# Properties

 Label 322.2.i.c Level $322$ Weight $2$ Character orbit 322.i Analytic conductor $2.571$ Analytic rank $0$ Dimension $20$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$322 = 2 \cdot 7 \cdot 23$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 322.i (of order $$11$$, degree $$10$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$2.57118294509$$ Analytic rank: $$0$$ Dimension: $$20$$ Relative dimension: $$2$$ over $$\Q(\zeta_{11})$$ Coefficient field: $$\mathbb{Q}[x]/(x^{20} - \cdots)$$ Defining polynomial: $$x^{20} - 9 x^{19} + 41 x^{18} - 119 x^{17} + 245 x^{16} - 404 x^{15} + 623 x^{14} - 898 x^{13} + 1048 x^{12} - 693 x^{11} + 859 x^{10} - 935 x^{9} + 620 x^{8} + 679 x^{7} + 1220 x^{6} + 2241 x^{5} + 1750 x^{4} + 1079 x^{3} + 534 x^{2} + 38 x + 1$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{11}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + \beta_{19} q^{2} + ( \beta_{1} + \beta_{2} + \beta_{4} + \beta_{5} - \beta_{6} + \beta_{8} - \beta_{10} - \beta_{16} ) q^{3} + \beta_{11} q^{4} + ( \beta_{1} + \beta_{2} + \beta_{4} + \beta_{9} - \beta_{10} ) q^{5} + ( -\beta_{5} - \beta_{12} ) q^{6} + ( -1 - \beta_{11} - \beta_{12} + \beta_{13} - \beta_{14} - \beta_{15} - \beta_{16} - \beta_{17} - \beta_{18} - \beta_{19} ) q^{7} + \beta_{17} q^{8} + ( -\beta_{3} + \beta_{4} + \beta_{5} - \beta_{7} - \beta_{9} - \beta_{13} + \beta_{17} ) q^{9} +O(q^{10})$$ $$q + \beta_{19} q^{2} + ( \beta_{1} + \beta_{2} + \beta_{4} + \beta_{5} - \beta_{6} + \beta_{8} - \beta_{10} - \beta_{16} ) q^{3} + \beta_{11} q^{4} + ( \beta_{1} + \beta_{2} + \beta_{4} + \beta_{9} - \beta_{10} ) q^{5} + ( -\beta_{5} - \beta_{12} ) q^{6} + ( -1 - \beta_{11} - \beta_{12} + \beta_{13} - \beta_{14} - \beta_{15} - \beta_{16} - \beta_{17} - \beta_{18} - \beta_{19} ) q^{7} + \beta_{17} q^{8} + ( -\beta_{3} + \beta_{4} + \beta_{5} - \beta_{7} - \beta_{9} - \beta_{13} + \beta_{17} ) q^{9} + ( \beta_{1} + \beta_{3} - \beta_{5} + \beta_{6} + \beta_{10} ) q^{10} + ( -\beta_{5} - \beta_{7} - \beta_{8} - \beta_{9} + \beta_{10} + \beta_{11} - \beta_{13} ) q^{11} + ( \beta_{10} - \beta_{18} ) q^{12} + ( \beta_{2} + \beta_{4} + \beta_{5} - 2 \beta_{7} - \beta_{8} - \beta_{9} - \beta_{11} - \beta_{12} + \beta_{13} - 2 \beta_{14} - \beta_{15} - 2 \beta_{16} - \beta_{19} ) q^{13} + \beta_{15} q^{14} + ( 1 - \beta_{1} + \beta_{4} - \beta_{7} - 2 \beta_{8} - \beta_{9} + \beta_{10} + \beta_{14} - \beta_{17} ) q^{15} + ( -1 - \beta_{11} - \beta_{12} + \beta_{13} - \beta_{14} - \beta_{15} - \beta_{16} - \beta_{17} - \beta_{18} - \beta_{19} ) q^{16} + ( \beta_{1} - \beta_{5} + \beta_{7} + \beta_{11} + \beta_{12} - 2 \beta_{14} + \beta_{15} + \beta_{16} + \beta_{18} - 2 \beta_{19} ) q^{17} + ( -1 + \beta_{2} + 2 \beta_{9} - \beta_{10} - \beta_{11} - \beta_{12} + \beta_{13} - \beta_{14} - \beta_{15} - \beta_{17} - \beta_{18} - \beta_{19} ) q^{18} + ( 2 \beta_{2} - 2 \beta_{3} + 2 \beta_{4} + 3 \beta_{5} - 2 \beta_{6} + 2 \beta_{8} + \beta_{9} - 4 \beta_{10} - 2 \beta_{11} - 2 \beta_{12} - 2 \beta_{14} - \beta_{15} - 2 \beta_{17} - \beta_{18} - 2 \beta_{19} ) q^{19} + ( -\beta_{1} - \beta_{2} + \beta_{3} - \beta_{4} - \beta_{5} + \beta_{6} - \beta_{7} - \beta_{8} - \beta_{9} + 2 \beta_{10} ) q^{20} + ( -\beta_{3} - \beta_{4} + \beta_{7} + \beta_{9} - \beta_{19} ) q^{21} + ( -\beta_{1} - \beta_{3} - \beta_{4} + \beta_{6} + \beta_{9} + \beta_{10} + \beta_{16} + \beta_{17} ) q^{22} + ( \beta_{1} + \beta_{2} + 2 \beta_{3} + \beta_{4} + \beta_{5} + 2 \beta_{6} - 3 \beta_{7} - 2 \beta_{11} - \beta_{12} + \beta_{13} - \beta_{14} - 3 \beta_{15} - \beta_{17} - \beta_{18} ) q^{23} + ( -1 - \beta_{7} ) q^{24} + ( 3 - 3 \beta_{1} - 2 \beta_{2} - \beta_{3} - \beta_{4} - \beta_{7} - 2 \beta_{8} - 2 \beta_{9} + 2 \beta_{10} + 3 \beta_{11} - 4 \beta_{13} + 4 \beta_{14} + 2 \beta_{15} + 2 \beta_{16} + \beta_{17} + \beta_{18} ) q^{25} + ( \beta_{2} - \beta_{3} - \beta_{4} + \beta_{6} + \beta_{7} + \beta_{8} + 2 \beta_{9} - \beta_{10} - \beta_{11} - 2 \beta_{12} + 2 \beta_{13} - \beta_{14} - \beta_{16} - \beta_{17} - \beta_{18} ) q^{26} + ( -\beta_{2} + \beta_{3} - 3 \beta_{5} + \beta_{6} - \beta_{7} - 3 \beta_{9} + \beta_{10} + 3 \beta_{11} + 2 \beta_{12} - 3 \beta_{13} + 2 \beta_{14} + 4 \beta_{15} + 3 \beta_{16} + 4 \beta_{17} + 4 \beta_{18} + 4 \beta_{19} ) q^{27} + \beta_{14} q^{28} + ( -2 \beta_{1} - 2 \beta_{2} - 2 \beta_{4} - \beta_{5} + 2 \beta_{6} - 2 \beta_{8} - \beta_{11} - 2 \beta_{12} - 2 \beta_{14} - \beta_{15} + \beta_{16} + \beta_{18} - 2 \beta_{19} ) q^{29} + ( 1 + \beta_{1} + 2 \beta_{2} + \beta_{4} + \beta_{5} + \beta_{6} - \beta_{7} + \beta_{8} + \beta_{9} - \beta_{10} + \beta_{11} + \beta_{12} - 2 \beta_{13} + \beta_{14} + \beta_{15} + \beta_{16} + \beta_{17} + \beta_{18} + 2 \beta_{19} ) q^{30} + ( \beta_{7} + 2 \beta_{8} + \beta_{9} - 3 \beta_{12} + 3 \beta_{13} - \beta_{15} - 3 \beta_{16} - 4 \beta_{17} - 3 \beta_{18} - \beta_{19} ) q^{31} + \beta_{15} q^{32} + ( 2 + 2 \beta_{1} + \beta_{3} - \beta_{5} + \beta_{7} + \beta_{8} + \beta_{11} + 2 \beta_{12} - 2 \beta_{13} + 3 \beta_{14} + 2 \beta_{15} + 3 \beta_{16} + 2 \beta_{17} + \beta_{19} ) q^{33} + ( 1 - \beta_{1} - \beta_{2} + \beta_{3} - \beta_{5} + \beta_{6} - \beta_{7} - \beta_{8} - \beta_{9} + 2 \beta_{10} - 2 \beta_{11} + \beta_{12} + 2 \beta_{13} + \beta_{14} + \beta_{17} + \beta_{18} ) q^{34} + ( -\beta_{3} + \beta_{5} - \beta_{6} + \beta_{7} + \beta_{8} + \beta_{9} - 2 \beta_{10} ) q^{35} + ( 2 \beta_{1} + \beta_{7} + \beta_{8} + \beta_{12} + \beta_{15} ) q^{36} + ( -1 - \beta_{1} + \beta_{3} + \beta_{4} + \beta_{5} - \beta_{7} - \beta_{8} - \beta_{9} + \beta_{12} + 2 \beta_{13} - 4 \beta_{15} - 2 \beta_{17} - \beta_{18} + \beta_{19} ) q^{37} + ( 1 + 3 \beta_{1} + 2 \beta_{2} + 2 \beta_{4} - 2 \beta_{6} + 2 \beta_{7} + 2 \beta_{8} + 2 \beta_{9} - 3 \beta_{10} + 2 \beta_{12} + \beta_{14} + 2 \beta_{15} + 2 \beta_{16} + 2 \beta_{19} ) q^{38} + ( 