Newspace parameters
| Level: | \( N \) | \(=\) | \( 27 = 3^{3} \) |
| Weight: | \( k \) | \(=\) | \( 14 \) |
| Character orbit: | \([\chi]\) | \(=\) | 27.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(28.9523508170\) |
| Analytic rank: | \(1\) |
| Dimension: | \(4\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{4} - \cdots)\) |
|
|
|
| Defining polynomial: |
\( x^{4} - x^{3} - 2080x^{2} - 9500x + 13552 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 2^{2}\cdot 3^{8}\cdot 13 \) |
| 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 \(1,\beta_1,\beta_2,\beta_3\) 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
\( x^{4} - x^{3} - 2080x^{2} - 9500x + 13552 \)
:
| \(\beta_{1}\) | \(=\) |
\( 3\nu - 1 \)
|
| \(\beta_{2}\) | \(=\) |
\( 9\nu^{2} - 72\nu - 9344 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 27\nu^{3} - 81\nu^{2} - 55386\nu - 136396 ) / 4 \)
|
| \(\nu\) | \(=\) |
\( ( \beta _1 + 1 ) / 3 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( \beta_{2} + 24\beta _1 + 9368 ) / 9 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 4\beta_{3} + 9\beta_{2} + 18678\beta _1 + 239170 ) / 27 \)
|
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 |
|
−157.508 | 0 | 16616.8 | −4600.12 | 0 | −68896.2 | −1.32698e6 | 0 | 724556. | ||||||||||||||||||||||||||||||
| 1.2 | −47.4017 | 0 | −5945.08 | −47381.4 | 0 | 480775. | 670121. | 0 | 2.24596e6 | |||||||||||||||||||||||||||||||
| 1.3 | −26.5760 | 0 | −7485.72 | 47111.1 | 0 | −360331. | 416651. | 0 | −1.25202e6 | |||||||||||||||||||||||||||||||
| 1.4 | 114.486 | 0 | 4914.99 | −6541.53 | 0 | 3023.86 | −375171. | 0 | −748912. | |||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(3\) | \( +1 \) |
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 | 27.14.a.b | ✓ | 4 |
| 3.b | odd | 2 | 1 | 27.14.a.e | yes | 4 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 27.14.a.b | ✓ | 4 | 1.a | even | 1 | 1 | trivial |
| 27.14.a.e | yes | 4 | 3.b | odd | 2 | 1 | |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{4} + 117T_{2}^{3} - 13590T_{2}^{2} - 1279800T_{2} - 22716288 \)
acting on \(S_{14}^{\mathrm{new}}(\Gamma_0(27))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} + 117 T^{3} + \cdots - 22716288 \)
|
| $3$ |
\( T^{4} \)
|
| $5$ |
\( T^{4} + \cdots - 67\!\cdots\!75 \)
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| $7$ |
\( T^{4} + \cdots + 36\!\cdots\!25 \)
|
| $11$ |
\( T^{4} + \cdots + 45\!\cdots\!25 \)
|
| $13$ |
\( T^{4} + \cdots + 61\!\cdots\!00 \)
|
| $17$ |
\( T^{4} + \cdots + 33\!\cdots\!36 \)
|
| $19$ |
\( T^{4} + \cdots + 36\!\cdots\!08 \)
|
| $23$ |
\( T^{4} + \cdots - 27\!\cdots\!84 \)
|
| $29$ |
\( T^{4} + \cdots - 10\!\cdots\!00 \)
|
| $31$ |
\( T^{4} + \cdots + 66\!\cdots\!01 \)
|
| $37$ |
\( T^{4} + \cdots + 51\!\cdots\!00 \)
|
| $41$ |
\( T^{4} + \cdots - 65\!\cdots\!00 \)
|
| $43$ |
\( T^{4} + \cdots - 54\!\cdots\!00 \)
|
| $47$ |
\( T^{4} + \cdots - 12\!\cdots\!40 \)
|
| $53$ |
\( T^{4} + \cdots - 90\!\cdots\!11 \)
|
| $59$ |
\( T^{4} + \cdots - 94\!\cdots\!00 \)
|
| $61$ |
\( T^{4} + \cdots + 13\!\cdots\!52 \)
|
| $67$ |
\( T^{4} + \cdots - 23\!\cdots\!00 \)
|
| $71$ |
\( T^{4} + \cdots - 14\!\cdots\!00 \)
|
| $73$ |
\( T^{4} + \cdots - 16\!\cdots\!75 \)
|
| $79$ |
\( T^{4} + \cdots - 38\!\cdots\!16 \)
|
| $83$ |
\( T^{4} + \cdots - 34\!\cdots\!19 \)
|
| $89$ |
\( T^{4} + \cdots + 23\!\cdots\!00 \)
|
| $97$ |
\( T^{4} + \cdots - 98\!\cdots\!75 \)
|
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