LMFDB - Embedded newform 3024.1.eg.b.1997.1

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 4.2.2709504.10

Embedding invariants

Embedding label			1997.1
Root			$-0.258819 - 0.965926i$ of defining polynomial
Character	$\chi$	$=$	3024.1997
Dual form			3024.1.eg.b.53.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-0.965926 + 0.258819i) q^{2} +(0.866025 - 0.500000i) q^{4} +(0.866025 + 0.500000i) q^{7} +(-0.707107 + 0.707107i) q^{8} +O(q^{10})$	\(q+(-0.965926 + 0.258819i) q^{2} +(0.866025 - 0.500000i) q^{4} +(0.866025 + 0.500000i) q^{7} +(-0.707107 + 0.707107i) q^{8} +(-0.258819 + 0.965926i) q^{11} +(-1.00000 - 1.00000i) q^{13} +(-0.965926 - 0.258819i) q^{14} +(0.500000 - 0.866025i) q^{16} +(-1.36603 + 0.366025i) q^{19} -1.00000i q^{22} +(-0.707107 + 1.22474i) q^{23} +(-0.866025 + 0.500000i) q^{25} +(1.22474 + 0.707107i) q^{26} +1.00000 q^{28} +(-0.707107 + 0.707107i) q^{29} +(0.500000 + 0.866025i) q^{31} +(-0.258819 + 0.965926i) q^{32} +(0.366025 + 1.36603i) q^{37} +(1.22474 - 0.707107i) q^{38} -1.41421 q^{41} +(0.258819 + 0.965926i) q^{44} +(0.366025 - 1.36603i) q^{46} +(0.500000 + 0.866025i) q^{49} +(0.707107 - 0.707107i) q^{50} +(-1.36603 - 0.366025i) q^{52} +(0.517638 - 1.93185i) q^{53} +(-0.965926 + 0.258819i) q^{56} +(0.500000 - 0.866025i) q^{58} +(0.258819 - 0.965926i) q^{59} +(-1.36603 + 0.366025i) q^{61} +(-0.707107 - 0.707107i) q^{62} -1.00000i q^{64} +1.41421 q^{71} +(0.866025 - 0.500000i) q^{73} +(-0.707107 - 1.22474i) q^{74} +(-1.00000 + 1.00000i) q^{76} +(-0.707107 + 0.707107i) q^{77} +(-0.500000 + 0.866025i) q^{79} +(1.36603 - 0.366025i) q^{82} +(0.707107 - 0.707107i) q^{83} +(-0.500000 - 0.866025i) q^{88} +(-0.366025 - 1.36603i) q^{91} +1.41421i q^{92} +1.00000 q^{97} +(-0.707107 - 0.707107i) q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q+O(q^{10}) $ 8 * q	$ 8 q - 8 q^{13} + 4 q^{16} - 4 q^{19} + 8 q^{28} + 4 q^{31} - 4 q^{37} - 4 q^{46} + 4 q^{49} - 4 q^{52} + 4 q^{58} - 4 q^{61} - 8 q^{76} - 4 q^{79} + 4 q^{82} - 4 q^{88} + 4 q^{91} + 8 q^{97}+O(q^{100}) $ 8 * q - 8 * q^13 + 4 * q^16 - 4 * q^19 + 8 * q^28 + 4 * q^31 - 4 * q^37 - 4 * q^46 + 4 * q^49 - 4 * q^52 + 4 * q^58 - 4 * q^61 - 8 * q^76 - 4 * q^79 + 4 * q^82 - 4 * q^88 + 4 * q^91 + 8 * q^97

Display 100 coefficients 10 coefficients

Character values

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

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

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 $

$2$	−0.965926	+	0.258819i	−0.965926	+	0.258819i
$2$	−0.965926	+	0.258819i	−0.965926	+	0.258819i
$3$		0			0
$3$		0			0
$4$	0.866025	−	0.500000i	0.866025	−	0.500000i
$5$		0			0		−0.258819	−	0.965926i	$-0.583333\pi$
$5$		0			0		0.258819	+	0.965926i	$0.416667\pi$
$6$		0			0
$7$	0.866025	+	0.500000i	0.866025	+	0.500000i
$7$	0.866025	+	0.500000i	0.866025	+	0.500000i
$8$	−0.707107	+	0.707107i	−0.707107	+	0.707107i
$9$		0			0
$10$		0			0
$11$	−0.258819	+	0.965926i	−0.258819	+	0.965926i	0.707107	+	0.707107i	$0.250000\pi$
$11$	−0.258819	+	0.965926i	−0.258819	+	0.965926i	−0.965926	+	0.258819i	$0.916667\pi$
$12$		0			0
$13$	−1.00000	−	1.00000i	−1.00000	−	1.00000i		−	1.00000i	$-0.5\pi$
$13$	−1.00000	−	1.00000i	−1.00000	−	1.00000i	−1.00000			$\pi$
$14$	−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	$-0.666667\pi$
$17$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$18$		0			0
$19$	−1.36603	+	0.366025i	−1.36603	+	0.366025i	−0.866025	−	0.500000i	$-0.833333\pi$
$19$	−1.36603	+	0.366025i	−1.36603	+	0.366025i	−0.500000	+	0.866025i	$0.666667\pi$
$20$		0			0
$21$		0			0
$22$		−	1.00000i		−	1.00000i
$23$	−0.707107	+	1.22474i	−0.707107	+	1.22474i	0.258819	+	0.965926i	$0.416667\pi$
$23$	−0.707107	+	1.22474i	−0.707107	+	1.22474i	−0.965926	+	0.258819i	$0.916667\pi$
$24$		0			0
$25$	−0.866025	+	0.500000i	−0.866025	+	0.500000i
$26$	1.22474	+	0.707107i	1.22474	+	0.707107i
$27$		0			0
$28$	1.00000			1.00000
$29$	−0.707107	+	0.707107i	−0.707107	+	0.707107i	−0.965926	−	0.258819i	$-0.916667\pi$
$29$	−0.707107	+	0.707107i	−0.707107	+	0.707107i	0.258819	+	0.965926i	$0.416667\pi$
$30$		0			0
$31$	0.500000	+	0.866025i	0.500000	+	0.866025i	1.00000			$0$
$31$	0.500000	+	0.866025i	0.500000	+	0.866025i	−0.500000	+	0.866025i	$0.666667\pi$
$32$	−0.258819	+	0.965926i	−0.258819	+	0.965926i
$33$		0			0
$34$		0			0
$35$		0			0
$36$		0			0
$37$	0.366025	+	1.36603i	0.366025	+	1.36603i	0.866025	+	0.500000i	$0.166667\pi$
$37$	0.366025	+	1.36603i	0.366025	+	1.36603i	−0.500000	+	0.866025i	$0.666667\pi$
$38$	1.22474	−	0.707107i	1.22474	−	0.707107i
$39$		0			0
$40$		0			0
$41$	−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
$43$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$43$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$44$	0.258819	+	0.965926i	0.258819	+	0.965926i
$45$		0			0
$46$	0.366025	−	1.36603i	0.366025	−	1.36603i
$47$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$47$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$48$		0			0
$49$	0.500000	+	0.866025i	0.500000	+	0.866025i
$50$	0.707107	−	0.707107i	0.707107	−	0.707107i
$51$		0			0
$52$	−1.36603	−	0.366025i	−1.36603	−	0.366025i
$53$	0.517638	−	1.93185i	0.517638	−	1.93185i	0.258819	−	0.965926i	$-0.416667\pi$
$53$	0.517638	−	1.93185i	0.517638	−	1.93185i	0.258819	−	0.965926i	$-0.416667\pi$
$54$		0			0
$55$		0			0
$56$	−0.965926	+	0.258819i	−0.965926	+	0.258819i
