Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [8619,2,Mod(1,8619)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(8619, 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("8619.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 8619 = 3 \cdot 13^{2} \cdot 17 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 8619.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: | \(68.8230615021\) |
Analytic rank: | \(0\) |
Dimension: | \(24\) |
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 | −2.45596 | 1.00000 | 4.03174 | 1.63451 | −2.45596 | 2.80061 | −4.98987 | 1.00000 | −4.01430 | ||||||||||||||||||
1.2 | −2.23909 | 1.00000 | 3.01354 | 1.14777 | −2.23909 | 0.922290 | −2.26942 | 1.00000 | −2.56995 | ||||||||||||||||||
1.3 | −2.21206 | 1.00000 | 2.89319 | 3.18902 | −2.21206 | −4.76462 | −1.97579 | 1.00000 | −7.05429 | ||||||||||||||||||
1.4 | −2.14195 | 1.00000 | 2.58796 | −1.23309 | −2.14195 | 3.10701 | −1.25939 | 1.00000 | 2.64121 | ||||||||||||||||||
1.5 | −1.22201 | 1.00000 | −0.506690 | 3.06594 | −1.22201 | 1.45560 | 3.06320 | 1.00000 | −3.74661 | ||||||||||||||||||
1.6 | −1.13447 | 1.00000 | −0.712982 | −0.951442 | −1.13447 | 0.520635 | 3.07779 | 1.00000 | 1.07938 | ||||||||||||||||||
1.7 | −0.796821 | 1.00000 | −1.36508 | 3.75576 | −0.796821 | 2.20249 | 2.68136 | 1.00000 | −2.99267 | ||||||||||||||||||
1.8 | −0.783477 | 1.00000 | −1.38616 | 2.97805 | −0.783477 | −3.18728 | 2.65298 | 1.00000 | −2.33323 | ||||||||||||||||||
1.9 | −0.466852 | 1.00000 | −1.78205 | −2.19007 | −0.466852 | −1.56308 | 1.76566 | 1.00000 | 1.02244 | ||||||||||||||||||
1.10 | 0.198248 | 1.00000 | −1.96070 | 1.60317 | 0.198248 | 4.39668 | −0.785201 | 1.00000 | 0.317825 | ||||||||||||||||||
1.11 | 0.266702 | 1.00000 | −1.92887 | −3.43095 | 0.266702 | 1.86360 | −1.04784 | 1.00000 | −0.915042 | ||||||||||||||||||
1.12 | 0.308133 | 1.00000 | −1.90505 | 0.0777101 | 0.308133 | −0.667754 | −1.20328 | 1.00000 | 0.0239450 | ||||||||||||||||||
1.13 | 0.697261 | 1.00000 | −1.51383 | 2.38011 | 0.697261 | 0.404514 | −2.45006 | 1.00000 | 1.65956 | ||||||||||||||||||
1.14 | 1.26831 | 1.00000 | −0.391391 | −1.73153 | 1.26831 | −0.989351 | −3.03302 | 1.00000 | −2.19612 | ||||||||||||||||||
1.15 | 1.28568 | 1.00000 | −0.347029 | 4.43274 | 1.28568 | −0.824336 | −3.01753 | 1.00000 | 5.69909 | ||||||||||||||||||
1.16 | 1.35602 | 1.00000 | −0.161218 | 0.183673 | 1.35602 | −3.62200 | −2.93065 | 1.00000 | 0.249063 | ||||||||||||||||||
1.17 | 1.72731 | 1.00000 | 0.983594 | −2.35153 | 1.72731 | 2.41996 | −1.75565 | 1.00000 | −4.06181 | ||||||||||||||||||
1.18 | 1.90134 | 1.00000 | 1.61508 | 3.96068 | 1.90134 | 1.76877 | −0.731855 | 1.00000 | 7.53059 | ||||||||||||||||||
1.19 | 2.43983 | 1.00000 | 3.95278 | −1.74573 | 2.43983 | 3.42450 | 4.76447 | 1.00000 | −4.25930 | ||||||||||||||||||
1.20 | 2.51571 | 1.00000 | 4.32879 | 2.36751 | 2.51571 | 4.50590 | 5.85857 | 1.00000 | 5.95597 | ||||||||||||||||||
See all 24 embeddings |
Atkin-Lehner signs
\( p \) | Sign |
---|---|
\(3\) | \(-1\) |
\(13\) | \(-1\) |
\(17\) | \(-1\) |
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 | 8619.2.a.bx | yes | 24 |
13.b | even | 2 | 1 | 8619.2.a.bs | ✓ | 24 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
8619.2.a.bs | ✓ | 24 | 13.b | even | 2 | 1 | |
8619.2.a.bx | yes | 24 | 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(8619))\):
\( T_{2}^{24} - 11 T_{2}^{23} + 22 T_{2}^{22} + 178 T_{2}^{21} - 820 T_{2}^{20} - 449 T_{2}^{19} + \cdots - 251 \) |
\( T_{5}^{24} - 23 T_{5}^{23} + 189 T_{5}^{22} - 394 T_{5}^{21} - 3501 T_{5}^{20} + 22876 T_{5}^{19} + \cdots + 770861 \) |
\( T_{7}^{24} - 10 T_{7}^{23} - 35 T_{7}^{22} + 666 T_{7}^{21} - 617 T_{7}^{20} - 16562 T_{7}^{19} + \cdots - 524581 \) |