# Properties

 Label 73.2.h.a Level $73$ Weight $2$ Character orbit 73.h Analytic conductor $0.583$ Analytic rank $0$ Dimension $20$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$73$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 73.h (of order $$12$$, degree $$4$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$0.582907934755$$ Analytic rank: $$0$$ Dimension: $$20$$ Relative dimension: $$5$$ over $$\Q(\zeta_{12})$$ Coefficient field: $$\mathbb{Q}[x]/(x^{20} + \cdots)$$ Defining polynomial: $$x^{20} + 28 x^{18} + 326 x^{16} + 2044 x^{14} + 7471 x^{12} + 16090 x^{10} + 19590 x^{8} + 12030 x^{6} + 2877 x^{4} + 230 x^{2} + 1$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{12}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q -\beta_{9} q^{2} + ( \beta_{2} + \beta_{6} - \beta_{8} - \beta_{11} + \beta_{13} - \beta_{16} ) q^{3} + ( -\beta_{2} - \beta_{7} - \beta_{10} + \beta_{16} ) q^{4} + ( -1 - \beta_{3} - \beta_{4} + \beta_{5} - \beta_{6} + \beta_{7} + \beta_{8} + \beta_{9} + \beta_{10} + \beta_{11} - \beta_{17} ) q^{5} + ( 1 - 2 \beta_{2} + \beta_{3} - \beta_{5} - \beta_{6} - \beta_{9} - \beta_{10} - \beta_{12} - \beta_{13} - \beta_{14} + \beta_{16} + \beta_{17} - \beta_{18} - \beta_{19} ) q^{6} + ( -1 + \beta_{2} + \beta_{3} + \beta_{4} + 2 \beta_{6} - \beta_{7} - \beta_{10} + \beta_{12} + \beta_{13} + \beta_{17} + \beta_{19} ) q^{7} + ( 1 - \beta_{1} + \beta_{4} - \beta_{5} - \beta_{6} - \beta_{8} ) q^{8} + ( -\beta_{2} - \beta_{3} - \beta_{4} - \beta_{6} + \beta_{8} + \beta_{10} - 2 \beta_{12} - \beta_{13} + \beta_{16} ) q^{9} +O(q^{10})$$ $$q -\beta_{9} q^{2} + ( \beta_{2} + \beta_{6} - \beta_{8} - \beta_{11} + \beta_{13} - \beta_{16} ) q^{3} + ( -\beta_{2} - \beta_{7} - \beta_{10} + \beta_{16} ) q^{4} + ( -1 - \beta_{3} - \beta_{4} + \beta_{5} - \beta_{6} + \beta_{7} + \beta_{8} + \beta_{9} + \beta_{10} + \beta_{11} - \beta_{17} ) q^{5} + ( 1 - 2 \beta_{2} + \beta_{3} - \beta_{5} - \beta_{6} - \beta_{9} - \beta_{10} - \beta_{12} - \beta_{13} - \beta_{14} + \beta_{16} + \beta_{17} - \beta_{18} - \beta_{19} ) q^{6} + ( -1 + \beta_{2} + \beta_{3} + \beta_{4} + 2 \beta_{6} - \beta_{7} - \beta_{10} + \beta_{12} + \beta_{13} + \beta_{17} + \beta_{19} ) q^{7} + ( 1 - \beta_{1} + \beta_{4} - \beta_{5} - \beta_{6} - \beta_{8} ) q^{8} + ( -\beta_{2} - \beta_{3} - \beta_{4} - \beta_{6} + \beta_{8} + \beta_{10} - 2 \beta_{12} - \beta_{13} + \beta_{16} ) q^{9} + ( -3 + \beta_{1} + \beta_{2} + \beta_{5} + 2 \beta_{6} + \beta_{7} - \beta_{8} + \beta_{9} - \beta_{10} + 2 \beta_{12} + \beta_{13} + \beta_{14} - \beta_{16} + \beta_{17} + \beta_{19} ) q^{10} + ( -1 + \beta_{1} + \beta_{2} + \beta_{3} + \beta_{5} + \beta_{6} + \beta_{9} + \beta_{10} + \beta_{11} + \beta_{12} - \beta_{13} + \beta_{14} - \beta_{16} - \beta_{17} + \beta_{18} - \beta_{19} ) q^{11} + ( 2 - \beta_{1} + 2 \beta_{4} - \beta_{5} - \beta_{7} + \beta_{8} - \beta_{9} - \beta_{10} - \beta_{11} - \beta_{14} + \beta_{16} + \beta_{17} + \beta_{19} ) q^{12} + ( -1 - \beta_{6} + \beta_{7} - \beta_{9} + \beta_{11} - \beta_{13} + \beta_{15} ) q^{13} + ( -1 + \beta_{1} + 3 \beta_{2} - \beta_{4} + \beta_{6} + \beta_{7} + \beta_{9} + 2 \beta_{10} - \beta_{11} + \beta_{14} - \beta_{15} - 2 \beta_{16} - \beta_{17} + \beta_{18} ) q^{14} + ( 1 + \beta_{1} - \beta_{3} - \beta_{6} + \beta_{9} + 2 \beta_{10} + \beta_{12} + \beta_{14} - \beta_{16} - \beta_{17} + \beta_{18} ) q^{15} + ( \beta_{3} + \beta_{5} + \beta_{6} - 2 \beta_{8} + \beta_{9} + \beta_{10} + \beta_{13} + \beta_{14} - \beta_{15} - \beta_{16} - 2 \beta_{17} - \beta_{19} ) q^{16} + ( 2 - \beta_{2} - \beta_{3} - \beta_{4} - \beta_{5} - 2 \beta_{6} + \beta_{8} + \beta_{10} - \beta_{11} - \beta_{12} - \beta_{13} - \beta_{14} - \beta_{17} - \beta_{18} - \beta_{19} ) q^{17} + ( -3 - \beta_{1} - \beta_{3} - 2 \beta_{4} + 3 \beta_{5} + \beta_{7} + 4 \beta_{8} + \beta_{9} - 2 \beta_{10} + 2 \beta_{11} + 2 \beta_{12} + \beta_{13} - \beta_{14} + 2 \beta_{15} + \beta_{16} - \beta_{17} + \beta_{18} + \beta_{19} ) q^{18} + ( 1 + \beta_{3} - 3 \beta_{5} - 4 \beta_{6} - \beta_{7} - 2 \beta_{9} + \beta_{10} - 2 \beta_{12} - 2 \beta_{13} - 2 \beta_{14} - \beta_{15} + \beta_{16} - \beta_{18} - 2 \beta_{19} ) q^{19} + ( 4 - \beta_{1} - 3 \beta_{2} + \beta_{3} + \beta_{4} - 4 \beta_{5} - 4 \beta_{6} - \beta_{7} + \beta_{9} + 2 \beta_{10} - 3 \beta_{12} - 2 \beta_{13} - \beta_{14} - \beta_{15} + 2 \beta_{16} + \beta_{17} - \beta_{18} - \beta_{19} ) q^{20} + ( 2 - 3 \beta_{1} - 2 \beta_{2} - 3 \beta_{3} - \beta_{4} + 2 \beta_{5} - \beta_{8} + 3 \beta_{9} + \beta_{10} + 2 \beta_{11} - \beta_{12} + 2 \beta_{13} + 2 \beta_{14} - \beta_{15} - \beta_{16} - \beta_{17} - \beta_{18} + \beta_{19} ) q^{21} + ( \beta_{4} + 2 \beta_{6} + 2 \beta_{7} - \beta_{11} + \beta_{15} + 2 \beta_{17} + \beta_{19} ) q^{22} + ( -2 + 3 \beta_{2} - \beta_{3} + 2 \beta_{5} + 2 \beta_{6} + 2 \beta_{8} - 2 \beta_{9} - 2 \beta_{10} - \beta_{11} + 2 \beta_{12} + 2 \beta_{15} + \beta_{17} + 2 \beta_{18} + 2 \beta_{19} ) q^{23} + ( \beta_{1} + 4 \beta_{2} - \beta_{3} - \beta_{4} + 2 \beta_{6} - 2 \beta_{8} - 4 \beta_{9} - 2 \beta_{11} - 2 \beta_{16} + \beta_{18} - \beta_{19} ) q^{24} + ( -\beta_{1} - 2 \beta_{2} - 2 \beta_{3} + \beta_{5} - 2 \beta_{7} - \beta_{9} - 2 \beta_{10} - 2 \beta_{12} + \beta_{16} + \beta_{17} ) q^{25} + ( -3 + \beta_{1} + 2 \beta_{4} + 2 \beta_{5} + 3 \beta_{6} + 3 \beta_{9} + \beta_{12} + \beta_{13} + \beta_{14} - \beta_{16} + \beta_{18} + \beta_{19} ) q^{26} + ( -2 - 2 \beta_{2} + \beta_{3} + \beta_{4} - 2 \beta_{6} + 4 \beta_{7} + 2 \beta_{8} + 2 \beta_{9} + \beta_{11} - \beta_{13} + \beta_{16} - \beta_{18} + \beta_{19} ) q^{27} + ( 2 \beta_{1} - 2 \beta_{2} + 3 \beta_{3} - \beta_{4} - 2 \beta_{5} - 2 \beta_{6} + \beta_{7} + \beta_{8} - \beta_{9} + \beta_{10} - \beta_{12} - \beta_{13} - 2 \beta_{14} + \beta_{16} - \beta_{17} - \beta_{18} - 2 \beta_{19} ) q^{28} + ( \beta_{1} - 2 \beta_{2} + \beta_{3} - \beta_{4} - 2 \beta_{5} + \beta_{9} - \beta_{15} + \beta_{17} - \beta_{18} ) q^{29} + ( 3 + \beta_{2} + 3 \beta_{3} + \beta_{4} + 2 \beta_{6} - 5 \beta_{7} - 5 \beta_{8} - \beta_{9} - 2 \beta_{10} - \beta_{11} + 2 \beta_{12} + 2 \beta_{13} + \beta_{15} - \beta_{19} ) q^{30} + ( \beta_{2} + \beta_{3} + 2 \beta_{4} + 2 \beta_{6} - \beta_{7} - 2 \beta_{10} - \beta_{11} + \beta_{12} + \beta_{13} + \beta_{14} - \beta_{15} + 2 \beta_{17} + \beta_{19} ) q^{31} + ( -1 + \beta_{2} - \beta_{6} + \beta_{8} - \beta_{9} - \beta_{10} + \beta_{12} + \beta_{15} + \beta_{17} - \beta_{18} + \beta_{19} ) q^{32} + ( 3 + \beta_{1} - 2 \beta_{2} + 3 \beta_{3} - 3 \beta_{5} + \beta_{6} - 2 \beta_{7} - 2 \beta_{8} + \beta_{9} + \beta_{10} + \beta_{11} - \beta_{12} - \beta_{14} - 2 \beta_{15} + \beta_{16} - 2 \beta_{17} - \beta_{18} - 2 \beta_{19} ) q^{33} + ( 2 - \beta_{1} - \beta_{2} - 2 \beta_{3} - \beta_{4} - \beta_{6} - \beta_{7} - \beta_{8} - 3 \beta_{9} - 2 \beta_{10} + \beta_{11} - \beta_{14} + \beta_{15} + 2 \beta_{16} + \beta_{17} - \beta_{18} ) q^{34} + ( -\beta_{1} + 2 \beta_{2} + \beta_{3} + \beta_{4} + 2 \beta_{6} - \beta_{7} - 3 \beta_{8} - \beta_{9} - 2 \beta_{10} - \beta_{11} + 2 \beta_{12} + \beta_{13} + \beta_{15} + \beta_{17} + \beta_{18} + \beta_{19} ) q^{35} + ( -2 + 6 \beta_{2} - 3 \beta_{3} + 3 \beta_{5} + 4 \beta_{6} - \beta_{8} - \beta_{9} - 2 \beta_{10} - 2 \beta_{11} + 2 \beta_{12} + 3 \beta_{13} + 3 \beta_{14} - \beta_{15} - 4 \beta_{16} + 2 \beta_{17} + \beta_{18} + 2 \beta_{19} ) q^{36} + ( \beta_{2} - \beta_{3} - \beta_{4} + \beta_{5} + 2 \beta_{6} - 2 \beta_{7} - \beta_{8} + \beta_{10} + \beta_{11} + \beta_{13} + \beta_{14} - \beta_{15} + \beta_{16} - 2 \beta_{17} + \beta_{18} ) q^{37} + ( -5 + \beta_{1} - \beta_{2} - 2 \beta_{3} + \beta_{4} + 2 \beta_{5} - \beta_{6} + 3 \beta_{7} + 2 \beta_{8} + \beta_{9} + \beta_{10} + 2 \beta_{11} - \beta_{12} - \beta_{13} - \beta_{17} ) q^{38} + ( -\beta_{2} + \beta_{3} - \beta_{4} + \beta_{5} + \beta_{6} - \beta_{8} - 2 \beta_{9} - 3 \beta_{10} + \beta_{11} + \beta_{12} - \beta_{13} - \beta_{14} + \beta_{15} + 2 \beta_{16} - \beta_{17} - \beta_{19} ) q^{39} + ( -4 - \beta_{3} - 2 \beta_{4} + 3 \beta_{5} - 2 \beta_{6} + 5 \beta_{7} + 5 \beta_{8} + 2 \beta_{9} + 2 \beta_{10} + 3 \beta_{11} + \beta_{12} - \beta_{16} - 3 \beta_{17} + \beta_{18} + \beta_{19} ) q^{40} + ( 1 - \beta_{1} - \beta_{2} + \beta_{4} - 3 \beta_{6} + \beta_{7} + \beta_{8} + \beta_{9} + \beta_{10} - \beta_{12} + \beta_{13} + \beta_{17} ) q^{41} + ( -2 + 3 \beta_{1} - 2 \beta_{2} + 5 \beta_{3} - \beta_{4} - 3 \beta_{5} - 2 \beta_{6} + \beta_{7} - 2 \beta_{8} + 2 \beta_{9} + 4 \beta_{10} + \beta_{11} - \beta_{12} + \beta_{14} - 3 \beta_{15} - \beta_{17} - 3 \beta_{18} - \beta_{19} ) q^{42} + ( 2 - \beta_{1} - 2 \beta_{2} + \beta_{3} + \beta_{4} - 2 \beta_{5} - \beta_{8} + \beta_{9} + \beta_{10} - \beta_{12} - \beta_{15} + \beta_{16} + \beta_{17} - \beta_{18} - \beta_{19} ) q^{43} + ( 3 - 2 \beta_{1} - 2 \beta_{2} - 3 \beta_{3} - 3 \beta_{4} + 4 \beta_{5} + 2 \beta_{6} - \beta_{7} + 3 \beta_{8} - \beta_{9} - \beta_{10} + 2 \beta_{11} + \beta_{12} - \beta_{13} + \beta_{14} + 3 \beta_{15} - \beta_{17} + 2 \beta_{18} + \beta_{19} ) q^{44} + ( 2 + 2 \beta_{1} + 5 \beta_{2} + 4 \beta_{3} + 6 \beta_{4} - 3 \beta_{5} + 2 \beta_{6} - 3 \beta_{7} - 3 \beta_{8} - 10 \beta_{9} - 3 \beta_{10} - 5 \beta_{11} - \beta_{13} - 2 \beta_{14} + 2 \beta_{15} + 4 \beta_{17} - \beta_{19} ) q^{45} + ( \beta_{1} + 4 \beta_{2} - 4 \beta_{3} - 5 \beta_{4} + \beta_{5} - 2 \beta_{6} + 2 \beta_{8} - 2 \beta_{9} + \beta_{10} - 2 \beta_{11} + \beta_{13} - 3 \beta_{16} - 2 \beta_{17} + \beta_{18} - \beta_{19} ) q^{46} + ( -2 - \beta_{1} - 2 \beta_{2} + \beta_{4} + 2 \beta_{6} + 2 \beta_{7} - 2 \beta_{8} - 2 \beta_{10} - \beta_{12} - \beta_{14} + \beta_{16} + \beta_{17} - \beta_{18} ) q^{47} + ( 4 - 8 \beta_{2} - 6 \beta_{5} - 7 \beta_{6} - 2 \beta_{7} - \beta_{9} + 2 \beta_{10} - 6 \beta_{12} - 4 \beta_{13} - 4 \beta_{14} - 2 \beta_{15} + 4 \beta_{16} - 2 \beta_{18} - 4 \beta_{19} ) q^{48} + ( \beta_{2} + \beta_{4} - \beta_{5} - 2 \beta_{10} - 3 \beta_{11} + \beta_{13} - 2 \beta_{14} + 2 \beta_{15} - \beta_{16} + 2 \beta_{17} - \beta_{18} + \beta_{19} ) q^{49} + ( 5 - 3 \beta_{1} - 6 \beta_{2} - 5 \beta_{5} - 5 \beta_{6} - \beta_{8} + 4 \beta_{9} + 2 \beta_{10} + 2 \beta_{11} - 5 \beta_{12} - 2 \beta_{13} - \beta_{14} - 3 \beta_{15} + 3 \beta_{16} - 3 \beta_{18} - 2 \beta_{19} ) q^{50} + ( -3 + 2 \beta_{1} + 5 \beta_{2} + \beta_{3} + 2 \beta_{4} + \beta_{6} + \beta_{8} - 2 \beta_{9} - 3 \beta_{11} + 2 \beta_{12} + \beta_{15} - 2 \beta_{16} + 2 \beta_{17} + 2 \beta_{18} + \beta_{19} ) q^{51} + ( 4 - 2 \beta_{1} - \beta_{3} - \beta_{4} + \beta_{5} + 5 \beta_{6} + \beta_{7} - 4 \beta_{8} + 4 \beta_{9} - \beta_{11} + 2 \beta_{13} + 2 \beta_{14} - 2 \beta_{16} + \beta_{17} + 2 \beta_{19} ) q^{52} + ( -1 + 2 \beta_{1} + 7 \beta_{2} + \beta_{3} + 3 \beta_{5} + 2 \beta_{6} - 2 \beta_{7} - \beta_{8} - \beta_{10} - 3 \beta_{11} + 4 \beta_{12} + 3 \beta_{13} + 2 \beta_{14} + \beta_{15} - 4 \beta_{16} - \beta_{17} + 2 \beta_{18} + 2 \beta_{19} ) q^{53} + ( -1 + 2 \beta_{2} - 3 \beta_{3} + 4 \beta_{5} + 7 \beta_{6} + 2 \beta_{7} + 5 \beta_{9} + 3 \beta_{10} + 3 \beta_{12} + 3 \beta_{13} + 3 \beta_{14} - 