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: | \(0\) |
Dimension: | \(121\) |
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 | −2.77548 | −0.417524 | 5.70327 | 0.239845 | 1.15883 | 2.35024 | −10.2783 | −2.82567 | −0.665684 | ||||||||||||||||||
1.2 | −2.70572 | −1.68319 | 5.32093 | 1.67453 | 4.55424 | 1.84178 | −8.98553 | −0.166876 | −4.53081 | ||||||||||||||||||
1.3 | −2.68420 | 3.02151 | 5.20492 | 3.70816 | −8.11034 | 3.11131 | −8.60266 | 6.12953 | −9.95344 | ||||||||||||||||||
1.4 | −2.66371 | −2.14217 | 5.09536 | −3.47946 | 5.70612 | −4.33919 | −8.24515 | 1.58889 | 9.26828 | ||||||||||||||||||
1.5 | −2.63308 | 2.70578 | 4.93313 | 2.79051 | −7.12456 | −5.09743 | −7.72317 | 4.32127 | −7.34766 | ||||||||||||||||||
1.6 | −2.58868 | −1.67883 | 4.70126 | −1.30271 | 4.34594 | −1.36507 | −6.99268 | −0.181538 | 3.37229 | ||||||||||||||||||
1.7 | −2.53489 | 1.89170 | 4.42568 | −3.70427 | −4.79525 | 1.26299 | −6.14885 | 0.578524 | 9.38992 | ||||||||||||||||||
1.8 | −2.51719 | 0.226046 | 4.33623 | −1.31460 | −0.569001 | −1.40872 | −5.88074 | −2.94890 | 3.30910 | ||||||||||||||||||
1.9 | −2.51330 | −2.00286 | 4.31669 | 3.06325 | 5.03379 | 0.908464 | −5.82253 | 1.01144 | −7.69887 | ||||||||||||||||||
1.10 | −2.48199 | 1.43528 | 4.16025 | 0.277666 | −3.56234 | 3.77920 | −5.36171 | −0.939973 | −0.689164 | ||||||||||||||||||
1.11 | −2.43959 | 2.16595 | 3.95162 | −1.55175 | −5.28403 | −2.38051 | −4.76117 | 1.69132 | 3.78564 | ||||||||||||||||||
1.12 | −2.40063 | 1.09881 | 3.76301 | 3.50161 | −2.63784 | −0.487426 | −4.23233 | −1.79261 | −8.40605 | ||||||||||||||||||
1.13 | −2.26922 | −1.02425 | 3.14937 | −0.261054 | 2.32424 | 2.61024 | −2.60816 | −1.95092 | 0.592389 | ||||||||||||||||||
1.14 | −2.25245 | −3.40363 | 3.07353 | −2.33953 | 7.66650 | 2.81637 | −2.41808 | 8.58466 | 5.26966 | ||||||||||||||||||
1.15 | −2.24229 | 2.77352 | 3.02787 | 0.258648 | −6.21903 | 3.59133 | −2.30477 | 4.69240 | −0.579964 | ||||||||||||||||||
1.16 | −2.20655 | 2.67875 | 2.86888 | −1.00223 | −5.91080 | −0.364832 | −1.91723 | 4.17568 | 2.21147 | ||||||||||||||||||
1.17 | −2.15090 | 0.00192410 | 2.62636 | −2.18392 | −0.00413853 | −0.0660171 | −1.34723 | −3.00000 | 4.69738 | ||||||||||||||||||
1.18 | −2.15009 | −0.229401 | 2.62287 | 4.13461 | 0.493233 | −0.301576 | −1.33923 | −2.94737 | −8.88976 | ||||||||||||||||||
1.19 | −2.08098 | −2.15379 | 2.33048 | 2.78537 | 4.48200 | −3.81134 | −0.687730 | 1.63882 | −5.79629 | ||||||||||||||||||
1.20 | −1.98791 | 3.32266 | 1.95179 | 2.27659 | −6.60516 | 3.45158 | 0.0958417 | 8.04010 | −4.52565 | ||||||||||||||||||
See next 80 embeddings (of 121 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.d | ✓ | 121 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
6001.2.a.d | ✓ | 121 | 1.a | even | 1 | 1 | trivial |