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Properties

Label 783.2.e.a
Level $783$
Weight $2$
Character orbit 783.e
Analytic conductor $6.252$
Analytic rank $0$
Dimension $22$
Inner twists $2$

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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [783,2,Mod(262,783)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(783, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([4, 0])) 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("783.262"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Level: \( N \) \(=\) \( 783 = 3^{3} \cdot 29 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 783.e (of order \(3\), degree \(2\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [22] 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: \(6.25228647827\)
Analytic rank: \(0\)
Dimension: \(22\)
Relative dimension: \(11\) over \(\Q(\zeta_{3})\)
Twist minimal: no (minimal twist has level 261)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 22 q + q^{2} - 5 q^{4} - q^{5} + 7 q^{7} - 6 q^{8} - 20 q^{10} + 3 q^{11} + 7 q^{13} - 10 q^{14} + 7 q^{16} - 2 q^{17} - 56 q^{19} - 4 q^{20} + 13 q^{22} + 4 q^{23} + 4 q^{25} + 12 q^{26} - 32 q^{28}+ \cdots - 86 q^{98}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
262.1 −1.13175 + 1.96024i 0 −1.56171 − 2.70495i 0.731092 + 1.26629i 0 0.881947 − 1.52758i 2.54284 0 −3.30965
262.2 −0.942480 + 1.63242i 0 −0.776536 − 1.34500i 0.325626 + 0.564001i 0 −0.0750596 + 0.130007i −0.842439 0 −1.22758
262.3 −0.788768 + 1.36619i 0 −0.244311 − 0.423159i −0.409773 − 0.709748i 0 −1.13728 + 1.96983i −2.38425 0 1.29286
262.4 −0.489415 + 0.847692i 0 0.520946 + 0.902304i 1.17228 + 2.03045i 0 0.172733 − 0.299183i −2.97750 0 −2.29493
262.5 −0.100158 + 0.173479i 0 0.979937 + 1.69730i −0.556635 − 0.964120i 0 0.359329 − 0.622375i −0.793225 0 0.223006
262.6 −0.0549262 + 0.0951349i 0 0.993966 + 1.72160i −1.31653 − 2.28029i 0 1.68615 − 2.92049i −0.438084 0 0.289247
262.7 0.391918 − 0.678822i 0 0.692800 + 1.19996i −0.728601 − 1.26197i 0 −2.16737 + 3.75400i 2.65376 0 −1.14221
262.8 0.540070 − 0.935429i 0 0.416648 + 0.721656i 1.91815 + 3.32233i 0 1.07867 − 1.86831i 3.06036 0 4.14374
262.9 0.764092 − 1.32345i 0 −0.167673 − 0.290418i −0.0756080 − 0.130957i 0 0.0289713 − 0.0501797i 2.54390 0 −0.231086
262.10 1.10224 − 1.90914i 0 −1.42988 − 2.47662i 0.463291 + 0.802444i 0 2.26860 − 3.92934i −1.89533 0 2.04264
262.11 1.20917 − 2.09435i 0 −1.92419 − 3.33280i −2.02330 − 3.50445i 0 0.403314 − 0.698560i −4.47002 0 −9.78605
523.1 −1.13175 − 1.96024i 0 −1.56171 + 2.70495i 0.731092 − 1.26629i 0 0.881947 + 1.52758i 2.54284 0 −3.30965
523.2 −0.942480 − 1.63242i 0 −0.776536 + 1.34500i 0.325626 − 0.564001i 0 −0.0750596 − 0.130007i −0.842439 0 −1.22758
523.3 −0.788768 − 1.36619i 0 −0.244311 + 0.423159i −0.409773 + 0.709748i 0 −1.13728 − 1.96983i −2.38425 0 1.29286
523.4 −0.489415 − 0.847692i 0 0.520946 − 0.902304i 1.17228 − 2.03045i 0 0.172733 + 0.299183i −2.97750 0 −2.29493
523.5 −0.100158 − 0.173479i 0 0.979937 − 1.69730i −0.556635 + 0.964120i 0 0.359329 + 0.622375i −0.793225 0 0.223006
523.6 −0.0549262 − 0.0951349i 0 0.993966 − 1.72160i −1.31653 + 2.28029i 0 1.68615 + 2.92049i −0.438084 0 0.289247
523.7 0.391918 + 0.678822i 0 0.692800 − 1.19996i −0.728601 + 1.26197i 0 −2.16737 − 3.75400i 2.65376 0 −1.14221
523.8 0.540070 + 0.935429i 0 0.416648 − 0.721656i 1.91815 − 3.32233i 0 1.07867 + 1.86831i 3.06036 0 4.14374
523.9 0.764092 + 1.32345i 0 −0.167673 + 0.290418i −0.0756080 + 0.130957i 0 0.0289713 + 0.0501797i 2.54390 0 −0.231086
See all 22 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 262.11
Significant digits:
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
9.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 783.2.e.a 22
3.b odd 2 1 261.2.e.a ✓ 22
9.c even 3 1 inner 783.2.e.a 22
9.c even 3 1 2349.2.a.e 11
9.d odd 6 1 261.2.e.a ✓ 22
9.d odd 6 1 2349.2.a.f 11
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
261.2.e.a ✓ 22 3.b odd 2 1
261.2.e.a ✓ 22 9.d odd 6 1
783.2.e.a 22 1.a even 1 1 trivial
783.2.e.a 22 9.c even 3 1 inner
2349.2.a.e 11 9.c even 3 1
2349.2.a.f 11 9.d odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{22} - T_{2}^{21} + 14 T_{2}^{20} - 9 T_{2}^{19} + 121 T_{2}^{18} - 66 T_{2}^{17} + 627 T_{2}^{16} + \cdots + 1 \) acting on \(S_{2}^{\mathrm{new}}(783, [\chi])\). Copy content Toggle raw display