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: | 6.6.7459857.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{6} - 3x^{5} - 6x^{4} + 13x^{3} + 12x^{2} - 12x - 8 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| 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 \(1,\beta_1,\ldots,\beta_{5}\) 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
\( x^{6} - 3x^{5} - 6x^{4} + 13x^{3} + 12x^{2} - 12x - 8 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{4} - 3\nu^{3} - 4\nu^{2} + 7\nu + 4 ) / 2 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( \nu^{5} - 3\nu^{4} - 6\nu^{3} + 13\nu^{2} + 8\nu - 8 ) / 4 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( \nu^{5} - 3\nu^{4} - 4\nu^{3} + 7\nu^{2} + 4\nu + 2 ) / 2 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 3\nu^{5} - 9\nu^{4} - 14\nu^{3} + 31\nu^{2} + 12\nu - 16 ) / 4 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{5} - \beta_{4} - \beta_{3} + \beta _1 + 3 \)
|
| \(\nu^{3}\) | \(=\) |
\( 3\beta_{5} - 2\beta_{4} - 5\beta_{3} + 5\beta _1 + 4 \)
|
| \(\nu^{4}\) | \(=\) |
\( 13\beta_{5} - 10\beta_{4} - 19\beta_{3} + 2\beta_{2} + 12\beta _1 + 20 \)
|
| \(\nu^{5}\) | \(=\) |
\( 44\beta_{5} - 29\beta_{4} - 70\beta_{3} + 6\beta_{2} + 45\beta _1 + 53 \)
|
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 |
|
−2.45779 | 0 | 4.04073 | −3.08026 | 0 | −2.65867 | −5.01568 | 0 | 7.57064 | ||||||||||||||||||||||||||||||||||||
| 1.2 | −0.777732 | 0 | −1.39513 | −2.37635 | 0 | −2.50138 | 2.64050 | 0 | 1.84816 | |||||||||||||||||||||||||||||||||||||
| 1.3 | −0.172976 | 0 | −1.97008 | 3.73656 | 0 | 3.03150 | 0.686728 | 0 | −0.646335 | |||||||||||||||||||||||||||||||||||||
| 1.4 | 1.57840 | 0 | 0.491360 | −1.67851 | 0 | 2.77928 | −2.38124 | 0 | −2.64936 | |||||||||||||||||||||||||||||||||||||
| 1.5 | 2.12503 | 0 | 2.51575 | 2.07094 | 0 | 4.84867 | 1.09598 | 0 | 4.40081 | |||||||||||||||||||||||||||||||||||||
| 1.6 | 2.70506 | 0 | 5.31738 | −1.67238 | 0 | 0.500591 | 8.97372 | 0 | −4.52391 | |||||||||||||||||||||||||||||||||||||
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.e | yes | 6 |
| 3.b | odd | 2 | 1 | 729.2.a.b | ✓ | 6 | |
| 9.c | even | 3 | 2 | 729.2.c.a | 12 | ||
| 9.d | odd | 6 | 2 | 729.2.c.d | 12 | ||
| 27.e | even | 9 | 2 | 729.2.e.k | 12 | ||
| 27.e | even | 9 | 2 | 729.2.e.l | 12 | ||
| 27.e | even | 9 | 2 | 729.2.e.u | 12 | ||
| 27.f | odd | 18 | 2 | 729.2.e.j | 12 | ||
| 27.f | odd | 18 | 2 | 729.2.e.s | 12 | ||
| 27.f | odd | 18 | 2 | 729.2.e.t | 12 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 729.2.a.b | ✓ | 6 | 3.b | odd | 2 | 1 | |
| 729.2.a.e | yes | 6 | 1.a | even | 1 | 1 | trivial |
| 729.2.c.a | 12 | 9.c | even | 3 | 2 | ||
| 729.2.c.d | 12 | 9.d | odd | 6 | 2 | ||
| 729.2.e.j | 12 | 27.f | odd | 18 | 2 | ||
| 729.2.e.k | 12 | 27.e | even | 9 | 2 | ||
| 729.2.e.l | 12 | 27.e | even | 9 | 2 | ||
| 729.2.e.s | 12 | 27.f | odd | 18 | 2 | ||
| 729.2.e.t | 12 | 27.f | odd | 18 | 2 | ||
| 729.2.e.u | 12 | 27.e | even | 9 | 2 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{6} - 3T_{2}^{5} - 6T_{2}^{4} + 21T_{2}^{3} - 18T_{2} - 3 \)
acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(729))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{6} - 3 T^{5} + \cdots - 3 \)
|
| $3$ |
\( T^{6} \)
|
| $5$ |
\( T^{6} + 3 T^{5} + \cdots + 159 \)
|
| $7$ |
\( T^{6} - 6 T^{5} + \cdots + 136 \)
|
| $11$ |
\( T^{6} + 6 T^{5} + \cdots + 888 \)
|
| $13$ |
\( T^{6} - 6 T^{5} + \cdots - 89 \)
|
| $17$ |
\( T^{6} + 9 T^{5} + \cdots + 459 \)
|
| $19$ |
\( T^{6} - 12 T^{5} + \cdots - 296 \)
|
| $23$ |
\( T^{6} + 12 T^{5} + \cdots + 456 \)
|
| $29$ |
\( T^{6} - 21 T^{5} + \cdots + 9879 \)
|
| $31$ |
\( T^{6} - 15 T^{5} + \cdots - 1016 \)
|
| $37$ |
\( T^{6} - 3 T^{5} + \cdots - 16109 \)
|
| $41$ |
\( T^{6} + 12 T^{5} + \cdots + 23109 \)
|
| $43$ |
\( T^{6} - 6 T^{5} + \cdots + 17344 \)
|
| $47$ |
\( T^{6} + 15 T^{5} + \cdots + 17736 \)
|
| $53$ |
\( T^{6} + 9 T^{5} + \cdots + 1944 \)
|
| $59$ |
\( T^{6} - 6 T^{5} + \cdots - 408 \)
|
| $61$ |
\( T^{6} - 24 T^{5} + \cdots + 1576 \)
|
| $67$ |
\( T^{6} - 15 T^{5} + \cdots - 17288 \)
|
| $71$ |
\( T^{6} - 180 T^{4} + \cdots + 29376 \)
|
| $73$ |
\( T^{6} - 12 T^{5} + \cdots + 57601 \)
|
| $79$ |
\( T^{6} - 24 T^{5} + \cdots - 93176 \)
|
| $83$ |
\( T^{6} + 6 T^{5} + \cdots + 4344 \)
|
| $89$ |
\( T^{6} + 9 T^{5} + \cdots - 123957 \)
|
| $97$ |
\( T^{6} + 21 T^{5} + \cdots - 18251 \)
|
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