Newspace parameters
| Level: | \( N \) | \(=\) | \( 342 = 2 \cdot 3^{2} \cdot 19 \) |
| Weight: | \( k \) | \(=\) | \( 6 \) |
| Character orbit: | \([\chi]\) | \(=\) | 342.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(54.8512663760\) |
| 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: |
\( x^{3} - 469x - 3180 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2\cdot 3 \) |
| Twist minimal: | no (minimal twist has level 114) |
| 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
\( x^{3} - 469x - 3180 \)
:
| \(\beta_{1}\) | \(=\) |
\( 3\nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - 10\nu - 313 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_1 ) / 3 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( 3\beta_{2} + 10\beta _1 + 939 ) / 3 \)
|
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 |
|
−4.00000 | 0 | 16.0000 | −95.0578 | 0 | 174.280 | −64.0000 | 0 | 380.231 | |||||||||||||||||||||||||||
| 1.2 | −4.00000 | 0 | 16.0000 | −68.3618 | 0 | −132.485 | −64.0000 | 0 | 273.447 | ||||||||||||||||||||||||||||
| 1.3 | −4.00000 | 0 | 16.0000 | 28.4196 | 0 | 83.2059 | −64.0000 | 0 | −113.679 | ||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \( +1 \) |
| \(3\) | \( -1 \) |
| \(19\) | \( -1 \) |
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 | 342.6.a.k | 3 | |
| 3.b | odd | 2 | 1 | 114.6.a.i | ✓ | 3 | |
| 12.b | even | 2 | 1 | 912.6.a.l | 3 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 114.6.a.i | ✓ | 3 | 3.b | odd | 2 | 1 | |
| 342.6.a.k | 3 | 1.a | even | 1 | 1 | trivial | |
| 912.6.a.l | 3 | 12.b | even | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{3} + 135T_{5}^{2} + 1854T_{5} - 184680 \)
acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(342))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T + 4)^{3} \)
|
| $3$ |
\( T^{3} \)
|
| $5$ |
\( T^{3} + 135 T^{2} + \cdots - 184680 \)
|
| $7$ |
\( T^{3} - 125 T^{2} + \cdots + 1921184 \)
|
| $11$ |
\( T^{3} + 221 T^{2} + \cdots - 25354512 \)
|
| $13$ |
\( T^{3} - 244 T^{2} + \cdots + 414492000 \)
|
| $17$ |
\( T^{3} - 445 T^{2} + \cdots + 316934316 \)
|
| $19$ |
\( (T - 361)^{3} \)
|
| $23$ |
\( T^{3} + \cdots + 6679070064 \)
|
| $29$ |
\( T^{3} + \cdots + 38726892720 \)
|
| $31$ |
\( T^{3} + \cdots + 74981122560 \)
|
| $37$ |
\( T^{3} + \cdots + 5055349760 \)
|
| $41$ |
\( T^{3} + \cdots - 1613836800 \)
|
| $43$ |
\( T^{3} + \cdots - 2255465600544 \)
|
| $47$ |
\( T^{3} + \cdots + 2285815727340 \)
|
| $53$ |
\( T^{3} + \cdots + 9670314783600 \)
|
| $59$ |
\( T^{3} + \cdots + 1707080878080 \)
|
| $61$ |
\( T^{3} + \cdots - 21159090875540 \)
|
| $67$ |
\( T^{3} + \cdots - 9772333248000 \)
|
| $71$ |
\( T^{3} + \cdots + 7472920674816 \)
|
| $73$ |
\( T^{3} + \cdots - 645951685732 \)
|
| $79$ |
\( T^{3} + \cdots - 4110338771200 \)
|
| $83$ |
\( T^{3} + \cdots + 38538989668512 \)
|
| $89$ |
\( T^{3} + \cdots + 4105491505056 \)
|
| $97$ |
\( T^{3} + \cdots - 890468330878040 \)
|
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