Newspace parameters
| Level: | \( N \) | \(=\) | \( 264 = 2^{3} \cdot 3 \cdot 11 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 264.f (of order \(2\), degree \(1\), minimal) |
Newform invariants
| Self dual: | no |
| Analytic conductor: | \(2.10805061336\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Coefficient field: | \(\Q(i, \sqrt{7})\) |
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|
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| Defining polynomial: |
\( x^{4} - 3x^{2} + 4 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| 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
\( x^{4} - 3x^{2} + 4 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - 1 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( \nu^{3} - \nu ) / 2 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
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| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + 1 \)
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| \(\nu^{3}\) | \(=\) |
\( 2\beta_{3} + \beta_1 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/264\mathbb{Z}\right)^\times\).
| \(n\) | \(89\) | \(133\) | \(145\) | \(199\) |
| \(\chi(n)\) | \(1\) | \(-1\) | \(1\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 133.1 |
|
0.500000 | − | 1.32288i | − | 1.00000i | −1.50000 | − | 1.32288i | − | 1.64575i | −1.32288 | − | 0.500000i | 1.64575 | −2.50000 | + | 1.32288i | −1.00000 | −2.17712 | − | 0.822876i | ||||||||||||||||||
| 133.2 | 0.500000 | − | 1.32288i | 1.00000i | −1.50000 | − | 1.32288i | − | 3.64575i | 1.32288 | + | 0.500000i | −3.64575 | −2.50000 | + | 1.32288i | −1.00000 | −4.82288 | − | 1.82288i | ||||||||||||||||||||
| 133.3 | 0.500000 | + | 1.32288i | − | 1.00000i | −1.50000 | + | 1.32288i | 3.64575i | 1.32288 | − | 0.500000i | −3.64575 | −2.50000 | − | 1.32288i | −1.00000 | −4.82288 | + | 1.82288i | ||||||||||||||||||||
| 133.4 | 0.500000 | + | 1.32288i | 1.00000i | −1.50000 | + | 1.32288i | 1.64575i | −1.32288 | + | 0.500000i | 1.64575 | −2.50000 | − | 1.32288i | −1.00000 | −2.17712 | + | 0.822876i | |||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 8.b | even | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 264.2.f.e | ✓ | 4 |
| 3.b | odd | 2 | 1 | 792.2.f.d | 4 | ||
| 4.b | odd | 2 | 1 | 1056.2.f.d | 4 | ||
| 8.b | even | 2 | 1 | inner | 264.2.f.e | ✓ | 4 |
| 8.d | odd | 2 | 1 | 1056.2.f.d | 4 | ||
| 12.b | even | 2 | 1 | 3168.2.f.d | 4 | ||
| 16.e | even | 4 | 1 | 8448.2.a.ba | 2 | ||
| 16.e | even | 4 | 1 | 8448.2.a.cc | 2 | ||
| 16.f | odd | 4 | 1 | 8448.2.a.bl | 2 | ||
| 16.f | odd | 4 | 1 | 8448.2.a.bp | 2 | ||
| 24.f | even | 2 | 1 | 3168.2.f.d | 4 | ||
| 24.h | odd | 2 | 1 | 792.2.f.d | 4 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 264.2.f.e | ✓ | 4 | 1.a | even | 1 | 1 | trivial |
| 264.2.f.e | ✓ | 4 | 8.b | even | 2 | 1 | inner |
| 792.2.f.d | 4 | 3.b | odd | 2 | 1 | ||
| 792.2.f.d | 4 | 24.h | odd | 2 | 1 | ||
| 1056.2.f.d | 4 | 4.b | odd | 2 | 1 | ||
| 1056.2.f.d | 4 | 8.d | odd | 2 | 1 | ||
| 3168.2.f.d | 4 | 12.b | even | 2 | 1 | ||
| 3168.2.f.d | 4 | 24.f | even | 2 | 1 | ||
| 8448.2.a.ba | 2 | 16.e | even | 4 | 1 | ||
| 8448.2.a.bl | 2 | 16.f | odd | 4 | 1 | ||
| 8448.2.a.bp | 2 | 16.f | odd | 4 | 1 | ||
| 8448.2.a.cc | 2 | 16.e | even | 4 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(264, [\chi])\):
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\( T_{5}^{4} + 16T_{5}^{2} + 36 \)
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\( T_{7}^{2} + 2T_{7} - 6 \)
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Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{2} - T + 2)^{2} \)
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| $3$ |
\( (T^{2} + 1)^{2} \)
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| $5$ |
\( T^{4} + 16T^{2} + 36 \)
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| $7$ |
\( (T^{2} + 2 T - 6)^{2} \)
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| $11$ |
\( (T^{2} + 1)^{2} \)
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| $13$ |
\( T^{4} + 16T^{2} + 36 \)
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| $17$ |
\( (T^{2} - 4 T - 24)^{2} \)
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| $19$ |
\( (T^{2} + 16)^{2} \)
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| $23$ |
\( (T^{2} + 2 T - 6)^{2} \)
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| $29$ |
\( T^{4} + 64T^{2} + 576 \)
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| $31$ |
\( (T - 8)^{4} \)
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| $37$ |
\( (T^{2} + 16)^{2} \)
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| $41$ |
\( (T - 2)^{4} \)
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| $43$ |
\( (T^{2} + 64)^{2} \)
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| $47$ |
\( (T^{2} - 14 T + 42)^{2} \)
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| $53$ |
\( T^{4} + 64T^{2} + 324 \)
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| $59$ |
\( (T^{2} + 64)^{2} \)
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| $61$ |
\( T^{4} + 64T^{2} + 324 \)
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| $67$ |
\( T^{4} + 184T^{2} + 1296 \)
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| $71$ |
\( (T^{2} - 10 T + 18)^{2} \)
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| $73$ |
\( (T^{2} + 4 T - 108)^{2} \)
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| $79$ |
\( (T^{2} + 6 T + 2)^{2} \)
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| $83$ |
\( (T^{2} + 28)^{2} \)
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| $89$ |
\( (T^{2} + 4 T - 108)^{2} \)
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| $97$ |
\( (T^{2} - 12 T + 8)^{2} \)
|
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