# Properties

 Label 350.2.h.c Level 350 Weight 2 Character orbit 350.h Analytic conductor 2.795 Analytic rank 0 Dimension 16 CM no Inner twists 2

# Related objects

## Newspace parameters

 Level: $$N$$ = $$350 = 2 \cdot 5^{2} \cdot 7$$ Weight: $$k$$ = $$2$$ Character orbit: $$[\chi]$$ = 350.h (of order $$5$$, degree $$4$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$2.79476407074$$ Analytic rank: $$0$$ Dimension: $$16$$ Relative dimension: $$4$$ over $$\Q(\zeta_{5})$$ Coefficient field: $$\mathbb{Q}[x]/(x^{16} + \cdots)$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$5$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{5}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{15}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

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

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{16} + 25 x^{14} + 241 x^{12} + 1145 x^{10} + 2841 x^{8} + 3600 x^{6} + 2156 x^{4} + 480 x^{2} + 16$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$21 \nu^{15} + 32 \nu^{14} + 487 \nu^{13} + 784 \nu^{12} + 4215 \nu^{11} + 7248 \nu^{10} + 17143 \nu^{9} + 31568 \nu^{8} + 33847 \nu^{7} + 65168 \nu^{6} + 30074 \nu^{5} + 55152 \nu^{4} + 10332 \nu^{3} + 14272 \nu^{2} + 976 \nu + 448$$$$)/128$$ $$\beta_{2}$$ $$=$$ $$($$$$9 \nu^{14} + 223 \nu^{12} + 2087 \nu^{10} + 9191 \nu^{8} + 19079 \nu^{6} + 16022 \nu^{4} + 4132 \nu^{2} + 192$$$$)/32$$ $$\beta_{3}$$ $$=$$ $$($$$$21 \nu^{15} - 32 \nu^{14} + 487 \nu^{13} - 784 \nu^{12} + 4215 \nu^{11} - 7248 \nu^{10} + 17143 \nu^{9} - 31568 \nu^{8} + 33847 \nu^{7} - 65168 \nu^{6} + 30074 \nu^{5} - 55152 \nu^{4} + 10332 \nu^{3} - 14272 \nu^{2} + 976 \nu - 448$$$$)/128$$ $$\beta_{4}$$ $$=$$ $$($$$$-10 \nu^{15} - 31 \nu^{14} - 228 \nu^{13} - 721 \nu^{12} - 1928 \nu^{11} - 6249 \nu^{10} - 7608 \nu^{9} - 25289 \nu^{8} - 14488 \nu^{7} - 48649 \nu^{6} - 12254 \nu^{5} - 39346 \nu^{4} - 3388 \nu^{3} - 9820 \nu^{2} + 136 \nu - 224$$$$)/64$$ $$\beta_{5}$$ $$=$$ $$($$$$-10 \nu^{15} + 31 \nu^{14} - 228 \nu^{13} + 721 \nu^{12} - 1928 \nu^{11} + 6249 \nu^{10} - 7608 \nu^{9} + 25289 \nu^{8} - 14488 \nu^{7} + 48649 \nu^{6} - 12254 \nu^{5} + 39346 \nu^{4} - 3388 \nu^{3} + 9820 \nu^{2} + 136 \nu + 224$$$$)/64$$ $$\beta_{6}$$ $$=$$ $$($$$$51 \nu^{15} + 32 \nu^{14} + 1219 \nu^{13} + 784 \nu^{12} + 10943 \nu^{11} + 7248 \nu^{10} + 46191 \nu^{9} + 31568 \nu^{8} + 92911 \nu^{7} + 65168 \nu^{6} + 78308 \nu^{5} + 55152 \nu^{4} + 21360 \nu^{3} + 14272 \nu^{2} + 1112 \nu + 384$$$$)/128$$ $$\beta_{7}$$ $$=$$ $$($$$$19 \nu^{15} + 60 \nu^{14} + 467 \nu^{13} + 1484 \nu^{12} + 4335 \nu^{11} + 13884 \nu^{10} + 18975 \nu^{9} + 61372 \nu^{8} + 39391 \nu^{7} + 129148 \nu^{6} + 33396 \nu^{5} + 112736 \nu^{4} + 7952 \nu^{3} + 31360 \nu^{2} - 616 \nu + 1248$$$$)/128$$ $$\beta_{8}$$ $$=$$ $$($$$$19 \nu^{15} - 60 \nu^{14} + 467 \nu^{13} - 1484 \nu^{12} + 4335 \nu^{11} - 13884 \nu^{10} + 18975 \nu^{9} - 61372 \nu^{8} + 39391 \nu^{7} - 129148 \nu^{6} + 33396 \nu^{5} - 112736 \nu^{4} + 7952 \nu^{3} - 31360 \nu^{2} - 616 \nu - 1248$$$$)/128$$ $$\beta_{9}$$ $$=$$ $$($$$$-3 \nu^{15} + 61 \nu^{14} - 53 \nu^{13} + 1447 \nu^{12} - 253 \nu^{11} + 12871 \nu^{10} + 163 \nu^{9} + 53799 \nu^{8} + 3683 \nu^{7} + 107399 \nu^{6} + 7966 \nu^{5} + 90562 \nu^{4} + 6052 \nu^{3} + 24748 \nu^{2} + 1248 \nu + 944$$$$)/64$$ $$\beta_{10}$$ $$=$$ $$($$$$65 \nu^{15} + 98 \nu^{14} + 1537 \nu^{13} + 2330 \nu^{12} + 13589 \nu^{11} + 20786 \nu^{10} + 56101 \nu^{9} + 87186 \nu^{8} + 108901 \nu^{7} + 174674 \nu^{6} + 85148 \nu^{5} + 147680 \nu^{4} + 18000 \nu^{3} + 40400 \nu^{2} - 376 \nu + 1456$$$$)/128$$ $$\beta_{11}$$ $$=$$ $$($$$$65 \nu^{15} - 98 \nu^{14} + 1537 \nu^{13} - 2330 \nu^{12} + 13589 \nu^{11} - 20786 \nu^{10} + 56101 \nu^{9} - 87186 \nu^{8} + 108901 \nu^{7} - 174674 \nu^{6} + 85148 \nu^{5} - 147680 \nu^{4} + 18000 \nu^{3} - 40400 \nu^{2} - 376 \nu - 1456$$$$)/128$$ $$\beta_{12}$$ $$=$$ $$($$$$51 \nu^{15} - 118 \nu^{14} + 1239 \nu^{13} - 2734 \nu^{12} + 11371 \nu^{11} - 23574 \nu^{10} + 49467 \nu^{9} - 94774 \nu^{8} + 103931 \nu^{7} - 180982 \nu^{6} + 94576 \nu^{5} - 145680 \nu^{4} + 30664 \nu^{3} - 37008 \nu^{2} + 2280 \nu - 1232$$$$)/128$$ $$\beta_{13}$$ $$=$$ $$($$$$64 \nu^{15} - 3 \nu^{14} + 1531 \nu^{13} - 59 \nu^{12} + 13757 \nu^{11} - 391 \nu^{10} + 58117 \nu^{9} - 1015 \nu^{8} + 116869 \nu^{7} - 983 \nu^{6} + 97957 \nu^{5} - 700 \nu^{4} + 25714 \nu^{3} - 384 \nu^{2} + 764 \nu + 104$$$$)/64$$ $$\beta_{14}$$ $$=$$ $$($$$$-39 \nu^{15} - 61 \nu^{14} - 906 \nu^{13} - 1447 \nu^{12} - 7832 \nu^{11} - 12871 \nu^{10} - 31504 \nu^{9} - 53799 \nu^{8} - 59696 \nu^{7} - 107399 \nu^{6} - 46225 \nu^{5} - 90562 \nu^{4} - 9794 \nu^{3} - 24748 \nu^{2} + 204 \nu - 912$$$$)/64$$ $$\beta_{15}$$ $$=$$ $$($$$$-67 \nu^{15} - 58 \nu^{14} - 1590 \nu^{13} - 1388 \nu^{12} - 14148 \nu^{11} - 12480 \nu^{10} - 59132 \nu^{9} - 52784 \nu^{8} - 117852 \nu^{7} - 106416 \nu^{6} - 98657 \nu^{5} - 89862 \nu^{4} - 26098 \nu^{3} - 24332 \nu^{2} - 596 \nu - 920$$$$)/64$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$2 \beta_{15} + \beta_{14} + \beta_{13} + \beta_{12} + \beta_{11} + 3 \beta_{10} + 2 \beta_{9} - 2 \beta_{8} - 3 \beta_{7} - 3 \beta_{5} - 4 \beta_{4} - 3 \beta_{3} + \beta_{2} - 3 \beta_{1} - 1$$$$)/5$$ $$\nu^{2}$$ $$=$$ $$\beta_{13} - \beta_{12} + \beta_{11} - \beta_{10} + \beta_{9} - \beta_{8} - \beta_{6} - \beta_{5} - \beta_{3} - \beta_{2} - 4$$ $$\nu^{3}$$ $$=$$ $$($$$$-2 \beta_{15} - 11 \beta_{14} - 6 \beta_{13} - 6 \beta_{12} - 16 \beta_{11} - 18 \beta_{10} - 17 \beta_{9} + 17 \beta_{8} + 23 \beta_{7} + 25 \beta_{6} + 18 \beta_{5} + 24 \beta_{4} + 33 \beta_{3} - \beta_{2} + 8 \beta_{1} + 21$$$$)/5$$ $$\nu^{4}$$ $$=$$ $$2 \beta_{14} - 8 \beta_{13} + 8 \beta_{12} - 10 \beta_{11} + 10 \beta_{10} - 10 \beta_{9} + 7 \beta_{8} + \beta_{7} + 10 \beta_{6} + 9 \beta_{5} - \beta_{4} + 14 \beta_{3} + 11 \beta_{2} - 4 \beta_{1} + 29$$ $$\nu^{5}$$ $$=$$ $$($$$$-38 \beta_{15} + 96 \beta_{14} + 41 \beta_{13} + 41 \beta_{12} + 151 \beta_{11} + 113 \beta_{10} + 137 \beta_{9} - 152 \beta_{8} - 193 \beta_{7} - 265 \beta_{6} - 123 \beta_{5} - 164 \beta_{4} - 298 \beta_{3} - 19 \beta_{2} - 33 \beta_{1} - 201$$$$)/5$$ $$\nu^{6}$$ $$=$$ $$-23 \beta_{14} + 66 \beta_{13} - 66 \beta_{12} + 86 \beta_{11} - 86 \beta_{10} + 89 \beta_{9} - 51 \beta_{8} - 15 \beta_{7} - 89 \beta_{6} - 81 \beta_{5} + 15 \beta_{4} - 140 \beta_{3} - 102 \beta_{2} + 51 \beta_{1} - 231$$ $$\nu^{7}$$ $$=$$ $$($$$$598 \beta_{15} - 846 \beta_{14} - 281 \beta_{13} - 281 \beta_{12} - 1351 \beta_{11} - 753 \beta_{10} - 1127 \beta_{9} + 1357 \beta_{8} + 1638 \beta_{7} + 2385 \beta_{6} + 943 \beta_{5} + 1224 \beta_{4} + 2603 \beta_{3} + 299 \beta_{2} + 218 \beta_{1} + 1756$$$$)/5$$ $$\nu^{8}$$ $$=$$ $$205 \beta_{14} - 564 \beta_{13} + 564 \beta_{12} - 720 \beta_{11} + 720 \beta_{10} - 769 \beta_{9} + 399 \beta_{8} + 165 \beta_{7} + 769 \beta_{6} + 726 \beta_{5} - 162 \beta_{4} + 1287 \beta_{3} + 917 \beta_{2} - 518 \beta_{1} + 1911$$ $$\nu^{9}$$ $$=$$ $$($$$$-6618 \beta_{15} + 7526 \beta_{14} + 1926 \beta_{13} + 1926 \beta_{12} + 11936 \beta_{11} + 5318 \beta_{10} + 9452 \beta_{9} - 12057 \beta_{8} - 13983 \beta_{7} - 20570 \beta_{6} - 7723 \beta_{5} - 9649 \beta_{4} - 22498 \beta_{3} - 