Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [950,2,Mod(3,950)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(950, base_ring=CyclotomicField(180))
chi = DirichletCharacter(H, H._module([63, 130]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("950.3");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 950 = 2 \cdot 5^{2} \cdot 19 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 950.bi (of order \(180\), degree \(48\), 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: | \(7.58578819202\) |
Analytic rank: | \(0\) |
Dimension: | \(2400\) |
Relative dimension: | \(50\) over \(\Q(\zeta_{180})\) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{180}]$ |
$q$-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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
3.1 | −0.974370 | − | 0.224951i | −2.97099 | + | 1.64685i | 0.898794 | + | 0.438371i | −0.797955 | − | 2.08884i | 3.26530 | − | 0.936311i | −0.389434 | − | 0.104349i | −0.777146 | − | 0.629320i | 4.52492 | − | 7.24138i | 0.307615 | + | 2.21481i |
3.2 | −0.974370 | − | 0.224951i | −2.56787 | + | 1.42339i | 0.898794 | + | 0.438371i | −1.74255 | + | 1.40125i | 2.82225 | − | 0.809267i | 3.37715 | + | 0.904904i | −0.777146 | − | 0.629320i | 2.97815 | − | 4.76603i | 2.01310 | − | 0.973350i |
3.3 | −0.974370 | − | 0.224951i | −2.43284 | + | 1.34854i | 0.898794 | + | 0.438371i | 2.14980 | − | 0.615105i | 2.67384 | − | 0.766712i | −0.384955 | − | 0.103148i | −0.777146 | − | 0.629320i | 2.51038 | − | 4.01744i | −2.23307 | + | 0.115740i |
3.4 | −0.974370 | − | 0.224951i | −2.06215 | + | 1.14307i | 0.898794 | + | 0.438371i | −1.23872 | + | 1.86160i | 2.26643 | − | 0.649887i | −4.76696 | − | 1.27730i | −0.777146 | − | 0.629320i | 1.35609 | − | 2.17019i | 1.62575 | − | 1.53524i |
3.5 | −0.974370 | − | 0.224951i | −1.69565 | + | 0.939914i | 0.898794 | + | 0.438371i | 1.96376 | − | 1.06941i | 1.86363 | − | 0.534386i | 4.14562 | + | 1.11082i | −0.777146 | − | 0.629320i | 0.402034 | − | 0.643389i | −2.15400 | + | 0.600253i |
3.6 | −0.974370 | − | 0.224951i | −1.63632 | + | 0.907028i | 0.898794 | + | 0.438371i | −2.13706 | − | 0.658015i | 1.79842 | − | 0.515689i | 2.60059 | + | 0.696825i | −0.777146 | − | 0.629320i | 0.265093 | − | 0.424238i | 1.93426 | + | 1.12188i |
3.7 | −0.974370 | − | 0.224951i | −1.33092 | + | 0.737740i | 0.898794 | + | 0.438371i | −1.85939 | − | 1.24204i | 1.46276 | − | 0.419440i | −2.53826 | − | 0.680125i | −0.777146 | − | 0.629320i | −0.362675 | + | 0.580401i | 1.53234 | + | 1.62848i |
3.8 | −0.974370 | − | 0.224951i | −1.15312 | + | 0.639184i | 0.898794 | + | 0.438371i | 1.05144 | − | 1.97344i | 1.26735 | − | 0.363406i | −4.51015 | − | 1.20849i | −0.777146 | − | 0.629320i | −0.668633 | + | 1.07004i | −1.46842 | + | 1.68634i |
3.9 | −0.974370 | − | 0.224951i | −0.957340 | + | 0.530662i | 0.898794 | + | 0.438371i | 1.21737 | + | 1.87564i | 1.05218 | − | 0.301707i | 1.58359 | + | 0.424321i | −0.777146 | − | 0.629320i | −0.954860 | + | 1.52810i | −0.764238 | − | 2.10141i |
3.10 | −0.974370 | − | 0.224951i | −0.697754 | + | 0.386771i | 0.898794 | + | 0.438371i | −1.12307 | + | 1.93357i | 0.766875 | − | 0.219898i | −1.17316 | − | 0.314348i | −0.777146 | − | 0.629320i | −1.25249 | + | 2.00440i | 1.52925 | − | 1.63138i |
