Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [729,2,Mod(1,729)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(729, 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("729.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 729 = 3^{6} \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 729.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: | \(5.82109430735\) |
| Analytic rank: | \(0\) |
| Dimension: | \(6\) |
| Coefficient field: | 6.6.1397493.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{6} - 3x^{5} - 3x^{4} + 10x^{3} + 3x^{2} - 6x + 1 \)
|
| Coefficient ring: | \(\Z[a_1, a_2]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 27) |
| 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 \(1,\beta_1,\ldots,\beta_{5}\) 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
\( x^{6} - 3x^{5} - 3x^{4} + 10x^{3} + 3x^{2} - 6x + 1 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - \nu - 2 \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{3} - 2\nu^{2} - 2\nu + 2 \)
|
| \(\beta_{4}\) | \(=\) |
\( \nu^{4} - 3\nu^{3} - \nu^{2} + 6\nu - 1 \)
|
| \(\beta_{5}\) | \(=\) |
\( \nu^{5} - 3\nu^{4} - 3\nu^{3} + 9\nu^{2} + 4\nu - 3 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + \beta _1 + 2 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{3} + 2\beta_{2} + 4\beta _1 + 2 \)
|
| \(\nu^{4}\) | \(=\) |
\( \beta_{4} + 3\beta_{3} + 7\beta_{2} + 7\beta _1 + 9 \)
|
| \(\nu^{5}\) | \(=\) |
\( \beta_{5} + 3\beta_{4} + 12\beta_{3} + 18\beta_{2} + 20\beta _1 + 18 \)
|
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 |
|
−1.68091 | 0 | 0.825466 | 1.12954 | 0 | −3.90892 | 1.97429 | 0 | −1.89866 | ||||||||||||||||||||||||||||||||||||
| 1.2 | −1.05432 | 0 | −0.888399 | 1.74579 | 0 | 2.45925 | 3.04531 | 0 | −1.84063 | |||||||||||||||||||||||||||||||||||||
| 1.3 | 0.415466 | 0 | −1.82739 | −2.21519 | 0 | −1.31963 | −1.59015 | 0 | −0.920335 | |||||||||||||||||||||||||||||||||||||
| 1.4 | 0.801527 | 0 | −1.35755 | 2.74984 | 0 | 2.37683 | −2.69117 | 0 | 2.20407 | |||||||||||||||||||||||||||||||||||||
| 1.5 | 2.11662 | 0 | 2.48009 | 2.68310 | 0 | 0.972333 | 1.01617 | 0 | 5.67911 | |||||||||||||||||||||||||||||||||||||
| 1.6 | 2.40162 | 0 | 3.76778 | −0.0930834 | 0 | −0.579861 | 4.24555 | 0 | −0.223551 | |||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(3\) | \( -1 \) |
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 | 729.2.a.d | 6 | |
| 3.b | odd | 2 | 1 | 729.2.a.a | 6 | ||
| 9.c | even | 3 | 2 | 729.2.c.b | 12 | ||
| 9.d | odd | 6 | 2 | 729.2.c.e | 12 | ||
| 27.e | even | 9 | 2 | 81.2.e.a | 12 | ||
| 27.e | even | 9 | 2 | 243.2.e.a | 12 | ||
| 27.e | even | 9 | 2 | 243.2.e.b | 12 | ||
| 27.f | odd | 18 | 2 | 27.2.e.a | ✓ | 12 | |
| 27.f | odd | 18 | 2 | 243.2.e.c | 12 | ||
| 27.f | odd | 18 | 2 | 243.2.e.d | 12 | ||
| 108.l | even | 18 | 2 | 432.2.u.c | 12 | ||
| 135.n | odd | 18 | 2 | 675.2.l.c | 12 | ||
| 135.q | even | 36 | 4 | 675.2.u.b | 24 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 27.2.e.a | ✓ | 12 | 27.f | odd | 18 | 2 | |
| 81.2.e.a | 12 | 27.e | even | 9 | 2 | ||
| 243.2.e.a | 12 | 27.e | even | 9 | 2 | ||
| 243.2.e.b | 12 | 27.e | even | 9 | 2 | ||
| 243.2.e.c | 12 | 27.f | odd | 18 | 2 | ||
| 243.2.e.d | 12 | 27.f | odd | 18 | 2 | ||
| 432.2.u.c | 12 | 108.l | even | 18 | 2 | ||
| 675.2.l.c | 12 | 135.n | odd | 18 | 2 | ||
| 675.2.u.b | 24 | 135.q | even | 36 | 4 | ||
| 729.2.a.a | 6 | 3.b | odd | 2 | 1 | ||
| 729.2.a.d | 6 | 1.a | even | 1 | 1 | trivial | |
| 729.2.c.b | 12 | 9.c | even | 3 | 2 | ||
| 729.2.c.e | 12 | 9.d | odd | 6 | 2 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{6} - 3T_{2}^{5} - 3T_{2}^{4} + 12T_{2}^{3} - 9T_{2} + 3 \)
acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(729))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{6} - 3 T^{5} + \cdots + 3 \)
|
| $3$ |
\( T^{6} \)
|
| $5$ |
\( T^{6} - 6 T^{5} + \cdots + 3 \)
|
| $7$ |
\( T^{6} - 15 T^{4} + \cdots - 17 \)
|
| $11$ |
\( T^{6} - 12 T^{5} + \cdots + 3 \)
|
| $13$ |
\( T^{6} - 24 T^{4} + \cdots + 1 \)
|
| $17$ |
\( T^{6} - 9 T^{5} + \cdots + 27 \)
|
| $19$ |
\( T^{6} - 3 T^{5} + \cdots + 19 \)
|
| $23$ |
\( T^{6} - 15 T^{5} + \cdots + 327 \)
|
| $29$ |
\( T^{6} - 12 T^{5} + \cdots - 213 \)
|
| $31$ |
\( T^{6} - 51 T^{4} + \cdots + 163 \)
|
| $37$ |
\( T^{6} - 3 T^{5} + \cdots + 4933 \)
|
| $41$ |
\( T^{6} - 15 T^{5} + \cdots + 3351 \)
|
| $43$ |
\( T^{6} - 96 T^{4} + \cdots + 1819 \)
|
| $47$ |
\( T^{6} - 21 T^{5} + \cdots + 6537 \)
|
| $53$ |
\( T^{6} - 9 T^{5} + \cdots - 12393 \)
|
| $59$ |
\( T^{6} - 24 T^{5} + \cdots - 13281 \)
|
| $61$ |
\( T^{6} + 9 T^{5} + \cdots + 16543 \)
|
| $67$ |
\( T^{6} + 9 T^{5} + \cdots - 2879 \)
|
| $71$ |
\( T^{6} - 27 T^{5} + \cdots + 27 \)
|
| $73$ |
\( T^{6} + 6 T^{5} + \cdots - 431 \)
|
| $79$ |
\( T^{6} - 177 T^{4} + \cdots + 1873 \)
|
| $83$ |
\( T^{6} - 12 T^{5} + \cdots - 83373 \)
|
| $89$ |
\( T^{6} - 9 T^{5} + \cdots + 32589 \)
|
| $97$ |
\( T^{6} - 204 T^{4} + \cdots - 8171 \)
|
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