LMFDB - Embedded newform 2028.1.v.c.695.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2028,1,Mod(587,2028)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2028, base_ring=CyclotomicField(12))

chi = DirichletCharacter(H, H._module([6, 6, 1]))

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

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

chi := DirichletCharacter("2028.587");

S:= CuspForms(chi, 1);

N := Newforms(S);

Level:	$ N $	$=$	$ 2028 = 2^{2} \cdot 3 \cdot 13^{2} $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	2028.v (of order $12$, degree $4$, not 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:	$1.01210384562$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\zeta_{12})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{2} + 1 $ x^4 - x^2 + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Projective image:	$D_{4}$
Projective field:	Galois closure of 4.2.8112.1

Embedding invariants

Embedding label			695.1
Root			$-0.866025 + 0.500000i$ of defining polynomial
Character	$\chi$	$=$	2028.695
Dual form			2028.1.v.c.1103.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q+(-0.866025 - 0.500000i) q^{2} +(0.866025 + 0.500000i) q^{3} +(0.500000 + 0.866025i) q^{4} +(1.00000 + 1.00000i) q^{5} +(-0.500000 - 0.866025i) q^{6} -1.00000i q^{8} +(0.500000 + 0.866025i) q^{9} +(-0.366025 - 1.36603i) q^{10} +(-1.36603 + 0.366025i) q^{11} +1.00000i q^{12} +(0.366025 + 1.36603i) q^{15} +(-0.500000 + 0.866025i) q^{16} -1.00000i q^{18} +(-0.366025 + 1.36603i) q^{20} +(1.36603 + 0.366025i) q^{22} +(0.500000 - 0.866025i) q^{24} +1.00000i q^{25} +1.00000i q^{27} +(0.366025 - 1.36603i) q^{30} +(0.866025 - 0.500000i) q^{32} +(-1.36603 - 0.366025i) q^{33} +(-0.500000 + 0.866025i) q^{36} +(1.00000 - 1.00000i) q^{40} +(1.36603 - 0.366025i) q^{41} +(-1.00000 - 1.00000i) q^{44} +(-0.366025 + 1.36603i) q^{45} +(1.00000 + 1.00000i) q^{47} +(-0.866025 + 0.500000i) q^{48} +(-0.866025 - 0.500000i) q^{49} +(0.500000 - 0.866025i) q^{50} +(0.500000 - 0.866025i) q^{54} +(-1.73205 - 1.00000i) q^{55} +(0.366025 - 1.36603i) q^{59} +(-1.00000 + 1.00000i) q^{60} -1.00000 q^{64} +(1.00000 + 1.00000i) q^{66} +(-1.36603 - 0.366025i) q^{71} +(0.866025 - 0.500000i) q^{72} +(-0.500000 + 0.866025i) q^{75} -2.00000i q^{79} +(-1.36603 + 0.366025i) q^{80} +(-0.500000 + 0.866025i) q^{81} +(-1.36603 - 0.366025i) q^{82} +(-1.00000 + 1.00000i) q^{83} +(0.366025 + 1.36603i) q^{88} +(0.366025 + 1.36603i) q^{89} +(1.00000 - 1.00000i) q^{90} +(-0.366025 - 1.36603i) q^{94} +1.00000 q^{96} +(0.500000 + 0.866025i) q^{98} +(-1.00000 - 1.00000i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 2 q^{4} + 4 q^{5} - 2 q^{6} + 2 q^{9} + 2 q^{10} - 2 q^{11} - 2 q^{15} - 2 q^{16} + 2 q^{20} + 2 q^{22} + 2 q^{24} - 2 q^{30} - 2 q^{33} - 2 q^{36} + 4 q^{40} + 2 q^{41} - 4 q^{44} + 2 q^{45} + 4 q^{47}+ \cdots - 4 q^{99}+O(q^{100}) $ 4 * q + 2 * q^4 + 4 * q^5 - 2 * q^6 + 2 * q^9 + 2 * q^10 - 2 * q^11 - 2 * q^15 - 2 * q^16 + 2 * q^20 + 2 * q^22 + 2 * q^24 - 2 * q^30 - 2 * q^33 - 2 * q^36 + 4 * q^40 + 2 * q^41 - 4 * q^44 + 2 * q^45 + 4 * q^47 + 2 * q^50 + 2 * q^54 - 2 * q^59 - 4 * q^60 - 4 * q^64 + 4 * q^66 - 2 * q^71 - 2 * q^75 - 2 * q^80 - 2 * q^81 - 2 * q^82 - 4 * q^83 - 2 * q^88 - 2 * q^89 + 4 * q^90 + 2 * q^94 + 4 * q^96 + 2 * q^98 - 4 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2028\mathbb{Z}\right)^\times$.

