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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [54,26,Mod(19,54)] 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("54.19"); S:= CuspForms(chi, 26); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(54, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([4])) N = Newforms(chi, 26, names="a")
 
Level: \( N \) \(=\) \( 54 = 2 \cdot 3^{3} \)
Weight: \( k \) \(=\) \( 26 \)
Character orbit: \([\chi]\) \(=\) 54.c (of order \(3\), degree \(2\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [26] 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: \(213.838261174\)
Analytic rank: \(0\)
Dimension: \(26\)
Relative dimension: \(13\) over \(\Q(\zeta_{3})\)
Twist minimal: no (minimal twist has level 18)
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.

\(\operatorname{Tr}(f)(q) = \) \( 26 q + 53248 q^{2} - 218103808 q^{4} - 49854096 q^{5} - 35213963498 q^{7} - 1786706395136 q^{8} - 408404754432 q^{10} + 1998346177329 q^{11} - 158355783504242 q^{13} + 144236394487808 q^{14} - 36\!\cdots\!28 q^{16}+ \cdots - 14\!\cdots\!64 q^{98}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
19.1 2048.00 − 3547.24i 0 −8.38861e6 − 1.45295e7i −5.33479e8 − 9.24012e8i 0 −1.00136e10 + 1.73441e10i −6.87195e10 0 −4.37026e12
19.2 2048.00 − 3547.24i 0 −8.38861e6 − 1.45295e7i −3.88234e8 − 6.72441e8i 0 −2.74095e10 + 4.74746e10i −6.87195e10 0 −3.18041e12
19.3 2048.00 − 3547.24i 0 −8.38861e6 − 1.45295e7i −3.09690e8 − 5.36399e8i 0 1.01463e10 − 1.75739e10i −6.87195e10 0 −2.53698e12
19.4 2048.00 − 3547.24i 0 −8.38861e6 − 1.45295e7i −2.67502e8 − 4.63326e8i 0 −2.72292e9 + 4.71623e9i −6.87195e10 0 −2.19137e12
19.5 2048.00 − 3547.24i 0 −8.38861e6 − 1.45295e7i −1.32723e8 − 2.29884e8i 0 1.71990e10 − 2.97895e10i −6.87195e10 0 −1.08727e12
19.6 2048.00 − 3547.24i 0 −8.38861e6 − 1.45295e7i −1.88246e7 − 3.26052e7i 0 2.43675e10 − 4.22058e10i −6.87195e10 0 −1.54211e11
19.7 2048.00 − 3547.24i 0 −8.38861e6 − 1.45295e7i 7.76671e6 + 1.34523e7i 0 −2.51258e10 + 4.35191e10i −6.87195e10 0 6.36249e10
19.8 2048.00 − 3547.24i 0 −8.38861e6 − 1.45295e7i 2.81571e7 + 4.87695e7i 0 2.24971e10 − 3.89661e10i −6.87195e10 0 2.30663e11
19.9 2048.00 − 3547.24i 0 −8.38861e6 − 1.45295e7i 1.71781e8 + 2.97534e8i 0 −2.30982e10 + 4.00073e10i −6.87195e10 0 1.40723e12
19.10 2048.00 − 3547.24i 0 −8.38861e6 − 1.45295e7i 2.67172e8 + 4.62756e8i 0 −2.63260e10 + 4.55979e10i −6.87195e10 0 2.18867e12
19.11 2048.00 − 3547.24i 0 −8.38861e6 − 1.45295e7i 2.82392e8 + 4.89117e8i 0 2.35560e10 − 4.08002e10i −6.87195e10 0 2.31335e12
19.12 2048.00 − 3547.24i 0 −8.38861e6 − 1.45295e7i 3.58489e8 + 6.20922e8i 0 2.51388e9 − 4.35418e9i −6.87195e10 0 2.93675e12
19.13 2048.00 − 3547.24i 0 −8.38861e6 − 1.45295e7i 5.09767e8 + 8.82942e8i 0 −3.19078e9 + 5.52659e9i −6.87195e10 0 4.17601e12
37.1 2048.00 + 3547.24i 0 −8.38861e6 + 1.45295e7i −5.33479e8 + 9.24012e8i 0 −1.00136e10 − 1.73441e10i −6.87195e10 0 −4.37026e12
37.2 2048.00 + 3547.24i 0 −8.38861e6 + 1.45295e7i −3.88234e8 + 6.72441e8i 0 −2.74095e10 − 4.74746e10i −6.87195e10 0 −3.18041e12
37.3 2048.00 + 3547.24i 0 −8.38861e6 + 1.45295e7i −3.09690e8 + 5.36399e8i 0 1.01463e10 + 1.75739e10i −6.87195e10 0 −2.53698e12
37.4 2048.00 + 3547.24i 0 −8.38861e6 + 1.45295e7i −2.67502e8 + 4.63326e8i 0 −2.72292e9 − 4.71623e9i −6.87195e10 0 −2.19137e12
37.5 2048.00 + 3547.24i 0 −8.38861e6 + 1.45295e7i −1.32723e8 + 2.29884e8i 0 1.71990e10 + 2.97895e10i −6.87195e10 0 −1.08727e12
37.6 2048.00 + 3547.24i 0 −8.38861e6 + 1.45295e7i −1.88246e7 + 3.26052e7i 0 2.43675e10 + 4.22058e10i −6.87195e10 0 −1.54211e11
37.7 2048.00 + 3547.24i 0 −8.38861e6 + 1.45295e7i 7.76671e6 − 1.34523e7i 0 −2.51258e10 − 4.35191e10i −6.87195e10 0 6.36249e10
See all 26 embeddings
\(n\): e.g. 2-40 or 80-90
Embeddings: e.g. 1-3 or 19.13
Significant digits:
Format:

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 54.26.c.b 26
3.b odd 2 1 18.26.c.b ✓ 26
9.c even 3 1 inner 54.26.c.b 26
9.d odd 6 1 18.26.c.b ✓ 26
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
18.26.c.b ✓ 26 3.b odd 2 1
18.26.c.b ✓ 26 9.d odd 6 1
54.26.c.b 26 1.a even 1 1 trivial
54.26.c.b 26 9.c even 3 1 inner