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: | \(0\) |
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 | −2.46283 | 0 | 4.06555 | 1.36934 | 0 | 1.00000 | −5.08709 | 0 | −3.37246 | ||||||||||||||||||
1.2 | −2.30531 | 0 | 3.31443 | 1.59289 | 0 | 1.00000 | −3.03017 | 0 | −3.67209 | ||||||||||||||||||
1.3 | −2.08373 | 0 | 2.34193 | 3.69527 | 0 | 1.00000 | −0.712484 | 0 | −7.69993 | ||||||||||||||||||
1.4 | −1.88628 | 0 | 1.55803 | −0.142020 | 0 | 1.00000 | 0.833669 | 0 | 0.267889 | ||||||||||||||||||
1.5 | −1.77354 | 0 | 1.14546 | −1.97901 | 0 | 1.00000 | 1.51557 | 0 | 3.50985 | ||||||||||||||||||
1.6 | −1.53990 | 0 | 0.371288 | 2.11956 | 0 | 1.00000 | 2.50805 | 0 | −3.26391 | ||||||||||||||||||
1.7 | −1.43332 | 0 | 0.0543949 | −1.43954 | 0 | 1.00000 | 2.78867 | 0 | 2.06331 | ||||||||||||||||||
1.8 | −0.961458 | 0 | −1.07560 | −3.76231 | 0 | 1.00000 | 2.95706 | 0 | 3.61730 | ||||||||||||||||||
1.9 | −0.803391 | 0 | −1.35456 | 4.10793 | 0 | 1.00000 | 2.69503 | 0 | −3.30028 | ||||||||||||||||||
1.10 | −0.463515 | 0 | −1.78515 | −0.316473 | 0 | 1.00000 | 1.75448 | 0 | 0.146690 | ||||||||||||||||||
1.11 | −0.195845 | 0 | −1.96164 | −2.12393 | 0 | 1.00000 | 0.775867 | 0 | 0.415961 | ||||||||||||||||||
1.12 | −0.0926744 | 0 | −1.99141 | 2.41602 | 0 | 1.00000 | 0.369901 | 0 | −0.223903 | ||||||||||||||||||
1.13 | 0.105569 | 0 | −1.98886 | −0.0478140 | 0 | 1.00000 | −0.421099 | 0 | −0.00504766 | ||||||||||||||||||
1.14 | 0.268830 | 0 | −1.92773 | 1.98449 | 0 | 1.00000 | −1.05589 | 0 | 0.533491 | ||||||||||||||||||
1.15 | 0.954909 | 0 | −1.08815 | 4.02967 | 0 | 1.00000 | −2.94890 | 0 | 3.84797 | ||||||||||||||||||
1.16 | 1.02676 | 0 | −0.945770 | −3.79727 | 0 | 1.00000 | −3.02459 | 0 | −3.89888 | ||||||||||||||||||
1.17 | 1.15245 | 0 | −0.671868 | −0.763383 | 0 | 1.00000 | −3.07918 | 0 | −0.879758 | ||||||||||||||||||
1.18 | 1.26564 | 0 | −0.398155 | 0.909945 | 0 | 1.00000 | −3.03520 | 0 | 1.15166 | ||||||||||||||||||
1.19 | 1.27285 | 0 | −0.379841 | −0.744587 | 0 | 1.00000 | −3.02919 | 0 | −0.947751 | ||||||||||||||||||
1.20 | 1.86596 | 0 | 1.48182 | 1.49355 | 0 | 1.00000 | −0.966907 | 0 | 2.78691 | ||||||||||||||||||
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.i | 27 | |
3.b | odd | 2 | 1 | 5103.2.a.f | 27 | ||
27.e | even | 9 | 2 | 189.2.v.a | ✓ | 54 | |
27.f | odd | 18 | 2 | 567.2.v.b | 54 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
189.2.v.a | ✓ | 54 | 27.e | even | 9 | 2 | |
567.2.v.b | 54 | 27.f | odd | 18 | 2 | ||
5103.2.a.f | 27 | 3.b | odd | 2 | 1 | ||
5103.2.a.i | 27 | 1.a | even | 1 | 1 | trivial |
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 \(S_{2}^{\mathrm{new}}(\Gamma_0(5103))\).