Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [6038,2,Mod(1,6038)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(6038, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("6038.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 6038 = 2 \cdot 3019 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 6038.a (trivial) |
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: | yes |
Analytic conductor: | \(48.2136727404\) |
Analytic rank: | \(0\) |
Dimension: | \(70\) |
Twist minimal: | yes |
Fricke sign: | \(-1\) |
Sato-Tate group: | $\mathrm{SU}(2)$ |
$q$-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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
1.1 | 1.00000 | −3.07733 | 1.00000 | 0.607795 | −3.07733 | 4.61887 | 1.00000 | 6.46998 | 0.607795 | ||||||||||||||||||
1.2 | 1.00000 | −3.00085 | 1.00000 | −0.777425 | −3.00085 | 3.31557 | 1.00000 | 6.00511 | −0.777425 | ||||||||||||||||||
1.3 | 1.00000 | −2.94455 | 1.00000 | 3.92217 | −2.94455 | 4.11176 | 1.00000 | 5.67038 | 3.92217 | ||||||||||||||||||
1.4 | 1.00000 | −2.93737 | 1.00000 | −3.45627 | −2.93737 | 2.34218 | 1.00000 | 5.62815 | −3.45627 | ||||||||||||||||||
1.5 | 1.00000 | −2.81205 | 1.00000 | 2.74866 | −2.81205 | −2.36582 | 1.00000 | 4.90764 | 2.74866 | ||||||||||||||||||
1.6 | 1.00000 | −2.80845 | 1.00000 | 1.85496 | −2.80845 | −0.963784 | 1.00000 | 4.88737 | 1.85496 | ||||||||||||||||||
1.7 | 1.00000 | −2.79976 | 1.00000 | −2.91282 | −2.79976 | −0.947335 | 1.00000 | 4.83868 | −2.91282 | ||||||||||||||||||
1.8 | 1.00000 | −2.73176 | 1.00000 | 1.22157 | −2.73176 | −0.448839 | 1.00000 | 4.46253 | 1.22157 | ||||||||||||||||||
1.9 | 1.00000 | −2.44970 | 1.00000 | −2.67633 | −2.44970 | 2.27602 | 1.00000 | 3.00102 | −2.67633 | ||||||||||||||||||
1.10 | 1.00000 | −2.34935 | 1.00000 | −0.768821 | −2.34935 | −1.35127 | 1.00000 | 2.51947 | −0.768821 | ||||||||||||||||||
1.11 | 1.00000 | −2.23514 | 1.00000 | 3.00438 | −2.23514 | 1.56958 | 1.00000 | 1.99587 | 3.00438 | ||||||||||||||||||
1.12 | 1.00000 | −2.08835 | 1.00000 | −0.884909 | −2.08835 | −2.93795 | 1.00000 | 1.36120 | −0.884909 | ||||||||||||||||||
1.13 | 1.00000 | −2.05123 | 1.00000 | 3.94879 | −2.05123 | 5.22685 | 1.00000 | 1.20754 | 3.94879 | ||||||||||||||||||
1.14 | 1.00000 | −1.76800 | 1.00000 | 1.61299 | −1.76800 | 3.26382 | 1.00000 | 0.125835 | 1.61299 | ||||||||||||||||||
1.15 | 1.00000 | −1.60124 | 1.00000 | −1.76472 | −1.60124 | −0.621570 | 1.00000 | −0.436015 | −1.76472 | ||||||||||||||||||
1.16 | 1.00000 | −1.50199 | 1.00000 | −3.07076 | −1.50199 | 4.02705 | 1.00000 | −0.744026 | −3.07076 | ||||||||||||||||||
1.17 | 1.00000 | −1.49174 | 1.00000 | 1.96904 | −1.49174 | 1.25123 | 1.00000 | −0.774726 | 1.96904 | ||||||||||||||||||
1.18 | 1.00000 | −1.19115 | 1.00000 | −1.23162 | −1.19115 | −2.44516 | 1.00000 | −1.58116 | −1.23162 | ||||||||||||||||||
1.19 | 1.00000 | −1.12952 | 1.00000 | −0.938247 | −1.12952 | 0.310012 | 1.00000 | −1.72418 | −0.938247 | ||||||||||||||||||
1.20 | 1.00000 | −1.00890 | 1.00000 | 1.23789 | −1.00890 | −4.49878 | 1.00000 | −1.98213 | 1.23789 | ||||||||||||||||||
See all 70 embeddings |
Atkin-Lehner signs
\( p \) | Sign |
---|---|
\(2\) | \(-1\) |
\(3019\) | \(1\) |
Inner twists
This newform does not admit any (nontrivial) inner twists.
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 6038.2.a.e | ✓ | 70 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
6038.2.a.e | ✓ | 70 | 1.a | even | 1 | 1 | trivial |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{70} - 25 T_{3}^{69} + 163 T_{3}^{68} + 1056 T_{3}^{67} - 17875 T_{3}^{66} + 29034 T_{3}^{65} + \cdots + 4147175912 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(6038))\).