Newspace parameters
| Level: | \( N \) | \(=\) | \( 936 = 2^{3} \cdot 3^{2} \cdot 13 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 936.g (of order \(2\), degree \(1\), minimal) |
Newform invariants
| Self dual: | no |
| Analytic conductor: | \(7.47399762919\) |
| Analytic rank: | \(1\) |
| Dimension: | \(4\) |
| Coefficient field: | \(\Q(\zeta_{12})\) |
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|
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| Defining polynomial: |
\( x^{4} - x^{2} + 1 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2 \) |
| Twist minimal: | no (minimal twist has level 104) |
| 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
| \(\beta_{1}\) | \(=\) |
\( \zeta_{12}^{2} + \zeta_{12} \)
|
| \(\beta_{2}\) | \(=\) |
\( \zeta_{12}^{3} \)
|
| \(\beta_{3}\) | \(=\) |
\( -\zeta_{12}^{2} + \zeta_{12} \)
|
| \(\zeta_{12}\) | \(=\) |
\( ( \beta_{3} + \beta_1 ) / 2 \)
|
| \(\zeta_{12}^{2}\) | \(=\) |
\( ( -\beta_{3} + \beta_1 ) / 2 \)
|
| \(\zeta_{12}^{3}\) | \(=\) |
\( \beta_{2} \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/936\mathbb{Z}\right)^\times\).
| \(n\) | \(145\) | \(209\) | \(469\) | \(703\) |
| \(\chi(n)\) | \(1\) | \(1\) | \(-1\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 469.1 |
|
−1.36603 | − | 0.366025i | 0 | 1.73205 | + | 1.00000i | 3.46410i | 0 | −4.73205 | −2.00000 | − | 2.00000i | 0 | 1.26795 | − | 4.73205i | ||||||||||||||||||||||
| 469.2 | −1.36603 | + | 0.366025i | 0 | 1.73205 | − | 1.00000i | − | 3.46410i | 0 | −4.73205 | −2.00000 | + | 2.00000i | 0 | 1.26795 | + | 4.73205i | ||||||||||||||||||||||
| 469.3 | 0.366025 | − | 1.36603i | 0 | −1.73205 | − | 1.00000i | 3.46410i | 0 | −1.26795 | −2.00000 | + | 2.00000i | 0 | 4.73205 | + | 1.26795i | |||||||||||||||||||||||
| 469.4 | 0.366025 | + | 1.36603i | 0 | −1.73205 | + | 1.00000i | − | 3.46410i | 0 | −1.26795 | −2.00000 | − | 2.00000i | 0 | 4.73205 | − | 1.26795i | ||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 8.b | even | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 936.2.g.b | 4 | |
| 3.b | odd | 2 | 1 | 104.2.b.b | ✓ | 4 | |
| 4.b | odd | 2 | 1 | 3744.2.g.b | 4 | ||
| 8.b | even | 2 | 1 | inner | 936.2.g.b | 4 | |
| 8.d | odd | 2 | 1 | 3744.2.g.b | 4 | ||
| 12.b | even | 2 | 1 | 416.2.b.b | 4 | ||
| 24.f | even | 2 | 1 | 416.2.b.b | 4 | ||
| 24.h | odd | 2 | 1 | 104.2.b.b | ✓ | 4 | |
| 48.i | odd | 4 | 1 | 3328.2.a.n | 2 | ||
| 48.i | odd | 4 | 1 | 3328.2.a.bd | 2 | ||
| 48.k | even | 4 | 1 | 3328.2.a.m | 2 | ||
| 48.k | even | 4 | 1 | 3328.2.a.bc | 2 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 104.2.b.b | ✓ | 4 | 3.b | odd | 2 | 1 | |
| 104.2.b.b | ✓ | 4 | 24.h | odd | 2 | 1 | |
| 416.2.b.b | 4 | 12.b | even | 2 | 1 | ||
| 416.2.b.b | 4 | 24.f | even | 2 | 1 | ||
| 936.2.g.b | 4 | 1.a | even | 1 | 1 | trivial | |
| 936.2.g.b | 4 | 8.b | even | 2 | 1 | inner | |
| 3328.2.a.m | 2 | 48.k | even | 4 | 1 | ||
| 3328.2.a.n | 2 | 48.i | odd | 4 | 1 | ||
| 3328.2.a.bc | 2 | 48.k | even | 4 | 1 | ||
| 3328.2.a.bd | 2 | 48.i | odd | 4 | 1 | ||
| 3744.2.g.b | 4 | 4.b | odd | 2 | 1 | ||
| 3744.2.g.b | 4 | 8.d | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{2} + 12 \)
acting on \(S_{2}^{\mathrm{new}}(936, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} + 2 T^{3} + \cdots + 4 \)
|
| $3$ |
\( T^{4} \)
|
| $5$ |
\( (T^{2} + 12)^{2} \)
|
| $7$ |
\( (T^{2} + 6 T + 6)^{2} \)
|
| $11$ |
\( T^{4} + 24T^{2} + 36 \)
|
| $13$ |
\( (T^{2} + 1)^{2} \)
|
| $17$ |
\( (T^{2} + 4 T - 8)^{2} \)
|
| $19$ |
\( T^{4} + 8T^{2} + 4 \)
|
| $23$ |
\( (T + 4)^{4} \)
|
| $29$ |
\( (T^{2} + 4)^{2} \)
|
| $31$ |
\( (T^{2} + 10 T + 22)^{2} \)
|
| $37$ |
\( T^{4} + 104T^{2} + 1936 \)
|
| $41$ |
\( (T^{2} + 4 T - 44)^{2} \)
|
| $43$ |
\( T^{4} + 56T^{2} + 16 \)
|
| $47$ |
\( (T^{2} + 10 T + 22)^{2} \)
|
| $53$ |
\( T^{4} + 128T^{2} + 1024 \)
|
| $59$ |
\( T^{4} + 104T^{2} + 4 \)
|
| $61$ |
\( T^{4} + 128T^{2} + 1024 \)
|
| $67$ |
\( T^{4} + 8T^{2} + 4 \)
|
| $71$ |
\( (T^{2} - 6 T - 18)^{2} \)
|
| $73$ |
\( (T^{2} + 8 T + 4)^{2} \)
|
| $79$ |
\( (T^{2} - 4 T - 8)^{2} \)
|
| $83$ |
\( T^{4} + 56T^{2} + 484 \)
|
| $89$ |
\( (T^{2} - 300)^{2} \)
|
| $97$ |
\( (T^{2} + 8 T - 92)^{2} \)
|
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