Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [6005,2,Mod(1,6005)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(6005, 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("6005.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 6005 = 5 \cdot 1201 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 6005.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: | \(47.9501664138\) |
Analytic rank: | \(1\) |
Dimension: | \(83\) |
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 | −2.64292 | 0.725582 | 4.98500 | −1.00000 | −1.91765 | −1.58856 | −7.88911 | −2.47353 | 2.64292 | ||||||||||||||||||
1.2 | −2.62595 | 0.137720 | 4.89563 | −1.00000 | −0.361646 | 0.826916 | −7.60380 | −2.98103 | 2.62595 | ||||||||||||||||||
1.3 | −2.46128 | −2.46204 | 4.05791 | −1.00000 | 6.05978 | 3.81143 | −5.06509 | 3.06165 | 2.46128 | ||||||||||||||||||
1.4 | −2.43937 | 1.48645 | 3.95052 | −1.00000 | −3.62599 | 1.16444 | −4.75805 | −0.790480 | 2.43937 | ||||||||||||||||||
1.5 | −2.43196 | −2.45366 | 3.91445 | −1.00000 | 5.96721 | −1.89640 | −4.65586 | 3.02044 | 2.43196 | ||||||||||||||||||
1.6 | −2.41534 | 2.54592 | 3.83384 | −1.00000 | −6.14926 | −0.649970 | −4.42935 | 3.48173 | 2.41534 | ||||||||||||||||||
1.7 | −2.36067 | 1.42227 | 3.57275 | −1.00000 | −3.35751 | −1.80334 | −3.71273 | −0.977139 | 2.36067 | ||||||||||||||||||
1.8 | −2.35753 | −1.70273 | 3.55796 | −1.00000 | 4.01424 | 1.43471 | −3.67294 | −0.100706 | 2.35753 | ||||||||||||||||||
1.9 | −2.30347 | 2.77237 | 3.30599 | −1.00000 | −6.38609 | 3.31837 | −3.00832 | 4.68605 | 2.30347 | ||||||||||||||||||
1.10 | −2.27002 | 2.83748 | 3.15300 | −1.00000 | −6.44114 | −2.52643 | −2.61733 | 5.05129 | 2.27002 | ||||||||||||||||||
1.11 | −2.26906 | −1.09200 | 3.14865 | −1.00000 | 2.47782 | −1.23548 | −2.60637 | −1.80753 | 2.26906 | ||||||||||||||||||
1.12 | −2.05565 | −1.34402 | 2.22569 | −1.00000 | 2.76284 | 1.91384 | −0.463948 | −1.19360 | 2.05565 | ||||||||||||||||||
1.13 | −1.98693 | −0.154242 | 1.94789 | −1.00000 | 0.306469 | −4.64687 | 0.103547 | −2.97621 | 1.98693 | ||||||||||||||||||
1.14 | −1.97272 | −2.98994 | 1.89163 | −1.00000 | 5.89832 | 0.991432 | 0.213787 | 5.93974 | 1.97272 | ||||||||||||||||||
1.15 | −1.95246 | 1.27997 | 1.81211 | −1.00000 | −2.49910 | 5.02234 | 0.366847 | −1.36166 | 1.95246 | ||||||||||||||||||
1.16 | −1.93139 | −1.99415 | 1.73027 | −1.00000 | 3.85147 | 0.495869 | 0.520963 | 0.976620 | 1.93139 | ||||||||||||||||||
1.17 | −1.68633 | −0.538907 | 0.843700 | −1.00000 | 0.908773 | −3.51769 | 1.94990 | −2.70958 | 1.68633 | ||||||||||||||||||
1.18 | −1.67860 | −2.56726 | 0.817709 | −1.00000 | 4.30941 | −1.41200 | 1.98460 | 3.59082 | 1.67860 | ||||||||||||||||||
1.19 | −1.67033 | 1.77820 | 0.789988 | −1.00000 | −2.97017 | 2.94591 | 2.02111 | 0.161995 | 1.67033 | ||||||||||||||||||
1.20 | −1.55566 | −0.523024 | 0.420073 | −1.00000 | 0.813646 | 2.34977 | 2.45783 | −2.72645 | 1.55566 | ||||||||||||||||||
See all 83 embeddings |
Atkin-Lehner signs
\( p \) | Sign |
---|---|
\(5\) | \(1\) |
\(1201\) | \(1\) |
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 | 6005.2.a.d | ✓ | 83 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
6005.2.a.d | ✓ | 83 | 1.a | even | 1 | 1 | trivial |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{83} - T_{2}^{82} - 113 T_{2}^{81} + 113 T_{2}^{80} + 6126 T_{2}^{79} - 6128 T_{2}^{78} + \cdots - 316 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(6005))\).