Newform orbit 784.4.a.v

• Label: 784.4.a.v
• Level: $784$
• Weight: $4$
• Character orbit: 784.a
• Self dual: yes
• Analytic conductor: $46.257$
• Analytic rank: $0$
• Dimension: $2$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 784 = 2^{4} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	784.a (trivial)

Newform invariants

Self dual:	yes
Analytic conductor:	\(46.2574974445\)
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 392)
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.

\(f(q)\)	\(=\)	\( q + \beta q^{3} + 10 \beta q^{5} - 25 q^{9}+O(q^{10}) \)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 50 q^{9}+O(q^{10}) \)

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

−14.1421

0

0

0

−25.0000

0

1.2

0

1.41421

0

14.1421

0

0

0

−25.0000

0

Atkin-Lehner signs

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

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	784.4.a.v		2
4.b	odd	2	1		392.4.a.f	✓	2
7.b	odd	2	1	inner	784.4.a.v		2
28.d	even	2	1		392.4.a.f	✓	2
28.f	even	6	2		392.4.i.k		4
28.g	odd	6	2		392.4.i.k		4

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
392.4.a.f	✓	2	4.b	odd	2	1
392.4.a.f	✓	2	28.d	even	2	1
392.4.i.k		4	28.f	even	6	2
392.4.i.k		4	28.g	odd	6	2
784.4.a.v		2	1.a	even	1	1	trivial
784.4.a.v		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 \(S_{4}^{\mathrm{new}}(\Gamma_0(784))\):

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

\( T_{5}^{2} - 200 \)

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	\( T^{2} \)
$3$	\( T^{2} - 2 \)
$5$	\( T^{2} - 200 \)
$7$	\( T^{2} \)
$11$	\( (T + 54)^{2} \)