Newform orbit 7744.2.a.cn

• Label: 7744.2.a.cn
• Level: $7744$
• Weight: $2$
• Character orbit: 7744.a
• Self dual: yes
• Analytic conductor: $61.836$
• Analytic rank: $0$
• Dimension: $2$
• CM discriminant: -11
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 7744 = 2^{6} \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	7744.a (trivial)

Newform invariants

Self dual:	yes
Analytic conductor:	\(61.8361513253\)
Analytic rank:	\(0\)
Dimension:	\(2\)
Coefficient field:	\(\Q(\sqrt{33}) \)
Defining polynomial:	\( x^{2} - x - 8 \)
Coefficient ring:	\(\Z[a_1, a_2, a_3]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	no (minimal twist has level 484)
Fricke sign:	\(-1\)
Sato-Tate group:	$N(\mathrm{U}(1))$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \frac{1}{2}(1 + \sqrt{33})\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\)	\(=\)	\( q + \beta q^{3} + (\beta - 2) q^{5} + (\beta + 5) q^{9}+O(q^{10}) \)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + q^{3} - 3 q^{5} + 11 q^{9}+O(q^{10}) \)

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

−2.37228
3.37228

0

−2.37228

0

−4.37228

0

0

0

2.62772

0

1.2

0

3.37228

0

1.37228

0

0

0

8.37228

0

Atkin-Lehner signs

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

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
11.b	odd	2	1	CM by \(\Q(\sqrt{-11}) \)

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	7744.2.a.cn		2
4.b	odd	2	1		7744.2.a.bx		2
8.b	even	2	1		1936.2.a.s		2
8.d	odd	2	1		484.2.a.d	✓	2
11.b	odd	2	1	CM	7744.2.a.cn		2
24.f	even	2	1		4356.2.a.n		2
44.c	even	2	1		7744.2.a.bx		2
88.b	odd	2	1		1936.2.a.s		2
88.g	even	2	1		484.2.a.d	✓	2
88.k	even	10	4		484.2.e.g		8
88.l	odd	10	4		484.2.e.g		8
264.p	odd	2	1		4356.2.a.n		2

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
484.2.a.d	✓	2	8.d	odd	2	1
484.2.a.d	✓	2	88.g	even	2	1
484.2.e.g		8	88.k	even	10	4
484.2.e.g		8	88.l	odd	10	4
1936.2.a.s		2	8.b	even	2	1
1936.2.a.s		2	88.b	odd	2	1
4356.2.a.n		2	24.f	even	2	1
4356.2.a.n		2	264.p	odd	2	1
7744.2.a.bx		2	4.b	odd	2	1
7744.2.a.bx		2	44.c	even	2	1
7744.2.a.cn		2	1.a	even	1	1	trivial
7744.2.a.cn		2	11.b	odd	2	1	CM

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(7744))\):

\( T_{3}^{2} - T_{3} - 8 \)

\( T_{5}^{2} + 3T_{5} - 6 \)

\( T_{7} \)

\( T_{13} \)

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	\( T^{2} \)
$3$	\( T^{2} - T - 8 \)
$5$	\( T^{2} + 3T - 6 \)
$7$	\( T^{2} \)
$11$	\( T^{2} \)