Properties

 Label 72.2.n.b Level 72 Weight 2 Character orbit 72.n Analytic conductor 0.575 Analytic rank 0 Dimension 16 CM no Inner twists 4

Related objects

Newspace parameters

 Level: $$N$$ = $$72 = 2^{3} \cdot 3^{2}$$ Weight: $$k$$ = $$2$$ Character orbit: $$[\chi]$$ = 72.n (of order $$6$$, degree $$2$$, minimal)

Newform invariants

 Self dual: no Analytic conductor: $$0.574922894553$$ Analytic rank: $$0$$ Dimension: $$16$$ Relative dimension: $$8$$ over $$\Q(\zeta_{6})$$ Coefficient field: $$\mathbb{Q}[x]/(x^{16} - \cdots)$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{2}\cdot 3^{2}$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-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.

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

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{16} - x^{15} + x^{14} + 2 x^{12} - 4 x^{11} - 8 x^{9} + 4 x^{8} - 16 x^{7} - 32 x^{5} + 32 x^{4} + 64 x^{2} - 128 x + 256$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$($$$$\nu^{15} - \nu^{14} - 3 \nu^{13} - 8 \nu^{12} + 2 \nu^{11} + 8 \nu^{10} - 16 \nu^{9} - 48 \nu^{8} - 12 \nu^{7} + 16 \nu^{6} + 16 \nu^{5} - 80 \nu^{4} + 256 \nu^{2} + 384 \nu - 128$$$$)/384$$ $$\beta_{3}$$ $$=$$ $$($$$$-\nu^{15} - \nu^{14} - 3 \nu^{13} - 6 \nu^{10} - 4 \nu^{9} - 4 \nu^{8} + 4 \nu^{7} + 8 \nu^{6} + 24 \nu^{4} + 80 \nu^{3} + 32 \nu^{2} + 32 \nu + 64$$$$)/192$$ $$\beta_{4}$$ $$=$$ $$($$$$\nu^{15} + \nu^{14} + 3 \nu^{13} + 4 \nu^{12} + 8 \nu^{11} + 6 \nu^{10} + 8 \nu^{9} + 4 \nu^{8} - 12 \nu^{7} - 24 \nu^{6} - 32 \nu^{5} - 72 \nu^{4} - 96 \nu^{3} - 128 \nu^{2} - 96 \nu - 256$$$$)/192$$ $$\beta_{5}$$ $$=$$ $$($$$$-\nu^{15} + \nu^{14} - \nu^{13} - 2 \nu^{11} + 4 \nu^{10} + 8 \nu^{8} - 4 \nu^{7} + 16 \nu^{6} + 32 \nu^{4} - 32 \nu^{3} - 64 \nu + 128$$$$)/128$$ $$\beta_{6}$$ $$=$$ $$($$$$-\nu^{15} - 3 \nu^{14} - 5 \nu^{13} - 12 \nu^{12} - 18 \nu^{11} - 28 \nu^{10} - 24 \nu^{9} - 24 \nu^{8} - 20 \nu^{7} + 64 \nu^{6} + 96 \nu^{5} + 160 \nu^{4} + 256 \nu^{3} + 384 \nu^{2} + 448 \nu + 512$$$$)/384$$ $$\beta_{7}$$ $$=$$ $$($$$$-\nu^{15} - \nu^{14} - 2 \nu^{13} - 3 \nu^{12} - 5 \nu^{11} - 4 \nu^{10} - 6 \nu^{9} + 8 \nu^{8} + 4 \nu^{7} + 16 \nu^{6} + 20 \nu^{5} + 64 \nu^{4} + 