Newform orbit 1536.1.e.a

• Label: 1536.1.e.a
• Level: $1536$
• Weight: $1$
• Character orbit: 1536.e
• Analytic conductor: $0.767$
• Analytic rank: $0$
• Dimension: $4$
• Projective image: $D_{4}$
• CM discriminant: -24
• Inner twists: $8$

Newspace parameters

Level:	\( N \)	\(=\)	\( 1536 = 2^{9} \cdot 3 \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	1536.e (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(0.766563859404\)
Analytic rank:	\(0\)
Dimension:	\(4\)
Coefficient field:	\(\Q(\zeta_{8})\)
Defining polynomial:	\( x^{4} + 1 \)
Coefficient ring:	\(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index:	\( 2 \)
Twist minimal:	yes
Projective image:	\(D_{4}\)
Projective field:	Galois closure of 4.2.4608.1

$q$-expansion

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

\(f(q)\)	\(=\)	\( q - \zeta_{8}^{2} q^{3} + ( - \zeta_{8}^{3} - \zeta_{8}) q^{5} + (\zeta_{8}^{3} - \zeta_{8}) q^{7} - q^{9} +O(q^{10}) \)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1536\mathbb{Z}\right)^\times\).

\(n\)	\(511\)	\(517\)	\(1025\)
\(\chi(n)\)	\(1\)	\(1\)	\(-1\)

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

0.707107	+	0.707107i
−0.707107	−	0.707107i
−0.707107	+	0.707107i
0.707107	−	0.707107i

0

−

1.00000i

0

−

1.41421i

0

−1.41421

0

−1.00000

0

1025.2

0

−

1.00000i

0

1.41421i

0

1.41421

0

−1.00000

0

1025.3

0

1.00000i

0

−

1.41421i

0

1.41421

0

−1.00000

0

1025.4

0

1.00000i

0

1.41421i

0

−1.41421

0

−1.00000

0

Inner twists

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

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	1536.1.e.a	✓	4
3.b	odd	2	1	inner	1536.1.e.a	✓	4
4.b	odd	2	1	inner	1536.1.e.a	✓	4
8.b	even	2	1	inner	1536.1.e.a	✓	4
8.d	odd	2	1	inner	1536.1.e.a	✓	4
12.b	even	2	1	inner	1536.1.e.a	✓	4
16.e	even	4	1		1536.1.h.a		2
16.e	even	4	1		1536.1.h.b		2
16.f	odd	4	1		1536.1.h.a		2
16.f	odd	4	1		1536.1.h.b		2
24.f	even	2	1	inner	1536.1.e.a	✓	4
24.h	odd	2	1	CM	1536.1.e.a	✓	4
32.g	even	8	2		3072.1.i.e		4
32.g	even	8	2		3072.1.i.h		4
32.h	odd	8	2		3072.1.i.e		4
32.h	odd	8	2		3072.1.i.h		4
48.i	odd	4	1		1536.1.h.a		2
48.i	odd	4	1		1536.1.h.b		2
48.k	even	4	1		1536.1.h.a		2
48.k	even	4	1		1536.1.h.b		2
96.o	even	8	2		3072.1.i.e		4
96.o	even	8	2		3072.1.i.h		4
96.p	odd	8	2		3072.1.i.e		4
96.p	odd	8	2		3072.1.i.h		4

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
1536.1.e.a	✓	4	1.a	even	1	1	trivial
1536.1.e.a	✓	4	3.b	odd	2	1	inner
1536.1.e.a	✓	4	4.b	odd	2	1	inner
1536.1.e.a	✓	4	8.b	even	2	1	inner
1536.1.e.a	✓	4	8.d	odd	2	1	inner
1536.1.e.a	✓	4	12.b	even	2	1	inner
1536.1.e.a	✓	4	24.f	even	2	1	inner
1536.1.e.a	✓	4	24.h	odd	2	1	CM
1536.1.h.a		2	16.e	even	4	1
1536.1.h.a		2	16.f	odd	4	1
1536.1.h.a		2	48.i	odd	4	1
1536.1.h.a		2	48.k	even	4	1
1536.1.h.b		2	16.e	even	4	1
1536.1.h.b		2	16.f	odd	4	1
1536.1.h.b		2	48.i	odd	4	1
1536.1.h.b		2	48.k	even	4	1
3072.1.i.e		4	32.g	even	8	2
3072.1.i.e		4	32.h	odd	8	2
3072.1.i.e		4	96.o	even	8	2
3072.1.i.e		4	96.p	odd	8	2
3072.1.i.h		4	32.g	even	8	2
3072.1.i.h		4	32.h	odd	8	2
3072.1.i.h		4	96.o	even	8	2
3072.1.i.h		4	96.p	odd	8	2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{2} + 2 \) acting on \(S_{1}^{\mathrm{new}}(1536, [\chi])\).

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	\( T^{4} \)
$3$	\( (T^{2} + 1)^{2} \)
$5$	\( (T^{2} + 2)^{2} \)
$7$	\( (T^{2} - 2)^{2} \)
$11$	\( T^{4} \)