Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [141,4,Mod(1,141)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(141, 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("141.1");
S:= CuspForms(chi, 4);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 141 = 3 \cdot 47 \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 141.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: | \(8.31926931081\) |
| Analytic rank: | \(1\) |
| Dimension: | \(5\) |
| Coefficient field: | 5.5.13374304.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{5} - 2x^{4} - 12x^{3} + 6x^{2} + 28x + 12 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 2^{2} \) |
| 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 \(1,\beta_1,\beta_2,\beta_3,\beta_4\) 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
\( x^{5} - 2x^{4} - 12x^{3} + 6x^{2} + 28x + 12 \)
:
| \(\beta_{1}\) | \(=\) |
\( 2\nu - 1 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{3} - 2\nu^{2} - 6\nu + 2 ) / 2 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( \nu^{4} - 3\nu^{3} - 8\nu^{2} + 12\nu + 10 ) / 2 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( -2\nu^{4} + 5\nu^{3} + 22\nu^{2} - 26\nu - 40 ) / 2 \)
|
| \(\nu\) | \(=\) |
\( ( \beta _1 + 1 ) / 2 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( \beta_{4} + 2\beta_{3} + \beta_{2} + 2\beta _1 + 11 ) / 2 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{4} + 2\beta_{3} + 3\beta_{2} + 5\beta _1 + 12 \)
|
| \(\nu^{4}\) | \(=\) |
\( 7\beta_{4} + 16\beta_{3} + 13\beta_{2} + 17\beta _1 + 64 \)
|
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 |
|
−5.57746 | 3.00000 | 23.1080 | −6.83668 | −16.7324 | −1.37128 | −84.2644 | 9.00000 | 38.1313 | |||||||||||||||||||||||||||||||||
| 1.2 | −1.89865 | 3.00000 | −4.39514 | −10.8872 | −5.69594 | 30.8292 | 23.5340 | 9.00000 | 20.6709 | ||||||||||||||||||||||||||||||||||
| 1.3 | −1.65586 | 3.00000 | −5.25814 | 12.4254 | −4.96757 | −35.6053 | 21.9536 | 9.00000 | −20.5748 | ||||||||||||||||||||||||||||||||||
| 1.4 | 1.35580 | 3.00000 | −6.16180 | −6.65510 | 4.06741 | −10.5903 | −19.2006 | 9.00000 | −9.02302 | ||||||||||||||||||||||||||||||||||
| 1.5 | 2.77616 | 3.00000 | −0.292956 | −22.0465 | 8.32847 | −9.26230 | −23.0225 | 9.00000 | −61.2044 | ||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(3\) | \(-1\) |
| \(47\) | \(1\) |
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 | 141.4.a.b | ✓ | 5 |
| 3.b | odd | 2 | 1 | 423.4.a.e | 5 | ||
| 4.b | odd | 2 | 1 | 2256.4.a.p | 5 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 141.4.a.b | ✓ | 5 | 1.a | even | 1 | 1 | trivial |
| 423.4.a.e | 5 | 3.b | odd | 2 | 1 | ||
| 2256.4.a.p | 5 | 4.b | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{5} + 5T_{2}^{4} - 11T_{2}^{3} - 43T_{2}^{2} + 14T_{2} + 66 \)
acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(141))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{5} + 5 T^{4} + \cdots + 66 \)
|
| $3$ |
\( (T - 3)^{5} \)
|
| $5$ |
\( T^{5} + 34 T^{4} + \cdots - 135696 \)
|
| $7$ |
\( T^{5} + 26 T^{4} + \cdots - 147648 \)
|
| $11$ |
\( T^{5} + 150 T^{4} + \cdots - 14660496 \)
|
| $13$ |
\( T^{5} + 50 T^{4} + \cdots - 5039232 \)
|
| $17$ |
\( T^{5} + 50 T^{4} + \cdots + 610058496 \)
|
| $19$ |
\( T^{5} + \cdots - 1868277024 \)
|
| $23$ |
\( T^{5} + \cdots - 7467876672 \)
|
| $29$ |
\( T^{5} + \cdots + 6168380944 \)
|
| $31$ |
\( T^{5} + \cdots - 70727362176 \)
|
| $37$ |
\( T^{5} + \cdots - 1699352926056 \)
|
| $41$ |
\( T^{5} + \cdots + 153287158912 \)
|
| $43$ |
\( T^{5} + \cdots + 431723453184 \)
|
| $47$ |
\( (T + 47)^{5} \)
|
| $53$ |
\( T^{5} + \cdots - 74706674464 \)
|
| $59$ |
\( T^{5} + \cdots + 14393619140352 \)
|
| $61$ |
\( T^{5} + \cdots + 42483676885216 \)
|
| $67$ |
\( T^{5} + \cdots - 3927954482944 \)
|
| $71$ |
\( T^{5} + \cdots - 10205710152192 \)
|
| $73$ |
\( T^{5} + \cdots + 159812135750528 \)
|
| $79$ |
\( T^{5} + \cdots - 114505059002304 \)
|
| $83$ |
\( T^{5} + \cdots - 7386482400000 \)
|
| $89$ |
\( T^{5} + \cdots + 11\!\cdots\!12 \)
|
| $97$ |
\( T^{5} + \cdots + 718735111176 \)
|
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