Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [7524,2,Mod(2089,7524)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(7524, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([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("7524.2089");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 7524 = 2^{2} \cdot 3^{2} \cdot 11 \cdot 19 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 7524.l (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: | \(60.0794424808\) |
Analytic rank: | \(0\) |
Dimension: | \(40\) |
Twist minimal: | no (minimal twist has level 2508) |
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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
2089.1 | 0 | 0 | 0 | −3.99412 | 0 | 4.30161i | 0 | 0 | 0 | ||||||||||||||||||
2089.2 | 0 | 0 | 0 | −3.99412 | 0 | − | 4.30161i | 0 | 0 | 0 | |||||||||||||||||
2089.3 | 0 | 0 | 0 | −3.99412 | 0 | 4.30161i | 0 | 0 | 0 | ||||||||||||||||||
2089.4 | 0 | 0 | 0 | −3.99412 | 0 | − | 4.30161i | 0 | 0 | 0 | |||||||||||||||||
2089.5 | 0 | 0 | 0 | −3.19888 | 0 | 1.12710i | 0 | 0 | 0 | ||||||||||||||||||
2089.6 | 0 | 0 | 0 | −3.19888 | 0 | 1.12710i | 0 | 0 | 0 | ||||||||||||||||||
2089.7 | 0 | 0 | 0 | −3.19888 | 0 | − | 1.12710i | 0 | 0 | 0 | |||||||||||||||||
2089.8 | 0 | 0 | 0 | −3.19888 | 0 | − | 1.12710i | 0 | 0 | 0 | |||||||||||||||||
2089.9 | 0 | 0 | 0 | −1.86812 | 0 | 2.45225i | 0 | 0 | 0 | ||||||||||||||||||
2089.10 | 0 | 0 | 0 | −1.86812 | 0 | 2.45225i | 0 | 0 | 0 | ||||||||||||||||||
2089.11 | 0 | 0 | 0 | −1.86812 | 0 | − | 2.45225i | 0 | 0 | 0 | |||||||||||||||||
2089.12 | 0 | 0 | 0 | −1.86812 | 0 | − | 2.45225i | 0 | 0 | 0 | |||||||||||||||||
2089.13 | 0 | 0 | 0 | −1.04890 | 0 | 2.18010i | 0 | 0 | 0 | ||||||||||||||||||
2089.14 | 0 | 0 | 0 | −1.04890 | 0 | − | 2.18010i | 0 | 0 | 0 | |||||||||||||||||
2089.15 | 0 | 0 | 0 | −1.04890 | 0 | 2.18010i | 0 | 0 | 0 | ||||||||||||||||||
2089.16 | 0 | 0 | 0 | −1.04890 | 0 | − | 2.18010i | 0 | 0 | 0 | |||||||||||||||||
2089.17 | 0 | 0 | 0 | 0.0720070 | 0 | 3.03803i | 0 | 0 | 0 | ||||||||||||||||||
2089.18 | 0 | 0 | 0 | 0.0720070 | 0 | − | 3.03803i | 0 | 0 | 0 | |||||||||||||||||
2089.19 | 0 | 0 | 0 | 0.0720070 | 0 | 3.03803i | 0 | 0 | 0 | ||||||||||||||||||
2089.20 | 0 | 0 | 0 | 0.0720070 | 0 | − | 3.03803i | 0 | 0 | 0 | |||||||||||||||||
See all 40 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
11.b | odd | 2 | 1 | inner |
19.b | odd | 2 | 1 | inner |
209.d | even | 2 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 7524.2.l.f | 40 | |
3.b | odd | 2 | 1 | 2508.2.l.a | ✓ | 40 | |
11.b | odd | 2 | 1 | inner | 7524.2.l.f | 40 | |
19.b | odd | 2 | 1 | inner | 7524.2.l.f | 40 | |
33.d | even | 2 | 1 | 2508.2.l.a | ✓ | 40 | |
57.d | even | 2 | 1 | 2508.2.l.a | ✓ | 40 | |
209.d | even | 2 | 1 | inner | 7524.2.l.f | 40 | |
627.b | odd | 2 | 1 | 2508.2.l.a | ✓ | 40 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
2508.2.l.a | ✓ | 40 | 3.b | odd | 2 | 1 | |
2508.2.l.a | ✓ | 40 | 33.d | even | 2 | 1 | |
2508.2.l.a | ✓ | 40 | 57.d | even | 2 | 1 | |
2508.2.l.a | ✓ | 40 | 627.b | odd | 2 | 1 | |
7524.2.l.f | 40 | 1.a | even | 1 | 1 | trivial | |
7524.2.l.f | 40 | 11.b | odd | 2 | 1 | inner | |
7524.2.l.f | 40 | 19.b | odd | 2 | 1 | inner | |
7524.2.l.f | 40 | 209.d | even | 2 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{10} - T_{5}^{9} - 31 T_{5}^{8} + 31 T_{5}^{7} + 296 T_{5}^{6} - 288 T_{5}^{5} - 955 T_{5}^{4} + \cdots + 24 \) acting on \(S_{2}^{\mathrm{new}}(7524, [\chi])\).