Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [6012,2,Mod(1,6012)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(6012, 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("6012.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 6012 = 2^{2} \cdot 3^{2} \cdot 167 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 6012.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: | \(48.0060616952\) |
| Analytic rank: | \(0\) |
| Dimension: | \(5\) |
| Coefficient field: | 5.5.161121.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{5} - x^{4} - 6x^{3} + 3x^{2} + 5x + 1 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 2004) |
| 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
\( x^{5} - x^{4} - 6x^{3} + 3x^{2} + 5x + 1 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{4} - \nu^{3} - 5\nu^{2} + 3\nu + 1 \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{4} - \nu^{3} - 6\nu^{2} + 3\nu + 4 \)
|
| \(\beta_{4}\) | \(=\) |
\( -2\nu^{4} + 3\nu^{3} + 11\nu^{2} - 11\nu - 6 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( -\beta_{3} + \beta_{2} + 3 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{4} + \beta_{3} + \beta_{2} + 5\beta _1 + 1 \)
|
| \(\nu^{4}\) | \(=\) |
\( \beta_{4} - 4\beta_{3} + 7\beta_{2} + 2\beta _1 + 15 \)
|
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 |
|
0 | 0 | 0 | −1.28459 | 0 | −1.96468 | 0 | 0 | 0 | |||||||||||||||||||||||||||||||||
| 1.2 | 0 | 0 | 0 | 0.614948 | 0 | −3.46328 | 0 | 0 | 0 | ||||||||||||||||||||||||||||||||||
| 1.3 | 0 | 0 | 0 | 1.38924 | 0 | −0.871845 | 0 | 0 | 0 | ||||||||||||||||||||||||||||||||||
| 1.4 | 0 | 0 | 0 | 1.85181 | 0 | 2.41580 | 0 | 0 | 0 | ||||||||||||||||||||||||||||||||||
| 1.5 | 0 | 0 | 0 | 4.42860 | 0 | 1.88401 | 0 | 0 | 0 | ||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \(-1\) |
| \(3\) | \(-1\) |
| \(167\) | \(-1\) |
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 | 6012.2.a.f | 5 | |
| 3.b | odd | 2 | 1 | 2004.2.a.b | ✓ | 5 | |
| 12.b | even | 2 | 1 | 8016.2.a.q | 5 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 2004.2.a.b | ✓ | 5 | 3.b | odd | 2 | 1 | |
| 6012.2.a.f | 5 | 1.a | even | 1 | 1 | trivial | |
| 8016.2.a.q | 5 | 12.b | even | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{5} - 7T_{5}^{4} + 11T_{5}^{3} + 6T_{5}^{2} - 21T_{5} + 9 \)
acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(6012))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{5} \)
|
| $3$ |
\( T^{5} \)
|
| $5$ |
\( T^{5} - 7 T^{4} + \cdots + 9 \)
|
| $7$ |
\( T^{5} + 2 T^{4} + \cdots + 27 \)
|
| $11$ |
\( T^{5} - 5 T^{4} + \cdots - 243 \)
|
| $13$ |
\( T^{5} + 8 T^{4} + \cdots + 27 \)
|
| $17$ |
\( T^{5} - 7 T^{4} + \cdots - 3 \)
|
| $19$ |
\( T^{5} - 2 T^{4} + \cdots - 601 \)
|
| $23$ |
\( T^{5} - 13 T^{4} + \cdots + 243 \)
|
| $29$ |
\( T^{5} - 11 T^{4} + \cdots - 27 \)
|
| $31$ |
\( T^{5} + 12 T^{4} + \cdots + 421 \)
|
| $37$ |
\( T^{5} + 7 T^{4} + \cdots - 31 \)
|
| $41$ |
\( T^{5} - 12 T^{4} + \cdots + 2151 \)
|
| $43$ |
\( T^{5} - 108 T^{3} + \cdots + 6057 \)
|
| $47$ |
\( T^{5} - 19 T^{4} + \cdots - 69093 \)
|
| $53$ |
\( T^{5} - 21 T^{4} + \cdots - 603 \)
|
| $59$ |
\( T^{5} - 7 T^{4} + \cdots + 573 \)
|
| $61$ |
\( T^{5} + 6 T^{4} + \cdots - 4133 \)
|
| $67$ |
\( T^{5} - 10 T^{4} + \cdots + 1611 \)
|
| $71$ |
\( T^{5} - 35 T^{4} + \cdots + 36999 \)
|
| $73$ |
\( T^{5} + 8 T^{4} + \cdots + 173 \)
|
| $79$ |
\( T^{5} - 198 T^{3} + \cdots - 7569 \)
|
| $83$ |
\( T^{5} - 11 T^{4} + \cdots + 423 \)
|
| $89$ |
\( T^{5} - 32 T^{4} + \cdots + 133587 \)
|
| $97$ |
\( T^{5} - 11 T^{4} + \cdots - 109 \)
|
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