Newform orbit 9075.2.a.bz

• Label: 9075.2.a.bz
• Level: $9075$
• Weight: $2$
• Character orbit: 9075.a
• Self dual: yes
• Analytic conductor: $72.464$
• Analytic rank: $0$
• Dimension: $2$
• CM: no
• Inner twists: $1$

Newspace parameters

Level:	\( N \)	\(=\)	\( 9075 = 3 \cdot 5^{2} \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	9075.a (trivial)

Newform invariants

Self dual:	yes
Analytic conductor:	\(72.4642398343\)
Analytic rank:	\(0\)
Dimension:	\(2\)
Coefficient field:	\(\Q(\sqrt{5}) \)
Defining polynomial:	\( x^{2} - x - 1 \)
Coefficient ring:	\(\Z[a_1, a_2]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	no (minimal twist has level 825)
Fricke sign:	\(-1\)
Sato-Tate group:	$\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \frac{1}{2}(1 + \sqrt{5})\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\)	\(=\)	\( q + \beta q^{2} + q^{3} + (\beta - 1) q^{4} + \beta q^{6} + 3 q^{7} + ( - 2 \beta + 1) q^{8} + q^{9}+O(q^{10}) \)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + q^{2} + 2 q^{3} - q^{4} + q^{6} + 6 q^{7} + 2 q^{9}+O(q^{10}) \)

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

−0.618034
1.61803

−0.618034

1.00000

−1.61803

0

−0.618034

3.00000

2.23607

1.00000

0

1.2

1.61803

1.00000

0.618034

0

1.61803

3.00000

−2.23607

1.00000

0

Atkin-Lehner signs

\( p \)	Sign
\(3\)	\(-1\)
\(5\)	\(1\)
\(11\)	\(1\)

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	9075.2.a.bz		2
5.b	even	2	1		9075.2.a.y		2
11.b	odd	2	1		9075.2.a.bb		2
11.c	even	5	2		825.2.n.a	✓	4
55.d	odd	2	1		9075.2.a.bu		2
55.j	even	10	2		825.2.n.e	yes	4
55.k	odd	20	4		825.2.bx.a		8

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
825.2.n.a	✓	4	11.c	even	5	2
825.2.n.e	yes	4	55.j	even	10	2
825.2.bx.a		8	55.k	odd	20	4
9075.2.a.y		2	5.b	even	2	1
9075.2.a.bb		2	11.b	odd	2	1
9075.2.a.bu		2	55.d	odd	2	1
9075.2.a.bz		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 \(S_{2}^{\mathrm{new}}(\Gamma_0(9075))\):

\( T_{2}^{2} - T_{2} - 1 \)

\( T_{7} - 3 \)

\( T_{13}^{2} - 3T_{13} - 9 \)

\( T_{17}^{2} - 6T_{17} - 11 \)

\( T_{19}^{2} + 10T_{19} + 20 \)

\( T_{23}^{2} - 13T_{23} + 41 \)

\( T_{37}^{2} - 11T_{37} + 19 \)

Hecke characteristic polynomials

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