LMFDB - Embedded newform 3024.1.eg.a.53.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3024,1,Mod(53,3024)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3024, base_ring=CyclotomicField(12))

chi = DirichletCharacter(H, H._module([0, 3, 6, 8]))

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

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

chi := DirichletCharacter("3024.53");

S:= CuspForms(chi, 1);

N := Newforms(S);

Level:	$ N $	$=$	$ 3024 = 2^{4} \cdot 3^{3} \cdot 7 $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	3024.eg (of order $12$, degree $4$, 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:	$1.50917259820$
Analytic rank:	$0$
Dimension:	$8$
Relative dimension:	$2$ over $\Q(\zeta_{12})$
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:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Projective image:	$S_{4}$
Projective field:	Galois closure of 4.2.2709504.12

Embedding invariants

Embedding label			53.1
Root			$0.258819 - 0.965926i$ of defining polynomial
Character	$\chi$	$=$	3024.53
Dual form			3024.1.eg.a.1997.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-0.258819 + 0.965926i) q^{2} +(-0.866025 - 0.500000i) q^{4} +(0.965926 + 0.258819i) q^{5} +1.00000i q^{7} +(0.707107 - 0.707107i) q^{8} +O(q^{10})$	\(q+(-0.258819 + 0.965926i) q^{2} +(-0.866025 - 0.500000i) q^{4} +(0.965926 + 0.258819i) q^{5} +1.00000i q^{7} +(0.707107 - 0.707107i) q^{8} +(-0.500000 + 0.866025i) q^{10} +(-0.965926 - 0.258819i) q^{14} +(0.500000 + 0.866025i) q^{16} +(1.36603 + 0.366025i) q^{19} +(-0.707107 - 0.707107i) q^{20} +(0.707107 + 1.22474i) q^{23} +(0.500000 - 0.866025i) q^{28} +(-0.707107 - 0.707107i) q^{29} +(1.00000 - 1.73205i) q^{31} +(-0.965926 + 0.258819i) q^{32} +(-0.258819 + 0.965926i) q^{35} +(-0.707107 + 1.22474i) q^{38} +(0.866025 - 0.500000i) q^{40} -1.41421 q^{41} +(1.00000 + 1.00000i) q^{43} +(-1.36603 + 0.366025i) q^{46} +(-1.22474 + 0.707107i) q^{47} -1.00000 q^{49} +(-0.258819 - 0.965926i) q^{53} +(0.707107 + 0.707107i) q^{56} +(0.866025 - 0.500000i) q^{58} +(0.258819 + 0.965926i) q^{59} +(1.41421 + 1.41421i) q^{62} -1.00000i q^{64} +(0.366025 + 1.36603i) q^{67} +(-0.866025 - 0.500000i) q^{70} +1.41421 q^{71} +(0.866025 + 0.500000i) q^{73} +(-1.00000 - 1.00000i) q^{76} +(-0.500000 - 0.866025i) q^{79} +(0.258819 + 0.965926i) q^{80} +(0.366025 - 1.36603i) q^{82} +(-1.22474 + 0.707107i) q^{86} -1.41421i q^{92} +(-0.366025 - 1.36603i) q^{94} +(1.22474 + 0.707107i) q^{95} +(0.258819 - 0.965926i) q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q+O(q^{10}) $ 8 * q	$ 8 q - 4 q^{10} + 4 q^{16} + 4 q^{19} + 4 q^{28} + 8 q^{31} + 8 q^{43} - 4 q^{46} - 8 q^{49} - 4 q^{67} - 8 q^{76} - 4 q^{79} - 4 q^{82} + 4 q^{94}+O(q^{100}) $ 8 * q - 4 * q^10 + 4 * q^16 + 4 * q^19 + 4 * q^28 + 8 * q^31 + 8 * q^43 - 4 * q^46 - 8 * q^49 - 4 * q^67 - 8 * q^76 - 4 * q^79 - 4 * q^82 + 4 * q^94

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/3024\mathbb{Z}\right)^\times$.

$n$	$757$	$785$	$1135$	$2593$
$\chi(n)$	$e\left(\frac{1}{4}\right)$	$-1$	$1$	$e\left(\frac{2}{3}\right)$

