Newspace parameters
| Level: | \( N \) | \(=\) | \( 567 = 3^{4} \cdot 7 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 567.f (of order \(3\), degree \(2\), not minimal) |
Newform invariants
| Self dual: | no |
| Analytic conductor: | \(4.52751779461\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Relative dimension: | \(2\) over \(\Q(\zeta_{3})\) |
| Coefficient field: | \(\Q(\zeta_{12})\) |
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| Defining polynomial: |
\( x^{4} - x^{2} + 1 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{4}]\) |
| Coefficient ring index: | \( 3 \) |
| Twist minimal: | no (minimal twist has level 189) |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{3}]$ |
$q$-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
| \(\beta_{1}\) | \(=\) |
\( \zeta_{12}^{2} \)
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| \(\beta_{2}\) | \(=\) |
\( \zeta_{12}^{3} + \zeta_{12} \)
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| \(\beta_{3}\) | \(=\) |
\( -\zeta_{12}^{3} + 2\zeta_{12} \)
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| \(\zeta_{12}\) | \(=\) |
\( ( \beta_{3} + \beta_{2} ) / 3 \)
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| \(\zeta_{12}^{2}\) | \(=\) |
\( \beta_1 \)
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| \(\zeta_{12}^{3}\) | \(=\) |
\( ( -\beta_{3} + 2\beta_{2} ) / 3 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/567\mathbb{Z}\right)^\times\).
| \(n\) | \(325\) | \(407\) |
| \(\chi(n)\) | \(1\) | \(-1 + \beta_{1}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 190.1 |
|
−0.866025 | − | 1.50000i | 0 | −0.500000 | + | 0.866025i | −0.866025 | + | 1.50000i | 0 | −0.500000 | − | 0.866025i | −1.73205 | 0 | 3.00000 | ||||||||||||||||||||||
| 190.2 | 0.866025 | + | 1.50000i | 0 | −0.500000 | + | 0.866025i | 0.866025 | − | 1.50000i | 0 | −0.500000 | − | 0.866025i | 1.73205 | 0 | 3.00000 | |||||||||||||||||||||||
| 379.1 | −0.866025 | + | 1.50000i | 0 | −0.500000 | − | 0.866025i | −0.866025 | − | 1.50000i | 0 | −0.500000 | + | 0.866025i | −1.73205 | 0 | 3.00000 | |||||||||||||||||||||||
| 379.2 | 0.866025 | − | 1.50000i | 0 | −0.500000 | − | 0.866025i | 0.866025 | + | 1.50000i | 0 | −0.500000 | + | 0.866025i | 1.73205 | 0 | 3.00000 | |||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 3.b | odd | 2 | 1 | inner |
| 9.c | even | 3 | 1 | inner |
| 9.d | odd | 6 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 567.2.f.k | 4 | |
| 3.b | odd | 2 | 1 | inner | 567.2.f.k | 4 | |
| 9.c | even | 3 | 1 | 189.2.a.e | ✓ | 2 | |
| 9.c | even | 3 | 1 | inner | 567.2.f.k | 4 | |
| 9.d | odd | 6 | 1 | 189.2.a.e | ✓ | 2 | |
| 9.d | odd | 6 | 1 | inner | 567.2.f.k | 4 | |
| 36.f | odd | 6 | 1 | 3024.2.a.bg | 2 | ||
| 36.h | even | 6 | 1 | 3024.2.a.bg | 2 | ||
| 45.h | odd | 6 | 1 | 4725.2.a.ba | 2 | ||
| 45.j | even | 6 | 1 | 4725.2.a.ba | 2 | ||
| 63.l | odd | 6 | 1 | 1323.2.a.t | 2 | ||
| 63.o | even | 6 | 1 | 1323.2.a.t | 2 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 189.2.a.e | ✓ | 2 | 9.c | even | 3 | 1 | |
| 189.2.a.e | ✓ | 2 | 9.d | odd | 6 | 1 | |
| 567.2.f.k | 4 | 1.a | even | 1 | 1 | trivial | |
| 567.2.f.k | 4 | 3.b | odd | 2 | 1 | inner | |
| 567.2.f.k | 4 | 9.c | even | 3 | 1 | inner | |
| 567.2.f.k | 4 | 9.d | odd | 6 | 1 | inner | |
| 1323.2.a.t | 2 | 63.l | odd | 6 | 1 | ||
| 1323.2.a.t | 2 | 63.o | even | 6 | 1 | ||
| 3024.2.a.bg | 2 | 36.f | odd | 6 | 1 | ||
| 3024.2.a.bg | 2 | 36.h | even | 6 | 1 | ||
| 4725.2.a.ba | 2 | 45.h | odd | 6 | 1 | ||
| 4725.2.a.ba | 2 | 45.j | even | 6 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(567, [\chi])\):
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\( T_{2}^{4} + 3T_{2}^{2} + 9 \)
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\( T_{5}^{4} + 3T_{5}^{2} + 9 \)
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Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} + 3T^{2} + 9 \)
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| $3$ |
\( T^{4} \)
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| $5$ |
\( T^{4} + 3T^{2} + 9 \)
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| $7$ |
\( (T^{2} + T + 1)^{2} \)
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| $11$ |
\( T^{4} + 3T^{2} + 9 \)
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| $13$ |
\( (T^{2} + 2 T + 4)^{2} \)
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| $17$ |
\( (T^{2} - 48)^{2} \)
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| $19$ |
\( (T - 5)^{4} \)
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| $23$ |
\( T^{4} + 3T^{2} + 9 \)
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| $29$ |
\( T^{4} + 108 T^{2} + 11664 \)
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| $31$ |
\( (T^{2} + 5 T + 25)^{2} \)
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| $37$ |
\( (T + 7)^{4} \)
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| $41$ |
\( T^{4} + 27T^{2} + 729 \)
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| $43$ |
\( (T^{2} - 4 T + 16)^{2} \)
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| $47$ |
\( T^{4} + 48T^{2} + 2304 \)
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| $53$ |
\( (T^{2} - 192)^{2} \)
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| $59$ |
\( T^{4} + 48T^{2} + 2304 \)
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| $61$ |
\( (T^{2} + 8 T + 64)^{2} \)
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| $67$ |
\( (T^{2} + 14 T + 196)^{2} \)
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| $71$ |
\( (T^{2} - 27)^{2} \)
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| $73$ |
\( (T + 4)^{4} \)
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| $79$ |
\( (T^{2} + 8 T + 64)^{2} \)
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| $83$ |
\( T^{4} + 108 T^{2} + 11664 \)
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| $89$ |
\( (T^{2} - 75)^{2} \)
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| $97$ |
\( (T^{2} - 4 T + 16)^{2} \)
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