Newspace parameters
| Level: | \( N \) | \(=\) | \( 216 = 2^{3} \cdot 3^{3} \) |
| Weight: | \( k \) | \(=\) | \( 3 \) |
| Character orbit: | \([\chi]\) | \(=\) | 216.h (of order \(2\), degree \(1\), minimal) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(5.88557371018\) |
| Analytic rank: | \(0\) |
| Dimension: | \(2\) |
| Coefficient field: | \(\Q(\sqrt{2}) \) |
| Defining polynomial: | \(x^{2} - 2\) |
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 2\cdot 3 \) |
| Twist minimal: | yes |
| Sato-Tate group: | $\mathrm{U}(1)[D_{2}]$ |
$q$-expansion
Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = 6\sqrt{2}\). We also show the integral \(q\)-expansion of the trace form.
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/216\mathbb{Z}\right)^\times\).
| \(n\) | \(55\) | \(109\) | \(137\) |
| \(\chi(n)\) | \(1\) | \(-1\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 53.1 |
|
−2.00000 | 0 | 4.00000 | −9.48528 | 0 | −3.48528 | −8.00000 | 0 | 18.9706 | ||||||||||||||||||||||||
| 53.2 | −2.00000 | 0 | 4.00000 | 7.48528 | 0 | 13.4853 | −8.00000 | 0 | −14.9706 | |||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 24.h | odd | 2 | 1 | CM by \(\Q(\sqrt{-6}) \) |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 216.3.h.a | ✓ | 2 |
| 3.b | odd | 2 | 1 | 216.3.h.b | yes | 2 | |
| 4.b | odd | 2 | 1 | 864.3.h.a | 2 | ||
| 8.b | even | 2 | 1 | 216.3.h.b | yes | 2 | |
| 8.d | odd | 2 | 1 | 864.3.h.b | 2 | ||
| 12.b | even | 2 | 1 | 864.3.h.b | 2 | ||
| 24.f | even | 2 | 1 | 864.3.h.a | 2 | ||
| 24.h | odd | 2 | 1 | CM | 216.3.h.a | ✓ | 2 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 216.3.h.a | ✓ | 2 | 1.a | even | 1 | 1 | trivial |
| 216.3.h.a | ✓ | 2 | 24.h | odd | 2 | 1 | CM |
| 216.3.h.b | yes | 2 | 3.b | odd | 2 | 1 | |
| 216.3.h.b | yes | 2 | 8.b | even | 2 | 1 | |
| 864.3.h.a | 2 | 4.b | odd | 2 | 1 | ||
| 864.3.h.a | 2 | 24.f | even | 2 | 1 | ||
| 864.3.h.b | 2 | 8.d | odd | 2 | 1 | ||
| 864.3.h.b | 2 | 12.b | even | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{2} + 2 T_{5} - 71 \) acting on \(S_{3}^{\mathrm{new}}(216, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ | \( ( 2 + T )^{2} \) |
| $3$ | \( T^{2} \) |
| $5$ | \( -71 + 2 T + T^{2} \) |
| $7$ | \( -47 - 10 T + T^{2} \) |
| $11$ | \( -263 - 10 T + T^{2} \) |
| $13$ | \( T^{2} \) |
| $17$ | \( T^{2} \) |
| $19$ | \( T^{2} \) |
| $23$ | \( T^{2} \) |
| $29$ | \( ( -50 + T )^{2} \) |
| $31$ | \( -1439 + 38 T + T^{2} \) |
| $37$ | \( T^{2} \) |
| $41$ | \( T^{2} \) |
| $43$ | \( T^{2} \) |
| $47$ | \( T^{2} \) |
| $53$ | \( 409 - 94 T + T^{2} \) |
| $59$ | \( ( 10 + T )^{2} \) |
| $61$ | \( T^{2} \) |
| $67$ | \( T^{2} \) |
| $71$ | \( T^{2} \) |
| $73$ | \( -13487 + 50 T + T^{2} \) |
| $79$ | \( ( 58 + T )^{2} \) |
| $83$ | \( -2711 + 134 T + T^{2} \) |
| $89$ | \( T^{2} \) |
| $97$ | \( 7873 - 190 T + T^{2} \) |
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