64 \nu^{3} + 80 \nu^{2} + 64 \nu + 128$$$$)/96$$ $$\beta_{8}$$ $$=$$ $$($$$$\nu^{15} + \nu^{14} + 7 \nu^{13} + 14 \nu^{12} + 22 \nu^{11} + 20 \nu^{10} + 24 \nu^{9} + 16 \nu^{8} + 20 \nu^{7} - 40 \nu^{6} - 160 \nu^{5} - 272 \nu^{4} - 160 \nu^{3} - 128 \nu^{2} - 384 \nu - 640$$$$)/384$$ $$\beta_{9}$$ $$=$$ $$($$$$\nu^{15} - 7 \nu^{14} - 9 \nu^{13} - 2 \nu^{12} - 10 \nu^{11} - 4 \nu^{10} - 16 \nu^{9} + 36 \nu^{7} + 88 \nu^{6} + 64 \nu^{5} + 112 \nu^{4} + 192 \nu^{3} + 448 \nu^{2} - 128$$$$)/384$$ $$\beta_{10}$$ $$=$$ $$($$$$-\nu^{15} - \nu^{14} - 7 \nu^{13} - 14 \nu^{12} - 22 \nu^{11} - 20 \nu^{10} - 24 \nu^{9} - 16 \nu^{8} - 20 \nu^{7} + 40 \nu^{6} + 160 \nu^{5} + 272 \nu^{4} + 160 \nu^{3} + 512 \nu^{2} + 384 \nu + 640$$$$)/384$$ $$\beta_{11}$$ $$=$$ $$($$$$\nu^{15} - \nu^{14} + 4 \nu^{11} + \nu^{10} + 4 \nu^{9} - 2 \nu^{8} + 8 \nu^{7} + 20 \nu^{5} - 52 \nu^{4} - 56 \nu^{3} - 48 \nu^{2} + 16 \nu - 288$$$$)/96$$ $$\beta_{12}$$ $$=$$ $$($$$$-\nu^{15} + \nu^{14} - 4 \nu^{11} - \nu^{10} - 4 \nu^{9} + 2 \nu^{8} - 8 \nu^{7} - 20 \nu^{5} + 52 \nu^{4} - 40 \nu^{3} + 48 \nu^{2} - 16 \nu + 192$$$$)/96$$ $$\beta_{13}$$ $$=$$ $$($$$$3 \nu^{15} - \nu^{14} + 9 \nu^{13} + 6 \nu^{12} + 26 \nu^{11} + 12 \nu^{10} + 16 \nu^{9} - 16 \nu^{8} + 44 \nu^{7} - 40 \nu^{6} - 32 \nu^{5} - 240 \nu^{4} - 64 \nu^{3} - 256 \nu^{2} - 128 \nu - 896$$$$)/384$$ $$\beta_{14}$$ $$=$$ $$($$$$-2 \nu^{15} + \nu^{14} - 5 \nu^{13} - 5 \nu^{12} - 16 \nu^{11} - 10 \nu^{10} - 28 \nu^{9} - 8 \nu^{8} - 32 \nu^{7} + 12 \nu^{6} + 64 \nu^{5} + 160 \nu^{4} + 96 \nu^{3} + 256 \nu^{2} + 256 \nu + 704$$$$)/192$$ $$\beta_{15}$$ $$=$$ $$($$$$-9 \nu^{15} + \nu^{14} - 9 \nu^{13} - 8 \nu^{12} - 42 \nu^{11} - 12 \nu^{10} - 24 \nu^{9} + 40 \nu^{8} - 52 \nu^{7} + 48 \nu^{6} + 96 \nu^{5} + 384 \nu^{4} + 192 \nu^{3} + 448 \nu^{2} + 64 \nu + 1664$$$$)/384$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{10} + \beta_{8}$$ $$\nu^{3}$$ $$=$$ $$-\beta_{12} - \beta_{11} - 1$$ $$\nu^{4}$$ $$=$$ $$-\beta_{14} + \beta_{13} + \beta_{12} - \beta_{11} + \beta_{10} + \beta_{9} - 2 \beta_{8} - \beta_{6} + 2 \beta_{4} + \beta_{3} + \beta_{1}$$ $$\nu^{5}$$ $$=$$ $$\beta_{15} + \beta_{14} + 2 \beta_{13} - \beta_{12} + \beta_{11} + \beta_{10} - \beta_{7} + \beta_{5} - \beta_{2}$$ $$\nu^{6}$$ $$=$$ $$-\beta_{15} + \beta_{13} - 2 \beta_{10} + \beta_{9} - \beta_{7} + 5 \beta_{6} + 5 \beta_{5} - \beta_{3} - \beta_{2} + \beta_{1}$$ $$\nu^{7}$$ $$=$$ $$-2 \beta_{15} + \beta_{14} + \beta_{13} + \beta_{12} + \beta_{11} + \beta_{10} - \beta_{9} + 2 \beta_{8} - 3 \beta_{6} - 10 \beta_{4} - \beta_{3} - 2 \beta_{2} + \beta_{1}$$ $$\nu^{8}$$ $$=$$ $$-3 \beta_{15} + 3 \beta_{14} - 4 \beta_{13} - 3 \beta_{12} + \beta_{11} + \beta_{10} - 2 \beta_{9} + 6 \beta_{8} + 7 \beta_{7} + \beta_{5} + 2 \beta_{4} - 4 \beta_{3} - 3 \beta_{2} + 2$$ $$\nu^{9}$$ $$=$$ $$5 \beta_{15} - 12 \beta_{14} - 5 \beta_{13} - 2 \beta_{12} - 2 \beta_{11} + 10 \beta_{10} - \beta_{9} - 7 \beta_{7} - \beta_{6} - \beta_{5} + 5 \beta_{3} + \beta_{2} + 7 \beta_{1} + 2$$ $$\nu^{10}$$ $$=$$ $$6 \beta_{15} - 3 \beta_{14} + 5 \beta_{13} - 3 \beta_{12} - 7 \beta_{11} - 3 \beta_{10} + 7 \beta_{9} - 10 \beta_{8} - 11 \beta_{6} + 6 \beta_{4} - 5 \beta_{3} + 10 \beta_{2} + 5 \beta_{1}$$ $$\nu^{11}$$ $$=$$ $$-7 \beta_{15} - \beta_{14} + 12 \beta_{13} + \beta_{12} - 3 \beta_{11} + 17 \beta_{10} - 2 \beta_{9} + 2 \beta_{8} + 11 \beta_{7} + 13 \beta_{5} + 26 \beta_{4} + 20 \beta_{3} + \beta_{2} + 26$$ $$\nu^{12}$$ $$=$$ $$9 \beta_{15} + 12 \beta_{14} + 15 \beta_{13} + 2 \beta_{12} + 2 \beta_{11} - 6 \beta_{10} + 19 \beta_{9} - 27 \beta_{7} + 3 \beta_{6} + 3 \beta_{5} + 9 \beta_{3} - 19 \beta_{2} + 27 \beta_{1} - 10$$ $$\nu^{13}$$ $$=$$ $$14 \beta_{15} - 3 \beta_{14} + 45 \beta_{13} + 13 \beta_{12} - 7 \beta_{11} - 11 \beta_{10} + 7 \beta_{9} - 10 \beta_{8} + 29 \beta_{6} + 14 \beta_{4} - 29 \beta_{3} - 6 \beta_{2} + 5 \beta_{1}$$ $$\nu^{14}$$ $$=$$ $$-47 \beta_{15} + 31 \beta_{14} - 28 \beta_{13} - 31 \beta_{12} - 11 \beta_{11} + 41 \beta_{10} - 42 \beta_{9} + 58 \beta_{8} - 13 \beta_{7} + 53 \beta_{5} - 78 \beta_{4} - 12 \beta_{3} - 31 \beta_{2} - 78$$ $$\nu^{15}$$ $$=$$ $$-55 \beta_{15} + 12 \beta_{14} - 65 \beta_{13} - 22 \beta_{12} - 22 \beta_{11} + 10 \beta_{10} + 19 \beta_{9} + 5 \beta_{7} - 13 \beta_{6} - 13 \beta_{5} - 55 \beta_{3} - 19 \beta_{2} - 5 \beta_{1} + 46$$

Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/72\mathbb{Z}\right)^\times$$.

