Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [4017,2,Mod(1,4017)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(4017, 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("4017.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 4017 = 3 \cdot 13 \cdot 103 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 4017.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: | \(32.0759064919\) |
Analytic rank: | \(0\) |
Dimension: | \(32\) |
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 | −2.81093 | 1.00000 | 5.90132 | −1.40450 | −2.81093 | 4.16883 | −10.9663 | 1.00000 | 3.94794 | ||||||||||||||||||
1.2 | −2.59984 | 1.00000 | 4.75917 | 1.47347 | −2.59984 | 1.66117 | −7.17341 | 1.00000 | −3.83079 | ||||||||||||||||||
1.3 | −2.39948 | 1.00000 | 3.75753 | −1.39162 | −2.39948 | −0.807842 | −4.21716 | 1.00000 | 3.33917 | ||||||||||||||||||
1.4 | −2.23055 | 1.00000 | 2.97534 | 2.00994 | −2.23055 | −4.12307 | −2.17553 | 1.00000 | −4.48326 | ||||||||||||||||||
1.5 | −2.21491 | 1.00000 | 2.90581 | 3.81514 | −2.21491 | 4.67656 | −2.00629 | 1.00000 | −8.45019 | ||||||||||||||||||
1.6 | −2.13224 | 1.00000 | 2.54643 | −4.09989 | −2.13224 | 4.42610 | −1.16513 | 1.00000 | 8.74193 | ||||||||||||||||||
1.7 | −1.74675 | 1.00000 | 1.05112 | 2.78436 | −1.74675 | −1.73589 | 1.65745 | 1.00000 | −4.86356 | ||||||||||||||||||
1.8 | −1.60531 | 1.00000 | 0.577033 | −3.03050 | −1.60531 | 3.88777 | 2.28431 | 1.00000 | 4.86490 | ||||||||||||||||||
1.9 | −1.46454 | 1.00000 | 0.144881 | 1.31201 | −1.46454 | 2.70261 | 2.71690 | 1.00000 | −1.92149 | ||||||||||||||||||
1.10 | −1.16220 | 1.00000 | −0.649300 | 0.840727 | −1.16220 | −4.11692 | 3.07901 | 1.00000 | −0.977089 | ||||||||||||||||||
1.11 | −1.05981 | 1.00000 | −0.876804 | −3.52707 | −1.05981 | −1.13469 | 3.04886 | 1.00000 | 3.73802 | ||||||||||||||||||
1.12 | −0.869920 | 1.00000 | −1.24324 | 3.55798 | −0.869920 | 3.07477 | 2.82136 | 1.00000 | −3.09516 | ||||||||||||||||||
1.13 | −0.628086 | 1.00000 | −1.60551 | −0.0316739 | −0.628086 | −0.526505 | 2.26457 | 1.00000 | 0.0198939 | ||||||||||||||||||
1.14 | −0.166477 | 1.00000 | −1.97229 | 3.74884 | −0.166477 | −0.361248 | 0.661293 | 1.00000 | −0.624094 | ||||||||||||||||||
1.15 | −0.0300119 | 1.00000 | −1.99910 | −0.348908 | −0.0300119 | −0.368688 | 0.120021 | 1.00000 | 0.0104714 | ||||||||||||||||||
1.16 | 0.135928 | 1.00000 | −1.98152 | −2.49648 | 0.135928 | 4.30504 | −0.541201 | 1.00000 | −0.339342 | ||||||||||||||||||
1.17 | 0.262075 | 1.00000 | −1.93132 | 0.855445 | 0.262075 | 3.34709 | −1.03030 | 1.00000 | 0.224191 | ||||||||||||||||||
1.18 | 0.397144 | 1.00000 | −1.84228 | −2.35712 | 0.397144 | −3.48052 | −1.52594 | 1.00000 | −0.936118 | ||||||||||||||||||
1.19 | 0.921096 | 1.00000 | −1.15158 | −1.44579 | 0.921096 | −0.134596 | −2.90291 | 1.00000 | −1.33171 | ||||||||||||||||||
1.20 | 1.15421 | 1.00000 | −0.667790 | 1.41297 | 1.15421 | 4.47597 | −3.07920 | 1.00000 | 1.63087 | ||||||||||||||||||
See all 32 embeddings |
Atkin-Lehner signs
\( p \) | Sign |
---|---|
\(3\) | \(-1\) |
\(13\) | \(-1\) |
\(103\) | \(-1\) |
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 | 4017.2.a.k | ✓ | 32 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
4017.2.a.k | ✓ | 32 | 1.a | even | 1 | 1 | trivial |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(4017))\):
\( T_{2}^{32} - 5 T_{2}^{31} - 40 T_{2}^{30} + 231 T_{2}^{29} + 663 T_{2}^{28} - 4748 T_{2}^{27} + \cdots - 192 \) |
\( T_{23}^{32} - 37 T_{23}^{31} + 308 T_{23}^{30} + 4413 T_{23}^{29} - 78221 T_{23}^{28} + \cdots + 10\!\cdots\!52 \) |