Newspace parameters
| Level: | \( N \) | \(=\) | \( 9984 = 2^{8} \cdot 3 \cdot 13 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 9984.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(79.7226413780\) |
| Analytic rank: | \(1\) |
| Dimension: | \(4\) |
| Coefficient field: | \(\Q(\zeta_{20})^+\) |
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| Defining polynomial: |
\( x^{4} - 5x^{2} + 5 \)
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| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2 \) |
| Twist minimal: | no (minimal twist has level 312) |
| 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 \(\nu = \zeta_{20} + \zeta_{20}^{-1}\):
| \(\beta_{1}\) | \(=\) |
\( \nu^{2} + \nu - 3 \)
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| \(\beta_{2}\) | \(=\) |
\( -\nu^{2} + \nu + 3 \)
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| \(\beta_{3}\) | \(=\) |
\( \nu^{3} - \nu^{2} - 3\nu + 3 \)
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| \(\nu\) | \(=\) |
\( ( \beta_{2} + \beta_1 ) / 2 \)
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| \(\nu^{2}\) | \(=\) |
\( ( -\beta_{2} + \beta _1 + 6 ) / 2 \)
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| \(\nu^{3}\) | \(=\) |
\( \beta_{3} + \beta_{2} + 2\beta_1 \)
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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 | −1.00000 | 0 | −2.52015 | 0 | −2.79360 | 0 | 1.00000 | 0 | ||||||||||||||||||||||||||||||
| 1.2 | 0 | −1.00000 | 0 | 0.442463 | 0 | 2.52015 | 0 | 1.00000 | 0 | |||||||||||||||||||||||||||||||
| 1.3 | 0 | −1.00000 | 0 | 1.28408 | 0 | −0.442463 | 0 | 1.00000 | 0 | |||||||||||||||||||||||||||||||
| 1.4 | 0 | −1.00000 | 0 | 2.79360 | 0 | −1.28408 | 0 | 1.00000 | 0 | |||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \( +1 \) |
| \(3\) | \( +1 \) |
| \(13\) | \( +1 \) |
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 | 9984.2.a.y | 4 | |
| 4.b | odd | 2 | 1 | 9984.2.a.bh | 4 | ||
| 8.b | even | 2 | 1 | 9984.2.a.bb | 4 | ||
| 8.d | odd | 2 | 1 | 9984.2.a.s | 4 | ||
| 16.e | even | 4 | 2 | 312.2.g.a | ✓ | 8 | |
| 16.f | odd | 4 | 2 | 1248.2.g.a | 8 | ||
| 48.i | odd | 4 | 2 | 936.2.g.d | 8 | ||
| 48.k | even | 4 | 2 | 3744.2.g.d | 8 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 312.2.g.a | ✓ | 8 | 16.e | even | 4 | 2 | |
| 936.2.g.d | 8 | 48.i | odd | 4 | 2 | ||
| 1248.2.g.a | 8 | 16.f | odd | 4 | 2 | ||
| 3744.2.g.d | 8 | 48.k | even | 4 | 2 | ||
| 9984.2.a.s | 4 | 8.d | odd | 2 | 1 | ||
| 9984.2.a.y | 4 | 1.a | even | 1 | 1 | trivial | |
| 9984.2.a.bb | 4 | 8.b | even | 2 | 1 | ||
| 9984.2.a.bh | 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(9984))\):
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\( T_{5}^{4} - 2T_{5}^{3} - 6T_{5}^{2} + 12T_{5} - 4 \)
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\( T_{7}^{4} + 2T_{7}^{3} - 6T_{7}^{2} - 12T_{7} - 4 \)
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\( T_{11}^{4} - 20T_{11}^{2} + 40T_{11} - 20 \)
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\( T_{19}^{4} - 6T_{19}^{3} - 14T_{19}^{2} + 84T_{19} - 4 \)
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\( T_{29}^{4} + 4T_{29}^{3} - 44T_{29}^{2} + 64T_{29} + 16 \)
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Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} \)
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| $3$ |
\( (T + 1)^{4} \)
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| $5$ |
\( T^{4} - 2 T^{3} + \cdots - 4 \)
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| $7$ |
\( T^{4} + 2 T^{3} + \cdots - 4 \)
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| $11$ |
\( T^{4} - 20 T^{2} + \cdots - 20 \)
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| $13$ |
\( (T + 1)^{4} \)
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| $17$ |
\( T^{4} + 8 T^{3} + \cdots + 16 \)
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| $19$ |
\( T^{4} - 6 T^{3} + \cdots - 4 \)
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| $23$ |
\( T^{4} - 4 T^{3} + \cdots + 16 \)
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| $29$ |
\( T^{4} + 4 T^{3} + \cdots + 16 \)
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| $31$ |
\( T^{4} + 2 T^{3} + \cdots + 316 \)
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| $37$ |
\( T^{4} - 8 T^{3} + \cdots + 976 \)
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| $41$ |
\( T^{4} + 18 T^{3} + \cdots - 2644 \)
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| $43$ |
\( T^{4} - 16 T^{3} + \cdots - 7024 \)
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| $47$ |
\( T^{4} - 12 T^{3} + \cdots - 964 \)
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| $53$ |
\( T^{4} + 4 T^{3} + \cdots + 1216 \)
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| $59$ |
\( T^{4} - 4 T^{3} + \cdots + 76 \)
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| $61$ |
\( (T^{2} + 10 T + 20)^{2} \)
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| $67$ |
\( T^{4} - 22 T^{3} + \cdots + 76 \)
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| $71$ |
\( T^{4} + 8 T^{3} + \cdots + 956 \)
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| $73$ |
\( T^{4} + 16 T^{3} + \cdots - 304 \)
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| $79$ |
\( T^{4} - 120 T^{2} + \cdots + 80 \)
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| $83$ |
\( T^{4} + 16 T^{3} + \cdots - 1604 \)
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| $89$ |
\( T^{4} + 30 T^{3} + \cdots - 2420 \)
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| $97$ |
\( T^{4} + 28 T^{3} + \cdots - 18544 \)
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