Newspace parameters
| Level: | \( N \) | \(=\) | \( 6724 = 2^{2} \cdot 41^{2} \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 6724.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(53.6914103191\) |
| Analytic rank: | \(1\) |
| Dimension: | \(3\) |
| Coefficient field: | 3.3.785.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{3} - x^{2} - 6x + 5 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | yes |
| 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
\( x^{3} - x^{2} - 6x + 5 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - 4 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + 4 \)
|
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 |
|
0 | −3.38849 | 0 | −3.38849 | 0 | 3.70488 | 0 | 8.48186 | 0 | |||||||||||||||||||||||||||
| 1.2 | 0 | −0.187284 | 0 | −0.187284 | 0 | −1.33949 | 0 | −2.96492 | 0 | ||||||||||||||||||||||||||||
| 1.3 | 0 | 1.57577 | 0 | 1.57577 | 0 | 4.63461 | 0 | −0.516938 | 0 | ||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \(-1\) |
| \(41\) | \(-1\) |
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 | 6724.2.a.a | ✓ | 3 |
| 41.b | even | 2 | 1 | 6724.2.a.b | yes | 3 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 6724.2.a.a | ✓ | 3 | 1.a | even | 1 | 1 | trivial |
| 6724.2.a.b | yes | 3 | 41.b | even | 2 | 1 | |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{3} + 2T_{3}^{2} - 5T_{3} - 1 \)
acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(6724))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{3} \)
|
| $3$ |
\( T^{3} + 2 T^{2} + \cdots - 1 \)
|
| $5$ |
\( T^{3} + 2 T^{2} + \cdots - 1 \)
|
| $7$ |
\( T^{3} - 7 T^{2} + \cdots + 23 \)
|
| $11$ |
\( T^{3} + 2 T^{2} + \cdots + 9 \)
|
| $13$ |
\( T^{3} + 2 T^{2} + \cdots + 9 \)
|
| $17$ |
\( T^{3} + 7 T^{2} + \cdots - 23 \)
|
| $19$ |
\( T^{3} - T^{2} + \cdots + 15 \)
|
| $23$ |
\( T^{3} + 3 T^{2} + \cdots - 47 \)
|
| $29$ |
\( T^{3} + 4T^{2} - T - 9 \)
|
| $31$ |
\( T^{3} - 2 T^{2} + \cdots + 40 \)
|
| $37$ |
\( T^{3} - 2 T^{2} + \cdots - 9 \)
|
| $41$ |
\( T^{3} \)
|
| $43$ |
\( T^{3} + 7 T^{2} + \cdots - 411 \)
|
| $47$ |
\( T^{3} + 15 T^{2} + \cdots - 125 \)
|
| $53$ |
\( T^{3} + T^{2} + \cdots - 117 \)
|
| $59$ |
\( T^{3} - 3 T^{2} + \cdots + 569 \)
|
| $61$ |
\( T^{3} - 16 T^{2} + \cdots - 45 \)
|
| $67$ |
\( T^{3} + 19 T^{2} + \cdots + 131 \)
|
| $71$ |
\( T^{3} + 2 T^{2} + \cdots - 40 \)
|
| $73$ |
\( T^{3} - 9 T^{2} + \cdots + 685 \)
|
| $79$ |
\( T^{3} - T^{2} + \cdots + 117 \)
|
| $83$ |
\( T^{3} + 3 T^{2} + \cdots - 15 \)
|
| $89$ |
\( T^{3} - 13 T^{2} + \cdots + 557 \)
|
| $97$ |
\( T^{3} - 17 T^{2} + \cdots - 135 \)
|
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