Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [966,2,Mod(25,966)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(966, base_ring=CyclotomicField(66))
chi = DirichletCharacter(H, H._module([0, 44, 6]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("966.25");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 966 = 2 \cdot 3 \cdot 7 \cdot 23 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 966.y (of order \(33\), degree \(20\), 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: | \(7.71354883526\) |
Analytic rank: | \(0\) |
Dimension: | \(160\) |
Relative dimension: | \(8\) over \(\Q(\zeta_{33})\) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{33}]$ |
$q$-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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
25.1 | −0.0475819 | + | 0.998867i | −0.928368 | − | 0.371662i | −0.995472 | − | 0.0950560i | −0.740155 | − | 3.05096i | 0.415415 | − | 0.909632i | 0.385204 | − | 2.61756i | 0.142315 | − | 0.989821i | 0.723734 | + | 0.690079i | 3.08272 | − | 0.594146i |
25.2 | −0.0475819 | + | 0.998867i | −0.928368 | − | 0.371662i | −0.995472 | − | 0.0950560i | −0.678320 | − | 2.79607i | 0.415415 | − | 0.909632i | −0.304541 | + | 2.62817i | 0.142315 | − | 0.989821i | 0.723734 | + | 0.690079i | 2.82518 | − | 0.544509i |
25.3 | −0.0475819 | + | 0.998867i | −0.928368 | − | 0.371662i | −0.995472 | − | 0.0950560i | −0.582238 | − | 2.40002i | 0.415415 | − | 0.909632i | −0.0849128 | + | 2.64439i | 0.142315 | − | 0.989821i | 0.723734 | + | 0.690079i | 2.42500 | − | 0.467381i |
25.4 | −0.0475819 | + | 0.998867i | −0.928368 | − | 0.371662i | −0.995472 | − | 0.0950560i | 0.0760992 | + | 0.313685i | 0.415415 | − | 0.909632i | 2.64554 | − | 0.0334955i | 0.142315 | − | 0.989821i | 0.723734 | + | 0.690079i | −0.316951 | + | 0.0610872i |
25.5 | −0.0475819 | + | 0.998867i | −0.928368 | − | 0.371662i | −0.995472 | − | 0.0950560i | 0.373397 | + | 1.53917i | 0.415415 | − | 0.909632i | −2.64405 | − | 0.0947728i | 0.142315 | − | 0.989821i | 0.723734 | + | 0.690079i | −1.55519 | + | 0.299738i |
25.6 | −0.0475819 | + | 0.998867i | −0.928368 | − | 0.371662i | −0.995472 | − | 0.0950560i | 0.445036 | + | 1.83446i | 0.415415 | − | 0.909632i | 2.08936 | − | 1.62314i | 0.142315 | − | 0.989821i | 0.723734 | + | 0.690079i | −1.85356 | + | 0.357245i |
25.7 | −0.0475819 | + | 0.998867i | −0.928368 | − | 0.371662i | −0.995472 | − | 0.0950560i | 0.534809 | + | 2.20451i | 0.415415 | − | 0.909632i | −2.08203 | − | 1.63252i | 0.142315 | − | 0.989821i | 0.723734 | + | 0.690079i | −2.22746 | + | 0.429308i |
25.8 | −0.0475819 | + | 0.998867i | −0.928368 | − | 0.371662i | −0.995472 | − | 0.0950560i | 0.779910 | + | 3.21483i | 0.415415 | − | 0.909632i | 0.591550 | + | 2.57877i | 0.142315 | − | 0.989821i | 0.723734 | + | 0.690079i | −3.24830 | + | 0.626059i |
121.1 | −0.580057 | − | 0.814576i | −0.235759 | − | 0.971812i | −0.327068 | + | 0.945001i | −0.150385 | − | 3.15698i | −0.654861 | + | 0.755750i | −2.56837 | + | 0.635207i | 0.959493 | − | 0.281733i | −0.888835 | + | 0.458227i | −2.48437 | + | 1.95373i |
121.2 | −0.580057 | − | 0.814576i | −0.235759 | − | 0.971812i | −0.327068 | + | 0.945001i | −0.128045 | − | 2.68799i | −0.654861 | + | 0.755750i | 0.622376 | + | 2.57151i | 0.959493 | − | 0.281733i | −0.888835 | + | 0.458227i | −2.11530 | + | 1.66349i |
