Newform orbit 6480.2.a.be

• Label: 6480.2.a.be
• Level: $6480$
• Weight: $2$
• Character orbit: 6480.a
• Self dual: yes
• Analytic conductor: $51.743$
• Analytic rank: $0$
• Dimension: $2$
• CM: no
• Inner twists: $1$

Newspace parameters

Level:	\( N \)	\(=\)	\( 6480 = 2^{4} \cdot 3^{4} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6480.a (trivial)

Newform invariants

Self dual:	yes
Analytic conductor:	\(51.7430605098\)
Analytic rank:	\(0\)
Dimension:	\(2\)
Coefficient field:	\(\Q(\sqrt{33}) \)
Defining polynomial:	\( x^{2} - x - 8 \)
Coefficient ring:	\(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	no (minimal twist has level 90)
Fricke sign:	\(-1\)
Sato-Tate group:	$\mathrm{SU}(2)$

$q$-expansion

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

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

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

3.37228
−2.37228

0

0

0

−1.00000

0

−3.37228

0

0

0

1.2

0

0

0

−1.00000

0

2.37228

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	6480.2.a.be		2
3.b	odd	2	1		6480.2.a.bn		2
4.b	odd	2	1		810.2.a.i		2
9.c	even	3	2		720.2.q.f		4
9.d	odd	6	2		2160.2.q.f		4
12.b	even	2	1		810.2.a.k		2
20.d	odd	2	1		4050.2.a.bw		2
20.e	even	4	2		4050.2.c.v		4
36.f	odd	6	2		90.2.e.c	✓	4
36.h	even	6	2		270.2.e.c		4
60.h	even	2	1		4050.2.a.bo		2
60.l	odd	4	2		4050.2.c.ba		4
180.n	even	6	2		1350.2.e.l		4
180.p	odd	6	2		450.2.e.j		4
180.v	odd	12	4		1350.2.j.f		8
180.x	even	12	4		450.2.j.g		8

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
90.2.e.c	✓	4	36.f	odd	6	2
270.2.e.c		4	36.h	even	6	2
450.2.e.j		4	180.p	odd	6	2
450.2.j.g		8	180.x	even	12	4
720.2.q.f		4	9.c	even	3	2
810.2.a.i		2	4.b	odd	2	1
810.2.a.k		2	12.b	even	2	1
1350.2.e.l		4	180.n	even	6	2
1350.2.j.f		8	180.v	odd	12	4
2160.2.q.f		4	9.d	odd	6	2
4050.2.a.bo		2	60.h	even	2	1
4050.2.a.bw		2	20.d	odd	2	1
4050.2.c.v		4	20.e	even	4	2
4050.2.c.ba		4	60.l	odd	4	2
6480.2.a.be		2	1.a	even	1	1	trivial
6480.2.a.bn		2	3.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 \(S_{2}^{\mathrm{new}}(\Gamma_0(6480))\):

\( T_{7}^{2} + T_{7} - 8 \)

\( T_{11}^{2} - 3T_{11} - 6 \)

\( T_{13}^{2} + 2T_{13} - 32 \)

\( T_{17}^{2} + 9T_{17} + 12 \)

\( T_{19}^{2} + T_{19} - 8 \)

\( T_{23}^{2} - 3T_{23} - 6 \)

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	\( T^{2} \)
$3$	\( T^{2} \)
$5$	\( (T + 1)^{2} \)
$7$	\( T^{2} + T - 8 \)
$11$	\( T^{2} - 3T - 6 \)