Newform orbit 2450.2.a.bp

• Label: 2450.2.a.bp
• Level: $2450$
• Weight: $2$
• Character orbit: 2450.a
• Self dual: yes
• Analytic conductor: $19.563$
• Analytic rank: $0$
• Dimension: $2$
• CM: no
• Inner twists: $2$

Newspace parameters

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

Newform invariants

Self dual:	yes
Analytic conductor:	\(19.5633484952\)
Analytic rank:	\(0\)
Dimension:	\(2\)
Coefficient field:	\(\Q(\sqrt{2}) \)
Defining polynomial:	\( x^{2} - 2 \)
Coefficient ring:	\(\Z[a_1, a_2, a_3]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	no (minimal twist has level 490)
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.

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

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

−1.41421
1.41421

1.00000

−1.41421

1.00000

0

−1.41421

0

1.00000

−1.00000

0

1.2

1.00000

1.41421

1.00000

0

1.41421

0

1.00000

−1.00000

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	2450.2.a.bp		2
5.b	even	2	1		2450.2.a.bk		2
5.c	odd	4	2		490.2.c.f	✓	4
7.b	odd	2	1	inner	2450.2.a.bp		2
35.c	odd	2	1		2450.2.a.bk		2
35.f	even	4	2		490.2.c.f	✓	4
35.k	even	12	4		490.2.i.e		8
35.l	odd	12	4		490.2.i.e		8

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
490.2.c.f	✓	4	5.c	odd	4	2
490.2.c.f	✓	4	35.f	even	4	2
490.2.i.e		8	35.k	even	12	4
490.2.i.e		8	35.l	odd	12	4
2450.2.a.bk		2	5.b	even	2	1
2450.2.a.bk		2	35.c	odd	2	1
2450.2.a.bp		2	1.a	even	1	1	trivial
2450.2.a.bp		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 \(S_{2}^{\mathrm{new}}(\Gamma_0(2450))\):

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

\( T_{11} \)

\( T_{13}^{2} - 18 \)

\( T_{17}^{2} - 32 \)

\( T_{19}^{2} - 18 \)

\( T_{23} - 6 \)

\( T_{37} - 6 \)

Hecke characteristic polynomials

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