121.3 | −0.580057 | − | 0.814576i | −0.235759 | − | 0.971812i | −0.327068 | + | 0.945001i | −0.0861989 | − | 1.80954i | −0.654861 | + | 0.755750i | 1.04990 | − | 2.42852i | 0.959493 | − | 0.281733i | −0.888835 | + | 0.458227i | −1.42401 | + | 1.11985i |
121.4 | −0.580057 | − | 0.814576i | −0.235759 | − | 0.971812i | −0.327068 | + | 0.945001i | 0.00766190 | + | 0.160843i | −0.654861 | + | 0.755750i | −0.825321 | + | 2.51373i | 0.959493 | − | 0.281733i | −0.888835 | + | 0.458227i | 0.126575 | − | 0.0995393i |
121.5 | −0.580057 | − | 0.814576i | −0.235759 | − | 0.971812i | −0.327068 | + | 0.945001i | 0.0236187 | + | 0.495818i | −0.654861 | + | 0.755750i | 2.48735 | − | 0.901724i | 0.959493 | − | 0.281733i | −0.888835 | + | 0.458227i | 0.390181 | − | 0.306842i |
121.6 | −0.580057 | − | 0.814576i | −0.235759 | − | 0.971812i | −0.327068 | + | 0.945001i | 0.106503 | + | 2.23577i | −0.654861 | + | 0.755750i | −2.31212 | − | 1.28612i | 0.959493 | − | 0.281733i | −0.888835 | + | 0.458227i | 1.75943 | − | 1.38363i |
121.7 | −0.580057 | − | 0.814576i | −0.235759 | − | 0.971812i | −0.327068 | + | 0.945001i | 0.122200 | + | 2.56530i | −0.654861 | + | 0.755750i | −0.168303 | + | 2.64039i | 0.959493 | − | 0.281733i | −0.888835 | + | 0.458227i | 2.01875 | − | 1.58756i |
121.8 | −0.580057 | − | 0.814576i | −0.235759 | − | 0.971812i | −0.327068 | + | 0.945001i | 0.170819 | + | 3.58593i | −0.654861 | + | 0.755750i | 2.64569 | + | 0.0174095i | 0.959493 | − | 0.281733i | −0.888835 | + | 0.458227i | 2.82193 | − | 2.21919i |
151.1 | 0.327068 | − | 0.945001i | 0.888835 | − | 0.458227i | −0.786053 | − | 0.618159i | −4.11027 | + | 0.392484i | −0.142315 | − | 0.989821i | −2.51634 | + | 0.817338i | −0.841254 | + | 0.540641i | 0.580057 | − | 0.814576i | −0.973442 | + | 4.01258i |
151.2 | 0.327068 | − | 0.945001i | 0.888835 | − | 0.458227i | −0.786053 | − | 0.618159i | −2.84187 | + | 0.271366i | −0.142315 | − | 0.989821i | 2.39273 | + | 1.12909i | −0.841254 | + | 0.540641i | 0.580057 | − | 0.814576i | −0.673045 | + | 2.77433i |
151.3 | 0.327068 | − | 0.945001i | 0.888835 | − | 0.458227i | −0.786053 | − | 0.618159i | −2.66247 | + | 0.254235i | −0.142315 | − | 0.989821i | 1.24716 | + | 2.33337i | −0.841254 | + | 0.540641i | 0.580057 | − | 0.814576i | −0.630556 | + | 2.59919i |
151.4 | 0.327068 | − | 0.945001i | 0.888835 | − | 0.458227i | −0.786053 | − | 0.618159i | −1.00054 | + | 0.0955401i | −0.142315 | − | 0.989821i | −2.50267 | − | 0.858289i | −0.841254 | + | 0.540641i | 0.580057 | − | 0.814576i | −0.236960 | + | 0.976761i |
See next 80 embeddings (of 160 total) |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
7.c | even | 3 | 1 | inner |
23.c | even | 11 | 1 | inner |
161.m | even | 33 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 966.2.y.a | ✓ | 160 |
7.c | even | 3 | 1 | inner | 966.2.y.a | ✓ | 160 |
23.c | even | 11 | 1 | inner | 966.2.y.a | ✓ | 160 |
161.m | even | 33 | 1 | inner | 966.2.y.a | ✓ | 160 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
966.2.y.a | ✓ | 160 | 1.a | even | 1 | 1 | trivial |
966.2.y.a | ✓ | 160 | 7.c | even | 3 | 1 | inner |
966.2.y.a | ✓ | 160 | 23.c | even | 11 | 1 | inner |
966.2.y.a | ✓ | 160 | 161.m | even | 33 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{160} - 2 T_{5}^{159} - 39 T_{5}^{158} + 2 T_{5}^{157} + 775 T_{5}^{156} + 2105 T_{5}^{155} + \cdots + 47\!\cdots\!81 \) acting on \(S_{2}^{\mathrm{new}}(966, [\chi])\).