Newform orbit 2548.1.bi.a

• Label: 2548.1.bi.a
• Level: $2548$
• Weight: $1$
• Character orbit: 2548.bi
• Analytic conductor: $1.272$
• Analytic rank: $0$
• Dimension: $2$
• Projective image: $D_{3}$
• CM discriminant: -4
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 2548 = 2^{2} \cdot 7^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	2548.bi (of order \(6\), degree \(2\), not minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(1.27161765219\)
Analytic rank:	\(0\)
Dimension:	\(2\)
Coefficient field:	\(\Q(\zeta_{6})\)
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)
Projective image:	\(D_{3}\)
Projective field:	Galois closure of 3.1.676.1
Artin image:	$C_6\times S_3$
Artin field:	Galois closure of \(\mathbb{Q}[x]/(x^{12} - \cdots)\)

$q$-expansion

The \(q\)-expansion and trace form are shown below.

\(f(q)\)	\(=\)	\( q + q^{2} + q^{4} + \zeta_{6}^{2} q^{5} + q^{8} - \zeta_{6} q^{9} +O(q^{10}) \)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{2} + 2 q^{4} - q^{5} + 2 q^{8} - q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(197\)	\(885\)	\(1275\)
\(\chi(n)\)	\(\zeta_{6}^{2}\)	\(\zeta_{6}^{2}\)	\(-1\)

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

\(\iota_m(\nu)\)

\( a_{2} \)

\( a_{3} \)

\( a_{4} \)

\( a_{5} \)

\( a_{6} \)

\( a_{7} \)

\( a_{8} \)

\( a_{9} \)

\( a_{10} \)

471.1

0.500000	+	0.866025i
0.500000	−	0.866025i

1.00000

0

1.00000

−0.500000

+

0.866025i

0

0

1.00000

−0.500000

−

0.866025i

−0.500000

+

0.866025i

1439.1

1.00000

0

1.00000

−0.500000

−

0.866025i

0

0

1.00000

−0.500000

+

0.866025i

−0.500000

−

0.866025i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
4.b	odd	2	1	CM by \(\Q(\sqrt{-1}) \)
91.h	even	3	1	inner
364.bi	odd	6	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	2548.1.bi.a		2
4.b	odd	2	1	CM	2548.1.bi.a		2
7.b	odd	2	1		2548.1.bi.b		2
7.c	even	3	1		2548.1.q.a		2
7.c	even	3	1		2548.1.bn.a		2
7.d	odd	6	1		52.1.j.a	✓	2
7.d	odd	6	1		2548.1.q.b		2
13.c	even	3	1		2548.1.q.a		2
21.g	even	6	1		468.1.br.a		2
28.d	even	2	1		2548.1.bi.b		2
28.f	even	6	1		52.1.j.a	✓	2
28.f	even	6	1		2548.1.q.b		2
28.g	odd	6	1		2548.1.q.a		2
28.g	odd	6	1		2548.1.bn.a		2
35.i	odd	6	1		1300.1.bc.a		2
35.k	even	12	2		1300.1.w.a		4
52.j	odd	6	1		2548.1.q.a		2
56.j	odd	6	1		832.1.bb.a		2
56.m	even	6	1		832.1.bb.a		2
84.j	odd	6	1		468.1.br.a		2
91.g	even	3	1		2548.1.bn.a		2
91.h	even	3	1	inner	2548.1.bi.a		2
91.l	odd	6	1		676.1.c.a		1
91.m	odd	6	1		52.1.j.a	✓	2
91.n	odd	6	1		2548.1.q.b		2
91.p	odd	6	1		676.1.j.a		2
91.s	odd	6	1		676.1.j.a		2
91.v	odd	6	1		676.1.c.b		1
91.v	odd	6	1		2548.1.bi.b		2
91.w	even	12	2		676.1.i.a		4
91.ba	even	12	2		676.1.b.a		2
91.bb	even	12	2		676.1.i.a		4
112.v	even	12	2		3328.1.v.b		4
112.x	odd	12	2		3328.1.v.b		4
140.s	even	6	1		1300.1.bc.a		2
140.x	odd	12	2		1300.1.w.a		4
273.bf	even	6	1		468.1.br.a		2
364.q	odd	6	1		2548.1.bn.a		2
364.v	even	6	1		2548.1.q.b		2
364.w	even	6	1		676.1.c.a		1
364.x	even	6	1		676.1.j.a		2
364.ba	even	6	1		676.1.c.b		1
364.ba	even	6	1		2548.1.bi.b		2
364.bi	odd	6	1	inner	2548.1.bi.a		2
364.bp	even	6	1		676.1.j.a		2
364.br	even	6	1		52.1.j.a	✓	2
364.bw	odd	12	2		676.1.i.a		4
364.bz	odd	12	2		676.1.b.a		2
364.cg	odd	12	2		676.1.i.a		4
455.bw	odd	6	1		1300.1.bc.a		2
455.db	even	12	2		1300.1.w.a		4
728.bf	even	6	1		832.1.bb.a		2
728.de	odd	6	1		832.1.bb.a		2
1092.cc	odd	6	1		468.1.br.a		2
1456.fj	odd	12	2		3328.1.v.b		4
1456.fm	even	12	2		3328.1.v.b		4
1820.bt	even	6	1		1300.1.bc.a		2
1820.er	odd	12	2		1300.1.w.a		4

