Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [155,2,Mod(9,155)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(155, base_ring=CyclotomicField(30))
chi = DirichletCharacter(H, H._module([15, 2]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("155.9");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 155 = 5 \cdot 31 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 155.u (of order \(30\), 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: | \(1.23768123133\) |
Analytic rank: | \(0\) |
Dimension: | \(112\) |
Relative dimension: | \(14\) over \(\Q(\zeta_{30})\) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{30}]$ |
$q$-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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
9.1 | −2.29327 | − | 0.745128i | 0.184013 | − | 0.865712i | 3.08583 | + | 2.24199i | 1.31362 | − | 1.80953i | −1.06706 | + | 1.84820i | 1.16536 | + | 2.61745i | −2.57144 | − | 3.53928i | 2.02504 | + | 0.901606i | −4.36082 | + | 3.17093i |
9.2 | −1.93608 | − | 0.629071i | −0.622841 | + | 2.93024i | 1.73465 | + | 1.26029i | 2.22054 | − | 0.263033i | 3.04920 | − | 5.28137i | −0.119320 | − | 0.267996i | −0.172473 | − | 0.237389i | −5.45772 | − | 2.42993i | −4.46462 | − | 0.887626i |
9.3 | −1.78210 | − | 0.579041i | 0.541453 | − | 2.54734i | 1.22257 | + | 0.888250i | −2.14934 | − | 0.616727i | −2.43994 | + | 4.22609i | −0.941056 | − | 2.11365i | 0.538385 | + | 0.741023i | −3.45511 | − | 1.53831i | 3.47323 | + | 2.34362i |
9.4 | −1.35835 | − | 0.441355i | −0.0117953 | + | 0.0554924i | 0.0322880 | + | 0.0234586i | 0.797789 | + | 2.08891i | 0.0405139 | − | 0.0701722i | −0.461336 | − | 1.03618i | 1.64551 | + | 2.26485i | 2.73770 | + | 1.21890i | −0.161727 | − | 3.18958i |
9.5 | −0.745432 | − | 0.242206i | −0.105056 | + | 0.494249i | −1.12103 | − | 0.814475i | −2.20838 | + | 0.350796i | 0.198022 | − | 0.342984i | 1.56835 | + | 3.52258i | 1.55979 | + | 2.14686i | 2.50739 | + | 1.11636i | 1.73116 | + | 0.273388i |
9.6 | −0.705654 | − | 0.229281i | −0.394461 | + | 1.85579i | −1.17266 | − | 0.851985i | −0.801322 | − | 2.08755i | 0.703850 | − | 1.21910i | −1.25976 | − | 2.82948i | 1.50438 | + | 2.07060i | −0.547724 | − | 0.243862i | 0.0868201 | + | 1.65682i |
9.7 | −0.577976 | − | 0.187796i | 0.589803 | − | 2.77480i | −1.31925 | − | 0.958488i | 2.22899 | + | 0.177758i | −0.861987 | + | 1.49301i | 0.0276502 | + | 0.0621035i | 1.29691 | + | 1.78504i | −4.61103 | − | 2.05296i | −1.25492 | − | 0.521335i |
9.8 | 0.577976 | + | 0.187796i | −0.589803 | + | 2.77480i | −1.31925 | − | 0.958488i | −0.960553 | + | 2.01924i | −0.861987 | + | 1.49301i | −0.0276502 | − | 0.0621035i | −1.29691 | − | 1.78504i | −4.61103 | − | 2.05296i | −0.934381 | + | 0.986685i |
9.9 | 0.705654 | + | 0.229281i | 0.394461 | − | 1.85579i | −1.17266 | − | 0.851985i | −1.40721 | − | 1.73774i | 0.703850 | − | 1.21910i | 1.25976 | + | 2.82948i | −1.50438 | − | 2.07060i | −0.547724 | − | 0.243862i | −0.594575 | − | 1.54889i |
9.10 | 0.745432 | + | 0.242206i | 0.105056 | − | 0.494249i | −1.12103 | − | 0.814475i | 1.40799 | − | 1.73712i | 0.198022 | − | 0.342984i | −1.56835 | − | 3.52258i | −1.55979 | − | 2.14686i | 2.50739 | + | 1.11636i | 1.47030 | − | 0.953879i |
