# Properties

 Label 552.2.f.c Level $552$ Weight $2$ Character orbit 552.f Analytic conductor $4.408$ Analytic rank $0$ Dimension $18$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$552 = 2^{3} \cdot 3 \cdot 23$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 552.f (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$4.40774219157$$ Analytic rank: $$0$$ Dimension: $$18$$ Coefficient field: $$\mathbb{Q}[x]/(x^{18} - \cdots)$$ Defining polynomial: $$x^{18} - 4 x^{16} - 2 x^{15} + 5 x^{14} + 2 x^{13} + 6 x^{12} + 24 x^{11} - 12 x^{10} - 88 x^{9} - 24 x^{8} + 96 x^{7} + 48 x^{6} + 32 x^{5} + 160 x^{4} - 128 x^{3} - 512 x^{2} + 512$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{7}$$ 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 $$1,\beta_1,\ldots,\beta_{17}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q -\beta_{3} q^{2} -\beta_{4} q^{3} -\beta_{2} q^{4} + \beta_{9} q^{5} -\beta_{1} q^{6} + ( 1 + \beta_{3} + \beta_{8} - \beta_{10} - \beta_{12} ) q^{7} + ( 1 - \beta_{2} - \beta_{13} + \beta_{14} + \beta_{16} ) q^{8} - q^{9} +O(q^{10})$$ $$q -\beta_{3} q^{2} -\beta_{4} q^{3} -\beta_{2} q^{4} + \beta_{9} q^{5} -\beta_{1} q^{6} + ( 1 + \beta_{3} + \beta_{8} - \beta_{10} - \beta_{12} ) q^{7} + ( 1 - \beta_{2} - \beta_{13} + \beta_{14} + \beta_{16} ) q^{8} - q^{9} + ( 1 + \beta_{1} - \beta_{3} - \beta_{4} + \beta_{5} - \beta_{7} - \beta_{8} + \beta_{10} + \beta_{11} - \beta_{13} + \beta_{14} - \beta_{15} + \beta_{16} ) q^{10} + ( -1 + \beta_{6} - \beta_{9} - \beta_{10} - \beta_{11} - \beta_{14} - \beta_{16} ) q^{11} + \beta_{8} q^{12} + ( 1 - \beta_{2} - \beta_{3} - \beta_{4} + \beta_{9} - \beta_{13} + \beta_{16} ) q^{13} + ( \beta_{1} + \beta_{2} - \beta_{3} - 2 \beta_{4} + \beta_{5} - \beta_{7} + \beta_{9} + \beta_{11} + \beta_{14} + \beta_{15} + \beta_{17} ) q^{14} -\beta_{15} q^{15} + ( \beta_{2} + \beta_{3} + \beta_{8} - 2 \beta_{10} - \beta_{11} - \beta_{14} - \beta_{16} ) q^{16} + ( 1 - \beta_{2} - \beta_{3} - \beta_{4} + \beta_{5} + \beta_{11} - \beta_{13} + \beta_{14} ) q^{17} + \beta_{3} q^{18} + ( 2 + 2 \beta_{1} - \beta_{3} + \beta_{5} - \beta_{8} + \beta_{11} - \beta_{12} + \beta_{14} + \beta_{15} + \beta_{16} + 2 \beta_{17} ) q^{19} + ( -4 - \beta_{1} + \beta_{2} + \beta_{3} + 3 \beta_{4} - \beta_{5} + \beta_{6} + \beta_{7} - \beta_{8} - \beta_{9} - \beta_{10} - \beta_{11} + \beta_{12} + \beta_{13} - \beta_{14} - \beta_{15} - 2 \beta_{16} ) q^{20} + ( 1 + \beta_{1} - \beta_{3} - \beta_{4} + \beta_{5} - \beta_{13} + \beta_{16} ) q^{21} + ( -1 - \beta_{1} - \beta_{2} + \beta_{4} - \beta_{9} - \beta_{10} - \beta_{14} + \beta_{15} ) q^{22} - q^{23} + ( 1 + \beta_{1} + \beta_{15} + \beta_{17} ) q^{24} + ( -\beta_{1} - 2 \beta_{3} - \beta_{4} + \beta_{6} + \beta_{7} + \beta_{8} + \beta_{9} - \beta_{13} + 2 \beta_{14} + \beta_{16} ) q^{25} + ( -\beta_{3} - 2 \beta_{4} + \beta_{5} + \beta_{6} - 2 \beta_{7} - \beta_{8} + \beta_{11} - 2 \beta_{13} + \beta_{14} - 2 \beta_{15} + \beta_{16} ) q^{26} + \beta_{4} q^{27} + ( 1 + \beta_{4} - 2 \beta_{6} + \beta_{9} + \beta_{10} + \beta_{13} + \beta_{16} + \beta_{17} ) q^{28} + ( 3 \beta_{1} + 2 \beta_{2} + 2 \beta_{3} + \beta_{4} + 2 \beta_{5} + \beta_{6} + \beta_{7} - \beta_{8} - \beta_{9} - \beta_{10} + \beta_{13} + \beta_{15} - \beta_{16} + 2 \beta_{17} ) q^{29} + ( -1 + \beta_{6} - \beta_{8} - \beta_{9} + \beta_{12} - \beta_{16} ) q^{30} + ( -4 + \beta_{1} + 2 \beta_{2} + \beta_{4} - \beta_{5} + \beta_{7} + \beta_{8} - \beta_{9} - \beta_{10} + \beta_{13} - 2 \beta_{14} + \beta_{15} - \beta_{16} ) q^{31} + ( 1 + \beta_{1} + \beta_{2} - 2 \beta_{3} - 4 \beta_{4} + \beta_{5} - \beta_{6} - \beta_{7} - \beta_{8} + 2 \beta_{11} + \beta_{12} + \beta_{15} + \beta_{17} ) q^{32} + ( -1 - \beta_{1} + \beta_{4} + \beta_{7} + \beta_{8} - \beta_{11} + \beta_{13} - \beta_{14} + \beta_{15} ) q^{33} + ( 4 + \beta_{1} - \beta_{2} + \beta_{4} + \beta_{5} + \beta_{7} + \beta_{8} - \beta_{10} + \beta_{14} + \beta_{15} + \beta_{16} + 2 \beta_{17} ) q^{34} + ( -1 - 4 \beta_{1} - \beta_{2} + \beta_{3} + 5 \beta_{4} - 3 \beta_{5} - \beta_{6} + \beta_{7} - 2 \beta_{8} - \beta_{9} - \beta_{12} + \beta_{13} - \beta_{16} ) q^{35} + \beta_{2} q^{36} + ( 1 - \beta_{2} + \beta_{3} + 3 \beta_{4} - \beta_{5} + \beta_{7} - 2 \beta_{8} - 2 \beta_{9} + \beta_{11} - \beta_{12} + \beta_{13} + \beta_{14} + \beta_{15} + 2 \beta_{17} ) q^{37} + ( -3 - \beta_{1} + 2 \beta_{4} + \beta_{6} + \beta_{9} - 2 \beta_{11} + \beta_{12} + \beta_{13} - \beta_{14} - \beta_{15} - \beta_{17} ) q^{38} + ( -1 + \beta_{10} - \beta_{15} ) q^{39} + ( 2 + \beta_{1} - \beta_{2} + 2 \beta_{3} - 2 \beta_{9} + \beta_{13} - \beta_{14} + \beta_{15} - \beta_{16} - \beta_{17} ) q^{40} + ( -3 \beta_{1} + \beta_{2} + \beta_{3} + 2 \beta_{4} + \beta_{6} + \beta_{7} + \beta_{8} - 2 \beta_{11} + 2 \beta_{13} - 2 \beta_{14} ) q^{41} + ( -2 - \beta_{1} + \beta_{2} + \beta_{4} + \beta_{6} - \beta_{8} - \beta_{10} + \beta_{12} - \beta_{14} - \beta_{15} - \beta_{16} ) q^{42} + ( 2 - \beta_{1} - \beta_{2} - \beta_{3} - 6 \beta_{4} - \beta_{5} - 2 \beta_{6} - \beta_{7} + \beta_{8} + \beta_{9} + 2 \beta_{10} + 2 \beta_{11} + 2 \beta_{14} + 2 \beta_{16} ) q^{43} + ( 1 - \beta_{2} + \beta_{3} - 2 \beta_{4} - \beta_{5} - \beta_{6} - \beta_{7} + 2 \beta_{8} + 2 \beta_{9} + \beta_{11} - \beta_{12} - \beta_{13} + 2 \beta_{14} + 2 \beta_{16} ) q^{44} -\beta_{9} q^{45} + \beta_{3} q^{46} + ( 3 + \beta_{1} - 2 \beta_{2} + 2 \beta_{3} - \beta_{4} - \beta_{6} - \beta_{7} + \beta_{8} + \beta_{9} - 2 \beta_{12} - \beta_{13} + 2 \beta_{14} + \beta_{16} ) q^{47} + ( -\beta_{3} - \beta_{11} - \beta_{14} + \beta_{16} ) q^{48} + ( -2 + 2 \beta_{2} + 2 \beta_{3} + \beta_{5} + \beta_{6} - 2 \beta_{7} - 2 \beta_{8} - 2 \beta_{9} - \beta_{10} + 2 \beta_{11} - 2 \beta_{14} - 3 \beta_{15} - 2 \beta_{16} ) q^{49} + ( 4 + \beta_{1} + 2 \beta_{3} + 2 \beta_{4} + 2 \beta_{5} - \beta_{7} + 2 \beta_{8} - 2 \beta_{10} - \beta_{11} - \beta_{12} + \beta_{13} - 2 \beta_{14} + \beta_{15} - 2 \beta_{16} + \beta_{17} ) q^{50} + ( -1 - \beta_{1} + \beta_{2} + \beta_{3} + \beta_{8} - \beta_{10} - \beta_{11} + \beta_{12} - \beta_{14} - \beta_{16} ) q^{51} + ( -6 - \beta_{1} + 2 \beta_{2} + 2 \beta_{3} + 2 \beta_{4} - \beta_{5} + \beta_{6} - \beta_{7} - \beta_{8} - 2 \beta_{9} - 2 \beta_{10} + \beta_{12} + \beta_{13} - \beta_{14} - \beta_{15} - 3 \beta_{16} + \beta_{17} ) q^{52} + ( -4 - \beta_{1} + \beta_{2} + 2 \beta_{3} - \beta_{5} + \beta_{6} - \beta_{11} + 2 \beta_{13} - \beta_{14} - \beta_{15} - 3 \beta_{16} - 2 \beta_{17} ) q^{53} + \beta_{1} q^{54} + ( \beta_{1} + \beta_{2} - \beta_{3} - \beta_{5} - \beta_{8} + \beta_{12} - 2 \beta_{15} ) q^{55} + ( -3 - \beta_{3} - 2 \beta_{4} - \beta_{5} + \beta_{6} + \beta_{7} - \beta_{8} + 2 \beta_{10} - \beta_{11} + 3 \beta_{12} - \beta_{13} - 2 \beta_{15} - 2 \beta_{17} ) q^{56} + ( -\beta_{3} + \beta_{5} - \beta_{8} - \beta_{9} + \beta_{11} + \beta_{12} - \beta_{14} - \beta_{16} ) q^{57} + ( -\beta_{1} + 2 \beta_{2} + 6 \beta_{4} + 2 \beta_{7} - \beta_{8} - 2 \beta_{11} + 2 \beta_{12} + 3 \beta_{13} - 3 \beta_{14} + \beta_{15} - \beta_{16} - \beta_{17} ) q^{58} + ( -3 - 2 \beta_{1} + 2 \beta_{2} - 2 \beta_{4} + 2 \beta_{9} - \beta_{10} - 2 \beta_{11} - 2 \beta_{14} - \beta_{15} - 2 \beta_{16} - 2 \beta_{17} ) q^{59} + ( 2 - \beta_{1} - \beta_{2} + \beta_{3} + 3 \beta_{4} - \beta_{5} - \beta_{6} + \beta_{7} + \beta_{8} - \beta_{9} - \beta_{10} - \beta_{11} - \beta_{12} + \beta_{13} - \beta_{14} + \beta_{15} ) q^{60} + ( 