Newspace parameters
| Level: | \( N \) | \(=\) | \( 143 = 11 \cdot 13 \) |
| Weight: | \( k \) | \(=\) | \( 3 \) |
| Character orbit: | \([\chi]\) | \(=\) | 143.d (of order \(2\), degree \(1\), minimal) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(3.89646778035\) |
| Analytic rank: | \(0\) |
| Dimension: | \(5\) |
| Coefficient field: | 5.5.63903125.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{5} - 15x^{3} + 45x - 20 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| 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 \(1,\beta_1,\beta_2,\beta_3,\beta_4\) 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
\( x^{5} - 15x^{3} + 45x - 20 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - 6 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( \nu^{3} - \nu^{2} - 9\nu + 6 ) / 2 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( \nu^{4} - 12\nu^{2} - 3\nu + 18 ) / 2 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + 6 \)
|
| \(\nu^{3}\) | \(=\) |
\( 2\beta_{3} + \beta_{2} + 9\beta_1 \)
|
| \(\nu^{4}\) | \(=\) |
\( 2\beta_{4} + 12\beta_{2} + 3\beta _1 + 54 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/143\mathbb{Z}\right)^\times\).
| \(n\) | \(67\) | \(79\) |
| \(\chi(n)\) | \(-1\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | |||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 142.1 |
|
−3.97171 | −0.213425 | 11.7745 | 0 | 0.847661 | 8.33233 | −30.8779 | −8.95445 | 0 | |||||||||||||||||||||||||||||||||
| 142.2 | −1.67899 | 5.63678 | −1.18099 | 0 | −9.46410 | −0.128156 | 8.69883 | 22.7732 | 0 | ||||||||||||||||||||||||||||||||||
| 142.3 | −0.775658 | −5.76868 | −3.39835 | 0 | 4.47452 | −13.3538 | 5.73859 | 24.2777 | 0 | ||||||||||||||||||||||||||||||||||
| 142.4 | 2.93403 | 3.69714 | 4.60855 | 0 | 10.8475 | −8.12497 | 1.78551 | 4.66887 | 0 | ||||||||||||||||||||||||||||||||||
| 142.5 | 3.49232 | −3.35182 | 8.19633 | 0 | −11.7056 | 13.2746 | 14.6549 | 2.23467 | 0 | ||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 143.d | odd | 2 | 1 | CM by \(\Q(\sqrt{-143}) \) |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 143.3.d.a | ✓ | 5 |
| 11.b | odd | 2 | 1 | 143.3.d.b | yes | 5 | |
| 13.b | even | 2 | 1 | 143.3.d.b | yes | 5 | |
| 143.d | odd | 2 | 1 | CM | 143.3.d.a | ✓ | 5 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 143.3.d.a | ✓ | 5 | 1.a | even | 1 | 1 | trivial |
| 143.3.d.a | ✓ | 5 | 143.d | odd | 2 | 1 | CM |
| 143.3.d.b | yes | 5 | 11.b | odd | 2 | 1 | |
| 143.3.d.b | yes | 5 | 13.b | even | 2 | 1 | |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{5} - 20T_{2}^{3} + 80T_{2} + 53 \)
acting on \(S_{3}^{\mathrm{new}}(143, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{5} - 20 T^{3} + \cdots + 53 \)
|
| $3$ |
\( T^{5} - 45 T^{3} + \cdots + 86 \)
|
| $5$ |
\( T^{5} \)
|
| $7$ |
\( T^{5} - 245 T^{3} + \cdots + 1538 \)
|
| $11$ |
\( (T - 11)^{5} \)
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| $13$ |
\( (T + 13)^{5} \)
|
| $17$ |
\( T^{5} \)
|
| $19$ |
\( T^{5} - 1805 T^{3} + \cdots + 4932794 \)
|
| $23$ |
\( T^{5} - 2645 T^{3} + \cdots - 9481298 \)
|
| $29$ |
\( T^{5} \)
|
| $31$ |
\( T^{5} \)
|
| $37$ |
\( T^{5} \)
|
| $41$ |
\( T^{5} - 8405 T^{3} + \cdots + 231619298 \)
|
| $43$ |
\( T^{5} \)
|
| $47$ |
\( T^{5} \)
|
| $53$ |
\( T^{5} - 14045 T^{3} + \cdots + 218470822 \)
|
| $59$ |
\( T^{5} \)
|
| $61$ |
\( T^{5} \)
|
| $67$ |
\( T^{5} \)
|
| $71$ |
\( T^{5} \)
|
| $73$ |
\( T^{5} + \cdots + 3615121442 \)
|
| $79$ |
\( T^{5} \)
|
| $83$ |
\( T^{5} + \cdots + 7348587002 \)
|
| $89$ |
\( T^{5} \)
|
| $97$ |
\( T^{5} \)
|
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