# Properties

 Label 138.3.c.a Level $138$ Weight $3$ Character orbit 138.c Analytic conductor $3.760$ Analytic rank $0$ Dimension $16$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$138 = 2 \cdot 3 \cdot 23$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 138.c (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$3.76022764817$$ Analytic rank: $$0$$ Dimension: $$16$$ Coefficient field: $$\mathbb{Q}[x]/(x^{16} - \cdots)$$ Defining polynomial: $$x^{16} - 4 x^{15} + 10 x^{14} + 8 x^{13} - 119 x^{12} + 416 x^{11} - 774 x^{10} - 1284 x^{9} + 7956 x^{8} - 11556 x^{7} - 62694 x^{6} + 303264 x^{5} - 780759 x^{4} + 472392 x^{3} + 5314410 x^{2} - 19131876 x + 43046721$$ Coefficient ring: $$\Z[a_1, \ldots, a_{23}]$$ Coefficient ring index: $$2^{8}\cdot 3$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q -\beta_{3} q^{2} + \beta_{1} q^{3} -2 q^{4} -\beta_{7} q^{5} + \beta_{11} q^{6} -\beta_{6} q^{7} + 2 \beta_{3} q^{8} + ( -1 + \beta_{5} - \beta_{8} + \beta_{10} ) q^{9} +O(q^{10})$$ $$q -\beta_{3} q^{2} + \beta_{1} q^{3} -2 q^{4} -\beta_{7} q^{5} + \beta_{11} q^{6} -\beta_{6} q^{7} + 2 \beta_{3} q^{8} + ( -1 + \beta_{5} - \beta_{8} + \beta_{10} ) q^{9} + ( -1 + \beta_{2} + \beta_{4} - \beta_{8} ) q^{10} + ( \beta_{2} - \beta_{4} + \beta_{5} - \beta_{8} + \beta_{11} + \beta_{12} + \beta_{13} ) q^{11} -2 \beta_{1} q^{12} + ( -1 + 2 \beta_{5} + \beta_{8} - \beta_{9} + \beta_{12} - \beta_{14} ) q^{13} + ( -2 \beta_{1} + \beta_{2} - \beta_{3} - \beta_{4} + \beta_{5} + \beta_{9} + \beta_{12} - \beta_{15} ) q^{14} + ( 2 + \beta_{2} - \beta_{3} - \beta_{6} - 2 \beta_{9} + 2 \beta_{10} + \beta_{13} - \beta_{14} + \beta_{15} ) q^{15} + 4 q^{16} + ( \beta_{1} - \beta_{2} + \beta_{4} - \beta_{5} - \beta_{7} - 2 \beta_{10} - \beta_{12} - \beta_{13} - \beta_{14} + \beta_{15} ) q^{17} + ( 1 - \beta_{1} + \beta_{9} - \beta_{12} - \beta_{14} ) q^{18} + ( 3 - \beta_{1} - \beta_{3} - \beta_{5} - \beta_{6} - \beta_{9} - \beta_{11} + \beta_{12} - \beta_{14} ) q^{19} + 2 \beta_{7} q^{20} + ( \beta_{2} + 4 \beta_{3} - \beta_{5} - \beta_{6} + 2 \beta_{9} + \beta_{10} - \beta_{12} - \beta_{13} + \beta_{14} - 2 \beta_{15} ) q^{21} + ( 2 - \beta_{1} + \beta_{3} + 2 \beta_{6} + \beta_{9} + 2 \beta_{11} + \beta_{13} + \beta_{14} ) q^{22} -\beta_{10} q^{23} -2 \beta_{11} q^{24} + ( -12 + \beta_{1} + \beta_{3} + \beta_{9} + \beta_{11} + 3 \beta_{12} + \beta_{14} ) q^{25} + ( 1 - 2 \beta_{1} + 2 \beta_{10} + \beta_{11} - \beta_{13} - 2 \beta_{14} + 2 \beta_{15} ) q^{26} + ( -6 + \beta_{1} - 4 \beta_{3} + 2 \beta_{4} - \beta_{5} + 2 \beta_{6} - 2 \beta_{7} + \beta_{9} + \beta_{10} - 3 \beta_{12} - \beta_{13} + \beta_{14} ) q^{27} + 2 \beta_{6} q^{28} + ( 3 \beta_{1} - 2 \beta_{2} + \beta_{3} + 2 \beta_{4} - 2 \beta_{5} + 2 \beta_{7} - \beta_{8} - 2 \beta_{10} + \beta_{11} - 2 \beta_{12} - 2 \beta_{13} + 2 \beta_{14} - 2 \beta_{15} ) q^{29} + ( -1 + \beta_{2} - \beta_{3} - \beta_{4} + \beta_{5} + \beta_{6} + \beta_{7} - \beta_{8} - \beta_{10} + 2 \beta_{11} - \beta_{12} + 2 \beta_{13} + 2 \beta_{14} + \beta_{15} ) q^{30} + ( 11 - \beta_{1} - 2 \beta_{2} + \beta_{3} - 2 \beta_{4} + 2 \beta_{6} + \beta_{8} + 2 \beta_{9} + 2 \beta_{11} + \beta_{13} + 2 \beta_{14} ) q^{31} -4 \beta_{3} q^{32} + ( -6 - 2 \beta_{1} - \beta_{2} - 7 \beta_{3} - 2 \beta_{4} + 3 \beta_{5} - \beta_{6} - \beta_{7} + \beta_{8} - \beta_{9} - \beta_{10} - 3 \beta_{11} - \beta_{12} - \beta_{14} - \beta_{15} ) q^{33} + ( -1 + \beta_{2} + \beta_{4} - 2 \beta_{5} - 2 \beta_{6} - \beta_{8} + 2 \beta_{12} ) q^{34} + ( 1 + 2 \beta_{2} - 5 \beta_{3} - 2 \beta_{4} + 2 \beta_{5} + 2 \beta_{7} - 4 \beta_{9} - 4 \beta_{10} + \beta_{11} + 2 \beta_{12} + 5 \beta_{13} + 4 \beta_{14} ) q^{35} + ( 2 - 2 \beta_{5} + 2 \beta_{8} - 2 \beta_{10} ) q^{36} + ( -12 + 3 \beta_{1} + \beta_{2} - \beta_{3} + \beta_{4} - \beta_{5} - 