Newspace parameters
| Level: | \( N \) | \(=\) | \( 132 = 2^{2} \cdot 3 \cdot 11 \) |
| Weight: | \( k \) | \(=\) | \( 10 \) |
| Character orbit: | \([\chi]\) | \(=\) | 132.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(67.9847303736\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{4} - \cdots)\) |
|
|
|
| Defining polynomial: |
\( x^{4} - x^{3} - 627166x^{2} - 19147766x + 32438978088 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{17}]\) |
| Coefficient ring index: | \( 2^{7}\cdot 3 \) |
| 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 \(1,\beta_1,\beta_2,\beta_3\) 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
\( x^{4} - x^{3} - 627166x^{2} - 19147766x + 32438978088 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( -67\nu^{3} + 28623\nu^{2} + 24096238\nu - 7992022302 ) / 7990455 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 206\nu^{3} + 150516\nu^{2} - 70270604\nu - 50232960624 ) / 7990455 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -541\nu^{3} - 7401\nu^{2} + 344168530\nu + 10258466295 ) / 1598091 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{3} + 5\beta_{2} - 25\beta _1 + 9 ) / 96 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( -\beta_{3} + 196\beta_{2} + 643\beta _1 + 1881723 ) / 6 \)
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| \(\nu^{3}\) | \(=\) |
\( ( 176405\beta_{3} + 1568977\beta_{2} - 8022509\beta _1 + 707101821 ) / 48 \)
|
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 |
|
0 | −81.0000 | 0 | −1433.61 | 0 | −112.787 | 0 | 6561.00 | 0 | ||||||||||||||||||||||||||||||
| 1.2 | 0 | −81.0000 | 0 | −443.654 | 0 | 11229.5 | 0 | 6561.00 | 0 | |||||||||||||||||||||||||||||||
| 1.3 | 0 | −81.0000 | 0 | −287.951 | 0 | −6234.62 | 0 | 6561.00 | 0 | |||||||||||||||||||||||||||||||
| 1.4 | 0 | −81.0000 | 0 | 2011.22 | 0 | −3684.12 | 0 | 6561.00 | 0 | |||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \( -1 \) |
| \(3\) | \( +1 \) |
| \(11\) | \( +1 \) |
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 | 132.10.a.b | ✓ | 4 |
| 3.b | odd | 2 | 1 | 396.10.a.d | 4 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 132.10.a.b | ✓ | 4 | 1.a | even | 1 | 1 | trivial |
| 396.10.a.d | 4 | 3.b | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{4} + 154T_{5}^{3} - 3178132T_{5}^{2} - 2183228680T_{5} - 368343888000 \)
acting on \(S_{10}^{\mathrm{new}}(\Gamma_0(132))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} \)
|
| $3$ |
\( (T + 81)^{4} \)
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| $5$ |
\( T^{4} + \cdots - 368343888000 \)
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| $7$ |
\( T^{4} + \cdots - 29091302463328 \)
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| $11$ |
\( (T + 14641)^{4} \)
|
| $13$ |
\( T^{4} + \cdots + 17\!\cdots\!08 \)
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| $17$ |
\( T^{4} + \cdots - 44\!\cdots\!84 \)
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| $19$ |
\( T^{4} + \cdots + 10\!\cdots\!60 \)
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| $23$ |
\( T^{4} + \cdots + 30\!\cdots\!24 \)
|
| $29$ |
\( T^{4} + \cdots + 31\!\cdots\!00 \)
|
| $31$ |
\( T^{4} + \cdots - 11\!\cdots\!00 \)
|
| $37$ |
\( T^{4} + \cdots - 29\!\cdots\!76 \)
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| $41$ |
\( T^{4} + \cdots + 14\!\cdots\!00 \)
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| $43$ |
\( T^{4} + \cdots + 33\!\cdots\!00 \)
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| $47$ |
\( T^{4} + \cdots - 10\!\cdots\!00 \)
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| $53$ |
\( T^{4} + \cdots - 88\!\cdots\!56 \)
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| $59$ |
\( T^{4} + \cdots - 26\!\cdots\!48 \)
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| $61$ |
\( T^{4} + \cdots - 79\!\cdots\!80 \)
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| $67$ |
\( T^{4} + \cdots - 40\!\cdots\!16 \)
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| $71$ |
\( T^{4} + \cdots - 23\!\cdots\!00 \)
|
| $73$ |
\( T^{4} + \cdots - 14\!\cdots\!40 \)
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| $79$ |
\( T^{4} + \cdots - 20\!\cdots\!64 \)
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| $83$ |
\( T^{4} + \cdots - 56\!\cdots\!88 \)
|
| $89$ |
\( T^{4} + \cdots - 15\!\cdots\!80 \)
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| $97$ |
\( T^{4} + \cdots - 86\!\cdots\!48 \)
|
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