$57$		0			0
$58$	0.500000	−	0.866025i	0.500000	−	0.866025i
$59$	0.258819	−	0.965926i	0.258819	−	0.965926i	−0.707107	−	0.707107i	$-0.750000\pi$
$59$	0.258819	−	0.965926i	0.258819	−	0.965926i	0.965926	−	0.258819i	$-0.0833333\pi$
$60$		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			0
$66$		0			0
$67$		0			0		−0.965926	−	0.258819i	$-0.916667\pi$
$67$		0			0		0.965926	+	0.258819i	$0.0833333\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	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		−	1.00000i	$-0.5\pi$
$73$	0.866025	−	0.500000i	0.866025	−	0.500000i	0.866025	+	0.500000i	$0.166667\pi$
$74$	−0.707107	−	1.22474i	−0.707107	−	1.22474i
$75$		0			0
$76$	−1.00000	+	1.00000i	−1.00000	+	1.00000i
$77$	−0.707107	+	0.707107i	−0.707107	+	0.707107i
$78$		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			0
$81$		0			0
$82$	1.36603	−	0.366025i	1.36603	−	0.366025i
$83$	0.707107	−	0.707107i	0.707107	−	0.707107i	−0.258819	−	0.965926i	$-0.583333\pi$
$83$	0.707107	−	0.707107i	0.707107	−	0.707107i	0.965926	+	0.258819i	$0.0833333\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$		0			0
$88$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$89$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$89$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$90$		0			0
$91$	−0.366025	−	1.36603i	−0.366025	−	1.36603i
$92$			1.41421i			1.41421i
$93$		0			0
$94$		0			0
$95$		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	$0.333333\pi$
$98$	−0.707107	−	0.707107i	−0.707107	−	0.707107i
$99$		0			0
$100$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$101$	−0.258819	+	0.965926i	−0.258819	+	0.965926i	0.707107	+	0.707107i	$0.250000\pi$
$101$	−0.258819	+	0.965926i	−0.258819	+	0.965926i	−0.965926	+	0.258819i	$0.916667\pi$
$102$		0			0
$103$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$103$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$104$	1.41421			1.41421
$105$		0			0
$106$			2.00000i			2.00000i
$107$		0			0		−0.258819	−	0.965926i	$-0.583333\pi$
$107$		0			0		0.258819	+	0.965926i	$0.416667\pi$
$108$		0			0
$109$		0			0		−0.965926	−	0.258819i	$-0.916667\pi$
$109$		0			0		0.965926	+	0.258819i	$0.0833333\pi$
$110$		0			0
$111$		0			0
$112$	0.866025	−	0.500000i	0.866025	−	0.500000i
$113$			1.41421i			1.41421i	0.707107	+	0.707107i	$0.250000\pi$
$113$			1.41421i			1.41421i	−0.707107	+	0.707107i	$0.750000\pi$
$114$		0			0
$115$		0			0
$116$	−0.258819	+	0.965926i	−0.258819	+	0.965926i
$117$		0			0
$118$			1.00000i			1.00000i
$119$		0			0
$120$		0			0
$121$		0			0
$122$	1.22474	−	0.707107i	1.22474	−	0.707107i
$123$		0			0
$124$	0.866025	+	0.500000i	0.866025	+	0.500000i
$125$		0			0
$126$		0			0
$127$		0			0			−	1.00000i	$-0.5\pi$
$127$		0			0				1.00000i	$0.5\pi$
$128$	0.258819	+	0.965926i	0.258819	+	0.965926i
$129$		0			0
$130$		0			0
$131$	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
$133$	−1.36603	−	0.366025i	−1.36603	−	0.366025i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−0.707107	−	1.22474i	−0.707107	−	1.22474i	−0.965926	−	0.258819i	$-0.916667\pi$
$137$	−0.707107	−	1.22474i	−0.707107	−	1.22474i	0.258819	−	0.965926i	$-0.416667\pi$
$138$		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			$0$
$140$		0			0
$141$		0			0
$142$	−1.36603	+	0.366025i	−1.36603	+	0.366025i
$143$	1.22474	−	0.707107i	1.22474	−	0.707107i
$144$		0			0
$145$		0			0
$146$	−0.707107	+	0.707107i	−0.707107	+	0.707107i
$147$		0			0
$148$	1.00000	+	1.00000i	1.00000	+	1.00000i
$149$	−0.965926	+	0.258819i	−0.965926	+	0.258819i	−0.707107	−	0.707107i	$-0.750000\pi$
$149$	−0.965926	+	0.258819i	−0.965926	+	0.258819i	−0.258819	+	0.965926i	$0.583333\pi$
$150$		0			0
$151$	−0.866025	+	0.500000i	−0.866025	+	0.500000i	−0.866025	−	0.500000i	$-0.833333\pi$
$151$	−0.866025	+	0.500000i	−0.866025	+	0.500000i			1.00000i	$0.5\pi$
$152$	0.707107	−	1.22474i	0.707107	−	1.22474i
$153$		0			0
$154$	0.500000	−	0.866025i	0.500000	−	0.866025i
$155$		0			0
$156$		0			0
$157$	−0.366025	+	1.36603i	−0.366025	+	1.36603i	0.500000	+	0.866025i	$0.333333\pi$
$157$	−0.366025	+	1.36603i	−0.366025	+	1.36603i	−0.866025	+	0.500000i	$0.833333\pi$
$158$	0.258819	−	0.965926i	0.258819	−	0.965926i
$159$		0			0
$160$		0			0
$161$	−1.22474	+	0.707107i	−1.22474	+	0.707107i
$162$		0			0
$163$		0			0		−0.258819	−	0.965926i	$-0.583333\pi$
$163$		0			0		0.258819	+	0.965926i	$0.416667\pi$
$164$	−1.22474	+	0.707107i	−1.22474	+	0.707107i
$165$		0			0
$166$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$167$	1.41421			1.41421			0.707107	−	0.707107i	$-0.250000\pi$
$167$	1.41421			1.41421			0.707107	+	0.707107i	$0.250000\pi$
$168$		0			0
$169$			1.00000i			1.00000i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−0.258819	−	0.965926i	−0.258819	−	0.965926i	−0.965926	−	0.258819i	$-0.916667\pi$
$173$	−0.258819	−	0.965926i	−0.258819	−	0.965926i	0.707107	−	0.707107i	$-0.250000\pi$
$174$		0			0
$175$	−1.00000			−1.00000
$176$	0.707107	+	0.707107i	0.707107	+	0.707107i
$177$		0			0
$178$		0			0
$179$	0.965926	+	0.258819i	0.965926	+	0.258819i	0.707107	−	0.707107i	$-0.250000\pi$
$179$	0.965926	+	0.258819i	0.965926	+	0.258819i	0.258819	+	0.965926i	$0.416667\pi$
$180$		0			0
$181$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$181$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$182$	0.707107	+	1.22474i	0.707107	+	1.22474i