$n$	$677$	$1015$	$1861$
$\chi(n)$	$-1$	$-1$	$e\left(\frac{5}{12}\right)$

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 $

$2$	−0.866025	−	0.500000i	−0.866025	−	0.500000i
$2$	−0.866025	−	0.500000i	−0.866025	−	0.500000i
$3$	0.866025	+	0.500000i	0.866025	+	0.500000i
$3$	0.866025	+	0.500000i	0.866025	+	0.500000i
$4$	0.500000	+	0.866025i	0.500000	+	0.866025i
$5$	1.00000	+	1.00000i	1.00000	+	1.00000i	1.00000			$0$
$5$	1.00000	+	1.00000i	1.00000	+	1.00000i			1.00000i	$0.5\pi$
$6$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$7$		0			0		0.258819	−	0.965926i	$-0.416667\pi$
$7$		0			0		−0.258819	+	0.965926i	$0.583333\pi$
$8$		−	1.00000i		−	1.00000i
$9$	0.500000	+	0.866025i	0.500000	+	0.866025i
$10$	−0.366025	−	1.36603i	−0.366025	−	1.36603i
$11$	−1.36603	+	0.366025i	−1.36603	+	0.366025i	−0.866025	−	0.500000i	$-0.833333\pi$
$11$	−1.36603	+	0.366025i	−1.36603	+	0.366025i	−0.500000	+	0.866025i	$0.666667\pi$
$12$			1.00000i			1.00000i
$13$		0			0
$13$		0			0
$14$		0			0
$15$	0.366025	+	1.36603i	0.366025	+	1.36603i
$16$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$17$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$17$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$18$		−	1.00000i		−	1.00000i
$19$		0			0		−0.965926	−	0.258819i	$-0.916667\pi$
$19$		0			0		0.965926	+	0.258819i	$0.0833333\pi$
$20$	−0.366025	+	1.36603i	−0.366025	+	1.36603i
$21$		0			0
$22$	1.36603	+	0.366025i	1.36603	+	0.366025i
$23$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$23$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$24$	0.500000	−	0.866025i	0.500000	−	0.866025i
$25$			1.00000i			1.00000i
$26$		0			0
$27$			1.00000i			1.00000i
$28$		0			0
$29$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$29$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$30$	0.366025	−	1.36603i	0.366025	−	1.36603i
$31$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$31$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$32$	0.866025	−	0.500000i	0.866025	−	0.500000i
$33$	−1.36603	−	0.366025i	−1.36603	−	0.366025i
$34$		0			0
$35$		0			0
$36$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$37$		0			0		−0.258819	−	0.965926i	$-0.583333\pi$
$37$		0			0		0.258819	+	0.965926i	$0.416667\pi$
$38$		0			0
$39$		0			0
$40$	1.00000	−	1.00000i	1.00000	−	1.00000i
$41$	1.36603	−	0.366025i	1.36603	−	0.366025i	0.500000	−	0.866025i	$-0.333333\pi$
$41$	1.36603	−	0.366025i	1.36603	−	0.366025i	0.866025	+	0.500000i	$0.166667\pi$
$42$		0			0
$43$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$43$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$44$	−1.00000	−	1.00000i	−1.00000	−	1.00000i
$45$	−0.366025	+	1.36603i	−0.366025	+	1.36603i
$46$		0			0
$47$	1.00000	+	1.00000i	1.00000	+	1.00000i	1.00000			$0$
$47$	1.00000	+	1.00000i	1.00000	+	1.00000i			1.00000i	$0.5\pi$
$48$	−0.866025	+	0.500000i	−0.866025	+	0.500000i
$49$	−0.866025	−	0.500000i	−0.866025	−	0.500000i
$50$	0.500000	−	0.866025i	0.500000	−	0.866025i
$51$		0			0
$52$		0			0
$53$		0			0		1.00000			$0$
$53$		0			0		−1.00000			$\pi$
$54$	0.500000	−	0.866025i	0.500000	−	0.866025i
$55$	−1.73205	−	1.00000i	−1.73205	−	1.00000i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	0.366025	−	1.36603i	0.366025	−	1.36603i	−0.500000	−	0.866025i	$-0.666667\pi$
$59$	0.366025	−	1.36603i	0.366025	−	1.36603i	0.866025	−	0.500000i	$-0.166667\pi$
$60$	−1.00000	+	1.00000i	−1.00000	+	1.00000i
$61$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$61$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$62$		0			0
$63$		0			0
$64$	−1.00000			−1.00000
$65$		0			0
$66$	1.00000	+	1.00000i	1.00000	+	1.00000i
$67$		0			0		−0.258819	−	0.965926i	$-0.583333\pi$
$67$		0			0		0.258819	+	0.965926i	$0.416667\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	−1.36603	−	0.366025i	−1.36603	−	0.366025i	−0.500000	−	0.866025i	$-0.666667\pi$
$71$	−1.36603	−	0.366025i	−1.36603	−	0.366025i	−0.866025	+	0.500000i	$0.833333\pi$
$72$	0.866025	−	0.500000i	0.866025	−	0.500000i
$73$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$73$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$74$		0			0
$75$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$76$		0			0
$77$		0			0
$78$		0			0
$79$		−	2.00000i		−	2.00000i		−	1.00000i	$-0.5\pi$
$79$		−	2.00000i		−	2.00000i		−	1.00000i	$-0.5\pi$
$80$	−1.36603	+	0.366025i	−1.36603	+	0.366025i
$81$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$82$	−1.36603	−	0.366025i	−1.36603	−	0.366025i
$83$	−1.00000	+	1.00000i	−1.00000	+	1.00000i			1.00000i	$0.5\pi$
$83$	−1.00000	+	1.00000i	−1.00000	+	1.00000i	−1.00000			$\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$		0			0
$88$	0.366025	+	1.36603i	0.366025	+	1.36603i
$89$	0.366025	+	1.36603i	0.366025	+	1.36603i	0.866025	+	0.500000i	$0.166667\pi$
$89$	0.366025	+	1.36603i	0.366025	+	1.36603i	−0.500000	+	0.866025i	$0.666667\pi$
$90$	1.00000	−	1.00000i	1.00000	−	1.00000i
$91$		0			0
$92$		0			0
$93$		0			0
$94$	−0.366025	−	1.36603i	−0.366025	−	1.36603i
$95$		0			0
$96$	1.00000			1.00000
$97$		0			0		0.258819	−	0.965926i	$-0.416667\pi$
$97$		0			0		−0.258819	+	0.965926i	$0.583333\pi$
$98$	0.500000	+	0.866025i	0.500000	+	0.866025i
$99$	−1.00000	−	1.00000i	−1.00000	−	1.00000i
$100$	−0.866025	+	0.500000i	−0.866025	+	0.500000i
$101$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$101$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$102$		0			0
$103$		0			0			−	1.00000i	$-0.5\pi$
$103$		0			0				1.00000i	$0.5\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$107$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$108$	−0.866025	+	0.500000i	−0.866025	+	0.500000i
$109$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$109$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$110$	1.00000	+	1.73205i	1.00000	+	1.73205i
$111$		0			0
$112$		0			0
$113$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$113$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$		0			0
$118$	−1.00000	+	1.00000i	−1.00000	+	1.00000i
$119$		0			0
$120$	1.36603	−	0.366025i	1.36603	−	0.366025i
$121$	0.866025	−	0.500000i	0.866025	−	0.500000i
$122$		0			0
$123$	1.36603	+	0.366025i	1.36603	+	0.366025i
$124$		0			0
$125$		0			0
$126$		0			0
$127$	1.00000	−	1.73205i	1.00000	−	1.73205i	0.500000	−	0.866025i	$-0.333333\pi$
$127$	1.00000	−	1.73205i	1.00000	−	1.73205i	0.500000	−	0.866025i	$-0.333333\pi$
$128$	0.866025	+	0.500000i	0.866025	+	0.500000i
$129$		0			0
$130$		0			0
$131$		0			0			−	1.00000i	$-0.5\pi$
$131$		0			0				1.00000i	$0.5\pi$
$132$	−0.366025	−	1.36603i	−0.366025	−	1.36603i
$133$		0			0
$134$		0			0
$135$	−1.00000	+	1.00000i	−1.00000	+	1.00000i
$136$		0			0
$137$	1.36603	+	0.366025i	1.36603	+	0.366025i	0.866025	−	0.500000i	$-0.166667\pi$
$137$	1.36603	+	0.366025i	1.36603	+	0.366025i	0.500000	+	0.866025i	$0.333333\pi$
$138$		0			0
$139$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$139$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$140$		0			0
$141$	0.366025	+	1.36603i	0.366025	+	1.36603i
$142$	1.00000	+	1.00000i	1.00000	+	1.00000i
