Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [288,3,Mod(53,288)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(288, base_ring=CyclotomicField(8))
chi = DirichletCharacter(H, H._module([0, 5, 4]))
N = Newforms(chi, 3, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("288.53");
S:= CuspForms(chi, 3);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 288 = 2^{5} \cdot 3^{2} \) |
Weight: | \( k \) | \(=\) | \( 3 \) |
Character orbit: | \([\chi]\) | \(=\) | 288.x (of order \(8\), degree \(4\), 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: | \(7.84743161358\) |
Analytic rank: | \(0\) |
Dimension: | \(64\) |
Relative dimension: | \(16\) over \(\Q(\zeta_{8})\) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{8}]$ |
$q$-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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
53.1 | −1.99481 | + | 0.143966i | 0 | 3.95855 | − | 0.574371i | −6.99476 | − | 2.89732i | 0 | −4.93873 | − | 4.93873i | −7.81387 | + | 1.71566i | 0 | 14.3703 | + | 4.77260i | ||||||
53.2 | −1.99180 | + | 0.180966i | 0 | 3.93450 | − | 0.720894i | 3.32592 | + | 1.37764i | 0 | 9.14542 | + | 9.14542i | −7.70627 | + | 2.14788i | 0 | −6.87386 | − | 2.14210i | ||||||
53.3 | −1.85072 | − | 0.758185i | 0 | 2.85031 | + | 2.80637i | 8.07100 | + | 3.34312i | 0 | −8.03674 | − | 8.03674i | −3.14737 | − | 7.35487i | 0 | −12.4024 | − | 12.3065i | ||||||
53.4 | −1.41770 | − | 1.41072i | 0 | 0.0197520 | + | 3.99995i | −3.22612 | − | 1.33630i | 0 | 2.13502 | + | 2.13502i | 5.61480 | − | 5.69860i | 0 | 2.68853 | + | 6.44563i | ||||||
53.5 | −1.35868 | + | 1.46765i | 0 | −0.307970 | − | 3.98813i | 0.365829 | + | 0.151531i | 0 | −1.83662 | − | 1.83662i | 6.27159 | + | 4.96660i | 0 | −0.719440 | + | 0.331025i | ||||||
53.6 | −0.527453 | + | 1.92919i | 0 | −3.44359 | − | 2.03512i | −6.57829 | − | 2.72482i | 0 | 6.20812 | + | 6.20812i | 5.74247 | − | 5.56992i | 0 | 8.72644 | − | 11.2536i | ||||||
53.7 | −0.524318 | − | 1.93005i | 0 | −3.45018 | + | 2.02392i | 2.81794 | + | 1.16723i | 0 | −2.45726 | − | 2.45726i | 5.71526 | + | 5.59784i | 0 | 0.775311 | − | 6.05075i | ||||||
53.8 | −0.0880022 | + | 1.99806i | 0 | −3.98451 | − | 0.351668i | 6.78731 | + | 2.81140i | 0 | −0.402227 | − | 0.402227i | 1.05330 | − | 7.93036i | 0 | −6.21465 | + | 13.3141i | ||||||
53.9 | 0.268531 | − | 1.98189i | 0 | −3.85578 | − | 1.06440i | 1.91277 | + | 0.792297i | 0 | 5.76529 | + | 5.76529i | −3.14492 | + | 7.35591i | 0 | 2.08389 | − | 3.57815i | ||||||
53.10 | 0.891593 | + | 1.79027i | 0 | −2.41013 | + | 3.19238i | −2.45719 | − | 1.01780i | 0 | −4.99207 | − | 4.99207i | −7.86407 | − | 1.46847i | 0 | −0.368674 | − | 5.30649i | ||||||
