Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [1376,2,Mod(1,1376)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1376, 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("1376.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 1376 = 2^{5} \cdot 43 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 1376.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: | \(10.9874153181\) |
| Analytic rank: | \(0\) |
| Dimension: | \(5\) |
| Coefficient field: | 5.5.792644.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{5} - x^{4} - 7x^{3} + 4x^{2} + 4x - 2 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 1 \) |
| 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
\( x^{5} - x^{4} - 7x^{3} + 4x^{2} + 4x - 2 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{4} - \nu^{3} - 7\nu^{2} + 3\nu + 3 \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{4} - 8\nu^{2} - 2\nu + 5 \)
|
| \(\beta_{4}\) | \(=\) |
\( 2\nu^{4} - \nu^{3} - 14\nu^{2} + 6 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{4} - \beta_{3} - \beta_{2} + \beta _1 + 2 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{4} - 2\beta_{2} + 6\beta_1 \)
|
| \(\nu^{4}\) | \(=\) |
\( 8\beta_{4} - 7\beta_{3} - 8\beta_{2} + 10\beta _1 + 11 \)
|
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 | −2.82909 | 0 | −2.25366 | 0 | 4.45706 | 0 | 5.00378 | 0 | |||||||||||||||||||||||||||||||||
| 1.2 | 0 | −0.827959 | 0 | 2.39697 | 0 | 3.49807 | 0 | −2.31448 | 0 | ||||||||||||||||||||||||||||||||||
| 1.3 | 0 | −0.466307 | 0 | −3.41528 | 0 | −0.908821 | 0 | −2.78256 | 0 | ||||||||||||||||||||||||||||||||||
| 1.4 | 0 | 0.782073 | 0 | 2.11845 | 0 | −1.82722 | 0 | −2.38836 | 0 | ||||||||||||||||||||||||||||||||||
| 1.5 | 0 | 2.34129 | 0 | 0.153517 | 0 | 2.78091 | 0 | 2.48163 | 0 | ||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \( +1 \) |
| \(43\) | \( -1 \) |
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 | 1376.2.a.f | ✓ | 5 |
| 4.b | odd | 2 | 1 | 1376.2.a.g | yes | 5 | |
| 8.b | even | 2 | 1 | 2752.2.a.ba | 5 | ||
| 8.d | odd | 2 | 1 | 2752.2.a.y | 5 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 1376.2.a.f | ✓ | 5 | 1.a | even | 1 | 1 | trivial |
| 1376.2.a.g | yes | 5 | 4.b | odd | 2 | 1 | |
| 2752.2.a.y | 5 | 8.d | odd | 2 | 1 | ||
| 2752.2.a.ba | 5 | 8.b | even | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{5} + T_{3}^{4} - 7T_{3}^{3} - 4T_{3}^{2} + 4T_{3} + 2 \)
acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(1376))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{5} \)
|
| $3$ |
\( T^{5} + T^{4} - 7 T^{3} + \cdots + 2 \)
|
| $5$ |
\( T^{5} + T^{4} - 13 T^{3} + \cdots - 6 \)
|
| $7$ |
\( T^{5} - 8 T^{4} + \cdots - 72 \)
|
| $11$ |
\( T^{5} - T^{4} + \cdots - 16 \)
|
| $13$ |
\( T^{5} - T^{4} - 21 T^{3} + \cdots + 4 \)
|
| $17$ |
\( T^{5} + 2 T^{4} + \cdots - 179 \)
|
| $19$ |
\( T^{5} - 13 T^{4} + \cdots + 688 \)
|
| $23$ |
\( T^{5} - 12 T^{4} + \cdots + 1743 \)
|
| $29$ |
\( T^{5} + 7 T^{4} + \cdots + 1494 \)
|
| $31$ |
\( T^{5} - 22 T^{4} + \cdots + 26701 \)
|
| $37$ |
\( T^{5} - 9 T^{4} + \cdots - 16 \)
|
| $41$ |
\( T^{5} + 2 T^{4} + \cdots + 8409 \)
|
| $43$ |
\( (T - 1)^{5} \)
|
| $47$ |
\( T^{5} - 23 T^{4} + \cdots + 2564 \)
|
| $53$ |
\( T^{5} + 5 T^{4} + \cdots + 4 \)
|
| $59$ |
\( T^{5} + 4 T^{4} + \cdots + 8336 \)
|
| $61$ |
\( T^{5} - 66 T^{3} + \cdots - 568 \)
|
| $67$ |
\( T^{5} - 11 T^{4} + \cdots + 6756 \)
|
| $71$ |
\( T^{5} - 24 T^{4} + \cdots + 192 \)
|
| $73$ |
\( T^{5} + 10 T^{4} + \cdots + 4584 \)
|
| $79$ |
\( T^{5} - 7 T^{4} + \cdots - 162412 \)
|
| $83$ |
\( T^{5} - 3 T^{4} + \cdots + 4 \)
|
| $89$ |
\( T^{5} - 358 T^{3} + \cdots + 1336 \)
|
| $97$ |
\( T^{5} + 6 T^{4} + \cdots - 33929 \)
|
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