Newspace parameters
| Level: | \( N \) | \(=\) | \( 63 = 3^{2} \cdot 7 \) |
| Weight: | \( k \) | \(=\) | \( 12 \) |
| Character orbit: | \([\chi]\) | \(=\) | 63.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(48.4056203753\) |
| Analytic rank: | \(1\) |
| Dimension: | \(3\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{3} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{3} - x^{2} - 815x - 6609 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 2^{2}\cdot 3 \) |
| Twist minimal: | no (minimal twist has level 21) |
| 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
\( x^{3} - x^{2} - 815x - 6609 \)
:
| \(\beta_{1}\) | \(=\) |
\( 2\nu - 1 \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - 14\nu - 539 \)
|
| \(\nu\) | \(=\) |
\( ( \beta _1 + 1 ) / 2 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + 7\beta _1 + 546 \)
|
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 |
|
−22.5406 | 0 | −1539.92 | −5853.63 | 0 | −16807.0 | 80874.0 | 0 | 131944. | |||||||||||||||||||||||||||
| 1.2 | 3.69438 | 0 | −2034.35 | 13175.4 | 0 | −16807.0 | −15081.7 | 0 | 48674.7 | ||||||||||||||||||||||||||||
| 1.3 | 86.8463 | 0 | 5494.27 | −10647.7 | 0 | −16807.0 | 299296. | 0 | −924715. | ||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(3\) | \(-1\) |
| \(7\) | \(1\) |
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 | 63.12.a.f | 3 | |
| 3.b | odd | 2 | 1 | 21.12.a.c | ✓ | 3 | |
| 21.c | even | 2 | 1 | 147.12.a.d | 3 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 21.12.a.c | ✓ | 3 | 3.b | odd | 2 | 1 | |
| 63.12.a.f | 3 | 1.a | even | 1 | 1 | trivial | |
| 147.12.a.d | 3 | 21.c | even | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{3} - 68T_{2}^{2} - 1720T_{2} + 7232 \)
acting on \(S_{12}^{\mathrm{new}}(\Gamma_0(63))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{3} - 68 T^{2} + \cdots + 7232 \)
|
| $3$ |
\( T^{3} \)
|
| $5$ |
\( T^{3} + \cdots - 821191152800 \)
|
| $7$ |
\( (T + 16807)^{3} \)
|
| $11$ |
\( T^{3} + \cdots + 49\!\cdots\!08 \)
|
| $13$ |
\( T^{3} + \cdots - 62\!\cdots\!56 \)
|
| $17$ |
\( T^{3} + \cdots + 13\!\cdots\!88 \)
|
| $19$ |
\( T^{3} + \cdots + 77\!\cdots\!60 \)
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| $23$ |
\( T^{3} + \cdots - 24\!\cdots\!12 \)
|
| $29$ |
\( T^{3} + \cdots - 65\!\cdots\!60 \)
|
| $31$ |
\( T^{3} + \cdots - 57\!\cdots\!16 \)
|
| $37$ |
\( T^{3} + \cdots - 71\!\cdots\!52 \)
|
| $41$ |
\( T^{3} + \cdots + 99\!\cdots\!00 \)
|
| $43$ |
\( T^{3} + \cdots - 68\!\cdots\!76 \)
|
| $47$ |
\( T^{3} + \cdots - 35\!\cdots\!04 \)
|
| $53$ |
\( T^{3} + \cdots - 13\!\cdots\!72 \)
|
| $59$ |
\( T^{3} + \cdots + 26\!\cdots\!80 \)
|
| $61$ |
\( T^{3} + \cdots + 41\!\cdots\!52 \)
|
| $67$ |
\( T^{3} + \cdots - 23\!\cdots\!96 \)
|
| $71$ |
\( T^{3} + \cdots + 15\!\cdots\!08 \)
|
| $73$ |
\( T^{3} + \cdots - 38\!\cdots\!72 \)
|
| $79$ |
\( T^{3} + \cdots - 13\!\cdots\!20 \)
|
| $83$ |
\( T^{3} + \cdots - 10\!\cdots\!92 \)
|
| $89$ |
\( T^{3} + \cdots - 56\!\cdots\!40 \)
|
| $97$ |
\( T^{3} + \cdots + 93\!\cdots\!48 \)
|
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