# Properties

 Label 27.2.e.a Level 27 Weight 2 Character orbit 27.e Analytic conductor 0.216 Analytic rank 0 Dimension 12 CM no Inner twists 2

# Related objects

## Newspace parameters

 Level: $$N$$ = $$27 = 3^{3}$$ Weight: $$k$$ = $$2$$ Character orbit: $$[\chi]$$ = 27.e (of order $$9$$, degree $$6$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$0.215596085457$$ Analytic rank: $$0$$ Dimension: $$12$$ Relative dimension: $$2$$ over $$\Q(\zeta_{9})$$ Coefficient field: 12.0.1952986685049.1 Coefficient ring: $$\Z[a_1, \ldots, a_{4}]$$ Coefficient ring index: $$3$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{9}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{11}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

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

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{12} - 6 x^{11} + 27 x^{10} - 80 x^{9} + 186 x^{8} - 330 x^{7} + 463 x^{6} - 504 x^{5} + 420 x^{4} - 258 x^{3} + 108 x^{2} - 27 x + 3$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu^{10} - 5 \nu^{9} + 22 \nu^{8} - 58 \nu^{7} + 127 \nu^{6} - 199 \nu^{5} + 249 \nu^{4} - 224 \nu^{3} + 145 \nu^{2} - 58 \nu + 9$$ $$\beta_{2}$$ $$=$$ $$3 \nu^{11} - 16 \nu^{10} + 71 \nu^{9} - 197 \nu^{8} + 445 \nu^{7} - 747 \nu^{6} + 1006 \nu^{5} - 1030 \nu^{4} + 803 \nu^{3} - 445 \nu^{2} + 155 \nu - 25$$ $$\beta_{3}$$ $$=$$ $$-6 \nu^{11} + 32 \nu^{10} - 140 \nu^{9} + 384 \nu^{8} - 849 \nu^{7} + 1390 \nu^{6} - 1805 \nu^{5} + 1762 \nu^{4} - 1285 \nu^{3} + 649 \nu^{2} - 195 \nu + 25$$ $$\beta_{4}$$ $$=$$ $$-9 \nu^{11} + 49 \nu^{10} - 216 \nu^{9} + 601 \nu^{8} - 1344 \nu^{7} + 2232 \nu^{6} - 2942 \nu^{5} + 2918 \nu^{4} - 2170 \nu^{3} + 1118 \nu^{2} - 348 \nu + 49$$ $$\beta_{5}$$ $$=$$ $$9 \nu^{11} - 50 \nu^{10} + 221 \nu^{9} - 623 \nu^{8} + 1402 \nu^{7} - 2360 \nu^{6} + 3144 \nu^{5} - 3178 \nu^{4} + 2411 \nu^{3} - 1286 \nu^{2} + 421 \nu - 62$$ $$\beta_{6}$$ $$=$$ $$11 \nu^{11} - 60 \nu^{10} + 265 \nu^{9} - 739 \nu^{8} + 1657 \nu^{7} - 2761 \nu^{6} + 3653 \nu^{5} - 3643 \nu^{4} + 2724 \nu^{3} - 1417 \nu^{2} + 442 \nu - 61$$ $$\beta_{7}$$ $$=$$ $$-16 \nu^{11} + 87 \nu^{10} - 383 \nu^{9} + 1064 \nu^{8} - 2375 \nu^{7} + 3936 \nu^{6} - 5176 \nu^{5} + 5122 \nu^{4} - 3802 \nu^{3} + 1958 \nu^{2} - 610 \nu + 85$$ $$\beta_{8}$$ $$=$$ $$-16 \nu^{11} + 89 \nu^{10} - 393 \nu^{9} + 1108 \nu^{8} - 2491 \nu^{7} + 4191 \nu^{6} - 5577 \nu^{5} + 5631 \nu^{4} - 4267 \nu^{3} + 2272 \nu^{2} - 742 \nu + 110$$ $$\beta_{9}$$ $$=$$ $$36 \nu^{11} - 198 \nu^{10} + 873 \nu^{9} - 2443 \nu^{8} + 5472 \nu^{7} - 9134 \nu^{6} + 12076 \nu^{5} - 12058 \nu^{4} + 9024 \nu^{3} - 4708 \nu^{2} + 1486 \nu - 209$$ $$\beta_{10}$$ $$=$$ $$-36 \nu^{11} + 198 \nu^{10} - 