Newform orbit 5292.2.f.a

• Label: 5292.2.f.a
• Level: $5292$
• Weight: $2$
• Character orbit: 5292.f
• Analytic conductor: $42.257$
• Analytic rank: $0$
• Dimension: $2$
• CM discriminant: -3
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 5292 = 2^{2} \cdot 3^{3} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5292.f (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(42.2568327497\)
Analytic rank:	\(0\)
Dimension:	\(2\)
Coefficient field:	\(\Q(\sqrt{-3}) \)
Defining polynomial:	\( x^{2} - x + 1 \)
Coefficient ring:	\(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index:	\( 2 \)
Twist minimal:	no (minimal twist has level 756)
Sato-Tate group:	$\mathrm{U}(1)[D_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \sqrt{-3}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\)	\(=\)	\( q+O(q^{10}) \)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/5292\mathbb{Z}\right)^\times\).

\(n\)	\(785\)	\(1081\)	\(2647\)
\(\chi(n)\)	\(-1\)	\(-1\)	\(1\)

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  

\(\iota_m(\nu)\)

\( a_{2} \)

\( a_{3} \)

\( a_{4} \)

\( a_{5} \)

\( a_{6} \)

\( a_{7} \)

\( a_{8} \)

\( a_{9} \)

\( a_{10} \)

2645.1

0.500000	+	0.866025i
0.500000	−	0.866025i

0

0

0

0

0

0

0

0

0

2645.2

0

0

0

0

0

0

0

0

0

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
3.b	odd	2	1	CM by \(\Q(\sqrt{-3}) \)
7.b	odd	2	1	inner
21.c	even	2	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	5292.2.f.a		2
3.b	odd	2	1	CM	5292.2.f.a		2
7.b	odd	2	1	inner	5292.2.f.a		2
7.c	even	3	1		756.2.t.b	✓	2
7.d	odd	6	1		756.2.t.b	✓	2
21.c	even	2	1	inner	5292.2.f.a		2
21.g	even	6	1		756.2.t.b	✓	2
21.h	odd	6	1		756.2.t.b	✓	2
63.g	even	3	1		2268.2.bm.c		2
63.h	even	3	1		2268.2.w.b		2
63.i	even	6	1		2268.2.w.b		2
63.j	odd	6	1		2268.2.w.b		2
63.k	odd	6	1		2268.2.bm.c		2
63.n	odd	6	1		2268.2.bm.c		2
63.s	even	6	1		2268.2.bm.c		2
63.t	odd	6	1		2268.2.w.b		2

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
756.2.t.b	✓	2	7.c	even	3	1
756.2.t.b	✓	2	7.d	odd	6	1
756.2.t.b	✓	2	21.g	even	6	1
756.2.t.b	✓	2	21.h	odd	6	1
2268.2.w.b		2	63.h	even	3	1
2268.2.w.b		2	63.i	even	6	1
2268.2.w.b		2	63.j	odd	6	1
2268.2.w.b		2	63.t	odd	6	1
2268.2.bm.c		2	63.g	even	3	1
2268.2.bm.c		2	63.k	odd	6	1
2268.2.bm.c		2	63.n	odd	6	1
2268.2.bm.c		2	63.s	even	6	1
5292.2.f.a		2	1.a	even	1	1	trivial
5292.2.f.a		2	3.b	odd	2	1	CM
5292.2.f.a		2	7.b	odd	2	1	inner
5292.2.f.a		2	21.c	even	2	1	inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(5292, [\chi])\):

\( T_{5} \)

\( T_{13}^{2} + 27 \)

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	\( T^{2} \)
$3$	\( T^{2} \)
$5$	\( T^{2} \)
$7$	\( T^{2} \)
$11$	\( T^{2} \)