Newspace parameters
| Level: | \( N \) | \(=\) | \( 342 = 2 \cdot 3^{2} \cdot 19 \) |
| Weight: | \( k \) | \(=\) | \( 10 \) |
| Character orbit: | \([\chi]\) | \(=\) | 342.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(176.142255968\) |
| 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: |
\( x^{3} - x^{2} - 14066x - 550320 \)
|
| 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
\( x^{3} - x^{2} - 14066x - 550320 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( -\nu^{2} + 233\nu + 9300 ) / 30 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{2} - 53\nu - 9360 ) / 30 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{2} + \beta _1 + 2 ) / 6 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( 233\beta_{2} + 53\beta _1 + 56266 ) / 6 \)
|
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 |
|
16.0000 | 0 | 256.000 | −941.371 | 0 | 3613.21 | 4096.00 | 0 | −15061.9 | |||||||||||||||||||||||||||
| 1.2 | 16.0000 | 0 | 256.000 | 1453.78 | 0 | 6607.88 | 4096.00 | 0 | 23260.5 | ||||||||||||||||||||||||||||
| 1.3 | 16.0000 | 0 | 256.000 | 1692.59 | 0 | −7806.09 | 4096.00 | 0 | 27081.4 | ||||||||||||||||||||||||||||
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 | 342.10.a.h | 3 | |
| 3.b | odd | 2 | 1 | 114.10.a.b | ✓ | 3 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 114.10.a.b | ✓ | 3 | 3.b | odd | 2 | 1 | |
| 342.10.a.h | 3 | 1.a | even | 1 | 1 | trivial | |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{3} - 2205T_{5}^{2} - 501246T_{5} + 2316389320 \)
acting on \(S_{10}^{\mathrm{new}}(\Gamma_0(342))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T - 16)^{3} \)
|
| $3$ |
\( T^{3} \)
|
| $5$ |
\( T^{3} + \cdots + 2316389320 \)
|
| $7$ |
\( T^{3} + \cdots + 186375772800 \)
|
| $11$ |
\( T^{3} + \cdots + 107848463647360 \)
|
| $13$ |
\( T^{3} + \cdots - 10\!\cdots\!92 \)
|
| $17$ |
\( T^{3} + \cdots - 16\!\cdots\!32 \)
|
| $19$ |
\( (T + 130321)^{3} \)
|
| $23$ |
\( T^{3} + \cdots - 35\!\cdots\!00 \)
|
| $29$ |
\( T^{3} + \cdots + 13\!\cdots\!80 \)
|
| $31$ |
\( T^{3} + \cdots + 10\!\cdots\!00 \)
|
| $37$ |
\( T^{3} + \cdots - 90\!\cdots\!56 \)
|
| $41$ |
\( T^{3} + \cdots + 47\!\cdots\!60 \)
|
| $43$ |
\( T^{3} + \cdots - 93\!\cdots\!68 \)
|
| $47$ |
\( T^{3} + \cdots - 87\!\cdots\!00 \)
|
| $53$ |
\( T^{3} + \cdots + 95\!\cdots\!32 \)
|
| $59$ |
\( T^{3} + \cdots + 30\!\cdots\!00 \)
|
| $61$ |
\( T^{3} + \cdots + 63\!\cdots\!88 \)
|
| $67$ |
\( T^{3} + \cdots + 23\!\cdots\!16 \)
|
| $71$ |
\( T^{3} + \cdots + 40\!\cdots\!80 \)
|
| $73$ |
\( T^{3} + \cdots + 85\!\cdots\!68 \)
|
| $79$ |
\( T^{3} + \cdots + 22\!\cdots\!80 \)
|
| $83$ |
\( T^{3} + \cdots - 34\!\cdots\!60 \)
|
| $89$ |
\( T^{3} + \cdots - 30\!\cdots\!20 \)
|
| $97$ |
\( T^{3} + \cdots + 13\!\cdots\!20 \)
|
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