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: | 4.4.4525.1 |
|
|
|
| Defining polynomial: |
\( x^{4} - x^{3} - 7x^{2} + 3x + 9 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 62) |
| 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 a root \(\nu\) of
\( x^{4} - x^{3} - 7x^{2} + 3x + 9 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{3} - \nu^{2} - 4\nu ) / 3 \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{2} - \nu - 4 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{3} + \beta _1 + 4 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{3} + 3\beta_{2} + 5\beta _1 + 4 \)
|
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 | −1.72069 | 1.00000 | −1.68148 | −1.72069 | −4.82335 | 1.00000 | −0.0392126 | −1.68148 | ||||||||||||||||||||||||||||||
| 1.2 | 1.00000 | −0.450384 | 1.00000 | 2.34677 | −0.450384 | −4.51880 | 1.00000 | −2.79715 | 2.34677 | |||||||||||||||||||||||||||||||
| 1.3 | 1.00000 | 2.10266 | 1.00000 | 0.681481 | 2.10266 | 2.82335 | 1.00000 | 1.42118 | 0.681481 | |||||||||||||||||||||||||||||||
| 1.4 | 1.00000 | 3.06842 | 1.00000 | −3.34677 | 3.06842 | 2.51880 | 1.00000 | 6.41519 | −3.34677 | |||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \( -1 \) |
| \(31\) | \( +1 \) |
Inner twists
This newform does not admit any (nontrivial) inner twists.
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 1922.2.a.r | 4 | |
| 31.b | odd | 2 | 1 | 1922.2.a.n | 4 | ||
| 31.d | even | 5 | 2 | 62.2.d.a | ✓ | 8 | |
| 93.l | odd | 10 | 2 | 558.2.i.i | 8 | ||
| 124.l | odd | 10 | 2 | 496.2.n.e | 8 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 62.2.d.a | ✓ | 8 | 31.d | even | 5 | 2 | |
| 496.2.n.e | 8 | 124.l | odd | 10 | 2 | ||
| 558.2.i.i | 8 | 93.l | odd | 10 | 2 | ||
| 1922.2.a.n | 4 | 31.b | odd | 2 | 1 | ||
| 1922.2.a.r | 4 | 1.a | even | 1 | 1 | trivial | |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{4} - 3T_{3}^{3} - 4T_{3}^{2} + 10T_{3} + 5 \)
acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(1922))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T - 1)^{4} \)
|
| $3$ |
\( T^{4} - 3 T^{3} + \cdots + 5 \)
|
| $5$ |
\( T^{4} + 2 T^{3} + \cdots + 9 \)
|
| $7$ |
\( T^{4} + 4 T^{3} + \cdots + 155 \)
|
| $11$ |
\( T^{4} - 9 T^{3} + \cdots - 45 \)
|
| $13$ |
\( T^{4} - 7 T^{3} + \cdots - 101 \)
|
| $17$ |
\( T^{4} - 4 T^{3} + \cdots + 9 \)
|
| $19$ |
\( T^{4} + 7 T^{3} + \cdots + 45 \)
|
| $23$ |
\( T^{4} + 11 T^{3} + \cdots + 9 \)
|
| $29$ |
\( T^{4} - 4 T^{3} + \cdots - 45 \)
|
| $31$ |
\( T^{4} \)
|
| $37$ |
\( T^{4} - 26 T^{3} + \cdots + 1145 \)
|
| $41$ |
\( T^{4} - 2 T^{3} + \cdots + 909 \)
|
| $43$ |
\( T^{4} - 4 T^{3} + \cdots - 661 \)
|
| $47$ |
\( T^{4} + 10 T^{3} + \cdots - 909 \)
|
| $53$ |
\( T^{4} - 6 T^{3} + \cdots + 171 \)
|
| $59$ |
\( T^{4} - T^{3} + \cdots - 855 \)
|
| $61$ |
\( T^{4} - 2 T^{3} + \cdots + 279 \)
|
| $67$ |
\( T^{4} + 13 T^{3} + \cdots - 139 \)
|
| $71$ |
\( T^{4} - 108 T^{2} + \cdots + 81 \)
|
| $73$ |
\( T^{4} - 13 T^{3} + \cdots - 2195 \)
|
| $79$ |
\( T^{4} + 21 T^{3} + \cdots - 3205 \)
|
| $83$ |
\( T^{4} - 14 T^{3} + \cdots + 1899 \)
|
| $89$ |
\( T^{4} - 38 T^{3} + \cdots + 45 \)
|
| $97$ |
\( T^{4} + 30 T^{3} + \cdots - 269 \)
|
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