Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [468,4,Mod(1,468)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(468, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0, 0]))
N = Newforms(chi, 4, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("468.1");
S:= CuspForms(chi, 4);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 468 = 2^{2} \cdot 3^{2} \cdot 13 \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 468.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: | \(27.6128938827\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Coefficient field: | 4.4.5126992.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{4} - 25x^{2} + 117 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{11}]\) |
| Coefficient ring index: | \( 2^{5}\cdot 3 \) |
| 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
\( x^{4} - 25x^{2} + 117 \)
:
| \(\beta_{1}\) | \(=\) |
\( 4\nu \)
|
| \(\beta_{2}\) | \(=\) |
\( 2\nu^{3} - 32\nu \)
|
| \(\beta_{3}\) | \(=\) |
\( 4\nu^{2} - 50 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_1 ) / 4 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( \beta_{3} + 50 ) / 4 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( \beta_{2} + 8\beta_1 ) / 2 \)
|
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 |
|
0 | 0 | 0 | −17.3274 | 0 | −25.0599 | 0 | 0 | 0 | ||||||||||||||||||||||||||||||
| 1.2 | 0 | 0 | 0 | −9.98801 | 0 | 25.0599 | 0 | 0 | 0 | |||||||||||||||||||||||||||||||
| 1.3 | 0 | 0 | 0 | 9.98801 | 0 | 25.0599 | 0 | 0 | 0 | |||||||||||||||||||||||||||||||
| 1.4 | 0 | 0 | 0 | 17.3274 | 0 | −25.0599 | 0 | 0 | 0 | |||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \( -1 \) |
| \(3\) | \( +1 \) |
| \(13\) | \( -1 \) |
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 | 468.4.a.h | ✓ | 4 |
| 3.b | odd | 2 | 1 | inner | 468.4.a.h | ✓ | 4 |
| 4.b | odd | 2 | 1 | 1872.4.a.bp | 4 | ||
| 12.b | even | 2 | 1 | 1872.4.a.bp | 4 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 468.4.a.h | ✓ | 4 | 1.a | even | 1 | 1 | trivial |
| 468.4.a.h | ✓ | 4 | 3.b | odd | 2 | 1 | inner |
| 1872.4.a.bp | 4 | 4.b | odd | 2 | 1 | ||
| 1872.4.a.bp | 4 | 12.b | even | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{4} - 400T_{5}^{2} + 29952 \)
acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(468))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} \)
|
| $3$ |
\( T^{4} \)
|
| $5$ |
\( T^{4} - 400 T^{2} + 29952 \)
|
| $7$ |
\( (T^{2} - 628)^{2} \)
|
| $11$ |
\( T^{4} - 3496 T^{2} + 151632 \)
|
| $13$ |
\( (T - 13)^{4} \)
|
| $17$ |
\( T^{4} - 17056 T^{2} + 45556992 \)
|
| $19$ |
\( (T^{2} - 48 T - 52)^{2} \)
|
| $23$ |
\( T^{4} - 11808 T^{2} + 21835008 \)
|
| $29$ |
\( T^{4} - 11808 T^{2} + 21835008 \)
|
| $31$ |
\( (T^{2} - 208 T + 5164)^{2} \)
|
| $37$ |
\( (T^{2} - 372 T - 5596)^{2} \)
|
| $41$ |
\( T^{4} - 120592 T^{2} + 382457088 \)
|
| $43$ |
\( (T^{2} - 312 T - 38464)^{2} \)
|
| $47$ |
\( T^{4} + \cdots + 10881788112 \)
|
| $53$ |
\( T^{4} - 240448 T^{2} + 80990208 \)
|
| $59$ |
\( T^{4} + \cdots + 107019845712 \)
|
| $61$ |
\( (T^{2} - 732 T - 26812)^{2} \)
|
| $67$ |
\( (T^{2} - 784 T + 102796)^{2} \)
|
| $71$ |
\( T^{4} + \cdots + 79116337872 \)
|
| $73$ |
\( (T - 518)^{4} \)
|
| $79$ |
\( (T - 308)^{4} \)
|
| $83$ |
\( T^{4} + \cdots + 559558623312 \)
|
| $89$ |
\( T^{4} + \cdots + 378108327168 \)
|
| $97$ |
\( (T^{2} - 636 T - 1596988)^{2} \)
|
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