Newspace parameters
| Level: | \( N \) | \(=\) | \( 666 = 2 \cdot 3^{2} \cdot 37 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 666.c (of order \(2\), degree \(1\), minimal) |
Newform invariants
| Self dual: | no |
| Analytic conductor: | \(5.31803677462\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Coefficient field: | \(\Q(i, \sqrt{21})\) |
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| Defining polynomial: |
\( x^{4} + 11x^{2} + 25 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{11}]\) |
| Coefficient ring index: | \( 2^{2} \) |
| 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
\( x^{4} + 11x^{2} + 25 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( \nu^{3} + 6\nu ) / 5 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{3} + 16\nu ) / 5 \)
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| \(\beta_{3}\) | \(=\) |
\( 2\nu^{2} + 11 \)
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| \(\nu\) | \(=\) |
\( ( \beta_{2} - \beta_1 ) / 2 \)
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| \(\nu^{2}\) | \(=\) |
\( ( \beta_{3} - 11 ) / 2 \)
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| \(\nu^{3}\) | \(=\) |
\( -3\beta_{2} + 8\beta_1 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/666\mathbb{Z}\right)^\times\).
| \(n\) | \(371\) | \(631\) |
| \(\chi(n)\) | \(1\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 73.1 |
|
− | 1.00000i | 0 | −1.00000 | 0 | 0 | 1.00000 | 1.00000i | 0 | 0 | |||||||||||||||||||||||||||||
| 73.2 | − | 1.00000i | 0 | −1.00000 | 0 | 0 | 1.00000 | 1.00000i | 0 | 0 | ||||||||||||||||||||||||||||||
| 73.3 | 1.00000i | 0 | −1.00000 | 0 | 0 | 1.00000 | − | 1.00000i | 0 | 0 | ||||||||||||||||||||||||||||||
| 73.4 | 1.00000i | 0 | −1.00000 | 0 | 0 | 1.00000 | − | 1.00000i | 0 | 0 | ||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 3.b | odd | 2 | 1 | inner |
| 37.b | even | 2 | 1 | inner |
| 111.d | odd | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 666.2.c.d | ✓ | 4 |
| 3.b | odd | 2 | 1 | inner | 666.2.c.d | ✓ | 4 |
| 4.b | odd | 2 | 1 | 5328.2.h.h | 4 | ||
| 12.b | even | 2 | 1 | 5328.2.h.h | 4 | ||
| 37.b | even | 2 | 1 | inner | 666.2.c.d | ✓ | 4 |
| 111.d | odd | 2 | 1 | inner | 666.2.c.d | ✓ | 4 |
| 148.b | odd | 2 | 1 | 5328.2.h.h | 4 | ||
| 444.g | even | 2 | 1 | 5328.2.h.h | 4 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 666.2.c.d | ✓ | 4 | 1.a | even | 1 | 1 | trivial |
| 666.2.c.d | ✓ | 4 | 3.b | odd | 2 | 1 | inner |
| 666.2.c.d | ✓ | 4 | 37.b | even | 2 | 1 | inner |
| 666.2.c.d | ✓ | 4 | 111.d | odd | 2 | 1 | inner |
| 5328.2.h.h | 4 | 4.b | odd | 2 | 1 | ||
| 5328.2.h.h | 4 | 12.b | even | 2 | 1 | ||
| 5328.2.h.h | 4 | 148.b | odd | 2 | 1 | ||
| 5328.2.h.h | 4 | 444.g | even | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(666, [\chi])\):
|
\( T_{5} \)
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\( T_{7} - 1 \)
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Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{2} + 1)^{2} \)
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| $3$ |
\( T^{4} \)
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| $5$ |
\( T^{4} \)
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| $7$ |
\( (T - 1)^{4} \)
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| $11$ |
\( (T^{2} - 21)^{2} \)
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| $13$ |
\( (T^{2} + 21)^{2} \)
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| $17$ |
\( (T^{2} + 9)^{2} \)
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| $19$ |
\( (T^{2} + 21)^{2} \)
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| $23$ |
\( (T^{2} + 9)^{2} \)
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| $29$ |
\( (T^{2} + 36)^{2} \)
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| $31$ |
\( T^{4} \)
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| $37$ |
\( (T^{2} - 8 T + 37)^{2} \)
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| $41$ |
\( (T^{2} - 84)^{2} \)
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| $43$ |
\( T^{4} \)
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| $47$ |
\( (T^{2} - 84)^{2} \)
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| $53$ |
\( (T^{2} - 21)^{2} \)
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| $59$ |
\( T^{4} \)
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| $61$ |
\( (T^{2} + 84)^{2} \)
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| $67$ |
\( (T + 2)^{4} \)
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| $71$ |
\( (T^{2} - 84)^{2} \)
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| $73$ |
\( (T - 7)^{4} \)
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| $79$ |
\( (T^{2} + 84)^{2} \)
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| $83$ |
\( (T^{2} - 21)^{2} \)
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| $89$ |
\( (T^{2} + 225)^{2} \)
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| $97$ |
\( (T^{2} + 84)^{2} \)
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