$143$		0			0
$144$	−1.00000			−1.00000
$145$		0			0
$146$		0			0
$147$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$148$		0			0
$149$	0.366025	−	1.36603i	0.366025	−	1.36603i	−0.500000	−	0.866025i	$-0.666667\pi$
$149$	0.366025	−	1.36603i	0.366025	−	1.36603i	0.866025	−	0.500000i	$-0.166667\pi$
$150$	0.866025	−	0.500000i	0.866025	−	0.500000i
$151$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$151$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		0			0
$156$		0			0
$157$	−2.00000			−2.00000			−1.00000			$\pi$
$157$	−2.00000			−2.00000			−1.00000			$\pi$
$158$	−1.00000	+	1.73205i	−1.00000	+	1.73205i
$159$		0			0
$160$	1.36603	+	0.366025i	1.36603	+	0.366025i
$161$		0			0
$162$	0.866025	−	0.500000i	0.866025	−	0.500000i
$163$		0			0		0.258819	−	0.965926i	$-0.416667\pi$
$163$		0			0		−0.258819	+	0.965926i	$0.583333\pi$
$164$	1.00000	+	1.00000i	1.00000	+	1.00000i
$165$	−1.00000	−	1.73205i	−1.00000	−	1.73205i
$166$	1.36603	−	0.366025i	1.36603	−	0.366025i
$167$	1.36603	−	0.366025i	1.36603	−	0.366025i	0.500000	−	0.866025i	$-0.333333\pi$
$167$	1.36603	−	0.366025i	1.36603	−	0.366025i	0.866025	+	0.500000i	$0.166667\pi$
$168$		0			0
$169$		0			0
$170$		0			0
$171$		0			0
$172$		0			0
$173$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$173$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$174$		0			0
$175$		0			0
$176$	0.366025	−	1.36603i	0.366025	−	1.36603i
$177$	1.00000	−	1.00000i	1.00000	−	1.00000i
$178$	0.366025	−	1.36603i	0.366025	−	1.36603i
$179$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$179$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$180$	−1.36603	+	0.366025i	−1.36603	+	0.366025i
$181$		−	2.00000i		−	2.00000i		−	1.00000i	$-0.5\pi$
$181$		−	2.00000i		−	2.00000i		−	1.00000i	$-0.5\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$		0			0
$186$		0			0
$187$		0			0
$188$	−0.366025	+	1.36603i	−0.366025	+	1.36603i
$189$		0			0
$190$		0			0
$191$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$191$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$192$	−0.866025	−	0.500000i	−0.866025	−	0.500000i
$193$		0			0		−0.258819	−	0.965926i	$-0.583333\pi$
$193$		0			0		0.258819	+	0.965926i	$0.416667\pi$
$194$		0			0
$195$		0			0
$196$		−	1.00000i		−	1.00000i
$197$	1.36603	−	0.366025i	1.36603	−	0.366025i	0.500000	−	0.866025i	$-0.333333\pi$
$197$	1.36603	−	0.366025i	1.36603	−	0.366025i	0.866025	+	0.500000i	$0.166667\pi$
$198$	0.366025	+	1.36603i	0.366025	+	1.36603i
$199$	1.00000	+	1.73205i	1.00000	+	1.73205i	0.500000	+	0.866025i	$0.333333\pi$
$199$	1.00000	+	1.73205i	1.00000	+	1.73205i	0.500000	+	0.866025i	$0.333333\pi$
$200$	1.00000			1.00000
$201$		0			0
$202$		0			0
$203$		0			0
$204$		0			0
$205$	1.73205	+	1.00000i	1.73205	+	1.00000i
$206$		0			0
$207$		0			0
$208$		0			0
$209$		0			0
$210$		0			0
$211$	1.73205	+	1.00000i	1.73205	+	1.00000i	0.866025	+	0.500000i	$0.166667\pi$
$211$	1.73205	+	1.00000i	1.73205	+	1.00000i	0.866025	+	0.500000i	$0.166667\pi$
$212$		0			0
$213$	−1.00000	−	1.00000i	−1.00000	−	1.00000i
$214$		0			0
$215$		0			0
$216$	1.00000			1.00000
$217$		0			0
$218$		0			0
$219$		0			0
$220$		−	2.00000i		−	2.00000i
$221$		0			0
$222$		0			0
$223$		0			0		−0.258819	−	0.965926i	$-0.583333\pi$
$223$		0			0		0.258819	+	0.965926i	$0.416667\pi$
$224$		0			0
$225$	−0.866025	+	0.500000i	−0.866025	+	0.500000i
$226$		0			0
$227$	−1.36603	−	0.366025i	−1.36603	−	0.366025i	−0.500000	−	0.866025i	$-0.666667\pi$
$227$	−1.36603	−	0.366025i	−1.36603	−	0.366025i	−0.866025	+	0.500000i	$0.833333\pi$
$228$		0			0
$229$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$229$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		0			0			−	1.00000i	$-0.5\pi$
$233$		0			0				1.00000i	$0.5\pi$
$234$		0			0
$235$			2.00000i			2.00000i
$236$	1.36603	−	0.366025i	1.36603	−	0.366025i
$237$	1.00000	−	1.73205i	1.00000	−	1.73205i
$238$		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			$0$
$240$	−1.36603	−	0.366025i	−1.36603	−	0.366025i
$241$		0			0		−0.965926	−	0.258819i	$-0.916667\pi$
$241$		0			0		0.965926	+	0.258819i	$0.0833333\pi$
$242$	−1.00000			−1.00000
$243$	−0.866025	+	0.500000i	−0.866025	+	0.500000i
$244$		0			0
$245$	−0.366025	−	1.36603i	−0.366025	−	1.36603i
$246$	−1.00000	−	1.00000i	−1.00000	−	1.00000i
$247$		0			0
$248$		0			0
$249$	−1.36603	+	0.366025i	−1.36603	+	0.366025i
$250$		0			0
$251$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$251$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$252$		0			0
$253$		0			0
$254$	−1.73205	+	1.00000i	−1.73205	+	1.00000i
$255$		0			0
$256$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$257$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$257$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$		0			0
$262$		0			0
$263$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$263$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$264$	−0.366025	+	1.36603i	−0.366025	+	1.36603i
$265$		0			0
$266$		0			0
$267$	−0.366025	+	1.36603i	−0.366025	+	1.36603i
$268$		0			0
$269$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$269$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$270$	1.36603	−	0.366025i	1.36603	−	0.366025i
$271$		0			0		0.965926	−	0.258819i	$-0.0833333\pi$
$271$		0			0		−0.965926	+	0.258819i	$0.916667\pi$
$272$		0			0
$273$		0			0
$274$	−1.00000	−	1.00000i	−1.00000	−	1.00000i
$275$	−0.366025	−	1.36603i	−0.366025	−	1.36603i
$276$		0			0
$277$	−1.73205	+	1.00000i	−1.73205	+	1.00000i	−0.866025	+	0.500000i	$0.833333\pi$
$277$	−1.73205	+	1.00000i	−1.73205	+	1.00000i	−0.866025	+	0.500000i	$0.833333\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	−1.00000	+	1.00000i	−1.00000	+	1.00000i			1.00000i	$0.5\pi$
$281$	−1.00000	+	1.00000i	−1.00000	+	1.00000i	−1.00000			$\pi$
$282$	0.366025	−	1.36603i	0.366025	−	1.36603i
$283$	−1.00000	+	1.73205i	−1.00000	+	1.73205i	−0.500000	+	0.866025i	$0.666667\pi$
$283$	−1.00000	+	1.73205i	−1.00000	+	1.73205i	−0.500000	+	0.866025i	$0.666667\pi$
$284$	−0.366025	−	1.36603i	−0.366025	−	1.36603i
$285$		0			0
$286$		0			0
$287$		0			0
$288$	0.866025	+	0.500000i	0.866025	+	0.500000i
$289$	0.500000	−	0.866025i	0.500000	−	0.866025i
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−1.36603	−	0.366025i	−1.36603	−	0.366025i	−0.500000	−	0.866025i	$-0.666667\pi$
$293$	−1.36603	−	0.366025i	−1.36603	−	0.366025i	−0.866025	+	0.500000i	$0.833333\pi$
$294$			1.00000i			1.00000i
$295$	1.73205	−	1.00000i	1.73205	−	1.00000i
$296$		0			0
$297$	−0.366025	−	1.36603i	−0.366025	−	1.36603i
$298$	−1.00000	+	1.00000i	−1.00000	+	1.00000i
$299$		0			0
$300$	−1.00000			−1.00000
$301$		0			0
$302$		0			0
$303$		0			0
$304$		0			0
$305$		0			0
$306$		0			0
$307$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$307$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$		0			0		1.00000			$0$
$311$		0			0		−1.00000			$\pi$
$312$		0			0
$313$		0			0			−	1.00000i	$-0.5\pi$
$313$		0			0				1.00000i	$0.5\pi$
$314$	1.73205	+	1.00000i	1.73205	+	1.00000i
$315$		0			0
$316$	1.73205	−	1.00000i	1.73205	−	1.00000i
