Newspace parameters
Level: | \( N \) | \(=\) | \( 585 = 3^{2} \cdot 5 \cdot 13 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 585.c (of order \(2\), degree \(1\), minimal) |
Newform invariants
Self dual: | no |
Analytic conductor: | \(4.67124851824\) |
Analytic rank: | \(0\) |
Dimension: | \(6\) |
Coefficient field: | 6.0.350464.1 |
Defining polynomial: |
\( x^{6} - 2x^{5} + 2x^{4} + 2x^{3} + 4x^{2} - 4x + 2 \)
|
Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
Coefficient ring index: | \( 2 \) |
Twist minimal: | no (minimal twist has level 65) |
Sato-Tate group: | $\mathrm{SU}(2)[C_{2}]$ |
$q$-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
\( x^{6} - 2x^{5} + 2x^{4} + 2x^{3} + 4x^{2} - 4x + 2 \)
:
\(\beta_{1}\) | \(=\) |
\( ( -\nu^{5} + 8\nu^{4} - 4\nu^{3} - \nu^{2} + 2\nu + 38 ) / 23 \)
|
\(\beta_{2}\) | \(=\) |
\( ( -5\nu^{5} + 17\nu^{4} - 20\nu^{3} - 5\nu^{2} + 10\nu + 29 ) / 23 \)
|
\(\beta_{3}\) | \(=\) |
\( ( 7\nu^{5} - 10\nu^{4} + 5\nu^{3} + 30\nu^{2} + 32\nu - 13 ) / 23 \)
|
\(\beta_{4}\) | \(=\) |
\( ( -11\nu^{5} + 19\nu^{4} - 21\nu^{3} - 11\nu^{2} - 70\nu + 27 ) / 23 \)
|
\(\beta_{5}\) | \(=\) |
\( ( -14\nu^{5} + 20\nu^{4} - 10\nu^{3} - 37\nu^{2} - 64\nu + 26 ) / 23 \)
|
\(\nu\) | \(=\) |
\( ( \beta_{5} - \beta_{4} + \beta_{3} + \beta_{2} - \beta _1 + 1 ) / 2 \)
|
\(\nu^{2}\) | \(=\) |
\( \beta_{5} + 2\beta_{3} \)
|
\(\nu^{3}\) | \(=\) |
\( 2\beta_{5} - \beta_{4} + 2\beta_{3} - \beta_{2} + 2\beta _1 - 2 \)
|
\(\nu^{4}\) | \(=\) |
\( -\beta_{2} + 5\beta _1 - 7 \)
|
\(\nu^{5}\) | \(=\) |
\( -8\beta_{5} + 3\beta_{4} - 9\beta_{3} - 3\beta_{2} + 8\beta _1 - 9 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/585\mathbb{Z}\right)^\times\).
\(n\) | \(326\) | \(352\) | \(496\) |
\(\chi(n)\) | \(1\) | \(-1\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
469.1 |
|
− | 2.67513i | 0 | −5.15633 | −1.67513 | + | 1.48119i | 0 | 0.806063i | 8.44358i | 0 | 3.96239 | + | 4.48119i | |||||||||||||||||||||||||||||||
469.2 | − | 1.53919i | 0 | −0.369102 | −0.539189 | − | 2.17009i | 0 | − | 1.70928i | − | 2.51026i | 0 | −3.34017 | + | 0.829914i | ||||||||||||||||||||||||||||||
469.3 | − | 1.21432i | 0 | 0.525428 | 2.21432 | + | 0.311108i | 0 | − | 2.90321i | − | 3.06668i | 0 | 0.377784 | − | 2.68889i | ||||||||||||||||||||||||||||||
469.4 | 1.21432i | 0 | 0.525428 | 2.21432 | − | 0.311108i | 0 | 2.90321i | 3.06668i | 0 | 0.377784 | + | 2.68889i | |||||||||||||||||||||||||||||||||
469.5 | 1.53919i | 0 | −0.369102 | −0.539189 | + | 2.17009i | 0 | 1.70928i | 2.51026i | 0 | −3.34017 | − | 0.829914i | |||||||||||||||||||||||||||||||||
469.6 | 2.67513i | 0 | −5.15633 | −1.67513 | − | 1.48119i | 0 | − | 0.806063i | − | 8.44358i | 0 | 3.96239 | − | 4.48119i | |||||||||||||||||||||||||||||||
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
5.b | even | 2 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 585.2.c.b | 6 | |
3.b | odd | 2 | 1 | 65.2.b.a | ✓ | 6 | |
5.b | even | 2 | 1 | inner | 585.2.c.b | 6 | |
5.c | odd | 4 | 1 | 2925.2.a.bf | 3 | ||
5.c | odd | 4 | 1 | 2925.2.a.bj | 3 | ||
12.b | even | 2 | 1 | 1040.2.d.c | 6 | ||
15.d | odd | 2 | 1 | 65.2.b.a | ✓ | 6 | |
15.e | even | 4 | 1 | 325.2.a.j | 3 | ||
15.e | even | 4 | 1 | 325.2.a.k | 3 | ||
39.d | odd | 2 | 1 | 845.2.b.c | 6 | ||
39.f | even | 4 | 1 | 845.2.d.a | 6 | ||
39.f | even | 4 | 1 | 845.2.d.b | 6 | ||
