Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [3204,2,Mod(3025,3204)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(3204, base_ring=CyclotomicField(2))
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("3204.3025");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 3204 = 2^{2} \cdot 3^{2} \cdot 89 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 3204.g (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: | \(25.5840688076\) |
| Analytic rank: | \(0\) |
| Dimension: | \(6\) |
| Coefficient field: | 6.0.4227136.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{6} + 6x^{4} + 7x^{2} + 1 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{23}]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | yes |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{2}]$ |
$q$-expansion
comment: q-expansion
sage: f.q_expansion() # note that sage often uses an isomorphic number field
gp: mfcoefs(f, 20)
Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.
Basis of coefficient ring in terms of a root \(\nu\) of
\( x^{6} + 6x^{4} + 7x^{2} + 1 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} + 2 \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{4} + 5\nu^{2} + 3 \)
|
| \(\beta_{4}\) | \(=\) |
\( -\nu^{5} - 5\nu^{3} - 3\nu \)
|
| \(\beta_{5}\) | \(=\) |
\( \nu^{5} + 6\nu^{3} + 6\nu \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} - 2 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{5} + \beta_{4} - 3\beta_1 \)
|
| \(\nu^{4}\) | \(=\) |
\( \beta_{3} - 5\beta_{2} + 7 \)
|
| \(\nu^{5}\) | \(=\) |
\( -5\beta_{5} - 6\beta_{4} + 12\beta_1 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3204\mathbb{Z}\right)^\times\).
| \(n\) | \(181\) | \(713\) | \(1603\) |
| \(\chi(n)\) | \(-1\) | \(1\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 3025.1 |
|
0 | 0 | 0 | −2.03293 | 0 | − | 4.99330i | 0 | 0 | 0 | |||||||||||||||||||||||||||||||||||
| 3025.2 | 0 | 0 | 0 | −2.03293 | 0 | 4.99330i | 0 | 0 | 0 | |||||||||||||||||||||||||||||||||||||
| 3025.3 | 0 | 0 | 0 | 3.25561 | 0 | − | 2.80888i | 0 | 0 | 0 | ||||||||||||||||||||||||||||||||||||
| 3025.4 | 0 | 0 | 0 | 3.25561 | 0 | 2.80888i | 0 | 0 | 0 | |||||||||||||||||||||||||||||||||||||
| 3025.5 | 0 | 0 | 0 | 3.77733 | 0 | − | 1.78246i | 0 | 0 | 0 | ||||||||||||||||||||||||||||||||||||
| 3025.6 | 0 | 0 | 0 | 3.77733 | 0 | 1.78246i | 0 | 0 | 0 | |||||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 89.b | even | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 3204.2.g.c | yes | 6 |
| 3.b | odd | 2 | 1 | 3204.2.g.b | ✓ | 6 | |
| 89.b | even | 2 | 1 | inner | 3204.2.g.c | yes | 6 |
| 267.b | odd | 2 | 1 | 3204.2.g.b | ✓ | 6 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 3204.2.g.b | ✓ | 6 | 3.b | odd | 2 | 1 | |
| 3204.2.g.b | ✓ | 6 | 267.b | odd | 2 | 1 | |
| 3204.2.g.c | yes | 6 | 1.a | even | 1 | 1 | trivial |
| 3204.2.g.c | yes | 6 | 89.b | even | 2 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{3} - 5T_{5}^{2} - 2T_{5} + 25 \)
acting on \(S_{2}^{\mathrm{new}}(3204, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{6} \)
|
| $3$ |
\( T^{6} \)
|
| $5$ |
\( (T^{3} - 5 T^{2} - 2 T + 25)^{2} \)
|
| $7$ |
\( T^{6} + 36 T^{4} + \cdots + 625 \)
|
| $11$ |
\( (T^{3} - 8 T^{2} - T + 75)^{2} \)
|
| $13$ |
\( T^{6} + 57 T^{4} + \cdots + 625 \)
|
| $17$ |
\( (T^{3} - 6 T^{2} + \cdots + 175)^{2} \)
|
| $19$ |
\( T^{6} + 88 T^{4} + \cdots + 5625 \)
|
| $23$ |
\( T^{6} + 67 T^{4} + \cdots + 3721 \)
|
| $29$ |
\( T^{6} + 88 T^{4} + \cdots + 4489 \)
|
| $31$ |
\( T^{6} + 103 T^{4} + \cdots + 625 \)
|
| $37$ |
\( T^{6} + 79 T^{4} + \cdots + 625 \)
|
| $41$ |
\( T^{6} + 87 T^{4} + \cdots + 49 \)
|
| $43$ |
\( T^{6} + 127 T^{4} + \cdots + 625 \)
|
| $47$ |
\( (T^{3} - 14 T^{2} + \cdots - 25)^{2} \)
|
| $53$ |
\( (T^{3} - 19 T^{2} + \cdots - 225)^{2} \)
|
| $59$ |
\( T^{6} + 214 T^{4} + \cdots + 218089 \)
|
| $61$ |
\( T^{6} + 126 T^{4} + \cdots + 15625 \)
|
| $67$ |
\( (T^{3} + 7 T^{2} + \cdots + 225)^{2} \)
|
| $71$ |
\( (T^{3} - 13 T^{2} + \cdots - 25)^{2} \)
|
| $73$ |
\( (T^{3} - 41 T + 101)^{2} \)
|
| $79$ |
\( (T^{3} + 11 T^{2} + \cdots - 313)^{2} \)
|
| $83$ |
\( T^{6} + 408 T^{4} + \cdots + 1575025 \)
|
| $89$ |
\( T^{6} + 38 T^{5} + \cdots + 704969 \)
|
| $97$ |
\( (T^{3} - 12 T^{2} + \cdots + 811)^{2} \)
|
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