# Properties

 Label 336.2.w.a Level 336 Weight 2 Character orbit 336.w Analytic conductor 2.683 Analytic rank 0 Dimension 20 CM no Inner twists 2

# Related objects

## Newspace parameters

 Level: $$N$$ = $$336 = 2^{4} \cdot 3 \cdot 7$$ Weight: $$k$$ = $$2$$ Character orbit: $$[\chi]$$ = 336.w (of order $$4$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$2.68297350792$$ Analytic rank: $$0$$ Dimension: $$20$$ Relative dimension: $$10$$ over $$\Q(i)$$ Coefficient field: $$\mathbb{Q}[x]/(x^{20} - \cdots)$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2^{7}$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{19}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

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

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{20} - 4 x^{19} + 8 x^{18} - 16 x^{17} + 35 x^{16} - 56 x^{15} + 64 x^{14} - 84 x^{13} + 125 x^{12} - 120 x^{11} + 100 x^{10} - 240 x^{9} + 500 x^{8} - 672 x^{7} + 1024 x^{6} - 1792 x^{5} + 2240 x^{4} - 2048 x^{3} + 2048 x^{2} - 2048 x + 1024$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$-2 \nu^{19} + 35 \nu^{18} + 12 \nu^{17} - 424 \nu^{16} + 978 \nu^{15} - 1247 \nu^{14} + 2640 \nu^{13} - 4952 \nu^{12} + 4162 \nu^{11} - 1249 \nu^{10} + 4632 \nu^{9} - 8180 \nu^{8} + 1616 \nu^{7} - 8740 \nu^{6} + 47552 \nu^{5} - 70528 \nu^{4} + 63520 \nu^{3} - 101568 \nu^{2} + 142464 \nu - 65792$$$$)/8960$$ $$\beta_{2}$$ $$=$$ $$($$$$\nu^{19} - 4 \nu^{18} - 34 \nu^{17} - 160 \nu^{16} + 519 \nu^{15} - 472 \nu^{14} + 890 \nu^{13} - 2444 \nu^{12} + 2505 \nu^{11} - 32 \nu^{10} + 1258 \nu^{9} - 5448 \nu^{8} + 1856 \nu^{7} - 400 \nu^{6} + 19104 \nu^{5} - 37312 \nu^{4} + 28032 \nu^{3} - 36352 \nu^{2} + 66560 \nu - 38912$$$$)/2560$$ $$\beta_{3}$$ $$=$$ $$($$$$-11 \nu^{19} + 7 \nu^{18} + 500 \nu^{17} - 1786 \nu^{16} + 3055 \nu^{15} - 4867 \nu^{14} + 9760 \nu^{13} - 13586 \nu^{12} + 9213 \nu^{11} - 6845 \nu^{10} + 17244 \nu^{9} - 20406 \nu^{8} + 14796 \nu^{7} - 60600 \nu^{6} + 156816 \nu^{5} - 206016 \nu^{4} + 211264 \nu^{3} - 297216 \nu^{2} + 327936 \nu - 156672$$$$)/17920$$ $$\beta_{4}$$ $$=$$ $$($$$$-3 \nu^{19} + 168 \nu^{18} + 74 \nu^{17} - 692 \nu^{16} + 151 \nu^{15} - 292 \nu^{14} + 3190 \nu^{13} - 3368 \nu^{12} - 2759 \nu^{11} + 972 \nu^{10} + 9230 \nu^{9} + 1156 \nu^{8} - 12724 \nu^{7} - 20320 \nu^{6} + 50888 \nu^{5} - 12720 \nu^{4} + 17216 \nu^{3} - 93440 \nu^{2} + 37632 \nu + 36608$$$$)/8960$$ $$\beta_{5}$$ $$=$$ $$($$$$-11 \nu^{19} + 14 \nu^{18} + 80 \nu^{17} - 176 \nu^{16} + 255 \nu^{15} - 562 \nu^{14} + 1024 \nu^{13} - 1140 \nu^{12} + 729 \nu^{11} - 958 \nu^{10} + 1620 \nu^{9} - 1240 \nu^{8} + 2308 \nu^{7} - 7288 \nu^{6} + 14240 \nu^{5} - 18752 \nu^{4} + 24448 \nu^{3} - 27520 \nu^{2} + 25088 \nu - 20480$$$$)/3584$$ $$\beta_{6}$$ $$=$$ $$($$$$37 \nu^{19} - 588 \nu^{18} + 1962 \nu^{17} - 3440 \nu^{16} + 5483 \nu^{15} - 10464 \nu^{14} + 14510 \nu^{13} - 10588 \nu^{12} + 8165 \nu^{11} - 18344 \nu^{10} + 23046 \nu^{9} - 20296 \nu^{8} + 67992 \nu^{7} - 168080 \nu^{6} + 225728 \nu^{5} - 235904 \nu^{4} + 319744 \nu^{3} - 350464 \nu^{2} + 179200 \nu - 11264$$$$)/17920$$ $$\beta_{7}$$ $$=$$ $$($$$$-269 \nu^{19} + 224 \nu^{18} + 382 \nu^{17} + 1464 \nu^{16} - 4707 \nu^{15} + 2284 \nu^{14} - 3110 \nu^{13} + 19956 \nu^{12} - 22877 \nu^{11} - 3844 \nu^{10} - 8270 \nu^{9} + 63008 \nu^{8} - 25912 \nu^{7} - 34320 \nu^{6} - 129376 \nu^{5} + 296960 \nu^{4} - 171392 \nu^{3} + 202240 \nu^{2} - 523264 \nu + 400384$$$$)/35840$$ $$\beta_{8}$$ $$=$$ $$($$$$-15 \nu^{19} + 84 \nu^{18} - 176 \nu^{17} + 320 \nu^{16} - 589 \nu^{15} + 864 \nu^{14} - 1032 \nu^{13} + 1052 \nu^{12} - 1139 \nu^{11} + 1360 \nu^{10} - 1940 \nu^{9} + 3904 \nu^{8} - 7340 \nu^{7} + 12752 \nu^{6} - 19232 \nu^{5} + 24544 \nu^{4} - 25024 \nu^{3} + 23808 \nu^{2} - 21504 \nu + 5632$$$$)/1792$$ $$\beta_{9}$$ $$=$$ $$($$$$86 \nu^{19} - 84 \nu^{18} - 19 \nu^{17} - 500 \nu^{16} + 884 \nu^{15} + 148 \nu^{14} + 195 \nu^{13} - 2944 \nu^{12} + 1620 \nu^{11} + 3328 \nu^{10} + 3173 \nu^{9} - 12848 \nu^{8} + 1926 \nu^{7} + 3280 \nu^{6} + 28664 \nu^{5} - 35872 \nu^{4} - 10208 \nu^{3} - 12672 \nu^{2} + 53760 \nu - 512$$$$)/8960$$ $$\beta_{10}$$ $$=$$ $$($$$$-179 \nu^{19} + 539 \nu^{18} - 1138 \nu^{17} + 2134 \nu^{16} - 3357 \nu^{15} + 4569 \nu^{14} - 5850 \nu^{13} + 5286 \nu^{12} - 4087 \nu^{11} + 6511 \nu^{10} - 14410 \nu^{9} + 22098 \nu^{8} - 38992 \nu^{7} + 83880 \nu^{6} - 116016 \nu^{5} + 115200 \nu^{4} - 114432 \nu^{3} + 142720 \nu^{2} - 66304 \nu - 34816$$$$)/17920$$ $$\beta_{11}$$ $$=$$ $$($$$$-202 \nu^{19} + 679 \nu^{18} - 1840 \nu^{17} + 4818 \nu^{16} - 8070 \nu^{15} + 11001 \nu^{14} - 17700 \nu^{13} + 24078 \nu^{12} - 18734 \nu^{11} + 14215 \nu^{10} - 36812 \nu^{9} + 60118 \nu^{8} - 71928 \nu^{7} + 154800 \nu^{6} - 325168 \nu^{5} + 415168 \nu^{4} - 394752 \nu^{3} + 496768 \nu^{2} - 517888 \nu + 216576$$$$)/17920$$ $$\beta_{12}$$ $$=$$ $$($$$$-451 \nu^{19} + 1176 \nu^{18} - 822 \nu^{17} + 1576 \nu^{16} - 4973 \nu^{15} + 3516 \nu^{14} + 2910 \nu^{13} + 2764 \nu^{12} - 14323 \nu^{11} + 1084 \nu^{10} + 1110 \nu^{9} + 52032 \nu^{8} - 80008 \nu^{7} + 23920 \nu^{6} - 41344 \nu^{5} + 135680 \nu^{4} - 37888 \nu^{3} - 84480 \nu^{2} - 32256 \nu + 70656$$$$)/35840$$ $$\beta_{13}$$ $$=$$ $$($$$$-237 \nu^{19} + 210 \nu^{18} + 162 \nu^{17} + 576 \nu^{16} - 1987 \nu^{15} + 578 \nu^{14} + 430 \nu^{13} + 7628 \nu^{12} - 11013 \nu^{11} - 394 \nu^{10} - 3618 \nu^{9} + 30480 \nu^{8} - 20744 \nu^{7} - 21320 \nu^{6} - 33968 \nu^{5} + 115232 \nu^{4} - 61440 \nu^{3} + 61312 \nu^{2} - 184576 \nu + 227328$$$$)/17920$$ $$\beta_{14}$$ $$=$$ $$($$$$-121 \nu^{19} + 224 \nu^{18} - 261 \nu^{17} + 1060 \nu^{16} - 2109 \nu^{15} + 1812 \nu^{14} - 2435 \nu^{13} + 5884 \nu^{12} - 5995 \nu^{11} + 872 \nu^{10} - 6673 \nu^{9} + 21248 \nu^{8} - 19426 \nu^{7} + 20240 \nu^{6} - 64504 \nu^{5} + 98592 \nu^{4} - 68192 \nu^{3} + 84352 \nu^{2} - 125440 \nu + 72192$$$$)/8960$$ $$\beta_{15}$$ $$=$$ $$($$$$-260 \nu^{19} + 707 \nu^{18} - 876 \nu^{17} + 2126 \nu^{16} - 4964 \nu^{15} + 5309 \nu^{14} - 4280 \nu^{13} + 9130 \nu^{12} - 13648 \nu^{11} + 5427 \nu^{10} - 7792 \nu^{9} + 41074 \nu^{8} - 61072 \nu^{7} + 59400 \nu^{6} - 118240 \nu^{5} + 202848 \nu^{4} - 163456 \nu^{3} + 122368 \nu^{2} - 175616 \nu + 105984$$$$)/17920$$ $$\beta_{16}$$ $$=$$ $$($$$$-38 \nu^{19} + 151 \nu^{18} - 300 \nu^{17} + 642 \nu^{16} - 1170 \nu^{15} + 1609 \nu^{14} - 1960 \nu^{13} + 2302 \nu^{12} - 2306 \nu^{11} + 2055 \nu^{10} - 3768 \nu^{9} + 7862 \nu^{8} - 13552 \nu^{7} + 23680 \nu^{6} - 38512 \nu^{5} + 48992 \nu^{4} - 47808 \nu^{3} + 49792 \nu^{2} - 41472 \nu + 11264$$$$)/2560$$ $$\beta_{17}$$ $$=$$ $$($$$$-393 \nu^{19} + 1204 \nu^{18} - 1814 \nu^{17} + 3736 \nu^{16} - 7631 \nu^{15} + 9320 \nu^{14} - 8130 \nu^{13} + 12532 \nu^{12} - 17953 \nu^{11} + 11888 \nu^{10} - 17186 \nu^{9} + 59008 \nu^{8} - 99600 \nu^{7} + 122960 \nu^{6} - 200672 \nu^{5} + 293344 \nu^{4} - 258432 \nu^{3} + 217344 \nu^{2} - 220416 \nu + 102400$$$$)/17920$$ $$\beta_{18}$$ $$=$$ $$($$$$-279 \nu^{19} + 938 \nu^{18} - 1595 \nu^{17} + 3236 \nu^{16} - 6495 \nu^{15} + 8362 \nu^{14} - 8285 \nu^{13} + 11436 \nu^{12} - 15073 \nu^{11} + 10890 \nu^{10} - 16239 \nu^{9} + 49296 \nu^{8} - 82666 \nu^{7} + 112520 \nu^{6} - 185056 \nu^{5} + 265536 \nu^{4} - 245344 \nu^{3} + 214976 \nu^{2} - 220416 \nu + 105472$$$$)/8960$$ $$\beta_{19}$$ $$=$$ $$($$$$-610 \nu^{19} + 1799 \nu^{18} - 2892 \nu^{17} + 6522 \nu^{16} - 13238 \nu^{15} + 15753 \nu^{14} - 16040 \nu^{13} + 26070 \nu^{12} - 31806 \nu^{11} + 17159 \nu^{10} - 32264 \nu^{9} + 104718 \nu^{8} - 160304 \nu^{7} + 204160 \nu^{6} - 375280 \nu^{5} + 551616 \nu^{4} - 482432 \nu^{3} + 425216 \nu^{2} - 476672 \nu + 252928$$$$)/17920$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$-\beta_{19} + \beta_{15} - \beta_{14} + \beta_{13} + \beta_{10} - \beta_{9} - \beta_{6} - \beta_{5} + \beta_{3} - \beta_{2} + \beta_{1}$$$$)/2$$ $$\nu^{2}$$ $$=$$ $$($$$$-\beta_{18} + \beta_{17} + 2 \beta_{16} - \beta_{15} + \beta_{13} - 2 \beta_{12} - \beta_{11} + \beta_{10} - \beta_{9} - \beta_{5} + \beta_{4} + \beta_{3} - \beta_{1}$$$$)/2$$ $$\nu^{3}$$ $$=$$ $$($$$$-\beta_{19} - \beta_{18} + 3 \beta_{17} + \beta_{14} - \beta_{11} - 2 \beta_{9} - 2 \beta_{7} - 2 \beta_{5} + \beta_{4} - 2 \beta_{3} + 2$$$$)/2$$ $$\nu^{4}$$ $$=$$ $$($$$$-\beta_{19} + \beta_{18} - \beta_{17} + 2 \beta_{16} + 2 \beta_{15} - 3 \beta_{14} + 2 \beta_{13} - 2 \beta_{12} + \beta_{11} - 2 \beta_{9} - \beta_{8} - 2 \beta_{7} + \beta_{6} + \beta_{4}$$$$)/2$$ $$\nu^{5}$$ $$=$$ $$($$$$-3 \beta_{18} + 3 \beta_{17} + 6 \beta_{16} - \beta_{15} + 4 \beta_{14} - 3 \beta_{13} - 3 \beta_{11} + \beta_{10} + 3 \beta_{9} + 2 \beta_{7} + 5 \beta_{6} - 3 \beta_{5} + \beta_{4} - 5 \beta_{3} + 3 \beta_{2} - \beta_{1} - 4$$$$)/2$$ $$\nu^{6}$$ $$=$$ $$($$$$\beta_{19} + 3 \beta_{18} + 2 \beta_{17} - 4 \beta_{16} - \beta_{15} + 5 \beta_{14} - 3 \beta_{13} - 2 \beta_{11} - 2 \beta_{10} + 