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Properties

Label 24.12.a.c
Level $24$
Weight $12$
Character orbit 24.a
Self dual yes
Analytic conductor $18.440$
Analytic rank $1$
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] = [24,12,Mod(1,24)] 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("24.1"); S:= CuspForms(chi, 12); N := Newforms(S);
 
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, 12, names="a")
 
Level: \( N \) \(=\) \( 24 = 2^{3} \cdot 3 \)
Weight: \( k \) \(=\) \( 12 \)
Character orbit: \([\chi]\) \(=\) 24.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [1,0,243,0,1870] 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: \(18.4402363334\)
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)
 
\(f(q)\) \(=\) \( q + 243 q^{3} + 1870 q^{5} - 72312 q^{7} + 59049 q^{9} + 147940 q^{11} - 1562858 q^{13} + 454410 q^{15} - 145774 q^{17} + 1096796 q^{19} - 17571816 q^{21} - 60014264 q^{23} - 45331225 q^{25} + 14348907 q^{27}+ \cdots + 8735709060 q^{99}+O(q^{100}) \) 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
0 243.000 0 1870.00 0 −72312.0 0 59049.0 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 24.12.a.c ✓ 1
3.b odd 2 1 72.12.a.a 1
4.b odd 2 1 48.12.a.b 1
8.b even 2 1 192.12.a.e 1
8.d odd 2 1 192.12.a.o 1
12.b even 2 1 144.12.a.g 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
24.12.a.c ✓ 1 1.a even 1 1 trivial
48.12.a.b 1 4.b odd 2 1
72.12.a.a 1 3.b odd 2 1
144.12.a.g 1 12.b even 2 1
192.12.a.e 1 8.b even 2 1
192.12.a.o 1 8.d odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5} - 1870 \) acting on \(S_{12}^{\mathrm{new}}(\Gamma_0(24))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T \) Copy content Toggle raw display
$3$ \( T - 243 \) Copy content Toggle raw display
$5$ \( T - 1870 \) Copy content Toggle raw display
$7$ \( T + 72312 \) Copy content Toggle raw display
$11$ \( T - 147940 \) Copy content Toggle raw display
$13$ \( T + 1562858 \) Copy content Toggle raw display
$17$ \( T + 145774 \) Copy content Toggle raw display
$19$ \( T - 1096796 \) Copy content Toggle raw display
$23$ \( T + 60014264 \) Copy content Toggle raw display
$29$ \( T + 19626954 \) Copy content Toggle raw display
$31$ \( T + 239950480 \) Copy content Toggle raw display
$37$ \( T - 488238078 \) Copy content Toggle raw display
$41$ \( T - 47066010 \) Copy content Toggle raw display
$43$ \( T - 428866948 \) Copy content Toggle raw display
$47$ \( T - 450903216 \) Copy content Toggle raw display
$53$ \( T - 4336685950 \) Copy content Toggle raw display
$59$ \( T + 8937556460 \) Copy content Toggle raw display
$61$ \( T - 4673884486 \) Copy content Toggle raw display
$67$ \( T - 7498937612 \) Copy content Toggle raw display
$71$ \( T + 27032101480 \) Copy content Toggle raw display
$73$ \( T - 11676141658 \) Copy content Toggle raw display
$79$ \( T - 2478876544 \) Copy content Toggle raw display
$83$ \( T - 42745596956 \) Copy content Toggle raw display
$89$ \( T + 93270772662 \) Copy content Toggle raw display
$97$ \( T - 118032786914 \) Copy content Toggle raw display
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