Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [529,4,Mod(1,529)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(529, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0]))
N = Newforms(chi, 4, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("529.1");
S:= CuspForms(chi, 4);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 529 = 23^{2} \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 529.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: | \(31.2120103930\) |
| Analytic rank: | \(1\) |
| Dimension: | \(6\) |
| Coefficient field: | 6.6.13191844032.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{6} - 39x^{4} + 276x^{2} - 192 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{17}]\) |
| Coefficient ring index: | \( 3 \) |
| 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 \(1,\beta_1,\ldots,\beta_{5}\) 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
\( x^{6} - 39x^{4} + 276x^{2} - 192 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( -\nu^{5} + 47\nu^{3} - 444\nu ) / 208 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -\nu^{4} + 47\nu^{2} - 288 ) / 52 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 3\nu^{4} - 89\nu^{2} + 188 ) / 52 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( -9\nu^{5} + 319\nu^{3} - 1188\nu ) / 208 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 5\nu^{5} - 183\nu^{3} + 1128\nu ) / 104 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{5} + \beta_{4} + \beta_1 ) / 3 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{3} + 3\beta_{2} + 13 \)
|
| \(\nu^{3}\) | \(=\) |
\( 9\beta_{5} + 7\beta_{4} + 27\beta_1 \)
|
| \(\nu^{4}\) | \(=\) |
\( 47\beta_{3} + 89\beta_{2} + 323 \)
|
| \(\nu^{5}\) | \(=\) |
\( 275\beta_{5} + 181\beta_{4} + 913\beta_1 \)
|
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 1.1 |
|
−4.84527 | 6.16096 | 15.4766 | −14.7737 | −29.8515 | 30.6409 | −36.2264 | 10.9574 | 71.5824 | ||||||||||||||||||||||||||||||||||||
| 1.2 | −4.84527 | 6.16096 | 15.4766 | 14.7737 | −29.8515 | −30.6409 | −36.2264 | 10.9574 | −71.5824 | |||||||||||||||||||||||||||||||||||||
| 1.3 | 0.580756 | −8.12174 | −7.66272 | −20.7157 | −4.71675 | 11.8446 | −9.09622 | 38.9626 | −12.0308 | |||||||||||||||||||||||||||||||||||||
| 1.4 | 0.580756 | −8.12174 | −7.66272 | 20.7157 | −4.71675 | −11.8446 | −9.09622 | 38.9626 | 12.0308 | |||||||||||||||||||||||||||||||||||||
| 1.5 | 4.26451 | −1.03922 | 10.1861 | −4.31248 | −4.43175 | 0.916304 | 9.32257 | −25.9200 | −18.3906 | |||||||||||||||||||||||||||||||||||||
| 1.6 | 4.26451 | −1.03922 | 10.1861 | 4.31248 | −4.43175 | −0.916304 | 9.32257 | −25.9200 | 18.3906 | |||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(23\) | \( -1 \) |
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 23.b | odd | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 529.4.a.j | ✓ | 6 |
| 23.b | odd | 2 | 1 | inner | 529.4.a.j | ✓ | 6 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 529.4.a.j | ✓ | 6 | 1.a | even | 1 | 1 | trivial |
| 529.4.a.j | ✓ | 6 | 23.b | odd | 2 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(529))\):
|
\( T_{2}^{3} - 21T_{2} + 12 \)
|
|
\( T_{3}^{3} + 3T_{3}^{2} - 48T_{3} - 52 \)
|
|
\( T_{5}^{6} - 666T_{5}^{4} + 105705T_{5}^{2} - 1741932 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{3} - 21 T + 12)^{2} \)
|
| $3$ |
\( (T^{3} + 3 T^{2} - 48 T - 52)^{2} \)
|
| $5$ |
\( T^{6} - 666 T^{4} + \cdots - 1741932 \)
|
| $7$ |
\( T^{6} - 1080 T^{4} + \cdots - 110592 \)
|
| $11$ |
\( T^{6} + \cdots - 1341206208 \)
|
| $13$ |
\( (T^{3} + 84 T^{2} + \cdots - 1948)^{2} \)
|
| $17$ |
\( T^{6} + \cdots - 3281271552 \)
|
| $19$ |
\( T^{6} + \cdots - 59359082688 \)
|
| $23$ |
\( T^{6} \)
|
| $29$ |
\( (T^{3} + 288 T^{2} + \cdots - 56688)^{2} \)
|
| $31$ |
\( (T^{3} - 96 T^{2} + \cdots + 12267392)^{2} \)
|
| $37$ |
\( T^{6} + \cdots - 7883993088 \)
|
| $41$ |
\( (T^{3} + 324 T^{2} + \cdots - 7698498)^{2} \)
|
| $43$ |
\( T^{6} + \cdots - 228202687426752 \)
|
| $47$ |
\( (T^{3} + 342 T^{2} + \cdots - 5178552)^{2} \)
|
| $53$ |
\( T^{6} + \cdots - 184087970695872 \)
|
| $59$ |
\( (T^{3} - 873 T^{2} + \cdots - 5440608)^{2} \)
|
| $61$ |
\( T^{6} + \cdots - 11\!\cdots\!12 \)
|
| $67$ |
\( T^{6} + \cdots - 13\!\cdots\!68 \)
|
| $71$ |
\( (T^{3} + 180 T^{2} + \cdots - 5909952)^{2} \)
|
| $73$ |
\( (T^{3} + 1839 T^{2} + \cdots + 200808689)^{2} \)
|
| $79$ |
\( T^{6} + \cdots - 33\!\cdots\!52 \)
|
| $83$ |
\( T^{6} + \cdots - 546901642513152 \)
|
| $89$ |
\( T^{6} + \cdots - 11\!\cdots\!48 \)
|
| $97$ |
\( T^{6} + \cdots - 11\!\cdots\!52 \)
|
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