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Properties

Label 24.22.d.a
Level $24$
Weight $22$
Character orbit 24.d
Analytic conductor $67.075$
Analytic rank $0$
Dimension $42$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [24,22,Mod(13,24)]
 
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, 0]))
 
N = Newforms(chi, 22, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("24.13");
 
S:= CuspForms(chi, 22);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 24 = 2^{3} \cdot 3 \)
Weight: \( k \) \(=\) \( 22 \)
Character orbit: \([\chi]\) \(=\) 24.d (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: no
Analytic conductor: \(67.0745626289\)
Analytic rank: \(0\)
Dimension: \(42\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

The dimension is sufficiently large that we do not compute an algebraic \(q\)-expansion, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 42 q - 574 q^{2} + 392288 q^{4} - 44759142 q^{6} - 1129900996 q^{7} - 2157413692 q^{8} - 146444944842 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 42 q - 574 q^{2} + 392288 q^{4} - 44759142 q^{6} - 1129900996 q^{7} - 2157413692 q^{8} - 146444944842 q^{9} + 31553345052 q^{10} - 274707993996 q^{12} + 1565450468972 q^{14} + 2306601562500 q^{15} - 3863799224840 q^{16} - 3481428994004 q^{17} + 2001414246174 q^{18} + 72162740641304 q^{20} - 372001656147968 q^{22} - 174381110476664 q^{23} - 137569089593988 q^{24} - 37\!\cdots\!46 q^{25}+ \cdots + 13\!\cdots\!98 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
13.1 −1432.06 − 215.300i − 59049.0i 2.00444e6 + 616646.i 3.89605e7i −1.27133e7 + 8.45618e7i −5.76846e8 −2.73772e9 − 1.31463e9i −3.48678e9 8.38821e9 − 5.57938e10i
13.2 −1432.06 + 215.300i 59049.0i 2.00444e6 − 616646.i − 3.89605e7i −1.27133e7 − 8.45618e7i −5.76846e8 −2.73772e9 + 1.31463e9i −3.48678e9 8.38821e9 + 5.57938e10i
13.3 −1425.38 − 255.822i − 59049.0i 1.96626e6 + 729286.i − 6.52910e6i −1.51060e7 + 8.41672e7i −1.01290e8 −2.61610e9 − 1.54252e9i −3.48678e9 −1.67029e9 + 9.30645e9i
13.4 −1425.38 + 255.822i 59049.0i 1.96626e6 − 729286.i 6.52910e6i −1.51060e7 − 8.41672e7i −1.01290e8 −2.61610e9 + 1.54252e9i −3.48678e9 −1.67029e9 − 9.30645e9i
13.5 −1327.20 − 579.385i 59049.0i 1.42578e6 + 1.53792e6i 3.76014e7i 3.42121e7 − 7.83699e7i 3.92911e8 −1.00125e9 − 2.86721e9i −3.48678e9 2.17857e10 − 4.99046e10i
13.6 −1327.20 + 579.385i − 59049.0i 1.42578e6 − 1.53792e6i − 3.76014e7i 3.42121e7 + 7.83699e7i 3.92911e8 −1.00125e9 + 2.86721e9i −3.48678e9 2.17857e10 + 4.99046e10i
13.7 −1304.11 − 629.633i 59049.0i 1.30428e6 + 1.64223e6i − 1.97665e7i 3.71792e7 − 7.70066e7i 1.00011e9 −6.66923e8 − 2.96287e9i −3.48678e9 −1.24457e10 + 2.57778e10i
13.8 −1304.11 + 629.633i − 59049.0i 1.30428e6 − 1.64223e6i 1.97665e7i 3.71792e7 + 7.70066e7i 1.00011e9 −6.66923e8 + 2.96287e9i −3.48678e9 −1.24457e10 − 2.57778e10i
13.9 −1040.85 − 1006.87i − 59049.0i 69588.1 + 2.09600e6i − 3.31530e7i −5.94545e7 + 6.14612e7i −9.65383e8 2.03796e9 − 2.25169e9i −3.48678e9 −3.33807e10 + 3.45073e10i
13.10 −1040.85 + 1006.87i 59049.0i 69588.1 − 2.09600e6i 3.31530e7i −5.94545e7 − 6.14612e7i −9.65383e8 2.03796e9 + 2.25169e9i −3.48678e9 −3.33807e10 − 3.45073e10i
13.11 −1000.56 − 1046.91i − 59049.0i −94895.0 + 2.09500e6i 1.17189e7i −6.18191e7 + 5.90823e7i 8.98360e8 2.28823e9 − 1.99684e9i −3.48678e9 1.22686e10 − 1.17255e10i
13.12 −1000.56 + 1046.91i 59049.0i −94895.0 − 2.09500e6i − 1.17189e7i −6.18191e7 − 5.90823e7i 8.98360e8 2.28823e9 + 1.99684e9i −3.48678e9 1.22686e10 + 1.17255e10i
13.13 −975.110 − 1070.66i 59049.0i −195474. + 2.08802e6i 1.04371e6i 6.32214e7 − 5.75793e7i −9.20837e8 2.42617e9 − 1.82676e9i −3.48678e9 1.11746e9 − 1.01773e9i
13.14 −975.110 + 1070.66i − 59049.0i −195474. − 2.08802e6i − 1.04371e6i 6.32214e7 + 5.75793e7i −9.20837e8 2.42617e9 + 1.82676e9i −3.48678e9 1.11746e9 + 1.01773e9i
13.15 −745.078 − 1241.78i 59049.0i −986870. + 1.85044e6i − 1.87294e7i 7.33257e7 − 4.39961e7i 4.95293e8 3.03313e9 − 1.53250e8i −3.48678e9 −2.32578e10 + 1.39549e10i
13.16 −745.078 + 1241.78i − 59049.0i −986870. − 1.85044e6i 1.87294e7i 7.33257e7 + 4.39961e7i 4.95293e8 3.03313e9 + 1.53250e8i −3.48678e9 −2.32578e10 − 1.39549e10i
13.17 −299.073 − 1416.94i − 59049.0i −1.91826e6 + 847535.i 2.45935e7i −8.36686e7 + 1.76600e7i −1.36613e9 1.77460e9 + 2.46458e9i −3.48678e9 3.48475e10 − 7.35527e9i
13.18 −299.073 + 1416.94i 59049.0i −1.91826e6 − 847535.i − 2.45935e7i −8.36686e7 − 1.76600e7i −1.36613e9 1.77460e9 − 2.46458e9i −3.48678e9 3.48475e10 + 7.35527e9i
13.19 −225.622 − 1430.47i − 59049.0i −1.99534e6 + 645491.i − 1.24189e7i −8.44679e7 + 1.33227e7i 4.04688e8 1.37355e9 + 2.70864e9i −3.48678e9 −1.77648e10 + 2.80197e9i
13.20 −225.622 + 1430.47i 59049.0i −1.99534e6 − 645491.i 1.24189e7i −8.44679e7 − 1.33227e7i 4.04688e8 1.37355e9 − 2.70864e9i −3.48678e9 −1.77648e10 − 2.80197e9i
See all 42 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 13.42
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
8.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 24.22.d.a ✓ 42
8.b even 2 1 inner 24.22.d.a ✓ 42
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
24.22.d.a ✓ 42 1.a even 1 1 trivial
24.22.d.a ✓ 42 8.b even 2 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{22}^{\mathrm{new}}(24, [\chi])\).