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Properties

Label 720.2.bl.b
Level $720$
Weight $2$
Character orbit 720.bl
Analytic conductor $5.749$
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] = [720,2,Mod(251,720)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(720, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([2, 1, 2, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("720.251");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 720 = 2^{4} \cdot 3^{2} \cdot 5 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 720.bl (of order \(4\), degree \(2\), 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: \(5.74922894553\)
Analytic rank: \(0\)
Dimension: \(24\)
Relative dimension: \(12\) over \(\Q(i)\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 24 q - 8 q^{4} - 8 q^{7}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 24 q - 8 q^{4} - 8 q^{7} - 4 q^{16} - 24 q^{19} + 12 q^{22} - 20 q^{28} - 12 q^{34} + 8 q^{37} - 20 q^{40} + 48 q^{43} + 12 q^{46} + 24 q^{49} + 4 q^{52} + 24 q^{55} + 48 q^{58} + 40 q^{61} + 16 q^{64} - 40 q^{67} - 20 q^{70} - 84 q^{76} - 12 q^{82} + 24 q^{85} - 132 q^{88} + 40 q^{91} + 60 q^{94} - 16 q^{97}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
251.1 −1.40272 − 0.179908i 0 1.93527 + 0.504724i 0.707107 − 0.707107i 0 1.61527 −2.62384 − 1.05616i 0 −1.11909 + 0.864661i
251.2 −1.16580 + 0.800573i 0 0.718165 − 1.86661i 0.707107 − 0.707107i 0 −4.13654 0.657124 + 2.75103i 0 −0.258252 + 1.39043i
251.3 −0.904816 + 1.08688i 0 −0.362616 − 1.96685i −0.707107 + 0.707107i 0 −2.69352 2.46583 + 1.38552i 0 −0.128739 − 1.40834i
251.4 −0.827528 − 1.14682i 0 −0.630396 + 1.89805i −0.707107 + 0.707107i 0 −0.527405 2.69840 − 0.847739i 0 1.39608 + 0.225774i
251.5 −0.348666 + 1.37056i 0 −1.75686 − 0.955734i −0.707107 + 0.707107i 0 4.49261 1.92245 − 2.07465i 0 −0.722588 − 1.21568i
251.6 −0.219596 + 1.39706i 0 −1.90356 − 0.613577i 0.707107 − 0.707107i 0 −0.750417 1.27522 − 2.52464i 0 0.832593 + 1.14315i
251.7 0.219596 − 1.39706i 0 −1.90356 − 0.613577i −0.707107 + 0.707107i 0 −0.750417 −1.27522 + 2.52464i 0 0.832593 + 1.14315i
251.8 0.348666 − 1.37056i 0 −1.75686 − 0.955734i 0.707107 − 0.707107i 0 4.49261 −1.92245 + 2.07465i 0 −0.722588 − 1.21568i
251.9 0.827528 + 1.14682i 0 −0.630396 + 1.89805i 0.707107 − 0.707107i 0 −0.527405 −2.69840 + 0.847739i 0 1.39608 + 0.225774i
251.10 0.904816 − 1.08688i 0 −0.362616 − 1.96685i 0.707107 − 0.707107i 0 −2.69352 −2.46583 − 1.38552i 0 −0.128739 − 1.40834i
251.11 1.16580 − 0.800573i 0 0.718165 − 1.86661i −0.707107 + 0.707107i 0 −4.13654 −0.657124 − 2.75103i 0 −0.258252 + 1.39043i
251.12 1.40272 + 0.179908i 0 1.93527 + 0.504724i −0.707107 + 0.707107i 0 1.61527 2.62384 + 1.05616i 0 −1.11909 + 0.864661i
611.1 −1.40272 + 0.179908i 0 1.93527 − 0.504724i 0.707107 + 0.707107i 0 1.61527 −2.62384 + 1.05616i 0 −1.11909 − 0.864661i
611.2 −1.16580 − 0.800573i 0 0.718165 + 1.86661i 0.707107 + 0.707107i 0 −4.13654 0.657124 − 2.75103i 0 −0.258252 − 1.39043i
611.3 −0.904816 − 1.08688i 0 −0.362616 + 1.96685i −0.707107 − 0.707107i 0 −2.69352 2.46583 − 1.38552i 0 −0.128739 + 1.40834i
611.4 −0.827528 + 1.14682i 0 −0.630396 − 1.89805i −0.707107 − 0.707107i 0 −0.527405 2.69840 + 0.847739i 0 1.39608 − 0.225774i
611.5 −0.348666 − 1.37056i 0 −1.75686 + 0.955734i −0.707107 − 0.707107i 0 4.49261 1.92245 + 2.07465i 0 −0.722588 + 1.21568i
611.6 −0.219596 − 1.39706i 0 −1.90356 + 0.613577i 0.707107 + 0.707107i 0 −0.750417 1.27522 + 2.52464i 0 0.832593 − 1.14315i
611.7 0.219596 + 1.39706i 0 −1.90356 + 0.613577i −0.707107 − 0.707107i 0 −0.750417 −1.27522 − 2.52464i 0 0.832593 − 1.14315i
611.8 0.348666 + 1.37056i 0 −1.75686 + 0.955734i 0.707107 + 0.707107i 0 4.49261 −1.92245 − 2.07465i 0 −0.722588 + 1.21568i
See all 24 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 251.12
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
16.f odd 4 1 inner
48.k even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 720.2.bl.b ✓ 24
3.b odd 2 1 inner 720.2.bl.b ✓ 24
4.b odd 2 1 2880.2.bl.b 24
12.b even 2 1 2880.2.bl.b 24
16.e even 4 1 2880.2.bl.b 24
16.f odd 4 1 inner 720.2.bl.b ✓ 24
48.i odd 4 1 2880.2.bl.b 24
48.k even 4 1 inner 720.2.bl.b ✓ 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
720.2.bl.b ✓ 24 1.a even 1 1 trivial
720.2.bl.b ✓ 24 3.b odd 2 1 inner
720.2.bl.b ✓ 24 16.f odd 4 1 inner
720.2.bl.b ✓ 24 48.k even 4 1 inner
2880.2.bl.b 24 4.b odd 2 1
2880.2.bl.b 24 12.b even 2 1
2880.2.bl.b 24 16.e even 4 1
2880.2.bl.b 24 48.i odd 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{6} + 2T_{7}^{5} - 22T_{7}^{4} - 48T_{7}^{3} + 48T_{7}^{2} + 96T_{7} + 32 \) acting on \(S_{2}^{\mathrm{new}}(720, [\chi])\). Copy content Toggle raw display