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 | −2.76408 | 1.20534 | − | 1.24385i | 5.64014 | 0.500000 | + | 0.866025i | −3.33166 | + | 3.43809i | 1.30005 | + | 2.25175i | −10.0616 | −0.0943127 | − | 2.99852i | −1.38204 | − | 2.39376i | ||||||
16.2 | −2.58098 | −1.21772 | − | 1.23173i | 4.66146 | 0.500000 | + | 0.866025i | 3.14292 | + | 3.17906i | −1.30127 | − | 2.25386i | −6.86918 | −0.0343020 | + | 2.99980i | −1.29049 | − | 2.23519i | ||||||
16.3 | −2.51255 | −0.575022 | + | 1.63381i | 4.31289 | 0.500000 | + | 0.866025i | 1.44477 | − | 4.10503i | 1.40071 | + | 2.42609i | −5.81124 | −2.33870 | − | 1.87896i | −1.25627 | − | 2.17593i | ||||||
16.4 | −2.12815 | 1.35704 | + | 1.07630i | 2.52901 | 0.500000 | + | 0.866025i | −2.88799 | − | 2.29054i | −0.470061 | − | 0.814170i | −1.12582 | 0.683136 | + | 2.92119i | −1.06407 | − | 1.84303i | ||||||
16.5 | −1.90412 | 0.150405 | + | 1.72551i | 1.62569 | 0.500000 | + | 0.866025i | −0.286390 | − | 3.28558i | −1.00653 | − | 1.74336i | 0.712733 | −2.95476 | + | 0.519050i | −0.952062 | − | 1.64902i | ||||||
16.6 | −1.85591 | −1.71867 | + | 0.214845i | 1.44441 | 0.500000 | + | 0.866025i | 3.18971 | − | 0.398733i | −0.968036 | − | 1.67669i | 1.03113 | 2.90768 | − | 0.738496i | −0.927956 | − | 1.60727i | ||||||
16.7 | −1.69989 | 1.65633 | − | 0.506513i | 0.889617 | 0.500000 | + | 0.866025i | −2.81558 | + | 0.861016i | 1.04879 | + | 1.81656i | 1.88753 | 2.48689 | − | 1.67791i | −0.849944 | − | 1.47215i | ||||||
16.8 | −1.66452 | −1.56929 | − | 0.733032i | 0.770642 | 0.500000 | + | 0.866025i | 2.61212 | + | 1.22015i | 2.37650 | + | 4.11622i | 2.04630 | 1.92533 | + | 2.30068i | −0.832262 | − | 1.44152i | ||||||
16.9 | −1.41906 | 0.192139 | − | 1.72136i | 0.0137247 | 0.500000 | + | 0.866025i | −0.272656 | + | 2.44271i | −0.0722641 | − | 0.125165i | 2.81864 | −2.92617 | − | 0.661480i | −0.709529 | − | 1.22894i | ||||||
16.10 | −0.763785 | 1.64087 | + | 0.554555i | −1.41663 | 0.500000 | + | 0.866025i | −1.25328 | − | 0.423561i | −2.34538 | − | 4.06232i | 2.60957 | 2.38494 | + | 1.81991i | −0.381893 | − | 0.661458i | ||||||
16.11 | −0.671737 | −1.45581 | + | 0.938409i | −1.54877 | 0.500000 | + | 0.866025i | 0.977923 | − | 0.630364i | −0.0474320 | − | 0.0821546i | 2.38384 | 1.23878 | − | 2.73229i | −0.335869 | − | 0.581742i | ||||||
16.12 | −0.657086 | 0.285448 | + | 1.70837i | −1.56824 | 0.500000 | + | 0.866025i | −0.187564 | − | 1.12254i | 1.88943 | + | 3.27259i | 2.34464 | −2.83704 | + | 0.975301i | −0.328543 | − | 0.569053i | ||||||
16.13 | −0.515516 | −0.626853 | − | 1.61464i | −1.73424 | 0.500000 | + | 0.866025i | 0.323152 | + | 0.832371i | −2.25375 | − | 3.90361i | 1.92506 | −2.21411 | + | 2.02428i | −0.257758 | − | 0.446450i | ||||||
16.14 | −0.0917973 | 0.451768 | − | 1.67210i | −1.99157 | 0.500000 | + | 0.866025i | −0.0414711 | + | 0.153494i | 2.14686 | + | 3.71846i | 0.366416 | −2.59181 | − | 1.51080i | −0.0458987 | − | 0.0794988i | ||||||
16.15 | 0.0143342 | −1.64723 | − | 0.535378i | −1.99979 | 0.500000 | + | 0.866025i | −0.0236117 | − | 0.00767421i | 0.299595 | + | 0.518914i | −0.0573338 | 2.42674 | + | 1.76378i | 0.00716709 | + | 0.0124138i | ||||||
16.16 | 0.279919 | 1.47779 | + | 0.903405i | −1.92165 | 0.500000 | + | 0.866025i | 0.413661 | + | 0.252880i | 0.459383 | + | 0.795675i | −1.09774 | 1.36772 | + | 2.67008i | 0.139959 | + | 0.242417i | ||||||
16.17 | 0.788554 | 1.20224 | − | 1.24684i | −1.37818 | 0.500000 | + | 0.866025i | 0.948031 | − | 0.983203i | −1.40210 | − | 2.42851i | −2.66388 | −0.109240 | − | 2.99801i | 0.394277 | + | 0.682908i | ||||||
16.18 | 0.809618 | −1.11759 | − | 1.32325i | −1.34452 | 0.500000 | + | 0.866025i | −0.904819 | − | 1.07133i | 0.662471 | + | 1.14743i | −2.70778 | −0.501999 | + | 2.95770i | 0.404809 | + | 0.701150i | ||||||
16.19 | 0.829649 | 0.333567 | + | 1.69963i | −1.31168 | 0.500000 | + | 0.866025i | 0.276744 | + | 1.41009i | −1.11914 | − | 1.93840i | −2.74753 | −2.77747 | + | 1.13388i | 0.414825 | + | 0.718497i | ||||||
16.20 | 1.31077 | 1.72189 | − | 0.187297i | −0.281879 | 0.500000 | + | 0.866025i | 2.25701 | − | 0.245503i | 1.76552 | + | 3.05796i | −2.99102 | 2.92984 | − | 0.645010i | 0.655386 | + | 1.13516i | ||||||
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.a | yes | 56 |
9.c | even | 3 | 1 | 585.2.k.a | ✓ | 56 | |
13.c | even | 3 | 1 | 585.2.k.a | ✓ | 56 | |
117.h | even | 3 | 1 | inner | 585.2.l.a | yes | 56 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
585.2.k.a | ✓ | 56 | 9.c | even | 3 | 1 | |
585.2.k.a | ✓ | 56 | 13.c | even | 3 | 1 | |
585.2.l.a | yes | 56 | 1.a | even | 1 | 1 | trivial |
585.2.l.a | yes | 56 | 117.h | even | 3 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{28} - 42 T_{2}^{26} - T_{2}^{25} + 775 T_{2}^{24} + 36 T_{2}^{23} - 8277 T_{2}^{22} - 555 T_{2}^{21} + 56743 T_{2}^{20} + 4808 T_{2}^{19} - 261793 T_{2}^{18} - 25862 T_{2}^{17} + 828958 T_{2}^{16} + 90349 T_{2}^{15} - 1804468 T_{2}^{14} + \cdots + 9 \)
acting on \(S_{2}^{\mathrm{new}}(585, [\chi])\).