1 - 3 \beta_{2} - 2 \beta_{5} - 2 \beta_{6} - 3 \beta_{9} + 2 \beta_{10} + \beta_{11} + \beta_{12} + \beta_{14} + 3 \beta_{15} + \beta_{16} + 3 \beta_{17} + \beta_{18} ) q^{39} + ( -\beta_{1} - \beta_{3} - \beta_{4} + \beta_{5} ) q^{40} + ( 1 - 3 \beta_{1} - \beta_{2} - 3 \beta_{4} - 2 \beta_{5} - 2 \beta_{8} - \beta_{9} + \beta_{10} + 4 \beta_{11} + 3 \beta_{12} - 2 \beta_{13} + 2 \beta_{15} + 3 \beta_{17} + 4 \beta_{18} + \beta_{19} ) q^{41} + ( -\beta_{2} - \beta_{11} ) q^{42} + ( -2 + 3 \beta_{1} + 2 \beta_{2} - 3 \beta_{3} + 3 \beta_{5} + 3 \beta_{7} + 2 \beta_{8} + 4 \beta_{9} - 4 \beta_{10} - 3 \beta_{11} - \beta_{12} + \beta_{13} - 3 \beta_{14} - 2 \beta_{15} - 3 \beta_{16} - 3 \beta_{17} - 3 \beta_{18} ) q^{43} + ( -1 + \beta_{1} + \beta_{5} - \beta_{6} + \beta_{8} + \beta_{9} - \beta_{10} - \beta_{11} + \beta_{13} - \beta_{14} - \beta_{15} - \beta_{16} - \beta_{17} - \beta_{18} - \beta_{19} ) q^{44} + ( 2 - 2 \beta_{2} + \beta_{3} - 3 \beta_{4} - 2 \beta_{5} + 2 \beta_{7} - \beta_{9} + 3 \beta_{10} - 2 \beta_{16} - 2 \beta_{17} - \beta_{18} - \beta_{19} ) q^{45} + ( -3 \beta_{4} - \beta_{5} + \beta_{6} + 2 \beta_{7} + \beta_{9} + \beta_{11} + \beta_{12} - 2 \beta_{14} + \beta_{15} - \beta_{17} + \beta_{19} ) q^{46} + ( -4 + \beta_{1} + \beta_{3} + 2 \beta_{4} - \beta_{6} + \beta_{7} - \beta_{9} - 2 \beta_{10} - 3 \beta_{16} - 3 \beta_{17} ) q^{47} + ( -\beta_{3} - \beta_{4} + \beta_{7} + \beta_{9} - \beta_{19} ) q^{48} + \beta_{16} q^{49} + ( 2 \beta_{2} - 2 \beta_{3} + \beta_{4} + 3 \beta_{5} - \beta_{6} + \beta_{8} + 2 \beta_{9} - 3 \beta_{10} - \beta_{11} + \beta_{12} - 3 \beta_{13} + \beta_{14} - \beta_{15} + 3 \beta_{16} + 2 \beta_{17} - \beta_{18} + 2 \beta_{19} ) q^{50} + ( 1 + \beta_{1} + 2 \beta_{2} + \beta_{4} + 4 \beta_{5} + 4 \beta_{8} + \beta_{9} - 2 \beta_{10} - \beta_{11} + 3 \beta_{12} + 3 \beta_{17} - \beta_{18} + \beta_{19} ) q^{51} + ( \beta_{1} - \beta_{2} + \beta_{3} - \beta_{6} + \beta_{7} + \beta_{8} + \beta_{11} + \beta_{14} + \beta_{15} - \beta_{16} - \beta_{18} + \beta_{19} ) q^{52} + ( 2 - 2 \beta_{1} - \beta_{2} - \beta_{3} - 2 \beta_{4} - \beta_{5} + \beta_{6} + 2 \beta_{7} - \beta_{8} + \beta_{9} + \beta_{10} + 4 \beta_{11} + 4 \beta_{12} - 3 \beta_{13} + 4 \beta_{14} + 4 \beta_{15} + 2 \beta_{16} + 4 \beta_{17} + 4 \beta_{18} + 3 \beta_{19} ) q^{53} + ( -3 \beta_{1} - \beta_{4} - \beta_{8} + 3 \beta_{10} - \beta_{12} + 2 \beta_{13} - 4 \beta_{15} - \beta_{16} - \beta_{17} - 2 \beta_{18} - 4 \beta_{19} ) q^{54} + ( \beta_{1} + \beta_{3} - 2 \beta_{4} - 2 \beta_{5} + \beta_{6} + \beta_{7} + \beta_{8} + \beta_{9} + \beta_{10} + \beta_{11} + \beta_{12} - \beta_{13} - \beta_{15} + \beta_{16} + \beta_{17} + \beta_{19} ) q^{55} -\beta_{13} q^{56} + ( -1 - 4 \beta_{1} - 4 \beta_{2} - 3 \beta_{8} - 3 \beta_{9} + 2 \beta_{10} - 2 \beta_{11} - \beta_{12} + 2 \beta_{13} + \beta_{14} - 2 \beta_{16} + \beta_{17} + \beta_{18} - 2 \beta_{19} ) q^{57} + ( 1 + 2 \beta_{5} + 2 \beta_{7} - \beta_{10} - 2 \beta_{11} + \beta_{12} + 2 \beta_{13} - \beta_{14} - \beta_{17} - 2 \beta_{18} ) q^{58} + ( -5 + 3 \beta_{1} + 3 \beta_{2} + 5 \beta_{3} + 3 \beta_{4} - 2 \beta_{5} - \beta_{6} - 2 \beta_{7} + \beta_{8} - 3 \beta_{9} - \beta_{10} + 2 \beta_{11} - \beta_{12} + 2 \beta_{13} - \beta_{14} - \beta_{15} - \beta_{16} - 5 \beta_{17} + 2 \beta_{19} ) q^{59} + ( \beta_{1} + \beta_{3} - \beta_{4} - \beta_{5} + \beta_{7} + \beta_{8} + \beta_{9} + \beta_{11} - \beta_{15} + \beta_{16} ) q^{60} + ( -3 - \beta_{1} - \beta_{2} - 2 \beta_{4} + \beta_{6} + 2 \beta_{7} - \beta_{8} + 2 \beta_{9} + \beta_{10} + \beta_{12} + 3 \beta_{13} - 3 \beta_{14} + \beta_{16} - \beta_{17} - 3 \beta_{18} ) q^{61} + ( 1 + \beta_{1} + \beta_{3} + \beta_{4} - 2 \beta_{6} - \beta_{7} - \beta_{9} + 3 \beta_{11} + \beta_{12} - 4 \beta_{13} + 3 \beta_{14} + 4 \beta_{15} + \beta_{16} + 4 \beta_{17} + \beta_{18} + 4 \beta_{19} ) q^{62} + ( 1 + \beta_{2} - \beta_{3} + 2 \beta_{5} - \beta_{13} ) q^{63} + \beta_{14} q^{64} + ( -4 \beta_{2} - 3 \beta_{3} - \beta_{4} - \beta_{5} - 4 \beta_{6} + 5 \beta_{7} - \beta_{8} - 2 \beta_{10} - \beta_{15} - \beta_{18} ) q^{65} + ( -2 - 2 \beta_{1} - 2 \beta_{2} + \beta_{3} - \beta_{4} - 2 \beta_{5} + \beta_{6} - \beta_{7} - 2 \beta_{8} - 2 \beta_{9} + 3 \beta_{10} - \beta_{11} + \beta_{12} - \beta_{13} - 2 \beta_{15} - \beta_{17} ) q^{66} + ( -5 + 2 \beta_{1} + \beta_{2} + \beta_{3} + \beta_{4} + \beta_{5} - \beta_{6} + \beta_{9} - 5 \beta_{11} + 2 \beta_{13} - 2 \beta_{14} - 4 \beta_{15} - 4 \beta_{16} + 2 \beta_{17} + 2 \beta_{18} - \beta_{19} ) q^{67} + ( \beta_{2} + \beta_{5} - \beta_{7} - \beta_{11} - \beta_{12} - \beta_{14} - \beta_{15} - 3 \beta_{16} - 3 \beta_{17} ) q^{68} + ( 2 - 3 \beta_{1} - \beta_{2} + 3 \beta_{3} + \beta_{4} - 3 \beta_{5} + \beta_{6} - 4 \beta_{7} - 4 \beta_{8} - 4 \beta_{9} + 4 \beta_{10} + 5 \beta_{11} - \beta_{12} - \beta_{13} + \beta_{14} + 3 \beta_{16} + 2 \beta_{17} + 5 \beta_{18} + 3 \beta_{19} ) q^{69} + ( \beta_{1} + \beta_{4} - \beta_{6} + \beta_{7} - \beta_{10} ) q^{70} + ( -3 + \beta_{1} + \beta_{2} + 2 \beta_{3} + 2 \beta_{4} + 2 \beta_{5} - 2 \beta_{6} - \beta_{7} + 3 \beta_{8} - \beta_{9} - 3 \beta_{10} - 3 \beta_{11} + \beta_{13} - \beta_{14} - \beta_{17} - \beta_{18} - 3 \beta_{19} ) q^{71} + ( -2 \beta_{1} - 2 \beta_{2} + \beta_{3} - \beta_{4} - 2 \beta_{5} + \beta_{6} - \beta_{7} - 2 \beta_{8} - \beta_{9} + 2 \beta_{10} + \beta_{14} + \beta_{18} ) q^{72} + ( 3 \beta_{2} + \beta_{3} + 2 \beta_{4} + 4 \beta_{5} - 2 \beta_{7} + 2 \beta_{8} + 2 \beta_{9} - \beta_{10} - 4 \beta_{11} + \beta_{17} + \beta_{19} ) q^{73} + ( 1 + \beta_{1} + 2 \beta_{2} + \beta_{4} + \beta_{5} + \beta_{8} - 2 \beta_{10} + 3 \beta_{11} + 2 \beta_{12} - 2 \beta_{13} - 2 \beta_{14} + 2 \beta_{15} + 2 \beta_{17} + 3 \beta_{18} + \beta_{19} ) q^{74} + ( 2 + 4 \beta_{1} - 3 \beta_{2} + 3 \beta_{3} - \beta_{5} - \beta_{6} + 4 \beta_{7} + \beta_{8} + \beta_{11} + 