Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [690,2,Mod(89,690)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(690, base_ring=CyclotomicField(22))
chi = DirichletCharacter(H, H._module([11, 11, 5]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("690.89");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 690 = 2 \cdot 3 \cdot 5 \cdot 23 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 690.n (of order \(22\), degree \(10\), 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: | \(5.50967773947\) |
Analytic rank: | \(0\) |
Dimension: | \(240\) |
Relative dimension: | \(24\) over \(\Q(\zeta_{22})\) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{22}]$ |
$q$-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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
89.1 | −0.142315 | − | 0.989821i | −1.68612 | + | 0.396228i | −0.959493 | + | 0.281733i | 2.06421 | + | 0.859668i | 0.632155 | + | 1.61257i | 0.955977 | − | 0.614369i | 0.415415 | + | 0.909632i | 2.68601 | − | 1.33618i | 0.557150 | − | 2.16554i |
89.2 | −0.142315 | − | 0.989821i | −1.67059 | + | 0.457309i | −0.959493 | + | 0.281733i | 1.26417 | − | 1.84441i | 0.690404 | + | 1.58850i | −1.66558 | + | 1.07040i | 0.415415 | + | 0.909632i | 2.58174 | − | 1.52795i | −2.00555 | − | 0.988819i |
89.3 | −0.142315 | − | 0.989821i | −1.66535 | + | 0.476026i | −0.959493 | + | 0.281733i | −2.23496 | + | 0.0702463i | 0.708185 | + | 1.58066i | 3.21136 | − | 2.06381i | 0.415415 | + | 0.909632i | 2.54680 | − | 1.58550i | 0.387600 | + | 2.20222i |
89.4 | −0.142315 | − | 0.989821i | −1.58165 | − | 0.705954i | −0.959493 | + | 0.281733i | −1.76486 | + | 1.37305i | −0.473676 | + | 1.66602i | −3.04110 | + | 1.95439i | 0.415415 | + | 0.909632i | 2.00326 | + | 2.23315i | 1.61024 | + | 1.55149i |
89.5 | −0.142315 | − | 0.989821i | −1.42695 | − | 0.981738i | −0.959493 | + | 0.281733i | −0.584056 | − | 2.15844i | −0.768669 | + | 1.55214i | −1.28454 | + | 0.825522i | 0.415415 | + | 0.909632i | 1.07238 | + | 2.80178i | −2.05335 | + | 0.885290i |
89.6 | −0.142315 | − | 0.989821i | −1.34594 | + | 1.09016i | −0.959493 | + | 0.281733i | −2.02703 | − | 0.944001i | 1.27061 | + | 1.17710i | −1.08447 | + | 0.696948i | 0.415415 | + | 0.909632i | 0.623112 | − | 2.93458i | −0.645916 | + | 2.14075i |
89.7 | −0.142315 | − | 0.989821i | −1.16282 | − | 1.28368i | −0.959493 | + | 0.281733i | 2.23565 | − | 0.0433117i | −1.10513 | + | 1.33367i | 1.95813 | − | 1.25842i | 0.415415 | + | 0.909632i | −0.295684 | + | 2.98539i | −0.361037 | − | 2.20673i |
89.8 | −0.142315 | − | 0.989821i | −0.943600 | + | 1.45245i | −0.959493 | + | 0.281733i | −0.0269160 | + | 2.23591i | 1.57196 | + | 0.727290i | −2.74772 | + | 1.76585i | 0.415415 | + | 0.909632i | −1.21924 | − | 2.74107i | 2.21698 | − | 0.291561i |
89.9 | −0.142315 | − | 0.989821i | −0.763868 | − | 1.55451i | −0.959493 | + | 0.281733i | −1.25961 | − | 1.84754i | −1.42998 | + | 0.977323i | 3.27194 | − | 2.10275i | 0.415415 | + | 0.909632i | −1.83301 | + | 2.37488i | −1.64947 | + | 1.50972i |
89.10 | −0.142315 | − | 0.989821i | −0.479764 | + | 1.66428i | −0.959493 | + | 0.281733i | 2.20932 | − | 0.344845i | 1.71562 | + | 0.238029i | 2.74772 | − | 1.76585i | 0.415415 | + | 0.909632i | −2.53965 | − | 1.59692i | −0.655753 | − | 2.13775i |
89.11 | −0.142315 | − | 0.989821i | −0.397135 | − | 1.68591i | −0.959493 | + | 0.281733i | 2.02447 | + | 0.949486i | −1.61223 | + | 0.633022i | −4.08214 | + | 2.62343i | 0.415415 | + | 0.909632i | −2.68457 | + | 1.33906i | 0.651709 | − | 2.13899i |
