Newspace parameters
Level: | \( N \) | \(=\) | \( 63 = 3^{2} \cdot 7 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 63.f (of order \(3\), degree \(2\), minimal) |
Newform invariants
Self dual: | no |
Analytic conductor: | \(0.503057532734\) |
Analytic rank: | \(0\) |
Dimension: | \(6\) |
Relative dimension: | \(3\) over \(\Q(\zeta_{3})\) |
Coefficient field: | \(\Q(\zeta_{18})\) |
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
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Defining polynomial: | \( x^{6} - x^{3} + 1 \) |
Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
Coefficient ring index: | \( 3 \) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{3}]$ |
$q$-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
\(\beta_{1}\) | \(=\) | \( \zeta_{18}^{3} \) |
\(\beta_{2}\) | \(=\) | \( \zeta_{18}^{5} + \zeta_{18} \) |
\(\beta_{3}\) | \(=\) | \( -\zeta_{18}^{4} + \zeta_{18}^{2} + \zeta_{18} \) |
\(\beta_{4}\) | \(=\) | \( -\zeta_{18}^{5} + \zeta_{18}^{4} \) |
\(\beta_{5}\) | \(=\) | \( -\zeta_{18}^{5} - \zeta_{18}^{4} + \zeta_{18} \) |
\(\zeta_{18}\) | \(=\) | \( ( \beta_{5} + \beta_{4} + 2\beta_{2} ) / 3 \) |
\(\zeta_{18}^{2}\) | \(=\) | \( ( -2\beta_{5} + \beta_{4} + 3\beta_{3} - \beta_{2} ) / 3 \) |
\(\zeta_{18}^{3}\) | \(=\) | \( \beta_1 \) |
\(\zeta_{18}^{4}\) | \(=\) | \( ( -\beta_{5} + 2\beta_{4} + \beta_{2} ) / 3 \) |
\(\zeta_{18}^{5}\) | \(=\) | \( ( -\beta_{5} - \beta_{4} + \beta_{2} ) / 3 \) |
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/63\mathbb{Z}\right)^\times\).
\(n\) | \(10\) | \(29\) |
\(\chi(n)\) | \(1\) | \(-\beta_{1}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
22.1 |
|
−1.26604 | − | 2.19285i | −1.11334 | − | 1.32683i | −2.20574 | + | 3.82045i | 0.439693 | − | 0.761570i | −1.50000 | + | 4.12122i | −0.500000 | − | 0.866025i | 6.10607 | −0.520945 | + | 2.95442i | −2.22668 | ||||||||||||||||||||||
22.2 | −0.673648 | − | 1.16679i | 1.70574 | − | 0.300767i | 0.0923963 | − | 0.160035i | −1.26604 | + | 2.19285i | −1.50000 | − | 1.78763i | −0.500000 | − | 0.866025i | −2.94356 | 2.81908 | − | 1.02606i | 3.41147 | |||||||||||||||||||||||
22.3 | 0.439693 | + | 0.761570i | −0.592396 | + | 1.62760i | 0.613341 | − | 1.06234i | −0.673648 | + | 1.16679i | −1.50000 | + | 0.264490i | −0.500000 | − | 0.866025i | 2.83750 | −2.29813 | − | 1.92836i | −1.18479 | |||||||||||||||||||||||
43.1 | −1.26604 | + | 2.19285i | −1.11334 | + | 1.32683i | −2.20574 | − | 3.82045i | 0.439693 | + | 0.761570i | −1.50000 | − | 4.12122i | −0.500000 | + | 0.866025i | 6.10607 | −0.520945 | − | 2.95442i | −2.22668 | |||||||||||||||||||||||
