Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [136,3,Mod(5,136)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(136, base_ring=CyclotomicField(16))
chi = DirichletCharacter(H, H._module([0, 8, 5]))
N = Newforms(chi, 3, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("136.5");
S:= CuspForms(chi, 3);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 136 = 2^{3} \cdot 17 \) |
Weight: | \( k \) | \(=\) | \( 3 \) |
Character orbit: | \([\chi]\) | \(=\) | 136.q (of order \(16\), degree \(8\), 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: | \(3.70573159530\) |
Analytic rank: | \(0\) |
Dimension: | \(272\) |
Relative dimension: | \(34\) over \(\Q(\zeta_{16})\) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{16}]$ |
$q$-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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
5.1 | −1.99615 | − | 0.123996i | −2.49131 | + | 1.66464i | 3.96925 | + | 0.495030i | −5.28331 | − | 1.05092i | 5.17944 | − | 3.01396i | 0.430227 | + | 2.16290i | −7.86185 | − | 1.48033i | −0.00856088 | + | 0.0206678i | 10.4160 | + | 2.75290i |
5.2 | −1.95431 | − | 0.425075i | 4.73781 | − | 3.16571i | 3.63862 | + | 1.66145i | −5.99545 | − | 1.19257i | −10.6048 | + | 4.17283i | −1.47940 | − | 7.43743i | −6.40474 | − | 4.79367i | 8.98103 | − | 21.6821i | 11.2100 | + | 4.87916i |
5.3 | −1.93868 | + | 0.491460i | 1.98785 | − | 1.32824i | 3.51693 | − | 1.90556i | 8.85748 | + | 1.76186i | −3.20102 | + | 3.55198i | −1.81827 | − | 9.14104i | −5.88169 | + | 5.42271i | −1.25682 | + | 3.03423i | −18.0377 | + | 0.937417i |
5.4 | −1.86396 | − | 0.725012i | 2.06162 | − | 1.37753i | 2.94871 | + | 2.70279i | 4.63458 | + | 0.921876i | −4.84150 | + | 1.07296i | 2.38099 | + | 11.9700i | −3.53674 | − | 7.17576i | −1.09147 | + | 2.63504i | −7.97032 | − | 5.07847i |
5.5 | −1.83427 | − | 0.797164i | −2.62580 | + | 1.75450i | 2.72906 | + | 2.92442i | 3.72340 | + | 0.740630i | 6.21503 | − | 1.12503i | −0.824370 | − | 4.14439i | −2.67457 | − | 7.53967i | 0.372384 | − | 0.899013i | −6.23929 | − | 4.32667i |
5.6 | −1.82287 | + | 0.822889i | 0.416587 | − | 0.278355i | 2.64571 | − | 3.00004i | −4.26113 | − | 0.847591i | −0.530329 | + | 0.850210i | −0.301354 | − | 1.51501i | −2.35408 | + | 7.64580i | −3.34809 | + | 8.08300i | 8.46495 | − | 1.96139i |
5.7 | −1.75423 | + | 0.960563i | −3.89540 | + | 2.60282i | 2.15464 | − | 3.37009i | 6.75022 | + | 1.34270i | 4.33324 | − | 8.30772i | 2.12232 | + | 10.6696i | −0.542538 | + | 7.98158i | 4.95529 | − | 11.9631i | −13.1312 | + | 4.12860i |
5.8 | −1.56990 | + | 1.23912i | 3.79009 | − | 2.53245i | 0.929162 | − | 3.89059i | −0.274813 | − | 0.0546637i | −2.81203 | + | 8.67207i | 1.94983 | + | 9.80246i | 3.36221 | + | 7.25917i | 4.50728 | − | 10.8815i | 0.499163 | − | 0.254710i |
5.9 | −1.39201 | − | 1.43608i | 1.32480 | − | 0.885206i | −0.124624 | + | 3.99806i | 0.988766 | + | 0.196678i | −3.11536 | − | 0.670305i | −1.85512 | − | 9.32633i | 5.91499 | − | 5.38636i | −2.47263 | + | 5.96947i | −1.09393 | − | 1.69372i |
5.10 | −1.20172 | − | 1.59871i | 1.03356 | − | 0.690603i | −1.11173 | + | 3.84240i | −6.72566 | − | 1.33782i | −2.34613 | − | 0.822447i | 1.20217 | + | 6.04371i | 7.47886 | − | 2.84018i | −2.85284 | + | 6.88735i | 5.94360 | + | 12.3600i |
