# Properties

 Label 1849.2.a.n Level $1849$ Weight $2$ Character orbit 1849.a Self dual yes Analytic conductor $14.764$ Analytic rank $1$ Dimension $18$ CM no Inner twists $1$

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## Newspace parameters

 Level: $$N$$ $$=$$ $$1849 = 43^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1849.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$14.7643393337$$ Analytic rank: $$1$$ Dimension: $$18$$ Coefficient field: $$\mathbb{Q}[x]/(x^{18} - \cdots)$$ Defining polynomial: $$x^{18} - 5 x^{17} - 15 x^{16} + 106 x^{15} + 47 x^{14} - 897 x^{13} + 364 x^{12} + 3855 x^{11} - 3223 x^{10} - 8851 x^{9} + 9909 x^{8} + 10400 x^{7} - 14483 x^{6} - 5118 x^{5} + 9945 x^{4} - 27 x^{3} - 2493 x^{2} + 513 x - 27$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 43) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-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.

 $$f(q)$$ $$=$$ $$q -\beta_{1} q^{2} -\beta_{14} q^{3} + ( 1 + \beta_{3} - \beta_{5} + \beta_{7} ) q^{4} + ( -1 - \beta_{4} - \beta_{5} - \beta_{10} - \beta_{13} ) q^{5} + ( -1 - \beta_{3} + \beta_{9} - \beta_{12} + \beta_{14} ) q^{6} + ( -2 + \beta_{1} + \beta_{2} - \beta_{4} - 2 \beta_{6} + \beta_{7} - \beta_{8} - \beta_{9} - \beta_{11} + \beta_{15} + \beta_{16} ) q^{7} + ( 1 - \beta_{1} - \beta_{2} - \beta_{3} + \beta_{4} + \beta_{5} + 2 \beta_{6} - 2 \beta_{7} + \beta_{8} + \beta_{9} + \beta_{11} - \beta_{15} - \beta_{16} - \beta_{17} ) q^{8} + ( 1 + \beta_{3} - \beta_{6} - \beta_{16} ) q^{9} +O(q^{10})$$ $$q -\beta_{1} q^{2} -\beta_{14} q^{3} + ( 1 + \beta_{3} - \beta_{5} + \beta_{7} ) q^{4} + ( -1 - \beta_{4} - \beta_{5} - \beta_{10} - \beta_{13} ) q^{5} + ( -1 - \beta_{3} + \beta_{9} - \beta_{12} + \beta_{14} ) q^{6} + ( -2 + \beta_{1} + \beta_{2} - \beta_{4} - 2 \beta_{6} + \beta_{7} - \beta_{8} - \beta_{9} - \beta_{11} + \beta_{15} + \beta_{16} ) q^{7} + ( 1 - \beta_{1} - \beta_{2} - \beta_{3} + \beta_{4} + \beta_{5} + 2 \beta_{6} - 2 \beta_{7} + \beta_{8} + \beta_{9} + \beta_{11} - \beta_{15} - \beta_{16} - \beta_{17} ) q^{8} + ( 1 + \beta_{3} - \beta_{6} - \beta_{16} ) q^{9} + ( 1 + \beta_{1} + \beta_{6} - \beta_{7} - \beta_{9} + \beta_{10} + \beta_{11} + \beta_{12} + \beta_{13} - \beta_{14} + \beta_{15} ) q^{10} + ( \beta_{1} + \beta_{4} + \beta_{5} + \beta_{6} + \beta_{14} - \beta_{15} ) q^{11} + ( \beta_{4} + 2 \beta_{5} + \beta_{6} + \beta_{7} + \beta_{10} + \beta_{13} - \beta_{14} + \beta_{17} ) q^{12} + ( -1 + \beta_{1} + \beta_{2} + \beta_{10} + \beta_{14} + \beta_{16} ) q^{13} + ( \beta_{1} + \beta_{2} + \beta_{5} + \beta_{6} + \beta_{10} - \beta_{13} + \beta_{17} ) q^{14} + ( -2 \beta_{3} + \beta_{4} + 2 \beta_{5} - \beta_{6} - 2 \beta_{7} - \beta_{8} + \beta_{11} + \beta_{13} + \beta_{14} ) q^{15} + ( -1 + \beta_{1} + \beta_{3} - \beta_{4} - 2 \beta_{5} - 2 \beta_{6} + 2 \beta_{7} - \beta_{9} - \beta_{10} - 2 \beta_{11} + \beta_{12} + \beta_{13} + \beta_{16} ) q^{16} + ( 1 - \beta_{1} - 2 \beta_{2} + \beta_{3} + \beta_{4} + \beta_{6} - \beta_{7} + \beta_{9} + \beta_{11} - \beta_{16} + \beta_{17} ) q^{17} + ( -1 - \beta_{2} - \beta_{5} + \beta_{6} - \beta_{7} + \beta_{8} - 2 \beta_{9} - \beta_{11} + \beta_{12} + \beta_{14} - \beta_{15} + \beta_{16} - \beta_{17} ) q^{18} + ( -2 \beta_{1} + \beta_{2} - \beta_{5} + \beta_{8} - \beta_{10} + \beta_{12} + \beta_{13} - \beta_{14} ) q^{19} + ( -2 - \beta_{1} - 2 \beta_{2} - 2 \beta_{6} - \beta_{7} + \beta_{9} - \beta_{10} - 2 \beta_{13} + \beta_{14} + \beta_{17} ) q^{20} + ( 1 - \beta_{1} - 3 \beta_{2} - \beta_{5} + \beta_{6} - \beta_{7} + 3 \beta_{8} - \beta_{9} - \beta_{10} - \beta_{11} - \beta_{16} - \beta_{17} ) q^{21} + ( -2 + \beta_{1} + \beta_{2} + \beta_{5} - 2 \beta_{6} - \beta_{7} - \beta_{8} - \beta_{11} + \beta_{13} + \beta_{14} - \beta_{17} ) q^{22} + ( -1 - \beta_{1} + \beta_{3} - \beta_{5} + 3 \beta_{7} + \beta_{8} - \beta_{9} - \beta_{11} - \beta_{13} ) q^{23} + ( -3 + \beta_{1} + \beta_{2} - \beta_{3} + \beta_{5} - 5 \beta_{6} + \beta_{7} - 3 \beta_{8} + 2 \beta_{9} - \beta_{10} - \beta_{11} - \beta_{12} - \beta_{13} + \beta_{14} + \beta_{16} ) q^{24} + ( \beta_{1} - \beta_{3} + \beta_{4} + 3 \beta_{5} + \beta_{6} - \beta_{7} + \beta_{9} + \beta_{10} - \beta_{12} + 2 \beta_{13} + 2 \beta_{14} - \beta_{15} ) q^{25} + ( -2 + \beta_{1} + \beta_{2} + \beta_{5} - \beta_{6} + 2 \beta_{7} + \beta_{8} - 2 \beta_{9} - \beta_{11} - \beta_{14} ) q^{26} + ( -3 + 2 \beta_{1} - \beta_{3} + \beta_{4} + 2 \beta_{5} - \beta_{6} + \beta_{7} - 2 \beta_{8} + \beta_{9} + \beta_{10} - \beta_{13} + \beta_{16} + \beta_{17} ) q^{27} + ( -3 + 2 \beta_{1} - \beta_{3} - 2 \beta_{6} - \beta_{9} + \beta_{13} + \beta_{14} + \beta_{16} ) q^{28} + ( -2 - 2 \beta_{2} - 2 \beta_{3} + \beta_{4} + 2 \beta_{5} + 2 \beta_{6} - 3 \beta_{7} - \beta_{8} + \beta_{9} + 2 \beta_{10} + \beta_{11} - \beta_{12} + \beta_{13} + \beta_{14} - \beta_{16} ) q^{29} + ( 2 + \beta_{1} - \beta_{2} + 3 \beta_{3} - \beta_{4} - \beta_{5} - \beta_{9} - 2 \beta_{10} - \beta_{11} - 2 \beta_{13} - \beta_{14} + \beta_{15} + 2 \beta_{17} ) q^{30} + ( 2 + \beta_{1} + \beta_{2} + 2 \beta_{4} + 2 \beta_{6} - \beta_{7} + \beta_{8} - \beta_{9} + \beta_{11} + \beta_{12} + \beta_{13} - \beta_{14} + \beta_{16} - \beta_{17} ) q^{31} + ( -1 + 2 \beta_{2} - \beta_{3} + 3 \beta_{5} + 2 \beta_{6} - 2 \beta_{7} - \beta_{8} + \beta_{9} + 2 \beta_{10} + \beta_{11} - \beta_{12} + \beta_{15} - 2 \beta_{16} + \beta_{17} ) q^{32} + ( -2 + \beta_{1} + 2 \beta_{2} + \beta_{3} - \beta_{4} - \beta_{5} + \beta_{6} + 2 \beta_{7} + \beta_{9} + \beta_{10} + \beta_{11} + \beta_{12} + \beta_{13} ) q^{33} + ( 1 - \beta_{1} - 2 \beta_{2} - 2 \beta_{3} + \beta_{4} + \beta_{5} + \beta_{6} - 4 \beta_{7} - \beta_{8} + \beta_{9} - 2 \beta_{10} + 3 \beta_{11} - \beta_{16} - \beta_{17} ) q^{34} + ( 1 - \beta_{1} - \beta_{2} + \beta_{6} + \beta_{7} + \beta_{8} + \beta_{9} + 2 \beta_{10} - \beta_{11} + \beta_{12} + \beta_{13} - \beta_{15} ) q^{35} + ( 4 + \beta_{1} + 2 \beta_{2} + 2 \beta_{3} - 2 \beta_{4} - 2 \beta_{5} + 2 \beta_{6} - \beta_{7} + \beta_{8} - \beta_{9} + \beta_{10} + \beta_{11} + \beta_{12} + \beta_{14} - \beta_{16} - \beta_{17} ) q^{36} + ( 2 + 2 \beta_{1} + \beta_{6} - 2 \beta_{7} + \beta_{8} - 2 \beta_{9} + \beta_{10} - \beta_{11} + \beta_{13} - \beta_{14} + \beta_{15} ) q^{37} + ( 2 + \beta_{1} - \beta_{2} - \beta_{3} + \beta_{4} + \beta_{7} + \beta_{9} - \beta_{10} + \beta_{11} - \beta_{12} - \beta_{13} ) q^{38} + ( -2 - 2 \beta_{1} - \beta_{5} + \beta_{6} + 2 \beta_{7} + 2 \beta_{8} - \beta_{9} + \beta_{12} + \beta_{14} - \beta_{15} ) q^{39} + ( 1 + 2 \beta_{1} + 2 \beta_{2} + \beta_{5} + \beta_{7} - 3 \beta_{9} - 2 \beta_{10} + 2 \beta_{11} + \beta_{13} + \beta_{15} ) q^{40} + ( 1 + \beta_{1} - 2 \beta_{3} + \beta_{4} + \beta_{5} + 5 \beta_{6} - 3 \beta_{7} - \beta_{8} + 2 \beta_{9} + 2 \beta_{10} + \beta_{11} + \beta_{13} - \beta_{15} ) q^{41} + ( 2 \beta_{2} + \beta_{3} - \beta_{4} - 2 \beta_{5} - 2 \beta_{6} + \beta_{7} - \beta_{8} + \beta_{10} + 3 \beta_{11} + 2 \beta_{12} - \beta_{15} ) q^{42} + ( 1 + \beta_{1} + 2 \beta_{3} - \beta_{4} - 2 \beta_{5} + \beta_{6} + \beta_{7} + \beta_{8} - 2 \beta_{9} + \beta_{10} - 3 \beta_{11} + \beta_{12} - 2 \beta_{14} ) q^{44} + ( 1 - \beta_{1} - \beta_{2} + 2 \beta_{3} - 2 \beta_{5} + 3 \beta_{6} - \beta_{7} + 2 \beta_{8} + \beta_{9} + \beta_{10} - \beta_{11} - \beta_{13} - \beta_{14} ) q^{45} + ( 2 + 2 \beta_{2} + \beta_{3} - \beta_{4} - \beta_{5} - 2 \beta_{6} + \beta_{7} + \beta_{10} + \beta_{11} + \beta_{12} + \beta_{13} + \beta_{14} + \beta_{16} - \beta_{17} ) q^{46} + ( -\beta_{1} - \beta_{2} - \beta_{4} + \beta_{5} - \beta_{8} + 2 \beta_{9} + \beta_{10} + 3 \beta_{11} - 2 \beta_{13} + \beta_{15} + \beta_{17} ) q^{47} + ( 4 + \beta_{1} + \beta_{2} + \beta_{4} + \beta_{5} + 7 \beta_{6} - 3 \beta_{7} + 2 \beta_{8} - 2 \beta_{9} + 2 \beta_{13} - \beta_{14} + \beta_{15} - \beta_{16} - \beta_{17} ) q^{48} + ( -1 + \beta_{2} + \beta_{3} + \beta_{5} - \beta_{6} + 3 \beta_{7} - 2 \beta_{8} + \beta_{9} + 2 \beta_{10} - \beta_{11} + \beta_{12} - \beta_{13} - \beta_{14} + \beta_{15} + \beta_{17} ) q^{49} + ( -4 + \beta_{1} + \beta_{2} - \beta_{3} + 2 \beta_{4} + 2 \beta_{5} - 3 \beta_{6} + 3 \beta_{7} - 2 \beta_{8} + 2 \beta_{9} + \beta_{10} - 4 \beta_{11} - 2 \beta_{12} - 2 \beta_{15} ) q^{50} + ( -2 + 2 \beta_{1} + 2 \beta_{2} + \beta_{5} + 2 \beta_{6} - 3 \beta_{8} + 2 \beta_{10} + 2 \beta_{11} + \beta_{15} + \beta_{16} + \beta_{17} ) q^{51} + ( -5 + 2 \beta_{1} + \beta_{2} - \beta_{3} - 2 \beta_{4} - \beta_{5} - 6 \beta_{6} + 2 \beta_{7} - \beta_{8} - \beta_{10} - 2 \beta_{11} - \beta_{12} - \beta_{13} + 2 \beta_{14} + \beta_{16} ) q^{52} + ( -3 \beta_{2} - 2 \beta_{3} + \beta_{4} + 2 \beta_{5} + 2 \beta_{6} - 4 \beta_{7} + \beta_{8} + \beta_{9} + 2 \beta_{11} - 3 \beta_{12} + 2 \beta_{14} - \beta_{15} - 3 \beta_{16} - \beta_{17} ) q^{53} + ( -4 + 3 \beta_{1} + 2 \beta_{2} - \beta_{3} - \beta_{4} + 3 \beta_{5} - \beta_{7} - \beta_{8} - \beta_{10} - \beta_{12} + 2 \beta_{13} + 3 \beta_{15} + \beta_{17} ) q^{54} + ( -3 - \beta_{3} - \beta_{4} - 2 \beta_{5} + 2 \beta_{7} + 2 \beta_{8} - \beta_{10} + 2 \beta_{11} - \beta_{13} + \beta_{14} - \beta_{15} - 2 \beta_{17} ) q^{55} + ( -3 - \beta_{1} - 2 \beta_{2} - \beta_{3} + \beta_{5} - 3 \beta_{7} + 2 \beta_{8} - 2 \beta_{10} + \beta_{11} - \beta_{12} - \beta_{14} - 2 \beta_{16} - \beta_{17} ) q^{56} + ( 4 - \beta_{1} + 2 \beta_{2} - 3 \beta_{5} + 3 \beta_{6} + 2 \beta_{8} - \beta_{9} - \beta_{10} + \beta_{12} + 2 \beta_{14} - \beta_{15} - \beta_{17} ) q^{57} + ( 2 + 2 \beta_{1} - 2 \beta_{2} + 4 \beta_{3} - 2 \beta_{4} - 2 \beta_{5} - \beta_{8} + \beta_{9} - \beta_{10} - 2 \beta_{11} - \beta_{13} - 2 \beta_{14} - \beta_{16} + 2 \beta_{17} ) q^{58} + ( 1 + \beta_{1} - \beta_{3} + \beta_{4} + 3 \beta_{5} - \beta_{7} + \beta_{9} - \beta_{10} + 2 \beta_{11} - \beta_{12} + \beta_{13} + \beta_{14} - \beta_{15} - \beta_{16} - \beta_{17} ) q^{59} + ( -6 + \beta_{1} + 2 \beta_{2} - 4 \beta_{3} + \beta_{5} - \beta_{6} - 3 \beta_{7} - 3 \beta_{8} + \beta_{10} + 4 \beta_{11} - \beta_{13} + \beta_{14} + \beta_{15} + \beta_{16} - 2 \beta_{17} ) q^{60} + ( -3 - \beta_{1} - 2 \beta_{2} - \beta_{4} - \beta_{5} - 2 \beta_{6} - \beta_{7} - \beta_{8} + \beta_{9} - 2 \beta_{12} - \beta_{13} - \beta_{14} + 2 \beta_{15} + \beta_{17} ) q^{61} + ( -4 - 2 \beta_{1} - \beta_{3} - 4 \beta_{6} + \beta_{7} + 2 \beta_{9} - 4 \beta_{10} + \beta_{11} - \beta_{12} - 3 \beta_{13} + 2 \beta_{14} - \beta_{15} ) q^{62} + ( -1 + \beta_{1} - 2 \beta_{2} - 4 \beta_{3} + 3 \beta_{4} + 3 \beta_{5} + \beta_{6} - 5 \beta_{7} - \beta_{8} - \beta_{10} + \beta_{11} - 2 \beta_{12} + 2 \beta_{13} + \beta_{15} + \beta_{16} - 2 \beta_{17} ) q^{63} + ( -6 + 6 \beta_{1} - \beta_{2} - \beta_{4} - 5 \beta_{6} + 2 \beta_{7} - 4 \beta_{8} - 2 \beta_{9} + \beta_{10} - 3 \beta_{11} - \beta_{12} - 3 \beta_{13} + \beta_{15} + 5 \beta_{16} + 2 \beta_{17} ) q^{64} + ( -2 - 2 \beta_{1} - \beta_{2} + 3 \beta_{3} - \beta_{4} - 2 \beta_{5} - 3 \beta_{6} + 5 \beta_{7} + \beta_{8} - 2 \beta_{10} - 3 \beta_{11} - \beta_{14} - 2 \beta_{16} + \beta_{17} ) q^{65} + ( -5 - 3 \beta_{2} - 2 \beta_{3} + \beta_{4} + \beta_{6} + \beta_{7} + 2 \beta_{8} + \beta_{9} - \beta_{10} - 3 \beta_{11} - \beta_{13} - \beta_{14} - \beta_{15} - \beta_{17} ) q^{66} + ( -1 + \beta_{2} - \beta_{3} + 2 \beta_{4} + 2 \beta_{5} - 4 \beta_{6} + 4 \beta_{7} - \beta_{8} - \beta_{9} - 3 \beta_{11} + 2 \beta_{13} - \beta_{14} + \beta_{15} - \beta_{16} ) q^{67} + ( 5 + 2 \beta_{3} - 2 \beta_{4} - 4 \beta_{5} + \beta_{6} - \beta_{9} - 4 \beta_{10} + 2 \beta_{12} - 2 \beta_{13} - 2 \beta_{14} + \beta_{15} + \beta_{16} ) q^{68} + ( -3 + \beta_{1} + \beta_{2} - \beta_{3} + 2 \beta_{5} - 3 \beta_{6} + \beta_{7} - \beta_{8} + \beta_{9} + \beta_{10} + \beta_{11} - \beta_{12} + 2 \beta_{13} + 2 \beta_{14} + \beta_{15} + \beta_{16} ) q^{69} + ( 1 - \beta_{1} + \beta_{2} + \beta_{3} - \beta_{4} - \beta_{5} + \beta_{6} + 3 \beta_{7} + \beta_{8} + \beta_{9} + \beta_{10} - 2 \beta_{11} - \beta_{12} + 2 \beta_{13} - 2 \beta_{16} - \beta_{17} ) q^{70} + ( -6 - \beta_{1} - 2 \beta_{2} - \beta_{6} + \beta_{7} + \beta_{8} + \beta_{9} - 2 \beta_{10} - \beta_{11} - 2 \beta_{12} - 2 \beta_{13} - 2 \beta_{16} + \beta_{17} ) q^{71} + ( -1 - 2 \beta_{1} - \beta_{2} + 3 \beta_{3} - 2 \beta_{4} - 4 \beta_{5} - 4 \beta_{6} + 4 \beta_{7} + \beta_{8} - 2 \beta_{9} - 3 \beta_{11} + 2 \beta_{12} - 2 \beta_{13} - 2 \beta_{14} ) q^{72} + ( 1 - 2 \beta_{2} - \beta_{3} - 2 \beta_{5} + \beta_{6} - 2 \beta_{7} + 3 \beta_{8} + \beta_{9} - \beta_{10} - 2 \beta_{11} + 2 \beta_{13} + 2 \beta_{14} - 2 \beta_{15} - \beta_{16} - 2 \beta_{17} ) q^{73} + ( -9 + \beta_{1} + \beta_{2} - 2 \beta_{3} - 2 \beta_{4} + \beta_{5} - 5 \beta_{6} - \beta_{7} - 2 \beta_{8} - \beta_{12} - 3 \beta_{13} + 2 \beta_{14} - \beta_{15} + \beta_{16} + \beta_{17} ) q^{74} + ( -1 + \beta_{2} + 2 \beta_{3} - \beta_{4} - 5 \beta_{5} + 5 \beta_{6} + 3 \beta_{7} + 2 \beta_{8} - 2 \beta_{9} + \beta_{10} - 2 \beta_{11} + \beta_{12} - \beta_{15} + \beta_{16} - 2 \beta_{17} ) q^{75} + ( -3 - \beta_{1} - 2 \beta_{2} - 2 \beta_{3} + \beta_{4} + 5 \beta_{5} + \beta_{6} - 2 \beta_{7} - 2 \beta_{8} + 2 \beta_{9} + \beta_{10} + 3 \beta_{11} - 2 \beta_{12} - \beta_{13} + \beta_{15} + \beta_{17} ) q^{76} + ( -1 - \beta_{1} + \beta_{4} + \beta_{5} - 3 \beta_{7} - 2 \beta_{8} + \beta_{9} - 3 \beta_{10} + 2 \beta_{11} - \beta_{12} + \beta_{14} + \beta_{16} + \beta_{17} ) q^{77} + ( 4 + 2 \beta_{1} + \beta_{2} + 3 \beta_{3} - 2 \beta_{4} - 3 \beta_{5} - 3 \beta_{6} + 2 \beta_{7} + \beta_{8} - \beta_{9} + \beta_{12} + \beta_{15} - \beta_{17} ) q^{78} + ( 1 - \beta_{1} + 3 \beta_{2} - \beta_{3} + \beta_{4} + \beta_{5} - 2 \beta_{6} - \beta_{7} + \beta_{9} + \beta_{10} - \beta_{11} - \beta_{12} + 2 \beta_{13} - \beta_{14} ) q^{79} + ( -3 + 2 \beta_{2} - 2 \beta_{3} - \beta_{5} - 3 \beta_{6} - \beta_{7} - 2 \beta_{8} + \beta_{9} - \beta_{10} - 3 \beta_{13} + \beta_{14} - \beta_{15} + 3 \beta_{16} - \beta_{17} ) q^{80} + ( 2 - 2 \beta_{1} + 2 \beta_{3} - 2 \beta_{4} - 3 \beta_{5} + 3 \beta_{6} - \beta_{7} + \beta_{8} + \beta_{9} + 3 \beta_{11} + 2 \beta_{12} - 2 \beta_{13} + \beta_{14} - \beta_{15} - \beta_{17} ) q^{81} + ( -2 - 2 \beta_{2} + \beta_{3} - \beta_{4} + \beta_{5} - \beta_{6} - 2 \beta_{8} + 3 \beta_{9} - \beta_{10} - 3 \beta_{11} - 2 \beta_{12} + \beta_{13} - 2 \beta_{14} + \beta_{15} - 2 \beta_{16} + \beta_{17} ) q^{82} + ( -3 - \beta_{1} + 2 \beta_{2} - 2 \beta_{4} - 4 \beta_{5} - 3 \beta_{6} + 3 \beta_{7} + \beta_{8} - \beta_{9} - 3 \beta_{10} - \beta_{11} + \beta_{12} + \beta_{14} - \beta_{15} + 2 \beta_{16} - 2 \beta_{17} ) q^{83} + ( 2 - \beta_{1} + 3 \beta_{3} - 2 \beta_{4} - 3 \beta_{5} + 5 \beta_{6} - \beta_{7} + 3 \beta_{8} - 3 \beta_{9} - \beta_{10} - \beta_{11} + \beta_{12} - 2 \beta_{13} + \beta_{15} ) q^{84} + ( -2 - \beta_{1} + \beta_{2} + 2 \beta_{3} + \beta_{4} - \beta_{5} - \beta_{6} + \beta_{7} - \beta_{8} + \beta_{9} + 2 \beta_{10} - \beta_{11} - 2 \beta_{12} - \beta_{13} - \beta_{14} + \beta_{15} ) q^{85} + ( -5 - 2 \beta_{2} + \beta_{3} - 2 \beta_{4} - 3 \beta_{7} - 3 \beta_{8} + 2 \beta_{9} - \beta_{10} - 2 \beta_{11} - 3 \beta_{12} - 3 \beta_{13} + 3 \beta_{14} + \beta_{15} + \beta_{16} + 2 \beta_{17} ) q^{87} + ( -\beta_{1} - 2 \beta_{3} - 2 \beta_{4} - \beta_{5} + 3 \beta_{6} - \beta_{7} + \beta_{8} + \beta_{9} + 3 \beta_{10} + 2 \beta_{11} - \beta_{12} - \beta_{13} + 2 \beta_{14} - \beta_{15} - \beta_{16} ) q^{88} + ( -4 - 2 \beta_{1} - 5 \beta_{2} - \beta_{6} + 2 \beta_{8} + \beta_{11} - \beta_{12} - 2 \beta_{13} + 2 \beta_{14} - \beta_{15} - \beta_{16} + \beta_{17} ) q^{89} + ( -2 + \beta_{1} + 2 \beta_{2} - 3 \beta_{3} + \beta_{4} + \beta_{5} - 2 \beta_{6} + 3 \beta_{7} - \beta_{8} - \beta_{9} + \beta_{10} + 2 \beta_{11} + 3 \beta_{13} - \beta_{15} - 2 \beta_{17} ) q^{90} + ( 1 - 3 \beta_{1} + \beta_{2} + \beta_{4} - \beta_{5} - \beta_{6} + 3 \beta_{7} - \beta_{8} + \beta_{9} - 2 \beta_{10} + 2 \beta_{11} + \beta_{14} - \beta_{15} - \beta_{16} + \beta_{17} ) q^{91} + ( 5 - 3 \beta_{1} - 2 \beta_{2} + \beta_{4} - \beta_{5} + 4 \beta_{6} - 2 \beta_{7} + 4 \beta_{8} - \beta_{9} - \beta_{10} - \beta_{11} + \beta_{12} - \beta_{14} - 2 \beta_{15} - \beta_{16} - \beta_{17} ) q^{92} + ( 3 - 2 \beta_{1} + 4 \beta_{3} - 2 \beta_{4} - 4 \beta_{5} - \beta_{6} + 6 \beta_{7} + 5 \beta_{8} - 2 \beta_{9} - 2 \beta_{10} - 2 \beta_{11} + 3 \beta_{12} + \beta_{13} - \beta_{16} ) q^{93} + ( 1 - \beta_{1} - \beta_{2} + \beta_{3} - \beta_{4} - 5 \beta_{5} - \beta_{6} + 5 \beta_{7} + \beta_{8} - 3 \beta_{9} - 3 \beta_{10} + 2 \beta_{12} - 2 \beta_{14} + 3 \beta_{15} + 2 \beta_{16} + \beta_{17} ) q^{94} + ( 4 + 2 \beta_{2} + 2 \beta_{4} + \beta_{6} - 3 \beta_{7} - \beta_{9} + \beta_{10} - \beta_{11} + 2 \beta_{12} + 3 \beta_{13} - 3 \beta_{14} + \beta_{15} - \beta_{17} ) q^{95} + ( -4 - \beta_{1} - 2 \beta_{2} + \beta_{3} - \beta_{4} - 4 \beta_{5} - 4 \beta_{6} + \beta_{7} - 2 \beta_{8} + \beta_{9} + 2 \beta_{12} - 3 \beta_{13} + \beta_{14} - 3 \beta_{15} + 2 \beta_{16} + \beta_{17} ) q^{96} + ( -2 \beta_{1} + \beta_{3} - \beta_{4} - 6 \beta_{5} - 2 \beta_{6} + 3 \beta_{7} + 2 \beta_{8} - \beta_{9} - 4 \beta_{10} + 3 \beta_{12} - \beta_{13} - 3 \beta_{14} - \beta_{15} - \beta_{17} ) q^{97} + ( -1 + 2 \beta_{1} + \beta_{2} + \beta_{3} - 2 \beta_{4} - \beta_{5} + 2 \beta_{7} - \beta_{8} - \beta_{9} - \beta_{10} - 2 \beta_{11} + \beta_{12} + 2 \beta_{13} + \beta_{14} + 2 \beta_{15} + 2 \beta_{16} ) q^{98} + ( 5 - 4 \beta_{1} + \beta_{2} + 3 \beta_{3} - 2 \beta_{5} + 2 \beta_{7} - \beta_{9} + 2 \beta_{12} - \beta_{13} - \beta_{14} + 2 \beta_{17} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$18q - 5q^{2} - 5q^{3} + 19q^{4} - 11q^{5} + 4q^{6} - 6q^{7} - 12q^{8} + 15q^{9} + O(q^{10})$$ $$18q - 5q^{2} - 5q^{3} + 19q^{4} - 11q^{5} + 4q^{6} - 6q^{7} - 12q^{8} + 15q^{9} - 7q^{10} - 2q^{11} - 16q^{12} - 7q^{13} - 4q^{14} + 3q^{15} + 9q^{16} - 11q^{17} - 25q^{18} - 31q^{19} - 25q^{20} - 19q^{22} - 11q^{23} + 18q^{24} + 9q^{25} - 27q^{26} - 23q^{27} - 20q^{28} - 37q^{29} + 17q^{30} - 12q^{31} - 39q^{32} - 38q^{33} - 14q^{34} + 16q^{35} + 47q^{36} + 19q^{37} + 56q^{38} - 46q^{39} + 6q^{40} - 7q^{41} + q^{42} + 7q^{44} - 23q^{45} + 47q^{46} - q^{47} - 15q^{48} - 6q^{49} + 3q^{50} - 38q^{51} + 15q^{52} + 3q^{53} - 67q^{54} - 28q^{55} - 81q^{56} + 46q^{57} + 34q^{58} + 17q^{59} - 83q^{60} - 28q^{61} - 33q^{62} - 26q^{63} + 10q^{64} - 16q^{65} - 72q^{66} + 18q^{67} + 53q^{68} - 7q^{69} + 34q^{70} - 86q^{71} + 2q^{72} + 27q^{73} - 79q^{74} - 31q^{75} - 59q^{76} - 43q^{77} + 91q^{78} + 17q^{79} - 8q^{80} - 10q^{81} - 13q^{82} - 12q^{83} - 32q^{84} - 28q^{85} - 43q^{87} + 23q^{88} - 51q^{89} + 10q^{90} + 20q^{91} + 18q^{92} + 30q^{93} + 15q^{94} - q^{95} - 20q^{96} - 19q^{97} + 5q^{98} + 38q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{18} - 5 x^{17} - 15 x^{16} + 106 x^{15} + 47 x^{14} - 897 x^{13} + 364 x^{12} + 3855 x^{11} - 3223 x^{10} - 8851 x^{9} + 9909 x^{8} + 10400 x^{7} - 14483 x^{6} - 5118 