43.2 | −0.673648 | + | 1.16679i | 1.70574 | + | 0.300767i | 0.0923963 | + | 0.160035i | −1.26604 | − | 2.19285i | −1.50000 | + | 1.78763i | −0.500000 | + | 0.866025i | −2.94356 | 2.81908 | + | 1.02606i | 3.41147 | |||||||||||||||||||||||
43.3 | 0.439693 | − | 0.761570i | −0.592396 | − | 1.62760i | 0.613341 | + | 1.06234i | −0.673648 | − | 1.16679i | −1.50000 | − | 0.264490i | −0.500000 | + | 0.866025i | 2.83750 | −2.29813 | + | 1.92836i | −1.18479 | |||||||||||||||||||||||
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
9.c | even | 3 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 63.2.f.a | ✓ | 6 |
3.b | odd | 2 | 1 | 189.2.f.b | 6 | ||
4.b | odd | 2 | 1 | 1008.2.r.h | 6 | ||
7.b | odd | 2 | 1 | 441.2.f.c | 6 | ||
7.c | even | 3 | 1 | 441.2.g.c | 6 | ||
7.c | even | 3 | 1 | 441.2.h.d | 6 | ||
7.d | odd | 6 | 1 | 441.2.g.b | 6 | ||
7.d | odd | 6 | 1 | 441.2.h.e | 6 | ||
9.c | even | 3 | 1 | inner | 63.2.f.a | ✓ | 6 |
9.c | even | 3 | 1 | 567.2.a.h | 3 | ||
9.d | odd | 6 | 1 | 189.2.f.b | 6 | ||
9.d | odd | 6 | 1 | 567.2.a.c | 3 | ||
12.b | even | 2 | 1 | 3024.2.r.k | 6 | ||
21.c | even | 2 | 1 | 1323.2.f.d | 6 | ||
21.g | even | 6 | 1 | 1323.2.g.e | 6 | ||
21.g | even | 6 | 1 | 1323.2.h.b | 6 | ||
21.h | odd | 6 | 1 | 1323.2.g.d | 6 | ||
21.h | odd | 6 | 1 | 1323.2.h.c | 6 | ||
36.f | odd | 6 | 1 | 1008.2.r.h | 6 | ||
36.f | odd | 6 | 1 | 9072.2.a.ca | 3 | ||
36.h | even | 6 | 1 | 3024.2.r.k | 6 | ||
36.h | even | 6 | 1 | 9072.2.a.bs | 3 | ||
63.g | even | 3 | 1 | 441.2.h.d | 6 | ||
63.h | even | 3 | 1 | 441.2.g.c | 6 | ||
63.i | even | 6 | 1 | 1323.2.g.e | 6 | ||
63.j | odd | 6 | 1 | 1323.2.g.d | 6 | ||
63.k | odd | 6 | 1 | 441.2.h.e | 6 | ||
63.l | odd | 6 | 1 | 441.2.f.c | 6 | ||
63.l | odd | 6 | 1 | 3969.2.a.q | 3 | ||
63.n | odd | 6 | 1 | 1323.2.h.c | 6 | ||
63.o | even | 6 | 1 | 1323.2.f.d | 6 | ||
63.o | even | 6 | 1 | 3969.2.a.l | 3 | ||
63.s | even | 6 | 1 | 1323.2.h.b | 6 | ||
63.t | odd | 6 | 1 | 441.2.g.b | 6 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
63.2.f.a | ✓ | 6 | 1.a | even | 1 | 1 | trivial |
63.2.f.a | ✓ | 6 | 9.c | even | 3 | 1 | inner |
189.2.f.b | 6 | 3.b | odd | 2 | 1 | ||
189.2.f.b | 6 | 9.d | odd | 6 | 1 | ||
441.2.f.c | 6 | 7.b | odd | 2 | 1 | ||
441.2.f.c | 6 | 63.l | odd | 6 | 1 | ||
441.2.g.b | 6 | 7.d | odd | 6 | 1 | ||
441.2.g.b | 6 | 63.t | odd | 6 | 1 | ||
441.2.g.c | 6 | 7.c | even | 3 | 1 | ||
441.2.g.c | 6 | 63.h | even | 3 | 1 | ||