1 + 2 \beta_{1} - 3 \beta_{2} - \beta_{3} - 3 \beta_{4} + \beta_{6} + 2 \beta_{9} + 2 \beta_{10} + \beta_{11} - \beta_{13} + \beta_{14} - \beta_{15} + 2 \beta_{16} - 2 \beta_{17} ) q^{61} + ( 2 + \beta_{1} + 2 \beta_{3} - 4 \beta_{4} - 2 \beta_{7} - \beta_{8} + 2 \beta_{9} + 2 \beta_{10} + 2 \beta_{11} - 2 \beta_{12} + \beta_{13} + \beta_{14} + \beta_{15} + \beta_{16} + \beta_{17} ) q^{62} + ( -1 - \beta_{3} - \beta_{8} + \beta_{10} + \beta_{12} ) q^{63} + ( 2 - 3 \beta_{1} - \beta_{2} - \beta_{3} - 2 \beta_{4} - 2 \beta_{5} - 2 \beta_{6} + 2 \beta_{9} + 2 \beta_{10} + \beta_{11} + \beta_{13} + 2 \beta_{14} - \beta_{15} + 2 \beta_{16} + \beta_{17} ) q^{64} + ( -4 - \beta_{1} - \beta_{2} - 2 \beta_{3} - 2 \beta_{4} + \beta_{5} + \beta_{6} + \beta_{9} + \beta_{11} - 2 \beta_{13} + 3 \beta_{14} - \beta_{15} + \beta_{16} ) q^{65} + ( 1 - \beta_{2} + \beta_{4} + \beta_{8} + \beta_{9} + \beta_{10} - \beta_{13} + \beta_{16} - \beta_{17} ) q^{66} + ( -2 \beta_{2} - \beta_{3} + 6 \beta_{4} + \beta_{6} + \beta_{7} - \beta_{8} - \beta_{11} - 2 \beta_{13} - \beta_{14} - \beta_{15} + \beta_{16} - 2 \beta_{17} ) q^{67} + ( 3 + \beta_{1} - 2 \beta_{3} + \beta_{4} + \beta_{8} + \beta_{9} - \beta_{10} - 2 \beta_{11} + 2 \beta_{12} - \beta_{14} + \beta_{15} ) q^{68} + \beta_{4} q^{69} + ( 2 \beta_{1} - 2 \beta_{2} + \beta_{3} - 6 \beta_{4} + \beta_{5} + \beta_{6} + 5 \beta_{8} - \beta_{11} - 2 \beta_{12} + \beta_{14} + \beta_{16} - 2 \beta_{17} ) q^{70} + ( 2 + 3 \beta_{1} - 2 \beta_{2} + 2 \beta_{3} - \beta_{4} - \beta_{5} - 2 \beta_{6} - \beta_{7} + \beta_{8} + \beta_{9} + \beta_{10} - 2 \beta_{12} - \beta_{13} + 2 \beta_{14} + \beta_{15} + \beta_{16} ) q^{71} + ( -1 + \beta_{2} + \beta_{13} - \beta_{14} - \beta_{16} ) q^{72} + ( -\beta_{1} + \beta_{2} + \beta_{3} + \beta_{6} + \beta_{8} + 2 \beta_{10} - \beta_{12} ) q^{73} + ( -1 - \beta_{1} - \beta_{2} + 2 \beta_{3} + \beta_{4} - \beta_{5} + \beta_{6} + \beta_{7} + 4 \beta_{8} + \beta_{9} - \beta_{10} - 4 \beta_{11} - \beta_{12} + 2 \beta_{13} - \beta_{14} + \beta_{15} - 2 \beta_{17} ) q^{74} + ( 1 - \beta_{1} + \beta_{2} + \beta_{3} + \beta_{4} - \beta_{6} + \beta_{7} - \beta_{8} - \beta_{10} + \beta_{15} + 2 \beta_{17} ) q^{75} + ( -2 - \beta_{2} + \beta_{3} + 3 \beta_{4} - \beta_{5} - \beta_{6} - \beta_{7} - 2 \beta_{8} - \beta_{9} + \beta_{10} + \beta_{11} + \beta_{12} - 2 \beta_{13} - 2 \beta_{15} - \beta_{16} - 3 \beta_{17} ) q^{76} + ( 2 + 2 \beta_{1} + 3 \beta_{2} - \beta_{3} - 3 \beta_{4} + 3 \beta_{5} - \beta_{6} + \beta_{9} - \beta_{10} - \beta_{13} + \beta_{15} + \beta_{16} + 2 \beta_{17} ) q^{77} + ( -2 + \beta_{3} + 2 \beta_{4} + 2 \beta_{6} + \beta_{7} - \beta_{8} - 2 \beta_{9} - \beta_{11} + \beta_{12} - \beta_{14} - \beta_{16} ) q^{78} + ( 3 + \beta_{1} - \beta_{2} - \beta_{3} + 2 \beta_{4} - 2 \beta_{5} - 2 \beta_{6} + 3 \beta_{7} + 3 \beta_{8} + \beta_{9} + \beta_{10} - 3 \beta_{11} + 2 \beta_{13} - \beta_{14} + 3 \beta_{15} + \beta_{16} ) q^{79} + ( 2 - 2 \beta_{1} - \beta_{2} - 2 \beta_{3} + 2 \beta_{4} - 2 \beta_{5} - 2 \beta_{6} + 2 \beta_{7} + \beta_{8} + 2 \beta_{9} - 2 \beta_{12} + 2 \beta_{14} + 2 \beta_{15} + 2 \beta_{16} ) q^{80} + q^{81} + ( -4 - \beta_{1} - 2 \beta_{3} + 4 \beta_{4} - \beta_{5} + \beta_{6} - \beta_{7} + \beta_{8} + 2 \beta_{10} + \beta_{12} - \beta_{13} - \beta_{14} - \beta_{15} - \beta_{16} - 3 \beta_{17} ) q^{82} + ( 1 + 2 \beta_{2} - 2 \beta_{3} + \beta_{5} - 2 \beta_{6} - \beta_{7} + \beta_{9} + \beta_{10} + \beta_{11} - \beta_{12} + \beta_{14} + \beta_{16} ) q^{83} + ( 1 + \beta_{1} - \beta_{4} - 2 \beta_{7} - \beta_{8} - \beta_{9} + \beta_{10} + 2 \beta_{11} + \beta_{14} - \beta_{15} ) q^{84} + ( -5 + 4 \beta_{3} + 