Newspace parameters
| Level: | \( N \) | \(=\) | \( 31 \) |
| Weight: | \( k \) | \(=\) | \( 3 \) |
| Character orbit: | \([\chi]\) | \(=\) | 31.b (of order \(2\), degree \(1\), minimal) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(0.844688819517\) |
| 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: | \(\Z[a_1, a_2]\) |
| Coefficient ring index: | \( 1 \) |
| 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
\( x^{3} - 6x - 1 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - 4 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + 4 \)
|
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 |
|
−3.97196 | 0 | 11.7765 | 4.47431 | 0 | 3.13471 | −30.8878 | 9.00000 | −17.7718 | |||||||||||||||||||||||||||
| 30.2 | 1.57653 | 0 | −1.51454 | 5.50787 | 0 | −13.3839 | −8.69386 | 9.00000 | 8.68335 | ||||||||||||||||||||||||||||
| 30.3 | 2.39543 | 0 | 1.73807 | −9.98218 | 0 | 10.2492 | −5.41830 | 9.00000 | −23.9116 | ||||||||||||||||||||||||||||
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 | 31.3.b.b | ✓ | 3 |
| 3.b | odd | 2 | 1 | 279.3.d.c | 3 | ||
| 4.b | odd | 2 | 1 | 496.3.e.c | 3 | ||
| 5.b | even | 2 | 1 | 775.3.d.d | 3 | ||
| 5.c | odd | 4 | 2 | 775.3.c.b | 6 | ||
| 31.b | odd | 2 | 1 | CM | 31.3.b.b | ✓ | 3 |
| 93.c | even | 2 | 1 | 279.3.d.c | 3 | ||
| 124.d | even | 2 | 1 | 496.3.e.c | 3 | ||
| 155.c | odd | 2 | 1 | 775.3.d.d | 3 | ||
| 155.f | even | 4 | 2 | 775.3.c.b | 6 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 31.3.b.b | ✓ | 3 | 1.a | even | 1 | 1 | trivial |
| 31.3.b.b | ✓ | 3 | 31.b | odd | 2 | 1 | CM |
| 279.3.d.c | 3 | 3.b | odd | 2 | 1 | ||
| 279.3.d.c | 3 | 93.c | even | 2 | 1 | ||
| 496.3.e.c | 3 | 4.b | odd | 2 | 1 | ||
| 496.3.e.c | 3 | 124.d | even | 2 | 1 | ||
| 775.3.c.b | 6 | 5.c | odd | 4 | 2 | ||
| 775.3.c.b | 6 | 155.f | even | 4 | 2 | ||
| 775.3.d.d | 3 | 5.b | even | 2 | 1 | ||
| 775.3.d.d | 3 | 155.c | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{3} - 12T_{2} + 15 \)
acting on \(S_{3}^{\mathrm{new}}(31, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{3} - 12T + 15 \)
|
| $3$ |
\( T^{3} \)
|
| $5$ |
\( T^{3} - 75T + 246 \)
|
| $7$ |
\( T^{3} - 147T + 430 \)
|
| $11$ |
\( T^{3} \)
|
| $13$ |
\( T^{3} \)
|
| $17$ |
\( T^{3} \)
|
| $19$ |
\( T^{3} - 1083T - 10618 \)
|
| $23$ |
\( T^{3} \)
|
| $29$ |
\( T^{3} \)
|
| $31$ |
\( (T + 31)^{3} \)
|
| $37$ |
\( T^{3} \)
|
| $41$ |
\( T^{3} - 5043T + 60558 \)
|
| $43$ |
\( T^{3} \)
|
| $47$ |
\( (T + 30)^{3} \)
|
| $53$ |
\( T^{3} \)
|
| $59$ |
\( T^{3} - 10443 T - 136842 \)
|
| $61$ |
\( T^{3} \)
|
| $67$ |
\( (T - 10)^{3} \)
|
| $71$ |
\( T^{3} - 15123 T - 284178 \)
|
| $73$ |
\( T^{3} \)
|
| $79$ |
\( T^{3} \)
|
| $83$ |
\( T^{3} \)
|
| $89$ |
\( T^{3} \)
|
| $97$ |
\( T^{3} - 28227 T - 1807490 \)
|
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