Newform orbit 3724.1.bc.b

• Label: 3724.1.bc.b
• Level: $3724$
• Weight: $1$
• Character orbit: 3724.bc
• Analytic conductor: $1.859$
• Analytic rank: $0$
• Dimension: $2$
• Projective image: $D_{3}$
• CM discriminant: -19
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 3724 = 2^{2} \cdot 7^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	3724.bc (of order \(6\), degree \(2\), not minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(1.85851810705\)
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 76)
Projective image:	\(D_{3}\)
Projective field:	Galois closure of 3.1.76.1
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.

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3724\mathbb{Z}\right)^\times\).

\(n\)	\(1863\)	\(3041\)	\(3137\)
\(\chi(n)\)	\(1\)	\(-\zeta_{6}\)	\(-1\)

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  

\(\iota_m(\nu)\)

\( a_{2} \)

\( a_{3} \)

\( a_{4} \)

\( a_{5} \)

\( a_{6} \)

\( a_{7} \)

\( a_{8} \)

\( a_{9} \)

\( a_{10} \)

569.1

0.500000	−	0.866025i
0.500000	+	0.866025i

0

0

0

−0.500000

+

0.866025i

0

0

0

−0.500000

+

0.866025i

0

949.1

0

0

0

−0.500000

−

0.866025i

0

0

0

−0.500000

−

0.866025i

0

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
19.b	odd	2	1	CM by \(\Q(\sqrt{-19}) \)
7.c	even	3	1	inner
133.r	odd	6	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	3724.1.bc.b		2
7.b	odd	2	1		3724.1.bc.c		2
7.c	even	3	1		3724.1.e.c		1
7.c	even	3	1	inner	3724.1.bc.b		2
7.d	odd	6	1		76.1.c.a	✓	1
7.d	odd	6	1		3724.1.bc.c		2
19.b	odd	2	1	CM	3724.1.bc.b		2
21.g	even	6	1		684.1.h.a		1
28.f	even	6	1		304.1.e.a		1
35.i	odd	6	1		1900.1.e.a		1
35.k	even	12	2		1900.1.g.a		2
56.j	odd	6	1		1216.1.e.a		1
56.m	even	6	1		1216.1.e.b		1
84.j	odd	6	1		2736.1.o.b		1
133.c	even	2	1		3724.1.bc.c		2
133.i	even	6	1		1444.1.h.a		2
133.k	odd	6	1		1444.1.h.a		2
133.o	even	6	1		76.1.c.a	✓	1
133.o	even	6	1		3724.1.bc.c		2
133.r	odd	6	1		3724.1.e.c		1
133.r	odd	6	1	inner	3724.1.bc.b		2
133.s	even	6	1		1444.1.h.a		2
133.t	odd	6	1		1444.1.h.a		2
133.x	odd	18	3		1444.1.j.a		6
133.z	odd	18	3		1444.1.j.a		6
133.bb	even	18	3		1444.1.j.a		6
133.bf	even	18	3		1444.1.j.a		6
399.s	odd	6	1		684.1.h.a		1
532.bh	odd	6	1		304.1.e.a		1
665.y	even	6	1		1900.1.e.a		1
665.ca	odd	12	2		1900.1.g.a		2
1064.bi	odd	6	1		1216.1.e.b		1
1064.cf	even	6	1		1216.1.e.a		1
1596.bl	even	6	1		2736.1.o.b		1

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
76.1.c.a	✓	1	7.d	odd	6	1
76.1.c.a	✓	1	133.o	even	6	1
304.1.e.a		1	28.f	even	6	1
304.1.e.a		1	532.bh	odd	6	1
684.1.h.a		1	21.g	even	6	1
684.1.h.a		1	399.s	odd	6	1
1216.1.e.a		1	56.j	odd	6	1
1216.1.e.a		1	1064.cf	even	6	1
1216.1.e.b		1	56.m	even	6	1
1216.1.e.b		1	1064.bi	odd	6	1
1444.1.h.a		2	133.i	even	6	1
1444.1.h.a		2	133.k	odd	6	1
1444.1.h.a		2	133.s	even	6	1
1444.1.h.a		2	133.t	odd	6	1
1444.1.j.a		6	133.x	odd	18	3
1444.1.j.a		6	133.z	odd	18	3
1444.1.j.a		6	133.bb	even	18	3
1444.1.j.a		6	133.bf	even	18	3
1900.1.e.a		1	35.i	odd	6	1
1900.1.e.a		1	665.y	even	6	1
1900.1.g.a		2	35.k	even	12	2
1900.1.g.a		2	665.ca	odd	12	2
2736.1.o.b		1	84.j	odd	6	1
2736.1.o.b		1	1596.bl	even	6	1
3724.1.e.c		1	7.c	even	3	1
3724.1.e.c		1	133.r	odd	6	1
3724.1.bc.b		2	1.a	even	1	1	trivial
3724.1.bc.b		2	7.c	even	3	1	inner
3724.1.bc.b		2	19.b	odd	2	1	CM
3724.1.bc.b		2	133.r	odd	6	1	inner
3724.1.bc.c		2	7.b	odd	2	1
3724.1.bc.c		2	7.d	odd	6	1
3724.1.bc.c		2	133.c	even	2	1
3724.1.bc.c		2	133.o	even	6	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{1}^{\mathrm{new}}(3724, [\chi])\):

\( T_{5}^{2} + T_{5} + 1 \)

\( T_{11}^{2} - T_{11} + 1 \)

Hecke characteristic polynomials

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