Coefficient data

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

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

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

$2$	−0.258819	+	0.965926i	−0.258819	+	0.965926i
$2$	−0.258819	+	0.965926i	−0.258819	+	0.965926i
$3$		0			0
$3$		0			0
$4$	−0.866025	−	0.500000i	−0.866025	−	0.500000i
$5$	0.965926	+	0.258819i	0.965926	+	0.258819i	0.707107	−	0.707107i	$-0.250000\pi$
$5$	0.965926	+	0.258819i	0.965926	+	0.258819i	0.258819	+	0.965926i	$0.416667\pi$
$6$		0			0
$7$			1.00000i			1.00000i
$7$			1.00000i			1.00000i
$8$	0.707107	−	0.707107i	0.707107	−	0.707107i
$9$		0			0
$10$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$11$		0			0		0.965926	−	0.258819i	$-0.0833333\pi$
$11$		0			0		−0.965926	+	0.258819i	$0.916667\pi$
$12$		0			0
$13$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$13$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$14$	−0.965926	−	0.258819i	−0.965926	−	0.258819i
$15$		0			0
$16$	0.500000	+	0.866025i	0.500000	+	0.866025i
$17$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$17$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$18$		0			0
$19$	1.36603	+	0.366025i	1.36603	+	0.366025i	0.866025	−	0.500000i	$-0.166667\pi$
$19$	1.36603	+	0.366025i	1.36603	+	0.366025i	0.500000	+	0.866025i	$0.333333\pi$
$20$	−0.707107	−	0.707107i	−0.707107	−	0.707107i
$21$		0			0
$22$		0			0
$23$	0.707107	+	1.22474i	0.707107	+	1.22474i	0.965926	+	0.258819i	$0.0833333\pi$
$23$	0.707107	+	1.22474i	0.707107	+	1.22474i	−0.258819	+	0.965926i	$0.583333\pi$
$24$		0			0
$25$		0			0
$26$		0			0
$27$		0			0
$28$	0.500000	−	0.866025i	0.500000	−	0.866025i
$29$	−0.707107	−	0.707107i	−0.707107	−	0.707107i	0.258819	−	0.965926i	$-0.416667\pi$
$29$	−0.707107	−	0.707107i	−0.707107	−	0.707107i	−0.965926	+	0.258819i	$0.916667\pi$
$30$		0			0
$31$	1.00000	−	1.73205i	1.00000	−	1.73205i	0.500000	−	0.866025i	$-0.333333\pi$
$31$	1.00000	−	1.73205i	1.00000	−	1.73205i	0.500000	−	0.866025i	$-0.333333\pi$
$32$	−0.965926	+	0.258819i	−0.965926	+	0.258819i
$33$		0			0
$34$		0			0
$35$	−0.258819	+	0.965926i	−0.258819	+	0.965926i
$36$		0			0
$37$		0			0		−0.965926	−	0.258819i	$-0.916667\pi$
$37$		0			0		0.965926	+	0.258819i	$0.0833333\pi$
$38$	−0.707107	+	1.22474i	−0.707107	+	1.22474i
$39$		0			0
$40$	0.866025	−	0.500000i	0.866025	−	0.500000i
$41$	−1.41421			−1.41421			−0.707107	−	0.707107i	$-0.750000\pi$
$41$	−1.41421			−1.41421			−0.707107	+	0.707107i	$0.750000\pi$
$42$		0			0
$43$	1.00000	+	1.00000i	1.00000	+	1.00000i	1.00000			$0$
$43$	1.00000	+	1.00000i	1.00000	+	1.00000i			1.00000i	$0.5\pi$
$44$		0			0
$45$		0			0
$46$	−1.36603	+	0.366025i	−1.36603	+	0.366025i
$47$	−1.22474	+	0.707107i	−1.22474	+	0.707107i	−0.965926	−	0.258819i	$-0.916667\pi$
$47$	−1.22474	+	0.707107i	−1.22474	+	0.707107i	−0.258819	+	0.965926i	$0.583333\pi$
$48$		0			0
$49$	−1.00000			−1.00000
$50$		0			0
$51$		0			0
$52$		0			0
$53$	−0.258819	−	0.965926i	−0.258819	−	0.965926i	−0.965926	−	0.258819i	$-0.916667\pi$
$53$	−0.258819	−	0.965926i	−0.258819	−	0.965926i	0.707107	−	0.707107i	$-0.250000\pi$
$54$		0			0
$55$		0			0
$56$	0.707107	+	0.707107i	0.707107	+	0.707107i
$57$		0			0
$58$	0.866025	−	0.500000i	0.866025	−	0.500000i
$59$	0.258819	+	0.965926i	0.258819	+	0.965926i	0.965926	+	0.258819i	$0.0833333\pi$
$59$	0.258819	+	0.965926i	0.258819	+	0.965926i	−0.707107	+	0.707107i	$0.750000\pi$
$60$		0			0
$61$		0			0		0.258819	−	0.965926i	$-0.416667\pi$
$61$		0			0		−0.258819	+	0.965926i	$0.583333\pi$
$62$	1.41421	+	1.41421i	1.41421	+	1.41421i
$63$		0			0
$64$		−	1.00000i		−	1.00000i
$65$		0			0
$66$		0			0
$67$	0.366025	+	1.36603i	0.366025	+	1.36603i	0.866025	+	0.500000i	$0.166667\pi$
$67$	0.366025	+	1.36603i	0.366025	+	1.36603i	−0.500000	+	0.866025i	$0.666667\pi$
$68$		0			0
$69$		0			0
$70$	−0.866025	−	0.500000i	−0.866025	−	0.500000i
$71$	1.41421			1.41421			0.707107	−	0.707107i	$-0.250000\pi$
$71$	1.41421			1.41421			0.707107	+	0.707107i	$0.250000\pi$
$72$		0			0
$73$	0.866025	+	0.500000i	0.866025	+	0.500000i	0.866025	−	0.500000i	$-0.166667\pi$
$73$	0.866025	+	0.500000i	0.866025	+	0.500000i			1.00000i	$0.5\pi$
$74$		0			0
$75$		0			0
$76$	−1.00000	−	1.00000i	−1.00000	−	1.00000i
$77$		0			0
$78$		0			0
$79$	−0.500000	−	0.866025i	−0.500000	−	0.866025i	0.500000	−	0.866025i	$-0.333333\pi$
$79$	−0.500000	−	0.866025i	−0.500000	−	0.866025i	−1.00000			$\pi$
$80$	0.258819	+	0.965926i	0.258819	+	0.965926i
$81$		0			0
$82$	0.366025	−	1.36603i	0.366025	−	1.36603i
$83$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$83$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$84$		0			0
$85$		0			0
$86$	−1.22474	+	0.707107i	−1.22474	+	0.707107i
$87$		0			0
$88$		0			0
$89$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$89$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$90$		0			0
$91$		0			0
$92$		−	1.41421i		−	1.41421i
$93$		0			0
$94$	−0.366025	−	1.36603i	−0.366025	−	1.36603i
$95$	1.22474	+	0.707107i	1.22474	+	0.707107i
$96$		0			0
$97$		0			0			−	1.00000i	$-0.5\pi$
$97$		0			0				1.00000i	$0.5\pi$
$98$	0.258819	−	0.965926i	0.258819	−	0.965926i
$99$		0			0
$100$		0			0
$101$	0.258819	+	0.965926i	0.258819	+	0.965926i	0.965926	+	0.258819i	$0.0833333\pi$
$101$	0.258819	+	0.965926i	0.258819	+	0.965926i	−0.707107	+	0.707107i	$0.750000\pi$
$102$		0			0
$103$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$103$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$104$		0			0
$105$		0			0
$106$	1.00000			1.00000
$107$	−0.965926	−	0.258819i	−0.965926	−	0.258819i	−0.258819	−	0.965926i	$-0.583333\pi$
$107$	−0.965926	−	0.258819i	−0.965926	−	0.258819i	−0.707107	+	0.707107i	$0.750000\pi$
$108$		0			0
$109$		0			0		0.965926	−	0.258819i	$-0.0833333\pi$
$109$		0			0		−0.965926	+	0.258819i	$0.916667\pi$
$110$		0			0
$111$		0			0
$112$	−0.866025	+	0.500000i	−0.866025	+	0.500000i
$113$		0			0		1.00000			$0$
$113$		0			0		−1.00000			$\pi$
$114$		0			0
$115$	0.366025	+	1.36603i	0.366025	+	1.36603i
$116$	0.258819	+	0.965926i	0.258819	+	0.965926i
$117$		0			0
$118$	−1.00000			−1.00000
$119$		0			0
$120$		0			0
$121$	0.866025	−	0.500000i	0.866025	−	0.500000i
$122$		0			0
$123$		0			0
$124$	−1.73205	+	1.00000i	−1.73205	+	1.00000i
$125$	−0.707107	−	0.707107i	−0.707107	−	0.707107i
$126$		0			0
$127$	−1.00000			−1.00000			−0.500000	−	0.866025i	$-0.666667\pi$
$127$	−1.00000			−1.00000			−0.500000	+	0.866025i	$0.666667\pi$
$128$	0.965926	+	0.258819i	0.965926	+	0.258819i
$129$		0			0
$130$		0			0
$131$		0			0		−0.965926	−	0.258819i	$-0.916667\pi$
$131$		0			0		0.965926	+	0.258819i	$0.0833333\pi$
$132$		0			0
$133$	−0.366025	+	1.36603i	−0.366025	+	1.36603i
$134$	−1.41421			−1.41421
$135$		0			0
$136$		0			0
$137$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$137$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$138$		0			0
$139$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$139$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$140$	0.707107	−	0.707107i	0.707107	−	0.707107i
$141$		0			0
$142$	−0.366025	+	1.36603i	−0.366025	+	1.36603i
$143$		0			0
$144$		0			0
$145$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$146$	−0.707107	+	0.707107i	−0.707107	+	0.707107i
$147$		0			0
$148$		0			0
