Newspace parameters
| Level: | \( N \) | \(=\) | \( 966 = 2 \cdot 3 \cdot 7 \cdot 23 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 966.l (of order \(6\), degree \(2\), minimal) |
Newform invariants
| Self dual: | no |
| Analytic conductor: | \(7.71354883526\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Relative dimension: | \(2\) over \(\Q(\zeta_{6})\) |
| Coefficient field: | \(\Q(\zeta_{12})\) |
|
|
|
| Defining polynomial: |
\( x^{4} - x^{2} + 1 \)
|
| Coefficient ring: | \(\Z[a_1, a_2]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | yes |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{6}]$ |
$q$-expansion
Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{12}\). We also show the integral \(q\)-expansion of the trace form.
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/966\mathbb{Z}\right)^\times\).
| \(n\) | \(323\) | \(829\) | \(925\) |
| \(\chi(n)\) | \(-1\) | \(\zeta_{12}^{2}\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 47.1 |
|
−0.866025 | − | 0.500000i | −0.866025 | − | 1.50000i | 0.500000 | + | 0.866025i | 0.866025 | − | 1.50000i | 1.73205i | 0.500000 | + | 2.59808i | − | 1.00000i | −1.50000 | + | 2.59808i | −1.50000 | + | 0.866025i | |||||||||||||||
| 47.2 | 0.866025 | + | 0.500000i | 0.866025 | + | 1.50000i | 0.500000 | + | 0.866025i | −0.866025 | + | 1.50000i | 1.73205i | 0.500000 | + | 2.59808i | 1.00000i | −1.50000 | + | 2.59808i | −1.50000 | + | 0.866025i | |||||||||||||||||
| 185.1 | −0.866025 | + | 0.500000i | −0.866025 | + | 1.50000i | 0.500000 | − | 0.866025i | 0.866025 | + | 1.50000i | − | 1.73205i | 0.500000 | − | 2.59808i | 1.00000i | −1.50000 | − | 2.59808i | −1.50000 | − | 0.866025i | ||||||||||||||||
| 185.2 | 0.866025 | − | 0.500000i | 0.866025 | − | 1.50000i | 0.500000 | − | 0.866025i | −0.866025 | − | 1.50000i | − | 1.73205i | 0.500000 | − | 2.59808i | − | 1.00000i | −1.50000 | − | 2.59808i | −1.50000 | − | 0.866025i | |||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 3.b | odd | 2 | 1 | inner |
| 7.d | odd | 6 | 1 | inner |
| 21.g | even | 6 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 966.2.l.a | ✓ | 4 |
| 3.b | odd | 2 | 1 | inner | 966.2.l.a | ✓ | 4 |
| 7.d | odd | 6 | 1 | inner | 966.2.l.a | ✓ | 4 |
| 21.g | even | 6 | 1 | inner | 966.2.l.a | ✓ | 4 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 966.2.l.a | ✓ | 4 | 1.a | even | 1 | 1 | trivial |
| 966.2.l.a | ✓ | 4 | 3.b | odd | 2 | 1 | inner |
| 966.2.l.a | ✓ | 4 | 7.d | odd | 6 | 1 | inner |
| 966.2.l.a | ✓ | 4 | 21.g | even | 6 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{4} + 3T_{5}^{2} + 9 \)
acting on \(S_{2}^{\mathrm{new}}(966, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} - T^{2} + 1 \)
|
| $3$ |
\( T^{4} + 3T^{2} + 9 \)
|
| $5$ |
\( T^{4} + 3T^{2} + 9 \)
|
| $7$ |
\( (T^{2} - T + 7)^{2} \)
|
| $11$ |
\( T^{4} \)
|
| $13$ |
\( (T^{2} + 3)^{2} \)
|
| $17$ |
\( T^{4} + 27T^{2} + 729 \)
|
| $19$ |
\( (T^{2} - 6 T + 12)^{2} \)
|
| $23$ |
\( T^{4} - T^{2} + 1 \)
|
| $29$ |
\( (T^{2} + 36)^{2} \)
|
| $31$ |
\( (T^{2} - 6 T + 12)^{2} \)
|
| $37$ |
\( (T^{2} + 4 T + 16)^{2} \)
|
| $41$ |
\( T^{4} \)
|
| $43$ |
\( (T - 4)^{4} \)
|
| $47$ |
\( T^{4} + 27T^{2} + 729 \)
|
| $53$ |
\( T^{4} - 81T^{2} + 6561 \)
|
| $59$ |
\( T^{4} + 108 T^{2} + 11664 \)
|
| $61$ |
\( (T^{2} - 24 T + 192)^{2} \)
|
| $67$ |
\( (T^{2} + 5 T + 25)^{2} \)
|
| $71$ |
\( (T^{2} + 225)^{2} \)
|
| $73$ |
\( (T^{2} - 21 T + 147)^{2} \)
|
| $79$ |
\( (T^{2} + 8 T + 64)^{2} \)
|
| $83$ |
\( (T^{2} - 48)^{2} \)
|
| $89$ |
\( T^{4} + 48T^{2} + 2304 \)
|
| $97$ |
\( (T^{2} + 108)^{2} \)
|
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