LMFDB - Embedded newform 1368.1.ee.a.1307.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1368,1,Mod(59,1368)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1368, base_ring=CyclotomicField(18))

chi = DirichletCharacter(H, H._module([9, 9, 15, 1]))

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

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

chi := DirichletCharacter("1368.59");

S:= CuspForms(chi, 1);

N := Newforms(S);

Level:	$ N $	$=$	$ 1368 = 2^{3} \cdot 3^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	1368.ee (of order $18$, degree $6$, minimal)

Newform invariants

comment: select newform

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

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

Self dual:	no
Analytic conductor:	$0.682720937282$
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:	yes
Projective image:	$D_{18}$
Projective field:	Galois closure of $\mathbb{Q}[x]/(x^{18} + \cdots)$

Embedding invariants

Embedding label			1307.1
Root			$-0.173648 - 0.984808i$ of defining polynomial
Character	$\chi$	$=$	1368.1307
Dual form			1368.1.ee.a.1211.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(0.766044 - 0.642788i) q^{2} -1.00000 q^{3} +(0.173648 - 0.984808i) q^{4} +(-0.766044 + 0.642788i) q^{6} +(-0.500000 - 0.866025i) q^{8} +1.00000 q^{9} +O(q^{10})$	\(q+(0.766044 - 0.642788i) q^{2} -1.00000 q^{3} +(0.173648 - 0.984808i) q^{4} +(-0.766044 + 0.642788i) q^{6} +(-0.500000 - 0.866025i) q^{8} +1.00000 q^{9} -0.684040i q^{11} +(-0.173648 + 0.984808i) q^{12} +(-0.939693 - 0.342020i) q^{16} +(0.592396 - 1.62760i) q^{17} +(0.766044 - 0.642788i) q^{18} +(-0.173648 + 0.984808i) q^{19} +(-0.439693 - 0.524005i) q^{22} +(0.500000 + 0.866025i) q^{24} +(-0.766044 - 0.642788i) q^{25} -1.00000 q^{27} +(-0.939693 + 0.342020i) q^{32} +0.684040i q^{33} +(-0.592396 - 1.62760i) q^{34} +(0.173648 - 0.984808i) q^{36} +(0.500000 + 0.866025i) q^{38} +(1.43969 - 1.20805i) q^{41} +(-0.347296 - 1.96962i) q^{43} +(-0.673648 - 0.118782i) q^{44} +(0.939693 + 0.342020i) q^{48} +(-0.500000 + 0.866025i) q^{49} -1.00000 q^{50} +(-0.592396 + 1.62760i) q^{51} +(-0.766044 + 0.642788i) q^{54} +(0.173648 - 0.984808i) q^{57} +(0.266044 + 1.50881i) q^{59} +(-0.500000 + 0.866025i) q^{64} +(0.439693 + 0.524005i) q^{66} +(-0.439693 + 0.524005i) q^{67} +(-1.50000 - 0.866025i) q^{68} +(-0.500000 - 0.866025i) q^{72} +(0.326352 + 1.85083i) q^{73} +(0.766044 + 0.642788i) q^{75} +(0.939693 + 0.342020i) q^{76} +1.00000 q^{81} +(0.326352 - 1.85083i) q^{82} +(1.11334 - 0.642788i) q^{83} +(-1.53209 - 1.28558i) q^{86} +(-0.592396 + 0.342020i) q^{88} +(-0.173648 + 0.984808i) q^{89} +(0.939693 - 0.342020i) q^{96} +(1.26604 + 1.50881i) q^{97} +(0.173648 + 0.984808i) q^{98} -0.684040i q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 6 q^{3} - 3 q^{8} + 6 q^{9}+O(q^{10}) $ 6 * q - 6 * q^3 - 3 * q^8 + 6 * q^9	$ 6 q - 6 q^{3} - 3 q^{8} + 6 q^{9} + 3 q^{22} + 3 q^{24} - 6 q^{27} + 3 q^{38} + 3 q^{41} - 3 q^{44} - 3 q^{49} - 6 q^{50} - 3 q^{59} - 3 q^{64} - 3 q^{66} + 3 q^{67} - 9 q^{68} - 3 q^{72} + 3 q^{73} + 6 q^{81} + 3 q^{82} + 3 q^{97}+O(q^{100}) $ 6 * q - 6 * q^3 - 3 * q^8 + 6 * q^9 + 3 * q^22 + 3 * q^24 - 6 * q^27 + 3 * q^38 + 3 * q^41 - 3 * q^44 - 3 * q^49 - 6 * q^50 - 3 * q^59 - 3 * q^64 - 3 * q^66 + 3 * q^67 - 9 * q^68 - 3 * q^72 + 3 * q^73 + 6 * q^81 + 3 * q^82 + 3 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1368\mathbb{Z}\right)^\times$.

$n$	$343$	$685$	$1009$	$1217$
$\chi(n)$	$-1$	$-1$	$e\left(\frac{11}{18}\right)$	$e\left(\frac{1}{6}\right)$

