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: | \(6\) |
| Coefficient field: | 6.6.1385718192.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{6} - 15x^{4} + 51x^{2} - 48 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{17}]\) |
| Coefficient ring index: | \( 2 \) |
| 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 \(1,\beta_1,\ldots,\beta_{5}\) 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
\( x^{6} - 15x^{4} + 51x^{2} - 48 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{5} - 11\nu^{3} + 7\nu ) / 4 \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{2} - 5 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( -\nu^{5} + 13\nu^{3} - 25\nu ) / 2 \)
|
| \(\beta_{5}\) | \(=\) |
\( \nu^{4} - 13\nu^{2} + 24 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{3} + 5 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{4} + 2\beta_{2} + 9\beta_1 \)
|
| \(\nu^{4}\) | \(=\) |
\( \beta_{5} + 13\beta_{3} + 41 \)
|
| \(\nu^{5}\) | \(=\) |
\( 11\beta_{4} + 26\beta_{2} + 92\beta_1 \)
|
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 | −3.25965 | 1.00000 | −0.377917 | −3.25965 | −1.00000 | 1.00000 | 7.62530 | −0.377917 | ||||||||||||||||||||||||||||||||||||
| 1.2 | 1.00000 | −1.64440 | 1.00000 | −2.33564 | −1.64440 | −1.00000 | 1.00000 | −0.295955 | −2.33564 | |||||||||||||||||||||||||||||||||||||
| 1.3 | 1.00000 | −1.29254 | 1.00000 | 3.92453 | −1.29254 | −1.00000 | 1.00000 | −1.32935 | 3.92453 | |||||||||||||||||||||||||||||||||||||
| 1.4 | 1.00000 | 1.29254 | 1.00000 | −3.92453 | 1.29254 | −1.00000 | 1.00000 | −1.32935 | −3.92453 | |||||||||||||||||||||||||||||||||||||
| 1.5 | 1.00000 | 1.64440 | 1.00000 | 2.33564 | 1.64440 | −1.00000 | 1.00000 | −0.295955 | 2.33564 | |||||||||||||||||||||||||||||||||||||
| 1.6 | 1.00000 | 3.25965 | 1.00000 | 0.377917 | 3.25965 | −1.00000 | 1.00000 | 7.62530 | 0.377917 | |||||||||||||||||||||||||||||||||||||
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.s | ✓ | 6 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 2002.2.a.s | ✓ | 6 | 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(2002))\):
|
\( T_{3}^{6} - 15T_{3}^{4} + 51T_{3}^{2} - 48 \)
|
|
\( T_{5}^{6} - 21T_{5}^{4} + 87T_{5}^{2} - 12 \)
|
|
\( T_{17}^{6} - 6T_{17}^{5} - 39T_{17}^{4} + 196T_{17}^{3} + 291T_{17}^{2} - 990T_{17} - 1328 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T - 1)^{6} \)
|
| $3$ |
\( T^{6} - 15 T^{4} + \cdots - 48 \)
|
| $5$ |
\( T^{6} - 21 T^{4} + \cdots - 12 \)
|
| $7$ |
\( (T + 1)^{6} \)
|
| $11$ |
\( (T - 1)^{6} \)
|
| $13$ |
\( (T - 1)^{6} \)
|
| $17$ |
\( T^{6} - 6 T^{5} + \cdots - 1328 \)
|
| $19$ |
\( T^{6} - 57 T^{4} + \cdots - 2640 \)
|
| $23$ |
\( T^{6} - 6 T^{5} + \cdots + 384 \)
|
| $29$ |
\( T^{6} - 18 T^{5} + \cdots + 2112 \)
|
| $31$ |
\( T^{6} + 6 T^{5} + \cdots + 2560 \)
|
| $37$ |
\( T^{6} - 6 T^{5} + \cdots - 800 \)
|
| $41$ |
\( T^{6} - 78 T^{4} + \cdots + 144 \)
|
| $43$ |
\( T^{6} - 18 T^{5} + \cdots + 6400 \)
|
| $47$ |
\( T^{6} - 6 T^{5} + \cdots + 384 \)
|
| $53$ |
\( T^{6} - 12 T^{5} + \cdots - 20564 \)
|
| $59$ |
\( T^{6} - 114 T^{4} + \cdots + 160 \)
|
| $61$ |
\( T^{6} - 18 T^{5} + \cdots + 17596 \)
|
| $67$ |
\( T^{6} - 12 T^{5} + \cdots + 309136 \)
|
| $71$ |
\( T^{6} + 12 T^{5} + \cdots + 698704 \)
|
| $73$ |
\( T^{6} - 30 T^{5} + \cdots + 111120 \)
|
| $79$ |
\( T^{6} - 213 T^{4} + \cdots - 77888 \)
|
| $83$ |
\( T^{6} + 12 T^{5} + \cdots + 1740976 \)
|
| $89$ |
\( T^{6} + 12 T^{5} + \cdots + 9328 \)
|
| $97$ |
\( T^{6} - 6 T^{5} + \cdots - 512 \)
|
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