Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [4018,2,Mod(1,4018)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(4018, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([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("4018.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 4018 = 2 \cdot 7^{2} \cdot 41 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 4018.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: | \(32.0838915322\) |
| Analytic rank: | \(0\) |
| Dimension: | \(6\) |
| Coefficient field: | 6.6.52046292.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{6} - x^{5} - 10x^{4} + 9x^{3} + 24x^{2} - 18x + 1 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{13}]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 574) |
| 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
\( x^{6} - x^{5} - 10x^{4} + 9x^{3} + 24x^{2} - 18x + 1 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - 4 \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{3} - 5\nu \)
|
| \(\beta_{4}\) | \(=\) |
\( \nu^{4} + \nu^{3} - 6\nu^{2} - 5\nu + 3 \)
|
| \(\beta_{5}\) | \(=\) |
\( \nu^{5} + \nu^{4} - 7\nu^{3} - 5\nu^{2} + 8\nu - 2 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + 4 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{3} + 5\beta_1 \)
|
| \(\nu^{4}\) | \(=\) |
\( \beta_{4} - \beta_{3} + 6\beta_{2} + 21 \)
|
| \(\nu^{5}\) | \(=\) |
\( \beta_{5} - \beta_{4} + 8\beta_{3} - \beta_{2} + 27\beta _1 + 1 \)
|
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 | −2.95121 | 1.00000 | −2.26461 | 2.95121 | 0 | −1.00000 | 5.70965 | 2.26461 | ||||||||||||||||||||||||||||||||||||
| 1.2 | −1.00000 | −2.30861 | 1.00000 | 0.374437 | 2.30861 | 0 | −1.00000 | 2.32969 | −0.374437 | |||||||||||||||||||||||||||||||||||||
| 1.3 | −1.00000 | −0.302513 | 1.00000 | 2.67551 | 0.302513 | 0 | −1.00000 | −2.90849 | −2.67551 | |||||||||||||||||||||||||||||||||||||
| 1.4 | −1.00000 | 0.486321 | 1.00000 | −0.616311 | −0.486321 | 0 | −1.00000 | −2.76349 | 0.616311 | |||||||||||||||||||||||||||||||||||||
| 1.5 | −1.00000 | 1.63447 | 1.00000 | 2.84627 | −1.63447 | 0 | −1.00000 | −0.328504 | −2.84627 | |||||||||||||||||||||||||||||||||||||
| 1.6 | −1.00000 | 2.44154 | 1.00000 | −3.01529 | −2.44154 | 0 | −1.00000 | 2.96114 | 3.01529 | |||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \(1\) |
| \(7\) | \(-1\) |
| \(41\) | \(1\) |
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 | 4018.2.a.bn | 6 | |
| 7.b | odd | 2 | 1 | 4018.2.a.bo | 6 | ||
| 7.d | odd | 6 | 2 | 574.2.e.g | ✓ | 12 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 574.2.e.g | ✓ | 12 | 7.d | odd | 6 | 2 | |
| 4018.2.a.bn | 6 | 1.a | even | 1 | 1 | trivial | |
| 4018.2.a.bo | 6 | 7.b | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(4018))\):
|
\( T_{3}^{6} + T_{3}^{5} - 11T_{3}^{4} - 5T_{3}^{3} + 30T_{3}^{2} - 4T_{3} - 4 \)
|
|
\( T_{5}^{6} - 15T_{5}^{4} - T_{5}^{3} + 56T_{5}^{2} + 12T_{5} - 12 \)
|
|
\( T_{11}^{6} - T_{11}^{5} - 38T_{11}^{4} + 7T_{11}^{3} + 320T_{11}^{2} - 48T_{11} - 684 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T + 1)^{6} \)
|
| $3$ |
\( T^{6} + T^{5} - 11 T^{4} + \cdots - 4 \)
|
| $5$ |
\( T^{6} - 15 T^{4} + \cdots - 12 \)
|
| $7$ |
\( T^{6} \)
|
| $11$ |
\( T^{6} - T^{5} + \cdots - 684 \)
|
| $13$ |
\( T^{6} + 4 T^{5} + \cdots - 5 \)
|
| $17$ |
\( T^{6} - T^{5} + \cdots - 36 \)
|
| $19$ |
\( T^{6} - 3 T^{5} + \cdots + 776 \)
|
| $23$ |
\( T^{6} - 21 T^{5} + \cdots - 600 \)
|
| $29$ |
\( T^{6} - 5 T^{5} + \cdots + 375 \)
|
| $31$ |
\( T^{6} + 3 T^{5} + \cdots - 304 \)
|
| $37$ |
\( T^{6} + 2 T^{5} + \cdots - 27256 \)
|
| $41$ |
\( (T + 1)^{6} \)
|
| $43$ |
\( T^{6} - 12 T^{5} + \cdots + 367 \)
|
| $47$ |
\( T^{6} - 18 T^{5} + \cdots - 60 \)
|
| $53$ |
\( T^{6} - 14 T^{5} + \cdots - 30684 \)
|
| $59$ |
\( T^{6} + 16 T^{5} + \cdots - 21249 \)
|
| $61$ |
\( T^{6} - 20 T^{5} + \cdots + 632 \)
|
| $67$ |
\( T^{6} + 13 T^{5} + \cdots - 49228 \)
|
| $71$ |
\( T^{6} - 11 T^{5} + \cdots + 1053807 \)
|
| $73$ |
\( T^{6} + T^{5} + \cdots - 5869 \)
|
| $79$ |
\( T^{6} - 19 T^{5} + \cdots - 66716 \)
|
| $83$ |
\( T^{6} + 15 T^{5} + \cdots + 2307 \)
|
| $89$ |
\( T^{6} - 14 T^{5} + \cdots + 39132 \)
|
| $97$ |
\( T^{6} + T^{5} + \cdots + 3824 \)
|
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