Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [8024,2,Mod(1,8024)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(8024, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 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("8024.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 8024 = 2^{3} \cdot 17 \cdot 59 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 8024.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: | \(64.0719625819\) |
Analytic rank: | \(0\) |
Dimension: | \(33\) |
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 | 0 | −3.44977 | 0 | 2.91669 | 0 | −3.05603 | 0 | 8.90091 | 0 | ||||||||||||||||||
1.2 | 0 | −3.36646 | 0 | 0.242538 | 0 | 4.57037 | 0 | 8.33308 | 0 | ||||||||||||||||||
1.3 | 0 | −3.33387 | 0 | −3.73347 | 0 | −3.52781 | 0 | 8.11466 | 0 | ||||||||||||||||||
1.4 | 0 | −2.83790 | 0 | 2.04357 | 0 | 3.25946 | 0 | 5.05368 | 0 | ||||||||||||||||||
1.5 | 0 | −2.77346 | 0 | 1.76876 | 0 | −4.75466 | 0 | 4.69208 | 0 | ||||||||||||||||||
1.6 | 0 | −2.56116 | 0 | 0.330730 | 0 | −0.779452 | 0 | 3.55957 | 0 | ||||||||||||||||||
1.7 | 0 | −2.22636 | 0 | −2.51166 | 0 | −2.56246 | 0 | 1.95667 | 0 | ||||||||||||||||||
1.8 | 0 | −2.18303 | 0 | −1.35778 | 0 | 2.12244 | 0 | 1.76561 | 0 | ||||||||||||||||||
1.9 | 0 | −2.08632 | 0 | −3.87621 | 0 | 3.88634 | 0 | 1.35275 | 0 | ||||||||||||||||||
1.10 | 0 | −1.52586 | 0 | 4.15650 | 0 | 0.772198 | 0 | −0.671760 | 0 | ||||||||||||||||||
1.11 | 0 | −1.48206 | 0 | 0.847914 | 0 | −1.54807 | 0 | −0.803509 | 0 | ||||||||||||||||||
1.12 | 0 | −1.13464 | 0 | 3.41042 | 0 | 3.81192 | 0 | −1.71258 | 0 | ||||||||||||||||||
1.13 | 0 | −0.973545 | 0 | −2.57461 | 0 | 0.184041 | 0 | −2.05221 | 0 | ||||||||||||||||||
1.14 | 0 | −0.882440 | 0 | 0.934682 | 0 | 3.89754 | 0 | −2.22130 | 0 | ||||||||||||||||||
1.15 | 0 | −0.772486 | 0 | 2.52530 | 0 | −1.43717 | 0 | −2.40327 | 0 | ||||||||||||||||||
1.16 | 0 | −0.481497 | 0 | −1.68499 | 0 | −3.55584 | 0 | −2.76816 | 0 | ||||||||||||||||||
1.17 | 0 | 0.0609456 | 0 | 3.92947 | 0 | −4.97459 | 0 | −2.99629 | 0 | ||||||||||||||||||
1.18 | 0 | 0.270435 | 0 | −0.155002 | 0 | −4.46551 | 0 | −2.92686 | 0 | ||||||||||||||||||
1.19 | 0 | 0.504210 | 0 | 1.10428 | 0 | −0.448564 | 0 | −2.74577 | 0 | ||||||||||||||||||
1.20 | 0 | 0.510308 | 0 | −0.733675 | 0 | 1.56284 | 0 | −2.73959 | 0 | ||||||||||||||||||
See all 33 embeddings |
Atkin-Lehner signs
\( p \) | Sign |
---|---|
\(2\) | \(-1\) |
\(17\) | \(1\) |
\(59\) | \(1\) |
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 | 8024.2.a.bc | ✓ | 33 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
8024.2.a.bc | ✓ | 33 | 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(8024))\):
\( T_{3}^{33} + 3 T_{3}^{32} - 69 T_{3}^{31} - 202 T_{3}^{30} + 2135 T_{3}^{29} + 6084 T_{3}^{28} - 39151 T_{3}^{27} - 108337 T_{3}^{26} + 473735 T_{3}^{25} + 1270129 T_{3}^{24} - 3985452 T_{3}^{23} - 10332869 T_{3}^{22} + \cdots - 148736 \) |
\( T_{5}^{33} - 3 T_{5}^{32} - 104 T_{5}^{31} + 316 T_{5}^{30} + 4767 T_{5}^{29} - 14780 T_{5}^{28} - 126644 T_{5}^{27} + 404789 T_{5}^{26} + 2156337 T_{5}^{25} - 7209643 T_{5}^{24} - 24538656 T_{5}^{23} + \cdots + 44679680 \) |
\( T_{7}^{33} - 162 T_{7}^{31} + T_{7}^{30} + 11770 T_{7}^{29} - 200 T_{7}^{28} - 507070 T_{7}^{27} + 16312 T_{7}^{26} + 14430070 T_{7}^{25} - 746624 T_{7}^{24} - 286000513 T_{7}^{23} + 21718475 T_{7}^{22} + \cdots - 87286906880 \) |