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: | \(4\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{4} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{4} - x^{3} - 141x^{2} + 200x + 3500 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{13}]\) |
| Coefficient ring index: | \( 2^{3}\cdot 3^{2} \) |
| 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 \(1,\beta_1,\beta_2,\beta_3\) 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
\( x^{4} - x^{3} - 141x^{2} + 200x + 3500 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( 7\nu^{3} + 3\nu^{2} - 617\nu + 250 ) / 40 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -\nu^{3} - 9\nu^{2} + 131\nu + 550 ) / 20 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( \nu^{3} + 9\nu^{2} - 71\nu - 575 ) / 5 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{3} + 4\beta_{2} + 5 ) / 12 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( 5\beta_{3} - 8\beta_{2} - 8\beta _1 + 845 ) / 12 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 43\beta_{3} + 178\beta_{2} + 36\beta _1 - 175 ) / 6 \)
|
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 | −30.4893 | 0 | 0 | 0 | ||||||||||||||||||||||||||||||
| 1.2 | 0 | 0 | 0 | −5.00000 | 0 | −2.40438 | 0 | 0 | 0 | |||||||||||||||||||||||||||||||
| 1.3 | 0 | 0 | 0 | −5.00000 | 0 | 11.2995 | 0 | 0 | 0 | |||||||||||||||||||||||||||||||
| 1.4 | 0 | 0 | 0 | −5.00000 | 0 | 35.5942 | 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.o | ✓ | 4 |
| 3.b | odd | 2 | 1 | 1080.4.a.p | yes | 4 | |
| 4.b | odd | 2 | 1 | 2160.4.a.bu | 4 | ||
| 12.b | even | 2 | 1 | 2160.4.a.bv | 4 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 1080.4.a.o | ✓ | 4 | 1.a | even | 1 | 1 | trivial |
| 1080.4.a.p | yes | 4 | 3.b | odd | 2 | 1 | |
| 2160.4.a.bu | 4 | 4.b | odd | 2 | 1 | ||
| 2160.4.a.bv | 4 | 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 \(S_{4}^{\mathrm{new}}(\Gamma_0(1080))\):
|
\( T_{7}^{4} - 14T_{7}^{3} - 1067T_{7}^{2} + 9792T_{7} + 29484 \)
|
|
\( T_{11}^{4} - 4T_{11}^{3} - 3749T_{11}^{2} + 45756T_{11} + 1807596 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} \)
|
| $3$ |
\( T^{4} \)
|
| $5$ |
\( (T + 5)^{4} \)
|
| $7$ |
\( T^{4} - 14 T^{3} + \cdots + 29484 \)
|
| $11$ |
\( T^{4} - 4 T^{3} + \cdots + 1807596 \)
|
| $13$ |
\( T^{4} - 30 T^{3} + \cdots + 18230751 \)
|
| $17$ |
\( T^{4} - 28 T^{3} + \cdots + 4462704 \)
|
| $19$ |
\( T^{4} - 78 T^{3} + \cdots + 21407584 \)
|
| $23$ |
\( T^{4} + 182 T^{3} + \cdots - 146544156 \)
|
| $29$ |
\( T^{4} + 202 T^{3} + \cdots - 561688740 \)
|
| $31$ |
\( T^{4} + 76 T^{3} + \cdots + 901737216 \)
|
| $37$ |
\( T^{4} - 302 T^{3} + \cdots + 9791964 \)
|
| $41$ |
\( T^{4} + 380 T^{3} + \cdots + 1440000 \)
|
| $43$ |
\( T^{4} - 178 T^{3} + \cdots - 372768336 \)
|
| $47$ |
\( T^{4} + 114 T^{3} + \cdots + 465218416 \)
|
| $53$ |
\( T^{4} + \cdots + 26357772096 \)
|
| $59$ |
\( T^{4} + \cdots + 3841737984 \)
|
| $61$ |
\( T^{4} + \cdots - 50264811540 \)
|
| $67$ |
\( T^{4} - 330 T^{3} + \cdots + 415421296 \)
|
| $71$ |
\( T^{4} - 1060 T^{3} + \cdots - 897110784 \)
|
| $73$ |
\( T^{4} + \cdots + 1120468716 \)
|
| $79$ |
\( T^{4} + \cdots + 364066942735 \)
|
| $83$ |
\( T^{4} + \cdots + 4935879504 \)
|
| $89$ |
\( T^{4} + \cdots + 16886941696 \)
|
| $97$ |
\( T^{4} + \cdots - 650133282260 \)
|
| show more | |
| show less |