Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [6400,2,Mod(1,6400)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("6400.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(6400, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 6400 = 2^{8} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6400.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [1,0,0,0,0,0,0,0,-3,0,0,0,4,0,0,0,2,0,0,0,0,0,0,0,0,0,0,0,-4, 0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(31)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(51.1042572936\)
Analytic rank: \(1\)
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

Copy content comment:q-expansion
 
Copy content sage:f.q_expansion() # note that sage often uses an isomorphic number field
 
Copy content gp:mfcoefs(f, 20)
 
\(f(q)\) \(=\) \( q - 3 q^{9} + 4 q^{13} + 2 q^{17} - 4 q^{29} - 12 q^{37} - 10 q^{41} - 7 q^{49} + 4 q^{53} + 12 q^{61} + 6 q^{73} + 9 q^{81} + 10 q^{89} + 18 q^{97}+O(q^{100}) \) Copy content Toggle raw display

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.

Copy content comment:embeddings in the coefficient field
 
Copy content gp:mfembed(f)
 
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 0 0 0 0 −3.00000 0
\(n\): e.g. 2-40 or 80-90
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \( +1 \)
\(5\) \( +1 \)

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 6400.2.a.m 1
4.b odd 2 1 CM 6400.2.a.m 1
5.b even 2 1 256.2.a.b 1
8.b even 2 1 6400.2.a.l 1
8.d odd 2 1 6400.2.a.l 1
15.d odd 2 1 2304.2.a.p 1
16.e even 4 2 3200.2.d.e 2
16.f odd 4 2 3200.2.d.e 2
20.d odd 2 1 256.2.a.b 1
40.e odd 2 1 256.2.a.c 1
40.f even 2 1 256.2.a.c 1
60.h even 2 1 2304.2.a.p 1
80.i odd 4 2 3200.2.f.c 2
80.j even 4 2 3200.2.f.d 2
80.k odd 4 2 128.2.b.b ✓ 2
80.q even 4 2 128.2.b.b ✓ 2
80.s even 4 2 3200.2.f.c 2
80.t odd 4 2 3200.2.f.d 2
120.i odd 2 1 2304.2.a.a 1
120.m even 2 1 2304.2.a.a 1
160.y odd 8 4 1024.2.e.k 4
160.z even 8 4 1024.2.e.k 4
240.t even 4 2 1152.2.d.d 2
240.bm odd 4 2 1152.2.d.d 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
128.2.b.b ✓ 2 80.k odd 4 2
128.2.b.b ✓ 2 80.q even 4 2
256.2.a.b 1 5.b even 2 1
256.2.a.b 1 20.d odd 2 1
256.2.a.c 1 40.e odd 2 1
256.2.a.c 1 40.f even 2 1
1024.2.e.k 4 160.y odd 8 4
1024.2.e.k 4 160.z even 8 4
1152.2.d.d 2 240.t even 4 2
1152.2.d.d 2 240.bm odd 4 2
2304.2.a.a 1 120.i odd 2 1
2304.2.a.a 1 120.m even 2 1
2304.2.a.p 1 15.d odd 2 1
2304.2.a.p 1 60.h even 2 1
3200.2.d.e 2 16.e even 4 2
3200.2.d.e 2 16.f odd 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 8.b even 2 1
6400.2.a.l 1 8.d odd 2 1
6400.2.a.m 1 1.a even 1 1 trivial
6400.2.a.m 1 4.b odd 2 1 CM

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(6400))\):

\( T_{3} \) Copy content Toggle raw display
\( T_{7} \) Copy content Toggle raw display
\( T_{11} \) Copy content Toggle raw display
\( T_{13} - 4 \) Copy content Toggle raw display
\( T_{17} - 2 \) Copy content Toggle raw display
\( T_{29} + 4 \) Copy content Toggle raw display
\( T_{31} \) Copy content Toggle raw display

Hecke characteristic polynomials

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