Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [54,5,Mod(53,54)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(54, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([1]))
N = Newforms(chi, 5, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("54.53");
S:= CuspForms(chi, 5);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 54 = 2 \cdot 3^{3} \) |
| Weight: | \( k \) | \(=\) | \( 5 \) |
| Character orbit: | \([\chi]\) | \(=\) | 54.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: | \(5.58197800653\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Coefficient field: | \(\Q(\zeta_{8})\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{4} + 1 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 2^{3}\cdot 3^{6} \) |
| 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
| \(\beta_{1}\) | \(=\) |
\( 27\zeta_{8}^{2} \)
|
| \(\beta_{2}\) | \(=\) |
\( 2\zeta_{8}^{3} + 2\zeta_{8} \)
|
| \(\beta_{3}\) | \(=\) |
\( -54\zeta_{8}^{3} + 54\zeta_{8} \)
|
| \(\zeta_{8}\) | \(=\) |
\( ( \beta_{3} + 27\beta_{2} ) / 108 \)
|
| \(\zeta_{8}^{2}\) | \(=\) |
\( ( \beta_1 ) / 27 \)
|
| \(\zeta_{8}^{3}\) | \(=\) |
\( ( -\beta_{3} + 27\beta_{2} ) / 108 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/54\mathbb{Z}\right)^\times\).
| \(n\) | \(29\) |
| \(\chi(n)\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 53.1 |
|
− | 2.82843i | 0 | −8.00000 | − | 18.5147i | 0 | −59.3675 | 22.6274i | 0 | −52.3675 | ||||||||||||||||||||||||||||
| 53.2 | − | 2.82843i | 0 | −8.00000 | 35.4853i | 0 | 93.3675 | 22.6274i | 0 | 100.368 | ||||||||||||||||||||||||||||||
| 53.3 | 2.82843i | 0 | −8.00000 | − | 35.4853i | 0 | 93.3675 | − | 22.6274i | 0 | 100.368 | |||||||||||||||||||||||||||||
| 53.4 | 2.82843i | 0 | −8.00000 | 18.5147i | 0 | −59.3675 | − | 22.6274i | 0 | −52.3675 | ||||||||||||||||||||||||||||||
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 | 54.5.b.b | ✓ | 4 |
| 3.b | odd | 2 | 1 | inner | 54.5.b.b | ✓ | 4 |
| 4.b | odd | 2 | 1 | 432.5.e.h | 4 | ||
| 5.b | even | 2 | 1 | 1350.5.d.c | 4 | ||
| 5.c | odd | 4 | 1 | 1350.5.b.a | 4 | ||
| 5.c | odd | 4 | 1 | 1350.5.b.c | 4 | ||
| 9.c | even | 3 | 2 | 162.5.d.c | 8 | ||
| 9.d | odd | 6 | 2 | 162.5.d.c | 8 | ||
| 12.b | even | 2 | 1 | 432.5.e.h | 4 | ||
| 15.d | odd | 2 | 1 | 1350.5.d.c | 4 | ||
| 15.e | even | 4 | 1 | 1350.5.b.a | 4 | ||
| 15.e | even | 4 | 1 | 1350.5.b.c | 4 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 54.5.b.b | ✓ | 4 | 1.a | even | 1 | 1 | trivial |
| 54.5.b.b | ✓ | 4 | 3.b | odd | 2 | 1 | inner |
| 162.5.d.c | 8 | 9.c | even | 3 | 2 | ||
| 162.5.d.c | 8 | 9.d | odd | 6 | 2 | ||
| 432.5.e.h | 4 | 4.b | odd | 2 | 1 | ||
| 432.5.e.h | 4 | 12.b | even | 2 | 1 | ||
| 1350.5.b.a | 4 | 5.c | odd | 4 | 1 | ||
| 1350.5.b.a | 4 | 15.e | even | 4 | 1 | ||
| 1350.5.b.c | 4 | 5.c | odd | 4 | 1 | ||
| 1350.5.b.c | 4 | 15.e | even | 4 | 1 | ||
| 1350.5.d.c | 4 | 5.b | even | 2 | 1 | ||
| 1350.5.d.c | 4 | 15.d | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{4} + 1602T_{5}^{2} + 431649 \)
acting on \(S_{5}^{\mathrm{new}}(54, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{2} + 8)^{2} \)
|
| $3$ |
\( T^{4} \)
|
| $5$ |
\( T^{4} + 1602 T^{2} + 431649 \)
|
| $7$ |
\( (T^{2} - 34 T - 5543)^{2} \)
|
| $11$ |
\( T^{4} + 15858 T^{2} + 41873841 \)
|
| $13$ |
\( (T^{2} + 260 T - 6428)^{2} \)
|
| $17$ |
\( T^{4} + \cdots + 8171436816 \)
|
| $19$ |
\( (T^{2} - 544 T + 50656)^{2} \)
|
| $23$ |
\( T^{4} + \cdots + 35205767424 \)
|
| $29$ |
\( T^{4} + \cdots + 41915191824 \)
|
| $31$ |
\( (T^{2} - 406 T - 1270991)^{2} \)
|
| $37$ |
\( (T^{2} + 536 T - 1071248)^{2} \)
|
| $41$ |
\( T^{4} + \cdots + 6248480092416 \)
|
| $43$ |
\( (T^{2} + 3548 T + 2937124)^{2} \)
|
| $47$ |
\( T^{4} + \cdots + 20387283891984 \)
|
| $53$ |
\( T^{4} + \cdots + 62602291689921 \)
|
| $59$ |
\( T^{4} + \cdots + 43368072677136 \)
|
| $61$ |
\( (T^{2} + 6968 T + 8779024)^{2} \)
|
| $67$ |
\( (T^{2} - 1612 T + 439684)^{2} \)
|
| $71$ |
\( T^{4} + \cdots + 848775738667536 \)
|
| $73$ |
\( (T^{2} - 4306 T + 2745841)^{2} \)
|
| $79$ |
\( (T^{2} + 5780 T + 3779812)^{2} \)
|
| $83$ |
\( T^{4} + \cdots + 507789764024481 \)
|
| $89$ |
\( T^{4} + \cdots + 191787378007824 \)
|
| $97$ |
\( (T^{2} - 790 T - 45006983)^{2} \)
|
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