3 \beta_{9} - 6 \beta_{7} + 4 \beta_{6} + \beta_{5} - 2 \beta_{4} - 6 \beta_{3} - \beta_{2} + \beta_{1} + 4$$$$)/2$$ $$\nu^{7}$$ $$=$$ $$($$$$-7 \beta_{19} + 5 \beta_{18} - 2 \beta_{16} + 9 \beta_{15} - 5 \beta_{14} - 3 \beta_{13} + 2 \beta_{12} + 2 \beta_{11} + 4 \beta_{10} + 3 \beta_{9} - \beta_{8} + 6 \beta_{7} + 2 \beta_{6} - \beta_{5} - 4 \beta_{4} - 2 \beta_{2} + 5 \beta_{1} + 8$$$$)/2$$ $$\nu^{8}$$ $$=$$ $$($$$$-6 \beta_{19} + 4 \beta_{18} + 3 \beta_{17} + 2 \beta_{16} - 7 \beta_{15} + 10 \beta_{14} - 3 \beta_{13} - 11 \beta_{11} + 4 \beta_{10} + \beta_{9} + 3 \beta_{8} + 10 \beta_{7} - \beta_{6} - 5 \beta_{5} - \beta_{4} - 8 \beta_{3} - \beta_{2} + 9 \beta_{1} - 14$$$$)/2$$ $$\nu^{9}$$ $$=$$ $$($$$$5 \beta_{19} - 5 \beta_{18} + 21 \beta_{17} - 2 \beta_{16} - 12 \beta_{15} + 15 \beta_{14} - 10 \beta_{13} - 22 \beta_{12} - 11 \beta_{11} - 2 \beta_{10} - 2 \beta_{9} + 8 \beta_{7} + 2 \beta_{6} + 4 \beta_{5} + 7 \beta_{4} - 14 \beta_{3} + 6 \beta_{2} + 2 \beta_{1} + 16$$$$)/2$$ $$\nu^{10}$$ $$=$$ $$($$$$-17 \beta_{19} + 6 \beta_{18} + 11 \beta_{17} - 12 \beta_{16} + 26 \beta_{15} - 7 \beta_{14} + 14 \beta_{13} - 18 \beta_{12} - 5 \beta_{11} - 7 \beta_{10} - 12 \beta_{9} + 20 \beta_{8} - 10 \beta_{7} - 4 \beta_{6} - 2 \beta_{5} - 7 \beta_{4} + 3 \beta_{3} - 7 \beta_{2} + 8 \beta_{1} + 8$$$$)/2$$ $$\nu^{11}$$ $$=$$ $$($$$$22 \beta_{19} - 10 \beta_{18} - 7 \beta_{17} + 50 \beta_{16} - 27 \beta_{15} - 6 \beta_{14} + 17 \beta_{13} - 42 \beta_{12} - 33 \beta_{11} - 16 \beta_{10} + \beta_{9} + 27 \beta_{8} + 12 \beta_{7} + 26 \beta_{6} + 17 \beta_{5} + 3 \beta_{4} - 18 \beta_{3} + 22 \beta_{2} - 19 \beta_{1} - 10$$$$)/2$$ $$\nu^{12}$$ $$=$$ $$($$$$31 \beta_{19} - 33 \beta_{18} + 60 \beta_{17} - 4 \beta_{16} - 41 \beta_{15} + 21 \beta_{14} - 5 \beta_{13} - 22 \beta_{12} - 40 \beta_{11} - 12 \beta_{10} + 7 \beta_{9} + 32 \beta_{8} - 20 \beta_{7} + 26 \beta_{6} - 9 \beta_{5} - 6 \beta_{4} - 44 \beta_{3} - \beta_{2} + \beta_{1} - 14$$$$)/2$$ $$\nu^{13}$$ $$=$$ $$($$$$\beta_{19} + 42 \beta_{18} + 14 \beta_{17} - 16 \beta_{16} + 13 \beta_{15} - 65 \beta_{14} + 41 \beta_{13} - 54 \beta_{12} + 4 \beta_{11} - 35 \beta_{10} - 5 \beta_{9} + 24 \beta_{8} - 66 \beta_{7} + 37 \beta_{6} + 7 \beta_{5} - 38 \beta_{4} + 47 \beta_{3} - 29 \beta_{2} - 29 \beta_{1} + 36$$$$)/2$$ $$\nu^{14}$$ $$=$$ $$($$$$-30 \beta_{19} + 11 \beta_{18} + 61 \beta_{17} + 64 \beta_{16} - 77 \beta_{15} + 8 \beta_{14} - 55 \beta_{13} + 86 \beta_{12} - 47 \beta_{11} - 25 \beta_{10} + 33 \beta_{9} - 28 \beta_{8} + 84 \beta_{7} + 40 \beta_{6} - 27 \beta_{5} - 17 \beta_{4} - 123 \beta_{3} + 38 \beta_{2} + 31 \beta_{1} - 76$$$$)/2$$ $$\nu^{15}$$ $$=$$ $$($$$$37 \beta_{19} + 49 \beta_{18} + 117 \beta_{17} - 76 \beta_{16} - 240 \beta_{15} + 95 \beta_{14} - 88 \beta_{13} + 28 \beta_{12} + 25 \beta_{11} - 60 \beta_{10} + 26 \beta_{9} - 120 \beta_{8} + 22 \beta_{7} - 8 \beta_{6} + 78 \beta_{5} + 35 \beta_{4} - 154 \beta_{3} + 8 \beta_{2} + 108 \beta_{1} - 138$$$$)/2$$ $$\nu^{16}$$ $$=$$ $$($$$$-95 \beta_{19} + 15 \beta_{18} + 217 \beta_{17} + 2 \beta_{16} + 42 \beta_{15} + 119 \beta_{14} - 182 \beta_{13} - 46 \beta_{12} + 115 \beta_{11} - 120 \beta_{10} + 98 \beta_{9} - 115 \beta_{8} + 98 \beta_{7} + 83 \beta_{6} + 8 \beta_{5} - 5 \beta_{4} - 20 \beta_{3} + 76 \beta_{2} + 36 \beta_{1} + 120$$$$)/2$$ $$\nu^{17}$$ $$=$$ $$($$$$-104 \beta_{19} + 143 \beta_{18} - 87 \beta_{17} + 66 \beta_{16} - 11 \beta_{15} + 164 \beta_{14} - 49 \beta_{13} + 168 \beta_{12} - 17 \beta_{11} - 253 \beta_{10} + 137 \beta_{9} - 24 \beta_{8} + 30 \beta_{7} - \beta_{6} + 207 \beta_{5} - 109 \beta_{4} - 199 \beta_{3} + 25 \beta_{2} + 157 \beta_{1} - 92$$$$)/2$$ $$\nu^{18}$$ $$=$$ $$($$$$235 \beta_{19} - 167 \beta_{18} + 30 \beta_{17} + 180 \beta_{16} - 459 \beta_{15} - 41 \beta_{14} - 49 \beta_{13} - 48 \beta_{12} + 106 \beta_{11} + 2 \beta_{10} + 201 \beta_{9} - 56 \beta_{8} + 286 \beta_{7} + 4 \beta_{6} + 435 \beta_{5} + 66 \beta_{4} + 6 \beta_{3} + 5 \beta_{2} + 99 \beta_{1} + 236$$$$)/2$$ $$\nu^{19}$$ $$=$$ $$($$$$-245 \beta_{19} - 129 \beta_{18} + 376 \beta_{17} - 6 \beta_{16} + 91 \beta_{15} - 31 \beta_{14} + 215 \beta_{13} - 218 \beta_{12} - 26 \beta_{11} - 4 \beta_{10} - 175 \beta_{9} + 365 \beta_{8} - 334 \beta_{7} - 74 \beta_{6} - 475 \beta_{5} - 116 \beta_{4} + 752 \beta_{3} - 470 \beta_{2} + 7 \beta_{1} + 344$$$$)/2$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/336\mathbb{Z}\right)^\times$$.