873 \nu^{9} + 2444 \nu^{8} - 5476 \nu^{7} + 9150 \nu^{6} - 12110 \nu^{5} + 12120 \nu^{4} - 9096 \nu^{3} + 4772 \nu^{2} - 1519 \nu + 217$$ $$\beta_{11}$$ $$=$$ $$-42 \nu^{11} + 231 \nu^{10} - 1019 \nu^{9} + 2853 \nu^{8} - 6396 \nu^{7} + 10689 \nu^{6} - 14157 \nu^{5} + 14172 \nu^{4} - 10648 \nu^{3} + 5589 \nu^{2} - 1785 \nu + 257$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{11} - \beta_{10} + \beta_{9} + \beta_{8} + \beta_{7} - 2 \beta_{6} + \beta_{5} - 2 \beta_{4} + \beta_{3} + \beta_{2} + \beta_{1} + 3$$$$)/3$$ $$\nu^{2}$$ $$=$$ $$($$$$\beta_{11} - \beta_{10} + \beta_{9} + 4 \beta_{8} - 2 \beta_{7} - 2 \beta_{6} + 4 \beta_{5} + \beta_{4} + \beta_{3} + \beta_{2} - 2 \beta_{1} - 6$$$$)/3$$ $$\nu^{3}$$ $$=$$ $$($$$$-5 \beta_{11} + 5 \beta_{10} - 5 \beta_{9} + \beta_{8} - 8 \beta_{7} + 7 \beta_{6} + 4 \beta_{5} + 10 \beta_{4} - 5 \beta_{3} - 5 \beta_{2} - 8 \beta_{1} - 18$$$$)/3$$ $$\nu^{4}$$ $$=$$ $$($$$$-11 \beta_{11} + 17 \beta_{10} - 5 \beta_{9} - 20 \beta_{8} + 4 \beta_{7} + 16 \beta_{6} - 14 \beta_{5} + 4 \beta_{4} - 14 \beta_{3} - 8 \beta_{2} + \beta_{1} + 6$$$$)/3$$ $$\nu^{5}$$ $$=$$ $$($$$$19 \beta_{11} - \beta_{10} + 31 \beta_{9} - 32 \beta_{8} + 43 \beta_{7} - 20 \beta_{6} - 41 \beta_{5} - 44 \beta_{4} + 10 \beta_{3} + 25 \beta_{2} + 40 \beta_{1} + 87$$$$)/3$$ $$\nu^{6}$$ $$=$$ $$($$$$85 \beta_{11} - 97 \beta_{10} + 55 \beta_{9} + 64 \beta_{8} + 10 \beta_{7} - 101 \beta_{6} + 19 \beta_{5} - 62 \beta_{4} + 91 \beta_{3} + 70 \beta_{2} + 31 \beta_{1} + 60$$$$)/3$$ $$\nu^{7}$$ $$=$$ $$($$$$-20 \beta_{11} - 118 \beta_{10} - 134 \beta_{9} + 244 \beta_{8} - 218 \beta_{7} + \beta_{6} + 232 \beta_{5} + 157 \beta_{4} + 52 \beta_{3} - 74 \beta_{2} - 179 \beta_{1} - 357$$$$)/3$$ $$\nu^{8}$$ $$=$$ $$($$$$-503 \beta_{11} + 386 \beta_{10} - 440 \beta_{9} - 47 \beta_{8} - 233 \beta_{7} + 514 \beta_{6} + 163 \beta_{5} + 466 \beta_{4} - 431 \beta_{3} - 461 \beta_{2} - 329 \beta_{1} - 639$$$$)/3$$ $$\nu^{9}$$ $$=$$ $$($$$$-425 \beta_{11} + 1076 \beta_{10} + 319 \beta_{9} - 1313 \beta_{8} + 955 \beta_{7} + 502 \beta_{6} - 1013 \beta_{5} - 332 \beta_{4} - 743 \beta_{3} - 59 \beta_{2} + 631 \beta_{1} + 1164$$$$)/3$$ $$\nu^{10}$$ $$=$$ $$($$$$2299 \beta_{11} - 862 \beta_{10} + 2725 \beta_{9} - 1193 \beta_{8} + 2104 \beta_{7} - 2135 \beta_{6} - 1907 \beta_{5} - 2705 \beta_{4} + 1495 \beta_{3} + 2425 \beta_{2} + 2344 \beta_{1} + 4356$$$$)/3$$ $$\nu^{11}$$ $$=$$ $$($$$$4708 \beta_{11} - 6628 \beta_{10} + 985 \beta_{9} + 5506 \beta_{8} - 3107 \beta_{7} - 4679 \beta_{6} + 3238 \beta_{5} - 992 \beta_{4} + 5476 \beta_{3} + 2770 \beta_{2} - 1043 \beta_{1} - 1698$$$$)/3$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/27\mathbb{Z}\right)^\times$$.

