Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [1045,2,Mod(1,1045)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1045, 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("1045.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 1045 = 5 \cdot 11 \cdot 19 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 1045.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: | \(8.34436701122\) |
| Analytic rank: | \(1\) |
| Dimension: | \(5\) |
| Coefficient field: | 5.5.36497.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{5} - 2x^{4} - 3x^{3} + 5x^{2} + x - 1 \)
|
| Coefficient ring: | \(\Z[a_1, a_2]\) |
| 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
\( x^{5} - 2x^{4} - 3x^{3} + 5x^{2} + x - 1 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - \nu - 2 \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{4} - \nu^{3} - 4\nu^{2} + 2\nu + 2 \)
|
| \(\beta_{4}\) | \(=\) |
\( \nu^{4} - 2\nu^{3} - 3\nu^{2} + 4\nu + 1 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + \beta _1 + 2 \)
|
| \(\nu^{3}\) | \(=\) |
\( -\beta_{4} + \beta_{3} + \beta_{2} + 3\beta _1 + 1 \)
|
| \(\nu^{4}\) | \(=\) |
\( -\beta_{4} + 2\beta_{3} + 5\beta_{2} + 5\beta _1 + 7 \)
|
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 |
|
−1.88661 | 1.31801 | 1.55930 | −1.00000 | −2.48658 | 0.927281 | 0.831437 | −1.26284 | 1.88661 | |||||||||||||||||||||||||||||||||
| 1.2 | −1.46888 | −1.50629 | 0.157597 | −1.00000 | 2.21255 | −2.37015 | 2.70626 | −0.731099 | 1.46888 | ||||||||||||||||||||||||||||||||||
| 1.3 | −0.584664 | 0.334185 | −1.65817 | −1.00000 | −0.195386 | 1.85355 | 2.13880 | −2.88832 | 0.584664 | ||||||||||||||||||||||||||||||||||
| 1.4 | 0.913732 | −2.55629 | −1.16509 | −1.00000 | −2.33576 | 3.50085 | −2.89205 | 3.53460 | −0.913732 | ||||||||||||||||||||||||||||||||||
| 1.5 | 2.02642 | −0.589625 | 2.10637 | −1.00000 | −1.19483 | −0.911544 | 0.215549 | −2.65234 | −2.02642 | ||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(5\) | \(1\) |
| \(11\) | \(-1\) |
| \(19\) | \(-1\) |
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 | 1045.2.a.e | ✓ | 5 |
| 3.b | odd | 2 | 1 | 9405.2.a.u | 5 | ||
| 5.b | even | 2 | 1 | 5225.2.a.i | 5 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 1045.2.a.e | ✓ | 5 | 1.a | even | 1 | 1 | trivial |
| 5225.2.a.i | 5 | 5.b | even | 2 | 1 | ||
| 9405.2.a.u | 5 | 3.b | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{5} + T_{2}^{4} - 5T_{2}^{3} - 5T_{2}^{2} + 4T_{2} + 3 \)
acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(1045))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{5} + T^{4} - 5 T^{3} + \cdots + 3 \)
|
| $3$ |
\( T^{5} + 3 T^{4} + \cdots + 1 \)
|
| $5$ |
\( (T + 1)^{5} \)
|
| $7$ |
\( T^{5} - 3 T^{4} + \cdots - 13 \)
|
| $11$ |
\( (T - 1)^{5} \)
|
| $13$ |
\( T^{5} + 3 T^{4} + \cdots - 299 \)
|
| $17$ |
\( T^{5} + 11 T^{4} + \cdots - 69 \)
|
| $19$ |
\( (T - 1)^{5} \)
|
| $23$ |
\( T^{5} - 74 T^{3} + \cdots + 1821 \)
|
| $29$ |
\( T^{5} + 15 T^{4} + \cdots - 1047 \)
|
| $31$ |
\( T^{5} + 9 T^{4} + \cdots - 3025 \)
|
| $37$ |
\( T^{5} - 11 T^{4} + \cdots - 79 \)
|
| $41$ |
\( T^{5} + 23 T^{4} + \cdots - 22467 \)
|
| $43$ |
\( T^{5} - 9 T^{4} + \cdots - 2783 \)
|
| $47$ |
\( T^{5} + 6 T^{4} + \cdots + 21621 \)
|
| $53$ |
\( T^{5} + 13 T^{4} + \cdots - 27 \)
|
| $59$ |
\( T^{5} + 21 T^{4} + \cdots - 6849 \)
|
| $61$ |
\( T^{5} + 31 T^{4} + \cdots + 3307 \)
|
| $67$ |
\( T^{5} - 195 T^{3} + \cdots + 1031 \)
|
| $71$ |
\( T^{5} + 28 T^{4} + \cdots - 141 \)
|
| $73$ |
\( T^{5} + 14 T^{4} + \cdots + 115699 \)
|
| $79$ |
\( T^{5} - 3 T^{4} + \cdots + 6125 \)
|
| $83$ |
\( T^{5} + 33 T^{4} + \cdots - 77949 \)
|
| $89$ |
\( T^{5} + 10 T^{4} + \cdots + 6909 \)
|
| $97$ |
\( T^{5} + 10 T^{4} + \cdots - 1597 \)
|
| show more | |
| show less |