Newspace parameters
| Level: | \( N \) | \(=\) | \( 14 = 2 \cdot 7 \) |
| Weight: | \( k \) | \(=\) | \( 48 \) |
| Character orbit: | \([\chi]\) | \(=\) | 14.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(195.870727717\) |
| Analytic rank: | \(0\) |
| Dimension: | \(6\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{6} - \cdots)\) |
|
|
|
| Defining polynomial: |
\( x^{6} - 2 x^{5} + \cdots - 12\!\cdots\!04 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{13}]\) |
| Coefficient ring index: | multiple of \( 2^{38}\cdot 3^{10}\cdot 5^{4}\cdot 7^{6}\cdot 11 \) |
| 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
\( x^{6} - 2 x^{5} + \cdots - 12\!\cdots\!04 \)
:
| \(\beta_{1}\) | \(=\) |
\( 2\nu - 1 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 14\!\cdots\!81 \nu^{5} + \cdots + 14\!\cdots\!44 ) / 82\!\cdots\!00 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 91\!\cdots\!69 \nu^{5} + \cdots + 19\!\cdots\!56 ) / 85\!\cdots\!00 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 39\!\cdots\!71 \nu^{5} + \cdots + 82\!\cdots\!04 ) / 25\!\cdots\!00 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 22\!\cdots\!57 \nu^{5} + \cdots - 54\!\cdots\!32 ) / 12\!\cdots\!00 \)
|
| \(\nu\) | \(=\) |
\( ( \beta _1 + 1 ) / 2 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( 5\beta_{5} - 6\beta_{4} + 216296\beta_{2} + 31256905256\beta _1 + 36616041983826691938817 ) / 4 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 82874430530 \beta_{5} + 694117179615 \beta_{4} - 1070672421429 \beta_{3} + \cdots + 11\!\cdots\!97 ) / 8 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( 32\!\cdots\!20 \beta_{5} + \cdots + 21\!\cdots\!10 ) / 16 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( 38\!\cdots\!50 \beta_{5} + \cdots + 22\!\cdots\!75 ) / 8 \)
|
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 |
|
−8.38861e6 | −2.49632e11 | 7.03687e13 | 8.52652e15 | 2.09407e18 | −2.73687e19 | −5.90296e20 | 3.57276e22 | −7.15256e22 | ||||||||||||||||||||||||||||||||||||
| 1.2 | −8.38861e6 | −2.08406e11 | 7.03687e13 | −4.20201e16 | 1.74824e18 | −2.73687e19 | −5.90296e20 | 1.68445e22 | 3.52490e23 | |||||||||||||||||||||||||||||||||||||
| 1.3 | −8.38861e6 | 9.30785e9 | 7.03687e13 | 1.53089e16 | −7.80799e16 | −2.73687e19 | −5.90296e20 | −2.65022e22 | −1.28421e23 | |||||||||||||||||||||||||||||||||||||
| 1.4 | −8.38861e6 | 7.45939e10 | 7.03687e13 | −2.75974e15 | −6.25739e17 | −2.73687e19 | −5.90296e20 | −2.10246e22 | 2.31504e22 | |||||||||||||||||||||||||||||||||||||
| 1.5 | −8.38861e6 | 1.88457e11 | 7.03687e13 | −3.51579e16 | −1.58089e18 | −2.73687e19 | −5.90296e20 | 8.92728e21 | 2.94926e23 | |||||||||||||||||||||||||||||||||||||
| 1.6 | −8.38861e6 | 2.72073e11 | 7.03687e13 | 3.64201e16 | −2.28231e18 | −2.73687e19 | −5.90296e20 | 4.74348e22 | −3.05514e23 | |||||||||||||||||||||||||||||||||||||
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 | 14.48.a.b | ✓ | 6 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 14.48.a.b | ✓ | 6 | 1.a | even | 1 | 1 | trivial |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{6} - 86392723256 T_{3}^{5} + \cdots + 18\!\cdots\!04 \)
acting on \(S_{48}^{\mathrm{new}}(\Gamma_0(14))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T + 8388608)^{6} \)
|
| $3$ |
\( T^{6} + \cdots + 18\!\cdots\!04 \)
|
| $5$ |
\( T^{6} + \cdots - 19\!\cdots\!00 \)
|
| $7$ |
\( (T + 27\!\cdots\!43)^{6} \)
|
| $11$ |
\( T^{6} + \cdots - 15\!\cdots\!36 \)
|
| $13$ |
\( T^{6} + \cdots - 87\!\cdots\!96 \)
|
| $17$ |
\( T^{6} + \cdots - 85\!\cdots\!24 \)
|
| $19$ |
\( T^{6} + \cdots - 14\!\cdots\!00 \)
|
| $23$ |
\( T^{6} + \cdots + 23\!\cdots\!00 \)
|
| $29$ |
\( T^{6} + \cdots + 89\!\cdots\!00 \)
|
| $31$ |
\( T^{6} + \cdots + 35\!\cdots\!84 \)
|
| $37$ |
\( T^{6} + \cdots - 10\!\cdots\!44 \)
|
| $41$ |
\( T^{6} + \cdots - 33\!\cdots\!16 \)
|
| $43$ |
\( T^{6} + \cdots + 15\!\cdots\!84 \)
|
| $47$ |
\( T^{6} + \cdots - 82\!\cdots\!04 \)
|
| $53$ |
\( T^{6} + \cdots + 40\!\cdots\!44 \)
|
| $59$ |
\( T^{6} + \cdots + 97\!\cdots\!00 \)
|
| $61$ |
\( T^{6} + \cdots - 94\!\cdots\!00 \)
|
| $67$ |
\( T^{6} + \cdots - 29\!\cdots\!84 \)
|
| $71$ |
\( T^{6} + \cdots - 36\!\cdots\!96 \)
|
| $73$ |
\( T^{6} + \cdots + 35\!\cdots\!00 \)
|
| $79$ |
\( T^{6} + \cdots + 12\!\cdots\!00 \)
|
| $83$ |
\( T^{6} + \cdots - 40\!\cdots\!56 \)
|
| $89$ |
\( T^{6} + \cdots + 22\!\cdots\!00 \)
|
| $97$ |
\( T^{6} + \cdots - 51\!\cdots\!00 \)
|
| show more | |
| show less |