Newspace parameters
| Level: | \( N \) | \(=\) | \( 444 = 2^{2} \cdot 3 \cdot 37 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 444.m (of order \(4\), degree \(2\), minimal) |
Newform invariants
| Self dual: | no |
| Analytic conductor: | \(3.54535784974\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Relative dimension: | \(2\) over \(\Q(i)\) |
| Coefficient field: | \(\Q(\zeta_{12})\) |
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| Defining polynomial: |
\( x^{4} - x^{2} + 1 \)
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| Coefficient ring: | \(\Z[a_1, \ldots, a_{13}]\) |
| Coefficient ring index: | \( 2^{2} \) |
| Twist minimal: | yes |
| Sato-Tate group: | $\mathrm{U}(1)[D_{4}]$ |
$q$-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
| \(\beta_{1}\) | \(=\) |
\( \zeta_{12}^{3} \)
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| \(\beta_{2}\) | \(=\) |
\( 2\zeta_{12}^{2} - 1 \)
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| \(\beta_{3}\) | \(=\) |
\( -\zeta_{12}^{3} + 2\zeta_{12} \)
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| \(\zeta_{12}\) | \(=\) |
\( ( \beta_{3} + \beta_1 ) / 2 \)
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| \(\zeta_{12}^{2}\) | \(=\) |
\( ( \beta_{2} + 1 ) / 2 \)
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| \(\zeta_{12}^{3}\) | \(=\) |
\( \beta_1 \)
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Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/444\mathbb{Z}\right)^\times\).
| \(n\) | \(149\) | \(223\) | \(409\) |
| \(\chi(n)\) | \(-1\) | \(1\) | \(\beta_{1}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 401.1 |
|
0 | − | 1.73205i | 0 | 0 | 0 | 3.46410 | 0 | −3.00000 | 0 | |||||||||||||||||||||||||||||
| 401.2 | 0 | 1.73205i | 0 | 0 | 0 | −3.46410 | 0 | −3.00000 | 0 | |||||||||||||||||||||||||||||||
| 413.1 | 0 | − | 1.73205i | 0 | 0 | 0 | −3.46410 | 0 | −3.00000 | 0 | ||||||||||||||||||||||||||||||
| 413.2 | 0 | 1.73205i | 0 | 0 | 0 | 3.46410 | 0 | −3.00000 | 0 | |||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 3.b | odd | 2 | 1 | CM by \(\Q(\sqrt{-3}) \) |
| 37.d | odd | 4 | 1 | inner |
| 111.g | even | 4 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 444.2.m.b | ✓ | 4 |
| 3.b | odd | 2 | 1 | CM | 444.2.m.b | ✓ | 4 |
| 37.d | odd | 4 | 1 | inner | 444.2.m.b | ✓ | 4 |
| 111.g | even | 4 | 1 | inner | 444.2.m.b | ✓ | 4 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 444.2.m.b | ✓ | 4 | 1.a | even | 1 | 1 | trivial |
| 444.2.m.b | ✓ | 4 | 3.b | odd | 2 | 1 | CM |
| 444.2.m.b | ✓ | 4 | 37.d | odd | 4 | 1 | inner |
| 444.2.m.b | ✓ | 4 | 111.g | even | 4 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(444, [\chi])\):
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\( T_{5} \)
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\( T_{7}^{2} - 12 \)
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Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} \)
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| $3$ |
\( (T^{2} + 3)^{2} \)
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| $5$ |
\( T^{4} \)
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| $7$ |
\( (T^{2} - 12)^{2} \)
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| $11$ |
\( T^{4} \)
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| $13$ |
\( T^{4} - 4 T^{3} + \cdots + 484 \)
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| $17$ |
\( T^{4} \)
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| $19$ |
\( T^{4} + 16 T^{3} + \cdots + 676 \)
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| $23$ |
\( T^{4} \)
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| $29$ |
\( T^{4} \)
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| $31$ |
\( T^{4} + 8 T^{3} + \cdots + 2116 \)
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| $37$ |
\( (T^{2} - 10 T + 37)^{2} \)
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| $41$ |
\( T^{4} \)
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| $43$ |
\( T^{4} + 16 T^{3} + \cdots + 484 \)
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| $47$ |
\( T^{4} \)
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| $53$ |
\( T^{4} \)
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| $59$ |
\( T^{4} \)
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| $61$ |
\( T^{4} - 28 T^{3} + \cdots + 5476 \)
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| $67$ |
\( (T^{2} + 256)^{2} \)
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| $71$ |
\( T^{4} \)
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| $73$ |
\( (T^{2} + 100)^{2} \)
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| $79$ |
\( T^{4} - 8 T^{3} + \cdots + 20164 \)
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| $83$ |
\( T^{4} \)
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| $89$ |
\( T^{4} \)
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| $97$ |
\( T^{4} - 28 T^{3} + \cdots + 4 \)
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