LMFDB - Embedded newform 2888.1.u.c.1867.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2888,1,Mod(99,2888)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2888, base_ring=CyclotomicField(18))

chi = DirichletCharacter(H, H._module([9, 9, 2]))

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

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

chi := DirichletCharacter("2888.99");

S:= CuspForms(chi, 1);

N := Newforms(S);

Level:	$ N $	$=$	$ 2888 = 2^{3} \cdot 19^{2} $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	2888.u (of order $18$, degree $6$, 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:	$1.44129975648$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	$\Q(\zeta_{18})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - x^{3} + 1 $ x^6 - x^3 + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 152)
Projective image:	$D_{3}$
Projective field:	Galois closure of 3.1.2888.1
Artin image:	$S_3\times C_9$
Artin field:	Galois closure of $\mathbb{Q}[x]/(x^{54} - \cdots)$

Embedding invariants

Embedding label			1867.1
Root			$-0.766044 + 0.642788i$ of defining polynomial
Character	$\chi$	$=$	2888.1867
Dual form			2888.1.u.c.99.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-0.939693 + 0.342020i) q^{2} +(-0.173648 + 0.984808i) q^{3} +(0.766044 - 0.642788i) q^{4} +(-0.173648 - 0.984808i) q^{6} +(-0.500000 + 0.866025i) q^{8} +O(q^{10})$	\(q+(-0.939693 + 0.342020i) q^{2} +(-0.173648 + 0.984808i) q^{3} +(0.766044 - 0.642788i) q^{4} +(-0.173648 - 0.984808i) q^{6} +(-0.500000 + 0.866025i) q^{8} +(0.500000 - 0.866025i) q^{11} +(0.500000 + 0.866025i) q^{12} +(0.173648 - 0.984808i) q^{16} +(-1.87939 + 0.684040i) q^{17} +(-0.173648 + 0.984808i) q^{22} +(-0.766044 - 0.642788i) q^{24} +(0.173648 + 0.984808i) q^{25} +(-0.500000 + 0.866025i) q^{27} +(0.173648 + 0.984808i) q^{32} +(0.766044 + 0.642788i) q^{33} +(1.53209 - 1.28558i) q^{34} +(-0.173648 + 0.984808i) q^{41} +(1.53209 + 1.28558i) q^{43} +(-0.173648 - 0.984808i) q^{44} +(0.939693 + 0.342020i) q^{48} +(-0.500000 + 0.866025i) q^{49} +(-0.500000 - 0.866025i) q^{50} +(-0.347296 - 1.96962i) q^{51} +(0.173648 - 0.984808i) q^{54} +(0.939693 - 0.342020i) q^{59} +(-0.500000 - 0.866025i) q^{64} +(-0.939693 - 0.342020i) q^{66} +(0.939693 + 0.342020i) q^{67} +(-1.00000 + 1.73205i) q^{68} +(-0.173648 + 0.984808i) q^{73} -1.00000 q^{75} +(-0.766044 - 0.642788i) q^{81} +(-0.173648 - 0.984808i) q^{82} +(0.500000 + 0.866025i) q^{83} +(-1.87939 - 0.684040i) q^{86} +(0.500000 + 0.866025i) q^{88} +(0.347296 + 1.96962i) q^{89} -1.00000 q^{96} +(0.939693 - 0.342020i) q^{97} +(0.173648 - 0.984808i) q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 3 q^{8}+O(q^{10}) $ 6 * q - 3 * q^8	$ 6 q - 3 q^{8} + 3 q^{11} + 3 q^{12} - 3 q^{27} - 3 q^{49} - 3 q^{50} - 3 q^{64} - 6 q^{68} - 6 q^{75} + 3 q^{83} + 3 q^{88} - 6 q^{96}+O(q^{100}) $ 6 * q - 3 * q^8 + 3 * q^11 + 3 * q^12 - 3 * q^27 - 3 * q^49 - 3 * q^50 - 3 * q^64 - 6 * q^68 - 6 * q^75 + 3 * q^83 + 3 * q^88 - 6 * q^96

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2888\mathbb{Z}\right)^\times$.