$149$	−0.965926	−	0.258819i	−0.965926	−	0.258819i	−0.258819	−	0.965926i	$-0.583333\pi$
$149$	−0.965926	−	0.258819i	−0.965926	−	0.258819i	−0.707107	+	0.707107i	$0.750000\pi$
$150$		0			0
$151$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$151$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$152$	1.22474	−	0.707107i	1.22474	−	0.707107i
$153$		0			0
$154$		0			0
$155$	1.41421	−	1.41421i	1.41421	−	1.41421i
$156$		0			0
$157$		0			0		0.965926	−	0.258819i	$-0.0833333\pi$
$157$		0			0		−0.965926	+	0.258819i	$0.916667\pi$
$158$	0.965926	−	0.258819i	0.965926	−	0.258819i
$159$		0			0
$160$	−1.00000			−1.00000
$161$	−1.22474	+	0.707107i	−1.22474	+	0.707107i
$162$		0			0
$163$	−1.36603	−	0.366025i	−1.36603	−	0.366025i	−0.500000	−	0.866025i	$-0.666667\pi$
$163$	−1.36603	−	0.366025i	−1.36603	−	0.366025i	−0.866025	+	0.500000i	$0.833333\pi$
$164$	1.22474	+	0.707107i	1.22474	+	0.707107i
$165$		0			0
$166$		0			0
$167$		0			0			−	1.00000i	$-0.5\pi$
$167$		0			0				1.00000i	$0.5\pi$
$168$		0			0
$169$			1.00000i			1.00000i
$170$		0			0
$171$		0			0
$172$	−0.366025	−	1.36603i	−0.366025	−	1.36603i
$173$	0.258819	−	0.965926i	0.258819	−	0.965926i	−0.707107	−	0.707107i	$-0.750000\pi$
$173$	0.258819	−	0.965926i	0.258819	−	0.965926i	0.965926	−	0.258819i	$-0.0833333\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$		0			0
$178$		0			0
$179$	0.965926	−	0.258819i	0.965926	−	0.258819i	0.258819	−	0.965926i	$-0.416667\pi$
$179$	0.965926	−	0.258819i	0.965926	−	0.258819i	0.707107	+	0.707107i	$0.250000\pi$
$180$		0			0
$181$	1.00000	+	1.00000i	1.00000	+	1.00000i	1.00000			$0$
$181$	1.00000	+	1.00000i	1.00000	+	1.00000i			1.00000i	$0.5\pi$
$182$		0			0
$183$		0			0
$184$	1.36603	+	0.366025i	1.36603	+	0.366025i
$185$		0			0
$186$		0			0
$187$		0			0
$188$	1.41421			1.41421
$189$		0			0
$190$	−1.00000	+	1.00000i	−1.00000	+	1.00000i
$191$	−1.22474	+	0.707107i	−1.22474	+	0.707107i	−0.965926	−	0.258819i	$-0.916667\pi$
$191$	−1.22474	+	0.707107i	−1.22474	+	0.707107i	−0.258819	+	0.965926i	$0.583333\pi$
$192$		0			0
$193$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$193$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$194$		0			0
$195$		0			0
$196$	0.866025	+	0.500000i	0.866025	+	0.500000i
$197$	−0.707107	+	0.707107i	−0.707107	+	0.707107i	−0.965926	−	0.258819i	$-0.916667\pi$
$197$	−0.707107	+	0.707107i	−0.707107	+	0.707107i	0.258819	+	0.965926i	$0.416667\pi$
$198$		0			0
$199$	−0.866025	−	0.500000i	−0.866025	−	0.500000i		−	1.00000i	$-0.5\pi$
$199$	−0.866025	−	0.500000i	−0.866025	−	0.500000i	−0.866025	+	0.500000i	$0.833333\pi$
$200$		0			0
$201$		0			0
$202$	−1.00000			−1.00000
$203$	0.707107	−	0.707107i	0.707107	−	0.707107i
$204$		0			0
$205$	−1.36603	−	0.366025i	−1.36603	−	0.366025i
$206$		0			0
$207$		0			0
$208$		0			0
$209$		0			0
$210$		0			0
$211$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$211$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$212$	−0.258819	+	0.965926i	−0.258819	+	0.965926i
$213$		0			0
$214$	0.500000	−	0.866025i	0.500000	−	0.866025i
$215$	0.707107	+	1.22474i	0.707107	+	1.22474i
$216$		0			0
$217$	1.73205	+	1.00000i	1.73205	+	1.00000i
$218$		0			0
$219$		0			0
$220$		0			0
$221$		0			0
$222$		0			0
$223$	1.00000			1.00000			0.500000	−	0.866025i	$-0.333333\pi$
$223$	1.00000			1.00000			0.500000	+	0.866025i	$0.333333\pi$
$224$	−0.258819	−	0.965926i	−0.258819	−	0.965926i
$225$		0			0
$226$		0			0
$227$	−0.965926	+	0.258819i	−0.965926	+	0.258819i	−0.707107	−	0.707107i	$-0.750000\pi$
$227$	−0.965926	+	0.258819i	−0.965926	+	0.258819i	−0.258819	+	0.965926i	$0.583333\pi$
$228$		0			0
$229$		0			0		−0.965926	−	0.258819i	$-0.916667\pi$
$229$		0			0		0.965926	+	0.258819i	$0.0833333\pi$
$230$	−1.41421			−1.41421
$231$		0			0
$232$	−1.00000			−1.00000
$233$	−0.707107	−	1.22474i	−0.707107	−	1.22474i	−0.965926	−	0.258819i	$-0.916667\pi$
$233$	−0.707107	−	1.22474i	−0.707107	−	1.22474i	0.258819	−	0.965926i	$-0.416667\pi$
$234$		0			0
$235$	−1.36603	+	0.366025i	−1.36603	+	0.366025i
$236$	0.258819	−	0.965926i	0.258819	−	0.965926i
$237$		0			0
$238$		0			0
$239$		−	1.41421i		−	1.41421i	−0.707107	−	0.707107i	$-0.750000\pi$
$239$		−	1.41421i		−	1.41421i	0.707107	−	0.707107i	$-0.250000\pi$
$240$		0			0
$241$	0.500000	−	0.866025i	0.500000	−	0.866025i	−0.500000	−	0.866025i	$-0.666667\pi$
$241$	0.500000	−	0.866025i	0.500000	−	0.866025i	1.00000			$0$
$242$	0.258819	+	0.965926i	0.258819	+	0.965926i
$243$		0			0
$244$		0			0
$245$	−0.965926	−	0.258819i	−0.965926	−	0.258819i
$246$		0			0
$247$		0			0
$248$	−0.517638	−	1.93185i	−0.517638	−	1.93185i
$249$		0			0
$250$	0.866025	−	0.500000i	0.866025	−	0.500000i
$251$	−0.707107	+	0.707107i	−0.707107	+	0.707107i	−0.965926	−	0.258819i	$-0.916667\pi$
$251$	−0.707107	+	0.707107i	−0.707107	+	0.707107i	0.258819	+	0.965926i	$0.416667\pi$
$252$		0			0
$253$		0			0
$254$	0.258819	−	0.965926i	0.258819	−	0.965926i
$255$		0			0
$256$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$257$	1.22474	−	0.707107i	1.22474	−	0.707107i	0.258819	−	0.965926i	$-0.416667\pi$
$257$	1.22474	−	0.707107i	1.22474	−	0.707107i	0.965926	+	0.258819i	$0.0833333\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$		0			0
$262$		0			0
$263$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$263$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$264$		0			0
$265$		−	1.00000i		−	1.00000i
$266$	−1.22474	−	0.707107i	−1.22474	−	0.707107i
$267$		0			0
$268$	0.366025	−	1.36603i	0.366025	−	1.36603i
$269$	0.965926	−	0.258819i	0.965926	−	0.258819i	0.258819	−	0.965926i	$-0.416667\pi$
$269$	0.965926	−	0.258819i	0.965926	−	0.258819i	0.707107	+	0.707107i	$0.250000\pi$
$270$		0			0
$271$	−0.500000	−	0.866025i	−0.500000	−	0.866025i	0.500000	−	0.866025i	$-0.333333\pi$
$271$	−0.500000	−	0.866025i	−0.500000	−	0.866025i	−1.00000			$\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$		0			0
$276$		0			0
$277$	−1.36603	+	0.366025i	−1.36603	+	0.366025i	−0.866025	−	0.500000i	$-0.833333\pi$
$277$	−1.36603	+	0.366025i	−1.36603	+	0.366025i	−0.500000	+	0.866025i	$0.666667\pi$
$278$		0			0
$279$		0			0
$280$	0.500000	+	0.866025i	0.500000	+	0.866025i
$281$		0			0			−	1.00000i	$-0.5\pi$
$281$		0			0				1.00000i	$0.5\pi$
$282$		0			0
$283$		0			0		−0.258819	−	0.965926i	$-0.583333\pi$
$283$		0			0		0.258819	+	0.965926i	$0.416667\pi$
$284$	−1.22474	−	0.707107i	−1.22474	−	0.707107i
$285$		0			0
$286$		0			0
$287$		−	1.41421i		−	1.41421i
$288$		0			0
$289$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$290$	0.965926	−	0.258819i	0.965926	−	0.258819i
$291$		0			0
$292$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$293$	0.707107	−	0.707107i	0.707107	−	0.707107i	−0.258819	−	0.965926i	$-0.583333\pi$
$293$	0.707107	−	0.707107i	0.707107	−	0.707107i	0.965926	+	0.258819i	$0.0833333\pi$
$294$		0			0
$295$			1.00000i			1.00000i
$296$		0			0
$297$		0			0
$298$	0.500000	−	0.866025i	0.500000	−	0.866025i
$299$		0			0
$300$		0			0
$301$	−1.00000	+	1.00000i	−1.00000	+	1.00000i
$302$		0			0
$303$		0			0
$304$	0.366025	+	1.36603i	0.366025	+	1.36603i
$305$		0			0
$306$		0			0
$307$	1.00000	−	1.00000i	1.00000	−	1.00000i		−	1.00000i	$-0.5\pi$
$307$	1.00000	−	1.00000i	1.00000	−	1.00000i	1.00000			$0$
$308$		0			0
$309$		0			0
