Newform orbit 966.2.a.p

• Label: 966.2.a.p
• Level: $966$
• Weight: $2$
• Character orbit: 966.a
• Self dual: yes
• Analytic conductor: $7.714$
• Analytic rank: $0$
• Dimension: $2$
• CM: no
• Inner twists: $1$

Newspace parameters

Level:	\( N \)	\(=\)	\( 966 = 2 \cdot 3 \cdot 7 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	966.a (trivial)

Newform invariants

Self dual:	yes
Analytic conductor:	\(7.71354883526\)
Analytic rank:	\(0\)
Dimension:	\(2\)
Coefficient field:	\(\Q(\sqrt{5}) \)
Defining polynomial:	\( x^{2} - x - 1 \)
Coefficient ring:	\(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index:	\( 2^{2} \)
Twist minimal:	yes
Fricke sign:	\(-1\)
Sato-Tate group:	$\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = 2\sqrt{5}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\)	\(=\)	\( q + q^{2} + q^{3} + q^{4} + 2 q^{5} + q^{6} + q^{7} + q^{8} + q^{9}+O(q^{10}) \)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{2} + 2 q^{3} + 2 q^{4} + 4 q^{5} + 2 q^{6} + 2 q^{7} + 2 q^{8} + 2 q^{9}+O(q^{10}) \)

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

1.61803
−0.618034

1.00000

1.00000

1.00000

2.00000

1.00000

1.00000

1.00000

1.00000

2.00000

1.2

1.00000

1.00000

1.00000

2.00000

1.00000

1.00000

1.00000

1.00000

2.00000

Atkin-Lehner signs

\( p \)	Sign
\(2\)	\(-1\)
\(3\)	\(-1\)
\(7\)	\(-1\)
\(23\)	\(1\)

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	966.2.a.p	✓	2
3.b	odd	2	1		2898.2.a.v		2
4.b	odd	2	1		7728.2.a.bd		2
7.b	odd	2	1		6762.2.a.bz		2

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
966.2.a.p	✓	2	1.a	even	1	1	trivial
2898.2.a.v		2	3.b	odd	2	1
6762.2.a.bz		2	7.b	odd	2	1
7728.2.a.bd		2	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 \(S_{2}^{\mathrm{new}}(\Gamma_0(966))\):

\( T_{5} - 2 \)

\( T_{11} \)

\( T_{13}^{2} - 20 \)

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	\( (T - 1)^{2} \)
$3$	\( (T - 1)^{2} \)
$5$	\( (T - 2)^{2} \)
$7$	\( (T - 1)^{2} \)
$11$	\( T^{2} \)