Newspace parameters
Level: | \( N \) | \(=\) | \( 585 = 3^{2} \cdot 5 \cdot 13 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 585.l (of order \(3\), degree \(2\), minimal) |
Newform invariants
Self dual: | no |
Analytic conductor: | \(4.67124851824\) |
Analytic rank: | \(0\) |
Dimension: | \(56\) |
Relative dimension: | \(28\) over \(\Q(\zeta_{3})\) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{3}]$ |
$q$-expansion
The dimension is sufficiently large that we do not compute an algebraic \(q\)-expansion, but we have computed the trace expansion.
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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
16.1 | −2.69141 | −1.04767 | + | 1.37927i | 5.24369 | −0.500000 | − | 0.866025i | 2.81971 | − | 3.71218i | −2.38650 | − | 4.13354i | −8.73010 | −0.804776 | − | 2.89004i | 1.34571 | + | 2.33083i | ||||||
16.2 | −2.57428 | −1.67879 | − | 0.426208i | 4.62691 | −0.500000 | − | 0.866025i | 4.32168 | + | 1.09718i | 1.24059 | + | 2.14876i | −6.76239 | 2.63669 | + | 1.43103i | 1.28714 | + | 2.22939i | ||||||
16.3 | −2.35889 | 1.71934 | − | 0.209467i | 3.56438 | −0.500000 | − | 0.866025i | −4.05574 | + | 0.494109i | −0.253078 | − | 0.438343i | −3.69022 | 2.91225 | − | 0.720288i | 1.17945 | + | 2.04286i | ||||||
16.4 | −2.27248 | 0.441019 | − | 1.67496i | 3.16415 | −0.500000 | − | 0.866025i | −1.00221 | + | 3.80631i | 1.99843 | + | 3.46138i | −2.64549 | −2.61100 | − | 1.47738i | 1.13624 | + | 1.96802i | ||||||
16.5 | −2.11041 | 1.03489 | − | 1.38889i | 2.45385 | −0.500000 | − | 0.866025i | −2.18404 | + | 2.93113i | −2.18806 | − | 3.78983i | −0.957805 | −0.858025 | − | 2.87468i | 1.05521 | + | 1.82767i | ||||||
16.6 | −1.62135 | −1.65325 | − | 0.516489i | 0.628764 | −0.500000 | − | 0.866025i | 2.68049 | + | 0.837407i | −1.96848 | − | 3.40951i | 2.22325 | 2.46648 | + | 1.70777i | 0.810673 | + | 1.40413i | ||||||
16.7 | −1.57635 | −0.683981 | + | 1.59128i | 0.484891 | −0.500000 | − | 0.866025i | 1.07820 | − | 2.50842i | 0.939234 | + | 1.62680i | 2.38835 | −2.06434 | − | 2.17681i | 0.788177 | + | 1.36516i | ||||||
16.8 | −1.54763 | 1.63645 | + | 0.567488i | 0.395159 | −0.500000 | − | 0.866025i | −2.53261 | − | 0.878261i | 1.17606 | + | 2.03700i | 2.48370 | 2.35592 | + | 1.85733i | 0.773815 | + | 1.34029i | ||||||
16.9 | −1.48791 | 0.736970 | + | 1.56744i | 0.213889 | −0.500000 | − | 0.866025i | −1.09655 | − | 2.33222i | −1.05683 | − | 1.83048i | 2.65758 | −1.91375 | + | 2.31032i | 0.743957 | + | 1.28857i | ||||||
16.10 | −1.30229 | −0.671232 | − | 1.59670i | −0.304039 | −0.500000 | − | 0.866025i | 0.874139 | + | 2.07937i | −0.414568 | − | 0.718052i | 3.00053 | −2.09890 | + | 2.14351i | 0.651145 | + | 1.12782i | ||||||
16.11 | −0.704886 | −1.01815 | − | 1.40121i | −1.50314 | −0.500000 | − | 0.866025i | 0.717678 | + | 0.987690i | 1.19422 | + | 2.06845i | 2.46931 | −0.926753 | + | 2.85327i | 0.352443 | + | 0.610449i | ||||||
