Newform orbit 2394.2.a.r

• Label: 2394.2.a.r
• Level: $2394$
• Weight: $2$
• Character orbit: 2394.a
• Self dual: yes
• Analytic conductor: $19.116$
• Analytic rank: $0$
• Dimension: $2$
• CM: no
• Inner twists: $1$

Newspace parameters

Level:	\( N \)	\(=\)	\( 2394 = 2 \cdot 3^{2} \cdot 7 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2394.a (trivial)

Newform invariants

Self dual:	yes
Analytic conductor:	\(19.1161862439\)
Analytic rank:	\(0\)
Dimension:	\(2\)
Coefficient field:	\(\Q(\sqrt{2}) \)
Defining polynomial:	\( x^{2} - 2 \)
Coefficient ring:	\(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index:	\( 2 \)
Twist minimal:	no (minimal twist has level 798)
Fricke sign:	\(-1\)
Sato-Tate group:	$\mathrm{SU}(2)$

$q$-expansion

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

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

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

−2.82843

0

1.00000

−1.00000

0

2.82843

1.2

−1.00000

0

1.00000

2.82843

0

1.00000

−1.00000

0

−2.82843

Atkin-Lehner signs

\( p \)	Sign
\(2\)	\(1\)
\(3\)	\(-1\)
\(7\)	\(-1\)
\(19\)	\(-1\)

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	2394.2.a.r		2
3.b	odd	2	1		798.2.a.l	✓	2
12.b	even	2	1		6384.2.a.bq		2
21.c	even	2	1		5586.2.a.bp		2

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
798.2.a.l	✓	2	3.b	odd	2	1
2394.2.a.r		2	1.a	even	1	1	trivial
5586.2.a.bp		2	21.c	even	2	1
6384.2.a.bq		2	12.b	even	2	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(2394))\):

\( T_{5}^{2} - 8 \)

\( T_{11} + 2 \)

\( T_{13}^{2} - 8 \)

\( T_{17} \)

Hecke characteristic polynomials

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