Newspace parameters
Level: | \( N \) | \(=\) | \( 567 = 3^{4} \cdot 7 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 567.a (trivial) |
Newform invariants
Self dual: | yes |
Analytic conductor: | \(4.52751779461\) |
Analytic rank: | \(0\) |
Dimension: | \(4\) |
Coefficient field: | \(\Q(\sqrt{3}, \sqrt{7})\) |
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
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Defining polynomial: | \( x^{4} - 5x^{2} + 1 \) |
Coefficient ring: | \(\Z[a_1, a_2]\) |
Coefficient ring index: | \( 1 \) |
Twist minimal: | yes |
Fricke sign: | \(-1\) |
Sato-Tate group: | $\mathrm{SU}(2)$ |
$q$-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 \( x^{4} - 5x^{2} + 1 \) :
\(\beta_{1}\) | \(=\) | \( \nu \) |
\(\beta_{2}\) | \(=\) | \( \nu^{2} - 3 \) |
\(\beta_{3}\) | \(=\) | \( \nu^{3} - 5\nu \) |
\(\nu\) | \(=\) | \( \beta_1 \) |
\(\nu^{2}\) | \(=\) | \( \beta_{2} + 3 \) |
\(\nu^{3}\) | \(=\) | \( \beta_{3} + 5\beta_1 \) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
1.1 |
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−2.18890 | 0 | 2.79129 | −0.913701 | 0 | −1.00000 | −1.73205 | 0 | 2.00000 | ||||||||||||||||||||||||||||||
1.2 | −0.456850 | 0 | −1.79129 | −4.37780 | 0 | −1.00000 | 1.73205 | 0 | 2.00000 | |||||||||||||||||||||||||||||||
1.3 | 0.456850 | 0 | −1.79129 | 4.37780 | 0 | −1.00000 | −1.73205 | 0 | 2.00000 | |||||||||||||||||||||||||||||||
1.4 | 2.18890 | 0 | 2.79129 | 0.913701 | 0 | −1.00000 | 1.73205 | 0 | 2.00000 | |||||||||||||||||||||||||||||||
Atkin-Lehner signs
\( p \) | Sign |
---|---|
\(3\) | \(-1\) |
\(7\) | \(1\) |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
3.b | odd | 2 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 567.2.a.i | ✓ | 4 |
3.b | odd | 2 | 1 | inner | 567.2.a.i | ✓ | 4 |
4.b | odd | 2 | 1 | 9072.2.a.ci | 4 | ||
7.b | odd | 2 | 1 | 3969.2.a.u | 4 | ||
9.c | even | 3 | 2 | 567.2.f.n | 8 | ||
9.d | odd | 6 | 2 | 567.2.f.n | 8 | ||
12.b | even | 2 | 1 | 9072.2.a.ci | 4 | ||
21.c | even | 2 | 1 | 3969.2.a.u | 4 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
567.2.a.i | ✓ | 4 | 1.a | even | 1 | 1 | trivial |
567.2.a.i | ✓ | 4 | 3.b | odd | 2 | 1 | inner |
567.2.f.n | 8 | 9.c | even | 3 | 2 | ||
567.2.f.n | 8 | 9.d | odd | 6 | 2 | ||
3969.2.a.u | 4 | 7.b | odd | 2 | 1 | ||
3969.2.a.u | 4 | 21.c | even | 2 | 1 | ||
9072.2.a.ci | 4 | 4.b | odd | 2 | 1 | ||
9072.2.a.ci | 4 | 12.b | even | 2 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{4} - 5T_{2}^{2} + 1 \)
acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(567))\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{4} - 5T^{2} + 1 \)
$3$
\( T^{4} \)
$5$
\( T^{4} - 20T^{2} + 16 \)
$7$
\( (T + 1)^{4} \)
$11$
\( (T^{2} - 7)^{2} \)
$13$
\( (T - 4)^{4} \)
$17$
\( (T^{2} - 12)^{2} \)
$19$
\( (T^{2} - 2 T - 20)^{2} \)
$23$
\( (T^{2} - 12)^{2} \)
$29$
\( T^{4} - 80T^{2} + 256 \)
$31$
\( (T^{2} - 84)^{2} \)
$37$
\( (T - 3)^{4} \)
$41$
\( T^{4} - 20T^{2} + 16 \)
$43$
\( (T^{2} + 8 T - 5)^{2} \)
$47$
\( T^{4} - 180T^{2} + 1296 \)
$53$
\( (T^{2} - 75)^{2} \)
$59$
\( (T^{2} - 12)^{2} \)
$61$
\( (T^{2} - 14 T + 28)^{2} \)
$67$
\( (T^{2} - 8 T - 5)^{2} \)
$71$
\( T^{4} - 150T^{2} + 2601 \)
$73$
\( (T^{2} - 12 T - 48)^{2} \)
$79$
\( (T^{2} + 8 T - 5)^{2} \)
$83$
\( T^{4} - 132T^{2} + 3600 \)
$89$
\( T^{4} - 80T^{2} + 256 \)
$97$
\( (T^{2} - 6 T - 12)^{2} \)
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