LMFDB - Embedded newform 760.1.bz.a.139.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [760,1,Mod(99,760)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(760, base_ring=CyclotomicField(18))

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

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

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

chi := DirichletCharacter("760.99");

S:= CuspForms(chi, 1);

N := Newforms(S);

Level:	$ N $	$=$	$ 760 = 2^{3} \cdot 5 \cdot 19 $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	760.bz (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:	$0.379289409601$
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:	$D_{9}$
Projective field:	Galois closure of 9.1.43477921384960000.1

Embedding invariants

Embedding label			139.1
Root			$0.939693 - 0.342020i$ of defining polynomial
Character	$\chi$	$=$	760.139
Dual form			760.1.bz.a.339.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(0.766044 - 0.642788i) q^{2} +(0.173648 - 0.984808i) q^{4} +(0.173648 + 0.984808i) q^{5} +(-0.173648 + 0.300767i) q^{7} +(-0.500000 - 0.866025i) q^{8} +(0.766044 + 0.642788i) q^{9} +O(q^{10})$	\(q+(0.766044 - 0.642788i) q^{2} +(0.173648 - 0.984808i) q^{4} +(0.173648 + 0.984808i) q^{5} +(-0.173648 + 0.300767i) q^{7} +(-0.500000 - 0.866025i) q^{8} +(0.766044 + 0.642788i) q^{9} +(0.766044 + 0.642788i) q^{10} +(-0.766044 - 1.32683i) q^{11} +(0.939693 - 0.342020i) q^{13} +(0.0603074 + 0.342020i) q^{14} +(-0.939693 - 0.342020i) q^{16} +1.00000 q^{18} +(0.766044 - 0.642788i) q^{19} +1.00000 q^{20} +(-1.43969 - 0.524005i) q^{22} +(-0.326352 + 1.85083i) q^{23} +(-0.939693 + 0.342020i) q^{25} +(0.500000 - 0.866025i) q^{26} +(0.266044 + 0.223238i) q^{28} +(-0.939693 + 0.342020i) q^{32} +(-0.326352 - 0.118782i) q^{35} +(0.766044 - 0.642788i) q^{36} -1.87939 q^{37} +(0.173648 - 0.984808i) q^{38} +(0.766044 - 0.642788i) q^{40} +(-1.43969 - 0.524005i) q^{41} +(-1.43969 + 0.524005i) q^{44} +(-0.500000 + 0.866025i) q^{45} +(0.939693 + 1.62760i) q^{46} +(-0.766044 - 0.642788i) q^{47} +(0.439693 + 0.761570i) q^{49} +(-0.500000 + 0.866025i) q^{50} +(-0.173648 - 0.984808i) q^{52} +(0.0603074 - 0.342020i) q^{53} +(1.17365 - 0.984808i) q^{55} +0.347296 q^{56} +(-0.766044 + 0.642788i) q^{59} +(-0.326352 + 0.118782i) q^{63} +(-0.500000 + 0.866025i) q^{64} +(0.500000 + 0.866025i) q^{65} +(-0.326352 + 0.118782i) q^{70} +(0.173648 - 0.984808i) q^{72} +(-1.43969 + 1.20805i) q^{74} +(-0.500000 - 0.866025i) q^{76} +0.532089 q^{77} +(0.173648 - 0.984808i) q^{80} +(0.173648 + 0.984808i) q^{81} +(-1.43969 + 0.524005i) q^{82} +(-0.766044 + 1.32683i) q^{88} +(1.76604 - 0.642788i) q^{89} +(0.173648 + 0.984808i) q^{90} +(-0.0603074 + 0.342020i) q^{91} +(1.76604 + 0.642788i) q^{92} -1.00000 q^{94} +(0.766044 + 0.642788i) q^{95} +(0.826352 + 0.300767i) q^{98} +(0.266044 - 1.50881i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 3 q^{8}+O(q^{10}) $ 6 * q - 3 * q^8	$ 6 q - 3 q^{8} + 6 q^{14} + 6 q^{18} + 6 q^{20} - 3 q^{22} - 3 q^{23} + 3 q^{26} - 3 q^{28} - 3 q^{35} - 3 q^{41} - 3 q^{44} - 3 q^{45} - 3 q^{49} - 3 q^{50} + 6 q^{53} + 6 q^{55} - 3 q^{63} - 3 q^{64} + 3 q^{65} - 3 q^{70} - 3 q^{74} - 3 q^{76} - 6 q^{77} - 3 q^{82} + 6 q^{89} - 6 q^{91} + 6 q^{92} - 6 q^{94} + 6 q^{98} - 3 q^{99}+O(q^{100}) $ 6 * q - 3 * q^8 + 6 * q^14 + 6 * q^18 + 6 * q^20 - 3 * q^22 - 3 * q^23 + 3 * q^26 - 3 * q^28 - 3 * q^35 - 3 * q^41 - 3 * q^44 - 3 * q^45 - 3 * q^49 - 3 * q^50 + 6 * q^53 + 6 * q^55 - 3 * q^63 - 3 * q^64 + 3 * q^65 - 3 * q^70 - 3 * q^74 - 3 * q^76 - 6 * q^77 - 3 * q^82 + 6 * q^89 - 6 * q^91 + 6 * q^92 - 6 * q^94 + 6 * q^98 - 3 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/760\mathbb{Z}\right)^\times$.

