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Properties

Label 792.2.k.a
Level $792$
Weight $2$
Character orbit 792.k
Analytic conductor $6.324$
Analytic rank $0$
Dimension $40$
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] = [792,2,Mod(683,792)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(792, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 1, 1, 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("792.683"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Level: \( N \) \(=\) \( 792 = 2^{3} \cdot 3^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 792.k (of order \(2\), degree \(1\), 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: \(6.32415184009\)
Analytic rank: \(0\)
Dimension: \(40\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 40 q - 8 q^{4} + 8 q^{10} - 8 q^{16} + 32 q^{19} + 40 q^{25} - 32 q^{28} + 32 q^{34} + 16 q^{40} - 24 q^{46} - 8 q^{49} + 40 q^{52} - 8 q^{58} + 16 q^{64} + 72 q^{70} + 32 q^{73} - 8 q^{76} - 64 q^{82}+ \cdots - 32 q^{97}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
683.1 −1.41385 − 0.0321228i 0 1.99794 + 0.0908336i −0.682984 0 2.58173i −2.82186 − 0.192604i 0 0.965636 + 0.0219394i
683.2 −1.41385 + 0.0321228i 0 1.99794 − 0.0908336i −0.682984 0 − 2.58173i −2.82186 + 0.192604i 0 0.965636 − 0.0219394i
683.3 −1.34782 − 0.428242i 0 1.63322 + 1.15438i 1.96913 0 1.38646i −1.70692 − 2.25531i 0 −2.65403 − 0.843265i
683.4 −1.34782 + 0.428242i 0 1.63322 − 1.15438i 1.96913 0 − 1.38646i −1.70692 + 2.25531i 0 −2.65403 + 0.843265i
683.5 −1.25027 − 0.660932i 0 1.12634 + 1.65268i −3.71620 0 1.27025i −0.315915 − 2.81073i 0 4.64625 + 2.45616i
683.6 −1.25027 + 0.660932i 0 1.12634 − 1.65268i −3.71620 0 − 1.27025i −0.315915 + 2.81073i 0 4.64625 − 2.45616i
683.7 −1.08198 − 0.910674i 0 0.341347 + 1.97066i −2.67216 0 2.05923i 1.42529 − 2.44306i 0 2.89121 + 2.43346i
683.8 −1.08198 + 0.910674i 0 0.341347 − 1.97066i −2.67216 0 − 2.05923i 1.42529 + 2.44306i 0 2.89121 − 2.43346i
683.9 −0.906629 − 1.08537i 0 −0.356048 + 1.96805i 1.01786 0 2.06769i 2.45886 − 1.39785i 0 −0.922818 − 1.10475i
683.10 −0.906629 + 1.08537i 0 −0.356048 − 1.96805i 1.01786 0 − 2.06769i 2.45886 + 1.39785i 0 −0.922818 + 1.10475i
683.11 −0.885984 − 1.10229i 0 −0.430065 + 1.95321i 3.67452 0 4.28843i 2.53403 − 1.25646i 0 −3.25556 − 4.05037i
683.12 −0.885984 + 1.10229i 0 −0.430065 − 1.95321i 3.67452 0 − 4.28843i 2.53403 + 1.25646i 0 −3.25556 + 4.05037i
683.13 −0.747372 − 1.20060i 0 −0.882871 + 1.79459i 1.25912 0 − 4.96906i 2.81441 − 0.281250i 0 −0.941027 − 1.51169i
683.14 −0.747372 + 1.20060i 0 −0.882871 − 1.79459i 1.25912 0 4.96906i 2.81441 + 0.281250i 0 −0.941027 + 1.51169i
683.15 −0.401488 − 1.35603i 0 −1.67761 + 1.08886i −0.138848 0 − 0.456841i 2.15006 + 1.83773i 0 0.0557460 + 0.188282i
683.16 −0.401488 + 1.35603i 0 −1.67761 − 1.08886i −0.138848 0 0.456841i 2.15006 − 1.83773i 0 0.0557460 − 0.188282i
683.17 −0.313193 − 1.37910i 0 −1.80382 + 0.863848i −1.89382 0 − 3.03135i 1.75627 + 2.21709i 0 0.593133 + 2.61177i
683.18 −0.313193 + 1.37910i 0 −1.80382 − 0.863848i −1.89382 0 3.03135i 1.75627 − 2.21709i 0 0.593133 − 2.61177i
683.19 −0.160590 − 1.40507i 0 −1.94842 + 0.451279i −3.86990 0 − 0.896188i 0.946974 + 2.66519i 0 0.621467 + 5.43747i
683.20 −0.160590 + 1.40507i 0 −1.94842 − 0.451279i −3.86990 0 0.896188i 0.946974 − 2.66519i 0 0.621467 − 5.43747i
See all 40 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 683.40
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
8.d odd 2 1 inner
24.f even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 792.2.k.a ✓ 40
3.b odd 2 1 inner 792.2.k.a ✓ 40
4.b odd 2 1 3168.2.k.a 40
8.b even 2 1 3168.2.k.a 40
8.d odd 2 1 inner 792.2.k.a ✓ 40
12.b even 2 1 3168.2.k.a 40
24.f even 2 1 inner 792.2.k.a ✓ 40
24.h odd 2 1 3168.2.k.a 40
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
792.2.k.a ✓ 40 1.a even 1 1 trivial
792.2.k.a ✓ 40 3.b odd 2 1 inner
792.2.k.a ✓ 40 8.d odd 2 1 inner
792.2.k.a ✓ 40 24.f even 2 1 inner
3168.2.k.a 40 4.b odd 2 1
3168.2.k.a 40 8.b even 2 1
3168.2.k.a 40 12.b even 2 1
3168.2.k.a 40 24.h odd 2 1

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(792, [\chi])\).