Newspace parameters
| Level: | \( N \) | \(=\) | \( 87 = 3 \cdot 29 \) |
| Weight: | \( k \) | \(=\) | \( 3 \) |
| Character orbit: | \([\chi]\) | \(=\) | 87.d (of order \(2\), degree \(1\), minimal) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(2.37057829993\) |
| Analytic rank: | \(0\) |
| Dimension: | \(3\) |
| Coefficient field: | 3.3.2349.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{3} - 12x - 13 \)
|
| 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
\( x^{3} - 12x - 13 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - 2\nu - 8 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + 2\beta _1 + 8 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/87\mathbb{Z}\right)^\times\).
| \(n\) | \(31\) | \(59\) |
| \(\chi(n)\) | \(-1\) | \(-1\) |
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.
| Label | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | |||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 86.1 |
|
−3.91423 | 3.00000 | 11.3212 | 0 | −11.7427 | −2.39248 | −28.6569 | 9.00000 | 0 | |||||||||||||||||||||||||||
| 86.2 | 1.24361 | 3.00000 | −2.45343 | 0 | 3.73083 | 13.1422 | −8.02556 | 9.00000 | 0 | ||||||||||||||||||||||||||||
| 86.3 | 2.67062 | 3.00000 | 3.13222 | 0 | 8.01186 | −10.7498 | −2.31751 | 9.00000 | 0 | ||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 87.d | odd | 2 | 1 | CM by \(\Q(\sqrt{-87}) \) |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 87.3.d.b | yes | 3 |
| 3.b | odd | 2 | 1 | 87.3.d.a | ✓ | 3 | |
| 29.b | even | 2 | 1 | 87.3.d.a | ✓ | 3 | |
| 87.d | odd | 2 | 1 | CM | 87.3.d.b | yes | 3 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 87.3.d.a | ✓ | 3 | 3.b | odd | 2 | 1 | |
| 87.3.d.a | ✓ | 3 | 29.b | even | 2 | 1 | |
| 87.3.d.b | yes | 3 | 1.a | even | 1 | 1 | trivial |
| 87.3.d.b | yes | 3 | 87.d | odd | 2 | 1 | CM |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{3} - 12T_{2} + 13 \)
acting on \(S_{3}^{\mathrm{new}}(87, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{3} - 12T + 13 \)
|
| $3$ |
\( (T - 3)^{3} \)
|
| $5$ |
\( T^{3} \)
|
| $7$ |
\( T^{3} - 147T - 338 \)
|
| $11$ |
\( T^{3} - 363T - 806 \)
|
| $13$ |
\( T^{3} - 507T - 3002 \)
|
| $17$ |
\( T^{3} - 867T + 9778 \)
|
| $19$ |
\( T^{3} \)
|
| $23$ |
\( T^{3} \)
|
| $29$ |
\( (T + 29)^{3} \)
|
| $31$ |
\( T^{3} \)
|
| $37$ |
\( T^{3} \)
|
| $41$ |
\( (T + 34)^{3} \)
|
| $43$ |
\( T^{3} \)
|
| $47$ |
\( T^{3} - 6627 T - 151502 \)
|
| $53$ |
\( T^{3} \)
|
| $59$ |
\( T^{3} \)
|
| $61$ |
\( T^{3} \)
|
| $67$ |
\( T^{3} - 13467 T - 267098 \)
|
| $71$ |
\( T^{3} \)
|
| $73$ |
\( T^{3} \)
|
| $79$ |
\( T^{3} \)
|
| $83$ |
\( T^{3} \)
|
| $89$ |
\( T^{3} - 23763 T + 71266 \)
|
| $97$ |
\( T^{3} \)
|
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