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Properties

Label 2970.2.t.a
Level $2970$
Weight $2$
Character orbit 2970.t
Analytic conductor $23.716$
Analytic rank $0$
Dimension $48$
Inner twists $2$

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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] = [2970,2,Mod(791,2970)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2970, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([1, 0, 3])) N = Newforms(chi, 2, names="a")
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("2970.791"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Level: \( N \) \(=\) \( 2970 = 2 \cdot 3^{3} \cdot 5 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2970.t (of order \(6\), degree \(2\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [48,-24] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(2)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(23.7155694003\)
Analytic rank: \(0\)
Dimension: \(48\)
Relative dimension: \(24\) over \(\Q(\zeta_{6})\)
Twist minimal: no (minimal twist has level 990)
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.

\(\operatorname{Tr}(f)(q) = \) \( 48 q - 24 q^{2} - 24 q^{4} + 48 q^{8} + 12 q^{11} - 24 q^{13} - 24 q^{16} - 12 q^{17} - 6 q^{22} - 36 q^{23} + 24 q^{25} - 24 q^{32} + 6 q^{34} + 6 q^{38} - 6 q^{41} - 30 q^{43} - 6 q^{44} + 24 q^{49} + 24 q^{50}+ \cdots - 48 q^{98}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
791.1 −0.500000 + 0.866025i 0 −0.500000 − 0.866025i −0.866025 + 0.500000i 0 −4.54797 − 2.62577i 1.00000 0 − 1.00000i
791.2 −0.500000 + 0.866025i 0 −0.500000 − 0.866025i −0.866025 + 0.500000i 0 3.86969 + 2.23417i 1.00000 0 − 1.00000i
791.3 −0.500000 + 0.866025i 0 −0.500000 − 0.866025i −0.866025 + 0.500000i 0 −3.44565 − 1.98935i 1.00000 0 − 1.00000i
791.4 −0.500000 + 0.866025i 0 −0.500000 − 0.866025i −0.866025 + 0.500000i 0 −0.0653962 − 0.0377565i 1.00000 0 − 1.00000i
791.5 −0.500000 + 0.866025i 0 −0.500000 − 0.866025i −0.866025 + 0.500000i 0 0.700188 + 0.404254i 1.00000 0 − 1.00000i
791.6 −0.500000 + 0.866025i 0 −0.500000 − 0.866025i −0.866025 + 0.500000i 0 −1.01491 − 0.585956i 1.00000 0 − 1.00000i
791.7 −0.500000 + 0.866025i 0 −0.500000 − 0.866025i −0.866025 + 0.500000i 0 1.10421 + 0.637517i 1.00000 0 − 1.00000i
791.8 −0.500000 + 0.866025i 0 −0.500000 − 0.866025i −0.866025 + 0.500000i 0 −2.35584 − 1.36015i 1.00000 0 − 1.00000i
791.9 −0.500000 + 0.866025i 0 −0.500000 − 0.866025i −0.866025 + 0.500000i 0 1.93804 + 1.11893i 1.00000 0 − 1.00000i
791.10 −0.500000 + 0.866025i 0 −0.500000 − 0.866025i −0.866025 + 0.500000i 0 −2.62958 − 1.51819i 1.00000 0 − 1.00000i
791.11 −0.500000 + 0.866025i 0 −0.500000 − 0.866025i −0.866025 + 0.500000i 0 2.83014 + 1.63398i 1.00000 0 − 1.00000i
791.12 −0.500000 + 0.866025i 0 −0.500000 − 0.866025i −0.866025 + 0.500000i 0 3.61708 + 2.08832i 1.00000 0 − 1.00000i
791.13 −0.500000 + 0.866025i 0 −0.500000 − 0.866025i 0.866025 − 0.500000i 0 −3.20594 − 1.85095i 1.00000 0 1.00000i
791.14 −0.500000 + 0.866025i 0 −0.500000 − 0.866025i 0.866025 − 0.500000i 0 2.96705 + 1.71303i 1.00000 0 1.00000i
791.15 −0.500000 + 0.866025i 0 −0.500000 − 0.866025i 0.866025 − 0.500000i 0 −1.95421 − 1.12826i 1.00000 0 1.00000i
791.16 −0.500000 + 0.866025i 0 −0.500000 − 0.866025i 0.866025 − 0.500000i 0 1.11114 + 0.641516i 1.00000 0 1.00000i
791.17 −0.500000 + 0.866025i 0 −0.500000 − 0.866025i 0.866025 − 0.500000i 0 1.26719 + 0.731611i 1.00000 0 1.00000i
791.18 −0.500000 + 0.866025i 0 −0.500000 − 0.866025i 0.866025 − 0.500000i 0 −0.0947420 − 0.0546993i 1.00000 0 1.00000i
791.19 −0.500000 + 0.866025i 0 −0.500000 − 0.866025i 0.866025 − 0.500000i 0 0.803327 + 0.463801i 1.00000 0 1.00000i
791.20 −0.500000 + 0.866025i 0 −0.500000 − 0.866025i 0.866025 − 0.500000i 0 0.915692 + 0.528675i 1.00000 0 1.00000i
See all 48 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 791.24
Significant digits:
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
99.g even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2970.2.t.a 48
3.b odd 2 1 990.2.t.b yes 48
9.c even 3 1 990.2.t.a ✓ 48
9.d odd 6 1 2970.2.t.b 48
11.b odd 2 1 2970.2.t.b 48
33.d even 2 1 990.2.t.a ✓ 48
99.g even 6 1 inner 2970.2.t.a 48
99.h odd 6 1 990.2.t.b yes 48
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
990.2.t.a ✓ 48 9.c even 3 1
990.2.t.a ✓ 48 33.d even 2 1
990.2.t.b yes 48 3.b odd 2 1
990.2.t.b yes 48 99.h odd 6 1
2970.2.t.a 48 1.a even 1 1 trivial
2970.2.t.a 48 99.g even 6 1 inner
2970.2.t.b 48 9.d odd 6 1
2970.2.t.b 48 11.b odd 2 1