 $$n$$ $$2$$ $$\chi(n)$$ $$\beta_{10}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
4.1
 0.5 − 1.27297i 0.5 − 0.0126039i 0.5 + 1.27297i 0.5 + 0.0126039i 0.5 − 1.68614i 0.5 + 1.00210i 0.5 + 0.258654i 0.5 − 2.22827i 0.5 − 0.258654i 0.5 + 2.22827i 0.5 + 1.68614i 0.5 − 1.00210i
−1.57954 0.574906i 1.45446 0.940501i 0.632343 + 0.530599i −0.196143 + 1.11238i −2.83808 + 0.649381i −2.99441 + 2.51261i 0.987144 + 1.70978i 1.23092 2.73584i 0.949332 1.64429i
4.2 0.753189 + 0.274138i −1.68842 0.386327i −1.03995 0.872619i −0.477505 + 2.70806i −1.16579 0.753837i 1.82076 1.52780i −1.34559 2.33062i 2.70150 + 1.30456i −1.10204 + 1.90878i
7.1 −1.57954 + 0.574906i 1.45446 + 0.940501i 0.632343 0.530599i −0.196143 1.11238i −2.83808 0.649381i −2.99441 2.51261i 0.987144 1.70978i 1.23092 + 2.73584i 0.949332 + 1.64429i
7.2 0.753189 0.274138i −1.68842 + 0.386327i −1.03995 + 0.872619i −0.477505 2.70806i −1.16579 + 0.753837i 1.82076 + 1.52780i −1.34559 + 2.33062i 2.70150 1.30456i −1.10204 1.90878i
13.1 −0.417037 2.36514i −0.210069 + 1.71926i −3.54056 + 1.28866i 0.0713060 + 0.0598329i 4.15390 0.220155i 0.544891 + 0.198324i 2.12277 + 3.67675i −2.91174 0.722330i 0.111776 0.193601i
13.2 0.183082 + 1.03831i −1.72962 0.0916693i 0.834822 0.303850i −1.33735 1.12217i −0.221481 1.81266i −2.31094 0.841112i 1.52266 + 2.63732i 2.98319 + 0.317107i 0.920313 1.59403i
16.1 −1.62143 1.36054i −0.986166 1.42389i 0.430663 + 2.44241i 2.52129 + 0.917674i −0.338267 + 3.65046i 0.168844 0.957561i 0.508086 0.880031i −1.05495 + 2.80839i −2.83955 4.91825i
16.2 −0.318266 0.267057i 0.159815 + 1.72466i −0.317323 1.79963i −2.08159 0.757639i 0.409719 0.591580i −0.229151 + 1.29958i −0.795075 + 1.37711i −2.94892 + 0.551252i 0.460168 + 0.797034i
22.1 −1.62143 + 1.36054i −0.986166 + 1.42389i 0.430663 2.44241i 2.52129 0.917674i −0.338267 3.65046i 0.168844 + 0.957561i 0.508086 + 0.880031i −1.05495 2.80839i −2.83955 + 4.91825i
22.2 −0.318266 + 0.267057i 0.159815 1.72466i −0.317323 + 1.79963i −2.08159 + 0.757639i 0.409719 + 0.591580i −0.229151 1.29958i −0.795075 1.37711i −2.94892 0.551252i 0.460168 0.797034i
25.1 −0.417037 + 2.36514i −0.210069 1.71926i −3.54056 1.28866i 0.0713060 0.0598329i 4.15390 + 0.220155i 0.544891 0.198324i 2.12277 3.67675i −2.91174 + 0.722330i 0.111776 + 0.193601i
25.2 0.183082 1.03831i −1.72962 + 0.0916693i 0.834822 + 0.303850i −1.33735 + 1.12217i −0.221481 + 1.81266i −2.31094 + 0.841112i 1.52266 2.63732i 2.98319 0.317107i 0.920313 + 1.59403i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 25.2 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