$310$	1.00000	+	1.73205i	1.00000	+	1.73205i
$311$	0.707107	−	1.22474i	0.707107	−	1.22474i	−0.258819	−	0.965926i	$-0.583333\pi$
$311$	0.707107	−	1.22474i	0.707107	−	1.22474i	0.965926	−	0.258819i	$-0.0833333\pi$
$312$		0			0
$313$	0.866025	−	0.500000i	0.866025	−	0.500000i		−	1.00000i	$-0.5\pi$
$313$	0.866025	−	0.500000i	0.866025	−	0.500000i	0.866025	+	0.500000i	$0.166667\pi$
$314$		0			0
$315$		0			0
$316$			1.00000i			1.00000i
$317$		0			0		−0.965926	−	0.258819i	$-0.916667\pi$
$317$		0			0		0.965926	+	0.258819i	$0.0833333\pi$
$318$		0			0
$319$		0			0
$320$	0.258819	−	0.965926i	0.258819	−	0.965926i
$321$		0			0
$322$	−0.366025	−	1.36603i	−0.366025	−	1.36603i
$323$		0			0
$324$		0			0
$325$		0			0
$326$	0.707107	−	1.22474i	0.707107	−	1.22474i
$327$		0			0
$328$	−1.00000	+	1.00000i	−1.00000	+	1.00000i
$329$	−0.707107	−	1.22474i	−0.707107	−	1.22474i
$330$		0			0
$331$		0			0		−0.965926	−	0.258819i	$-0.916667\pi$
$331$		0			0		0.965926	+	0.258819i	$0.0833333\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$			1.41421i			1.41421i
$336$		0			0
$337$	1.00000			1.00000			0.500000	−	0.866025i	$-0.333333\pi$
$337$	1.00000			1.00000			0.500000	+	0.866025i	$0.333333\pi$
$338$	−0.965926	−	0.258819i	−0.965926	−	0.258819i
$339$		0			0
$340$		0			0
$341$		0			0
$342$		0			0
$343$		−	1.00000i		−	1.00000i
$344$	1.41421			1.41421
$345$		0			0
$346$	0.866025	+	0.500000i	0.866025	+	0.500000i
$347$	0.258819	+	0.965926i	0.258819	+	0.965926i	0.965926	+	0.258819i	$0.0833333\pi$
$347$	0.258819	+	0.965926i	0.258819	+	0.965926i	−0.707107	+	0.707107i	$0.750000\pi$
$348$		0			0
$349$	1.00000	−	1.00000i	1.00000	−	1.00000i		−	1.00000i	$-0.5\pi$
$349$	1.00000	−	1.00000i	1.00000	−	1.00000i	1.00000			$0$
$350$		0			0
$351$		0			0
$352$		0			0
$353$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$353$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$354$		0			0
$355$	1.36603	+	0.366025i	1.36603	+	0.366025i
$356$		0			0
$357$		0			0
$358$			1.00000i			1.00000i
$359$	−0.707107	−	1.22474i	−0.707107	−	1.22474i	−0.965926	−	0.258819i	$-0.916667\pi$
$359$	−0.707107	−	1.22474i	−0.707107	−	1.22474i	0.258819	−	0.965926i	$-0.416667\pi$
$360$		0			0
$361$	0.866025	+	0.500000i	0.866025	+	0.500000i
$362$	−1.22474	+	0.707107i	−1.22474	+	0.707107i
$363$		0			0
$364$		0			0
$365$	0.707107	+	0.707107i	0.707107	+	0.707107i
$366$		0			0
$367$	0.500000	−	0.866025i	0.500000	−	0.866025i	−0.500000	−	0.866025i	$-0.666667\pi$
$367$	0.500000	−	0.866025i	0.500000	−	0.866025i	1.00000			$0$
$368$	−0.707107	+	1.22474i	−0.707107	+	1.22474i
$369$		0			0
$370$		0			0
$371$	0.965926	−	0.258819i	0.965926	−	0.258819i
$372$		0			0
$373$	−0.366025	+	1.36603i	−0.366025	+	1.36603i	0.500000	+	0.866025i	$0.333333\pi$
$373$	−0.366025	+	1.36603i	−0.366025	+	1.36603i	−0.866025	+	0.500000i	$0.833333\pi$
$374$		0			0
$375$		0			0
$376$	−0.366025	+	1.36603i	−0.366025	+	1.36603i
$377$		0			0
$378$		0			0
$379$	−1.00000	−	1.00000i	−1.00000	−	1.00000i		−	1.00000i	$-0.5\pi$
$379$	−1.00000	−	1.00000i	−1.00000	−	1.00000i	−1.00000			$\pi$
$380$	−0.707107	−	1.22474i	−0.707107	−	1.22474i
$381$		0			0
$382$	−0.366025	−	1.36603i	−0.366025	−	1.36603i
$383$	1.22474	−	0.707107i	1.22474	−	0.707107i	0.258819	−	0.965926i	$-0.416667\pi$
$383$	1.22474	−	0.707107i	1.22474	−	0.707107i	0.965926	+	0.258819i	$0.0833333\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−0.258819	−	0.965926i	−0.258819	−	0.965926i	−0.965926	−	0.258819i	$-0.916667\pi$
$389$	−0.258819	−	0.965926i	−0.258819	−	0.965926i	0.707107	−	0.707107i	$-0.250000\pi$
$390$		0			0
$391$		0			0
$392$	−0.707107	+	0.707107i	−0.707107	+	0.707107i
$393$		0			0
$394$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$395$	−0.258819	−	0.965926i	−0.258819	−	0.965926i
$396$		0			0
$397$		0			0		0.258819	−	0.965926i	$-0.416667\pi$
$397$		0			0		−0.258819	+	0.965926i	$0.583333\pi$
$398$	0.707107	−	0.707107i	0.707107	−	0.707107i
$399$		0			0
$400$		0			0
$401$	1.22474	−	0.707107i	1.22474	−	0.707107i	0.258819	−	0.965926i	$-0.416667\pi$
$401$	1.22474	−	0.707107i	1.22474	−	0.707107i	0.965926	+	0.258819i	$0.0833333\pi$
$402$		0			0
$403$		0			0
$404$	0.258819	−	0.965926i	0.258819	−	0.965926i
$405$		0			0
$406$	0.500000	+	0.866025i	0.500000	+	0.866025i
$407$		0			0
$408$		0			0
$409$	−0.866025	−	0.500000i	−0.866025	−	0.500000i		−	1.00000i	$-0.5\pi$
$409$	−0.866025	−	0.500000i	−0.866025	−	0.500000i	−0.866025	+	0.500000i	$0.833333\pi$
$410$	0.707107	−	1.22474i	0.707107	−	1.22474i
$411$		0			0
$412$		0			0
$413$	−0.965926	+	0.258819i	−0.965926	+	0.258819i
$414$		0			0
$415$		0			0
$416$		0			0
$417$		0			0
$418$		0			0
$419$	1.41421	+	1.41421i	1.41421	+	1.41421i	0.707107	+	0.707107i	$0.250000\pi$
$419$	1.41421	+	1.41421i	1.41421	+	1.41421i	0.707107	+	0.707107i	$0.250000\pi$
$420$		0			0
$421$	−1.00000	−	1.00000i	−1.00000	−	1.00000i		−	1.00000i	$-0.5\pi$
$421$	−1.00000	−	1.00000i	−1.00000	−	1.00000i	−1.00000			$\pi$
$422$		0			0
$423$		0			0
$424$	−0.866025	−	0.500000i	−0.866025	−	0.500000i
$425$		0			0
$426$		0			0
$427$		0			0
$428$	0.707107	+	0.707107i	0.707107	+	0.707107i
$429$		0			0
$430$	−1.36603	+	0.366025i	−1.36603	+	0.366025i
$431$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$431$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$432$		0			0
$433$	1.00000			1.00000			0.500000	−	0.866025i	$-0.333333\pi$
$433$	1.00000			1.00000			0.500000	+	0.866025i	$0.333333\pi$
$434$	−1.41421	+	1.41421i	−1.41421	+	1.41421i
$435$		0			0
$436$		0			0
$437$	0.517638	+	1.93185i	0.517638	+	1.93185i
$438$		0			0
$439$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$439$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	0.965926	+	0.258819i	0.965926	+	0.258819i	0.707107	−	0.707107i	$-0.250000\pi$
$443$	0.965926	+	0.258819i	0.965926	+	0.258819i	0.258819	+	0.965926i	$0.416667\pi$
$444$		0			0
$445$		0			0
$446$	−0.258819	+	0.965926i	−0.258819	+	0.965926i
$447$		0			0
$448$	1.00000			1.00000
$449$		−	1.41421i		−	1.41421i	−0.707107	−	0.707107i	$-0.750000\pi$
$449$		−	1.41421i		−	1.41421i	0.707107	−	0.707107i	$-0.250000\pi$
$450$		0			0
$451$		0			0
$452$		0			0
$453$		0			0
$454$		−	1.00000i		−	1.00000i
$455$		0			0
$456$		0			0
$457$	0.866025	−	0.500000i	0.866025	−	0.500000i		−	1.00000i	$-0.5\pi$
$457$	0.866025	−	0.500000i	0.866025	−	0.500000i	0.866025	+	0.500000i	$0.166667\pi$
$458$		0			0
$459$		0			0
$460$	0.366025	−	1.36603i	0.366025	−	1.36603i
$461$	−0.707107	−	0.707107i	−0.707107	−	0.707107i	0.258819	−	0.965926i	$-0.416667\pi$
$461$	−0.707107	−	0.707107i	−0.707107	−	0.707107i	−0.965926	+	0.258819i	$0.916667\pi$
$462$		0			0
$463$	−1.00000			−1.00000			−0.500000	−	0.866025i	$-0.666667\pi$
$463$	−1.00000			−1.00000			−0.500000	+	0.866025i	$0.666667\pi$
$464$	0.258819	−	0.965926i	0.258819	−	0.965926i
$465$		0			0
$466$	1.36603	−	0.366025i	1.36603	−	0.366025i
$467$	−0.258819	+	0.965926i	−0.258819	+	0.965926i	0.707107	+	0.707107i	$0.250000\pi$
$467$	−0.258819	+	0.965926i	−0.258819	+	0.965926i	−0.965926	+	0.258819i	$0.916667\pi$
$468$		0			0
$469$	−1.36603	+	0.366025i	−1.36603	+	0.366025i
$470$		−	1.41421i		−	1.41421i
$471$		0			0
$472$	0.866025	+	0.500000i	0.866025	+	0.500000i
$473$		0			0
$474$		0			0
$475$		0			0
$476$		0			0
$477$		0			0