$n$	$191$	$381$	$401$	$457$
$\chi(n)$	$-1$	$-1$	$e\left(\frac{7}{9}\right)$	$-1$

Coefficient data

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

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

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

$2$	0.766044	−	0.642788i	0.766044	−	0.642788i
$2$	0.766044	−	0.642788i	0.766044	−	0.642788i
$3$		0			0		−0.939693	−	0.342020i	$-0.888889\pi$
$3$		0			0		0.939693	+	0.342020i	$0.111111\pi$
$4$	0.173648	−	0.984808i	0.173648	−	0.984808i
$5$	0.173648	+	0.984808i	0.173648	+	0.984808i
$5$	0.173648	+	0.984808i	0.173648	+	0.984808i
$6$		0			0
$7$	−0.173648	+	0.300767i	−0.173648	+	0.300767i	−0.939693	−	0.342020i	$-0.888889\pi$
$7$	−0.173648	+	0.300767i	−0.173648	+	0.300767i	0.766044	+	0.642788i	$0.222222\pi$
$8$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$9$	0.766044	+	0.642788i	0.766044	+	0.642788i
$10$	0.766044	+	0.642788i	0.766044	+	0.642788i
$11$	−0.766044	−	1.32683i	−0.766044	−	1.32683i	−0.939693	−	0.342020i	$-0.888889\pi$
$11$	−0.766044	−	1.32683i	−0.766044	−	1.32683i	0.173648	−	0.984808i	$-0.444444\pi$
$12$		0			0
$13$	0.939693	−	0.342020i	0.939693	−	0.342020i	0.173648	−	0.984808i	$-0.444444\pi$
$13$	0.939693	−	0.342020i	0.939693	−	0.342020i	0.766044	+	0.642788i	$0.222222\pi$
$14$	0.0603074	+	0.342020i	0.0603074	+	0.342020i
$15$		0			0
$16$	−0.939693	−	0.342020i	−0.939693	−	0.342020i
$17$		0			0		0.766044	−	0.642788i	$-0.222222\pi$
$17$		0			0		−0.766044	+	0.642788i	$0.777778\pi$
$18$	1.00000			1.00000
$19$	0.766044	−	0.642788i	0.766044	−	0.642788i
$19$	0.766044	−	0.642788i	0.766044	−	0.642788i
$20$	1.00000			1.00000
$21$		0			0
$22$	−1.43969	−	0.524005i	−1.43969	−	0.524005i
$23$	−0.326352	+	1.85083i	−0.326352	+	1.85083i	0.173648	+	0.984808i	$0.444444\pi$
$23$	−0.326352	+	1.85083i	−0.326352	+	1.85083i	−0.500000	+	0.866025i	$0.666667\pi$
$24$		0			0
$25$	−0.939693	+	0.342020i	−0.939693	+	0.342020i
$26$	0.500000	−	0.866025i	0.500000	−	0.866025i
$27$		0			0
$28$	0.266044	+	0.223238i	0.266044	+	0.223238i
$29$		0			0		−0.766044	−	0.642788i	$-0.777778\pi$
$29$		0			0		0.766044	+	0.642788i	$0.222222\pi$
$30$		0			0
$31$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$31$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$32$	−0.939693	+	0.342020i	−0.939693	+	0.342020i
$33$		0			0
$34$		0			0
$35$	−0.326352	−	0.118782i	−0.326352	−	0.118782i
$36$	0.766044	−	0.642788i	0.766044	−	0.642788i
$37$	−1.87939			−1.87939			−0.939693	−	0.342020i	$-0.888889\pi$
$37$	−1.87939			−1.87939			−0.939693	+	0.342020i	$0.888889\pi$
$38$	0.173648	−	0.984808i	0.173648	−	0.984808i
$39$		0			0
$40$	0.766044	−	0.642788i	0.766044	−	0.642788i
$41$	−1.43969	−	0.524005i	−1.43969	−	0.524005i	−0.500000	−	0.866025i	$-0.666667\pi$
$41$	−1.43969	−	0.524005i	−1.43969	−	0.524005i	−0.939693	+	0.342020i	$0.888889\pi$
$42$		0			0
$43$		0			0		−0.173648	−	0.984808i	$-0.555556\pi$
$43$		0			0		0.173648	+	0.984808i	$0.444444\pi$
$44$	−1.43969	+	0.524005i	−1.43969	+	0.524005i
$45$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$46$	0.939693	+	1.62760i	0.939693	+	1.62760i
$47$	−0.766044	−	0.642788i	−0.766044	−	0.642788i	0.173648	−	0.984808i	$-0.444444\pi$
$47$	−0.766044	−	0.642788i	−0.766044	−	0.642788i	−0.939693	+	0.342020i	$0.888889\pi$
$48$		0			0
$49$	0.439693	+	0.761570i	0.439693	+	0.761570i
$50$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$51$		0			0
$52$	−0.173648	−	0.984808i	−0.173648	−	0.984808i
$53$	0.0603074	−	0.342020i	0.0603074	−	0.342020i	−0.939693	−	0.342020i	$-0.888889\pi$
$53$	0.0603074	−	0.342020i	0.0603074	−	0.342020i	1.00000			$0$
$54$		0			0
$55$	1.17365	−	0.984808i	1.17365	−	0.984808i
$56$	0.347296			0.347296
$57$		0			0
$58$		0			0
$59$	−0.766044	+	0.642788i	−0.766044	+	0.642788i	−0.939693	−	0.342020i	$-0.888889\pi$
$59$	−0.766044	+	0.642788i	−0.766044	+	0.642788i	0.173648	+	0.984808i	$0.444444\pi$
$60$		0			0
$61$		0			0		0.173648	−	0.984808i	$-0.444444\pi$
$61$		0			0		−0.173648	+	0.984808i	$0.555556\pi$
$62$		0			0
$63$	−0.326352	+	0.118782i	−0.326352	+	0.118782i
$64$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$65$	0.500000	+	0.866025i	0.500000	+	0.866025i
$66$		0			0
$67$		0			0		−0.766044	−	0.642788i	$-0.777778\pi$
$67$		0			0		0.766044	+	0.642788i	$0.222222\pi$
$68$		0			0
$69$		0			0
$70$	−0.326352	+	0.118782i	−0.326352	+	0.118782i
$71$		0			0		−0.173648	−	0.984808i	$-0.555556\pi$
$71$		0			0		0.173648	+	0.984808i	$0.444444\pi$
$72$	0.173648	−	0.984808i	0.173648	−	0.984808i
$73$		0			0		−0.939693	−	0.342020i	$-0.888889\pi$
$73$		0			0		0.939693	+	0.342020i	$0.111111\pi$
$74$	−1.43969	+	1.20805i	−1.43969	+	1.20805i
$75$		0			0
$76$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$77$	0.532089			0.532089
$78$		0			0
$79$		0			0		−0.939693	−	0.342020i	$-0.888889\pi$
$79$		0			0		0.939693	+	0.342020i	$0.111111\pi$
$80$	0.173648	−	0.984808i	0.173648	−	0.984808i
$81$	0.173648	+	0.984808i	0.173648	+	0.984808i
$82$	−1.43969	+	0.524005i	−1.43969	+	0.524005i
$83$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$83$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$		0			0
$88$	−0.766044	+	1.32683i	−0.766044	+	1.32683i
$89$	1.76604	−	0.642788i	1.76604	−	0.642788i	0.766044	−	0.642788i	$-0.222222\pi$
$89$	1.76604	−	0.642788i	1.76604	−	0.642788i	1.00000			$0$
$90$	0.173648	+	0.984808i	0.173648	+	0.984808i
$91$	−0.0603074	+	0.342020i	−0.0603074	+	0.342020i
$92$	1.76604	+	0.642788i	1.76604	+	0.642788i
$93$		0			0
$94$	−1.00000			−1.00000
$95$	0.766044	+	0.642788i	0.766044	+	0.642788i
$96$		0			0
$97$		0			0		0.766044	−	0.642788i	$-0.222222\pi$
$97$		0			0		−0.766044	+	0.642788i	$0.777778\pi$
$98$	0.826352	+	0.300767i	0.826352	+	0.300767i
$99$	0.266044	−	1.50881i	0.266044	−	1.50881i
$100$	0.173648	+	0.984808i	0.173648	+	0.984808i
$101$		0			0		0.939693	−	0.342020i	$-0.111111\pi$
$101$		0			0		−0.939693	+	0.342020i	$0.888889\pi$
