Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [8032,2,Mod(1,8032)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(8032, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([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("8032.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 8032 = 2^{5} \cdot 251 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 8032.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: | \(64.1358429035\) |
Analytic rank: | \(1\) |
Dimension: | \(30\) |
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 | −3.26665 | 0 | −1.74507 | 0 | −2.44649 | 0 | 7.67103 | 0 | ||||||||||||||||||
1.2 | 0 | −3.07132 | 0 | 2.45715 | 0 | −4.52565 | 0 | 6.43300 | 0 | ||||||||||||||||||
1.3 | 0 | −2.85699 | 0 | 3.33276 | 0 | 0.481170 | 0 | 5.16239 | 0 | ||||||||||||||||||
1.4 | 0 | −2.83439 | 0 | −0.941648 | 0 | −0.892503 | 0 | 5.03379 | 0 | ||||||||||||||||||
1.5 | 0 | −2.61806 | 0 | 3.85373 | 0 | 0.108922 | 0 | 3.85426 | 0 | ||||||||||||||||||
1.6 | 0 | −2.45889 | 0 | 2.64078 | 0 | 1.82573 | 0 | 3.04612 | 0 | ||||||||||||||||||
1.7 | 0 | −2.41541 | 0 | −2.02020 | 0 | 4.66334 | 0 | 2.83423 | 0 | ||||||||||||||||||
1.8 | 0 | −1.92470 | 0 | −1.32031 | 0 | −3.41056 | 0 | 0.704453 | 0 | ||||||||||||||||||
1.9 | 0 | −1.66369 | 0 | 0.543435 | 0 | 2.23287 | 0 | −0.232125 | 0 | ||||||||||||||||||
1.10 | 0 | −1.32607 | 0 | 2.50636 | 0 | −4.14411 | 0 | −1.24155 | 0 | ||||||||||||||||||
1.11 | 0 | −1.18444 | 0 | −0.0964158 | 0 | 1.78364 | 0 | −1.59711 | 0 | ||||||||||||||||||
1.12 | 0 | −1.07677 | 0 | −3.37253 | 0 | −5.03712 | 0 | −1.84058 | 0 | ||||||||||||||||||
1.13 | 0 | −0.700584 | 0 | −3.32459 | 0 | 0.693647 | 0 | −2.50918 | 0 | ||||||||||||||||||
1.14 | 0 | −0.679540 | 0 | −1.36350 | 0 | 2.99119 | 0 | −2.53823 | 0 | ||||||||||||||||||
1.15 | 0 | −0.486172 | 0 | −3.69982 | 0 | −0.0825276 | 0 | −2.76364 | 0 | ||||||||||||||||||
1.16 | 0 | 0.0692070 | 0 | 0.775693 | 0 | 4.19948 | 0 | −2.99521 | 0 | ||||||||||||||||||
1.17 | 0 | 0.0957135 | 0 | 3.87548 | 0 | −3.14340 | 0 | −2.99084 | 0 | ||||||||||||||||||
1.18 | 0 | 0.513479 | 0 | 3.65851 | 0 | 0.146814 | 0 | −2.73634 | 0 | ||||||||||||||||||
1.19 | 0 | 0.797542 | 0 | −1.59101 | 0 | −1.45275 | 0 | −2.36393 | 0 | ||||||||||||||||||
1.20 | 0 | 1.12351 | 0 | 0.837743 | 0 | 0.914625 | 0 | −1.73772 | 0 | ||||||||||||||||||
See all 30 embeddings |
Atkin-Lehner signs
\( p \) | Sign |
---|---|
\(2\) | \(1\) |
\(251\) | \(1\) |
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 | 8032.2.a.h | ✓ | 30 |
4.b | odd | 2 | 1 | 8032.2.a.i | yes | 30 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
8032.2.a.h | ✓ | 30 | 1.a | even | 1 | 1 | trivial |
8032.2.a.i | yes | 30 | 4.b | odd | 2 | 1 |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(8032))\):
\( T_{3}^{30} + 3 T_{3}^{29} - 58 T_{3}^{28} - 173 T_{3}^{27} + 1490 T_{3}^{26} + 4418 T_{3}^{25} + \cdots - 14616 \) |
\( T_{7}^{30} + 13 T_{7}^{29} - 33 T_{7}^{28} - 1047 T_{7}^{27} - 1293 T_{7}^{26} + 35345 T_{7}^{25} + \cdots - 533733 \) |
\( T_{11}^{30} + 13 T_{11}^{29} - 101 T_{11}^{28} - 1880 T_{11}^{27} + 2741 T_{11}^{26} + 117520 T_{11}^{25} + \cdots + 6097731584 \) |