LMFDB - Embedded newform 1368.1.ce.a.949.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1368,1,Mod(493,1368)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1368, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 3, 4, 3]))

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

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

chi := DirichletCharacter("1368.493");

S:= CuspForms(chi, 1);

N := Newforms(S);

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

Newform invariants

comment: select newform

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

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

Self dual:	no
Analytic conductor:	$0.682720937282$
Analytic rank:	$0$
Dimension:	$6$
Relative dimension:	$3$ over $\Q(\zeta_{6})$
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:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Projective image:	$D_{9}$
Projective field:	Galois closure of 9.1.283680450809856.6

Embedding invariants

Embedding label			949.1
Root			$-0.766044 - 0.642788i$ of defining polynomial
Character	$\chi$	$=$	1368.949
Dual form			1368.1.ce.a.493.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-0.500000 - 0.866025i) q^{2} +(-0.939693 + 0.342020i) q^{3} +(-0.500000 + 0.866025i) q^{4} +(0.766044 + 0.642788i) q^{6} +(0.939693 + 1.62760i) q^{7} +1.00000 q^{8} +(0.766044 - 0.642788i) q^{9} +O(q^{10})$	\(q+(-0.500000 - 0.866025i) q^{2} +(-0.939693 + 0.342020i) q^{3} +(-0.500000 + 0.866025i) q^{4} +(0.766044 + 0.642788i) q^{6} +(0.939693 + 1.62760i) q^{7} +1.00000 q^{8} +(0.766044 - 0.642788i) q^{9} +(0.173648 - 0.984808i) q^{12} +(-0.766044 + 1.32683i) q^{13} +(0.939693 - 1.62760i) q^{14} +(-0.500000 - 0.866025i) q^{16} -1.87939 q^{17} +(-0.939693 - 0.342020i) q^{18} +1.00000 q^{19} +(-1.43969 - 1.20805i) q^{21} +(-0.173648 + 0.300767i) q^{23} +(-0.939693 + 0.342020i) q^{24} +(-0.500000 - 0.866025i) q^{25} +1.53209 q^{26} +(-0.500000 + 0.866025i) q^{27} -1.87939 q^{28} +(-0.173648 - 0.300767i) q^{29} +(-0.500000 + 0.866025i) q^{32} +(0.939693 + 1.62760i) q^{34} +(0.173648 + 0.984808i) q^{36} -1.00000 q^{37} +(-0.500000 - 0.866025i) q^{38} +(0.266044 - 1.50881i) q^{39} +(-0.326352 + 1.85083i) q^{42} +0.347296 q^{46} +(0.500000 + 0.866025i) q^{47} +(0.766044 + 0.642788i) q^{48} +(-1.26604 + 2.19285i) q^{49} +(-0.500000 + 0.866025i) q^{50} +(1.76604 - 0.642788i) q^{51} +(-0.766044 - 1.32683i) q^{52} +1.53209 q^{53} +1.00000 q^{54} +(0.939693 + 1.62760i) q^{56} +(-0.939693 + 0.342020i) q^{57} +(-0.173648 + 0.300767i) q^{58} +(-0.766044 + 1.32683i) q^{59} +(1.76604 + 0.642788i) q^{63} +1.00000 q^{64} +(-0.173648 + 0.300767i) q^{67} +(0.939693 - 1.62760i) q^{68} +(0.0603074 - 0.342020i) q^{69} +(0.766044 - 0.642788i) q^{72} +0.347296 q^{73} +(0.500000 + 0.866025i) q^{74} +(0.766044 + 0.642788i) q^{75} +(-0.500000 + 0.866025i) q^{76} +(-1.43969 + 0.524005i) q^{78} +(0.173648 - 0.984808i) q^{81} +(1.76604 - 0.642788i) q^{84} +(0.266044 + 0.223238i) q^{87} -2.87939 q^{91} +(-0.173648 - 0.300767i) q^{92} +(0.500000 - 0.866025i) q^{94} +(0.173648 - 0.984808i) q^{96} +2.53209 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 3 q^{2} - 3 q^{4} + 6 q^{8}+O(q^{10}) $ 6 * q - 3 * q^2 - 3 * q^4 + 6 * q^8	$ 6 q - 3 q^{2} - 3 q^{4} + 6 q^{8} - 3 q^{16} + 6 q^{19} - 3 q^{21} - 3 q^{25} - 3 q^{27} - 3 q^{32} - 6 q^{37} - 3 q^{38} - 3 q^{39} - 3 q^{42} + 3 q^{47} - 3 q^{49} - 3 q^{50} + 6 q^{51} + 6 q^{54} + 6 q^{63} + 6 q^{64} + 6 q^{69} + 3 q^{74} - 3 q^{76} - 3 q^{78} + 6 q^{84} - 3 q^{87} - 6 q^{91} + 3 q^{94} + 6 q^{98}+O(q^{100}) $ 6 * q - 3 * q^2 - 3 * q^4 + 6 * q^8 - 3 * q^16 + 6 * q^19 - 3 * q^21 - 3 * q^25 - 3 * q^27 - 3 * q^32 - 6 * q^37 - 3 * q^38 - 3 * q^39 - 3 * q^42 + 3 * q^47 - 3 * q^49 - 3 * q^50 + 6 * q^51 + 6 * q^54 + 6 * q^63 + 6 * q^64 + 6 * q^69 + 3 * q^74 - 3 * q^76 - 3 * q^78 + 6 * q^84 - 3 * q^87 - 6 * q^91 + 3 * q^94 + 6 * q^98

