Newform orbit 256.10.b.g

• Label: 256.10.b.g
• Level: $256$
• Weight: $10$
• Character orbit: 256.b
• Analytic conductor: $131.849$
• Analytic rank: $0$
• Dimension: $2$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 256 = 2^{8} \)
Weight:	\( k \)	\(=\)	\( 10 \)
Character orbit:	\([\chi]\)	\(=\)	256.b (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(131.849174058\)
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 2)
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.

\(f(q)\)	\(=\)	\( q + 78 \beta q^{3} + 435 \beta q^{5} + 952 q^{7} - 4653 q^{9}+O(q^{10}) \)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 1904 q^{7} - 9306 q^{9}+O(q^{10}) \)

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

−

156.000i

0

−

870.000i

0

952.000

0

−4653.00

0

129.2

0

156.000i

0

870.000i

0

952.000

0

−4653.00

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	256.10.b.g		2
4.b	odd	2	1		256.10.b.e		2
8.b	even	2	1	inner	256.10.b.g		2
8.d	odd	2	1		256.10.b.e		2
16.e	even	4	1		2.10.a.a	✓	1
16.e	even	4	1		64.10.a.h		1
16.f	odd	4	1		16.10.a.d		1
16.f	odd	4	1		64.10.a.b		1
48.i	odd	4	1		18.10.a.a		1
48.k	even	4	1		144.10.a.d		1
80.i	odd	4	1		50.10.b.a		2
80.j	even	4	1		400.10.c.d		2
80.k	odd	4	1		400.10.a.b		1
80.q	even	4	1		50.10.a.c		1
80.s	even	4	1		400.10.c.d		2
80.t	odd	4	1		50.10.b.a		2
112.l	odd	4	1		98.10.a.c		1
112.w	even	12	2		98.10.c.c		2
112.x	odd	12	2		98.10.c.b		2
144.w	odd	12	2		162.10.c.i		2
144.x	even	12	2		162.10.c.b		2
176.l	odd	4	1		242.10.a.a		1
208.p	even	4	1		338.10.a.a		1

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
2.10.a.a	✓	1	16.e	even	4	1
16.10.a.d		1	16.f	odd	4	1
18.10.a.a		1	48.i	odd	4	1
50.10.a.c		1	80.q	even	4	1
50.10.b.a		2	80.i	odd	4	1
50.10.b.a		2	80.t	odd	4	1
64.10.a.b		1	16.f	odd	4	1
64.10.a.h		1	16.e	even	4	1
98.10.a.c		1	112.l	odd	4	1
98.10.c.b		2	112.x	odd	12	2
98.10.c.c		2	112.w	even	12	2
144.10.a.d		1	48.k	even	4	1
162.10.c.b		2	144.x	even	12	2
162.10.c.i		2	144.w	odd	12	2
242.10.a.a		1	176.l	odd	4	1
256.10.b.e		2	4.b	odd	2	1
256.10.b.e		2	8.d	odd	2	1
256.10.b.g		2	1.a	even	1	1	trivial
256.10.b.g		2	8.b	even	2	1	inner
338.10.a.a		1	208.p	even	4	1
400.10.a.b		1	80.k	odd	4	1
400.10.c.d		2	80.j	even	4	1
400.10.c.d		2	80.s	even	4	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{10}^{\mathrm{new}}(256, [\chi])\):

\( T_{3}^{2} + 24336 \)

\( T_{7} - 952 \)

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	\( T^{2} \)
$3$	\( T^{2} + 24336 \)
$5$	\( T^{2} + 756900 \)
$7$	\( (T - 952)^{2} \)
$11$	\( T^{2} + 3152597904 \)