 $$n$$ $$85$$ $$113$$ $$127$$ $$241$$ $$\chi(n)$$ $$-\beta_{15}$$ $$1$$ $$1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
85.1
 1.10050 + 0.888196i 1.07232 − 0.922026i −0.603113 − 1.27916i 0.752388 + 1.19746i −1.32974 − 0.481443i 1.35964 − 0.389081i −0.545677 − 1.30470i −0.423640 + 1.34927i −0.792380 − 1.17138i 1.40970 + 0.112864i 1.10050 − 0.888196i 1.07232 + 0.922026i −0.603113 + 1.27916i 0.752388 − 1.19746i −1.32974 + 0.481443i 1.35964 + 0.389081i −0.545677 + 1.30470i −0.423640 − 1.34927i −0.792380 + 1.17138i 1.40970 − 0.112864i
−1.33348 + 0.470984i −0.707107 + 0.707107i 1.55635 1.25610i 2.01011 + 2.01011i 0.609878 1.27595i 1.00000i −1.48376 + 2.40800i 1.00000i −3.62718 1.73372i
85.2 −1.32599 0.491687i −0.707107 + 0.707107i 1.51649 + 1.30394i −1.17321 1.17321i 1.28529 0.589940i 1.00000i −1.36971 2.47465i 1.00000i 0.978808 + 2.13251i
85.3 −1.19435 + 0.757321i 0.707107 0.707107i 0.852931 1.80901i −0.894131 0.894131i −0.309024 + 1.38004i 1.00000i 0.351303 + 2.80653i 1.00000i 1.74505 + 0.390759i
85.4 −0.684092 1.23775i 0.707107 0.707107i −1.06404 + 1.69347i 1.13147 + 1.13147i −1.35895 0.391494i 1.00000i 2.82398 + 0.158525i 1.00000i 0.626447 2.17451i
85.5 −0.244399 1.39294i −0.707107 + 0.707107i −1.88054 + 0.680863i −3.00215 3.00215i 1.15777 + 0.812138i 1.00000i 1.40800 + 2.45307i 1.00000i −3.44808 + 4.91552i
85.6 0.196445 1.40050i 0.707107 0.707107i −1.92282 0.550244i −1.69093 1.69093i −0.851398 1.12921i 1.00000i −1.14835 + 2.58482i 1.00000i −2.70032 + 2.03597i
85.7 0.783676 + 1.17722i −0.707107 + 0.707107i −0.771704 + 1.84512i 0.134119 + 0.134119i −1.38656 0.278279i 1.00000i −2.77688 + 0.537511i 1.00000i −0.0527819 + 0.262993i
85.8 1.13998 + 0.836924i 0.707107 0.707107i 0.599117 + 1.90816i 1.18844 + 1.18844i 1.39788 0.214294i 1.00000i −0.913999 + 2.67668i 1.00000i 0.360165 + 2.34943i
85.9 1.24912 0.663101i 0.707107 0.707107i 1.12060 1.65658i 1.67936 + 1.67936i 0.414377 1.35214i 1.00000i 0.301275 2.81234i 1.00000i 3.21131 + 0.984136i
85.10 1.41309 0.0564773i −0.707107 + 0.707107i 1.99362 0.159615i 0.616911 + 0.616911i −0.959267 + 1.03914i 1.00000i 2.80814 0.338143i 1.00000i 0.906590 + 0.836907i
253.1 −1.33348 0.470984i −0.707107 0.707107i 1.55635 + 1.25610i 2.01011 2.01011i 0.609878 + 1.27595i 1.00000i −1.48376 2.40800i 1.00000i −3.62718 + 1.73372i
253.2 −1.32599 + 0.491687i −0.707107 0.707107i 1.51649 1.30394i −1.17321 + 1.17321i 1.28529 + 0.589940i 1.00000i −1.36971 + 2.47465i 1.00000i 0.978808 2.13251i
253.3 −1.19435 0.757321i 0.707107 + 0.707107i 0.852931 + 1.80901i −0.894131 + 0.894131i −0.309024 1.38004i 1.00000i 0.351303 2.80653i 1.00000i 1.74505 0.390759i
253.4 −0.684092 + 1.23775i 0.707107 + 0.707107i −1.06404 1.69347i 1.13147 1.13147i −1.35895 + 0.391494i 1.00000i 2.82398 0.158525i 1.00000i 0.626447 + 2.17451i
253.5 −0.244399 + 1.39294i −0.707107 0.707107i −1.88054 0.680863i −3.00215 + 3.00215i 1.15777 0.812138i 1.00000i 1.40800 2.45307i 1.00000i −3.44808 4.91552i
253.6 0.196445 + 1.40050i 0.707107 + 0.707107i −1.92282 + 0.550244i −1.69093 + 1.69093i −0.851398 + 1.12921i 1.00000i −1.14835 2.58482i 1.00000i −2.70032 2.03597i
253.7 0.783676 1.17722i −0.707107 0.707107i −0.771704 1.84512i 0.134119 0.134119i −1.38656 + 0.278279i 1.00000i −2.77688 0.537511i 1.00000i −0.0527819 0.262993i
253.8 1.13998 0.836924i 0.707107 + 0.707107i 0.599117 1.90816i 1.18844 1.18844i 1.39788 + 0.214294i 1.00000i −0.913999 2.67668i 1.00000i 0.360165 2.34943i
253.9 1.24912 + 0.663101i 0.707107 + 0.707107i 1.12060 + 1.65658i 1.67936 1.67936i 0.414377 + 1.35214i 1.00000i 0.301275 + 2.81234i 1.00000i 3.21131 0.984136i
253.10 1.41309 + 0.0564773i −0.707107 0.707107i 1.99362 + 0.159615i 0.616911 0.616911i −0.959267 1.03914i 1.00000i 2.80814 + 0.338143i 1.00000i 0.906590 0.836907i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 253.10 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
16.e even 4 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 336.2.w.a 20
4.b odd 2 1 1344.2.w.a 20
8.b even 2 1 2688.2.w.a 20
8.d odd 2 1 2688.2.w.b 20
16.e even 4 1 inner 336.2.w.a 20
16.e even 4 1 2688.2.w.a 20
16.f odd 4 1 1344.2.w.a 20
16.f odd 4 1 2688.2.w.b 20

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
336.2.w.a 20 1.a even 1 1 trivial
336.2.w.a 20 16.e even 4 1 inner
1344.2.w.a 20 4.b odd 2 1
1344.2.w.a 20 16.f odd 4 1
2688.2.w.a 20 8.b even 2 1
2688.2.w.a 20 16.e even 4 1
2688.2.w.b 20 8.d odd 2 1
2688.2.w.b 20 16.f odd 4 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{20} - \cdots$$ acting on $$S_{2}^{\mathrm{new}}(336, [\chi])$$.

## Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ $$1 - 2 T^{2} + 2 T^{4} + 4 T^{6} - 8 T^{7} - 4 T^{8} - 16 T^{12} - 64 T^{13} + 64 T^{14} + 128 T^{16} - 512 T^{18} + 1024 T^{20}$$
$3$ $$( 1 + T^{4} )^{5}$$
$5$ $$1 - 24 T^{3} + 10 T^{4} + 24 T^{5} + 288 T^{6} - 320 T^{7} + 485 T^{8} - 2720 T^{9} + 5088 T^{10} - 20208 T^{11} + 28392 T^{12} - 21904 T^{13} + 280352 T^{14} - 374560 T^{15} + 85402 T^{16} - 2818304 T^{17} + 5181408 T^{18} - 7099744 T^{19} + 30439356 T^{20} - 35498720 T^{21} + 129535200 T^{22} - 352288000 T^{23} + 53376250 T^{24} - 1170500000 T^{25} + 4380500000 T^{26} - 1711250000 T^{27} + 11090625000 T^{28} - 39468750000 T^{29} + 49687500000 T^{30} - 132812500000 T^{31} + 118408203125 T^{32} - 390625000000 T^{33} + 1757812500000 T^{34} + 732421875000 T^{35} + 1525878906250 T^{36} - 18310546875000 T^{37} + 95367431640625 T^{40}$$
$7$ $$( 1 + T^{2} )^{10}$$
$11$ $$1 - 12 T + 72 T^{2} - 404 T^{3} + 2214 T^{4} - 9404 T^{5} + 35048 T^{6} - 135156 T^{7} + 456709 T^{8} - 1318720 T^{9} + 4165952 T^{10} - 13629648 T^{11} + 41111256 T^{12} - 141076784 T^{13} + 543949056 T^{14} - 1968344544 T^{15} + 6732047034 T^{16} - 23311377560 T^{17} + 80768058832 T^{18} - 261514302488 T^{19} + 834968255588 T^{20} - 2876657327368 T^{21} + 9772935118672 T^{22} - 31027443532360 T^{23} + 98563900624794 T^{24} - 317003857155744 T^{25} + 963638933596416 T^{26} - 2749187413938064 T^{27} + 8812562832664536 T^{28} - 32137997030742768 T^{29} + 108054065891385152 T^{30} - 376246206268137920 T^{31} + 1433348485503871189 T^{32} - 4665951682525138236 T^{33} + 13309472167425430568 T^{34} - 39282841785184782004 T^{35} +$$$$10\!\cdots\!54$$$$T^{36} -$$$$20\!\cdots\!84$$$$T^{37} +$$$$40\!\cdots\!32$$$$T^{38} -$$$$73\!\cdots\!92$$$$T^{39} +$$$$67\!\cdots\!01$$$$T^{40}$$
$13$ $$1 - 80 T^{3} - 118 T^{4} + 1616 T^{5} + 3200 T^{6} + 22560 T^{7} - 156451 T^{8} - 297120 T^{9} - 121472 T^{10} + 6082400 T^{11} + 43092280 T^{12} - 92564192 T^{13} - 225951616 T^{14} - 2287891296 T^{15} + 3788315570 T^{16} + 30959844064 T^{17} + 31244658048 T^{18} + 7310121920 T^{19} - 2235843760900 T^{20} + 95031584960 T^{21} + 5280347210112 T^{22} + 68018777408608 T^{23} + 108198080994770 T^{24} - 849478022965728 T^{25} - 1090625293673344 T^{26} - 5808265775303264 T^{27} + 35151696633933880 T^{28} + 64500806986335200 T^{29} - 16745946721881728 T^{30} - 532486696276273440 T^{31} - 3645008715497274931 T^{32} + 6832862404721227680 T^{33} + 12599604434237724800 T^{34} + 82716403110770663312 T^{35} - 78519159883615221238 T^{36} -$$$$69\!\cdots\!40$$$$T^{37} +$$$$19\!\cdots\!01$$$$T^{40}$$
$17$ $$( 1 + 94 T^{2} + 88 T^{3} + 4293 T^{4} + 6432 T^{5} + 134312 T^{6} + 219984 T^{7} + 3234206 T^{8} + 4965576 T^{9} + 61650188 T^{10} + 84414792 T^{11} + 934685534 T^{12} + 1080781392 T^{13} + 11217872552 T^{14} + 9132520224 T^{15} + 103622583717 T^{16} + 36109803224 T^{17} + 655721199454 T^{18} + 2015993900449 T^{20} )^{2}$$
$19$ $$1 - 8 T + 32 T^{2} - 120 T^{3} + 170 T^{4} + 664 T^{5} - 3552 T^{6} + 30376 T^{7} - 137251 T^{8} + 141472 T^{9} + 439424 T^{10} - 1977376 T^{11} + 92609976 T^{12} - 734762848 T^{13} + 3144939136 T^{14} - 11850068256 T^{15} + 18020441650 T^{16} + 30523407824 T^{17} - 82150980672 T^{18} + 1163681013680 T^{19} - 8113217288772 T^{20} + 22109939259920 T^{21} - 29656504022592 T^{22} + 209360054264816 T^{23} + 2348441976269650 T^{24} - 29341942158613344 T^{25} + 147956432344498816 T^{26} - 656783744694352672 T^{27} + 1572847365621497016 T^{28} - 638074909083447904 T^{29} + 2694137659267946624 T^{30} + 16480109906848838368 T^{31} -$$$$30\!\cdots\!11$$$$T^{32} +$$$$12\!\cdots\!84$$$$T^{33} -$$$$28\!\cdots\!92$$$$T^{34} +$$$$10\!\cdots\!36$$$$T^{35} +$$$$49\!\cdots\!70$$$$T^{36} -$$$$65\!\cdots\!80$$$$T^{37} +$$$$33\!\cdots\!12$$$$T^{38} -$$$$15\!\cdots\!32$$$$T^{39} +$$$$37\!\cdots\!01$$$$T^{40}$$