Display 100 coefficients 10 coefficients

Character values

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

$n$	$343$	$685$	$1009$	$1217$
$\chi(n)$	$1$	$-1$	$-1$	$e\left(\frac{1}{3}\right)$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$2$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$3$	−0.939693	+	0.342020i	−0.939693	+	0.342020i
$3$	−0.939693	+	0.342020i	−0.939693	+	0.342020i
$4$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$5$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$5$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$6$	0.766044	+	0.642788i	0.766044	+	0.642788i
$7$	0.939693	+	1.62760i	0.939693	+	1.62760i	0.766044	+	0.642788i	$0.222222\pi$
$7$	0.939693	+	1.62760i	0.939693	+	1.62760i	0.173648	+	0.984808i	$0.444444\pi$
$8$	1.00000			1.00000
$9$	0.766044	−	0.642788i	0.766044	−	0.642788i
$10$		0			0
$11$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$11$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$12$	0.173648	−	0.984808i	0.173648	−	0.984808i
$13$	−0.766044	+	1.32683i	−0.766044	+	1.32683i	0.173648	+	0.984808i	$0.444444\pi$
$13$	−0.766044	+	1.32683i	−0.766044	+	1.32683i	−0.939693	+	0.342020i	$0.888889\pi$
$14$	0.939693	−	1.62760i	0.939693	−	1.62760i
$15$		0			0
$16$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$17$	−1.87939			−1.87939			−0.939693	−	0.342020i	$-0.888889\pi$
$17$	−1.87939			−1.87939			−0.939693	+	0.342020i	$0.888889\pi$
$18$	−0.939693	−	0.342020i	−0.939693	−	0.342020i
$19$	1.00000			1.00000
$19$	1.00000			1.00000
$20$		0			0
$21$	−1.43969	−	1.20805i	−1.43969	−	1.20805i
$22$		0			0
$23$	−0.173648	+	0.300767i	−0.173648	+	0.300767i	−0.939693	−	0.342020i	$-0.888889\pi$
$23$	−0.173648	+	0.300767i	−0.173648	+	0.300767i	0.766044	+	0.642788i	$0.222222\pi$
$24$	−0.939693	+	0.342020i	−0.939693	+	0.342020i
$25$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$26$	1.53209			1.53209
$27$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$28$	−1.87939			−1.87939
$29$	−0.173648	−	0.300767i	−0.173648	−	0.300767i	0.766044	−	0.642788i	$-0.222222\pi$
$29$	−0.173648	−	0.300767i	−0.173648	−	0.300767i	−0.939693	+	0.342020i	$0.888889\pi$
$30$		0			0
$31$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$31$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$32$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$33$		0			0
$34$	0.939693	+	1.62760i	0.939693	+	1.62760i
$35$		0			0
$36$	0.173648	+	0.984808i	0.173648	+	0.984808i
$37$	−1.00000			−1.00000			−0.500000	−	0.866025i	$-0.666667\pi$
$37$	−1.00000			−1.00000			−0.500000	+	0.866025i	$0.666667\pi$
$38$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$39$	0.266044	−	1.50881i	0.266044	−	1.50881i
$40$		0			0
$41$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$41$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$42$	−0.326352	+	1.85083i	−0.326352	+	1.85083i
$43$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$43$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$44$		0			0
$45$		0			0
$46$	0.347296			0.347296
$47$	0.500000	+	0.866025i	0.500000	+	0.866025i	1.00000			$0$
$47$	0.500000	+	0.866025i	0.500000	+	0.866025i	−0.500000	+	0.866025i	$0.666667\pi$
$48$	0.766044	+	0.642788i	0.766044	+	0.642788i
$49$	−1.26604	+	2.19285i	−1.26604	+	2.19285i
$50$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$51$	1.76604	−	0.642788i	1.76604	−	0.642788i
$52$	−0.766044	−	1.32683i	−0.766044	−	1.32683i
$53$	1.53209			1.53209			0.766044	−	0.642788i	$-0.222222\pi$
$53$	1.53209			1.53209			0.766044	+	0.642788i	$0.222222\pi$
$54$	1.00000			1.00000
$55$		0			0
$56$	0.939693	+	1.62760i	0.939693	+	1.62760i
$57$	−0.939693	+	0.342020i	−0.939693	+	0.342020i
$58$	−0.173648	+	0.300767i	−0.173648	+	0.300767i
$59$	−0.766044	+	1.32683i	−0.766044	+	1.32683i	0.173648	+	0.984808i	$0.444444\pi$
$59$	−0.766044	+	1.32683i	−0.766044	+	1.32683i	−0.939693	+	0.342020i	$0.888889\pi$
$60$		0			0
$61$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$61$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$62$		0			0
$63$	1.76604	+	0.642788i	1.76604	+	0.642788i
$64$	1.00000			1.00000
$65$		0			0
$66$		0			0
$67$	−0.173648	+	0.300767i	−0.173648	+	0.300767i	−0.939693	−	0.342020i	$-0.888889\pi$
$67$	−0.173648	+	0.300767i	−0.173648	+	0.300767i	0.766044	+	0.642788i	$0.222222\pi$
$68$	0.939693	−	1.62760i	0.939693	−	1.62760i
$69$	0.0603074	−	0.342020i	0.0603074	−	0.342020i
$70$		0			0
$71$		0			0		1.00000			$0$
$71$		0			0		−1.00000			$\pi$
$72$	0.766044	−	0.642788i	0.766044	−	0.642788i
$73$	0.347296			0.347296			0.173648	−	0.984808i	$-0.444444\pi$
$73$	0.347296			0.347296			0.173648	+	0.984808i	$0.444444\pi$
$74$	0.500000	+	0.866025i	0.500000	+	0.866025i
$75$	0.766044	+	0.642788i	0.766044	+	0.642788i
$76$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$77$		0			0
$78$	−1.43969	+	0.524005i	−1.43969	+	0.524005i
$79$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$79$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$80$		0			0
$81$	0.173648	−	0.984808i	0.173648	−	0.984808i
$82$		0			0
$83$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$83$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$84$	1.76604	−	0.642788i	1.76604	−	0.642788i
$85$		0			0
$86$		0			0
$87$	0.266044	+	0.223238i	0.266044	+	0.223238i
$88$		0			0
$89$		0			0		1.00000			$0$
$89$		0			0		−1.00000			$\pi$
$90$		0			0
$91$	−2.87939			−2.87939
$92$	−0.173648	−	0.300767i	−0.173648	−	0.300767i
$93$		0			0
$94$	0.500000	−	0.866025i	0.500000	−	0.866025i
$95$		0			0
$96$	0.173648	−	0.984808i	0.173648	−	0.984808i
$97$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$97$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$98$	2.53209			2.53209
$99$		0			0
$100$	1.00000			1.00000
$101$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$101$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$102$	−1.43969	−	1.20805i	−1.43969	−	1.20805i
$103$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$103$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$104$	−0.766044	+	1.32683i	−0.766044	+	1.32683i
$105$		0			0
$106$	−0.766044	−	1.32683i	−0.766044	−	1.32683i
$107$	0.347296			0.347296			0.173648	−	0.984808i	$-0.444444\pi$
$107$	0.347296			0.347296			0.173648	+	0.984808i	$0.444444\pi$
$108$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$109$	0.347296			0.347296			0.173648	−	0.984808i	$-0.444444\pi$
$109$	0.347296			0.347296			0.173648	+	0.984808i	$0.444444\pi$
$110$		0			0
$111$	0.939693	−	0.342020i	0.939693	−	0.342020i
$112$	0.939693	−	1.62760i	0.939693	−	1.62760i
$113$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$113$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$114$	0.766044	+	0.642788i	0.766044	+	0.642788i
$115$		0			0
$116$	0.347296			0.347296
$117$	0.266044	+	1.50881i	0.266044	+	1.50881i
$118$	1.53209			1.53209
$119$	−1.76604	−	3.05888i	−1.76604	−	3.05888i
$120$		0			0
$121$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$122$		0			0
$123$		0			0
$124$		0			0
$125$		0			0
$126$	−0.326352	−	1.85083i	−0.326352	−	1.85083i
$127$		0			0		1.00000			$0$
$127$		0			0		−1.00000			$\pi$
$128$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$129$		0			0
$130$		0			0
$131$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$131$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$132$		0			0
$133$	0.939693	+	1.62760i	0.939693	+	1.62760i
$134$	0.347296			0.347296
$135$		0			0
