Newform orbit 8281.2.a.n

• Label: 8281.2.a.n
• Level: $8281$
• Weight: $2$
• Character orbit: 8281.a
• Self dual: yes
• Analytic conductor: $66.124$
• Analytic rank: $0$
• Dimension: $2$
• CM: no
• Inner twists: $1$

Newspace parameters

Level:	\( N \)	\(=\)	\( 8281 = 7^{2} \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8281.a (trivial)

Newform invariants

Self dual:	yes
Analytic conductor:	\(66.1241179138\)
Analytic rank:	\(0\)
Dimension:	\(2\)
Coefficient field:	\(\Q(\sqrt{5}) \)
Defining polynomial:	\( x^{2} - x - 1 \)
Coefficient ring:	\(\Z[a_1, a_2]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	no (minimal twist has level 91)
Fricke sign:	\(-1\)
Sato-Tate group:	$\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \frac{1}{2}(1 + \sqrt{5})\). We also show the integral \(q\)-expansion of the trace form.

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

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

1.61803
−0.618034

−2.61803

2.61803

4.85410

2.61803

−6.85410

0

−7.47214

3.85410

−6.85410

1.2

−0.381966

0.381966

−1.85410

0.381966

−0.145898

0

1.47214

−2.85410

−0.145898

Atkin-Lehner signs

\( p \)	Sign
\(7\)	\(-1\)
\(13\)	\(1\)

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	8281.2.a.n		2
7.b	odd	2	1		1183.2.a.c		2
13.b	even	2	1		8281.2.a.bb		2
13.e	even	6	2		637.2.f.c		4
91.b	odd	2	1		1183.2.a.g		2
91.i	even	4	2		1183.2.c.c		4
91.k	even	6	2		637.2.h.f		4
91.l	odd	6	2		637.2.h.g		4
91.p	odd	6	2		637.2.g.b		4
91.t	odd	6	2		91.2.f.a	✓	4
91.u	even	6	2		637.2.g.c		4
273.u	even	6	2		819.2.o.c		4
364.bc	even	6	2		1456.2.s.h		4

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
91.2.f.a	✓	4	91.t	odd	6	2
637.2.f.c		4	13.e	even	6	2
637.2.g.b		4	91.p	odd	6	2
637.2.g.c		4	91.u	even	6	2
637.2.h.f		4	91.k	even	6	2
637.2.h.g		4	91.l	odd	6	2
819.2.o.c		4	273.u	even	6	2
1183.2.a.c		2	7.b	odd	2	1
1183.2.a.g		2	91.b	odd	2	1
1183.2.c.c		4	91.i	even	4	2
1456.2.s.h		4	364.bc	even	6	2
8281.2.a.n		2	1.a	even	1	1	trivial
8281.2.a.bb		2	13.b	even	2	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(8281))\):

\( T_{2}^{2} + 3T_{2} + 1 \)

\( T_{3}^{2} - 3T_{3} + 1 \)

\( T_{5}^{2} - 3T_{5} + 1 \)

\( T_{11}^{2} - 3T_{11} - 9 \)

\( T_{17}^{2} + 6T_{17} - 11 \)

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	\( T^{2} + 3T + 1 \)
$3$	\( T^{2} - 3T + 1 \)
$5$	\( T^{2} - 3T + 1 \)
$7$	\( T^{2} \)
$11$	\( T^{2} - 3T - 9 \)