Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [3204,2,Mod(1369,3204)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(3204, base_ring=CyclotomicField(4))
chi = DirichletCharacter(H, H._module([0, 0, 1]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("3204.1369");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 3204 = 2^{2} \cdot 3^{2} \cdot 89 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 3204.k (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: | \(25.5840688076\) |
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 algebraic \(q\)-expansion of this newform has not been computed, 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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
1369.1 | 0 | 0 | 0 | − | 3.94020i | 0 | −1.53250 | − | 1.53250i | 0 | 0 | 0 | |||||||||||||||
1369.2 | 0 | 0 | 0 | − | 2.73593i | 0 | 0.607916 | + | 0.607916i | 0 | 0 | 0 | |||||||||||||||
1369.3 | 0 | 0 | 0 | − | 2.27309i | 0 | 0.309364 | + | 0.309364i | 0 | 0 | 0 | |||||||||||||||
1369.4 | 0 | 0 | 0 | − | 2.13968i | 0 | 3.05356 | + | 3.05356i | 0 | 0 | 0 | |||||||||||||||
1369.5 | 0 | 0 | 0 | − | 1.31044i | 0 | 1.75135 | + | 1.75135i | 0 | 0 | 0 | |||||||||||||||
1369.6 | 0 | 0 | 0 | − | 1.23576i | 0 | −2.18969 | − | 2.18969i | 0 | 0 | 0 | |||||||||||||||
1369.7 | 0 | 0 | 0 | 1.23576i | 0 | −2.18969 | − | 2.18969i | 0 | 0 | 0 | ||||||||||||||||
1369.8 | 0 | 0 | 0 | 1.31044i | 0 | 1.75135 | + | 1.75135i | 0 | 0 | 0 | ||||||||||||||||
1369.9 | 0 | 0 | 0 | 2.13968i | 0 | 3.05356 | + | 3.05356i | 0 | 0 | 0 | ||||||||||||||||
1369.10 | 0 | 0 | 0 | 2.27309i | 0 | 0.309364 | + | 0.309364i | 0 | 0 | 0 | ||||||||||||||||
1369.11 | 0 | 0 | 0 | 2.73593i | 0 | 0.607916 | + | 0.607916i | 0 | 0 | 0 | ||||||||||||||||
1369.12 | 0 | 0 | 0 | 3.94020i | 0 | −1.53250 | − | 1.53250i | 0 | 0 | 0 | ||||||||||||||||
1657.1 | 0 | 0 | 0 | − | 3.94020i | 0 | −1.53250 | + | 1.53250i | 0 | 0 | 0 | |||||||||||||||
1657.2 | 0 | 0 | 0 | − | 2.73593i | 0 | 0.607916 | − | 0.607916i | 0 | 0 | 0 | |||||||||||||||
1657.3 | 0 | 0 | 0 | − | 2.27309i | 0 | 0.309364 | − | 0.309364i | 0 | 0 | 0 | |||||||||||||||
1657.4 | 0 | 0 | 0 | − | 2.13968i | 0 | 3.05356 | − | 3.05356i | 0 | 0 | 0 | |||||||||||||||
1657.5 | 0 | 0 | 0 | − | 1.31044i | 0 | 1.75135 | − | 1.75135i | 0 | 0 | 0 | |||||||||||||||
1657.6 | 0 | 0 | 0 | − | 1.23576i | 0 | −2.18969 | + | 2.18969i | 0 | 0 | 0 | |||||||||||||||
1657.7 | 0 | 0 | 0 | 1.23576i | 0 | −2.18969 | + | 2.18969i | 0 | 0 | 0 | ||||||||||||||||
1657.8 | 0 | 0 | 0 | 1.31044i | 0 | 1.75135 | − | 1.75135i | 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 |
89.c | even | 4 | 1 | inner |
267.e | odd | 4 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 3204.2.k.d | ✓ | 24 |
3.b | odd | 2 | 1 | inner | 3204.2.k.d | ✓ | 24 |
89.c | even | 4 | 1 | inner | 3204.2.k.d | ✓ | 24 |
267.e | odd | 4 | 1 | inner | 3204.2.k.d | ✓ | 24 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
3204.2.k.d | ✓ | 24 | 1.a | even | 1 | 1 | trivial |
3204.2.k.d | ✓ | 24 | 3.b | odd | 2 | 1 | inner |
3204.2.k.d | ✓ | 24 | 89.c | even | 4 | 1 | inner |
3204.2.k.d | ✓ | 24 | 267.e | odd | 4 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{12} + 36T_{5}^{10} + 473T_{5}^{8} + 2944T_{5}^{6} + 9144T_{5}^{4} + 13316T_{5}^{2} + 7209 \) acting on \(S_{2}^{\mathrm{new}}(3204, [\chi])\).