Newspace parameters
Level: | \( N \) | \(=\) | \( 59 \) |
Weight: | \( k \) | \(=\) | \( 3 \) |
Character orbit: | \([\chi]\) | \(=\) | 59.b (of order \(2\), degree \(1\), minimal) |
Newform invariants
Self dual: | yes |
Analytic conductor: | \(1.60763355973\) |
Analytic rank: | \(0\) |
Dimension: | \(3\) |
Coefficient field: | 3.3.1593.1 |
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
Defining polynomial: | \( x^{3} - 9x - 7 \) |
Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
Coefficient ring index: | \( 1 \) |
Twist minimal: | yes |
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 \( x^{3} - 9x - 7 \) :
\(\beta_{1}\) | \(=\) | \( \nu \) |
\(\beta_{2}\) | \(=\) | \( \nu^{2} - \nu - 6 \) |
\(\nu\) | \(=\) | \( \beta_1 \) |
\(\nu^{2}\) | \(=\) | \( \beta_{2} + \beta _1 + 6 \) |
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/59\mathbb{Z}\right)^\times\).
\(n\) | \(2\) |
\(\chi(n)\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | |||||||||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
58.1 |
|
0 | −5.28638 | 4.00000 | 9.72800 | 0 | −1.06258 | 0 | 18.9458 | 0 | |||||||||||||||||||||||||||
58.2 | 0 | 0.185421 | 4.00000 | −2.85789 | 0 | 12.6207 | 0 | −8.96562 | 0 | ||||||||||||||||||||||||||||
58.3 | 0 | 5.10096 | 4.00000 | −6.87011 | 0 | −11.5581 | 0 | 17.0198 | 0 | ||||||||||||||||||||||||||||
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
59.b | odd | 2 | 1 | CM by \(\Q(\sqrt{-59}) \) |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 59.3.b.a | ✓ | 3 |
3.b | odd | 2 | 1 | 531.3.c.a | 3 | ||
4.b | odd | 2 | 1 | 944.3.h.b | 3 | ||
59.b | odd | 2 | 1 | CM | 59.3.b.a | ✓ | 3 |
177.d | even | 2 | 1 | 531.3.c.a | 3 | ||
236.c | even | 2 | 1 | 944.3.h.b | 3 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
59.3.b.a | ✓ | 3 | 1.a | even | 1 | 1 | trivial |
59.3.b.a | ✓ | 3 | 59.b | odd | 2 | 1 | CM |
531.3.c.a | 3 | 3.b | odd | 2 | 1 | ||
531.3.c.a | 3 | 177.d | even | 2 | 1 | ||
944.3.h.b | 3 | 4.b | odd | 2 | 1 | ||
944.3.h.b | 3 | 236.c | even | 2 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2} \)
acting on \(S_{3}^{\mathrm{new}}(59, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{3} \)
$3$
\( T^{3} - 27T + 5 \)
$5$
\( T^{3} - 75T - 191 \)
$7$
\( T^{3} - 147T - 155 \)
$11$
\( T^{3} \)
$13$
\( T^{3} \)
$17$
\( (T + 25)^{3} \)
$19$
\( T^{3} - 1083T - 443 \)
$23$
\( T^{3} \)
$29$
\( T^{3} - 2523T + 23497 \)
$31$
\( T^{3} \)
$37$
\( T^{3} \)
$41$
\( T^{3} - 5043 T - 137783 \)
$43$
\( T^{3} \)
$47$
\( T^{3} \)
$53$
\( T^{3} - 8427 T + 190825 \)
$59$
\( (T + 59)^{3} \)
$61$
\( T^{3} \)
$67$
\( T^{3} \)
$71$
\( (T - 83)^{3} \)
$73$
\( T^{3} \)
$79$
\( T^{3} - 18723 T + 288853 \)
$83$
\( T^{3} \)
$89$
\( T^{3} \)
$97$
\( T^{3} \)
show more
show less