Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [2759,1,Mod(2758,2759)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2759, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([1, 1]))
N = Newforms(chi, 1, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("2759.2758");
S:= CuspForms(chi, 1);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 2759 = 31 \cdot 89 \) |
| Weight: | \( k \) | \(=\) | \( 1 \) |
| Character orbit: | \([\chi]\) | \(=\) | 2759.d (of order \(2\), degree \(1\), minimal) |
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: | \(1.37692036985\) |
| Analytic rank: | \(0\) |
| Dimension: | \(9\) |
| Coefficient field: | \(\Q(\zeta_{54})^+\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{9} - 9x^{7} + 27x^{5} - 30x^{3} + 9x - 1 \)
|
| Coefficient ring: | \(\Z[a_1, a_2]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | yes |
| Projective image: | \(D_{27}\) |
| Projective field: | Galois closure of \(\mathbb{Q}[x]/(x^{27} - \cdots)\) |
$q$-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 \(1,\beta_1,\ldots,\beta_{8}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.
Basis of coefficient ring in terms of \(\nu = \zeta_{54} + \zeta_{54}^{-1}\):
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - 2 \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{3} - 3\nu \)
|
| \(\beta_{4}\) | \(=\) |
\( \nu^{4} - 4\nu^{2} + 2 \)
|
| \(\beta_{5}\) | \(=\) |
\( \nu^{5} - 5\nu^{3} + 5\nu \)
|
| \(\beta_{6}\) | \(=\) |
\( \nu^{6} - 6\nu^{4} + 9\nu^{2} - 2 \)
|
| \(\beta_{7}\) | \(=\) |
\( \nu^{7} - 7\nu^{5} + 14\nu^{3} - 7\nu \)
|
| \(\beta_{8}\) | \(=\) |
\( \nu^{8} - 8\nu^{6} + 20\nu^{4} - 16\nu^{2} + 2 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + 2 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{3} + 3\beta_1 \)
|
| \(\nu^{4}\) | \(=\) |
\( \beta_{4} + 4\beta_{2} + 6 \)
|
| \(\nu^{5}\) | \(=\) |
\( \beta_{5} + 5\beta_{3} + 10\beta_1 \)
|
| \(\nu^{6}\) | \(=\) |
\( \beta_{6} + 6\beta_{4} + 15\beta_{2} + 20 \)
|
| \(\nu^{7}\) | \(=\) |
\( \beta_{7} + 7\beta_{5} + 21\beta_{3} + 35\beta_1 \)
|
| \(\nu^{8}\) | \(=\) |
\( \beta_{8} + 8\beta_{6} + 28\beta_{4} + 56\beta_{2} + 70 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2759\mathbb{Z}\right)^\times\).
| \(n\) | \(1336\) | \(1427\) |
| \(\chi(n)\) | \(-1\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | |||||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 2758.1 |
|
−1.98648 | 0.792160 | 2.94609 | −0.573606 | −1.57361 | 0 | −3.86586 | −0.372483 | 1.13946 | |||||||||||||||||||||||||||||||||||||||||||||
| 2758.2 | −1.67098 | 1.78727 | 1.79216 | −1.98648 | −2.98648 | 0 | −1.32368 | 2.19432 | 3.31935 | ||||||||||||||||||||||||||||||||||||||||||||||
| 2758.3 | −1.37248 | −0.573606 | 0.883710 | 1.78727 | 0.787265 | 0 | 0.159606 | −0.670976 | −2.45299 | ||||||||||||||||||||||||||||||||||||||||||||||
