Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [546,4,Mod(79,546)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(546, base_ring=CyclotomicField(6))
chi = DirichletCharacter(H, H._module([0, 2, 0]))
N = Newforms(chi, 4, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("546.79");
S:= CuspForms(chi, 4);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 546 = 2 \cdot 3 \cdot 7 \cdot 13 \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 546.i (of order \(3\), 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: | \(32.2150428631\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Relative dimension: | \(2\) over \(\Q(\zeta_{3})\) |
| Coefficient field: | \(\Q(\sqrt{-3}, \sqrt{-5})\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{4} - 5x^{2} + 25 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 3 \) |
| Twist minimal: | yes |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{3}]$ |
$q$-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 \(1,\beta_1,\beta_2,\beta_3\) 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
\( x^{4} - 5x^{2} + 25 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( \nu^{2} ) / 5 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{3} + 5\nu ) / 5 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -\nu^{3} + 10\nu ) / 5 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{3} + \beta_{2} ) / 3 \)
|
| \(\nu^{2}\) | \(=\) |
\( 5\beta_1 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( -5\beta_{3} + 10\beta_{2} ) / 3 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/546\mathbb{Z}\right)^\times\).
| \(n\) | \(157\) | \(365\) | \(379\) |
| \(\chi(n)\) | \(-\beta_{1}\) | \(1\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 79.1 |
|
−1.00000 | + | 1.73205i | 1.50000 | + | 2.59808i | −2.00000 | − | 3.46410i | −0.436492 | + | 0.756026i | −6.00000 | −10.8095 | − | 15.0385i | 8.00000 | −4.50000 | + | 7.79423i | −0.872983 | − | 1.51205i | ||||||||||||||||
| 79.2 | −1.00000 | + | 1.73205i | 1.50000 | + | 2.59808i | −2.00000 | − | 3.46410i | 3.43649 | − | 5.95218i | −6.00000 | 0.809475 | + | 18.5026i | 8.00000 | −4.50000 | + | 7.79423i | 6.87298 | + | 11.9044i | |||||||||||||||||
| 235.1 | −1.00000 | − | 1.73205i | 1.50000 | − | 2.59808i | −2.00000 | + | 3.46410i | −0.436492 | − | 0.756026i | −6.00000 | −10.8095 | + | 15.0385i | 8.00000 | −4.50000 | − | 7.79423i | −0.872983 | + | 1.51205i | |||||||||||||||||
| 235.2 | −1.00000 | − | 1.73205i | 1.50000 | − | 2.59808i | −2.00000 | + | 3.46410i | 3.43649 | + | 5.95218i | −6.00000 | 0.809475 | − | 18.5026i | 8.00000 | −4.50000 | − | 7.79423i | 6.87298 | − | 11.9044i | |||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 7.c | even | 3 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 546.4.i.a | ✓ | 4 |
| 7.c | even | 3 | 1 | inner | 546.4.i.a | ✓ | 4 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 546.4.i.a | ✓ | 4 | 1.a | even | 1 | 1 | trivial |
| 546.4.i.a | ✓ | 4 | 7.c | even | 3 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{4} - 6T_{5}^{3} + 42T_{5}^{2} + 36T_{5} + 36 \)
acting on \(S_{4}^{\mathrm{new}}(546, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{2} + 2 T + 4)^{2} \)
|
| $3$ |
\( (T^{2} - 3 T + 9)^{2} \)
|
| $5$ |
\( T^{4} - 6 T^{3} + \cdots + 36 \)
|
| $7$ |
\( T^{4} + 20 T^{3} + \cdots + 117649 \)
|
| $11$ |
\( T^{4} + 36 T^{3} + \cdots + 2601 \)
|
| $13$ |
\( (T - 13)^{4} \)
|
| $17$ |
\( T^{4} + 120 T^{3} + \cdots + 5688225 \)
|
| $19$ |
\( T^{4} + 118 T^{3} + \cdots + 1745041 \)
|
| $23$ |
\( T^{4} - 42 T^{3} + \cdots + 8608356 \)
|
| $29$ |
\( (T^{2} + 54 T - 231)^{2} \)
|
| $31$ |
\( T^{4} + \cdots + 1896079936 \)
|
| $37$ |
\( T^{4} + 154 T^{3} + \cdots + 25060036 \)
|
| $41$ |
\( (T^{2} - 546 T + 56154)^{2} \)
|
| $43$ |
\( (T^{2} + 200 T - 146060)^{2} \)
|
| $47$ |
\( T^{4} + \cdots + 27549692361 \)
|
| $53$ |
\( T^{4} + \cdots + 9768357225 \)
|
| $59$ |
\( T^{4} + \cdots + 8075718225 \)
|
| $61$ |
\( T^{4} + \cdots + 2328159001 \)
|
| $67$ |
\( T^{4} + \cdots + 2848890625 \)
|
| $71$ |
\( (T^{2} + 66 T - 10671)^{2} \)
|
| $73$ |
\( T^{4} + \cdots + 6913922500 \)
|
| $79$ |
\( T^{4} + \cdots + 3485249296 \)
|
| $83$ |
\( (T^{2} + 816 T - 28476)^{2} \)
|
| $89$ |
\( T^{4} + \cdots + 2327864238756 \)
|
| $97$ |
\( (T^{2} - 1486 T + 200914)^{2} \)
|
| show more | |
| show less |