Newspace parameters
| Level: | \( N \) | \(=\) | \( 79 \) |
| Weight: | \( k \) | \(=\) | \( 3 \) |
| Character orbit: | \([\chi]\) | \(=\) | 79.b (of order \(2\), degree \(1\), minimal) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(2.15259408845\) |
| 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: | \( 1 \) |
| 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
\( x^{5} - 10x^{3} + 20x - 7 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - 4 \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{3} - \nu^{2} - 6\nu + 4 \)
|
| \(\beta_{4}\) | \(=\) |
\( \nu^{4} - 8\nu^{2} - 2\nu + 8 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + 4 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{3} + \beta_{2} + 6\beta_1 \)
|
| \(\nu^{4}\) | \(=\) |
\( \beta_{4} + 8\beta_{2} + 2\beta _1 + 24 \)
|
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 |
|
−3.85843 | 0 | 10.8875 | −7.26375 | 0 | 0 | −26.5749 | 9.00000 | 28.0267 | |||||||||||||||||||||||||||||||||
| 78.2 | −2.19545 | 0 | 0.820009 | 1.83666 | 0 | 0 | 6.98152 | 9.00000 | −4.03229 | ||||||||||||||||||||||||||||||||||
| 78.3 | −0.189190 | 0 | −3.96421 | 9.91634 | 0 | 0 | 1.50675 | 9.00000 | −1.87607 | ||||||||||||||||||||||||||||||||||
| 78.4 | 2.50157 | 0 | 2.25784 | 4.29198 | 0 | 0 | −4.35813 | 9.00000 | 10.7367 | ||||||||||||||||||||||||||||||||||
| 78.5 | 3.74151 | 0 | 9.99886 | −8.78122 | 0 | 0 | 22.4448 | 9.00000 | −32.8550 | ||||||||||||||||||||||||||||||||||
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 | 79.3.b.a | ✓ | 5 |
| 3.b | odd | 2 | 1 | 711.3.d.a | 5 | ||
| 4.b | odd | 2 | 1 | 1264.3.e.a | 5 | ||
| 79.b | odd | 2 | 1 | CM | 79.3.b.a | ✓ | 5 |
| 237.b | even | 2 | 1 | 711.3.d.a | 5 | ||
| 316.d | even | 2 | 1 | 1264.3.e.a | 5 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 79.3.b.a | ✓ | 5 | 1.a | even | 1 | 1 | trivial |
| 79.3.b.a | ✓ | 5 | 79.b | odd | 2 | 1 | CM |
| 711.3.d.a | 5 | 3.b | odd | 2 | 1 | ||
| 711.3.d.a | 5 | 237.b | even | 2 | 1 | ||
| 1264.3.e.a | 5 | 4.b | odd | 2 | 1 | ||
| 1264.3.e.a | 5 | 316.d | even | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{5} - 20T_{2}^{3} + 80T_{2} + 15 \)
acting on \(S_{3}^{\mathrm{new}}(79, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{5} - 20 T^{3} + \cdots + 15 \)
|
| $3$ |
\( T^{5} \)
|
| $5$ |
\( T^{5} - 125 T^{3} + \cdots - 4986 \)
|
| $7$ |
\( T^{5} \)
|
| $11$ |
\( T^{5} - 605 T^{3} + \cdots + 110502 \)
|
| $13$ |
\( T^{5} - 845 T^{3} + \cdots - 731210 \)
|
| $17$ |
\( T^{5} \)
|
| $19$ |
\( T^{5} - 1805 T^{3} + \cdots + 806902 \)
|
| $23$ |
\( T^{5} - 2645 T^{3} + \cdots + 7035630 \)
|
| $29$ |
\( T^{5} \)
|
| $31$ |
\( T^{5} - 4805 T^{3} + \cdots - 2835202 \)
|
| $37$ |
\( T^{5} \)
|
| $41$ |
\( T^{5} \)
|
| $43$ |
\( T^{5} \)
|
| $47$ |
\( T^{5} \)
|
| $53$ |
\( T^{5} \)
|
| $59$ |
\( T^{5} \)
|
| $61$ |
\( T^{5} \)
|
| $67$ |
\( T^{5} + \cdots - 2280156970 \)
|
| $71$ |
\( T^{5} \)
|
| $73$ |
\( T^{5} + \cdots - 4143230930 \)
|
| $79$ |
\( (T + 79)^{5} \)
|
| $83$ |
\( (T + 150)^{5} \)
|
| $89$ |
\( T^{5} + \cdots - 6410928498 \)
|
| $97$ |
\( T^{5} + \cdots + 13372478270 \)
|
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