Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [1386,2,Mod(701,1386)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1386, base_ring=CyclotomicField(10))
chi = DirichletCharacter(H, H._module([5, 0, 3]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("1386.701");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 1386 = 2 \cdot 3^{2} \cdot 7 \cdot 11 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 1386.bu (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: | \(11.0672657201\) |
Analytic rank: | \(0\) |
Dimension: | \(48\) |
Relative dimension: | \(12\) 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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
701.1 | −0.809017 | + | 0.587785i | 0 | 0.309017 | − | 0.951057i | −2.53437 | + | 3.48826i | 0 | 0.951057 | + | 0.309017i | 0.309017 | + | 0.951057i | 0 | − | 4.31173i | |||||||
701.2 | −0.809017 | + | 0.587785i | 0 | 0.309017 | − | 0.951057i | −2.22800 | + | 3.06659i | 0 | 0.951057 | + | 0.309017i | 0.309017 | + | 0.951057i | 0 | − | 3.79051i | |||||||
701.3 | −0.809017 | + | 0.587785i | 0 | 0.309017 | − | 0.951057i | −1.14907 | + | 1.58155i | 0 | −0.951057 | − | 0.309017i | 0.309017 | + | 0.951057i | 0 | − | 1.95491i | |||||||
701.4 | −0.809017 | + | 0.587785i | 0 | 0.309017 | − | 0.951057i | −1.04276 | + | 1.43524i | 0 | −0.951057 | − | 0.309017i | 0.309017 | + | 0.951057i | 0 | − | 1.77406i | |||||||
701.5 | −0.809017 | + | 0.587785i | 0 | 0.309017 | − | 0.951057i | −0.450005 | + | 0.619379i | 0 | 0.951057 | + | 0.309017i | 0.309017 | + | 0.951057i | 0 | − | 0.765594i | |||||||
701.6 | −0.809017 | + | 0.587785i | 0 | 0.309017 | − | 0.951057i | 0.0348544 | − | 0.0479729i | 0 | 0.951057 | + | 0.309017i | 0.309017 | + | 0.951057i | 0 | 0.0592978i | ||||||||
701.7 | −0.809017 | + | 0.587785i | 0 | 0.309017 | − | 0.951057i | 0.741879 | − | 1.02111i | 0 | −0.951057 | − | 0.309017i | 0.309017 | + | 0.951057i | 0 | 1.26216i | ||||||||
701.8 | −0.809017 | + | 0.587785i | 0 | 0.309017 | − | 0.951057i | 0.943712 | − | 1.29891i | 0 | −0.951057 | − | 0.309017i | 0.309017 | + | 0.951057i | 0 | 1.60554i | ||||||||
701.9 | −0.809017 | + | 0.587785i | 0 | 0.309017 | − | 0.951057i | 1.15641 | − | 1.59166i | 0 | −0.951057 | − | 0.309017i | 0.309017 | + | 0.951057i | 0 | 1.96740i | ||||||||
701.10 | −0.809017 | + | 0.587785i | 0 | 0.309017 | − | 0.951057i | 1.24209 | − | 1.70960i | 0 | 0.951057 | + | 0.309017i | 0.309017 | + | 0.951057i | 0 | 2.11318i | ||||||||
701.11 | −0.809017 | + | 0.587785i | 0 | 0.309017 | − | 0.951057i | 1.25194 | − | 1.72315i | 0 | −0.951057 | − | 0.309017i | 0.309017 | + | 0.951057i | 0 | 2.12993i | ||||||||
701.12 | −0.809017 | + | 0.587785i | 0 | 0.309017 | − | 0.951057i | 2.03332 | − | 2.79863i | 0 | 0.951057 | + | 0.309017i | 0.309017 | + | 0.951057i | 0 | 3.45929i | ||||||||
827.1 | 0.309017 | − | 0.951057i | 0 | −0.809017 | − | 0.587785i | −4.02756 | + | 1.30863i | 0 | 0.587785 | − | 0.809017i | −0.809017 | + | 0.587785i | 0 | 4.23483i | ||||||||
827.2 | 0.309017 | − | 0.951057i | 0 | −0.809017 | − | 0.587785i | −3.13150 | + | 1.01749i | 0 | −0.587785 | + | 0.809017i | −0.809017 | + | 0.587785i | 0 | 3.29265i | ||||||||
827.3 | 0.309017 | − | 0.951057i | 0 | −0.809017 | − | 0.587785i | −2.17923 | + | 0.708075i | 0 | 0.587785 | − | 0.809017i | −0.809017 | + | 0.587785i | 0 | 2.29138i | ||||||||
827.4 | 0.309017 | − | 0.951057i | 0 | −0.809017 | − | 0.587785i | −1.81910 | + | 0.591063i | 0 | 0.587785 | − | 0.809017i | −0.809017 | + | 0.587785i | 0 | 1.91272i | ||||||||
827.5 | 0.309017 | − | 0.951057i | 0 | −0.809017 | − | 0.587785i | −1.69500 | + | 0.550738i | 0 | −0.587785 | + | 0.809017i | −0.809017 | + | 0.587785i | 0 | 1.78222i | ||||||||
827.6 | 0.309017 | − | 0.951057i | 0 | −0.809017 | − | 0.587785i | 0.411682 | − | 0.133764i | 0 | −0.587785 | + | 0.809017i | −0.809017 | + | 0.587785i | 0 | − | 0.432868i | |||||||
827.7 | 0.309017 | − | 0.951057i | 0 | −0.809017 | − | 0.587785i | 0.599043 | − | 0.194641i | 0 | −0.587785 | + | 0.809017i | −0.809017 | + | 0.587785i | 0 | − | 0.629871i | |||||||
827.8 | 0.309017 | − | 0.951057i | 0 | −0.809017 | − | 0.587785i | 0.678594 | − | 0.220489i | 0 | 0.587785 | − | 0.809017i | −0.809017 | + | 0.587785i | 0 | − | 0.713516i | |||||||
See all 48 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
33.f | even | 10 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 1386.2.bu.a | ✓ | 48 |
3.b | odd | 2 | 1 | 1386.2.bu.b | yes | 48 | |
11.d | odd | 10 | 1 | 1386.2.bu.b | yes | 48 | |
33.f | even | 10 | 1 | inner | 1386.2.bu.a | ✓ | 48 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
1386.2.bu.a | ✓ | 48 | 1.a | even | 1 | 1 | trivial |
1386.2.bu.a | ✓ | 48 | 33.f | even | 10 | 1 | inner |
1386.2.bu.b | yes | 48 | 3.b | odd | 2 | 1 | |
1386.2.bu.b | yes | 48 | 11.d | odd | 10 | 1 |