LMFDB - Embedded newform 1456.1.bl.a.1035.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1456,1,Mod(83,1456)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1456, base_ring=CyclotomicField(4))

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

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

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

chi := DirichletCharacter("1456.83");

S:= CuspForms(chi, 1);

N := Newforms(S);

Level:	$ N $	$=$	$ 1456 = 2^{4} \cdot 7 \cdot 13 $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	1456.bl (of order $4$, degree $2$, minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$0.726638658394$
Analytic rank:	$0$
Dimension:	$4$
Relative dimension:	$2$ over $\Q(i)$
Coefficient field:	$\Q(\zeta_{8})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 1 $ x^4 + 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Projective image:	$S_{4}$
Projective field:	Galois closure of 4.2.31496192.1

Embedding invariants

Embedding label			1035.2
Root			$0.707107 + 0.707107i$ of defining polynomial
Character	$\chi$	$=$	1456.1035
Dual form			1456.1.bl.a.83.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{2} +(0.707107 - 0.707107i) q^{3} +1.00000 q^{4} +(-0.707107 + 0.707107i) q^{6} +(0.707107 + 0.707107i) q^{7} -1.00000 q^{8} +O(q^{10})$	\(q-1.00000 q^{2} +(0.707107 - 0.707107i) q^{3} +1.00000 q^{4} +(-0.707107 + 0.707107i) q^{6} +(0.707107 + 0.707107i) q^{7} -1.00000 q^{8} -1.00000i q^{11} +(0.707107 - 0.707107i) q^{12} +(0.707107 - 0.707107i) q^{13} +(-0.707107 - 0.707107i) q^{14} +1.00000 q^{16} -1.41421 q^{19} +1.00000 q^{21} +1.00000i q^{22} -1.00000i q^{23} +(-0.707107 + 0.707107i) q^{24} -1.00000 q^{25} +(-0.707107 + 0.707107i) q^{26} +(0.707107 + 0.707107i) q^{27} +(0.707107 + 0.707107i) q^{28} +(1.00000 - 1.00000i) q^{29} +(0.707107 + 0.707107i) q^{31} -1.00000 q^{32} +(-0.707107 - 0.707107i) q^{33} +1.00000 q^{37} +1.41421 q^{38} -1.00000i q^{39} +(0.707107 + 0.707107i) q^{41} -1.00000 q^{42} +(1.00000 + 1.00000i) q^{43} -1.00000i q^{44} +1.00000i q^{46} +(-0.707107 + 0.707107i) q^{47} +(0.707107 - 0.707107i) q^{48} +1.00000i q^{49} +1.00000 q^{50} +(0.707107 - 0.707107i) q^{52} +(-0.707107 - 0.707107i) q^{54} +(-0.707107 - 0.707107i) q^{56} +(-1.00000 + 1.00000i) q^{57} +(-1.00000 + 1.00000i) q^{58} +(-0.707107 - 0.707107i) q^{61} +(-0.707107 - 0.707107i) q^{62} +1.00000 q^{64} +(0.707107 + 0.707107i) q^{66} -1.00000i q^{67} +(-0.707107 - 0.707107i) q^{69} +(-1.00000 - 1.00000i) q^{71} +(-0.707107 + 0.707107i) q^{73} -1.00000 q^{74} +(-0.707107 + 0.707107i) q^{75} -1.41421 q^{76} +(0.707107 - 0.707107i) q^{77} +1.00000i q^{78} -1.00000i q^{79} +1.00000 q^{81} +(-0.707107 - 0.707107i) q^{82} -1.41421 q^{83} +1.00000 q^{84} +(-1.00000 - 1.00000i) q^{86} -1.41421i q^{87} +1.00000i q^{88} +1.00000 q^{91} -1.00000i q^{92} +1.00000 q^{93} +(0.707107 - 0.707107i) q^{94} +(-0.707107 + 0.707107i) q^{96} +(-0.707107 + 0.707107i) q^{97} -1.00000i q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 4 q^{2} + 4 q^{4} - 4 q^{8}+O(q^{10}) $ 4 * q - 4 * q^2 + 4 * q^4 - 4 * q^8	$ 4 q - 4 q^{2} + 4 q^{4} - 4 q^{8} + 4 q^{16} + 4 q^{21} - 4 q^{25} + 4 q^{29} - 4 q^{32} + 4 q^{37} - 4 q^{42} + 4 q^{43} + 4 q^{50} - 4 q^{57} - 4 q^{58} + 4 q^{64} - 4 q^{71} - 4 q^{74} + 4 q^{81} + 4 q^{84} - 4 q^{86} + 4 q^{91} + 4 q^{93}+O(q^{100}) $ 4 * q - 4 * q^2 + 4 * q^4 - 4 * q^8 + 4 * q^16 + 4 * q^21 - 4 * q^25 + 4 * q^29 - 4 * q^32 + 4 * q^37 - 4 * q^42 + 4 * q^43 + 4 * q^50 - 4 * q^57 - 4 * q^58 + 4 * q^64 - 4 * q^71 - 4 * q^74 + 4 * q^81 + 4 * q^84 - 4 * q^86 + 4 * q^91 + 4 * q^93

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1456\mathbb{Z}\right)^\times$.