16.12 | −0.695911 | −1.64450 | + | 0.543713i | −1.51571 | −0.500000 | − | 0.866025i | 1.14442 | − | 0.378376i | −0.108191 | − | 0.187393i | 2.44662 | 2.40875 | − | 1.78827i | 0.347956 | + | 0.602677i | ||||||
16.13 | −0.443328 | 1.61582 | − | 0.623809i | −1.80346 | −0.500000 | − | 0.866025i | −0.716337 | + | 0.276552i | −1.51185 | − | 2.61860i | 1.68618 | 2.22172 | − | 2.01592i | 0.221664 | + | 0.383934i | ||||||
16.14 | −0.0299620 | 0.971767 | + | 1.43376i | −1.99910 | −0.500000 | − | 0.866025i | −0.0291161 | − | 0.0429584i | −1.27470 | − | 2.20784i | 0.119821 | −1.11134 | + | 2.78656i | 0.0149810 | + | 0.0259479i | ||||||
16.15 | 0.0303319 | 1.64405 | − | 0.545074i | −1.99908 | −0.500000 | − | 0.866025i | 0.0498671 | − | 0.0165331i | 1.78412 | + | 3.09018i | −0.121300 | 2.40579 | − | 1.79226i | −0.0151659 | − | 0.0262682i | ||||||
16.16 | 0.198180 | −0.761883 | + | 1.55549i | −1.96072 | −0.500000 | − | 0.866025i | −0.150990 | + | 0.308265i | −0.807590 | − | 1.39879i | −0.784935 | −1.83907 | − | 2.37019i | −0.0990898 | − | 0.171629i | ||||||
16.17 | 0.915325 | −0.253923 | + | 1.71334i | −1.16218 | −0.500000 | − | 0.866025i | −0.232422 | + | 1.56826i | 1.99723 | + | 3.45930i | −2.89442 | −2.87105 | − | 0.870111i | −0.457662 | − | 0.792694i | ||||||
16.18 | 0.919862 | −0.842111 | − | 1.51355i | −1.15385 | −0.500000 | − | 0.866025i | −0.774626 | − | 1.39226i | 1.23537 | + | 2.13972i | −2.90111 | −1.58170 | + | 2.54916i | −0.459931 | − | 0.796624i | ||||||
16.19 | 1.01088 | −1.70428 | − | 0.308944i | −0.978128 | −0.500000 | − | 0.866025i | −1.72281 | − | 0.312304i | −0.443377 | − | 0.767952i | −3.01052 | 2.80911 | + | 1.05305i | −0.505439 | − | 0.875445i | ||||||
16.20 | 1.18191 | 0.368619 | − | 1.69237i | −0.603097 | −0.500000 | − | 0.866025i | 0.435674 | − | 2.00022i | −0.282481 | − | 0.489272i | −3.07662 | −2.72824 | − | 1.24768i | −0.590953 | − | 1.02356i | ||||||
See all 56 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
117.h | even | 3 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 585.2.l.b | yes | 56 |
9.c | even | 3 | 1 | 585.2.k.b | ✓ | 56 | |
13.c | even | 3 | 1 | 585.2.k.b | ✓ | 56 | |
117.h | even | 3 | 1 | inner | 585.2.l.b | yes | 56 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
585.2.k.b | ✓ | 56 | 9.c | even | 3 | 1 | |
585.2.k.b | ✓ | 56 | 13.c | even | 3 | 1 | |
585.2.l.b | yes | 56 | 1.a | even | 1 | 1 | trivial |
585.2.l.b | yes | 56 | 117.h | even | 3 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{28} - 42 T_{2}^{26} - T_{2}^{25} + 779 T_{2}^{24} + 36 T_{2}^{23} - 8413 T_{2}^{22} - 559 T_{2}^{21} + 58743 T_{2}^{20} + 4914 T_{2}^{19} - 278499 T_{2}^{18} - 27018 T_{2}^{17} + 916404 T_{2}^{16} + 97093 T_{2}^{15} - 2102586 T_{2}^{14} + \cdots + 9 \)
acting on \(S_{2}^{\mathrm{new}}(585, [\chi])\).