$317$	−1.00000	−	1.00000i	−1.00000	−	1.00000i		−	1.00000i	$-0.5\pi$
$317$	−1.00000	−	1.00000i	−1.00000	−	1.00000i	−1.00000			$\pi$
$318$		0			0
$319$		0			0
$320$	−1.00000	−	1.00000i	−1.00000	−	1.00000i
$321$		0			0
$322$		0			0
$323$		0			0
$324$	−1.00000			−1.00000
$325$		0			0
$326$		0			0
$327$		0			0
$328$	−0.366025	−	1.36603i	−0.366025	−	1.36603i
$329$		0			0
$330$			2.00000i			2.00000i
$331$		0			0		−0.965926	−	0.258819i	$-0.916667\pi$
$331$		0			0		0.965926	+	0.258819i	$0.0833333\pi$
$332$	−1.36603	−	0.366025i	−1.36603	−	0.366025i
$333$		0			0
$334$	−1.36603	−	0.366025i	−1.36603	−	0.366025i
$335$		0			0
$336$		0			0
$337$		0			0		1.00000			$0$
$337$		0			0		−1.00000			$\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$		0			0
$342$		0			0
$343$		0			0
$344$		0			0
$345$		0			0
$346$		0			0
$347$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$347$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$348$		0			0
$349$		0			0		−0.258819	−	0.965926i	$-0.583333\pi$
$349$		0			0		0.258819	+	0.965926i	$0.416667\pi$
$350$		0			0
$351$		0			0
$352$	−1.00000	+	1.00000i	−1.00000	+	1.00000i
$353$	−1.36603	+	0.366025i	−1.36603	+	0.366025i	−0.866025	−	0.500000i	$-0.833333\pi$
$353$	−1.36603	+	0.366025i	−1.36603	+	0.366025i	−0.500000	+	0.866025i	$0.666667\pi$
$354$	−1.36603	+	0.366025i	−1.36603	+	0.366025i
$355$	−1.00000	−	1.73205i	−1.00000	−	1.73205i
$356$	−1.00000	+	1.00000i	−1.00000	+	1.00000i
$357$		0			0
$358$		0			0
$359$	−1.00000	−	1.00000i	−1.00000	−	1.00000i		−	1.00000i	$-0.5\pi$
$359$	−1.00000	−	1.00000i	−1.00000	−	1.00000i	−1.00000			$\pi$
$360$	1.36603	+	0.366025i	1.36603	+	0.366025i
$361$	0.866025	+	0.500000i	0.866025	+	0.500000i
$362$	−1.00000	+	1.73205i	−1.00000	+	1.73205i
$363$	1.00000			1.00000
$364$		0			0
$365$		0			0
$366$		0			0
$367$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$367$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$368$		0			0
$369$	1.00000	+	1.00000i	1.00000	+	1.00000i
$370$		0			0
$371$		0			0
$372$		0			0
$373$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$373$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$374$		0			0
$375$		0			0
$376$	1.00000	−	1.00000i	1.00000	−	1.00000i
$377$		0			0
$378$		0			0
$379$		0			0		−0.258819	−	0.965926i	$-0.583333\pi$
$379$		0			0		0.258819	+	0.965926i	$0.416667\pi$
$380$		0			0
$381$	1.73205	−	1.00000i	1.73205	−	1.00000i
$382$		0			0
$383$	1.36603	+	0.366025i	1.36603	+	0.366025i	0.866025	−	0.500000i	$-0.166667\pi$
$383$	1.36603	+	0.366025i	1.36603	+	0.366025i	0.500000	+	0.866025i	$0.333333\pi$
$384$	0.500000	+	0.866025i	0.500000	+	0.866025i
$385$		0			0
$386$		0			0
$387$		0			0
$388$		0			0
$389$		0			0			−	1.00000i	$-0.5\pi$
$389$		0			0				1.00000i	$0.5\pi$
$390$		0			0
$391$		0			0
$392$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$393$		0			0
$394$	−1.36603	−	0.366025i	−1.36603	−	0.366025i
$395$	2.00000	−	2.00000i	2.00000	−	2.00000i
$396$	0.366025	−	1.36603i	0.366025	−	1.36603i
$397$		0			0		−0.965926	−	0.258819i	$-0.916667\pi$
$397$		0			0		0.965926	+	0.258819i	$0.0833333\pi$
$398$		−	2.00000i		−	2.00000i
$399$		0			0
$400$	−0.866025	−	0.500000i	−0.866025	−	0.500000i
$401$	−0.366025	−	1.36603i	−0.366025	−	1.36603i	−0.866025	−	0.500000i	$-0.833333\pi$
$401$	−0.366025	−	1.36603i	−0.366025	−	1.36603i	0.500000	−	0.866025i	$-0.333333\pi$
$402$		0			0
$403$		0			0
$404$		0			0
$405$	−1.36603	+	0.366025i	−1.36603	+	0.366025i
$406$		0			0
$407$		0			0
$408$		0			0
$409$		0			0		0.258819	−	0.965926i	$-0.416667\pi$
$409$		0			0		−0.258819	+	0.965926i	$0.583333\pi$
$410$	−1.00000	−	1.73205i	−1.00000	−	1.73205i
$411$	1.00000	+	1.00000i	1.00000	+	1.00000i
$412$		0			0
$413$		0			0
$414$		0			0
$415$	−2.00000			−2.00000
$416$		0			0
$417$		0			0
$418$		0			0
$419$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$419$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$420$		0			0
$421$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$421$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$422$	−1.00000	−	1.73205i	−1.00000	−	1.73205i
$423$	−0.366025	+	1.36603i	−0.366025	+	1.36603i
$424$		0			0
$425$		0			0
$426$	0.366025	+	1.36603i	0.366025	+	1.36603i
$427$		0			0
$428$		0			0
$429$		0			0
$430$		0			0
$431$	−0.366025	−	1.36603i	−0.366025	−	1.36603i	−0.866025	−	0.500000i	$-0.833333\pi$
$431$	−0.366025	−	1.36603i	−0.366025	−	1.36603i	0.500000	−	0.866025i	$-0.333333\pi$
$432$	−0.866025	−	0.500000i	−0.866025	−	0.500000i
$433$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$433$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$		0			0
$438$		0			0
$439$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$439$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$440$	−1.00000	+	1.73205i	−1.00000	+	1.73205i
$441$		−	1.00000i		−	1.00000i
$442$		0			0
$443$		0			0			−	1.00000i	$-0.5\pi$
$443$		0			0				1.00000i	$0.5\pi$
$444$		0			0
$445$	−1.00000	+	1.73205i	−1.00000	+	1.73205i
$446$		0			0
$447$	1.00000	−	1.00000i	1.00000	−	1.00000i
$448$		0			0
$449$	1.36603	+	0.366025i	1.36603	+	0.366025i	0.866025	−	0.500000i	$-0.166667\pi$
$449$	1.36603	+	0.366025i	1.36603	+	0.366025i	0.500000	+	0.866025i	$0.333333\pi$
$450$	1.00000			1.00000
$451$	−1.73205	+	1.00000i	−1.73205	+	1.00000i
$452$		0			0
$453$		0			0
$454$	1.00000	+	1.00000i	1.00000	+	1.00000i
$455$		0			0
$456$		0			0
$457$		0			0		0.965926	−	0.258819i	$-0.0833333\pi$
$457$		0			0		−0.965926	+	0.258819i	$0.916667\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	0.366025	−	1.36603i	0.366025	−	1.36603i	−0.500000	−	0.866025i	$-0.666667\pi$
$461$	0.366025	−	1.36603i	0.366025	−	1.36603i	0.866025	−	0.500000i	$-0.166667\pi$
$462$		0			0
$463$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$463$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$		0			0		1.00000			$0$
$467$		0			0		−1.00000			$\pi$
$468$		0			0
$469$		0			0
$470$	1.00000	−	1.73205i	1.00000	−	1.73205i
$471$	−1.73205	−	1.00000i	−1.73205	−	1.00000i
$472$	−1.36603	−	0.366025i	−1.36603	−	0.366025i
$473$		0			0
$474$	−1.73205	+	1.00000i	−1.73205	+	1.00000i
$475$		0			0
$476$		0			0
$477$		0			0
$478$	−1.36603	+	0.366025i	−1.36603	+	0.366025i
$479$	−1.36603	+	0.366025i	−1.36603	+	0.366025i	−0.866025	−	0.500000i	$-0.833333\pi$
$479$	−1.36603	+	0.366025i	−1.36603	+	0.366025i	−0.500000	+	0.866025i	$0.666667\pi$
$480$	1.00000	+	1.00000i	1.00000	+	1.00000i
$481$		0			0
$482$		0			0
$483$		0			0
$484$	0.866025	+	0.500000i	0.866025	+	0.500000i
$485$		0			0
$486$	1.00000			1.00000
$487$		0			0		−0.965926	−	0.258819i	$-0.916667\pi$
$487$		0			0		0.965926	+	0.258819i	$0.0833333\pi$
$488$		0			0
$489$		0			0
$490$	−0.366025	+	1.36603i	−0.366025	+	1.36603i
$491$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$491$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$492$	0.366025	+	1.36603i	0.366025	+	1.36603i
$493$		0			0
$494$		0			0
$495$		−	2.00000i		−	2.00000i
$496$		0			0
$497$		0			0
$498$	1.36603	+	0.366025i	1.36603	+	0.366025i
$499$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$499$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$500$		0			0
