Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [504,2,Mod(323,504)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(504, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([1, 1, 1, 0]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("504.323");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 504 = 2^{3} \cdot 3^{2} \cdot 7 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 504.j (of order \(2\), degree \(1\), 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: | \(4.02446026187\) |
Analytic rank: | \(0\) |
Dimension: | \(24\) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{2}]$ |
$q$-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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
323.1 | −1.40979 | − | 0.111820i | 0 | 1.97499 | + | 0.315285i | −0.472391 | 0 | 1.00000i | −2.74906 | − | 0.665328i | 0 | 0.665969 | + | 0.0528227i | ||||||||||
323.2 | −1.40979 | + | 0.111820i | 0 | 1.97499 | − | 0.315285i | −0.472391 | 0 | − | 1.00000i | −2.74906 | + | 0.665328i | 0 | 0.665969 | − | 0.0528227i | |||||||||
323.3 | −1.25516 | − | 0.651592i | 0 | 1.15086 | + | 1.63571i | −1.82464 | 0 | 1.00000i | −0.378695 | − | 2.80296i | 0 | 2.29021 | + | 1.18892i | ||||||||||
323.4 | −1.25516 | + | 0.651592i | 0 | 1.15086 | − | 1.63571i | −1.82464 | 0 | − | 1.00000i | −0.378695 | + | 2.80296i | 0 | 2.29021 | − | 1.18892i | |||||||||
323.5 | −0.937511 | − | 1.05881i | 0 | −0.242146 | + | 1.98529i | −4.34020 | 0 | − | 1.00000i | 2.32905 | − | 1.60484i | 0 | 4.06898 | + | 4.59543i | |||||||||
323.6 | −0.937511 | + | 1.05881i | 0 | −0.242146 | − | 1.98529i | −4.34020 | 0 | 1.00000i | 2.32905 | + | 1.60484i | 0 | 4.06898 | − | 4.59543i | ||||||||||
323.7 | −0.619645 | − | 1.27124i | 0 | −1.23208 | + | 1.57543i | −0.753720 | 0 | − | 1.00000i | 2.76619 | + | 0.590057i | 0 | 0.467039 | + | 0.958156i | |||||||||
323.8 | −0.619645 | + | 1.27124i | 0 | −1.23208 | − | 1.57543i | −0.753720 | 0 | 1.00000i | 2.76619 | − | 0.590057i | 0 | 0.467039 | − | 0.958156i | ||||||||||
323.9 | −0.386593 | − | 1.36035i | 0 | −1.70109 | + | 1.05180i | 3.11999 | 0 | 1.00000i | 2.08845 | + | 1.90746i | 0 | −1.20617 | − | 4.24428i | ||||||||||
323.10 | −0.386593 | + | 1.36035i | 0 | −1.70109 | − | 1.05180i | 3.11999 | 0 | − | 1.00000i | 2.08845 | − | 1.90746i | 0 | −1.20617 | + | 4.24428i | |||||||||
323.11 | −0.157273 | − | 1.40544i | 0 | −1.95053 | + | 0.442076i | 1.81873 | 0 | − | 1.00000i | 0.928077 | + | 2.67183i | 0 | −0.286037 | − | 2.55612i | |||||||||
323.12 | −0.157273 | + | 1.40544i | 0 | −1.95053 | − | 0.442076i | 1.81873 | 0 | 1.00000i | 0.928077 | − | 2.67183i | 0 | −0.286037 | + | 2.55612i | ||||||||||
323.13 | 0.157273 | − | 1.40544i | 0 | −1.95053 | − | 0.442076i | −1.81873 | 0 | 1.00000i | −0.928077 | + | 2.67183i | 0 | −0.286037 | + | 2.55612i | ||||||||||
323.14 | 0.157273 | + | 1.40544i | 0 | −1.95053 | + | 0.442076i | −1.81873 | 0 | − | 1.00000i | −0.928077 | − | 2.67183i | 0 | −0.286037 | − | 2.55612i | |||||||||
323.15 | 0.386593 | − | 1.36035i | 0 | −1.70109 | − | 1.05180i | −3.11999 | 0 | − | 1.00000i | −2.08845 | + | 1.90746i | 0 | −1.20617 | + | 4.24428i | |||||||||
323.16 | 0.386593 | + | 1.36035i | 0 | −1.70109 | + | 1.05180i | −3.11999 | 0 | 1.00000i | −2.08845 | − | 1.90746i | 0 | −1.20617 | − | 4.24428i | ||||||||||
323.17 | 0.619645 | − | 1.27124i | 0 | −1.23208 | − | 1.57543i | 0.753720 | 0 | 1.00000i | −2.76619 | + | 0.590057i | 0 | 0.467039 | − | 0.958156i | ||||||||||
323.18 | 0.619645 | + | 1.27124i | 0 | −1.23208 | + | 1.57543i | 0.753720 | 0 | − | 1.00000i | −2.76619 | − | 0.590057i | 0 | 0.467039 | + | 0.958156i | |||||||||
323.19 | 0.937511 | − | 1.05881i | 0 | −0.242146 | − | 1.98529i | 4.34020 | 0 | 1.00000i | −2.32905 | − | 1.60484i | 0 | 4.06898 | − | 4.59543i | ||||||||||
323.20 | 0.937511 | + | 1.05881i | 0 | −0.242146 | + | 1.98529i | 4.34020 | 0 | − | 1.00000i | −2.32905 | + | 1.60484i | 0 | 4.06898 | + | 4.59543i | |||||||||
See all 24 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
3.b | odd | 2 | 1 | inner |
8.d | odd | 2 | 1 | inner |
24.f | even | 2 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 504.2.j.a | ✓ | 24 |
3.b | odd | 2 | 1 | inner | 504.2.j.a | ✓ | 24 |
4.b | odd | 2 | 1 | 2016.2.j.a | 24 | ||
8.b | even | 2 | 1 | 2016.2.j.a | 24 | ||
8.d | odd | 2 | 1 | inner | 504.2.j.a | ✓ | 24 |
12.b | even | 2 | 1 | 2016.2.j.a | 24 | ||
24.f | even | 2 | 1 | inner | 504.2.j.a | ✓ | 24 |
24.h | odd | 2 | 1 | 2016.2.j.a | 24 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
504.2.j.a | ✓ | 24 | 1.a | even | 1 | 1 | trivial |
504.2.j.a | ✓ | 24 | 3.b | odd | 2 | 1 | inner |
504.2.j.a | ✓ | 24 | 8.d | odd | 2 | 1 | inner |
504.2.j.a | ✓ | 24 | 24.f | even | 2 | 1 | inner |
2016.2.j.a | 24 | 4.b | odd | 2 | 1 | ||
2016.2.j.a | 24 | 8.b | even | 2 | 1 | ||
2016.2.j.a | 24 | 12.b | even | 2 | 1 | ||
2016.2.j.a | 24 | 24.h | odd | 2 | 1 |
Hecke kernels
This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(504, [\chi])\).