Newform orbit 5202.2.a.t

• Label: 5202.2.a.t
• Level: $5202$
• Weight: $2$
• Character orbit: 5202.a
• Self dual: yes
• Analytic conductor: $41.538$
• Analytic rank: $1$
• Dimension: $2$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 5202 = 2 \cdot 3^{2} \cdot 17^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5202.a (trivial)

Newform invariants

Self dual:	yes
Analytic conductor:	\(41.5381791315\)
Analytic rank:	\(1\)
Dimension:	\(2\)
Coefficient field:	\(\Q(\sqrt{2}) \)
Defining polynomial:	\( x^{2} - 2 \)
Coefficient ring:	\(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	no (minimal twist has level 306)
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.

\(f(q)\)	\(=\)	\( q - q^{2} + q^{4} + \beta q^{5} - 3 \beta q^{7} - q^{8} +O(q^{10}) \)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{2} + 2 q^{4} - 2 q^{8}+O(q^{10}) \)

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

−1.41421
1.41421

−1.00000

0

1.00000

−1.41421

0

4.24264

−1.00000

0

1.41421

1.2

−1.00000

0

1.00000

1.41421

0

−4.24264

−1.00000

0

−1.41421

Atkin-Lehner signs

\( p \)	Sign
\(2\)	\(1\)
\(3\)	\(1\)
\(17\)	\(1\)

Inner twists

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

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	5202.2.a.t		2
3.b	odd	2	1		5202.2.a.bd		2
17.b	even	2	1	inner	5202.2.a.t		2
17.c	even	4	2		306.2.b.c	yes	2
51.c	odd	2	1		5202.2.a.bd		2
51.f	odd	4	2		306.2.b.b	✓	2
68.f	odd	4	2		2448.2.c.i		2
204.l	even	4	2		2448.2.c.k		2

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
306.2.b.b	✓	2	51.f	odd	4	2
306.2.b.c	yes	2	17.c	even	4	2
2448.2.c.i		2	68.f	odd	4	2
2448.2.c.k		2	204.l	even	4	2
5202.2.a.t		2	1.a	even	1	1	trivial
5202.2.a.t		2	17.b	even	2	1	inner
5202.2.a.bd		2	3.b	odd	2	1
5202.2.a.bd		2	51.c	odd	2	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(5202))\):

\( T_{5}^{2} - 2 \)

\( T_{7}^{2} - 18 \)

\( T_{23}^{2} - 50 \)

\( T_{47} + 12 \)

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	\( (T + 1)^{2} \)
$3$	\( T^{2} \)
$5$	\( T^{2} - 2 \)
$7$	\( T^{2} - 18 \)
$11$	\( T^{2} - 8 \)