Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [966,2,Mod(461,966)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(966, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([1, 1, 0]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("966.461");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 966 = 2 \cdot 3 \cdot 7 \cdot 23 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 966.f (of order \(2\), degree \(1\), 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: | \(24\) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{2}]$ |
$q$-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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
461.1 | − | 1.00000i | −1.72384 | − | 0.168468i | −1.00000 | 1.25574 | −0.168468 | + | 1.72384i | −1.98144 | − | 1.75325i | 1.00000i | 2.94324 | + | 0.580822i | − | 1.25574i | ||||||||
461.2 | − | 1.00000i | −1.66919 | − | 0.462387i | −1.00000 | −1.95027 | −0.462387 | + | 1.66919i | −0.602835 | + | 2.57616i | 1.00000i | 2.57240 | + | 1.54363i | 1.95027i | |||||||||
461.3 | − | 1.00000i | −1.62776 | + | 0.591954i | −1.00000 | 2.12506 | 0.591954 | + | 1.62776i | 2.32692 | + | 1.25915i | 1.00000i | 2.29918 | − | 1.92711i | − | 2.12506i | ||||||||
461.4 | − | 1.00000i | −0.950180 | − | 1.44816i | −1.00000 | 0.575987 | −1.44816 | + | 0.950180i | 2.10044 | − | 1.60877i | 1.00000i | −1.19432 | + | 2.75202i | − | 0.575987i | ||||||||
461.5 | − | 1.00000i | −0.740879 | − | 1.56560i | −1.00000 | −3.67536 | −1.56560 | + | 0.740879i | −0.229372 | − | 2.63579i | 1.00000i | −1.90220 | + | 2.31984i | 3.67536i | |||||||||
461.6 | − | 1.00000i | −0.375300 | + | 1.69090i | −1.00000 | −0.513446 | 1.69090 | + | 0.375300i | −2.61371 | + | 0.410499i | 1.00000i | −2.71830 | − | 1.26919i | 0.513446i | |||||||||
461.7 | − | 1.00000i | 0.375300 | − | 1.69090i | −1.00000 | 0.513446 | −1.69090 | − | 0.375300i | −2.61371 | − | 0.410499i | 1.00000i | −2.71830 | − | 1.26919i | − | 0.513446i | ||||||||
461.8 | − | 1.00000i | 0.740879 | + | 1.56560i | −1.00000 | 3.67536 | 1.56560 | − | 0.740879i | −0.229372 | + | 2.63579i | 1.00000i | −1.90220 | + | 2.31984i | − | 3.67536i | ||||||||
461.9 | − | 1.00000i | 0.950180 | + | 1.44816i | −1.00000 | −0.575987 | 1.44816 | − | 0.950180i | 2.10044 | + | 1.60877i | 1.00000i | −1.19432 | + | 2.75202i | 0.575987i | |||||||||
461.10 | − | 1.00000i | 1.62776 | − | 0.591954i | −1.00000 | −2.12506 | −0.591954 | − | 1.62776i | 2.32692 | − | 1.25915i | 1.00000i | 2.29918 | − | 1.92711i | 2.12506i | |||||||||
461.11 | − | 1.00000i | 1.66919 | + | 0.462387i | −1.00000 | 1.95027 | 0.462387 | − | 1.66919i | −0.602835 | − | 2.57616i | 1.00000i | 2.57240 | + | 1.54363i | − | 1.95027i | ||||||||
461.12 | − | 1.00000i | 1.72384 | + | 0.168468i | −1.00000 | −1.25574 | 0.168468 | − | 1.72384i | −1.98144 | + | 1.75325i | 1.00000i | 2.94324 | + | 0.580822i | 1.25574i | |||||||||
461.13 | 1.00000i | −1.72384 | + | 0.168468i | −1.00000 | 1.25574 | −0.168468 | − | 1.72384i | −1.98144 | + | 1.75325i | − | 1.00000i | 2.94324 | − | 0.580822i | 1.25574i | |||||||||
461.14 | 1.00000i | −1.66919 | + | 0.462387i | −1.00000 | −1.95027 | −0.462387 | − | 1.66919i | −0.602835 | − | 2.57616i | − | 1.00000i | 2.57240 | − | 1.54363i | − | 1.95027i | ||||||||
461.15 | 1.00000i | −1.62776 | − | 0.591954i | −1.00000 | 2.12506 | 0.591954 | − | 1.62776i | 2.32692 | − | 1.25915i | − | 1.00000i | 2.29918 | + | 1.92711i | 2.12506i | |||||||||
461.16 | 1.00000i | −0.950180 | + | 1.44816i | −1.00000 | 0.575987 | −1.44816 | − | 0.950180i | 2.10044 | + | 1.60877i | − | 1.00000i | −1.19432 | − | 2.75202i | 0.575987i | |||||||||
461.17 | 1.00000i | −0.740879 | + | 1.56560i | −1.00000 | −3.67536 | −1.56560 | − | 0.740879i | −0.229372 | + | 2.63579i | − | 1.00000i | −1.90220 | − | 2.31984i | − | 3.67536i | ||||||||
461.18 | 1.00000i | −0.375300 | − | 1.69090i | −1.00000 | −0.513446 | 1.69090 | − | 0.375300i | −2.61371 | − | 0.410499i | − | 1.00000i | −2.71830 | + | 1.26919i | − | 0.513446i | ||||||||
461.19 | 1.00000i | 0.375300 | + | 1.69090i | −1.00000 | 0.513446 | −1.69090 | + | 0.375300i | −2.61371 | + | 0.410499i | − | 1.00000i | −2.71830 | + | 1.26919i | 0.513446i | |||||||||
461.20 | 1.00000i | 0.740879 | − | 1.56560i | −1.00000 | 3.67536 | 1.56560 | + | 0.740879i | −0.229372 | − | 2.63579i | − | 1.00000i | −1.90220 | − | 2.31984i | 3.67536i | |||||||||
See all 24 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
3.b | odd | 2 | 1 | inner |
7.b | odd | 2 | 1 | inner |
21.c | even | 2 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 966.2.f.b | ✓ | 24 |
3.b | odd | 2 | 1 | inner | 966.2.f.b | ✓ | 24 |
7.b | odd | 2 | 1 | inner | 966.2.f.b | ✓ | 24 |
21.c | even | 2 | 1 | inner | 966.2.f.b | ✓ | 24 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
966.2.f.b | ✓ | 24 | 1.a | even | 1 | 1 | trivial |
966.2.f.b | ✓ | 24 | 3.b | odd | 2 | 1 | inner |
966.2.f.b | ✓ | 24 | 7.b | odd | 2 | 1 | inner |
966.2.f.b | ✓ | 24 | 21.c | even | 2 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{12} - 24T_{5}^{10} + 178T_{5}^{8} - 536T_{5}^{6} + 640T_{5}^{4} - 256T_{5}^{2} + 32 \) acting on \(S_{2}^{\mathrm{new}}(966, [\chi])\).