$136$	−1.87939			−1.87939
$137$	−0.766044	−	1.32683i	−0.766044	−	1.32683i	−0.939693	−	0.342020i	$-0.888889\pi$
$137$	−0.766044	−	1.32683i	−0.766044	−	1.32683i	0.173648	−	0.984808i	$-0.444444\pi$
$138$	−0.326352	+	0.118782i	−0.326352	+	0.118782i
$139$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$139$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$140$		0			0
$141$	−0.766044	−	0.642788i	−0.766044	−	0.642788i
$142$		0			0
$143$		0			0
$144$	−0.939693	−	0.342020i	−0.939693	−	0.342020i
$145$		0			0
$146$	−0.173648	−	0.300767i	−0.173648	−	0.300767i
$147$	0.439693	−	2.49362i	0.439693	−	2.49362i
$148$	0.500000	−	0.866025i	0.500000	−	0.866025i
$149$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$149$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$150$	0.173648	−	0.984808i	0.173648	−	0.984808i
$151$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$151$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$152$	1.00000			1.00000
$153$	−1.43969	+	1.20805i	−1.43969	+	1.20805i
$154$		0			0
$155$		0			0
$156$	1.17365	+	0.984808i	1.17365	+	0.984808i
$157$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$157$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$158$		0			0
$159$	−1.43969	+	0.524005i	−1.43969	+	0.524005i
$160$		0			0
$161$	−0.652704			−0.652704
$162$	−0.939693	+	0.342020i	−0.939693	+	0.342020i
$163$		0			0		1.00000			$0$
$163$		0			0		−1.00000			$\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$167$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$168$	−1.43969	−	1.20805i	−1.43969	−	1.20805i
$169$	−0.673648	−	1.16679i	−0.673648	−	1.16679i
$170$		0			0
$171$	0.766044	−	0.642788i	0.766044	−	0.642788i
$172$		0			0
$173$	0.500000	+	0.866025i	0.500000	+	0.866025i	1.00000			$0$
$173$	0.500000	+	0.866025i	0.500000	+	0.866025i	−0.500000	+	0.866025i	$0.666667\pi$
$174$	0.0603074	−	0.342020i	0.0603074	−	0.342020i
$175$	0.939693	−	1.62760i	0.939693	−	1.62760i
$176$		0			0
$177$	0.266044	−	1.50881i	0.266044	−	1.50881i
$178$		0			0
$179$	−1.00000			−1.00000			−0.500000	−	0.866025i	$-0.666667\pi$
$179$	−1.00000			−1.00000			−0.500000	+	0.866025i	$0.666667\pi$
$180$		0			0
$181$	2.00000			2.00000			1.00000			$0$
$181$	2.00000			2.00000			1.00000			$0$
$182$	1.43969	+	2.49362i	1.43969	+	2.49362i
$183$		0			0
$184$	−0.173648	+	0.300767i	−0.173648	+	0.300767i
$185$		0			0
$186$		0			0
$187$		0			0
$188$	−1.00000			−1.00000
$189$	−1.87939			−1.87939
$190$		0			0
$191$	0.939693	+	1.62760i	0.939693	+	1.62760i	0.766044	+	0.642788i	$0.222222\pi$
$191$	0.939693	+	1.62760i	0.939693	+	1.62760i	0.173648	+	0.984808i	$0.444444\pi$
$192$	−0.939693	+	0.342020i	−0.939693	+	0.342020i
$193$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$193$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$194$		0			0
$195$		0			0
$196$	−1.26604	−	2.19285i	−1.26604	−	2.19285i
$197$		0			0		1.00000			$0$
$197$		0			0		−1.00000			$\pi$
$198$		0			0
$199$	0.347296			0.347296			0.173648	−	0.984808i	$-0.444444\pi$
$199$	0.347296			0.347296			0.173648	+	0.984808i	$0.444444\pi$
$200$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$201$	0.0603074	−	0.342020i	0.0603074	−	0.342020i
$202$		0			0
$203$	0.326352	−	0.565258i	0.326352	−	0.565258i
$204$	−0.326352	+	1.85083i	−0.326352	+	1.85083i
$205$		0			0
$206$		0			0
$207$	0.0603074	+	0.342020i	0.0603074	+	0.342020i
$208$	1.53209			1.53209
$209$		0			0
$210$		0			0
$211$	0.939693	−	1.62760i	0.939693	−	1.62760i	0.173648	−	0.984808i	$-0.444444\pi$
$211$	0.939693	−	1.62760i	0.939693	−	1.62760i	0.766044	−	0.642788i	$-0.222222\pi$
$212$	−0.766044	+	1.32683i	−0.766044	+	1.32683i
$213$		0			0
$214$	−0.173648	−	0.300767i	−0.173648	−	0.300767i
$215$		0			0
$216$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$217$		0			0
$218$	−0.173648	−	0.300767i	−0.173648	−	0.300767i
$219$	−0.326352	+	0.118782i	−0.326352	+	0.118782i
$220$		0			0
$221$	1.43969	−	2.49362i	1.43969	−	2.49362i
$222$	−0.766044	−	0.642788i	−0.766044	−	0.642788i
$223$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$223$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$224$	−1.87939			−1.87939
$225$	−0.939693	−	0.342020i	−0.939693	−	0.342020i
$226$		0			0
$227$	0.939693	+	1.62760i	0.939693	+	1.62760i	0.766044	+	0.642788i	$0.222222\pi$
$227$	0.939693	+	1.62760i	0.939693	+	1.62760i	0.173648	+	0.984808i	$0.444444\pi$
$228$	0.173648	−	0.984808i	0.173648	−	0.984808i
$229$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$229$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$230$		0			0
$231$		0			0
$232$	−0.173648	−	0.300767i	−0.173648	−	0.300767i
$233$	2.00000			2.00000			1.00000			$0$
$233$	2.00000			2.00000			1.00000			$0$
$234$	1.17365	−	0.984808i	1.17365	−	0.984808i
$235$		0			0
$236$	−0.766044	−	1.32683i	−0.766044	−	1.32683i
$237$		0			0
$238$	−1.76604	+	3.05888i	−1.76604	+	3.05888i
$239$	−0.766044	+	1.32683i	−0.766044	+	1.32683i	0.173648	+	0.984808i	$0.444444\pi$
$239$	−0.766044	+	1.32683i	−0.766044	+	1.32683i	−0.939693	+	0.342020i	$0.888889\pi$
$240$		0			0
$241$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$241$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$242$	1.00000			1.00000
$243$	0.173648	+	0.984808i	0.173648	+	0.984808i
$244$		0			0
$245$		0			0
$246$		0			0
$247$	−0.766044	+	1.32683i	−0.766044	+	1.32683i
$248$		0			0
$249$		0			0
$250$		0			0
$251$		0			0		1.00000			$0$
$251$		0			0		−1.00000			$\pi$
$252$	−1.43969	+	1.20805i	−1.43969	+	1.20805i
$253$		0			0
$254$		0			0
$255$		0			0
$256$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$257$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$257$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$258$		0			0
$259$	−0.939693	−	1.62760i	−0.939693	−	1.62760i
$260$		0			0
$261$	−0.326352	−	0.118782i	−0.326352	−	0.118782i
$262$		0			0
$263$	0.500000	+	0.866025i	0.500000	+	0.866025i	1.00000			$0$
$263$	0.500000	+	0.866025i	0.500000	+	0.866025i	−0.500000	+	0.866025i	$0.666667\pi$
$264$		0			0
$265$		0			0
$266$	0.939693	−	1.62760i	0.939693	−	1.62760i
$267$		0			0
$268$	−0.173648	−	0.300767i	−0.173648	−	0.300767i
$269$	−1.00000			−1.00000			−0.500000	−	0.866025i	$-0.666667\pi$
$269$	−1.00000			−1.00000			−0.500000	+	0.866025i	$0.666667\pi$
$270$		0			0
$271$	1.53209			1.53209			0.766044	−	0.642788i	$-0.222222\pi$
$271$	1.53209			1.53209			0.766044	+	0.642788i	$0.222222\pi$
$272$	0.939693	+	1.62760i	0.939693	+	1.62760i
$273$	2.70574	−	0.984808i	2.70574	−	0.984808i
$274$	−0.766044	+	1.32683i	−0.766044	+	1.32683i
$275$		0			0
$276$	0.266044	+	0.223238i	0.266044	+	0.223238i
$277$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$277$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$281$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$282$	−0.173648	+	0.984808i	−0.173648	+	0.984808i
$283$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$283$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$	0.173648	+	0.984808i	0.173648	+	0.984808i
$289$	2.53209			2.53209
$290$		0			0
$291$		0			0
$292$	−0.173648	+	0.300767i	−0.173648	+	0.300767i
$293$	0.939693	−	1.62760i	0.939693	−	1.62760i	0.173648	−	0.984808i	$-0.444444\pi$
$293$	0.939693	−	1.62760i	0.939693	−	1.62760i	0.766044	−	0.642788i	$-0.222222\pi$
$294$	−2.37939	+	0.866025i	−2.37939	+	0.866025i
$295$		0			0
$296$	−1.00000			−1.00000
$297$		0			0
$298$		0			0
$299$	−0.266044	−	0.460802i	−0.266044	−	0.460802i
