Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [102,5,Mod(47,102)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(102, base_ring=CyclotomicField(4))
chi = DirichletCharacter(H, H._module([2, 1]))
N = Newforms(chi, 5, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("102.47");
S:= CuspForms(chi, 5);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 102 = 2 \cdot 3 \cdot 17 \) |
Weight: | \( k \) | \(=\) | \( 5 \) |
Character orbit: | \([\chi]\) | \(=\) | 102.e (of order \(4\), degree \(2\), 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: | \(10.5437362346\) |
Analytic rank: | \(0\) |
Dimension: | \(48\) |
Relative dimension: | \(24\) over \(\Q(i)\) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{4}]$ |
$q$-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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
47.1 | −2.82843 | −8.80790 | + | 1.84956i | 8.00000 | 29.6453 | − | 29.6453i | 24.9125 | − | 5.23133i | −54.3772 | + | 54.3772i | −22.6274 | 74.1583 | − | 32.5814i | −83.8495 | + | 83.8495i | ||||||
47.2 | −2.82843 | −7.85161 | − | 4.39911i | 8.00000 | −30.6085 | + | 30.6085i | 22.2077 | + | 12.4426i | −1.55384 | + | 1.55384i | −22.6274 | 42.2956 | + | 69.0802i | 86.5740 | − | 86.5740i | ||||||
47.3 | −2.82843 | −7.71559 | + | 4.63353i | 8.00000 | −1.85553 | + | 1.85553i | 21.8230 | − | 13.1056i | 25.3003 | − | 25.3003i | −22.6274 | 38.0607 | − | 71.5009i | 5.24824 | − | 5.24824i | ||||||
47.4 | −2.82843 | −5.06624 | − | 7.43863i | 8.00000 | 5.10481 | − | 5.10481i | 14.3295 | + | 21.0396i | −37.6419 | + | 37.6419i | −22.6274 | −29.6665 | + | 75.3717i | −14.4386 | + | 14.4386i | ||||||
47.5 | −2.82843 | −1.72878 | − | 8.83240i | 8.00000 | 22.2248 | − | 22.2248i | 4.88972 | + | 24.9818i | 54.5002 | − | 54.5002i | −22.6274 | −75.0227 | + | 30.5385i | −62.8612 | + | 62.8612i | ||||||
47.6 | −2.82843 | −0.433615 | + | 8.98955i | 8.00000 | −21.7545 | + | 21.7545i | 1.22645 | − | 25.4263i | −17.2401 | + | 17.2401i | −22.6274 | −80.6240 | − | 7.79601i | 61.5311 | − | 61.5311i | ||||||
47.7 | −2.82843 | 0.612626 | + | 8.97913i | 8.00000 | 22.9323 | − | 22.9323i | −1.73277 | − | 25.3968i | 22.7811 | − | 22.7811i | −22.6274 | −80.2494 | + | 11.0017i | −64.8624 | + | 64.8624i | ||||||
47.8 | −2.82843 | 1.53089 | − | 8.86884i | 8.00000 | −3.11497 | + | 3.11497i | −4.33000 | + | 25.0849i | −30.3117 | + | 30.3117i | −22.6274 | −76.3128 | − | 27.1544i | 8.81045 | − | 8.81045i | ||||||
47.9 | −2.82843 | 6.09014 | − | 6.62647i | 8.00000 | −17.8296 | + | 17.8296i | −17.2255 | + | 18.7425i | 8.63785 | − | 8.63785i | −22.6274 | −6.82034 | − | 80.7123i | 50.4297 | − | 50.4297i | ||||||
47.10 | −2.82843 | 7.47777 | + | 5.00829i | 8.00000 | 5.40912 | − | 5.40912i | −21.1503 | − | 14.1656i | −54.1428 | + | 54.1428i | −22.6274 | 30.8340 | + | 74.9017i | −15.2993 | + | 15.2993i | ||||||
