Newspace parameters
| Level: | \( N \) | \(=\) | \( 184 = 2^{3} \cdot 23 \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 184.h (of order \(2\), degree \(1\), minimal) |
Newform invariants
| Self dual: | no |
| Analytic conductor: | \(10.8563514411\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Coefficient field: | \(\Q(\sqrt{-3}, \sqrt{-23})\) |
|
|
|
| Defining polynomial: |
\( x^{4} - x^{3} - 5x^{2} - 6x + 36 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 2 \) |
| Twist minimal: | yes |
| 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 in terms of a root \(\nu\) of
\( x^{4} - x^{3} - 5x^{2} - 6x + 36 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{3} + 5\nu^{2} - 5\nu - 21 ) / 15 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -\nu^{3} + 5\nu + 6 ) / 5 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{3} + 3\beta_{2} + 3 \)
|
| \(\nu^{3}\) | \(=\) |
\( -5\beta_{3} + 5\beta _1 + 6 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/184\mathbb{Z}\right)^\times\).
| \(n\) | \(47\) | \(93\) | \(97\) |
| \(\chi(n)\) | \(-1\) | \(-1\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 91.1 |
|
−2.82666 | − | 0.100080i | −6.30662 | 7.97997 | + | 0.565785i | 0 | 17.8267 | + | 0.631168i | 0 | −22.5000 | − | 2.39792i | 12.7735 | 0 | ||||||||||||||||||||||
| 91.2 | −2.82666 | + | 0.100080i | −6.30662 | 7.97997 | − | 0.565785i | 0 | 17.8267 | − | 0.631168i | 0 | −22.5000 | + | 2.39792i | 12.7735 | 0 | |||||||||||||||||||||||
| 91.3 | 1.32666 | − | 2.49800i | 10.3066 | −4.47997 | − | 6.62796i | 0 | 13.6733 | − | 25.7459i | 0 | −22.5000 | + | 2.39792i | 79.2265 | 0 | |||||||||||||||||||||||
| 91.4 | 1.32666 | + | 2.49800i | 10.3066 | −4.47997 | + | 6.62796i | 0 | 13.6733 | + | 25.7459i | 0 | −22.5000 | − | 2.39792i | 79.2265 | 0 | |||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 23.b | odd | 2 | 1 | CM by \(\Q(\sqrt{-23}) \) |
| 8.d | odd | 2 | 1 | inner |
| 184.h | even | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 184.4.h.b | ✓ | 4 |
| 4.b | odd | 2 | 1 | 736.4.h.b | 4 | ||
| 8.b | even | 2 | 1 | 736.4.h.b | 4 | ||
| 8.d | odd | 2 | 1 | inner | 184.4.h.b | ✓ | 4 |
| 23.b | odd | 2 | 1 | CM | 184.4.h.b | ✓ | 4 |
| 92.b | even | 2 | 1 | 736.4.h.b | 4 | ||
| 184.e | odd | 2 | 1 | 736.4.h.b | 4 | ||
| 184.h | even | 2 | 1 | inner | 184.4.h.b | ✓ | 4 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 184.4.h.b | ✓ | 4 | 1.a | even | 1 | 1 | trivial |
| 184.4.h.b | ✓ | 4 | 8.d | odd | 2 | 1 | inner |
| 184.4.h.b | ✓ | 4 | 23.b | odd | 2 | 1 | CM |
| 184.4.h.b | ✓ | 4 | 184.h | even | 2 | 1 | inner |
| 736.4.h.b | 4 | 4.b | odd | 2 | 1 | ||
| 736.4.h.b | 4 | 8.b | even | 2 | 1 | ||
| 736.4.h.b | 4 | 92.b | even | 2 | 1 | ||
| 736.4.h.b | 4 | 184.e | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{2} - 4T_{3} - 65 \)
acting on \(S_{4}^{\mathrm{new}}(184, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} + 3 T^{3} + \cdots + 64 \)
|
| $3$ |
\( (T^{2} - 4 T - 65)^{2} \)
|
| $5$ |
\( T^{4} \)
|
| $7$ |
\( T^{4} \)
|
| $11$ |
\( T^{4} \)
|
| $13$ |
\( T^{4} + 9870 T^{2} + 10751841 \)
|
| $17$ |
\( T^{4} \)
|
| $19$ |
\( T^{4} \)
|
| $23$ |
\( (T^{2} + 12167)^{2} \)
|
| $29$ |
\( T^{4} + \cdots + 3039868225 \)
|
| $31$ |
\( T^{4} + \cdots + 7840217025 \)
|
| $37$ |
\( T^{4} \)
|
| $41$ |
\( (T^{2} - 426 T - 25287)^{2} \)
|
| $43$ |
\( T^{4} \)
|
| $47$ |
\( T^{4} + \cdots + 10306107361 \)
|
| $53$ |
\( T^{4} \)
|
| $59$ |
\( (T + 396)^{4} \)
|
| $61$ |
\( T^{4} \)
|
| $67$ |
\( T^{4} \)
|
| $71$ |
\( T^{4} + \cdots + 1050758254225 \)
|
| $73$ |
\( (T^{2} - 1226 T + 336025)^{2} \)
|
| $79$ |
\( T^{4} \)
|
| $83$ |
\( T^{4} \)
|
| $89$ |
\( T^{4} \)
|
| $97$ |
\( T^{4} \)
|
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