LMFDB - Embedded newform 1148.1.o.d.163.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1148,1,Mod(163,1148)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1148, base_ring=CyclotomicField(6))

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

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

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

chi := DirichletCharacter("1148.163");

S:= CuspForms(chi, 1);

N := Newforms(S);

Level:	$ N $	$=$	$ 1148 = 2^{2} \cdot 7 \cdot 41 $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	1148.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:	$0.572926634503$
Analytic rank:	$0$
Dimension:	$8$
Relative dimension:	$4$ over $\Q(\zeta_{6})$
Coefficient field:	$\Q(\zeta_{24})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - x^{4} + 1 $ x^8 - x^4 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Projective image:	$D_{12}$
Projective field:	Galois closure of $\mathbb{Q}[x]/(x^{12} - \cdots)$

Embedding invariants

Embedding label			163.4
Root			$-0.258819 + 0.965926i$ of defining polynomial
Character	$\chi$	$=$	1148.163
Dual form			1148.1.o.d.655.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(0.500000 - 0.866025i) q^{2} +(0.965926 + 1.67303i) q^{3} +(-0.500000 - 0.866025i) q^{4} +(0.866025 - 1.50000i) q^{5} +1.93185 q^{6} +(-0.258819 - 0.965926i) q^{7} -1.00000 q^{8} +(-1.36603 + 2.36603i) q^{9} +O(q^{10})$	\(q+(0.500000 - 0.866025i) q^{2} +(0.965926 + 1.67303i) q^{3} +(-0.500000 - 0.866025i) q^{4} +(0.866025 - 1.50000i) q^{5} +1.93185 q^{6} +(-0.258819 - 0.965926i) q^{7} -1.00000 q^{8} +(-1.36603 + 2.36603i) q^{9} +(-0.866025 - 1.50000i) q^{10} +(0.258819 + 0.448288i) q^{11} +(0.965926 - 1.67303i) q^{12} +(-0.965926 - 0.258819i) q^{14} +3.34607 q^{15} +(-0.500000 + 0.866025i) q^{16} +(1.36603 + 2.36603i) q^{18} +(-0.258819 + 0.448288i) q^{19} -1.73205 q^{20} +(1.36603 - 1.36603i) q^{21} +0.517638 q^{22} +(-0.965926 - 1.67303i) q^{24} +(-1.00000 - 1.73205i) q^{25} -3.34607 q^{27} +(-0.707107 + 0.707107i) q^{28} +(1.67303 - 2.89778i) q^{30} +(0.500000 + 0.866025i) q^{32} +(-0.500000 + 0.866025i) q^{33} +(-1.67303 - 0.448288i) q^{35} +2.73205 q^{36} +(0.258819 + 0.448288i) q^{38} +(-0.866025 + 1.50000i) q^{40} +1.00000 q^{41} +(-0.500000 - 1.86603i) q^{42} +(0.258819 - 0.448288i) q^{44} +(2.36603 + 4.09808i) q^{45} +(-0.707107 + 1.22474i) q^{47} -1.93185 q^{48} +(-0.866025 + 0.500000i) q^{49} -2.00000 q^{50} +(-1.67303 + 2.89778i) q^{54} +0.896575 q^{55} +(0.258819 + 0.965926i) q^{56} -1.00000 q^{57} +(-1.67303 - 2.89778i) q^{60} +(-0.500000 + 0.866025i) q^{61} +(2.63896 + 0.707107i) q^{63} +1.00000 q^{64} +(0.500000 + 0.866025i) q^{66} +(0.707107 + 1.22474i) q^{67} +(-1.22474 + 1.22474i) q^{70} -1.93185 q^{71} +(1.36603 - 2.36603i) q^{72} +(1.93185 - 3.34607i) q^{75} +0.517638 q^{76} +(0.366025 - 0.366025i) q^{77} +(0.965926 - 1.67303i) q^{79} +(0.866025 + 1.50000i) q^{80} +(-1.86603 - 3.23205i) q^{81} +(0.500000 - 0.866025i) q^{82} +(-1.86603 - 0.500000i) q^{84} +(-0.258819 - 0.448288i) q^{88} +4.73205 q^{90} +(0.707107 + 1.22474i) q^{94} +(0.448288 + 0.776457i) q^{95} +(-0.965926 + 1.67303i) q^{96} +1.00000i q^{98} -1.41421 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 4 q^{2} - 4 q^{4} - 8 q^{8} - 4 q^{9}+O(q^{10}) $ 8 * q + 4 * q^2 - 4 * q^4 - 8 * q^8 - 4 * q^9	$ 8 q + 4 q^{2} - 4 q^{4} - 8 q^{8} - 4 q^{9} - 4 q^{16} + 4 q^{18} + 4 q^{21} - 8 q^{25} + 4 q^{32} - 4 q^{33} + 8 q^{36} + 8 q^{41} - 4 q^{42} + 12 q^{45} - 16 q^{50} - 8 q^{57} - 4 q^{61} + 8 q^{64} + 4 q^{66} + 4 q^{72} - 4 q^{77} - 8 q^{81} + 4 q^{82} - 8 q^{84} + 24 q^{90}+O(q^{100}) $ 8 * q + 4 * q^2 - 4 * q^4 - 8 * q^8 - 4 * q^9 - 4 * q^16 + 4 * q^18 + 4 * q^21 - 8 * q^25 + 4 * q^32 - 4 * q^33 + 8 * q^36 + 8 * q^41 - 4 * q^42 + 12 * q^45 - 16 * q^50 - 8 * q^57 - 4 * q^61 + 8 * q^64 + 4 * q^66 + 4 * q^72 - 4 * q^77 - 8 * q^81 + 4 * q^82 - 8 * q^84 + 24 * q^90

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1148\mathbb{Z}\right)^\times$.

