Newspace parameters
| Level: | \( N \) | \(=\) | \( 528 = 2^{4} \cdot 3 \cdot 11 \) |
| Weight: | \( k \) | \(=\) | \( 8 \) |
| Character orbit: | \([\chi]\) | \(=\) | 528.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(164.939293456\) |
| Analytic rank: | \(1\) |
| Dimension: | \(3\) |
| Coefficient field: | 3.3.115512.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{3} - x^{2} - 70x - 194 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{17}]\) |
| Coefficient ring index: | \( 2^{6}\cdot 3 \) |
| Twist minimal: | no (minimal twist has level 33) |
| 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
\( x^{3} - x^{2} - 70x - 194 \)
:
| \(\beta_{1}\) | \(=\) |
\( 6\nu^{2} - 48\nu - 266 \)
|
| \(\beta_{2}\) | \(=\) |
\( -10\nu^{2} + 48\nu + 454 \)
|
| \(\nu\) | \(=\) |
\( ( -3\beta_{2} - 5\beta _1 + 32 ) / 96 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( -\beta_{2} - \beta _1 + 188 ) / 4 \)
|
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 | 27.0000 | 0 | −505.769 | 0 | −1401.07 | 0 | 729.000 | 0 | |||||||||||||||||||||||||||
| 1.2 | 0 | 27.0000 | 0 | −22.9029 | 0 | 1466.13 | 0 | 729.000 | 0 | ||||||||||||||||||||||||||||
| 1.3 | 0 | 27.0000 | 0 | 84.6717 | 0 | −1679.06 | 0 | 729.000 | 0 | ||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \(-1\) |
| \(3\) | \(-1\) |
| \(11\) | \(-1\) |
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 | 528.8.a.o | 3 | |
| 4.b | odd | 2 | 1 | 33.8.a.d | ✓ | 3 | |
| 12.b | even | 2 | 1 | 99.8.a.e | 3 | ||
| 44.c | even | 2 | 1 | 363.8.a.e | 3 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 33.8.a.d | ✓ | 3 | 4.b | odd | 2 | 1 | |
| 99.8.a.e | 3 | 12.b | even | 2 | 1 | ||
| 363.8.a.e | 3 | 44.c | even | 2 | 1 | ||
| 528.8.a.o | 3 | 1.a | even | 1 | 1 | trivial | |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{3} + 444T_{5}^{2} - 33180T_{5} - 980800 \)
acting on \(S_{8}^{\mathrm{new}}(\Gamma_0(528))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{3} \)
|
| $3$ |
\( (T - 27)^{3} \)
|
| $5$ |
\( T^{3} + 444 T^{2} + \cdots - 980800 \)
|
| $7$ |
\( T^{3} + \cdots - 3449053112 \)
|
| $11$ |
\( (T - 1331)^{3} \)
|
| $13$ |
\( T^{3} + \cdots + 665759180384 \)
|
| $17$ |
\( T^{3} + \cdots - 11730861043168 \)
|
| $19$ |
\( T^{3} + \cdots + 5608943166816 \)
|
| $23$ |
\( T^{3} + \cdots - 200462606267008 \)
|
| $29$ |
\( T^{3} + \cdots + 61407931779072 \)
|
| $31$ |
\( T^{3} + \cdots - 19\!\cdots\!96 \)
|
| $37$ |
\( T^{3} + \cdots + 11\!\cdots\!64 \)
|
| $41$ |
\( T^{3} + \cdots - 12\!\cdots\!52 \)
|
| $43$ |
\( T^{3} + \cdots + 20\!\cdots\!12 \)
|
| $47$ |
\( T^{3} + \cdots - 94\!\cdots\!80 \)
|
| $53$ |
\( T^{3} + \cdots + 29\!\cdots\!44 \)
|
| $59$ |
\( T^{3} + \cdots - 38\!\cdots\!00 \)
|
| $61$ |
\( T^{3} + \cdots - 45\!\cdots\!60 \)
|
| $67$ |
\( T^{3} + \cdots + 13\!\cdots\!00 \)
|
| $71$ |
\( T^{3} + \cdots - 13\!\cdots\!84 \)
|
| $73$ |
\( T^{3} + \cdots + 26\!\cdots\!96 \)
|
| $79$ |
\( T^{3} + \cdots + 11\!\cdots\!12 \)
|
| $83$ |
\( T^{3} + \cdots - 81\!\cdots\!28 \)
|
| $89$ |
\( T^{3} + \cdots - 26\!\cdots\!48 \)
|
| $97$ |
\( T^{3} + \cdots + 24\!\cdots\!84 \)
|
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