Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [2310,2,Mod(769,2310)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2310, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 1, 1, 1]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("2310.769");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 2310 = 2 \cdot 3 \cdot 5 \cdot 7 \cdot 11 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 2310.d (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: | \(18.4454428669\) |
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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
769.1 | −1.00000 | 1.00000 | 1.00000 | −2.23546 | − | 0.0523022i | −1.00000 | 2.19471 | − | 1.47758i | −1.00000 | 1.00000 | 2.23546 | + | 0.0523022i | ||||||||||||
769.2 | −1.00000 | 1.00000 | 1.00000 | −2.23546 | + | 0.0523022i | −1.00000 | 2.19471 | + | 1.47758i | −1.00000 | 1.00000 | 2.23546 | − | 0.0523022i | ||||||||||||
769.3 | −1.00000 | 1.00000 | 1.00000 | −1.92186 | − | 1.14300i | −1.00000 | −0.641067 | + | 2.56691i | −1.00000 | 1.00000 | 1.92186 | + | 1.14300i | ||||||||||||
769.4 | −1.00000 | 1.00000 | 1.00000 | −1.92186 | + | 1.14300i | −1.00000 | −0.641067 | − | 2.56691i | −1.00000 | 1.00000 | 1.92186 | − | 1.14300i | ||||||||||||
769.5 | −1.00000 | 1.00000 | 1.00000 | −1.84766 | − | 1.25943i | −1.00000 | −2.39421 | − | 1.12594i | −1.00000 | 1.00000 | 1.84766 | + | 1.25943i | ||||||||||||
769.6 | −1.00000 | 1.00000 | 1.00000 | −1.84766 | + | 1.25943i | −1.00000 | −2.39421 | + | 1.12594i | −1.00000 | 1.00000 | 1.84766 | − | 1.25943i | ||||||||||||
769.7 | −1.00000 | 1.00000 | 1.00000 | −1.09857 | − | 1.94760i | −1.00000 | 1.89343 | − | 1.84795i | −1.00000 | 1.00000 | 1.09857 | + | 1.94760i | ||||||||||||
769.8 | −1.00000 | 1.00000 | 1.00000 | −1.09857 | + | 1.94760i | −1.00000 | 1.89343 | + | 1.84795i | −1.00000 | 1.00000 | 1.09857 | − | 1.94760i | ||||||||||||
769.9 | −1.00000 | 1.00000 | 1.00000 | −0.295999 | − | 2.21639i | −1.00000 | −0.465453 | + | 2.60449i | −1.00000 | 1.00000 | 0.295999 | + | 2.21639i | ||||||||||||
769.10 | −1.00000 | 1.00000 | 1.00000 | −0.295999 | + | 2.21639i | −1.00000 | −0.465453 | − | 2.60449i | −1.00000 | 1.00000 | 0.295999 | − | 2.21639i | ||||||||||||
769.11 | −1.00000 | 1.00000 | 1.00000 | 0.145802 | − | 2.23131i | −1.00000 | 2.61462 | + | 0.404680i | −1.00000 | 1.00000 | −0.145802 | + | 2.23131i | ||||||||||||
769.12 | −1.00000 | 1.00000 | 1.00000 | 0.145802 | + | 2.23131i | −1.00000 | 2.61462 | − | 0.404680i | −1.00000 | 1.00000 | −0.145802 | − | 2.23131i | ||||||||||||
769.13 | −1.00000 | 1.00000 | 1.00000 | 0.356638 | − | 2.20744i | −1.00000 | −0.788235 | − | 2.52561i | −1.00000 | 1.00000 | −0.356638 | + | 2.20744i | ||||||||||||
769.14 | −1.00000 | 1.00000 | 1.00000 | 0.356638 | + | 2.20744i | −1.00000 | −0.788235 | + | 2.52561i | −1.00000 | 1.00000 | −0.356638 | − | 2.20744i | ||||||||||||
769.15 | −1.00000 | 1.00000 | 1.00000 | 0.548027 | − | 2.16787i | −1.00000 | −2.59668 | + | 0.507194i | −1.00000 | 1.00000 | −0.548027 | + | 2.16787i | ||||||||||||
769.16 | −1.00000 | 1.00000 | 1.00000 | 0.548027 | + | 2.16787i | −1.00000 | −2.59668 | − | 0.507194i | −1.00000 | 1.00000 | −0.548027 | − | 2.16787i | ||||||||||||
769.17 | −1.00000 | 1.00000 | 1.00000 | 1.20144 | − | 1.88588i | −1.00000 | −0.238908 | + | 2.63494i | −1.00000 | 1.00000 | −1.20144 | + | 1.88588i | ||||||||||||
769.18 | −1.00000 | 1.00000 | 1.00000 | 1.20144 | + | 1.88588i | −1.00000 | −0.238908 | − | 2.63494i | −1.00000 | 1.00000 | −1.20144 | − | 1.88588i | ||||||||||||
769.19 | −1.00000 | 1.00000 | 1.00000 | 1.77744 | − | 1.35673i | −1.00000 | −0.178779 | − | 2.63970i | −1.00000 | 1.00000 | −1.77744 | + | 1.35673i | ||||||||||||
769.20 | −1.00000 | 1.00000 | 1.00000 | 1.77744 | + | 1.35673i | −1.00000 | −0.178779 | + | 2.63970i | −1.00000 | 1.00000 | −1.77744 | − | 1.35673i | ||||||||||||
See all 24 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
385.h | even | 2 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 2310.2.d.b | yes | 24 |
5.b | even | 2 | 1 | 2310.2.d.c | yes | 24 | |
7.b | odd | 2 | 1 | 2310.2.d.a | ✓ | 24 | |
11.b | odd | 2 | 1 | 2310.2.d.d | yes | 24 | |
35.c | odd | 2 | 1 | 2310.2.d.d | yes | 24 | |
55.d | odd | 2 | 1 | 2310.2.d.a | ✓ | 24 | |
77.b | even | 2 | 1 | 2310.2.d.c | yes | 24 | |
385.h | even | 2 | 1 | inner | 2310.2.d.b | yes | 24 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
2310.2.d.a | ✓ | 24 | 7.b | odd | 2 | 1 | |
2310.2.d.a | ✓ | 24 | 55.d | odd | 2 | 1 | |
2310.2.d.b | yes | 24 | 1.a | even | 1 | 1 | trivial |
2310.2.d.b | yes | 24 | 385.h | even | 2 | 1 | inner |
2310.2.d.c | yes | 24 | 5.b | even | 2 | 1 | |
2310.2.d.c | yes | 24 | 77.b | even | 2 | 1 | |
2310.2.d.d | yes | 24 | 11.b | odd | 2 | 1 | |
2310.2.d.d | yes | 24 | 35.c | odd | 2 | 1 |