$102$		0			0
$103$	−0.173648	−	0.300767i	−0.173648	−	0.300767i	0.766044	−	0.642788i	$-0.222222\pi$
$103$	−0.173648	−	0.300767i	−0.173648	−	0.300767i	−0.939693	+	0.342020i	$0.888889\pi$
$104$	−0.766044	−	0.642788i	−0.766044	−	0.642788i
$105$		0			0
$106$	−0.173648	−	0.300767i	−0.173648	−	0.300767i
$107$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$107$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$108$		0			0
$109$		0			0		−0.173648	−	0.984808i	$-0.555556\pi$
$109$		0			0		0.173648	+	0.984808i	$0.444444\pi$
$110$	0.266044	−	1.50881i	0.266044	−	1.50881i
$111$		0			0
$112$	0.266044	−	0.223238i	0.266044	−	0.223238i
$113$		0			0		1.00000			$0$
$113$		0			0		−1.00000			$\pi$
$114$		0			0
$115$	−1.87939			−1.87939
$116$		0			0
$117$	0.939693	+	0.342020i	0.939693	+	0.342020i
$118$	−0.173648	+	0.984808i	−0.173648	+	0.984808i
$119$		0			0
$120$		0			0
$121$	−0.673648	+	1.16679i	−0.673648	+	1.16679i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$126$	−0.173648	+	0.300767i	−0.173648	+	0.300767i
$127$	1.76604	−	0.642788i	1.76604	−	0.642788i	0.766044	−	0.642788i	$-0.222222\pi$
$127$	1.76604	−	0.642788i	1.76604	−	0.642788i	1.00000			$0$
$128$	0.173648	+	0.984808i	0.173648	+	0.984808i
$129$		0			0
$130$	0.939693	+	0.342020i	0.939693	+	0.342020i
$131$	−1.43969	+	1.20805i	−1.43969	+	1.20805i	−0.500000	+	0.866025i	$0.666667\pi$
$131$	−1.43969	+	1.20805i	−1.43969	+	1.20805i	−0.939693	+	0.342020i	$0.888889\pi$
$132$		0			0
$133$	0.0603074	+	0.342020i	0.0603074	+	0.342020i
$134$		0			0
$135$		0			0
$136$		0			0
$137$		0			0		0.173648	−	0.984808i	$-0.444444\pi$
$137$		0			0		−0.173648	+	0.984808i	$0.555556\pi$
$138$		0			0
$139$	0.939693	−	0.342020i	0.939693	−	0.342020i	0.173648	−	0.984808i	$-0.444444\pi$
$139$	0.939693	−	0.342020i	0.939693	−	0.342020i	0.766044	+	0.642788i	$0.222222\pi$
$140$	−0.173648	+	0.300767i	−0.173648	+	0.300767i
$141$		0			0
$142$		0			0
$143$	−1.17365	−	0.984808i	−1.17365	−	0.984808i
$144$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$145$		0			0
$146$		0			0
$147$		0			0
$148$	−0.326352	+	1.85083i	−0.326352	+	1.85083i
$149$		0			0		−0.939693	−	0.342020i	$-0.888889\pi$
$149$		0			0		0.939693	+	0.342020i	$0.111111\pi$
$150$		0			0
$151$		0			0		1.00000			$0$
$151$		0			0		−1.00000			$\pi$
$152$	−0.939693	−	0.342020i	−0.939693	−	0.342020i
$153$		0			0
$154$	0.407604	−	0.342020i	0.407604	−	0.342020i
$155$		0			0
$156$		0			0
$157$	−0.326352	−	1.85083i	−0.326352	−	1.85083i	−0.500000	−	0.866025i	$-0.666667\pi$
$157$	−0.326352	−	1.85083i	−0.326352	−	1.85083i	0.173648	−	0.984808i	$-0.444444\pi$
$158$		0			0
$159$		0			0
$160$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$161$	−0.500000	−	0.419550i	−0.500000	−	0.419550i
$162$	0.766044	+	0.642788i	0.766044	+	0.642788i
$163$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$163$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$164$	−0.766044	+	1.32683i	−0.766044	+	1.32683i
$165$		0			0
$166$		0			0
$167$	0.266044	−	1.50881i	0.266044	−	1.50881i	−0.500000	−	0.866025i	$-0.666667\pi$
$167$	0.266044	−	1.50881i	0.266044	−	1.50881i	0.766044	−	0.642788i	$-0.222222\pi$
$168$		0			0
$169$		0			0
$170$		0			0
$171$	1.00000			1.00000
$172$		0			0
$173$	1.17365	−	0.984808i	1.17365	−	0.984808i	0.173648	−	0.984808i	$-0.444444\pi$
$173$	1.17365	−	0.984808i	1.17365	−	0.984808i	1.00000			$0$
$174$		0			0
$175$	0.0603074	−	0.342020i	0.0603074	−	0.342020i
$176$	0.266044	+	1.50881i	0.266044	+	1.50881i
$177$		0			0
$178$	0.939693	−	1.62760i	0.939693	−	1.62760i
$179$	0.939693	+	1.62760i	0.939693	+	1.62760i	0.766044	+	0.642788i	$0.222222\pi$
$179$	0.939693	+	1.62760i	0.939693	+	1.62760i	0.173648	+	0.984808i	$0.444444\pi$
$180$	0.766044	+	0.642788i	0.766044	+	0.642788i
$181$		0			0		−0.766044	−	0.642788i	$-0.777778\pi$
$181$		0			0		0.766044	+	0.642788i	$0.222222\pi$
$182$	0.173648	+	0.300767i	0.173648	+	0.300767i
$183$		0			0
$184$	1.76604	−	0.642788i	1.76604	−	0.642788i
$185$	−0.326352	−	1.85083i	−0.326352	−	1.85083i
$186$		0			0
$187$		0			0
$188$	−0.766044	+	0.642788i	−0.766044	+	0.642788i
$189$		0			0
$190$	1.00000			1.00000
$191$		0			0		1.00000			$0$
$191$		0			0		−1.00000			$\pi$
$192$		0			0
$193$		0			0		−0.939693	−	0.342020i	$-0.888889\pi$
$193$		0			0		0.939693	+	0.342020i	$0.111111\pi$
$194$		0			0
$195$		0			0
$196$	0.826352	−	0.300767i	0.826352	−	0.300767i
$197$	−0.766044	+	1.32683i	−0.766044	+	1.32683i	0.173648	+	0.984808i	$0.444444\pi$
$197$	−0.766044	+	1.32683i	−0.766044	+	1.32683i	−0.939693	+	0.342020i	$0.888889\pi$
$198$	−0.766044	−	1.32683i	−0.766044	−	1.32683i
$199$		0			0		−0.766044	−	0.642788i	$-0.777778\pi$
$199$		0			0		0.766044	+	0.642788i	$0.222222\pi$
$200$	0.766044	+	0.642788i	0.766044	+	0.642788i
$201$		0			0
$202$		0			0
$203$		0			0
$204$		0			0
$205$	0.266044	−	1.50881i	0.266044	−	1.50881i
$206$	−0.326352	−	0.118782i	−0.326352	−	0.118782i
$207$	−1.43969	+	1.20805i	−1.43969	+	1.20805i
$208$	−1.00000			−1.00000
$209$	−1.43969	−	0.524005i	−1.43969	−	0.524005i
$210$		0			0
$211$	1.17365	−	0.984808i	1.17365	−	0.984808i	0.173648	−	0.984808i	$-0.444444\pi$
$211$	1.17365	−	0.984808i	1.17365	−	0.984808i	1.00000			$0$
$212$	−0.326352	−	0.118782i	−0.326352	−	0.118782i
$213$		0			0
$214$		0			0
$215$		0			0
$216$		0			0
$217$		0			0
$218$		0			0
$219$		0			0
$220$	−0.766044	−	1.32683i	−0.766044	−	1.32683i
$221$		0			0
$222$		0			0
$223$	0.266044	+	1.50881i	0.266044	+	1.50881i	0.766044	+	0.642788i	$0.222222\pi$
$223$	0.266044	+	1.50881i	0.266044	+	1.50881i	−0.500000	+	0.866025i	$0.666667\pi$
$224$	0.0603074	−	0.342020i	0.0603074	−	0.342020i
$225$	−0.939693	−	0.342020i	−0.939693	−	0.342020i
$226$		0			0
$227$		0			0		1.00000			$0$
$227$		0			0		−1.00000			$\pi$
$228$		0			0
$229$		0			0		1.00000			$0$
$229$		0			0		−1.00000			$\pi$
$230$	−1.43969	+	1.20805i	−1.43969	+	1.20805i
$231$		0			0
$232$		0			0