 $$n$$ $$37$$ $$55$$ $$65$$ $$\chi(n)$$ $$-1$$ $$1$$ $$-1 - \beta_{4}$$

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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
13.1
 1.05026 − 0.947078i −0.179748 − 1.40274i 1.41411 − 0.0174668i −1.12494 − 0.857038i 0.820200 + 1.15207i 0.587625 + 1.28635i −1.34532 + 0.436011i −0.722180 + 1.21592i 1.05026 + 0.947078i −0.179748 + 1.40274i 1.41411 + 0.0174668i −1.12494 + 0.857038i 0.820200 − 1.15207i 0.587625 − 1.28635i −1.34532 − 0.436011i −0.722180 − 1.21592i
−1.34532 + 0.436011i 1.52768 + 0.816201i 1.61979 1.17315i −0.602794 0.348023i −2.41110 0.431967i 0.795065 + 1.37709i −1.66763 + 2.28452i 1.66763 + 2.49379i 0.962695 + 0.205379i
13.2 −1.12494 0.857038i −0.986088 + 1.42395i 0.530970 + 1.92823i 1.19115 + 0.687709i 2.32967 0.756738i 1.80469 + 3.12581i 1.05526 2.62420i −1.05526 2.80828i −0.750573 1.79449i
13.3 −0.722180 + 1.21592i 0.294546 1.70682i −0.956913 1.75622i 3.17262 + 1.83171i 1.86264 + 1.59078i −0.191926 0.332426i 2.82649 + 0.104780i −2.82649 1.00547i −4.51841 + 2.53482i
13.4 −0.179748 1.40274i 0.986088 1.42395i −1.93538 + 0.504281i −1.19115 0.687709i −2.17468 1.12728i 1.80469 + 3.12581i 1.05526 + 2.62420i −1.05526 2.80828i −0.750573 + 1.79449i
13.5 0.587625 + 1.28635i 1.69028 0.378078i −1.30939 + 1.51178i −1.97542 1.14051i 1.47959 + 1.95213i −0.907824 1.57240i −2.71411 0.795980i 2.71411 1.27812i 0.306290 3.21128i
13.6 0.820200 + 1.15207i −1.69028 + 0.378078i −0.654545 + 1.88986i 1.97542 + 1.14051i −1.82194 1.63723i −0.907824 1.57240i −2.71411 + 0.795980i 2.71411 1.27812i 0.306290 + 3.21128i
13.7 1.05026 0.947078i −1.52768 0.816201i 0.206086 1.98935i 0.602794 + 0.348023i −2.37747 + 0.589613i 0.795065 + 1.37709i −1.66763 2.28452i 1.66763 + 2.49379i 0.962695 0.205379i
13.8 1.41411 0.0174668i −0.294546 + 1.70682i 1.99939 0.0493999i −3.17262 1.83171i −0.386706 + 2.41877i −0.191926 0.332426i 2.82649 0.104780i −2.82649 1.00547i −4.51841 2.53482i
61.1 −1.34532 0.436011i 1.52768 0.816201i 1.61979 + 1.17315i −0.602794 + 0.348023i −2.41110 + 0.431967i 0.795065 1.37709i −1.66763 2.28452i 1.66763 2.49379i 0.962695 0.205379i
61.2 −1.12494 + 0.857038i −0.986088 1.42395i 0.530970 1.92823i 1.19115 0.687709i 2.32967 + 0.756738i 1.80469 3.12581i 1.05526 + 2.62420i −1.05526 + 2.80828i −0.750573 + 1.79449i
61.3 −0.722180 1.21592i 0.294546 + 1.70682i −0.956913 + 1.75622i 3.17262 1.83171i 1.86264 1.59078i −0.191926 + 0.332426i 2.82649 0.104780i −2.82649 + 1.00547i −4.51841 2.53482i
