Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [966,2,Mod(59,966)]
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([33, 11, 42]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("966.59");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 966 = 2 \cdot 3 \cdot 7 \cdot 23 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 966.bd (of order \(66\), 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: | \(1280\) |
Relative dimension: | \(64\) over \(\Q(\zeta_{66})\) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{66}]$ |
$q$-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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
59.1 | −0.945001 | − | 0.327068i | −1.72725 | − | 0.128835i | 0.786053 | + | 0.618159i | −2.25085 | + | 0.214930i | 1.59012 | + | 0.686678i | −2.10733 | − | 1.59974i | −0.540641 | − | 0.841254i | 2.96680 | + | 0.445061i | 2.19735 | + | 0.533071i |
59.2 | −0.945001 | − | 0.327068i | −1.72300 | + | 0.176876i | 0.786053 | + | 0.618159i | −3.77287 | + | 0.360266i | 1.68608 | + | 0.396389i | 1.25466 | + | 2.32934i | −0.540641 | − | 0.841254i | 2.93743 | − | 0.609512i | 3.68320 | + | 0.893535i |
59.3 | −0.945001 | − | 0.327068i | −1.71282 | − | 0.257380i | 0.786053 | + | 0.618159i | −0.220141 | + | 0.0210209i | 1.53444 | + | 0.803434i | 2.60042 | + | 0.487642i | −0.540641 | − | 0.841254i | 2.86751 | + | 0.881693i | 0.214909 | + | 0.0521363i |
59.4 | −0.945001 | − | 0.327068i | −1.66777 | + | 0.467492i | 0.786053 | + | 0.618159i | 3.60829 | − | 0.344550i | 1.72894 | + | 0.103693i | −2.58647 | − | 0.556950i | −0.540641 | − | 0.841254i | 2.56290 | − | 1.55934i | −3.52253 | − | 0.854556i |
59.5 | −0.945001 | − | 0.327068i | −1.49813 | − | 0.869251i | 0.786053 | + | 0.618159i | 2.42488 | − | 0.231548i | 1.13143 | + | 1.31143i | −1.25611 | + | 2.32856i | −0.540641 | − | 0.841254i | 1.48881 | + | 2.60451i | −2.36724 | − | 0.574287i |
59.6 | −0.945001 | − | 0.327068i | −1.40527 | + | 1.01253i | 0.786053 | + | 0.618159i | 1.29830 | − | 0.123973i | 1.65915 | − | 0.497225i | −0.324812 | − | 2.62574i | −0.540641 | − | 0.841254i | 0.949560 | − | 2.84576i | −1.26744 | − | 0.307478i |
59.7 | −0.945001 | − | 0.327068i | −1.38916 | + | 1.03452i | 0.786053 | + | 0.618159i | −0.847161 | + | 0.0808940i | 1.65112 | − | 0.523271i | −1.22496 | + | 2.34510i | −0.540641 | − | 0.841254i | 0.859542 | − | 2.87423i | 0.827025 | + | 0.200634i |
59.8 | −0.945001 | − | 0.327068i | −1.30303 | − | 1.14110i | 0.786053 | + | 0.618159i | 0.986781 | − | 0.0942262i | 0.858148 | + | 1.50452i | −0.514065 | + | 2.59533i | −0.540641 | − | 0.841254i | 0.395779 | + | 2.97378i | −0.963327 | − | 0.233701i |
59.9 | −0.945001 | − | 0.327068i | −1.02432 | − | 1.39670i | 0.786053 | + | 0.618159i | −3.14301 | + | 0.300121i | 0.511165 | + | 1.65491i | 1.51112 | − | 2.17176i | −0.540641 | − | 0.841254i | −0.901547 | + | 2.86133i | 3.06831 | + | 0.744364i |
59.10 | −0.945001 | − | 0.327068i | −0.968062 | − | 1.43626i | 0.786053 | + | 0.618159i | −0.185246 | + | 0.0176888i | 0.445064 | + | 1.67389i | −0.860764 | − | 2.50182i | −0.540641 | − | 0.841254i | −1.12571 | + | 2.78079i | 0.180843 | + | 0.0438720i |
59.11 | −0.945001 | − | 0.327068i | −0.755746 | + | 1.55848i | 0.786053 | + | 0.618159i | −0.102884 | + | 0.00982421i | 1.22391 | − | 1.22558i | 2.61358 | − | 0.411367i | −0.540641 | − | 0.841254i | −1.85770 | − | 2.35562i | 0.100438 | + | 0.0243661i |
