000			−1.00000
$41$		0			0		1.00000			$0$
$41$		0			0		−1.00000			$\pi$
$42$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$43$		0			0			−	1.00000i	$-0.5\pi$
$43$		0			0				1.00000i	$0.5\pi$
$44$	0.866025	−	1.50000i	0.866025	−	1.50000i
$45$	0.866025	+	0.500000i	0.866025	+	0.500000i
$46$		0			0
$47$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$47$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$48$		−	1.00000i		−	1.00000i
$49$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$50$	−0.866025	−	0.500000i	−0.866025	−	0.500000i
$51$		0			0
$52$		0			0
$53$	0.866025	−	0.500000i	0.866025	−	0.500000i		−	1.00000i	$-0.5\pi$
$53$	0.866025	−	0.500000i	0.866025	−	0.500000i	0.866025	+	0.500000i	$0.166667\pi$
$54$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$55$	1.50000	−	0.866025i	1.50000	−	0.866025i
$56$	−0.866025	+	0.500000i	−0.866025	+	0.500000i
$57$		0			0
$58$	1.50000	+	0.866025i	1.50000	+	0.866025i
$59$	0.866025	+	1.50000i	0.866025	+	1.50000i	0.866025	+	0.500000i	$0.166667\pi$
$59$	0.866025	+	1.50000i	0.866025	+	1.50000i			1.00000i	$0.5\pi$
$60$	0.500000	−	0.866025i	0.500000	−	0.866025i
$61$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$61$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$62$		−	1.00000i		−	1.00000i
$63$	1.00000			1.00000
$64$	−1.00000			−1.00000
$65$		0			0
$66$	0.866025	+	1.50000i	0.866025	+	1.50000i
$67$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$67$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$68$		0			0
$69$		0			0
$70$	−1.00000			−1.00000
$71$		0			0		1.00000			$0$
$71$		0			0		−1.00000			$\pi$
$72$	0.866025	+	0.500000i	0.866025	+	0.500000i
$73$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$73$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$74$		0			0
$75$	0.866025	−	0.500000i	0.866025	−	0.500000i
$76$		0			0
$77$	0.866025	−	1.50000i	0.866025	−	1.50000i
$78$		0			0
$79$	0.500000	−	0.866025i	0.500000	−	0.866025i	−0.500000	−	0.866025i	$-0.666667\pi$
$79$	0.500000	−	0.866025i	0.500000	−	0.866025i	1.00000			$0$
$80$	−0.866025	−	0.500000i	−0.866025	−	0.500000i
$81$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$82$		0			0
$83$		−	1.00000i		−	1.00000i	−0.866025	−	0.500000i	$-0.833333\pi$
$83$		−	1.00000i		−	1.00000i	0.866025	−	0.500000i	$-0.166667\pi$
$84$		−	1.00000i		−	1.00000i
$85$		0			0
$86$		0			0
$87$	−1.50000	+	0.866025i	−1.50000	+	0.866025i
$88$	1.50000	−	0.866025i	1.50000	−	0.866025i
$89$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$89$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$90$	0.500000	+	0.866025i	0.500000	+	0.866025i
$91$		0			0
$92$		0			0
$93$	0.866025	+	0.500000i	0.866025	+	0.500000i
$94$		0			0
$95$		0			0
$96$	0.500000	−	0.866025i	0.500000	−	0.866025i
$97$		−	1.73205i		−	1.73205i	−0.500000	−	0.866025i	$-0.666667\pi$
$97$		−	1.73205i		−	1.73205i	0.500000	−	0.866025i	$-0.333333\pi$
$98$	−0.866025	+	0.500000i	−0.866025	+	0.500000i
$99$	−1.73205			−1.73205
$100$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$101$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$101$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$102$		0			0
$103$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$103$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$104$		0			0
$105$	0.500000	−	0.866025i	0.500000	−	0.866025i
$106$	1.00000			1.00000
$107$	−0.866025	−	0.500000i	−0.866025	−	0.500000i		−	1.00000i	$-0.5\pi$
$107$	−0.866025	−	0.500000i	−0.866025	−	0.500000i	−0.866025	+	0.500000i	$0.833333\pi$
$108$	−0.866025	+	0.500000i	−0.866025	+	0.500000i
$109$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$109$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$110$	1.73205			1.73205
$111$		0			0
$112$	−1.00000			−1.00000
$113$		0			0			−	1.00000i	$-0.5\pi$
$113$		0			0				1.00000i	$0.5\pi$
$114$		0			0
$115$		0			0
$116$	0.866025	+	1.50000i	0.866025	+	1.50000i
$117$		0			0
$118$			1.73205i			1.73205i
$119$		0			0
$120$	0.866025	−	0.500000i	0.866025	−	0.500000i
$121$	−1.00000	+	1.73205i	−1.00000	+	1.73205i
$122$		0			0
$123$		0			0
$124$	0.500000	−	0.866025i	0.500000	−	0.866025i
$125$		−	1.00000i		−	1.00000i
$126$	0.866025	+	0.500000i	0.866025	+	0.500000i
$127$			1.73205i			1.73205i	0.500000	+	0.866025i	$0.333333\pi$
$127$			1.73205i			1.73205i	−0.500000	+	0.866025i	$0.666667\pi$
$128$	−0.866025	−	0.500000i	−0.866025	−	0.500000i
$129$		0			0
$130$		0			0
$131$	0.866025	−	1.50000i	0.866025	−	1.50000i		−	1.00000i	$-0.5\pi$
$131$	0.866025	−	1.50000i	0.866025	−	1.50000i	0.866025	−	0.500000i	$-0.166667\pi$
$132$			1.73205i			1.73205i
$133$		0			0
$134$		0			0
$135$	−1.00000			−1.00000
$136$		0			0
$137$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$137$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$138$		0			0
$139$		0			0		1.00000			$0$
$139$		0			0		−1.00000			$\pi$
$140$	−0.866025	−	0.500000i	−0.866025	−	0.500000i
$141$		0			0
$142$		0			0
$143$		0			0
$144$	0.500000	+	0.866025i	0.500000	+	0.866025i
$145$			1.73205i			1.73205i
$146$		0			0
$147$		−	1.00000i		−	1.00000i
$148$		0			0
$149$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$149$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$150$	1.00000			1.00000
$151$	0.500000	+	0.866025i	0.500000	+	0.866025i	1.00000			$0$
$151$	0.500000	+	0.866025i	0.500000	+	0.866025i	−0.500000	+	0.866025i	$0.666667\pi$
$152$		0			0
$153$		0			0
$154$	1.50000	−	0.866025i	1.50000	−	0.866025i
$155$	0.866025	−	0.500000i	0.866025	−	0.500000i
$156$		0			0
