Newspace parameters
| Level: | \( N \) | \(=\) | \( 1976 = 2^{3} \cdot 13 \cdot 19 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 1976.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(15.7784394394\) |
| Analytic rank: | \(1\) |
| Dimension: | \(4\) |
| Coefficient field: | 4.4.26825.1 |
|
|
|
| Defining polynomial: |
\( x^{4} - 2x^{3} - 5x^{2} + 5x + 5 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| 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
\( x^{4} - 2x^{3} - 5x^{2} + 5x + 5 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
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| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - \nu - 3 \)
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| \(\beta_{3}\) | \(=\) |
\( \nu^{3} - 2\nu^{2} - 3\nu + 3 \)
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| \(\nu\) | \(=\) |
\( \beta_1 \)
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| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + \beta _1 + 3 \)
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| \(\nu^{3}\) | \(=\) |
\( \beta_{3} + 2\beta_{2} + 5\beta _1 + 3 \)
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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 | −2.92520 | 0 | 2.63158 | 0 | −0.706381 | 0 | 5.55677 | 0 | ||||||||||||||||||||||||||||||
| 1.2 | 0 | −1.45612 | 0 | −2.33583 | 0 | 2.79195 | 0 | −0.879714 | 0 | |||||||||||||||||||||||||||||||
| 1.3 | 0 | 0.696894 | 0 | −1.81744 | 0 | 0.120551 | 0 | −2.51434 | 0 | |||||||||||||||||||||||||||||||
| 1.4 | 0 | 1.68442 | 0 | 1.52170 | 0 | −4.20612 | 0 | −0.162720 | 0 | |||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \( +1 \) |
| \(13\) | \( +1 \) |
| \(19\) | \( +1 \) |
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 | 1976.2.a.f | ✓ | 4 |
| 4.b | odd | 2 | 1 | 3952.2.a.w | 4 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 1976.2.a.f | ✓ | 4 | 1.a | even | 1 | 1 | trivial |
| 3952.2.a.w | 4 | 4.b | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(1976))\):
|
\( T_{3}^{4} + 2T_{3}^{3} - 5T_{3}^{2} - 5T_{3} + 5 \)
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\( T_{5}^{4} - 9T_{5}^{2} - T_{5} + 17 \)
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Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} \)
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| $3$ |
\( T^{4} + 2 T^{3} + \cdots + 5 \)
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| $5$ |
\( T^{4} - 9T^{2} - T + 17 \)
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| $7$ |
\( T^{4} + 2 T^{3} + \cdots + 1 \)
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| $11$ |
\( T^{4} - 4 T^{3} + \cdots + 8 \)
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| $13$ |
\( (T + 1)^{4} \)
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| $17$ |
\( T^{4} + 7 T^{3} + \cdots + 5 \)
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| $19$ |
\( (T + 1)^{4} \)
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| $23$ |
\( T^{4} + 4 T^{3} + \cdots + 10 \)
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| $29$ |
\( T^{4} - 9 T^{3} + \cdots - 112 \)
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| $31$ |
\( T^{4} + 9 T^{3} + \cdots - 112 \)
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| $37$ |
\( T^{4} - 7 T^{3} + \cdots + 5 \)
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| $41$ |
\( T^{4} + 5 T^{3} + \cdots + 2482 \)
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| $43$ |
\( T^{4} - 4 T^{3} + \cdots + 29 \)
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| $47$ |
\( T^{4} + 2 T^{3} + \cdots + 739 \)
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| $53$ |
\( T^{4} - 5 T^{3} + \cdots + 2 \)
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| $59$ |
\( T^{4} + 26 T^{3} + \cdots + 290 \)
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| $61$ |
\( T^{4} + 14 T^{3} + \cdots + 80 \)
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| $67$ |
\( T^{4} + 11 T^{3} + \cdots + 20 \)
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| $71$ |
\( T^{4} + 22 T^{3} + \cdots - 857 \)
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| $73$ |
\( T^{4} + 3 T^{3} + \cdots + 40 \)
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| $79$ |
\( T^{4} + 16 T^{3} + \cdots + 3676 \)
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| $83$ |
\( T^{4} + 18 T^{3} + \cdots - 16 \)
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| $89$ |
\( T^{4} + 15 T^{3} + \cdots - 16200 \)
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| $97$ |
\( T^{4} + 16 T^{3} + \cdots - 3824 \)
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