Newspace parameters
Level: | \( N \) | \(=\) | \( 135 = 3^{3} \cdot 5 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 135.e (of order \(3\), degree \(2\), not minimal) |
Newform invariants
Self dual: | no |
Analytic conductor: | \(1.07798042729\) |
Analytic rank: | \(0\) |
Dimension: | \(6\) |
Relative dimension: | \(3\) over \(\Q(\zeta_{3})\) |
Coefficient field: | 6.0.954288.1 |
Defining polynomial: |
\( x^{6} - x^{5} - 2x^{4} + 3x^{3} - 6x^{2} - 9x + 27 \)
|
Coefficient ring: | \(\Z[a_1, \ldots, a_{4}]\) |
Coefficient ring index: | \( 3 \) |
Twist minimal: | no (minimal twist has level 45) |
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
\( x^{6} - x^{5} - 2x^{4} + 3x^{3} - 6x^{2} - 9x + 27 \)
:
\(\beta_{1}\) | \(=\) |
\( ( -\nu^{5} + 4\nu^{4} - \nu^{3} + 9\nu^{2} - 21\nu - 9 ) / 27 \)
|
\(\beta_{2}\) | \(=\) |
\( ( -\nu^{5} + 4\nu^{4} - \nu^{3} - 18\nu^{2} + 33\nu - 9 ) / 27 \)
|
\(\beta_{3}\) | \(=\) |
\( ( -2\nu^{5} - \nu^{4} - 2\nu^{3} + 12\nu + 36 ) / 27 \)
|
\(\beta_{4}\) | \(=\) |
\( ( -2\nu^{5} - \nu^{4} + 7\nu^{3} + 9\nu^{2} + 12\nu + 9 ) / 27 \)
|
\(\beta_{5}\) | \(=\) |
\( ( 4\nu^{5} + 2\nu^{4} - 5\nu^{3} + 18\nu^{2} + 3\nu - 72 ) / 27 \)
|
\(\nu\) | \(=\) |
\( ( \beta_{5} + \beta_{4} + \beta_{3} + \beta_{2} - \beta _1 + 1 ) / 3 \)
|
\(\nu^{2}\) | \(=\) |
\( ( 2\beta_{5} + 2\beta_{4} + 2\beta_{3} - \beta_{2} + \beta _1 + 2 ) / 3 \)
|
\(\nu^{3}\) | \(=\) |
\( ( -2\beta_{5} + 7\beta_{4} - 11\beta_{3} + \beta_{2} - \beta _1 + 7 ) / 3 \)
|
\(\nu^{4}\) | \(=\) |
\( ( 2\beta_{5} + 2\beta_{4} - 7\beta_{3} + 8\beta_{2} + 10\beta _1 + 20 ) / 3 \)
|
\(\nu^{5}\) | \(=\) |
\( ( 7\beta_{5} - 2\beta_{4} - 20\beta_{3} + \beta_{2} - 10\beta _1 + 43 ) / 3 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/135\mathbb{Z}\right)^\times\).
\(n\) | \(56\) | \(82\) |
\(\chi(n)\) | \(-1 + \beta_{3}\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
46.1 |
|
−1.04307 | + | 1.80664i | 0 | −1.17597 | − | 2.03684i | 0.500000 | + | 0.866025i | 0 | −2.04307 | + | 3.53869i | 0.734191 | 0 | −2.08613 | ||||||||||||||||||||||||||||
46.2 | 0.285997 | − | 0.495361i | 0 | 0.836412 | + | 1.44871i | 0.500000 | + | 0.866025i | 0 | −0.714003 | + | 1.23669i | 2.10083 | 0 | 0.571993 | |||||||||||||||||||||||||||||
46.3 | 1.25707 | − | 2.17731i | 0 | −2.16044 | − | 3.74200i | 0.500000 | + | 0.866025i | 0 | 0.257068 | − | 0.445256i | −5.83502 | 0 | 2.51414 | |||||||||||||||||||||||||||||
91.1 | −1.04307 | − | 1.80664i | 0 | −1.17597 | + | 2.03684i | 0.500000 | − | 0.866025i | 0 | −2.04307 | − | 3.53869i | 0.734191 | 0 | −2.08613 | |||||||||||||||||||||||||||||
91.2 | 0.285997 | + | 0.495361i | 0 | 0.836412 | − | 1.44871i | 0.500000 | − | 0.866025i | 0 | −0.714003 | − | 1.23669i | 2.10083 | 0 | 0.571993 | |||||||||||||||||||||||||||||
91.3 | 1.25707 | + | 2.17731i | 0 | −2.16044 | + | 3.74200i | 0.500000 | − | 0.866025i | 0 | 0.257068 | + | 0.445256i | −5.83502 | 0 | 2.51414 | |||||||||||||||||||||||||||||
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 | 135.2.e.b | 6 | |
3.b | odd | 2 | 1 | 45.2.e.b | ✓ | 6 | |
4.b | odd | 2 | 1 | 2160.2.q.k | 6 | ||
5.b | even | 2 | 1 | 675.2.e.b | 6 | ||
5.c | odd | 4 | 2 | 675.2.k.b | 12 | ||
9.c | even | 3 | 1 | inner | 135.2.e.b | 6 | |
9.c | even | 3 | 1 | 405.2.a.i | 3 | ||
