Newspace parameters
Level: | \( N \) | \(=\) | \( 2008 = 2^{3} \cdot 251 \) |
Weight: | \( k \) | \(=\) | \( 1 \) |
Character orbit: | \([\chi]\) | \(=\) | 2008.c (of order \(2\), degree \(1\), minimal) |
Newform invariants
Self dual: | yes |
Analytic conductor: | \(1.00212254537\) |
Analytic rank: | \(0\) |
Dimension: | \(3\) |
Coefficient field: | \(\Q(\zeta_{14})^+\) |
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
Defining polynomial: | \( x^{3} - x^{2} - 2x + 1 \) |
Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
Coefficient ring index: | \( 1 \) |
Twist minimal: | yes |
Projective image: | \(D_{7}\) |
Projective field: | Galois closure of 7.1.8096384512.1 |
Artin image: | $D_7$ |
Artin field: | Galois closure of 7.1.8096384512.1 |
$q$-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 \(\nu = \zeta_{14} + \zeta_{14}^{-1}\):
\(\beta_{1}\) | \(=\) | \( \nu \) |
\(\beta_{2}\) | \(=\) | \( \nu^{2} - 2 \) |
\(\nu\) | \(=\) | \( \beta_1 \) |
\(\nu^{2}\) | \(=\) | \( \beta_{2} + 2 \) |
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2008\mathbb{Z}\right)^\times\).
\(n\) | \(257\) | \(503\) | \(1005\) |
\(\chi(n)\) | \(-1\) | \(1\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | |||||||||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
501.1 |
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1.00000 | 0 | 1.00000 | 0 | 0 | −1.80194 | 1.00000 | 1.00000 | 0 | |||||||||||||||||||||||||||
501.2 | 1.00000 | 0 | 1.00000 | 0 | 0 | −0.445042 | 1.00000 | 1.00000 | 0 | ||||||||||||||||||||||||||||
501.3 | 1.00000 | 0 | 1.00000 | 0 | 0 | 1.24698 | 1.00000 | 1.00000 | 0 | ||||||||||||||||||||||||||||
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
2008.c | odd | 2 | 1 | CM by \(\Q(\sqrt{-502}) \) |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 2008.1.c.b | yes | 3 |
8.b | even | 2 | 1 | 2008.1.c.a | ✓ | 3 | |
251.b | odd | 2 | 1 | 2008.1.c.a | ✓ | 3 | |
2008.c | odd | 2 | 1 | CM | 2008.1.c.b | yes | 3 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
2008.1.c.a | ✓ | 3 | 8.b | even | 2 | 1 | |
2008.1.c.a | ✓ | 3 | 251.b | odd | 2 | 1 | |
2008.1.c.b | yes | 3 | 1.a | even | 1 | 1 | trivial |
2008.1.c.b | yes | 3 | 2008.c | odd | 2 | 1 | CM |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{11}^{3} + T_{11}^{2} - 2T_{11} - 1 \)
acting on \(S_{1}^{\mathrm{new}}(2008, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( (T - 1)^{3} \)
$3$
\( T^{3} \)
$5$
\( T^{3} \)
$7$
\( T^{3} + T^{2} - 2T - 1 \)
$11$
\( T^{3} + T^{2} - 2T - 1 \)
$13$
\( T^{3} \)
$17$
\( T^{3} + T^{2} - 2T - 1 \)
$19$
\( T^{3} + T^{2} - 2T - 1 \)
$23$
\( T^{3} + T^{2} - 2T - 1 \)
$29$
\( T^{3} + T^{2} - 2T - 1 \)
$31$
\( T^{3} + T^{2} - 2T - 1 \)
$37$
\( T^{3} + T^{2} - 2T - 1 \)
$41$
\( T^{3} + T^{2} - 2T - 1 \)
$43$
\( T^{3} + T^{2} - 2T - 1 \)
$47$
\( T^{3} \)
$53$
\( T^{3} + T^{2} - 2T - 1 \)
$59$
\( T^{3} + T^{2} - 2T - 1 \)
$61$
\( T^{3} + T^{2} - 2T - 1 \)
$67$
\( T^{3} \)
$71$
\( T^{3} \)
$73$
\( T^{3} + T^{2} - 2T - 1 \)
$79$
\( T^{3} + T^{2} - 2T - 1 \)
$83$
\( T^{3} \)
$89$
\( T^{3} + T^{2} - 2T - 1 \)
$97$
\( T^{3} \)
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