Newform orbit 585.2.j.c

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

Newspace parameters

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

Newform invariants

Self dual:	no
Analytic conductor:	\(4.67124851824\)
Analytic rank:	\(0\)
Dimension:	\(4\)
Relative dimension:	\(2\) over \(\Q(\zeta_{3})\)
Coefficient field:	\(\Q(\zeta_{12})\)
Defining polynomial:	\( x^{4} - x^{2} + 1 \)
Coefficient ring:	\(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index:	\( 3 \)
Twist minimal:	no (minimal twist has level 195)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\)	\(=\)	\( q + (\beta_{2} - \beta_1) q^{2} + (2 \beta_{3} - 2 \beta_{2} + 2 \beta_1 - 2) q^{4} - q^{5} + (\beta_{3} - \beta_{2}) q^{7} + ( - 2 \beta_{3} + 6) q^{8}+O(q^{10}) \)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 2 q^{2} - 4 q^{4} - 4 q^{5} + 24 q^{8}+O(q^{10}) \)

Basis of coefficient ring

\(\beta_{1}\)	\(=\)	\( \zeta_{12}^{2} \)
\(\beta_{2}\)	\(=\)	\( \zeta_{12}^{3} + \zeta_{12} \)
\(\beta_{3}\)	\(=\)	\( -\zeta_{12}^{3} + 2\zeta_{12} \)

Character values

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

\(n\)	\(326\)	\(352\)	\(496\)
\(\chi(n)\)	\(1\)	\(1\)	\(-1 + \beta_{1}\)

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  

\(\iota_m(\nu)\)

\( a_{2} \)

\( a_{3} \)

\( a_{4} \)

\( a_{5} \)

\( a_{6} \)

\( a_{7} \)

\( a_{8} \)

\( a_{9} \)

\( a_{10} \)

406.1

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

−1.36603

−

2.36603i

0

−2.73205

+

4.73205i

−1.00000

0

−0.866025

+

1.50000i

9.46410

0

1.36603

+

2.36603i

406.2

0.366025

+

0.633975i

0

0.732051

−

1.26795i

−1.00000

0

0.866025

−

1.50000i

2.53590

0

−0.366025

−

0.633975i

451.1

−1.36603

+

2.36603i

0

−2.73205

−

4.73205i

−1.00000

0

−0.866025

−

1.50000i

9.46410

0

1.36603

−

2.36603i

451.2

0.366025

−

0.633975i

0

0.732051

+

1.26795i

−1.00000

0

0.866025

+

1.50000i

2.53590

0

−0.366025

+

0.633975i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
13.c	even	3	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	585.2.j.c		4
3.b	odd	2	1		195.2.i.c	✓	4
13.c	even	3	1	inner	585.2.j.c		4
13.c	even	3	1		7605.2.a.bj		2
13.e	even	6	1		7605.2.a.z		2
15.d	odd	2	1		975.2.i.j		4
15.e	even	4	1		975.2.bb.a		4
15.e	even	4	1		975.2.bb.h		4
39.h	odd	6	1		2535.2.a.r		2
39.i	odd	6	1		195.2.i.c	✓	4
39.i	odd	6	1		2535.2.a.o		2
195.x	odd	6	1		975.2.i.j		4
195.bl	even	12	1		975.2.bb.a		4
195.bl	even	12	1		975.2.bb.h		4

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
195.2.i.c	✓	4	3.b	odd	2	1
195.2.i.c	✓	4	39.i	odd	6	1
585.2.j.c		4	1.a	even	1	1	trivial
585.2.j.c		4	13.c	even	3	1	inner
975.2.i.j		4	15.d	odd	2	1
975.2.i.j		4	195.x	odd	6	1
975.2.bb.a		4	15.e	even	4	1
975.2.bb.a		4	195.bl	even	12	1
975.2.bb.h		4	15.e	even	4	1
975.2.bb.h		4	195.bl	even	12	1
2535.2.a.o		2	39.i	odd	6	1
2535.2.a.r		2	39.h	odd	6	1
7605.2.a.z		2	13.e	even	6	1
7605.2.a.bj		2	13.c	even	3	1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{4} + 2T_{2}^{3} + 6T_{2}^{2} - 4T_{2} + 4 \) acting on \(S_{2}^{\mathrm{new}}(585, [\chi])\).

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	T^4 + 2*T^3 + 6*T^2 - 4*T + 4
$3$	\( T^{4} \)
$5$	\( (T + 1)^{4} \)
$7$	\( T^{4} + 3T^{2} + 9 \)
$11$	\( (T^{2} + 2 T + 4)^{2} \)