Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [960,4,Mod(241,960)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(960, base_ring=CyclotomicField(4))
chi = DirichletCharacter(H, H._module([0, 1, 0, 0]))
N = Newforms(chi, 4, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("960.241");
S:= CuspForms(chi, 4);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 960 = 2^{6} \cdot 3 \cdot 5 \) |
Weight: | \( k \) | \(=\) | \( 4 \) |
Character orbit: | \([\chi]\) | \(=\) | 960.s (of order \(4\), degree \(2\), not 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: | \(56.6418336055\) |
Analytic rank: | \(0\) |
Dimension: | \(44\) |
Relative dimension: | \(22\) over \(\Q(i)\) |
Twist minimal: | no (minimal twist has level 240) |
Sato-Tate group: | $\mathrm{SU}(2)[C_{4}]$ |
$q$-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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
241.1 | 0 | −2.12132 | + | 2.12132i | 0 | 3.53553 | + | 3.53553i | 0 | − | 27.4299i | 0 | − | 9.00000i | 0 | ||||||||||||
241.2 | 0 | −2.12132 | + | 2.12132i | 0 | 3.53553 | + | 3.53553i | 0 | 26.0663i | 0 | − | 9.00000i | 0 | |||||||||||||
241.3 | 0 | −2.12132 | + | 2.12132i | 0 | 3.53553 | + | 3.53553i | 0 | − | 25.3745i | 0 | − | 9.00000i | 0 | ||||||||||||
241.4 | 0 | −2.12132 | + | 2.12132i | 0 | 3.53553 | + | 3.53553i | 0 | 19.3290i | 0 | − | 9.00000i | 0 | |||||||||||||
241.5 | 0 | −2.12132 | + | 2.12132i | 0 | 3.53553 | + | 3.53553i | 0 | 17.9506i | 0 | − | 9.00000i | 0 | |||||||||||||
241.6 | 0 | −2.12132 | + | 2.12132i | 0 | 3.53553 | + | 3.53553i | 0 | 14.2753i | 0 | − | 9.00000i | 0 | |||||||||||||
241.7 | 0 | −2.12132 | + | 2.12132i | 0 | 3.53553 | + | 3.53553i | 0 | − | 12.0545i | 0 | − | 9.00000i | 0 | ||||||||||||
241.8 | 0 | −2.12132 | + | 2.12132i | 0 | 3.53553 | + | 3.53553i | 0 | − | 6.35344i | 0 | − | 9.00000i | 0 | ||||||||||||
241.9 | 0 | −2.12132 | + | 2.12132i | 0 | 3.53553 | + | 3.53553i | 0 | 7.14794i | 0 | − | 9.00000i | 0 | |||||||||||||
241.10 | 0 | −2.12132 | + | 2.12132i | 0 | 3.53553 | + | 3.53553i | 0 | − | 13.5482i | 0 | − | 9.00000i | 0 | ||||||||||||
241.11 | 0 | −2.12132 | + | 2.12132i | 0 | 3.53553 | + | 3.53553i | 0 | − | 14.0086i | 0 | − | 9.00000i | 0 | ||||||||||||
241.12 | 0 | 2.12132 | − | 2.12132i | 0 | −3.53553 | − | 3.53553i | 0 | − | 33.3112i | 0 | − | 9.00000i | 0 | ||||||||||||
241.13 | 0 | 2.12132 | − | 2.12132i | 0 | −3.53553 | − | 3.53553i | 0 | 32.0383i | 0 | − | 9.00000i | 0 | |||||||||||||
241.14 | 0 | 2.12132 | − | 2.12132i | 0 | −3.53553 | − | 3.53553i | 0 | 24.6196i | 0 | − | 9.00000i | 0 | |||||||||||||
241.15 | 0 | 2.12132 | − | 2.12132i | 0 | −3.53553 | − | 3.53553i | 0 | − | 21.2989i | 0 | − | 9.00000i | 0 | ||||||||||||
241.16 | 0 | 2.12132 | − | 2.12132i | 0 | −3.53553 | − | 3.53553i | 0 | − | 16.8574i | 0 | − | 9.00000i | 0 | ||||||||||||
241.17 | 0 | 2.12132 | − | 2.12132i | 0 | −3.53553 | − | 3.53553i | 0 | − | 16.3904i | 0 | − | 9.00000i | 0 | ||||||||||||
241.18 | 0 | 2.12132 | − | 2.12132i | 0 | −3.53553 | − | 3.53553i | 0 | 15.9493i | 0 | − | 9.00000i | 0 | |||||||||||||
241.19 | 0 | 2.12132 | − | 2.12132i | 0 | −3.53553 | − | 3.53553i | 0 | 10.1380i | 0 | − | 9.00000i | 0 | |||||||||||||
241.20 | 0 | 2.12132 | − | 2.12132i | 0 | −3.53553 | − | 3.53553i | 0 | 5.63046i | 0 | − | 9.00000i | 0 | |||||||||||||
See all 44 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
16.e | even | 4 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 960.4.s.a | 44 | |
4.b | odd | 2 | 1 | 240.4.s.a | ✓ | 44 | |
16.e | even | 4 | 1 | inner | 960.4.s.a | 44 | |
16.f | odd | 4 | 1 | 240.4.s.a | ✓ | 44 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
240.4.s.a | ✓ | 44 | 4.b | odd | 2 | 1 | |
240.4.s.a | ✓ | 44 | 16.f | odd | 4 | 1 | |
960.4.s.a | 44 | 1.a | even | 1 | 1 | trivial | |
960.4.s.a | 44 | 16.e | even | 4 | 1 | inner |