$23$ $$1 - 228 T^{2} + 24654 T^{4} - 1671172 T^{6} + 79227605 T^{8} - 2786970288 T^{10} + 76106467832 T^{12} - 1708777858416 T^{14} + 34629617977562 T^{16} - 710121266972024 T^{18} + 15689370725592020 T^{20} - 375654150228200696 T^{22} + 9690786924458927642 T^{24} -$$$$25\!\cdots\!24$$$$T^{26} +$$$$59\!\cdots\!92$$$$T^{28} -$$$$11\!\cdots\!12$$$$T^{30} +$$$$17\!\cdots\!05$$$$T^{32} -$$$$19\!\cdots\!48$$$$T^{34} +$$$$15\!\cdots\!94$$$$T^{36} -$$$$73\!\cdots\!32$$$$T^{38} +$$$$17\!\cdots\!01$$$$T^{40}$$
$29$ $$1 - 12 T + 72 T^{2} - 660 T^{3} + 5062 T^{4} - 19812 T^{5} + 91080 T^{6} - 552988 T^{7} + 898845 T^{8} + 3230928 T^{9} - 3305184 T^{10} + 180759600 T^{11} - 2401479032 T^{12} + 11590609840 T^{13} - 61731406432 T^{14} + 542524821712 T^{15} - 2180038368078 T^{16} + 2872412612888 T^{17} - 25762237441424 T^{18} + 2546561760168 T^{19} + 1284480158865124 T^{20} + 73850291044872 T^{21} - 21666041688237584 T^{22} + 70055271215725432 T^{23} - 1541899717012575918 T^{24} + 11127807454333267088 T^{25} - 36719280183883000672 T^{26} +$$$$19\!\cdots\!60$$$$T^{27} -$$$$12\!\cdots\!52$$$$T^{28} +$$$$26\!\cdots\!00$$$$T^{29} -$$$$13\!\cdots\!84$$$$T^{30} +$$$$39\!\cdots\!12$$$$T^{31} +$$$$31\!\cdots\!45$$$$T^{32} -$$$$56\!\cdots\!32$$$$T^{33} +$$$$27\!\cdots\!80$$$$T^{34} -$$$$17\!\cdots\!88$$$$T^{35} +$$$$12\!\cdots\!02$$$$T^{36} -$$$$47\!\cdots\!40$$$$T^{37} +$$$$15\!\cdots\!92$$$$T^{38} -$$$$73\!\cdots\!28$$$$T^{39} +$$$$17\!\cdots\!01$$$$T^{40}$$
$31$ $$( 1 + 134 T^{2} - 64 T^{3} + 7637 T^{4} + 64 T^{5} + 250776 T^{6} + 647232 T^{7} + 5787898 T^{8} + 46731200 T^{9} + 141742500 T^{10} + 1448667200 T^{11} + 5562169978 T^{12} + 19281688512 T^{13} + 231596902296 T^{14} + 1832265664 T^{15} + 6777865611797 T^{16} - 1760807303104 T^{17} + 114287399017094 T^{18} + 819628286980801 T^{20} )^{2}$$
$37$ $$1 - 12 T + 72 T^{2} - 692 T^{3} + 4646 T^{4} - 25508 T^{5} + 211016 T^{6} - 1455420 T^{7} + 12729021 T^{8} - 102763568 T^{9} + 668772640 T^{10} - 4907491152 T^{11} + 30685175816 T^{12} - 181058288208 T^{13} + 1246622676896 T^{14} - 7968453274800 T^{15} + 52782260071730 T^{16} - 346974533424104 T^{17} + 2138919388551792 T^{18} - 13427579452955416 T^{19} + 82729716000045604 T^{20} - 496820439759350392 T^{21} + 2928180642927403248 T^{22} - 17575301041531139912 T^{23} + 98922453318293568530 T^{24} -$$$$55\!\cdots\!00$$$$T^{25} +$$$$31\!\cdots\!64$$$$T^{26} -$$$$17\!\cdots\!64$$$$T^{27} +$$$$10\!\cdots\!36$$$$T^{28} -$$$$63\!\cdots\!04$$$$T^{29} +$$$$32\!\cdots\!60$$$$T^{30} -$$$$18\!\cdots\!84$$$$T^{31} +$$$$83\!\cdots\!01$$$$T^{32} -$$$$35\!\cdots\!40$$$$T^{33} +$$$$19\!\cdots\!24$$$$T^{34} -$$$$85\!\cdots\!44$$$$T^{35} +$$$$57\!\cdots\!86$$$$T^{36} -$$$$31\!\cdots\!64$$$$T^{37} +$$$$12\!\cdots\!88$$$$T^{38} -$$$$74\!\cdots\!76$$$$T^{39} +$$$$23\!\cdots\!01$$$$T^{40}$$
$41$ $$1 - 444 T^{2} + 98574 T^{4} - 14512124 T^{6} + 1588488149 T^{8} - 137675000112 T^{10} + 9846636632440 T^{12} - 599501624736304 T^{14} + 31880388630283994 T^{16} - 1512585137889846120 T^{18} + 64990434418739739988 T^{20} -$$$$25\!\cdots\!20$$$$T^{22} +$$$$90\!\cdots\!34$$$$T^{24} -$$$$28\!\cdots\!64$$$$T^{26} +$$$$78\!\cdots\!40$$$$T^{28} -$$$$18\!\cdots\!12$$$$T^{30} +$$$$35\!\cdots\!69$$$$T^{32} -$$$$55\!\cdots\!64$$$$T^{34} +$$$$62\!\cdots\!34$$$$T^{36} -$$$$47\!\cdots\!24$$$$T^{38} +$$$$18\!\cdots\!01$$$$T^{40}$$
$43$ $$1 - 4 T + 8 T^{2} + 300 T^{3} + 7078 T^{4} - 44284 T^{5} + 165512 T^{6} + 1626836 T^{7} + 19994909 T^{8} - 198352400 T^{9} + 930543520 T^{10} + 1629993648 T^{11} + 32879446344 T^{12} - 523812434096 T^{13} + 2532563214112 T^{14} - 8844397053040 T^{15} + 52614175288210 T^{16} - 1034528298704760 T^{17} + 4610401791613808 T^{18} - 37116758465801176 T^{19} + 100269344691342500 T^{20} - 1596020614029450568 T^{21} + 8524632912693930992 T^{22} - 82252241445119353320 T^{23} +$$$$17\!\cdots\!10$$$$T^{24} -$$$$13\!\cdots\!20$$$$T^{25} +$$$$16\!\cdots\!88$$$$T^{26} -$$$$14\!\cdots\!72$$$$T^{27} +$$$$38\!\cdots\!44$$$$T^{28} +$$$$81\!\cdots\!64$$$$T^{29} +$$$$20\!\cdots\!80$$$$T^{30} -$$$$18\!\cdots\!00$$$$T^{31} +$$$$79\!\cdots\!09$$$$T^{32} +$$$$27\!\cdots\!48$$$$T^{33} +$$$$12\!\cdots\!88$$$$T^{34} -$$$$14\!\cdots\!88$$$$T^{35} +$$$$96\!\cdots\!78$$$$T^{36} +$$$$17\!\cdots\!00$$$$T^{37} +$$$$20\!\cdots\!92$$$$T^{38} -$$$$43\!\cdots\!28$$$$T^{39} +$$$$46\!\cdots\!01$$$$T^{40}$$