4 \beta_{4} - \beta_{5} + 3 \beta_{6} + \beta_{7} - 4 \beta_{9} - \beta_{10} - 2 \beta_{11} + \beta_{12} + 2 \beta_{13} - 2 \beta_{14} - \beta_{15} - 4 \beta_{16} - 2 \beta_{17} ) q^{85} + ( 1 - 4 \beta_{1} + \beta_{3} - 3 \beta_{4} - 2 \beta_{5} - \beta_{6} + 2 \beta_{8} + 2 \beta_{9} + \beta_{10} - \beta_{11} - \beta_{12} + 2 \beta_{14} + \beta_{17} ) q^{86} + ( \beta_{1} - 2 \beta_{3} + \beta_{4} - \beta_{6} + \beta_{7} - \beta_{8} - \beta_{9} - \beta_{10} + 2 \beta_{12} + \beta_{13} - 2 \beta_{14} + \beta_{15} - \beta_{16} ) q^{87} + ( 1 + 2 \beta_{1} + 3 \beta_{2} + \beta_{3} - 6 \beta_{4} + \beta_{5} - \beta_{6} - 3 \beta_{7} + 2 \beta_{9} + \beta_{11} - \beta_{12} - \beta_{13} + 2 \beta_{14} + 2 \beta_{17} ) q^{88} + ( 5 - \beta_{1} - 2 \beta_{2} + \beta_{3} + 3 \beta_{4} - \beta_{5} - 2 \beta_{6} + 3 \beta_{7} + 4 \beta_{8} + \beta_{9} - 2 \beta_{10} - 4 \beta_{11} - \beta_{12} + 3 \beta_{13} - 2 \beta_{14} + 3 \beta_{15} + \beta_{16} ) q^{89} + ( -1 - \beta_{1} + \beta_{3} + \beta_{4} - \beta_{5} + \beta_{7} + \beta_{8} - \beta_{10} - \beta_{11} + \beta_{13} - \beta_{14} + \beta_{15} - \beta_{16} ) q^{90} + ( -2 - 5 \beta_{1} - \beta_{2} + 2 \beta_{3} + 2 \beta_{4} - 3 \beta_{5} - \beta_{6} + 2 \beta_{7} - 2 \beta_{8} - \beta_{9} - \beta_{11} - \beta_{14} - \beta_{15} - \beta_{16} - 2 \beta_{17} ) q^{91} + \beta_{2} q^{92} + ( -\beta_{1} - 2 \beta_{3} + 3 \beta_{4} - \beta_{6} - \beta_{8} + \beta_{9} + \beta_{10} - \beta_{12} - \beta_{13} - \beta_{15} + \beta_{16} - 2 \beta_{17} ) q^{93} + ( -2 + \beta_{1} + 2 \beta_{2} - \beta_{3} - 2 \beta_{4} + 2 \beta_{5} + \beta_{7} - 2 \beta_{10} - \beta_{11} + \beta_{12} - \beta_{13} + \beta_{15} + \beta_{17} ) q^{94} + ( 1 + \beta_{1} - \beta_{2} + \beta_{3} - 2 \beta_{4} + \beta_{5} - 2 \beta_{7} - \beta_{8} + \beta_{10} + 2 \beta_{11} - \beta_{12} - 2 \beta_{13} + 2 \beta_{14} + \beta_{15} ) q^{95} + ( -5 - 2 \beta_{1} - \beta_{5} + \beta_{6} - \beta_{7} - \beta_{8} + \beta_{12} - \beta_{13} - \beta_{14} - 2 \beta_{15} - \beta_{16} - 2 \beta_{17} ) q^{96} + ( -4 - 2 \beta_{1} + 2 \beta_{2} + 2 \beta_{6} - 2 \beta_{7} - 4 \beta_{8} + 2 \beta_{10} + 2 \beta_{12} - 2 \beta_{15} ) q^{97} + ( -6 + 2 \beta_{2} - 2 \beta_{5} + 2 \beta_{6} - \beta_{7} - 3 \beta_{8} - 2 \beta_{9} + 2 \beta_{10} + 3 \beta_{11} + \beta_{12} + 2 \beta_{13} + 3 \beta_{14} - \beta_{16} + 2 \beta_{17} ) q^{98} + ( 1 - \beta_{6} + \beta_{9} + \beta_{10} + \beta_{11} + \beta_{14} + \beta_{16} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$18q - 8q^{4} + 8q^{7} - 6q^{8} - 18q^{9} + O(q^{10})$$ $$18q - 8q^{4} + 8q^{7} - 6q^{8} - 18q^{9} + 12q^{10} - 4q^{12} - 14q^{14} + 8q^{15} + 12q^{16} - 8q^{17} - 16q^{20} - 30q^{22} - 18q^{23} + 6q^{24} - 22q^{25} + 8q^{26} - 2q^{28} + 4q^{30} - 44q^{31} + 10q^{32} - 24q^{33} + 18q^{34} + 8q^{36} - 20q^{38} - 8q^{39} + 40q^{40} + 28q^{41} + 6q^{42} - 26q^{44} + 42q^{49} + 60q^{50} - 36q^{52} + 40q^{55} - 2q^{56} + 12q^{57} + 52q^{58} + 16q^{60} + 24q^{62} - 8q^{63} + 16q^{64} - 104q^{65} + 2q^{66} + 54q^{68} - 48q^{70} - 24q^{71} + 6q^{72} + 12q^{73} - 22q^{74} - 4q^{78} + 8q^{79} - 32q^{80} + 18q^{81} - 20q^{82} + 34q^{84} + 12q^{87} + 10q^{88} + 24q^{89} - 12q^{90} + 8q^{92} - 56q^{94} - 16q^{95} - 30q^{96} + 12q^{97} - 48q^{98} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{18} - 4 x^{16} - 2 x^{15} + 5 x^{14} + 2 x^{13} + 6 x^{12} + 24 x^{11} - 12 x^{10} - 88 x^{9} - 24 x^{8} + 96 x^{7} + 48 x^{6} + 32 x^{5} + 160 x^{4} - 128 x^{3} - 512 x^{2} + 512$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$\nu^{2}$$ $$\beta_{3}$$ $$=$$ $$($$$$\nu^{17} + 2 \nu^{16} - 2 \nu^{14} + \nu^{13} + 4 \nu^{12} - 18 \nu^{11} - 12 \nu^{10} + 28 \nu^{9} + 32 \nu^{8} + 8 \nu^{7} + 48 \nu^{6} + 16 \nu^{5} - 192 \nu^{4} - 224 \nu^{3} + 192 \nu^{2} + 384 \nu + 256$$$$)/256$$ $$\beta_{4}$$ $$=$$ $$($$$$-\nu^{17} + 2 \nu^{16} + 8 \nu^{15} + 2 \nu^{14} - 9 \nu^{13} + 2 \nu^{11} - 60 \nu^{10} - 12 \nu^{9} + 144 \nu^{8} + 88 \nu^{7} - 80 \nu^{6} + 48 \nu^{5} - 544 \nu^{3} - 320 \nu^{2} + 896 \nu + 768$$$$)/512$$ $$\beta_{5}$$ $$=$$ $$($$$$\nu^{17} + 2 \nu^{16} - 4 \nu^{15} - 10 \nu^{14} + \nu^{13} + 12 \nu^{12} + 10 \nu^{11} + 36 \nu^{10} + 36 \nu^{9} - 112 \nu^{8} - 200 \nu^{7} + 48 \nu^{6} + 240 \nu^{5} + 128 \nu^{4} + 224 \nu^{3} + 192 \nu^{2} - 768 \nu - 1024$$$$)/256$$ $$\beta_{6}$$ $$=$$ $$($$$$\nu^{17} - 2 \nu^{16} - 4 \nu^{15} + 6 \nu^{14} + 9 \nu^{13} - 8 \nu^{12} + 2 \nu^{11} + 12 \nu^{10} - 60 \nu^{9} - 64 \nu^{8} + 152 \nu^{7} + 144 \nu^{6} - 144 \nu^{5} - 64 \nu^{4} + 96 \nu^{3} - 448 \nu^{2} - 256 \nu + 1024$$$$)/256$$ $$\beta_{7}$$ $$=$$ $$($$$$\nu^{17} + \nu^{16} - 2 \nu^{15} - 6 \nu^{14} - 5 \nu^{13} + 3 \nu^{12} + 18 \nu^{11} + 34 \nu^{10} - 8 \nu^{9} - 52 \nu^{8} - 72 \nu^{7} - 40 \nu^{6} + 64 \nu^{5} + 208 \nu^{4} + 160 \nu^{3} - 160 \nu^{2} - 320 \nu$$$$)/128$$ $$\beta_{8}$$ $$=$$ $$($$$$\nu^{17} + 2 \nu^{16} - 2 \nu^{14} + \nu^{13} - 12 \nu^{12} - 18 \nu^{11} + 20 \nu^{10} + 60 \nu^{9} + 16 \nu^{8} - 24 \nu^{7} - 16 \nu^{6} - 112 \nu^{5} - 192 \nu^{4} + 160 \nu^{3} + 448 \nu^{2} + 128 \nu - 256$$$$)/128$$ $$\beta_{9}$$ $$=$$ $$($$$$-5 \nu^{17} - 6 \nu^{16} + 8 \nu^{15} + 10 \nu^{14} + 3 \nu^{13} + 48 \nu^{12} + 42 \nu^{11} - 172 \nu^{10} - 300 \nu^{9} + 48 \nu^{8} + 312 \nu^{7} + 176 \nu^{6} + 496 \nu^{5} + 640 \nu^{4} - 1312 \nu^{3} - 2624 \nu^{2} + 384 \nu + 2304$$$$)/512$$ $$\beta_{10}$$ $$=$$ $$($$$$9 \nu^{17} + 18 \nu^{16} - 24 \nu^{15} - 66 \nu^{14} - 7 \nu^{13} + 20 \nu^{12} + 6 \nu^{11} + 276 \nu^{10} + 412 \nu^{9} - 416 \nu^{8} - 1080 \nu^{7} - 144 \nu^{6} + 336 \nu^{5} - 192 \nu^{4} + 2080 \nu^{3} + 3008 \nu^{2} - 1920 \nu - 4352$$$$)/512$$ $$\beta_{11}$$ $$=$$ $$($$$$-7 \nu^{17} - 18 \nu^{16} + 24 \nu^{15} + 62 \nu^{14} + 33 \nu^{13} - 50 \nu^{11} - 372 \nu^{10} - 468 \nu^{9} + 496 \nu^{8} + 1256 \nu^{7} + 464 \nu^{6} - 176 \nu^{5} - 640 \nu^{4} - 3040 \nu^{3} - 3264 \nu^{2} + 3200 \nu + 5888$$$$)/512$$ $$\beta_{12}$$ $$=$$ $$($$$$-5 \nu^{17} - 4 \nu^{16} + 20 \nu^{15} + 34 \nu^{14} - \nu^{13} - 30 \nu^{12} - 54 \nu^{11} - 136 \nu^{10} - 68 \nu^{9} + 344 \nu^{8} + 504 \nu^{7} - 32 \nu^{6} - 432 \nu^{5} - 416 \nu^{4} - 800 \nu^{3} - 384 \nu^{2} + 1536 \nu + 1792$$$$)/256$$ $$\beta_{13}$$ $$=$$ $$($$$$-13 \nu^{17} - 22 \nu^{16} + 24 \nu^{15} + 74 \nu^{14} + 43 \nu^{13} + 16 \nu^{12} - 54 \nu^{11} - 412 \nu^{10} - 476 \nu^{9} + 464 \nu^{8} + 1272 \nu^{7} + 496 \nu^{6} - 16 \nu^{5} - 640 \nu^{4} - 3232 \nu^{3} - 3136 \nu^{2} + 2944 \nu + 4864$$$$)/512$$ $$\beta_{14}$$ $$=$$ $$($$$$2 \nu^{17} + 5 \nu^{16} - 12 \nu^{14} - 20 \nu^{13} - 19 \nu^{12} + 14 \nu^{11} + 98 \nu^{10} + 132 \nu^{9} - 44 \nu^{8} - 232 \nu^{7} - 248 \nu^{6} - 128 \nu^{5} + 176 \nu^{4} + 768 \nu^{3} + 736 \nu^{2} - 512 \nu - 1024$$$$)/128$$ $$\beta_{15}$$ $$=$$ $$($$$$-15 \nu^{17} - 18 \nu^{16} + 40 \nu^{15} + 78 \nu^{14} + 9 \nu^{13} - 16 \nu^{12} - 82 \nu^{11} - 436 \nu^{10} - 388 \nu^{9} + 752 \nu^{8} + 1320 \nu^{7} + 272 \nu^{6} - 432 \nu^{5} - 768 \nu^{4} - 2912 \nu^{3} - 2496 \nu^{2} + 3200 \nu + 4864$$$$)/512$$ $$\beta_{16}$$ $$=$$ $$($$$$-13 \nu^{17} - 26 \nu^{16} + 24 \nu^{15} + 106 \nu^{14} + 67 \nu^{13} - 4 \nu^{12} - 126 \nu^{11} - 516 \nu^{10} - 588 \nu^{9} + 640 \nu^{8} + 1752 \nu^{7} + 848 \nu^{6} - 400 \nu^{5} - 1344 \nu^{4} - 4000 \nu^{3} - 3008 \nu^{2} + 3968 \nu + 6400$$$$)/512$$ $$\beta_{17}$$ $$=$$ $$($$$$15 \nu^{17} + 18 \nu^{16} - 40 \nu^{15} - 78 \nu^{14} - 9 \nu^{13} + 16 \nu^{12} + 82 \nu^{11} + 436 \nu^{10} + 