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: | \(1\) |
Dimension: | \(54\) |
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 | −3.41141 | 1.00000 | 0.211160 | −3.41141 | −3.35690 | 1.00000 | 8.63772 | 0.211160 | ||||||||||||||||||
1.2 | 1.00000 | −3.37741 | 1.00000 | −2.92787 | −3.37741 | −0.403159 | 1.00000 | 8.40689 | −2.92787 | ||||||||||||||||||
1.3 | 1.00000 | −3.34616 | 1.00000 | 0.269513 | −3.34616 | 1.67569 | 1.00000 | 8.19680 | 0.269513 | ||||||||||||||||||
1.4 | 1.00000 | −3.17799 | 1.00000 | −0.0583780 | −3.17799 | −5.13519 | 1.00000 | 7.09963 | −0.0583780 | ||||||||||||||||||
1.5 | 1.00000 | −2.98721 | 1.00000 | 2.38189 | −2.98721 | −0.220104 | 1.00000 | 5.92341 | 2.38189 | ||||||||||||||||||
1.6 | 1.00000 | −2.82052 | 1.00000 | −0.526511 | −2.82052 | 2.78989 | 1.00000 | 4.95531 | −0.526511 | ||||||||||||||||||
1.7 | 1.00000 | −2.77973 | 1.00000 | 3.72096 | −2.77973 | −2.52302 | 1.00000 | 4.72690 | 3.72096 | ||||||||||||||||||
1.8 | 1.00000 | −2.56435 | 1.00000 | −4.01554 | −2.56435 | −3.36655 | 1.00000 | 3.57587 | −4.01554 | ||||||||||||||||||
1.9 | 1.00000 | −2.48349 | 1.00000 | 3.70736 | −2.48349 | −1.05138 | 1.00000 | 3.16770 | 3.70736 | ||||||||||||||||||
1.10 | 1.00000 | −2.40402 | 1.00000 | −1.73865 | −2.40402 | 4.25060 | 1.00000 | 2.77932 | −1.73865 | ||||||||||||||||||
1.11 | 1.00000 | −2.32590 | 1.00000 | 3.29375 | −2.32590 | −5.00561 | 1.00000 | 2.40980 | 3.29375 | ||||||||||||||||||
1.12 | 1.00000 | −2.27315 | 1.00000 | −2.64091 | −2.27315 | 0.657447 | 1.00000 | 2.16722 | −2.64091 | ||||||||||||||||||
1.13 | 1.00000 | −2.11795 | 1.00000 | 1.57843 | −2.11795 | 1.37381 | 1.00000 | 1.48570 | 1.57843 | ||||||||||||||||||
1.14 | 1.00000 | −2.02410 | 1.00000 | −4.35432 | −2.02410 | −1.05145 | 1.00000 | 1.09700 | −4.35432 | ||||||||||||||||||
1.15 | 1.00000 | −1.99905 | 1.00000 | −3.02278 | −1.99905 | −5.14057 | 1.00000 | 0.996217 | −3.02278 | ||||||||||||||||||
1.16 | 1.00000 | −1.93892 | 1.00000 | 0.524165 | −1.93892 | −1.38269 | 1.00000 | 0.759396 | 0.524165 | ||||||||||||||||||
1.17 | 1.00000 | −1.77991 | 1.00000 | −0.815447 | −1.77991 | −3.69164 | 1.00000 | 0.168091 | −0.815447 | ||||||||||||||||||
1.18 | 1.00000 | −1.61049 | 1.00000 | 1.72129 | −1.61049 | 1.31284 | 1.00000 | −0.406311 | 1.72129 | ||||||||||||||||||
1.19 | 1.00000 | −1.56837 | 1.00000 | −1.56733 | −1.56837 | 2.76703 | 1.00000 | −0.540224 | −1.56733 | ||||||||||||||||||
1.20 | 1.00000 | −1.51709 | 1.00000 | 2.00493 | −1.51709 | 2.93208 | 1.00000 | −0.698426 | 2.00493 | ||||||||||||||||||
See all 54 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.b | ✓ | 54 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
6038.2.a.b | ✓ | 54 | 1.a | even | 1 | 1 | trivial |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{54} + 21 T_{3}^{53} + 120 T_{3}^{52} - 503 T_{3}^{51} - 8000 T_{3}^{50} - 14065 T_{3}^{49} + \cdots + 5806105 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(6038))\).