Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [3040,2,Mod(1521,3040)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(3040, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 1, 0, 0]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("3040.1521");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 3040 = 2^{5} \cdot 5 \cdot 19 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 3040.f (of order \(2\), degree \(1\), 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: | \(24.2745222145\) |
Analytic rank: | \(0\) |
Dimension: | \(28\) |
Twist minimal: | no (minimal twist has level 760) |
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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
1521.1 | 0 | − | 3.19922i | 0 | − | 1.00000i | 0 | 0.255739 | 0 | −7.23503 | 0 | ||||||||||||||||
1521.2 | 0 | − | 2.72205i | 0 | 1.00000i | 0 | 1.81393 | 0 | −4.40958 | 0 | |||||||||||||||||
1521.3 | 0 | − | 2.50151i | 0 | 1.00000i | 0 | −0.698520 | 0 | −3.25757 | 0 | |||||||||||||||||
1521.4 | 0 | − | 2.48816i | 0 | 1.00000i | 0 | −3.58757 | 0 | −3.19093 | 0 | |||||||||||||||||
1521.5 | 0 | − | 2.02371i | 0 | − | 1.00000i | 0 | −1.28404 | 0 | −1.09538 | 0 | ||||||||||||||||
1521.6 | 0 | − | 1.92849i | 0 | − | 1.00000i | 0 | −0.210397 | 0 | −0.719076 | 0 | ||||||||||||||||
1521.7 | 0 | − | 1.74129i | 0 | 1.00000i | 0 | 4.26759 | 0 | −0.0320903 | 0 | |||||||||||||||||
1521.8 | 0 | − | 1.71180i | 0 | − | 1.00000i | 0 | 2.02692 | 0 | 0.0697451 | 0 | ||||||||||||||||
1521.9 | 0 | − | 1.53286i | 0 | − | 1.00000i | 0 | 0.194868 | 0 | 0.650334 | 0 | ||||||||||||||||
1521.10 | 0 | − | 0.823852i | 0 | 1.00000i | 0 | 0.523009 | 0 | 2.32127 | 0 | |||||||||||||||||
1521.11 | 0 | − | 0.782983i | 0 | − | 1.00000i | 0 | 3.28528 | 0 | 2.38694 | 0 | ||||||||||||||||
1521.12 | 0 | − | 0.496172i | 0 | 1.00000i | 0 | −1.35432 | 0 | 2.75381 | 0 | |||||||||||||||||
1521.13 | 0 | − | 0.485865i | 0 | 1.00000i | 0 | −3.31744 | 0 | 2.76394 | 0 | |||||||||||||||||
1521.14 | 0 | − | 0.0798397i | 0 | − | 1.00000i | 0 | −3.91506 | 0 | 2.99363 | 0 | ||||||||||||||||
1521.15 | 0 | 0.0798397i | 0 | 1.00000i | 0 | −3.91506 | 0 | 2.99363 | 0 | ||||||||||||||||||
1521.16 | 0 | 0.485865i | 0 | − | 1.00000i | 0 | −3.31744 | 0 | 2.76394 | 0 | |||||||||||||||||
1521.17 | 0 | 0.496172i | 0 | − | 1.00000i | 0 | −1.35432 | 0 | 2.75381 | 0 | |||||||||||||||||
1521.18 | 0 | 0.782983i | 0 | 1.00000i | 0 | 3.28528 | 0 | 2.38694 | 0 | ||||||||||||||||||
1521.19 | 0 | 0.823852i | 0 | − | 1.00000i | 0 | 0.523009 | 0 | 2.32127 | 0 | |||||||||||||||||
1521.20 | 0 | 1.53286i | 0 | 1.00000i | 0 | 0.194868 | 0 | 0.650334 | 0 | ||||||||||||||||||
See all 28 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
8.b | even | 2 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 3040.2.f.a | 28 | |
4.b | odd | 2 | 1 | 760.2.f.a | ✓ | 28 | |
8.b | even | 2 | 1 | inner | 3040.2.f.a | 28 | |
8.d | odd | 2 | 1 | 760.2.f.a | ✓ | 28 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
760.2.f.a | ✓ | 28 | 4.b | odd | 2 | 1 | |
760.2.f.a | ✓ | 28 | 8.d | odd | 2 | 1 | |
3040.2.f.a | 28 | 1.a | even | 1 | 1 | trivial | |
3040.2.f.a | 28 | 8.b | even | 2 | 1 | inner |