Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [140,3,Mod(11,140)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(140, base_ring=CyclotomicField(6))
chi = DirichletCharacter(H, H._module([3, 0, 4]))
N = Newforms(chi, 3, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("140.11");
S:= CuspForms(chi, 3);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 140 = 2^{2} \cdot 5 \cdot 7 \) |
Weight: | \( k \) | \(=\) | \( 3 \) |
Character orbit: | \([\chi]\) | \(=\) | 140.t (of order \(6\), 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: | \(3.81472370104\) |
Analytic rank: | \(0\) |
Dimension: | \(64\) |
Relative dimension: | \(32\) over \(\Q(\zeta_{6})\) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{6}]$ |
$q$-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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
11.1 | −1.98057 | − | 0.278096i | −3.59758 | − | 2.07707i | 3.84533 | + | 1.10158i | 1.11803 | + | 1.93649i | 6.54764 | + | 5.11425i | −5.74208 | + | 4.00357i | −7.30960 | − | 3.25112i | 4.12840 | + | 7.15060i | −1.67582 | − | 4.14628i |
11.2 | −1.97671 | + | 0.304335i | 0.0515150 | + | 0.0297422i | 3.81476 | − | 1.20316i | −1.11803 | − | 1.93649i | −0.110882 | − | 0.0431139i | 6.36655 | + | 2.90982i | −7.17451 | + | 3.53927i | −4.49823 | − | 7.79116i | 2.79937 | + | 3.48762i |
11.3 | −1.84602 | − | 0.769551i | 4.25107 | + | 2.45436i | 2.81558 | + | 2.84121i | 1.11803 | + | 1.93649i | −5.95882 | − | 7.80221i | 4.48383 | + | 5.37543i | −3.01117 | − | 7.41167i | 7.54776 | + | 13.0731i | −0.573685 | − | 4.43519i |
11.4 | −1.83325 | + | 0.799499i | −3.67944 | − | 2.12433i | 2.72160 | − | 2.93136i | −1.11803 | − | 1.93649i | 8.44374 | + | 0.952710i | −2.95680 | − | 6.34487i | −2.64575 | + | 7.54983i | 4.52554 | + | 7.83847i | 3.59786 | + | 2.65620i |
11.5 | −1.78604 | − | 0.900030i | −0.304256 | − | 0.175662i | 2.37989 | + | 3.21498i | −1.11803 | − | 1.93649i | 0.385312 | + | 0.587580i | −3.95608 | + | 5.77489i | −1.35700 | − | 7.88407i | −4.43829 | − | 7.68734i | 0.253954 | + | 4.46492i |
11.6 | −1.64012 | + | 1.14455i | 2.36556 | + | 1.36575i | 1.38001 | − | 3.75441i | 1.11803 | + | 1.93649i | −5.44298 | + | 0.467488i | −2.85107 | + | 6.39308i | 2.03371 | + | 7.73719i | −0.769427 | − | 1.33269i | −4.05012 | − | 1.89644i |
11.7 | −1.60242 | + | 1.19677i | −3.02406 | − | 1.74594i | 1.13550 | − | 3.83545i | 1.11803 | + | 1.93649i | 6.93529 | − | 0.821360i | 6.46101 | + | 2.69358i | 2.77058 | + | 7.50492i | 1.59661 | + | 2.76541i | −4.10909 | − | 1.76505i |
11.8 | −1.59436 | − | 1.20748i | 3.26501 | + | 1.88505i | 1.08398 | + | 3.85032i | −1.11803 | − | 1.93649i | −2.92944 | − | 6.94789i | 0.821623 | − | 6.95161i | 2.92093 | − | 7.44769i | 2.60684 | + | 4.51519i | −0.555726 | + | 4.43747i |
11.9 | −1.39243 | + | 1.43566i | 4.67673 | + | 2.70011i | −0.122262 | − | 3.99813i | −1.11803 | − | 1.93649i | −10.3885 | + | 2.95449i | 6.64462 | − | 2.20205i | 5.91021 | + | 5.39160i | 10.0812 | + | 17.4612i | 4.33694 | + | 1.09131i |
11.10 | −1.21722 | − | 1.58694i | −0.0501702 | − | 0.0289658i | −1.03674 | + | 3.86331i | 1.11803 | + | 1.93649i | 0.0151014 | + | 0.114875i | −5.51812 | − | 4.30701i | 7.39278 | − | 3.05726i | −4.49832 | − | 7.79132i | 1.71220 | − | 4.13139i |
