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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [18,26,Mod(7,18)] 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("18.7"); S:= CuspForms(chi, 26); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(18, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([4])) N = Newforms(chi, 26, names="a")
 
Level: \( N \) \(=\) \( 18 = 2 \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 26 \)
Character orbit: \([\chi]\) \(=\) 18.c (of order \(3\), degree \(2\), 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: \(71.2794203914\)
Analytic rank: \(0\)
Dimension: \(26\)
Relative dimension: \(13\) over \(\Q(\zeta_{3})\)
Twist minimal: yes
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.

\(\operatorname{Tr}(f)(q) = \) \( 26 q - 53248 q^{2} + 151593 q^{3} - 218103808 q^{4} + 49854096 q^{5} + 768700416 q^{6} - 35213963498 q^{7} + 1786706395136 q^{8} - 906685440585 q^{9} - 408404754432 q^{10} - 1998346177329 q^{11} - 5691905409024 q^{12}+ \cdots + 19\!\cdots\!26 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
7.1 −2048.00 + 3547.24i −916998. − 80022.4i −8.38861e6 − 1.45295e7i 3.09690e8 + 5.36399e8i 2.16187e9 − 3.08893e9i 1.01463e10 − 1.75739e10i 6.87195e10 8.34481e11 + 1.46761e11i −2.53698e12
7.2 −2048.00 + 3547.24i −912391. − 121783.i −8.38861e6 − 1.45295e7i −2.67172e8 − 4.62756e8i 2.30057e9 − 2.98706e9i −2.63260e10 + 4.55979e10i 6.87195e10 8.17626e11 + 2.22227e11i 2.18867e12
7.3 −2048.00 + 3547.24i −683105. − 616973.i −8.38861e6 − 1.45295e7i −2.82392e8 − 4.89117e8i 3.58755e9 − 1.15958e9i 2.35560e10 − 4.08002e10i 6.87195e10 8.59761e10 + 8.42915e11i 2.31335e12
7.4 −2048.00 + 3547.24i −539595. + 745739.i −8.38861e6 − 1.45295e7i 1.32723e8 + 2.29884e8i −1.54022e9 − 3.44134e9i 1.71990e10 − 2.97895e10i 6.87195e10 −2.64964e11 − 8.04793e11i −1.08727e12
7.5 −2048.00 + 3547.24i −374466. − 840871.i −8.38861e6 − 1.45295e7i 5.33479e8 + 9.24012e8i 3.74968e9 + 3.93782e8i −1.00136e10 + 1.73441e10i 6.87195e10 −5.66839e11 + 6.29755e11i −4.37026e12
7.6 −2048.00 + 3547.24i −210133. + 896177.i −8.38861e6 − 1.45295e7i −5.09767e8 − 8.82942e8i −2.74860e9 − 2.58076e9i −3.19078e9 + 5.52659e9i 6.87195e10 −7.58977e11 − 3.76632e11i 4.17601e12
7.7 −2048.00 + 3547.24i 30075.0 − 919991.i −8.38861e6 − 1.45295e7i −1.71781e8 − 2.97534e8i 3.20184e9 + 1.99083e9i −2.30982e10 + 4.00073e10i 6.87195e10 −8.45480e11 − 5.53375e10i 1.40723e12
7.8 −2048.00 + 3547.24i 195863. + 899403.i −8.38861e6 − 1.45295e7i 3.88234e8 + 6.72441e8i −3.59153e9 − 1.14720e9i −2.74095e10 + 4.74746e10i 6.87195e10 −7.70564e11 + 3.52320e11i −3.18041e12
