Newspace parameters
| Level: | \( N \) | \(=\) | \( 462 = 2 \cdot 3 \cdot 7 \cdot 11 \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 462.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(27.2588824227\) |
| Analytic rank: | \(0\) |
| Dimension: | \(3\) |
| Coefficient field: | 3.3.843032.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{3} - x^{2} - 192x + 1028 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 2 \) |
| 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
\( x^{3} - x^{2} - 192x + 1028 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} + 7\nu - 131 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} - 7\beta _1 + 131 \)
|
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 |
|
2.00000 | 3.00000 | 4.00000 | −11.5708 | 6.00000 | 7.00000 | 8.00000 | 9.00000 | −23.1416 | |||||||||||||||||||||||||||
| 1.2 | 2.00000 | 3.00000 | 4.00000 | 10.6646 | 6.00000 | 7.00000 | 8.00000 | 9.00000 | 21.3292 | ||||||||||||||||||||||||||||
| 1.3 | 2.00000 | 3.00000 | 4.00000 | 13.9062 | 6.00000 | 7.00000 | 8.00000 | 9.00000 | 27.8124 | ||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \(-1\) |
| \(3\) | \(-1\) |
| \(7\) | \(-1\) |
| \(11\) | \(-1\) |
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 | 462.4.a.s | ✓ | 3 |
| 3.b | odd | 2 | 1 | 1386.4.a.bd | 3 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 462.4.a.s | ✓ | 3 | 1.a | even | 1 | 1 | trivial |
| 1386.4.a.bd | 3 | 3.b | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{3} - 13T_{5}^{2} - 136T_{5} + 1716 \)
acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(462))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T - 2)^{3} \)
|
| $3$ |
\( (T - 3)^{3} \)
|
| $5$ |
\( T^{3} - 13 T^{2} + \cdots + 1716 \)
|
| $7$ |
\( (T - 7)^{3} \)
|
| $11$ |
\( (T - 11)^{3} \)
|
| $13$ |
\( T^{3} - 33 T^{2} + \cdots + 37772 \)
|
| $17$ |
\( T^{3} - 70 T^{2} + \cdots + 786984 \)
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| $19$ |
\( T^{3} - 37 T^{2} + \cdots + 136808 \)
|
| $23$ |
\( T^{3} - 64 T^{2} + \cdots + 147840 \)
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| $29$ |
\( T^{3} - 45 T^{2} + \cdots + 3169500 \)
|
| $31$ |
\( T^{3} - 126 T^{2} + \cdots + 1383856 \)
|
| $37$ |
\( T^{3} - 61 T^{2} + \cdots + 1356476 \)
|
| $41$ |
\( T^{3} - 8 T^{2} + \cdots - 713328 \)
|
| $43$ |
\( T^{3} - 80 T^{2} + \cdots - 5722112 \)
|
| $47$ |
\( T^{3} - 19 T^{2} + \cdots + 4638600 \)
|
| $53$ |
\( T^{3} - 112 T^{2} + \cdots - 57465648 \)
|
| $59$ |
\( T^{3} - 95 T^{2} + \cdots + 4158000 \)
|
| $61$ |
\( T^{3} - 50 T^{2} + \cdots - 47052728 \)
|
| $67$ |
\( T^{3} + 239 T^{2} + \cdots - 76288304 \)
|
| $71$ |
\( T^{3} + 268 T^{2} + \cdots + 199473984 \)
|
| $73$ |
\( T^{3} - 271 T^{2} + \cdots + 163495580 \)
|
| $79$ |
\( T^{3} + 88 T^{2} + \cdots - 77425920 \)
|
| $83$ |
\( T^{3} + 262 T^{2} + \cdots - 66268944 \)
|
| $89$ |
\( T^{3} + 662 T^{2} + \cdots - 28363608 \)
|
| $97$ |
\( T^{3} + 12 T^{2} + \cdots - 992528320 \)
|
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