27.e even 9 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 27.2.e.a 12
3.b odd 2 1 81.2.e.a 12
4.b odd 2 1 432.2.u.c 12
5.b even 2 1 675.2.l.c 12
5.c odd 4 2 675.2.u.b 24
9.c even 3 1 243.2.e.c 12
9.c even 3 1 243.2.e.d 12
9.d odd 6 1 243.2.e.a 12
9.d odd 6 1 243.2.e.b 12
27.e even 9 1 inner 27.2.e.a 12
27.e even 9 1 243.2.e.c 12
27.e even 9 1 243.2.e.d 12
27.e even 9 1 729.2.a.a 6
27.e even 9 2 729.2.c.e 12
27.f odd 18 1 81.2.e.a 12
27.f odd 18 1 243.2.e.a 12
27.f odd 18 1 243.2.e.b 12
27.f odd 18 1 729.2.a.d 6
27.f odd 18 2 729.2.c.b 12
108.j odd 18 1 432.2.u.c 12
135.p even 18 1 675.2.l.c 12
135.r odd 36 2 675.2.u.b 24

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
27.2.e.a 12 1.a even 1 1 trivial
27.2.e.a 12 27.e even 9 1 inner
81.2.e.a 12 3.b odd 2 1
81.2.e.a 12 27.f odd 18 1
243.2.e.a 12 9.d odd 6 1
243.2.e.a 12 27.f odd 18 1
243.2.e.b 12 9.d odd 6 1
243.2.e.b 12 27.f odd 18 1
243.2.e.c 12 9.c even 3 1
243.2.e.c 12 27.e even 9 1
243.2.e.d 12 9.c even 3 1
243.2.e.d 12 27.e even 9 1
432.2.u.c 12 4.b odd 2 1
432.2.u.c 12 108.j odd 18 1
675.2.l.c 12 5.b even 2 1
675.2.l.c 12 135.p even 18 1
675.2.u.b 24 5.c odd 4 2
675.2.u.b 24 135.r odd 36 2
729.2.a.a 6 27.e even 9 1
729.2.a.d 6 27.f odd 18 1
729.2.c.b 12 27.f odd 18 2
729.2.c.e 12 27.e even 9 2

## Hecke kernels

This newform subspace is the entire newspace $$S_{2}^{\mathrm{new}}(27, [\chi])$$.

## Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ $$( 1 + 3 T + 3 T^{2} - 3 T^{4} - 6 T^{5} - 11 T^{6} - 12 T^{7} - 12 T^{8} + 48 T^{10} + 96 T^{11} + 64 T^{12} )( 1 + 3 T + 9 T^{2} + 18 T^{3} + 36 T^{4} + 57 T^{5} + 91 T^{6} + 114 T^{7} + 144 T^{8} + 144 T^{9} + 144 T^{10} + 96 T^{11} + 64 T^{12} )$$
$3$ $$1 + 6 T + 18 T^{2} + 39 T^{3} + 63 T^{4} + 81 T^{5} + 117 T^{6} + 243 T^{7} + 567 T^{8} + 1053 T^{9} + 1458 T^{10} + 1458 T^{11} + 729 T^{12}$$
$5$ $$1 + 3 T + 3 T^{2} + 18 T^{3} + 87 T^{4} + 147 T^{5} + 323 T^{6} + 1368 T^{7} + 3096 T^{8} + 5562 T^{9} + 16272 T^{10} + 41310 T^{11} + 82629 T^{12} + 206550 T^{13} + 406800 T^{14} + 695250 T^{15} + 1935000 T^{16} + 4275000 T^{17} + 5046875 T^{18} + 11484375 T^{19} + 33984375 T^{20} + 35156250 T^{21} + 29296875 T^{22} + 146484375 T^{23} + 244140625 T^{24}$$
$7$ $$1 + 6 T + 12 T^{2} - 11 T^{3} - 213 T^{4} - 678 T^{5} - 224 T^{6} + 3942 T^{7} + 15255 T^{8} + 25135 T^{9} - 22044 T^{10} - 210732 T^{11} - 647141 T^{12} - 1475124 T^{13} - 1080156 T^{14} + 8621305 T^{15} + 36627255 T^{16} + 66253194 T^{17} - 26353376 T^{18} - 558362154 T^{19} - 1227902613 T^{20} - 443889677 T^{21} + 3389702988 T^{22} + 11863960458 T^{23} + 13841287201 T^{24}$$