$233$		0			0		−0.173648	−	0.984808i	$-0.555556\pi$
$233$		0			0		0.173648	+	0.984808i	$0.444444\pi$
$234$	0.939693	−	0.342020i	0.939693	−	0.342020i
$235$	0.500000	−	0.866025i	0.500000	−	0.866025i
$236$	0.500000	+	0.866025i	0.500000	+	0.866025i
$237$		0			0
$238$		0			0
$239$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$239$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$240$		0			0
$241$	0.939693	−	0.342020i	0.939693	−	0.342020i	0.173648	−	0.984808i	$-0.444444\pi$
$241$	0.939693	−	0.342020i	0.939693	−	0.342020i	0.766044	+	0.642788i	$0.222222\pi$
$242$	0.233956	+	1.32683i	0.233956	+	1.32683i
$243$		0			0
$244$		0			0
$245$	−0.673648	+	0.565258i	−0.673648	+	0.565258i
$246$		0			0
$247$	0.500000	−	0.866025i	0.500000	−	0.866025i
$248$		0			0
$249$		0			0
$250$	−0.939693	−	0.342020i	−0.939693	−	0.342020i
$251$	−0.173648	+	0.984808i	−0.173648	+	0.984808i	0.766044	+	0.642788i	$0.222222\pi$
$251$	−0.173648	+	0.984808i	−0.173648	+	0.984808i	−0.939693	+	0.342020i	$0.888889\pi$
$252$	0.0603074	+	0.342020i	0.0603074	+	0.342020i
$253$	2.70574	−	0.984808i	2.70574	−	0.984808i
$254$	0.939693	−	1.62760i	0.939693	−	1.62760i
$255$		0			0
$256$	0.766044	+	0.642788i	0.766044	+	0.642788i
$257$		0			0		−0.766044	−	0.642788i	$-0.777778\pi$
$257$		0			0		0.766044	+	0.642788i	$0.222222\pi$
$258$		0			0
$259$	0.326352	−	0.565258i	0.326352	−	0.565258i
$260$	0.939693	−	0.342020i	0.939693	−	0.342020i
$261$		0			0
$262$	−0.326352	+	1.85083i	−0.326352	+	1.85083i
$263$	−1.43969	−	0.524005i	−1.43969	−	0.524005i	−0.500000	−	0.866025i	$-0.666667\pi$
$263$	−1.43969	−	0.524005i	−1.43969	−	0.524005i	−0.939693	+	0.342020i	$0.888889\pi$
$264$		0			0
$265$	0.347296			0.347296
$266$	0.266044	+	0.223238i	0.266044	+	0.223238i
$267$		0			0
$268$		0			0
$269$		0			0		−0.939693	−	0.342020i	$-0.888889\pi$
$269$		0			0		0.939693	+	0.342020i	$0.111111\pi$
$270$		0			0
$271$		0			0		−0.173648	−	0.984808i	$-0.555556\pi$
$271$		0			0		0.173648	+	0.984808i	$0.444444\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	1.17365	+	0.984808i	1.17365	+	0.984808i
$276$		0			0
$277$	0.500000	+	0.866025i	0.500000	+	0.866025i	1.00000			$0$
$277$	0.500000	+	0.866025i	0.500000	+	0.866025i	−0.500000	+	0.866025i	$0.666667\pi$
$278$	0.500000	−	0.866025i	0.500000	−	0.866025i
$279$		0			0
$280$	0.0603074	+	0.342020i	0.0603074	+	0.342020i
$281$	0.0603074	−	0.342020i	0.0603074	−	0.342020i	−0.939693	−	0.342020i	$-0.888889\pi$
$281$	0.0603074	−	0.342020i	0.0603074	−	0.342020i	1.00000			$0$
$282$		0			0
$283$		0			0		0.766044	−	0.642788i	$-0.222222\pi$
$283$		0			0		−0.766044	+	0.642788i	$0.777778\pi$
$284$		0			0
$285$		0			0
$286$	−1.53209			−1.53209
$287$	0.407604	−	0.342020i	0.407604	−	0.342020i
$288$	−0.939693	−	0.342020i	−0.939693	−	0.342020i
$289$	0.173648	−	0.984808i	0.173648	−	0.984808i
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−0.173648	−	0.300767i	−0.173648	−	0.300767i	0.766044	−	0.642788i	$-0.222222\pi$
$293$	−0.173648	−	0.300767i	−0.173648	−	0.300767i	−0.939693	+	0.342020i	$0.888889\pi$
$294$		0			0
$295$	−0.766044	−	0.642788i	−0.766044	−	0.642788i
$296$	0.939693	+	1.62760i	0.939693	+	1.62760i
$297$		0			0
$298$		0			0
$299$	0.326352	+	1.85083i	0.326352	+	1.85083i
$300$		0			0
$301$		0			0
$302$		0			0
$303$		0			0
$304$	−0.939693	+	0.342020i	−0.939693	+	0.342020i
$305$		0			0
$306$		0			0
$307$		0			0		−0.939693	−	0.342020i	$-0.888889\pi$
$307$		0			0		0.939693	+	0.342020i	$0.111111\pi$
$308$	0.0923963	−	0.524005i	0.0923963	−	0.524005i
$309$		0			0
$310$		0			0
$311$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$311$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$312$		0			0
$313$		0			0		−0.766044	−	0.642788i	$-0.777778\pi$
$313$		0			0		0.766044	+	0.642788i	$0.222222\pi$
$314$	−1.43969	−	1.20805i	−1.43969	−	1.20805i
$315$	−0.173648	−	0.300767i	−0.173648	−	0.300767i
$316$		0			0
$317$	−1.43969	+	0.524005i	−1.43969	+	0.524005i	−0.939693	−	0.342020i	$-0.888889\pi$
$317$	−1.43969	+	0.524005i	−1.43969	+	0.524005i	−0.500000	+	0.866025i	$0.666667\pi$
$318$		0			0
$319$		0			0
$320$	−0.939693	−	0.342020i	−0.939693	−	0.342020i
$321$		0			0
$322$	−0.652704			−0.652704
$323$		0			0
$324$	1.00000			1.00000
$325$	−0.766044	+	0.642788i	−0.766044	+	0.642788i
$326$		0			0
$327$		0			0
$328$	0.266044	+	1.50881i	0.266044	+	1.50881i
$329$	0.326352	−	0.118782i	0.326352	−	0.118782i
$330$		0			0
$331$	−0.173648	−	0.300767i	−0.173648	−	0.300767i	0.766044	−	0.642788i	$-0.222222\pi$
$331$	−0.173648	−	0.300767i	−0.173648	−	0.300767i	−0.939693	+	0.342020i	$0.888889\pi$
$332$		0			0
$333$	−1.43969	−	1.20805i	−1.43969	−	1.20805i
$334$	−0.766044	−	1.32683i	−0.766044	−	1.32683i
$335$		0			0
$336$		0			0
$337$		0			0		−0.173648	−	0.984808i	$-0.555556\pi$
$337$		0			0		0.173648	+	0.984808i	$0.444444\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$		0			0
$342$	0.766044	−	0.642788i	0.766044	−	0.642788i
$343$	−0.652704			−0.652704
$344$		0			0
$345$		0			0
$346$	0.266044	−	1.50881i	0.266044	−	1.50881i
$347$		0			0		−0.173648	−	0.984808i	$-0.555556\pi$
$347$		0			0		0.173648	+	0.984808i	$0.444444\pi$
$348$		0			0
$349$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$349$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$350$	−0.173648	−	0.300767i	−0.173648	−	0.300767i
$351$		0			0
$352$	1.17365	+	0.984808i	1.17365	+	0.984808i
$353$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$353$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$354$		0			0
$355$		0			0
$356$	−0.326352	−	1.85083i	−0.326352	−	1.85083i
$357$		0			0
$358$	1.76604	+	0.642788i	1.76604	+	0.642788i
$359$		0			0		0.766044	−	0.642788i	$-0.222222\pi$
$359$		0			0		−0.766044	+	0.642788i	$0.777778\pi$
$360$	1.00000			1.00000
$361$	0.173648	−	0.984808i	0.173648	−	0.984808i
$362$		0			0
$363$		0			0
$364$	0.326352	+	0.118782i	0.326352	+	0.118782i
$365$		0			0
$366$		0			0
