Newspace parameters
Level: | \( N \) | \(=\) | \( 207 = 3^{2} \cdot 23 \) |
Weight: | \( k \) | \(=\) | \( 5 \) |
Character orbit: | \([\chi]\) | \(=\) | 207.d (of order \(2\), degree \(1\), minimal) |
Newform invariants
Self dual: | yes |
Analytic conductor: | \(21.3975823584\) |
Analytic rank: | \(0\) |
Dimension: | \(3\) |
Coefficient field: | 3.3.621.1 |
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
Defining polynomial: | \( x^{3} - 6x - 3 \) |
Coefficient ring: | \(\Z[a_1, \ldots, a_{13}]\) |
Coefficient ring index: | \( 3 \) |
Twist minimal: | no (minimal twist has level 23) |
Sato-Tate group: | $\mathrm{U}(1)[D_{2}]$ |
$q$-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 a root \(\nu\) of \( x^{3} - 6x - 3 \) :
\(\beta_{1}\) | \(=\) | \( \nu^{2} - 4 \) |
\(\beta_{2}\) | \(=\) | \( -\nu^{2} + 3\nu + 4 \) |
\(\nu\) | \(=\) | \( ( \beta_{2} + \beta_1 ) / 3 \) |
\(\nu^{2}\) | \(=\) | \( \beta _1 + 4 \) |
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/207\mathbb{Z}\right)^\times\).
\(n\) | \(28\) | \(47\) |
\(\chi(n)\) | \(-1\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | |||||||||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
91.1 |
|
−5.87897 | 0 | 18.5623 | 0 | 0 | 0 | −15.0635 | 0 | 0 | |||||||||||||||||||||||||||
91.2 | −1.75927 | 0 | −12.9050 | 0 | 0 | 0 | 50.8517 | 0 | 0 | ||||||||||||||||||||||||||||
91.3 | 7.63824 | 0 | 42.3427 | 0 | 0 | 0 | 201.212 | 0 | 0 | ||||||||||||||||||||||||||||
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
23.b | odd | 2 | 1 | CM by \(\Q(\sqrt{-23}) \) |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 207.5.d.a | 3 | |
3.b | odd | 2 | 1 | 23.5.b.a | ✓ | 3 | |
12.b | even | 2 | 1 | 368.5.f.a | 3 | ||
23.b | odd | 2 | 1 | CM | 207.5.d.a | 3 | |
69.c | even | 2 | 1 | 23.5.b.a | ✓ | 3 | |
276.h | odd | 2 | 1 | 368.5.f.a | 3 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
23.5.b.a | ✓ | 3 | 3.b | odd | 2 | 1 | |
23.5.b.a | ✓ | 3 | 69.c | even | 2 | 1 | |
207.5.d.a | 3 | 1.a | even | 1 | 1 | trivial | |
207.5.d.a | 3 | 23.b | odd | 2 | 1 | CM | |
368.5.f.a | 3 | 12.b | even | 2 | 1 | ||
368.5.f.a | 3 | 276.h | odd | 2 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{3} - 48T_{2} - 79 \)
acting on \(S_{5}^{\mathrm{new}}(207, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{3} - 48T - 79 \)
$3$
\( T^{3} \)
$5$
\( T^{3} \)
$7$
\( T^{3} \)
$11$
\( T^{3} \)
$13$
\( T^{3} - 85683 T + 8482894 \)
$17$
\( T^{3} \)
$19$
\( T^{3} \)
$23$
\( (T + 529)^{3} \)
$29$
\( T^{3} - 2121843 T - 244330126 \)
$31$
\( T^{3} - 2770563 T - 1677025154 \)
$37$
\( T^{3} \)
$41$
\( T^{3} - 8477283 T - 7596282526 \)
$43$
\( T^{3} \)
$47$
\( T^{3} - 14639043 T + 20606906306 \)
$53$
\( T^{3} \)
$59$
\( (T - 6286)^{3} \)
$61$
\( T^{3} \)
$67$
\( T^{3} \)
$71$
\( T^{3} - 76235043 T + 188893891874 \)
$73$
\( T^{3} - 85194723 T - 223017449186 \)
$79$
\( T^{3} \)
$83$
\( T^{3} \)
$89$
\( T^{3} \)
$97$
\( T^{3} \)
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