Newspace parameters
| Level: | \( N \) | \(=\) | \( 1984 = 2^{6} \cdot 31 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 1984.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(15.8423197610\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Coefficient field: | 4.4.13968.1 |
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|
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| Defining polynomial: |
\( x^{4} - 2x^{3} - 7x^{2} + 8x + 4 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 992) |
| 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
\( x^{4} - 2x^{3} - 7x^{2} + 8x + 4 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{2} - \nu - 4 ) / 2 \)
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| \(\beta_{3}\) | \(=\) |
\( ( \nu^{3} - \nu^{2} - 6\nu + 2 ) / 2 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( 2\beta_{2} + \beta _1 + 4 \)
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| \(\nu^{3}\) | \(=\) |
\( 2\beta_{3} + 2\beta_{2} + 7\beta _1 + 2 \)
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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 | −0.732051 | 0 | −1.38651 | 0 | −5.17452 | 0 | −2.46410 | 0 | ||||||||||||||||||||||||||||||
| 1.2 | 0 | −0.732051 | 0 | 0.386509 | 0 | 1.44247 | 0 | −2.46410 | 0 | |||||||||||||||||||||||||||||||
| 1.3 | 0 | 2.73205 | 0 | −3.27743 | 0 | −0.878184 | 0 | 4.46410 | 0 | |||||||||||||||||||||||||||||||
| 1.4 | 0 | 2.73205 | 0 | 2.27743 | 0 | 0.610235 | 0 | 4.46410 | 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 | 1984.2.a.z | 4 | |
| 4.b | odd | 2 | 1 | 1984.2.a.y | 4 | ||
| 8.b | even | 2 | 1 | 992.2.a.e | ✓ | 4 | |
| 8.d | odd | 2 | 1 | 992.2.a.f | yes | 4 | |
| 24.f | even | 2 | 1 | 8928.2.a.bn | 4 | ||
| 24.h | odd | 2 | 1 | 8928.2.a.bm | 4 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 992.2.a.e | ✓ | 4 | 8.b | even | 2 | 1 | |
| 992.2.a.f | yes | 4 | 8.d | odd | 2 | 1 | |
| 1984.2.a.y | 4 | 4.b | odd | 2 | 1 | ||
| 1984.2.a.z | 4 | 1.a | even | 1 | 1 | trivial | |
| 8928.2.a.bm | 4 | 24.h | odd | 2 | 1 | ||
| 8928.2.a.bn | 4 | 24.f | even | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(1984))\):
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\( T_{3}^{2} - 2T_{3} - 2 \)
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\( T_{5}^{4} + 2T_{5}^{3} - 7T_{5}^{2} - 8T_{5} + 4 \)
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\( T_{7}^{4} + 4T_{7}^{3} - 7T_{7}^{2} - 4T_{7} + 4 \)
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\( T_{19}^{4} - 8T_{19}^{3} - 3T_{19}^{2} + 112T_{19} - 128 \)
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Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} \)
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| $3$ |
\( (T^{2} - 2 T - 2)^{2} \)
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| $5$ |
\( T^{4} + 2 T^{3} + \cdots + 4 \)
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| $7$ |
\( T^{4} + 4 T^{3} + \cdots + 4 \)
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| $11$ |
\( T^{4} - 10 T^{3} + \cdots - 8 \)
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| $13$ |
\( T^{4} - 40T^{2} + 388 \)
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| $17$ |
\( T^{4} - 2 T^{3} + \cdots + 184 \)
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| $19$ |
\( T^{4} - 8 T^{3} + \cdots - 128 \)
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| $23$ |
\( T^{4} + 2 T^{3} + \cdots + 64 \)
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| $29$ |
\( T^{4} - 40 T^{2} + \cdots + 4 \)
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| $31$ |
\( (T + 1)^{4} \)
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| $37$ |
\( T^{4} - 10 T^{3} + \cdots - 488 \)
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| $41$ |
\( T^{4} - 2 T^{3} + \cdots - 104 \)
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| $43$ |
\( T^{4} - 24 T^{3} + \cdots - 716 \)
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| $47$ |
\( T^{4} - 100 T^{2} + \cdots - 416 \)
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| $53$ |
\( T^{4} - 2 T^{3} + \cdots - 128 \)
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| $59$ |
\( T^{4} - 20 T^{3} + \cdots - 5864 \)
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| $61$ |
\( T^{4} - 40T^{2} + 388 \)
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| $67$ |
\( T^{4} - 24 T^{3} + \cdots - 1472 \)
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| $71$ |
\( T^{4} - 16 T^{3} + \cdots + 256 \)
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| $73$ |
\( (T^{2} - 4 T - 44)^{2} \)
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| $79$ |
\( T^{4} - 6 T^{3} + \cdots + 144 \)
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| $83$ |
\( T^{4} - 6 T^{3} + \cdots + 56536 \)
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| $89$ |
\( T^{4} + 18 T^{3} + \cdots + 1048 \)
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| $97$ |
\( T^{4} + 6 T^{3} + \cdots - 872 \)
|
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