3 \beta_{16} - 2 \beta_{17} + 3 \beta_{18} + 3 \beta_{19} ) q^{54} + ( \beta_{2} - 2 \beta_{4} + \beta_{5} + 2 \beta_{9} - 2 \beta_{11} + 2 \beta_{13} - \beta_{15} - \beta_{16} - 2 \beta_{17} - \beta_{18} ) q^{55} + ( -1 - \beta_{1} + 2 \beta_{2} - \beta_{3} + 2 \beta_{4} + 6 \beta_{5} + 6 \beta_{6} - 5 \beta_{7} + 2 \beta_{8} - 4 \beta_{10} + \beta_{11} + 3 \beta_{12} + 2 \beta_{14} + 2 \beta_{15} + \beta_{16} + 3 \beta_{18} + 2 \beta_{19} ) q^{56} + ( -1 - 2 \beta_{2} - 3 \beta_{3} - 6 \beta_{4} + 2 \beta_{5} - 4 \beta_{6} + \beta_{7} + 8 \beta_{8} + 4 \beta_{9} + 2 \beta_{10} + 4 \beta_{11} - 2 \beta_{17} ) q^{57} + ( 7 - 2 \beta_{1} - 3 \beta_{2} + 2 \beta_{3} + 2 \beta_{4} - 3 \beta_{5} - 7 \beta_{6} - 3 \beta_{7} + 3 \beta_{8} - \beta_{9} - 2 \beta_{12} - 3 \beta_{13} - 2 \beta_{14} + \beta_{15} + 4 \beta_{16} - \beta_{17} - \beta_{19} ) q^{58} + ( 3 - 3 \beta_{1} - 8 \beta_{2} - 5 \beta_{5} - 4 \beta_{6} + \beta_{7} + \beta_{8} + \beta_{9} + 2 \beta_{10} + 3 \beta_{11} - 5 \beta_{12} - 3 \beta_{13} - 2 \beta_{14} - 2 \beta_{15} + 4 \beta_{16} + \beta_{17} - 3 \beta_{18} - 3 \beta_{19} ) q^{59} + ( -1 - \beta_{1} + 2 \beta_{2} - \beta_{3} + 2 \beta_{4} + 2 \beta_{5} - \beta_{6} + 3 \beta_{7} + 4 \beta_{8} - \beta_{9} - 2 \beta_{10} + 2 \beta_{11} + 2 \beta_{12} - \beta_{13} + \beta_{14} + 2 \beta_{15} + \beta_{17} + \beta_{18} + 2 \beta_{19} ) q^{60} + ( 3 - \beta_{1} - 2 \beta_{2} - 2 \beta_{3} + 2 \beta_{4} + 2 \beta_{5} + \beta_{6} - 2 \beta_{7} + \beta_{8} - 3 \beta_{9} - 2 \beta_{10} + 2 \beta_{11} + \beta_{12} - \beta_{13} + 2 \beta_{15} + \beta_{16} + \beta_{17} + 2 \beta_{18} + 2 \beta_{19} ) q^{61} + ( 5 - \beta_{1} - 2 \beta_{2} - \beta_{4} - 6 \beta_{5} - 5 \beta_{6} + \beta_{7} - \beta_{8} + 6 \beta_{10} - \beta_{11} - 5 \beta_{12} - 2 \beta_{13} - \beta_{14} - 2 \beta_{15} + \beta_{16} - 2 \beta_{17} - \beta_{18} - 2 \beta_{19} ) q^{62} + ( -1 - 6 \beta_{2} - 5 \beta_{3} - 4 \beta_{4} - \beta_{5} - 4 \beta_{6} + 5 \beta_{7} - 3 \beta_{8} + \beta_{9} + 2 \beta_{10} + \beta_{11} - 2 \beta_{12} - 2 \beta_{13} - \beta_{17} - 2 \beta_{19} ) q^{63} + ( -2 + \beta_{1} - \beta_{3} + 2 \beta_{4} - \beta_{5} - 2 \beta_{6} - 2 \beta_{8} + \beta_{10} - \beta_{11} - \beta_{18} - \beta_{19} ) q^{64} + ( -7 + 2 \beta_{1} + 3 \beta_{2} + \beta_{3} - \beta_{4} + 4 \beta_{5} + 4 \beta_{6} - 2 \beta_{10} + 4 \beta_{12} + 2 \beta_{13} - 2 \beta_{16} + \beta_{18} + \beta_{19} ) q^{65} + ( -5 + \beta_{1} + 5 \beta_{2} + 3 \beta_{3} + 3 \beta_{4} + 3 \beta_{5} + 4 \beta_{6} + \beta_{7} - 2 \beta_{8} - 2 \beta_{10} + \beta_{11} + 3 \beta_{12} + 2 \beta_{14} + \beta_{15} - \beta_{16} + \beta_{17} + \beta_{18} + \beta_{19} ) q^{66} + ( -3 + \beta_{2} + \beta_{3} + \beta_{5} + \beta_{6} + 3 \beta_{9} + 2 \beta_{10} + 3 \beta_{12} + \beta_{13} + \beta_{14} - 3 \beta_{16} - \beta_{17} + \beta_{18} + \beta_{19} ) q^{67} + ( -2 + 2 \beta_{1} + 2 \beta_{2} - 2 \beta_{3} - 2 \beta_{5} - 4 \beta_{6} - \beta_{7} - \beta_{8} - 2 \beta_{9} + \beta_{10} - \beta_{12} - 3 \beta_{13} - 2 \beta_{14} + \beta_{16} + \beta_{17} - \beta_{18} ) q^{68} + ( -4 + 2 \beta_{1} + 8 \beta_{2} + 3 \beta_{3} + \beta_{4} + 4 \beta_{5} + 10 \beta_{6} + \beta_{7} - 3 \beta_{8} + \beta_{9} + 2 \beta_{10} + \beta_{11} + 4 \beta_{12} + 2 \beta_{13} + 4 \beta_{14} - 2 \beta_{15} - 4 \beta_{16} - \beta_{17} + 2 \beta_{18} + 2 \beta_{19} ) q^{69} + ( -1 + 2 \beta_{1} + 6 \beta_{2} + \beta_{3} + \beta_{5} - 2 \beta_{6} + 2 \beta_{7} + 2 \beta_{8} - 4 \beta_{9} + 2 \beta_{10} - 2 \beta_{11} - 4 \beta_{16} - 2 \beta_{17} + \beta_{18} - \beta_{19} ) q^{70} + ( 2 + \beta_{2} + 4 \beta_{3} + 2 \beta_{4} - 2 \beta_{5} - 2 \beta_{7} + 2 \beta_{9} + 2 \beta_{10} - 2 \beta_{11} + 4 \beta_{13} + 2 \beta_{14} - 3 \beta_{15} - 3 \beta_{16} - \beta_{17} - \beta_{18} - 2 \beta_{19} ) q^{71} + ( -1 + \beta_{1} - 5 \beta_{2} + 5 \beta_{3} - 3 \beta_{4} - 8 \beta_{5} - 6 \beta_{6} + 3 \beta_{10} - 6 \beta_{12} - 3 \beta_{13} - 3 \beta_{14} - 3 \beta_{15} + 3 \beta_{16} - 4 \beta_{18} - 4 \beta_{19} ) q^{72} + ( 1 - 2 \beta_{1} + 2 \beta_{3} + 3 \beta_{4} - \beta_{5} + 3 \beta_{6} - \beta_{7} - 3 \beta_{8} - 2 \beta_{9} - \beta_{10} + 2 \beta_{15} + 2 \beta_{16} - \beta_{17} - \beta_{19} ) q^{73} + ( -10 + 3 \beta_{1} + 7 \beta_{2} + 4 \beta_{3} - \beta_{4} + \beta_{5} + \beta_{6} - 3 \beta_{8} - 2 \beta_{10} + 4 \beta_{12} + 2 \beta_{13} + \beta_{14} + \beta_{15} - 2 \beta_{16} ) q^{74} + ( -6 - \beta_{1} - 5 \beta_{2} - 6 \beta_{3} - 6 \beta_{4} + 2 \beta_{5} - 7 \beta_{6} + 8 \beta_{7} + 10 \beta_{8} + 3 \beta_{9} + 4 \beta_{11} - 2 \beta_{12} - 5 \beta_{13} - \beta_{14} + \beta_{15} + 4 \beta_{16} - \beta_{17} + \beta_{19} ) q^{75} + ( 2 + \beta_{1} - 2 \beta_{2} + \beta_{4} - \beta_{5} + 6 \beta_{6} - 4 \beta_{7} - 6 \beta_{8} + 6 \beta_{9} + 2 \beta_{10} + \beta_{11} - \beta_{13} + 2 \beta_{14} - 2 \beta_{15} - \beta_{16} + 2 \beta_{17} + \beta_{18} - \beta_{19} ) q^{76} + ( 2 + 2 \beta_{1} - \beta_{2} - \beta_{3} - \beta_{4} - 4 \beta_{5} - 2 \beta_{6} + 3 \beta_{7} - 6 \beta_{9} + \beta_{11} - 2 \beta_{12} - 6 \beta_{13} - 4 \beta_{14} + 4 \beta_{16} + \beta_{17} - 4 \beta_{19} ) q^{77} + ( -1 - \beta_{1} + 3 \beta_{2} - 2 \beta_{3} + 4 \beta_{4} - \beta_{5} - \beta_{6} - 3 \beta_{9} - 2 \beta_{10} - \beta_{11} - \beta_{13} + \beta_{14} + 3 \beta_{17} + 2 \beta_{19} ) q^{78} + ( 4 - 5 \beta_{2} + 2 \beta_{3} - 2 \beta_{5} - 3 \beta_{6} + 2 \beta_{7} + \beta_{9} - 2 \beta_{12} - \beta_{13} - \beta_{14} - \beta_{15} + \beta_{16} - \beta_{17} - \beta_{19} ) q^{79} + ( 1 + 2 \beta_{1} - \beta_{2} - 2 \beta_{3} - 2 \beta_{4} + 2 \beta_{5} + \beta_{6} - 