$n$	$1445$	$2167$	$2529$
$\chi(n)$	$-1$	$-1$	$e\left(\frac{8}{9}\right)$

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 $

$2$	−0.939693	+	0.342020i	−0.939693	+	0.342020i
$2$	−0.939693	+	0.342020i	−0.939693	+	0.342020i
$3$	−0.173648	+	0.984808i	−0.173648	+	0.984808i	0.766044	+	0.642788i	$0.222222\pi$
$3$	−0.173648	+	0.984808i	−0.173648	+	0.984808i	−0.939693	+	0.342020i	$0.888889\pi$
$4$	0.766044	−	0.642788i	0.766044	−	0.642788i
$5$		0			0		−0.766044	−	0.642788i	$-0.777778\pi$
$5$		0			0		0.766044	+	0.642788i	$0.222222\pi$
$6$	−0.173648	−	0.984808i	−0.173648	−	0.984808i
$7$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$7$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$8$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$9$		0			0
$10$		0			0
$11$	0.500000	−	0.866025i	0.500000	−	0.866025i	−0.500000	−	0.866025i	$-0.666667\pi$
$11$	0.500000	−	0.866025i	0.500000	−	0.866025i	1.00000			$0$
$12$	0.500000	+	0.866025i	0.500000	+	0.866025i
$13$		0			0		−0.173648	−	0.984808i	$-0.555556\pi$
$13$		0			0		0.173648	+	0.984808i	$0.444444\pi$
$14$		0			0
$15$		0			0
$16$	0.173648	−	0.984808i	0.173648	−	0.984808i
$17$	−1.87939	+	0.684040i	−1.87939	+	0.684040i	−0.939693	+	0.342020i	$0.888889\pi$
$17$	−1.87939	+	0.684040i	−1.87939	+	0.684040i	−0.939693	+	0.342020i	$0.888889\pi$
$18$		0			0
$19$		0			0
$19$		0			0
$20$		0			0
$21$		0			0
$22$	−0.173648	+	0.984808i	−0.173648	+	0.984808i
$23$		0			0		0.766044	−	0.642788i	$-0.222222\pi$
$23$		0			0		−0.766044	+	0.642788i	$0.777778\pi$
$24$	−0.766044	−	0.642788i	−0.766044	−	0.642788i
$25$	0.173648	+	0.984808i	0.173648	+	0.984808i
$26$		0			0
$27$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$28$		0			0
$29$		0			0		−0.939693	−	0.342020i	$-0.888889\pi$
$29$		0			0		0.939693	+	0.342020i	$0.111111\pi$
$30$		0			0
$31$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$31$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$32$	0.173648	+	0.984808i	0.173648	+	0.984808i
$33$	0.766044	+	0.642788i	0.766044	+	0.642788i
$34$	1.53209	−	1.28558i	1.53209	−	1.28558i
$35$		0			0
$36$		0			0
$37$		0			0		1.00000			$0$
$37$		0			0		−1.00000			$\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	−0.173648	+	0.984808i	−0.173648	+	0.984808i	0.766044	+	0.642788i	$0.222222\pi$
$41$	−0.173648	+	0.984808i	−0.173648	+	0.984808i	−0.939693	+	0.342020i	$0.888889\pi$
$42$		0			0
$43$	1.53209	+	1.28558i	1.53209	+	1.28558i	0.766044	+	0.642788i	$0.222222\pi$
$43$	1.53209	+	1.28558i	1.53209	+	1.28558i	0.766044	+	0.642788i	$0.222222\pi$
$44$	−0.173648	−	0.984808i	−0.173648	−	0.984808i
$45$		0			0
$46$		0			0
$47$		0			0		−0.939693	−	0.342020i	$-0.888889\pi$
$47$		0			0		0.939693	+	0.342020i	$0.111111\pi$
$48$	0.939693	+	0.342020i	0.939693	+	0.342020i
$49$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$50$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$51$	−0.347296	−	1.96962i	−0.347296	−	1.96962i
$52$		0			0
$53$		0			0		0.766044	−	0.642788i	$-0.222222\pi$
$53$		0			0		−0.766044	+	0.642788i	$0.777778\pi$
$54$	0.173648	−	0.984808i	0.173648	−	0.984808i
$55$		0			0
$56$		0			0
$57$		0			0
$58$		0			0
$59$	0.939693	−	0.342020i	0.939693	−	0.342020i	0.173648	−	0.984808i	$-0.444444\pi$
$59$	0.939693	−	0.342020i	0.939693	−	0.342020i	0.766044	+	0.642788i	$0.222222\pi$
$60$		0			0
$61$		0			0		0.766044	−	0.642788i	$-0.222222\pi$
$61$		0			0		−0.766044	+	0.642788i	$0.777778\pi$
$62$		0			0
$63$		0			0
$64$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$65$		0			0
$66$	−0.939693	−	0.342020i	−0.939693	−	0.342020i
$67$	0.939693	+	0.342020i	0.939693	+	0.342020i	0.766044	−	0.642788i	$-0.222222\pi$
$67$	0.939693	+	0.342020i	0.939693	+	0.342020i	0.173648	+	0.984808i	$0.444444\pi$
$68$	−1.00000	+	1.73205i	−1.00000	+	1.73205i
$69$		0			0
$70$		0			0
$71$		0			0		−0.766044	−	0.642788i	$-0.777778\pi$
$71$		0			0		0.766044	+	0.642788i	$0.222222\pi$
$72$		0			0
$73$	−0.173648	+	0.984808i	−0.173648	+	0.984808i	0.766044	+	0.642788i	$0.222222\pi$
$73$	−0.173648	+	0.984808i	−0.173648	+	0.984808i	−0.939693	+	0.342020i	$0.888889\pi$
$74$		0			0
$75$	−1.00000			−1.00000
$76$		0			0
$77$		0			0
$78$		0			0
$79$		0			0		0.173648	−	0.984808i	$-0.444444\pi$
$79$		0			0		−0.173648	+	0.984808i	$0.555556\pi$
$80$		0			0
$81$	−0.766044	−	0.642788i	−0.766044	−	0.642788i
$82$	−0.173648	−	0.984808i	−0.173648	−	0.984808i
$83$	0.500000	+	0.866025i	0.500000	+	0.866025i	1.00000			$0$
$83$	0.500000	+	0.866025i	0.500000	+	0.866025i	−0.500000	+	0.866025i	$0.666667\pi$
$84$		0			0
$85$		0			0
$86$	−1.87939	−	0.684040i	−1.87939	−	0.684040i
$87$		0			0
$88$	0.500000	+	0.866025i	0.500000	+	0.866025i
$89$	0.347296	+	1.96962i	0.347296	+	1.96962i	0.173648	+	0.984808i	$0.444444\pi$
$89$	0.347296	+	1.96962i	0.347296	+	1.96962i	0.173648	+	0.984808i	$0.444444\pi$
$90$		0			0
$91$		0			0
$92$		0			0
$93$		0			0
$94$		0			0
$95$		0			0
$96$	−1.00000			−1.00000
$97$	0.939693	−	0.342020i	0.939693	−	0.342020i	0.173648	−	0.984808i	$-0.444444\pi$
$97$	0.939693	−	0.342020i	0.939693	−	0.342020i	0.766044	+	0.642788i	$0.222222\pi$
$98$	0.173648	−	0.984808i	0.173648	−	0.984808i
$99$		0			0
$100$	0.766044	+	0.642788i	0.766044	+	0.642788i
$101$		0			0		−0.173648	−	0.984808i	$-0.555556\pi$
$101$		0			0		0.173648	+	0.984808i	$0.444444\pi$
$102$	1.00000	+	1.73205i	1.00000	+	1.73205i
$103$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$103$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−1.00000	−	1.73205i	−1.00000	−	1.73205i	−0.500000	−	0.866025i	$-0.666667\pi$
$107$	−1.00000	−	1.73205i	−1.00000	−	1.73205i	−0.500000	−	0.866025i	$-0.666667\pi$
$108$	0.173648	+	0.984808i	0.173648	+	0.984808i
$109$		0			0		−0.766044	−	0.642788i	$-0.777778\pi$
$109$		0			0		0.766044	+	0.642788i	$0.222222\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−1.00000			−1.00000			−0.500000	−	0.866025i	$-0.666667\pi$
$113$	−1.00000			−1.00000			−0.500000	+	0.866025i	$0.666667\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$		0			0
$118$	−0.766044	+	0.642788i	−0.766044	+	0.642788i
$119$		0			0
$120$		0			0
$121$		0			0
$122$		0			0
$123$	−0.939693	−	0.342020i	−0.939693	−	0.342020i
$124$		0			0
$125$		0			0
$126$		0			0
$127$		0			0		−0.173648	−	0.984808i	$-0.555556\pi$
$127$		0			0		0.173648	+	0.984808i	$0.444444\pi$
$128$	0.766044	+	0.642788i	0.766044	+	0.642788i
$129$	−1.53209	+	1.28558i	−1.53209	+	1.28558i
$130$		0			0
$131$	0.939693	−	0.342020i	0.939693	−	0.342020i	0.173648	−	0.984808i	$-0.444444\pi$
$131$	0.939693	−	0.342020i	0.939693	−	0.342020i	0.766044	+	0.642788i	$0.222222\pi$
$132$	1.00000			1.00000
$133$		0			0
$134$	−1.00000			−1.00000
$135$		0			0
$136$	0.347296	−	1.96962i	0.347296	−	1.96962i
$137$	−0.766044	+	0.642788i	−0.766044	+	0.642788i	−0.939693	−	0.342020i	$-0.888889\pi$
$137$	−0.766044	+	0.642788i	−0.766044	+	0.642788i	0.173648	+	0.984808i	$0.444444\pi$
$138$		0			0
$139$	−0.173648	−	0.984808i	−0.173648	−	0.984808i	−0.939693	−	0.342020i	$-0.888889\pi$
$139$	−0.173648	−	0.984808i	−0.173648	−	0.984808i	0.766044	−	0.642788i	$-0.222222\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		0			0
$144$		0			0
$145$		0			0
$146$	−0.173648	−	0.984808i	−0.173648	−	0.984808i
$147$	−0.766044	−	0.642788i	−0.766044	−	0.642788i
$148$		0			0
$149$		0			0		0.173648	−	0.984808i	$-0.444444\pi$
$149$		0			0		−0.173648	+	0.984808i	$0.555556\pi$