3309 \beta_{2} - 1928 \beta_{1} - 15011$$$$)/5$$ $$\nu^{10}$$ $$=$$ $$-1702 \beta_{14} + 4893 \beta_{13} - 4893 \beta_{12} + 6019 \beta_{11} - 6019 \beta_{10} + 6595 \beta_{9} - 3270 \beta_{8} - 1623 \beta_{7} - 6595 \beta_{6} - 6461 \beta_{5} + 1568 \beta_{4} - 11466 \beta_{3} - 8147 \beta_{2} + 4871 \beta_{1} - 16113$$ $$\nu^{11}$$ $$=$$ $$($$$$65408 \beta_{15} - 66901 \beta_{14} - 13346 \beta_{13} - 13346 \beta_{12} - 104836 \beta_{11} - 39428 \beta_{10} - 80247 \beta_{9} + 106537 \beta_{8} + 119883 \beta_{7} + 175435 \beta_{6} + 65323 \beta_{5} + 78669 \beta_{4} + 193948 \beta_{3} + 32704 \beta_{2} + 18513 \beta_{1} + 127841$$$$)/5$$ $$\nu^{12}$$ $$=$$ $$13890 \beta_{14} - 42651 \beta_{13} + 42651 \beta_{12} - 50591 \beta_{11} + 50591 \beta_{10} - 56541 \beta_{9} + 27491 \beta_{8} + 15160 \beta_{7} + 56541 \beta_{6} + 57119 \beta_{5} - 14468 \beta_{4} + 100789 \beta_{3} + 71861 \beta_{2} - 44248 \beta_{1} + 137304$$ $$\nu^{13}$$ $$=$$ $$($$$$-613478 \beta_{15} + 592041 \beta_{14} + 94356 \beta_{13} + 94356 \beta_{12} + 917336 \beta_{11} + 303858 \beta_{10} + 686397 \beta_{9} - 936817 \beta_{8} - 1031173 \beta_{7} - 1495205 \beta_{6} - 560628 \beta_{5} - 654984 \beta_{4} - 1672383 \beta_{3} - 306739 \beta_{2} - 177178 \beta_{1} - 1090801$$$$)/5$$ $$\nu^{14}$$ $$=$$ $$-113642 \beta_{14} + 372004 \beta_{13} - 372004 \beta_{12} + 428106 \beta_{11} - 428106 \beta_{10} + 485646 \beta_{9} - 234245 \beta_{8} - 137759 \beta_{7} - 485646 \beta_{6} - 502307 \beta_{5} + 130303 \beta_{4} - 880482 \beta_{3} - 630709 \beta_{2} + 394836 \beta_{1} - 1177335$$ $$\nu^{15}$$ $$=$$ $$($$$$5592498 \beta_{15} - 5214016 \beta_{14} - 684591 \beta_{13} - 684591 \beta_{12} - 8006061 \beta_{11} - 2413563 \beta_{10} - 5898607 \beta_{9} + 8206512 \beta_{8} + 8891103 \beta_{7} + 12777335 \beta_{6} + 4842053 \beta_{5} + 5526644 \beta_{4} + 14436358 \beta_{3} + 2796249 \beta_{2} + 1659023 \beta_{1} + 9337971$$$$)/5$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/350\mathbb{Z}\right)^\times$$.

 $$n$$ $$101$$ $$127$$ $$\chi(n)$$ $$1$$ $$-\beta_{1}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
71.1
 2.46620i − 1.07895i − 1.93398i − 0.628834i − 0.200203i 2.94531i − 1.92944i 1.08645i 0.200203i − 2.94531i 1.92944i − 1.08645i − 2.46620i 1.07895i 1.93398i 0.628834i
0.309017 0.951057i −1.78422 1.29631i −0.809017 0.587785i −0.298278 2.21608i −1.78422 + 1.29631i −1.00000 −0.809017 + 0.587785i 0.575968 + 1.77265i −2.19979 0.401128i
71.2 0.309017 0.951057i −1.46563 1.06484i −0.809017 0.587785i −1.01402 + 1.99293i −1.46563 + 1.06484i −1.00000 −0.809017 + 0.587785i 0.0871311 + 0.268162i 1.58203 + 1.58024i
71.3 0.309017 0.951057i 0.689113 + 0.500670i −0.809017 0.587785i −1.78357 1.34866i 0.689113 0.500670i −1.00000 −0.809017 + 0.587785i −0.702844 2.16313i −1.83381 + 1.27951i
71.4 0.309017 0.951057i 2.25172 + 1.63597i −0.809017 0.587785i 2.09587 0.779319i 2.25172 1.63597i −1.00000 −0.809017 + 0.587785i 1.46680 + 4.51433i −0.0935175 2.23411i
141.1 −0.809017 0.587785i −0.491708 1.51332i 0.309017 + 0.951057i 2.15788 + 0.586125i −0.491708 + 1.51332i −1.00000 0.309017 0.951057i 0.378688 0.275133i −1.40125 1.74256i
141.2 −0.809017 0.587785i −0.0305168 0.0939212i 0.309017 + 0.951057i −1.05684 + 1.97056i −0.0305168 + 0.0939212i −1.00000 0.309017 0.951057i 2.41916 1.75762i 2.01326 0.973023i
141.3 −0.809017 0.587785i 0.315945 + 0.972380i 0.309017 + 0.951057i −2.00668 0.986533i 0.315945 0.972380i −1.00000 0.309017 0.951057i 1.58135 1.14892i 1.04357 + 1.97762i
141.4 −0.809017 0.587785i 1.01530 + 3.12476i 0.309017 + 0.951057i −0.0943702 + 2.23408i 1.01530 3.12476i −1.00000 0.309017 0.951057i −6.30625 + 4.58176i 1.38950 1.75194i