$300$	−0.939693	+	0.342020i	−0.939693	+	0.342020i
$301$		0			0
$302$		0			0
$303$		0			0
$304$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$305$		0			0
$306$	1.76604	+	0.642788i	1.76604	+	0.642788i
$307$	−1.00000			−1.00000			−0.500000	−	0.866025i	$-0.666667\pi$
$307$	−1.00000			−1.00000			−0.500000	+	0.866025i	$0.666667\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	0.939693	−	1.62760i	0.939693	−	1.62760i	0.173648	−	0.984808i	$-0.444444\pi$
$311$	0.939693	−	1.62760i	0.939693	−	1.62760i	0.766044	−	0.642788i	$-0.222222\pi$
$312$	0.266044	−	1.50881i	0.266044	−	1.50881i
$313$	−0.766044	−	1.32683i	−0.766044	−	1.32683i	−0.939693	−	0.342020i	$-0.888889\pi$
$313$	−0.766044	−	1.32683i	−0.766044	−	1.32683i	0.173648	−	0.984808i	$-0.444444\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−0.766044	−	1.32683i	−0.766044	−	1.32683i	−0.939693	−	0.342020i	$-0.888889\pi$
$317$	−0.766044	−	1.32683i	−0.766044	−	1.32683i	0.173648	−	0.984808i	$-0.444444\pi$
$318$	1.17365	+	0.984808i	1.17365	+	0.984808i
$319$		0			0
$320$		0			0
$321$	−0.326352	+	0.118782i	−0.326352	+	0.118782i
$322$	0.326352	+	0.565258i	0.326352	+	0.565258i
$323$	−1.87939			−1.87939
$324$	0.766044	+	0.642788i	0.766044	+	0.642788i
$325$	1.53209			1.53209
$326$		0			0
$327$	−0.326352	+	0.118782i	−0.326352	+	0.118782i
$328$		0			0
$329$	−0.939693	+	1.62760i	−0.939693	+	1.62760i
$330$		0			0
$331$	−0.766044	−	1.32683i	−0.766044	−	1.32683i	−0.939693	−	0.342020i	$-0.888889\pi$
$331$	−0.766044	−	1.32683i	−0.766044	−	1.32683i	0.173648	−	0.984808i	$-0.444444\pi$
$332$		0			0
$333$	−0.766044	+	0.642788i	−0.766044	+	0.642788i
$334$		0			0
$335$		0			0
$336$	−0.326352	+	1.85083i	−0.326352	+	1.85083i
$337$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$337$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$338$	−0.673648	+	1.16679i	−0.673648	+	1.16679i
$339$		0			0
$340$		0			0
$341$		0			0
$342$	−0.939693	−	0.342020i	−0.939693	−	0.342020i
$343$	−2.87939			−2.87939
$344$		0			0
$345$		0			0
$346$	0.500000	−	0.866025i	0.500000	−	0.866025i
$347$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$347$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$348$	−0.326352	+	0.118782i	−0.326352	+	0.118782i
$349$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$349$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$350$	−1.87939			−1.87939
$351$	−0.766044	−	1.32683i	−0.766044	−	1.32683i
$352$		0			0
$353$	−0.173648	−	0.300767i	−0.173648	−	0.300767i	0.766044	−	0.642788i	$-0.222222\pi$
$353$	−0.173648	−	0.300767i	−0.173648	−	0.300767i	−0.939693	+	0.342020i	$0.888889\pi$
$354$	−1.43969	+	0.524005i	−1.43969	+	0.524005i
$355$		0			0
$356$		0			0
$357$	2.70574	+	2.27038i	2.70574	+	2.27038i
$358$	0.500000	+	0.866025i	0.500000	+	0.866025i
$359$	−1.87939			−1.87939			−0.939693	−	0.342020i	$-0.888889\pi$
$359$	−1.87939			−1.87939			−0.939693	+	0.342020i	$0.888889\pi$
$360$		0			0
$361$	1.00000			1.00000
$362$	−1.00000	−	1.73205i	−1.00000	−	1.73205i
$363$	0.173648	−	0.984808i	0.173648	−	0.984808i
$364$	1.43969	−	2.49362i	1.43969	−	2.49362i
$365$		0			0
$366$		0			0
$367$	0.500000	+	0.866025i	0.500000	+	0.866025i	1.00000			$0$
$367$	0.500000	+	0.866025i	0.500000	+	0.866025i	−0.500000	+	0.866025i	$0.666667\pi$
$368$	0.347296			0.347296
$369$		0			0
$370$		0			0
$371$	1.43969	+	2.49362i	1.43969	+	2.49362i
$372$		0			0
$373$	0.939693	−	1.62760i	0.939693	−	1.62760i	0.173648	−	0.984808i	$-0.444444\pi$
$373$	0.939693	−	1.62760i	0.939693	−	1.62760i	0.766044	−	0.642788i	$-0.222222\pi$
$374$		0			0
$375$		0			0
$376$	0.500000	+	0.866025i	0.500000	+	0.866025i
$377$	0.532089			0.532089
$378$	0.939693	+	1.62760i	0.939693	+	1.62760i
$379$	1.53209			1.53209			0.766044	−	0.642788i	$-0.222222\pi$
$379$	1.53209			1.53209			0.766044	+	0.642788i	$0.222222\pi$
$380$		0			0
$381$		0			0
$382$	0.939693	−	1.62760i	0.939693	−	1.62760i
$383$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$383$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$384$	0.766044	+	0.642788i	0.766044	+	0.642788i
$385$		0			0
$386$		0			0
$387$		0			0
$388$		0			0
$389$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$389$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$390$		0			0
$391$	0.326352	−	0.565258i	0.326352	−	0.565258i
$392$	−1.26604	+	2.19285i	−1.26604	+	2.19285i
$393$		0			0
$394$		0			0
$395$		0			0
$396$		0			0
$397$		0			0		1.00000			$0$
$397$		0			0		−1.00000			$\pi$
$398$	−0.173648	−	0.300767i	−0.173648	−	0.300767i
$399$	−1.43969	−	1.20805i	−1.43969	−	1.20805i
$400$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$401$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$401$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$402$	−0.326352	+	0.118782i	−0.326352	+	0.118782i
$403$		0			0
$404$		0			0
$405$		0			0
$406$	−0.652704			−0.652704
$407$		0			0
$408$	1.76604	−	0.642788i	1.76604	−	0.642788i
$409$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$409$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$410$		0			0
$411$	1.17365	+	0.984808i	1.17365	+	0.984808i
$412$		0			0
$413$	−2.87939			−2.87939
$414$	0.266044	−	0.223238i	0.266044	−	0.223238i
$415$		0			0
$416$	−0.766044	−	1.32683i	−0.766044	−	1.32683i
$417$		0			0
$418$		0			0
$419$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$419$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$420$		0			0
$421$	0.939693	+	1.62760i	0.939693	+	1.62760i	0.766044	+	0.642788i	$0.222222\pi$
$421$	0.939693	+	1.62760i	0.939693	+	1.62760i	0.173648	+	0.984808i	$0.444444\pi$
$422$	−1.87939			−1.87939
$423$	0.939693	+	0.342020i	0.939693	+	0.342020i
$424$	1.53209			1.53209
$425$	0.939693	+	1.62760i	0.939693	+	1.62760i
$426$		0			0
$427$		0			0
$428$	−0.173648	+	0.300767i	−0.173648	+	0.300767i
$429$		0			0
$430$		0			0
$431$		0			0		1.00000			$0$
$431$		0			0		−1.00000			$\pi$
$432$	1.00000			1.00000
$433$		0			0		1.00000			$0$
$433$		0			0		−1.00000			$\pi$
$434$		0			0
$435$		0			0
$436$	−0.173648	+	0.300767i	−0.173648	+	0.300767i
$437$	−0.173648	+	0.300767i	−0.173648	+	0.300767i
$438$	0.266044	+	0.223238i	0.266044	+	0.223238i
$439$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$439$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$440$		0			0
$441$	0.439693	+	2.49362i	0.439693	+	2.49362i
$442$	−2.87939			−2.87939
$443$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$443$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$444$	−0.173648	+	0.984808i	−0.173648	+	0.984808i
$445$		0			0
$446$		0			0
$447$		0			0
$448$	0.939693	+	1.62760i	0.939693	+	1.62760i
$449$		0			0		1.00000			$0$
$449$		0			0		−1.00000			$\pi$
$450$	0.173648	+	0.984808i	0.173648	+	0.984808i
$451$		0			0
$452$		0			0
$453$		0			0
$454$	0.939693	−	1.62760i	0.939693	−	1.62760i
$455$		0			0
$456$	−0.939693	+	0.342020i	−0.939693	+	0.342020i
$457$	0.939693	+	1.62760i	0.939693	+	1.62760i	0.766044	+	0.642788i	$0.222222\pi$
$457$	0.939693	+	1.62760i	0.939693	+	1.62760i	0.173648	+	0.984808i	$0.444444\pi$
$458$		0			0
$459$	0.939693	−	1.62760i	0.939693	−	1.62760i
$460$		0			0
$461$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$461$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$462$		0			0
$463$	0.500000	−	0.866025i	0.500000	−	0.866025i	−0.500000	−	0.866025i	$-0.666667\pi$
$463$	0.500000	−	0.866025i	0.500000	−	0.866025i	1.00000			$0$
$464$	−0.173648	+	0.300767i	−0.173648	+	0.300767i
$465$		0			0