$150$	0.939693	−	0.342020i	0.939693	−	0.342020i
$151$		0			0		1.00000			$0$
$151$		0			0		−1.00000			$\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		0			0
$156$		0			0
$157$		0			0		−0.766044	−	0.642788i	$-0.777778\pi$
$157$		0			0		0.766044	+	0.642788i	$0.222222\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$	0.939693	+	0.342020i	0.939693	+	0.342020i
$163$	0.500000	−	0.866025i	0.500000	−	0.866025i	−0.500000	−	0.866025i	$-0.666667\pi$
$163$	0.500000	−	0.866025i	0.500000	−	0.866025i	1.00000			$0$
$164$	0.500000	+	0.866025i	0.500000	+	0.866025i
$165$		0			0
$166$	−0.766044	−	0.642788i	−0.766044	−	0.642788i
$167$		0			0		0.766044	−	0.642788i	$-0.222222\pi$
$167$		0			0		−0.766044	+	0.642788i	$0.777778\pi$
$168$		0			0
$169$	−0.939693	+	0.342020i	−0.939693	+	0.342020i
$170$		0			0
$171$		0			0
$172$	2.00000			2.00000
$173$		0			0		0.939693	−	0.342020i	$-0.111111\pi$
$173$		0			0		−0.939693	+	0.342020i	$0.888889\pi$
$174$		0			0
$175$		0			0
$176$	−0.766044	−	0.642788i	−0.766044	−	0.642788i
$177$	0.173648	+	0.984808i	0.173648	+	0.984808i
$178$	−1.00000	−	1.73205i	−1.00000	−	1.73205i
$179$	0.500000	−	0.866025i	0.500000	−	0.866025i	−0.500000	−	0.866025i	$-0.666667\pi$
$179$	0.500000	−	0.866025i	0.500000	−	0.866025i	1.00000			$0$
$180$		0			0
$181$		0			0		−0.939693	−	0.342020i	$-0.888889\pi$
$181$		0			0		0.939693	+	0.342020i	$0.111111\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$		0			0
$186$		0			0
$187$	−0.347296	+	1.96962i	−0.347296	+	1.96962i
$188$		0			0
$189$		0			0
$190$		0			0
$191$		0			0		1.00000			$0$
$191$		0			0		−1.00000			$\pi$
$192$	0.939693	−	0.342020i	0.939693	−	0.342020i
$193$	0.347296	−	1.96962i	0.347296	−	1.96962i	0.173648	−	0.984808i	$-0.444444\pi$
$193$	0.347296	−	1.96962i	0.347296	−	1.96962i	0.173648	−	0.984808i	$-0.444444\pi$
$194$	−0.766044	+	0.642788i	−0.766044	+	0.642788i
$195$		0			0
$196$	0.173648	+	0.984808i	0.173648	+	0.984808i
$197$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$197$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$198$		0			0
$199$		0			0		−0.939693	−	0.342020i	$-0.888889\pi$
$199$		0			0		0.939693	+	0.342020i	$0.111111\pi$
$200$	−0.939693	−	0.342020i	−0.939693	−	0.342020i
$201$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$202$		0			0
$203$		0			0
$204$	−1.53209	−	1.28558i	−1.53209	−	1.28558i
$205$		0			0
$206$		0			0
$207$		0			0
$208$		0			0
$209$		0			0
$210$		0			0
$211$	−1.87939	+	0.684040i	−1.87939	+	0.684040i	−0.939693	+	0.342020i	$0.888889\pi$
$211$	−1.87939	+	0.684040i	−1.87939	+	0.684040i	−0.939693	+	0.342020i	$0.888889\pi$
$212$		0			0
$213$		0			0
$214$	1.53209	+	1.28558i	1.53209	+	1.28558i
$215$		0			0
$216$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$217$		0			0
$218$		0			0
$219$	−0.939693	−	0.342020i	−0.939693	−	0.342020i
$220$		0			0
$221$		0			0
$222$		0			0
$223$		0			0		−0.766044	−	0.642788i	$-0.777778\pi$
$223$		0			0		0.766044	+	0.642788i	$0.222222\pi$
$224$		0			0
$225$		0			0
$226$	0.939693	−	0.342020i	0.939693	−	0.342020i
$227$	−1.00000			−1.00000			−0.500000	−	0.866025i	$-0.666667\pi$
$227$	−1.00000			−1.00000			−0.500000	+	0.866025i	$0.666667\pi$
$228$		0			0
$229$		0			0		1.00000			$0$
$229$		0			0		−1.00000			$\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−0.766044	−	0.642788i	−0.766044	−	0.642788i	0.173648	−	0.984808i	$-0.444444\pi$
$233$	−0.766044	−	0.642788i	−0.766044	−	0.642788i	−0.939693	+	0.342020i	$0.888889\pi$
$234$		0			0
$235$		0			0
$236$	0.500000	−	0.866025i	0.500000	−	0.866025i
$237$		0			0
$238$		0			0
$239$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$239$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$240$		0			0
$241$	−0.173648	−	0.984808i	−0.173648	−	0.984808i	−0.939693	−	0.342020i	$-0.888889\pi$
$241$	−0.173648	−	0.984808i	−0.173648	−	0.984808i	0.766044	−	0.642788i	$-0.222222\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$		0			0
$246$	1.00000			1.00000
$247$		0			0
$248$		0			0
$249$	−0.939693	+	0.342020i	−0.939693	+	0.342020i
$250$		0			0
$251$	−0.766044	+	0.642788i	−0.766044	+	0.642788i	−0.939693	−	0.342020i	$-0.888889\pi$
$251$	−0.766044	+	0.642788i	−0.766044	+	0.642788i	0.173648	+	0.984808i	$0.444444\pi$
$252$		0			0
$253$		0			0
$254$		0			0
$255$		0			0
$256$	−0.939693	−	0.342020i	−0.939693	−	0.342020i
$257$	0.939693	+	0.342020i	0.939693	+	0.342020i	0.766044	−	0.642788i	$-0.222222\pi$
$257$	0.939693	+	0.342020i	0.939693	+	0.342020i	0.173648	+	0.984808i	$0.444444\pi$
$258$	1.00000	−	1.73205i	1.00000	−	1.73205i
$259$		0			0
$260$		0			0
$261$		0			0
$262$	−0.766044	+	0.642788i	−0.766044	+	0.642788i
$263$		0			0		0.173648	−	0.984808i	$-0.444444\pi$
$263$		0			0		−0.173648	+	0.984808i	$0.555556\pi$
$264$	−0.939693	+	0.342020i	−0.939693	+	0.342020i
$265$		0			0
$266$		0			0
$267$	−2.00000			−2.00000
$268$	0.939693	−	0.342020i	0.939693	−	0.342020i
$269$		0			0		0.173648	−	0.984808i	$-0.444444\pi$
$269$		0			0		−0.173648	+	0.984808i	$0.555556\pi$
$270$		0			0
$271$		0			0		−0.766044	−	0.642788i	$-0.777778\pi$
$271$		0			0		0.766044	+	0.642788i	$0.222222\pi$
$272$	0.347296	+	1.96962i	0.347296	+	1.96962i
$273$		0			0
$274$	0.500000	−	0.866025i	0.500000	−	0.866025i
$275$	0.939693	+	0.342020i	0.939693	+	0.342020i
$276$		0			0
$277$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$277$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$278$	0.500000	+	0.866025i	0.500000	+	0.866025i
$279$		0			0
$280$		0			0
$281$	−0.766044	+	0.642788i	−0.766044	+	0.642788i	−0.939693	−	0.342020i	$-0.888889\pi$
$281$	−0.766044	+	0.642788i	−0.766044	+	0.642788i	0.173648	+	0.984808i	$0.444444\pi$
$282$		0			0
$283$	0.939693	−	0.342020i	0.939693	−	0.342020i	0.173648	−	0.984808i	$-0.444444\pi$
$283$	0.939693	−	0.342020i	0.939693	−	0.342020i	0.766044	+	0.642788i	$0.222222\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$	2.29813	−	1.92836i	2.29813	−	1.92836i
$290$		0			0
$291$	0.173648	+	0.984808i	0.173648	+	0.984808i
$292$	0.500000	+	0.866025i	0.500000	+	0.866025i
$293$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$293$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$294$	0.939693	+	0.342020i	0.939693	+	0.342020i
$295$		0			0
$296$		0			0
$297$	0.500000	+	0.866025i	0.500000	+	0.866025i
$298$		0			0
$299$		0			0
$300$	−0.766044	+	0.642788i	−0.766044	+	0.642788i
$301$		0			0
$302$		0			0
$303$		0			0
$304$		0			0
$305$		0			0
$306$		0			0
$307$	−0.173648	+	0.984808i	−0.173648	+	0.984808i	0.766044	+	0.642788i	$0.222222\pi$
$307$	−0.173648	+	0.984808i	−0.173648	+	0.984808i	−0.939693	+	0.342020i	$0.888889\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$311$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$312$		0			0
$313$	0.939693	+	0.342020i	0.939693	+	0.342020i	0.766044	−	0.642788i	$-0.222222\pi$
$313$	0.939693	+	0.342020i	0.939693	+	0.342020i	0.173648	+	0.984808i	$0.444444\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$		0			0		−0.173648	−	0.984808i	$-0.555556\pi$
$317$		0			0		0.173648	+	0.984808i	$0.444444\pi$
$318$		0			0
$319$		0			0
$320$		0			0
$321$	1.87939	−	0.684040i	1.87939	−	0.684040i
$322$		0			0
$323$		0			0
$324$	−1.00000			−1.00000
$325$		0			0
$326$	−0.173648	+	0.984808i	−0.173648	+	0.984808i
$327$		0			0
$328$	−0.766044	−	0.642788i	−0.766044	−	0.642788i
$329$		0			0
$330$		0			0