$183$		0			0
$184$	−0.366025	−	1.36603i	−0.366025	−	1.36603i
$185$		0			0
$186$		0			0
$187$		0			0
$188$		0			0
$189$		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			$\pi$
$194$	−0.965926	+	0.258819i	−0.965926	+	0.258819i
$195$		0			0
$196$	0.866025	+	0.500000i	0.866025	+	0.500000i
$197$	0.707107	+	0.707107i	0.707107	+	0.707107i	0.965926	−	0.258819i	$-0.0833333\pi$
$197$	0.707107	+	0.707107i	0.707107	+	0.707107i	−0.258819	+	0.965926i	$0.583333\pi$
$198$		0			0
$199$	−0.866025	+	0.500000i	−0.866025	+	0.500000i	−0.866025	−	0.500000i	$-0.833333\pi$
$199$	−0.866025	+	0.500000i	−0.866025	+	0.500000i			1.00000i	$0.5\pi$
$200$	0.258819	−	0.965926i	0.258819	−	0.965926i
$201$		0			0
$202$		−	1.00000i		−	1.00000i
$203$	−0.965926	+	0.258819i	−0.965926	+	0.258819i
$204$		0			0
$205$		0			0
$206$		0			0
$207$		0			0
$208$	−1.36603	+	0.366025i	−1.36603	+	0.366025i
$209$		−	1.41421i		−	1.41421i
$210$		0			0
$211$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$211$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$212$	−0.517638	−	1.93185i	−0.517638	−	1.93185i
$213$		0			0
$214$		0			0
$215$		0			0
$216$		0			0
$217$			1.00000i			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	$-0.666667\pi$
$223$	−1.00000			−1.00000			−0.500000	+	0.866025i	$0.666667\pi$
$224$	−0.707107	+	0.707107i	−0.707107	+	0.707107i
$225$		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			0		0.965926	−	0.258819i	$-0.0833333\pi$
$229$		0			0		−0.965926	+	0.258819i	$0.916667\pi$
$230$		0			0
$231$		0			0
$232$		−	1.00000i		−	1.00000i
$233$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$233$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$234$		0			0
$235$		0			0
$236$	−0.258819	−	0.965926i	−0.258819	−	0.965926i
$237$		0			0
$238$		0			0
$239$		0			0		1.00000			$0$
$239$		0			0		−1.00000			$\pi$
$240$		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			0
$246$		0			0
$247$	1.73205	+	1.00000i	1.73205	+	1.00000i
$248$	−0.965926	−	0.258819i	−0.965926	−	0.258819i
$249$		0			0
$250$		0			0
$251$	0.707107	+	0.707107i	0.707107	+	0.707107i	0.965926	−	0.258819i	$-0.0833333\pi$
$251$	0.707107	+	0.707107i	0.707107	+	0.707107i	−0.258819	+	0.965926i	$0.583333\pi$
$252$		0			0
$253$	−1.00000	−	1.00000i	−1.00000	−	1.00000i
$254$		0			0
$255$		0			0
$256$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$257$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$257$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$258$		0			0
$259$	−0.366025	+	1.36603i	−0.366025	+	1.36603i
$260$		0			0
$261$		0			0
$262$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$263$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$263$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$264$		0			0
$265$		0			0
$266$	1.41421			1.41421
$267$		0			0
$268$		0			0
$269$	−0.965926	−	0.258819i	−0.965926	−	0.258819i	−0.258819	−	0.965926i	$-0.583333\pi$
$269$	−0.965926	−	0.258819i	−0.965926	−	0.258819i	−0.707107	+	0.707107i	$0.750000\pi$
$270$		0			0
$271$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$271$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$272$		0			0
$273$		0			0
$274$	1.00000	+	1.00000i	1.00000	+	1.00000i
$275$	−0.258819	−	0.965926i	−0.258819	−	0.965926i
$276$		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
$280$		0			0
$281$		0			0			−	1.00000i	$-0.5\pi$
$281$		0			0				1.00000i	$0.5\pi$
$282$		0			0
$283$	1.36603	+	0.366025i	1.36603	+	0.366025i	0.866025	−	0.500000i	$-0.166667\pi$
$283$	1.36603	+	0.366025i	1.36603	+	0.366025i	0.500000	+	0.866025i	$0.333333\pi$
$284$	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			0
$291$		0			0
$292$	0.500000	−	0.866025i	0.500000	−	0.866025i
$293$	−0.707107	−	0.707107i	−0.707107	−	0.707107i	0.258819	−	0.965926i	$-0.416667\pi$
$293$	−0.707107	−	0.707107i	−0.707107	−	0.707107i	−0.965926	+	0.258819i	$0.916667\pi$
$294$		0			0
$295$		0			0
$296$	−1.22474	−	0.707107i	−1.22474	−	0.707107i
$297$		0			0
$298$	0.866025	−	0.500000i	0.866025	−	0.500000i
$299$	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
$304$	−0.366025	+	1.36603i	−0.366025	+	1.36603i
$305$		0			0
$306$		0			0
$307$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$307$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$308$	−0.258819	+	0.965926i	−0.258819	+	0.965926i
$309$		0			0
$310$		0			0
$311$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$311$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$312$		0			0
$313$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$313$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$314$		−	1.41421i		−	1.41421i
$315$		0			0
$316$			1.00000i			1.00000i
$317$	−0.258819	−	0.965926i	−0.258819	−	0.965926i	−0.965926	−	0.258819i	$-0.916667\pi$
$317$	−0.258819	−	0.965926i	−0.258819	−	0.965926i	0.707107	−	0.707107i	$-0.250000\pi$
$318$		0			0
$319$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$320$		0			0
$321$		0			0
$322$	1.00000	−	1.00000i	1.00000	−	1.00000i
$323$		0			0
$324$		0			0
$325$	1.36603	+	0.366025i	1.36603	+	0.366025i
$326$		0			0
$327$		0			0
$328$	1.00000	−	1.00000i	1.00000	−	1.00000i
$329$		0			0
$330$		0			0
$331$		0			0		0.965926	−	0.258819i	$-0.0833333\pi$
$331$		0			0		−0.965926	+	0.258819i	$0.916667\pi$
$332$	0.258819	−	0.965926i	0.258819	−	0.965926i
$333$		0			0
$334$	−1.36603	+	0.366025i	−1.36603	+	0.366025i
$335$		0			0
