Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [165,3,Mod(26,165)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(165, base_ring=CyclotomicField(10))
chi = DirichletCharacter(H, H._module([5, 0, 2]))
N = Newforms(chi, 3, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("165.26");
S:= CuspForms(chi, 3);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 165 = 3 \cdot 5 \cdot 11 \) |
Weight: | \( k \) | \(=\) | \( 3 \) |
Character orbit: | \([\chi]\) | \(=\) | 165.q (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: | \(4.49592436194\) |
Analytic rank: | \(0\) |
Dimension: | \(128\) |
Relative dimension: | \(32\) 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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
26.1 | −2.30025 | + | 3.16603i | −2.99037 | + | 0.240127i | −3.49650 | − | 10.7611i | 1.31433 | + | 1.80902i | 6.11838 | − | 10.0200i | −1.29634 | − | 3.98973i | 27.2253 | + | 8.84603i | 8.88468 | − | 1.43614i | −8.75069 | ||
26.2 | −2.20174 | + | 3.03043i | 0.615997 | − | 2.93608i | −3.09980 | − | 9.54021i | −1.31433 | − | 1.80902i | 7.54132 | + | 8.33121i | 2.97551 | + | 9.15766i | 21.4859 | + | 6.98121i | −8.24109 | − | 3.61723i | 8.37591 | ||
26.3 | −2.04878 | + | 2.81990i | 2.94142 | + | 0.589968i | −2.51827 | − | 7.75045i | −1.31433 | − | 1.80902i | −7.68995 | + | 7.08579i | −3.41857 | − | 10.5213i | 13.7549 | + | 4.46924i | 8.30388 | + | 3.47068i | 7.79401 | ||
26.4 | −1.84427 | + | 2.53842i | 2.61041 | − | 1.47844i | −1.80619 | − | 5.55887i | 1.31433 | + | 1.80902i | −1.06139 | + | 9.35296i | −0.0257605 | − | 0.0792825i | 5.50549 | + | 1.78884i | 4.62843 | − | 7.71865i | −7.01603 | ||
26.5 | −1.73462 | + | 2.38749i | −2.66585 | − | 1.37596i | −1.45517 | − | 4.47855i | −1.31433 | − | 1.80902i | 7.90931 | − | 3.97794i | −0.926259 | − | 2.85073i | 1.98999 | + | 0.646587i | 5.21349 | + | 7.33618i | 6.59887 | ||
26.6 | −1.61829 | + | 2.22739i | 0.208595 | + | 2.99274i | −1.10632 | − | 3.40491i | 1.31433 | + | 1.80902i | −7.00356 | − | 4.37850i | −3.69283 | − | 11.3654i | −1.09940 | − | 0.357218i | −8.91298 | + | 1.24854i | −6.15635 | ||
26.7 | −1.59966 | + | 2.20175i | −0.632879 | − | 2.93248i | −1.05270 | − | 3.23988i | 1.31433 | + | 1.80902i | 7.46898 | + | 3.29755i | −0.283495 | − | 0.872508i | −1.53586 | − | 0.499032i | −8.19893 | + | 3.71182i | −6.08548 | ||
26.8 | −1.46951 | + | 2.02260i | −1.97655 | + | 2.25682i | −0.695407 | − | 2.14024i | −1.31433 | − | 1.80902i | −1.66009 | − | 7.31420i | 0.101982 | + | 0.313867i | −4.16008 | − | 1.35169i | −1.18647 | − | 8.92145i | 5.59034 | ||
26.9 | −1.30528 | + | 1.79656i | −2.99034 | + | 0.240553i | −0.287817 | − | 0.885809i | 1.31433 | + | 1.80902i | 3.47106 | − | 5.68633i | 4.07959 | + | 12.5557i | −6.48085 | − | 2.10576i | 8.88427 | − | 1.43867i | −4.96558 | ||
26.10 | −1.14753 | + | 1.57945i | 2.62795 | + | 1.44702i | 0.0582553 | + | 0.179291i | −1.31433 | − | 1.80902i | −5.30115 | + | 2.49021i | 1.64114 | + | 5.05091i | −7.77703 | − | 2.52691i | 4.81228 | + | 7.60539i | 4.36548 | ||
