Newspace parameters
| Level: | \( N \) | \(=\) | \( 9 = 3^{2} \) |
| Weight: | \( k \) | \(=\) | \( 38 \) |
| Character orbit: | \([\chi]\) | \(=\) | 9.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(78.0426343121\) |
| Analytic rank: | \(0\) |
| 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: |
\( x^{3} - x^{2} - 2495042360x + 9241471873200 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 2^{9}\cdot 3^{5}\cdot 5 \) |
| Twist minimal: | no (minimal twist has level 3) |
| 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
\( x^{3} - x^{2} - 2495042360x + 9241471873200 \)
:
| \(\beta_{1}\) | \(=\) |
\( 12\nu - 4 \)
|
| \(\beta_{2}\) | \(=\) |
\( 72\nu^{2} + 399936\nu - 119762166616 \)
|
| \(\nu\) | \(=\) |
\( ( \beta _1 + 4 ) / 12 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( \beta_{2} - 33328\beta _1 + 119762033304 ) / 72 \)
|
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 |
|
−516864. | 0 | 1.29709e11 | −3.13693e12 | 0 | −3.10972e15 | 3.99525e15 | 0 | 1.62136e18 | |||||||||||||||||||||||||||
| 1.2 | 148328. | 0 | −1.15438e11 | −7.60742e11 | 0 | 2.59260e15 | −3.75086e16 | 0 | −1.12839e17 | ||||||||||||||||||||||||||||
| 1.3 | 679444. | 0 | 3.24205e11 | 1.35264e13 | 0 | −4.10476e15 | 1.26897e17 | 0 | 9.19042e18 | ||||||||||||||||||||||||||||
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 | 9.38.a.b | 3 | |
| 3.b | odd | 2 | 1 | 3.38.a.a | ✓ | 3 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 3.38.a.a | ✓ | 3 | 3.b | odd | 2 | 1 | |
| 9.38.a.b | 3 | 1.a | even | 1 | 1 | trivial | |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{3} - 310908T_{2}^{2} - 327064838400T_{2} + 52089706281959424 \)
acting on \(S_{38}^{\mathrm{new}}(\Gamma_0(9))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{3} + \cdots + 52\!\cdots\!24 \)
|
| $3$ |
\( T^{3} \)
|
| $5$ |
\( T^{3} + \cdots - 32\!\cdots\!00 \)
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| $7$ |
\( T^{3} + \cdots - 33\!\cdots\!80 \)
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| $11$ |
\( T^{3} + \cdots + 31\!\cdots\!04 \)
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| $13$ |
\( T^{3} + \cdots + 15\!\cdots\!08 \)
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| $17$ |
\( T^{3} + \cdots + 72\!\cdots\!92 \)
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| $19$ |
\( T^{3} + \cdots + 83\!\cdots\!00 \)
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| $23$ |
\( T^{3} + \cdots + 88\!\cdots\!20 \)
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| $29$ |
\( T^{3} + \cdots - 24\!\cdots\!00 \)
|
| $31$ |
\( T^{3} + \cdots + 58\!\cdots\!00 \)
|
| $37$ |
\( T^{3} + \cdots + 13\!\cdots\!60 \)
|
| $41$ |
\( T^{3} + \cdots + 28\!\cdots\!80 \)
|
| $43$ |
\( T^{3} + \cdots + 10\!\cdots\!44 \)
|
| $47$ |
\( T^{3} + \cdots + 13\!\cdots\!08 \)
|
| $53$ |
\( T^{3} + \cdots - 46\!\cdots\!20 \)
|
| $59$ |
\( T^{3} + \cdots + 61\!\cdots\!00 \)
|
| $61$ |
\( T^{3} + \cdots - 25\!\cdots\!12 \)
|
| $67$ |
\( T^{3} + \cdots + 88\!\cdots\!08 \)
|
| $71$ |
\( T^{3} + \cdots + 49\!\cdots\!08 \)
|
| $73$ |
\( T^{3} + \cdots + 45\!\cdots\!00 \)
|
| $79$ |
\( T^{3} + \cdots + 12\!\cdots\!00 \)
|
| $83$ |
\( T^{3} + \cdots + 58\!\cdots\!48 \)
|
| $89$ |
\( T^{3} + \cdots + 50\!\cdots\!00 \)
|
| $97$ |
\( T^{3} + \cdots + 41\!\cdots\!28 \)
|
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