$501$	1.36603	+	0.366025i	1.36603	+	0.366025i
$502$		0			0
$503$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$503$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$		0			0
$508$	2.00000			2.00000
$509$	−1.36603	+	0.366025i	−1.36603	+	0.366025i	−0.866025	−	0.500000i	$-0.833333\pi$
$509$	−1.36603	+	0.366025i	−1.36603	+	0.366025i	−0.500000	+	0.866025i	$0.666667\pi$
$510$		0			0
$511$		0			0
$512$			1.00000i			1.00000i
$513$		0			0
$514$		0			0
$515$		0			0
$516$		0			0
$517$	−1.73205	−	1.00000i	−1.73205	−	1.00000i
$518$		0			0
$519$		0			0
$520$		0			0
$521$		0			0		1.00000			$0$
$521$		0			0		−1.00000			$\pi$
$522$		0			0
$523$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$523$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$		0			0
$528$	1.00000	−	1.00000i	1.00000	−	1.00000i
$529$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$530$		0			0
$531$	1.36603	−	0.366025i	1.36603	−	0.366025i
$532$		0			0
$533$		0			0
$534$	1.00000	−	1.00000i	1.00000	−	1.00000i
$535$		0			0
$536$		0			0
$537$		0			0
$538$		0			0
$539$	1.36603	+	0.366025i	1.36603	+	0.366025i
$540$	−1.36603	−	0.366025i	−1.36603	−	0.366025i
$541$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$541$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$542$		0			0
$543$	1.00000	−	1.73205i	1.00000	−	1.73205i
$544$		0			0
$545$		0			0
$546$		0			0
$547$		0			0		1.00000			$0$
$547$		0			0		−1.00000			$\pi$
$548$	0.366025	+	1.36603i	0.366025	+	1.36603i
$549$		0			0
$550$	−0.366025	+	1.36603i	−0.366025	+	1.36603i
$551$		0			0
$552$		0			0
$553$		0			0
$554$	2.00000			2.00000
$555$		0			0
$556$		0			0
$557$	−0.366025	−	1.36603i	−0.366025	−	1.36603i	−0.866025	−	0.500000i	$-0.833333\pi$
$557$	−0.366025	−	1.36603i	−0.366025	−	1.36603i	0.500000	−	0.866025i	$-0.333333\pi$
$558$		0			0
$559$		0			0
$560$		0			0
$561$		0			0
$562$	1.36603	−	0.366025i	1.36603	−	0.366025i
$563$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$563$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$564$	−1.00000	+	1.00000i	−1.00000	+	1.00000i
$565$		0			0
$566$	1.73205	−	1.00000i	1.73205	−	1.00000i
$567$		0			0
$568$	−0.366025	+	1.36603i	−0.366025	+	1.36603i
$569$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$569$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$570$		0			0
$571$	2.00000			2.00000			1.00000			$0$
$571$	2.00000			2.00000			1.00000			$0$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$577$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$577$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$578$	−0.866025	+	0.500000i	−0.866025	+	0.500000i
$579$		0			0
$580$		0			0
$581$		0			0
$582$		0			0
$583$		0			0
$584$		0			0
$585$		0			0
$586$	1.00000	+	1.00000i	1.00000	+	1.00000i
$587$	0.366025	+	1.36603i	0.366025	+	1.36603i	0.866025	+	0.500000i	$0.166667\pi$
$587$	0.366025	+	1.36603i	0.366025	+	1.36603i	−0.500000	+	0.866025i	$0.666667\pi$
$588$	0.500000	−	0.866025i	0.500000	−	0.866025i
$589$		0			0
$590$	−2.00000			−2.00000
$591$	1.36603	+	0.366025i	1.36603	+	0.366025i
$592$		0			0
$593$	−1.00000	+	1.00000i	−1.00000	+	1.00000i			1.00000i	$0.5\pi$
$593$	−1.00000	+	1.00000i	−1.00000	+	1.00000i	−1.00000			$\pi$
$594$	−0.366025	+	1.36603i	−0.366025	+	1.36603i
$595$		0			0
$596$	1.36603	−	0.366025i	1.36603	−	0.366025i
$597$			2.00000i			2.00000i
$598$		0			0
$599$		0			0			−	1.00000i	$-0.5\pi$
$599$		0			0				1.00000i	$0.5\pi$
$600$	0.866025	+	0.500000i	0.866025	+	0.500000i
$601$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$601$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	1.36603	+	0.366025i	1.36603	+	0.366025i
$606$		0			0
$607$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$607$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		0			0
$612$		0			0
$613$		0			0		0.965926	−	0.258819i	$-0.0833333\pi$
$613$		0			0		−0.965926	+	0.258819i	$0.916667\pi$
$614$		0			0
$615$	1.00000	+	1.73205i	1.00000	+	1.73205i
$616$		0			0
$617$	−0.366025	+	1.36603i	−0.366025	+	1.36603i	0.500000	+	0.866025i	$0.333333\pi$
$617$	−0.366025	+	1.36603i	−0.366025	+	1.36603i	−0.866025	+	0.500000i	$0.833333\pi$
$618$		0			0
$619$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$619$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$		0			0
$625$	1.00000			1.00000
$626$		0			0
$627$		0			0
$628$	−1.00000	−	1.73205i	−1.00000	−	1.73205i
$629$		0			0
$630$		0			0
$631$		0			0		0.258819	−	0.965926i	$-0.416667\pi$
$631$		0			0		−0.258819	+	0.965926i	$0.583333\pi$
$632$	−2.00000			−2.00000
$633$	1.00000	+	1.73205i	1.00000	+	1.73205i
$634$	0.366025	+	1.36603i	0.366025	+	1.36603i
$635$	2.73205	−	0.732051i	2.73205	−	0.732051i
$636$		0			0
$637$		0			0
$638$		0			0
$639$	−0.366025	−	1.36603i	−0.366025	−	1.36603i
$640$	0.366025	+	1.36603i	0.366025	+	1.36603i
$641$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$641$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$642$		0			0
$643$		0			0		−0.965926	−	0.258819i	$-0.916667\pi$
$643$		0			0		0.965926	+	0.258819i	$0.0833333\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$647$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$648$	0.866025	+	0.500000i	0.866025	+	0.500000i
$649$			2.00000i			2.00000i
$650$		0			0
$651$		0			0
$652$		0			0
$653$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$653$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$654$		0			0
$655$		0			0
$656$	−0.366025	+	1.36603i	−0.366025	+	1.36603i
$657$		0			0
$658$		0			0
$659$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$659$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$660$	1.00000	−	1.73205i	1.00000	−	1.73205i
$661$		0			0		−0.258819	−	0.965926i	$-0.583333\pi$
$661$		0			0		0.258819	+	0.965926i	$0.416667\pi$
$662$		0			0
$663$		0			0
$664$	1.00000	+	1.00000i	1.00000	+	1.00000i
$665$		0			0
$666$		0			0
$667$		0			0
$668$	1.00000	+	1.00000i	1.00000	+	1.00000i
$669$		0			0
$670$		0			0
$671$		0			0
$672$		0			0
$673$	−1.73205	−	1.00000i	−1.73205	−	1.00000i	−0.866025	−	0.500000i	$-0.833333\pi$
$673$	−1.73205	−	1.00000i	−1.73205	−	1.00000i	−0.866025	−	0.500000i	$-0.833333\pi$
$674$		0			0
$675$	−1.00000			−1.00000
$676$		0			0
$677$		0			0		1.00000			$0$
$677$		0			0		−1.00000			$\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$	−1.00000	−	1.00000i	−1.00000	−	1.00000i
$682$		0			0
$683$	−0.366025	+	1.36603i	−0.366025	+	1.36603i	0.500000	+	0.866025i	$0.333333\pi$
$683$	−0.366025	+	1.36603i	−0.366025	+	1.36603i	−0.866025	+	0.500000i	$0.833333\pi$
$684$		0			0
$685$	1.00000	+	1.73205i	1.00000	+	1.73205i
$686$		0			0
$687$		0			0
$688$		0			0
$689$		0			0
$690$		0			0
$691$		0			0		−0.258819	−	0.965926i	$-0.583333\pi$
$691$		0			0		0.258819	+	0.965926i	$0.416667\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$		0			0
$696$		0			0
$697$		0			0
$698$		0			0
$699$		0			0
$700$		0			0
$701$		0			0			−	1.00000i	$-0.5\pi$
$701$		0			0				1.00000i	$0.5\pi$
$702$		0			0
$703$		0			0
$704$	1.36603	−	0.366025i	1.36603	−	0.366025i
$705$	−1.00000	+	1.73205i	−1.00000	+	1.73205i