211.1 −0.809017 + 0.587785i −0.491708 + 1.51332i 0.309017 0.951057i 2.15788 0.586125i −0.491708 1.51332i −1.00000 0.309017 + 0.951057i 0.378688 + 0.275133i −1.40125 + 1.74256i
211.2 −0.809017 + 0.587785i −0.0305168 + 0.0939212i 0.309017 0.951057i −1.05684 1.97056i −0.0305168 0.0939212i −1.00000 0.309017 + 0.951057i 2.41916 + 1.75762i 2.01326 + 0.973023i
211.3 −0.809017 + 0.587785i 0.315945 0.972380i 0.309017 0.951057i −2.00668 + 0.986533i 0.315945 + 0.972380i −1.00000 0.309017 + 0.951057i 1.58135 + 1.14892i 1.04357 1.97762i
211.4 −0.809017 + 0.587785i 1.01530 3.12476i 0.309017 0.951057i −0.0943702 2.23408i 1.01530 + 3.12476i −1.00000 0.309017 + 0.951057i −6.30625 4.58176i 1.38950 + 1.75194i
281.1 0.309017 + 0.951057i −1.78422 + 1.29631i −0.809017 + 0.587785i −0.298278 + 2.21608i −1.78422 1.29631i −1.00000 −0.809017 0.587785i 0.575968 1.77265i −2.19979 + 0.401128i
281.2 0.309017 + 0.951057i −1.46563 + 1.06484i −0.809017 + 0.587785i −1.01402 1.99293i −1.46563 1.06484i −1.00000 −0.809017 0.587785i 0.0871311 0.268162i 1.58203 1.58024i
281.3 0.309017 + 0.951057i 0.689113 0.500670i −0.809017 + 0.587785i −1.78357 + 1.34866i 0.689113 + 0.500670i −1.00000 −0.809017 0.587785i −0.702844 + 2.16313i −1.83381 1.27951i
281.4 0.309017 + 0.951057i 2.25172 1.63597i −0.809017 + 0.587785i 2.09587 + 0.779319i 2.25172 + 1.63597i −1.00000 −0.809017 0.587785i 1.46680 4.51433i −0.0935175 + 2.23411i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 281.4 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
25.d even 5 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 350.2.h.c 16
25.d even 5 1 inner 350.2.h.c 16
25.d even 5 1 8750.2.a.u 8
25.e even 10 1 8750.2.a.s 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
350.2.h.c 16 1.a even 1 1 trivial
350.2.h.c 16 25.d even 5 1 inner
8750.2.a.s 8 25.e even 10 1
8750.2.a.u 8 25.d even 5 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{16} - \cdots$$ acting on $$S_{2}^{\mathrm{new}}(350, [\chi])$$.

## Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ $$( 1 + T + T^{2} + T^{3} + T^{4} )^{4}$$
$3$ $$1 - T - 5 T^{2} + 12 T^{3} - 11 T^{4} - 34 T^{5} + 135 T^{6} - 166 T^{7} - 223 T^{8} + 1011 T^{9} - 1180 T^{10} - 315 T^{11} + 5026 T^{12} - 9385 T^{13} + 3426 T^{14} + 18392 T^{15} - 50450 T^{16} + 55176 T^{17} + 30834 T^{18} - 253395 T^{19} + 407106 T^{20} - 76545 T^{21} - 860220 T^{22} + 2211057 T^{23} - 1463103 T^{24} - 3267378 T^{25} + 7971615 T^{26} - 6022998 T^{27} - 5845851 T^{28} + 19131876 T^{29} - 23914845 T^{30} - 14348907 T^{31} + 43046721 T^{32}$$
$5$ $$1 + 4 T + 11 T^{2} - 6 T^{3} - 84 T^{4} - 280 T^{5} - 130 T^{6} + 1050 T^{7} + 4525 T^{8} + 5250 T^{9} - 3250 T^{10} - 35000 T^{11} - 52500 T^{12} - 18750 T^{13} + 171875 T^{14} + 312500 T^{15} + 390625 T^{16}$$
$7$ $$( 1 + T )^{16}$$
$11$ $$1 - 7 T + 4 T^{2} + T^{3} + 367 T^{4} - 671 T^{5} - 874 T^{6} - 12433 T^{7} + 46158 T^{8} + 27843 T^{9} + 284444 T^{10} - 2592183 T^{11} + 1138482 T^{12} + 4725054 T^{13} + 79059510 T^{14} - 177853274 T^{15} - 239193093 T^{16} - 1956386014 T^{17} + 9566200710 T^{18} + 6289046874 T^{19} + 16668514962 T^{20} - 417473664333 T^{21} + 503909897084 T^{22} + 542581302153 T^{23} + 9894377229198 T^{24} - 29316363642203 T^{25} - 22669309101274 T^{26} - 191444130979981 T^{27} + 1151803214256607 T^{28} + 34522712143931 T^{29} + 1518999334332964 T^{30} - 29240737185909557 T^{31} + 45949729863572161 T^{32}$$
