Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [4030,2,Mod(1,4030)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(4030, 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("4030.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 4030 = 2 \cdot 5 \cdot 13 \cdot 31 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 4030.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: | \(32.1797120146\) |
| Analytic rank: | \(1\) |
| Dimension: | \(6\) |
| Coefficient field: | 6.6.10369693.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{6} - 3x^{5} - 4x^{4} + 12x^{3} + 5x^{2} - 7x + 1 \)
|
| 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 \(1,\beta_1,\ldots,\beta_{5}\) 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
\( x^{6} - 3x^{5} - 4x^{4} + 12x^{3} + 5x^{2} - 7x + 1 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{3} - \nu^{2} - 4\nu \)
|
| \(\beta_{3}\) | \(=\) |
\( -\nu^{3} + 2\nu^{2} + 3\nu - 3 \)
|
| \(\beta_{4}\) | \(=\) |
\( \nu^{5} - 3\nu^{4} - 4\nu^{3} + 11\nu^{2} + 6\nu - 4 \)
|
| \(\beta_{5}\) | \(=\) |
\( \nu^{5} - 2\nu^{4} - 6\nu^{3} + 8\nu^{2} + 10\nu - 4 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{3} + \beta_{2} + \beta _1 + 3 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{3} + 2\beta_{2} + 5\beta _1 + 3 \)
|
| \(\nu^{4}\) | \(=\) |
\( \beta_{5} - \beta_{4} + 5\beta_{3} + 7\beta_{2} + 9\beta _1 + 15 \)
|
| \(\nu^{5}\) | \(=\) |
\( 3\beta_{5} - 2\beta_{4} + 8\beta_{3} + 18\beta_{2} + 30\beta _1 + 28 \)
|
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 | −2.66311 | 1.00000 | 1.00000 | −2.66311 | −4.14258 | 1.00000 | 4.09215 | 1.00000 | ||||||||||||||||||||||||||||||||||||
| 1.2 | 1.00000 | −2.06862 | 1.00000 | 1.00000 | −2.06862 | 0.701675 | 1.00000 | 1.27917 | 1.00000 | |||||||||||||||||||||||||||||||||||||
| 1.3 | 1.00000 | −0.827755 | 1.00000 | 1.00000 | −0.827755 | 0.429039 | 1.00000 | −2.31482 | 1.00000 | |||||||||||||||||||||||||||||||||||||
| 1.4 | 1.00000 | −0.482624 | 1.00000 | 1.00000 | −0.482624 | −0.948993 | 1.00000 | −2.76707 | 1.00000 | |||||||||||||||||||||||||||||||||||||
| 1.5 | 1.00000 | 1.31687 | 1.00000 | 1.00000 | 1.31687 | −4.24970 | 1.00000 | −1.26586 | 1.00000 | |||||||||||||||||||||||||||||||||||||
| 1.6 | 1.00000 | 1.72524 | 1.00000 | 1.00000 | 1.72524 | −1.78945 | 1.00000 | −0.0235611 | 1.00000 | |||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \(-1\) |
| \(5\) | \(-1\) |
| \(13\) | \(1\) |
| \(31\) | \(1\) |
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 | 4030.2.a.g | ✓ | 6 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 4030.2.a.g | ✓ | 6 | 1.a | even | 1 | 1 | trivial |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{6} + 3T_{3}^{5} - 4T_{3}^{4} - 14T_{3}^{3} + 2T_{3}^{2} + 14T_{3} + 5 \)
acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(4030))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T - 1)^{6} \)
|
| $3$ |
\( T^{6} + 3 T^{5} + \cdots + 5 \)
|
| $5$ |
\( (T - 1)^{6} \)
|
| $7$ |
\( T^{6} + 10 T^{5} + \cdots + 9 \)
|
| $11$ |
\( (T^{3} + 5 T^{2} - 9)^{2} \)
|
| $13$ |
\( (T + 1)^{6} \)
|
| $17$ |
\( T^{6} + 4 T^{5} + \cdots + 11221 \)
|
| $19$ |
\( T^{6} + 7 T^{5} + \cdots + 9527 \)
|
| $23$ |
\( T^{6} + 3 T^{5} + \cdots - 20205 \)
|
| $29$ |
\( T^{6} + 4 T^{5} + \cdots - 811 \)
|
| $31$ |
\( (T + 1)^{6} \)
|
| $37$ |
\( T^{6} + 2 T^{5} + \cdots + 441 \)
|
| $41$ |
\( T^{6} + 10 T^{5} + \cdots + 25 \)
|
| $43$ |
\( T^{6} + 15 T^{5} + \cdots + 152315 \)
|
| $47$ |
\( T^{6} + 24 T^{5} + \cdots - 3215 \)
|
| $53$ |
\( T^{6} - 2 T^{5} + \cdots + 14063 \)
|
| $59$ |
\( T^{6} + 25 T^{5} + \cdots + 204055 \)
|
| $61$ |
\( T^{6} - 9 T^{5} + \cdots - 13625 \)
|
| $67$ |
\( T^{6} + 26 T^{5} + \cdots + 113813 \)
|
| $71$ |
\( T^{6} + 38 T^{5} + \cdots - 567547 \)
|
| $73$ |
\( T^{6} + T^{5} + \cdots + 4937 \)
|
| $79$ |
\( T^{6} - 151 T^{4} + \cdots + 14125 \)
|
| $83$ |
\( T^{6} + 18 T^{5} + \cdots - 11565 \)
|
| $89$ |
\( T^{6} - 4 T^{5} + \cdots + 27135 \)
|
| $97$ |
\( T^{6} + 21 T^{5} + \cdots + 35639 \)
|
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