Newspace parameters
| Level: | \( N \) | \(=\) | \( 666 = 2 \cdot 3^{2} \cdot 37 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 666.f (of order \(3\), degree \(2\), minimal) |
Newform invariants
| Self dual: | no |
| Analytic conductor: | \(5.31803677462\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Relative dimension: | \(2\) over \(\Q(\zeta_{3})\) |
| Coefficient field: | \(\Q(\sqrt{-3}, \sqrt{11})\) |
|
|
|
| Defining polynomial: |
\( x^{4} + 11x^{2} + 121 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | yes |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{3}]$ |
$q$-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
\( x^{4} + 11x^{2} + 121 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{2} ) / 11 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( \nu^{3} ) / 11 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( 11\beta_{2} \)
|
| \(\nu^{3}\) | \(=\) |
\( 11\beta_{3} \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/666\mathbb{Z}\right)^\times\).
| \(n\) | \(371\) | \(631\) |
| \(\chi(n)\) | \(1\) | \(-1 - \beta_{2}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 343.1 |
|
0.500000 | − | 0.866025i | 0 | −0.500000 | − | 0.866025i | −0.500000 | − | 0.866025i | 0 | −1.15831 | − | 2.00626i | −1.00000 | 0 | −1.00000 | ||||||||||||||||||||||
| 343.2 | 0.500000 | − | 0.866025i | 0 | −0.500000 | − | 0.866025i | −0.500000 | − | 0.866025i | 0 | 2.15831 | + | 3.73831i | −1.00000 | 0 | −1.00000 | |||||||||||||||||||||||
| 433.1 | 0.500000 | + | 0.866025i | 0 | −0.500000 | + | 0.866025i | −0.500000 | + | 0.866025i | 0 | −1.15831 | + | 2.00626i | −1.00000 | 0 | −1.00000 | |||||||||||||||||||||||
| 433.2 | 0.500000 | + | 0.866025i | 0 | −0.500000 | + | 0.866025i | −0.500000 | + | 0.866025i | 0 | 2.15831 | − | 3.73831i | −1.00000 | 0 | −1.00000 | |||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 37.c | even | 3 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 666.2.f.i | yes | 4 |
| 3.b | odd | 2 | 1 | 666.2.f.h | ✓ | 4 | |
| 37.c | even | 3 | 1 | inner | 666.2.f.i | yes | 4 |
| 111.i | odd | 6 | 1 | 666.2.f.h | ✓ | 4 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 666.2.f.h | ✓ | 4 | 3.b | odd | 2 | 1 | |
| 666.2.f.h | ✓ | 4 | 111.i | odd | 6 | 1 | |
| 666.2.f.i | yes | 4 | 1.a | even | 1 | 1 | trivial |
| 666.2.f.i | yes | 4 | 37.c | even | 3 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(666, [\chi])\):
|
\( T_{5}^{2} + T_{5} + 1 \)
|
|
\( T_{7}^{4} - 2T_{7}^{3} + 14T_{7}^{2} + 20T_{7} + 100 \)
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Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{2} - T + 1)^{2} \)
|
| $3$ |
\( T^{4} \)
|
| $5$ |
\( (T^{2} + T + 1)^{2} \)
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| $7$ |
\( T^{4} - 2 T^{3} + \cdots + 100 \)
|
| $11$ |
\( (T^{2} - 2 T - 10)^{2} \)
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| $13$ |
\( (T^{2} + 2 T + 4)^{2} \)
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| $17$ |
\( T^{4} + 11T^{2} + 121 \)
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| $19$ |
\( T^{4} - 6 T^{3} + \cdots + 4 \)
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| $23$ |
\( (T + 4)^{4} \)
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| $29$ |
\( (T^{2} - 6 T - 35)^{2} \)
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| $31$ |
\( (T^{2} - 2 T - 10)^{2} \)
|
| $37$ |
\( T^{4} + 4 T^{3} + \cdots + 1369 \)
|
| $41$ |
\( T^{4} + 16 T^{3} + \cdots + 2809 \)
|
| $43$ |
\( (T^{2} + 6 T - 2)^{2} \)
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| $47$ |
\( (T^{2} + 2 T - 10)^{2} \)
|
| $53$ |
\( T^{4} - 4 T^{3} + \cdots + 1600 \)
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| $59$ |
\( T^{4} - 4 T^{3} + \cdots + 1600 \)
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| $61$ |
\( T^{4} - 8 T^{3} + \cdots + 25 \)
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| $67$ |
\( T^{4} + 6 T^{3} + \cdots + 4 \)
|
| $71$ |
\( T^{4} - 6 T^{3} + \cdots + 4 \)
|
| $73$ |
\( (T^{2} - 12 T - 8)^{2} \)
|
| $79$ |
\( T^{4} - 6 T^{3} + \cdots + 8100 \)
|
| $83$ |
\( T^{4} - 4 T^{3} + \cdots + 1600 \)
|
| $89$ |
\( T^{4} + 12 T^{3} + \cdots + 625 \)
|
| $97$ |
\( (T^{2} - 14 T + 5)^{2} \)
|
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