x^{5} + 9945 x^{4} - 27 x^{3} - 2493 x^{2} + 513 x - 27$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$($$$$13275 \nu^{17} + 49216 \nu^{16} - 638270 \nu^{15} - 817698 \nu^{14} + 10040797 \nu^{13} + 4293380 \nu^{12} - 76161915 \nu^{11} - 2151884 \nu^{10} + 314282184 \nu^{9} - 52495987 \nu^{8} - 726303448 \nu^{7} + 175925085 \nu^{6} + 916736762 \nu^{5} - 215589719 \nu^{4} - 580027476 \nu^{3} + 104938560 \nu^{2} + 140920983 \nu - 16390890$$$$)/1312164$$ $$\beta_{3}$$ $$=$$ $$($$$$20710 \nu^{17} - 38599 \nu^{16} - 543260 \nu^{15} + 949666 \nu^{14} + 5791002 \nu^{13} - 9559915 \nu^{12} - 31639730 \nu^{11} + 50308357 \nu^{10} + 90932936 \nu^{9} - 146365724 \nu^{8} - 120576325 \nu^{7} + 228076724 \nu^{6} + 27615381 \nu^{5} - 167553770 \nu^{4} + 64429719 \nu^{3} + 40814856 \nu^{2} - 32487966 \nu - 7389$$$$)/1312164$$ $$\beta_{4}$$ $$=$$ $$($$$$-44003 \nu^{17} + 173548 \nu^{16} + 771312 \nu^{15} - 3533264 \nu^{14} - 4677973 \nu^{13} + 28270866 \nu^{12} + 8945071 \nu^{11} - 112470798 \nu^{10} + 16877036 \nu^{9} + 233856473 \nu^{8} - 86897862 \nu^{7} - 253491883 \nu^{6} + 100861486 \nu^{5} + 150145347 \nu^{4} - 26028240 \nu^{3} - 48095238 \nu^{2} - 7993449 \nu + 401490$$$$)/1312164$$ $$\beta_{5}$$ $$=$$ $$($$$$-55583 \nu^{17} + 218229 \nu^{16} + 1056380 \nu^{15} - 4620818 \nu^{14} - 7782617 \nu^{13} + 39149731 \nu^{12} + 28825423 \nu^{11} - 169440829 \nu^{10} - 64968220 \nu^{9} + 397852707 \nu^{8} + 127874509 \nu^{7} - 500960689 \nu^{6} - 228333277 \nu^{5} + 317741963 \nu^{4} + 221535819 \nu^{3} - 89122374 \nu^{2} - 73666089 \nu + 9437373$$$$)/1312164$$ $$\beta_{6}$$ $$=$$ $$($$$$69430 \nu^{17} - 284265 \nu^{16} - 1359184 \nu^{15} + 6326890 \nu^{14} + 10180162 \nu^{13} - 57120029 \nu^{12} - 35687486 \nu^{11} + 268538099 \nu^{10} + 55159664 \nu^{9} - 702822648 \nu^{8} - 19113311 \nu^{7} + 1015190912 \nu^{6} - 24846673 \nu^{5} - 746262274 \nu^{4} + 11271585 \nu^{3} + 218833080 \nu^{2} + 3726342 \nu - 4659615$$$$)/1312164$$ $$\beta_{7}$$ $$=$$ $$($$$$-76293 \nu^{17} + 256828 \nu^{16} + 1599640 \nu^{15} - 5570484 \nu^{14} - 13573619 \nu^{13} + 48709646 \nu^{12} + 60465153 \nu^{11} - 219749186 \nu^{10} - 155901156 \nu^{9} + 544218431 \nu^{8} + 248450834 \nu^{7} - 729037413 \nu^{6} - 255948658 \nu^{5} + 485295733 \nu^{4} + 157106100 \nu^{3} - 128625066 \nu^{2} - 41178123 \nu + 5508270$$$$)/1312164$$ $$\beta_{8}$$ $$=$$ $$($$$$59046 \nu^{17} - 244357 \nu^{16} - 1095040 \nu^{15} + 5288955 \nu^{14} + 7351847 \nu^{13} - 46037891 \nu^{12} - 18893697 \nu^{11} + 206093975 \nu^{10} - 6203235 \nu^{9} - 504355352 \nu^{8} + 123911218 \nu^{7} + 663208641 \nu^{6} - 230450510 \nu^{5} - 424393738 \nu^{4} + 170060541 \nu^{3} + 94679412 \nu^{2} - 43668981 \nu + 4297068$$$$)/656082$$ $$\beta_{9}$$ $$=$$ $$($$$$124637 \nu^{17} - 455245 \nu^{16} - 2516574 \nu^{15} + 9987848 \nu^{14} + 19725175 \nu^{13} - 88235805 \nu^{12} - 74360329 \nu^{11} + 401793495 \nu^{10} + 131050912 \nu^{9} - 1004438171 \nu^{8} - 64409787 \nu^{7} + 1360900177 \nu^{6} - 94828159 \nu^{5} - 915839985 \nu^{4} + 130684977 \nu^{3} + 231376572 \nu^{2} - 43334415 \nu + 2059911$$$$)/1312164$$ $$\beta_{10}$$ $$=$$ $$($$$$141246 \nu^{17} - 576685 \nu^{16} - 2561026 \nu^{15} + 12268074 \nu^{14} + 16443476 \nu^{13} - 104501855 \nu^{12} - 36379716 \nu^{11} + 454974713 \nu^{10} - 48490122 \nu^{9} - 1073642906 \nu^{8} + 361330573 \nu^{7} + 1347595896 \nu^{6} - 608721671 \nu^{5} - 817237726 \nu^{4} + 426518757 \nu^{3} + 172094370 \nu^{2} - 109231326 \nu + 8750907$$$$)/1312164$$ $$\beta_{11}$$ $$=$$ $$($$$$167003 \nu^{17} - 649353 \nu^{16} - 3148820 \nu^{15} + 13928240 \nu^{14} + 21852299 \nu^{13} - 119904103 \nu^{12} - 61900753 \nu^{11} + 529610989 \nu^{10} + 16191022 \nu^{9} - 1276405047 \nu^{8} + 277767581 \nu^{7} + 1655953135 \nu^{6} - 561275045 \nu^{5} - 1061265551 \nu^{4} + 411372357 \nu^{3} + 250284606 \nu^{2} - 99718389 \nu + 6009885$$$$)/1312164$$ $$\beta_{12}$$ $$=$$ $$($$$$94434 \nu^{17} - 367211 \nu^{16} - 1822472 \nu^{15} + 8017242 \nu^{14} + 13158388 \nu^{13} - 70696261 \nu^{12} - 40633854 \nu^{11} + 322516021 \nu^{10} + 25631112 \nu^{9} - 810758590 \nu^{8} + 147391919 \nu^{7} + 1105999782 \nu^{6} - 361418209 \nu^{5} - 742219478 \nu^{4} + 311539797 \nu^{3} + 175378248 \nu^{2} - 91873764 \nu + 9088893$$$$)/656082$$ $$\beta_{13}$$ $$=$$ $$($$$$204816 \nu^{17} - 823249 \nu^{16} - 3821242 \nu^{15} + 17751510 \nu^{14} + 25899626 \nu^{13} - 153706883 \nu^{12} - 68256762 \nu^{11} + 683247641 \nu^{10} - 11491806 \nu^{9} - 1657489664 \nu^{8} + 404042737 \nu^{7} + 2161286394 \nu^{6} - 747127031 \nu^{5} - 1383174472 \nu^{4} + 526887669 \nu^{3} + 318068358 \nu^{2} - 126336348 \nu + 11305827$$$$)/1312164$$ $$\beta_{14}$$ $$=$$ $$($$$$245597 \nu^{17} - 1067329 \nu^{16} - 4433508 \nu^{15} + 23271404 \nu^{14} + 27818161 \nu^{13} - 204864951 \nu^{12} - 52751011 \nu^{11} + 932572533 \nu^{10} - 152533502 \nu^{9} - 2337456797 \nu^{8} + 859608609 \nu^{7} + 3175200109 \nu^{6} - 1460957353 \nu^{5} - 2117529921 \nu^{4} + 1061084985 \nu^{3} + 497629530 \nu^{2} - 274838571 \nu + 24053769$$$$)/1312164$$ $$\beta_{15}$$ $$=$$ $$($$$$-253199 \nu^{17} + 944792 \nu^{16} + 4923820 \nu^{15} - 20322290 \nu^{14} - 36421791 \nu^{13} + 175651034 \nu^{12} + 122991985 \nu^{11} - 780467606 \nu^{10} - 157879042 \nu^{9} + 1898220181 \nu^{8} - 84082444 \nu^{7} - 2498571613 \nu^{6} + 382011948 \nu^{5} + 1642014991 \nu^{4} - 290653932 \nu^{3} - 412418418 \nu^{2} + 59409549 \nu - 2620152$$$$)/1312164$$ $$\beta_{16}$$ $$=$$ $$($$$$160886 \nu^{17} - 603273 \nu^{16} - 3199685 \nu^{15} + 13198295 \nu^{14} + 24565100 \nu^{13} - 116487784 \nu^{12} - 89543356 \nu^{11} + 531312130 \nu^{10} + 146650858 \nu^{9} - 1335019437 \nu^{8} - 46035718 \nu^{7} + 1826041525 \nu^{6} - 138676475 \nu^{5} - 1248308096 \nu^{4} + 145462917 \nu^{3} + 327551883 \nu^{2} - 38129769 \nu - 572175$$$$)/656082$$ $$\beta_{17}$$ $$=$$ $$($$$$249517 \nu^{17} - 1016292 \nu^{16} - 4644175 \nu^{15} + 21964198 \nu^{14} + 31393966 \nu^{13} - 190931687 \nu^{12} - 82343444 \nu^{11} + 853971125 \nu^{10} - 15885937 \nu^{9} - 2090623248 \nu^{8} + 495031120 \nu^{7} + 2759880713 \nu^{6} - 915641593 \nu^{5} - 1791688513 \nu^{4} + 651327492 \nu^{3} + 420139959 \nu^{2} - 160192611 \nu + 13820985$$$$)/656082$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{7} - \beta_{5} + \beta_{3} + 3$$ $$\nu^{3}$$ $$=$$ $$\beta_{17} + \beta_{16} + \beta_{15} - \beta_{11} - \beta_{9} - \beta_{8} + 