Newspace parameters
Level: | \( N \) | \(=\) | \( 572 = 2^{2} \cdot 11 \cdot 13 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 572.f (of order \(2\), degree \(1\), minimal) |
Newform invariants
Self dual: | no |
Analytic conductor: | \(4.56744299562\) |
Analytic rank: | \(0\) |
Dimension: | \(4\) |
Coefficient field: | \(\Q(i, \sqrt{21})\) |
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
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Defining polynomial: | \( x^{4} + 11x^{2} + 25 \) |
Coefficient ring: | \(\Z[a_1, \ldots, a_{11}]\) |
Coefficient ring index: | \( 1 \) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{2}]$ |
$q$-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 \( x^{4} + 11x^{2} + 25 \) :
\(\beta_{1}\) | \(=\) | \( \nu \) |
\(\beta_{2}\) | \(=\) | \( ( \nu^{3} + 6\nu ) / 5 \) |
\(\beta_{3}\) | \(=\) | \( \nu^{2} + 6 \) |
\(\nu\) | \(=\) | \( \beta_1 \) |
\(\nu^{2}\) | \(=\) | \( \beta_{3} - 6 \) |
\(\nu^{3}\) | \(=\) | \( 5\beta_{2} - 6\beta_1 \) |
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/572\mathbb{Z}\right)^\times\).
\(n\) | \(287\) | \(353\) | \(365\) |
\(\chi(n)\) | \(1\) | \(-1\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
441.1 |
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0 | −1.79129 | 0 | 0 | 0 | − | 0.791288i | 0 | 0.208712 | 0 | |||||||||||||||||||||||||||||
441.2 | 0 | −1.79129 | 0 | 0 | 0 | 0.791288i | 0 | 0.208712 | 0 | |||||||||||||||||||||||||||||||
441.3 | 0 | 2.79129 | 0 | 0 | 0 | − | 3.79129i | 0 | 4.79129 | 0 | ||||||||||||||||||||||||||||||
441.4 | 0 | 2.79129 | 0 | 0 | 0 | 3.79129i | 0 | 4.79129 | 0 | |||||||||||||||||||||||||||||||
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
13.b | even | 2 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 572.2.f.b | ✓ | 4 |
3.b | odd | 2 | 1 | 5148.2.e.b | 4 | ||
4.b | odd | 2 | 1 | 2288.2.j.f | 4 | ||
13.b | even | 2 | 1 | inner | 572.2.f.b | ✓ | 4 |
13.d | odd | 4 | 1 | 7436.2.a.i | 2 | ||
13.d | odd | 4 | 1 | 7436.2.a.j | 2 | ||
39.d | odd | 2 | 1 | 5148.2.e.b | 4 | ||
52.b | odd | 2 | 1 | 2288.2.j.f | 4 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
572.2.f.b | ✓ | 4 | 1.a | even | 1 | 1 | trivial |
572.2.f.b | ✓ | 4 | 13.b | even | 2 | 1 | inner |
2288.2.j.f | 4 | 4.b | odd | 2 | 1 | ||
2288.2.j.f | 4 | 52.b | odd | 2 | 1 | ||
5148.2.e.b | 4 | 3.b | odd | 2 | 1 | ||
5148.2.e.b | 4 | 39.d | odd | 2 | 1 | ||
7436.2.a.i | 2 | 13.d | odd | 4 | 1 | ||
7436.2.a.j | 2 | 13.d | odd | 4 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{2} - T_{3} - 5 \)
acting on \(S_{2}^{\mathrm{new}}(572, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{4} \)
$3$
\( (T^{2} - T - 5)^{2} \)
$5$
\( T^{4} \)
$7$
\( T^{4} + 15T^{2} + 9 \)
$11$
\( (T^{2} + 1)^{2} \)
$13$
\( (T^{2} - 4 T + 13)^{2} \)
$17$
\( (T^{2} + 6 T - 12)^{2} \)
$19$
\( T^{4} + 51T^{2} + 225 \)
$23$
\( (T^{2} - 3 T - 3)^{2} \)
$29$
\( (T^{2} - 6 T - 12)^{2} \)
$31$
\( T^{4} + 60T^{2} + 144 \)
$37$
\( (T^{2} + 84)^{2} \)
$41$
\( T^{4} + 99T^{2} + 2025 \)
$43$
\( (T - 2)^{4} \)
$47$
\( (T^{2} + 36)^{2} \)
$53$
\( (T^{2} + 9 T + 15)^{2} \)
$59$
\( (T^{2} + 36)^{2} \)
$61$
\( (T - 10)^{4} \)
$67$
\( T^{4} + 240T^{2} + 2304 \)
$71$
\( T^{4} + 396 T^{2} + 32400 \)
$73$
\( T^{4} + 267 T^{2} + 16641 \)
$79$
\( (T + 8)^{4} \)
$83$
\( T^{4} + 135T^{2} + 729 \)
$89$
\( (T^{2} + 36)^{2} \)
$97$
\( T^{4} + 60T^{2} + 144 \)
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