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 $

$2$	0.766044	−	0.642788i	0.766044	−	0.642788i
$2$	0.766044	−	0.642788i	0.766044	−	0.642788i
$3$	−1.00000			−1.00000
$3$	−1.00000			−1.00000
$4$	0.173648	−	0.984808i	0.173648	−	0.984808i
$5$		0			0		0.342020	−	0.939693i	$-0.388889\pi$
$5$		0			0		−0.342020	+	0.939693i	$0.611111\pi$
$6$	−0.766044	+	0.642788i	−0.766044	+	0.642788i
$7$		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
$9$	1.00000			1.00000
$10$		0			0
$11$		−	0.684040i		−	0.684040i	−0.939693	−	0.342020i	$-0.888889\pi$
$11$		−	0.684040i		−	0.684040i	0.939693	−	0.342020i	$-0.111111\pi$
$12$	−0.173648	+	0.984808i	−0.173648	+	0.984808i
$13$		0			0		−0.342020	−	0.939693i	$-0.611111\pi$
$13$		0			0		0.342020	+	0.939693i	$0.388889\pi$
$14$		0			0
$15$		0			0
$16$	−0.939693	−	0.342020i	−0.939693	−	0.342020i
$17$	0.592396	−	1.62760i	0.592396	−	1.62760i	−0.173648	−	0.984808i	$-0.555556\pi$
$17$	0.592396	−	1.62760i	0.592396	−	1.62760i	0.766044	−	0.642788i	$-0.222222\pi$
$18$	0.766044	−	0.642788i	0.766044	−	0.642788i
$19$	−0.173648	+	0.984808i	−0.173648	+	0.984808i
$19$	−0.173648	+	0.984808i	−0.173648	+	0.984808i
$20$		0			0
$21$		0			0
$22$	−0.439693	−	0.524005i	−0.439693	−	0.524005i
$23$		0			0		−0.984808	−	0.173648i	$-0.944444\pi$
$23$		0			0		0.984808	+	0.173648i	$0.0555556\pi$
$24$	0.500000	+	0.866025i	0.500000	+	0.866025i
$25$	−0.766044	−	0.642788i	−0.766044	−	0.642788i
$26$		0			0
$27$	−1.00000			−1.00000
$28$		0			0
$29$		0			0		0.173648	−	0.984808i	$-0.444444\pi$
$29$		0			0		−0.173648	+	0.984808i	$0.555556\pi$
$30$		0			0
$31$		0			0			−	1.00000i	$-0.5\pi$
$31$		0			0				1.00000i	$0.5\pi$
$32$	−0.939693	+	0.342020i	−0.939693	+	0.342020i
$33$			0.684040i			0.684040i
$34$	−0.592396	−	1.62760i	−0.592396	−	1.62760i
$35$		0			0
$36$	0.173648	−	0.984808i	0.173648	−	0.984808i
$37$		0			0			−	1.00000i	$-0.5\pi$
$37$		0			0				1.00000i	$0.5\pi$
$38$	0.500000	+	0.866025i	0.500000	+	0.866025i
$39$		0			0
$40$		0			0
$41$	1.43969	−	1.20805i	1.43969	−	1.20805i	0.500000	−	0.866025i	$-0.333333\pi$
$41$	1.43969	−	1.20805i	1.43969	−	1.20805i	0.939693	−	0.342020i	$-0.111111\pi$
$42$		0			0
$43$	−0.347296	−	1.96962i	−0.347296	−	1.96962i	−0.173648	−	0.984808i	$-0.555556\pi$
$43$	−0.347296	−	1.96962i	−0.347296	−	1.96962i	−0.173648	−	0.984808i	$-0.555556\pi$
$44$	−0.673648	−	0.118782i	−0.673648	−	0.118782i
$45$		0			0
$46$		0			0
$47$		0			0		−0.984808	−	0.173648i	$-0.944444\pi$
$47$		0			0		0.984808	+	0.173648i	$0.0555556\pi$
$48$	0.939693	+	0.342020i	0.939693	+	0.342020i
$49$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$50$	−1.00000			−1.00000
$51$	−0.592396	+	1.62760i	−0.592396	+	1.62760i
$52$		0			0
$53$		0			0		−0.766044	−	0.642788i	$-0.777778\pi$
$53$		0			0		0.766044	+	0.642788i	$0.222222\pi$
$54$	−0.766044	+	0.642788i	−0.766044	+	0.642788i
$55$		0			0
$56$		0			0
$57$	0.173648	−	0.984808i	0.173648	−	0.984808i
$58$		0			0
$59$	0.266044	+	1.50881i	0.266044	+	1.50881i	0.766044	+	0.642788i	$0.222222\pi$
$59$	0.266044	+	1.50881i	0.266044	+	1.50881i	−0.500000	+	0.866025i	$0.666667\pi$
$60$		0			0
$61$		0			0		0.939693	−	0.342020i	$-0.111111\pi$
$61$		0			0		−0.939693	+	0.342020i	$0.888889\pi$
$62$		0			0
$63$		0			0
$64$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$65$		0			0
$66$	0.439693	+	0.524005i	0.439693	+	0.524005i
$67$	−0.439693	+	0.524005i	−0.439693	+	0.524005i	−0.939693	−	0.342020i	$-0.888889\pi$
$67$	−0.439693	+	0.524005i	−0.439693	+	0.524005i	0.500000	+	0.866025i	$0.333333\pi$
$68$	−1.50000	−	0.866025i	−1.50000	−	0.866025i
$69$		0			0
$70$		0			0
$71$		0			0		0.766044	−	0.642788i	$-0.222222\pi$
$71$		0			0		−0.766044	+	0.642788i	$0.777778\pi$
$72$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$73$	0.326352	+	1.85083i	0.326352	+	1.85083i	0.500000	+	0.866025i	$0.333333\pi$
$73$	0.326352	+	1.85083i	0.326352	+	1.85083i	−0.173648	+	0.984808i	$0.555556\pi$
$74$		0			0
$75$	0.766044	+	0.642788i	0.766044	+	0.642788i
$76$	0.939693	+	0.342020i	0.939693	+	0.342020i
$77$		0			0
$78$		0			0
$79$		0			0		0.342020	−	0.939693i	$-0.388889\pi$
$79$		0			0		−0.342020	+	0.939693i	$0.611111\pi$
$80$		0			0
$81$	1.00000			1.00000
$82$	0.326352	−	1.85083i	0.326352	−	1.85083i
$83$	1.11334	−	0.642788i	1.11334	−	0.642788i	0.173648	−	0.984808i	$-0.444444\pi$
$83$	1.11334	−	0.642788i	1.11334	−	0.642788i	0.939693	+	0.342020i	$0.111111\pi$
$84$		0			0
$85$		0			0
$86$	−1.53209	−	1.28558i	−1.53209	−	1.28558i
$87$		0			0
$88$	−0.592396	+	0.342020i	−0.592396	+	0.342020i
$89$	−0.173648	+	0.984808i	−0.173648	+	0.984808i	0.766044	+	0.642788i	$0.222222\pi$
$89$	−0.173648	+	0.984808i	−0.173648	+	0.984808i	−0.939693	+	0.342020i	$0.888889\pi$
$90$		0			0
$91$		0			0
$92$		0			0
$93$		0			0
$94$		0			0
$95$		0			0
$96$	0.939693	−	0.342020i	0.939693	−	0.342020i
$97$	1.26604	+	1.50881i	1.26604	+	1.50881i	0.766044	+	0.642788i	$0.222222\pi$
$97$	1.26604	+	1.50881i	1.26604	+	1.50881i	0.500000	+	0.866025i	$0.333333\pi$
$98$	0.173648	+	0.984808i	0.173648	+	0.984808i
$99$		−	0.684040i		−	0.684040i
$100$	−0.766044	+	0.642788i	−0.766044	+	0.642788i
$101$		0			0		0.642788	−	0.766044i	$-0.277778\pi$
$101$		0			0		−0.642788	+	0.766044i	$0.722222\pi$
$102$	0.592396	+	1.62760i	0.592396	+	1.62760i
$103$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$103$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	0.500000	−	0.866025i	0.500000	−	0.866025i	−0.500000	−	0.866025i	$-0.666667\pi$
$107$	0.500000	−	0.866025i	0.500000	−	0.866025i	1.00000			$0$
$108$	−0.173648	+	0.984808i	−0.173648	+	0.984808i
$109$		0			0		−0.642788	−	0.766044i	$-0.722222\pi$
$109$		0			0		0.642788	+	0.766044i	$0.277778\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	0.173648	+	0.300767i	0.173648	+	0.300767i	0.939693	−	0.342020i	$-0.111111\pi$
$113$	0.173648	+	0.300767i	0.173648	+	0.300767i	−0.766044	+	0.642788i	$0.777778\pi$
$114$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$115$		0			0
$116$		0			0
$117$		0			0
$118$	1.17365	+	0.984808i	1.17365	+	0.984808i
$119$		0			0
$120$		0			0
$121$	0.532089			0.532089
$122$		0			0
$123$	−1.43969	+	1.20805i	−1.43969	+	1.20805i
$124$		0			0
$125$		0			0
$126$		0			0
$127$		0			0		−0.984808	−	0.173648i	$-0.944444\pi$
$127$		0			0		0.984808	+	0.173648i	$0.0555556\pi$
$128$	0.173648	+	0.984808i	0.173648	+	0.984808i
$129$	0.347296	+	1.96962i	0.347296	+	1.96962i
$130$		0			0
$131$	0.673648	−	0.118782i	0.673648	−	0.118782i	0.173648	−	0.984808i	$-0.444444\pi$
$131$	0.673648	−	0.118782i	0.673648	−	0.118782i	0.500000	+	0.866025i	$0.333333\pi$
$132$	0.673648	+	0.118782i	0.673648	+	0.118782i
$133$		0			0
$134$			0.684040i			0.684040i
$135$		0			0
$136$	−1.70574	+	0.300767i	−1.70574	+	0.300767i
$137$	−0.439693	−	1.20805i	−0.439693	−	1.20805i	−0.939693	−	0.342020i	$-0.888889\pi$
$137$	−0.439693	−	1.20805i	−0.439693	−	1.20805i	0.500000	−	0.866025i	$-0.333333\pi$
$138$		0			0
$139$	−0.326352	+	0.118782i	−0.326352	+	0.118782i	−0.500000	−	0.866025i	$-0.666667\pi$
