Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [8041,2,Mod(1,8041)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(8041, 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("8041.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 8041 = 11 \cdot 17 \cdot 43 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 8041.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: | \(64.2077082653\) |
Analytic rank: | \(1\) |
Dimension: | \(60\) |
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 | −2.78000 | −1.81181 | 5.72839 | 1.28422 | 5.03683 | 3.89203 | −10.3649 | 0.282655 | −3.57012 | ||||||||||||||||||
1.2 | −2.66007 | −2.34749 | 5.07599 | −2.77927 | 6.24449 | 0.942946 | −8.18236 | 2.51070 | 7.39305 | ||||||||||||||||||
1.3 | −2.54628 | 1.17861 | 4.48352 | 1.63315 | −3.00107 | −0.708864 | −6.32373 | −1.61088 | −4.15845 | ||||||||||||||||||
1.4 | −2.53062 | 2.95421 | 4.40406 | −1.92556 | −7.47600 | 2.94104 | −6.08377 | 5.72738 | 4.87287 | ||||||||||||||||||
1.5 | −2.51152 | −1.72847 | 4.30774 | −3.07277 | 4.34109 | −4.07006 | −5.79593 | −0.0123937 | 7.71734 | ||||||||||||||||||
1.6 | −2.42660 | 2.76366 | 3.88839 | −0.843592 | −6.70630 | −2.86536 | −4.58236 | 4.63781 | 2.04706 | ||||||||||||||||||
1.7 | −2.38636 | −2.80769 | 3.69472 | 2.77009 | 6.70017 | −2.49366 | −4.04422 | 4.88313 | −6.61043 | ||||||||||||||||||
1.8 | −2.36766 | 0.323518 | 3.60581 | −3.56257 | −0.765980 | 3.82937 | −3.80202 | −2.89534 | 8.43496 | ||||||||||||||||||
1.9 | −2.17121 | 0.274303 | 2.71415 | −0.625848 | −0.595570 | −3.36796 | −1.55058 | −2.92476 | 1.35885 | ||||||||||||||||||
1.10 | −2.12431 | 0.433471 | 2.51268 | 3.47058 | −0.920826 | 2.89316 | −1.08909 | −2.81210 | −7.37258 | ||||||||||||||||||
1.11 | −2.07976 | −1.74571 | 2.32541 | −0.111413 | 3.63066 | −0.659970 | −0.676772 | 0.0475049 | 0.231713 | ||||||||||||||||||
1.12 | −1.94215 | 1.51854 | 1.77194 | 3.53502 | −2.94923 | −1.41025 | 0.442922 | −0.694041 | −6.86554 | ||||||||||||||||||
1.13 | −1.83493 | 1.92990 | 1.36698 | 0.407070 | −3.54123 | 0.437202 | 1.16155 | 0.724496 | −0.746946 | ||||||||||||||||||
1.14 | −1.57436 | 0.0662834 | 0.478605 | −1.53153 | −0.104354 | 0.748849 | 2.39522 | −2.99561 | 2.41117 | ||||||||||||||||||
1.15 | −1.55010 | 2.25408 | 0.402812 | −2.41752 | −3.49405 | 2.04645 | 2.47580 | 2.08087 | 3.74740 | ||||||||||||||||||
1.16 | −1.50893 | −3.04602 | 0.276872 | −1.87360 | 4.59624 | −2.31151 | 2.60008 | 6.27825 | 2.82713 | ||||||||||||||||||
1.17 | −1.49915 | −0.148971 | 0.247445 | −2.56328 | 0.223329 | 2.63900 | 2.62734 | −2.97781 | 3.84274 | ||||||||||||||||||
1.18 | −1.33974 | 3.18054 | −0.205105 | 0.454728 | −4.26109 | −3.38234 | 2.95426 | 7.11585 | −0.609216 | ||||||||||||||||||
1.19 | −1.32373 | −3.06819 | −0.247749 | 3.29067 | 4.06144 | −2.61144 | 2.97540 | 6.41379 | −4.35594 | ||||||||||||||||||
1.20 | −1.23062 | −0.724033 | −0.485567 | 2.55129 | 0.891011 | −2.06260 | 3.05880 | −2.47578 | −3.13968 | ||||||||||||||||||
See all 60 embeddings |
Atkin-Lehner signs
\( p \) | Sign |
---|---|
\(11\) | \(-1\) |
\(17\) | \(-1\) |
\(43\) | \(1\) |
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 | 8041.2.a.c | ✓ | 60 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
8041.2.a.c | ✓ | 60 | 1.a | even | 1 | 1 | trivial |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{60} + 9 T_{2}^{59} - 41 T_{2}^{58} - 593 T_{2}^{57} + 275 T_{2}^{56} + 18073 T_{2}^{55} + \cdots - 2858 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(8041))\).