# Properties

 Label 546.2.bd.a Level $546$ Weight $2$ Character orbit 546.bd Analytic conductor $4.360$ Analytic rank $0$ Dimension $16$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$546 = 2 \cdot 3 \cdot 7 \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 546.bd (of order $$6$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$4.35983195036$$ Analytic rank: $$0$$ Dimension: $$16$$ Relative dimension: $$8$$ over $$\Q(\zeta_{6})$$ Coefficient field: $$\mathbb{Q}[x]/(x^{16} + \cdots)$$ Defining polynomial: $$x^{16} + 26 x^{14} + 249 x^{12} + 1144 x^{10} + 2766 x^{8} + 3554 x^{6} + 2260 x^{4} + 564 x^{2} + 9$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ 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.

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

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{16} + 26 x^{14} + 249 x^{12} + 1144 x^{10} + 2766 x^{8} + 3554 x^{6} + 2260 x^{4} + 564 x^{2} + 9$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$36 \nu^{14} + 811 \nu^{12} + 6086 \nu^{10} + 18652 \nu^{8} + 24037 \nu^{6} + 8001 \nu^{4} - 9025 \nu^{2} + 3190 \nu - 879$$$$)/6380$$ $$\beta_{2}$$ $$=$$ $$($$$$-36 \nu^{14} - 811 \nu^{12} - 6086 \nu^{10} - 18652 \nu^{8} - 24037 \nu^{6} - 8001 \nu^{4} + 9025 \nu^{2} + 3190 \nu + 879$$$$)/6380$$ $$\beta_{3}$$ $$=$$ $$($$$$293 \nu^{15} + 7726 \nu^{13} + 75390 \nu^{11} + 353450 \nu^{9} + 866394 \nu^{7} + 1113433 \nu^{5} + 686183 \nu^{3} + 138177 \nu + 9570$$$$)/19140$$ $$\beta_{4}$$ $$=$$ $$($$$$1810 \nu^{15} - 1059 \nu^{14} + 37922 \nu^{13} - 24681 \nu^{12} + 233613 \nu^{11} - 196947 \nu^{10} + 284221 \nu^{9} - 675429 \nu^{8} - 1397898 \nu^{7} - 1070775 \nu^{6} - 4152250 \nu^{5} - 767694 \nu^{4} - 3598583 \nu^{3} - 133557 \nu^{2} - 1022514 \nu + 143010$$$$)/57420$$ $$\beta_{5}$$ $$=$$ $$($$$$1160 \nu^{15} - 5139 \nu^{14} + 28507 \nu^{13} - 122868 \nu^{12} + 247428 \nu^{11} - 1020780 \nu^{10} + 954071 \nu^{9} - 3698685 \nu^{8} + 1657872 \nu^{7} - 6096447 \nu^{6} + 929305 \nu^{5} - 4110039 \nu^{4} - 481603 \nu^{3} - 706149 \nu^{2} - 501294 \nu + 46764$$$$)/57420$$ $$\beta_{6}$$ $$=$$ $$($$$$-2366 \nu^{15} - 1386 \nu^{14} - 64129 \nu^{13} - 37125 \nu^{12} - 646218 \nu^{11} - 369567 \nu^{10} - 3102776 \nu^{9} - 1763784 \nu^{8} - 7486245 \nu^{7} - 4299966 \nu^{6} - 8597341 \nu^{5} - 5095431 \nu^{4} - 3830873 \nu^{3} - 2387484 \nu^{2} - 341565 \nu - 227898$$$$)/57420$$ $$\beta_{7}$$ $$=$$ $$($$$$56 \nu^{14} + 1363 \nu^{12} + 11694 \nu^{10} + 44963 \nu^{8} + 82794 \nu^{6} + 70141 \nu^{4} + 21947 \nu^{2} + 312$$$$)/330$$ $$\beta_{8}$$ $$=$$ $$($$$$3283 \nu^{15} + 1326 \nu^{14} + 77831 \nu^{13} + 28968 \nu^{12} + 635805 \nu^{11} + 200349 \nu^{10} + 2219830 \nu^{9} + 462333 \nu^{8} + 3323844 \nu^{7} - 35856 \nu^{6} + 1523558 \nu^{5} - 1224534 \nu^{4} - 489347 \nu^{3} - 1040973 \nu^{2} - 253563 \nu - 73368$$$$)/57420$$ $$\beta_{9}$$ $$=$$ $$($$$$3002 \nu^{15} + 3813 \nu^{14} + 72502 \nu^{13} + 93900 \nu^{12} + 614547 \nu^{11} + 820431 \nu^{10} + 2326109 \nu^{9} + 3236352 \nu^{8} + 4265154 \nu^{7} + 6132303 \nu^{6} + 3934552 \nu^{5} + 5334573 \nu^{4} + 2050805 \nu^{3} + 1747122 \nu^{2} + 662262 \nu + 55314$$$$)/57420$$ $$\beta_{10}$$ $$=$$ $$($$$$-3002 \nu^{15} + 3813 \nu^{14} - 72502 \nu^{13} + 93900 \nu^{12} - 614547 \nu^{11} + 820431 \nu^{10} - 2326109 \nu^{9} + 3236352 \nu^{8} - 4265154 \nu^{7} + 6132303 \nu^{6} - 3934552 \nu^{5} + 5334573 \nu^{4} - 2050805 \nu^{3} + 1747122 \nu^{2} - 662262 \nu + 55314$$$$)/57420$$ $$\beta_{11}$$ $$=$$ $$($$$$-967 \nu^{15} - 2236 \nu^{14} - 22892 \nu^{13} - 55547 \nu^{12} - 188022 \nu^{11} - 492990 \nu^{10} - 682489 \nu^{9} - 1999345 \nu^{8} - 1226019 \nu^{7} - 3960288 \nu^{6} - 1257647 \nu^{5} - 3629531 \nu^{4} - 888655 \nu^{3} - 1158361 \nu^{2} - 321042 \nu + 8766$$$$)/19140$$ $$\beta_{12}$$ $$=$$ $$($$$$3016 \nu^{15} + 7260 \nu^{14} + 73544 \nu^{13} + 177375 \nu^{12} + 632403 \nu^{11} + 1532520 \nu^{10} + 2432926 \nu^{9} + 5970030 \nu^{8} + 4430475 \nu^{7} + 11250855 \nu^{6} + 3515786 \nu^{5} + 9901155 \nu^{4} + 713893 \nu^{3} + 3308415 \nu^{2} - 208365 \nu + 103455$$$$)/57420$$ $$\beta_{13}$$ $$=$$ $$($$$$3016 \nu^{15} - 7260 \nu^{14} + 73544 \nu^{13} - 177375 \nu^{12} + 632403 \nu^{11} - 1532520 \nu^{10} + 2432926 \nu^{9} - 5970030 \nu^{8} + 4430475 \nu^{7} - 11250855 \nu^{6} + 3515786 \nu^{5} - 9901155 \nu^{4} + 713893 \nu^{3} - 3308415 \nu^{2} - 208365 \nu - 103455$$$$)/57420$$ $$\beta_{14}$$ $$=$$ $$($$$$2127 \nu^{15} - 1624 \nu^{14} + 51399 \nu^{13} - 39527 \nu^{12} + 435450 \nu^{11} - 339126 \nu^{10} + 1636560 \nu^{9} - 1303927 \nu^{8} + 2883891 \nu^{7} - 2401026 \nu^{6} + 2186952 \nu^{5} - 2034089 \nu^{4} + 416622 \nu^{3} - 636463 \nu^{2} - 103692 \nu - 18618$$$$)/19140$$ $$\beta_{15}$$ $$=$$ $$($$$$2761 \nu^{15} - 170 \nu^{14} + 68783 \nu^{13} - 6364 \nu^{12} + 614229 \nu^{11} - 87081 \nu^{10} + 2527723 \nu^{9} - 541307 \nu^{8} + 5198259 \nu^{7} - 1571214 \nu^{6} + 5295356 \nu^{5} - 2003620 \nu^{4} + 2403577 \nu^{3} - 878459 \nu^{2} + 375012 \nu - 35352$$$$)/19140$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{2} + \beta_{1}$$ $$\nu^{2}$$ $$=$$ $$-\beta_{12} - \beta_{10} - \beta_{9} + \beta_{7} - \beta_{6} - \beta_{5} + \beta_{4} - \beta_{1} - 