Newspace parameters
Level: | \( N \) | \(=\) | \( 2007 = 3^{2} \cdot 223 \) |
Weight: | \( k \) | \(=\) | \( 1 \) |
Character orbit: | \([\chi]\) | \(=\) | 2007.d (of order \(2\), degree \(1\), minimal) |
Newform invariants
Self dual: | yes |
Analytic conductor: | \(1.00162348035\) |
Analytic rank: | \(0\) |
Dimension: | \(6\) |
Coefficient field: | \(\Q(\zeta_{28})^+\) |
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
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Defining polynomial: | \( x^{6} - 7x^{4} + 14x^{2} - 7 \) |
Coefficient ring: | \(\Z[a_1, a_2]\) |
Coefficient ring index: | \( 1 \) |
Twist minimal: | yes |
Projective image: | \(D_{14}\) |
Projective field: | Galois closure of \(\mathbb{Q}[x]/(x^{14} - \cdots)\) |
$q$-expansion
Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.
Basis of coefficient ring in terms of \(\nu = \zeta_{28} + \zeta_{28}^{-1}\):
\(\beta_{1}\) | \(=\) | \( \nu \) |
\(\beta_{2}\) | \(=\) | \( \nu^{2} - 2 \) |
\(\beta_{3}\) | \(=\) | \( \nu^{3} - 3\nu \) |
\(\beta_{4}\) | \(=\) | \( \nu^{4} - 5\nu^{2} + 5 \) |
\(\beta_{5}\) | \(=\) | \( \nu^{5} - 5\nu^{3} + 5\nu \) |
\(\nu\) | \(=\) | \( \beta_1 \) |
\(\nu^{2}\) | \(=\) | \( \beta_{2} + 2 \) |
\(\nu^{3}\) | \(=\) | \( \beta_{3} + 3\beta_1 \) |
\(\nu^{4}\) | \(=\) | \( \beta_{4} + 5\beta_{2} + 5 \) |
\(\nu^{5}\) | \(=\) | \( \beta_{5} + 5\beta_{3} + 10\beta_1 \) |
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2007\mathbb{Z}\right)^\times\).
\(n\) | \(226\) | \(893\) |
\(\chi(n)\) | \(-1\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
1783.1 |
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−1.94986 | 0 | 2.80194 | 0 | 0 | −1.24698 | −3.51352 | 0 | 0 | ||||||||||||||||||||||||||||||||||||
1783.2 | −1.56366 | 0 | 1.44504 | 0 | 0 | 1.80194 | −0.695895 | 0 | 0 | |||||||||||||||||||||||||||||||||||||
1783.3 | −0.867767 | 0 | −0.246980 | 0 | 0 | 0.445042 | 1.08209 | 0 | 0 | |||||||||||||||||||||||||||||||||||||
1783.4 | 0.867767 | 0 | −0.246980 | 0 | 0 | 0.445042 | −1.08209 | 0 | 0 | |||||||||||||||||||||||||||||||||||||
1783.5 | 1.56366 | 0 | 1.44504 | 0 | 0 | 1.80194 | 0.695895 | 0 | 0 | |||||||||||||||||||||||||||||||||||||
1783.6 | 1.94986 | 0 | 2.80194 | 0 | 0 | −1.24698 | 3.51352 | 0 | 0 | |||||||||||||||||||||||||||||||||||||
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
223.b | odd | 2 | 1 | CM by \(\Q(\sqrt{-223}) \) |
3.b | odd | 2 | 1 | inner |
669.c | even | 2 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 2007.1.d.c | ✓ | 6 |
3.b | odd | 2 | 1 | inner | 2007.1.d.c | ✓ | 6 |
223.b | odd | 2 | 1 | CM | 2007.1.d.c | ✓ | 6 |
669.c | even | 2 | 1 | inner | 2007.1.d.c | ✓ | 6 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
2007.1.d.c | ✓ | 6 | 1.a | even | 1 | 1 | trivial |
2007.1.d.c | ✓ | 6 | 3.b | odd | 2 | 1 | inner |
2007.1.d.c | ✓ | 6 | 223.b | odd | 2 | 1 | CM |
2007.1.d.c | ✓ | 6 | 669.c | even | 2 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{6} - 7T_{2}^{4} + 14T_{2}^{2} - 7 \)
acting on \(S_{1}^{\mathrm{new}}(2007, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{6} - 7 T^{4} + 14 T^{2} - 7 \)
$3$
\( T^{6} \)
$5$
\( T^{6} \)
$7$
\( (T^{3} - T^{2} - 2 T + 1)^{2} \)
$11$
\( T^{6} \)
$13$
\( T^{6} \)
$17$
\( T^{6} - 7 T^{4} + 14 T^{2} - 7 \)
$19$
\( (T^{3} + T^{2} - 2 T - 1)^{2} \)
$23$
\( T^{6} \)
$29$
\( T^{6} - 7 T^{4} + 14 T^{2} - 7 \)
$31$
\( (T^{3} - T^{2} - 2 T + 1)^{2} \)
$37$
\( (T^{3} + T^{2} - 2 T - 1)^{2} \)
$41$
\( T^{6} - 7 T^{4} + 14 T^{2} - 7 \)
$43$
\( (T^{3} + T^{2} - 2 T - 1)^{2} \)
$47$
\( T^{6} - 7 T^{4} + 14 T^{2} - 7 \)
$53$
\( T^{6} - 7 T^{4} + 14 T^{2} - 7 \)
$59$
\( T^{6} \)
$61$
\( T^{6} \)
$67$
\( T^{6} \)
$71$
\( T^{6} \)
$73$
\( (T^{3} - T^{2} - 2 T + 1)^{2} \)
$79$
\( T^{6} \)
$83$
\( T^{6} - 7 T^{4} + 14 T^{2} - 7 \)
$89$
\( T^{6} - 7 T^{4} + 14 T^{2} - 7 \)
$97$
\( T^{6} \)
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