Newform orbit 585.2.bt.a

• Label: 585.2.bt.a
• Level: $585$
• Weight: $2$
• Character orbit: 585.bt
• Analytic conductor: $4.671$
• Analytic rank: $0$
• Dimension: $4$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 585 = 3^{2} \cdot 5 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	585.bt (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(4.67124851824\)
Analytic rank:	\(0\)
Dimension:	\(4\)
Relative dimension:	\(2\) over \(\Q(\zeta_{6})\)
Coefficient field:	\(\Q(\zeta_{12})\)
Defining polynomial:	\( x^{4} - x^{2} + 1 \)
Coefficient ring:	\(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{12}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\)	\(=\)	\( q + 2 \zeta_{12} q^{2} + ( - \zeta_{12}^{2} + 2) q^{3} + 2 \zeta_{12}^{2} q^{4} + ( - \zeta_{12}^{3} + \zeta_{12}) q^{5} + ( - 2 \zeta_{12}^{3} + 4 \zeta_{12}) q^{6} + 2 \zeta_{12} q^{7} + ( - 3 \zeta_{12}^{2} + 3) q^{9}+O(q^{10}) \)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 6 q^{3} + 4 q^{4} + 6 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/585\mathbb{Z}\right)^\times\).

\(n\)	\(326\)	\(352\)	\(496\)
\(\chi(n)\)	\(-\zeta_{12}^{2}\)	\(1\)	\(-1\)

Embeddings

For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.

For more information on an embedded modular form you can click on its label.

Label

\(\iota_m(\nu)\)

\( a_{2} \)

\( a_{3} \)

\( a_{4} \)

\( a_{5} \)

\( a_{6} \)

\( a_{7} \)

\( a_{8} \)

\( a_{9} \)

\( a_{10} \)

376.1

−0.866025	−	0.500000i
0.866025	+	0.500000i
−0.866025	+	0.500000i
0.866025	−	0.500000i

−1.73205

−

1.00000i

1.50000

−

0.866025i

1.00000

+

1.73205i

−0.866025

+

0.500000i

−3.46410

−1.73205

−

1.00000i

0

1.50000

−

2.59808i

2.00000

376.2

1.73205

+

1.00000i

1.50000

−

0.866025i

1.00000

+

1.73205i

0.866025

−

0.500000i

3.46410

1.73205

+

1.00000i

0

1.50000

−

2.59808i

2.00000

571.1

−1.73205

+

1.00000i

1.50000

+

0.866025i

1.00000

−

1.73205i

−0.866025

−

0.500000i

−3.46410

−1.73205

+

1.00000i

0

1.50000

+

2.59808i

2.00000

571.2

1.73205

−

1.00000i

1.50000

+

0.866025i

1.00000

−

1.73205i

0.866025

+

0.500000i

3.46410

1.73205

−

1.00000i

0

1.50000

+

2.59808i

2.00000

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
9.c	even	3	1	inner
13.b	even	2	1	inner
117.t	even	6	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	585.2.bt.a	✓	4
9.c	even	3	1	inner	585.2.bt.a	✓	4
13.b	even	2	1	inner	585.2.bt.a	✓	4
117.t	even	6	1	inner	585.2.bt.a	✓	4

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
585.2.bt.a	✓	4	1.a	even	1	1	trivial
585.2.bt.a	✓	4	9.c	even	3	1	inner
585.2.bt.a	✓	4	13.b	even	2	1	inner
585.2.bt.a	✓	4	117.t	even	6	1	inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{4} - 4T_{2}^{2} + 16 \) acting on \(S_{2}^{\mathrm{new}}(585, [\chi])\).

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	\( T^{4} - 4T^{2} + 16 \)
$3$	\( (T^{2} - 3 T + 3)^{2} \)
$5$	\( T^{4} - T^{2} + 1 \)
$7$	\( T^{4} - 4T^{2} + 16 \)
$11$	\( T^{4} - 36T^{2} + 1296 \)