4 \beta_{12} - 2 \beta_{13} + 3 \beta_{14} + \beta_{15} + \beta_{16} + \beta_{18} + 3 \beta_{19} ) q^{75} + ( \beta_{1} - \beta_{2} + 2 \beta_{3} + \beta_{4} - 3 \beta_{5} + \beta_{6} - \beta_{7} - \beta_{8} - 2 \beta_{9} + 3 \beta_{10} + 2 \beta_{11} + 2 \beta_{12} - \beta_{13} + 2 \beta_{14} + 2 \beta_{18} + \beta_{19} ) q^{76} + ( 1 - \beta_{1} - \beta_{2} + \beta_{6} - \beta_{8} + \beta_{10} + \beta_{14} ) q^{77} + ( -2 - 3 \beta_{1} - 2 \beta_{3} - 2 \beta_{7} - 3 \beta_{8} + 2 \beta_{10} - 3 \beta_{11} - 2 \beta_{12} + 2 \beta_{13} - 3 \beta_{15} - 3 \beta_{16} - 2 \beta_{17} - 2 \beta_{18} - 2 \beta_{19} ) q^{78} + ( -1 - 3 \beta_{1} + \beta_{2} - 3 \beta_{3} + \beta_{4} + 4 \beta_{5} + \beta_{6} - \beta_{7} - \beta_{9} + \beta_{10} - 5 \beta_{11} - 5 \beta_{12} + 3 \beta_{13} - 2 \beta_{14} - 5 \beta_{15} - 2 \beta_{16} - \beta_{17} - 5 \beta_{19} ) q^{79} + ( -\beta_{3} + \beta_{5} - \beta_{6} + \beta_{7} + \beta_{8} + \beta_{9} - 2 \beta_{10} ) q^{80} + ( -4 - 2 \beta_{1} - 2 \beta_{2} + 4 \beta_{3} - 5 \beta_{5} + 4 \beta_{6} - 5 \beta_{7} - 2 \beta_{8} - 2 \beta_{9} + 4 \beta_{10} - 3 \beta_{11} - 4 \beta_{12} + 3 \beta_{13} + 4 \beta_{14} + 2 \beta_{16} + 4 \beta_{17} + 2 \beta_{19} ) q^{81} + ( 1 - \beta_{1} - 3 \beta_{3} + 3 \beta_{5} - \beta_{6} + 2 \beta_{7} + 2 \beta_{8} - \beta_{10} - 2 \beta_{11} - 3 \beta_{12} + 3 \beta_{13} - \beta_{14} - 3 \beta_{15} - \beta_{16} + \beta_{17} - 2 \beta_{19} ) q^{82} + ( 2 + 2 \beta_{1} + 3 \beta_{3} - 3 \beta_{4} - 3 \beta_{5} + 2 \beta_{7} + 2 \beta_{8} + 3 \beta_{9} + \beta_{10} + 6 \beta_{11} + 3 \beta_{12} - \beta_{13} + 5 \beta_{15} + 6 \beta_{16} + \beta_{17} + 2 \beta_{18} + 3 \beta_{19} ) q^{83} + ( -\beta_{8} - \beta_{17} ) q^{84} + ( 1 - 3 \beta_{1} + \beta_{2} - 7 \beta_{3} - 3 \beta_{4} + 7 \beta_{5} - 6 \beta_{6} + 3 \beta_{7} + 4 \beta_{8} + 4 \beta_{9} - 7 \beta_{10} + 2 \beta_{11} - 2 \beta_{13} + 2 \beta_{14} + \beta_{16} + 2 \beta_{19} ) q^{85} + ( \beta_{1} - 3 \beta_{2} + 3 \beta_{3} - 3 \beta_{5} + \beta_{6} + \beta_{7} - \beta_{8} + 3 \beta_{11} + \beta_{14} + 3 \beta_{15} + 2 \beta_{16} + 2 \beta_{18} + \beta_{19} ) q^{86} + ( 1 - 3 \beta_{1} + \beta_{3} - 2 \beta_{4} + 3 \beta_{9} + 2 \beta_{11} + \beta_{12} + 6 \beta_{13} + \beta_{14} - 6 \beta_{15} + \beta_{17} + 2 \beta_{18} + \beta_{19} ) q^{87} + ( -\beta_{2} - \beta_{3} - \beta_{4} - \beta_{5} + \beta_{7} - \beta_{8} + \beta_{15} + \beta_{18} ) q^{88} + ( 2 + 3 \beta_{1} - 2 \beta_{2} - \beta_{3} + 3 \beta_{4} + 3 \beta_{5} - 3 \beta_{6} + \beta_{7} - 2 \beta_{8} - 3 \beta_{10} - \beta_{11} + 3 \beta_{12} - 3 \beta_{13} + 3 \beta_{14} + 2 \beta_{15} + 2 \beta_{16} - \beta_{17} + 3 \beta_{18} ) q^{89} + ( 1 - 4 \beta_{1} - 3 \beta_{2} - \beta_{3} - \beta_{4} - 2 \beta_{7} - 2 \beta_{8} - 3 \beta_{9} + 2 \beta_{10} + \beta_{11} - 2 \beta_{13} + 2 \beta_{14} + 2 \beta_{15} + 2 \beta_{16} + 2 \beta_{17} + 2 \beta_{18} + 4 \beta_{19} ) q^{90} + ( -1 - \beta_{1} + \beta_{2} - \beta_{3} + \beta_{5} + \beta_{6} + \beta_{9} - \beta_{11} - \beta_{12} - \beta_{14} - \beta_{15} - 2 \beta_{18} - 2 \beta_{19} ) q^{91} + ( 1 - 2 \beta_{1} - 3 \beta_{2} - \beta_{4} + \beta_{7} - 2 \beta_{9} + \beta_{10} + 2 \beta_{11} + \beta_{12} + \beta_{13} + 2 \beta_{14} + \beta_{15} + \beta_{16} + 2 \beta_{17} + 2 \beta_{18} + \beta_{19} ) q^{92} + ( -3 + \beta_{1} + 2 \beta_{2} - 4 \beta_{3} + 2 \beta_{5} - \beta_{6} + 3 \beta_{7} + 4 \beta_{9} - 4 \beta_{11} - 4 \beta_{12} - \beta_{14} - \beta_{15} - 5 \beta_{16} - 5 \beta_{17} - 4 \beta_{18} - 4 \beta_{19} ) q^{93} + ( 3 + \beta_{2} + 2 \beta_{3} + 2 \beta_{4} - \beta_{5} + \beta_{6} - \beta_{7} - \beta_{8} - 2 \beta_{9} + \beta_{10} + 3 \beta_{11} - 3 \beta_{13} + 3 \beta_{14} + 3 \beta_{15} + 3 \beta_{16} + 3 \beta_{17} + 3 \beta_{18} - \beta_{19} ) q^{94} + ( 4 - 2 \beta_{1} - 2 \beta_{2} + 2 \beta_{3} - 2 \beta_{5} - 2 \beta_{7} - 2 \beta_{8} - 4 \beta_{9} + 4 \beta_{10} - 4 \beta_{11} - \beta_{12} + \beta_{13} + 2 \beta_{14} + 4 \beta_{15} - 4 \beta_{16} - 4 \beta_{17} + 2 \beta_{18} ) q^{95} + ( -\beta_{2} - \beta_{11} ) q^{96} + ( 2 - 4 \beta_{1} - 5 \beta_{2} - 4 \beta_{4} - 4 \beta_{5} - 4 \beta_{8} - \beta_{9} + 5 \beta_{10} + 3 \beta_{11} + 5 \beta_{12} - 5 \beta_{13} + 9 \beta_{14} + 5 \beta_{15} + 5 \beta_{17} + 3 \beta_{18} + 2 \beta_{19} ) q^{97} + \beta_{12} q^{98} + ( 1 + \beta_{1} + 2 \beta_{2} - \beta_{3} + \beta_{4} + 2 \beta_{5} + \beta_{6} - \beta_{7} + 2 \beta_{8} + \beta_{9} - 2 \beta_{10} + \beta_{11} + 2 \beta_{12} - \beta_{13} + \beta_{14} + \beta_{15} + \beta_{16} + \beta_{17} + 2 \beta_{18} + \beta_{19} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$20q - 2q^{2} + 4q^{3} - 2q^{4} - 5q^{5} + 4q^{6} - 2q^{7} - 2q^{8} - 12q^{9} + O(q^{10})$$ $$20q - 2q^{2} + 4q^{3} - 2q^{4} - 5q^{5} + 4q^{6} - 2q^{7} - 2q^{8} - 12q^{9} + 6q^{10} + 6q^{11} + 4q^{12} + 14q^{13} - 2q^{14} + 13q^{15} - 2q^{16} + 7q^{17} - 12q^{18} - 2q^{19} + 6q^{20} + 4q^{21} - 16q^{22} + 9q^{23} - 18q^{24} + 23q^{25} + 3q^{26} - 47q^{27} - 2q^{28} + 2q^{29} - 20q^{30} + 28q^{31} - 2q^{32} + 26q^{33} + 18q^{34} - 5q^{35} + 10q^{36} - 17q^{37} + 9q^{38} + 36q^{39} - 5q^{40} - 7q^{41} + 4q^{42} - 17q^{43} + 6q^{44} + 90q^{45} + 9q^{46} - 68q^{47} + 4q^{48} - 2q^{49} - 43q^{50} - 15q^{51} + 14q^{52} - 35q^{53} + 19q^{54} + 8q^{55} - 2q^{56} - 20q^{57} + 24q^{58} - 60q^{59} + 13q^{60} - 48q^{61} - 5q^{62} + 10q^{63} - 2q^{64} + 40q^{65} - 29q^{66} - 57q^{67} + 18q^{68} + 4q^{69} + 6q^{70} - 39q^{71} - q^{72} - 30q^{73} - 17q^{74} + 57q^{75} - 2q^{76} + 6q^{77} - 19q^{78} - 18q^{79} - 5q^{80} - 90q^{81} + 26q^{82} + 28q^{83} + 4q^{84} - 13q^{85} - 6q^{86} + 13q^{87} + 6q^{88} + 34q^{89} - 20q^{90} - 30q^{91} - 2q^{92} - 16q^{93} + 9q^{94} + 110q^{95} + 4q^{96} - 2q^{98} - 28q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{20} - 9 x^{19} + 41 x^{18} - 119 x^{17} + 245 x^{16} - 404 x^{15} + 623 x^{14} - 898 x^{13} + 1048 x^{12} - 693 x^{11} + 859 x^{10} - 935 x^{9} + 620 x^{8} + 679 x^{7} + 1220 x^{6} + 2241 x^{5} + 1750 x^{4} + 1079 x^{3} + 534 x^{2} + 38 x + 1$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$($$$$-$$$$20\!