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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [108,3,Mod(5,108)] 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("108.5"); S:= CuspForms(chi, 3); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(108, base_ring=CyclotomicField(18)) chi = DirichletCharacter(H, H._module([0, 5])) N = Newforms(chi, 3, names="a")
 
Level: \( N \) \(=\) \( 108 = 2^{2} \cdot 3^{3} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 108.k (of order \(18\), degree \(6\), 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: \(2.94278685509\)
Analytic rank: \(0\)
Dimension: \(36\)
Relative dimension: \(6\) over \(\Q(\zeta_{18})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{18}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 36 q - 9 q^{5} + 6 q^{9} + 36 q^{11} + 45 q^{15} + 42 q^{21} - 18 q^{23} - 9 q^{25} - 18 q^{29} + 45 q^{31} - 153 q^{33} - 243 q^{35} - 123 q^{39} - 198 q^{41} + 90 q^{43} - 333 q^{45} - 243 q^{47} + 72 q^{49}+ \cdots - 945 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
5.1 0 −2.92720 − 0.656882i 0 0.740753 + 2.03520i 0 1.08248 + 6.13906i 0 8.13701 + 3.84565i 0
5.2 0 −1.95183 + 2.27824i 0 −3.25030 − 8.93012i 0 0.410040 + 2.32545i 0 −1.38071 − 8.89346i 0
5.3 0 −0.727525 − 2.91045i 0 0.0686711 + 0.188672i 0 −1.47862 − 8.38565i 0 −7.94141 + 4.23485i 0
5.4 0 1.02038 + 2.82114i 0 1.26650 + 3.47969i 0 −0.0728181 − 0.412972i 0 −6.91763 + 5.75729i 0
5.5 0 2.70588 − 1.29546i 0 1.65461 + 4.54600i 0 1.68621 + 9.56295i 0 5.64356 − 7.01072i 0
5.6 0 2.99363 − 0.195330i 0 −2.50118 − 6.87194i 0 −1.62729 − 9.22884i 0 8.92369 − 1.16949i 0
29.1 0 −2.78257 + 1.12129i 0 −2.69546 − 3.21232i 0 11.1367 − 4.05342i 0 6.48544 − 6.24012i 0
29.2 0 −2.59609 − 1.50344i 0 −0.298552 − 0.355800i 0 −10.1488 + 3.69384i 0 4.47934 + 7.80612i 0
29.3 0 −0.776129 + 2.89787i 0 2.68656 + 3.20172i 0 −4.88621 + 1.77844i 0 −7.79525 − 4.49824i 0
29.4 0 0.0634556 − 2.99933i 0 5.64904 + 6.73227i 0 4.05297 − 1.47516i 0 −8.99195 − 0.380648i 0
29.5 0 1.49775 − 2.59937i 0 −5.00278 − 5.96208i 0 −3.39388 + 1.23527i 0 −4.51349 − 7.78642i 0
29.6 0 2.88785 + 0.812609i 0 0.980262 + 1.16823i 0 3.23920 − 1.17897i 0 7.67933 + 4.69338i 0
41.1 0 −2.78257 − 1.12129i 0 −2.69546 + 3.21232i 0 11.1367 + 4.05342i 0 6.48544 + 6.24012i 0
41.2 0 −2.59609 + 1.50344i 0 −0.298552 + 0.355800i 0 −10.1488 − 3.69384i 0 4.47934 − 7.80612i 0
41.3 0 −0.776129 − 2.89787i 0 2.68656 − 3.20172i 0 −4.88621 − 1.77844i 0 −7.79525 + 4.49824i 0
41.4 0 0.0634556 + 2.99933i 0 5.64904 − 6.73227i 0 4.05297 + 1.47516i 0 −8.99195 + 0.380648i 0
41.5 0 1.49775 + 2.59937i 0 −5.00278 + 5.96208i 0 −3.39388 − 1.23527i 0 −4.51349 + 7.78642i 0
41.6 0 2.88785 − 0.812609i 0 0.980262 − 1.16823i 0 3.23920 + 1.17897i 0 7.67933 − 4.69338i 0
65.1 0 −2.92720 + 0.656882i 0 0.740753 − 2.03520i 0 1.08248 − 6.13906i 0 8.13701 − 3.84565i 0
65.2 0 −1.95183 − 2.27824i 0 −3.25030 + 8.93012i 0 0.410040 − 2.32545i 0 −1.38071 + 8.89346i 0
See all 36 embeddings
\(n\): e.g. 2-40 or 80-90
Embeddings: e.g. 1-3 or 5.6
Significant digits:
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
27.f odd 18 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 108.3.k.a ✓ 36
3.b odd 2 1 324.3.k.a 36
4.b odd 2 1 432.3.bc.b 36
27.e even 9 1 324.3.k.a 36
27.e even 9 1 2916.3.c.b 36
27.f odd 18 1 inner 108.3.k.a ✓ 36
27.f odd 18 1 2916.3.c.b 36
108.l even 18 1 432.3.bc.b 36
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
108.3.k.a ✓ 36 1.a even 1 1 trivial
108.3.k.a ✓ 36 27.f odd 18 1 inner
324.3.k.a 36 3.b odd 2 1
324.3.k.a 36 27.e even 9 1
432.3.bc.b 36 4.b odd 2 1
432.3.bc.b 36 108.l even 18 1
2916.3.c.b 36 27.e even 9 1
2916.3.c.b 36 27.f odd 18 1

Hecke kernels

This newform subspace is the entire newspace \(S_{3}^{\mathrm{new}}(108, [\chi])\).