# Properties

 Label 720.2.z.g Level $720$ Weight $2$ Character orbit 720.z Analytic conductor $5.749$ Analytic rank $0$ Dimension $18$ 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.z (of order $$4$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$5.74922894553$$ Analytic rank: $$0$$ Dimension: $$18$$ Relative dimension: $$9$$ over $$\Q(i)$$ Coefficient field: $$\mathbb{Q}[x]/(x^{18} + \cdots)$$ Defining polynomial: $$x^{18} + 2 x^{16} - 4 x^{15} - 5 x^{14} - 14 x^{13} - 10 x^{12} + 6 x^{11} + 37 x^{10} + 70 x^{9} + 74 x^{8} + 24 x^{7} - 80 x^{6} - 224 x^{5} - 160 x^{4} - 256 x^{3} + 256 x^{2} + 512$$ Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$2^{6}$$ 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 $$1,\beta_1,\ldots,\beta_{17}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q -\beta_{1} q^{2} + \beta_{13} q^{4} -\beta_{17} q^{5} + ( -\beta_{1} - \beta_{2} - \beta_{7} + \beta_{10} - \beta_{11} - \beta_{12} + \beta_{16} ) q^{7} + ( 1 - \beta_{1} + \beta_{2} + \beta_{3} - \beta_{4} + \beta_{5} + \beta_{6} - \beta_{8} + \beta_{9} - \beta_{10} + \beta_{12} - \beta_{13} + \beta_{15} - \beta_{16} ) q^{8} +O(q^{10})$$ $$q -\beta_{1} q^{2} + \beta_{13} q^{4} -\beta_{17} q^{5} + ( -\beta_{1} - \beta_{2} - \beta_{7} + \beta_{10} - \beta_{11} - \beta_{12} + \beta_{16} ) q^{7} + ( 1 - \beta_{1} + \beta_{2} + \beta_{3} - \beta_{4} + \beta_{5} + \beta_{6} - \beta_{8} + \beta_{9} - \beta_{10} + \beta_{12} - \beta_{13} + \beta_{15} - \beta_{16} ) q^{8} + ( -\beta_{4} + \beta_{6} + \beta_{11} - \beta_{12} - \beta_{13} - \beta_{14} ) q^{10} + ( -1 + \beta_{1} - \beta_{2} - \beta_{3} + \beta_{4} - \beta_{5} - \beta_{6} + \beta_{11} + \beta_{13} - \beta_{15} ) q^{11} + ( \beta_{1} - \beta_{2} - \beta_{4} - \beta_{5} - \beta_{6} - \beta_{8} + \beta_{10} + \beta_{11} + \beta_{12} - \beta_{13} + 2 \beta_{15} - \beta_{16} + \beta_{17} ) q^{13} + ( 1 + \beta_{1} - \beta_{4} - \beta_{5} - \beta_{7} + 2 \beta_{10} + \beta_{15} + \beta_{16} + \beta_{17} ) q^{14} + ( \beta_{4} - \beta_{8} + \beta_{13} + \beta_{14} - \beta_{15} + \beta_{16} ) q^{16} + ( 1 + \beta_{3} - \beta_{4} + 2 \beta_{6} - \beta_{7} + \beta_{10} - 2 \beta_{12} + \beta_{16} ) q^{17} + ( -1 + \beta_{2} + \beta_{4} + \beta_{6} + \beta_{8} + \beta_{9} - \beta_{10} + \beta_{13} - \beta_{14} - \beta_{15} - \beta_{17} ) q^{19} + ( 1 - \beta_{1} - 2 \beta_{2} - \beta_{6} - \beta_{7} - \beta_{9} - \beta_{11} - 2 \beta_{12} + \beta_{14} ) q^{20} + ( -\beta_{1} - \beta_{4} + 2 \beta_{5} + \beta_{6} + \beta_{8} - \beta_{13} - \beta_{14} - \beta_{15} - \beta_{16} ) q^{22} + ( -\beta_{1} - \beta_{3} - \beta_{6} + \beta_{8} + \beta_{9} - \beta_{10} - \beta_{11} + \beta_{13} + \beta_{14} + 2 \beta_{15} + \beta_{17} ) q^{23} + ( -1 + \beta_{1} - \beta_{2} - \beta_{3} + \beta_{5} - \beta_{6} + \beta_{7} + \beta_{12} + \beta_{13} + \beta_{14} + \beta_{17} ) q^{25} + ( 2 - \beta_{4} + \beta_{5} + \beta_{6} - \beta_{9} + \beta_{10} - \beta_{11} - \beta_{13} - \beta_{15} + 2 \beta_{16} - \beta_{17} ) q^{26} + ( 3 - \beta_{1} - \beta_{2} - \beta_{3} + \beta_{10} - 2 \beta_{11} + \beta_{14} + \beta_{15} + \beta_{16} + \beta_{17} ) q^{28} + ( -1 - \beta_{10} + 2 \beta_{12} ) q^{29} + ( 2 \beta_{1} + \beta_{2} + \beta_{6} - \beta_{7} + \beta_{8} - \beta_{9} - \beta_{12} + \beta_{13} - \beta_{14} - 2 \beta_{15} + 2 \beta_{16} - \beta_{17} ) q^{31} + ( -2 + \beta_{2} - \beta_{3} + \beta_{4} + 2 \beta_{5} + \beta_{6} + \beta_{8} + \beta_{9} - 2 \beta_{10} + \beta_{12} + \beta_{13} + \beta_{14} - \beta_{16} + \beta_{17} ) q^{32} + ( 2 - \beta_{1} - \beta_{2} - \beta_{3} - 2 \beta_{7} + \beta_{9} - 2 \beta_{11} - \beta_{12} + 2 \beta_{14} + \beta_{15} + \beta_{17} ) q^{34} + ( -1 + \beta_{1} + \beta_{4} + \beta_{5} + \beta_{7} + \beta_{8} - \beta_{9} + \beta_{11} - \beta_{12} - \beta_{14} - 3 \beta_{15} - \beta_{17} ) q^{35} + ( 3 \beta_{1} + 2 \beta_{2} - \beta_{5} + \beta_{9} - \beta_{10} + \beta_{11} + 2 \beta_{12} + 2 \beta_{13} + \beta_{14} - \beta_{16} ) q^{37} + ( -2 - \beta_{1} - \beta_{2} + \beta_{3} - \beta_{4} - \beta_{8} + \beta_{9} + 2 \beta_{10} - \beta_{12} - \beta_{13} - \beta_{14} - \beta_{16} + \beta_{17} ) q^{38} + ( 3 + \beta_{1} + 2 \beta_{2} - \beta_{4} - \beta_{5} - \beta_{7} + \beta_{8} + \beta_{11} + \beta_{12} - \beta_{17} ) q^{40} + ( 3 \beta_{1} + \beta_{2} - \beta_{4} - \beta_{5} - \beta_{6} - \beta_{8} + \beta_{10} + \beta_{11} + \beta_{12} + \beta_{13} - \beta_{16} + \beta_{17} ) q^{41} + ( \beta_{1} + \beta_{2} + \beta_{5} + \beta_{6} - \beta_{7} + 2 \beta_{9} + \beta_{10} - \beta_{11} + \beta_{12} + \beta_{13} + 2 \beta_{14} ) q^{43} + ( 2 - \beta_{2} + \beta_{3} + \beta_{4} - 2 \beta_{5} - \beta_{6} - \beta_{7} - \beta_{8} - \beta_{9} + \beta_{10} + \beta_{11} - \beta_{14} - \beta_{17} ) q^{44} + ( 1 + \beta_{1} - 2 \beta_{3} - \beta_{4} - \beta_{5} - 2 \beta_{6} + \beta_{7} - 2 \beta_{8} + 2 \beta_{12} + \beta_{15} - \beta_{16} + \beta_{17} ) q^{46} + ( -2 - \beta_{1} + 2 \beta_{2} + \beta_{6} + \beta_{8} + \beta_{9} + \beta_{10} - \beta_{11} + \beta_{13} - \beta_{14} + \beta_{16} - \beta_{17} ) q^{47} + ( 2 \beta_{1} - \beta_{2} - \beta_{4} + \beta_{6} - 2 \beta_{7} - \beta_{9} + \beta_{10} - \beta_{12} - \beta_{13} - \beta_{14} + 2 \beta_{16} ) q^{49} + ( \beta_{1} + \beta_{2} - \beta_{3} + \beta_{5} - \beta_{8} - 3 \beta_{10} + \beta_{11} + 3 \beta_{12} - 2 \beta_{13} - \beta_{14} + \beta_{15} - \beta_{16} ) q^{50} + ( -\beta_{2} - \beta_{3} + \beta_{4} - 2 \beta_{5} - \beta_{6} - \beta_{7} + \beta_{8} - \beta_{9} - \beta_{10} + \beta_{11} - 2 \beta_{12} - \beta_{14} + 2 \beta_{15} + 2 \beta_{16} + \beta_{17} ) q^{52} + ( -2 + \beta_{1} - \beta_{2} - 2 \beta_{3} + \beta_{4} + \beta_{5} + \beta_{6} + 2 \beta_{7} + \beta_{8} - \beta_{10} + \beta_{11} + \beta_{12} + \beta_{13} - 2 \beta_{15} - \beta_{16} + \beta_{17} ) q^{53} + ( -1 + \beta_{1} - 2 \beta_{2} - 2 \beta_{3} + 2 \beta_{4} - \beta_{5} - 2 \beta_{6} + 2 \beta_{7} - \beta_{9} - \beta_{10} - \beta_{12} + 2 \beta_{13} + \beta_{15} - \beta_{16} + \beta_{17} ) q^{55} + ( \beta_{2} - \beta_{3} - 2 \beta_{5} - \beta_{6} - \beta_{7} - \beta_{9} + \beta_{10} - \beta_{11} + \beta_{13} + \beta_{15} + 3 \beta_{16} + \beta_{17} ) q^{56} + ( \beta_{1} - \beta_{6} + 2 \beta_{10} - 2 \beta_{14} + 2 \beta_{16} ) q^{58} + ( 1 + 2 \beta_{1} - \beta_{2} - \beta_{4} + \beta_{6} + \beta_{8} - \beta_{9} + \beta_{10} - 2 \beta_{11} - 2 \beta_{12} + \beta_{13} + \beta_{14} - \beta_{15} + 2 \beta_{16} - \beta_{17} ) q^{59} + ( 1 - 4 \beta_{1} + 2 \beta_{3} - 2 \beta_{4} + 2 \beta_{7} - 2 \beta_{10} + \beta_{11} - \beta_{14} - 2 \beta_{16} - \beta_{17} ) q^{61} + ( 2 - 2 \beta_{1} - \beta_{2} + \beta_{3} - \beta_{6} + \beta_{9} - \beta_{12} - 2 \beta_{13} + 3 \beta_{15} - 2 \beta_{16} + \beta_{17} ) q^{62} + ( 2 \beta_{1} - 2 \beta_{3} + 2 \beta_{4} - 2 \beta_{6} - 2 \beta_{8} + 2 \beta_{10} + 2 \beta_{12} - 2 \beta_{16} ) q^{64} + ( -2 + 3 \beta_{1} - \beta_{2} + \beta_{4} - 3 \beta_{5} - 5 \beta_{6} + 2 \beta_{11} + \beta_{12} + \beta_{13} - \beta_{14} + 2 \beta_{15} - \beta_{16} + \beta_{17} ) q^{65} + ( -3 \beta_{1} - \beta_{2} + \beta_{5} - \beta_{6} + \beta_{7} + \beta_{10} - \beta_{11} - 3 \beta_{12} - \beta_{13} - 2 \beta_{15} ) q^{67} + ( 4 + 2 \beta_{1} + \beta_{2} - 3 \beta_{3} + \beta_{4} - 2 \beta_{5} - \beta_{6} - \beta_{7} + \beta_{8} - \beta_{9} + 3 \beta_{10} - \beta_{11} + 2 \beta_{13} + \beta_{14} - 2 \beta_{15} + 2 \beta_{16} + \beta_{17} ) q^{68} + ( -3 - \beta_{1} - \beta_{2} + \beta_{3} + \beta_{4} + \beta_{8} - 2 \beta_{10} + 2 \beta_{11} - \beta_{13} - \beta_{14} - 3 \beta_{16} - \beta_{17} ) q^{70} + ( -2 - \beta_{2} + \beta_{6} + \beta_{7} - \beta_{8} + \beta_{9} - \beta_{12} + \beta_{13} - \beta_{14} - \beta_{17} ) q^{71} + ( -1 - \beta_{3} + \beta_{4} - 2 \beta_{6} + \beta_{7} - \beta_{10} - 2 \beta_{12} - \beta_{16} ) q^{73} + ( 2 - 2 \beta_{1} + \beta_{2} - \beta_{3} + \beta_{4} + 3 \beta_{5} + 2 \beta_{6} + 2 \beta_{7} + 2 \beta_{9} - 3 \beta_{10} - \beta_{11} + \beta_{12} - \beta_{13} - 2 \beta_{15} - 2 \beta_{16} ) q^{74} + ( 2 \beta_{1} + 2 \beta_{3} + 2 \beta_{6} - \beta_{7} - \beta_{10} - \beta_{11} + \beta_{12} + \beta_{13} + 2 \beta_{14} - 2 \beta_{15} + \beta_{16} ) q^{76} + ( -\beta_{2} + 2 \beta_{3} + \beta_{4} + 2 \beta_{5} + \beta_{6} + 2 \beta_{7} - \beta_{8} + \beta_{9} - 2 \beta_{10} + 2 \beta_{11} + \beta_{12} + \beta_{13} - \beta_{14} - 2 \beta_{15} - 2 \beta_{16} - \beta_{17} ) q^{77} + ( 4 - \beta_{1} + 2 \beta_{2} + 2 \beta_{3} - 4 \beta_{4} - \beta_{5} + 4 \beta_{6} - \beta_{8} - \beta_{9} + \beta_{10} - \beta_{11} - 2 \beta_{13} + \beta_{14} + 2 \beta_{15} + \beta_{16} - \beta_{17} ) q^{79} + ( 4 - 4 \beta_{1} - 2 \beta_{2} - 2 \beta_{3} + \beta_{4} + 2 \beta_{7} + \beta_{8} - 2 \beta_{12} + \beta_{13} - \beta_{14} - \beta_{15} - \beta_{16} ) q^{80} + ( 4 - 2 \beta_{1} + 2 \beta_{2} + 2 \beta_{3} + \beta_{4} + 3 \beta_{5} + 3 \beta_{6} + 2 \beta_{7} + \beta_{9} - 5 \beta_{10} - \beta_{11} + 2 \beta_{12} - \beta_{13} + 2 \beta_{14} - \beta_{15} - 2 \beta_{16} - \beta_{17} ) q^{82} + ( -2 + 3 \beta_{1} - \beta_{2} - 3 \beta_{3} + 2 \beta_{4} - 3 \beta_{5} - 3 \beta_{6} - \beta_{7} + 3 \beta_{10} - 3 \beta_{11} - \beta_{12} + \beta_{13} + 3 \beta_{16} ) q^{83} + ( -1 - 2 \beta_{1} - \beta_{2} - 2 \beta_{3} + 3 \beta_{4} + 2 \beta_{5} - \beta_{6} - \beta_{9} - 2 \beta_{10} + \beta_{11} + \beta_{12} - \beta_{13} - 2 \beta_{15} - 2 \beta_{16} - \beta_{17} ) q^{85} + ( 1 + 2 \beta_{1} - 4 \beta_{3} + 2 \beta_{4} + 2 \beta_{6} + \beta_{9} + \beta_{12} - 2 \beta_{15} + 2 \beta_{17} ) q^{86} + ( -4 \beta_{1} + \beta_{2} + 3 \beta_{3} - 3 \beta_{4} + 2 \beta_{5} + 3 \beta_{6} + \beta_{8} + \beta_{9} + 4 \beta_{10} - \beta_{12} - \beta_{13} - \beta_{14} + \beta_{16} - \beta_{17} ) q^{88} + ( 2 + 3 \beta_{1} + 2 \beta_{2} + 4 \beta_{3} + \beta_{5} + 2 \beta_{6} + 2 \beta_{7} - \beta_{8} - \beta_{9} - \beta_{10} + \beta_{11} - 2 \beta_{13} + \beta_{14} + 2 \beta_{15} - \beta_{16} - \beta_{17} ) q^{89} + ( 3 \beta_{1} - \beta_{3} + 2 \beta_{4} - \beta_{5} + 2 \beta_{6} - \beta_{8} - \beta_{9} + \beta_{10} + \beta_{11} - \beta_{14} - 2 \beta_{15} + 2 \beta_{16} - \beta_{17} ) q^{91} + ( 3 - \beta_{1} + 3 \beta_{2} + 3 \beta_{3} + 2 \beta_{5} + 2 \beta_{6} + 2 \beta_{7} - 2 \beta_{9} - 5 \beta_{10} + \beta_{14} - \beta_{15} + \beta_{16} - \beta_{17} ) q^{92} + ( -1 + \beta_{1} - \beta_{2} + \beta_{3} - \beta_{4} - \beta_{5} + \beta_{6} + \beta_{7} - 2 \beta_{8} + \beta_{9} - \beta_{12} + 2 \beta_{15} - \beta_{16} + 2 \beta_{17} ) q^{94} + ( -1 + 3 \beta_{1} + \beta_{2} - 2 \beta_{3} + 2 \beta_{4} + \beta_{5} + 3 \beta_{6} - \beta_{7} + \beta_{8} - 3 \beta_{10} - 2 \beta_{11} + 3 \beta_{13} + 3 \beta_{14} - 3 \beta_{15} + \beta_{16} ) q^{95} + ( 1 + \beta_{1} + 2 \beta_{2} + 3 \beta_{3} - \beta_{4} - \beta_{5} + \beta_{7} - \beta_{8} + \beta_{9} + 3 \beta_{11} + 2 \beta_{12} - 2 \beta_{13} - \beta_{14} + 2 \beta_{15} + \beta_{17} ) q^{97} + ( 4 - 2 \beta_{2} - \beta_{4} - \beta_{5} - 2 \beta_{6} - 2 \beta_{7} - \beta_{9} + 5 \beta_{10} - \beta_{11} - 2 \beta_{12} - 3 \beta_{13} + 3 \beta_{15} + 2 \beta_{16} + \beta_{17} ) q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$18q + 4q^{4} - 2q^{5} + 2q^{7} + 12q^{8} + O(q^{10})$$ $$18q + 4q^{4} - 2q^{5} + 2q^{7} + 12q^{8} + 2q^{11} + 12q^{14} + 6q^{17} - 2q^{19} + 12q^{20} + 12q^{22} + 2q^{23} - 6q^{25} + 16q^{26} + 40q^{28} - 14q^{29} - 20q^{32} + 28q^{34} - 2q^{35} - 24q^{38} + 44q^{40} + 44q^{44} + 12q^{46} - 38q^{47} + 8q^{50} + 8q^{52} - 12q^{53} - 6q^{55} - 20q^{56} + 20q^{58} - 10q^{59} + 14q^{61} + 40q^{62} + 16q^{64} + 60q^{68} - 28q^{70} - 24q^{71} - 14q^{73} + 48q^{74} - 16q^{76} + 44q^{77} - 16q^{79} + 92q^{80} + 48q^{82} - 40q^{83} + 14q^{85} + 36q^{86} - 8q^{88} - 12q^{89} + 8q^{92} - 28q^{94} - 34q^{95} + 18q^{97} + 56q^{98} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{18} + 2 x^{16} - 4 x^{15} - 5 x^{14} - 14 x^{13} - 10 x^{12} + 6 x^{11} + 37 x^{10} + 