7.9 −2048.00 + 3547.24i 451927. − 801904.i −8.38861e6 − 1.45295e7i −2.81571e7 − 4.87695e7i 1.91900e9 + 3.24539e9i 2.24971e10 − 3.89661e10i 6.87195e10 −4.38813e11 − 7.24804e11i 2.30663e11
7.10 −2048.00 + 3547.24i 473097. + 789600.i −8.38861e6 − 1.45295e7i 1.88246e7 + 3.26052e7i −3.76980e9 + 6.10888e7i 2.43675e10 − 4.22058e10i 6.87195e10 −3.99647e11 + 7.47115e11i −1.54211e11
7.11 −2048.00 + 3547.24i 771680. − 501795.i −8.38861e6 − 1.45295e7i 2.67502e8 + 4.63326e8i 1.99588e8 + 3.76501e9i −2.72292e9 + 4.71623e9i 6.87195e10 3.43692e11 − 7.74451e11i −2.19137e12
7.12 −2048.00 + 3547.24i 877537. + 277881.i −8.38861e6 − 1.45295e7i −7.76671e6 − 1.34523e7i −2.78291e9 + 2.54373e9i −2.51258e10 + 4.35191e10i 6.87195e10 6.92852e11 + 4.87702e11i 6.36249e10
7.13 −2048.00 + 3547.24i 912305. + 122428.i −8.38861e6 − 1.45295e7i −3.58489e8 − 6.20922e8i −2.30268e9 + 2.98543e9i 2.51388e9 − 4.35418e9i 6.87195e10 8.17312e11 + 2.23382e11i 2.93675e12
13.1 −2048.00 − 3547.24i −916998. + 80022.4i −8.38861e6 + 1.45295e7i 3.09690e8 − 5.36399e8i 2.16187e9 + 3.08893e9i 1.01463e10 + 1.75739e10i 6.87195e10 8.34481e11 − 1.46761e11i −2.53698e12
13.2 −2048.00 − 3547.24i −912391. + 121783.i −8.38861e6 + 1.45295e7i −2.67172e8 + 4.62756e8i 2.30057e9 + 2.98706e9i −2.63260e10 − 4.55979e10i 6.87195e10 8.17626e11 − 2.22227e11i 2.18867e12
13.3 −2048.00 − 3547.24i −683105. + 616973.i −8.38861e6 + 1.45295e7i −2.82392e8 + 4.89117e8i 3.58755e9 + 1.15958e9i 2.35560e10 + 4.08002e10i 6.87195e10 8.59761e10 − 8.42915e11i 2.31335e12
13.4 −2048.00 − 3547.24i −539595. − 745739.i −8.38861e6 + 1.45295e7i 1.32723e8 − 2.29884e8i −1.54022e9 + 3.44134e9i 1.71990e10 + 2.97895e10i 6.87195e10 −2.64964e11 + 8.04793e11i −1.08727e12
13.5 −2048.00 − 3547.24i −374466. + 840871.i −8.38861e6 + 1.45295e7i 5.33479e8 − 9.24012e8i 3.74968e9 − 3.93782e8i −1.00136e10 − 1.73441e10i 6.87195e10 −5.66839e11 − 6.29755e11i −4.37026e12
13.6 −2048.00 − 3547.24i −210133. − 896177.i −8.38861e6 + 1.45295e7i −5.09767e8 + 8.82942e8i −2.74860e9 + 2.58076e9i −3.19078e9 − 5.52659e9i 6.87195e10 −7.58977e11 + 3.76632e11i 4.17601e12
13.7 −2048.00 − 3547.24i 30075.0 + 919991.i −8.38861e6 + 1.45295e7i −1.71781e8 + 2.97534e8i 3.20184e9 − 1.99083e9i −2.30982e10 − 4.00073e10i 6.87195e10 −8.45480e11 + 5.53375e10i 1.40723e12
See all 26 embeddings
\(n\): e.g. 2-40 or 80-90
Embeddings: e.g. 1-3 or 7.13
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 18.26.c.b ✓ 26
3.b odd 2 1 54.26.c.b 26
9.c even 3 1 inner 18.26.c.b ✓ 26
9.d odd 6 1 54.26.c.b 26
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
18.26.c.b ✓ 26 1.a even 1 1 trivial
18.26.c.b ✓ 26 9.c even 3 1 inner
54.26.c.b 26 3.b odd 2 1
54.26.c.b 26 9.d odd 6 1