2 \beta_{6} - 3 \beta_{8} + \beta_{9} - 3 \beta_{11} + \beta_{12} - 2 \beta_{13} + \beta_{14} ) q^{37} + ( \beta_{1} + \beta_{2} - 2 \beta_{3} - \beta_{4} + \beta_{5} - 2 \beta_{9} + 2 \beta_{10} + \beta_{12} + 3 \beta_{13} + \beta_{14} + \beta_{15} ) q^{38} + ( 12 + \beta_{1} - 4 \beta_{2} + 12 \beta_{3} - \beta_{5} + 4 \beta_{6} + 3 \beta_{8} + \beta_{9} + 3 \beta_{10} + \beta_{11} - 5 \beta_{12} - 3 \beta_{13} - \beta_{14} + 2 \beta_{15} ) q^{39} + ( 2 - 2 \beta_{2} - 2 \beta_{4} + 2 \beta_{8} ) q^{40} + ( -4 + 7 \beta_{1} - 2 \beta_{2} + 7 \beta_{3} + 2 \beta_{4} - 2 \beta_{5} - 4 \beta_{7} + \beta_{8} + 2 \beta_{9} + 2 \beta_{10} - 5 \beta_{11} - 2 \beta_{12} - 2 \beta_{14} ) q^{41} + ( 8 - 2 \beta_{1} + \beta_{2} - 2 \beta_{3} - \beta_{4} + 2 \beta_{5} + \beta_{6} + \beta_{7} - \beta_{9} - 3 \beta_{10} - 2 \beta_{15} ) q^{42} + ( 6 + 2 \beta_{1} - \beta_{2} - \beta_{4} - \beta_{5} - \beta_{6} + 3 \beta_{8} - 2 \beta_{9} - \beta_{11} + 7 \beta_{12} - \beta_{13} - 2 \beta_{14} ) q^{43} + ( -2 \beta_{2} + 2 \beta_{4} - 2 \beta_{5} + 2 \beta_{8} - 2 \beta_{11} - 2 \beta_{12} - 2 \beta_{13} ) q^{44} + ( 4 + 4 \beta_{1} - 5 \beta_{2} + 5 \beta_{3} - 3 \beta_{5} - \beta_{6} + 3 \beta_{8} - 3 \beta_{9} - \beta_{10} + \beta_{11} - 5 \beta_{12} + 4 \beta_{13} + 3 \beta_{14} + \beta_{15} ) q^{45} + \beta_{12} q^{46} + ( -2 + \beta_{1} - 2 \beta_{2} + 2 \beta_{4} - 2 \beta_{5} + 2 \beta_{7} + \beta_{8} + 4 \beta_{9} - 2 \beta_{10} - 3 \beta_{11} - 2 \beta_{12} - 4 \beta_{13} - 4 \beta_{14} ) q^{47} + 4 \beta_{1} q^{48} + ( 10 + 5 \beta_{1} - 3 \beta_{3} - 3 \beta_{9} - 7 \beta_{11} + 3 \beta_{12} - 4 \beta_{13} - 3 \beta_{14} ) q^{49} + ( -2 \beta_{1} + 11 \beta_{3} + 2 \beta_{9} + 6 \beta_{10} - 2 \beta_{13} - 2 \beta_{15} ) q^{50} + ( -1 + \beta_{1} - 6 \beta_{3} + \beta_{4} - 2 \beta_{5} - 5 \beta_{6} + 2 \beta_{7} - \beta_{8} - \beta_{9} + 2 \beta_{10} - 2 \beta_{11} - 4 \beta_{13} - 4 \beta_{14} + 3 \beta_{15} ) q^{51} + ( 2 - 4 \beta_{5} - 2 \beta_{8} + 2 \beta_{9} - 2 \beta_{12} + 2 \beta_{14} ) q^{52} + ( 2 - 8 \beta_{1} + 5 \beta_{3} - 3 \beta_{7} + 3 \beta_{8} + \beta_{9} - 4 \beta_{10} - \beta_{11} - 3 \beta_{13} + 4 \beta_{14} - 5 \beta_{15} ) q^{53} + ( -14 + 3 \beta_{1} + 5 \beta_{3} + 4 \beta_{4} - 2 \beta_{5} - 2 \beta_{6} + 2 \beta_{7} - 2 \beta_{8} - \beta_{9} - 4 \beta_{10} - 2 \beta_{11} - 3 \beta_{12} - 3 \beta_{13} - \beta_{14} ) q^{54} + ( -10 - 4 \beta_{1} - 4 \beta_{3} + 2 \beta_{5} + 4 \beta_{6} - 6 \beta_{8} + 2 \beta_{9} - 4 \beta_{11} - 2 \beta_{12} + 2 \beta_{14} ) q^{55} + ( 4 \beta_{1} - 2 \beta_{2} + 2 \beta_{3} + 2 \beta_{4} - 2 \beta_{5} - 2 \beta_{9} - 2 \beta_{12} + 2 \beta_{15} ) q^{56} + ( -6 + 3 \beta_{1} - 3 \beta_{2} - 5 \beta_{3} - 3 \beta_{4} - 3 \beta_{5} + 3 \beta_{7} + \beta_{8} - \beta_{9} + \beta_{11} + \beta_{12} + 2 \beta_{13} + \beta_{14} ) q^{57} + ( 1 - 6 \beta_{1} - 2 \beta_{2} - 4 \beta_{3} - 2 \beta_{4} + 4 \beta_{5} - 4 \beta_{6} + 2 \beta_{8} - 4 \beta_{9} - 3 \beta_{11} + 2 \beta_{12} + \beta_{13} - 4 \beta_{14} ) q^{58} + ( 1 - 6 \beta_{1} + 5 \beta_{3} + 2 \beta_{7} + 8 \beta_{8} - 4 \beta_{9} - 7 \beta_{11} + 3 \beta_{13} + 4 \beta_{15} ) q^{59} + ( -4 - 2 \beta_{2} + 2 \beta_{3} + 2 \beta_{6} + 4 \beta_{9} - 4 \beta_{10} - 2 \beta_{13} + 2 \beta_{14} - 2 \beta_{15} ) q^{60} + ( 8 - 4 \beta_{1} - 2 \beta_{5} + 4 \beta_{8} - 4 \beta_{9} + 2 \beta_{11} + 2 \beta_{12} + 2 \beta_{13} - 4 \beta_{14} ) q^{61} + ( -1 - 2 \beta_{2} - 8 \beta_{3} + 2 \beta_{4} - 2 \beta_{5} - 4 \beta_{7} + 2 \beta_{8} + 2 \beta_{9} - 3 \beta_{11} - 2 \beta_{12} - 3 \beta_{13} - 2 \beta_{15} ) q^{62} + ( 7 - 3 \beta_{1} + 7 \beta_{2} + 2 \beta_{3} + 2 \beta_{4} + 3 \beta_{5} + \beta_{6} + 4 \beta_{7} - 6 \beta_{8} + \beta_{9} - 2 \beta_{10} - \beta_{11} + \beta_{12} + 5 \beta_{13} + 4 \beta_{14} - 5 \beta_{15} ) q^{63} -8 q^{64} + ( 6 + 4 \beta_{1} - \beta_{3} + 2 \beta_{7} - 5 \beta_{8} - 11 \beta_{9} + 4 \beta_{10} + 11 \beta_{11} + 5 \beta_{13} + 2 \beta_{14} + 9 \beta_{15} ) q^{65} + ( -11 + \beta_{1} + 2 \beta_{2} + 5 \beta_{3} + 2 \beta_{5} + \beta_{6} - 3 \beta_{7} - \beta_{8} - 3 \beta_{10} + 2 \beta_{12} + \beta_{13} - 3 \beta_{14} + 2 \beta_{15} ) q^{66} + ( -11 + 5 \beta_{1} - \beta_{2} + 11 \beta_{3} - \beta_{4} - 2 \beta_{5} + 7 \beta_{6} + 7 \beta_{8} + 5 \beta_{9} + 14 \beta_{11} - 4 \beta_{12} + 3 \beta_{13} + 5 \beta_{14} ) q^{67} + ( -2 \beta_{1} + 2 \beta_{2} - 2 \beta_{4} + 2 \beta_{5} + 2 \beta_{7} + 4 \beta_{10} + 2 \beta_{12} + 2 \beta_{13} + 2 \beta_{14} - 2 \beta_{15} ) q^{68} + ( -\beta_{1} - \beta_{3} - \beta_{4} + \beta_{5} - \beta_{6} + \beta_{7} + \beta_{8} - \beta_{11} + \beta_{12} - \beta_{13} - \beta_{14} ) q^{69} + ( -14 - 2 \beta_{2} - 2 \beta_{4} + 8 \beta_{5} + 4 \beta_{6} + 2 \beta_{9} + 4 \beta_{12} + 2 \beta_{14} ) q^{70} + ( -3 - 9 \beta_{1} + 6 \beta_{2} + 5 \beta_{3} - 6 \beta_{4} + 6 \beta_{5} + 4 \beta_{7} + \beta_{8} + 8 \beta_{9} - 2 \beta_{10} - 4 \beta_{11} + 6 \beta_{12} + \beta_{13} - 4 \beta_{14} - 4 \beta_{15} ) q^{71} + ( -2 + 2 \beta_{1} - 2 \beta_{9} + 2 \beta_{12} + 2 \beta_{14} ) q^{72} + ( 26 + 7 \beta_{1} - 2 \beta_{2} + 3 \beta_{3} - 2 \beta_{4} - 4 \beta_{6} + 3 \beta_{8} + 2 \beta_{9} + \beta_{11} - 6 \beta_{12} - 2 \beta_{13} + 2 \beta_{14} ) q^{73} + ( -2 + 3 \beta_{1} + 2 \beta_{2} + 7 \beta_{3} - 2 \beta_{4} + 2 \beta_{5} + 2 \beta_{7} - 4 \beta_{8} + 3 \beta_{9} + 2 \beta_{10} + 2 \beta_{11} + 2 \beta_{12} + \beta_{13} + \beta_{14} - 4 \beta_{15} ) q^{74} + ( 7 - 13 \beta_{1} + 4 \beta_{2} + \beta_{3} - 6 \beta_{4} + 2 \beta_{5} + 2 \beta_{6} - 6 \beta_{7} - 6 \beta_{8} + 6 \beta_{9} + 4 \beta_{10} + 3 \beta_{11} + 4 \beta_{12} - \beta_{13} - 2 \beta_{15} ) q^{75} + ( -6 + 2 \beta_{1} + 2 \beta_{3} + 2 \beta_{5} + 2 \beta_{6} + 2 \beta_{9} + 2 \beta_{11} - 2 \beta_{12} + 2 \beta_{14} ) q^{76} + ( 2 - 4 \beta_{1} + 2 \beta_{2} - 26 \beta_{3} - 2 \beta_{4} + 2 \beta_{5} - 4 \beta_{7} - 2 \beta_{8} + 4 \beta_{10} + 4 \beta_{11} + 2 \beta_{12} + 2 \beta_{14} - 2 \beta_{15} ) q^{77} + ( 21 + 5 \beta_{1} - 4 \beta_{2} - 9 \beta_{3} + 4 \beta_{4} - 8 \beta_{5} - 4 \beta_{6} - 4 \beta_{7} - 2 \beta_{8} - \beta_{9} - 6 \beta_{10} - 2 \beta_{11} - 7 \beta_{12} - 4 \beta_{13} - \beta_{14} + 2 \beta_{15} ) q^{78} + ( 19 - 17 \beta_{1} + 2 \beta_{2} - 5 \beta_{3} + 2 \beta_{4} + \beta_{5} + 3 \beta_{6} - 10 \beta_{8} + 3 \beta_{9} + \beta_{11} - \beta_{12} + 6 \beta_{13} + 3 \beta_{14} ) q^{79} -4 \beta_{7} q^{80} + ( 11 - 3 \beta_{1} + 6 \beta_{2} + 3 \beta_{3} + 6 \beta_{4} + \beta_{5} - 4 \beta_{8} - 6 \beta_{9} + \beta_{10} + 9 \beta_{11} + 6 \beta_{13} + 6 \beta_{15} ) q^{81} + ( 11 + 8 \beta_{1} + 4 \beta_{2} + 6 \beta_{3} + 4 \beta_{4} - 4 \beta_{5} - 4 \beta_{6} + 4 \beta_{8} - 2 \beta_{9} + 5 \beta_{11} - 2 \beta_{12} - \beta_{13} - 2 \beta_{14} ) q^{82} + ( 8 \beta_{1} - 3 \beta_{2} - 10 \beta_{3} + 3 \beta_{4} - 3 \beta_{5} - 7 \beta_{8} + 2 \beta_{9} + 7 \beta_{11} - 3 \beta_{12} - 5 \beta_{13} - 6 \beta_{14} + 4 \beta_{15} ) q^{83} + ( -2 \beta_{2} - 8 \beta_{3} + 2 \beta_{5} + 2 \beta_{6} - 4 \beta_{9} - 2 \beta_{10} + 2 \beta_{12} + 2 \beta_{13} - 2 \beta_{14} + 4 \beta_{15} ) q^{84} + ( -37 + 11 \beta_{1} + 6 \beta_{2} + 3 \beta_{3} + 6 \beta_{4} - 2 \beta_{5} - 4 \beta_{6} - 4 \beta_{8} + \beta_{9} - \beta_{11} - \beta_{12} - 4 \beta_{13} + \beta_{14} ) q^{85} + ( 2 + \beta_{1} + \beta_{2} - 4 \beta_{3} - \beta_{4} + \beta_{5} - 2 \beta_{7} - 2 \beta_{8} - 4 \beta_{9} + 14 \beta_{10} + 4 \beta_{11} + \beta_{12} + 3 \beta_{13} + \beta_{14} + 3 \beta_{15} ) q^{86} + ( -19 - 5 \beta_{1} + 4 \beta_{2} - 11 \beta_{3} + 3 \beta_{5} + 