Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [8002,2,Mod(1,8002)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(8002, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("8002.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 8002 = 2 \cdot 4001 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 8002.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: | \(63.8962916974\) |
Analytic rank: | \(1\) |
Dimension: | \(69\) |
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 | 1.00000 | −3.43715 | 1.00000 | −2.28285 | −3.43715 | −3.85963 | 1.00000 | 8.81400 | −2.28285 | ||||||||||||||||||
1.2 | 1.00000 | −3.40667 | 1.00000 | 2.18421 | −3.40667 | −1.24057 | 1.00000 | 8.60538 | 2.18421 | ||||||||||||||||||
1.3 | 1.00000 | −3.22228 | 1.00000 | −3.24951 | −3.22228 | 4.15498 | 1.00000 | 7.38311 | −3.24951 | ||||||||||||||||||
1.4 | 1.00000 | −3.09800 | 1.00000 | −2.64538 | −3.09800 | 3.07801 | 1.00000 | 6.59762 | −2.64538 | ||||||||||||||||||
1.5 | 1.00000 | −3.04857 | 1.00000 | −0.106661 | −3.04857 | −0.613952 | 1.00000 | 6.29376 | −0.106661 | ||||||||||||||||||
1.6 | 1.00000 | −3.03319 | 1.00000 | −3.55972 | −3.03319 | −4.22109 | 1.00000 | 6.20023 | −3.55972 | ||||||||||||||||||
1.7 | 1.00000 | −2.87355 | 1.00000 | 2.00628 | −2.87355 | 0.474155 | 1.00000 | 5.25731 | 2.00628 | ||||||||||||||||||
1.8 | 1.00000 | −2.80104 | 1.00000 | 0.676703 | −2.80104 | 3.76033 | 1.00000 | 4.84585 | 0.676703 | ||||||||||||||||||
1.9 | 1.00000 | −2.76808 | 1.00000 | 4.07666 | −2.76808 | 2.61903 | 1.00000 | 4.66227 | 4.07666 | ||||||||||||||||||
1.10 | 1.00000 | −2.73719 | 1.00000 | −4.28393 | −2.73719 | 2.06732 | 1.00000 | 4.49223 | −4.28393 | ||||||||||||||||||
1.11 | 1.00000 | −2.64782 | 1.00000 | −0.675974 | −2.64782 | −0.999289 | 1.00000 | 4.01094 | −0.675974 | ||||||||||||||||||
1.12 | 1.00000 | −2.62997 | 1.00000 | −0.133179 | −2.62997 | −2.98290 | 1.00000 | 3.91674 | −0.133179 | ||||||||||||||||||
1.13 | 1.00000 | −2.42898 | 1.00000 | −1.63165 | −2.42898 | 3.93412 | 1.00000 | 2.89993 | −1.63165 | ||||||||||||||||||
1.14 | 1.00000 | −2.38948 | 1.00000 | 1.42184 | −2.38948 | −5.14183 | 1.00000 | 2.70964 | 1.42184 | ||||||||||||||||||
1.15 | 1.00000 | −2.15867 | 1.00000 | 0.496482 | −2.15867 | −0.313465 | 1.00000 | 1.65985 | 0.496482 | ||||||||||||||||||
1.16 | 1.00000 | −2.04234 | 1.00000 | −4.37946 | −2.04234 | −3.34038 | 1.00000 | 1.17114 | −4.37946 | ||||||||||||||||||
1.17 | 1.00000 | −2.01476 | 1.00000 | 2.07868 | −2.01476 | −1.64070 | 1.00000 | 1.05926 | 2.07868 | ||||||||||||||||||
1.18 | 1.00000 | −1.99879 | 1.00000 | 2.05838 | −1.99879 | 2.31041 | 1.00000 | 0.995149 | 2.05838 | ||||||||||||||||||
1.19 | 1.00000 | −1.96903 | 1.00000 | 3.30115 | −1.96903 | 1.85739 | 1.00000 | 0.877064 | 3.30115 | ||||||||||||||||||
1.20 | 1.00000 | −1.96349 | 1.00000 | 0.136598 | −1.96349 | −0.645165 | 1.00000 | 0.855285 | 0.136598 | ||||||||||||||||||
See all 69 embeddings |
Atkin-Lehner signs
\( p \) | Sign |
---|---|
\(2\) | \(-1\) |
\(4001\) | \(-1\) |
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 | 8002.2.a.d | ✓ | 69 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
8002.2.a.d | ✓ | 69 | 1.a | even | 1 | 1 | trivial |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{69} + 25 T_{3}^{68} + 182 T_{3}^{67} - 583 T_{3}^{66} - 14542 T_{3}^{65} - 42332 T_{3}^{64} + \cdots + 388376961 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(8002))\).