$478$	1.36603	+	0.366025i	1.36603	+	0.366025i
$479$	1.22474	+	0.707107i	1.22474	+	0.707107i	0.965926	−	0.258819i	$-0.0833333\pi$
$479$	1.22474	+	0.707107i	1.22474	+	0.707107i	0.258819	+	0.965926i	$0.416667\pi$
$480$		0			0
$481$		0			0
$482$	0.707107	+	0.707107i	0.707107	+	0.707107i
$483$		0			0
$484$	−1.00000			−1.00000
$485$		0			0
$486$		0			0
$487$	−0.866025	−	0.500000i	−0.866025	−	0.500000i		−	1.00000i	$-0.5\pi$
$487$	−0.866025	−	0.500000i	−0.866025	−	0.500000i	−0.866025	+	0.500000i	$0.833333\pi$
$488$		0			0
$489$		0			0
$490$	0.500000	−	0.866025i	0.500000	−	0.866025i
$491$	−0.707107	+	0.707107i	−0.707107	+	0.707107i	−0.965926	−	0.258819i	$-0.916667\pi$
$491$	−0.707107	+	0.707107i	−0.707107	+	0.707107i	0.258819	+	0.965926i	$0.416667\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$		0			0
$496$	2.00000			2.00000
$497$			1.41421i			1.41421i
$498$		0			0
$499$		0			0		0.258819	−	0.965926i	$-0.416667\pi$
$499$		0			0		−0.258819	+	0.965926i	$0.583333\pi$
$500$	0.258819	+	0.965926i	0.258819	+	0.965926i
$501$		0			0
$502$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$503$	−1.41421			−1.41421			−0.707107	−	0.707107i	$-0.750000\pi$
$503$	−1.41421			−1.41421			−0.707107	+	0.707107i	$0.750000\pi$
$504$		0			0
$505$			1.00000i			1.00000i
$506$		0			0
$507$		0			0
$508$	0.866025	+	0.500000i	0.866025	+	0.500000i
$509$		0			0		−0.965926	−	0.258819i	$-0.916667\pi$
$509$		0			0		0.965926	+	0.258819i	$0.0833333\pi$
$510$		0			0
$511$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$512$	−0.707107	−	0.707107i	−0.707107	−	0.707107i
$513$		0			0
$514$	0.366025	+	1.36603i	0.366025	+	1.36603i
$515$		0			0
$516$		0			0
$517$		0			0
$518$		0			0
$519$		0			0
$520$		0			0
$521$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$521$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$522$		0			0
$523$	−0.366025	+	1.36603i	−0.366025	+	1.36603i	0.500000	+	0.866025i	$0.333333\pi$
$523$	−0.366025	+	1.36603i	−0.366025	+	1.36603i	−0.866025	+	0.500000i	$0.833333\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$		0			0
$528$		0			0
$529$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$530$	0.965926	+	0.258819i	0.965926	+	0.258819i
$531$		0			0
$532$	1.00000	−	1.00000i	1.00000	−	1.00000i
$533$		0			0
$534$		0			0
$535$	−0.866025	−	0.500000i	−0.866025	−	0.500000i
$536$	1.22474	+	0.707107i	1.22474	+	0.707107i
$537$		0			0
$538$			1.00000i			1.00000i
$539$		0			0
$540$		0			0
$541$	−1.36603	−	0.366025i	−1.36603	−	0.366025i	−0.500000	−	0.866025i	$-0.666667\pi$
$541$	−1.36603	−	0.366025i	−1.36603	−	0.366025i	−0.866025	+	0.500000i	$0.833333\pi$
$542$	0.965926	−	0.258819i	0.965926	−	0.258819i
$543$		0			0
$544$		0			0
$545$		0			0
$546$		0			0
$547$	1.00000	−	1.00000i	1.00000	−	1.00000i		−	1.00000i	$-0.5\pi$
$547$	1.00000	−	1.00000i	1.00000	−	1.00000i	1.00000			$0$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	−0.707107	−	1.22474i	−0.707107	−	1.22474i
$552$		0			0
$553$	0.866025	−	0.500000i	0.866025	−	0.500000i
$554$		−	1.41421i		−	1.41421i
$555$		0			0
$556$		0			0
$557$		0			0		−0.258819	−	0.965926i	$-0.583333\pi$
$557$		0			0		0.258819	+	0.965926i	$0.416667\pi$
$558$		0			0
$559$		0			0
$560$	−0.965926	+	0.258819i	−0.965926	+	0.258819i
$561$		0			0
$562$		0			0
$563$	0.965926	−	0.258819i	0.965926	−	0.258819i	0.258819	−	0.965926i	$-0.416667\pi$
$563$	0.965926	−	0.258819i	0.965926	−	0.258819i	0.707107	+	0.707107i	$0.250000\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$		0			0
$568$	1.00000	−	1.00000i	1.00000	−	1.00000i
$569$	0.707107	+	1.22474i	0.707107	+	1.22474i	0.965926	+	0.258819i	$0.0833333\pi$
$569$	0.707107	+	1.22474i	0.707107	+	1.22474i	−0.258819	+	0.965926i	$0.583333\pi$
$570$		0			0
$571$		0			0		−0.258819	−	0.965926i	$-0.583333\pi$
$571$		0			0		0.258819	+	0.965926i	$0.416667\pi$
$572$		0			0
$573$		0			0
$574$	1.36603	+	0.366025i	1.36603	+	0.366025i
$575$		0			0
$576$		0			0
$577$	−0.500000	+	0.866025i	−0.500000	+	0.866025i	0.500000	+	0.866025i	$0.333333\pi$
$577$	−0.500000	+	0.866025i	−0.500000	+	0.866025i	−1.00000			$\pi$
$578$	0.965926	−	0.258819i	0.965926	−	0.258819i
$579$		0			0
$580$			1.00000i			1.00000i
$581$		0			0
$582$		0			0
$583$		0			0
$584$	0.965926	−	0.258819i	0.965926	−	0.258819i
$585$		0			0
$586$	0.500000	+	0.866025i	0.500000	+	0.866025i
$587$	0.707107	−	0.707107i	0.707107	−	0.707107i	−0.258819	−	0.965926i	$-0.583333\pi$
$587$	0.707107	−	0.707107i	0.707107	−	0.707107i	0.965926	+	0.258819i	$0.0833333\pi$
$588$		0			0
$589$	2.00000	−	2.00000i	2.00000	−	2.00000i
$590$	−0.965926	−	0.258819i	−0.965926	−	0.258819i
$591$		0			0
$592$		0			0
$593$	1.22474	−	0.707107i	1.22474	−	0.707107i	0.258819	−	0.965926i	$-0.416667\pi$
$593$	1.22474	−	0.707107i	1.22474	−	0.707107i	0.965926	+	0.258819i	$0.0833333\pi$
$594$		0			0
$595$		0			0
$596$	0.707107	+	0.707107i	0.707107	+	0.707107i
$597$		0			0
$598$		0			0
$599$	−0.707107	+	1.22474i	−0.707107	+	1.22474i	0.258819	+	0.965926i	$0.416667\pi$
$599$	−0.707107	+	1.22474i	−0.707107	+	1.22474i	−0.965926	+	0.258819i	$0.916667\pi$
$600$		0			0
$601$		−	1.00000i		−	1.00000i	−0.866025	−	0.500000i	$-0.833333\pi$
$601$		−	1.00000i		−	1.00000i	0.866025	−	0.500000i	$-0.166667\pi$
$602$	−0.707107	−	1.22474i	−0.707107	−	1.22474i
$603$		0			0
$604$		0			0
$605$	0.965926	−	0.258819i	0.965926	−	0.258819i
$606$		0			0
$607$	0.500000	+	0.866025i	0.500000	+	0.866025i	1.00000			$0$
$607$	0.500000	+	0.866025i	0.500000	+	0.866025i	−0.500000	+	0.866025i	$0.666667\pi$
$608$	−1.41421			−1.41421
$609$		0			0
$610$		0			0
$611$		0			0
$612$		0			0
$613$		0			0		−0.258819	−	0.965926i	$-0.583333\pi$
$613$		0			0		0.258819	+	0.965926i	$0.416667\pi$
$614$	0.707107	+	1.22474i	0.707107	+	1.22474i
$615$		0			0
$616$		0			0
$617$		0			0			−	1.00000i	$-0.5\pi$
$617$		0			0				1.00000i	$0.5\pi$
$618$		0			0
$619$		0			0		−0.258819	−	0.965926i	$-0.583333\pi$
$619$		0			0		0.258819	+	0.965926i	$0.416667\pi$
$620$	−1.93185	+	0.517638i	−1.93185	+	0.517638i
$621$		0			0
$622$	1.00000	+	1.00000i	1.00000	+	1.00000i
$623$		0			0
$624$		0			0
$625$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$626$	0.258819	+	0.965926i	0.258819	+	0.965926i
$627$		0			0
$628$		0			0
$629$		0			0
$630$		0			0
$631$		0			0		1.00000			$0$
$631$		0			0		−1.00000			$\pi$
$632$	−0.965926	−	0.258819i	−0.965926	−	0.258819i
$633$		0			0
$634$		0			0
$635$	−0.965926	−	0.258819i	−0.965926	−	0.258819i
$636$		0			0
$637$		0			0
$638$		0			0
$639$		0			0
$640$	0.866025	+	0.500000i	0.866025	+	0.500000i
$641$	−1.22474	−	0.707107i	−1.22474	−	0.707107i	−0.258819	−	0.965926i	$-0.583333\pi$
$641$	−1.22474	−	0.707107i	−1.22474	−	0.707107i	−0.965926	+	0.258819i	$0.916667\pi$
$642$		0			0
$643$	−1.00000	+	1.00000i	−1.00000	+	1.00000i			1.00000i	$0.5\pi$
$643$	−1.00000	+	1.00000i	−1.00000	+	1.00000i	−1.00000			$\pi$
$644$	1.41421			1.41421
$645$		0			0
$646$		0			0
$647$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$647$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$648$		0			0
$649$		0			0
$650$		0			0
$651$		0			0
$652$	1.00000	+	1.00000i	1.00000	+	1.00000i
$653$	0.258819	−	0.965926i	0.258819	−	0.965926i	−0.707107	−	0.707107i	$-0.750000\pi$
$653$	0.258819	−	0.965926i	0.258819	−	0.965926i	0.965926	−	0.258819i	$-0.0833333\pi$