8 \beta_{6} + 6 \beta_{7} - \beta_{8} + 11 \beta_{9} + 5 \beta_{10} + 4 \beta_{11} + 5 \beta_{12} + 2 \beta_{13} + \beta_{14} - 2 \beta_{15} ) q^{87} + ( -4 + 2 \beta_{1} - 2 \beta_{3} - 4 \beta_{6} - 2 \beta_{9} - 4 \beta_{11} - 2 \beta_{13} - 2 \beta_{14} ) q^{88} + ( -6 + 3 \beta_{1} + 3 \beta_{2} - 19 \beta_{3} - 3 \beta_{4} + 3 \beta_{5} + \beta_{7} + \beta_{8} + 5 \beta_{9} - 2 \beta_{10} - 7 \beta_{11} + 3 \beta_{12} + 4 \beta_{13} - 7 \beta_{14} + 2 \beta_{15} ) q^{89} + ( -5 \beta_{1} + \beta_{2} - \beta_{3} - \beta_{4} + 6 \beta_{5} - 5 \beta_{6} - 5 \beta_{7} + 6 \beta_{8} - 5 \beta_{10} - 4 \beta_{11} + 2 \beta_{12} - \beta_{13} - \beta_{14} - 2 \beta_{15} ) q^{90} + ( -3 - 17 \beta_{1} + 8 \beta_{2} - \beta_{3} + 8 \beta_{4} - \beta_{5} - 4 \beta_{6} - 6 \beta_{8} - 3 \beta_{9} + 7 \beta_{11} - 11 \beta_{12} + 8 \beta_{13} - 3 \beta_{14} ) q^{91} + 2 \beta_{10} q^{92} + ( -14 + 8 \beta_{1} + 4 \beta_{2} + 2 \beta_{3} + 2 \beta_{4} + \beta_{5} - 2 \beta_{6} - 8 \beta_{7} - \beta_{8} + 2 \beta_{9} - \beta_{10} - 4 \beta_{11} + 6 \beta_{12} - 4 \beta_{13} - 6 \beta_{14} - 2 \beta_{15} ) q^{93} + ( 7 + 4 \beta_{1} - 2 \beta_{2} + 2 \beta_{3} - 2 \beta_{4} - 8 \beta_{5} - 4 \beta_{6} + 6 \beta_{8} - 2 \beta_{9} + \beta_{11} + 2 \beta_{12} - \beta_{13} - 2 \beta_{14} ) q^{94} + ( 5 - 16 \beta_{1} + 2 \beta_{2} - 7 \beta_{3} - 2 \beta_{4} + 2 \beta_{5} - 2 \beta_{7} - 4 \beta_{8} + 8 \beta_{9} - 2 \beta_{10} + 9 \beta_{11} + 2 \beta_{12} - 11 \beta_{13} - 4 \beta_{14} - 4 \beta_{15} ) q^{95} + 4 \beta_{11} q^{96} + ( 8 + 7 \beta_{1} - \beta_{2} + 3 \beta_{3} - \beta_{4} + 3 \beta_{5} + 6 \beta_{6} + \beta_{8} + 3 \beta_{9} + \beta_{11} + 3 \beta_{12} - 2 \beta_{13} + 3 \beta_{14} ) q^{97} + ( 14 \beta_{1} - 11 \beta_{3} - 8 \beta_{8} - 6 \beta_{9} + 6 \beta_{10} + 8 \beta_{11} + 6 \beta_{13} + 6 \beta_{15} ) q^{98} + ( 1 - 5 \beta_{1} - 4 \beta_{2} + 4 \beta_{3} + 5 \beta_{4} - 2 \beta_{5} + 4 \beta_{7} + 11 \beta_{8} - 5 \beta_{9} + 4 \beta_{10} - 4 \beta_{12} - 6 \beta_{13} - 2 \beta_{14} - \beta_{15} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$16q + 4q^{3} - 32q^{4} - 8q^{6} - 4q^{9} + O(q^{10})$$ $$16q + 4q^{3} - 32q^{4} - 8q^{6} - 4q^{9} - 8q^{12} - 8q^{13} + 28q^{15} + 64q^{16} + 16q^{18} + 40q^{19} + 4q^{21} + 16q^{22} + 16q^{24} - 192q^{25} - 80q^{27} - 24q^{30} + 136q^{31} - 84q^{33} - 16q^{34} + 8q^{36} - 136q^{37} + 156q^{39} + 128q^{42} + 72q^{43} + 4q^{45} + 16q^{48} + 224q^{49} - 4q^{51} + 16q^{52} - 176q^{54} - 96q^{55} - 160q^{57} - 56q^{60} + 48q^{61} + 204q^{63} - 128q^{64} - 144q^{66} - 304q^{67} - 176q^{70} - 32q^{72} + 408q^{73} + 68q^{75} - 80q^{76} + 328q^{78} + 312q^{79} + 164q^{81} + 160q^{82} - 8q^{84} - 464q^{85} - 268q^{87} - 32q^{88} + 32q^{90} - 72q^{91} - 108q^{93} - 32q^{96} + 168q^{97} - 60q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{16} - 4 x^{15} + 10 x^{14} + 8 x^{13} - 119 x^{12} + 416 x^{11} - 774 x^{10} - 1284 x^{9} + 7956 x^{8} - 11556 x^{7} - 62694 x^{6} + 303264 x^{5} - 780759 x^{4} + 472392 x^{3} + 5314410 x^{2} - 19131876 x + 43046721$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$($$$$-217 \nu^{15} + 3982 \nu^{14} + 2141 \nu^{13} - 71198 \nu^{12} + 44174 \nu^{11} - 255260 \nu^{10} + 1371528 \nu^{9} - 1191198 \nu^{8} + 441054 \nu^{7} + 13141494 \nu^{6} + 55581390 \nu^{5} + 119343132 \nu^{4} + 286905969 \nu^{3} - 2262049092 \nu^{2} - 1802116431 \nu + 33461651124$$$$)/ 5739562800$$ $$\beta_{3}$$ $$=$$ $$($$$$-1049 \nu^{15} - 2761 \nu^{14} - 563 \nu^{13} + 46859 \nu^{12} - 119312 \nu^{11} + 380240 \nu^{10} + 322866 \nu^{9} - 756006 \nu^{8} + 11963718 \nu^{7} + 6408018 \nu^{6} - 33057720 \nu^{5} + 188472744 \nu^{4} - 81861597 \nu^{3} - 1739642589 \nu^{2} + 3525048153 \nu - 31342795857$$$$)/ 17218688400$$ $$\beta_{4}$$ $$=$$ $$($$$$-2257 \nu^{15} - 24758 \nu^{14} + 36191 \nu^{13} - 18308 \nu^{12} + 241214 \nu^{11} - 213620 \nu^{10} - 4661172 \nu^{9} + 16826262 \nu^{8} + 6454494 \nu^{7} + 7348374 \nu^{6} - 151678170 \nu^{5} + 1126937772 \nu^{4} + 161853309 \nu^{3} + 12382102908 \nu^{2} - 21789612441 \nu - 62628196086$$$$)/ 17218688400$$ $$\beta_{5}$$ $$=$$ $$($$$$-449 \nu^{15} - 598 \nu^{14} - 6497 \nu^{13} - 17650 \nu^{12} + 70810 \nu^{11} - 149110 \nu^{10} + 90666 \nu^{9} + 2237178 \nu^{8} - 5056056 \nu^{7} + 8100432 \nu^{6} + 64314810 \nu^{5} - 83681910 \nu^{4} + 205195275 \nu^{3} + 2727119016 \nu^{2} - 2316551319 \nu + 6782250042$$$$)/ 3443737680$$ $$\beta_{6}$$ $$=$$ $$($$$$-329 \nu^{15} - 1708 \nu^{14} + 3703 \nu^{13} + 2120 \nu^{12} - 22220 \nu^{11} + 275480 \nu^{10} + 28926 \nu^{9} - 281292 \nu^{8} + 5112954 \nu^{7} - 18822348 \nu^{6} - 48507660 \nu^{5} + 95586480 \nu^{4} - 75221865 \nu^{3} - 581986944 \nu^{2} + 1600168851 \nu - 10005971148$$$$)/ 2295825120$$ $$\beta_{7}$$ $$=$$ $$($$$$-6992 \nu^{15} - 36013 \nu^{14} + 58996 \nu^{13} - 326953 \nu^{12} + 578104 \nu^{11} + 243020 \nu^{10} - 9067572 \nu^{9} + 12700002 \nu^{8} + 95667444 \nu^{7} - 230079906 \nu^{6} + 173132640 \nu^{5} + 1510916652 \nu^{4} - 8620602876 \nu^{3} + 25348968063 \nu^{2} - 31427294976 \nu - 137983872681$$$$)/ 34437376800$$ $$\beta_{8}$$ $$=$$ $$($$$$667 \nu^{15} - 3082 \nu^{14} + 874 \nu^{13} - 4717 \nu^{12} + 31606 \nu^{11} - 37600 \nu^{10} + 135612 \nu^{9} - 21642 \nu^{8} + 787716 \nu^{7} + 11190096 \nu^{6} - 15734250 \nu^{5} + 68257728 \nu^{4} - 87910839 \nu^{3} + 399643632 \nu^{2} - 581396454 \nu + 1095299901$$$$)/ 2869781400$$ $$\beta_{9}$$ $$=$$ $$($$$$-4039 \nu^{15} - 3041 \nu^{14} + 49277 \nu^{13} - 178841 \nu^{12} + 468248 \nu^{11} - 1872680 \nu^{10} - 1437354 \nu^{9} + 15826074 \nu^{8} - 24568722 \nu^{7} - 57363822 \nu^{6} + 490471200 \nu^{5} - 480603456 \nu^{4} - 146093787 \nu^{3} + 12571473051 \nu^{2} - 49895401167 \nu + 72064993923$$$$)/ 11479125600$$ $$\beta_{10}$$ $$=$$ $$($$$$6247 \nu^{15} - 15502 \nu^{14} + 37729 \nu^{13} + 59948 \nu^{12} - 164414 \nu^{11} + 519950 \nu^{10} + 360342 \nu^{9} - 11315742 \nu^{8} + 30006576 \nu^{7} + 26638416 \nu^{6} - 415979550 \nu^{5} + 827955918 \nu^{4} - 1553441409 \nu^{3} + 5980955112 \nu^{2} + 8094377871 \nu - 10120762404$$$$)/ 17218688400$$ $$\beta_{11}$$ $$=$$ $$($$$$773 \nu^{15} - 1103 \nu^{14} - 6139 \nu^{13} + 27127 \nu^{12} - 90736 \nu^{11} + 54340 \nu^{10} + 233658 \nu^{9} - 2256618 \nu^{8} + 634914 \nu^{7} + 10980414 \nu^{6} - 56288520 \nu^{5} + 100097532 \nu^{4} + 138233709 \nu^{3} - 1011096027 \nu^{2} + 5712459309 \nu - 5017334481$$$$)/ 1913187600$$ $$\beta_{12}$$ $$=$$ $$($$$$-559 \nu^{15} + 2326 \nu^{14} - 199 \nu^{13} - 6164 \nu^{12} + 75908 \nu^{11} - 242444 \nu^{10} + 218034 \nu^{9} + 1818708 \nu^{8} - 5902866 \nu^{7} - 2124252 \nu^{6} + 27924588 \nu^{5} - 155361564 \nu^{4} + 155580993 \nu^{3} + 125065782 \nu^{2} - 4627256787 \nu + 9795520512$$$$)/ 1147912560$$ $$\beta_{13}$$ $$=$$ $$($$$$13753 \nu^{15} - 64453 \nu^{14} + 112681 \nu^{13} + 104957 \nu^{12} - 1214876 \nu^{11} + 4647440 \nu^{10} - 7222662 \nu^{9} - 14753058 \nu^{8} + 102614814 \nu^{7} - 51256206 \nu^{6} - 804558420 \nu^{5} + 3873270312 \nu^{4} - 9041523831 \nu^{3} + 5760052803 \nu^{2} + 57432297429 \nu - 214176568851$$$$)/ 17218688400$$ $$\beta_{14}$$ $$=$$ $$($$$$-3461 \nu^{15} + 18971 \nu^{14} - 24797 \nu^{13} - 41569 \nu^{12} + 437512 \nu^{11} - 1480000 \nu^{10} + 1322274 \nu^{9} + 10243146 \nu^{8} - 28339758 \nu^{7} + 3611682 \nu^{6} + 339059520 \nu^{5} - 1125850104 \nu^{4} + 2185208847 \nu^{3} + 564764319 \nu^{2} - 24577197633 \nu + 61746535467$$$$)/ 3826375200$$ $$\beta_{15}$$ $$=$$ $$($$$$-49627 \nu^{15} + 159457 \nu^{14} + 8801 \nu^{13} - 1090223 \nu^{12} + 5845844 \nu^{11} - 15027140 \nu^{10} + 5408478 \nu^{9} + 112033842 \nu^{8} - 318871206 \nu^{7} - 