70 x^{9} + 74 x^{8} + 24 x^{7} - 80 x^{6} - 224 x^{5} - 160 x^{4} - 256 x^{3} + 256 x^{2} + 512$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$129 \nu^{17} + 124 \nu^{16} + 398 \nu^{15} - 116 \nu^{14} - 797 \nu^{13} - 2778 \nu^{12} - 4526 \nu^{11} - 4402 \nu^{10} - 123 \nu^{9} + 9450 \nu^{8} + 21502 \nu^{7} + 29424 \nu^{6} + 25048 \nu^{5} - 368 \nu^{4} - 22176 \nu^{3} - 64960 \nu^{2} - 44800 \nu - 58624$$$$)/1280$$ $$\beta_{2}$$ $$=$$ $$($$$$-16 \nu^{17} - 19 \nu^{16} - 53 \nu^{15} + 2 \nu^{14} + 88 \nu^{13} + 331 \nu^{12} + 559 \nu^{11} + 580 \nu^{10} + 98 \nu^{9} - 1033 \nu^{8} - 2471 \nu^{7} - 3408 \nu^{6} - 2972 \nu^{5} - 44 \nu^{4} + 2444 \nu^{3} + 7152 \nu^{2} + 4832 \nu + 6048$$$$)/160$$ $$\beta_{3}$$ $$=$$ $$($$$$-75 \nu^{17} - 89 \nu^{16} - 248 \nu^{15} - 6 \nu^{14} + 375 \nu^{13} + 1487 \nu^{12} + 2550 \nu^{11} + 2676 \nu^{10} + 583 \nu^{9} - 4379 \nu^{8} - 10894 \nu^{7} - 15406 \nu^{6} - 13500 \nu^{5} - 848 \nu^{4} + 9920 \nu^{3} + 31296 \nu^{2} + 21696 \nu + 28416$$$$)/640$$ $$\beta_{4}$$ $$=$$ $$($$$$-229 \nu^{17} - 258 \nu^{16} - 706 \nu^{15} + 120 \nu^{14} + 1377 \nu^{13} + 4800 \nu^{12} + 7846 \nu^{11} + 7678 \nu^{10} + 331 \nu^{9} - 15784 \nu^{8} - 35846 \nu^{7} - 48500 \nu^{6} - 40528 \nu^{5} + 1200 \nu^{4} + 37376 \nu^{3} + 102976 \nu^{2} + 71296 \nu + 89600$$$$)/1280$$ $$\beta_{5}$$ $$=$$ $$($$$$-178 \nu^{17} - 217 \nu^{16} - 614 \nu^{15} + 6 \nu^{14} + 954 \nu^{13} + 3793 \nu^{12} + 6492 \nu^{11} + 6810 \nu^{10} + 1284 \nu^{9} - 11749 \nu^{8} - 28628 \nu^{7} - 40294 \nu^{6} - 35116 \nu^{5} - 1872 \nu^{4} + 28832 \nu^{3} + 83776 \nu^{2} + 58816 \nu + 75264$$$$)/640$$ $$\beta_{6}$$ $$=$$ $$($$$$545 \nu^{17} + 684 \nu^{16} + 1918 \nu^{15} + 156 \nu^{14} - 2685 \nu^{13} - 11242 \nu^{12} - 19710 \nu^{11} - 21346 \nu^{10} - 5723 \nu^{9} + 32794 \nu^{8} + 84014 \nu^{7} + 120736 \nu^{6} + 108280 \nu^{5} + 10928 \nu^{4} - 79200 \nu^{3} - 245696 \nu^{2} - 175616 \nu - 222976$$$$)/1280$$ $$\beta_{7}$$ $$=$$ $$($$$$871 \nu^{17} + 1090 \nu^{16} + 3110 \nu^{15} + 352 \nu^{14} - 4043 \nu^{13} - 17564 \nu^{12} - 31194 \nu^{11} - 34194 \nu^{10} - 10465 \nu^{9} + 49524 \nu^{8} + 130042 \nu^{7} + 188932 \nu^{6} + 171792 \nu^{5} + 21456 \nu^{4} - 117504 \nu^{3} - 381376 \nu^{2} - 268416 \nu - 351232$$$$)/1280$$ $$\beta_{8}$$ $$=$$ $$($$$$228 \nu^{17} + 359 \nu^{16} + 1013 \nu^{15} + 672 \nu^{14} - 134 \nu^{13} - 3459 \nu^{12} - 7827 \nu^{11} - 11038 \nu^{10} - 8848 \nu^{9} + 2381 \nu^{8} + 21455 \nu^{7} + 40462 \nu^{6} + 46806 \nu^{5} + 23236 \nu^{4} - 752 \nu^{3} - 61744 \nu^{2} - 43104 \nu - 71232$$$$)/320$$ $$\beta_{9}$$ $$=$$ $$($$$$-1017 \nu^{17} - 1536 \nu^{16} - 4342 \nu^{15} - 2508 \nu^{14} + 1381 \nu^{13} + 16526 \nu^{12} + 35438 \nu^{11} + 47762 \nu^{10} + 34427 \nu^{9} - 20334 \nu^{8} - 108430 \nu^{7} - 191488 \nu^{6} - 210344 \nu^{5} - 89664 \nu^{4} + 31808 \nu^{3} + 315776 \nu^{2} + 226816 \nu + 345088$$$$)/1280$$ $$\beta_{10}$$ $$=$$ $$($$$$129 \nu^{17} + 186 \nu^{16} + 524 \nu^{15} + 232 \nu^{14} - 321 \nu^{13} - 2280 \nu^{12} - 4568 \nu^{11} - 5762 \nu^{10} - 3435 \nu^{9} + 4196 \nu^{8} + 15672 \nu^{7} + 25600 \nu^{6} + 26316 \nu^{5} + 8624 \nu^{4} - 8592 \nu^{3} - 45504 \nu^{2} - 32320 \nu - 46208$$$$)/128$$ $$\beta_{11}$$ $$=$$ $$($$$$-799 \nu^{17} - 1073 \nu^{16} - 3066 \nu^{15} - 930 \nu^{14} + 2727 \nu^{13} + 14975 \nu^{12} + 28416 \nu^{11} + 33828 \nu^{10} + 16191 \nu^{9} - 34559 \nu^{8} - 107056 \nu^{7} - 165670 \nu^{6} - 161448 \nu^{5} - 39080 \nu^{4} + 77216 \nu^{3} + 313696 \nu^{2} + 222336 \nu + 305280$$$$)/640$$ $$\beta_{12}$$ $$=$$ $$($$$$1799 \nu^{17} + 2608 \nu^{16} + 7346 \nu^{15} + 3380 \nu^{14} - 4267 \nu^{13} - 31490 \nu^{12} - 63546 \nu^{11} - 80798 \nu^{10} - 49301 \nu^{9} + 56114 \nu^{8} + 215386 \nu^{7} + 354480 \nu^{6} + 366728 \nu^{5} + 123840 \nu^{4} - 112896 \nu^{3} - 624896 \nu^{2} - 443136 \nu - 636160$$$$)/1280$$ $$\beta_{13}$$ $$=$$ $$($$$$93 \nu^{17} + 133 \nu^{16} + 374 \nu^{15} + 162 \nu^{14} - 237 \nu^{13} - 1639 \nu^{12} - 3268 \nu^{11} - 4104 \nu^{10} - 2417 \nu^{9} + 3063 \nu^{8} + 11252 \nu^{7} + 18318 \nu^{6} + 18760 \nu^{5} + 6024 \nu^{4} - 6240 \nu^{3} - 32672 \nu^{2} - 23168 \nu - 33024$$$$)/64$$ $$\beta_{14}$$ $$=$$ $$($$$$-981 \nu^{17} - 1344 \nu^{16} - 3828 \nu^{15} - 1348 \nu^{14} + 3013 \nu^{13} + 17886 \nu^{12} + 34564 \nu^{11} + 41990 \nu^{10} + 21883 \nu^{9} - 38378 \nu^{8} - 125876 \nu^{7} - 198348 \nu^{6} - 196852 \nu^{5} - 53344 \nu^{4} + 82784 \nu^{3} + 368512 \nu^{2} + 259392 \nu + 363008$$$$)/640$$ $$\beta_{15}$$ $$=$$ $$($$$$1999 \nu^{17} + 2830 \nu^{16} + 8030 \nu^{15} + 3408 \nu^{14} - 5187 \nu^{13} - 35416 \nu^{12} - 70426 \nu^{11} - 88226 \nu^{10} - 51425 \nu^{9} + 67296 \nu^{8} + 244618 \nu^{7} + 397028 \nu^{6} + 405248 \nu^{5} + 128064 \nu^{4} - 138656 \nu^{3} - 713344 \nu^{2} - 504704 \nu - 718848$$$$)/1280$$ $$\beta_{16}$$ $$=$$ $$($$$$-1039 \nu^{17} - 1484 \nu^{16} - 4198 \nu^{15} - 1844 \nu^{14} + 2547 \nu^{13} + 18238 \nu^{12} + 36566 \nu^{11} + 46182 \nu^{10} + 27653 \nu^{9} - 33350 \nu^{8} - 125222 \nu^{7} - 205144 \nu^{6} - 211368 \nu^{5} - 69792 \nu^{4} + 67376 \nu^{3} + 364800 \nu^{2} + 258560 \nu + 372864$$$$)/640$$ $$\beta_{17}$$ $$=$$ $$($$$$-593 \nu^{17} - 812 \nu^{16} - 2294 \nu^{15} - 774 \nu^{14} + 1929 \nu^{13} + 10958 \nu^{12} + 20982 \nu^{11} + 25220 \nu^{10} + 12619 \nu^{9} - 24294 \nu^{8} - 77418 \nu^{7} - 120714 \nu^{6} - 118616 \nu^{5} - 30132 \nu^{4} + 53272 \nu^{3} + 225296 \nu^{2} + 158976 \nu + 218944$$$$)/320$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$-\beta_{16} - \beta_{13} + \beta_{12} + \beta_{11} - \beta_{10} + \beta_{7}$$$$)/2$$ $$\nu^{2}$$ $$=$$ $$($$$$\beta_{16} - \beta_{15} + \beta_{14} + \beta_{13} - 2 \beta_{11} + \beta_{8} + \beta_{4} - 2 \beta_{3}$$$$)/2$$ $$\nu^{3}$$ $$=$$ $$($$$$-\beta_{15} - \beta_{14} + \beta_{12} + \beta_{11} + \beta_{10} - \beta_{8} - \beta_{7} + \beta_{4} - 2 \beta_{3} + 2 \beta_{1}$$$$)/2$$ $$\nu^{4}$$ $$=$$ $$($$$$\beta_{17} - \beta_{14} - 2 \beta_{13} + 2 \beta_{12} + \beta_{11} + \beta_{10} - \beta_{9} - \beta_{8} + \beta_{7} - \beta_{6} - 3 \beta_{4} + \beta_{3} + \beta_{2} + 4$$$$)/2$$ $$\nu^{5}$$ $$=$$ $$($$$$-2 \beta_{17} + \beta_{16} - 5 \beta_{15} + \beta_{14} + \beta_{13} - 2 \beta_{9} + 3 \beta_{8} + 2 \beta_{7} + 2 \beta_{6} + 3 \beta_{4} + 4 \beta_{1} + 4$$$$)/2$$ $$\nu^{6}$$ $$=$$ $$($$$$\beta_{17} - 3 \beta_{16} + \beta_{15} + 2 \beta_{14} + \beta_{13} + 4 \beta_{12} + \beta_{11} - 9 \beta_{10} + 3 \beta_{9} + 2 \beta_{8} + 3 \beta_{7} + 3 \beta_{6} + 4 \beta_{4} - 5 \beta_{3} + 5 \beta_{2} + 4$$$$)/2$$ $$\nu^{7}$$ $$=$$ $$($$$$2 \beta_{17} + \beta_{16} - 2 \beta_{14} - 7 \beta_{13} - \beta_{12} - 3 \beta_{11} + 11 \beta_{10} - 2 \beta_{9} - 2 \beta_{8} - \beta_{7} - 2 \beta_{6} + 4 \beta_{5} - 2 \beta_{4} - 8 \beta_{3} - 8 \beta_{2} - 6 \beta_{1}$$$$)/2$$ $$\nu^{8}$$ $$=$$ $$($$$$3 \beta_{16} - 11 \beta_{15} - \beta_{14} + 7 \beta_{13} + 12 \beta_{12} - 8 \beta_{10} - 4 \beta_{9} - 3 \beta_{8} + 2 \beta_{7} - 6 \beta_{6} - 4 \beta_{5} + 5 \beta_{4} - 4 \beta_{3} + 10 \beta_{1}$$$$)/2$$ $$\nu^{9}$$ $$=$$ $$($$$$-8 \beta_{17} + 4 \beta_{16} - 13 \beta_{15} - 7 \beta_{14} - 6 \beta_{13} - \beta_{12} + 3 \beta_{11} + 7 \beta_{10} - 2 \beta_{9} + 9 \beta_{8} - 7 \beta_{7} + 12 \beta_{6} - 2 \beta_{5} - \beta_{4} + 2 \beta_{3} + 18 \beta_{2} + 10 \beta_{1} + 22$$$$)/2$$ $$\nu^{10}$$ $$=$$ $$($$$$-\beta_{17} - 6 \beta_{16} - 6 \beta_{15} + 7 \beta_{14} + 4 \beta_{12} + 9 \beta_{11} - 3 \beta_{10} - 7 \beta_{9} - \beta_{8} + 27 \beta_{7} - 9 \beta_{6} + 6 \beta_{5} + 11 \beta_{4} - 11 \beta_{3} + \beta_{2} + 2 \beta_{1} + 24$$$$)/2$$ $$\nu^{11}$$ $$=$$ $$($$$$18 \beta_{17} + 17 \beta_{16} + 5 \beta_{15} + 3 \beta_{14} - 3 \beta_{13} + 10 \beta_{12} + 2 \beta_{10} - 14 \beta_{9} + 15 \beta_{8} + 8 \beta_{7} + 22 \beta_{6} - 16 \beta_{5} + 5 \beta_{4} - 26 \beta_{3} - 16 \beta_{2} + 12 \beta_{1} + 12$$$$)/2$$ $$\nu^{12}$$ $$=$$ $$($$$$-9 \beta_{17} - 21 \beta_{16} - 47 \beta_{15} - 20 \beta_{14} - 3 \beta_{13} + 