$654$		0			0
$655$		0			0
$656$	−0.707107	−	1.22474i	−0.707107	−	1.22474i
$657$		0			0
$658$	1.36603	−	0.366025i	1.36603	−	0.366025i
$659$	−0.707107	−	0.707107i	−0.707107	−	0.707107i	0.258819	−	0.965926i	$-0.416667\pi$
$659$	−0.707107	−	0.707107i	−0.707107	−	0.707107i	−0.965926	+	0.258819i	$0.916667\pi$
$660$		0			0
$661$		0			0		−0.258819	−	0.965926i	$-0.583333\pi$
$661$		0			0		0.258819	+	0.965926i	$0.416667\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	−0.707107	+	1.22474i	−0.707107	+	1.22474i
$666$		0			0
$667$	0.366025	−	1.36603i	0.366025	−	1.36603i
$668$		0			0
$669$		0			0
$670$	−1.36603	−	0.366025i	−1.36603	−	0.366025i
$671$		0			0
$672$		0			0
$673$	−1.00000			−1.00000			−0.500000	−	0.866025i	$-0.666667\pi$
$673$	−1.00000			−1.00000			−0.500000	+	0.866025i	$0.666667\pi$
$674$	−0.258819	+	0.965926i	−0.258819	+	0.965926i
$675$		0			0
$676$	0.500000	−	0.866025i	0.500000	−	0.866025i
$677$	−0.965926	−	0.258819i	−0.965926	−	0.258819i	−0.258819	−	0.965926i	$-0.583333\pi$
$677$	−0.965926	−	0.258819i	−0.965926	−	0.258819i	−0.707107	+	0.707107i	$0.750000\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$		0			0
$682$		0			0
$683$	0.258819	+	0.965926i	0.258819	+	0.965926i	0.965926	+	0.258819i	$0.0833333\pi$
$683$	0.258819	+	0.965926i	0.258819	+	0.965926i	−0.707107	+	0.707107i	$0.750000\pi$
$684$		0			0
$685$		0			0
$686$	0.965926	+	0.258819i	0.965926	+	0.258819i
$687$		0			0
$688$	−0.366025	+	1.36603i	−0.366025	+	1.36603i
$689$		0			0
$690$		0			0
$691$		0			0		0.258819	−	0.965926i	$-0.416667\pi$
$691$		0			0		−0.258819	+	0.965926i	$0.583333\pi$
$692$	−0.707107	+	0.707107i	−0.707107	+	0.707107i
$693$		0			0
$694$	−1.00000			−1.00000
$695$		0			0
$696$		0			0
$697$		0			0
$698$	0.707107	+	1.22474i	0.707107	+	1.22474i
$699$		0			0
$700$		0			0
$701$	0.707107	+	0.707107i	0.707107	+	0.707107i	0.965926	−	0.258819i	$-0.0833333\pi$
$701$	0.707107	+	0.707107i	0.707107	+	0.707107i	−0.258819	+	0.965926i	$0.583333\pi$
$702$		0			0
$703$		0			0
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−0.965926	+	0.258819i	−0.965926	+	0.258819i
$708$		0			0
$709$	−0.366025	+	1.36603i	−0.366025	+	1.36603i	0.500000	+	0.866025i	$0.333333\pi$
$709$	−0.366025	+	1.36603i	−0.366025	+	1.36603i	−0.866025	+	0.500000i	$0.833333\pi$
$710$	−0.707107	+	1.22474i	−0.707107	+	1.22474i
$711$		0			0
$712$		0			0
$713$	2.82843			2.82843
$714$		0			0
$715$		0			0
$716$	−0.965926	−	0.258819i	−0.965926	−	0.258819i
$717$		0			0
$718$	1.36603	−	0.366025i	1.36603	−	0.366025i
$719$	−1.22474	+	0.707107i	−1.22474	+	0.707107i	−0.965926	−	0.258819i	$-0.916667\pi$
$719$	−1.22474	+	0.707107i	−1.22474	+	0.707107i	−0.258819	+	0.965926i	$0.583333\pi$
$720$		0			0
$721$		0			0
$722$	−0.707107	+	0.707107i	−0.707107	+	0.707107i
$723$		0			0
$724$	−0.366025	−	1.36603i	−0.366025	−	1.36603i
$725$		0			0
$726$		0			0
$727$		−	1.00000i		−	1.00000i	−0.866025	−	0.500000i	$-0.833333\pi$
$727$		−	1.00000i		−	1.00000i	0.866025	−	0.500000i	$-0.166667\pi$
$728$		0			0
$729$		0			0
$730$	−0.866025	+	0.500000i	−0.866025	+	0.500000i
$731$		0			0
$732$		0			0
$733$	−1.36603	−	0.366025i	−1.36603	−	0.366025i	−0.500000	−	0.866025i	$-0.666667\pi$
$733$	−1.36603	−	0.366025i	−1.36603	−	0.366025i	−0.866025	+	0.500000i	$0.833333\pi$
$734$	0.707107	+	0.707107i	0.707107	+	0.707107i
$735$		0			0
$736$	−1.00000	−	1.00000i	−1.00000	−	1.00000i
$737$		0			0
$738$		0			0
$739$	0.366025	+	1.36603i	0.366025	+	1.36603i	0.866025	+	0.500000i	$0.166667\pi$
$739$	0.366025	+	1.36603i	0.366025	+	1.36603i	−0.500000	+	0.866025i	$0.666667\pi$
$740$		0			0
$741$		0			0
$742$			1.00000i			1.00000i
$743$		0			0			−	1.00000i	$-0.5\pi$
$743$		0			0				1.00000i	$0.5\pi$
$744$		0			0
$745$	−0.866025	−	0.500000i	−0.866025	−	0.500000i
$746$	−1.22474	−	0.707107i	−1.22474	−	0.707107i
$747$		0			0
$748$		0			0
$749$	0.258819	−	0.965926i	0.258819	−	0.965926i
$750$		0			0
$751$	−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			$\pi$
$752$	−1.22474	−	0.707107i	−1.22474	−	0.707107i
$753$		0			0
$754$		0			0
$755$		0			0
$756$		0			0
$757$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$757$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$758$	1.22474	−	0.707107i	1.22474	−	0.707107i
$759$		0			0
$760$	1.36603	−	0.366025i	1.36603	−	0.366025i
$761$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$761$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$762$		0			0
$763$		0			0
$764$	1.41421			1.41421
$765$		0			0
$766$	0.366025	+	1.36603i	0.366025	+	1.36603i
$767$		0			0
$768$		0			0
$769$		0			0			−	1.00000i	$-0.5\pi$
$769$		0			0				1.00000i	$0.5\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$		0			0		0.965926	−	0.258819i	$-0.0833333\pi$
$773$		0			0		−0.965926	+	0.258819i	$0.916667\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$		0			0
$778$	1.00000			1.00000
$779$	−1.93185	−	0.517638i	−1.93185	−	0.517638i
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$785$		0			0
$786$		0			0
$787$		0			0		0.965926	−	0.258819i	$-0.0833333\pi$
$787$		0			0		−0.965926	+	0.258819i	$0.916667\pi$
$788$	0.965926	−	0.258819i	0.965926	−	0.258819i
$789$		0			0
$790$	1.00000			1.00000
$791$		0			0
$792$		0			0
$793$		0			0
$794$		0			0
$795$		0			0
$796$	0.500000	+	0.866025i	0.500000	+	0.866025i
$797$	1.41421	+	1.41421i	1.41421	+	1.41421i	0.707107	+	0.707107i	$0.250000\pi$
$797$	1.41421	+	1.41421i	1.41421	+	1.41421i	0.707107	+	0.707107i	$0.250000\pi$
$798$		0			0
$799$		0			0
$800$		0			0
$801$		0			0
$802$	0.366025	+	1.36603i	0.366025	+	1.36603i
$803$		0			0
$804$		0			0
$805$	−1.36603	+	0.366025i	−1.36603	+	0.366025i
$806$		0			0
$807$		0			0
$808$	0.866025	+	0.500000i	0.866025	+	0.500000i
$809$	0.707107	−	1.22474i	0.707107	−	1.22474i	−0.258819	−	0.965926i	$-0.583333\pi$
$809$	0.707107	−	1.22474i	0.707107	−	1.22474i	0.965926	−	0.258819i	$-0.0833333\pi$
$810$		0			0
$811$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$811$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$812$	−0.965926	+	0.258819i	−0.965926	+	0.258819i
$813$		0			0
$814$		0			0
$815$	−1.22474	−	0.707107i	−1.22474	−	0.707107i
$816$		0			0
$817$	1.00000	+	1.73205i	1.00000	+	1.73205i
$818$	0.707107	−	0.707107i	0.707107	−	0.707107i
$819$		0			0
$820$	1.00000	+	1.00000i	1.00000	+	1.00000i
$821$	−0.965926	−	0.258819i	−0.965926	−	0.258819i	−0.258819	−	0.965926i	$-0.583333\pi$
$821$	−0.965926	−	0.258819i	−0.965926	−	0.258819i	−0.707107	+	0.707107i	$0.750000\pi$
$822$		0			0
$823$	0.866025	+	0.500000i	0.866025	+	0.500000i	0.866025	−	0.500000i	$-0.166667\pi$
$823$	0.866025	+	0.500000i	0.866025	+	0.500000i			1.00000i	$0.5\pi$
$824$		0			0
$825$		0			0
$826$		−	1.00000i		−	1.00000i
$827$	0.707107	−	0.707107i	0.707107	−	0.707107i	−0.258819	−	0.965926i	$-0.583333\pi$
$827$	0.707107	−	0.707107i	0.707107	−	0.707107i	0.965926	+	0.258819i	$0.0833333\pi$
$828$		0			0
$829$	0.366025	+	1.36603i	0.366025	+	1.36603i	0.866025	+	0.500000i	$0.166667\pi$
$829$	0.366025	+	1.36603i	0.366025	+	1.36603i	−0.500000	+	0.866025i	$0.666667\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$		0			0
$834$		0			0
$835$		0			0
$836$		0			0
$837$		0			0
$838$	−1.73205	+	1.00000i	−1.73205	+	1.00000i
