Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [912,2,Mod(67,912)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(912, base_ring=CyclotomicField(36))
chi = DirichletCharacter(H, H._module([18, 27, 0, 34]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("912.67");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 912 = 2^{4} \cdot 3 \cdot 19 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 912.cn (of order \(36\), degree \(12\), 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: | \(7.28235666434\) |
Analytic rank: | \(0\) |
Dimension: | \(960\) |
Relative dimension: | \(80\) over \(\Q(\zeta_{36})\) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{36}]$ |
$q$-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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
67.1 | −1.41411 | − | 0.0169281i | −0.996195 | − | 0.0871557i | 1.99943 | + | 0.0478763i | 2.68717 | − | 1.25305i | 1.40726 | + | 0.140112i | 0.522822 | − | 0.905554i | −2.82660 | − | 0.101549i | 0.984808 | + | 0.173648i | −3.82117 | + | 1.72646i |
67.2 | −1.41324 | − | 0.0523593i | 0.996195 | + | 0.0871557i | 1.99452 | + | 0.147993i | 0.825343 | − | 0.384864i | −1.40330 | − | 0.175332i | −1.56239 | + | 2.70614i | −2.81099 | − | 0.313582i | 0.984808 | + | 0.173648i | −1.18656 | + | 0.500692i |
67.3 | −1.39604 | + | 0.225998i | −0.996195 | − | 0.0871557i | 1.89785 | − | 0.631005i | −3.02036 | + | 1.40842i | 1.41042 | − | 0.103465i | 0.0696086 | − | 0.120566i | −2.50687 | + | 1.30982i | 0.984808 | + | 0.173648i | 3.89824 | − | 2.64880i |
67.4 | −1.39537 | − | 0.230080i | 0.996195 | + | 0.0871557i | 1.89413 | + | 0.642094i | 1.80595 | − | 0.842128i | −1.37001 | − | 0.350819i | 0.971129 | − | 1.68204i | −2.49528 | − | 1.33176i | 0.984808 | + | 0.173648i | −2.71373 | + | 0.759570i |
67.5 | −1.38343 | + | 0.293462i | 0.996195 | + | 0.0871557i | 1.82776 | − | 0.811970i | −1.56884 | + | 0.731561i | −1.40374 | + | 0.171772i | 2.36990 | − | 4.10479i | −2.29030 | + | 1.65968i | 0.984808 | + | 0.173648i | 1.95569 | − | 1.47246i |
67.6 | −1.38263 | − | 0.297198i | −0.996195 | − | 0.0871557i | 1.82335 | + | 0.821833i | −1.83893 | + | 0.857507i | 1.35147 | + | 0.416572i | −2.27979 | + | 3.94871i | −2.27677 | − | 1.67819i | 0.984808 | + | 0.173648i | 2.79741 | − | 0.639090i |
67.7 | −1.36953 | + | 0.352702i | −0.996195 | − | 0.0871557i | 1.75120 | − | 0.966068i | 0.690327 | − | 0.321905i | 1.39505 | − | 0.231997i | 1.45929 | − | 2.52757i | −2.05758 | + | 1.94071i | 0.984808 | + | 0.173648i | −0.831884 | + | 0.684336i |
67.8 | −1.36898 | + | 0.354799i | 0.996195 | + | 0.0871557i | 1.74824 | − | 0.971427i | −0.640238 | + | 0.298548i | −1.39470 | + | 0.234134i | −1.64256 | + | 2.84500i | −2.04865 | + | 1.95014i | 0.984808 | + | 0.173648i | 0.770552 | − | 0.635863i |
67.9 | −1.36646 | + | 0.364412i | −0.996195 | − | 0.0871557i | 1.73441 | − | 0.995906i | 1.59591 | − | 0.744183i | 1.39302 | − | 0.243931i | −0.878950 | + | 1.52239i | −2.00707 | + | 1.99290i | 0.984808 | + | 0.173648i | −1.90955 | + | 1.59846i |
67.10 | −1.32152 | − | 0.503585i | 0.996195 | + | 0.0871557i | 1.49280 | + | 1.33099i | −3.76611 | + | 1.75616i | −1.27260 | − | 0.616846i | 0.782781 | − | 1.35582i | −1.30250 | − | 2.51068i | 0.984808 | + | 0.173648i | 5.86135 | − | 0.424243i |
