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: | \(1\) |
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 | −2.74671 | −1.00000 | 5.54443 | −3.14409 | 2.74671 | −4.71348 | −9.73553 | 1.00000 | 8.63591 | ||||||||||||||||||
1.2 | −2.70695 | −1.00000 | 5.32758 | 2.36895 | 2.70695 | 2.88217 | −9.00759 | 1.00000 | −6.41264 | ||||||||||||||||||
1.3 | −2.67819 | −1.00000 | 5.17269 | −0.816148 | 2.67819 | 2.66447 | −8.49705 | 1.00000 | 2.18580 | ||||||||||||||||||
1.4 | −2.21324 | −1.00000 | 2.89842 | 2.02862 | 2.21324 | −4.79566 | −1.98841 | 1.00000 | −4.48982 | ||||||||||||||||||
1.5 | −2.06452 | −1.00000 | 2.26226 | −0.587747 | 2.06452 | −1.74034 | −0.541432 | 1.00000 | 1.21342 | ||||||||||||||||||
1.6 | −1.82913 | −1.00000 | 1.34571 | 4.14175 | 1.82913 | −1.02190 | 1.19679 | 1.00000 | −7.57579 | ||||||||||||||||||
1.7 | −1.77190 | −1.00000 | 1.13963 | −2.58005 | 1.77190 | 2.67810 | 1.52449 | 1.00000 | 4.57158 | ||||||||||||||||||
1.8 | −1.38482 | −1.00000 | −0.0822870 | −0.467175 | 1.38482 | −3.78384 | 2.88358 | 1.00000 | 0.646950 | ||||||||||||||||||
1.9 | −1.29020 | −1.00000 | −0.335394 | 1.27850 | 1.29020 | 0.146656 | 3.01312 | 1.00000 | −1.64951 | ||||||||||||||||||
1.10 | −0.985882 | −1.00000 | −1.02804 | −3.07225 | 0.985882 | 1.70297 | 2.98529 | 1.00000 | 3.02888 | ||||||||||||||||||
1.11 | −0.856339 | −1.00000 | −1.26668 | 1.28941 | 0.856339 | 1.75524 | 2.79739 | 1.00000 | −1.10417 | ||||||||||||||||||
1.12 | −0.633751 | −1.00000 | −1.59836 | 3.72594 | 0.633751 | −0.819859 | 2.28046 | 1.00000 | −2.36132 | ||||||||||||||||||
1.13 | −0.451110 | −1.00000 | −1.79650 | −1.20422 | 0.451110 | 5.19076 | 1.71264 | 1.00000 | 0.543237 | ||||||||||||||||||
1.14 | −0.0180837 | −1.00000 | −1.99967 | −4.11271 | 0.0180837 | −4.60383 | 0.0723287 | 1.00000 | 0.0743729 | ||||||||||||||||||
1.15 | 0.343002 | −1.00000 | −1.88235 | 1.20148 | −0.343002 | −3.15145 | −1.33165 | 1.00000 | 0.412112 | ||||||||||||||||||
1.16 | 0.563268 | −1.00000 | −1.68273 | 2.43057 | −0.563268 | 2.01416 | −2.07436 | 1.00000 | 1.36906 | ||||||||||||||||||
1.17 | 0.626060 | −1.00000 | −1.60805 | −4.21209 | −0.626060 | 2.03004 | −2.25885 | 1.00000 | −2.63702 | ||||||||||||||||||
1.18 | 1.34842 | −1.00000 | −0.181762 | −1.77320 | −1.34842 | 1.31404 | −2.94193 | 1.00000 | −2.39102 | ||||||||||||||||||
1.19 | 1.62615 | −1.00000 | 0.644365 | 0.607200 | −1.62615 | 4.96474 | −2.20447 | 1.00000 | 0.987398 | ||||||||||||||||||
1.20 | 1.81336 | −1.00000 | 1.28829 | 3.11371 | −1.81336 | −0.776263 | −1.29059 | 1.00000 | 5.64629 | ||||||||||||||||||
See all 25 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.h | ✓ | 25 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
4017.2.a.h | ✓ | 25 | 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(4017))\):
\( T_{2}^{25} + 4 T_{2}^{24} - 31 T_{2}^{23} - 137 T_{2}^{22} + 394 T_{2}^{21} + 2025 T_{2}^{20} - 2550 T_{2}^{19} - 16916 T_{2}^{18} + 7684 T_{2}^{17} + 87742 T_{2}^{16} + 2511 T_{2}^{15} - 291899 T_{2}^{14} - 103146 T_{2}^{13} + \cdots - 64 \) |
\( T_{23}^{25} + 47 T_{23}^{24} + 760 T_{23}^{23} + 1953 T_{23}^{22} - 86260 T_{23}^{21} - 964550 T_{23}^{20} + 42074 T_{23}^{19} + 61178887 T_{23}^{18} + 305164092 T_{23}^{17} - 1196214094 T_{23}^{16} + \cdots + 6539299170560 \) |