$139$	−0.326352	+	0.118782i	−0.326352	+	0.118782i	0.173648	+	0.984808i	$0.444444\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		0			0
$144$	−0.939693	−	0.342020i	−0.939693	−	0.342020i
$145$		0			0
$146$	1.43969	+	1.20805i	1.43969	+	1.20805i
$147$	0.500000	−	0.866025i	0.500000	−	0.866025i
$148$		0			0
$149$		0			0		−0.642788	−	0.766044i	$-0.722222\pi$
$149$		0			0		0.642788	+	0.766044i	$0.277778\pi$
$150$	1.00000			1.00000
$151$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$151$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$152$	0.939693	−	0.342020i	0.939693	−	0.342020i
$153$	0.592396	−	1.62760i	0.592396	−	1.62760i
$154$		0			0
$155$		0			0
$156$		0			0
$157$		0			0		−0.939693	−	0.342020i	$-0.888889\pi$
$157$		0			0		0.939693	+	0.342020i	$0.111111\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$	0.766044	−	0.642788i	0.766044	−	0.642788i
$163$	0.939693	+	1.62760i	0.939693	+	1.62760i	0.766044	+	0.642788i	$0.222222\pi$
$163$	0.939693	+	1.62760i	0.939693	+	1.62760i	0.173648	+	0.984808i	$0.444444\pi$
$164$	−0.939693	−	1.62760i	−0.939693	−	1.62760i
$165$		0			0
$166$	0.439693	−	1.20805i	0.439693	−	1.20805i
$167$		0			0		0.173648	−	0.984808i	$-0.444444\pi$
$167$		0			0		−0.173648	+	0.984808i	$0.555556\pi$
$168$		0			0
$169$	−0.766044	+	0.642788i	−0.766044	+	0.642788i
$170$		0			0
$171$	−0.173648	+	0.984808i	−0.173648	+	0.984808i
$172$	−2.00000			−2.00000
$173$		0			0		0.766044	−	0.642788i	$-0.222222\pi$
$173$		0			0		−0.766044	+	0.642788i	$0.777778\pi$
$174$		0			0
$175$		0			0
$176$	−0.233956	+	0.642788i	−0.233956	+	0.642788i
$177$	−0.266044	−	1.50881i	−0.266044	−	1.50881i
$178$	0.500000	+	0.866025i	0.500000	+	0.866025i
$179$	0.766044	+	1.32683i	0.766044	+	1.32683i	0.939693	+	0.342020i	$0.111111\pi$
$179$	0.766044	+	1.32683i	0.766044	+	1.32683i	−0.173648	+	0.984808i	$0.555556\pi$
$180$		0			0
$181$		0			0		−0.342020	−	0.939693i	$-0.611111\pi$
$181$		0			0		0.342020	+	0.939693i	$0.388889\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$		0			0
$186$		0			0
$187$	−1.11334	−	0.405223i	−1.11334	−	0.405223i
$188$		0			0
$189$		0			0
$190$		0			0
$191$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$191$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$192$	0.500000	−	0.866025i	0.500000	−	0.866025i
$193$		0			0		0.766044	−	0.642788i	$-0.222222\pi$
$193$		0			0		−0.766044	+	0.642788i	$0.777778\pi$
$194$	1.93969	+	0.342020i	1.93969	+	0.342020i
$195$		0			0
$196$	0.766044	+	0.642788i	0.766044	+	0.642788i
$197$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$197$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$198$	−0.439693	−	0.524005i	−0.439693	−	0.524005i
$199$		0			0		0.939693	−	0.342020i	$-0.111111\pi$
$199$		0			0		−0.939693	+	0.342020i	$0.888889\pi$
$200$	−0.173648	+	0.984808i	−0.173648	+	0.984808i
$201$	0.439693	−	0.524005i	0.439693	−	0.524005i
$202$		0			0
$203$		0			0
$204$	1.50000	+	0.866025i	1.50000	+	0.866025i
$205$		0			0
$206$		0			0
$207$		0			0
$208$		0			0
$209$	0.673648	+	0.118782i	0.673648	+	0.118782i
$210$		0			0
$211$	1.70574	−	0.300767i	1.70574	−	0.300767i	0.766044	−	0.642788i	$-0.222222\pi$
$211$	1.70574	−	0.300767i	1.70574	−	0.300767i	0.939693	+	0.342020i	$0.111111\pi$
$212$		0			0
$213$		0			0
$214$	−0.173648	−	0.984808i	−0.173648	−	0.984808i
$215$		0			0
$216$	0.500000	+	0.866025i	0.500000	+	0.866025i
$217$		0			0
$218$		0			0
$219$	−0.326352	−	1.85083i	−0.326352	−	1.85083i
$220$		0			0
$221$		0			0
$222$		0			0
$223$		0			0		0.984808	−	0.173648i	$-0.0555556\pi$
$223$		0			0		−0.984808	+	0.173648i	$0.944444\pi$
$224$		0			0
$225$	−0.766044	−	0.642788i	−0.766044	−	0.642788i
$226$	0.326352	+	0.118782i	0.326352	+	0.118782i
$227$	0.173648	−	0.300767i	0.173648	−	0.300767i	−0.766044	−	0.642788i	$-0.777778\pi$
$227$	0.173648	−	0.300767i	0.173648	−	0.300767i	0.939693	+	0.342020i	$0.111111\pi$
$228$	−0.939693	−	0.342020i	−0.939693	−	0.342020i
$229$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$229$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−1.26604	−	1.50881i	−1.26604	−	1.50881i	−0.766044	−	0.642788i	$-0.777778\pi$
$233$	−1.26604	−	1.50881i	−1.26604	−	1.50881i	−0.500000	−	0.866025i	$-0.666667\pi$
$234$		0			0
$235$		0			0
$236$	1.53209			1.53209
$237$		0			0
$238$		0			0
$239$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$239$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$240$		0			0
$241$	−1.26604	+	1.50881i	−1.26604	+	1.50881i	−0.500000	+	0.866025i	$0.666667\pi$
$241$	−1.26604	+	1.50881i	−1.26604	+	1.50881i	−0.766044	+	0.642788i	$0.777778\pi$
$242$	0.407604	−	0.342020i	0.407604	−	0.342020i
$243$	−1.00000			−1.00000
$244$		0			0
$245$		0			0
$246$	−0.326352	+	1.85083i	−0.326352	+	1.85083i
$247$		0			0
$248$		0			0
$249$	−1.11334	+	0.642788i	−1.11334	+	0.642788i
$250$		0			0
$251$	−1.26604	+	1.50881i	−1.26604	+	1.50881i	−0.500000	+	0.866025i	$0.666667\pi$
$251$	−1.26604	+	1.50881i	−1.26604	+	1.50881i	−0.766044	+	0.642788i	$0.777778\pi$
$252$		0			0
$253$		0			0
$254$		0			0
$255$		0			0
$256$	0.766044	+	0.642788i	0.766044	+	0.642788i
$257$	0.266044	+	0.223238i	0.266044	+	0.223238i	0.766044	−	0.642788i	$-0.222222\pi$
$257$	0.266044	+	0.223238i	0.266044	+	0.223238i	−0.500000	+	0.866025i	$0.666667\pi$
$258$	1.53209	+	1.28558i	1.53209	+	1.28558i
$259$		0			0
$260$		0			0
$261$		0			0
$262$	0.439693	−	0.524005i	0.439693	−	0.524005i
$263$		0			0		0.342020	−	0.939693i	$-0.388889\pi$
$263$		0			0		−0.342020	+	0.939693i	$0.611111\pi$
$264$	0.592396	−	0.342020i	0.592396	−	0.342020i
$265$		0			0
$266$		0			0
$267$	0.173648	−	0.984808i	0.173648	−	0.984808i
$268$	0.439693	+	0.524005i	0.439693	+	0.524005i
$269$		0			0		−0.173648	−	0.984808i	$-0.555556\pi$
$269$		0			0		0.173648	+	0.984808i	$0.444444\pi$
$270$		0			0
$271$		0			0		0.766044	−	0.642788i	$-0.222222\pi$
$271$		0			0		−0.766044	+	0.642788i	$0.777778\pi$
$272$	−1.11334	+	1.32683i	−1.11334	+	1.32683i
$273$		0			0
$274$	−1.11334	−	0.642788i	−1.11334	−	0.642788i
$275$	−0.439693	+	0.524005i	−0.439693	+	0.524005i
$276$		0			0
$277$		0			0		1.00000			$0$
$277$		0			0		−1.00000			$\pi$
$278$	−0.173648	+	0.300767i	−0.173648	+	0.300767i
$279$		0			0
$280$		0			0
$281$	−1.76604	+	0.642788i	−1.76604	+	0.642788i	−0.766044	+	0.642788i	$0.777778\pi$
$281$	−1.76604	+	0.642788i	−1.76604	+	0.642788i	−1.00000			$1.00000\pi$
$282$		0			0
$283$	0.0603074	+	0.342020i	0.0603074	+	0.342020i	1.00000			$0$
$283$	0.0603074	+	0.342020i	0.0603074	+	0.342020i	−0.939693	+	0.342020i	$0.888889\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$	−0.939693	+	0.342020i	−0.939693	+	0.342020i
$289$	−1.53209	−	1.28558i	−1.53209	−	1.28558i
$290$		0			0
$291$	−1.26604	−	1.50881i	−1.26604	−	1.50881i
$292$	1.87939			1.87939
$293$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$293$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$294$	−0.173648	−	0.984808i	−0.173648	−	0.984808i
$295$		0			0
$296$		0			0
$297$			0.684040i			0.684040i
$298$		0			0
$299$		0			0