$367$	0.939693	−	0.342020i	0.939693	−	0.342020i	0.173648	−	0.984808i	$-0.444444\pi$
$367$	0.939693	−	0.342020i	0.939693	−	0.342020i	0.766044	+	0.642788i	$0.222222\pi$
$368$	0.939693	−	1.62760i	0.939693	−	1.62760i
$369$	−0.766044	−	1.32683i	−0.766044	−	1.32683i
$370$	−1.43969	−	1.20805i	−1.43969	−	1.20805i
$371$	0.0923963	+	0.0775297i	0.0923963	+	0.0775297i
$372$		0			0
$373$	−0.173648	+	0.300767i	−0.173648	+	0.300767i	−0.939693	−	0.342020i	$-0.888889\pi$
$373$	−0.173648	+	0.300767i	−0.173648	+	0.300767i	0.766044	+	0.642788i	$0.222222\pi$
$374$		0			0
$375$		0			0
$376$	−0.173648	+	0.984808i	−0.173648	+	0.984808i
$377$		0			0
$378$		0			0
$379$	−1.00000			−1.00000			−0.500000	−	0.866025i	$-0.666667\pi$
$379$	−1.00000			−1.00000			−0.500000	+	0.866025i	$0.666667\pi$
$380$	0.766044	−	0.642788i	0.766044	−	0.642788i
$381$		0			0
$382$		0			0
$383$	0.939693	+	0.342020i	0.939693	+	0.342020i	0.766044	−	0.642788i	$-0.222222\pi$
$383$	0.939693	+	0.342020i	0.939693	+	0.342020i	0.173648	+	0.984808i	$0.444444\pi$
$384$		0			0
$385$	0.0923963	+	0.524005i	0.0923963	+	0.524005i
$386$		0			0
$387$		0			0
$388$		0			0
$389$		0			0		−0.766044	−	0.642788i	$-0.777778\pi$
$389$		0			0		0.766044	+	0.642788i	$0.222222\pi$
$390$		0			0
$391$		0			0
$392$	0.439693	−	0.761570i	0.439693	−	0.761570i
$393$		0			0
$394$	0.266044	+	1.50881i	0.266044	+	1.50881i
$395$		0			0
$396$	−1.43969	−	0.524005i	−1.43969	−	0.524005i
$397$	1.17365	−	0.984808i	1.17365	−	0.984808i	0.173648	−	0.984808i	$-0.444444\pi$
$397$	1.17365	−	0.984808i	1.17365	−	0.984808i	1.00000			$0$
$398$		0			0
$399$		0			0
$400$	1.00000			1.00000
$401$	1.53209	−	1.28558i	1.53209	−	1.28558i	0.766044	−	0.642788i	$-0.222222\pi$
$401$	1.53209	−	1.28558i	1.53209	−	1.28558i	0.766044	−	0.642788i	$-0.222222\pi$
$402$		0			0
$403$		0			0
$404$		0			0
$405$	−0.939693	+	0.342020i	−0.939693	+	0.342020i
$406$		0			0
$407$	1.43969	+	2.49362i	1.43969	+	2.49362i
$408$		0			0
$409$	0.266044	+	0.223238i	0.266044	+	0.223238i	0.766044	−	0.642788i	$-0.222222\pi$
$409$	0.266044	+	0.223238i	0.266044	+	0.223238i	−0.500000	+	0.866025i	$0.666667\pi$
$410$	−0.766044	−	1.32683i	−0.766044	−	1.32683i
$411$		0			0
$412$	−0.326352	+	0.118782i	−0.326352	+	0.118782i
$413$	−0.0603074	−	0.342020i	−0.0603074	−	0.342020i
$414$	−0.326352	+	1.85083i	−0.326352	+	1.85083i
$415$		0			0
$416$	−0.766044	+	0.642788i	−0.766044	+	0.642788i
$417$		0			0
$418$	−1.43969	+	0.524005i	−1.43969	+	0.524005i
$419$	0.347296			0.347296			0.173648	−	0.984808i	$-0.444444\pi$
$419$	0.347296			0.347296			0.173648	+	0.984808i	$0.444444\pi$
$420$		0			0
$421$		0			0		−0.939693	−	0.342020i	$-0.888889\pi$
$421$		0			0		0.939693	+	0.342020i	$0.111111\pi$
$422$	0.266044	−	1.50881i	0.266044	−	1.50881i
$423$	−0.173648	−	0.984808i	−0.173648	−	0.984808i
$424$	−0.326352	+	0.118782i	−0.326352	+	0.118782i
$425$		0			0
$426$		0			0
$427$		0			0
$428$		0			0
$429$		0			0
$430$		0			0
$431$		0			0		0.939693	−	0.342020i	$-0.111111\pi$
$431$		0			0		−0.939693	+	0.342020i	$0.888889\pi$
$432$		0			0
$433$		0			0		0.173648	−	0.984808i	$-0.444444\pi$
$433$		0			0		−0.173648	+	0.984808i	$0.555556\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	0.939693	+	1.62760i	0.939693	+	1.62760i
$438$		0			0
$439$		0			0		0.766044	−	0.642788i	$-0.222222\pi$
$439$		0			0		−0.766044	+	0.642788i	$0.777778\pi$
$440$	−1.43969	−	0.524005i	−1.43969	−	0.524005i
$441$	−0.152704	+	0.866025i	−0.152704	+	0.866025i
$442$		0			0
$443$		0			0		0.939693	−	0.342020i	$-0.111111\pi$
$443$		0			0		−0.939693	+	0.342020i	$0.888889\pi$
$444$		0			0
$445$	0.939693	+	1.62760i	0.939693	+	1.62760i
$446$	1.17365	+	0.984808i	1.17365	+	0.984808i
$447$		0			0
$448$	−0.173648	−	0.300767i	−0.173648	−	0.300767i
$449$	−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	$0.888889\pi$
$450$	−0.939693	+	0.342020i	−0.939693	+	0.342020i
$451$	0.407604	+	2.31164i	0.407604	+	2.31164i
$452$		0			0
$453$		0			0
$454$		0			0
$455$	−0.347296			−0.347296
$456$		0			0
$457$		0			0		1.00000			$0$
$457$		0			0		−1.00000			$\pi$
$458$		0			0
$459$		0			0
$460$	−0.326352	+	1.85083i	−0.326352	+	1.85083i
$461$		0			0		−0.173648	−	0.984808i	$-0.555556\pi$
$461$		0			0		0.173648	+	0.984808i	$0.444444\pi$
$462$		0			0
$463$	−0.766044	+	1.32683i	−0.766044	+	1.32683i	0.173648	+	0.984808i	$0.444444\pi$
$463$	−0.766044	+	1.32683i	−0.766044	+	1.32683i	−0.939693	+	0.342020i	$0.888889\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$467$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$468$	0.500000	−	0.866025i	0.500000	−	0.866025i
$469$		0			0
$470$	−0.173648	−	0.984808i	−0.173648	−	0.984808i
$471$		0			0
$472$	0.939693	+	0.342020i	0.939693	+	0.342020i
$473$		0			0
$474$		0			0
$475$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$476$		0			0
$477$	0.266044	−	0.223238i	0.266044	−	0.223238i
$478$		0			0
$479$		0			0		0.173648	−	0.984808i	$-0.444444\pi$
$479$		0			0		−0.173648	+	0.984808i	$0.555556\pi$
$480$		0			0
$481$	−1.76604	+	0.642788i	−1.76604	+	0.642788i
$482$	0.500000	−	0.866025i	0.500000	−	0.866025i
$483$		0			0
$484$	1.03209	+	0.866025i	1.03209	+	0.866025i
$485$		0			0
$486$		0			0
$487$	−0.173648	+	0.300767i	−0.173648	+	0.300767i	−0.939693	−	0.342020i	$-0.888889\pi$
$487$	−0.173648	+	0.300767i	−0.173648	+	0.300767i	0.766044	+	0.642788i	$0.222222\pi$
$488$		0			0
$489$		0			0
$490$	−0.152704	+	0.866025i	−0.152704	+	0.866025i
$491$	1.76604	+	0.642788i	1.76604	+	0.642788i	1.00000			$0$
$491$	1.76604	+	0.642788i	1.76604	+	0.642788i	0.766044	+	0.642788i	$0.222222\pi$
$492$		0			0
$493$		0			0
$494$	−0.173648	−	0.984808i	−0.173648	−	0.984808i
$495$	1.53209			1.53209
$496$		0			0
$497$		0			0
$498$		0			0
$499$	0.0603074	+	0.342020i	0.0603074	+	0.342020i	1.00000			$0$
$499$	0.0603074	+	0.342020i	0.0603074	+	0.342020i	−0.939693	+	0.342020i	$0.888889\pi$
$500$	−0.939693	+	0.342020i	−0.939693	+	0.342020i
$501$		0			0
