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: | \(1\) |
| Dimension: | \(6\) |
| Coefficient field: | \(\Q(\zeta_{36})^+\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{6} - 6x^{4} + 9x^{2} - 3 \)
|
| 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 \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.
Basis of coefficient ring in terms of \(\nu = \zeta_{36} + \zeta_{36}^{-1}\):
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - 2 \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{3} - 3\nu \)
|
| \(\beta_{4}\) | \(=\) |
\( \nu^{4} - 5\nu^{2} + 4 \)
|
| \(\beta_{5}\) | \(=\) |
\( \nu^{5} - 5\nu^{3} + 4\nu \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + 2 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{3} + 3\beta_1 \)
|
| \(\nu^{4}\) | \(=\) |
\( \beta_{4} + 5\beta_{2} + 6 \)
|
| \(\nu^{5}\) | \(=\) |
\( \beta_{5} + 5\beta_{3} + 11\beta_1 \)
|
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 |
|
−1.96962 | 0 | 1.87939 | 3.70167 | 0 | −2.34730 | 0.237565 | 0 | −7.29086 | ||||||||||||||||||||||||||||||||||||
| 1.2 | −1.28558 | 0 | −0.347296 | −0.446476 | 0 | −3.53209 | 3.01763 | 0 | 0.573978 | |||||||||||||||||||||||||||||||||||||
| 1.3 | −0.684040 | 0 | −1.53209 | −1.04801 | 0 | −0.120615 | 2.41609 | 0 | 0.716881 | |||||||||||||||||||||||||||||||||||||
| 1.4 | 0.684040 | 0 | −1.53209 | 1.04801 | 0 | −0.120615 | −2.41609 | 0 | 0.716881 | |||||||||||||||||||||||||||||||||||||
| 1.5 | 1.28558 | 0 | −0.347296 | 0.446476 | 0 | −3.53209 | −3.01763 | 0 | 0.573978 | |||||||||||||||||||||||||||||||||||||
| 1.6 | 1.96962 | 0 | 1.87939 | −3.70167 | 0 | −2.34730 | −0.237565 | 0 | −7.29086 | |||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(3\) | \(1\) |
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 3.b | odd | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 729.2.a.c | ✓ | 6 |
| 3.b | odd | 2 | 1 | inner | 729.2.a.c | ✓ | 6 |
| 9.c | even | 3 | 2 | 729.2.c.c | 12 | ||
| 9.d | odd | 6 | 2 | 729.2.c.c | 12 | ||
| 27.e | even | 9 | 2 | 729.2.e.m | 12 | ||
| 27.e | even | 9 | 2 | 729.2.e.q | 12 | ||
| 27.e | even | 9 | 2 | 729.2.e.r | 12 | ||
| 27.f | odd | 18 | 2 | 729.2.e.m | 12 | ||
| 27.f | odd | 18 | 2 | 729.2.e.q | 12 | ||
| 27.f | odd | 18 | 2 | 729.2.e.r | 12 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 729.2.a.c | ✓ | 6 | 1.a | even | 1 | 1 | trivial |
| 729.2.a.c | ✓ | 6 | 3.b | odd | 2 | 1 | inner |
| 729.2.c.c | 12 | 9.c | even | 3 | 2 | ||
| 729.2.c.c | 12 | 9.d | odd | 6 | 2 | ||
| 729.2.e.m | 12 | 27.e | even | 9 | 2 | ||
| 729.2.e.m | 12 | 27.f | odd | 18 | 2 | ||
| 729.2.e.q | 12 | 27.e | even | 9 | 2 | ||
| 729.2.e.q | 12 | 27.f | odd | 18 | 2 | ||
| 729.2.e.r | 12 | 27.e | even | 9 | 2 | ||
| 729.2.e.r | 12 | 27.f | odd | 18 | 2 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{6} - 6T_{2}^{4} + 9T_{2}^{2} - 3 \)
acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(729))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{6} - 6 T^{4} + \cdots - 3 \)
|
| $3$ |
\( T^{6} \)
|
| $5$ |
\( T^{6} - 15 T^{4} + \cdots - 3 \)
|
| $7$ |
\( (T^{3} + 6 T^{2} + 9 T + 1)^{2} \)
|
| $11$ |
\( T^{6} - 42 T^{4} + \cdots - 1083 \)
|
| $13$ |
\( (T^{3} + 6 T^{2} - 9 T - 71)^{2} \)
|
| $17$ |
\( T^{6} - 81 T^{4} + \cdots - 9747 \)
|
| $19$ |
\( (T^{3} + 12 T^{2} + \cdots + 19)^{2} \)
|
| $23$ |
\( T^{6} - 114 T^{4} + \cdots - 867 \)
|
| $29$ |
\( T^{6} - 51 T^{4} + \cdots - 4107 \)
|
| $31$ |
\( (T^{3} + 15 T^{2} + \cdots + 73)^{2} \)
|
| $37$ |
\( (T^{3} + 3 T^{2} - 6 T - 17)^{2} \)
|
| $41$ |
\( T^{6} - 87 T^{4} + \cdots - 8427 \)
|
| $43$ |
\( (T^{3} + 6 T^{2} - 8)^{2} \)
|
| $47$ |
\( T^{6} - 15 T^{4} + \cdots - 3 \)
|
| $53$ |
\( T^{6} - 108 T^{4} + \cdots - 15552 \)
|
| $59$ |
\( T^{6} - 186 T^{4} + \cdots - 3 \)
|
| $61$ |
\( (T^{3} + 6 T^{2} + \cdots - 296)^{2} \)
|
| $67$ |
\( (T^{3} - 3 T^{2} + \cdots - 251)^{2} \)
|
| $71$ |
\( T^{6} - 72 T^{4} + \cdots - 1728 \)
|
| $73$ |
\( (T^{3} - 6 T^{2} - 69 T - 89)^{2} \)
|
| $79$ |
\( (T^{3} + 24 T^{2} + \cdots - 107)^{2} \)
|
| $83$ |
\( T^{6} - 438 T^{4} + \cdots - 1550883 \)
|
| $89$ |
\( T^{6} - 387 T^{4} + \cdots - 7803 \)
|
| $97$ |
\( (T^{3} + 6 T^{2} + \cdots - 647)^{2} \)
|
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