34 \beta_{12} - 21 \beta_{11} - 41 \beta_{10} + 29 \beta_{9} + 6 \beta_{8} - 3 \beta_{7} + 15 \beta_{6} + 56 \beta_{5} + 24 \beta_{4} - 21 \beta_{3} + 19 \beta_{2} - 26 \beta_{1} - 28$$$$)/2$$ $$\nu^{13}$$ $$=$$ $$($$$$-14 \beta_{17} + 3 \beta_{16} - 24 \beta_{15} - 8 \beta_{14} - 3 \beta_{13} - 25 \beta_{12} - \beta_{11} + 39 \beta_{10} - 40 \beta_{9} - 22 \beta_{8} - 13 \beta_{7} - 50 \beta_{6} - 50 \beta_{5} + 8 \beta_{4} - 26 \beta_{3} - 22 \beta_{2} - 6 \beta_{1} + 114$$$$)/2$$ $$\nu^{14}$$ $$=$$ $$($$$$-42 \beta_{17} + 25 \beta_{16} - 47 \beta_{15} - 7 \beta_{14} - 49 \beta_{13} + 72 \beta_{12} + 76 \beta_{11} - 26 \beta_{10} - 70 \beta_{9} + 13 \beta_{8} + 56 \beta_{7} + 20 \beta_{6} - 14 \beta_{5} - 5 \beta_{4} + 14 \beta_{3} + 72 \beta_{2} + 90 \beta_{1}$$$$)/2$$ $$\nu^{15}$$ $$=$$ $$($$$$32 \beta_{17} + 160 \beta_{16} - 75 \beta_{15} + 13 \beta_{14} + 104 \beta_{13} - 33 \beta_{12} - 117 \beta_{11} + 91 \beta_{10} + 4 \beta_{9} + 117 \beta_{8} + 21 \beta_{7} + 40 \beta_{6} - 8 \beta_{5} + 103 \beta_{4} - 130 \beta_{3} + 28 \beta_{2} + 58 \beta_{1} + 116$$$$)/2$$ $$\nu^{16}$$ $$=$$ $$($$$$59 \beta_{17} - 196 \beta_{16} - 80 \beta_{15} - 103 \beta_{14} - 58 \beta_{13} + 82 \beta_{12} + 187 \beta_{11} - 57 \beta_{10} + 33 \beta_{9} - 51 \beta_{8} + 83 \beta_{7} + 121 \beta_{6} + 80 \beta_{5} + 79 \beta_{4} - 77 \beta_{3} - 61 \beta_{2} - 60 \beta_{1} + 8$$$$)/2$$ $$\nu^{17}$$ $$=$$ $$($$$$-54 \beta_{17} - 61 \beta_{16} - 63 \beta_{15} - 5 \beta_{14} - 121 \beta_{13} + 52 \beta_{12} - 112 \beta_{11} - 284 \beta_{10} - 134 \beta_{9} + 37 \beta_{8} + 82 \beta_{7} - 34 \beta_{6} + 100 \beta_{5} - 143 \beta_{4} + 8 \beta_{3} - 4 \beta_{2} - 284 \beta_{1} + 228$$$$)/2$$

## Character values

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

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

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
163.1
 −1.08900 + 0.902261i 0.482716 + 1.32928i −0.635486 − 1.26339i 1.41303 + 0.0578659i 0.235136 − 1.39453i 1.41323 − 0.0526497i −1.37691 − 0.322680i 0.0376504 + 1.41371i −0.480367 + 1.33013i −1.08900 − 0.902261i 0.482716 − 1.32928i −0.635486 + 1.26339i 1.41303 − 0.0578659i 0.235136 + 1.39453i 1.41323 + 0.0526497i −1.37691 + 0.322680i 0.0376504 − 1.41371i −0.480367 − 1.33013i
−1.41267 0.0660953i 0 1.99126 + 0.186742i 2.00635 0.987189i 0 1.55426 1.55426i −2.80065 0.395417i 0 −2.89956 + 1.26196i
163.2 −1.19301 + 0.759419i 0 0.846564 1.81200i −2.17104 + 0.535339i 0 −2.13436 + 2.13436i 0.366101 + 2.80463i 0 2.18353 2.28740i
163.3 −0.828280 1.14628i 0 −0.627905 + 1.89888i 0.245325 + 2.22257i 0 −0.343872 + 0.343872i 2.69672 0.853049i 0 2.34448 2.12212i
163.4 −0.567819 + 1.29521i 0 −1.35516 1.47090i 1.42182 1.72581i 0 −1.60205 + 1.60205i 2.67461 0.920026i 0 1.42795 + 2.82151i
163.5 −0.430311 1.34716i 0 −1.62967 + 1.15939i 0.177336 2.22902i 0 −0.115101 + 0.115101i 2.26315 + 1.69652i 0 −3.07916 + 0.720273i
163.6 0.516777 1.31641i 0 −1.46588 1.36058i −2.07160 0.841703i 0 −1.13975 + 1.13975i −2.54862 + 1.22659i 0 −2.17858 + 2.29211i
163.7 1.23576 + 0.687667i 0 1.05423 + 1.69959i −0.832020 2.07551i 0 2.83610 2.83610i 0.134028 + 2.82525i 0 0.399079 3.13699i
163.8 1.29924 0.558542i 0 1.37606 1.45136i −1.49107 + 1.66635i 0 2.40368 2.40368i 0.977191 2.65426i 0 −1.00653 + 2.99782i
163.9 1.38031 0.307817i 0 1.81050 0.849763i 1.71489 + 1.43498i 0 −0.458895 + 0.458895i 2.23747 1.73024i 0 2.80878 + 1.45284i
667.1 −1.41267 + 0.0660953i 0 1.99126 0.186742i 2.00635 + 0.987189i 0 1.55426 + 1.55426i −2.80065 + 0.395417i 0 −2.89956 1.26196i
667.2 −1.19301 0.759419i 0 0.846564 + 1.81200i −2.17104 0.535339i 0 −2.13436 2.13436i 0.366101 2.80463i 0 2.18353 + 2.28740i
667.3 −0.828280 + 1.14628i 0 −0.627905 1.89888i 0.245325 2.22257i 0 −0.343872 0.343872i 2.69672 + 0.853049i 0 2.34448 + 2.12212i
667.4 −0.567819 1.29521i 0 −1.35516 + 1.47090i 1.42182 + 1.72581i 0 −1.60205 1.60205i 2.67461 + 0.920026i 0 1.42795 2.82151i