$839$		0			0			−	1.00000i	$-0.5\pi$
$839$		0			0				1.00000i	$0.5\pi$
$840$		0			0
$841$		0			0
$842$	1.22474	−	0.707107i	1.22474	−	0.707107i
$843$		0			0
$844$		0			0
$845$	−0.258819	+	0.965926i	−0.258819	+	0.965926i
$846$		0			0
$847$	0.500000	+	0.866025i	0.500000	+	0.866025i
$848$	0.707107	−	0.707107i	0.707107	−	0.707107i
$849$		0			0
$850$		0			0
$851$		0			0
$852$		0			0
$853$	−1.00000	−	1.00000i	−1.00000	−	1.00000i		−	1.00000i	$-0.5\pi$
$853$	−1.00000	−	1.00000i	−1.00000	−	1.00000i	−1.00000			$\pi$
$854$		0			0
$855$		0			0
$856$	−0.866025	+	0.500000i	−0.866025	+	0.500000i
$857$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$857$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$858$		0			0
$859$	−0.366025	+	1.36603i	−0.366025	+	1.36603i	0.500000	+	0.866025i	$0.333333\pi$
$859$	−0.366025	+	1.36603i	−0.366025	+	1.36603i	−0.866025	+	0.500000i	$0.833333\pi$
$860$		−	1.41421i		−	1.41421i
$861$		0			0
$862$		0			0
$863$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$863$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$864$		0			0
$865$	0.500000	−	0.866025i	0.500000	−	0.866025i
$866$	−0.258819	+	0.965926i	−0.258819	+	0.965926i
$867$		0			0
$868$	−1.00000	−	1.73205i	−1.00000	−	1.73205i
$869$		0			0
$870$		0			0
$871$		0			0
$872$		0			0
$873$		0			0
$874$	−2.00000			−2.00000
$875$	0.707107	−	0.707107i	0.707107	−	0.707107i
$876$		0			0
$877$		0			0		0.258819	−	0.965926i	$-0.416667\pi$
$877$		0			0		−0.258819	+	0.965926i	$0.583333\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$		0			0		1.00000			$0$
$881$		0			0		−1.00000			$\pi$
$882$		0			0
$883$	1.00000	−	1.00000i	1.00000	−	1.00000i		−	1.00000i	$-0.5\pi$
$883$	1.00000	−	1.00000i	1.00000	−	1.00000i	1.00000			$0$
$884$		0			0
$885$		0			0
$886$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$887$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$887$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$888$		0			0
$889$		−	1.00000i		−	1.00000i
$890$		0			0
$891$		0			0
$892$	−0.866025	−	0.500000i	−0.866025	−	0.500000i
$893$	−1.93185	+	0.517638i	−1.93185	+	0.517638i
$894$		0			0
$895$	1.00000			1.00000
$896$	−0.258819	+	0.965926i	−0.258819	+	0.965926i
$897$		0			0
$898$	1.36603	+	0.366025i	1.36603	+	0.366025i
$899$	−1.93185	+	0.517638i	−1.93185	+	0.517638i
$900$		0			0
$901$		0			0
$902$		0			0
$903$		0			0
$904$		0			0
$905$	0.707107	+	1.22474i	0.707107	+	1.22474i
$906$		0			0
$907$		0			0		−0.258819	−	0.965926i	$-0.583333\pi$
$907$		0			0		0.258819	+	0.965926i	$0.416667\pi$
$908$	0.965926	+	0.258819i	0.965926	+	0.258819i
$909$		0			0
$910$		0			0
$911$		0			0		1.00000			$0$
$911$		0			0		−1.00000			$\pi$
$912$		0			0
$913$		0			0
$914$	0.258819	+	0.965926i	0.258819	+	0.965926i
$915$		0			0
$916$		0			0
$917$		0			0
$918$		0			0
$919$	−0.866025	+	0.500000i	−0.866025	+	0.500000i	−0.866025	−	0.500000i	$-0.833333\pi$
$919$	−0.866025	+	0.500000i	−0.866025	+	0.500000i			1.00000i	$0.5\pi$
$920$	1.22474	+	0.707107i	1.22474	+	0.707107i
$921$		0			0
$922$	0.866025	−	0.500000i	0.866025	−	0.500000i
$923$		0			0
$924$		0			0
$925$		0			0
$926$	0.258819	−	0.965926i	0.258819	−	0.965926i
$927$		0			0
$928$	0.866025	+	0.500000i	0.866025	+	0.500000i
$929$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$929$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$930$		0			0
$931$	−1.36603	−	0.366025i	−1.36603	−	0.366025i
$932$			1.41421i			1.41421i
$933$		0			0
$934$	−0.866025	−	0.500000i	−0.866025	−	0.500000i
$935$		0			0
$936$		0			0
$937$		−	1.00000i		−	1.00000i	−0.866025	−	0.500000i	$-0.833333\pi$
$937$		−	1.00000i		−	1.00000i	0.866025	−	0.500000i	$-0.166667\pi$
$938$		−	1.41421i		−	1.41421i
$939$		0			0
$940$	1.36603	+	0.366025i	1.36603	+	0.366025i
$941$	−1.93185	+	0.517638i	−1.93185	+	0.517638i	−0.965926	+	0.258819i	$0.916667\pi$
$941$	−1.93185	+	0.517638i	−1.93185	+	0.517638i	−0.965926	+	0.258819i	$0.916667\pi$
$942$		0			0
$943$	−1.00000	−	1.73205i	−1.00000	−	1.73205i
$944$	−0.707107	+	0.707107i	−0.707107	+	0.707107i
$945$		0			0
$946$		0			0
$947$	0.258819	−	0.965926i	0.258819	−	0.965926i	−0.707107	−	0.707107i	$-0.750000\pi$
$947$	0.258819	−	0.965926i	0.258819	−	0.965926i	0.965926	−	0.258819i	$-0.0833333\pi$
$948$		0			0
$949$		0			0
$950$		0			0
$951$		0			0
$952$		0			0
$953$	−1.41421			−1.41421			−0.707107	−	0.707107i	$-0.750000\pi$
$953$	−1.41421			−1.41421			−0.707107	+	0.707107i	$0.750000\pi$
$954$		0			0
$955$	−1.36603	+	0.366025i	−1.36603	+	0.366025i
$956$	−0.707107	+	1.22474i	−0.707107	+	1.22474i
$957$		0			0
$958$	−1.00000	+	1.00000i	−1.00000	+	1.00000i
$959$		0			0
$960$		0			0
$961$	−1.50000	−	2.59808i	−1.50000	−	2.59808i
$962$		0			0
$963$		0			0
$964$	−0.866025	+	0.500000i	−0.866025	+	0.500000i
$965$		0			0
$966$		0			0
$967$			1.00000i			1.00000i	0.866025	+	0.500000i	$0.166667\pi$
$967$			1.00000i			1.00000i	−0.866025	+	0.500000i	$0.833333\pi$
$968$	0.258819	−	0.965926i	0.258819	−	0.965926i
$969$		0			0
$970$		0			0
$971$	−0.965926	−	0.258819i	−0.965926	−	0.258819i	−0.258819	−	0.965926i	$-0.583333\pi$
$971$	−0.965926	−	0.258819i	−0.965926	−	0.258819i	−0.707107	+	0.707107i	$0.750000\pi$
$972$		0			0
$973$		0			0
$974$	0.707107	−	0.707107i	0.707107	−	0.707107i
$975$		0			0
$976$		0			0
$977$	−1.22474	−	0.707107i	−1.22474	−	0.707107i	−0.258819	−	0.965926i	$-0.583333\pi$
$977$	−1.22474	−	0.707107i	−1.22474	−	0.707107i	−0.965926	+	0.258819i	$0.916667\pi$
$978$		0			0
$979$		0			0
$980$	0.707107	+	0.707107i	0.707107	+	0.707107i
$981$		0			0
$982$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$983$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$983$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$984$		0			0
$985$	−0.866025	+	0.500000i	−0.866025	+	0.500000i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−0.517638	+	1.93185i	−0.517638	+	1.93185i
$990$		0			0
$991$	0.500000	−	0.866025i	0.500000	−	0.866025i	−0.500000	−	0.866025i	$-0.666667\pi$
$991$	0.500000	−	0.866025i	0.500000	−	0.866025i	1.00000			$0$
$992$	−0.517638	+	1.93185i	−0.517638	+	1.93185i
$993$		0			0
$994$	−1.36603	−	0.366025i	−1.36603	−	0.366025i
$995$	−0.707107	−	0.707107i	−0.707107	−	0.707107i
$996$		0			0
$997$		0			0		−0.258819	−	0.965926i	$-0.583333\pi$
$997$		0			0		0.258819	+	0.965926i	$0.416667\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3024.1.eg.a.53.1	✓	8
3.2	odd	2	inner	3024.1.eg.a.53.2	yes	8
7.2	even	3	inner	3024.1.eg.a.485.2	yes	8
16.13	even	4	inner	3024.1.eg.a.1565.1	yes	8
21.2	odd	6	inner	3024.1.eg.a.485.1	yes	8
48.29	odd	4	inner	3024.1.eg.a.1565.2	yes	8
112.93	even	12	inner	3024.1.eg.a.1997.2	yes	8
336.317	odd	12	inner	3024.1.eg.a.1997.1	yes	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3024.1.eg.a.53.1	✓	8	1.1	even	1	trivial
3024.1.eg.a.53.2	yes	8	3.2	odd	2	inner
3024.1.eg.a.485.1	yes	8	21.2	odd	6	inner
3024.1.eg.a.485.2	yes	8	7.2	even	3	inner
3024.1.eg.a.1565.1	yes	8	16.13	even	4	inner
3024.1.eg.a.1565.2	yes	8	48.29	odd	4	inner
3024.1.eg.a.1997.1	yes	8	336.317	odd	12	inner
3024.1.eg.a.1997.2	yes	8	112.93	even	12	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:	\( N \)	\(=\)	\( 3024 = 2^{4} \cdot 3^{3} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	3024.eg (of order \(12\), degree \(4\), minimal)

