Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [324,2,Mod(13,324)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(324, base_ring=CyclotomicField(54))
chi = DirichletCharacter(H, H._module([0, 8]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("324.13");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 324 = 2^{2} \cdot 3^{4} \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 324.m (of order \(27\), degree \(18\), 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: | \(2.58715302549\) |
Analytic rank: | \(0\) |
Dimension: | \(162\) |
Relative dimension: | \(9\) over \(\Q(\zeta_{27})\) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{27}]$ |
$q$-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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
13.1 | 0 | −1.73190 | − | 0.0227943i | 0 | −1.16619 | − | 3.89534i | 0 | 1.26893 | + | 2.94171i | 0 | 2.99896 | + | 0.0789550i | 0 | ||||||||||
13.2 | 0 | −1.48240 | + | 0.895824i | 0 | 0.0229196 | + | 0.0765569i | 0 | −1.39529 | − | 3.23464i | 0 | 1.39500 | − | 2.65593i | 0 | ||||||||||
13.3 | 0 | −1.37167 | − | 1.05760i | 0 | 0.465459 | + | 1.55474i | 0 | 0.273810 | + | 0.634762i | 0 | 0.762961 | + | 2.90136i | 0 | ||||||||||
13.4 | 0 | −0.429742 | + | 1.67789i | 0 | 0.575939 | + | 1.92377i | 0 | 0.833877 | + | 1.93314i | 0 | −2.63064 | − | 1.44212i | 0 | ||||||||||
13.5 | 0 | −0.355946 | − | 1.69508i | 0 | −0.474145 | − | 1.58376i | 0 | −0.187557 | − | 0.434806i | 0 | −2.74661 | + | 1.20671i | 0 | ||||||||||
13.6 | 0 | 1.19354 | − | 1.25517i | 0 | 0.462190 | + | 1.54382i | 0 | 1.59904 | + | 3.70700i | 0 | −0.150919 | − | 2.99620i | 0 | ||||||||||
13.7 | 0 | 1.26242 | + | 1.18587i | 0 | −0.909501 | − | 3.03794i | 0 | 0.410517 | + | 0.951685i | 0 | 0.187417 | + | 2.99414i | 0 | ||||||||||
13.8 | 0 | 1.50798 | + | 0.852058i | 0 | 1.16819 | + | 3.90202i | 0 | −1.69197 | − | 3.92242i | 0 | 1.54799 | + | 2.56977i | 0 | ||||||||||
13.9 | 0 | 1.55696 | − | 0.758860i | 0 | −0.491276 | − | 1.64098i | 0 | −1.11136 | − | 2.57643i | 0 | 1.84826 | − | 2.36303i | 0 | ||||||||||
25.1 | 0 | −1.73190 | + | 0.0227943i | 0 | −1.16619 | + | 3.89534i | 0 | 1.26893 | − | 2.94171i | 0 | 2.99896 | − | 0.0789550i | 0 | ||||||||||
25.2 | 0 | −1.48240 | − | 0.895824i | 0 | 0.0229196 | − | 0.0765569i | 0 | −1.39529 | + | 3.23464i | 0 | 1.39500 | + | 2.65593i | 0 | ||||||||||
25.3 | 0 | −1.37167 | + | 1.05760i | 0 | 0.465459 | − | 1.55474i | 0 | 0.273810 | − | 0.634762i | 0 | 0.762961 | − | 2.90136i | 0 | ||||||||||
25.4 | 0 | −0.429742 | − | 1.67789i | 0 | 0.575939 | − | 1.92377i | 0 | 0.833877 | − | 1.93314i | 0 | −2.63064 | + | 1.44212i | 0 | ||||||||||
25.5 | 0 | −0.355946 | + | 1.69508i | 0 | −0.474145 | + | 1.58376i | 0 | −0.187557 | + | 0.434806i | 0 | −2.74661 | − | 1.20671i | 0 | ||||||||||
25.6 | 0 | 1.19354 | + | 1.25517i | 0 | 0.462190 | − | 1.54382i | 0 | 1.59904 | − | 3.70700i | 0 | −0.150919 | + | 2.99620i | 0 | ||||||||||
25.7 | 0 | 1.26242 | − | 1.18587i | 0 | −0.909501 | + | 3.03794i | 0 | 0.410517 | − | 0.951685i | 0 | 0.187417 | − | 2.99414i | 0 | ||||||||||
25.8 | 0 | 1.50798 | − | 0.852058i | 0 | 1.16819 | − | 3.90202i | 0 | −1.69197 | + | 3.92242i | 0 | 1.54799 | − | 2.56977i | 0 | ||||||||||
25.9 | 0 | 1.55696 | + | 0.758860i | 0 | −0.491276 | + | 1.64098i | 0 | −1.11136 | + | 2.57643i | 0 | 1.84826 | + | 2.36303i | 0 | ||||||||||
49.1 | 0 | −1.71035 | + | 0.273337i | 0 | 0.101566 | − | 0.235457i | 0 | 0.0645556 | + | 1.10838i | 0 | 2.85057 | − | 0.935001i | 0 | ||||||||||
49.2 | 0 | −1.50806 | − | 0.851908i | 0 | −1.59595 | + | 3.69982i | 0 | −0.240156 | − | 4.12332i | 0 | 1.54850 | + | 2.56946i | 0 | ||||||||||
See next 80 embeddings (of 162 total) |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
81.g | even | 27 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 324.2.m.a | ✓ | 162 |
3.b | odd | 2 | 1 | 972.2.m.a | 162 | ||
81.g | even | 27 | 1 | inner | 324.2.m.a | ✓ | 162 |
81.h | odd | 54 | 1 | 972.2.m.a | 162 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
324.2.m.a | ✓ | 162 | 1.a | even | 1 | 1 | trivial |
324.2.m.a | ✓ | 162 | 81.g | even | 27 | 1 | inner |
972.2.m.a | 162 | 3.b | odd | 2 | 1 | ||
972.2.m.a | 162 | 81.h | odd | 54 | 1 |
Hecke kernels
This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(324, [\chi])\).