$706$	1.36603	+	0.366025i	1.36603	+	0.366025i
$707$		0			0
$708$	1.36603	+	0.366025i	1.36603	+	0.366025i
$709$		0			0		−0.965926	−	0.258819i	$-0.916667\pi$
$709$		0			0		0.965926	+	0.258819i	$0.0833333\pi$
$710$			2.00000i			2.00000i
$711$	1.73205	−	1.00000i	1.73205	−	1.00000i
$712$	1.36603	−	0.366025i	1.36603	−	0.366025i
$713$		0			0
$714$		0			0
$715$		0			0
$716$		0			0
$717$	1.36603	−	0.366025i	1.36603	−	0.366025i
$718$	0.366025	+	1.36603i	0.366025	+	1.36603i
$719$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$719$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$720$	−1.00000	−	1.00000i	−1.00000	−	1.00000i
$721$		0			0
$722$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$723$		0			0
$724$	1.73205	−	1.00000i	1.73205	−	1.00000i
$725$		0			0
$726$	−0.866025	−	0.500000i	−0.866025	−	0.500000i
$727$	2.00000			2.00000			1.00000			$0$
$727$	2.00000			2.00000			1.00000			$0$
$728$		0			0
$729$	−1.00000			−1.00000
$730$		0			0
$731$		0			0
$732$		0			0
$733$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$733$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$734$		0			0
$735$	0.366025	−	1.36603i	0.366025	−	1.36603i
$736$		0			0
$737$		0			0
$738$	−0.366025	−	1.36603i	−0.366025	−	1.36603i
$739$		0			0		0.965926	−	0.258819i	$-0.0833333\pi$
$739$		0			0		−0.965926	+	0.258819i	$0.916667\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	−0.366025	−	1.36603i	−0.366025	−	1.36603i	−0.866025	−	0.500000i	$-0.833333\pi$
$743$	−0.366025	−	1.36603i	−0.366025	−	1.36603i	0.500000	−	0.866025i	$-0.333333\pi$
$744$		0			0
$745$	1.73205	−	1.00000i	1.73205	−	1.00000i
$746$		0			0
$747$	−1.36603	−	0.366025i	−1.36603	−	0.366025i
$748$		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	$0.166667\pi$
$752$	−1.36603	+	0.366025i	−1.36603	+	0.366025i
$753$		0			0
$754$		0			0
$755$		0			0
$756$		0			0
$757$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$757$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−1.36603	−	0.366025i	−1.36603	−	0.366025i	−0.500000	−	0.866025i	$-0.666667\pi$
$761$	−1.36603	−	0.366025i	−1.36603	−	0.366025i	−0.866025	+	0.500000i	$0.833333\pi$
$762$	−2.00000			−2.00000
$763$		0			0
$764$		0			0
$765$		0			0
$766$	−1.00000	−	1.00000i	−1.00000	−	1.00000i
$767$		0			0
$768$		−	1.00000i		−	1.00000i
$769$		0			0		0.965926	−	0.258819i	$-0.0833333\pi$
$769$		0			0		−0.965926	+	0.258819i	$0.916667\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−0.366025	+	1.36603i	−0.366025	+	1.36603i	0.500000	+	0.866025i	$0.333333\pi$
$773$	−0.366025	+	1.36603i	−0.366025	+	1.36603i	−0.866025	+	0.500000i	$0.833333\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$		0			0
$778$		0			0
$779$		0			0
$780$		0			0
$781$	2.00000			2.00000
$782$		0			0
$783$		0			0
$784$	0.866025	−	0.500000i	0.866025	−	0.500000i
$785$	−2.00000	−	2.00000i	−2.00000	−	2.00000i
$786$		0			0
$787$		0			0		0.258819	−	0.965926i	$-0.416667\pi$
$787$		0			0		−0.258819	+	0.965926i	$0.583333\pi$
$788$	1.00000	+	1.00000i	1.00000	+	1.00000i
$789$		0			0
$790$	−2.73205	+	0.732051i	−2.73205	+	0.732051i
$791$		0			0
$792$	−1.00000	+	1.00000i	−1.00000	+	1.00000i
$793$		0			0
$794$		0			0
$795$		0			0
$796$	−1.00000	+	1.73205i	−1.00000	+	1.73205i
$797$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$797$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$798$		0			0
$799$		0			0
$800$	0.500000	+	0.866025i	0.500000	+	0.866025i
$801$	−1.00000	+	1.00000i	−1.00000	+	1.00000i
$802$	−0.366025	+	1.36603i	−0.366025	+	1.36603i
$803$		0			0
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$809$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$810$	1.36603	+	0.366025i	1.36603	+	0.366025i
$811$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$811$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$		0			0
$816$		0			0
$817$		0			0
$818$		0			0
$819$		0			0
$820$			2.00000i			2.00000i
$821$	−1.36603	+	0.366025i	−1.36603	+	0.366025i	−0.866025	−	0.500000i	$-0.833333\pi$
$821$	−1.36603	+	0.366025i	−1.36603	+	0.366025i	−0.500000	+	0.866025i	$0.666667\pi$
$822$	−0.366025	−	1.36603i	−0.366025	−	1.36603i
$823$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$823$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$824$		0			0
$825$	0.366025	−	1.36603i	0.366025	−	1.36603i
$826$		0			0
$827$	1.00000	+	1.00000i	1.00000	+	1.00000i	1.00000			$0$
$827$	1.00000	+	1.00000i	1.00000	+	1.00000i			1.00000i	$0.5\pi$
$828$		0			0
$829$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$829$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$830$	1.73205	+	1.00000i	1.73205	+	1.00000i
$831$	−2.00000			−2.00000
$832$		0			0
$833$		0			0
$834$		0			0
$835$	1.73205	+	1.00000i	1.73205	+	1.00000i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	0.366025	−	1.36603i	0.366025	−	1.36603i	−0.500000	−	0.866025i	$-0.666667\pi$
$839$	0.366025	−	1.36603i	0.366025	−	1.36603i	0.866025	−	0.500000i	$-0.166667\pi$
$840$		0			0
$841$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$842$		0			0
$843$	−1.36603	+	0.366025i	−1.36603	+	0.366025i
$844$			2.00000i			2.00000i
$845$		0			0
$846$	1.00000	−	1.00000i	1.00000	−	1.00000i
$847$		0			0
$848$		0			0
$849$	−1.73205	+	1.00000i	−1.73205	+	1.00000i
$850$		0			0
$851$		0			0
$852$	0.366025	−	1.36603i	0.366025	−	1.36603i
$853$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$853$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$		0			0			−	1.00000i	$-0.5\pi$
$857$		0			0				1.00000i	$0.5\pi$
$858$		0			0
$859$			2.00000i			2.00000i			1.00000i	$0.5\pi$
$859$			2.00000i			2.00000i			1.00000i	$0.5\pi$
$860$		0			0
$861$		0			0
$862$	−0.366025	+	1.36603i	−0.366025	+	1.36603i
$863$	−1.00000	+	1.00000i	−1.00000	+	1.00000i			1.00000i	$0.5\pi$
$863$	−1.00000	+	1.00000i	−1.00000	+	1.00000i	−1.00000			$\pi$
$864$	0.500000	+	0.866025i	0.500000	+	0.866025i
$865$		0			0
$866$		0			0
$867$	0.866025	−	0.500000i	0.866025	−	0.500000i
$868$		0			0
$869$	0.732051	+	2.73205i	0.732051	+	2.73205i
$870$		0			0
$871$		0			0
$872$		0			0
$873$		0			0
$874$		0			0
$875$		0			0
$876$		0			0
$877$		0			0		0.258819	−	0.965926i	$-0.416667\pi$
$877$		0			0		−0.258819	+	0.965926i	$0.583333\pi$
$878$		0			0
$879$	−1.00000	−	1.00000i	−1.00000	−	1.00000i
$880$	1.73205	−	1.00000i	1.73205	−	1.00000i
$881$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$881$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$882$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$883$	−2.00000			−2.00000			−1.00000			$\pi$
$883$	−2.00000			−2.00000			−1.00000			$\pi$
$884$		0			0
$885$	2.00000			2.00000
$886$		0			0
$887$		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
$890$	1.73205	−	1.00000i	1.73205	−	1.00000i
$891$	0.366025	−	1.36603i	0.366025	−	1.36603i
$892$		0			0
$893$		0			0
$894$	−1.36603	+	0.366025i	−1.36603	+	0.366025i
$895$		0			0
$896$		0			0
$897$		0			0
$898$	−1.00000	−	1.00000i	−1.00000	−	1.00000i
$899$		0			0
$900$	−0.866025	−	0.500000i	−0.866025	−	0.500000i
$901$		0			0
$902$	2.00000			2.00000
$903$		0			0
$904$		0			0
$905$	2.00000	−	2.00000i	2.00000	−	2.00000i