61.4 −0.179748 + 1.40274i 0.986088 + 1.42395i −1.93538 0.504281i −1.19115 + 0.687709i −2.17468 + 1.12728i 1.80469 3.12581i 1.05526 2.62420i −1.05526 + 2.80828i −0.750573 1.79449i
61.5 0.587625 1.28635i 1.69028 + 0.378078i −1.30939 1.51178i −1.97542 + 1.14051i 1.47959 1.95213i −0.907824 + 1.57240i −2.71411 + 0.795980i 2.71411 + 1.27812i 0.306290 + 3.21128i
61.6 0.820200 1.15207i −1.69028 0.378078i −0.654545 1.88986i 1.97542 1.14051i −1.82194 + 1.63723i −0.907824 + 1.57240i −2.71411 0.795980i 2.71411 + 1.27812i 0.306290 3.21128i
61.7 1.05026 + 0.947078i −1.52768 + 0.816201i 0.206086 + 1.98935i 0.602794 0.348023i −2.37747 0.589613i 0.795065 1.37709i −1.66763 + 2.28452i 1.66763 2.49379i 0.962695 + 0.205379i
61.8 1.41411 + 0.0174668i −0.294546 1.70682i 1.99939 + 0.0493999i −3.17262 + 1.83171i −0.386706 2.41877i −0.191926 + 0.332426i 2.82649 + 0.104780i −2.82649 + 1.00547i −4.51841 + 2.53482i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 61.8 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
8.b even 2 1 inner
9.c even 3 1 inner
72.n even 6 1 inner

Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 72.2.n.b 16
3.b odd 2 1 216.2.n.b 16
4.b odd 2 1 288.2.r.b 16
8.b even 2 1 inner 72.2.n.b 16
8.d odd 2 1 288.2.r.b 16
9.c even 3 1 inner 72.2.n.b 16
9.c even 3 1 648.2.d.j 8
9.d odd 6 1 216.2.n.b 16
9.d odd 6 1 648.2.d.k 8
12.b even 2 1 864.2.r.b 16
24.f even 2 1 864.2.r.b 16
24.h odd 2 1 216.2.n.b 16
36.f odd 6 1 288.2.r.b 16
36.f odd 6 1 2592.2.d.j 8
36.h even 6 1 864.2.r.b 16
36.h even 6 1 2592.2.d.k 8
72.j odd 6 1 216.2.n.b 16
72.j odd 6 1 648.2.d.k 8
72.l even 6 1 864.2.r.b 16
72.l even 6 1 2592.2.d.k 8
72.n even 6 1 inner 72.2.n.b 16
72.n even 6 1 648.2.d.j 8
72.p odd 6 1 288.2.r.b 16
72.p odd 6 1 2592.2.d.j 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
72.2.n.b 16 1.a even 1 1 trivial
72.2.n.b 16 8.b even 2 1 inner
72.2.n.b 16 9.c even 3 1 inner
72.2.n.b 16 72.n even 6 1 inner
216.2.n.b 16 3.b odd 2 1
216.2.n.b 16 9.d odd 6 1
216.2.n.b 16 24.h odd 2 1
216.2.n.b 16 72.j odd 6 1
288.2.r.b 16 4.b odd 2 1
288.2.r.b 16 8.d odd 2 1
288.2.r.b 16 36.f odd 6 1
288.2.r.b 16 72.p odd 6 1
648.2.d.j 8 9.c even 3 1
648.2.d.j 8 72.n even 6 1
648.2.d.k 8 9.d odd 6 1
648.2.d.k 8 72.j odd 6 1
864.2.r.b 16 12.b even 2 1
864.2.r.b 16 24.f even 2 1
864.2.r.b 16 36.h even 6 1
864.2.r.b 16 72.l even 6 1
2592.2.d.j 8 36.f odd 6 1
2592.2.d.j 8 72.p odd 6 1
2592.2.d.k 8 36.h even 6 1
2592.2.d.k 8 72.l even 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{16} - \cdots$$ acting on $$S_{2}^{\mathrm{new}}(72, [\chi])$$.

