Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [441,2,Mod(25,441)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(441, base_ring=CyclotomicField(42))
chi = DirichletCharacter(H, H._module([28, 16]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("441.25");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 441 = 3^{2} \cdot 7^{2} \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 441.y (of order \(21\), 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: | \(3.52140272914\) |
Analytic rank: | \(0\) |
Dimension: | \(648\) |
Relative dimension: | \(54\) over \(\Q(\zeta_{21})\) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{21}]$ |
$q$-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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
25.1 | −0.620557 | + | 2.71884i | −1.73004 | + | 0.0835206i | −5.20506 | − | 2.50662i | −0.446610 | − | 1.13794i | 0.846508 | − | 4.75552i | 2.43233 | + | 1.04104i | 6.56762 | − | 8.23553i | 2.98605 | − | 0.288987i | 3.37104 | − | 0.508102i |
25.2 | −0.603411 | + | 2.64372i | 0.233708 | + | 1.71621i | −4.82320 | − | 2.32273i | −0.317680 | − | 0.809436i | −4.67820 | − | 0.417724i | −2.63556 | − | 0.231947i | 5.66957 | − | 7.10941i | −2.89076 | + | 0.802183i | 2.33161 | − | 0.351434i |
25.3 | −0.596104 | + | 2.61170i | 0.768590 | − | 1.55218i | −4.66370 | − | 2.24592i | 1.23942 | + | 3.15800i | 3.59567 | + | 2.93259i | 1.25551 | + | 2.32888i | 5.30523 | − | 6.65255i | −1.81854 | − | 2.38598i | −8.98657 | + | 1.35451i |
25.4 | −0.533217 | + | 2.33617i | 1.50562 | + | 0.856211i | −3.37146 | − | 1.62361i | 0.810456 | + | 2.06501i | −2.80308 | + | 3.06086i | 1.78727 | − | 1.95082i | 2.60267 | − | 3.26364i | 1.53381 | + | 2.57826i | −5.25637 | + | 0.792271i |
25.5 | −0.524991 | + | 2.30014i | −1.18605 | − | 1.26225i | −3.21307 | − | 1.54733i | 1.00605 | + | 2.56336i | 3.52602 | − | 2.06541i | −1.26982 | − | 2.32111i | 2.30393 | − | 2.88904i | −0.186564 | + | 2.99419i | −6.42425 | + | 0.968300i |
25.6 | −0.524692 | + | 2.29882i | 1.63349 | + | 0.575929i | −3.20735 | − | 1.54458i | −1.57024 | − | 4.00089i | −2.18104 | + | 3.45293i | 2.51000 | + | 0.836609i | 2.29328 | − | 2.87568i | 2.33661 | + | 1.88155i | 10.0212 | − | 1.51046i |
25.7 | −0.523139 | + | 2.29202i | 1.32112 | − | 1.12011i | −3.17776 | − | 1.53033i | −0.871294 | − | 2.22002i | 1.87618 | + | 3.61401i | −2.54574 | + | 0.720548i | 2.23835 | − | 2.80680i | 0.490715 | − | 2.95959i | 5.54415 | − | 0.835646i |
25.8 | −0.502908 | + | 2.20339i | −0.976404 | − | 1.43061i | −2.80006 | − | 1.34844i | −0.712094 | − | 1.81439i | 3.64322 | − | 1.43193i | −2.27396 | + | 1.35244i | 1.56106 | − | 1.95751i | −1.09327 | + | 2.79370i | 4.35591 | − | 0.656548i |
25.9 | −0.474737 | + | 2.07996i | −1.49636 | + | 0.872292i | −2.29891 | − | 1.10710i | −0.320939 | − | 0.817739i | −1.10395 | − | 3.52648i | −0.354961 | − | 2.62183i | 0.733726 | − | 0.920064i | 1.47821 | − | 2.61053i | 1.85322 | − | 0.279328i |
25.10 | −0.464927 | + | 2.03698i | 0.999043 | − | 1.41489i | −2.13119 | − | 1.02633i | −0.231038 | − | 0.588676i | 2.41762 | + | 2.69285i | 1.57123 | − | 2.12867i | 0.476059 | − | 0.596959i | −1.00382 | − | 2.82707i | 1.30654 | − | 0.196929i |
