Newspace parameters
| Level: | \( N \) | \(=\) | \( 63 = 3^{2} \cdot 7 \) |
| Weight: | \( k \) | \(=\) | \( 6 \) |
| Character orbit: | \([\chi]\) | \(=\) | 63.p (of order \(6\), degree \(2\), minimal) |
Newform invariants
| Self dual: | no |
| Analytic conductor: | \(10.1041806482\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Relative dimension: | \(2\) over \(\Q(\zeta_{6})\) |
| Coefficient field: | \(\Q(\sqrt{-2}, \sqrt{-3})\) |
|
|
|
| Defining polynomial: |
\( x^{4} - 2x^{2} + 4 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | yes |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{6}]$ |
$q$-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
\( x^{4} - 2x^{2} + 4 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{2} ) / 2 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( \nu^{3} ) / 2 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( 2\beta_{2} \)
|
| \(\nu^{3}\) | \(=\) |
\( 2\beta_{3} \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/63\mathbb{Z}\right)^\times\).
| \(n\) | \(10\) | \(29\) |
| \(\chi(n)\) | \(1 - \beta_{2}\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 17.1 |
|
−2.44949 | + | 1.41421i | 0 | −12.0000 | + | 20.7846i | 25.7196 | + | 44.5477i | 0 | 122.500 | + | 42.4352i | − | 158.392i | 0 | −126.000 | − | 72.7461i | |||||||||||||||||||
| 17.2 | 2.44949 | − | 1.41421i | 0 | −12.0000 | + | 20.7846i | −25.7196 | − | 44.5477i | 0 | 122.500 | + | 42.4352i | 158.392i | 0 | −126.000 | − | 72.7461i | |||||||||||||||||||||
| 26.1 | −2.44949 | − | 1.41421i | 0 | −12.0000 | − | 20.7846i | 25.7196 | − | 44.5477i | 0 | 122.500 | − | 42.4352i | 158.392i | 0 | −126.000 | + | 72.7461i | |||||||||||||||||||||
| 26.2 | 2.44949 | + | 1.41421i | 0 | −12.0000 | − | 20.7846i | −25.7196 | + | 44.5477i | 0 | 122.500 | − | 42.4352i | − | 158.392i | 0 | −126.000 | + | 72.7461i | ||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 3.b | odd | 2 | 1 | inner |
| 7.d | odd | 6 | 1 | inner |
| 21.g | even | 6 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 63.6.p.a | ✓ | 4 |
| 3.b | odd | 2 | 1 | inner | 63.6.p.a | ✓ | 4 |
| 7.c | even | 3 | 1 | 441.6.c.a | 4 | ||
| 7.d | odd | 6 | 1 | inner | 63.6.p.a | ✓ | 4 |
| 7.d | odd | 6 | 1 | 441.6.c.a | 4 | ||
| 21.g | even | 6 | 1 | inner | 63.6.p.a | ✓ | 4 |
| 21.g | even | 6 | 1 | 441.6.c.a | 4 | ||
| 21.h | odd | 6 | 1 | 441.6.c.a | 4 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 63.6.p.a | ✓ | 4 | 1.a | even | 1 | 1 | trivial |
| 63.6.p.a | ✓ | 4 | 3.b | odd | 2 | 1 | inner |
| 63.6.p.a | ✓ | 4 | 7.d | odd | 6 | 1 | inner |
| 63.6.p.a | ✓ | 4 | 21.g | even | 6 | 1 | inner |
| 441.6.c.a | 4 | 7.c | even | 3 | 1 | ||
| 441.6.c.a | 4 | 7.d | odd | 6 | 1 | ||
| 441.6.c.a | 4 | 21.g | even | 6 | 1 | ||
| 441.6.c.a | 4 | 21.h | odd | 6 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{4} - 8T_{2}^{2} + 64 \)
acting on \(S_{6}^{\mathrm{new}}(63, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} - 8T^{2} + 64 \)
|
| $3$ |
\( T^{4} \)
|
| $5$ |
\( T^{4} + 2646 T^{2} + 7001316 \)
|
| $7$ |
\( (T^{2} - 245 T + 16807)^{2} \)
|
| $11$ |
\( T^{4} + \cdots + 111931731844 \)
|
| $13$ |
\( (T^{2} + 1326675)^{2} \)
|
| $17$ |
\( T^{4} + \cdots + 20248151616 \)
|
| $19$ |
\( (T^{2} + 3003 T + 3006003)^{2} \)
|
| $23$ |
\( T^{4} + \cdots + 451783527040000 \)
|
| $29$ |
\( (T^{2} + 21177032)^{2} \)
|
| $31$ |
\( (T^{2} + 3927 T + 5140443)^{2} \)
|
| $37$ |
\( (T^{2} + 203 T + 41209)^{2} \)
|
| $41$ |
\( (T^{2} - 7350)^{2} \)
|
| $43$ |
\( (T - 4697)^{4} \)
|
| $47$ |
\( T^{4} + \cdots + 33\!\cdots\!36 \)
|
| $53$ |
\( T^{4} + \cdots + 30\!\cdots\!24 \)
|
| $59$ |
\( T^{4} + \cdots + 611345405160000 \)
|
| $61$ |
\( (T^{2} + 30198 T + 303973068)^{2} \)
|
| $67$ |
\( (T^{2} + 30197 T + 911858809)^{2} \)
|
| $71$ |
\( (T^{2} + 15691202)^{2} \)
|
| $73$ |
\( (T^{2} - 92379 T + 2844626547)^{2} \)
|
| $79$ |
\( (T^{2} - 39385 T + 1551178225)^{2} \)
|
| $83$ |
\( (T^{2} - 4040103606)^{2} \)
|
| $89$ |
\( T^{4} + \cdots + 12\!\cdots\!16 \)
|
| $97$ |
\( (T^{2} + 4248300)^{2} \)
|
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