Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [105,4,Mod(52,105)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(105, base_ring=CyclotomicField(12))
chi = DirichletCharacter(H, H._module([0, 3, 2]))
N = Newforms(chi, 4, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("105.52");
S:= CuspForms(chi, 4);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 105 = 3 \cdot 5 \cdot 7 \) |
Weight: | \( k \) | \(=\) | \( 4 \) |
Character orbit: | \([\chi]\) | \(=\) | 105.u (of order \(12\), 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: | \(6.19520055060\) |
Analytic rank: | \(0\) |
Dimension: | \(96\) |
Relative dimension: | \(24\) over \(\Q(\zeta_{12})\) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{12}]$ |
$q$-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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
52.1 | −1.41620 | + | 5.28532i | −2.89778 | + | 0.776457i | −19.0007 | − | 10.9701i | 11.0616 | − | 1.62545i | − | 16.4153i | 0.587638 | − | 18.5109i | 53.9362 | − | 53.9362i | 7.79423 | − | 4.50000i | −7.07431 | + | 60.7657i | |
52.2 | −1.31873 | + | 4.92156i | 2.89778 | − | 0.776457i | −15.5545 | − | 8.98042i | −0.535886 | + | 11.1675i | 15.2855i | −13.3869 | + | 12.7980i | 35.8873 | − | 35.8873i | 7.79423 | − | 4.50000i | −54.2548 | − | 17.3643i | ||
52.3 | −1.24190 | + | 4.63484i | −2.89778 | + | 0.776457i | −13.0112 | − | 7.51204i | −10.1233 | − | 4.74540i | − | 14.3950i | −3.69659 | + | 18.1476i | 23.8323 | − | 23.8323i | 7.79423 | − | 4.50000i | 34.5663 | − | 41.0266i | |
52.4 | −1.17018 | + | 4.36716i | 2.89778 | − | 0.776457i | −10.7746 | − | 6.22070i | −6.73223 | − | 8.92620i | 13.5637i | −1.57380 | − | 18.4533i | 14.1991 | − | 14.1991i | 7.79423 | − | 4.50000i | 46.8601 | − | 18.9555i | ||
52.5 | −1.14346 | + | 4.26746i | 2.89778 | − | 0.776457i | −9.97548 | − | 5.75935i | 9.24082 | − | 6.29343i | 13.2540i | 15.1987 | + | 10.5830i | 10.9924 | − | 10.9924i | 7.79423 | − | 4.50000i | 16.2904 | + | 46.6311i | ||
52.6 | −0.871352 | + | 3.25193i | −2.89778 | + | 0.776457i | −2.88758 | − | 1.66715i | 6.94823 | + | 8.75911i | − | 10.0999i | 10.8772 | + | 14.9896i | −11.1071 | + | 11.1071i | 7.79423 | − | 4.50000i | −34.5384 | + | 14.9629i | |
52.7 | −0.624851 | + | 2.33198i | −2.89778 | + | 0.776457i | 1.88053 | + | 1.08572i | −9.58456 | + | 5.75641i | − | 7.24272i | −5.69491 | − | 17.6229i | −17.3639 | + | 17.3639i | 7.79423 | − | 4.50000i | −7.43490 | − | 25.9479i | |
52.8 | −0.529124 | + | 1.97472i | 2.89778 | − | 0.776457i | 3.30867 | + | 1.91026i | −11.1089 | + | 1.26224i | 6.13313i | −16.3965 | + | 8.61137i | −17.0877 | + | 17.0877i | 7.79423 | − | 4.50000i | 3.38539 | − | 22.6047i | ||
52.9 | −0.514613 | + | 1.92056i | −2.89778 | + | 0.776457i | 3.50448 | + | 2.02331i | 5.44490 | − | 9.76489i | − | 5.96493i | 14.3387 | − | 11.7218i | −16.9369 | + | 16.9369i | 7.79423 | − | 4.50000i | 15.9520 | + | 15.4824i | |
52.10 | −0.465010 | + | 1.73544i | 2.89778 | − | 0.776457i | 4.13268 | + | 2.38600i | 2.32023 | + | 10.9369i | 5.38999i | 13.1952 | − | 12.9956i | −16.2260 | + | 16.2260i | 7.79423 | − | 4.50000i | −20.0594 | − | 1.05916i | ||