$157$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$157$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$158$	0.866025	−	0.500000i	0.866025	−	0.500000i
$159$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$160$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$161$		0			0
$162$		−	1.00000i		−	1.00000i
$163$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$163$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$164$		0			0
$165$	−0.866025	+	1.50000i	−0.866025	+	1.50000i
$166$	0.500000	−	0.866025i	0.500000	−	0.866025i
$167$		0			0			−	1.00000i	$-0.5\pi$
$167$		0			0				1.00000i	$0.5\pi$
$168$	0.500000	−	0.866025i	0.500000	−	0.866025i
$169$	−1.00000			−1.00000
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−1.73205	−	1.00000i	−1.73205	−	1.00000i	−0.866025	−	0.500000i	$-0.833333\pi$
$173$	−1.73205	−	1.00000i	−1.73205	−	1.00000i	−0.866025	−	0.500000i	$-0.833333\pi$
$174$	−1.73205			−1.73205
$175$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$176$	1.73205			1.73205
$177$	−1.50000	−	0.866025i	−1.50000	−	0.866025i
$178$		0			0
$179$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$179$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$180$			1.00000i			1.00000i
$181$		0			0		1.00000			$0$
$181$		0			0		−1.00000			$\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$		0			0
$186$	0.500000	+	0.866025i	0.500000	+	0.866025i
$187$		0			0
$188$		0			0
$189$	−0.866025	+	0.500000i	−0.866025	+	0.500000i
$190$		0			0
$191$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$191$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$192$	0.866025	−	0.500000i	0.866025	−	0.500000i
$193$	−1.50000	+	0.866025i	−1.50000	+	0.866025i	−0.500000	+	0.866025i	$0.666667\pi$
$193$	−1.50000	+	0.866025i	−1.50000	+	0.866025i	−1.00000			$\pi$
$194$	0.866025	−	1.50000i	0.866025	−	1.50000i
$195$		0			0
$196$	−1.00000			−1.00000
$197$			2.00000i			2.00000i			1.00000i	$0.5\pi$
$197$			2.00000i			2.00000i			1.00000i	$0.5\pi$
$198$	−1.50000	−	0.866025i	−1.50000	−	0.866025i
$199$	1.00000	+	1.73205i	1.00000	+	1.73205i	0.500000	+	0.866025i	$0.333333\pi$
$199$	1.00000	+	1.73205i	1.00000	+	1.73205i	0.500000	+	0.866025i	$0.333333\pi$
$200$		−	1.00000i		−	1.00000i
$201$		0			0
$202$		0			0
$203$	0.866025	+	1.50000i	0.866025	+	1.50000i
$204$		0			0
$205$		0			0
$206$		0			0
$207$		0			0
$208$		0			0
$209$		0			0
$210$	0.866025	−	0.500000i	0.866025	−	0.500000i
$211$		0			0		1.00000			$0$
$211$		0			0		−1.00000			$\pi$
$212$	0.866025	+	0.500000i	0.866025	+	0.500000i
$213$		0			0
$214$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$215$		0			0
$216$	−1.00000			−1.00000
$217$	0.500000	−	0.866025i	0.500000	−	0.866025i
$218$		0			0
$219$		0			0
$220$	1.50000	+	0.866025i	1.50000	+	0.866025i
$221$		0			0
$222$		0			0
$223$		−	1.73205i		−	1.73205i	−0.500000	−	0.866025i	$-0.666667\pi$
$223$		−	1.73205i		−	1.73205i	0.500000	−	0.866025i	$-0.333333\pi$
$224$	−0.866025	−	0.500000i	−0.866025	−	0.500000i
$225$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$226$		0			0
$227$	−0.866025	+	0.500000i	−0.866025	+	0.500000i	−0.866025	−	0.500000i	$-0.833333\pi$
$227$	−0.866025	+	0.500000i	−0.866025	+	0.500000i			1.00000i	$0.5\pi$
$228$		0			0
$229$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$229$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$230$		0			0
$231$			1.73205i			1.73205i
$232$			1.73205i			1.73205i
$233$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$233$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$234$		0			0
$235$		0			0
$236$	−0.866025	+	1.50000i	−0.866025	+	1.50000i
$237$			1.00000i			1.00000i
$238$		0			0
$239$		0			0		1.00000			$0$
$239$		0			0		−1.00000			$\pi$
$240$	1.00000			1.00000
$241$	−0.500000	−	0.866025i	−0.500000	−	0.866025i	0.500000	−	0.866025i	$-0.333333\pi$
$241$	−0.500000	−	0.866025i	−0.500000	−	0.866025i	−1.00000			$\pi$
$242$	−1.73205	+	1.00000i	−1.73205	+	1.00000i
$243$	0.866025	+	0.500000i	0.866025	+	0.500000i
$244$		0			0
$245$	−0.866025	−	0.500000i	−0.866025	−	0.500000i
$246$		0			0
$247$		0			0
$248$	0.866025	−	0.500000i	0.866025	−	0.500000i
$249$	0.500000	+	0.866025i	0.500000	+	0.866025i
$250$	0.500000	−	0.866025i	0.500000	−	0.866025i
$251$	−1.73205			−1.73205			−0.866025	−	0.500000i	$-0.833333\pi$
$251$	−1.73205			−1.73205			−0.866025	+	0.500000i	$0.833333\pi$
$252$	0.500000	+	0.866025i	0.500000	+	0.866025i
$253$		0			0
$254$	−0.866025	+	1.50000i	−0.866025	+	1.50000i
$255$		0			0
$256$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$257$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$257$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$	0.866025	−	1.50000i	0.866025	−	1.50000i
$262$	1.50000	−	0.866025i	1.50000	−	0.866025i
$263$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$263$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$264$	−0.866025	+	1.50000i	−0.866025	+	1.50000i
$265$	0.500000	+	0.866025i	0.500000	+	0.866025i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−0.866025	−	1.50000i	−0.866025	−	1.50000i	−0.866025	−	0.500000i	$-0.833333\pi$
$269$	−0.866025	−	1.50000i	−0.866025	−	1.50000i		−	1.00000i	$-0.5\pi$
$270$	−0.866025	−	0.500000i	−0.866025	−	0.500000i
$271$	0.500000	−	0.866025i	0.500000	−	0.866025i	−0.500000	−	0.866025i	$-0.666667\pi$
$271$	0.500000	−	0.866025i	0.500000	−	0.866025i	1.00000			$0$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	0.866025	+	1.50000i	0.866025	+	1.50000i
$276$		0			0
