Newform orbit 512.2.a.f

• Label: 512.2.a.f
• Level: $512$
• Weight: $2$
• Character orbit: 512.a
• Self dual: yes
• Analytic conductor: $4.088$
• Analytic rank: $0$
• Dimension: $2$
• CM discriminant: -8
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 512 = 2^{9} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	512.a (trivial)

Newform invariants

Self dual:	yes
Analytic conductor:	\(4.08834058349\)
Analytic rank:	\(0\)
Dimension:	\(2\)
Coefficient field:	\(\Q(\sqrt{2}) \)
Defining polynomial:	\( x^{2} - 2 \)
Coefficient ring:	\(\Z[a_1, a_2, a_3]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	yes
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.

\(f(q)\)	\(=\)	\( q + (\beta + 2) q^{3} + (4 \beta + 3) q^{9}+O(q^{10}) \)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 4 q^{3} + 6 q^{9}+O(q^{10}) \)

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

0.585786

0

0

0

0

0

−2.65685

0

1.2

0

3.41421

0

0

0

0

0

8.65685

0

Atkin-Lehner signs

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

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
8.d	odd	2	1	CM by \(\Q(\sqrt{-2}) \)

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	512.2.a.f	yes	2
3.b	odd	2	1		4608.2.a.i		2
4.b	odd	2	1		512.2.a.a	✓	2
8.b	even	2	1		512.2.a.a	✓	2
8.d	odd	2	1	CM	512.2.a.f	yes	2
12.b	even	2	1		4608.2.a.k		2
16.e	even	4	2		512.2.b.c		4
16.f	odd	4	2		512.2.b.c		4
24.f	even	2	1		4608.2.a.i		2
24.h	odd	2	1		4608.2.a.k		2
32.g	even	8	2		1024.2.e.g		4
32.g	even	8	2		1024.2.e.o		4
32.h	odd	8	2		1024.2.e.g		4
32.h	odd	8	2		1024.2.e.o		4
48.i	odd	4	2		4608.2.d.k		4
48.k	even	4	2		4608.2.d.k		4

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
512.2.a.a	✓	2	4.b	odd	2	1
512.2.a.a	✓	2	8.b	even	2	1
512.2.a.f	yes	2	1.a	even	1	1	trivial
512.2.a.f	yes	2	8.d	odd	2	1	CM
512.2.b.c		4	16.e	even	4	2
512.2.b.c		4	16.f	odd	4	2
1024.2.e.g		4	32.g	even	8	2
1024.2.e.g		4	32.h	odd	8	2
1024.2.e.o		4	32.g	even	8	2
1024.2.e.o		4	32.h	odd	8	2
4608.2.a.i		2	3.b	odd	2	1
4608.2.a.i		2	24.f	even	2	1
4608.2.a.k		2	12.b	even	2	1
4608.2.a.k		2	24.h	odd	2	1
4608.2.d.k		4	48.i	odd	4	2
4608.2.d.k		4	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 \(S_{2}^{\mathrm{new}}(\Gamma_0(512))\):

\( T_{3}^{2} - 4T_{3} + 2 \)

\( T_{5} \)

\( T_{7} \)

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	\( T^{2} \)
$3$	\( T^{2} - 4T + 2 \)
$5$	\( T^{2} \)
$7$	\( T^{2} \)
$11$	\( T^{2} - 4T - 14 \)