Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [5103,2,Mod(1,5103)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(5103, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([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("5103.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 5103 = 3^{6} \cdot 7 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 5103.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: | \(40.7476601515\) |
Analytic rank: | \(1\) |
Dimension: | \(27\) |
Twist minimal: | no (minimal twist has level 189) |
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 | −2.78307 | 0 | 5.74550 | 2.42866 | 0 | 1.00000 | −10.4240 | 0 | −6.75913 | ||||||||||||||||||
1.2 | −2.68770 | 0 | 5.22375 | −1.17018 | 0 | 1.00000 | −8.66449 | 0 | 3.14510 | ||||||||||||||||||
1.3 | −2.61362 | 0 | 4.83099 | −3.50881 | 0 | 1.00000 | −7.39913 | 0 | 9.17068 | ||||||||||||||||||
1.4 | −2.46956 | 0 | 4.09871 | 2.30541 | 0 | 1.00000 | −5.18288 | 0 | −5.69333 | ||||||||||||||||||
1.5 | −2.31907 | 0 | 3.37807 | −2.55206 | 0 | 1.00000 | −3.19584 | 0 | 5.91841 | ||||||||||||||||||
1.6 | −2.29809 | 0 | 3.28121 | −3.71728 | 0 | 1.00000 | −2.94435 | 0 | 8.54265 | ||||||||||||||||||
1.7 | −1.91771 | 0 | 1.67760 | 2.81660 | 0 | 1.00000 | 0.618268 | 0 | −5.40142 | ||||||||||||||||||
1.8 | −1.86596 | 0 | 1.48182 | −1.49355 | 0 | 1.00000 | 0.966907 | 0 | 2.78691 | ||||||||||||||||||
1.9 | −1.27285 | 0 | −0.379841 | 0.744587 | 0 | 1.00000 | 3.02919 | 0 | −0.947751 | ||||||||||||||||||
1.10 | −1.26564 | 0 | −0.398155 | −0.909945 | 0 | 1.00000 | 3.03520 | 0 | 1.15166 | ||||||||||||||||||
1.11 | −1.15245 | 0 | −0.671868 | 0.763383 | 0 | 1.00000 | 3.07918 | 0 | −0.879758 | ||||||||||||||||||
1.12 | −1.02676 | 0 | −0.945770 | 3.79727 | 0 | 1.00000 | 3.02459 | 0 | −3.89888 | ||||||||||||||||||
1.13 | −0.954909 | 0 | −1.08815 | −4.02967 | 0 | 1.00000 | 2.94890 | 0 | 3.84797 | ||||||||||||||||||
1.14 | −0.268830 | 0 | −1.92773 | −1.98449 | 0 | 1.00000 | 1.05589 | 0 | 0.533491 | ||||||||||||||||||
1.15 | −0.105569 | 0 | −1.98886 | 0.0478140 | 0 | 1.00000 | 0.421099 | 0 | −0.00504766 | ||||||||||||||||||
1.16 | 0.0926744 | 0 | −1.99141 | −2.41602 | 0 | 1.00000 | −0.369901 | 0 | −0.223903 | ||||||||||||||||||
1.17 | 0.195845 | 0 | −1.96164 | 2.12393 | 0 | 1.00000 | −0.775867 | 0 | 0.415961 | ||||||||||||||||||
1.18 | 0.463515 | 0 | −1.78515 | 0.316473 | 0 | 1.00000 | −1.75448 | 0 | 0.146690 | ||||||||||||||||||
1.19 | 0.803391 | 0 | −1.35456 | −4.10793 | 0 | 1.00000 | −2.69503 | 0 | −3.30028 | ||||||||||||||||||
1.20 | 0.961458 | 0 | −1.07560 | 3.76231 | 0 | 1.00000 | −2.95706 | 0 | 3.61730 | ||||||||||||||||||
See all 27 embeddings |
Atkin-Lehner signs
\( p \) | Sign |
---|---|
\(3\) | \(-1\) |
\(7\) | \(-1\) |
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 | 5103.2.a.f | 27 | |
3.b | odd | 2 | 1 | 5103.2.a.i | 27 | ||
27.e | even | 9 | 2 | 567.2.v.b | 54 | ||
27.f | odd | 18 | 2 | 189.2.v.a | ✓ | 54 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
189.2.v.a | ✓ | 54 | 27.f | odd | 18 | 2 | |
567.2.v.b | 54 | 27.e | even | 9 | 2 | ||
5103.2.a.f | 27 | 1.a | even | 1 | 1 | trivial | |
5103.2.a.i | 27 | 3.b | odd | 2 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{27} + 9 T_{2}^{26} - 222 T_{2}^{24} - 459 T_{2}^{23} + 2133 T_{2}^{22} + 7362 T_{2}^{21} + \cdots + 27 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(5103))\).