25.11 | −0.436037 | + | 1.91040i | 0.660299 | + | 1.60125i | −1.65757 | − | 0.798243i | 0.532974 | + | 1.35800i | −3.34695 | + | 0.563231i | 0.00125911 | + | 2.64575i | −0.195769 | + | 0.245486i | −2.12801 | + | 2.11461i | −2.82672 | + | 0.426059i |
25.12 | −0.416291 | + | 1.82389i | −0.611111 | + | 1.62066i | −1.35134 | − | 0.650771i | −0.0698517 | − | 0.177979i | −2.70151 | − | 1.78927i | 2.35192 | + | 1.21181i | −0.583357 | + | 0.731506i | −2.25309 | − | 1.98081i | 0.353693 | − | 0.0533107i |
25.13 | −0.395139 | + | 1.73122i | 1.72082 | + | 0.196905i | −1.03904 | − | 0.500374i | 1.06496 | + | 2.71348i | −1.02085 | + | 2.90131i | −2.62846 | + | 0.301952i | −0.937490 | + | 1.17557i | 2.92246 | + | 0.677677i | −5.11842 | + | 0.771478i |
25.14 | −0.322970 | + | 1.41503i | −1.49665 | − | 0.871802i | −0.0960492 | − | 0.0462548i | −1.42246 | − | 3.62437i | 1.71700 | − | 1.83623i | 1.71510 | − | 2.01455i | −1.71341 | + | 2.14855i | 1.47992 | + | 2.60956i | 5.58799 | − | 0.842254i |
25.15 | −0.316624 | + | 1.38722i | −1.35055 | + | 1.08445i | −0.0221929 | − | 0.0106875i | −1.32101 | − | 3.36589i | −1.07675 | − | 2.21687i | −1.86586 | + | 1.87578i | −1.75247 | + | 2.19753i | 0.647946 | − | 2.92919i | 5.08750 | − | 0.766817i |
25.16 | −0.302994 | + | 1.32750i | −1.73138 | − | 0.0482649i | 0.131481 | + | 0.0633181i | 1.28284 | + | 3.26863i | 0.588668 | − | 2.28378i | 2.63656 | + | 0.220319i | −1.82183 | + | 2.28450i | 2.99534 | + | 0.167130i | −4.72781 | + | 0.712603i |
25.17 | −0.279438 | + | 1.22430i | −0.359714 | − | 1.69429i | 0.381117 | + | 0.183536i | 0.462772 | + | 1.17912i | 2.17483 | + | 0.0330508i | 2.42137 | − | 1.06628i | −1.89714 | + | 2.37894i | −2.74121 | + | 1.21892i | −1.57292 | + | 0.237079i |
25.18 | −0.263181 | + | 1.15307i | 1.58469 | − | 0.699101i | 0.541631 | + | 0.260836i | −0.0198537 | − | 0.0505863i | 0.389051 | + | 2.01125i | 1.10457 | + | 2.40415i | −1.91814 | + | 2.40527i | 2.02252 | − | 2.21572i | 0.0635547 | − | 0.00957933i |
25.19 | −0.234242 | + | 1.02628i | −0.449041 | + | 1.67283i | 0.803556 | + | 0.386972i | 0.973235 | + | 2.47976i | −1.61161 | − | 0.852689i | −1.43872 | − | 2.22038i | −1.89803 | + | 2.38005i | −2.59672 | − | 1.50234i | −2.77290 | + | 0.417948i |
25.20 | −0.222631 | + | 0.975409i | −0.211834 | − | 1.71905i | 0.900079 | + | 0.433455i | 0.692294 | + | 1.76394i | 1.72394 | + | 0.176088i | −0.942297 | + | 2.47226i | −1.87078 | + | 2.34588i | −2.91025 | + | 0.728307i | −1.87469 | + | 0.282563i |
See next 80 embeddings (of 648 total) |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
441.y | even | 21 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 441.2.y.a | ✓ | 648 |
9.c | even | 3 | 1 | 441.2.z.a | yes | 648 | |
49.g | even | 21 | 1 | 441.2.z.a | yes | 648 | |
441.y | even | 21 | 1 | inner | 441.2.y.a | ✓ | 648 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
441.2.y.a | ✓ | 648 | 1.a | even | 1 | 1 | trivial |
441.2.y.a | ✓ | 648 | 441.y | even | 21 | 1 | inner |
441.2.z.a | yes | 648 | 9.c | even | 3 | 1 | |
441.2.z.a | yes | 648 | 49.g | even | 21 | 1 |
Hecke kernels
This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(441, [\chi])\).