Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [47,2,Mod(1,47)]
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([0]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("47.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 47 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 47.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: | \(0.375296889500\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Coefficient field: | 4.4.1957.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{4} - 4x^{2} - x + 1 \)
|
| Coefficient ring: | \(\Z[a_1, a_2]\) |
| Coefficient ring index: | \( 1 \) |
| 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
\( x^{4} - 4x^{2} - x + 1 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{3} - 4\nu - 1 \)
|
| \(\beta_{3}\) | \(=\) |
\( -\nu^{3} + \nu^{2} + 3\nu - 1 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{3} + \beta_{2} + \beta _1 + 2 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{2} + 4\beta _1 + 1 \)
|
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 |
|
−2.26608 | 0.824788 | 3.13511 | 1.49494 | −1.86903 | 3.21243 | −2.57226 | −2.31972 | −3.38764 | ||||||||||||||||||||||||||||||
| 1.2 | 0.283841 | 2.23925 | −1.91943 | −4.25351 | 0.635593 | 2.44658 | −1.11250 | 2.01426 | −1.20732 | |||||||||||||||||||||||||||||||
| 1.3 | 0.673363 | −0.188279 | −1.54658 | 3.15283 | −0.126780 | −3.31128 | −2.38814 | −2.96455 | 2.12300 | |||||||||||||||||||||||||||||||
| 1.4 | 2.30887 | −2.87576 | 3.33090 | −2.39425 | −6.63978 | 1.65227 | 3.07289 | 5.27002 | −5.52803 | |||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(47\) | \(-1\) |
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 | 47.2.a.a | ✓ | 4 |
| 3.b | odd | 2 | 1 | 423.2.a.k | 4 | ||
| 4.b | odd | 2 | 1 | 752.2.a.h | 4 | ||
| 5.b | even | 2 | 1 | 1175.2.a.f | 4 | ||
| 5.c | odd | 4 | 2 | 1175.2.c.e | 8 | ||
| 7.b | odd | 2 | 1 | 2303.2.a.h | 4 | ||
| 8.b | even | 2 | 1 | 3008.2.a.q | 4 | ||
| 8.d | odd | 2 | 1 | 3008.2.a.p | 4 | ||
| 11.b | odd | 2 | 1 | 5687.2.a.s | 4 | ||
| 12.b | even | 2 | 1 | 6768.2.a.bv | 4 | ||
| 13.b | even | 2 | 1 | 7943.2.a.h | 4 | ||
| 47.b | odd | 2 | 1 | 2209.2.a.e | 4 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 47.2.a.a | ✓ | 4 | 1.a | even | 1 | 1 | trivial |
| 423.2.a.k | 4 | 3.b | odd | 2 | 1 | ||
| 752.2.a.h | 4 | 4.b | odd | 2 | 1 | ||
| 1175.2.a.f | 4 | 5.b | even | 2 | 1 | ||
| 1175.2.c.e | 8 | 5.c | odd | 4 | 2 | ||
| 2209.2.a.e | 4 | 47.b | odd | 2 | 1 | ||
| 2303.2.a.h | 4 | 7.b | odd | 2 | 1 | ||
| 3008.2.a.p | 4 | 8.d | odd | 2 | 1 | ||
| 3008.2.a.q | 4 | 8.b | even | 2 | 1 | ||
| 5687.2.a.s | 4 | 11.b | odd | 2 | 1 | ||
| 6768.2.a.bv | 4 | 12.b | even | 2 | 1 | ||
| 7943.2.a.h | 4 | 13.b | even | 2 | 1 | ||
Hecke kernels
This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(\Gamma_0(47))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} - T^{3} - 5 T^{2} + \cdots - 1 \)
|
| $3$ |
\( T^{4} - 7 T^{2} + \cdots + 1 \)
|
| $5$ |
\( T^{4} + 2 T^{3} + \cdots + 48 \)
|
| $7$ |
\( T^{4} - 4 T^{3} + \cdots - 43 \)
|
| $11$ |
\( T^{4} + 6 T^{3} + \cdots - 48 \)
|
| $13$ |
\( T^{4} - 8 T^{3} + \cdots + 48 \)
|
| $17$ |
\( T^{4} - 6 T^{3} + \cdots + 141 \)
|
| $19$ |
\( T^{4} - 16 T^{2} + \cdots + 16 \)
|
| $23$ |
\( T^{4} + 6 T^{3} + \cdots - 16 \)
|
| $29$ |
\( T^{4} + 10 T^{3} + \cdots - 16 \)
|
| $31$ |
\( T^{4} + 8 T^{3} + \cdots + 48 \)
|
| $37$ |
\( T^{4} - 10 T^{3} + \cdots + 9 \)
|
| $41$ |
\( T^{4} - 6 T^{3} + \cdots - 16 \)
|
| $43$ |
\( T^{4} - 2 T^{3} + \cdots + 432 \)
|
| $47$ |
\( (T - 1)^{4} \)
|
| $53$ |
\( T^{4} + 6 T^{3} + \cdots + 2429 \)
|
| $59$ |
\( T^{4} - 4 T^{3} + \cdots - 519 \)
|
| $61$ |
\( T^{4} + 6 T^{3} + \cdots + 337 \)
|
| $67$ |
\( T^{4} - 10 T^{3} + \cdots + 3184 \)
|
| $71$ |
\( T^{4} + 12 T^{3} + \cdots + 657 \)
|
| $73$ |
\( T^{4} - 22 T^{3} + \cdots - 7664 \)
|
| $79$ |
\( T^{4} - 20 T^{3} + \cdots - 47 \)
|
| $83$ |
\( T^{4} - 20 T^{3} + \cdots - 256 \)
|
| $89$ |
\( T^{4} + 6 T^{3} + \cdots + 4841 \)
|
| $97$ |
\( T^{4} - 30 T^{3} + \cdots - 14307 \)
|
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