2 \beta_{7} + 2 \beta_{8} + 5 \beta_{9} + 3 \beta_{10} + \beta_{11} + \beta_{12} + \beta_{13} + 3 \beta_{14} - \beta_{15} - 2 \beta_{16} - 2 \beta_{17} + \beta_{18} ) q^{80} + ( -1 + \beta_{1} + 7 \beta_{2} - \beta_{3} + 5 \beta_{5} + \beta_{6} - 3 \beta_{8} - \beta_{10} - \beta_{11} + 4 \beta_{12} + 2 \beta_{13} + 2 \beta_{14} + 2 \beta_{15} - 2 \beta_{16} + 2 \beta_{18} + 2 \beta_{19} ) q^{81} + ( -\beta_{1} - 2 \beta_{2} - \beta_{3} + 3 \beta_{5} + 4 \beta_{6} + 2 \beta_{8} - 2 \beta_{10} + \beta_{11} + 2 \beta_{12} + \beta_{13} - \beta_{16} + \beta_{18} + \beta_{19} ) q^{82} + ( -4 + \beta_{1} + 2 \beta_{2} + \beta_{3} - \beta_{4} + \beta_{5} + 3 \beta_{6} + \beta_{7} - 2 \beta_{8} + \beta_{9} - \beta_{11} + 2 \beta_{12} + 3 \beta_{13} + \beta_{14} - \beta_{15} - \beta_{16} + 2 \beta_{19} ) q^{83} + ( -3 \beta_{1} - 3 \beta_{2} + 3 \beta_{3} + 7 \beta_{4} - \beta_{5} - 5 \beta_{7} + 5 \beta_{8} + \beta_{9} - 6 \beta_{10} - \beta_{11} + \beta_{12} + 2 \beta_{13} - 3 \beta_{14} + 3 \beta_{15} + 5 \beta_{16} + 4 \beta_{17} - \beta_{18} + 4 \beta_{19} ) q^{84} + ( 4 - 4 \beta_{2} - 2 \beta_{3} - \beta_{5} - 4 \beta_{6} + 5 \beta_{8} + 2 \beta_{9} + 3 \beta_{10} + 2 \beta_{11} - 4 \beta_{12} - 3 \beta_{13} + \beta_{14} - \beta_{15} + \beta_{16} - \beta_{17} - \beta_{18} - 2 \beta_{19} ) q^{85} + ( -\beta_{1} + \beta_{3} - \beta_{4} + \beta_{5} + \beta_{7} + 2 \beta_{8} - 2 \beta_{9} + \beta_{11} + \beta_{12} - \beta_{14} + \beta_{15} - \beta_{17} + \beta_{18} + \beta_{19} ) q^{86} + ( 2 - 2 \beta_{1} - 2 \beta_{2} + 4 \beta_{3} + 4 \beta_{4} - \beta_{5} - 2 \beta_{7} - 2 \beta_{8} - 5 \beta_{10} - 2 \beta_{11} + 2 \beta_{12} + 3 \beta_{13} - \beta_{14} + 3 \beta_{15} + 2 \beta_{16} + 3 \beta_{17} - 2 \beta_{18} + 2 \beta_{19} ) q^{87} + ( -1 + 3 \beta_{2} - 2 \beta_{3} - 2 \beta_{4} + \beta_{5} + \beta_{6} - 2 \beta_{7} + 2 \beta_{8} - 3 \beta_{9} - 2 \beta_{10} - \beta_{11} + 2 \beta_{12} + 3 \beta_{13} + \beta_{15} + 2 \beta_{17} + 3 \beta_{19} ) q^{88} + ( -3 - 2 \beta_{2} - 2 \beta_{3} - 2 \beta_{4} - 2 \beta_{5} - 2 \beta_{6} + 5 \beta_{7} + 2 \beta_{9} + \beta_{10} - 2 \beta_{12} - 2 \beta_{14} - \beta_{15} + \beta_{16} + \beta_{17} - 3 \beta_{18} + 2 \beta_{19} ) q^{89} + ( 1 - 4 \beta_{1} - 6 \beta_{2} - 4 \beta_{3} + 3 \beta_{4} + 8 \beta_{6} - 9 \beta_{7} - 3 \beta_{8} - \beta_{9} - 4 \beta_{10} - 4 \beta_{12} - 2 \beta_{14} + 4 \beta_{16} + 2 \beta_{17} + 2 \beta_{18} ) q^{90} + ( 2 + \beta_{1} - 2 \beta_{3} - 2 \beta_{4} + \beta_{5} - \beta_{6} - \beta_{7} - \beta_{9} + \beta_{10} - \beta_{12} - 4 \beta_{13} + 2 \beta_{14} - \beta_{17} + 2 \beta_{18} - 2 \beta_{19} ) q^{91} + ( -1 + \beta_{1} - 4 \beta_{2} + \beta_{3} - 3 \beta_{5} - 9 \beta_{6} - 3 \beta_{7} - 3 \beta_{9} - 3 \beta_{10} - 2 \beta_{12} - 2 \beta_{13} - 2 \beta_{14} + \beta_{15} + 4 \beta_{16} + 2 \beta_{17} - 3 \beta_{18} - 2 \beta_{19} ) q^{92} + ( 1 - \beta_{1} + 3 \beta_{2} - 3 \beta_{4} + 3 \beta_{5} + 2 \beta_{6} - 3 \beta_{8} + 3 \beta_{9} + \beta_{11} + 2 \beta_{12} + 4 \beta_{13} + 2 \beta_{14} - \beta_{15} - 4 \beta_{16} - \beta_{17} - \beta_{18} ) q^{93} + ( 4 - 2 \beta_{1} - 5 \beta_{2} - 2 \beta_{5} + 2 \beta_{7} - \beta_{8} + 5 \beta_{9} + 3 \beta_{10} + \beta_{11} - 3 \beta_{12} - \beta_{13} + \beta_{14} - 3 \beta_{15} - \beta_{18} ) q^{94} + ( -3 + \beta_{1} + 6 \beta_{2} + 4 \beta_{3} + 3 \beta_{4} + 7 \beta_{5} + 11 \beta_{6} - 4 \beta_{8} - 3 \beta_{9} - 5 \beta_{10} - 2 \beta_{11} + 7 \beta_{12} + 6 \beta_{13} + 2 \beta_{14} + 4 \beta_{15} - 5 \beta_{16} - 2 \beta_{17} + 3 \beta_{18} + 2 \beta_{19} ) q^{95} + ( 2 - 2 \beta_{1} + \beta_{2} - \beta_{4} + 6 \beta_{5} + 4 \beta_{6} - 3 \beta_{7} - 4 \beta_{8} - 3 \beta_{10} + 2 \beta_{11} + 4 \beta_{12} + 2 \beta_{13} + 2 \beta_{14} + 2 \beta_{15} - \beta_{16} - 2 \beta_{17} + 2 \beta_{18} ) q^{96} + ( -\beta_{1} - 3 \beta_{3} - 2 \beta_{4} - \beta_{5} + \beta_{6} - \beta_{8} - 2 \beta_{9} - 2 \beta_{10} - \beta_{11} + \beta_{13} - \beta_{14} + \beta_{15} + \beta_{16} + 2 \beta_{17} - 2 \beta_{18} + 2 \beta_{19} ) q^{97} + ( 6 - 2 \beta_{1} - \beta_{2} - 3 \beta_{3} + \beta_{6} + 4 \beta_{7} - \beta_{9} - 2 \beta_{12} - \beta_{13} - \beta_{14} - \beta_{15} - \beta_{16} + \beta_{17} - \beta_{19} ) q^{98} + ( 6 - 2 \beta_{1} - 7 \beta_{2} + 3 \beta_{4} - 7 \beta_{5} - 6 \beta_{6} - 2 \beta_{7} + 2 \beta_{8} + 4 \beta_{9} + 4 \beta_{10} - 2 \beta_{11} - 6 \beta_{12} - 2 \beta_{14} - 3 \beta_{15} - 4 \beta_{18} - 2 \beta_{19} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$20q - 4q^{2} - 8q^{4} - 4q^{5} + 6q^{6} - 2q^{7} + 12q^{8} - 32q^{9} + O(q^{10})$$ $$20q - 4q^{2} - 8q^{4} - 4q^{5} + 6q^{6} - 2q^{7} + 12q^{8} - 32q^{9} - 12q^{10} - 6q^{11} + 30q^{12} - 16q^{13} - 8q^{14} + 8q^{15} - 4q^{16} + 8q^{17} + 4q^{18} - 12q^{19} + 8q^{20} + 24q^{21} + 8q^{22} - 6q^{23} - 36q^{25} - 36q^{26} - 12q^{28} - 6q^{29} + 34q^{30} + 20q^{31} - 6q^{32} + 34q^{33} + 36q^{34} + 18q^{35} + 18q^{36} - 8q^{37} - 66q^{38} + 28q^{39} - 2q^{40} + 10q^{41} - 56q^{42} + 12q^{43} + 34q^{44} - 4q^{45} - 20q^{47} - 48q^{48} + 30q^{50} - 36q^{51} + 80q^{52} + 24q^{53} + 24q^{54} + 10q^{55} + 10q^{57} + 54q^{58} - 18q^{59} + 50q^{60} + 42q^{61} - 12q^{62} - 48q^{63} - 56q^{64} - 44q^{65} - 10q^{66} - 42q^{67} - 44q^{68} + 24q^{69} + 4q^{71} - 112q^{72} - 16q^{73} - 96q^{74} - 52q^{75} + 52q^{77} - 12q^{78} + 54q^{79} - 2q^{80} + 60q^{81} + 32q^{82} - 30q^{83} - 16q^{84} + 6q^{85} + 16q^{86} + 32q^{87} + 2q^{88} - 22q^{89} - 110q^{90} - 8q^{91} - 78q^{92} + 78q^{93} + 38q^{94} + 38q^{95} + 72q^{96} + 138q^{98} - 32q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{20} + 