Newform orbit 1911.1.bc.a

• Label: 1911.1.bc.a
• Level: $1911$
• Weight: $1$
• Character orbit: 1911.bc
• Analytic conductor: $0.954$
• Analytic rank: $0$
• Dimension: $2$
• Projective image: $D_{6}$
• CM discriminant: -3
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 1911 = 3 \cdot 7^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	1911.bc (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(0.953713239142\)
Analytic rank:	\(0\)
Dimension:	\(2\)
Coefficient field:	\(\Q(\zeta_{6})\)
Defining polynomial:	\( x^{2} - x + 1 \)
Coefficient ring:	\(\Z[a_1, a_2, a_3]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	no (minimal twist has level 273)
Projective image:	\(D_{6}\)
Projective field:	Galois closure of 6.0.24069811311.4

$q$-expansion

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

\(f(q)\)	\(=\)	\( q - \zeta_{6} q^{3} - \zeta_{6}^{2} q^{4} + \zeta_{6}^{2} q^{9} +O(q^{10}) \)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - q^{3} + q^{4} - q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1911\mathbb{Z}\right)^\times\).

\(n\)	\(638\)	\(1471\)	\(1522\)
\(\chi(n)\)	\(-1\)	\(-\zeta_{6}^{2}\)	\(1\)

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

\(\iota_m(\nu)\)

\( a_{2} \)

\( a_{3} \)

\( a_{4} \)

\( a_{5} \)

\( a_{6} \)

\( a_{7} \)

\( a_{8} \)

\( a_{9} \)

\( a_{10} \)

491.1

0.500000	+	0.866025i
0.500000	−	0.866025i

0

−0.500000

−

0.866025i

0.500000

−

0.866025i

0

0

0

0

−0.500000

+

0.866025i

0

1226.1

0

−0.500000

+

0.866025i

0.500000

+

0.866025i

0

0

0

0

−0.500000

−

0.866025i

0

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
3.b	odd	2	1	CM by \(\Q(\sqrt{-3}) \)
13.e	even	6	1	inner
39.h	odd	6	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	1911.1.bc.a		2
3.b	odd	2	1	CM	1911.1.bc.a		2
7.b	odd	2	1		1911.1.bc.b		2
7.c	even	3	1		1911.1.x.a		2
7.c	even	3	1		1911.1.bp.a		2
7.d	odd	6	1		273.1.x.a	✓	2
7.d	odd	6	1		273.1.bp.a	yes	2
13.e	even	6	1	inner	1911.1.bc.a		2
21.c	even	2	1		1911.1.bc.b		2
21.g	even	6	1		273.1.x.a	✓	2
21.g	even	6	1		273.1.bp.a	yes	2
21.h	odd	6	1		1911.1.x.a		2
21.h	odd	6	1		1911.1.bp.a		2
39.h	odd	6	1	inner	1911.1.bc.a		2
91.k	even	6	1		1911.1.x.a		2
91.l	odd	6	1		273.1.x.a	✓	2
91.l	odd	6	1		3549.1.w.a		2
91.m	odd	6	1		3549.1.w.a		2
91.m	odd	6	1		3549.1.bp.a		2
91.p	odd	6	1		273.1.bp.a	yes	2
91.p	odd	6	1		3549.1.w.b		2
91.s	odd	6	1		3549.1.x.a		2
91.s	odd	6	1		3549.1.bp.a		2
91.t	odd	6	1		1911.1.bc.b		2
91.u	even	6	1		1911.1.bp.a		2
91.v	odd	6	1		3549.1.w.b		2
91.v	odd	6	1		3549.1.x.a		2
91.w	even	12	2		3549.1.s.c		4
91.w	even	12	2		3549.1.bk.e		4
91.ba	even	12	2		3549.1.bk.e		4
91.ba	even	12	2		3549.1.bm.b		4
91.bb	even	12	2		3549.1.s.c		4
91.bb	even	12	2		3549.1.bm.b		4
273.r	even	6	1		3549.1.w.b		2
273.r	even	6	1		3549.1.x.a		2
273.u	even	6	1		1911.1.bc.b		2
273.x	odd	6	1		1911.1.bp.a		2
273.y	even	6	1		273.1.bp.a	yes	2
273.y	even	6	1		3549.1.w.b		2
273.ba	even	6	1		3549.1.x.a		2
273.ba	even	6	1		3549.1.bp.a		2
273.bf	even	6	1		3549.1.w.a		2
273.bf	even	6	1		3549.1.bp.a		2
273.bp	odd	6	1		1911.1.x.a		2
273.br	even	6	1		273.1.x.a	✓	2
273.br	even	6	1		3549.1.w.a		2
273.bs	odd	12	2		3549.1.bk.e		4
273.bs	odd	12	2		3549.1.bm.b		4
273.cb	odd	12	2		3549.1.s.c		4
273.cb	odd	12	2		3549.1.bm.b		4
273.ch	odd	12	2		3549.1.s.c		4
273.ch	odd	12	2		3549.1.bk.e		4

