Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [1080,4,Mod(1,1080)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1080, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0, 0, 0]))
N = Newforms(chi, 4, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("1080.1");
S:= CuspForms(chi, 4);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 1080 = 2^{3} \cdot 3^{3} \cdot 5 \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 1080.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: | \(63.7220628062\) |
| Analytic rank: | \(0\) |
| Dimension: | \(3\) |
| Coefficient field: | 3.3.47977.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{3} - x^{2} - 60x - 44 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{11}]\) |
| Coefficient ring index: | \( 2\cdot 3^{2} \) |
| 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 a basis \(1,\beta_1,\beta_2\) 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
\( x^{3} - x^{2} - 60x - 44 \)
:
| \(\beta_{1}\) | \(=\) |
\( 6\nu - 2 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 3\nu^{2} - 9\nu - 118 ) / 4 \)
|
| \(\nu\) | \(=\) |
\( ( \beta _1 + 2 ) / 6 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( 8\beta_{2} + 3\beta _1 + 242 ) / 6 \)
|
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 |
|
0 | 0 | 0 | −5.00000 | 0 | −24.3916 | 0 | 0 | 0 | |||||||||||||||||||||||||||
| 1.2 | 0 | 0 | 0 | −5.00000 | 0 | 9.46549 | 0 | 0 | 0 | ||||||||||||||||||||||||||||
| 1.3 | 0 | 0 | 0 | −5.00000 | 0 | 23.9261 | 0 | 0 | 0 | ||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \(-1\) |
| \(3\) | \(-1\) |
| \(5\) | \(1\) |
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 | 1080.4.a.h | ✓ | 3 |
| 3.b | odd | 2 | 1 | 1080.4.a.n | yes | 3 | |
| 4.b | odd | 2 | 1 | 2160.4.a.bf | 3 | ||
| 12.b | even | 2 | 1 | 2160.4.a.bn | 3 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 1080.4.a.h | ✓ | 3 | 1.a | even | 1 | 1 | trivial |
| 1080.4.a.n | yes | 3 | 3.b | odd | 2 | 1 | |
| 2160.4.a.bf | 3 | 4.b | odd | 2 | 1 | ||
| 2160.4.a.bn | 3 | 12.b | even | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(1080))\):
|
\( T_{7}^{3} - 9T_{7}^{2} - 588T_{7} + 5524 \)
|
|
\( T_{11}^{3} + 18T_{11}^{2} - 3081T_{11} - 76426 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{3} \)
|
| $3$ |
\( T^{3} \)
|
| $5$ |
\( (T + 5)^{3} \)
|
| $7$ |
\( T^{3} - 9 T^{2} + \cdots + 5524 \)
|
| $11$ |
\( T^{3} + 18 T^{2} + \cdots - 76426 \)
|
| $13$ |
\( T^{3} + 21 T^{2} + \cdots - 30901 \)
|
| $17$ |
\( T^{3} + 84 T^{2} + \cdots - 86732 \)
|
| $19$ |
\( T^{3} - 21 T^{2} + \cdots + 138464 \)
|
| $23$ |
\( T^{3} + 48 T^{2} + \cdots - 2038 \)
|
| $29$ |
\( T^{3} - 36 T^{2} + \cdots + 3034586 \)
|
| $31$ |
\( T^{3} - 324 T^{2} + \cdots + 9213176 \)
|
| $37$ |
\( T^{3} - 33 T^{2} + \cdots + 2851956 \)
|
| $41$ |
\( T^{3} + 114 T^{2} + \cdots - 1847232 \)
|
| $43$ |
\( T^{3} - 282 T^{2} + \cdots + 1113236 \)
|
| $47$ |
\( T^{3} + 282 T^{2} + \cdots + 11238296 \)
|
| $53$ |
\( T^{3} + 222 T^{2} + \cdots + 5908896 \)
|
| $59$ |
\( (T - 92)^{3} \)
|
| $61$ |
\( T^{3} - 303 T^{2} + \cdots + 13330220 \)
|
| $67$ |
\( T^{3} - 1035 T^{2} + \cdots + 849676336 \)
|
| $71$ |
\( T^{3} + 510 T^{2} + \cdots - 11439360 \)
|
| $73$ |
\( T^{3} - 447 T^{2} + \cdots + 78574428 \)
|
| $79$ |
\( T^{3} - 777 T^{2} + \cdots - 14000175 \)
|
| $83$ |
\( T^{3} + 78 T^{2} + \cdots + 87546568 \)
|
| $89$ |
\( T^{3} - 324 T^{2} + \cdots - 92454912 \)
|
| $97$ |
\( T^{3} - 1191 T^{2} + \cdots + 31736348 \)
|
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