Newform orbit 2450.4.a.bx

• Label: 2450.4.a.bx
• Level: $2450$
• Weight: $4$
• Character orbit: 2450.a
• Self dual: yes
• Analytic conductor: $144.555$
• Analytic rank: $1$
• Dimension: $2$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 2450 = 2 \cdot 5^{2} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	2450.a (trivial)

Newform invariants

Self dual:	yes
Analytic conductor:	\(144.554679514\)
Analytic rank:	\(1\)
Dimension:	\(2\)
Coefficient field:	\(\Q(\sqrt{2}) \)
Defining polynomial:	\( x^{2} - 2 \)
Coefficient ring:	\(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	no (minimal twist has level 98)
Fricke sign:	\(-1\)
Sato-Tate group:	$\mathrm{SU}(2)$

$q$-expansion

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

\(f(q)\)	\(=\)	\( q + 2 q^{2} + 5 \beta q^{3} + 4 q^{4} + 10 \beta q^{6} + 8 q^{8} + 23 q^{9}+O(q^{10}) \)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 4 q^{2} + 8 q^{4} + 16 q^{8} + 46 q^{9}+O(q^{10}) \)

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

−1.41421
1.41421

2.00000

−7.07107

4.00000

0

−14.1421

0

8.00000

23.0000

0

1.2

2.00000

7.07107

4.00000

0

14.1421

0

8.00000

23.0000

0

Atkin-Lehner signs

\( p \)	Sign
\(2\)	\(-1\)
\(5\)	\(1\)
\(7\)	\(1\)

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
7.b	odd	2	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	2450.4.a.bx		2
5.b	even	2	1		98.4.a.g	✓	2
7.b	odd	2	1	inner	2450.4.a.bx		2
15.d	odd	2	1		882.4.a.bg		2
20.d	odd	2	1		784.4.a.y		2
35.c	odd	2	1		98.4.a.g	✓	2
35.i	odd	6	2		98.4.c.h		4
35.j	even	6	2		98.4.c.h		4
105.g	even	2	1		882.4.a.bg		2
105.o	odd	6	2		882.4.g.ba		4
105.p	even	6	2		882.4.g.ba		4
140.c	even	2	1		784.4.a.y		2

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
98.4.a.g	✓	2	5.b	even	2	1
98.4.a.g	✓	2	35.c	odd	2	1
98.4.c.h		4	35.i	odd	6	2
98.4.c.h		4	35.j	even	6	2
784.4.a.y		2	20.d	odd	2	1
784.4.a.y		2	140.c	even	2	1
882.4.a.bg		2	15.d	odd	2	1
882.4.a.bg		2	105.g	even	2	1
882.4.g.ba		4	105.o	odd	6	2
882.4.g.ba		4	105.p	even	6	2
2450.4.a.bx		2	1.a	even	1	1	trivial
2450.4.a.bx		2	7.b	odd	2	1	inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(2450))\):

\( T_{3}^{2} - 50 \)

\( T_{11} + 14 \)

\( T_{19}^{2} - 2 \)

\( T_{23} + 140 \)

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	\( (T - 2)^{2} \)
$3$	\( T^{2} - 50 \)
$5$	\( T^{2} \)
$7$	\( T^{2} \)
$11$	\( (T + 14)^{2} \)