Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [6036,2,Mod(1,6036)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(6036, 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("6036.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 6036 = 2^{2} \cdot 3 \cdot 503 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 6036.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: | \(48.1977026600\) |
Analytic rank: | \(0\) |
Dimension: | \(26\) |
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 | −1.00000 | 0 | −4.35282 | 0 | 4.52941 | 0 | 1.00000 | 0 | ||||||||||||||||||
1.2 | 0 | −1.00000 | 0 | −3.83715 | 0 | 0.0390685 | 0 | 1.00000 | 0 | ||||||||||||||||||
1.3 | 0 | −1.00000 | 0 | −3.00702 | 0 | 4.42560 | 0 | 1.00000 | 0 | ||||||||||||||||||
1.4 | 0 | −1.00000 | 0 | −2.74974 | 0 | 1.46678 | 0 | 1.00000 | 0 | ||||||||||||||||||
1.5 | 0 | −1.00000 | 0 | −2.60444 | 0 | −1.53102 | 0 | 1.00000 | 0 | ||||||||||||||||||
1.6 | 0 | −1.00000 | 0 | −2.57233 | 0 | −0.722776 | 0 | 1.00000 | 0 | ||||||||||||||||||
1.7 | 0 | −1.00000 | 0 | −2.38993 | 0 | 0.588107 | 0 | 1.00000 | 0 | ||||||||||||||||||
1.8 | 0 | −1.00000 | 0 | −1.79097 | 0 | −4.44185 | 0 | 1.00000 | 0 | ||||||||||||||||||
1.9 | 0 | −1.00000 | 0 | −1.78585 | 0 | −2.07926 | 0 | 1.00000 | 0 | ||||||||||||||||||
1.10 | 0 | −1.00000 | 0 | −0.970233 | 0 | 3.74274 | 0 | 1.00000 | 0 | ||||||||||||||||||
1.11 | 0 | −1.00000 | 0 | −0.461317 | 0 | −3.07386 | 0 | 1.00000 | 0 | ||||||||||||||||||
1.12 | 0 | −1.00000 | 0 | −0.244833 | 0 | 2.69533 | 0 | 1.00000 | 0 | ||||||||||||||||||
1.13 | 0 | −1.00000 | 0 | 0.0581310 | 0 | −2.78315 | 0 | 1.00000 | 0 | ||||||||||||||||||
1.14 | 0 | −1.00000 | 0 | 0.344611 | 0 | 2.72948 | 0 | 1.00000 | 0 | ||||||||||||||||||
1.15 | 0 | −1.00000 | 0 | 0.754498 | 0 | −4.03319 | 0 | 1.00000 | 0 | ||||||||||||||||||
1.16 | 0 | −1.00000 | 0 | 1.62125 | 0 | 5.08723 | 0 | 1.00000 | 0 | ||||||||||||||||||
1.17 | 0 | −1.00000 | 0 | 1.66086 | 0 | −2.19144 | 0 | 1.00000 | 0 | ||||||||||||||||||
1.18 | 0 | −1.00000 | 0 | 1.90366 | 0 | −3.61918 | 0 | 1.00000 | 0 | ||||||||||||||||||
1.19 | 0 | −1.00000 | 0 | 2.09204 | 0 | −1.11995 | 0 | 1.00000 | 0 | ||||||||||||||||||
1.20 | 0 | −1.00000 | 0 | 2.53458 | 0 | 3.60576 | 0 | 1.00000 | 0 | ||||||||||||||||||
See all 26 embeddings |
Atkin-Lehner signs
\( p \) | Sign |
---|---|
\(2\) | \(-1\) |
\(3\) | \(1\) |
\(503\) | \(1\) |
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 | 6036.2.a.i | ✓ | 26 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
6036.2.a.i | ✓ | 26 | 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(6036))\):
\( T_{5}^{26} - 6 T_{5}^{25} - 71 T_{5}^{24} + 466 T_{5}^{23} + 2080 T_{5}^{22} - 15504 T_{5}^{21} - 32313 T_{5}^{20} + 291304 T_{5}^{19} + 277197 T_{5}^{18} - 3426026 T_{5}^{17} - 1085536 T_{5}^{16} + 26362034 T_{5}^{15} + \cdots + 696354 \) |
\( T_{7}^{26} - 5 T_{7}^{25} - 112 T_{7}^{24} + 544 T_{7}^{23} + 5501 T_{7}^{22} - 25487 T_{7}^{21} - 157024 T_{7}^{20} + 675426 T_{7}^{19} + 2910247 T_{7}^{18} - 11179876 T_{7}^{17} - 36993674 T_{7}^{16} + \cdots - 124612608 \) |