398863386 \nu^{6} + 4508840700 \nu^{5} - 8900282268 \nu^{4} + 11421710289 \nu^{3} + 40699936593 \nu^{2} - 354190014711 \nu + 673542477549$$$$)/ 34437376800$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{10} - \beta_{8} + \beta_{5} - 1$$ $$\nu^{3}$$ $$=$$ $$\beta_{14} - \beta_{13} - 3 \beta_{12} + \beta_{10} + \beta_{9} - 2 \beta_{7} + 2 \beta_{6} - \beta_{5} + 2 \beta_{4} - 4 \beta_{3} + \beta_{1} - 6$$ $$\nu^{4}$$ $$=$$ $$6 \beta_{15} + 6 \beta_{13} + 9 \beta_{11} + \beta_{10} - 6 \beta_{9} - 4 \beta_{8} + \beta_{5} + 6 \beta_{4} + 3 \beta_{3} + 6 \beta_{2} - 3 \beta_{1} + 11$$ $$\nu^{5}$$ $$=$$ $$12 \beta_{15} + 4 \beta_{14} + 11 \beta_{13} - 18 \beta_{12} + 6 \beta_{11} - 8 \beta_{10} + 4 \beta_{9} + 15 \beta_{8} - 8 \beta_{7} - 4 \beta_{6} - 4 \beta_{5} - 10 \beta_{4} + 41 \beta_{3} - 6 \beta_{2} + 16 \beta_{1} + 3$$ $$\nu^{6}$$ $$=$$ $$-18 \beta_{15} - 30 \beta_{14} - 36 \beta_{13} + 6 \beta_{12} - 90 \beta_{11} + 4 \beta_{10} - 24 \beta_{9} + 8 \beta_{8} - 24 \beta_{7} - 96 \beta_{6} + 10 \beta_{5} + 30 \beta_{4} + 156 \beta_{3} + 18 \beta_{2} - 6 \beta_{1} + 5$$ $$\nu^{7}$$ $$=$$ $$-84 \beta_{15} + 184 \beta_{14} + 2 \beta_{13} - 114 \beta_{12} - 60 \beta_{11} + 160 \beta_{10} + 40 \beta_{9} + 18 \beta_{8} + 136 \beta_{7} + 20 \beta_{6} - 112 \beta_{5} - 10 \beta_{4} + 320 \beta_{3} + 42 \beta_{2} + 103 \beta_{1} + 504$$ $$\nu^{8}$$ $$=$$ $$-330 \beta_{15} + 552 \beta_{14} + 624 \beta_{13} + 408 \beta_{12} - 18 \beta_{11} - 281 \beta_{10} + 66 \beta_{9} - 187 \beta_{8} - 48 \beta_{7} + 240 \beta_{6} + 325 \beta_{5} + 90 \beta_{4} + 174 \beta_{3} + 282 \beta_{2} + 324 \beta_{1} + 485$$ $$\nu^{9}$$ $$=$$ $$373 \beta_{14} + 1001 \beta_{13} + 1311 \beta_{12} + 918 \beta_{11} + 1255 \beta_{10} + 373 \beta_{9} + 138 \beta_{8} - 350 \beta_{7} + 1466 \beta_{6} + 2201 \beta_{5} - 1540 \beta_{4} + 4658 \beta_{3} + 1134 \beta_{2} + 253 \beta_{1} - 7086$$ $$\nu^{10}$$ $$=$$ $$996 \beta_{15} - 1512 \beta_{14} + 564 \beta_{13} - 1512 \beta_{12} - 6201 \beta_{11} - 809 \beta_{10} - 3804 \beta_{9} + 176 \beta_{8} - 4752 \beta_{7} + 1296 \beta_{6} + 4645 \beta_{5} - 300 \beta_{4} - 717 \beta_{3} - 1164 \beta_{2} - 7761 \beta_{1} + 13763$$ $$\nu^{11}$$ $$=$$ $$12360 \beta_{15} - 3632 \beta_{14} - 2275 \beta_{13} - 5220 \beta_{12} - 4836 \beta_{11} + 6904 \beta_{10} - 18752 \beta_{9} + 24927 \beta_{8} + 352 \beta_{7} + 248 \beta_{6} - 10192 \beta_{5} + 5732 \beta_{4} - 1741 \beta_{3} - 7476 \beta_{2} + 23632 \beta_{1} + 16527$$ $$\nu^{12}$$ $$=$$ $$28260 \beta_{15} + 6468 \beta_{14} + 21384 \beta_{13} + 348 \beta_{12} + 29556 \beta_{11} + 30712 \beta_{10} - 16008 \beta_{9} + 11384 \beta_{8} - 10032 \beta_{7} - 7296 \beta_{6} - 19700 \beta_{5} - 492 \beta_{4} + 146280 \beta_{3} - 39636 \beta_{2} + 31980 \beta_{1} + 259997$$ $$\nu^{13}$$ $$=$$ $$4488 \beta_{15} - 35240 \beta_{14} - 24460 \beta_{13} - 14388 \beta_{12} - 127104 \beta_{11} + 112984 \beta_{10} + 73480 \beta_{9} - 27756 \beta_{8} - 132704 \beta_{7} - 36088 \beta_{6} - 106600 \beta_{5} + 27476 \beta_{4} + 208544 \beta_{3} - 2244 \beta_{2} + 307273 \beta_{1} - 404640$$ $$\nu^{14}$$ $$=$$ $$49764 \beta_{15} + 133464 \beta_{14} + 227136 \beta_{13} + 78888 \beta_{12} - 95004 \beta_{11} + 406537 \beta_{10} - 343644 \beta_{9} - 795625 \beta_{8} - 10704 \beta_{7} - 206544 \beta_{6} + 86509 \beta_{5} + 146124 \beta_{4} - 218364 \beta_{3} + 272748 \beta_{2} - 125208 \beta_{1} - 1093681$$ $$\nu^{15}$$ $$=$$ $$-46080 \beta_{15} + 1249717 \beta_{14} + 1304399 \beta_{13} - 491403 \beta_{12} + 430236 \beta_{11} + 296113 \beta_{10} - 1355 \beta_{9} + 744564 \beta_{8} - 371762 \beta_{7} + 568538 \beta_{6} - 255217 \beta_{5} - 414586 \beta_{4} - 4009216 \beta_{3} - 56988 \beta_{2} - 686339 \beta_{1} - 4803294$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/138\mathbb{Z}\right)^\times$$.