$466$	−1.00000	−	1.73205i	−1.00000	−	1.73205i
$467$		0			0		1.00000			$0$
$467$		0			0		−1.00000			$\pi$
$468$	−1.43969	−	0.524005i	−1.43969	−	0.524005i
$469$	−0.652704			−0.652704
$470$		0			0
$471$		0			0
$472$	−0.766044	+	1.32683i	−0.766044	+	1.32683i
$473$		0			0
$474$		0			0
$475$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$476$	3.53209			3.53209
$477$	1.17365	−	0.984808i	1.17365	−	0.984808i
$478$	1.53209			1.53209
$479$	0.500000	+	0.866025i	0.500000	+	0.866025i	1.00000			$0$
$479$	0.500000	+	0.866025i	0.500000	+	0.866025i	−0.500000	+	0.866025i	$0.666667\pi$
$480$		0			0
$481$	0.766044	−	1.32683i	0.766044	−	1.32683i
$482$		0			0
$483$	0.613341	−	0.223238i	0.613341	−	0.223238i
$484$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$485$		0			0
$486$	0.766044	−	0.642788i	0.766044	−	0.642788i
$487$		0			0		1.00000			$0$
$487$		0			0		−1.00000			$\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$491$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$492$		0			0
$493$	0.326352	+	0.565258i	0.326352	+	0.565258i
$494$	1.53209			1.53209
$495$		0			0
$496$		0			0
$497$		0			0
$498$		0			0
$499$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$499$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	1.53209			1.53209			0.766044	−	0.642788i	$-0.222222\pi$
$503$	1.53209			1.53209			0.766044	+	0.642788i	$0.222222\pi$
$504$	1.76604	+	0.642788i	1.76604	+	0.642788i
$505$		0			0
$506$		0			0
$507$	1.03209	+	0.866025i	1.03209	+	0.866025i
$508$		0			0
$509$	0.500000	−	0.866025i	0.500000	−	0.866025i	−0.500000	−	0.866025i	$-0.666667\pi$
$509$	0.500000	−	0.866025i	0.500000	−	0.866025i	1.00000			$0$
$510$		0			0
$511$	0.326352	+	0.565258i	0.326352	+	0.565258i
$512$	1.00000			1.00000
$513$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$514$		0			0
$515$		0			0
$516$		0			0
$517$		0			0
$518$	−0.939693	+	1.62760i	−0.939693	+	1.62760i
$519$	−0.766044	−	0.642788i	−0.766044	−	0.642788i
$520$		0			0
$521$		0			0		1.00000			$0$
$521$		0			0		−1.00000			$\pi$
$522$	0.0603074	+	0.342020i	0.0603074	+	0.342020i
$523$	1.53209			1.53209			0.766044	−	0.642788i	$-0.222222\pi$
$523$	1.53209			1.53209			0.766044	+	0.642788i	$0.222222\pi$
$524$		0			0
$525$	−0.326352	+	1.85083i	−0.326352	+	1.85083i
$526$	0.500000	−	0.866025i	0.500000	−	0.866025i
$527$		0			0
$528$		0			0
$529$	0.439693	+	0.761570i	0.439693	+	0.761570i
$530$		0			0
$531$	0.266044	+	1.50881i	0.266044	+	1.50881i
$532$	−1.87939			−1.87939
$533$		0			0
$534$		0			0
$535$		0			0
$536$	−0.173648	+	0.300767i	−0.173648	+	0.300767i
$537$	0.939693	−	0.342020i	0.939693	−	0.342020i
$538$	0.500000	+	0.866025i	0.500000	+	0.866025i
$539$		0			0
$540$		0			0
$541$		0			0		1.00000			$0$
$541$		0			0		−1.00000			$\pi$
$542$	−0.766044	−	1.32683i	−0.766044	−	1.32683i
$543$	−1.87939	+	0.684040i	−1.87939	+	0.684040i
$544$	0.939693	−	1.62760i	0.939693	−	1.62760i
$545$		0			0
$546$	−2.20574	−	1.85083i	−2.20574	−	1.85083i
$547$	0.500000	+	0.866025i	0.500000	+	0.866025i	1.00000			$0$
$547$	0.500000	+	0.866025i	0.500000	+	0.866025i	−0.500000	+	0.866025i	$0.666667\pi$
$548$	1.53209			1.53209
$549$		0			0
$550$		0			0
$551$	−0.173648	−	0.300767i	−0.173648	−	0.300767i
$552$	0.0603074	−	0.342020i	0.0603074	−	0.342020i
$553$		0			0
$554$		0			0
$555$		0			0
$556$		0			0
$557$		0			0		1.00000			$0$
$557$		0			0		−1.00000			$\pi$
$558$		0			0
$559$		0			0
$560$		0			0
$561$		0			0
$562$		0			0
$563$	0.500000	−	0.866025i	0.500000	−	0.866025i	−0.500000	−	0.866025i	$-0.666667\pi$
$563$	0.500000	−	0.866025i	0.500000	−	0.866025i	1.00000			$0$
$564$	0.939693	−	0.342020i	0.939693	−	0.342020i
$565$		0			0
$566$		0			0
$567$	1.76604	−	0.642788i	1.76604	−	0.642788i
$568$		0			0
$569$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$569$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$570$		0			0
$571$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$571$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$572$		0			0
$573$	−1.43969	−	1.20805i	−1.43969	−	1.20805i
$574$		0			0
$575$	0.347296			0.347296
$576$	0.766044	−	0.642788i	0.766044	−	0.642788i
$577$	−1.87939			−1.87939			−0.939693	−	0.342020i	$-0.888889\pi$
$577$	−1.87939			−1.87939			−0.939693	+	0.342020i	$0.888889\pi$
$578$	−1.26604	−	2.19285i	−1.26604	−	2.19285i
$579$		0			0
$580$		0			0
$581$		0			0
$582$		0			0
$583$		0			0
$584$	0.347296			0.347296
$585$		0			0
$586$	−1.87939			−1.87939
$587$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$587$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$588$	1.93969	+	1.62760i	1.93969	+	1.62760i
$589$		0			0
$590$		0			0
$591$		0			0
$592$	0.500000	+	0.866025i	0.500000	+	0.866025i
$593$	−1.00000			−1.00000			−0.500000	−	0.866025i	$-0.666667\pi$
$593$	−1.00000			−1.00000			−0.500000	+	0.866025i	$0.666667\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$	−0.326352	+	0.118782i	−0.326352	+	0.118782i
$598$	−0.266044	+	0.460802i	−0.266044	+	0.460802i
$599$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$599$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$600$	0.766044	+	0.642788i	0.766044	+	0.642788i
$601$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$601$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$602$		0			0
$603$	0.0603074	+	0.342020i	0.0603074	+	0.342020i
$604$		0			0
$605$		0			0
$606$		0			0
$607$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$607$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$608$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$609$	−0.113341	+	0.642788i	−0.113341	+	0.642788i
$610$		0			0
$611$	−1.53209			−1.53209
$612$	−0.326352	−	1.85083i	−0.326352	−	1.85083i
$613$		0			0		1.00000			$0$
$613$		0			0		−1.00000			$\pi$
$614$	0.500000	+	0.866025i	0.500000	+	0.866025i
$615$		0			0
$616$		0			0
$617$	0.500000	−	0.866025i	0.500000	−	0.866025i	−0.500000	−	0.866025i	$-0.666667\pi$
$617$	0.500000	−	0.866025i	0.500000	−	0.866025i	1.00000			$0$
$618$		0			0
$619$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$619$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$620$		0			0
$621$	−0.173648	−	0.300767i	−0.173648	−	0.300767i
$622$	−1.87939			−1.87939
$623$		0			0
$624$	−1.43969	+	0.524005i	−1.43969	+	0.524005i
$625$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$626$	−0.766044	+	1.32683i	−0.766044	+	1.32683i
$627$		0			0
$628$		0			0
$629$	1.87939			1.87939
$630$		0			0
$631$	−1.00000			−1.00000			−0.500000	−	0.866025i	$-0.666667\pi$
$631$	−1.00000			−1.00000			−0.500000	+	0.866025i	$0.666667\pi$
$632$		0			0
$633$	−0.326352	+	1.85083i	−0.326352	+	1.85083i
$634$	−0.766044	+	1.32683i	−0.766044	+	1.32683i
$635$		0			0
$636$	0.266044	−	1.50881i	0.266044	−	1.50881i
$637$	−1.93969	−	3.35965i	−1.93969	−	3.35965i
$638$		0			0
$639$		0			0
$640$		0			0
$641$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$641$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$642$	0.266044	+	0.223238i	0.266044	+	0.223238i
$643$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$643$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$644$	0.326352	−	0.565258i	0.326352	−	0.565258i
$645$		0			0
$646$	0.939693	+	1.62760i	0.939693	+	1.62760i
$647$	−1.87939			−1.87939			−0.939693	−	0.342020i	$-0.888889\pi$
$647$	−1.87939			−1.87939			−0.939693	+	0.342020i	$0.888889\pi$
$648$	0.173648	−	0.984808i	0.173648	−	0.984808i
$649$		0			0
$650$	−0.766044	−	1.32683i	−0.766044	−	1.32683i
