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 | −2.74549 | −1.00000 | 5.53773 | 0.596155 | 2.74549 | −1.94658 | −9.71281 | 1.00000 | −1.63674 | ||||||||||||||||||
1.2 | −2.66982 | −1.00000 | 5.12792 | −3.00815 | 2.66982 | 2.19544 | −8.35096 | 1.00000 | 8.03119 | ||||||||||||||||||
1.3 | −2.38645 | −1.00000 | 3.69513 | 2.94139 | 2.38645 | −4.88857 | −4.04533 | 1.00000 | −7.01947 | ||||||||||||||||||
1.4 | −2.31427 | −1.00000 | 3.35587 | 3.52277 | 2.31427 | 3.16201 | −3.13785 | 1.00000 | −8.15265 | ||||||||||||||||||
1.5 | −2.25716 | −1.00000 | 3.09477 | −2.92277 | 2.25716 | −0.587921 | −2.47106 | 1.00000 | 6.59715 | ||||||||||||||||||
1.6 | −2.08104 | −1.00000 | 2.33074 | 1.74075 | 2.08104 | −0.124046 | −0.688276 | 1.00000 | −3.62257 | ||||||||||||||||||
1.7 | −1.54007 | −1.00000 | 0.371810 | −0.217078 | 1.54007 | 2.62202 | 2.50752 | 1.00000 | 0.334314 | ||||||||||||||||||
1.8 | −1.47736 | −1.00000 | 0.182601 | −3.64531 | 1.47736 | −4.01724 | 2.68496 | 1.00000 | 5.38545 | ||||||||||||||||||
1.9 | −1.46195 | −1.00000 | 0.137285 | −2.88545 | 1.46195 | 1.16184 | 2.72319 | 1.00000 | 4.21838 | ||||||||||||||||||
1.10 | −0.726163 | −1.00000 | −1.47269 | 0.449507 | 0.726163 | 4.04793 | 2.52174 | 1.00000 | −0.326416 | ||||||||||||||||||
1.11 | −0.590079 | −1.00000 | −1.65181 | 0.503953 | 0.590079 | −2.98562 | 2.15486 | 1.00000 | −0.297372 | ||||||||||||||||||
1.12 | −0.307676 | −1.00000 | −1.90534 | −1.88076 | 0.307676 | −3.18485 | 1.20158 | 1.00000 | 0.578664 | ||||||||||||||||||
1.13 | −0.150477 | −1.00000 | −1.97736 | 2.78105 | 0.150477 | 2.48525 | 0.598501 | 1.00000 | −0.418485 | ||||||||||||||||||
1.14 | 0.444143 | −1.00000 | −1.80274 | 0.495430 | −0.444143 | 2.63750 | −1.68896 | 1.00000 | 0.220041 | ||||||||||||||||||
1.15 | 0.823398 | −1.00000 | −1.32202 | −3.52626 | −0.823398 | −0.677842 | −2.73534 | 1.00000 | −2.90351 | ||||||||||||||||||
1.16 | 0.831544 | −1.00000 | −1.30853 | −3.33918 | −0.831544 | −4.05766 | −2.75119 | 1.00000 | −2.77667 | ||||||||||||||||||
1.17 | 0.883592 | −1.00000 | −1.21927 | 3.49819 | −0.883592 | −3.21070 | −2.84452 | 1.00000 | 3.09097 | ||||||||||||||||||
1.18 | 1.25624 | −1.00000 | −0.421872 | 0.726022 | −1.25624 | 0.867951 | −3.04244 | 1.00000 | 0.912055 | ||||||||||||||||||
1.19 | 1.44489 | −1.00000 | 0.0877157 | 4.08746 | −1.44489 | −1.49545 | −2.76305 | 1.00000 | 5.90594 | ||||||||||||||||||
1.20 | 1.46693 | −1.00000 | 0.151895 | 0.636277 | −1.46693 | −0.403322 | −2.71105 | 1.00000 | 0.933377 | ||||||||||||||||||
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.i | ✓ | 25 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
4017.2.a.i | ✓ | 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(4017))\):
\( T_{2}^{25} + 2 T_{2}^{24} - 37 T_{2}^{23} - 69 T_{2}^{22} + 600 T_{2}^{21} + 1023 T_{2}^{20} + \cdots + 576 \) |
\( T_{23}^{25} + 49 T_{23}^{24} + 922 T_{23}^{23} + 6773 T_{23}^{22} - 22026 T_{23}^{21} + \cdots + 86801359872 \) |