Newspace parameters
| Level: | \( N \) | \(=\) | \( 444 = 2^{2} \cdot 3 \cdot 37 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 444.g (of order \(2\), degree \(1\), minimal) |
Newform invariants
| Self dual: | no |
| Analytic conductor: | \(3.54535784974\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Coefficient field: | \(\Q(\sqrt{3}, \sqrt{-7})\) |
|
|
|
| Defining polynomial: |
\( x^{4} - 2x^{3} - x^{2} + 2x + 22 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 1 \) |
| 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
\( x^{4} - 2x^{3} - x^{2} + 2x + 22 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( \nu^{3} + 8\nu^{2} - 16\nu - 6 ) / 19 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -2\nu^{3} + 3\nu^{2} - 6\nu + 12 ) / 19 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -2\nu^{3} + 3\nu^{2} + 13\nu - 7 ) / 19 \)
|
| \(\nu\) | \(=\) |
\( \beta_{3} - \beta_{2} + 1 \)
|
| \(\nu^{2}\) | \(=\) |
\( 2\beta_{3} - \beta_{2} + 2\beta _1 + 2 \)
|
| \(\nu^{3}\) | \(=\) |
\( -8\beta_{2} + 3\beta _1 + 6 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/444\mathbb{Z}\right)^\times\).
| \(n\) | \(149\) | \(223\) | \(409\) |
| \(\chi(n)\) | \(-1\) | \(-1\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 443.1 |
|
−0.500000 | − | 1.32288i | −1.73205 | −1.50000 | + | 1.32288i | −2.00000 | 0.866025 | + | 2.29129i | 2.64575i | 2.50000 | + | 1.32288i | 3.00000 | 1.00000 | + | 2.64575i | ||||||||||||||||||||
| 443.2 | −0.500000 | − | 1.32288i | 1.73205 | −1.50000 | + | 1.32288i | −2.00000 | −0.866025 | − | 2.29129i | 2.64575i | 2.50000 | + | 1.32288i | 3.00000 | 1.00000 | + | 2.64575i | |||||||||||||||||||||
| 443.3 | −0.500000 | + | 1.32288i | −1.73205 | −1.50000 | − | 1.32288i | −2.00000 | 0.866025 | − | 2.29129i | − | 2.64575i | 2.50000 | − | 1.32288i | 3.00000 | 1.00000 | − | 2.64575i | ||||||||||||||||||||
| 443.4 | −0.500000 | + | 1.32288i | 1.73205 | −1.50000 | − | 1.32288i | −2.00000 | −0.866025 | + | 2.29129i | − | 2.64575i | 2.50000 | − | 1.32288i | 3.00000 | 1.00000 | − | 2.64575i | ||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 4.b | odd | 2 | 1 | inner |
| 111.d | odd | 2 | 1 | inner |
| 444.g | even | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 444.2.g.a | ✓ | 4 |
| 3.b | odd | 2 | 1 | 444.2.g.b | yes | 4 | |
| 4.b | odd | 2 | 1 | inner | 444.2.g.a | ✓ | 4 |
| 12.b | even | 2 | 1 | 444.2.g.b | yes | 4 | |
| 37.b | even | 2 | 1 | 444.2.g.b | yes | 4 | |
| 111.d | odd | 2 | 1 | inner | 444.2.g.a | ✓ | 4 |
| 148.b | odd | 2 | 1 | 444.2.g.b | yes | 4 | |
| 444.g | even | 2 | 1 | inner | 444.2.g.a | ✓ | 4 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 444.2.g.a | ✓ | 4 | 1.a | even | 1 | 1 | trivial |
| 444.2.g.a | ✓ | 4 | 4.b | odd | 2 | 1 | inner |
| 444.2.g.a | ✓ | 4 | 111.d | odd | 2 | 1 | inner |
| 444.2.g.a | ✓ | 4 | 444.g | even | 2 | 1 | inner |
| 444.2.g.b | yes | 4 | 3.b | odd | 2 | 1 | |
| 444.2.g.b | yes | 4 | 12.b | even | 2 | 1 | |
| 444.2.g.b | yes | 4 | 37.b | even | 2 | 1 | |
| 444.2.g.b | yes | 4 | 148.b | odd | 2 | 1 | |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5} + 2 \)
acting on \(S_{2}^{\mathrm{new}}(444, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{2} + T + 2)^{2} \)
|
| $3$ |
\( (T^{2} - 3)^{2} \)
|
| $5$ |
\( (T + 2)^{4} \)
|
| $7$ |
\( (T^{2} + 7)^{2} \)
|
| $11$ |
\( (T^{2} - 3)^{2} \)
|
| $13$ |
\( (T^{2} + 21)^{2} \)
|
| $17$ |
\( (T - 5)^{4} \)
|
| $19$ |
\( (T^{2} - 3)^{2} \)
|
| $23$ |
\( (T^{2} + 7)^{2} \)
|
| $29$ |
\( (T - 4)^{4} \)
|
| $31$ |
\( (T^{2} - 12)^{2} \)
|
| $37$ |
\( (T^{2} - 8 T + 37)^{2} \)
|
| $41$ |
\( (T^{2} + 84)^{2} \)
|
| $43$ |
\( (T^{2} - 12)^{2} \)
|
| $47$ |
\( (T^{2} - 108)^{2} \)
|
| $53$ |
\( (T^{2} + 189)^{2} \)
|
| $59$ |
\( (T^{2} + 28)^{2} \)
|
| $61$ |
\( (T^{2} + 84)^{2} \)
|
| $67$ |
\( (T^{2} + 112)^{2} \)
|
| $71$ |
\( (T^{2} - 12)^{2} \)
|
| $73$ |
\( (T - 9)^{4} \)
|
| $79$ |
\( (T^{2} - 192)^{2} \)
|
| $83$ |
\( (T^{2} - 75)^{2} \)
|
| $89$ |
\( (T + 5)^{4} \)
|
| $97$ |
\( T^{4} \)
|
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