Newform orbit 162.4.a.h

• Label: 162.4.a.h
• Level: $162$
• Weight: $4$
• Character orbit: 162.a
• Self dual: yes
• Analytic conductor: $9.558$
• Analytic rank: $0$
• Dimension: $2$
• CM: no
• Inner twists: $1$

Newspace parameters

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

Newform invariants

Self dual:	yes
Analytic conductor:	\(9.55830942093\)
Analytic rank:	\(0\)
Dimension:	\(2\)
Coefficient field:	\(\Q(\sqrt{3}) \)
Defining polynomial:	\( x^{2} - 3 \)
Coefficient ring:	\(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index:	\( 3 \)
Twist minimal:	yes
Fricke sign:	\(1\)
Sato-Tate group:	$\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = 3\sqrt{3}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\)	\(=\)	\( q + 2 q^{2} + 4 q^{4} + (\beta + 6) q^{5} + (2 \beta + 8) q^{7} + 8 q^{8}+O(q^{10}) \)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 4 q^{2} + 8 q^{4} + 12 q^{5} + 16 q^{7} + 16 q^{8}+O(q^{10}) \)

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

−1.73205
1.73205

2.00000

0

4.00000

0.803848

0

−2.39230

8.00000

0

1.60770

1.2

2.00000

0

4.00000

11.1962

0

18.3923

8.00000

0

22.3923

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	162.4.a.h	yes	2
3.b	odd	2	1		162.4.a.e	✓	2
4.b	odd	2	1		1296.4.a.s		2
9.c	even	3	2		162.4.c.i		4
9.d	odd	6	2		162.4.c.j		4
12.b	even	2	1		1296.4.a.j		2

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
162.4.a.e	✓	2	3.b	odd	2	1
162.4.a.h	yes	2	1.a	even	1	1	trivial
162.4.c.i		4	9.c	even	3	2
162.4.c.j		4	9.d	odd	6	2
1296.4.a.j		2	12.b	even	2	1
1296.4.a.s		2	4.b	odd	2	1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{2} - 12T_{5} + 9 \) acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(162))\).

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	\( (T - 2)^{2} \)
$3$	\( T^{2} \)
$5$	\( T^{2} - 12T + 9 \)
$7$	\( T^{2} - 16T - 44 \)
$11$	\( T^{2} - 36T - 1404 \)