53.11 | 1.02913 | − | 1.71490i | 0 | −1.88179 | − | 3.52971i | 0.0174241 | + | 0.00721732i | 0 | −8.15851 | − | 8.15851i | −7.98972 | − | 0.405443i | 0 | 0.0303087 | − | 0.0224532i | ||||||
53.12 | 1.23426 | − | 1.57372i | 0 | −0.953195 | − | 3.88477i | −8.50340 | − | 3.52222i | 0 | 5.30972 | + | 5.30972i | −7.29003 | − | 3.29476i | 0 | −16.0384 | + | 9.03463i | ||||||
53.13 | 1.71363 | + | 1.03125i | 0 | 1.87307 | + | 3.53435i | 6.91716 | + | 2.86518i | 0 | 2.33712 | + | 2.33712i | −0.435032 | + | 7.98816i | 0 | 8.89876 | + | 12.0432i | ||||||
53.14 | 1.78948 | + | 0.893172i | 0 | 2.40449 | + | 3.19663i | −4.02591 | − | 1.66758i | 0 | 3.15434 | + | 3.15434i | 1.44765 | + | 7.86793i | 0 | −5.71485 | − | 6.57994i | ||||||
53.15 | 1.85043 | − | 0.758899i | 0 | 2.84815 | − | 2.80857i | 4.62310 | + | 1.91495i | 0 | 3.34124 | + | 3.34124i | 3.13886 | − | 7.35850i | 0 | 10.0079 | + | 0.0350087i | ||||||
53.16 | 1.97643 | − | 0.306155i | 0 | 3.81254 | − | 1.21019i | −3.05280 | − | 1.26451i | 0 | −6.57413 | − | 6.57413i | 7.16470 | − | 3.55908i | 0 | −6.42077 | − | 1.56458i | ||||||
125.1 | −1.99481 | − | 0.143966i | 0 | 3.95855 | + | 0.574371i | −6.99476 | + | 2.89732i | 0 | −4.93873 | + | 4.93873i | −7.81387 | − | 1.71566i | 0 | 14.3703 | − | 4.77260i | ||||||
125.2 | −1.99180 | − | 0.180966i | 0 | 3.93450 | + | 0.720894i | 3.32592 | − | 1.37764i | 0 | 9.14542 | − | 9.14542i | −7.70627 | − | 2.14788i | 0 | −6.87386 | + | 2.14210i | ||||||
125.3 | −1.85072 | + | 0.758185i | 0 | 2.85031 | − | 2.80637i | 8.07100 | − | 3.34312i | 0 | −8.03674 | + | 8.03674i | −3.14737 | + | 7.35487i | 0 | −12.4024 | + | 12.3065i | ||||||
125.4 | −1.41770 | + | 1.41072i | 0 | 0.0197520 | − | 3.99995i | −3.22612 | + | 1.33630i | 0 | 2.13502 | − | 2.13502i | 5.61480 | + | 5.69860i | 0 | 2.68853 | − | 6.44563i | ||||||
See all 64 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
96.p | odd | 8 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 288.3.x.b | yes | 64 |
3.b | odd | 2 | 1 | 288.3.x.a | ✓ | 64 | |
32.g | even | 8 | 1 | 288.3.x.a | ✓ | 64 | |
96.p | odd | 8 | 1 | inner | 288.3.x.b | yes | 64 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
288.3.x.a | ✓ | 64 | 3.b | odd | 2 | 1 | |
288.3.x.a | ✓ | 64 | 32.g | even | 8 | 1 | |
288.3.x.b | yes | 64 | 1.a | even | 1 | 1 | trivial |
288.3.x.b | yes | 64 | 96.p | odd | 8 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{64} + 32 T_{5}^{61} - 11456 T_{5}^{59} - 8192 T_{5}^{58} + 17536 T_{5}^{57} + 62953216 T_{5}^{56} + 2693888 T_{5}^{55} - 133727232 T_{5}^{54} - 6721296640 T_{5}^{53} + \cdots + 12\!\cdots\!76 \)
acting on \(S_{3}^{\mathrm{new}}(288, [\chi])\).