$331$	0.500000	−	0.866025i	0.500000	−	0.866025i	−0.500000	−	0.866025i	$-0.666667\pi$
$331$	0.500000	−	0.866025i	0.500000	−	0.866025i	1.00000			$0$
$332$	0.939693	+	0.342020i	0.939693	+	0.342020i
$333$		0			0
$334$		0			0
$335$		0			0
$336$		0			0
$337$	−0.766044	−	0.642788i	−0.766044	−	0.642788i	0.173648	−	0.984808i	$-0.444444\pi$
$337$	−0.766044	−	0.642788i	−0.766044	−	0.642788i	−0.939693	+	0.342020i	$0.888889\pi$
$338$	0.766044	−	0.642788i	0.766044	−	0.642788i
$339$	0.173648	−	0.984808i	0.173648	−	0.984808i
$340$		0			0
$341$		0			0
$342$		0			0
$343$		0			0
$344$	−1.87939	+	0.684040i	−1.87939	+	0.684040i
$345$		0			0
$346$		0			0
$347$	−0.766044	−	0.642788i	−0.766044	−	0.642788i	0.173648	−	0.984808i	$-0.444444\pi$
$347$	−0.766044	−	0.642788i	−0.766044	−	0.642788i	−0.939693	+	0.342020i	$0.888889\pi$
$348$		0			0
$349$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$349$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$350$		0			0
$351$		0			0
$352$	0.939693	+	0.342020i	0.939693	+	0.342020i
$353$	0.500000	−	0.866025i	0.500000	−	0.866025i	−0.500000	−	0.866025i	$-0.666667\pi$
$353$	0.500000	−	0.866025i	0.500000	−	0.866025i	1.00000			$0$
$354$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$355$		0			0
$356$	1.53209	+	1.28558i	1.53209	+	1.28558i
$357$		0			0
$358$	−0.173648	+	0.984808i	−0.173648	+	0.984808i
$359$		0			0		0.939693	−	0.342020i	$-0.111111\pi$
$359$		0			0		−0.939693	+	0.342020i	$0.888889\pi$
$360$		0			0
$361$		0			0
$362$		0			0
$363$		0			0
$364$		0			0
$365$		0			0
$366$		0			0
$367$		0			0		−0.173648	−	0.984808i	$-0.555556\pi$
$367$		0			0		0.173648	+	0.984808i	$0.444444\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		0			0
$372$		0			0
$373$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$373$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$374$	−0.347296	−	1.96962i	−0.347296	−	1.96962i
$375$		0			0
$376$		0			0
$377$		0			0
$378$		0			0
$379$	2.00000			2.00000			1.00000			$0$
$379$	2.00000			2.00000			1.00000			$0$
$380$		0			0
$381$		0			0
$382$		0			0
$383$		0			0		0.173648	−	0.984808i	$-0.444444\pi$
$383$		0			0		−0.173648	+	0.984808i	$0.555556\pi$
$384$	−0.766044	+	0.642788i	−0.766044	+	0.642788i
$385$		0			0
$386$	0.347296	+	1.96962i	0.347296	+	1.96962i
$387$		0			0
$388$	0.500000	−	0.866025i	0.500000	−	0.866025i
$389$		0			0		−0.939693	−	0.342020i	$-0.888889\pi$
$389$		0			0		0.939693	+	0.342020i	$0.111111\pi$
$390$		0			0
$391$		0			0
$392$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$393$	0.173648	+	0.984808i	0.173648	+	0.984808i
$394$		0			0
$395$		0			0
$396$		0			0
$397$		0			0		0.939693	−	0.342020i	$-0.111111\pi$
$397$		0			0		−0.939693	+	0.342020i	$0.888889\pi$
$398$		0			0
$399$		0			0
$400$	1.00000			1.00000
$401$	0.939693	−	0.342020i	0.939693	−	0.342020i	0.173648	−	0.984808i	$-0.444444\pi$
$401$	0.939693	−	0.342020i	0.939693	−	0.342020i	0.766044	+	0.642788i	$0.222222\pi$
$402$	0.173648	−	0.984808i	0.173648	−	0.984808i
$403$		0			0
$404$		0			0
$405$		0			0
$406$		0			0
$407$		0			0
$408$	1.87939	+	0.684040i	1.87939	+	0.684040i
$409$	0.939693	+	0.342020i	0.939693	+	0.342020i	0.766044	−	0.642788i	$-0.222222\pi$
$409$	0.939693	+	0.342020i	0.939693	+	0.342020i	0.173648	+	0.984808i	$0.444444\pi$
$410$		0			0
$411$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$412$		0			0
$413$		0			0
$414$		0			0
$415$		0			0
$416$		0			0
$417$	1.00000			1.00000
$418$		0			0
$419$	2.00000			2.00000			1.00000			$0$
$419$	2.00000			2.00000			1.00000			$0$
$420$		0			0
$421$		0			0		0.173648	−	0.984808i	$-0.444444\pi$
$421$		0			0		−0.173648	+	0.984808i	$0.555556\pi$
$422$	1.53209	−	1.28558i	1.53209	−	1.28558i
$423$		0			0
$424$		0			0
$425$	−1.00000	−	1.73205i	−1.00000	−	1.73205i
$426$		0			0
$427$		0			0
$428$	−1.87939	−	0.684040i	−1.87939	−	0.684040i
$429$		0			0
$430$		0			0
$431$		0			0		−0.173648	−	0.984808i	$-0.555556\pi$
$431$		0			0		0.173648	+	0.984808i	$0.444444\pi$
$432$	0.766044	+	0.642788i	0.766044	+	0.642788i
$433$	1.53209	−	1.28558i	1.53209	−	1.28558i	0.766044	−	0.642788i	$-0.222222\pi$
$433$	1.53209	−	1.28558i	1.53209	−	1.28558i	0.766044	−	0.642788i	$-0.222222\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$		0			0
$438$	1.00000			1.00000
$439$		0			0		0.939693	−	0.342020i	$-0.111111\pi$
$439$		0			0		−0.939693	+	0.342020i	$0.888889\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−0.173648	−	0.984808i	−0.173648	−	0.984808i	−0.939693	−	0.342020i	$-0.888889\pi$
$443$	−0.173648	−	0.984808i	−0.173648	−	0.984808i	0.766044	−	0.642788i	$-0.222222\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$		0			0
$448$		0			0
$449$	0.500000	+	0.866025i	0.500000	+	0.866025i	1.00000			$0$
$449$	0.500000	+	0.866025i	0.500000	+	0.866025i	−0.500000	+	0.866025i	$0.666667\pi$
$450$		0			0
$451$	0.766044	+	0.642788i	0.766044	+	0.642788i
$452$	−0.766044	+	0.642788i	−0.766044	+	0.642788i
$453$		0			0
$454$	0.939693	−	0.342020i	0.939693	−	0.342020i
$455$		0			0
$456$		0			0
$457$	−1.00000			−1.00000			−0.500000	−	0.866025i	$-0.666667\pi$
$457$	−1.00000			−1.00000			−0.500000	+	0.866025i	$0.666667\pi$
$458$		0			0
$459$	0.347296	−	1.96962i	0.347296	−	1.96962i
$460$		0			0
$461$		0			0		−0.766044	−	0.642788i	$-0.777778\pi$
$461$		0			0		0.766044	+	0.642788i	$0.222222\pi$
$462$		0			0
$463$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$463$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$464$		0			0
$465$		0			0
$466$	0.939693	+	0.342020i	0.939693	+	0.342020i
$467$	0.500000	−	0.866025i	0.500000	−	0.866025i	−0.500000	−	0.866025i	$-0.666667\pi$
$467$	0.500000	−	0.866025i	0.500000	−	0.866025i	1.00000			$0$
$468$		0			0
$469$		0			0
$470$		0			0
$471$		0			0
$472$	−0.173648	+	0.984808i	−0.173648	+	0.984808i
$473$	1.87939	−	0.684040i	1.87939	−	0.684040i
$474$		0			0
$475$		0			0
$476$		0			0
$477$		0			0
$478$		0			0
$479$		0			0		0.766044	−	0.642788i	$-0.222222\pi$
$479$		0			0		−0.766044	+	0.642788i	$0.777778\pi$
$480$		0			0
$481$		0			0
$482$	0.500000	+	0.866025i	0.500000	+	0.866025i
$483$		0			0
$484$		0			0
$485$		0			0
$486$		0			0
$487$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$487$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$488$		0			0
$489$	0.766044	+	0.642788i	0.766044	+	0.642788i
$490$		0			0
$491$	0.347296	−	1.96962i	0.347296	−	1.96962i	0.173648	−	0.984808i	$-0.444444\pi$
$491$	0.347296	−	1.96962i	0.347296	−	1.96962i	0.173648	−	0.984808i	$-0.444444\pi$
$492$	−0.939693	+	0.342020i	−0.939693	+	0.342020i
$493$		0			0
$494$		0			0
$495$		0			0
$496$		0			0
$497$		0			0
$498$	0.766044	−	0.642788i	0.766044	−	0.642788i
$499$	−0.766044	−	0.642788i	−0.766044	−	0.642788i	0.173648	−	0.984808i	$-0.444444\pi$
$499$	−0.766044	−	0.642788i	−0.766044	−	0.642788i	−0.939693	+	0.342020i	$0.888889\pi$
$500$		0			0
$501$		0			0
$502$	0.500000	−	0.866025i	0.500000	−	0.866025i
$503$		0			0		−0.939693	−	0.342020i	$-0.888889\pi$
$503$		0			0		0.939693	+	0.342020i	$0.111111\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$	−0.173648	−	0.984808i	−0.173648	−	0.984808i
$508$		0			0
$509$		0			0		0.766044	−	0.642788i	$-0.222222\pi$
$509$		0			0		−0.766044	+	0.642788i	$0.777778\pi$
$510$		0			0
$511$		0			0
$512$	1.00000			1.00000
$513$		0			0
$514$	−1.00000			−1.00000
$515$		0			0
$516$	−0.347296	+	1.96962i	−0.347296	+	1.96962i
$517$		0			0
$518$		0			0
$519$		0			0
$520$		0			0
