Newform orbit 256.5.d.a

• Label: 256.5.d.a
• Level: $256$
• Weight: $5$
• Character orbit: 256.d
• Analytic conductor: $26.463$
• Analytic rank: $0$
• Dimension: $2$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 256 = 2^{8} \)
Weight:	\( k \)	\(=\)	\( 5 \)
Character orbit:	\([\chi]\)	\(=\)	256.d (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(26.4627105495\)
Analytic rank:	\(0\)
Dimension:	\(2\)
Coefficient field:	\(\Q(\sqrt{-1}) \)
Defining polynomial:	\( x^{2} + 1 \)
Coefficient ring:	\(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index:	\( 2 \)
Twist minimal:	no (minimal twist has level 32)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = 2i\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\)	\(=\)	\( q - 12 q^{3} - 19 \beta q^{5} + 4 \beta q^{7} + 63 q^{9} +O(q^{10}) \)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 24 q^{3} + 126 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/256\mathbb{Z}\right)^\times\).

\(n\)	\(5\)	\(255\)
\(\chi(n)\)	\(-1\)	\(-1\)

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  

\(\iota_m(\nu)\)

\( a_{2} \)

\( a_{3} \)

\( a_{4} \)

\( a_{5} \)

\( a_{6} \)

\( a_{7} \)

\( a_{8} \)

\( a_{9} \)

\( a_{10} \)

127.1

		1.00000i
	−	1.00000i

0

−12.0000

0

−

38.0000i

0

8.00000i

0

63.0000

0

127.2

0

−12.0000

0

38.0000i

0

−

8.00000i

0

63.0000

0

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
8.d	odd	2	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	256.5.d.a		2
4.b	odd	2	1		256.5.d.e		2
8.b	even	2	1		256.5.d.e		2
8.d	odd	2	1	inner	256.5.d.a		2
16.e	even	4	1		32.5.c.a	✓	2
16.e	even	4	1		64.5.c.d		2
16.f	odd	4	1		32.5.c.a	✓	2
16.f	odd	4	1		64.5.c.d		2
48.i	odd	4	1		288.5.g.b		2
48.i	odd	4	1		576.5.g.e		2
48.k	even	4	1		288.5.g.b		2
48.k	even	4	1		576.5.g.e		2
80.i	odd	4	1		800.5.h.d		2
80.j	even	4	1		800.5.h.d		2
80.k	odd	4	1		800.5.b.a		2
80.q	even	4	1		800.5.b.a		2
80.s	even	4	1		800.5.h.a		2
80.t	odd	4	1		800.5.h.a		2

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
32.5.c.a	✓	2	16.e	even	4	1
32.5.c.a	✓	2	16.f	odd	4	1
64.5.c.d		2	16.e	even	4	1
64.5.c.d		2	16.f	odd	4	1
256.5.d.a		2	1.a	even	1	1	trivial
256.5.d.a		2	8.d	odd	2	1	inner
256.5.d.e		2	4.b	odd	2	1
256.5.d.e		2	8.b	even	2	1
288.5.g.b		2	48.i	odd	4	1
288.5.g.b		2	48.k	even	4	1
576.5.g.e		2	48.i	odd	4	1
576.5.g.e		2	48.k	even	4	1
800.5.b.a		2	80.k	odd	4	1
800.5.b.a		2	80.q	even	4	1
800.5.h.a		2	80.s	even	4	1
800.5.h.a		2	80.t	odd	4	1
800.5.h.d		2	80.i	odd	4	1
800.5.h.d		2	80.j	even	4	1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3} + 12 \) acting on \(S_{5}^{\mathrm{new}}(256, [\chi])\).

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	\( T^{2} \)
$3$	\( (T + 12)^{2} \)
$5$	\( T^{2} + 1444 \)
$7$	\( T^{2} + 64 \)
$11$	\( (T - 212)^{2} \)