Newform orbit 2772.2.a.n

• Label: 2772.2.a.n
• Level: $2772$
• Weight: $2$
• Character orbit: 2772.a
• Self dual: yes
• Analytic conductor: $22.135$
• Analytic rank: $0$
• Dimension: $2$
• CM: no
• Inner twists: $1$

Newspace parameters

Level:	\( N \)	\(=\)	\( 2772 = 2^{2} \cdot 3^{2} \cdot 7 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2772.a (trivial)

Newform invariants

Self dual:	yes
Analytic conductor:	\(22.1345314403\)
Analytic rank:	\(0\)
Dimension:	\(2\)
Coefficient field:	\(\Q(\sqrt{6}) \)
Defining polynomial:	\( x^{2} - 6 \)
Coefficient ring:	\(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	no (minimal twist has level 308)
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.

\(f(q)\)	\(=\)	\( q - 2 q^{5} - q^{7}+O(q^{10}) \)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 4 q^{5} - 2 q^{7}+O(q^{10}) \)

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

0

0

0

−2.00000

0

−1.00000

0

0

0

1.2

0

0

0

−2.00000

0

−1.00000

0

0

0

Atkin-Lehner signs

\( p \)	Sign
\(2\)	\(-1\)
\(3\)	\(-1\)
\(7\)	\(1\)
\(11\)	\(-1\)

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	2772.2.a.n		2
3.b	odd	2	1		308.2.a.b	✓	2
12.b	even	2	1		1232.2.a.n		2
15.d	odd	2	1		7700.2.a.s		2
15.e	even	4	2		7700.2.e.j		4
21.c	even	2	1		2156.2.a.c		2
21.g	even	6	2		2156.2.i.i		4
21.h	odd	6	2		2156.2.i.e		4
24.f	even	2	1		4928.2.a.bq		2
24.h	odd	2	1		4928.2.a.bp		2
33.d	even	2	1		3388.2.a.h		2
84.h	odd	2	1		8624.2.a.bj		2

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
308.2.a.b	✓	2	3.b	odd	2	1
1232.2.a.n		2	12.b	even	2	1
2156.2.a.c		2	21.c	even	2	1
2156.2.i.e		4	21.h	odd	6	2
2156.2.i.i		4	21.g	even	6	2
2772.2.a.n		2	1.a	even	1	1	trivial
3388.2.a.h		2	33.d	even	2	1
4928.2.a.bp		2	24.h	odd	2	1
4928.2.a.bq		2	24.f	even	2	1
7700.2.a.s		2	15.d	odd	2	1
7700.2.e.j		4	15.e	even	4	2
8624.2.a.bj		2	84.h	odd	2	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(2772))\):

\( T_{5} + 2 \)

\( T_{13}^{2} - 4T_{13} - 2 \)

Hecke characteristic polynomials

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