$47$ $$( 1 + 254 T^{2} - 128 T^{3} + 33997 T^{4} - 26752 T^{5} + 3110696 T^{6} - 2769024 T^{7} + 212743186 T^{8} - 186077312 T^{9} + 11293892148 T^{10} - 8745633664 T^{11} + 469949697874 T^{12} - 287488378752 T^{13} + 15179204167976 T^{14} - 6135437627264 T^{15} + 366460983540013 T^{16} - 64847759419264 T^{17} + 6048066812087294 T^{18} + 52599132235830049 T^{20} )^{2}$$
$53$ $$1 + 36 T + 648 T^{2} + 8284 T^{3} + 81222 T^{4} + 595308 T^{5} + 3111560 T^{6} + 8277492 T^{7} - 42534915 T^{8} - 754029808 T^{9} - 6543023328 T^{10} - 50633847440 T^{11} - 420252241016 T^{12} - 3590853938064 T^{13} - 27232240956512 T^{14} - 163759336672624 T^{15} - 644108178966222 T^{16} - 225724213191624 T^{17} + 21359012626695920 T^{18} + 241077022660459848 T^{19} + 1941342482920284900 T^{20} + 12777082201004371944 T^{21} + 59997466468388839280 T^{22} - 33605143687329406248 T^{23} -$$$$50\!\cdots\!82$$$$T^{24} -$$$$68\!\cdots\!32$$$$T^{25} -$$$$60\!\cdots\!48$$$$T^{26} -$$$$42\!\cdots\!68$$$$T^{27} -$$$$26\!\cdots\!76$$$$T^{28} -$$$$16\!\cdots\!20$$$$T^{29} -$$$$11\!\cdots\!72$$$$T^{30} -$$$$69\!\cdots\!76$$$$T^{31} -$$$$20\!\cdots\!15$$$$T^{32} +$$$$21\!\cdots\!16$$$$T^{33} +$$$$42\!\cdots\!40$$$$T^{34} +$$$$43\!\cdots\!56$$$$T^{35} +$$$$31\!\cdots\!62$$$$T^{36} +$$$$17\!\cdots\!92$$$$T^{37} +$$$$70\!\cdots\!72$$$$T^{38} +$$$$20\!\cdots\!12$$$$T^{39} +$$$$30\!\cdots\!01$$$$T^{40}$$
$59$ $$1 + 64 T^{3} + 17498 T^{4} + 16320 T^{5} + 2048 T^{6} + 964352 T^{7} + 139305821 T^{8} + 265011968 T^{9} + 159053824 T^{10} + 7117236224 T^{11} + 675196772280 T^{12} + 1865453307392 T^{13} + 2177063692288 T^{14} + 46993822767872 T^{15} + 2378495821379602 T^{16} + 7365185499789056 T^{17} + 13796808199690240 T^{18} + 308926250030895744 T^{19} + 7830691115365441372 T^{20} + 18226648751822848896 T^{21} + 48026689343121725440 T^{22} +$$$$15\!\cdots\!24$$$$T^{23} +$$$$28\!\cdots\!22$$$$T^{24} +$$$$33\!\cdots\!28$$$$T^{25} +$$$$91\!\cdots\!08$$$$T^{26} +$$$$46\!\cdots\!48$$$$T^{27} +$$$$99\!\cdots\!80$$$$T^{28} +$$$$61\!\cdots\!36$$$$T^{29} +$$$$81\!\cdots\!24$$$$T^{30} +$$$$79\!\cdots\!12$$$$T^{31} +$$$$24\!\cdots\!01$$$$T^{32} +$$$$10\!\cdots\!08$$$$T^{33} +$$$$12\!\cdots\!28$$$$T^{34} +$$$$59\!\cdots\!80$$$$T^{35} +$$$$37\!\cdots\!18$$$$T^{36} +$$$$81\!\cdots\!16$$$$T^{37} +$$$$26\!\cdots\!01$$$$T^{40}$$
$61$ $$1 - 8 T + 32 T^{2} - 24 T^{3} - 22 T^{4} + 28776 T^{5} - 229216 T^{6} + 1701688 T^{7} - 7327587 T^{8} + 193356480 T^{9} - 944223232 T^{10} + 7441249856 T^{11} + 25030907320 T^{12} - 252322194176 T^{13} + 6350522522880 T^{14} - 30700611174144 T^{15} + 72405460654514 T^{16} + 1378721064691056 T^{17} + 10443615781913408 T^{18} + 4859962566983248 T^{19} + 374225284282585404 T^{20} + 296457716585978128 T^{21} + 38860694324499791168 T^{22} +$$$$31\!\cdots\!36$$$$T^{23} +$$$$10\!\cdots\!74$$$$T^{24} -$$$$25\!\cdots\!44$$$$T^{25} +$$$$32\!\cdots\!80$$$$T^{26} -$$$$79\!\cdots\!96$$$$T^{27} +$$$$47\!\cdots\!20$$$$T^{28} +$$$$87\!\cdots\!96$$$$T^{29} -$$$$67\!\cdots\!32$$$$T^{30} +$$$$84\!\cdots\!80$$$$T^{31} -$$$$19\!\cdots\!27$$$$T^{32} +$$$$27\!\cdots\!28$$$$T^{33} -$$$$22\!\cdots\!56$$$$T^{34} +$$$$17\!\cdots\!76$$$$T^{35} -$$$$80\!\cdots\!42$$$$T^{36} -$$$$53\!\cdots\!04$$$$T^{37} +$$$$43\!\cdots\!92$$$$T^{38} -$$$$66\!\cdots\!28$$$$T^{39} +$$$$50\!\cdots\!01$$$$T^{40}$$
$67$ $$1 + 12 T + 72 T^{2} + 444 T^{3} + 4102 T^{4} + 54484 T^{5} + 457032 T^{6} + 4150628 T^{7} - 7647939 T^{8} - 292442192 T^{9} - 1506268000 T^{10} - 8570912144 T^{11} - 68934819128 T^{12} - 1058425781488 T^{13} - 9188787191264 T^{14} - 93290480178608 T^{15} + 105617726671378 T^{16} + 7399255991086056 T^{17} + 46962879527876336 T^{18} + 365051212101087880 T^{19} + 2665710348810285028 T^{20} + 24458431210772887960 T^{21} +$$$$21\!\cdots\!04$$$$T^{22} +$$$$22\!\cdots\!28$$$$T^{23} +$$$$21\!\cdots\!38$$$$T^{24} -$$$$12\!\cdots\!56$$$$T^{25} -$$$$83\!\cdots\!16$$$$T^{26} -$$$$64\!\cdots\!24$$$$T^{27} -$$$$27\!\cdots\!48$$$$T^{28} -$$$$23\!\cdots\!68$$$$T^{29} -$$$$27\!\cdots\!00$$$$T^{30} -$$$$35\!\cdots\!36$$$$T^{31} -$$$$62\!\cdots\!79$$$$T^{32} +$$$$22\!\cdots\!36$$$$T^{33} +$$$$16\!\cdots\!28$$$$T^{34} +$$$$13\!\cdots\!12$$$$T^{35} +$$$$67\!\cdots\!62$$$$T^{36} +$$$$49\!\cdots\!88$$$$T^{37} +$$$$53\!\cdots\!48$$$$T^{38} +$$$$59\!\cdots\!36$$$$T^{39} +$$$$33\!\cdots\!01$$$$T^{40}$$
