Newspace parameters
| Level: | \( N \) | \(=\) | \( 1984 = 2^{6} \cdot 31 \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 1984.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(117.059789451\) |
| Analytic rank: | \(0\) |
| Dimension: | \(5\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{5} - \cdots)\) |
|
|
|
| Defining polynomial: |
\( x^{5} - 2x^{4} - 32x^{3} + 19x^{2} + 228x + 172 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 2^{3} \) |
| Twist minimal: | no (minimal twist has level 31) |
| Fricke sign: | \(+1\) |
| Sato-Tate group: | $\mathrm{SU}(2)$ |
$q$-expansion
Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3,\beta_4\) 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
\( x^{5} - 2x^{4} - 32x^{3} + 19x^{2} + 228x + 172 \)
:
| \(\beta_{1}\) | \(=\) |
\( 2\nu - 1 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{4} - 10\nu^{3} - 4\nu^{2} + 155\nu - 24 ) / 26 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -5\nu^{4} + 24\nu^{3} + 176\nu^{2} - 463\nu - 1362 ) / 104 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 17\nu^{4} - 40\nu^{3} - 432\nu^{2} + 451\nu + 1698 ) / 104 \)
|
| \(\nu\) | \(=\) |
\( ( \beta _1 + 1 ) / 2 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( \beta_{4} + 5\beta_{3} + 2\beta_{2} + 3\beta _1 + 54 ) / 4 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 3\beta_{4} + 7\beta_{3} - 4\beta_{2} + 21\beta _1 + 60 ) / 2 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( 32\beta_{4} + 80\beta_{3} + 16\beta_{2} + 61\beta _1 + 601 ) / 2 \)
|
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 |
|
0 | −8.71716 | 0 | 10.4546 | 0 | −5.68067 | 0 | 48.9889 | 0 | |||||||||||||||||||||||||||||||||
| 1.2 | 0 | −5.22185 | 0 | −10.8334 | 0 | 25.9132 | 0 | 0.267763 | 0 | ||||||||||||||||||||||||||||||||||
| 1.3 | 0 | 0.0377000 | 0 | −13.5012 | 0 | −3.86558 | 0 | −26.9986 | 0 | ||||||||||||||||||||||||||||||||||
| 1.4 | 0 | 2.52950 | 0 | 17.8984 | 0 | 6.14402 | 0 | −20.6016 | 0 | ||||||||||||||||||||||||||||||||||
| 1.5 | 0 | 7.37181 | 0 | −19.0185 | 0 | −13.5109 | 0 | 27.3436 | 0 | ||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \( +1 \) |
| \(31\) | \( +1 \) |
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 | 1984.4.a.r | 5 | |
| 4.b | odd | 2 | 1 | 1984.4.a.s | 5 | ||
| 8.b | even | 2 | 1 | 31.4.a.b | ✓ | 5 | |
| 8.d | odd | 2 | 1 | 496.4.a.i | 5 | ||
| 24.h | odd | 2 | 1 | 279.4.a.h | 5 | ||
| 40.f | even | 2 | 1 | 775.4.a.f | 5 | ||
| 56.h | odd | 2 | 1 | 1519.4.a.c | 5 | ||
| 248.g | odd | 2 | 1 | 961.4.a.e | 5 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 31.4.a.b | ✓ | 5 | 8.b | even | 2 | 1 | |
| 279.4.a.h | 5 | 24.h | odd | 2 | 1 | ||
| 496.4.a.i | 5 | 8.d | odd | 2 | 1 | ||
| 775.4.a.f | 5 | 40.f | even | 2 | 1 | ||
| 961.4.a.e | 5 | 248.g | odd | 2 | 1 | ||
| 1519.4.a.c | 5 | 56.h | odd | 2 | 1 | ||
| 1984.4.a.r | 5 | 1.a | even | 1 | 1 | trivial | |
| 1984.4.a.s | 5 | 4.b | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{5} + 4T_{3}^{4} - 74T_{3}^{3} - 188T_{3}^{2} + 856T_{3} - 32 \)
acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(1984))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{5} \)
|
| $3$ |
\( T^{5} + 4 T^{4} + \cdots - 32 \)
|
| $5$ |
\( T^{5} + 15 T^{4} + \cdots + 520516 \)
|
| $7$ |
\( T^{5} - 9 T^{4} + \cdots + 47236 \)
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| $11$ |
\( T^{5} + 88 T^{4} + \cdots + 76793648 \)
|
| $13$ |
\( T^{5} - 28 T^{4} + \cdots - 85935616 \)
|
| $17$ |
\( T^{5} - 138 T^{4} + \cdots + 845793728 \)
|
| $19$ |
\( T^{5} - 43 T^{4} + \cdots + 885299824 \)
|
| $23$ |
\( T^{5} + \cdots - 1477525504 \)
|
| $29$ |
\( T^{5} + \cdots - 6492808496 \)
|
| $31$ |
\( (T + 31)^{5} \)
|
| $37$ |
\( T^{5} + \cdots + 315180705232 \)
|
| $41$ |
\( T^{5} + \cdots + 19192433688 \)
|
| $43$ |
\( T^{5} + \cdots + 154176970896 \)
|
| $47$ |
\( T^{5} + \cdots + 197408306432 \)
|
| $53$ |
\( T^{5} + \cdots - 79406336128 \)
|
| $59$ |
\( T^{5} + \cdots - 19804492743336 \)
|
| $61$ |
\( T^{5} + \cdots - 922927740352 \)
|
| $67$ |
\( T^{5} + \cdots + 22262005628928 \)
|
| $71$ |
\( T^{5} + \cdots - 69573929276736 \)
|
| $73$ |
\( T^{5} + \cdots - 1852644259168 \)
|
| $79$ |
\( T^{5} + \cdots - 59985571648 \)
|
| $83$ |
\( T^{5} + \cdots - 657431598704 \)
|
| $89$ |
\( T^{5} + \cdots + 68090734165536 \)
|
| $97$ |
\( T^{5} + \cdots - 31148274036888 \)
|
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