Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [111,3,Mod(82,111)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(111, base_ring=CyclotomicField(12))
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("111.82");
S:= CuspForms(chi, 3);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 111 = 3 \cdot 37 \) |
Weight: | \( k \) | \(=\) | \( 3 \) |
Character orbit: | \([\chi]\) | \(=\) | 111.l (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: | \(3.02453093440\) |
Analytic rank: | \(0\) |
Dimension: | \(24\) |
Relative dimension: | \(6\) 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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
82.1 | −3.81682 | − | 1.02271i | 1.50000 | + | 0.866025i | 10.0581 | + | 5.80704i | 3.12011 | − | 0.836031i | −4.83954 | − | 4.83954i | −4.42663 | + | 7.66714i | −21.2746 | − | 21.2746i | 1.50000 | + | 2.59808i | −12.7639 | ||
82.2 | −2.51039 | − | 0.672656i | 1.50000 | + | 0.866025i | 2.38548 | + | 1.37726i | −2.86514 | + | 0.767711i | −3.18304 | − | 3.18304i | 4.00759 | − | 6.94135i | 2.28887 | + | 2.28887i | 1.50000 | + | 2.59808i | 7.70901 | ||
82.3 | −0.839390 | − | 0.224914i | 1.50000 | + | 0.866025i | −2.81011 | − | 1.62242i | −2.87694 | + | 0.770875i | −1.06430 | − | 1.06430i | −5.46857 | + | 9.47184i | 4.45178 | + | 4.45178i | 1.50000 | + | 2.59808i | 2.58826 | ||
82.4 | −0.118812 | − | 0.0318357i | 1.50000 | + | 0.866025i | −3.45100 | − | 1.99244i | 7.22425 | − | 1.93573i | −0.150648 | − | 0.150648i | 0.350980 | − | 0.607914i | 0.694498 | + | 0.694498i | 1.50000 | + | 2.59808i | −0.919956 | ||
82.5 | 2.32451 | + | 0.622850i | 1.50000 | + | 0.866025i | 1.55129 | + | 0.895636i | 3.55969 | − | 0.953815i | 2.94736 | + | 2.94736i | 1.46050 | − | 2.52967i | −3.75850 | − | 3.75850i | 1.50000 | + | 2.59808i | 8.86860 | ||
82.6 | 3.09488 | + | 0.829271i | 1.50000 | + | 0.866025i | 5.42650 | + | 3.13299i | −3.56389 | + | 0.954941i | 3.92415 | + | 3.92415i | −1.12003 | + | 1.93996i | 5.13384 | + | 5.13384i | 1.50000 | + | 2.59808i | −11.8217 | ||
88.1 | −3.81682 | + | 1.02271i | 1.50000 | − | 0.866025i | 10.0581 | − | 5.80704i | 3.12011 | + | 0.836031i | −4.83954 | + | 4.83954i | −4.42663 | − | 7.66714i | −21.2746 | + | 21.2746i | 1.50000 | − | 2.59808i | −12.7639 | ||
88.2 | −2.51039 | + | 0.672656i | 1.50000 | − | 0.866025i | 2.38548 | − | 1.37726i | −2.86514 | − | 0.767711i | −3.18304 | + | 3.18304i | 4.00759 | + | 6.94135i | 2.28887 | − | 2.28887i | 1.50000 | − | 2.59808i | 7.70901 | ||
88.3 | −0.839390 | + | 0.224914i | 1.50000 | − | 0.866025i | −2.81011 | + | 1.62242i | −2.87694 | − | 0.770875i | −1.06430 | + | 1.06430i | −5.46857 | − | 9.47184i | 4.45178 | − | 4.45178i | 1.50000 | − | 2.59808i | 2.58826 | ||
88.4 | −0.118812 | + | 0.0318357i | 1.50000 | − | 0.866025i | −3.45100 | + | 1.99244i | 7.22425 | + | 1.93573i | −0.150648 | + | 0.150648i | 0.350980 | + | 0.607914i | 0.694498 | − | 0.694498i | 1.50000 | − | 2.59808i | −0.919956 | ||
