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Properties

Label 580.2.s.a
Level $580$
Weight $2$
Character orbit 580.s
Analytic conductor $4.631$
Analytic rank $0$
Dimension $30$
Inner twists $2$

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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [580,2,Mod(133,580)] 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("580.133"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(580, base_ring=CyclotomicField(4)) chi = DirichletCharacter(H, H._module([0, 3, 3])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 580 = 2^{2} \cdot 5 \cdot 29 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 580.s (of order \(4\), degree \(2\), 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: \(30\)
Relative dimension: \(15\) over \(\Q(i)\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 30 q + 38 q^{9} - 4 q^{11} - 6 q^{13} + 6 q^{15} - 4 q^{21} - 2 q^{25} + 12 q^{27} - 4 q^{31} + 4 q^{33} - 16 q^{35} - 24 q^{37} - 12 q^{39} + 10 q^{41} - 16 q^{43} - 20 q^{45} - 32 q^{47} - 18 q^{53} - 26 q^{55}+ \cdots - 16 q^{97}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
133.1 0 −3.33306 0 −1.73594 − 1.40943i 0 2.63647 − 2.63647i 0 8.10929 0
133.2 0 −2.93055 0 0.953330 + 2.02266i 0 −1.87865 + 1.87865i 0 5.58815 0
133.3 0 −1.94358 0 −1.60038 − 1.56166i 0 −2.43648 + 2.43648i 0 0.777490 0
133.4 0 −1.61432 0 1.66951 − 1.48752i 0 1.96009 − 1.96009i 0 −0.393986 0
133.5 0 −1.59876 0 −0.354774 + 2.20774i 0 2.73334 − 2.73334i 0 −0.443956 0
133.6 0 −1.57681 0 1.09241 − 1.95106i 0 −1.37454 + 1.37454i 0 −0.513683 0
133.7 0 −0.357354 0 −1.56439 + 1.59771i 0 −1.34106 + 1.34106i 0 −2.87230 0
133.8 0 −0.121921 0 2.20227 + 0.387294i 0 −2.56056 + 2.56056i 0 −2.98514 0
133.9 0 0.548138 0 −2.23196 + 0.135478i 0 1.10894 − 1.10894i 0 −2.69955 0
133.10 0 0.938623 0 2.12400 + 0.699017i 0 0.222978 − 0.222978i 0 −2.11899 0
133.11 0 0.993874 0 −0.775222 − 2.09739i 0 1.61651 − 1.61651i 0 −2.01221 0
133.12 0 2.30110 0 −0.312235 + 2.21416i 0 −2.56511 + 2.56511i 0 2.29505 0
133.13 0 2.68138 0 1.12534 + 1.93226i 0 3.30434 − 3.30434i 0 4.18982 0
133.14 0 2.99066 0 1.63787 − 1.52229i 0 −1.61250 + 1.61250i 0 5.94404 0
133.15 0 3.02258 0 −2.22983 − 0.166966i 0 0.186244 − 0.186244i 0 6.13596 0
157.1 0 −3.33306 0 −1.73594 + 1.40943i 0 2.63647 + 2.63647i 0 8.10929 0
157.2 0 −2.93055 0 0.953330 − 2.02266i 0 −1.87865 − 1.87865i 0 5.58815 0
157.3 0 −1.94358 0 −1.60038 + 1.56166i 0 −2.43648 − 2.43648i 0 0.777490 0
157.4 0 −1.61432 0 1.66951 + 1.48752i 0 1.96009 + 1.96009i 0 −0.393986 0
157.5 0 −1.59876 0 −0.354774 − 2.20774i 0 2.73334 + 2.73334i 0 −0.443956 0
See all 30 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 133.15
Significant digits:
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
145.e even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 580.2.s.a yes 30
5.b even 2 1 2900.2.s.d 30
5.c odd 4 1 580.2.j.a ✓ 30
5.c odd 4 1 2900.2.j.d 30
29.c odd 4 1 580.2.j.a ✓ 30
145.e even 4 1 inner 580.2.s.a yes 30
145.f odd 4 1 2900.2.j.d 30
145.j even 4 1 2900.2.s.d 30
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
580.2.j.a ✓ 30 5.c odd 4 1
580.2.j.a ✓ 30 29.c odd 4 1
580.2.s.a yes 30 1.a even 1 1 trivial
580.2.s.a yes 30 145.e even 4 1 inner
2900.2.j.d 30 5.c odd 4 1
2900.2.j.d 30 145.f odd 4 1
2900.2.s.d 30 5.b even 2 1
2900.2.s.d 30 145.j even 4 1

Hecke kernels

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