$n$	$493$	$575$	$785$
$\chi(n)$	$e\left(\frac{1}{3}\right)$	$-1$	$-1$

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 $

$2$	0.500000	−	0.866025i	0.500000	−	0.866025i
$2$	0.500000	−	0.866025i	0.500000	−	0.866025i
$3$	0.965926	+	1.67303i	0.965926	+	1.67303i	0.707107	+	0.707107i	$0.250000\pi$
$3$	0.965926	+	1.67303i	0.965926	+	1.67303i	0.258819	+	0.965926i	$0.416667\pi$
$4$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$5$	0.866025	−	1.50000i	0.866025	−	1.50000i		−	1.00000i	$-0.5\pi$
$5$	0.866025	−	1.50000i	0.866025	−	1.50000i	0.866025	−	0.500000i	$-0.166667\pi$
$6$	1.93185			1.93185
$7$	−0.258819	−	0.965926i	−0.258819	−	0.965926i
$7$	−0.258819	−	0.965926i	−0.258819	−	0.965926i
$8$	−1.00000			−1.00000
$9$	−1.36603	+	2.36603i	−1.36603	+	2.36603i
$10$	−0.866025	−	1.50000i	−0.866025	−	1.50000i
$11$	0.258819	+	0.448288i	0.258819	+	0.448288i	0.965926	−	0.258819i	$-0.0833333\pi$
$11$	0.258819	+	0.448288i	0.258819	+	0.448288i	−0.707107	+	0.707107i	$0.750000\pi$
$12$	0.965926	−	1.67303i	0.965926	−	1.67303i
$13$		0			0		1.00000			$0$
$13$		0			0		−1.00000			$\pi$
$14$	−0.965926	−	0.258819i	−0.965926	−	0.258819i
$15$	3.34607			3.34607
$16$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$17$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$17$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$18$	1.36603	+	2.36603i	1.36603	+	2.36603i
$19$	−0.258819	+	0.448288i	−0.258819	+	0.448288i	−0.965926	−	0.258819i	$-0.916667\pi$
$19$	−0.258819	+	0.448288i	−0.258819	+	0.448288i	0.707107	+	0.707107i	$0.250000\pi$
$20$	−1.73205			−1.73205
$21$	1.36603	−	1.36603i	1.36603	−	1.36603i
$22$	0.517638			0.517638
$23$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$23$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$24$	−0.965926	−	1.67303i	−0.965926	−	1.67303i
$25$	−1.00000	−	1.73205i	−1.00000	−	1.73205i
$26$		0			0
$27$	−3.34607			−3.34607
$28$	−0.707107	+	0.707107i	−0.707107	+	0.707107i
$29$		0			0		1.00000			$0$
$29$		0			0		−1.00000			$\pi$
$30$	1.67303	−	2.89778i	1.67303	−	2.89778i
$31$		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
$33$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$34$		0			0
$35$	−1.67303	−	0.448288i	−1.67303	−	0.448288i
$36$	2.73205			2.73205
$37$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$37$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$38$	0.258819	+	0.448288i	0.258819	+	0.448288i
$39$		0			0
$40$	−0.866025	+	1.50000i	−0.866025	+	1.50000i
$41$	1.00000			1.00000
$41$	1.00000			1.00000
$42$	−0.500000	−	1.86603i	−0.500000	−	1.86603i
$43$		0			0		1.00000			$0$
$43$		0			0		−1.00000			$\pi$
$44$	0.258819	−	0.448288i	0.258819	−	0.448288i
$45$	2.36603	+	4.09808i	2.36603	+	4.09808i
$46$		0			0
$47$	−0.707107	+	1.22474i	−0.707107	+	1.22474i	0.258819	+	0.965926i	$0.416667\pi$
$47$	−0.707107	+	1.22474i	−0.707107	+	1.22474i	−0.965926	+	0.258819i	$0.916667\pi$
$48$	−1.93185			−1.93185
$49$	−0.866025	+	0.500000i	−0.866025	+	0.500000i
$50$	−2.00000			−2.00000
$51$		0			0
$52$		0			0
$53$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$53$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$54$	−1.67303	+	2.89778i	−1.67303	+	2.89778i
$55$	0.896575			0.896575
$56$	0.258819	+	0.965926i	0.258819	+	0.965926i
$57$	−1.00000			−1.00000
$58$		0			0
$59$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$59$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$60$	−1.67303	−	2.89778i	−1.67303	−	2.89778i
$61$	−0.500000	+	0.866025i	−0.500000	+	0.866025i	0.500000	+	0.866025i	$0.333333\pi$
$61$	−0.500000	+	0.866025i	−0.500000	+	0.866025i	−1.00000			$\pi$
$62$		0			0
$63$	2.63896	+	0.707107i	2.63896	+	0.707107i
$64$	1.00000			1.00000
$65$		0			0
$66$	0.500000	+	0.866025i	0.500000	+	0.866025i
$67$	0.707107	+	1.22474i	0.707107	+	1.22474i	0.965926	+	0.258819i	$0.0833333\pi$
$67$	0.707107	+	1.22474i	0.707107	+	1.22474i	−0.258819	+	0.965926i	$0.583333\pi$
$68$		0			0
$69$		0			0
$70$	−1.22474	+	1.22474i	−1.22474	+	1.22474i
$71$	−1.93185			−1.93185			−0.965926	−	0.258819i	$-0.916667\pi$
$71$	−1.93185			−1.93185			−0.965926	+	0.258819i	$0.916667\pi$
$72$	1.36603	−	2.36603i	1.36603	−	2.36603i
$73$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$73$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$74$		0			0
$75$	1.93185	−	3.34607i	1.93185	−	3.34607i
$76$	0.517638			0.517638
$77$	0.366025	−	0.366025i	0.366025	−	0.366025i
$78$		0			0
$79$	0.965926	−	1.67303i	0.965926	−	1.67303i	0.258819	−	0.965926i	$-0.416667\pi$
$79$	0.965926	−	1.67303i	0.965926	−	1.67303i	0.707107	−	0.707107i	$-0.250000\pi$
$80$	0.866025	+	1.50000i	0.866025	+	1.50000i
$81$	−1.86603	−	3.23205i	−1.86603	−	3.23205i
$82$	0.500000	−	0.866025i	0.500000	−	0.866025i
$83$		0			0		1.00000			$0$
$83$		0			0		−1.00000			$\pi$
$84$	−1.86603	−	0.500000i	−1.86603	−	0.500000i
$85$		0			0
$86$		0			0
$87$		0			0
$88$	−0.258819	−	0.448288i	−0.258819	−	0.448288i
$89$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$89$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$90$	4.73205			4.73205
$91$		0			0
$92$		0			0
$93$		0			0
$94$	0.707107	+	1.22474i	0.707107	+	1.22474i
$95$	0.448288	+	0.776457i	0.448288	+	0.776457i
$96$	−0.965926	+	1.67303i	−0.965926	+	1.67303i
$97$		0			0		1.00000			$0$
$97$		0			0		−1.00000			$\pi$
$98$			1.00000i			1.00000i
$99$	−1.41421			−1.41421
$100$	−1.00000	+	1.73205i	−1.00000	+	1.73205i
$101$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$101$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$102$		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
$105$	−0.866025	−	3.23205i	−0.866025	−	3.23205i
$106$		0			0
$107$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$107$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$108$	1.67303	+	2.89778i	1.67303	+	2.89778i
$109$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$109$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$110$	0.448288	−	0.776457i	0.448288	−	0.776457i
$111$		0			0
$112$	0.965926	+	0.258819i	0.965926	+	0.258819i
$113$	−1.73205			−1.73205			−0.866025	−	0.500000i	$-0.833333\pi$
$113$	−1.73205			−1.73205			−0.866025	+	0.500000i	$0.833333\pi$
$114$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$115$		0			0
$116$		0			0
$117$		0			0
$118$		0			0
$119$		0			0
$120$	−3.34607			−3.34607
$121$	0.366025	−	0.633975i	0.366025	−	0.633975i
$122$	0.500000	+	0.866025i	0.500000	+	0.866025i
$123$	0.965926	+	1.67303i	0.965926	+	1.67303i
$124$		0			0
$125$	−1.73205			−1.73205
$126$	1.93185	−	1.93185i	1.93185	−	1.93185i
$127$		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	$-0.333333\pi$
$131$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$132$	1.00000			1.00000
$133$	0.500000	+	0.133975i	0.500000	+	0.133975i
$134$	1.41421			1.41421
$135$	−2.89778	+	5.01910i	−2.89778	+	5.01910i
$136$		0			0
$137$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$137$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$138$		0			0