28 x^{18} + 326 x^{16} + 2044 x^{14} + 7471 x^{12} + 16090 x^{10} + 19590 x^{8} + 12030 x^{6} + 2877 x^{4} + 230 x^{2} + 1$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu^{2} + 3$$ $$\beta_{2}$$ $$=$$ $$($$$$451 \nu^{18} + 11451 \nu^{16} + 117529 \nu^{14} + 622481 \nu^{12} + 1794474 \nu^{10} + 2697392 \nu^{8} + 1750626 \nu^{6} + 153068 \nu^{4} - 105869 \nu^{2} - 2647$$$$)/31760$$ $$\beta_{3}$$ $$=$$ $$($$$$-27 \nu^{18} - 763 \nu^{16} - 8941 \nu^{14} - 56168 \nu^{12} - 204075 \nu^{10} - 430435 \nu^{8} - 497369 \nu^{6} - 265903 \nu^{4} - 35795 \nu^{2} + 794 \nu - 66$$$$)/1588$$ $$\beta_{4}$$ $$=$$ $$($$$$27 \nu^{18} + 763 \nu^{16} + 8941 \nu^{14} + 56168 \nu^{12} + 204075 \nu^{10} + 430435 \nu^{8} + 497369 \nu^{6} + 265903 \nu^{4} + 35795 \nu^{2} + 794 \nu + 66$$$$)/1588$$ $$\beta_{5}$$ $$=$$ $$($$$$-1203 \nu^{18} - 33643 \nu^{16} - 391137 \nu^{14} - 2446443 \nu^{12} - 8894572 \nu^{10} - 18911266 \nu^{8} - 22279608 \nu^{6} - 12451734 \nu^{4} - 2076383 \nu^{2} + 7940 \nu - 73739$$$$)/15880$$ $$\beta_{6}$$ $$=$$ $$($$$$2647 \nu^{19} + 3573 \nu^{18} + 74567 \nu^{17} + 98853 \nu^{16} + 874373 \nu^{15} + 1131847 \nu^{14} + 5527997 \nu^{13} + 6931223 \nu^{12} + 20398218 \nu^{11} + 24489342 \nu^{10} + 44384704 \nu^{9} + 50136336 \nu^{8} + 54552122 \nu^{7} + 56236638 \nu^{6} + 33594036 \nu^{5} + 29430404 \nu^{4} + 7768487 \nu^{3} + 4307053 \nu^{2} + 502941 \nu + 74239$$$$)/63520$$ $$\beta_{7}$$ $$=$$ $$($$$$-66 \nu^{19} - 1821 \nu^{17} - 20753 \nu^{15} - 125963 \nu^{13} - 436918 \nu^{11} - 857865 \nu^{9} - 862505 \nu^{7} - 296611 \nu^{5} + 76021 \nu^{3} + 20615 \nu + 794$$$$)/1588$$ $$\beta_{8}$$ $$=$$ $$($$$$-2647 \nu^{19} + 3573 \nu^{18} - 74567 \nu^{17} + 98853 \nu^{16} - 874373 \nu^{15} + 1131847 \nu^{14} - 5527997 \nu^{13} + 6931223 \nu^{12} - 20398218 \nu^{11} + 24489342 \nu^{10} - 44384704 \nu^{9} + 50136336 \nu^{8} - 54552122 \nu^{7} + 56236638 \nu^{6} - 33594036 \nu^{5} + 29430404 \nu^{4} - 7768487 \nu^{3} + 4307053 \nu^{2} - 502941 \nu + 74239$$$$)/63520$$ $$\beta_{9}$$ $$=$$ $$($$$$3573 \nu^{19} + 451 \nu^{18} + 98853 \nu^{17} + 11451 \nu^{16} + 1131847 \nu^{15} + 117529 \nu^{14} + 6931223 \nu^{13} + 622481 \nu^{12} + 24489342 \nu^{11} + 1794474 \nu^{10} + 50136336 \nu^{9} + 2697392 \nu^{8} + 56236638 \nu^{7} + 1750626 \nu^{6} + 29430404 \nu^{5} + 153068 \nu^{4} + 4307053 \nu^{3} - 105869 \nu^{2} + 74239 \nu - 2647$$$$)/63520$$ $$\beta_{10}$$ $$=$$ $$($$$$-3339 \nu^{19} - 452 \nu^{18} - 93299 \nu^{17} - 12832 \nu^{16} - 1084001 \nu^{15} - 152208 \nu^{14} - 6785589 \nu^{13} - 977112 \nu^{12} - 24801766 \nu^{11} - 3669388 \nu^{10} - 53644268 \nu^{9} - 8104844 \nu^{8} - 66295654 \nu^{7} - 9973552 \nu^{6} - 42453552 \nu^{5} - 5933076 \nu^{4} - 11405379 \nu^{3} - 1245492 \nu^{2} - 1014557 \nu - 116676$$$$)/31760$$ $$\beta_{11}$$ $$=$$ $$($$$$-3339 \nu^{19} + 452 \nu^{18} - 93299 \nu^{17} + 12832 \nu^{16} - 1084001 \nu^{15} + 152208 \nu^{14} - 6785589 \nu^{13} + 977112 \nu^{12} - 24801766 \nu^{11} + 3669388 \nu^{10} - 53644268 \nu^{9} + 8104844 \nu^{8} - 66295654 \nu^{7} + 9973552 \nu^{6} - 42453552 \nu^{5} + 5933076 \nu^{4} - 11405379 \nu^{3} + 1245492 \nu^{2} - 1014557 \nu + 116676$$$$)/31760$$ $$\beta_{12}$$ $$=$$ $$($$$$-4043 \nu^{19} + 1532 \nu^{18} - 112723 \nu^{17} + 43352 \nu^{16} - 1304837 \nu^{15} + 509848 \nu^{14} - 8114373 \nu^{13} + 3223832 \nu^{12} - 29297602 \nu^{11} + 11832388 \nu^{10} - 61867436 \nu^{9} + 25306364 \nu^{8} - 72700298 \nu^{7} + 29677752 \nu^{6} - 41304924 \nu^{5} + 15854596 \nu^{4} - 7772083 \nu^{3} + 1772132 \nu^{2} - 456949 \nu - 103004$$$$)/31760$$ $$\beta_{13}$$ $$=$$ $$($$$$-4659 \nu^{19} - 129719 \nu^{17} - 1499061 \nu^{15} - 9304849 \nu^{13} - 33540126 \nu^{11} - 70801568 \nu^{9} - 83545754 \nu^{7} - 48369892 \nu^{5} - 9757919 \nu^{3} + 15880 \nu^{2} - 411697 \nu + 47640$$$$)/31760$$ $$\beta_{14}$$ $$=$$ $$($$$$-5363 \nu^{19} - 464 \nu^{18} - 149143 \nu^{17} - 13524 \nu^{16} - 1719897 \nu^{15} - 163416 \nu^{14} - 10633633 \nu^{13} - 1056244 \nu^{12} - 38035962 \nu^{11} - 3920476 \nu^{10} - 79024736 \nu^{9} - 8283268 \nu^{8} - 89950398 \nu^{7} - 9065184 \nu^{6} - 47237144 \nu^{5} - 3698192 \nu^{4} - 6235783 \nu^{3} + 395236 \nu^{2} + 34751 \nu + 110908$$$$)/31760$$ $$\beta_{15}$$ $$=$$ $$($$$$5363 \nu^{19} - 464 \nu^{18} + 149143 \nu^{17} - 13524 \nu^{16} + 1719897 \nu^{15} - 163416 \nu^{14} + 10633633 \nu^{13} - 1056244 \nu^{12} + 38035962 \nu^{11} - 3920476 \nu^{10} + 79024736 \nu^{9} - 8283268 \nu^{8} + 89950398 \nu^{7} - 9065184 \nu^{6} + 47237144 \nu^{5} - 3698192 \nu^{4} + 6235783 \nu^{3} + 395236 \nu^{2} - 34751 \nu + 110908$$$$)/31760$$ $$\beta_{16}$$ $$=$$ $$($$$$-6759 \nu^{19} - \nu^{18} - 187299 \nu^{17} - 1381 \nu^{16} - 2150361 \nu^{15} - 34679 \nu^{14} - 13220009 \nu^{13} - 354631 \nu^{12} - 46935346 \nu^{11} - 1874914 \nu^{10} - 96507468 \nu^{9} - 5407452 \nu^{8} - 108098574 \nu^{7} - 8222926 \nu^{6} - 54932152 \nu^{5} - 5780008 \nu^{4} - 6128219 \nu^{3} - 1335481 \nu^{2} + 223663 \nu - 71683$$$$)/31760$$ $$\beta_{17}$$ $$=$$ $$($$$$-8421 \nu^{19} - 1473 \nu^{18} - 235501 \nu^{17} - 41273 \nu^{16} - 2737959 \nu^{15} - 480547 \nu^{14} - 17137011 \nu^{13} - 3008123 \nu^{12} - 62504174 \nu^{11} - 10935322 \nu^{10} - 134264612 \nu^{9} - 23215616 \nu^{8} - 162972246 \nu^{7} - 27253298 \nu^{6} - 99719248 \nu^{5} - 15110764 \nu^{4} - 23649801 \nu^{3} - 2434333 \nu^{2} - 1758783 \nu - 74399$$$$)/31760$$ $$\beta_{18}$$ $$=$$ $$($$$$-6267 \nu^{19} - 1006 \nu^{18} - 174807 \nu^{17} - 27576 \nu^{16} - 2024313 \nu^{15} - 312344 \nu^{14} - 12591827 \nu^{13} - 1888426 \nu^{12} - 45458108 \nu^{11} - 6570954 \nu^{10} - 95897414 \nu^{9} - 