Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [600,2,Mod(169,600)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(600, base_ring=CyclotomicField(10))
chi = DirichletCharacter(H, H._module([0, 0, 0, 9]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("600.169");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 600 = 2^{3} \cdot 3 \cdot 5^{2} \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 600.bc (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: | \(4.79102412128\) |
Analytic rank: | \(0\) |
Dimension: | \(24\) |
Relative dimension: | \(6\) 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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
169.1 | 0 | −0.587785 | − | 0.809017i | 0 | −2.23321 | + | 0.113099i | 0 | 1.02996i | 0 | −0.309017 | + | 0.951057i | 0 | ||||||||||||
169.2 | 0 | −0.587785 | − | 0.809017i | 0 | −0.454498 | − | 2.18939i | 0 | − | 4.18140i | 0 | −0.309017 | + | 0.951057i | 0 | |||||||||||
169.3 | 0 | −0.587785 | − | 0.809017i | 0 | 2.18770 | − | 0.462550i | 0 | 1.81748i | 0 | −0.309017 | + | 0.951057i | 0 | ||||||||||||
169.4 | 0 | 0.587785 | + | 0.809017i | 0 | −1.69051 | − | 1.46362i | 0 | 0.184115i | 0 | −0.309017 | + | 0.951057i | 0 | ||||||||||||
169.5 | 0 | 0.587785 | + | 0.809017i | 0 | −0.696192 | + | 2.12493i | 0 | 2.74003i | 0 | −0.309017 | + | 0.951057i | 0 | ||||||||||||
169.6 | 0 | 0.587785 | + | 0.809017i | 0 | 1.88670 | − | 1.20015i | 0 | 2.21404i | 0 | −0.309017 | + | 0.951057i | 0 | ||||||||||||
289.1 | 0 | −0.951057 | − | 0.309017i | 0 | −2.14870 | − | 0.618935i | 0 | 0.783294i | 0 | 0.809017 | + | 0.587785i | 0 | ||||||||||||
289.2 | 0 | −0.951057 | − | 0.309017i | 0 | −0.580331 | + | 2.15945i | 0 | − | 4.32657i | 0 | 0.809017 | + | 0.587785i | 0 | |||||||||||
289.3 | 0 | −0.951057 | − | 0.309017i | 0 | 2.22903 | − | 0.177242i | 0 | 3.48278i | 0 | 0.809017 | + | 0.587785i | 0 | ||||||||||||
289.4 | 0 | 0.951057 | + | 0.309017i | 0 | −2.02922 | + | 0.939285i | 0 | 0.466908i | 0 | 0.809017 | + | 0.587785i | 0 | ||||||||||||
289.5 | 0 | 0.951057 | + | 0.309017i | 0 | −0.633377 | − | 2.14449i | 0 | − | 1.68627i | 0 | 0.809017 | + | 0.587785i | 0 | |||||||||||
289.6 | 0 | 0.951057 | + | 0.309017i | 0 | 2.16260 | + | 0.568475i | 0 | 3.63100i | 0 | 0.809017 | + | 0.587785i | 0 | ||||||||||||
409.1 | 0 | −0.951057 | + | 0.309017i | 0 | −2.14870 | + | 0.618935i | 0 | − | 0.783294i | 0 | 0.809017 | − | 0.587785i | 0 | |||||||||||
409.2 | 0 | −0.951057 | + | 0.309017i | 0 | −0.580331 | − | 2.15945i | 0 | 4.32657i | 0 | 0.809017 | − | 0.587785i | 0 | ||||||||||||
409.3 | 0 | −0.951057 | + | 0.309017i | 0 | 2.22903 | + | 0.177242i | 0 | − | 3.48278i | 0 | 0.809017 | − | 0.587785i | 0 | |||||||||||
409.4 | 0 | 0.951057 | − | 0.309017i | 0 | −2.02922 | − | 0.939285i | 0 | − | 0.466908i | 0 | 0.809017 | − | 0.587785i | 0 | |||||||||||
409.5 | 0 | 0.951057 | − | 0.309017i | 0 | −0.633377 | + | 2.14449i | 0 | 1.68627i | 0 | 0.809017 | − | 0.587785i | 0 | ||||||||||||
409.6 | 0 | 0.951057 | − | 0.309017i | 0 | 2.16260 | − | 0.568475i | 0 | − | 3.63100i | 0 | 0.809017 | − | 0.587785i | 0 | |||||||||||
529.1 | 0 | −0.587785 | + | 0.809017i | 0 | −2.23321 | − | 0.113099i | 0 | − | 1.02996i | 0 | −0.309017 | − | 0.951057i | 0 | |||||||||||
529.2 | 0 | −0.587785 | + | 0.809017i | 0 | −0.454498 | + | 2.18939i | 0 | 4.18140i | 0 | −0.309017 | − | 0.951057i | 0 | ||||||||||||
See all 24 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
25.e | even | 10 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 600.2.bc.b | ✓ | 24 |
25.e | even | 10 | 1 | inner | 600.2.bc.b | ✓ | 24 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
600.2.bc.b | ✓ | 24 | 1.a | even | 1 | 1 | trivial |
600.2.bc.b | ✓ | 24 | 25.e | even | 10 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{24} + 82 T_{7}^{22} + 2823 T_{7}^{20} + 53224 T_{7}^{18} + 602215 T_{7}^{16} + 4226114 T_{7}^{14} + \cdots + 87025 \) acting on \(S_{2}^{\mathrm{new}}(600, [\chi])\).