Newform orbit 8640.2.a.cy

• Label: 8640.2.a.cy
• Level: $8640$
• Weight: $2$
• Character orbit: 8640.a
• Self dual: yes
• Analytic conductor: $68.991$
• Analytic rank: $0$
• Dimension: $2$
• CM: no
• Inner twists: $1$

Newspace parameters

Level:	\( N \)	\(=\)	\( 8640 = 2^{6} \cdot 3^{3} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8640.a (trivial)

Newform invariants

Self dual:	yes
Analytic conductor:	\(68.9907473464\)
Analytic rank:	\(0\)
Dimension:	\(2\)
Coefficient field:	\(\Q(\sqrt{13}) \)
Defining polynomial:	\( x^{2} - x - 3 \)
Coefficient ring:	\(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index:	\( 2 \)
Twist minimal:	no (minimal twist has level 135)
Fricke sign:	\(-1\)
Sato-Tate group:	$\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \sqrt{13}\). We also show the integral \(q\)-expansion of the trace form.

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

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

2.30278
−1.30278

0

0

0

1.00000

0

−4.60555

0

0

0

1.2

0

0

0

1.00000

0

2.60555

0

0

0

Atkin-Lehner signs

\( p \)	Sign
\(2\)	\(-1\)
\(3\)	\(-1\)
\(5\)	\(-1\)

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	8640.2.a.cy		2
3.b	odd	2	1		8640.2.a.ck		2
4.b	odd	2	1		8640.2.a.df		2
8.b	even	2	1		2160.2.a.y		2
8.d	odd	2	1		135.2.a.d	yes	2
12.b	even	2	1		8640.2.a.cr		2
24.f	even	2	1		135.2.a.c	✓	2
24.h	odd	2	1		2160.2.a.ba		2
40.e	odd	2	1		675.2.a.k		2
40.k	even	4	2		675.2.b.h		4
56.e	even	2	1		6615.2.a.v		2
72.l	even	6	2		405.2.e.k		4
72.p	odd	6	2		405.2.e.j		4
120.m	even	2	1		675.2.a.p		2
120.q	odd	4	2		675.2.b.i		4
168.e	odd	2	1		6615.2.a.p		2

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
135.2.a.c	✓	2	24.f	even	2	1
135.2.a.d	yes	2	8.d	odd	2	1
405.2.e.j		4	72.p	odd	6	2
405.2.e.k		4	72.l	even	6	2
675.2.a.k		2	40.e	odd	2	1
675.2.a.p		2	120.m	even	2	1
675.2.b.h		4	40.k	even	4	2
675.2.b.i		4	120.q	odd	4	2
2160.2.a.y		2	8.b	even	2	1
2160.2.a.ba		2	24.h	odd	2	1
6615.2.a.p		2	168.e	odd	2	1
6615.2.a.v		2	56.e	even	2	1
8640.2.a.ck		2	3.b	odd	2	1
8640.2.a.cr		2	12.b	even	2	1
8640.2.a.cy		2	1.a	even	1	1	trivial
8640.2.a.df		2	4.b	odd	2	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(8640))\):

\( T_{7}^{2} + 2T_{7} - 12 \)

\( T_{11}^{2} + 2T_{11} - 12 \)

\( T_{13}^{2} + 6T_{13} - 4 \)

\( T_{17}^{2} - 4T_{17} - 9 \)

\( T_{19}^{2} - 13 \)

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	\( T^{2} \)
$3$	\( T^{2} \)
$5$	\( (T - 1)^{2} \)
$7$	\( T^{2} + 2T - 12 \)
$11$	\( T^{2} + 2T - 12 \)