Newform orbit 5184.2.c.j

• Label: 5184.2.c.j
• Level: $5184$
• Weight: $2$
• Character orbit: 5184.c
• Analytic conductor: $41.394$
• Analytic rank: $0$
• Dimension: $8$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 5184 = 2^{6} \cdot 3^{4} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5184.c (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(41.3944484078\)
Analytic rank:	\(0\)
Dimension:	\(8\)
Coefficient field:	8.0.170772624.1
Defining polynomial:	\( x^{8} - 3x^{7} + 5x^{6} - 6x^{5} + 6x^{4} - 12x^{3} + 20x^{2} - 24x + 16 \)
Coefficient ring:	\(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index:	\( 2^{6} \)
Twist minimal:	no (minimal twist has level 36)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 3x^{7} + 5x^{6} - 6x^{5} + 6x^{4} - 12x^{3} + 20x^{2} - 24x + 16 \) :

\(\beta_{1}\)	\(=\)	\( ( \nu^{6} - \nu^{5} - \nu^{4} + 2\nu^{2} - 8\nu + 4 ) / 4 \)
\(\beta_{2}\)	\(=\)	\( ( -\nu^{5} + \nu^{4} - \nu^{3} + 2\nu^{2} - 4\nu + 6 ) / 2 \)
\(\beta_{3}\)	\(=\)	\( ( \nu^{7} - \nu^{6} + \nu^{5} - 2\nu^{4} - 8\nu^{2} + 4\nu - 4 ) / 4 \)
\(\beta_{4}\)	\(=\)	\( ( \nu^{7} - 2\nu^{6} + 4\nu^{5} - 3\nu^{4} + 2\nu^{3} - 6\nu^{2} + 12\nu - 16 ) / 4 \)
\(\beta_{5}\)	\(=\)	\( ( -\nu^{7} + 4\nu^{6} - 6\nu^{5} + 5\nu^{4} - 6\nu^{3} + 14\nu^{2} - 20\nu + 24 ) / 4 \)
\(\beta_{6}\)	\(=\)	\( ( 3\nu^{7} - 4\nu^{6} + 8\nu^{5} - 5\nu^{4} + 8\nu^{3} - 22\nu^{2} + 20\nu - 32 ) / 4 \)
\(\beta_{7}\)	\(=\)	\( ( 3\nu^{7} - 5\nu^{6} + 7\nu^{5} - 6\nu^{4} + 10\nu^{3} - 24\nu^{2} + 28\nu - 28 ) / 4 \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/5184\mathbb{Z}\right)^\times\).

\(n\)	\(325\)	\(1217\)	\(2431\)
\(\chi(n)\)	\(1\)	\(-1\)	\(-1\)

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  

\(\iota_m(\nu)\)

\( a_{2} \)

\( a_{3} \)

\( a_{4} \)

\( a_{5} \)

\( a_{6} \)

\( a_{7} \)

\( a_{8} \)

\( a_{9} \)

\( a_{10} \)

5183.1

−1.02187	−	0.977642i
0.335728	+	1.37379i
1.41203	+	0.0786378i
0.774115	+	1.18353i
0.774115	−	1.18353i
1.41203	−	0.0786378i
0.335728	−	1.37379i
−1.02187	+	0.977642i