$502$	0.500000	+	0.866025i	0.500000	+	0.866025i
$503$	1.17365	+	0.984808i	1.17365	+	0.984808i	1.00000			$0$
$503$	1.17365	+	0.984808i	1.17365	+	0.984808i	0.173648	+	0.984808i	$0.444444\pi$
$504$	0.266044	+	0.223238i	0.266044	+	0.223238i
$505$		0			0
$506$	1.43969	−	2.49362i	1.43969	−	2.49362i
$507$		0			0
$508$	−0.326352	−	1.85083i	−0.326352	−	1.85083i
$509$		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			0
$514$		0			0
$515$	0.266044	−	0.223238i	0.266044	−	0.223238i
$516$		0			0
$517$	−0.266044	+	1.50881i	−0.266044	+	1.50881i
$518$	−0.113341	−	0.642788i	−0.113341	−	0.642788i
$519$		0			0
$520$	0.500000	−	0.866025i	0.500000	−	0.866025i
$521$	−1.00000	−	1.73205i	−1.00000	−	1.73205i	−0.500000	−	0.866025i	$-0.666667\pi$
$521$	−1.00000	−	1.73205i	−1.00000	−	1.73205i	−0.500000	−	0.866025i	$-0.666667\pi$
$522$		0			0
$523$		0			0		−0.766044	−	0.642788i	$-0.777778\pi$
$523$		0			0		0.766044	+	0.642788i	$0.222222\pi$
$524$	0.939693	+	1.62760i	0.939693	+	1.62760i
$525$		0			0
$526$	−1.43969	+	0.524005i	−1.43969	+	0.524005i
$527$		0			0
$528$		0			0
$529$	−2.37939	−	0.866025i	−2.37939	−	0.866025i
$530$	0.266044	−	0.223238i	0.266044	−	0.223238i
$531$	−1.00000			−1.00000
$532$	0.347296			0.347296
$533$	−1.53209			−1.53209
$534$		0			0
$535$		0			0
$536$		0			0
$537$		0			0
$538$		0			0
$539$	0.673648	−	1.16679i	0.673648	−	1.16679i
$540$		0			0
$541$		0			0		−0.766044	−	0.642788i	$-0.777778\pi$
$541$		0			0		0.766044	+	0.642788i	$0.222222\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$		0			0
$546$		0			0
$547$		0			0		0.173648	−	0.984808i	$-0.444444\pi$
$547$		0			0		−0.173648	+	0.984808i	$0.555556\pi$
$548$		0			0
$549$		0			0
$550$	1.53209			1.53209
$551$		0			0
$552$		0			0
$553$		0			0
$554$	0.939693	+	0.342020i	0.939693	+	0.342020i
$555$		0			0
$556$	−0.173648	−	0.984808i	−0.173648	−	0.984808i
$557$	−0.326352	+	0.118782i	−0.326352	+	0.118782i	−0.500000	−	0.866025i	$-0.666667\pi$
$557$	−0.326352	+	0.118782i	−0.326352	+	0.118782i	0.173648	+	0.984808i	$0.444444\pi$
$558$		0			0
$559$		0			0
$560$	0.266044	+	0.223238i	0.266044	+	0.223238i
$561$		0			0
$562$	−0.173648	−	0.300767i	−0.173648	−	0.300767i
$563$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$563$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$	−0.326352	−	0.118782i	−0.326352	−	0.118782i
$568$		0			0
$569$	0.347296			0.347296			0.173648	−	0.984808i	$-0.444444\pi$
$569$	0.347296			0.347296			0.173648	+	0.984808i	$0.444444\pi$
$570$		0			0
$571$	−1.00000			−1.00000			−0.500000	−	0.866025i	$-0.666667\pi$
$571$	−1.00000			−1.00000			−0.500000	+	0.866025i	$0.666667\pi$
$572$	−1.17365	+	0.984808i	−1.17365	+	0.984808i
$573$		0			0
$574$	0.0923963	−	0.524005i	0.0923963	−	0.524005i
$575$	−0.326352	−	1.85083i	−0.326352	−	1.85083i
$576$	−0.939693	+	0.342020i	−0.939693	+	0.342020i
$577$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$577$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$578$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$579$		0			0
$580$		0			0
$581$		0			0
$582$		0			0
$583$	−0.500000	+	0.181985i	−0.500000	+	0.181985i
$584$		0			0
$585$	−0.173648	+	0.984808i	−0.173648	+	0.984808i
$586$	−0.326352	−	0.118782i	−0.326352	−	0.118782i
$587$		0			0		0.766044	−	0.642788i	$-0.222222\pi$
$587$		0			0		−0.766044	+	0.642788i	$0.777778\pi$
$588$		0			0
$589$		0			0
$590$	−1.00000			−1.00000
$591$		0			0
$592$	1.76604	+	0.642788i	1.76604	+	0.642788i
$593$		0			0		0.173648	−	0.984808i	$-0.444444\pi$
$593$		0			0		−0.173648	+	0.984808i	$0.555556\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$		0			0
$598$	1.43969	+	1.20805i	1.43969	+	1.20805i
$599$		0			0		−0.766044	−	0.642788i	$-0.777778\pi$
$599$		0			0		0.766044	+	0.642788i	$0.222222\pi$
$600$		0			0
$601$	−0.766044	+	1.32683i	−0.766044	+	1.32683i	0.173648	+	0.984808i	$0.444444\pi$
$601$	−0.766044	+	1.32683i	−0.766044	+	1.32683i	−0.939693	+	0.342020i	$0.888889\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−1.26604	−	0.460802i	−1.26604	−	0.460802i
$606$		0			0
$607$	0.347296			0.347296			0.173648	−	0.984808i	$-0.444444\pi$
$607$	0.347296			0.347296			0.173648	+	0.984808i	$0.444444\pi$
$608$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$609$		0			0
$610$		0			0
$611$	−0.939693	−	0.342020i	−0.939693	−	0.342020i
$612$		0			0
$613$	0.266044	+	1.50881i	0.266044	+	1.50881i	0.766044	+	0.642788i	$0.222222\pi$
$613$	0.266044	+	1.50881i	0.266044	+	1.50881i	−0.500000	+	0.866025i	$0.666667\pi$
$614$		0			0
$615$		0			0
$616$	−0.266044	−	0.460802i	−0.266044	−	0.460802i
$617$		0			0		−0.766044	−	0.642788i	$-0.777778\pi$
$617$		0			0		0.766044	+	0.642788i	$0.222222\pi$
$618$		0			0
$619$	−0.173648	−	0.300767i	−0.173648	−	0.300767i	0.766044	−	0.642788i	$-0.222222\pi$
$619$	−0.173648	−	0.300767i	−0.173648	−	0.300767i	−0.939693	+	0.342020i	$0.888889\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	−0.113341	+	0.642788i	−0.113341	+	0.642788i
$624$		0			0
$625$	0.766044	−	0.642788i	0.766044	−	0.642788i
$626$		0			0
$627$		0			0
$628$	−1.87939			−1.87939
$629$		0			0
$630$	−0.326352	−	0.118782i	−0.326352	−	0.118782i
$631$		0			0		0.173648	−	0.984808i	$-0.444444\pi$
$631$		0			0		−0.173648	+	0.984808i	$0.555556\pi$
$632$		0			0
$633$		0			0
$634$	−0.766044	+	1.32683i	−0.766044	+	1.32683i
$635$	0.939693	+	1.62760i	0.939693	+	1.62760i
$636$		0			0
$637$	0.673648	+	0.565258i	0.673648	+	0.565258i
$638$		0			0
$639$		0			0
$640$	−0.939693	+	0.342020i	−0.939693	+	0.342020i
$641$	−0.173648	−	0.984808i	−0.173648	−	0.984808i	−0.939693	−	0.342020i	$-0.888889\pi$
$641$	−0.173648	−	0.984808i	−0.173648	−	0.984808i	0.766044	−	0.642788i	$-0.222222\pi$
$642$		0			0
$643$		0			0		−0.939693	−	0.342020i	$-0.888889\pi$
$643$		0			0		0.939693	+	0.342020i	$0.111111\pi$
$644$	−0.500000	+	0.419550i	−0.500000	+	0.419550i
$645$		0			0
$646$		0			0
$647$	−1.87939			−1.87939			−0.939693	−	0.342020i	$-0.888889\pi$
