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: | \(1\) |
Dimension: | \(3\) |
Coefficient field: | \(\Q(\zeta_{18})^+\) |
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
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Defining polynomial: | \( x^{3} - 3x - 1 \) |
Coefficient ring: | \(\Z[a_1, a_2]\) |
Coefficient ring index: | \( 1 \) |
Twist minimal: | no (minimal twist has level 63) |
Fricke sign: | \(1\) |
Sato-Tate group: | $\mathrm{SU}(2)$ |
$q$-expansion
Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.
Basis of coefficient ring in terms of \(\nu = \zeta_{18} + \zeta_{18}^{-1}\):
\(\beta_{1}\) | \(=\) | \( \nu \) |
\(\beta_{2}\) | \(=\) | \( \nu^{2} - 2 \) |
\(\nu\) | \(=\) | \( \beta_1 \) |
\(\nu^{2}\) | \(=\) | \( \beta_{2} + 2 \) |
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 |
|
−2.53209 | 0 | 4.41147 | 0.879385 | 0 | 1.00000 | −6.10607 | 0 | −2.22668 | |||||||||||||||||||||||||||
1.2 | −1.34730 | 0 | −0.184793 | −2.53209 | 0 | 1.00000 | 2.94356 | 0 | 3.41147 | ||||||||||||||||||||||||||||
1.3 | 0.879385 | 0 | −1.22668 | −1.34730 | 0 | 1.00000 | −2.83750 | 0 | −1.18479 | ||||||||||||||||||||||||||||
Atkin-Lehner signs
\( p \) | Sign |
---|---|
\(3\) | \(-1\) |
\(7\) | \(-1\) |
Inner twists
This newform does not admit any (nontrivial) inner twists.
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 567.2.a.c | 3 | |
3.b | odd | 2 | 1 | 567.2.a.h | 3 | ||
4.b | odd | 2 | 1 | 9072.2.a.bs | 3 | ||
7.b | odd | 2 | 1 | 3969.2.a.l | 3 | ||
9.c | even | 3 | 2 | 189.2.f.b | 6 | ||
9.d | odd | 6 | 2 | 63.2.f.a | ✓ | 6 | |
12.b | even | 2 | 1 | 9072.2.a.ca | 3 | ||
21.c | even | 2 | 1 | 3969.2.a.q | 3 | ||
36.f | odd | 6 | 2 | 3024.2.r.k | 6 | ||
36.h | even | 6 | 2 | 1008.2.r.h | 6 | ||
63.g | even | 3 | 2 | 1323.2.g.d | 6 | ||
63.h | even | 3 | 2 | 1323.2.h.c | 6 | ||
63.i | even | 6 | 2 | 441.2.h.e | 6 | ||
63.j | odd | 6 | 2 | 441.2.h.d | 6 | ||
63.k | odd | 6 | 2 | 1323.2.g.e | 6 | ||
63.l | odd | 6 | 2 | 1323.2.f.d | 6 | ||
63.n | odd | 6 | 2 | 441.2.g.c | 6 | ||
63.o | even | 6 | 2 | 441.2.f.c | 6 | ||
63.s | even | 6 | 2 | 441.2.g.b | 6 | ||
63.t | odd | 6 | 2 | 1323.2.h.b | 6 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
63.2.f.a | ✓ | 6 | 9.d | odd | 6 | 2 | |
189.2.f.b | 6 | 9.c | even | 3 | 2 | ||
441.2.f.c | 6 | 63.o | even | 6 | 2 | ||
441.2.g.b | 6 | 63.s | even | 6 | 2 | ||
441.2.g.c | 6 | 63.n | odd | 6 | 2 | ||
441.2.h.d | 6 | 63.j | odd | 6 | 2 | ||
441.2.h.e | 6 | 63.i | even | 6 | 2 | ||
567.2.a.c | 3 | 1.a | even | 1 | 1 | trivial | |
567.2.a.h | 3 | 3.b | odd | 2 | 1 | ||
1008.2.r.h | 6 | 36.h | even | 6 | 2 | ||
1323.2.f.d | 6 | 63.l | odd | 6 | 2 | ||
1323.2.g.d | 6 | 63.g | even | 3 | 2 | ||
1323.2.g.e | 6 | 63.k | odd | 6 | 2 | ||
1323.2.h.b | 6 | 63.t | odd | 6 | 2 | ||
1323.2.h.c | 6 | 63.h | even | 3 | 2 | ||
3024.2.r.k | 6 | 36.f | odd | 6 | 2 | ||
3969.2.a.l | 3 | 7.b | odd | 2 | 1 | ||
3969.2.a.q | 3 | 21.c | even | 2 | 1 | ||
9072.2.a.bs | 3 | 4.b | odd | 2 | 1 | ||
9072.2.a.ca | 3 | 12.b | even | 2 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{3} + 3T_{2}^{2} - 3 \)
acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(567))\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{3} + 3T^{2} - 3 \)
$3$
\( T^{3} \)
$5$
\( T^{3} + 3T^{2} - 3 \)
$7$
\( (T - 1)^{3} \)
$11$
\( T^{3} + 6 T^{2} + 9 T + 3 \)
$13$
\( T^{3} + 3 T^{2} - 33 T - 107 \)
$17$
\( T^{3} + 6 T^{2} + 9 T + 3 \)
$19$
\( T^{3} + 3 T^{2} - 6 T - 17 \)
$23$
\( T^{3} + 12 T^{2} + 27 T - 3 \)
$29$
\( T^{3} + 9 T^{2} - 36 T - 333 \)
$31$
\( T^{3} + 3 T^{2} - 78 T - 323 \)
$37$
\( T^{3} + 3 T^{2} - 78 T - 323 \)
$41$
\( T^{3} - 9T + 9 \)
$43$
\( T^{3} + 3 T^{2} - 6 T + 1 \)
$47$
\( T^{3} + 3 T^{2} - 54 T + 51 \)
$53$
\( T^{3} + 6 T^{2} - 9 T + 3 \)
$59$
\( T^{3} - 3 T^{2} - 72 T - 51 \)
$61$
\( T^{3} - 6 T^{2} - 15 T + 19 \)
$67$
\( T^{3} + 12 T^{2} + 21 T - 17 \)
$71$
\( T^{3} + 9 T^{2} - 54 T + 27 \)
$73$
\( T^{3} + 21 T^{2} + 84 T - 269 \)
$79$
\( T^{3} + 21 T^{2} + 120 T + 181 \)
$83$
\( T^{3} - 18 T^{2} + 45 T - 9 \)
$89$
\( T^{3} + 12 T^{2} - 63 T - 813 \)
$97$
\( T^{3} + 3 T^{2} - 168 T - 323 \)
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