Newspace parameters
| Level: | \( N \) | \(=\) | \( 27 = 3^{3} \) |
| Weight: | \( k \) | \(=\) | \( 10 \) |
| Character orbit: | \([\chi]\) | \(=\) | 27.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(13.9059675764\) |
| Analytic rank: | \(1\) |
| Dimension: | \(3\) |
| Coefficient field: | 3.3.177113.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{3} - x^{2} - 118x + 136 \)
|
| Coefficient ring: | \(\Z[a_1, a_2]\) |
| Coefficient ring index: | \( 2\cdot 3^{4} \) |
| 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 \(1,\beta_1,\beta_2\) 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
\( x^{3} - x^{2} - 118x + 136 \)
:
| \(\beta_{1}\) | \(=\) |
\( 3\nu - 1 \)
|
| \(\beta_{2}\) | \(=\) |
\( 9\nu^{2} + 6\nu - 713 \)
|
| \(\nu\) | \(=\) |
\( ( \beta _1 + 1 ) / 3 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( \beta_{2} - 2\beta _1 + 711 ) / 9 \)
|
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 |
|
−32.7960 | 0 | 563.577 | −1315.38 | 0 | 5158.54 | −1691.52 | 0 | 43139.3 | |||||||||||||||||||||||||||
| 1.2 | 3.46285 | 0 | −500.009 | 1404.01 | 0 | 1665.57 | −3504.44 | 0 | 4861.88 | ||||||||||||||||||||||||||||
| 1.3 | 32.3331 | 0 | 533.432 | −2071.63 | 0 | −10517.1 | 692.954 | 0 | −66982.2 | ||||||||||||||||||||||||||||
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 | 27.10.a.c | yes | 3 |
| 3.b | odd | 2 | 1 | 27.10.a.b | ✓ | 3 | |
| 9.c | even | 3 | 2 | 81.10.c.g | 6 | ||
| 9.d | odd | 6 | 2 | 81.10.c.h | 6 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 27.10.a.b | ✓ | 3 | 3.b | odd | 2 | 1 | |
| 27.10.a.c | yes | 3 | 1.a | even | 1 | 1 | trivial |
| 81.10.c.g | 6 | 9.c | even | 3 | 2 | ||
| 81.10.c.h | 6 | 9.d | odd | 6 | 2 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{3} - 3T_{2}^{2} - 1062T_{2} + 3672 \)
acting on \(S_{10}^{\mathrm{new}}(\Gamma_0(27))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{3} - 3 T^{2} + \cdots + 3672 \)
|
| $3$ |
\( T^{3} \)
|
| $5$ |
\( T^{3} + \cdots - 3825897975 \)
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| $7$ |
\( T^{3} + \cdots + 90362069875 \)
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| $11$ |
\( T^{3} + \cdots + 30446445345165 \)
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| $13$ |
\( T^{3} + \cdots + 955953747392320 \)
|
| $17$ |
\( T^{3} + \cdots + 78\!\cdots\!84 \)
|
| $19$ |
\( T^{3} + \cdots + 44\!\cdots\!64 \)
|
| $23$ |
\( T^{3} + \cdots - 45\!\cdots\!72 \)
|
| $29$ |
\( T^{3} + \cdots - 42\!\cdots\!80 \)
|
| $31$ |
\( T^{3} + \cdots - 62\!\cdots\!31 \)
|
| $37$ |
\( T^{3} + \cdots - 10\!\cdots\!80 \)
|
| $41$ |
\( T^{3} + \cdots + 36\!\cdots\!80 \)
|
| $43$ |
\( T^{3} + \cdots + 45\!\cdots\!60 \)
|
| $47$ |
\( T^{3} + \cdots - 10\!\cdots\!60 \)
|
| $53$ |
\( T^{3} + \cdots - 13\!\cdots\!97 \)
|
| $59$ |
\( T^{3} + \cdots - 52\!\cdots\!40 \)
|
| $61$ |
\( T^{3} + \cdots - 11\!\cdots\!48 \)
|
| $67$ |
\( T^{3} + \cdots - 19\!\cdots\!00 \)
|
| $71$ |
\( T^{3} + \cdots - 98\!\cdots\!40 \)
|
| $73$ |
\( T^{3} + \cdots - 64\!\cdots\!45 \)
|
| $79$ |
\( T^{3} + \cdots + 83\!\cdots\!08 \)
|
| $83$ |
\( T^{3} + \cdots - 21\!\cdots\!07 \)
|
| $89$ |
\( T^{3} + \cdots + 10\!\cdots\!00 \)
|
| $97$ |
\( T^{3} + \cdots - 26\!\cdots\!55 \)
|
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