Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [54,8,Mod(1,54)]
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([0]))
N = Newforms(chi, 8, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("54.1");
S:= CuspForms(chi, 8);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 54 = 2 \cdot 3^{3} \) |
| Weight: | \( k \) | \(=\) | \( 8 \) |
| Character orbit: | \([\chi]\) | \(=\) | 54.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: | \(16.8687913761\) |
| Analytic rank: | \(0\) |
| Dimension: | \(2\) |
| Coefficient field: | \(\Q(\sqrt{329}) \) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{2} - x - 82 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 2\cdot 3^{3} \) |
| 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 \(\beta = 27\sqrt{329}\). 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 |
|
8.00000 | 0 | 64.0000 | −513.736 | 0 | 929.736 | 512.000 | 0 | −4109.89 | ||||||||||||||||||||||||
| 1.2 | 8.00000 | 0 | 64.0000 | 465.736 | 0 | −49.7356 | 512.000 | 0 | 3725.89 | |||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \(-1\) |
| \(3\) | \(-1\) |
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 | 54.8.a.h | yes | 2 |
| 3.b | odd | 2 | 1 | 54.8.a.g | ✓ | 2 | |
| 4.b | odd | 2 | 1 | 432.8.a.k | 2 | ||
| 9.c | even | 3 | 2 | 162.8.c.m | 4 | ||
| 9.d | odd | 6 | 2 | 162.8.c.p | 4 | ||
| 12.b | even | 2 | 1 | 432.8.a.p | 2 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 54.8.a.g | ✓ | 2 | 3.b | odd | 2 | 1 | |
| 54.8.a.h | yes | 2 | 1.a | even | 1 | 1 | trivial |
| 162.8.c.m | 4 | 9.c | even | 3 | 2 | ||
| 162.8.c.p | 4 | 9.d | odd | 6 | 2 | ||
| 432.8.a.k | 2 | 4.b | odd | 2 | 1 | ||
| 432.8.a.p | 2 | 12.b | even | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{2} + 48T_{5} - 239265 \)
acting on \(S_{8}^{\mathrm{new}}(\Gamma_0(54))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T - 8)^{2} \)
|
| $3$ |
\( T^{2} \)
|
| $5$ |
\( T^{2} + 48T - 239265 \)
|
| $7$ |
\( T^{2} - 880T - 46241 \)
|
| $11$ |
\( T^{2} - 7230 T - 2281599 \)
|
| $13$ |
\( T^{2} - 8560 T - 77618000 \)
|
| $17$ |
\( T^{2} - 25704 T + 69237504 \)
|
| $19$ |
\( T^{2} - 37048 T - 832082324 \)
|
| $23$ |
\( T^{2} + 59628 T + 792938196 \)
|
| $29$ |
\( T^{2} + \cdots + 6257180700 \)
|
| $31$ |
\( T^{2} + \cdots - 4403209961 \)
|
| $37$ |
\( T^{2} + \cdots - 179439584684 \)
|
| $41$ |
\( T^{2} + \cdots + 61090498764 \)
|
| $43$ |
\( T^{2} + \cdots + 146018088316 \)
|
| $47$ |
\( T^{2} + \cdots + 45928723716 \)
|
| $53$ |
\( T^{2} + \cdots + 463161939111 \)
|
| $59$ |
\( T^{2} + \cdots + 383589380304 \)
|
| $61$ |
\( T^{2} + \cdots + 2269257806656 \)
|
| $67$ |
\( T^{2} + \cdots + 71859687676 \)
|
| $71$ |
\( T^{2} + \cdots - 5355488275200 \)
|
| $73$ |
\( T^{2} + \cdots - 5381037627791 \)
|
| $79$ |
\( T^{2} + \cdots - 2609413236800 \)
|
| $83$ |
\( T^{2} + \cdots + 19347454919025 \)
|
| $89$ |
\( T^{2} + \cdots - 20597975704836 \)
|
| $97$ |
\( T^{2} + \cdots - 51902270530991 \)
|
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