3.11 | −0.974370 | − | 0.224951i | −0.607413 | + | 0.336695i | 0.898794 | + | 0.438371i | 2.22562 | − | 0.215946i | 0.667585 | − | 0.191427i | −0.710217 | − | 0.190302i | −0.777146 | − | 0.629320i | −1.33417 | + | 2.13512i | −2.21715 | − | 0.290244i |
3.12 | −0.974370 | − | 0.224951i | −0.195966 | + | 0.108626i | 0.898794 | + | 0.438371i | 0.0267602 | − | 2.23591i | 0.215379 | − | 0.0617590i | 2.91467 | + | 0.780983i | −0.777146 | − | 0.629320i | −1.56315 | + | 2.50157i | −0.529044 | + | 2.17258i |
3.13 | −0.974370 | − | 0.224951i | −0.00117903 | 0.000653545i | 0.898794 | + | 0.438371i | 1.22275 | + | 1.87213i | 0.00129582 | 0.000371572i | −2.92963 | − | 0.784993i | −0.777146 | − | 0.629320i | −1.58976 | + | 2.54414i | −0.770274 | − | 2.09921i | ||
3.14 | −0.974370 | − | 0.224951i | 0.344356 | − | 0.190880i | 0.898794 | + | 0.438371i | −0.109124 | − | 2.23340i | −0.378469 | + | 0.108524i | 2.03298 | + | 0.544736i | −0.777146 | − | 0.629320i | −1.50761 | + | 2.41268i | −0.396079 | + | 2.20071i |
3.15 | −0.974370 | − | 0.224951i | 0.407545 | − | 0.225906i | 0.898794 | + | 0.438371i | 2.17316 | + | 0.526646i | −0.447918 | + | 0.128438i | −2.14541 | − | 0.574860i | −0.777146 | − | 0.629320i | −1.47470 | + | 2.36001i | −1.99900 | − | 1.00200i |
3.16 | −0.974370 | − | 0.224951i | 0.979229 | − | 0.542796i | 0.898794 | + | 0.438371i | −2.02665 | + | 0.944813i | −1.07623 | + | 0.308605i | 4.34720 | + | 1.16483i | −0.777146 | − | 0.629320i | −0.925495 | + | 1.48110i | 2.18725 | − | 0.464699i |
3.17 | −0.974370 | − | 0.224951i | 1.03787 | − | 0.575301i | 0.898794 | + | 0.438371i | 0.253929 | + | 2.22160i | −1.14069 | + | 0.327086i | 4.09315 | + | 1.09676i | −0.777146 | − | 0.629320i | −0.843553 | + | 1.34997i | 0.252331 | − | 2.22179i |
3.18 | −0.974370 | − | 0.224951i | 1.07952 | − | 0.598386i | 0.898794 | + | 0.438371i | −0.707459 | − | 2.12120i | −1.18646 | + | 0.340211i | −1.79990 | − | 0.482282i | −0.777146 | − | 0.629320i | −0.782466 | + | 1.25221i | 0.212160 | + | 2.22598i |
3.19 | −0.974370 | − | 0.224951i | 1.12268 | − | 0.622309i | 0.898794 | + | 0.438371i | −2.20793 | + | 0.353612i | −1.23389 | + | 0.353812i | −2.85847 | − | 0.765924i | −0.777146 | − | 0.629320i | −0.716626 | + | 1.14684i | 2.23089 | + | 0.152128i |
3.20 | −0.974370 | − | 0.224951i | 1.72849 | − | 0.958117i | 0.898794 | + | 0.438371i | 1.89512 | − | 1.18681i | −1.89972 | + | 0.544735i | 0.768108 | + | 0.205814i | −0.777146 | − | 0.629320i | 0.479928 | − | 0.768045i | −2.11352 | + | 0.730084i |
See next 80 embeddings (of 2400 total) |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
19.f | odd | 18 | 1 | inner |
25.f | odd | 20 | 1 | inner |
475.bi | even | 180 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 950.2.bi.a | ✓ | 2400 |
19.f | odd | 18 | 1 | inner | 950.2.bi.a | ✓ | 2400 |
25.f | odd | 20 | 1 | inner | 950.2.bi.a | ✓ | 2400 |
475.bi | even | 180 | 1 | inner | 950.2.bi.a | ✓ | 2400 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
950.2.bi.a | ✓ | 2400 | 1.a | even | 1 | 1 | trivial |
950.2.bi.a | ✓ | 2400 | 19.f | odd | 18 | 1 | inner |
950.2.bi.a | ✓ | 2400 | 25.f | odd | 20 | 1 | inner |
950.2.bi.a | ✓ | 2400 | 475.bi | even | 180 | 1 | inner |
Hecke kernels
This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(950, [\chi])\).