 $$n$$ $$47$$ $$97$$ $$\chi(n)$$ $$-1$$ $$1$$

## Embeddings

For each embedding $$\iota_m$$ of the coefficient field, the values $$\iota_m(a_n)$$ are shown below.

For more information on an embedded modular form you can click on its label.

Label $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
47.1
 −2.99749 − 0.122585i −2.52762 − 1.61590i −0.752365 − 2.90413i −0.0515370 + 2.99956i 1.30755 + 2.70006i 1.62763 − 2.52009i 2.42141 − 1.77110i 2.97243 + 0.405752i −2.99749 + 0.122585i −2.52762 + 1.61590i −0.752365 + 2.90413i −0.0515370 − 2.99956i 1.30755 − 2.70006i 1.62763 + 2.52009i 2.42141 + 1.77110i 2.97243 − 0.405752i
1.41421i −2.99749 0.122585i −2.00000 4.36583i −0.173361 + 4.23910i 5.99969 2.82843i 8.96995 + 0.734895i 6.17421
47.2 1.41421i −2.52762 1.61590i −2.00000 7.30416i −2.28522 + 3.57460i −0.709880 2.82843i 3.77777 + 8.16875i −10.3296
47.3 1.41421i −0.752365 2.90413i −2.00000 9.12709i −4.10705 + 1.06400i −11.5474 2.82843i −7.86789 + 4.36992i 12.9076
47.4 1.41421i −0.0515370 + 2.99956i −2.00000 2.42657i 4.24201 + 0.0728843i −2.87957 2.82843i −8.99469 0.309176i 3.43169
47.5 1.41421i 1.30755 + 2.70006i −2.00000 8.14461i 3.81846 1.84916i 7.21035 2.82843i −5.58060 + 7.06094i −11.5182
47.6 1.41421i 1.62763 2.52009i −2.00000 1.81026i −3.56394 2.30181i 9.47484 2.82843i −3.70167 8.20351i −2.56010
47.7 1.41421i 2.42141 1.77110i −2.00000 4.98213i −2.50471 3.42438i −12.5950 2.82843i 2.72641 8.57710i −7.04580
47.8 1.41421i 2.97243 + 0.405752i −2.00000 6.32168i 0.573819 4.20366i 5.04690 2.82843i 8.67073 + 2.41214i 8.94020
47.9 1.41421i −2.99749 + 0.122585i −2.00000 4.36583i −0.173361 4.23910i 5.99969 2.82843i 8.96995 0.734895i 6.17421
47.10 1.41421i −2.52762 + 1.61590i −2.00000 7.30416i −2.28522 3.57460i −0.709880 2.82843i 3.77777 8.16875i −10.3296
47.11 1.41421i −0.752365 + 2.90413i −2.00000 9.12709i −4.10705 1.06400i −11.5474 2.82843i −7.86789 4.36992i 12.9076
47.12 1.41421i −0.0515370 2.99956i −2.00000 2.42657i 4.24201 0.0728843i −2.87957 2.82843i −8.99469 + 0.309176i 3.43169
47.13 1.41421i 1.30755 2.70006i −2.00000 8.14461i 3.81846 + 1.84916i 7.21035 2.82843i −5.58060 7.06094i −11.5182
47.14 1.41421i 1.62763 + 2.52009i −2.00000 1.81026i −3.56394 + 2.30181i 9.47484 2.82843i −3.70167 + 8.20351i −2.56010
47.15 1.41421i 2.42141 + 1.77110i −2.00000 4.98213i −2.50471 + 3.42438i −12.5950 2.82843i 2.72641 + 8.57710i −7.04580
47.16 1.41421i 2.97243 0.405752i −2.00000 6.32168i 0.573819 + 4.20366i 5.04690 2.82843i 8.67073 2.41214i 8.94020
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 47.16 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 138.3.c.a 16
3.b odd 2 1 inner 138.3.c.a 16
4.b odd 2 1 1104.3.g.c 16
12.b even 2 1 1104.3.g.c 16

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
138.3.c.a 16 1.a even 1 1 trivial
138.3.c.a 16 3.b odd 2 1 inner
1104.3.g.c 16 4.b odd 2 1
1104.3.g.c 16 12.b even 2 1

## Hecke kernels

This newform subspace is the entire newspace $$S_{3}^{\mathrm{new}}(138, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 2 + T^{2} )^{8}$$
$3$ $$43046721 - 19131876 T + 5314410 T^{2} + 472392 T^{3} - 780759 T^{4} + 303264 T^{5} - 62694 T^{6} - 11556 T^{7} + 7956 T^{8} - 1284 T^{9} - 774 T^{10} + 416 T^{11} - 119 T^{12} + 8 T^{13} + 10 T^{14} - 4 T^{15} + T^{16}$$
$5$ $$107557761600 + 68684262912 T^{2} + 15063480736 T^{4} + 1485983552 T^{6} + 76670644 T^{8} + 2204176 T^{10} + 35404 T^{12} + 296 T^{14} + T^{16}$$
$7$ $$( 615000 + 807440 T - 151440 T^{2} - 84216 T^{3} + 17822 T^{4} + 664 T^{5} - 252 T^{6} + T^{8} )^{2}$$
