Newspace parameters
Level: | \( N \) | \(=\) | \( 799 = 17 \cdot 47 \) |
Weight: | \( k \) | \(=\) | \( 1 \) |
Character orbit: | \([\chi]\) | \(=\) | 799.e (of order \(4\), degree \(2\), minimal) |
Newform invariants
Self dual: | no |
Analytic conductor: | \(0.398752945094\) |
Analytic rank: | \(0\) |
Dimension: | \(2\) |
Coefficient field: | \(\Q(i)\) |
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
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Defining polynomial: | \( x^{2} + 1 \) |
Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
Coefficient ring index: | \( 1 \) |
Twist minimal: | yes |
Projective image: | \(D_{4}\) |
Projective field: | Galois closure of 4.2.230911.1 |
$q$-expansion
The \(q\)-expansion and trace form are shown below.
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/799\mathbb{Z}\right)^\times\).
\(n\) | \(52\) | \(377\) |
\(\chi(n)\) | \(-1\) | \(-i\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
140.1 |
|
− | 2.00000i | 1.00000 | − | 1.00000i | −3.00000 | 0 | −2.00000 | − | 2.00000i | −1.00000 | − | 1.00000i | 4.00000i | − | 1.00000i | 0 | ||||||||||||||||
234.1 | 2.00000i | 1.00000 | + | 1.00000i | −3.00000 | 0 | −2.00000 | + | 2.00000i | −1.00000 | + | 1.00000i | − | 4.00000i | 1.00000i | 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.c | even | 4 | 1 | inner |
799.e | odd | 4 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 799.1.e.a | ✓ | 2 |
17.c | even | 4 | 1 | inner | 799.1.e.a | ✓ | 2 |
47.b | odd | 2 | 1 | CM | 799.1.e.a | ✓ | 2 |
799.e | odd | 4 | 1 | inner | 799.1.e.a | ✓ | 2 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
799.1.e.a | ✓ | 2 | 1.a | even | 1 | 1 | trivial |
799.1.e.a | ✓ | 2 | 17.c | even | 4 | 1 | inner |
799.1.e.a | ✓ | 2 | 47.b | odd | 2 | 1 | CM |
799.1.e.a | ✓ | 2 | 799.e | odd | 4 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{2} + 4 \)
acting on \(S_{1}^{\mathrm{new}}(799, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{2} + 4 \)
$3$
\( T^{2} - 2T + 2 \)
$5$
\( T^{2} \)
$7$
\( T^{2} + 2T + 2 \)
$11$
\( T^{2} \)
$13$
\( T^{2} \)
$17$
\( (T - 1)^{2} \)
$19$
\( T^{2} \)
$23$
\( T^{2} \)
$29$
\( T^{2} \)
$31$
\( T^{2} \)
$37$
\( T^{2} + 2T + 2 \)
$41$
\( T^{2} \)
$43$
\( T^{2} \)
$47$
\( (T + 1)^{2} \)
$53$
\( T^{2} \)
$59$
\( T^{2} + 4 \)
$61$
\( T^{2} - 2T + 2 \)
$67$
\( T^{2} \)
$71$
\( T^{2} - 2T + 2 \)
$73$
\( T^{2} \)
$79$
\( T^{2} - 2T + 2 \)
$83$
\( T^{2} \)
$89$
\( (T - 2)^{2} \)
$97$
\( T^{2} + 2T + 2 \)
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