Newform orbit 1323.2.a.u

• Label: 1323.2.a.u
• Level: $1323$
• Weight: $2$
• Character orbit: 1323.a
• Self dual: yes
• Analytic conductor: $10.564$
• Analytic rank: $0$
• Dimension: $2$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 1323 = 3^{3} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1323.a (trivial)

Newform invariants

Self dual:	yes
Analytic conductor:	\(10.5642081874\)
Analytic rank:	\(0\)
Dimension:	\(2\)
Coefficient field:	\(\Q(\sqrt{6}) \)
Defining polynomial:	\( x^{2} - 6 \)
Coefficient ring:	\(\Z[a_1, a_2]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	no (minimal twist has level 189)
Fricke sign:	\(-1\)
Sato-Tate group:	$\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \sqrt{6}\). We also show the integral \(q\)-expansion of the trace form.

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

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

−2.44949
2.44949

−2.44949

0

4.00000

2.44949

0

0

−4.89898

0

−6.00000

1.2

2.44949

0

4.00000

−2.44949

0

0

4.89898

0

−6.00000

Atkin-Lehner signs

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

Inner twists

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

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	1323.2.a.u		2
3.b	odd	2	1	inner	1323.2.a.u		2
7.b	odd	2	1		1323.2.a.v		2
7.d	odd	6	2		189.2.e.d	✓	4
21.c	even	2	1		1323.2.a.v		2
21.g	even	6	2		189.2.e.d	✓	4
63.i	even	6	2		567.2.h.g		4
63.k	odd	6	2		567.2.g.g		4
63.s	even	6	2		567.2.g.g		4
63.t	odd	6	2		567.2.h.g		4

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
189.2.e.d	✓	4	7.d	odd	6	2
189.2.e.d	✓	4	21.g	even	6	2
567.2.g.g		4	63.k	odd	6	2
567.2.g.g		4	63.s	even	6	2
567.2.h.g		4	63.i	even	6	2
567.2.h.g		4	63.t	odd	6	2
1323.2.a.u		2	1.a	even	1	1	trivial
1323.2.a.u		2	3.b	odd	2	1	inner
1323.2.a.v		2	7.b	odd	2	1
1323.2.a.v		2	21.c	even	2	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(1323))\):

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

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

\( T_{13} - 4 \)

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	\( T^{2} - 6 \)
$3$	\( T^{2} \)
$5$	\( T^{2} - 6 \)
$7$	\( T^{2} \)
$11$	\( T^{2} - 24 \)