Newform orbit 3267.1.b.e

• Label: 3267.1.b.e
• Level: $3267$
• Weight: $1$
• Character orbit: 3267.b
• Self dual: yes
• Analytic conductor: $1.630$
• Analytic rank: $0$
• Dimension: $2$
• Projective image: $D_{6}$
• CM discriminant: -3
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 3267 = 3^{3} \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	3267.b (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual:	yes
Analytic conductor:	\(1.63044539627\)
Analytic rank:	\(0\)
Dimension:	\(2\)
Coefficient field:	\(\Q(\sqrt{3}) \)
Defining polynomial:	\( x^{2} - 3 \)
Coefficient ring:	\(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	yes
Projective image:	\(D_{6}\)
Projective field:	Galois closure of 6.0.117406179.1
Artin image:	$D_{12}$
Artin field:	Galois closure of \(\mathbb{Q}[x]/(x^{12} + \cdots)\)

$q$-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.

\(f(q)\)	\(=\)	\( q + q^{4} - \beta q^{7} +O(q^{10}) \)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{4}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3267\mathbb{Z}\right)^\times\).

\(n\)	\(244\)	\(3026\)
\(\chi(n)\)	\(1\)	\(-1\)

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  

\(\iota_m(\nu)\)

\( a_{2} \)

\( a_{3} \)

\( a_{4} \)

\( a_{5} \)

\( a_{6} \)

\( a_{7} \)

\( a_{8} \)

\( a_{9} \)

\( a_{10} \)

485.1

1.73205
−1.73205

0

0

1.00000

0

0

−1.73205

0

0

0

485.2

0

0

1.00000

0

0

1.73205

0

0

0

Inner twists

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

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	3267.1.b.e	✓	2
3.b	odd	2	1	CM	3267.1.b.e	✓	2
11.b	odd	2	1	inner	3267.1.b.e	✓	2
11.c	even	5	4		3267.1.m.c		8
11.d	odd	10	4		3267.1.m.c		8
33.d	even	2	1	inner	3267.1.b.e	✓	2
33.f	even	10	4		3267.1.m.c		8
33.h	odd	10	4		3267.1.m.c		8

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
3267.1.b.e	✓	2	1.a	even	1	1	trivial
3267.1.b.e	✓	2	3.b	odd	2	1	CM
3267.1.b.e	✓	2	11.b	odd	2	1	inner
3267.1.b.e	✓	2	33.d	even	2	1	inner
3267.1.m.c		8	11.c	even	5	4
3267.1.m.c		8	11.d	odd	10	4
3267.1.m.c		8	33.f	even	10	4
3267.1.m.c		8	33.h	odd	10	4

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{1}^{\mathrm{new}}(3267, [\chi])\):

\( T_{2} \)

\( T_{7}^{2} - 3 \)

Hecke characteristic polynomials

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