Newspace parameters
Level: | \( N \) | \(=\) | \( 1003 = 17 \cdot 59 \) |
Weight: | \( k \) | \(=\) | \( 1 \) |
Character orbit: | \([\chi]\) | \(=\) | 1003.f (of order \(4\), degree \(2\), minimal) |
Newform invariants
Self dual: | no |
Analytic conductor: | \(0.500562207671\) |
Analytic rank: | \(0\) |
Dimension: | \(4\) |
Relative dimension: | \(2\) over \(\Q(i)\) |
Coefficient field: | \(\Q(\zeta_{12})\) |
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
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Defining polynomial: | \( x^{4} - x^{2} + 1 \) |
Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
Coefficient ring index: | \( 2 \) |
Twist minimal: | yes |
Projective image: | \(D_{12}\) |
Projective field: | Galois closure of \(\mathbb{Q}[x]/(x^{12} - \cdots)\) |
$q$-expansion
The \(q\)-expansion and trace form are shown below.
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1003\mathbb{Z}\right)^\times\).
\(n\) | \(120\) | \(768\) |
\(\chi(n)\) | \(-1\) | \(-\zeta_{12}^{3}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
353.1 |
|
0 | −0.366025 | − | 0.366025i | −1.00000 | 1.36603 | + | 1.36603i | 0 | −0.366025 | + | 0.366025i | 0 | − | 0.732051i | 0 | |||||||||||||||||||||||
353.2 | 0 | 1.36603 | + | 1.36603i | −1.00000 | −0.366025 | − | 0.366025i | 0 | 1.36603 | − | 1.36603i | 0 | 2.73205i | 0 | |||||||||||||||||||||||||
412.1 | 0 | −0.366025 | + | 0.366025i | −1.00000 | 1.36603 | − | 1.36603i | 0 | −0.366025 | − | 0.366025i | 0 | 0.732051i | 0 | |||||||||||||||||||||||||
412.2 | 0 | 1.36603 | − | 1.36603i | −1.00000 | −0.366025 | + | 0.366025i | 0 | 1.36603 | + | 1.36603i | 0 | − | 2.73205i | 0 | ||||||||||||||||||||||||
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
59.b | odd | 2 | 1 | CM by \(\Q(\sqrt{-59}) \) |
17.c | even | 4 | 1 | inner |
1003.f | odd | 4 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 1003.1.f.b | ✓ | 4 |
17.c | even | 4 | 1 | inner | 1003.1.f.b | ✓ | 4 |
59.b | odd | 2 | 1 | CM | 1003.1.f.b | ✓ | 4 |
1003.f | odd | 4 | 1 | inner | 1003.1.f.b | ✓ | 4 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
1003.1.f.b | ✓ | 4 | 1.a | even | 1 | 1 | trivial |
1003.1.f.b | ✓ | 4 | 17.c | even | 4 | 1 | inner |
1003.1.f.b | ✓ | 4 | 59.b | odd | 2 | 1 | CM |
1003.1.f.b | ✓ | 4 | 1003.f | odd | 4 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{4} - 2T_{3}^{3} + 2T_{3}^{2} + 2T_{3} + 1 \)
acting on \(S_{1}^{\mathrm{new}}(1003, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{4} \)
$3$
\( T^{4} - 2 T^{3} + 2 T^{2} + 2 T + 1 \)
$5$
\( T^{4} - 2 T^{3} + 2 T^{2} + 2 T + 1 \)
$7$
\( T^{4} - 2 T^{3} + 2 T^{2} + 2 T + 1 \)
$11$
\( T^{4} \)
$13$
\( T^{4} \)
$17$
\( (T^{2} + 1)^{2} \)
$19$
\( (T^{2} + 1)^{2} \)
$23$
\( T^{4} \)
$29$
\( T^{4} + 2 T^{3} + 2 T^{2} - 2 T + 1 \)
$31$
\( T^{4} \)
$37$
\( T^{4} \)
$41$
\( T^{4} - 2 T^{3} + 2 T^{2} + 2 T + 1 \)
$43$
\( T^{4} \)
$47$
\( T^{4} \)
$53$
\( (T^{2} + 1)^{2} \)
$59$
\( (T^{2} + 1)^{2} \)
$61$
\( T^{4} \)
$67$
\( T^{4} \)
$71$
\( (T^{2} + 2 T + 2)^{2} \)
$73$
\( T^{4} \)
$79$
\( T^{4} - 2 T^{3} + 2 T^{2} + 2 T + 1 \)
$83$
\( T^{4} \)
$89$
\( T^{4} \)
$97$
\( T^{4} \)
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