Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [54,3,Mod(5,54)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(54, base_ring=CyclotomicField(18))
chi = DirichletCharacter(H, H._module([5]))
N = Newforms(chi, 3, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("54.5");
S:= CuspForms(chi, 3);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 54 = 2 \cdot 3^{3} \) |
Weight: | \( k \) | \(=\) | \( 3 \) |
Character orbit: | \([\chi]\) | \(=\) | 54.f (of order \(18\), degree \(6\), 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: | \(1.47139342755\) |
Analytic rank: | \(0\) |
Dimension: | \(36\) |
Relative dimension: | \(6\) over \(\Q(\zeta_{18})\) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{18}]$ |
$q$-expansion
The algebraic \(q\)-expansion of this newform has not been computed, 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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
5.1 | −0.909039 | − | 1.08335i | −1.44503 | + | 2.62905i | −0.347296 | + | 1.96962i | 0.696976 | + | 1.91493i | 4.16177 | − | 0.824439i | 2.23222 | + | 12.6596i | 2.44949 | − | 1.41421i | −4.82380 | − | 7.59809i | 1.44096 | − | 2.49581i |
5.2 | −0.909039 | − | 1.08335i | 1.30968 | − | 2.69902i | −0.347296 | + | 1.96962i | −1.54033 | − | 4.23203i | −4.11454 | + | 1.03467i | 0.223815 | + | 1.26932i | 2.44949 | − | 1.41421i | −5.56946 | − | 7.06974i | −3.18455 | + | 5.51580i |
5.3 | −0.909039 | − | 1.08335i | 2.80846 | + | 1.05478i | −0.347296 | + | 1.96962i | 2.86430 | + | 7.86960i | −1.41030 | − | 4.00138i | −1.95208 | − | 11.0708i | 2.44949 | − | 1.41421i | 6.77487 | + | 5.92462i | 5.92177 | − | 10.2568i |
5.4 | 0.909039 | + | 1.08335i | −2.95590 | + | 0.512490i | −0.347296 | + | 1.96962i | 2.71293 | + | 7.45370i | −3.24224 | − | 2.73640i | 0.0787775 | + | 0.446769i | −2.44949 | + | 1.41421i | 8.47471 | − | 3.02974i | −5.60882 | + | 9.71475i |
5.5 | 0.909039 | + | 1.08335i | 1.54884 | + | 2.56926i | −0.347296 | + | 1.96962i | −1.07911 | − | 2.96482i | −1.37545 | + | 4.01349i | −0.250410 | − | 1.42015i | −2.44949 | + | 1.41421i | −4.20220 | + | 7.95874i | 2.23099 | − | 3.86419i |
5.6 | 0.909039 | + | 1.08335i | 2.14542 | − | 2.09694i | −0.347296 | + | 1.96962i | 0.387127 | + | 1.06362i | 4.22200 | + | 0.418040i | −0.332318 | − | 1.88467i | −2.44949 | + | 1.41421i | 0.205664 | − | 8.99765i | −0.800362 | + | 1.38627i |
11.1 | −0.909039 | + | 1.08335i | −1.44503 | − | 2.62905i | −0.347296 | − | 1.96962i | 0.696976 | − | 1.91493i | 4.16177 | + | 0.824439i | 2.23222 | − | 12.6596i | 2.44949 | + | 1.41421i | −4.82380 | + | 7.59809i | 1.44096 | + | 2.49581i |
11.2 | −0.909039 | + | 1.08335i | 1.30968 | + | 2.69902i | −0.347296 | − | 1.96962i | −1.54033 | + | 4.23203i | −4.11454 | − | 1.03467i | 0.223815 | − | 1.26932i | 2.44949 | + | 1.41421i | −5.56946 | + | 7.06974i | −3.18455 | − | 5.51580i |
11.3 | −0.909039 | + | 1.08335i | 2.80846 | − | 1.05478i | −0.347296 | − | 1.96962i | 2.86430 | − | 7.86960i | −1.41030 | + | 4.00138i | −1.95208 | + | 11.0708i | 2.44949 | + | 1.41421i | 6.77487 | − | 5.92462i | 5.92177 | + | 10.2568i |
11.4 | 0.909039 | − | 1.08335i | −2.95590 | − | 0.512490i | −0.347296 | − | 1.96962i | 2.71293 | − | 7.45370i | −3.24224 | + | 2.73640i | 0.0787775 | − | 0.446769i | −2.44949 | − | 1.41421i | 8.47471 | + | 3.02974i | −5.60882 | − | 9.71475i |