\cdots\!71$$$$\nu^{19} +$$$$18\!\cdots\!73$$$$\nu^{18} -$$$$79\!\cdots\!96$$$$\nu^{17} +$$$$19\!\cdots\!51$$$$\nu^{16} -$$$$28\!\cdots\!26$$$$\nu^{15} +$$$$18\!\cdots\!94$$$$\nu^{14} +$$$$11\!\cdots\!02$$$$\nu^{13} -$$$$52\!\cdots\!92$$$$\nu^{12} +$$$$15\!\cdots\!68$$$$\nu^{11} -$$$$39\!\cdots\!96$$$$\nu^{10} +$$$$49\!\cdots\!57$$$$\nu^{9} -$$$$37\!\cdots\!13$$$$\nu^{8} +$$$$42\!\cdots\!79$$$$\nu^{7} -$$$$85\!\cdots\!84$$$$\nu^{6} +$$$$51\!\cdots\!86$$$$\nu^{5} -$$$$52\!\cdots\!52$$$$\nu^{4} -$$$$90\!\cdots\!07$$$$\nu^{3} +$$$$16\!\cdots\!40$$$$\nu^{2} +$$$$21\!\cdots\!28$$$$\nu +$$$$25\!\cdots\!26$$$$)/$$$$38\!\cdots\!89$$ $$\beta_{3}$$ $$=$$ $$($$$$34\!\cdots\!09$$$$\nu^{19} -$$$$25\!\cdots\!40$$$$\nu^{18} +$$$$98\!\cdots\!87$$$$\nu^{17} -$$$$22\!\cdots\!61$$$$\nu^{16} +$$$$32\!\cdots\!09$$$$\nu^{15} -$$$$36\!\cdots\!66$$$$\nu^{14} +$$$$48\!\cdots\!43$$$$\nu^{13} -$$$$40\!\cdots\!30$$$$\nu^{12} -$$$$55\!\cdots\!55$$$$\nu^{11} +$$$$27\!\cdots\!87$$$$\nu^{10} -$$$$10\!\cdots\!36$$$$\nu^{9} +$$$$31\!\cdots\!07$$$$\nu^{8} -$$$$64\!\cdots\!57$$$$\nu^{7} +$$$$82\!\cdots\!27$$$$\nu^{6} +$$$$57\!\cdots\!25$$$$\nu^{5} +$$$$17\!\cdots\!51$$$$\nu^{4} +$$$$12\!\cdots\!30$$$$\nu^{3} +$$$$11\!\cdots\!89$$$$\nu^{2} +$$$$62\!\cdots\!68$$$$\nu +$$$$11\!\cdots\!14$$$$)/$$$$38\!\cdots\!89$$ $$\beta_{4}$$ $$=$$ $$($$$$54\!\cdots\!91$$$$\nu^{19} -$$$$72\!\cdots\!29$$$$\nu^{18} +$$$$43\!\cdots\!86$$$$\nu^{17} -$$$$15\!\cdots\!42$$$$\nu^{16} +$$$$38\!\cdots\!71$$$$\nu^{15} -$$$$68\!\cdots\!55$$$$\nu^{14} +$$$$10\!\cdots\!82$$$$\nu^{13} -$$$$14\!\cdots\!30$$$$\nu^{12} +$$$$19\!\cdots\!04$$$$\nu^{11} -$$$$17\!\cdots\!82$$$$\nu^{10} +$$$$67\!\cdots\!58$$$$\nu^{9} -$$$$12\!\cdots\!91$$$$\nu^{8} +$$$$18\!\cdots\!81$$$$\nu^{7} +$$$$85\!\cdots\!45$$$$\nu^{6} -$$$$23\!\cdots\!43$$$$\nu^{5} -$$$$17\!\cdots\!06$$$$\nu^{4} -$$$$36\!\cdots\!17$$$$\nu^{3} -$$$$27\!\cdots\!63$$$$\nu^{2} -$$$$90\!\cdots\!27$$$$\nu -$$$$25\!\cdots\!44$$$$)/$$$$38\!\cdots\!89$$ $$\beta_{5}$$ $$=$$ $$($$$$-$$$$91\!\cdots\!12$$$$\nu^{19} +$$$$92\!\cdots\!59$$$$\nu^{18} -$$$$46\!\cdots\!49$$$$\nu^{17} +$$$$14\!\cdots\!26$$$$\nu^{16} -$$$$34\!\cdots\!14$$$$\nu^{15} +$$$$61\!\cdots\!98$$$$\nu^{14} -$$$$96\!\cdots\!48$$$$\nu^{13} +$$$$14\!\cdots\!61$$$$\nu^{12} -$$$$18\!\cdots\!68$$$$\nu^{11} +$$$$17\!\cdots\!79$$$$\nu^{10} -$$$$15\!\cdots\!28$$$$\nu^{9} +$$$$18\!\cdots\!86$$$$\nu^{8} -$$$$18\!\cdots\!89$$$$\nu^{7} +$$$$32\!\cdots\!98$$$$\nu^{6} -$$$$73\!\cdots\!90$$$$\nu^{5} -$$$$65\!\cdots\!09$$$$\nu^{4} -$$$$71\!\cdots\!71$$$$\nu^{3} +$$$$12\!\cdots\!41$$$$\nu^{2} -$$$$19\!\cdots\!57$$$$\nu +$$$$97\!\cdots\!18$$$$)/$$$$38\!\cdots\!89$$ $$\beta_{6}$$ $$=$$ $$($$$$11\!\cdots\!53$$$$\nu^{19} -$$$$11\!\cdots\!20$$$$\nu^{18} +$$$$57\!\cdots\!38$$$$\nu^{17} -$$$$18\!\cdots\!04$$$$\nu^{16} +$$$$41\!\cdots\!41$$$$\nu^{15} -$$$$74\!\cdots\!78$$$$\nu^{14} +$$$$12\!\cdots\!48$$$$\nu^{13} -$$$$18\!\cdots\!18$$$$\nu^{12} +$$$$23\!\cdots\!25$$$$\nu^{11} -$$$$21\!\cdots\!03$$$$\nu^{10} +$$$$21\!\cdots\!65$$$$\nu^{9} -$$$$22\!\cdots\!46$$$$\nu^{8} +$$$$19\!\cdots\!66$$$$\nu^{7} -$$$$66\!\cdots\!46$$$$\nu^{6} +$$$$11\!\cdots\!62$$$$\nu^{5} +$$$$18\!\cdots\!44$$$$\nu^{4} +$$$$61\!\cdots\!73$$$$\nu^{3} +$$$$70\!\cdots\!20$$$$\nu^{2} +$$$$44\!\cdots\!39$$$$\nu +$$$$31\!\cdots\!48$$$$)/$$$$38\!\cdots\!89$$ $$\beta_{7}$$ $$=$$ $$($$$$13\!\cdots\!80$$$$\nu^{19} -$$$$12\!\cdots\!79$$$$\nu^{18} +$$$$63\!\cdots\!95$$$$\nu^{17} -$$$$19\!\cdots\!72$$$$\nu^{16} +$$$$44\!\cdots\!37$$$$\nu^{15} -$$$$77\!\cdots\!65$$$$\nu^{14} +$$$$12\!\cdots\!51$$$$\nu^{13} -$$$$17\!\cdots\!78$$$$\nu^{12} +$$$$22\!\cdots\!17$$$$\nu^{11} -$$$$18\!\cdots\!08$$$$\nu^{10} +$$$$16\!\cdots\!35$$$$\nu^{9} -$$$$19\!\cdots\!86$$$$\nu^{8} +$$$$17\!\cdots\!10$$$$\nu^{7} +$$$$36\!\cdots\!65$$$$\nu^{6} +$$$$65\!\cdots\!24$$$$\nu^{5} +$$$$16\!\cdots\!35$$$$\nu^{4} +$$$$38\!\cdots\!53$$$$\nu^{3} +$$$$99\!\cdots\!05$$$$\nu^{2} +$$$$65\!\cdots\!70$$$$\nu -$$$$17\!\cdots\!59$$$$)/$$$$38\!\cdots\!89$$ $$\beta_{8}$$ $$=$$ $$($$$$15\!\cdots\!46$$$$\nu^{19} -$$$$14\!\cdots\!69$$$$\nu^{18} +$$$$62\!\cdots\!97$$$$\nu^{17} -$$$$17\!\cdots\!07$$$$\nu^{16} +$$$$36\!\cdots\!49$$$$\nu^{15} -$$$$60\!\cdots\!63$$$$\nu^{14} +$$$$95\!\cdots\!96$$$$\nu^{13} -$$$$13\!\cdots\!54$$$$\nu^{12} +$$$$15\!\cdots\!21$$$$\nu^{11} -$$$$10\!\cdots\!43$$$$\nu^{10} +$$$$15\!\cdots\!37$$$$\nu^{9} -$$$$14\!\cdots\!77$$$$\nu^{8} +$$$$79\!\cdots\!97$$$$\nu^{7} +$$$$99\!\cdots\!72$$$$\nu^{6} +$$$$25\!\cdots\!22$$$$\nu^{5} +$$$$42\!\cdots\!88$$$$\nu^{4} +$$$$32\!\cdots\!45$$$$\nu^{3} +$$$$24\!\cdots\!66$$$$\nu^{2} +$$$$83\!\cdots\!74$$$$\nu +$$$$23\!\cdots\!14$$$$)/$$$$38\!\cdots\!89$$ $$\beta_{9}$$ $$=$$ $$($$$$17\!\cdots\!05$$$$\nu^{19} -$$$$16\!\cdots\!29$$$$\nu^{18} +$$$$77\!\cdots\!84$$$$\nu^{17} -$$$$23\!\cdots\!78$$$$\nu^{16} +$$$$49\!\cdots\!33$$$$\nu^{15} -$$$$81\!\cdots\!86$$$$\nu^{14} +$$$$12\!\cdots\!44$$$$\nu^{13} -$$$$17\!\cdots\!34$$$$\nu^{12} +$$$$21\!\cdots\!68$$$$\nu^{11} -$$$$14\!\cdots\!17$$$$\nu^{10} +$$$$13\!\cdots\!99$$$$\nu^{9} -$$$$16\!\cdots\!97$$$$\nu^{8} +$$$$12\!\cdots\!63$$$$\nu^{7} +$$$$13\!\cdots\!15$$$$\nu^{6} +$$$$94\!\cdots\!62$$$$\nu^{5} +$$$$32\!\cdots\!98$$$$\nu^{4} +$$$$14\!\cdots\!26$$$$\nu^{3} +$$$$93\!\cdots\!17$$$$\nu^{2} +$$$$60\!\cdots\!56$$$$\nu +$$$$35\!\cdots\!58$$$$)/$$$$38\!\cdots\!89$$ $$\beta_{10}$$ $$=$$ $$($$$$-$$$$18\!\cdots\!01$$$$\nu^{19} +$$$$16\!\cdots\!24$$$$\nu^{18} -$$$$74\!\cdots\!80$$$$\nu^{17} +$$$$21\!\cdots\!93$$$$\nu^{16} -$$$$42\!\cdots\!23$$$$\nu^{15} +$$$$69\!\cdots\!98$$$$\nu^{14} -$$$$10\!\cdots\!43$$$$\nu^{13} +$$$$15\!\cdots\!87$$$$\nu^{12} -$$$$17\!\cdots\!86$$$$\nu^{11} +$$$$10\!\cdots\!43$$$$\nu^{10} -$$$$14\!\cdots\!33$$$$\nu^{9} +$$$$16\!