Newspace parameters
| Level: | \( N \) | \(=\) | \( 27 = 3^{3} \) |
| Weight: | \( k \) | \(=\) | \( 12 \) |
| Character orbit: | \([\chi]\) | \(=\) | 27.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(20.7452658751\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{4} - \cdots)\) |
|
|
|
| Defining polynomial: |
\( x^{4} - x^{3} - 793x^{2} + 3505x + 73960 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 2^{2}\cdot 3^{8} \) |
| 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
\( x^{4} - x^{3} - 793x^{2} + 3505x + 73960 \)
:
| \(\beta_{1}\) | \(=\) |
\( 3\nu - 1 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 27\nu^{3} + 216\nu^{2} - 15051\nu - 27016 ) / 16 \)
|
| \(\beta_{3}\) | \(=\) |
\( 9\nu^{2} + 48\nu - 3583 \)
|
| \(\nu\) | \(=\) |
\( ( \beta _1 + 1 ) / 3 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( \beta_{3} - 16\beta _1 + 3567 ) / 9 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( -24\beta_{3} + 16\beta_{2} + 5401\beta _1 - 53575 ) / 27 \)
|
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 |
|
−65.3950 | 0 | 2228.50 | 12582.1 | 0 | 52013.1 | −11803.9 | 0 | −822804. | ||||||||||||||||||||||||||||||
| 1.2 | −40.2029 | 0 | −431.728 | −8008.14 | 0 | −73745.9 | 99692.2 | 0 | 321950. | |||||||||||||||||||||||||||||||
| 1.3 | 26.9518 | 0 | −1321.60 | −1017.63 | 0 | 23889.2 | −90816.8 | 0 | −27426.9 | |||||||||||||||||||||||||||||||
| 1.4 | 87.6460 | 0 | 5633.83 | 8503.70 | 0 | −14232.4 | 314283. | 0 | 745315. | |||||||||||||||||||||||||||||||
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 | 27.12.a.e | yes | 4 |
| 3.b | odd | 2 | 1 | 27.12.a.c | ✓ | 4 | |
| 9.c | even | 3 | 2 | 81.12.c.h | 8 | ||
| 9.d | odd | 6 | 2 | 81.12.c.j | 8 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 27.12.a.c | ✓ | 4 | 3.b | odd | 2 | 1 | |
| 27.12.a.e | yes | 4 | 1.a | even | 1 | 1 | trivial |
| 81.12.c.h | 8 | 9.c | even | 3 | 2 | ||
| 81.12.c.j | 8 | 9.d | odd | 6 | 2 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{4} - 9T_{2}^{3} - 7110T_{2}^{2} - 51840T_{2} + 6210432 \)
acting on \(S_{12}^{\mathrm{new}}(\Gamma_0(27))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} - 9 T^{3} + \cdots + 6210432 \)
|
| $3$ |
\( T^{4} \)
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| $5$ |
\( T^{4} + \cdots + 871926684580125 \)
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| $7$ |
\( T^{4} + \cdots + 13\!\cdots\!25 \)
|
| $11$ |
\( T^{4} + \cdots + 13\!\cdots\!25 \)
|
| $13$ |
\( T^{4} + \cdots - 20\!\cdots\!00 \)
|
| $17$ |
\( T^{4} + \cdots - 69\!\cdots\!84 \)
|
| $19$ |
\( T^{4} + \cdots - 16\!\cdots\!92 \)
|
| $23$ |
\( T^{4} + \cdots - 13\!\cdots\!84 \)
|
| $29$ |
\( T^{4} + \cdots + 63\!\cdots\!00 \)
|
| $31$ |
\( T^{4} + \cdots + 12\!\cdots\!61 \)
|
| $37$ |
\( T^{4} + \cdots - 30\!\cdots\!00 \)
|
| $41$ |
\( T^{4} + \cdots + 26\!\cdots\!00 \)
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| $43$ |
\( T^{4} + \cdots - 72\!\cdots\!00 \)
|
| $47$ |
\( T^{4} + \cdots + 41\!\cdots\!40 \)
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| $53$ |
\( T^{4} + \cdots - 16\!\cdots\!31 \)
|
| $59$ |
\( T^{4} + \cdots - 26\!\cdots\!00 \)
|
| $61$ |
\( T^{4} + \cdots + 19\!\cdots\!32 \)
|
| $67$ |
\( T^{4} + \cdots + 43\!\cdots\!00 \)
|
| $71$ |
\( T^{4} + \cdots - 61\!\cdots\!00 \)
|
| $73$ |
\( T^{4} + \cdots + 73\!\cdots\!25 \)
|
| $79$ |
\( T^{4} + \cdots + 10\!\cdots\!44 \)
|
| $83$ |
\( T^{4} + \cdots - 26\!\cdots\!79 \)
|
| $89$ |
\( T^{4} + \cdots - 10\!\cdots\!00 \)
|
| $97$ |
\( T^{4} + \cdots - 12\!\cdots\!75 \)
|
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