2 \beta_{7} - 2 \beta_{6} - \beta_{5} - \beta_{4} + \beta_{3} + \beta_{2} + 5 \beta_{1} - 1$$ $$\nu^{4}$$ $$=$$ $$\beta_{16} + \beta_{13} + \beta_{12} - 2 \beta_{11} - \beta_{10} - \beta_{9} + 8 \beta_{7} - 2 \beta_{6} - 8 \beta_{5} - \beta_{4} + 7 \beta_{3} + \beta_{1} + 13$$ $$\nu^{5}$$ $$=$$ $$7 \beta_{17} + 10 \beta_{16} + 7 \beta_{15} + \beta_{12} - 9 \beta_{11} - 2 \beta_{10} - 9 \beta_{9} - 7 \beta_{8} + 18 \beta_{7} - 18 \beta_{6} - 11 \beta_{5} - 8 \beta_{4} + 9 \beta_{3} + 6 \beta_{2} + 28 \beta_{1} - 7$$ $$\nu^{6}$$ $$=$$ $$2 \beta_{17} + 15 \beta_{16} + \beta_{15} + 7 \beta_{13} + 9 \beta_{12} - 23 \beta_{11} - 9 \beta_{10} - 12 \beta_{9} - 4 \beta_{8} + 58 \beta_{7} - 25 \beta_{6} - 56 \beta_{5} - 11 \beta_{4} + 46 \beta_{3} - \beta_{2} + 16 \beta_{1} + 60$$ $$\nu^{7}$$ $$=$$ $$45 \beta_{17} + 82 \beta_{16} + 42 \beta_{15} - \beta_{14} - 3 \beta_{13} + 14 \beta_{12} - 73 \beta_{11} - 24 \beta_{10} - 67 \beta_{9} - 46 \beta_{8} + 142 \beta_{7} - 138 \beta_{6} - 98 \beta_{5} - 56 \beta_{4} + 72 \beta_{3} + 29 \beta_{2} + 167 \beta_{1} - 43$$ $$\nu^{8}$$ $$=$$ $$30 \beta_{17} + 152 \beta_{16} + 10 \beta_{15} + 33 \beta_{13} + 69 \beta_{12} - 206 \beta_{11} - 73 \beta_{10} - 108 \beta_{9} - 52 \beta_{8} + 424 \beta_{7} - 240 \beta_{6} - 394 \beta_{5} - 94 \beta_{4} + 306 \beta_{3} - 14 \beta_{2} + 161 \beta_{1} + 278$$ $$\nu^{9}$$ $$=$$ $$296 \beta_{17} + 635 \beta_{16} + 240 \beta_{15} - 10 \beta_{14} - 55 \beta_{13} + 136 \beta_{12} - 579 \beta_{11} - 221 \beta_{10} - 472 \beta_{9} - 315 \beta_{8} + 1085 \beta_{7} - 1031 \beta_{6} - 808 \beta_{5} - 387 \beta_{4} + 556 \beta_{3} + 121 \beta_{2} + 1042 \beta_{1} - 271$$ $$\nu^{10}$$ $$=$$ $$316 \beta_{17} + 1342 \beta_{16} + 69 \beta_{15} + 10 \beta_{14} + 78 \beta_{13} + 510 \beta_{12} - 1692 \beta_{11} - 590 \beta_{10} - 868 \beta_{9} - 503 \beta_{8} + 3135 \beta_{7} - 2093 \beta_{6} - 2827 \beta_{5} - 749 \beta_{4} + 2078 \beta_{3} - 146 \beta_{2} + 1385 \beta_{1} + 1229$$ $$\nu^{11}$$ $$=$$ $$2009 \beta_{17} + 4827 \beta_{16} + 1330 \beta_{15} - 40 \beta_{14} - 677 \beta_{13} + 1147 \beta_{12} - 4541 \beta_{11} - 1862 \beta_{10} - 3259 \beta_{9} - 2253 \beta_{8} + 8199 \beta_{7} - 7724 \beta_{6} - 6419 \beta_{5} - 2710 \beta_{4} + 4214 \beta_{3} + 374 \beta_{2} + 6742 \beta_{1} - 1846$$ $$\nu^{12}$$ $$=$$ $$2884 \beta_{17} + 11118 \beta_{16} + 381 \beta_{15} + 236 \beta_{14} - 629 \beta_{13} + 3722 \beta_{12} - 13363 \beta_{11} - 4774 \beta_{10} - 6582 \beta_{9} - 4401 \beta_{8} + 23307 \beta_{7} - 17419 \beta_{6} - 20616 \beta_{5} - 5820 \beta_{4} + 14367 \beta_{3} - 1373 \beta_{2} + 11104 \beta_{1} + 4724$$ $$\nu^{13}$$ $$=$$ $$13986 \beta_{17} + 36498 \beta_{16} + 7132 \beta_{15} + 284 \beta_{14} - 7029 \beta_{13} + 9034 \beta_{12} - 35326 \beta_{11} - 15080 \beta_{10} - 22338 \beta_{9} - 16633 \beta_{8} + 61687 \beta_{7} - 58350 \beta_{6} - 49990 \beta_{5} - 19326 \beta_{4} + 31561 \beta_{3} - 93 \beta_{2} + 44941 \beta_{1} - 13667$$ $$\nu^{14}$$ $$=$$ $$24429 \beta_{17} + 89053 \beta_{16} + 1512 \beta_{15} + 3507 \beta_{14} - 12885 \beta_{13} + 27005 \beta_{12} - 103453 \beta_{11} - 38486 \beta_{10} - 48265 \beta_{9} - 36816 \beta_{8} + 173685 \beta_{7} - 141204 \beta_{6} - 152080 \beta_{5} - 44738 \beta_{4} + 100734 \beta_{3} - 12230 \beta_{2} + 85859 \beta_{1} + 10825$$ $$\nu^{15}$$ $$=$$ $$99222 \beta_{17} + 275782 \beta_{16} + 36415 \beta_{15} + 8100 \beta_{14} - 66561 \beta_{13} + 68573 \beta_{12} - 273052 \beta_{11} - 119674 \beta_{10} - 152784 \beta_{9} - 125404 \beta_{8} + 463302 \beta_{7} - 444135 \beta_{6} - 384896 \beta_{5} - 140118 \beta_{4} + 234512 \beta_{3} - 16495 \beta_{2} + 306917 \beta_{1} - 107636$$ $$\nu^{16}$$ $$=$$ $$198097 \beta_{17} + 699601 \beta_{16} + 156 \beta_{15} + 42115 \beta_{14} - 150414 \beta_{13} + 195355 \beta_{12} - 792366 \beta_{11} - 308182 \beta_{10} - 346727 \beta_{9} - 300668 \beta_{8} + 1296038 \beta_{7} - 1126248 \beta_{6} - 1130729 \beta_{5} - 342050 \beta_{4} + 713907 \beta_{3} - 105095 \beta_{2} + 650835 \beta_{1} - 58659$$ $$\nu^{17}$$ $$=$$ $$713751 \beta_{17} + 2085523 \beta_{16} + 170041 \beta_{15} + 112366 \beta_{14} - 595897 \beta_{13} + 509658 \beta_{12} - 2099810 \beta_{11} - 938905 \beta_{10} - 1045038 \beta_{9} - 958176 \beta_{8} + 3477606 \beta_{7} - 3398743 \beta_{6} - 2943361 \beta_{5} - 1029194 \beta_{4} + 1733440 \beta_{3} - 210984 \beta_{2} + 2137240 \beta_{1} - 877416$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1
 2.75930 2.61223 2.31259 2.16300 1.78421 1.69598 1.10272 1.04477 0.847401 0.131407 0.0953887 −0.800597 −1.09100 −1.18520 −1.70036 −2.17399 −2.27976 −2.31808
−2.75930 −1.44592 5.61371 0.916508 3.98973 1.94308 −9.97129 −0.909309 −2.52892
1.2 −2.61223 1.92275 4.82373 −2.98756 −5.02267 0.679767 −7.37622 0.696981 7.80417
1.3 −2.31259 −2.69989 3.34809 −2.53973 6.24374 2.18729 −3.11758 4.28939 5.87337
1.4 −2.16300 −2.91800 2.67858 2.73005 6.31164 −2.02167 −1.46776 5.51474 −5.90511
1.5 −1.78421 2.96672 1.18339 −0.596316 −5.29325 −4.36308 1.45700 5.80145 1.06395
1.6 −1.69598 −1.71311 0.876333 0.385406 2.90540 −3.49171 1.90571 −0.0652397 −0.653640
1.7 −1.10272 −0.645986 −0.784019 −3.32240 0.712338 −2.46546 3.06998 −2.58270 3.66366
1.8 −1.04477 2.52412 −0.908449 −0.499809 −2.63714 3.30668 3.03867 3.37119 0.522187
1.9 −0.847401 0.0176385 −1.28191 3.43657 −0.0149469 0.269043 2.78110 −2.99969 −2.91216
1.10 −0.131407 1.32082 −1.98273 1.78074 −0.173564 0.0149856 0.523357 −1.25544 −0.234000
1.11 −0.0953887 −2.36384 −1.99090 −3.81314 0.225484 −2.61962 0.380687 2.58774 0.363731
1.12 0.800597 −0.744601 −1.35904 1.40135 −0.596125 4.75856 −2.68924 −2.44557 1.12192
1.13 1.09100 1.45952 −0.809726 1.35087 1.59233 0.216325 −3.06540 −0.869803 1.47379
1.14 1.18520 −2.30227 −0.595300 −3.24275 −2.72866 2.76837 −3.07595 2.30047 −3.84331
1.15 1.70036 1.15587 0.891224 −1.76341 1.96539 0.594565 −1.88532 −1.66397 −2.99843
1.16 2.17399 0.327650 2.72622 −0.0263127 0.712306 −3.12927 1.57880 −2.89265 −0.0572034
1.17 2.27976 −3.20997 3.19730 −0.137674 −7.31797 −1.86944 2.72957 7.30393 −0.313864
1.18 2.31808 1.34851 3.37350 −4.07240 3.12595 −2.77841 3.18389 −1.18153 −9.44015
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1.18 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$43$$ $$1$$

## Inner twists

This newform does not admit any (nontrivial) inner twists.