$71$ $$1 - 588 T^{2} + 187310 T^{4} - 41853292 T^{6} + 7286062485 T^{8} - 1044227363664 T^{10} + 127464606182264 T^{12} - 13544991508155920 T^{14} + 1271322412181875034 T^{16} -$$$$10\!\cdots\!40$$$$T^{18} +$$$$79\!\cdots\!36$$$$T^{20} -$$$$53\!\cdots\!40$$$$T^{22} +$$$$32\!\cdots\!54$$$$T^{24} -$$$$17\!\cdots\!20$$$$T^{26} +$$$$82\!\cdots\!04$$$$T^{28} -$$$$33\!\cdots\!64$$$$T^{30} +$$$$11\!\cdots\!85$$$$T^{32} -$$$$34\!\cdots\!52$$$$T^{34} +$$$$78\!\cdots\!10$$$$T^{36} -$$$$12\!\cdots\!68$$$$T^{38} +$$$$10\!\cdots\!01$$$$T^{40}$$
$73$ $$1 - 588 T^{2} + 185438 T^{4} - 41247404 T^{6} + 7184467213 T^{8} - 1034846312688 T^{10} + 127434000595944 T^{12} - 13711360420506544 T^{14} + 1308031080629694098 T^{16} -$$$$11\!\cdots\!16$$$$T^{18} +$$$$85\!\cdots\!04$$$$T^{20} -$$$$59\!\cdots\!64$$$$T^{22} +$$$$37\!\cdots\!18$$$$T^{24} -$$$$20\!\cdots\!16$$$$T^{26} +$$$$10\!\cdots\!64$$$$T^{28} -$$$$44\!\cdots\!12$$$$T^{30} +$$$$16\!\cdots\!73$$$$T^{32} -$$$$50\!\cdots\!36$$$$T^{34} +$$$$12\!\cdots\!18$$$$T^{36} -$$$$20\!\cdots\!72$$$$T^{38} +$$$$18\!\cdots\!01$$$$T^{40}$$
$79$ $$( 1 - 12 T + 410 T^{2} - 3636 T^{3} + 71661 T^{4} - 457104 T^{5} + 6962296 T^{6} - 26461552 T^{7} + 422059634 T^{8} - 425195336 T^{9} + 24696310492 T^{10} - 33590431544 T^{11} + 2634074175794 T^{12} - 13046577136528 T^{13} + 271181993145976 T^{14} - 1406534788208496 T^{15} + 17419890150090381 T^{16} - 69825413073674124 T^{17} + 622014612061690010 T^{18} - 1438219151791419828 T^{19} + 9468276082626847201 T^{20} )^{2}$$
$83$ $$1 - 40 T + 800 T^{2} - 11352 T^{3} + 130714 T^{4} - 1275720 T^{5} + 10891552 T^{6} - 81158136 T^{7} + 533032573 T^{8} - 3591219744 T^{9} + 28582322304 T^{10} - 250649734880 T^{11} + 2277897381816 T^{12} - 18560960241440 T^{13} + 98547001707136 T^{14} + 326327140527904 T^{15} - 18693002705618542 T^{16} + 320247589539147216 T^{17} - 4017303386407089728 T^{18} + 43457553116792444976 T^{19} -$$$$41\!\cdots\!72$$$$T^{20} +$$$$36\!\cdots\!08$$$$T^{21} -$$$$27\!\cdots\!92$$$$T^{22} +$$$$18\!\cdots\!92$$$$T^{23} -$$$$88\!\cdots\!82$$$$T^{24} +$$$$12\!\cdots\!72$$$$T^{25} +$$$$32\!\cdots\!84$$$$T^{26} -$$$$50\!\cdots\!80$$$$T^{27} +$$$$51\!\cdots\!56$$$$T^{28} -$$$$46\!\cdots\!40$$$$T^{29} +$$$$44\!\cdots\!96$$$$T^{30} -$$$$46\!\cdots\!48$$$$T^{31} +$$$$56\!\cdots\!53$$$$T^{32} -$$$$72\!\cdots\!68$$$$T^{33} +$$$$80\!\cdots\!08$$$$T^{34} -$$$$77\!\cdots\!40$$$$T^{35} +$$$$66\!\cdots\!34$$$$T^{36} -$$$$47\!\cdots\!96$$$$T^{37} +$$$$27\!\cdots\!00$$$$T^{38} -$$$$11\!\cdots\!80$$$$T^{39} +$$$$24\!\cdots\!01$$$$T^{40}$$
$89$ $$1 - 700 T^{2} + 256046 T^{4} - 64249660 T^{6} + 12305003221 T^{8} - 1904804480368 T^{10} + 247648665045240 T^{12} - 27922948227608560 T^{14} + 2822449388027483546 T^{16} -$$$$26\!\cdots\!68$$$$T^{18} +$$$$23\!\cdots\!96$$$$T^{20} -$$$$21\!\cdots\!28$$$$T^{22} +$$$$17\!\cdots\!86$$$$T^{24} -$$$$13\!\cdots\!60$$$$T^{26} +$$$$97\!\cdots\!40$$$$T^{28} -$$$$59\!\cdots\!68$$$$T^{30} +$$$$30\!\cdots\!41$$$$T^{32} -$$$$12\!\cdots\!60$$$$T^{34} +$$$$39\!\cdots\!06$$$$T^{36} -$$$$85\!\cdots\!00$$$$T^{38} +$$$$97\!\cdots\!01$$$$T^{40}$$
$97$ $$( 1 + 36 T + 1006 T^{2} + 17860 T^{3} + 282157 T^{4} + 3451280 T^{5} + 41738920 T^{6} + 425104976 T^{7} + 4638925202 T^{8} + 43768470968 T^{9} + 461725986516 T^{10} + 4245541683896 T^{11} + 43647647225618 T^{12} + 387981833760848 T^{13} + 3695116577316520 T^{14} + 29637315682178960 T^{15} + 235028881994751853 T^{16} + 1443057360779098180 T^{17} + 7884458195943222766 T^{18} + 27368318111564347812 T^{19} + 73742412689492826049 T^{20} )^{2}$$