Newspace parameters
Level: | \( N \) | \(=\) | \( 160 = 2^{5} \cdot 5 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 160.c (of order \(2\), degree \(1\), minimal) |
Newform invariants
Self dual: | no |
Analytic conductor: | \(1.27760643234\) |
Analytic rank: | \(0\) |
Dimension: | \(4\) |
Coefficient field: | \(\Q(i, \sqrt{5})\) |
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
Defining polynomial: | \( x^{4} + 3x^{2} + 1 \) |
Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
Coefficient ring index: | \( 2^{4} \) |
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 \(1,\beta_1,\beta_2,\beta_3\) 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 \( x^{4} + 3x^{2} + 1 \) :
\(\beta_{1}\) | \(=\) | \( 2\nu \) |
\(\beta_{2}\) | \(=\) | \( 2\nu^{2} + 3 \) |
\(\beta_{3}\) | \(=\) | \( 4\nu^{3} + 10\nu \) |
\(\nu\) | \(=\) | \( ( \beta_1 ) / 2 \) |
\(\nu^{2}\) | \(=\) | \( ( \beta_{2} - 3 ) / 2 \) |
\(\nu^{3}\) | \(=\) | \( ( \beta_{3} - 5\beta_1 ) / 4 \) |
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/160\mathbb{Z}\right)^\times\).
\(n\) | \(31\) | \(97\) | \(101\) |
\(\chi(n)\) | \(1\) | \(-1\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
129.1 |
|
0 | − | 3.23607i | 0 | 2.23607 | 0 | 0.763932i | 0 | −7.47214 | 0 | |||||||||||||||||||||||||||||
129.2 | 0 | − | 1.23607i | 0 | −2.23607 | 0 | − | 5.23607i | 0 | 1.47214 | 0 | |||||||||||||||||||||||||||||
129.3 | 0 | 1.23607i | 0 | −2.23607 | 0 | 5.23607i | 0 | 1.47214 | 0 | |||||||||||||||||||||||||||||||
129.4 | 0 | 3.23607i | 0 | 2.23607 | 0 | − | 0.763932i | 0 | −7.47214 | 0 | ||||||||||||||||||||||||||||||
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
20.d | odd | 2 | 1 | CM by \(\Q(\sqrt{-5}) \) |
4.b | odd | 2 | 1 | inner |
5.b | even | 2 | 1 | inner |
Twists
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{4} + 12T_{3}^{2} + 16 \)
acting on \(S_{2}^{\mathrm{new}}(160, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{4} \)
$3$
\( T^{4} + 12T^{2} + 16 \)
$5$
\( (T^{2} - 5)^{2} \)
$7$
\( T^{4} + 28T^{2} + 16 \)
$11$
\( T^{4} \)
$13$
\( T^{4} \)
$17$
\( T^{4} \)
$19$
\( T^{4} \)
$23$
\( T^{4} + 92T^{2} + 1936 \)
$29$
\( (T - 6)^{4} \)
$31$
\( T^{4} \)
$37$
\( T^{4} \)
$41$
\( (T^{2} - 20)^{2} \)
$43$
\( T^{4} + 172T^{2} + 5776 \)
$47$
\( T^{4} + 188T^{2} + 16 \)
$53$
\( T^{4} \)
$59$
\( T^{4} \)
$61$
\( (T^{2} - 180)^{2} \)
$67$
\( T^{4} + 268 T^{2} + 13456 \)
$71$
\( T^{4} \)
$73$
\( T^{4} \)
$79$
\( T^{4} \)
$83$
\( T^{4} + 332T^{2} + 5776 \)
$89$
\( (T + 6)^{4} \)
$97$
\( T^{4} \)
show more
show less