Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [125,2,Mod(4,125)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(125, base_ring=CyclotomicField(50))
chi = DirichletCharacter(H, H._module([1]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("125.4");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 125 = 5^{3} \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 125.h (of order \(50\), degree \(20\), 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: | \(0.998130025266\) |
Analytic rank: | \(0\) |
Dimension: | \(240\) |
Relative dimension: | \(12\) over \(\Q(\zeta_{50})\) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{50}]$ |
$q$-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 | −2.69510 | − | 0.169561i | 2.45880 | + | 1.15702i | 5.25058 | + | 0.663302i | 1.42401 | − | 1.72400i | −6.43051 | − | 3.53521i | −0.679948 | + | 0.935868i | −8.73317 | − | 1.66594i | 2.79471 | + | 3.37822i | −4.13018 | + | 4.40490i |
4.2 | −2.48068 | − | 0.156071i | −1.52292 | − | 0.716634i | 4.14517 | + | 0.523657i | −0.268080 | + | 2.21994i | 3.66604 | + | 2.01542i | 0.423640 | − | 0.583091i | −5.31799 | − | 1.01446i | −0.106538 | − | 0.128782i | 1.01149 | − | 5.46511i |
4.3 | −1.80542 | − | 0.113587i | −2.06171 | − | 0.970167i | 1.26239 | + | 0.159477i | 0.397505 | − | 2.20045i | 3.61205 | + | 1.98574i | −2.29756 | + | 3.16232i | 1.29286 | + | 0.246625i | 1.39715 | + | 1.68887i | −0.967604 | + | 3.92758i |
4.4 | −1.58276 | − | 0.0995789i | 0.718817 | + | 0.338250i | 0.510990 | + | 0.0645531i | −1.79303 | − | 1.33605i | −1.10403 | − | 0.606948i | 2.20198 | − | 3.03077i | 2.31325 | + | 0.441277i | −1.50999 | − | 1.82526i | 2.70490 | + | 2.29320i |
4.5 | −0.921064 | − | 0.0579484i | 1.90310 | + | 0.895532i | −1.13923 | − | 0.143918i | 1.22430 | + | 1.87112i | −1.70099 | − | 0.935125i | −0.820926 | + | 1.12991i | 2.85404 | + | 0.544437i | 0.907551 | + | 1.09704i | −1.01923 | − | 1.79437i |
4.6 | −0.303433 | − | 0.0190904i | −0.819682 | − | 0.385713i | −1.89252 | − | 0.239081i | 2.06070 | − | 0.868045i | 0.241355 | + | 0.132686i | 1.71112 | − | 2.35515i | 1.16698 | + | 0.222614i | −1.38917 | − | 1.67922i | −0.641857 | + | 0.224054i |
4.7 | −0.0569284 | − | 0.00358163i | −1.13144 | − | 0.532413i | −1.98100 | − | 0.250259i | −1.71095 | + | 1.43967i | 0.0625039 | + | 0.0343618i | −1.60389 | + | 2.20757i | 0.223940 | + | 0.0427189i | −0.915589 | − | 1.10676i | 0.102558 | − | 0.0758299i |
4.8 | 0.893875 | + | 0.0562379i | 2.85666 | + | 1.34424i | −1.18838 | − | 0.150127i | −2.22874 | − | 0.180863i | 2.47790 | + | 1.36224i | 1.29134 | − | 1.77738i | −2.81338 | − | 0.536680i | 4.44123 | + | 5.36853i | −1.98205 | − | 0.287009i |
4.9 | 1.24554 | + | 0.0783628i | 1.16976 | + | 0.550449i | −0.438999 | − | 0.0554585i | 1.46862 | − | 1.68617i | 1.41385 | + | 0.777273i | −0.902275 | + | 1.24187i | −2.99424 | − | 0.571182i | −0.846918 | − | 1.02375i | 1.96135 | − | 1.98511i |
4.10 | 1.33093 | + | 0.0837352i | −2.88787 | − | 1.35893i | −0.219857 | − | 0.0277744i | −1.37708 | − | 1.76172i | −3.72977 | − | 2.05046i | 1.78965 | − | 2.46324i | −2.91018 | − | 0.555146i | 4.58082 | + | 5.53726i | −1.68528 | − | 2.46004i |