$336$		0			0
$337$	1.00000			1.00000			0.500000	−	0.866025i	$-0.333333\pi$
$337$	1.00000			1.00000			0.500000	+	0.866025i	$0.333333\pi$
$338$	−0.258819	−	0.965926i	−0.258819	−	0.965926i
$339$		0			0
$340$		0			0
$341$	−0.965926	+	0.258819i	−0.965926	+	0.258819i
$342$		0			0
$343$			1.00000i			1.00000i
$344$		0			0
$345$		0			0
$346$	0.500000	+	0.866025i	0.500000	+	0.866025i
$347$	−0.258819	+	0.965926i	−0.258819	+	0.965926i	0.707107	+	0.707107i	$0.250000\pi$
$347$	−0.258819	+	0.965926i	−0.258819	+	0.965926i	−0.965926	+	0.258819i	$0.916667\pi$
$348$		0			0
$349$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$349$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$350$	0.965926	−	0.258819i	0.965926	−	0.258819i
$351$		0			0
$352$	−0.866025	−	0.500000i	−0.866025	−	0.500000i
$353$	1.22474	−	0.707107i	1.22474	−	0.707107i	0.258819	−	0.965926i	$-0.416667\pi$
$353$	1.22474	−	0.707107i	1.22474	−	0.707107i	0.965926	+	0.258819i	$0.0833333\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		0			0
$358$	−1.00000			−1.00000
$359$	0.707107	−	1.22474i	0.707107	−	1.22474i	−0.258819	−	0.965926i	$-0.583333\pi$
$359$	0.707107	−	1.22474i	0.707107	−	1.22474i	0.965926	−	0.258819i	$-0.0833333\pi$
$360$		0			0
$361$	0.866025	−	0.500000i	0.866025	−	0.500000i
$362$		0			0
$363$		0			0
$364$	−1.00000	−	1.00000i	−1.00000	−	1.00000i
$365$		0			0
$366$		0			0
$367$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$367$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$368$	0.707107	+	1.22474i	0.707107	+	1.22474i
$369$		0			0
$370$		0			0
$371$	1.41421	−	1.41421i	1.41421	−	1.41421i
$372$		0			0
$373$		0			0		0.965926	−	0.258819i	$-0.0833333\pi$
$373$		0			0		−0.965926	+	0.258819i	$0.916667\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	1.41421			1.41421
$378$		0			0
$379$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$379$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	1.22474	+	0.707107i	1.22474	+	0.707107i	0.965926	−	0.258819i	$-0.0833333\pi$
$383$	1.22474	+	0.707107i	1.22474	+	0.707107i	0.258819	+	0.965926i	$0.416667\pi$
$384$		0			0
$385$		0			0
$386$	0.707107	+	0.707107i	0.707107	+	0.707107i
$387$		0			0
$388$	0.866025	−	0.500000i	0.866025	−	0.500000i
$389$	−0.258819	+	0.965926i	−0.258819	+	0.965926i	0.707107	+	0.707107i	$0.250000\pi$
$389$	−0.258819	+	0.965926i	−0.258819	+	0.965926i	−0.965926	+	0.258819i	$0.916667\pi$
$390$		0			0
$391$		0			0
$392$	−0.965926	−	0.258819i	−0.965926	−	0.258819i
$393$		0			0
$394$	−0.866025	−	0.500000i	−0.866025	−	0.500000i
$395$		0			0
$396$		0			0
$397$		0			0		−0.258819	−	0.965926i	$-0.583333\pi$
$397$		0			0		0.258819	+	0.965926i	$0.416667\pi$
$398$	0.707107	−	0.707107i	0.707107	−	0.707107i
$399$		0			0
$400$			1.00000i			1.00000i
$401$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$401$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$402$		0			0
$403$	0.366025	−	1.36603i	0.366025	−	1.36603i
$404$	0.258819	+	0.965926i	0.258819	+	0.965926i
$405$		0			0
$406$	0.866025	−	0.500000i	0.866025	−	0.500000i
$407$	−1.41421			−1.41421
$408$		0			0
$409$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$409$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	0.707107	−	0.707107i	0.707107	−	0.707107i
$414$		0			0
$415$		0			0
$416$	1.22474	−	0.707107i	1.22474	−	0.707107i
$417$		0			0
$418$	0.366025	+	1.36603i	0.366025	+	1.36603i
$419$	−1.41421	+	1.41421i	−1.41421	+	1.41421i	−0.707107	+	0.707107i	$0.750000\pi$
$419$	−1.41421	+	1.41421i	−1.41421	+	1.41421i	−0.707107	+	0.707107i	$0.750000\pi$
$420$		0			0
$421$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$421$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$422$		0			0
$423$		0			0
$424$	1.00000	+	1.73205i	1.00000	+	1.73205i
$425$		0			0
$426$		0			0
$427$	−1.36603	−	0.366025i	−1.36603	−	0.366025i
$428$		0			0
$429$		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
$433$	1.00000			1.00000			0.500000	−	0.866025i	$-0.333333\pi$
$433$	1.00000			1.00000			0.500000	+	0.866025i	$0.333333\pi$
$434$	−0.258819	−	0.965926i	−0.258819	−	0.965926i
$435$		0			0
$436$		0			0
$437$	0.517638	−	1.93185i	0.517638	−	1.93185i
$438$		0			0
$439$	0.866025	+	0.500000i	0.866025	+	0.500000i	0.866025	−	0.500000i	$-0.166667\pi$
$439$	0.866025	+	0.500000i	0.866025	+	0.500000i			1.00000i	$0.5\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−0.965926	+	0.258819i	−0.965926	+	0.258819i	−0.707107	−	0.707107i	$-0.750000\pi$
$443$	−0.965926	+	0.258819i	−0.965926	+	0.258819i	−0.258819	+	0.965926i	$0.583333\pi$
$444$		0			0
$445$		0			0
$446$	0.965926	−	0.258819i	0.965926	−	0.258819i
$447$		0			0
$448$	0.500000	−	0.866025i	0.500000	−	0.866025i
$449$		0			0		1.00000			$0$
$449$		0			0		−1.00000			$\pi$
$450$		0			0
$451$	0.366025	−	1.36603i	0.366025	−	1.36603i
$452$	0.707107	+	1.22474i	0.707107	+	1.22474i
$453$		0			0
$454$	−1.00000			−1.00000
$455$		0			0
$456$		0			0
$457$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$457$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−0.707107	+	0.707107i	−0.707107	+	0.707107i	−0.965926	−	0.258819i	$-0.916667\pi$
$461$	−0.707107	+	0.707107i	−0.707107	+	0.707107i	0.258819	+	0.965926i	$0.416667\pi$
$462$		0			0
$463$	1.00000			1.00000			0.500000	−	0.866025i	$-0.333333\pi$
$463$	1.00000			1.00000			0.500000	+	0.866025i	$0.333333\pi$
$464$	0.258819	+	0.965926i	0.258819	+	0.965926i
