Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [57,4,Mod(4,57)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(57, base_ring=CyclotomicField(18))
chi = DirichletCharacter(H, H._module([0, 2]))
N = Newforms(chi, 4, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("57.4");
S:= CuspForms(chi, 4);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 57 = 3 \cdot 19 \) |
Weight: | \( k \) | \(=\) | \( 4 \) |
Character orbit: | \([\chi]\) | \(=\) | 57.i (of order \(9\), degree \(6\), 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: | \(3.36310887033\) |
Analytic rank: | \(0\) |
Dimension: | \(36\) |
Relative dimension: | \(6\) over \(\Q(\zeta_{9})\) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{9}]$ |
$q$-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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
4.1 | −4.48747 | − | 1.63331i | −0.520945 | − | 2.95442i | 11.3414 | + | 9.51654i | 13.4789 | − | 11.3101i | −2.48775 | + | 14.1088i | 2.15951 | − | 3.74039i | −16.2488 | − | 28.1437i | −8.45723 | + | 3.07818i | −78.9591 | + | 28.7388i |
4.2 | −3.29747 | − | 1.20018i | −0.520945 | − | 2.95442i | 3.30450 | + | 2.77281i | −11.1815 | + | 9.38240i | −1.82804 | + | 10.3673i | 8.00421 | − | 13.8637i | 6.46774 | + | 11.2025i | −8.45723 | + | 3.07818i | 48.1312 | − | 17.5183i |
4.3 | −0.465112 | − | 0.169287i | −0.520945 | − | 2.95442i | −5.94068 | − | 4.98483i | −5.05805 | + | 4.24421i | −0.257847 | + | 1.46233i | −15.4910 | + | 26.8311i | 3.89906 | + | 6.75337i | −8.45723 | + | 3.07818i | 3.07105 | − | 1.11777i |
4.4 | 1.20418 | + | 0.438285i | −0.520945 | − | 2.95442i | −4.87040 | − | 4.08675i | 1.07347 | − | 0.900745i | 0.667569 | − | 3.78597i | 13.8796 | − | 24.0401i | −9.19951 | − | 15.9340i | −8.45723 | + | 3.07818i | 1.68743 | − | 0.614173i |
4.5 | 3.52088 | + | 1.28150i | −0.520945 | − | 2.95442i | 4.62603 | + | 3.88170i | 16.1205 | − | 13.5267i | 1.95190 | − | 11.0698i | −13.6365 | + | 23.6191i | −3.67406 | − | 6.36365i | −8.45723 | + | 3.07818i | 74.0930 | − | 26.9677i |
4.6 | 4.96468 | + | 1.80700i | −0.520945 | − | 2.95442i | 15.2545 | + | 12.8000i | −7.48978 | + | 6.28467i | 2.75231 | − | 15.6091i | 8.01316 | − | 13.8792i | 31.4708 | + | 54.5091i | −8.45723 | + | 3.07818i | −48.5407 | + | 17.6674i |
16.1 | −3.78921 | − | 3.17953i | 2.81908 | − | 1.02606i | 2.85956 | + | 16.2174i | 3.37491 | − | 19.1401i | −13.9445 | − | 5.07537i | −4.62456 | − | 8.00998i | 20.9422 | − | 36.2730i | 6.89440 | − | 5.78509i | −73.6446 | + | 61.7951i |
16.2 | −3.15086 | − | 2.64388i | 2.81908 | − | 1.02606i | 1.54860 | + | 8.78252i | −3.48122 | + | 19.7430i | −11.5953 | − | 4.22034i | 2.91460 | + | 5.04824i | 1.88795 | − | 3.27003i | 6.89440 | − | 5.78509i | 63.1670 | − | 53.0034i |
16.3 | −1.08875 | − | 0.913570i | 2.81908 | − | 1.02606i | −1.03842 | − | 5.88916i | 0.0582200 | − | 0.330182i | −4.00665 | − | 1.45830i | −7.44302 | − | 12.8917i | −9.93464 | + | 17.2073i | 6.89440 | − | 5.78509i | −0.365031 | + | 0.306298i |
16.4 | 1.05960 | + | 0.889114i | 2.81908 | − | 1.02606i | −1.05695 | − | 5.99424i | 1.90436 | − | 10.8002i | 3.89939 | + | 1.41926i | 1.68783 | + | 2.92340i | 9.74248 | − | 16.8745i | 6.89440 | − | 5.78509i | 11.6205 | − | 9.75073i |
