Newspace parameters
| Level: | \( N \) | \(=\) | \( 14 = 2 \cdot 7 \) |
| Weight: | \( k \) | \(=\) | \( 50 \) |
| Character orbit: | \([\chi]\) | \(=\) | 14.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(212.892687139\) |
| Analytic rank: | \(0\) |
| Dimension: | \(6\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{6} - \cdots)\) |
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|
|
| Defining polynomial: |
\( x^{6} - 2 x^{5} + \cdots + 43\!\cdots\!00 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{11}]\) |
| Coefficient ring index: | \( 2^{36}\cdot 3^{12}\cdot 5^{6}\cdot 7^{6} \) |
| 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 \(1,\beta_1,\ldots,\beta_{5}\) 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
\( x^{6} - 2 x^{5} + \cdots + 43\!\cdots\!00 \)
:
| \(\beta_{1}\) | \(=\) |
\( 2\nu - 1 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 67\!\cdots\!27 \nu^{5} + \cdots - 54\!\cdots\!80 ) / 52\!\cdots\!80 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 14\!\cdots\!61 \nu^{5} + \cdots - 64\!\cdots\!00 ) / 21\!\cdots\!00 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 16\!\cdots\!73 \nu^{5} + \cdots + 16\!\cdots\!00 ) / 17\!\cdots\!60 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( - 80\!\cdots\!23 \nu^{5} + \cdots - 23\!\cdots\!00 ) / 26\!\cdots\!00 \)
|
| \(\nu\) | \(=\) |
\( ( \beta _1 + 1 ) / 2 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( 38\beta_{5} + 10\beta_{4} + 7\beta_{3} + 797080\beta_{2} + 50492114262\beta _1 + 270900902078175922139419 ) / 4 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( - 2196588358129 \beta_{5} - 8324804170631 \beta_{4} - 15716888531501 \beta_{3} + \cdots + 13\!\cdots\!63 ) / 8 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( 23\!\cdots\!97 \beta_{5} + \cdots + 13\!\cdots\!26 ) / 16 \)
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| \(\nu^{5}\) | \(=\) |
\( ( - 54\!\cdots\!24 \beta_{5} + \cdots - 36\!\cdots\!37 ) / 8 \)
|
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 |
|
1.67772e7 | −6.34751e11 | 2.81475e14 | 1.14301e17 | −1.06494e19 | −1.91581e20 | 4.72237e21 | 1.63610e23 | 1.91765e24 | ||||||||||||||||||||||||||||||||||||
| 1.2 | 1.67772e7 | −4.81157e11 | 2.81475e14 | −9.77002e16 | −8.07247e18 | −1.91581e20 | 4.72237e21 | −7.78761e21 | −1.63914e24 | |||||||||||||||||||||||||||||||||||||
| 1.3 | 1.67772e7 | 1.12609e11 | 2.81475e14 | −1.14808e16 | 1.88926e18 | −1.91581e20 | 4.72237e21 | −2.26619e23 | −1.92616e23 | |||||||||||||||||||||||||||||||||||||
| 1.4 | 1.67772e7 | 3.20530e11 | 2.81475e14 | −1.37317e17 | 5.37760e18 | −1.91581e20 | 4.72237e21 | −1.36560e23 | −2.30380e24 | |||||||||||||||||||||||||||||||||||||
| 1.5 | 1.67772e7 | 4.04355e11 | 2.81475e14 | 2.21202e17 | 6.78395e18 | −1.91581e20 | 4.72237e21 | −7.57966e22 | 3.71115e24 | |||||||||||||||||||||||||||||||||||||
| 1.6 | 1.67772e7 | 8.78703e11 | 2.81475e14 | 1.48490e17 | 1.47422e19 | −1.91581e20 | 4.72237e21 | 5.32820e23 | 2.49126e24 | |||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \( -1 \) |
| \(7\) | \( +1 \) |
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 | 14.50.a.c | ✓ | 6 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 14.50.a.c | ✓ | 6 | 1.a | even | 1 | 1 | trivial |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{6} - 600288540488 T_{3}^{5} + \cdots + 39\!\cdots\!00 \)
acting on \(S_{50}^{\mathrm{new}}(\Gamma_0(14))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T - 16777216)^{6} \)
|
| $3$ |
\( T^{6} + \cdots + 39\!\cdots\!00 \)
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| $5$ |
\( T^{6} + \cdots - 57\!\cdots\!00 \)
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| $7$ |
\( (T + 19\!\cdots\!01)^{6} \)
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| $11$ |
\( T^{6} + \cdots + 17\!\cdots\!00 \)
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| $13$ |
\( T^{6} + \cdots + 95\!\cdots\!96 \)
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| $17$ |
\( T^{6} + \cdots - 69\!\cdots\!76 \)
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| $19$ |
\( T^{6} + \cdots + 85\!\cdots\!00 \)
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| $23$ |
\( T^{6} + \cdots + 26\!\cdots\!00 \)
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| $29$ |
\( T^{6} + \cdots - 15\!\cdots\!24 \)
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| $31$ |
\( T^{6} + \cdots - 45\!\cdots\!96 \)
|
| $37$ |
\( T^{6} + \cdots - 13\!\cdots\!00 \)
|
| $41$ |
\( T^{6} + \cdots - 43\!\cdots\!16 \)
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| $43$ |
\( T^{6} + \cdots - 89\!\cdots\!64 \)
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| $47$ |
\( T^{6} + \cdots + 97\!\cdots\!64 \)
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| $53$ |
\( T^{6} + \cdots + 20\!\cdots\!04 \)
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| $59$ |
\( T^{6} + \cdots - 37\!\cdots\!56 \)
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| $61$ |
\( T^{6} + \cdots - 11\!\cdots\!24 \)
|
| $67$ |
\( T^{6} + \cdots - 16\!\cdots\!96 \)
|
| $71$ |
\( T^{6} + \cdots - 28\!\cdots\!00 \)
|
| $73$ |
\( T^{6} + \cdots + 84\!\cdots\!24 \)
|
| $79$ |
\( T^{6} + \cdots + 38\!\cdots\!84 \)
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| $83$ |
\( T^{6} + \cdots - 18\!\cdots\!96 \)
|
| $89$ |
\( T^{6} + \cdots + 34\!\cdots\!00 \)
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| $97$ |
\( T^{6} + \cdots - 64\!\cdots\!00 \)
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