Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [145,2,Mod(3,145)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(145, base_ring=CyclotomicField(28))
chi = DirichletCharacter(H, H._module([21, 5]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("145.3");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 145 = 5 \cdot 29 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 145.t (of order \(28\), degree \(12\), 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: | \(1.15783082931\) |
Analytic rank: | \(0\) |
Dimension: | \(156\) |
Relative dimension: | \(13\) over \(\Q(\zeta_{28})\) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{28}]$ |
$q$-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 | −2.43135 | + | 0.554939i | 1.81404 | + | 0.873594i | 3.80156 | − | 1.83074i | −0.151265 | + | 2.23095i | −4.89535 | − | 1.11733i | 3.00411 | − | 1.05118i | −4.32739 | + | 3.45098i | 0.657093 | + | 0.823969i | −0.870262 | − | 5.50815i |
3.2 | −2.31914 | + | 0.529329i | −1.77490 | − | 0.854745i | 3.29630 | − | 1.58741i | −2.07065 | − | 0.844052i | 4.56868 | + | 1.04277i | 0.698030 | − | 0.244251i | −3.08471 | + | 2.45997i | 0.549197 | + | 0.688671i | 5.24891 | + | 0.861425i |
3.3 | −2.02391 | + | 0.461944i | −1.48024 | − | 0.712845i | 2.08088 | − | 1.00210i | 1.88201 | + | 1.20750i | 3.32516 | + | 0.758947i | −3.33902 | + | 1.16837i | −0.502501 | + | 0.400731i | −0.187511 | − | 0.235132i | −4.36681 | − | 1.57449i |
3.4 | −1.62317 | + | 0.370478i | 1.59768 | + | 0.769400i | 0.695489 | − | 0.334930i | 1.95368 | − | 1.08773i | −2.87835 | − | 0.656964i | 0.649579 | − | 0.227298i | 1.59855 | − | 1.27480i | 0.0901230 | + | 0.113011i | −2.76817 | + | 2.48936i |
3.5 | −0.718317 | + | 0.163951i | −0.427973 | − | 0.206101i | −1.31284 | + | 0.632230i | −0.265444 | − | 2.22026i | 0.341211 | + | 0.0778792i | 1.22364 | − | 0.428169i | 1.99147 | − | 1.58814i | −1.72979 | − | 2.16908i | 0.554686 | + | 1.55133i |
3.6 | −0.469443 | + | 0.107147i | 0.197855 | + | 0.0952819i | −1.59304 | + | 0.767168i | −1.09670 | + | 1.94865i | −0.103091 | − | 0.0235298i | −3.11661 | + | 1.09055i | 1.41857 | − | 1.13127i | −1.84040 | − | 2.30779i | 0.306044 | − | 1.03229i |
3.7 | −0.426538 | + | 0.0973546i | −2.42758 | − | 1.16906i | −1.62948 | + | 0.784717i | 1.36135 | + | 1.77390i | 1.14927 | + | 0.262313i | 4.53036 | − | 1.58524i | 1.30275 | − | 1.03891i | 2.65596 | + | 3.33047i | −0.753366 | − | 0.624102i |
3.8 | 0.0671767 | − | 0.0153326i | 2.71909 | + | 1.30944i | −1.79766 | + | 0.865707i | −2.19083 | + | 0.447510i | 0.202737 | + | 0.0462733i | 3.00352 | − | 1.05098i | −0.215230 | + | 0.171640i | 3.80833 | + | 4.77549i | −0.140311 | + | 0.0636534i |
3.9 | 0.898066 | − | 0.204978i | −2.27912 | − | 1.09757i | −1.03743 | + | 0.499601i | −2.08580 | − | 0.805888i | −2.27178 | − | 0.518518i | −0.941288 | + | 0.329371i | −2.26966 | + | 1.80999i | 2.11927 | + | 2.65748i | −2.03837 | − | 0.296199i |
3.10 | 1.40869 | − | 0.321524i | 0.596213 | + | 0.287121i | 0.0790869 | − | 0.0380862i | 1.12791 | + | 1.93076i | 0.932195 | + | 0.212767i | 0.878334 | − | 0.307342i | −2.16020 | + | 1.72270i | −1.59744 | − | 2.00312i | 2.20966 | + | 2.35718i |
3.11 | 1.68122 | − | 0.383728i | 1.75904 | + | 0.847111i | 0.877317 | − | 0.422494i | −0.646560 | − | 2.14055i | 3.28240 | + | 0.749187i | −2.98844 | + | 1.04570i | −1.38363 | + | 1.10341i | 0.506171 | + | 0.634719i | −1.90840 | − | 3.35064i |
