Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [633,3,Mod(421,633)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(633, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 1]))
N = Newforms(chi, 3, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("633.421");
S:= CuspForms(chi, 3);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 633 = 3 \cdot 211 \) |
Weight: | \( k \) | \(=\) | \( 3 \) |
Character orbit: | \([\chi]\) | \(=\) | 633.c (of order \(2\), degree \(1\), 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: | \(17.2480007340\) |
Analytic rank: | \(0\) |
Dimension: | \(70\) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{2}]$ |
$q$-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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
421.1 | − | 3.89386i | 1.73205i | −11.1622 | −8.01349 | 6.74437 | 2.08342i | 27.8885i | −3.00000 | 31.2034i | |||||||||||||||||
421.2 | − | 3.86519i | 1.73205i | −10.9397 | 4.52341 | 6.69470 | − | 6.15822i | 26.8232i | −3.00000 | − | 17.4838i | |||||||||||||||
421.3 | − | 3.85406i | − | 1.73205i | −10.8538 | 3.36541 | −6.67544 | 6.97346i | 26.4150i | −3.00000 | − | 12.9705i | |||||||||||||||
421.4 | − | 3.66343i | − | 1.73205i | −9.42072 | −6.96010 | −6.34525 | − | 4.69902i | 19.8584i | −3.00000 | 25.4978i | |||||||||||||||
421.5 | − | 3.59880i | − | 1.73205i | −8.95137 | 0.151845 | −6.23331 | − | 11.0053i | 17.8190i | −3.00000 | − | 0.546461i | ||||||||||||||
421.6 | − | 3.48277i | 1.73205i | −8.12969 | 9.79345 | 6.03233 | 12.7499i | 14.3828i | −3.00000 | − | 34.1083i | ||||||||||||||||
421.7 | − | 3.41191i | 1.73205i | −7.64115 | 0.482983 | 5.90961 | − | 1.24753i | 12.4233i | −3.00000 | − | 1.64790i | |||||||||||||||
421.8 | − | 3.20877i | − | 1.73205i | −6.29622 | 5.84037 | −5.55776 | 1.24166i | 7.36803i | −3.00000 | − | 18.7404i | |||||||||||||||
421.9 | − | 3.20778i | − | 1.73205i | −6.28988 | −7.87944 | −5.55605 | 10.1392i | 7.34545i | −3.00000 | 25.2756i | ||||||||||||||||
421.10 | − | 3.14566i | − | 1.73205i | −5.89516 | 2.80095 | −5.44844 | − | 0.445584i | 5.96154i | −3.00000 | − | 8.81083i | ||||||||||||||
421.11 | − | 3.12474i | 1.73205i | −5.76401 | −2.63058 | 5.41221 | 11.5486i | 5.51207i | −3.00000 | 8.21987i | |||||||||||||||||
421.12 | − | 3.10766i | 1.73205i | −5.65757 | −8.33156 | 5.38263 | − | 5.74125i | 5.15117i | −3.00000 | 25.8917i | ||||||||||||||||
421.13 | − | 2.58659i | 1.73205i | −2.69047 | −3.76006 | 4.48011 | − | 13.5456i | − | 3.38722i | −3.00000 | 9.72574i | |||||||||||||||
421.14 | − | 2.57146i | − | 1.73205i | −2.61241 | 0.177066 | −4.45390 | 9.58955i | − | 3.56814i | −3.00000 | − | 0.455318i | ||||||||||||||
421.15 | − | 2.52983i | 1.73205i | −2.40004 | 2.80670 | 4.38179 | 2.89007i | − | 4.04762i | −3.00000 | − | 7.10047i | |||||||||||||||
421.16 | − | 2.52154i | 1.73205i | −2.35816 | 7.67033 | 4.36743 | − | 9.91383i | − | 4.13996i | −3.00000 | − | 19.3410i | ||||||||||||||
421.17 | − | 2.49430i | − | 1.73205i | −2.22151 | −0.0577969 | −4.32025 | − | 9.93281i | − | 4.43607i | −3.00000 | 0.144163i | ||||||||||||||
421.18 | − | 2.32666i | − | 1.73205i | −1.41336 | −6.54120 | −4.02990 | − | 1.49967i | − | 6.01823i | −3.00000 | 15.2192i | ||||||||||||||
421.19 | − | 2.21937i | 1.73205i | −0.925614 | −3.95657 | 3.84407 | 6.99995i | − | 6.82321i | −3.00000 | 8.78110i | ||||||||||||||||
421.20 | − | 2.15041i | 1.73205i | −0.624268 | 5.51087 | 3.72462 | − | 1.27747i | − | 7.25921i | −3.00000 | − | 11.8506i | ||||||||||||||
See all 70 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
211.b | odd | 2 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 633.3.c.a | ✓ | 70 |
211.b | odd | 2 | 1 | inner | 633.3.c.a | ✓ | 70 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
633.3.c.a | ✓ | 70 | 1.a | even | 1 | 1 | trivial |
633.3.c.a | ✓ | 70 | 211.b | odd | 2 | 1 | inner |
Hecke kernels
This newform subspace is the entire newspace \(S_{3}^{\mathrm{new}}(633, [\chi])\).