$465$		0			0
$466$		0			0
$467$	−0.258819	−	0.965926i	−0.258819	−	0.965926i	−0.965926	−	0.258819i	$-0.916667\pi$
$467$	−0.258819	−	0.965926i	−0.258819	−	0.965926i	0.707107	−	0.707107i	$-0.250000\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$		0			0
$472$	0.500000	+	0.866025i	0.500000	+	0.866025i
$473$		0			0
$474$		0			0
$475$	1.00000	−	1.00000i	1.00000	−	1.00000i
$476$		0			0
$477$		0			0
$478$		0			0
$479$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$479$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$480$		0			0
$481$	1.00000	−	1.73205i	1.00000	−	1.73205i
$482$	0.707107	+	0.707107i	0.707107	+	0.707107i
$483$		0			0
$484$		0			0
$485$		0			0
$486$		0			0
$487$	−0.866025	+	0.500000i	−0.866025	+	0.500000i	−0.866025	−	0.500000i	$-0.833333\pi$
$487$	−0.866025	+	0.500000i	−0.866025	+	0.500000i			1.00000i	$0.5\pi$
$488$	0.707107	−	1.22474i	0.707107	−	1.22474i
$489$		0			0
$490$		0			0
$491$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$491$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$492$		0			0
$493$		0			0
$494$	−1.93185	−	0.517638i	−1.93185	−	0.517638i
$495$		0			0
$496$	1.00000			1.00000
$497$	1.22474	+	0.707107i	1.22474	+	0.707107i
$498$		0			0
$499$		0			0		−0.258819	−	0.965926i	$-0.583333\pi$
$499$		0			0		0.258819	+	0.965926i	$0.416667\pi$
$500$		0			0
$501$		0			0
$502$	−0.866025	−	0.500000i	−0.866025	−	0.500000i
$503$		0			0			−	1.00000i	$-0.5\pi$
$503$		0			0				1.00000i	$0.5\pi$
$504$		0			0
$505$		0			0
$506$	1.22474	+	0.707107i	1.22474	+	0.707107i
$507$		0			0
$508$		0			0
$509$	0.258819	+	0.965926i	0.258819	+	0.965926i	0.965926	+	0.258819i	$0.0833333\pi$
$509$	0.258819	+	0.965926i	0.258819	+	0.965926i	−0.707107	+	0.707107i	$0.750000\pi$
$510$		0			0
$511$	1.00000			1.00000
$512$	0.707107	+	0.707107i	0.707107	+	0.707107i
$513$		0			0
$514$		0			0
$515$		0			0
$516$		0			0
$517$		0			0
$518$		−	1.41421i		−	1.41421i
$519$		0			0
$520$		0			0
$521$	−0.707107	−	1.22474i	−0.707107	−	1.22474i	−0.965926	−	0.258819i	$-0.916667\pi$
$521$	−0.707107	−	1.22474i	−0.707107	−	1.22474i	0.258819	−	0.965926i	$-0.416667\pi$
$522$		0			0
$523$	−0.366025	−	1.36603i	−0.366025	−	1.36603i	−0.866025	−	0.500000i	$-0.833333\pi$
$523$	−0.366025	−	1.36603i	−0.366025	−	1.36603i	0.500000	−	0.866025i	$-0.333333\pi$
$524$	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			0
$531$		0			0
$532$	−1.36603	+	0.366025i	−1.36603	+	0.366025i
$533$	1.41421	+	1.41421i	1.41421	+	1.41421i
$534$		0			0
$535$		0			0
$536$		0			0
$537$		0			0
$538$	1.00000			1.00000
$539$	−0.965926	+	0.258819i	−0.965926	+	0.258819i
$540$		0			0
$541$		0			0		−0.258819	−	0.965926i	$-0.583333\pi$
$541$		0			0		0.258819	+	0.965926i	$0.416667\pi$
$542$		0			0
$543$		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			$\pi$
$548$	−1.22474	−	0.707107i	−1.22474	−	0.707107i
$549$		0			0
$550$	0.500000	+	0.866025i	0.500000	+	0.866025i
$551$	0.707107	−	1.22474i	0.707107	−	1.22474i
$552$		0			0
$553$	−0.866025	+	0.500000i	−0.866025	+	0.500000i
$554$		0			0
$555$		0			0
$556$	0.366025	−	1.36603i	0.366025	−	1.36603i
$557$	0.965926	+	0.258819i	0.965926	+	0.258819i	0.707107	−	0.707107i	$-0.250000\pi$
$557$	0.965926	+	0.258819i	0.965926	+	0.258819i	0.258819	+	0.965926i	$0.416667\pi$
$558$		0			0
$559$		0			0
$560$		0			0
$561$		0			0
$562$		0			0
$563$		0			0		0.258819	−	0.965926i	$-0.416667\pi$
$563$		0			0		−0.258819	+	0.965926i	$0.583333\pi$
$564$		0			0
$565$		0			0
$566$	−1.41421			−1.41421
$567$		0			0
$568$	−1.00000	+	1.00000i	−1.00000	+	1.00000i
$569$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$569$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$570$		0			0
$571$		0			0		0.258819	−	0.965926i	$-0.416667\pi$
$571$		0			0		−0.258819	+	0.965926i	$0.583333\pi$
$572$	0.707107	−	1.22474i	0.707107	−	1.22474i
$573$		0			0
$574$	1.36603	+	0.366025i	1.36603	+	0.366025i
$575$		−	1.41421i		−	1.41421i
$576$		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			$\pi$
$578$	0.258819	−	0.965926i	0.258819	−	0.965926i
$579$		0			0
$580$		0			0
$581$	0.965926	−	0.258819i	0.965926	−	0.258819i
$582$		0			0
$583$	1.73205	+	1.00000i	1.73205	+	1.00000i
$584$	−0.258819	+	0.965926i	−0.258819	+	0.965926i
$585$		0			0
$586$	0.866025	+	0.500000i	0.866025	+	0.500000i
$587$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$587$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$588$		0			0
$589$	−1.00000	−	1.00000i	−1.00000	−	1.00000i
$590$		0			0
$591$		0			0
$592$	1.36603	+	0.366025i	1.36603	+	0.366025i
$593$	−1.22474	−	0.707107i	−1.22474	−	0.707107i	−0.258819	−	0.965926i	$-0.583333\pi$
$593$	−1.22474	−	0.707107i	−1.22474	−	0.707107i	−0.965926	+	0.258819i	$0.916667\pi$
$594$		0			0
$595$		0			0
$596$	−0.707107	+	0.707107i	−0.707107	+	0.707107i
$597$		0			0
$598$	−1.73205	+	1.00000i	−1.73205	+	1.00000i
$599$	0.707107	+	1.22474i	0.707107	+	1.22474i	0.965926	+	0.258819i	$0.0833333\pi$
$599$	0.707107	+	1.22474i	0.707107	+	1.22474i	−0.258819	+	0.965926i	$0.583333\pi$
$600$		0			0
$601$		−	2.00000i		−	2.00000i		−	1.00000i	$-0.5\pi$
$601$		−	2.00000i		−	2.00000i		−	1.00000i	$-0.5\pi$
$602$		0			0
$603$		0			0
$604$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$605$		0			0
$606$		0			0
$607$	−0.500000	+	0.866025i	−0.500000	+	0.866025i	0.500000	+	0.866025i	$0.333333\pi$
$607$	−0.500000	+	0.866025i	−0.500000	+	0.866025i	−1.00000			$\pi$
