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Properties

Label 24.13.h.a
Level $24$
Weight $13$
Character orbit 24.h
Self dual yes
Analytic conductor $21.936$
Analytic rank $0$
Dimension $1$
CM discriminant -24
Inner twists $2$

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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [24,13,Mod(5,24)] 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("24.5"); S:= CuspForms(chi, 13); 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, 1, 1])) N = Newforms(chi, 13, names="a")
 
Level: \( N \) \(=\) \( 24 = 2^{3} \cdot 3 \)
Weight: \( k \) \(=\) \( 13 \)
Character orbit: \([\chi]\) \(=\) 24.h (of order \(2\), degree \(1\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [1,-64] 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: \(21.9358516146\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

$q$-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)
 
\(f(q)\) \(=\) \( q - 64 q^{2} - 729 q^{3} + 4096 q^{4} + 11086 q^{5} + 46656 q^{6} - 14398 q^{7} - 262144 q^{8} + 531441 q^{9} - 709504 q^{10} - 3373778 q^{11} - 2985984 q^{12} + 921472 q^{14} - 8081694 q^{15} + 16777216 q^{16}+ \cdots - 1792963954098 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/24\mathbb{Z}\right)^\times\).

\(n\) \(7\) \(13\) \(17\)
\(\chi(n)\) \(1\) \(-1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
5.1
0
−64.0000 −729.000 4096.00 11086.0 46656.0 −14398.0 −262144. 531441. −709504.
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
24.h odd 2 1 CM by \(\Q(\sqrt{-6}) \)

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 24.13.h.a ✓ 1
3.b odd 2 1 24.13.h.b yes 1
4.b odd 2 1 96.13.h.b 1
8.b even 2 1 24.13.h.b yes 1
8.d odd 2 1 96.13.h.a 1
12.b even 2 1 96.13.h.a 1
24.f even 2 1 96.13.h.b 1
24.h odd 2 1 CM 24.13.h.a ✓ 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
24.13.h.a ✓ 1 1.a even 1 1 trivial
24.13.h.a ✓ 1 24.h odd 2 1 CM
24.13.h.b yes 1 3.b odd 2 1
24.13.h.b yes 1 8.b even 2 1
96.13.h.a 1 8.d odd 2 1
96.13.h.a 1 12.b even 2 1
96.13.h.b 1 4.b odd 2 1
96.13.h.b 1 24.f even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5} - 11086 \) acting on \(S_{13}^{\mathrm{new}}(24, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T + 64 \) Copy content Toggle raw display
$3$ \( T + 729 \) Copy content Toggle raw display
$5$ \( T - 11086 \) Copy content Toggle raw display
$7$ \( T + 14398 \) Copy content Toggle raw display
$11$ \( T + 3373778 \) Copy content Toggle raw display
$13$ \( T \) Copy content Toggle raw display
$17$ \( T \) Copy content Toggle raw display
$19$ \( T \) Copy content Toggle raw display
$23$ \( T \) Copy content Toggle raw display
$29$ \( T - 1188324142 \) Copy content Toggle raw display
$31$ \( T - 1215113762 \) Copy content Toggle raw display
$37$ \( T \) Copy content Toggle raw display
$41$ \( T \) Copy content Toggle raw display
$43$ \( T \) Copy content Toggle raw display
$47$ \( T \) Copy content Toggle raw display
$53$ \( T - 42850627342 \) Copy content Toggle raw display
$59$ \( T - 73663302382 \) Copy content Toggle raw display
$61$ \( T \) Copy content Toggle raw display
$67$ \( T \) Copy content Toggle raw display
$71$ \( T \) Copy content Toggle raw display
$73$ \( T - 152079469922 \) Copy content Toggle raw display
$79$ \( T - 307388924642 \) Copy content Toggle raw display
$83$ \( T - 521912747662 \) Copy content Toggle raw display
$89$ \( T \) Copy content Toggle raw display
$97$ \( T - 571683047042 \) Copy content Toggle raw display
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