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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,-43,0,0,0,-974] 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: \(0\)
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)
 
\(f(q)\) \(=\) \( q - 43 q^{3} - 974 q^{7} - 338 q^{9} - 87 q^{11} + 14828 q^{13} - 35571 q^{17} - 20615 q^{19} + 41882 q^{21} - 22218 q^{23} + 108575 q^{27} - 5760 q^{29} - 302942 q^{31} + 3741 q^{33} - 199366 q^{37} - 637604 q^{39}+ \cdots + 29406 q^{99}+O(q^{100}) \) 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 −43.0000 0 0 0 −974.000 0 −338.000 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.f 1
4.b odd 2 1 50.8.a.h yes 1
5.b even 2 1 400.8.a.o 1
5.c odd 4 2 400.8.c.g 2
12.b even 2 1 450.8.a.j 1
20.d odd 2 1 50.8.a.a ✓ 1
20.e even 4 2 50.8.b.e 2
60.h even 2 1 450.8.a.q 1
60.l odd 4 2 450.8.c.i 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
50.8.a.a ✓ 1 20.d odd 2 1
50.8.a.h yes 1 4.b odd 2 1
50.8.b.e 2 20.e even 4 2
400.8.a.f 1 1.a even 1 1 trivial
400.8.a.o 1 5.b even 2 1
400.8.c.g 2 5.c odd 4 2
450.8.a.j 1 12.b even 2 1
450.8.a.q 1 60.h even 2 1
450.8.c.i 2 60.l odd 4 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3} + 43 \) 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
$3$ \( T + 43 \) Copy content Toggle raw display
$5$ \( T \) Copy content Toggle raw display
$7$ \( T + 974 \) Copy content Toggle raw display
$11$ \( T + 87 \) Copy content Toggle raw display
$13$ \( T - 14828 \) Copy content Toggle raw display
$17$ \( T + 35571 \) Copy content Toggle raw display
$19$ \( T + 20615 \) Copy content Toggle raw display
$23$ \( T + 22218 \) Copy content Toggle raw display
$29$ \( T + 5760 \) Copy content Toggle raw display
$31$ \( T + 302942 \) Copy content Toggle raw display
$37$ \( T + 199366 \) Copy content Toggle raw display
$41$ \( T + 668523 \) Copy content Toggle raw display
$43$ \( T - 143212 \) Copy content Toggle raw display
$47$ \( T - 338316 \) Copy content Toggle raw display
$53$ \( T + 1094322 \) Copy content Toggle raw display
$59$ \( T - 2135520 \) Copy content Toggle raw display
$61$ \( T + 1939318 \) Copy content Toggle raw display
$67$ \( T - 3348751 \) Copy content Toggle raw display
$71$ \( T + 3005652 \) Copy content Toggle raw display
$73$ \( T + 3048397 \) Copy content Toggle raw display
$79$ \( T + 5485130 \) Copy content Toggle raw display
$83$ \( T + 5205933 \) Copy content Toggle raw display
$89$ \( T + 832665 \) Copy content Toggle raw display
$97$ \( T - 5314754 \) Copy content Toggle raw display
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