$521$	0.500000	−	0.866025i	0.500000	−	0.866025i	−0.500000	−	0.866025i	$-0.666667\pi$
$521$	0.500000	−	0.866025i	0.500000	−	0.866025i	1.00000			$0$
$522$		0			0
$523$	−1.87939	−	0.684040i	−1.87939	−	0.684040i	−0.939693	−	0.342020i	$-0.888889\pi$
$523$	−1.87939	−	0.684040i	−1.87939	−	0.684040i	−0.939693	−	0.342020i	$-0.888889\pi$
$524$	0.500000	−	0.866025i	0.500000	−	0.866025i
$525$		0			0
$526$		0			0
$527$		0			0
$528$	0.766044	−	0.642788i	0.766044	−	0.642788i
$529$	0.173648	−	0.984808i	0.173648	−	0.984808i
$530$		0			0
$531$		0			0
$532$		0			0
$533$		0			0
$534$	1.87939	−	0.684040i	1.87939	−	0.684040i
$535$		0			0
$536$	−0.766044	+	0.642788i	−0.766044	+	0.642788i
$537$	0.766044	+	0.642788i	0.766044	+	0.642788i
$538$		0			0
$539$	0.500000	+	0.866025i	0.500000	+	0.866025i
$540$		0			0
$541$		0			0		−0.939693	−	0.342020i	$-0.888889\pi$
$541$		0			0		0.939693	+	0.342020i	$0.111111\pi$
$542$		0			0
$543$		0			0
$544$	−1.00000	−	1.73205i	−1.00000	−	1.73205i
$545$		0			0
$546$		0			0
$547$	1.53209	−	1.28558i	1.53209	−	1.28558i	0.766044	−	0.642788i	$-0.222222\pi$
$547$	1.53209	−	1.28558i	1.53209	−	1.28558i	0.766044	−	0.642788i	$-0.222222\pi$
$548$	−0.173648	+	0.984808i	−0.173648	+	0.984808i
$549$		0			0
$550$	−1.00000			−1.00000
$551$		0			0
$552$		0			0
$553$		0			0
$554$		0			0
$555$		0			0
$556$	−0.766044	−	0.642788i	−0.766044	−	0.642788i
$557$		0			0		−0.173648	−	0.984808i	$-0.555556\pi$
$557$		0			0		0.173648	+	0.984808i	$0.444444\pi$
$558$		0			0
$559$		0			0
$560$		0			0
$561$	−1.87939	−	0.684040i	−1.87939	−	0.684040i
$562$	0.500000	−	0.866025i	0.500000	−	0.866025i
$563$	0.500000	+	0.866025i	0.500000	+	0.866025i	1.00000			$0$
$563$	0.500000	+	0.866025i	0.500000	+	0.866025i	−0.500000	+	0.866025i	$0.666667\pi$
$564$		0			0
$565$		0			0
$566$	−0.766044	+	0.642788i	−0.766044	+	0.642788i
$567$		0			0
$568$		0			0
$569$	2.00000			2.00000			1.00000			$0$
$569$	2.00000			2.00000			1.00000			$0$
$570$		0			0
$571$	−1.00000			−1.00000			−0.500000	−	0.866025i	$-0.666667\pi$
$571$	−1.00000			−1.00000			−0.500000	+	0.866025i	$0.666667\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$		0			0
$577$	0.500000	+	0.866025i	0.500000	+	0.866025i	1.00000			$0$
$577$	0.500000	+	0.866025i	0.500000	+	0.866025i	−0.500000	+	0.866025i	$0.666667\pi$
$578$	−1.50000	+	2.59808i	−1.50000	+	2.59808i
$579$	1.87939	+	0.684040i	1.87939	+	0.684040i
$580$		0			0
$581$		0			0
$582$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$583$		0			0
$584$	−0.766044	−	0.642788i	−0.766044	−	0.642788i
$585$		0			0
$586$		0			0
$587$	−1.87939	+	0.684040i	−1.87939	+	0.684040i	−0.939693	+	0.342020i	$0.888889\pi$
$587$	−1.87939	+	0.684040i	−1.87939	+	0.684040i	−0.939693	+	0.342020i	$0.888889\pi$
$588$	−1.00000			−1.00000
$589$		0			0
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−0.766044	+	0.642788i	−0.766044	+	0.642788i	−0.939693	−	0.342020i	$-0.888889\pi$
$593$	−0.766044	+	0.642788i	−0.766044	+	0.642788i	0.173648	+	0.984808i	$0.444444\pi$
$594$	−0.766044	−	0.642788i	−0.766044	−	0.642788i
$595$		0			0
$596$		0			0
$597$		0			0
$598$		0			0
$599$		0			0		−0.939693	−	0.342020i	$-0.888889\pi$
$599$		0			0		0.939693	+	0.342020i	$0.111111\pi$
$600$	0.500000	−	0.866025i	0.500000	−	0.866025i
$601$	0.500000	+	0.866025i	0.500000	+	0.866025i	1.00000			$0$
$601$	0.500000	+	0.866025i	0.500000	+	0.866025i	−0.500000	+	0.866025i	$0.666667\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$		0			0
$606$		0			0
$607$		0			0		1.00000			$0$
$607$		0			0		−1.00000			$\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		0			0
$612$		0			0
$613$		0			0		−0.766044	−	0.642788i	$-0.777778\pi$
$613$		0			0		0.766044	+	0.642788i	$0.222222\pi$
$614$	−0.173648	−	0.984808i	−0.173648	−	0.984808i
$615$		0			0
$616$		0			0
$617$	0.939693	+	0.342020i	0.939693	+	0.342020i	0.766044	−	0.642788i	$-0.222222\pi$
$617$	0.939693	+	0.342020i	0.939693	+	0.342020i	0.173648	+	0.984808i	$0.444444\pi$
$618$		0			0
$619$	−1.00000	+	1.73205i	−1.00000	+	1.73205i	−0.500000	+	0.866025i	$0.666667\pi$
$619$	−1.00000	+	1.73205i	−1.00000	+	1.73205i	−0.500000	+	0.866025i	$0.666667\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$		0			0
$625$	−0.939693	+	0.342020i	−0.939693	+	0.342020i
$626$	−1.00000			−1.00000
$627$		0			0
$628$		0			0
$629$		0			0
$630$		0			0
$631$		0			0		0.766044	−	0.642788i	$-0.222222\pi$
$631$		0			0		−0.766044	+	0.642788i	$0.777778\pi$
$632$		0			0
$633$	−0.347296	−	1.96962i	−0.347296	−	1.96962i
$634$		0			0
$635$		0			0
$636$		0			0
$637$		0			0
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−0.766044	−	0.642788i	−0.766044	−	0.642788i	0.173648	−	0.984808i	$-0.444444\pi$
$641$	−0.766044	−	0.642788i	−0.766044	−	0.642788i	−0.939693	+	0.342020i	$0.888889\pi$
$642$	−1.53209	+	1.28558i	−1.53209	+	1.28558i
$643$	−0.173648	+	0.984808i	−0.173648	+	0.984808i	0.766044	+	0.642788i	$0.222222\pi$
$643$	−0.173648	+	0.984808i	−0.173648	+	0.984808i	−0.939693	+	0.342020i	$0.888889\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		0			0		1.00000			$0$
$647$		0			0		−1.00000			$\pi$
$648$	0.939693	−	0.342020i	0.939693	−	0.342020i
$649$	0.173648	−	0.984808i	0.173648	−	0.984808i
$650$		0			0
$651$		0			0
$652$	−0.173648	−	0.984808i	−0.173648	−	0.984808i
$653$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$653$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$654$		0			0
$655$		0			0
$656$	0.939693	+	0.342020i	0.939693	+	0.342020i
$657$		0			0
$658$		0			0
$659$	0.347296	+	1.96962i	0.347296	+	1.96962i	0.173648	+	0.984808i	$0.444444\pi$
$659$	0.347296	+	1.96962i	0.347296	+	1.96962i	0.173648	+	0.984808i	$0.444444\pi$
$660$		0			0
$661$		0			0		0.766044	−	0.642788i	$-0.222222\pi$
$661$		0			0		−0.766044	+	0.642788i	$0.777778\pi$
$662$	−0.173648	+	0.984808i	−0.173648	+	0.984808i
$663$		0			0
$664$	−1.00000			−1.00000
$665$		0			0
$666$		0			0
$667$		0			0
$668$		0			0
$669$		0			0
$670$		0			0
$671$		0			0
$672$		0			0
$673$	−1.00000	+	1.73205i	−1.00000	+	1.73205i	−0.500000	+	0.866025i	$0.666667\pi$
$673$	−1.00000	+	1.73205i	−1.00000	+	1.73205i	−0.500000	+	0.866025i	$0.666667\pi$
$674$	0.939693	+	0.342020i	0.939693	+	0.342020i
$675$	−0.939693	−	0.342020i	−0.939693	−	0.342020i
$676$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$677$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$677$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$678$	0.173648	+	0.984808i	0.173648	+	0.984808i
$679$		0			0
$680$		0			0
$681$	0.173648	−	0.984808i	0.173648	−	0.984808i
$682$		0			0
$683$	2.00000			2.00000			1.00000			$0$
$683$	2.00000			2.00000			1.00000			$0$
$684$		0			0
$685$		0			0
$686$		0			0
$687$		0			0
$688$	1.53209	−	1.28558i	1.53209	−	1.28558i
$689$		0			0
$690$		0			0
$691$	−1.00000	−	1.73205i	−1.00000	−	1.73205i	−0.500000	−	0.866025i	$-0.666667\pi$
$691$	−1.00000	−	1.73205i	−1.00000	−	1.73205i	−0.500000	−	0.866025i	$-0.666667\pi$
$692$		0			0
$693$		0			0
$694$	0.939693	+	0.342020i	0.939693	+	0.342020i
$695$		0			0
$696$		0			0
$697$	−0.347296	−	1.96962i	−0.347296	−	1.96962i
$698$		0			0
$699$	0.766044	−	0.642788i	0.766044	−	0.642788i
$700$		0			0
$701$		0			0		0.939693	−	0.342020i	$-0.111111\pi$
$701$		0			0		−0.939693	+	0.342020i	$0.888889\pi$
$702$		0			0
$703$		0			0
$704$	−1.00000			−1.00000
$705$		0			0
$706$	−0.173648	+	0.984808i	−0.173648	+	0.984808i
$707$		0			0