$906$		0			0
$907$	−1.00000	+	1.73205i	−1.00000	+	1.73205i	−0.500000	+	0.866025i	$0.666667\pi$
$907$	−1.00000	+	1.73205i	−1.00000	+	1.73205i	−0.500000	+	0.866025i	$0.666667\pi$
$908$	−0.366025	−	1.36603i	−0.366025	−	1.36603i
$909$		0			0
$910$		0			0
$911$		0			0			−	1.00000i	$-0.5\pi$
$911$		0			0				1.00000i	$0.5\pi$
$912$		0			0
$913$	1.00000	−	1.73205i	1.00000	−	1.73205i
$914$		0			0
$915$		0			0
$916$		0			0
$917$		0			0
$918$		0			0
$919$	−1.73205	+	1.00000i	−1.73205	+	1.00000i	−0.866025	+	0.500000i	$0.833333\pi$
$919$	−1.73205	+	1.00000i	−1.73205	+	1.00000i	−0.866025	+	0.500000i	$0.833333\pi$
$920$		0			0
$921$		0			0
$922$	−1.00000	+	1.00000i	−1.00000	+	1.00000i
$923$		0			0
$924$		0			0
$925$		0			0
$926$		0			0
$927$		0			0
$928$		0			0
$929$	0.366025	−	1.36603i	0.366025	−	1.36603i	−0.500000	−	0.866025i	$-0.666667\pi$
$929$	0.366025	−	1.36603i	0.366025	−	1.36603i	0.866025	−	0.500000i	$-0.166667\pi$
$930$		0			0
$931$		0			0
$932$		0			0
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$		0			0			−	1.00000i	$-0.5\pi$
$937$		0			0				1.00000i	$0.5\pi$
$938$		0			0
$939$		0			0
$940$	−1.73205	+	1.00000i	−1.73205	+	1.00000i
$941$	1.00000	+	1.00000i	1.00000	+	1.00000i	1.00000			$0$
$941$	1.00000	+	1.00000i	1.00000	+	1.00000i			1.00000i	$0.5\pi$
$942$	1.00000	+	1.73205i	1.00000	+	1.73205i
$943$		0			0
$944$	1.00000	+	1.00000i	1.00000	+	1.00000i
$945$		0			0
$946$		0			0
$947$	1.36603	−	0.366025i	1.36603	−	0.366025i	0.500000	−	0.866025i	$-0.333333\pi$
$947$	1.36603	−	0.366025i	1.36603	−	0.366025i	0.866025	+	0.500000i	$0.166667\pi$
$948$	2.00000			2.00000
$949$		0			0
$950$		0			0
$951$	−0.366025	−	1.36603i	−0.366025	−	1.36603i
$952$		0			0
$953$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$953$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$954$		0			0
$955$		0			0
$956$	1.36603	+	0.366025i	1.36603	+	0.366025i
$957$		0			0
$958$	1.36603	+	0.366025i	1.36603	+	0.366025i
$959$		0			0
$960$	−0.366025	−	1.36603i	−0.366025	−	1.36603i
$961$		−	1.00000i		−	1.00000i
$962$		0			0
$963$		0			0
$964$		0			0
$965$		0			0
$966$		0			0
$967$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$967$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$968$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$969$		0			0
$970$		0			0
$971$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$971$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$972$	−0.866025	−	0.500000i	−0.866025	−	0.500000i
$973$		0			0
$974$		0			0
$975$		0			0
$976$		0			0
$977$	1.36603	−	0.366025i	1.36603	−	0.366025i	0.500000	−	0.866025i	$-0.333333\pi$
$977$	1.36603	−	0.366025i	1.36603	−	0.366025i	0.866025	+	0.500000i	$0.166667\pi$
$978$		0			0
$979$	−1.00000	−	1.73205i	−1.00000	−	1.73205i
$980$	1.00000	−	1.00000i	1.00000	−	1.00000i
$981$		0			0
$982$		0			0
$983$	−1.00000	−	1.00000i	−1.00000	−	1.00000i		−	1.00000i	$-0.5\pi$
$983$	−1.00000	−	1.00000i	−1.00000	−	1.00000i	−1.00000			$\pi$
$984$	0.366025	−	1.36603i	0.366025	−	1.36603i
$985$	1.73205	+	1.00000i	1.73205	+	1.00000i
$986$		0			0
$987$		0			0
$988$		0			0
$989$		0			0
$990$	−1.00000	+	1.73205i	−1.00000	+	1.73205i
$991$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$991$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	−0.732051	+	2.73205i	−0.732051	+	2.73205i
$996$	−1.00000	−	1.00000i	−1.00000	−	1.00000i
$997$	1.00000	+	1.73205i	1.00000	+	1.73205i	0.500000	+	0.866025i	$0.333333\pi$
$997$	1.00000	+	1.73205i	1.00000	+	1.73205i	0.500000	+	0.866025i	$0.333333\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2028.1.v.c.695.1		4
3.2	odd	2		2028.1.v.b.695.1		4
4.3	odd	2		2028.1.v.a.695.1		4
12.11	even	2		2028.1.v.d.695.1		4
13.2	odd	12		2028.1.v.a.1103.1		4
13.3	even	3	inner	2028.1.v.c.587.1		4
13.4	even	6		2028.1.l.b.239.1	yes	2
13.5	odd	4		2028.1.v.a.995.1		4
13.6	odd	12		2028.1.l.d.1451.1	yes	2
13.7	odd	12		2028.1.l.a.1451.1	yes	2
13.8	odd	4		2028.1.v.d.995.1		4
13.9	even	3		2028.1.l.c.239.1	yes	2
13.10	even	6		2028.1.v.b.587.1		4
13.11	odd	12		2028.1.v.d.1103.1		4
13.12	even	2		2028.1.v.b.695.1		4
39.2	even	12		2028.1.v.d.1103.1		4
39.5	even	4		2028.1.v.d.995.1		4
39.8	even	4		2028.1.v.a.995.1		4
39.11	even	12		2028.1.v.a.1103.1		4
39.17	odd	6		2028.1.l.c.239.1	yes	2
39.20	even	12		2028.1.l.d.1451.1	yes	2
39.23	odd	6	inner	2028.1.v.c.587.1		4
39.29	odd	6		2028.1.v.b.587.1		4
39.32	even	12		2028.1.l.a.1451.1	yes	2
39.35	odd	6		2028.1.l.b.239.1	yes	2
39.38	odd	2	CM	2028.1.v.c.695.1		4
52.3	odd	6		2028.1.v.a.587.1		4
52.7	even	12		2028.1.l.b.1451.1	yes	2
52.11	even	12		2028.1.v.b.1103.1		4
52.15	even	12	inner	2028.1.v.c.1103.1		4
52.19	even	12		2028.1.l.c.1451.1	yes	2
52.23	odd	6		2028.1.v.d.587.1		4
52.31	even	4	inner	2028.1.v.c.995.1		4
52.35	odd	6		2028.1.l.d.239.1	yes	2
52.43	odd	6		2028.1.l.a.239.1	✓	2
52.47	even	4		2028.1.v.b.995.1		4
52.51	odd	2		2028.1.v.d.695.1		4
156.11	odd	12	inner	2028.1.v.c.1103.1		4
156.23	even	6		2028.1.v.a.587.1		4
156.35	even	6		2028.1.l.a.239.1	✓	2
156.47	odd	4	inner	2028.1.v.c.995.1		4
156.59	odd	12		2028.1.l.c.1451.1	yes	2
156.71	odd	12		2028.1.l.b.1451.1	yes	2
156.83	odd	4		2028.1.v.b.995.1		4
156.95	even	6		2028.1.l.d.239.1	yes	2
156.107	even	6		2028.1.v.d.587.1		4
156.119	odd	12		2028.1.v.b.1103.1		4
156.155	even	2		2028.1.v.a.695.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2028.1.l.a.239.1	✓	2	52.43	odd	6
2028.1.l.a.239.1	✓	2	156.35	even	6
2028.1.l.a.1451.1	yes	2	13.7	odd	12
2028.1.l.a.1451.1	yes	2	39.32	even	12
2028.1.l.b.239.1	yes	2	13.4	even	6
2028.1.l.b.239.1	yes	2	39.35	odd	6
2028.1.l.b.1451.1	yes	2	52.7	even	12
2028.1.l.b.1451.1	yes	2	156.71	odd	12
2028.1.l.c.239.1	yes	2	13.9	even	3
2028.1.l.c.239.1	yes	2	39.17	odd	6
2028.1.l.c.1451.1	yes	2	52.19	even	12
2028.1.l.c.1451.1	yes	2	156.59	odd	12
2028.1.l.d.239.1	yes	2	52.35	odd	6
2028.1.l.d.239.1	yes	2	156.95	even	6
2028.1.l.d.1451.1	yes	2	13.6	odd	12
2028.1.l.d.1451.1	yes	2	39.20	even	12
2028.1.v.a.587.1		4	52.3	odd	6
2028.1.v.a.587.1		4	156.23	even	6
2028.1.v.a.695.1		4	4.3	odd	2
2028.1.v.a.695.1		4	156.155	even	2
2028.1.v.a.995.1		4	13.5	odd	4
2028.1.v.a.995.1		4	39.8	even	4
2028.1.v.a.1103.1		4	13.2	odd	12
2028.1.v.a.1103.1		4	39.11	even	12
2028.1.v.b.587.1		4	13.10	even	6
2028.1.v.b.587.1		4	39.29	odd	6
2028.1.v.b.695.1		4	3.2	odd	2
2028.1.v.b.695.1		4	13.12	even	2
2028.1.v.b.995.1		4	52.47	even	4
2028.1.v.b.995.1		4	156.83	odd	4
2028.1.v.b.1103.1		4	52.11	even	12
2028.1.v.b.1103.1		4	156.119	odd	12
2028.1.v.c.587.1		4	13.3	even	3	inner
2028.1.v.c.587.1		4	39.23	odd	6	inner
2028.1.v.c.695.1		4	1.1	even	1	trivial
2028.1.v.c.695.1		4	39.38	odd	2	CM
2028.1.v.c.995.1		4	52.31	even	4	inner
2028.1.v.c.995.1		4	156.47	odd	4	inner
2028.1.v.c.1103.1		4	52.15	even	12	inner
2028.1.v.c.1103.1		4	156.11	odd	12	inner
2028.1.v.d.587.1		4	52.23	odd	6
2028.1.v.d.587.1		4	156.107	even	6
2028.1.v.d.695.1		4	12.11	even	2
2028.1.v.d.695.1		4	52.51	odd	2
2028.1.v.d.995.1		4	13.8	odd	4
2028.1.v.d.995.1		4	39.5	even	4
2028.1.v.d.1103.1		4	13.11	odd	12
2028.1.v.d.1103.1		4	39.2	even	12

