Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [6028,2,Mod(1,6028)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(6028, 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("6028.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 6028 = 2^{2} \cdot 11 \cdot 137 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 6028.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: | \(48.1338223384\) |
Analytic rank: | \(0\) |
Dimension: | \(29\) |
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 | −2.97866 | 0 | −3.46603 | 0 | 0.700745 | 0 | 5.87240 | 0 | ||||||||||||||||||
1.2 | 0 | −2.90851 | 0 | 0.776599 | 0 | −2.27589 | 0 | 5.45943 | 0 | ||||||||||||||||||
1.3 | 0 | −2.65731 | 0 | 2.47175 | 0 | 0.711504 | 0 | 4.06129 | 0 | ||||||||||||||||||
1.4 | 0 | −2.38462 | 0 | −1.71860 | 0 | 3.80618 | 0 | 2.68641 | 0 | ||||||||||||||||||
1.5 | 0 | −2.00219 | 0 | 0.977497 | 0 | −3.03319 | 0 | 1.00878 | 0 | ||||||||||||||||||
1.6 | 0 | −1.84910 | 0 | −0.463178 | 0 | −1.96260 | 0 | 0.419175 | 0 | ||||||||||||||||||
1.7 | 0 | −1.36461 | 0 | 2.36459 | 0 | 1.37904 | 0 | −1.13783 | 0 | ||||||||||||||||||
1.8 | 0 | −1.28547 | 0 | 2.42568 | 0 | 5.14778 | 0 | −1.34758 | 0 | ||||||||||||||||||
1.9 | 0 | −1.14725 | 0 | 4.20688 | 0 | −0.775275 | 0 | −1.68381 | 0 | ||||||||||||||||||
1.10 | 0 | −0.886923 | 0 | −0.640762 | 0 | 1.27780 | 0 | −2.21337 | 0 | ||||||||||||||||||
1.11 | 0 | −0.774336 | 0 | −2.66500 | 0 | 0.769081 | 0 | −2.40040 | 0 | ||||||||||||||||||
1.12 | 0 | −0.191141 | 0 | −3.65958 | 0 | 2.84676 | 0 | −2.96346 | 0 | ||||||||||||||||||
1.13 | 0 | 0.213524 | 0 | 2.27425 | 0 | 4.54069 | 0 | −2.95441 | 0 | ||||||||||||||||||
1.14 | 0 | 0.780624 | 0 | 0.801470 | 0 | −4.30496 | 0 | −2.39063 | 0 | ||||||||||||||||||
1.15 | 0 | 0.785650 | 0 | 1.28190 | 0 | −3.55486 | 0 | −2.38275 | 0 | ||||||||||||||||||
1.16 | 0 | 1.04417 | 0 | −2.78682 | 0 | −0.254204 | 0 | −1.90970 | 0 | ||||||||||||||||||
1.17 | 0 | 1.09615 | 0 | 3.71316 | 0 | 3.72361 | 0 | −1.79846 | 0 | ||||||||||||||||||
1.18 | 0 | 1.58135 | 0 | 3.56375 | 0 | 0.0558785 | 0 | −0.499338 | 0 | ||||||||||||||||||
1.19 | 0 | 1.60026 | 0 | −1.38303 | 0 | 0.488214 | 0 | −0.439167 | 0 | ||||||||||||||||||
1.20 | 0 | 1.76391 | 0 | −2.22231 | 0 | 2.22847 | 0 | 0.111381 | 0 | ||||||||||||||||||
See all 29 embeddings |
Atkin-Lehner signs
\( p \) | Sign |
---|---|
\(2\) | \(-1\) |
\(11\) | \(-1\) |
\(137\) | \(-1\) |
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 | 6028.2.a.f | ✓ | 29 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
6028.2.a.f | ✓ | 29 | 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(6028))\):
\( T_{3}^{29} - 14 T_{3}^{28} + 33 T_{3}^{27} + 408 T_{3}^{26} - 2274 T_{3}^{25} - 2691 T_{3}^{24} + \cdots - 382200 \) |
\( T_{5}^{29} - 9 T_{5}^{28} - 50 T_{5}^{27} + 650 T_{5}^{26} + 599 T_{5}^{25} - 20234 T_{5}^{24} + \cdots + 188778816 \) |