Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [966,2,Mod(47,966)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(966, base_ring=CyclotomicField(6))
chi = DirichletCharacter(H, H._module([3, 5, 0]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("966.47");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 966 = 2 \cdot 3 \cdot 7 \cdot 23 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 966.l (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: | \(7.71354883526\) |
Analytic rank: | \(0\) |
Dimension: | \(56\) |
Relative dimension: | \(28\) 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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
47.1 | −0.866025 | − | 0.500000i | −1.72955 | − | 0.0930981i | 0.500000 | + | 0.866025i | 1.85941 | − | 3.22059i | 1.45128 | + | 0.945399i | −1.67325 | + | 2.04945i | − | 1.00000i | 2.98267 | + | 0.322035i | −3.22059 | + | 1.85941i | |
47.2 | −0.866025 | − | 0.500000i | −1.63883 | − | 0.560573i | 0.500000 | + | 0.866025i | −0.519026 | + | 0.898980i | 1.13898 | + | 1.30488i | 1.91946 | − | 1.82090i | − | 1.00000i | 2.37152 | + | 1.83737i | 0.898980 | − | 0.519026i | |
47.3 | −0.866025 | − | 0.500000i | −1.43165 | + | 0.974877i | 0.500000 | + | 0.866025i | −0.723421 | + | 1.25300i | 1.72728 | − | 0.128444i | −1.91613 | − | 1.82440i | − | 1.00000i | 1.09923 | − | 2.79136i | 1.25300 | − | 0.723421i | |
47.4 | −0.866025 | − | 0.500000i | −1.24955 | + | 1.19943i | 0.500000 | + | 0.866025i | −1.89177 | + | 3.27664i | 1.68186 | − | 0.413957i | 2.63339 | − | 0.255430i | − | 1.00000i | 0.122757 | − | 2.99749i | 3.27664 | − | 1.89177i | |
47.5 | −0.866025 | − | 0.500000i | −1.13768 | − | 1.30602i | 0.500000 | + | 0.866025i | 0.890465 | − | 1.54233i | 0.332245 | + | 1.69989i | −2.00001 | − | 1.73204i | − | 1.00000i | −0.411389 | + | 2.97166i | −1.54233 | + | 0.890465i | |
47.6 | −0.866025 | − | 0.500000i | −0.319643 | − | 1.70230i | 0.500000 | + | 0.866025i | 0.224543 | − | 0.388920i | −0.574332 | + | 1.63406i | 2.31951 | − | 1.27274i | − | 1.00000i | −2.79566 | + | 1.08826i | −0.388920 | + | 0.224543i | |
47.7 | −0.866025 | − | 0.500000i | −0.165511 | + | 1.72412i | 0.500000 | + | 0.866025i | −0.592581 | + | 1.02638i | 1.00540 | − | 1.41038i | −2.13419 | + | 1.56372i | − | 1.00000i | −2.94521 | − | 0.570723i | 1.02638 | − | 0.592581i | |
47.8 | −0.866025 | − | 0.500000i | 0.110462 | + | 1.72852i | 0.500000 | + | 0.866025i | −0.0289992 | + | 0.0502281i | 0.768599 | − | 1.55218i | 1.41346 | − | 2.23654i | − | 1.00000i | −2.97560 | + | 0.381874i | 0.0502281 | − | 0.0289992i | |
47.9 | −0.866025 | − | 0.500000i | 0.185627 | + | 1.72208i | 0.500000 | + | 0.866025i | 1.97297 | − | 3.41729i | 0.700280 | − | 1.58417i | 1.45851 | + | 2.20743i | − | 1.00000i | −2.93109 | + | 0.639328i | −3.41729 | + | 1.97297i | |
47.10 | −0.866025 | − | 0.500000i | 0.454222 | − | 1.67143i | 0.500000 | + | 0.866025i | 2.02138 | − | 3.50113i | −1.22908 | + | 1.22039i | 2.64571 | + | 0.0151948i | − | 1.00000i | −2.58737 | − | 1.51840i | −3.50113 | + | 2.02138i | |
