Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [76,4,Mod(5,76)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(76, base_ring=CyclotomicField(18))
chi = DirichletCharacter(H, H._module([0, 16]))
N = Newforms(chi, 4, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("76.5");
S:= CuspForms(chi, 4);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 76 = 2^{2} \cdot 19 \) |
Weight: | \( k \) | \(=\) | \( 4 \) |
Character orbit: | \([\chi]\) | \(=\) | 76.i (of order \(9\), degree \(6\), 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: | \(4.48414516044\) |
Analytic rank: | \(0\) |
Dimension: | \(30\) |
Relative dimension: | \(5\) over \(\Q(\zeta_{9})\) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{9}]$ |
$q$-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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
5.1 | 0 | −1.53481 | + | 8.70432i | 0 | 10.3935 | + | 8.72119i | 0 | −7.14447 | − | 12.3746i | 0 | −48.0379 | − | 17.4843i | 0 | ||||||||||
5.2 | 0 | −0.580234 | + | 3.29067i | 0 | −11.9065 | − | 9.99077i | 0 | −13.7659 | − | 23.8433i | 0 | 14.8799 | + | 5.41582i | 0 | ||||||||||
5.3 | 0 | −0.267608 | + | 1.51768i | 0 | −1.63306 | − | 1.37030i | 0 | 13.0616 | + | 22.6234i | 0 | 23.1400 | + | 8.42226i | 0 | ||||||||||
5.4 | 0 | 0.629037 | − | 3.56745i | 0 | 12.9592 | + | 10.8741i | 0 | −4.22583 | − | 7.31935i | 0 | 13.0407 | + | 4.74643i | 0 | ||||||||||
5.5 | 0 | 1.42726 | − | 8.09439i | 0 | −6.74894 | − | 5.66304i | 0 | 0.266189 | + | 0.461053i | 0 | −38.1104 | − | 13.8710i | 0 | ||||||||||
9.1 | 0 | −7.47864 | + | 6.27533i | 0 | −3.19346 | − | 1.16233i | 0 | 4.69206 | − | 8.12689i | 0 | 11.8619 | − | 67.2720i | 0 | ||||||||||
9.2 | 0 | −1.65540 | + | 1.38904i | 0 | 8.99353 | + | 3.27338i | 0 | −0.0466734 | + | 0.0808407i | 0 | −3.87760 | + | 21.9910i | 0 | ||||||||||
9.3 | 0 | −0.487209 | + | 0.408817i | 0 | −11.6536 | − | 4.24157i | 0 | −11.7723 | + | 20.3903i | 0 | −4.61826 | + | 26.1915i | 0 | ||||||||||
9.4 | 0 | 3.64834 | − | 3.06132i | 0 | −10.2365 | − | 3.72578i | 0 | 16.4303 | − | 28.4581i | 0 | −0.749805 | + | 4.25235i | 0 | ||||||||||
9.5 | 0 | 6.23895 | − | 5.23510i | 0 | 12.3313 | + | 4.48821i | 0 | −4.34015 | + | 7.51736i | 0 | 6.82973 | − | 38.7333i | 0 | ||||||||||
17.1 | 0 | −7.47864 | − | 6.27533i | 0 | −3.19346 | + | 1.16233i | 0 | 4.69206 | + | 8.12689i | 0 | 11.8619 | + | 67.2720i | 0 | ||||||||||
17.2 | 0 | −1.65540 | − | 1.38904i | 0 | 8.99353 | − | 3.27338i | 0 | −0.0466734 | − | 0.0808407i | 0 | −3.87760 | − | 21.9910i | 0 | ||||||||||
17.3 | 0 | −0.487209 | − | 0.408817i | 0 | −11.6536 | + | 4.24157i | 0 | −11.7723 | − | 20.3903i | 0 | −4.61826 | − | 26.1915i | 0 | ||||||||||
17.4 | 0 | 3.64834 | + | 3.06132i | 0 | −10.2365 | + | 3.72578i | 0 | 16.4303 | + | 28.4581i | 0 | −0.749805 | − | 4.25235i | 0 | ||||||||||
17.5 | 0 | 6.23895 | + | 5.23510i | 0 | 12.3313 | − | 4.48821i | 0 | −4.34015 | − | 7.51736i | 0 | 6.82973 | + | 38.7333i | 0 | ||||||||||
25.1 | 0 | −8.30252 | − | 3.02187i | 0 | 2.68735 | + | 15.2407i | 0 | 13.2377 | − | 22.9284i | 0 | 39.1169 | + | 32.8230i | 0 | ||||||||||
25.2 | 0 | −3.29897 | − | 1.20073i | 0 | −0.0631039 | − | 0.357880i | 0 | −13.5658 | + | 23.4967i | 0 | −11.2417 | − | 9.43292i | 0 | ||||||||||
25.3 | 0 | −1.97311 | − | 0.718152i | 0 | −1.01111 | − | 5.73427i | 0 | 2.37147 | − | 4.10751i | 0 | −17.3058 | − | 14.5213i | 0 | ||||||||||
25.4 | 0 | 5.87446 | + | 2.13813i | 0 | −3.07163 | − | 17.4201i | 0 | 12.3773 | − | 21.4380i | 0 | 9.25452 | + | 7.76546i | 0 | ||||||||||
25.5 | 0 | 6.26044 | + | 2.27861i | 0 | 2.15308 | + | 12.2107i | 0 | −4.57544 | + | 7.92489i | 0 | 13.3179 | + | 11.1750i | 0 | ||||||||||
See all 30 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
19.e | even | 9 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 76.4.i.a | ✓ | 30 |
19.e | even | 9 | 1 | inner | 76.4.i.a | ✓ | 30 |
19.e | even | 9 | 1 | 1444.4.a.j | 15 | ||
19.f | odd | 18 | 1 | 1444.4.a.k | 15 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
76.4.i.a | ✓ | 30 | 1.a | even | 1 | 1 | trivial |
76.4.i.a | ✓ | 30 | 19.e | even | 9 | 1 | inner |
1444.4.a.j | 15 | 19.e | even | 9 | 1 | ||
1444.4.a.k | 15 | 19.f | odd | 18 | 1 |
Hecke kernels
This newform subspace is the entire newspace \(S_{4}^{\mathrm{new}}(76, [\chi])\).