Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [2259,2,Mod(1,2259)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2259, 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("2259.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 2259 = 3^{2} \cdot 251 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 2259.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: | \(18.0382058166\) |
| Analytic rank: | \(1\) |
| Dimension: | \(6\) |
| Coefficient field: | 6.6.4448597.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{6} - 2x^{5} - 5x^{4} + 8x^{3} + 3x^{2} - 5x + 1 \)
|
| Coefficient ring: | \(\Z[a_1, a_2]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 753) |
| 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
\( x^{6} - 2x^{5} - 5x^{4} + 8x^{3} + 3x^{2} - 5x + 1 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( -\nu^{5} + 2\nu^{4} + 5\nu^{3} - 7\nu^{2} - 4\nu + 2 \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{5} - 2\nu^{4} - 5\nu^{3} + 8\nu^{2} + 3\nu - 4 \)
|
| \(\beta_{4}\) | \(=\) |
\( 2\nu^{5} - 3\nu^{4} - 11\nu^{3} + 10\nu^{2} + 9\nu - 5 \)
|
| \(\beta_{5}\) | \(=\) |
\( -2\nu^{5} + 3\nu^{4} + 12\nu^{3} - 11\nu^{2} - 13\nu + 6 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{3} + \beta_{2} + \beta _1 + 2 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{5} + \beta_{4} + \beta_{3} + \beta_{2} + 5\beta _1 + 1 \)
|
| \(\nu^{4}\) | \(=\) |
\( \beta_{5} + 2\beta_{4} + 5\beta_{3} + 7\beta_{2} + 8\beta _1 + 10 \)
|
| \(\nu^{5}\) | \(=\) |
\( 7\beta_{5} + 9\beta_{4} + 8\beta_{3} + 11\beta_{2} + 30\beta _1 + 13 \)
|
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 |
|
−2.39355 | 0 | 3.72909 | −1.35011 | 0 | 2.09887 | −4.13865 | 0 | 3.23156 | ||||||||||||||||||||||||||||||||||||
| 1.2 | −2.08701 | 0 | 2.35562 | 1.95981 | 0 | 1.76307 | −0.742187 | 0 | −4.09014 | |||||||||||||||||||||||||||||||||||||
| 1.3 | −1.06759 | 0 | −0.860261 | −1.10264 | 0 | 0.583929 | 3.05357 | 0 | 1.17717 | |||||||||||||||||||||||||||||||||||||
| 1.4 | 0.645721 | 0 | −1.58304 | 2.40149 | 0 | −3.46836 | −2.31365 | 0 | 1.55069 | |||||||||||||||||||||||||||||||||||||
| 1.5 | 1.18052 | 0 | −0.606384 | −2.61742 | 0 | 4.35918 | −3.07688 | 0 | −3.08990 | |||||||||||||||||||||||||||||||||||||
| 1.6 | 1.72191 | 0 | 0.964982 | 0.708880 | 0 | −0.336703 | −1.78221 | 0 | 1.22063 | |||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(3\) | \( -1 \) |
| \(251\) | \( -1 \) |
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 | 2259.2.a.g | 6 | |
| 3.b | odd | 2 | 1 | 753.2.a.g | ✓ | 6 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 753.2.a.g | ✓ | 6 | 3.b | odd | 2 | 1 | |
| 2259.2.a.g | 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(2259))\):
|
\( T_{2}^{6} + 2T_{2}^{5} - 6T_{2}^{4} - 9T_{2}^{3} + 12T_{2}^{2} + 8T_{2} - 7 \)
|
|
\( T_{5}^{6} - 10T_{5}^{4} + 25T_{5}^{2} + 4T_{5} - 13 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{6} + 2 T^{5} + \cdots - 7 \)
|
| $3$ |
\( T^{6} \)
|
| $5$ |
\( T^{6} - 10 T^{4} + \cdots - 13 \)
|
| $7$ |
\( T^{6} - 5 T^{5} + \cdots + 11 \)
|
| $11$ |
\( T^{6} + 3 T^{5} + \cdots + 553 \)
|
| $13$ |
\( T^{6} - 11 T^{5} + \cdots + 511 \)
|
| $17$ |
\( T^{6} + 6 T^{5} + \cdots - 47 \)
|
| $19$ |
\( T^{6} + 11 T^{5} + \cdots + 13 \)
|
| $23$ |
\( T^{6} + 23 T^{5} + \cdots - 1991 \)
|
| $29$ |
\( T^{6} + 17 T^{5} + \cdots - 329 \)
|
| $31$ |
\( T^{6} + 7 T^{5} + \cdots - 5093 \)
|
| $37$ |
\( T^{6} + 18 T^{5} + \cdots + 40459 \)
|
| $41$ |
\( T^{6} + 10 T^{5} + \cdots + 62293 \)
|
| $43$ |
\( T^{6} - 11 T^{5} + \cdots + 38963 \)
|
| $47$ |
\( T^{6} - 11 T^{5} + \cdots - 1 \)
|
| $53$ |
\( T^{6} + 10 T^{5} + \cdots + 17333 \)
|
| $59$ |
\( T^{6} + 23 T^{5} + \cdots - 65497 \)
|
| $61$ |
\( T^{6} + 7 T^{5} + \cdots - 18989 \)
|
| $67$ |
\( T^{6} - 8 T^{5} + \cdots - 2089 \)
|
| $71$ |
\( T^{6} + 27 T^{5} + \cdots + 18091 \)
|
| $73$ |
\( T^{6} + 32 T^{5} + \cdots - 277717 \)
|
| $79$ |
\( T^{6} + T^{5} + \cdots - 15769 \)
|
| $83$ |
\( T^{6} - 2 T^{5} + \cdots + 11011 \)
|
| $89$ |
\( T^{6} + 50 T^{5} + \cdots - 893639 \)
|
| $97$ |
\( T^{6} + 21 T^{5} + \cdots + 226471 \)
|
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