11.5 | 0.909039 | − | 1.08335i | 1.54884 | − | 2.56926i | −0.347296 | − | 1.96962i | −1.07911 | + | 2.96482i | −1.37545 | − | 4.01349i | −0.250410 | + | 1.42015i | −2.44949 | − | 1.41421i | −4.20220 | − | 7.95874i | 2.23099 | + | 3.86419i |
11.6 | 0.909039 | − | 1.08335i | 2.14542 | + | 2.09694i | −0.347296 | − | 1.96962i | 0.387127 | − | 1.06362i | 4.22200 | − | 0.418040i | −0.332318 | + | 1.88467i | −2.44949 | − | 1.41421i | 0.205664 | + | 8.99765i | −0.800362 | − | 1.38627i |
23.1 | −0.483690 | + | 1.32893i | −2.93655 | − | 0.613727i | −1.53209 | − | 1.28558i | −7.71206 | − | 1.35984i | 2.23598 | − | 3.60561i | −0.690206 | + | 0.579152i | 2.44949 | − | 1.41421i | 8.24668 | + | 3.60448i | 5.53738 | − | 9.59102i |
23.2 | −0.483690 | + | 1.32893i | −0.570511 | + | 2.94525i | −1.53209 | − | 1.28558i | 3.98669 | + | 0.702961i | −3.63807 | − | 2.18276i | −10.1193 | + | 8.49107i | 2.44949 | − | 1.41421i | −8.34903 | − | 3.36060i | −2.86250 | + | 4.95800i |
23.3 | −0.483690 | + | 1.32893i | −0.391734 | − | 2.97431i | −1.53209 | − | 1.28558i | 7.52350 | + | 1.32660i | 4.14212 | + | 0.918059i | 4.40811 | − | 3.69885i | 2.44949 | − | 1.41421i | −8.69309 | + | 2.33028i | −5.40199 | + | 9.35652i |
23.4 | 0.483690 | − | 1.32893i | −1.47299 | − | 2.61348i | −1.53209 | − | 1.28558i | −2.99623 | − | 0.528316i | −4.18560 | + | 0.693383i | 6.30359 | − | 5.28934i | −2.44949 | + | 1.41421i | −4.66059 | + | 7.69928i | −2.15134 | + | 3.72623i |
23.5 | 0.483690 | − | 1.32893i | 0.320982 | + | 2.98278i | −1.53209 | − | 1.28558i | 5.90332 | + | 1.04091i | 4.11915 | + | 1.01618i | 5.59840 | − | 4.69762i | −2.44949 | + | 1.41421i | −8.79394 | + | 1.91484i | 4.23867 | − | 7.34160i |
23.6 | 0.483690 | − | 1.32893i | 2.82413 | − | 1.01208i | −1.53209 | − | 1.28558i | 0.891042 | + | 0.157115i | 0.0210163 | − | 4.24259i | −5.50064 | + | 4.61559i | −2.44949 | + | 1.41421i | 6.95137 | − | 5.71650i | 0.639781 | − | 1.10813i |
29.1 | −1.39273 | − | 0.245576i | −2.97067 | − | 0.418457i | 1.87939 | + | 0.684040i | 5.54906 | + | 6.61311i | 4.03458 | + | 1.31232i | 7.83131 | − | 2.85036i | −2.44949 | − | 1.41421i | 8.64979 | + | 2.48620i | −6.10431 | − | 10.5730i |
29.2 | −1.39273 | − | 0.245576i | −2.10518 | + | 2.13733i | 1.87939 | + | 0.684040i | −5.37469 | − | 6.40531i | 3.45683 | − | 2.45974i | −8.61655 | + | 3.13617i | −2.44949 | − | 1.41421i | −0.136395 | − | 8.99897i | 5.91250 | + | 10.2407i |
See all 36 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
27.f | odd | 18 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 54.3.f.a | ✓ | 36 |
3.b | odd | 2 | 1 | 162.3.f.a | 36 | ||
4.b | odd | 2 | 1 | 432.3.bc.c | 36 | ||
27.e | even | 9 | 1 | 162.3.f.a | 36 | ||
27.e | even | 9 | 1 | 1458.3.b.c | 36 | ||
27.f | odd | 18 | 1 | inner | 54.3.f.a | ✓ | 36 |
27.f | odd | 18 | 1 | 1458.3.b.c | 36 | ||
108.l | even | 18 | 1 | 432.3.bc.c | 36 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
54.3.f.a | ✓ | 36 | 1.a | even | 1 | 1 | trivial |
54.3.f.a | ✓ | 36 | 27.f | odd | 18 | 1 | inner |
162.3.f.a | 36 | 3.b | odd | 2 | 1 | ||
162.3.f.a | 36 | 27.e | even | 9 | 1 | ||
432.3.bc.c | 36 | 4.b | odd | 2 | 1 | ||
432.3.bc.c | 36 | 108.l | even | 18 | 1 | ||
1458.3.b.c | 36 | 27.e | even | 9 | 1 | ||
1458.3.b.c | 36 | 27.f | odd | 18 | 1 |
Hecke kernels
This newform subspace is the entire newspace \(S_{3}^{\mathrm{new}}(54, [\chi])\).