0

0

0

−

2.52434i

0

−

1.27582i

0

0

0

5183.2

0

0

0

−

2.52434i

0

1.27582i

0

0

0

5183.3

0

0

0

−

0.792287i

0

−

2.71519i

0

0

0

5183.4

0

0

0

−

0.792287i

0

2.71519i

0

0

0

5183.5

0

0

0

0.792287i

0

−

2.71519i

0

0

0

5183.6

0

0

0

0.792287i

0

2.71519i

0

0

0

5183.7

0

0

0

2.52434i

0

−

1.27582i

0

0

0

5183.8

0

0

0

2.52434i

0

1.27582i

0

0

0

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
3.b	odd	2	1	inner
4.b	odd	2	1	inner
12.b	even	2	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	5184.2.c.j		8
3.b	odd	2	1	inner	5184.2.c.j		8
4.b	odd	2	1	inner	5184.2.c.j		8
8.b	even	2	1		324.2.b.b		8
8.d	odd	2	1		324.2.b.b		8
9.c	even	3	1		576.2.s.f		8
9.c	even	3	1		1728.2.s.f		8
9.d	odd	6	1		576.2.s.f		8
9.d	odd	6	1		1728.2.s.f		8
12.b	even	2	1	inner	5184.2.c.j		8
24.f	even	2	1		324.2.b.b		8
24.h	odd	2	1		324.2.b.b		8
36.f	odd	6	1		576.2.s.f		8
36.f	odd	6	1		1728.2.s.f		8
36.h	even	6	1		576.2.s.f		8
36.h	even	6	1		1728.2.s.f		8
72.j	odd	6	1		36.2.h.a	✓	8
72.j	odd	6	1		108.2.h.a		8
72.l	even	6	1		36.2.h.a	✓	8
72.l	even	6	1		108.2.h.a		8
72.n	even	6	1		36.2.h.a	✓	8
72.n	even	6	1		108.2.h.a		8
72.p	odd	6	1		36.2.h.a	✓	8
72.p	odd	6	1		108.2.h.a		8
360.z	odd	6	1		900.2.r.c		8
360.bd	even	6	1		900.2.r.c		8
360.bh	odd	6	1		900.2.r.c		8
360.bk	even	6	1		900.2.r.c		8
360.bo	even	12	2		900.2.o.a		16
360.br	even	12	2		900.2.o.a		16
360.bt	odd	12	2		900.2.o.a		16
360.bu	odd	12	2		900.2.o.a		16

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
36.2.h.a	✓	8	72.j	odd	6	1
36.2.h.a	✓	8	72.l	even	6	1
36.2.h.a	✓	8	72.n	even	6	1
36.2.h.a	✓	8	72.p	odd	6	1
108.2.h.a		8	72.j	odd	6	1
108.2.h.a		8	72.l	even	6	1
108.2.h.a		8	72.n	even	6	1
108.2.h.a		8	72.p	odd	6	1
324.2.b.b		8	8.b	even	2	1
324.2.b.b		8	8.d	odd	2	1
324.2.b.b		8	24.f	even	2	1
324.2.b.b		8	24.h	odd	2	1
576.2.s.f		8	9.c	even	3	1
576.2.s.f		8	9.d	odd	6	1
576.2.s.f		8	36.f	odd	6	1
576.2.s.f		8	36.h	even	6	1
900.2.o.a		16	360.bo	even	12	2
900.2.o.a		16	360.br	even	12	2
900.2.o.a		16	360.bt	odd	12	2
900.2.o.a		16	360.bu	odd	12	2
900.2.r.c		8	360.z	odd	6	1
900.2.r.c		8	360.bd	even	6	1
900.2.r.c		8	360.bh	odd	6	1
900.2.r.c		8	360.bk	even	6	1
1728.2.s.f		8	9.c	even	3	1
1728.2.s.f		8	9.d	odd	6	1
1728.2.s.f		8	36.f	odd	6	1
1728.2.s.f		8	36.h	even	6	1
5184.2.c.j		8	1.a	even	1	1	trivial
5184.2.c.j		8	3.b	odd	2	1	inner
5184.2.c.j		8	4.b	odd	2	1	inner
5184.2.c.j		8	12.b	even	2	1	inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(5184, [\chi])\):

\( T_{5}^{4} + 7T_{5}^{2} + 4 \)

\( T_{7}^{4} + 9T_{7}^{2} + 12 \)

\( T_{11}^{4} - 12T_{11}^{2} + 3 \)

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	\( T^{8} \)
$3$	\( T^{8} \)
$5$	\( (T^{4} + 7 T^{2} + 4)^{2} \)
$7$	\( (T^{4} + 9 T^{2} + 12)^{2} \)
$11$	\( (T^{4} - 12 T^{2} + 3)^{2} \)