Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [287,3,Mod(44,287)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(287, base_ring=CyclotomicField(24))
chi = DirichletCharacter(H, H._module([8, 9]))
N = Newforms(chi, 3, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("287.44");
S:= CuspForms(chi, 3);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 287 = 7 \cdot 41 \) |
Weight: | \( k \) | \(=\) | \( 3 \) |
Character orbit: | \([\chi]\) | \(=\) | 287.v (of order \(24\), degree \(8\), 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: | \(7.82018358714\) |
Analytic rank: | \(0\) |
Dimension: | \(432\) |
Relative dimension: | \(54\) over \(\Q(\zeta_{24})\) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{24}]$ |
$q$-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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
44.1 | −1.00113 | − | 3.73626i | 4.33433 | − | 0.570626i | −9.49326 | + | 5.48094i | 6.12199 | − | 1.64038i | −6.47122 | − | 15.6229i | −6.59685 | − | 2.34128i | 19.0416 | + | 19.0416i | 9.76749 | − | 2.61719i | −12.2578 | − | 21.2311i |
44.2 | −0.964481 | − | 3.59949i | −2.15102 | + | 0.283187i | −8.56202 | + | 4.94329i | 4.97373 | − | 1.33271i | 3.09394 | + | 7.46944i | 6.47657 | + | 2.65595i | 15.5112 | + | 15.5112i | −4.14666 | + | 1.11109i | −9.59414 | − | 16.6175i |
44.3 | −0.953963 | − | 3.56024i | 3.93802 | − | 0.518450i | −8.30116 | + | 4.79268i | −6.13448 | + | 1.64373i | −5.60253 | − | 13.5257i | 2.91136 | + | 6.36584i | 14.5570 | + | 14.5570i | 6.54587 | − | 1.75396i | 11.7041 | + | 20.2722i |
44.4 | −0.934785 | − | 3.48866i | 0.164201 | − | 0.0216175i | −7.83285 | + | 4.52230i | −1.38302 | + | 0.370580i | −0.228909 | − | 0.552635i | 0.440489 | − | 6.98613i | 12.8833 | + | 12.8833i | −8.66684 | + | 2.32227i | 2.58566 | + | 4.47849i |
44.5 | −0.895161 | − | 3.34079i | −2.33888 | + | 0.307920i | −6.89545 | + | 3.98109i | −6.16719 | + | 1.65249i | 3.12237 | + | 7.53807i | −5.68361 | + | 4.08615i | 9.69000 | + | 9.69000i | −3.31778 | + | 0.888995i | 11.0413 | + | 19.1240i |
44.6 | −0.845521 | − | 3.15553i | −5.30986 | + | 0.699057i | −5.77835 | + | 3.33613i | −4.43760 | + | 1.18905i | 6.69550 | + | 16.1644i | 6.89536 | + | 1.20580i | 6.17295 | + | 6.17295i | 19.0126 | − | 5.09442i | 7.50417 | + | 12.9976i |
44.7 | −0.822999 | − | 3.07147i | −0.665898 | + | 0.0876672i | −5.29251 | + | 3.05563i | 6.18603 | − | 1.65754i | 0.817301 | + | 1.97314i | −2.80548 | + | 6.41321i | 4.74713 | + | 4.74713i | −8.25760 | + | 2.21262i | −10.1822 | − | 17.6361i |
44.8 | −0.805923 | − | 3.00775i | 2.67346 | − | 0.351967i | −4.93293 | + | 2.84803i | −4.41031 | + | 1.18174i | −3.21323 | − | 7.75742i | 6.30583 | − | 3.03916i | 3.73441 | + | 3.73441i | −1.66984 | + | 0.447433i | 7.10875 | + | 12.3127i |
44.9 | −0.743556 | − | 2.77499i | −4.02310 | + | 0.529651i | −3.68360 | + | 2.12672i | 5.12912 | − | 1.37434i | 4.46118 | + | 10.7702i | −1.22522 | − | 6.89194i | 0.514865 | + | 0.514865i | 7.21145 | − | 1.93230i | −7.62758 | − | 13.2113i |
44.10 | −0.711148 | − | 2.65404i | 4.57597 | − | 0.602438i | −3.07410 | + | 1.77483i | 5.06808 | − | 1.35799i | −4.85309 | − | 11.7164i | 5.42371 | + | 4.42530i | −0.874947 | − | 0.874947i | 11.8832 | − | 3.18410i | −7.20831 | − | 12.4852i |
