Newspace parameters
| Level: | \( N \) | \(=\) | \( 79 \) |
| Weight: | \( k \) | \(=\) | \( 5 \) |
| Character orbit: | \([\chi]\) | \(=\) | 79.b (of order \(2\), degree \(1\), minimal) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(8.16622708362\) |
| Analytic rank: | \(0\) |
| Dimension: | \(5\) |
| Coefficient field: | 5.5.19503125.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{5} - 10x^{3} + 20x - 7 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 3^{2} \) |
| Twist minimal: | yes |
| Sato-Tate group: | $\mathrm{U}(1)[D_{2}]$ |
$q$-expansion
Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3,\beta_4\) 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
\( x^{5} - 10x^{3} + 20x - 7 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu^{3} + \nu^{2} - 6\nu - 4 \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{4} - 8\nu^{2} + 8 \)
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| \(\beta_{3}\) | \(=\) |
\( -\nu^{3} + 2\nu^{2} + 6\nu - 8 \)
|
| \(\beta_{4}\) | \(=\) |
\( \nu^{4} - 8\nu^{2} - 3\nu + 8 \)
|
| \(\nu\) | \(=\) |
\( ( -\beta_{4} + \beta_{2} ) / 3 \)
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| \(\nu^{2}\) | \(=\) |
\( ( \beta_{3} + \beta _1 + 12 ) / 3 \)
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| \(\nu^{3}\) | \(=\) |
\( ( -6\beta_{4} - \beta_{3} + 6\beta_{2} + 2\beta_1 ) / 3 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( 8\beta_{3} + 3\beta_{2} + 8\beta _1 + 72 ) / 3 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/79\mathbb{Z}\right)^\times\).
| \(n\) | \(3\) |
| \(\chi(n)\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | |||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 78.1 |
|
−7.96421 | 0 | 47.4286 | 48.3337 | 0 | 0 | −250.304 | 81.0000 | −384.940 | |||||||||||||||||||||||||||||||||
| 78.2 | −3.17999 | 0 | −5.88766 | −46.6267 | 0 | 0 | 69.6026 | 81.0000 | 148.272 | ||||||||||||||||||||||||||||||||||
| 78.3 | −1.74216 | 0 | −12.9649 | −31.5789 | 0 | 0 | 50.4614 | 81.0000 | 55.0155 | ||||||||||||||||||||||||||||||||||
| 78.4 | 5.99886 | 0 | 19.9864 | 27.1098 | 0 | 0 | 23.9137 | 81.0000 | 162.628 | ||||||||||||||||||||||||||||||||||
| 78.5 | 6.88749 | 0 | 31.4376 | 2.76205 | 0 | 0 | 106.326 | 81.0000 | 19.0236 | ||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 79.b | odd | 2 | 1 | CM by \(\Q(\sqrt{-79}) \) |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 79.5.b.a | ✓ | 5 |
| 79.b | odd | 2 | 1 | CM | 79.5.b.a | ✓ | 5 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 79.5.b.a | ✓ | 5 | 1.a | even | 1 | 1 | trivial |
| 79.5.b.a | ✓ | 5 | 79.b | odd | 2 | 1 | CM |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{5} - 80T_{2}^{3} + 1280T_{2} + 1823 \)
acting on \(S_{5}^{\mathrm{new}}(79, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{5} - 80 T^{3} + \cdots + 1823 \)
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| $3$ |
\( T^{5} \)
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| $5$ |
\( T^{5} - 3125 T^{3} + \cdots - 5328946 \)
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| $7$ |
\( T^{5} \)
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| $11$ |
\( T^{5} + \cdots + 39664157198 \)
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| $13$ |
\( T^{5} + \cdots - 258951080402 \)
|
| $17$ |
\( T^{5} \)
|
| $19$ |
\( T^{5} + \cdots + 11611041677998 \)
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| $23$ |
\( T^{5} + \cdots + 33352932930398 \)
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| $29$ |
\( T^{5} \)
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| $31$ |
\( T^{5} + \cdots + 16\!\cdots\!98 \)
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| $37$ |
\( T^{5} \)
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| $41$ |
\( T^{5} \)
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| $43$ |
\( T^{5} \)
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| $47$ |
\( T^{5} \)
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| $53$ |
\( T^{5} \)
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| $59$ |
\( T^{5} \)
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| $61$ |
\( T^{5} \)
|
| $67$ |
\( T^{5} + \cdots - 15\!\cdots\!02 \)
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| $71$ |
\( T^{5} \)
|
| $73$ |
\( T^{5} + \cdots - 85\!\cdots\!02 \)
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| $79$ |
\( (T - 6241)^{5} \)
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| $83$ |
\( (T - 8722)^{5} \)
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| $89$ |
\( T^{5} + \cdots + 21\!\cdots\!98 \)
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| $97$ |
\( T^{5} + \cdots - 31\!\cdots\!02 \)
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