$608$		−	1.41421i		−	1.41421i
$609$		0			0
$610$		0			0
$611$		0			0
$612$		0			0
$613$		0			0		0.258819	−	0.965926i	$-0.416667\pi$
$613$		0			0		−0.258819	+	0.965926i	$0.583333\pi$
$614$		0			0
$615$		0			0
$616$		−	1.00000i		−	1.00000i
$617$	1.41421			1.41421			0.707107	−	0.707107i	$-0.250000\pi$
$617$	1.41421			1.41421			0.707107	+	0.707107i	$0.250000\pi$
$618$		0			0
$619$	1.36603	+	0.366025i	1.36603	+	0.366025i	0.866025	−	0.500000i	$-0.166667\pi$
$619$	1.36603	+	0.366025i	1.36603	+	0.366025i	0.500000	+	0.866025i	$0.333333\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$		0			0
$625$	0.500000	−	0.866025i	0.500000	−	0.866025i
$626$		0			0
$627$		0			0
$628$	0.366025	+	1.36603i	0.366025	+	1.36603i
$629$		0			0
$630$		0			0
$631$		−	1.00000i		−	1.00000i	−0.866025	−	0.500000i	$-0.833333\pi$
$631$		−	1.00000i		−	1.00000i	0.866025	−	0.500000i	$-0.166667\pi$
$632$	−0.258819	−	0.965926i	−0.258819	−	0.965926i
$633$		0			0
$634$	0.500000	+	0.866025i	0.500000	+	0.866025i
$635$		0			0
$636$		0			0
$637$	0.366025	−	1.36603i	0.366025	−	1.36603i
$638$	0.707107	+	0.707107i	0.707107	+	0.707107i
$639$		0			0
$640$		0			0
$641$	1.22474	−	0.707107i	1.22474	−	0.707107i	0.258819	−	0.965926i	$-0.416667\pi$
$641$	1.22474	−	0.707107i	1.22474	−	0.707107i	0.965926	+	0.258819i	$0.0833333\pi$
$642$		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			$\pi$
$644$	−0.707107	+	1.22474i	−0.707107	+	1.22474i
$645$		0			0
$646$		0			0
$647$	0.707107	+	1.22474i	0.707107	+	1.22474i	0.965926	+	0.258819i	$0.0833333\pi$
$647$	0.707107	+	1.22474i	0.707107	+	1.22474i	−0.258819	+	0.965926i	$0.583333\pi$
$648$		0			0
$649$	0.866025	+	0.500000i	0.866025	+	0.500000i
$650$	−1.41421			−1.41421
$651$		0			0
$652$		0			0
$653$		0			0		0.965926	−	0.258819i	$-0.0833333\pi$
$653$		0			0		−0.965926	+	0.258819i	$0.916667\pi$
$654$		0			0
$655$		0			0
$656$	−0.707107	+	1.22474i	−0.707107	+	1.22474i
$657$		0			0
$658$		0			0
$659$	0.707107	−	0.707107i	0.707107	−	0.707107i	−0.258819	−	0.965926i	$-0.583333\pi$
$659$	0.707107	−	0.707107i	0.707107	−	0.707107i	0.965926	+	0.258819i	$0.0833333\pi$
$660$		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
$664$			1.00000i			1.00000i
$665$		0			0
$666$		0			0
$667$	−0.366025	−	1.36603i	−0.366025	−	1.36603i
$668$	1.22474	−	0.707107i	1.22474	−	0.707107i
$669$		0			0
$670$		0			0
$671$		−	1.41421i		−	1.41421i
$672$		0			0
$673$	2.00000			2.00000			1.00000			$0$
$673$	2.00000			2.00000			1.00000			$0$
$674$	−0.965926	+	0.258819i	−0.965926	+	0.258819i
$675$		0			0
$676$	0.500000	+	0.866025i	0.500000	+	0.866025i
$677$	0.965926	−	0.258819i	0.965926	−	0.258819i	0.258819	−	0.965926i	$-0.416667\pi$
$677$	0.965926	−	0.258819i	0.965926	−	0.258819i	0.707107	+	0.707107i	$0.250000\pi$
$678$		0			0
$679$	0.866025	+	0.500000i	0.866025	+	0.500000i
$680$		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	$-0.750000\pi$
$683$	0.258819	−	0.965926i	0.258819	−	0.965926i	0.965926	−	0.258819i	$-0.0833333\pi$
$684$		0			0
$685$		0			0
$686$	−0.258819	−	0.965926i	−0.258819	−	0.965926i
$687$		0			0
$688$		0			0
$689$	−2.44949	+	1.41421i	−2.44949	+	1.41421i
$690$		0			0
$691$	1.36603	−	0.366025i	1.36603	−	0.366025i	0.500000	−	0.866025i	$-0.333333\pi$
$691$	1.36603	−	0.366025i	1.36603	−	0.366025i	0.866025	+	0.500000i	$0.166667\pi$
$692$	−0.707107	−	0.707107i	−0.707107	−	0.707107i
$693$		0			0
$694$		−	1.00000i		−	1.00000i
$695$		0			0
$696$		0			0
$697$		0			0
$698$		0			0
$699$		0			0
$700$	−0.866025	+	0.500000i	−0.866025	+	0.500000i
$701$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$701$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$702$		0			0
$703$	−1.00000	−	1.73205i	−1.00000	−	1.73205i
$704$	0.965926	+	0.258819i	0.965926	+	0.258819i
$705$		0			0
$706$	−1.00000	+	1.00000i	−1.00000	+	1.00000i
$707$	−0.707107	+	0.707107i	−0.707107	+	0.707107i
$708$		0			0
$709$		0			0		0.965926	−	0.258819i	$-0.0833333\pi$
$709$		0			0		−0.965926	+	0.258819i	$0.916667\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	−1.41421			−1.41421
$714$		0			0
$715$		0			0
$716$	0.965926	−	0.258819i	0.965926	−	0.258819i
$717$		0			0
$718$	−0.366025	+	1.36603i	−0.366025	+	1.36603i
$719$	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			0
$725$	0.258819	−	0.965926i	0.258819	−	0.965926i
$726$		0			0
$727$		0			0		1.00000			$0$
$727$		0			0		−1.00000			$\pi$
$728$	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			0
$735$		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	0.500000	+	0.866025i	$0.333333\pi$
$739$	−0.366025	+	1.36603i	−0.366025	+	1.36603i	−0.866025	+	0.500000i	$0.833333\pi$
$740$		0			0
$741$		0			0
$742$	−1.00000	+	1.73205i	−1.00000	+	1.73205i
$743$		0			0			−	1.00000i	$-0.5\pi$
$743$		0			0				1.00000i	$0.5\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$751$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$752$		0			0
$753$		0			0
$754$	−1.36603	+	0.366025i	−1.36603	+	0.366025i
$755$		0			0
$756$		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			$\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−0.707107	+	1.22474i	−0.707107	+	1.22474i	0.258819	+	0.965926i	$0.416667\pi$
$761$	−0.707107	+	1.22474i	−0.707107	+	1.22474i	−0.965926	+	0.258819i	$0.916667\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$		0			0
$766$	−1.36603	−	0.366025i	−1.36603	−	0.366025i
