Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [289,4,Mod(288,289)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(289, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([1]))
N = Newforms(chi, 4, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("289.288");
S:= CuspForms(chi, 4);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 289 = 17^{2} \) |
Weight: | \( k \) | \(=\) | \( 4 \) |
Character orbit: | \([\chi]\) | \(=\) | 289.b (of order \(2\), degree \(1\), not minimal) |
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: | no |
Analytic conductor: | \(17.0515519917\) |
Analytic rank: | \(0\) |
Dimension: | \(24\) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{2}]$ |
$q$-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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
288.1 | −4.44354 | − | 0.537336i | 11.7450 | − | 17.2592i | 2.38767i | − | 6.40763i | −16.6411 | 26.7113 | 76.6918i | |||||||||||||||
288.2 | −4.44354 | 0.537336i | 11.7450 | 17.2592i | − | 2.38767i | 6.40763i | −16.6411 | 26.7113 | − | 76.6918i | ||||||||||||||||
288.3 | −4.42326 | 9.44971i | 11.5652 | − | 8.63042i | − | 41.7985i | − | 12.6513i | −15.7698 | −62.2970 | 38.1746i | |||||||||||||||
288.4 | −4.42326 | − | 9.44971i | 11.5652 | 8.63042i | 41.7985i | 12.6513i | −15.7698 | −62.2970 | − | 38.1746i | ||||||||||||||||
288.5 | −4.28857 | − | 2.85039i | 10.3918 | 6.36629i | 12.2241i | 29.4308i | −10.2575 | 18.8753 | − | 27.3023i | ||||||||||||||||
288.6 | −4.28857 | 2.85039i | 10.3918 | − | 6.36629i | − | 12.2241i | − | 29.4308i | −10.2575 | 18.8753 | 27.3023i | |||||||||||||||
288.7 | −3.25752 | − | 6.51757i | 2.61146 | 19.1073i | 21.2311i | − | 22.0995i | 17.5533 | −15.4787 | − | 62.2426i | |||||||||||||||
288.8 | −3.25752 | 6.51757i | 2.61146 | − | 19.1073i | − | 21.2311i | 22.0995i | 17.5533 | −15.4787 | 62.2426i | ||||||||||||||||
288.9 | −1.26162 | 6.09538i | −6.40830 | − | 13.2627i | − | 7.69007i | − | 27.2593i | 18.1779 | −10.1536 | 16.7325i | |||||||||||||||
288.10 | −1.26162 | − | 6.09538i | −6.40830 | 13.2627i | 7.69007i | 27.2593i | 18.1779 | −10.1536 | − | 16.7325i | ||||||||||||||||
288.11 | −0.447590 | 9.51519i | −7.79966 | − | 11.7425i | − | 4.25890i | − | 2.27797i | 7.07177 | −63.5388 | 5.25584i | |||||||||||||||
288.12 | −0.447590 | − | 9.51519i | −7.79966 | 11.7425i | 4.25890i | 2.27797i | 7.07177 | −63.5388 | − | 5.25584i | ||||||||||||||||
288.13 | 0.820352 | 0.130026i | −7.32702 | 8.70710i | 0.106667i | − | 11.4024i | −12.5736 | 26.9831 | 7.14288i | |||||||||||||||||
288.14 | 0.820352 | − | 0.130026i | −7.32702 | − | 8.70710i | − | 0.106667i | 11.4024i | −12.5736 | 26.9831 | − | 7.14288i | ||||||||||||||
288.15 | 1.07125 | 5.02012i | −6.85243 | − | 4.32398i | 5.37778i | 4.44673i | −15.9106 | 1.79838 | − | 4.63204i | ||||||||||||||||
288.16 | 1.07125 | − | 5.02012i | −6.85243 | 4.32398i | − | 5.37778i | − | 4.44673i | −15.9106 | 1.79838 | 4.63204i | |||||||||||||||
288.17 | 2.16473 | − | 9.14133i | −3.31395 | − | 16.2298i | − | 19.7885i | 12.2246i | −24.4916 | −56.5639 | − | 35.1331i | ||||||||||||||
288.18 | 2.16473 | 9.14133i | −3.31395 | 16.2298i | 19.7885i | − | 12.2246i | −24.4916 | −56.5639 | 35.1331i | |||||||||||||||||
288.19 | 3.73276 | − | 3.65511i | 5.93351 | − | 14.5154i | − | 13.6437i | − | 12.9694i | −7.71372 | 13.6402 | − | 54.1823i | |||||||||||||
288.20 | 3.73276 | 3.65511i | 5.93351 | 14.5154i | 13.6437i | 12.9694i | −7.71372 | 13.6402 | 54.1823i | ||||||||||||||||||
See all 24 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
17.b | even | 2 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 289.4.b.f | 24 | |
17.b | even | 2 | 1 | inner | 289.4.b.f | 24 | |
17.c | even | 4 | 1 | 289.4.a.h | ✓ | 12 | |
17.c | even | 4 | 1 | 289.4.a.i | yes | 12 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
289.4.a.h | ✓ | 12 | 17.c | even | 4 | 1 | |
289.4.a.i | yes | 12 | 17.c | even | 4 | 1 | |
289.4.b.f | 24 | 1.a | even | 1 | 1 | trivial | |
289.4.b.f | 24 | 17.b | even | 2 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{12} - 72 T_{2}^{10} - 17 T_{2}^{9} + 1872 T_{2}^{8} + 627 T_{2}^{7} - 20922 T_{2}^{6} - 5163 T_{2}^{5} + 93255 T_{2}^{4} - 4607 T_{2}^{3} - 117822 T_{2}^{2} + 21960 T_{2} + 29352 \)
acting on \(S_{4}^{\mathrm{new}}(289, [\chi])\).