Newspace parameters
| Level: | \( N \) | \(=\) | \( 1452 = 2^{2} \cdot 3 \cdot 11^{2} \) |
| Weight: | \( k \) | \(=\) | \( 3 \) |
| Character orbit: | \([\chi]\) | \(=\) | 1452.e (of order \(2\), degree \(1\), minimal) |
Newform invariants
| Self dual: | no |
| Analytic conductor: | \(39.5641343851\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Coefficient field: | \(\Q(\sqrt{-3}, \sqrt{-11})\) |
|
|
|
| Defining polynomial: |
\( x^{4} - x^{3} - 2x^{2} - 3x + 9 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 2^{4} \) |
| 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
\( x^{4} - x^{3} - 2x^{2} - 3x + 9 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( \nu^{3} + 2\nu^{2} - 7 ) / 2 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -5\nu^{3} - 4\nu^{2} + 10\nu + 21 ) / 6 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 4\nu^{3} - \nu^{2} + \nu - 15 ) / 3 \)
|
| \(\nu\) | \(=\) |
\( ( 3\beta_{3} + 9\beta_{2} + 7\beta _1 + 8 ) / 16 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( -5\beta_{3} + \beta_{2} + 15\beta _1 + 24 ) / 16 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 5\beta_{3} - \beta_{2} + \beta _1 + 32 ) / 8 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1452\mathbb{Z}\right)^\times\).
| \(n\) | \(485\) | \(727\) | \(1333\) |
| \(\chi(n)\) | \(-1\) | \(1\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 485.1 |
|
0 | −2.68614 | − | 1.33591i | 0 | − | 5.84096i | 0 | 0 | 0 | 5.43070 | + | 7.17687i | 0 | |||||||||||||||||||||||||
| 485.2 | 0 | −2.68614 | + | 1.33591i | 0 | 5.84096i | 0 | 0 | 0 | 5.43070 | − | 7.17687i | 0 | |||||||||||||||||||||||||||
| 485.3 | 0 | 0.186141 | − | 2.99422i | 0 | 4.10891i | 0 | 0 | 0 | −8.93070 | − | 1.11469i | 0 | |||||||||||||||||||||||||||
| 485.4 | 0 | 0.186141 | + | 2.99422i | 0 | − | 4.10891i | 0 | 0 | 0 | −8.93070 | + | 1.11469i | 0 | ||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 11.b | odd | 2 | 1 | CM by \(\Q(\sqrt{-11}) \) |
| 3.b | odd | 2 | 1 | inner |
| 33.d | even | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 1452.3.e.g | ✓ | 4 |
| 3.b | odd | 2 | 1 | inner | 1452.3.e.g | ✓ | 4 |
| 11.b | odd | 2 | 1 | CM | 1452.3.e.g | ✓ | 4 |
| 33.d | even | 2 | 1 | inner | 1452.3.e.g | ✓ | 4 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 1452.3.e.g | ✓ | 4 | 1.a | even | 1 | 1 | trivial |
| 1452.3.e.g | ✓ | 4 | 3.b | odd | 2 | 1 | inner |
| 1452.3.e.g | ✓ | 4 | 11.b | odd | 2 | 1 | CM |
| 1452.3.e.g | ✓ | 4 | 33.d | even | 2 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{3}^{\mathrm{new}}(1452, [\chi])\):
|
\( T_{5}^{4} + 51T_{5}^{2} + 576 \)
|
|
\( T_{7} \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} \)
|
| $3$ |
\( T^{4} + 5 T^{3} + \cdots + 81 \)
|
| $5$ |
\( T^{4} + 51T^{2} + 576 \)
|
| $7$ |
\( T^{4} \)
|
| $11$ |
\( T^{4} \)
|
| $13$ |
\( T^{4} \)
|
| $17$ |
\( T^{4} \)
|
| $19$ |
\( T^{4} \)
|
| $23$ |
\( T^{4} + 2283 T^{2} + 484416 \)
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| $29$ |
\( T^{4} \)
|
| $31$ |
\( (T^{2} + 37 T - 1514)^{2} \)
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| $37$ |
\( (T^{2} + 25 T - 3482)^{2} \)
|
| $41$ |
\( T^{4} \)
|
| $43$ |
\( T^{4} \)
|
| $47$ |
\( (T^{2} + 6336)^{2} \)
|
| $53$ |
\( (T^{2} + 6336)^{2} \)
|
| $59$ |
\( T^{4} + 18411 T^{2} + 63489024 \)
|
| $61$ |
\( T^{4} \)
|
| $67$ |
\( (T^{2} - 35 T - 12242)^{2} \)
|
| $71$ |
\( T^{4} + 27771 T^{2} + 159971904 \)
|
| $73$ |
\( T^{4} \)
|
| $79$ |
\( T^{4} \)
|
| $83$ |
\( T^{4} \)
|
| $89$ |
\( T^{4} + 25251 T^{2} + 2214144 \)
|
| $97$ |
\( (T^{2} + 95 T - 19202)^{2} \)
|
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