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ $$1 - T + T^{2} + 2 T^{4} - 4 T^{5} - 8 T^{7} + 4 T^{8} - 16 T^{9} - 32 T^{11} + 32 T^{12} + 64 T^{14} - 128 T^{15} + 256 T^{16}$$
$3$ $$1 - T^{2} - 2 T^{4} + 9 T^{6} + 18 T^{8} + 81 T^{10} - 162 T^{12} - 729 T^{14} + 6561 T^{16}$$
$5$ $$1 + 19 T^{2} + 176 T^{4} + 1031 T^{6} + 3893 T^{8} + 4928 T^{10} - 61926 T^{12} - 631122 T^{14} - 3717344 T^{16} - 15778050 T^{18} - 38703750 T^{20} + 77000000 T^{22} + 1520703125 T^{24} + 10068359375 T^{26} + 42968750000 T^{28} + 115966796875 T^{30} + 152587890625 T^{32}$$
$7$ $$( 1 - 3 T - 14 T^{2} + 39 T^{3} + 139 T^{4} - 252 T^{5} - 1208 T^{6} + 666 T^{7} + 9424 T^{8} + 4662 T^{9} - 59192 T^{10} - 86436 T^{11} + 333739 T^{12} + 655473 T^{13} - 1647086 T^{14} - 2470629 T^{15} + 5764801 T^{16} )^{2}$$
$11$ $$1 + 48 T^{2} + 1090 T^{4} + 17592 T^{6} + 242041 T^{8} + 2632140 T^{10} + 21031138 T^{12} + 158095260 T^{14} + 1558598596 T^{16} + 19129526460 T^{18} + 307916891458 T^{20} + 4662996570540 T^{22} + 51883637916121 T^{24} + 456291173580792 T^{26} + 3420886930625890 T^{28} + 18227992011995568 T^{30} + 45949729863572161 T^{32}$$
$13$ $$1 + 51 T^{2} + 1288 T^{4} + 16911 T^{6} + 73645 T^{8} - 1059264 T^{10} - 5456606 T^{12} + 411982806 T^{14} + 9084740848 T^{16} + 69625094214 T^{18} - 155846123966 T^{20} - 5112865008576 T^{22} + 60074488948045 T^{24} + 2331324955658439 T^{26} + 30007933637755528 T^{28} + 200806195670663739 T^{30} + 665416609183179841 T^{32}$$
$17$ $$( 1 + 7 T + 66 T^{2} + 309 T^{3} + 1702 T^{4} + 5253 T^{5} + 19074 T^{6} + 34391 T^{7} + 83521 T^{8} )^{4}$$
$19$ $$( 1 - 69 T^{2} + 2306 T^{4} - 53763 T^{6} + 1069146 T^{8} - 19408443 T^{10} + 300520226 T^{12} - 3246165789 T^{14} + 16983563041 T^{16} )^{2}$$
$23$ $$( 1 + 5 T - 32 T^{2} + 45 T^{3} + 1385 T^{4} - 3040 T^{5} - 11142 T^{6} + 48640 T^{7} - 123716 T^{8} + 1118720 T^{9} - 5894118 T^{10} - 36987680 T^{11} + 387579785 T^{12} + 289635435 T^{13} - 4737148448 T^{14} + 17024127235 T^{15} + 78310985281 T^{16} )^{2}$$