667.5 −0.430311 + 1.34716i 0 −1.62967 1.15939i 0.177336 + 2.22902i 0 −0.115101 0.115101i 2.26315 1.69652i 0 −3.07916 0.720273i
667.6 0.516777 + 1.31641i 0 −1.46588 + 1.36058i −2.07160 + 0.841703i 0 −1.13975 1.13975i −2.54862 1.22659i 0 −2.17858 2.29211i
667.7 1.23576 0.687667i 0 1.05423 1.69959i −0.832020 + 2.07551i 0 2.83610 + 2.83610i 0.134028 2.82525i 0 0.399079 + 3.13699i
667.8 1.29924 + 0.558542i 0 1.37606 + 1.45136i −1.49107 1.66635i 0 2.40368 + 2.40368i 0.977191 + 2.65426i 0 −1.00653 2.99782i
667.9 1.38031 + 0.307817i 0 1.81050 + 0.849763i 1.71489 1.43498i 0 −0.458895 0.458895i 2.23747 + 1.73024i 0 2.80878 1.45284i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 667.9 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
80.s 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.z.g 18
3.b odd 2 1 80.2.s.b yes 18
5.c odd 4 1 720.2.bd.g 18
12.b even 2 1 320.2.s.b 18
15.d odd 2 1 400.2.s.d 18
15.e even 4 1 80.2.j.b 18
15.e even 4 1 400.2.j.d 18
16.f odd 4 1 720.2.bd.g 18
24.f even 2 1 640.2.s.c 18
24.h odd 2 1 640.2.s.d 18
48.i odd 4 1 320.2.j.b 18
48.i odd 4 1 640.2.j.c 18
48.k even 4 1 80.2.j.b 18
48.k even 4 1 640.2.j.d 18
60.h even 2 1 1600.2.s.d 18
60.l odd 4 1 320.2.j.b 18
60.l odd 4 1 1600.2.j.d 18
80.s even 4 1 inner 720.2.z.g 18
120.q odd 4 1 640.2.j.c 18
120.w even 4 1 640.2.j.d 18
240.t even 4 1 400.2.j.d 18
240.z odd 4 1 80.2.s.b yes 18
240.bb even 4 1 320.2.s.b 18
240.bd odd 4 1 400.2.s.d 18
240.bd odd 4 1 640.2.s.d 18
240.bf even 4 1 640.2.s.c 18
240.bf even 4 1 1600.2.s.d 18
240.bm odd 4 1 1600.2.j.d 18

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
80.2.j.b 18 15.e even 4 1
80.2.j.b 18 48.k even 4 1
80.2.s.b yes 18 3.b odd 2 1
80.2.s.b yes 18 240.z odd 4 1
320.2.j.b 18 48.i odd 4 1
320.2.j.b 18 60.l odd 4 1
320.2.s.b 18 12.b even 2 1
320.2.s.b 18 240.bb even 4 1
400.2.j.d 18 15.e even 4 1
400.2.j.d 18 240.t even 4 1
400.2.s.d 18 15.d odd 2 1
400.2.s.d 18 240.bd odd 4 1
640.2.j.c 18 48.i odd 4 1
640.2.j.c 18 120.q odd 4 1
640.2.j.d 18 48.k even 4 1
640.2.j.d 18 120.w even 4 1
640.2.s.c 18 24.f even 2 1
640.2.s.c 18 240.bf even 4 1
640.2.s.d 18 24.h odd 2 1
640.2.s.d 18 240.bd odd 4 1
720.2.z.g 18 1.a even 1 1 trivial
720.2.z.g 18 80.s even 4 1 inner
720.2.bd.g 18 5.c odd 4 1
720.2.bd.g 18 16.f odd 4 1
1600.2.j.d 18 60.l odd 4 1
1600.2.j.d 18 240.bm odd 4 1
1600.2.s.d 18 60.h even 2 1
1600.2.s.d 18 240.bf even 4 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(720, [\chi])$$:

 $$T_{7}^{18} - \cdots$$ $$T_{11}^{18} - \cdots$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$512 - 256 T^{2} - 256 T^{3} + 64 T^{4} + 192 T^{5} + 32 T^{6} - 32 T^{7} - 24 T^{8} + 8 T^{9} - 12 T^{10} - 8 T^{11} + 4 T^{12} + 12 T^{13} + 2 T^{14} - 4 T^{15} - 2 T^{16} + T^{18}$$
$3$ $$T^{18}$$
$5$ $$1953125 + 781250 T + 390625 T^{2} + 250000 T^{3} + 112500 T^{4} + 65000 T^{5} + 38500 T^{6} + 17200 T^{7} + 7550 T^{8} + 2572 T^{9} + 1510 T^{10} + 688 T^{11} + 308 T^{12} + 104 T^{13} + 36 T^{14} + 16 T^{15} + 5 T^{16} + 2 T^{17} + T^{18}$$
$7$ $$288 + 4128 T + 29584 T^{2} + 108160 T^{3} + 239360 T^{4} + 315488 T^{5} + 244448 T^{6} + 85184 T^{7} + 17328 T^{8} + 11648 T^{9} + 14648 T^{10} + 4160 T^{11} + 352 T^{12} - 120 T^{13} + 200 T^{14} + 32 T^{15} + 2 T^{16} - 2 T^{17} + T^{18}$$
$11$ $$5431808 + 4060672 T + 1517824 T^{2} + 4217856 T^{3} + 18015744 T^{4} + 19945472 T^{5} + 11514112 T^{6} + 2158336 T^{7} + 289472 T^{8} + 209344 T^{9} + 182240 T^{10} + 27968 T^{11} + 1376 T^{12} + 64 T^{13} + 848 T^{14} + 80 T^{15} + 2 T^{16} - 2 T^{17} + T^{18}$$
$13$ $$67108864 + 131948800 T^{2} + 105268224 T^{4} + 44025600 T^{6} + 10452224 T^{8} + 1437280 T^{10} + 113344 T^{12} + 4976 T^{14} + 112 T^{16} + T^{18}$$
$17$ $$512 - 1536 T + 2304 T^{2} + 186368 T^{3} + 1493504 T^{4} + 5352960 T^{5} + 11139328 T^{6} + 12464640 T^{7} + 6499264 T^{8} - 984640 T^{9} + 14816 T^{10} + 36480 T^{11} + 66080 T^{12} - 19232 T^{13} + 2768 T^{14} - 32 T^{15} + 18 T^{16} - 6 T^{17} + T^{18}$$
