Newform orbit 9408.2.a.bg

• Label: 9408.2.a.bg
• Level: $9408$
• Weight: $2$
• Character orbit: 9408.a
• Self dual: yes
• Analytic conductor: $75.123$
• Analytic rank: $1$
• Dimension: $1$
• CM: no
• Inner twists: $1$

Newspace parameters

Level:	\( N \)	\(=\)	\( 9408 = 2^{6} \cdot 3 \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	9408.a (trivial)

Newform invariants

Self dual:	yes
Analytic conductor:	\(75.1232582216\)
Analytic rank:	\(1\)
Dimension:	\(1\)
Coefficient field:	\(\mathbb{Q}\)
Coefficient ring:	\(\mathbb{Z}\)
Coefficient ring index:	\( 1 \)
Twist minimal:	no (minimal twist has level 21)
Fricke sign:	\(+1\)
Sato-Tate group:	$\mathrm{SU}(2)$

$q$-expansion

\(f(q)\)

\(=\)

\( q - q^{3} + 2 q^{5} + q^{9}+O(q^{10}) \)

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  

\(\iota_m(\nu)\)

\( a_{2} \)

\( a_{3} \)

\( a_{4} \)

\( a_{5} \)

\( a_{6} \)

\( a_{7} \)

\( a_{8} \)

\( a_{9} \)

\( a_{10} \)

1.1

0

0

−1.00000

0

2.00000

0

0

0

1.00000

0

Atkin-Lehner signs

\( p \)	Sign
\(2\)	\( +1 \)
\(3\)	\( +1 \)
\(7\)	\( +1 \)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	9408.2.a.bg		1
4.b	odd	2	1		9408.2.a.cv		1
7.b	odd	2	1		9408.2.a.bz		1
7.c	even	3	2		1344.2.q.m		2
8.b	even	2	1		147.2.a.c		1
8.d	odd	2	1		2352.2.a.d		1
24.f	even	2	1		7056.2.a.bp		1
24.h	odd	2	1		441.2.a.b		1
28.d	even	2	1		9408.2.a.k		1
28.g	odd	6	2		1344.2.q.c		2
40.f	even	2	1		3675.2.a.a		1
56.e	even	2	1		2352.2.a.w		1
56.h	odd	2	1		147.2.a.b		1
56.j	odd	6	2		147.2.e.a		2
56.k	odd	6	2		336.2.q.f		2
56.m	even	6	2		2352.2.q.c		2
56.p	even	6	2		21.2.e.a	✓	2
168.e	odd	2	1		7056.2.a.m		1
168.i	even	2	1		441.2.a.a		1
168.s	odd	6	2		63.2.e.b		2
168.v	even	6	2		1008.2.s.d		2
168.ba	even	6	2		441.2.e.e		2
280.c	odd	2	1		3675.2.a.c		1
280.bf	even	6	2		525.2.i.e		2
280.bt	odd	12	4		525.2.r.e		4
504.w	even	6	2		567.2.g.a		2
504.bi	odd	6	2		567.2.h.a		2
504.cq	even	6	2		567.2.h.f		2
504.db	odd	6	2		567.2.g.f		2

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
21.2.e.a	✓	2	56.p	even	6	2
63.2.e.b		2	168.s	odd	6	2
147.2.a.b		1	56.h	odd	2	1
147.2.a.c		1	8.b	even	2	1
147.2.e.a		2	56.j	odd	6	2
336.2.q.f		2	56.k	odd	6	2
441.2.a.a		1	168.i	even	2	1
441.2.a.b		1	24.h	odd	2	1
441.2.e.e		2	168.ba	even	6	2
525.2.i.e		2	280.bf	even	6	2
525.2.r.e		4	280.bt	odd	12	4
567.2.g.a		2	504.w	even	6	2
567.2.g.f		2	504.db	odd	6	2
567.2.h.a		2	504.bi	odd	6	2
567.2.h.f		2	504.cq	even	6	2
1008.2.s.d		2	168.v	even	6	2
1344.2.q.c		2	28.g	odd	6	2
1344.2.q.m		2	7.c	even	3	2
2352.2.a.d		1	8.d	odd	2	1
2352.2.a.w		1	56.e	even	2	1
2352.2.q.c		2	56.m	even	6	2
3675.2.a.a		1	40.f	even	2	1
3675.2.a.c		1	280.c	odd	2	1
7056.2.a.m		1	168.e	odd	2	1
7056.2.a.bp		1	24.f	even	2	1
9408.2.a.k		1	28.d	even	2	1
9408.2.a.bg		1	1.a	even	1	1	trivial
9408.2.a.bz		1	7.b	odd	2	1
9408.2.a.cv		1	4.b	odd	2	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(9408))\):

\( T_{5} - 2 \)

\( T_{11} - 2 \)

\( T_{13} + 1 \)

\( T_{17} \)

\( T_{19} + 1 \)

\( T_{31} - 9 \)

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	\( T \)
$3$	\( T + 1 \)
$5$	\( T - 2 \)
$7$	\( T \)
$11$	\( T - 2 \)