Newspace parameters
| Level: | \( N \) | \(=\) | \( 31 \) |
| Weight: | \( k \) | \(=\) | \( 9 \) |
| Character orbit: | \([\chi]\) | \(=\) | 31.b (of order \(2\), degree \(1\), minimal) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(12.6287369119\) |
| Analytic rank: | \(0\) |
| Dimension: | \(3\) |
| Coefficient field: | 3.3.837.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{3} - 6x - 1 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 3 \) |
| Twist minimal: | yes |
| Sato-Tate group: | $\mathrm{U}(1)[D_{2}]$ |
$q$-expansion
Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2\) 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
\( x^{3} - 6x - 1 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu^{2} + \nu - 4 \)
|
| \(\beta_{2}\) | \(=\) |
\( -\nu^{2} + 2\nu + 4 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{2} + \beta_1 ) / 3 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( -\beta_{2} + 2\beta _1 + 12 ) / 3 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/31\mathbb{Z}\right)^\times\).
| \(n\) | \(3\) |
| \(\chi(n)\) | \(-1\) |
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.
| Label | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | |||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 30.1 |
|
−26.8837 | 0 | 466.731 | 1214.52 | 0 | −4752.36 | −5665.22 | 6561.00 | −32650.7 | |||||||||||||||||||||||||||
| 30.2 | −1.58987 | 0 | −253.472 | −863.353 | 0 | 1779.79 | 809.994 | 6561.00 | 1372.62 | ||||||||||||||||||||||||||||
| 30.3 | 28.4735 | 0 | 554.741 | −351.166 | 0 | 2972.58 | 8506.22 | 6561.00 | −9998.93 | ||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 31.b | odd | 2 | 1 | CM by \(\Q(\sqrt{-31}) \) |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 31.9.b.a | ✓ | 3 |
| 31.b | odd | 2 | 1 | CM | 31.9.b.a | ✓ | 3 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 31.9.b.a | ✓ | 3 | 1.a | even | 1 | 1 | trivial |
| 31.9.b.a | ✓ | 3 | 31.b | odd | 2 | 1 | CM |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{3} - 768T_{2} - 1217 \)
acting on \(S_{9}^{\mathrm{new}}(31, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{3} - 768T - 1217 \)
|
| $3$ |
\( T^{3} \)
|
| $5$ |
\( T^{3} - 1171875 T - 368217506 \)
|
| $7$ |
\( T^{3} + \cdots + 25142615998 \)
|
| $11$ |
\( T^{3} \)
|
| $13$ |
\( T^{3} \)
|
| $17$ |
\( T^{3} \)
|
| $19$ |
\( T^{3} + \cdots + 40\!\cdots\!78 \)
|
| $23$ |
\( T^{3} \)
|
| $29$ |
\( T^{3} \)
|
| $31$ |
\( (T - 923521)^{3} \)
|
| $37$ |
\( T^{3} \)
|
| $41$ |
\( T^{3} + \cdots + 11\!\cdots\!38 \)
|
| $43$ |
\( T^{3} \)
|
| $47$ |
\( (T - 2616962)^{3} \)
|
| $53$ |
\( T^{3} \)
|
| $59$ |
\( T^{3} + \cdots - 74\!\cdots\!62 \)
|
| $61$ |
\( T^{3} \)
|
| $67$ |
\( (T - 38516642)^{3} \)
|
| $71$ |
\( T^{3} + \cdots + 20\!\cdots\!18 \)
|
| $73$ |
\( T^{3} \)
|
| $79$ |
\( T^{3} \)
|
| $83$ |
\( T^{3} \)
|
| $89$ |
\( T^{3} \)
|
| $97$ |
\( T^{3} + \cdots - 11\!\cdots\!82 \)
|
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