$651$		0			0
$652$		0			0
$653$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$653$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$654$	0.266044	+	0.223238i	0.266044	+	0.223238i
$655$		0			0
$656$		0			0
$657$	0.266044	−	0.223238i	0.266044	−	0.223238i
$658$	1.87939			1.87939
$659$	−0.766044	−	1.32683i	−0.766044	−	1.32683i	−0.939693	−	0.342020i	$-0.888889\pi$
$659$	−0.766044	−	1.32683i	−0.766044	−	1.32683i	0.173648	−	0.984808i	$-0.444444\pi$
$660$		0			0
$661$	−0.173648	+	0.300767i	−0.173648	+	0.300767i	−0.939693	−	0.342020i	$-0.888889\pi$
$661$	−0.173648	+	0.300767i	−0.173648	+	0.300767i	0.766044	+	0.642788i	$0.222222\pi$
$662$	−0.766044	+	1.32683i	−0.766044	+	1.32683i
$663$	−0.500000	+	2.83564i	−0.500000	+	2.83564i
$664$		0			0
$665$		0			0
$666$	0.939693	+	0.342020i	0.939693	+	0.342020i
$667$	0.120615			0.120615
$668$		0			0
$669$		0			0
$670$		0			0
$671$		0			0
$672$	1.76604	−	0.642788i	1.76604	−	0.642788i
$673$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$673$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$674$		0			0
$675$	1.00000			1.00000
$676$	1.34730			1.34730
$677$	0.939693	+	1.62760i	0.939693	+	1.62760i	0.766044	+	0.642788i	$0.222222\pi$
$677$	0.939693	+	1.62760i	0.939693	+	1.62760i	0.173648	+	0.984808i	$0.444444\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$	−1.43969	−	1.20805i	−1.43969	−	1.20805i
$682$		0			0
$683$	−1.00000			−1.00000			−0.500000	−	0.866025i	$-0.666667\pi$
$683$	−1.00000			−1.00000			−0.500000	+	0.866025i	$0.666667\pi$
$684$	0.173648	+	0.984808i	0.173648	+	0.984808i
$685$		0			0
$686$	1.43969	+	2.49362i	1.43969	+	2.49362i
$687$		0			0
$688$		0			0
$689$	−1.17365	+	2.03282i	−1.17365	+	2.03282i
$690$		0			0
$691$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$691$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$692$	−1.00000			−1.00000
$693$		0			0
$694$		0			0
$695$		0			0
$696$	0.266044	+	0.223238i	0.266044	+	0.223238i
$697$		0			0
$698$		0			0
$699$	−1.87939	+	0.684040i	−1.87939	+	0.684040i
$700$	0.939693	+	1.62760i	0.939693	+	1.62760i
$701$		0			0		1.00000			$0$
$701$		0			0		−1.00000			$\pi$
$702$	−0.766044	+	1.32683i	−0.766044	+	1.32683i
$703$	−1.00000			−1.00000
$704$		0			0
$705$		0			0
$706$	−0.173648	+	0.300767i	−0.173648	+	0.300767i
$707$		0			0
$708$	1.17365	+	0.984808i	1.17365	+	0.984808i
$709$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$709$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$		0			0
$714$	0.613341	−	3.47843i	0.613341	−	3.47843i
$715$		0			0
$716$	0.500000	−	0.866025i	0.500000	−	0.866025i
$717$	0.266044	−	1.50881i	0.266044	−	1.50881i
$718$	0.939693	+	1.62760i	0.939693	+	1.62760i
$719$	0.347296			0.347296			0.173648	−	0.984808i	$-0.444444\pi$
$719$	0.347296			0.347296			0.173648	+	0.984808i	$0.444444\pi$
$720$		0			0
$721$		0			0
$722$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$723$		0			0
$724$	−1.00000	+	1.73205i	−1.00000	+	1.73205i
$725$	−0.173648	+	0.300767i	−0.173648	+	0.300767i
$726$	−0.939693	+	0.342020i	−0.939693	+	0.342020i
$727$	−0.766044	−	1.32683i	−0.766044	−	1.32683i	−0.939693	−	0.342020i	$-0.888889\pi$
$727$	−0.766044	−	1.32683i	−0.766044	−	1.32683i	0.173648	−	0.984808i	$-0.444444\pi$
$728$	−2.87939			−2.87939
$729$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$730$		0			0
$731$		0			0
$732$		0			0
$733$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$733$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$734$	0.500000	−	0.866025i	0.500000	−	0.866025i
$735$		0			0
$736$	−0.173648	−	0.300767i	−0.173648	−	0.300767i
$737$		0			0
$738$		0			0
$739$		0			0		1.00000			$0$
$739$		0			0		−1.00000			$\pi$
$740$		0			0
$741$	0.266044	−	1.50881i	0.266044	−	1.50881i
$742$	1.43969	−	2.49362i	1.43969	−	2.49362i
$743$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$743$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$744$		0			0
$745$		0			0
$746$	−1.87939			−1.87939
$747$		0			0
$748$		0			0
$749$	0.326352	+	0.565258i	0.326352	+	0.565258i
$750$		0			0
$751$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$751$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$752$	0.500000	−	0.866025i	0.500000	−	0.866025i
$753$		0			0
$754$	−0.266044	−	0.460802i	−0.266044	−	0.460802i
$755$		0			0
$756$	0.939693	−	1.62760i	0.939693	−	1.62760i
$757$		0			0		1.00000			$0$
$757$		0			0		−1.00000			$\pi$
$758$	−0.766044	−	1.32683i	−0.766044	−	1.32683i
$759$		0			0
$760$		0			0
$761$	0.939693	−	1.62760i	0.939693	−	1.62760i	0.173648	−	0.984808i	$-0.444444\pi$
$761$	0.939693	−	1.62760i	0.939693	−	1.62760i	0.766044	−	0.642788i	$-0.222222\pi$
$762$		0			0
$763$	0.326352	+	0.565258i	0.326352	+	0.565258i
$764$	−1.87939			−1.87939
$765$		0			0
$766$		0			0
$767$	−1.17365	−	2.03282i	−1.17365	−	2.03282i
$768$	0.173648	−	0.984808i	0.173648	−	0.984808i
$769$	−0.173648	+	0.300767i	−0.173648	+	0.300767i	−0.939693	−	0.342020i	$-0.888889\pi$
$769$	−0.173648	+	0.300767i	−0.173648	+	0.300767i	0.766044	+	0.642788i	$0.222222\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−1.87939			−1.87939			−0.939693	−	0.342020i	$-0.888889\pi$
$773$	−1.87939			−1.87939			−0.939693	+	0.342020i	$0.888889\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$	1.43969	+	1.20805i	1.43969	+	1.20805i
$778$		0			0
$779$		0			0
$780$		0			0
$781$		0			0
$782$	−0.652704			−0.652704
$783$	0.347296			0.347296
$784$	2.53209			2.53209
$785$		0			0
$786$		0			0
$787$	−0.173648	+	0.300767i	−0.173648	+	0.300767i	−0.939693	−	0.342020i	$-0.888889\pi$
$787$	−0.173648	+	0.300767i	−0.173648	+	0.300767i	0.766044	+	0.642788i	$0.222222\pi$
$788$		0			0
$789$	−0.766044	−	0.642788i	−0.766044	−	0.642788i
$790$		0			0
$791$		0			0
$792$		0			0
$793$		0			0
$794$		0			0
$795$		0			0
$796$	−0.173648	+	0.300767i	−0.173648	+	0.300767i
$797$	−0.766044	+	1.32683i	−0.766044	+	1.32683i	0.173648	+	0.984808i	$0.444444\pi$
$797$	−0.766044	+	1.32683i	−0.766044	+	1.32683i	−0.939693	+	0.342020i	$0.888889\pi$
$798$	−0.326352	+	1.85083i	−0.326352	+	1.85083i
$799$	−0.939693	−	1.62760i	−0.939693	−	1.62760i
$800$	1.00000			1.00000
$801$		0			0
$802$		0			0
$803$		0			0
$804$	0.266044	+	0.223238i	0.266044	+	0.223238i
$805$		0			0
$806$		0			0
$807$	0.939693	−	0.342020i	0.939693	−	0.342020i
$808$		0			0
$809$	1.53209			1.53209			0.766044	−	0.642788i	$-0.222222\pi$
$809$	1.53209			1.53209			0.766044	+	0.642788i	$0.222222\pi$
$810$		0			0
$811$	0.347296			0.347296			0.173648	−	0.984808i	$-0.444444\pi$
$811$	0.347296			0.347296			0.173648	+	0.984808i	$0.444444\pi$
$812$	0.326352	+	0.565258i	0.326352	+	0.565258i
$813$	−1.43969	+	0.524005i	−1.43969	+	0.524005i
$814$		0			0
$815$		0			0
$816$	−1.43969	−	1.20805i	−1.43969	−	1.20805i
$817$		0			0
$818$		0			0
$819$	−2.20574	+	1.85083i	−2.20574	+	1.85083i
$820$		0			0
$821$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$821$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$822$	0.266044	−	1.50881i	0.266044	−	1.50881i
$823$	−0.766044	+	1.32683i	−0.766044	+	1.32683i	0.173648	+	0.984808i	$0.444444\pi$
$823$	−0.766044	+	1.32683i	−0.766044	+	1.32683i	−0.939693	+	0.342020i	$0.888889\pi$
$824$		0			0
$825$		0			0
$826$	1.43969	+	2.49362i	1.43969	+	2.49362i
$827$	1.53209			1.53209			0.766044	−	0.642788i	$-0.222222\pi$
$827$	1.53209			1.53209			0.766044	+	0.642788i	$0.222222\pi$
$828$	−0.326352	−	0.118782i	−0.326352	−	0.118782i
$829$	−1.87939			−1.87939			−0.939693	−	0.342020i	$-0.888889\pi$
