Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [8042,2,Mod(1,8042)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(8042, 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("8042.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 8042 = 2 \cdot 4021 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 8042.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: | \(64.2156933055\) |
Analytic rank: | \(1\) |
Dimension: | \(82\) |
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 | −3.40861 | 1.00000 | −1.73046 | 3.40861 | −5.08691 | −1.00000 | 8.61865 | 1.73046 | ||||||||||||||||||
1.2 | −1.00000 | −3.35064 | 1.00000 | 3.93349 | 3.35064 | 2.37767 | −1.00000 | 8.22677 | −3.93349 | ||||||||||||||||||
1.3 | −1.00000 | −3.32799 | 1.00000 | −3.39172 | 3.32799 | 1.47014 | −1.00000 | 8.07554 | 3.39172 | ||||||||||||||||||
1.4 | −1.00000 | −3.26666 | 1.00000 | 1.36820 | 3.26666 | −2.10777 | −1.00000 | 7.67104 | −1.36820 | ||||||||||||||||||
1.5 | −1.00000 | −3.16571 | 1.00000 | −2.70471 | 3.16571 | 0.275737 | −1.00000 | 7.02173 | 2.70471 | ||||||||||||||||||
1.6 | −1.00000 | −3.05101 | 1.00000 | −2.07881 | 3.05101 | −1.72566 | −1.00000 | 6.30864 | 2.07881 | ||||||||||||||||||
1.7 | −1.00000 | −2.97766 | 1.00000 | 2.45317 | 2.97766 | −2.95290 | −1.00000 | 5.86643 | −2.45317 | ||||||||||||||||||
1.8 | −1.00000 | −2.93784 | 1.00000 | 2.35767 | 2.93784 | −3.95734 | −1.00000 | 5.63089 | −2.35767 | ||||||||||||||||||
1.9 | −1.00000 | −2.89017 | 1.00000 | 1.86034 | 2.89017 | −1.49646 | −1.00000 | 5.35311 | −1.86034 | ||||||||||||||||||
1.10 | −1.00000 | −2.84903 | 1.00000 | −1.49595 | 2.84903 | −3.40125 | −1.00000 | 5.11699 | 1.49595 | ||||||||||||||||||
1.11 | −1.00000 | −2.83011 | 1.00000 | −3.27582 | 2.83011 | 1.55181 | −1.00000 | 5.00950 | 3.27582 | ||||||||||||||||||
1.12 | −1.00000 | −2.79380 | 1.00000 | −0.886780 | 2.79380 | 2.62201 | −1.00000 | 4.80532 | 0.886780 | ||||||||||||||||||
1.13 | −1.00000 | −2.65806 | 1.00000 | 4.11943 | 2.65806 | −4.31253 | −1.00000 | 4.06529 | −4.11943 | ||||||||||||||||||
1.14 | −1.00000 | −2.65492 | 1.00000 | 1.59716 | 2.65492 | 4.41180 | −1.00000 | 4.04858 | −1.59716 | ||||||||||||||||||
1.15 | −1.00000 | −2.63165 | 1.00000 | 1.97422 | 2.63165 | 0.692516 | −1.00000 | 3.92558 | −1.97422 | ||||||||||||||||||
1.16 | −1.00000 | −2.37311 | 1.00000 | 2.96553 | 2.37311 | 1.19055 | −1.00000 | 2.63165 | −2.96553 | ||||||||||||||||||
1.17 | −1.00000 | −2.27252 | 1.00000 | 1.18592 | 2.27252 | 3.11806 | −1.00000 | 2.16434 | −1.18592 | ||||||||||||||||||
1.18 | −1.00000 | −2.21914 | 1.00000 | −2.28006 | 2.21914 | 5.09322 | −1.00000 | 1.92459 | 2.28006 | ||||||||||||||||||
1.19 | −1.00000 | −1.99235 | 1.00000 | −2.03322 | 1.99235 | −0.500887 | −1.00000 | 0.969445 | 2.03322 | ||||||||||||||||||
1.20 | −1.00000 | −1.88117 | 1.00000 | 0.290138 | 1.88117 | 2.48398 | −1.00000 | 0.538819 | −0.290138 | ||||||||||||||||||
See all 82 embeddings |
Atkin-Lehner signs
\( p \) | Sign |
---|---|
\(2\) | \(1\) |
\(4021\) | \(1\) |
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 | 8042.2.a.b | ✓ | 82 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
8042.2.a.b | ✓ | 82 | 1.a | even | 1 | 1 | trivial |