3$$ $$\nu^{3}$$ $$=$$ $$-\beta_{15} + 2 \beta_{14} + \beta_{13} + \beta_{12} + \beta_{11} - 2 \beta_{10} - 2 \beta_{8} + \beta_{7} - 3 \beta_{5} - \beta_{4} - 2 \beta_{3} - 7 \beta_{2} - 7 \beta_{1} + 4$$ $$\nu^{4}$$ $$=$$ $$\beta_{15} - 3 \beta_{13} + 10 \beta_{12} + \beta_{11} + 12 \beta_{10} + 13 \beta_{9} - \beta_{8} - 14 \beta_{7} + 9 \beta_{6} + 9 \beta_{5} - 8 \beta_{4} - \beta_{3} - 5 \beta_{2} + 12 \beta_{1} + 17$$ $$\nu^{5}$$ $$=$$ $$16 \beta_{15} - 30 \beta_{14} - 11 \beta_{13} - 14 \beta_{12} - 16 \beta_{11} + 25 \beta_{10} + 25 \beta_{8} - 15 \beta_{7} - 3 \beta_{6} + 38 \beta_{5} + 13 \beta_{4} + 7 \beta_{3} + 60 \beta_{2} + 57 \beta_{1} - 44$$ $$\nu^{6}$$ $$=$$ $$-13 \beta_{15} + 34 \beta_{13} - 94 \beta_{12} - 13 \beta_{11} - 129 \beta_{10} - 144 \beta_{9} + 15 \beta_{8} + 147 \beta_{7} - 86 \beta_{6} - 88 \beta_{5} + 73 \beta_{4} + 15 \beta_{3} + 78 \beta_{2} - 138 \beta_{1} - 133$$ $$\nu^{7}$$ $$=$$ $$-203 \beta_{15} + 376 \beta_{14} + 94 \beta_{13} + 146 \beta_{12} + 203 \beta_{11} - 274 \beta_{10} + 9 \beta_{9} - 265 \beta_{8} + 188 \beta_{7} + 52 \beta_{6} - 416 \beta_{5} - 151 \beta_{4} + 53 \beta_{3} - 564 \beta_{2} - 512 \beta_{1} + 445$$ $$\nu^{8}$$ $$=$$ $$151 \beta_{15} - 342 \beta_{13} + 906 \beta_{12} + 151 \beta_{11} + 1371 \beta_{10} + 1568 \beta_{9} - 197 \beta_{8} - 1499 \beta_{7} + 866 \beta_{6} + 912 \beta_{5} - 715 \beta_{4} - 197 \beta_{3} - 984 \beta_{2} + 1548 \beta_{1} + 1218$$ $$\nu^{9}$$ $$=$$ $$2381 \beta_{15} - 4420 \beta_{14} - 774 \beta_{13} - 1456 \beta_{12} - 2381 \beta_{11} + 2942 \beta_{10} - 177 \beta_{9} + 2765 \beta_{8} - 2210 \beta_{7} - 682 \beta_{6} + 4464 \beta_{5} + 1699 \beta_{4} - 1457 \beta_{3} + 5587 \beta_{2} + 4905 \beta_{1} - 4563$$ $$\nu^{10}$$ $$=$$ $$-1699 \beta_{15} + 3442 \beta_{13} - 9029 \beta_{12} - 1699 \beta_{11} - 14642 \beta_{10} - 17029 \beta_{9} + 2387 \beta_{8} + 15466 \beta_{7} - 8985 \beta_{6} - 9673 \beta_{5} + 7286 \beta_{4} + 2387 \beta_{3} + 11520 \beta_{2} - 17107 \beta_{1} - 12021$$ $$\nu^{11}$$ $$=$$ $$-26928 \beta_{15} + 50214 \beta_{14} + 6547 \beta_{13} + 14669 \beta_{12} + 26928 \beta_{11} - 31562 \beta_{10} + 2465 \beta_{9} - 29097 \beta_{8} + 25107 \beta_{7} + 8122 \beta_{6} - 47903 \beta_{5} - 18806 \beta_{4} + 21569 \beta_{3} - 57202 \beta_{2} - 49080 \beta_{1} + 47677$$ $$\nu^{12}$$ $$=$$ $$18806 \beta_{15} - 35301 \beta_{13} + 92503 \beta_{12} + 18806 \beta_{11} + 157140 \beta_{10} + 184712 \beta_{9} - 27572 \beta_{8} - 162020 \beta_{7} + 94814 \beta_{6} + 103580 \beta_{5} - 76008 \beta_{4} - 27572 \beta_{3} - 130201 \beta_{2} + 187403 \beta_{1} + 123298$$ $$\nu^{13}$$ $$=$$ $$298798 \beta_{15} - 559224 \beta_{14} - 58196 \beta_{13} - 150785 \beta_{12} - 298798 \beta_{11} + 339383 \beta_{10} - 30263 \beta_{9} + 309120 \beta_{8} - 279612 \beta_{7} - 92589 \beta_{6} + 515329 \beta_{5} + 206209 \beta_{4} - 271116 \beta_{3} + 598080 \beta_{2} + 505491 \beta_{1} - 504823$$ $$\nu^{14}$$ $$=$$ $$-206209 \beta_{15} + 368523 \beta_{13} - 966603 \beta_{12} - 206209 \beta_{11} - 1691819 \beta_{10} - 2001694 \beta_{9} + 309875 \beta_{8} + 1717197 \beta_{7} - 1010498 \beta_{6} - 1114164 \beta_{5} + 804289 \beta_{4} + 309875 \beta_{3} + 1444383 \beta_{2} - 2042463 \beta_{1} - 1292175$$ $$\nu^{15}$$ $$=$$ $$-3280637 \beta_{15} + 6155108 \beta_{14} + 545170 \beta_{13} + 1577135 \beta_{12} + 3280637 \beta_{11} - 3657117 \beta_{10} + 350644 \beta_{9} - 3306473 \beta_{8} + 3077554 \beta_{7} + 1031965 \beta_{6} - 5555145 \beta_{5} - 2248672 \beta_{4} + 3178193 \beta_{3} - 6336270 \beta_{2} - 5304305 \beta_{1} + 5390366$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/546\mathbb{Z}\right)^\times$$.

 $$n$$ $$157$$ $$365$$ $$379$$ $$\chi(n)$$ $$-\beta_{3}$$ $$1$$ $$1 - \beta_{3}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
121.1
 − 0.130758i 0.960282i − 2.54804i 1.45057i 1.38609i 0.809195i 3.28902i − 1.75225i 0.130758i − 0.960282i 2.54804i − 1.45057i − 1.38609i − 0.809195i − 3.28902i 1.75225i
−0.866025 + 0.500000i −1.00000 0.500000 0.866025i −0.813575 0.469718i 0.866025 0.500000i 2.13374 1.56433i 1.00000i 1.00000 0.939436
121.2 −0.866025 + 0.500000i −1.00000 0.500000 0.866025i −0.620092 0.358010i 0.866025 0.500000i −2.52418 + 0.792781i 1.00000i 1.00000 0.716021
121.3 −0.866025 + 0.500000i −1.00000 0.500000 0.866025i 0.825077 + 0.476358i 0.866025 0.500000i −1.08980 + 2.41088i 1.00000i 1.00000 −0.952717
121.4 −0.866025 + 0.500000i −1.00000 0.500000 0.866025i 2.34064 + 1.35137i 0.866025 0.500000i 2.61422 0.407273i 1.00000i 1.00000 −2.70274
121.5 0.866025 0.500000i −1.00000 0.500000 0.866025i −3.80406 2.19627i −0.866025 + 0.500000i −0.227820 + 2.63592i 1.00000i 1.00000 −4.39255
121.6 0.866025 0.500000i −1.00000 0.500000 0.866025i −0.594123 0.343017i −0.866025 + 0.500000i 2.63424 0.246484i 1.00000i 1.00000 −0.686034
121.7 0.866025 0.500000i −1.00000 0.500000 0.866025i −0.152918 0.0882870i −0.866025 + 0.500000i −0.623746 2.57117i 1.00000i 1.00000 −0.176574
121.8 0.866025 0.500000i −1.00000 0.500000 0.866025i 2.81905 + 1.62758i −0.866025 + 0.500000i 1.08335 + 2.41379i 1.00000i 1.00000 3.25515
