# Properties

 Label 720.2.t.c Level $720$ Weight $2$ Character orbit 720.t Analytic conductor $5.749$ Analytic rank $0$ Dimension $16$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$720 = 2^{4} \cdot 3^{2} \cdot 5$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 720.t (of order $$4$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$5.74922894553$$ Analytic rank: $$0$$ Dimension: $$16$$ Relative dimension: $$8$$ over $$\Q(i)$$ Coefficient field: $$\mathbb{Q}[x]/(x^{16} - \cdots)$$ Defining polynomial: $$x^{16} - 4 x^{15} + 4 x^{14} + 7 x^{12} - 8 x^{11} - 28 x^{10} + 28 x^{9} + 17 x^{8} + 56 x^{7} - 112 x^{6} - 64 x^{5} + 112 x^{4} + 256 x^{2} - 512 x + 256$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{5}$$ Twist minimal: no (minimal twist has level 80) Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{15}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_{9} q^{2} + ( \beta_{1} - \beta_{5} - \beta_{14} ) q^{4} -\beta_{4} q^{5} + ( -\beta_{1} - \beta_{2} + \beta_{3} + \beta_{6} + \beta_{7} - \beta_{12} - \beta_{13} + \beta_{14} ) q^{7} + ( \beta_{3} + \beta_{4} + \beta_{7} + \beta_{8} - \beta_{9} - \beta_{10} - \beta_{13} ) q^{8} +O(q^{10})$$ $$q + \beta_{9} q^{2} + ( \beta_{1} - \beta_{5} - \beta_{14} ) q^{4} -\beta_{4} q^{5} + ( -\beta_{1} - \beta_{2} + \beta_{3} + \beta_{6} + \beta_{7} - \beta_{12} - \beta_{13} + \beta_{14} ) q^{7} + ( \beta_{3} + \beta_{4} + \beta_{7} + \beta_{8} - \beta_{9} - \beta_{10} - \beta_{13} ) q^{8} -\beta_{11} q^{10} + ( 1 + \beta_{1} + \beta_{4} - \beta_{6} - \beta_{8} - \beta_{9} + \beta_{10} + \beta_{11} - \beta_{14} + \beta_{15} ) q^{11} + ( 1 + \beta_{2} + \beta_{4} - \beta_{5} + \beta_{7} + 2 \beta_{8} - \beta_{9} - \beta_{13} - \beta_{14} + \beta_{15} ) q^{13} + ( -1 - \beta_{2} + \beta_{4} + \beta_{5} + \beta_{6} + \beta_{9} + \beta_{10} - \beta_{12} + \beta_{14} - \beta_{15} ) q^{14} + ( 1 + \beta_{2} - \beta_{3} + 2 \beta_{4} + \beta_{5} + \beta_{6} - \beta_{7} - \beta_{12} - \beta_{13} + \beta_{15} ) q^{16} + ( 2 \beta_{1} + \beta_{2} + \beta_{3} + 2 \beta_{4} - 2 \beta_{5} - \beta_{6} - \beta_{7} - \beta_{8} + \beta_{10} + \beta_{11} + \beta_{12} + \beta_{15} ) q^{17} + ( -1 + \beta_{1} + \beta_{4} + \beta_{6} - 2 \beta_{7} - \beta_{8} + \beta_{9} + \beta_{10} + \beta_{11} + \beta_{14} + \beta_{15} ) q^{19} + ( -1 + \beta_{1} - \beta_{6} - \beta_{7} - \beta_{8} + \beta_{12} ) q^{20} + ( -1 - 2 \beta_{1} - \beta_{2} + \beta_{3} + 2 \beta_{4} + \beta_{5} + 3 \beta_{6} + \beta_{7} + \beta_{8} - 2 \beta_{10} + \beta_{11} - \beta_{12} - \beta_{13} + 2 \beta_{14} + \beta_{15} ) q^{22} + ( -\beta_{1} - \beta_{2} + \beta_{4} - \beta_{7} + \beta_{8} + \beta_{11} - \beta_{13} + \beta_{14} ) q^{23} + \beta_{10} q^{25} + ( 1 + 3 \beta_{2} - \beta_{3} + \beta_{5} + 3 \beta_{6} + \beta_{7} + 2 \beta_{10} - 2 \beta_{11} - \beta_{12} - \beta_{13} + \beta_{15} ) q^{26} + ( -\beta_{2} - \beta_{3} + 2 \beta_{4} - 2 \beta_{6} + \beta_{8} - 2 \beta_{10} + 2 \beta_{11} + \beta_{13} + \beta_{14} - \beta_{15} ) q^{28} + ( 1 - 2 \beta_{2} + 2 \beta_{3} - 2 \beta_{5} - 2 \beta_{9} - 3 \beta_{10} - 2 \beta_{13} ) q^{29} + ( 2 \beta_{1} - 2 \beta_{5} - 2 \beta_{6} + 2 \beta_{8} ) q^{31} + ( 1 + \beta_{2} - \beta_{3} - 2 \beta_{4} + \beta_{5} - \beta_{6} + \beta_{7} + \beta_{8} + \beta_{9} + \beta_{11} - \beta_{12} + \beta_{13} - \beta_{14} + \beta_{15} ) q^{32} + ( 1 - \beta_{2} + 3 \beta_{3} - 2 \beta_{4} - \beta_{5} + \beta_{6} + \beta_{7} + 2 \beta_{8} - 2 \beta_{10} + 2 \beta_{11} - \beta_{12} - \beta_{13} - \beta_{15} ) q^{34} + ( \beta_{1} + \beta_{3} + \beta_{4} - \beta_{5} + \beta_{8} - \beta_{13} + \beta_{15} ) q^{35} + ( -2 + 2 \beta_{1} + 2 \beta_{2} - 2 \beta_{6} - 2 \beta_{8} - 2 \beta_{9} + 2 \beta_{12} + 2 \beta_{14} ) q^{37} + ( -1 - \beta_{2} + \beta_{3} - 2 \beta_{4} - \beta_{5} - \beta_{6} + \beta_{7} + 3 \beta_{8} - 2 \beta_{10} + \beta_{11} - \beta_{12} + \beta_{13} - \beta_{15} ) q^{38} + ( \beta_{1} + \beta_{3} + \beta_{4} - \beta_{5} - 2 \beta_{6} - \beta_{7} - \beta_{9} - \beta_{10} + \beta_{11} + \beta_{15} ) q^{40} + ( -2 \beta_{1} - \beta_{2} - \beta_{3} + \beta_{4} + 2 \beta_{6} - \beta_{7} + \beta_{8} + 2 \beta_{9} - \beta_{10} - \beta_{11} + \beta_{12} + 2 \beta_{14} + \beta_{15} ) q^{41} + ( 1 - 3 \beta_{4} + \beta_{5} - \beta_{8} - \beta_{9} + \beta_{10} - \beta_{12} - \beta_{13} - \beta_{14} + 2 \beta_{15} ) q^{43} + ( -2 - \beta_{1} - 2 \beta_{3} + \beta_{5} + 4 \beta_{6} + 2 \beta_{8} + \beta_{9} + 2 \beta_{10} - 2 \beta_{12} + \beta_{13} - \beta_{14} - \beta_{15} ) q^{44} + ( 2 \beta_{2} - \beta_{3} - \beta_{4} - 2 \beta_{6} + \beta_{7} + \beta_{9} + 3 \beta_{10} - 2 \beta_{12} + \beta_{13} + \beta_{14} ) q^{46} + ( 1 + \beta_{1} + \beta_{2} - \beta_{3} + \beta_{5} - 3 \beta_{6} - 3 \beta_{7} - 2 \beta_{8} - \beta_{9} + \beta_{10} - 2 \beta_{11} - \beta_{12} + 2 \beta_{14} + 2 \beta_{15} ) q^{47} + ( -3 + 2 \beta_{1} - \beta_{2} + \beta_{3} + 3 \beta_{4} - 3 \beta_{7} - \beta_{8} - 2 \beta_{9} - \beta_{10} - \beta_{11} + \beta_{12} + 2 \beta_{14} + \beta_{15} ) q^{49} + \beta_{14} q^{50} + ( 3 + 2 \beta_{1} + \beta_{2} - \beta_{3} - 2 \beta_{4} + \beta_{5} + \beta_{6} - \beta_{7} + \beta_{8} + \beta_{9} - \beta_{11} + \beta_{12} + \beta_{13} - \beta_{14} + \beta_{15} ) q^{52} + ( -1 + 2 \beta_{1} - \beta_{2} - \beta_{4} + \beta_{5} - 2 \beta_{6} - \beta_{7} - 2 \beta_{8} - 3 \beta_{9} - 2 \beta_{10} + 2 \beta_{11} + 2 \beta_{12} - \beta_{13} - \beta_{14} + \beta_{15} ) q^{53} + ( \beta_{1} - \beta_{2} - \beta_{4} - \beta_{7} - \beta_{8} + \beta_{9} - \beta_{10} + \beta_{11} - \beta_{14} ) q^{55} + ( 1 - 3 \beta_{1} + \beta_{2} - \beta_{3} + 2 \beta_{4} + \beta_{6} + 3 \beta_{7} + \beta_{9} + 2 \beta_{11} - \beta_{12} + 2 \beta_{13} - \beta_{14} - 2 \beta_{15} ) q^{56} + ( -2 + 2 \beta_{3} + 2 \beta_{4} + 2 \beta_{5} - 2 \beta_{6} + \beta_{9} + 2 \beta_{10} - 2 \beta_{12} - 2 \beta_{13} + \beta_{14} ) q^{58} + ( 1 + \beta_{1} + 2 \beta_{2} - \beta_{4} - 2 \beta_{5} - \beta_{6} - \beta_{8} + \beta_{9} + 3 \beta_{10} - \beta_{11} + 2 \beta_{13} - \beta_{14} + \beta_{15} ) q^{59} + ( -2 \beta_{1} - 2 \beta_{2} + 2 \beta_{3} + 2 \beta_{5} - 2 \beta_{10} + 2 \beta_{11} + 2 \beta_{13} + 4 \beta_{14} ) q^{61} + ( 2 \beta_{2} + 2 \beta_{4} + 2 \beta_{7} - 2 \beta_{10} - 2 \beta_{11} - 2 \beta_{15} ) q^{62} + ( -2 \beta_{1} - 2 \beta_{3} + 6 \beta_{6} + 2 \beta_{7} - 2 \beta_{10} - 4 \beta_{11} - 2 \beta_{14} ) q^{64} + ( -\beta_{2} + \beta_{3} - \beta_{6} - \beta_{7} - \beta_{8} - 2 \beta_{9} - \beta_{10} + \beta_{11} + \beta_{12} + \beta_{15} ) q^{65} + ( 3 + 2 \beta_{1} + 2 \beta_{2} - 3 \beta_{3} + 2 \beta_{4} - \beta_{5} - 5 \beta_{6} - 2 \beta_{7} + \beta_{9} - \beta_{10} - \beta_{11} - \beta_{13} - \beta_{14} + 2 \beta_{15} ) q^{67} + ( 3 - 2 \beta_{1} - \beta_{2} - \beta_{3} + \beta_{5} + \beta_{6} + \beta_{7} + 3 \beta_{8} + \beta_{9} - 