Newspace parameters
| Level: | \( N \) | \(=\) | \( 129 = 3 \cdot 43 \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 129.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(7.61124639074\) |
| Analytic rank: | \(1\) |
| Dimension: | \(5\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{5} - \cdots)\) |
|
|
|
| Defining polynomial: |
\( x^{5} - 2x^{4} - 27x^{3} + 20x^{2} + 162x + 104 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | yes |
| Fricke sign: | \(-1\) |
| Sato-Tate group: | $\mathrm{SU}(2)$ |
$q$-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
\( x^{5} - 2x^{4} - 27x^{3} + 20x^{2} + 162x + 104 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{4} - 4\nu^{3} - 17\nu^{2} + 54\nu + 44 ) / 6 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( \nu^{4} - 2\nu^{3} - 23\nu^{2} + 26\nu + 82 ) / 6 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( -\nu^{4} + 4\nu^{3} + 23\nu^{2} - 60\nu - 110 ) / 6 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{4} + \beta_{2} + \beta _1 + 11 \)
|
| \(\nu^{3}\) | \(=\) |
\( 3\beta_{4} + 3\beta_{3} + 17\beta _1 + 14 \)
|
| \(\nu^{4}\) | \(=\) |
\( 29\beta_{4} + 12\beta_{3} + 23\beta_{2} + 31\beta _1 + 199 \)
|
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | |||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 1.1 |
|
−4.93121 | −3.00000 | 16.3169 | −10.2967 | 14.7936 | 7.71379 | −41.0122 | 9.00000 | 50.7754 | |||||||||||||||||||||||||||||||||
| 1.2 | −2.86717 | −3.00000 | 0.220685 | −3.69072 | 8.60152 | 29.3827 | 22.3046 | 9.00000 | 10.5819 | ||||||||||||||||||||||||||||||||||
| 1.3 | −1.79854 | −3.00000 | −4.76524 | 15.8745 | 5.39563 | −13.8629 | 22.9588 | 9.00000 | −28.5510 | ||||||||||||||||||||||||||||||||||
| 1.4 | 2.44175 | −3.00000 | −2.03788 | 4.14545 | −7.32524 | −13.2388 | −24.5099 | 9.00000 | 10.1221 | ||||||||||||||||||||||||||||||||||
| 1.5 | 4.15519 | −3.00000 | 9.26557 | −18.0325 | −12.4656 | −27.9948 | 5.25868 | 9.00000 | −74.9284 | ||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(3\) | \( +1 \) |
| \(43\) | \( -1 \) |
Inner twists
This newform does not admit any (nontrivial) inner twists.
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 129.4.a.d | ✓ | 5 |
| 3.b | odd | 2 | 1 | 387.4.a.f | 5 | ||
| 4.b | odd | 2 | 1 | 2064.4.a.r | 5 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 129.4.a.d | ✓ | 5 | 1.a | even | 1 | 1 | trivial |
| 387.4.a.f | 5 | 3.b | odd | 2 | 1 | ||
| 2064.4.a.r | 5 | 4.b | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{5} + 3T_{2}^{4} - 25T_{2}^{3} - 63T_{2}^{2} + 118T_{2} + 258 \)
acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(129))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{5} + 3 T^{4} + \cdots + 258 \)
|
| $3$ |
\( (T + 3)^{5} \)
|
| $5$ |
\( T^{5} + 12 T^{4} + \cdots + 45096 \)
|
| $7$ |
\( T^{5} + 18 T^{4} + \cdots + 1164496 \)
|
| $11$ |
\( T^{5} + 60 T^{4} + \cdots - 2126064 \)
|
| $13$ |
\( T^{5} - 42 T^{4} + \cdots - 522764846 \)
|
| $17$ |
\( T^{5} + 72 T^{4} + \cdots - 39886104 \)
|
| $19$ |
\( T^{5} + \cdots + 5134231024 \)
|
| $23$ |
\( T^{5} + 414 T^{4} + \cdots + 702472416 \)
|
| $29$ |
\( T^{5} + \cdots - 6514826112 \)
|
| $31$ |
\( T^{5} + \cdots + 19376796672 \)
|
| $37$ |
\( T^{5} + \cdots - 24575768064 \)
|
| $41$ |
\( T^{5} + \cdots + 20700418368 \)
|
| $43$ |
\( (T - 43)^{5} \)
|
| $47$ |
\( T^{5} + \cdots + 292016295936 \)
|
| $53$ |
\( T^{5} + \cdots - 52460054016 \)
|
| $59$ |
\( T^{5} + \cdots - 43217175118848 \)
|
| $61$ |
\( T^{5} + \cdots + 1322936822272 \)
|
| $67$ |
\( T^{5} + \cdots - 74608174759952 \)
|
| $71$ |
\( T^{5} + \cdots + 25283107955328 \)
|
| $73$ |
\( T^{5} + \cdots - 3409901483904 \)
|
| $79$ |
\( T^{5} + \cdots + 804463538176 \)
|
| $83$ |
\( T^{5} + \cdots - 12720053735496 \)
|
| $89$ |
\( T^{5} + \cdots + 215973388228032 \)
|
| $97$ |
\( T^{5} + \cdots - 28\!\cdots\!74 \)
|
| show more | |
| show less |