Newspace parameters
| Level: | \( N \) | \(=\) | \( 2304 = 2^{8} \cdot 3^{2} \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 2304.f (of order \(2\), degree \(1\), not minimal) |
Newform invariants
| Self dual: | no |
| Analytic conductor: | \(18.3975326257\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Coefficient field: | \(\Q(\zeta_{8})\) |
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| Defining polynomial: |
\( x^{4} + 1 \)
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| Coefficient ring: | \(\Z[a_1, \ldots, a_{17}]\) |
| Coefficient ring index: | \( 2^{2}\cdot 3^{2} \) |
| Twist minimal: | no (minimal twist has level 144) |
| Sato-Tate group: | $\mathrm{U}(1)[D_{2}]$ |
$q$-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
| \(\beta_{1}\) | \(=\) |
\( 2\zeta_{8}^{2} \)
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| \(\beta_{2}\) | \(=\) |
\( 3\zeta_{8}^{3} + 3\zeta_{8} \)
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| \(\beta_{3}\) | \(=\) |
\( -3\zeta_{8}^{3} + 3\zeta_{8} \)
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| \(\zeta_{8}\) | \(=\) |
\( ( \beta_{3} + \beta_{2} ) / 6 \)
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| \(\zeta_{8}^{2}\) | \(=\) |
\( ( \beta_1 ) / 2 \)
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| \(\zeta_{8}^{3}\) | \(=\) |
\( ( -\beta_{3} + \beta_{2} ) / 6 \)
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Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2304\mathbb{Z}\right)^\times\).
| \(n\) | \(1279\) | \(1793\) | \(2053\) |
| \(\chi(n)\) | \(-1\) | \(-1\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 1151.1 |
|
0 | 0 | 0 | −4.24264 | 0 | 0 | 0 | 0 | 0 | ||||||||||||||||||||||||||||||
| 1151.2 | 0 | 0 | 0 | −4.24264 | 0 | 0 | 0 | 0 | 0 | |||||||||||||||||||||||||||||||
| 1151.3 | 0 | 0 | 0 | 4.24264 | 0 | 0 | 0 | 0 | 0 | |||||||||||||||||||||||||||||||
| 1151.4 | 0 | 0 | 0 | 4.24264 | 0 | 0 | 0 | 0 | 0 | |||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 4.b | odd | 2 | 1 | CM by \(\Q(\sqrt{-1}) \) |
| 3.b | odd | 2 | 1 | inner |
| 8.b | even | 2 | 1 | inner |
| 8.d | odd | 2 | 1 | inner |
| 12.b | even | 2 | 1 | inner |
| 24.f | even | 2 | 1 | inner |
| 24.h | odd | 2 | 1 | inner |
Twists
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(2304, [\chi])\):
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\( T_{5}^{2} - 18 \)
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\( T_{7} \)
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\( T_{23} \)
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\( T_{43} \)
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Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} \)
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| $3$ |
\( T^{4} \)
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| $5$ |
\( (T^{2} - 18)^{2} \)
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| $7$ |
\( T^{4} \)
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| $11$ |
\( T^{4} \)
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| $13$ |
\( (T^{2} + 16)^{2} \)
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| $17$ |
\( (T^{2} + 18)^{2} \)
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| $19$ |
\( T^{4} \)
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| $23$ |
\( T^{4} \)
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| $29$ |
\( (T^{2} - 18)^{2} \)
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| $31$ |
\( T^{4} \)
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| $37$ |
\( (T^{2} + 4)^{2} \)
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| $41$ |
\( (T^{2} + 162)^{2} \)
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| $43$ |
\( T^{4} \)
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| $47$ |
\( T^{4} \)
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| $53$ |
\( (T^{2} - 162)^{2} \)
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| $59$ |
\( T^{4} \)
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| $61$ |
\( (T^{2} + 100)^{2} \)
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| $67$ |
\( T^{4} \)
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| $71$ |
\( T^{4} \)
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| $73$ |
\( (T + 16)^{4} \)
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| $79$ |
\( T^{4} \)
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| $83$ |
\( T^{4} \)
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| $89$ |
\( (T^{2} + 18)^{2} \)
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| $97$ |
\( (T + 8)^{4} \)
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