Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [1110,4,Mod(1,1110)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1110, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([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("1110.1");
S:= CuspForms(chi, 4);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 1110 = 2 \cdot 3 \cdot 5 \cdot 37 \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 1110.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: | \(65.4921201064\) |
| Analytic rank: | \(0\) |
| Dimension: | \(5\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{5} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{5} - x^{4} - 539x^{3} - 390x^{2} + 37800x + 48000 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{13}]\) |
| Coefficient ring index: | \( 2 \) |
| 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 \(1,\beta_1,\beta_2,\beta_3,\beta_4\) 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
\( x^{5} - x^{4} - 539x^{3} - 390x^{2} + 37800x + 48000 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 63\nu^{4} - 343\nu^{3} - 24677\nu^{2} + 77350\nu + 52200 ) / 69800 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -81\nu^{4} + 441\nu^{3} + 41699\nu^{2} - 99450\nu - 2201000 ) / 69800 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( -11\nu^{4} - 134\nu^{3} + 5749\nu^{2} + 57070\nu - 223500 ) / 8725 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( 7\beta_{3} + 9\beta_{2} + 214 \)
|
| \(\nu^{3}\) | \(=\) |
\( -45\beta_{4} + 52\beta_{3} + 4\beta_{2} + 364\beta _1 + 484 \)
|
| \(\nu^{4}\) | \(=\) |
\( -245\beta_{4} + 3025\beta_{3} + 4655\beta_{2} + 754\beta _1 + 85630 \)
|
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 |
|
−2.00000 | 3.00000 | 4.00000 | −5.00000 | −6.00000 | −25.8499 | −8.00000 | 9.00000 | 10.0000 | |||||||||||||||||||||||||||||||||
| 1.2 | −2.00000 | 3.00000 | 4.00000 | −5.00000 | −6.00000 | −16.9260 | −8.00000 | 9.00000 | 10.0000 | ||||||||||||||||||||||||||||||||||
| 1.3 | −2.00000 | 3.00000 | 4.00000 | −5.00000 | −6.00000 | 3.24266 | −8.00000 | 9.00000 | 10.0000 | ||||||||||||||||||||||||||||||||||
| 1.4 | −2.00000 | 3.00000 | 4.00000 | −5.00000 | −6.00000 | 18.4337 | −8.00000 | 9.00000 | 10.0000 | ||||||||||||||||||||||||||||||||||
| 1.5 | −2.00000 | 3.00000 | 4.00000 | −5.00000 | −6.00000 | 30.0995 | −8.00000 | 9.00000 | 10.0000 | ||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \(1\) |
| \(3\) | \(-1\) |
| \(5\) | \(1\) |
| \(37\) | \(-1\) |
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 | 1110.4.a.n | ✓ | 5 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 1110.4.a.n | ✓ | 5 | 1.a | even | 1 | 1 | trivial |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{7}^{5} - 9T_{7}^{4} - 1065T_{7}^{3} + 6013T_{7}^{2} + 234660T_{7} - 787200 \)
acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(1110))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T + 2)^{5} \)
|
| $3$ |
\( (T - 3)^{5} \)
|
| $5$ |
\( (T + 5)^{5} \)
|
| $7$ |
\( T^{5} - 9 T^{4} + \cdots - 787200 \)
|
| $11$ |
\( T^{5} + 40 T^{4} + \cdots + 21344856 \)
|
| $13$ |
\( T^{5} - 68 T^{4} + \cdots - 19026008 \)
|
| $17$ |
\( T^{5} + 92 T^{4} + \cdots + 32752068 \)
|
| $19$ |
\( T^{5} - 71 T^{4} + \cdots + 754306864 \)
|
| $23$ |
\( T^{5} - 62 T^{4} + \cdots - 194507520 \)
|
| $29$ |
\( T^{5} + \cdots + 1404172164 \)
|
| $31$ |
\( T^{5} - 169 T^{4} + \cdots + 608186464 \)
|
| $37$ |
\( (T - 37)^{5} \)
|
| $41$ |
\( T^{5} + \cdots + 518708986476 \)
|
| $43$ |
\( T^{5} + \cdots + 448908176784 \)
|
| $47$ |
\( T^{5} + \cdots - 60711579648 \)
|
| $53$ |
\( T^{5} + \cdots - 629882102016 \)
|
| $59$ |
\( T^{5} + \cdots - 170951870784 \)
|
| $61$ |
\( T^{5} + \cdots - 88997762164256 \)
|
| $67$ |
\( T^{5} + \cdots - 22614391424 \)
|
| $71$ |
\( T^{5} + \cdots - 13809061382400 \)
|
| $73$ |
\( T^{5} + \cdots - 2227813427328 \)
|
| $79$ |
\( T^{5} + \cdots + 144409053797632 \)
|
| $83$ |
\( T^{5} + \cdots + 581142928370400 \)
|
| $89$ |
\( T^{5} + \cdots + 404211104445864 \)
|
| $97$ |
\( T^{5} + \cdots - 350059393596704 \)
|
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