Properties

Label 24.9.h.a
Level $24$
Weight $9$
Character orbit 24.h
Self dual yes
Analytic conductor $9.777$
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] = [24,9,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, 9, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("24.5");
 
S:= CuspForms(chi, 9);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 24 = 2^{3} \cdot 3 \)
Weight: \( k \) \(=\) \( 9 \)
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: \(9.77708664147\)
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)
 
\(f(q)\) \(=\) \( q - 16 q^{2} - 81 q^{3} + 256 q^{4} - 866 q^{5} + 1296 q^{6} - 4798 q^{7} - 4096 q^{8} + 6561 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - 16 q^{2} - 81 q^{3} + 256 q^{4} - 866 q^{5} + 1296 q^{6} - 4798 q^{7} - 4096 q^{8} + 6561 q^{9} + 13856 q^{10} + 9118 q^{11} - 20736 q^{12} + 76768 q^{14} + 70146 q^{15} + 65536 q^{16} - 104976 q^{18} - 221696 q^{20} + 388638 q^{21} - 145888 q^{22} + 331776 q^{24} + 359331 q^{25} - 531441 q^{27} - 1228288 q^{28} + 745438 q^{29} - 1122336 q^{30} - 1618558 q^{31} - 1048576 q^{32} - 738558 q^{33} + 4155068 q^{35} + 1679616 q^{36} + 3547136 q^{40} - 6218208 q^{42} + 2334208 q^{44} - 5681826 q^{45} - 5308416 q^{48} + 17256003 q^{49} - 5749296 q^{50} + 5425438 q^{53} + 8503056 q^{54} - 7896188 q^{55} + 19652608 q^{56} - 11927008 q^{58} - 22852322 q^{59} + 17957376 q^{60} + 25896928 q^{62} - 31479678 q^{63} + 16777216 q^{64} + 11816928 q^{66} - 66481088 q^{70} - 26873856 q^{72} + 9756482 q^{73} - 29105811 q^{75} - 43748164 q^{77} + 5237762 q^{79} - 56754176 q^{80} + 43046721 q^{81} + 77460958 q^{83} + 99491328 q^{84} - 60380478 q^{87} - 37347328 q^{88} + 90909216 q^{90} + 131103198 q^{93} + 84934656 q^{96} + 121608962 q^{97} - 276096048 q^{98} + 59823198 q^{99}+O(q^{100}) \) 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
−16.0000 −81.0000 256.000 −866.000 1296.00 −4798.00 −4096.00 6561.00 13856.0
\(n\): 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 24.9.h.a 1
3.b odd 2 1 24.9.h.b yes 1
4.b odd 2 1 96.9.h.b 1
8.b even 2 1 24.9.h.b yes 1
8.d odd 2 1 96.9.h.a 1
12.b even 2 1 96.9.h.a 1
24.f even 2 1 96.9.h.b 1
24.h odd 2 1 CM 24.9.h.a 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
24.9.h.a 1 1.a even 1 1 trivial
24.9.h.a 1 24.h odd 2 1 CM
24.9.h.b yes 1 3.b odd 2 1
24.9.h.b yes 1 8.b even 2 1
96.9.h.a 1 8.d odd 2 1
96.9.h.a 1 12.b even 2 1
96.9.h.b 1 4.b odd 2 1
96.9.h.b 1 24.f even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5} + 866 \) acting on \(S_{9}^{\mathrm{new}}(24, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T + 16 \) Copy content Toggle raw display
$3$ \( T + 81 \) Copy content Toggle raw display
$5$ \( T + 866 \) Copy content Toggle raw display
$7$ \( T + 4798 \) Copy content Toggle raw display
$11$ \( T - 9118 \) 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
$29$ \( T - 745438 \) Copy content Toggle raw display
$31$ \( T + 1618558 \) 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
$53$ \( T - 5425438 \) Copy content Toggle raw display
$59$ \( T + 22852322 \) 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
$73$ \( T - 9756482 \) Copy content Toggle raw display
$79$ \( T - 5237762 \) Copy content Toggle raw display
$83$ \( T - 77460958 \) Copy content Toggle raw display
$89$ \( T \) Copy content Toggle raw display
$97$ \( T - 121608962 \) Copy content Toggle raw display
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