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Properties

Label 2548.2.bb.g
Level $2548$
Weight $2$
Character orbit 2548.bb
Analytic conductor $20.346$
Analytic rank $0$
Dimension $24$
Inner twists $4$

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Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [2548,2,Mod(569,2548)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("2548.569"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2548, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 2, 5])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 2548 = 2^{2} \cdot 7^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2548.bb (of order \(6\), degree \(2\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [24,0,0,0,0,0,0,0,-8,0,0,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(13)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(20.3458824350\)
Analytic rank: \(0\)
Dimension: \(24\)
Relative dimension: \(12\) over \(\Q(\zeta_{6})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

The algebraic \(q\)-expansion of this newform has not been computed, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 24 q - 8 q^{9} - 12 q^{15} - 32 q^{23} + 24 q^{29} - 28 q^{39} + 4 q^{43} - 40 q^{51} + 24 q^{53} + 32 q^{65} - 24 q^{71} + 24 q^{79} + 4 q^{81} + 12 q^{85} + 40 q^{95}+O(q^{100}) \) 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.

Copy content comment:embeddings in the coefficient field
 
Copy content gp:mfembed(f)
 
Label   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
569.1 0 −1.49140 + 2.58319i 0 −0.626773 − 0.361868i 0 0 0 −2.94857 − 5.10707i 0
569.2 0 −1.31183 + 2.27215i 0 2.89137 + 1.66933i 0 0 0 −1.94179 − 3.36327i 0
569.3 0 −0.937653 + 1.62406i 0 0.192850 + 0.111342i 0 0 0 −0.258387 − 0.447540i 0
569.4 0 −0.614889 + 1.06502i 0 −2.34090 − 1.35152i 0 0 0 0.743824 + 1.28834i 0
569.5 0 −0.450600 + 0.780462i 0 1.46259 + 0.844424i 0 0 0 1.09392 + 1.89472i 0
569.6 0 −0.307410 + 0.532450i 0 −2.46799 − 1.42489i 0 0 0 1.31100 + 2.27072i 0
569.7 0 0.307410 − 0.532450i 0 2.46799 + 1.42489i 0 0 0 1.31100 + 2.27072i 0
569.8 0 0.450600 − 0.780462i 0 −1.46259 − 0.844424i 0 0 0 1.09392 + 1.89472i 0
569.9 0 0.614889 − 1.06502i 0 2.34090 + 1.35152i 0 0 0 0.743824 + 1.28834i 0
569.10 0 0.937653 − 1.62406i 0 −0.192850 − 0.111342i 0 0 0 −0.258387 − 0.447540i 0
569.11 0 1.31183 − 2.27215i 0 −2.89137 − 1.66933i 0 0 0 −1.94179 − 3.36327i 0
569.12 0 1.49140 − 2.58319i 0 0.626773 + 0.361868i 0 0 0 −2.94857 − 5.10707i 0
1733.1 0 −1.49140 − 2.58319i 0 −0.626773 + 0.361868i 0 0 0 −2.94857 + 5.10707i 0
1733.2 0 −1.31183 − 2.27215i 0 2.89137 − 1.66933i 0 0 0 −1.94179 + 3.36327i 0
1733.3 0 −0.937653 − 1.62406i 0 0.192850 − 0.111342i 0 0 0 −0.258387 + 0.447540i 0
1733.4 0 −0.614889 − 1.06502i 0 −2.34090 + 1.35152i 0 0 0 0.743824 − 1.28834i 0
1733.5 0 −0.450600 − 0.780462i 0 1.46259 − 0.844424i 0 0 0 1.09392 − 1.89472i 0
1733.6 0 −0.307410 − 0.532450i 0 −2.46799 + 1.42489i 0 0 0 1.31100 − 2.27072i 0
1733.7 0 0.307410 + 0.532450i 0 2.46799 − 1.42489i 0 0 0 1.31100 − 2.27072i 0
1733.8 0 0.450600 + 0.780462i 0 −1.46259 + 0.844424i 0 0 0 1.09392 − 1.89472i 0
See all 24 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 569.12
Significant digits:
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.b odd 2 1 inner
91.k even 6 1 inner
91.l odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2548.2.bb.g 24
7.b odd 2 1 inner 2548.2.bb.g 24
7.c even 3 1 2548.2.u.f ✓ 24
7.c even 3 1 2548.2.bq.g 24
7.d odd 6 1 2548.2.u.f ✓ 24
7.d odd 6 1 2548.2.bq.g 24
13.e even 6 1 2548.2.bq.g 24
91.k even 6 1 inner 2548.2.bb.g 24
91.l odd 6 1 inner 2548.2.bb.g 24
91.p odd 6 1 2548.2.u.f ✓ 24
91.t odd 6 1 2548.2.bq.g 24
91.u even 6 1 2548.2.u.f ✓ 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2548.2.u.f ✓ 24 7.c even 3 1
2548.2.u.f ✓ 24 7.d odd 6 1
2548.2.u.f ✓ 24 91.p odd 6 1
2548.2.u.f ✓ 24 91.u even 6 1
2548.2.bb.g 24 1.a even 1 1 trivial
2548.2.bb.g 24 7.b odd 2 1 inner
2548.2.bb.g 24 91.k even 6 1 inner
2548.2.bb.g 24 91.l odd 6 1 inner
2548.2.bq.g 24 7.c even 3 1
2548.2.bq.g 24 7.d odd 6 1
2548.2.bq.g 24 13.e even 6 1
2548.2.bq.g 24 91.t odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{24} + 22 T_{3}^{22} + 313 T_{3}^{20} + 2618 T_{3}^{18} + 15820 T_{3}^{16} + 61492 T_{3}^{14} + \cdots + 10000 \) acting on \(S_{2}^{\mathrm{new}}(2548, [\chi])\). Copy content Toggle raw display