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 | −0.786833 | − | 3.24337i | 0.415415 | − | 0.909632i | −2.25109 | − | 1.39018i | −0.142315 | + | 0.989821i | 0.723734 | + | 0.690079i | −3.27714 | + | 0.631616i |
25.2 | 0.0475819 | − | 0.998867i | 0.928368 | + | 0.371662i | −0.995472 | − | 0.0950560i | −0.781948 | − | 3.22324i | 0.415415 | − | 0.909632i | 2.33065 | − | 1.25222i | −0.142315 | + | 0.989821i | 0.723734 | + | 0.690079i | −3.25679 | + | 0.627695i |
25.3 | 0.0475819 | − | 0.998867i | 0.928368 | + | 0.371662i | −0.995472 | − | 0.0950560i | −0.363961 | − | 1.50027i | 0.415415 | − | 0.909632i | −0.661406 | + | 2.56175i | −0.142315 | + | 0.989821i | 0.723734 | + | 0.690079i | −1.51589 | + | 0.292163i |
25.4 | 0.0475819 | − | 0.998867i | 0.928368 | + | 0.371662i | −0.995472 | − | 0.0950560i | −0.167957 | − | 0.692328i | 0.415415 | − | 0.909632i | 2.14841 | − | 1.54413i | −0.142315 | + | 0.989821i | 0.723734 | + | 0.690079i | −0.699536 | + | 0.134824i |
25.5 | 0.0475819 | − | 0.998867i | 0.928368 | + | 0.371662i | −0.995472 | − | 0.0950560i | −0.130603 | − | 0.538354i | 0.415415 | − | 0.909632i | −2.29097 | + | 1.32342i | −0.142315 | + | 0.989821i | 0.723734 | + | 0.690079i | −0.543959 | + | 0.104840i |
25.6 | 0.0475819 | − | 0.998867i | 0.928368 | + | 0.371662i | −0.995472 | − | 0.0950560i | 0.112949 | + | 0.465581i | 0.415415 | − | 0.909632i | 1.20408 | + | 2.35588i | −0.142315 | + | 0.989821i | 0.723734 | + | 0.690079i | 0.470428 | − | 0.0906676i |
25.7 | 0.0475819 | − | 0.998867i | 0.928368 | + | 0.371662i | −0.995472 | − | 0.0950560i | 0.415021 | + | 1.71074i | 0.415415 | − | 0.909632i | −0.322897 | − | 2.62597i | −0.142315 | + | 0.989821i | 0.723734 | + | 0.690079i | 1.72855 | − | 0.333151i |
25.8 | 0.0475819 | − | 0.998867i | 0.928368 | + | 0.371662i | −0.995472 | − | 0.0950560i | 0.968835 | + | 3.99359i | 0.415415 | − | 0.909632i | 2.64295 | − | 0.121825i | −0.142315 | + | 0.989821i | 0.723734 | + | 0.690079i | 4.03517 | − | 0.777715i |
121.1 | 0.580057 | + | 0.814576i | 0.235759 | + | 0.971812i | −0.327068 | + | 0.945001i | −0.182773 | − | 3.83688i | −0.654861 | + | 0.755750i | −1.41671 | + | 2.23449i | −0.959493 | + | 0.281733i | −0.888835 | + | 0.458227i | 3.01941 | − | 2.37449i |
121.2 | 0.580057 | + | 0.814576i | 0.235759 | + | 0.971812i | −0.327068 | + | 0.945001i | −0.180250 | − | 3.78390i | −0.654861 | + | 0.755750i | 1.94204 | + | 1.79680i | −0.959493 | + | 0.281733i | −0.888835 | + | 0.458227i | 2.97772 | − | 2.34171i |
121.3 | 0.580057 | + | 0.814576i | 0.235759 | + | 0.971812i | −0.327068 | + | 0.945001i | −0.0817448 | − | 1.71603i | −0.654861 | + | 0.755750i | −2.45748 | − | 0.980189i | −0.959493 | + | 0.281733i | −0.888835 | + | 0.458227i | 1.35042 | − | 1.06199i |