11.11 | −1.05997 | + | 1.69602i | 0.809998 | + | 0.467653i | −1.75294 | − | 3.59544i | −1.11803 | − | 1.93649i | −1.65172 | + | 0.878074i | −6.99108 | + | 0.353216i | 7.95599 | + | 0.838020i | −4.06260 | − | 7.03663i | 4.46940 | + | 0.156411i |
11.12 | −0.803353 | + | 1.83156i | −1.11560 | − | 0.644090i | −2.70925 | − | 2.94278i | 1.11803 | + | 1.93649i | 2.07591 | − | 1.52585i | 1.11922 | − | 6.90995i | 7.56638 | − | 2.59807i | −3.67030 | − | 6.35714i | −4.44498 | + | 0.492064i |
11.13 | −0.765717 | − | 1.84761i | 0.0501702 | + | 0.0289658i | −2.82735 | + | 2.82950i | 1.11803 | + | 1.93649i | 0.0151014 | − | 0.114875i | 5.51812 | + | 4.30701i | 7.39278 | + | 3.05726i | −4.49832 | − | 7.79132i | 2.72179 | − | 3.54850i |
11.14 | −0.422124 | + | 1.95495i | −3.79087 | − | 2.18866i | −3.64362 | − | 1.65046i | −1.11803 | − | 1.93649i | 5.87893 | − | 6.48706i | 2.80797 | + | 6.41212i | 4.76462 | − | 6.42638i | 5.08046 | + | 8.79962i | 4.25768 | − | 1.36826i |
11.15 | −0.248528 | − | 1.98450i | −3.26501 | − | 1.88505i | −3.87647 | + | 0.986407i | −1.11803 | − | 1.93649i | −2.92944 | + | 6.94789i | −0.821623 | + | 6.95161i | 2.92093 | + | 7.44769i | 2.60684 | + | 4.51519i | −3.56510 | + | 2.70001i |
11.16 | 0.0819421 | + | 1.99832i | 4.93310 | + | 2.84813i | −3.98657 | + | 0.327493i | 1.11803 | + | 1.93649i | −5.28724 | + | 10.0913i | −6.48796 | − | 2.62800i | −0.981105 | − | 7.93961i | 11.7237 | + | 20.3060i | −3.77812 | + | 2.39287i |
11.17 | 0.113572 | − | 1.99677i | 0.304256 | + | 0.175662i | −3.97420 | − | 0.453554i | −1.11803 | − | 1.93649i | 0.385312 | − | 0.587580i | 3.95608 | − | 5.77489i | −1.35700 | + | 7.88407i | −4.43829 | − | 7.68734i | −3.99371 | + | 2.01253i |
11.18 | 0.256560 | − | 1.98348i | −4.25107 | − | 2.45436i | −3.86835 | − | 1.01776i | 1.11803 | + | 1.93649i | −5.95882 | + | 7.80221i | −4.48383 | − | 5.37543i | −3.01117 | + | 7.41167i | 7.54776 | + | 13.0731i | 4.12783 | − | 1.72077i |
11.19 | 0.553243 | + | 1.92196i | −2.00413 | − | 1.15708i | −3.38784 | + | 2.12662i | 1.11803 | + | 1.93649i | 1.11510 | − | 4.49200i | −5.87344 | + | 3.80825i | −5.96157 | − | 5.33476i | −1.82232 | − | 3.15635i | −3.10331 | + | 3.22016i |
11.20 | 0.749447 | − | 1.85427i | 3.59758 | + | 2.07707i | −2.87666 | − | 2.77936i | 1.11803 | + | 1.93649i | 6.54764 | − | 5.11425i | 5.74208 | − | 4.00357i | −7.30960 | + | 3.25112i | 4.12840 | + | 7.15060i | 4.42869 | − | 0.621842i |
See all 64 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
4.b | odd | 2 | 1 | inner |
7.c | even | 3 | 1 | inner |
28.g | odd | 6 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 140.3.t.a | ✓ | 64 |
4.b | odd | 2 | 1 | inner | 140.3.t.a | ✓ | 64 |
7.c | even | 3 | 1 | inner | 140.3.t.a | ✓ | 64 |
28.g | odd | 6 | 1 | inner | 140.3.t.a | ✓ | 64 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
140.3.t.a | ✓ | 64 | 1.a | even | 1 | 1 | trivial |
140.3.t.a | ✓ | 64 | 4.b | odd | 2 | 1 | inner |
140.3.t.a | ✓ | 64 | 7.c | even | 3 | 1 | inner |
140.3.t.a | ✓ | 64 | 28.g | odd | 6 | 1 | inner |
Hecke kernels
This newform subspace is the entire newspace \(S_{3}^{\mathrm{new}}(140, [\chi])\).