Newspace parameters
| Level: | \( N \) | \(=\) | \( 126 = 2 \cdot 3^{2} \cdot 7 \) |
| Weight: | \( k \) | \(=\) | \( 14 \) |
| Character orbit: | \([\chi]\) | \(=\) | 126.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(135.110970479\) |
| Analytic rank: | \(1\) |
| Dimension: | \(3\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{3} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{3} - x^{2} - 166540x + 26034700 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{11}]\) |
| Coefficient ring index: | \( 2^{4}\cdot 3^{4}\cdot 7 \) |
| 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
\( x^{3} - x^{2} - 166540x + 26034700 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( 9\nu^{2} + 1821\nu - 999850 ) / 5 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 18\nu^{2} + 4482\nu - 1999980 ) / 5 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{2} - 2\beta _1 + 56 ) / 168 \)
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| \(\nu^{2}\) | \(=\) |
\( ( -607\beta_{2} + 1494\beta _1 + 55957608 ) / 504 \)
|
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 |
|
64.0000 | 0 | 4096.00 | −23421.3 | 0 | 117649. | 262144. | 0 | −1.49897e6 | |||||||||||||||||||||||||||
| 1.2 | 64.0000 | 0 | 4096.00 | 15323.1 | 0 | 117649. | 262144. | 0 | 980680. | ||||||||||||||||||||||||||||
| 1.3 | 64.0000 | 0 | 4096.00 | 36724.2 | 0 | 117649. | 262144. | 0 | 2.35035e6 | ||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \(-1\) |
| \(3\) | \(1\) |
| \(7\) | \(-1\) |
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 | 126.14.a.q | yes | 3 |
| 3.b | odd | 2 | 1 | 126.14.a.n | ✓ | 3 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 126.14.a.n | ✓ | 3 | 3.b | odd | 2 | 1 | |
| 126.14.a.q | yes | 3 | 1.a | even | 1 | 1 | trivial |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{3} - 28626T_{5}^{2} - 656288460T_{5} + 13179876039000 \)
acting on \(S_{14}^{\mathrm{new}}(\Gamma_0(126))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T - 64)^{3} \)
|
| $3$ |
\( T^{3} \)
|
| $5$ |
\( T^{3} + \cdots + 13179876039000 \)
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| $7$ |
\( (T - 117649)^{3} \)
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| $11$ |
\( T^{3} + \cdots - 11\!\cdots\!40 \)
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| $13$ |
\( T^{3} + \cdots - 79\!\cdots\!72 \)
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| $17$ |
\( T^{3} + \cdots - 23\!\cdots\!16 \)
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| $19$ |
\( T^{3} + \cdots + 70\!\cdots\!12 \)
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| $23$ |
\( T^{3} + \cdots + 31\!\cdots\!40 \)
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| $29$ |
\( T^{3} + \cdots + 27\!\cdots\!16 \)
|
| $31$ |
\( T^{3} + \cdots - 15\!\cdots\!92 \)
|
| $37$ |
\( T^{3} + \cdots + 17\!\cdots\!20 \)
|
| $41$ |
\( T^{3} + \cdots - 51\!\cdots\!40 \)
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| $43$ |
\( T^{3} + \cdots - 35\!\cdots\!48 \)
|
| $47$ |
\( T^{3} + \cdots + 58\!\cdots\!80 \)
|
| $53$ |
\( T^{3} + \cdots - 14\!\cdots\!16 \)
|
| $59$ |
\( T^{3} + \cdots - 24\!\cdots\!48 \)
|
| $61$ |
\( T^{3} + \cdots - 28\!\cdots\!64 \)
|
| $67$ |
\( T^{3} + \cdots - 20\!\cdots\!00 \)
|
| $71$ |
\( T^{3} + \cdots - 17\!\cdots\!40 \)
|
| $73$ |
\( T^{3} + \cdots + 18\!\cdots\!20 \)
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| $79$ |
\( T^{3} + \cdots - 62\!\cdots\!72 \)
|
| $83$ |
\( T^{3} + \cdots + 19\!\cdots\!80 \)
|
| $89$ |
\( T^{3} + \cdots + 14\!\cdots\!92 \)
|
| $97$ |
\( T^{3} + \cdots + 16\!\cdots\!88 \)
|
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