Newspace parameters
| Level: | \( N \) | \(=\) | \( 63 = 3^{2} \cdot 7 \) |
| Weight: | \( k \) | \(=\) | \( 3 \) |
| Character orbit: | \([\chi]\) | \(=\) | 63.j (of order \(6\), degree \(2\), minimal) |
Newform invariants
| Self dual: | no |
| Analytic conductor: | \(1.71662566547\) |
| Analytic rank: | \(0\) |
| Dimension: | \(6\) |
| Relative dimension: | \(3\) over \(\Q(\zeta_{6})\) |
| Coefficient field: | 6.0.63369648.1 |
| Defining polynomial: |
\( x^{6} - x^{5} + 12x^{4} + 17x^{3} + 118x^{2} + 33x + 9 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 2^{2} \) |
| 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 \(1,\beta_1,\ldots,\beta_{5}\) 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
\( x^{6} - x^{5} + 12x^{4} + 17x^{3} + 118x^{2} + 33x + 9 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( 2\nu^{5} - 24\nu^{4} + 288\nu^{3} - 236\nu^{2} - 66\nu + 2241 ) / 1449 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 44\nu^{5} - 45\nu^{4} + 540\nu^{3} + 604\nu^{2} + 5310\nu + 36 ) / 1449 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 2\nu^{5} - 3\nu^{4} + 36\nu^{3} + 16\nu^{2} + 354\nu + 99 ) / 63 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 83\nu^{5} - 30\nu^{4} + 843\nu^{3} + 2281\nu^{2} + 9819\nu + 5337 ) / 1449 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 32\nu^{5} - 62\nu^{4} + 422\nu^{3} + 249\nu^{2} + 3130\nu - 1335 ) / 483 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{3} - \beta_{2} - \beta_1 ) / 2 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{5} + 2\beta_{4} + \beta_{3} - 7\beta_{2} - 6 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( -2\beta_{5} + 2\beta_{4} + 13\beta _1 - 33 ) / 2 \)
|
| \(\nu^{4}\) | \(=\) |
\( -24\beta_{5} - 12\beta_{4} - 19\beta_{3} + 94\beta_{2} + 19\beta _1 - 24 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( -52\beta_{5} - 104\beta_{4} - 187\beta_{3} + 571\beta_{2} + 519 ) / 2 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/63\mathbb{Z}\right)^\times\).
| \(n\) | \(10\) | \(29\) |
| \(\chi(n)\) | \(-1 - \beta_{2}\) | \(1 + \beta_{2}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 11.1 |
|
− | 3.09257i | 3.00000 | −5.56399 | −5.67824 | − | 3.27834i | − | 9.27771i | 6.63848 | + | 2.22048i | 4.83675i | 9.00000 | −10.1385 | + | 17.5604i | ||||||||||||||||||||||||||||
| 11.2 | 1.03435i | 3.00000 | 2.93011 | −2.10422 | − | 1.21487i | 3.10306i | −4.75661 | − | 5.13563i | 7.16819i | 9.00000 | 1.25661 | − | 2.17651i | |||||||||||||||||||||||||||||||
| 11.3 | 3.79027i | 3.00000 | −10.3661 | 0.282467 | + | 0.163082i | 11.3708i | −2.88187 | + | 6.37925i | − | 24.1293i | 9.00000 | −0.618126 | + | 1.07063i | ||||||||||||||||||||||||||||||
| 23.1 | − | 3.79027i | 3.00000 | −10.3661 | 0.282467 | − | 0.163082i | − | 11.3708i | −2.88187 | − | 6.37925i | 24.1293i | 9.00000 | −0.618126 | − | 1.07063i | |||||||||||||||||||||||||||||
| 23.2 | − | 1.03435i | 3.00000 | 2.93011 | −2.10422 | + | 1.21487i | − | 3.10306i | −4.75661 | + | 5.13563i | − | 7.16819i | 9.00000 | 1.25661 | + | 2.17651i | ||||||||||||||||||||||||||||
