Newform orbit 5200.2.a.bu

• Label: 5200.2.a.bu
• Level: $5200$
• Weight: $2$
• Character orbit: 5200.a
• Self dual: yes
• Analytic conductor: $41.522$
• Analytic rank: $1$
• Dimension: $2$
• CM: no
• Inner twists: $1$

Newspace parameters

Level:	\( N \)	\(=\)	\( 5200 = 2^{4} \cdot 5^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5200.a (trivial)

Newform invariants

Self dual:	yes
Analytic conductor:	\(41.5222090511\)
Analytic rank:	\(1\)
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 65)
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.

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

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

−1.41421
1.41421

0

−1.41421

0

0

0

4.82843

0

−1.00000

0

1.2

0

1.41421

0

0

0

−0.828427

0

−1.00000

0

Atkin-Lehner signs

\( p \)	Sign
\(2\)	\(-1\)
\(5\)	\(1\)
\(13\)	\(-1\)

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	5200.2.a.bu		2
4.b	odd	2	1		325.2.a.i		2
5.b	even	2	1		1040.2.a.j		2
12.b	even	2	1		2925.2.a.u		2
15.d	odd	2	1		9360.2.a.cd		2
20.d	odd	2	1		65.2.a.b	✓	2
20.e	even	4	2		325.2.b.f		4
40.e	odd	2	1		4160.2.a.bf		2
40.f	even	2	1		4160.2.a.z		2
52.b	odd	2	1		4225.2.a.r		2
60.h	even	2	1		585.2.a.m		2
60.l	odd	4	2		2925.2.c.r		4
140.c	even	2	1		3185.2.a.j		2
220.g	even	2	1		7865.2.a.j		2
260.g	odd	2	1		845.2.a.g		2
260.u	even	4	2		845.2.c.b		4
260.v	odd	6	2		845.2.e.h		4
260.w	odd	6	2		845.2.e.c		4
260.bc	even	12	4		845.2.m.f		8
780.d	even	2	1		7605.2.a.x		2

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
65.2.a.b	✓	2	20.d	odd	2	1
325.2.a.i		2	4.b	odd	2	1
325.2.b.f		4	20.e	even	4	2
585.2.a.m		2	60.h	even	2	1
845.2.a.g		2	260.g	odd	2	1
845.2.c.b		4	260.u	even	4	2
845.2.e.c		4	260.w	odd	6	2
845.2.e.h		4	260.v	odd	6	2
845.2.m.f		8	260.bc	even	12	4
1040.2.a.j		2	5.b	even	2	1
2925.2.a.u		2	12.b	even	2	1
2925.2.c.r		4	60.l	odd	4	2
3185.2.a.j		2	140.c	even	2	1
4160.2.a.z		2	40.f	even	2	1
4160.2.a.bf		2	40.e	odd	2	1
4225.2.a.r		2	52.b	odd	2	1
5200.2.a.bu		2	1.a	even	1	1	trivial
7605.2.a.x		2	780.d	even	2	1
7865.2.a.j		2	220.g	even	2	1
9360.2.a.cd		2	15.d	odd	2	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(5200))\):

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

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

\( T_{11}^{2} + 4T_{11} + 2 \)

Hecke characteristic polynomials

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