3.12 | 2.15407 | − | 0.491652i | −2.28313 | − | 1.09950i | 2.59634 | − | 1.25033i | 2.11300 | − | 0.731603i | −5.45857 | − | 1.24588i | −0.579985 | + | 0.202945i | 1.52312 | − | 1.21464i | 2.13330 | + | 2.67508i | 4.19184 | − | 2.61478i |
3.13 | 2.44273 | − | 0.557537i | −0.416303 | − | 0.200481i | 3.85413 | − | 1.85605i | −1.91761 | + | 1.15012i | −1.12869 | − | 0.257616i | −0.222008 | + | 0.0776841i | 4.46194 | − | 3.55828i | −1.73735 | − | 2.17857i | −4.04296 | + | 3.87857i |
27.1 | −2.03189 | + | 1.62038i | 0.433809 | + | 1.90064i | 1.05791 | − | 4.63499i | 0.920670 | + | 2.03774i | −3.96120 | − | 3.15895i | −1.75507 | + | 2.79318i | 3.10566 | + | 6.44896i | −0.721337 | + | 0.347377i | −5.17260 | − | 2.64862i |
27.2 | −1.72889 | + | 1.37874i | 0.0324304 | + | 0.142087i | 0.643081 | − | 2.81752i | 1.02293 | − | 1.98837i | −0.251970 | − | 0.200939i | 0.555045 | − | 0.883349i | 0.853902 | + | 1.77315i | 2.68377 | − | 1.29244i | 0.972925 | + | 4.84802i |
27.3 | −1.49240 | + | 1.19015i | −0.343037 | − | 1.50294i | 0.365759 | − | 1.60249i | −1.92139 | + | 1.14380i | 2.30068 | + | 1.83473i | 1.10422 | − | 1.75735i | −0.295090 | − | 0.612760i | 0.561738 | − | 0.270519i | 1.50618 | − | 3.99374i |
27.4 | −0.892134 | + | 0.711453i | −0.324743 | − | 1.42279i | −0.155304 | + | 0.680432i | 1.31414 | + | 1.80916i | 1.30196 | + | 1.03828i | −0.495384 | + | 0.788400i | −1.33574 | − | 2.77369i | 0.784033 | − | 0.377570i | −2.45952 | − | 0.679062i |
27.5 | −0.554273 | + | 0.442018i | 0.339722 | + | 1.48842i | −0.333203 | + | 1.45986i | −1.82434 | − | 1.29297i | −0.846207 | − | 0.674828i | −2.04713 | + | 3.25799i | −1.07579 | − | 2.23391i | 0.602920 | − | 0.290351i | 1.58270 | − | 0.0897329i |
27.6 | −0.553028 | + | 0.441025i | 0.668619 | + | 2.92941i | −0.333705 | + | 1.46206i | 2.21821 | − | 0.282015i | −1.66171 | − | 1.32517i | 1.47214 | − | 2.34289i | −1.07407 | − | 2.23033i | −5.43148 | + | 2.61566i | −1.10236 | + | 1.13425i |
27.7 | −0.0214476 | + | 0.0171039i | −0.243851 | − | 1.06838i | −0.444874 | + | 1.94912i | −0.0575944 | − | 2.23533i | 0.0235034 | + | 0.0187434i | 2.24458 | − | 3.57223i | −0.0476010 | − | 0.0988446i | 1.62093 | − | 0.780601i | 0.0394680 | + | 0.0469572i |
See next 80 embeddings (of 156 total) |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
145.t | even | 28 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 145.2.t.a | yes | 156 |
5.b | even | 2 | 1 | 725.2.bd.b | 156 | ||
5.c | odd | 4 | 1 | 145.2.o.a | ✓ | 156 | |
5.c | odd | 4 | 1 | 725.2.y.b | 156 | ||
29.f | odd | 28 | 1 | 145.2.o.a | ✓ | 156 | |
145.o | even | 28 | 1 | 725.2.bd.b | 156 | ||
145.s | odd | 28 | 1 | 725.2.y.b | 156 | ||
145.t | even | 28 | 1 | inner | 145.2.t.a | yes | 156 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
145.2.o.a | ✓ | 156 | 5.c | odd | 4 | 1 | |
145.2.o.a | ✓ | 156 | 29.f | odd | 28 | 1 | |
145.2.t.a | yes | 156 | 1.a | even | 1 | 1 | trivial |
145.2.t.a | yes | 156 | 145.t | even | 28 | 1 | inner |
725.2.y.b | 156 | 5.c | odd | 4 | 1 | ||
725.2.y.b | 156 | 145.s | odd | 28 | 1 | ||
725.2.bd.b | 156 | 5.b | even | 2 | 1 | ||
725.2.bd.b | 156 | 145.o | even | 28 | 1 |
Hecke kernels
This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(145, [\chi])\).