Newspace parameters
| Level: | \( N \) | \(=\) | \( 79 \) |
| Weight: | \( k \) | \(=\) | \( 9 \) |
| Character orbit: | \([\chi]\) | \(=\) | 79.b (of order \(2\), degree \(1\), minimal) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(32.1829101948\) |
| 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: | \( 3^{2}\cdot 7^{2} \) |
| 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
\( x^{5} - 10x^{3} + 20x - 7 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu^{4} - 8\nu^{2} + 8 \)
|
| \(\beta_{2}\) | \(=\) |
\( 3\nu^{2} - 12 \)
|
| \(\beta_{3}\) | \(=\) |
\( 7\nu^{3} - 5\nu^{2} - 42\nu + 20 \)
|
| \(\beta_{4}\) | \(=\) |
\( -5\nu^{4} + 40\nu^{2} + 21\nu - 40 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{4} + 5\beta_1 ) / 21 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( \beta_{2} + 12 ) / 3 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 6\beta_{4} + 3\beta_{3} + 5\beta_{2} + 30\beta_1 ) / 21 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( 8\beta_{2} + 3\beta _1 + 72 ) / 3 \)
|
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 |
|
−28.9649 | 0 | 582.964 | −252.771 | 0 | 0 | −9470.48 | 6561.00 | 7321.48 | |||||||||||||||||||||||||||||||||
| 78.2 | −21.8877 | 0 | 223.070 | 924.049 | 0 | 0 | 720.770 | 6561.00 | −20225.3 | ||||||||||||||||||||||||||||||||||
| 78.3 | 3.98638 | 0 | −240.109 | −515.057 | 0 | 0 | −1977.68 | 6561.00 | −2053.21 | ||||||||||||||||||||||||||||||||||
| 78.4 | 15.4376 | 0 | −17.6816 | −1242.37 | 0 | 0 | −4224.98 | 6561.00 | −19179.2 | ||||||||||||||||||||||||||||||||||
| 78.5 | 31.4286 | 0 | 731.757 | 1086.15 | 0 | 0 | 14952.4 | 6561.00 | 34136.2 | ||||||||||||||||||||||||||||||||||
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 | 79.9.b.a | ✓ | 5 |
| 79.b | odd | 2 | 1 | CM | 79.9.b.a | ✓ | 5 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 79.9.b.a | ✓ | 5 | 1.a | even | 1 | 1 | trivial |
| 79.9.b.a | ✓ | 5 | 79.b | odd | 2 | 1 | CM |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{5} - 1280T_{2}^{3} + 327680T_{2} - 1226177 \)
acting on \(S_{9}^{\mathrm{new}}(79, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{5} - 1280 T^{3} + \cdots - 1226177 \)
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| $3$ |
\( T^{5} \)
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| $5$ |
\( T^{5} + \cdots + 162337197810334 \)
|
| $7$ |
\( T^{5} \)
|
| $11$ |
\( T^{5} + \cdots - 22\!\cdots\!02 \)
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| $13$ |
\( T^{5} + \cdots - 29\!\cdots\!02 \)
|
| $17$ |
\( T^{5} \)
|
| $19$ |
\( T^{5} + \cdots - 59\!\cdots\!02 \)
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| $23$ |
\( T^{5} + \cdots + 23\!\cdots\!98 \)
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| $29$ |
\( T^{5} \)
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| $31$ |
\( T^{5} + \cdots - 13\!\cdots\!02 \)
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| $37$ |
\( T^{5} \)
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| $41$ |
\( T^{5} \)
|
| $43$ |
\( T^{5} \)
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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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| $61$ |
\( T^{5} \)
|
| $67$ |
\( T^{5} + \cdots + 42\!\cdots\!98 \)
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| $71$ |
\( T^{5} \)
|
| $73$ |
\( T^{5} + \cdots - 36\!\cdots\!02 \)
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| $79$ |
\( (T - 38950081)^{5} \)
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| $83$ |
\( (T + 18843358)^{5} \)
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| $89$ |
\( T^{5} + \cdots + 14\!\cdots\!98 \)
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| $97$ |
\( T^{5} + \cdots + 98\!\cdots\!98 \)
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