\(f(q)\)	\(=\)	\(q+(-0.258819 + 0.965926i) q^{2} +(-0.866025 - 0.500000i) q^{4} +(0.965926 + 0.258819i) q^{5} +1.00000i q^{7} +(0.707107 - 0.707107i) q^{8} +O(q^{10})\)	\(q+(-0.258819 + 0.965926i) q^{2} +(-0.866025 - 0.500000i) q^{4} +(0.965926 + 0.258819i) q^{5} +1.00000i q^{7} +(0.707107 - 0.707107i) q^{8} +(-0.500000 + 0.866025i) q^{10} +(-0.965926 - 0.258819i) q^{14} +(0.500000 + 0.866025i) q^{16} +(1.36603 + 0.366025i) q^{19} +(-0.707107 - 0.707107i) q^{20} +(0.707107 + 1.22474i) q^{23} +(0.500000 - 0.866025i) q^{28} +(-0.707107 - 0.707107i) q^{29} +(1.00000 - 1.73205i) q^{31} +(-0.965926 + 0.258819i) q^{32} +(-0.258819 + 0.965926i) q^{35} +(-0.707107 + 1.22474i) q^{38} +(0.866025 - 0.500000i) q^{40} -1.41421 q^{41} +(1.00000 + 1.00000i) q^{43} +(-1.36603 + 0.366025i) q^{46} +(-1.22474 + 0.707107i) q^{47} -1.00000 q^{49} +(-0.258819 - 0.965926i) q^{53} +(0.707107 + 0.707107i) q^{56} +(0.866025 - 0.500000i) q^{58} +(0.258819 + 0.965926i) q^{59} +(1.41421 + 1.41421i) q^{62} -1.00000i q^{64} +(0.366025 + 1.36603i) q^{67} +(-0.866025 - 0.500000i) q^{70} +1.41421 q^{71} +(0.866025 + 0.500000i) q^{73} +(-1.00000 - 1.00000i) q^{76} +(-0.500000 - 0.866025i) q^{79} +(0.258819 + 0.965926i) q^{80} +(0.366025 - 1.36603i) q^{82} +(-1.22474 + 0.707107i) q^{86} -1.41421i q^{92} +(-0.366025 - 1.36603i) q^{94} +(1.22474 + 0.707107i) q^{95} +(0.258819 - 0.965926i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q+O(q^{10}) \) 8 * q	\( 8 q - 4 q^{10} + 4 q^{16} + 4 q^{19} + 4 q^{28} + 8 q^{31} + 8 q^{43} - 4 q^{46} - 8 q^{49} - 4 q^{67} - 8 q^{76} - 4 q^{79} - 4 q^{82} + 4 q^{94}+O(q^{100}) \) 8 * q - 4 * q^10 + 4 * q^16 + 4 * q^19 + 4 * q^28 + 8 * q^31 + 8 * q^43 - 4 * q^46 - 8 * q^49 - 4 * q^67 - 8 * q^76 - 4 * q^79 - 4 * q^82 + 4 * q^94

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