$829$	−1.87939			−1.87939			−0.939693	+	0.342020i	$0.888889\pi$
$830$		0			0
$831$		0			0
$832$	−0.766044	+	1.32683i	−0.766044	+	1.32683i
$833$	2.37939	−	4.12122i	2.37939	−	4.12122i
$834$		0			0
$835$		0			0
$836$		0			0
$837$		0			0
$838$		0			0
$839$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$839$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$840$		0			0
$841$	0.439693	−	0.761570i	0.439693	−	0.761570i
$842$	0.939693	−	1.62760i	0.939693	−	1.62760i
$843$		0			0
$844$	0.939693	+	1.62760i	0.939693	+	1.62760i
$845$		0			0
$846$	−0.173648	−	0.984808i	−0.173648	−	0.984808i
$847$	−1.87939			−1.87939
$848$	−0.766044	−	1.32683i	−0.766044	−	1.32683i
$849$		0			0
$850$	0.939693	−	1.62760i	0.939693	−	1.62760i
$851$	0.173648	−	0.300767i	0.173648	−	0.300767i
$852$		0			0
$853$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$853$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$854$		0			0
$855$		0			0
$856$	0.347296			0.347296
$857$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$857$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$858$		0			0
$859$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$859$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		0			0		1.00000			$0$
$863$		0			0		−1.00000			$\pi$
$864$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$865$		0			0
$866$		0			0
$867$	−2.37939	+	0.866025i	−2.37939	+	0.866025i
$868$		0			0
$869$		0			0
$870$		0			0
$871$	−0.266044	−	0.460802i	−0.266044	−	0.460802i
$872$	0.347296			0.347296
$873$		0			0
$874$	0.347296			0.347296
$875$		0			0
$876$	0.0603074	−	0.342020i	0.0603074	−	0.342020i
$877$	0.939693	−	1.62760i	0.939693	−	1.62760i	0.173648	−	0.984808i	$-0.444444\pi$
$877$	0.939693	−	1.62760i	0.939693	−	1.62760i	0.766044	−	0.642788i	$-0.222222\pi$
$878$		0			0
$879$	−0.326352	+	1.85083i	−0.326352	+	1.85083i
$880$		0			0
$881$	2.00000			2.00000			1.00000			$0$
$881$	2.00000			2.00000			1.00000			$0$
$882$	1.93969	−	1.62760i	1.93969	−	1.62760i
$883$		0			0		1.00000			$0$
$883$		0			0		−1.00000			$\pi$
$884$	1.43969	+	2.49362i	1.43969	+	2.49362i
$885$		0			0
$886$		0			0
$887$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$887$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$888$	0.939693	−	0.342020i	0.939693	−	0.342020i
$889$		0			0
$890$		0			0
$891$		0			0
$892$		0			0
$893$	0.500000	+	0.866025i	0.500000	+	0.866025i
$894$		0			0
$895$		0			0
$896$	0.939693	−	1.62760i	0.939693	−	1.62760i
$897$	0.407604	+	0.342020i	0.407604	+	0.342020i
$898$		0			0
$899$		0			0
$900$	0.766044	−	0.642788i	0.766044	−	0.642788i
$901$	−2.87939			−2.87939
$902$		0			0
$903$		0			0
$904$		0			0
$905$		0			0
$906$		0			0
$907$	0.939693	+	1.62760i	0.939693	+	1.62760i	0.766044	+	0.642788i	$0.222222\pi$
$907$	0.939693	+	1.62760i	0.939693	+	1.62760i	0.173648	+	0.984808i	$0.444444\pi$
$908$	−1.87939			−1.87939
$909$		0			0
$910$		0			0
$911$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$911$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$912$	0.766044	+	0.642788i	0.766044	+	0.642788i
$913$		0			0
$914$	0.939693	−	1.62760i	0.939693	−	1.62760i
$915$		0			0
$916$		0			0
$917$		0			0
$918$	−1.87939			−1.87939
$919$	0.347296			0.347296			0.173648	−	0.984808i	$-0.444444\pi$
$919$	0.347296			0.347296			0.173648	+	0.984808i	$0.444444\pi$
$920$		0			0
$921$	0.939693	−	0.342020i	0.939693	−	0.342020i
$922$		0			0
$923$		0			0
$924$		0			0
$925$	0.500000	+	0.866025i	0.500000	+	0.866025i
$926$	−1.00000			−1.00000
$927$		0			0
$928$	0.347296			0.347296
$929$	0.939693	+	1.62760i	0.939693	+	1.62760i	0.766044	+	0.642788i	$0.222222\pi$
$929$	0.939693	+	1.62760i	0.939693	+	1.62760i	0.173648	+	0.984808i	$0.444444\pi$
$930$		0			0
$931$	−1.26604	+	2.19285i	−1.26604	+	2.19285i
$932$	−1.00000	+	1.73205i	−1.00000	+	1.73205i
$933$	−0.326352	+	1.85083i	−0.326352	+	1.85083i
$934$		0			0
$935$		0			0
$936$	0.266044	+	1.50881i	0.266044	+	1.50881i
$937$	−1.87939			−1.87939			−0.939693	−	0.342020i	$-0.888889\pi$
$937$	−1.87939			−1.87939			−0.939693	+	0.342020i	$0.888889\pi$
$938$	0.326352	+	0.565258i	0.326352	+	0.565258i
$939$	1.17365	+	0.984808i	1.17365	+	0.984808i
$940$		0			0
$941$	−0.766044	+	1.32683i	−0.766044	+	1.32683i	0.173648	+	0.984808i	$0.444444\pi$
$941$	−0.766044	+	1.32683i	−0.766044	+	1.32683i	−0.939693	+	0.342020i	$0.888889\pi$
$942$		0			0
$943$		0			0
$944$	1.53209			1.53209
$945$		0			0
$946$		0			0
$947$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$947$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$948$		0			0
$949$	−0.266044	+	0.460802i	−0.266044	+	0.460802i
$950$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$951$	1.17365	+	0.984808i	1.17365	+	0.984808i
$952$	−1.76604	−	3.05888i	−1.76604	−	3.05888i
$953$		0			0		1.00000			$0$
$953$		0			0		−1.00000			$\pi$
$954$	−1.43969	−	0.524005i	−1.43969	−	0.524005i
$955$		0			0
$956$	−0.766044	−	1.32683i	−0.766044	−	1.32683i
$957$		0			0
$958$	0.500000	−	0.866025i	0.500000	−	0.866025i
$959$	1.43969	−	2.49362i	1.43969	−	2.49362i
$960$		0			0
$961$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$962$	−1.53209			−1.53209
$963$	0.266044	−	0.223238i	0.266044	−	0.223238i
$964$		0			0
$965$		0			0
$966$	−0.500000	−	0.419550i	−0.500000	−	0.419550i
$967$	0.500000	−	0.866025i	0.500000	−	0.866025i	−0.500000	−	0.866025i	$-0.666667\pi$
$967$	0.500000	−	0.866025i	0.500000	−	0.866025i	1.00000			$0$
$968$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$969$	1.76604	−	0.642788i	1.76604	−	0.642788i
$970$		0			0
$971$	−1.00000			−1.00000			−0.500000	−	0.866025i	$-0.666667\pi$
$971$	−1.00000			−1.00000			−0.500000	+	0.866025i	$0.666667\pi$
$972$	−0.939693	−	0.342020i	−0.939693	−	0.342020i
$973$		0			0
$974$		0			0
$975$	−1.43969	+	0.524005i	−1.43969	+	0.524005i
$976$		0			0
$977$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$977$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$978$		0			0
$979$		0			0
$980$		0			0
$981$	0.266044	−	0.223238i	0.266044	−	0.223238i
$982$		0			0
$983$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$983$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$984$		0			0
$985$		0			0
$986$	0.326352	−	0.565258i	0.326352	−	0.565258i
$987$	0.326352	−	1.85083i	0.326352	−	1.85083i
$988$	−0.766044	−	1.32683i	−0.766044	−	1.32683i
$989$		0			0
$990$		0			0
$991$		0			0		1.00000			$0$
$991$		0			0		−1.00000			$\pi$
$992$		0			0
$993$	1.17365	+	0.984808i	1.17365	+	0.984808i
$994$		0			0
$995$		0			0
$996$		0			0
$997$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$997$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$998$		0			0
$999$	0.500000	−	0.866025i	0.500000	−	0.866025i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1368.1.ce.a.949.1	yes	6
8.5	even	2		1368.1.ce.b.949.3	yes	6
9.7	even	3	inner	1368.1.ce.a.493.1	✓	6
19.18	odd	2		1368.1.ce.b.949.3	yes	6
72.61	even	6		1368.1.ce.b.493.3	yes	6
152.37	odd	2	CM	1368.1.ce.a.949.1	yes	6
171.151	odd	6		1368.1.ce.b.493.3	yes	6
1368.493	odd	6	inner	1368.1.ce.a.493.1	✓	6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1368.1.ce.a.493.1	✓	6	9.7	even	3	inner
1368.1.ce.a.493.1	✓	6	1368.493	odd	6	inner
1368.1.ce.a.949.1	yes	6	1.1	even	1	trivial
1368.1.ce.a.949.1	yes	6	152.37	odd	2	CM
1368.1.ce.b.493.3	yes	6	72.61	even	6
1368.1.ce.b.493.3	yes	6	171.151	odd	6
1368.1.ce.b.949.3	yes	6	8.5	even	2
1368.1.ce.b.949.3	yes	6	19.18	odd	2