$277$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$277$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$278$		0			0
$279$	−1.00000			−1.00000
$280$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$281$		0			0		1.00000			$0$
$281$		0			0		−1.00000			$\pi$
$282$		0			0
$283$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$283$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$			1.00000i			1.00000i
$289$	0.500000	−	0.866025i	0.500000	−	0.866025i
$290$	−0.866025	+	1.50000i	−0.866025	+	1.50000i
$291$	0.866025	+	1.50000i	0.866025	+	1.50000i
$292$		0			0
$293$			1.00000i			1.00000i	0.866025	+	0.500000i	$0.166667\pi$
$293$			1.00000i			1.00000i	−0.866025	+	0.500000i	$0.833333\pi$
$294$	0.500000	−	0.866025i	0.500000	−	0.866025i
$295$	−1.50000	+	0.866025i	−1.50000	+	0.866025i
$296$		0			0
$297$	1.50000	−	0.866025i	1.50000	−	0.866025i
$298$		0			0
$299$		0			0
$300$	0.866025	+	0.500000i	0.866025	+	0.500000i
$301$		0			0
$302$			1.00000i			1.00000i
$303$		0			0
$304$		0			0
$305$		0			0
$306$		0			0
$307$		0			0			−	1.00000i	$-0.5\pi$
$307$		0			0				1.00000i	$0.5\pi$
$308$	1.73205			1.73205
$309$		0			0
$310$	1.00000			1.00000
$311$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$311$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$312$		0			0
$313$	−1.50000	−	0.866025i	−1.50000	−	0.866025i	−0.500000	−	0.866025i	$-0.666667\pi$
$313$	−1.50000	−	0.866025i	−1.50000	−	0.866025i	−1.00000			$\pi$
$314$		0			0
$315$			1.00000i			1.00000i
$316$	1.00000			1.00000
$317$	−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	$0.833333\pi$
$318$	−0.866025	+	0.500000i	−0.866025	+	0.500000i
$319$	−1.50000	−	2.59808i	−1.50000	−	2.59808i
$320$		−	1.00000i		−	1.00000i
$321$	1.00000			1.00000
$322$		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			0
$329$		0			0
$330$	−1.50000	+	0.866025i	−1.50000	+	0.866025i
$331$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$331$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$332$	0.866025	−	0.500000i	0.866025	−	0.500000i
$333$		0			0
$334$		0			0
$335$		0			0
$336$	0.866025	−	0.500000i	0.866025	−	0.500000i
$337$		−	1.73205i		−	1.73205i	−0.500000	−	0.866025i	$-0.666667\pi$
$337$		−	1.73205i		−	1.73205i	0.500000	−	0.866025i	$-0.333333\pi$
$338$	−0.866025	−	0.500000i	−0.866025	−	0.500000i
$339$		0			0
$340$		0			0
$341$	−0.866025	+	1.50000i	−0.866025	+	1.50000i
$342$		0			0
$343$	−1.00000			−1.00000
$344$		0			0
$345$		0			0
$346$	−1.00000	−	1.73205i	−1.00000	−	1.73205i
$347$	1.73205	−	1.00000i	1.73205	−	1.00000i	0.866025	−	0.500000i	$-0.166667\pi$
$347$	1.73205	−	1.00000i	1.73205	−	1.00000i	0.866025	−	0.500000i	$-0.166667\pi$
$348$	−1.50000	−	0.866025i	−1.50000	−	0.866025i
$349$		0			0		1.00000			$0$
$349$		0			0		−1.00000			$\pi$
$350$		−	1.00000i		−	1.00000i
$351$		0			0
$352$	1.50000	+	0.866025i	1.50000	+	0.866025i
$353$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$353$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$354$	−0.866025	−	1.50000i	−0.866025	−	1.50000i
$355$		0			0
$356$		0			0
$357$		0			0
$358$		0			0
$359$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$359$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$360$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$361$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$362$		0			0
$363$		−	2.00000i		−	2.00000i
$364$		0			0
$365$		0			0
$366$		0			0
$367$	1.50000	−	0.866025i	1.50000	−	0.866025i	0.500000	−	0.866025i	$-0.333333\pi$
$367$	1.50000	−	0.866025i	1.50000	−	0.866025i	1.00000			$0$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	0.866025	+	0.500000i	0.866025	+	0.500000i
$372$			1.00000i			1.00000i
$373$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$373$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$374$		0			0
$375$	0.500000	+	0.866025i	0.500000	+	0.866025i
$376$		0			0
$377$		0			0
$378$	−1.00000			−1.00000
$379$		0			0		1.00000			$0$
$379$		0			0		−1.00000			$\pi$
$380$		0			0
$381$	−0.866025	−	1.50000i	−0.866025	−	1.50000i
$382$		0			0
$383$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$383$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$384$	1.00000			1.00000
$385$	1.50000	+	0.866025i	1.50000	+	0.866025i
$386$	−1.73205			−1.73205
$387$		0			0
$388$	1.50000	−	0.866025i	1.50000	−	0.866025i
$389$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$389$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$390$		0			0
$391$		0			0
$392$	−0.866025	−	0.500000i	−0.866025	−	0.500000i
$393$			1.73205i			1.73205i
$394$	−1.00000	+	1.73205i	−1.00000	+	1.73205i
$395$	0.866025	+	0.500000i	0.866025	+	0.500000i
$396$	−0.866025	−	1.50000i	−0.866025	−	1.50000i
$397$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$397$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$398$			2.00000i			2.00000i
$399$		0			0
$400$	0.500000	−	0.866025i	0.500000	−	0.866025i
$401$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$401$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$402$		0			0
$403$		0			0
$404$		0			0
$405$	0.866025	−	0.500000i	0.866025	−	0.500000i
$406$			1.73205i			1.73205i
$407$		0			0
$408$		0			0
$409$	0.500000	+	0.866025i	0.500000	+	0.866025i	1.00000			$0$
$409$	0.500000	+	0.866025i	0.500000	+	0.866025i	−0.500000	+	0.866025i	$0.666667\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	−0.866025	+	1.50000i	−0.866025	+	1.50000i
$414$		0			0
$415$	1.00000			1.00000
$416$		0			0
$417$		0			0
$418$		0			0
$419$		0			0			−	1.00000i	$-0.5\pi$
$419$		0			0				1.00000i	$0.5\pi$
