Newform orbit 784.4.a.ba

• Label: 784.4.a.ba
• Level: $784$
• Weight: $4$
• Character orbit: 784.a
• Self dual: yes
• Analytic conductor: $46.257$
• Analytic rank: $0$
• Dimension: $2$
• CM: no
• Inner twists: $1$

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:	\(\Q(\sqrt{37}) \)
Defining polynomial:	\( x^{2} - x - 9 \)
Coefficient ring:	\(\Z[a_1, a_2, a_3]\)
Coefficient ring index:	\( 2 \)
Twist minimal:	no (minimal twist has level 28)
Fricke sign:	\(1\)
Sato-Tate group:	$\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \sqrt{37}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\)	\(=\)	\( q - \beta q^{3} + ( - 2 \beta + 7) q^{5} + 10 q^{9} +O(q^{10}) \)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 14 q^{5} + 20 q^{9}+O(q^{10}) \)

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

3.54138
−2.54138

0

−6.08276

0

−5.16553

0

0

0

10.0000

0

1.2

0

6.08276

0

19.1655

0

0

0

10.0000

0

Atkin-Lehner signs

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

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	784.4.a.ba		2
4.b	odd	2	1		196.4.a.g		2
7.b	odd	2	1		784.4.a.u		2
7.d	odd	6	2		112.4.i.d		4
12.b	even	2	1		1764.4.a.n		2
28.d	even	2	1		196.4.a.e		2
28.f	even	6	2		28.4.e.a	✓	4
28.g	odd	6	2		196.4.e.g		4
56.j	odd	6	2		448.4.i.g		4
56.m	even	6	2		448.4.i.h		4
84.h	odd	2	1		1764.4.a.z		2
84.j	odd	6	2		252.4.k.c		4
84.n	even	6	2		1764.4.k.ba		4
140.s	even	6	2		700.4.i.g		4
140.x	odd	12	4		700.4.r.d		8

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
28.4.e.a	✓	4	28.f	even	6	2
112.4.i.d		4	7.d	odd	6	2
196.4.a.e		2	28.d	even	2	1
196.4.a.g		2	4.b	odd	2	1
196.4.e.g		4	28.g	odd	6	2
252.4.k.c		4	84.j	odd	6	2
448.4.i.g		4	56.j	odd	6	2
448.4.i.h		4	56.m	even	6	2
700.4.i.g		4	140.s	even	6	2
700.4.r.d		8	140.x	odd	12	4
784.4.a.u		2	7.b	odd	2	1
784.4.a.ba		2	1.a	even	1	1	trivial
1764.4.a.n		2	12.b	even	2	1
1764.4.a.z		2	84.h	odd	2	1
1764.4.k.ba		4	84.n	even	6	2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(784))\):

\( T_{3}^{2} - 37 \)

\( T_{5}^{2} - 14T_{5} - 99 \)

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	\( T^{2} \)
$3$	\( T^{2} - 37 \)
$5$	\( T^{2} - 14T - 99 \)
$7$	\( T^{2} \)
$11$	\( T^{2} + 32T - 1557 \)