Newspace parameters
| Level: | \( N \) | \(=\) | \( 882 = 2 \cdot 3^{2} \cdot 7^{2} \) |
| Weight: | \( k \) | \(=\) | \( 6 \) |
| Character orbit: | \([\chi]\) | \(=\) | 882.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(141.458529075\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Coefficient field: | \(\Q(\sqrt{2}, \sqrt{793})\) |
|
|
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| Defining polynomial: |
\( x^{4} - 2x^{3} - 399x^{2} + 400x + 38414 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{17}]\) |
| Coefficient ring index: | \( 2\cdot 3^{2}\cdot 7^{2} \) |
| Twist minimal: | no (minimal twist has level 98) |
| 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
\( x^{4} - 2x^{3} - 399x^{2} + 400x + 38414 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( -14\nu^{3} + 21\nu^{2} + 2849\nu - 1428 ) / 785 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -84\nu^{3} + 126\nu^{2} + 50064\nu - 25053 ) / 785 \)
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| \(\beta_{3}\) | \(=\) |
\( ( 7\nu^{3} + 1167\nu^{2} - 2602\nu - 234786 ) / 785 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{2} - 6\beta _1 + 21 ) / 42 \)
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| \(\nu^{2}\) | \(=\) |
\( ( 28\beta_{3} + \beta_{2} + 8\beta _1 + 8421 ) / 42 \)
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| \(\nu^{3}\) | \(=\) |
\( ( 42\beta_{3} + 205\beta_{2} - 3564\beta _1 + 12621 ) / 42 \)
|
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 |
|
−4.00000 | 0 | 16.0000 | −84.4857 | 0 | 0 | −64.0000 | 0 | 337.943 | ||||||||||||||||||||||||||||||
| 1.2 | −4.00000 | 0 | 16.0000 | −34.9882 | 0 | 0 | −64.0000 | 0 | 139.953 | |||||||||||||||||||||||||||||||
| 1.3 | −4.00000 | 0 | 16.0000 | 34.9882 | 0 | 0 | −64.0000 | 0 | −139.953 | |||||||||||||||||||||||||||||||
| 1.4 | −4.00000 | 0 | 16.0000 | 84.4857 | 0 | 0 | −64.0000 | 0 | −337.943 | |||||||||||||||||||||||||||||||
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 | 882.6.a.bv | 4 | |
| 3.b | odd | 2 | 1 | 98.6.a.i | ✓ | 4 | |
| 7.b | odd | 2 | 1 | inner | 882.6.a.bv | 4 | |
| 12.b | even | 2 | 1 | 784.6.a.bg | 4 | ||
| 21.c | even | 2 | 1 | 98.6.a.i | ✓ | 4 | |
| 21.g | even | 6 | 2 | 98.6.c.i | 8 | ||
| 21.h | odd | 6 | 2 | 98.6.c.i | 8 | ||
| 84.h | odd | 2 | 1 | 784.6.a.bg | 4 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 98.6.a.i | ✓ | 4 | 3.b | odd | 2 | 1 | |
| 98.6.a.i | ✓ | 4 | 21.c | even | 2 | 1 | |
| 98.6.c.i | 8 | 21.g | even | 6 | 2 | ||
| 98.6.c.i | 8 | 21.h | odd | 6 | 2 | ||
| 784.6.a.bg | 4 | 12.b | even | 2 | 1 | ||
| 784.6.a.bg | 4 | 84.h | odd | 2 | 1 | ||
| 882.6.a.bv | 4 | 1.a | even | 1 | 1 | trivial | |
| 882.6.a.bv | 4 | 7.b | odd | 2 | 1 | inner | |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(882))\):
|
\( T_{5}^{4} - 8362T_{5}^{2} + 8737936 \)
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\( T_{11}^{2} + 314T_{11} - 325064 \)
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Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T + 4)^{4} \)
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| $3$ |
\( T^{4} \)
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| $5$ |
\( T^{4} - 8362 T^{2} + 8737936 \)
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| $7$ |
\( T^{4} \)
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| $11$ |
\( (T^{2} + 314 T - 325064)^{2} \)
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| $13$ |
\( T^{4} - 720594 T^{2} + 112021056 \)
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| $17$ |
\( T^{4} + \cdots + 445030415236 \)
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| $19$ |
\( T^{4} + \cdots + 15587240932624 \)
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| $23$ |
\( (T^{2} + 1312 T - 5165072)^{2} \)
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| $29$ |
\( (T^{2} + 8866 T + 19301776)^{2} \)
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| $31$ |
\( T^{4} + \cdots + 19507839565824 \)
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| $37$ |
\( (T^{2} - 1466 T - 2610128)^{2} \)
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| $41$ |
\( T^{4} + \cdots + 18\!\cdots\!96 \)
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| $43$ |
\( (T^{2} - 4918 T - 22280072)^{2} \)
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| $47$ |
\( T^{4} + \cdots + 12\!\cdots\!96 \)
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| $53$ |
\( (T^{2} + 776 T - 504835028)^{2} \)
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| $59$ |
\( T^{4} + \cdots + 43\!\cdots\!64 \)
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| $61$ |
\( T^{4} + \cdots + 15\!\cdots\!36 \)
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| $67$ |
\( (T^{2} - 6852 T - 101569536)^{2} \)
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| $71$ |
\( (T^{2} - 28332 T - 3437738496)^{2} \)
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| $73$ |
\( T^{4} + \cdots + 71\!\cdots\!84 \)
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| $79$ |
\( (T^{2} + 109148 T + 2741915488)^{2} \)
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| $83$ |
\( T^{4} + \cdots + 28\!\cdots\!96 \)
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| $89$ |
\( T^{4} + \cdots + 12\!\cdots\!64 \)
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| $97$ |
\( T^{4} + \cdots + 32780823506116 \)
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