Newspace parameters
| Level: | \( N \) | \(=\) | \( 1859 = 11 \cdot 13^{2} \) |
| Weight: | \( k \) | \(=\) | \( 1 \) |
| Character orbit: | \([\chi]\) | \(=\) | 1859.i (of order \(6\), degree \(2\), not minimal) |
Newform invariants
| Self dual: | no |
| Analytic conductor: | \(0.927761858485\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Relative dimension: | \(2\) over \(\Q(\zeta_{6})\) |
| Coefficient field: | \(\Q(\sqrt{-3}, \sqrt{5})\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{4} - x^{3} + 2x^{2} + x + 1 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{4}]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 143) |
| Projective image: | \(D_{5}\) |
| Projective field: | Galois closure of 5.1.20449.1 |
| Artin image: | $C_3\times D_5$ |
| Artin field: | Galois closure of \(\mathbb{Q}[x]/(x^{15} - \cdots)\) |
$q$-expansion
Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3\) 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
\( x^{4} - x^{3} + 2x^{2} + x + 1 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{3} + 1 ) / 2 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -\nu^{3} + 2\nu^{2} - 2\nu - 1 ) / 2 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{3} + \beta_{2} + \beta_1 \)
|
| \(\nu^{3}\) | \(=\) |
\( 2\beta_{2} - 1 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1859\mathbb{Z}\right)^\times\).
| \(n\) | \(508\) | \(1354\) |
| \(\chi(n)\) | \(-1\) | \(1 + \beta_{3}\) |
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.
| Label | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 868.1 |
|
−0.309017 | − | 0.535233i | −0.309017 | − | 0.535233i | 0.309017 | − | 0.535233i | 0 | −0.190983 | + | 0.330792i | 0.809017 | − | 1.40126i | −1.00000 | 0.309017 | − | 0.535233i | 0 | ||||||||||||||||||
| 868.2 | 0.809017 | + | 1.40126i | 0.809017 | + | 1.40126i | −0.809017 | + | 1.40126i | 0 | −1.30902 | + | 2.26728i | −0.309017 | + | 0.535233i | −1.00000 | −0.809017 | + | 1.40126i | 0 | |||||||||||||||||||
| 1330.1 | −0.309017 | + | 0.535233i | −0.309017 | + | 0.535233i | 0.309017 | + | 0.535233i | 0 | −0.190983 | − | 0.330792i | 0.809017 | + | 1.40126i | −1.00000 | 0.309017 | + | 0.535233i | 0 | |||||||||||||||||||
| 1330.2 | 0.809017 | − | 1.40126i | 0.809017 | − | 1.40126i | −0.809017 | − | 1.40126i | 0 | −1.30902 | − | 2.26728i | −0.309017 | − | 0.535233i | −1.00000 | −0.809017 | − | 1.40126i | 0 | |||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 143.d | odd | 2 | 1 | CM by \(\Q(\sqrt{-143}) \) |
| 13.c | even | 3 | 1 | inner |
| 143.i | odd | 6 | 1 | inner |
Twists
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{4} - T_{2}^{3} + 2T_{2}^{2} + T_{2} + 1 \)
acting on \(S_{1}^{\mathrm{new}}(1859, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} - T^{3} + 2 T^{2} + \cdots + 1 \)
|
| $3$ |
\( T^{4} - T^{3} + 2 T^{2} + \cdots + 1 \)
|
| $5$ |
\( T^{4} \)
|
| $7$ |
\( T^{4} - T^{3} + 2 T^{2} + \cdots + 1 \)
|
| $11$ |
\( (T^{2} + T + 1)^{2} \)
|
| $13$ |
\( T^{4} \)
|
| $17$ |
\( T^{4} \)
|
| $19$ |
\( T^{4} - T^{3} + 2 T^{2} + \cdots + 1 \)
|
| $23$ |
\( T^{4} - T^{3} + 2 T^{2} + \cdots + 1 \)
|
| $29$ |
\( T^{4} \)
|
| $31$ |
\( T^{4} \)
|
| $37$ |
\( T^{4} \)
|
| $41$ |
\( T^{4} - T^{3} + 2 T^{2} + \cdots + 1 \)
|
| $43$ |
\( T^{4} \)
|
| $47$ |
\( T^{4} \)
|
| $53$ |
\( (T^{2} + T - 1)^{2} \)
|
| $59$ |
\( T^{4} \)
|
| $61$ |
\( T^{4} \)
|
| $67$ |
\( T^{4} \)
|
| $71$ |
\( T^{4} \)
|
| $73$ |
\( (T^{2} + T - 1)^{2} \)
|
| $79$ |
\( T^{4} \)
|
| $83$ |
\( (T^{2} + T - 1)^{2} \)
|
| $89$ |
\( T^{4} \)
|
| $97$ |
\( T^{4} \)
|
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