Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [1476,2,Mod(161,1476)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1476, base_ring=CyclotomicField(8))
chi = DirichletCharacter(H, H._module([0, 4, 7]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("1476.161");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 1476 = 2^{2} \cdot 3^{2} \cdot 41 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 1476.x (of order \(8\), degree \(4\), 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: | \(11.7859193383\) |
Analytic rank: | \(0\) |
Dimension: | \(28\) |
Relative dimension: | \(7\) over \(\Q(\zeta_{8})\) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{8}]$ |
$q$-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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
161.1 | 0 | 0 | 0 | −2.81045 | + | 2.81045i | 0 | −1.84108 | − | 0.762601i | 0 | 0 | 0 | ||||||||||||||
161.2 | 0 | 0 | 0 | −1.96503 | + | 1.96503i | 0 | 1.04671 | + | 0.433563i | 0 | 0 | 0 | ||||||||||||||
161.3 | 0 | 0 | 0 | −0.160559 | + | 0.160559i | 0 | −1.58587 | − | 0.656890i | 0 | 0 | 0 | ||||||||||||||
161.4 | 0 | 0 | 0 | 0.659514 | − | 0.659514i | 0 | 2.33734 | + | 0.968156i | 0 | 0 | 0 | ||||||||||||||
161.5 | 0 | 0 | 0 | 1.08307 | − | 1.08307i | 0 | −3.46391 | − | 1.43480i | 0 | 0 | 0 | ||||||||||||||
161.6 | 0 | 0 | 0 | 1.75916 | − | 1.75916i | 0 | 4.68497 | + | 1.94058i | 0 | 0 | 0 | ||||||||||||||
161.7 | 0 | 0 | 0 | 2.43430 | − | 2.43430i | 0 | −1.17816 | − | 0.488012i | 0 | 0 | 0 | ||||||||||||||
413.1 | 0 | 0 | 0 | −2.56900 | + | 2.56900i | 0 | 0.926138 | − | 2.23590i | 0 | 0 | 0 | ||||||||||||||
413.2 | 0 | 0 | 0 | −0.991747 | + | 0.991747i | 0 | −1.53086 | + | 3.69582i | 0 | 0 | 0 | ||||||||||||||
413.3 | 0 | 0 | 0 | −0.790254 | + | 0.790254i | 0 | 0.489069 | − | 1.18072i | 0 | 0 | 0 | ||||||||||||||
413.4 | 0 | 0 | 0 | −0.358870 | + | 0.358870i | 0 | 0.130495 | − | 0.315043i | 0 | 0 | 0 | ||||||||||||||
413.5 | 0 | 0 | 0 | 1.44287 | − | 1.44287i | 0 | −0.365476 | + | 0.882337i | 0 | 0 | 0 | ||||||||||||||
413.6 | 0 | 0 | 0 | 1.95510 | − | 1.95510i | 0 | −1.33274 | + | 3.21753i | 0 | 0 | 0 | ||||||||||||||
413.7 | 0 | 0 | 0 | 2.31190 | − | 2.31190i | 0 | 1.68338 | − | 4.06403i | 0 | 0 | 0 | ||||||||||||||
629.1 | 0 | 0 | 0 | −2.56900 | − | 2.56900i | 0 | 0.926138 | + | 2.23590i | 0 | 0 | 0 | ||||||||||||||
629.2 | 0 | 0 | 0 | −0.991747 | − | 0.991747i | 0 | −1.53086 | − | 3.69582i | 0 | 0 | 0 | ||||||||||||||
629.3 | 0 | 0 | 0 | −0.790254 | − | 0.790254i | 0 | 0.489069 | + | 1.18072i | 0 | 0 | 0 | ||||||||||||||
629.4 | 0 | 0 | 0 | −0.358870 | − | 0.358870i | 0 | 0.130495 | + | 0.315043i | 0 | 0 | 0 | ||||||||||||||
629.5 | 0 | 0 | 0 | 1.44287 | + | 1.44287i | 0 | −0.365476 | − | 0.882337i | 0 | 0 | 0 | ||||||||||||||
629.6 | 0 | 0 | 0 | 1.95510 | + | 1.95510i | 0 | −1.33274 | − | 3.21753i | 0 | 0 | 0 | ||||||||||||||
See all 28 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
123.i | even | 8 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 1476.2.x.b | yes | 28 |
3.b | odd | 2 | 1 | 1476.2.x.a | ✓ | 28 | |
41.e | odd | 8 | 1 | 1476.2.x.a | ✓ | 28 | |
123.i | even | 8 | 1 | inner | 1476.2.x.b | yes | 28 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
1476.2.x.a | ✓ | 28 | 3.b | odd | 2 | 1 | |
1476.2.x.a | ✓ | 28 | 41.e | odd | 8 | 1 | |
1476.2.x.b | yes | 28 | 1.a | even | 1 | 1 | trivial |
1476.2.x.b | yes | 28 | 123.i | even | 8 | 1 | inner |