Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [164,3,Mod(23,164)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(164, base_ring=CyclotomicField(10))
chi = DirichletCharacter(H, H._module([5, 9]))
N = Newforms(chi, 3, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("164.23");
S:= CuspForms(chi, 3);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 164 = 2^{2} \cdot 41 \) |
Weight: | \( k \) | \(=\) | \( 3 \) |
Character orbit: | \([\chi]\) | \(=\) | 164.l (of order \(10\), 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: | \(4.46867633551\) |
Analytic rank: | \(0\) |
Dimension: | \(160\) |
Relative dimension: | \(40\) over \(\Q(\zeta_{10})\) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{10}]$ |
$q$-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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
23.1 | −1.99597 | − | 0.126903i | 2.39245 | 3.96779 | + | 0.506589i | −2.73540 | + | 1.98738i | −4.77525 | − | 0.303609i | −2.96576 | + | 9.12767i | −7.85530 | − | 1.51466i | −3.27620 | 5.71198 | − | 3.61963i | ||||
23.2 | −1.98941 | − | 0.205567i | 5.80392 | 3.91548 | + | 0.817912i | −0.392502 | + | 0.285169i | −11.5464 | − | 1.19309i | 3.50661 | − | 10.7922i | −7.62136 | − | 2.43205i | 24.6855 | 0.839467 | − | 0.486633i | ||||
23.3 | −1.94462 | + | 0.467387i | −3.71133 | 3.56310 | − | 1.81778i | −4.22575 | + | 3.07019i | 7.21714 | − | 1.73463i | −1.55229 | + | 4.77745i | −6.07927 | + | 5.20024i | 4.77400 | 6.78252 | − | 7.94541i | ||||
23.4 | −1.94108 | + | 0.481888i | 0.0526689 | 3.53557 | − | 1.87076i | 6.01711 | − | 4.37168i | −0.102234 | + | 0.0253805i | 1.71992 | − | 5.29336i | −5.96132 | + | 5.33504i | −8.99723 | −9.57301 | + | 11.3853i | ||||
23.5 | −1.88686 | − | 0.663140i | −5.42935 | 3.12049 | + | 2.50251i | 2.11565 | − | 1.53711i | 10.2444 | + | 3.60042i | 2.69024 | − | 8.27972i | −4.22842 | − | 6.79121i | 20.4779 | −5.01127 | + | 1.49734i | ||||
23.6 | −1.87660 | − | 0.691635i | −1.80399 | 3.04328 | + | 2.59585i | 1.98179 | − | 1.43986i | 3.38537 | + | 1.24770i | −1.53649 | + | 4.72884i | −3.91565 | − | 6.97622i | −5.74562 | −4.71489 | + | 1.33136i | ||||
23.7 | −1.82820 | − | 0.810985i | 0.174784 | 2.68461 | + | 2.96528i | −5.60132 | + | 4.06960i | −0.319540 | − | 0.141747i | 3.43849 | − | 10.5826i | −2.50319 | − | 7.59829i | −8.96945 | 13.5407 | − | 2.89744i | ||||
23.8 | −1.59233 | + | 1.21015i | 4.23702 | 1.07105 | − | 3.85394i | 2.10237 | − | 1.52746i | −6.74676 | + | 5.12745i | −2.22561 | + | 6.84972i | 2.95838 | + | 7.43290i | 8.95238 | −1.49921 | + | 4.97643i | ||||
23.9 | −1.56964 | − | 1.23944i | 3.24329 | 0.927554 | + | 3.89097i | 7.04687 | − | 5.11985i | −5.09081 | − | 4.01988i | −0.790554 | + | 2.43307i | 3.36671 | − | 7.25708i | 1.51893 | −17.4068 | − | 0.697870i | ||||
23.10 | −1.49088 | + | 1.33315i | −3.04044 | 0.445422 | − | 3.97512i | 0.357384 | − | 0.259655i | 4.53292 | − | 4.05337i | 1.22026 | − | 3.75559i | 4.63537 | + | 6.52023i | 0.244298 | −0.186657 | + | 0.863560i | ||||
