Newform orbit 6.22.a.a

• Label: 6.22.a.a
• Level: $6$
• Weight: $22$
• Character orbit: 6.a
• Self dual: yes
• Analytic conductor: $16.769$
• Analytic rank: $0$
• Dimension: $1$
• CM: no
• Inner twists: $1$

Newspace parameters

Level:	\( N \)	\(=\)	\( 6 = 2 \cdot 3 \)
Weight:	\( k \)	\(=\)	\( 22 \)
Character orbit:	\([\chi]\)	\(=\)	6.a (trivial)

Newform invariants

Self dual:	yes
Analytic conductor:	\(16.7686406572\)
Analytic rank:	\(0\)
Dimension:	\(1\)
Coefficient field:	\(\mathbb{Q}\)
Coefficient ring:	\(\mathbb{Z}\)
Coefficient ring index:	\( 1 \)
Twist minimal:	yes
Fricke sign:	\(-1\)
Sato-Tate group:	$\mathrm{SU}(2)$

$q$-expansion

\(f(q)\)

\(=\)

\( q - 1024 q^{2} + 59049 q^{3} + 1048576 q^{4} + 26444550 q^{5} - 60466176 q^{6} + 166115864 q^{7} - 1073741824 q^{8} + 3486784401 q^{9}+O(q^{10}) \)

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

−1024.00

59049.0

1.04858e6

2.64446e7

−6.04662e7

1.66116e8

−1.07374e9

3.48678e9

−2.70792e10

Atkin-Lehner signs

\( p \)	Sign
\(2\)	\(1\)
\(3\)	\(-1\)

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	6.22.a.a	✓	1
3.b	odd	2	1		18.22.a.d		1
4.b	odd	2	1		48.22.a.c		1

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
6.22.a.a	✓	1	1.a	even	1	1	trivial
18.22.a.d		1	3.b	odd	2	1
48.22.a.c		1	4.b	odd	2	1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5} - 26444550 \) acting on \(S_{22}^{\mathrm{new}}(\Gamma_0(6))\).

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	\( T + 1024 \)
$3$	\( T - 59049 \)
$5$	\( T - 26444550 \)
$7$	\( T - 166115864 \)
$11$	\( T + 104878761780 \)