Newspace parameters
Level: | \( N \) | \(=\) | \( 756 = 2^{2} \cdot 3^{3} \cdot 7 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 756.j (of order \(3\), degree \(2\), not minimal) |
Newform invariants
Self dual: | no |
Analytic conductor: | \(6.03669039281\) |
Analytic rank: | \(0\) |
Dimension: | \(6\) |
Relative dimension: | \(3\) over \(\Q(\zeta_{3})\) |
Coefficient field: | 6.0.309123.1 |
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
Defining polynomial: | \( x^{6} - 3x^{5} + 10x^{4} - 15x^{3} + 19x^{2} - 12x + 3 \) |
Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
Coefficient ring index: | \( 3 \) |
Twist minimal: | no (minimal twist has level 252) |
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 in terms of a root \(\nu\) of \( x^{6} - 3x^{5} + 10x^{4} - 15x^{3} + 19x^{2} - 12x + 3 \) :
\(\beta_{1}\) | \(=\) | \( \nu^{2} - \nu + 2 \) |
\(\beta_{2}\) | \(=\) | \( ( -\nu^{5} + \nu^{4} - 8\nu^{3} + 5\nu^{2} - 18\nu + 6 ) / 3 \) |
\(\beta_{3}\) | \(=\) | \( \nu^{4} - 2\nu^{3} + 6\nu^{2} - 5\nu + 3 \) |
\(\beta_{4}\) | \(=\) | \( ( -2\nu^{5} + 5\nu^{4} - 16\nu^{3} + 19\nu^{2} - 21\nu + 9 ) / 3 \) |
\(\beta_{5}\) | \(=\) | \( ( 2\nu^{5} - 5\nu^{4} + 19\nu^{3} - 22\nu^{2} + 30\nu - 9 ) / 3 \) |
\(\nu\) | \(=\) | \( ( -2\beta_{5} - \beta_{4} - \beta_{3} - 2\beta_{2} + \beta _1 + 2 ) / 3 \) |
\(\nu^{2}\) | \(=\) | \( ( -2\beta_{5} - \beta_{4} - \beta_{3} - 2\beta_{2} + 4\beta _1 - 4 ) / 3 \) |
\(\nu^{3}\) | \(=\) | \( ( 7\beta_{5} + 5\beta_{4} + 2\beta_{3} + 4\beta_{2} + \beta _1 - 10 ) / 3 \) |
\(\nu^{4}\) | \(=\) | \( ( 16\beta_{5} + 11\beta_{4} + 8\beta_{3} + 10\beta_{2} - 17\beta _1 + 5 ) / 3 \) |
\(\nu^{5}\) | \(=\) | \( ( -14\beta_{5} - 16\beta_{4} + 5\beta_{3} - 5\beta_{2} - 23\beta _1 + 47 ) / 3 \) |
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/756\mathbb{Z}\right)^\times\).