| 23.3 | 3.09257i | 3.00000 | −5.56399 | −5.67824 | + | 3.27834i | 9.27771i | 6.63848 | − | 2.22048i | − | 4.83675i | 9.00000 | −10.1385 | − | 17.5604i | ||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 63.j | odd | 6 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 63.3.j.a | ✓ | 6 |
| 3.b | odd | 2 | 1 | 189.3.j.a | 6 | ||
| 7.b | odd | 2 | 1 | 441.3.j.c | 6 | ||
| 7.c | even | 3 | 1 | 63.3.n.a | yes | 6 | |
| 7.c | even | 3 | 1 | 441.3.r.b | 6 | ||
| 7.d | odd | 6 | 1 | 441.3.n.c | 6 | ||
| 7.d | odd | 6 | 1 | 441.3.r.c | 6 | ||
| 9.c | even | 3 | 1 | 189.3.n.a | 6 | ||
| 9.d | odd | 6 | 1 | 63.3.n.a | yes | 6 | |
| 21.h | odd | 6 | 1 | 189.3.n.a | 6 | ||
| 63.h | even | 3 | 1 | 189.3.j.a | 6 | ||
| 63.i | even | 6 | 1 | 441.3.j.c | 6 | ||
| 63.j | odd | 6 | 1 | inner | 63.3.j.a | ✓ | 6 |
| 63.n | odd | 6 | 1 | 441.3.r.b | 6 | ||
| 63.o | even | 6 | 1 | 441.3.n.c | 6 | ||
| 63.s | even | 6 | 1 | 441.3.r.c | 6 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 63.3.j.a | ✓ | 6 | 1.a | even | 1 | 1 | trivial |
| 63.3.j.a | ✓ | 6 | 63.j | odd | 6 | 1 | inner |
| 63.3.n.a | yes | 6 | 7.c | even | 3 | 1 | |
| 63.3.n.a | yes | 6 | 9.d | odd | 6 | 1 | |
| 189.3.j.a | 6 | 3.b | odd | 2 | 1 | ||
| 189.3.j.a | 6 | 63.h | even | 3 | 1 | ||
| 189.3.n.a | 6 | 9.c | even | 3 | 1 | ||
| 189.3.n.a | 6 | 21.h | odd | 6 | 1 | ||
| 441.3.j.c | 6 | 7.b | odd | 2 | 1 | ||
| 441.3.j.c | 6 | 63.i | even | 6 | 1 | ||
| 441.3.n.c | 6 | 7.d | odd | 6 | 1 | ||
| 441.3.n.c | 6 | 63.o | even | 6 | 1 | ||
| 441.3.r.b | 6 | 7.c | even | 3 | 1 | ||
| 441.3.r.b | 6 | 63.n | odd | 6 | 1 | ||
| 441.3.r.c | 6 | 7.d | odd | 6 | 1 | ||
| 441.3.r.c | 6 | 63.s | even | 6 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{6} + 25T_{2}^{4} + 163T_{2}^{2} + 147 \)
acting on \(S_{3}^{\mathrm{new}}(63, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{6} + 25 T^{4} + 163 T^{2} + \cdots + 147 \)
$3$
\( (T - 3)^{6} \)
$5$
\( T^{6} + 15 T^{5} + 88 T^{4} + 195 T^{3} + \cdots + 27 \)
$7$
\( T^{6} + 2 T^{5} - T^{4} - 532 T^{3} + \cdots + 117649 \)
$11$
\( T^{6} + 9 T^{5} - 188 T^{4} + \cdots + 1982907 \)
$13$
\( T^{6} - 11 T^{5} + 182 T^{4} + \cdots + 499849 \)
$17$
\( T^{6} - 33 T^{5} - 252 T^{4} + \cdots + 31434507 \)
$19$
\( T^{6} + 19 T^{5} + 422 T^{4} + \cdots + 214369 \)
$23$
\( T^{6} - 15 T^{5} - 84 T^{4} + \cdots + 128547 \)
$29$
\( T^{6} + 51 T^{5} + 1144 T^{4} + \cdots + 694083 \)
$31$
\( (T^{3} + 46 T^{2} - 324 T - 4632)^{2} \)
$37$
\( T^{6} - 7 T^{5} + 2230 T^{4} + \cdots + 564110001 \)
$41$
\( T^{6} + 27 T^{5} + \cdots + 1387137027 \)
$43$
\( T^{6} + 99 T^{5} + \cdots + 432265681 \)
$47$
\( T^{6} + 10128 T^{4} + \cdots + 29274835968 \)
$53$
\( T^{6} - 45 T^{5} + \cdots + 5141134827 \)
$59$
\( T^{6} + 4896 T^{4} + \cdots + 813189888 \)
$61$
\( (T^{3} - 22 T^{2} - 4996 T + 160696)^{2} \)
$67$
\( (T^{3} + 98 T^{2} - 1156 T - 210392)^{2} \)
$71$
\( T^{6} + 5568 T^{4} + \cdots + 109734912 \)
$73$
\( T^{6} + 101 T^{5} + \cdots + 653628591729 \)
$79$
\( (T^{3} - 90 T^{2} + 2268 T - 16712)^{2} \)
$83$
\( T^{6} - 99 T^{5} + \cdots + 165842832483 \)
$89$
\( T^{6} + 243 T^{5} + \cdots + 15857178627 \)
$97$
\( T^{6} + 161 T^{5} + \cdots + 36908941689 \)
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