$139$		0			0		1.00000			$0$
$139$		0			0		−1.00000			$\pi$
$140$	0.448288	+	1.67303i	0.448288	+	1.67303i
$141$	−2.73205			−2.73205
$142$	−0.965926	+	1.67303i	−0.965926	+	1.67303i
$143$		0			0
$144$	−1.36603	−	2.36603i	−1.36603	−	2.36603i
$145$		0			0
$146$		0			0
$147$	−1.67303	−	0.965926i	−1.67303	−	0.965926i
$148$		0			0
$149$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$149$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$150$	−1.93185	−	3.34607i	−1.93185	−	3.34607i
$151$	−0.707107	−	1.22474i	−0.707107	−	1.22474i	−0.965926	−	0.258819i	$-0.916667\pi$
$151$	−0.707107	−	1.22474i	−0.707107	−	1.22474i	0.258819	−	0.965926i	$-0.416667\pi$
$152$	0.258819	−	0.448288i	0.258819	−	0.448288i
$153$		0			0
$154$	−0.133975	−	0.500000i	−0.133975	−	0.500000i
$155$		0			0
$156$		0			0
$157$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$157$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$158$	−0.965926	−	1.67303i	−0.965926	−	1.67303i
$159$		0			0
$160$	1.73205			1.73205
$161$		0			0
$162$	−3.73205			−3.73205
$163$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$163$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$164$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$165$	0.866025	+	1.50000i	0.866025	+	1.50000i
$166$		0			0
$167$	−1.41421			−1.41421			−0.707107	−	0.707107i	$-0.750000\pi$
$167$	−1.41421			−1.41421			−0.707107	+	0.707107i	$0.750000\pi$
$168$	−1.36603	+	1.36603i	−1.36603	+	1.36603i
$169$	1.00000			1.00000
$170$		0			0
$171$	−0.707107	−	1.22474i	−0.707107	−	1.22474i
$172$		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
$175$	−1.41421	+	1.41421i	−1.41421	+	1.41421i
$176$	−0.517638			−0.517638
$177$		0			0
$178$		0			0
$179$	−0.965926	−	1.67303i	−0.965926	−	1.67303i	−0.707107	−	0.707107i	$-0.750000\pi$
$179$	−0.965926	−	1.67303i	−0.965926	−	1.67303i	−0.258819	−	0.965926i	$-0.583333\pi$
$180$	2.36603	−	4.09808i	2.36603	−	4.09808i
$181$		0			0		1.00000			$0$
$181$		0			0		−1.00000			$\pi$
$182$		0			0
$183$	−1.93185			−1.93185
$184$		0			0
$185$		0			0
$186$		0			0
$187$		0			0
$188$	1.41421			1.41421
$189$	0.866025	+	3.23205i	0.866025	+	3.23205i
$190$	0.896575			0.896575
$191$	0.707107	−	1.22474i	0.707107	−	1.22474i	−0.258819	−	0.965926i	$-0.583333\pi$
$191$	0.707107	−	1.22474i	0.707107	−	1.22474i	0.965926	−	0.258819i	$-0.0833333\pi$
$192$	0.965926	+	1.67303i	0.965926	+	1.67303i
$193$		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
$196$	0.866025	+	0.500000i	0.866025	+	0.500000i
$197$	1.00000			1.00000			0.500000	−	0.866025i	$-0.333333\pi$
$197$	1.00000			1.00000			0.500000	+	0.866025i	$0.333333\pi$
$198$	−0.707107	+	1.22474i	−0.707107	+	1.22474i
$199$	0.707107	+	1.22474i	0.707107	+	1.22474i	0.965926	+	0.258819i	$0.0833333\pi$
$199$	0.707107	+	1.22474i	0.707107	+	1.22474i	−0.258819	+	0.965926i	$0.583333\pi$
$200$	1.00000	+	1.73205i	1.00000	+	1.73205i
$201$	−1.36603	+	2.36603i	−1.36603	+	2.36603i
$202$		0			0
$203$		0			0
$204$		0			0
$205$	0.866025	−	1.50000i	0.866025	−	1.50000i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−0.267949			−0.267949
$210$	−3.23205	−	0.866025i	−3.23205	−	0.866025i
$211$	1.93185			1.93185			0.965926	−	0.258819i	$-0.0833333\pi$
$211$	1.93185			1.93185			0.965926	+	0.258819i	$0.0833333\pi$
$212$		0			0
$213$	−1.86603	−	3.23205i	−1.86603	−	3.23205i
$214$		0			0
$215$		0			0
$216$	3.34607			3.34607
$217$		0			0
$218$		0			0
$219$		0			0
$220$	−0.448288	−	0.776457i	−0.448288	−	0.776457i
$221$		0			0
$222$		0			0
$223$		0			0		1.00000			$0$
$223$		0			0		−1.00000			$\pi$
$224$	0.707107	−	0.707107i	0.707107	−	0.707107i
$225$	5.46410			5.46410
$226$	−0.866025	+	1.50000i	−0.866025	+	1.50000i
$227$	−0.258819	−	0.448288i	−0.258819	−	0.448288i	0.707107	−	0.707107i	$-0.250000\pi$
$227$	−0.258819	−	0.448288i	−0.258819	−	0.448288i	−0.965926	+	0.258819i	$0.916667\pi$
$228$	0.500000	+	0.866025i	0.500000	+	0.866025i
$229$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$229$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$230$		0			0
$231$	0.965926	+	0.258819i	0.965926	+	0.258819i
$232$		0			0
$233$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$233$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$234$		0			0
$235$	1.22474	+	2.12132i	1.22474	+	2.12132i
$236$		0			0
$237$	3.73205			3.73205
$238$		0			0
$239$	0.517638			0.517638			0.258819	−	0.965926i	$-0.416667\pi$
$239$	0.517638			0.517638			0.258819	+	0.965926i	$0.416667\pi$
$240$	−1.67303	+	2.89778i	−1.67303	+	2.89778i
$241$	1.00000	+	1.73205i	1.00000	+	1.73205i	0.500000	+	0.866025i	$0.333333\pi$
$241$	1.00000	+	1.73205i	1.00000	+	1.73205i	0.500000	+	0.866025i	$0.333333\pi$
$242$	−0.366025	−	0.633975i	−0.366025	−	0.633975i
$243$	1.93185	−	3.34607i	1.93185	−	3.34607i
$244$	1.00000			1.00000
$245$			1.73205i			1.73205i
$246$	1.93185			1.93185
$247$		0			0
$248$		0			0
$249$		0			0
$250$	−0.866025	+	1.50000i	−0.866025	+	1.50000i
$251$		0			0		1.00000			$0$
$251$		0			0		−1.00000			$\pi$
$252$	−0.707107	−	2.63896i	−0.707107	−	2.63896i
$253$		0			0
$254$		0			0
$255$		0			0
$256$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$257$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$257$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$		0			0
$262$		0			0
$263$	−0.707107	−	1.22474i	−0.707107	−	1.22474i	−0.965926	−	0.258819i	$-0.916667\pi$
$263$	−0.707107	−	1.22474i	−0.707107	−	1.22474i	0.258819	−	0.965926i	$-0.416667\pi$
$264$	0.500000	−	0.866025i	0.500000	−	0.866025i
$265$		0			0
$266$	0.366025	−	0.366025i	0.366025	−	0.366025i
$267$		0			0
$268$	0.707107	−	1.22474i	0.707107	−	1.22474i
$269$	−0.500000	−	0.866025i	−0.500000	−	0.866025i	0.500000	−	0.866025i	$-0.333333\pi$
$269$	−0.500000	−	0.866025i	−0.500000	−	0.866025i	−1.00000			$\pi$
$270$	2.89778	+	5.01910i	2.89778	+	5.01910i
$271$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$271$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	0.517638	−	0.896575i	0.517638	−	0.896575i
$276$		0			0
$277$	−0.866025	−	1.50000i	−0.866025	−	1.50000i	−0.866025	−	0.500000i	$-0.833333\pi$
$277$	−0.866025	−	1.50000i	−0.866025	−	1.50000i		−	1.00000i	$-0.5\pi$
$278$		0			0
$279$		0			0
$280$	1.67303	+	0.448288i	1.67303	+	0.448288i
$281$		0			0		1.00000			$0$
$281$		0			0		−1.00000			$\pi$
$282$	−1.36603	+	2.36603i	−1.36603	+	2.36603i
$283$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$283$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$284$	0.965926	+	1.67303i	0.965926	+	1.67303i
$285$	−0.866025	+	1.50000i	−0.866025	+	1.50000i
$286$		0			0
$287$	−0.258819	−	0.965926i	−0.258819	−	0.965926i
$288$	−2.73205			−2.73205
$289$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$290$		0			0
$291$		0			0
$292$		0			0
$293$		0			0		1.00000			$0$
$293$		0			0		−1.00000			$\pi$
$294$	−1.67303	+	0.965926i	−1.67303	+	0.965926i
$295$		0			0
$296$		0			0
$297$	−0.866025	−	1.50000i	−0.866025	−	1.50000i
$298$		0			0
$299$		0			0
$300$	−3.86370			−3.86370
$301$		0			0