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1849.2.a.n 18
43.b odd 2 1 1849.2.a.o 18
43.g even 21 2 43.2.g.a 36
129.o odd 42 2 387.2.y.c 36
172.o odd 42 2 688.2.bg.c 36

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
43.2.g.a 36 43.g even 21 2
387.2.y.c 36 129.o odd 42 2
688.2.bg.c 36 172.o odd 42 2
1849.2.a.n 18 1.a even 1 1 trivial
1849.2.a.o 18 43.b odd 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{18} + \cdots$$ acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(1849))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 + 5 T + 21 T^{2} + 64 T^{3} + 179 T^{4} + 437 T^{5} + 1008 T^{6} + 2147 T^{7} + 4381 T^{8} + 8465 T^{9} + 15785 T^{10} + 28194 T^{11} + 48769 T^{12} + 81286 T^{13} + 131557 T^{14} + 205927 T^{15} + 313527 T^{16} + 462513 T^{17} + 664265 T^{18} + 925026 T^{19} + 1254108 T^{20} + 1647416 T^{21} + 2104912 T^{22} + 2601152 T^{23} + 3121216 T^{24} + 3608832 T^{25} + 4040960 T^{26} + 4334080 T^{27} + 4486144 T^{28} + 4397056 T^{29} + 4128768 T^{30} + 3579904 T^{31} + 2932736 T^{32} + 2097152 T^{33} + 1376256 T^{34} + 655360 T^{35} + 262144 T^{36}$$
$3$ $$1 + 5 T + 32 T^{2} + 121 T^{3} + 487 T^{4} + 1519 T^{5} + 4810 T^{6} + 12993 T^{7} + 35099 T^{8} + 84694 T^{9} + 203425 T^{10} + 447800 T^{11} + 980330 T^{12} + 1997474 T^{13} + 4049010 T^{14} + 7709756 T^{15} + 14612100 T^{16} + 26142572 T^{17} + 46568083 T^{18} + 78427716 T^{19} + 131508900 T^{20} + 208163412 T^{21} + 327969810 T^{22} + 485386182 T^{23} + 714660570 T^{24} + 979338600 T^{25} + 1334671425 T^{26} + 1667032002 T^{27} + 2072560851 T^{28} + 2301670971 T^{29} + 2556231210 T^{30} + 2421776637 T^{31} + 2329305903 T^{32} + 1736217747 T^{33} + 1377495072 T^{34} + 645700815 T^{35} + 387420489 T^{36}$$
$5$ $$1 + 11 T + 101 T^{2} + 669 T^{3} + 3903 T^{4} + 19466 T^{5} + 88477 T^{6} + 363160 T^{7} + 1386460 T^{8} + 4905936 T^{9} + 16359692 T^{10} + 51316565 T^{11} + 153036579 T^{12} + 433243564 T^{13} + 1172911578 T^{14} + 3031457583 T^{15} + 7519728391 T^{16} + 17863052655 T^{17} + 40796547233 T^{18} + 89315263275 T^{19} + 187993209775 T^{20} + 378932197875 T^{21} + 733069736250 T^{22} + 1353886137500 T^{23} + 2391196546875 T^{24} + 4009106640625 T^{25} + 6390504687500 T^{26} + 9581906250000 T^{27} + 13539648437500 T^{28} + 17732421875000 T^{29} + 21600830078125 T^{30} + 23762207031250 T^{31} + 23822021484375 T^{32} + 20416259765625 T^{33} + 15411376953125 T^{34} + 8392333984375 T^{35} + 3814697265625 T^{36}$$
$7$ $$1 + 6 T + 84 T^{2} + 404 T^{3} + 3274 T^{4} + 13241 T^{5} + 80350 T^{6} + 281196 T^{7} + 1411311 T^{8} + 4354137 T^{9} + 19080222 T^{10} + 52630458 T^{11} + 208882075 T^{12} + 522279147 T^{13} + 1931335668 T^{14} + 4450614354 T^{15} + 15661259403 T^{16} + 33919735838 T^{17} + 114695007659 T^{18} + 237438150866 T^{19} + 767401710747 T^{20} + 1526560723422 T^{21} + 4637136938868 T^{22} + 8777945623629 T^{23} + 24574767241675 T^{24} + 43343445272694 T^{25} + 109993682865822 T^{26} + 175705133322159 T^{27} + 398660426141439 T^{28} + 556016370824628 T^{29} + 1112147426600350 T^{30} + 1282907386799087 T^{31} + 2220502340507626 T^{32} + 1918014850016972 T^{33} + 2791566167846484 T^{34} + 1395783083923242 T^{35} + 1628413597910449 T^{36}$$
$11$ $$1 + 2 T + 106 T^{2} + 229 T^{3} + 5690 T^{4} + 12747 T^{5} + 204282 T^{6} + 465265 T^{7} + 5488286 T^{8} + 12572724 T^{9} + 117435749 T^{10} + 268589062 T^{11} + 2084590259 T^{12} + 4722228468 T^{13} + 31603117559 T^{14} + 70103709703 T^{15} + 417787467273 T^{16} + 892536995699 T^{17} + 4878827245269 T^{18} + 9817906952689 T^{19} + 50552283540033 T^{20} + 93308037614693 T^{21} + 462701244181319 T^{22} + 760519616999868 T^{23} + 3692978803824299 T^{24} + 5234040979923602 T^{25} + 25173395745036869 T^{26} + 29645825525380284 T^{27} + 142352004313723886 T^{28} + 132745534426826915 T^{29} + 641124425653319322 T^{30} + 440061011698688457 T^{31} + 2160776553088641290 T^{32} + 956589830796184079 T^{33} + 4870671365538649066 T^{34} + 1010894056998587542 T^{35} + 5559917313492231481 T^{36}$$
$13$ $$1 + 7 T + 177 T^{2} + 1018 T^{3} + 14604 T^{4} + 71860 T^{5} + 763549 T^{6} + 3294581 T^{7} + 28732485 T^{8} + 110474722 T^{9} + 834095319 T^{10} + 2891021534 T^{11} + 19507460320 T^{12} + 61516713994 T^{13} + 378392228455 T^{14} + 1094212986566 T^{15} + 6206713401763 T^{16} + 16564106110855 T^{17} + 87079098238593 T^{18} + 215333379441115 T^{19} + 1048934564897947 T^{20} + 2403985931485502 T^{21} + 10807260436903255 T^{22} + 22840725288974242 T^{23} + 94158785039718880 T^{24} + 181407313873565078 T^{25} + 680397175950594999 T^{26} + 1171529120181349306 T^{27} + 3961017049174014765 T^{28} + 5904417583146813497 T^{29} + 17789229597185245069 T^{30} + 21764605159719300580 T^{31} + 57501444736752416556 T^{32} + 52107239088344390626 T^{33} +$$$$11\!\cdots\!57$$$$T^{34} + 60552911435669365531 T^{35} +$$$$11\!\cdots\!29$$$$T^{36}$$
$17$ $$1 + 11 T + 231 T^{2} + 2107 T^{3} + 25583 T^{4} + 199091 T^{5} + 1812738 T^{6} + 12335903 T^{7} + 92694466 T^{8} + 562106618 T^{9} + 3654967661 T^{10} + 20022851568 T^{11} + 115745414311 T^{12} + 578272697713 T^{13} + 3021053869261 T^{14} + 13847422994218 T^{15} + 66040420965504 T^{16} + 278550252870720 T^{17} + 1219711875918623 T^{18} + 4735354298802240 T^{19} + 19085681659030656 T^{20} + 68032389170593034 T^{21} + 252321440214547981 T^{22} + 821064537756687041 T^{23} + 2793812924365349959 T^{24} + 8216150342089089264 T^{25} + 25496167857835115501 T^{26} + 66659030193530357146 T^{27} +$$$$18\!\cdots\!34$$$$T^{28} +$$$$42\!\cdots\!99$$$$T^{29} +$$$$10\!\cdots\!18$$$$T^{30} +$$$$19\!\cdots\!67$$$$T^{31} +$$$$43\!\cdots\!07$$$$T^{32} +$$$$60\!\cdots\!51$$$$T^{33} +$$$$11\!\cdots\!11$$$$T^{34} +$$$$90\!\cdots\!47$$$$T^{35} +$$$$14\!\cdots\!09$$$$T^{36}$$
$19$ $$1 + 31 T + 675 T^{2} + 10622 T^{3} + 138940 T^{4} + 1530782 T^{5} + 14850476 T^{6} + 127843653 T^{7} + 997668299 T^{8} + 7093671340 T^{9} + 46564190903 T^{10} + 283329671122 T^{11} + 1614507118332 T^{12} + 8652670079090 T^{13} + 44043302083234 T^{14} + 213996678807527 T^{15} + 1002183404208665 T^{16} + 4542498121139650 T^{17} + 20064380211625821 T^{18} + 86307464301653350 T^{19} + 361788208919328065 T^{20} + 1467803219940827693 T^{21} + 5739767170789138114 T^{22} + 21424867730164669910 T^{23} + 75955909762700190492 T^{24} +$$$$25\!\cdots\!58$$$$T^{25} +$$$$79\!\cdots\!23$$$$T^{26} +$$$$22\!\cdots\!60$$$$T^{27} +$$$$61\!\cdots\!99$$$$T^{28} +$$$$14\!\cdots\!07$$$$T^{29} +$$$$32\!\cdots\!36$$$$T^{30} +$$$$64\!\cdots\!38$$$$T^{31} +$$$$11\!\cdots\!40$$$$T^{32} +$$$$16\!\cdots\!78$$$$T^{33} +$$$$19\!\cdots\!75$$$$T^{34} +$$$$16\!\cdots\!09$$$$T^{35} +$$$$10\!\cdots\!41$$$$T^{36}$$
$23$ $$1 + 11 T + 345 T^{2} + 3132 T^{3} + 54949 T^{4} + 426589 T^{5} + 5467252 T^{6} + 37174733 T^{7} + 385954917 T^{8} + 2338057961 T^{9} + 20753234723 T^{10} + 113433389569 T^{11} + 889059106742 T^{12} + 4426383841877 T^{13} + 31268375908075 T^{14} + 142781111925693 T^{15} + 920860147265066 T^{16} + 3872880455307088 T^{17} + 22970106119407687 T^{18} + 89076250472063024 T^{19} + 487135017903219914 T^{20} + 1737217788799906731 T^{21} + 8750173582491616075 T^{22} + 28489724655978135811 T^{23} +$$$$13\!\cdots\!38$$$$T^{24} +$$$$38\!\cdots\!43$$$$T^{25} +$$$$16\!\cdots\!63$$$$T^{26} +$$$$42\!\cdots\!43$$$$T^{27} +$$$$15\!\cdots\!33$$$$T^{28} +$$$$35\!\cdots\!91$$$$T^{29} +$$$$11\!\cdots\!92$$$$T^{30} +$$$$21\!\cdots\!87$$$$T^{31} +$$$$63\!\cdots\!41$$$$T^{32} +$$$$83\!\cdots\!24$$$$T^{33} +$$$$21\!\cdots\!45$$$$T^{34} +$$$$15\!\cdots\!33$$$$T^{35} +$$$$32\!\cdots\!69$$$$T^{36}$$
