Newspace parameters
| Level: | \( N \) | \(=\) | \( 9 = 3^{2} \) |
| Weight: | \( k \) | \(=\) | \( 44 \) |
| Character orbit: | \([\chi]\) | \(=\) | 9.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(105.399355811\) |
| 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: |
\( x^{3} - 11258260111x - 264759545317170 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{11}]\) |
| Coefficient ring index: | \( 2^{11}\cdot 3^{5}\cdot 5\cdot 7 \) |
| Twist minimal: | no (minimal twist has level 1) |
| 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
\( x^{3} - 11258260111x - 264759545317170 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( -6\nu^{2} + 343074\nu + 45033040444 ) / 8369 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 44520\nu^{2} + 3528276360\nu - 334145160094480 ) / 8369 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{2} + 7420\beta_1 ) / 725760 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( 57179\beta_{2} - 588046060\beta _1 + 5447196572106240 ) / 725760 \)
|
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 |
|
−3.62707e6 | 0 | 4.35955e12 | −6.19839e14 | 0 | 2.58688e18 | 1.60917e19 | 0 | 2.24820e21 | |||||||||||||||||||||||||||
| 1.2 | 1.18359e6 | 0 | −7.39521e12 | 6.39116e14 | 0 | −1.72730e18 | −1.91639e19 | 0 | 7.56451e20 | ||||||||||||||||||||||||||||
| 1.3 | 4.65343e6 | 0 | 1.28583e13 | −5.54482e14 | 0 | −5.57604e17 | 1.89031e19 | 0 | −2.58024e21 | ||||||||||||||||||||||||||||
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 | 9.44.a.b | 3 | |
| 3.b | odd | 2 | 1 | 1.44.a.a | ✓ | 3 | |
| 12.b | even | 2 | 1 | 16.44.a.c | 3 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 1.44.a.a | ✓ | 3 | 3.b | odd | 2 | 1 | |
| 9.44.a.b | 3 | 1.a | even | 1 | 1 | trivial | |
| 16.44.a.c | 3 | 12.b | even | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{3} - 2209944T_{2}^{2} - 15663522502656T_{2} + 19976984434430705664 \)
acting on \(S_{44}^{\mathrm{new}}(\Gamma_0(9))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{3} + \cdots + 19\!\cdots\!64 \)
|
| $3$ |
\( T^{3} \)
|
| $5$ |
\( T^{3} + \cdots - 21\!\cdots\!00 \)
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| $7$ |
\( T^{3} + \cdots - 24\!\cdots\!84 \)
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| $11$ |
\( T^{3} + \cdots - 32\!\cdots\!32 \)
|
| $13$ |
\( T^{3} + \cdots - 50\!\cdots\!72 \)
|
| $17$ |
\( T^{3} + \cdots - 28\!\cdots\!76 \)
|
| $19$ |
\( T^{3} + \cdots - 21\!\cdots\!00 \)
|
| $23$ |
\( T^{3} + \cdots - 36\!\cdots\!48 \)
|
| $29$ |
\( T^{3} + \cdots - 73\!\cdots\!00 \)
|
| $31$ |
\( T^{3} + \cdots - 17\!\cdots\!88 \)
|
| $37$ |
\( T^{3} + \cdots + 56\!\cdots\!96 \)
|
| $41$ |
\( T^{3} + \cdots - 27\!\cdots\!52 \)
|
| $43$ |
\( T^{3} + \cdots - 49\!\cdots\!12 \)
|
| $47$ |
\( T^{3} + \cdots - 86\!\cdots\!56 \)
|
| $53$ |
\( T^{3} + \cdots + 90\!\cdots\!92 \)
|
| $59$ |
\( T^{3} + \cdots - 78\!\cdots\!00 \)
|
| $61$ |
\( T^{3} + \cdots - 87\!\cdots\!68 \)
|
| $67$ |
\( T^{3} + \cdots + 42\!\cdots\!76 \)
|
| $71$ |
\( T^{3} + \cdots - 18\!\cdots\!72 \)
|
| $73$ |
\( T^{3} + \cdots - 31\!\cdots\!52 \)
|
| $79$ |
\( T^{3} + \cdots - 26\!\cdots\!00 \)
|
| $83$ |
\( T^{3} + \cdots + 17\!\cdots\!32 \)
|
| $89$ |
\( T^{3} + \cdots + 10\!\cdots\!00 \)
|
| $97$ |
\( T^{3} + \cdots - 26\!\cdots\!44 \)
|
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