Newspace parameters
| Level: | \( N \) | \(=\) | \( 2057 = 11^{2} \cdot 17 \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 2057.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(121.366928882\) |
| Analytic rank: | \(1\) |
| Dimension: | \(4\) |
| Coefficient field: | 4.4.22000.1 |
|
|
|
| Defining polynomial: |
\( x^{4} - 15x^{2} + 55 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 2 \) |
| Twist minimal: | no (minimal twist has level 187) |
| 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
\( x^{4} - 15x^{2} + 55 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu^{3} - 8\nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} + \nu - 8 \)
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| \(\beta_{3}\) | \(=\) |
\( -2\nu^{2} + 15 \)
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| \(\nu\) | \(=\) |
\( ( \beta_{3} + 2\beta_{2} + 1 ) / 2 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( -\beta_{3} + 15 ) / 2 \)
|
| \(\nu^{3}\) | \(=\) |
\( 4\beta_{3} + 8\beta_{2} + \beta _1 + 4 \)
|
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 |
|
−4.13195 | −5.55368 | 9.07297 | −2.07498 | 22.9475 | 7.72946 | −4.43347 | 3.84339 | 8.57370 | ||||||||||||||||||||||||||||||
| 1.2 | −3.17935 | 2.14429 | 2.10824 | −14.3194 | −6.81744 | 7.87813 | 18.7319 | −22.4020 | 45.5264 | |||||||||||||||||||||||||||||||
| 1.3 | −0.0567223 | −2.90822 | −7.99678 | 7.08336 | 0.164961 | −24.8224 | 0.907375 | −18.5422 | −0.401784 | |||||||||||||||||||||||||||||||
| 1.4 | 5.36801 | 0.317615 | 20.8156 | −0.688953 | 1.70496 | −6.78519 | 68.7942 | −26.8991 | −3.69831 | |||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(11\) | \( -1 \) |
| \(17\) | \( +1 \) |
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 | 2057.4.a.g | 4 | |
| 11.b | odd | 2 | 1 | 187.4.a.c | ✓ | 4 | |
| 33.d | even | 2 | 1 | 1683.4.a.e | 4 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 187.4.a.c | ✓ | 4 | 11.b | odd | 2 | 1 | |
| 1683.4.a.e | 4 | 33.d | even | 2 | 1 | ||
| 2057.4.a.g | 4 | 1.a | even | 1 | 1 | trivial | |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(2057))\):
|
\( T_{2}^{4} + 2T_{2}^{3} - 26T_{2}^{2} - 72T_{2} - 4 \)
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\( T_{3}^{4} + 6T_{3}^{3} - 4T_{3}^{2} - 34T_{3} + 11 \)
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\( T_{5}^{4} + 10T_{5}^{3} - 80T_{5}^{2} - 270T_{5} - 145 \)
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Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} + 2 T^{3} + \cdots - 4 \)
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| $3$ |
\( T^{4} + 6 T^{3} + \cdots + 11 \)
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| $5$ |
\( T^{4} + 10 T^{3} + \cdots - 145 \)
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| $7$ |
\( T^{4} + 16 T^{3} + \cdots + 10256 \)
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| $11$ |
\( T^{4} \)
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| $13$ |
\( T^{4} - 10 T^{3} + \cdots + 160380 \)
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| $17$ |
\( (T + 17)^{4} \)
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| $19$ |
\( T^{4} - 114 T^{3} + \cdots + 29119996 \)
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| $23$ |
\( T^{4} + 258 T^{3} + \cdots - 10141489 \)
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| $29$ |
\( T^{4} - 10 T^{3} + \cdots + 534934180 \)
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| $31$ |
\( T^{4} - 14 T^{3} + \cdots + 54054171 \)
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| $37$ |
\( T^{4} + \cdots + 2373625351 \)
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| $41$ |
\( T^{4} + \cdots + 1487647936 \)
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| $43$ |
\( T^{4} + \cdots - 2213137904 \)
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| $47$ |
\( T^{4} + \cdots + 3725923696 \)
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| $53$ |
\( T^{4} - 636 T^{3} + \cdots + 34991296 \)
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| $59$ |
\( T^{4} + 412 T^{3} + \cdots + 197861121 \)
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| $61$ |
\( T^{4} + \cdots + 32055831296 \)
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| $67$ |
\( T^{4} + \cdots - 2268239959 \)
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| $71$ |
\( T^{4} + \cdots + 12642213631 \)
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| $73$ |
\( T^{4} + \cdots + 57365296516 \)
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| $79$ |
\( T^{4} - 598 T^{3} + \cdots - 10775844 \)
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| $83$ |
\( T^{4} + \cdots + 62502614416 \)
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| $89$ |
\( T^{4} + \cdots - 39747946519 \)
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| $97$ |
\( T^{4} + \cdots - 1077654262225 \)
|
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