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Properties

Label 24.22.f.b
Level $24$
Weight $22$
Character orbit 24.f
Analytic conductor $67.075$
Analytic rank $0$
Dimension $80$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [24,22,Mod(11,24)]
 
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([1, 1, 1]))
 
N = Newforms(chi, 22, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("24.11");
 
S:= CuspForms(chi, 22);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 24 = 2^{3} \cdot 3 \)
Weight: \( k \) \(=\) \( 22 \)
Character orbit: \([\chi]\) \(=\) 24.f (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: \(80\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 80 q + 71808 q^{3} + 2982260 q^{4} - 346652136 q^{6} + 15764030304 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 80 q + 71808 q^{3} + 2982260 q^{4} - 346652136 q^{6} + 15764030304 q^{9} - 8532785112 q^{10} - 563812835292 q^{12} - 12550257321688 q^{16} + 23322230265720 q^{18} + 59712367669312 q^{19} + 67957211435872 q^{22} + 488194083606216 q^{24} + 80\!\cdots\!96 q^{25}+ \cdots + 19\!\cdots\!44 q^{99}+O(q^{100}) \) 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 \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
11.1 −1447.69 − 36.5854i 83218.8 − 59455.7i 2.09448e6 + 105929.i 2.15453e7 −1.22650e8 + 8.30290e7i 7.78742e8i −3.02828e9 − 2.29979e8i 3.39039e9 − 9.89567e9i −3.11909e10 − 7.88241e8i
11.2 −1447.69 + 36.5854i 83218.8 + 59455.7i 2.09448e6 − 105929.i 2.15453e7 −1.22650e8 − 8.30290e7i − 7.78742e8i −3.02828e9 + 2.29979e8i 3.39039e9 + 9.89567e9i −3.11909e10 + 7.88241e8i
11.3 −1447.32 − 49.2747i 27542.1 − 98497.6i 2.09230e6 + 142632.i −3.60058e7 −4.47156e7 + 1.41200e8i 5.30004e8i −3.02119e9 − 3.09531e8i −8.94322e9 − 5.42566e9i 5.21117e10 + 1.77417e9i
11.4 −1447.32 + 49.2747i 27542.1 + 98497.6i 2.09230e6 − 142632.i −3.60058e7 −4.47156e7 − 1.41200e8i − 5.30004e8i −3.02119e9 + 3.09531e8i −8.94322e9 + 5.42566e9i 5.21117e10 − 1.77417e9i
11.5 −1432.54 − 212.110i −18617.4 − 100567.i 2.00717e6 + 607710.i 1.24708e7 5.33885e6 + 1.48015e8i − 1.44834e9i −2.74644e9 − 1.29631e9i −9.76714e9 + 3.74460e9i −1.78649e10 − 2.64519e9i
11.6 −1432.54 + 212.110i −18617.4 + 100567.i 2.00717e6 − 607710.i 1.24708e7 5.33885e6 − 1.48015e8i 1.44834e9i −2.74644e9 + 1.29631e9i −9.76714e9 − 3.74460e9i −1.78649e10 + 2.64519e9i
11.7 −1407.10 − 342.382i −94251.2 − 39712.2i 1.86270e6 + 963530.i −1.78035e7 1.19024e8 + 8.81488e7i 2.30554e8i −2.29111e9 − 1.99354e9i 7.30624e9 + 7.48584e9i 2.50513e10 + 6.09560e9i
11.8 −1407.10 + 342.382i −94251.2 + 39712.2i 1.86270e6 − 963530.i −1.78035e7 1.19024e8 − 8.81488e7i − 2.30554e8i −2.29111e9 + 1.99354e9i 7.30624e9 − 7.48584e9i 2.50513e10 − 6.09560e9i
