Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [5445,2,Mod(1,5445)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(5445, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0, 0]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("5445.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 5445 = 3^{2} \cdot 5 \cdot 11^{2} \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 5445.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: | \(43.4785439006\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Coefficient field: | 4.4.5725.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{4} - x^{3} - 8x^{2} + 6x + 11 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 165) |
| 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
\( x^{4} - x^{3} - 8x^{2} + 6x + 11 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{3} - 5\nu + 1 ) / 3 \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{2} - 4 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{3} + 4 \)
|
| \(\nu^{3}\) | \(=\) |
\( 3\beta_{2} + 5\beta _1 - 1 \)
|
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 |
|
−2.48008 | 0 | 4.15081 | 1.00000 | 0 | 4.39482 | −5.33418 | 0 | −2.48008 | ||||||||||||||||||||||||||||||
| 1.2 | −0.933531 | 0 | −1.12852 | 1.00000 | 0 | 2.04108 | 2.92057 | 0 | −0.933531 | |||||||||||||||||||||||||||||||
| 1.3 | 1.86205 | 0 | 1.46722 | 1.00000 | 0 | −2.63089 | −0.992053 | 0 | 1.86205 | |||||||||||||||||||||||||||||||
| 1.4 | 2.55157 | 0 | 4.51049 | 1.00000 | 0 | 4.19499 | 6.40567 | 0 | 2.55157 | |||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(3\) | \(-1\) |
| \(5\) | \(-1\) |
| \(11\) | \(-1\) |
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 | 5445.2.a.br | 4 | |
| 3.b | odd | 2 | 1 | 1815.2.a.r | 4 | ||
| 11.b | odd | 2 | 1 | 5445.2.a.bk | 4 | ||
| 11.d | odd | 10 | 2 | 495.2.n.b | 8 | ||
| 15.d | odd | 2 | 1 | 9075.2.a.dg | 4 | ||
| 33.d | even | 2 | 1 | 1815.2.a.v | 4 | ||
| 33.f | even | 10 | 2 | 165.2.m.b | ✓ | 8 | |
| 165.d | even | 2 | 1 | 9075.2.a.cq | 4 | ||
| 165.r | even | 10 | 2 | 825.2.n.i | 8 | ||
| 165.u | odd | 20 | 4 | 825.2.bx.g | 16 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 165.2.m.b | ✓ | 8 | 33.f | even | 10 | 2 | |
| 495.2.n.b | 8 | 11.d | odd | 10 | 2 | ||
| 825.2.n.i | 8 | 165.r | even | 10 | 2 | ||
| 825.2.bx.g | 16 | 165.u | odd | 20 | 4 | ||
| 1815.2.a.r | 4 | 3.b | odd | 2 | 1 | ||
| 1815.2.a.v | 4 | 33.d | even | 2 | 1 | ||
| 5445.2.a.bk | 4 | 11.b | odd | 2 | 1 | ||
| 5445.2.a.br | 4 | 1.a | even | 1 | 1 | trivial | |
| 9075.2.a.cq | 4 | 165.d | even | 2 | 1 | ||
| 9075.2.a.dg | 4 | 15.d | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(5445))\):
|
\( T_{2}^{4} - T_{2}^{3} - 8T_{2}^{2} + 6T_{2} + 11 \)
|
|
\( T_{7}^{4} - 8T_{7}^{3} + 8T_{7}^{2} + 57T_{7} - 99 \)
|
|
\( T_{23}^{4} - T_{23}^{3} - 39T_{23}^{2} + 29T_{23} + 341 \)
|
|
\( T_{53}^{4} + 2T_{53}^{3} - 82T_{53}^{2} - 83T_{53} + 1711 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} - T^{3} + \cdots + 11 \)
|
| $3$ |
\( T^{4} \)
|
| $5$ |
\( (T - 1)^{4} \)
|
| $7$ |
\( T^{4} - 8 T^{3} + \cdots - 99 \)
|
| $11$ |
\( T^{4} \)
|
| $13$ |
\( T^{4} - 15 T^{3} + \cdots + 99 \)
|
| $17$ |
\( T^{4} + 6 T^{3} + \cdots + 99 \)
|
| $19$ |
\( T^{4} - T^{3} - 10 T^{2} + \cdots + 9 \)
|
| $23$ |
\( T^{4} - T^{3} + \cdots + 341 \)
|
| $29$ |
\( T^{4} - 17 T^{3} + \cdots - 619 \)
|
| $31$ |
\( T^{4} - 15 T^{3} + \cdots - 25 \)
|
| $37$ |
\( T^{4} + T^{3} + \cdots + 1151 \)
|
| $41$ |
\( T^{4} + 12 T^{3} + \cdots + 11 \)
|
| $43$ |
\( T^{4} - 14 T^{3} + \cdots + 9 \)
|
| $47$ |
\( T^{4} + 14 T^{3} + \cdots - 3509 \)
|
| $53$ |
\( T^{4} + 2 T^{3} + \cdots + 1711 \)
|
| $59$ |
\( T^{4} - 11 T^{3} + \cdots - 99 \)
|
| $61$ |
\( T^{4} + T^{3} + \cdots + 1111 \)
|
| $67$ |
\( T^{4} - 5 T^{3} + \cdots + 1049 \)
|
| $71$ |
\( T^{4} - 3 T^{3} + \cdots - 821 \)
|
| $73$ |
\( T^{4} - 45 T^{3} + \cdots - 151 \)
|
| $79$ |
\( T^{4} - 229 T^{2} + \cdots - 841 \)
|
| $83$ |
\( T^{4} - 15 T^{3} + \cdots - 1975 \)
|
| $89$ |
\( T^{4} + 2 T^{3} + \cdots + 99 \)
|
| $97$ |
\( T^{4} + 26 T^{3} + \cdots + 891 \)
|
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