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: | \(6\) |
| Coefficient field: | 6.6.22931361.1 |
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| Defining polynomial: |
\( x^{6} - 2x^{5} - 7x^{4} + 12x^{3} + 10x^{2} - 11x + 2 \)
|
| Coefficient ring: | \(\Z[a_1, a_2]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 117) |
| Fricke sign: | \(-1\) |
| Sato-Tate group: | $\mathrm{SU}(2)$ |
$q$-expansion
Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{5}\) 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
\( x^{6} - 2x^{5} - 7x^{4} + 12x^{3} + 10x^{2} - 11x + 2 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{5} - \nu^{4} - 7\nu^{3} + 5\nu^{2} + 10\nu - 3 \)
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| \(\beta_{3}\) | \(=\) |
\( \nu^{5} - 2\nu^{4} - 7\nu^{3} + 11\nu^{2} + 11\nu - 7 \)
|
| \(\beta_{4}\) | \(=\) |
\( \nu^{5} - 2\nu^{4} - 8\nu^{3} + 11\nu^{2} + 16\nu - 6 \)
|
| \(\beta_{5}\) | \(=\) |
\( \nu^{5} - 2\nu^{4} - 8\nu^{3} + 12\nu^{2} + 15\nu - 9 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{5} - \beta_{4} + \beta _1 + 3 \)
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| \(\nu^{3}\) | \(=\) |
\( -\beta_{4} + \beta_{3} + 5\beta _1 + 1 \)
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| \(\nu^{4}\) | \(=\) |
\( 6\beta_{5} - 6\beta_{4} - \beta_{3} + \beta_{2} + 7\beta _1 + 14 \)
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| \(\nu^{5}\) | \(=\) |
\( \beta_{5} - 8\beta_{4} + 6\beta_{3} + 2\beta_{2} + 27\beta _1 + 9 \)
|
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 |
|
−2.20550 | 0 | 2.86424 | 3.60740 | 0 | −1.08012 | −1.90608 | 0 | −7.95614 | ||||||||||||||||||||||||||||||||||||
| 1.2 | −1.27336 | 0 | −0.378549 | 0.268529 | 0 | −1.15524 | 3.02875 | 0 | −0.341935 | |||||||||||||||||||||||||||||||||||||
| 1.3 | 0.258181 | 0 | −1.93334 | −3.47282 | 0 | −4.42331 | −1.01551 | 0 | −0.896615 | |||||||||||||||||||||||||||||||||||||
| 1.4 | 0.496094 | 0 | −1.75389 | 1.32688 | 0 | 3.12929 | −1.86228 | 0 | 0.658258 | |||||||||||||||||||||||||||||||||||||
| 1.5 | 2.21970 | 0 | 2.92706 | 2.26088 | 0 | 0.366055 | 2.05779 | 0 | 5.01848 | |||||||||||||||||||||||||||||||||||||
| 1.6 | 2.50489 | 0 | 4.27448 | −0.990881 | 0 | 3.16333 | 5.69734 | 0 | −2.48205 | |||||||||||||||||||||||||||||||||||||
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.m | 6 | |
| 3.b | odd | 2 | 1 | 1053.2.a.l | 6 | ||
| 9.c | even | 3 | 2 | 351.2.e.c | 12 | ||
| 9.d | odd | 6 | 2 | 117.2.e.c | ✓ | 12 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 117.2.e.c | ✓ | 12 | 9.d | odd | 6 | 2 | |
| 351.2.e.c | 12 | 9.c | even | 3 | 2 | ||
| 1053.2.a.l | 6 | 3.b | odd | 2 | 1 | ||
| 1053.2.a.m | 6 | 1.a | even | 1 | 1 | trivial | |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{6} - 2T_{2}^{5} - 7T_{2}^{4} + 12T_{2}^{3} + 10T_{2}^{2} - 11T_{2} + 2 \)
acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(1053))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{6} - 2 T^{5} + \cdots + 2 \)
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| $3$ |
\( T^{6} \)
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| $5$ |
\( T^{6} - 3 T^{5} + \cdots + 10 \)
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| $7$ |
\( T^{6} - 21 T^{4} + \cdots - 20 \)
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| $11$ |
\( T^{6} - 7 T^{5} + \cdots - 8 \)
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| $13$ |
\( (T + 1)^{6} \)
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| $17$ |
\( T^{6} - 14 T^{5} + \cdots + 5 \)
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| $19$ |
\( T^{6} + 3 T^{5} + \cdots + 196 \)
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| $23$ |
\( T^{6} - 17 T^{5} + \cdots + 2161 \)
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| $29$ |
\( T^{6} - 14 T^{5} + \cdots + 29470 \)
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| $31$ |
\( T^{6} - 6 T^{5} + \cdots + 302 \)
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| $37$ |
\( T^{6} + 9 T^{5} + \cdots + 16562 \)
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| $41$ |
\( T^{6} + 4 T^{5} + \cdots - 28330 \)
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| $43$ |
\( T^{6} + 3 T^{5} + \cdots - 2239 \)
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| $47$ |
\( T^{6} - 9 T^{5} + \cdots - 74 \)
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| $53$ |
\( T^{6} - 28 T^{5} + \cdots - 6983 \)
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| $59$ |
\( T^{6} - 17 T^{5} + \cdots - 170854 \)
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| $61$ |
\( T^{6} + 6 T^{5} + \cdots + 17687 \)
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| $67$ |
\( T^{6} + 12 T^{5} + \cdots + 2588 \)
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| $71$ |
\( T^{6} + T^{5} + \cdots - 195956 \)
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| $73$ |
\( T^{6} + 9 T^{5} + \cdots + 5782 \)
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| $79$ |
\( T^{6} + 6 T^{5} + \cdots + 4801 \)
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| $83$ |
\( T^{6} - 28 T^{5} + \cdots - 5686 \)
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| $89$ |
\( T^{6} + 9 T^{5} + \cdots + 78148 \)
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| $97$ |
\( T^{6} - 228 T^{4} + \cdots - 244 \)
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