$647$	−1.87939			−1.87939			−0.939693	+	0.342020i	$0.888889\pi$
$648$	0.766044	−	0.642788i	0.766044	−	0.642788i
$649$	1.43969	+	0.524005i	1.43969	+	0.524005i
$650$	−0.173648	+	0.984808i	−0.173648	+	0.984808i
$651$		0			0
$652$		0			0
$653$	−0.173648	+	0.300767i	−0.173648	+	0.300767i	−0.939693	−	0.342020i	$-0.888889\pi$
$653$	−0.173648	+	0.300767i	−0.173648	+	0.300767i	0.766044	+	0.642788i	$0.222222\pi$
$654$		0			0
$655$	−1.43969	−	1.20805i	−1.43969	−	1.20805i
$656$	1.17365	+	0.984808i	1.17365	+	0.984808i
$657$		0			0
$658$	0.173648	−	0.300767i	0.173648	−	0.300767i
$659$	−1.43969	+	0.524005i	−1.43969	+	0.524005i	−0.939693	−	0.342020i	$-0.888889\pi$
$659$	−1.43969	+	0.524005i	−1.43969	+	0.524005i	−0.500000	+	0.866025i	$0.666667\pi$
$660$		0			0
$661$		0			0		0.173648	−	0.984808i	$-0.444444\pi$
$661$		0			0		−0.173648	+	0.984808i	$0.555556\pi$
$662$	−0.326352	−	0.118782i	−0.326352	−	0.118782i
$663$		0			0
$664$		0			0
$665$	−0.326352	+	0.118782i	−0.326352	+	0.118782i
$666$	−1.87939			−1.87939
$667$		0			0
$668$	−1.43969	−	0.524005i	−1.43969	−	0.524005i
$669$		0			0
$670$		0			0
$671$		0			0
$672$		0			0
$673$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$673$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	0.939693	−	1.62760i	0.939693	−	1.62760i	0.173648	−	0.984808i	$-0.444444\pi$
$677$	0.939693	−	1.62760i	0.939693	−	1.62760i	0.766044	−	0.642788i	$-0.222222\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$		0			0
$682$		0			0
$683$		0			0		1.00000			$0$
$683$		0			0		−1.00000			$\pi$
$684$	0.173648	−	0.984808i	0.173648	−	0.984808i
$685$		0			0
$686$	−0.500000	+	0.419550i	−0.500000	+	0.419550i
$687$		0			0
$688$		0			0
$689$	−0.0603074	−	0.342020i	−0.0603074	−	0.342020i
$690$		0			0
$691$	0.939693	−	1.62760i	0.939693	−	1.62760i	0.173648	−	0.984808i	$-0.444444\pi$
$691$	0.939693	−	1.62760i	0.939693	−	1.62760i	0.766044	−	0.642788i	$-0.222222\pi$
$692$	−0.766044	−	1.32683i	−0.766044	−	1.32683i
$693$	0.407604	+	0.342020i	0.407604	+	0.342020i
$694$		0			0
$695$	0.500000	+	0.866025i	0.500000	+	0.866025i
$696$		0			0
$697$		0			0
$698$		0			0
$699$		0			0
$700$	−0.326352	−	0.118782i	−0.326352	−	0.118782i
$701$		0			0		0.766044	−	0.642788i	$-0.222222\pi$
$701$		0			0		−0.766044	+	0.642788i	$0.777778\pi$
$702$		0			0
$703$	−1.43969	+	1.20805i	−1.43969	+	1.20805i
$704$	1.53209			1.53209
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$		0			0		0.939693	−	0.342020i	$-0.111111\pi$
$709$		0			0		−0.939693	+	0.342020i	$0.888889\pi$
$710$		0			0
$711$		0			0
$712$	−1.43969	−	1.20805i	−1.43969	−	1.20805i
$713$		0			0
$714$		0			0
$715$	0.766044	−	1.32683i	0.766044	−	1.32683i
$716$	1.76604	−	0.642788i	1.76604	−	0.642788i
$717$		0			0
$718$		0			0
$719$		0			0		−0.939693	−	0.342020i	$-0.888889\pi$
$719$		0			0		0.939693	+	0.342020i	$0.111111\pi$
$720$	0.766044	−	0.642788i	0.766044	−	0.642788i
$721$	0.120615			0.120615
$722$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$723$		0			0
$724$		0			0
$725$		0			0
$726$		0			0
$727$	0.347296	+	1.96962i	0.347296	+	1.96962i	0.173648	+	0.984808i	$0.444444\pi$
$727$	0.347296	+	1.96962i	0.347296	+	1.96962i	0.173648	+	0.984808i	$0.444444\pi$
$728$	0.326352	−	0.118782i	0.326352	−	0.118782i
$729$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$730$		0			0
$731$		0			0
$732$		0			0
$733$	−0.173648	−	0.300767i	−0.173648	−	0.300767i	0.766044	−	0.642788i	$-0.222222\pi$
$733$	−0.173648	−	0.300767i	−0.173648	−	0.300767i	−0.939693	+	0.342020i	$0.888889\pi$
$734$	0.500000	−	0.866025i	0.500000	−	0.866025i
$735$		0			0
$736$	−0.326352	−	1.85083i	−0.326352	−	1.85083i
$737$		0			0
$738$	−1.43969	−	0.524005i	−1.43969	−	0.524005i
$739$	−1.43969	+	1.20805i	−1.43969	+	1.20805i	−0.500000	+	0.866025i	$0.666667\pi$
$739$	−1.43969	+	1.20805i	−1.43969	+	1.20805i	−0.939693	+	0.342020i	$0.888889\pi$
$740$	−1.87939			−1.87939
$741$		0			0
$742$	0.120615			0.120615
$743$	0.266044	−	0.223238i	0.266044	−	0.223238i	−0.500000	−	0.866025i	$-0.666667\pi$
$743$	0.266044	−	0.223238i	0.266044	−	0.223238i	0.766044	+	0.642788i	$0.222222\pi$
$744$		0			0
$745$		0			0
$746$	0.0603074	+	0.342020i	0.0603074	+	0.342020i
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$		0			0		−0.766044	−	0.642788i	$-0.777778\pi$
$751$		0			0		0.766044	+	0.642788i	$0.222222\pi$
$752$	0.500000	+	0.866025i	0.500000	+	0.866025i
$753$		0			0
$754$		0			0
$755$		0			0
$756$		0			0
$757$	−1.43969	−	0.524005i	−1.43969	−	0.524005i	−0.500000	−	0.866025i	$-0.666667\pi$
$757$	−1.43969	−	0.524005i	−1.43969	−	0.524005i	−0.939693	+	0.342020i	$0.888889\pi$
$758$	−0.766044	+	0.642788i	−0.766044	+	0.642788i
$759$		0			0
$760$	0.173648	−	0.984808i	0.173648	−	0.984808i
$761$	1.53209			1.53209			0.766044	−	0.642788i	$-0.222222\pi$
$761$	1.53209			1.53209			0.766044	+	0.642788i	$0.222222\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$		0			0
$766$	0.939693	−	0.342020i	0.939693	−	0.342020i
$767$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$768$		0			0
$769$	−0.766044	−	0.642788i	−0.766044	−	0.642788i	0.173648	−	0.984808i	$-0.444444\pi$
$769$	−0.766044	−	0.642788i	−0.766044	−	0.642788i	−0.939693	+	0.342020i	$0.888889\pi$
$770$	0.407604	+	0.342020i	0.407604	+	0.342020i
$771$		0			0
$772$		0			0
$773$	1.76604	−	0.642788i	1.76604	−	0.642788i	0.766044	−	0.642788i	$-0.222222\pi$
$773$	1.76604	−	0.642788i	1.76604	−	0.642788i	1.00000			$0$
$774$		0			0
$775$		0			0
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−1.43969	+	0.524005i	−1.43969	+	0.524005i
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$	−0.152704	−	0.866025i	−0.152704	−	0.866025i
$785$	1.76604	−	0.642788i	1.76604	−	0.642788i
$786$		0			0
$787$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$787$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$788$	1.17365	+	0.984808i	1.17365	+	0.984808i
$789$		0			0
$790$		0			0
$791$		0			0

