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: | \(33\) |
Twist minimal: | no (minimal twist has level 241) |
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 | −5.50714 | 0 | 22.3286 | 4.56136 | 0 | 16.8108 | −78.9099 | 0 | −25.1200 | ||||||||||||||||||
1.2 | −5.34167 | 0 | 20.5335 | −15.6482 | 0 | −35.3555 | −66.9498 | 0 | 83.5875 | ||||||||||||||||||
1.3 | −5.14541 | 0 | 18.4753 | −9.13578 | 0 | 11.4940 | −53.8997 | 0 | 47.0074 | ||||||||||||||||||
1.4 | −5.06398 | 0 | 17.6439 | 7.53582 | 0 | −4.42337 | −48.8363 | 0 | −38.1612 | ||||||||||||||||||
1.5 | −4.68737 | 0 | 13.9714 | 20.5680 | 0 | 17.2543 | −27.9902 | 0 | −96.4100 | ||||||||||||||||||
1.6 | −4.50393 | 0 | 12.2854 | −16.5922 | 0 | 27.9465 | −19.3010 | 0 | 74.7302 | ||||||||||||||||||
1.7 | −4.28480 | 0 | 10.3595 | −5.61150 | 0 | −3.11768 | −10.1099 | 0 | 24.0441 | ||||||||||||||||||
1.8 | −4.22357 | 0 | 9.83857 | −18.6441 | 0 | −0.526916 | −7.76535 | 0 | 78.7445 | ||||||||||||||||||
1.9 | −4.21161 | 0 | 9.73762 | 10.4748 | 0 | 24.1777 | −7.31816 | 0 | −44.1159 | ||||||||||||||||||
1.10 | −2.89955 | 0 | 0.407364 | −15.7141 | 0 | −3.82589 | 22.0152 | 0 | 45.5639 | ||||||||||||||||||
1.11 | −2.52337 | 0 | −1.63263 | 2.94898 | 0 | −33.1605 | 24.3066 | 0 | −7.44136 | ||||||||||||||||||
1.12 | −2.45382 | 0 | −1.97876 | 9.95086 | 0 | 24.2976 | 24.4861 | 0 | −24.4176 | ||||||||||||||||||
1.13 | −1.89779 | 0 | −4.39839 | −10.1327 | 0 | 24.3749 | 23.5296 | 0 | 19.2298 | ||||||||||||||||||
1.14 | −1.71404 | 0 | −5.06206 | 2.30722 | 0 | 0.735682 | 22.3889 | 0 | −3.95467 | ||||||||||||||||||
1.15 | −0.975987 | 0 | −7.04745 | 19.7613 | 0 | −7.10811 | 14.6861 | 0 | −19.2868 | ||||||||||||||||||
1.16 | −0.664281 | 0 | −7.55873 | −16.8228 | 0 | 18.5044 | 10.3354 | 0 | 11.1751 | ||||||||||||||||||
1.17 | −0.270698 | 0 | −7.92672 | 11.6718 | 0 | 13.4994 | 4.31133 | 0 | −3.15954 | ||||||||||||||||||
1.18 | −0.0980206 | 0 | −7.99039 | 13.5070 | 0 | −15.0904 | 1.56739 | 0 | −1.32396 | ||||||||||||||||||
1.19 | 0.139152 | 0 | −7.98064 | −14.6080 | 0 | −24.3125 | −2.22374 | 0 | −2.03273 | ||||||||||||||||||
1.20 | 0.788730 | 0 | −7.37790 | 5.19898 | 0 | −27.1369 | −12.1290 | 0 | 4.10059 | ||||||||||||||||||
See all 33 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.e | 33 | |
3.b | odd | 2 | 1 | 241.4.a.b | ✓ | 33 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
241.4.a.b | ✓ | 33 | 3.b | odd | 2 | 1 | |
2169.4.a.e | 33 | 1.a | even | 1 | 1 | trivial |