$11$ $$1 - 3 T - 15 T^{2} + 126 T^{3} - 201 T^{4} - 1488 T^{5} + 7145 T^{6} - 1530 T^{7} - 61974 T^{8} + 202716 T^{9} - 19692 T^{10} - 1304451 T^{11} + 4526883 T^{12} - 14348961 T^{13} - 2382732 T^{14} + 269814996 T^{15} - 907361334 T^{16} - 246408030 T^{17} + 12657803345 T^{18} - 28996910448 T^{19} - 43086135081 T^{20} + 297101409066 T^{21} - 389061369015 T^{22} - 855935011833 T^{23} + 3138428376721 T^{24}$$
$13$ $$1 + 6 T + 48 T^{2} + 214 T^{3} + 1488 T^{4} + 5928 T^{5} + 32329 T^{6} + 112023 T^{7} + 560277 T^{8} + 1799710 T^{9} + 8467593 T^{10} + 25055985 T^{11} + 112181629 T^{12} + 325727805 T^{13} + 1431023217 T^{14} + 3953962870 T^{15} + 16002071397 T^{16} + 41593355739 T^{17} + 156045908161 T^{18} + 371973208776 T^{19} + 1213807312848 T^{20} + 2269362865822 T^{21} + 6617207608752 T^{22} + 10752962364222 T^{23} + 23298085122481 T^{24}$$
$17$ $$1 - 9 T - 30 T^{2} + 423 T^{3} + 1029 T^{4} - 14184 T^{5} - 23521 T^{6} + 296649 T^{7} + 637560 T^{8} - 4620213 T^{9} - 12537675 T^{10} + 28264410 T^{11} + 250681641 T^{12} + 480494970 T^{13} - 3623388075 T^{14} - 22699106469 T^{15} + 53249648760 T^{16} + 421199159193 T^{17} - 567739760449 T^{18} - 5820243737832 T^{19} + 7178054406789 T^{20} + 50162671758231 T^{21} - 60479817013470 T^{22} - 308447066768697 T^{23} + 582622237229761 T^{24}$$
$19$ $$1 + 3 T - 75 T^{2} - 242 T^{3} + 3012 T^{4} + 9714 T^{5} - 85589 T^{6} - 257166 T^{7} + 1946502 T^{8} + 4391737 T^{9} - 39399504 T^{10} - 33490578 T^{11} + 763159453 T^{12} - 636320982 T^{13} - 14223220944 T^{14} + 30122924083 T^{15} + 253670087142 T^{16} - 636768475434 T^{17} - 4026609908909 T^{18} + 8683070072646 T^{19} + 51154491879492 T^{20} - 78090422862518 T^{21} - 459829969335075 T^{22} + 349470776694657 T^{23} + 2213314919066161 T^{24}$$
$23$ $$1 + 12 T + 48 T^{2} + 153 T^{3} - 336 T^{4} - 12228 T^{5} - 51922 T^{6} - 116820 T^{7} - 165330 T^{8} + 4324509 T^{9} + 14509764 T^{10} - 2453454 T^{11} + 107316369 T^{12} - 56429442 T^{13} + 7675665156 T^{14} + 52616301003 T^{15} - 46266112530 T^{16} - 751893589260 T^{17} - 7686319428658 T^{18} - 41634205565916 T^{19} - 26312491054416 T^{20} + 275576357203839 T^{21} + 1988472538255152 T^{22} + 11433717094967124 T^{23} + 21914624432020321 T^{24}$$
$29$ $$1 + 6 T + 21 T^{2} + 252 T^{3} + 249 T^{4} - 984 T^{5} + 18431 T^{6} + 29592 T^{7} + 680634 T^{8} + 5882274 T^{9} + 10161684 T^{10} + 17557326 T^{11} + 254066229 T^{12} + 509162454 T^{13} + 8545976244 T^{14} + 143462780586 T^{15} + 481399496154 T^{16} + 606965921208 T^{17} + 10963188629351 T^{18} - 16973878288056 T^{19} + 124561356827289 T^{20} + 3655800785918988 T^{21} + 8834851899304221 T^{22} + 73203058594234974 T^{23} + 353814783205469041 T^{24}$$
