Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [27,8,Mod(1,27)]
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([0]))
N = Newforms(chi, 8, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("27.1");
S:= CuspForms(chi, 8);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 27 = 3^{3} \) |
| Weight: | \( k \) | \(=\) | \( 8 \) |
| Character orbit: | \([\chi]\) | \(=\) | 27.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: | \(8.43439568807\) |
| Analytic rank: | \(1\) |
| Dimension: | \(2\) |
| Coefficient field: | \(\Q(\sqrt{65}) \) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{2} - x - 16 \)
|
| Coefficient ring: | \(\Z[a_1, a_2]\) |
| Coefficient ring index: | \( 3 \) |
| 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 \(\beta = \frac{1}{2}(-1 + 3\sqrt{65})\). We also show the integral \(q\)-expansion of the trace form.
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 |
|
−16.5934 | 0 | 147.340 | −114.187 | 0 | 1438.40 | −320.924 | 0 | 1894.75 | ||||||||||||||||||||||||
| 1.2 | 7.59339 | 0 | −70.3405 | −65.8132 | 0 | −738.405 | −1506.08 | 0 | −499.745 | |||||||||||||||||||||||||
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 | 27.8.a.b | ✓ | 2 |
| 3.b | odd | 2 | 1 | 27.8.a.e | yes | 2 | |
| 4.b | odd | 2 | 1 | 432.8.a.j | 2 | ||
| 9.c | even | 3 | 2 | 81.8.c.h | 4 | ||
| 9.d | odd | 6 | 2 | 81.8.c.d | 4 | ||
| 12.b | even | 2 | 1 | 432.8.a.q | 2 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 27.8.a.b | ✓ | 2 | 1.a | even | 1 | 1 | trivial |
| 27.8.a.e | yes | 2 | 3.b | odd | 2 | 1 | |
| 81.8.c.d | 4 | 9.d | odd | 6 | 2 | ||
| 81.8.c.h | 4 | 9.c | even | 3 | 2 | ||
| 432.8.a.j | 2 | 4.b | odd | 2 | 1 | ||
| 432.8.a.q | 2 | 12.b | even | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{2} + 9T_{2} - 126 \)
acting on \(S_{8}^{\mathrm{new}}(\Gamma_0(27))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{2} + 9T - 126 \)
|
| $3$ |
\( T^{2} \)
|
| $5$ |
\( T^{2} + 180T + 7515 \)
|
| $7$ |
\( T^{2} - 700 T - 1062125 \)
|
| $11$ |
\( T^{2} + 10890 T + 29414025 \)
|
| $13$ |
\( T^{2} + 5480 T - 68308400 \)
|
| $17$ |
\( T^{2} + 16416 T - 741669696 \)
|
| $19$ |
\( T^{2} - 16024 T - 142216916 \)
|
| $23$ |
\( T^{2} + \cdots - 3904044444 \)
|
| $29$ |
\( T^{2} + \cdots + 5122403100 \)
|
| $31$ |
\( T^{2} + \cdots - 28528424669 \)
|
| $37$ |
\( T^{2} + \cdots - 16337003900 \)
|
| $41$ |
\( T^{2} + \cdots + 83424185100 \)
|
| $43$ |
\( T^{2} + \cdots + 207456229900 \)
|
| $47$ |
\( T^{2} + \cdots + 410995953540 \)
|
| $53$ |
\( T^{2} + \cdots + 1583691044571 \)
|
| $59$ |
\( T^{2} + \cdots - 647792463600 \)
|
| $61$ |
\( T^{2} + \cdots + 17550467776 \)
|
| $67$ |
\( T^{2} + \cdots + 15698927500 \)
|
| $71$ |
\( T^{2} + \cdots - 3541802241600 \)
|
| $73$ |
\( T^{2} + \cdots - 3239699519975 \)
|
| $79$ |
\( T^{2} + \cdots + 19219527915904 \)
|
| $83$ |
\( T^{2} + \cdots - 1694562670359 \)
|
| $89$ |
\( T^{2} + \cdots + 6381093451500 \)
|
| $97$ |
\( T^{2} + \cdots - 155863647601775 \)
|
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