361.1 −0.866025 0.500000i −1.00000 0.500000 + 0.866025i −0.813575 + 0.469718i 0.866025 + 0.500000i 2.13374 + 1.56433i 1.00000i 1.00000 0.939436
361.2 −0.866025 0.500000i −1.00000 0.500000 + 0.866025i −0.620092 + 0.358010i 0.866025 + 0.500000i −2.52418 0.792781i 1.00000i 1.00000 0.716021
361.3 −0.866025 0.500000i −1.00000 0.500000 + 0.866025i 0.825077 0.476358i 0.866025 + 0.500000i −1.08980 2.41088i 1.00000i 1.00000 −0.952717
361.4 −0.866025 0.500000i −1.00000 0.500000 + 0.866025i 2.34064 1.35137i 0.866025 + 0.500000i 2.61422 + 0.407273i 1.00000i 1.00000 −2.70274
361.5 0.866025 + 0.500000i −1.00000 0.500000 + 0.866025i −3.80406 + 2.19627i −0.866025 0.500000i −0.227820 2.63592i 1.00000i 1.00000 −4.39255
361.6 0.866025 + 0.500000i −1.00000 0.500000 + 0.866025i −0.594123 + 0.343017i −0.866025 0.500000i 2.63424 + 0.246484i 1.00000i 1.00000 −0.686034
361.7 0.866025 + 0.500000i −1.00000 0.500000 + 0.866025i −0.152918 + 0.0882870i −0.866025 0.500000i −0.623746 + 2.57117i 1.00000i 1.00000 −0.176574
361.8 0.866025 + 0.500000i −1.00000 0.500000 + 0.866025i 2.81905 1.62758i −0.866025 0.500000i 1.08335 2.41379i 1.00000i 1.00000 3.25515
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 361.8 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
91.u even 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 546.2.bd.a 16
3.b odd 2 1 1638.2.cr.a 16
7.c even 3 1 546.2.bm.a yes 16
13.e even 6 1 546.2.bm.a yes 16
21.h odd 6 1 1638.2.dt.a 16
39.h odd 6 1 1638.2.dt.a 16
91.u even 6 1 inner 546.2.bd.a 16
273.x odd 6 1 1638.2.cr.a 16

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
546.2.bd.a 16 1.a even 1 1 trivial
546.2.bd.a 16 91.u even 6 1 inner
546.2.bm.a yes 16 7.c even 3 1
546.2.bm.a yes 16 13.e even 6 1
1638.2.cr.a 16 3.b odd 2 1
1638.2.cr.a 16 273.x odd 6 1
1638.2.dt.a 16 21.h odd 6 1
1638.2.dt.a 16 39.h odd 6 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{16} - \cdots$$ acting on $$S_{2}^{\mathrm{new}}(546, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 1 - T^{2} + T^{4} )^{4}$$
$3$ $$( 1 + T )^{16}$$
$5$ $$9 + 126 T + 708 T^{2} + 1680 T^{3} + 1375 T^{4} - 1146 T^{5} - 2168 T^{6} + 1110 T^{7} + 3608 T^{8} + 1212 T^{9} - 1112 T^{10} - 390 T^{11} + 335 T^{12} - 20 T^{14} + T^{16}$$
$7$ $$5764801 - 6588344 T + 3882417 T^{2} - 1949612 T^{3} + 773122 T^{4} - 195510 T^{5} + 20531 T^{6} + 9422 T^{7} - 6261 T^{8} + 1346 T^{9} + 419 T^{10} - 570 T^{11} + 322 T^{12} - 116 T^{13} + 33 T^{14} - 8 T^{15} + T^{16}$$
$11$ $$1089 + 203262 T^{2} + 463621 T^{4} + 364580 T^{6} + 123150 T^{8} + 20446 T^{10} + 1716 T^{12} + 68 T^{14} + T^{16}$$