388 \nu^{9} - 752 \nu^{8} - 1320 \nu^{7} - 272 \nu^{6} + 432 \nu^{5} + 768 \nu^{4} + 3424 \nu^{3} + 2496 \nu^{2} - 3712 \nu - 5376$$$$)/512$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{2}$$ $$\nu^{3}$$ $$=$$ $$\beta_{17} + \beta_{15} + \beta_{1} + 1$$ $$\nu^{4}$$ $$=$$ $$-\beta_{16} - \beta_{14} - \beta_{11} - 2 \beta_{10} + \beta_{8} + \beta_{3} + \beta_{2}$$ $$\nu^{5}$$ $$=$$ $$2 \beta_{17} + \beta_{16} + 2 \beta_{15} + \beta_{14} + \beta_{13} - \beta_{12} + \beta_{8} + \beta_{7} - \beta_{6} + \beta_{5} + 2 \beta_{1} + 5$$ $$\nu^{6}$$ $$=$$ $$-\beta_{17} - 2 \beta_{16} + \beta_{15} - 2 \beta_{14} - \beta_{13} - \beta_{11} - 2 \beta_{10} - 2 \beta_{9} + 2 \beta_{6} + 2 \beta_{5} + 2 \beta_{4} + \beta_{3} + \beta_{2} + 3 \beta_{1} - 2$$ $$\nu^{7}$$ $$=$$ $$3 \beta_{17} - 2 \beta_{16} + 3 \beta_{15} + 4 \beta_{13} - \beta_{12} - 2 \beta_{11} - 2 \beta_{10} + 2 \beta_{8} + \beta_{7} + \beta_{6} - \beta_{5} + 4 \beta_{3} + 4 \beta_{2} - \beta_{1} - 5$$ $$\nu^{8}$$ $$=$$ $$2 \beta_{17} - 3 \beta_{16} + 4 \beta_{15} - 5 \beta_{14} - 3 \beta_{11} - 4 \beta_{10} - 6 \beta_{9} + \beta_{8} + 4 \beta_{7} + 6 \beta_{4} + \beta_{3} - \beta_{2} - 2 \beta_{1}$$ $$\nu^{9}$$ $$=$$ $$2 \beta_{17} - 7 \beta_{16} + 2 \beta_{15} - 3 \beta_{14} + 5 \beta_{13} - \beta_{12} - 4 \beta_{11} - 8 \beta_{10} - 4 \beta_{9} + \beta_{8} - 3 \beta_{7} + 3 \beta_{6} + \beta_{5} + 4 \beta_{4} + 8 \beta_{3} - 2 \beta_{1} + 1$$ $$\nu^{10}$$ $$=$$ $$-\beta_{17} - 6 \beta_{16} + \beta_{15} - 6 \beta_{14} + 7 \beta_{13} + 4 \beta_{12} - 5 \beta_{11} - 2 \beta_{10} - 10 \beta_{9} + 12 \beta_{7} + 2 \beta_{6} + 2 \beta_{5} + 2 \beta_{4} + 5 \beta_{3} + \beta_{2} + 3 \beta_{1} - 10$$ $$\nu^{11}$$ $$=$$ $$-\beta_{17} - 10 \beta_{16} - \beta_{15} - 8 \beta_{14} + 4 \beta_{13} + 3 \beta_{12} - 2 \beta_{11} - 2 \beta_{10} - 16 \beta_{9} - 10 \beta_{8} - 3 \beta_{7} + 13 \beta_{6} + 3 \beta_{5} + 32 \beta_{4} - 4 \beta_{3} + 8 \beta_{2} - 5 \beta_{1} - 25$$ $$\nu^{12}$$ $$=$$ $$6 \beta_{17} - 19 \beta_{16} - 13 \beta_{14} + 12 \beta_{13} + 16 \beta_{12} - 7 \beta_{11} - 4 \beta_{10} - 14 \beta_{9} - 19 \beta_{8} + 4 \beta_{7} + 8 \beta_{6} - 8 \beta_{5} - 2 \beta_{4} + 29 \beta_{3} + 7 \beta_{2} - 14 \beta_{1} - 48$$ $$\nu^{13}$$ $$=$$ $$2 \beta_{17} - 27 \beta_{16} - 14 \beta_{15} - 31 \beta_{14} + 17 \beta_{13} + 19 \beta_{12} - 12 \beta_{11} - 16 \beta_{10} - 20 \beta_{9} + 9 \beta_{8} + 9 \beta_{7} + 7 \beta_{6} - 3 \beta_{5} + 36 \beta_{4} - 8 \beta_{3} + 12 \beta_{2} - 34 \beta_{1} - 27$$ $$\nu^{14}$$ $$=$$ $$7 \beta_{17} + 22 \beta_{16} - 15 \beta_{15} + 6 \beta_{14} - 9 \beta_{13} + 20 \beta_{12} - \beta_{11} + 6 \beta_{10} - 18 \beta_{9} - 40 \beta_{8} - 4 \beta_{7} + 2 \beta_{6} + 18 \beta_{5} + 26 \beta_{4} + \beta_{3} - 31 \beta_{2} + 11 \beta_{1} + 22$$ $$\nu^{15}$$ $$=$$ $$-21 \beta_{17} - 50 \beta_{16} - 37 \beta_{15} - 16 \beta_{14} + 12 \beta_{13} + 55 \beta_{12} + 38 \beta_{11} + 14 \beta_{10} - 16 \beta_{9} - 30 \beta_{8} - 7 \beta_{7} + 25 \beta_{6} + 39 \beta_{5} - 16 \beta_{4} + 4 \beta_{3} + 52 \beta_{2} + 7 \beta_{1} - 101$$ $$\nu^{16}$$ $$=$$ $$2 \beta_{17} + 45 \beta_{16} - 28 \beta_{15} - 5 \beta_{14} + 40 \beta_{12} - 51 \beta_{11} + 44 \beta_{10} + 26 \beta_{9} - 7 \beta_{8} + 28 \beta_{7} + 40 \beta_{6} - 8 \beta_{5} + 134 \beta_{4} - 31 \beta_{3} - \beta_{2} - 50 \beta_{1} - 128$$ $$\nu^{17}$$ $$=$$ $$50 \beta_{17} + \beta_{16} - 30 \beta_{15} + 21 \beta_{14} - 67 \beta_{13} + 31 \beta_{12} + 124 \beta_{11} - 24 \beta_{10} - 20 \beta_{9} - 79 \beta_{8} - 67 \beta_{7} - 29 \beta_{6} + 33 \beta_{5} - 44 \beta_{4} + 56 \beta_{3} + 8 \beta_{2} - 50 \beta_{1} - 55$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/552\mathbb{Z}\right)^\times$$.

