Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [4002,2,Mod(1,4002)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(4002, 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("4002.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 4002 = 2 \cdot 3 \cdot 23 \cdot 29 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 4002.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: | \(31.9561308889\) |
| Analytic rank: | \(0\) |
| Dimension: | \(6\) |
| Coefficient field: | 6.6.61157024.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{6} - 2x^{5} - 8x^{4} + 8x^{3} + 17x^{2} - 4x - 4 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{11}]\) |
| 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
\( x^{6} - 2x^{5} - 8x^{4} + 8x^{3} + 17x^{2} - 4x - 4 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( \nu^{4} - 3\nu^{3} - 3\nu^{2} + 9\nu ) / 2 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -\nu^{4} + 3\nu^{3} + 5\nu^{2} - 9\nu - 6 ) / 2 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( \nu^{5} - 2\nu^{4} - 6\nu^{3} + 4\nu^{2} + 7\nu + 2 ) / 2 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( \nu^{5} - 3\nu^{4} - 5\nu^{3} + 11\nu^{2} + 8\nu - 2 ) / 2 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( \nu^{5} - 3\nu^{4} - 5\nu^{3} + 13\nu^{2} + 4\nu - 8 ) / 2 \)
|
| \(\nu\) | \(=\) |
\( ( -\beta_{5} + \beta_{4} + \beta_{2} + \beta_1 ) / 2 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + \beta _1 + 3 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( -5\beta_{5} + 3\beta_{4} + 2\beta_{3} + 9\beta_{2} + 7\beta _1 + 8 ) / 2 \)
|
| \(\nu^{4}\) | \(=\) |
\( -3\beta_{5} + 3\beta_{3} + 12\beta_{2} + 11\beta _1 + 21 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( -35\beta_{5} + 11\beta_{4} + 28\beta_{3} + 87\beta_{2} + 71\beta _1 + 104 ) / 2 \)
|
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 | −1.56155 | −1.00000 | −3.98403 | −1.00000 | 1.00000 | 1.56155 | ||||||||||||||||||||||||||||||||||||
| 1.2 | −1.00000 | 1.00000 | 1.00000 | −1.56155 | −1.00000 | −1.09168 | −1.00000 | 1.00000 | 1.56155 | |||||||||||||||||||||||||||||||||||||
| 1.3 | −1.00000 | 1.00000 | 1.00000 | −1.56155 | −1.00000 | 1.95260 | −1.00000 | 1.00000 | 1.56155 | |||||||||||||||||||||||||||||||||||||
| 1.4 | −1.00000 | 1.00000 | 1.00000 | 2.56155 | −1.00000 | −1.96335 | −1.00000 | 1.00000 | −2.56155 | |||||||||||||||||||||||||||||||||||||
| 1.5 | −1.00000 | 1.00000 | 1.00000 | 2.56155 | −1.00000 | 3.26093 | −1.00000 | 1.00000 | −2.56155 | |||||||||||||||||||||||||||||||||||||
| 1.6 | −1.00000 | 1.00000 | 1.00000 | 2.56155 | −1.00000 | 3.82553 | −1.00000 | 1.00000 | −2.56155 | |||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \(1\) |
| \(3\) | \(-1\) |
| \(23\) | \(-1\) |
| \(29\) | \(-1\) |
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 | 4002.2.a.bf | ✓ | 6 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 4002.2.a.bf | ✓ | 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(4002))\):
|
\( T_{5}^{2} - T_{5} - 4 \)
|
|
\( T_{7}^{6} - 2T_{7}^{5} - 23T_{7}^{4} + 40T_{7}^{3} + 128T_{7}^{2} - 124T_{7} - 208 \)
|
|
\( T_{11}^{6} - 7T_{11}^{5} + 6T_{11}^{4} + 38T_{11}^{3} - 48T_{11}^{2} - 48T_{11} + 32 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T + 1)^{6} \)
|
| $3$ |
\( (T - 1)^{6} \)
|
| $5$ |
\( (T^{2} - T - 4)^{3} \)
|
| $7$ |
\( T^{6} - 2 T^{5} + \cdots - 208 \)
|
| $11$ |
\( T^{6} - 7 T^{5} + \cdots + 32 \)
|
| $13$ |
\( T^{6} - 3 T^{5} + \cdots - 16 \)
|
| $17$ |
\( T^{6} - 8 T^{5} + \cdots - 32 \)
|
| $19$ |
\( T^{6} + 4 T^{5} + \cdots + 344 \)
|
| $23$ |
\( (T - 1)^{6} \)
|
| $29$ |
\( (T - 1)^{6} \)
|
| $31$ |
\( T^{6} - T^{5} + \cdots - 4064 \)
|
| $37$ |
\( T^{6} - 7 T^{5} + \cdots - 1912 \)
|
| $41$ |
\( T^{6} - 13 T^{5} + \cdots + 47752 \)
|
| $43$ |
\( T^{6} - 39 T^{4} + \cdots + 200 \)
|
| $47$ |
\( T^{6} - 22 T^{5} + \cdots - 1024 \)
|
| $53$ |
\( T^{6} - 10 T^{5} + \cdots - 1024 \)
|
| $59$ |
\( T^{6} - 17 T^{5} + \cdots + 15712 \)
|
| $61$ |
\( T^{6} - T^{5} + \cdots + 5024 \)
|
| $67$ |
\( T^{6} - 3 T^{5} + \cdots - 24896 \)
|
| $71$ |
\( T^{6} - 11 T^{5} + \cdots + 24544 \)
|
| $73$ |
\( T^{6} - 156 T^{4} + \cdots - 64 \)
|
| $79$ |
\( T^{6} - 2 T^{5} + \cdots - 4387072 \)
|
| $83$ |
\( T^{6} - 16 T^{5} + \cdots - 15200 \)
|
| $89$ |
\( T^{6} - 16 T^{5} + \cdots + 16384 \)
|
| $97$ |
\( T^{6} - 2 T^{5} + \cdots - 175744 \)
|
| show more | |
| show less |