121.4 | 0.580057 | + | 0.814576i | 0.235759 | + | 0.971812i | −0.327068 | + | 0.945001i | −0.0809367 | − | 1.69907i | −0.654861 | + | 0.755750i | 1.60083 | − | 2.10650i | −0.959493 | + | 0.281733i | −0.888835 | + | 0.458227i | 1.33707 | − | 1.05149i |
121.5 | 0.580057 | + | 0.814576i | 0.235759 | + | 0.971812i | −0.327068 | + | 0.945001i | 0.0478985 | + | 1.00551i | −0.654861 | + | 0.755750i | −0.794478 | − | 2.52365i | −0.959493 | + | 0.281733i | −0.888835 | + | 0.458227i | −0.791284 | + | 0.622272i |
121.6 | 0.580057 | + | 0.814576i | 0.235759 | + | 0.971812i | −0.327068 | + | 0.945001i | 0.0802489 | + | 1.68463i | −0.654861 | + | 0.755750i | 1.50657 | + | 2.17491i | −0.959493 | + | 0.281733i | −0.888835 | + | 0.458227i | −1.32571 | + | 1.04255i |
121.7 | 0.580057 | + | 0.814576i | 0.235759 | + | 0.971812i | −0.327068 | + | 0.945001i | 0.132422 | + | 2.77988i | −0.654861 | + | 0.755750i | −1.83001 | + | 1.91077i | −0.959493 | + | 0.281733i | −0.888835 | + | 0.458227i | −2.18761 | + | 1.72035i |
121.8 | 0.580057 | + | 0.814576i | 0.235759 | + | 0.971812i | −0.327068 | + | 0.945001i | 0.140981 | + | 2.95955i | −0.654861 | + | 0.755750i | 2.61692 | − | 0.389498i | −0.959493 | + | 0.281733i | −0.888835 | + | 0.458227i | −2.32900 | + | 1.83155i |
151.1 | −0.327068 | + | 0.945001i | −0.888835 | + | 0.458227i | −0.786053 | − | 0.618159i | −4.06127 | + | 0.387805i | −0.142315 | − | 0.989821i | −0.441487 | − | 2.60866i | 0.841254 | − | 0.540641i | 0.580057 | − | 0.814576i | 0.961837 | − | 3.96475i |
151.2 | −0.327068 | + | 0.945001i | −0.888835 | + | 0.458227i | −0.786053 | − | 0.618159i | −2.76706 | + | 0.264222i | −0.142315 | − | 0.989821i | 2.35671 | + | 1.20246i | 0.841254 | − | 0.540641i | 0.580057 | − | 0.814576i | 0.655327 | − | 2.70130i |
151.3 | −0.327068 | + | 0.945001i | −0.888835 | + | 0.458227i | −0.786053 | − | 0.618159i | −0.688242 | + | 0.0657192i | −0.142315 | − | 0.989821i | −0.529942 | + | 2.59213i | 0.841254 | − | 0.540641i | 0.580057 | − | 0.814576i | 0.162997 | − | 0.671884i |
151.4 | −0.327068 | + | 0.945001i | −0.888835 | + | 0.458227i | −0.786053 | − | 0.618159i | −0.527443 | + | 0.0503647i | −0.142315 | − | 0.989821i | −2.62119 | − | 0.359704i | 0.841254 | − | 0.540641i | 0.580057 | − | 0.814576i | 0.124915 | − | 0.514906i |
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.d | ✓ | 160 |
7.c | even | 3 | 1 | inner | 966.2.y.d | ✓ | 160 |
23.c | even | 11 | 1 | inner | 966.2.y.d | ✓ | 160 |
161.m | even | 33 | 1 | inner | 966.2.y.d | ✓ | 160 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
966.2.y.d | ✓ | 160 | 1.a | even | 1 | 1 | trivial |
966.2.y.d | ✓ | 160 | 7.c | even | 3 | 1 | inner |
966.2.y.d | ✓ | 160 | 23.c | even | 11 | 1 | inner |
966.2.y.d | ✓ | 160 | 161.m | even | 33 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{160} + 2 T_{5}^{159} - 25 T_{5}^{158} - 78 T_{5}^{157} + 193 T_{5}^{156} + 2045 T_{5}^{155} + \cdots + 10\!\cdots\!01 \) acting on \(S_{2}^{\mathrm{new}}(966, [\chi])\).