Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [644,3,Mod(137,644)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(644, base_ring=CyclotomicField(6))
chi = DirichletCharacter(H, H._module([0, 4, 3]))
N = Newforms(chi, 3, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("644.137");
S:= CuspForms(chi, 3);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 644 = 2^{2} \cdot 7 \cdot 23 \) |
Weight: | \( k \) | \(=\) | \( 3 \) |
Character orbit: | \([\chi]\) | \(=\) | 644.l (of order \(6\), degree \(2\), 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: | \(17.5477290248\) |
Analytic rank: | \(0\) |
Dimension: | \(64\) |
Relative dimension: | \(32\) over \(\Q(\zeta_{6})\) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{6}]$ |
$q$-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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
137.1 | 0 | −2.90811 | + | 5.03700i | 0 | −7.84842 | + | 4.53128i | 0 | −2.06919 | − | 6.68719i | 0 | −12.4142 | − | 21.5021i | 0 | ||||||||||
137.2 | 0 | −2.90811 | + | 5.03700i | 0 | 7.84842 | − | 4.53128i | 0 | 2.06919 | + | 6.68719i | 0 | −12.4142 | − | 21.5021i | 0 | ||||||||||
137.3 | 0 | −2.46727 | + | 4.27343i | 0 | 3.99932 | − | 2.30901i | 0 | −0.938284 | − | 6.93683i | 0 | −7.67483 | − | 13.2932i | 0 | ||||||||||
137.4 | 0 | −2.46727 | + | 4.27343i | 0 | −3.99932 | + | 2.30901i | 0 | 0.938284 | + | 6.93683i | 0 | −7.67483 | − | 13.2932i | 0 | ||||||||||
137.5 | 0 | −2.13349 | + | 3.69531i | 0 | −0.685850 | + | 0.395976i | 0 | 6.87030 | + | 1.34124i | 0 | −4.60352 | − | 7.97353i | 0 | ||||||||||
137.6 | 0 | −2.13349 | + | 3.69531i | 0 | 0.685850 | − | 0.395976i | 0 | −6.87030 | − | 1.34124i | 0 | −4.60352 | − | 7.97353i | 0 | ||||||||||
137.7 | 0 | −1.58316 | + | 2.74211i | 0 | 2.93433 | − | 1.69414i | 0 | −6.74912 | + | 1.85726i | 0 | −0.512791 | − | 0.888179i | 0 | ||||||||||
137.8 | 0 | −1.58316 | + | 2.74211i | 0 | −2.93433 | + | 1.69414i | 0 | 6.74912 | − | 1.85726i | 0 | −0.512791 | − | 0.888179i | 0 | ||||||||||
137.9 | 0 | −1.40566 | + | 2.43468i | 0 | 3.68913 | − | 2.12992i | 0 | 1.00033 | − | 6.92816i | 0 | 0.548214 | + | 0.949535i | 0 | ||||||||||
137.10 | 0 | −1.40566 | + | 2.43468i | 0 | −3.68913 | + | 2.12992i | 0 | −1.00033 | + | 6.92816i | 0 | 0.548214 | + | 0.949535i | 0 | ||||||||||
137.11 | 0 | −0.711722 | + | 1.23274i | 0 | −5.51457 | + | 3.18384i | 0 | −0.907879 | − | 6.94088i | 0 | 3.48690 | + | 6.03950i | 0 | ||||||||||
137.12 | 0 | −0.711722 | + | 1.23274i | 0 | 5.51457 | − | 3.18384i | 0 | 0.907879 | + | 6.94088i | 0 | 3.48690 | + | 6.03950i | 0 | ||||||||||
137.13 | 0 | −0.690959 | + | 1.19678i | 0 | −6.75903 | + | 3.90232i | 0 | −6.41594 | + | 2.79924i | 0 | 3.54515 | + | 6.14038i | 0 | ||||||||||
137.14 | 0 | −0.690959 | + | 1.19678i | 0 | 6.75903 | − | 3.90232i | 0 | 6.41594 | − | 2.79924i | 0 | 3.54515 | + | 6.14038i | 0 | ||||||||||
137.15 | 0 | −0.618470 | + | 1.07122i | 0 | 5.81540 | − | 3.35753i | 0 | −2.40948 | + | 6.57224i | 0 | 3.73499 | + | 6.46919i | 0 | ||||||||||
137.16 | 0 | −0.618470 | + | 1.07122i | 0 | −5.81540 | + | 3.35753i | 0 | 2.40948 | − | 6.57224i | 0 | 3.73499 | + | 6.46919i | 0 | ||||||||||
137.17 | 0 | 0.324570 | − | 0.562173i | 0 | −0.620474 | + | 0.358231i | 0 | −5.98238 | − | 3.63471i | 0 | 4.28931 | + | 7.42930i | 0 | ||||||||||
137.18 | 0 | 0.324570 | − | 0.562173i | 0 | 0.620474 | − | 0.358231i | 0 | 5.98238 | + | 3.63471i | 0 | 4.28931 | + | 7.42930i | 0 | ||||||||||
137.19 | 0 | 0.426440 | − | 0.738616i | 0 | −5.42556 | + | 3.13245i | 0 | 3.96981 | + | 5.76547i | 0 | 4.13630 | + | 7.16428i | 0 | ||||||||||
137.20 | 0 | 0.426440 | − | 0.738616i | 0 | 5.42556 | − | 3.13245i | 0 | −3.96981 | − | 5.76547i | 0 | 4.13630 | + | 7.16428i | 0 | ||||||||||
See all 64 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
7.c | even | 3 | 1 | inner |
23.b | odd | 2 | 1 | inner |
161.f | odd | 6 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 644.3.l.a | ✓ | 64 |
7.c | even | 3 | 1 | inner | 644.3.l.a | ✓ | 64 |
23.b | odd | 2 | 1 | inner | 644.3.l.a | ✓ | 64 |
161.f | odd | 6 | 1 | inner | 644.3.l.a | ✓ | 64 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
644.3.l.a | ✓ | 64 | 1.a | even | 1 | 1 | trivial |
644.3.l.a | ✓ | 64 | 7.c | even | 3 | 1 | inner |
644.3.l.a | ✓ | 64 | 23.b | odd | 2 | 1 | inner |
644.3.l.a | ✓ | 64 | 161.f | odd | 6 | 1 | inner |
Hecke kernels
This newform subspace is the entire newspace \(S_{3}^{\mathrm{new}}(644, [\chi])\).