39.h | odd | 6 | 2 | 845.2.n.g | 12 | ||
39.i | odd | 6 | 2 | 845.2.n.f | 12 | ||
39.k | even | 12 | 2 | 845.2.l.d | 12 | ||
39.k | even | 12 | 2 | 845.2.l.e | 12 | ||
60.h | even | 2 | 1 | 1040.2.d.c | 6 | ||
60.l | odd | 4 | 1 | 5200.2.a.cb | 3 | ||
60.l | odd | 4 | 1 | 5200.2.a.cj | 3 | ||
195.e | odd | 2 | 1 | 845.2.b.c | 6 | ||
195.n | even | 4 | 1 | 845.2.d.a | 6 | ||
195.n | even | 4 | 1 | 845.2.d.b | 6 | ||
195.s | even | 4 | 1 | 4225.2.a.ba | 3 | ||
195.s | even | 4 | 1 | 4225.2.a.bh | 3 | ||
195.x | odd | 6 | 2 | 845.2.n.f | 12 | ||
195.y | odd | 6 | 2 | 845.2.n.g | 12 | ||
195.bh | even | 12 | 2 | 845.2.l.d | 12 | ||
195.bh | even | 12 | 2 | 845.2.l.e | 12 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
65.2.b.a | ✓ | 6 | 3.b | odd | 2 | 1 | |
65.2.b.a | ✓ | 6 | 15.d | odd | 2 | 1 | |
325.2.a.j | 3 | 15.e | even | 4 | 1 | ||
325.2.a.k | 3 | 15.e | even | 4 | 1 | ||
585.2.c.b | 6 | 1.a | even | 1 | 1 | trivial | |
585.2.c.b | 6 | 5.b | even | 2 | 1 | inner | |
845.2.b.c | 6 | 39.d | odd | 2 | 1 | ||
845.2.b.c | 6 | 195.e | odd | 2 | 1 | ||
845.2.d.a | 6 | 39.f | even | 4 | 1 | ||
845.2.d.a | 6 | 195.n | even | 4 | 1 | ||
845.2.d.b | 6 | 39.f | even | 4 | 1 | ||
845.2.d.b | 6 | 195.n | even | 4 | 1 | ||
845.2.l.d | 12 | 39.k | even | 12 | 2 | ||
845.2.l.d | 12 | 195.bh | even | 12 | 2 | ||
845.2.l.e | 12 | 39.k | even | 12 | 2 | ||
845.2.l.e | 12 | 195.bh | even | 12 | 2 | ||
845.2.n.f | 12 | 39.i | odd | 6 | 2 | ||
845.2.n.f | 12 | 195.x | odd | 6 | 2 | ||
845.2.n.g | 12 | 39.h | odd | 6 | 2 | ||
845.2.n.g | 12 | 195.y | odd | 6 | 2 | ||
1040.2.d.c | 6 | 12.b | even | 2 | 1 | ||
1040.2.d.c | 6 | 60.h | even | 2 | 1 | ||
2925.2.a.bf | 3 | 5.c | odd | 4 | 1 | ||
2925.2.a.bj | 3 | 5.c | odd | 4 | 1 | ||
4225.2.a.ba | 3 | 195.s | even | 4 | 1 | ||
4225.2.a.bh | 3 | 195.s | even | 4 | 1 | ||
5200.2.a.cb | 3 | 60.l | odd | 4 | 1 | ||
5200.2.a.cj | 3 | 60.l | odd | 4 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{6} + 11T_{2}^{4} + 31T_{2}^{2} + 25 \)
acting on \(S_{2}^{\mathrm{new}}(585, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{6} + 11 T^{4} + 31 T^{2} + 25 \)
$3$
\( T^{6} \)
$5$
\( T^{6} - T^{4} - 16 T^{3} - 5 T^{2} + \cdots + 125 \)
$7$
\( T^{6} + 12 T^{4} + 32 T^{2} + 16 \)
$11$
\( (T^{3} - 6 T^{2} + 8 T + 2)^{2} \)
$13$
\( (T^{2} + 1)^{3} \)
$17$
\( T^{6} + 44 T^{4} + 112 T^{2} + \cdots + 64 \)
$19$
\( (T^{3} - 4 T + 2)^{2} \)
$23$
\( T^{6} + 72 T^{4} + 1436 T^{2} + \cdots + 7396 \)
$29$
\( (T^{3} - 6 T^{2} - 36 T + 108)^{2} \)
$31$
\( (T^{3} + 10 T^{2} + 20 T - 26)^{2} \)
$37$
\( T^{6} + 56 T^{4} + 784 T^{2} + \cdots + 2704 \)
$41$
\( (T^{3} - 4 T^{2} - 32 T - 32)^{2} \)
$43$
\( T^{6} + 128 T^{4} + 5452 T^{2} + \cdots + 77284 \)
$47$
\( T^{6} + 44 T^{4} + 384 T^{2} + \cdots + 400 \)
$53$
\( T^{6} + 144 T^{4} + 6464 T^{2} + \cdots + 92416 \)
$59$
\( (T^{3} + 8 T^{2} - 40 T - 262)^{2} \)
$61$
\( (T^{3} - 6 T^{2} - 16 T - 4)^{2} \)
$67$
\( T^{6} + 220 T^{4} + 15680 T^{2} + \cdots + 364816 \)
$71$
\( (T^{3} - 12 T^{2} - 88 T + 754)^{2} \)
$73$
\( T^{6} + 248 T^{4} + 15568 T^{2} + \cdots + 55696 \)
$79$
\( (T^{3} - 16 T^{2} + 24 T + 16)^{2} \)
$83$
\( T^{6} + 180 T^{4} + 9200 T^{2} + \cdots + 99856 \)
$89$
\( (T^{3} - 10 T^{2} - 52 T + 200)^{2} \)
$97$
\( T^{6} + 364 T^{4} + 12656 T^{2} + \cdots + 40000 \)
show more
show less