Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [29,3,Mod(2,29)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(29, base_ring=CyclotomicField(28))
chi = DirichletCharacter(H, H._module([1]))
N = Newforms(chi, 3, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("29.2");
S:= CuspForms(chi, 3);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 29 \) |
Weight: | \( k \) | \(=\) | \( 3 \) |
Character orbit: | \([\chi]\) | \(=\) | 29.f (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: | \(0.790192766645\) |
Analytic rank: | \(0\) |
Dimension: | \(48\) |
Relative dimension: | \(4\) 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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
2.1 | −3.56136 | − | 0.401269i | 1.78374 | + | 1.12080i | 8.62256 | + | 1.96804i | 5.24106 | − | 4.17960i | −5.90279 | − | 4.70732i | 2.11977 | + | 9.28734i | −16.3872 | − | 5.73413i | −1.97942 | − | 4.11031i | −20.3424 | + | 12.7820i |
2.2 | −1.68783 | − | 0.190173i | −3.78594 | − | 2.37886i | −1.08711 | − | 0.248126i | 0.141728 | − | 0.113024i | 5.93762 | + | 4.73509i | −1.55116 | − | 6.79606i | 8.20045 | + | 2.86946i | 4.76938 | + | 9.90371i | −0.260707 | + | 0.163813i |
2.3 | 0.415096 | + | 0.0467701i | 2.68233 | + | 1.68542i | −3.72959 | − | 0.851255i | 0.738700 | − | 0.589093i | 1.03460 | + | 0.825065i | −0.577468 | − | 2.53005i | −3.08546 | − | 1.07965i | 0.449302 | + | 0.932984i | 0.334184 | − | 0.209981i |
2.4 | 2.58169 | + | 0.290886i | −2.11401 | − | 1.32832i | 2.68078 | + | 0.611870i | −2.49104 | + | 1.98654i | −5.07133 | − | 4.04425i | 1.30161 | + | 5.70272i | −3.06598 | − | 1.07283i | −1.20034 | − | 2.49253i | −7.00893 | + | 4.40400i |
3.1 | −3.24379 | − | 2.03821i | −2.23035 | + | 0.780434i | 4.63233 | + | 9.61914i | −5.12808 | + | 1.17045i | 8.82547 | + | 2.01436i | −6.56728 | − | 3.16264i | 2.86375 | − | 25.4165i | −2.67109 | + | 2.13013i | 19.0200 | + | 6.65539i |
3.2 | −1.29187 | − | 0.811733i | 2.15095 | − | 0.752648i | −0.725528 | − | 1.50658i | 3.36294 | − | 0.767569i | −3.38968 | − | 0.773673i | −0.255710 | − | 0.123143i | −0.968958 | + | 8.59974i | −2.97640 | + | 2.37360i | −4.96753 | − | 1.73821i |
3.3 | 1.44470 | + | 0.907767i | 1.22940 | − | 0.430186i | −0.472410 | − | 0.980970i | −8.15497 | + | 1.86132i | 2.16663 | + | 0.494518i | 7.53715 | + | 3.62970i | 0.972146 | − | 8.62804i | −5.71012 | + | 4.55367i | −13.4711 | − | 4.71376i |
3.4 | 2.05804 | + | 1.29315i | −2.93183 | + | 1.02589i | 0.827740 | + | 1.71882i | 5.87720 | − | 1.34143i | −7.36043 | − | 1.67997i | −9.36468 | − | 4.50979i | 0.569384 | − | 5.05343i | 0.506667 | − | 0.404053i | 13.8301 | + | 4.83938i |
8.1 | −2.32673 | − | 0.814157i | 0.184193 | − | 1.63476i | 1.62348 | + | 1.29468i | −3.83207 | − | 7.95737i | −1.75952 | + | 3.65367i | 2.23777 | + | 2.80607i | 2.52264 | + | 4.01476i | 6.13584 | + | 1.40047i | 2.43763 | + | 21.6345i |
8.2 | −2.20133 | − | 0.770280i | −0.552425 | + | 4.90291i | 1.12521 | + | 0.897324i | 2.64264 | + | 5.48750i | 4.99268 | − | 10.3674i | −3.22936 | − | 4.04949i | 3.17747 | + | 5.05691i | −14.9590 | − | 3.41428i | −1.59042 | − | 14.1154i |