 $$n$$ $$97$$ $$185$$ $$277$$ $$415$$ $$\chi(n)$$ $$1$$ $$1$$ $$-1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
277.1
 0.238365 − 1.39398i 0.238365 + 1.39398i −0.809892 + 1.15934i −0.809892 − 1.15934i −1.30337 + 0.548841i −1.30337 − 0.548841i 1.37219 − 0.342174i 1.37219 + 0.342174i 1.39509 + 0.231756i 1.39509 − 0.231756i −1.38678 − 0.277192i −1.38678 + 0.277192i 1.26873 + 0.624769i 1.26873 − 0.624769i −1.06785 − 0.927201i −1.06785 + 0.927201i 0.293513 + 1.38342i 0.293513 − 1.38342i
−1.39398 0.238365i 1.00000i 1.88636 + 0.664552i 4.40892i −0.238365 + 1.39398i 4.93691 −2.47115 1.37602i −1.00000 1.05093 6.14595i
277.2 −1.39398 + 0.238365i 1.00000i 1.88636 0.664552i 4.40892i −0.238365 1.39398i 4.93691 −2.47115 + 1.37602i −1.00000 1.05093 + 6.14595i
277.3 −1.15934 0.809892i 1.00000i 0.688150 + 1.87788i 3.56420i 0.809892 1.15934i −2.96155 0.723081 2.73444i −1.00000 2.88661 4.13212i
277.4 −1.15934 + 0.809892i 1.00000i 0.688150 1.87788i 3.56420i 0.809892 + 1.15934i −2.96155 0.723081 + 2.73444i −1.00000 2.88661 + 4.13212i
277.5 −0.548841 1.30337i 1.00000i −1.39755 + 1.43069i 0.625627i 1.30337 0.548841i 0.327807 2.63174 + 1.03630i −1.00000 0.815424 0.343370i
277.6 −0.548841 + 1.30337i 1.00000i −1.39755 1.43069i 0.625627i 1.30337 + 0.548841i 0.327807 2.63174 1.03630i −1.00000 0.815424 + 0.343370i
277.7 −0.342174 1.37219i 1.00000i −1.76583 + 0.939057i 0.957976i −1.37219 + 0.342174i 2.14538 1.89279 + 2.10175i −1.00000 1.31453 0.327794i
277.8 −0.342174 + 1.37219i 1.00000i −1.76583 0.939057i 0.957976i −1.37219 0.342174i 2.14538 1.89279 2.10175i −1.00000 1.31453 + 0.327794i
277.9 0.231756 1.39509i 1.00000i −1.89258 0.646644i 2.83774i −1.39509 0.231756i 0.890209 −1.34075 + 2.49046i −1.00000 −3.95892 0.657665i
277.10 0.231756 + 1.39509i 1.00000i −1.89258 + 0.646644i 2.83774i −1.39509 + 0.231756i 0.890209 −1.34075 2.49046i −1.00000 −3.95892 + 0.657665i
277.11 0.277192 1.38678i 1.00000i −1.84633 0.768809i 0.208104i 1.38678 + 0.277192i 4.69627 −1.57796 + 2.34735i −1.00000 −0.288595 0.0576846i
277.12 0.277192 + 1.38678i 1.00000i −1.84633 + 0.768809i 0.208104i 1.38678 0.277192i 4.69627 −1.57796 2.34735i −1.00000 −0.288595 + 0.0576846i
277.13 0.624769 1.26873i 1.00000i −1.21933 1.58532i 3.50951i −1.26873 0.624769i −4.50530 −2.77313 + 0.556532i −1.00000 4.45260 + 2.19263i
277.14 0.624769 + 1.26873i 1.00000i −1.21933 + 1.58532i 3.50951i −1.26873 + 0.624769i −4.50530 −2.77313 0.556532i −1.00000 4.45260 2.19263i
277.15 0.927201 1.06785i 1.00000i −0.280598 1.98022i 0.619156i 1.06785 + 0.927201i −1.72166 −2.37474 1.53642i −1.00000 −0.661164 0.574082i
277.16 0.927201 + 1.06785i 1.00000i −0.280598 + 1.98022i 0.619156i 1.06785 0.927201i −1.72166 −2.37474 + 1.53642i −1.00000 −0.661164 + 0.574082i
277.17 1.38342 0.293513i 1.00000i 1.82770 0.812103i 1.32391i −0.293513 1.38342i 0.191941 2.29011 1.65993i −1.00000 0.388583 + 1.83152i
277.18 1.38342 + 0.293513i 1.00000i 1.82770 + 0.812103i 1.32391i −0.293513 + 1.38342i 0.191941 2.29011 + 1.65993i −1.00000 0.388583 1.83152i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 277.18 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

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 552.2.f.c 18
4.b odd 2 1 2208.2.f.c 18
8.b even 2 1 inner 552.2.f.c 18
8.d odd 2 1 2208.2.f.c 18

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
552.2.f.c 18 1.a even 1 1 trivial
552.2.f.c 18 8.b even 2 1 inner
2208.2.f.c 18 4.b odd 2 1
2208.2.f.c 18 8.d odd 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{18} + \cdots$$ acting on $$S_{2}^{\mathrm{new}}(552, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$512 + 512 T^{2} + 128 T^{3} + 160 T^{4} + 96 T^{5} - 16 T^{6} - 24 T^{8} - 24 T^{9} - 12 T^{10} - 2 T^{12} + 6 T^{13} + 5 T^{14} + 2 T^{15} + 4 T^{16} + T^{18}$$