$708$	0.766044	+	0.642788i	0.766044	+	0.642788i
$709$		0			0		−0.173648	−	0.984808i	$-0.555556\pi$
$709$		0			0		0.173648	+	0.984808i	$0.444444\pi$
$710$		0			0
$711$		0			0
$712$	−1.87939	−	0.684040i	−1.87939	−	0.684040i
$713$		0			0
$714$		0			0
$715$		0			0
$716$	−0.173648	−	0.984808i	−0.173648	−	0.984808i
$717$		0			0
$718$		0			0
$719$		0			0		0.173648	−	0.984808i	$-0.444444\pi$
$719$		0			0		−0.173648	+	0.984808i	$0.555556\pi$
$720$		0			0
$721$		0			0
$722$		0			0
$723$	1.00000			1.00000
$724$		0			0
$725$		0			0
$726$		0			0
$727$		0			0		−0.766044	−	0.642788i	$-0.777778\pi$
$727$		0			0		0.766044	+	0.642788i	$0.222222\pi$
$728$		0			0
$729$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$730$		0			0
$731$	−3.75877	−	1.36808i	−3.75877	−	1.36808i
$732$		0			0
$733$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$733$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	0.766044	−	0.642788i	0.766044	−	0.642788i
$738$		0			0
$739$	0.939693	−	0.342020i	0.939693	−	0.342020i	0.173648	−	0.984808i	$-0.444444\pi$
$739$	0.939693	−	0.342020i	0.939693	−	0.342020i	0.766044	+	0.642788i	$0.222222\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$		0			0		0.939693	−	0.342020i	$-0.111111\pi$
$743$		0			0		−0.939693	+	0.342020i	$0.888889\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$		0			0
$748$	1.00000	+	1.73205i	1.00000	+	1.73205i
$749$		0			0
$750$		0			0
$751$		0			0		−0.939693	−	0.342020i	$-0.888889\pi$
$751$		0			0		0.939693	+	0.342020i	$0.111111\pi$
$752$		0			0
$753$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$754$		0			0
$755$		0			0
$756$		0			0
$757$		0			0		0.173648	−	0.984808i	$-0.444444\pi$
$757$		0			0		−0.173648	+	0.984808i	$0.555556\pi$
$758$	−1.87939	+	0.684040i	−1.87939	+	0.684040i
$759$		0			0
$760$		0			0
$761$	−1.00000			−1.00000			−0.500000	−	0.866025i	$-0.666667\pi$
$761$	−1.00000			−1.00000			−0.500000	+	0.866025i	$0.666667\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$		0			0
$766$		0			0
$767$		0			0
$768$	0.500000	−	0.866025i	0.500000	−	0.866025i
$769$	−1.87939	−	0.684040i	−1.87939	−	0.684040i	−0.939693	−	0.342020i	$-0.888889\pi$
$769$	−1.87939	−	0.684040i	−1.87939	−	0.684040i	−0.939693	−	0.342020i	$-0.888889\pi$
$770$		0			0
$771$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$772$	−1.00000	−	1.73205i	−1.00000	−	1.73205i
$773$		0			0		−0.173648	−	0.984808i	$-0.555556\pi$
$773$		0			0		0.173648	+	0.984808i	$0.444444\pi$
$774$		0			0
$775$		0			0
$776$	−0.173648	+	0.984808i	−0.173648	+	0.984808i
$777$		0			0
$778$		0			0
$779$		0			0
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$	0.766044	+	0.642788i	0.766044	+	0.642788i
$785$		0			0
$786$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$787$	0.500000	−	0.866025i	0.500000	−	0.866025i	−0.500000	−	0.866025i	$-0.666667\pi$
$787$	0.500000	−	0.866025i	0.500000	−	0.866025i	1.00000			$0$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$		0			0
$794$		0			0
$795$		0			0
$796$		0			0
$797$		0			0		1.00000			$0$
$797$		0			0		−1.00000			$\pi$
$798$		0			0
$799$		0			0
$800$	−0.939693	+	0.342020i	−0.939693	+	0.342020i
$801$		0			0
$802$	−0.766044	+	0.642788i	−0.766044	+	0.642788i
$803$	0.766044	+	0.642788i	0.766044	+	0.642788i
$804$	0.173648	+	0.984808i	0.173648	+	0.984808i
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$	0.500000	−	0.866025i	0.500000	−	0.866025i	−0.500000	−	0.866025i	$-0.666667\pi$
$809$	0.500000	−	0.866025i	0.500000	−	0.866025i	1.00000			$0$
$810$		0			0
$811$	0.347296	+	1.96962i	0.347296	+	1.96962i	0.173648	+	0.984808i	$0.444444\pi$
$811$	0.347296	+	1.96962i	0.347296	+	1.96962i	0.173648	+	0.984808i	$0.444444\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$		0			0
$816$	−2.00000			−2.00000
$817$		0			0
$818$	−1.00000			−1.00000
$819$		0			0
$820$		0			0
$821$		0			0		0.766044	−	0.642788i	$-0.222222\pi$
$821$		0			0		−0.766044	+	0.642788i	$0.777778\pi$
$822$	0.766044	+	0.642788i	0.766044	+	0.642788i
$823$		0			0		−0.173648	−	0.984808i	$-0.555556\pi$
$823$		0			0		0.173648	+	0.984808i	$0.444444\pi$
$824$		0			0
$825$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$826$		0			0
$827$	0.939693	+	0.342020i	0.939693	+	0.342020i	0.766044	−	0.642788i	$-0.222222\pi$
$827$	0.939693	+	0.342020i	0.939693	+	0.342020i	0.173648	+	0.984808i	$0.444444\pi$
$828$		0			0
$829$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$829$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	0.347296	−	1.96962i	0.347296	−	1.96962i
$834$	−0.939693	+	0.342020i	−0.939693	+	0.342020i
$835$		0			0
$836$		0			0
$837$		0			0
$838$	−1.87939	+	0.684040i	−1.87939	+	0.684040i
$839$		0			0		0.173648	−	0.984808i	$-0.444444\pi$
$839$		0			0		−0.173648	+	0.984808i	$0.555556\pi$
$840$		0			0
$841$	0.766044	+	0.642788i	0.766044	+	0.642788i
$842$		0			0
$843$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$844$	−1.00000	+	1.73205i	−1.00000	+	1.73205i
$845$		0			0
$846$		0			0
$847$		0			0
$848$		0			0
$849$	0.173648	+	0.984808i	0.173648	+	0.984808i
$850$	1.53209	+	1.28558i	1.53209	+	1.28558i
$851$		0			0
$852$		0			0
$853$		0			0		0.939693	−	0.342020i	$-0.111111\pi$
$853$		0			0		−0.939693	+	0.342020i	$0.888889\pi$
$854$		0			0
$855$		0			0
$856$	2.00000			2.00000
$857$	0.939693	−	0.342020i	0.939693	−	0.342020i	0.173648	−	0.984808i	$-0.444444\pi$
$857$	0.939693	−	0.342020i	0.939693	−	0.342020i	0.766044	+	0.642788i	$0.222222\pi$
$858$		0			0
$859$	−0.766044	+	0.642788i	−0.766044	+	0.642788i	−0.939693	−	0.342020i	$-0.888889\pi$
$859$	−0.766044	+	0.642788i	−0.766044	+	0.642788i	0.173648	+	0.984808i	$0.444444\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$863$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$864$	−0.939693	−	0.342020i	−0.939693	−	0.342020i
$865$		0			0
$866$	−1.00000	+	1.73205i	−1.00000	+	1.73205i
$867$	1.50000	+	2.59808i	1.50000	+	2.59808i
$868$		0			0
$869$		0			0
$870$		0			0
$871$		0			0
$872$		0			0
$873$		0			0
$874$		0			0
$875$		0			0
$876$	−0.939693	+	0.342020i	−0.939693	+	0.342020i
$877$		0			0		0.173648	−	0.984808i	$-0.444444\pi$
$877$		0			0		−0.173648	+	0.984808i	$0.555556\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	0.500000	+	0.866025i	0.500000	+	0.866025i	1.00000			$0$
$881$	0.500000	+	0.866025i	0.500000	+	0.866025i	−0.500000	+	0.866025i	$0.666667\pi$
$882$		0			0
$883$	0.939693	+	0.342020i	0.939693	+	0.342020i	0.766044	−	0.642788i	$-0.222222\pi$
$883$	0.939693	+	0.342020i	0.939693	+	0.342020i	0.173648	+	0.984808i	$0.444444\pi$
$884$		0			0
$885$		0			0
$886$	0.500000	+	0.866025i	0.500000	+	0.866025i
$887$		0			0		−0.173648	−	0.984808i	$-0.555556\pi$
$887$		0			0		0.173648	+	0.984808i	$0.444444\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$	−0.939693	+	0.342020i	−0.939693	+	0.342020i
$892$		0			0
$893$		0			0
$894$		0			0
$895$		0			0
$896$		0			0
$897$		0			0
$898$	−0.766044	−	0.642788i	−0.766044	−	0.642788i
$899$		0			0
$900$		0			0
$901$		0			0
$902$	−0.939693	−	0.342020i	−0.939693	−	0.342020i
$903$		0			0
$904$	0.500000	−	0.866025i	0.500000	−	0.866025i
$905$		0			0
$906$		0			0
$907$	−0.766044	−	0.642788i	−0.766044	−	0.642788i	0.173648	−	0.984808i	$-0.444444\pi$
$907$	−0.766044	−	0.642788i	−0.766044	−	0.642788i	−0.939693	+	0.342020i	$0.888889\pi$
$908$	−0.766044	+	0.642788i	−0.766044	+	0.642788i
$909$		0			0
$910$		0			0
$911$		0			0		1.00000			$0$
