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 | −2.77236 | 0 | 5.68599 | −0.395080 | 0 | −1.00000 | −10.2189 | 0 | 1.09530 | ||||||||||||||||||
1.2 | −2.68576 | 0 | 5.21330 | −3.42783 | 0 | −1.00000 | −8.63015 | 0 | 9.20631 | ||||||||||||||||||
1.3 | −2.48866 | 0 | 4.19343 | −0.208083 | 0 | −1.00000 | −5.45871 | 0 | 0.517848 | ||||||||||||||||||
1.4 | −2.20469 | 0 | 2.86065 | −3.24369 | 0 | −1.00000 | −1.89747 | 0 | 7.15133 | ||||||||||||||||||
1.5 | −2.18188 | 0 | 2.76060 | 3.00756 | 0 | −1.00000 | −1.65953 | 0 | −6.56213 | ||||||||||||||||||
1.6 | −1.82086 | 0 | 1.31554 | 3.49209 | 0 | −1.00000 | 1.24631 | 0 | −6.35861 | ||||||||||||||||||
1.7 | −1.79040 | 0 | 1.20554 | 2.01328 | 0 | −1.00000 | 1.42241 | 0 | −3.60457 | ||||||||||||||||||
1.8 | −1.54487 | 0 | 0.386629 | −3.18169 | 0 | −1.00000 | 2.49245 | 0 | 4.91531 | ||||||||||||||||||
1.9 | −1.50314 | 0 | 0.259440 | −2.84172 | 0 | −1.00000 | 2.61631 | 0 | 4.27151 | ||||||||||||||||||
1.10 | −1.19996 | 0 | −0.560087 | −0.751497 | 0 | −1.00000 | 3.07201 | 0 | 0.901769 | ||||||||||||||||||
1.11 | −0.823658 | 0 | −1.32159 | 1.60534 | 0 | −1.00000 | 2.73585 | 0 | −1.32225 | ||||||||||||||||||
1.12 | −0.602205 | 0 | −1.63735 | −2.85254 | 0 | −1.00000 | 2.19043 | 0 | 1.71781 | ||||||||||||||||||
1.13 | −0.322238 | 0 | −1.89616 | 1.71564 | 0 | −1.00000 | 1.25549 | 0 | −0.552843 | ||||||||||||||||||
1.14 | −0.172037 | 0 | −1.97040 | 2.10998 | 0 | −1.00000 | 0.683057 | 0 | −0.362995 | ||||||||||||||||||
1.15 | 0.0565901 | 0 | −1.99680 | −4.09472 | 0 | −1.00000 | −0.226179 | 0 | −0.231721 | ||||||||||||||||||
1.16 | 0.530796 | 0 | −1.71826 | 1.05743 | 0 | −1.00000 | −1.97363 | 0 | 0.561280 | ||||||||||||||||||
1.17 | 0.705157 | 0 | −1.50275 | 1.95134 | 0 | −1.00000 | −2.46999 | 0 | 1.37600 | ||||||||||||||||||
1.18 | 0.723918 | 0 | −1.47594 | −2.56098 | 0 | −1.00000 | −2.51630 | 0 | −1.85394 | ||||||||||||||||||
1.19 | 1.01560 | 0 | −0.968555 | 0.00534434 | 0 | −1.00000 | −3.01487 | 0 | 0.00542772 | ||||||||||||||||||
1.20 | 1.12052 | 0 | −0.744424 | −3.99580 | 0 | −1.00000 | −3.07520 | 0 | −4.47740 | ||||||||||||||||||
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.g | 27 | |
3.b | odd | 2 | 1 | 5103.2.a.h | 27 | ||
27.e | even | 9 | 2 | 189.2.v.b | ✓ | 54 | |
27.f | odd | 18 | 2 | 567.2.v.a | 54 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
189.2.v.b | ✓ | 54 | 27.e | even | 9 | 2 | |
567.2.v.a | 54 | 27.f | odd | 18 | 2 | ||
5103.2.a.g | 27 | 1.a | even | 1 | 1 | trivial | |
5103.2.a.h | 27 | 3.b | odd | 2 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{27} + 3 T_{2}^{26} - 36 T_{2}^{25} - 110 T_{2}^{24} + 561 T_{2}^{23} + 1755 T_{2}^{22} + \cdots - 71 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(5103))\).