59.12 | −0.945001 | − | 0.327068i | −0.689077 | − | 1.58908i | 0.786053 | + | 0.618159i | 3.18017 | − | 0.303670i | 0.131442 | + | 1.72706i | −2.23440 | − | 1.41685i | −0.540641 | − | 0.841254i | −2.05034 | + | 2.19000i | −3.10459 | − | 0.753165i |
59.13 | −0.945001 | − | 0.327068i | −0.321085 | + | 1.70203i | 0.786053 | + | 0.618159i | −2.90323 | + | 0.277225i | 0.860105 | − | 1.50340i | −2.41573 | + | 1.07900i | −0.540641 | − | 0.841254i | −2.79381 | − | 1.09299i | 2.83423 | + | 0.687576i |
59.14 | −0.945001 | − | 0.327068i | −0.133141 | + | 1.72693i | 0.786053 | + | 0.618159i | 3.12559 | − | 0.298458i | 0.690641 | − | 1.58840i | 0.569043 | + | 2.58383i | −0.540641 | − | 0.841254i | −2.96455 | − | 0.459850i | −3.05130 | − | 0.740238i |
59.15 | −0.945001 | − | 0.327068i | −0.0481888 | − | 1.73138i | 0.786053 | + | 0.618159i | −1.37494 | + | 0.131291i | −0.520741 | + | 1.65192i | 2.07811 | + | 1.63751i | −0.540641 | − | 0.841254i | −2.99536 | + | 0.166866i | 1.34226 | + | 0.325630i |
59.16 | −0.945001 | − | 0.327068i | −0.0112042 | − | 1.73201i | 0.786053 | + | 0.618159i | 3.78210 | − | 0.361147i | −0.555899 | + | 1.64042i | 2.61932 | − | 0.373018i | −0.540641 | − | 0.841254i | −2.99975 | + | 0.0388116i | −3.69221 | − | 0.895721i |
59.17 | −0.945001 | − | 0.327068i | −0.00143408 | + | 1.73205i | 0.786053 | + | 0.618159i | −2.94984 | + | 0.281676i | 0.567853 | − | 1.63632i | −1.51302 | − | 2.17043i | −0.540641 | − | 0.841254i | −3.00000 | − | 0.00496781i | 2.87973 | + | 0.698615i |
59.18 | −0.945001 | − | 0.327068i | 0.129478 | + | 1.72720i | 0.786053 | + | 0.618159i | 2.15460 | − | 0.205739i | 0.442557 | − | 1.67456i | 0.734488 | − | 2.54176i | −0.540641 | − | 0.841254i | −2.96647 | + | 0.447269i | −2.10339 | − | 0.510276i |
59.19 | −0.945001 | − | 0.327068i | 0.238443 | − | 1.71556i | 0.786053 | + | 0.618159i | −0.133592 | + | 0.0127565i | −0.786433 | + | 1.54322i | −2.64105 | − | 0.157712i | −0.540641 | − | 0.841254i | −2.88629 | − | 0.818126i | 0.130417 | + | 0.0316389i |
59.20 | −0.945001 | − | 0.327068i | 0.974657 | + | 1.43180i | 0.786053 | + | 0.618159i | −3.35223 | + | 0.320099i | −0.452757 | − | 1.67183i | 2.60360 | + | 0.470376i | −0.540641 | − | 0.841254i | −1.10009 | + | 2.79102i | 3.27255 | + | 0.793912i |
See next 80 embeddings (of 1280 total) |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
3.b | odd | 2 | 1 | inner |
7.d | odd | 6 | 1 | inner |
21.g | even | 6 | 1 | inner |
23.c | even | 11 | 1 | inner |
69.h | odd | 22 | 1 | inner |
161.n | odd | 66 | 1 | inner |
483.bb | even | 66 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 966.2.bd.a | ✓ | 1280 |
3.b | odd | 2 | 1 | inner | 966.2.bd.a | ✓ | 1280 |
7.d | odd | 6 | 1 | inner | 966.2.bd.a | ✓ | 1280 |
21.g | even | 6 | 1 | inner | 966.2.bd.a | ✓ | 1280 |
23.c | even | 11 | 1 | inner | 966.2.bd.a | ✓ | 1280 |
69.h | odd | 22 | 1 | inner | 966.2.bd.a | ✓ | 1280 |
161.n | odd | 66 | 1 | inner | 966.2.bd.a | ✓ | 1280 |
483.bb | even | 66 | 1 | inner | 966.2.bd.a | ✓ | 1280 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
966.2.bd.a | ✓ | 1280 | 1.a | even | 1 | 1 | trivial |
966.2.bd.a | ✓ | 1280 | 3.b | odd | 2 | 1 | inner |
966.2.bd.a | ✓ | 1280 | 7.d | odd | 6 | 1 | inner |
966.2.bd.a | ✓ | 1280 | 21.g | even | 6 | 1 | inner |
966.2.bd.a | ✓ | 1280 | 23.c | even | 11 | 1 | inner |
966.2.bd.a | ✓ | 1280 | 69.h | odd | 22 | 1 | inner |
966.2.bd.a | ✓ | 1280 | 161.n | odd | 66 | 1 | inner |
966.2.bd.a | ✓ | 1280 | 483.bb | even | 66 | 1 | inner |
Hecke kernels
This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(966, [\chi])\).