Newspace parameters
| Level: | \( N \) | \(=\) | \( 243 = 3^{5} \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 243.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(1.94036476912\) |
| Analytic rank: | \(0\) |
| Dimension: | \(3\) |
| Coefficient field: | \(\Q(\zeta_{18})^+\) |
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|
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| Defining polynomial: |
\( x^{3} - 3x - 1 \)
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| Coefficient ring: | \(\Z[a_1, a_2]\) |
| Coefficient ring index: | \( 1 \) |
| 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 \(\nu = \zeta_{18} + \zeta_{18}^{-1}\):
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - 2 \)
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| \(\nu\) | \(=\) |
\( \beta_1 \)
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| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + 2 \)
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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 |
|
−0.879385 | 0 | −1.22668 | 3.87939 | 0 | −2.18479 | 2.83750 | 0 | −3.41147 | |||||||||||||||||||||||||||
| 1.2 | 1.34730 | 0 | −0.184793 | 1.65270 | 0 | 2.41147 | −2.94356 | 0 | 2.22668 | ||||||||||||||||||||||||||||
| 1.3 | 2.53209 | 0 | 4.41147 | 0.467911 | 0 | −3.22668 | 6.10607 | 0 | 1.18479 | ||||||||||||||||||||||||||||
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 | 243.2.a.f | yes | 3 |
| 3.b | odd | 2 | 1 | 243.2.a.e | ✓ | 3 | |
| 4.b | odd | 2 | 1 | 3888.2.a.bk | 3 | ||
| 5.b | even | 2 | 1 | 6075.2.a.bq | 3 | ||
| 9.c | even | 3 | 2 | 243.2.c.e | 6 | ||
| 9.d | odd | 6 | 2 | 243.2.c.f | 6 | ||
| 12.b | even | 2 | 1 | 3888.2.a.bd | 3 | ||
| 15.d | odd | 2 | 1 | 6075.2.a.bv | 3 | ||
| 27.e | even | 9 | 2 | 729.2.e.b | 6 | ||
| 27.e | even | 9 | 2 | 729.2.e.c | 6 | ||
| 27.e | even | 9 | 2 | 729.2.e.i | 6 | ||
| 27.f | odd | 18 | 2 | 729.2.e.a | 6 | ||
| 27.f | odd | 18 | 2 | 729.2.e.g | 6 | ||
| 27.f | odd | 18 | 2 | 729.2.e.h | 6 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 243.2.a.e | ✓ | 3 | 3.b | odd | 2 | 1 | |
| 243.2.a.f | yes | 3 | 1.a | even | 1 | 1 | trivial |
| 243.2.c.e | 6 | 9.c | even | 3 | 2 | ||
| 243.2.c.f | 6 | 9.d | odd | 6 | 2 | ||
| 729.2.e.a | 6 | 27.f | odd | 18 | 2 | ||
| 729.2.e.b | 6 | 27.e | even | 9 | 2 | ||
| 729.2.e.c | 6 | 27.e | even | 9 | 2 | ||
| 729.2.e.g | 6 | 27.f | odd | 18 | 2 | ||
| 729.2.e.h | 6 | 27.f | odd | 18 | 2 | ||
| 729.2.e.i | 6 | 27.e | even | 9 | 2 | ||
| 3888.2.a.bd | 3 | 12.b | even | 2 | 1 | ||
| 3888.2.a.bk | 3 | 4.b | odd | 2 | 1 | ||
| 6075.2.a.bq | 3 | 5.b | even | 2 | 1 | ||
| 6075.2.a.bv | 3 | 15.d | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(243))\):
|
\( T_{2}^{3} - 3T_{2}^{2} + 3 \)
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\( T_{7}^{3} + 3T_{7}^{2} - 6T_{7} - 17 \)
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Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{3} - 3T^{2} + 3 \)
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| $3$ |
\( T^{3} \)
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| $5$ |
\( T^{3} - 6 T^{2} + \cdots - 3 \)
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| $7$ |
\( T^{3} + 3 T^{2} + \cdots - 17 \)
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| $11$ |
\( T^{3} - 3 T^{2} + \cdots + 3 \)
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| $13$ |
\( T^{3} + 3 T^{2} + \cdots - 17 \)
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| $17$ |
\( (T - 3)^{3} \)
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| $19$ |
\( T^{3} + 3 T^{2} + \cdots + 1 \)
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| $23$ |
\( T^{3} - 6 T^{2} + \cdots + 51 \)
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| $29$ |
\( T^{3} - 12 T^{2} + \cdots + 57 \)
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| $31$ |
\( T^{3} + 12 T^{2} + \cdots + 19 \)
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| $37$ |
\( T^{3} + 3 T^{2} + \cdots + 1 \)
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| $41$ |
\( T^{3} + 3 T^{2} + \cdots - 219 \)
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| $43$ |
\( T^{3} + 12 T^{2} + \cdots + 19 \)
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| $47$ |
\( T^{3} + 6 T^{2} + \cdots - 267 \)
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| $53$ |
\( T^{3} - 18 T^{2} + \cdots - 81 \)
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| $59$ |
\( T^{3} + 21 T^{2} + \cdots + 321 \)
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| $61$ |
\( T^{3} - 6 T^{2} + \cdots - 53 \)
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| $67$ |
\( T^{3} - 6 T^{2} + \cdots + 109 \)
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| $71$ |
\( T^{3} + 9 T^{2} + \cdots - 999 \)
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| $73$ |
\( T^{3} - 6 T^{2} + \cdots + 397 \)
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| $79$ |
\( T^{3} - 6 T^{2} + \cdots - 53 \)
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| $83$ |
\( T^{3} + 6 T^{2} + \cdots - 51 \)
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| $89$ |
\( T^{3} - 189T + 999 \)
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| $97$ |
\( T^{3} - 15 T^{2} + \cdots + 19 \)
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