67.11 | −1.31402 | − | 0.522838i | −0.996195 | − | 0.0871557i | 1.45328 | + | 1.37404i | −1.75107 | + | 0.816536i | 1.26345 | + | 0.635372i | −0.825112 | + | 1.42914i | −1.19124 | − | 2.56534i | 0.984808 | + | 0.173648i | 2.72785 | − | 0.157418i |
67.12 | −1.27607 | + | 0.609618i | 0.996195 | + | 0.0871557i | 1.25673 | − | 1.55584i | 3.88530 | − | 1.81174i | −1.32435 | + | 0.496082i | 0.817991 | − | 1.41680i | −0.655215 | + | 2.75149i | 0.984808 | + | 0.173648i | −3.85346 | + | 4.68047i |
67.13 | −1.22595 | − | 0.705018i | 0.996195 | + | 0.0871557i | 1.00590 | + | 1.72863i | −0.977951 | + | 0.456026i | −1.15984 | − | 0.809183i | −0.576210 | + | 0.998025i | −0.0144655 | − | 2.82839i | 0.984808 | + | 0.173648i | 1.52042 | + | 0.130408i |
67.14 | −1.18996 | + | 0.764195i | 0.996195 | + | 0.0871557i | 0.832013 | − | 1.81872i | −1.97149 | + | 0.919319i | −1.25204 | + | 0.657575i | 0.262357 | − | 0.454416i | 0.399795 | + | 2.80003i | 0.984808 | + | 0.173648i | 1.64345 | − | 2.60055i |
67.15 | −1.16246 | + | 0.805415i | −0.996195 | − | 0.0871557i | 0.702613 | − | 1.87252i | −2.99568 | + | 1.39691i | 1.22823 | − | 0.701035i | 0.753682 | − | 1.30542i | 0.691398 | + | 2.74262i | 0.984808 | + | 0.173648i | 2.35726 | − | 4.03661i |
67.16 | −1.12072 | − | 0.862545i | 0.996195 | + | 0.0871557i | 0.512032 | + | 1.93335i | 2.97317 | − | 1.38641i | −1.04128 | − | 0.956940i | −0.314742 | + | 0.545150i | 1.09375 | − | 2.60839i | 0.984808 | + | 0.173648i | −4.52794 | − | 1.01071i |
67.17 | −1.10948 | − | 0.876960i | −0.996195 | − | 0.0871557i | 0.461884 | + | 1.94594i | 1.24808 | − | 0.581989i | 1.02882 | + | 0.970320i | 1.65003 | − | 2.85793i | 1.19406 | − | 2.56403i | 0.984808 | + | 0.173648i | −1.89510 | − | 0.448811i |
67.18 | −1.09515 | + | 0.894789i | −0.996195 | − | 0.0871557i | 0.398706 | − | 1.95986i | 1.67888 | − | 0.782875i | 1.16897 | − | 0.795935i | −1.05656 | + | 1.83001i | 1.31701 | + | 2.50309i | 0.984808 | + | 0.173648i | −1.13812 | + | 2.35961i |
67.19 | −1.06269 | − | 0.933102i | −0.996195 | − | 0.0871557i | 0.258640 | + | 1.98321i | −0.264715 | + | 0.123438i | 0.977326 | + | 1.02217i | −0.251352 | + | 0.435354i | 1.57568 | − | 2.34888i | 0.984808 | + | 0.173648i | 0.396492 | + | 0.115828i |
67.20 | −0.992328 | + | 1.00761i | −0.996195 | − | 0.0871557i | −0.0305710 | − | 1.99977i | −0.810516 | + | 0.377950i | 1.07637 | − | 0.917292i | −2.00575 | + | 3.47406i | 2.04533 | + | 1.95362i | 0.984808 | + | 0.173648i | 0.423470 | − | 1.19174i |
See next 80 embeddings (of 960 total) |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
16.f | odd | 4 | 1 | inner |
19.f | odd | 18 | 1 | inner |
304.bg | even | 36 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 912.2.cn.a | ✓ | 960 |
16.f | odd | 4 | 1 | inner | 912.2.cn.a | ✓ | 960 |
19.f | odd | 18 | 1 | inner | 912.2.cn.a | ✓ | 960 |
304.bg | even | 36 | 1 | inner | 912.2.cn.a | ✓ | 960 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
912.2.cn.a | ✓ | 960 | 1.a | even | 1 | 1 | trivial |
912.2.cn.a | ✓ | 960 | 16.f | odd | 4 | 1 | inner |
912.2.cn.a | ✓ | 960 | 19.f | odd | 18 | 1 | inner |
912.2.cn.a | ✓ | 960 | 304.bg | even | 36 | 1 | inner |
Hecke kernels
This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(912, [\chi])\).