$300$	0.766044	−	0.642788i	0.766044	−	0.642788i
$301$		0			0
$302$		0			0
$303$		0			0
$304$	0.500000	−	0.866025i	0.500000	−	0.866025i
$305$		0			0
$306$	−0.592396	−	1.62760i	−0.592396	−	1.62760i
$307$	−1.26604	+	0.223238i	−1.26604	+	0.223238i	−0.766044	−	0.642788i	$-0.777778\pi$
$307$	−1.26604	+	0.223238i	−1.26604	+	0.223238i	−0.500000	+	0.866025i	$0.666667\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$		0			0			−	1.00000i	$-0.5\pi$
$311$		0			0				1.00000i	$0.5\pi$
$312$		0			0
$313$	−0.0603074	+	0.342020i	−0.0603074	+	0.342020i	0.939693	+	0.342020i	$0.111111\pi$
$313$	−0.0603074	+	0.342020i	−0.0603074	+	0.342020i	−1.00000			$\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$		0			0		−0.766044	−	0.642788i	$-0.777778\pi$
$317$		0			0		0.766044	+	0.642788i	$0.222222\pi$
$318$		0			0
$319$		0			0
$320$		0			0
$321$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$322$		0			0
$323$	1.50000	+	0.866025i	1.50000	+	0.866025i
$324$	0.173648	−	0.984808i	0.173648	−	0.984808i
$325$		0			0
$326$	1.76604	+	0.642788i	1.76604	+	0.642788i
$327$		0			0
$328$	−1.76604	−	0.642788i	−1.76604	−	0.642788i
$329$		0			0
$330$		0			0
$331$		−	1.96962i		−	1.96962i	−0.173648	−	0.984808i	$-0.555556\pi$
$331$		−	1.96962i		−	1.96962i	0.173648	−	0.984808i	$-0.444444\pi$
$332$	−0.439693	−	1.20805i	−0.439693	−	1.20805i
$333$		0			0
$334$		0			0
$335$		0			0
$336$		0			0
$337$	0.439693	−	1.20805i	0.439693	−	1.20805i	−0.500000	−	0.866025i	$-0.666667\pi$
$337$	0.439693	−	1.20805i	0.439693	−	1.20805i	0.939693	−	0.342020i	$-0.111111\pi$
$338$	−0.173648	+	0.984808i	−0.173648	+	0.984808i
$339$	−0.173648	−	0.300767i	−0.173648	−	0.300767i
$340$		0			0
$341$		0			0
$342$	0.500000	+	0.866025i	0.500000	+	0.866025i
$343$		0			0
$344$	−1.53209	+	1.28558i	−1.53209	+	1.28558i
$345$		0			0
$346$		0			0
$347$	0.673648	−	1.85083i	0.673648	−	1.85083i	0.173648	−	0.984808i	$-0.444444\pi$
$347$	0.673648	−	1.85083i	0.673648	−	1.85083i	0.500000	−	0.866025i	$-0.333333\pi$
$348$		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
$352$	0.233956	+	0.642788i	0.233956	+	0.642788i
$353$		−	1.96962i		−	1.96962i	−0.173648	−	0.984808i	$-0.555556\pi$
$353$		−	1.96962i		−	1.96962i	0.173648	−	0.984808i	$-0.444444\pi$
$354$	−1.17365	−	0.984808i	−1.17365	−	0.984808i
$355$		0			0
$356$	0.939693	+	0.342020i	0.939693	+	0.342020i
$357$		0			0
$358$	1.43969	+	0.524005i	1.43969	+	0.524005i
$359$		0			0		0.342020	−	0.939693i	$-0.388889\pi$
$359$		0			0		−0.342020	+	0.939693i	$0.611111\pi$
$360$		0			0
$361$	−0.939693	−	0.342020i	−0.939693	−	0.342020i
$362$		0			0
$363$	−0.532089			−0.532089
$364$		0			0
$365$		0			0
$366$		0			0
$367$		0			0		−0.766044	−	0.642788i	$-0.777778\pi$
$367$		0			0		0.766044	+	0.642788i	$0.222222\pi$
$368$		0			0
$369$	1.43969	−	1.20805i	1.43969	−	1.20805i
$370$		0			0
$371$		0			0
$372$		0			0
$373$		0			0			−	1.00000i	$-0.5\pi$
$373$		0			0				1.00000i	$0.5\pi$
$374$	−1.11334	+	0.405223i	−1.11334	+	0.405223i
$375$		0			0
$376$		0			0
$377$		0			0
$378$		0			0
$379$		−	1.73205i		−	1.73205i	−0.500000	−	0.866025i	$-0.666667\pi$
$379$		−	1.73205i		−	1.73205i	0.500000	−	0.866025i	$-0.333333\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$		0			0		0.766044	−	0.642788i	$-0.222222\pi$
$383$		0			0		−0.766044	+	0.642788i	$0.777778\pi$
$384$	−0.173648	−	0.984808i	−0.173648	−	0.984808i
$385$		0			0
$386$		0			0
$387$	−0.347296	−	1.96962i	−0.347296	−	1.96962i
$388$	1.70574	−	0.984808i	1.70574	−	0.984808i
$389$		0			0		−0.984808	−	0.173648i	$-0.944444\pi$
$389$		0			0		0.984808	+	0.173648i	$0.0555556\pi$
$390$		0			0
$391$		0			0
$392$	1.00000			1.00000
$393$	−0.673648	+	0.118782i	−0.673648	+	0.118782i
$394$		0			0
$395$		0			0
$396$	−0.673648	−	0.118782i	−0.673648	−	0.118782i
$397$		0			0		−0.939693	−	0.342020i	$-0.888889\pi$
$397$		0			0		0.939693	+	0.342020i	$0.111111\pi$
$398$		0			0
$399$		0			0
$400$	0.500000	+	0.866025i	0.500000	+	0.866025i
$401$	0.326352	+	1.85083i	0.326352	+	1.85083i	0.500000	+	0.866025i	$0.333333\pi$
$401$	0.326352	+	1.85083i	0.326352	+	1.85083i	−0.173648	+	0.984808i	$0.555556\pi$
$402$		−	0.684040i		−	0.684040i
$403$		0			0
$404$		0			0
$405$		0			0
$406$		0			0
$407$		0			0
$408$	1.70574	−	0.300767i	1.70574	−	0.300767i
$409$	−0.826352	+	0.984808i	−0.826352	+	0.984808i	0.173648	+	0.984808i	$0.444444\pi$
$409$	−0.826352	+	0.984808i	−0.826352	+	0.984808i	−1.00000			$\pi$
$410$		0			0
$411$	0.439693	+	1.20805i	0.439693	+	1.20805i
$412$		0			0
$413$		0			0
$414$		0			0
$415$		0			0
$416$		0			0
$417$	0.326352	−	0.118782i	0.326352	−	0.118782i
$418$	0.592396	−	0.342020i	0.592396	−	0.342020i
$419$	1.50000	−	0.866025i	1.50000	−	0.866025i	0.500000	−	0.866025i	$-0.333333\pi$
$419$	1.50000	−	0.866025i	1.50000	−	0.866025i	1.00000			$0$
$420$		0			0
$421$		0			0		0.342020	−	0.939693i	$-0.388889\pi$
$421$		0			0		−0.342020	+	0.939693i	$0.611111\pi$
$422$	1.11334	−	1.32683i	1.11334	−	1.32683i
$423$		0			0
$424$		0			0
$425$	−1.50000	+	0.866025i	−1.50000	+	0.866025i
$426$		0			0
$427$		0			0
$428$	−0.766044	−	0.642788i	−0.766044	−	0.642788i
$429$		0			0
$430$		0			0
$431$		0			0		0.173648	−	0.984808i	$-0.444444\pi$
$431$		0			0		−0.173648	+	0.984808i	$0.555556\pi$
$432$	0.939693	+	0.342020i	0.939693	+	0.342020i
$433$	1.11334	−	1.32683i	1.11334	−	1.32683i	0.173648	−	0.984808i	$-0.444444\pi$
$433$	1.11334	−	1.32683i	1.11334	−	1.32683i	0.939693	−	0.342020i	$-0.111111\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$		0			0
$438$	−1.43969	−	1.20805i	−1.43969	−	1.20805i
$439$		0			0		−0.642788	−	0.766044i	$-0.722222\pi$
$439$		0			0		0.642788	+	0.766044i	$0.277778\pi$
$440$		0			0
$441$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$442$		0			0
$443$	−0.826352	+	0.984808i	−0.826352	+	0.984808i	0.173648	+	0.984808i	$0.444444\pi$
$443$	−0.826352	+	0.984808i	−0.826352	+	0.984808i	−1.00000			$\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$		0			0
$448$		0			0
$449$	−0.766044	+	1.32683i	−0.766044	+	1.32683i	0.173648	+	0.984808i	$0.444444\pi$
$449$	−0.766044	+	1.32683i	−0.766044	+	1.32683i	−0.939693	+	0.342020i	$0.888889\pi$
$450$	−1.00000			−1.00000
$451$	−0.826352	−	0.984808i	−0.826352	−	0.984808i
$452$	0.326352	−	0.118782i	0.326352	−	0.118782i
$453$		0			0
$454$	−0.0603074	−	0.342020i	−0.0603074	−	0.342020i
$455$		0			0
$456$	−0.939693	+	0.342020i	−0.939693	+	0.342020i
$457$	−0.173648	+	0.300767i	−0.173648	+	0.300767i	−0.939693	−	0.342020i	$-0.888889\pi$
$457$	−0.173648	+	0.300767i	−0.173648	+	0.300767i	0.766044	+	0.642788i	$0.222222\pi$
$458$		0			0
$459$	−0.592396	+	1.62760i	−0.592396	+	1.62760i
$460$		0			0
$461$		0			0		0.984808	−	0.173648i	$-0.0555556\pi$
$461$		0			0		−0.984808	+	0.173648i	$0.944444\pi$
$462$		0			0
$463$		0			0		1.00000			$0$
$463$		0			0		−1.00000			$\pi$
$464$		0			0
$465$		0			0
$466$	−1.93969	−	0.342020i	−1.93969	−	0.342020i
$467$	−0.592396	+	0.342020i	−0.592396	+	0.342020i	−0.766044	−	0.642788i	$-0.777778\pi$
