Newform orbit 1053.2.e.m

• Label: 1053.2.e.m
• Level: $1053$
• Weight: $2$
• Character orbit: 1053.e
• Analytic conductor: $8.408$
• Analytic rank: $0$
• Dimension: $4$
• 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:	\(0\)
Dimension:	\(4\)
Relative dimension:	\(2\) over \(\Q(\zeta_{3})\)
Coefficient field:	\(\Q(\sqrt{2}, \sqrt{-3})\)
Defining polynomial:	\( x^{4} + 2x^{2} + 4 \)
Coefficient ring:	\(\Z[a_1, \ldots, a_{4}]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	no (minimal twist has level 39)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\)	\(=\)	\( q + (\beta_{2} + \beta_1 + 1) q^{2} + (2 \beta_{3} + \beta_{2} + 2 \beta_1) q^{4} + (2 \beta_{3} + 2 \beta_1) q^{5} + 2 \beta_1 q^{7} + (\beta_{3} - 3) q^{8}+O(q^{10}) \)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 2 q^{2} - 2 q^{4} - 12 q^{8}+O(q^{10}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} + 2x^{2} + 4 \) :

\(\beta_{1}\)	\(=\)	\( \nu \)
\(\beta_{2}\)	\(=\)	\( ( \nu^{2} ) / 2 \)
\(\beta_{3}\)	\(=\)	\( ( \nu^{3} ) / 2 \)

Character values

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

\(n\)	\(326\)	\(730\)
\(\chi(n)\)	\(\beta_{2}\)	\(1\)

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

−0.707107	−	1.22474i
0.707107	+	1.22474i
−0.707107	+	1.22474i
0.707107	−	1.22474i

−0.207107

−

0.358719i

0

0.914214

−

1.58346i

1.41421

−

2.44949i

0

−1.41421

−

2.44949i

−1.58579

0

−1.17157

352.2

1.20711

+

2.09077i

0

−1.91421

+

3.31552i

−1.41421

+

2.44949i

0

1.41421

+

2.44949i

−4.41421

0

−6.82843

703.1

−0.207107

+

0.358719i

0

0.914214

+

1.58346i

1.41421

+

2.44949i

0

−1.41421

+

2.44949i

−1.58579

0

−1.17157

703.2

1.20711

−

2.09077i

0

−1.91421

−

3.31552i

−1.41421

−

2.44949i

0

1.41421

−

2.44949i

−4.41421

0

−6.82843

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.m		4
3.b	odd	2	1		1053.2.e.e		4
9.c	even	3	1		39.2.a.b	✓	2
9.c	even	3	1	inner	1053.2.e.m		4
9.d	odd	6	1		117.2.a.c		2
9.d	odd	6	1		1053.2.e.e		4
36.f	odd	6	1		624.2.a.k		2
36.h	even	6	1		1872.2.a.w		2
45.h	odd	6	1		2925.2.a.v		2
45.j	even	6	1		975.2.a.l		2
45.k	odd	12	2		975.2.c.h		4
45.l	even	12	2		2925.2.c.u		4
63.l	odd	6	1		1911.2.a.h		2
63.o	even	6	1		5733.2.a.u		2
72.j	odd	6	1		7488.2.a.cl		2
72.l	even	6	1		7488.2.a.co		2
72.n	even	6	1		2496.2.a.bf		2
72.p	odd	6	1		2496.2.a.bi		2
99.h	odd	6	1		4719.2.a.p		2
117.f	even	3	1		507.2.e.h		4
117.h	even	3	1		507.2.e.h		4
117.l	even	6	1		507.2.e.d		4
117.n	odd	6	1		1521.2.a.f		2
117.r	even	6	1		507.2.e.d		4
117.t	even	6	1		507.2.a.h		2
117.w	odd	12	2		507.2.j.f		8
117.y	odd	12	2		507.2.b.e		4
117.z	even	12	2		1521.2.b.j		4
117.bb	odd	12	2		507.2.j.f		8
468.bg	odd	6	1		8112.2.a.bm		2

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
39.2.a.b	✓	2	9.c	even	3	1
117.2.a.c		2	9.d	odd	6	1
507.2.a.h		2	117.t	even	6	1
507.2.b.e		4	117.y	odd	12	2
507.2.e.d		4	117.l	even	6	1
507.2.e.d		4	117.r	even	6	1
507.2.e.h		4	117.f	even	3	1
507.2.e.h		4	117.h	even	3	1
507.2.j.f		8	117.w	odd	12	2
507.2.j.f		8	117.bb	odd	12	2
624.2.a.k		2	36.f	odd	6	1
975.2.a.l		2	45.j	even	6	1
975.2.c.h		4	45.k	odd	12	2
1053.2.e.e		4	3.b	odd	2	1
1053.2.e.e		4	9.d	odd	6	1
1053.2.e.m		4	1.a	even	1	1	trivial
1053.2.e.m		4	9.c	even	3	1	inner
1521.2.a.f		2	117.n	odd	6	1
1521.2.b.j		4	117.z	even	12	2
1872.2.a.w		2	36.h	even	6	1
1911.2.a.h		2	63.l	odd	6	1
2496.2.a.bf		2	72.n	even	6	1
2496.2.a.bi		2	72.p	odd	6	1
2925.2.a.v		2	45.h	odd	6	1
2925.2.c.u		4	45.l	even	12	2
4719.2.a.p		2	99.h	odd	6	1
5733.2.a.u		2	63.o	even	6	1
7488.2.a.cl		2	72.j	odd	6	1
7488.2.a.co		2	72.l	even	6	1
8112.2.a.bm		2	468.bg	odd	6	1

Hecke kernels

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

\( T_{2}^{4} - 2T_{2}^{3} + 5T_{2}^{2} + 2T_{2} + 1 \)

\( T_{5}^{4} + 8T_{5}^{2} + 64 \)

\( T_{7}^{4} + 8T_{7}^{2} + 64 \)

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	T^4 - 2*T^3 + 5*T^2 + 2*T + 1
$3$	\( T^{4} \)
$5$	\( T^{4} + 8T^{2} + 64 \)
$7$	\( T^{4} + 8T^{2} + 64 \)
$11$	\( (T^{2} - 2 T + 4)^{2} \)