$3$ $$( 1 + T^{2} )^{9}$$
$5$ $$256 + 7744 T^{2} + 46976 T^{4} + 112608 T^{6} + 119952 T^{8} + 57236 T^{10} + 11920 T^{12} + 1188 T^{14} + 56 T^{16} + T^{18}$$
$7$ $$( 64 - 584 T + 1432 T^{2} - 508 T^{3} - 852 T^{4} + 306 T^{5} + 120 T^{6} - 34 T^{7} - 4 T^{8} + T^{9} )^{2}$$
$11$ $$200704 + 2520064 T^{2} + 5872640 T^{4} + 5381632 T^{6} + 2286144 T^{8} + 488208 T^{10} + 54736 T^{12} + 3212 T^{14} + 92 T^{16} + T^{18}$$
$13$ $$750321664 + 989728768 T^{2} + 507574272 T^{4} + 137212672 T^{6} + 22003072 T^{8} + 2199568 T^{10} + 138208 T^{12} + 5288 T^{14} + 112 T^{16} + T^{18}$$
$17$ $$( 6976 + 41016 T - 11992 T^{2} - 13756 T^{3} + 3156 T^{4} + 1450 T^{5} - 220 T^{6} - 66 T^{7} + 4 T^{8} + T^{9} )^{2}$$
$19$ $$3650093056 + 5162595392 T^{2} + 2714809856 T^{4} + 689282272 T^{6} + 96499328 T^{8} + 7916404 T^{10} + 386384 T^{12} + 10932 T^{14} + 164 T^{16} + T^{18}$$
$23$ $$( 1 + T )^{18}$$
$29$ $$440664064 + 9944629248 T^{2} + 36173955072 T^{4} + 24085655552 T^{6} + 3945244672 T^{8} + 205356544 T^{10} + 4980416 T^{12} + 62576 T^{14} + 396 T^{16} + T^{18}$$
$31$ $$( 67072 + 408832 T + 620672 T^{2} + 312128 T^{3} + 21248 T^{4} - 14016 T^{5} - 2280 T^{6} + 28 T^{7} + 22 T^{8} + T^{9} )^{2}$$
$37$ $$2801579659264 + 2403984479232 T^{2} + 772450724864 T^{4} + 116454083072 T^{6} + 8674510144 T^{8} + 336818704 T^{10} + 7004240 T^{12} + 78348 T^{14} + 444 T^{16} + T^{18}$$
$41$ $$( 16384 - 770048 T + 579584 T^{2} + 40192 T^{3} - 65184 T^{4} + 2288 T^{5} + 1784 T^{6} - 108 T^{7} - 14 T^{8} + T^{9} )^{2}$$
$43$ $$23406089184256 + 20509127222848 T^{2} + 4007041613056 T^{4} + 356959096800 T^{6} + 17428596800 T^{8} + 498760852 T^{10} + 8506464 T^{12} + 84628 T^{14} + 452 T^{16} + T^{18}$$
$47$ $$( -1608704 + 1446720 T + 152000 T^{2} - 207328 T^{3} - 3568 T^{4} + 9748 T^{5} + 32 T^{6} - 172 T^{7} + T^{9} )^{2}$$
$53$ $$3540726016 + 261918950976 T^{2} + 171488090496 T^{4} + 41809709920 T^{6} + 4680419024 T^{8} + 247108084 T^{10} + 6419392 T^{12} + 81380 T^{14} + 472 T^{16} + T^{18}$$
$59$ $$1609619535364096 + 747042929217536 T^{2} + 113756870215680 T^{4} + 6659973192448 T^{6} + 197530984576 T^{8} + 3362870288 T^{10} + 34427104 T^{12} + 209960 T^{14} + 704 T^{16} + T^{18}$$
$61$ $$1294060757979136 + 736698105050112 T^{2} + 129570600946688 T^{4} + 7873567952384 T^{6} + 237058972736 T^{8} + 4042354448 T^{10} + 40815824 T^{12} + 240876 T^{14} + 764 T^{16} + T^{18}$$
$67$ $$20495829581824 + 11151744463424 T^{2} + 2392011559168 T^{4} + 260189261920 T^{6} + 15486382784 T^{8} + 516539124 T^{10} + 9715568 T^{12} + 100212 T^{14} + 516 T^{16} + T^{18}$$
$71$ $$( 2633728 + 1160192 T - 1509376 T^{2} - 160512 T^{3} + 115520 T^{4} + 10960 T^{5} - 2544 T^{6} - 244 T^{7} + 12 T^{8} + T^{9} )^{2}$$
$73$ $$( 34816 - 185344 T + 139776 T^{2} + 22272 T^{3} - 36704 T^{4} + 3248 T^{5} + 1816 T^{6} - 252 T^{7} - 6 T^{8} + T^{9} )^{2}$$
$79$ $$( -371744 + 250680 T + 284680 T^{2} - 134556 T^{3} - 51716 T^{4} + 19138 T^{5} + 1444 T^{6} - 330 T^{7} - 4 T^{8} + T^{9} )^{2}$$
$83$ $$28955763675136 + 11680746968064 T^{2} + 1971731293184 T^{4} + 181283657216 T^{6} + 9907958592 T^{8} + 330048784 T^{10} + 6614224 T^{12} + 75788 T^{14} + 444 T^{16} + T^{18}$$
$89$ $$( -203802464 + 155595064 T + 22670248 T^{2} - 5223244 T^{3} - 508276 T^{4} + 68442 T^{5} + 4216 T^{6} - 418 T^{7} - 12 T^{8} + T^{9} )^{2}$$
$97$ $$( -53447168 - 57511168 T - 20548864 T^{2} - 2154368 T^{3} + 272896 T^{4} + 61248 T^{5} + 144 T^{6} - 424 T^{7} - 6 T^{8} + T^{9} )^{2}$$