Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [4009,2,Mod(1,4009)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(4009, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([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("4009.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 4009 = 19 \cdot 211 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 4009.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: | \(32.0120261703\) |
Analytic rank: | \(0\) |
Dimension: | \(82\) |
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 | −2.71228 | −0.799829 | 5.35647 | 2.45685 | 2.16936 | −1.47623 | −9.10370 | −2.36027 | −6.66368 | ||||||||||||||||||
1.2 | −2.64355 | −0.244924 | 4.98837 | −2.38993 | 0.647469 | −3.32754 | −7.89992 | −2.94001 | 6.31792 | ||||||||||||||||||
1.3 | −2.62936 | 3.14041 | 4.91352 | 2.46626 | −8.25726 | 1.15889 | −7.66068 | 6.86218 | −6.48468 | ||||||||||||||||||
1.4 | −2.62201 | −2.46509 | 4.87496 | −1.74208 | 6.46350 | −2.61324 | −7.53817 | 3.07667 | 4.56776 | ||||||||||||||||||
1.5 | −2.40525 | 0.482964 | 3.78524 | 1.80217 | −1.16165 | −1.90696 | −4.29395 | −2.76675 | −4.33466 | ||||||||||||||||||
1.6 | −2.39399 | 1.52449 | 3.73121 | −2.51919 | −3.64962 | −1.00997 | −4.14450 | −0.675931 | 6.03093 | ||||||||||||||||||
1.7 | −2.39206 | −0.112128 | 3.72195 | 0.840987 | 0.268217 | 2.66760 | −4.11902 | −2.98743 | −2.01169 | ||||||||||||||||||
1.8 | −2.27080 | −1.94834 | 3.15654 | 0.0772315 | 4.42430 | 2.91283 | −2.62627 | 0.796044 | −0.175377 | ||||||||||||||||||
1.9 | −2.22938 | 2.96636 | 2.97012 | −0.893171 | −6.61314 | 4.38195 | −2.16277 | 5.79930 | 1.99121 | ||||||||||||||||||
1.10 | −2.21501 | 1.93892 | 2.90626 | 3.78369 | −4.29473 | 0.133386 | −2.00737 | 0.759427 | −8.38090 | ||||||||||||||||||
1.11 | −2.16486 | 3.17577 | 2.68660 | 0.570979 | −6.87507 | −2.83752 | −1.48639 | 7.08548 | −1.23609 | ||||||||||||||||||
1.12 | −2.09995 | −2.73690 | 2.40979 | 2.29135 | 5.74735 | −1.74327 | −0.860544 | 4.49062 | −4.81171 | ||||||||||||||||||
1.13 | −1.93317 | −1.52394 | 1.73714 | −2.99074 | 2.94604 | −0.667472 | 0.508147 | −0.677595 | 5.78161 | ||||||||||||||||||
1.14 | −1.90886 | 1.35744 | 1.64375 | −1.48428 | −2.59115 | 2.77982 | 0.680038 | −1.15737 | 2.83328 | ||||||||||||||||||
1.15 | −1.89977 | −2.93667 | 1.60912 | −0.296578 | 5.57898 | 2.59029 | 0.742589 | 5.62400 | 0.563430 | ||||||||||||||||||
1.16 | −1.69115 | −1.97765 | 0.859975 | −4.28235 | 3.34450 | 4.00531 | 1.92795 | 0.911117 | 7.24208 | ||||||||||||||||||
1.17 | −1.66851 | −1.79206 | 0.783934 | 2.47432 | 2.99007 | 0.566931 | 2.02902 | 0.211475 | −4.12843 | ||||||||||||||||||
1.18 | −1.53574 | 2.00602 | 0.358512 | 1.47205 | −3.08073 | −4.88163 | 2.52091 | 1.02410 | −2.26069 | ||||||||||||||||||
1.19 | −1.53259 | 1.29314 | 0.348820 | 3.28950 | −1.98185 | 4.48074 | 2.53058 | −1.32779 | −5.04144 | ||||||||||||||||||
1.20 | −1.51298 | 0.189655 | 0.289107 | −0.503823 | −0.286944 | −1.74123 | 2.58855 | −2.96403 | 0.762274 | ||||||||||||||||||
See all 82 embeddings |
Atkin-Lehner signs
\( p \) | Sign |
---|---|
\(19\) | \(-1\) |
\(211\) | \(1\) |
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 | 4009.2.a.e | ✓ | 82 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
4009.2.a.e | ✓ | 82 | 1.a | even | 1 | 1 | trivial |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{82} - 15 T_{2}^{81} - 14 T_{2}^{80} + 1301 T_{2}^{79} - 3930 T_{2}^{78} - 50065 T_{2}^{77} + \cdots - 1538721 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(4009))\).