Newspace parameters
| Level: | \( N \) | \(=\) | \( 63 = 3^{2} \cdot 7 \) |
| Weight: | \( k \) | \(=\) | \( 6 \) |
| Character orbit: | \([\chi]\) | \(=\) | 63.c (of order \(2\), degree \(1\), minimal) |
Newform invariants
| Self dual: | no |
| Analytic conductor: | \(10.1041806482\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Coefficient field: | \(\Q(\sqrt{-2}, \sqrt{7})\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{4} + 8x^{2} + 9 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{11}]\) |
| Coefficient ring index: | \( 3 \) |
| Twist minimal: | yes |
| Sato-Tate group: | $\mathrm{U}(1)[D_{2}]$ |
$q$-expansion
Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3\) 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
\( x^{4} + 8x^{2} + 9 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} + 4 \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{3} + 6\nu \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} - 4 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{3} - 6\beta_1 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/63\mathbb{Z}\right)^\times\).
| \(n\) | \(10\) | \(29\) |
| \(\chi(n)\) | \(-1\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 62.1 |
|
− | 9.64900i | 0 | −61.1033 | 0 | 0 | −129.642 | 280.817i | 0 | 0 | |||||||||||||||||||||||||||||
| 62.2 | − | 5.90735i | 0 | −2.89674 | 0 | 0 | 129.642 | − | 171.923i | 0 | 0 | |||||||||||||||||||||||||||||
| 62.3 | 5.90735i | 0 | −2.89674 | 0 | 0 | 129.642 | 171.923i | 0 | 0 | |||||||||||||||||||||||||||||||
| 62.4 | 9.64900i | 0 | −61.1033 | 0 | 0 | −129.642 | − | 280.817i | 0 | 0 | ||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 7.b | odd | 2 | 1 | CM by \(\Q(\sqrt{-7}) \) |
| 3.b | odd | 2 | 1 | inner |
| 21.c | even | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 63.6.c.a | ✓ | 4 |
| 3.b | odd | 2 | 1 | inner | 63.6.c.a | ✓ | 4 |
| 4.b | odd | 2 | 1 | 1008.6.k.a | 4 | ||
| 7.b | odd | 2 | 1 | CM | 63.6.c.a | ✓ | 4 |
| 12.b | even | 2 | 1 | 1008.6.k.a | 4 | ||
| 21.c | even | 2 | 1 | inner | 63.6.c.a | ✓ | 4 |
| 28.d | even | 2 | 1 | 1008.6.k.a | 4 | ||
| 84.h | odd | 2 | 1 | 1008.6.k.a | 4 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 63.6.c.a | ✓ | 4 | 1.a | even | 1 | 1 | trivial |
| 63.6.c.a | ✓ | 4 | 3.b | odd | 2 | 1 | inner |
| 63.6.c.a | ✓ | 4 | 7.b | odd | 2 | 1 | CM |
| 63.6.c.a | ✓ | 4 | 21.c | even | 2 | 1 | inner |
| 1008.6.k.a | 4 | 4.b | odd | 2 | 1 | ||
| 1008.6.k.a | 4 | 12.b | even | 2 | 1 | ||
| 1008.6.k.a | 4 | 28.d | even | 2 | 1 | ||
| 1008.6.k.a | 4 | 84.h | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{4} + 128T_{2}^{2} + 3249 \)
acting on \(S_{6}^{\mathrm{new}}(63, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} + 128T^{2} + 3249 \)
|
| $3$ |
\( T^{4} \)
|
| $5$ |
\( T^{4} \)
|
| $7$ |
\( (T^{2} - 16807)^{2} \)
|
| $11$ |
\( T^{4} + \cdots + 100062138276 \)
|
| $13$ |
\( T^{4} \)
|
| $17$ |
\( T^{4} \)
|
| $19$ |
\( T^{4} \)
|
| $23$ |
\( T^{4} + \cdots + 135713599545924 \)
|
| $29$ |
\( T^{4} + \cdots + 144125451311076 \)
|
| $31$ |
\( T^{4} \)
|
| $37$ |
\( (T^{2} - 198414832)^{2} \)
|
| $41$ |
\( T^{4} \)
|
| $43$ |
\( (T^{2} - 450018268)^{2} \)
|
| $47$ |
\( T^{4} \)
|
| $53$ |
\( T^{4} + \cdots + 54\!\cdots\!96 \)
|
| $59$ |
\( T^{4} \)
|
| $61$ |
\( T^{4} \)
|
| $67$ |
\( (T + 69364)^{4} \)
|
| $71$ |
\( T^{4} + \cdots + 12\!\cdots\!76 \)
|
| $73$ |
\( T^{4} \)
|
| $79$ |
\( (T - 80168)^{4} \)
|
| $83$ |
\( T^{4} \)
|
| $89$ |
\( T^{4} \)
|
| $97$ |
\( T^{4} \)
|
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