Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [15,8,Mod(1,15)] 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("15.1"); S:= CuspForms(chi, 8); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(15, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 8, names="a")
 
Level: \( N \) \(=\) \( 15 = 3 \cdot 5 \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 15.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [1,-22] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(2)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(4.68577538226\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
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)
 
\(f(q)\) \(=\) \( q - 22 q^{2} + 27 q^{3} + 356 q^{4} - 125 q^{5} - 594 q^{6} - 420 q^{7} - 5016 q^{8} + 729 q^{9} + 2750 q^{10} - 2944 q^{11} + 9612 q^{12} - 11006 q^{13} + 9240 q^{14} - 3375 q^{15} + 64784 q^{16} - 16546 q^{17}+ \cdots - 2146176 q^{99}+O(q^{100}) \) 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
−22.0000 27.0000 356.000 −125.000 −594.000 −420.000 −5016.00 729.000 2750.00
\(n\): e.g. 2-40 or 80-90
Significant digits:
Format:

Atkin-Lehner signs

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

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 15.8.a.a ✓ 1
3.b odd 2 1 45.8.a.g 1
4.b odd 2 1 240.8.a.c 1
5.b even 2 1 75.8.a.c 1
5.c odd 4 2 75.8.b.a 2
15.d odd 2 1 225.8.a.a 1
15.e even 4 2 225.8.b.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
15.8.a.a ✓ 1 1.a even 1 1 trivial
45.8.a.g 1 3.b odd 2 1
75.8.a.c 1 5.b even 2 1
75.8.b.a 2 5.c odd 4 2
225.8.a.a 1 15.d odd 2 1
225.8.b.a 2 15.e even 4 2
240.8.a.c 1 4.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2} + 22 \) acting on \(S_{8}^{\mathrm{new}}(\Gamma_0(15))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T + 22 \) 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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