4.11 | 2.07048 | + | 0.130264i | −0.974456 | − | 0.458544i | 2.28570 | + | 0.288751i | 0.707806 | + | 2.12109i | −1.95786 | − | 1.07634i | 1.45498 | − | 2.00261i | 0.619230 | + | 0.118124i | −1.17297 | − | 1.41788i | 1.18920 | + | 4.48388i |
4.12 | 2.31243 | + | 0.145486i | 0.181456 | + | 0.0853865i | 3.34195 | + | 0.422186i | −2.20265 | − | 0.385146i | 0.407181 | + | 0.223850i | −1.85539 | + | 2.55372i | 3.11468 | + | 0.594156i | −1.88664 | − | 2.28055i | −5.03744 | − | 1.21108i |
9.1 | −2.48947 | − | 1.17145i | 0.834394 | − | 0.0524956i | 3.55029 | + | 4.29156i | −2.14236 | + | 0.640526i | −2.13869 | − | 0.846768i | −3.54615 | + | 1.15221i | −2.44251 | − | 9.51294i | −2.28289 | + | 0.288396i | 6.08369 | + | 0.915114i |
9.2 | −1.67272 | − | 0.787121i | −3.20210 | + | 0.201459i | 0.903578 | + | 1.09224i | −2.18650 | − | 0.468215i | 5.51479 | + | 2.18346i | 2.31306 | − | 0.751558i | 0.267779 | + | 1.04293i | 7.23652 | − | 0.914185i | 3.28885 | + | 2.50423i |
9.3 | −1.61529 | − | 0.760100i | 2.82599 | − | 0.177796i | 0.756578 | + | 0.914545i | 2.02990 | − | 0.937822i | −4.69996 | − | 1.86084i | −1.81397 | + | 0.589394i | 0.360971 | + | 1.40589i | 4.97829 | − | 0.628904i | −3.99172 | − | 0.0280665i |
9.4 | −1.57955 | − | 0.743281i | 1.06748 | − | 0.0671603i | 0.667670 | + | 0.807075i | 0.0877868 | + | 2.23434i | −1.73606 | − | 0.687356i | 4.47078 | − | 1.45264i | 0.413537 | + | 1.61062i | −1.84134 | + | 0.232615i | 1.52208 | − | 3.59451i |
9.5 | −0.926191 | − | 0.435832i | −1.75492 | + | 0.110410i | −0.606968 | − | 0.733699i | 1.67095 | + | 1.48591i | 1.67351 | + | 0.662589i | −3.38254 | + | 1.09905i | 0.751522 | + | 2.92699i | 0.0912001 | − | 0.0115212i | −0.900008 | − | 2.10449i |
9.6 | −0.694701 | − | 0.326902i | −0.172742 | + | 0.0108680i | −0.899103 | − | 1.08683i | −0.949236 | − | 2.02459i | 0.123557 | + | 0.0489196i | −2.28461 | + | 0.742313i | 0.651196 | + | 2.53624i | −2.94662 | + | 0.372245i | −0.00240546 | + | 1.71679i |
9.7 | 0.129181 | + | 0.0607882i | 2.44002 | − | 0.153513i | −1.26186 | − | 1.52532i | −1.95252 | − | 1.08980i | 0.324537 | + | 0.128493i | 2.93692 | − | 0.954262i | −0.141297 | − | 0.550317i | 2.95378 | − | 0.373150i | −0.185983 | − | 0.259472i |
9.8 | 0.379582 | + | 0.178618i | −1.61444 | + | 0.101572i | −1.16267 | − | 1.40543i | 1.99812 | − | 1.00375i | −0.630955 | − | 0.249813i | 4.50478 | − | 1.46369i | −0.398949 | − | 1.55380i | −0.380236 | + | 0.0480350i | 0.937737 | − | 0.0241068i |
See next 80 embeddings (of 240 total) |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
125.h | even | 50 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 125.2.h.a | ✓ | 240 |
5.b | even | 2 | 1 | 625.2.h.a | 240 | ||
5.c | odd | 4 | 2 | 625.2.g.b | 480 | ||
125.g | even | 25 | 1 | 625.2.h.a | 240 | ||
125.h | even | 50 | 1 | inner | 125.2.h.a | ✓ | 240 |
125.i | odd | 100 | 2 | 625.2.g.b | 480 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
125.2.h.a | ✓ | 240 | 1.a | even | 1 | 1 | trivial |
125.2.h.a | ✓ | 240 | 125.h | even | 50 | 1 | inner |
625.2.g.b | 480 | 5.c | odd | 4 | 2 | ||
625.2.g.b | 480 | 125.i | odd | 100 | 2 | ||
625.2.h.a | 240 | 5.b | even | 2 | 1 | ||
625.2.h.a | 240 | 125.g | even | 25 | 1 |
Hecke kernels
This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(125, [\chi])\).