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
273.1.x.a	✓	2	7.d	odd	6	1
273.1.x.a	✓	2	21.g	even	6	1
273.1.x.a	✓	2	91.l	odd	6	1
273.1.x.a	✓	2	273.br	even	6	1
273.1.bp.a	yes	2	7.d	odd	6	1
273.1.bp.a	yes	2	21.g	even	6	1
273.1.bp.a	yes	2	91.p	odd	6	1
273.1.bp.a	yes	2	273.y	even	6	1
1911.1.x.a		2	7.c	even	3	1
1911.1.x.a		2	21.h	odd	6	1
1911.1.x.a		2	91.k	even	6	1
1911.1.x.a		2	273.bp	odd	6	1
1911.1.bc.a		2	1.a	even	1	1	trivial
1911.1.bc.a		2	3.b	odd	2	1	CM
1911.1.bc.a		2	13.e	even	6	1	inner
1911.1.bc.a		2	39.h	odd	6	1	inner
1911.1.bc.b		2	7.b	odd	2	1
1911.1.bc.b		2	21.c	even	2	1
1911.1.bc.b		2	91.t	odd	6	1
1911.1.bc.b		2	273.u	even	6	1
1911.1.bp.a		2	7.c	even	3	1
1911.1.bp.a		2	21.h	odd	6	1
1911.1.bp.a		2	91.u	even	6	1
1911.1.bp.a		2	273.x	odd	6	1
3549.1.s.c		4	91.w	even	12	2
3549.1.s.c		4	91.bb	even	12	2
3549.1.s.c		4	273.cb	odd	12	2
3549.1.s.c		4	273.ch	odd	12	2
3549.1.w.a		2	91.l	odd	6	1
3549.1.w.a		2	91.m	odd	6	1
3549.1.w.a		2	273.bf	even	6	1
3549.1.w.a		2	273.br	even	6	1
3549.1.w.b		2	91.p	odd	6	1
3549.1.w.b		2	91.v	odd	6	1
3549.1.w.b		2	273.r	even	6	1
3549.1.w.b		2	273.y	even	6	1
3549.1.x.a		2	91.s	odd	6	1
3549.1.x.a		2	91.v	odd	6	1
3549.1.x.a		2	273.r	even	6	1
3549.1.x.a		2	273.ba	even	6	1
3549.1.bk.e		4	91.w	even	12	2
3549.1.bk.e		4	91.ba	even	12	2
3549.1.bk.e		4	273.bs	odd	12	2
3549.1.bk.e		4	273.ch	odd	12	2
3549.1.bm.b		4	91.ba	even	12	2
3549.1.bm.b		4	91.bb	even	12	2
3549.1.bm.b		4	273.bs	odd	12	2
3549.1.bm.b		4	273.cb	odd	12	2
3549.1.bp.a		2	91.m	odd	6	1
3549.1.bp.a		2	91.s	odd	6	1
3549.1.bp.a		2	273.ba	even	6	1
3549.1.bp.a		2	273.bf	even	6	1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{19}^{2} + 3T_{19} + 3 \) acting on \(S_{1}^{\mathrm{new}}(1911, [\chi])\).

Hecke characteristic polynomials

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