$13$ $$1 - 9 T + 23 T^{2} - 65 T^{3} + 420 T^{4} + 793 T^{5} - 8537 T^{6} + 3579 T^{7} - 93425 T^{8} + 576885 T^{9} + 290947 T^{10} + 84117 T^{11} - 7580374 T^{12} - 72282820 T^{13} + 134961730 T^{14} - 124529416 T^{15} + 2958255499 T^{16} - 1618882408 T^{17} + 22808532370 T^{18} - 158805355540 T^{19} - 216503061814 T^{20} + 31232053281 T^{21} + 1404345598123 T^{22} + 36198678229545 T^{23} - 76209642609425 T^{24} + 37953503255967 T^{25} - 1176897944914913 T^{26} + 1421183192471341 T^{27} + 9785195751442020 T^{28} - 19686881928496445 T^{29} + 90559656871083647 T^{30} - 460673037126816813 T^{31} + 665416609183179841 T^{32}$$
$17$ $$1 + 2 T - 81 T^{2} - 164 T^{3} + 2667 T^{4} + 5428 T^{5} - 41998 T^{6} - 90626 T^{7} + 212692 T^{8} + 1088886 T^{9} + 3872547 T^{10} - 26863856 T^{11} - 147904226 T^{12} + 680142794 T^{13} + 4324416006 T^{14} - 6144073958 T^{15} - 92016330155 T^{16} - 104449257286 T^{17} + 1249756225734 T^{18} + 3341541546922 T^{19} - 12353108859746 T^{20} - 38142833988592 T^{21} + 93473870418243 T^{22} + 446812036288278 T^{23} + 1483687801641172 T^{24} - 10747144895417122 T^{25} - 84667711831057102 T^{26} + 186027853157831924 T^{27} + 1553853506691772587 T^{28} - 1624350797396573668 T^{29} - 13638603951311475249 T^{30} + 5724846103019631586 T^{31} + 48661191875666868481 T^{32}$$
$19$ $$1 + 24 T + 296 T^{2} + 2592 T^{3} + 17982 T^{4} + 98712 T^{5} + 406844 T^{6} + 1030944 T^{7} - 1201257 T^{8} - 34550296 T^{9} - 246931384 T^{10} - 1149072336 T^{11} - 3482962948 T^{12} - 1595520248 T^{13} + 65096881880 T^{14} + 533938597512 T^{15} + 2732959874037 T^{16} + 10144833352728 T^{17} + 23499974358680 T^{18} - 10943673381032 T^{19} - 453903214346308 T^{20} - 2845216862097264 T^{21} - 11617104506829304 T^{22} - 30883533168484744 T^{23} - 20401623987942537 T^{24} + 332672945899073376 T^{25} + 2494387520588790044 T^{26} + 11498986436360993928 T^{27} + 39799828874647707102 T^{28} +$$$$10\!\cdots\!28$$$$T^{29} +$$$$23\!\cdots\!16$$$$T^{30} +$$$$36\!\cdots\!76$$$$T^{31} +$$$$28\!\cdots\!81$$$$T^{32}$$
$23$ $$1 + 5 T - 22 T^{2} - 25 T^{3} + 495 T^{4} - 7425 T^{5} - 19220 T^{6} + 207375 T^{7} - 35370 T^{8} - 1776625 T^{9} + 23723034 T^{10} - 27251505 T^{11} - 556173278 T^{12} + 857611800 T^{13} - 314171630 T^{14} - 4900274400 T^{15} + 245436206455 T^{16} - 112706311200 T^{17} - 166196792270 T^{18} + 10434562770600 T^{19} - 155640086288798 T^{20} - 175400033446215 T^{21} + 3511860427967226 T^{22} - 6049098009776375 T^{23} - 2769859549388970 T^{24} + 373514033170889625 T^{25} - 796217545526333780 T^{26} - 7074612452510907975 T^{27} + 10847739093850058895 T^{28} - 12600909048411684575 T^{29} -$$$$25\!\cdots\!98$$$$T^{30} +$$$$13\!\cdots\!35$$$$T^{31} +$$$$61\!\cdots\!61$$$$T^{32}$$
$29$ $$1 - 20 T + 201 T^{2} - 1320 T^{3} + 7525 T^{4} - 37240 T^{5} + 89750 T^{6} + 615040 T^{7} - 8311490 T^{8} + 61959320 T^{9} - 387993087 T^{10} + 1800586880 T^{11} - 3797715622 T^{12} - 15900561440 T^{13} + 250739424210 T^{14} - 2172262465880 T^{15} + 13894638163425 T^{16} - 62995611510520 T^{17} + 210871855760610 T^{18} - 387798792960160 T^{19} - 2686052102843782 T^{20} + 36932105783125120 T^{21} - 230787336534381927 T^{22} + 1068790606189749880 T^{23} - 4157793058861221890 T^{24} + 8922475060998469760 T^{25} + 37758474188693039750 T^{26} -$$$$45\!\cdots\!60$$$$T^{27} +$$$$26\!\cdots\!25$$$$T^{28} -$$$$13\!\cdots\!80$$$$T^{29} +$$$$59\!\cdots\!81$$$$T^{30} -$$$$17\!\cdots\!80$$$$T^{31} +$$$$25\!\cdots\!21$$$$T^{32}$$
