Newform orbit 1053.2.e.d

• Label: 1053.2.e.d
• Level: $1053$
• Weight: $2$
• Character orbit: 1053.e
• Analytic conductor: $8.408$
• Analytic rank: $0$
• 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:	\(0\)
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:	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 primitive root of unity \(\zeta_{6}\). We also show the integral \(q\)-expansion of the trace form.

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

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

2.00000

+

3.46410i

3.00000

0

2.00000

703.1

0.500000

−

0.866025i

0

0.500000

+

0.866025i

1.00000

+

1.73205i

0

2.00000

−

3.46410i

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.d		2
3.b	odd	2	1		1053.2.e.b		2
9.c	even	3	1		117.2.a.a		1
9.c	even	3	1	inner	1053.2.e.d		2
9.d	odd	6	1		39.2.a.a	✓	1
9.d	odd	6	1		1053.2.e.b		2
36.f	odd	6	1		1872.2.a.h		1
36.h	even	6	1		624.2.a.i		1
45.h	odd	6	1		975.2.a.f		1
45.j	even	6	1		2925.2.a.p		1
45.k	odd	12	2		2925.2.c.e		2
45.l	even	12	2		975.2.c.f		2
63.l	odd	6	1		5733.2.a.e		1
63.o	even	6	1		1911.2.a.f		1
72.j	odd	6	1		2496.2.a.q		1
72.l	even	6	1		2496.2.a.e		1
72.n	even	6	1		7488.2.a.bl		1
72.p	odd	6	1		7488.2.a.by		1
99.g	even	6	1		4719.2.a.c		1
117.k	odd	6	1		507.2.e.a		2
117.m	odd	6	1		507.2.e.b		2
117.n	odd	6	1		507.2.a.a		1
117.t	even	6	1		1521.2.a.e		1
117.u	odd	6	1		507.2.e.a		2
117.v	odd	6	1		507.2.e.b		2
117.x	even	12	2		507.2.j.e		4
117.y	odd	12	2		1521.2.b.b		2
117.z	even	12	2		507.2.b.a		2
117.bc	even	12	2		507.2.j.e		4
468.x	even	6	1		8112.2.a.s		1

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
39.2.a.a	✓	1	9.d	odd	6	1
117.2.a.a		1	9.c	even	3	1
507.2.a.a		1	117.n	odd	6	1
507.2.b.a		2	117.z	even	12	2
507.2.e.a		2	117.k	odd	6	1
507.2.e.a		2	117.u	odd	6	1
507.2.e.b		2	117.m	odd	6	1
507.2.e.b		2	117.v	odd	6	1
507.2.j.e		4	117.x	even	12	2
507.2.j.e		4	117.bc	even	12	2
624.2.a.i		1	36.h	even	6	1
975.2.a.f		1	45.h	odd	6	1
975.2.c.f		2	45.l	even	12	2
1053.2.e.b		2	3.b	odd	2	1
1053.2.e.b		2	9.d	odd	6	1
1053.2.e.d		2	1.a	even	1	1	trivial
1053.2.e.d		2	9.c	even	3	1	inner
1521.2.a.e		1	117.t	even	6	1
1521.2.b.b		2	117.y	odd	12	2
1872.2.a.h		1	36.f	odd	6	1
1911.2.a.f		1	63.o	even	6	1
2496.2.a.e		1	72.l	even	6	1
2496.2.a.q		1	72.j	odd	6	1
2925.2.a.p		1	45.j	even	6	1
2925.2.c.e		2	45.k	odd	12	2
4719.2.a.c		1	99.g	even	6	1
5733.2.a.e		1	63.l	odd	6	1
7488.2.a.bl		1	72.n	even	6	1
7488.2.a.by		1	72.p	odd	6	1
8112.2.a.s		1	468.x	even	6	1

Hecke kernels

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

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

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

\( T_{7}^{2} - 4T_{7} + 16 \)

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	\( T^{2} - T + 1 \)
$3$	\( T^{2} \)
$5$	\( T^{2} - 2T + 4 \)
$7$	\( T^{2} - 4T + 16 \)
$11$	\( T^{2} - 4T + 16 \)