Newform orbit 810.2.a.k

• Label: 810.2.a.k
• Level: $810$
• Weight: $2$
• Character orbit: 810.a
• Self dual: yes
• Analytic conductor: $6.468$
• Analytic rank: $0$
• Dimension: $2$
• CM: no
• Inner twists: $1$

Newspace parameters

Level:	\( N \)	\(=\)	\( 810 = 2 \cdot 3^{4} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	810.a (trivial)

Newform invariants

Self dual:	yes
Analytic conductor:	\(6.46788256372\)
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.

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

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

1.00000

0

1.00000

1.00000

0

−2.37228

1.00000

0

1.00000

1.2

1.00000

0

1.00000

1.00000

0

3.37228

1.00000

0

1.00000

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	810.2.a.k		2
3.b	odd	2	1		810.2.a.i		2
4.b	odd	2	1		6480.2.a.bn		2
5.b	even	2	1		4050.2.a.bo		2
5.c	odd	4	2		4050.2.c.ba		4
9.c	even	3	2		270.2.e.c		4
9.d	odd	6	2		90.2.e.c	✓	4
12.b	even	2	1		6480.2.a.be		2
15.d	odd	2	1		4050.2.a.bw		2
15.e	even	4	2		4050.2.c.v		4
36.f	odd	6	2		2160.2.q.f		4
36.h	even	6	2		720.2.q.f		4
45.h	odd	6	2		450.2.e.j		4
45.j	even	6	2		1350.2.e.l		4
45.k	odd	12	4		1350.2.j.f		8
45.l	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	9.d	odd	6	2
270.2.e.c		4	9.c	even	3	2
450.2.e.j		4	45.h	odd	6	2
450.2.j.g		8	45.l	even	12	4
720.2.q.f		4	36.h	even	6	2
810.2.a.i		2	3.b	odd	2	1
810.2.a.k		2	1.a	even	1	1	trivial
1350.2.e.l		4	45.j	even	6	2
1350.2.j.f		8	45.k	odd	12	4
2160.2.q.f		4	36.f	odd	6	2
4050.2.a.bo		2	5.b	even	2	1
4050.2.a.bw		2	15.d	odd	2	1
4050.2.c.v		4	15.e	even	4	2
4050.2.c.ba		4	5.c	odd	4	2
6480.2.a.be		2	12.b	even	2	1
6480.2.a.bn		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 \(S_{2}^{\mathrm{new}}(\Gamma_0(810))\):

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

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

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

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

Hecke characteristic polynomials

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