Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [229,2,Mod(1,229)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(229, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("229.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 229 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 229.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: | \(1.82857420629\) |
| Analytic rank: | \(1\) |
| Dimension: | \(6\) |
| Coefficient field: | 6.6.1868969.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{6} - 6x^{4} - x^{3} + 8x^{2} + x - 2 \)
|
| Coefficient ring: | \(\Z[a_1, a_2]\) |
| 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
\( x^{6} - 6x^{4} - x^{3} + 8x^{2} + x - 2 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - 2 \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{4} - 5\nu^{2} - \nu + 3 \)
|
| \(\beta_{4}\) | \(=\) |
\( -\nu^{4} + \nu^{3} + 4\nu^{2} - 2\nu - 2 \)
|
| \(\beta_{5}\) | \(=\) |
\( \nu^{5} - 5\nu^{3} - \nu^{2} + 4\nu \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + 2 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{4} + \beta_{3} + \beta_{2} + 3\beta _1 + 1 \)
|
| \(\nu^{4}\) | \(=\) |
\( \beta_{3} + 5\beta_{2} + \beta _1 + 7 \)
|
| \(\nu^{5}\) | \(=\) |
\( \beta_{5} + 5\beta_{4} + 5\beta_{3} + 6\beta_{2} + 11\beta _1 + 7 \)
|
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 |
|
−2.60592 | 1.77374 | 4.79081 | −2.08320 | −4.62222 | −2.85836 | −7.27261 | 0.146159 | 5.42865 | ||||||||||||||||||||||||||||||||||||
| 1.2 | −2.15745 | −2.75286 | 2.65458 | −1.11279 | 5.93915 | 4.77597 | −1.41221 | 4.57825 | 2.40079 | |||||||||||||||||||||||||||||||||||||
| 1.3 | −1.46497 | −1.26066 | 0.146125 | 2.24211 | 1.84682 | −2.25582 | 2.71586 | −1.41074 | −3.28461 | |||||||||||||||||||||||||||||||||||||
| 1.4 | 0.117922 | 0.490067 | −1.98609 | −1.53742 | 0.0577898 | −3.56063 | −0.470049 | −2.75983 | −0.181297 | |||||||||||||||||||||||||||||||||||||
| 1.5 | 0.765656 | −2.57995 | −1.41377 | 2.90016 | −1.97535 | −2.50593 | −2.61377 | 3.65613 | 2.22053 | |||||||||||||||||||||||||||||||||||||
| 1.6 | 1.34475 | −1.67034 | −0.191642 | −3.40885 | −2.24619 | 1.40478 | −2.94722 | −0.209969 | −4.58406 | |||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(229\) | \(1\) |
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 | 229.2.a.b | ✓ | 6 |
| 3.b | odd | 2 | 1 | 2061.2.a.d | 6 | ||
| 4.b | odd | 2 | 1 | 3664.2.a.q | 6 | ||
| 5.b | even | 2 | 1 | 5725.2.a.f | 6 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 229.2.a.b | ✓ | 6 | 1.a | even | 1 | 1 | trivial |
| 2061.2.a.d | 6 | 3.b | odd | 2 | 1 | ||
| 3664.2.a.q | 6 | 4.b | odd | 2 | 1 | ||
| 5725.2.a.f | 6 | 5.b | even | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{6} + 4T_{2}^{5} - 12T_{2}^{3} - 3T_{2}^{2} + 9T_{2} - 1 \)
acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(229))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{6} + 4 T^{5} + \cdots - 1 \)
|
| $3$ |
\( T^{6} + 6 T^{5} + \cdots + 13 \)
|
| $5$ |
\( T^{6} + 3 T^{5} + \cdots + 79 \)
|
| $7$ |
\( T^{6} + 5 T^{5} + \cdots + 386 \)
|
| $11$ |
\( T^{6} + 22 T^{5} + \cdots + 853 \)
|
| $13$ |
\( T^{6} - T^{5} + \cdots + 46 \)
|
| $17$ |
\( T^{6} - 6 T^{5} + \cdots - 17 \)
|
| $19$ |
\( T^{6} + 19 T^{5} + \cdots - 1157 \)
|
| $23$ |
\( T^{6} + 10 T^{5} + \cdots + 1996 \)
|
| $29$ |
\( T^{6} + 7 T^{5} + \cdots + 4394 \)
|
| $31$ |
\( T^{6} + 3 T^{5} + \cdots - 500 \)
|
| $37$ |
\( T^{6} - 2 T^{5} + \cdots - 59758 \)
|
| $41$ |
\( T^{6} + 12 T^{5} + \cdots - 298 \)
|
| $43$ |
\( T^{6} + 9 T^{5} + \cdots - 315859 \)
|
| $47$ |
\( T^{6} + 4 T^{5} + \cdots + 132082 \)
|
| $53$ |
\( T^{6} - 5 T^{5} + \cdots - 2614 \)
|
| $59$ |
\( T^{6} + 32 T^{5} + \cdots - 4612 \)
|
| $61$ |
\( T^{6} + 6 T^{5} + \cdots + 428339 \)
|
| $67$ |
\( T^{6} - 2 T^{5} + \cdots + 87014 \)
|
| $71$ |
\( T^{6} + 32 T^{5} + \cdots + 4399 \)
|
| $73$ |
\( T^{6} - 10 T^{5} + \cdots + 9134 \)
|
| $79$ |
\( T^{6} + 5 T^{5} + \cdots - 147178 \)
|
| $83$ |
\( T^{6} + 14 T^{5} + \cdots - 50033 \)
|
| $89$ |
\( T^{6} + 4 T^{5} + \cdots - 5158 \)
|
| $97$ |
\( T^{6} - 22 T^{5} + \cdots + 7199 \)
|
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