Properties

Label 256.2.a.c
Level $256$
Weight $2$
Character orbit 256.a
Self dual yes
Analytic conductor $2.044$
Analytic rank $0$
Dimension $1$
CM discriminant -4
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 256 = 2^{8} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 256.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(2.04417029174\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 128)
Fricke sign: \(-1\)
Sato-Tate group: $N(\mathrm{U}(1))$

$q$-expansion

\(f(q)\) \(=\) \( q + 4q^{5} - 3q^{9} + O(q^{10}) \) \( q + 4q^{5} - 3q^{9} + 4q^{13} - 2q^{17} + 11q^{25} + 4q^{29} - 12q^{37} - 10q^{41} - 12q^{45} - 7q^{49} + 4q^{53} - 12q^{61} + 16q^{65} - 6q^{73} + 9q^{81} - 8q^{85} + 10q^{89} - 18q^{97} + O(q^{100}) \)

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
0
0 0 0 4.00000 0 0 0 −3.00000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

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 \(\Q(\sqrt{-1}) \)

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 256.2.a.c 1
3.b odd 2 1 2304.2.a.a 1
4.b odd 2 1 CM 256.2.a.c 1
5.b even 2 1 6400.2.a.l 1
8.b even 2 1 256.2.a.b 1
8.d odd 2 1 256.2.a.b 1
12.b even 2 1 2304.2.a.a 1
16.e even 4 2 128.2.b.b 2
16.f odd 4 2 128.2.b.b 2
20.d odd 2 1 6400.2.a.l 1
24.f even 2 1 2304.2.a.p 1
24.h odd 2 1 2304.2.a.p 1
32.g even 8 4 1024.2.e.k 4
32.h odd 8 4 1024.2.e.k 4
40.e odd 2 1 6400.2.a.m 1
40.f even 2 1 6400.2.a.m 1
48.i odd 4 2 1152.2.d.d 2
48.k even 4 2 1152.2.d.d 2
80.i odd 4 2 3200.2.f.c 2
80.j even 4 2 3200.2.f.d 2
80.k odd 4 2 3200.2.d.e 2
80.q even 4 2 3200.2.d.e 2
80.s even 4 2 3200.2.f.c 2
80.t odd 4 2 3200.2.f.d 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
128.2.b.b 2 16.e even 4 2
128.2.b.b 2 16.f odd 4 2
256.2.a.b 1 8.b even 2 1
256.2.a.b 1 8.d odd 2 1
256.2.a.c 1 1.a even 1 1 trivial
256.2.a.c 1 4.b odd 2 1 CM
1024.2.e.k 4 32.g even 8 4
1024.2.e.k 4 32.h odd 8 4
1152.2.d.d 2 48.i odd 4 2
1152.2.d.d 2 48.k even 4 2
2304.2.a.a 1 3.b odd 2 1
2304.2.a.a 1 12.b even 2 1
2304.2.a.p 1 24.f even 2 1
2304.2.a.p 1 24.h odd 2 1
3200.2.d.e 2 80.k odd 4 2
3200.2.d.e 2 80.q even 4 2
3200.2.f.c 2 80.i odd 4 2
3200.2.f.c 2 80.s even 4 2
3200.2.f.d 2 80.j even 4 2
3200.2.f.d 2 80.t odd 4 2
6400.2.a.l 1 5.b even 2 1
6400.2.a.l 1 20.d odd 2 1
6400.2.a.m 1 40.e odd 2 1
6400.2.a.m 1 40.f even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(256))\):

\( T_{3} \)
\( T_{5} - 4 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T \)
$3$ \( T \)
$5$ \( -4 + T \)
$7$ \( T \)
$11$ \( T \)
$13$ \( -4 + T \)
$17$ \( 2 + T \)
$19$ \( T \)
$23$ \( T \)
$29$ \( -4 + T \)
$31$ \( T \)
$37$ \( 12 + T \)
$41$ \( 10 + T \)
$43$ \( T \)
$47$ \( T \)
$53$ \( -4 + T \)
$59$ \( T \)
$61$ \( 12 + T \)
$67$ \( T \)
$71$ \( T \)
$73$ \( 6 + T \)
$79$ \( T \)
$83$ \( T \)
$89$ \( -10 + T \)
$97$ \( 18 + T \)
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