Newspace parameters
| Level: | \( N \) | \(=\) | \( 633 = 3 \cdot 211 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 633.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(5.05453044795\) |
| Analytic rank: | \(1\) |
| Dimension: | \(4\) |
| Coefficient field: | 4.4.725.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{4} - x^{3} - 3x^{2} + x + 1 \)
|
| 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 \(1,\beta_1,\beta_2,\beta_3\) 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
\( x^{4} - x^{3} - 3x^{2} + x + 1 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - \nu - 1 \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{3} - \nu^{2} - 2\nu + 1 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + \beta _1 + 1 \)
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| \(\nu^{3}\) | \(=\) |
\( \beta_{3} + \beta_{2} + 3\beta_1 \)
|
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 |
|
−2.19353 | 1.00000 | 2.81156 | −0.262360 | −2.19353 | −4.35567 | −1.78018 | 1.00000 | 0.575493 | ||||||||||||||||||||||||||||||
| 1.2 | −1.29496 | 1.00000 | −0.323071 | −1.47726 | −1.29496 | −0.904706 | 3.00829 | 1.00000 | 1.91300 | |||||||||||||||||||||||||||||||
| 1.3 | 0.294963 | 1.00000 | −1.91300 | 1.09529 | 0.294963 | −3.47726 | −1.15419 | 1.00000 | 0.323071 | |||||||||||||||||||||||||||||||
| 1.4 | 1.19353 | 1.00000 | −0.575493 | −2.35567 | 1.19353 | −2.26236 | −3.07392 | 1.00000 | −2.81156 | |||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(3\) | \(-1\) |
| \(211\) | \(-1\) |
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 | 633.2.a.b | ✓ | 4 |
| 3.b | odd | 2 | 1 | 1899.2.a.g | 4 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 633.2.a.b | ✓ | 4 | 1.a | even | 1 | 1 | trivial |
| 1899.2.a.g | 4 | 3.b | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{4} + 2T_{2}^{3} - 2T_{2}^{2} - 3T_{2} + 1 \)
acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(633))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} + 2 T^{3} + \cdots + 1 \)
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| $3$ |
\( (T - 1)^{4} \)
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| $5$ |
\( T^{4} + 3 T^{3} + \cdots - 1 \)
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| $7$ |
\( T^{4} + 11 T^{3} + \cdots + 31 \)
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| $11$ |
\( T^{4} + T^{3} - 29 T^{2} + \cdots + 1 \)
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| $13$ |
\( T^{4} + 9 T^{3} + \cdots - 19 \)
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| $17$ |
\( T^{4} + T^{3} + \cdots + 19 \)
|
| $19$ |
\( (T^{2} + T - 11)^{2} \)
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| $23$ |
\( T^{4} + 15 T^{3} + \cdots - 121 \)
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| $29$ |
\( T^{4} - 2 T^{3} + \cdots + 49 \)
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| $31$ |
\( T^{4} + 10 T^{3} + \cdots - 1 \)
|
| $37$ |
\( T^{4} + 13 T^{3} + \cdots + 11 \)
|
| $41$ |
\( T^{4} - 7 T^{3} + \cdots - 281 \)
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| $43$ |
\( T^{4} + 11 T^{3} + \cdots + 11 \)
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| $47$ |
\( T^{4} - T^{3} + \cdots + 211 \)
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| $53$ |
\( T^{4} + 6 T^{3} + \cdots - 1 \)
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| $59$ |
\( T^{4} - 141 T^{2} + \cdots + 2879 \)
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| $61$ |
\( T^{4} + 18 T^{3} + \cdots - 31 \)
|
| $67$ |
\( T^{4} + 6 T^{3} + \cdots + 4769 \)
|
| $71$ |
\( T^{4} + 9 T^{3} + \cdots - 5599 \)
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| $73$ |
\( T^{4} - 7 T^{3} + \cdots - 3091 \)
|
| $79$ |
\( T^{4} + 21 T^{3} + \cdots + 89 \)
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| $83$ |
\( T^{4} + 8 T^{3} + \cdots - 2729 \)
|
| $89$ |
\( T^{4} - 3 T^{3} + \cdots - 109 \)
|
| $97$ |
\( T^{4} + 9 T^{3} + \cdots - 7789 \)
|
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