$420$	1.00000			1.00000
$421$		0			0		1.00000			$0$
$421$		0			0		−1.00000			$\pi$
$422$		0			0
$423$		0			0
$424$	0.500000	+	0.866025i	0.500000	+	0.866025i
$425$		0			0
$426$		0			0
$427$		0			0
$428$		−	1.00000i		−	1.00000i
$429$		0			0
$430$		0			0
$431$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$431$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$432$	−0.866025	−	0.500000i	−0.866025	−	0.500000i
$433$		0			0		1.00000			$0$
$433$		0			0		−1.00000			$\pi$
$434$	0.866025	−	0.500000i	0.866025	−	0.500000i
$435$	−0.866025	−	1.50000i	−0.866025	−	1.50000i
$436$		0			0
$437$		0			0
$438$		0			0
$439$	−0.500000	+	0.866025i	−0.500000	+	0.866025i	0.500000	+	0.866025i	$0.333333\pi$
$439$	−0.500000	+	0.866025i	−0.500000	+	0.866025i	−1.00000			$\pi$
$440$	0.866025	+	1.50000i	0.866025	+	1.50000i
$441$	0.500000	+	0.866025i	0.500000	+	0.866025i
$442$		0			0
$443$	−0.866025	−	0.500000i	−0.866025	−	0.500000i		−	1.00000i	$-0.5\pi$
$443$	−0.866025	−	0.500000i	−0.866025	−	0.500000i	−0.866025	+	0.500000i	$0.833333\pi$
$444$		0			0
$445$		0			0
$446$	0.866025	−	1.50000i	0.866025	−	1.50000i
$447$		0			0
$448$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$449$		0			0		1.00000			$0$
$449$		0			0		−1.00000			$\pi$
$450$	−0.866025	+	0.500000i	−0.866025	+	0.500000i
$451$		0			0
$452$		0			0
$453$	−0.866025	−	0.500000i	−0.866025	−	0.500000i
$454$	−1.00000			−1.00000
$455$		0			0
$456$		0			0
$457$	1.50000	+	0.866025i	1.50000	+	0.866025i	1.00000			$0$
$457$	1.50000	+	0.866025i	1.50000	+	0.866025i	0.500000	+	0.866025i	$0.333333\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$		0			0			−	1.00000i	$-0.5\pi$
$461$		0			0				1.00000i	$0.5\pi$
$462$	−0.866025	+	1.50000i	−0.866025	+	1.50000i
$463$		0			0		1.00000			$0$
$463$		0			0		−1.00000			$\pi$
$464$	−0.866025	+	1.50000i	−0.866025	+	1.50000i
$465$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$466$		0			0
$467$	1.73205	+	1.00000i	1.73205	+	1.00000i	0.866025	+	0.500000i	$0.166667\pi$
$467$	1.73205	+	1.00000i	1.73205	+	1.00000i	0.866025	+	0.500000i	$0.166667\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$		0			0
$472$	−1.50000	+	0.866025i	−1.50000	+	0.866025i
$473$		0			0
$474$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$475$		0			0
$476$		0			0
$477$		−	1.00000i		−	1.00000i
$478$		0			0
$479$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$479$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$480$	0.866025	+	0.500000i	0.866025	+	0.500000i
$481$		0			0
$482$		−	1.00000i		−	1.00000i
$483$		0			0
$484$	−2.00000			−2.00000
$485$	1.73205			1.73205
$486$	0.500000	+	0.866025i	0.500000	+	0.866025i
$487$	−1.50000	+	0.866025i	−1.50000	+	0.866025i	−0.500000	+	0.866025i	$0.666667\pi$
$487$	−1.50000	+	0.866025i	−1.50000	+	0.866025i	−1.00000			$\pi$
$488$		0			0
$489$		0			0
$490$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$491$	−1.73205			−1.73205			−0.866025	−	0.500000i	$-0.833333\pi$
$491$	−1.73205			−1.73205			−0.866025	+	0.500000i	$0.833333\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$		−	1.73205i		−	1.73205i
$496$	1.00000			1.00000
$497$		0			0
$498$			1.00000i			1.00000i
$499$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$499$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$500$	0.866025	−	0.500000i	0.866025	−	0.500000i
$501$		0			0
$502$	−1.50000	−	0.866025i	−1.50000	−	0.866025i
$503$		0			0			−	1.00000i	$-0.5\pi$
$503$		0			0				1.00000i	$0.5\pi$
$504$			1.00000i			1.00000i
$505$		0			0
$506$		0			0
$507$	0.866025	−	0.500000i	0.866025	−	0.500000i
$508$	−1.50000	+	0.866025i	−1.50000	+	0.866025i
$509$	−0.866025	+	1.50000i	−0.866025	+	1.50000i			1.00000i	$0.5\pi$
$509$	−0.866025	+	1.50000i	−0.866025	+	1.50000i	−0.866025	+	0.500000i	$0.833333\pi$
$510$		0			0
$511$		0			0
$512$		−	1.00000i		−	1.00000i
$513$		0			0
$514$		0			0
$515$		0			0
$516$		0			0
$517$		0			0
$518$		0			0
$519$	2.00000			2.00000
$520$		0			0
$521$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$521$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$522$	1.50000	−	0.866025i	1.50000	−	0.866025i
$523$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$523$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$524$	1.73205			1.73205
$525$	0.866025	+	0.500000i	0.866025	+	0.500000i
$526$		0			0
$527$		0			0
$528$	−1.50000	+	0.866025i	−1.50000	+	0.866025i
$529$	0.500000	+	0.866025i	0.500000	+	0.866025i
$530$			1.00000i			1.00000i
$531$	1.73205			1.73205
$532$		0			0
$533$		0			0
$534$		0			0
$535$	0.500000	−	0.866025i	0.500000	−	0.866025i
$536$		0			0
$537$		0			0
$538$		−	1.73205i		−	1.73205i
$539$	1.73205			1.73205
$540$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$541$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$541$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$542$	0.866025	−	0.500000i	0.866025	−	0.500000i
$543$		0			0
$544$		0			0
$545$		0			0
$546$		0			0
$547$		0			0			−	1.00000i	$-0.5\pi$
$547$		0			0				1.00000i	$0.5\pi$
$548$		0			0
$549$		0			0
$550$			1.73205i			1.73205i
$551$		0			0
$552$		0			0
$553$	1.00000			1.00000
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−0.866025	+	0.500000i	−0.866025	+	0.500000i	−0.866025	−	0.500000i	$-0.833333\pi$
$557$	−0.866025	+	0.500000i	−0.866025	+	0.500000i			1.00000i	$0.5\pi$
$558$	−0.866025	−	0.500000i	−0.866025	−	0.500000i
$559$		0			0
$560$		−	1.00000i		−	1.00000i
$561$		0			0
$562$		0			0
$563$	0.866025	−	0.500000i	0.866025	−	0.500000i		−	1.00000i	$-0.5\pi$
