Newspace parameters
| Level: | \( N \) | \(=\) | \( 546 = 2 \cdot 3 \cdot 7 \cdot 13 \) |
| Weight: | \( k \) | \(=\) | \( 8 \) |
| Character orbit: | \([\chi]\) | \(=\) | 546.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(170.562223914\) |
| 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: |
\( x^{3} - x^{2} - 3417x + 10260 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{17}]\) |
| Coefficient ring index: | \( 2^{2}\cdot 3^{2} \) |
| 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
\( x^{3} - x^{2} - 3417x + 10260 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( \nu^{2} - 13\nu - 2274 ) / 7 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{2} + 71\nu - 2302 ) / 7 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{2} - \beta _1 + 4 ) / 12 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( 13\beta_{2} + 71\beta _1 + 27340 ) / 12 \)
|
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 |
|
8.00000 | −27.0000 | 64.0000 | −212.151 | −216.000 | 343.000 | 512.000 | 729.000 | −1697.21 | |||||||||||||||||||||||||||
| 1.2 | 8.00000 | −27.0000 | 64.0000 | 156.340 | −216.000 | 343.000 | 512.000 | 729.000 | 1250.72 | ||||||||||||||||||||||||||||
| 1.3 | 8.00000 | −27.0000 | 64.0000 | 406.810 | −216.000 | 343.000 | 512.000 | 729.000 | 3254.48 | ||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \(-1\) |
| \(3\) | \(1\) |
| \(7\) | \(-1\) |
| \(13\) | \(-1\) |
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 | 546.8.a.d | ✓ | 3 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 546.8.a.d | ✓ | 3 | 1.a | even | 1 | 1 | trivial |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{3} - 351T_{5}^{2} - 55872T_{5} + 13492980 \)
acting on \(S_{8}^{\mathrm{new}}(\Gamma_0(546))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T - 8)^{3} \)
|
| $3$ |
\( (T + 27)^{3} \)
|
| $5$ |
\( T^{3} - 351 T^{2} + \cdots + 13492980 \)
|
| $7$ |
\( (T - 343)^{3} \)
|
| $11$ |
\( T^{3} + \cdots - 1490327424 \)
|
| $13$ |
\( (T - 2197)^{3} \)
|
| $17$ |
\( T^{3} + \cdots - 10078817461200 \)
|
| $19$ |
\( T^{3} + \cdots - 7940611216688 \)
|
| $23$ |
\( T^{3} + \cdots - 91724659215984 \)
|
| $29$ |
\( T^{3} + \cdots + 309834203001300 \)
|
| $31$ |
\( T^{3} + \cdots - 13\!\cdots\!00 \)
|
| $37$ |
\( T^{3} + \cdots + 14\!\cdots\!40 \)
|
| $41$ |
\( T^{3} + \cdots - 98\!\cdots\!00 \)
|
| $43$ |
\( T^{3} + \cdots - 31\!\cdots\!84 \)
|
| $47$ |
\( T^{3} + \cdots - 81\!\cdots\!08 \)
|
| $53$ |
\( T^{3} + \cdots - 10\!\cdots\!80 \)
|
| $59$ |
\( T^{3} + \cdots + 19\!\cdots\!00 \)
|
| $61$ |
\( T^{3} + \cdots + 89\!\cdots\!20 \)
|
| $67$ |
\( T^{3} + \cdots + 30\!\cdots\!60 \)
|
| $71$ |
\( T^{3} + \cdots + 27\!\cdots\!80 \)
|
| $73$ |
\( T^{3} + \cdots + 22\!\cdots\!00 \)
|
| $79$ |
\( T^{3} + \cdots + 47\!\cdots\!56 \)
|
| $83$ |
\( T^{3} + \cdots - 10\!\cdots\!16 \)
|
| $89$ |
\( T^{3} + \cdots - 34\!\cdots\!08 \)
|
| $97$ |
\( T^{3} + \cdots + 43\!\cdots\!88 \)
|
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