Newform orbit 9.24.a.a

• Label: 9.24.a.a
• Level: $9$
• Weight: $24$
• Character orbit: 9.a
• Self dual: yes
• Analytic conductor: $30.168$
• Analytic rank: $1$
• Dimension: $1$
• CM: no
• Inner twists: $1$

Newspace parameters

Level:	\( N \)	\(=\)	\( 9 = 3^{2} \)
Weight:	\( k \)	\(=\)	\( 24 \)
Character orbit:	\([\chi]\)	\(=\)	9.a (trivial)

Newform invariants

Self dual:	yes
Analytic conductor:	\(30.1683633611\)
Analytic rank:	\(1\)
Dimension:	\(1\)
Coefficient field:	\(\mathbb{Q}\)
Coefficient ring:	\(\mathbb{Z}\)
Coefficient ring index:	\( 1 \)
Twist minimal:	no (minimal twist has level 3)
Fricke sign:	\(-1\)
Sato-Tate group:	$\mathrm{SU}(2)$

$q$-expansion

\(f(q)\)

\(=\)

\( q - 1128 q^{2} - 7116224 q^{4} + 48863730 q^{5} - 1723688680 q^{7} + 17489450496 q^{8}+O(q^{10}) \)

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

−1128.00

0

−7.11622e6

4.88637e7

0

−1.72369e9

1.74895e10

0

−5.51183e10

Atkin-Lehner signs

\( p \)	Sign
\(3\)	\(-1\)

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	9.24.a.a		1
3.b	odd	2	1		3.24.a.a	✓	1
12.b	even	2	1		48.24.a.a		1

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
3.24.a.a	✓	1	3.b	odd	2	1
9.24.a.a		1	1.a	even	1	1	trivial
48.24.a.a		1	12.b	even	2	1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2} + 1128 \) acting on \(S_{24}^{\mathrm{new}}(\Gamma_0(9))\).

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	\( T + 1128 \)
$3$	\( T \)
$5$	\( T - 48863730 \)
$7$	\( T + 1723688680 \)
$11$	\( T - 1428263180124 \)