$563$	0.866025	−	0.500000i	0.866025	−	0.500000i	0.866025	+	0.500000i	$0.166667\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$	0.500000	−	0.866025i	0.500000	−	0.866025i
$568$		0			0
$569$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$569$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$570$		0			0
$571$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$571$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$577$	−1.50000	+	0.866025i	−1.50000	+	0.866025i	−0.500000	+	0.866025i	$0.666667\pi$
$577$	−1.50000	+	0.866025i	−1.50000	+	0.866025i	−1.00000			$\pi$
$578$	0.866025	−	0.500000i	0.866025	−	0.500000i
$579$	0.866025	−	1.50000i	0.866025	−	1.50000i
$580$	−1.50000	+	0.866025i	−1.50000	+	0.866025i
$581$	0.866025	−	0.500000i	0.866025	−	0.500000i
$582$			1.73205i			1.73205i
$583$	−1.50000	−	0.866025i	−1.50000	−	0.866025i
$584$		0			0
$585$		0			0
$586$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$587$			1.00000i			1.00000i	0.866025	+	0.500000i	$0.166667\pi$
$587$			1.00000i			1.00000i	−0.866025	+	0.500000i	$0.833333\pi$
$588$	0.866025	−	0.500000i	0.866025	−	0.500000i
$589$		0			0
$590$	−1.73205			−1.73205
$591$	−1.00000	−	1.73205i	−1.00000	−	1.73205i
$592$		0			0
$593$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$593$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$594$	1.73205			1.73205
$595$		0			0
$596$		0			0
$597$	−1.73205	−	1.00000i	−1.73205	−	1.00000i
$598$		0			0
$599$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$599$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$600$	0.500000	+	0.866025i	0.500000	+	0.866025i
$601$	−1.00000			−1.00000			−0.500000	−	0.866025i	$-0.666667\pi$
$601$	−1.00000			−1.00000			−0.500000	+	0.866025i	$0.666667\pi$
$602$		0			0
$603$		0			0
$604$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$605$	−1.73205	−	1.00000i	−1.73205	−	1.00000i
$606$		0			0
$607$	1.50000	+	0.866025i	1.50000	+	0.866025i	1.00000			$0$
$607$	1.50000	+	0.866025i	1.50000	+	0.866025i	0.500000	+	0.866025i	$0.333333\pi$
$608$		0			0
$609$	−1.50000	−	0.866025i	−1.50000	−	0.866025i
$610$		0			0
$611$		0			0
$612$		0			0
$613$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$613$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$614$		0			0
$615$		0			0
$616$	1.50000	+	0.866025i	1.50000	+	0.866025i
$617$		0			0			−	1.00000i	$-0.5\pi$
$617$		0			0				1.00000i	$0.5\pi$
$618$		0			0
$619$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$619$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$620$	0.866025	+	0.500000i	0.866025	+	0.500000i
$621$		0			0
$622$		0			0
$623$		0			0
$624$		0			0
$625$	1.00000			1.00000
$626$	−0.866025	−	1.50000i	−0.866025	−	1.50000i
$627$		0			0
$628$		0			0
$629$		0			0
$630$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$631$	1.00000			1.00000			0.500000	−	0.866025i	$-0.333333\pi$
$631$	1.00000			1.00000			0.500000	+	0.866025i	$0.333333\pi$
$632$	0.866025	+	0.500000i	0.866025	+	0.500000i
$633$		0			0
$634$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$635$	−1.73205			−1.73205
$636$	−1.00000			−1.00000
$637$		0			0
$638$		−	3.00000i		−	3.00000i
$639$		0			0
$640$	0.500000	−	0.866025i	0.500000	−	0.866025i
$641$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$641$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$642$	0.866025	+	0.500000i	0.866025	+	0.500000i
$643$		0			0			−	1.00000i	$-0.5\pi$
$643$		0			0				1.00000i	$0.5\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$647$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$648$	0.866025	−	0.500000i	0.866025	−	0.500000i
$649$	1.50000	−	2.59808i	1.50000	−	2.59808i
$650$		0			0
$651$			1.00000i			1.00000i
$652$		0			0
$653$	0.866025	+	0.500000i	0.866025	+	0.500000i	0.866025	−	0.500000i	$-0.166667\pi$
$653$	0.866025	+	0.500000i	0.866025	+	0.500000i			1.00000i	$0.5\pi$
$654$		0			0
$655$	1.50000	+	0.866025i	1.50000	+	0.866025i
$656$		0			0
$657$		0			0
$658$		0			0
$659$		0			0			−	1.00000i	$-0.5\pi$
$659$		0			0				1.00000i	$0.5\pi$
$660$	−1.73205			−1.73205
$661$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$661$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$662$		0			0
$663$		0			0
$664$	1.00000			1.00000
$665$		0			0
$666$		0			0
$667$		0			0
$668$		0			0
$669$	0.866025	+	1.50000i	0.866025	+	1.50000i
$670$		0			0
$671$		0			0
$672$	1.00000			1.00000
$673$		−	1.73205i		−	1.73205i	−0.500000	−	0.866025i	$-0.666667\pi$
$673$		−	1.73205i		−	1.73205i	0.500000	−	0.866025i	$-0.333333\pi$
$674$	0.866025	−	1.50000i	0.866025	−	1.50000i
$675$		−	1.00000i		−	1.00000i
$676$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$677$	−0.866025	−	0.500000i	−0.866025	−	0.500000i		−	1.00000i	$-0.5\pi$
$677$	−0.866025	−	0.500000i	−0.866025	−	0.500000i	−0.866025	+	0.500000i	$0.833333\pi$
$678$		0			0
$679$	1.50000	−	0.866025i	1.50000	−	0.866025i
$680$		0			0
$681$	0.500000	−	0.866025i	0.500000	−	0.866025i
$682$	−1.50000	+	0.866025i	−1.50000	+	0.866025i
$683$	−0.866025	+	0.500000i	−0.866025	+	0.500000i	−0.866025	−	0.500000i	$-0.833333\pi$
$683$	−0.866025	+	0.500000i	−0.866025	+	0.500000i			1.00000i	$0.5\pi$
$684$		0			0
$685$		0			0
$686$	−0.866025	−	0.500000i	−0.866025	−	0.500000i
$687$		0			0
$688$		0			0
$689$		0			0
$690$		0			0
$691$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$691$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$692$		−	2.00000i		−	2.00000i
$693$	−0.866025	−	1.50000i	−0.866025	−	1.50000i
$694$	2.00000			2.00000
$695$		0			0
$696$	−0.866025	−	1.50000i	−0.866025	−	1.50000i