\cdots\!97$$$$\nu^{8} -$$$$10\!\cdots\!69$$$$\nu^{7} -$$$$13\!\cdots\!61$$$$\nu^{6} -$$$$23\!\cdots\!44$$$$\nu^{5} -$$$$39\!\cdots\!14$$$$\nu^{4} -$$$$37\!\cdots\!01$$$$\nu^{3} -$$$$17\!\cdots\!10$$$$\nu^{2} -$$$$82\!\cdots\!95$$$$\nu -$$$$33\!\cdots\!96$$$$)/$$$$38\!\cdots\!89$$ $$\beta_{11}$$ $$=$$ $$($$$$-$$$$97\!\cdots\!18$$$$\nu^{19} +$$$$86\!\cdots\!50$$$$\nu^{18} -$$$$38\!\cdots\!79$$$$\nu^{17} +$$$$11\!\cdots\!93$$$$\nu^{16} -$$$$22\!\cdots\!84$$$$\nu^{15} +$$$$35\!\cdots\!58$$$$\nu^{14} -$$$$54\!\cdots\!16$$$$\nu^{13} +$$$$77\!\cdots\!16$$$$\nu^{12} -$$$$87\!\cdots\!03$$$$\nu^{11} +$$$$48\!\cdots\!06$$$$\nu^{10} -$$$$66\!\cdots\!83$$$$\nu^{9} +$$$$75\!\cdots\!02$$$$\nu^{8} -$$$$41\!\cdots\!74$$$$\nu^{7} -$$$$84\!\cdots\!11$$$$\nu^{6} -$$$$11\!\cdots\!62$$$$\nu^{5} -$$$$22\!\cdots\!28$$$$\nu^{4} -$$$$17\!\cdots\!09$$$$\nu^{3} -$$$$10\!\cdots\!93$$$$\nu^{2} -$$$$50\!\cdots\!71$$$$\nu -$$$$56\!\cdots\!41$$$$)/$$$$38\!\cdots\!89$$ $$\beta_{12}$$ $$=$$ $$($$$$11\!\cdots\!14$$$$\nu^{19} -$$$$10\!\cdots\!35$$$$\nu^{18} +$$$$48\!\cdots\!14$$$$\nu^{17} -$$$$14\!\cdots\!53$$$$\nu^{16} +$$$$29\!\cdots\!91$$$$\nu^{15} -$$$$47\!\cdots\!65$$$$\nu^{14} +$$$$73\!\cdots\!88$$$$\nu^{13} -$$$$10\!\cdots\!15$$$$\nu^{12} +$$$$12\!\cdots\!02$$$$\nu^{11} -$$$$81\!\cdots\!47$$$$\nu^{10} +$$$$98\!\cdots\!39$$$$\nu^{9} -$$$$10\!\cdots\!54$$$$\nu^{8} +$$$$69\!\cdots\!73$$$$\nu^{7} +$$$$86\!\cdots\!63$$$$\nu^{6} +$$$$13\!\cdots\!53$$$$\nu^{5} +$$$$25\!\cdots\!49$$$$\nu^{4} +$$$$18\!\cdots\!49$$$$\nu^{3} +$$$$11\!\cdots\!76$$$$\nu^{2} +$$$$51\!\cdots\!87$$$$\nu -$$$$17\!\cdots\!36$$$$)/$$$$38\!\cdots\!89$$ $$\beta_{13}$$ $$=$$ $$($$$$23\!\cdots\!14$$$$\nu^{19} -$$$$20\!\cdots\!72$$$$\nu^{18} +$$$$95\!\cdots\!43$$$$\nu^{17} -$$$$28\!\cdots\!63$$$$\nu^{16} +$$$$58\!\cdots\!37$$$$\nu^{15} -$$$$96\!\cdots\!05$$$$\nu^{14} +$$$$14\!\cdots\!85$$$$\nu^{13} -$$$$21\!\cdots\!68$$$$\nu^{12} +$$$$25\!\cdots\!26$$$$\nu^{11} -$$$$17\!\cdots\!23$$$$\nu^{10} +$$$$20\!\cdots\!69$$$$\nu^{9} -$$$$23\!\cdots\!27$$$$\nu^{8} +$$$$15\!\cdots\!57$$$$\nu^{7} +$$$$14\!\cdots\!09$$$$\nu^{6} +$$$$27\!\cdots\!08$$$$\nu^{5} +$$$$49\!\cdots\!52$$$$\nu^{4} +$$$$36\!\cdots\!12$$$$\nu^{3} +$$$$21\!\cdots\!61$$$$\nu^{2} +$$$$98\!\cdots\!10$$$$\nu +$$$$41\!\cdots\!58$$$$)/$$$$38\!\cdots\!89$$ $$\beta_{14}$$ $$=$$ $$($$$$-$$$$25\!\cdots\!26$$$$\nu^{19} +$$$$23\!\cdots\!63$$$$\nu^{18} -$$$$10\!\cdots\!93$$$$\nu^{17} +$$$$30\!\cdots\!98$$$$\nu^{16} -$$$$62\!\cdots\!19$$$$\nu^{15} +$$$$10\!\cdots\!78$$$$\nu^{14} -$$$$16\!\cdots\!04$$$$\nu^{13} +$$$$23\!\cdots\!50$$$$\nu^{12} -$$$$27\!\cdots\!40$$$$\nu^{11} +$$$$17\!\cdots\!86$$$$\nu^{10} -$$$$22\!\cdots\!30$$$$\nu^{9} +$$$$24\!\cdots\!67$$$$\nu^{8} -$$$$16\!\cdots\!33$$$$\nu^{7} -$$$$17\!\cdots\!75$$$$\nu^{6} -$$$$32\!\cdots\!04$$$$\nu^{5} -$$$$57\!\cdots\!80$$$$\nu^{4} -$$$$45\!\cdots\!52$$$$\nu^{3} -$$$$27\!\cdots\!61$$$$\nu^{2} -$$$$13\!\cdots\!44$$$$\nu -$$$$76\!\cdots\!60$$$$)/$$$$38\!\cdots\!89$$ $$\beta_{15}$$ $$=$$ $$($$$$30\!\cdots\!29$$$$\nu^{19} -$$$$27\!\cdots\!46$$$$\nu^{18} +$$$$12\!\cdots\!28$$$$\nu^{17} -$$$$37\!\cdots\!57$$$$\nu^{16} +$$$$76\!\cdots\!52$$$$\nu^{15} -$$$$12\!\cdots\!06$$$$\nu^{14} +$$$$19\!\cdots\!97$$$$\nu^{13} -$$$$28\!\cdots\!12$$$$\nu^{12} +$$$$33\!\cdots\!44$$$$\nu^{11} -$$$$23\!\cdots\!33$$$$\nu^{10} +$$$$27\!\cdots\!41$$$$\nu^{9} -$$$$30\!\cdots\!27$$$$\nu^{8} +$$$$21\!\cdots\!79$$$$\nu^{7} +$$$$18\!\cdots\!42$$$$\nu^{6} +$$$$36\!\cdots\!72$$$$\nu^{5} +$$$$64\!\cdots\!85$$$$\nu^{4} +$$$$48\!\cdots\!62$$$$\nu^{3} +$$$$28\!\cdots\!25$$$$\nu^{2} +$$$$13\!\cdots\!90$$$$\nu +$$$$25\!\cdots\!17$$$$)/$$$$38\!\cdots\!89$$ $$\beta_{16}$$ $$=$$ $$($$$$31\!\cdots\!48$$$$\nu^{19} -$$$$28\!\cdots\!85$$$$\nu^{18} +$$$$13\!\cdots\!88$$$$\nu^{17} -$$$$38\!\cdots\!50$$$$\nu^{16} +$$$$80\!\cdots\!64$$$$\nu^{15} -$$$$13\!\cdots\!33$$$$\nu^{14} +$$$$20\!\cdots\!82$$$$\nu^{13} -$$$$29\!\cdots\!52$$$$\nu^{12} +$$$$35\!\cdots\!22$$$$\nu^{11} -$$$$24\!\cdots\!89$$$$\nu^{10} +$$$$29\!\cdots\!35$$$$\nu^{9} -$$$$31\!\cdots\!45$$$$\nu^{8} +$$$$22\!\cdots\!06$$$$\nu^{7} +$$$$19\!\cdots\!26$$$$\nu^{6} +$$$$39\!\cdots\!06$$$$\nu^{5} +$$$$70\!\cdots\!06$$$$\nu^{4} +$$$$54\!\cdots\!56$$$$\nu^{3} +$$$$33\!\cdots\!19$$$$\nu^{2} +$$$$16\!\cdots\!12$$$$\nu +$$$$77\!\cdots\!85$$$$)/$$$$38\!\cdots\!89$$ $$\beta_{17}$$ $$=$$ $$($$$$-$$$$33\!\cdots\!96$$$$\nu^{19} +$$$$30\!\cdots\!65$$$$\nu^{18} -$$$$13\!\cdots\!60$$$$\nu^{17} +$$$$40\!\cdots\!04$$$$\nu^{16} -$$$$84\!\cdots\!13$$$$\nu^{15} +$$$$14\!\cdots\!07$$$$\nu^{14} -$$$$21\!\cdots\!06$$$$\nu^{13} +$$$$31\!\cdots\!51$$$$\nu^{12} -$$$$36\!\cdots\!95$$$$\nu^{11} +$$$$25\!\cdots\!14$$$$\nu^{10} -$$$$29\!\cdots\!07$$$$\nu^{9} +$$$$32\!\cdots\!93$$$$\nu^{8} -$$$$22\!\cdots\!17$$$$\nu^{7} -$$$$21\!\cdots\!15$$$$\nu^{6} -$$$$39\!\cdots\!59$$$$\nu^{5} -$$$$73\!\cdots\!92$$$$\nu^{4} -$$$$55\!\cdots\!86$$$$\nu^{3} -$$$$32\!\cdots\!83$$$$\nu^{2} -$$$$16\!\cdots\!54$$$$\nu -$$$$44\!\cdots\!53$$$$)/$$$$38\!\cdots\!89$$ $$\beta_{18}$$ $$=$$ $$($$$$-$$$$35\!\cdots\!58$$$$\nu^{19} +$$$$31\!\cdots\!27$$$$\nu^{18} -$$$$14\!\cdots\!07$$$$\nu^{17} +$$$$42\!\cdots\!86$$$$\nu^{16} -$$$$88\!\cdots\!88$$$$\nu^{15} +$$$$14\!\cdots\!65$$$$\nu^{14} -$$$$22\!\cdots\!20$$$$\nu^{13} +$$$$32\!\cdots\!28$$$$\nu^{12} -$$$$38\!\cdots\!18$$$$\nu^{11} +$$$$26\!\cdots\!62$$$$\nu^{10} -$$$$31\!\cdots\!39$$$$\nu^{9} +$$$$34\!\cdots\!29$$$$\nu^{8} -$$$$23\!\cdots\!57$$$$\nu^{7} -$$$$22\!\cdots\!19$$$$\nu^{6} -$$$$41\!\cdots\!45$$$$\nu^{5} -$$$$77\!\cdots\!16$$$$\nu^{4} -$$$$58\!\cdots\!02$$$$\nu^{3} -$$$$36\!\cdots\!56$$$$\nu^{2} -$$$$17\!\cdots\!55$$$$\nu -$$$$72\!\cdots\!48$$$$)/$$$$38\!\cdots\!89$$ $$\beta_{19}$$ $$=$$ $$($$$$35\!\cdots\!62$$$$\nu^{19} -$$$$31\!\cdots\!60$$$$\nu^{18} +$$$$14\!\cdots\!12$$$$\nu^{17} -$$$$42\!\cdots\!58$$$$\nu^{16} +$$$$87\!\cdots\!51$$$$\nu^{15} -$$$$14\!\cdots\!10$$$$\nu^{14} +$$$$22\!\cdots\!72$$$$\nu^{13} -$$$$32\!\cdots\!03$$$$\nu^{12} +$$$$37\!