Newform orbit 7938.2.a.bq

• Label: 7938.2.a.bq
• Level: $7938$
• Weight: $2$
• Character orbit: 7938.a
• Self dual: yes
• Analytic conductor: $63.385$
• Analytic rank: $0$
• Dimension: $2$
• CM: no
• Inner twists: $1$

Newspace parameters

Level:	\( N \)	\(=\)	\( 7938 = 2 \cdot 3^{4} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	7938.a (trivial)

Newform invariants

Self dual:	yes
Analytic conductor:	\(63.3852491245\)
Analytic rank:	\(0\)
Dimension:	\(2\)
Coefficient field:	\(\Q(\sqrt{2}) \)
Defining polynomial:	\( x^{2} - 2 \)
Coefficient ring:	\(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index:	\( 3 \)
Twist minimal:	no (minimal twist has level 1134)
Fricke sign:	\(-1\)
Sato-Tate group:	$\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = 3\sqrt{2}\). We also show the integral \(q\)-expansion of the trace form.

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

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

0

0

0

1.00000

0

0

1.2

1.00000

0

1.00000

0

0

0

1.00000

0

0

Atkin-Lehner signs

\( p \)	Sign
\(2\)	\(-1\)
\(3\)	\(1\)
\(7\)	\(1\)

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	7938.2.a.bq		2
3.b	odd	2	1		7938.2.a.bk		2
7.b	odd	2	1		7938.2.a.bp		2
7.c	even	3	2		1134.2.g.i	✓	4
21.c	even	2	1		7938.2.a.bj		2
21.h	odd	6	2		1134.2.g.j	yes	4
63.g	even	3	2		1134.2.h.r		4
63.h	even	3	2		1134.2.e.s		4
63.j	odd	6	2		1134.2.e.r		4
63.n	odd	6	2		1134.2.h.s		4

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
1134.2.e.r		4	63.j	odd	6	2
1134.2.e.s		4	63.h	even	3	2
1134.2.g.i	✓	4	7.c	even	3	2
1134.2.g.j	yes	4	21.h	odd	6	2
1134.2.h.r		4	63.g	even	3	2
1134.2.h.s		4	63.n	odd	6	2
7938.2.a.bj		2	21.c	even	2	1
7938.2.a.bk		2	3.b	odd	2	1
7938.2.a.bp		2	7.b	odd	2	1
7938.2.a.bq		2	1.a	even	1	1	trivial

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(7938))\):

\( T_{5} \)

\( T_{11}^{2} - 18 \)

\( T_{13}^{2} - 4T_{13} - 14 \)

\( T_{17} \)

\( T_{23}^{2} - 6T_{23} - 9 \)

Hecke characteristic polynomials

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