LMFDB - Embedded newform 1008.1.dt.a.163.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1008,1,Mod(163,1008)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1008, base_ring=CyclotomicField(12))

chi = DirichletCharacter(H, H._module([6, 9, 0, 4]))

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

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

chi := DirichletCharacter("1008.163");

S:= CuspForms(chi, 1);

N := Newforms(S);

Level:	$ N $	$=$	$ 1008 = 2^{4} \cdot 3^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	1008.dt (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:	$0.503057532734$
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 4.2.301056.1

Embedding invariants

Embedding label			163.2
Root			$0.258819 - 0.965926i$ of defining polynomial
Character	$\chi$	$=$	1008.163
Dual form			1008.1.dt.a.235.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(0.258819 + 0.965926i) q^{2} +(-0.866025 + 0.500000i) q^{4} +(-0.258819 - 0.965926i) q^{5} +(-0.500000 - 0.866025i) q^{7} +(-0.707107 - 0.707107i) q^{8} +O(q^{10})$	\(q+(0.258819 + 0.965926i) q^{2} +(-0.866025 + 0.500000i) q^{4} +(-0.258819 - 0.965926i) q^{5} +(-0.500000 - 0.866025i) q^{7} +(-0.707107 - 0.707107i) q^{8} +(0.866025 - 0.500000i) q^{10} +(0.258819 - 0.965926i) q^{11} +(1.00000 + 1.00000i) q^{13} +(0.707107 - 0.707107i) q^{14} +(0.500000 - 0.866025i) q^{16} +(0.707107 + 0.707107i) q^{20} +1.00000 q^{22} +(0.707107 - 1.22474i) q^{23} +(-0.707107 + 1.22474i) q^{26} +(0.866025 + 0.500000i) q^{28} +(-0.707107 - 0.707107i) q^{29} +(0.866025 - 0.500000i) q^{31} +(0.965926 + 0.258819i) q^{32} +(-0.707107 + 0.707107i) q^{35} +(-0.500000 + 0.866025i) q^{40} +1.41421i q^{41} +(0.258819 + 0.965926i) q^{44} +(1.36603 + 0.366025i) q^{46} +(-1.22474 - 0.707107i) q^{47} +(-0.500000 + 0.866025i) q^{49} +(-1.36603 - 0.366025i) q^{52} +(-0.965926 - 0.258819i) q^{53} -1.00000 q^{55} +(-0.258819 + 0.965926i) q^{56} +(0.500000 - 0.866025i) q^{58} +(-0.258819 + 0.965926i) q^{59} +(-1.36603 + 0.366025i) q^{61} +(0.707107 + 0.707107i) q^{62} +1.00000i q^{64} +(0.707107 - 1.22474i) q^{65} +(-0.866025 - 0.500000i) q^{70} +1.41421 q^{71} +(-0.965926 + 0.258819i) q^{77} +(-0.866025 - 0.500000i) q^{79} +(-0.965926 - 0.258819i) q^{80} +(-1.36603 + 0.366025i) q^{82} +(-0.707107 + 0.707107i) q^{83} +(-0.866025 + 0.500000i) q^{88} +(1.22474 + 0.707107i) q^{89} +(0.366025 - 1.36603i) q^{91} +1.41421i q^{92} +(0.366025 - 1.36603i) q^{94} +1.00000 q^{97} +(-0.965926 - 0.258819i) q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 4 q^{7}+O(q^{10}) $ 8 * q - 4 * q^7	$ 8 q - 4 q^{7} + 8 q^{13} + 4 q^{16} + 8 q^{22} - 4 q^{40} + 4 q^{46} - 4 q^{49} - 4 q^{52} - 8 q^{55} + 4 q^{58} - 4 q^{61} - 4 q^{82} - 4 q^{91} - 4 q^{94} + 8 q^{97}+O(q^{100}) $ 8 * q - 4 * q^7 + 8 * q^13 + 4 * q^16 + 8 * q^22 - 4 * q^40 + 4 * q^46 - 4 * q^49 - 4 * q^52 - 8 * q^55 + 4 * q^58 - 4 * q^61 - 4 * q^82 - 4 * q^91 - 4 * q^94 + 8 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1008\mathbb{Z}\right)^\times$.