5.11 | −1.18540 | + | 1.61084i | −2.34076 | + | 1.56405i | −1.18963 | − | 3.81900i | 0.703425 | + | 0.139920i | 0.255314 | − | 5.62462i | −1.92692 | − | 9.68727i | 7.56201 | + | 2.61075i | −0.411234 | + | 0.992807i | −1.05923 | + | 0.967245i |
5.12 | −0.962010 | − | 1.75344i | −4.61315 | + | 3.08241i | −2.14907 | + | 3.37365i | −4.96080 | − | 0.986765i | 9.84269 | + | 5.12355i | 0.206012 | + | 1.03569i | 7.98290 | + | 0.522780i | 8.33574 | − | 20.1243i | 3.04211 | + | 9.64772i |
5.13 | −0.809891 | + | 1.82868i | 0.402825 | − | 0.269159i | −2.68815 | − | 2.96207i | 0.164157 | + | 0.0326529i | 0.165962 | + | 0.954627i | 1.23928 | + | 6.23029i | 7.59379 | − | 2.51683i | −3.35433 | + | 8.09807i | −0.192661 | + | 0.273746i |
5.14 | −0.559622 | − | 1.92011i | 4.61315 | − | 3.08241i | −3.37365 | + | 2.14907i | 4.96080 | + | 0.986765i | −8.50018 | − | 7.13277i | 0.206012 | + | 1.03569i | 6.01442 | + | 5.27510i | 8.33574 | − | 20.1243i | −0.881479 | − | 10.0775i |
5.15 | −0.301940 | + | 1.97708i | 2.58349 | − | 1.72623i | −3.81766 | − | 1.19392i | −9.15038 | − | 1.82012i | 2.63284 | + | 5.62898i | −1.42493 | − | 7.16362i | 3.51317 | − | 7.18732i | 0.250398 | − | 0.604514i | 6.36139 | − | 17.5414i |
5.16 | −0.295486 | + | 1.97805i | 3.57309 | − | 2.38746i | −3.82538 | − | 1.16897i | 5.68249 | + | 1.13032i | 3.66673 | + | 7.77323i | −1.07342 | − | 5.39645i | 3.44263 | − | 7.22138i | 3.62286 | − | 8.74635i | −3.91492 | + | 10.9063i |
5.17 | −0.280710 | − | 1.98020i | −1.03356 | + | 0.690603i | −3.84240 | + | 1.11173i | 6.72566 | + | 1.33782i | 1.65767 | + | 1.85280i | 1.20217 | + | 6.04371i | 3.28004 | + | 7.29667i | −2.85284 | + | 6.88735i | 0.761189 | − | 13.6937i |
5.18 | −0.0311599 | − | 1.99976i | −1.32480 | + | 0.885206i | −3.99806 | + | 0.124624i | −0.988766 | − | 0.196678i | 1.81148 | + | 2.62170i | −1.85512 | − | 9.32633i | 0.373797 | + | 7.99126i | −2.47263 | + | 5.96947i | −0.362498 | + | 1.98342i |
5.19 | 0.108110 | + | 1.99708i | −3.69899 | + | 2.47159i | −3.97662 | + | 0.431807i | −1.02401 | − | 0.203689i | −5.33584 | − | 7.11996i | 0.149919 | + | 0.753693i | −1.29226 | − | 7.89494i | 4.12963 | − | 9.96982i | 0.296076 | − | 2.06705i |
5.20 | 0.506868 | + | 1.93471i | −0.155374 | + | 0.103818i | −3.48617 | + | 1.96128i | 6.64253 | + | 1.32128i | −0.279611 | − | 0.247982i | 0.946911 | + | 4.76044i | −5.56153 | − | 5.75060i | −3.43079 | + | 8.28265i | 0.810599 | + | 13.5211i |
See next 80 embeddings (of 272 total) |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
8.b | even | 2 | 1 | inner |
17.e | odd | 16 | 1 | inner |
136.q | odd | 16 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 136.3.q.a | ✓ | 272 |
8.b | even | 2 | 1 | inner | 136.3.q.a | ✓ | 272 |
17.e | odd | 16 | 1 | inner | 136.3.q.a | ✓ | 272 |
136.q | odd | 16 | 1 | inner | 136.3.q.a | ✓ | 272 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
136.3.q.a | ✓ | 272 | 1.a | even | 1 | 1 | trivial |
136.3.q.a | ✓ | 272 | 8.b | even | 2 | 1 | inner |
136.3.q.a | ✓ | 272 | 17.e | odd | 16 | 1 | inner |
136.3.q.a | ✓ | 272 | 136.q | odd | 16 | 1 | inner |
Hecke kernels
This newform subspace is the entire newspace \(S_{3}^{\mathrm{new}}(136, [\chi])\).