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Properties

Label 232.2.g.b
Level $232$
Weight $2$
Character orbit 232.g
Analytic conductor $1.853$
Analytic rank $0$
Dimension $24$
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] = [232,2,Mod(173,232)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(232, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("232.173");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 232 = 2^{3} \cdot 29 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 232.g (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: \(1.85252932689\)
Analytic rank: \(0\)
Dimension: \(24\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 24 q - 4 q^{6} - 24 q^{7} + 24 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 24 q - 4 q^{6} - 24 q^{7} + 24 q^{9} - 4 q^{16} + 20 q^{20} + 16 q^{22} - 8 q^{23} - 48 q^{25} + 28 q^{28} + 8 q^{30} - 8 q^{33} + 36 q^{34} - 36 q^{36} - 4 q^{38} - 68 q^{42} - 24 q^{49} - 40 q^{52} + 32 q^{54} + 8 q^{57} + 20 q^{58} + 12 q^{62} - 48 q^{63} - 72 q^{64} + 32 q^{65} + 80 q^{71} + 40 q^{74} - 60 q^{78} + 32 q^{80} + 32 q^{81} + 20 q^{82} - 40 q^{86} + 24 q^{87} - 20 q^{88} - 20 q^{92} + 80 q^{94} - 68 q^{96}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
173.1 −1.36166 − 0.381937i 1.14600 1.70825 + 1.04014i − 3.91119i −1.56046 − 0.437700i 2.64656 −1.92879 − 2.06876i −1.68668 −1.49383 + 5.32572i
173.2 −1.36166 + 0.381937i 1.14600 1.70825 − 1.04014i 3.91119i −1.56046 + 0.437700i 2.64656 −1.92879 + 2.06876i −1.68668 −1.49383 − 5.32572i
173.3 −1.28674 − 0.586780i −0.818289 1.31138 + 1.51006i 1.48028i 1.05292 + 0.480156i −1.43730 −0.801322 − 2.71254i −2.33040 0.868599 − 1.90473i
173.4 −1.28674 + 0.586780i −0.818289 1.31138 − 1.51006i − 1.48028i 1.05292 − 0.480156i −1.43730 −0.801322 + 2.71254i −2.33040 0.868599 + 1.90473i
173.5 −1.12715 − 0.854125i 2.30071 0.540940 + 1.92546i 1.53806i −2.59325 − 1.96510i −0.395331 1.03486 − 2.63231i 2.29328 1.31369 − 1.73362i
173.6 −1.12715 + 0.854125i 2.30071 0.540940 − 1.92546i − 1.53806i −2.59325 + 1.96510i −0.395331 1.03486 + 2.63231i 2.29328 1.31369 + 1.73362i
173.7 −1.01559 − 0.984159i −2.87565 0.0628634 + 1.99901i − 3.40866i 2.92050 + 2.83010i −4.10145 1.90350 − 2.09205i 5.26938 −3.35467 + 3.46182i
173.8 −1.01559 + 0.984159i −2.87565 0.0628634 − 1.99901i 3.40866i 2.92050 − 2.83010i −4.10145 1.90350 + 2.09205i 5.26938 −3.35467 − 3.46182i
173.9 −0.324429 − 1.37650i −0.0554430 −1.78949 + 0.893152i 2.55444i 0.0179873 + 0.0763171i −3.12725 1.80998 + 2.17347i −2.99693 3.51619 − 0.828736i
173.10 −0.324429 + 1.37650i −0.0554430 −1.78949 − 0.893152i − 2.55444i 0.0179873 − 0.0763171i −3.12725 1.80998 − 2.17347i −2.99693 3.51619 + 0.828736i
173.11 −0.288151 − 1.38455i 2.90712 −1.83394 + 0.797917i − 2.00039i −0.837690 − 4.02504i 0.414764 1.63320 + 2.30925i 5.45135 −2.76963 + 0.576414i
173.12 −0.288151 + 1.38455i 2.90712 −1.83394 − 0.797917i 2.00039i −0.837690 + 4.02504i 0.414764 1.63320 − 2.30925i 5.45135 −2.76963 − 0.576414i
173.13 0.288151 − 1.38455i −2.90712 −1.83394 − 0.797917i 2.00039i −0.837690 + 4.02504i 0.414764 −1.63320 + 2.30925i 5.45135 2.76963 + 0.576414i
173.14 0.288151 + 1.38455i −2.90712 −1.83394 + 0.797917i − 2.00039i −0.837690 − 4.02504i 0.414764 −1.63320 − 2.30925i 5.45135 2.76963 − 0.576414i
173.15 0.324429 − 1.37650i 0.0554430 −1.78949 − 0.893152i − 2.55444i 0.0179873 − 0.0763171i −3.12725 −1.80998 + 2.17347i −2.99693 −3.51619 − 0.828736i
173.16 0.324429 + 1.37650i 0.0554430 −1.78949 + 0.893152i 2.55444i 0.0179873 + 0.0763171i −3.12725 −1.80998 − 2.17347i −2.99693 −3.51619 + 0.828736i
173.17 1.01559 − 0.984159i 2.87565 0.0628634 − 1.99901i 3.40866i 2.92050 − 2.83010i −4.10145 −1.90350 − 2.09205i 5.26938 3.35467 + 3.46182i
173.18 1.01559 + 0.984159i 2.87565 0.0628634 + 1.99901i − 3.40866i 2.92050 + 2.83010i −4.10145 −1.90350 + 2.09205i 5.26938 3.35467 − 3.46182i
173.19 1.12715 − 0.854125i −2.30071 0.540940 − 1.92546i − 1.53806i −2.59325 + 1.96510i −0.395331 −1.03486 − 2.63231i 2.29328 −1.31369 − 1.73362i
173.20 1.12715 + 0.854125i −2.30071 0.540940 + 1.92546i 1.53806i −2.59325 − 1.96510i −0.395331 −1.03486 + 2.63231i 2.29328 −1.31369 + 1.73362i
See all 24 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 173.24
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 232.2.g.b ✓ 24
4.b odd 2 1 928.2.g.b 24
8.b even 2 1 inner 232.2.g.b ✓ 24
8.d odd 2 1 928.2.g.b 24
29.b even 2 1 inner 232.2.g.b ✓ 24
116.d odd 2 1 928.2.g.b 24
232.b odd 2 1 928.2.g.b 24
232.g even 2 1 inner 232.2.g.b ✓ 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
232.2.g.b ✓ 24 1.a even 1 1 trivial
232.2.g.b ✓ 24 8.b even 2 1 inner
232.2.g.b ✓ 24 29.b even 2 1 inner
232.2.g.b ✓ 24 232.g even 2 1 inner
928.2.g.b 24 4.b odd 2 1
928.2.g.b 24 8.d odd 2 1
928.2.g.b 24 116.d odd 2 1
928.2.g.b 24 232.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{12} - 24T_{3}^{10} + 203T_{3}^{8} - 704T_{3}^{6} + 875T_{3}^{4} - 328T_{3}^{2} + 1 \) acting on \(S_{2}^{\mathrm{new}}(232, [\chi])\). Copy content Toggle raw display