Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [2002,2,Mod(1,2002)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2002, 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("2002.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 2002 = 2 \cdot 7 \cdot 11 \cdot 13 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 2002.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: | \(15.9860504847\) |
| Analytic rank: | \(0\) |
| Dimension: | \(5\) |
| Coefficient field: | 5.5.1590832.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{5} - x^{4} - 7x^{3} + 3x^{2} + 11x + 1 \)
|
| 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
\( x^{5} - x^{4} - 7x^{3} + 3x^{2} + 11x + 1 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - \nu - 3 \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{3} - \nu^{2} - 4\nu + 1 \)
|
| \(\beta_{4}\) | \(=\) |
\( \nu^{4} - \nu^{3} - 5\nu^{2} + 2\nu + 3 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + \beta _1 + 3 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{3} + \beta_{2} + 5\beta _1 + 2 \)
|
| \(\nu^{4}\) | \(=\) |
\( \beta_{4} + \beta_{3} + 6\beta_{2} + 8\beta _1 + 14 \)
|
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 | −1.60084 | 1.00000 | 2.42526 | −1.60084 | 1.00000 | 1.00000 | −0.437326 | 2.42526 | |||||||||||||||||||||||||||||||||
| 1.2 | 1.00000 | −0.689953 | 1.00000 | −2.78935 | −0.689953 | 1.00000 | 1.00000 | −2.52397 | −2.78935 | ||||||||||||||||||||||||||||||||||
| 1.3 | 1.00000 | 1.09383 | 1.00000 | 2.36568 | 1.09383 | 1.00000 | 1.00000 | −1.80354 | 2.36568 | ||||||||||||||||||||||||||||||||||
| 1.4 | 1.00000 | 2.23749 | 1.00000 | 3.52352 | 2.23749 | 1.00000 | 1.00000 | 2.00634 | 3.52352 | ||||||||||||||||||||||||||||||||||
| 1.5 | 1.00000 | 2.95947 | 1.00000 | −1.52512 | 2.95947 | 1.00000 | 1.00000 | 5.75849 | −1.52512 | ||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \(-1\) |
| \(7\) | \(-1\) |
| \(11\) | \(-1\) |
| \(13\) | \(1\) |
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 | 2002.2.a.r | ✓ | 5 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 2002.2.a.r | ✓ | 5 | 1.a | even | 1 | 1 | trivial |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(2002))\):
|
\( T_{3}^{5} - 4T_{3}^{4} - T_{3}^{3} + 14T_{3}^{2} - 3T_{3} - 8 \)
|
|
\( T_{5}^{5} - 4T_{5}^{4} - 9T_{5}^{3} + 42T_{5}^{2} + 9T_{5} - 86 \)
|
|
\( T_{17}^{5} - 10T_{17}^{4} + 11T_{17}^{3} + 128T_{17}^{2} - 357T_{17} + 184 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T - 1)^{5} \)
|
| $3$ |
\( T^{5} - 4 T^{4} + \cdots - 8 \)
|
| $5$ |
\( T^{5} - 4 T^{4} + \cdots - 86 \)
|
| $7$ |
\( (T - 1)^{5} \)
|
| $11$ |
\( (T - 1)^{5} \)
|
| $13$ |
\( (T + 1)^{5} \)
|
| $17$ |
\( T^{5} - 10 T^{4} + \cdots + 184 \)
|
| $19$ |
\( T^{5} - 12 T^{4} + \cdots + 464 \)
|
| $23$ |
\( T^{5} + 4 T^{4} + \cdots + 1648 \)
|
| $29$ |
\( T^{5} + 2 T^{4} + \cdots + 304 \)
|
| $31$ |
\( T^{5} + 6 T^{4} + \cdots + 320 \)
|
| $37$ |
\( T^{5} + 6 T^{4} + \cdots + 22240 \)
|
| $41$ |
\( T^{5} - 12 T^{4} + \cdots - 1288 \)
|
| $43$ |
\( T^{5} - 12 T^{4} + \cdots - 8 \)
|
| $47$ |
\( T^{5} - 12 T^{4} + \cdots + 3440 \)
|
| $53$ |
\( T^{5} + 2 T^{4} + \cdots + 3386 \)
|
| $59$ |
\( T^{5} + 2 T^{4} + \cdots + 9304 \)
|
| $61$ |
\( T^{5} - 2 T^{4} + \cdots - 1250 \)
|
| $67$ |
\( T^{5} - 4 T^{4} + \cdots - 6316 \)
|
| $71$ |
\( T^{5} + 4 T^{4} + \cdots + 94714 \)
|
| $73$ |
\( T^{5} - 10 T^{4} + \cdots + 6040 \)
|
| $79$ |
\( T^{5} + 2 T^{4} + \cdots - 82 \)
|
| $83$ |
\( T^{5} - 20 T^{4} + \cdots - 14224 \)
|
| $89$ |
\( T^{5} - 14 T^{4} + \cdots - 50632 \)
|
| $97$ |
\( T^{5} + 28 T^{4} + \cdots + 63616 \)
|
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