Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [3249,1,Mod(127,3249)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(3249, base_ring=CyclotomicField(18))
chi = DirichletCharacter(H, H._module([0, 5]))
N = Newforms(chi, 1, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("3249.127");
S:= CuspForms(chi, 1);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 3249 = 3^{2} \cdot 19^{2} \) |
| Weight: | \( k \) | \(=\) | \( 1 \) |
| Character orbit: | \([\chi]\) | \(=\) | 3249.ba (of order \(18\), degree \(6\), not 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: | \(1.62146222604\) |
| Analytic rank: | \(0\) |
| Dimension: | \(6\) |
| Coefficient field: | \(\Q(\zeta_{18})\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{6} - x^{3} + 1 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{4}]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 171) |
| Projective image: | \(D_{2}\) |
| Projective field: | Galois closure of \(\Q(\sqrt{-3}, \sqrt{-19})\) |
| Artin image: | $D_4\times C_9$ |
| Artin field: | Galois closure of \(\mathbb{Q}[x]/(x^{36} - \cdots)\) |
$q$-expansion
comment: q-expansion
sage: f.q_expansion() # note that sage often uses an isomorphic number field
gp: mfcoefs(f, 20)
The \(q\)-expansion and trace form are shown below.
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3249\mathbb{Z}\right)^\times\).
| \(n\) | \(362\) | \(2890\) |
| \(\chi(n)\) | \(1\) | \(\zeta_{18}^{7}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 127.1 |
|
0 | 0 | 0.173648 | − | 0.984808i | 0 | 0 | 1.00000 | − | 1.73205i | 0 | 0 | 0 | ||||||||||||||||||||||||||||||||
| 262.1 | 0 | 0 | 0.766044 | + | 0.642788i | 0 | 0 | 1.00000 | − | 1.73205i | 0 | 0 | 0 | |||||||||||||||||||||||||||||||||
| 307.1 | 0 | 0 | 0.173648 | + | 0.984808i | 0 | 0 | 1.00000 | + | 1.73205i | 0 | 0 | 0 | |||||||||||||||||||||||||||||||||
| 694.1 | 0 | 0 | −0.939693 | + | 0.342020i | 0 | 0 | 1.00000 | − | 1.73205i | 0 | 0 | 0 | |||||||||||||||||||||||||||||||||
| 838.1 | 0 | 0 | −0.939693 | − | 0.342020i | 0 | 0 | 1.00000 | + | 1.73205i | 0 | 0 | 0 | |||||||||||||||||||||||||||||||||
| 3187.1 | 0 | 0 | 0.766044 | − | 0.642788i | 0 | 0 | 1.00000 | + | 1.73205i | 0 | 0 | 0 | |||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 3.b | odd | 2 | 1 | CM by \(\Q(\sqrt{-3}) \) |
| 19.b | odd | 2 | 1 | CM by \(\Q(\sqrt{-19}) \) |
| 57.d | even | 2 | 1 | RM by \(\Q(\sqrt{57}) \) |
| 19.c | even | 3 | 2 | inner |
| 19.d | odd | 6 | 2 | inner |
| 19.e | even | 9 | 3 | inner |
| 19.f | odd | 18 | 3 | inner |
| 57.f | even | 6 | 2 | inner |
| 57.h | odd | 6 | 2 | inner |
| 57.j | even | 18 | 3 | inner |
| 57.l | odd | 18 | 3 | inner |
Twists
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{1}^{\mathrm{new}}(3249, [\chi])\):
|
\( T_{7}^{2} - 2T_{7} + 4 \)
|
|
\( T_{13} \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{6} \)
|
| $3$ |
\( T^{6} \)
|
| $5$ |
\( T^{6} \)
|
| $7$ |
\( (T^{2} - 2 T + 4)^{3} \)
|
| $11$ |
\( T^{6} \)
|
| $13$ |
\( T^{6} \)
|
| $17$ |
\( T^{6} \)
|
| $19$ |
\( T^{6} \)
|
| $23$ |
\( T^{6} \)
|
| $29$ |
\( T^{6} \)
|
| $31$ |
\( T^{6} \)
|
| $37$ |
\( T^{6} \)
|
| $41$ |
\( T^{6} \)
|
| $43$ |
\( T^{6} + 8T^{3} + 64 \)
|
| $47$ |
\( T^{6} \)
|
| $53$ |
\( T^{6} \)
|
| $59$ |
\( T^{6} \)
|
| $61$ |
\( T^{6} - 8T^{3} + 64 \)
|
| $67$ |
\( T^{6} \)
|
| $71$ |
\( T^{6} \)
|
| $73$ |
\( T^{6} + 8T^{3} + 64 \)
|
| $79$ |
\( T^{6} \)
|
| $83$ |
\( T^{6} \)
|
| $89$ |
\( T^{6} \)
|
| $97$ |
\( T^{6} \)
|
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