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: | \(8\) |
| Coefficient field: | 8.8.316689813504.1 |
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|
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| Defining polynomial: |
\( x^{8} - 15x^{6} + 68x^{4} - 87x^{2} + 25 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2 \) |
| 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 \(1,\beta_1,\ldots,\beta_{7}\) 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
\( x^{8} - 15x^{6} + 68x^{4} - 87x^{2} + 25 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - 4 \)
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| \(\beta_{3}\) | \(=\) |
\( ( -\nu^{7} + 10\nu^{5} - 18\nu^{3} - 23\nu ) / 10 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( -\nu^{7} + 10\nu^{5} - 28\nu^{3} + 37\nu ) / 10 \)
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| \(\beta_{5}\) | \(=\) |
\( \nu^{4} - 7\nu^{2} + 4 \)
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| \(\beta_{6}\) | \(=\) |
\( ( \nu^{6} - 10\nu^{4} + 24\nu^{2} - 11 ) / 2 \)
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| \(\beta_{7}\) | \(=\) |
\( ( -\nu^{7} + 15\nu^{5} - 63\nu^{3} + 52\nu ) / 5 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
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| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + 4 \)
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| \(\nu^{3}\) | \(=\) |
\( -\beta_{4} + \beta_{3} + 6\beta_1 \)
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| \(\nu^{4}\) | \(=\) |
\( \beta_{5} + 7\beta_{2} + 24 \)
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| \(\nu^{5}\) | \(=\) |
\( \beta_{7} - 9\beta_{4} + 7\beta_{3} + 39\beta_1 \)
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| \(\nu^{6}\) | \(=\) |
\( 2\beta_{6} + 10\beta_{5} + 46\beta_{2} + 155 \)
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| \(\nu^{7}\) | \(=\) |
\( 10\beta_{7} - 72\beta_{4} + 42\beta_{3} + 259\beta_1 \)
|
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 1.1 |
|
−2.61501 | 0 | 4.83829 | −1.12586 | 0 | −1.63517 | −7.42216 | 0 | 2.94412 | ||||||||||||||||||||||||||||||||||||||||||
| 1.2 | −2.51527 | 0 | 4.32660 | −2.22119 | 0 | 4.09627 | −5.85205 | 0 | 5.58691 | |||||||||||||||||||||||||||||||||||||||||||
| 1.3 | −1.19641 | 0 | −0.568591 | −3.67961 | 0 | −1.89868 | 3.07310 | 0 | 4.40234 | |||||||||||||||||||||||||||||||||||||||||||
| 1.4 | −0.635374 | 0 | −1.59630 | 3.04290 | 0 | 2.43758 | 2.28499 | 0 | −1.93338 | |||||||||||||||||||||||||||||||||||||||||||
| 1.5 | 0.635374 | 0 | −1.59630 | −3.04290 | 0 | 2.43758 | −2.28499 | 0 | −1.93338 | |||||||||||||||||||||||||||||||||||||||||||
| 1.6 | 1.19641 | 0 | −0.568591 | 3.67961 | 0 | −1.89868 | −3.07310 | 0 | 4.40234 | |||||||||||||||||||||||||||||||||||||||||||
| 1.7 | 2.51527 | 0 | 4.32660 | 2.22119 | 0 | 4.09627 | 5.85205 | 0 | 5.58691 | |||||||||||||||||||||||||||||||||||||||||||
| 1.8 | 2.61501 | 0 | 4.83829 | 1.12586 | 0 | −1.63517 | 7.42216 | 0 | 2.94412 | |||||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(3\) | \( -1 \) |
| \(13\) | \( +1 \) |
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 3.b | odd | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 1053.2.a.n | ✓ | 8 |
| 3.b | odd | 2 | 1 | inner | 1053.2.a.n | ✓ | 8 |
| 9.c | even | 3 | 2 | 1053.2.e.s | 16 | ||
| 9.d | odd | 6 | 2 | 1053.2.e.s | 16 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 1053.2.a.n | ✓ | 8 | 1.a | even | 1 | 1 | trivial |
| 1053.2.a.n | ✓ | 8 | 3.b | odd | 2 | 1 | inner |
| 1053.2.e.s | 16 | 9.c | even | 3 | 2 | ||
| 1053.2.e.s | 16 | 9.d | odd | 6 | 2 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{8} - 15T_{2}^{6} + 68T_{2}^{4} - 87T_{2}^{2} + 25 \)
acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(1053))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{8} - 15 T^{6} + \cdots + 25 \)
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| $3$ |
\( T^{8} \)
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| $5$ |
\( T^{8} - 29 T^{6} + \cdots + 784 \)
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| $7$ |
\( (T^{4} - 3 T^{3} - 10 T^{2} + \cdots + 31)^{2} \)
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| $11$ |
\( T^{8} - 91 T^{6} + \cdots + 17689 \)
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| $13$ |
\( (T + 1)^{8} \)
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| $17$ |
\( T^{8} - 54 T^{6} + \cdots + 256 \)
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| $19$ |
\( (T^{4} - 28 T^{2} + 12 T + 7)^{2} \)
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| $23$ |
\( T^{8} - 61 T^{6} + \cdots + 40000 \)
|
| $29$ |
\( T^{8} - 103 T^{6} + \cdots + 85849 \)
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| $31$ |
\( (T^{4} + T^{3} - 79 T^{2} + \cdots + 100)^{2} \)
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| $37$ |
\( (T^{4} - 13 T^{3} + \cdots + 64)^{2} \)
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| $41$ |
\( T^{8} - 273 T^{6} + \cdots + 3136 \)
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| $43$ |
\( (T^{4} - 22 T^{3} + \cdots - 2276)^{2} \)
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| $47$ |
\( T^{8} - 110 T^{6} + \cdots + 3136 \)
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| $53$ |
\( T^{8} - 250 T^{6} + \cdots + 2704 \)
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| $59$ |
\( T^{8} - 144 T^{6} + \cdots + 182329 \)
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| $61$ |
\( (T^{4} + T^{3} - 124 T^{2} + \cdots - 1091)^{2} \)
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| $67$ |
\( (T^{4} - 31 T^{3} + \cdots - 2396)^{2} \)
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| $71$ |
\( T^{8} - 256 T^{6} + \cdots + 436921 \)
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| $73$ |
\( (T^{4} - 181 T^{2} + \cdots + 304)^{2} \)
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| $79$ |
\( (T^{4} - 24 T^{3} + \cdots - 356)^{2} \)
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| $83$ |
\( T^{8} - 288 T^{6} + \cdots + 7070281 \)
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| $89$ |
\( T^{8} - 545 T^{6} + \cdots + 23541904 \)
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| $97$ |
\( (T^{4} + 2 T^{3} + \cdots + 544)^{2} \)
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