Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [3234,2,Mod(2155,3234)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(3234, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 1, 1]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("3234.2155");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 3234 = 2 \cdot 3 \cdot 7^{2} \cdot 11 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 3234.e (of order \(2\), degree \(1\), 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: | \(25.8236200137\) |
Analytic rank: | \(0\) |
Dimension: | \(24\) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{2}]$ |
$q$-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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
2155.1 | − | 1.00000i | − | 1.00000i | −1.00000 | − | 3.51377i | −1.00000 | 0 | 1.00000i | −1.00000 | −3.51377 | |||||||||||||||
2155.2 | − | 1.00000i | − | 1.00000i | −1.00000 | − | 2.84462i | −1.00000 | 0 | 1.00000i | −1.00000 | −2.84462 | |||||||||||||||
2155.3 | − | 1.00000i | − | 1.00000i | −1.00000 | − | 1.98215i | −1.00000 | 0 | 1.00000i | −1.00000 | −1.98215 | |||||||||||||||
2155.4 | − | 1.00000i | − | 1.00000i | −1.00000 | − | 1.58222i | −1.00000 | 0 | 1.00000i | −1.00000 | −1.58222 | |||||||||||||||
2155.5 | − | 1.00000i | − | 1.00000i | −1.00000 | − | 0.942563i | −1.00000 | 0 | 1.00000i | −1.00000 | −0.942563 | |||||||||||||||
2155.6 | − | 1.00000i | − | 1.00000i | −1.00000 | − | 0.500309i | −1.00000 | 0 | 1.00000i | −1.00000 | −0.500309 | |||||||||||||||
2155.7 | − | 1.00000i | − | 1.00000i | −1.00000 | 0.313884i | −1.00000 | 0 | 1.00000i | −1.00000 | 0.313884 | ||||||||||||||||
2155.8 | − | 1.00000i | − | 1.00000i | −1.00000 | 0.357203i | −1.00000 | 0 | 1.00000i | −1.00000 | 0.357203 | ||||||||||||||||
2155.9 | − | 1.00000i | − | 1.00000i | −1.00000 | 0.516505i | −1.00000 | 0 | 1.00000i | −1.00000 | 0.516505 | ||||||||||||||||
2155.10 | − | 1.00000i | − | 1.00000i | −1.00000 | 2.66749i | −1.00000 | 0 | 1.00000i | −1.00000 | 2.66749 | ||||||||||||||||
2155.11 | − | 1.00000i | − | 1.00000i | −1.00000 | 3.45711i | −1.00000 | 0 | 1.00000i | −1.00000 | 3.45711 | ||||||||||||||||
2155.12 | − | 1.00000i | − | 1.00000i | −1.00000 | 4.05345i | −1.00000 | 0 | 1.00000i | −1.00000 | 4.05345 | ||||||||||||||||
2155.13 | 1.00000i | 1.00000i | −1.00000 | − | 4.05345i | −1.00000 | 0 | − | 1.00000i | −1.00000 | 4.05345 | ||||||||||||||||
2155.14 | 1.00000i | 1.00000i | −1.00000 | − | 3.45711i | −1.00000 | 0 | − | 1.00000i | −1.00000 | 3.45711 | ||||||||||||||||
2155.15 | 1.00000i | 1.00000i | −1.00000 | − | 2.66749i | −1.00000 | 0 | − | 1.00000i | −1.00000 | 2.66749 | ||||||||||||||||
2155.16 | 1.00000i | 1.00000i | −1.00000 | − | 0.516505i | −1.00000 | 0 | − | 1.00000i | −1.00000 | 0.516505 | ||||||||||||||||
2155.17 | 1.00000i | 1.00000i | −1.00000 | − | 0.357203i | −1.00000 | 0 | − | 1.00000i | −1.00000 | 0.357203 | ||||||||||||||||
2155.18 | 1.00000i | 1.00000i | −1.00000 | − | 0.313884i | −1.00000 | 0 | − | 1.00000i | −1.00000 | 0.313884 | ||||||||||||||||
2155.19 | 1.00000i | 1.00000i | −1.00000 | 0.500309i | −1.00000 | 0 | − | 1.00000i | −1.00000 | −0.500309 | |||||||||||||||||
2155.20 | 1.00000i | 1.00000i | −1.00000 | 0.942563i | −1.00000 | 0 | − | 1.00000i | −1.00000 | −0.942563 | |||||||||||||||||
See all 24 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
77.b | even | 2 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 3234.2.e.c | ✓ | 24 |
7.b | odd | 2 | 1 | 3234.2.e.d | yes | 24 | |
11.b | odd | 2 | 1 | 3234.2.e.d | yes | 24 | |
77.b | even | 2 | 1 | inner | 3234.2.e.c | ✓ | 24 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
3234.2.e.c | ✓ | 24 | 1.a | even | 1 | 1 | trivial |
3234.2.e.c | ✓ | 24 | 77.b | even | 2 | 1 | inner |
3234.2.e.d | yes | 24 | 7.b | odd | 2 | 1 | |
3234.2.e.d | yes | 24 | 11.b | odd | 2 | 1 |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(3234, [\chi])\):
\( T_{5}^{24} + 64 T_{5}^{22} + 1696 T_{5}^{20} + 24160 T_{5}^{18} + 201024 T_{5}^{16} + 995200 T_{5}^{14} + \cdots + 1024 \) |
\( T_{13}^{12} - 76 T_{13}^{10} + 48 T_{13}^{9} + 2134 T_{13}^{8} - 2432 T_{13}^{7} - 26512 T_{13}^{6} + \cdots - 207224 \) |