Newform orbit 256.6.b.g

• Label: 256.6.b.g
• Level: $256$
• Weight: $6$
• Character orbit: 256.b
• Analytic conductor: $41.058$
• Analytic rank: $0$
• Dimension: $2$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 256 = 2^{8} \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	256.b (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(41.0582578721\)
Analytic rank:	\(0\)
Dimension:	\(2\)
Coefficient field:	\(\Q(\sqrt{-1}) \)
Defining polynomial:	\( x^{2} + 1 \)
Coefficient ring:	\(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index:	\( 2 \)
Twist minimal:	no (minimal twist has level 4)
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.

\(f(q)\)	\(=\)	\( q + 6 \beta q^{3} + 27 \beta q^{5} + 88 q^{7} + 99 q^{9}+O(q^{10}) \)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 176 q^{7} + 198 q^{9}+O(q^{10}) \)

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  

\(\iota_m(\nu)\)

\( a_{2} \)

\( a_{3} \)

\( a_{4} \)

\( a_{5} \)

\( a_{6} \)

\( a_{7} \)

\( a_{8} \)

\( a_{9} \)

\( a_{10} \)

129.1

	−	1.00000i
		1.00000i

0

−

12.0000i

0

−

54.0000i

0

88.0000

0

99.0000

0

129.2

0

12.0000i

0

54.0000i

0

88.0000

0

99.0000

0

Inner twists

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

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	256.6.b.g		2
4.b	odd	2	1		256.6.b.c		2
8.b	even	2	1	inner	256.6.b.g		2
8.d	odd	2	1		256.6.b.c		2
16.e	even	4	1		4.6.a.a	✓	1
16.e	even	4	1		64.6.a.f		1
16.f	odd	4	1		16.6.a.b		1
16.f	odd	4	1		64.6.a.b		1
48.i	odd	4	1		36.6.a.a		1
48.i	odd	4	1		576.6.a.bc		1
48.k	even	4	1		144.6.a.c		1
48.k	even	4	1		576.6.a.bd		1
80.i	odd	4	1		100.6.c.b		2
80.j	even	4	1		400.6.c.f		2
80.k	odd	4	1		400.6.a.d		1
80.q	even	4	1		100.6.a.b		1
80.s	even	4	1		400.6.c.f		2
80.t	odd	4	1		100.6.c.b		2
112.j	even	4	1		784.6.a.d		1
112.l	odd	4	1		196.6.a.e		1
112.w	even	12	2		196.6.e.g		2
112.x	odd	12	2		196.6.e.d		2
144.w	odd	12	2		324.6.e.d		2
144.x	even	12	2		324.6.e.a		2
176.l	odd	4	1		484.6.a.a		1
208.m	odd	4	1		676.6.d.a		2
208.p	even	4	1		676.6.a.a		1
208.r	odd	4	1		676.6.d.a		2
240.bb	even	4	1		900.6.d.a		2
240.bf	even	4	1		900.6.d.a		2
240.bm	odd	4	1		900.6.a.h		1

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
4.6.a.a	✓	1	16.e	even	4	1
16.6.a.b		1	16.f	odd	4	1
36.6.a.a		1	48.i	odd	4	1
64.6.a.b		1	16.f	odd	4	1
64.6.a.f		1	16.e	even	4	1
100.6.a.b		1	80.q	even	4	1
100.6.c.b		2	80.i	odd	4	1
100.6.c.b		2	80.t	odd	4	1
144.6.a.c		1	48.k	even	4	1
196.6.a.e		1	112.l	odd	4	1
196.6.e.d		2	112.x	odd	12	2
196.6.e.g		2	112.w	even	12	2
256.6.b.c		2	4.b	odd	2	1
256.6.b.c		2	8.d	odd	2	1
256.6.b.g		2	1.a	even	1	1	trivial
256.6.b.g		2	8.b	even	2	1	inner
324.6.e.a		2	144.x	even	12	2
324.6.e.d		2	144.w	odd	12	2
400.6.a.d		1	80.k	odd	4	1
400.6.c.f		2	80.j	even	4	1
400.6.c.f		2	80.s	even	4	1
484.6.a.a		1	176.l	odd	4	1
576.6.a.bc		1	48.i	odd	4	1
576.6.a.bd		1	48.k	even	4	1
676.6.a.a		1	208.p	even	4	1
676.6.d.a		2	208.m	odd	4	1
676.6.d.a		2	208.r	odd	4	1
784.6.a.d		1	112.j	even	4	1
900.6.a.h		1	240.bm	odd	4	1
900.6.d.a		2	240.bb	even	4	1
900.6.d.a		2	240.bf	even	4	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{6}^{\mathrm{new}}(256, [\chi])\):

\( T_{3}^{2} + 144 \)

\( T_{7} - 88 \)

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	\( T^{2} \)
$3$	\( T^{2} + 144 \)
$5$	\( T^{2} + 2916 \)
$7$	\( (T - 88)^{2} \)
$11$	\( T^{2} + 291600 \)