\cdots\!60$$$$\nu^{11} -$$$$25\!\cdots\!52$$$$\nu^{10} +$$$$30\!\cdots\!40$$$$\nu^{9} -$$$$33\!\cdots\!62$$$$\nu^{8} +$$$$22\!\cdots\!86$$$$\nu^{7} +$$$$23\!\cdots\!27$$$$\nu^{6} +$$$$40\!\cdots\!02$$$$\nu^{5} +$$$$77\!\cdots\!85$$$$\nu^{4} +$$$$58\!\cdots\!09$$$$\nu^{3} +$$$$35\!\cdots\!20$$$$\nu^{2} +$$$$17\!\cdots\!34$$$$\nu +$$$$60\!\cdots\!93$$$$)/$$$$38\!\cdots\!89$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{19} + 2 \beta_{17} + \beta_{15} - \beta_{10} + \beta_{1}$$ $$\nu^{3}$$ $$=$$ $$-\beta_{18} + \beta_{17} + \beta_{14} + \beta_{13} - \beta_{12} - \beta_{11} - 3 \beta_{10} - \beta_{9} - \beta_{8} - \beta_{7} + \beta_{6} - \beta_{5} + \beta_{3} - 1$$ $$\nu^{4}$$ $$=$$ $$-5 \beta_{19} - 5 \beta_{18} - 4 \beta_{17} - 5 \beta_{15} + 4 \beta_{14} + 5 \beta_{13} - 4 \beta_{12} - 5 \beta_{11} - 5 \beta_{10} - 2 \beta_{9} - 2 \beta_{8} - 2 \beta_{5} + \beta_{3} + 5 \beta_{2} - \beta_{1} - 5$$ $$\nu^{5}$$ $$=$$ $$-8 \beta_{19} - 8 \beta_{18} - 8 \beta_{17} - 8 \beta_{15} + 7 \beta_{14} + 7 \beta_{12} - 7 \beta_{11} - 6 \beta_{10} - 2 \beta_{9} - \beta_{8} + 7 \beta_{7} - 6 \beta_{6} - 3 \beta_{5} - \beta_{4} - 7 \beta_{3} + 20 \beta_{2}$$ $$\nu^{6}$$ $$=$$ $$8 \beta_{19} - 3 \beta_{15} + 8 \beta_{14} - 26 \beta_{13} + 49 \beta_{12} - 3 \beta_{11} + 7 \beta_{8} + 18 \beta_{7} - 7 \beta_{6} - 11 \beta_{5} - 10 \beta_{4} - 45 \beta_{3} + 35 \beta_{2} + 8 \beta_{1} + 26$$ $$\nu^{7}$$ $$=$$ $$95 \beta_{19} + 49 \beta_{18} + 50 \beta_{17} - 3 \beta_{16} + 4 \beta_{15} - 50 \beta_{13} + 95 \beta_{12} - 3 \beta_{11} + 35 \beta_{10} + 21 \beta_{9} - 2 \beta_{8} + 32 \beta_{7} + 46 \beta_{6} - 67 \beta_{5} - 67 \beta_{4} - 143 \beta_{3} - 2 \beta_{1} + 49$$ $$\nu^{8}$$ $$=$$ $$229 \beta_{19} + 86 \beta_{18} + 86 \beta_{17} - 90 \beta_{16} - 90 \beta_{15} - 56 \beta_{14} + 56 \beta_{13} - 57 \beta_{11} + 158 \beta_{10} + 131 \beta_{9} - 158 \beta_{8} + 95 \beta_{7} + 287 \beta_{6} - 287 \beta_{5} - 301 \beta_{4} - 301 \beta_{3} - 206 \beta_{2} - 170 \beta_{1} - 57$$ $$\nu^{9}$$ $$=$$ $$-289 \beta_{18} - 299 \beta_{17} - 590 \beta_{16} - 622 \beta_{15} - 289 \beta_{14} + 577 \beta_{13} - 577 \beta_{12} - 299 \beta_{11} + 471 \beta_{10} + 436 \beta_{9} - 649 \beta_{8} + 472 \beta_{7} + 907 \beta_{6} - 743 \beta_{5} - 907 \beta_{4} - 472 \beta_{3} - 649 \beta_{2} - 743 \beta_{1} - 622$$ $$\nu^{10}$$ $$=$$ $$-1320 \beta_{19} - 1698 \beta_{18} - 1722 \beta_{17} - 1992 \beta_{16} - 1722 \beta_{15} - 907 \beta_{14} + 1320 \beta_{13} - 1698 \beta_{12} - 907 \beta_{11} + 1078 \beta_{10} + 863 \beta_{9} - 1226 \beta_{8} + 1837 \beta_{7} + 1969 \beta_{6} - 1078 \beta_{5} - 1837 \beta_{4} - 611 \beta_{3} - 974 \beta_{2} - 1837 \beta_{1} - 1992$$ $$\nu^{11}$$ $$=$$ $$-3463 \beta_{19} - 3463 \beta_{18} - 3926 \beta_{17} - 3926 \beta_{16} - 1941 \beta_{15} - 1941 \beta_{14} - 1865 \beta_{12} - 1865 \beta_{11} + 2091 \beta_{10} + 739 \beta_{9} + 4637 \beta_{7} + 3252 \beta_{6} - 413 \beta_{5} - 2091 \beta_{4} - 739 \beta_{3} - 413 \beta_{2} - 3252 \beta_{1} - 3834$$ $$\nu^{12}$$ $$=$$ $$-2513 \beta_{19} - 4826 \beta_{17} - 2830 \beta_{16} + 1911 \beta_{15} - 2830 \beta_{14} - 7709 \beta_{13} + 1911 \beta_{12} - 2513 \beta_{11} + 4454 \beta_{10} - 370 \beta_{9} + 7361 \beta_{8} + 6991 \beta_{7} + 4454 \beta_{6} + 1185 \beta_{5} + 370 \beta_{4} - 815 \beta_{3} + 370 \beta_{2} - 5523 \beta_{1} - 4826$$ $$\nu^{13}$$ $$=$$ $$8662 \beta_{19} + 20731 \beta_{18} - 4222 \beta_{17} + 8101 \beta_{16} + 8662 \beta_{15} - 4084 \beta_{14} - 20731 \beta_{13} + 8101 \beta_{12} + 13475 \beta_{10} + 3895 \beta_{9} + 23759 \beta_{8} + 3895 \beta_{7} + 5586 \beta_{6} + 4794 \beta_{4} - 5586 \beta_{2} - 13475 \beta_{1} - 4084$$ $$\nu^{14}$$ $$=$$ $$24658 \beta_{19} + 61892 \beta_{18} - 17370 \beta_{17} + 24658 \beta_{16} - 17370 \beta_{14} - 16578 \beta_{13} - 3283 \beta_{12} + 16578 \beta_{11} + 44628 \beta_{10} + 39723 \beta_{9} + 39723 \beta_{8} - 4767 \beta_{7} + 6147 \beta_{6} - 4767 \beta_{5} + 6147 \beta_{3} - 36601 \beta_{2} - 36601 \beta_{1} - 3283$$ $$\nu^{15}$$ $$=$$ $$-4267 \beta_{19} + 78204 \beta_{18} - 79211 \beta_{17} - 45997 \beta_{15} - 84351 \beta_{14} + 45997 \beta_{13} - 79211 \beta_{12} + 78204 \beta_{11} + 117110 \beta_{10} + 146105 \beta_{9} + 33023 \beta_{8} + 33023 \beta_{5} - 23278 \beta_{4} + 41264 \beta_{3} - 117110 \beta_{2} - 64542 \beta_{1} - 4267$$ $$\nu^{16}$$ $$=$$ $$-166320 \beta_{19} - 27491 \beta_{18} - 166320 \beta_{17} - 136360 \beta_{16} - 27491 \beta_{15} - 263215 \beta_{14} + 136360 \beta_{13} - 263215 \beta_{12} + 239937 \beta_{11} + 192054 \beta_{10} + 270306 \beta_{9} + 32557 \beta_{8} + 12974 \beta_{7} - 45531 \beta_{6} + 302863 \beta_{5} + 32557 \beta_{4} + 224611 \beta_{3} - 249762 \beta_{2}$$ $$\nu^{17}$$ $$=$$ $$-462360 \beta_{19} - 205192 \beta_{18} - 205192 \beta_{16} + 540448 \beta_{15} - 462360 \beta_{14} - 33354 \beta_{13} - 519271 \beta_{12} + 540448 \beta_{11} + 300102 \beta_{8} - 262398 \beta_{7} - 300102 \beta_{6} + 1204037 \beta_{5} + 666303 \beta_{4} + 1012960 \beta_{3} - 346657 \beta_{2} + 403905 \beta_{1} + 33354$$ $$\nu^{18}$$ $$=$$ $$-655610 \beta_{19} + 253547 \beta_{18} + 998887 \beta_{17} + 966405 \beta_{16} + 2549581 \beta_{15} - 998887 \beta_{13} - 655610 \beta_{12} + 966405 \beta_{11} - 1212922 \beta_{10} - 1649575 \beta_{9} + 1546378 \beta_{8} - 2006883 \beta_{7} - 1570230 \beta_{6} + 3219805 \beta_{5} + 3219805 \beta_{4} + 3623373 \beta_{3} + 1546378 \beta_{1} + 253547$$ $$\nu^{19}$$ $$=$$ $$-497532 \beta_{19} + 3125841 \beta_{18} + 3125841 \beta_{17} + 6415758 \beta_{16} + 6415758 \beta_{15} + 2862497 \beta_{14} - 2862497 \beta_{13} + 1927538 \beta_{11} - 4817365 \beta_{10} - 5619036 \beta_{9} + 4817365 \beta_{8} - 7776269 \beta_{7} - 6815618 \beta_{6} + 6815618 \beta_{5} + 10048369 \beta_{4} + 10048369 \beta_{3} + 2272100 \beta_{2} + 4429333 \beta_{1} + 1927538$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/322\mathbb{Z}\right)^\times$$.

