Newform orbit 722.2.a.j

• Label: 722.2.a.j
• Level: $722$
• Weight: $2$
• Character orbit: 722.a
• Self dual: yes
• Analytic conductor: $5.765$
• Analytic rank: $0$
• Dimension: $2$
• CM: no
• Inner twists: $1$

Newspace parameters

Level:	\( N \)	\(=\)	\( 722 = 2 \cdot 19^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	722.a (trivial)

Newform invariants

Self dual:	yes
Analytic conductor:	\(5.76519902594\)
Analytic rank:	\(0\)
Dimension:	\(2\)
Coefficient field:	\(\Q(\sqrt{7}) \)
Defining polynomial:	\( x^{2} - 7 \)
Coefficient ring:	\(\Z[a_1, a_2, a_3]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	no (minimal twist has level 38)
Fricke sign:	\(-1\)
Sato-Tate group:	$\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \sqrt{7}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\)	\(=\)	\( q + q^{2} + \beta q^{3} + q^{4} + ( - \beta + 1) q^{5} + \beta q^{6} + (\beta + 1) q^{7} + q^{8} + 4 q^{9}+O(q^{10}) \)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{2} + 2 q^{4} + 2 q^{5} + 2 q^{7} + 2 q^{8} + 8 q^{9}+O(q^{10}) \)

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

−2.64575
2.64575

1.00000

−2.64575

1.00000

3.64575

−2.64575

−1.64575

1.00000

4.00000

3.64575

1.2

1.00000

2.64575

1.00000

−1.64575

2.64575

3.64575

1.00000

4.00000

−1.64575

Atkin-Lehner signs

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

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	722.2.a.j		2
3.b	odd	2	1		6498.2.a.ba		2
4.b	odd	2	1		5776.2.a.ba		2
19.b	odd	2	1		722.2.a.g		2
19.c	even	3	2		38.2.c.b	✓	4
19.d	odd	6	2		722.2.c.j		4
19.e	even	9	6		722.2.e.n		12
19.f	odd	18	6		722.2.e.o		12
57.d	even	2	1		6498.2.a.bg		2
57.h	odd	6	2		342.2.g.f		4
76.d	even	2	1		5776.2.a.z		2
76.g	odd	6	2		304.2.i.e		4
95.i	even	6	2		950.2.e.k		4
95.m	odd	12	4		950.2.j.g		8
152.k	odd	6	2		1216.2.i.k		4
152.p	even	6	2		1216.2.i.l		4
228.m	even	6	2		2736.2.s.v		4

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
38.2.c.b	✓	4	19.c	even	3	2
304.2.i.e		4	76.g	odd	6	2
342.2.g.f		4	57.h	odd	6	2
722.2.a.g		2	19.b	odd	2	1
722.2.a.j		2	1.a	even	1	1	trivial
722.2.c.j		4	19.d	odd	6	2
722.2.e.n		12	19.e	even	9	6
722.2.e.o		12	19.f	odd	18	6
950.2.e.k		4	95.i	even	6	2
950.2.j.g		8	95.m	odd	12	4
1216.2.i.k		4	152.k	odd	6	2
1216.2.i.l		4	152.p	even	6	2
2736.2.s.v		4	228.m	even	6	2
5776.2.a.z		2	76.d	even	2	1
5776.2.a.ba		2	4.b	odd	2	1
6498.2.a.ba		2	3.b	odd	2	1
6498.2.a.bg		2	57.d	even	2	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(722))\):

\( T_{3}^{2} - 7 \)

\( T_{5}^{2} - 2T_{5} - 6 \)

\( T_{7}^{2} - 2T_{7} - 6 \)

\( T_{13} - 2 \)

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	\( (T - 1)^{2} \)
$3$	\( T^{2} - 7 \)
$5$	\( T^{2} - 2T - 6 \)
$7$	\( T^{2} - 2T - 6 \)
$11$	\( T^{2} + 4T - 3 \)