Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [6039,2,Mod(1,6039)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(6039, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0, 0]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("6039.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 6039 = 3^{2} \cdot 11 \cdot 61 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 6039.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: | \(48.2216577807\) |
Analytic rank: | \(1\) |
Dimension: | \(25\) |
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 | −2.77875 | 0 | 5.72143 | 2.25657 | 0 | 0.879084 | −10.3409 | 0 | −6.27043 | ||||||||||||||||||
1.2 | −2.62068 | 0 | 4.86797 | −4.12800 | 0 | −0.639358 | −7.51602 | 0 | 10.8182 | ||||||||||||||||||
1.3 | −2.50682 | 0 | 4.28416 | −0.826559 | 0 | 1.21549 | −5.72598 | 0 | 2.07204 | ||||||||||||||||||
1.4 | −2.28499 | 0 | 3.22117 | −2.86039 | 0 | 3.60040 | −2.79036 | 0 | 6.53597 | ||||||||||||||||||
1.5 | −2.22076 | 0 | 2.93178 | −0.179885 | 0 | −5.01600 | −2.06925 | 0 | 0.399482 | ||||||||||||||||||
1.6 | −1.79702 | 0 | 1.22928 | 4.13243 | 0 | 4.22858 | 1.38500 | 0 | −7.42606 | ||||||||||||||||||
1.7 | −1.77462 | 0 | 1.14926 | 1.38341 | 0 | −3.00047 | 1.50973 | 0 | −2.45503 | ||||||||||||||||||
1.8 | −1.65004 | 0 | 0.722641 | 2.19434 | 0 | −0.319690 | 2.10770 | 0 | −3.62075 | ||||||||||||||||||
1.9 | −1.56609 | 0 | 0.452630 | −1.36136 | 0 | 3.35990 | 2.42332 | 0 | 2.13201 | ||||||||||||||||||
1.10 | −0.926278 | 0 | −1.14201 | −3.01280 | 0 | 2.23888 | 2.91037 | 0 | 2.79069 | ||||||||||||||||||
1.11 | −0.702784 | 0 | −1.50609 | 2.41068 | 0 | 0.380608 | 2.46403 | 0 | −1.69419 | ||||||||||||||||||
1.12 | −0.226426 | 0 | −1.94873 | −3.14756 | 0 | −2.84661 | 0.894095 | 0 | 0.712690 | ||||||||||||||||||
1.13 | −0.110609 | 0 | −1.98777 | 0.0761036 | 0 | 3.79648 | 0.441081 | 0 | −0.00841771 | ||||||||||||||||||
1.14 | −0.0892703 | 0 | −1.99203 | −0.250715 | 0 | −4.16429 | 0.356370 | 0 | 0.0223814 | ||||||||||||||||||
1.15 | −0.0238689 | 0 | −1.99943 | 2.56388 | 0 | 0.894211 | 0.0954618 | 0 | −0.0611970 | ||||||||||||||||||
1.16 | 0.363962 | 0 | −1.86753 | −0.137887 | 0 | 3.90557 | −1.40764 | 0 | −0.0501855 | ||||||||||||||||||
1.17 | 1.04470 | 0 | −0.908610 | 2.79878 | 0 | −4.32870 | −3.03861 | 0 | 2.92388 | ||||||||||||||||||
1.18 | 1.06662 | 0 | −0.862332 | −3.84706 | 0 | −1.30714 | −3.05301 | 0 | −4.10333 | ||||||||||||||||||
1.19 | 1.07776 | 0 | −0.838436 | −3.04682 | 0 | 3.44893 | −3.05915 | 0 | −3.28374 | ||||||||||||||||||
1.20 | 1.76243 | 0 | 1.10617 | 2.60834 | 0 | 0.445474 | −1.57532 | 0 | 4.59703 | ||||||||||||||||||
See all 25 embeddings |
Atkin-Lehner signs
\( p \) | Sign |
---|---|
\(3\) | \(1\) |
\(11\) | \(1\) |
\(61\) | \(1\) |
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 | 6039.2.a.n | ✓ | 25 |
3.b | odd | 2 | 1 | 6039.2.a.o | yes | 25 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
6039.2.a.n | ✓ | 25 | 1.a | even | 1 | 1 | trivial |
6039.2.a.o | yes | 25 | 3.b | odd | 2 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{25} + 5 T_{2}^{24} - 25 T_{2}^{23} - 155 T_{2}^{22} + 226 T_{2}^{21} + 2056 T_{2}^{20} - 607 T_{2}^{19} + \cdots + 1 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(6039))\).