$467$	−0.592396	+	0.342020i	−0.592396	+	0.342020i	0.173648	+	0.984808i	$0.444444\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$		0			0
$472$	1.17365	−	0.984808i	1.17365	−	0.984808i
$473$	−1.34730	+	0.237565i	−1.34730	+	0.237565i
$474$		0			0
$475$	0.766044	−	0.642788i	0.766044	−	0.642788i
$476$		0			0
$477$		0			0
$478$		0			0
$479$		0			0		−0.342020	−	0.939693i	$-0.611111\pi$
$479$		0			0		0.342020	+	0.939693i	$0.388889\pi$
$480$		0			0
$481$		0			0
$482$			1.96962i			1.96962i
$483$		0			0
$484$	0.0923963	−	0.524005i	0.0923963	−	0.524005i
$485$		0			0
$486$	−0.766044	+	0.642788i	−0.766044	+	0.642788i
$487$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$487$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$488$		0			0
$489$	−0.939693	−	1.62760i	−0.939693	−	1.62760i
$490$		0			0
$491$		0			0		0.766044	−	0.642788i	$-0.222222\pi$
$491$		0			0		−0.766044	+	0.642788i	$0.777778\pi$
$492$	0.939693	+	1.62760i	0.939693	+	1.62760i
$493$		0			0
$494$		0			0
$495$		0			0
$496$		0			0
$497$		0			0
$498$	−0.439693	+	1.20805i	−0.439693	+	1.20805i
$499$	1.43969	+	0.524005i	1.43969	+	0.524005i	0.939693	−	0.342020i	$-0.111111\pi$
$499$	1.43969	+	0.524005i	1.43969	+	0.524005i	0.500000	+	0.866025i	$0.333333\pi$
$500$		0			0
$501$		0			0
$502$			1.96962i			1.96962i
$503$		0			0		−0.342020	−	0.939693i	$-0.611111\pi$
$503$		0			0		0.342020	+	0.939693i	$0.388889\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$	0.766044	−	0.642788i	0.766044	−	0.642788i
$508$		0			0
$509$		0			0		0.173648	−	0.984808i	$-0.444444\pi$
$509$		0			0		−0.173648	+	0.984808i	$0.555556\pi$
$510$		0			0
$511$		0			0
$512$	1.00000			1.00000
$513$	0.173648	−	0.984808i	0.173648	−	0.984808i
$514$	0.347296			0.347296
$515$		0			0
$516$	2.00000			2.00000
$517$		0			0
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−0.766044	−	1.32683i	−0.766044	−	1.32683i	−0.939693	−	0.342020i	$-0.888889\pi$
$521$	−0.766044	−	1.32683i	−0.766044	−	1.32683i	0.173648	−	0.984808i	$-0.444444\pi$
$522$		0			0
$523$		0			0		0.939693	−	0.342020i	$-0.111111\pi$
$523$		0			0		−0.939693	+	0.342020i	$0.888889\pi$
$524$		−	0.684040i		−	0.684040i
$525$		0			0
$526$		0			0
$527$		0			0
$528$	0.233956	−	0.642788i	0.233956	−	0.642788i
$529$	0.939693	+	0.342020i	0.939693	+	0.342020i
$530$		0			0
$531$	0.266044	+	1.50881i	0.266044	+	1.50881i
$532$		0			0
$533$		0			0
$534$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$535$		0			0
$536$	0.673648	+	0.118782i	0.673648	+	0.118782i
$537$	−0.766044	−	1.32683i	−0.766044	−	1.32683i
$538$		0			0
$539$	0.592396	+	0.342020i	0.592396	+	0.342020i
$540$		0			0
$541$		0			0		0.939693	−	0.342020i	$-0.111111\pi$
$541$		0			0		−0.939693	+	0.342020i	$0.888889\pi$
$542$		0			0
$543$		0			0
$544$			1.73205i			1.73205i
$545$		0			0
$546$		0			0
$547$		0			0		0.939693	−	0.342020i	$-0.111111\pi$
$547$		0			0		−0.939693	+	0.342020i	$0.888889\pi$
$548$	−1.26604	+	0.223238i	−1.26604	+	0.223238i
$549$		0			0
$550$			0.684040i			0.684040i
$551$		0			0
$552$		0			0
$553$		0			0
$554$		0			0
$555$		0			0
$556$	0.0603074	+	0.342020i	0.0603074	+	0.342020i
$557$		0			0		−0.984808	−	0.173648i	$-0.944444\pi$
$557$		0			0		0.984808	+	0.173648i	$0.0555556\pi$
$558$		0			0
$559$		0			0
$560$		0			0
$561$	1.11334	+	0.405223i	1.11334	+	0.405223i
$562$	−0.939693	+	1.62760i	−0.939693	+	1.62760i
$563$	−1.87939			−1.87939			−0.939693	−	0.342020i	$-0.888889\pi$
$563$	−1.87939			−1.87939			−0.939693	+	0.342020i	$0.888889\pi$
$564$		0			0
$565$		0			0
$566$	0.266044	+	0.223238i	0.266044	+	0.223238i
$567$		0			0
$568$		0			0
$569$	0.500000	−	0.866025i	0.500000	−	0.866025i	−0.500000	−	0.866025i	$-0.666667\pi$
$569$	0.500000	−	0.866025i	0.500000	−	0.866025i	1.00000			$0$
$570$		0			0
$571$	0.766044	+	1.32683i	0.766044	+	1.32683i	0.939693	+	0.342020i	$0.111111\pi$
$571$	0.766044	+	1.32683i	0.766044	+	1.32683i	−0.173648	+	0.984808i	$0.555556\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$577$	−0.766044	+	1.32683i	−0.766044	+	1.32683i	0.173648	+	0.984808i	$0.444444\pi$
$577$	−0.766044	+	1.32683i	−0.766044	+	1.32683i	−0.939693	+	0.342020i	$0.888889\pi$
$578$	−2.00000			−2.00000
$579$		0			0
$580$		0			0
$581$		0			0
$582$	−1.93969	−	0.342020i	−1.93969	−	0.342020i
$583$		0			0
$584$	1.43969	−	1.20805i	1.43969	−	1.20805i
$585$		0			0
$586$		0			0
$587$		0			0		0.766044	−	0.642788i	$-0.222222\pi$
$587$		0			0		−0.766044	+	0.642788i	$0.777778\pi$
$588$	−0.766044	−	0.642788i	−0.766044	−	0.642788i
$589$		0			0
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−1.26604	+	1.50881i	−1.26604	+	1.50881i	−0.500000	+	0.866025i	$0.666667\pi$
$593$	−1.26604	+	1.50881i	−1.26604	+	1.50881i	−0.766044	+	0.642788i	$0.777778\pi$
$594$	0.439693	+	0.524005i	0.439693	+	0.524005i
$595$		0			0
$596$		0			0
$597$		0			0
$598$		0			0
$599$		0			0		−0.766044	−	0.642788i	$-0.777778\pi$
$599$		0			0		0.766044	+	0.642788i	$0.222222\pi$
$600$	0.173648	−	0.984808i	0.173648	−	0.984808i
$601$	1.11334	−	0.642788i	1.11334	−	0.642788i	0.173648	−	0.984808i	$-0.444444\pi$
$601$	1.11334	−	0.642788i	1.11334	−	0.642788i	0.939693	+	0.342020i	$0.111111\pi$
$602$		0			0
$603$	−0.439693	+	0.524005i	−0.439693	+	0.524005i
$604$		0			0
$605$		0			0
$606$		0			0
$607$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$607$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$608$	−0.173648	−	0.984808i	−0.173648	−	0.984808i
$609$		0			0
$610$		0			0
$611$		0			0
$612$	−1.50000	−	0.866025i	−1.50000	−	0.866025i
$613$		0			0		0.766044	−	0.642788i	$-0.222222\pi$
$613$		0			0		−0.766044	+	0.642788i	$0.777778\pi$
$614$	−0.826352	+	0.984808i	−0.826352	+	0.984808i
$615$		0			0
$616$		0			0
$617$	0.826352	−	0.984808i	0.826352	−	0.984808i	−0.173648	−	0.984808i	$-0.555556\pi$
$617$	0.826352	−	0.984808i	0.826352	−	0.984808i	1.00000			$0$
$618$		0			0
$619$	1.00000			1.00000			0.500000	−	0.866025i	$-0.333333\pi$
$619$	1.00000			1.00000			0.500000	+	0.866025i	$0.333333\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$		0			0
$625$	0.173648	+	0.984808i	0.173648	+	0.984808i
$626$	0.173648	+	0.300767i	0.173648	+	0.300767i
$627$	−0.673648	−	0.118782i	−0.673648	−	0.118782i
$628$		0			0
$629$		0			0
$630$		0			0
$631$		0			0		−0.766044	−	0.642788i	$-0.777778\pi$
$631$		0			0		0.766044	+	0.642788i	$0.222222\pi$
$632$		0			0
$633$	−1.70574	+	0.300767i	−1.70574	+	0.300767i
$634$		0			0
$635$		0			0
$636$		0			0
$637$		0			0
$638$		0			0
$639$		0			0
$640$		0			0
$641$	0.0603074	+	0.342020i	0.0603074	+	0.342020i	1.00000			$0$
$641$	0.0603074	+	0.342020i	0.0603074	+	0.342020i	−0.939693	+	0.342020i	$0.888889\pi$
$642$	0.173648	+	0.984808i	0.173648	+	0.984808i
$643$	1.17365	−	0.984808i	1.17365	−	0.984808i	0.173648	−	0.984808i	$-0.444444\pi$
$643$	1.17365	−	0.984808i	1.17365	−	0.984808i	1.00000			$0$
$644$		0			0
$645$		0			0
$646$	1.70574	−	0.300767i	1.70574	−	0.300767i
$647$		0			0			−	1.00000i	$-0.5\pi$
$647$		0			0				1.00000i	$0.5\pi$
$648$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$649$	1.03209	−	0.181985i	1.03209	−	0.181985i
