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: | \(\mathbb{Q}[x]/(x^{6} - \cdots)\) |
|
|
|
| Defining polynomial: |
\( x^{6} - 2 x^{5} + \cdots - 62\!\cdots\!00 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{13}]\) |
| Coefficient ring index: | \( 2^{33}\cdot 3^{11}\cdot 5^{3}\cdot 7^{6} \) |
| 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
\( x^{6} - 2 x^{5} + \cdots - 62\!\cdots\!00 \)
:
| \(\beta_{1}\) | \(=\) |
\( 2\nu - 1 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 15\!\cdots\!49 \nu^{5} + \cdots - 88\!\cdots\!20 ) / 20\!\cdots\!00 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( - 75\!\cdots\!69 \nu^{5} + \cdots + 45\!\cdots\!80 ) / 51\!\cdots\!00 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( - 11\!\cdots\!93 \nu^{5} + \cdots + 69\!\cdots\!00 ) / 22\!\cdots\!00 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( - 48\!\cdots\!61 \nu^{5} + \cdots + 29\!\cdots\!40 ) / 22\!\cdots\!00 \)
|
| \(\nu\) | \(=\) |
\( ( \beta _1 + 1 ) / 2 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( -11\beta_{4} - 56\beta_{3} - 181601\beta_{2} + 7929500891\beta _1 + 480661604359064410657 ) / 4 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( - 9713280105 \beta_{5} + 315585534118 \beta_{4} - 1472110413152 \beta_{3} + \cdots + 38\!\cdots\!13 ) / 8 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( - 43\!\cdots\!40 \beta_{5} + \cdots + 34\!\cdots\!74 ) / 16 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( - 47\!\cdots\!95 \beta_{5} + \cdots + 16\!\cdots\!35 ) / 8 \)
|
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 | −3.31306e10 | 4.39805e12 | 9.93092e14 | 6.94798e16 | −5.58546e17 | −9.22337e18 | 7.69378e20 | −2.08266e21 | ||||||||||||||||||||||||||||||||||||
| 1.2 | −2.09715e6 | −1.95024e10 | 4.39805e12 | −1.73111e14 | 4.08994e16 | −5.58546e17 | −9.22337e18 | 5.20852e19 | 3.63040e20 | |||||||||||||||||||||||||||||||||||||
| 1.3 | −2.09715e6 | −5.75608e9 | 4.39805e12 | −8.77444e14 | 1.20714e16 | −5.58546e17 | −9.22337e18 | −2.95124e20 | 1.84013e21 | |||||||||||||||||||||||||||||||||||||
| 1.4 | −2.09715e6 | 2.06706e10 | 4.39805e12 | −3.66872e14 | −4.33493e16 | −5.58546e17 | −9.22337e18 | 9.90153e19 | 7.69386e20 | |||||||||||||||||||||||||||||||||||||
| 1.5 | −2.09715e6 | 2.12548e10 | 4.39805e12 | −5.00494e14 | −4.45746e16 | −5.58546e17 | −9.22337e18 | 1.23511e20 | 1.04961e21 | |||||||||||||||||||||||||||||||||||||
| 1.6 | −2.09715e6 | 2.23517e10 | 4.39805e12 | 2.02638e15 | −4.68749e16 | −5.58546e17 | −9.22337e18 | 1.71341e20 | −4.24964e21 | |||||||||||||||||||||||||||||||||||||
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.c | ✓ | 6 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 14.44.a.c | ✓ | 6 | 1.a | even | 1 | 1 | trivial |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{6} - 5888073464 T_{3}^{5} + \cdots - 36\!\cdots\!96 \)
acting on \(S_{44}^{\mathrm{new}}(\Gamma_0(14))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T + 2097152)^{6} \)
|
| $3$ |
\( T^{6} + \cdots - 36\!\cdots\!96 \)
|
| $5$ |
\( T^{6} + \cdots + 56\!\cdots\!00 \)
|
| $7$ |
\( (T + 55\!\cdots\!07)^{6} \)
|
| $11$ |
\( T^{6} + \cdots + 16\!\cdots\!04 \)
|
| $13$ |
\( T^{6} + \cdots + 30\!\cdots\!44 \)
|
| $17$ |
\( T^{6} + \cdots + 19\!\cdots\!56 \)
|
| $19$ |
\( T^{6} + \cdots + 46\!\cdots\!00 \)
|
| $23$ |
\( T^{6} + \cdots + 90\!\cdots\!00 \)
|
| $29$ |
\( T^{6} + \cdots + 17\!\cdots\!00 \)
|
| $31$ |
\( T^{6} + \cdots + 39\!\cdots\!04 \)
|
| $37$ |
\( T^{6} + \cdots + 13\!\cdots\!56 \)
|
| $41$ |
\( T^{6} + \cdots - 28\!\cdots\!96 \)
|
| $43$ |
\( T^{6} + \cdots - 15\!\cdots\!56 \)
|
| $47$ |
\( T^{6} + \cdots + 18\!\cdots\!36 \)
|
| $53$ |
\( T^{6} + \cdots + 93\!\cdots\!84 \)
|
| $59$ |
\( T^{6} + \cdots + 56\!\cdots\!00 \)
|
| $61$ |
\( T^{6} + \cdots + 78\!\cdots\!00 \)
|
| $67$ |
\( T^{6} + \cdots - 12\!\cdots\!64 \)
|
| $71$ |
\( T^{6} + \cdots + 56\!\cdots\!24 \)
|
| $73$ |
\( T^{6} + \cdots - 32\!\cdots\!00 \)
|
| $79$ |
\( T^{6} + \cdots - 50\!\cdots\!00 \)
|
| $83$ |
\( T^{6} + \cdots + 45\!\cdots\!04 \)
|
| $89$ |
\( T^{6} + \cdots + 42\!\cdots\!00 \)
|
| $97$ |
\( T^{6} + \cdots + 59\!\cdots\!00 \)
|
| show more | |
| show less |