88.5 | 2.32451 | − | 0.622850i | 1.50000 | − | 0.866025i | 1.55129 | − | 0.895636i | 3.55969 | + | 0.953815i | 2.94736 | − | 2.94736i | 1.46050 | + | 2.52967i | −3.75850 | + | 3.75850i | 1.50000 | − | 2.59808i | 8.86860 | ||
88.6 | 3.09488 | − | 0.829271i | 1.50000 | − | 0.866025i | 5.42650 | − | 3.13299i | −3.56389 | − | 0.954941i | 3.92415 | − | 3.92415i | −1.12003 | − | 1.93996i | 5.13384 | − | 5.13384i | 1.50000 | − | 2.59808i | −11.8217 | ||
97.1 | −0.693785 | − | 2.58924i | 1.50000 | − | 0.866025i | −2.75873 | + | 1.59276i | −0.209120 | + | 0.780447i | −3.28303 | − | 3.28303i | −3.61609 | − | 6.26326i | −1.54383 | − | 1.54383i | 1.50000 | − | 2.59808i | 2.16585 | ||
97.2 | −0.682073 | − | 2.54553i | 1.50000 | − | 0.866025i | −2.55040 | + | 1.47247i | 2.05541 | − | 7.67090i | −3.22760 | − | 3.22760i | 4.48853 | + | 7.77437i | −1.96605 | − | 1.96605i | 1.50000 | − | 2.59808i | −20.9284 | ||
97.3 | −0.103369 | − | 0.385779i | 1.50000 | − | 0.866025i | 3.32596 | − | 1.92024i | −2.38266 | + | 8.89221i | −0.489148 | − | 0.489148i | 6.18137 | + | 10.7065i | −2.21423 | − | 2.21423i | 1.50000 | − | 2.59808i | 3.67672 | ||
97.4 | −0.0210891 | − | 0.0787054i | 1.50000 | − | 0.866025i | 3.45835 | − | 1.99668i | 0.222135 | − | 0.829019i | −0.0997945 | − | 0.0997945i | −2.82588 | − | 4.89456i | −0.460548 | − | 0.460548i | 1.50000 | − | 2.59808i | −0.0699329 | ||
97.5 | 0.544686 | + | 2.03280i | 1.50000 | − | 0.866025i | −0.371475 | + | 0.214471i | 1.21483 | − | 4.53380i | 2.57748 | + | 2.57748i | 1.71728 | + | 2.97442i | 5.31413 | + | 5.31413i | 1.50000 | − | 2.59808i | 9.87799 | ||
97.6 | 0.821656 | + | 3.06646i | 1.50000 | − | 0.866025i | −5.26396 | + | 3.03915i | −1.49867 | + | 5.59311i | 3.88812 | + | 3.88812i | −0.749067 | − | 1.29742i | −4.66537 | − | 4.66537i | 1.50000 | − | 2.59808i | −18.3824 | ||
103.1 | −0.693785 | + | 2.58924i | 1.50000 | + | 0.866025i | −2.75873 | − | 1.59276i | −0.209120 | − | 0.780447i | −3.28303 | + | 3.28303i | −3.61609 | + | 6.26326i | −1.54383 | + | 1.54383i | 1.50000 | + | 2.59808i | 2.16585 | ||
103.2 | −0.682073 | + | 2.54553i | 1.50000 | + | 0.866025i | −2.55040 | − | 1.47247i | 2.05541 | + | 7.67090i | −3.22760 | + | 3.22760i | 4.48853 | − | 7.77437i | −1.96605 | + | 1.96605i | 1.50000 | + | 2.59808i | −20.9284 | ||
See all 24 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
37.g | odd | 12 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 111.3.l.b | ✓ | 24 |
3.b | odd | 2 | 1 | 333.3.bb.b | 24 | ||
37.g | odd | 12 | 1 | inner | 111.3.l.b | ✓ | 24 |
111.m | even | 12 | 1 | 333.3.bb.b | 24 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
111.3.l.b | ✓ | 24 | 1.a | even | 1 | 1 | trivial |
111.3.l.b | ✓ | 24 | 37.g | odd | 12 | 1 | inner |
333.3.bb.b | 24 | 3.b | odd | 2 | 1 | ||
333.3.bb.b | 24 | 111.m | even | 12 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{24} + 4 T_{2}^{23} - T_{2}^{22} - 8 T_{2}^{21} - 114 T_{2}^{20} - 562 T_{2}^{19} + 109 T_{2}^{18} + \cdots + 169 \) acting on \(S_{3}^{\mathrm{new}}(111, [\chi])\).