Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [47,5,Mod(46,47)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(47, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([1]))
N = Newforms(chi, 5, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("47.46");
S:= CuspForms(chi, 5);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 47 \) |
| Weight: | \( k \) | \(=\) | \( 5 \) |
| Character orbit: | \([\chi]\) | \(=\) | 47.b (of order \(2\), degree \(1\), minimal) |
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: | \(4.85838826494\) |
| Analytic rank: | \(0\) |
| Dimension: | \(5\) |
| Coefficient field: | 5.5.6903125.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{5} - 10x^{3} + 20x - 9 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | yes |
| Sato-Tate group: | $\mathrm{U}(1)[D_{2}]$ |
$q$-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
\( x^{5} - 10x^{3} + 20x - 9 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - 4 \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{3} - \nu^{2} - 6\nu + 4 \)
|
| \(\beta_{4}\) | \(=\) |
\( \nu^{4} - 8\nu^{2} - 2\nu + 8 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + 4 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{3} + \beta_{2} + 6\beta_1 \)
|
| \(\nu^{4}\) | \(=\) |
\( \beta_{4} + 8\beta_{2} + 2\beta _1 + 24 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/47\mathbb{Z}\right)^\times\).
| \(n\) | \(5\) |
| \(\chi(n)\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | |||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 46.1 |
|
−7.95378 | 16.8188 | 47.2627 | 0 | −133.773 | 18.1910 | −248.657 | 201.871 | 0 | |||||||||||||||||||||||||||||||||
| 46.2 | −3.27450 | −17.3762 | −5.27766 | 0 | 56.8985 | −85.9624 | 69.6737 | 220.934 | 0 | ||||||||||||||||||||||||||||||||||
| 46.3 | −1.64121 | −9.83710 | −13.3064 | 0 | 16.1448 | 97.2051 | 48.0980 | 15.7684 | 0 | ||||||||||||||||||||||||||||||||||
| 46.4 | 5.93003 | 11.2966 | 19.1653 | 0 | 66.9891 | −71.3187 | 18.7703 | 46.6129 | 0 | ||||||||||||||||||||||||||||||||||
| 46.5 | 6.93946 | −0.902016 | 32.1561 | 0 | −6.25950 | 41.8851 | 112.115 | −80.1864 | 0 | ||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 47.b | odd | 2 | 1 | CM by \(\Q(\sqrt{-47}) \) |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 47.5.b.a | ✓ | 5 |
| 3.b | odd | 2 | 1 | 423.5.d.a | 5 | ||
| 4.b | odd | 2 | 1 | 752.5.g.a | 5 | ||
| 47.b | odd | 2 | 1 | CM | 47.5.b.a | ✓ | 5 |
| 141.c | even | 2 | 1 | 423.5.d.a | 5 | ||
| 188.b | even | 2 | 1 | 752.5.g.a | 5 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 47.5.b.a | ✓ | 5 | 1.a | even | 1 | 1 | trivial |
| 47.5.b.a | ✓ | 5 | 47.b | odd | 2 | 1 | CM |
| 423.5.d.a | 5 | 3.b | odd | 2 | 1 | ||
| 423.5.d.a | 5 | 141.c | even | 2 | 1 | ||
| 752.5.g.a | 5 | 4.b | odd | 2 | 1 | ||
| 752.5.g.a | 5 | 188.b | even | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{5} - 80T_{2}^{3} + 1280T_{2} + 1759 \)
acting on \(S_{5}^{\mathrm{new}}(47, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{5} - 80 T^{3} + \cdots + 1759 \)
|
| $3$ |
\( T^{5} - 405 T^{3} + \cdots + 29294 \)
|
| $5$ |
\( T^{5} \)
|
| $7$ |
\( T^{5} - 12005 T^{3} + \cdots - 454063586 \)
|
| $11$ |
\( T^{5} \)
|
| $13$ |
\( T^{5} \)
|
| $17$ |
\( T^{5} + \cdots - 1403260974146 \)
|
| $19$ |
\( T^{5} \)
|
| $23$ |
\( T^{5} \)
|
| $29$ |
\( T^{5} \)
|
| $31$ |
\( T^{5} \)
|
| $37$ |
\( T^{5} + \cdots - 93\!\cdots\!98 \)
|
| $41$ |
\( T^{5} \)
|
| $43$ |
\( T^{5} \)
|
| $47$ |
\( (T - 2209)^{5} \)
|
| $53$ |
\( T^{5} + \cdots + 21\!\cdots\!02 \)
|
| $59$ |
\( T^{5} + \cdots + 98\!\cdots\!86 \)
|
| $61$ |
\( T^{5} + \cdots + 12\!\cdots\!98 \)
|
| $67$ |
\( T^{5} \)
|
| $71$ |
\( T^{5} + \cdots - 28\!\cdots\!94 \)
|
| $73$ |
\( T^{5} \)
|
| $79$ |
\( T^{5} + \cdots - 49\!\cdots\!02 \)
|
| $83$ |
\( (T + 13294)^{5} \)
|
| $89$ |
\( T^{5} + \cdots - 13\!\cdots\!02 \)
|
| $97$ |
\( T^{5} + \cdots - 40\!\cdots\!98 \)
|
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