Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [240,8,Mod(1,240)] 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("240.1"); S:= CuspForms(chi, 8); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(240, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 8, names="a")
 
Level: \( N \) \(=\) \( 240 = 2^{4} \cdot 3 \cdot 5 \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 240.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [1,0,-27,0,-125,0,420] 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: \(74.9724061162\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 15)
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)
 
\(f(q)\) \(=\) \( q - 27 q^{3} - 125 q^{5} + 420 q^{7} + 729 q^{9} + 2944 q^{11} - 11006 q^{13} + 3375 q^{15} - 16546 q^{17} + 25364 q^{19} - 11340 q^{21} + 5880 q^{23} + 15625 q^{25} - 19683 q^{27} + 163042 q^{29} + 201600 q^{31}+ \cdots + 2146176 q^{99}+O(q^{100}) \) 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 −27.0000 0 −125.000 0 420.000 0 729.000 0
\(n\): e.g. 2-40 or 80-90
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \( -1 \)
\(3\) \( +1 \)
\(5\) \( +1 \)

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 240.8.a.c 1
4.b odd 2 1 15.8.a.a ✓ 1
12.b even 2 1 45.8.a.g 1
20.d odd 2 1 75.8.a.c 1
20.e even 4 2 75.8.b.a 2
60.h even 2 1 225.8.a.a 1
60.l odd 4 2 225.8.b.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
15.8.a.a ✓ 1 4.b odd 2 1
45.8.a.g 1 12.b even 2 1
75.8.a.c 1 20.d odd 2 1
75.8.b.a 2 20.e even 4 2
225.8.a.a 1 60.h even 2 1
225.8.b.a 2 60.l odd 4 2
240.8.a.c 1 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7} - 420 \) acting on \(S_{8}^{\mathrm{new}}(\Gamma_0(240))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T \) Copy content Toggle raw display
$3$ \( T + 27 \) Copy content Toggle raw display
$5$ \( T + 125 \) Copy content Toggle raw display
$7$ \( T - 420 \) Copy content Toggle raw display
$11$ \( T - 2944 \) Copy content Toggle raw display
$13$ \( T + 11006 \) Copy content Toggle raw display
$17$ \( T + 16546 \) Copy content Toggle raw display
$19$ \( T - 25364 \) Copy content Toggle raw display
$23$ \( T - 5880 \) Copy content Toggle raw display
$29$ \( T - 163042 \) Copy content Toggle raw display
$31$ \( T - 201600 \) Copy content Toggle raw display
$37$ \( T - 120530 \) Copy content Toggle raw display
$41$ \( T + 115910 \) Copy content Toggle raw display
$43$ \( T - 19148 \) Copy content Toggle raw display
$47$ \( T + 841016 \) Copy content Toggle raw display
$53$ \( T - 501890 \) Copy content Toggle raw display
$59$ \( T - 1586176 \) Copy content Toggle raw display
$61$ \( T + 372962 \) Copy content Toggle raw display
$67$ \( T + 4561044 \) Copy content Toggle raw display
$71$ \( T + 1512832 \) Copy content Toggle raw display
$73$ \( T + 1522910 \) Copy content Toggle raw display
$79$ \( T + 4231920 \) Copy content Toggle raw display
$83$ \( T - 1854204 \) Copy content Toggle raw display
$89$ \( T + 6888174 \) Copy content Toggle raw display
$97$ \( T - 3700034 \) Copy content Toggle raw display
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