Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [1444,4,Mod(1,1444)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1444, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([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("1444.1");
S:= CuspForms(chi, 4);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 1444 = 2^{2} \cdot 19^{2} \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 1444.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: | \(85.1987580483\) |
| Analytic rank: | \(1\) |
| Dimension: | \(5\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{5} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{5} - 2x^{4} - 86x^{3} + 192x^{2} + 768x - 1008 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 2^{2} \) |
| Twist minimal: | no (minimal twist has level 76) |
| 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 \(1,\beta_1,\beta_2,\beta_3,\beta_4\) 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
\( x^{5} - 2x^{4} - 86x^{3} + 192x^{2} + 768x - 1008 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{3} + 2\nu^{2} - 66\nu - 36 ) / 6 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( \nu^{4} - 82\nu^{2} + 36\nu + 552 ) / 24 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( -\nu^{3} + 4\nu^{2} + 66\nu - 180 ) / 6 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{4} + \beta_{2} + 36 \)
|
| \(\nu^{3}\) | \(=\) |
\( -2\beta_{4} + 4\beta_{2} + 66\beta _1 - 36 \)
|
| \(\nu^{4}\) | \(=\) |
\( 82\beta_{4} + 24\beta_{3} + 82\beta_{2} - 36\beta _1 + 2400 \)
|
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 | −9.41967 | 0 | 3.81538 | 0 | 30.9131 | 0 | 61.7302 | 0 | |||||||||||||||||||||||||||||||||
| 1.2 | 0 | −5.13017 | 0 | −15.9628 | 0 | −31.8733 | 0 | −0.681390 | 0 | ||||||||||||||||||||||||||||||||||
| 1.3 | 0 | −2.15366 | 0 | 21.2570 | 0 | −18.8370 | 0 | −22.3618 | 0 | ||||||||||||||||||||||||||||||||||
| 1.4 | 0 | 1.83230 | 0 | −4.97538 | 0 | 19.2102 | 0 | −23.6427 | 0 | ||||||||||||||||||||||||||||||||||
| 1.5 | 0 | 7.87119 | 0 | −0.134134 | 0 | −9.41300 | 0 | 34.9557 | 0 | ||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \(-1\) |
| \(19\) | \(1\) |
Inner twists
This newform does not admit any (nontrivial) inner twists.
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 1444.4.a.f | 5 | |
| 19.b | odd | 2 | 1 | 1444.4.a.g | 5 | ||
| 19.c | even | 3 | 2 | 76.4.e.a | ✓ | 10 | |
| 57.h | odd | 6 | 2 | 684.4.k.c | 10 | ||
| 76.g | odd | 6 | 2 | 304.4.i.f | 10 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 76.4.e.a | ✓ | 10 | 19.c | even | 3 | 2 | |
| 304.4.i.f | 10 | 76.g | odd | 6 | 2 | ||
| 684.4.k.c | 10 | 57.h | odd | 6 | 2 | ||
| 1444.4.a.f | 5 | 1.a | even | 1 | 1 | trivial | |
| 1444.4.a.g | 5 | 19.b | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{5} + 7T_{3}^{4} - 68T_{3}^{3} - 428T_{3}^{2} + 139T_{3} + 1501 \)
acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(1444))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{5} \)
|
| $3$ |
\( T^{5} + 7 T^{4} + \cdots + 1501 \)
|
| $5$ |
\( T^{5} - 4 T^{4} + \cdots + 864 \)
|
| $7$ |
\( T^{5} + 10 T^{4} + \cdots + 3356160 \)
|
| $11$ |
\( T^{5} + 25 T^{4} + \cdots - 14618352 \)
|
| $13$ |
\( T^{5} - 56 T^{4} + \cdots - 53188296 \)
|
| $17$ |
\( T^{5} + 32 T^{4} + \cdots + 511488 \)
|
| $19$ |
\( T^{5} \)
|
| $23$ |
\( T^{5} + 184 T^{4} + \cdots - 156908460 \)
|
| $29$ |
\( T^{5} + \cdots + 27533998656 \)
|
| $31$ |
\( T^{5} + \cdots - 11017558272 \)
|
| $37$ |
\( T^{5} + \cdots - 309775727920 \)
|
| $41$ |
\( T^{5} + \cdots - 1523115846765 \)
|
| $43$ |
\( T^{5} + \cdots - 784723578720 \)
|
| $47$ |
\( T^{5} + \cdots + 370249383552 \)
|
| $53$ |
\( T^{5} + \cdots - 31399904184 \)
|
| $59$ |
\( T^{5} + \cdots + 19038048363531 \)
|
| $61$ |
\( T^{5} + \cdots + 483304559848 \)
|
| $67$ |
\( T^{5} + \cdots + 5591565262089 \)
|
| $71$ |
\( T^{5} + \cdots - 101121274130616 \)
|
| $73$ |
\( T^{5} + \cdots + 3035011555725 \)
|
| $79$ |
\( T^{5} + \cdots - 2798859893056 \)
|
| $83$ |
\( T^{5} + \cdots + 138587802934320 \)
|
| $89$ |
\( T^{5} + \cdots - 35101938991104 \)
|
| $97$ |
\( T^{5} + \cdots - 2728426897445 \)
|
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