Newspace parameters
| Level: | \( N \) | \(=\) | \( 9 = 3^{2} \) |
| Weight: | \( k \) | \(=\) | \( 34 \) |
| Character orbit: | \([\chi]\) | \(=\) | 9.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(62.0845459929\) |
| 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: |
\( x^{3} - x^{2} - 5357605x + 842871622 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 2^{11}\cdot 3^{6}\cdot 11 \) |
| 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
\( x^{3} - x^{2} - 5357605x + 842871622 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( -4\nu^{2} + 1788\nu + 14286352 ) / 105 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 14876\nu^{2} + 41250588\nu - 53146909808 ) / 105 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{2} + 3719\beta _1 + 152064 ) / 456192 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( 149\beta_{2} - 3437549\beta _1 + 543132615168 ) / 152064 \)
|
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 |
|
−167611. | 0 | 1.95036e10 | 4.35148e11 | 0 | −1.26073e14 | −1.82925e15 | 0 | −7.29356e16 | |||||||||||||||||||||||||||
| 1.2 | −61269.1 | 0 | −4.83603e9 | −2.12383e11 | 0 | 1.58203e14 | 8.22597e14 | 0 | 1.30125e16 | ||||||||||||||||||||||||||||
| 1.3 | 92260.3 | 0 | −7.79766e7 | 3.77234e10 | 0 | −2.13696e13 | −7.99704e14 | 0 | 3.48037e15 | ||||||||||||||||||||||||||||
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 | 9.34.a.c | 3 | |
| 3.b | odd | 2 | 1 | 3.34.a.b | ✓ | 3 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 3.34.a.b | ✓ | 3 | 3.b | odd | 2 | 1 | |
| 9.34.a.c | 3 | 1.a | even | 1 | 1 | trivial | |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{3} + 136620T_{2}^{2} - 10847167488T_{2} - 947456640811008 \)
acting on \(S_{34}^{\mathrm{new}}(\Gamma_0(9))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{3} + \cdots - 947456640811008 \)
|
| $3$ |
\( T^{3} \)
|
| $5$ |
\( T^{3} + \cdots + 34\!\cdots\!00 \)
|
| $7$ |
\( T^{3} + \cdots - 42\!\cdots\!00 \)
|
| $11$ |
\( T^{3} + \cdots + 68\!\cdots\!92 \)
|
| $13$ |
\( T^{3} + \cdots + 13\!\cdots\!36 \)
|
| $17$ |
\( T^{3} + \cdots + 25\!\cdots\!48 \)
|
| $19$ |
\( T^{3} + \cdots - 57\!\cdots\!20 \)
|
| $23$ |
\( T^{3} + \cdots + 43\!\cdots\!00 \)
|
| $29$ |
\( T^{3} + \cdots - 87\!\cdots\!00 \)
|
| $31$ |
\( T^{3} + \cdots - 34\!\cdots\!00 \)
|
| $37$ |
\( T^{3} + \cdots - 75\!\cdots\!00 \)
|
| $41$ |
\( T^{3} + \cdots + 14\!\cdots\!00 \)
|
| $43$ |
\( T^{3} + \cdots - 38\!\cdots\!96 \)
|
| $47$ |
\( T^{3} + \cdots - 14\!\cdots\!92 \)
|
| $53$ |
\( T^{3} + \cdots + 39\!\cdots\!00 \)
|
| $59$ |
\( T^{3} + \cdots + 38\!\cdots\!80 \)
|
| $61$ |
\( T^{3} + \cdots - 13\!\cdots\!08 \)
|
| $67$ |
\( T^{3} + \cdots + 13\!\cdots\!32 \)
|
| $71$ |
\( T^{3} + \cdots - 84\!\cdots\!52 \)
|
| $73$ |
\( T^{3} + \cdots + 16\!\cdots\!00 \)
|
| $79$ |
\( T^{3} + \cdots - 51\!\cdots\!00 \)
|
| $83$ |
\( T^{3} + \cdots + 16\!\cdots\!84 \)
|
| $89$ |
\( T^{3} + \cdots - 96\!\cdots\!20 \)
|
| $97$ |
\( T^{3} + \cdots - 22\!\cdots\!08 \)
|
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