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: | \(32\) |
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 | −3.16628 | 0 | −1.06664 | 0 | −0.836052 | 0 | 7.02534 | 0 | ||||||||||||||||||
1.2 | 0 | −3.01806 | 0 | 3.78696 | 0 | −1.07498 | 0 | 6.10866 | 0 | ||||||||||||||||||
1.3 | 0 | −2.97402 | 0 | −3.33527 | 0 | 3.60888 | 0 | 5.84478 | 0 | ||||||||||||||||||
1.4 | 0 | −2.96706 | 0 | 3.51833 | 0 | 3.62712 | 0 | 5.80344 | 0 | ||||||||||||||||||
1.5 | 0 | −2.65698 | 0 | −1.37353 | 0 | −4.51033 | 0 | 4.05953 | 0 | ||||||||||||||||||
1.6 | 0 | −2.35639 | 0 | 1.88610 | 0 | −3.68573 | 0 | 2.55257 | 0 | ||||||||||||||||||
1.7 | 0 | −2.29360 | 0 | −3.85032 | 0 | 1.95004 | 0 | 2.26061 | 0 | ||||||||||||||||||
1.8 | 0 | −2.23251 | 0 | 3.10540 | 0 | −3.44467 | 0 | 1.98408 | 0 | ||||||||||||||||||
1.9 | 0 | −1.67764 | 0 | −0.206110 | 0 | −1.52712 | 0 | −0.185517 | 0 | ||||||||||||||||||
1.10 | 0 | −1.26190 | 0 | 1.28218 | 0 | −0.0440837 | 0 | −1.40761 | 0 | ||||||||||||||||||
1.11 | 0 | −1.21753 | 0 | 0.238125 | 0 | 5.25396 | 0 | −1.51763 | 0 | ||||||||||||||||||
1.12 | 0 | −1.11553 | 0 | −3.13935 | 0 | 0.425117 | 0 | −1.75560 | 0 | ||||||||||||||||||
1.13 | 0 | −0.551342 | 0 | 0.616724 | 0 | 1.02317 | 0 | −2.69602 | 0 | ||||||||||||||||||
1.14 | 0 | −0.394982 | 0 | −0.737434 | 0 | −1.31966 | 0 | −2.84399 | 0 | ||||||||||||||||||
1.15 | 0 | −0.321300 | 0 | 3.25819 | 0 | 3.32621 | 0 | −2.89677 | 0 | ||||||||||||||||||
1.16 | 0 | −0.159636 | 0 | −1.22368 | 0 | 3.18149 | 0 | −2.97452 | 0 | ||||||||||||||||||
1.17 | 0 | 0.0290013 | 0 | 3.75701 | 0 | −1.56335 | 0 | −2.99916 | 0 | ||||||||||||||||||
1.18 | 0 | 0.147881 | 0 | −3.84317 | 0 | −0.153864 | 0 | −2.97813 | 0 | ||||||||||||||||||
1.19 | 0 | 0.792010 | 0 | −0.983575 | 0 | 1.99855 | 0 | −2.37272 | 0 | ||||||||||||||||||
1.20 | 0 | 0.804465 | 0 | 3.90815 | 0 | 3.10524 | 0 | −2.35284 | 0 | ||||||||||||||||||
See all 32 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.bb | ✓ | 32 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
8024.2.a.bb | ✓ | 32 | 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(8024))\):
\( T_{3}^{32} - 68 T_{3}^{30} - 3 T_{3}^{29} + 2057 T_{3}^{28} + 163 T_{3}^{27} - 36519 T_{3}^{26} + \cdots + 1152 \) |
\( T_{5}^{32} - 8 T_{5}^{31} - 67 T_{5}^{30} + 658 T_{5}^{29} + 1751 T_{5}^{28} - 23833 T_{5}^{27} + \cdots - 3207168 \) |
\( T_{7}^{32} + 3 T_{7}^{31} - 126 T_{7}^{30} - 375 T_{7}^{29} + 7013 T_{7}^{28} + 20576 T_{7}^{27} + \cdots - 82313216 \) |