Newspace parameters
| Level: | \( N \) | \(=\) | \( 50 = 2 \cdot 5^{2} \) |
| Weight: | \( k \) | \(=\) | \( 22 \) |
| Character orbit: | \([\chi]\) | \(=\) | 50.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(139.738672144\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{4} - \cdots)\) |
|
|
|
| Defining polynomial: |
\( x^{4} - 315576783x^{2} - 792388522562x + 1396510961585520 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{11}]\) |
| Coefficient ring index: | \( 2^{9}\cdot 3^{2}\cdot 5^{7}\cdot 7^{2} \) |
| 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
\( x^{4} - 315576783x^{2} - 792388522562x + 1396510961585520 \)
:
| \(\beta_{1}\) | \(=\) |
\( 10\nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -1000\nu^{3} + 2629900\nu^{2} + 296822573000\nu + 179323701115650 ) / 620757 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 5000\nu^{3} + 48926200\nu^{2} - 1717939611760\nu - 10691443359814800 ) / 6828327 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_1 ) / 10 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( 11\beta_{3} + 5\beta_{2} + 37668\beta _1 + 15778839150 ) / 100 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 289289\beta_{3} - 489262\beta_{2} + 30672888032\beta _1 + 594291391921500 ) / 1000 \)
|
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 |
|
−1024.00 | −185808. | 1.04858e6 | 0 | 1.90268e8 | −4.88079e8 | −1.07374e9 | 2.40644e10 | 0 | ||||||||||||||||||||||||||||||
| 1.2 | −1024.00 | −62614.8 | 1.04858e6 | 0 | 6.41176e7 | 1.35870e9 | −1.07374e9 | −6.53974e9 | 0 | |||||||||||||||||||||||||||||||
| 1.3 | −1024.00 | −12234.5 | 1.04858e6 | 0 | 1.25282e7 | −9.06115e8 | −1.07374e9 | −1.03107e10 | 0 | |||||||||||||||||||||||||||||||
| 1.4 | −1024.00 | 163894. | 1.04858e6 | 0 | −1.67827e8 | −1.27262e8 | −1.07374e9 | 1.64008e10 | 0 | |||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \( +1 \) |
| \(5\) | \( -1 \) |
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 | 50.22.a.i | ✓ | 4 |
| 5.b | even | 2 | 1 | 50.22.a.j | yes | 4 | |
| 5.c | odd | 4 | 2 | 50.22.b.h | 8 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 50.22.a.i | ✓ | 4 | 1.a | even | 1 | 1 | trivial |
| 50.22.a.j | yes | 4 | 5.b | even | 2 | 1 | |
| 50.22.b.h | 8 | 5.c | odd | 4 | 2 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{4} + 96764T_{3}^{3} - 28046451414T_{3}^{2} - 2262585387673116T_{3} - 23328791599976124939 \)
acting on \(S_{22}^{\mathrm{new}}(\Gamma_0(50))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T + 1024)^{4} \)
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| $3$ |
\( T^{4} + \cdots - 23\!\cdots\!39 \)
|
| $5$ |
\( T^{4} \)
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| $7$ |
\( T^{4} + \cdots - 76\!\cdots\!24 \)
|
| $11$ |
\( T^{4} + \cdots - 48\!\cdots\!59 \)
|
| $13$ |
\( T^{4} + \cdots + 86\!\cdots\!76 \)
|
| $17$ |
\( T^{4} + \cdots + 54\!\cdots\!21 \)
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| $19$ |
\( T^{4} + \cdots - 34\!\cdots\!75 \)
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| $23$ |
\( T^{4} + \cdots + 19\!\cdots\!56 \)
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| $29$ |
\( T^{4} + \cdots - 27\!\cdots\!00 \)
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| $31$ |
\( T^{4} + \cdots + 88\!\cdots\!36 \)
|
| $37$ |
\( T^{4} + \cdots - 61\!\cdots\!44 \)
|
| $41$ |
\( T^{4} + \cdots + 61\!\cdots\!61 \)
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| $43$ |
\( T^{4} + \cdots - 42\!\cdots\!64 \)
|
| $47$ |
\( T^{4} + \cdots - 67\!\cdots\!04 \)
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| $53$ |
\( T^{4} + \cdots - 11\!\cdots\!84 \)
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| $59$ |
\( T^{4} + \cdots - 84\!\cdots\!00 \)
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| $61$ |
\( T^{4} + \cdots - 19\!\cdots\!44 \)
|
| $67$ |
\( T^{4} + \cdots + 83\!\cdots\!41 \)
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| $71$ |
\( T^{4} + \cdots + 12\!\cdots\!76 \)
|
| $73$ |
\( T^{4} + \cdots - 26\!\cdots\!39 \)
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| $79$ |
\( T^{4} + \cdots - 20\!\cdots\!00 \)
|
| $83$ |
\( T^{4} + \cdots - 29\!\cdots\!39 \)
|
| $89$ |
\( T^{4} + \cdots + 53\!\cdots\!25 \)
|
| $97$ |
\( T^{4} + \cdots + 33\!\cdots\!16 \)
|
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