Newform orbit 2736.1.bk.b

• Label: 2736.1.bk.b
• Level: $2736$
• Weight: $1$
• Character orbit: 2736.bk
• Analytic conductor: $1.365$
• Analytic rank: $0$
• Dimension: $2$
• Projective image: $D_{6}$
• CM discriminant: -3
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 2736 = 2^{4} \cdot 3^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	2736.bk (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(1.36544187456\)
Analytic rank:	\(0\)
Dimension:	\(2\)
Coefficient field:	\(\Q(\zeta_{6})\)
Defining polynomial:	\( x^{2} - x + 1 \)
Coefficient ring:	\(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	yes
Projective image:	\(D_{6}\)
Projective field:	Galois closure of 6.2.225194688.1

$q$-expansion

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

\(f(q)\)	\(=\)	\( q + (\zeta_{6}^{2} + \zeta_{6}) q^{7}+O(q^{10}) \)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2736\mathbb{Z}\right)^\times\).

\(n\)	\(1009\)	\(1217\)	\(1711\)	\(2053\)
\(\chi(n)\)	\(-\zeta_{6}\)	\(1\)	\(-1\)	\(1\)

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

\(\iota_m(\nu)\)

\( a_{2} \)

\( a_{3} \)

\( a_{4} \)

\( a_{5} \)

\( a_{6} \)

\( a_{7} \)

\( a_{8} \)

\( a_{9} \)

\( a_{10} \)

847.1

0.500000	+	0.866025i
0.500000	−	0.866025i

0

0

0

0

0

1.73205i

0

0

0

2287.1

0

0

0

0

0

−

1.73205i

0

0

0

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
3.b	odd	2	1	CM by \(\Q(\sqrt{-3}) \)
76.g	odd	6	1	inner
228.m	even	6	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	2736.1.bk.b	yes	2
3.b	odd	2	1	CM	2736.1.bk.b	yes	2
4.b	odd	2	1		2736.1.bk.a	✓	2
12.b	even	2	1		2736.1.bk.a	✓	2
19.c	even	3	1		2736.1.bk.a	✓	2
57.h	odd	6	1		2736.1.bk.a	✓	2
76.g	odd	6	1	inner	2736.1.bk.b	yes	2
228.m	even	6	1	inner	2736.1.bk.b	yes	2

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
2736.1.bk.a	✓	2	4.b	odd	2	1
2736.1.bk.a	✓	2	12.b	even	2	1
2736.1.bk.a	✓	2	19.c	even	3	1
2736.1.bk.a	✓	2	57.h	odd	6	1
2736.1.bk.b	yes	2	1.a	even	1	1	trivial
2736.1.bk.b	yes	2	3.b	odd	2	1	CM
2736.1.bk.b	yes	2	76.g	odd	6	1	inner
2736.1.bk.b	yes	2	228.m	even	6	1	inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{43}^{2} - 3T_{43} + 3 \) acting on \(S_{1}^{\mathrm{new}}(2736, [\chi])\).

Hecke characteristic polynomials

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