Newform orbit 117.2.a.b

• Label: 117.2.a.b
• Level: $117$
• Weight: $2$
• Character orbit: 117.a
• Self dual: yes
• Analytic conductor: $0.934$
• Analytic rank: $0$
• Dimension: $2$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 117 = 3^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	117.a (trivial)

Newform invariants

Self dual:	yes
Analytic conductor:	\(0.934249703649\)
Analytic rank:	\(0\)
Dimension:	\(2\)
Coefficient field:	\(\Q(\sqrt{3}) \)
Defining polynomial:	\( x^{2} - 3 \)
Coefficient ring:	\(\Z[a_1, a_2]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	yes
Fricke sign:	\(-1\)
Sato-Tate group:	$\mathrm{SU}(2)$

$q$-expansion

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

\(f(q)\)	\(=\)	\( q + \beta q^{2} + q^{4} + 2 q^{7} - \beta q^{8} +O(q^{10}) \)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{4} + 4 q^{7}+O(q^{10}) \)

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

−1.73205

0

1.00000

0

0

2.00000

1.73205

0

0

1.2

1.73205

0

1.00000

0

0

2.00000

−1.73205

0

0

Atkin-Lehner signs

\( p \)	Sign
\(3\)	\(1\)
\(13\)	\(-1\)

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
3.b	odd	2	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	117.2.a.b	✓	2
3.b	odd	2	1	inner	117.2.a.b	✓	2
4.b	odd	2	1		1872.2.a.v		2
5.b	even	2	1		2925.2.a.y		2
5.c	odd	4	2		2925.2.c.s		4
7.b	odd	2	1		5733.2.a.t		2
8.b	even	2	1		7488.2.a.cq		2
8.d	odd	2	1		7488.2.a.cj		2
9.c	even	3	2		1053.2.e.i		4
9.d	odd	6	2		1053.2.e.i		4
12.b	even	2	1		1872.2.a.v		2
13.b	even	2	1		1521.2.a.j		2
13.d	odd	4	2		1521.2.b.i		4
15.d	odd	2	1		2925.2.a.y		2
15.e	even	4	2		2925.2.c.s		4
21.c	even	2	1		5733.2.a.t		2
24.f	even	2	1		7488.2.a.cj		2
24.h	odd	2	1		7488.2.a.cq		2
39.d	odd	2	1		1521.2.a.j		2
39.f	even	4	2		1521.2.b.i		4

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
117.2.a.b	✓	2	1.a	even	1	1	trivial
117.2.a.b	✓	2	3.b	odd	2	1	inner
1053.2.e.i		4	9.c	even	3	2
1053.2.e.i		4	9.d	odd	6	2
1521.2.a.j		2	13.b	even	2	1
1521.2.a.j		2	39.d	odd	2	1
1521.2.b.i		4	13.d	odd	4	2
1521.2.b.i		4	39.f	even	4	2
1872.2.a.v		2	4.b	odd	2	1
1872.2.a.v		2	12.b	even	2	1
2925.2.a.y		2	5.b	even	2	1
2925.2.a.y		2	15.d	odd	2	1
2925.2.c.s		4	5.c	odd	4	2
2925.2.c.s		4	15.e	even	4	2
5733.2.a.t		2	7.b	odd	2	1
5733.2.a.t		2	21.c	even	2	1
7488.2.a.cj		2	8.d	odd	2	1
7488.2.a.cj		2	24.f	even	2	1
7488.2.a.cq		2	8.b	even	2	1
7488.2.a.cq		2	24.h	odd	2	1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{2} - 3 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(117))\).

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	\( T^{2} - 3 \)
$3$	\( T^{2} \)
$5$	\( T^{2} \)
$7$	\( (T - 2)^{2} \)
$11$	\( T^{2} - 12 \)