$31$ $$1 - 3 T + 84 T^{2} - 434 T^{3} + 5601 T^{4} - 30963 T^{5} + 266473 T^{6} - 1627992 T^{7} + 11453211 T^{8} - 69240287 T^{9} + 408317577 T^{10} - 2527882269 T^{11} + 13547586181 T^{12} - 78364350339 T^{13} + 392393191497 T^{14} - 2062737390017 T^{15} + 10577280875931 T^{16} - 46608028794792 T^{17} + 236495768387113 T^{18} - 851873070718893 T^{19} + 4777042700707041 T^{20} - 11474796017731214 T^{21} + 68848776106387284 T^{22} - 76225430689214493 T^{23} + 787662783788549761 T^{24}$$
$37$ $$1 + 3 T - 156 T^{2} - 107 T^{3} + 13731 T^{4} - 9132 T^{5} - 864755 T^{6} + 641043 T^{7} + 43249536 T^{8} - 18536771 T^{9} - 1953626739 T^{10} + 269355786 T^{11} + 78884071369 T^{12} + 9966164082 T^{13} - 2674515005691 T^{14} - 938943061463 T^{15} + 81056593639296 T^{16} + 44452458227151 T^{17} - 2218724740814795 T^{18} - 866917901978556 T^{19} + 48229855381789251 T^{20} - 13905906158073239 T^{21} - 750139162097184444 T^{22} + 533752865338381239 T^{23} + 6582952005840035281 T^{24}$$
$41$ $$1 - 15 T + 93 T^{2} - 90 T^{3} - 2460 T^{4} + 12513 T^{5} + 27971 T^{6} - 441396 T^{7} + 3206862 T^{8} - 12736494 T^{9} - 41813613 T^{10} + 1731506832 T^{11} - 16389887967 T^{12} + 70991780112 T^{13} - 70288683453 T^{14} - 877811902974 T^{15} + 9061825571982 T^{16} - 51138463696596 T^{17} + 132865165725011 T^{18} + 2436960229072953 T^{19} - 19642916063637660 T^{20} - 29464374095456490 T^{21} + 1248307315844173293 T^{22} - 8254935475743726615 T^{23} + 22563490300366186081 T^{24}$$
$43$ $$1 - 3 T - 60 T^{2} + 16 T^{3} + 606 T^{4} + 5874 T^{5} + 128269 T^{6} - 73818 T^{7} - 5307417 T^{8} - 22910987 T^{9} - 79924800 T^{10} + 580797228 T^{11} + 14492410483 T^{12} + 24974280804 T^{13} - 147780955200 T^{14} - 1821583843409 T^{15} - 18145002547017 T^{16} - 10851869245374 T^{17} + 810834916932181 T^{18} + 1596662521642518 T^{19} + 7083049368226206 T^{20} + 8041481790989488 T^{21} - 1296688938797054940 T^{22} - 2787881218413668121 T^{23} + 39959630797262576401 T^{24}$$
$47$ $$1 + 15 T + 111 T^{2} + 873 T^{3} + 6828 T^{4} + 69612 T^{5} + 654227 T^{6} + 4732173 T^{7} + 31522707 T^{8} + 170376048 T^{9} + 1258782219 T^{10} + 11485670769 T^{11} + 82734051465 T^{12} + 539826526143 T^{13} + 2780649921771 T^{14} + 17688952431504 T^{15} + 153820754416467 T^{16} + 1085300249810211 T^{17} + 7052053707045683 T^{18} + 35267048661670356 T^{19} + 162583465326504108 T^{20} + 977000903018715591 T^{21} + 5838503678177135439 T^{22} + 37082388226260184545 T^{23} +$$$$11\!\cdots\!41$$$$T^{24}$$
