Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [6040,2,Mod(1,6040)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(6040, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 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("6040.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 6040 = 2^{3} \cdot 5 \cdot 151 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 6040.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: | \(48.2296428209\) |
Analytic rank: | \(0\) |
Dimension: | \(23\) |
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 | 0 | −3.35416 | 0 | −1.00000 | 0 | 2.39898 | 0 | 8.25038 | 0 | ||||||||||||||||||
1.2 | 0 | −3.04170 | 0 | −1.00000 | 0 | −1.27793 | 0 | 6.25193 | 0 | ||||||||||||||||||
1.3 | 0 | −2.47861 | 0 | −1.00000 | 0 | −4.84176 | 0 | 3.14349 | 0 | ||||||||||||||||||
1.4 | 0 | −2.42679 | 0 | −1.00000 | 0 | −1.20395 | 0 | 2.88931 | 0 | ||||||||||||||||||
1.5 | 0 | −2.28476 | 0 | −1.00000 | 0 | 4.31809 | 0 | 2.22015 | 0 | ||||||||||||||||||
1.6 | 0 | −1.86185 | 0 | −1.00000 | 0 | −1.14229 | 0 | 0.466481 | 0 | ||||||||||||||||||
1.7 | 0 | −1.63718 | 0 | −1.00000 | 0 | 2.53297 | 0 | −0.319632 | 0 | ||||||||||||||||||
1.8 | 0 | −1.02367 | 0 | −1.00000 | 0 | 3.11653 | 0 | −1.95210 | 0 | ||||||||||||||||||
1.9 | 0 | −1.00618 | 0 | −1.00000 | 0 | 0.737056 | 0 | −1.98759 | 0 | ||||||||||||||||||
1.10 | 0 | −0.579257 | 0 | −1.00000 | 0 | −2.62931 | 0 | −2.66446 | 0 | ||||||||||||||||||
1.11 | 0 | −0.0712098 | 0 | −1.00000 | 0 | 0.143994 | 0 | −2.99493 | 0 | ||||||||||||||||||
1.12 | 0 | −0.0188980 | 0 | −1.00000 | 0 | 4.41010 | 0 | −2.99964 | 0 | ||||||||||||||||||
1.13 | 0 | 0.0845518 | 0 | −1.00000 | 0 | −0.761778 | 0 | −2.99285 | 0 | ||||||||||||||||||
1.14 | 0 | 1.08343 | 0 | −1.00000 | 0 | −5.06106 | 0 | −1.82619 | 0 | ||||||||||||||||||
1.15 | 0 | 1.30289 | 0 | −1.00000 | 0 | −2.66956 | 0 | −1.30249 | 0 | ||||||||||||||||||
1.16 | 0 | 1.31203 | 0 | −1.00000 | 0 | −4.08175 | 0 | −1.27858 | 0 | ||||||||||||||||||
1.17 | 0 | 1.69908 | 0 | −1.00000 | 0 | 2.43676 | 0 | −0.113117 | 0 | ||||||||||||||||||
1.18 | 0 | 1.83763 | 0 | −1.00000 | 0 | 1.58536 | 0 | 0.376872 | 0 | ||||||||||||||||||
1.19 | 0 | 2.26595 | 0 | −1.00000 | 0 | 0.518518 | 0 | 2.13452 | 0 | ||||||||||||||||||
1.20 | 0 | 2.58192 | 0 | −1.00000 | 0 | 4.44988 | 0 | 3.66631 | 0 | ||||||||||||||||||
See all 23 embeddings |
Atkin-Lehner signs
\( p \) | Sign |
---|---|
\(2\) | \(-1\) |
\(5\) | \(1\) |
\(151\) | \(1\) |
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 | 6040.2.a.r | ✓ | 23 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
6040.2.a.r | ✓ | 23 | 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(6040))\):
\( T_{3}^{23} - 2 T_{3}^{22} - 48 T_{3}^{21} + 90 T_{3}^{20} + 978 T_{3}^{19} - 1705 T_{3}^{18} - 11083 T_{3}^{17} + \cdots - 32 \) |
\( T_{7}^{23} - 3 T_{7}^{22} - 96 T_{7}^{21} + 285 T_{7}^{20} + 3815 T_{7}^{19} - 10958 T_{7}^{18} + \cdots + 5934848 \) |
\( T_{11}^{23} - 5 T_{11}^{22} - 142 T_{11}^{21} + 722 T_{11}^{20} + 8376 T_{11}^{19} - 43585 T_{11}^{18} + \cdots + 6585892864 \) |