Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [6010,2,Mod(1,6010)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(6010, 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("6010.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 6010 = 2 \cdot 5 \cdot 601 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 6010.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: | \(47.9900916148\) |
Analytic rank: | \(1\) |
Dimension: | \(22\) |
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 | 1.00000 | −3.26920 | 1.00000 | −1.00000 | −3.26920 | 0.898513 | 1.00000 | 7.68765 | −1.00000 | ||||||||||||||||||
1.2 | 1.00000 | −3.13304 | 1.00000 | −1.00000 | −3.13304 | −3.32735 | 1.00000 | 6.81597 | −1.00000 | ||||||||||||||||||
1.3 | 1.00000 | −2.63651 | 1.00000 | −1.00000 | −2.63651 | 2.17132 | 1.00000 | 3.95120 | −1.00000 | ||||||||||||||||||
1.4 | 1.00000 | −2.29642 | 1.00000 | −1.00000 | −2.29642 | −4.86493 | 1.00000 | 2.27356 | −1.00000 | ||||||||||||||||||
1.5 | 1.00000 | −1.98091 | 1.00000 | −1.00000 | −1.98091 | −1.31383 | 1.00000 | 0.924017 | −1.00000 | ||||||||||||||||||
1.6 | 1.00000 | −1.88167 | 1.00000 | −1.00000 | −1.88167 | −3.66727 | 1.00000 | 0.540685 | −1.00000 | ||||||||||||||||||
1.7 | 1.00000 | −1.55148 | 1.00000 | −1.00000 | −1.55148 | 2.51624 | 1.00000 | −0.592901 | −1.00000 | ||||||||||||||||||
1.8 | 1.00000 | −1.51693 | 1.00000 | −1.00000 | −1.51693 | −1.44279 | 1.00000 | −0.698933 | −1.00000 | ||||||||||||||||||
1.9 | 1.00000 | −1.30771 | 1.00000 | −1.00000 | −1.30771 | 4.54753 | 1.00000 | −1.28989 | −1.00000 | ||||||||||||||||||
1.10 | 1.00000 | −1.30184 | 1.00000 | −1.00000 | −1.30184 | 0.511613 | 1.00000 | −1.30521 | −1.00000 | ||||||||||||||||||
1.11 | 1.00000 | −0.310973 | 1.00000 | −1.00000 | −0.310973 | 1.47909 | 1.00000 | −2.90330 | −1.00000 | ||||||||||||||||||
1.12 | 1.00000 | 0.0740024 | 1.00000 | −1.00000 | 0.0740024 | 2.79514 | 1.00000 | −2.99452 | −1.00000 | ||||||||||||||||||
1.13 | 1.00000 | 0.214164 | 1.00000 | −1.00000 | 0.214164 | −0.767476 | 1.00000 | −2.95413 | −1.00000 | ||||||||||||||||||
1.14 | 1.00000 | 0.530792 | 1.00000 | −1.00000 | 0.530792 | −3.74336 | 1.00000 | −2.71826 | −1.00000 | ||||||||||||||||||
1.15 | 1.00000 | 0.724060 | 1.00000 | −1.00000 | 0.724060 | −1.42945 | 1.00000 | −2.47574 | −1.00000 | ||||||||||||||||||
1.16 | 1.00000 | 1.17759 | 1.00000 | −1.00000 | 1.17759 | 1.86080 | 1.00000 | −1.61328 | −1.00000 | ||||||||||||||||||
1.17 | 1.00000 | 1.34202 | 1.00000 | −1.00000 | 1.34202 | 2.38251 | 1.00000 | −1.19898 | −1.00000 | ||||||||||||||||||
1.18 | 1.00000 | 1.73340 | 1.00000 | −1.00000 | 1.73340 | −0.557678 | 1.00000 | 0.00467827 | −1.00000 | ||||||||||||||||||
1.19 | 1.00000 | 2.04538 | 1.00000 | −1.00000 | 2.04538 | −3.30792 | 1.00000 | 1.18360 | −1.00000 | ||||||||||||||||||
1.20 | 1.00000 | 2.14756 | 1.00000 | −1.00000 | 2.14756 | −1.75294 | 1.00000 | 1.61200 | −1.00000 | ||||||||||||||||||
See all 22 embeddings |
Atkin-Lehner signs
\( p \) | Sign |
---|---|
\(2\) | \(-1\) |
\(5\) | \(1\) |
\(601\) | \(-1\) |
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 | 6010.2.a.f | ✓ | 22 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
6010.2.a.f | ✓ | 22 | 1.a | even | 1 | 1 | trivial |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{22} + 6 T_{3}^{21} - 21 T_{3}^{20} - 179 T_{3}^{19} + 105 T_{3}^{18} + 2208 T_{3}^{17} + \cdots - 140 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(6010))\).