\(n\) | \(29\) | \(325\) | \(379\) |
\(\chi(n)\) | \(-\beta_{4}\) | \(1\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
253.1 |
|
0 | 0 | 0 | −0.849814 | + | 1.47192i | 0 | 0.500000 | + | 0.866025i | 0 | 0 | 0 | ||||||||||||||||||||||||||||||||
253.2 | 0 | 0 | 0 | 0.119562 | − | 0.207087i | 0 | 0.500000 | + | 0.866025i | 0 | 0 | 0 | |||||||||||||||||||||||||||||||||
253.3 | 0 | 0 | 0 | 1.23025 | − | 2.13086i | 0 | 0.500000 | + | 0.866025i | 0 | 0 | 0 | |||||||||||||||||||||||||||||||||
505.1 | 0 | 0 | 0 | −0.849814 | − | 1.47192i | 0 | 0.500000 | − | 0.866025i | 0 | 0 | 0 | |||||||||||||||||||||||||||||||||
505.2 | 0 | 0 | 0 | 0.119562 | + | 0.207087i | 0 | 0.500000 | − | 0.866025i | 0 | 0 | 0 | |||||||||||||||||||||||||||||||||
505.3 | 0 | 0 | 0 | 1.23025 | + | 2.13086i | 0 | 0.500000 | − | 0.866025i | 0 | 0 | 0 | |||||||||||||||||||||||||||||||||
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 | 756.2.j.b | 6 | |
3.b | odd | 2 | 1 | 252.2.j.a | ✓ | 6 | |
4.b | odd | 2 | 1 | 3024.2.r.j | 6 | ||
7.b | odd | 2 | 1 | 5292.2.j.d | 6 | ||
7.c | even | 3 | 1 | 5292.2.i.f | 6 | ||
7.c | even | 3 | 1 | 5292.2.l.e | 6 | ||
7.d | odd | 6 | 1 | 5292.2.i.e | 6 | ||
7.d | odd | 6 | 1 | 5292.2.l.f | 6 | ||
9.c | even | 3 | 1 | inner | 756.2.j.b | 6 | |
9.c | even | 3 | 1 | 2268.2.a.h | 3 | ||
9.d | odd | 6 | 1 | 252.2.j.a | ✓ | 6 | |
9.d | odd | 6 | 1 | 2268.2.a.i | 3 | ||
12.b | even | 2 | 1 | 1008.2.r.j | 6 | ||
21.c | even | 2 | 1 | 1764.2.j.e | 6 | ||
21.g | even | 6 | 1 | 1764.2.i.d | 6 | ||
21.g | even | 6 | 1 | 1764.2.l.f | 6 | ||
21.h | odd | 6 | 1 | 1764.2.i.g | 6 | ||
21.h | odd | 6 | 1 | 1764.2.l.e | 6 | ||
36.f | odd | 6 | 1 | 3024.2.r.j | 6 | ||
36.f | odd | 6 | 1 | 9072.2.a.bv | 3 | ||
36.h | even | 6 | 1 | 1008.2.r.j | 6 | ||
36.h | even | 6 | 1 | 9072.2.a.by | 3 | ||
63.g | even | 3 | 1 | 5292.2.i.f | 6 | ||
63.h | even | 3 | 1 | 5292.2.l.e | 6 | ||
63.i | even | 6 | 1 | 1764.2.l.f | 6 | ||
63.j | odd | 6 | 1 | 1764.2.l.e | 6 | ||
63.k | odd | 6 | 1 | 5292.2.i.e | 6 | ||
63.l | odd | 6 | 1 | 5292.2.j.d | 6 | ||
63.n | odd | 6 | 1 | 1764.2.i.g | 6 | ||
63.o | even | 6 | 1 | 1764.2.j.e | 6 | ||
63.s | even | 6 | 1 | 1764.2.i.d | 6 | ||
63.t | odd | 6 | 1 | 5292.2.l.f | 6 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
252.2.j.a | ✓ | 6 | 3.b | odd | 2 | 1 | |
252.2.j.a | ✓ | 6 | 9.d | odd | 6 | 1 | |
756.2.j.b | 6 | 1.a | even | 1 | 1 | trivial | |
756.2.j.b | 6 | 9.c | even | 3 | 1 | inner | |
1008.2.r.j | 6 | 12.b | even | 2 | 1 | ||
1008.2.r.j | 6 | 36.h | even | 6 | 1 | ||
