Newform orbit 162.2.a.b

• Label: 162.2.a.b
• Level: $162$
• Weight: $2$
• Character orbit: 162.a
• Self dual: yes
• Analytic conductor: $1.294$
• Analytic rank: $0$
• Dimension: $1$
• CM: no
• Inner twists: $1$

Newspace parameters

Level:	\( N \)	\(=\)	\( 162 = 2 \cdot 3^{4} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	162.a (trivial)

Newform invariants

Self dual:	yes
Analytic conductor:	\(1.29357651274\)
Analytic rank:	\(0\)
Dimension:	\(1\)
Coefficient field:	\(\mathbb{Q}\)
Coefficient ring:	\(\mathbb{Z}\)
Coefficient ring index:	\( 1 \)
Twist minimal:	no (minimal twist has level 18)
Fricke sign:	\(-1\)
Sato-Tate group:	$\mathrm{SU}(2)$

$q$-expansion

\(f(q)\)

\(=\)

\( q - q^{2} + q^{4} + 2 q^{7} - q^{8}+O(q^{10}) \)

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

−1.00000

0

1.00000

0

0

2.00000

−1.00000

0

0

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	162.2.a.b		1
3.b	odd	2	1		162.2.a.c		1
4.b	odd	2	1		1296.2.a.f		1
5.b	even	2	1		4050.2.a.v		1
5.c	odd	4	2		4050.2.c.r		2
7.b	odd	2	1		7938.2.a.i		1
8.b	even	2	1		5184.2.a.q		1
8.d	odd	2	1		5184.2.a.p		1
9.c	even	3	2		54.2.c.a		2
9.d	odd	6	2		18.2.c.a	✓	2
12.b	even	2	1		1296.2.a.g		1
15.d	odd	2	1		4050.2.a.c		1
15.e	even	4	2		4050.2.c.c		2
21.c	even	2	1		7938.2.a.x		1
24.f	even	2	1		5184.2.a.o		1
24.h	odd	2	1		5184.2.a.r		1
36.f	odd	6	2		432.2.i.b		2
36.h	even	6	2		144.2.i.c		2
45.h	odd	6	2		450.2.e.i		2
45.j	even	6	2		1350.2.e.c		2
45.k	odd	12	4		1350.2.j.a		4
45.l	even	12	4		450.2.j.e		4
63.g	even	3	2		2646.2.h.h		2
63.h	even	3	2		2646.2.e.b		2
63.i	even	6	2		882.2.e.g		2
63.j	odd	6	2		882.2.e.i		2
63.k	odd	6	2		2646.2.h.i		2
63.l	odd	6	2		2646.2.f.g		2
63.n	odd	6	2		882.2.h.c		2
63.o	even	6	2		882.2.f.d		2
63.s	even	6	2		882.2.h.b		2
63.t	odd	6	2		2646.2.e.c		2
72.j	odd	6	2		576.2.i.g		2
72.l	even	6	2		576.2.i.a		2
72.n	even	6	2		1728.2.i.e		2
72.p	odd	6	2		1728.2.i.f		2

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
18.2.c.a	✓	2	9.d	odd	6	2
54.2.c.a		2	9.c	even	3	2
144.2.i.c		2	36.h	even	6	2
162.2.a.b		1	1.a	even	1	1	trivial
162.2.a.c		1	3.b	odd	2	1
432.2.i.b		2	36.f	odd	6	2
450.2.e.i		2	45.h	odd	6	2
450.2.j.e		4	45.l	even	12	4
576.2.i.a		2	72.l	even	6	2
576.2.i.g		2	72.j	odd	6	2
882.2.e.g		2	63.i	even	6	2
882.2.e.i		2	63.j	odd	6	2
882.2.f.d		2	63.o	even	6	2
882.2.h.b		2	63.s	even	6	2
882.2.h.c		2	63.n	odd	6	2
1296.2.a.f		1	4.b	odd	2	1
1296.2.a.g		1	12.b	even	2	1
1350.2.e.c		2	45.j	even	6	2
1350.2.j.a		4	45.k	odd	12	4
1728.2.i.e		2	72.n	even	6	2
1728.2.i.f		2	72.p	odd	6	2
2646.2.e.b		2	63.h	even	3	2
2646.2.e.c		2	63.t	odd	6	2
2646.2.f.g		2	63.l	odd	6	2
2646.2.h.h		2	63.g	even	3	2
2646.2.h.i		2	63.k	odd	6	2
4050.2.a.c		1	15.d	odd	2	1
4050.2.a.v		1	5.b	even	2	1
4050.2.c.c		2	15.e	even	4	2
4050.2.c.r		2	5.c	odd	4	2
5184.2.a.o		1	24.f	even	2	1
5184.2.a.p		1	8.d	odd	2	1
5184.2.a.q		1	8.b	even	2	1
5184.2.a.r		1	24.h	odd	2	1
7938.2.a.i		1	7.b	odd	2	1
7938.2.a.x		1	21.c	even	2	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(162))\):

\( T_{5} \)

\( T_{11} - 3 \)

Hecke characteristic polynomials

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