89.12 | −0.142315 | − | 0.989821i | 0.0512012 | − | 1.73129i | −0.959493 | + | 0.281733i | 0.182307 | + | 2.22862i | −1.72096 | + | 0.195709i | 1.96759 | − | 1.26449i | 0.415415 | + | 0.909632i | −2.99476 | − | 0.177289i | 2.17999 | − | 0.497618i |
89.13 | −0.142315 | − | 0.989821i | 0.0575173 | + | 1.73110i | −0.959493 | + | 0.281733i | −1.22287 | − | 1.87206i | 1.70529 | − | 0.303292i | 1.08447 | − | 0.696948i | 0.415415 | + | 0.909632i | −2.99338 | + | 0.199136i | −1.67897 | + | 1.47684i |
89.14 | −0.142315 | − | 0.989821i | 0.277794 | − | 1.70963i | −0.959493 | + | 0.281733i | −1.79300 | + | 1.33609i | −1.73176 | − | 0.0316608i | −0.483302 | + | 0.310599i | 0.415415 | + | 0.909632i | −2.84566 | − | 0.949849i | 1.57766 | + | 1.58461i |
89.15 | −0.142315 | − | 0.989821i | 0.730817 | + | 1.57032i | −0.959493 | + | 0.281733i | −0.248537 | − | 2.22221i | 1.45033 | − | 0.946859i | −3.21136 | + | 2.06381i | 0.415415 | + | 0.909632i | −1.93181 | + | 2.29523i | −2.16422 | + | 0.562261i |
89.16 | −0.142315 | − | 0.989821i | 0.748393 | + | 1.56202i | −0.959493 | + | 0.281733i | −1.64573 | + | 1.51379i | 1.43961 | − | 0.963074i | 1.66558 | − | 1.07040i | 0.415415 | + | 0.909632i | −1.87982 | + | 2.33801i | 1.73260 | + | 1.41354i |
89.17 | −0.142315 | − | 0.989821i | 0.804725 | + | 1.53376i | −0.959493 | + | 0.281733i | 1.14469 | + | 1.92086i | 1.40362 | − | 1.01481i | −0.955977 | + | 0.614369i | 0.415415 | + | 0.909632i | −1.70483 | + | 2.46851i | 1.73840 | − | 1.40640i |
89.18 | −0.142315 | − | 0.989821i | 1.11013 | − | 1.32951i | −0.959493 | + | 0.281733i | 1.06732 | − | 1.96490i | −1.47397 | − | 0.909626i | 0.483302 | − | 0.310599i | 0.415415 | + | 0.909632i | −0.535201 | − | 2.95187i | −2.09680 | − | 0.776821i |
89.19 | −0.142315 | − | 0.989821i | 1.27489 | − | 1.17245i | −0.959493 | + | 0.281733i | 2.23188 | − | 0.136714i | −1.34195 | − | 1.09506i | −1.96759 | + | 1.26449i | 0.415415 | + | 0.909632i | 0.250714 | − | 2.98951i | −0.452953 | − | 2.18971i |
89.20 | −0.142315 | − | 0.989821i | 1.53419 | − | 0.803900i | −0.959493 | + | 0.281733i | 1.22793 | + | 1.86874i | −1.01406 | − | 1.40417i | 4.08214 | − | 2.62343i | 0.415415 | + | 0.909632i | 1.70749 | − | 2.46667i | 1.67496 | − | 1.48138i |
See next 80 embeddings (of 240 total) |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
15.d | odd | 2 | 1 | inner |
23.d | odd | 22 | 1 | inner |
345.n | even | 22 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 690.2.n.a | ✓ | 240 |
3.b | odd | 2 | 1 | 690.2.n.b | yes | 240 | |
5.b | even | 2 | 1 | 690.2.n.b | yes | 240 | |
15.d | odd | 2 | 1 | inner | 690.2.n.a | ✓ | 240 |
23.d | odd | 22 | 1 | inner | 690.2.n.a | ✓ | 240 |
69.g | even | 22 | 1 | 690.2.n.b | yes | 240 | |
115.i | odd | 22 | 1 | 690.2.n.b | yes | 240 | |
345.n | even | 22 | 1 | inner | 690.2.n.a | ✓ | 240 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
690.2.n.a | ✓ | 240 | 1.a | even | 1 | 1 | trivial |
690.2.n.a | ✓ | 240 | 15.d | odd | 2 | 1 | inner |
690.2.n.a | ✓ | 240 | 23.d | odd | 22 | 1 | inner |
690.2.n.a | ✓ | 240 | 345.n | even | 22 | 1 | inner |
690.2.n.b | yes | 240 | 3.b | odd | 2 | 1 | |
690.2.n.b | yes | 240 | 5.b | even | 2 | 1 | |
690.2.n.b | yes | 240 | 69.g | even | 22 | 1 | |
690.2.n.b | yes | 240 | 115.i | odd | 22 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator \( T_{17}^{120} - 83 T_{17}^{118} + 176 T_{17}^{117} + 6090 T_{17}^{116} - 20438 T_{17}^{115} + \cdots + 23\!\cdots\!84 \) acting on \(S_{2}^{\mathrm{new}}(690, [\chi])\).