Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [3420,2,Mod(1673,3420)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(3420, base_ring=CyclotomicField(4))
chi = DirichletCharacter(H, H._module([0, 2, 3, 0]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("3420.1673");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 3420 = 2^{2} \cdot 3^{2} \cdot 5 \cdot 19 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 3420.x (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: | \(27.3088374913\) |
Analytic rank: | \(0\) |
Dimension: | \(24\) |
Relative dimension: | \(12\) over \(\Q(i)\) |
Twist minimal: | yes |
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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
1673.1 | 0 | 0 | 0 | −2.20020 | − | 0.398923i | 0 | −2.63796 | + | 2.63796i | 0 | 0 | 0 | ||||||||||||||
1673.2 | 0 | 0 | 0 | −1.98287 | − | 1.03354i | 0 | 2.54134 | − | 2.54134i | 0 | 0 | 0 | ||||||||||||||
1673.3 | 0 | 0 | 0 | −1.04895 | − | 1.97477i | 0 | 0.585090 | − | 0.585090i | 0 | 0 | 0 | ||||||||||||||
1673.4 | 0 | 0 | 0 | −0.925456 | + | 2.03557i | 0 | 1.92964 | − | 1.92964i | 0 | 0 | 0 | ||||||||||||||
1673.5 | 0 | 0 | 0 | −0.401341 | + | 2.19976i | 0 | −0.0703440 | + | 0.0703440i | 0 | 0 | 0 | ||||||||||||||
1673.6 | 0 | 0 | 0 | −0.330936 | + | 2.21144i | 0 | −2.34776 | + | 2.34776i | 0 | 0 | 0 | ||||||||||||||
1673.7 | 0 | 0 | 0 | 0.330936 | − | 2.21144i | 0 | −2.34776 | + | 2.34776i | 0 | 0 | 0 | ||||||||||||||
1673.8 | 0 | 0 | 0 | 0.401341 | − | 2.19976i | 0 | −0.0703440 | + | 0.0703440i | 0 | 0 | 0 | ||||||||||||||
1673.9 | 0 | 0 | 0 | 0.925456 | − | 2.03557i | 0 | 1.92964 | − | 1.92964i | 0 | 0 | 0 | ||||||||||||||
1673.10 | 0 | 0 | 0 | 1.04895 | + | 1.97477i | 0 | 0.585090 | − | 0.585090i | 0 | 0 | 0 | ||||||||||||||
1673.11 | 0 | 0 | 0 | 1.98287 | + | 1.03354i | 0 | 2.54134 | − | 2.54134i | 0 | 0 | 0 | ||||||||||||||
1673.12 | 0 | 0 | 0 | 2.20020 | + | 0.398923i | 0 | −2.63796 | + | 2.63796i | 0 | 0 | 0 | ||||||||||||||
2357.1 | 0 | 0 | 0 | −2.20020 | + | 0.398923i | 0 | −2.63796 | − | 2.63796i | 0 | 0 | 0 | ||||||||||||||
2357.2 | 0 | 0 | 0 | −1.98287 | + | 1.03354i | 0 | 2.54134 | + | 2.54134i | 0 | 0 | 0 | ||||||||||||||
2357.3 | 0 | 0 | 0 | −1.04895 | + | 1.97477i | 0 | 0.585090 | + | 0.585090i | 0 | 0 | 0 | ||||||||||||||
2357.4 | 0 | 0 | 0 | −0.925456 | − | 2.03557i | 0 | 1.92964 | + | 1.92964i | 0 | 0 | 0 | ||||||||||||||
2357.5 | 0 | 0 | 0 | −0.401341 | − | 2.19976i | 0 | −0.0703440 | − | 0.0703440i | 0 | 0 | 0 | ||||||||||||||
2357.6 | 0 | 0 | 0 | −0.330936 | − | 2.21144i | 0 | −2.34776 | − | 2.34776i | 0 | 0 | 0 | ||||||||||||||
2357.7 | 0 | 0 | 0 | 0.330936 | + | 2.21144i | 0 | −2.34776 | − | 2.34776i | 0 | 0 | 0 | ||||||||||||||
2357.8 | 0 | 0 | 0 | 0.401341 | + | 2.19976i | 0 | −0.0703440 | − | 0.0703440i | 0 | 0 | 0 | ||||||||||||||
See all 24 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
3.b | odd | 2 | 1 | inner |
5.c | odd | 4 | 1 | inner |
15.e | even | 4 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 3420.2.x.c | ✓ | 24 |
3.b | odd | 2 | 1 | inner | 3420.2.x.c | ✓ | 24 |
5.c | odd | 4 | 1 | inner | 3420.2.x.c | ✓ | 24 |
15.e | even | 4 | 1 | inner | 3420.2.x.c | ✓ | 24 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
3420.2.x.c | ✓ | 24 | 1.a | even | 1 | 1 | trivial |
3420.2.x.c | ✓ | 24 | 3.b | odd | 2 | 1 | inner |
3420.2.x.c | ✓ | 24 | 5.c | odd | 4 | 1 | inner |
3420.2.x.c | ✓ | 24 | 15.e | even | 4 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{7}^{12} - 10 T_{7}^{9} + 269 T_{7}^{8} - 106 T_{7}^{7} + 50 T_{7}^{6} - 1552 T_{7}^{5} + 16440 T_{7}^{4} - 16020 T_{7}^{3} + 7688 T_{7}^{2} + 1240 T_{7} + 100 \)
acting on \(S_{2}^{\mathrm{new}}(3420, [\chi])\).