$53$ $$( 1 + 9 T + 210 T^{2} + 1872 T^{3} + 23856 T^{4} + 168327 T^{5} + 1634317 T^{6} + 8921331 T^{7} + 67011504 T^{8} + 278697744 T^{9} + 1657001010 T^{10} + 3763759437 T^{11} + 22164361129 T^{12} )^{2}$$
$59$ $$1 + 12 T + 192 T^{2} + 2349 T^{3} + 25089 T^{4} + 223824 T^{5} + 1972808 T^{6} + 12709350 T^{7} + 71501877 T^{8} + 339681357 T^{9} - 107943444 T^{10} - 13247965206 T^{11} - 122980417173 T^{12} - 781629947154 T^{13} - 375751128564 T^{14} + 69763417419303 T^{15} + 866414055786597 T^{16} + 9086223139495650 T^{17} + 83214094211233928 T^{18} + 557019929938127856 T^{19} + 3683828849054809569 T^{20} + 20349377178020451711 T^{21} + 98134416633723148992 T^{22} +$$$$36\!\cdots\!08$$$$T^{23} +$$$$17\!\cdots\!81$$$$T^{24}$$
$61$ $$1 - 12 T - 51 T^{2} + 583 T^{3} + 2127 T^{4} + 45474 T^{5} - 455363 T^{6} - 2399139 T^{7} + 11507670 T^{8} + 53383966 T^{9} + 844033821 T^{10} - 2578276122 T^{11} - 56950876769 T^{12} - 157274843442 T^{13} + 3140649847941 T^{14} + 12117145986646 T^{15} + 159333369100470 T^{16} - 2026303924984839 T^{17} - 23460472230148043 T^{18} + 142913087725218954 T^{19} + 407761454745216687 T^{20} + 6817687172122304203 T^{21} - 36380488494807012651 T^{22} -$$$$52\!\cdots\!32$$$$T^{23} +$$$$26\!\cdots\!21$$$$T^{24}$$
$67$ $$1 + 15 T + 255 T^{2} + 2968 T^{3} + 36174 T^{4} + 397221 T^{5} + 4107115 T^{6} + 41367024 T^{7} + 386429292 T^{8} + 3556146616 T^{9} + 31289775603 T^{10} + 264760435272 T^{11} + 2216964278029 T^{12} + 17738949163224 T^{13} + 140459802681867 T^{14} + 1069557324668008 T^{15} + 7786983421036332 T^{16} + 55850657704271568 T^{17} + 371522978282032435 T^{18} + 2407441924578007383 T^{19} + 14689092167933931534 T^{20} + 80748994088203402696 T^{21} +$$$$46\!\cdots\!95$$$$T^{22} +$$$$18\!\cdots\!45$$$$T^{23} +$$$$81\!\cdots\!61$$$$T^{24}$$
$71$ $$1 - 27 T + 78 T^{2} + 2565 T^{3} + 13071 T^{4} - 524664 T^{5} - 751711 T^{6} + 30297321 T^{7} + 410765508 T^{8} - 3391054713 T^{9} - 30034133541 T^{10} + 13624108308 T^{11} + 3600759258249 T^{12} + 967311689868 T^{13} - 151402067180181 T^{14} - 1213695783384543 T^{15} + 10438242055098948 T^{16} + 54663315804868671 T^{17} - 96294392526538831 T^{18} - 4771882122782055624 T^{19} + 8440644406913342031 T^{20} +$$$$11\!\cdots\!15$$$$T^{21} +$$$$25\!\cdots\!78$$$$T^{22} -$$$$62\!\cdots\!17$$$$T^{23} +$$$$16\!\cdots\!41$$$$T^{24}$$