$697$		0			0
$698$		0			0
$699$		0			0
$700$	0.500000	−	0.866025i	0.500000	−	0.866025i
$701$	1.73205			1.73205			0.866025	−	0.500000i	$-0.166667\pi$
$701$	1.73205			1.73205			0.866025	+	0.500000i	$0.166667\pi$
$702$		0			0
$703$		0			0
$704$	0.866025	+	1.50000i	0.866025	+	1.50000i
$705$		0			0
$706$		0			0
$707$		0			0
$708$		−	1.73205i		−	1.73205i
$709$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$709$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$710$		0			0
$711$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$712$		0			0
$713$		0			0
$714$		0			0
$715$		0			0
$716$		0			0
$717$		0			0
$718$		0			0
$719$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$719$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$720$	−0.866025	+	0.500000i	−0.866025	+	0.500000i
$721$		0			0
$722$		−	1.00000i		−	1.00000i
$723$	0.866025	+	0.500000i	0.866025	+	0.500000i
$724$		0			0
$725$	−1.73205			−1.73205
$726$	1.00000	−	1.73205i	1.00000	−	1.73205i
$727$			1.73205i			1.73205i	0.500000	+	0.866025i	$0.333333\pi$
$727$			1.73205i			1.73205i	−0.500000	+	0.866025i	$0.666667\pi$
$728$		0			0
$729$	−1.00000			−1.00000
$730$		0			0
$731$		0			0
$732$		0			0
$733$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$733$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$734$	1.73205			1.73205
$735$	1.00000			1.00000
$736$		0			0
$737$		0			0
$738$		0			0
$739$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$739$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$740$		0			0
$741$		0			0
$742$	0.500000	+	0.866025i	0.500000	+	0.866025i
$743$		0			0			−	1.00000i	$-0.5\pi$
$743$		0			0				1.00000i	$0.5\pi$
$744$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$745$		0			0
$746$		0			0
$747$	−0.866025	−	0.500000i	−0.866025	−	0.500000i
$748$		0			0
$749$		−	1.00000i		−	1.00000i
$750$			1.00000i			1.00000i
$751$	−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			0
$753$	1.50000	−	0.866025i	1.50000	−	0.866025i
$754$		0			0
$755$	−0.866025	+	0.500000i	−0.866025	+	0.500000i
$756$	−0.866025	−	0.500000i	−0.866025	−	0.500000i
$757$		0			0			−	1.00000i	$-0.5\pi$
$757$		0			0				1.00000i	$0.5\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$761$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$762$		−	1.73205i		−	1.73205i
$763$		0			0
$764$		0			0
$765$		0			0
$766$		0			0
$767$		0			0
$768$	0.866025	+	0.500000i	0.866025	+	0.500000i
$769$	1.00000			1.00000			0.500000	−	0.866025i	$-0.333333\pi$
$769$	1.00000			1.00000			0.500000	+	0.866025i	$0.333333\pi$
$770$	0.866025	+	1.50000i	0.866025	+	1.50000i
$771$		0			0
$772$	−1.50000	−	0.866025i	−1.50000	−	0.866025i
$773$	1.73205	−	1.00000i	1.73205	−	1.00000i	0.866025	−	0.500000i	$-0.166667\pi$
$773$	1.73205	−	1.00000i	1.73205	−	1.00000i	0.866025	−	0.500000i	$-0.166667\pi$
$774$		0			0
$775$	0.500000	+	0.866025i	0.500000	+	0.866025i
$776$	1.73205			1.73205
$777$		0			0
$778$		0			0
$779$		0			0
$780$		0			0
$781$		0			0
$782$		0			0
$783$			1.73205i			1.73205i
$784$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$785$		0			0
$786$	−0.866025	+	1.50000i	−0.866025	+	1.50000i
$787$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$787$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$788$	−1.73205	+	1.00000i	−1.73205	+	1.00000i
$789$		0			0
$790$	0.500000	+	0.866025i	0.500000	+	0.866025i
$791$		0			0
$792$		−	1.73205i		−	1.73205i
$793$		0			0
$794$		0			0
$795$	−0.866025	−	0.500000i	−0.866025	−	0.500000i
$796$	−1.00000	+	1.73205i	−1.00000	+	1.73205i
$797$			1.00000i			1.00000i	0.866025	+	0.500000i	$0.166667\pi$
$797$			1.00000i			1.00000i	−0.866025	+	0.500000i	$0.833333\pi$
$798$		0			0
$799$		0			0
$800$	0.866025	−	0.500000i	0.866025	−	0.500000i
$801$		0			0
$802$		0			0
$803$		0			0
$804$		0			0
$805$		0			0
$806$		0			0
$807$	1.50000	+	0.866025i	1.50000	+	0.866025i
$808$		0			0
$809$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$809$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$810$	1.00000			1.00000
$811$		0			0		1.00000			$0$
$811$		0			0		−1.00000			$\pi$
$812$	−0.866025	+	1.50000i	−0.866025	+	1.50000i
$813$			1.00000i			1.00000i
$814$		0			0
$815$		0			0
$816$		0			0
$817$		0			0
$818$			1.00000i			1.00000i
$819$		0			0
$820$		0			0
$821$	−0.866025	+	1.50000i	−0.866025	+	1.50000i			1.00000i	$0.5\pi$
$821$	−0.866025	+	1.50000i	−0.866025	+	1.50000i	−0.866025	+	0.500000i	$0.833333\pi$
$822$		0			0
$823$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$823$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$824$		0			0
$825$	−1.50000	−	0.866025i	−1.50000	−	0.866025i
$826$	−1.50000	+	0.866025i	−1.50000	+	0.866025i
$827$		−	1.00000i		−	1.00000i	−0.866025	−	0.500000i	$-0.833333\pi$
$827$		−	1.00000i		−	1.00000i	0.866025	−	0.500000i	$-0.166667\pi$
$828$		0			0
$829$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$829$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$830$	0.866025	+	0.500000i	0.866025	+	0.500000i
$831$		0			0
$832$		0			0
$833$		0			0
$834$		0			0
$835$		0			0
$836$		0			0
$837$	0.866025	−	0.500000i	0.866025	−	0.500000i
$838$		0			0
$839$		0			0		1.00000			$0$
$839$		0			0		−1.00000			$\pi$
$840$	0.866025	+	0.500000i	0.866025	+	0.500000i
$841$	2.00000			2.00000
$842$		0			0
$843$		0			0
$844$		0			0
$845$		−	1.00000i		−	1.00000i
$846$		0			0
$847$	−2.00000			−2.00000
$848$			1.00000i			1.00000i
$849$		0			0
$850$		0			0
$851$		0			0
$852$		0			0
$853$		0			0			−	1.00000i	$-0.5\pi$
$853$		0			0				1.00000i	$0.5\pi$
