Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [400,8,Mod(1,400)] 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("400.1"); S:= CuspForms(chi, 8); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(400, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 8, names="a")
 
Level: \( N \) \(=\) \( 400 = 2^{4} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 400.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [1,0,57,0,0,0,-1174] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(124.954010194\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 50)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-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)
 
\(f(q)\) \(=\) \( q + 57 q^{3} - 1174 q^{7} + 1062 q^{9} + 7563 q^{11} - 5372 q^{13} - 24021 q^{17} + 51235 q^{19} - 66918 q^{21} - 57618 q^{23} - 64125 q^{27} + 47040 q^{29} + 192358 q^{31} + 431091 q^{33} - 197066 q^{37}+ \cdots + 8031906 q^{99}+O(q^{100}) \) 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 57.0000 0 0 0 −1174.00 0 1062.00 0
\(n\): e.g. 2-40 or 80-90
Significant digits:
Format:

Atkin-Lehner signs

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

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 400.8.a.q 1
4.b odd 2 1 50.8.a.f yes 1
5.b even 2 1 400.8.a.d 1
5.c odd 4 2 400.8.c.d 2
12.b even 2 1 450.8.a.l 1
20.d odd 2 1 50.8.a.c ✓ 1
20.e even 4 2 50.8.b.b 2
60.h even 2 1 450.8.a.p 1
60.l odd 4 2 450.8.c.q 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
50.8.a.c ✓ 1 20.d odd 2 1
50.8.a.f yes 1 4.b odd 2 1
50.8.b.b 2 20.e even 4 2
400.8.a.d 1 5.b even 2 1
400.8.a.q 1 1.a even 1 1 trivial
400.8.c.d 2 5.c odd 4 2
450.8.a.l 1 12.b even 2 1
450.8.a.p 1 60.h even 2 1
450.8.c.q 2 60.l odd 4 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3} - 57 \) acting on \(S_{8}^{\mathrm{new}}(\Gamma_0(400))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T \) Copy content Toggle raw display
$3$ \( T - 57 \) Copy content Toggle raw display
$5$ \( T \) Copy content Toggle raw display
$7$ \( T + 1174 \) Copy content Toggle raw display
$11$ \( T - 7563 \) Copy content Toggle raw display
$13$ \( T + 5372 \) Copy content Toggle raw display
$17$ \( T + 24021 \) Copy content Toggle raw display
$19$ \( T - 51235 \) Copy content Toggle raw display
$23$ \( T + 57618 \) Copy content Toggle raw display
$29$ \( T - 47040 \) Copy content Toggle raw display
$31$ \( T - 192358 \) Copy content Toggle raw display
$37$ \( T + 197066 \) Copy content Toggle raw display
$41$ \( T + 237723 \) Copy content Toggle raw display
$43$ \( T - 653012 \) Copy content Toggle raw display
$47$ \( T + 826884 \) Copy content Toggle raw display
$53$ \( T + 569022 \) Copy content Toggle raw display
$59$ \( T + 1501080 \) Copy content Toggle raw display
$61$ \( T + 2068918 \) Copy content Toggle raw display
$67$ \( T + 3444349 \) Copy content Toggle raw display
$71$ \( T + 4121052 \) Copy content Toggle raw display
$73$ \( T - 83653 \) Copy content Toggle raw display
$79$ \( T + 1454030 \) Copy content Toggle raw display
$83$ \( T - 1626567 \) Copy content Toggle raw display
$89$ \( T - 6004335 \) Copy content Toggle raw display
$97$ \( T + 3411746 \) Copy content Toggle raw display
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