Newspace parameters
| Level: | \( N \) | \(=\) | \( 1922 = 2 \cdot 31^{2} \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 1922.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(15.3472472685\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Coefficient field: | \(\Q(\zeta_{16})^+\) |
|
|
|
| Defining polynomial: |
\( x^{4} - 4x^{2} + 2 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | yes |
| Fricke sign: | \(-1\) |
| Sato-Tate group: | $\mathrm{SU}(2)$ |
$q$-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 \(\nu = \zeta_{16} + \zeta_{16}^{-1}\):
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - 2 \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{3} - 3\nu \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + 2 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{3} + 3\beta_1 \)
|
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 1.1 |
|
−1.00000 | −1.84776 | 1.00000 | 2.82843 | 1.84776 | −0.828427 | −1.00000 | 0.414214 | −2.82843 | ||||||||||||||||||||||||||||||
| 1.2 | −1.00000 | −0.765367 | 1.00000 | −2.82843 | 0.765367 | 4.82843 | −1.00000 | −2.41421 | 2.82843 | |||||||||||||||||||||||||||||||
| 1.3 | −1.00000 | 0.765367 | 1.00000 | −2.82843 | −0.765367 | 4.82843 | −1.00000 | −2.41421 | 2.82843 | |||||||||||||||||||||||||||||||
| 1.4 | −1.00000 | 1.84776 | 1.00000 | 2.82843 | −1.84776 | −0.828427 | −1.00000 | 0.414214 | −2.82843 | |||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \( +1 \) |
| \(31\) | \( -1 \) |
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 31.b | odd | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 1922.2.a.j | ✓ | 4 |
| 31.b | odd | 2 | 1 | inner | 1922.2.a.j | ✓ | 4 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 1922.2.a.j | ✓ | 4 | 1.a | even | 1 | 1 | trivial |
| 1922.2.a.j | ✓ | 4 | 31.b | odd | 2 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{4} - 4T_{3}^{2} + 2 \)
acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(1922))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T + 1)^{4} \)
|
| $3$ |
\( T^{4} - 4T^{2} + 2 \)
|
| $5$ |
\( (T^{2} - 8)^{2} \)
|
| $7$ |
\( (T^{2} - 4 T - 4)^{2} \)
|
| $11$ |
\( T^{4} - 52T^{2} + 578 \)
|
| $13$ |
\( T^{4} - 40T^{2} + 8 \)
|
| $17$ |
\( T^{4} - 36T^{2} + 162 \)
|
| $19$ |
\( (T - 2)^{4} \)
|
| $23$ |
\( T^{4} - 32T^{2} + 128 \)
|
| $29$ |
\( T^{4} - 72T^{2} + 648 \)
|
| $31$ |
\( T^{4} \)
|
| $37$ |
\( T^{4} - 72T^{2} + 648 \)
|
| $41$ |
\( (T^{2} + 8 T - 2)^{2} \)
|
| $43$ |
\( T^{4} - 20T^{2} + 2 \)
|
| $47$ |
\( (T^{2} - 12 T + 4)^{2} \)
|
| $53$ |
\( T^{4} - 40T^{2} + 392 \)
|
| $59$ |
\( (T^{2} - 8 T - 34)^{2} \)
|
| $61$ |
\( T^{4} - 200T^{2} + 5000 \)
|
| $67$ |
\( (T^{2} - 32)^{2} \)
|
| $71$ |
\( (T^{2} - 72)^{2} \)
|
| $73$ |
\( T^{4} - 212T^{2} + 578 \)
|
| $79$ |
\( T^{4} - 16T^{2} + 32 \)
|
| $83$ |
\( T^{4} - 36T^{2} + 162 \)
|
| $89$ |
\( T^{4} - 116T^{2} + 2 \)
|
| $97$ |
\( (T^{2} - 16 T + 56)^{2} \)
|
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