Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [27,7,Mod(26,27)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(27, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([1]))
N = Newforms(chi, 7, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("27.26");
S:= CuspForms(chi, 7);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 27 = 3^{3} \) |
| Weight: | \( k \) | \(=\) | \( 7 \) |
| Character orbit: | \([\chi]\) | \(=\) | 27.b (of order \(2\), degree \(1\), minimal) |
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: | no |
| Analytic conductor: | \(6.21146025774\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Coefficient field: | \(\Q(i, \sqrt{41})\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{4} + 21x^{2} + 100 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 3^{8} \) |
| Twist minimal: | yes |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{2}]$ |
$q$-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 \(1,\beta_1,\beta_2,\beta_3\) 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
\( x^{4} + 21x^{2} + 100 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( 3\nu^{3} + 3\nu ) / 10 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 36\nu^{3} + 441\nu ) / 5 \)
|
| \(\beta_{3}\) | \(=\) |
\( 27\nu^{2} + 284 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{2} - 24\beta_1 ) / 81 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( \beta_{3} - 284 ) / 27 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( -\beta_{2} + 294\beta_1 ) / 81 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/27\mathbb{Z}\right)^\times\).
| \(n\) | \(2\) |
| \(\chi(n)\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 26.1 |
|
− | 14.1047i | 0 | −134.942 | 137.419i | 0 | −514.769 | 1000.62i | 0 | 1938.25 | |||||||||||||||||||||||||||||
| 26.2 | − | 5.10469i | 0 | 37.9422 | − | 60.5813i | 0 | 176.769 | − | 520.383i | 0 | −309.248 | ||||||||||||||||||||||||||||
| 26.3 | 5.10469i | 0 | 37.9422 | 60.5813i | 0 | 176.769 | 520.383i | 0 | −309.248 | |||||||||||||||||||||||||||||||
| 26.4 | 14.1047i | 0 | −134.942 | − | 137.419i | 0 | −514.769 | − | 1000.62i | 0 | 1938.25 | |||||||||||||||||||||||||||||
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 | 27.7.b.c | ✓ | 4 |
| 3.b | odd | 2 | 1 | inner | 27.7.b.c | ✓ | 4 |
| 4.b | odd | 2 | 1 | 432.7.e.j | 4 | ||
| 9.c | even | 3 | 2 | 81.7.d.e | 8 | ||
| 9.d | odd | 6 | 2 | 81.7.d.e | 8 | ||
| 12.b | even | 2 | 1 | 432.7.e.j | 4 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 27.7.b.c | ✓ | 4 | 1.a | even | 1 | 1 | trivial |
| 27.7.b.c | ✓ | 4 | 3.b | odd | 2 | 1 | inner |
| 81.7.d.e | 8 | 9.c | even | 3 | 2 | ||
| 81.7.d.e | 8 | 9.d | odd | 6 | 2 | ||
| 432.7.e.j | 4 | 4.b | odd | 2 | 1 | ||
| 432.7.e.j | 4 | 12.b | even | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{4} + 225T_{2}^{2} + 5184 \)
acting on \(S_{7}^{\mathrm{new}}(27, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} + 225T^{2} + 5184 \)
|
| $3$ |
\( T^{4} \)
|
| $5$ |
\( T^{4} + 22554 T^{2} + 69305625 \)
|
| $7$ |
\( (T^{2} + 338 T - 90995)^{2} \)
|
| $11$ |
\( T^{4} + \cdots + 7962639894225 \)
|
| $13$ |
\( (T^{2} + 1724 T + 264820)^{2} \)
|
| $17$ |
\( T^{4} + \cdots + 11074279462416 \)
|
| $19$ |
\( (T^{2} + 1832 T - 106761344)^{2} \)
|
| $23$ |
\( T^{4} + \cdots + 35\!\cdots\!00 \)
|
| $29$ |
\( T^{4} + \cdots + 74\!\cdots\!00 \)
|
| $31$ |
\( (T^{2} + 76622 T + 1321276621)^{2} \)
|
| $37$ |
\( (T^{2} - 64480 T + 931817200)^{2} \)
|
| $41$ |
\( T^{4} + \cdots + 11\!\cdots\!00 \)
|
| $43$ |
\( (T^{2} - 63004 T + 934510900)^{2} \)
|
| $47$ |
\( T^{4} + \cdots + 26\!\cdots\!24 \)
|
| $53$ |
\( T^{4} + \cdots + 27\!\cdots\!49 \)
|
| $59$ |
\( T^{4} + \cdots + 29\!\cdots\!00 \)
|
| $61$ |
\( (T^{2} - 83848 T - 3024618224)^{2} \)
|
| $67$ |
\( (T^{2} - 4 T - 145189284620)^{2} \)
|
| $71$ |
\( T^{4} + \cdots + 74\!\cdots\!00 \)
|
| $73$ |
\( (T^{2} + 940838 T + 135725893465)^{2} \)
|
| $79$ |
\( (T^{2} - 594364 T + 40563605380)^{2} \)
|
| $83$ |
\( T^{4} + \cdots + 11\!\cdots\!89 \)
|
| $89$ |
\( T^{4} + \cdots + 52\!\cdots\!00 \)
|
| $97$ |
\( (T^{2} - 487606 T - 680628953255)^{2} \)
|
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