Newspace parameters
| Level: | \( N \) | \(=\) | \( 354 = 2 \cdot 3 \cdot 59 \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 354.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(20.8866761420\) |
| Analytic rank: | \(0\) |
| Dimension: | \(3\) |
| Coefficient field: | 3.3.18989.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{3} - x^{2} - 17x + 14 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 3 \) |
| 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 \(1,\beta_1,\beta_2\) 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
\( x^{3} - x^{2} - 17x + 14 \)
:
| \(\beta_{1}\) | \(=\) |
\( 3\nu - 1 \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - 12 \)
|
| \(\nu\) | \(=\) |
\( ( \beta _1 + 1 ) / 3 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + 12 \)
|
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 |
|
−2.00000 | 3.00000 | 4.00000 | −18.8846 | −6.00000 | 32.1162 | −8.00000 | 9.00000 | 37.7692 | |||||||||||||||||||||||||||
| 1.2 | −2.00000 | 3.00000 | 4.00000 | −17.7453 | −6.00000 | −13.9633 | −8.00000 | 9.00000 | 35.4906 | ||||||||||||||||||||||||||||
| 1.3 | −2.00000 | 3.00000 | 4.00000 | 8.62992 | −6.00000 | 7.84707 | −8.00000 | 9.00000 | −17.2598 | ||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \(1\) |
| \(3\) | \(-1\) |
| \(59\) | \(-1\) |
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 | 354.4.a.f | ✓ | 3 |
| 3.b | odd | 2 | 1 | 1062.4.a.m | 3 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 354.4.a.f | ✓ | 3 | 1.a | even | 1 | 1 | trivial |
| 1062.4.a.m | 3 | 3.b | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{3} + 28T_{5}^{2} + 19T_{5} - 2892 \)
acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(354))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T + 2)^{3} \)
|
| $3$ |
\( (T - 3)^{3} \)
|
| $5$ |
\( T^{3} + 28 T^{2} + \cdots - 2892 \)
|
| $7$ |
\( T^{3} - 26 T^{2} + \cdots + 3519 \)
|
| $11$ |
\( T^{3} + 5 T^{2} + \cdots - 7017 \)
|
| $13$ |
\( T^{3} + 19 T^{2} + \cdots + 11889 \)
|
| $17$ |
\( T^{3} - 20 T^{2} + \cdots - 80151 \)
|
| $19$ |
\( T^{3} - 121 T^{2} + \cdots - 43748 \)
|
| $23$ |
\( T^{3} + 7 T^{2} + \cdots - 363426 \)
|
| $29$ |
\( T^{3} + 121 T^{2} + \cdots + 1370598 \)
|
| $31$ |
\( T^{3} - 651 T^{2} + \cdots - 8689496 \)
|
| $37$ |
\( T^{3} - 618 T^{2} + \cdots - 697589 \)
|
| $41$ |
\( T^{3} - 314 T^{2} + \cdots + 3105543 \)
|
| $43$ |
\( T^{3} + 265 T^{2} + \cdots - 7316469 \)
|
| $47$ |
\( T^{3} - 179 T^{2} + \cdots + 38723196 \)
|
| $53$ |
\( T^{3} - 252 T^{2} + \cdots - 22429872 \)
|
| $59$ |
\( (T - 59)^{3} \)
|
| $61$ |
\( T^{3} - 997 T^{2} + \cdots - 20451722 \)
|
| $67$ |
\( T^{3} - 460 T^{2} + \cdots + 210188668 \)
|
| $71$ |
\( T^{3} - 1263 T^{2} + \cdots - 38789037 \)
|
| $73$ |
\( T^{3} - 1543 T^{2} + \cdots + 715729492 \)
|
| $79$ |
\( T^{3} - 1719 T^{2} + \cdots + 667248523 \)
|
| $83$ |
\( T^{3} + 224 T^{2} + \cdots - 629913 \)
|
| $89$ |
\( T^{3} - 1347 T^{2} + \cdots - 72989964 \)
|
| $97$ |
\( T^{3} - 1200 T^{2} + \cdots + 36576076 \)
|
| show more | |
| show less |