26.11 | −0.971166 | + | 1.33669i | 2.00130 | + | 2.23491i | 0.392477 | + | 1.20792i | 1.31433 | + | 1.80902i | −4.93098 | + | 0.504659i | 1.32808 | + | 4.08740i | −8.28130 | − | 2.69076i | −0.989618 | + | 8.94543i | −3.69453 | ||
26.12 | −0.881600 | + | 1.21342i | 1.06116 | − | 2.80605i | 0.540903 | + | 1.66473i | −1.31433 | − | 1.80902i | 2.46939 | + | 3.76145i | −3.68024 | − | 11.3266i | −8.20270 | − | 2.66522i | −6.74786 | − | 5.95536i | 3.35381 | ||
26.13 | −0.462120 | + | 0.636053i | −0.392907 | − | 2.97416i | 1.04506 | + | 3.21636i | −1.31433 | − | 1.80902i | 2.07329 | + | 1.12451i | 3.54884 | + | 10.9222i | −5.51962 | − | 1.79343i | −8.69125 | + | 2.33714i | 1.75801 | ||
26.14 | −0.356298 | + | 0.490402i | −1.91621 | − | 2.30827i | 1.12252 | + | 3.45477i | 1.31433 | + | 1.80902i | 1.81473 | − | 0.117281i | −0.651691 | − | 2.00570i | −4.40019 | − | 1.42971i | −1.65626 | + | 8.84629i | −1.35544 | ||
26.15 | −0.0434364 | + | 0.0597850i | −0.712635 | + | 2.91413i | 1.23438 | + | 3.79903i | −1.31433 | − | 1.80902i | −0.143267 | − | 0.169184i | 1.56117 | + | 4.80478i | −0.561868 | − | 0.182562i | −7.98430 | − | 4.15342i | 0.165242 | ||
26.16 | −0.00589937 | + | 0.00811978i | −2.77949 | − | 1.12890i | 1.23604 | + | 3.80413i | −1.31433 | − | 1.80902i | 0.0255637 | − | 0.0159090i | −1.26110 | − | 3.88128i | −0.0763620 | − | 0.0248115i | 6.45115 | + | 6.27556i | 0.0224425 | ||
26.17 | 0.00589937 | − | 0.00811978i | 2.91221 | + | 0.720441i | 1.23604 | + | 3.80413i | 1.31433 | + | 1.80902i | 0.0230300 | − | 0.0193964i | −1.26110 | − | 3.88128i | 0.0763620 | + | 0.0248115i | 7.96193 | + | 4.19615i | 0.0224425 | ||
26.18 | 0.0434364 | − | 0.0597850i | −1.13635 | + | 2.77646i | 1.23438 | + | 3.79903i | 1.31433 | + | 1.80902i | 0.116632 | + | 0.188536i | 1.56117 | + | 4.80478i | 0.561868 | + | 0.182562i | −6.41742 | − | 6.31005i | 0.165242 | ||
26.19 | 0.356298 | − | 0.490402i | 2.90702 | − | 0.741111i | 1.12252 | + | 3.45477i | −1.31433 | − | 1.80902i | 0.672322 | − | 1.68966i | −0.651691 | − | 2.00570i | 4.40019 | + | 1.42971i | 7.90151 | − | 4.30885i | −1.35544 | ||
26.20 | 0.462120 | − | 0.636053i | 2.06604 | − | 2.17520i | 1.04506 | + | 3.21636i | 1.31433 | + | 1.80902i | −0.428787 | − | 2.31931i | 3.54884 | + | 10.9222i | 5.51962 | + | 1.79343i | −0.462994 | − | 8.98808i | 1.75801 | ||
See next 80 embeddings (of 128 total) |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
3.b | odd | 2 | 1 | inner |
11.c | even | 5 | 1 | inner |
33.h | odd | 10 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 165.3.q.a | ✓ | 128 |
3.b | odd | 2 | 1 | inner | 165.3.q.a | ✓ | 128 |
11.c | even | 5 | 1 | inner | 165.3.q.a | ✓ | 128 |
33.h | odd | 10 | 1 | inner | 165.3.q.a | ✓ | 128 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
165.3.q.a | ✓ | 128 | 1.a | even | 1 | 1 | trivial |
165.3.q.a | ✓ | 128 | 3.b | odd | 2 | 1 | inner |
165.3.q.a | ✓ | 128 | 11.c | even | 5 | 1 | inner |
165.3.q.a | ✓ | 128 | 33.h | odd | 10 | 1 | inner |
Hecke kernels
This newform subspace is the entire newspace \(S_{3}^{\mathrm{new}}(165, [\chi])\).