$19$ $$4608 - 339456 T + 12503296 T^{2} - 7467008 T^{3} + 2215424 T^{4} + 335360 T^{5} + 13924096 T^{6} - 9073664 T^{7} + 2905024 T^{8} + 1873856 T^{9} + 675296 T^{10} - 48384 T^{11} + 16480 T^{12} + 10080 T^{13} + 2864 T^{14} + 64 T^{15} + 2 T^{16} + 2 T^{17} + T^{18}$$
$23$ $$17700587552 - 17587696352 T + 8737762576 T^{2} + 758796096 T^{3} + 938524160 T^{4} - 784929440 T^{5} + 332892064 T^{6} + 13833888 T^{7} + 11039216 T^{8} - 7905920 T^{9} + 2752216 T^{10} - 54000 T^{11} + 9120 T^{12} - 7896 T^{13} + 3480 T^{14} - 24 T^{15} + 2 T^{16} - 2 T^{17} + T^{18}$$
$29$ $$82330112 + 372641280 T + 843321600 T^{2} + 687951872 T^{3} + 281020928 T^{4} + 16436736 T^{5} + 70118656 T^{6} + 45759488 T^{7} + 14859968 T^{8} + 1819968 T^{9} + 693088 T^{10} + 360192 T^{11} + 111008 T^{12} + 15584 T^{13} + 1616 T^{14} + 320 T^{15} + 98 T^{16} + 14 T^{17} + T^{18}$$
$31$ $$16384 + 1150976 T^{2} + 16687104 T^{4} + 32610304 T^{6} + 17532416 T^{8} + 3648384 T^{10} + 327040 T^{12} + 12832 T^{14} + 196 T^{16} + T^{18}$$
$37$ $$574297214976 + 457920768256 T^{2} + 131428787200 T^{4} + 18630090496 T^{6} + 1481216256 T^{8} + 69964768 T^{10} + 1988288 T^{12} + 32944 T^{14} + 288 T^{16} + T^{18}$$
$41$ $$242788765696 + 208906121472 T^{2} + 68836776960 T^{4} + 11517059840 T^{6} + 1077600512 T^{8} + 58367328 T^{10} + 1831744 T^{12} + 32176 T^{14} + 288 T^{16} + T^{18}$$
$43$ $$337207844416 + 627531510928 T^{2} + 248751738624 T^{4} + 38727202720 T^{6} + 3041915424 T^{8} + 132726008 T^{10} + 3310976 T^{12} + 46520 T^{14} + 340 T^{16} + T^{18}$$
$47$ $$16870640672 + 178795286432 T + 947437476496 T^{2} + 866389389184 T^{3} + 419056561664 T^{4} + 124183363808 T^{5} + 28973153120 T^{6} + 7620275328 T^{7} + 2411169968 T^{8} + 643585920 T^{9} + 126078328 T^{10} + 18512576 T^{11} + 2503968 T^{12} + 394472 T^{13} + 64680 T^{14} + 8272 T^{15} + 722 T^{16} + 38 T^{17} + T^{18}$$
$53$ $$( -220832 + 334608 T - 44032 T^{2} - 85984 T^{3} + 10768 T^{4} + 5720 T^{5} - 480 T^{6} - 136 T^{7} + 6 T^{8} + T^{9} )^{2}$$
$59$ $$144166720393728 + 63266193922560 T + 13881883705600 T^{2} - 2799567403008 T^{3} + 1079531100672 T^{4} + 262683543040 T^{5} + 38509938944 T^{6} - 5387559424 T^{7} + 1811701952 T^{8} + 356104896 T^{9} + 37394528 T^{10} - 1999488 T^{11} + 826528 T^{12} + 162272 T^{13} + 16016 T^{14} - 96 T^{15} + 50 T^{16} + 10 T^{17} + T^{18}$$
$61$ $$121236758528 - 124325191680 T + 63746150400 T^{2} + 35916963840 T^{3} + 28713865216 T^{4} - 15011409920 T^{5} + 5616384000 T^{6} + 2393885696 T^{7} + 974385920 T^{8} - 231668992 T^{9} + 33596800 T^{10} + 3516032 T^{11} + 740800 T^{12} - 135872 T^{13} + 14496 T^{14} + 720 T^{15} + 98 T^{16} - 14 T^{17} + T^{18}$$
$67$ $$555525752896 + 1362082550416 T^{2} + 742701558272 T^{4} + 154793099680 T^{6} + 14366974496 T^{8} + 616717432 T^{10} + 12278976 T^{12} + 120376 T^{14} + 564 T^{16} + T^{18}$$
$71$ $$( 27648 - 72640 T - 110336 T^{2} - 8704 T^{3} + 23136 T^{4} + 3344 T^{5} - 1408 T^{6} - 152 T^{7} + 12 T^{8} + T^{9} )^{2}$$
$73$ $$35535647232 + 3228962304 T + 146700544 T^{2} + 5453305856 T^{3} + 14687183360 T^{4} + 5125904896 T^{5} + 823567616 T^{6} - 175353344 T^{7} + 212641728 T^{8} + 49164608 T^{9} + 5990112 T^{10} - 758144 T^{11} + 628896 T^{12} + 125856 T^{13} + 13072 T^{14} - 544 T^{15} + 98 T^{16} + 14 T^{17} + T^{18}$$
$79$ $$( 45002752 + 3950848 T - 7267840 T^{2} - 485376 T^{3} + 296064 T^{4} + 27488 T^{5} - 2976 T^{6} - 320 T^{7} + 8 T^{8} + T^{9} )^{2}$$
$83$ $$( 8413744 + 2612884 T - 2578680 T^{2} - 329896 T^{3} + 197504 T^{4} + 7292 T^{5} - 3612 T^{6} - 136 T^{7} + 20 T^{8} + T^{9} )^{2}$$
$89$ $$( 251904 + 5727232 T - 4338688 T^{2} - 1356288 T^{3} + 330368 T^{4} + 55104 T^{5} - 2752 T^{6} - 448 T^{7} + 6 T^{8} + T^{9} )^{2}$$
$97$ $$380349381734912 + 440719049317888 T + 255335343974656 T^{2} + 73928552845312 T^{3} + 10687493231104 T^{4} + 76758095360 T^{5} + 98986508544 T^{6} + 49913260032 T^{7} + 9402360768 T^{8} - 513287616 T^{9} + 21224416 T^{10} + 9460992 T^{11} + 2501024 T^{12} - 238368 T^{13} + 11856 T^{14} + 512 T^{15} + 162 T^{16} - 18 T^{17} + T^{18}$$