Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [8013,2,Mod(1,8013)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(8013, 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("8013.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 8013 = 3 \cdot 2671 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 8013.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: | \(63.9841271397\) |
Analytic rank: | \(1\) |
Dimension: | \(94\) |
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 | −2.78585 | 1.00000 | 5.76095 | 3.26911 | −2.78585 | −3.26627 | −10.4774 | 1.00000 | −9.10723 | ||||||||||||||||||
1.2 | −2.73477 | 1.00000 | 5.47898 | −0.238588 | −2.73477 | 1.07409 | −9.51422 | 1.00000 | 0.652485 | ||||||||||||||||||
1.3 | −2.64959 | 1.00000 | 5.02034 | −2.74590 | −2.64959 | −5.01160 | −8.00268 | 1.00000 | 7.27553 | ||||||||||||||||||
1.4 | −2.64705 | 1.00000 | 5.00689 | −1.00090 | −2.64705 | −3.26297 | −7.95941 | 1.00000 | 2.64944 | ||||||||||||||||||
1.5 | −2.57838 | 1.00000 | 4.64804 | 1.93561 | −2.57838 | 2.52768 | −6.82766 | 1.00000 | −4.99075 | ||||||||||||||||||
1.6 | −2.56682 | 1.00000 | 4.58855 | 3.72342 | −2.56682 | −2.49598 | −6.64435 | 1.00000 | −9.55734 | ||||||||||||||||||
1.7 | −2.55292 | 1.00000 | 4.51739 | 3.29007 | −2.55292 | 1.28784 | −6.42669 | 1.00000 | −8.39927 | ||||||||||||||||||
1.8 | −2.51426 | 1.00000 | 4.32151 | −2.69736 | −2.51426 | −1.28921 | −5.83689 | 1.00000 | 6.78186 | ||||||||||||||||||
1.9 | −2.46856 | 1.00000 | 4.09380 | 0.0867826 | −2.46856 | 0.912467 | −5.16868 | 1.00000 | −0.214228 | ||||||||||||||||||
1.10 | −2.45728 | 1.00000 | 4.03824 | 0.781238 | −2.45728 | −3.06583 | −5.00853 | 1.00000 | −1.91972 | ||||||||||||||||||
1.11 | −2.33315 | 1.00000 | 3.44357 | −2.85647 | −2.33315 | −3.40806 | −3.36806 | 1.00000 | 6.66457 | ||||||||||||||||||
1.12 | −2.27625 | 1.00000 | 3.18130 | 1.70680 | −2.27625 | −0.170344 | −2.68893 | 1.00000 | −3.88509 | ||||||||||||||||||
1.13 | −2.15734 | 1.00000 | 2.65411 | 1.74315 | −2.15734 | 2.90909 | −1.41114 | 1.00000 | −3.76056 | ||||||||||||||||||
1.14 | −2.15634 | 1.00000 | 2.64980 | −1.53192 | −2.15634 | −0.507117 | −1.40120 | 1.00000 | 3.30334 | ||||||||||||||||||
1.15 | −2.06033 | 1.00000 | 2.24497 | 1.16597 | −2.06033 | 0.252514 | −0.504726 | 1.00000 | −2.40228 | ||||||||||||||||||
1.16 | −2.05369 | 1.00000 | 2.21762 | −2.14256 | −2.05369 | −0.368331 | −0.446932 | 1.00000 | 4.40015 | ||||||||||||||||||
1.17 | −2.05347 | 1.00000 | 2.21673 | −2.14654 | −2.05347 | −0.521799 | −0.445046 | 1.00000 | 4.40785 | ||||||||||||||||||
1.18 | −2.03943 | 1.00000 | 2.15929 | −1.46688 | −2.03943 | 4.57052 | −0.324864 | 1.00000 | 2.99161 | ||||||||||||||||||
1.19 | −1.99976 | 1.00000 | 1.99902 | −2.89200 | −1.99976 | 2.76984 | 0.00195305 | 1.00000 | 5.78329 | ||||||||||||||||||
1.20 | −1.92790 | 1.00000 | 1.71679 | 3.85366 | −1.92790 | −4.10922 | 0.545993 | 1.00000 | −7.42947 | ||||||||||||||||||
See all 94 embeddings |
Atkin-Lehner signs
\( p \) | Sign |
---|---|
\(3\) | \(-1\) |
\(2671\) | \(-1\) |
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 | 8013.2.a.a | ✓ | 94 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
8013.2.a.a | ✓ | 94 | 1.a | even | 1 | 1 | trivial |