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Properties

Label 783.2.k.f
Level $783$
Weight $2$
Character orbit 783.k
Analytic conductor $6.252$
Analytic rank $0$
Dimension $60$
Inner twists $4$

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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [783,2,Mod(82,783)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(783, base_ring=CyclotomicField(14)) chi = DirichletCharacter(H, H._module([0, 4])) 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("783.82"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Level: \( N \) \(=\) \( 783 = 3^{3} \cdot 29 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 783.k (of order \(7\), degree \(6\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [60,0,0,-10,0,0,4,0,0,4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(10)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(6.25228647827\)
Analytic rank: \(0\)
Dimension: \(60\)
Relative dimension: \(10\) over \(\Q(\zeta_{7})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{7}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 60 q - 10 q^{4} + 4 q^{7} + 4 q^{10} - 24 q^{13} - 26 q^{16} + 4 q^{19} - 8 q^{22} - 16 q^{25} + 112 q^{28} - 4 q^{31} + 26 q^{34} - 18 q^{37} - 78 q^{40} - 8 q^{43} + 72 q^{46} + 14 q^{49} - 12 q^{52}+ \cdots + 8 q^{97}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
82.1 −1.62244 − 2.03447i 0 −1.06173 + 4.65174i −1.57981 − 1.98102i 0 −0.0818290 − 0.358516i 6.49744 − 3.12900i 0 −1.46718 + 6.42814i
82.2 −1.32704 − 1.66405i 0 −0.563000 + 2.46666i 1.44699 + 1.81447i 0 −0.311446 − 1.36453i 1.01653 − 0.489534i 0 1.09916 − 4.81573i
82.3 −0.869054 − 1.08976i 0 0.0127219 − 0.0557384i −1.34944 − 1.69215i 0 1.02284 + 4.48135i −2.58344 + 1.24412i 0 −0.671295 + 2.94113i
82.4 −0.620079 − 0.777554i 0 0.224949 − 0.985566i −0.913886 − 1.14598i 0 −0.762091 − 3.33894i −2.69790 + 1.29924i 0 −0.324378 + 1.42119i
82.5 −0.545907 − 0.684546i 0 0.274453 − 1.20246i 2.54463 + 3.19087i 0 0.410005 + 1.79635i −2.55068 + 1.22834i 0 0.795163 − 3.48384i
82.6 0.545907 + 0.684546i 0 0.274453 − 1.20246i −2.54463 − 3.19087i 0 0.410005 + 1.79635i 2.55068 − 1.22834i 0 0.795163 − 3.48384i
82.7 0.620079 + 0.777554i 0 0.224949 − 0.985566i 0.913886 + 1.14598i 0 −0.762091 − 3.33894i 2.69790 − 1.29924i 0 −0.324378 + 1.42119i
82.8 0.869054 + 1.08976i 0 0.0127219 − 0.0557384i 1.34944 + 1.69215i 0 1.02284 + 4.48135i 2.58344 − 1.24412i 0 −0.671295 + 2.94113i
82.9 1.32704 + 1.66405i 0 −0.563000 + 2.46666i −1.44699 − 1.81447i 0 −0.311446 − 1.36453i −1.01653 + 0.489534i 0 1.09916 − 4.81573i
82.10 1.62244 + 2.03447i 0 −1.06173 + 4.65174i 1.57981 + 1.98102i 0 −0.0818290 − 0.358516i −6.49744 + 3.12900i 0 −1.46718 + 6.42814i
136.1 −2.21804 + 1.06815i 0 2.53177 − 3.17474i 0.935499 − 0.450513i 0 2.91878 + 3.66004i −1.12884 + 4.94577i 0 −1.59376 + 1.99851i
136.2 −2.07439 + 0.998975i 0 2.05818 − 2.58087i −3.28060 + 1.57985i 0 −0.480272 − 0.602242i −0.666575 + 2.92046i 0 5.22702 − 6.55447i
136.3 −1.42552 + 0.686496i 0 0.313861 − 0.393569i 1.33026 − 0.640620i 0 0.300750 + 0.377129i 0.526918 − 2.30858i 0 −1.45653 + 1.82644i
136.4 −0.959163 + 0.461908i 0 −0.540346 + 0.677573i −0.748493 + 0.360455i 0 −2.74812 − 3.44603i 0.679091 − 2.97529i 0 0.551429 − 0.691470i
136.5 −0.0355415 + 0.0171159i 0 −1.24601 + 1.56245i 2.93813 − 1.41493i 0 1.13234 + 1.41991i 0.0350984 − 0.153776i 0 −0.0802077 + 0.100577i
136.6 0.0355415 − 0.0171159i 0 −1.24601 + 1.56245i −2.93813 + 1.41493i 0 1.13234 + 1.41991i −0.0350984 + 0.153776i 0 −0.0802077 + 0.100577i
136.7 0.959163 − 0.461908i 0 −0.540346 + 0.677573i 0.748493 − 0.360455i 0 −2.74812 − 3.44603i −0.679091 + 2.97529i 0 0.551429 − 0.691470i
136.8 1.42552 − 0.686496i 0 0.313861 − 0.393569i −1.33026 + 0.640620i 0 0.300750 + 0.377129i −0.526918 + 2.30858i 0 −1.45653 + 1.82644i
136.9 2.07439 − 0.998975i 0 2.05818 − 2.58087i 3.28060 − 1.57985i 0 −0.480272 − 0.602242i 0.666575 − 2.92046i 0 5.22702 − 6.55447i
136.10 2.21804 − 1.06815i 0 2.53177 − 3.17474i −0.935499 + 0.450513i 0 2.91878 + 3.66004i 1.12884 − 4.94577i 0 −1.59376 + 1.99851i
See all 60 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 82.10
Significant digits:
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
29.d even 7 1 inner
87.j odd 14 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 783.2.k.f ✓ 60
3.b odd 2 1 inner 783.2.k.f ✓ 60
29.d even 7 1 inner 783.2.k.f ✓ 60
87.j odd 14 1 inner 783.2.k.f ✓ 60
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
783.2.k.f ✓ 60 1.a even 1 1 trivial
783.2.k.f ✓ 60 3.b odd 2 1 inner
783.2.k.f ✓ 60 29.d even 7 1 inner
783.2.k.f ✓ 60 87.j odd 14 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{60} + 15 T_{2}^{58} + 154 T_{2}^{56} + 1289 T_{2}^{54} + 9533 T_{2}^{52} + 69070 T_{2}^{50} + \cdots + 41209 \) acting on \(S_{2}^{\mathrm{new}}(783, [\chi])\). Copy content Toggle raw display