Newspace parameters
| Level: | \( N \) | \(=\) | \( 799 = 17 \cdot 47 \) |
| Weight: | \( k \) | \(=\) | \( 1 \) |
| Character orbit: | \([\chi]\) | \(=\) | 799.c (of order \(2\), degree \(1\), minimal) |
Newform invariants
| Self dual: | no |
| Analytic conductor: | \(0.398752945094\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Coefficient field: | \(\Q(\zeta_{10})\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{4} - x^{3} + x^{2} - x + 1 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{17}]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | yes |
| Projective image: | \(D_{10}\) |
| Projective field: | Galois closure of 10.2.6928449225617.1 |
$q$-expansion
The \(q\)-expansion and trace form are shown below.
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/799\mathbb{Z}\right)^\times\).
| \(n\) | \(52\) | \(377\) |
| \(\chi(n)\) | \(-1\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 798.1 |
|
−0.618034 | − | 1.17557i | −0.618034 | 0 | 0.726543i | − | 1.90211i | 1.00000 | −0.381966 | 0 | ||||||||||||||||||||||||||||
| 798.2 | −0.618034 | 1.17557i | −0.618034 | 0 | − | 0.726543i | 1.90211i | 1.00000 | −0.381966 | 0 | ||||||||||||||||||||||||||||||
| 798.3 | 1.61803 | − | 1.90211i | 1.61803 | 0 | − | 3.07768i | 1.17557i | 1.00000 | −2.61803 | 0 | |||||||||||||||||||||||||||||
| 798.4 | 1.61803 | 1.90211i | 1.61803 | 0 | 3.07768i | − | 1.17557i | 1.00000 | −2.61803 | 0 | ||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 47.b | odd | 2 | 1 | CM by \(\Q(\sqrt{-47}) \) |
| 17.b | even | 2 | 1 | inner |
| 799.c | odd | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 799.1.c.d | ✓ | 4 |
| 17.b | even | 2 | 1 | inner | 799.1.c.d | ✓ | 4 |
| 47.b | odd | 2 | 1 | CM | 799.1.c.d | ✓ | 4 |
| 799.c | odd | 2 | 1 | inner | 799.1.c.d | ✓ | 4 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 799.1.c.d | ✓ | 4 | 1.a | even | 1 | 1 | trivial |
| 799.1.c.d | ✓ | 4 | 17.b | even | 2 | 1 | inner |
| 799.1.c.d | ✓ | 4 | 47.b | odd | 2 | 1 | CM |
| 799.1.c.d | ✓ | 4 | 799.c | odd | 2 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{2} - T_{2} - 1 \)
acting on \(S_{1}^{\mathrm{new}}(799, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( (T^{2} - T - 1)^{2} \)
$3$
\( T^{4} + 5T^{2} + 5 \)
$5$
\( T^{4} \)
$7$
\( T^{4} + 5T^{2} + 5 \)
$11$
\( T^{4} \)
$13$
\( T^{4} \)
$17$
\( T^{4} + T^{3} + T^{2} + T + 1 \)
$19$
\( T^{4} \)
$23$
\( T^{4} \)
$29$
\( T^{4} \)
$31$
\( T^{4} \)
$37$
\( T^{4} + 5T^{2} + 5 \)
$41$
\( T^{4} \)
$43$
\( T^{4} \)
$47$
\( (T - 1)^{4} \)
$53$
\( (T^{2} + T - 1)^{2} \)
$59$
\( (T^{2} - T - 1)^{2} \)
$61$
\( T^{4} + 5T^{2} + 5 \)
$67$
\( T^{4} \)
$71$
\( T^{4} + 5T^{2} + 5 \)
$73$
\( T^{4} \)
$79$
\( T^{4} + 5T^{2} + 5 \)
$83$
\( (T - 2)^{4} \)
$89$
\( (T^{2} + T - 1)^{2} \)
$97$
\( T^{4} + 5T^{2} + 5 \)
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