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:	\( N \)	\(=\)	\( 2028 = 2^{2} \cdot 3 \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	2028.v (of order \(12\), degree \(4\), not minimal)

\(f(q)\)	\(=\)	\(q+(-0.866025 - 0.500000i) q^{2} +(0.866025 + 0.500000i) q^{3} +(0.500000 + 0.866025i) q^{4} +(1.00000 + 1.00000i) q^{5} +(-0.500000 - 0.866025i) q^{6} -1.00000i q^{8} +(0.500000 + 0.866025i) q^{9} +(-0.366025 - 1.36603i) q^{10} +(-1.36603 + 0.366025i) q^{11} +1.00000i q^{12} +(0.366025 + 1.36603i) q^{15} +(-0.500000 + 0.866025i) q^{16} -1.00000i q^{18} +(-0.366025 + 1.36603i) q^{20} +(1.36603 + 0.366025i) q^{22} +(0.500000 - 0.866025i) q^{24} +1.00000i q^{25} +1.00000i q^{27} +(0.366025 - 1.36603i) q^{30} +(0.866025 - 0.500000i) q^{32} +(-1.36603 - 0.366025i) q^{33} +(-0.500000 + 0.866025i) q^{36} +(1.00000 - 1.00000i) q^{40} +(1.36603 - 0.366025i) q^{41} +(-1.00000 - 1.00000i) q^{44} +(-0.366025 + 1.36603i) q^{45} +(1.00000 + 1.00000i) q^{47} +(-0.866025 + 0.500000i) q^{48} +(-0.866025 - 0.500000i) q^{49} +(0.500000 - 0.866025i) q^{50} +(0.500000 - 0.866025i) q^{54} +(-1.73205 - 1.00000i) q^{55} +(0.366025 - 1.36603i) q^{59} +(-1.00000 + 1.00000i) q^{60} -1.00000 q^{64} +(1.00000 + 1.00000i) q^{66} +(-1.36603 - 0.366025i) q^{71} +(0.866025 - 0.500000i) q^{72} +(-0.500000 + 0.866025i) q^{75} -2.00000i q^{79} +(-1.36603 + 0.366025i) q^{80} +(-0.500000 + 0.866025i) q^{81} +(-1.36603 - 0.366025i) q^{82} +(-1.00000 + 1.00000i) q^{83} +(0.366025 + 1.36603i) q^{88} +(0.366025 + 1.36603i) q^{89} +(1.00000 - 1.00000i) q^{90} +(-0.366025 - 1.36603i) q^{94} +1.00000 q^{96} +(0.500000 + 0.866025i) q^{98} +(-1.00000 - 1.00000i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 2 q^{4} + 4 q^{5} - 2 q^{6} + 2 q^{9} + 2 q^{10} - 2 q^{11} - 2 q^{15} - 2 q^{16} + 2 q^{20} + 2 q^{22} + 2 q^{24} - 2 q^{30} - 2 q^{33} - 2 q^{36} + 4 q^{40} + 2 q^{41} - 4 q^{44} + 2 q^{45} + 4 q^{47}+ \cdots - 4 q^{99}+O(q^{100}) \) 4 * q + 2 * q^4 + 4 * q^5 - 2 * q^6 + 2 * q^9 + 2 * q^10 - 2 * q^11 - 2 * q^15 - 2 * q^16 + 2 * q^20 + 2 * q^22 + 2 * q^24 - 2 * q^30 - 2 * q^33 - 2 * q^36 + 4 * q^40 + 2 * q^41 - 4 * q^44 + 2 * q^45 + 4 * q^47 + 2 * q^50 + 2 * q^54 - 2 * q^59 - 4 * q^60 - 4 * q^64 + 4 * q^66 - 2 * q^71 - 2 * q^75 - 2 * q^80 - 2 * q^81 - 2 * q^82 - 4 * q^83 - 2 * q^88 - 2 * q^89 + 4 * q^90 + 2 * q^94 + 4 * q^96 + 2 * q^98 - 4 * q^99

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