| 2758.4 | −0.573606 | 1.94609 | −0.670976 | −0.116290 | −1.11629 | 0 | 0.958482 | 2.78727 | 0.0667045 | ||||||||||||||||||||||||||||||||||||||||||||||
| 2758.5 | −0.116290 | −1.67098 | −0.986477 | 1.19432 | 0.194317 | 0 | 0.231007 | 1.79216 | −0.138887 | ||||||||||||||||||||||||||||||||||||||||||||||
| 2758.6 | 0.792160 | 1.19432 | −0.372483 | 1.94609 | 0.946090 | 0 | −1.08723 | 0.426394 | 1.54161 | ||||||||||||||||||||||||||||||||||||||||||||||
| 2758.7 | 1.19432 | −1.98648 | 0.426394 | −1.37248 | −2.37248 | 0 | −0.685068 | 2.94609 | −1.63918 | ||||||||||||||||||||||||||||||||||||||||||||||
| 2758.8 | 1.78727 | −0.116290 | 2.19432 | 0.792160 | −0.207840 | 0 | 2.13456 | −0.986477 | 1.41580 | ||||||||||||||||||||||||||||||||||||||||||||||
| 2758.9 | 1.94609 | −1.37248 | 2.78727 | −1.67098 | −2.67098 | 0 | 3.47818 | 0.883710 | −3.25187 | ||||||||||||||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 2759.d | odd | 2 | 1 | CM by \(\Q(\sqrt{-2759}) \) |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 2759.1.d.e | ✓ | 9 |
| 31.b | odd | 2 | 1 | 2759.1.d.f | yes | 9 | |
| 89.b | even | 2 | 1 | 2759.1.d.f | yes | 9 | |
| 2759.d | odd | 2 | 1 | CM | 2759.1.d.e | ✓ | 9 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 2759.1.d.e | ✓ | 9 | 1.a | even | 1 | 1 | trivial |
| 2759.1.d.e | ✓ | 9 | 2759.d | odd | 2 | 1 | CM |
| 2759.1.d.f | yes | 9 | 31.b | odd | 2 | 1 | |
| 2759.1.d.f | yes | 9 | 89.b | even | 2 | 1 | |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{1}^{\mathrm{new}}(2759, [\chi])\):
|
\( T_{2}^{9} - 9T_{2}^{7} + 27T_{2}^{5} - 30T_{2}^{3} + 9T_{2} + 1 \)
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\( T_{3}^{9} - 9T_{3}^{7} + 27T_{3}^{5} - 30T_{3}^{3} + 9T_{3} + 1 \)
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Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{9} - 9 T^{7} + \cdots + 1 \)
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| $3$ |
\( T^{9} - 9 T^{7} + \cdots + 1 \)
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| $5$ |
\( T^{9} - 9 T^{7} + \cdots + 1 \)
|
| $7$ |
\( T^{9} \)
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| $11$ |
\( T^{9} \)
|
| $13$ |
\( T^{9} - 9 T^{7} + \cdots + 1 \)
|
| $17$ |
\( T^{9} \)
|
| $19$ |
\( T^{9} \)
|
| $23$ |
\( T^{9} - 9 T^{7} + \cdots + 1 \)
|
| $29$ |
\( T^{9} - 9 T^{7} + \cdots + 1 \)
|
| $31$ |
\( (T - 1)^{9} \)
|
| $37$ |
\( T^{9} - 9 T^{7} + \cdots + 1 \)
|
| $41$ |
\( T^{9} \)
|
| $43$ |
\( T^{9} - 9 T^{7} + \cdots + 1 \)
|
| $47$ |
\( T^{9} - 9 T^{7} + \cdots + 1 \)
|
| $53$ |
\( T^{9} \)
|
| $59$ |
\( T^{9} \)
|
| $61$ |
\( T^{9} - 9 T^{7} + \cdots + 1 \)
|
| $67$ |
\( (T + 1)^{9} \)
|
| $71$ |
\( T^{9} - 9 T^{7} + \cdots + 1 \)
|
| $73$ |
\( T^{9} \)
|
| $79$ |
\( T^{9} \)
|
| $83$ |
\( T^{9} - 9 T^{7} + \cdots + 1 \)
|
| $89$ |
\( (T - 1)^{9} \)
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| $97$ |
\( (T^{3} - 3 T + 1)^{3} \)
|
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