Newform orbit 2548.2.i.g

• Label: 2548.2.i.g
• Level: $2548$
• Weight: $2$
• Character orbit: 2548.i
• Analytic conductor: $20.346$
• Analytic rank: $0$
• Dimension: $2$
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 2548 = 2^{2} \cdot 7^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2548.i (of order \(3\), degree \(2\), not minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(20.3458824350\)
Analytic rank:	\(0\)
Dimension:	\(2\)
Coefficient field:	\(\Q(\sqrt{-3}) \)
Defining polynomial:	\( x^{2} - x + 1 \)
Coefficient ring:	\(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	no (minimal twist has level 52)
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.

\(f(q)\)	\(=\)	\( q + 2 \zeta_{6} q^{3} + 3 \zeta_{6} q^{5} + (\zeta_{6} - 1) q^{9}+O(q^{10}) \)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{3} + 3 q^{5} - q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2548\mathbb{Z}\right)^\times\).

\(n\)	\(197\)	\(885\)	\(1275\)
\(\chi(n)\)	\(-\zeta_{6}\)	\(-\zeta_{6}\)	\(1\)

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  

\(\iota_m(\nu)\)

\( a_{2} \)

\( a_{3} \)

\( a_{4} \)

\( a_{5} \)

\( a_{6} \)

\( a_{7} \)

\( a_{8} \)

\( a_{9} \)

\( a_{10} \)

165.1

0.500000	+	0.866025i
0.500000	−	0.866025i

0

1.00000

+

1.73205i

0

1.50000

+

2.59808i

0

0

0

−0.500000

+

0.866025i

0

1745.1

0

1.00000

−

1.73205i

0

1.50000

−

2.59808i

0

0

0

−0.500000

−

0.866025i

0

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
91.h	even	3	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	2548.2.i.g		2
7.b	odd	2	1		2548.2.i.b		2
7.c	even	3	1		52.2.e.b	✓	2
7.c	even	3	1		2548.2.l.b		2
7.d	odd	6	1		2548.2.k.a		2
7.d	odd	6	1		2548.2.l.g		2
13.c	even	3	1		2548.2.l.b		2
21.h	odd	6	1		468.2.l.d		2
28.g	odd	6	1		208.2.i.a		2
35.j	even	6	1		1300.2.i.b		2
35.l	odd	12	2		1300.2.bb.d		4
56.k	odd	6	1		832.2.i.i		2
56.p	even	6	1		832.2.i.c		2
84.n	even	6	1		1872.2.t.m		2
91.g	even	3	1		52.2.e.b	✓	2
91.h	even	3	1		676.2.a.a		1
91.h	even	3	1	inner	2548.2.i.g		2
91.k	even	6	1		676.2.a.b		1
91.m	odd	6	1		2548.2.k.a		2
91.n	odd	6	1		2548.2.l.g		2
91.r	even	6	1		676.2.e.d		2
91.u	even	6	1		676.2.e.d		2
91.v	odd	6	1		2548.2.i.b		2
91.x	odd	12	2		676.2.d.a		2
91.z	odd	12	2		676.2.h.d		4
91.bd	odd	12	2		676.2.h.d		4
273.s	odd	6	1		6084.2.a.o		1
273.bm	odd	6	1		468.2.l.d		2
273.bp	odd	6	1		6084.2.a.c		1
273.bv	even	12	2		6084.2.b.k		2
364.q	odd	6	1		208.2.i.a		2
364.bi	odd	6	1		2704.2.a.l		1
364.bk	odd	6	1		2704.2.a.m		1
364.ca	even	12	2		2704.2.f.i		2
455.bm	even	6	1		1300.2.i.b		2
455.cx	odd	12	2		1300.2.bb.d		4
728.bg	even	6	1		832.2.i.c		2
728.di	odd	6	1		832.2.i.i		2
1092.dc	even	6	1		1872.2.t.m		2

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
52.2.e.b	✓	2	7.c	even	3	1
52.2.e.b	✓	2	91.g	even	3	1
208.2.i.a		2	28.g	odd	6	1
208.2.i.a		2	364.q	odd	6	1
468.2.l.d		2	21.h	odd	6	1
468.2.l.d		2	273.bm	odd	6	1
676.2.a.a		1	91.h	even	3	1
676.2.a.b		1	91.k	even	6	1
676.2.d.a		2	91.x	odd	12	2
676.2.e.d		2	91.r	even	6	1
676.2.e.d		2	91.u	even	6	1
676.2.h.d		4	91.z	odd	12	2
676.2.h.d		4	91.bd	odd	12	2
832.2.i.c		2	56.p	even	6	1
832.2.i.c		2	728.bg	even	6	1
832.2.i.i		2	56.k	odd	6	1
832.2.i.i		2	728.di	odd	6	1
1300.2.i.b		2	35.j	even	6	1
1300.2.i.b		2	455.bm	even	6	1
1300.2.bb.d		4	35.l	odd	12	2
1300.2.bb.d		4	455.cx	odd	12	2
1872.2.t.m		2	84.n	even	6	1
1872.2.t.m		2	1092.dc	even	6	1
2548.2.i.b		2	7.b	odd	2	1
2548.2.i.b		2	91.v	odd	6	1
2548.2.i.g		2	1.a	even	1	1	trivial
2548.2.i.g		2	91.h	even	3	1	inner
2548.2.k.a		2	7.d	odd	6	1
2548.2.k.a		2	91.m	odd	6	1
2548.2.l.b		2	7.c	even	3	1
2548.2.l.b		2	13.c	even	3	1
2548.2.l.g		2	7.d	odd	6	1
2548.2.l.g		2	91.n	odd	6	1
2704.2.a.l		1	364.bi	odd	6	1
2704.2.a.m		1	364.bk	odd	6	1
2704.2.f.i		2	364.ca	even	12	2
6084.2.a.c		1	273.bp	odd	6	1
6084.2.a.o		1	273.s	odd	6	1
6084.2.b.k		2	273.bv	even	12	2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(2548, [\chi])\):

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

\( T_{5}^{2} - 3T_{5} + 9 \)

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	\( T^{2} \)
$3$	\( T^{2} - 2T + 4 \)
$5$	\( T^{2} - 3T + 9 \)
$7$	\( T^{2} \)
$11$	\( T^{2} \)