9.11 | 1.35835 | + | 0.441355i | 0.0117953 | − | 0.0554924i | 0.0322880 | + | 0.0234586i | 1.41015 | + | 1.73536i | 0.0405139 | − | 0.0701722i | 0.461336 | + | 1.03618i | −1.64551 | − | 2.26485i | 2.73770 | + | 1.21890i | 1.14957 | + | 2.97960i |
9.12 | 1.78210 | + | 0.579041i | −0.541453 | + | 2.54734i | 1.22257 | + | 0.888250i | 0.540567 | − | 2.16974i | −2.43994 | + | 4.22609i | 0.941056 | + | 2.11365i | −0.538385 | − | 0.741023i | −3.45511 | − | 1.53831i | 2.21972 | − | 3.55370i |
9.13 | 1.93608 | + | 0.629071i | 0.622841 | − | 2.93024i | 1.73465 | + | 1.26029i | −1.33807 | + | 1.79153i | 3.04920 | − | 5.28137i | 0.119320 | + | 0.267996i | 0.172473 | + | 0.237389i | −5.45772 | − | 2.42993i | −3.71760 | + | 2.62681i |
9.14 | 2.29327 | + | 0.745128i | −0.184013 | + | 0.865712i | 3.08583 | + | 2.24199i | −2.22391 | + | 0.232862i | −1.06706 | + | 1.84820i | −1.16536 | − | 2.61745i | 2.57144 | + | 3.53928i | 2.02504 | + | 0.901606i | −5.27354 | − | 1.12308i |
14.1 | −2.52575 | − | 0.820665i | 2.29523 | + | 2.06664i | 4.08788 | + | 2.97002i | 0.752855 | − | 2.10552i | −4.10116 | − | 7.10343i | 1.50439 | + | 0.158118i | −4.76557 | − | 6.55924i | 0.683521 | + | 6.50327i | −3.62945 | + | 4.70017i |
14.2 | −2.02472 | − | 0.657871i | −0.0273246 | − | 0.0246032i | 2.04866 | + | 1.48844i | −2.14173 | + | 0.642634i | 0.0391389 | + | 0.0677905i | 3.48503 | + | 0.366291i | −0.666072 | − | 0.916769i | −0.313444 | − | 2.98222i | 4.75918 | + | 0.107831i |
14.3 | −1.99742 | − | 0.649000i | −1.07358 | − | 0.966660i | 1.95043 | + | 1.41707i | 2.22162 | + | 0.253777i | 1.51703 | + | 2.62758i | 1.57417 | + | 0.165452i | −0.507205 | − | 0.698108i | −0.0954328 | − | 0.907983i | −4.27280 | − | 1.94873i |
14.4 | −1.38693 | − | 0.450639i | 0.888812 | + | 0.800290i | 0.102453 | + | 0.0744362i | −2.05304 | − | 0.886010i | −0.872074 | − | 1.51048i | −4.30036 | − | 0.451986i | 1.60578 | + | 2.21017i | −0.164063 | − | 1.56095i | 2.44815 | + | 2.15401i |
14.5 | −1.04247 | − | 0.338718i | 1.69229 | + | 1.52374i | −0.646026 | − | 0.469365i | 1.68153 | + | 1.47393i | −1.24804 | − | 2.16166i | −0.717171 | − | 0.0753778i | 1.80304 | + | 2.48167i | 0.228460 | + | 2.17366i | −1.25369 | − | 2.10609i |
14.6 | −0.944189 | − | 0.306786i | −1.97801 | − | 1.78101i | −0.820658 | − | 0.596243i | −0.582501 | + | 2.15886i | 1.32123 | + | 2.28844i | −0.852316 | − | 0.0895820i | 1.75902 | + | 2.42108i | 0.426951 | + | 4.06217i | 1.21230 | − | 1.85967i |
See next 80 embeddings (of 112 total) |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
5.b | even | 2 | 1 | inner |
31.g | even | 15 | 1 | inner |
155.u | even | 30 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 155.2.u.a | ✓ | 112 |
5.b | even | 2 | 1 | inner | 155.2.u.a | ✓ | 112 |
5.c | odd | 4 | 2 | 775.2.bl.f | 112 | ||
31.g | even | 15 | 1 | inner | 155.2.u.a | ✓ | 112 |
155.u | even | 30 | 1 | inner | 155.2.u.a | ✓ | 112 |
155.w | odd | 60 | 2 | 775.2.bl.f | 112 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
155.2.u.a | ✓ | 112 | 1.a | even | 1 | 1 | trivial |
155.2.u.a | ✓ | 112 | 5.b | even | 2 | 1 | inner |
155.2.u.a | ✓ | 112 | 31.g | even | 15 | 1 | inner |
155.2.u.a | ✓ | 112 | 155.u | even | 30 | 1 | inner |
775.2.bl.f | 112 | 5.c | odd | 4 | 2 | ||
775.2.bl.f | 112 | 155.w | odd | 60 | 2 |
Hecke kernels
This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(155, [\chi])\).