47.11 | −0.866025 | − | 0.500000i | 0.633338 | − | 1.61211i | 0.500000 | + | 0.866025i | −1.42388 | + | 2.46624i | −1.35454 | + | 1.07946i | 0.837175 | + | 2.50981i | − | 1.00000i | −2.19777 | − | 2.04201i | 2.46624 | − | 1.42388i | |
47.12 | −0.866025 | − | 0.500000i | 1.55959 | − | 0.753450i | 0.500000 | + | 0.866025i | 0.416548 | − | 0.721483i | −1.72737 | − | 0.127287i | −0.928188 | + | 2.47759i | − | 1.00000i | 1.86463 | − | 2.35014i | −0.721483 | + | 0.416548i | |
47.13 | −0.866025 | − | 0.500000i | 1.56059 | + | 0.751362i | 0.500000 | + | 0.866025i | −1.92458 | + | 3.33347i | −0.975834 | − | 1.43100i | 0.526310 | − | 2.59287i | − | 1.00000i | 1.87091 | + | 2.34514i | 3.33347 | − | 1.92458i | |
47.14 | −0.866025 | − | 0.500000i | 1.66857 | + | 0.464617i | 0.500000 | + | 0.866025i | 0.584963 | − | 1.01319i | −1.21272 | − | 1.23666i | −0.101753 | + | 2.64379i | − | 1.00000i | 2.56826 | + | 1.55049i | −1.01319 | + | 0.584963i | |
47.15 | 0.866025 | + | 0.500000i | −1.66351 | + | 0.482430i | 0.500000 | + | 0.866025i | 1.89177 | − | 3.27664i | −1.68186 | − | 0.413957i | 2.63339 | − | 0.255430i | 1.00000i | 2.53452 | − | 1.60505i | 3.27664 | − | 1.89177i | ||
47.16 | 0.866025 | + | 0.500000i | −1.57589 | − | 0.718726i | 0.500000 | + | 0.866025i | 0.592581 | − | 1.02638i | −1.00540 | − | 1.41038i | −2.13419 | + | 1.56372i | 1.00000i | 1.96687 | + | 2.26527i | 1.02638 | − | 0.592581i | ||
47.17 | 0.866025 | + | 0.500000i | −1.56009 | + | 0.752405i | 0.500000 | + | 0.866025i | 0.723421 | − | 1.25300i | −1.72728 | − | 0.128444i | −1.91613 | − | 1.82440i | 1.00000i | 1.86777 | − | 2.34764i | 1.25300 | − | 0.723421i | ||
47.18 | 0.866025 | + | 0.500000i | −1.44172 | − | 0.959926i | 0.500000 | + | 0.866025i | 0.0289992 | − | 0.0502281i | −0.768599 | − | 1.55218i | 1.41346 | − | 2.23654i | 1.00000i | 1.15709 | + | 2.76788i | 0.0502281 | − | 0.0289992i | ||
47.19 | 0.866025 | + | 0.500000i | −1.39855 | − | 1.02180i | 0.500000 | + | 0.866025i | −1.97297 | + | 3.41729i | −0.700280 | − | 1.58417i | 1.45851 | + | 2.20743i | 1.00000i | 0.911868 | + | 2.85806i | −3.41729 | + | 1.97297i | ||
47.20 | 0.866025 | + | 0.500000i | −0.784148 | + | 1.54438i | 0.500000 | + | 0.866025i | −1.85941 | + | 3.22059i | −1.45128 | + | 0.945399i | −1.67325 | + | 2.04945i | 1.00000i | −1.77022 | − | 2.42205i | −3.22059 | + | 1.85941i | ||
See all 56 embeddings |
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 |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 966.2.l.c | ✓ | 56 |
3.b | odd | 2 | 1 | inner | 966.2.l.c | ✓ | 56 |
7.d | odd | 6 | 1 | inner | 966.2.l.c | ✓ | 56 |
21.g | even | 6 | 1 | inner | 966.2.l.c | ✓ | 56 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
966.2.l.c | ✓ | 56 | 1.a | even | 1 | 1 | trivial |
966.2.l.c | ✓ | 56 | 3.b | odd | 2 | 1 | inner |
966.2.l.c | ✓ | 56 | 7.d | odd | 6 | 1 | inner |
966.2.l.c | ✓ | 56 | 21.g | even | 6 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{56} + 93 T_{5}^{54} + 4930 T_{5}^{52} + 178669 T_{5}^{50} + 4895734 T_{5}^{48} + \cdots + 1536953616 \) acting on \(S_{2}^{\mathrm{new}}(966, [\chi])\).