$29$ $$1 + 123 T^{2} + 7912 T^{4} + 340647 T^{6} + 10067965 T^{8} + 144056448 T^{10} - 3705324014 T^{12} - 328820871018 T^{14} - 12137186779472 T^{16} - 276538352526138 T^{18} - 2620705273945934 T^{20} + 85688134810823808 T^{22} + 5036463377066894365 T^{24} +$$$$14\!\cdots\!47$$$$T^{26} +$$$$27\!\cdots\!92$$$$T^{28} +$$$$36\!\cdots\!63$$$$T^{30} +$$$$25\!\cdots\!21$$$$T^{32}$$
$31$ $$( 1 + 5 T - 48 T^{2} - 155 T^{3} + 1121 T^{4} - 2040 T^{5} - 44678 T^{6} + 79040 T^{7} + 1805724 T^{8} + 2450240 T^{9} - 42935558 T^{10} - 60773640 T^{11} + 1035267041 T^{12} - 4437518405 T^{13} - 42600176688 T^{14} + 137563070555 T^{15} + 852891037441 T^{16} )^{2}$$
$37$ $$( 1 - 144 T^{2} + 12668 T^{4} - 733824 T^{6} + 31784838 T^{8} - 1004605056 T^{10} + 23741871548 T^{12} - 369464602896 T^{14} + 3512479453921 T^{16} )^{2}$$
$41$ $$( 1 + 4 T - 74 T^{2} + 288 T^{3} + 4517 T^{4} - 22796 T^{5} - 11538 T^{6} + 738968 T^{7} - 2462804 T^{8} + 30297688 T^{9} - 19395378 T^{10} - 1571123116 T^{11} + 12763962437 T^{12} + 33366585888 T^{13} - 351507713834 T^{14} + 779017095524 T^{15} + 7984925229121 T^{16} )^{2}$$
$43$ $$1 + 324 T^{2} + 58258 T^{4} + 7305024 T^{6} + 705172105 T^{8} + 54954910764 T^{10} + 3558033934834 T^{12} + 194436327001344 T^{14} + 9048929543300068 T^{16} + 359512768625485056 T^{18} + 12164209974444414034 T^{20} +$$$$34\!\cdots\!36$$$$T^{22} +$$$$82\!\cdots\!05$$$$T^{24} +$$$$15\!\cdots\!76$$$$T^{26} +$$$$23\!\cdots\!58$$$$T^{28} +$$$$23\!\cdots\!76$$$$T^{30} +$$$$13\!\cdots\!01$$$$T^{32}$$
$47$ $$( 1 - 3 T - 98 T^{2} + 219 T^{3} + 3523 T^{4} + 1116 T^{5} - 253856 T^{6} - 226218 T^{7} + 18909448 T^{8} - 10632246 T^{9} - 560767904 T^{10} + 115866468 T^{11} + 17191116163 T^{12} + 50226556533 T^{13} - 1056363102242 T^{14} - 1519869361389 T^{15} + 23811286661761 T^{16} )^{2}$$
$53$ $$( 1 - 264 T^{2} + 35516 T^{4} - 3123720 T^{6} + 194863110 T^{8} - 8774529480 T^{10} + 280238323196 T^{12} - 5851391338056 T^{14} + 62259690411361 T^{16} )^{2}$$
$59$ $$1 + 364 T^{2} + 70130 T^{4} + 9549968 T^{6} + 1028964713 T^{8} + 92991430988 T^{10} + 7291973853618 T^{12} + 506327334867240 T^{14} + 31497451193778532 T^{16} + 1762525452672862440 T^{18} + 88359479586850462098 T^{20} +$$$$39\!\cdots\!08$$$$T^{22} +$$$$15\!\cdots\!73$$$$T^{24} +$$$$48\!\cdots\!68$$$$T^{26} +$$$$12\!\cdots\!30$$$$T^{28} +$$$$22\!\cdots\!04$$$$T^{30} +$$$$21\!\cdots\!41$$$$T^{32}$$