$854$		0			0
$855$		0			0
$856$	0.500000	−	0.866025i	0.500000	−	0.866025i
$857$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$857$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$858$		0			0
$859$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$859$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$863$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$864$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$865$	1.00000	−	1.73205i	1.00000	−	1.73205i
$866$		0			0
$867$			1.00000i			1.00000i
$868$	1.00000			1.00000
$869$	−1.73205			−1.73205
$870$		−	1.73205i		−	1.73205i
$871$		0			0
$872$		0			0
$873$	−1.50000	−	0.866025i	−1.50000	−	0.866025i
$874$		0			0
$875$	0.866025	−	0.500000i	0.866025	−	0.500000i
$876$		0			0
$877$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$877$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$878$	−0.866025	+	0.500000i	−0.866025	+	0.500000i
$879$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$880$			1.73205i			1.73205i
$881$		0			0		1.00000			$0$
$881$		0			0		−1.00000			$\pi$
$882$			1.00000i			1.00000i
$883$		0			0			−	1.00000i	$-0.5\pi$
$883$		0			0				1.00000i	$0.5\pi$
$884$		0			0
$885$	0.866025	−	1.50000i	0.866025	−	1.50000i
$886$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$887$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$887$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$888$		0			0
$889$	−1.50000	+	0.866025i	−1.50000	+	0.866025i
$890$		0			0
$891$	−0.866025	+	1.50000i	−0.866025	+	1.50000i
$892$	1.50000	−	0.866025i	1.50000	−	0.866025i
$893$		0			0
$894$		0			0
$895$		0			0
$896$		−	1.00000i		−	1.00000i
$897$		0			0
$898$		0			0
$899$	−0.866025	−	1.50000i	−0.866025	−	1.50000i
$900$	−1.00000			−1.00000
$901$		0			0
$902$		0			0
$903$		0			0
$904$		0			0
$905$		0			0
$906$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$907$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$907$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$908$	−0.866025	−	0.500000i	−0.866025	−	0.500000i
$909$		0			0
$910$		0			0
$911$		0			0		1.00000			$0$
$911$		0			0		−1.00000			$\pi$
$912$		0			0
$913$	−1.50000	+	0.866025i	−1.50000	+	0.866025i
$914$	0.866025	+	1.50000i	0.866025	+	1.50000i
$915$		0			0
$916$		0			0
$917$	1.73205			1.73205
$918$		0			0
$919$	−1.00000	+	1.73205i	−1.00000	+	1.73205i	−0.500000	+	0.866025i	$0.666667\pi$
$919$	−1.00000	+	1.73205i	−1.00000	+	1.73205i	−0.500000	+	0.866025i	$0.666667\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$		0			0
$924$	−1.50000	+	0.866025i	−1.50000	+	0.866025i
$925$		0			0
$926$		0			0
$927$		0			0
$928$	−1.50000	+	0.866025i	−1.50000	+	0.866025i
$929$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$929$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$930$	−0.866025	+	0.500000i	−0.866025	+	0.500000i
$931$		0			0
$932$		0			0
$933$		0			0
$934$	1.00000	+	1.73205i	1.00000	+	1.73205i
$935$		0			0
$936$		0			0
$937$			1.73205i			1.73205i	0.500000	+	0.866025i	$0.333333\pi$
$937$			1.73205i			1.73205i	−0.500000	+	0.866025i	$0.666667\pi$
$938$		0			0
$939$	1.73205			1.73205
$940$		0			0
$941$	0.866025	+	1.50000i	0.866025	+	1.50000i	0.866025	+	0.500000i	$0.166667\pi$
$941$	0.866025	+	1.50000i	0.866025	+	1.50000i			1.00000i	$0.5\pi$
$942$		0			0
$943$		0			0
$944$	−1.73205			−1.73205
$945$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$946$		0			0
$947$	−1.73205	−	1.00000i	−1.73205	−	1.00000i	−0.866025	−	0.500000i	$-0.833333\pi$
$947$	−1.73205	−	1.00000i	−1.73205	−	1.00000i	−0.866025	−	0.500000i	$-0.833333\pi$
$948$	−0.866025	+	0.500000i	−0.866025	+	0.500000i
$949$		0			0
$950$		0			0
$951$	1.00000			1.00000
$952$		0			0
$953$		0			0			−	1.00000i	$-0.5\pi$
$953$		0			0				1.00000i	$0.5\pi$
$954$	0.500000	−	0.866025i	0.500000	−	0.866025i
$955$		0			0
$956$		0			0
$957$	2.59808	+	1.50000i	2.59808	+	1.50000i
$958$		0			0
$959$		0			0
$960$	0.500000	+	0.866025i	0.500000	+	0.866025i
$961$		0			0
$962$		0			0
$963$	−0.866025	+	0.500000i	−0.866025	+	0.500000i
$964$	0.500000	−	0.866025i	0.500000	−	0.866025i
$965$	−0.866025	−	1.50000i	−0.866025	−	1.50000i
$966$		0			0
$967$		−	1.73205i		−	1.73205i	−0.500000	−	0.866025i	$-0.666667\pi$
$967$		−	1.73205i		−	1.73205i	0.500000	−	0.866025i	$-0.333333\pi$
$968$	−1.73205	−	1.00000i	−1.73205	−	1.00000i
$969$		0			0
$970$	1.50000	+	0.866025i	1.50000	+	0.866025i
$971$	−0.866025	+	1.50000i	−0.866025	+	1.50000i			1.00000i	$0.5\pi$
$971$	−0.866025	+	1.50000i	−0.866025	+	1.50000i	−0.866025	+	0.500000i	$0.833333\pi$
$972$			1.00000i			1.00000i
$973$		0			0
$974$	−1.73205			−1.73205
$975$		0			0
$976$		0			0
$977$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$977$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$978$		0			0
$979$		0			0
$980$		−	1.00000i		−	1.00000i
$981$		0			0
$982$	−1.50000	−	0.866025i	−1.50000	−	0.866025i
$983$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$983$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$984$		0			0
$985$	−2.00000			−2.00000
$986$		0			0
$987$		0			0
$988$		0			0
$989$		0			0
$990$	0.866025	−	1.50000i	0.866025	−	1.50000i
$991$	0.500000	+	0.866025i	0.500000	+	0.866025i	1.00000			$0$
$991$	0.500000	+	0.866025i	0.500000	+	0.866025i	−0.500000	+	0.866025i	$0.666667\pi$
$992$	0.866025	+	0.500000i	0.866025	+	0.500000i
$993$		0			0
$994$		0			0
$995$	−1.73205	+	1.00000i	−1.73205	+	1.00000i
$996$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$997$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$997$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	840.1.cg.b.149.2	yes	4
