Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [8038,2,Mod(1,8038)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(8038, 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("8038.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 8038 = 2 \cdot 4019 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 8038.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: | \(64.1837531447\) |
Analytic rank: | \(1\) |
Dimension: | \(84\) |
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 | −3.36115 | 1.00000 | −3.51543 | 3.36115 | −3.19269 | −1.00000 | 8.29735 | 3.51543 | ||||||||||||||||||
1.2 | −1.00000 | −3.35998 | 1.00000 | 2.98972 | 3.35998 | 3.64714 | −1.00000 | 8.28947 | −2.98972 | ||||||||||||||||||
1.3 | −1.00000 | −3.33190 | 1.00000 | −1.90041 | 3.33190 | −1.90946 | −1.00000 | 8.10156 | 1.90041 | ||||||||||||||||||
1.4 | −1.00000 | −3.26564 | 1.00000 | 1.02149 | 3.26564 | −2.73396 | −1.00000 | 7.66439 | −1.02149 | ||||||||||||||||||
1.5 | −1.00000 | −3.22723 | 1.00000 | −4.01107 | 3.22723 | 2.38054 | −1.00000 | 7.41504 | 4.01107 | ||||||||||||||||||
1.6 | −1.00000 | −3.18728 | 1.00000 | −3.96813 | 3.18728 | 3.34102 | −1.00000 | 7.15875 | 3.96813 | ||||||||||||||||||
1.7 | −1.00000 | −2.97502 | 1.00000 | 1.80363 | 2.97502 | −0.809668 | −1.00000 | 5.85072 | −1.80363 | ||||||||||||||||||
1.8 | −1.00000 | −2.90498 | 1.00000 | −2.22164 | 2.90498 | −2.98182 | −1.00000 | 5.43889 | 2.22164 | ||||||||||||||||||
1.9 | −1.00000 | −2.89385 | 1.00000 | 0.388076 | 2.89385 | −0.0680600 | −1.00000 | 5.37435 | −0.388076 | ||||||||||||||||||
1.10 | −1.00000 | −2.71506 | 1.00000 | −2.98033 | 2.71506 | 3.18004 | −1.00000 | 4.37153 | 2.98033 | ||||||||||||||||||
1.11 | −1.00000 | −2.71043 | 1.00000 | 1.06338 | 2.71043 | 1.86356 | −1.00000 | 4.34646 | −1.06338 | ||||||||||||||||||
1.12 | −1.00000 | −2.66947 | 1.00000 | 0.308435 | 2.66947 | −4.20139 | −1.00000 | 4.12607 | −0.308435 | ||||||||||||||||||
1.13 | −1.00000 | −2.63531 | 1.00000 | 2.31488 | 2.63531 | −0.771560 | −1.00000 | 3.94487 | −2.31488 | ||||||||||||||||||
1.14 | −1.00000 | −2.62072 | 1.00000 | −0.614579 | 2.62072 | 2.08468 | −1.00000 | 3.86818 | 0.614579 | ||||||||||||||||||
1.15 | −1.00000 | −2.53354 | 1.00000 | 1.39163 | 2.53354 | 4.71879 | −1.00000 | 3.41884 | −1.39163 | ||||||||||||||||||
1.16 | −1.00000 | −2.39740 | 1.00000 | −1.80214 | 2.39740 | −2.60681 | −1.00000 | 2.74755 | 1.80214 | ||||||||||||||||||
1.17 | −1.00000 | −2.39489 | 1.00000 | 4.12569 | 2.39489 | 3.01891 | −1.00000 | 2.73549 | −4.12569 | ||||||||||||||||||
1.18 | −1.00000 | −2.25732 | 1.00000 | −3.39963 | 2.25732 | 3.01257 | −1.00000 | 2.09550 | 3.39963 | ||||||||||||||||||
1.19 | −1.00000 | −2.22339 | 1.00000 | −0.973252 | 2.22339 | −3.76604 | −1.00000 | 1.94347 | 0.973252 | ||||||||||||||||||
1.20 | −1.00000 | −1.99033 | 1.00000 | −1.73876 | 1.99033 | 5.09180 | −1.00000 | 0.961408 | 1.73876 | ||||||||||||||||||
See all 84 embeddings |
Atkin-Lehner signs
\( p \) | Sign |
---|---|
\(2\) | \(1\) |
\(4019\) | \(1\) |
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 | 8038.2.a.c | ✓ | 84 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
8038.2.a.c | ✓ | 84 | 1.a | even | 1 | 1 | trivial |