Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [592,2,Mod(339,592)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(592, base_ring=CyclotomicField(4))
chi = DirichletCharacter(H, H._module([2, 3, 3]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("592.339");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 592 = 2^{4} \cdot 37 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 592.s (of order \(4\), degree \(2\), 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: | \(4.72714379966\) |
Analytic rank: | \(0\) |
Dimension: | \(148\) |
Relative dimension: | \(74\) over \(\Q(i)\) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{4}]$ |
$q$-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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
339.1 | −1.40895 | − | 0.121888i | 1.82659 | + | 1.82659i | 1.97029 | + | 0.343469i | 2.73488 | −2.35094 | − | 2.79622i | 0.0550223 | −2.73417 | − | 0.724086i | 3.67289i | −3.85331 | − | 0.333349i | ||||||
339.2 | −1.40004 | − | 0.199708i | −1.00580 | − | 1.00580i | 1.92023 | + | 0.559198i | 3.89641 | 1.20729 | + | 1.60902i | −2.11242 | −2.57673 | − | 1.16639i | − | 0.976747i | −5.45514 | − | 0.778143i | |||||
339.3 | −1.39780 | + | 0.214867i | −1.10115 | − | 1.10115i | 1.90766 | − | 0.600681i | −1.71699 | 1.77579 | + | 1.30259i | −0.420453 | −2.53746 | + | 1.24952i | − | 0.574923i | 2.40000 | − | 0.368925i | |||||
339.4 | −1.39081 | − | 0.256242i | −2.29968 | − | 2.29968i | 1.86868 | + | 0.712764i | −2.15315 | 2.60914 | + | 3.78768i | −4.08815 | −2.41633 | − | 1.47015i | 7.57707i | 2.99461 | + | 0.551726i | ||||||
339.5 | −1.38151 | + | 0.302357i | −0.641946 | − | 0.641946i | 1.81716 | − | 0.835419i | 1.23635 | 1.08095 | + | 0.692761i | 3.44294 | −2.25784 | + | 1.70357i | − | 2.17581i | −1.70804 | + | 0.373819i | |||||
339.6 | −1.38027 | − | 0.307994i | 2.14376 | + | 2.14376i | 1.81028 | + | 0.850229i | −3.28719 | −2.29869 | − | 3.61922i | 4.29351 | −2.23680 | − | 1.73110i | 6.19139i | 4.53720 | + | 1.01244i | ||||||
339.7 | −1.37121 | − | 0.346101i | −0.415690 | − | 0.415690i | 1.76043 | + | 0.949154i | −1.23281 | 0.426127 | + | 0.713868i | 0.515574 | −2.08541 | − | 1.91077i | − | 2.65440i | 1.69045 | + | 0.426678i | |||||
339.8 | −1.36814 | + | 0.358024i | 0.814535 | + | 0.814535i | 1.74364 | − | 0.979658i | −3.64825 | −1.40603 | − | 0.822778i | −2.22907 | −2.03481 | + | 1.96458i | − | 1.67306i | 4.99133 | − | 1.30616i | |||||
339.9 | −1.35541 | + | 0.403572i | 1.14471 | + | 1.14471i | 1.67426 | − | 1.09401i | 1.84909 | −2.01352 | − | 1.08957i | 3.26353 | −1.82779 | + | 2.15851i | − | 0.379294i | −2.50627 | + | 0.746241i | |||||
339.10 | −1.30913 | + | 0.534969i | 0.704988 | + | 0.704988i | 1.42762 | − | 1.40068i | 0.949460 | −1.30006 | − | 0.545771i | −3.64867 | −1.11961 | + | 2.59740i | − | 2.00598i | −1.24296 | + | 0.507931i | |||||
339.11 | −1.30119 | − | 0.553996i | 0.972092 | + | 0.972092i | 1.38618 | + | 1.44171i | −1.78988 | −0.726338 | − | 1.80341i | −2.26792 | −1.00497 | − | 2.64387i | − | 1.11008i | 2.32897 | + | 0.991587i | |||||
339.12 | −1.19953 | − | 0.749083i | −1.62819 | − | 1.62819i | 0.877750 | + | 1.79710i | 1.61926 | 0.733417 | + | 3.17272i | 0.443847 | 0.293285 | − | 2.81318i | 2.30203i | −1.94236 | − | 1.21296i | ||||||
339.13 | −1.18524 | + | 0.771493i | −2.05999 | − | 2.05999i | 0.809597 | − | 1.82881i | 3.07499 | 4.03085 | + | 0.852317i | −1.16594 | 0.451348 | + | 2.79218i | 5.48709i | −3.64461 | + | 2.37234i | ||||||
339.14 | −1.17336 | + | 0.789447i | −1.74500 | − | 1.74500i | 0.753548 | − | 1.85261i | −3.24438 | 3.42509 | + | 0.669927i | 2.87009 | 0.578353 | + | 2.76867i | 3.09002i | 3.80682 | − | 2.56126i | ||||||
339.15 | −1.13524 | − | 0.843348i | −1.40169 | − | 1.40169i | 0.577529 | + | 1.91480i | −4.05599 | 0.409139 | + | 2.77336i | 4.34286 | 0.959209 | − | 2.66081i | 0.929469i | 4.60452 | + | 3.42061i | ||||||
339.16 | −1.13093 | − | 0.849111i | 1.98293 | + | 1.98293i | 0.558020 | + | 1.92058i | 2.31249 | −0.558833 | − | 3.92628i | −3.56433 | 0.999699 | − | 2.64587i | 4.86399i | −2.61527 | − | 1.96356i | ||||||
339.17 | −1.09621 | + | 0.893492i | 2.36762 | + | 2.36762i | 0.403344 | − | 1.95891i | −1.01012 | −4.71086 | − | 0.479956i | −2.40965 | 1.30812 | + | 2.50775i | 8.21129i | 1.10731 | − | 0.902538i | ||||||
339.18 | −1.08578 | − | 0.906132i | 0.326280 | + | 0.326280i | 0.357848 | + | 1.96773i | 3.56659 | −0.0586163 | − | 0.649922i | 3.97694 | 1.39447 | − | 2.46078i | − | 2.78708i | −3.87254 | − | 3.23181i | |||||
339.19 | −1.08040 | + | 0.912541i | 1.23308 | + | 1.23308i | 0.334537 | − | 1.97182i | −1.94551 | −2.45746 | − | 0.206986i | 4.34289 | 1.43794 | + | 2.43564i | 0.0409839i | 2.10193 | − | 1.77536i | ||||||
339.20 | −1.03953 | + | 0.958838i | 0.444118 | + | 0.444118i | 0.161261 | − | 1.99349i | 3.03578 | −0.887513 | − | 0.0358386i | −2.92719 | 1.74380 | + | 2.22692i | − | 2.60552i | −3.15580 | + | 2.91082i | |||||
See next 80 embeddings (of 148 total) |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
592.s | even | 4 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 592.2.s.a | yes | 148 |
16.f | odd | 4 | 1 | 592.2.m.a | ✓ | 148 | |
37.d | odd | 4 | 1 | 592.2.m.a | ✓ | 148 | |
592.s | even | 4 | 1 | inner | 592.2.s.a | yes | 148 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
592.2.m.a | ✓ | 148 | 16.f | odd | 4 | 1 | |
592.2.m.a | ✓ | 148 | 37.d | odd | 4 | 1 | |
592.2.s.a | yes | 148 | 1.a | even | 1 | 1 | trivial |
592.2.s.a | yes | 148 | 592.s | even | 4 | 1 | inner |
Hecke kernels
This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(592, [\chi])\).