$13$ $$815730721 + 627485170 T + 111016607 T^{2} - 28960854 T^{3} - 14451866 T^{4} - 3792022 T^{5} - 781287 T^{6} + 232960 T^{7} + 163851 T^{8} + 17920 T^{9} - 4623 T^{10} - 1726 T^{11} - 506 T^{12} - 78 T^{13} + 23 T^{14} + 10 T^{15} + T^{16}$$
$17$ $$729 + 30618 T + 1234953 T^{2} + 2139210 T^{3} + 3527497 T^{4} + 1816560 T^{5} + 1436184 T^{6} + 27968 T^{7} + 469665 T^{8} - 5454 T^{9} + 37050 T^{10} - 162 T^{11} + 2160 T^{12} - 4 T^{13} + 54 T^{14} + T^{16}$$
$19$ $$50936769 + 225306000 T^{2} + 258776362 T^{4} + 70036928 T^{6} + 7706091 T^{8} + 422464 T^{10} + 12186 T^{12} + 176 T^{14} + T^{16}$$
$23$ $$55935441 - 135429732 T + 451056357 T^{2} + 251246232 T^{3} + 291981087 T^{4} + 130044672 T^{5} + 106883922 T^{6} + 47568240 T^{7} + 20568661 T^{8} + 5065840 T^{9} + 1084222 T^{10} + 144994 T^{11} + 20626 T^{12} + 2068 T^{13} + 250 T^{14} + 16 T^{15} + T^{16}$$
$29$ $$845588241 + 2338126074 T + 6682606677 T^{2} + 752910714 T^{3} + 2033942643 T^{4} + 777506310 T^{5} + 470194938 T^{6} + 104710404 T^{7} + 28746529 T^{8} + 3634502 T^{9} + 819338 T^{10} + 78974 T^{11} + 15874 T^{12} + 784 T^{13} + 146 T^{14} + 4 T^{15} + T^{16}$$
$31$ $$5963391729 - 8371590984 T - 120481959 T^{2} + 5668545912 T^{3} + 945192091 T^{4} - 2628610344 T^{5} + 934633838 T^{6} - 73152960 T^{7} - 22174851 T^{8} + 3419496 T^{9} + 581374 T^{10} - 160170 T^{11} + 5094 T^{12} + 1416 T^{13} - 70 T^{14} - 12 T^{15} + T^{16}$$
$37$ $$18014203089 - 233434501344 T + 1076992316235 T^{2} - 890018930592 T^{3} + 319891238307 T^{4} - 50453705088 T^{5} - 113745870 T^{6} + 1033630146 T^{7} - 15456987 T^{8} - 38129130 T^{9} + 6135750 T^{10} - 212706 T^{11} - 30870 T^{12} + 1980 T^{13} + 234 T^{14} - 30 T^{15} + T^{16}$$
$41$ $$24389381241 + 49136706414 T + 32623530423 T^{2} - 754806966 T^{3} - 3962896775 T^{4} - 144591792 T^{5} + 353842224 T^{6} + 39397266 T^{7} - 14175063 T^{8} - 2078556 T^{9} + 428258 T^{10} + 80166 T^{11} - 4692 T^{12} - 1404 T^{13} + 30 T^{14} + 18 T^{15} + T^{16}$$
$43$ $$3870808348969 + 2804065975006 T + 1666204105117 T^{2} + 536390504046 T^{3} + 170241737333 T^{4} + 40453591852 T^{5} + 10024795764 T^{6} + 1941903776 T^{7} + 366695941 T^{8} + 55620622 T^{9} + 8320602 T^{10} + 1026662 T^{11} + 118268 T^{12} + 10212 T^{13} + 730 T^{14} + 32 T^{15} + T^{16}$$
$47$ $$24307197434289 - 24417462095334 T + 8716448907600 T^{2} - 542824551192 T^{3} - 319301019236 T^{4} + 37702562436 T^{5} + 15365240918 T^{6} - 4225637976 T^{7} + 211792059 T^{8} + 50329134 T^{9} - 5223668 T^{10} - 996288 T^{11} + 282285 T^{12} - 31218 T^{13} + 1925 T^{14} - 66 T^{15} + T^{16}$$