$911$		0			0		−1.00000			$\pi$
$912$		0			0
$913$	1.00000			1.00000
$914$	0.939693	−	0.342020i	0.939693	−	0.342020i
$915$		0			0
$916$		0			0
$917$		0			0
$918$	0.347296	+	1.96962i	0.347296	+	1.96962i
$919$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$919$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$920$		0			0
$921$	−0.939693	−	0.342020i	−0.939693	−	0.342020i
$922$		0			0
$923$		0			0
$924$		0			0
$925$		0			0
$926$		0			0
$927$		0			0
$928$		0			0
$929$	0.939693	−	0.342020i	0.939693	−	0.342020i	0.173648	−	0.984808i	$-0.444444\pi$
$929$	0.939693	−	0.342020i	0.939693	−	0.342020i	0.766044	+	0.642788i	$0.222222\pi$
$930$		0			0
$931$		0			0
$932$	−1.00000			−1.00000
$933$		0			0
$934$	−0.173648	+	0.984808i	−0.173648	+	0.984808i
$935$		0			0
$936$		0			0
$937$	−0.173648	−	0.984808i	−0.173648	−	0.984808i	−0.939693	−	0.342020i	$-0.888889\pi$
$937$	−0.173648	−	0.984808i	−0.173648	−	0.984808i	0.766044	−	0.642788i	$-0.222222\pi$
$938$		0			0
$939$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$940$		0			0
$941$		0			0		−0.939693	−	0.342020i	$-0.888889\pi$
$941$		0			0		0.939693	+	0.342020i	$0.111111\pi$
$942$		0			0
$943$		0			0
$944$	−0.173648	−	0.984808i	−0.173648	−	0.984808i
$945$		0			0
$946$	−1.53209	+	1.28558i	−1.53209	+	1.28558i
$947$	0.347296	−	1.96962i	0.347296	−	1.96962i	0.173648	−	0.984808i	$-0.444444\pi$
$947$	0.347296	−	1.96962i	0.347296	−	1.96962i	0.173648	−	0.984808i	$-0.444444\pi$
$948$		0			0
$949$		0			0
$950$		0			0
$951$		0			0
$952$		0			0
$953$	−0.173648	+	0.984808i	−0.173648	+	0.984808i	0.766044	+	0.642788i	$0.222222\pi$
$953$	−0.173648	+	0.984808i	−0.173648	+	0.984808i	−0.939693	+	0.342020i	$0.888889\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$962$		0			0
$963$		0			0
$964$	−0.766044	−	0.642788i	−0.766044	−	0.642788i
$965$		0			0
$966$		0			0
$967$		0			0		0.939693	−	0.342020i	$-0.111111\pi$
$967$		0			0		−0.939693	+	0.342020i	$0.888889\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	0.939693	−	0.342020i	0.939693	−	0.342020i	0.173648	−	0.984808i	$-0.444444\pi$
$971$	0.939693	−	0.342020i	0.939693	−	0.342020i	0.766044	+	0.642788i	$0.222222\pi$
$972$		0			0
$973$		0			0
$974$		0			0
$975$		0			0
$976$		0			0
$977$	0.500000	−	0.866025i	0.500000	−	0.866025i	−0.500000	−	0.866025i	$-0.666667\pi$
$977$	0.500000	−	0.866025i	0.500000	−	0.866025i	1.00000			$0$
$978$	−0.939693	−	0.342020i	−0.939693	−	0.342020i
$979$	1.87939	+	0.684040i	1.87939	+	0.684040i
$980$		0			0
$981$		0			0
$982$	0.347296	+	1.96962i	0.347296	+	1.96962i
$983$		0			0		−0.766044	−	0.642788i	$-0.777778\pi$
$983$		0			0		0.766044	+	0.642788i	$0.222222\pi$
$984$	0.766044	−	0.642788i	0.766044	−	0.642788i
$985$		0			0
$986$		0			0
$987$		0			0
$988$		0			0
$989$		0			0
$990$		0			0
$991$		0			0		0.173648	−	0.984808i	$-0.444444\pi$
$991$		0			0		−0.173648	+	0.984808i	$0.555556\pi$
$992$		0			0
$993$	0.766044	+	0.642788i	0.766044	+	0.642788i
$994$		0			0
$995$		0			0
$996$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$997$		0			0		−0.939693	−	0.342020i	$-0.888889\pi$
$997$		0			0		0.939693	+	0.342020i	$0.111111\pi$
$998$	0.939693	+	0.342020i	0.939693	+	0.342020i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2888.1.u.c.1867.1		6
8.3	odd	2	CM	2888.1.u.c.1867.1		6
19.2	odd	18		2888.1.f.a.723.1		1
19.3	odd	18		2888.1.k.a.2819.1		2
19.4	even	9	inner	2888.1.u.c.99.1		6
19.5	even	9		152.1.k.a.11.1	✓	2
19.6	even	9	inner	2888.1.u.c.2555.1		6
19.7	even	3	inner	2888.1.u.c.2411.1		6
19.8	odd	6		2888.1.u.d.1859.1		6
19.9	even	9	inner	2888.1.u.c.595.1		6
19.10	odd	18		2888.1.u.d.595.1		6
19.11	even	3	inner	2888.1.u.c.1859.1		6
19.12	odd	6		2888.1.u.d.2411.1		6
19.13	odd	18		2888.1.u.d.2555.1		6
19.14	odd	18		2888.1.k.a.2595.1		2
19.15	odd	18		2888.1.u.d.99.1		6
19.16	even	9		152.1.k.a.83.1	yes	2
19.17	even	9		2888.1.f.b.723.1		1
19.18	odd	2		2888.1.u.d.1867.1		6
57.5	odd	18		1368.1.bz.a.163.1		2
57.35	odd	18		1368.1.bz.a.235.1		2
76.35	odd	18		608.1.o.a.463.1		2
76.43	odd	18		608.1.o.a.239.1		2
95.24	even	18		3800.1.bd.c.3051.1		2
95.43	odd	36		3800.1.bn.b.2899.2		4
95.54	even	18		3800.1.bd.c.1451.1		2
95.62	odd	36		3800.1.bn.b.2899.1		4
95.73	odd	36		3800.1.bn.b.1299.1		4
95.92	odd	36		3800.1.bn.b.1299.2		4
152.3	even	18		2888.1.k.a.2819.1		2
152.5	even	18		608.1.o.a.239.1		2
152.11	odd	6	inner	2888.1.u.c.1859.1		6
152.27	even	6		2888.1.u.d.1859.1		6
152.35	odd	18		152.1.k.a.83.1	yes	2
152.43	odd	18		152.1.k.a.11.1	✓	2
152.51	even	18		2888.1.u.d.2555.1		6
152.59	even	18		2888.1.f.a.723.1		1
152.67	even	18		2888.1.u.d.595.1		6
152.75	even	2		2888.1.u.d.1867.1		6
152.83	odd	6	inner	2888.1.u.c.2411.1		6
152.91	even	18		2888.1.u.d.99.1		6
152.99	odd	18	inner	2888.1.u.c.99.1		6
152.107	even	6		2888.1.u.d.2411.1		6
152.123	odd	18	inner	2888.1.u.c.595.1		6
152.131	odd	18		2888.1.f.b.723.1		1
152.139	odd	18	inner	2888.1.u.c.2555.1		6
152.147	even	18		2888.1.k.a.2595.1		2
152.149	even	18		608.1.o.a.463.1		2
456.35	even	18		1368.1.bz.a.235.1		2
456.347	even	18		1368.1.bz.a.163.1		2
760.43	even	36		3800.1.bn.b.2899.2		4
760.187	even	36		3800.1.bn.b.1299.2		4
760.339	odd	18		3800.1.bd.c.1451.1		2
760.347	even	36		3800.1.bn.b.2899.1		4
760.499	odd	18		3800.1.bd.c.3051.1		2
760.643	even	36		3800.1.bn.b.1299.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
152.1.k.a.11.1	✓	2	19.5	even	9
152.1.k.a.11.1	✓	2	152.43	odd	18
152.1.k.a.83.1	yes	2	19.16	even	9
152.1.k.a.83.1	yes	2	152.35	odd	18
608.1.o.a.239.1		2	76.43	odd	18
608.1.o.a.239.1		2	152.5	even	18
608.1.o.a.463.1		2	76.35	odd	18
608.1.o.a.463.1		2	152.149	even	18
1368.1.bz.a.163.1		2	57.5	odd	18
1368.1.bz.a.163.1		2	456.347	even	18
1368.1.bz.a.235.1		2	57.35	odd	18
1368.1.bz.a.235.1		2	456.35	even	18
2888.1.f.a.723.1		1	19.2	odd	18
2888.1.f.a.723.1		1	152.59	even	18
2888.1.f.b.723.1		1	19.17	even	9
2888.1.f.b.723.1		1	152.131	odd	18
2888.1.k.a.2595.1		2	19.14	odd	18
2888.1.k.a.2595.1		2	152.147	even	18
2888.1.k.a.2819.1		2	19.3	odd	18
2888.1.k.a.2819.1		2	152.3	even	18
2888.1.u.c.99.1		6	19.4	even	9	inner
2888.1.u.c.99.1		6	152.99	odd	18	inner
2888.1.u.c.595.1		6	19.9	even	9	inner
2888.1.u.c.595.1		6	152.123	odd	18	inner
2888.1.u.c.1859.1		6	19.11	even	3	inner
2888.1.u.c.1859.1		6	152.11	odd	6	inner
2888.1.u.c.1867.1		6	1.1	even	1	trivial
2888.1.u.c.1867.1		6	8.3	odd	2	CM
2888.1.u.c.2411.1		6	19.7	even	3	inner
2888.1.u.c.2411.1		6	152.83	odd	6	inner
2888.1.u.c.2555.1		6	19.6	even	9	inner
2888.1.u.c.2555.1		6	152.139	odd	18	inner
2888.1.u.d.99.1		6	19.15	odd	18
2888.1.u.d.99.1		6	152.91	even	18
2888.1.u.d.595.1		6	19.10	odd	18
2888.1.u.d.595.1		6	152.67	even	18
2888.1.u.d.1859.1		6	19.8	odd	6
2888.1.u.d.1859.1		6	152.27	even	6
2888.1.u.d.1867.1		6	19.18	odd	2
2888.1.u.d.1867.1		6	152.75	even	2
2888.1.u.d.2411.1		6	19.12	odd	6
2888.1.u.d.2411.1		6	152.107	even	6
2888.1.u.d.2555.1		6	19.13	odd	18
2888.1.u.d.2555.1		6	152.51	even	18
3800.1.bd.c.1451.1		2	95.54	even	18
3800.1.bd.c.1451.1		2	760.339	odd	18
3800.1.bd.c.3051.1		2	95.24	even	18
3800.1.bd.c.3051.1		2	760.499	odd	18
3800.1.bn.b.1299.1		4	95.73	odd	36
3800.1.bn.b.1299.1		4	760.643	even	36
3800.1.bn.b.1299.2		4	95.92	odd	36
3800.1.bn.b.1299.2		4	760.187	even	36
3800.1.bn.b.2899.1		4	95.62	odd	36
3800.1.bn.b.2899.1		4	760.347	even	36
3800.1.bn.b.2899.2		4	95.43	odd	36
3800.1.bn.b.2899.2		4	760.43	even	36

