Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [543,2,Mod(1,543)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(543, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([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("543.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 543 = 3 \cdot 181 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 543.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: | \(4.33587682976\) |
| Analytic rank: | \(0\) |
| Dimension: | \(8\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{8} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{8} - 3x^{7} - 6x^{6} + 21x^{5} + 5x^{4} - 35x^{3} + 10x^{2} + 4x - 1 \)
|
| 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 \(1,\beta_1,\ldots,\beta_{7}\) 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
\( x^{8} - 3x^{7} - 6x^{6} + 21x^{5} + 5x^{4} - 35x^{3} + 10x^{2} + 4x - 1 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - \nu - 2 \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{3} - \nu^{2} - 4\nu + 2 \)
|
| \(\beta_{4}\) | \(=\) |
\( \nu^{7} - 3\nu^{6} - 6\nu^{5} + 21\nu^{4} + 5\nu^{3} - 35\nu^{2} + 10\nu + 3 \)
|
| \(\beta_{5}\) | \(=\) |
\( -\nu^{7} + 3\nu^{6} + 7\nu^{5} - 21\nu^{4} - 13\nu^{3} + 34\nu^{2} + 4\nu - 3 \)
|
| \(\beta_{6}\) | \(=\) |
\( \nu^{7} - 3\nu^{6} - 7\nu^{5} + 22\nu^{4} + 12\nu^{3} - 39\nu^{2} - \nu + 5 \)
|
| \(\beta_{7}\) | \(=\) |
\( 2\nu^{7} - 5\nu^{6} - 14\nu^{5} + 35\nu^{4} + 23\nu^{3} - 58\nu^{2} + 6 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + \beta _1 + 2 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{3} + \beta_{2} + 5\beta_1 \)
|
| \(\nu^{4}\) | \(=\) |
\( \beta_{6} + \beta_{5} + \beta_{3} + 6\beta_{2} + 7\beta _1 + 8 \)
|
| \(\nu^{5}\) | \(=\) |
\( \beta_{5} + \beta_{4} + 8\beta_{3} + 9\beta_{2} + 27\beta _1 + 2 \)
|
| \(\nu^{6}\) | \(=\) |
\( \beta_{7} + 7\beta_{6} + 9\beta_{5} + 10\beta_{3} + 35\beta_{2} + 46\beta _1 + 36 \)
|
| \(\nu^{7}\) | \(=\) |
\( 3\beta_{7} + 12\beta_{5} + 7\beta_{4} + 52\beta_{3} + 63\beta_{2} + 153\beta _1 + 19 \)
|
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 |
|
−2.09416 | −1.00000 | 2.38551 | 1.57030 | 2.09416 | −3.68416 | −0.807326 | 1.00000 | −3.28847 | ||||||||||||||||||||||||||||||||||||||||||
| 1.2 | −1.57409 | −1.00000 | 0.477759 | 2.61079 | 1.57409 | 3.43627 | 2.39614 | 1.00000 | −4.10962 | |||||||||||||||||||||||||||||||||||||||||||
| 1.3 | −0.330181 | −1.00000 | −1.89098 | −3.90200 | 0.330181 | −4.54089 | 1.28473 | 1.00000 | 1.28837 | |||||||||||||||||||||||||||||||||||||||||||
| 1.4 | 0.216541 | −1.00000 | −1.95311 | −1.13532 | −0.216541 | 1.21084 | −0.856012 | 1.00000 | −0.245843 | |||||||||||||||||||||||||||||||||||||||||||
| 1.5 | 0.495668 | −1.00000 | −1.75431 | 3.58208 | −0.495668 | −1.17967 | −1.86089 | 1.00000 | 1.77552 | |||||||||||||||||||||||||||||||||||||||||||
| 1.6 | 1.49117 | −1.00000 | 0.223595 | −1.54064 | −1.49117 | 2.68646 | −2.64893 | 1.00000 | −2.29736 | |||||||||||||||||||||||||||||||||||||||||||
| 1.7 | 2.31010 | −1.00000 | 3.33658 | 3.44398 | −2.31010 | 1.03443 | 3.08764 | 1.00000 | 7.95596 | |||||||||||||||||||||||||||||||||||||||||||
| 1.8 | 2.48495 | −1.00000 | 4.17495 | 0.370811 | −2.48495 | −1.96328 | 5.40464 | 1.00000 | 0.921444 | |||||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(3\) | \( +1 \) |
| \(181\) | \( -1 \) |
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 | 543.2.a.d | ✓ | 8 |
| 3.b | odd | 2 | 1 | 1629.2.a.e | 8 | ||
| 4.b | odd | 2 | 1 | 8688.2.a.bf | 8 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 543.2.a.d | ✓ | 8 | 1.a | even | 1 | 1 | trivial |
| 1629.2.a.e | 8 | 3.b | odd | 2 | 1 | ||
| 8688.2.a.bf | 8 | 4.b | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{8} - 3T_{2}^{7} - 6T_{2}^{6} + 21T_{2}^{5} + 5T_{2}^{4} - 35T_{2}^{3} + 10T_{2}^{2} + 4T_{2} - 1 \)
acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(543))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{8} - 3 T^{7} + \cdots - 1 \)
|
| $3$ |
\( (T + 1)^{8} \)
|
| $5$ |
\( T^{8} - 5 T^{7} + \cdots - 128 \)
|
| $7$ |
\( T^{8} + 3 T^{7} + \cdots + 448 \)
|
| $11$ |
\( T^{8} - 4 T^{7} + \cdots + 64 \)
|
| $13$ |
\( T^{8} - 13 T^{7} + \cdots - 6298 \)
|
| $17$ |
\( T^{8} - 27 T^{7} + \cdots + 640 \)
|
| $19$ |
\( T^{8} + 10 T^{7} + \cdots - 188416 \)
|
| $23$ |
\( T^{8} - 3 T^{7} + \cdots + 3872 \)
|
| $29$ |
\( T^{8} - 16 T^{7} + \cdots - 16928 \)
|
| $31$ |
\( T^{8} + T^{7} + \cdots - 11200 \)
|
| $37$ |
\( T^{8} - 3 T^{7} + \cdots + 218 \)
|
| $41$ |
\( T^{8} - 20 T^{7} + \cdots - 1030 \)
|
| $43$ |
\( T^{8} + 4 T^{7} + \cdots - 8348 \)
|
| $47$ |
\( T^{8} - 12 T^{7} + \cdots - 78848 \)
|
| $53$ |
\( T^{8} - 23 T^{7} + \cdots - 98 \)
|
| $59$ |
\( T^{8} - 7 T^{7} + \cdots + 591104 \)
|
| $61$ |
\( T^{8} - T^{7} + \cdots + 1262368 \)
|
| $67$ |
\( T^{8} + 25 T^{7} + \cdots + 189700 \)
|
| $71$ |
\( T^{8} - 11 T^{7} + \cdots - 111136 \)
|
| $73$ |
\( T^{8} - 23 T^{7} + \cdots - 1191058 \)
|
| $79$ |
\( T^{8} + 26 T^{7} + \cdots - 126104 \)
|
| $83$ |
\( T^{8} - 16 T^{7} + \cdots - 208076 \)
|
| $89$ |
\( T^{8} - 14 T^{7} + \cdots - 15046 \)
|
| $97$ |
\( T^{8} - 28 T^{7} + \cdots + 420704 \)
|
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