47.11 | −2.82843 | 8.00369 | + | 4.11595i | 8.00000 | −16.0342 | + | 16.0342i | −22.6378 | − | 11.6417i | 61.3672 | − | 61.3672i | −22.6274 | 47.1180 | + | 65.8855i | 45.3515 | − | 45.3515i | ||||||
47.12 | −2.82843 | 8.71706 | − | 2.23896i | 8.00000 | 28.5084 | − | 28.5084i | −24.6556 | + | 6.33274i | 22.6809 | − | 22.6809i | −22.6274 | 70.9741 | − | 39.0343i | −80.6339 | + | 80.6339i | ||||||
47.13 | 2.82843 | −8.86884 | + | 1.53089i | 8.00000 | 3.11497 | − | 3.11497i | −25.0849 | + | 4.33000i | −30.3117 | + | 30.3117i | 22.6274 | 76.3128 | − | 27.1544i | 8.81045 | − | 8.81045i | ||||||
47.14 | 2.82843 | −8.83240 | − | 1.72878i | 8.00000 | −22.2248 | + | 22.2248i | −24.9818 | − | 4.88972i | 54.5002 | − | 54.5002i | 22.6274 | 75.0227 | + | 30.5385i | −62.8612 | + | 62.8612i | ||||||
47.15 | 2.82843 | −7.43863 | − | 5.06624i | 8.00000 | −5.10481 | + | 5.10481i | −21.0396 | − | 14.3295i | −37.6419 | + | 37.6419i | 22.6274 | 29.6665 | + | 75.3717i | −14.4386 | + | 14.4386i | ||||||
47.16 | 2.82843 | −6.62647 | + | 6.09014i | 8.00000 | 17.8296 | − | 17.8296i | −18.7425 | + | 17.2255i | 8.63785 | − | 8.63785i | 22.6274 | 6.82034 | − | 80.7123i | 50.4297 | − | 50.4297i | ||||||
47.17 | 2.82843 | −4.39911 | − | 7.85161i | 8.00000 | 30.6085 | − | 30.6085i | −12.4426 | − | 22.2077i | −1.55384 | + | 1.55384i | 22.6274 | −42.2956 | + | 69.0802i | 86.5740 | − | 86.5740i | ||||||
47.18 | 2.82843 | −2.23896 | + | 8.71706i | 8.00000 | −28.5084 | + | 28.5084i | −6.33274 | + | 24.6556i | 22.6809 | − | 22.6809i | 22.6274 | −70.9741 | − | 39.0343i | −80.6339 | + | 80.6339i | ||||||
47.19 | 2.82843 | 1.84956 | − | 8.80790i | 8.00000 | −29.6453 | + | 29.6453i | 5.23133 | − | 24.9125i | −54.3772 | + | 54.3772i | 22.6274 | −74.1583 | − | 32.5814i | −83.8495 | + | 83.8495i | ||||||
47.20 | 2.82843 | 4.11595 | + | 8.00369i | 8.00000 | 16.0342 | − | 16.0342i | 11.6417 | + | 22.6378i | 61.3672 | − | 61.3672i | 22.6274 | −47.1180 | + | 65.8855i | 45.3515 | − | 45.3515i | ||||||
See all 48 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
3.b | odd | 2 | 1 | inner |
17.c | even | 4 | 1 | inner |
51.f | odd | 4 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 102.5.e.a | ✓ | 48 |
3.b | odd | 2 | 1 | inner | 102.5.e.a | ✓ | 48 |
17.c | even | 4 | 1 | inner | 102.5.e.a | ✓ | 48 |
51.f | odd | 4 | 1 | inner | 102.5.e.a | ✓ | 48 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
102.5.e.a | ✓ | 48 | 1.a | even | 1 | 1 | trivial |
102.5.e.a | ✓ | 48 | 3.b | odd | 2 | 1 | inner |
102.5.e.a | ✓ | 48 | 17.c | even | 4 | 1 | inner |
102.5.e.a | ✓ | 48 | 51.f | odd | 4 | 1 | inner |
Hecke kernels
This newform subspace is the entire newspace \(S_{5}^{\mathrm{new}}(102, [\chi])\).