$31$ $$1 - 7 T - 52 T^{2} + 434 T^{3} - 611 T^{4} + 9420 T^{5} + 20954 T^{6} - 1040008 T^{7} + 3045078 T^{8} - 3464302 T^{9} + 23457032 T^{10} + 1136716529 T^{11} - 7254596780 T^{12} + 2110239704 T^{13} - 18287520106 T^{14} - 556338309516 T^{15} + 9102576366561 T^{16} - 17246487594996 T^{17} - 17574306821866 T^{18} + 62866151021864 T^{19} - 6699772472862380 T^{20} + 32543229152936879 T^{21} + 20818202245334792 T^{22} - 95312004089965522 T^{23} + 2597119734508765398 T^{24} - 27497418564075125368 T^{25} + 17174491125395704154 T^{26} +$$$$23\!\cdots\!20$$$$T^{27} -$$$$48\!\cdots\!71$$$$T^{28} +$$$$10\!\cdots\!94$$$$T^{29} -$$$$39\!\cdots\!92$$$$T^{30} -$$$$16\!\cdots\!57$$$$T^{31} +$$$$72\!\cdots\!81$$$$T^{32}$$
$37$ $$1 + 6 T - 86 T^{2} - 202 T^{3} + 4956 T^{4} - 10220 T^{5} - 180710 T^{6} + 1119974 T^{7} + 3483905 T^{8} - 45334964 T^{9} - 27988086 T^{10} + 613887092 T^{11} + 1192594492 T^{12} + 19789376570 T^{13} - 169429782588 T^{14} - 709747280326 T^{15} + 8113479874525 T^{16} - 26260649372062 T^{17} - 231949372362972 T^{18} + 1002391291400210 T^{19} + 2235114085721212 T^{20} + 42569360110503044 T^{21} - 71809771387563174 T^{22} - 4303733232276978212 T^{23} + 12237144731912641505 T^{24} +$$$$14\!\cdots\!98$$$$T^{25} -$$$$86\!\cdots\!90$$$$T^{26} -$$$$18\!\cdots\!60$$$$T^{27} +$$$$32\!\cdots\!36$$$$T^{28} -$$$$49\!\cdots\!94$$$$T^{29} -$$$$77\!\cdots\!54$$$$T^{30} +$$$$20\!\cdots\!58$$$$T^{31} +$$$$12\!\cdots\!41$$$$T^{32}$$
$41$ $$1 - 9 T + 209 T^{2} - 1888 T^{3} + 27988 T^{4} - 215873 T^{5} + 2568095 T^{6} - 17950187 T^{7} + 184170237 T^{8} - 1171290382 T^{9} + 11003655253 T^{10} - 64730717249 T^{11} + 572386438742 T^{12} - 3170831215026 T^{13} + 26747011255536 T^{14} - 140954342581060 T^{15} + 1145771417437051 T^{16} - 5779128045823460 T^{17} + 44961725920556016 T^{18} - 218536858170806946 T^{19} + 1617427275526032662 T^{20} - 7499454988474311049 T^{21} + 52268509483777227973 T^{22} -$$$$22\!\cdots\!42$$$$T^{23} +$$$$14\!\cdots\!77$$$$T^{24} -$$$$58\!\cdots\!07$$$$T^{25} +$$$$34\!\cdots\!95$$$$T^{26} -$$$$11\!\cdots\!93$$$$T^{27} +$$$$63\!\cdots\!28$$$$T^{28} -$$$$17\!\cdots\!48$$$$T^{29} +$$$$79\!\cdots\!49$$$$T^{30} -$$$$13\!\cdots\!09$$$$T^{31} +$$$$63\!\cdots\!41$$$$T^{32}$$
$43$ $$( 1 + 11 T + 307 T^{2} + 2726 T^{3} + 41567 T^{4} + 303824 T^{5} + 3314561 T^{6} + 20111097 T^{7} + 173316472 T^{8} + 864777171 T^{9} + 6128623289 T^{10} + 24156134768 T^{11} + 142109301167 T^{12} + 400745015618 T^{13} + 1940658456043 T^{14} + 2990004722177 T^{15} + 11688200277601 T^{16} )^{2}$$
$47$ $$1 + 40 T + 860 T^{2} + 13620 T^{3} + 177582 T^{4} + 1962120 T^{5} + 18728240 T^{6} + 157365740 T^{7} + 1162186863 T^{8} + 7330805360 T^{9} + 36645561260 T^{10} + 98824008780 T^{11} - 685267268196 T^{12} - 14848564608900 T^{13} - 164307010116600 T^{14} - 1429293953116700 T^{15} - 10561394484169275 T^{16} - 67176815796484900 T^{17} - 362954185347569400 T^{18} - 1541622523389824700 T^{19} - 3343885668537925476 T^{20} + 22664792985417161460 T^{21} +$$$$39\!\cdots\!40$$$$T^{22} +$$$$37\!\cdots\!80$$$$T^{23} +$$$$27\!\cdots\!43$$$$T^{24} +$$$$17\!\cdots\!80$$$$T^{25} +$$$$98\!\cdots\!60$$$$T^{26} +$$$$48\!\cdots\!60$$$$T^{27} +$$$$20\!\cdots\!62$$$$T^{28} +$$$$74\!\cdots\!40$$$$T^{29} +$$$$22\!\cdots\!40$$$$T^{30} +$$$$48\!\cdots\!20$$$$T^{31} +$$$$56\!\cdots\!21$$$$T^{32}$$
$53$ $$1 + 24 T + 131 T^{2} - 2082 T^{3} - 29943 T^{4} - 25594 T^{5} + 1376548 T^{6} + 284722 T^{7} - 80364198 T^{8} + 348867682 T^{9} + 9354078043 T^{10} + 4903415212 T^{11} - 484269386646 T^{12} - 422646277832 T^{13} + 23628062037094 T^{14} - 16685214230384 T^{15} - 1553788861344115 T^{16} - 884316354210352 T^{17} + 66371226262197046 T^{18} - 62922309904794664 T^{19} - 3821118394211916726 T^{20} + 2050586141966039516 T^{21} +$$$$20\!\cdots\!47$$$$T^{22} +$$$$40\!\cdots\!34$$$$T^{23} -$$$$50\!\cdots\!78$$$$T^{24} +$$$$93\!\cdots\!26$$$$T^{25} +$$$$24\!\cdots\!52$$$$T^{26} -$$$$23\!\cdots\!18$$$$T^{27} -$$$$14\!\cdots\!63$$$$T^{28} -$$$$54\!\cdots\!86$$$$T^{29} +$$$$18\!\cdots\!39$$$$T^{30} +$$$$17\!\cdots\!68$$$$T^{31} +$$$$38\!\cdots\!21$$$$T^{32}$$
