Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [4012,2,Mod(1,4012)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(4012, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([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("4012.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 4012 = 2^{2} \cdot 17 \cdot 59 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 4012.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: | \(32.0359812909\) |
Analytic rank: | \(0\) |
Dimension: | \(21\) |
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 | −2.75827 | 0 | 0.760103 | 0 | 0.427165 | 0 | 4.60804 | 0 | ||||||||||||||||||
1.2 | 0 | −2.52360 | 0 | −3.46979 | 0 | 1.96229 | 0 | 3.36854 | 0 | ||||||||||||||||||
1.3 | 0 | −2.50436 | 0 | 2.93974 | 0 | −0.515336 | 0 | 3.27181 | 0 | ||||||||||||||||||
1.4 | 0 | −1.96950 | 0 | −1.93705 | 0 | −2.26754 | 0 | 0.878948 | 0 | ||||||||||||||||||
1.5 | 0 | −1.89300 | 0 | −2.37849 | 0 | 4.40428 | 0 | 0.583446 | 0 | ||||||||||||||||||
1.6 | 0 | −1.58435 | 0 | 2.56834 | 0 | 3.40444 | 0 | −0.489844 | 0 | ||||||||||||||||||
1.7 | 0 | −1.03698 | 0 | 0.719057 | 0 | −1.71880 | 0 | −1.92468 | 0 | ||||||||||||||||||
1.8 | 0 | 0.0134244 | 0 | 3.34756 | 0 | −3.13615 | 0 | −2.99982 | 0 | ||||||||||||||||||
1.9 | 0 | 0.188519 | 0 | −3.21916 | 0 | −0.179540 | 0 | −2.96446 | 0 | ||||||||||||||||||
1.10 | 0 | 0.210829 | 0 | 0.593138 | 0 | −1.60715 | 0 | −2.95555 | 0 | ||||||||||||||||||
1.11 | 0 | 0.461334 | 0 | 1.69592 | 0 | 4.42760 | 0 | −2.78717 | 0 | ||||||||||||||||||
1.12 | 0 | 0.765447 | 0 | −3.22340 | 0 | −4.55619 | 0 | −2.41409 | 0 | ||||||||||||||||||
1.13 | 0 | 0.967697 | 0 | −3.19923 | 0 | 3.75414 | 0 | −2.06356 | 0 | ||||||||||||||||||
1.14 | 0 | 1.90340 | 0 | 1.03137 | 0 | 0.216287 | 0 | 0.622916 | 0 | ||||||||||||||||||
1.15 | 0 | 1.97354 | 0 | 3.25764 | 0 | 2.36478 | 0 | 0.894844 | 0 | ||||||||||||||||||
1.16 | 0 | 2.22137 | 0 | −1.27900 | 0 | 0.396716 | 0 | 1.93446 | 0 | ||||||||||||||||||
1.17 | 0 | 2.29806 | 0 | 4.01491 | 0 | 4.07651 | 0 | 2.28106 | 0 | ||||||||||||||||||
1.18 | 0 | 2.45518 | 0 | −1.72896 | 0 | −4.96392 | 0 | 3.02793 | 0 | ||||||||||||||||||
1.19 | 0 | 3.11094 | 0 | 3.12050 | 0 | −1.63406 | 0 | 6.67797 | 0 | ||||||||||||||||||
1.20 | 0 | 3.31779 | 0 | −0.252330 | 0 | 4.50552 | 0 | 8.00775 | 0 | ||||||||||||||||||
See all 21 embeddings |
Atkin-Lehner signs
\( p \) | Sign |
---|---|
\(2\) | \(-1\) |
\(17\) | \(-1\) |
\(59\) | \(-1\) |
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 | 4012.2.a.j | ✓ | 21 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
4012.2.a.j | ✓ | 21 | 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(4012))\):
\( T_{3}^{21} - 9 T_{3}^{20} - 4 T_{3}^{19} + 250 T_{3}^{18} - 437 T_{3}^{17} - 2642 T_{3}^{16} + 7806 T_{3}^{15} + 12170 T_{3}^{14} - 58757 T_{3}^{13} - 10873 T_{3}^{12} + 229457 T_{3}^{11} - 116876 T_{3}^{10} - 459309 T_{3}^{9} + \cdots + 32 \) |
\( T_{5}^{21} - T_{5}^{20} - 65 T_{5}^{19} + 54 T_{5}^{18} + 1805 T_{5}^{17} - 1228 T_{5}^{16} - 27934 T_{5}^{15} + 15419 T_{5}^{14} + 263489 T_{5}^{13} - 119039 T_{5}^{12} - 1556390 T_{5}^{11} + 606335 T_{5}^{10} + \cdots - 408928 \) |