 $$n$$ $$185$$ $$281$$ $$\chi(n)$$ $$1$$ $$\beta_{19}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
29.1
 0.140215 + 0.975217i −0.0997080 − 0.693485i 1.61117 + 1.85938i −0.753480 − 0.869562i −0.637182 − 0.187094i 2.47844 + 0.727734i 1.61117 − 1.85938i −0.753480 + 0.869562i 1.45482 + 0.934959i −0.0394097 − 0.0253271i 1.45482 − 0.934959i −0.0394097 + 0.0253271i −0.637182 + 0.187094i 2.47844 − 0.727734i 0.140215 − 0.975217i −0.0997080 + 0.693485i −0.643071 + 1.40813i 0.988210 − 2.16388i −0.643071 − 1.40813i 0.988210 + 2.16388i
0.415415 0.909632i 0.0141569 + 0.00415683i −0.654861 0.755750i −3.50493 + 2.25248i 0.00966216 0.0111507i −0.142315 + 0.989821i −0.959493 + 0.281733i −2.52358 1.62180i 0.592928 + 4.12391i
29.2 0.415415 0.909632i 1.63173 + 0.479119i −0.654861 0.755750i 2.49238 1.60176i 1.11367 1.28524i −0.142315 + 0.989821i −0.959493 + 0.281733i −0.0907762 0.0583384i −0.421636 2.93254i
71.1 0.841254 + 0.540641i −0.207825 1.44545i 0.415415 + 0.909632i 1.07127 + 0.314552i 0.606638 1.32835i −0.654861 + 0.755750i −0.142315 + 0.989821i 0.832335 0.244396i 0.731146 + 0.843788i
71.2 0.841254 + 0.540641i 0.306062 + 2.12871i 0.415415 + 0.909632i −0.500990 0.147104i −0.893390 + 1.95625i −0.654861 + 0.755750i −0.142315 + 0.989821i −1.55924 + 0.457833i −0.341929 0.394607i
85.1 −0.654861 + 0.755750i −1.39992 + 0.899671i −0.142315 0.989821i −0.109762 0.240346i 0.236824 1.64714i −0.959493 + 0.281733i 0.841254 + 0.540641i −0.0958901 + 0.209970i 0.253520 + 0.0744403i
85.2 −0.654861 + 0.755750i 1.33176 0.855871i −0.142315 0.989821i 0.426940 + 0.934869i −0.225294 + 1.56696i −0.959493 + 0.281733i 0.841254 + 0.540641i −0.205171 + 0.449262i −0.986114 0.289549i
127.1 0.841254 0.540641i −0.207825 + 1.44545i 0.415415 0.909632i 1.07127 0.314552i 0.606638 + 1.32835i −0.654861 0.755750i −0.142315 0.989821i 0.832335 + 0.244396i 0.731146 0.843788i
127.2 0.841254 0.540641i 0.306062 2.12871i 0.415415 0.909632i −0.500990 + 0.147104i −0.893390 1.95625i −0.654861 0.755750i −0.142315 0.989821i −1.55924 0.457833i −0.341929 + 0.394607i
141.1 −0.142315 0.989821i −1.13381 + 2.48271i −0.959493 + 0.281733i −2.36481 + 2.72913i 2.61880 + 0.768948i 0.841254 0.540641i 0.415415 + 0.909632i −2.91372 3.36261i 3.03790 + 1.95234i
141.2 −0.142315 0.989821i −0.395954 + 0.867019i −0.959493 + 0.281733i 0.0640602 0.0739294i 0.914544 + 0.268534i 0.841254 0.540641i 0.415415 + 0.909632i 1.36964 + 1.58065i −0.0822937 0.0528869i
169.1 −0.142315 + 0.989821i −1.13381 2.48271i −0.959493 0.281733i −2.36481 2.72913i 2.61880 0.768948i 0.841254 + 0.540641i 0.415415 0.909632i −2.91372 + 3.36261i 3.03790 1.95234i
169.2 −0.142315 + 0.989821i −0.395954 0.867019i −0.959493 0.281733i 0.0640602 + 0.0739294i 0.914544 0.268534i 0.841254 + 0.540641i 0.415415 0.909632i 1.36964 1.58065i −0.0822937 + 0.0528869i
197.1 −0.654861 0.755750i −1.39992 0.899671i −0.142315 + 0.989821i −0.109762 + 0.240346i 0.236824 + 1.64714i −0.959493 0.281733i 0.841254 0.540641i −0.0958901 0.209970i 0.253520 0.0744403i
197.2 −0.654861 0.755750i 1.33176 + 0.855871i −0.142315 + 0.989821i 0.426940 0.934869i −0.225294 1.56696i −0.959493 0.281733i 0.841254 0.540641i −0.205171 0.449262i −0.986114 + 0.289549i
211.1 0.415415 + 0.909632i 0.0141569 0.00415683i −0.654861 + 0.755750i −3.50493 2.25248i 0.00966216 + 0.0111507i −0.142315 0.989821i −0.959493 0.281733i −2.52358 + 1.62180i 0.592928 4.12391i
211.2 0.415415 + 0.909632i 1.63173 0.479119i −0.654861 + 0.755750i 2.49238 + 1.60176i 1.11367 + 1.28524i −0.142315 0.989821i −0.959493 0.281733i −0.0907762 + 0.0583384i −0.421636 + 2.93254i
225.1 −0.959493 0.281733i −0.358877 + 0.414166i 0.841254 + 0.540641i 0.138179 0.961058i 0.461024 0.296282i 0.415415 + 0.909632i −0.654861 0.755750i 0.384204 + 2.67220i −0.403343 + 0.883198i
225.2 −0.959493 0.281733i 2.21268 2.55356i 0.841254 + 0.540641i −0.212341 + 1.47686i −2.84247 + 1.82674i 0.415415 + 0.909632i −0.654861 0.755750i −1.19781 8.33096i 0.619820 1.35722i
239.1 −0.959493 + 0.281733i −0.358877 0.414166i 0.841254 0.540641i 0.138179 + 0.961058i 0.461024 + 0.296282i 0.415415 0.909632i −0.654861 + 0.755750i 0.384204 2.67220i −0.403343 0.883198i
239.2 −0.959493 + 0.281733i 2.21268 + 2.55356i 0.841254 0.540641i −0.212341 1.47686i −2.84247 1.82674i 0.415415 0.909632i −0.654861 + 0.755750i −1.19781 + 8.33096i 0.619820 + 1.35722i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 239.2 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
23.c even 11 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 322.2.i.c 20
23.c even 11 1 inner 322.2.i.c 20
23.c even 11 1 7406.2.a.bp 10
23.d odd 22 1 7406.2.a.bo 10

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
322.2.i.c 20 1.a even 1 1 trivial
322.2.i.c 20 23.c even 11 1 inner
7406.2.a.bo 10 23.d odd 22 1
7406.2.a.bp 10 23.c even 11 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{20} - \cdots$$ acting on $$S_{2}^{\mathrm{new}}(322, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} )^{2}$$
$3$ $$1 - 128 T + 4329 T^{2} + 9015 T^{3} + 15947 T^{4} - 490 T^{5} + 4945 T^{6} - 12368 T^{7} + 8075 T^{8} - 4477 T^{9} + 4643 T^{10} - 1694 T^{11} + 705 T^{12} - 777 T^{13} + 479 T^{14} - 258 T^{15} + 151 T^{16} - 39 T^{17} + 17 T^{18} - 4 T^{19} + T^{20}$$
$5$ $$1 - 9 T + 63 T^{2} + 244 T^{3} + 1576 T^{4} + 1840 T^{5} - 1071 T^{6} + 2215 T^{7} + 1752 T^{8} - 7810 T^{9} + 9406 T^{10} - 9042 T^{11} + 5460 T^{12} - 1697 T^{13} + 1253 T^{14} + 327 T^{15} + 12 T^{16} - 43 T^{17} + 6 T^{18} + 5 T^{19} + T^{20}$$
$7$ $$( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} )^{2}$$
$11$ $$192721 - 5268 T + 806677 T^{2} - 2175003 T^{3} + 2174777 T^{4} - 2320077 T^{5} + 2201044 T^{6} - 669727 T^{7} - 44423 T^{8} - 200376 T^{9} + 181930 T^{10} - 59323 T^{11} + 31176 T^{12} - 12604 T^{13} + 4791 T^{14} - 1431 T^{15} + 462 T^{16} - 101 T^{17} + 31 T^{18} - 6 T^{19} + T^{20}$$
$13$ $$1804635361 - 4789647788 T + 10314836968 T^{2} - 9450181266 T^{3} + 5567192461 T^{4} - 1863534052 T^{5} + 174476244 T^{6} + 196554533 T^{7} - 114849060 T^{8} + 26322879 T^{9} + 3359401 T^{10} - 4757181 T^{11} + 1898955 T^{12} - 441722 T^{13} + 62061 T^{14} - 3657 T^{15} + 51 T^{16} - 183 T^{17} + 79 T^{18} - 14 T^{19} + T^{20}$$
$17$ $$1617407089 - 438123998 T - 1252626270 T^{2} + 1746115717 T^{3} - 365182876 T^{4} - 870530248 T^{5} + 1292361252 T^{6} - 919775930 T^{7} + 474954954 T^{8} - 177654785 T^{9} + 54259052 T^{10} - 14262446 T^{11} + 3724436 T^{12} - 1000960 T^{13} + 223636 T^{14} - 34689 T^{15} + 4243 T^{16} - 533 T^{17} + 61 T^{18} - 7 T^{19} + T^{20}$$