$59$ $$1 + 17 T + 231 T^{2} + 2711 T^{3} + 31383 T^{4} + 368824 T^{5} + 3899415 T^{6} + 38857474 T^{7} + 369688167 T^{8} + 3403580961 T^{9} + 31220539482 T^{10} + 272706121363 T^{11} + 2333967246102 T^{12} + 19121516358498 T^{13} + 154947098550224 T^{14} + 1243271597158160 T^{15} + 9651740778329686 T^{16} + 73353024232331440 T^{17} + 539370850053329744 T^{18} + 3927157908191960742 T^{19} + 28281523683193776822 T^{20} +$$$$19\!\cdots\!37$$$$T^{21} +$$$$13\!\cdots\!62$$$$T^{22} +$$$$84\!\cdots\!59$$$$T^{23} +$$$$54\!\cdots\!07$$$$T^{24} +$$$$33\!\cdots\!86$$$$T^{25} +$$$$19\!\cdots\!15$$$$T^{26} +$$$$11\!\cdots\!16$$$$T^{27} +$$$$55\!\cdots\!23$$$$T^{28} +$$$$28\!\cdots\!69$$$$T^{29} +$$$$14\!\cdots\!91$$$$T^{30} +$$$$62\!\cdots\!83$$$$T^{31} +$$$$21\!\cdots\!41$$$$T^{32}$$
$61$ $$1 - 2 T + 178 T^{2} - 346 T^{3} + 11144 T^{4} + 40340 T^{5} + 160139 T^{6} + 8917862 T^{7} - 14033607 T^{8} + 339988058 T^{9} + 800783942 T^{10} - 13436388236 T^{11} + 187167906365 T^{12} - 898375851426 T^{13} + 479169344749 T^{14} + 14288394224174 T^{15} - 732267975983454 T^{16} + 871592047674614 T^{17} + 1782989131811029 T^{18} - 203914249132524906 T^{19} + 2591497071832677965 T^{20} - 11348323802925515036 T^{21} + 41256688474117311062 T^{22} +$$$$10\!\cdots\!18$$$$T^{23} -$$$$26\!\cdots\!67$$$$T^{24} +$$$$10\!\cdots\!42$$$$T^{25} +$$$$11\!\cdots\!39$$$$T^{26} +$$$$17\!\cdots\!40$$$$T^{27} +$$$$29\!\cdots\!24$$$$T^{28} -$$$$56\!\cdots\!26$$$$T^{29} +$$$$17\!\cdots\!98$$$$T^{30} -$$$$12\!\cdots\!02$$$$T^{31} +$$$$36\!\cdots\!61$$$$T^{32}$$
$67$ $$1 + 14 T - 5 T^{2} - 1002 T^{3} - 7675 T^{4} - 114318 T^{5} - 864154 T^{6} + 5397658 T^{7} + 76521194 T^{8} + 18318554 T^{9} + 2123751863 T^{10} + 51556521834 T^{11} + 75655239118 T^{12} - 659111383092 T^{13} + 11489865587542 T^{14} - 146859077325800 T^{15} - 3515784432557183 T^{16} - 9839558180828600 T^{17} + 51578006622476038 T^{18} - 198236316912899196 T^{19} + 1524537877750751278 T^{20} + 69607754557677086238 T^{21} +$$$$19\!\cdots\!47$$$$T^{22} +$$$$11\!\cdots\!42$$$$T^{23} +$$$$31\!\cdots\!54$$$$T^{24} +$$$$14\!\cdots\!26$$$$T^{25} -$$$$15\!\cdots\!46$$$$T^{26} -$$$$13\!\cdots\!94$$$$T^{27} -$$$$62\!\cdots\!75$$$$T^{28} -$$$$54\!\cdots\!74$$$$T^{29} -$$$$18\!\cdots\!45$$$$T^{30} +$$$$34\!\cdots\!02$$$$T^{31} +$$$$16\!\cdots\!81$$$$T^{32}$$
$71$ $$1 - 7 T - 131 T^{2} - 114 T^{3} + 21322 T^{4} - 49561 T^{5} - 1032319 T^{6} + 2461237 T^{7} + 92179373 T^{8} - 675870262 T^{9} - 2152863231 T^{10} + 44390382057 T^{11} - 83384161648 T^{12} - 2913469018956 T^{13} + 14917443172720 T^{14} + 129116899378886 T^{15} - 1673894927111283 T^{16} + 9167299855900906 T^{17} + 75198831033681520 T^{18} - 1042762610043560916 T^{19} - 2118931716251410288 T^{20} + 80090430209343155007 T^{21} -$$$$27\!\cdots\!51$$$$T^{22} -$$$$61\!\cdots\!42$$$$T^{23} +$$$$59\!\cdots\!53$$$$T^{24} +$$$$11\!\cdots\!47$$$$T^{25} -$$$$33\!\cdots\!19$$$$T^{26} -$$$$11\!\cdots\!31$$$$T^{27} +$$$$34\!\cdots\!02$$$$T^{28} -$$$$13\!\cdots\!54$$$$T^{29} -$$$$10\!\cdots\!11$$$$T^{30} -$$$$41\!\cdots\!57$$$$T^{31} +$$$$41\!\cdots\!21$$$$T^{32}$$
