Newspace parameters
| Level: | \( N \) | \(=\) | \( 14 = 2 \cdot 7 \) |
| Weight: | \( k \) | \(=\) | \( 44 \) |
| Character orbit: | \([\chi]\) | \(=\) | 14.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(163.954553484\) |
| Analytic rank: | \(0\) |
| Dimension: | \(6\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{6} - \cdots)\) |
|
|
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| Defining polynomial: |
\( x^{6} - 2 x^{5} + \cdots - 17\!\cdots\!00 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{13}]\) |
| Coefficient ring index: | \( 2^{31}\cdot 3^{9}\cdot 5^{3}\cdot 7^{6} \) |
| 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
\( x^{6} - 2 x^{5} + \cdots - 17\!\cdots\!00 \)
:
| \(\beta_{1}\) | \(=\) |
\( 2\nu - 1 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 11\!\cdots\!47 \nu^{5} + \cdots - 32\!\cdots\!00 ) / 36\!\cdots\!50 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( - 55\!\cdots\!84 \nu^{5} + \cdots - 87\!\cdots\!25 ) / 20\!\cdots\!75 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( - 39\!\cdots\!29 \nu^{5} + \cdots + 27\!\cdots\!50 ) / 12\!\cdots\!50 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 87\!\cdots\!27 \nu^{5} + \cdots - 35\!\cdots\!00 ) / 36\!\cdots\!50 \)
|
| \(\nu\) | \(=\) |
\( ( \beta _1 + 1 ) / 2 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( \beta_{4} + \beta_{3} + 18417\beta_{2} - 1958625924\beta _1 + 362021973265852304016 ) / 4 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( - 16894461593 \beta_{5} - 44518608977 \beta_{4} - 2939952353 \beta_{3} + 772048377091208 \beta_{2} + \cdots - 70\!\cdots\!16 ) / 8 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( - 11\!\cdots\!71 \beta_{5} + \cdots + 22\!\cdots\!22 ) / 16 \)
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| \(\nu^{5}\) | \(=\) |
\( ( - 42\!\cdots\!11 \beta_{5} + \cdots - 37\!\cdots\!40 ) / 8 \)
|
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 |
|
2.09715e6 | −2.46339e10 | 4.39805e12 | 1.65516e14 | −5.16609e16 | 5.58546e17 | 9.22337e18 | 2.78570e20 | 3.47112e20 | ||||||||||||||||||||||||||||||||||||
| 1.2 | 2.09715e6 | −6.47205e9 | 4.39805e12 | 1.80534e15 | −1.35729e16 | 5.58546e17 | 9.22337e18 | −2.86370e20 | 3.78608e21 | |||||||||||||||||||||||||||||||||||||
| 1.3 | 2.09715e6 | −5.44731e9 | 4.39805e12 | −1.56089e15 | −1.14238e16 | 5.58546e17 | 9.22337e18 | −2.98584e20 | −3.27343e21 | |||||||||||||||||||||||||||||||||||||
| 1.4 | 2.09715e6 | 1.16279e10 | 4.39805e12 | 7.04460e14 | 2.43856e16 | 5.58546e17 | 9.22337e18 | −1.93048e20 | 1.47736e21 | |||||||||||||||||||||||||||||||||||||
| 1.5 | 2.09715e6 | 2.36158e10 | 4.39805e12 | −1.55500e15 | 4.95259e16 | 5.58546e17 | 9.22337e18 | 2.29448e20 | −3.26107e21 | |||||||||||||||||||||||||||||||||||||
| 1.6 | 2.09715e6 | 3.07447e10 | 4.39805e12 | 1.08735e15 | 6.44763e16 | 5.58546e17 | 9.22337e18 | 6.16978e20 | 2.28034e21 | |||||||||||||||||||||||||||||||||||||
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 | 14.44.a.d | ✓ | 6 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 14.44.a.d | ✓ | 6 | 1.a | even | 1 | 1 | trivial |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{6} - 29435191576 T_{3}^{5} + \cdots - 73\!\cdots\!96 \)
acting on \(S_{44}^{\mathrm{new}}(\Gamma_0(14))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T - 2097152)^{6} \)
|
| $3$ |
\( T^{6} + \cdots - 73\!\cdots\!96 \)
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| $5$ |
\( T^{6} + \cdots + 55\!\cdots\!00 \)
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| $7$ |
\( (T - 55\!\cdots\!07)^{6} \)
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| $11$ |
\( T^{6} + \cdots - 72\!\cdots\!56 \)
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| $13$ |
\( T^{6} + \cdots + 51\!\cdots\!64 \)
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| $17$ |
\( T^{6} + \cdots - 64\!\cdots\!04 \)
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| $19$ |
\( T^{6} + \cdots + 11\!\cdots\!00 \)
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| $23$ |
\( T^{6} + \cdots - 13\!\cdots\!00 \)
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| $29$ |
\( T^{6} + \cdots - 10\!\cdots\!00 \)
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| $31$ |
\( T^{6} + \cdots + 12\!\cdots\!84 \)
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| $37$ |
\( T^{6} + \cdots + 59\!\cdots\!16 \)
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| $41$ |
\( T^{6} + \cdots - 12\!\cdots\!96 \)
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| $43$ |
\( T^{6} + \cdots + 43\!\cdots\!84 \)
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| $47$ |
\( T^{6} + \cdots - 68\!\cdots\!44 \)
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| $53$ |
\( T^{6} + \cdots - 17\!\cdots\!96 \)
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| $59$ |
\( T^{6} + \cdots - 11\!\cdots\!00 \)
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| $61$ |
\( T^{6} + \cdots - 44\!\cdots\!00 \)
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| $67$ |
\( T^{6} + \cdots + 16\!\cdots\!36 \)
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| $71$ |
\( T^{6} + \cdots - 19\!\cdots\!36 \)
|
| $73$ |
\( T^{6} + \cdots - 29\!\cdots\!00 \)
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| $79$ |
\( T^{6} + \cdots - 20\!\cdots\!00 \)
|
| $83$ |
\( T^{6} + \cdots + 14\!\cdots\!04 \)
|
| $89$ |
\( T^{6} + \cdots - 20\!\cdots\!00 \)
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| $97$ |
\( T^{6} + \cdots - 53\!\cdots\!00 \)
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