2 \beta_{10} - 3 \beta_{11} - 3 \beta_{12} - 3 \beta_{13} - \beta_{14} - \beta_{15} ) q^{68} + ( -1 + \beta_{1} + \beta_{2} + \beta_{4} + \beta_{6} + \beta_{8} - \beta_{10} - \beta_{11} - \beta_{12} - \beta_{13} ) q^{70} + ( -2 \beta_{1} - 2 \beta_{2} - 6 \beta_{4} + 4 \beta_{6} + 2 \beta_{7} - 2 \beta_{8} + 2 \beta_{11} + 2 \beta_{13} + 2 \beta_{14} - 4 \beta_{15} ) q^{71} + ( -4 \beta_{1} - \beta_{2} + \beta_{3} + 2 \beta_{4} + 3 \beta_{6} + \beta_{7} + 3 \beta_{8} - 2 \beta_{9} - 3 \beta_{10} - \beta_{11} - \beta_{12} - 4 \beta_{13} + 4 \beta_{14} + \beta_{15} ) q^{73} + ( 4 + 2 \beta_{3} + 2 \beta_{4} - 2 \beta_{7} - 2 \beta_{10} + 2 \beta_{11} + 2 \beta_{13} ) q^{74} + ( -3 \beta_{1} - 2 \beta_{3} + \beta_{5} + 2 \beta_{6} + 2 \beta_{7} - \beta_{9} - 2 \beta_{10} - 4 \beta_{11} - \beta_{13} - \beta_{14} - 3 \beta_{15} ) q^{76} + ( -1 - 2 \beta_{1} - \beta_{2} - 2 \beta_{3} + \beta_{4} + \beta_{5} + 2 \beta_{6} + \beta_{7} + 2 \beta_{8} - \beta_{9} - 2 \beta_{11} + \beta_{13} + 3 \beta_{14} + \beta_{15} ) q^{77} + ( 2 - 2 \beta_{1} - 2 \beta_{3} + 2 \beta_{4} + 4 \beta_{6} - 2 \beta_{11} - 2 \beta_{12} - 2 \beta_{14} ) q^{79} + ( -1 - 2 \beta_{1} - \beta_{2} + \beta_{3} - 2 \beta_{4} + \beta_{5} + \beta_{6} + \beta_{7} - 2 \beta_{10} - \beta_{12} - \beta_{13} - \beta_{15} ) q^{80} + ( -5 + 4 \beta_{1} + 3 \beta_{2} - \beta_{3} - 2 \beta_{4} - 3 \beta_{5} - \beta_{6} - \beta_{7} - \beta_{8} + 2 \beta_{9} + 6 \beta_{10} - \beta_{11} + \beta_{12} + 3 \beta_{13} - 2 \beta_{14} - \beta_{15} ) q^{82} + ( -1 + \beta_{3} + 2 \beta_{4} - 3 \beta_{5} - 3 \beta_{6} + 2 \beta_{8} - 3 \beta_{9} + \beta_{10} + \beta_{11} - 3 \beta_{13} - \beta_{14} + 2 \beta_{15} ) q^{83} + ( -1 - \beta_{2} - \beta_{4} - \beta_{5} - \beta_{7} + \beta_{9} - 2 \beta_{10} + 2 \beta_{12} + \beta_{13} - \beta_{14} - \beta_{15} ) q^{85} + ( -3 - \beta_{1} - \beta_{2} + \beta_{4} + 2 \beta_{5} + 3 \beta_{6} - \beta_{10} - 4 \beta_{11} - \beta_{12} - \beta_{13} + 2 \beta_{14} + 2 \beta_{15} ) q^{86} + ( 1 - 2 \beta_{1} + \beta_{2} - 3 \beta_{3} - 2 \beta_{4} + \beta_{5} - \beta_{6} + 3 \beta_{7} + 3 \beta_{8} - 3 \beta_{9} - \beta_{11} + \beta_{12} + \beta_{13} + \beta_{14} - \beta_{15} ) q^{88} + ( 2 \beta_{1} + 2 \beta_{2} - 2 \beta_{3} - 2 \beta_{4} - 2 \beta_{7} - 2 \beta_{9} + 2 \beta_{12} - 2 \beta_{14} ) q^{89} + ( 4 - 4 \beta_{1} - 2 \beta_{2} + 2 \beta_{4} - 2 \beta_{5} + 4 \beta_{6} + 4 \beta_{7} + 6 \beta_{9} + 2 \beta_{10} + 2 \beta_{11} + 2 \beta_{13} - 4 \beta_{15} ) q^{91} + ( 3 - 2 \beta_{1} - 2 \beta_{2} - 4 \beta_{4} + \beta_{5} + 3 \beta_{6} + 3 \beta_{7} - 2 \beta_{8} + \beta_{9} - 2 \beta_{10} - \beta_{11} + \beta_{12} + 2 \beta_{13} - 4 \beta_{15} ) q^{92} + ( -4 - 2 \beta_{2} + \beta_{3} - 5 \beta_{4} + 2 \beta_{5} - 4 \beta_{6} - \beta_{7} - 4 \beta_{8} + 3 \beta_{9} - 3 \beta_{10} + 2 \beta_{12} + 3 \beta_{13} + \beta_{14} - 2 \beta_{15} ) q^{94} + ( -2 - \beta_{1} - \beta_{2} - \beta_{4} + 2 \beta_{6} + \beta_{7} + \beta_{8} + \beta_{9} - \beta_{10} - \beta_{11} - \beta_{14} - 2 \beta_{15} ) q^{95} + ( 2 - 2 \beta_{1} - \beta_{2} + \beta_{3} - \beta_{6} - \beta_{7} - 3 \beta_{8} - 4 \beta_{9} - \beta_{10} + 3 \beta_{11} + \beta_{12} - 2 \beta_{14} + 3 \beta_{15} ) q^{97} + ( -5 - \beta_{2} + \beta_{3} + \beta_{5} - 5 \beta_{6} - 3 \beta_{7} + \beta_{8} - \beta_{9} - 4 \beta_{10} + 3 \beta_{11} + \beta_{12} + \beta_{13} + 2 \beta_{14} - \beta_{15} ) q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$16q - 4q^{4} + O(q^{10})$$ $$16q - 4q^{4} + 4q^{10} + 8q^{11} - 4q^{14} + 16q^{16} - 8q^{19} - 8q^{20} - 20q^{22} + 16q^{26} - 4q^{28} + 16q^{29} + 16q^{34} - 16q^{37} - 20q^{38} + 8q^{43} - 40q^{44} - 4q^{46} + 40q^{47} - 16q^{49} + 4q^{50} + 56q^{52} - 16q^{53} - 16q^{56} - 12q^{58} + 8q^{59} + 16q^{61} + 8q^{62} - 16q^{64} + 40q^{67} + 48q^{68} - 8q^{70} + 72q^{74} - 16q^{77} + 16q^{79} - 16q^{80} - 76q^{82} - 40q^{83} - 16q^{85} - 28q^{86} + 32q^{91} + 52q^{92} - 36q^{94} - 32q^{95} - 60q^{98} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{16} - 4 x^{15} + 4 x^{14} + 7 x^{12} - 8 x^{11} - 28 x^{10} + 28 x^{9} + 17 x^{8} + 56 x^{7} - 112 x^{6} - 64 x^{5} + 112 x^{4} + 256 x^{2} - 512 x + 256$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$-\nu^{15} - 70 \nu^{14} + 120 \nu^{13} + 144 \nu^{12} + 89 \nu^{11} - 606 \nu^{10} - 968 \nu^{9} + 916 \nu^{8} + 2103 \nu^{7} + 1774 \nu^{6} - 4044 \nu^{5} - 4664 \nu^{4} + 1600 \nu^{3} + 4832 \nu^{2} + 7296 \nu - 9472$$$$)/384$$ $$\beta_{2}$$ $$=$$ $$($$$$-51 \nu^{15} - 68 \nu^{14} + 320 \nu^{13} + 176 \nu^{12} - 277 \nu^{11} - 1368 \nu^{10} - 408 \nu^{9} + 3116 \nu^{8} + 2797 \nu^{7} - 1640 \nu^{6} - 10212 \nu^{5} - 1920 \nu^{4} + 10256 \nu^{3} + 11392 \nu^{2} + 3200 \nu - 25216$$$$)/2688$$ $$\beta_{3}$$ $$=$$ $$($$$$-2 \nu^{15} + \nu^{14} + 6 \nu^{13} + 3 \nu^{12} - 14 \nu^{11} - 33 \nu^{10} + 14 \nu^{9} + 77 \nu^{8} + 54 \nu^{7} - 103 \nu^{6} - 210 \nu^{5} + 23 \nu^{4} + 236 \nu^{3} + 244 \nu^{2} - 192 \nu - 320$$$$)/48$$ $$\beta_{4}$$ $$=$$ $$($$$$-163 \nu^{15} + 58 \nu^{14} + 376 \nu^{13} + 568 \nu^{12} - 501 \nu^{11} - 2502 \nu^{10} - 632 \nu^{9} + 3284 \nu^{8} + 6101 \nu^{7} - 1962 \nu^{6} - 10212 \nu^{5} - 4496 \nu^{4} + 4544 \nu^{3} + 16768 \nu^{2} - 6656 \nu - 7296$$$$)/2688$$ $$\beta_{5}$$ $$=$$ $$($$$$121 \nu^{15} - 65 \nu^{14} - 264 \nu^{13} - 372 \nu^{12} + 487 \nu^{11} + 1725 \nu^{10} + 44 \nu^{9} - 2584 \nu^{8} - 4155 \nu^{7} + 2459 \nu^{6} + 7608 \nu^{5} + 1808 \nu^{4} - 4600 \nu^{3} - 11840 \nu^{2} + 5760 \nu + 5056$$$$)/1344$$ $$\beta_{6}$$ $$=$$ $$($$$$-397 \nu^{15} + 502 \nu^{14} + 772 \nu^{13} + 664 \nu^{12} - 2523 \nu^{11} - 5226 \nu^{10} + 3292 \nu^{9} + 9932 \nu^{8} + 10139 \nu^{7} - 16854 \nu^{6} - 24480 \nu^{5} + 4720 \nu^{4} + 21776 \nu^{3} + 40576 \nu^{2} - 42176 \nu - 8832$$$$)/2688$$ $$\beta_{7}$$ $$=$$ $$($$$$-61 \nu^{15} + 82 \nu^{14} + 124 \nu^{13} + 112 \nu^{12} - 411 \nu^{11} - 870 \nu^{10} + 532 \nu^{9} + 1652 \nu^{8} + 1739 \nu^{7} - 2778 \nu^{6} - 4152 \nu^{5} + 568 \nu^{4} + 3680 \nu^{3} + 6688 \nu^{2} - 6848 \nu - 1536$$$$)/384$$ $$\beta_{8}$$ $$=$$ $$($$$$177 \nu^{15} - 604 \nu^{14} + 240 \nu^{13} + 328 \nu^{12} + 1607 \nu^{11} - 368 \nu^{10} - 6088 \nu^{9} + 132 \nu^{8} + 4721 \nu^{7} + 15472 \nu^{6} - 8772 \nu^{5} - 23112 \nu^{4} + 272 \nu^{3} + 4064 \nu^{2} + 55040 \nu - 48256$$$$)/896$$ $$\beta_{9}$$ $$=$$ $$($$$$-81 \nu^{15} + 184 \nu^{14} + 56 \nu^{13} + 8 \nu^{12} - 679 \nu^{11} - 588 \nu^{10} + 1680 \nu^{9} + 1388 \nu^{8} + 319 \nu^{7} - 5492 \nu^{6} - 1668 \nu^{5} + 5256 \nu^{4} + 3344 \nu^{3} + 4384 \nu^{2} - 16384 \nu + 7808$$$$)/384$$ $$\beta_{10}$$ $$=$$ $$($$$$396 \nu^{15} - 1201 \nu^{14} + 256 \nu^{13} + 460 \nu^{12} + 3508 \nu^{11} + 489 \nu^{10} - 11700 \nu^{9} - 2228 \nu^{8} + 6320 \nu^{7} + 32015 \nu^{6} - 9900 \nu^{5} - 42864 \nu^{4} - 5840 \nu^{3} - 1504 \nu^{2} + 108736 \nu - 83072$$$$)/1344$$ $$\beta_{11}$$ $$=$$ $$($$$$827 \nu^{15} - 2780 \nu^{14} + 904 \nu^{13} + 1480 \nu^{12} + 7821 \nu^{11} - 744 \nu^{10} - 27824 \nu^{9} - 2020 \nu^{8} + 19451 \nu^{7} + 73032 \nu^{6} - 33492 \nu^{5} - 103592 \nu^{4} - 5968 \nu^{3} + 12448 \nu^{2} + 250624 \nu - 210048$$$$)/2688$$ $$\beta_{12}$$ $$=$$ $$($$$$-526 \nu^{15} + 1205 \nu^{14} + 300 \nu^{13} - 24 \nu^{12} - 4258 \nu^{11} - 3477 \nu^{10} + 11080 \nu^{9} + 8152 \nu^{8} + 1038 \nu^{7} - 35603 \nu^{6} - 8256 \nu^{5} + 36196 \nu^{4} + 20080 \nu^{3} + 25520 \nu^{2} - 109056 \nu + 53696$$$$)/1344$$ $$\beta_{13}$$ $$=$$ $$($$$$-206 \nu^{15} + 633 \nu^{14} - 178 \nu^{13} - 268 \nu^{12} - 1770 \nu^{11} - 57 \nu^{10} + 6150 \nu^{9} + 628 \nu^{8} - 3802 \nu^{7} - 16439 \nu^{6} + 6690 \nu^{5} + 22936 \nu^{4} + 1328 \nu^{3} - 1056 \nu^{2} - 57248 \nu + 46400$$$$)/448$$ $$\beta_{14}$$ $$=$$ $$($$$$1431 \nu^{15} - 3748 \nu^{14} + 136 \nu^{13} + 856 \nu^{12} + 11857 \nu^{11} + 5328 \nu^{10} - 35760 \nu^{9} - 14180 \nu^{8} + 10487 \nu^{7} + 104912 \nu^{6} - 7908 \nu^{5} - 125304 \nu^{4} - 36944 \nu^{3} - 37024 \nu^{2} + 342400 \nu - 219776$$$$)/2688$$ $$\beta_{15}$$ $$=$$ $$($$$$-2295 \nu^{15} + 6418 \nu^{14} - 580 \nu^{13} - 1936 \nu^{12} - 19969 \nu^{11} - 7086 \nu^{10} + 61860 \nu^{9} + 21500 \nu^{8} - 22367 \nu^{7} - 177554 \nu^{6} + 22704 \nu^{5} + 217848 \nu^{4} + 55184 \nu^{3} + 48736 \nu^{2} - 579520 \nu + 392960$$$$)/2688$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$-\beta_{12} - \beta_{11} + \beta_{8} + \beta_{7} + \beta_{6} - \beta_{3} + \beta_{2} - \beta_{1} + 1$$$$)/2$$ $$\nu^{2}$$ $$=$$ $$($$$$\beta_{15} - \beta_{13} - \beta_{12} - \beta_{7} + \beta_{6} + \beta_{5} + 2 \beta_{4} - \beta_{3} + \beta_{2} + 1$$$$)/2$$ $$\nu^{3}$$ $$=$$ $$($$$$-\beta_{15} - 2 \beta_{14} - \beta_{13} - \beta_{11} - 2 \beta_{10} - 2 \beta_{9} + \beta_{8} - \beta_{5} - 2 \beta_{4} + \beta_{1} + 2$$$$)/2$$ $$\nu^{4}$$ $$=$$ $$($$$$3 \beta_{15} + 3 \beta_{14} - \beta_{13} - 2 \beta_{10} - 3 \beta_{9} + 2 \beta_{8} - 2 \beta_{7} - \beta_{5} + 2 \beta_{4} + 2 \beta_{2} + \beta_{1}$$$$)/2$$ $$\nu^{5}$$ $$=$$ $$($$$$\beta_{15} - 5 \beta_{13} - \beta_{12} - 4 \beta_{11} - 4 \beta_{10} - 4 \beta_{9} + 2 \beta_{8} - \beta_{7} + 5 \beta_{6} + \beta_{5} + 4 \beta_{4} - \beta_{3} - \beta_{2} - 2 \beta_{1} - 5$$$$)/2$$ $$\nu^{6}$$ $$=$$ $$($$$$\beta_{14} - 2 \beta_{13} + \beta_{12} + 2 \beta_{11} - 6 \beta_{10} - 3 \beta_{9} - 4 \beta_{8} - 5 \beta_{7} + 3 \beta_{6} - 2 \beta_{5} - 4 \beta_{4} + 5 \beta_{3} - 3 \beta_{2} + 5 \beta_{1} - 5$$$$)/2$$ $$\nu^{7}$$ $$=$$ $$($$$$4 \beta_{15} + 2 \beta_{14} - 4 \beta_{13} + 7 \beta_{12} - \beta_{11} + 2 \beta_{10} - 10 \beta_{9} - \beta_{8} - 5 \beta_{7} - 5 \beta_{6} - 8 \beta_{5} - 4 \beta_{4} + \beta_{3} - 3 \beta_{2} + 5 \beta_{1} - 7$$$$)/2$$ $$\nu^{8}$$ $$=$$ $$($$$$\beta_{15} + 10 \beta_{14} - 5 \beta_{13} + \beta_{12} + 4 \beta_{11} - 18 \beta_{10} - 2 \beta_{9} - \beta_{7} + 21 \beta_{6} + 3 \beta_{5} + 2 \beta_{4} + 7 \beta_{3} - 17 \beta_{2} - 14 \beta_{1} - 9$$$$)/2$$ $$\nu^{9}$$ $$=$$ $$($$$$-5 \beta_{15} - 2 \beta_{14} - 7 \beta_{13} + 2 \beta_{12} - \beta_{11} + 2 \beta_{10} - 2 \beta_{9} - 25 \beta_{8} + 6 \beta_{7} + 4 \beta_{6} - 5 \beta_{5} - 12 \beta_{4} + 6 \beta_{3} - 6 \beta_{2} + \beta_{1} - 16$$$$)/2$$ $$\nu^{10}$$ $$=$$ $$($$$$-9 \beta_{15} - 3 \beta_{14} + 11 \beta_{13} + 22 \beta_{12} + 30 \beta_{11} + 2 \beta_{10} + 17 \beta_{9} - 16 \beta_{8} - 16 \beta_{7} - 6 \beta_{6} - 15 \beta_{5} - 34 \beta_{4} + 2 \beta_{3} - 24 \beta_{2} + \beta_{1} + 6$$$$)/2$$ $$\nu^{11}$$ $$=$$ $$($$$$-19 \beta_{15} + 4 \beta_{14} + 5 \beta_{13} + 19 \beta_{12} + 12 \beta_{11} - 20 \beta_{10} - 16 \beta_{9} - 14 \beta_{8} + 39 \beta_{7} + 23 \beta_{6} + 9 \beta_{5} - 18 \beta_{4} - 3 \beta_{3} - 23 \beta_{2} - 40 \beta_{1} + 27$$$$)/2$$ $$\nu^{12}$$ $$=$$ $$($$$$-22 \beta_{15} - \beta_{14} + 8 \beta_{13} - 19 \beta_{12} + 36 \beta_{11} - 48 \beta_{10} + 57 \beta_{9} - 18 \beta_{8} + 29 \beta_{7} + \beta_{6} - 16 \beta_{4} - 5 \beta_{3} - 7 \beta_{2} - 31 \beta_{1} - 5$$$$)/2$$ $$\nu^{13}$$ $$=$$ $$($$$$-58 \beta_{15} - 34 \beta_{14} + 26 \beta_{13} + 75 \beta_{12} + 23 \beta_{11} + 38 \beta_{10} + 42 \beta_{9} - 63 \beta_{8} - 9 \beta_{7} - 49 \beta_{6} + 6 \beta_{5} - 40 \beta_{4} - 41 \beta_{3} + 19 \beta_{2} + 23 \beta_{1} + 53$$$$)/2$$ $$\nu^{14}$$ $$=$$ $$($$$$-101 \beta_{15} - 32 \beta_{14} + 97 \beta_{13} + 29 \beta_{12} + 60 \beta_{11} - 112 \beta_{10} + 32 \beta_{9} + 68 \beta_{8} + 85 \beta_{7} + 31 \beta_{6} + 23 \beta_{5} - 134 \beta_{4} - 51 \beta_{3} - 13 \beta_{2} - 48 \beta_{1} + 67$$$$)/2$$ $$\nu^{15}$$ $$=$$ $$($$$$69 \beta_{15} + 74 \beta_{14} - 51 \beta_{13} - 16 \beta_{12} - 11 \beta_{11} - 62 \beta_{10} + 2 \beta_{9} + 19 \beta_{8} + 40 \beta_{7} - 144 \beta_{6} + 69 \beta_{5} + 178 \beta_{4} - 56 \beta_{3} + 128 \beta_{2} + 43 \beta_{1} - 26$$$$)/2$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/720\mathbb{Z}\right)^\times$$.

 $$n$$ $$181$$ $$271$$ $$577$$ $$641$$ $$\chi(n)$$ $$\beta_{10}$$ $$1$$ $$1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
181.1
 1.38652 − 0.278517i −0.296075 − 1.38287i −0.966675 − 1.03225i −1.39563 + 0.228522i 1.32070 + 0.505727i 1.21331 − 0.726558i −0.530822 + 1.31081i 1.26868 − 0.624862i 1.38652 + 0.278517i −0.296075 + 1.38287i −0.966675 + 1.03225i −1.39563 − 0.228522i 1.32070 − 0.505727i 1.21331 + 0.726558i −0.530822 − 1.31081i 1.26868 + 0.624862i
−1.40727 + 0.139945i 0 1.96083 0.393883i −0.707107 0.707107i 0 0.982011i −2.70430 + 0.828709i 0 1.09405 + 0.896135i
181.2 −1.09971 0.889181i 0 0.418713 + 1.95568i 0.707107 + 0.707107i 0 2.66881i 1.27849 2.52299i 0 −0.148864 1.40636i
181.3 −0.562546 + 1.29751i 0 −1.36708 1.45982i −0.707107 0.707107i 0 1.73696i 2.66319 0.952595i 0 1.31526 0.519701i
181.4 −0.114638 1.40956i 0 −1.97372 + 0.323179i −0.707107 0.707107i 0 0.690576i 0.681804 + 2.74502i 0 −0.915648 + 1.07777i
181.5 0.257150 1.39064i 0 −1.86775 0.715205i 0.707107 + 0.707107i 0 2.89402i −1.47488 + 2.41345i 0 1.16516 0.801497i
181.6 0.376912 + 1.36306i 0 −1.71587 + 1.02751i 0.707107 + 0.707107i 0 4.50961i −2.04729 1.95156i 0 −0.697313 + 1.23035i
181.7 1.17275 0.790349i 0 0.750696 1.85377i 0.707107 + 0.707107i 0 2.73482i −0.584744 2.76732i 0 1.38812 + 0.270400i
181.8 1.37735 0.320793i 0 1.79418 0.883688i −0.707107 0.707107i 0 4.02840i 2.18774 1.79271i 0 −1.20077 0.747098i
541.1 −1.40727 0.139945i 0 1.96083 + 0.393883i −0.707107 + 0.707107i 0 0.982011i −2.70430 0.828709i 0 1.09405 0.896135i
541.2 −1.09971 + 0.889181i 0 0.418713 1.95568i 0.707107 0.707107i 0 2.66881i 1.27849 + 2.52299i 0 −0.148864 + 1.40636i
541.3 −0.562546 1.29751i 0 −1.36708 + 1.45982i −0.707107 + 0.707107i 0 1.73696i 2.66319 + 0.952595i 0 1.31526 + 0.519701i
541.4 −0.114638 + 1.40956i 0 −1.97372 0.323179i −0.707107 + 0.707107i 0 0.690576i 0.681804 2.74502i 0 −0.915648 1.07777i
