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: | \(113\) |
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 | −2.78095 | −2.69588 | 5.73371 | 2.30800 | 7.49712 | 4.50681 | −10.3833 | 4.26777 | −6.41845 | ||||||||||||||||||
1.2 | −2.76448 | −0.301476 | 5.64235 | −1.88471 | 0.833425 | 2.57260 | −10.0692 | −2.90911 | 5.21024 | ||||||||||||||||||
1.3 | −2.72078 | −0.920111 | 5.40267 | −2.98260 | 2.50342 | −1.43517 | −9.25792 | −2.15340 | 8.11501 | ||||||||||||||||||
1.4 | −2.69808 | 2.45523 | 5.27961 | −1.37053 | −6.62441 | 5.08280 | −8.84865 | 3.02818 | 3.69780 | ||||||||||||||||||
1.5 | −2.64384 | −0.876989 | 4.98991 | 2.40505 | 2.31862 | 1.36446 | −7.90487 | −2.23089 | −6.35858 | ||||||||||||||||||
1.6 | −2.63611 | 2.68277 | 4.94905 | −3.99469 | −7.07208 | −3.60242 | −7.77401 | 4.19728 | 10.5304 | ||||||||||||||||||
1.7 | −2.63080 | 1.97535 | 4.92113 | 2.13953 | −5.19675 | 1.65541 | −7.68492 | 0.902000 | −5.62867 | ||||||||||||||||||
1.8 | −2.58651 | −0.844986 | 4.69006 | −0.771482 | 2.18557 | −2.98561 | −6.95787 | −2.28600 | 1.99545 | ||||||||||||||||||
1.9 | −2.50694 | 0.525806 | 4.28474 | −1.87738 | −1.31816 | 4.48031 | −5.72770 | −2.72353 | 4.70647 | ||||||||||||||||||
1.10 | −2.49624 | 1.48745 | 4.23122 | −3.58200 | −3.71302 | 1.79069 | −5.56966 | −0.787505 | 8.94153 | ||||||||||||||||||
1.11 | −2.49146 | −2.38691 | 4.20738 | −2.02282 | 5.94689 | 1.32918 | −5.49960 | 2.69734 | 5.03978 | ||||||||||||||||||
1.12 | −2.49070 | 2.79631 | 4.20359 | 2.44678 | −6.96477 | −1.91297 | −5.48850 | 4.81934 | −6.09421 | ||||||||||||||||||
1.13 | −2.30912 | −2.56184 | 3.33205 | −3.64906 | 5.91560 | −1.49745 | −3.07586 | 3.56301 | 8.42614 | ||||||||||||||||||
1.14 | −2.26295 | −1.98147 | 3.12095 | 3.45141 | 4.48397 | −1.45595 | −2.53665 | 0.926221 | −7.81036 | ||||||||||||||||||
1.15 | −2.22225 | −0.120604 | 2.93840 | 1.74504 | 0.268012 | −3.61581 | −2.08537 | −2.98545 | −3.87792 | ||||||||||||||||||
1.16 | −2.20707 | 2.35834 | 2.87116 | 2.79748 | −5.20501 | −1.23289 | −1.92271 | 2.56174 | −6.17423 | ||||||||||||||||||
1.17 | −2.18121 | 0.0139647 | 2.75766 | −0.236603 | −0.0304599 | 2.78463 | −1.65261 | −2.99980 | 0.516079 | ||||||||||||||||||
1.18 | −2.15299 | 2.37258 | 2.63537 | −0.231510 | −5.10813 | 3.32403 | −1.36794 | 2.62912 | 0.498439 | ||||||||||||||||||
1.19 | −2.14256 | −2.11238 | 2.59055 | 1.25903 | 4.52589 | 1.44603 | −1.26528 | 1.46215 | −2.69753 | ||||||||||||||||||
1.20 | −2.09226 | 2.92082 | 2.37755 | −1.12068 | −6.11112 | −1.50686 | −0.789928 | 5.53120 | 2.34475 | ||||||||||||||||||
See next 80 embeddings (of 113 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.a | ✓ | 113 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
6001.2.a.a | ✓ | 113 | 1.a | even | 1 | 1 | trivial |