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: | \(0\) |
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 | −3.01325 | 0 | 0.418153 | 0 | −0.933114 | 0 | 6.07967 | 0 | ||||||||||||||||||
1.2 | 0 | −2.74123 | 0 | −4.14653 | 0 | −3.66203 | 0 | 4.51434 | 0 | ||||||||||||||||||
1.3 | 0 | −2.60599 | 0 | 0.822843 | 0 | 3.53536 | 0 | 3.79120 | 0 | ||||||||||||||||||
1.4 | 0 | −2.50495 | 0 | 0.621089 | 0 | 1.81379 | 0 | 3.27480 | 0 | ||||||||||||||||||
1.5 | 0 | −2.26361 | 0 | −0.762700 | 0 | −1.25553 | 0 | 2.12395 | 0 | ||||||||||||||||||
1.6 | 0 | −1.83124 | 0 | 4.45088 | 0 | 3.13811 | 0 | 0.353447 | 0 | ||||||||||||||||||
1.7 | 0 | −1.73095 | 0 | 0.367690 | 0 | 0.998919 | 0 | −0.00381480 | 0 | ||||||||||||||||||
1.8 | 0 | −1.62030 | 0 | −2.98882 | 0 | −1.58362 | 0 | −0.374619 | 0 | ||||||||||||||||||
1.9 | 0 | −1.26148 | 0 | 2.94335 | 0 | −0.461818 | 0 | −1.40867 | 0 | ||||||||||||||||||
1.10 | 0 | −1.19215 | 0 | −0.449856 | 0 | −2.62487 | 0 | −1.57879 | 0 | ||||||||||||||||||
1.11 | 0 | −1.13597 | 0 | −1.24613 | 0 | 4.57789 | 0 | −1.70957 | 0 | ||||||||||||||||||
1.12 | 0 | −0.685335 | 0 | −1.94665 | 0 | 1.34986 | 0 | −2.53032 | 0 | ||||||||||||||||||
1.13 | 0 | −0.222021 | 0 | 0.0345121 | 0 | −5.16899 | 0 | −2.95071 | 0 | ||||||||||||||||||
1.14 | 0 | 0.193782 | 0 | 1.98943 | 0 | 0.152201 | 0 | −2.96245 | 0 | ||||||||||||||||||
1.15 | 0 | 0.369540 | 0 | −3.94367 | 0 | 3.05773 | 0 | −2.86344 | 0 | ||||||||||||||||||
1.16 | 0 | 0.692263 | 0 | 3.18782 | 0 | −3.79509 | 0 | −2.52077 | 0 | ||||||||||||||||||
1.17 | 0 | 0.824754 | 0 | −2.00155 | 0 | 1.90493 | 0 | −2.31978 | 0 | ||||||||||||||||||
1.18 | 0 | 0.933880 | 0 | 0.0426876 | 0 | −2.12724 | 0 | −2.12787 | 0 | ||||||||||||||||||
1.19 | 0 | 1.21978 | 0 | 2.91757 | 0 | 2.62203 | 0 | −1.51215 | 0 | ||||||||||||||||||
1.20 | 0 | 1.73507 | 0 | −3.61768 | 0 | −3.69185 | 0 | 0.0104735 | 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.j | yes | 30 |
4.b | odd | 2 | 1 | 8032.2.a.g | ✓ | 30 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
8032.2.a.g | ✓ | 30 | 4.b | odd | 2 | 1 | |
8032.2.a.j | yes | 30 | 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(8032))\):
\( T_{3}^{30} - 9 T_{3}^{29} - 22 T_{3}^{28} + 411 T_{3}^{27} - 250 T_{3}^{26} - 8070 T_{3}^{25} + 14563 T_{3}^{24} + 88631 T_{3}^{23} - 235560 T_{3}^{22} - 588389 T_{3}^{21} + 2126360 T_{3}^{20} + 2318358 T_{3}^{19} + \cdots - 138600 \) |
\( T_{7}^{30} - 5 T_{7}^{29} - 113 T_{7}^{28} + 589 T_{7}^{27} + 5493 T_{7}^{26} - 30333 T_{7}^{25} - 150636 T_{7}^{24} + 902625 T_{7}^{23} + 2555283 T_{7}^{22} - 17259808 T_{7}^{21} - 27386124 T_{7}^{20} + \cdots + 3343679661 \) |
\( T_{11}^{30} - 31 T_{11}^{29} + 307 T_{11}^{28} + 164 T_{11}^{27} - 23879 T_{11}^{26} + 130424 T_{11}^{25} + 404151 T_{11}^{24} - 6100396 T_{11}^{23} + 9645227 T_{11}^{22} + 111502172 T_{11}^{21} + \cdots + 829293330432 \) |