Newspace parameters
| Level: | \( N \) | \(=\) | \( 1188 = 2^{2} \cdot 3^{3} \cdot 11 \) |
| Weight: | \( k \) | \(=\) | \( 3 \) |
| Character orbit: | \([\chi]\) | \(=\) | 1188.e (of order \(2\), degree \(1\), minimal) |
Newform invariants
| Self dual: | no |
| Analytic conductor: | \(32.3706554060\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Coefficient field: | \(\Q(\sqrt{-2}, \sqrt{-11})\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{4} - 2x^{3} + 11x^{2} - 10x + 3 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{11}]\) |
| Coefficient ring index: | \( 2^{4} \) |
| 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
\( x^{4} - 2x^{3} + 11x^{2} - 10x + 3 \)
:
| \(\beta_{1}\) | \(=\) |
\( 2\nu^{2} - 2\nu + 10 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 4\nu^{3} - 6\nu^{2} + 38\nu - 18 ) / 3 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 4\nu^{3} - 6\nu^{2} + 44\nu - 21 ) / 3 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{3} - \beta_{2} + 1 ) / 2 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( \beta_{3} - \beta_{2} + \beta _1 - 9 ) / 2 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( -16\beta_{3} + 19\beta_{2} + 3\beta _1 - 28 ) / 4 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1188\mathbb{Z}\right)^\times\).
| \(n\) | \(353\) | \(541\) | \(595\) |
| \(\chi(n)\) | \(-1\) | \(1\) | \(1\) |
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 | 0 | 0 | − | 6.14505i | 0 | 2.00000 | 0 | 0 | 0 | |||||||||||||||||||||||||||||
| 485.2 | 0 | 0 | 0 | − | 0.488198i | 0 | 2.00000 | 0 | 0 | 0 | ||||||||||||||||||||||||||||||
| 485.3 | 0 | 0 | 0 | 0.488198i | 0 | 2.00000 | 0 | 0 | 0 | |||||||||||||||||||||||||||||||
| 485.4 | 0 | 0 | 0 | 6.14505i | 0 | 2.00000 | 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 | 1188.3.e.b | ✓ | 4 |
| 3.b | odd | 2 | 1 | inner | 1188.3.e.b | ✓ | 4 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 1188.3.e.b | ✓ | 4 | 1.a | even | 1 | 1 | trivial |
| 1188.3.e.b | ✓ | 4 | 3.b | odd | 2 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{4} + 38T_{5}^{2} + 9 \)
acting on \(S_{3}^{\mathrm{new}}(1188, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} \)
|
| $3$ |
\( T^{4} \)
|
| $5$ |
\( T^{4} + 38T^{2} + 9 \)
|
| $7$ |
\( (T - 2)^{4} \)
|
| $11$ |
\( (T^{2} + 11)^{2} \)
|
| $13$ |
\( (T^{2} + 10 T - 63)^{2} \)
|
| $17$ |
\( T^{4} + 278 T^{2} + 13689 \)
|
| $19$ |
\( (T^{2} - 24 T - 208)^{2} \)
|
| $23$ |
\( T^{4} + 1334 T^{2} + 13689 \)
|
| $29$ |
\( T^{4} + 3096 T^{2} + 571536 \)
|
| $31$ |
\( (T^{2} - 20 T - 252)^{2} \)
|
| $37$ |
\( (T^{2} - 44 T + 132)^{2} \)
|
| $41$ |
\( T^{4} + 2358 T^{2} + 363609 \)
|
| $43$ |
\( (T^{2} - 5632)^{2} \)
|
| $47$ |
\( (T^{2} + 7200)^{2} \)
|
| $53$ |
\( T^{4} + 3014 T^{2} + 184041 \)
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| $59$ |
\( T^{4} + 3744 T^{2} + 1679616 \)
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| $61$ |
\( (T^{2} + 42 T + 353)^{2} \)
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| $67$ |
\( (T^{2} + 42 T - 8359)^{2} \)
|
| $71$ |
\( T^{4} + 8352 T^{2} + 1016064 \)
|
| $73$ |
\( (T^{2} + 112 T + 1728)^{2} \)
|
| $79$ |
\( (T^{2} - 82 T - 5447)^{2} \)
|
| $83$ |
\( T^{4} + 18902 T^{2} + 9566649 \)
|
| $89$ |
\( T^{4} + 15128 T^{2} + 10575504 \)
|
| $97$ |
\( (T^{2} + 6 T - 343)^{2} \)
|
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