16.5 | 2.57889 | + | 2.16395i | 2.81908 | − | 1.02606i | 0.578834 | + | 3.28273i | −1.62337 | + | 9.20660i | 9.49045 | + | 3.45424i | 10.2044 | + | 17.6746i | 7.85512 | − | 13.6055i | 6.89440 | − | 5.78509i | −24.1091 | + | 20.2299i |
16.6 | 4.12428 | + | 3.46068i | 2.81908 | − | 1.02606i | 3.64417 | + | 20.6671i | 0.929607 | − | 5.27206i | 15.1775 | + | 5.52417i | −18.0524 | − | 31.2677i | −34.9573 | + | 60.5478i | 6.89440 | − | 5.78509i | 22.0789 | − | 18.5264i |
25.1 | −3.78921 | + | 3.17953i | 2.81908 | + | 1.02606i | 2.85956 | − | 16.2174i | 3.37491 | + | 19.1401i | −13.9445 | + | 5.07537i | −4.62456 | + | 8.00998i | 20.9422 | + | 36.2730i | 6.89440 | + | 5.78509i | −73.6446 | − | 61.7951i |
25.2 | −3.15086 | + | 2.64388i | 2.81908 | + | 1.02606i | 1.54860 | − | 8.78252i | −3.48122 | − | 19.7430i | −11.5953 | + | 4.22034i | 2.91460 | − | 5.04824i | 1.88795 | + | 3.27003i | 6.89440 | + | 5.78509i | 63.1670 | + | 53.0034i |
25.3 | −1.08875 | + | 0.913570i | 2.81908 | + | 1.02606i | −1.03842 | + | 5.88916i | 0.0582200 | + | 0.330182i | −4.00665 | + | 1.45830i | −7.44302 | + | 12.8917i | −9.93464 | − | 17.2073i | 6.89440 | + | 5.78509i | −0.365031 | − | 0.306298i |
25.4 | 1.05960 | − | 0.889114i | 2.81908 | + | 1.02606i | −1.05695 | + | 5.99424i | 1.90436 | + | 10.8002i | 3.89939 | − | 1.41926i | 1.68783 | − | 2.92340i | 9.74248 | + | 16.8745i | 6.89440 | + | 5.78509i | 11.6205 | + | 9.75073i |
25.5 | 2.57889 | − | 2.16395i | 2.81908 | + | 1.02606i | 0.578834 | − | 3.28273i | −1.62337 | − | 9.20660i | 9.49045 | − | 3.45424i | 10.2044 | − | 17.6746i | 7.85512 | + | 13.6055i | 6.89440 | + | 5.78509i | −24.1091 | − | 20.2299i |
25.6 | 4.12428 | − | 3.46068i | 2.81908 | + | 1.02606i | 3.64417 | − | 20.6671i | 0.929607 | + | 5.27206i | 15.1775 | − | 5.52417i | −18.0524 | + | 31.2677i | −34.9573 | − | 60.5478i | 6.89440 | + | 5.78509i | 22.0789 | + | 18.5264i |
28.1 | −0.810730 | − | 4.59788i | −2.29813 | + | 1.92836i | −12.9657 | + | 4.71912i | −4.59346 | − | 1.67188i | 10.7295 | + | 9.00316i | −13.4140 | + | 23.2337i | 13.5344 | + | 23.4422i | 1.56283 | − | 8.86327i | −3.96306 | + | 22.4756i |
28.2 | −0.443311 | − | 2.51414i | −2.29813 | + | 1.92836i | 1.39316 | − | 0.507068i | 12.9233 | + | 4.70368i | 5.86696 | + | 4.92297i | 7.81757 | − | 13.5404i | −12.1041 | − | 20.9650i | 1.56283 | − | 8.86327i | 6.09670 | − | 34.5761i |
See all 36 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
19.e | even | 9 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 57.4.i.b | ✓ | 36 |
3.b | odd | 2 | 1 | 171.4.u.c | 36 | ||
19.e | even | 9 | 1 | inner | 57.4.i.b | ✓ | 36 |
19.e | even | 9 | 1 | 1083.4.a.s | 18 | ||
19.f | odd | 18 | 1 | 1083.4.a.t | 18 | ||
57.l | odd | 18 | 1 | 171.4.u.c | 36 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
57.4.i.b | ✓ | 36 | 1.a | even | 1 | 1 | trivial |
57.4.i.b | ✓ | 36 | 19.e | even | 9 | 1 | inner |
171.4.u.c | 36 | 3.b | odd | 2 | 1 | ||
171.4.u.c | 36 | 57.l | odd | 18 | 1 | ||
1083.4.a.s | 18 | 19.e | even | 9 | 1 | ||
1083.4.a.t | 18 | 19.f | odd | 18 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{36} - 3 T_{2}^{35} + 28 T_{2}^{33} + 27 T_{2}^{32} - 1005 T_{2}^{31} + 23712 T_{2}^{30} - 15201 T_{2}^{29} - 216288 T_{2}^{28} - 201152 T_{2}^{27} + 3705891 T_{2}^{26} + 13346883 T_{2}^{25} + \cdots + 24\!\cdots\!16 \)
acting on \(S_{4}^{\mathrm{new}}(57, [\chi])\).