Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [2768,2,Mod(1,2768)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2768, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0, 0]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("2768.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 2768 = 2^{4} \cdot 173 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 2768.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: | \(22.1025912795\) |
| Analytic rank: | \(0\) |
| Dimension: | \(5\) |
| Coefficient field: | 5.5.2075621.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{5} - 2x^{4} - 10x^{3} + 11x^{2} + 29x - 1 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 346) |
| 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
\( x^{5} - 2x^{4} - 10x^{3} + 11x^{2} + 29x - 1 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - \nu - 4 \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{3} - \nu^{2} - 6\nu \)
|
| \(\beta_{4}\) | \(=\) |
\( ( \nu^{4} - 3\nu^{3} - 5\nu^{2} + 12\nu + 5 ) / 2 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + \beta _1 + 4 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{3} + \beta_{2} + 7\beta _1 + 4 \)
|
| \(\nu^{4}\) | \(=\) |
\( 2\beta_{4} + 3\beta_{3} + 8\beta_{2} + 14\beta _1 + 27 \)
|
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 | −2.14616 | 0 | −0.659088 | 0 | 1.09305 | 0 | 1.60598 | 0 | |||||||||||||||||||||||||||||||||
| 1.2 | 0 | −1.61479 | 0 | 3.12507 | 0 | 2.34740 | 0 | −0.392458 | 0 | ||||||||||||||||||||||||||||||||||
| 1.3 | 0 | 0.965943 | 0 | 2.33152 | 0 | −2.70138 | 0 | −2.06695 | 0 | ||||||||||||||||||||||||||||||||||
| 1.4 | 0 | 2.72084 | 0 | 3.51762 | 0 | 3.19976 | 0 | 4.40299 | 0 | ||||||||||||||||||||||||||||||||||
| 1.5 | 0 | 3.07416 | 0 | −3.31511 | 0 | −1.93883 | 0 | 6.45044 | 0 | ||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \( -1 \) |
| \(173\) | \( +1 \) |
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 | 2768.2.a.i | 5 | |
| 4.b | odd | 2 | 1 | 346.2.a.e | ✓ | 5 | |
| 12.b | even | 2 | 1 | 3114.2.a.p | 5 | ||
| 20.d | odd | 2 | 1 | 8650.2.a.m | 5 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 346.2.a.e | ✓ | 5 | 4.b | odd | 2 | 1 | |
| 2768.2.a.i | 5 | 1.a | even | 1 | 1 | trivial | |
| 3114.2.a.p | 5 | 12.b | even | 2 | 1 | ||
| 8650.2.a.m | 5 | 20.d | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{5} - 3T_{3}^{4} - 8T_{3}^{3} + 21T_{3}^{2} + 18T_{3} - 28 \)
acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(2768))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{5} \)
|
| $3$ |
\( T^{5} - 3 T^{4} + \cdots - 28 \)
|
| $5$ |
\( T^{5} - 5 T^{4} + \cdots - 56 \)
|
| $7$ |
\( T^{5} - 2 T^{4} + \cdots - 43 \)
|
| $11$ |
\( T^{5} + 5 T^{4} + \cdots + 48 \)
|
| $13$ |
\( T^{5} - 9 T^{4} + \cdots + 108 \)
|
| $17$ |
\( T^{5} - 6 T^{4} + \cdots + 96 \)
|
| $19$ |
\( T^{5} - 5 T^{4} + \cdots - 324 \)
|
| $23$ |
\( T^{5} - 2 T^{4} + \cdots - 128 \)
|
| $29$ |
\( T^{5} - 5 T^{4} + \cdots - 612 \)
|
| $31$ |
\( T^{5} - 8 T^{4} + \cdots + 224 \)
|
| $37$ |
\( T^{5} - 7 T^{4} + \cdots - 4 \)
|
| $41$ |
\( T^{5} - 12 T^{4} + \cdots + 971 \)
|
| $43$ |
\( T^{5} - 13 T^{4} + \cdots - 3264 \)
|
| $47$ |
\( T^{5} - 68 T^{3} + \cdots - 1632 \)
|
| $53$ |
\( T^{5} - 3 T^{4} + \cdots + 632 \)
|
| $59$ |
\( T^{5} - T^{4} + \cdots - 3252 \)
|
| $61$ |
\( T^{5} - 17 T^{4} + \cdots - 5848 \)
|
| $67$ |
\( T^{5} - 21 T^{4} + \cdots - 16136 \)
|
| $71$ |
\( T^{5} - 4 T^{4} + \cdots - 28903 \)
|
| $73$ |
\( T^{5} + 14 T^{4} + \cdots + 4363 \)
|
| $79$ |
\( T^{5} - 2 T^{4} + \cdots - 12149 \)
|
| $83$ |
\( T^{5} - 13 T^{4} + \cdots + 21696 \)
|
| $89$ |
\( T^{5} - 18 T^{4} + \cdots - 53113 \)
|
| $97$ |
\( T^{5} + 6 T^{4} + \cdots - 768 \)
|
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