541.5 0.257150 + 1.39064i 0 −1.86775 + 0.715205i 0.707107 0.707107i 0 2.89402i −1.47488 2.41345i 0 1.16516 + 0.801497i
541.6 0.376912 1.36306i 0 −1.71587 1.02751i 0.707107 0.707107i 0 4.50961i −2.04729 + 1.95156i 0 −0.697313 1.23035i
541.7 1.17275 + 0.790349i 0 0.750696 + 1.85377i 0.707107 0.707107i 0 2.73482i −0.584744 + 2.76732i 0 1.38812 0.270400i
541.8 1.37735 + 0.320793i 0 1.79418 + 0.883688i −0.707107 + 0.707107i 0 4.02840i 2.18774 + 1.79271i 0 −1.20077 + 0.747098i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 541.8 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
16.e even 4 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 720.2.t.c 16
3.b odd 2 1 80.2.l.a 16
4.b odd 2 1 2880.2.t.c 16
12.b even 2 1 320.2.l.a 16
15.d odd 2 1 400.2.l.h 16
15.e even 4 1 400.2.q.g 16
15.e even 4 1 400.2.q.h 16
16.e even 4 1 inner 720.2.t.c 16
16.f odd 4 1 2880.2.t.c 16
24.f even 2 1 640.2.l.a 16
24.h odd 2 1 640.2.l.b 16
48.i odd 4 1 80.2.l.a 16
48.i odd 4 1 640.2.l.b 16
48.k even 4 1 320.2.l.a 16
48.k even 4 1 640.2.l.a 16
60.h even 2 1 1600.2.l.i 16
60.l odd 4 1 1600.2.q.g 16
60.l odd 4 1 1600.2.q.h 16
96.o even 8 1 5120.2.a.t 8
96.o even 8 1 5120.2.a.u 8
96.p odd 8 1 5120.2.a.s 8
96.p odd 8 1 5120.2.a.v 8
240.t even 4 1 1600.2.l.i 16
240.z odd 4 1 1600.2.q.h 16
240.bb even 4 1 400.2.q.g 16
240.bd odd 4 1 1600.2.q.g 16
240.bf even 4 1 400.2.q.h 16
240.bm odd 4 1 400.2.l.h 16

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
80.2.l.a 16 3.b odd 2 1
80.2.l.a 16 48.i odd 4 1
320.2.l.a 16 12.b even 2 1
320.2.l.a 16 48.k even 4 1
400.2.l.h 16 15.d odd 2 1
400.2.l.h 16 240.bm odd 4 1
400.2.q.g 16 15.e even 4 1
400.2.q.g 16 240.bb even 4 1
400.2.q.h 16 15.e even 4 1
400.2.q.h 16 240.bf even 4 1
640.2.l.a 16 24.f even 2 1
640.2.l.a 16 48.k even 4 1
640.2.l.b 16 24.h odd 2 1
640.2.l.b 16 48.i odd 4 1
720.2.t.c 16 1.a even 1 1 trivial
720.2.t.c 16 16.e even 4 1 inner
1600.2.l.i 16 60.h even 2 1
1600.2.l.i 16 240.t even 4 1
1600.2.q.g 16 60.l odd 4 1
1600.2.q.g 16 240.bd odd 4 1
1600.2.q.h 16 60.l odd 4 1
1600.2.q.h 16 240.z odd 4 1
2880.2.t.c 16 4.b odd 2 1
2880.2.t.c 16 16.f odd 4 1
5120.2.a.s 8 96.p odd 8 1
5120.2.a.t 8 96.o even 8 1
5120.2.a.u 8 96.o even 8 1
5120.2.a.v 8 96.p odd 8 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{7}^{16} + \cdots$$ acting on $$S_{2}^{\mathrm{new}}(720, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$256 + 128 T^{2} - 32 T^{4} - 16 T^{6} + 16 T^{7} + 4 T^{8} + 8 T^{9} - 4 T^{10} - 2 T^{12} + 2 T^{14} + T^{16}$$
$3$ $$T^{16}$$
$5$ $$( 1 + T^{4} )^{4}$$
$7$ $$204304 + 811008 T^{2} + 1033536 T^{4} + 549632 T^{6} + 145224 T^{8} + 20736 T^{10} + 1616 T^{12} + 64 T^{14} + T^{16}$$
$11$ $$1290496 - 799744 T + 247808 T^{2} - 848384 T^{3} + 3958016 T^{4} - 3673856 T^{5} + 1795584 T^{6} - 446848 T^{7} + 139616 T^{8} - 82368 T^{9} + 40320 T^{10} - 9568 T^{11} + 1232 T^{12} - 80 T^{13} + 32 T^{14} - 8 T^{15} + T^{16}$$
$13$ $$20647936 + 46530560 T + 52428800 T^{2} + 27869184 T^{3} + 9146368 T^{4} + 3137536 T^{5} + 2654208 T^{6} + 1400832 T^{7} + 415872 T^{8} + 49152 T^{9} + 8192 T^{10} + 4352 T^{11} + 1600 T^{12} + 128 T^{13} + T^{16}$$
$17$ $$( 13888 + 5120 T - 7744 T^{2} - 1536 T^{3} + 1408 T^{4} + 64 T^{5} - 72 T^{6} + T^{8} )^{2}$$
$19$ $$614656 + 4164608 T + 14108672 T^{2} + 26513920 T^{3} + 30308608 T^{4} + 19398912 T^{5} + 7595520 T^{6} + 1921408 T^{7} + 731744 T^{8} + 363072 T^{9} + 132480 T^{10} + 27488 T^{11} + 3216 T^{12} + 176 T^{13} + 32 T^{14} + 8 T^{15} + T^{16}$$
