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: | 3.3.3368.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{3} - x^{2} - 15x + 11 \)
|
| 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
\( x^{3} - x^{2} - 15x + 11 \)
:
| \(\beta_{1}\) | \(=\) |
\( 2\nu - 1 \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - 10 \)
|
| \(\nu\) | \(=\) |
\( ( \beta _1 + 1 ) / 2 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + 10 \)
|
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 |
|
−7.52601 | −9.00000 | 24.6408 | 0 | 67.7341 | 234.126 | 55.3856 | 81.0000 | 0 | |||||||||||||||||||||||||||
| 1.2 | 1.44737 | −9.00000 | −29.9051 | 0 | −13.0263 | −41.5023 | −89.5997 | 81.0000 | 0 | ||||||||||||||||||||||||||||
| 1.3 | 8.07863 | −9.00000 | 33.2643 | 0 | −72.7077 | 39.3760 | 10.2141 | 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.i | 3 | |
| 5.b | even | 2 | 1 | 165.6.a.b | ✓ | 3 | |
| 15.d | odd | 2 | 1 | 495.6.a.d | 3 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 165.6.a.b | ✓ | 3 | 5.b | even | 2 | 1 | |
| 495.6.a.d | 3 | 15.d | odd | 2 | 1 | ||
| 825.6.a.i | 3 | 1.a | even | 1 | 1 | trivial | |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{3} - 2T_{2}^{2} - 60T_{2} + 88 \)
acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(825))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{3} - 2 T^{2} + \cdots + 88 \)
|
| $3$ |
\( (T + 9)^{3} \)
|
| $5$ |
\( T^{3} \)
|
| $7$ |
\( T^{3} - 232 T^{2} + \cdots + 382608 \)
|
| $11$ |
\( (T - 121)^{3} \)
|
| $13$ |
\( T^{3} + 450 T^{2} + \cdots - 22659488 \)
|
| $17$ |
\( T^{3} - 334 T^{2} + \cdots + 57782448 \)
|
| $19$ |
\( T^{3} + \cdots - 1630951200 \)
|
| $23$ |
\( T^{3} + \cdots + 24275701568 \)
|
| $29$ |
\( T^{3} + \cdots + 65949214584 \)
|
| $31$ |
\( T^{3} + \cdots - 211578448896 \)
|
| $37$ |
\( T^{3} + \cdots + 431879868536 \)
|
| $41$ |
\( T^{3} + \cdots - 7803557208 \)
|
| $43$ |
\( T^{3} + \cdots - 1659712050000 \)
|
| $47$ |
\( T^{3} + \cdots - 52162385088 \)
|
| $53$ |
\( T^{3} + \cdots - 2687939232856 \)
|
| $59$ |
\( T^{3} + \cdots - 3633753791296 \)
|
| $61$ |
\( T^{3} + \cdots - 12904038746056 \)
|
| $67$ |
\( T^{3} + \cdots + 40648408406912 \)
|
| $71$ |
\( T^{3} + \cdots + 8578136735360 \)
|
| $73$ |
\( T^{3} + \cdots + 144432126809632 \)
|
| $79$ |
\( T^{3} + \cdots + 351884592248992 \)
|
| $83$ |
\( T^{3} + \cdots - 5794291383408 \)
|
| $89$ |
\( T^{3} + \cdots - 246103360939432 \)
|
| $97$ |
\( T^{3} + \cdots - 179909862970168 \)
|
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