Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [63,10,Mod(1,63)] 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("63.1"); S:= CuspForms(chi, 10); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(63, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 10, names="a")
 
Level: \( N \) \(=\) \( 63 = 3^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 10 \)
Character orbit: \([\chi]\) \(=\) 63.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [1,24] 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: \(32.4472576783\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 21)
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)
 
\(f(q)\) \(=\) \( q + 24 q^{2} + 64 q^{4} + 144 q^{5} + 2401 q^{7} - 10752 q^{8} + 3456 q^{10} + 15030 q^{11} - 151486 q^{13} + 57624 q^{14} - 290816 q^{16} + 350448 q^{17} - 691108 q^{19} + 9216 q^{20} + 360720 q^{22}+ \cdots + 138355224 q^{98}+O(q^{100}) \) 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
24.0000 0 64.0000 144.000 0 2401.00 −10752.0 0 3456.00
\(n\): e.g. 2-40 or 80-90
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \( -1 \)
\(7\) \( -1 \)

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 63.10.a.a 1
3.b odd 2 1 21.10.a.a ✓ 1
12.b even 2 1 336.10.a.d 1
21.c even 2 1 147.10.a.b 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
21.10.a.a ✓ 1 3.b odd 2 1
63.10.a.a 1 1.a even 1 1 trivial
147.10.a.b 1 21.c even 2 1
336.10.a.d 1 12.b even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2} - 24 \) acting on \(S_{10}^{\mathrm{new}}(\Gamma_0(63))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T - 24 \) Copy content Toggle raw display
$3$ \( T \) Copy content Toggle raw display
$5$ \( T - 144 \) Copy content Toggle raw display
$7$ \( T - 2401 \) Copy content Toggle raw display
$11$ \( T - 15030 \) Copy content Toggle raw display
$13$ \( T + 151486 \) Copy content Toggle raw display
$17$ \( T - 350448 \) Copy content Toggle raw display
$19$ \( T + 691108 \) Copy content Toggle raw display
$23$ \( T + 892458 \) Copy content Toggle raw display
$29$ \( T + 1648518 \) Copy content Toggle raw display
$31$ \( T + 3734296 \) Copy content Toggle raw display
$37$ \( T + 11471902 \) Copy content Toggle raw display
$41$ \( T + 13985724 \) Copy content Toggle raw display
$43$ \( T - 16794524 \) Copy content Toggle raw display
$47$ \( T - 14012052 \) Copy content Toggle raw display
$53$ \( T - 97439910 \) Copy content Toggle raw display
$59$ \( T + 110798304 \) Copy content Toggle raw display
$61$ \( T + 93816682 \) Copy content Toggle raw display
$67$ \( T + 122446456 \) Copy content Toggle raw display
$71$ \( T + 206197398 \) Copy content Toggle raw display
$73$ \( T - 250337558 \) Copy content Toggle raw display
$79$ \( T + 38314852 \) Copy content Toggle raw display
$83$ \( T - 514086924 \) Copy content Toggle raw display
$89$ \( T - 1061294916 \) Copy content Toggle raw display
$97$ \( T + 73841578 \) Copy content Toggle raw display
show more
show less