Newform orbit 1225.2.a.g

• Label: 1225.2.a.g
• Level: $1225$
• Weight: $2$
• Character orbit: 1225.a
• Self dual: yes
• Analytic conductor: $9.782$
• Analytic rank: $1$
• Dimension: $1$
• CM: no
• Inner twists: $1$

Newspace parameters

Level:	\( N \)	\(=\)	\( 1225 = 5^{2} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1225.a (trivial)

Newform invariants

Self dual:	yes
Analytic conductor:	\(9.78167424761\)
Analytic rank:	\(1\)
Dimension:	\(1\)
Coefficient field:	\(\mathbb{Q}\)
Coefficient ring:	\(\mathbb{Z}\)
Coefficient ring index:	\( 1 \)
Twist minimal:	no (minimal twist has level 35)
Fricke sign:	\(+1\)
Sato-Tate group:	$\mathrm{SU}(2)$

$q$-expansion

\(f(q)\)

\(=\)

\( q + q^{2} + q^{3} - q^{4} + q^{6} - 3 q^{8} - 2 q^{9}+O(q^{10}) \)

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

0

1.00000

1.00000

−1.00000

0

1.00000

0

−3.00000

−2.00000

0

Atkin-Lehner signs

\( p \)	Sign
\(5\)	\( -1 \)
\(7\)	\( -1 \)

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	1225.2.a.g		1
5.b	even	2	1		1225.2.a.b		1
5.c	odd	4	2		245.2.b.b		2
7.b	odd	2	1		1225.2.a.f		1
7.d	odd	6	2		175.2.e.a		2
15.e	even	4	2		2205.2.d.e		2
35.c	odd	2	1		1225.2.a.d		1
35.f	even	4	2		245.2.b.c		2
35.i	odd	6	2		175.2.e.b		2
35.k	even	12	4		35.2.j.a	✓	4
35.l	odd	12	4		245.2.j.c		4
105.k	odd	4	2		2205.2.d.d		2
105.w	odd	12	4		315.2.bf.a		4
140.x	odd	12	4		560.2.bw.b		4

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
35.2.j.a	✓	4	35.k	even	12	4
175.2.e.a		2	7.d	odd	6	2
175.2.e.b		2	35.i	odd	6	2
245.2.b.b		2	5.c	odd	4	2
245.2.b.c		2	35.f	even	4	2
245.2.j.c		4	35.l	odd	12	4
315.2.bf.a		4	105.w	odd	12	4
560.2.bw.b		4	140.x	odd	12	4
1225.2.a.b		1	5.b	even	2	1
1225.2.a.d		1	35.c	odd	2	1
1225.2.a.f		1	7.b	odd	2	1
1225.2.a.g		1	1.a	even	1	1	trivial
2205.2.d.d		2	105.k	odd	4	2
2205.2.d.e		2	15.e	even	4	2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(1225))\):

\( T_{2} - 1 \)

\( T_{3} - 1 \)

Hecke characteristic polynomials

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