Newspace parameters
| Level: | \( N \) | \(=\) | \( 288 = 2^{5} \cdot 3^{2} \) |
| Weight: | \( k \) | \(=\) | \( 5 \) |
| Character orbit: | \([\chi]\) | \(=\) | 288.e (of order \(2\), degree \(1\), minimal) |
Newform invariants
| Self dual: | no |
| Analytic conductor: | \(29.7705493681\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Coefficient field: | \(\Q(\sqrt{-2}, \sqrt{7})\) |
|
|
|
| Defining polynomial: |
\( x^{4} + 8x^{2} + 9 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{17}]\) |
| Coefficient ring index: | \( 2^{8} \) |
| 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
\( x^{4} + 8x^{2} + 9 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( \nu^{3} + 5\nu ) / 3 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 8\nu^{3} + 88\nu ) / 3 \)
|
| \(\beta_{3}\) | \(=\) |
\( 16\nu^{2} + 64 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{2} - 8\beta_1 ) / 16 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( \beta_{3} - 64 ) / 16 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( -5\beta_{2} + 88\beta_1 ) / 16 \)
|
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 161.1 |
|
0 | 0 | 0 | − | 42.6612i | 0 | −66.3320 | 0 | 0 | 0 | |||||||||||||||||||||||||||||
| 161.2 | 0 | 0 | 0 | − | 17.2053i | 0 | 18.3320 | 0 | 0 | 0 | ||||||||||||||||||||||||||||||
| 161.3 | 0 | 0 | 0 | 17.2053i | 0 | 18.3320 | 0 | 0 | 0 | |||||||||||||||||||||||||||||||
| 161.4 | 0 | 0 | 0 | 42.6612i | 0 | −66.3320 | 0 | 0 | 0 | |||||||||||||||||||||||||||||||
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 | 288.5.e.c | ✓ | 4 |
| 3.b | odd | 2 | 1 | inner | 288.5.e.c | ✓ | 4 |
| 4.b | odd | 2 | 1 | 288.5.e.e | yes | 4 | |
| 8.b | even | 2 | 1 | 576.5.e.k | 4 | ||
| 8.d | odd | 2 | 1 | 576.5.e.m | 4 | ||
| 12.b | even | 2 | 1 | 288.5.e.e | yes | 4 | |
| 24.f | even | 2 | 1 | 576.5.e.m | 4 | ||
| 24.h | odd | 2 | 1 | 576.5.e.k | 4 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 288.5.e.c | ✓ | 4 | 1.a | even | 1 | 1 | trivial |
| 288.5.e.c | ✓ | 4 | 3.b | odd | 2 | 1 | inner |
| 288.5.e.e | yes | 4 | 4.b | odd | 2 | 1 | |
| 288.5.e.e | yes | 4 | 12.b | even | 2 | 1 | |
| 576.5.e.k | 4 | 8.b | even | 2 | 1 | ||
| 576.5.e.k | 4 | 24.h | odd | 2 | 1 | ||
| 576.5.e.m | 4 | 8.d | odd | 2 | 1 | ||
| 576.5.e.m | 4 | 24.f | even | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{5}^{\mathrm{new}}(288, [\chi])\):
|
\( T_{5}^{4} + 2116T_{5}^{2} + 538756 \)
|
|
\( T_{7}^{2} + 48T_{7} - 1216 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} \)
|
| $3$ |
\( T^{4} \)
|
| $5$ |
\( T^{4} + 2116 T^{2} + 538756 \)
|
| $7$ |
\( (T^{2} + 48 T - 1216)^{2} \)
|
| $11$ |
\( T^{4} + 70912 T^{2} + 844251136 \)
|
| $13$ |
\( (T^{2} - 128 T - 12032)^{2} \)
|
| $17$ |
\( T^{4} + 68868 T^{2} + 904686084 \)
|
| $19$ |
\( (T^{2} + 576 T + 54272)^{2} \)
|
| $23$ |
\( T^{4} + \cdots + 11478122496 \)
|
| $29$ |
\( T^{4} + \cdots + 21003545476 \)
|
| $31$ |
\( (T^{2} + 1392 T - 162496)^{2} \)
|
| $37$ |
\( (T^{2} - 1628 T - 369596)^{2} \)
|
| $41$ |
\( T^{4} + \cdots + 65963703556 \)
|
| $43$ |
\( (T^{2} + 4320 T + 3798272)^{2} \)
|
| $47$ |
\( T^{4} + \cdots + 12535905009664 \)
|
| $53$ |
\( T^{4} + \cdots + 9710665045636 \)
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| $59$ |
\( T^{4} + \cdots + 25191726186496 \)
|
| $61$ |
\( (T^{2} + 156 T - 25798716)^{2} \)
|
| $67$ |
\( (T^{2} + 11808 T + 34505984)^{2} \)
|
| $71$ |
\( T^{4} + \cdots + 6827058282496 \)
|
| $73$ |
\( (T^{2} + 8064 T + 15998976)^{2} \)
|
| $79$ |
\( (T^{2} + 6672 T - 20569792)^{2} \)
|
| $83$ |
\( T^{4} + \cdots + 202642741149696 \)
|
| $89$ |
\( T^{4} + \cdots + 23\!\cdots\!64 \)
|
| $97$ |
\( (T^{2} + 5632 T - 32390144)^{2} \)
|
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