52.11 | −0.121182 | + | 0.452258i | 2.89778 | − | 0.776457i | 6.73835 | + | 3.89039i | 10.5131 | − | 3.80453i | 1.40463i | −5.91256 | + | 17.5511i | −5.22463 | + | 5.22463i | 7.79423 | − | 4.50000i | 0.446628 | + | 5.21568i | ||
52.12 | 0.0422777 | − | 0.157783i | −2.89778 | + | 0.776457i | 6.90510 | + | 3.98666i | 9.25781 | + | 6.26841i | 0.490046i | −18.5183 | − | 0.270750i | 1.84500 | − | 1.84500i | 7.79423 | − | 4.50000i | 1.38044 | − | 1.19571i | ||
52.13 | 0.121038 | − | 0.451720i | −2.89778 | + | 0.776457i | 6.73880 | + | 3.89065i | −10.1913 | − | 4.59752i | 1.40296i | 16.4236 | + | 8.55956i | 5.21859 | − | 5.21859i | 7.79423 | − | 4.50000i | −3.31032 | + | 4.04714i | ||
52.14 | 0.131114 | − | 0.489325i | 2.89778 | − | 0.776457i | 6.70596 | + | 3.87169i | 1.00965 | − | 11.1347i | − | 1.51976i | −6.72481 | − | 17.2562i | 5.63944 | − | 5.63944i | 7.79423 | − | 4.50000i | −5.31609 | − | 1.95396i | |
52.15 | 0.366039 | − | 1.36607i | −2.89778 | + | 0.776457i | 5.19603 | + | 2.99993i | −4.85914 | − | 10.0692i | 4.24279i | −18.5166 | − | 0.368999i | 14.0004 | − | 14.0004i | 7.79423 | − | 4.50000i | −15.5339 | + | 2.95223i | ||
52.16 | 0.503088 | − | 1.87755i | 2.89778 | − | 0.776457i | 3.65611 | + | 2.11085i | −11.1743 | + | 0.367232i | − | 5.83135i | 18.4739 | − | 1.30903i | 16.7983 | − | 16.7983i | 7.79423 | − | 4.50000i | −4.93216 | + | 21.1651i | |
52.17 | 0.662502 | − | 2.47249i | 2.89778 | − | 0.776457i | 1.25390 | + | 0.723938i | 4.16719 | + | 10.3747i | − | 7.67914i | 6.62391 | + | 17.2952i | 17.1006 | − | 17.1006i | 7.79423 | − | 4.50000i | 28.4121 | − | 3.43007i | |
52.18 | 0.791608 | − | 2.95432i | −2.89778 | + | 0.776457i | −1.17317 | − | 0.677329i | 2.51558 | + | 10.8937i | 9.17562i | 5.89275 | − | 17.5578i | 14.3720 | − | 14.3720i | 7.79423 | − | 4.50000i | 34.1747 | + | 1.19167i | ||
52.19 | 0.815920 | − | 3.04505i | −2.89778 | + | 0.776457i | −1.67842 | − | 0.969037i | 10.1281 | − | 4.73506i | 9.45741i | 13.1472 | + | 13.0442i | 13.5128 | − | 13.5128i | 7.79423 | − | 4.50000i | −6.15476 | − | 34.7042i | ||
52.20 | 1.00662 | − | 3.75676i | 2.89778 | − | 0.776457i | −6.17173 | − | 3.56325i | 10.7872 | + | 2.93879i | − | 11.6678i | −14.4985 | − | 11.5236i | 2.40224 | − | 2.40224i | 7.79423 | − | 4.50000i | 21.8989 | − | 37.5666i | |
See all 96 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
5.c | odd | 4 | 1 | inner |
7.d | odd | 6 | 1 | inner |
35.k | even | 12 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 105.4.u.a | ✓ | 96 |
5.c | odd | 4 | 1 | inner | 105.4.u.a | ✓ | 96 |
7.d | odd | 6 | 1 | inner | 105.4.u.a | ✓ | 96 |
35.k | even | 12 | 1 | inner | 105.4.u.a | ✓ | 96 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
105.4.u.a | ✓ | 96 | 1.a | even | 1 | 1 | trivial |
105.4.u.a | ✓ | 96 | 5.c | odd | 4 | 1 | inner |
105.4.u.a | ✓ | 96 | 7.d | odd | 6 | 1 | inner |
105.4.u.a | ✓ | 96 | 35.k | even | 12 | 1 | inner |
Hecke kernels
This newform subspace is the entire newspace \(S_{4}^{\mathrm{new}}(105, [\chi])\).