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:	\( N \)	\(=\)	\( 2888 = 2^{3} \cdot 19^{2} \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	2888.u (of order \(18\), degree \(6\), not minimal)

\(f(q)\)	\(=\)	\(q+(-0.939693 + 0.342020i) q^{2} +(-0.173648 + 0.984808i) q^{3} +(0.766044 - 0.642788i) q^{4} +(-0.173648 - 0.984808i) q^{6} +(-0.500000 + 0.866025i) q^{8} +O(q^{10})\)	\(q+(-0.939693 + 0.342020i) q^{2} +(-0.173648 + 0.984808i) q^{3} +(0.766044 - 0.642788i) q^{4} +(-0.173648 - 0.984808i) q^{6} +(-0.500000 + 0.866025i) q^{8} +(0.500000 - 0.866025i) q^{11} +(0.500000 + 0.866025i) q^{12} +(0.173648 - 0.984808i) q^{16} +(-1.87939 + 0.684040i) q^{17} +(-0.173648 + 0.984808i) q^{22} +(-0.766044 - 0.642788i) q^{24} +(0.173648 + 0.984808i) q^{25} +(-0.500000 + 0.866025i) q^{27} +(0.173648 + 0.984808i) q^{32} +(0.766044 + 0.642788i) q^{33} +(1.53209 - 1.28558i) q^{34} +(-0.173648 + 0.984808i) q^{41} +(1.53209 + 1.28558i) q^{43} +(-0.173648 - 0.984808i) q^{44} +(0.939693 + 0.342020i) q^{48} +(-0.500000 + 0.866025i) q^{49} +(-0.500000 - 0.866025i) q^{50} +(-0.347296 - 1.96962i) q^{51} +(0.173648 - 0.984808i) q^{54} +(0.939693 - 0.342020i) q^{59} +(-0.500000 - 0.866025i) q^{64} +(-0.939693 - 0.342020i) q^{66} +(0.939693 + 0.342020i) q^{67} +(-1.00000 + 1.73205i) q^{68} +(-0.173648 + 0.984808i) q^{73} -1.00000 q^{75} +(-0.766044 - 0.642788i) q^{81} +(-0.173648 - 0.984808i) q^{82} +(0.500000 + 0.866025i) q^{83} +(-1.87939 - 0.684040i) q^{86} +(0.500000 + 0.866025i) q^{88} +(0.347296 + 1.96962i) q^{89} -1.00000 q^{96} +(0.939693 - 0.342020i) q^{97} +(0.173648 - 0.984808i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 3 q^{8}+O(q^{10}) \) 6 * q - 3 * q^8	\( 6 q - 3 q^{8} + 3 q^{11} + 3 q^{12} - 3 q^{27} - 3 q^{49} - 3 q^{50} - 3 q^{64} - 6 q^{68} - 6 q^{75} + 3 q^{83} + 3 q^{88} - 6 q^{96}+O(q^{100}) \) 6 * q - 3 * q^8 + 3 * q^11 + 3 * q^12 - 3 * q^27 - 3 * q^49 - 3 * q^50 - 3 * q^64 - 6 * q^68 - 6 * q^75 + 3 * q^83 + 3 * q^88 - 6 * q^96

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