Newspace parameters
| Level: | \( N \) | \(=\) | \( 7938 = 2 \cdot 3^{4} \cdot 7^{2} \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 7938.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(63.3852491245\) |
| Analytic rank: | \(1\) |
| Dimension: | \(4\) |
| Coefficient field: | \(\Q(\sqrt{16 +6 \sqrt{3}})\) |
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|
|
| Defining polynomial: |
\( x^{4} - 2x^{3} - 8x^{2} + 6 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{11}]\) |
| Coefficient ring index: | \( 2 \) |
| 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
\( x^{4} - 2x^{3} - 8x^{2} + 6 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu^{3} - 3\nu^{2} - 4\nu + 3 \)
|
| \(\beta_{2}\) | \(=\) |
\( -\nu^{3} + 3\nu^{2} + 6\nu - 4 \)
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| \(\beta_{3}\) | \(=\) |
\( -\nu^{3} + 4\nu^{2} + 2\nu - 7 \)
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| \(\nu\) | \(=\) |
\( ( \beta_{2} + \beta _1 + 1 ) / 2 \)
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| \(\nu^{2}\) | \(=\) |
\( \beta_{3} + \beta_{2} + 2\beta _1 + 5 \)
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| \(\nu^{3}\) | \(=\) |
\( 3\beta_{3} + 5\beta_{2} + 9\beta _1 + 14 \)
|
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 |
|
−1.00000 | 0 | 1.00000 | −3.23483 | 0 | 0 | −1.00000 | 0 | 3.23483 | ||||||||||||||||||||||||||||||
| 1.2 | −1.00000 | 0 | 1.00000 | −1.88040 | 0 | 0 | −1.00000 | 0 | 1.88040 | |||||||||||||||||||||||||||||||
| 1.3 | −1.00000 | 0 | 1.00000 | 1.88040 | 0 | 0 | −1.00000 | 0 | −1.88040 | |||||||||||||||||||||||||||||||
| 1.4 | −1.00000 | 0 | 1.00000 | 3.23483 | 0 | 0 | −1.00000 | 0 | −3.23483 | |||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \( +1 \) |
| \(3\) | \( -1 \) |
| \(7\) | \( -1 \) |
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 7.b | odd | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 7938.2.a.cf | ✓ | 4 |
| 3.b | odd | 2 | 1 | 7938.2.a.cs | yes | 4 | |
| 7.b | odd | 2 | 1 | inner | 7938.2.a.cf | ✓ | 4 |
| 21.c | even | 2 | 1 | 7938.2.a.cs | yes | 4 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 7938.2.a.cf | ✓ | 4 | 1.a | even | 1 | 1 | trivial |
| 7938.2.a.cf | ✓ | 4 | 7.b | odd | 2 | 1 | inner |
| 7938.2.a.cs | yes | 4 | 3.b | odd | 2 | 1 | |
| 7938.2.a.cs | yes | 4 | 21.c | even | 2 | 1 | |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(7938))\):
|
\( T_{5}^{4} - 14T_{5}^{2} + 37 \)
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\( T_{11}^{2} + 2T_{11} - 2 \)
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\( T_{13}^{4} - 50T_{13}^{2} + 37 \)
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\( T_{17}^{4} - 42T_{17}^{2} + 333 \)
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\( T_{23}^{2} + 6T_{23} + 6 \)
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Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T + 1)^{4} \)
|
| $3$ |
\( T^{4} \)
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| $5$ |
\( T^{4} - 14T^{2} + 37 \)
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| $7$ |
\( T^{4} \)
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| $11$ |
\( (T^{2} + 2 T - 2)^{2} \)
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| $13$ |
\( T^{4} - 50T^{2} + 37 \)
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| $17$ |
\( T^{4} - 42T^{2} + 333 \)
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| $19$ |
\( T^{4} - 32T^{2} + 148 \)
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| $23$ |
\( (T^{2} + 6 T + 6)^{2} \)
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| $29$ |
\( (T + 1)^{4} \)
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| $31$ |
\( T^{4} - 80T^{2} + 148 \)
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| $37$ |
\( (T^{2} - 27)^{2} \)
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| $41$ |
\( T^{4} - 168T^{2} + 5328 \)
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| $43$ |
\( (T^{2} - 8 T - 32)^{2} \)
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| $47$ |
\( T^{4} - 32T^{2} + 148 \)
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| $53$ |
\( (T^{2} - 4 T - 104)^{2} \)
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| $59$ |
\( T^{4} - 320 T^{2} + 25012 \)
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| $61$ |
\( T^{4} - 146T^{2} + 37 \)
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| $67$ |
\( (T^{2} - 2 T - 74)^{2} \)
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| $71$ |
\( (T^{2} + 8 T - 32)^{2} \)
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| $73$ |
\( T^{4} - 14T^{2} + 37 \)
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| $79$ |
\( (T^{2} + 10 T - 50)^{2} \)
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| $83$ |
\( T^{4} - 96T^{2} + 1332 \)
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| $89$ |
\( T^{4} - 170T^{2} + 6253 \)
|
| $97$ |
\( T^{4} - 200T^{2} + 592 \)
|
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