Newform orbit 1024.2.b.f

• Label: 1024.2.b.f
• Level: $1024$
• Weight: $2$
• Character orbit: 1024.b
• Analytic conductor: $8.177$
• Analytic rank: $0$
• Dimension: $2$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 1024 = 2^{10} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1024.b (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(8.17668116698\)
Analytic rank:	\(0\)
Dimension:	\(2\)
Coefficient field:	\(\Q(\sqrt{-2}) \)
Defining polynomial:	\( x^{2} + 2 \)
Coefficient ring:	\(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	no (minimal twist has level 512)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

$q$-expansion

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

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1024\mathbb{Z}\right)^\times\).

\(n\)	\(5\)	\(1023\)
\(\chi(n)\)	\(-1\)	\(1\)

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  

\(\iota_m(\nu)\)

\( a_{2} \)

\( a_{3} \)

\( a_{4} \)

\( a_{5} \)

\( a_{6} \)

\( a_{7} \)

\( a_{8} \)

\( a_{9} \)

\( a_{10} \)

513.1

	−	1.41421i
		1.41421i

0

−

2.82843i

0

1.41421i

0

4.00000

0

−5.00000

0

513.2

0

2.82843i

0

−

1.41421i

0

4.00000

0

−5.00000

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	1024.2.b.f		2
4.b	odd	2	1		1024.2.b.a		2
8.b	even	2	1	inner	1024.2.b.f		2
8.d	odd	2	1		1024.2.b.a		2
16.e	even	4	2		1024.2.a.a		2
16.f	odd	4	2		1024.2.a.f		2
32.g	even	8	2		512.2.e.b	yes	2
32.g	even	8	2		512.2.e.g	yes	2
32.h	odd	8	2		512.2.e.a	✓	2
32.h	odd	8	2		512.2.e.h	yes	2
48.i	odd	4	2		9216.2.a.b		2
48.k	even	4	2		9216.2.a.u		2
96.o	even	8	2		4608.2.k.f		2
96.o	even	8	2		4608.2.k.s		2
96.p	odd	8	2		4608.2.k.j		2
96.p	odd	8	2		4608.2.k.o		2

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
512.2.e.a	✓	2	32.h	odd	8	2
512.2.e.b	yes	2	32.g	even	8	2
512.2.e.g	yes	2	32.g	even	8	2
512.2.e.h	yes	2	32.h	odd	8	2
1024.2.a.a		2	16.e	even	4	2
1024.2.a.f		2	16.f	odd	4	2
1024.2.b.a		2	4.b	odd	2	1
1024.2.b.a		2	8.d	odd	2	1
1024.2.b.f		2	1.a	even	1	1	trivial
1024.2.b.f		2	8.b	even	2	1	inner
4608.2.k.f		2	96.o	even	8	2
4608.2.k.j		2	96.p	odd	8	2
4608.2.k.o		2	96.p	odd	8	2
4608.2.k.s		2	96.o	even	8	2
9216.2.a.b		2	48.i	odd	4	2
9216.2.a.u		2	48.k	even	4	2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1024, [\chi])\):

\( T_{3}^{2} + 8 \)

\( T_{5}^{2} + 2 \)

\( T_{7} - 4 \)

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	\( T^{2} \)
$3$	\( T^{2} + 8 \)
$5$	\( T^{2} + 2 \)
$7$	\( (T - 4)^{2} \)
$11$	\( T^{2} + 8 \)