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:	\( N \)	\(=\)	\( 760 = 2^{3} \cdot 5 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	760.bz (of order \(18\), degree \(6\), minimal)

\(f(q)\)	\(=\)	\(q+(0.766044 - 0.642788i) q^{2} +(0.173648 - 0.984808i) q^{4} +(0.173648 + 0.984808i) q^{5} +(-0.173648 + 0.300767i) q^{7} +(-0.500000 - 0.866025i) q^{8} +(0.766044 + 0.642788i) q^{9} +O(q^{10})\)	\(q+(0.766044 - 0.642788i) q^{2} +(0.173648 - 0.984808i) q^{4} +(0.173648 + 0.984808i) q^{5} +(-0.173648 + 0.300767i) q^{7} +(-0.500000 - 0.866025i) q^{8} +(0.766044 + 0.642788i) q^{9} +(0.766044 + 0.642788i) q^{10} +(-0.766044 - 1.32683i) q^{11} +(0.939693 - 0.342020i) q^{13} +(0.0603074 + 0.342020i) q^{14} +(-0.939693 - 0.342020i) q^{16} +1.00000 q^{18} +(0.766044 - 0.642788i) q^{19} +1.00000 q^{20} +(-1.43969 - 0.524005i) q^{22} +(-0.326352 + 1.85083i) q^{23} +(-0.939693 + 0.342020i) q^{25} +(0.500000 - 0.866025i) q^{26} +(0.266044 + 0.223238i) q^{28} +(-0.939693 + 0.342020i) q^{32} +(-0.326352 - 0.118782i) q^{35} +(0.766044 - 0.642788i) q^{36} -1.87939 q^{37} +(0.173648 - 0.984808i) q^{38} +(0.766044 - 0.642788i) q^{40} +(-1.43969 - 0.524005i) q^{41} +(-1.43969 + 0.524005i) q^{44} +(-0.500000 + 0.866025i) q^{45} +(0.939693 + 1.62760i) q^{46} +(-0.766044 - 0.642788i) q^{47} +(0.439693 + 0.761570i) q^{49} +(-0.500000 + 0.866025i) q^{50} +(-0.173648 - 0.984808i) q^{52} +(0.0603074 - 0.342020i) q^{53} +(1.17365 - 0.984808i) q^{55} +0.347296 q^{56} +(-0.766044 + 0.642788i) q^{59} +(-0.326352 + 0.118782i) q^{63} +(-0.500000 + 0.866025i) q^{64} +(0.500000 + 0.866025i) q^{65} +(-0.326352 + 0.118782i) q^{70} +(0.173648 - 0.984808i) q^{72} +(-1.43969 + 1.20805i) q^{74} +(-0.500000 - 0.866025i) q^{76} +0.532089 q^{77} +(0.173648 - 0.984808i) q^{80} +(0.173648 + 0.984808i) q^{81} +(-1.43969 + 0.524005i) q^{82} +(-0.766044 + 1.32683i) q^{88} +(1.76604 - 0.642788i) q^{89} +(0.173648 + 0.984808i) q^{90} +(-0.0603074 + 0.342020i) q^{91} +(1.76604 + 0.642788i) q^{92} -1.00000 q^{94} +(0.766044 + 0.642788i) q^{95} +(0.826352 + 0.300767i) q^{98} +(0.266044 - 1.50881i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 3 q^{8}+O(q^{10}) \) 6 * q - 3 * q^8	\( 6 q - 3 q^{8} + 6 q^{14} + 6 q^{18} + 6 q^{20} - 3 q^{22} - 3 q^{23} + 3 q^{26} - 3 q^{28} - 3 q^{35} - 3 q^{41} - 3 q^{44} - 3 q^{45} - 3 q^{49} - 3 q^{50} + 6 q^{53} + 6 q^{55} - 3 q^{63} - 3 q^{64} + 3 q^{65} - 3 q^{70} - 3 q^{74} - 3 q^{76} - 6 q^{77} - 3 q^{82} + 6 q^{89} - 6 q^{91} + 6 q^{92} - 6 q^{94} + 6 q^{98} - 3 q^{99}+O(q^{100}) \) 6 * q - 3 * q^8 + 6 * q^14 + 6 * q^18 + 6 * q^20 - 3 * q^22 - 3 * q^23 + 3 * q^26 - 3 * q^28 - 3 * q^35 - 3 * q^41 - 3 * q^44 - 3 * q^45 - 3 * q^49 - 3 * q^50 + 6 * q^53 + 6 * q^55 - 3 * q^63 - 3 * q^64 + 3 * q^65 - 3 * q^70 - 3 * q^74 - 3 * q^76 - 6 * q^77 - 3 * q^82 + 6 * q^89 - 6 * q^91 + 6 * q^92 - 6 * q^94 + 6 * q^98 - 3 * q^99

Introduction

L-functions

Modular forms

Varieties

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Groups

Database

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