23.11 | −1.47996 | + | 1.34526i | 2.13105 | 0.380577 | − | 3.98185i | −7.01448 | + | 5.09632i | −3.15388 | + | 2.86681i | 1.33584 | − | 4.11128i | 4.79337 | + | 6.40497i | −4.45862 | 3.52532 | − | 16.9786i | ||||
23.12 | −1.21580 | − | 1.58803i | 0.677342 | −1.04367 | + | 3.86144i | −1.99860 | + | 1.45207i | −0.823511 | − | 1.07564i | 0.934712 | − | 2.87675i | 7.40098 | − | 3.03736i | −8.54121 | 4.73583 | + | 1.40842i | ||||
23.13 | −0.995129 | − | 1.73485i | −3.65070 | −2.01944 | + | 3.45281i | 2.46264 | − | 1.78921i | 3.63292 | + | 6.33344i | −1.94656 | + | 5.99089i | 7.99972 | + | 0.0674398i | 4.32763 | −5.55467 | − | 2.49182i | ||||
23.14 | −0.981376 | − | 1.74267i | 5.08911 | −2.07380 | + | 3.42043i | −6.37097 | + | 4.62878i | −4.99432 | − | 8.86863i | −2.93081 | + | 9.02010i | 7.99586 | + | 0.257229i | 16.8990 | 14.3188 | + | 6.55993i | ||||
23.15 | −0.888883 | + | 1.79162i | −1.25447 | −2.41978 | − | 3.18507i | 4.13475 | − | 3.00407i | 1.11507 | − | 2.24752i | −3.84198 | + | 11.8244i | 7.85732 | − | 1.50415i | −7.42631 | 1.70683 | + | 10.0781i | ||||
23.16 | −0.568924 | + | 1.91737i | 4.43662 | −3.35265 | − | 2.18168i | 1.63991 | − | 1.19147i | −2.52410 | + | 8.50667i | 2.22047 | − | 6.83389i | 6.09050 | − | 5.18708i | 10.6836 | 1.35150 | + | 3.82218i | ||||
23.17 | −0.491389 | + | 1.93869i | 0.0745071 | −3.51707 | − | 1.90530i | 1.56299 | − | 1.13558i | −0.0366119 | + | 0.144447i | 2.91894 | − | 8.98359i | 5.42205 | − | 5.88229i | −8.99445 | 1.43351 | + | 3.58818i | ||||
23.18 | −0.288297 | + | 1.97911i | −4.83617 | −3.83377 | − | 1.14114i | −5.05367 | + | 3.67170i | 1.39425 | − | 9.57132i | −1.02466 | + | 3.15358i | 3.36371 | − | 7.25847i | 14.3885 | −5.80976 | − | 11.0603i | ||||
23.19 | −0.230366 | − | 1.98669i | −5.08911 | −3.89386 | + | 0.915333i | −6.37097 | + | 4.62878i | 1.17236 | + | 10.1105i | 2.93081 | − | 9.02010i | 2.71550 | + | 7.52503i | 16.8990 | 10.6636 | + | 11.5908i | ||||
23.20 | −0.214645 | − | 1.98845i | 3.65070 | −3.90785 | + | 0.853622i | 2.46264 | − | 1.78921i | −0.783606 | − | 7.25923i | 1.94656 | − | 5.99089i | 2.53619 | + | 7.58734i | 4.32763 | −4.08635 | − | 4.51279i | ||||
See next 80 embeddings (of 160 total) |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
4.b | odd | 2 | 1 | inner |
41.f | even | 10 | 1 | inner |
164.l | odd | 10 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 164.3.l.a | ✓ | 160 |
4.b | odd | 2 | 1 | inner | 164.3.l.a | ✓ | 160 |
41.f | even | 10 | 1 | inner | 164.3.l.a | ✓ | 160 |
164.l | odd | 10 | 1 | inner | 164.3.l.a | ✓ | 160 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
164.3.l.a | ✓ | 160 | 1.a | even | 1 | 1 | trivial |
164.3.l.a | ✓ | 160 | 4.b | odd | 2 | 1 | inner |
164.3.l.a | ✓ | 160 | 41.f | even | 10 | 1 | inner |
164.3.l.a | ✓ | 160 | 164.l | odd | 10 | 1 | inner |
Hecke kernels
This newform subspace is the entire newspace \(S_{3}^{\mathrm{new}}(164, [\chi])\).