Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [8014,2,Mod(1,8014)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(8014, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("8014.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 8014 = 2 \cdot 4007 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 8014.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: | \(63.9921121799\) |
Analytic rank: | \(0\) |
Dimension: | \(91\) |
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 | −1.00000 | −3.44800 | 1.00000 | −2.43856 | 3.44800 | 4.30446 | −1.00000 | 8.88870 | 2.43856 | ||||||||||||||||||
1.2 | −1.00000 | −3.41500 | 1.00000 | 2.43339 | 3.41500 | −1.50978 | −1.00000 | 8.66224 | −2.43339 | ||||||||||||||||||
1.3 | −1.00000 | −3.27811 | 1.00000 | 4.00106 | 3.27811 | −4.58549 | −1.00000 | 7.74601 | −4.00106 | ||||||||||||||||||
1.4 | −1.00000 | −3.27677 | 1.00000 | −0.135024 | 3.27677 | −2.05105 | −1.00000 | 7.73722 | 0.135024 | ||||||||||||||||||
1.5 | −1.00000 | −3.21498 | 1.00000 | −2.38557 | 3.21498 | −0.918897 | −1.00000 | 7.33611 | 2.38557 | ||||||||||||||||||
1.6 | −1.00000 | −3.13974 | 1.00000 | 3.30555 | 3.13974 | 0.656006 | −1.00000 | 6.85799 | −3.30555 | ||||||||||||||||||
1.7 | −1.00000 | −3.09151 | 1.00000 | −2.61298 | 3.09151 | −3.49285 | −1.00000 | 6.55743 | 2.61298 | ||||||||||||||||||
1.8 | −1.00000 | −3.06251 | 1.00000 | 1.40510 | 3.06251 | 4.44564 | −1.00000 | 6.37896 | −1.40510 | ||||||||||||||||||
1.9 | −1.00000 | −2.93164 | 1.00000 | −4.27136 | 2.93164 | −1.93753 | −1.00000 | 5.59453 | 4.27136 | ||||||||||||||||||
1.10 | −1.00000 | −2.86981 | 1.00000 | −0.806947 | 2.86981 | −4.79468 | −1.00000 | 5.23582 | 0.806947 | ||||||||||||||||||
1.11 | −1.00000 | −2.86622 | 1.00000 | 0.269496 | 2.86622 | 2.82265 | −1.00000 | 5.21519 | −0.269496 | ||||||||||||||||||
1.12 | −1.00000 | −2.85455 | 1.00000 | −2.30346 | 2.85455 | 3.17509 | −1.00000 | 5.14844 | 2.30346 | ||||||||||||||||||
1.13 | −1.00000 | −2.66657 | 1.00000 | 3.59792 | 2.66657 | 3.19859 | −1.00000 | 4.11059 | −3.59792 | ||||||||||||||||||
1.14 | −1.00000 | −2.62836 | 1.00000 | 2.05043 | 2.62836 | −5.05998 | −1.00000 | 3.90825 | −2.05043 | ||||||||||||||||||
1.15 | −1.00000 | −2.35351 | 1.00000 | 2.15005 | 2.35351 | 1.12674 | −1.00000 | 2.53903 | −2.15005 | ||||||||||||||||||
1.16 | −1.00000 | −2.32734 | 1.00000 | 3.86368 | 2.32734 | −1.21196 | −1.00000 | 2.41649 | −3.86368 | ||||||||||||||||||
1.17 | −1.00000 | −2.31549 | 1.00000 | 0.682396 | 2.31549 | −3.17347 | −1.00000 | 2.36151 | −0.682396 | ||||||||||||||||||
1.18 | −1.00000 | −2.25449 | 1.00000 | 0.777840 | 2.25449 | 1.47835 | −1.00000 | 2.08272 | −0.777840 | ||||||||||||||||||
1.19 | −1.00000 | −2.03768 | 1.00000 | −1.30464 | 2.03768 | −1.43874 | −1.00000 | 1.15214 | 1.30464 | ||||||||||||||||||
1.20 | −1.00000 | −1.99318 | 1.00000 | 0.808905 | 1.99318 | 3.91873 | −1.00000 | 0.972766 | −0.808905 | ||||||||||||||||||
See all 91 embeddings |
Atkin-Lehner signs
\( p \) | Sign |
---|---|
\(2\) | \(1\) |
\(4007\) | \(-1\) |
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 | 8014.2.a.e | ✓ | 91 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
8014.2.a.e | ✓ | 91 | 1.a | even | 1 | 1 | trivial |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{91} + 2 T_{3}^{90} - 196 T_{3}^{89} - 388 T_{3}^{88} + 18477 T_{3}^{87} + 36194 T_{3}^{86} + \cdots + 115514277888 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(8014))\).