Properties

Label 256.2.a.e
Level $256$
Weight $2$
Character orbit 256.a
Self dual yes
Analytic conductor $2.044$
Analytic rank $0$
Dimension $2$
CM discriminant -8
Inner twists $4$

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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: \(2\)
Coefficient field: \(\Q(\sqrt{2}) \)
Defining polynomial: \(x^{2} - 2\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 128)
Fricke sign: \(-1\)
Sato-Tate group: $N(\mathrm{U}(1))$

$q$-expansion

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

\(f(q)\) \(=\) \( q + 2 \beta q^{3} + 5 q^{9} +O(q^{10})\) \( q + 2 \beta q^{3} + 5 q^{9} -2 \beta q^{11} + 6 q^{17} -6 \beta q^{19} -5 q^{25} + 4 \beta q^{27} -8 q^{33} + 6 q^{41} + 6 \beta q^{43} -7 q^{49} + 12 \beta q^{51} -24 q^{57} -10 \beta q^{59} -6 \beta q^{67} + 2 q^{73} -10 \beta q^{75} + q^{81} + 2 \beta q^{83} + 18 q^{89} -10 q^{97} -10 \beta q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 10q^{9} + O(q^{10}) \) \( 2q + 10q^{9} + 12q^{17} - 10q^{25} - 16q^{33} + 12q^{41} - 14q^{49} - 48q^{57} + 4q^{73} + 2q^{81} + 36q^{89} - 20q^{97} + O(q^{100}) \)

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 −2.82843 0 0 0 0 0 5.00000 0
1.2 0 2.82843 0 0 0 0 0 5.00000 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
8.d odd 2 1 CM by \(\Q(\sqrt{-2}) \)
4.b odd 2 1 inner
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 256.2.a.e 2
3.b odd 2 1 2304.2.a.t 2
4.b odd 2 1 inner 256.2.a.e 2
5.b even 2 1 6400.2.a.by 2
8.b even 2 1 inner 256.2.a.e 2
8.d odd 2 1 CM 256.2.a.e 2
12.b even 2 1 2304.2.a.t 2
16.e even 4 2 128.2.b.a 2
16.f odd 4 2 128.2.b.a 2
20.d odd 2 1 6400.2.a.by 2
24.f even 2 1 2304.2.a.t 2
24.h odd 2 1 2304.2.a.t 2
32.g even 8 2 1024.2.e.a 2
32.g even 8 2 1024.2.e.f 2
32.h odd 8 2 1024.2.e.a 2
32.h odd 8 2 1024.2.e.f 2
40.e odd 2 1 6400.2.a.by 2
40.f even 2 1 6400.2.a.by 2
48.i odd 4 2 1152.2.d.c 2
48.k even 4 2 1152.2.d.c 2
80.i odd 4 2 3200.2.f.o 4
80.j even 4 2 3200.2.f.o 4
80.k odd 4 2 3200.2.d.c 2
80.q even 4 2 3200.2.d.c 2
80.s even 4 2 3200.2.f.o 4
80.t odd 4 2 3200.2.f.o 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
128.2.b.a 2 16.e even 4 2
128.2.b.a 2 16.f odd 4 2
256.2.a.e 2 1.a even 1 1 trivial
256.2.a.e 2 4.b odd 2 1 inner
256.2.a.e 2 8.b even 2 1 inner
256.2.a.e 2 8.d odd 2 1 CM
1024.2.e.a 2 32.g even 8 2
1024.2.e.a 2 32.h odd 8 2
1024.2.e.f 2 32.g even 8 2
1024.2.e.f 2 32.h odd 8 2
1152.2.d.c 2 48.i odd 4 2
1152.2.d.c 2 48.k even 4 2
2304.2.a.t 2 3.b odd 2 1
2304.2.a.t 2 12.b even 2 1
2304.2.a.t 2 24.f even 2 1
2304.2.a.t 2 24.h odd 2 1
3200.2.d.c 2 80.k odd 4 2
3200.2.d.c 2 80.q even 4 2
3200.2.f.o 4 80.i odd 4 2
3200.2.f.o 4 80.j even 4 2
3200.2.f.o 4 80.s even 4 2
3200.2.f.o 4 80.t odd 4 2
6400.2.a.by 2 5.b even 2 1
6400.2.a.by 2 20.d odd 2 1
6400.2.a.by 2 40.e odd 2 1
6400.2.a.by 2 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(256))\):

\( T_{3}^{2} - 8 \)
\( T_{5} \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( -8 + T^{2} \)
$5$ \( T^{2} \)
$7$ \( T^{2} \)
$11$ \( -8 + T^{2} \)
$13$ \( T^{2} \)
$17$ \( ( -6 + T )^{2} \)
$19$ \( -72 + T^{2} \)
$23$ \( T^{2} \)
$29$ \( T^{2} \)
$31$ \( T^{2} \)
$37$ \( T^{2} \)
$41$ \( ( -6 + T )^{2} \)
$43$ \( -72 + T^{2} \)
$47$ \( T^{2} \)
$53$ \( T^{2} \)
$59$ \( -200 + T^{2} \)
$61$ \( T^{2} \)
$67$ \( -72 + T^{2} \)
$71$ \( T^{2} \)
$73$ \( ( -2 + T )^{2} \)
$79$ \( T^{2} \)
$83$ \( -8 + T^{2} \)
$89$ \( ( -18 + T )^{2} \)
$97$ \( ( 10 + T )^{2} \)
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