$29$ $$1 + 37 T + 947 T^{2} + 17923 T^{3} + 283912 T^{4} + 3855141 T^{5} + 46586206 T^{6} + 507248339 T^{7} + 5061572456 T^{8} + 46624628122 T^{9} + 400155735430 T^{10} + 3214306403195 T^{11} + 24300142007900 T^{12} + 173384015282340 T^{13} + 1171644165735371 T^{14} + 7510398411648064 T^{15} + 45763756699519426 T^{16} + 265249439219992246 T^{17} + 1463952283209284935 T^{18} + 7692233737379775134 T^{19} + 38487319384295837266 T^{20} +$$$$18\!\cdots\!96$$$$T^{21} +$$$$82\!\cdots\!51$$$$T^{22} +$$$$35\!\cdots\!60$$$$T^{23} +$$$$14\!\cdots\!00$$$$T^{24} +$$$$55\!\cdots\!55$$$$T^{25} +$$$$20\!\cdots\!30$$$$T^{26} +$$$$67\!\cdots\!18$$$$T^{27} +$$$$21\!\cdots\!56$$$$T^{28} +$$$$61\!\cdots\!31$$$$T^{29} +$$$$16\!\cdots\!46$$$$T^{30} +$$$$39\!\cdots\!49$$$$T^{31} +$$$$84\!\cdots\!72$$$$T^{32} +$$$$15\!\cdots\!27$$$$T^{33} +$$$$23\!\cdots\!87$$$$T^{34} +$$$$26\!\cdots\!33$$$$T^{35} +$$$$21\!\cdots\!61$$$$T^{36}$$
$31$ $$1 + 12 T + 394 T^{2} + 4062 T^{3} + 73790 T^{4} + 667459 T^{5} + 8810142 T^{6} + 71153719 T^{7} + 759179657 T^{8} + 5555787033 T^{9} + 50683565295 T^{10} + 340153856696 T^{11} + 2745099275027 T^{12} + 17046603704779 T^{13} + 124394247414918 T^{14} + 718550937993414 T^{15} + 4809074473741562 T^{16} + 25882288865469751 T^{17} + 160293089661061235 T^{18} + 802350954829562281 T^{19} + 4621520569265641082 T^{20} + 21406350993761796474 T^{21} +$$$$11\!\cdots\!78$$$$T^{22} +$$$$48\!\cdots\!29$$$$T^{23} +$$$$24\!\cdots\!87$$$$T^{24} +$$$$93\!\cdots\!56$$$$T^{25} +$$$$43\!\cdots\!95$$$$T^{26} +$$$$14\!\cdots\!43$$$$T^{27} +$$$$62\!\cdots\!57$$$$T^{28} +$$$$18\!\cdots\!89$$$$T^{29} +$$$$69\!\cdots\!62$$$$T^{30} +$$$$16\!\cdots\!69$$$$T^{31} +$$$$55\!\cdots\!90$$$$T^{32} +$$$$95\!\cdots\!62$$$$T^{33} +$$$$28\!\cdots\!14$$$$T^{34} +$$$$27\!\cdots\!32$$$$T^{35} +$$$$69\!\cdots\!41$$$$T^{36}$$
$37$ $$1 - 19 T + 589 T^{2} - 8420 T^{3} + 151063 T^{4} - 1759614 T^{5} + 23459736 T^{6} - 233272537 T^{7} + 2542189646 T^{8} - 22257059122 T^{9} + 208488795367 T^{10} - 1643111338752 T^{11} + 13649989650958 T^{12} - 98401118005386 T^{13} + 740427345007403 T^{14} - 4935637642077440 T^{15} + 34126021579140508 T^{16} - 211580817085833068 T^{17} + 1356370300983331895 T^{18} - 7828490232175823516 T^{19} + 46718523541843355452 T^{20} -$$$$25\!\cdots\!20$$$$T^{21} +$$$$13\!\cdots\!83$$$$T^{22} -$$$$68\!\cdots\!02$$$$T^{23} +$$$$35\!\cdots\!22$$$$T^{24} -$$$$15\!\cdots\!16$$$$T^{25} +$$$$73\!\cdots\!07$$$$T^{26} -$$$$28\!\cdots\!94$$$$T^{27} +$$$$12\!\cdots\!54$$$$T^{28} -$$$$41\!\cdots\!81$$$$T^{29} +$$$$15\!\cdots\!16$$$$T^{30} -$$$$42\!\cdots\!58$$$$T^{31} +$$$$13\!\cdots\!07$$$$T^{32} -$$$$28\!\cdots\!60$$$$T^{33} +$$$$72\!\cdots\!49$$$$T^{34} -$$$$86\!\cdots\!23$$$$T^{35} +$$$$16\!\cdots\!29$$$$T^{36}$$
$41$ $$1 + 7 T + 420 T^{2} + 3240 T^{3} + 90202 T^{4} + 716786 T^{5} + 13103701 T^{6} + 102786964 T^{7} + 1433407248 T^{8} + 10828886149 T^{9} + 124839503390 T^{10} + 895792488941 T^{11} + 8953259514722 T^{12} + 60526589311090 T^{13} + 540233339209753 T^{14} + 3422358652721154 T^{15} + 27793767354912440 T^{16} + 164244156889528625 T^{17} + 1228158616200818539 T^{18} + 6734010432470673625 T^{19} + 46721322923607811640 T^{20} +$$$$23\!\cdots\!34$$$$T^{21} +$$$$15\!\cdots\!33$$$$T^{22} +$$$$70\!\cdots\!90$$$$T^{23} +$$$$42\!\cdots\!02$$$$T^{24} +$$$$17\!\cdots\!21$$$$T^{25} +$$$$99\!\cdots\!90$$$$T^{26} +$$$$35\!\cdots\!89$$$$T^{27} +$$$$19\!\cdots\!48$$$$T^{28} +$$$$56\!\cdots\!24$$$$T^{29} +$$$$29\!\cdots\!81$$$$T^{30} +$$$$66\!\cdots\!06$$$$T^{31} +$$$$34\!\cdots\!22$$$$T^{32} +$$$$50\!\cdots\!40$$$$T^{33} +$$$$26\!\cdots\!20$$$$T^{34} +$$$$18\!\cdots\!67$$$$T^{35} +$$$$10\!\cdots\!21$$$$T^{36}$$
$43$ 1
$47$ $$1 + T + 441 T^{2} + 378 T^{3} + 100243 T^{4} + 64059 T^{5} + 15507867 T^{6} + 6421735 T^{7} + 1823396570 T^{8} + 389880529 T^{9} + 172695386616 T^{10} + 9060656475 T^{11} + 13634569947126 T^{12} - 876954772658 T^{13} + 916565401866601 T^{14} - 124355291786094 T^{15} + 53146338562582634 T^{16} - 8694613538868321 T^{17} + 2676993076546814875 T^{18} - 408646836326811087 T^{19} +$$$$11\!\cdots\!06$$$$T^{20} - 12910939459107637362 T^{21} +$$$$44\!\cdots\!81$$$$T^{22} -$$$$20\!\cdots\!06$$$$T^{23} +$$$$14\!\cdots\!54$$$$T^{24} +$$$$45\!\cdots\!25$$$$T^{25} +$$$$41\!\cdots\!76$$$$T^{26} +$$$$43\!\cdots\!43$$$$T^{27} +$$$$95\!\cdots\!30$$$$T^{28} +$$$$15\!\cdots\!05$$$$T^{29} +$$$$18\!\cdots\!47$$$$T^{30} +$$$$34\!\cdots\!93$$$$T^{31} +$$$$25\!\cdots\!67$$$$T^{32} +$$$$45\!\cdots\!54$$$$T^{33} +$$$$25\!\cdots\!61$$$$T^{34} +$$$$26\!\cdots\!87$$$$T^{35} +$$$$12\!\cdots\!89$$$$T^{36}$$
$53$ $$1 - 3 T + 471 T^{2} - 1056 T^{3} + 113523 T^{4} - 153549 T^{5} + 18344467 T^{6} - 6960585 T^{7} + 2222494860 T^{8} + 1426226391 T^{9} + 215187452214 T^{10} + 355484655993 T^{11} + 17400446510730 T^{12} + 44440317183162 T^{13} + 1215028415653965 T^{14} + 3904928969645238 T^{15} + 75147380191168470 T^{16} + 262691650979577669 T^{17} + 4185046851497965019 T^{18} + 13922657501917616457 T^{19} +$$$$21\!\cdots\!30$$$$T^{20} +$$$$58\!\cdots\!26$$$$T^{21} +$$$$95\!\cdots\!65$$$$T^{22} +$$$$18\!\cdots\!66$$$$T^{23} +$$$$38\!\cdots\!70$$$$T^{24} +$$$$41\!\cdots\!41$$$$T^{25} +$$$$13\!\cdots\!54$$$$T^{26} +$$$$47\!\cdots\!03$$$$T^{27} +$$$$38\!\cdots\!40$$$$T^{28} -$$$$64\!\cdots\!45$$$$T^{29} +$$$$90\!\cdots\!47$$$$T^{30} -$$$$39\!\cdots\!77$$$$T^{31} +$$$$15\!\cdots\!87$$$$T^{32} -$$$$77\!\cdots\!92$$$$T^{33} +$$$$18\!\cdots\!91$$$$T^{34} -$$$$61\!\cdots\!39$$$$T^{35} +$$$$10\!\cdots\!89$$$$T^{36}$$
$59$ $$1 - 17 T + 890 T^{2} - 12969 T^{3} + 371964 T^{4} - 4775447 T^{5} + 98199544 T^{6} - 1131507793 T^{7} + 18510827194 T^{8} - 193828091271 T^{9} + 2662524759788 T^{10} - 25536895733756 T^{11} + 304262792505393 T^{12} - 2684476478739574 T^{13} + 28341034676025816 T^{14} - 230279921162601801 T^{15} + 2186359093053923584 T^{16} - 16332723780371798493 T^{17} +$$$$14\!\cdots\!85$$$$T^{18} -$$$$96\!\cdots\!87$$$$T^{19} +$$$$76\!\cdots\!04$$$$T^{20} -$$$$47\!\cdots\!79$$$$T^{21} +$$$$34\!\cdots\!76$$$$T^{22} -$$$$19\!\cdots\!26$$$$T^{23} +$$$$12\!\cdots\!13$$$$T^{24} -$$$$63\!\cdots\!64$$$$T^{25} +$$$$39\!\cdots\!48$$$$T^{26} -$$$$16\!\cdots\!69$$$$T^{27} +$$$$94\!\cdots\!94$$$$T^{28} -$$$$34\!\cdots\!87$$$$T^{29} +$$$$17\!\cdots\!64$$$$T^{30} -$$$$50\!\cdots\!13$$$$T^{31} +$$$$23\!\cdots\!04$$$$T^{32} -$$$$47\!\cdots\!31$$$$T^{33} +$$$$19\!\cdots\!90$$$$T^{34} -$$$$21\!\cdots\!23$$$$T^{35} +$$$$75\!\cdots\!21$$$$T^{36}$$
