Newform orbit 4050.2.a.bl

• Label: 4050.2.a.bl
• Level: $4050$
• Weight: $2$
• Character orbit: 4050.a
• Self dual: yes
• Analytic conductor: $32.339$
• Analytic rank: $0$
• Dimension: $2$
• CM: no
• Inner twists: $1$

Newspace parameters

Level:	\( N \)	\(=\)	\( 4050 = 2 \cdot 3^{4} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4050.a (trivial)

Newform invariants

Self dual:	yes
Analytic conductor:	\(32.3394128186\)
Analytic rank:	\(0\)
Dimension:	\(2\)
Coefficient field:	\(\Q(\sqrt{6}) \)
Defining polynomial:	\( x^{2} - 6 \)
Coefficient ring:	\(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	no (minimal twist has level 450)
Fricke sign:	\(-1\)
Sato-Tate group:	$\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \sqrt{6}\). We also show the integral \(q\)-expansion of the trace form.

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

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

−2.44949
2.44949

−1.00000

0

1.00000

0

0

−4.44949

−1.00000

0

0

1.2

−1.00000

0

1.00000

0

0

0.449490

−1.00000

0

0

Atkin-Lehner signs

\( p \)	Sign
\(2\)	\( +1 \)
\(3\)	\( -1 \)
\(5\)	\( +1 \)

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	4050.2.a.bl		2
3.b	odd	2	1		4050.2.a.bu		2
5.b	even	2	1		4050.2.a.by		2
5.c	odd	4	2		4050.2.c.w		4
9.c	even	3	2		1350.2.e.n		4
9.d	odd	6	2		450.2.e.l	✓	4
15.d	odd	2	1		4050.2.a.br		2
15.e	even	4	2		4050.2.c.y		4
45.h	odd	6	2		450.2.e.m	yes	4
45.j	even	6	2		1350.2.e.k		4
45.k	odd	12	4		1350.2.j.g		8
45.l	even	12	4		450.2.j.f		8

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
450.2.e.l	✓	4	9.d	odd	6	2
450.2.e.m	yes	4	45.h	odd	6	2
450.2.j.f		8	45.l	even	12	4
1350.2.e.k		4	45.j	even	6	2
1350.2.e.n		4	9.c	even	3	2
1350.2.j.g		8	45.k	odd	12	4
4050.2.a.bl		2	1.a	even	1	1	trivial
4050.2.a.br		2	15.d	odd	2	1
4050.2.a.bu		2	3.b	odd	2	1
4050.2.a.by		2	5.b	even	2	1
4050.2.c.w		4	5.c	odd	4	2
4050.2.c.y		4	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 \(S_{2}^{\mathrm{new}}(\Gamma_0(4050))\):

\( T_{7}^{2} + 4T_{7} - 2 \)

\( T_{11}^{2} - 24 \)

\( T_{13}^{2} + 4T_{13} - 2 \)

\( T_{17}^{2} - 24 \)

\( T_{23}^{2} - 6 \)

\( T_{41} + 9 \)

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	\( (T + 1)^{2} \)
$3$	\( T^{2} \)
$5$	\( T^{2} \)
$7$	\( T^{2} + 4T - 2 \)
$11$	\( T^{2} - 24 \)