13208432 \nu^{8} - 112370812 \nu^{7} - 14496816 \nu^{6} - 63418686 \nu^{5} - 7387928 \nu^{4} - 11672507 \nu^{3} - 1002656 \nu^{2} - 551931 \nu + 13862$$$$)/15880$$ $$\beta_{19}$$ $$=$$ $$($$$$6267 \nu^{19} - 1006 \nu^{18} + 174807 \nu^{17} - 27576 \nu^{16} + 2024313 \nu^{15} - 312344 \nu^{14} + 12591827 \nu^{13} - 1888426 \nu^{12} + 45458108 \nu^{11} - 6570954 \nu^{10} + 95897414 \nu^{9} - 13208432 \nu^{8} + 112370812 \nu^{7} - 14496816 \nu^{6} + 63418686 \nu^{5} - 7387928 \nu^{4} + 11672507 \nu^{3} - 1002656 \nu^{2} + 551931 \nu + 13862$$$$)/15880$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{4} + \beta_{3}$$ $$\nu^{2}$$ $$=$$ $$\beta_{1} - 3$$ $$\nu^{3}$$ $$=$$ $$-\beta_{16} - \beta_{15} + \beta_{14} + \beta_{13} - \beta_{11} - \beta_{8} - 2 \beta_{7} + \beta_{6} - 4 \beta_{4} - 4 \beta_{3} + \beta_{2} + 1$$ $$\nu^{4}$$ $$=$$ $$\beta_{19} + \beta_{18} + \beta_{8} + \beta_{6} + \beta_{2} - 6 \beta_{1} + 14$$ $$\nu^{5}$$ $$=$$ $$8 \beta_{16} + 8 \beta_{15} - 8 \beta_{14} - 6 \beta_{13} + 7 \beta_{11} - \beta_{10} + 8 \beta_{8} + 14 \beta_{7} - 8 \beta_{6} + 20 \beta_{4} + 20 \beta_{3} - 8 \beta_{2} - \beta_{1} - 7$$ $$\nu^{6}$$ $$=$$ $$-9 \beta_{19} - 9 \beta_{18} + \beta_{11} - \beta_{10} - 8 \beta_{8} - 8 \beta_{6} + \beta_{5} - 2 \beta_{4} + \beta_{3} - 10 \beta_{2} + 35 \beta_{1} - 74$$ $$\nu^{7}$$ $$=$$ $$\beta_{19} - \beta_{18} + 2 \beta_{17} - 55 \beta_{16} - 54 \beta_{15} + 54 \beta_{14} + 35 \beta_{13} - 44 \beta_{11} + 11 \beta_{10} + 2 \beta_{9} - 54 \beta_{8} - 80 \beta_{7} + 54 \beta_{6} - \beta_{5} - 108 \beta_{4} - 109 \beta_{3} + 54 \beta_{2} + 10 \beta_{1} + 40$$ $$\nu^{8}$$ $$=$$ $$63 \beta_{19} + 63 \beta_{18} + \beta_{16} - \beta_{13} - 2 \beta_{12} - 10 \beta_{11} + 11 \beta_{10} + 52 \beta_{8} + 50 \beta_{6} - 12 \beta_{5} + 25 \beta_{4} - 15 \beta_{3} + 74 \beta_{2} - 207 \beta_{1} + 415$$ $$\nu^{9}$$ $$=$$ $$-11 \beta_{19} + 11 \beta_{18} - 24 \beta_{17} + 364 \beta_{16} + 346 \beta_{15} - 346 \beta_{14} - 208 \beta_{13} + 271 \beta_{11} - 93 \beta_{10} - 28 \beta_{9} + 352 \beta_{8} + 432 \beta_{7} - 352 \beta_{6} + 12 \beta_{5} + 606 \beta_{4} + 618 \beta_{3} - 350 \beta_{2} - 78 \beta_{1} - 216$$ $$\nu^{10}$$ $$=$$ $$-409 \beta_{19} - 409 \beta_{18} - 12 \beta_{16} + 12 \beta_{13} + 24 \beta_{12} + 71 \beta_{11} - 83 \beta_{10} - 318 \beta_{8} - 294 \beta_{6} + 97 \beta_{5} - 220 \beta_{4} + 147 \beta_{3} - 504 \beta_{2} + 1236 \beta_{1} - 2388$$ $$\nu^{11}$$ $$=$$ $$83 \beta_{19} - 83 \beta_{18} + 190 \beta_{17} - 2376 \beta_{16} - 2175 \beta_{15} + 2175 \beta_{14} + 1260 \beta_{13} - 1657 \beta_{11} + 719 \beta_{10} + 282 \beta_{9} - 2271 \beta_{8} - 2282 \beta_{7} + 2271 \beta_{6} - 95 \beta_{5} - 3479 \beta_{4} - 3574 \beta_{3} + 2235 \beta_{2} + 558 \beta_{1} + 1141$$ $$\nu^{12}$$ $$=$$ $$2588 \beta_{19} + 2588 \beta_{18} + 87 \beta_{16} + 8 \beta_{15} + 8 \beta_{14} - 87 \beta_{13} - 174 \beta_{12} - 440 \beta_{11} + 527 \beta_{10} + 1901 \beta_{8} + 1727 \beta_{6} - 652 \beta_{5} + 1687 \beta_{4} - 1209 \beta_{3} + 3351 \beta_{2} - 7418 \beta_{1} + 13886$$ $$\nu^{13}$$ $$=$$ $$-535 \beta_{19} + 535 \beta_{18} - 1244 \beta_{17} + 15395 \beta_{16} + 13569 \beta_{15} - 13569 \beta_{14} - 7741 \beta_{13} + 10085 \beta_{11} - 5310 \beta_{10} - 2460 \beta_{9} + 14571 \beta_{8} + 11900 \beta_{7} - 14571 \beta_{6} + 622 \beta_{5} + 20280 \beta_{4} + 20902 \beta_{3} - 14165 \beta_{2} - 3827 \beta_{1} - 5950$$ $$\nu^{14}$$ $$=$$ $$-16245 \beta_{19} - 16245 \beta_{18} - 456 \beta_{16} - 166 \beta_{15} - 166 \beta_{14} + 456 \beta_{13} + 912 \beta_{12} + 2536 \beta_{11} - 2992 \beta_{10} - 11275 \beta_{8} - 10363 \beta_{6} + 3890 \beta_{5} - 12068 \beta_{4} + 9090 \beta_{3} - 22127 \beta_{2} + 44643 \beta_{1} - 81084$$ $$\nu^{15}$$ $$=$$ $$3158 \beta_{19} - 3158 \beta_{18} + 7228 \beta_{17} - 99211 \beta_{16} - 84369 \beta_{15} + 84369 \beta_{14} + 47989 \beta_{13} - 61178 \beta_{11} + 38033 \beta_{10} + 19672 \beta_{9} - 93051 \beta_{8} - 61308 \beta_{7} + 93051 \beta_{6} - 3614 \beta_{5} - 119515 \beta_{4} - 123129 \beta_{3} + 89375 \beta_{2} + 25611 \beta_{1} + 30654$$ $$\nu^{16}$$ $$=$$ $$101818 \beta_{19} + 101818 \beta_{18} + 1484 \beta_{16} + 2130 \beta_{15} + 2130 \beta_{14} - 1484 \beta_{13} - 2968 \beta_{12} - 13927 \beta_{11} + 15411 \beta_{10} + 66827 \beta_{8} + 63859 \beta_{6} - 20987 \beta_{5} + 82940 \beta_{4} - 64921 \beta_{3} + 145657 \beta_{2} - 269132 \beta_{1} + 474291$$ $$\nu^{17}$$ $$=$$ $$-17541 \beta_{19} + 17541 \beta_{18} - 38050 \beta_{17} + 636446 \beta_{16} + 523667 \beta_{15} - 523667 \beta_{14} - 299044 \beta_{13} + 370304 \beta_{11} - 266142 \beta_{10} - 148186 \beta_{9} + 591557 \beta_{8} + 311120 \beta_{7} - 591557 \beta_{6} + 19025 \beta_{5} + 710110 \beta_{4} + 729135 \beta_{3} - 562353 \beta_{2} - 168701 \beta_{1} - 155560$$ $$\nu^{18}$$ $$=$$ $$-638655 \beta_{19} - 638655 \beta_{18} + 2839 \beta_{16} - 21864 \beta_{15} - 21864 \beta_{14} - 2839 \beta_{13} - 5678 \beta_{12} + 73463 \beta_{11} - 70624 \beta_{10} - 397345 \beta_{8} - 403023 \beta_{6} + 100989 \beta_{5} - 555833 \beta_{4} + 449166 \beta_{3} - 955865 \beta_{2} + 1624590 \beta_{1} - 2776677$$ $$\nu^{19}$$ $$=$$ $$92488 \beta_{19} - 92488 \beta_{18} + 179298 \beta_{17} - 4066659 \beta_{16} - 3246636 \beta_{15} + 3246636 \beta_{14} + 1868337 \beta_{13} - 2238542 \beta_{11} + 1828117 \beta_{10} + 1068894 \beta_{9} - 3744638 \beta_{8} - 1546074 \beta_{7} + 3744638 \beta_{6} - 89649 \beta_{5} - 4246042 \beta_{4} - 4335691 \beta_{3} + 3532212 \beta_{2} + 1099161 \beta_{1} + 773037$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/73\mathbb{Z}\right)^\times$$.

