Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [54,7,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, 7, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("54.53");
S:= CuspForms(chi, 7);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 54 = 2 \cdot 3^{3} \) |
| Weight: | \( k \) | \(=\) | \( 7 \) |
| 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: | \(12.4229205155\) |
| 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: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2^{5}\cdot 3^{6} \) |
| 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
| \(\beta_{1}\) | \(=\) |
\( 27\zeta_{8}^{2} \)
|
| \(\beta_{2}\) | \(=\) |
\( 4\zeta_{8}^{3} + 4\zeta_{8} \)
|
| \(\beta_{3}\) | \(=\) |
\( -108\zeta_{8}^{3} + 108\zeta_{8} \)
|
| \(\zeta_{8}\) | \(=\) |
\( ( \beta_{3} + 27\beta_{2} ) / 216 \)
|
| \(\zeta_{8}^{2}\) | \(=\) |
\( ( \beta_1 ) / 27 \)
|
| \(\zeta_{8}^{3}\) | \(=\) |
\( ( -\beta_{3} + 27\beta_{2} ) / 216 \)
|
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 |
|
− | 5.65685i | 0 | −32.0000 | − | 219.853i | 0 | 361.735 | 181.019i | 0 | −1243.68 | ||||||||||||||||||||||||||||
| 53.2 | − | 5.65685i | 0 | −32.0000 | 50.1472i | 0 | 56.2649 | 181.019i | 0 | 283.675 | ||||||||||||||||||||||||||||||
| 53.3 | 5.65685i | 0 | −32.0000 | − | 50.1472i | 0 | 56.2649 | − | 181.019i | 0 | 283.675 | |||||||||||||||||||||||||||||
| 53.4 | 5.65685i | 0 | −32.0000 | 219.853i | 0 | 361.735 | − | 181.019i | 0 | −1243.68 | ||||||||||||||||||||||||||||||
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 | 54.7.b.c | ✓ | 4 |
| 3.b | odd | 2 | 1 | inner | 54.7.b.c | ✓ | 4 |
| 4.b | odd | 2 | 1 | 432.7.e.h | 4 | ||
| 9.c | even | 3 | 2 | 162.7.d.e | 8 | ||
| 9.d | odd | 6 | 2 | 162.7.d.e | 8 | ||
| 12.b | even | 2 | 1 | 432.7.e.h | 4 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 54.7.b.c | ✓ | 4 | 1.a | even | 1 | 1 | trivial |
| 54.7.b.c | ✓ | 4 | 3.b | odd | 2 | 1 | inner |
| 162.7.d.e | 8 | 9.c | even | 3 | 2 | ||
| 162.7.d.e | 8 | 9.d | odd | 6 | 2 | ||
| 432.7.e.h | 4 | 4.b | odd | 2 | 1 | ||
| 432.7.e.h | 4 | 12.b | even | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{4} + 50850T_{5}^{2} + 121550625 \)
acting on \(S_{7}^{\mathrm{new}}(54, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{2} + 32)^{2} \)
|
| $3$ |
\( T^{4} \)
|
| $5$ |
\( T^{4} + 50850 T^{2} + 121550625 \)
|
| $7$ |
\( (T^{2} - 418 T + 20353)^{2} \)
|
| $11$ |
\( T^{4} + \cdots + 11429961921 \)
|
| $13$ |
\( (T^{2} - 220 T - 11278652)^{2} \)
|
| $17$ |
\( T^{4} + \cdots + 14397198164496 \)
|
| $19$ |
\( (T^{2} + 3344 T - 46566464)^{2} \)
|
| $23$ |
\( T^{4} + \cdots + 152996317012224 \)
|
| $29$ |
\( T^{4} + \cdots + 13\!\cdots\!04 \)
|
| $31$ |
\( (T^{2} - 80086 T + 1598193049)^{2} \)
|
| $37$ |
\( (T^{2} + 75272 T + 1154355088)^{2} \)
|
| $41$ |
\( T^{4} + \cdots + 54\!\cdots\!16 \)
|
| $43$ |
\( (T^{2} - 45076 T - 9469797404)^{2} \)
|
| $47$ |
\( T^{4} + \cdots + 89\!\cdots\!44 \)
|
| $53$ |
\( T^{4} + \cdots + 34\!\cdots\!21 \)
|
| $59$ |
\( T^{4} + \cdots + 39\!\cdots\!36 \)
|
| $61$ |
\( (T^{2} - 292072 T + 19055672464)^{2} \)
|
| $67$ |
\( (T^{2} + 383396 T + 33623197636)^{2} \)
|
| $71$ |
\( T^{4} + \cdots + 11\!\cdots\!56 \)
|
| $73$ |
\( (T^{2} - 787378 T + 154378808689)^{2} \)
|
| $79$ |
\( (T^{2} - 161500 T - 575746690748)^{2} \)
|
| $83$ |
\( T^{4} + \cdots + 78\!\cdots\!21 \)
|
| $89$ |
\( T^{4} + \cdots + 24\!\cdots\!64 \)
|
| $97$ |
\( (T^{2} - 2216470 T + 810384440953)^{2} \)
|
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