Newspace parameters
| Level: | \( N \) | \(=\) | \( 288 = 2^{5} \cdot 3^{2} \) |
| Weight: | \( k \) | \(=\) | \( 6 \) |
| Character orbit: | \([\chi]\) | \(=\) | 288.d (of order \(2\), degree \(1\), not minimal) |
Newform invariants
| Self dual: | no |
| Analytic conductor: | \(46.1905401061\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Coefficient field: | \(\Q(\sqrt{-54 +2 \sqrt{73}})\) |
|
|
|
| Defining polynomial: |
\( x^{4} - x^{3} - 2x^{2} - 8x + 64 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{11}]\) |
| Coefficient ring index: | \( 2^{12} \) |
| Twist minimal: | no (minimal twist has level 8) |
| 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
\( x^{4} - x^{3} - 2x^{2} - 8x + 64 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( \nu^{3} + 3\nu^{2} + 2\nu - 12 ) / 2 \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{3} - 5\nu^{2} + 10\nu - 4 \)
|
| \(\beta_{3}\) | \(=\) |
\( -4\nu^{3} + 4\nu^{2} + 40\nu + 16 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{3} + 2\beta_{2} + 4\beta _1 + 16 ) / 64 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( \beta_{3} - 6\beta_{2} + 20\beta _1 + 80 ) / 64 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( -5\beta_{3} + 14\beta_{2} + 60\beta _1 + 496 ) / 64 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/288\mathbb{Z}\right)^\times\).
| \(n\) | \(37\) | \(65\) | \(127\) |
| \(\chi(n)\) | \(-1\) | \(1\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 145.1 |
|
0 | 0 | 0 | − | 73.9600i | 0 | 112.704 | 0 | 0 | 0 | |||||||||||||||||||||||||||||
| 145.2 | 0 | 0 | 0 | − | 1.38521i | 0 | −160.704 | 0 | 0 | 0 | ||||||||||||||||||||||||||||||
| 145.3 | 0 | 0 | 0 | 1.38521i | 0 | −160.704 | 0 | 0 | 0 | |||||||||||||||||||||||||||||||
| 145.4 | 0 | 0 | 0 | 73.9600i | 0 | 112.704 | 0 | 0 | 0 | |||||||||||||||||||||||||||||||
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 | 288.6.d.b | 4 | |
| 3.b | odd | 2 | 1 | 32.6.b.a | 4 | ||
| 4.b | odd | 2 | 1 | 72.6.d.b | 4 | ||
| 8.b | even | 2 | 1 | inner | 288.6.d.b | 4 | |
| 8.d | odd | 2 | 1 | 72.6.d.b | 4 | ||
| 12.b | even | 2 | 1 | 8.6.b.a | ✓ | 4 | |
| 15.d | odd | 2 | 1 | 800.6.d.a | 4 | ||
| 15.e | even | 4 | 2 | 800.6.f.a | 8 | ||
| 24.f | even | 2 | 1 | 8.6.b.a | ✓ | 4 | |
| 24.h | odd | 2 | 1 | 32.6.b.a | 4 | ||
| 48.i | odd | 4 | 2 | 256.6.a.n | 4 | ||
| 48.k | even | 4 | 2 | 256.6.a.k | 4 | ||
| 60.h | even | 2 | 1 | 200.6.d.a | 4 | ||
| 60.l | odd | 4 | 2 | 200.6.f.a | 8 | ||
| 120.i | odd | 2 | 1 | 800.6.d.a | 4 | ||
| 120.m | even | 2 | 1 | 200.6.d.a | 4 | ||
| 120.q | odd | 4 | 2 | 200.6.f.a | 8 | ||
| 120.w | even | 4 | 2 | 800.6.f.a | 8 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 8.6.b.a | ✓ | 4 | 12.b | even | 2 | 1 | |
| 8.6.b.a | ✓ | 4 | 24.f | even | 2 | 1 | |
| 32.6.b.a | 4 | 3.b | odd | 2 | 1 | ||
| 32.6.b.a | 4 | 24.h | odd | 2 | 1 | ||
| 72.6.d.b | 4 | 4.b | odd | 2 | 1 | ||
| 72.6.d.b | 4 | 8.d | odd | 2 | 1 | ||
| 200.6.d.a | 4 | 60.h | even | 2 | 1 | ||
| 200.6.d.a | 4 | 120.m | even | 2 | 1 | ||
| 200.6.f.a | 8 | 60.l | odd | 4 | 2 | ||
| 200.6.f.a | 8 | 120.q | odd | 4 | 2 | ||
| 256.6.a.k | 4 | 48.k | even | 4 | 2 | ||
| 256.6.a.n | 4 | 48.i | odd | 4 | 2 | ||
| 288.6.d.b | 4 | 1.a | even | 1 | 1 | trivial | |
| 288.6.d.b | 4 | 8.b | even | 2 | 1 | inner | |
| 800.6.d.a | 4 | 15.d | odd | 2 | 1 | ||
| 800.6.d.a | 4 | 120.i | odd | 2 | 1 | ||
| 800.6.f.a | 8 | 15.e | even | 4 | 2 | ||
| 800.6.f.a | 8 | 120.w | even | 4 | 2 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{4} + 5472T_{5}^{2} + 10496 \)
acting on \(S_{6}^{\mathrm{new}}(288, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} \)
|
| $3$ |
\( T^{4} \)
|
| $5$ |
\( T^{4} + 5472 T^{2} + 10496 \)
|
| $7$ |
\( (T^{2} + 48 T - 18112)^{2} \)
|
| $11$ |
\( T^{4} + \cdots + 5520765456 \)
|
| $13$ |
\( T^{4} + \cdots + 7999305984 \)
|
| $17$ |
\( (T^{2} + 100 T - 72252)^{2} \)
|
| $19$ |
\( T^{4} + \cdots + 120994976016 \)
|
| $23$ |
\( (T^{2} - 1168 T - 2817216)^{2} \)
|
| $29$ |
\( T^{4} + \cdots + 535633608132864 \)
|
| $31$ |
\( (T^{2} - 6464 T + 7754752)^{2} \)
|
| $37$ |
\( T^{4} + \cdots + 306881230162176 \)
|
| $41$ |
\( (T^{2} - 2284 T - 85109148)^{2} \)
|
| $43$ |
\( T^{4} + \cdots + 23\!\cdots\!56 \)
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| $47$ |
\( (T^{2} + 27360 T + 132648192)^{2} \)
|
| $53$ |
\( T^{4} + \cdots + 76\!\cdots\!44 \)
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| $59$ |
\( T^{4} + \cdots + 22\!\cdots\!36 \)
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| $61$ |
\( T^{4} + \cdots + 41\!\cdots\!16 \)
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| $67$ |
\( T^{4} + \cdots + 32\!\cdots\!84 \)
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| $71$ |
\( (T^{2} - 103344 T + 2609278272)^{2} \)
|
| $73$ |
\( (T^{2} - 19988 T - 1604540316)^{2} \)
|
| $79$ |
\( (T^{2} - 123936 T + 3701816576)^{2} \)
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| $83$ |
\( T^{4} + \cdots + 72\!\cdots\!56 \)
|
| $89$ |
\( (T^{2} - 42316 T - 6875717724)^{2} \)
|
| $97$ |
\( (T^{2} + 49788 T - 783309052)^{2} \)
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