Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [6041,2,Mod(1,6041)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(6041, 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("6041.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 6041 = 7 \cdot 863 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 6041.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: | \(48.2376278611\) |
Analytic rank: | \(1\) |
Dimension: | \(101\) |
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 | −2.71484 | 1.06768 | 5.37033 | 2.50113 | −2.89858 | −1.00000 | −9.14989 | −1.86006 | −6.79016 | ||||||||||||||||||
1.2 | −2.68338 | −0.794633 | 5.20055 | −0.431207 | 2.13231 | −1.00000 | −8.58829 | −2.36856 | 1.15709 | ||||||||||||||||||
1.3 | −2.67103 | −2.02602 | 5.13442 | 3.05924 | 5.41156 | −1.00000 | −8.37215 | 1.10475 | −8.17132 | ||||||||||||||||||
1.4 | −2.59007 | −1.31514 | 4.70847 | −2.80668 | 3.40631 | −1.00000 | −7.01513 | −1.27040 | 7.26950 | ||||||||||||||||||
1.5 | −2.44864 | −2.39074 | 3.99581 | −0.961081 | 5.85406 | −1.00000 | −4.88702 | 2.71565 | 2.35334 | ||||||||||||||||||
1.6 | −2.44717 | 0.719096 | 3.98863 | −3.39767 | −1.75975 | −1.00000 | −4.86651 | −2.48290 | 8.31466 | ||||||||||||||||||
1.7 | −2.41856 | 2.09832 | 3.84945 | −1.02982 | −5.07493 | −1.00000 | −4.47302 | 1.40297 | 2.49069 | ||||||||||||||||||
1.8 | −2.41637 | −2.07802 | 3.83883 | 1.82876 | 5.02127 | −1.00000 | −4.44330 | 1.31818 | −4.41896 | ||||||||||||||||||
1.9 | −2.41440 | 0.0560859 | 3.82935 | 0.0819711 | −0.135414 | −1.00000 | −4.41678 | −2.99685 | −0.197911 | ||||||||||||||||||
1.10 | −2.41346 | 2.83250 | 3.82477 | 1.74211 | −6.83611 | −1.00000 | −4.40400 | 5.02306 | −4.20452 | ||||||||||||||||||
1.11 | −2.37665 | 1.96222 | 3.64847 | −3.56190 | −4.66352 | −1.00000 | −3.91784 | 0.850318 | 8.46540 | ||||||||||||||||||
1.12 | −2.34195 | 3.16467 | 3.48474 | −1.79579 | −7.41150 | −1.00000 | −3.47719 | 7.01511 | 4.20566 | ||||||||||||||||||
1.13 | −2.15616 | −2.99968 | 2.64901 | −1.14985 | 6.46779 | −1.00000 | −1.39937 | 5.99811 | 2.47926 | ||||||||||||||||||
1.14 | −2.14147 | −2.96762 | 2.58591 | 4.08817 | 6.35507 | −1.00000 | −1.25470 | 5.80675 | −8.75470 | ||||||||||||||||||
1.15 | −2.08521 | 1.06538 | 2.34812 | 1.41345 | −2.22154 | −1.00000 | −0.725898 | −1.86497 | −2.94734 | ||||||||||||||||||
1.16 | −2.00569 | 2.11059 | 2.02278 | 3.12491 | −4.23318 | −1.00000 | −0.0456843 | 1.45460 | −6.26760 | ||||||||||||||||||
1.17 | −1.98854 | −0.0463309 | 1.95428 | −0.0496669 | 0.0921307 | −1.00000 | 0.0909182 | −2.99785 | 0.0987645 | ||||||||||||||||||
1.18 | −1.87092 | −1.43026 | 1.50033 | 1.61115 | 2.67590 | −1.00000 | 0.934843 | −0.954356 | −3.01432 | ||||||||||||||||||
1.19 | −1.83237 | 1.94542 | 1.35759 | 1.21630 | −3.56474 | −1.00000 | 1.17714 | 0.784666 | −2.22871 | ||||||||||||||||||
1.20 | −1.79653 | 2.70133 | 1.22753 | 2.96874 | −4.85303 | −1.00000 | 1.38776 | 4.29718 | −5.33345 | ||||||||||||||||||
See next 80 embeddings (of 101 total) |
Atkin-Lehner signs
\( p \) | Sign |
---|---|
\(7\) | \(1\) |
\(863\) | \(1\) |
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 | 6041.2.a.d | ✓ | 101 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
6041.2.a.d | ✓ | 101 | 1.a | even | 1 | 1 | trivial |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{101} - 3 T_{2}^{100} - 139 T_{2}^{99} + 423 T_{2}^{98} + 9322 T_{2}^{97} - 28815 T_{2}^{96} + \cdots + 403759 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(6041))\).