$61$ $$1 + 323 T^{2} + 52152 T^{4} + 6035887 T^{6} + 584536925 T^{8} + 49803739968 T^{10} + 3801157073266 T^{12} + 264102902456582 T^{14} + 16823476283802768 T^{16} + 982726900040941622 T^{18} + 52630216452466386706 T^{20} +$$$$25\!\cdots\!48$$$$T^{22} +$$$$11\!\cdots\!25$$$$T^{24} +$$$$43\!\cdots\!87$$$$T^{26} +$$$$13\!\cdots\!92$$$$T^{28} +$$$$31\!\cdots\!43$$$$T^{30} +$$$$36\!\cdots\!61$$$$T^{32}$$
$67$ $$1 + 296 T^{2} + 39450 T^{4} + 3796936 T^{6} + 362561921 T^{8} + 33090030780 T^{10} + 2631719645962 T^{12} + 193036342550180 T^{14} + 13423375489686516 T^{16} + 866540141707758020 T^{18} + 53032101023857423402 T^{20} +$$$$29\!\cdots\!20$$$$T^{22} +$$$$14\!\cdots\!61$$$$T^{24} +$$$$69\!\cdots\!64$$$$T^{26} +$$$$32\!\cdots\!50$$$$T^{28} +$$$$10\!\cdots\!84$$$$T^{30} +$$$$16\!\cdots\!81$$$$T^{32}$$
$71$ $$( 1 - 18 T + 332 T^{2} - 3582 T^{3} + 36198 T^{4} - 254322 T^{5} + 1673612 T^{6} - 6442398 T^{7} + 25411681 T^{8} )^{4}$$
$73$ $$( 1 + 11 T + 278 T^{2} + 2325 T^{3} + 29966 T^{4} + 169725 T^{5} + 1481462 T^{6} + 4279187 T^{7} + 28398241 T^{8} )^{4}$$
$79$ $$( 1 + 15 T - 80 T^{2} - 1305 T^{3} + 13273 T^{4} + 59400 T^{5} - 1907150 T^{6} - 913560 T^{7} + 190569148 T^{8} - 72171240 T^{9} - 11902523150 T^{10} + 29286516600 T^{11} + 516984425113 T^{12} - 4015558600695 T^{13} - 19446996441680 T^{14} + 288058634792385 T^{15} + 1517108809906561 T^{16} )^{2}$$
$83$ $$1 + 559 T^{2} + 168260 T^{4} + 35967119 T^{6} + 6055752221 T^{8} + 843016361960 T^{10} + 99713806040442 T^{12} + 10182388433060610 T^{14} + 904920089638581976 T^{16} + 70146473915354542290 T^{18} +$$$$47\!\cdots\!82$$$$T^{20} +$$$$27\!\cdots\!40$$$$T^{22} +$$$$13\!\cdots\!61$$$$T^{24} +$$$$55\!\cdots\!31$$$$T^{26} +$$$$17\!\cdots\!60$$$$T^{28} +$$$$41\!\cdots\!11$$$$T^{30} +$$$$50\!\cdots\!81$$$$T^{32}$$
$89$ $$( 1 - 16 T + 408 T^{2} - 4116 T^{3} + 56278 T^{4} - 366324 T^{5} + 3231768 T^{6} - 11279504 T^{7} + 62742241 T^{8} )^{4}$$
$97$ $$( 1 - 254 T^{2} + 1560 T^{3} + 35209 T^{4} - 273780 T^{5} - 2055806 T^{6} + 15575820 T^{7} + 104325124 T^{8} + 1510854540 T^{9} - 19343078654 T^{10} - 249871613940 T^{11} + 3117027454729 T^{12} + 13396250800920 T^{13} - 211574889251966 T^{14} + 7837433594376961 T^{16} )^{2}$$