Newspace parameters
| Level: | \( N \) | \(=\) | \( 825 = 3 \cdot 5^{2} \cdot 11 \) |
| Weight: | \( k \) | \(=\) | \( 6 \) |
| Character orbit: | \([\chi]\) | \(=\) | 825.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(132.316651346\) |
| Analytic rank: | \(0\) |
| Dimension: | \(3\) |
| Coefficient field: | 3.3.788.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{3} - x^{2} - 7x - 3 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2^{2} \) |
| Twist minimal: | no (minimal twist has level 165) |
| 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
\( x^{3} - x^{2} - 7x - 3 \)
:
| \(\beta_{1}\) | \(=\) |
\( 2\nu - 1 \)
|
| \(\beta_{2}\) | \(=\) |
\( 2\nu^{2} - 4\nu - 9 \)
|
| \(\nu\) | \(=\) |
\( ( \beta _1 + 1 ) / 2 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( \beta_{2} + 2\beta _1 + 11 ) / 2 \)
|
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 |
|
−6.55890 | 9.00000 | 11.0192 | 0 | −59.0301 | −146.487 | 137.611 | 81.0000 | 0 | |||||||||||||||||||||||||||
| 1.2 | −1.08127 | 9.00000 | −30.8308 | 0 | −9.73147 | 139.508 | 67.9374 | 81.0000 | 0 | ||||||||||||||||||||||||||||
| 1.3 | 5.64018 | 9.00000 | −0.188384 | 0 | 50.7616 | −145.021 | −181.548 | 81.0000 | 0 | ||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(3\) | \(-1\) |
| \(5\) | \(1\) |
| \(11\) | \(1\) |
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 | 825.6.a.h | 3 | |
| 5.b | even | 2 | 1 | 165.6.a.d | ✓ | 3 | |
| 15.d | odd | 2 | 1 | 495.6.a.c | 3 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 165.6.a.d | ✓ | 3 | 5.b | even | 2 | 1 | |
| 495.6.a.c | 3 | 15.d | odd | 2 | 1 | ||
| 825.6.a.h | 3 | 1.a | even | 1 | 1 | trivial | |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{3} + 2T_{2}^{2} - 36T_{2} - 40 \)
acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(825))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{3} + 2 T^{2} + \cdots - 40 \)
|
| $3$ |
\( (T - 9)^{3} \)
|
| $5$ |
\( T^{3} \)
|
| $7$ |
\( T^{3} + 152 T^{2} + \cdots - 2963664 \)
|
| $11$ |
\( (T + 121)^{3} \)
|
| $13$ |
\( T^{3} - 546 T^{2} + \cdots + 7691848 \)
|
| $17$ |
\( T^{3} + \cdots + 1079472216 \)
|
| $19$ |
\( T^{3} - 1808 T^{2} + \cdots + 729480096 \)
|
| $23$ |
\( T^{3} + 4288 T^{2} + \cdots + 857355136 \)
|
| $29$ |
\( T^{3} - 5582 T^{2} + \cdots - 329440872 \)
|
| $31$ |
\( T^{3} + \cdots - 5126546304 \)
|
| $37$ |
\( T^{3} + \cdots - 244700027368 \)
|
| $41$ |
\( T^{3} + \cdots + 945181300968 \)
|
| $43$ |
\( T^{3} + \cdots - 1240285492944 \)
|
| $47$ |
\( T^{3} + \cdots + 18016103040 \)
|
| $53$ |
\( T^{3} + \cdots + 985333601848 \)
|
| $59$ |
\( T^{3} + \cdots + 2567903224000 \)
|
| $61$ |
\( T^{3} + \cdots + 4408473611240 \)
|
| $67$ |
\( T^{3} + \cdots + 41154720036800 \)
|
| $71$ |
\( T^{3} + \cdots + 117803610062464 \)
|
| $73$ |
\( T^{3} + \cdots + 163971863295832 \)
|
| $79$ |
\( T^{3} + \cdots - 554174036768 \)
|
| $83$ |
\( T^{3} + \cdots - 3029676562224 \)
|
| $89$ |
\( T^{3} + \cdots + 96218735089528 \)
|
| $97$ |
\( T^{3} + \cdots + 251540556847112 \)
|
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