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: | \(24\) |
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 | −3.28894 | 0 | 1.00000 | 0 | 4.04677 | 0 | 7.81711 | 0 | ||||||||||||||||||
1.2 | 0 | −3.20286 | 0 | 1.00000 | 0 | −3.67370 | 0 | 7.25831 | 0 | ||||||||||||||||||
1.3 | 0 | −3.03909 | 0 | 1.00000 | 0 | −1.69120 | 0 | 6.23604 | 0 | ||||||||||||||||||
1.4 | 0 | −2.50341 | 0 | 1.00000 | 0 | −2.70762 | 0 | 3.26705 | 0 | ||||||||||||||||||
1.5 | 0 | −2.45370 | 0 | 1.00000 | 0 | −1.79083 | 0 | 3.02066 | 0 | ||||||||||||||||||
1.6 | 0 | −2.29572 | 0 | 1.00000 | 0 | 4.99611 | 0 | 2.27034 | 0 | ||||||||||||||||||
1.7 | 0 | −2.03886 | 0 | 1.00000 | 0 | 1.67649 | 0 | 1.15693 | 0 | ||||||||||||||||||
1.8 | 0 | −1.08470 | 0 | 1.00000 | 0 | 4.94762 | 0 | −1.82343 | 0 | ||||||||||||||||||
1.9 | 0 | −0.875996 | 0 | 1.00000 | 0 | −1.42220 | 0 | −2.23263 | 0 | ||||||||||||||||||
1.10 | 0 | −0.559764 | 0 | 1.00000 | 0 | −0.837547 | 0 | −2.68666 | 0 | ||||||||||||||||||
1.11 | 0 | 0.0496662 | 0 | 1.00000 | 0 | 1.60144 | 0 | −2.99753 | 0 | ||||||||||||||||||
1.12 | 0 | 0.161432 | 0 | 1.00000 | 0 | −4.19061 | 0 | −2.97394 | 0 | ||||||||||||||||||
1.13 | 0 | 0.225898 | 0 | 1.00000 | 0 | 2.54851 | 0 | −2.94897 | 0 | ||||||||||||||||||
1.14 | 0 | 0.549309 | 0 | 1.00000 | 0 | −0.285162 | 0 | −2.69826 | 0 | ||||||||||||||||||
1.15 | 0 | 0.568728 | 0 | 1.00000 | 0 | −5.03281 | 0 | −2.67655 | 0 | ||||||||||||||||||
1.16 | 0 | 1.21987 | 0 | 1.00000 | 0 | 0.325030 | 0 | −1.51192 | 0 | ||||||||||||||||||
1.17 | 0 | 1.49843 | 0 | 1.00000 | 0 | 1.73584 | 0 | −0.754697 | 0 | ||||||||||||||||||
1.18 | 0 | 1.94109 | 0 | 1.00000 | 0 | 3.18961 | 0 | 0.767827 | 0 | ||||||||||||||||||
1.19 | 0 | 2.40154 | 0 | 1.00000 | 0 | 4.37326 | 0 | 2.76742 | 0 | ||||||||||||||||||
1.20 | 0 | 2.57003 | 0 | 1.00000 | 0 | −3.15345 | 0 | 3.60506 | 0 | ||||||||||||||||||
See all 24 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.s | ✓ | 24 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
6040.2.a.s | ✓ | 24 | 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(6040))\):
\( T_{3}^{24} - 2 T_{3}^{23} - 54 T_{3}^{22} + 108 T_{3}^{21} + 1242 T_{3}^{20} - 2487 T_{3}^{19} - 15853 T_{3}^{18} + 31874 T_{3}^{17} + 122485 T_{3}^{16} - 248751 T_{3}^{15} - 584720 T_{3}^{14} + 1214112 T_{3}^{13} + \cdots + 512 \) |
\( T_{7}^{24} - 3 T_{7}^{23} - 110 T_{7}^{22} + 329 T_{7}^{21} + 5053 T_{7}^{20} - 15020 T_{7}^{19} - 126624 T_{7}^{18} + 372827 T_{7}^{17} + 1902930 T_{7}^{16} - 5527710 T_{7}^{15} - 17830623 T_{7}^{14} + \cdots + 58789888 \) |
\( T_{11}^{24} - 17 T_{11}^{23} - 45 T_{11}^{22} + 2247 T_{11}^{21} - 6181 T_{11}^{20} - 109944 T_{11}^{19} + 604245 T_{11}^{18} + 2170225 T_{11}^{17} - 21327696 T_{11}^{16} + 533851 T_{11}^{15} + 356758849 T_{11}^{14} + \cdots + 628228096 \) |