44.11 | −0.669740 | − | 2.49950i | 1.94687 | − | 0.256310i | −2.33487 | + | 1.34804i | −2.15743 | + | 0.578081i | −1.94454 | − | 4.69454i | −6.88381 | − | 1.27009i | −2.38587 | − | 2.38587i | −4.96873 | + | 1.33137i | 2.88983 | + | 5.00534i |
44.12 | −0.605293 | − | 2.25898i | 5.84002 | − | 0.768853i | −1.27252 | + | 0.734692i | −4.08855 | + | 1.09552i | −5.27175 | − | 12.7271i | −1.90551 | − | 6.73565i | −4.18485 | − | 4.18485i | 24.8213 | − | 6.65085i | 4.94953 | + | 8.57284i |
44.13 | −0.601567 | − | 2.24508i | −4.60009 | + | 0.605614i | −1.21439 | + | 0.701128i | 1.97238 | − | 0.528497i | 4.12691 | + | 9.96325i | −6.51687 | + | 2.55546i | −4.26942 | − | 4.26942i | 12.1007 | − | 3.24239i | −2.37303 | − | 4.11021i |
44.14 | −0.585405 | − | 2.18476i | 2.68487 | − | 0.353470i | −0.966375 | + | 0.557937i | 7.59297 | − | 2.03453i | −2.34399 | − | 5.65888i | 2.78312 | − | 6.42295i | −4.61274 | − | 4.61274i | −1.60972 | + | 0.431323i | −8.88992 | − | 15.3978i |
44.15 | −0.518206 | − | 1.93397i | −0.899559 | + | 0.118429i | −0.00760404 | + | 0.00439020i | −3.55516 | + | 0.952603i | 0.695196 | + | 1.67835i | 5.90938 | + | 3.75223i | −5.65063 | − | 5.65063i | −7.89815 | + | 2.11630i | 3.68461 | + | 6.38194i |
44.16 | −0.472662 | − | 1.76400i | −3.59669 | + | 0.473513i | 0.575817 | − | 0.332448i | −8.51355 | + | 2.28120i | 2.53530 | + | 6.12074i | −2.12311 | − | 6.67026i | −6.02395 | − | 6.02395i | 4.01862 | − | 1.07679i | 8.04807 | + | 13.9397i |
44.17 | −0.472042 | − | 1.76168i | 0.937698 | − | 0.123450i | 0.583396 | − | 0.336824i | 2.21585 | − | 0.593735i | −0.660113 | − | 1.59365i | 2.05508 | + | 6.69153i | −6.02733 | − | 6.02733i | −7.82930 | + | 2.09785i | −2.09195 | − | 3.62336i |
44.18 | −0.432281 | − | 1.61329i | −1.08853 | + | 0.143308i | 1.04825 | − | 0.605209i | −2.34666 | + | 0.628787i | 0.701750 | + | 1.69417i | 5.57803 | − | 4.22914i | −6.15357 | − | 6.15357i | −7.52897 | + | 2.01738i | 2.02884 | + | 3.51405i |
44.19 | −0.358555 | − | 1.33815i | 2.81232 | − | 0.370250i | 1.80203 | − | 1.04040i | −9.33412 | + | 2.50107i | −1.50382 | − | 3.63054i | −3.19103 | + | 6.23036i | −5.95670 | − | 5.95670i | −0.921246 | + | 0.246847i | 6.69359 | + | 11.5936i |
44.20 | −0.345775 | − | 1.29045i | 0.635300 | − | 0.0836388i | 1.91841 | − | 1.10759i | 7.27076 | − | 1.94819i | −0.327602 | − | 0.790901i | −5.53537 | − | 4.28481i | −5.87132 | − | 5.87132i | −8.29672 | + | 2.22310i | −5.02809 | − | 8.70890i |
See next 80 embeddings (of 432 total) |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
7.c | even | 3 | 1 | inner |
41.e | odd | 8 | 1 | inner |
287.v | odd | 24 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 287.3.v.a | ✓ | 432 |
7.c | even | 3 | 1 | inner | 287.3.v.a | ✓ | 432 |
41.e | odd | 8 | 1 | inner | 287.3.v.a | ✓ | 432 |
287.v | odd | 24 | 1 | inner | 287.3.v.a | ✓ | 432 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
287.3.v.a | ✓ | 432 | 1.a | even | 1 | 1 | trivial |
287.3.v.a | ✓ | 432 | 7.c | even | 3 | 1 | inner |
287.3.v.a | ✓ | 432 | 41.e | odd | 8 | 1 | inner |
287.3.v.a | ✓ | 432 | 287.v | odd | 24 | 1 | inner |
Hecke kernels
This newform subspace is the entire newspace \(S_{3}^{\mathrm{new}}(287, [\chi])\).