Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [63,12,Mod(1,63)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(63, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0]))
N = Newforms(chi, 12, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("63.1");
S:= CuspForms(chi, 12);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 63 = 3^{2} \cdot 7 \) |
| Weight: | \( k \) | \(=\) | \( 12 \) |
| Character orbit: | \([\chi]\) | \(=\) | 63.a (trivial) |
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: | yes |
| Analytic conductor: | \(48.4056203753\) |
| Analytic rank: | \(1\) |
| Dimension: | \(4\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{4} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{4} - 403x^{2} + 2352 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 2^{5}\cdot 3^{2}\cdot 7 \) |
| Twist minimal: | yes |
| Fricke sign: | \(-1\) |
| Sato-Tate group: | $\mathrm{SU}(2)$ |
$q$-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 \(1,\beta_1,\beta_2,\beta_3\) 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
\( x^{4} - 403x^{2} + 2352 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( -\nu^{3} + 403\nu ) / 14 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 3\nu^{3} - 621\nu ) / 14 \)
|
| \(\beta_{3}\) | \(=\) |
\( 12\nu^{2} - 2418 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{2} + 3\beta_1 ) / 42 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( \beta_{3} + 2418 ) / 12 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 403\beta_{2} + 621\beta_1 ) / 42 \)
|
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 1.1 |
|
−69.0284 | 0 | 2716.92 | 4997.94 | 0 | 16807.0 | −46174.5 | 0 | −345000. | ||||||||||||||||||||||||||||||
| 1.2 | −8.43086 | 0 | −1976.92 | 4732.63 | 0 | 16807.0 | 33933.6 | 0 | −39900.2 | |||||||||||||||||||||||||||||||
| 1.3 | 8.43086 | 0 | −1976.92 | −4732.63 | 0 | 16807.0 | −33933.6 | 0 | −39900.2 | |||||||||||||||||||||||||||||||
| 1.4 | 69.0284 | 0 | 2716.92 | −4997.94 | 0 | 16807.0 | 46174.5 | 0 | −345000. | |||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(3\) | \( +1 \) |
| \(7\) | \( -1 \) |
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 3.b | odd | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 63.12.a.h | ✓ | 4 |
| 3.b | odd | 2 | 1 | inner | 63.12.a.h | ✓ | 4 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 63.12.a.h | ✓ | 4 | 1.a | even | 1 | 1 | trivial |
| 63.12.a.h | ✓ | 4 | 3.b | odd | 2 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{4} - 4836T_{2}^{2} + 338688 \)
acting on \(S_{12}^{\mathrm{new}}(\Gamma_0(63))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} - 4836 T^{2} + 338688 \)
|
| $3$ |
\( T^{4} \)
|
| $5$ |
\( T^{4} + \cdots + 559483589070000 \)
|
| $7$ |
\( (T - 16807)^{4} \)
|
| $11$ |
\( T^{4} + \cdots + 13\!\cdots\!00 \)
|
| $13$ |
\( (T^{2} + 837716 T - 679603452332)^{2} \)
|
| $17$ |
\( T^{4} + \cdots + 28\!\cdots\!48 \)
|
| $19$ |
\( (T^{2} + \cdots - 44269232310800)^{2} \)
|
| $23$ |
\( T^{4} + \cdots + 45\!\cdots\!00 \)
|
| $29$ |
\( T^{4} + \cdots + 62\!\cdots\!00 \)
|
| $31$ |
\( (T^{2} + \cdots - 84\!\cdots\!04)^{2} \)
|
| $37$ |
\( (T^{2} + \cdots - 291108465643244)^{2} \)
|
| $41$ |
\( T^{4} + \cdots + 19\!\cdots\!68 \)
|
| $43$ |
\( (T^{2} + \cdots + 80\!\cdots\!28)^{2} \)
|
| $47$ |
\( T^{4} + \cdots + 15\!\cdots\!52 \)
|
| $53$ |
\( T^{4} + \cdots + 61\!\cdots\!88 \)
|
| $59$ |
\( T^{4} + \cdots + 17\!\cdots\!72 \)
|
| $61$ |
\( (T^{2} + \cdots - 10\!\cdots\!00)^{2} \)
|
| $67$ |
\( (T^{2} + \cdots - 86\!\cdots\!12)^{2} \)
|
| $71$ |
\( T^{4} + \cdots + 15\!\cdots\!28 \)
|
| $73$ |
\( (T^{2} + \cdots + 16\!\cdots\!28)^{2} \)
|
| $79$ |
\( (T^{2} + \cdots + 44\!\cdots\!76)^{2} \)
|
| $83$ |
\( T^{4} + \cdots + 23\!\cdots\!00 \)
|
| $89$ |
\( T^{4} + \cdots + 15\!\cdots\!08 \)
|
| $97$ |
\( (T^{2} + \cdots - 20\!\cdots\!44)^{2} \)
|
| show more | |
| show less |