$302$	−1.41421			−1.41421
$303$		0			0
$304$	−0.258819	−	0.448288i	−0.258819	−	0.448288i
$305$	0.866025	+	1.50000i	0.866025	+	1.50000i
$306$		0			0
$307$		0			0		1.00000			$0$
$307$		0			0		−1.00000			$\pi$
$308$	−0.500000	−	0.133975i	−0.500000	−	0.133975i
$309$		0			0
$310$		0			0
$311$	0.707107	+	1.22474i	0.707107	+	1.22474i	0.965926	+	0.258819i	$0.0833333\pi$
$311$	0.707107	+	1.22474i	0.707107	+	1.22474i	−0.258819	+	0.965926i	$0.583333\pi$
$312$		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
$315$	3.34607	−	3.34607i	3.34607	−	3.34607i
$316$	−1.93185			−1.93185
$317$		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
$320$	0.866025	−	1.50000i	0.866025	−	1.50000i
$321$		0			0
$322$		0			0
$323$		0			0
$324$	−1.86603	+	3.23205i	−1.86603	+	3.23205i
$325$		0			0
$326$		0			0
$327$		0			0
$328$	−1.00000			−1.00000
$329$	1.36603	+	0.366025i	1.36603	+	0.366025i
$330$	1.73205			1.73205
$331$	0.707107	−	1.22474i	0.707107	−	1.22474i	−0.258819	−	0.965926i	$-0.583333\pi$
$331$	0.707107	−	1.22474i	0.707107	−	1.22474i	0.965926	−	0.258819i	$-0.0833333\pi$
$332$		0			0
$333$		0			0
$334$	−0.707107	+	1.22474i	−0.707107	+	1.22474i
$335$	2.44949			2.44949
$336$	0.500000	+	1.86603i	0.500000	+	1.86603i
$337$	1.73205			1.73205			0.866025	−	0.500000i	$-0.166667\pi$
$337$	1.73205			1.73205			0.866025	+	0.500000i	$0.166667\pi$
$338$	0.500000	−	0.866025i	0.500000	−	0.866025i
$339$	−1.67303	−	2.89778i	−1.67303	−	2.89778i
$340$		0			0
$341$		0			0
$342$	−1.41421			−1.41421
$343$	0.707107	+	0.707107i	0.707107	+	0.707107i
$344$		0			0
$345$		0			0
$346$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$347$	−0.258819	−	0.448288i	−0.258819	−	0.448288i	0.707107	−	0.707107i	$-0.250000\pi$
$347$	−0.258819	−	0.448288i	−0.258819	−	0.448288i	−0.965926	+	0.258819i	$0.916667\pi$
$348$		0			0
$349$		0			0			−	1.00000i	$-0.5\pi$
$349$		0			0				1.00000i	$0.5\pi$
$350$	0.517638	+	1.93185i	0.517638	+	1.93185i
$351$		0			0
$352$	−0.258819	+	0.448288i	−0.258819	+	0.448288i
$353$	−0.866025	−	1.50000i	−0.866025	−	1.50000i	−0.866025	−	0.500000i	$-0.833333\pi$
$353$	−0.866025	−	1.50000i	−0.866025	−	1.50000i		−	1.00000i	$-0.5\pi$
$354$		0			0
$355$	−1.67303	+	2.89778i	−1.67303	+	2.89778i
$356$		0			0
$357$		0			0
$358$	−1.93185			−1.93185
$359$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$359$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$360$	−2.36603	−	4.09808i	−2.36603	−	4.09808i
$361$	0.366025	+	0.633975i	0.366025	+	0.633975i
$362$		0			0
$363$	1.41421			1.41421
$364$		0			0
$365$		0			0
$366$	−0.965926	+	1.67303i	−0.965926	+	1.67303i
$367$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$367$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$368$		0			0
$369$	−1.36603	+	2.36603i	−1.36603	+	2.36603i
$370$		0			0
$371$		0			0
$372$		0			0
$373$	0.500000	−	0.866025i	0.500000	−	0.866025i	−0.500000	−	0.866025i	$-0.666667\pi$
$373$	0.500000	−	0.866025i	0.500000	−	0.866025i	1.00000			$0$
$374$		0			0
$375$	−1.67303	−	2.89778i	−1.67303	−	2.89778i
$376$	0.707107	−	1.22474i	0.707107	−	1.22474i
$377$		0			0
$378$	3.23205	+	0.866025i	3.23205	+	0.866025i
$379$		0			0		1.00000			$0$
$379$		0			0		−1.00000			$\pi$
$380$	0.448288	−	0.776457i	0.448288	−	0.776457i
$381$		0			0
$382$	−0.707107	−	1.22474i	−0.707107	−	1.22474i
$383$	−0.965926	+	1.67303i	−0.965926	+	1.67303i	−0.258819	+	0.965926i	$0.583333\pi$
$383$	−0.965926	+	1.67303i	−0.965926	+	1.67303i	−0.707107	+	0.707107i	$0.750000\pi$
$384$	1.93185			1.93185
$385$	−0.232051	−	0.866025i	−0.232051	−	0.866025i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	0.500000	+	0.866025i	0.500000	+	0.866025i	1.00000			$0$
$389$	0.500000	+	0.866025i	0.500000	+	0.866025i	−0.500000	+	0.866025i	$0.666667\pi$
$390$		0			0
$391$		0			0
$392$	0.866025	−	0.500000i	0.866025	−	0.500000i
$393$		0			0
$394$	0.500000	−	0.866025i	0.500000	−	0.866025i
$395$	−1.67303	−	2.89778i	−1.67303	−	2.89778i
$396$	0.707107	+	1.22474i	0.707107	+	1.22474i
$397$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$397$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$398$	1.41421			1.41421
$399$	0.258819	+	0.965926i	0.258819	+	0.965926i
$400$	2.00000			2.00000
$401$	−0.866025	+	1.50000i	−0.866025	+	1.50000i			1.00000i	$0.5\pi$
$401$	−0.866025	+	1.50000i	−0.866025	+	1.50000i	−0.866025	+	0.500000i	$0.833333\pi$
$402$	1.36603	+	2.36603i	1.36603	+	2.36603i
$403$		0			0
$404$		0			0
$405$	−6.46410			−6.46410
$406$		0			0
$407$		0			0
$408$		0			0
$409$	0.866025	+	1.50000i	0.866025	+	1.50000i	0.866025	+	0.500000i	$0.166667\pi$
$409$	0.866025	+	1.50000i	0.866025	+	1.50000i			1.00000i	$0.5\pi$
$410$	−0.866025	−	1.50000i	−0.866025	−	1.50000i
$411$		0			0
$412$		0			0
$413$		0			0
$414$		0			0
$415$		0			0
$416$		0			0
$417$		0			0
$418$	−0.133975	+	0.232051i	−0.133975	+	0.232051i
$419$		0			0		1.00000			$0$
$419$		0			0		−1.00000			$\pi$
$420$	−2.36603	+	2.36603i	−2.36603	+	2.36603i
$421$		0			0		1.00000			$0$
$421$		0			0		−1.00000			$\pi$
$422$	0.965926	−	1.67303i	0.965926	−	1.67303i
$423$	−1.93185	−	3.34607i	−1.93185	−	3.34607i
$424$		0			0
$425$		0			0
$426$	−3.73205			−3.73205
$427$	0.965926	+	0.258819i	0.965926	+	0.258819i
$428$		0			0
$429$		0			0
$430$		0			0
$431$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$431$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$432$	1.67303	−	2.89778i	1.67303	−	2.89778i
$433$	−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			0
$439$	−0.965926	+	1.67303i	−0.965926	+	1.67303i	−0.258819	+	0.965926i	$0.583333\pi$
$439$	−0.965926	+	1.67303i	−0.965926	+	1.67303i	−0.707107	+	0.707107i	$0.750000\pi$
$440$	−0.896575			−0.896575
$441$		−	2.73205i		−	2.73205i
$442$		0			0
$443$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$443$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$		0			0
$448$	−0.258819	−	0.965926i	−0.258819	−	0.965926i
$449$	1.00000			1.00000			0.500000	−	0.866025i	$-0.333333\pi$
$449$	1.00000			1.00000			0.500000	+	0.866025i	$0.333333\pi$
$450$	2.73205	−	4.73205i	2.73205	−	4.73205i
$451$	0.258819	+	0.448288i	0.258819	+	0.448288i
$452$	0.866025	+	1.50000i	0.866025	+	1.50000i
$453$	1.36603	−	2.36603i	1.36603	−	2.36603i
$454$	−0.517638			−0.517638
$455$		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
$461$	1.73205			1.73205			0.866025	−	0.500000i	$-0.166667\pi$
$461$	1.73205			1.73205			0.866025	+	0.500000i	$0.166667\pi$
$462$	0.707107	−	0.707107i	0.707107	−	0.707107i
$463$	−1.93185			−1.93185			−0.965926	−	0.258819i	$-0.916667\pi$
$463$	−1.93185			−1.93185			−0.965926	+	0.258819i	$0.916667\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$467$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$468$		0			0
$469$	1.00000	−	1.00000i	1.00000	−	1.00000i
$470$	2.44949			2.44949
$471$		0			0
$472$		0			0
$473$		0			0
$474$	1.86603	−	3.23205i	1.86603	−	3.23205i
$475$	1.03528			1.03528
$476$		0			0
$477$		0			0
$478$	0.258819	−	0.448288i	0.258819	−	0.448288i