8.3 | −0.0310749 | − | 0.0108736i | 0.529545 | − | 4.69985i | −3.12648 | − | 2.49328i | 3.79007 | + | 7.87017i | −0.0675598 | + | 0.140289i | 3.62612 | + | 4.54701i | 0.140107 | + | 0.222980i | −13.0338 | − | 2.97487i | −0.0321993 | − | 0.285777i |
8.4 | 1.65381 | + | 0.578694i | −0.186386 | + | 1.65422i | −0.727117 | − | 0.579856i | −0.825315 | − | 1.71379i | −1.26553 | + | 2.62791i | −1.24782 | − | 1.56472i | −4.59573 | − | 7.31406i | 6.07265 | + | 1.38604i | −0.373160 | − | 3.31188i |
10.1 | −3.24379 | + | 2.03821i | −2.23035 | − | 0.780434i | 4.63233 | − | 9.61914i | −5.12808 | − | 1.17045i | 8.82547 | − | 2.01436i | −6.56728 | + | 3.16264i | 2.86375 | + | 25.4165i | −2.67109 | − | 2.13013i | 19.0200 | − | 6.65539i |
10.2 | −1.29187 | + | 0.811733i | 2.15095 | + | 0.752648i | −0.725528 | + | 1.50658i | 3.36294 | + | 0.767569i | −3.38968 | + | 0.773673i | −0.255710 | + | 0.123143i | −0.968958 | − | 8.59974i | −2.97640 | − | 2.37360i | −4.96753 | + | 1.73821i |
10.3 | 1.44470 | − | 0.907767i | 1.22940 | + | 0.430186i | −0.472410 | + | 0.980970i | −8.15497 | − | 1.86132i | 2.16663 | − | 0.494518i | 7.53715 | − | 3.62970i | 0.972146 | + | 8.62804i | −5.71012 | − | 4.55367i | −13.4711 | + | 4.71376i |
10.4 | 2.05804 | − | 1.29315i | −2.93183 | − | 1.02589i | 0.827740 | − | 1.71882i | 5.87720 | + | 1.34143i | −7.36043 | + | 1.67997i | −9.36468 | + | 4.50979i | 0.569384 | + | 5.05343i | 0.506667 | + | 0.404053i | 13.8301 | − | 4.83938i |
11.1 | −2.32673 | + | 0.814157i | 0.184193 | + | 1.63476i | 1.62348 | − | 1.29468i | −3.83207 | + | 7.95737i | −1.75952 | − | 3.65367i | 2.23777 | − | 2.80607i | 2.52264 | − | 4.01476i | 6.13584 | − | 1.40047i | 2.43763 | − | 21.6345i |
11.2 | −2.20133 | + | 0.770280i | −0.552425 | − | 4.90291i | 1.12521 | − | 0.897324i | 2.64264 | − | 5.48750i | 4.99268 | + | 10.3674i | −3.22936 | + | 4.04949i | 3.17747 | − | 5.05691i | −14.9590 | + | 3.41428i | −1.59042 | + | 14.1154i |
11.3 | −0.0310749 | + | 0.0108736i | 0.529545 | + | 4.69985i | −3.12648 | + | 2.49328i | 3.79007 | − | 7.87017i | −0.0675598 | − | 0.140289i | 3.62612 | − | 4.54701i | 0.140107 | − | 0.222980i | −13.0338 | + | 2.97487i | −0.0321993 | + | 0.285777i |
11.4 | 1.65381 | − | 0.578694i | −0.186386 | − | 1.65422i | −0.727117 | + | 0.579856i | −0.825315 | + | 1.71379i | −1.26553 | − | 2.62791i | −1.24782 | + | 1.56472i | −4.59573 | + | 7.31406i | 6.07265 | − | 1.38604i | −0.373160 | + | 3.31188i |
See all 48 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
29.f | odd | 28 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 29.3.f.a | ✓ | 48 |
3.b | odd | 2 | 1 | 261.3.s.a | 48 | ||
29.f | odd | 28 | 1 | inner | 29.3.f.a | ✓ | 48 |
87.k | even | 28 | 1 | 261.3.s.a | 48 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
29.3.f.a | ✓ | 48 | 1.a | even | 1 | 1 | trivial |
29.3.f.a | ✓ | 48 | 29.f | odd | 28 | 1 | inner |
261.3.s.a | 48 | 3.b | odd | 2 | 1 | ||
261.3.s.a | 48 | 87.k | even | 28 | 1 |
Hecke kernels
This newform subspace is the entire newspace \(S_{3}^{\mathrm{new}}(29, [\chi])\).