Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [3360,2,Mod(2239,3360)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(3360, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([1, 0, 0, 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("3360.2239");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 3360 = 2^{5} \cdot 3 \cdot 5 \cdot 7 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 3360.q (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: | \(26.8297350792\) |
Analytic rank: | \(0\) |
Dimension: | \(48\) |
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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
2239.1 | 0 | − | 1.00000i | 0 | −0.863650 | + | 2.06255i | 0 | 1.23155 | − | 2.34164i | 0 | −1.00000 | 0 | |||||||||||||
2239.2 | 0 | 1.00000i | 0 | −0.863650 | − | 2.06255i | 0 | 1.23155 | + | 2.34164i | 0 | −1.00000 | 0 | ||||||||||||||
2239.3 | 0 | − | 1.00000i | 0 | 1.38844 | − | 1.75278i | 0 | −2.64036 | − | 0.168780i | 0 | −1.00000 | 0 | |||||||||||||
2239.4 | 0 | 1.00000i | 0 | 1.38844 | + | 1.75278i | 0 | −2.64036 | + | 0.168780i | 0 | −1.00000 | 0 | ||||||||||||||
2239.5 | 0 | − | 1.00000i | 0 | 1.90730 | − | 1.16714i | 0 | −0.473165 | − | 2.60310i | 0 | −1.00000 | 0 | |||||||||||||
2239.6 | 0 | 1.00000i | 0 | 1.90730 | + | 1.16714i | 0 | −0.473165 | + | 2.60310i | 0 | −1.00000 | 0 | ||||||||||||||
2239.7 | 0 | − | 1.00000i | 0 | −2.22073 | + | 0.261479i | 0 | 2.49772 | − | 0.872569i | 0 | −1.00000 | 0 | |||||||||||||
2239.8 | 0 | 1.00000i | 0 | −2.22073 | − | 0.261479i | 0 | 2.49772 | + | 0.872569i | 0 | −1.00000 | 0 | ||||||||||||||
2239.9 | 0 | − | 1.00000i | 0 | 1.38844 | + | 1.75278i | 0 | 2.64036 | − | 0.168780i | 0 | −1.00000 | 0 | |||||||||||||
2239.10 | 0 | 1.00000i | 0 | 1.38844 | − | 1.75278i | 0 | 2.64036 | + | 0.168780i | 0 | −1.00000 | 0 | ||||||||||||||
2239.11 | 0 | − | 1.00000i | 0 | −0.500413 | + | 2.17935i | 0 | −2.63707 | − | 0.214187i | 0 | −1.00000 | 0 | |||||||||||||
2239.12 | 0 | 1.00000i | 0 | −0.500413 | − | 2.17935i | 0 | −2.63707 | + | 0.214187i | 0 | −1.00000 | 0 | ||||||||||||||
2239.13 | 0 | − | 1.00000i | 0 | 2.21304 | − | 0.320091i | 0 | −1.58416 | + | 2.11906i | 0 | −1.00000 | 0 | |||||||||||||
2239.14 | 0 | 1.00000i | 0 | 2.21304 | + | 0.320091i | 0 | −1.58416 | − | 2.11906i | 0 | −1.00000 | 0 | ||||||||||||||
2239.15 | 0 | − | 1.00000i | 0 | −0.863650 | − | 2.06255i | 0 | −1.23155 | − | 2.34164i | 0 | −1.00000 | 0 | |||||||||||||
2239.16 | 0 | 1.00000i | 0 | −0.863650 | + | 2.06255i | 0 | −1.23155 | + | 2.34164i | 0 | −1.00000 | 0 | ||||||||||||||
2239.17 | 0 | − | 1.00000i | 0 | −2.22073 | − | 0.261479i | 0 | −2.49772 | − | 0.872569i | 0 | −1.00000 | 0 | |||||||||||||
2239.18 | 0 | 1.00000i | 0 | −2.22073 | + | 0.261479i | 0 | −2.49772 | + | 0.872569i | 0 | −1.00000 | 0 | ||||||||||||||
2239.19 | 0 | − | 1.00000i | 0 | −0.742720 | + | 2.10912i | 0 | 1.31337 | + | 2.29675i | 0 | −1.00000 | 0 | |||||||||||||
2239.20 | 0 | 1.00000i | 0 | −0.742720 | − | 2.10912i | 0 | 1.31337 | − | 2.29675i | 0 | −1.00000 | 0 | ||||||||||||||
See all 48 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
4.b | odd | 2 | 1 | inner |
35.c | odd | 2 | 1 | inner |
140.c | even | 2 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 3360.2.q.b | yes | 48 |
4.b | odd | 2 | 1 | inner | 3360.2.q.b | yes | 48 |
5.b | even | 2 | 1 | 3360.2.q.a | ✓ | 48 | |
7.b | odd | 2 | 1 | 3360.2.q.a | ✓ | 48 | |
20.d | odd | 2 | 1 | 3360.2.q.a | ✓ | 48 | |
28.d | even | 2 | 1 | 3360.2.q.a | ✓ | 48 | |
35.c | odd | 2 | 1 | inner | 3360.2.q.b | yes | 48 |
140.c | even | 2 | 1 | inner | 3360.2.q.b | yes | 48 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
3360.2.q.a | ✓ | 48 | 5.b | even | 2 | 1 | |
3360.2.q.a | ✓ | 48 | 7.b | odd | 2 | 1 | |
3360.2.q.a | ✓ | 48 | 20.d | odd | 2 | 1 | |
3360.2.q.a | ✓ | 48 | 28.d | even | 2 | 1 | |
3360.2.q.b | yes | 48 | 1.a | even | 1 | 1 | trivial |
3360.2.q.b | yes | 48 | 4.b | odd | 2 | 1 | inner |
3360.2.q.b | yes | 48 | 35.c | odd | 2 | 1 | inner |
3360.2.q.b | yes | 48 | 140.c | even | 2 | 1 | inner |