$650$		0			0
$651$		0			0
$652$	1.76604	−	0.642788i	1.76604	−	0.642788i
$653$		0			0			−	1.00000i	$-0.5\pi$
$653$		0			0				1.00000i	$0.5\pi$
$654$		0			0
$655$		0			0
$656$	−1.76604	+	0.642788i	−1.76604	+	0.642788i
$657$	0.326352	+	1.85083i	0.326352	+	1.85083i
$658$		0			0
$659$	0.766044	+	0.642788i	0.766044	+	0.642788i	0.939693	−	0.342020i	$-0.111111\pi$
$659$	0.766044	+	0.642788i	0.766044	+	0.642788i	−0.173648	+	0.984808i	$0.555556\pi$
$660$		0			0
$661$		0			0		−0.984808	−	0.173648i	$-0.944444\pi$
$661$		0			0		0.984808	+	0.173648i	$0.0555556\pi$
$662$	−1.26604	−	1.50881i	−1.26604	−	1.50881i
$663$		0			0
$664$	−1.11334	−	0.642788i	−1.11334	−	0.642788i
$665$		0			0
$666$		0			0
$667$		0			0
$668$		0			0
$669$		0			0
$670$		0			0
$671$		0			0
$672$		0			0
$673$			1.73205i			1.73205i	0.500000	+	0.866025i	$0.333333\pi$
$673$			1.73205i			1.73205i	−0.500000	+	0.866025i	$0.666667\pi$
$674$	−0.439693	−	1.20805i	−0.439693	−	1.20805i
$675$	0.766044	+	0.642788i	0.766044	+	0.642788i
$676$	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	$0.333333\pi$
$678$	−0.326352	−	0.118782i	−0.326352	−	0.118782i
$679$		0			0
$680$		0			0
$681$	−0.173648	+	0.300767i	−0.173648	+	0.300767i
$682$		0			0
$683$	−2.00000			−2.00000			−1.00000			$\pi$
$683$	−2.00000			−2.00000			−1.00000			$\pi$
$684$	0.939693	+	0.342020i	0.939693	+	0.342020i
$685$		0			0
$686$		0			0
$687$		0			0
$688$	−0.347296	+	1.96962i	−0.347296	+	1.96962i
$689$		0			0
$690$		0			0
$691$	−0.500000	−	0.866025i	−0.500000	−	0.866025i	0.500000	−	0.866025i	$-0.333333\pi$
$691$	−0.500000	−	0.866025i	−0.500000	−	0.866025i	−1.00000			$\pi$
$692$		0			0
$693$		0			0
$694$	−0.673648	−	1.85083i	−0.673648	−	1.85083i
$695$		0			0
$696$		0			0
$697$	−1.11334	−	3.05888i	−1.11334	−	3.05888i
$698$		0			0
$699$	1.26604	+	1.50881i	1.26604	+	1.50881i
$700$		0			0
$701$		0			0		0.342020	−	0.939693i	$-0.388889\pi$
$701$		0			0		−0.342020	+	0.939693i	$0.611111\pi$
$702$		0			0
$703$		0			0
$704$	0.592396	+	0.342020i	0.592396	+	0.342020i
$705$		0			0
$706$	−1.26604	−	1.50881i	−1.26604	−	1.50881i
$707$		0			0
$708$	−1.53209			−1.53209
$709$		0			0		−0.766044	−	0.642788i	$-0.777778\pi$
$709$		0			0		0.766044	+	0.642788i	$0.222222\pi$
$710$		0			0
$711$		0			0
$712$	0.939693	−	0.342020i	0.939693	−	0.342020i
$713$		0			0
$714$		0			0
$715$		0			0
$716$	1.43969	−	0.524005i	1.43969	−	0.524005i
$717$		0			0
$718$		0			0
$719$		0			0		0.984808	−	0.173648i	$-0.0555556\pi$
$719$		0			0		−0.984808	+	0.173648i	$0.944444\pi$
$720$		0			0
$721$		0			0
$722$	−0.939693	+	0.342020i	−0.939693	+	0.342020i
$723$	1.26604	−	1.50881i	1.26604	−	1.50881i
$724$		0			0
$725$		0			0
$726$	−0.407604	+	0.342020i	−0.407604	+	0.342020i
$727$		0			0		−0.173648	−	0.984808i	$-0.555556\pi$
$727$		0			0		0.173648	+	0.984808i	$0.444444\pi$
$728$		0			0
$729$	1.00000			1.00000
$730$		0			0
$731$	−3.41147	−	0.601535i	−3.41147	−	0.601535i
$732$		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
$737$	0.358441	+	0.300767i	0.358441	+	0.300767i
$738$	0.326352	−	1.85083i	0.326352	−	1.85083i
$739$	1.76604	+	0.642788i	1.76604	+	0.642788i	1.00000			$0$
$739$	1.76604	+	0.642788i	1.76604	+	0.642788i	0.766044	+	0.642788i	$0.222222\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$		0			0		−0.173648	−	0.984808i	$-0.555556\pi$
$743$		0			0		0.173648	+	0.984808i	$0.444444\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$	1.11334	−	0.642788i	1.11334	−	0.642788i
$748$	−0.592396	+	1.02606i	−0.592396	+	1.02606i
$749$		0			0
$750$		0			0
$751$		0			0		0.642788	−	0.766044i	$-0.277778\pi$
$751$		0			0		−0.642788	+	0.766044i	$0.722222\pi$
$752$		0			0
$753$	1.26604	−	1.50881i	1.26604	−	1.50881i
$754$		0			0
$755$		0			0
$756$		0			0
$757$		0			0		−0.173648	−	0.984808i	$-0.555556\pi$
$757$		0			0		0.173648	+	0.984808i	$0.444444\pi$
$758$	−1.11334	−	1.32683i	−1.11334	−	1.32683i
$759$		0			0
$760$		0			0
$761$	0.592396	−	0.342020i	0.592396	−	0.342020i	−0.173648	−	0.984808i	$-0.555556\pi$
$761$	0.592396	−	0.342020i	0.592396	−	0.342020i	0.766044	+	0.642788i	$0.222222\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$		0			0
$766$		0			0
$767$		0			0
$768$	−0.766044	−	0.642788i	−0.766044	−	0.642788i
$769$	0.766044	+	0.642788i	0.766044	+	0.642788i	0.939693	−	0.342020i	$-0.111111\pi$
$769$	0.766044	+	0.642788i	0.766044	+	0.642788i	−0.173648	+	0.984808i	$0.555556\pi$
$770$		0			0
$771$	−0.266044	−	0.223238i	−0.266044	−	0.223238i
$772$		0			0
$773$		0			0		0.173648	−	0.984808i	$-0.444444\pi$
$773$		0			0		−0.173648	+	0.984808i	$0.555556\pi$
$774$	−1.53209	−	1.28558i	−1.53209	−	1.28558i
$775$		0			0
$776$	0.673648	−	1.85083i	0.673648	−	1.85083i
$777$		0			0
$778$		0			0
$779$	0.939693	+	1.62760i	0.939693	+	1.62760i
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$	0.766044	−	0.642788i	0.766044	−	0.642788i
$785$		0			0
$786$	−0.439693	+	0.524005i	−0.439693	+	0.524005i
$787$	−1.70574	−	0.984808i	−1.70574	−	0.984808i	−0.939693	−	0.342020i	$-0.888889\pi$
$787$	−1.70574	−	0.984808i	−1.70574	−	0.984808i	−0.766044	−	0.642788i	$-0.777778\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		0			0
$792$	−0.592396	+	0.342020i	−0.592396	+	0.342020i
$793$		0			0
$794$		0			0
$795$		0			0
$796$		0			0
$797$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$797$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$798$		0			0
$799$		0			0
$800$	0.939693	+	0.342020i	0.939693	+	0.342020i
$801$	−0.173648	+	0.984808i	−0.173648	+	0.984808i
$802$	1.43969	+	1.20805i	1.43969	+	1.20805i
$803$	1.26604	−	0.223238i	1.26604	−	0.223238i
$804$	−0.439693	−	0.524005i	−0.439693	−	0.524005i
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$	1.11334	−	0.642788i	1.11334	−	0.642788i	0.173648	−	0.984808i	$-0.444444\pi$
$809$	1.11334	−	0.642788i	1.11334	−	0.642788i	0.939693	+	0.342020i	$0.111111\pi$
$810$		0			0
$811$	1.70574	+	0.300767i	1.70574	+	0.300767i	0.939693	−	0.342020i	$-0.111111\pi$
$811$	1.70574	+	0.300767i	1.70574	+	0.300767i	0.766044	+	0.642788i	$0.222222\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$		0			0
$816$	1.11334	−	1.32683i	1.11334	−	1.32683i
$817$	2.00000			2.00000
$818$			1.28558i			1.28558i
$819$		0			0
$820$		0			0
$821$		0			0		−0.342020	−	0.939693i	$-0.611111\pi$
$821$		0			0		0.342020	+	0.939693i	$0.388889\pi$
$822$	1.11334	+	0.642788i	1.11334	+	0.642788i
$823$		0			0		0.939693	−	0.342020i	$-0.111111\pi$
$823$		0			0		−0.939693	+	0.342020i	$0.888889\pi$
$824$		0			0
$825$	0.439693	−	0.524005i	0.439693	−	0.524005i
$826$		0			0
$827$	0.326352	−	0.118782i	0.326352	−	0.118782i	−0.173648	−	0.984808i	$-0.555556\pi$
$827$	0.326352	−	0.118782i	0.326352	−	0.118782i	0.500000	+	0.866025i	$0.333333\pi$
$828$		0			0
$829$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$829$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	1.11334	+	1.32683i	1.11334	+	1.32683i
$834$	0.173648	−	0.300767i	0.173648	−	0.300767i
$835$		0			0
$836$	0.233956	−	0.642788i	0.233956	−	0.642788i