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.ce (of order \(6\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(-0.500000 - 0.866025i) q^{2} +(-0.939693 + 0.342020i) q^{3} +(-0.500000 + 0.866025i) q^{4} +(0.766044 + 0.642788i) q^{6} +(0.939693 + 1.62760i) q^{7} +1.00000 q^{8} +(0.766044 - 0.642788i) q^{9} +O(q^{10})\)	\(q+(-0.500000 - 0.866025i) q^{2} +(-0.939693 + 0.342020i) q^{3} +(-0.500000 + 0.866025i) q^{4} +(0.766044 + 0.642788i) q^{6} +(0.939693 + 1.62760i) q^{7} +1.00000 q^{8} +(0.766044 - 0.642788i) q^{9} +(0.173648 - 0.984808i) q^{12} +(-0.766044 + 1.32683i) q^{13} +(0.939693 - 1.62760i) q^{14} +(-0.500000 - 0.866025i) q^{16} -1.87939 q^{17} +(-0.939693 - 0.342020i) q^{18} +1.00000 q^{19} +(-1.43969 - 1.20805i) q^{21} +(-0.173648 + 0.300767i) q^{23} +(-0.939693 + 0.342020i) q^{24} +(-0.500000 - 0.866025i) q^{25} +1.53209 q^{26} +(-0.500000 + 0.866025i) q^{27} -1.87939 q^{28} +(-0.173648 - 0.300767i) q^{29} +(-0.500000 + 0.866025i) q^{32} +(0.939693 + 1.62760i) q^{34} +(0.173648 + 0.984808i) q^{36} -1.00000 q^{37} +(-0.500000 - 0.866025i) q^{38} +(0.266044 - 1.50881i) q^{39} +(-0.326352 + 1.85083i) q^{42} +0.347296 q^{46} +(0.500000 + 0.866025i) q^{47} +(0.766044 + 0.642788i) q^{48} +(-1.26604 + 2.19285i) q^{49} +(-0.500000 + 0.866025i) q^{50} +(1.76604 - 0.642788i) q^{51} +(-0.766044 - 1.32683i) q^{52} +1.53209 q^{53} +1.00000 q^{54} +(0.939693 + 1.62760i) q^{56} +(-0.939693 + 0.342020i) q^{57} +(-0.173648 + 0.300767i) q^{58} +(-0.766044 + 1.32683i) q^{59} +(1.76604 + 0.642788i) q^{63} +1.00000 q^{64} +(-0.173648 + 0.300767i) q^{67} +(0.939693 - 1.62760i) q^{68} +(0.0603074 - 0.342020i) q^{69} +(0.766044 - 0.642788i) q^{72} +0.347296 q^{73} +(0.500000 + 0.866025i) q^{74} +(0.766044 + 0.642788i) q^{75} +(-0.500000 + 0.866025i) q^{76} +(-1.43969 + 0.524005i) q^{78} +(0.173648 - 0.984808i) q^{81} +(1.76604 - 0.642788i) q^{84} +(0.266044 + 0.223238i) q^{87} -2.87939 q^{91} +(-0.173648 - 0.300767i) q^{92} +(0.500000 - 0.866025i) q^{94} +(0.173648 - 0.984808i) q^{96} +2.53209 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 3 q^{2} - 3 q^{4} + 6 q^{8}+O(q^{10}) \) 6 * q - 3 * q^2 - 3 * q^4 + 6 * q^8	\( 6 q - 3 q^{2} - 3 q^{4} + 6 q^{8} - 3 q^{16} + 6 q^{19} - 3 q^{21} - 3 q^{25} - 3 q^{27} - 3 q^{32} - 6 q^{37} - 3 q^{38} - 3 q^{39} - 3 q^{42} + 3 q^{47} - 3 q^{49} - 3 q^{50} + 6 q^{51} + 6 q^{54} + 6 q^{63} + 6 q^{64} + 6 q^{69} + 3 q^{74} - 3 q^{76} - 3 q^{78} + 6 q^{84} - 3 q^{87} - 6 q^{91} + 3 q^{94} + 6 q^{98}+O(q^{100}) \) 6 * q - 3 * q^2 - 3 * q^4 + 6 * q^8 - 3 * q^16 + 6 * q^19 - 3 * q^21 - 3 * q^25 - 3 * q^27 - 3 * q^32 - 6 * q^37 - 3 * q^38 - 3 * q^39 - 3 * q^42 + 3 * q^47 - 3 * q^49 - 3 * q^50 + 6 * q^51 + 6 * q^54 + 6 * q^63 + 6 * q^64 + 6 * q^69 + 3 * q^74 - 3 * q^76 - 3 * q^78 + 6 * q^84 - 3 * q^87 - 6 * q^91 + 3 * q^94 + 6 * q^98

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