Newspace parameters
Level: | \( N \) | \(=\) | \( 2003 \) |
Weight: | \( k \) | \(=\) | \( 1 \) |
Character orbit: | \([\chi]\) | \(=\) | 2003.b (of order \(2\), degree \(1\), minimal) |
Newform invariants
Self dual: | yes |
Analytic conductor: | \(0.999627220304\) |
Analytic rank: | \(0\) |
Dimension: | \(3\) |
Coefficient field: | \(\Q(\zeta_{18})^+\) |
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
Defining polynomial: | \( x^{3} - 3x - 1 \) |
Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
Coefficient ring index: | \( 1 \) |
Twist minimal: | yes |
Projective image: | \(D_{9}\) |
Projective field: | Galois closure of 9.1.16096216216081.1 |
Artin image: | $D_9$ |
Artin field: | Galois closure of 9.1.16096216216081.1 |
$q$-expansion
Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.
Basis of coefficient ring in terms of \(\nu = \zeta_{18} + \zeta_{18}^{-1}\):
\(\beta_{1}\) | \(=\) | \( \nu \) |
\(\beta_{2}\) | \(=\) | \( \nu^{2} - 2 \) |
\(\nu\) | \(=\) | \( \beta_1 \) |
\(\nu^{2}\) | \(=\) | \( \beta_{2} + 2 \) |
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2003\mathbb{Z}\right)^\times\).
\(n\) | \(5\) |
\(\chi(n)\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | |||||||||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
2002.1 |
|
0 | −1.87939 | 1.00000 | 0 | 0 | 0 | 0 | 2.53209 | 0 | |||||||||||||||||||||||||||
2002.2 | 0 | 0.347296 | 1.00000 | 0 | 0 | 0 | 0 | −0.879385 | 0 | ||||||||||||||||||||||||||||
2002.3 | 0 | 1.53209 | 1.00000 | 0 | 0 | 0 | 0 | 1.34730 | 0 | ||||||||||||||||||||||||||||
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
2003.b | odd | 2 | 1 | CM by \(\Q(\sqrt{-2003}) \) |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 2003.1.b.b | ✓ | 3 |
2003.b | odd | 2 | 1 | CM | 2003.1.b.b | ✓ | 3 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
2003.1.b.b | ✓ | 3 | 1.a | even | 1 | 1 | trivial |
2003.1.b.b | ✓ | 3 | 2003.b | odd | 2 | 1 | CM |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{3} - 3T_{3} + 1 \)
acting on \(S_{1}^{\mathrm{new}}(2003, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{3} \)
$3$
\( T^{3} - 3T + 1 \)
$5$
\( T^{3} \)
$7$
\( T^{3} \)
$11$
\( T^{3} \)
$13$
\( T^{3} - 3T + 1 \)
$17$
\( T^{3} \)
$19$
\( (T + 1)^{3} \)
$23$
\( T^{3} \)
$29$
\( T^{3} \)
$31$
\( T^{3} \)
$37$
\( T^{3} \)
$41$
\( T^{3} \)
$43$
\( T^{3} \)
$47$
\( T^{3} - 3T + 1 \)
$53$
\( (T + 1)^{3} \)
$59$
\( T^{3} - 3T + 1 \)
$61$
\( T^{3} \)
$67$
\( T^{3} \)
$71$
\( T^{3} \)
$73$
\( T^{3} - 3T + 1 \)
$79$
\( T^{3} - 3T + 1 \)
$83$
\( T^{3} \)
$89$
\( (T + 1)^{3} \)
$97$
\( T^{3} \)
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