Newspace parameters
| Level: | \( N \) | \(=\) | \( 8208 = 2^{4} \cdot 3^{3} \cdot 19 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 8208.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(65.5412099791\) |
| Analytic rank: | \(1\) |
| Dimension: | \(3\) |
| Coefficient field: | 3.3.321.1 |
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|
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| Defining polynomial: |
\( x^{3} - x^{2} - 4x + 1 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 513) |
| 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
\( x^{3} - x^{2} - 4x + 1 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - \nu - 3 \)
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| \(\nu\) | \(=\) |
\( \beta_1 \)
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| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + \beta _1 + 3 \)
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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 | 0 | 0 | −0.460505 | 0 | 0.593579 | 0 | 0 | 0 | |||||||||||||||||||||||||||
| 1.2 | 0 | 0 | 0 | 1.76088 | 0 | −3.18194 | 0 | 0 | 0 | ||||||||||||||||||||||||||||
| 1.3 | 0 | 0 | 0 | 3.69963 | 0 | 1.58836 | 0 | 0 | 0 | ||||||||||||||||||||||||||||
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 | 8208.2.a.bp | 3 | |
| 3.b | odd | 2 | 1 | 8208.2.a.bf | 3 | ||
| 4.b | odd | 2 | 1 | 513.2.a.f | yes | 3 | |
| 12.b | even | 2 | 1 | 513.2.a.e | ✓ | 3 | |
| 76.d | even | 2 | 1 | 9747.2.a.x | 3 | ||
| 228.b | odd | 2 | 1 | 9747.2.a.ba | 3 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 513.2.a.e | ✓ | 3 | 12.b | even | 2 | 1 | |
| 513.2.a.f | yes | 3 | 4.b | odd | 2 | 1 | |
| 8208.2.a.bf | 3 | 3.b | odd | 2 | 1 | ||
| 8208.2.a.bp | 3 | 1.a | even | 1 | 1 | trivial | |
| 9747.2.a.x | 3 | 76.d | even | 2 | 1 | ||
| 9747.2.a.ba | 3 | 228.b | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(8208))\):
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\( T_{5}^{3} - 5T_{5}^{2} + 4T_{5} + 3 \)
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\( T_{7}^{3} + T_{7}^{2} - 6T_{7} + 3 \)
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\( T_{11}^{3} + 8T_{11}^{2} + 15T_{11} - 3 \)
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Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{3} \)
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| $3$ |
\( T^{3} \)
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| $5$ |
\( T^{3} - 5 T^{2} + \cdots + 3 \)
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| $7$ |
\( T^{3} + T^{2} - 6T + 3 \)
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| $11$ |
\( T^{3} + 8 T^{2} + \cdots - 3 \)
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| $13$ |
\( T^{3} + T^{2} - 6T + 3 \)
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| $17$ |
\( T^{3} - 7 T^{2} + \cdots + 207 \)
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| $19$ |
\( (T - 1)^{3} \)
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| $23$ |
\( T^{3} + 9 T^{2} + \cdots - 101 \)
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| $29$ |
\( T^{3} + 6 T^{2} + \cdots - 569 \)
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| $31$ |
\( T^{3} - 2 T^{2} + \cdots + 9 \)
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| $37$ |
\( T^{3} + 11 T^{2} + \cdots - 9 \)
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| $41$ |
\( T^{3} - 5 T^{2} + \cdots + 3 \)
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| $43$ |
\( T^{3} + 9 T^{2} + \cdots - 811 \)
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| $47$ |
\( (T + 9)^{3} \)
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| $53$ |
\( T^{3} - 2 T^{2} + \cdots + 837 \)
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| $59$ |
\( T^{3} - T^{2} + \cdots - 21 \)
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| $61$ |
\( T^{3} - 7 T^{2} + \cdots + 1 \)
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| $67$ |
\( T^{3} - 12 T^{2} + \cdots + 707 \)
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| $71$ |
\( T^{3} - 69T + 189 \)
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| $73$ |
\( T^{3} + 3 T^{2} + \cdots - 43 \)
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| $79$ |
\( T^{3} + 7 T^{2} + \cdots + 81 \)
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| $83$ |
\( T^{3} + 5 T^{2} + \cdots + 531 \)
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| $89$ |
\( T^{3} + 4 T^{2} + \cdots + 99 \)
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| $97$ |
\( T^{3} + 23 T^{2} + \cdots - 101 \)
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