$479$	0.258819	+	0.448288i	0.258819	+	0.448288i	0.965926	−	0.258819i	$-0.0833333\pi$
$479$	0.258819	+	0.448288i	0.258819	+	0.448288i	−0.707107	+	0.707107i	$0.750000\pi$
$480$	1.67303	+	2.89778i	1.67303	+	2.89778i
$481$		0			0
$482$	2.00000			2.00000
$483$		0			0
$484$	−0.732051			−0.732051
$485$		0			0
$486$	−1.93185	−	3.34607i	−1.93185	−	3.34607i
$487$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$487$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$488$	0.500000	−	0.866025i	0.500000	−	0.866025i
$489$		0			0
$490$	1.50000	+	0.866025i	1.50000	+	0.866025i
$491$		0			0		1.00000			$0$
$491$		0			0		−1.00000			$\pi$
$492$	0.965926	−	1.67303i	0.965926	−	1.67303i
$493$		0			0
$494$		0			0
$495$	−1.22474	+	2.12132i	−1.22474	+	2.12132i
$496$		0			0
$497$	0.500000	+	1.86603i	0.500000	+	1.86603i
$498$		0			0
$499$	−0.707107	+	1.22474i	−0.707107	+	1.22474i	0.258819	+	0.965926i	$0.416667\pi$
$499$	−0.707107	+	1.22474i	−0.707107	+	1.22474i	−0.965926	+	0.258819i	$0.916667\pi$
$500$	0.866025	+	1.50000i	0.866025	+	1.50000i
$501$	−1.36603	−	2.36603i	−1.36603	−	2.36603i
$502$		0			0
$503$	0.517638			0.517638			0.258819	−	0.965926i	$-0.416667\pi$
$503$	0.517638			0.517638			0.258819	+	0.965926i	$0.416667\pi$
$504$	−2.63896	−	0.707107i	−2.63896	−	0.707107i
$505$		0			0
$506$		0			0
$507$	0.965926	+	1.67303i	0.965926	+	1.67303i
$508$		0			0
$509$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$509$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$510$		0			0
$511$		0			0
$512$	−1.00000			−1.00000
$513$	0.866025	−	1.50000i	0.866025	−	1.50000i
$514$		0			0
$515$		0			0
$516$		0			0
$517$	−0.732051			−0.732051
$518$		0			0
$519$	1.93185			1.93185
$520$		0			0
$521$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$521$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$522$		0			0
$523$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$523$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$524$		0			0
$525$	−3.73205	−	1.00000i	−3.73205	−	1.00000i
$526$	−1.41421			−1.41421
$527$		0			0
$528$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$529$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$530$		0			0
$531$		0			0
$532$	−0.133975	−	0.500000i	−0.133975	−	0.500000i
$533$		0			0
$534$		0			0
$535$		0			0
$536$	−0.707107	−	1.22474i	−0.707107	−	1.22474i
$537$	1.86603	−	3.23205i	1.86603	−	3.23205i
$538$	−1.00000			−1.00000
$539$	−0.448288	−	0.258819i	−0.448288	−	0.258819i
$540$	5.79555			5.79555
$541$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$541$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$		0			0
$546$		0			0
$547$	1.41421			1.41421			0.707107	−	0.707107i	$-0.250000\pi$
$547$	1.41421			1.41421			0.707107	+	0.707107i	$0.250000\pi$
$548$		0			0
$549$	−1.36603	−	2.36603i	−1.36603	−	2.36603i
$550$	−0.517638	−	0.896575i	−0.517638	−	0.896575i
$551$		0			0
$552$		0			0
$553$	−1.86603	−	0.500000i	−1.86603	−	0.500000i
$554$	−1.73205			−1.73205
$555$		0			0
$556$		0			0
$557$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$557$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$558$		0			0
$559$		0			0
$560$	1.22474	−	1.22474i	1.22474	−	1.22474i
$561$		0			0
$562$		0			0
$563$	−0.965926	−	1.67303i	−0.965926	−	1.67303i	−0.707107	−	0.707107i	$-0.750000\pi$
$563$	−0.965926	−	1.67303i	−0.965926	−	1.67303i	−0.258819	−	0.965926i	$-0.583333\pi$
$564$	1.36603	+	2.36603i	1.36603	+	2.36603i
$565$	−1.50000	+	2.59808i	−1.50000	+	2.59808i
$566$		0			0
$567$	−2.63896	+	2.63896i	−2.63896	+	2.63896i
$568$	1.93185			1.93185
$569$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$569$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$570$	0.866025	+	1.50000i	0.866025	+	1.50000i
$571$	−0.965926	−	1.67303i	−0.965926	−	1.67303i	−0.707107	−	0.707107i	$-0.750000\pi$
$571$	−0.965926	−	1.67303i	−0.965926	−	1.67303i	−0.258819	−	0.965926i	$-0.583333\pi$
$572$		0			0
$573$	2.73205			2.73205
$574$	−0.965926	−	0.258819i	−0.965926	−	0.258819i
$575$		0			0
$576$	−1.36603	+	2.36603i	−1.36603	+	2.36603i
$577$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$577$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$578$	0.500000	+	0.866025i	0.500000	+	0.866025i
$579$		0			0
$580$		0			0
$581$		0			0
$582$		0			0
$583$		0			0
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−0.517638			−0.517638			−0.258819	−	0.965926i	$-0.583333\pi$
$587$	−0.517638			−0.517638			−0.258819	+	0.965926i	$0.583333\pi$
$588$			1.93185i			1.93185i
$589$		0			0
$590$		0			0
$591$	0.965926	+	1.67303i	0.965926	+	1.67303i
$592$		0			0
$593$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$593$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$594$	−1.73205			−1.73205
$595$		0			0
$596$		0			0
$597$	−1.36603	+	2.36603i	−1.36603	+	2.36603i
$598$		0			0
$599$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$599$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$600$	−1.93185	+	3.34607i	−1.93185	+	3.34607i
$601$		0			0		1.00000			$0$
$601$		0			0		−1.00000			$\pi$
$602$		0			0
$603$	−3.86370			−3.86370
$604$	−0.707107	+	1.22474i	−0.707107	+	1.22474i
$605$	−0.633975	−	1.09808i	−0.633975	−	1.09808i
$606$		0			0
$607$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$607$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$608$	−0.517638			−0.517638
$609$		0			0
$610$	1.73205			1.73205
$611$		0			0
$612$		0			0
$613$	0.866025	+	1.50000i	0.866025	+	1.50000i	0.866025	+	0.500000i	$0.166667\pi$
$613$	0.866025	+	1.50000i	0.866025	+	1.50000i			1.00000i	$0.5\pi$
$614$		0			0
$615$	3.34607			3.34607
$616$	−0.366025	+	0.366025i	−0.366025	+	0.366025i
$617$	−1.00000			−1.00000			−0.500000	−	0.866025i	$-0.666667\pi$
$617$	−1.00000			−1.00000			−0.500000	+	0.866025i	$0.666667\pi$
$618$		0			0
$619$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$619$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$620$		0			0
$621$		0			0
$622$	1.41421			1.41421
$623$		0			0
$624$		0			0
$625$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$626$		0			0
$627$	−0.258819	−	0.448288i	−0.258819	−	0.448288i
$628$		0			0
$629$		0			0
$630$	−1.22474	−	4.57081i	−1.22474	−	4.57081i
$631$		0			0		1.00000			$0$
$631$		0			0		−1.00000			$\pi$
$632$	−0.965926	+	1.67303i	−0.965926	+	1.67303i
$633$	1.86603	+	3.23205i	1.86603	+	3.23205i
$634$		0			0
$635$		0			0
$636$		0			0
$637$		0			0
$638$		0			0
$639$	2.63896	−	4.57081i	2.63896	−	4.57081i
$640$	−0.866025	−	1.50000i	−0.866025	−	1.50000i
$641$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$641$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$642$		0			0
$643$	1.93185			1.93185			0.965926	−	0.258819i	$-0.0833333\pi$
$643$	1.93185			1.93185			0.965926	+	0.258819i	$0.0833333\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$647$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$648$	1.86603	+	3.23205i	1.86603	+	3.23205i
$649$		0			0
$650$		0			0
$651$		0			0
$652$		0			0
