Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [528,2,Mod(79,528)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(528, base_ring=CyclotomicField(10))
chi = DirichletCharacter(H, H._module([5, 0, 0, 1]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("528.79");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 528 = 2^{4} \cdot 3 \cdot 11 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 528.ba (of order \(10\), degree \(4\), 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: | \(4.21610122672\) |
Analytic rank: | \(0\) |
Dimension: | \(32\) |
Relative dimension: | \(8\) over \(\Q(\zeta_{10})\) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{10}]$ |
$q$-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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
79.1 | 0 | −0.587785 | − | 0.809017i | 0 | −1.12408 | − | 3.45956i | 0 | −1.00359 | − | 0.729149i | 0 | −0.309017 | + | 0.951057i | 0 | ||||||||||
79.2 | 0 | −0.587785 | − | 0.809017i | 0 | −0.448465 | − | 1.38023i | 0 | 3.53832 | + | 2.57074i | 0 | −0.309017 | + | 0.951057i | 0 | ||||||||||
79.3 | 0 | −0.587785 | − | 0.809017i | 0 | 0.279257 | + | 0.859465i | 0 | −1.89104 | − | 1.37392i | 0 | −0.309017 | + | 0.951057i | 0 | ||||||||||
79.4 | 0 | −0.587785 | − | 0.809017i | 0 | 0.293287 | + | 0.902645i | 0 | −0.643691 | − | 0.467669i | 0 | −0.309017 | + | 0.951057i | 0 | ||||||||||
79.5 | 0 | 0.587785 | + | 0.809017i | 0 | −1.12408 | − | 3.45956i | 0 | 1.00359 | + | 0.729149i | 0 | −0.309017 | + | 0.951057i | 0 | ||||||||||
79.6 | 0 | 0.587785 | + | 0.809017i | 0 | −0.448465 | − | 1.38023i | 0 | −3.53832 | − | 2.57074i | 0 | −0.309017 | + | 0.951057i | 0 | ||||||||||
79.7 | 0 | 0.587785 | + | 0.809017i | 0 | 0.279257 | + | 0.859465i | 0 | 1.89104 | + | 1.37392i | 0 | −0.309017 | + | 0.951057i | 0 | ||||||||||
79.8 | 0 | 0.587785 | + | 0.809017i | 0 | 0.293287 | + | 0.902645i | 0 | 0.643691 | + | 0.467669i | 0 | −0.309017 | + | 0.951057i | 0 | ||||||||||
127.1 | 0 | −0.587785 | + | 0.809017i | 0 | −1.12408 | + | 3.45956i | 0 | −1.00359 | + | 0.729149i | 0 | −0.309017 | − | 0.951057i | 0 | ||||||||||
127.2 | 0 | −0.587785 | + | 0.809017i | 0 | −0.448465 | + | 1.38023i | 0 | 3.53832 | − | 2.57074i | 0 | −0.309017 | − | 0.951057i | 0 | ||||||||||
127.3 | 0 | −0.587785 | + | 0.809017i | 0 | 0.279257 | − | 0.859465i | 0 | −1.89104 | + | 1.37392i | 0 | −0.309017 | − | 0.951057i | 0 | ||||||||||
127.4 | 0 | −0.587785 | + | 0.809017i | 0 | 0.293287 | − | 0.902645i | 0 | −0.643691 | + | 0.467669i | 0 | −0.309017 | − | 0.951057i | 0 | ||||||||||
127.5 | 0 | 0.587785 | − | 0.809017i | 0 | −1.12408 | + | 3.45956i | 0 | 1.00359 | − | 0.729149i | 0 | −0.309017 | − | 0.951057i | 0 | ||||||||||
127.6 | 0 | 0.587785 | − | 0.809017i | 0 | −0.448465 | + | 1.38023i | 0 | −3.53832 | + | 2.57074i | 0 | −0.309017 | − | 0.951057i | 0 | ||||||||||
127.7 | 0 | 0.587785 | − | 0.809017i | 0 | 0.279257 | − | 0.859465i | 0 | 1.89104 | − | 1.37392i | 0 | −0.309017 | − | 0.951057i | 0 | ||||||||||
127.8 | 0 | 0.587785 | − | 0.809017i | 0 | 0.293287 | − | 0.902645i | 0 | 0.643691 | − | 0.467669i | 0 | −0.309017 | − | 0.951057i | 0 | ||||||||||
271.1 | 0 | −0.951057 | − | 0.309017i | 0 | −3.14510 | + | 2.28505i | 0 | −0.322237 | − | 0.991745i | 0 | 0.809017 | + | 0.587785i | 0 | ||||||||||
271.2 | 0 | −0.951057 | − | 0.309017i | 0 | −1.49073 | + | 1.08308i | 0 | 0.0221271 | + | 0.0681003i | 0 | 0.809017 | + | 0.587785i | 0 | ||||||||||
271.3 | 0 | −0.951057 | − | 0.309017i | 0 | 0.377817 | − | 0.274500i | 0 | 1.34031 | + | 4.12505i | 0 | 0.809017 | + | 0.587785i | 0 | ||||||||||
271.4 | 0 | −0.951057 | − | 0.309017i | 0 | 3.25801 | − | 2.36708i | 0 | −1.04020 | − | 3.20141i | 0 | 0.809017 | + | 0.587785i | 0 | ||||||||||
See all 32 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
4.b | odd | 2 | 1 | inner |
11.d | odd | 10 | 1 | inner |
44.g | even | 10 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 528.2.ba.c | ✓ | 32 |
4.b | odd | 2 | 1 | inner | 528.2.ba.c | ✓ | 32 |
11.d | odd | 10 | 1 | inner | 528.2.ba.c | ✓ | 32 |
44.g | even | 10 | 1 | inner | 528.2.ba.c | ✓ | 32 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
528.2.ba.c | ✓ | 32 | 1.a | even | 1 | 1 | trivial |
528.2.ba.c | ✓ | 32 | 4.b | odd | 2 | 1 | inner |
528.2.ba.c | ✓ | 32 | 11.d | odd | 10 | 1 | inner |
528.2.ba.c | ✓ | 32 | 44.g | even | 10 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{16} + 4 T_{5}^{15} + 11 T_{5}^{14} - 4 T_{5}^{13} + 108 T_{5}^{12} + 784 T_{5}^{11} + 4879 T_{5}^{10} + \cdots + 3721 \) acting on \(S_{2}^{\mathrm{new}}(528, [\chi])\).