Newform orbit 384.1.h.a

• Label: 384.1.h.a
• Level: $384$
• Weight: $1$
• Character orbit: 384.h
• Self dual: yes
• Analytic conductor: $0.192$
• Analytic rank: $0$
• Dimension: $1$
• Projective image: $D_{2}$
• CM/RM discs: -8, -24, 12
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 384 = 2^{7} \cdot 3 \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	384.h (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual:	yes
Analytic conductor:	\(0.191640964851\)
Analytic rank:	\(0\)
Dimension:	\(1\)
Coefficient field:	\(\mathbb{Q}\)
Coefficient ring:	\(\mathbb{Z}\)
Coefficient ring index:	\( 1 \)
Twist minimal:	yes
Projective image:	\(D_{2}\)
Projective field:	Galois closure of \(\Q(\sqrt{-2}, \sqrt{3})\)
Artin image:	$D_4$
Artin field:	Galois closure of 4.0.3072.1

$q$-expansion

\(f(q)\)

\(=\)

\( q - q^{3} + q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/384\mathbb{Z}\right)^\times\).

\(n\)	\(127\)	\(133\)	\(257\)
\(\chi(n)\)	\(1\)	\(-1\)	\(-1\)

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

\(\iota_m(\nu)\)

\( a_{2} \)

\( a_{3} \)

\( a_{4} \)

\( a_{5} \)

\( a_{6} \)

\( a_{7} \)

\( a_{8} \)

\( a_{9} \)

\( a_{10} \)

65.1

0

0

−1.00000

0

0

0

0

0

1.00000

0

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
8.d	odd	2	1	CM by \(\Q(\sqrt{-2}) \)
12.b	even	2	1	RM by \(\Q(\sqrt{3}) \)
24.h	odd	2	1	CM by \(\Q(\sqrt{-6}) \)

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	384.1.h.a	✓	1
3.b	odd	2	1		384.1.h.b	yes	1
4.b	odd	2	1		384.1.h.b	yes	1
8.b	even	2	1		384.1.h.b	yes	1
8.d	odd	2	1	CM	384.1.h.a	✓	1
12.b	even	2	1	RM	384.1.h.a	✓	1
16.e	even	4	2		768.1.e.c		2
16.f	odd	4	2		768.1.e.c		2
24.f	even	2	1		384.1.h.b	yes	1
24.h	odd	2	1	CM	384.1.h.a	✓	1
32.g	even	8	4		3072.1.i.f		4
32.h	odd	8	4		3072.1.i.f		4
48.i	odd	4	2		768.1.e.c		2
48.k	even	4	2		768.1.e.c		2
96.o	even	8	4		3072.1.i.f		4
96.p	odd	8	4		3072.1.i.f		4

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
384.1.h.a	✓	1	1.a	even	1	1	trivial
384.1.h.a	✓	1	8.d	odd	2	1	CM
384.1.h.a	✓	1	12.b	even	2	1	RM
384.1.h.a	✓	1	24.h	odd	2	1	CM
384.1.h.b	yes	1	3.b	odd	2	1
384.1.h.b	yes	1	4.b	odd	2	1
384.1.h.b	yes	1	8.b	even	2	1
384.1.h.b	yes	1	24.f	even	2	1
768.1.e.c		2	16.e	even	4	2
768.1.e.c		2	16.f	odd	4	2
768.1.e.c		2	48.i	odd	4	2
768.1.e.c		2	48.k	even	4	2
3072.1.i.f		4	32.g	even	8	4
3072.1.i.f		4	32.h	odd	8	4
3072.1.i.f		4	96.o	even	8	4
3072.1.i.f		4	96.p	odd	8	4

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{11} - 2 \) acting on \(S_{1}^{\mathrm{new}}(384, [\chi])\).

Hecke characteristic polynomials

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