Newform orbit 7605.2.a.y

• Label: 7605.2.a.y
• 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:	\(\Q(\sqrt{3}) \)
Defining polynomial:	\( x^{2} - 3 \)
Coefficient ring:	\(\Z[a_1, a_2]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	no (minimal twist has level 195)
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.

\(f(q)\)	\(=\)	\( q + (\beta - 1) q^{2} + ( - 2 \beta + 2) q^{4} - q^{5} + ( - 2 \beta - 1) q^{7} + (2 \beta - 6) q^{8}+O(q^{10}) \)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{2} + 4 q^{4} - 2 q^{5} - 2 q^{7} - 12 q^{8}+O(q^{10}) \)

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

−2.73205

0

5.46410

−1.00000

0

2.46410

−9.46410

0

2.73205

1.2

0.732051

0

−1.46410

−1.00000

0

−4.46410

−2.53590

0

−0.732051

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.y		2
3.b	odd	2	1		2535.2.a.s		2
13.b	even	2	1		7605.2.a.bk		2
13.f	odd	12	2		585.2.bu.a		4
39.d	odd	2	1		2535.2.a.n		2
39.k	even	12	2		195.2.bb.a	✓	4
195.bc	odd	12	2		975.2.w.f		4
195.bh	even	12	2		975.2.bc.h		4
195.bn	odd	12	2		975.2.w.a		4

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
195.2.bb.a	✓	4	39.k	even	12	2
585.2.bu.a		4	13.f	odd	12	2
975.2.w.a		4	195.bn	odd	12	2
975.2.w.f		4	195.bc	odd	12	2
975.2.bc.h		4	195.bh	even	12	2
2535.2.a.n		2	39.d	odd	2	1
2535.2.a.s		2	3.b	odd	2	1
7605.2.a.y		2	1.a	even	1	1	trivial
7605.2.a.bk		2	13.b	even	2	1

Hecke kernels

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

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

\( T_{7}^{2} + 2T_{7} - 11 \)

\( T_{11}^{2} - 12 \)

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	\( T^{2} + 2T - 2 \)
$3$	\( T^{2} \)
$5$	\( (T + 1)^{2} \)
$7$	\( T^{2} + 2T - 11 \)
$11$	\( T^{2} - 12 \)