Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [1078,4,Mod(1,1078)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1078, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([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("1078.1");
S:= CuspForms(chi, 4);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 1078 = 2 \cdot 7^{2} \cdot 11 \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 1078.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: | \(63.6040589862\) |
| 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: |
\( x^{5} - x^{4} - 69x^{3} + 112x^{2} + 1142x - 2439 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{13}]\) |
| Coefficient ring index: | \( 3^{2} \) |
| Twist minimal: | no (minimal twist has level 154) |
| 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
\( x^{5} - x^{4} - 69x^{3} + 112x^{2} + 1142x - 2439 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} + \nu - 28 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( \nu^{4} + 7\nu^{3} - 49\nu^{2} - 226\nu + 630 ) / 9 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( -2\nu^{4} - 5\nu^{3} + 98\nu^{2} + 146\nu - 972 ) / 9 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} - \beta _1 + 28 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{4} + 2\beta_{3} + 34\beta _1 - 32 \)
|
| \(\nu^{4}\) | \(=\) |
\( -7\beta_{4} - 5\beta_{3} + 49\beta_{2} - 61\beta _1 + 966 \)
|
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 | −4.79629 | 4.00000 | 15.3932 | −9.59257 | 0 | 8.00000 | −3.99564 | 30.7864 | |||||||||||||||||||||||||||||||||
| 1.2 | 2.00000 | −3.33376 | 4.00000 | −0.884782 | −6.66752 | 0 | 8.00000 | −15.8861 | −1.76956 | ||||||||||||||||||||||||||||||||||
| 1.3 | 2.00000 | 4.33584 | 4.00000 | −16.2080 | 8.67167 | 0 | 8.00000 | −8.20051 | −32.4161 | ||||||||||||||||||||||||||||||||||
| 1.4 | 2.00000 | 6.82792 | 4.00000 | 4.13676 | 13.6558 | 0 | 8.00000 | 19.6205 | 8.27351 | ||||||||||||||||||||||||||||||||||
| 1.5 | 2.00000 | 7.96628 | 4.00000 | 17.5628 | 15.9326 | 0 | 8.00000 | 36.4617 | 35.1257 | ||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \(-1\) |
| \(7\) | \(1\) |
| \(11\) | \(-1\) |
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 | 1078.4.a.ba | 5 | |
| 7.b | odd | 2 | 1 | 1078.4.a.x | 5 | ||
| 7.c | even | 3 | 2 | 154.4.e.a | ✓ | 10 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 154.4.e.a | ✓ | 10 | 7.c | even | 3 | 2 | |
| 1078.4.a.x | 5 | 7.b | odd | 2 | 1 | ||
| 1078.4.a.ba | 5 | 1.a | even | 1 | 1 | trivial | |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{5} - 11T_{3}^{4} - 21T_{3}^{3} + 422T_{3}^{2} - 22T_{3} - 3771 \)
acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(1078))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T - 2)^{5} \)
|
| $3$ |
\( T^{5} - 11 T^{4} + \cdots - 3771 \)
|
| $5$ |
\( T^{5} - 20 T^{4} + \cdots - 16038 \)
|
| $7$ |
\( T^{5} \)
|
| $11$ |
\( (T - 11)^{5} \)
|
| $13$ |
\( T^{5} - 61 T^{4} + \cdots - 1235871 \)
|
| $17$ |
\( T^{5} - 185 T^{4} + \cdots + 332949708 \)
|
| $19$ |
\( T^{5} + \cdots - 1564558578 \)
|
| $23$ |
\( T^{5} + \cdots + 4224218166 \)
|
| $29$ |
\( T^{5} + \cdots + 9654511473 \)
|
| $31$ |
\( T^{5} + \cdots - 212495502192 \)
|
| $37$ |
\( T^{5} + \cdots - 10247301168 \)
|
| $41$ |
\( T^{5} + \cdots + 4184213166 \)
|
| $43$ |
\( T^{5} + \cdots - 1000873584272 \)
|
| $47$ |
\( T^{5} + \cdots + 144351307332 \)
|
| $53$ |
\( T^{5} + \cdots - 656305010838 \)
|
| $59$ |
\( T^{5} + \cdots + 29893593336903 \)
|
| $61$ |
\( T^{5} + \cdots - 2096364631896 \)
|
| $67$ |
\( T^{5} + \cdots + 8490116169624 \)
|
| $71$ |
\( T^{5} + \cdots + 611193718962 \)
|
| $73$ |
\( T^{5} + \cdots + 1417430363598 \)
|
| $79$ |
\( T^{5} + \cdots + 143819196507 \)
|
| $83$ |
\( T^{5} + \cdots - 957399906264228 \)
|
| $89$ |
\( T^{5} + \cdots + 22987584282096 \)
|
| $97$ |
\( T^{5} + \cdots + 31101556401363 \)
|
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