Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [324,4,Mod(37,324)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(324, base_ring=CyclotomicField(18))
chi = DirichletCharacter(H, H._module([0, 14]))
N = Newforms(chi, 4, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("324.37");
S:= CuspForms(chi, 4);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 324 = 2^{2} \cdot 3^{4} \) |
Weight: | \( k \) | \(=\) | \( 4 \) |
Character orbit: | \([\chi]\) | \(=\) | 324.i (of order \(9\), degree \(6\), not minimal) |
Newform invariants
comment: select newform
sage: f = N[0] # Warning: the index may be different
gp: f = lf[1] \\ Warning: the index may be different
Self dual: | no |
Analytic conductor: | \(19.1166188419\) |
Analytic rank: | \(0\) |
Dimension: | \(54\) |
Relative dimension: | \(9\) over \(\Q(\zeta_{9})\) |
Twist minimal: | no (minimal twist has level 108) |
Sato-Tate group: | $\mathrm{SU}(2)[C_{9}]$ |
$q$-expansion
The dimension is sufficiently large that we do not compute an algebraic \(q\)-expansion, but we have computed the trace expansion.
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.
comment: embeddings in the coefficient field
gp: mfembed(f)
Label | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
37.1 | 0 | 0 | 0 | −19.7528 | + | 7.18944i | 0 | 0.723940 | + | 4.10567i | 0 | 0 | 0 | ||||||||||||||
37.2 | 0 | 0 | 0 | −13.2490 | + | 4.82225i | 0 | 3.17294 | + | 17.9946i | 0 | 0 | 0 | ||||||||||||||
37.3 | 0 | 0 | 0 | −11.1609 | + | 4.06223i | 0 | −4.28132 | − | 24.2806i | 0 | 0 | 0 | ||||||||||||||
37.4 | 0 | 0 | 0 | −5.21695 | + | 1.89881i | 0 | 1.58868 | + | 9.00985i | 0 | 0 | 0 | ||||||||||||||
37.5 | 0 | 0 | 0 | 4.27250 | − | 1.55506i | 0 | −4.69568 | − | 26.6305i | 0 | 0 | 0 | ||||||||||||||
37.6 | 0 | 0 | 0 | 4.80495 | − | 1.74886i | 0 | 5.74341 | + | 32.5725i | 0 | 0 | 0 | ||||||||||||||
37.7 | 0 | 0 | 0 | 6.50317 | − | 2.36696i | 0 | 1.34685 | + | 7.63837i | 0 | 0 | 0 | ||||||||||||||
37.8 | 0 | 0 | 0 | 10.0092 | − | 3.64305i | 0 | −2.90933 | − | 16.4997i | 0 | 0 | 0 | ||||||||||||||
37.9 | 0 | 0 | 0 | 13.0875 | − | 4.76347i | 0 | −0.689494 | − | 3.91032i | 0 | 0 | 0 | ||||||||||||||
73.1 | 0 | 0 | 0 | −15.7004 | + | 13.1742i | 0 | −8.81657 | + | 3.20897i | 0 | 0 | 0 | ||||||||||||||
73.2 | 0 | 0 | 0 | −8.18465 | + | 6.86774i | 0 | −24.4722 | + | 8.90714i | 0 | 0 | 0 | ||||||||||||||
73.3 | 0 | 0 | 0 | −7.52940 | + | 6.31792i | 0 | 24.9076 | − | 9.06562i | 0 | 0 | 0 | ||||||||||||||
73.4 | 0 | 0 | 0 | −6.54095 | + | 5.48851i | 0 | 11.9650 | − | 4.35489i | 0 | 0 | 0 | ||||||||||||||
73.5 | 0 | 0 | 0 | 1.73722 | − | 1.45770i | 0 | −5.32680 | + | 1.93880i | 0 | 0 | 0 | ||||||||||||||
73.6 | 0 | 0 | 0 | 5.18194 | − | 4.34817i | 0 | 16.8240 | − | 6.12342i | 0 | 0 | 0 | ||||||||||||||
73.7 | 0 | 0 | 0 | 5.19451 | − | 4.35871i | 0 | −17.8009 | + | 6.47901i | 0 | 0 | 0 | ||||||||||||||
73.8 | 0 | 0 | 0 | 12.8040 | − | 10.7439i | 0 | −20.8033 | + | 7.57178i | 0 | 0 | 0 | ||||||||||||||
73.9 | 0 | 0 | 0 | 14.9393 | − | 12.5356i | 0 | 23.5233 | − | 8.56177i | 0 | 0 | 0 | ||||||||||||||
145.1 | 0 | 0 | 0 | −3.27839 | + | 18.5927i | 0 | 4.82725 | − | 4.05055i | 0 | 0 | 0 | ||||||||||||||
145.2 | 0 | 0 | 0 | −2.26368 | + | 12.8380i | 0 | −3.04053 | + | 2.55131i | 0 | 0 | 0 | ||||||||||||||
See all 54 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
27.e | even | 9 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 324.4.i.a | 54 | |
3.b | odd | 2 | 1 | 108.4.i.a | ✓ | 54 | |
27.e | even | 9 | 1 | inner | 324.4.i.a | 54 | |
27.f | odd | 18 | 1 | 108.4.i.a | ✓ | 54 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
108.4.i.a | ✓ | 54 | 3.b | odd | 2 | 1 | |
108.4.i.a | ✓ | 54 | 27.f | odd | 18 | 1 | |
324.4.i.a | 54 | 1.a | even | 1 | 1 | trivial | |
324.4.i.a | 54 | 27.e | even | 9 | 1 | inner |
Hecke kernels
This newform subspace is the entire newspace \(S_{4}^{\mathrm{new}}(324, [\chi])\).