Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [6001,2,Mod(1,6001)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(6001, 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("6001.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 6001 = 17 \cdot 353 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 6001.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: | \(47.9182262530\) |
Analytic rank: | \(1\) |
Dimension: | \(114\) |
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 | −2.79736 | 1.48474 | 5.82522 | 0.639507 | −4.15334 | 0.101329 | −10.7005 | −0.795561 | −1.78893 | ||||||||||||||||||
1.2 | −2.75915 | 3.04798 | 5.61289 | −2.57270 | −8.40983 | −0.275912 | −9.96848 | 6.29019 | 7.09847 | ||||||||||||||||||
1.3 | −2.75487 | −1.38395 | 5.58934 | 1.31563 | 3.81261 | −4.81832 | −9.88817 | −1.08468 | −3.62439 | ||||||||||||||||||
1.4 | −2.73023 | −3.10574 | 5.45415 | −3.71411 | 8.47938 | 1.88828 | −9.43061 | 6.64562 | 10.1404 | ||||||||||||||||||
1.5 | −2.69250 | 0.578947 | 5.24958 | 4.35539 | −1.55882 | −1.96736 | −8.74952 | −2.66482 | −11.7269 | ||||||||||||||||||
1.6 | −2.68479 | 0.238039 | 5.20809 | −3.86690 | −0.639084 | 2.11314 | −8.61305 | −2.94334 | 10.3818 | ||||||||||||||||||
1.7 | −2.61975 | −2.31872 | 4.86309 | 0.216177 | 6.07447 | 3.38406 | −7.50057 | 2.37647 | −0.566328 | ||||||||||||||||||
1.8 | −2.56595 | 1.01181 | 4.58410 | 2.09874 | −2.59626 | 3.20271 | −6.63067 | −1.97623 | −5.38525 | ||||||||||||||||||
1.9 | −2.39336 | 0.882221 | 3.72817 | 0.123030 | −2.11147 | −1.77301 | −4.13613 | −2.22169 | −0.294456 | ||||||||||||||||||
1.10 | −2.39289 | −2.96659 | 3.72594 | 3.31044 | 7.09873 | 0.664206 | −4.12999 | 5.80065 | −7.92153 | ||||||||||||||||||
1.11 | −2.38632 | −0.731332 | 3.69453 | 1.80073 | 1.74519 | −2.93364 | −4.04370 | −2.46515 | −4.29713 | ||||||||||||||||||
1.12 | −2.36622 | −1.16111 | 3.59897 | −3.78541 | 2.74743 | −1.24944 | −3.78352 | −1.65183 | 8.95709 | ||||||||||||||||||
1.13 | −2.31496 | −0.975130 | 3.35906 | −3.23089 | 2.25739 | 5.09566 | −3.14618 | −2.04912 | 7.47940 | ||||||||||||||||||
1.14 | −2.31236 | 3.26187 | 3.34699 | −1.15202 | −7.54261 | −3.58394 | −3.11471 | 7.63982 | 2.66388 | ||||||||||||||||||
1.15 | −2.29894 | 0.676824 | 3.28511 | −3.14176 | −1.55598 | −3.61058 | −2.95439 | −2.54191 | 7.22271 | ||||||||||||||||||
1.16 | −2.28982 | −3.10307 | 3.24328 | 2.16467 | 7.10548 | −1.63956 | −2.84688 | 6.62906 | −4.95671 | ||||||||||||||||||
1.17 | −2.18122 | 1.64876 | 2.75771 | −2.64324 | −3.59630 | −4.67385 | −1.65274 | −0.281605 | 5.76549 | ||||||||||||||||||
1.18 | −2.12375 | 2.77384 | 2.51031 | −3.45627 | −5.89093 | 3.25214 | −1.08377 | 4.69417 | 7.34024 | ||||||||||||||||||
1.19 | −2.10824 | −2.66682 | 2.44469 | −3.46040 | 5.62230 | −1.09952 | −0.937516 | 4.11192 | 7.29537 | ||||||||||||||||||
1.20 | −2.09110 | −1.81594 | 2.37271 | 1.01486 | 3.79732 | 2.98461 | −0.779370 | 0.297648 | −2.12218 | ||||||||||||||||||
See next 80 embeddings (of 114 total) |
Atkin-Lehner signs
\( p \) | Sign |
---|---|
\(17\) | \(-1\) |
\(353\) | \(-1\) |
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 | 6001.2.a.b | ✓ | 114 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
6001.2.a.b | ✓ | 114 | 1.a | even | 1 | 1 | trivial |