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: | \(57\) |
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 | −3.40132 | 1.00000 | −0.259244 | 3.40132 | 0.796720 | −1.00000 | 8.56895 | 0.259244 | ||||||||||||||||||
1.2 | −1.00000 | −3.09911 | 1.00000 | −3.06696 | 3.09911 | −3.37062 | −1.00000 | 6.60450 | 3.06696 | ||||||||||||||||||
1.3 | −1.00000 | −3.09194 | 1.00000 | −2.28798 | 3.09194 | 0.749696 | −1.00000 | 6.56008 | 2.28798 | ||||||||||||||||||
1.4 | −1.00000 | −2.99307 | 1.00000 | 2.62099 | 2.99307 | −1.28357 | −1.00000 | 5.95844 | −2.62099 | ||||||||||||||||||
1.5 | −1.00000 | −2.94445 | 1.00000 | −1.42608 | 2.94445 | 1.57265 | −1.00000 | 5.66979 | 1.42608 | ||||||||||||||||||
1.6 | −1.00000 | −2.85836 | 1.00000 | −3.68447 | 2.85836 | 2.06392 | −1.00000 | 5.17024 | 3.68447 | ||||||||||||||||||
1.7 | −1.00000 | −2.75036 | 1.00000 | 0.139827 | 2.75036 | 0.205180 | −1.00000 | 4.56449 | −0.139827 | ||||||||||||||||||
1.8 | −1.00000 | −2.68115 | 1.00000 | 1.78050 | 2.68115 | −4.24119 | −1.00000 | 4.18856 | −1.78050 | ||||||||||||||||||
1.9 | −1.00000 | −2.46647 | 1.00000 | 3.27492 | 2.46647 | 3.14216 | −1.00000 | 3.08346 | −3.27492 | ||||||||||||||||||
1.10 | −1.00000 | −2.39662 | 1.00000 | −1.22776 | 2.39662 | −4.40908 | −1.00000 | 2.74377 | 1.22776 | ||||||||||||||||||
1.11 | −1.00000 | −2.25784 | 1.00000 | 0.481380 | 2.25784 | −0.819021 | −1.00000 | 2.09784 | −0.481380 | ||||||||||||||||||
1.12 | −1.00000 | −2.15416 | 1.00000 | 1.67498 | 2.15416 | 2.16308 | −1.00000 | 1.64039 | −1.67498 | ||||||||||||||||||
1.13 | −1.00000 | −2.12089 | 1.00000 | −1.22072 | 2.12089 | −2.75912 | −1.00000 | 1.49816 | 1.22072 | ||||||||||||||||||
1.14 | −1.00000 | −1.79555 | 1.00000 | 3.22751 | 1.79555 | −1.51507 | −1.00000 | 0.223993 | −3.22751 | ||||||||||||||||||
1.15 | −1.00000 | −1.71541 | 1.00000 | 1.06645 | 1.71541 | 4.85447 | −1.00000 | −0.0573586 | −1.06645 | ||||||||||||||||||
1.16 | −1.00000 | −1.68023 | 1.00000 | −2.51181 | 1.68023 | 3.66839 | −1.00000 | −0.176828 | 2.51181 | ||||||||||||||||||
1.17 | −1.00000 | −1.57568 | 1.00000 | 1.43358 | 1.57568 | 2.19919 | −1.00000 | −0.517233 | −1.43358 | ||||||||||||||||||
1.18 | −1.00000 | −1.49224 | 1.00000 | −4.20825 | 1.49224 | −4.82054 | −1.00000 | −0.773233 | 4.20825 | ||||||||||||||||||
1.19 | −1.00000 | −1.42937 | 1.00000 | −1.03254 | 1.42937 | −0.605000 | −1.00000 | −0.956891 | 1.03254 | ||||||||||||||||||
1.20 | −1.00000 | −1.21264 | 1.00000 | 1.98344 | 1.21264 | −0.999136 | −1.00000 | −1.52951 | −1.98344 | ||||||||||||||||||
See all 57 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.c | ✓ | 57 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
6038.2.a.c | ✓ | 57 | 1.a | even | 1 | 1 | trivial |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{57} + 5 T_{3}^{56} - 98 T_{3}^{55} - 515 T_{3}^{54} + 4458 T_{3}^{53} + 24847 T_{3}^{52} + \cdots - 33684056 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(6038))\).