1764.2.i.d | 6 | 21.g | even | 6 | 1 | ||
1764.2.i.d | 6 | 63.s | even | 6 | 1 | ||
1764.2.i.g | 6 | 21.h | odd | 6 | 1 | ||
1764.2.i.g | 6 | 63.n | odd | 6 | 1 | ||
1764.2.j.e | 6 | 21.c | even | 2 | 1 | ||
1764.2.j.e | 6 | 63.o | even | 6 | 1 | ||
1764.2.l.e | 6 | 21.h | odd | 6 | 1 | ||
1764.2.l.e | 6 | 63.j | odd | 6 | 1 | ||
1764.2.l.f | 6 | 21.g | even | 6 | 1 | ||
1764.2.l.f | 6 | 63.i | even | 6 | 1 | ||
2268.2.a.h | 3 | 9.c | even | 3 | 1 | ||
2268.2.a.i | 3 | 9.d | odd | 6 | 1 | ||
3024.2.r.j | 6 | 4.b | odd | 2 | 1 | ||
3024.2.r.j | 6 | 36.f | odd | 6 | 1 | ||
5292.2.i.e | 6 | 7.d | odd | 6 | 1 | ||
5292.2.i.e | 6 | 63.k | odd | 6 | 1 | ||
5292.2.i.f | 6 | 7.c | even | 3 | 1 | ||
5292.2.i.f | 6 | 63.g | even | 3 | 1 | ||
5292.2.j.d | 6 | 7.b | odd | 2 | 1 | ||
5292.2.j.d | 6 | 63.l | odd | 6 | 1 | ||
5292.2.l.e | 6 | 7.c | even | 3 | 1 | ||
5292.2.l.e | 6 | 63.h | even | 3 | 1 | ||
5292.2.l.f | 6 | 7.d | odd | 6 | 1 | ||
5292.2.l.f | 6 | 63.t | odd | 6 | 1 | ||
9072.2.a.bv | 3 | 36.f | odd | 6 | 1 | ||
9072.2.a.by | 3 | 36.h | even | 6 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{6} - T_{5}^{5} + 5T_{5}^{4} + 2T_{5}^{3} + 17T_{5}^{2} - 4T_{5} + 1 \)
acting on \(S_{2}^{\mathrm{new}}(756, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{6} \)
$3$
\( T^{6} \)
$5$
\( T^{6} - T^{5} + 5 T^{4} + 2 T^{3} + 17 T^{2} + \cdots + 1 \)
$7$
\( (T^{2} - T + 1)^{3} \)
$11$
\( T^{6} - 2 T^{5} + 29 T^{4} + \cdots + 3481 \)
$13$
\( T^{6} + 3 T^{5} + 42 T^{4} - 45 T^{3} + \cdots + 729 \)
$17$
\( (T^{3} + 2 T^{2} - 19 T - 47)^{2} \)
$19$
\( (T^{3} - 3 T^{2} - 24 T + 79)^{2} \)
$23$
\( T^{6} - 14 T^{5} + 143 T^{4} + \cdots + 961 \)
$29$
\( T^{6} - T^{5} + 41 T^{4} - 178 T^{3} + \cdots + 11881 \)
$31$
\( T^{6} - 3 T^{5} + 45 T^{4} + 54 T^{3} + \cdots + 729 \)
$37$
\( (T^{3} + 3 T^{2} - 30 T - 23)^{2} \)
$41$
\( T^{6} + 33 T^{4} - 18 T^{3} + 1089 T^{2} + \cdots + 81 \)
$43$
\( T^{6} + 3 T^{5} + 105 T^{4} + \cdots + 78961 \)
$47$
\( T^{6} - 21 T^{5} + 321 T^{4} + \cdots + 42849 \)
$53$
\( (T^{3} + 6 T^{2} + 3 T - 9)^{2} \)
$59$
\( T^{6} - 31 T^{5} + 653 T^{4} + \cdots + 978121 \)
$61$
\( T^{6} + 6 T^{5} + 231 T^{4} + \cdots + 1185921 \)
$67$
\( T^{6} + 6 T^{5} + 51 T^{4} - 104 T^{3} + \cdots + 49 \)
$71$
\( (T^{3} + 17 T^{2} - 94 T - 1907)^{2} \)
$73$
\( (T^{3} + 3 T^{2} - 24 T - 79)^{2} \)
$79$
\( T^{6} - 9 T^{5} + 123 T^{4} + \cdots + 136161 \)
$83$
\( T^{6} - 20 T^{5} + 287 T^{4} + \cdots + 16129 \)
$89$
\( (T^{3} + 12 T^{2} - 51 T - 711)^{2} \)
$97$
\( T^{6} - 9 T^{5} + 225 T^{4} + \cdots + 5329 \)
show more
show less