Newform orbit 1024.4.b.c

• Label: 1024.4.b.c
• Level: $1024$
• Weight: $4$
• Character orbit: 1024.b
• Analytic conductor: $60.418$
• Analytic rank: $0$
• Dimension: $2$
• CM discriminant: -4
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 1024 = 2^{10} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1024.b (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(60.4179558459\)
Analytic rank:	\(0\)
Dimension:	\(2\)
Coefficient field:	\(\Q(\sqrt{-2}) \)
Defining polynomial:	\( x^{2} + 2 \)
Coefficient ring:	\(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	no (minimal twist has level 512)
Sato-Tate group:	$\mathrm{U}(1)[D_{2}]$

$q$-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.

\(f(q)\)	\(=\)	\( q + 9 \beta q^{5} + 27 q^{9}+O(q^{10}) \)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 54 q^{9}+O(q^{10}) \)

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

0

0

−

12.7279i

0

0

0

27.0000

0

513.2

0

0

0

12.7279i

0

0

0

27.0000

0

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
4.b	odd	2	1	CM by \(\Q(\sqrt{-1}) \)
8.b	even	2	1	inner
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	1024.4.b.c		2
4.b	odd	2	1	CM	1024.4.b.c		2
8.b	even	2	1	inner	1024.4.b.c		2
8.d	odd	2	1	inner	1024.4.b.c		2
16.e	even	4	2		1024.4.a.c		2
16.f	odd	4	2		1024.4.a.c		2
32.g	even	8	2		512.4.e.d	✓	2
32.g	even	8	2		512.4.e.e	yes	2
32.h	odd	8	2		512.4.e.d	✓	2
32.h	odd	8	2		512.4.e.e	yes	2

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
512.4.e.d	✓	2	32.g	even	8	2
512.4.e.d	✓	2	32.h	odd	8	2
512.4.e.e	yes	2	32.g	even	8	2
512.4.e.e	yes	2	32.h	odd	8	2
1024.4.a.c		2	16.e	even	4	2
1024.4.a.c		2	16.f	odd	4	2
1024.4.b.c		2	1.a	even	1	1	trivial
1024.4.b.c		2	4.b	odd	2	1	CM
1024.4.b.c		2	8.b	even	2	1	inner
1024.4.b.c		2	8.d	odd	2	1	inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(1024, [\chi])\):

\( T_{3} \)

\( T_{5}^{2} + 162 \)

\( T_{7} \)

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	\( T^{2} \)
$3$	\( T^{2} \)
$5$	\( T^{2} + 162 \)
$7$	\( T^{2} \)
$11$	\( T^{2} \)