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Properties

Label 580.2.z.b
Level $580$
Weight $2$
Character orbit 580.z
Analytic conductor $4.631$
Analytic rank $0$
Dimension $36$
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(121,580)] 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("580.121"); 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, 0, 11])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 580 = 2^{2} \cdot 5 \cdot 29 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 580.z (of order \(14\), degree \(6\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [36] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(4.63132331723\)
Analytic rank: \(0\)
Dimension: \(36\)
Relative dimension: \(6\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 36 q + 6 q^{5} - 4 q^{7} + 6 q^{9} + 4 q^{13} + 7 q^{21} - 10 q^{23} - 6 q^{25} + 63 q^{27} + 5 q^{29} + 7 q^{31} + 6 q^{33} + 4 q^{35} + 49 q^{39} - 35 q^{43} - 6 q^{45} - 28 q^{47} - 46 q^{49} - 2 q^{51}+ \cdots + 63 q^{97}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
121.1 0 −2.60833 + 0.595335i 0 −0.623490 − 0.781831i 0 0.296568 + 1.29935i 0 3.74607 − 1.80401i 0
121.2 0 −1.28442 + 0.293161i 0 −0.623490 − 0.781831i 0 −1.14027 − 4.99586i 0 −1.13910 + 0.548564i 0
121.3 0 −0.706830 + 0.161329i 0 −0.623490 − 0.781831i 0 0.634892 + 2.78164i 0 −2.22933 + 1.07359i 0
121.4 0 0.170057 − 0.0388144i 0 −0.623490 − 0.781831i 0 0.0981377 + 0.429969i 0 −2.67549 + 1.28845i 0
121.5 0 2.07651 − 0.473950i 0 −0.623490 − 0.781831i 0 −0.877698 − 3.84544i 0 1.38436 − 0.666675i 0
121.6 0 3.19903 − 0.730158i 0 −0.623490 − 0.781831i 0 0.944301 + 4.13725i 0 6.99775 − 3.36994i 0
241.1 0 −2.66599 + 2.12606i 0 0.900969 − 0.433884i 0 1.14653 + 1.43770i 0 1.91983 − 8.41131i 0
241.2 0 −1.56649 + 1.24924i 0 0.900969 − 0.433884i 0 −0.367280 − 0.460555i 0 0.225745 − 0.989054i 0
241.3 0 −0.520764 + 0.415295i 0 0.900969 − 0.433884i 0 −2.54025 − 3.18538i 0 −0.568838 + 2.49224i 0
241.4 0 −0.327706 + 0.261337i 0 0.900969 − 0.433884i 0 1.39270 + 1.74638i 0 −0.628469 + 2.75350i 0
241.5 0 1.17413 − 0.936338i 0 0.900969 − 0.433884i 0 2.40726 + 3.01861i 0 −0.165709 + 0.726018i 0
241.6 0 2.38236 − 1.89987i 0 0.900969 − 0.433884i 0 −1.06946 − 1.34105i 0 1.39858 − 6.12758i 0
341.1 0 −1.44551 + 3.00163i 0 0.222521 + 0.974928i 0 −1.41373 − 0.680817i 0 −5.04983 − 6.33229i 0
341.2 0 −0.470771 + 0.977565i 0 0.222521 + 0.974928i 0 1.15843 + 0.557871i 0 1.13646 + 1.42508i 0
341.3 0 −0.167872 + 0.348591i 0 0.222521 + 0.974928i 0 −3.26144 − 1.57063i 0 1.77714 + 2.22846i 0
341.4 0 0.698366 − 1.45017i 0 0.222521 + 0.974928i 0 1.04362 + 0.502582i 0 0.255186 + 0.319993i 0
341.5 0 0.929991 − 1.93115i 0 0.222521 + 0.974928i 0 3.09706 + 1.49146i 0 −0.993972 − 1.24640i 0
341.6 0 1.13424 − 2.35528i 0 0.222521 + 0.974928i 0 −3.54936 − 1.70928i 0 −2.39038 − 2.99744i 0
361.1 0 −2.66599 − 2.12606i 0 0.900969 + 0.433884i 0 1.14653 − 1.43770i 0 1.91983 + 8.41131i 0
361.2 0 −1.56649 − 1.24924i 0 0.900969 + 0.433884i 0 −0.367280 + 0.460555i 0 0.225745 + 0.989054i 0
See all 36 embeddings
\(n\): e.g. 2-40 or 80-90
Embeddings: e.g. 1-3 or 121.6
Significant digits:
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
29.e 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.z.b ✓ 36
29.e even 14 1 inner 580.2.z.b ✓ 36
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
580.2.z.b ✓ 36 1.a even 1 1 trivial
580.2.z.b ✓ 36 29.e even 14 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{36} - 12 T_{3}^{34} - 21 T_{3}^{33} + 144 T_{3}^{32} + 189 T_{3}^{31} - 1091 T_{3}^{30} + \cdots + 117649 \) acting on \(S_{2}^{\mathrm{new}}(580, [\chi])\). Copy content Toggle raw display