11.9 −1356.69 − 506.515i −91907.9 + 44869.7i 1.58404e6 + 1.37436e6i 3.41502e7 1.47417e8 − 1.43212e7i 1.23766e8i −1.45290e9 − 2.66692e9i 6.43378e9 − 8.24775e9i −4.63311e10 − 1.72976e10i
11.10 −1356.69 + 506.515i −91907.9 − 44869.7i 1.58404e6 − 1.37436e6i 3.41502e7 1.47417e8 + 1.43212e7i − 1.23766e8i −1.45290e9 + 2.66692e9i 6.43378e9 + 8.24775e9i −4.63311e10 + 1.72976e10i
11.11 −1278.89 − 679.406i −43276.9 + 92668.5i 1.17397e6 + 1.73777e6i −3.88324e6 1.18306e8 − 8.91103e7i − 7.66700e8i −3.20723e8 − 3.02002e9i −6.71457e9 − 8.02082e9i 4.96624e9 + 2.63830e9i
11.12 −1278.89 + 679.406i −43276.9 − 92668.5i 1.17397e6 − 1.73777e6i −3.88324e6 1.18306e8 + 8.91103e7i 7.66700e8i −3.20723e8 + 3.02002e9i −6.71457e9 + 8.02082e9i 4.96624e9 − 2.63830e9i
11.13 −1259.59 − 714.555i 99544.4 − 23479.2i 1.07597e6 + 1.80009e6i −1.95847e7 −1.42162e8 − 4.15557e7i − 3.09866e8i −6.90211e7 − 3.03622e9i 9.35780e9 − 4.67445e9i 2.46687e10 + 1.39943e10i
11.14 −1259.59 + 714.555i 99544.4 + 23479.2i 1.07597e6 − 1.80009e6i −1.95847e7 −1.42162e8 + 4.15557e7i 3.09866e8i −6.90211e7 + 3.03622e9i 9.35780e9 + 4.67445e9i 2.46687e10 − 1.39943e10i
11.15 −1239.13 − 749.479i 71611.8 + 73021.3i 973714. + 1.85740e6i −183918. −3.40080e7 − 1.44154e8i 8.83191e8i 1.85527e8 − 3.03133e9i −2.03868e8 + 1.04584e10i 2.27898e8 + 1.37843e8i
11.16 −1239.13 + 749.479i 71611.8 − 73021.3i 973714. − 1.85740e6i −183918. −3.40080e7 + 1.44154e8i − 8.83191e8i 1.85527e8 + 3.03133e9i −2.03868e8 − 1.04584e10i 2.27898e8 − 1.37843e8i
11.17 −1162.26 − 863.883i 2559.62 − 102244.i 604564. + 2.00812e6i 2.80198e7 −9.13017e7 + 1.16623e8i 6.76763e8i 1.03212e9 − 2.85624e9i −1.04472e10 − 5.23411e8i −3.25664e10 − 2.42058e10i
11.18 −1162.26 + 863.883i 2559.62 + 102244.i 604564. − 2.00812e6i 2.80198e7 −9.13017e7 − 1.16623e8i − 6.76763e8i 1.03212e9 + 2.85624e9i −1.04472e10 + 5.23411e8i −3.25664e10 + 2.42058e10i
11.19 −1052.87 − 994.292i −76264.6 − 68147.4i 119920. + 2.09372e6i 1.46521e6 1.25383e7 + 1.47580e8i 5.38101e8i 1.95551e9 − 2.32365e9i 1.17221e9 + 1.03945e10i −1.54268e9 − 1.45685e9i
11.20 −1052.87 + 994.292i −76264.6 + 68147.4i 119920. − 2.09372e6i 1.46521e6 1.25383e7 − 1.47580e8i − 5.38101e8i 1.95551e9 + 2.32365e9i 1.17221e9 − 1.03945e10i −1.54268e9 + 1.45685e9i
See all 80 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 11.80
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
8.d odd 2 1 inner
24.f 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.f.b ✓ 80
3.b odd 2 1 inner 24.22.f.b ✓ 80
8.d odd 2 1 inner 24.22.f.b ✓ 80
24.f even 2 1 inner 24.22.f.b ✓ 80
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
24.22.f.b ✓ 80 1.a even 1 1 trivial
24.22.f.b ✓ 80 3.b odd 2 1 inner
24.22.f.b ✓ 80 8.d odd 2 1 inner
24.22.f.b ✓ 80 24.f even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{40} + \cdots + 15\!\cdots\!00 \) acting on \(S_{22}^{\mathrm{new}}(24, [\chi])\). Copy content Toggle raw display