Newform orbit 1053.2.e.a

• Label: 1053.2.e.a
• Level: $1053$
• Weight: $2$
• Character orbit: 1053.e
• Analytic conductor: $8.408$
• Analytic rank: $1$
• Dimension: $2$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 1053 = 3^{4} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1053.e (of order \(3\), degree \(2\), not minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(8.40824733284\)
Analytic rank:	\(1\)
Dimension:	\(2\)
Coefficient field:	\(\Q(\sqrt{-3}) \)
Defining polynomial:	\( x^{2} - x + 1 \)
Coefficient ring:	\(\Z[a_1, a_2]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{6}\). We also show the integral \(q\)-expansion of the trace form.

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1053\mathbb{Z}\right)^\times\).

\(n\)	\(326\)	\(730\)
\(\chi(n)\)	\(-\zeta_{6}\)	\(1\)

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  

\(\iota_m(\nu)\)

\( a_{2} \)

\( a_{3} \)

\( a_{4} \)

\( a_{5} \)

\( a_{6} \)

\( a_{7} \)

\( a_{8} \)

\( a_{9} \)

\( a_{10} \)

352.1

0.500000	−	0.866025i
0.500000	+	0.866025i

−0.500000

−

0.866025i

0

0.500000

−

0.866025i

−1.00000

+

1.73205i

0

−2.50000

−

4.33013i

−3.00000

0

2.00000

703.1

−0.500000

+

0.866025i

0

0.500000

+

0.866025i

−1.00000

−

1.73205i

0

−2.50000

+

4.33013i

−3.00000

0

2.00000

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
9.c	even	3	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	1053.2.e.a		2
3.b	odd	2	1		1053.2.e.c		2
9.c	even	3	1		1053.2.a.c	yes	1
9.c	even	3	1	inner	1053.2.e.a		2
9.d	odd	6	1		1053.2.a.b	✓	1
9.d	odd	6	1		1053.2.e.c		2

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
1053.2.a.b	✓	1	9.d	odd	6	1
1053.2.a.c	yes	1	9.c	even	3	1
1053.2.e.a		2	1.a	even	1	1	trivial
1053.2.e.a		2	9.c	even	3	1	inner
1053.2.e.c		2	3.b	odd	2	1
1053.2.e.c		2	9.d	odd	6	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1053, [\chi])\):

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

\( T_{5}^{2} + 2T_{5} + 4 \)

\( T_{7}^{2} + 5T_{7} + 25 \)

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	\( T^{2} + T + 1 \)
$3$	\( T^{2} \)
$5$	\( T^{2} + 2T + 4 \)
$7$	\( T^{2} + 5T + 25 \)
$11$	\( T^{2} + T + 1 \)