Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [2169,4,Mod(1,2169)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2169, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0]))
N = Newforms(chi, 4, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("2169.1");
S:= CuspForms(chi, 4);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 2169 = 3^{2} \cdot 241 \) |
Weight: | \( k \) | \(=\) | \( 4 \) |
Character orbit: | \([\chi]\) | \(=\) | 2169.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: | \(127.975142802\) |
Analytic rank: | \(1\) |
Dimension: | \(29\) |
Twist minimal: | no (minimal twist has level 723) |
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 | −5.53188 | 0 | 22.6017 | 4.41322 | 0 | −24.6542 | −80.7747 | 0 | −24.4134 | ||||||||||||||||||
1.2 | −5.40987 | 0 | 21.2667 | −19.5426 | 0 | 5.91074 | −71.7711 | 0 | 105.723 | ||||||||||||||||||
1.3 | −5.00185 | 0 | 17.0185 | −7.18086 | 0 | 2.32627 | −45.1091 | 0 | 35.9176 | ||||||||||||||||||
1.4 | −4.92480 | 0 | 16.2537 | −0.995320 | 0 | 35.5796 | −40.6479 | 0 | 4.90176 | ||||||||||||||||||
1.5 | −4.04552 | 0 | 8.36624 | 17.6002 | 0 | −8.19403 | −1.48162 | 0 | −71.2021 | ||||||||||||||||||
1.6 | −3.42298 | 0 | 3.71677 | −8.82688 | 0 | −24.4656 | 14.6614 | 0 | 30.2142 | ||||||||||||||||||
1.7 | −3.42059 | 0 | 3.70041 | −14.1333 | 0 | 16.0369 | 14.7071 | 0 | 48.3443 | ||||||||||||||||||
1.8 | −3.34560 | 0 | 3.19303 | −17.6990 | 0 | 18.2648 | 16.0822 | 0 | 59.2136 | ||||||||||||||||||
1.9 | −2.93958 | 0 | 0.641136 | 15.8431 | 0 | −32.0159 | 21.6320 | 0 | −46.5721 | ||||||||||||||||||
1.10 | −2.75926 | 0 | −0.386494 | 13.3161 | 0 | −12.5857 | 23.1405 | 0 | −36.7427 | ||||||||||||||||||
1.11 | −2.61315 | 0 | −1.17143 | 6.42911 | 0 | 15.9906 | 23.9664 | 0 | −16.8002 | ||||||||||||||||||
1.12 | −1.18901 | 0 | −6.58626 | −19.7501 | 0 | −30.4473 | 17.3432 | 0 | 23.4831 | ||||||||||||||||||
1.13 | −1.17966 | 0 | −6.60840 | 5.20729 | 0 | 19.7482 | 17.2329 | 0 | −6.14283 | ||||||||||||||||||
1.14 | −1.14345 | 0 | −6.69252 | 0.465287 | 0 | −21.2063 | 16.8002 | 0 | −0.532033 | ||||||||||||||||||
1.15 | −1.12035 | 0 | −6.74482 | −13.1582 | 0 | 29.4550 | 16.5193 | 0 | 14.7417 | ||||||||||||||||||
1.16 | 0.364523 | 0 | −7.86712 | −5.79603 | 0 | 11.9832 | −5.78393 | 0 | −2.11279 | ||||||||||||||||||
1.17 | 1.16445 | 0 | −6.64406 | 10.5306 | 0 | 9.63072 | −17.0522 | 0 | 12.2623 | ||||||||||||||||||
1.18 | 1.28359 | 0 | −6.35240 | −0.508995 | 0 | −35.7971 | −18.4226 | 0 | −0.653341 | ||||||||||||||||||
1.19 | 1.68254 | 0 | −5.16907 | −2.08567 | 0 | 30.8603 | −22.1574 | 0 | −3.50922 | ||||||||||||||||||
1.20 | 1.68847 | 0 | −5.14906 | −22.1888 | 0 | −11.1315 | −22.2018 | 0 | −37.4651 | ||||||||||||||||||
See all 29 embeddings |
Atkin-Lehner signs
\( p \) | Sign |
---|---|
\(3\) | \(-1\) |
\(241\) | \(1\) |
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 | 2169.4.a.c | 29 | |
3.b | odd | 2 | 1 | 723.4.a.b | ✓ | 29 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
723.4.a.b | ✓ | 29 | 3.b | odd | 2 | 1 | |
2169.4.a.c | 29 | 1.a | even | 1 | 1 | trivial |