$653$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$653$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$654$		0			0
$655$		0			0
$656$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$657$		0			0
$658$	1.00000	−	1.00000i	1.00000	−	1.00000i
$659$	1.41421			1.41421			0.707107	−	0.707107i	$-0.250000\pi$
$659$	1.41421			1.41421			0.707107	+	0.707107i	$0.250000\pi$
$660$	0.866025	−	1.50000i	0.866025	−	1.50000i
$661$	0.866025	+	1.50000i	0.866025	+	1.50000i	0.866025	+	0.500000i	$0.166667\pi$
$661$	0.866025	+	1.50000i	0.866025	+	1.50000i			1.00000i	$0.5\pi$
$662$	−0.707107	−	1.22474i	−0.707107	−	1.22474i
$663$		0			0
$664$		0			0
$665$	0.633975	−	0.633975i	0.633975	−	0.633975i
$666$		0			0
$667$		0			0
$668$	0.707107	+	1.22474i	0.707107	+	1.22474i
$669$		0			0
$670$	1.22474	−	2.12132i	1.22474	−	2.12132i
$671$	−0.517638			−0.517638
$672$	1.86603	+	0.500000i	1.86603	+	0.500000i
$673$		0			0		1.00000			$0$
$673$		0			0		−1.00000			$\pi$
$674$	0.866025	−	1.50000i	0.866025	−	1.50000i
$675$	3.34607	+	5.79555i	3.34607	+	5.79555i
$676$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$677$	−0.866025	+	1.50000i	−0.866025	+	1.50000i			1.00000i	$0.5\pi$
$677$	−0.866025	+	1.50000i	−0.866025	+	1.50000i	−0.866025	+	0.500000i	$0.833333\pi$
$678$	−3.34607			−3.34607
$679$		0			0
$680$		0			0
$681$	0.500000	−	0.866025i	0.500000	−	0.866025i
$682$		0			0
$683$	−0.707107	−	1.22474i	−0.707107	−	1.22474i	−0.965926	−	0.258819i	$-0.916667\pi$
$683$	−0.707107	−	1.22474i	−0.707107	−	1.22474i	0.258819	−	0.965926i	$-0.416667\pi$
$684$	−0.707107	+	1.22474i	−0.707107	+	1.22474i
$685$		0			0
$686$	0.965926	−	0.258819i	0.965926	−	0.258819i
$687$		0			0
$688$		0			0
$689$		0			0
$690$		0			0
$691$	0.965926	−	1.67303i	0.965926	−	1.67303i	0.258819	−	0.965926i	$-0.416667\pi$
$691$	0.965926	−	1.67303i	0.965926	−	1.67303i	0.707107	−	0.707107i	$-0.250000\pi$
$692$	−1.00000			−1.00000
$693$	0.366025	+	1.36603i	0.366025	+	1.36603i
$694$	−0.517638			−0.517638
$695$		0			0
$696$		0			0
$697$		0			0
$698$		0			0
$699$		0			0
$700$	1.93185	+	0.517638i	1.93185	+	0.517638i
$701$	−1.73205			−1.73205			−0.866025	−	0.500000i	$-0.833333\pi$
$701$	−1.73205			−1.73205			−0.866025	+	0.500000i	$0.833333\pi$
$702$		0			0
$703$		0			0
$704$	0.258819	+	0.448288i	0.258819	+	0.448288i
$705$	−2.36603	+	4.09808i	−2.36603	+	4.09808i
$706$	−1.73205			−1.73205
$707$		0			0
$708$		0			0
$709$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$709$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$710$	1.67303	+	2.89778i	1.67303	+	2.89778i
$711$	2.63896	+	4.57081i	2.63896	+	4.57081i
$712$		0			0
$713$		0			0
$714$		0			0
$715$		0			0
$716$	−0.965926	+	1.67303i	−0.965926	+	1.67303i
$717$	0.500000	+	0.866025i	0.500000	+	0.866025i
$718$		0			0
$719$	−0.258819	+	0.448288i	−0.258819	+	0.448288i	−0.965926	−	0.258819i	$-0.916667\pi$
$719$	−0.258819	+	0.448288i	−0.258819	+	0.448288i	0.707107	+	0.707107i	$0.250000\pi$
$720$	−4.73205			−4.73205
$721$		0			0
$722$	0.732051			0.732051
$723$	−1.93185	+	3.34607i	−1.93185	+	3.34607i
$724$		0			0
$725$		0			0
$726$	0.707107	−	1.22474i	0.707107	−	1.22474i
$727$	1.41421			1.41421			0.707107	−	0.707107i	$-0.250000\pi$
$727$	1.41421			1.41421			0.707107	+	0.707107i	$0.250000\pi$
$728$		0			0
$729$	3.73205			3.73205
$730$		0			0
$731$		0			0
$732$	0.965926	+	1.67303i	0.965926	+	1.67303i
$733$	1.00000	−	1.73205i	1.00000	−	1.73205i	0.500000	−	0.866025i	$-0.333333\pi$
$733$	1.00000	−	1.73205i	1.00000	−	1.73205i	0.500000	−	0.866025i	$-0.333333\pi$
$734$		0			0
$735$	−2.89778	+	1.67303i	−2.89778	+	1.67303i
$736$		0			0
$737$	−0.366025	+	0.633975i	−0.366025	+	0.633975i
$738$	1.36603	+	2.36603i	1.36603	+	2.36603i
$739$		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			0
$743$		0			0		1.00000			$0$
$743$		0			0		−1.00000			$\pi$
$744$		0			0
$745$		0			0
$746$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$747$		0			0
$748$		0			0
$749$		0			0
$750$	−3.34607			−3.34607
$751$	0.258819	−	0.448288i	0.258819	−	0.448288i	−0.707107	−	0.707107i	$-0.750000\pi$
$751$	0.258819	−	0.448288i	0.258819	−	0.448288i	0.965926	+	0.258819i	$0.0833333\pi$
$752$	−0.707107	−	1.22474i	−0.707107	−	1.22474i
$753$		0			0
$754$		0			0
$755$	−2.44949			−2.44949
$756$	2.36603	−	2.36603i	2.36603	−	2.36603i
$757$		0			0		1.00000			$0$
$757$		0			0		−1.00000			$\pi$
$758$		0			0
$759$		0			0
$760$	−0.448288	−	0.776457i	−0.448288	−	0.776457i
$761$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$761$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$762$		0			0
$763$		0			0
$764$	−1.41421			−1.41421
$765$		0			0
$766$	0.965926	+	1.67303i	0.965926	+	1.67303i
$767$		0			0
$768$	0.965926	−	1.67303i	0.965926	−	1.67303i
$769$	−1.00000			−1.00000			−0.500000	−	0.866025i	$-0.666667\pi$
$769$	−1.00000			−1.00000			−0.500000	+	0.866025i	$0.666667\pi$
$770$	−0.866025	−	0.232051i	−0.866025	−	0.232051i
$771$		0			0
$772$		0			0
$773$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$773$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$		0			0
$778$	1.00000			1.00000
$779$	−0.258819	+	0.448288i	−0.258819	+	0.448288i
$780$		0			0
$781$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$782$		0			0
$783$		0			0
$784$		−	1.00000i		−	1.00000i
$785$		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
$789$	1.36603	−	2.36603i	1.36603	−	2.36603i
$790$	−3.34607			−3.34607
$791$	0.448288	+	1.67303i	0.448288	+	1.67303i
$792$	1.41421			1.41421
$793$		0			0
$794$		0			0
$795$		0			0
$796$	0.707107	−	1.22474i	0.707107	−	1.22474i
$797$	−2.00000			−2.00000			−1.00000			$\pi$
$797$	−2.00000			−2.00000			−1.00000			$\pi$
$798$	0.965926	+	0.258819i	0.965926	+	0.258819i
$799$		0			0
$800$	1.00000	−	1.73205i	1.00000	−	1.73205i
$801$		0			0
$802$	0.866025	+	1.50000i	0.866025	+	1.50000i
$803$		0			0
$804$	2.73205			2.73205
$805$		0			0
$806$		0			0
$807$	0.965926	−	1.67303i	0.965926	−	1.67303i
$808$		0			0
$809$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$809$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$810$	−3.23205	+	5.59808i	−3.23205	+	5.59808i
$811$		0			0		1.00000			$0$
$811$		0			0		−1.00000			$\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$		0			0
$816$		0			0
$817$		0			0
$818$	1.73205			1.73205
$819$		0			0
$820$	−1.73205			−1.73205
$821$	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	$-0.166667\pi$
$822$		0			0
$823$	−0.258819	−	0.448288i	−0.258819	−	0.448288i	0.707107	−	0.707107i	$-0.250000\pi$
$823$	−0.258819	−	0.448288i	−0.258819	−	0.448288i	−0.965926	+	0.258819i	$0.916667\pi$
$824$		0			0
$825$	2.00000			2.00000
$826$		0			0
$827$	−1.41421			−1.41421			−0.707107	−	0.707107i	$-0.750000\pi$
$827$	−1.41421			−1.41421			−0.707107	+	0.707107i	$0.750000\pi$
$828$		0			0
$829$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$829$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$830$		0			0
$831$	1.67303	−	2.89778i	1.67303	−	2.89778i
$832$		0			0
$833$		0			0
$834$		0			0
$835$	−1.22474	+	2.12132i	−1.22474	+	2.12132i