 $$n$$ $$5$$ $$\chi(n)$$ $$\beta_{6}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
3.1
 − 2.49160i − 1.44785i − 0.467725i 0.0678852i 2.33929i − 2.43584i − 1.96659i − 0.405053i 1.18677i 1.62072i 2.49160i 1.44785i 0.467725i − 0.0678852i − 2.33929i 2.43584i 1.96659i 0.405053i − 1.18677i − 1.62072i
−1.24580 + 2.15779i 2.90855i −2.10403 3.64429i 1.39079 + 0.372661i −6.27603 3.62347i 2.87065 2.87065i 5.50161 −5.45964 −2.53677 + 2.53677i
3.2 −0.723926 + 1.25388i 0.458687i −0.0481386 0.0833785i 2.03807 + 0.546100i 0.575137 + 0.332055i −2.38436 + 2.38436i −2.75631 2.78961 −2.16016 + 2.16016i
3.3 −0.233863 + 0.405062i 2.48355i 0.890617 + 1.54259i −0.733428 0.196522i 1.00599 + 0.580811i 3.06980 3.06980i −1.76858 −3.16804 0.251125 0.251125i
3.4 0.0339426 0.0587903i 1.73613i 0.997696 + 1.72806i −2.59207 0.694543i 0.102067 + 0.0589286i 0.823696 0.823696i 0.271228 −0.0141334 −0.128814 + 0.128814i
3.5 1.16965 2.02589i 1.29757i −1.73614 3.00709i −1.10336 0.295646i 2.62873 + 1.51770i −0.549659 + 0.549659i −3.44410 1.31631 −1.88949 + 1.88949i
24.1 −1.21792 2.10950i 0.455408i −1.96665 + 3.40634i −0.913508 3.40926i −0.960682 + 0.554650i −1.73085 + 1.73085i 4.70921 2.79260 −6.07924 + 6.07924i
24.2 −0.983297 1.70312i 3.07168i −0.933744 + 1.61729i 0.710245 + 2.65067i 5.23144 3.02037i 1.20747 1.20747i −0.260597 −6.43521 3.81603 3.81603i
24.3 −0.202527 0.350787i 0.0458652i 0.917966 1.58996i 0.0766784 + 0.286168i 0.0160889 0.00928892i 0.00171153 0.00171153i −1.55376 2.99790 0.0848543 0.0848543i
24.4 0.593383 + 1.02777i 3.06395i 0.295794 0.512330i −1.03319 3.85590i −3.14904 + 1.81810i −1.17070 + 1.17070i 3.07561 −6.38782 3.34990 3.34990i
24.5 0.810359 + 1.40358i 2.72609i −0.313364 + 0.542762i 0.159770 + 0.596269i 3.82630 2.20911i −3.13775 + 3.13775i 2.22569 −4.43157 −0.707442 + 0.707442i
49.1 −1.24580 2.15779i 2.90855i −2.10403 + 3.64429i 1.39079 0.372661i −6.27603 + 3.62347i 2.87065 + 2.87065i 5.50161 −5.45964 −2.53677 2.53677i
49.2 −0.723926 1.25388i 0.458687i −0.0481386 + 0.0833785i 2.03807 0.546100i 0.575137 0.332055i −2.38436 2.38436i −2.75631 2.78961 −2.16016 2.16016i
49.3 −0.233863 0.405062i 2.48355i 0.890617 1.54259i −0.733428 + 0.196522i 1.00599 0.580811i 3.06980 + 3.06980i −1.76858 −3.16804 0.251125 + 0.251125i
49.4 0.0339426 + 0.0587903i 1.73613i 0.997696 1.72806i −2.59207 + 0.694543i 0.102067 0.0589286i 0.823696 + 0.823696i 0.271228 −0.0141334 −0.128814 0.128814i
49.5 1.16965 + 2.02589i 1.29757i −1.73614 + 3.00709i −1.10336 + 0.295646i 2.62873 1.51770i −0.549659 0.549659i −3.44410 1.31631 −1.88949 1.88949i
70.1 −1.21792 + 2.10950i 0.455408i −1.96665 3.40634i −0.913508 + 3.40926i −0.960682 0.554650i −1.73085 1.73085i 4.70921 2.79260 −6.07924 6.07924i
70.2 −0.983297 + 1.70312i 3.07168i −0.933744 1.61729i 0.710245 2.65067i 5.23144 + 3.02037i 1.20747 + 1.20747i −0.260597 −6.43521 3.81603 + 3.81603i
70.3 −0.202527 + 0.350787i 0.0458652i 0.917966 + 1.58996i 0.0766784 0.286168i 0.0160889 + 0.00928892i 0.00171153 + 0.00171153i −1.55376 2.99790 0.0848543 + 0.0848543i
70.4 0.593383 1.02777i 3.06395i 0.295794 + 0.512330i −1.03319 + 3.85590i −3.14904 1.81810i −1.17070 1.17070i 3.07561 −6.38782 3.34990 + 3.34990i
70.5 0.810359 1.40358i 2.72609i −0.313364 0.542762i 0.159770 0.596269i 3.82630 + 2.20911i −3.13775 3.13775i 2.22569 −4.43157 −0.707442 0.707442i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 70.5 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
73.h even 12 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 73.2.h.a 20
3.b odd 2 1 657.2.be.c 20
73.h even 12 1 inner 73.2.h.a 20
73.j odd 24 2 5329.2.a.m 20
219.o odd 12 1 657.2.be.c 20

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
73.2.h.a 20 1.a even 1 1 trivial
73.2.h.a 20 73.h even 12 1 inner
657.2.be.c 20 3.b odd 2 1
657.2.be.c 20 219.o odd 12 1
5329.2.a.m 20 73.j odd 24 2

## Hecke kernels

This newform subspace is the entire newspace $$S_{2}^{\mathrm{new}}(73, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 - 10 T + 165 T^{2} + 798 T^{3} + 3419 T^{4} + 6232 T^{5} + 10772 T^{6} + 8408 T^{7} + 12215 T^{8} + 8218 T^{9} + 8873 T^{10} + 4506 T^{11} + 3794 T^{12} + 1680 T^{13} + 1126 T^{14} + 386 T^{15} + 195 T^{16} + 52 T^{17} + 22 T^{18} + 4 T^{19} + T^{20}$$
$3$ $$16 + 7784 T^{2} + 85257 T^{4} + 295236 T^{6} + 341328 T^{8} + 184282 T^{10} + 53842 T^{12} + 9064 T^{14} + 881 T^{16} + 46 T^{18} + T^{20}$$
$5$ $$2500 + 500 T + 23750 T^{2} + 58110 T^{3} + 35599 T^{4} + 79024 T^{5} + 106593 T^{6} - 98978 T^{7} - 142417 T^{8} + 37580 T^{9} + 54476 T^{10} - 2858 T^{11} - 545 T^{12} - 1110 T^{13} - 1218 T^{14} - 96 T^{15} + 34 T^{16} + 46 T^{17} + 26 T^{18} + 4 T^{19} + T^{20}$$
$7$ $$16 - 9328 T + 2719112 T^{2} + 3469036 T^{3} + 2216729 T^{4} - 1122432 T^{5} + 2562856 T^{6} + 2247638 T^{7} + 1010296 T^{8} - 129264 T^{9} + 282106 T^{10} + 227870 T^{11} + 90802 T^{12} + 2254 T^{13} + 1444 T^{14} + 1322 T^{15} + 609 T^{16} + 6 T^{17} + 2 T^{18} + 2 T^{19} + T^{20}$$
$11$ $$40806544 + 89917488 T - 30071772 T^{2} - 175611456 T^{3} - 136039663 T^{4} - 22782292 T^{5} + 293568084 T^{6} + 200554156 T^{7} + 24974539 T^{8} - 6589966 T^{9} - 2249992 T^{10} - 1180184 T^{11} - 214916 T^{12} + 37830 T^{13} + 19196 T^{14} + 8854 T^{15} + 2343 T^{16} + 380 T^{17} + 84 T^{18} + 6 T^{19} + T^{20}$$
$13$ $$47734281 + 6674094 T + 797694912 T^{2} + 1902458712 T^{3} + 2767511590 T^{4} + 2706545708 T^{5} + 1734134864 T^{6} + 751450966 T^{7} + 232328042 T^{8} + 50737204 T^{9} + 5860605 T^{10} - 703444 T^{11} - 567504 T^{12} - 165578 T^{13} - 21072 T^{14} + 3326 T^{15} + 1988 T^{16} + 512 T^{17} + 110 T^{18} + 16 T^{19} + T^{20}$$
$17$ $$28561 - 3584828 T + 224974472 T^{2} - 1093833320 T^{3} + 2718066472 T^{4} - 3722465578 T^{5} + 2958419000 T^{6} - 1056142524 T^{7} + 188899652 T^{8} - 16441804 T^{9} + 25232290 T^{10} - 8033820 T^{11} + 1257276 T^{12} + 2064 T^{13} + 52496 T^{14} - 15326 T^{15} + 2216 T^{16} + 40 T^{17} + 32 T^{18} - 8 T^{19} + T^{20}$$
$19$ $$10234964224 + 51476706432 T + 86830437808 T^{2} + 2664711288 T^{3} - 31690204647 T^{4} - 3431679372 T^{5} + 8033946824 T^{6} + 1367603184 T^{7} - 1073676882 T^{8} - 213647424 T^{9} + 104765034 T^{10} + 23896488 T^{11} - 4813177 T^{12} - 1386900 T^{13} + 146870 T^{14} + 59688 T^{15} + 775 T^{16} - 972 T^{17} - 33 T^{18} + 12 T^{19} + T^{20}$$
