Newspace parameters
| Level: | \( N \) | \(=\) | \( 9 = 3^{2} \) |
| Weight: | \( k \) | \(=\) | \( 40 \) |
| Character orbit: | \([\chi]\) | \(=\) | 9.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(86.7055962508\) |
| 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: |
\( x^{3} - 216694123x - 94580724378 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 2^{10}\cdot 3^{6}\cdot 5 \) |
| 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
\( x^{3} - 216694123x - 94580724378 \)
:
| \(\beta_{1}\) | \(=\) |
\( 72\nu \)
|
| \(\beta_{2}\) | \(=\) |
\( 324\nu^{2} - 212148\nu - 46805930568 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_1 ) / 72 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( 2\beta_{2} + 5893\beta _1 + 93611861136 ) / 648 \)
|
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 |
|
−674802. | 0 | −9.43987e10 | 4.30989e13 | 0 | 4.52904e16 | 4.34676e17 | 0 | −2.90832e19 | |||||||||||||||||||||||||||
| 1.2 | 337546. | 0 | −4.35818e11 | −2.60351e13 | 0 | −1.06972e16 | −3.32677e17 | 0 | −8.78806e18 | ||||||||||||||||||||||||||||
| 1.3 | 1.44426e6 | 0 | 1.53612e12 | −2.64208e13 | 0 | −2.07513e16 | 1.42456e18 | 0 | −3.81584e19 | ||||||||||||||||||||||||||||
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 | 9.40.a.d | 3 | |
| 3.b | odd | 2 | 1 | 3.40.a.a | ✓ | 3 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 3.40.a.a | ✓ | 3 | 3.b | odd | 2 | 1 | |
| 9.40.a.d | 3 | 1.a | even | 1 | 1 | trivial | |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{3} - 1107000T_{2}^{2} - 714859333632T_{2} + 328967845897568256 \)
acting on \(S_{40}^{\mathrm{new}}(\Gamma_0(9))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{3} + \cdots + 32\!\cdots\!56 \)
|
| $3$ |
\( T^{3} \)
|
| $5$ |
\( T^{3} + \cdots - 29\!\cdots\!00 \)
|
| $7$ |
\( T^{3} + \cdots - 10\!\cdots\!00 \)
|
| $11$ |
\( T^{3} + \cdots - 33\!\cdots\!28 \)
|
| $13$ |
\( T^{3} + \cdots - 44\!\cdots\!68 \)
|
| $17$ |
\( T^{3} + \cdots - 18\!\cdots\!16 \)
|
| $19$ |
\( T^{3} + \cdots - 12\!\cdots\!80 \)
|
| $23$ |
\( T^{3} + \cdots + 27\!\cdots\!00 \)
|
| $29$ |
\( T^{3} + \cdots + 45\!\cdots\!60 \)
|
| $31$ |
\( T^{3} + \cdots - 17\!\cdots\!00 \)
|
| $37$ |
\( T^{3} + \cdots - 23\!\cdots\!00 \)
|
| $41$ |
\( T^{3} + \cdots - 73\!\cdots\!00 \)
|
| $43$ |
\( T^{3} + \cdots + 12\!\cdots\!88 \)
|
| $47$ |
\( T^{3} + \cdots + 73\!\cdots\!64 \)
|
| $53$ |
\( T^{3} + \cdots - 42\!\cdots\!00 \)
|
| $59$ |
\( T^{3} + \cdots - 10\!\cdots\!40 \)
|
| $61$ |
\( T^{3} + \cdots + 60\!\cdots\!12 \)
|
| $67$ |
\( T^{3} + \cdots - 37\!\cdots\!64 \)
|
| $71$ |
\( T^{3} + \cdots - 63\!\cdots\!92 \)
|
| $73$ |
\( T^{3} + \cdots - 91\!\cdots\!00 \)
|
| $79$ |
\( T^{3} + \cdots + 66\!\cdots\!00 \)
|
| $83$ |
\( T^{3} + \cdots + 52\!\cdots\!28 \)
|
| $89$ |
\( T^{3} + \cdots + 67\!\cdots\!00 \)
|
| $97$ |
\( T^{3} + \cdots + 36\!\cdots\!16 \)
|
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