$767$	−1.22474	+	0.707107i	−1.22474	+	0.707107i
$768$		0			0
$769$	−1.00000			−1.00000			−0.500000	−	0.866025i	$-0.666667\pi$
$769$	−1.00000			−1.00000			−0.500000	+	0.866025i	$0.666667\pi$
$770$		0			0
$771$		0			0
$772$	−0.866025	−	0.500000i	−0.866025	−	0.500000i
$773$		0			0		−0.965926	−	0.258819i	$-0.916667\pi$
$773$		0			0		0.965926	+	0.258819i	$0.0833333\pi$
$774$		0			0
$775$	−0.866025	−	0.500000i	−0.866025	−	0.500000i
$776$	−0.707107	+	0.707107i	−0.707107	+	0.707107i
$777$		0			0
$778$		−	1.00000i		−	1.00000i
$779$	1.93185	−	0.517638i	1.93185	−	0.517638i
$780$		0			0
$781$	−0.366025	+	1.36603i	−0.366025	+	1.36603i
$782$		0			0
$783$		0			0
$784$	1.00000			1.00000
$785$		0			0
$786$		0			0
$787$		0			0		−0.965926	−	0.258819i	$-0.916667\pi$
$787$		0			0		0.965926	+	0.258819i	$0.0833333\pi$
$788$	0.965926	+	0.258819i	0.965926	+	0.258819i
$789$		0			0
$790$		0			0
$791$	−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
$796$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$797$	0.707107	−	0.707107i	0.707107	−	0.707107i	−0.258819	−	0.965926i	$-0.583333\pi$
$797$	0.707107	−	0.707107i	0.707107	−	0.707107i	0.965926	+	0.258819i	$0.0833333\pi$
$798$		0			0
$799$		0			0
$800$	−0.258819	−	0.965926i	−0.258819	−	0.965926i
$801$		0			0
$802$		0			0
$803$	0.258819	+	0.965926i	0.258819	+	0.965926i
$804$		0			0
$805$		0			0
$806$			1.41421i			1.41421i
$807$		0			0
$808$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$809$	0.707107	+	1.22474i	0.707107	+	1.22474i	0.965926	+	0.258819i	$0.0833333\pi$
$809$	0.707107	+	1.22474i	0.707107	+	1.22474i	−0.258819	+	0.965926i	$0.583333\pi$
$810$		0			0
$811$	1.00000	−	1.00000i	1.00000	−	1.00000i		−	1.00000i	$-0.5\pi$
$811$	1.00000	−	1.00000i	1.00000	−	1.00000i	1.00000			$0$
$812$	−0.707107	+	0.707107i	−0.707107	+	0.707107i
$813$		0			0
$814$	1.36603	−	0.366025i	1.36603	−	0.366025i
$815$		0			0
$816$		0			0
$817$		0			0
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−0.965926	+	0.258819i	−0.965926	+	0.258819i	−0.707107	−	0.707107i	$-0.750000\pi$
$821$	−0.965926	+	0.258819i	−0.965926	+	0.258819i	−0.258819	+	0.965926i	$0.583333\pi$
$822$		0			0
$823$	−0.866025	+	0.500000i	−0.866025	+	0.500000i	−0.866025	−	0.500000i	$-0.833333\pi$
$823$	−0.866025	+	0.500000i	−0.866025	+	0.500000i			1.00000i	$0.5\pi$
$824$		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			0
$831$		0			0
$832$	−1.00000	+	1.00000i	−1.00000	+	1.00000i
$833$		0			0
$834$		0			0
$835$		0			0
$836$	−0.707107	−	1.22474i	−0.707107	−	1.22474i
$837$		0			0
$838$	1.00000	−	1.73205i	1.00000	−	1.73205i
$839$	−1.41421			−1.41421			−0.707107	−	0.707107i	$-0.750000\pi$
$839$	−1.41421			−1.41421			−0.707107	+	0.707107i	$0.750000\pi$
$840$		0			0
$841$		0			0
$842$		0			0
$843$		0			0
$844$		0			0
$845$		0			0
$846$		0			0
$847$		0			0
$848$	−1.41421	−	1.41421i	−1.41421	−	1.41421i
$849$		0			0
$850$		0			0
$851$	−1.93185	−	0.517638i	−1.93185	−	0.517638i
$852$		0			0
$853$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$853$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$854$	1.41421			1.41421
$855$		0			0
$856$		0			0
$857$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$857$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$858$		0			0
$859$		0			0		0.965926	−	0.258819i	$-0.0833333\pi$
$859$		0			0		−0.965926	+	0.258819i	$0.916667\pi$
$860$		0			0
$861$		0			0
$862$	1.00000	−	1.00000i	1.00000	−	1.00000i
$863$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$863$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$864$		0			0
$865$		0			0
$866$	−0.965926	+	0.258819i	−0.965926	+	0.258819i
$867$		0			0
$868$	0.500000	+	0.866025i	0.500000	+	0.866025i
$869$	−0.707107	−	0.707107i	−0.707107	−	0.707107i
$870$		0			0
$871$		0			0
$872$		0			0
$873$		0			0
$874$			2.00000i			2.00000i
$875$		0			0
$876$		0			0
$877$	1.36603	−	0.366025i	1.36603	−	0.366025i	0.500000	−	0.866025i	$-0.333333\pi$
$877$	1.36603	−	0.366025i	1.36603	−	0.366025i	0.866025	+	0.500000i	$0.166667\pi$
$878$	−0.965926	−	0.258819i	−0.965926	−	0.258819i
$879$		0			0
$880$		0			0
$881$			1.41421i			1.41421i	0.707107	+	0.707107i	$0.250000\pi$
$881$			1.41421i			1.41421i	−0.707107	+	0.707107i	$0.750000\pi$
$882$		0			0
$883$	1.00000	+	1.00000i	1.00000	+	1.00000i	1.00000			$0$
$883$	1.00000	+	1.00000i	1.00000	+	1.00000i			1.00000i	$0.5\pi$
$884$		0			0
$885$		0			0
$886$	0.866025	−	0.500000i	0.866025	−	0.500000i
$887$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$887$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$		0			0
$892$	−0.866025	+	0.500000i	−0.866025	+	0.500000i
$893$		0			0
$894$		0			0
$895$		0			0
$896$	−0.258819	+	0.965926i	−0.258819	+	0.965926i
$897$		0			0
$898$		0			0
$899$	−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
$905$		0			0
$906$		0			0
$907$		0			0		0.258819	−	0.965926i	$-0.416667\pi$
$907$		0			0		−0.258819	+	0.965926i	$0.583333\pi$
$908$	0.965926	−	0.258819i	0.965926	−	0.258819i
$909$		0			0
$910$		0			0
$911$			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			0
$915$		0			0
$916$		0			0
$917$	−0.258819	+	0.965926i	−0.258819	+	0.965926i
$918$		0			0
$919$	0.866025	+	0.500000i	0.866025	+	0.500000i	0.866025	−	0.500000i	$-0.166667\pi$
$919$	0.866025	+	0.500000i	0.866025	+	0.500000i			1.00000i	$0.5\pi$
$920$		0			0
$921$		0			0
$922$	0.500000	−	0.866025i	0.500000	−	0.866025i
