Newspace parameters
| Level: | \( N \) | \(=\) | \( 504 = 2^{3} \cdot 3^{2} \cdot 7 \) |
| Weight: | \( k \) | \(=\) | \( 8 \) |
| Character orbit: | \([\chi]\) | \(=\) | 504.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(157.442052844\) |
| Analytic rank: | \(0\) |
| Dimension: | \(3\) |
| Coefficient field: | 3.3.3109313.1 |
|
|
|
| Defining polynomial: |
\( x^{3} - x^{2} - 384x + 1148 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{11}]\) |
| Coefficient ring index: | \( 2^{4}\cdot 3^{3} \) |
| Twist minimal: | no (minimal twist has level 56) |
| Fricke sign: | \(+1\) |
| Sato-Tate group: | $\mathrm{SU}(2)$ |
$q$-expansion
Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2\) 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
\( x^{3} - x^{2} - 384x + 1148 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( 3\nu^{2} + 99\nu - 802 ) / 4 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -9\nu^{2} - 9\nu + 2310 ) / 4 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{2} + 3\beta _1 + 24 ) / 72 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( -11\beta_{2} - \beta _1 + 6152 ) / 24 \)
|
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 | 0 | −438.788 | 0 | 343.000 | 0 | 0 | 0 | |||||||||||||||||||||||||||
| 1.2 | 0 | 0 | 0 | −164.370 | 0 | 343.000 | 0 | 0 | 0 | ||||||||||||||||||||||||||||
| 1.3 | 0 | 0 | 0 | 465.158 | 0 | 343.000 | 0 | 0 | 0 | ||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \( +1 \) |
| \(3\) | \( -1 \) |
| \(7\) | \( -1 \) |
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 | 504.8.a.k | 3 | |
| 3.b | odd | 2 | 1 | 56.8.a.e | ✓ | 3 | |
| 12.b | even | 2 | 1 | 112.8.a.k | 3 | ||
| 21.c | even | 2 | 1 | 392.8.a.f | 3 | ||
| 24.f | even | 2 | 1 | 448.8.a.x | 3 | ||
| 24.h | odd | 2 | 1 | 448.8.a.u | 3 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 56.8.a.e | ✓ | 3 | 3.b | odd | 2 | 1 | |
| 112.8.a.k | 3 | 12.b | even | 2 | 1 | ||
| 392.8.a.f | 3 | 21.c | even | 2 | 1 | ||
| 448.8.a.u | 3 | 24.h | odd | 2 | 1 | ||
| 448.8.a.x | 3 | 24.f | even | 2 | 1 | ||
| 504.8.a.k | 3 | 1.a | even | 1 | 1 | trivial | |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{3} + 138T_{5}^{2} - 208440T_{5} - 33548800 \)
acting on \(S_{8}^{\mathrm{new}}(\Gamma_0(504))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{3} \)
|
| $3$ |
\( T^{3} \)
|
| $5$ |
\( T^{3} + 138 T^{2} + \cdots - 33548800 \)
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| $7$ |
\( (T - 343)^{3} \)
|
| $11$ |
\( T^{3} + \cdots - 84330160128 \)
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| $13$ |
\( T^{3} + \cdots + 72413129088 \)
|
| $17$ |
\( T^{3} + \cdots - 9430557450200 \)
|
| $19$ |
\( T^{3} + \cdots + 1167232610288 \)
|
| $23$ |
\( T^{3} + \cdots - 44561003855872 \)
|
| $29$ |
\( T^{3} + \cdots - 45\!\cdots\!32 \)
|
| $31$ |
\( T^{3} + \cdots + 35\!\cdots\!40 \)
|
| $37$ |
\( T^{3} + \cdots - 12\!\cdots\!92 \)
|
| $41$ |
\( T^{3} + \cdots - 31\!\cdots\!48 \)
|
| $43$ |
\( T^{3} + \cdots - 25\!\cdots\!88 \)
|
| $47$ |
\( T^{3} + \cdots + 96\!\cdots\!44 \)
|
| $53$ |
\( T^{3} + \cdots - 95\!\cdots\!16 \)
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| $59$ |
\( T^{3} + \cdots + 15\!\cdots\!96 \)
|
| $61$ |
\( T^{3} + \cdots - 35\!\cdots\!40 \)
|
| $67$ |
\( T^{3} + \cdots - 64\!\cdots\!84 \)
|
| $71$ |
\( T^{3} + \cdots + 87\!\cdots\!96 \)
|
| $73$ |
\( T^{3} + \cdots - 21\!\cdots\!88 \)
|
| $79$ |
\( T^{3} + \cdots - 13\!\cdots\!08 \)
|
| $83$ |
\( T^{3} + \cdots - 14\!\cdots\!88 \)
|
| $89$ |
\( T^{3} + \cdots + 25\!\cdots\!16 \)
|
| $97$ |
\( T^{3} + \cdots - 68\!\cdots\!68 \)
|
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