Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [1620,3,Mod(973,1620)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1620, base_ring=CyclotomicField(4))
chi = DirichletCharacter(H, H._module([0, 0, 3]))
N = Newforms(chi, 3, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("1620.973");
S:= CuspForms(chi, 3);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 1620 = 2^{2} \cdot 3^{4} \cdot 5 \) |
Weight: | \( k \) | \(=\) | \( 3 \) |
Character orbit: | \([\chi]\) | \(=\) | 1620.l (of order \(4\), 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: | \(44.1418028264\) |
Analytic rank: | \(0\) |
Dimension: | \(24\) |
Relative dimension: | \(12\) over \(\Q(i)\) |
Twist minimal: | no (minimal twist has level 180) |
Sato-Tate group: | $\mathrm{SU}(2)[C_{4}]$ |
$q$-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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
973.1 | 0 | 0 | 0 | −4.98831 | − | 0.341776i | 0 | 7.68868 | − | 7.68868i | 0 | 0 | 0 | ||||||||||||||
973.2 | 0 | 0 | 0 | −3.97841 | − | 3.02858i | 0 | −7.24392 | + | 7.24392i | 0 | 0 | 0 | ||||||||||||||
973.3 | 0 | 0 | 0 | −3.87041 | + | 3.16543i | 0 | 3.43863 | − | 3.43863i | 0 | 0 | 0 | ||||||||||||||
973.4 | 0 | 0 | 0 | −3.12092 | + | 3.90639i | 0 | −9.07494 | + | 9.07494i | 0 | 0 | 0 | ||||||||||||||
973.5 | 0 | 0 | 0 | −2.01955 | + | 4.57399i | 0 | 2.94505 | − | 2.94505i | 0 | 0 | 0 | ||||||||||||||
973.6 | 0 | 0 | 0 | −1.81205 | − | 4.66009i | 0 | −0.519255 | + | 0.519255i | 0 | 0 | 0 | ||||||||||||||
973.7 | 0 | 0 | 0 | 0.555312 | − | 4.96907i | 0 | 3.23469 | − | 3.23469i | 0 | 0 | 0 | ||||||||||||||
973.8 | 0 | 0 | 0 | 2.81568 | + | 4.13182i | 0 | −4.37804 | + | 4.37804i | 0 | 0 | 0 | ||||||||||||||
973.9 | 0 | 0 | 0 | 3.08973 | − | 3.93110i | 0 | −2.44070 | + | 2.44070i | 0 | 0 | 0 | ||||||||||||||
973.10 | 0 | 0 | 0 | 3.37743 | + | 3.68687i | 0 | 7.75406 | − | 7.75406i | 0 | 0 | 0 | ||||||||||||||
973.11 | 0 | 0 | 0 | 4.95548 | + | 0.665771i | 0 | 1.80677 | − | 1.80677i | 0 | 0 | 0 | ||||||||||||||
973.12 | 0 | 0 | 0 | 4.99601 | − | 0.199651i | 0 | −3.21104 | + | 3.21104i | 0 | 0 | 0 | ||||||||||||||
1297.1 | 0 | 0 | 0 | −4.98831 | + | 0.341776i | 0 | 7.68868 | + | 7.68868i | 0 | 0 | 0 | ||||||||||||||
1297.2 | 0 | 0 | 0 | −3.97841 | + | 3.02858i | 0 | −7.24392 | − | 7.24392i | 0 | 0 | 0 | ||||||||||||||
1297.3 | 0 | 0 | 0 | −3.87041 | − | 3.16543i | 0 | 3.43863 | + | 3.43863i | 0 | 0 | 0 | ||||||||||||||
1297.4 | 0 | 0 | 0 | −3.12092 | − | 3.90639i | 0 | −9.07494 | − | 9.07494i | 0 | 0 | 0 | ||||||||||||||
1297.5 | 0 | 0 | 0 | −2.01955 | − | 4.57399i | 0 | 2.94505 | + | 2.94505i | 0 | 0 | 0 | ||||||||||||||
1297.6 | 0 | 0 | 0 | −1.81205 | + | 4.66009i | 0 | −0.519255 | − | 0.519255i | 0 | 0 | 0 | ||||||||||||||
1297.7 | 0 | 0 | 0 | 0.555312 | + | 4.96907i | 0 | 3.23469 | + | 3.23469i | 0 | 0 | 0 | ||||||||||||||
1297.8 | 0 | 0 | 0 | 2.81568 | − | 4.13182i | 0 | −4.37804 | − | 4.37804i | 0 | 0 | 0 | ||||||||||||||
See all 24 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
5.c | odd | 4 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 1620.3.l.f | 24 | |
3.b | odd | 2 | 1 | 1620.3.l.g | 24 | ||
5.c | odd | 4 | 1 | inner | 1620.3.l.f | 24 | |
9.c | even | 3 | 2 | 180.3.u.a | ✓ | 48 | |
9.d | odd | 6 | 2 | 540.3.v.a | 48 | ||
15.e | even | 4 | 1 | 1620.3.l.g | 24 | ||
45.k | odd | 12 | 2 | 180.3.u.a | ✓ | 48 | |
45.l | even | 12 | 2 | 540.3.v.a | 48 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
180.3.u.a | ✓ | 48 | 9.c | even | 3 | 2 | |
180.3.u.a | ✓ | 48 | 45.k | odd | 12 | 2 | |
540.3.v.a | 48 | 9.d | odd | 6 | 2 | ||
540.3.v.a | 48 | 45.l | even | 12 | 2 | ||
1620.3.l.f | 24 | 1.a | even | 1 | 1 | trivial | |
1620.3.l.f | 24 | 5.c | odd | 4 | 1 | inner | |
1620.3.l.g | 24 | 3.b | odd | 2 | 1 | ||
1620.3.l.g | 24 | 15.e | even | 4 | 1 |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{3}^{\mathrm{new}}(1620, [\chi])\):
\( T_{7}^{24} - 310 T_{7}^{21} + 34980 T_{7}^{20} - 44088 T_{7}^{19} + 48050 T_{7}^{18} - 4294854 T_{7}^{17} + 309277434 T_{7}^{16} - 560018860 T_{7}^{15} + 622491612 T_{7}^{14} + \cdots + 69\!\cdots\!44 \) |
\( T_{11}^{12} + 6 T_{11}^{11} - 735 T_{11}^{10} - 3870 T_{11}^{9} + 171969 T_{11}^{8} + 667344 T_{11}^{7} - 16173989 T_{11}^{6} - 37660398 T_{11}^{5} + 649389456 T_{11}^{4} + 623750130 T_{11}^{3} + \cdots + 807941620 \) |