Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [8023,2,Mod(1,8023)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(8023, 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("8023.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 8023 = 71 \cdot 113 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 8023.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: | \(64.0639775417\) |
Analytic rank: | \(0\) |
Dimension: | \(165\) |
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 | −2.77651 | 2.27500 | 5.70902 | 0.808124 | −6.31657 | 0.0647974 | −10.2981 | 2.17564 | −2.24377 | ||||||||||||||||||
1.2 | −2.72160 | −0.981025 | 5.40712 | −1.25328 | 2.66996 | −1.09644 | −9.27282 | −2.03759 | 3.41092 | ||||||||||||||||||
1.3 | −2.70370 | −2.31695 | 5.31001 | 3.41961 | 6.26435 | 0.731020 | −8.94928 | 2.36827 | −9.24561 | ||||||||||||||||||
1.4 | −2.64042 | 3.27801 | 4.97182 | 3.57654 | −8.65532 | 2.80854 | −7.84685 | 7.74534 | −9.44356 | ||||||||||||||||||
1.5 | −2.62951 | 0.0659678 | 4.91432 | 1.30562 | −0.173463 | −3.45826 | −7.66322 | −2.99565 | −3.43314 | ||||||||||||||||||
1.6 | −2.60713 | −0.132138 | 4.79710 | −1.04101 | 0.344501 | 4.69646 | −7.29239 | −2.98254 | 2.71405 | ||||||||||||||||||
1.7 | −2.57888 | −1.57055 | 4.65062 | −2.72522 | 4.05026 | −1.77487 | −6.83564 | −0.533376 | 7.02803 | ||||||||||||||||||
1.8 | −2.52899 | 1.55552 | 4.39577 | −1.46602 | −3.93388 | 0.126533 | −6.05887 | −0.580366 | 3.70754 | ||||||||||||||||||
1.9 | −2.51204 | 1.13882 | 4.31034 | 2.43582 | −2.86077 | −2.08264 | −5.80366 | −1.70308 | −6.11887 | ||||||||||||||||||
1.10 | −2.49309 | −1.90773 | 4.21552 | 0.893200 | 4.75615 | 0.632761 | −5.52349 | 0.639435 | −2.22683 | ||||||||||||||||||
1.11 | −2.44821 | 2.59661 | 3.99372 | −2.59026 | −6.35704 | −1.59374 | −4.88104 | 3.74237 | 6.34149 | ||||||||||||||||||
1.12 | −2.38214 | 3.08941 | 3.67457 | 1.88389 | −7.35940 | 1.49844 | −3.98905 | 6.54446 | −4.48768 | ||||||||||||||||||
1.13 | −2.35257 | −0.106381 | 3.53459 | 1.14622 | 0.250270 | −0.376719 | −3.61023 | −2.98868 | −2.69657 | ||||||||||||||||||
1.14 | −2.33295 | −2.77061 | 3.44267 | −1.81617 | 6.46371 | 3.36128 | −3.36567 | 4.67630 | 4.23705 | ||||||||||||||||||
1.15 | −2.29416 | −0.794120 | 3.26316 | 4.14537 | 1.82184 | 3.97505 | −2.89789 | −2.36937 | −9.51013 | ||||||||||||||||||
1.16 | −2.29110 | 0.370553 | 3.24912 | −0.295381 | −0.848972 | 0.894029 | −2.86186 | −2.86269 | 0.676746 | ||||||||||||||||||
1.17 | −2.25574 | 0.317472 | 3.08836 | −3.70971 | −0.716135 | −3.94071 | −2.45506 | −2.89921 | 8.36814 | ||||||||||||||||||
1.18 | −2.20645 | 2.85395 | 2.86841 | −2.79248 | −6.29710 | 1.10201 | −1.91611 | 5.14504 | 6.16145 | ||||||||||||||||||
1.19 | −2.16974 | −2.66568 | 2.70776 | −3.38518 | 5.78382 | −1.43929 | −1.53565 | 4.10584 | 7.34496 | ||||||||||||||||||
1.20 | −2.14746 | −2.07357 | 2.61157 | 2.08017 | 4.45290 | −3.04722 | −1.31333 | 1.29970 | −4.46708 | ||||||||||||||||||
See next 80 embeddings (of 165 total) |
Atkin-Lehner signs
\( p \) | Sign |
---|---|
\(71\) | \(-1\) |
\(113\) | \(1\) |
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 | 8023.2.a.d | ✓ | 165 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
8023.2.a.d | ✓ | 165 | 1.a | even | 1 | 1 | trivial |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{165} - 22 T_{2}^{164} - 6 T_{2}^{163} + 3630 T_{2}^{162} - 18995 T_{2}^{161} - 265620 T_{2}^{160} + \cdots + 787040721 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(8023))\).