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Properties

Label 1240.2.bm.a
Level $1240$
Weight $2$
Character orbit 1240.bm
Analytic conductor $9.901$
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] = [1240,2,Mod(129,1240)] 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("1240.129"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1240, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 0, 3, 4])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 1240 = 2^{3} \cdot 5 \cdot 31 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1240.bm (of order \(6\), 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: \(9.90144985064\)
Analytic rank: \(0\)
Dimension: \(96\)
Relative dimension: \(48\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 96 q + 48 q^{9} + 12 q^{11} - 20 q^{15} - 8 q^{21} + 8 q^{25} - 24 q^{29} - 12 q^{31} + 4 q^{35} + 16 q^{39} + 60 q^{49} + 12 q^{51} + 2 q^{55} - 8 q^{59} + 24 q^{61} - 4 q^{69} + 8 q^{71} + 10 q^{75} + 84 q^{79}+ \cdots - 56 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
129.1 0 −2.93193 − 1.69275i 0 1.34090 − 1.78941i 0 −1.71429 − 0.989747i 0 4.23082 + 7.32799i 0
129.2 0 −2.85813 − 1.65014i 0 −0.812038 + 2.08341i 0 4.41525 + 2.54915i 0 3.94592 + 6.83454i 0
129.3 0 −2.73522 − 1.57918i 0 −1.84059 − 1.26974i 0 −1.78971 − 1.03329i 0 3.48762 + 6.04073i 0
129.4 0 −2.37824 − 1.37308i 0 2.12977 + 0.681248i 0 −3.56474 − 2.05811i 0 2.27069 + 3.93295i 0
129.5 0 −2.37481 − 1.37110i 0 0.168346 + 2.22972i 0 −1.13648 − 0.656146i 0 2.25981 + 3.91411i 0
129.6 0 −2.15418 − 1.24371i 0 −0.298019 − 2.21612i 0 3.03290 + 1.75105i 0 1.59365 + 2.76028i 0
129.7 0 −2.04398 − 1.18009i 0 1.73978 − 1.40470i 0 2.49105 + 1.43821i 0 1.28522 + 2.22607i 0
129.8 0 −1.98728 − 1.14736i 0 −2.11933 − 0.713055i 0 −0.636841 − 0.367680i 0 1.13286 + 1.96217i 0
129.9 0 −1.96809 − 1.13628i 0 2.11469 + 0.726689i 0 2.63463 + 1.52110i 0 1.08224 + 1.87450i 0
129.10 0 −1.93450 − 1.11688i 0 1.63121 + 1.52943i 0 0.440222 + 0.254162i 0 0.994857 + 1.72314i 0
129.11 0 −1.87851 − 1.08456i 0 −1.69077 + 1.46331i 0 −2.91592 − 1.68351i 0 0.852530 + 1.47662i 0
129.12 0 −1.71744 − 0.991562i 0 0.323977 − 2.21247i 0 0.685764 + 0.395926i 0 0.466389 + 0.807810i 0
129.13 0 −1.51820 − 0.876531i 0 −2.18524 − 0.474063i 0 2.19364 + 1.26650i 0 0.0366149 + 0.0634189i 0
129.14 0 −1.40720 − 0.812450i 0 2.20245 + 0.386291i 0 3.22236 + 1.86043i 0 −0.179851 − 0.311511i 0
129.15 0 −1.33572 − 0.771179i 0 1.72053 − 1.42820i 0 −3.04063 − 1.75551i 0 −0.310567 − 0.537918i 0
129.16 0 −1.05633 − 0.609871i 0 −1.43197 − 1.71740i 0 −2.33623 − 1.34883i 0 −0.756116 − 1.30963i 0
129.17 0 −0.974079 − 0.562385i 0 −1.08755 + 1.95377i 0 2.08447 + 1.20347i 0 −0.867447 − 1.50246i 0
129.18 0 −0.752453 − 0.434429i 0 −1.88544 + 1.20214i 0 −4.19131 − 2.41986i 0 −1.12254 − 1.94430i 0
129.19 0 −0.615013 − 0.355078i 0 0.890025 − 2.05131i 0 −2.78589 − 1.60843i 0 −1.24784 − 2.16132i 0
129.20 0 −0.450527 − 0.260112i 0 0.335380 + 2.21077i 0 −1.19305 − 0.688810i 0 −1.36468 − 2.36370i 0
See all 96 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 129.48
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
31.c even 3 1 inner
155.j even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1240.2.bm.a ✓ 96
5.b even 2 1 inner 1240.2.bm.a ✓ 96
31.c even 3 1 inner 1240.2.bm.a ✓ 96
155.j even 6 1 inner 1240.2.bm.a ✓ 96
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1240.2.bm.a ✓ 96 1.a even 1 1 trivial
1240.2.bm.a ✓ 96 5.b even 2 1 inner
1240.2.bm.a ✓ 96 31.c even 3 1 inner
1240.2.bm.a ✓ 96 155.j even 6 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(1240, [\chi])\).