$73$ $$1 - 5 T - 184 T^{2} - 660 T^{3} + 18729 T^{4} + 101130 T^{5} + 583946 T^{6} + 1517480 T^{7} - 99548154 T^{8} - 2205366530 T^{9} - 3106828272 T^{10} + 113345857735 T^{11} + 1007191618776 T^{12} + 4851951748650 T^{13} + 43053806124718 T^{14} - 510880245844670 T^{15} - 9238219037249483 T^{16} - 37294257946660910 T^{17} + 229433732838622222 T^{18} + 1887491713404577050 T^{19} + 28602470323180973016 T^{20} +$$$$23\!\cdots\!55$$$$T^{21} -$$$$47\!\cdots\!08$$$$T^{22} -$$$$24\!\cdots\!10$$$$T^{23} -$$$$80\!\cdots\!74$$$$T^{24} +$$$$89\!\cdots\!40$$$$T^{25} +$$$$25\!\cdots\!54$$$$T^{26} +$$$$31\!\cdots\!10$$$$T^{27} +$$$$42\!\cdots\!09$$$$T^{28} -$$$$11\!\cdots\!80$$$$T^{29} -$$$$22\!\cdots\!56$$$$T^{30} -$$$$44\!\cdots\!85$$$$T^{31} +$$$$65\!\cdots\!61$$$$T^{32}$$
$79$ $$1 - 20 T + 72 T^{2} + 240 T^{3} + 3616 T^{4} + 52220 T^{5} - 1218607 T^{6} - 3975630 T^{7} + 89503171 T^{8} + 339499660 T^{9} + 541594996 T^{10} - 86409648250 T^{11} - 192249239601 T^{12} + 5478908409570 T^{13} + 38059098256319 T^{14} + 34437029350830 T^{15} - 7141755789421758 T^{16} + 2720525318715570 T^{17} + 237526832217686879 T^{18} + 2701315523345983230 T^{19} - 7488123454647357681 T^{20} -$$$$26\!\cdots\!50$$$$T^{21} +$$$$13\!\cdots\!16$$$$T^{22} +$$$$65\!\cdots\!40$$$$T^{23} +$$$$13\!\cdots\!31$$$$T^{24} -$$$$47\!\cdots\!70$$$$T^{25} -$$$$11\!\cdots\!07$$$$T^{26} +$$$$39\!\cdots\!80$$$$T^{27} +$$$$21\!\cdots\!56$$$$T^{28} +$$$$11\!\cdots\!60$$$$T^{29} +$$$$26\!\cdots\!32$$$$T^{30} -$$$$58\!\cdots\!80$$$$T^{31} +$$$$23\!\cdots\!21$$$$T^{32}$$
$83$ $$1 - 17 T - 70 T^{2} + 2509 T^{3} - 7605 T^{4} + 46949 T^{5} - 1021496 T^{6} - 21178921 T^{7} + 317998444 T^{8} + 567110603 T^{9} - 13415734458 T^{10} - 13379267583 T^{11} - 1499536740482 T^{12} + 16468540679846 T^{13} + 120096814767848 T^{14} - 1133667594299970 T^{15} - 1727008891553813 T^{16} - 94094410326897510 T^{17} + 827346956935704872 T^{18} + 9416497469707104802 T^{19} - 71165495981088450722 T^{20} - 52701478783009375869 T^{21} -$$$$43\!\cdots\!02$$$$T^{22} +$$$$15\!\cdots\!81$$$$T^{23} +$$$$71\!\cdots\!04$$$$T^{24} -$$$$39\!\cdots\!63$$$$T^{25} -$$$$15\!\cdots\!04$$$$T^{26} +$$$$60\!\cdots\!83$$$$T^{27} -$$$$81\!\cdots\!05$$$$T^{28} +$$$$22\!\cdots\!67$$$$T^{29} -$$$$51\!\cdots\!30$$$$T^{30} -$$$$10\!\cdots\!19$$$$T^{31} +$$$$50\!\cdots\!81$$$$T^{32}$$
$89$ $$1 - 27 T - 73 T^{2} + 5791 T^{3} + 20328 T^{4} - 941897 T^{5} - 2350459 T^{6} + 90406805 T^{7} + 102589649 T^{8} - 6634669403 T^{9} + 39515779247 T^{10} + 388019762329 T^{11} - 7800813705650 T^{12} - 29920799249404 T^{13} + 911312691296408 T^{14} + 1205098080379952 T^{15} - 84382434058483177 T^{16} + 107253729153815728 T^{17} + 7218507827758847768 T^{18} - 21093235926053088476 T^{19} -$$$$48\!\cdots\!50$$$$T^{20} +$$$$21\!\cdots\!21$$$$T^{21} +$$$$19\!\cdots\!67$$$$T^{22} -$$$$29\!\cdots\!87$$$$T^{23} +$$$$40\!\cdots\!69$$$$T^{24} +$$$$31\!\cdots\!45$$$$T^{25} -$$$$73\!\cdots\!59$$$$T^{26} -$$$$26\!\cdots\!33$$$$T^{27} +$$$$50\!\cdots\!88$$$$T^{28} +$$$$12\!\cdots\!79$$$$T^{29} -$$$$14\!\cdots\!93$$$$T^{30} -$$$$47\!\cdots\!23$$$$T^{31} +$$$$15\!\cdots\!61$$$$T^{32}$$
$97$ $$1 - 6 T - 226 T^{2} + 3602 T^{3} + 30196 T^{4} - 833820 T^{5} - 1681890 T^{6} + 150728346 T^{7} - 383845095 T^{8} - 20809464676 T^{9} + 138289506334 T^{10} + 2169694559068 T^{11} - 24997900175828 T^{12} - 165816523144770 T^{13} + 3403873171920092 T^{14} + 6407107447256686 T^{15} - 366221833845958995 T^{16} + 621489422383898542 T^{17} + 32027042674596145628 T^{18} -$$$$15\!\cdots\!10$$$$T^{19} -$$$$22\!\cdots\!68$$$$T^{20} +$$$$18\!\cdots\!76$$$$T^{21} +$$$$11\!\cdots\!86$$$$T^{22} -$$$$16\!\cdots\!88$$$$T^{23} -$$$$30\!\cdots\!95$$$$T^{24} +$$$$11\!\cdots\!82$$$$T^{25} -$$$$12\!\cdots\!10$$$$T^{26} -$$$$59\!\cdots\!60$$$$T^{27} +$$$$20\!\cdots\!36$$$$T^{28} +$$$$24\!\cdots\!54$$$$T^{29} -$$$$14\!\cdots\!94$$$$T^{30} -$$$$37\!\cdots\!58$$$$T^{31} +$$$$61\!\cdots\!21$$$$T^{32}$$