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
52.1.j.a	✓	2	7.d	odd	6	1
52.1.j.a	✓	2	28.f	even	6	1
52.1.j.a	✓	2	91.m	odd	6	1
52.1.j.a	✓	2	364.br	even	6	1
468.1.br.a		2	21.g	even	6	1
468.1.br.a		2	84.j	odd	6	1
468.1.br.a		2	273.bf	even	6	1
468.1.br.a		2	1092.cc	odd	6	1
676.1.b.a		2	91.ba	even	12	2
676.1.b.a		2	364.bz	odd	12	2
676.1.c.a		1	91.l	odd	6	1
676.1.c.a		1	364.w	even	6	1
676.1.c.b		1	91.v	odd	6	1
676.1.c.b		1	364.ba	even	6	1
676.1.i.a		4	91.w	even	12	2
676.1.i.a		4	91.bb	even	12	2
676.1.i.a		4	364.bw	odd	12	2
676.1.i.a		4	364.cg	odd	12	2
676.1.j.a		2	91.p	odd	6	1
676.1.j.a		2	91.s	odd	6	1
676.1.j.a		2	364.x	even	6	1
676.1.j.a		2	364.bp	even	6	1
832.1.bb.a		2	56.j	odd	6	1
832.1.bb.a		2	56.m	even	6	1
832.1.bb.a		2	728.bf	even	6	1
832.1.bb.a		2	728.de	odd	6	1
1300.1.w.a		4	35.k	even	12	2
1300.1.w.a		4	140.x	odd	12	2
1300.1.w.a		4	455.db	even	12	2
1300.1.w.a		4	1820.er	odd	12	2
1300.1.bc.a		2	35.i	odd	6	1
1300.1.bc.a		2	140.s	even	6	1
1300.1.bc.a		2	455.bw	odd	6	1
1300.1.bc.a		2	1820.bt	even	6	1
2548.1.q.a		2	7.c	even	3	1
2548.1.q.a		2	13.c	even	3	1
2548.1.q.a		2	28.g	odd	6	1
2548.1.q.a		2	52.j	odd	6	1
2548.1.q.b		2	7.d	odd	6	1
2548.1.q.b		2	28.f	even	6	1
2548.1.q.b		2	91.n	odd	6	1
2548.1.q.b		2	364.v	even	6	1
2548.1.bi.a		2	1.a	even	1	1	trivial
2548.1.bi.a		2	4.b	odd	2	1	CM
2548.1.bi.a		2	91.h	even	3	1	inner
2548.1.bi.a		2	364.bi	odd	6	1	inner
2548.1.bi.b		2	7.b	odd	2	1
2548.1.bi.b		2	28.d	even	2	1
2548.1.bi.b		2	91.v	odd	6	1
2548.1.bi.b		2	364.ba	even	6	1
2548.1.bn.a		2	7.c	even	3	1
2548.1.bn.a		2	28.g	odd	6	1
2548.1.bn.a		2	91.g	even	3	1
2548.1.bn.a		2	364.q	odd	6	1
3328.1.v.b		4	112.v	even	12	2
3328.1.v.b		4	112.x	odd	12	2
3328.1.v.b		4	1456.fj	odd	12	2
3328.1.v.b		4	1456.fm	even	12	2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{2} + T_{5} + 1 \) acting on \(S_{1}^{\mathrm{new}}(2548, [\chi])\).

Hecke characteristic polynomials

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