Newspace parameters
| Level: | \( N \) | \(=\) | \( 1053 = 3^{4} \cdot 13 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 1053.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(8.40824733284\) |
| Analytic rank: | \(0\) |
| Dimension: | \(3\) |
| Coefficient field: | 3.3.621.1 |
|
|
|
| Defining polynomial: |
\( x^{3} - 6x - 3 \)
|
| Coefficient ring: | \(\Z[a_1, a_2]\) |
| 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 \(1,\beta_1,\beta_2\) 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
\( x^{3} - 6x - 3 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - \nu - 4 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + \beta _1 + 4 \)
|
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 |
|
−2.14510 | 0 | 2.60147 | −1.60147 | 0 | 0.601466 | −1.29021 | 0 | 3.43531 | |||||||||||||||||||||||||||
| 1.2 | −0.523976 | 0 | −1.72545 | 2.72545 | 0 | −3.72545 | 1.95205 | 0 | −1.42807 | ||||||||||||||||||||||||||||
| 1.3 | 2.66908 | 0 | 5.12398 | −4.12398 | 0 | 3.12398 | 8.33816 | 0 | −11.0072 | ||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(3\) | \( +1 \) |
| \(13\) | \( -1 \) |
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 | 1053.2.a.g | ✓ | 3 |
| 3.b | odd | 2 | 1 | 1053.2.a.h | yes | 3 | |
| 9.c | even | 3 | 2 | 1053.2.e.o | 6 | ||
| 9.d | odd | 6 | 2 | 1053.2.e.n | 6 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 1053.2.a.g | ✓ | 3 | 1.a | even | 1 | 1 | trivial |
| 1053.2.a.h | yes | 3 | 3.b | odd | 2 | 1 | |
| 1053.2.e.n | 6 | 9.d | odd | 6 | 2 | ||
| 1053.2.e.o | 6 | 9.c | even | 3 | 2 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{3} - 6T_{2} - 3 \)
acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(1053))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{3} - 6T - 3 \)
|
| $3$ |
\( T^{3} \)
|
| $5$ |
\( T^{3} + 3 T^{2} + \cdots - 18 \)
|
| $7$ |
\( T^{3} - 12T + 7 \)
|
| $11$ |
\( T^{3} - 6T - 3 \)
|
| $13$ |
\( (T - 1)^{3} \)
|
| $17$ |
\( T^{3} + 6 T^{2} + \cdots - 6 \)
|
| $19$ |
\( (T - 5)^{3} \)
|
| $23$ |
\( T^{3} - 3 T^{2} + \cdots - 48 \)
|
| $29$ |
\( T^{3} - 6T - 3 \)
|
| $31$ |
\( T^{3} - 3 T^{2} + \cdots + 4 \)
|
| $37$ |
\( T^{3} + 3 T^{2} + \cdots + 28 \)
|
| $41$ |
\( T^{3} - 3 T^{2} + \cdots - 12 \)
|
| $43$ |
\( T^{3} - 6 T^{2} + \cdots + 46 \)
|
| $47$ |
\( T^{3} + 6 T^{2} + \cdots - 144 \)
|
| $53$ |
\( T^{3} - 9 T^{2} + \cdots + 108 \)
|
| $59$ |
\( T^{3} - 9 T^{2} + \cdots + 453 \)
|
| $61$ |
\( T^{3} + 12 T^{2} + \cdots - 1367 \)
|
| $67$ |
\( T^{3} - 15 T^{2} + \cdots + 1468 \)
|
| $71$ |
\( T^{3} - 3 T^{2} + \cdots + 3 \)
|
| $73$ |
\( T^{3} - 12 T^{2} + \cdots + 652 \)
|
| $79$ |
\( T^{3} + 6 T^{2} + \cdots + 166 \)
|
| $83$ |
\( T^{3} - 3 T^{2} + \cdots - 441 \)
|
| $89$ |
\( T^{3} + 3 T^{2} + \cdots - 138 \)
|
| $97$ |
\( (T - 8)^{3} \)
|
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