441.2.h.d | 6 | 7.c | even | 3 | 1 | ||
441.2.h.d | 6 | 63.g | even | 3 | 1 | ||
441.2.h.e | 6 | 7.d | odd | 6 | 1 | ||
441.2.h.e | 6 | 63.k | odd | 6 | 1 | ||
567.2.a.c | 3 | 9.d | odd | 6 | 1 | ||
567.2.a.h | 3 | 9.c | even | 3 | 1 | ||
1008.2.r.h | 6 | 4.b | odd | 2 | 1 | ||
1008.2.r.h | 6 | 36.f | odd | 6 | 1 | ||
1323.2.f.d | 6 | 21.c | even | 2 | 1 | ||
1323.2.f.d | 6 | 63.o | even | 6 | 1 | ||
1323.2.g.d | 6 | 21.h | odd | 6 | 1 | ||
1323.2.g.d | 6 | 63.j | odd | 6 | 1 | ||
1323.2.g.e | 6 | 21.g | even | 6 | 1 | ||
1323.2.g.e | 6 | 63.i | even | 6 | 1 | ||
1323.2.h.b | 6 | 21.g | even | 6 | 1 | ||
1323.2.h.b | 6 | 63.s | even | 6 | 1 | ||
1323.2.h.c | 6 | 21.h | odd | 6 | 1 | ||
1323.2.h.c | 6 | 63.n | odd | 6 | 1 | ||
3024.2.r.k | 6 | 12.b | even | 2 | 1 | ||
3024.2.r.k | 6 | 36.h | even | 6 | 1 | ||
3969.2.a.l | 3 | 63.o | even | 6 | 1 | ||
3969.2.a.q | 3 | 63.l | odd | 6 | 1 | ||
9072.2.a.bs | 3 | 36.h | even | 6 | 1 | ||
9072.2.a.ca | 3 | 36.f | odd | 6 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{6} + 3T_{2}^{5} + 9T_{2}^{4} + 6T_{2}^{3} + 9T_{2}^{2} + 9 \)
acting on \(S_{2}^{\mathrm{new}}(63, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{6} + 3 T^{5} + 9 T^{4} + 6 T^{3} + \cdots + 9 \)
$3$
\( T^{6} - 9T^{3} + 27 \)
$5$
\( T^{6} + 3 T^{5} + 9 T^{4} + 6 T^{3} + \cdots + 9 \)
$7$
\( (T^{2} + T + 1)^{3} \)
$11$
\( T^{6} + 6 T^{5} + 27 T^{4} + 48 T^{3} + \cdots + 9 \)
$13$
\( T^{6} - 3 T^{5} + 42 T^{4} + \cdots + 11449 \)
$17$
\( (T^{3} - 6 T^{2} + 9 T - 3)^{2} \)
$19$
\( (T^{3} + 3 T^{2} - 6 T - 17)^{2} \)
$23$
\( T^{6} + 12 T^{5} + 117 T^{4} + 330 T^{3} + \cdots + 9 \)
$29$
\( T^{6} + 9 T^{5} + 117 T^{4} + \cdots + 110889 \)
$31$
\( T^{6} - 3 T^{5} + 87 T^{4} + \cdots + 104329 \)
$37$
\( (T^{3} + 3 T^{2} - 78 T - 323)^{2} \)
$41$
\( T^{6} + 9 T^{4} - 18 T^{3} + 81 T^{2} + \cdots + 81 \)
$43$
\( T^{6} - 3 T^{5} + 15 T^{4} + 20 T^{3} + \cdots + 1 \)
$47$
\( T^{6} + 3 T^{5} + 63 T^{4} + \cdots + 2601 \)
$53$
\( (T^{3} - 6 T^{2} - 9 T - 3)^{2} \)
$59$
\( T^{6} - 3 T^{5} + 81 T^{4} + \cdots + 2601 \)
$61$
\( T^{6} + 6 T^{5} + 51 T^{4} - 52 T^{3} + \cdots + 361 \)
$67$
\( T^{6} - 12 T^{5} + 123 T^{4} + \cdots + 289 \)
$71$
\( (T^{3} - 9 T^{2} - 54 T - 27)^{2} \)
$73$
\( (T^{3} + 21 T^{2} + 84 T - 269)^{2} \)
$79$
\( T^{6} - 21 T^{5} + 321 T^{4} + \cdots + 32761 \)
$83$
\( T^{6} - 18 T^{5} + 279 T^{4} + \cdots + 81 \)
$89$
\( (T^{3} - 12 T^{2} - 63 T + 813)^{2} \)
$97$
\( T^{6} - 3 T^{5} + 177 T^{4} + \cdots + 104329 \)
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