9.d | odd | 6 | 1 | 45.2.e.b | ✓ | 6 | |
9.d | odd | 6 | 1 | 405.2.a.j | 3 | ||
12.b | even | 2 | 1 | 720.2.q.i | 6 | ||
15.d | odd | 2 | 1 | 225.2.e.b | 6 | ||
15.e | even | 4 | 2 | 225.2.k.b | 12 | ||
36.f | odd | 6 | 1 | 2160.2.q.k | 6 | ||
36.f | odd | 6 | 1 | 6480.2.a.bs | 3 | ||
36.h | even | 6 | 1 | 720.2.q.i | 6 | ||
36.h | even | 6 | 1 | 6480.2.a.bv | 3 | ||
45.h | odd | 6 | 1 | 225.2.e.b | 6 | ||
45.h | odd | 6 | 1 | 2025.2.a.n | 3 | ||
45.j | even | 6 | 1 | 675.2.e.b | 6 | ||
45.j | even | 6 | 1 | 2025.2.a.o | 3 | ||
45.k | odd | 12 | 2 | 675.2.k.b | 12 | ||
45.k | odd | 12 | 2 | 2025.2.b.m | 6 | ||
45.l | even | 12 | 2 | 225.2.k.b | 12 | ||
45.l | even | 12 | 2 | 2025.2.b.l | 6 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
45.2.e.b | ✓ | 6 | 3.b | odd | 2 | 1 | |
45.2.e.b | ✓ | 6 | 9.d | odd | 6 | 1 | |
135.2.e.b | 6 | 1.a | even | 1 | 1 | trivial | |
135.2.e.b | 6 | 9.c | even | 3 | 1 | inner | |
225.2.e.b | 6 | 15.d | odd | 2 | 1 | ||
225.2.e.b | 6 | 45.h | odd | 6 | 1 | ||
225.2.k.b | 12 | 15.e | even | 4 | 2 | ||
225.2.k.b | 12 | 45.l | even | 12 | 2 | ||
405.2.a.i | 3 | 9.c | even | 3 | 1 | ||
405.2.a.j | 3 | 9.d | odd | 6 | 1 | ||
675.2.e.b | 6 | 5.b | even | 2 | 1 | ||
675.2.e.b | 6 | 45.j | even | 6 | 1 | ||
675.2.k.b | 12 | 5.c | odd | 4 | 2 | ||
675.2.k.b | 12 | 45.k | odd | 12 | 2 | ||
720.2.q.i | 6 | 12.b | even | 2 | 1 | ||
720.2.q.i | 6 | 36.h | even | 6 | 1 | ||
2025.2.a.n | 3 | 45.h | odd | 6 | 1 | ||
2025.2.a.o | 3 | 45.j | even | 6 | 1 | ||
2025.2.b.l | 6 | 45.l | even | 12 | 2 | ||
2025.2.b.m | 6 | 45.k | odd | 12 | 2 | ||
2160.2.q.k | 6 | 4.b | odd | 2 | 1 | ||
2160.2.q.k | 6 | 36.f | odd | 6 | 1 | ||
6480.2.a.bs | 3 | 36.f | odd | 6 | 1 | ||
6480.2.a.bv | 3 | 36.h | even | 6 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{6} - T_{2}^{5} + 6T_{2}^{4} - T_{2}^{3} + 28T_{2}^{2} - 15T_{2} + 9 \)
acting on \(S_{2}^{\mathrm{new}}(135, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{6} - T^{5} + 6 T^{4} - T^{3} + 28 T^{2} + \cdots + 9 \)
$3$
\( T^{6} \)
$5$
\( (T^{2} - T + 1)^{3} \)
$7$
\( T^{6} + 5 T^{5} + 22 T^{4} + 21 T^{3} + \cdots + 9 \)
$11$
\( T^{6} + 2 T^{5} + 12 T^{4} + 8 T^{3} + \cdots + 144 \)
$13$
\( T^{6} + 4 T^{5} + 20 T^{4} - 8 T^{3} + \cdots + 16 \)
$17$
\( (T^{3} - 2 T^{2} - 8 T + 12)^{2} \)
$19$
\( (T^{3} - 4 T^{2} - 4 T + 4)^{2} \)
$23$
\( T^{6} - 3 T^{5} + 42 T^{4} + \cdots + 13689 \)
$29$
\( T^{6} + 7 T^{5} + 78 T^{4} + \cdots + 2601 \)
$31$
\( T^{6} + 8 T^{5} + 124 T^{4} + \cdots + 219024 \)
$37$
\( (T^{3} - 6 T^{2} - 12 T + 4)^{2} \)
$41$
\( T^{6} + 13 T^{5} + 150 T^{4} + 241 T^{3} + \cdots + 9 \)
$43$
\( T^{6} + 10 T^{5} + 104 T^{4} + \cdots + 16 \)
$47$
\( T^{6} - 13 T^{5} + 180 T^{4} + \cdots + 136161 \)
$53$
\( (T^{3} - 2 T^{2} - 20 T + 24)^{2} \)
$59$
\( T^{6} + 2 T^{5} + 24 T^{4} + 8 T^{3} + \cdots + 576 \)
$61$
\( T^{6} + T^{5} + 38 T^{4} - 179 T^{3} + \cdots + 5041 \)
$67$
\( T^{6} + 11 T^{5} + 160 T^{4} + \cdots + 257049 \)
$71$
\( (T^{3} - 10 T^{2} - 92 T + 708)^{2} \)
$73$
\( (T^{3} + 8 T^{2} - 64 T - 128)^{2} \)
$79$
\( T^{6} + 2 T^{5} + 88 T^{4} - 216 T^{3} + \cdots + 576 \)
$83$
\( T^{6} + 15 T^{5} + 198 T^{4} + \cdots + 6561 \)
$89$
\( (T - 3)^{6} \)
$97$
\( T^{6} - 18 T^{5} + 360 T^{4} + \cdots + 1700416 \)
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