Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [1338,2,Mod(1,1338)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1338, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([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("1338.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 1338 = 2 \cdot 3 \cdot 223 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 1338.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: | \(10.6839837904\) |
| Analytic rank: | \(0\) |
| Dimension: | \(5\) |
| Coefficient field: | 5.5.356173.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{5} - 2x^{4} - 7x^{3} + 9x^{2} + 14x - 4 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{11}]\) |
| 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
\( x^{5} - 2x^{4} - 7x^{3} + 9x^{2} + 14x - 4 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{3} - \nu^{2} - 4\nu + 1 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( \nu^{4} - 2\nu^{3} - 3\nu^{2} + 5\nu - 2 ) / 2 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( \nu^{4} - 2\nu^{3} - 5\nu^{2} + 7\nu + 4 ) / 2 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( -\beta_{4} + \beta_{3} + \beta _1 + 3 \)
|
| \(\nu^{3}\) | \(=\) |
\( -\beta_{4} + \beta_{3} + \beta_{2} + 5\beta _1 + 2 \)
|
| \(\nu^{4}\) | \(=\) |
\( -5\beta_{4} + 7\beta_{3} + 2\beta_{2} + 8\beta _1 + 15 \)
|
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 |
|
−1.00000 | 1.00000 | 1.00000 | −3.02013 | −1.00000 | −3.77138 | −1.00000 | 1.00000 | 3.02013 | |||||||||||||||||||||||||||||||||
| 1.2 | −1.00000 | 1.00000 | 1.00000 | 0.686428 | −1.00000 | 2.87549 | −1.00000 | 1.00000 | −0.686428 | ||||||||||||||||||||||||||||||||||
| 1.3 | −1.00000 | 1.00000 | 1.00000 | 1.43386 | −1.00000 | −1.87103 | −1.00000 | 1.00000 | −1.43386 | ||||||||||||||||||||||||||||||||||
| 1.4 | −1.00000 | 1.00000 | 1.00000 | 1.54501 | −1.00000 | −1.28984 | −1.00000 | 1.00000 | −1.54501 | ||||||||||||||||||||||||||||||||||
| 1.5 | −1.00000 | 1.00000 | 1.00000 | 4.35484 | −1.00000 | 3.05676 | −1.00000 | 1.00000 | −4.35484 | ||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \(1\) |
| \(3\) | \(-1\) |
| \(223\) | \(1\) |
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 | 1338.2.a.h | ✓ | 5 |
| 3.b | odd | 2 | 1 | 4014.2.a.r | 5 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 1338.2.a.h | ✓ | 5 | 1.a | even | 1 | 1 | trivial |
| 4014.2.a.r | 5 | 3.b | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{5} - 5T_{5}^{4} - 4T_{5}^{3} + 41T_{5}^{2} - 54T_{5} + 20 \)
acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(1338))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T + 1)^{5} \)
|
| $3$ |
\( (T - 1)^{5} \)
|
| $5$ |
\( T^{5} - 5 T^{4} + \cdots + 20 \)
|
| $7$ |
\( T^{5} + T^{4} + \cdots + 80 \)
|
| $11$ |
\( T^{5} - 9 T^{4} + \cdots - 67 \)
|
| $13$ |
\( T^{5} - 24 T^{3} + \cdots + 13 \)
|
| $17$ |
\( T^{5} - 6 T^{4} + \cdots - 64 \)
|
| $19$ |
\( T^{5} + 4 T^{4} + \cdots - 53 \)
|
| $23$ |
\( T^{5} - 16 T^{4} + \cdots + 932 \)
|
| $29$ |
\( T^{5} - 8 T^{4} + \cdots - 71 \)
|
| $31$ |
\( T^{5} + T^{4} + \cdots - 500 \)
|
| $37$ |
\( T^{5} + 2 T^{4} + \cdots + 9104 \)
|
| $41$ |
\( T^{5} - 4 T^{4} + \cdots - 5620 \)
|
| $43$ |
\( T^{5} - 3 T^{4} + \cdots - 42977 \)
|
| $47$ |
\( T^{5} - 18 T^{4} + \cdots + 3371 \)
|
| $53$ |
\( T^{5} - 26 T^{4} + \cdots + 1213 \)
|
| $59$ |
\( T^{5} - 21 T^{4} + \cdots - 65603 \)
|
| $61$ |
\( T^{5} + 20 T^{4} + \cdots - 15961 \)
|
| $67$ |
\( T^{5} + 5 T^{4} + \cdots + 1076 \)
|
| $71$ |
\( T^{5} - 17 T^{4} + \cdots + 5008 \)
|
| $73$ |
\( T^{5} - 5 T^{4} + \cdots - 3025 \)
|
| $79$ |
\( T^{5} + 21 T^{4} + \cdots - 41435 \)
|
| $83$ |
\( T^{5} - 11 T^{4} + \cdots + 21232 \)
|
| $89$ |
\( T^{5} + 5 T^{4} + \cdots - 100 \)
|
| $97$ |
\( T^{5} + 11 T^{4} + \cdots - 5804 \)
|
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