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Properties

Label 24.8.a.c
Level $24$
Weight $8$
Character orbit 24.a
Self dual yes
Analytic conductor $7.497$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

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Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [24,8,Mod(1,24)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(24, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 8, names="a")
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("24.1"); S:= CuspForms(chi, 8); N := Newforms(S);
 
Level: \( N \) \(=\) \( 24 = 2^{3} \cdot 3 \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 24.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [1,0,27,0,110] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(7.49724061162\)
Analytic rank: \(0\)
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)
 
\(f(q)\) \(=\) \( q + 27 q^{3} + 110 q^{5} + 504 q^{7} + 729 q^{9} + 3812 q^{11} + 9574 q^{13} + 2970 q^{15} + 26098 q^{17} - 38308 q^{19} + 13608 q^{21} - 71128 q^{23} - 66025 q^{25} + 19683 q^{27} + 74262 q^{29} - 275680 q^{31}+ \cdots + 2778948 q^{99}+O(q^{100}) \) 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 0 110.000 0 504.000 0 729.000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

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

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 24.8.a.c ✓ 1
3.b odd 2 1 72.8.a.b 1
4.b odd 2 1 48.8.a.c 1
5.b even 2 1 600.8.a.a 1
5.c odd 4 2 600.8.f.d 2
8.b even 2 1 192.8.a.c 1
8.d odd 2 1 192.8.a.k 1
12.b even 2 1 144.8.a.d 1
24.f even 2 1 576.8.a.s 1
24.h odd 2 1 576.8.a.t 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
24.8.a.c ✓ 1 1.a even 1 1 trivial
48.8.a.c 1 4.b odd 2 1
72.8.a.b 1 3.b odd 2 1
144.8.a.d 1 12.b even 2 1
192.8.a.c 1 8.b even 2 1
192.8.a.k 1 8.d odd 2 1
576.8.a.s 1 24.f even 2 1
576.8.a.t 1 24.h odd 2 1
600.8.a.a 1 5.b even 2 1
600.8.f.d 2 5.c odd 4 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5} - 110 \) acting on \(S_{8}^{\mathrm{new}}(\Gamma_0(24))\). 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
$5$ \( T - 110 \) Copy content Toggle raw display
$7$ \( T - 504 \) Copy content Toggle raw display
$11$ \( T - 3812 \) Copy content Toggle raw display
$13$ \( T - 9574 \) Copy content Toggle raw display
$17$ \( T - 26098 \) Copy content Toggle raw display
$19$ \( T + 38308 \) Copy content Toggle raw display
$23$ \( T + 71128 \) Copy content Toggle raw display
$29$ \( T - 74262 \) Copy content Toggle raw display
$31$ \( T + 275680 \) Copy content Toggle raw display
$37$ \( T + 266610 \) Copy content Toggle raw display
$41$ \( T - 684762 \) Copy content Toggle raw display
$43$ \( T - 245956 \) Copy content Toggle raw display
$47$ \( T - 478800 \) Copy content Toggle raw display
$53$ \( T + 569410 \) Copy content Toggle raw display
$59$ \( T + 1525324 \) Copy content Toggle raw display
$61$ \( T + 2640458 \) Copy content Toggle raw display
$67$ \( T - 1416236 \) Copy content Toggle raw display
$71$ \( T + 3511304 \) Copy content Toggle raw display
$73$ \( T - 4738618 \) Copy content Toggle raw display
$79$ \( T - 4661488 \) Copy content Toggle raw display
$83$ \( T + 5729252 \) Copy content Toggle raw display
$89$ \( T - 11993514 \) Copy content Toggle raw display
$97$ \( T - 7150754 \) Copy content Toggle raw display
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