Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [2008,4,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, 4, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("2008.1");
S:= CuspForms(chi, 4);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 2008 = 2^{3} \cdot 251 \) |
Weight: | \( k \) | \(=\) | \( 4 \) |
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: | \(118.475835292\) |
Analytic rank: | \(1\) |
Dimension: | \(50\) |
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 | −10.0846 | 0 | −5.89682 | 0 | −15.9708 | 0 | 74.6997 | 0 | ||||||||||||||||||
1.2 | 0 | −9.57494 | 0 | 9.33217 | 0 | 9.70777 | 0 | 64.6794 | 0 | ||||||||||||||||||
1.3 | 0 | −9.54342 | 0 | −4.49657 | 0 | 15.0683 | 0 | 64.0770 | 0 | ||||||||||||||||||
1.4 | 0 | −9.42753 | 0 | 13.1979 | 0 | 10.2624 | 0 | 61.8784 | 0 | ||||||||||||||||||
1.5 | 0 | −9.21311 | 0 | −18.3228 | 0 | 29.9868 | 0 | 57.8813 | 0 | ||||||||||||||||||
1.6 | 0 | −8.87446 | 0 | 18.9783 | 0 | −28.1514 | 0 | 51.7560 | 0 | ||||||||||||||||||
1.7 | 0 | −8.24401 | 0 | −1.55575 | 0 | −34.0369 | 0 | 40.9638 | 0 | ||||||||||||||||||
1.8 | 0 | −7.41853 | 0 | 20.3771 | 0 | −19.4363 | 0 | 28.0346 | 0 | ||||||||||||||||||
1.9 | 0 | −6.82298 | 0 | −6.53711 | 0 | 25.6622 | 0 | 19.5530 | 0 | ||||||||||||||||||
1.10 | 0 | −6.36012 | 0 | −4.50955 | 0 | 33.5053 | 0 | 13.4511 | 0 | ||||||||||||||||||
1.11 | 0 | −5.72839 | 0 | 2.64954 | 0 | −7.10776 | 0 | 5.81447 | 0 | ||||||||||||||||||
1.12 | 0 | −5.67915 | 0 | −18.9564 | 0 | −22.1236 | 0 | 5.25275 | 0 | ||||||||||||||||||
1.13 | 0 | −5.21628 | 0 | 8.24798 | 0 | −5.99529 | 0 | 0.209617 | 0 | ||||||||||||||||||
1.14 | 0 | −5.14880 | 0 | −17.2771 | 0 | −32.9456 | 0 | −0.489833 | 0 | ||||||||||||||||||
1.15 | 0 | −4.81107 | 0 | −12.5553 | 0 | 7.59971 | 0 | −3.85362 | 0 | ||||||||||||||||||
1.16 | 0 | −4.13499 | 0 | −12.0479 | 0 | 2.03666 | 0 | −9.90185 | 0 | ||||||||||||||||||
1.17 | 0 | −3.99286 | 0 | 10.3257 | 0 | −21.1888 | 0 | −11.0570 | 0 | ||||||||||||||||||
1.18 | 0 | −3.56612 | 0 | −9.75482 | 0 | −29.2365 | 0 | −14.2828 | 0 | ||||||||||||||||||
1.19 | 0 | −3.12416 | 0 | 16.2839 | 0 | 22.5023 | 0 | −17.2396 | 0 | ||||||||||||||||||
1.20 | 0 | −3.03774 | 0 | 15.0797 | 0 | 11.9749 | 0 | −17.7721 | 0 | ||||||||||||||||||
See all 50 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 | 2008.4.a.c | ✓ | 50 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
2008.4.a.c | ✓ | 50 | 1.a | even | 1 | 1 | trivial |