Newspace parameters
| Level: | \( N \) | \(=\) | \( 927 = 3^{2} \cdot 103 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 927.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(7.40213226737\) |
| Analytic rank: | \(1\) |
| Dimension: | \(3\) |
| Coefficient field: | 3.3.148.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{3} - x^{2} - 3x + 1 \)
|
| Coefficient ring: | \(\Z[a_1, a_2]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 309) |
| Fricke sign: | \(1\) |
| Sato-Tate group: | $\mathrm{SU}(2)$ |
$q$-expansion
Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2\) 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
\( x^{3} - x^{2} - 3x + 1 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - \nu - 2 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + \beta _1 + 2 \)
|
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 |
|
−2.17009 | 0 | 2.70928 | −2.17009 | 0 | 0.630898 | −1.53919 | 0 | 4.70928 | |||||||||||||||||||||||||||
| 1.2 | −0.311108 | 0 | −1.90321 | −0.311108 | 0 | 1.52543 | 1.21432 | 0 | 0.0967881 | ||||||||||||||||||||||||||||
| 1.3 | 1.48119 | 0 | 0.193937 | 1.48119 | 0 | −4.15633 | −2.67513 | 0 | 2.19394 | ||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(3\) | \(-1\) |
| \(103\) | \(-1\) |
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 | 927.2.a.c | 3 | |
| 3.b | odd | 2 | 1 | 309.2.a.b | ✓ | 3 | |
| 12.b | even | 2 | 1 | 4944.2.a.v | 3 | ||
| 15.d | odd | 2 | 1 | 7725.2.a.p | 3 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 309.2.a.b | ✓ | 3 | 3.b | odd | 2 | 1 | |
| 927.2.a.c | 3 | 1.a | even | 1 | 1 | trivial | |
| 4944.2.a.v | 3 | 12.b | even | 2 | 1 | ||
| 7725.2.a.p | 3 | 15.d | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{3} + T_{2}^{2} - 3T_{2} - 1 \)
acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(927))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{3} + T^{2} - 3T - 1 \)
|
| $3$ |
\( T^{3} \)
|
| $5$ |
\( T^{3} + T^{2} - 3T - 1 \)
|
| $7$ |
\( T^{3} + 2 T^{2} + \cdots + 4 \)
|
| $11$ |
\( T^{3} + 8 T^{2} + \cdots + 4 \)
|
| $13$ |
\( T^{3} + 3 T^{2} + \cdots - 31 \)
|
| $17$ |
\( T^{3} + 4 T^{2} + \cdots - 16 \)
|
| $19$ |
\( T^{3} - 12 T^{2} + \cdots + 16 \)
|
| $23$ |
\( T^{3} + 7 T^{2} + \cdots + 5 \)
|
| $29$ |
\( T^{3} - 28T - 52 \)
|
| $31$ |
\( T^{3} - 7 T^{2} + \cdots + 19 \)
|
| $37$ |
\( T^{3} + 6 T^{2} + \cdots - 148 \)
|
| $41$ |
\( T^{3} + 2 T^{2} + \cdots + 52 \)
|
| $43$ |
\( T^{3} + T^{2} - 9T - 13 \)
|
| $47$ |
\( T^{3} + 12 T^{2} + \cdots - 20 \)
|
| $53$ |
\( T^{3} - 8 T^{2} + \cdots - 4 \)
|
| $59$ |
\( T^{3} + 15 T^{2} + \cdots + 17 \)
|
| $61$ |
\( T^{3} + 19 T^{2} + \cdots - 607 \)
|
| $67$ |
\( T^{3} + 3 T^{2} + \cdots - 191 \)
|
| $71$ |
\( T^{3} + 8 T^{2} + \cdots - 368 \)
|
| $73$ |
\( T^{3} + 12 T^{2} + \cdots - 20 \)
|
| $79$ |
\( T^{3} - 2 T^{2} + \cdots + 20 \)
|
| $83$ |
\( T^{3} + 17 T^{2} + \cdots + 125 \)
|
| $89$ |
\( T^{3} + 24 T^{2} + \cdots - 556 \)
|
| $97$ |
\( T^{3} - 3 T^{2} + \cdots - 449 \)
|
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