Newform orbit 1216.1.e.a

• Label: 1216.1.e.a
• Level: $1216$
• Weight: $1$
• Character orbit: 1216.e
• Self dual: yes
• Analytic conductor: $0.607$
• Analytic rank: $0$
• Dimension: $1$
• Projective image: $D_{3}$
• CM discriminant: -19
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 1216 = 2^{6} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	1216.e (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual:	yes
Analytic conductor:	\(0.606863055362\)
Analytic rank:	\(0\)
Dimension:	\(1\)
Coefficient field:	\(\mathbb{Q}\)
Coefficient ring:	\(\mathbb{Z}\)
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:	$D_6$
Artin field:	Galois closure of 6.2.739328.3

$q$-expansion

\(f(q)\)

\(=\)

\( q + q^{5} - q^{7} + q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1216\mathbb{Z}\right)^\times\).

\(n\)	\(191\)	\(705\)	\(837\)
\(\chi(n)\)	\(1\)	\(-1\)	\(1\)

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

\(\iota_m(\nu)\)

\( a_{2} \)

\( a_{3} \)

\( a_{4} \)

\( a_{5} \)

\( a_{6} \)

\( a_{7} \)

\( a_{8} \)

\( a_{9} \)

\( a_{10} \)

1025.1

0

0

0

0

1.00000

0

−1.00000

0

1.00000

0

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
19.b	odd	2	1	CM by \(\Q(\sqrt{-19}) \)

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	1216.1.e.a		1
4.b	odd	2	1		1216.1.e.b		1
8.b	even	2	1		76.1.c.a	✓	1
8.d	odd	2	1		304.1.e.a		1
19.b	odd	2	1	CM	1216.1.e.a		1
24.f	even	2	1		2736.1.o.b		1
24.h	odd	2	1		684.1.h.a		1
40.f	even	2	1		1900.1.e.a		1
40.i	odd	4	2		1900.1.g.a		2
56.h	odd	2	1		3724.1.e.c		1
56.j	odd	6	2		3724.1.bc.b		2
56.p	even	6	2		3724.1.bc.c		2
76.d	even	2	1		1216.1.e.b		1
152.b	even	2	1		304.1.e.a		1
152.g	odd	2	1		76.1.c.a	✓	1
152.l	odd	6	2		1444.1.h.a		2
152.p	even	6	2		1444.1.h.a		2
152.s	odd	18	6		1444.1.j.a		6
152.t	even	18	6		1444.1.j.a		6
456.l	odd	2	1		2736.1.o.b		1
456.p	even	2	1		684.1.h.a		1
760.b	odd	2	1		1900.1.e.a		1
760.t	even	4	2		1900.1.g.a		2
1064.f	even	2	1		3724.1.e.c		1
1064.br	odd	6	2		3724.1.bc.c		2
1064.cf	even	6	2		3724.1.bc.b		2

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
76.1.c.a	✓	1	8.b	even	2	1
76.1.c.a	✓	1	152.g	odd	2	1
304.1.e.a		1	8.d	odd	2	1
304.1.e.a		1	152.b	even	2	1
684.1.h.a		1	24.h	odd	2	1
684.1.h.a		1	456.p	even	2	1
1216.1.e.a		1	1.a	even	1	1	trivial
1216.1.e.a		1	19.b	odd	2	1	CM
1216.1.e.b		1	4.b	odd	2	1
1216.1.e.b		1	76.d	even	2	1
1444.1.h.a		2	152.l	odd	6	2
1444.1.h.a		2	152.p	even	6	2
1444.1.j.a		6	152.s	odd	18	6
1444.1.j.a		6	152.t	even	18	6
1900.1.e.a		1	40.f	even	2	1
1900.1.e.a		1	760.b	odd	2	1
1900.1.g.a		2	40.i	odd	4	2
1900.1.g.a		2	760.t	even	4	2
2736.1.o.b		1	24.f	even	2	1
2736.1.o.b		1	456.l	odd	2	1
3724.1.e.c		1	56.h	odd	2	1
3724.1.e.c		1	1064.f	even	2	1
3724.1.bc.b		2	56.j	odd	6	2
3724.1.bc.b		2	1064.cf	even	6	2
3724.1.bc.c		2	56.p	even	6	2
3724.1.bc.c		2	1064.br	odd	6	2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7} + 1 \) acting on \(S_{1}^{\mathrm{new}}(1216, [\chi])\).

Hecke characteristic polynomials

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