Newform orbit 1024.2.a.d

• Label: 1024.2.a.d
• Level: $1024$
• Weight: $2$
• Character orbit: 1024.a
• Self dual: yes
• Analytic conductor: $8.177$
• Analytic rank: $0$
• Dimension: $2$
• CM discriminant: -4
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 1024 = 2^{10} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1024.a (trivial)

Newform invariants

Self dual:	yes
Analytic conductor:	\(8.17668116698\)
Analytic rank:	\(0\)
Dimension:	\(2\)
Coefficient field:	\(\Q(\sqrt{2}) \)
Defining polynomial:	\( x^{2} - 2 \)
Coefficient ring:	\(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	no (minimal twist has level 512)
Fricke sign:	\(-1\)
Sato-Tate group:	$N(\mathrm{U}(1))$

$q$-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 + 3*b * q^5 - 3 * q^9 + b * q^13 + 8 * q^17 + 13 * q^25 - 7*b * q^29 + 5*b * q^37 + 8 * q^41 - 9*b * q^45 - 7 * q^49 - 5*b * q^53 - b * q^61 + 6 * q^65 + 6 * q^73 + 9 * q^81 + 24*b * q^85 - 10 * q^89 + 8 * q^97
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 6 q^{9} + 16 q^{17} + 26 q^{25} + 16 q^{41} - 14 q^{49} + 12 q^{65} + 12 q^{73} + 18 q^{81} - 20 q^{89} + 16 q^{97}+O(q^{100}) \)

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  

\(\iota_m(\nu)\)

\( a_{2} \)

\( a_{3} \)

\( a_{4} \)

\( a_{5} \)

\( a_{6} \)

\( a_{7} \)

\( a_{8} \)

\( a_{9} \)

\( a_{10} \)

1.1

−1.41421
1.41421

0

0

0

−4.24264

0

0

0

−3.00000

0

1.2

0

0

0

4.24264

0

0

0

−3.00000

0

Atkin-Lehner signs

\( p \)	Sign
\(2\)	\( -1 \)

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	1024.2.a.d		2
3.b	odd	2	1		9216.2.a.g		2
4.b	odd	2	1	CM	1024.2.a.d		2
8.b	even	2	1	inner	1024.2.a.d		2
8.d	odd	2	1	inner	1024.2.a.d		2
12.b	even	2	1		9216.2.a.g		2
16.e	even	4	2		1024.2.b.d		2
16.f	odd	4	2		1024.2.b.d		2
24.f	even	2	1		9216.2.a.g		2
24.h	odd	2	1		9216.2.a.g		2
32.g	even	8	2		512.2.e.c	✓	2
32.g	even	8	2		512.2.e.f	yes	2
32.h	odd	8	2		512.2.e.c	✓	2
32.h	odd	8	2		512.2.e.f	yes	2
96.o	even	8	2		4608.2.k.b		2
96.o	even	8	2		4608.2.k.w		2
96.p	odd	8	2		4608.2.k.b		2
96.p	odd	8	2		4608.2.k.w		2

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
512.2.e.c	✓	2	32.g	even	8	2
512.2.e.c	✓	2	32.h	odd	8	2
512.2.e.f	yes	2	32.g	even	8	2
512.2.e.f	yes	2	32.h	odd	8	2
1024.2.a.d		2	1.a	even	1	1	trivial
1024.2.a.d		2	4.b	odd	2	1	CM
1024.2.a.d		2	8.b	even	2	1	inner
1024.2.a.d		2	8.d	odd	2	1	inner
1024.2.b.d		2	16.e	even	4	2
1024.2.b.d		2	16.f	odd	4	2
4608.2.k.b		2	96.o	even	8	2
4608.2.k.b		2	96.p	odd	8	2
4608.2.k.w		2	96.o	even	8	2
4608.2.k.w		2	96.p	odd	8	2
9216.2.a.g		2	3.b	odd	2	1
9216.2.a.g		2	12.b	even	2	1
9216.2.a.g		2	24.f	even	2	1
9216.2.a.g		2	24.h	odd	2	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(1024))\):

\( T_{3} \)

\( T_{5}^{2} - 18 \)

\( T_{7} \)

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	\( T^{2} \)
$3$	\( T^{2} \)
$5$	\( T^{2} - 18 \)
$7$	\( T^{2} \)
$11$	\( T^{2} \)