Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [4140,3,Mod(4049,4140)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(4140, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 1, 1, 0]))
N = Newforms(chi, 3, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("4140.4049");
S:= CuspForms(chi, 3);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 4140 = 2^{2} \cdot 3^{2} \cdot 5 \cdot 23 \) |
Weight: | \( k \) | \(=\) | \( 3 \) |
Character orbit: | \([\chi]\) | \(=\) | 4140.c (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: | \(112.806829445\) |
Analytic rank: | \(0\) |
Dimension: | \(88\) |
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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
4049.1 | 0 | 0 | 0 | −4.99275 | − | 0.269219i | 0 | − | 13.6272i | 0 | 0 | 0 | |||||||||||||||
4049.2 | 0 | 0 | 0 | −4.99275 | + | 0.269219i | 0 | 13.6272i | 0 | 0 | 0 | ||||||||||||||||
4049.3 | 0 | 0 | 0 | −4.95880 | − | 0.640555i | 0 | 5.90914i | 0 | 0 | 0 | ||||||||||||||||
4049.4 | 0 | 0 | 0 | −4.95880 | + | 0.640555i | 0 | − | 5.90914i | 0 | 0 | 0 | |||||||||||||||
4049.5 | 0 | 0 | 0 | −4.81534 | − | 1.34629i | 0 | 4.38152i | 0 | 0 | 0 | ||||||||||||||||
4049.6 | 0 | 0 | 0 | −4.81534 | + | 1.34629i | 0 | − | 4.38152i | 0 | 0 | 0 | |||||||||||||||
4049.7 | 0 | 0 | 0 | −4.79317 | − | 1.42321i | 0 | − | 6.33514i | 0 | 0 | 0 | |||||||||||||||
4049.8 | 0 | 0 | 0 | −4.79317 | + | 1.42321i | 0 | 6.33514i | 0 | 0 | 0 | ||||||||||||||||
4049.9 | 0 | 0 | 0 | −4.77264 | − | 1.49061i | 0 | 2.56481i | 0 | 0 | 0 | ||||||||||||||||
4049.10 | 0 | 0 | 0 | −4.77264 | + | 1.49061i | 0 | − | 2.56481i | 0 | 0 | 0 | |||||||||||||||
4049.11 | 0 | 0 | 0 | −4.60834 | − | 1.93989i | 0 | − | 1.03288i | 0 | 0 | 0 | |||||||||||||||
4049.12 | 0 | 0 | 0 | −4.60834 | + | 1.93989i | 0 | 1.03288i | 0 | 0 | 0 | ||||||||||||||||
4049.13 | 0 | 0 | 0 | −4.49878 | − | 2.18197i | 0 | − | 10.0913i | 0 | 0 | 0 | |||||||||||||||
4049.14 | 0 | 0 | 0 | −4.49878 | + | 2.18197i | 0 | 10.0913i | 0 | 0 | 0 | ||||||||||||||||
4049.15 | 0 | 0 | 0 | −4.41871 | − | 2.33986i | 0 | 11.6062i | 0 | 0 | 0 | ||||||||||||||||
4049.16 | 0 | 0 | 0 | −4.41871 | + | 2.33986i | 0 | − | 11.6062i | 0 | 0 | 0 | |||||||||||||||
4049.17 | 0 | 0 | 0 | −3.90943 | − | 3.11711i | 0 | 7.62373i | 0 | 0 | 0 | ||||||||||||||||
4049.18 | 0 | 0 | 0 | −3.90943 | + | 3.11711i | 0 | − | 7.62373i | 0 | 0 | 0 | |||||||||||||||
4049.19 | 0 | 0 | 0 | −3.78150 | − | 3.27113i | 0 | − | 2.52965i | 0 | 0 | 0 | |||||||||||||||
4049.20 | 0 | 0 | 0 | −3.78150 | + | 3.27113i | 0 | 2.52965i | 0 | 0 | 0 | ||||||||||||||||
See all 88 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
3.b | odd | 2 | 1 | inner |
5.b | even | 2 | 1 | inner |
15.d | odd | 2 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 4140.3.c.a | ✓ | 88 |
3.b | odd | 2 | 1 | inner | 4140.3.c.a | ✓ | 88 |
5.b | even | 2 | 1 | inner | 4140.3.c.a | ✓ | 88 |
15.d | odd | 2 | 1 | inner | 4140.3.c.a | ✓ | 88 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
4140.3.c.a | ✓ | 88 | 1.a | even | 1 | 1 | trivial |
4140.3.c.a | ✓ | 88 | 3.b | odd | 2 | 1 | inner |
4140.3.c.a | ✓ | 88 | 5.b | even | 2 | 1 | inner |
4140.3.c.a | ✓ | 88 | 15.d | odd | 2 | 1 | inner |
Hecke kernels
This newform subspace is the entire newspace \(S_{3}^{\mathrm{new}}(4140, [\chi])\).