Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [3420,2,Mod(1189,3420)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(3420, base_ring=CyclotomicField(6))
chi = DirichletCharacter(H, H._module([0, 0, 3, 4]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("3420.1189");
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.bj (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: | \(27.3088374913\) |
Analytic rank: | \(0\) |
Dimension: | \(40\) |
Relative dimension: | \(20\) 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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
1189.1 | 0 | 0 | 0 | −2.22922 | + | 0.174920i | 0 | − | 3.23724i | 0 | 0 | 0 | |||||||||||||||
1189.2 | 0 | 0 | 0 | −2.22845 | − | 0.184465i | 0 | 3.28838i | 0 | 0 | 0 | ||||||||||||||||
1189.3 | 0 | 0 | 0 | −2.22321 | − | 0.239488i | 0 | − | 1.46405i | 0 | 0 | 0 | |||||||||||||||
1189.4 | 0 | 0 | 0 | −1.75804 | + | 1.38178i | 0 | 2.95630i | 0 | 0 | 0 | ||||||||||||||||
1189.5 | 0 | 0 | 0 | −1.42631 | + | 1.72210i | 0 | 2.19629i | 0 | 0 | 0 | ||||||||||||||||
1189.6 | 0 | 0 | 0 | −1.31901 | − | 1.80561i | 0 | 1.46405i | 0 | 0 | 0 | ||||||||||||||||
1189.7 | 0 | 0 | 0 | −1.27397 | − | 1.83766i | 0 | − | 3.28838i | 0 | 0 | 0 | |||||||||||||||
1189.8 | 0 | 0 | 0 | −0.963123 | − | 2.01802i | 0 | 3.23724i | 0 | 0 | 0 | ||||||||||||||||
1189.9 | 0 | 0 | 0 | −0.778229 | + | 2.09627i | 0 | − | 2.19629i | 0 | 0 | 0 | |||||||||||||||
1189.10 | 0 | 0 | 0 | −0.317638 | + | 2.21339i | 0 | − | 2.95630i | 0 | 0 | 0 | |||||||||||||||
1189.11 | 0 | 0 | 0 | 0.317638 | − | 2.21339i | 0 | − | 2.95630i | 0 | 0 | 0 | |||||||||||||||
1189.12 | 0 | 0 | 0 | 0.778229 | − | 2.09627i | 0 | − | 2.19629i | 0 | 0 | 0 | |||||||||||||||
1189.13 | 0 | 0 | 0 | 0.963123 | + | 2.01802i | 0 | 3.23724i | 0 | 0 | 0 | ||||||||||||||||
1189.14 | 0 | 0 | 0 | 1.27397 | + | 1.83766i | 0 | − | 3.28838i | 0 | 0 | 0 | |||||||||||||||
1189.15 | 0 | 0 | 0 | 1.31901 | + | 1.80561i | 0 | 1.46405i | 0 | 0 | 0 | ||||||||||||||||
1189.16 | 0 | 0 | 0 | 1.42631 | − | 1.72210i | 0 | 2.19629i | 0 | 0 | 0 | ||||||||||||||||
1189.17 | 0 | 0 | 0 | 1.75804 | − | 1.38178i | 0 | 2.95630i | 0 | 0 | 0 | ||||||||||||||||
1189.18 | 0 | 0 | 0 | 2.22321 | + | 0.239488i | 0 | − | 1.46405i | 0 | 0 | 0 | |||||||||||||||
1189.19 | 0 | 0 | 0 | 2.22845 | + | 0.184465i | 0 | 3.28838i | 0 | 0 | 0 | ||||||||||||||||
1189.20 | 0 | 0 | 0 | 2.22922 | − | 0.174920i | 0 | − | 3.23724i | 0 | 0 | 0 | |||||||||||||||
See all 40 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
3.b | odd | 2 | 1 | inner |
5.b | even | 2 | 1 | inner |
15.d | odd | 2 | 1 | inner |
19.c | even | 3 | 1 | inner |
57.h | odd | 6 | 1 | inner |
95.i | even | 6 | 1 | inner |
285.n | odd | 6 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 3420.2.bj.e | ✓ | 40 |
3.b | odd | 2 | 1 | inner | 3420.2.bj.e | ✓ | 40 |
5.b | even | 2 | 1 | inner | 3420.2.bj.e | ✓ | 40 |
15.d | odd | 2 | 1 | inner | 3420.2.bj.e | ✓ | 40 |
19.c | even | 3 | 1 | inner | 3420.2.bj.e | ✓ | 40 |
57.h | odd | 6 | 1 | inner | 3420.2.bj.e | ✓ | 40 |
95.i | even | 6 | 1 | inner | 3420.2.bj.e | ✓ | 40 |
285.n | odd | 6 | 1 | inner | 3420.2.bj.e | ✓ | 40 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
3420.2.bj.e | ✓ | 40 | 1.a | even | 1 | 1 | trivial |
3420.2.bj.e | ✓ | 40 | 3.b | odd | 2 | 1 | inner |
3420.2.bj.e | ✓ | 40 | 5.b | even | 2 | 1 | inner |
3420.2.bj.e | ✓ | 40 | 15.d | odd | 2 | 1 | inner |
3420.2.bj.e | ✓ | 40 | 19.c | even | 3 | 1 | inner |
3420.2.bj.e | ✓ | 40 | 57.h | odd | 6 | 1 | inner |
3420.2.bj.e | ✓ | 40 | 95.i | even | 6 | 1 | inner |
3420.2.bj.e | ✓ | 40 | 285.n | odd | 6 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(3420, [\chi])\):
\( T_{7}^{10} + 37T_{7}^{8} + 519T_{7}^{6} + 3387T_{7}^{4} + 9996T_{7}^{2} + 10240 \) |
\( T_{11}^{10} - 45T_{11}^{8} + 550T_{11}^{6} - 2068T_{11}^{4} + 1316T_{11}^{2} - 64 \) |