Newspace parameters
| Level: | \( N \) | \(=\) | \( 14 = 2 \cdot 7 \) |
| Weight: | \( k \) | \(=\) | \( 42 \) |
| Character orbit: | \([\chi]\) | \(=\) | 14.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(149.060338639\) |
| Analytic rank: | \(0\) |
| Dimension: | \(6\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{6} - \cdots)\) |
|
|
|
| Defining polynomial: |
\( x^{6} - 2 x^{5} + \cdots - 10\!\cdots\!00 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{11}]\) |
| Coefficient ring index: | \( 2^{30}\cdot 3^{10}\cdot 5^{3}\cdot 7^{6} \) |
| 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
\( x^{6} - 2 x^{5} + \cdots - 10\!\cdots\!00 \)
:
| \(\beta_{1}\) | \(=\) |
\( 2\nu - 1 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( - 44\!\cdots\!91 \nu^{5} + \cdots - 89\!\cdots\!50 ) / 17\!\cdots\!50 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 18\!\cdots\!52 \nu^{5} + \cdots - 55\!\cdots\!25 ) / 97\!\cdots\!25 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 40\!\cdots\!47 \nu^{5} + \cdots - 16\!\cdots\!50 ) / 39\!\cdots\!50 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( - 20\!\cdots\!27 \nu^{5} + \cdots - 80\!\cdots\!00 ) / 72\!\cdots\!50 \)
|
| \(\nu\) | \(=\) |
\( ( \beta _1 + 1 ) / 2 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( \beta_{4} + 41265\beta_{2} + 1446703378\beta _1 + 44872267367890140626 ) / 4 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( - 1444584488 \beta_{5} - 2830342942 \beta_{4} + 3492887883 \beta_{3} + 73516160039754 \beta_{2} + \cdots + 64\!\cdots\!62 ) / 8 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( - 11\!\cdots\!57 \beta_{5} + \cdots + 30\!\cdots\!74 ) / 16 \)
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| \(\nu^{5}\) | \(=\) |
\( ( - 57\!\cdots\!89 \beta_{5} + \cdots + 16\!\cdots\!12 ) / 8 \)
|
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 |
|
−1.04858e6 | −9.14521e9 | 1.09951e12 | 9.22250e13 | 9.58945e15 | 7.97923e16 | −1.15292e18 | 4.71619e19 | −9.67049e19 | ||||||||||||||||||||||||||||||||||||
| 1.2 | −1.04858e6 | −6.17994e9 | 1.09951e12 | −3.30318e14 | 6.48013e15 | 7.97923e16 | −1.15292e18 | 1.71863e18 | 3.46363e20 | |||||||||||||||||||||||||||||||||||||
| 1.3 | −1.04858e6 | −3.60118e9 | 1.09951e12 | 2.62316e14 | 3.77611e15 | 7.97923e16 | −1.15292e18 | −2.35045e19 | −2.75058e20 | |||||||||||||||||||||||||||||||||||||
| 1.4 | −1.04858e6 | 1.77814e8 | 1.09951e12 | −2.40373e14 | −1.86452e14 | 7.97923e16 | −1.15292e18 | −3.64414e19 | 2.52050e20 | |||||||||||||||||||||||||||||||||||||
| 1.5 | −1.04858e6 | 7.61188e9 | 1.09951e12 | −9.47383e13 | −7.98164e15 | 7.97923e16 | −1.15292e18 | 2.14677e19 | 9.93403e19 | |||||||||||||||||||||||||||||||||||||
| 1.6 | −1.04858e6 | 8.79653e9 | 1.09951e12 | 1.32726e14 | −9.22383e15 | 7.97923e16 | −1.15292e18 | 4.09059e19 | −1.39173e20 | |||||||||||||||||||||||||||||||||||||
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 | 14.42.a.d | ✓ | 6 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 14.42.a.d | ✓ | 6 | 1.a | even | 1 | 1 | trivial |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{6} + 2340113048 T_{3}^{5} + \cdots - 24\!\cdots\!00 \)
acting on \(S_{42}^{\mathrm{new}}(\Gamma_0(14))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T + 1048576)^{6} \)
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| $3$ |
\( T^{6} + \cdots - 24\!\cdots\!00 \)
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| $5$ |
\( T^{6} + \cdots - 24\!\cdots\!00 \)
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| $7$ |
\( (T - 79\!\cdots\!01)^{6} \)
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| $11$ |
\( T^{6} + \cdots - 14\!\cdots\!00 \)
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| $13$ |
\( T^{6} + \cdots + 18\!\cdots\!36 \)
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| $17$ |
\( T^{6} + \cdots - 34\!\cdots\!56 \)
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| $19$ |
\( T^{6} + \cdots - 97\!\cdots\!00 \)
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| $23$ |
\( T^{6} + \cdots + 28\!\cdots\!00 \)
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| $29$ |
\( T^{6} + \cdots - 50\!\cdots\!44 \)
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| $31$ |
\( T^{6} + \cdots + 74\!\cdots\!84 \)
|
| $37$ |
\( T^{6} + \cdots - 12\!\cdots\!00 \)
|
| $41$ |
\( T^{6} + \cdots - 14\!\cdots\!76 \)
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| $43$ |
\( T^{6} + \cdots + 10\!\cdots\!36 \)
|
| $47$ |
\( T^{6} + \cdots - 44\!\cdots\!36 \)
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| $53$ |
\( T^{6} + \cdots + 68\!\cdots\!04 \)
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| $59$ |
\( T^{6} + \cdots + 21\!\cdots\!64 \)
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| $61$ |
\( T^{6} + \cdots - 12\!\cdots\!24 \)
|
| $67$ |
\( T^{6} + \cdots - 32\!\cdots\!76 \)
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| $71$ |
\( T^{6} + \cdots + 72\!\cdots\!00 \)
|
| $73$ |
\( T^{6} + \cdots - 20\!\cdots\!16 \)
|
| $79$ |
\( T^{6} + \cdots + 78\!\cdots\!84 \)
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| $83$ |
\( T^{6} + \cdots - 54\!\cdots\!16 \)
|
| $89$ |
\( T^{6} + \cdots - 70\!\cdots\!00 \)
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| $97$ |
\( T^{6} + \cdots + 81\!\cdots\!00 \)
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