$61$ $$1 + 28 T + 1019 T^{2} + 20072 T^{3} + 437347 T^{4} + 6824646 T^{5} + 112590320 T^{6} + 1473831616 T^{7} + 20079161609 T^{8} + 228034178555 T^{9} + 2684926553124 T^{10} + 27078104395535 T^{11} + 283831171459274 T^{12} + 2590252054230334 T^{13} + 24721592527244465 T^{14} + 207570410437628343 T^{15} + 1836081624449708055 T^{16} + 14378049045728950908 T^{17} +$$$$11\!\cdots\!89$$$$T^{18} +$$$$87\!\cdots\!88$$$$T^{19} +$$$$68\!\cdots\!55$$$$T^{20} +$$$$47\!\cdots\!83$$$$T^{21} +$$$$34\!\cdots\!65$$$$T^{22} +$$$$21\!\cdots\!34$$$$T^{23} +$$$$14\!\cdots\!14$$$$T^{24} +$$$$85\!\cdots\!35$$$$T^{25} +$$$$51\!\cdots\!44$$$$T^{26} +$$$$26\!\cdots\!55$$$$T^{27} +$$$$14\!\cdots\!09$$$$T^{28} +$$$$64\!\cdots\!76$$$$T^{29} +$$$$29\!\cdots\!20$$$$T^{30} +$$$$11\!\cdots\!26$$$$T^{31} +$$$$43\!\cdots\!27$$$$T^{32} +$$$$12\!\cdots\!72$$$$T^{33} +$$$$37\!\cdots\!59$$$$T^{34} +$$$$62\!\cdots\!88$$$$T^{35} +$$$$13\!\cdots\!81$$$$T^{36}$$
$67$ $$1 - 18 T + 747 T^{2} - 11021 T^{3} + 266098 T^{4} - 3386261 T^{5} + 61776616 T^{6} - 700286712 T^{7} + 10647673611 T^{8} - 109744348215 T^{9} + 1459937506974 T^{10} - 13858055653902 T^{11} + 165766321224565 T^{12} - 1460245745643108 T^{13} + 15966332829424392 T^{14} - 131058934756843494 T^{15} + 1323312512220230094 T^{16} - 10136102287940402999 T^{17} + 95100333637183309469 T^{18} -$$$$67\!\cdots\!33$$$$T^{19} +$$$$59\!\cdots\!66$$$$T^{20} -$$$$39\!\cdots\!22$$$$T^{21} +$$$$32\!\cdots\!32$$$$T^{22} -$$$$19\!\cdots\!56$$$$T^{23} +$$$$14\!\cdots\!85$$$$T^{24} -$$$$83\!\cdots\!46$$$$T^{25} +$$$$59\!\cdots\!34$$$$T^{26} -$$$$29\!\cdots\!05$$$$T^{27} +$$$$19\!\cdots\!39$$$$T^{28} -$$$$85\!\cdots\!96$$$$T^{29} +$$$$50\!\cdots\!76$$$$T^{30} -$$$$18\!\cdots\!07$$$$T^{31} +$$$$97\!\cdots\!42$$$$T^{32} -$$$$27\!\cdots\!03$$$$T^{33} +$$$$12\!\cdots\!07$$$$T^{34} -$$$$19\!\cdots\!86$$$$T^{35} +$$$$74\!\cdots\!09$$$$T^{36}$$
$71$ $$1 + 86 T + 4381 T^{2} + 161393 T^{3} + 4751593 T^{4} + 117237598 T^{5} + 2504196342 T^{6} + 47287231405 T^{7} + 802182116929 T^{8} + 12371512985841 T^{9} + 175156069890510 T^{10} + 2294291010191878 T^{11} + 27986288362685489 T^{12} + 319629084849093298 T^{13} + 3433341088218201345 T^{14} + 34811199581426803904 T^{15} +$$$$33\!\cdots\!95$$$$T^{16} +$$$$30\!\cdots\!98$$$$T^{17} +$$$$26\!\cdots\!75$$$$T^{18} +$$$$21\!\cdots\!58$$$$T^{19} +$$$$16\!\cdots\!95$$$$T^{20} +$$$$12\!\cdots\!44$$$$T^{21} +$$$$87\!\cdots\!45$$$$T^{22} +$$$$57\!\cdots\!98$$$$T^{23} +$$$$35\!\cdots\!69$$$$T^{24} +$$$$20\!\cdots\!98$$$$T^{25} +$$$$11\!\cdots\!10$$$$T^{26} +$$$$56\!\cdots\!71$$$$T^{27} +$$$$26\!\cdots\!29$$$$T^{28} +$$$$10\!\cdots\!55$$$$T^{29} +$$$$41\!\cdots\!22$$$$T^{30} +$$$$13\!\cdots\!78$$$$T^{31} +$$$$39\!\cdots\!33$$$$T^{32} +$$$$94\!\cdots\!43$$$$T^{33} +$$$$18\!\cdots\!01$$$$T^{34} +$$$$25\!\cdots\!26$$$$T^{35} +$$$$21\!\cdots\!61$$$$T^{36}$$
$73$ $$1 - 27 T + 1007 T^{2} - 19815 T^{3} + 447765 T^{4} - 7156446 T^{5} + 124212868 T^{6} - 1702517030 T^{7} + 24770220974 T^{8} - 300583815481 T^{9} + 3832197589506 T^{10} - 42031917749538 T^{11} + 482153954860024 T^{12} - 4847925257622480 T^{13} + 50891805726542784 T^{14} - 473582143472657046 T^{15} + 4599037907896224962 T^{16} - 39832962881131374483 T^{17} +$$$$36\!\cdots\!59$$$$T^{18} -$$$$29\!\cdots\!59$$$$T^{19} +$$$$24\!\cdots\!98$$$$T^{20} -$$$$18\!\cdots\!82$$$$T^{21} +$$$$14\!\cdots\!44$$$$T^{22} -$$$$10\!\cdots\!40$$$$T^{23} +$$$$72\!\cdots\!36$$$$T^{24} -$$$$46\!\cdots\!86$$$$T^{25} +$$$$30\!\cdots\!86$$$$T^{26} -$$$$17\!\cdots\!53$$$$T^{27} +$$$$10\!\cdots\!26$$$$T^{28} -$$$$53\!\cdots\!10$$$$T^{29} +$$$$28\!\cdots\!28$$$$T^{30} -$$$$11\!\cdots\!18$$$$T^{31} +$$$$54\!\cdots\!85$$$$T^{32} -$$$$17\!\cdots\!55$$$$T^{33} +$$$$65\!\cdots\!27$$$$T^{34} -$$$$12\!\cdots\!31$$$$T^{35} +$$$$34\!\cdots\!69$$$$T^{36}$$
$79$ $$1 - 17 T + 946 T^{2} - 13939 T^{3} + 432350 T^{4} - 5643790 T^{5} + 128080842 T^{6} - 1504658537 T^{7} + 27779437033 T^{8} - 297208185691 T^{9} + 4716645703113 T^{10} - 46373717355855 T^{11} + 653486932362430 T^{12} - 5942510278530939 T^{13} + 75878465997252205 T^{14} - 640670189666756761 T^{15} + 7507412874523164847 T^{16} - 58935399487024227850 T^{17} +$$$$63\!\cdots\!79$$$$T^{18} -$$$$46\!\cdots\!50$$$$T^{19} +$$$$46\!\cdots\!27$$$$T^{20} -$$$$31\!\cdots\!79$$$$T^{21} +$$$$29\!\cdots\!05$$$$T^{22} -$$$$18\!\cdots\!61$$$$T^{23} +$$$$15\!\cdots\!30$$$$T^{24} -$$$$89\!\cdots\!45$$$$T^{25} +$$$$71\!\cdots\!93$$$$T^{26} -$$$$35\!\cdots\!29$$$$T^{27} +$$$$26\!\cdots\!33$$$$T^{28} -$$$$11\!\cdots\!23$$$$T^{29} +$$$$75\!\cdots\!22$$$$T^{30} -$$$$26\!\cdots\!10$$$$T^{31} +$$$$15\!\cdots\!50$$$$T^{32} -$$$$40\!\cdots\!61$$$$T^{33} +$$$$21\!\cdots\!66$$$$T^{34} -$$$$30\!\cdots\!03$$$$T^{35} +$$$$14\!\cdots\!61$$$$T^{36}$$
$83$ $$1 + 12 T + 1078 T^{2} + 11305 T^{3} + 561490 T^{4} + 5215328 T^{5} + 188751070 T^{6} + 1566682212 T^{7} + 46085293241 T^{8} + 344087059159 T^{9} + 8713394246075 T^{10} + 58850822346822 T^{11} + 1327322439513751 T^{12} + 8154166109275604 T^{13} + 167189447742982401 T^{14} + 939442501420537189 T^{15} + 17709135064744613010 T^{16} + 91493035467322566291 T^{17} +$$$$15\!\cdots\!39$$$$T^{18} +$$$$75\!\cdots\!53$$$$T^{19} +$$$$12\!\cdots\!90$$$$T^{20} +$$$$53\!\cdots\!43$$$$T^{21} +$$$$79\!\cdots\!21$$$$T^{22} +$$$$32\!\cdots\!72$$$$T^{23} +$$$$43\!\cdots\!19$$$$T^{24} +$$$$15\!\cdots\!94$$$$T^{25} +$$$$19\!\cdots\!75$$$$T^{26} +$$$$64\!\cdots\!77$$$$T^{27} +$$$$71\!\cdots\!09$$$$T^{28} +$$$$20\!\cdots\!04$$$$T^{29} +$$$$20\!\cdots\!70$$$$T^{30} +$$$$46\!\cdots\!64$$$$T^{31} +$$$$41\!\cdots\!10$$$$T^{32} +$$$$69\!\cdots\!35$$$$T^{33} +$$$$54\!\cdots\!18$$$$T^{34} +$$$$50\!\cdots\!76$$$$T^{35} +$$$$34\!\cdots\!09$$$$T^{36}$$
$89$ $$1 + 51 T + 2269 T^{2} + 69808 T^{3} + 1913596 T^{4} + 43812335 T^{5} + 915214729 T^{6} + 16990491798 T^{7} + 292605997565 T^{8} + 4611751813072 T^{9} + 68223773399243 T^{10} + 939340780074915 T^{11} + 12244142926058626 T^{12} + 150150675566213465 T^{13} + 1754271192106775274 T^{14} + 19419208919876480401 T^{15} +$$$$20\!\cdots\!07$$$$T^{16} +$$$$20\!\cdots\!40$$$$T^{17} +$$$$20\!\cdots\!57$$$$T^{18} +$$$$18\!\cdots\!60$$$$T^{19} +$$$$16\!\cdots\!47$$$$T^{20} +$$$$13\!\cdots\!69$$$$T^{21} +$$$$11\!\cdots\!34$$$$T^{22} +$$$$83\!\cdots\!85$$$$T^{23} +$$$$60\!\cdots\!86$$$$T^{24} +$$$$41\!\cdots\!35$$$$T^{25} +$$$$26\!\cdots\!83$$$$T^{26} +$$$$16\!\cdots\!48$$$$T^{27} +$$$$91\!\cdots\!65$$$$T^{28} +$$$$47\!\cdots\!22$$$$T^{29} +$$$$22\!\cdots\!09$$$$T^{30} +$$$$96\!\cdots\!15$$$$T^{31} +$$$$37\!\cdots\!36$$$$T^{32} +$$$$12\!\cdots\!92$$$$T^{33} +$$$$35\!\cdots\!09$$$$T^{34} +$$$$70\!\cdots\!79$$$$T^{35} +$$$$12\!\cdots\!81$$$$T^{36}$$
$97$ $$1 + 19 T + 1209 T^{2} + 19154 T^{3} + 675661 T^{4} + 9241955 T^{5} + 237136112 T^{6} + 2876020341 T^{7} + 59712863582 T^{8} + 655928587660 T^{9} + 11654954223677 T^{10} + 117944609895823 T^{11} + 1856060149182267 T^{12} + 17528861806014983 T^{13} + 250072895629370224 T^{14} + 2222533021348576604 T^{15} + 29240177165801608601 T^{16} +$$$$24\!\cdots\!45$$$$T^{17} +$$$$30\!\cdots\!63$$$$T^{18} +$$$$23\!\cdots\!65$$$$T^{19} +$$$$27\!\cdots\!09$$$$T^{20} +$$$$20\!\cdots\!92$$$$T^{21} +$$$$22\!\cdots\!44$$$$T^{22} +$$$$15\!\cdots\!31$$$$T^{23} +$$$$15\!\cdots\!43$$$$T^{24} +$$$$95\!\cdots\!99$$$$T^{25} +$$$$91\!\cdots\!97$$$$T^{26} +$$$$49\!\cdots\!20$$$$T^{27} +$$$$44\!\cdots\!18$$$$T^{28} +$$$$20\!\cdots\!73$$$$T^{29} +$$$$16\!\cdots\!92$$$$T^{30} +$$$$62\!\cdots\!35$$$$T^{31} +$$$$44\!\cdots\!09$$$$T^{32} +$$$$12\!\cdots\!22$$$$T^{33} +$$$$74\!\cdots\!89$$$$T^{34} +$$$$11\!\cdots\!03$$$$T^{35} +$$$$57\!\cdots\!89$$$$T^{36}$$
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