Newspace parameters
| Level: | \( N \) | \(=\) | \( 324 = 2^{2} \cdot 3^{4} \) |
| Weight: | \( k \) | \(=\) | \( 8 \) |
| Character orbit: | \([\chi]\) | \(=\) | 324.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(101.212748257\) |
| Analytic rank: | \(1\) |
| Dimension: | \(3\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{3} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{3} - x^{2} - 3156x + 15456 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2^{3}\cdot 3^{2} \) |
| 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
\( x^{3} - x^{2} - 3156x + 15456 \)
:
| \(\beta_{1}\) | \(=\) |
\( 6\nu - 2 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 3\nu^{2} + 15\nu - 6318 ) / 5 \)
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| \(\nu\) | \(=\) |
\( ( \beta _1 + 2 ) / 6 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( 10\beta_{2} - 5\beta _1 + 12626 ) / 6 \)
|
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 | 0 | 0 | −369.024 | 0 | −543.064 | 0 | 0 | 0 | |||||||||||||||||||||||||||
| 1.2 | 0 | 0 | 0 | 8.56533 | 0 | 1272.25 | 0 | 0 | 0 | ||||||||||||||||||||||||||||
| 1.3 | 0 | 0 | 0 | 303.458 | 0 | −615.184 | 0 | 0 | 0 | ||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \(-1\) |
| \(3\) | \(1\) |
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 | 324.8.a.a | ✓ | 3 |
| 3.b | odd | 2 | 1 | 324.8.a.b | yes | 3 | |
| 9.c | even | 3 | 2 | 324.8.e.l | 6 | ||
| 9.d | odd | 6 | 2 | 324.8.e.k | 6 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 324.8.a.a | ✓ | 3 | 1.a | even | 1 | 1 | trivial |
| 324.8.a.b | yes | 3 | 3.b | odd | 2 | 1 | |
| 324.8.e.k | 6 | 9.d | odd | 6 | 2 | ||
| 324.8.e.l | 6 | 9.c | even | 3 | 2 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{3} + 57T_{5}^{2} - 112545T_{5} + 959175 \)
acting on \(S_{8}^{\mathrm{new}}(\Gamma_0(324))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{3} \)
|
| $3$ |
\( T^{3} \)
|
| $5$ |
\( T^{3} + 57 T^{2} + \cdots + 959175 \)
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| $7$ |
\( T^{3} - 114 T^{2} + \cdots - 425038976 \)
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| $11$ |
\( T^{3} + \cdots - 8367221376 \)
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| $13$ |
\( T^{3} + \cdots - 8072122625 \)
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| $17$ |
\( T^{3} + \cdots + 4481557482501 \)
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| $19$ |
\( T^{3} + \cdots - 26298145598912 \)
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| $23$ |
\( T^{3} + \cdots - 36525198713472 \)
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| $29$ |
\( T^{3} + \cdots + 14\!\cdots\!23 \)
|
| $31$ |
\( T^{3} + \cdots - 55\!\cdots\!96 \)
|
| $37$ |
\( T^{3} + \cdots + 15\!\cdots\!75 \)
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| $41$ |
\( T^{3} + \cdots + 17\!\cdots\!04 \)
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| $43$ |
\( T^{3} + \cdots - 31\!\cdots\!00 \)
|
| $47$ |
\( T^{3} + \cdots + 19\!\cdots\!00 \)
|
| $53$ |
\( T^{3} + \cdots - 35\!\cdots\!68 \)
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| $59$ |
\( T^{3} + \cdots - 93\!\cdots\!52 \)
|
| $61$ |
\( T^{3} + \cdots - 71\!\cdots\!25 \)
|
| $67$ |
\( T^{3} + \cdots - 43\!\cdots\!16 \)
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| $71$ |
\( T^{3} + \cdots + 43\!\cdots\!76 \)
|
| $73$ |
\( T^{3} + \cdots - 16\!\cdots\!67 \)
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| $79$ |
\( T^{3} + \cdots - 45\!\cdots\!00 \)
|
| $83$ |
\( T^{3} + \cdots - 59\!\cdots\!00 \)
|
| $89$ |
\( T^{3} + \cdots + 16\!\cdots\!77 \)
|
| $97$ |
\( T^{3} + \cdots - 20\!\cdots\!00 \)
|
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