Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [4012,2,Mod(237,4012)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(4012, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 1, 0]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("4012.237");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 4012 = 2^{2} \cdot 17 \cdot 59 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 4012.b (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: | \(32.0359812909\) |
Analytic rank: | \(0\) |
Dimension: | \(40\) |
Twist minimal: | yes |
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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
237.1 | 0 | − | 3.27953i | 0 | 4.21610i | 0 | 0.547734i | 0 | −7.75529 | 0 | |||||||||||||||||
237.2 | 0 | − | 3.07288i | 0 | − | 1.34148i | 0 | 0.0167420i | 0 | −6.44259 | 0 | ||||||||||||||||
237.3 | 0 | − | 2.97688i | 0 | − | 2.24434i | 0 | − | 1.80545i | 0 | −5.86181 | 0 | |||||||||||||||
237.4 | 0 | − | 2.95725i | 0 | 2.24490i | 0 | 1.49294i | 0 | −5.74532 | 0 | |||||||||||||||||
237.5 | 0 | − | 2.44713i | 0 | − | 1.66009i | 0 | 3.65944i | 0 | −2.98842 | 0 | ||||||||||||||||
237.6 | 0 | − | 2.32869i | 0 | − | 4.13581i | 0 | − | 2.86231i | 0 | −2.42278 | 0 | |||||||||||||||
237.7 | 0 | − | 2.23721i | 0 | 0.538464i | 0 | − | 0.0679637i | 0 | −2.00513 | 0 | ||||||||||||||||
237.8 | 0 | − | 2.21522i | 0 | 2.69018i | 0 | − | 2.88510i | 0 | −1.90721 | 0 | ||||||||||||||||
237.9 | 0 | − | 2.18094i | 0 | − | 0.351694i | 0 | 3.15514i | 0 | −1.75651 | 0 | ||||||||||||||||
237.10 | 0 | − | 1.86007i | 0 | 2.89135i | 0 | 5.10759i | 0 | −0.459864 | 0 | |||||||||||||||||
237.11 | 0 | − | 1.67902i | 0 | − | 0.714826i | 0 | 1.81204i | 0 | 0.180890 | 0 | ||||||||||||||||
237.12 | 0 | − | 1.51601i | 0 | − | 1.74989i | 0 | − | 0.991299i | 0 | 0.701704 | 0 | |||||||||||||||
237.13 | 0 | − | 1.46162i | 0 | 3.03678i | 0 | − | 1.90612i | 0 | 0.863667 | 0 | ||||||||||||||||
237.14 | 0 | − | 1.18944i | 0 | − | 0.609716i | 0 | − | 4.31103i | 0 | 1.58523 | 0 | |||||||||||||||
237.15 | 0 | − | 1.02370i | 0 | 3.54148i | 0 | 0.166941i | 0 | 1.95203 | 0 | |||||||||||||||||
237.16 | 0 | − | 0.622620i | 0 | 0.657734i | 0 | − | 2.30157i | 0 | 2.61234 | 0 | ||||||||||||||||
237.17 | 0 | − | 0.510359i | 0 | − | 1.04821i | 0 | 1.04964i | 0 | 2.73953 | 0 | ||||||||||||||||
237.18 | 0 | − | 0.409767i | 0 | 0.188112i | 0 | − | 3.66007i | 0 | 2.83209 | 0 | ||||||||||||||||
237.19 | 0 | − | 0.346659i | 0 | − | 2.94947i | 0 | − | 4.18117i | 0 | 2.87983 | 0 | |||||||||||||||
237.20 | 0 | − | 0.0489925i | 0 | − | 3.24065i | 0 | 0.683113i | 0 | 2.99760 | 0 | ||||||||||||||||
See all 40 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
17.b | even | 2 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 4012.2.b.a | ✓ | 40 |
17.b | even | 2 | 1 | inner | 4012.2.b.a | ✓ | 40 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
4012.2.b.a | ✓ | 40 | 1.a | even | 1 | 1 | trivial |
4012.2.b.a | ✓ | 40 | 17.b | even | 2 | 1 | inner |