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 | −2.75387 | 1.00000 | 5.58381 | 3.90153 | −2.75387 | 0.781129 | −9.86935 | 1.00000 | −10.7443 | ||||||||||||||||||
1.2 | −2.67949 | 1.00000 | 5.17965 | −3.25805 | −2.67949 | 0.886771 | −8.51984 | 1.00000 | 8.72990 | ||||||||||||||||||
1.3 | −2.65538 | 1.00000 | 5.05105 | −1.68203 | −2.65538 | −3.96430 | −8.10171 | 1.00000 | 4.46642 | ||||||||||||||||||
1.4 | −2.55932 | 1.00000 | 4.55014 | −2.63190 | −2.55932 | 2.75923 | −6.52662 | 1.00000 | 6.73589 | ||||||||||||||||||
1.5 | −1.93942 | 1.00000 | 1.76133 | 0.142452 | −1.93942 | 3.99599 | 0.462872 | 1.00000 | −0.276273 | ||||||||||||||||||
1.6 | −1.85552 | 1.00000 | 1.44297 | 3.38665 | −1.85552 | −2.21342 | 1.03358 | 1.00000 | −6.28400 | ||||||||||||||||||
1.7 | −1.79771 | 1.00000 | 1.23177 | 3.18677 | −1.79771 | 1.45324 | 1.38105 | 1.00000 | −5.72890 | ||||||||||||||||||
1.8 | −1.78664 | 1.00000 | 1.19208 | −1.81642 | −1.78664 | −4.42230 | 1.44346 | 1.00000 | 3.24528 | ||||||||||||||||||
1.9 | −1.48812 | 1.00000 | 0.214502 | −3.58678 | −1.48812 | 0.0236485 | 2.65704 | 1.00000 | 5.33757 | ||||||||||||||||||
1.10 | −1.42411 | 1.00000 | 0.0280857 | −0.882122 | −1.42411 | 4.13066 | 2.80822 | 1.00000 | 1.25624 | ||||||||||||||||||
1.11 | −1.08109 | 1.00000 | −0.831238 | −0.942662 | −1.08109 | −2.40511 | 3.06083 | 1.00000 | 1.01911 | ||||||||||||||||||
1.12 | −0.772884 | 1.00000 | −1.40265 | −0.606956 | −0.772884 | 1.27278 | 2.62985 | 1.00000 | 0.469107 | ||||||||||||||||||
1.13 | −0.365601 | 1.00000 | −1.86634 | 0.232769 | −0.365601 | −3.73203 | 1.41354 | 1.00000 | −0.0851006 | ||||||||||||||||||
1.14 | −0.238219 | 1.00000 | −1.94325 | 2.19571 | −0.238219 | 5.28038 | 0.939357 | 1.00000 | −0.523060 | ||||||||||||||||||
1.15 | −0.163737 | 1.00000 | −1.97319 | 3.16137 | −0.163737 | 4.62100 | 0.650558 | 1.00000 | −0.517632 | ||||||||||||||||||
1.16 | −0.0270033 | 1.00000 | −1.99927 | −4.41470 | −0.0270033 | 1.75798 | 0.107994 | 1.00000 | 0.119212 | ||||||||||||||||||
1.17 | 0.310907 | 1.00000 | −1.90334 | 2.19968 | 0.310907 | −3.79510 | −1.21358 | 1.00000 | 0.683896 | ||||||||||||||||||
1.18 | 0.609784 | 1.00000 | −1.62816 | 2.30670 | 0.609784 | −1.80421 | −2.21240 | 1.00000 | 1.40659 | ||||||||||||||||||
1.19 | 0.934190 | 1.00000 | −1.12729 | −1.38248 | 0.934190 | −2.39473 | −2.92148 | 1.00000 | −1.29149 | ||||||||||||||||||
1.20 | 1.11274 | 1.00000 | −0.761813 | −3.97825 | 1.11274 | −2.83032 | −3.07318 | 1.00000 | −4.42675 | ||||||||||||||||||
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.l | ✓ | 32 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
4017.2.a.l | ✓ | 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(4017))\):
\( T_{2}^{32} - 5 T_{2}^{31} - 42 T_{2}^{30} + 241 T_{2}^{29} + 733 T_{2}^{28} - 5160 T_{2}^{27} + \cdots + 448 \) |
\( T_{23}^{32} - 53 T_{23}^{31} + 944 T_{23}^{30} - 1055 T_{23}^{29} - 190237 T_{23}^{28} + \cdots + 58\!\cdots\!00 \) |