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: | \(0\) |
Dimension: | \(106\) |
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 | −2.68955 | −1.00000 | 5.23367 | −0.143035 | 2.68955 | 0.169841 | −8.69711 | 1.00000 | 0.384699 | ||||||||||||||||||
1.2 | −2.68470 | −1.00000 | 5.20764 | 3.29964 | 2.68470 | 2.53834 | −8.61155 | 1.00000 | −8.85857 | ||||||||||||||||||
1.3 | −2.57806 | −1.00000 | 4.64638 | −0.00185853 | 2.57806 | 4.00944 | −6.82252 | 1.00000 | 0.00479139 | ||||||||||||||||||
1.4 | −2.57103 | −1.00000 | 4.61019 | 2.04961 | 2.57103 | −2.14469 | −6.71087 | 1.00000 | −5.26961 | ||||||||||||||||||
1.5 | −2.57051 | −1.00000 | 4.60751 | −0.585495 | 2.57051 | −2.49504 | −6.70262 | 1.00000 | 1.50502 | ||||||||||||||||||
1.6 | −2.45069 | −1.00000 | 4.00587 | 0.123141 | 2.45069 | −0.667497 | −4.91577 | 1.00000 | −0.301780 | ||||||||||||||||||
1.7 | −2.44992 | −1.00000 | 4.00213 | 3.65851 | 2.44992 | −1.10640 | −4.90506 | 1.00000 | −8.96306 | ||||||||||||||||||
1.8 | −2.43314 | −1.00000 | 3.92015 | 4.32607 | 2.43314 | 4.27561 | −4.67199 | 1.00000 | −10.5259 | ||||||||||||||||||
1.9 | −2.35891 | −1.00000 | 3.56446 | −3.22536 | 2.35891 | −2.62511 | −3.69041 | 1.00000 | 7.60832 | ||||||||||||||||||
1.10 | −2.32341 | −1.00000 | 3.39824 | 0.581135 | 2.32341 | 1.71987 | −3.24868 | 1.00000 | −1.35022 | ||||||||||||||||||
1.11 | −2.30949 | −1.00000 | 3.33375 | 0.518255 | 2.30949 | 1.93365 | −3.08028 | 1.00000 | −1.19691 | ||||||||||||||||||
1.12 | −2.30696 | −1.00000 | 3.32206 | −3.03521 | 2.30696 | 1.29495 | −3.04993 | 1.00000 | 7.00211 | ||||||||||||||||||
1.13 | −2.24374 | −1.00000 | 3.03438 | 2.67702 | 2.24374 | −0.787815 | −2.32089 | 1.00000 | −6.00654 | ||||||||||||||||||
1.14 | −2.16031 | −1.00000 | 2.66695 | −0.855629 | 2.16031 | 2.67269 | −1.44083 | 1.00000 | 1.84843 | ||||||||||||||||||
1.15 | −2.12375 | −1.00000 | 2.51030 | −3.37551 | 2.12375 | −2.22714 | −1.08375 | 1.00000 | 7.16873 | ||||||||||||||||||
1.16 | −2.09214 | −1.00000 | 2.37707 | −2.25066 | 2.09214 | −3.42799 | −0.788875 | 1.00000 | 4.70871 | ||||||||||||||||||
1.17 | −1.85225 | −1.00000 | 1.43083 | 2.49048 | 1.85225 | −3.91286 | 1.05424 | 1.00000 | −4.61299 | ||||||||||||||||||
1.18 | −1.84122 | −1.00000 | 1.39011 | −1.17426 | 1.84122 | 1.69712 | 1.12295 | 1.00000 | 2.16208 | ||||||||||||||||||
1.19 | −1.83416 | −1.00000 | 1.36415 | −3.29261 | 1.83416 | −0.883876 | 1.16624 | 1.00000 | 6.03918 | ||||||||||||||||||
1.20 | −1.79051 | −1.00000 | 1.20593 | −0.580006 | 1.79051 | 0.129222 | 1.42179 | 1.00000 | 1.03851 | ||||||||||||||||||
See next 80 embeddings (of 106 total) |
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.b | ✓ | 106 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
8013.2.a.b | ✓ | 106 | 1.a | even | 1 | 1 | trivial |