Newform orbit 7605.2.a.bg

• Label: 7605.2.a.bg
• Level: $7605$
• Weight: $2$
• Character orbit: 7605.a
• Self dual: yes
• Analytic conductor: $60.726$
• Analytic rank: $1$
• Dimension: $2$
• CM: no
• Inner twists: $1$

Newspace parameters

Level:	\( N \)	\(=\)	\( 7605 = 3^{2} \cdot 5 \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	7605.a (trivial)

Newform invariants

Self dual:	yes
Analytic conductor:	\(60.7262307372\)
Analytic rank:	\(1\)
Dimension:	\(2\)
Coefficient field:	\(\Q(\sqrt{13}) \)
Defining polynomial:	\( x^{2} - x - 3 \)
Coefficient ring:	\(\Z[a_1, a_2]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	no (minimal twist has level 65)
Fricke sign:	\(1\)
Sato-Tate group:	$\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \frac{1}{2}(1 + \sqrt{13})\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\)	\(=\)	\( q + \beta q^{2} + (\beta + 1) q^{4} + q^{5} - q^{7} + 3 q^{8}+O(q^{10}) \)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + q^{2} + 3 q^{4} + 2 q^{5} - 2 q^{7} + 6 q^{8}+O(q^{10}) \)

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

−1.30278
2.30278

−1.30278

0

−0.302776

1.00000

0

−1.00000

3.00000

0

−1.30278

1.2

2.30278

0

3.30278

1.00000

0

−1.00000

3.00000

0

2.30278

Atkin-Lehner signs

\( p \)	Sign
\(3\)	\(-1\)
\(5\)	\(-1\)
\(13\)	\(1\)

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	7605.2.a.bg		2
3.b	odd	2	1		845.2.a.c		2
13.b	even	2	1		7605.2.a.bb		2
13.c	even	3	2		585.2.j.d		4
15.d	odd	2	1		4225.2.a.x		2
39.d	odd	2	1		845.2.a.f		2
39.f	even	4	2		845.2.c.d		4
39.h	odd	6	2		845.2.e.d		4
39.i	odd	6	2		65.2.e.b	✓	4
39.k	even	12	4		845.2.m.d		8
156.p	even	6	2		1040.2.q.o		4
195.e	odd	2	1		4225.2.a.t		2
195.x	odd	6	2		325.2.e.a		4
195.bl	even	12	4		325.2.o.b		8

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
65.2.e.b	✓	4	39.i	odd	6	2
325.2.e.a		4	195.x	odd	6	2
325.2.o.b		8	195.bl	even	12	4
585.2.j.d		4	13.c	even	3	2
845.2.a.c		2	3.b	odd	2	1
845.2.a.f		2	39.d	odd	2	1
845.2.c.d		4	39.f	even	4	2
845.2.e.d		4	39.h	odd	6	2
845.2.m.d		8	39.k	even	12	4
1040.2.q.o		4	156.p	even	6	2
4225.2.a.t		2	195.e	odd	2	1
4225.2.a.x		2	15.d	odd	2	1
7605.2.a.bb		2	13.b	even	2	1
7605.2.a.bg		2	1.a	even	1	1	trivial

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(7605))\):

\( T_{2}^{2} - T_{2} - 3 \)

\( T_{7} + 1 \)

\( T_{11}^{2} - 4T_{11} - 9 \)

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	\( T^{2} - T - 3 \)
$3$	\( T^{2} \)
$5$	\( (T - 1)^{2} \)
$7$	\( (T + 1)^{2} \)
$11$	\( T^{2} - 4T - 9 \)