$11$ $$429981696 + 19659718656 T^{2} + 153698052096 T^{4} + 26482199040 T^{6} + 1458721936 T^{8} + 27647040 T^{10} + 227032 T^{12} + 816 T^{14} + T^{16}$$
$13$ $$( 960369700 + 84016240 T - 40723308 T^{2} - 154336 T^{3} + 339877 T^{4} - 1852 T^{5} - 1014 T^{6} + 4 T^{7} + T^{8} )^{2}$$
$17$ $$1616871764598336 + 36691278674199552 T^{2} + 1398146884576416 T^{4} + 20806810285760 T^{6} + 159621561140 T^{8} + 686358224 T^{10} + 1654156 T^{12} + 2056 T^{14} + T^{16}$$
$19$ $$( -780768 - 12924928 T + 9971312 T^{2} - 1157472 T^{3} - 92862 T^{4} + 20280 T^{5} - 708 T^{6} - 20 T^{7} + T^{8} )^{2}$$
$23$ $$( 23 + T^{2} )^{8}$$
$29$ $$21\!\cdots\!00$$$$+$$$$16\!\cdots\!00$$$$T^{2} + 3360349114642739520 T^{4} + 18213599301631712 T^{6} + 43384256578081 T^{8} + 51751748540 T^{10} + 31403878 T^{12} + 9116 T^{14} + T^{16}$$
$31$ $$( 145917216 + 261414912 T + 112185808 T^{2} - 5830536 T^{3} - 1518167 T^{4} + 110396 T^{5} - 774 T^{6} - 68 T^{7} + T^{8} )^{2}$$
$37$ $$( 44098389696 + 10724263936 T + 494610656 T^{2} - 63314656 T^{3} - 7469124 T^{4} - 265944 T^{5} - 2008 T^{6} + 68 T^{7} + T^{8} )^{2}$$
$41$ $$69\!\cdots\!24$$$$+$$$$21\!\cdots\!80$$$$T^{2} +$$$$89\!\cdots\!16$$$$T^{4} + 1445420297087810496 T^{6} + 1207196939364001 T^{8} + 562158988412 T^{10} + 145391910 T^{12} + 19196 T^{14} + T^{16}$$
$43$ $$( -2868882576000 + 323016705920 T + 3298290688 T^{2} - 1035167728 T^{3} + 19314674 T^{4} + 371360 T^{5} - 9004 T^{6} - 36 T^{7} + T^{8} )^{2}$$
$47$ $$37\!\cdots\!00$$$$+$$$$49\!\cdots\!40$$$$T^{2} +$$$$22\!\cdots\!84$$$$T^{4} + 470215617462811656 T^{6} + 527799014326849 T^{8} + 320990412740 T^{10} + 103943454 T^{12} + 16508 T^{14} + T^{16}$$
$53$ $$32\!\cdots\!00$$$$+$$$$26\!\cdots\!08$$$$T^{2} +$$$$75\!\cdots\!36$$$$T^{4} + 9087088497690725056 T^{6} + 5451365720916548 T^{8} + 1750514790736 T^{10} + 306448916 T^{12} + 27592 T^{14} + T^{16}$$
$59$ $$30\!\cdots\!84$$$$+$$$$24\!\cdots\!72$$$$T^{2} +$$$$49\!\cdots\!00$$$$T^{4} + 40223351979725939712 T^{6} + 17125501055541264 T^{8} + 4087009408928 T^{10} + 544544024 T^{12} + 37288 T^{14} + T^{16}$$
$61$ $$( -20431830161152 + 2332551584256 T - 52592155648 T^{2} - 1807959168 T^{3} + 54787680 T^{4} + 409824 T^{5} - 13952 T^{6} - 24 T^{7} + T^{8} )^{2}$$
$67$ $$( 3918307649688 - 2288587493520 T + 96790422672 T^{2} + 5961510472 T^{3} - 37400482 T^{4} - 2548304 T^{5} - 11804 T^{6} + 152 T^{7} + T^{8} )^{2}$$
$71$ $$26\!\cdots\!76$$$$+$$$$23\!\cdots\!32$$$$T^{2} +$$$$65\!\cdots\!24$$$$T^{4} + 61578517830468371712 T^{6} + 39156617209594545 T^{8} + 9372407334948 T^{10} + 1027986166 T^{12} + 52292 T^{14} + T^{16}$$
$73$ $$( 171604877785344 + 10185154221312 T - 9427624800 T^{2} - 7228346960 T^{3} - 16677639 T^{4} + 1980532 T^{5} - 1586 T^{6} - 204 T^{7} + T^{8} )^{2}$$
$79$ $$( 173075158110240 + 9926033782464 T - 159903289200 T^{2} - 10274412944 T^{3} + 86281986 T^{4} + 2635096 T^{5} - 17012 T^{6} - 156 T^{7} + T^{8} )^{2}$$
$83$ $$11\!\cdots\!84$$$$+$$$$43\!\cdots\!76$$$$T^{2} +$$$$26\!\cdots\!04$$$$T^{4} + 6743299886971114496 T^{6} + 7945520547421072 T^{8} + 4247598879424 T^{10} + 818729368 T^{12} + 51824 T^{14} + T^{16}$$
$89$ $$12\!\cdots\!00$$$$+$$$$52\!\cdots\!00$$$$T^{2} +$$$$16\!\cdots\!04$$$$T^{4} + 45009436477443568128 T^{6} + 35887368340223172 T^{8} + 11039901990480 T^{10} + 1358607604 T^{12} + 64472 T^{14} + T^{16}$$
$97$ $$( -44369259840 + 977375554560 T - 22337326176 T^{2} - 1878829664 T^{3} + 37806236 T^{4} + 739352 T^{5} - 12312 T^{6} - 84 T^{7} + T^{8} )^{2}$$