3.2	odd	2	inner	840.1.cg.b.149.1	yes	4
4.3	odd	2		3360.1.dm.a.3089.2		4
5.4	even	2		840.1.cg.a.149.1	✓	4
7.4	even	3		840.1.cg.a.389.1	yes	4
8.3	odd	2		3360.1.dm.a.3089.1		4
8.5	even	2	inner	840.1.cg.b.149.1	yes	4
12.11	even	2		3360.1.dm.a.3089.1		4
15.14	odd	2		840.1.cg.a.149.2	yes	4
20.19	odd	2		3360.1.dm.b.3089.1		4
21.11	odd	6		840.1.cg.a.389.2	yes	4
24.5	odd	2	CM	840.1.cg.b.149.2	yes	4
24.11	even	2		3360.1.dm.a.3089.2		4
28.11	odd	6		3360.1.dm.b.1649.1		4
35.4	even	6	inner	840.1.cg.b.389.2	yes	4
40.19	odd	2		3360.1.dm.b.3089.2		4
40.29	even	2		840.1.cg.a.149.2	yes	4
56.11	odd	6		3360.1.dm.b.1649.2		4
56.53	even	6		840.1.cg.a.389.2	yes	4
60.59	even	2		3360.1.dm.b.3089.2		4
84.11	even	6		3360.1.dm.b.1649.2		4
105.74	odd	6	inner	840.1.cg.b.389.1	yes	4
120.29	odd	2		840.1.cg.a.149.1	✓	4
120.59	even	2		3360.1.dm.b.3089.1		4
140.39	odd	6		3360.1.dm.a.1649.2		4
168.11	even	6		3360.1.dm.b.1649.1		4
168.53	odd	6		840.1.cg.a.389.1	yes	4
280.109	even	6	inner	840.1.cg.b.389.1	yes	4
280.179	odd	6		3360.1.dm.a.1649.1		4
420.179	even	6		3360.1.dm.a.1649.1		4
840.179	even	6		3360.1.dm.a.1649.2		4
840.389	odd	6	inner	840.1.cg.b.389.2	yes	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
840.1.cg.a.149.1	✓	4	5.4	even	2
840.1.cg.a.149.1	✓	4	120.29	odd	2
840.1.cg.a.149.2	yes	4	15.14	odd	2
840.1.cg.a.149.2	yes	4	40.29	even	2
840.1.cg.a.389.1	yes	4	7.4	even	3
840.1.cg.a.389.1	yes	4	168.53	odd	6
840.1.cg.a.389.2	yes	4	21.11	odd	6
840.1.cg.a.389.2	yes	4	56.53	even	6
840.1.cg.b.149.1	yes	4	3.2	odd	2	inner
840.1.cg.b.149.1	yes	4	8.5	even	2	inner
840.1.cg.b.149.2	yes	4	1.1	even	1	trivial
840.1.cg.b.149.2	yes	4	24.5	odd	2	CM
840.1.cg.b.389.1	yes	4	105.74	odd	6	inner
840.1.cg.b.389.1	yes	4	280.109	even	6	inner
840.1.cg.b.389.2	yes	4	35.4	even	6	inner
840.1.cg.b.389.2	yes	4	840.389	odd	6	inner
3360.1.dm.a.1649.1		4	280.179	odd	6
3360.1.dm.a.1649.1		4	420.179	even	6
3360.1.dm.a.1649.2		4	140.39	odd	6
3360.1.dm.a.1649.2		4	840.179	even	6
3360.1.dm.a.3089.1		4	8.3	odd	2
3360.1.dm.a.3089.1		4	12.11	even	2
3360.1.dm.a.3089.2		4	4.3	odd	2
3360.1.dm.a.3089.2		4	24.11	even	2
3360.1.dm.b.1649.1		4	28.11	odd	6
3360.1.dm.b.1649.1		4	168.11	even	6
3360.1.dm.b.1649.2		4	56.11	odd	6
3360.1.dm.b.1649.2		4	84.11	even	6
3360.1.dm.b.3089.1		4	20.19	odd	2
3360.1.dm.b.3089.1		4	120.59	even	2
3360.1.dm.b.3089.2		4	40.19	odd	2
3360.1.dm.b.3089.2		4	60.59	even	2

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:	\( N \)	\(=\)	\( 840 = 2^{3} \cdot 3 \cdot 5 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	840.cg (of order \(6\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(0.866025 + 0.500000i) q^{2} +(-0.866025 + 0.500000i) q^{3} +(0.500000 + 0.866025i) q^{4} +1.00000i q^{5} -1.00000 q^{6} +(0.500000 + 0.866025i) q^{7} +1.00000i q^{8} +(0.500000 - 0.866025i) q^{9} +O(q^{10})\)	\(q+(0.866025 + 0.500000i) q^{2} +(-0.866025 + 0.500000i) q^{3} +(0.500000 + 0.866025i) q^{4} +1.00000i q^{5} -1.00000 q^{6} +(0.500000 + 0.866025i) q^{7} +1.00000i q^{8} +(0.500000 - 0.866025i) q^{9} +(-0.500000 + 0.866025i) q^{10} +(-0.866025 - 1.50000i) q^{11} +(-0.866025 - 0.500000i) q^{12} +1.00000i q^{14} +(-0.500000 - 0.866025i) q^{15} +(-0.500000 + 0.866025i) q^{16} +(0.866025 - 0.500000i) q^{18} +(-0.866025 + 0.500000i) q^{20} +(-0.866025 - 0.500000i) q^{21} -1.73205i q^{22} +(-0.500000 - 0.866025i) q^{24} -1.00000 q^{25} +1.00000i q^{27} +(-0.500000 + 0.866025i) q^{28} +1.73205 q^{29} -1.00000i q^{30} +(-0.500000 - 0.866025i) q^{31} +(-0.866025 + 0.500000i) q^{32} +(1.50000 + 0.866025i) q^{33} +(-0.866025 + 0.500000i) q^{35} +1.00000 q^{36} -1.00000 q^{40} +(-0.500000 - 0.866025i) q^{42} +(0.866025 - 1.50000i) q^{44} +(0.866025 + 0.500000i) q^{45} -1.00000i q^{48} +(-0.500000 + 0.866025i) q^{49} +(-0.866025 - 0.500000i) q^{50} +(0.866025 - 0.500000i) q^{53} +(-0.500000 + 0.866025i) q^{54} +(1.50000 - 0.866025i) q^{55} +(-0.866025 + 0.500000i) q^{56} +(1.50000 + 0.866025i) q^{58} +(0.866025 + 1.50000i) q^{59} +(0.500000 - 0.866025i) q^{60} -1.00000i q^{62} +1.00000 q^{63} -1.00000 q^{64} +(0.866025 + 1.50000i) q^{66} -1.00000 q^{70} +(0.866025 + 0.500000i) q^{72} +(0.866025 - 0.500000i) q^{75} +(0.866025 - 1.50000i) q^{77} +(0.500000 - 0.866025i) q^{79} +(-0.866025 - 0.500000i) q^{80} +(-0.500000 - 0.866025i) q^{81} -1.00000i q^{83} -1.00000i q^{84} +(-1.50000 + 0.866025i) q^{87} +(1.50000 - 0.866025i) q^{88} +(0.500000 + 0.866025i) q^{90} +(0.866025 + 0.500000i) q^{93} +(0.500000 - 0.866025i) q^{96} -1.73205i q^{97} +(-0.866025 + 0.500000i) q^{98} -1.73205 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 2 q^{4} - 4 q^{6} + 2 q^{7} + 2 q^{9}+O(q^{10}) \) 4 * q + 2 * q^4 - 4 * q^6 + 2 * q^7 + 2 * q^9	\( 4 q + 2 q^{4} - 4 q^{6} + 2 q^{7} + 2 q^{9} - 2 q^{10} - 2 q^{15} - 2 q^{16} - 2 q^{24} - 4 q^{25} - 2 q^{28} - 2 q^{31} + 6 q^{33} + 4 q^{36} - 4 q^{40} - 2 q^{42} - 2 q^{49} - 2 q^{54} + 6 q^{55} + 6 q^{58} + 2 q^{60} + 4 q^{63} - 4 q^{64} - 4 q^{70} + 2 q^{79} - 2 q^{81} - 6 q^{87} + 6 q^{88} + 2 q^{90} + 2 q^{96}+O(q^{100}) \) 4 * q + 2 * q^4 - 4 * q^6 + 2 * q^7 + 2 * q^9 - 2 * q^10 - 2 * q^15 - 2 * q^16 - 2 * q^24 - 4 * q^25 - 2 * q^28 - 2 * q^31 + 6 * q^33 + 4 * q^36 - 4 * q^40 - 2 * q^42 - 2 * q^49 - 2 * q^54 + 6 * q^55 + 6 * q^58 + 2 * q^60 + 4 * q^63 - 4 * q^64 - 4 * q^70 + 2 * q^79 - 2 * q^81 - 6 * q^87 + 6 * q^88 + 2 * q^90 + 2 * q^96

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