$836$	0.133975	+	0.232051i	0.133975	+	0.232051i
$837$		0			0
$838$		0			0
$839$	1.93185			1.93185			0.965926	−	0.258819i	$-0.0833333\pi$
$839$	1.93185			1.93185			0.965926	+	0.258819i	$0.0833333\pi$
$840$	0.866025	+	3.23205i	0.866025	+	3.23205i
$841$	1.00000			1.00000
$842$		0			0
$843$		0			0
$844$	−0.965926	−	1.67303i	−0.965926	−	1.67303i
$845$	0.866025	−	1.50000i	0.866025	−	1.50000i
$846$	−3.86370			−3.86370
$847$	−0.707107	−	0.189469i	−0.707107	−	0.189469i
$848$		0			0
$849$		0			0
$850$		0			0
$851$		0			0
$852$	−1.86603	+	3.23205i	−1.86603	+	3.23205i
$853$	−1.00000			−1.00000			−0.500000	−	0.866025i	$-0.666667\pi$
$853$	−1.00000			−1.00000			−0.500000	+	0.866025i	$0.666667\pi$
$854$	0.707107	−	0.707107i	0.707107	−	0.707107i
$855$	−2.44949			−2.44949
$856$		0			0
$857$	−0.866025	−	1.50000i	−0.866025	−	1.50000i	−0.866025	−	0.500000i	$-0.833333\pi$
$857$	−0.866025	−	1.50000i	−0.866025	−	1.50000i		−	1.00000i	$-0.5\pi$
$858$		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
$861$	1.36603	−	1.36603i	1.36603	−	1.36603i
$862$		0			0
$863$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$863$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$864$	−1.67303	−	2.89778i	−1.67303	−	2.89778i
$865$	−0.866025	−	1.50000i	−0.866025	−	1.50000i
$866$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$867$	−1.93185			−1.93185
$868$		0			0
$869$	1.00000			1.00000
$870$		0			0
$871$		0			0
$872$		0			0
$873$		0			0
$874$		0			0
$875$	0.448288	+	1.67303i	0.448288	+	1.67303i
$876$		0			0
$877$	−0.500000	+	0.866025i	−0.500000	+	0.866025i	0.500000	+	0.866025i	$0.333333\pi$
$877$	−0.500000	+	0.866025i	−0.500000	+	0.866025i	−1.00000			$\pi$
$878$	0.965926	+	1.67303i	0.965926	+	1.67303i
$879$		0			0
$880$	−0.448288	+	0.776457i	−0.448288	+	0.776457i
$881$	−1.00000			−1.00000			−0.500000	−	0.866025i	$-0.666667\pi$
$881$	−1.00000			−1.00000			−0.500000	+	0.866025i	$0.666667\pi$
$882$	−2.36603	−	1.36603i	−2.36603	−	1.36603i
$883$	1.41421			1.41421			0.707107	−	0.707107i	$-0.250000\pi$
$883$	1.41421			1.41421			0.707107	+	0.707107i	$0.250000\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	0.258819	−	0.448288i	0.258819	−	0.448288i	−0.707107	−	0.707107i	$-0.750000\pi$
$887$	0.258819	−	0.448288i	0.258819	−	0.448288i	0.965926	+	0.258819i	$0.0833333\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$	0.965926	−	1.67303i	0.965926	−	1.67303i
$892$		0			0
$893$	−0.366025	−	0.633975i	−0.366025	−	0.633975i
$894$		0			0
$895$	−3.34607			−3.34607
$896$	−0.965926	−	0.258819i	−0.965926	−	0.258819i
$897$		0			0
$898$	0.500000	−	0.866025i	0.500000	−	0.866025i
$899$		0			0
$900$	−2.73205	−	4.73205i	−2.73205	−	4.73205i
$901$		0			0
$902$	0.517638			0.517638
$903$		0			0
$904$	1.73205			1.73205
$905$		0			0
$906$	−1.36603	−	2.36603i	−1.36603	−	2.36603i
$907$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$907$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$908$	−0.258819	+	0.448288i	−0.258819	+	0.448288i
$909$		0			0
$910$		0			0
$911$		0			0		1.00000			$0$
$911$		0			0		−1.00000			$\pi$
$912$	0.500000	−	0.866025i	0.500000	−	0.866025i
$913$		0			0
$914$		0			0
$915$	−1.67303	+	2.89778i	−1.67303	+	2.89778i
$916$		0			0
$917$		0			0
$918$		0			0
$919$	0.258819	−	0.448288i	0.258819	−	0.448288i	−0.707107	−	0.707107i	$-0.750000\pi$
$919$	0.258819	−	0.448288i	0.258819	−	0.448288i	0.965926	+	0.258819i	$0.0833333\pi$
$920$		0			0
$921$		0			0
$922$	0.866025	−	1.50000i	0.866025	−	1.50000i
$923$		0			0
$924$	−0.258819	−	0.965926i	−0.258819	−	0.965926i
$925$		0			0
$926$	−0.965926	+	1.67303i	−0.965926	+	1.67303i
$927$		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
$931$		−	0.517638i		−	0.517638i
$932$		0			0
$933$	−1.36603	+	2.36603i	−1.36603	+	2.36603i
$934$		0			0
$935$		0			0
$936$		0			0
$937$		0			0		1.00000			$0$
$937$		0			0		−1.00000			$\pi$
$938$	−0.366025	−	1.36603i	−0.366025	−	1.36603i
$939$		0			0
$940$	1.22474	−	2.12132i	1.22474	−	2.12132i
$941$	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
$945$	5.59808	+	1.50000i	5.59808	+	1.50000i
$946$		0			0
$947$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$947$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$948$	−1.86603	−	3.23205i	−1.86603	−	3.23205i
$949$		0			0
$950$	0.517638	−	0.896575i	0.517638	−	0.896575i
$951$		0			0
$952$		0			0
$953$		0			0			−	1.00000i	$-0.5\pi$
$953$		0			0				1.00000i	$0.5\pi$
$954$		0			0
$955$	−1.22474	−	2.12132i	−1.22474	−	2.12132i
$956$	−0.258819	−	0.448288i	−0.258819	−	0.448288i
$957$		0			0
$958$	0.517638			0.517638
$959$		0			0
$960$	3.34607			3.34607
$961$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$962$		0			0
$963$		0			0
$964$	1.00000	−	1.73205i	1.00000	−	1.73205i
$965$		0			0
$966$		0			0
$967$	−0.517638			−0.517638			−0.258819	−	0.965926i	$-0.583333\pi$
$967$	−0.517638			−0.517638			−0.258819	+	0.965926i	$0.583333\pi$
$968$	−0.366025	+	0.633975i	−0.366025	+	0.633975i
$969$		0			0
$970$		0			0
$971$	−0.707107	+	1.22474i	−0.707107	+	1.22474i	0.258819	+	0.965926i	$0.416667\pi$
$971$	−0.707107	+	1.22474i	−0.707107	+	1.22474i	−0.965926	+	0.258819i	$0.916667\pi$
$972$	−3.86370			−3.86370
$973$		0			0
$974$		0			0
$975$		0			0
$976$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$977$		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
$980$	1.50000	−	0.866025i	1.50000	−	0.866025i
$981$		0			0
$982$		0			0
$983$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$983$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$984$	−0.965926	−	1.67303i	−0.965926	−	1.67303i
$985$	0.866025	−	1.50000i	0.866025	−	1.50000i
$986$		0			0
$987$	0.707107	+	2.63896i	0.707107	+	2.63896i
$988$		0			0
$989$		0			0
$990$	1.22474	+	2.12132i	1.22474	+	2.12132i
$991$	−0.258819	−	0.448288i	−0.258819	−	0.448288i	0.707107	−	0.707107i	$-0.250000\pi$
$991$	−0.258819	−	0.448288i	−0.258819	−	0.448288i	−0.965926	+	0.258819i	$0.916667\pi$
$992$		0			0
$993$	2.73205			2.73205
$994$	1.86603	+	0.500000i	1.86603	+	0.500000i
$995$	2.44949			2.44949
$996$		0			0
$997$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$997$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$998$	0.707107	+	1.22474i	0.707107	+	1.22474i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1148.1.o.d.163.4	yes	8
4.3	odd	2	inner	1148.1.o.d.163.1	✓	8
7.4	even	3	inner	1148.1.o.d.655.4	yes	8
28.11	odd	6	inner	1148.1.o.d.655.1	yes	8
41.40	even	2	inner	1148.1.o.d.163.1	✓	8
164.163	odd	2	CM	1148.1.o.d.163.4	yes	8
287.81	even	6	inner	1148.1.o.d.655.1	yes	8
1148.655	odd	6	inner	1148.1.o.d.655.4	yes	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1148.1.o.d.163.1	✓	8	4.3	odd	2	inner
1148.1.o.d.163.1	✓	8	41.40	even	2	inner
1148.1.o.d.163.4	yes	8	1.1	even	1	trivial
1148.1.o.d.163.4	yes	8	164.163	odd	2	CM
1148.1.o.d.655.1	yes	8	28.11	odd	6	inner
1148.1.o.d.655.1	yes	8	287.81	even	6	inner
1148.1.o.d.655.4	yes	8	7.4	even	3	inner
1148.1.o.d.655.4	yes	8	1148.655	odd	6	inner

