Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [2008,2,Mod(1,2008)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2008, 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("2008.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 2008 = 2^{3} \cdot 251 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 2008.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: | \(16.0339607259\) |
Analytic rank: | \(0\) |
Dimension: | \(23\) |
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 | −3.39399 | 0 | −1.84335 | 0 | −1.40053 | 0 | 8.51918 | 0 | ||||||||||||||||||
1.2 | 0 | −3.22331 | 0 | 3.75978 | 0 | 2.77250 | 0 | 7.38975 | 0 | ||||||||||||||||||
1.3 | 0 | −3.19806 | 0 | −2.92729 | 0 | 4.80027 | 0 | 7.22760 | 0 | ||||||||||||||||||
1.4 | 0 | −2.74485 | 0 | 2.14962 | 0 | −3.54966 | 0 | 4.53421 | 0 | ||||||||||||||||||
1.5 | 0 | −2.21536 | 0 | 3.09704 | 0 | −4.67984 | 0 | 1.90784 | 0 | ||||||||||||||||||
1.6 | 0 | −2.13165 | 0 | 0.257423 | 0 | 0.578263 | 0 | 1.54395 | 0 | ||||||||||||||||||
1.7 | 0 | −1.28390 | 0 | 3.72143 | 0 | −0.978625 | 0 | −1.35160 | 0 | ||||||||||||||||||
1.8 | 0 | −1.19399 | 0 | −1.76720 | 0 | −4.20476 | 0 | −1.57440 | 0 | ||||||||||||||||||
1.9 | 0 | −1.15579 | 0 | −4.09592 | 0 | −0.621313 | 0 | −1.66414 | 0 | ||||||||||||||||||
1.10 | 0 | −0.347951 | 0 | −1.66668 | 0 | 3.92094 | 0 | −2.87893 | 0 | ||||||||||||||||||
1.11 | 0 | −0.201914 | 0 | −1.79819 | 0 | 3.34181 | 0 | −2.95923 | 0 | ||||||||||||||||||
1.12 | 0 | −0.0812979 | 0 | 4.08000 | 0 | 4.55548 | 0 | −2.99339 | 0 | ||||||||||||||||||
1.13 | 0 | 0.259421 | 0 | −1.96371 | 0 | −4.76268 | 0 | −2.93270 | 0 | ||||||||||||||||||
1.14 | 0 | 0.452441 | 0 | 1.91039 | 0 | 1.96124 | 0 | −2.79530 | 0 | ||||||||||||||||||
1.15 | 0 | 1.39383 | 0 | −3.54571 | 0 | −3.34862 | 0 | −1.05724 | 0 | ||||||||||||||||||
1.16 | 0 | 1.63778 | 0 | 2.87336 | 0 | −2.53418 | 0 | −0.317671 | 0 | ||||||||||||||||||
1.17 | 0 | 1.96174 | 0 | 0.807414 | 0 | 3.34299 | 0 | 0.848432 | 0 | ||||||||||||||||||
1.18 | 0 | 2.51727 | 0 | −0.472191 | 0 | 4.16870 | 0 | 3.33662 | 0 | ||||||||||||||||||
1.19 | 0 | 2.53334 | 0 | 1.50187 | 0 | 0.478852 | 0 | 3.41783 | 0 | ||||||||||||||||||
1.20 | 0 | 2.67736 | 0 | 4.29688 | 0 | −4.01755 | 0 | 4.16827 | 0 | ||||||||||||||||||
See all 23 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 | 2008.2.a.d | ✓ | 23 |
4.b | odd | 2 | 1 | 4016.2.a.m | 23 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
2008.2.a.d | ✓ | 23 | 1.a | even | 1 | 1 | trivial |
4016.2.a.m | 23 | 4.b | odd | 2 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{23} - 2 T_{3}^{22} - 55 T_{3}^{21} + 107 T_{3}^{20} + 1286 T_{3}^{19} - 2405 T_{3}^{18} + \cdots - 1408 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(2008))\).