$837$		0			0
$838$	0.592396	−	1.62760i	0.592396	−	1.62760i
$839$		0			0		−0.939693	−	0.342020i	$-0.888889\pi$
$839$		0			0		0.939693	+	0.342020i	$0.111111\pi$
$840$		0			0
$841$	−0.939693	−	0.342020i	−0.939693	−	0.342020i
$842$		0			0
$843$	1.76604	−	0.642788i	1.76604	−	0.642788i
$844$		−	1.73205i		−	1.73205i
$845$		0			0
$846$		0			0
$847$		0			0
$848$		0			0
$849$	−0.0603074	−	0.342020i	−0.0603074	−	0.342020i
$850$	−0.592396	+	1.62760i	−0.592396	+	1.62760i
$851$		0			0
$852$		0			0
$853$		0			0		0.766044	−	0.642788i	$-0.222222\pi$
$853$		0			0		−0.766044	+	0.642788i	$0.777778\pi$
$854$		0			0
$855$		0			0
$856$	−1.00000			−1.00000
$857$	−1.43969	+	1.20805i	−1.43969	+	1.20805i	−0.500000	+	0.866025i	$0.666667\pi$
$857$	−1.43969	+	1.20805i	−1.43969	+	1.20805i	−0.939693	+	0.342020i	$0.888889\pi$
$858$		0			0
$859$	0.326352	−	1.85083i	0.326352	−	1.85083i	−0.173648	−	0.984808i	$-0.555556\pi$
$859$	0.326352	−	1.85083i	0.326352	−	1.85083i	0.500000	−	0.866025i	$-0.333333\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$863$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$864$	0.939693	−	0.342020i	0.939693	−	0.342020i
$865$		0			0
$866$		−	1.73205i		−	1.73205i
$867$	1.53209	+	1.28558i	1.53209	+	1.28558i
$868$		0			0
$869$		0			0
$870$		0			0
$871$		0			0
$872$		0			0
$873$	1.26604	+	1.50881i	1.26604	+	1.50881i
$874$		0			0
$875$		0			0
$876$	−1.87939			−1.87939
$877$		0			0		−0.642788	−	0.766044i	$-0.722222\pi$
$877$		0			0		0.642788	+	0.766044i	$0.277778\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	1.11334	+	0.642788i	1.11334	+	0.642788i	0.939693	−	0.342020i	$-0.111111\pi$
$881$	1.11334	+	0.642788i	1.11334	+	0.642788i	0.173648	+	0.984808i	$0.444444\pi$
$882$	0.173648	+	0.984808i	0.173648	+	0.984808i
$883$	−1.76604	+	0.642788i	−1.76604	+	0.642788i	−0.766044	+	0.642788i	$0.777778\pi$
$883$	−1.76604	+	0.642788i	−1.76604	+	0.642788i	−1.00000			$1.00000\pi$
$884$		0			0
$885$		0			0
$886$			1.28558i			1.28558i
$887$		0			0		0.939693	−	0.342020i	$-0.111111\pi$
$887$		0			0		−0.939693	+	0.342020i	$0.888889\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$		−	0.684040i		−	0.684040i
$892$		0			0
$893$		0			0
$894$		0			0
$895$		0			0
$896$		0			0
$897$		0			0
$898$	0.266044	+	1.50881i	0.266044	+	1.50881i
$899$		0			0
$900$	−0.766044	+	0.642788i	−0.766044	+	0.642788i
$901$		0			0
$902$	−1.26604	−	0.223238i	−1.26604	−	0.223238i
$903$		0			0
$904$	0.173648	−	0.300767i	0.173648	−	0.300767i
$905$		0			0
$906$		0			0
$907$	−1.26604	+	0.223238i	−1.26604	+	0.223238i	−0.766044	−	0.642788i	$-0.777778\pi$
$907$	−1.26604	+	0.223238i	−1.26604	+	0.223238i	−0.500000	+	0.866025i	$0.666667\pi$
$908$	−0.266044	−	0.223238i	−0.266044	−	0.223238i
$909$		0			0
$910$		0			0
$911$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$911$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$912$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$913$	−0.439693	−	0.761570i	−0.439693	−	0.761570i
$914$	0.0603074	+	0.342020i	0.0603074	+	0.342020i
$915$		0			0
$916$		0			0
$917$		0			0
$918$	0.592396	+	1.62760i	0.592396	+	1.62760i
$919$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$919$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$920$		0			0
$921$	1.26604	−	0.223238i	1.26604	−	0.223238i
$922$		0			0
$923$		0			0
$924$		0			0
$925$		0			0
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−1.26604	−	1.50881i	−1.26604	−	1.50881i	−0.766044	−	0.642788i	$-0.777778\pi$
$929$	−1.26604	−	1.50881i	−1.26604	−	1.50881i	−0.500000	−	0.866025i	$-0.666667\pi$
$930$		0			0
$931$	−0.766044	−	0.642788i	−0.766044	−	0.642788i
$932$	−1.70574	+	0.984808i	−1.70574	+	0.984808i
$933$		0			0
$934$	−0.233956	+	0.642788i	−0.233956	+	0.642788i
$935$		0			0
$936$		0			0
$937$	0.326352	−	1.85083i	0.326352	−	1.85083i	−0.173648	−	0.984808i	$-0.555556\pi$
$937$	0.326352	−	1.85083i	0.326352	−	1.85083i	0.500000	−	0.866025i	$-0.333333\pi$
$938$		0			0
$939$	0.0603074	−	0.342020i	0.0603074	−	0.342020i
$940$		0			0
$941$		0			0		−0.766044	−	0.642788i	$-0.777778\pi$
$941$		0			0		0.766044	+	0.642788i	$0.222222\pi$
$942$		0			0
$943$		0			0
$944$	0.266044	−	1.50881i	0.266044	−	1.50881i
$945$		0			0
$946$	−0.879385	+	1.04801i	−0.879385	+	1.04801i
$947$	0.592396	−	1.62760i	0.592396	−	1.62760i	−0.173648	−	0.984808i	$-0.555556\pi$
$947$	0.592396	−	1.62760i	0.592396	−	1.62760i	0.766044	−	0.642788i	$-0.222222\pi$
$948$		0			0
$949$		0			0
$950$	0.173648	−	0.984808i	0.173648	−	0.984808i
$951$		0			0
$952$		0			0
$953$	0.326352	+	1.85083i	0.326352	+	1.85083i	0.500000	+	0.866025i	$0.333333\pi$
$953$	0.326352	+	1.85083i	0.326352	+	1.85083i	−0.173648	+	0.984808i	$0.555556\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	−1.00000			−1.00000
$962$		0			0
$963$	0.500000	−	0.866025i	0.500000	−	0.866025i
$964$	1.26604	+	1.50881i	1.26604	+	1.50881i
$965$		0			0
$966$		0			0
$967$		0			0		−0.173648	−	0.984808i	$-0.555556\pi$
$967$		0			0		0.173648	+	0.984808i	$0.444444\pi$
$968$	−0.266044	−	0.460802i	−0.266044	−	0.460802i
$969$	−1.50000	−	0.866025i	−1.50000	−	0.866025i
$970$		0			0
$971$	−1.76604	−	0.642788i	−1.76604	−	0.642788i	−0.766044	−	0.642788i	$-0.777778\pi$
$971$	−1.76604	−	0.642788i	−1.76604	−	0.642788i	−1.00000			$\pi$
$972$	−0.173648	+	0.984808i	−0.173648	+	0.984808i
$973$		0			0
$974$		0			0
$975$		0			0
$976$		0			0
$977$	0.173648	−	0.300767i	0.173648	−	0.300767i	−0.766044	−	0.642788i	$-0.777778\pi$
$977$	0.173648	−	0.300767i	0.173648	−	0.300767i	0.939693	+	0.342020i	$0.111111\pi$
$978$	−1.76604	−	0.642788i	−1.76604	−	0.642788i
$979$	0.673648	+	0.118782i	0.673648	+	0.118782i
$980$		0			0
$981$		0			0
$982$		0			0
$983$		0			0		−0.173648	−	0.984808i	$-0.555556\pi$
$983$		0			0		0.173648	+	0.984808i	$0.444444\pi$
$984$	1.76604	+	0.642788i	1.76604	+	0.642788i
$985$		0			0
$986$		0			0
$987$		0			0
$988$		0			0
$989$		0			0
$990$		0			0
$991$		0			0		0.984808	−	0.173648i	$-0.0555556\pi$
$991$		0			0		−0.984808	+	0.173648i	$0.944444\pi$
$992$		0			0
$993$			1.96962i			1.96962i
$994$		0			0
$995$		0			0
$996$	0.439693	+	1.20805i	0.439693	+	1.20805i
$997$		0			0		0.173648	−	0.984808i	$-0.444444\pi$
$997$		0			0		−0.173648	+	0.984808i	$0.555556\pi$
$998$	1.43969	−	0.524005i	1.43969	−	0.524005i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1368.1.ee.a.1307.1	yes	6
8.3	odd	2	CM	1368.1.ee.a.1307.1	yes	6
9.5	odd	6		1368.1.dk.a.851.1	yes	6
19.14	odd	18		1368.1.dk.a.299.1	✓	6
72.59	even	6		1368.1.dk.a.851.1	yes	6
152.147	even	18		1368.1.dk.a.299.1	✓	6
171.14	even	18	inner	1368.1.ee.a.1211.1	yes	6
1368.1211	odd	18	inner	1368.1.ee.a.1211.1	yes	6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1368.1.dk.a.299.1	✓	6	19.14	odd	18
1368.1.dk.a.299.1	✓	6	152.147	even	18
1368.1.dk.a.851.1	yes	6	9.5	odd	6
1368.1.dk.a.851.1	yes	6	72.59	even	6
1368.1.ee.a.1211.1	yes	6	171.14	even	18	inner
1368.1.ee.a.1211.1	yes	6	1368.1211	odd	18	inner
1368.1.ee.a.1307.1	yes	6	1.1	even	1	trivial
1368.1.ee.a.1307.1	yes	6	8.3	odd	2	CM

