Properties

Label 24.5.h.b
Level $24$
Weight $5$
Character orbit 24.h
Self dual yes
Analytic conductor $2.481$
Analytic rank $0$
Dimension $1$
CM discriminant -24
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [24,5,Mod(5,24)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(24, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1, 1]))
 
N = Newforms(chi, 5, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("24.5");
 
S:= CuspForms(chi, 5);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 24 = 2^{3} \cdot 3 \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 24.h (of order \(2\), degree \(1\), minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(2.48087911401\)
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

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \( q + 4 q^{2} + 9 q^{3} + 16 q^{4} - 46 q^{5} + 36 q^{6} + 2 q^{7} + 64 q^{8} + 81 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + 4 q^{2} + 9 q^{3} + 16 q^{4} - 46 q^{5} + 36 q^{6} + 2 q^{7} + 64 q^{8} + 81 q^{9} - 184 q^{10} - 142 q^{11} + 144 q^{12} + 8 q^{14} - 414 q^{15} + 256 q^{16} + 324 q^{18} - 736 q^{20} + 18 q^{21} - 568 q^{22} + 576 q^{24} + 1491 q^{25} + 729 q^{27} + 32 q^{28} + 818 q^{29} - 1656 q^{30} - 478 q^{31} + 1024 q^{32} - 1278 q^{33} - 92 q^{35} + 1296 q^{36} - 2944 q^{40} + 72 q^{42} - 2272 q^{44} - 3726 q^{45} + 2304 q^{48} - 2397 q^{49} + 5964 q^{50} + 3218 q^{53} + 2916 q^{54} + 6532 q^{55} + 128 q^{56} + 3272 q^{58} - 6862 q^{59} - 6624 q^{60} - 1912 q^{62} + 162 q^{63} + 4096 q^{64} - 5112 q^{66} - 368 q^{70} + 5184 q^{72} - 8158 q^{73} + 13419 q^{75} - 284 q^{77} - 9118 q^{79} - 11776 q^{80} + 6561 q^{81} + 4178 q^{83} + 288 q^{84} + 7362 q^{87} - 9088 q^{88} - 14904 q^{90} - 4302 q^{93} + 9216 q^{96} + 17282 q^{97} - 9588 q^{98} - 11502 q^{99}+O(q^{100}) \) 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.

comment: embeddings in the coefficient field
 
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
4.00000 9.00000 16.0000 −46.0000 36.0000 2.00000 64.0000 81.0000 −184.000
\(n\): 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 24.5.h.b yes 1
3.b odd 2 1 24.5.h.a 1
4.b odd 2 1 96.5.h.a 1
8.b even 2 1 24.5.h.a 1
8.d odd 2 1 96.5.h.b 1
12.b even 2 1 96.5.h.b 1
24.f even 2 1 96.5.h.a 1
24.h odd 2 1 CM 24.5.h.b yes 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
24.5.h.a 1 3.b odd 2 1
24.5.h.a 1 8.b even 2 1
24.5.h.b yes 1 1.a even 1 1 trivial
24.5.h.b yes 1 24.h odd 2 1 CM
96.5.h.a 1 4.b odd 2 1
96.5.h.a 1 24.f even 2 1
96.5.h.b 1 8.d odd 2 1
96.5.h.b 1 12.b even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5} + 46 \) acting on \(S_{5}^{\mathrm{new}}(24, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T - 4 \) Copy content Toggle raw display
$3$ \( T - 9 \) Copy content Toggle raw display
$5$ \( T + 46 \) Copy content Toggle raw display
$7$ \( T - 2 \) Copy content Toggle raw display
$11$ \( T + 142 \) 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
$29$ \( T - 818 \) Copy content Toggle raw display
$31$ \( T + 478 \) 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
$53$ \( T - 3218 \) Copy content Toggle raw display
$59$ \( T + 6862 \) 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
$73$ \( T + 8158 \) Copy content Toggle raw display
$79$ \( T + 9118 \) Copy content Toggle raw display
$83$ \( T - 4178 \) Copy content Toggle raw display
$89$ \( T \) Copy content Toggle raw display
$97$ \( T - 17282 \) Copy content Toggle raw display
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