$23$ $$454201344 - 748818432 T - 311049216 T^{2} + 1191250944 T^{3} + 279983872 T^{4} - 1358396160 T^{5} + 242790016 T^{6} + 586533504 T^{7} - 68911232 T^{8} - 158118144 T^{9} + 8706272 T^{10} + 28957056 T^{11} + 4473840 T^{12} - 1207008 T^{13} - 214952 T^{14} + 38640 T^{15} + 7480 T^{16} - 636 T^{17} - 94 T^{18} + 6 T^{19} + T^{20}$$
$29$ $$1349166361 + 2245733340 T + 22234858059 T^{2} - 1594523868 T^{3} + 54896889143 T^{4} - 77163507742 T^{5} + 49244463756 T^{6} - 19897468838 T^{7} + 4691595499 T^{8} - 237633622 T^{9} - 165776785 T^{10} + 50297260 T^{11} - 4424150 T^{12} - 1207704 T^{13} + 257162 T^{14} + 6490 T^{15} - 693 T^{16} - 328 T^{17} - 66 T^{18} + 6 T^{19} + T^{20}$$
$31$ $$992016 + 2151360 T - 12576204 T^{2} - 9690660 T^{3} + 37854253 T^{4} - 90015020 T^{5} + 656374541 T^{6} - 1448811854 T^{7} + 1451031490 T^{8} - 860059172 T^{9} + 404929861 T^{10} - 162967272 T^{11} + 51508209 T^{12} - 12440274 T^{13} + 2364787 T^{14} - 355124 T^{15} + 42334 T^{16} - 4250 T^{17} + 347 T^{18} - 20 T^{19} + T^{20}$$
$37$ $$31307409721 + 68604558470 T + 255904853372 T^{2} + 100811561760 T^{3} + 639195650067 T^{4} + 203935395770 T^{5} + 1153007643148 T^{6} - 449451848884 T^{7} + 187574854482 T^{8} - 34255217688 T^{9} + 7537373331 T^{10} - 783600500 T^{11} + 159222309 T^{12} - 10833972 T^{13} + 2563358 T^{14} - 56766 T^{15} + 27981 T^{16} + 256 T^{17} + 253 T^{18} + 8 T^{19} + T^{20}$$
$41$ $$41424867961 + 251036763648 T + 1298817390143 T^{2} + 1617712811740 T^{3} + 2023345200368 T^{4} - 567382964258 T^{5} + 432110061773 T^{6} - 87311216214 T^{7} + 39911297359 T^{8} - 7813659108 T^{9} + 2353546864 T^{10} - 353917580 T^{11} + 76549335 T^{12} - 9167632 T^{13} + 1669313 T^{14} - 159504 T^{15} + 23536 T^{16} - 1610 T^{17} + 211 T^{18} - 10 T^{19} + T^{20}$$
$43$ $$144 + 1296 T + 5832 T^{2} - 584820 T^{3} + 15427297 T^{4} - 52566552 T^{5} + 89645616 T^{6} - 2484456 T^{7} + 8639538 T^{8} - 23173560 T^{9} + 29115576 T^{10} - 4160436 T^{11} + 347555 T^{12} - 99912 T^{13} + 90864 T^{14} - 14208 T^{15} + 1170 T^{16} - 96 T^{17} + 72 T^{18} - 12 T^{19} + T^{20}$$
$47$ $$16151359744 + 72088380416 T + 103524407696 T^{2} + 62483819224 T^{3} + 44324606825 T^{4} - 24215747370 T^{5} + 2440149532 T^{6} - 2980006486 T^{7} + 219049888 T^{8} + 114886344 T^{9} + 35782066 T^{10} + 24429740 T^{11} + 1929343 T^{12} - 912884 T^{13} - 238832 T^{14} - 25336 T^{15} + 3237 T^{16} + 1446 T^{17} + 215 T^{18} + 20 T^{19} + T^{20}$$
$53$ $$145666505900644 - 244182916276396 T + 192536006835842 T^{2} - 99800292233118 T^{3} + 39031208402361 T^{4} - 11979627142488 T^{5} + 2962126131852 T^{6} - 601292732066 T^{7} + 95251196528 T^{8} - 10325031456 T^{9} + 350900816 T^{10} + 172051882 T^{11} - 43410637 T^{12} + 4834198 T^{13} - 262986 T^{14} - 95192 T^{15} + 30999 T^{16} - 3054 T^{17} + 255 T^{18} - 24 T^{19} + T^{20}$$
$59$ $$26510882872384 + 80228879088992 T + 52823033347976 T^{2} - 10241251015252 T^{3} + 22316075754809 T^{4} - 6291938445530 T^{5} + 2186162626734 T^{6} - 654994988396 T^{7} + 94009337053 T^{8} - 12383271084 T^{9} + 225275018 T^{10} + 358796194 T^{11} - 6085444 T^{12} + 4322150 T^{13} - 386910 T^{14} - 132476 T^{15} + 5877 T^{16} - 216 T^{17} + 6 T^{18} + 18 T^{19} + T^{20}$$
$61$ $$214069942479578761 + 54780831110459526 T - 36667937174910336 T^{2} - 10579167435977928 T^{3} + 7262256468927463 T^{4} - 861105808641006 T^{5} - 136048281050216 T^{6} + 33012885277350 T^{7} + 1634581346824 T^{8} - 898694994816 T^{9} + 32830870945 T^{10} + 11312164458 T^{11} - 888999065 T^{12} - 104349528 T^{13} + 15027388 T^{14} + 123462 T^{15} - 103313 T^{16} + 2394 T^{17} + 531 T^{18} - 42 T^{19} + T^{20}$$
$67$ $$353935892217856 + 782660361560064 T + 685147875069696 T^{2} + 239368296063456 T^{3} - 10060979990295 T^{4} - 25174512956472 T^{5} + 2114025520508 T^{6} + 5890935486720 T^{7} + 1963883006783 T^{8} + 235025149350 T^{9} - 9387256312 T^{10} - 4483155852 T^{11} + 5845264 T^{12} + 94553298 T^{13} + 8722808 T^{14} - 247374 T^{15} - 68025 T^{16} + 336 T^{17} + 596 T^{18} + 42 T^{19} + T^{20}$$
$71$ $$196571636496 + 568965474288 T + 3229322851452 T^{2} - 5865702741492 T^{3} + 10821159849121 T^{4} - 5572215471676 T^{5} + 2758100317962 T^{6} - 766143184676 T^{7} + 266935343657 T^{8} - 62766913356 T^{9} + 16193992774 T^{10} - 2617163500 T^{11} + 437918280 T^{12} - 45401992 T^{13} + 6121246 T^{14} - 475080 T^{15} + 58325 T^{16} - 2480 T^{17} + 270 T^{18} - 4 T^{19} + T^{20}$$
$73$ $$4297625829703557649 + 941945387332286608 T + 75807248638043614 T^{2} + 15510547520812188 T^{3} + 4290325315293150 T^{4} + 562934298792778 T^{5} + 40363783048868 T^{6} + 7458657952530 T^{7} + 1488813131682 T^{8} + 132922763356 T^{9} + 9378350075 T^{10} + 1820859772 T^{11} + 279379458 T^{12} + 19173090 T^{13} + 1421348 T^{14} + 271546 T^{15} + 28350 T^{16} + 1404 T^{17} + 94 T^{18} + 16 T^{19} + T^{20}$$
$79$ $$8234808929239104 - 4609666569580704 T - 1017565097104872 T^{2} + 1051092957694140 T^{3} + 233964577845457 T^{4} - 408707927092986 T^{5} + 164140569272186 T^{6} - 31942453910160 T^{7} + 2448848953733 T^{8} + 190968901992 T^{9} - 41067885106 T^{10} - 3664904262 T^{11} + 1570229560 T^{12} - 150181002 T^{13} - 33422 T^{14} + 910920 T^{15} - 8455 T^{16} - 10908 T^{17} + 1174 T^{18} - 54 T^{19} + T^{20}$$
$83$ $$8041149118864 + 20302477157040 T + 25630079272200 T^{2} - 8526080557716 T^{3} + 1706347701665 T^{4} + 425182760482 T^{5} + 154885979922 T^{6} - 51165067858 T^{7} + 12107430895 T^{8} + 2431776622 T^{9} + 173485160 T^{10} - 30031096 T^{11} + 20218237 T^{12} + 4635936 T^{13} + 442448 T^{14} + 22418 T^{15} + 13767 T^{16} + 3454 T^{17} + 450 T^{18} + 30 T^{19} + T^{20}$$
$89$ $$142102635503761 + 7085986565468 T + 66625142836541 T^{2} + 15122470498056 T^{3} + 24016254639035 T^{4} + 5120967281072 T^{5} + 3709744845724 T^{6} + 930388467804 T^{7} + 422240082711 T^{8} + 82982946228 T^{9} + 21094638913 T^{10} + 2690251180 T^{11} + 529568666 T^{12} + 56801346 T^{13} + 8909874 T^{14} + 708626 T^{15} + 84603 T^{16} + 5700 T^{17} + 526 T^{18} + 22 T^{19} + T^{20}$$
$97$ $$3690728950129 + 19593997090482 T^{2} + 13098241106013 T^{4} + 2894528391800 T^{6} + 278703039362 T^{8} + 14227313324 T^{10} + 421729570 T^{12} + 7476344 T^{14} + 77709 T^{16} + 434 T^{18} + T^{20}$$