Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [8045,2,Mod(1,8045)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(8045, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([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("8045.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 8045 = 5 \cdot 1609 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 8045.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: | \(64.2396484261\) |
Analytic rank: | \(0\) |
Dimension: | \(142\) |
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 | −2.77343 | 2.65510 | 5.69191 | 1.00000 | −7.36373 | 0.492807 | −10.2393 | 4.04956 | −2.77343 | ||||||||||||||||||
1.2 | −2.68924 | −0.0494506 | 5.23199 | 1.00000 | 0.132984 | 2.47489 | −8.69158 | −2.99755 | −2.68924 | ||||||||||||||||||
1.3 | −2.64935 | 2.86876 | 5.01907 | 1.00000 | −7.60037 | 4.90549 | −7.99858 | 5.22980 | −2.64935 | ||||||||||||||||||
1.4 | −2.64700 | −1.25206 | 5.00659 | 1.00000 | 3.31419 | 4.95452 | −7.95843 | −1.43235 | −2.64700 | ||||||||||||||||||
1.5 | −2.63029 | −2.20415 | 4.91841 | 1.00000 | 5.79754 | 3.10194 | −7.67624 | 1.85826 | −2.63029 | ||||||||||||||||||
1.6 | −2.60196 | 1.36677 | 4.77022 | 1.00000 | −3.55629 | 2.24153 | −7.20801 | −1.13194 | −2.60196 | ||||||||||||||||||
1.7 | −2.59468 | −0.861703 | 4.73239 | 1.00000 | 2.23585 | −3.11370 | −7.08968 | −2.25747 | −2.59468 | ||||||||||||||||||
1.8 | −2.58612 | 1.58438 | 4.68803 | 1.00000 | −4.09739 | −2.71567 | −6.95157 | −0.489749 | −2.58612 | ||||||||||||||||||
1.9 | −2.51445 | −1.12018 | 4.32245 | 1.00000 | 2.81662 | 0.310792 | −5.83967 | −1.74521 | −2.51445 | ||||||||||||||||||
1.10 | −2.48735 | −2.65295 | 4.18689 | 1.00000 | 6.59881 | 2.41015 | −5.43957 | 4.03816 | −2.48735 | ||||||||||||||||||
1.11 | −2.46500 | 2.31843 | 4.07621 | 1.00000 | −5.71491 | −0.590235 | −5.11785 | 2.37510 | −2.46500 | ||||||||||||||||||
1.12 | −2.39887 | −2.11291 | 3.75456 | 1.00000 | 5.06858 | 1.17746 | −4.20895 | 1.46438 | −2.39887 | ||||||||||||||||||
1.13 | −2.28247 | 0.951173 | 3.20967 | 1.00000 | −2.17102 | −3.35163 | −2.76104 | −2.09527 | −2.28247 | ||||||||||||||||||
1.14 | −2.20465 | −2.96481 | 2.86047 | 1.00000 | 6.53637 | 0.166097 | −1.89703 | 5.79013 | −2.20465 | ||||||||||||||||||
1.15 | −2.19407 | 0.184631 | 2.81393 | 1.00000 | −0.405094 | 1.35112 | −1.78582 | −2.96591 | −2.19407 | ||||||||||||||||||
1.16 | −2.18359 | 0.444286 | 2.76808 | 1.00000 | −0.970139 | −0.684124 | −1.67716 | −2.80261 | −2.18359 | ||||||||||||||||||
1.17 | −2.16169 | −1.95272 | 2.67292 | 1.00000 | 4.22118 | −2.55513 | −1.45466 | 0.813116 | −2.16169 | ||||||||||||||||||
1.18 | −2.15729 | −0.777397 | 2.65392 | 1.00000 | 1.67707 | 3.51600 | −1.41069 | −2.39565 | −2.15729 | ||||||||||||||||||
1.19 | −2.15373 | 3.04079 | 2.63855 | 1.00000 | −6.54904 | −4.47994 | −1.37525 | 6.24641 | −2.15373 | ||||||||||||||||||
1.20 | −2.15214 | 2.27833 | 2.63171 | 1.00000 | −4.90329 | 4.73674 | −1.35954 | 2.19079 | −2.15214 | ||||||||||||||||||
See next 80 embeddings (of 142 total) |
Atkin-Lehner signs
\( p \) | Sign |
---|---|
\(5\) | \(-1\) |
\(1609\) | \(1\) |
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 | 8045.2.a.e | ✓ | 142 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
8045.2.a.e | ✓ | 142 | 1.a | even | 1 | 1 | trivial |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{142} - 21 T_{2}^{141} + 3039 T_{2}^{139} - 15763 T_{2}^{138} - 192561 T_{2}^{137} + \cdots + 124007569699 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(8045))\).