$73$ $$1 - 6 T - 228 T^{2} + 2296 T^{3} + 24945 T^{4} - 381255 T^{5} - 980072 T^{6} + 40200363 T^{7} - 102286134 T^{8} - 2648934335 T^{9} + 21743689350 T^{10} + 78452536893 T^{11} - 2017821540323 T^{12} + 5727035193189 T^{13} + 115872120546150 T^{14} - 1030480488198695 T^{15} - 2904746284290294 T^{16} + 83338230563588259 T^{17} - 148318437827512808 T^{18} - 4211875922398326735 T^{19} + 20117146992297850545 T^{20} +$$$$13\!\cdots\!48$$$$T^{21} -$$$$97\!\cdots\!72$$$$T^{22} -$$$$18\!\cdots\!62$$$$T^{23} +$$$$22\!\cdots\!21$$$$T^{24}$$
$79$ $$1 + 42 T + 813 T^{2} + 9520 T^{3} + 72840 T^{4} + 356811 T^{5} + 973207 T^{6} - 893781 T^{7} - 62793603 T^{8} - 1355379536 T^{9} - 23955645108 T^{10} - 329298299862 T^{11} - 3388931313773 T^{12} - 26014565689098 T^{13} - 149507181119028 T^{14} - 668254971049904 T^{15} - 2445815923131843 T^{16} - 2750214545354619 T^{17} + 236574413325225847 T^{18} + 6852165969260378949 T^{19} +$$$$11\!\cdots\!40$$$$T^{20} +$$$$11\!\cdots\!80$$$$T^{21} +$$$$76\!\cdots\!13$$$$T^{22} +$$$$31\!\cdots\!18$$$$T^{23} +$$$$59\!\cdots\!41$$$$T^{24}$$
$83$ $$1 - 39 T + 912 T^{2} - 16200 T^{3} + 251079 T^{4} - 3515997 T^{5} + 45358019 T^{6} - 541131408 T^{7} + 6100532325 T^{8} - 65514800025 T^{9} + 671478204717 T^{10} - 6541571603403 T^{11} + 60933732837525 T^{12} - 542950443082449 T^{13} + 4625813352295413 T^{14} - 37460510961894675 T^{15} + 289521021350726325 T^{16} - 2131538609315815344 T^{17} + 14829367667138196011 T^{18} - 95410273871375563119 T^{19} +$$$$56\!\cdots\!39$$$$T^{20} -$$$$30\!\cdots\!00$$$$T^{21} +$$$$14\!\cdots\!88$$$$T^{22} -$$$$50\!\cdots\!13$$$$T^{23} +$$$$10\!\cdots\!61$$$$T^{24}$$
$89$ $$1 - 9 T - 273 T^{2} + 2772 T^{3} + 38802 T^{4} - 449316 T^{5} - 3561871 T^{6} + 54551502 T^{7} + 157767516 T^{8} - 4371660207 T^{9} + 3816883044 T^{10} + 152630961444 T^{11} - 900621732009 T^{12} + 13584155568516 T^{13} + 30233530591524 T^{14} - 3081884924468583 T^{15} + 9898687510843356 T^{16} + 304618830200242398 T^{17} - 1770183247816548031 T^{18} - 19873846469919508164 T^{19} +$$$$15\!\cdots\!62$$$$T^{20} +$$$$97\!\cdots\!48$$$$T^{21} -$$$$85\!\cdots\!73$$$$T^{22} -$$$$24\!\cdots\!01$$$$T^{23} +$$$$24\!\cdots\!21$$$$T^{24}$$
$97$ $$1 - 3 T + 102 T^{2} - 1010 T^{3} + 15132 T^{4} - 13512 T^{5} + 1127323 T^{6} - 7762230 T^{7} + 23311719 T^{8} + 575063737 T^{9} + 1596748254 T^{10} + 54554445012 T^{11} - 1453572795209 T^{12} + 5291781166164 T^{13} + 15023804321886 T^{14} + 524845146039001 T^{15} + 2063769721944039 T^{16} - 66656910163093110 T^{17} + 939028499512575067 T^{18} - 1091746419868262856 T^{19} +$$$$11\!\cdots\!52$$$$T^{20} -$$$$76\!\cdots\!70$$$$T^{21} +$$$$75\!\cdots\!98$$$$T^{22} -$$$$21\!\cdots\!59$$$$T^{23} +$$$$69\!\cdots\!41$$$$T^{24}$$