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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [324,3,Mod(17,324)] 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("324.17"); S:= CuspForms(chi, 3); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(324, base_ring=CyclotomicField(18)) chi = DirichletCharacter(H, H._module([0, 11])) N = Newforms(chi, 3, names="a")
 
Level: \( N \) \(=\) \( 324 = 2^{2} \cdot 3^{4} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 324.k (of order \(18\), degree \(6\), not 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: \(8.82836056527\)
Analytic rank: \(0\)
Dimension: \(36\)
Relative dimension: \(6\) over \(\Q(\zeta_{18})\)
Twist minimal: no (minimal twist has level 108)
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.

\(\operatorname{Tr}(f)(q) = \) \( 36 q + 9 q^{5} - 36 q^{11} + 18 q^{23} - 9 q^{25} + 18 q^{29} + 45 q^{31} + 243 q^{35} + 198 q^{41} + 90 q^{43} + 243 q^{47} + 72 q^{49} - 252 q^{59} - 144 q^{61} - 747 q^{65} + 108 q^{67} - 324 q^{71}+ \cdots - 216 q^{97}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
17.1 0 0 0 −7.10446 − 1.25271i 0 −3.36440 + 2.82306i 0 0 0
17.2 0 0 0 −3.29223 − 0.580508i 0 3.84473 − 3.22611i 0 0 0
17.3 0 0 0 −2.92426 − 0.515626i 0 0.715829 − 0.600652i 0 0 0
17.4 0 0 0 4.32861 + 0.763252i 0 −2.73772 + 2.29722i 0 0 0
17.5 0 0 0 5.13754 + 0.905886i 0 −8.93347 + 7.49607i 0 0 0
17.6 0 0 0 7.65293 + 1.34942i 0 10.4750 − 8.78960i 0 0 0
89.1 0 0 0 −5.64904 − 6.73227i 0 4.05297 − 1.47516i 0 0 0
89.2 0 0 0 −2.68656 − 3.20172i 0 −4.88621 + 1.77844i 0 0 0
89.3 0 0 0 −0.980262 − 1.16823i 0 3.23920 − 1.17897i 0 0 0
89.4 0 0 0 0.298552 + 0.355800i 0 −10.1488 + 3.69384i 0 0 0
89.5 0 0 0 2.69546 + 3.21232i 0 11.1367 − 4.05342i 0 0 0
89.6 0 0 0 5.00278 + 5.96208i 0 −3.39388 + 1.23527i 0 0 0
125.1 0 0 0 −1.65461 − 4.54600i 0 1.68621 + 9.56295i 0 0 0
125.2 0 0 0 −1.26650 − 3.47969i 0 −0.0728181 − 0.412972i 0 0 0
125.3 0 0 0 −0.740753 − 2.03520i 0 1.08248 + 6.13906i 0 0 0
125.4 0 0 0 −0.0686711 − 0.188672i 0 −1.47862 − 8.38565i 0 0 0
125.5 0 0 0 2.50118 + 6.87194i 0 −1.62729 − 9.22884i 0 0 0
125.6 0 0 0 3.25030 + 8.93012i 0 0.410040 + 2.32545i 0 0 0
197.1 0 0 0 −1.65461 + 4.54600i 0 1.68621 − 9.56295i 0 0 0
197.2 0 0 0 −1.26650 + 3.47969i 0 −0.0728181 + 0.412972i 0 0 0
See all 36 embeddings
\(n\): e.g. 2-40 or 80-90
Embeddings: e.g. 1-3 or 17.6
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 324.3.k.a 36
3.b odd 2 1 108.3.k.a ✓ 36
12.b even 2 1 432.3.bc.b 36
27.e even 9 1 108.3.k.a ✓ 36
27.e even 9 1 2916.3.c.b 36
27.f odd 18 1 inner 324.3.k.a 36
27.f odd 18 1 2916.3.c.b 36
108.j odd 18 1 432.3.bc.b 36
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
108.3.k.a ✓ 36 3.b odd 2 1
108.3.k.a ✓ 36 27.e even 9 1
324.3.k.a 36 1.a even 1 1 trivial
324.3.k.a 36 27.f odd 18 1 inner
432.3.bc.b 36 12.b even 2 1
432.3.bc.b 36 108.j odd 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 \(S_{3}^{\mathrm{new}}(324, [\chi])\).