Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [8021,2,Mod(1,8021)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(8021, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([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("8021.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 8021 = 13 \cdot 617 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 8021.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: | \(64.0480074613\) |
Analytic rank: | \(1\) |
Dimension: | \(140\) |
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 | −2.74251 | −0.873664 | 5.52139 | −0.639723 | 2.39604 | −2.37570 | −9.65746 | −2.23671 | 1.75445 | ||||||||||||||||||
1.2 | −2.72092 | 2.93492 | 5.40338 | −0.339682 | −7.98568 | −0.916249 | −9.26032 | 5.61377 | 0.924245 | ||||||||||||||||||
1.3 | −2.66546 | 1.20306 | 5.10470 | −3.76588 | −3.20671 | −2.98135 | −8.27547 | −1.55265 | 10.0378 | ||||||||||||||||||
1.4 | −2.62889 | 1.78605 | 4.91107 | 1.43411 | −4.69533 | 3.37960 | −7.65290 | 0.189969 | −3.77012 | ||||||||||||||||||
1.5 | −2.61802 | −0.518785 | 4.85402 | −1.93655 | 1.35819 | 1.44000 | −7.47189 | −2.73086 | 5.06994 | ||||||||||||||||||
1.6 | −2.58382 | −0.446323 | 4.67613 | 3.71919 | 1.15322 | −4.18348 | −6.91464 | −2.80080 | −9.60973 | ||||||||||||||||||
1.7 | −2.55171 | −0.135477 | 4.51121 | −1.86622 | 0.345697 | −2.01089 | −6.40787 | −2.98165 | 4.76206 | ||||||||||||||||||
1.8 | −2.51611 | −0.357802 | 4.33083 | 1.59960 | 0.900270 | 0.472398 | −5.86464 | −2.87198 | −4.02477 | ||||||||||||||||||
1.9 | −2.49455 | −1.79114 | 4.22277 | −0.0823474 | 4.46808 | 3.62956 | −5.54480 | 0.208181 | 0.205419 | ||||||||||||||||||
1.10 | −2.49397 | −2.62257 | 4.21989 | 0.472078 | 6.54061 | 1.90995 | −5.53635 | 3.87786 | −1.17735 | ||||||||||||||||||
1.11 | −2.47091 | 2.70447 | 4.10542 | −1.18660 | −6.68251 | −3.35758 | −5.20231 | 4.31416 | 2.93198 | ||||||||||||||||||
1.12 | −2.46538 | −1.17968 | 4.07807 | 2.88368 | 2.90835 | −1.57366 | −5.12323 | −1.60835 | −7.10935 | ||||||||||||||||||
1.13 | −2.42591 | −2.98941 | 3.88502 | −2.94726 | 7.25203 | −3.20190 | −4.57289 | 5.93657 | 7.14978 | ||||||||||||||||||
1.14 | −2.40289 | 0.734522 | 3.77388 | 0.776538 | −1.76498 | 4.43052 | −4.26243 | −2.46048 | −1.86594 | ||||||||||||||||||
1.15 | −2.35872 | 2.44520 | 3.56358 | 2.52211 | −5.76755 | 2.39189 | −3.68804 | 2.97900 | −5.94896 | ||||||||||||||||||
1.16 | −2.33878 | −3.04534 | 3.46990 | −1.89662 | 7.12240 | 2.91891 | −3.43778 | 6.27412 | 4.43579 | ||||||||||||||||||
1.17 | −2.27005 | 2.30637 | 3.15311 | −4.44364 | −5.23557 | 0.498318 | −2.61762 | 2.31934 | 10.0873 | ||||||||||||||||||
1.18 | −2.16089 | 1.85142 | 2.66943 | −2.55060 | −4.00072 | 2.44468 | −1.44657 | 0.427768 | 5.51156 | ||||||||||||||||||
1.19 | −2.13500 | 1.79448 | 2.55821 | 1.65943 | −3.83121 | −2.70571 | −1.19178 | 0.220161 | −3.54288 | ||||||||||||||||||
1.20 | −2.13485 | −1.44998 | 2.55757 | −1.78016 | 3.09549 | −0.238675 | −1.19032 | −0.897552 | 3.80036 | ||||||||||||||||||
See next 80 embeddings (of 140 total) |
Atkin-Lehner signs
\( p \) | Sign |
---|---|
\(13\) | \(1\) |
\(617\) | \(1\) |
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 | 8021.2.a.b | ✓ | 140 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
8021.2.a.b | ✓ | 140 | 1.a | even | 1 | 1 | trivial |