$53$ $$689748521121 - 389237251392 T + 320399410950 T^{2} - 101343009648 T^{3} + 68495393026 T^{4} - 22871615102 T^{5} + 7545464746 T^{6} - 1444002202 T^{7} + 261948253 T^{8} - 29397920 T^{9} + 3974186 T^{10} - 321650 T^{11} + 42019 T^{12} - 1954 T^{13} + 229 T^{14} - 2 T^{15} + T^{16}$$
$59$ $$63426911409 - 137058159564 T + 50453741322 T^{2} + 104302583496 T^{3} + 17268058845 T^{4} - 14616496440 T^{5} - 2823362790 T^{6} + 1990641204 T^{7} + 1002175036 T^{8} + 198279660 T^{9} + 19046766 T^{10} + 534240 T^{11} - 51547 T^{12} - 2376 T^{13} + 366 T^{14} + 36 T^{15} + T^{16}$$
$61$ $$( -2816 + 5888 T + 24960 T^{2} + 14592 T^{3} - 224 T^{4} - 1200 T^{5} - 156 T^{6} + 4 T^{7} + T^{8} )^{2}$$
$67$ $$59619415692321 + 25032524425614 T^{2} + 2679933282399 T^{4} + 116584132584 T^{6} + 2526460285 T^{8} + 30027608 T^{10} + 199671 T^{12} + 698 T^{14} + T^{16}$$
$71$ $$610341027216561 - 1096953326616492 T + 602891034017400 T^{2} + 97567384514976 T^{3} - 15147190490049 T^{4} - 2522510071326 T^{5} + 292923808596 T^{6} + 41313331044 T^{7} - 2548554516 T^{8} - 400741830 T^{9} + 15662484 T^{10} + 2858598 T^{11} - 18693 T^{12} - 11160 T^{13} - 72 T^{14} + 30 T^{15} + T^{16}$$
$73$ $$4284129113721081 + 1891523913852150 T + 181056304529817 T^{2} - 42970769322450 T^{3} - 7407240542996 T^{4} + 1211693176248 T^{5} + 401805924479 T^{6} + 28880778672 T^{7} - 1675889709 T^{8} - 290298816 T^{9} + 4915447 T^{10} + 2349576 T^{11} + 66096 T^{12} - 7038 T^{13} - 283 T^{14} + 18 T^{15} + T^{16}$$
$79$ $$1257624369 + 2519078742 T + 7260138876 T^{2} - 2763353436 T^{3} + 5283824647 T^{4} + 376447506 T^{5} + 932604096 T^{6} + 28485086 T^{7} + 102182940 T^{8} + 13373676 T^{9} + 3324984 T^{10} + 306798 T^{11} + 55203 T^{12} + 5060 T^{13} + 516 T^{14} + 24 T^{15} + T^{16}$$
$83$ $$4016014566553089 + 424910533723542 T^{2} + 18726082612965 T^{4} + 450095163060 T^{6} + 6458734966 T^{8} + 56648614 T^{10} + 296316 T^{12} + 844 T^{14} + T^{16}$$
$89$ $$25608042872721 - 11588658331950 T - 5403656958030 T^{2} + 3236458963500 T^{3} + 1969110015336 T^{4} + 241139942730 T^{5} - 30777258534 T^{6} - 7217360688 T^{7} + 705231385 T^{8} + 310157970 T^{9} + 31180348 T^{10} + 571644 T^{11} - 98835 T^{12} - 3234 T^{13} + 511 T^{14} + 42 T^{15} + T^{16}$$
$97$ $$11594767041 + 112518955692 T + 383888415408 T^{2} + 193273582080 T^{3} - 14621940542 T^{4} - 27315611616 T^{5} + 2623979814 T^{6} + 2751107274 T^{7} + 294651795 T^{8} - 35766252 T^{9} - 4739482 T^{10} + 373218 T^{11} + 56787 T^{12} - 1710 T^{13} - 273 T^{14} + 6 T^{15} + T^{16}$$