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:	\( N \)	\(=\)	\( 1368 = 2^{3} \cdot 3^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	1368.ee (of order \(18\), degree \(6\), minimal)

\(f(q)\)	\(=\)	\(q+(0.766044 - 0.642788i) q^{2} -1.00000 q^{3} +(0.173648 - 0.984808i) q^{4} +(-0.766044 + 0.642788i) q^{6} +(-0.500000 - 0.866025i) q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q+(0.766044 - 0.642788i) q^{2} -1.00000 q^{3} +(0.173648 - 0.984808i) q^{4} +(-0.766044 + 0.642788i) q^{6} +(-0.500000 - 0.866025i) q^{8} +1.00000 q^{9} -0.684040i q^{11} +(-0.173648 + 0.984808i) q^{12} +(-0.939693 - 0.342020i) q^{16} +(0.592396 - 1.62760i) q^{17} +(0.766044 - 0.642788i) q^{18} +(-0.173648 + 0.984808i) q^{19} +(-0.439693 - 0.524005i) q^{22} +(0.500000 + 0.866025i) q^{24} +(-0.766044 - 0.642788i) q^{25} -1.00000 q^{27} +(-0.939693 + 0.342020i) q^{32} +0.684040i q^{33} +(-0.592396 - 1.62760i) q^{34} +(0.173648 - 0.984808i) q^{36} +(0.500000 + 0.866025i) q^{38} +(1.43969 - 1.20805i) q^{41} +(-0.347296 - 1.96962i) q^{43} +(-0.673648 - 0.118782i) q^{44} +(0.939693 + 0.342020i) q^{48} +(-0.500000 + 0.866025i) q^{49} -1.00000 q^{50} +(-0.592396 + 1.62760i) q^{51} +(-0.766044 + 0.642788i) q^{54} +(0.173648 - 0.984808i) q^{57} +(0.266044 + 1.50881i) q^{59} +(-0.500000 + 0.866025i) q^{64} +(0.439693 + 0.524005i) q^{66} +(-0.439693 + 0.524005i) q^{67} +(-1.50000 - 0.866025i) q^{68} +(-0.500000 - 0.866025i) q^{72} +(0.326352 + 1.85083i) q^{73} +(0.766044 + 0.642788i) q^{75} +(0.939693 + 0.342020i) q^{76} +1.00000 q^{81} +(0.326352 - 1.85083i) q^{82} +(1.11334 - 0.642788i) q^{83} +(-1.53209 - 1.28558i) q^{86} +(-0.592396 + 0.342020i) q^{88} +(-0.173648 + 0.984808i) q^{89} +(0.939693 - 0.342020i) q^{96} +(1.26604 + 1.50881i) q^{97} +(0.173648 + 0.984808i) q^{98} -0.684040i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 6 q^{3} - 3 q^{8} + 6 q^{9}+O(q^{10}) \) 6 * q - 6 * q^3 - 3 * q^8 + 6 * q^9	\( 6 q - 6 q^{3} - 3 q^{8} + 6 q^{9} + 3 q^{22} + 3 q^{24} - 6 q^{27} + 3 q^{38} + 3 q^{41} - 3 q^{44} - 3 q^{49} - 6 q^{50} - 3 q^{59} - 3 q^{64} - 3 q^{66} + 3 q^{67} - 9 q^{68} - 3 q^{72} + 3 q^{73} + 6 q^{81} + 3 q^{82} + 3 q^{97}+O(q^{100}) \) 6 * q - 6 * q^3 - 3 * q^8 + 6 * q^9 + 3 * q^22 + 3 * q^24 - 6 * q^27 + 3 * q^38 + 3 * q^41 - 3 * q^44 - 3 * q^49 - 6 * q^50 - 3 * q^59 - 3 * q^64 - 3 * q^66 + 3 * q^67 - 9 * q^68 - 3 * q^72 + 3 * q^73 + 6 * q^81 + 3 * q^82 + 3 * q^97

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