$23$ $$1731856 + 5740288 T^{2} + 5719232 T^{4} + 2620928 T^{6} + 622088 T^{8} + 77504 T^{10} + 4784 T^{12} + 128 T^{14} + T^{16}$$
$29$ $$3017085184 + 4042700800 T + 2708480000 T^{2} + 456489984 T^{3} - 5714688 T^{4} - 1816064 T^{5} + 37230592 T^{6} + 2768128 T^{7} - 199840 T^{8} - 351616 T^{9} + 198144 T^{10} - 18624 T^{11} + 1104 T^{12} - 288 T^{13} + 128 T^{14} - 16 T^{15} + T^{16}$$
$31$ $$( -20224 + 58368 T - 26112 T^{2} - 4096 T^{3} + 2848 T^{4} + 64 T^{5} - 96 T^{6} + T^{8} )^{2}$$
$37$ $$18939904 + 236191744 T + 1472724992 T^{2} + 2707357696 T^{3} + 2705047552 T^{4} + 1380728832 T^{5} + 381124608 T^{6} + 40185856 T^{7} + 22554112 T^{8} + 9370624 T^{9} + 2144256 T^{10} + 249088 T^{11} + 16320 T^{12} + 704 T^{13} + 128 T^{14} + 16 T^{15} + T^{16}$$
$41$ $$110660014336 + 325786435584 T^{2} + 63304144128 T^{4} + 5024061440 T^{6} + 206041952 T^{8} + 4686848 T^{10} + 59088 T^{12} + 384 T^{14} + T^{16}$$
$43$ $$53640976 + 331044800 T + 1021520000 T^{2} + 1500385792 T^{3} + 1191507520 T^{4} + 280880992 T^{5} + 26364032 T^{6} - 1854880 T^{7} + 5444360 T^{8} + 950192 T^{9} + 76832 T^{10} - 18816 T^{11} + 5328 T^{12} + 440 T^{13} + 32 T^{14} - 8 T^{15} + T^{16}$$
$47$ $$( 575044 - 693376 T + 244768 T^{2} - 584 T^{3} - 14936 T^{4} + 2392 T^{5} - 8 T^{6} - 20 T^{7} + T^{8} )^{2}$$
$53$ $$383725735936 + 196640112640 T + 50384076800 T^{2} + 10552958976 T^{3} + 26109784064 T^{4} + 13942091776 T^{5} + 3861454848 T^{6} + 498471936 T^{7} + 43316352 T^{8} + 6504448 T^{9} + 1892352 T^{10} + 236416 T^{11} + 15552 T^{12} + 448 T^{13} + 128 T^{14} + 16 T^{15} + T^{16}$$
$59$ $$12227051776 + 30451745792 T + 37920376832 T^{2} + 27384356352 T^{3} + 12226302208 T^{4} + 3038184192 T^{5} + 314344960 T^{6} - 20658560 T^{7} + 7403616 T^{8} + 2562240 T^{9} + 291200 T^{10} - 65632 T^{11} + 8464 T^{12} + 272 T^{13} + 32 T^{14} - 8 T^{15} + T^{16}$$
$61$ $$1393986371584 + 1092943347712 T + 428456542208 T^{2} - 95024578560 T^{3} + 27549499392 T^{4} + 9643622400 T^{5} + 2332164096 T^{6} - 240730112 T^{7} + 17876992 T^{8} + 4136960 T^{9} + 1425408 T^{10} - 176640 T^{11} + 11520 T^{12} + 384 T^{13} + 128 T^{14} - 16 T^{15} + T^{16}$$
$67$ $$46120451769616 - 26534918246592 T + 7633293466752 T^{2} - 2087109786752 T^{3} + 1043974444608 T^{4} - 475604544352 T^{5} + 148072661120 T^{6} - 31795927072 T^{7} + 4867387016 T^{8} - 537576688 T^{9} + 43098400 T^{10} - 2604960 T^{11} + 147664 T^{12} - 10680 T^{13} + 800 T^{14} - 40 T^{15} + T^{16}$$
$71$ $$3333516427264 + 2007385505792 T^{2} + 408856297472 T^{4} + 33695596544 T^{6} + 1144522752 T^{8} + 18817024 T^{10} + 157440 T^{12} + 640 T^{14} + T^{16}$$
$73$ $$15847788544 + 28362989568 T^{2} + 17757564928 T^{4} + 4377603072 T^{6} + 321195136 T^{8} + 9035648 T^{10} + 108992 T^{12} + 560 T^{14} + T^{16}$$
$79$ $$( 4352 + 31232 T - 61952 T^{2} - 4992 T^{3} + 5856 T^{4} + 352 T^{5} - 160 T^{6} - 8 T^{7} + T^{8} )^{2}$$
$83$ $$2050640656 - 6041972416 T + 8900981888 T^{2} - 5315507456 T^{3} + 1230069056 T^{4} + 140621856 T^{5} + 1135671424 T^{6} - 379938272 T^{7} + 64272392 T^{8} + 9891472 T^{9} + 315168 T^{10} - 36928 T^{11} + 37520 T^{12} + 7976 T^{13} + 800 T^{14} + 40 T^{15} + T^{16}$$
$89$ $$684153962496 + 380947267584 T^{2} + 69045698560 T^{4} + 5734359040 T^{6} + 244188672 T^{8} + 5576192 T^{10} + 68032 T^{12} + 416 T^{14} + T^{16}$$
$97$ $$( -8549312 + 7621376 T - 1675968 T^{2} - 78720 T^{3} + 47936 T^{4} - 416 T^{5} - 440 T^{6} + T^{8} )^{2}$$