Newform orbit 2704.2.a.u

• Label: 2704.2.a.u
• Level: $2704$
• Weight: $2$
• Character orbit: 2704.a
• Self dual: yes
• Analytic conductor: $21.592$
• Analytic rank: $1$
• Dimension: $2$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 2704 = 2^{4} \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2704.a (trivial)

Newform invariants

Self dual:	yes
Analytic conductor:	\(21.5915487066\)
Analytic rank:	\(1\)
Dimension:	\(2\)
Coefficient field:	\(\Q(\sqrt{3}) \)
Defining polynomial:	\( x^{2} - 3 \)
Coefficient ring:	\(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	no (minimal twist has level 52)
Fricke sign:	\(1\)
Sato-Tate group:	$\mathrm{SU}(2)$

$q$-expansion

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

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

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

1.73205
−1.73205

0

1.00000

0

0

0

−1.73205

0

−2.00000

0

1.2

0

1.00000

0

0

0

1.73205

0

−2.00000

0

Atkin-Lehner signs

\( p \)	Sign
\(2\)	\(-1\)
\(13\)	\(-1\)

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
13.b	even	2	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	2704.2.a.u		2
4.b	odd	2	1		676.2.a.f		2
12.b	even	2	1		6084.2.a.t		2
13.b	even	2	1	inner	2704.2.a.u		2
13.d	odd	4	2		2704.2.f.h		2
13.f	odd	12	2		208.2.w.a		2
39.k	even	12	2		1872.2.by.e		2
52.b	odd	2	1		676.2.a.f		2
52.f	even	4	2		676.2.d.b		2
52.i	odd	6	2		676.2.e.e		4
52.j	odd	6	2		676.2.e.e		4
52.l	even	12	2		52.2.h.a	✓	2
52.l	even	12	2		676.2.h.a		2
104.u	even	12	2		832.2.w.b		2
104.x	odd	12	2		832.2.w.c		2
156.h	even	2	1		6084.2.a.t		2
156.l	odd	4	2		6084.2.b.d		2
156.v	odd	12	2		468.2.t.a		2
260.bc	even	12	2		1300.2.y.a		2
260.be	odd	12	2		1300.2.ba.a		4
260.bl	odd	12	2		1300.2.ba.a		4
364.bt	even	12	2		2548.2.bq.a		2
364.bv	odd	12	2		2548.2.u.a		2
364.bz	odd	12	2		2548.2.bb.a		2
364.ca	even	12	2		2548.2.bb.b		2
364.cg	odd	12	2		2548.2.bq.b		2

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
52.2.h.a	✓	2	52.l	even	12	2
208.2.w.a		2	13.f	odd	12	2
468.2.t.a		2	156.v	odd	12	2
676.2.a.f		2	4.b	odd	2	1
676.2.a.f		2	52.b	odd	2	1
676.2.d.b		2	52.f	even	4	2
676.2.e.e		4	52.i	odd	6	2
676.2.e.e		4	52.j	odd	6	2
676.2.h.a		2	52.l	even	12	2
832.2.w.b		2	104.u	even	12	2
832.2.w.c		2	104.x	odd	12	2
1300.2.y.a		2	260.bc	even	12	2
1300.2.ba.a		4	260.be	odd	12	2
1300.2.ba.a		4	260.bl	odd	12	2
1872.2.by.e		2	39.k	even	12	2
2548.2.u.a		2	364.bv	odd	12	2
2548.2.bb.a		2	364.bz	odd	12	2
2548.2.bb.b		2	364.ca	even	12	2
2548.2.bq.a		2	364.bt	even	12	2
2548.2.bq.b		2	364.cg	odd	12	2
2704.2.a.u		2	1.a	even	1	1	trivial
2704.2.a.u		2	13.b	even	2	1	inner
2704.2.f.h		2	13.d	odd	4	2
6084.2.a.t		2	12.b	even	2	1
6084.2.a.t		2	156.h	even	2	1
6084.2.b.d		2	156.l	odd	4	2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(2704))\):

\( T_{3} - 1 \)

\( T_{5} \)

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

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	\( T^{2} \)
$3$	\( (T - 1)^{2} \)
$5$	\( T^{2} \)
$7$	\( T^{2} - 3 \)
$11$	\( T^{2} - 27 \)