Newspace parameters
| Level: | \( N \) | \(=\) | \( 9 = 3^{2} \) |
| Weight: | \( k \) | \(=\) | \( 26 \) |
| Character orbit: | \([\chi]\) | \(=\) | 9.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(35.6397101957\) |
| Analytic rank: | \(0\) |
| Dimension: | \(3\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{3} - \cdots)\) |
|
|
|
| Defining polynomial: |
\( x^{3} - x^{2} - 783420x + 148321440 \)
|
| Coefficient ring: | \(\Z[a_1, a_2]\) |
| Coefficient ring index: | \( 2^{6}\cdot 3^{5}\cdot 5^{2} \) |
| 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
\( x^{3} - x^{2} - 783420x + 148321440 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( \nu^{2} + 455\nu - 522432 ) / 42 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -2227\nu^{2} + 1708315\nu + 1162548864 ) / 42 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{2} + 2227\beta _1 + 21600 ) / 64800 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( -91\beta_{2} + 341663\beta _1 + 6768753120 ) / 12960 \)
|
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 |
|
−8744.24 | 0 | 4.29073e7 | −4.98342e7 | 0 | 5.50645e10 | −8.17835e10 | 0 | 4.35762e11 | |||||||||||||||||||||||||||
| 1.2 | 1864.10 | 0 | −3.00796e7 | −6.53169e8 | 0 | −4.71716e10 | −1.18620e11 | 0 | −1.21757e12 | ||||||||||||||||||||||||||||
| 1.3 | 10558.1 | 0 | 7.79199e7 | 8.66156e8 | 0 | −1.75156e10 | 4.68416e11 | 0 | 9.14500e12 | ||||||||||||||||||||||||||||
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 | 9.26.a.c | 3 | |
| 3.b | odd | 2 | 1 | 3.26.a.b | ✓ | 3 | |
| 12.b | even | 2 | 1 | 48.26.a.i | 3 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 3.26.a.b | ✓ | 3 | 3.b | odd | 2 | 1 | |
| 9.26.a.c | 3 | 1.a | even | 1 | 1 | trivial | |
| 48.26.a.i | 3 | 12.b | even | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{3} - 3678T_{2}^{2} - 88941600T_{2} + 172099067904 \)
acting on \(S_{26}^{\mathrm{new}}(\Gamma_0(9))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{3} + \cdots + 172099067904 \)
|
| $3$ |
\( T^{3} \)
|
| $5$ |
\( T^{3} + \cdots - 28\!\cdots\!00 \)
|
| $7$ |
\( T^{3} + \cdots - 45\!\cdots\!80 \)
|
| $11$ |
\( T^{3} + \cdots + 18\!\cdots\!84 \)
|
| $13$ |
\( T^{3} + \cdots + 43\!\cdots\!48 \)
|
| $17$ |
\( T^{3} + \cdots + 75\!\cdots\!12 \)
|
| $19$ |
\( T^{3} + \cdots + 12\!\cdots\!00 \)
|
| $23$ |
\( T^{3} + \cdots - 17\!\cdots\!80 \)
|
| $29$ |
\( T^{3} + \cdots - 69\!\cdots\!00 \)
|
| $31$ |
\( T^{3} + \cdots + 15\!\cdots\!00 \)
|
| $37$ |
\( T^{3} + \cdots + 78\!\cdots\!60 \)
|
| $41$ |
\( T^{3} + \cdots + 20\!\cdots\!80 \)
|
| $43$ |
\( T^{3} + \cdots + 18\!\cdots\!44 \)
|
| $47$ |
\( T^{3} + \cdots + 12\!\cdots\!28 \)
|
| $53$ |
\( T^{3} + \cdots + 14\!\cdots\!80 \)
|
| $59$ |
\( T^{3} + \cdots + 28\!\cdots\!00 \)
|
| $61$ |
\( T^{3} + \cdots + 16\!\cdots\!48 \)
|
| $67$ |
\( T^{3} + \cdots + 19\!\cdots\!88 \)
|
| $71$ |
\( T^{3} + \cdots + 19\!\cdots\!28 \)
|
| $73$ |
\( T^{3} + \cdots - 36\!\cdots\!00 \)
|
| $79$ |
\( T^{3} + \cdots - 90\!\cdots\!00 \)
|
| $83$ |
\( T^{3} + \cdots - 45\!\cdots\!72 \)
|
| $89$ |
\( T^{3} + \cdots + 44\!\cdots\!00 \)
|
| $97$ |
\( T^{3} + \cdots - 69\!\cdots\!52 \)
|
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