$n$	$561$	$911$	$1093$	$1249$
$\chi(n)$	$e\left(\frac{1}{4}\right)$	$-1$	$e\left(\frac{1}{4}\right)$	$-1$

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 $

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

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1456.1.bl.a.1035.2	yes	4
7.6	odd	2	inner	1456.1.bl.a.1035.1	yes	4
13.5	odd	4		1456.1.bb.a.811.2	yes	4
16.3	odd	4		1456.1.bb.a.307.1	✓	4
91.83	even	4		1456.1.bb.a.811.1	yes	4
112.83	even	4		1456.1.bb.a.307.2	yes	4
208.83	even	4	inner	1456.1.bl.a.83.1	yes	4
1456.83	odd	4	inner	1456.1.bl.a.83.2	yes	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1456.1.bb.a.307.1	✓	4	16.3	odd	4
1456.1.bb.a.307.2	yes	4	112.83	even	4
1456.1.bb.a.811.1	yes	4	91.83	even	4
1456.1.bb.a.811.2	yes	4	13.5	odd	4
1456.1.bl.a.83.1	yes	4	208.83	even	4	inner
1456.1.bl.a.83.2	yes	4	1456.83	odd	4	inner
1456.1.bl.a.1035.1	yes	4	7.6	odd	2	inner
1456.1.bl.a.1035.2	yes	4	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 1456 = 2^{4} \cdot 7 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	1456.bl (of order \(4\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} +(0.707107 - 0.707107i) q^{3} +1.00000 q^{4} +(-0.707107 + 0.707107i) q^{6} +(0.707107 + 0.707107i) q^{7} -1.00000 q^{8} +O(q^{10})\)	\(q-1.00000 q^{2} +(0.707107 - 0.707107i) q^{3} +1.00000 q^{4} +(-0.707107 + 0.707107i) q^{6} +(0.707107 + 0.707107i) q^{7} -1.00000 q^{8} -1.00000i q^{11} +(0.707107 - 0.707107i) q^{12} +(0.707107 - 0.707107i) q^{13} +(-0.707107 - 0.707107i) q^{14} +1.00000 q^{16} -1.41421 q^{19} +1.00000 q^{21} +1.00000i q^{22} -1.00000i q^{23} +(-0.707107 + 0.707107i) q^{24} -1.00000 q^{25} +(-0.707107 + 0.707107i) q^{26} +(0.707107 + 0.707107i) q^{27} +(0.707107 + 0.707107i) q^{28} +(1.00000 - 1.00000i) q^{29} +(0.707107 + 0.707107i) q^{31} -1.00000 q^{32} +(-0.707107 - 0.707107i) q^{33} +1.00000 q^{37} +1.41421 q^{38} -1.00000i q^{39} +(0.707107 + 0.707107i) q^{41} -1.00000 q^{42} +(1.00000 + 1.00000i) q^{43} -1.00000i q^{44} +1.00000i q^{46} +(-0.707107 + 0.707107i) q^{47} +(0.707107 - 0.707107i) q^{48} +1.00000i q^{49} +1.00000 q^{50} +(0.707107 - 0.707107i) q^{52} +(-0.707107 - 0.707107i) q^{54} +(-0.707107 - 0.707107i) q^{56} +(-1.00000 + 1.00000i) q^{57} +(-1.00000 + 1.00000i) q^{58} +(-0.707107 - 0.707107i) q^{61} +(-0.707107 - 0.707107i) q^{62} +1.00000 q^{64} +(0.707107 + 0.707107i) q^{66} -1.00000i q^{67} +(-0.707107 - 0.707107i) q^{69} +(-1.00000 - 1.00000i) q^{71} +(-0.707107 + 0.707107i) q^{73} -1.00000 q^{74} +(-0.707107 + 0.707107i) q^{75} -1.41421 q^{76} +(0.707107 - 0.707107i) q^{77} +1.00000i q^{78} -1.00000i q^{79} +1.00000 q^{81} +(-0.707107 - 0.707107i) q^{82} -1.41421 q^{83} +1.00000 q^{84} +(-1.00000 - 1.00000i) q^{86} -1.41421i q^{87} +1.00000i q^{88} +1.00000 q^{91} -1.00000i q^{92} +1.00000 q^{93} +(0.707107 - 0.707107i) q^{94} +(-0.707107 + 0.707107i) q^{96} +(-0.707107 + 0.707107i) q^{97} -1.00000i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{2} + 4 q^{4} - 4 q^{8}+O(q^{10}) \) 4 * q - 4 * q^2 + 4 * q^4 - 4 * q^8	\( 4 q - 4 q^{2} + 4 q^{4} - 4 q^{8} + 4 q^{16} + 4 q^{21} - 4 q^{25} + 4 q^{29} - 4 q^{32} + 4 q^{37} - 4 q^{42} + 4 q^{43} + 4 q^{50} - 4 q^{57} - 4 q^{58} + 4 q^{64} - 4 q^{71} - 4 q^{74} + 4 q^{81} + 4 q^{84} - 4 q^{86} + 4 q^{91} + 4 q^{93}+O(q^{100}) \) 4 * q - 4 * q^2 + 4 * q^4 - 4 * q^8 + 4 * q^16 + 4 * q^21 - 4 * q^25 + 4 * q^29 - 4 * q^32 + 4 * q^37 - 4 * q^42 + 4 * q^43 + 4 * q^50 - 4 * q^57 - 4 * q^58 + 4 * q^64 - 4 * q^71 - 4 * q^74 + 4 * q^81 + 4 * q^84 - 4 * q^86 + 4 * q^91 + 4 * q^93

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