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: | \(1\) |
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 | −3.35813 | 0 | −1.00000 | 0 | −3.21673 | 0 | 8.27706 | 0 | ||||||||||||||||||
1.2 | 0 | −3.21791 | 0 | −1.00000 | 0 | 3.89140 | 0 | 7.35496 | 0 | ||||||||||||||||||
1.3 | 0 | −3.04643 | 0 | −1.00000 | 0 | −4.80583 | 0 | 6.28073 | 0 | ||||||||||||||||||
1.4 | 0 | −2.90197 | 0 | −1.00000 | 0 | 0.345703 | 0 | 5.42142 | 0 | ||||||||||||||||||
1.5 | 0 | −2.61857 | 0 | −1.00000 | 0 | 4.97833 | 0 | 3.85689 | 0 | ||||||||||||||||||
1.6 | 0 | −2.12620 | 0 | −1.00000 | 0 | −4.25186 | 0 | 1.52074 | 0 | ||||||||||||||||||
1.7 | 0 | −1.37961 | 0 | −1.00000 | 0 | −1.90795 | 0 | −1.09667 | 0 | ||||||||||||||||||
1.8 | 0 | −1.37145 | 0 | −1.00000 | 0 | 3.46098 | 0 | −1.11913 | 0 | ||||||||||||||||||
1.9 | 0 | −1.32609 | 0 | −1.00000 | 0 | 2.45028 | 0 | −1.24148 | 0 | ||||||||||||||||||
1.10 | 0 | −0.877804 | 0 | −1.00000 | 0 | −0.669807 | 0 | −2.22946 | 0 | ||||||||||||||||||
1.11 | 0 | −0.739679 | 0 | −1.00000 | 0 | 1.72019 | 0 | −2.45288 | 0 | ||||||||||||||||||
1.12 | 0 | −0.717713 | 0 | −1.00000 | 0 | −4.74749 | 0 | −2.48489 | 0 | ||||||||||||||||||
1.13 | 0 | −0.618582 | 0 | −1.00000 | 0 | −3.29286 | 0 | −2.61736 | 0 | ||||||||||||||||||
1.14 | 0 | 0.512071 | 0 | −1.00000 | 0 | −0.0450824 | 0 | −2.73778 | 0 | ||||||||||||||||||
1.15 | 0 | 1.23778 | 0 | −1.00000 | 0 | 2.85194 | 0 | −1.46790 | 0 | ||||||||||||||||||
1.16 | 0 | 1.34680 | 0 | −1.00000 | 0 | −1.73300 | 0 | −1.18612 | 0 | ||||||||||||||||||
1.17 | 0 | 1.71820 | 0 | −1.00000 | 0 | 3.71115 | 0 | −0.0477919 | 0 | ||||||||||||||||||
1.18 | 0 | 2.01176 | 0 | −1.00000 | 0 | −2.39634 | 0 | 1.04718 | 0 | ||||||||||||||||||
1.19 | 0 | 2.34338 | 0 | −1.00000 | 0 | 0.957310 | 0 | 2.49143 | 0 | ||||||||||||||||||
1.20 | 0 | 2.46975 | 0 | −1.00000 | 0 | −2.30951 | 0 | 3.09967 | 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.q | ✓ | 23 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
6040.2.a.q | ✓ | 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(6040))\):
\( T_{3}^{23} + 4 T_{3}^{22} - 45 T_{3}^{21} - 188 T_{3}^{20} + 846 T_{3}^{19} + 3771 T_{3}^{18} - 8547 T_{3}^{17} - 42303 T_{3}^{16} + 48752 T_{3}^{15} + 291821 T_{3}^{14} - 139453 T_{3}^{13} - 1280053 T_{3}^{12} + \cdots + 155000 \) |
\( T_{7}^{23} + 9 T_{7}^{22} - 74 T_{7}^{21} - 857 T_{7}^{20} + 1757 T_{7}^{19} + 34264 T_{7}^{18} + 336 T_{7}^{17} - 750187 T_{7}^{16} - 801360 T_{7}^{15} + 9793378 T_{7}^{14} + 17263757 T_{7}^{13} - 77324914 T_{7}^{12} + \cdots + 27713536 \) |
\( T_{11}^{23} + 17 T_{11}^{22} - 20 T_{11}^{21} - 1838 T_{11}^{20} - 6534 T_{11}^{19} + 70413 T_{11}^{18} + 451128 T_{11}^{17} - 958945 T_{11}^{16} - 12358265 T_{11}^{15} - 5968462 T_{11}^{14} + 164667377 T_{11}^{13} + \cdots + 2081775616 \) |