Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [252,8,Mod(1,252)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(252, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0, 0]))
N = Newforms(chi, 8, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("252.1");
S:= CuspForms(chi, 8);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 252 = 2^{2} \cdot 3^{2} \cdot 7 \) |
| Weight: | \( k \) | \(=\) | \( 8 \) |
| Character orbit: | \([\chi]\) | \(=\) | 252.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: | \(78.7210264220\) |
| 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: |
\( x^{4} - x^{3} - 14292x^{2} + 540043x + 5027477 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{13}]\) |
| Coefficient ring index: | \( 2^{8}\cdot 3^{2} \) |
| 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
\( x^{4} - x^{3} - 14292x^{2} + 540043x + 5027477 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( -\nu^{3} - 70\nu^{2} + 12238\nu + 102865 ) / 729 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 8\nu^{3} + 344\nu^{2} - 81704\nu + 716620 ) / 729 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 8\nu^{2} + 696\nu - 57344 ) / 27 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{3} + \beta_{2} + 8\beta _1 + 12 ) / 48 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( 25\beta_{3} - 29\beta_{2} - 232\beta _1 + 114340 ) / 16 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 1747\beta_{3} + 4582\beta_{2} + 27908\beta _1 - 4731756 ) / 12 \)
|
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 | −537.098 | 0 | −343.000 | 0 | 0 | 0 | ||||||||||||||||||||||||||||||
| 1.2 | 0 | 0 | 0 | −6.16284 | 0 | −343.000 | 0 | 0 | 0 | |||||||||||||||||||||||||||||||
| 1.3 | 0 | 0 | 0 | 6.16284 | 0 | −343.000 | 0 | 0 | 0 | |||||||||||||||||||||||||||||||
| 1.4 | 0 | 0 | 0 | 537.098 | 0 | −343.000 | 0 | 0 | 0 | |||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \(-1\) |
| \(3\) | \(1\) |
| \(7\) | \(1\) |
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 | 252.8.a.g | ✓ | 4 |
| 3.b | odd | 2 | 1 | inner | 252.8.a.g | ✓ | 4 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 252.8.a.g | ✓ | 4 | 1.a | even | 1 | 1 | trivial |
| 252.8.a.g | ✓ | 4 | 3.b | odd | 2 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{4} - 288512T_{5}^{2} + 10956400 \)
acting on \(S_{8}^{\mathrm{new}}(\Gamma_0(252))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} \)
|
| $3$ |
\( T^{4} \)
|
| $5$ |
\( T^{4} - 288512 T^{2} + 10956400 \)
|
| $7$ |
\( (T + 343)^{4} \)
|
| $11$ |
\( T^{4} + \cdots + 31940258996464 \)
|
| $13$ |
\( (T^{2} - 6636 T + 7501140)^{2} \)
|
| $17$ |
\( T^{4} + \cdots + 68\!\cdots\!00 \)
|
| $19$ |
\( (T^{2} + 4592 T - 1008535760)^{2} \)
|
| $23$ |
\( T^{4} + \cdots + 55\!\cdots\!96 \)
|
| $29$ |
\( T^{4} + \cdots + 29\!\cdots\!76 \)
|
| $31$ |
\( (T^{2} + 222768 T - 17934158160)^{2} \)
|
| $37$ |
\( (T^{2} + 426404 T - 61976917196)^{2} \)
|
| $41$ |
\( T^{4} + \cdots + 34\!\cdots\!00 \)
|
| $43$ |
\( (T^{2} - 75112 T - 702655967600)^{2} \)
|
| $47$ |
\( T^{4} + \cdots + 14\!\cdots\!16 \)
|
| $53$ |
\( T^{4} + \cdots + 98\!\cdots\!00 \)
|
| $59$ |
\( T^{4} + \cdots + 21\!\cdots\!16 \)
|
| $61$ |
\( (T^{2} + 651420 T - 273620692044)^{2} \)
|
| $67$ |
\( (T^{2} + \cdots + 1461826311376)^{2} \)
|
| $71$ |
\( T^{4} + \cdots + 25\!\cdots\!24 \)
|
| $73$ |
\( (T^{2} + \cdots - 18614489191580)^{2} \)
|
| $79$ |
\( (T^{2} + \cdots + 17187103673280)^{2} \)
|
| $83$ |
\( T^{4} + \cdots + 11\!\cdots\!44 \)
|
| $89$ |
\( T^{4} + \cdots + 65\!\cdots\!56 \)
|
| $97$ |
\( (T^{2} + \cdots + 79763151012580)^{2} \)
|
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