Properties

Label 24.22.a.d
Level $24$
Weight $22$
Character orbit 24.a
Self dual yes
Analytic conductor $67.075$
Analytic rank $0$
Dimension $3$
CM no
Inner twists $1$

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Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [24,22,Mod(1,24)]
 
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, 0, 0]))
 
N = Newforms(chi, 22, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("24.1");
 
S:= CuspForms(chi, 22);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 24 = 2^{3} \cdot 3 \)
Weight: \( k \) \(=\) \( 22 \)
Character orbit: \([\chi]\) \(=\) 24.a (trivial)

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: \(67.0745626289\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: \(\mathbb{Q}[x]/(x^{3} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - x^{2} - 12529199x - 17012391021 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{21}\cdot 3^{4}\cdot 7 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + 59049 q^{3} + ( - \beta_1 + 1760166) q^{5} + ( - \beta_{2} - 7 \beta_1 + 284180792) q^{7} + 3486784401 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + 59049 q^{3} + ( - \beta_1 + 1760166) q^{5} + ( - \beta_{2} - 7 \beta_1 + 284180792) q^{7} + 3486784401 q^{9} + ( - 56 \beta_{2} - 1042 \beta_1 + 20830252556) q^{11} + ( - 433 \beta_{2} - 11856 \beta_1 + 67921735934) q^{13} + ( - 59049 \beta_1 + 103936042134) q^{15} + (1730 \beta_{2} - 209590 \beta_1 + 231942606642) q^{17} + (32642 \beta_{2} + \cdots + 1651834707732) q^{19}+ \cdots + ( - 195259926456 \beta_{2} + \cdots + 72\!\cdots\!56) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + 177147 q^{3} + 5280498 q^{5} + 852542376 q^{7} + 10460353203 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 3 q + 177147 q^{3} + 5280498 q^{5} + 852542376 q^{7} + 10460353203 q^{9} + 62490757668 q^{11} + 203765207802 q^{13} + 311808126402 q^{15} + 695827819926 q^{17} + 4955504123196 q^{19} + 50341774760424 q^{21} + 150867407938152 q^{23} + 678194854969869 q^{25} + 617673396283947 q^{27} + 32\!\cdots\!46 q^{29}+ \cdots + 21\!\cdots\!68 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{3} - x^{2} - 12529199x - 17012391021 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 576\nu^{2} - 1212288\nu - 4810808512 ) / 11 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 8064\nu^{2} - 13187328\nu - 67352580736 ) / 11 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{2} - 14\beta _1 + 114688 ) / 344064 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 451\beta_{2} - 4906\beta _1 + 615835213824 ) / 73728 \) 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.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
−2135.93
4086.16
−1949.23
0 59049.0 0 −3.51850e7 0 2.43346e8 0 3.48678e9 0
1.2 0 59049.0 0 1.51338e7 0 −8.40758e8 0 3.48678e9 0
1.3 0 59049.0 0 2.53316e7 0 1.44995e9 0 3.48678e9 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 24.22.a.d 3
3.b odd 2 1 72.22.a.c 3
4.b odd 2 1 48.22.a.j 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
24.22.a.d 3 1.a even 1 1 trivial
48.22.a.j 3 4.b odd 2 1
72.22.a.c 3 3.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{3} - 5280498T_{5}^{2} - 1040411335225620T_{5} + 13488669456493277401000 \) acting on \(S_{22}^{\mathrm{new}}(\Gamma_0(24))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{3} \) Copy content Toggle raw display
$3$ \( (T - 59049)^{3} \) Copy content Toggle raw display
$5$ \( T^{3} + \cdots + 13\!\cdots\!00 \) Copy content Toggle raw display
$7$ \( T^{3} + \cdots + 29\!\cdots\!00 \) Copy content Toggle raw display
$11$ \( T^{3} + \cdots - 18\!\cdots\!60 \) Copy content Toggle raw display
$13$ \( T^{3} + \cdots - 64\!\cdots\!24 \) Copy content Toggle raw display
$17$ \( T^{3} + \cdots + 13\!\cdots\!12 \) Copy content Toggle raw display
$19$ \( T^{3} + \cdots - 18\!\cdots\!64 \) Copy content Toggle raw display
$23$ \( T^{3} + \cdots - 60\!\cdots\!52 \) Copy content Toggle raw display
$29$ \( T^{3} + \cdots + 20\!\cdots\!76 \) Copy content Toggle raw display
$31$ \( T^{3} + \cdots - 88\!\cdots\!20 \) Copy content Toggle raw display
$37$ \( T^{3} + \cdots - 10\!\cdots\!76 \) Copy content Toggle raw display
$41$ \( T^{3} + \cdots - 91\!\cdots\!60 \) Copy content Toggle raw display
$43$ \( T^{3} + \cdots - 11\!\cdots\!04 \) Copy content Toggle raw display
$47$ \( T^{3} + \cdots - 45\!\cdots\!00 \) Copy content Toggle raw display
$53$ \( T^{3} + \cdots - 78\!\cdots\!20 \) Copy content Toggle raw display
$59$ \( T^{3} + \cdots + 25\!\cdots\!60 \) Copy content Toggle raw display
$61$ \( T^{3} + \cdots + 12\!\cdots\!96 \) Copy content Toggle raw display
$67$ \( T^{3} + \cdots + 46\!\cdots\!16 \) Copy content Toggle raw display
$71$ \( T^{3} + \cdots - 43\!\cdots\!20 \) Copy content Toggle raw display
$73$ \( T^{3} + \cdots + 27\!\cdots\!00 \) Copy content Toggle raw display
$79$ \( T^{3} + \cdots - 33\!\cdots\!64 \) Copy content Toggle raw display
$83$ \( T^{3} + \cdots - 54\!\cdots\!88 \) Copy content Toggle raw display
$89$ \( T^{3} + \cdots + 14\!\cdots\!88 \) Copy content Toggle raw display
$97$ \( T^{3} + \cdots + 12\!\cdots\!08 \) Copy content Toggle raw display
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