Newspace parameters
| Level: | \( N \) | \(=\) | \( 669 = 3 \cdot 223 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 669.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(5.34199189522\) |
| Analytic rank: | \(1\) |
| Dimension: | \(3\) |
| Coefficient field: | 3.3.257.1 |
|
|
|
| Defining polynomial: |
\( x^{3} - x^{2} - 4x + 3 \)
|
| Coefficient ring: | \(\Z[a_1, a_2]\) |
| 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 \(1,\beta_1,\beta_2\) 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
\( x^{3} - x^{2} - 4x + 3 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - 3 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + 3 \)
|
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 |
|
−2.19869 | −1.00000 | 2.83424 | 4.39738 | 2.19869 | −2.19869 | −1.83424 | 1.00000 | −9.66849 | |||||||||||||||||||||||||||
| 1.2 | −0.713538 | −1.00000 | −1.49086 | 1.42708 | 0.713538 | −0.713538 | 2.49086 | 1.00000 | −1.01827 | ||||||||||||||||||||||||||||
| 1.3 | 1.91223 | −1.00000 | 1.65662 | −3.82446 | −1.91223 | 1.91223 | −0.656620 | 1.00000 | −7.31324 | ||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(3\) | \( +1 \) |
| \(223\) | \( +1 \) |
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 | 669.2.a.f | ✓ | 3 |
| 3.b | odd | 2 | 1 | 2007.2.a.g | 3 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 669.2.a.f | ✓ | 3 | 1.a | even | 1 | 1 | trivial |
| 2007.2.a.g | 3 | 3.b | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{3} + T_{2}^{2} - 4T_{2} - 3 \)
acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(669))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{3} + T^{2} - 4T - 3 \)
|
| $3$ |
\( (T + 1)^{3} \)
|
| $5$ |
\( T^{3} - 2 T^{2} + \cdots + 24 \)
|
| $7$ |
\( T^{3} + T^{2} - 4T - 3 \)
|
| $11$ |
\( T^{3} + 8 T^{2} + \cdots - 40 \)
|
| $13$ |
\( T^{3} + 4 T^{2} + \cdots - 8 \)
|
| $17$ |
\( T^{3} + 5 T^{2} + \cdots - 193 \)
|
| $19$ |
\( T^{3} + 11 T^{2} + \cdots + 19 \)
|
| $23$ |
\( T^{3} + 8 T^{2} + \cdots - 72 \)
|
| $29$ |
\( T^{3} + 9 T^{2} + \cdots - 189 \)
|
| $31$ |
\( T^{3} - 13 T^{2} + \cdots - 25 \)
|
| $37$ |
\( T^{3} - T^{2} + \cdots - 15 \)
|
| $41$ |
\( T^{3} + 23 T^{2} + \cdots + 225 \)
|
| $43$ |
\( T^{3} - 7 T^{2} + \cdots + 81 \)
|
| $47$ |
\( T^{3} + T^{2} + \cdots + 477 \)
|
| $53$ |
\( T^{3} + 19 T^{2} + \cdots + 201 \)
|
| $59$ |
\( T^{3} + 16 T^{2} + \cdots - 320 \)
|
| $61$ |
\( T^{3} - 8 T^{2} + \cdots + 536 \)
|
| $67$ |
\( T^{3} + 2 T^{2} + \cdots + 216 \)
|
| $71$ |
\( T^{3} + 20 T^{2} + \cdots - 1600 \)
|
| $73$ |
\( T^{3} - 3 T^{2} + \cdots - 37 \)
|
| $79$ |
\( T^{3} - 16 T^{2} + \cdots + 600 \)
|
| $83$ |
\( T^{3} + T^{2} + \cdots + 25 \)
|
| $89$ |
\( T^{3} - 9 T^{2} + \cdots + 361 \)
|
| $97$ |
\( T^{3} - 22 T^{2} + \cdots - 152 \)
|
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