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:	\( N \)	\(=\)	\( 1148 = 2^{2} \cdot 7 \cdot 41 \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	1148.o (of order \(6\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(0.500000 - 0.866025i) q^{2} +(0.965926 + 1.67303i) q^{3} +(-0.500000 - 0.866025i) q^{4} +(0.866025 - 1.50000i) q^{5} +1.93185 q^{6} +(-0.258819 - 0.965926i) q^{7} -1.00000 q^{8} +(-1.36603 + 2.36603i) q^{9} +O(q^{10})\)	\(q+(0.500000 - 0.866025i) q^{2} +(0.965926 + 1.67303i) q^{3} +(-0.500000 - 0.866025i) q^{4} +(0.866025 - 1.50000i) q^{5} +1.93185 q^{6} +(-0.258819 - 0.965926i) q^{7} -1.00000 q^{8} +(-1.36603 + 2.36603i) q^{9} +(-0.866025 - 1.50000i) q^{10} +(0.258819 + 0.448288i) q^{11} +(0.965926 - 1.67303i) q^{12} +(-0.965926 - 0.258819i) q^{14} +3.34607 q^{15} +(-0.500000 + 0.866025i) q^{16} +(1.36603 + 2.36603i) q^{18} +(-0.258819 + 0.448288i) q^{19} -1.73205 q^{20} +(1.36603 - 1.36603i) q^{21} +0.517638 q^{22} +(-0.965926 - 1.67303i) q^{24} +(-1.00000 - 1.73205i) q^{25} -3.34607 q^{27} +(-0.707107 + 0.707107i) q^{28} +(1.67303 - 2.89778i) q^{30} +(0.500000 + 0.866025i) q^{32} +(-0.500000 + 0.866025i) q^{33} +(-1.67303 - 0.448288i) q^{35} +2.73205 q^{36} +(0.258819 + 0.448288i) q^{38} +(-0.866025 + 1.50000i) q^{40} +1.00000 q^{41} +(-0.500000 - 1.86603i) q^{42} +(0.258819 - 0.448288i) q^{44} +(2.36603 + 4.09808i) q^{45} +(-0.707107 + 1.22474i) q^{47} -1.93185 q^{48} +(-0.866025 + 0.500000i) q^{49} -2.00000 q^{50} +(-1.67303 + 2.89778i) q^{54} +0.896575 q^{55} +(0.258819 + 0.965926i) q^{56} -1.00000 q^{57} +(-1.67303 - 2.89778i) q^{60} +(-0.500000 + 0.866025i) q^{61} +(2.63896 + 0.707107i) q^{63} +1.00000 q^{64} +(0.500000 + 0.866025i) q^{66} +(0.707107 + 1.22474i) q^{67} +(-1.22474 + 1.22474i) q^{70} -1.93185 q^{71} +(1.36603 - 2.36603i) q^{72} +(1.93185 - 3.34607i) q^{75} +0.517638 q^{76} +(0.366025 - 0.366025i) q^{77} +(0.965926 - 1.67303i) q^{79} +(0.866025 + 1.50000i) q^{80} +(-1.86603 - 3.23205i) q^{81} +(0.500000 - 0.866025i) q^{82} +(-1.86603 - 0.500000i) q^{84} +(-0.258819 - 0.448288i) q^{88} +4.73205 q^{90} +(0.707107 + 1.22474i) q^{94} +(0.448288 + 0.776457i) q^{95} +(-0.965926 + 1.67303i) q^{96} +1.00000i q^{98} -1.41421 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 4 q^{2} - 4 q^{4} - 8 q^{8} - 4 q^{9}+O(q^{10}) \) 8 * q + 4 * q^2 - 4 * q^4 - 8 * q^8 - 4 * q^9	\( 8 q + 4 q^{2} - 4 q^{4} - 8 q^{8} - 4 q^{9} - 4 q^{16} + 4 q^{18} + 4 q^{21} - 8 q^{25} + 4 q^{32} - 4 q^{33} + 8 q^{36} + 8 q^{41} - 4 q^{42} + 12 q^{45} - 16 q^{50} - 8 q^{57} - 4 q^{61} + 8 q^{64} + 4 q^{66} + 4 q^{72} - 4 q^{77} - 8 q^{81} + 4 q^{82} - 8 q^{84} + 24 q^{90}+O(q^{100}) \) 8 * q + 4 * q^2 - 4 * q^4 - 8 * q^8 - 4 * q^9 - 4 * q^16 + 4 * q^18 + 4 * q^21 - 8 * q^25 + 4 * q^32 - 4 * q^33 + 8 * q^36 + 8 * q^41 - 4 * q^42 + 12 * q^45 - 16 * q^50 - 8 * q^57 - 4 * q^61 + 8 * q^64 + 4 * q^66 + 4 * q^72 - 4 * q^77 - 8 * q^81 + 4 * q^82 - 8 * q^84 + 24 * q^90

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