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Properties

Label 580.2.x.a
Level $580$
Weight $2$
Character orbit 580.x
Analytic conductor $4.631$
Analytic rank $0$
Dimension $96$
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] = [580,2,Mod(9,580)] 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("580.9"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(580, base_ring=CyclotomicField(14)) chi = DirichletCharacter(H, H._module([0, 7, 5])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 580 = 2^{2} \cdot 5 \cdot 29 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 580.x (of order \(14\), degree \(6\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(0)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(4.63132331723\)
Analytic rank: \(0\)
Dimension: \(96\)
Relative dimension: \(16\) over \(\Q(\zeta_{14})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{14}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 96 q + 2 q^{5} - 28 q^{9} - 14 q^{15} - 28 q^{21} + 18 q^{25} + 22 q^{29} - 14 q^{31} + 30 q^{35} + 14 q^{39} - 15 q^{45} + 42 q^{49} - 24 q^{51} - 56 q^{55} - 60 q^{59} - 56 q^{61} - 29 q^{65} + 56 q^{69}+ \cdots - 112 q^{95}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
9.1 0 −1.93075 − 2.42108i 0 1.65502 − 1.50363i 0 −1.99315 + 1.58948i 0 −1.46629 + 6.42424i 0
9.2 0 −1.64624 − 2.06432i 0 −0.428962 + 2.19454i 0 3.72130 − 2.96764i 0 −0.883752 + 3.87197i 0
9.3 0 −1.55546 − 1.95048i 0 −2.20238 − 0.386705i 0 −0.0450912 + 0.0359590i 0 −0.717371 + 3.14301i 0
9.4 0 −1.45588 − 1.82562i 0 −0.0969974 − 2.23396i 0 1.65100 − 1.31663i 0 −0.545727 + 2.39099i 0
9.5 0 −1.06517 − 1.33568i 0 0.744061 + 2.10864i 0 −1.76552 + 1.40795i 0 0.0181082 − 0.0793374i 0
9.6 0 −0.917690 − 1.15075i 0 −2.15614 + 0.592515i 0 −3.36588 + 2.68420i 0 0.185499 − 0.812726i 0
9.7 0 −0.170611 − 0.213940i 0 2.18620 − 0.469585i 0 2.61285 − 2.08368i 0 0.650901 − 2.85178i 0
9.8 0 −0.136400 − 0.171040i 0 −0.806756 − 2.08546i 0 −0.370217 + 0.295238i 0 0.656913 − 2.87812i 0
9.9 0 0.136400 + 0.171040i 0 1.12747 + 1.93101i 0 0.370217 − 0.295238i 0 0.656913 − 2.87812i 0
9.10 0 0.170611 + 0.213940i 0 1.73021 − 1.41646i 0 −2.61285 + 2.08368i 0 0.650901 − 2.85178i 0
9.11 0 0.917690 + 1.15075i 0 −1.80758 + 1.31631i 0 3.36588 − 2.68420i 0 0.185499 − 0.812726i 0
9.12 0 1.06517 + 1.33568i 0 −1.18469 − 1.89645i 0 1.76552 − 1.40795i 0 0.0181082 − 0.0793374i 0
9.13 0 1.45588 + 1.82562i 0 1.68611 + 1.46869i 0 −1.65100 + 1.31663i 0 −0.545727 + 2.39099i 0
9.14 0 1.55546 + 1.95048i 0 −1.07082 + 1.96299i 0 0.0450912 − 0.0359590i 0 −0.717371 + 3.14301i 0
9.15 0 1.64624 + 2.06432i 0 −1.98321 − 1.03290i 0 −3.72130 + 2.96764i 0 −0.883752 + 3.87197i 0
9.16 0 1.93075 + 2.42108i 0 2.20747 − 0.356450i 0 1.99315 − 1.58948i 0 −1.46629 + 6.42424i 0
109.1 0 −3.04767 − 1.46768i 0 2.11444 − 0.727409i 0 1.27917 − 2.65623i 0 5.26372 + 6.60050i 0
109.2 0 −2.53539 − 1.22098i 0 −1.53333 + 1.62755i 0 −0.759876 + 1.57790i 0 3.06694 + 3.84583i 0
109.3 0 −1.74366 − 0.839705i 0 −2.08637 − 0.804396i 0 1.49816 − 3.11096i 0 0.464793 + 0.582832i 0
109.4 0 −1.63386 − 0.786825i 0 2.21373 + 0.315301i 0 −2.15642 + 4.47786i 0 0.179929 + 0.225624i 0
See all 96 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 9.16
Significant digits:
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
29.e even 14 1 inner
145.l even 14 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 580.2.x.a ✓ 96
5.b even 2 1 inner 580.2.x.a ✓ 96
29.e even 14 1 inner 580.2.x.a ✓ 96
145.l even 14 1 inner 580.2.x.a ✓ 96
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
580.2.x.a ✓ 96 1.a even 1 1 trivial
580.2.x.a ✓ 96 5.b even 2 1 inner
580.2.x.a ✓ 96 29.e even 14 1 inner
580.2.x.a ✓ 96 145.l even 14 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(580, [\chi])\).