$n$	$127$	$577$	$757$	$785$
$\chi(n)$	$-1$	$e\left(\frac{1}{3}\right)$	$e\left(\frac{3}{4}\right)$	$1$

Coefficient data

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

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

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

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

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1008.1.dt.a.163.2	yes	8
3.2	odd	2	inner	1008.1.dt.a.163.1	✓	8
7.4	even	3	inner	1008.1.dt.a.739.1	yes	8
16.11	odd	4	inner	1008.1.dt.a.667.1	yes	8
21.11	odd	6	inner	1008.1.dt.a.739.2	yes	8
48.11	even	4	inner	1008.1.dt.a.667.2	yes	8
112.11	odd	12	inner	1008.1.dt.a.235.2	yes	8
336.11	even	12	inner	1008.1.dt.a.235.1	yes	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1008.1.dt.a.163.1	✓	8	3.2	odd	2	inner
1008.1.dt.a.163.2	yes	8	1.1	even	1	trivial
1008.1.dt.a.235.1	yes	8	336.11	even	12	inner
1008.1.dt.a.235.2	yes	8	112.11	odd	12	inner
1008.1.dt.a.667.1	yes	8	16.11	odd	4	inner
1008.1.dt.a.667.2	yes	8	48.11	even	4	inner
1008.1.dt.a.739.1	yes	8	7.4	even	3	inner
1008.1.dt.a.739.2	yes	8	21.11	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:	\( N \)	\(=\)	\( 1008 = 2^{4} \cdot 3^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	1008.dt (of order \(12\), degree \(4\), minimal)

\(f(q)\)	\(=\)	\(q+(0.258819 + 0.965926i) q^{2} +(-0.866025 + 0.500000i) q^{4} +(-0.258819 - 0.965926i) q^{5} +(-0.500000 - 0.866025i) q^{7} +(-0.707107 - 0.707107i) q^{8} +O(q^{10})\)	\(q+(0.258819 + 0.965926i) q^{2} +(-0.866025 + 0.500000i) q^{4} +(-0.258819 - 0.965926i) q^{5} +(-0.500000 - 0.866025i) q^{7} +(-0.707107 - 0.707107i) q^{8} +(0.866025 - 0.500000i) q^{10} +(0.258819 - 0.965926i) q^{11} +(1.00000 + 1.00000i) q^{13} +(0.707107 - 0.707107i) q^{14} +(0.500000 - 0.866025i) q^{16} +(0.707107 + 0.707107i) q^{20} +1.00000 q^{22} +(0.707107 - 1.22474i) q^{23} +(-0.707107 + 1.22474i) q^{26} +(0.866025 + 0.500000i) q^{28} +(-0.707107 - 0.707107i) q^{29} +(0.866025 - 0.500000i) q^{31} +(0.965926 + 0.258819i) q^{32} +(-0.707107 + 0.707107i) q^{35} +(-0.500000 + 0.866025i) q^{40} +1.41421i q^{41} +(0.258819 + 0.965926i) q^{44} +(1.36603 + 0.366025i) q^{46} +(-1.22474 - 0.707107i) q^{47} +(-0.500000 + 0.866025i) q^{49} +(-1.36603 - 0.366025i) q^{52} +(-0.965926 - 0.258819i) q^{53} -1.00000 q^{55} +(-0.258819 + 0.965926i) q^{56} +(0.500000 - 0.866025i) q^{58} +(-0.258819 + 0.965926i) q^{59} +(-1.36603 + 0.366025i) q^{61} +(0.707107 + 0.707107i) q^{62} +1.00000i q^{64} +(0.707107 - 1.22474i) q^{65} +(-0.866025 - 0.500000i) q^{70} +1.41421 q^{71} +(-0.965926 + 0.258819i) q^{77} +(-0.866025 - 0.500000i) q^{79} +(-0.965926 - 0.258819i) q^{80} +(-1.36603 + 0.366025i) q^{82} +(-0.707107 + 0.707107i) q^{83} +(-0.866025 + 0.500000i) q^{88} +(1.22474 + 0.707107i) q^{89} +(0.366025 - 1.36603i) q^{91} +1.41421i q^{92} +(0.366025 - 1.36603i) q^{94} +1.00000 q^{97} +(-0.965926 - 0.258819i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 4 q^{7}+O(q^{10}) \) 8 * q - 4 * q^7	\( 8 q - 4 q^{7} + 8 q^{13} + 4 q^{16} + 8 q^{22} - 4 q^{40} + 4 q^{46} - 4 q^{49} - 4 q^{52} - 8 q^{55} + 4 q^{58} - 4 q^{61} - 4 q^{82} - 4 q^{91} - 4 q^{94} + 8 q^{97}+O(q^{100}) \) 8 * q - 4 * q^7 + 8 * q^13 + 4 * q^16 + 8 * q^22 - 4 * q^40 + 4 * q^46 - 4 * q^49 - 4 * q^52 - 8 * q^55 + 4 * q^58 - 4 * q^61 - 4 * q^82 - 4 * q^91 - 4 * q^94 + 8 * q^97

Introduction

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