$923$	−1.41421	−	1.41421i	−1.41421	−	1.41421i
$924$		0			0
$925$	−1.00000	−	1.00000i	−1.00000	−	1.00000i
$926$	−0.965926	+	0.258819i	−0.965926	+	0.258819i
$927$		0			0
$928$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$929$	−1.22474	−	0.707107i	−1.22474	−	0.707107i	−0.258819	−	0.965926i	$-0.583333\pi$
$929$	−1.22474	−	0.707107i	−1.22474	−	0.707107i	−0.965926	+	0.258819i	$0.916667\pi$
$930$		0			0
$931$	−1.00000	−	1.00000i	−1.00000	−	1.00000i
$932$		0			0
$933$		0			0
$934$	0.500000	+	0.866025i	0.500000	+	0.866025i
$935$		0			0
$936$		0			0
$937$		0			0		1.00000			$0$
$937$		0			0		−1.00000			$\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	0.965926	+	0.258819i	0.965926	+	0.258819i	0.707107	−	0.707107i	$-0.250000\pi$
$941$	0.965926	+	0.258819i	0.965926	+	0.258819i	0.258819	+	0.965926i	$0.416667\pi$
$942$		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	0.965926	+	0.258819i	$0.0833333\pi$
$947$	0.258819	+	0.965926i	0.258819	+	0.965926i	−0.707107	+	0.707107i	$0.750000\pi$
$948$		0			0
$949$	−1.36603	−	0.366025i	−1.36603	−	0.366025i
$950$	−0.707107	+	1.22474i	−0.707107	+	1.22474i
$951$		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
$955$		0			0
$956$		0			0
$957$		0			0
$958$		0			0
$959$		−	1.41421i		−	1.41421i
$960$		0			0
$961$		0			0
$962$	−0.517638	+	1.93185i	−0.517638	+	1.93185i
$963$		0			0
$964$	−0.866025	−	0.500000i	−0.866025	−	0.500000i
$965$		0			0
$966$		0			0
$967$		0			0		1.00000			$0$
$967$		0			0		−1.00000			$\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$		0			0		−0.258819	−	0.965926i	$-0.583333\pi$
$971$		0			0		0.258819	+	0.965926i	$0.416667\pi$
$972$		0			0
$973$	1.36603	−	0.366025i	1.36603	−	0.366025i
$974$	0.707107	−	0.707107i	0.707107	−	0.707107i
$975$		0			0
$976$	−0.366025	+	1.36603i	−0.366025	+	1.36603i
$977$	1.22474	−	0.707107i	1.22474	−	0.707107i	0.258819	−	0.965926i	$-0.416667\pi$
$977$	1.22474	−	0.707107i	1.22474	−	0.707107i	0.965926	+	0.258819i	$0.0833333\pi$
$978$		0			0
$979$		0			0
$980$		0			0
$981$		0			0
$982$		0			0
$983$	−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
$988$	2.00000			2.00000
$989$		0			0
$990$		0			0
$991$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$991$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$992$	−0.965926	+	0.258819i	−0.965926	+	0.258819i
$993$		0			0
$994$	−1.36603	−	0.366025i	−1.36603	−	0.366025i
$995$		0			0
$996$		0			0
$997$		0			0		0.258819	−	0.965926i	$-0.416667\pi$
$997$		0			0		−0.258819	+	0.965926i	$0.583333\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3024.1.eg.b.1997.1	yes	8
3.2	odd	2	inner	3024.1.eg.b.1997.2	yes	8
7.4	even	3	inner	3024.1.eg.b.1565.2	yes	8
16.5	even	4	inner	3024.1.eg.b.485.1	yes	8
21.11	odd	6	inner	3024.1.eg.b.1565.1	yes	8
48.5	odd	4	inner	3024.1.eg.b.485.2	yes	8
112.53	even	12	inner	3024.1.eg.b.53.2	yes	8
336.53	odd	12	inner	3024.1.eg.b.53.1	✓	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3024.1.eg.b.53.1	✓	8	336.53	odd	12	inner
3024.1.eg.b.53.2	yes	8	112.53	even	12	inner
3024.1.eg.b.485.1	yes	8	16.5	even	4	inner
3024.1.eg.b.485.2	yes	8	48.5	odd	4	inner
3024.1.eg.b.1565.1	yes	8	21.11	odd	6	inner
3024.1.eg.b.1565.2	yes	8	7.4	even	3	inner
3024.1.eg.b.1997.1	yes	8	1.1	even	1	trivial
3024.1.eg.b.1997.2	yes	8	3.2	odd	2	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)

\(f(q)\)	\(=\)	\(q+(-0.965926 + 0.258819i) q^{2} +(0.866025 - 0.500000i) q^{4} +(0.866025 + 0.500000i) q^{7} +(-0.707107 + 0.707107i) q^{8} +O(q^{10})\)	\(q+(-0.965926 + 0.258819i) q^{2} +(0.866025 - 0.500000i) q^{4} +(0.866025 + 0.500000i) q^{7} +(-0.707107 + 0.707107i) q^{8} +(-0.258819 + 0.965926i) q^{11} +(-1.00000 - 1.00000i) q^{13} +(-0.965926 - 0.258819i) q^{14} +(0.500000 - 0.866025i) q^{16} +(-1.36603 + 0.366025i) q^{19} -1.00000i q^{22} +(-0.707107 + 1.22474i) q^{23} +(-0.866025 + 0.500000i) q^{25} +(1.22474 + 0.707107i) q^{26} +1.00000 q^{28} +(-0.707107 + 0.707107i) q^{29} +(0.500000 + 0.866025i) q^{31} +(-0.258819 + 0.965926i) q^{32} +(0.366025 + 1.36603i) q^{37} +(1.22474 - 0.707107i) q^{38} -1.41421 q^{41} +(0.258819 + 0.965926i) q^{44} +(0.366025 - 1.36603i) q^{46} +(0.500000 + 0.866025i) q^{49} +(0.707107 - 0.707107i) q^{50} +(-1.36603 - 0.366025i) q^{52} +(0.517638 - 1.93185i) q^{53} +(-0.965926 + 0.258819i) q^{56} +(0.500000 - 0.866025i) q^{58} +(0.258819 - 0.965926i) q^{59} +(-1.36603 + 0.366025i) q^{61} +(-0.707107 - 0.707107i) q^{62} -1.00000i q^{64} +1.41421 q^{71} +(0.866025 - 0.500000i) q^{73} +(-0.707107 - 1.22474i) q^{74} +(-1.00000 + 1.00000i) q^{76} +(-0.707107 + 0.707107i) q^{77} +(-0.500000 + 0.866025i) q^{79} +(1.36603 - 0.366025i) q^{82} +(0.707107 - 0.707107i) q^{83} +(-0.500000 - 0.866025i) q^{88} +(-0.366025 - 1.36603i) q^{91} +1.41421i q^{92} +1.00000 q^{97} +(-0.707107 - 0.707107i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q+O(q^{10}) \) 8 * q	\( 8 q - 8 q^{13} + 4 q^{16} - 4 q^{19} + 8 q^{28} + 4 q^{31} - 4 q^{37} - 4 q^{46} + 4 q^{49} - 4 q^{52} + 4 q^{58} - 4 q^{61} - 8 q^{76} - 4 q^{79} + 4 q^{82} - 4 q^{88} + 4 q^{91} + 8 q^{97}+O(q^{100}) \) 8 * q - 8 * q^13 + 4 * q^16 - 4 * q^19 + 8 * q^28 + 4 * q^31 - 4 * q^37 - 4 * q^46 + 4 * q^49 - 4 * q^52 + 4 * q^58 - 4 * q^61 - 8 * q^76 - 4 * q^79 + 4 * q^82 - 4 * q^88 + 4 * q^91 + 8 * q^97

Introduction

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