$19$ $$4829833009 + 24655798175 T + 136493696047 T^{2} + 133708489923 T^{3} + 62019512126 T^{4} + 11353862785 T^{5} + 4026591522 T^{6} - 1344918053 T^{7} + 322391099 T^{8} + 94235108 T^{9} + 42538618 T^{10} + 9656075 T^{11} - 897736 T^{12} - 368983 T^{13} + 10485 T^{14} + 6524 T^{15} + 836 T^{16} - 268 T^{17} + 12 T^{18} + 2 T^{19} + T^{20}$$
$23$ $$41426511213649 - 16210373953167 T + 3680616308207 T^{2} - 286005337548 T^{3} - 36712900472 T^{4} + 3649406481 T^{5} - 1906556733 T^{6} + 1026627126 T^{7} - 245097338 T^{8} + 19899186 T^{9} + 4764991 T^{10} + 865182 T^{11} - 463322 T^{12} + 84378 T^{13} - 6813 T^{14} + 567 T^{15} - 248 T^{16} - 84 T^{17} + 47 T^{18} - 9 T^{19} + T^{20}$$
$29$ $$31841190481 + 75142397305 T + 220298629872 T^{2} - 64886270588 T^{3} + 776727201994 T^{4} - 888319799471 T^{5} + 432706766392 T^{6} - 104194058469 T^{7} + 19457629711 T^{8} - 4615024898 T^{9} + 1035622743 T^{10} - 197482362 T^{11} + 31124757 T^{12} - 3065932 T^{13} + 255802 T^{14} - 32456 T^{15} + 3955 T^{16} - 592 T^{17} + 40 T^{18} - 2 T^{19} + T^{20}$$
$31$ $$1216734745249 - 1665193599169 T + 4241708283559 T^{2} - 5928408703225 T^{3} + 5517087990633 T^{4} - 3505935631742 T^{5} + 1626235173438 T^{6} - 580710880164 T^{7} + 183242536042 T^{8} - 50959139440 T^{9} + 11520968801 T^{10} - 2019613574 T^{11} + 274316887 T^{12} - 29419113 T^{13} + 2452936 T^{14} - 151905 T^{15} + 11684 T^{16} - 2201 T^{17} + 338 T^{18} - 28 T^{19} + T^{20}$$
$37$ $$66617158609 + 720745916822 T + 2686440155425 T^{2} + 3900555098707 T^{3} + 3511612421651 T^{4} + 2159960397524 T^{5} + 960527379532 T^{6} + 314589647225 T^{7} + 77495543041 T^{8} + 14382406937 T^{9} + 1971488948 T^{10} + 146979598 T^{11} - 8838268 T^{12} - 4438285 T^{13} - 543148 T^{14} - 25019 T^{15} + 7456 T^{16} + 1505 T^{17} + 225 T^{18} + 17 T^{19} + T^{20}$$
$41$ $$214543778119969 - 336643588872900 T + 139781504818063 T^{2} - 6145655033359 T^{3} + 14414788209999 T^{4} - 6006506068634 T^{5} + 1356398205131 T^{6} - 219340822336 T^{7} + 67208152377 T^{8} - 5460843408 T^{9} + 1432308955 T^{10} - 154733221 T^{11} + 18870483 T^{12} - 4063797 T^{13} + 53094 T^{14} - 36899 T^{15} + 10068 T^{16} + 991 T^{17} + 177 T^{18} + 7 T^{19} + T^{20}$$
$43$ $$4261696046298889 + 3738384755603487 T + 1773647368000750 T^{2} + 482223225951918 T^{3} + 73991440236164 T^{4} - 453439929415 T^{5} - 3342866167212 T^{6} - 594881231329 T^{7} - 10992244439 T^{8} + 4605906877 T^{9} + 3724188777 T^{10} + 1441889033 T^{11} + 295700343 T^{12} + 37259659 T^{13} + 4575312 T^{14} + 419935 T^{15} + 40940 T^{16} + 3098 T^{17} + 256 T^{18} + 17 T^{19} + T^{20}$$
$47$ $$( -27697 + 64120 T + 116810 T^{2} - 147366 T^{3} - 228381 T^{4} - 86514 T^{5} - 9349 T^{6} + 1288 T^{7} + 397 T^{8} + 34 T^{9} + T^{10} )^{2}$$
$53$ $$1359470641 - 1740016232 T + 12615633110 T^{2} - 5611190977 T^{3} + 14835334420 T^{4} + 6653755180 T^{5} + 7295632802 T^{6} - 1244393824 T^{7} + 1275004564 T^{8} + 308301839 T^{9} + 53613296 T^{10} + 22446412 T^{11} + 787048 T^{12} - 2070016 T^{13} - 478526 T^{14} + 8531 T^{15} + 22247 T^{16} + 4855 T^{17} + 553 T^{18} + 35 T^{19} + T^{20}$$
$59$ $$45473767379215681 + 54735975481725065 T + 26078378587289842 T^{2} + 2854992817261761 T^{3} + 306810303700109 T^{4} - 327561305638721 T^{5} - 32791013479312 T^{6} - 8205443613543 T^{7} + 1205878663363 T^{8} + 638044209506 T^{9} + 153286374891 T^{10} + 24643278746 T^{11} + 3282806234 T^{12} + 395961437 T^{13} + 46546736 T^{14} + 5051586 T^{15} + 467366 T^{16} + 33851 T^{17} + 1774 T^{18} + 60 T^{19} + T^{20}$$
$61$ $$1267651723185409 + 288931733271222 T + 392421997413310 T^{2} + 170870595103688 T^{3} + 17607214659637 T^{4} + 1827682782885 T^{5} + 7303457082180 T^{6} + 4923523873860 T^{7} + 1763302581958 T^{8} + 430492943573 T^{9} + 80625692016 T^{10} + 12628226600 T^{11} + 1782513709 T^{12} + 236815761 T^{13} + 29302336 T^{14} + 3219037 T^{15} + 298049 T^{16} + 22126 T^{17} + 1242 T^{18} + 48 T^{19} + T^{20}$$
$67$ $$266368806414481 + 1664829876281602 T + 4167692064903735 T^{2} + 3675639467783764 T^{3} + 1657406841715570 T^{4} + 438459197568866 T^{5} + 74284378558409 T^{6} + 10848159277785 T^{7} + 2525694060766 T^{8} + 611610280556 T^{9} + 155514872074 T^{10} + 30825721355 T^{11} + 5206303417 T^{12} + 714459054 T^{13} + 81609145 T^{14} + 7748489 T^{15} + 605110 T^{16} + 37693 T^{17} + 1772 T^{18} + 57 T^{19} + T^{20}$$
$71$ $$7137751125649 + 6972153809818 T + 6913377207773 T^{2} + 2795595707768 T^{3} - 92178971544 T^{4} - 832618571256 T^{5} - 389392011262 T^{6} - 35102706196 T^{7} + 54960736232 T^{8} + 33909826907 T^{9} + 11570216516 T^{10} + 2748047863 T^{11} + 523703457 T^{12} + 79004539 T^{13} + 9065029 T^{14} + 856428 T^{15} + 78364 T^{16} + 7570 T^{17} + 702 T^{18} + 39 T^{19} + T^{20}$$
$73$ $$50793658619089 + 19741270971980 T + 22395905250617 T^{2} - 2670181841711 T^{3} + 3499470665295 T^{4} + 1977284745820 T^{5} + 474259559161 T^{6} - 57035625996 T^{7} - 1654418349 T^{8} - 1573116655 T^{9} + 1099009913 T^{10} - 71650216 T^{11} - 21833307 T^{12} - 146429 T^{13} + 928821 T^{14} + 225752 T^{15} + 43527 T^{16} + 4973 T^{17} + 437 T^{18} + 30 T^{19} + T^{20}$$
$79$ $$726855781029169 - 993429048037437 T + 1014355846630500 T^{2} - 443031623960640 T^{3} + 234519490950239 T^{4} - 20247889962795 T^{5} + 8021463433214 T^{6} - 3493921217919 T^{7} + 791105876253 T^{8} - 109147390814 T^{9} - 10483288936 T^{10} + 2489419512 T^{11} + 437162603 T^{12} + 39377465 T^{13} + 5300189 T^{14} + 287781 T^{15} - 4287 T^{16} + 46 T^{17} + 159 T^{18} + 18 T^{19} + T^{20}$$
$83$ $$1528396860228879169 - 2543414689097477818 T + 1747235725078080228 T^{2} - 654777601050519830 T^{3} + 157783958405295497 T^{4} - 27710434691116616 T^{5} + 4015133076961538 T^{6} - 483840982432718 T^{7} + 50410130360663 T^{8} - 4886645001557 T^{9} + 455612770768 T^{10} - 43010657008 T^{11} + 3928089679 T^{12} - 304839767 T^{13} + 23784381 T^{14} - 1540362 T^{15} + 134860 T^{16} - 9342 T^{17} + 703 T^{18} - 28 T^{19} + T^{20}$$
$89$ $$24726861484468441 + 22111534640690106 T + 15489992122404845 T^{2} + 5240586145655279 T^{3} + 1359029110923399 T^{4} + 245115518900929 T^{5} + 55007864461866 T^{6} + 6268659186085 T^{7} + 729859072343 T^{8} + 61848080993 T^{9} + 34407355642 T^{10} - 4925799734 T^{11} + 272915877 T^{12} + 90340593 T^{13} - 13198227 T^{14} - 24590 T^{15} + 118812 T^{16} - 10555 T^{17} + 636 T^{18} - 34 T^{19} + T^{20}$$
$97$ $$1962561500851249 - 7905083181439455 T + 12590422095138528 T^{2} - 10240464011652963 T^{3} + 5017539712532246 T^{4} - 1604356195564067 T^{5} + 376624351969442 T^{6} - 64903959322851 T^{7} + 9188952410301 T^{8} - 839707131857 T^{9} + 62914527534 T^{10} + 1527440002 T^{11} - 219657281 T^{12} + 47501795 T^{13} + 2103682 T^{14} - 79244 T^{15} + 17517 T^{16} + 209 T^{17} + 103 T^{18} + T^{20}$$